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\begin{document}
\begin{abstract} For each universal genus-$g$ polarization $\mu$ of degree $d$, we construct a universal tropical Jacobian $J_{\mu,g}^{trop}$ as a generalized cone complex over the moduli space of stable pointed genus-$g$ tropical curves. We show several properties of the space $J_{\mu,g}^{trop}$. In particular, we prove that the natural compactification of $J_{\mu,g}^{trop}$ is the tropicalization of the Esteves' compactified universal Jacobian over the moduli space of stable pointed genus-$g$ curves. \end{abstract}
\maketitle \noindent MSC (2010): 14T05, 14H10, 14H40.\\
Keywords: Tropical curve, algebraic curve, universal Jacobian, skeleton.
\tableofcontents
\section{Introduction}
\subsection{History and motivation}
A recent approach to study moduli spaces in algebraic geometry is the construction of a suitable tropical analogue from which one can extract useful combinatorial and topological properties. The construction of the moduli space of stable tropical curves by Mikhalkin \cite{M}, and Brannetti, Melo and Viviani \cite{BMV}, and of its compactification by Caporaso \cite{Caporaso}, revealed strong similarities with the moduli space of Deligne-Mumford stable curves: these spaces share the same dimension and have dual stratifications. These analogies were made rigorous by Abramovich, Caporaso and Payne \cite{ACP}. They proved that the skeleton of the Berkovich analytification of the moduli stack of stable curves is isomorphic to the compactified moduli space of stable tropical curves. In the last few years, these type of constructions have been carried out also for other important moduli spaces: Cavalieri, Hampe, Markwig and Ranganathan \cite{CHMR}, and Ulirsch \cite{U} for weighted stable curves; Ranganathan \cite{R1} and \cite{R2} for rational stable maps; Cavalieri, Markwig and Ranganathan \cite{CMR} for admissible covers; M\"oeller, Ulirsch and Werner \cite{MUW} for effective divisors; Caporaso, Melo and Pacini \cite{CMP} for spin curves. \par
Another central moduli space in algebraic geometry is the Jacobian of an algebraic curve, and its compactifications. The construction of a compactified Jacobian for a curve dates back to Igusa \cite{I56}. Since then, several compactified Jacobians have been constructed. Mayer and Mumford \cite{MM} introduced for the first time the use of torsion-free rank-$1$ sheaves to describe the boundary points, and the technique of Geometric Invariant Theory (GIT). This technique was later employed by D'Souza \cite{Ds} to compactify Jacobians of integral curves, and by Oda and Seshadri \cite{OS}, who first considered the case of reducible curves. Caporaso \cite{C} and, soon after, Pandharipande \cite{Pand}, used GIT to construct a universal compactified Jacobian over the moduli space of stable curves. Finally, Altman and Kleiman \cite{AK}, and Esteves \cite{Es01}, constructed a compactified relative Jacobian for families of reduced curves, without using GIT. This compactification employs the notion of quasistability for torsion-free rank-$1$ sheaves with respect to a polarization, and depends on the choice of a section of the family of curves.
Recently, Caporaso \cite{C08}, Melo \cite{M11}, \cite{M15}, and Kass and Pagani \cite{KP}, carried out a stacky version of theses Jacobians. In particular Melo \cite{M15} constructed a Deligne-Mumford stack $\overline{\mathcal{J}}_{\mu,g}$ over the moduli space $\overline{\mathcal M}_{g,1}$ of stable pointed genus-$g$ curves, following the work of Esteves \cite{Es01}. The Jacobian $\overline{\mathcal{J}}_{\mu,g}$, which we call the Esteves' universal Jacobian, compactifies the relative degree-$d$ Jacobian $\mathcal{J}_{d,g,1}$ over $\mathcal M_{g,1}$ and parametrizes torsion-free rank-$1$ sheaves which are quasistable with respect to a polarization $\mu$.\par
The goal of this paper is the construction of a universal tropical Jacobian and the study of its interplay with algebraic geometry. This is the first of a series of papers dedicated to the subject. In this paper we construct the universal tropical Jacobian $J_{\mu,g}^{trop}$ as a generalized cone complex and prove that there is an isomorphism between its natural compactification and the skeleton of the Esteves' universal compactified Jacobian $\overline{\mathcal{J}}_{\mu,g}$. This isomorphism is compatible with the maps of tropicalization and retraction to the skeleton. In the forthcoming paper \cite{APfuture}, we analyze the universal tropical Abel map from $M_{g,n}^{\text{trop}}$ to $J_{\mu,g}^\text{trop}$ and use it to resolve the universal Abel map for nodal curves having $\overline{\mathcal{J}}_{\mu,g}$ as a target. \par
It is worth noting that Baker and Rabinoff \cite{BR} already studied the problem of the tropicalization of the Jacobian of a curve. They prove that the skeleton of the analytification of the Jacobian of a curve over an algebraically closed, complete, non Archimedean valuation field is isomorphic to the usual tropical Jacobian defined in \cite{MZ} and \cite{BF}. Our analysis can be seen as an extension of this result to the universal setting.\par
A comment about the difference between the pointed and non-pointed case is due.
The Jacobian $\overline{\mathcal{J}}_{\mu,g}$ is a toroidal Deligne-Mumford stack proper over $\overline{\mathcal M}_{g,1}$. This allows us to apply the setting and results of \cite[Section 6]{ACP}. The known degree-$d$ universal Jacobian stacks over $\overline{\mathcal M}_g$ are not separated when $\gcd(d-g+1,2g-2)\neq 1$, so in these cases a different approach is needed to tropicalize these compactified Jacobians. On the other hand, if $\gcd(d-g+1,2g-2)=1$, the quasistability condition does not depend on the chosen marked point, and our approach for constructing $\overline{J}^\text{trop}_{\mu,g}$ can be employed, essentially in the same way, to build a universal tropical Jacobian over $M^\text{trop}_g$.
\subsection{The results}
Let $(X,p_0)$ be a pointed tropical curve and $\mu$ be a degree-$d$ polarization on $X$, i.e., a function $\mu\colon X\to \mathbb R$ such that $\sum_{p\in X}\mu(p)=d$, where the sum is finite. We introduce the notion of $(p_0,\mu)$-quasistability for degree-$d$ divisors on $X$ imposing that certain inequalities hold for every tropical subcurve of $X$. We are able to prove the following theorem, which is the tropical analogue of a similar statement on $(p_0,\mu)$-quasistable torsion-free rank-$1$ sheaves appearing in \cite{Es01}.
\begin{Thm*}[\ref{thm:quasistable}] Every degree-$d$ divisor on a tropical curve is equivalent to a unique $(p_0,\mu)$-quasistable degree-$d$ divisor. \end{Thm*}
This is also a generalization of the same result for break divisors in degree $g$ (see \cite{MZ} and \cite[Theorem 1.1]{ABKS}). In fact, a break divisor is $(p_0,\mu)$-quasistable for the so-called degree-$g$ canonical polarization $\mu$ (see \cite[Introduction]{CC}, \cite{Shen}, and Example \ref{exa:pol}). As for break divisors, quasistable divisors are well-behaved when varying the tropical curves under specializations. \par
Then we define a polyhedral complex $J_{p_0,\mu}^\text{trop}(X)$ parametrizing $(p_0,\mu)$-quasistable divisors on the tropical curve $X$. This polyhedral complex is the tropical analogue of the compactified Jacobian constructed by Oda-Seshadri \cite{OS} and Esteves \cite{Es01}. In Theorem \ref{thm:jX} we prove that $J_{p_0,\mu}^\text{trop}(X)$ is homeomorphic to the usual tropical Jacobian $J^\text{trop}(X)$ of $X$. When $\mu$ is the canonical degree-$g$ polarization, the polyhedral complex $J_{p_0,\mu}^\text{trop}(X)$ is the same as the one constructed by An, Baker, Kuperberg and Shokrieh \cite{ABKS}.\par
These polyhedral complexes can be cast together to form a generalized cone complex $J^\text{trop}_{\mu,g}$ parametrizing tuples $(X,p_0,\mathcal D)$ where $(X,p_0)$ is a stable pointed genus-$g$ tropical curve and $\mathcal D$ is a $(p_0,\mu)$-quasistable divisor on $X$. Here $\mu$ is a universal genus-$g$ polarization of degree-$d$, meaning that $\mu$ is compatible with specializations of genus-$g$ graphs. We prove that the generalized cone complex $J^\text{trop}_{\mu,g}$ is a universal tropical Jacobian over $M^\text{trop}_{g,1}$, as stated by the following theorem.
\begin{Thm*}[\ref{thm:maintrop}] The generalized cone complex $J^\text{trop}_{\mu,g}$ has pure dimension $4g-2$ and is connected in codimension $1$. The natural forgetful map $\pi^{trop}\colon J^\text{trop}_{\mu,g}\to M^\text{trop}_{g,1}$ is a map of generalized cone complexes and for every equivalence class of a stable pointed genus-$g$ tropical curve $X$, we have a homeomorphism \[ (\pi^{trop})^{-1}([X])\cong J^\text{trop}(X)/\textnormal{Aut}(X). \] \end{Thm*}
The above setup can be also given for graphs with $1$ leg. In fact, our starting point is the notion of $(v_0,\mu)$-quasistability for divisors on graphs with legs. In this case, the main objects of study have the natural structure of poset, instead of polyhedral/cone complex. The two frameworks are closely related, since every pointed tropical curve $(X,p_0)$ has a (not unique) model $(\Gamma_X,v_0)$ which is a graph with $1$ leg. There is a ranked poset $\mathcal{QD}_{v_0,\mu}(\Gamma_X)$ of $(v_0,\mu)$-quasistable (pseudo-)divisors on $\Gamma_X$, and a natural continuous map $J^{\text{trop}}_{p_0,\mu}(X)\to \mathcal{QD}_{v_0,\mu}(\Gamma_X)$ (see Corollary \ref{cor:quasiquasi} and Remark \ref{rem:cont}). We also define a universal ranked poset $\mathcal{QD}_{\mu, g}$ and a continuous map $J^\text{trop}_{\mu,g}\to \mathcal{QD}_{\mu, g}$, where $\mathcal{QD}_{\mu, g}$ parametrizes tuples $(\Gamma,v_0,D)$, with $(\Gamma,v_0)$ a stable genus-$g$ graph with $1$ leg and $D$ a $(v_0,\mu)$-quasistable (pseudo-)divisor on $\Gamma$. The properties of $\mathcal{QD}_{\mu, g}$ are essentially the same as the ones in the above theorem (see Theorem \ref{thm:maingraph}). \par Next, we study the skeleton of the Esteves' universal compactified Jacobian $\overline{\mathcal{J}}_{\mu, g}$. The first issue we need to address is the description of the natural stratification induced by the embedding $\mathcal{J}_{d,g,1}\subset \overline{\mathcal{J}}_{\mu, g}$. Recently, the analogous problem for other compactified Jacobians has been studied by Caporaso and Christ \cite{CC}. They give a combinatorially interesting incarnation for a stratification of the Caporaso's compactified Jacobians in degree $g-1$ and $g$.
A natural stratification for the Esteves' compactified Jacobian $\mathcal{J}_{p_0,\mu}(X)$ of a fixed nodal curve $X$ is described by Melo and Viviani \cite{MV}. In Proposition \ref{prop:stratification} we show that their stratification can be refined to a graded stratification of $\overline{\mathcal{J}}_{\mu,g}$ by the poset $\mathcal{QD}_{\mu, g}$. We give a useful description of the strata as quotient stacks in Proposition \ref{prop:stratades}. These are the key results for our analysis of the skeleton $\overline{\Sigma}(\overline{\mathcal{J}}_{\mu,g})$ of $\overline{\mathcal{J}}_{\mu,g}$, which allows us to prove the following theorem. \begin{Thm*}[\ref{thm:tropj}] There is an isomorphism of extended generalized cone complexes \[ \Phi_{\overline{\mathcal{J}}_{\mu,g}}\colon \overline{\Sigma}(\overline{\mathcal{J}}_{\mu,g})\to \overline{J}^{\text{trop}}_{\mu,g}. \] Moreover, the following diagram is commutative \begin{eqnarray*} \SelectTips{cm}{11} \begin{xy} <16pt,0pt>: \xymatrix{ \overline{\mathcal{J}}_{\mu,g}^{an} \ar@/^2pc/[rr]^{\text{trop}_{\overline{\mathcal{J}}_{\mu,g}}} \ar[d] \ar[r]^{{\bf p}_{\overline{\mathcal{J}}_{\mu,g}}\;}
& \ar[r]^{{\Phi}_{\overline{\mathcal{J}}_{\mu,g}}} \ar[d] \overline{\Sigma}(\overline{\mathcal{J}}_{g,n}) & \overline{J}_{\mu,g}^{trop} \ar[d] \\ \overline{\mathcal M}_{g,1}^{an} \ar@/_2pc/[rr]^{\text{trop}_{\overline{\mathcal M}_{g,1}}} \ar[r]^{{\bf p}_{\overline{ {\mathcal M}}_{g,1}}\;} & \ar[r]^{\cong} \overline{\Sigma}(\overline{{\mathcal M}}_{g,1}) & \overline{M}_{g,1}^{trop}
} \end{xy} \end{eqnarray*} where the vertical maps are forgetful maps, ${\bf p}_{-}$ denotes the retractions to the skeleton and $\text{trop}_{-}$ denotes the tropicalization maps. \end{Thm*}
In short, in Sections \ref{sec:preliminaries} and \ref{sec:trop} we introduce the background material and the technical tools about graphs, posets, and tropical curves. In Section \ref{sec:quasigraph} we study quasistability for pseudo-divisors on graphs. In Section \ref{sec:univtropJ} we introduce the Jacobian of quasistable divisors on tropical curves and study its properties. Finally, in Section \ref{sec:skeleton}, we prove the result on the skeleton of Esteves' universal compactified Jacobian.
\section{Graphs and posets} \label{sec:preliminaries}
\subsection{Graphs} \label{subsec:graphs}
Given a graph $\Gamma$, we denote by $E(\Gamma)$ the set of edges and by $V(\Gamma)$ the set of vertices of $\Gamma$. If $\mathcal E\subset E(\Gamma)$ and $v$ is a vertex of $\Gamma$, we define the \emph{valence of $v$ in $\mathcal E$}, denoted by $\text{val}_\mathcal E(v)$, as the number of edges in $\mathcal E$ incident to $v$ (with loops counting twice). In the case that $\mathcal E=E(\Gamma)$ we simply write $\text{val}(v)$ and call it the \emph{valence} of $v$. We say that $\Gamma$ is \emph{$k$-regular} if every vertex of $\Gamma$ have valence $k$; we say that $\Gamma$ is a \emph{circular graph} if $\Gamma$ is $2$-regular and connected. A \emph{cycle} on $\Gamma$ is a circular subgraph of $\Gamma$. \par
A \emph{digraph} (directed graph) is a graph $\Gamma$ where each edge has a orientation, i.e., there are functions $s,t\colon E(\Gamma)\to V(\Gamma)$ called \emph{source} and \emph{target} and each edge is oriented from the source to the target. We denote a digraph by $\overrightarrow{\Gamma}$ and by $\Gamma$ its underlying graph. We let $E(\overrightarrow{\Gamma})$ be the set of oriented edges of $\overrightarrow{\Gamma}$ and we set $V(\overrightarrow{\Gamma}):=V(\Gamma)$.
\par
Let $\Gamma$ be a graph. Fix disjoint subsets $V, W\subset V(\Gamma)$. We define $E(V,W)$ as the set of edges joining a vertex in $V$ with one in $W$. More generally if $V$ and $W$ have nonempty intersection, we define $E(V,W):=E(V\setminus W, W\setminus V)$. We set $V^c:=V(\Gamma)\setminus V$. If $E(V,V^c)$ is nonempty, it is called a \emph{cut} of $\Gamma$. We set $\delta_{\Gamma,V}:=|E(V,V^c)|$. When no confusion may arise, we simply write $\delta_V$ instead of $\delta_{\Gamma,V}$. \par
Fix a subset $\mathcal E\subset E(\Gamma)$. We define the graphs $\Gamma/\mathcal E$ and $\Gamma_\mathcal E$ as the graphs obtained by the contraction of edges in $\mathcal E$ and by the removal of edges in $\mathcal E$, respectively. We say that $\mathcal E$ is \emph{nondisconnecting} if $\Gamma_\mathcal E$ is connected. Note that $V(\Gamma_\mathcal E)=V(\Gamma)$ and $E(\Gamma_\mathcal E)=E(\Gamma)\setminus\mathcal E$. Also there is a natural surjection $V(\Gamma)\to V(\Gamma/\mathcal E)$ and a natural identification $E(\Gamma/\mathcal E)=E(\Gamma)\setminus\mathcal E$. Recall that a graph $\Gamma$ specializes to a graph $\Gamma'$ if there exists $\mathcal E\subset E(\Gamma)$ such that $\Gamma'$ is isomorphic to $\Gamma/\mathcal E$. We denote a specialization of $\Gamma$ to $\Gamma'$ by $\iota\colon \Gamma\rightarrow\Gamma'$. A specialization $\iota\colon \Gamma\rightarrow\Gamma'$ comes equipped with a surjective map $\iota^V\colon V(\Gamma)\rightarrow V(\Gamma')$ and an injective map $\iota^E\colon E(\Gamma')\rightarrow E(\Gamma)$. We usually write $\iota=\iota^V$ and we will see $E(\Gamma')$ as a subset of $E(\Gamma)$ via $\iota^E$. A similar notion of specialization can be given for digraphs. \par
We also define the graph $\Gamma^\mathcal E$ as the graph obtained from $\Gamma$ by adding exactly one vertex in the interior of each edge in $\mathcal E$. We call $\Gamma^\mathcal E$ the \emph{$\mathcal E$-subdivision} of $\Gamma$. Note that there is a natural inclusion $V(\Gamma)\subset V(\Gamma^\mathcal E)$. We call a vertex in $V(\Gamma^\mathcal E)\setminus V(\Gamma)$ an \emph{exceptional vertex}. We set $\Gamma^{(2)}:=\Gamma^{E(\Gamma)}$. \par
More generally, we say that a graph $\Gamma'$ is a \emph{refinement} of the graph $\Gamma$ if $\Gamma'$ is obtained from $\Gamma$ by successive subdivisions. In other words, there is an inclusion $a\colon V(\Gamma)\to V(\Gamma')$ and a surjection $b\colon E(\Gamma')\to E(\Gamma)$ such that for any edge $e$ of $\Gamma$ there exist distinct vertices $x_0,\ldots, x_n\in V(\Gamma')$ and distinct edges $e_1,\ldots, e_n\in E(\Gamma')$ such that
\begin{enumerate} \item $x_0=a(v_0)$, $x_n=a(v_1)$ and $x_{i}\notin\text{Im}(a)$ for every $i=1,\ldots,n-1$, where $v_0$ and $v_1$ are precisely the vertices of $\Gamma$ incident to $e$; \item $b^{-1}(e)=\{e_1,\ldots, e_n\}$; \item the vertices in $V(\Gamma')$ incident to $e_i$ are precisely $x_{i-1}$ and $x_i$, for $i=1,\dots,n$. \end{enumerate} We say that the edges $e_1,\ldots, e_n$ (respectively, $x_1,\ldots, x_{n-1}$) are the edges (respectively, the exceptional vertices) \emph{over} $e$.\par
Let $\iota\colon \Gamma\to\Gamma'$ be a specialization and $\mathcal E'\subset E(\Gamma')$ and $\mathcal E\subset E(\Gamma)$ be sets such that $\mathcal E'\subset\mathcal E\cap E(\Gamma')$. We call a specialization $\iota^\mathcal E\colon\Gamma^\mathcal E\rightarrow\Gamma'^{\mathcal E'}$ \emph{compatible with $\iota$} if the following diagrams \[ \SelectTips{cm}{11} \begin{xy} <16pt,0pt>: \xymatrix{ V(\Gamma)\ar[d]\ar[r]^{\iota} &V(\Gamma')\ar[d]&& E(\Gamma')\ar[r]^\iota &E(\Gamma)\\
V(\Gamma^\mathcal E)\ar[r]^{\iota^\mathcal E}& V(\Gamma'^{\mathcal E'})&&E(\Gamma'^{\mathcal E'})\ar[u]\ar[r]^{\iota^\mathcal E} & E(\Gamma^\mathcal E)\ar[u] } \end{xy} \]
are commutative. If $\iota$ is the identity of $\Gamma$, we just call $\iota^\mathcal E$ \emph{compatible}. Note that there are $2^{|\mathcal E\setminus\mathcal E'|}$ compatible specializations of $\Gamma^\mathcal E$ to $\Gamma^{\mathcal E'}$.
A divisor $D$ on the graph $\Gamma$ is a function $D\colon V(\Gamma)\to \mathbb{Z}$. The degree of $D$ is the integer $\deg D:=\sum_{v\in V(\Gamma)}D(v)$. The set of divisors on $\Gamma$ forms an Abelian group denoted by $\text{Div}(\Gamma)$. Given a subset $\mathcal E\subset E(\Gamma)$ and a divisor $D$ on $\Gamma$, we define $D^\mathcal E$ as the divisor on $\Gamma^\mathcal E$ such that \[ D^\mathcal E(v)=\begin{cases}
\begin{array}{ll}
D(v),&\text{ if }v\in V(\Gamma);\\
0,&\text{ if }v\in V(\Gamma^\mathcal E)\setminus V(\Gamma).
\end{array}
\end{cases} \] We also define a divisor $D_\mathcal E$ on $\Gamma_\mathcal E$ as $D_\mathcal E(v):=D(v)$, for every $v\in V(\Gamma_\mathcal E)=V(\Gamma)$.\par
A \emph{pseudo-divisor} on the graph $\Gamma$ is a pair $(\mathcal E,D)$ where $\mathcal E\subset E(\Gamma)$ and $D$ is a divisor on $\Gamma^\mathcal E$ such that $D(v)=-1$ for every exceptional vertex $v\in V(\Gamma^\mathcal E)$. If $\mathcal E=\emptyset$, then $(\mathcal E,D)$ is just a divisor of $\Gamma$. Since every divisor $D$ on $\Gamma^\mathcal E$ can be lifted to a divisor on $\Gamma^{(2)}$, it is equivalent to define a pseudo-divisor on $\Gamma$ as a divisor $D$ on $\Gamma^{(2)}$ such that $D(v)=0,-1$ for every exceptional vertex $v\in V(\Gamma^{(2)})$.\par
Let $\Gamma$ and $\Gamma'$ be graphs. Given a specialization $\iota\colon\Gamma\rightarrow\Gamma'$ and a divisor $D$ on $\Gamma$, we define the divisor $\iota_*(D)$ on $\Gamma'$ taking $v'\in V(\Gamma')$ to \[ \iota_*(D)(v'):=\sum_{v\in\iota^{-1}(v')}D(v). \]
We say that a pair $(\Gamma,D)$ \emph{specializes} to a pair $(\Gamma',D')$, where $D$ is a divisor on $\Gamma$ and $D'$ is a divisor on $\Gamma'$, if there exists a specialization of graphs $\iota\colon\Gamma\rightarrow\Gamma'$ such that $D'=\iota_*(D)$; we denote by $\iota\colon(\Gamma,D)\rightarrow(\Gamma',D')$ a specialization of pairs. Note that, given a specialization $\iota\colon \Gamma\rightarrow\Gamma'$ and a subset $\mathcal E$ of $E(\Gamma)$, there exists an induced specialization $\iota^\mathcal E\colon \Gamma^\mathcal E\to\Gamma'^{\mathcal E'}$, where $\mathcal E':=\mathcal E\cap E(\Gamma')$; in this case, if $(\mathcal E,D)$ is a pseudo-divisor on $\Gamma$, we define the pseudo-divisor $\iota_*(\mathcal E,D)$ on $\Gamma'$ as $\iota_*(\mathcal E,D):=(\mathcal E',\iota_*^\mathcal E(D))$.
Given pseudo-divisors $(\mathcal E,D)$ on $\Gamma$ and $(\mathcal E',D')$ on $\Gamma'$, we say that $(\Gamma,\mathcal E, D)$ \emph{specializes} to $(\Gamma',\mathcal E',D')$ if there exists a specialization $\iota\colon\Gamma\to\Gamma'$, such that $\mathcal E'\subset \mathcal E\cap E(\Gamma')$ and there is a specialization $\iota^\mathcal E\colon \Gamma^\mathcal E\rightarrow\Gamma'^{\mathcal E'}$, compatible with $\iota$, such that $\iota^\mathcal E_*(D)=(D')$. We denote by $\iota\colon (\Gamma,\mathcal E,D)\rightarrow (\Gamma',\mathcal E',D')$ such a specialization. \par
Let $\overrightarrow{\Gamma}$ be a digraph. A \emph{directed path} on $\overrightarrow{\Gamma}$ is a sequence $v_1, e_1, v_2,\ldots, e_n, v_{n+1}$ such that $s(e_i)=v_i$ and $t(e_i)=v_{i+1}$ for every $i=1,\dots,n$; a \emph{directed cycle} on $\overrightarrow{\Gamma}$ is a directed path on $\overrightarrow{\Gamma}$ such that $v_{n+1}=v_1$. A \emph{cycle} on $\overrightarrow{\Gamma}$ is just a cycle on $\Gamma$. \par
A \emph{source} (respectively, \emph{sink}) of $\overrightarrow{\Gamma}$ is a vertex in $V(\overrightarrow{\Gamma})$ such that $t(e)\neq v$ (respectively, $s(e)\neq v$) for every $e\in E(\overrightarrow{\Gamma})$. We say that $\overrightarrow{\Gamma}$ is {\emph{acyclic} if it has no directed cycles. It is a well known result that every acyclic (finite) digraph has at least a source and a sink. A \emph{flow} on $\overrightarrow{\Gamma}$ is a function $\phi \colon E(\overrightarrow{\Gamma})\to \mathbb{Z}_{\geq 0}$. We say that $\phi$ is \emph{acyclic} if the digraph $\ora{\Gamma}/S$ is acyclic, where $S:=\{e\in E(\ora{\Gamma});\phi(e)=0\}$. Moreover, we say that a flow $\phi$ is \emph{positive} if $\phi(e)>0$ for all $e\in E(\ora{\Gamma})$. Abusing terminology, we will say that a flow $\phi$ on a graph $\Gamma$ is a pair $(\ora{\Gamma},\phi)$ of an orientation on $\Gamma$ and a flow $\phi$ on $\ora{\Gamma}$.\par
Given a graph $\Gamma$ and a ring $A$, we define \[ C_0(\Gamma,A):=\bigoplus_{v\in V(\Gamma)}A\cdot v\quad\text{and}\quad C_1(\Gamma,A):=\bigoplus_{e\in E(\Gamma)}A\cdot e. \] Fix a orientation on $\Gamma$, i.e., choose a digraph $\ora{\Gamma}$ with $\Gamma$ as underlying graph. We define the differential operator $d\colon C_0(\ora{\Gamma},A)\to C_1(\ora{\Gamma}, A)$ as the linear operator taking a generator $v$ of $C_0(\ora{\Gamma},A)$ to \[ d(v):=\underset{t(e)=v}{\sum_{e\in{E(\ora{\Gamma})}}}e-\underset{s(e)=v}{\sum_{e\in{E(\ora{\Gamma})}}}e. \] The adjoint of $d$ is the linear operator $d^*\colon C_1(\ora{\Gamma},A)\to C_0(\ora{\Gamma},A)$ taking a generator $e$ of $C_1(\ora{\Gamma},A)$ to \[ d^*(e):=t(e)-s(e). \] The space of $1$-cycles of $\ora{\Gamma}$ (over $A$) is defined as $H_1(\ora{\Gamma},A):=\ker(d^*)$.\par We have a natural identification between $C_0(\Gamma,\mathbb{Z})$ and $\text{Div}(\Gamma)$. Note that the composition $d^*d\colon C_0(\Gamma,A)\to C_0(\Gamma,A)$ does not depend on the choice of the orientation. We define the group of principal divisors on $\Gamma$ as the subgroup $\text{Prin}(\Gamma):=\text{Im}(d^*d)$ of $\text{Div}(\Gamma)$. Given $D,D'\in \text{Div}(\Gamma)$, we say that $D$ is \emph{equivalent} to $D'$ if $D-D'\in \text{Prin}(\Gamma)$.\par
Given a flow $\phi$ on $\ora{\Gamma}$, we define the divisor associated to $\phi$ as the image $\textnormal{div}(\phi)=d^*(\phi)$, where $\phi$ is seen as an element of $C_1(\ora{\Gamma},\mathbb{Z})$. \par
A \emph{(vertex) weighted graph} is a pair $(\Gamma,w)$, where $\Gamma$ is a connected graph and $w$ is a function $w\colon V(\Gamma)\to\mathbb{Z}_{\geq0}$, called the \emph{weight function}. The genus of a weighted graph $(\Gamma,w)$ is defined as $g(\Gamma):=\sum_{v\in V(\Gamma)} w(v)+b_1(\Gamma)$, where $b_1(\Gamma)$ is the first Betti number of $\Gamma$. \par
A \emph{graph with legs} indexed by the finite set $L$ (the set of legs) is the data of a graph $\Gamma$ and a map $\text{leg}_\Gamma\colon L\to V(\Gamma)$. Usually, we will simply write $\Gamma$ for a graph with legs and we denote by $L(\Gamma)$ its set of legs. We denote by $L(v)$ the set of legs incident to $v$, i.e., $L(v):=\text{leg}_\Gamma^{-1}(v)$. A \emph{graph with $n$ legs} is a graph with legs $\Gamma$ such that $L(\Gamma)=I_n:=\{0,1,\ldots,n-1\}$. If $\Gamma$ is a graph with $n$ legs, we will always set $v_0:=\text{leg}_\Gamma(0)\in V(\Gamma)$. We will denote by $(\Gamma,v_0)$ a graph with $1$ leg.
If $(\Gamma,w,\text{leg}_{\Gamma})$ and $(\Gamma',w',\text{leg}_{\Gamma'})$ are weighted graphs with legs, we say that a specialization $\iota\colon\Gamma\rightarrow\Gamma'$ is a \emph{specialization of weighted graphs with legs} if, for every $v'\in V(\Gamma')$, we have $w'(v')=g(\iota^{-1}(v))$, and $\text{leg}_{\Gamma}=\text{leg}_{\Gamma'}\circ \iota^V$.
A weighted graph with $n$ legs $\Gamma$ is \emph{stable} if $\text{val}(v)+2w(v)+|L(v)|\geq3$ for every vertex $v\in V(\Gamma)$. A \emph{tree} is a connected graph whose first Betti number is $0$ or, equivalently, a connected graph whose number of edges is equal to the number of its vertices minus one. A \emph{tree-like} graph is a graph that becomes a tree after contracting all the loops.
A \emph{spanning tree} $T$ of a graph $\Gamma$ is a subset $T\subset E(\Gamma)$ such that $\Gamma_{E(\Gamma)\setminus T}$ is a tree. Note that, since a tree is connected, it follows that $\Gamma_{E(\Gamma)\setminus T}$ is connected, and hence every vertex of $\Gamma$ must be incident to an edge in $T$.
\begin{Lem} \label{lem:tree}
Let $\Gamma$ be a connected graph. Let $T$ and $T'$ be distinct spanning trees of $\Gamma$. Then, there are spanning trees $T_0,T_1,\ldots, T_n$ of $\Gamma$, such that $T_0=T$, $T_n=T'$ and $|T_{i-1}\cap T_i|=|V(\Gamma)|-2$, for every $i=1,\dots,n$. \end{Lem} \begin{proof}
The idea is to take an edge $e$ in $T'\setminus T$ and construct a spanning tree $T_1$ by removing an edge $e'$ from $T\setminus T'$ and adding $e$. It is clear that $|T\cap T_1|=|T|-1=|V(\Gamma)|-2$ and the result will follow by iterating the reasoning.\par Set $S:=T\cap T'$ and consider $e\in T'\setminus T$. Note that $e$ is not a loop. Hence, there is a unique cycle $\gamma$ on $\Gamma$ such that $e\in E(\gamma)$ and $\emptyset\ne E(\gamma)\setminus \{e\}\subset T$. Note that $\Gamma_{E(\Gamma)\setminus (S\cup\{e\})}$ does not have cycles, hence $E(\gamma)\setminus\{e\}$ is not contained in $S$. If we choose $e'$ as any edge in $E(\gamma)\setminus (S\cup \{e\})$, it follows that $(T\cup\{e\})\setminus \{e'\}$ is a spanning tree. \end{proof}
The category whose objects are weighted graphs with $n$ legs of genus $g$ and whose morphisms are specializations is denoted by $\mathbf{Graph}_{g,n}$. The poset $\mathcal{SG}_{g,n}$ will be the set whose elements are isomorphism classes of weighted graphs with $n$ legs of genus $g$, where the partial order is given by specialization.
\subsection{Partially ordered sets} \label{sub:tropicalcurves}
A \emph{poset} (\emph{partially ordered set}) is a pair $(S,\leq)$ where $S$ is a set and $\leq$ is a partial order on $S$. In this paper we will only consider finite posets. A function $f\colon S\to S'$ is called \emph{order preserving} if $x\leq y$ implies $f(x)\leq f(y)$. A subset $T\subset S$ is called a \emph{lower set} (respectively, \emph{upper set}) if $x\leq y$ (respectively, $y\leq x$) and $y\in T$ implies that $x\in T$. A poset $S$ can be given two natural topologies, one where the closed sets are the lower sets, and the other where the closed sets are the upper sets. In this paper we choose the topology induced by the lower sets.\par
A function $f\colon S\to S'$ between posets is continuous if and only if it is order preserving. Moreover, a continuous function $f$ is closed if for every $y_1, y_2\in S'$ with $y_1\leq y_2$ and $y_2=f(x_2)$, there exists $x_1\in S$ such that $x_1\le x_2$ and $f(x_1)=y_1$.\par Let $S$ be a poset. A \emph{chain} in $S$ is a sequence $x_0<x_1<\ldots < x_n$. We call $n$ the \emph{length} of the chain, and we say that $x_0$ (respectively, $x_n$) is the \emph{starting point} (respectively, \emph{ending point}) of the chain. We say that $S$ is \emph{ranked} (of length $n$) if every maximal chain has length $n$. It is easy to see that the maximal length of chains in $S$ is precisely the Krull dimension of $S$ as a topological space. Therefore, if $S$ is ranked then $S$ will be pure dimensional. For every $x\in S$ we define the \emph{dimension} of $x$ as the maximum length for a chain ending in $x$; in other words, the dimension of $x$ is precisely the Krull dimension of $\overline{\{x\}}$. If $S$ is a ranked poset of length $n$, then we define the \emph{codimension} of $x\in S$ as $n-\dim_S(x)$, i.e., the length of all maximal chains starting from $x$. A poset $S$ is \emph{graded} if it has a function $\text{rk}\colon S\to \mathbb{N}$, called \emph{rank function}, such that $\text{rk}(x)=\text{rk}(y)+1$ whenever $y<x$ and there is no $z\in S$ with $y<z<x$. Every ranked poset is graded with rank function given by the dimension.\par
Let $S$ be a ranked poset $S$. We say that $S$ is \emph{connected in codimension one} if for every maximal elements $y,y'\in S$ there are two sequences of elements $x_1,\ldots, x_n\in S$ and $y_0,\ldots, y_n\in S$ such that
\begin{enumerate}[label=(\roman*)]
\item $x_i$ has codimension $1$ for every $i=1,\ldots, n$.
\item $y_i$ is maximal for every $i=0,\ldots,n$.
\item $y_0=y$ and $y_n=y'$.
\item $x_{i+1}< y_i$ for every $i=0,\ldots,n-1$ and $x_i<y_i$ for every $i=1,\ldots, n$.
\end{enumerate} We call these sequences a \emph{path in codimension $1$} from $y$ to $y'$.
\section{Tropical curves and their moduli}\label{sec:trop} \subsection{Tropical curves}
A \emph{metric graph} is a pair $(\Gamma,\ell)$ where $\Gamma$ is a graph and $\ell$ is a function $\ell\colon E(\Gamma)\to \mathbb{R}_{>0}$ called the length function. If $\ora{\Gamma}$ is an orientation on $\Gamma$, we define the \emph{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 $(\ell(e),e)\sim t(e)$. The tropical curve $X$ has a natural topology and its connected components have a natural structure of metric space. Note that the definition of $X$ does not depend on the chosen orientation. For two points $p,q\in I_e$ we denote by $\overline{pq}$ (respectively, $\overrightarrow{pq}$) the interval (respectively, oriented interval) $[p,q]\subset I_e$.\par
We say that the tropical curves $X$ and $Y$ are \emph{isomorphic} if there is a bijection between $X$ and $Y$ that is an isometry over each connected component of $X$. We say that $\Gamma$ (respectively, $\overrightarrow{\Gamma}$) is a \emph{model} (respectively, \emph{directed model}) of $X$ if there exists a length function $\ell$ on $\Gamma$ such that $X$ and the tropical curve associated to $(\Gamma,\ell)$ are isomorphic. Sometimes, we will use interchangeably the notion of tropical curve and of its isomorphism class. If $X$ and $Y$ are tropical curves and $\Gamma_X$ and $\Gamma_Y$ are models for $X$ and $Y$, we say that the pairs $(X,\Gamma_X)$ and $(Y,\Gamma_Y)$ are \emph{isomorphic} if there exists an isomorphism $f\colon X\to Y$ of tropical curves such that $f(V(\Gamma_X))=V(\Gamma_Y)$ and such that $f$ induces an isomorphism of graphs between $\Gamma_X$ and $\Gamma_Y$. \par
Let $X$ be a tropical curve. If $X$ is connected, then every model of $X$ is a connected graph. Conversely, if a model of $X$ is connected, then so is $X$. Note that $X$, as topological space, might fail to be pure dimensional: this happens, for instance, if $\Gamma$ has isolated vertices.\par
We say that a tropical curve $Y$ is a \emph{tropical subcurve} of $X$ if there exists an injection $Y\subset X$ that is an isometry over each connected component of $Y$. In this case, if $\Gamma_Y$ is a model of $Y$, one can choose a model $\Gamma_X$ of $X$ such that $\Gamma_Y$ is a subgraph of $\Gamma_X$. Conversely, if $\Gamma'$ is a subgraph of a model $\Gamma_X$ of $X$, then $\Gamma'$ induces a tropical subcurve $Y$ of $X$. Note that a tropical subcurve can be a single point. If $Y$ and $Z$ are tropical subcurves of $X$ then $Y\cap Z$ and $Y\cup Z$ are also tropical subcurves of $X$.
From now on all tropical curves will be connected, and in particular pure dimensional, while we will allow possibly nonconnected (and hence possibly non pure dimensional) tropical subcurves. \par
Let $X$ be a tropical curve with a model $\Gamma_X$, and let $Y\subset X$ be a tropical subcurve of $X$. Then, there exists a minimal refinement $\Gamma_{X,Y}$ of $\Gamma_X$ such that $Y$ is induced by a subgraph $\Gamma_Y$ of $\Gamma_{X,Y}$. We define \[ \delta_{X,Y}:=\sum_{v\in V(\Gamma_Y)}\text{val}_{E(\Gamma_{X,Y})\setminus E(\Gamma_Y)}(v) \] The definition of $\delta_{X,Y}$ does not depend on the choice of the model $\Gamma_X$ of $X$. When no confusion may arise we will simply write $\delta_Y$ instead of $\delta_{X,Y}$.
We also define the set \[ \text{Out}(Y):=E(V(\Gamma_Y),V(\Gamma_{X,Y})\setminus V(\Gamma_Y)). \] Equivalently, the set $\text{Out}(Y)$ is the cut in $\Gamma_{X,Y}$ induced by $V(\Gamma_Y)$. The definition of the set $\text{Out}(Y)$ depends on the choice of the model $\Gamma_X$ (see the next example). \par \begin{Exa} \label{exa:out} Let $X$ be a tropical curve as in Figure \ref{fig:out}, with model $\Gamma_X$ whose vertices are the points $p_0,p_1,p_2,p_3,p_4$. Let $Y$ be the subcurve of $X$, given by \[ Y:=\overline{p_0q_0}\cup \overline{p_0q_1}\cup \overline{p_0q_3}\cup \overline{p_3q_4}\cup \{q_2\}. \] Then we have \[ \text{Out}(Y)=\{\overline{q_0p_2},\overline{q_2p_1},\overline{q_3p_1},\overline{q_4p_4}\}. \] Note that the edge $\overline{q_1q_2}$ does not belong to $\text{Out}(Y)$. If we choose a model for $X$ with vertices the points $p_0,p_1,p_3,p_4$, then the edge $\overline{q_0p_1}$ will be in $\text{Out}(Y)$ instead of $\overline{q_0p_2}$. On the other hand, if we choose a model with vertices $p_0,p_1,p_2,p_3,p_4,p_5$, for some $p_5$ in the interior of $\overline{q_1q_2}$, then the edges $\overline{q_1p_5}$ and $\overline{q_2p_5}$ will be in $\text{Out}(Y)$. \begin{figure}
\caption{An example of the set $\text{Out}(Y)$.}
\label{fig:out}
\end{figure} \end{Exa}
Let $X$ be a tropical curve, $\Gamma_X$ a model of $X$, and $\ell$ the induced metric on $\Gamma_X$. A specialization $\iota\colon\Gamma_X\rightarrow\Gamma'$ induces a metric $\ell'$ on $\Gamma'$ defined as \[ \ell'\colon E(\Gamma')\stackrel{\iota_*}{\hookrightarrow} E(\Gamma)\stackrel{\ell}{\to} \mathbb R_{>0}. \] Let $Y$ be the tropical curve associated to $(\Gamma',\ell')$. Then there exists an induced function $\iota\colon X\to Y$ that is constant on the edges of $\Gamma_X$ contracted by $\iota$. We call this function a \emph{specialization of $X$ to $Y$}.\par
Let $X$ be a tropical curve and $(\ora{\Gamma},\ell)$ be a directed model of $X$. Given $e\in E(\ora{\Gamma})$ and $a\in\mathbb R$ with $0\le a\le\ell(e)$, we let $p_{a,e}$ be the point of $X$ lying on $e$ whose distance from $s(e)$ is $a$. A \emph{divisor} on $X$ is a map $\mathcal D\colon X\to \mathbb{Z}$ such that $\mathcal D(p)\neq0$ for finitely many points $p\in X$. We define the \emph{support} of $\mathcal D$ as the set of points $p$ of $X$ such that $\mathcal D(p)\neq0$ and denote it by $\text{supp}(\mathcal D)$.
If $\Gamma$ is a model of $X$, then every divisor on $\Gamma$ can be seen as a divisor on $X$. Given a divisor $\mathcal D$ on $X$, the degree of $\mathcal D$ is the integer $\deg \mathcal D:=\sum_{p\in X} \mathcal D(p)$. We say that $\mathcal D$ is \emph{effective} if $\mathcal D(p)\ge0$ for every $p\in X$. We let $\text{Div}(X)$ be the Abelian group of divisors on $X$; we denote by $\text{Div}^d(X)$ the subset of degree-$d$ divisors on $X$. If $X$ and $Y$ are tropical curves and if $\mathcal D$ and $\mathcal D'$ are divisors on $X$ and $Y$, respectively, we say that the pairs $(X,\mathcal D)$ and $(Y, \mathcal D')$ are \emph{isomorphic} if there is an isomorphisms $f\colon X\to Y$ of tropical curves such that $\mathcal D(p)=\mathcal D'(f(p))$ for every $p\in X$.\par
A \emph{rational function} on $X$ is a continuous, piecewise linear function $f\colon X\rightarrow \mathbb R$ with integer slopes. We say that a rational function $f$ on $\Gamma$ has \emph{slope $s$ over} $\ora{p_{e,a_1},p_{e,a_2}}$, for $a_1,a_2\in\mathbb R$ and $0\leq a_1< a_2\leq \ell(e)$, if the restriction of $f$ to the locus of points $p_{a,e}$ for $a_1\leq a \leq a_2$ is linear and has slope $s$.
A \emph{principal divisor} on $\Gamma$ is a divisor \[ \textnormal{div}_X(f):=\sum_{p\in X} \textnormal{ord}_p(f) p\in \text{Div}(X), \] where $f$ is a rational function on $X$ and $\textnormal{ord}_p(f)$ is the sum of the incoming slopes of $f$ at $p$. A principal divisor has degree zero. The \emph{support} of a rational function $f$ on $X$ is defined as the set $\text{supp}(f)=\{p\in X;\textnormal{ord}_p(f)\neq 0\}$. We denote by $\text{Prin}(X)$ the subgroup of $\text{Div}(X)$ of principal divisors. Given divisors $\mathcal D_1,\mathcal D_2\in\text{Div}(X)$, we say that $\mathcal D_1$ and $\mathcal D_2$ are \emph{equivalent} if $\mathcal D_1-\mathcal D_2\in \text{Prin}(X)$. The \emph{Picard group} $\Pic(X)$ of $X$ is \[ \Pic(X):=\text{Div}(X)/\text{Prin}(X). \] The \emph{degree-$d$ Picard group} of $X$ is defined as $\Pic^d(X):=\text{Div}^d(X)/\text{Prin}(X)$.\par
Let $f$ be a rational function on the tropical curve $X$ and $\Gamma$ be a model of $X$ such that $\text{supp}(f)\subset V(\Gamma)$. Then $f$ is linear over each edge of $\Gamma$. If $f$ is nowhere constant, then it induces an orientation $\ora{\Gamma}$ on $\Gamma$, such that $f$ has always positive slopes. In this case, we can define a positive flow $\phi_f$ on $\ora{\Gamma}$ where $\phi_f(e)$ is equal to the slope of $f$ over $e$, for every $e\in E(\ora\Gamma)$. It is clear that $\textnormal{div}_X(f)=\textnormal{div}(\phi_f)$, where $\textnormal{div}(\phi_f)$ is seen as a divisor on $X$. Note that the orientation $\ora{\Gamma}$ on $\Gamma$ induced by a rational function $f$ is acyclic, because there are no strictly increasing functions on the circle $S^1$. If $f$ is constant on a subset $\mathcal E\subset E(\Gamma)$, then we can contract all edges in $\mathcal E$ and get a specialization $\iota\colon X\to Y$ of tropical curves. Clearly, $f$ induces a nowhere constant rational function on $Y$. Hence $f$ induces an acyclic orientation $\ora{\Gamma/\mathcal E}$ on $\Gamma/\mathcal E$ and a positive flow $\phi_f$ on $\Gamma/\mathcal E$.\par
Fix a model $\Gamma_X$ for the tropical curve $X$. For every tropical subcurve $Y\subset X$ and for every $\ell\in\mathbb R_{\geq0}$ such that $\ell\leq \min_{e\in\text{Out}(Y)}\{\ell(e)\}$, we define the divisor $\mathcal D_{Y,\ell}$ on $X$ taking $p\in X$ to \begin{equation} \label{eq:chip} \mathcal D_{Y,\ell}(p)=\begin{cases}
\begin{array}{ll}
-|\{e\in \text{Out}(Y); p=p_{e,0}\}| &\text{if $p\in Y$}; \\
|\{e\in \text{Out}(Y); p=p_{e,\ell}\}| &\text{if $p\notin Y$},
\end{array}
\end{cases} \end{equation} where we consider the edges $e\in\text{Out}(Y)$ oriented in the direction away from $Y$. The divisor $\mathcal D_{Y,\ell}$ is principal, since we have $\mathcal D_{Y,\ell}=\textnormal{div}_X(f)$ where $f$ has slope $1$ over $\overrightarrow{p_{e,0}p_{e,\ell}}$ for every edge $e\in\text{Out}(Y)$ and has slope $0$ everywhere else. Note that $f$ is well defined because the edges in $\text{Out}(Y)$ form a cut of $\Gamma_{X,Y}$, while the definition of $\mathcal D_{Y,\ell}$ depends on the choice of the model $\Gamma_X$ of $X$. We call $\mathcal D_{Y,\ell}$ a \emph{chip-firing divisor emanating from $Y$ with length $\ell$}.
\begin{Exa} \label{exa:chip} We maintain the definitions in Example \ref{exa:out}, where the model $\Gamma_X$ has vertices $p_0,p_1,p_2,p_3,p_4$. Assume that the edges $\overline{q_0p_2}$, $\overline{q_2q_5}$, $\overline{q_3p_1}$ and $\overline{q_4q_6}$ have all the same length $\ell$. Hence \[ \mathcal D_{Y,\ell}=(p_2-q_0)+(q_5-q_2)+(p_1-q_3)+(q_6-q_4). \] \begin{figure}
\caption{An example of a chip firing divisor.}
\label{fig:chip}
\end{figure}
\end{Exa}
\begin{Lem} \label{lem:valP} If $f$ is a nowhere constant rational function on a tropical curve $X$, then there exists a point $p$ in $X$ such that $\textnormal{ord}_p(f)\geq\delta_{X,p}$. \end{Lem} \begin{proof} The rational function $f$ induces a positive flow on a directed model $\ora{\Gamma_X}$ of $X$. The result follows from choosing $p$ to be a sink of $\ora{\Gamma_X}$. \end{proof} \begin{Rem} \label{rem:uniquef} Let $f$ and $f'$ be rational functions on a tropical curve $X$. Then $\textnormal{div}(f)=\textnormal{div}(f')$ if and only if $f-f'$ is a constant function. Indeed, it suffices to show that if $\textnormal{div}(f)=0$, then $f$ is constant. This follows by contracting the edges where $f$ is constant: if the resulting tropical curve is not a point, then using Lemma \ref{lem:valP} we see that $\textnormal{div}(f)\ne 0$. \end{Rem}
Let $X$ be a tropical curve. Given a directed model $\ora{\Gamma_X}$ of $X$, a \emph{$1$-form} on $X$ is a formal sum $\omega=\sum_{e\in E(\ora{\Gamma_X})}\omega_e\cdot de$, where $\omega_e\in\mathbb R$. We say that $\omega$ is \emph{harmonic} if for every $v\in V(\ora{\Gamma_X})$, \[ \underset{s(e)=v}{\sum_{e\in E(\ora{\Gamma_X})}}\omega_e=\underset{t(e)=v}{\sum_{e\in E(\ora{\Gamma_X})}}\omega_e. \] Let $\Omega(X)$ be the real vector space of harmonic $1$-forms and $\Omega(X)^\vee$ be its dual. Note that $\Omega(X)$ does not depend on the choice of $\ora{\Gamma_X}$. Given edges $e,e'\in E(\ora{\Gamma_X})$ and points $p_{e',a_1},p_{e',a_2}\in e'$ for $a_1,a_2\in\mathbb R$, we define the \emph{integration of $de$ over} $\overrightarrow{p_{e',a_1}p_{e',a_2}}$ as \[ \int_{p_{e',a_1}}^{p_{e',a_2}}de=\begin{cases}
\begin{array}{ll}
a_2-a_1,&\text{ if $e'=e$;}\\
0,&\text{ otherwise.}
\end{array}
\end{cases} \] There is a natural isomorphism (see \cite[Lemma 2.1]{BF}) \begin{align*} H_1(\Gamma_X,\mathbb R)&\to \Omega(X)^\vee\\
\gamma&\mapsto \int_\gamma\colon \end{align*} and hence $H_1(\Gamma_X,\mathbb{Z})$ can be viewed as a lattice in $\Omega(X)^\vee$. The \emph{Jacobian} of the tropical curve $X$ is defined as the real torus \begin{equation}\label{eq:Jtropdef} J^{\text{trop}}(X)=\Omega(X)^\vee/H_1(\Gamma_X,\mathbb{Z}). \end{equation}
Note that the definition of the Jacobian does not depend on the chosen directed model of $X$.
Fix a point $p_0$ in $X$ and assume that $p_0$ is a vertex of $\ora{\Gamma_X}$. Let $p_{e_1,a_1},\ldots,p_{e_d,a_d}$ be points of $X$. Choose a path $\gamma_i$ on $\ora{\Gamma_X}$ from $p_0$ to $s(e_i)$ for every $i=1,\dots,d$. One can define a map \begin{align*} \alpha\colon X^d&\longrightarrow \Omega(X)^\vee\\
(p_{e_i,a_i})_{i=1,\ldots,d}&\longmapsto \sum_{i=1}^{d} \left(\int_{\gamma_i}+\int_{s(e_i)}^{p_{e_i,a_i}}\right). \end{align*}
The \emph{degree-$d$ Abel-Jacobi map} of the tropical curve $X$ is the composition of $\alpha$ with the quotient map $\Omega(X)^\vee\to J^\text{trop}(X)$. Note that, while the map $\alpha$ may depend on the choices of the paths $\gamma_1,\dots,\gamma_d$, the degree-$d$ Abel-Jacobi map does not.
An \emph{$n$-pointed tropical curve} is a pair $(X,\text{leg})$ where $\text{leg}\colon I_n=\{0,\dots,n-1\} \to X$ is a function. For every $p\in X$, we set $\ell(p):=|\text{leg}^{-1}(p)|$. A \emph{weight} on a tropical curve $X$ is a function $w\colon X\to \mathbb{Z}_{\geq0}$, such that $w(p)\neq 0$ only for finitely many $p\in X$. Given an $n$-pointed weighted tropical curve $X$, we define \[ V(X):=\{p\in X; \text{ either }\delta_{X,p}\neq 2,\text{or }\ell(p)\geq1,\text{or }w(p)\geq 1\}. \]
Note that $V(X)\subset V(\Gamma)$ for every model $\Gamma$ of $X$; if $V(X)=V(\Gamma)$, we say that $\Gamma$ is a \emph{minimal model} of $X$. We say that an $n$-pointed weighted tropical curve $(X,w,\text{leg})$ is \emph{stable} if $\delta_{X,p}+2w(p)+\ell(p)\ge 3$ for every point $p\in X$ such that $\delta_{X,p}\leq1$. The genus $g(X,w)$ of a weighted tropical curve $(X,w)$ is defined by the formula \[ 2g(X,w)-2:=\sum_{p\in X}(2w(p)-2+\delta_{X,p}). \] The sum on right hand side of the last formula is finite, since $(\delta_{X,p}, w(p))\neq (2,0)$ only for finitely many $p\in X$.
If $(X,w,\text{leg})$ is a stable $n$-pointed weighted tropical curve, then there exists a unique stable weighted graph with $n$ legs $\Gamma_{st}$ that is a model of $X$ with induced weights and legs satisfying the following property: for every $p\in X$ such that either $w(p)\neq0$ or $\ell(p)\neq0$, then $p\in V(\Gamma_{st})$. We call $\Gamma_{st}$ the \emph{stable model} of $X$.\par
For the remainder of the paper we will usually omit to denote the weight $w$ of a weighted tropical curve.
\subsection{Tropical moduli spaces}
Given a finite set $S\subset \mathbb R^n$ we define \[
\textnormal{cone}(S):=\left\{\sum_{s\in S}\lambda_ss|\lambda_s\in \mathbb R_{\ge0}\right\}. \] A subset $\sigma\subset \mathbb R^n$ is called a \emph{polyhedral cone} if $\sigma=\textnormal{cone}(S)$ for some finite set $S\subset \mathbb{R}^n$. If there exists $S\subset \mathbb{Z}^n$ with $\sigma=\textnormal{cone}(S)$ then $\sigma$ is called \emph{rational}.\par
Every polyhedral cone is the intersection of finitely many closed half-spaces. The \emph{dimension} of $\sigma$, denoted $\dim(\sigma)$, is the dimension of the minimal linear subspace containing $\sigma$. The \emph{relative interior} $\sigma^\circ$ is the interior of $\sigma$ inside this linear subspace. A \emph{face} of $\sigma$ is the intersection of $\sigma$ with some linear subspace $H\subset \mathbb{R}^n$ of codimension one such that $\sigma$ is contained in one of the closed half-spaces determined by $H$. A face of $\sigma$ is also a polyhedral cone. If $\tau$ is a face of $\sigma$ then we write $\tau\prec \sigma$. In the sequel we will use the terminology \emph{cone} to mean \emph{rational polyhedral cone}. \par
A morphism $f\colon \tau\to\sigma$ between cones $\tau\subset \mathbb R^m$ and $\sigma\subset \mathbb R^n$ is the restriction to $\tau$ of an integral linear transformation $T\colon\mathbb R^n\to \mathbb R^m$ such that $T(\tau)\subset\sigma$. We say that $f$ is an isomorphism if there exists an inverse morphism $f^{-1}\colon\sigma\to\tau$.
A morphism $f\colon\tau\to\sigma$ is called a \emph{face morphism} if $f$ is an isomorphism between $\tau$ and a (not necessarily proper) face of $\sigma$.\par
A \emph{generalized cone complex} $\Sigma$ is the colimit (as topological space) of a finite diagram $\mathbf{D}$ of cones with face morphisms. \par
We say that $\sigma\in \mathbf{D}$ is \emph{maximal} if there is no proper face morphism $f\colon\sigma\to\tau$ in $\mathbf{D}$. We say that $\Sigma$ is \emph{pure dimensional} if all maximal cones in $\mathbf{D}$ have the same dimension. We say that $\Sigma$ is \emph{connected through codimension one} if for every pair $\sigma,\sigma'$ of maximal cones there are two sequences of cones $\tau_1,\ldots,\tau_n\in \mathbf{D}$ and $\sigma_0,\ldots, \sigma_n\in \mathbf{D}$ such that
\begin{enumerate}[label=(\roman*)]
\item $\tau_i$ has codimension $1$ for every $i=1,\ldots, n$;
\item $\sigma_i$ is maximal for every $i=0,\ldots,n$;
\item $\sigma_0=\sigma$ and $\sigma_n=\sigma'$;
\item there exists a face morphism $\tau_{i+1}\to \sigma_i$ in $\mathbf{D}$ for every $i=0,\ldots,n-1$, and a face morphism $\tau_i\to\sigma_i$ in $\mathbf{D}$ for every $i=1,\ldots, n$.
\end{enumerate}\par
A \emph{morphism of generalized cone complexes} is a continuous map of topological spaces $f\colon\Sigma\to\Sigma'$ such that for every cone $\sigma\in \mathbf{D}$ there exists a cone $\sigma'\in \mathbf{D}'$ such that the induced map $\sigma\to\Sigma'$ factors through a cone morphism $\sigma\to\sigma'$. \par
A \emph{polyhedron} $\mathcal{P}\subset \mathbb R^n$ is an intersection of a finite number of half-spaces of $\mathbb R^n$. A face of a polyhedron $\mathcal{P}$ is the intersection of $\mathcal{P}$ and a hyperplane $H$ such that $\mathcal{P}$ is contained in a closed half-space determined by $H$. A morphism $f\colon \mathcal{P}\rightarrow \mathcal{P}'$ between polyhedra $\mathcal{P}\subset \mathbb R^n$ and $\mathcal{P}'\subset \mathbb R^m$ is the restriction to $\mathcal{P}$ of an affine map $T\colon\mathbb R^n\to\mathbb R^m$ such that $T(\mathcal{P})\subset \mathcal{P}'$. A morphism $f\colon \mathcal{P}\to \mathcal{P}'$ of polyhedra is a \emph{face morphism} if the image of $f(\mathcal{P})$ is a face of $\mathcal{P}'$ and $f$ is an isometry. A \emph{polyhedral complex} is the colimit (as topological space) of a finite poset $\mathbf{D}$ of polyhedra with face morphisms. \par
For a graph $\Gamma$, the open cone $\mathbb{R}_{> 0}^{|E(\Gamma)|}$ parametrizes all possible choices for the lengths of the edges of $\Gamma$. Hence, $M_\Gamma^{\text{trop}}:=\mathbb R^{E(\Gamma)}_{>0}/\textnormal{Aut}(\Gamma)$ parametrizes isomorphism classes of pairs $(X,\Gamma_X)$, where $X$ is a tropical curve and $\Gamma_X$ is a model of $X$ isomorphic to $\Gamma$. We will identify $E(\Gamma)$ with the canonical basis of $\mathbb{R}^{|E(\Gamma)|}$. If $\iota\colon \Gamma\rightarrow \Gamma'$ is a specialization, then there exists an inclusion $\mathbb R^{E(\Gamma')}\subset \mathbb R^{E(\Gamma)}$ induced by the inclusion $E(\Gamma')\subset E(\Gamma)$. If $\Gamma'$ is a refinement of $\Gamma$, there exists a map \begin{equation} \label{eq:hat} f\colon\mathbb R^{E(\Gamma')}\to \mathbb R^{E(\Gamma)} \end{equation} given by $f((x_{e'})_{{e'}\in E(\Gamma')})=(y_e)_{e\in E(\Gamma)}$, with \[ y_e=\underset{b(e')=e}{\sum_{e'\in E(\Gamma')}}x_{e'}. \] (Recall that $b\colon E(\Gamma')\to E(\Gamma)$ is the surjection induced by the refinement $\Gamma'$.) \par
If $X$ is a stable $n$-pointed genus-$g$ tropical curve, then $X$ admits exactly one stable model $\Gamma_X$. Hence the moduli space $M_{g,n}^{\text{trop}}$ of stable $n$-pointed tropical curves of genus $g$ is the generalized cone complex given as the colimit of the diagram whose cones are $\mathbb{R}_{\geq0}^{|E(\Gamma)|}$, where $\Gamma$ runs through all stable genus-$g$ weighted graphs with $n$ legs, with face morphisms specified by specializations. More precisely, if $\Gamma$ specializes to $\Gamma'$, then $\mathbb{R}_{\geq 0}^{|E(\Gamma')|}$ is a face of $\mathbb{R}_{\geq 0}^{|E(\Gamma)|}$ via the inclusion $E(\Gamma')\subset E(\Gamma)$. For more details about $M_{g,n+1}^{\text{trop}}$ and its compactification $\overline{M}_{g,n+1}^{\textnormal{trop}}$, see \cite[Section 2]{M}, \cite[Sections 2.1 and 3.2]{BMV}, \cite[Section 3]{Caporaso1}, \cite[Section 3]{Caporaso}, and \cite[Section 4]{ACP}.\par
\section{Quasistability on graphs}\label{sec:quasigraph}
In this section we introduce the notion of quasistability for pseudo-divisors on graphs. We will study several properties of the poset formed by these divisors, in particular how it behaves under the operations of contraction and deletion of edges. All graphs will be considered connected unless otherwise specified.
Let $\Gamma$ be a graph. Let $d$ be an integer. A \emph{degree-$d$ polarization} on $\Gamma$ is a function $\mu\colon V(\Gamma)\to\mathbb R$ such that $\sum_{v\in V(\Gamma)}\mu(v)=d$.
Let $\mu$ be a degree-$d$ polarization on $\Gamma$. For every $V\subset V(\Gamma)$ we set $\mu(V):=\sum_{v\in V}\mu(v)$. Given a degree-$d$ divisor $D$ on $\Gamma$, we define \[
\beta_D(V):=\deg(D|_V)-\mu(V)+\frac{\delta_V}{2}. \] If $\iota\colon\Gamma\rightarrow\Gamma'$ is a specialization, then there is a induced degree-$d$ polarization $\iota_*(\mu)$ on $\Gamma'$ defined as \[ \iota_*(\mu)(v'):=\underset{v \in \iota^{-1}(v')}{\sum_{v\in V(\Gamma)}}\mu(v). \]
If $\mathcal E\subset E(\Gamma)$ is a nondisconnecting subset of edges, then there is an induced degree-$(d+|\mathcal E|)$ polarization $\mu_\mathcal E$ on $\Gamma_\mathcal E$ (the graph induced by removing the edges $\mathcal E$ in $E(\Gamma)$) given as \[ \mu_\mathcal E(v):=\mu(v)+\frac{1}{2}\text{val}_\mathcal E(v). \] Given a subdivision $\Gamma^{\mathcal E}$ of $\Gamma$ for some $\mathcal E\subset E(\Gamma)$, there is an induced degree-$d$ polarization $\mu^\mathcal E$ on $\Gamma^\mathcal E$ given as \[ \mu^\mathcal E(v):= \begin{cases} \begin{array}{ll} \mu(v)& \text{if}\;v\in V(\Gamma);\\ 0&\text{otherwise.} \end{array} \end{cases} \]
We now prove a numerical result concerning the function $\beta_D$, which is, essentially, the same statement as in \cite[Lemma 3]{Es01} (in the case of invertible sheaves). \begin{Lem} \label{lem:cap} Given subsets $V$ and $W$ of $V(\Gamma)$, we have \[
\beta_D(V\cup W)+\beta_D(V\cap W)=\beta_D(V)+\beta_D(W)-|E(V,W)|. \] In particular, $\beta_D(V)+\beta_D(V^c)=\delta_V$. \end{Lem} \begin{proof} It is sufficient to prove that \[
\delta_{V\cup W}+\delta_{V\cap W}=\delta_V+\delta_W-2|E(V,W)|. \] We can check that the last equality holds by proving that each edge $e\in E(\Gamma)$ is counted the same amount of times in each side. In the table below we enumerate all cases (up to symmetry). We write $z_1$ and $z_2$ for the (possibly coincident) vertices incident to $e$. \[
\begin{array}{l|c|c|c|c|c}
& \delta_{V\cup W} &\delta_{V\cap W}&\delta_{V}&\delta_W& |E(V,W)|\\ \hline z_1,z_2\notin V\cup W&0&0&0&0&0\\ z_1\in V\setminus W, z_2\notin V\cup W&1&0&1&0&0\\ z_1,z_2\in V\setminus W &0&0&0&0&0\\ z_1\in V\setminus W, z_2\in W\setminus V&0&0&1&1&1\\ z_1\in V\cap W, z_2\notin V\cup W&1&1&1&1&0\\ z_1\in V\cap W, z_2\in V\setminus W&0&1&0&1&0\\ z_1,z_2\in V\cap W&0&0&0&0&0\\ \end{array} \] This concludes the proof. \end{proof}
Let $\Gamma$ be a graph and $\mu$ a degree-$d$ polarization on $\Gamma$. We say that a degree-$d$ divisor $D$ on $\Gamma$ is $\mu$-\emph{semistable} if $\beta_D(V)\geq0$ for every $V\subset V(\Gamma)$. Given $v_0\in V(\Gamma)$, we say that a degree-$d$ divisor $D$ on $\Gamma$ is $(v_0,\mu)$-\emph{quasistable} if $\beta_D(V)\geq0$ for every $V\subsetneq V(\Gamma)$, with strict inequality if $v_0\in V$. We say that a pseudo-divisor $(\mathcal E,D)$ is \emph{$\mu$-semistable} (respectively, $(v_0,\mu)$-\emph{quasistable}) if $D$ is $\mu^\mathcal E$-semistable (respectively, $(v_0,\mu^\mathcal E)$-quasistable) on $\Gamma^\mathcal E$. Clearly every $(v_0,\mu)$-quasistable pseudo-divisor is $\mu$-semistable.\par Note that if $\Gamma$ is not connected, then the condition of quasistability can not be satisfied by any divisor on $\Gamma$.
\begin{Rem} \label{rem:Vc} By Lemma \ref{lem:cap}, we have that $\beta_D(V)+\beta_D(V^c)=\delta_V$, hence $\beta_D(V)\geq 0$ if and only if $\beta_D(V^c)\leq \delta_{V^c}$. We deduce that a divisor $D$ on $\Gamma$ is $(v_0,\mu)$-quasistable if and only if $\beta_D(V)\leq\delta_V$ for every nonempty $V\subsetneq V(\Gamma)$, with strict inequality if $v_0\notin V$. In the sequel, we will freely use both characterizations of quasistability. \end{Rem}
The following is the combinatorial translation of a result due to Esteves about quasistable invertible sheaves on nodal curves. \begin{Thm} \label{thm:esteves} Every divisor $D$ on $\Gamma$ is equivalent to a unique $(v_0,\mu)$-quasistable divisor. \end{Thm} \begin{proof} This follows from \cite[Proposition 27 and Theorem 32]{Es01}. Alternatively, this is also a consequence of Theorem \ref{thm:quasistable} (which gives the analogue statement for tropical curves). \end{proof} \begin{Rem} \label{rem:subdivision} It is easy to see that if $D$ is a $(v_0,\mu^\mathcal E)$-quasistable divisor on $\Gamma^\mathcal E$ then $D(v)=0, -1$ for every exceptional vertex $v\in V(\Gamma^\mathcal E)$. Moreover if $\widehat{\Gamma}$ is a refinement of $\Gamma$ then $\mu$ induces a polarization $\widehat{\mu}$ in $\widehat{\Gamma}$ given by \[ \widehat{\mu}(v)=\begin{cases}
\begin{array}{ll}
0,&\text{ if $v$ is exceptional;}\\
\mu(v),& \text{ if $v\in V(\Gamma)$},
\end{array}
\end{cases}
\] where we view $V(\Gamma)$ as a subset of $V(\widehat{\Gamma})$ via the injection $a\colon V(\Gamma)\rightarrow V(\widehat{\Gamma})$ induced by the refinement (see Subsection \ref{subsec:graphs}). If $D$ is a $(v_0,\widehat{\mu})$-quasistable divisor on $\widehat{\Gamma}$ then for every edge $e\in E(\Gamma)$, we have that $D(v)=0$ for all but at most one exceptional vertex $v$ over $e$; if such a vertex $v$ over $e$ exists, then $D(v)=-1$. Hence, every $(v_0,\widehat{\mu})$-quasistable divisor $D$ of $\widehat{\Gamma}$ induces a $(v_0,\mu)$-quasistable pseudo-divisor $(\mathcal E',D')$ on $\Gamma$. \end{Rem}
\begin{Rem}
The fact that the degree of quasistable divisors are $0$ or $-1$ on exceptional vertices follows from the choice of which inequalities are strict in the definition of quasistability. We could have defined quasistability by asking that $-D$ is $(v_0,-\mu)$-quasistable. In this case quasistable divisors would have degree $0$ or $1$ on the exceptional vertices. All the constructions and results of the paper would hold in this alternative setting. \end{Rem}
In the next result we study how quasistability behaves under any specialization and deletion of nondisconnecting edges. This is an important tool to establish an interplay between quasistable pseudo-divisors on a graph and quasistable divisors on spanning connected subgraphs (possibly changing the polarization).
\begin{Prop} \label{prop:spec} Let $\Gamma$ be a graph, $v_0$ a vertex of $\Gamma$, and $\mu$ a degree-$d$ polarization on $\Gamma$. Assume that $\iota\colon\Gamma\rightarrow\Gamma'$ is a specialization of graphs. For every pseudo-divisor $(\mathcal E,D)$ on $\Gamma$ of degree-$d$, the following properties hold: \begin{enumerate} \item[(i)] if $(\mathcal E,D)$ is $(v_0,\mu)$-quasistable then $\iota_*(\mathcal E,D)$ is an $(\iota(v_0),\iota_*(\mu))$-quasistable pseudo-divisor on $\Gamma'$; \item[(ii)] $(\mathcal E,D)$ is $(v_0,\mu)$-quasistable if and only if $\mathcal E$ is nondisconnecting and $D_\mathcal E$ is a $(v_0,\mu_\mathcal E)$-quasistable divisor on $\Gamma_\mathcal E$. \end{enumerate} \end{Prop} \begin{proof} Let us start proving \emph{(i)}. Upon switching $\Gamma$ with $\Gamma^\mathcal E$, we can assume that $\mathcal E=\emptyset$. We need to show that for every $V'\subsetneq V(\Gamma')$ we have $\beta_{\iota_*(D)}(V')\geq0$, with strict inequality if $\iota(v_0)\in V'$. We know that \[
\deg(\iota_*(D)|_{V'})=\deg(D|_{\iota^{-1}(V')}),\quad \iota_*(\mu)(V')=\mu(\iota^{-1}(V')),\quad\delta_{\Gamma',V'}= \delta_{\Gamma,\iota^{-1}(V')}, \] which readily implies $\beta_{\iota_*(D)}(V)=\beta_D(\iota^{-1}(V))$. Moreover, $\iota(v_0)\in V'$ if and only if $v_0\in \iota^{-1}(V')$, hence the result follows.\par Now we move to the proof of \emph{(ii)}. First we show that if $(\mathcal E,D)$ is $(v_0,\mu)$-quasistable then $\mathcal E$ is nondisconnecting. Assume, by contradiction, that $\mathcal E$ is disconnecting, and let $V$ be a subset of $V(\Gamma)$ such that the cut $E(V,V^c)$ is contained in $\mathcal E$. Note that \[
\deg(D|_V)-\mu(V)+\deg(D|_{V^c})-\mu(V^c)=\delta_V, \] because $D(v)=-1$ for each exceptional vertex $v\in V(\Gamma^\mathcal E)$ (by the definition of pseudo-divisor). Then \[
\left(\deg(D|_V)-\mu(V)-\frac{\delta_V}{2}\right)+\left(\deg(D|_{V^c})-\mu(V^c)-\frac{\delta_{V^c}}{2}\right)=0. \] However, since $(\mathcal E,D)$ is $(v_0,\mu)$-quasistable, we must have \[
\deg(D|_V)-\mu(V)-\frac{\delta_V}{2}\leq 0 \quad \text{ and } \quad
\deg(D|_{V^c})-\mu(V^c)-\frac{\delta_{V}}{2}\leq 0, \]
hence the two inequalities are in fact equalities, which contradicts that one of them must be strict.\par Now, we assume $\mathcal E$ nondisconnecting and we show the equivalence of the quasistability conditions. Fix $V\subset V(\Gamma)=V(\Gamma_\mathcal E)$. We have $\deg(D_\mathcal E|_V)=\deg(D|_V)$ and, if we let $\kappa_\mathcal E(V)=\sum_{v\in V}\text{val}_\mathcal E(v)$, then \[ \mu_\mathcal E(V)=\mu(V)+\frac{1}{2}\kappa_\mathcal E(V)=\mu^\mathcal E(V)+\frac{1}{2}\kappa_\mathcal E(V). \]
Moreover we have $\delta_{\Gamma_\mathcal E,V}=\delta_{\Gamma^\mathcal E,V}-\kappa_\mathcal E(V)$. These equalities imply that \begin{multline*}
\deg(D|_V)-\mu^\mathcal E(V)\pm\frac{\delta_{\Gamma^\mathcal E,V}}{2}=\\=\deg(D_\mathcal E|_V)-(\mu_\mathcal E(V)-\frac{1}{2}\kappa_\mathcal E(V))\pm\frac{\delta_{\Gamma_\mathcal E,V}+\kappa_\mathcal E(V)}{2} \end{multline*} and hence \begin{equation} \label{eq:DDE+}
\deg(D|_V)-\mu^\mathcal E(V)+\frac{\delta_{\Gamma^\mathcal E,V}}{2}=\deg(D_\mathcal E|_V)-\mu_\mathcal E(V)+\frac{\delta_{\Gamma_\mathcal E,V}}{2}+\kappa_\mathcal E(V) \end{equation} and \begin{equation} \label{eq:DDE-}
\deg(D|_V)-\mu^\mathcal E(V)-\frac{\delta_{\Gamma^\mathcal E,V}}{2}=\deg(D_\mathcal E|_V)-\mu_\mathcal E(V)-\frac{\delta_{\Gamma_\mathcal E,V}}{2}. \end{equation} If $(D,\mathcal E)$ is $(v_0,\mu^\mathcal E)$-quasistable, then the left hand side of Equation \eqref{eq:DDE-} is less or equal than $0$, and equality can only hold if $v_0\in V$. This means that $D_\mathcal E$ is $(v_0,\mu_\mathcal E)$-quasistable.\par Now assume that $D_\mathcal E$ is $(v_0,\mu_\mathcal E)$-quasistable. Fix $V'\subset V(\Gamma^\mathcal E)$ and define $V=V'\cap V(\Gamma)$. This means that every vertex in $V'\setminus V$ is exceptional. Then \[
\deg(D|_{V'})=\deg(D|_V)-|V'\setminus V|\quad\text{and}\quad \mu^\mathcal E(V')=\mu^\mathcal E(V). \] Combining with Equation \eqref{eq:DDE+}, we get \begin{multline*}
\deg(D|_{V'})-\mu^\mathcal E(V')+\frac{\delta_{\Gamma^\mathcal E,V'}}{2}=\\
=\deg(D_\mathcal E|_V)-\mu_\mathcal E(V)+\frac{\delta_{\Gamma_\mathcal E,V}}{2}+\frac{\delta_{\Gamma^\mathcal E,V'}-\delta_{\Gamma^\mathcal E,V}}{2}-|V'\setminus V|+\kappa_\mathcal E(V). \end{multline*}
Now, to prove that $(\mathcal E,D)$ is $(v_0,\mu)$-quasistable it is sufficient that \[
\delta_{\Gamma^\mathcal E,V'}+2\kappa_\mathcal E(V)\geq\delta_{\Gamma^\mathcal E,V}+2|V'\setminus V|. \] We check that the last inequality holds by proving that the contribution of each edge $e\in E(\Gamma)$ to the left hand side is greater or equal than its contribution to the right hand side. Any $e\notin \mathcal E$ contributes the same in both sides. In the table below, we enumerate the remaining cases. We write $w_1$ and $w_2$ for the (possibly coincident) vertices incident to $e$, and $w_e$ for the exceptional vertex over $e$. \[
\begin{array}{l|c|c|c|c}
& \delta_{\Gamma^\mathcal E,V'} &\kappa_\mathcal E(V)&\delta_{\Gamma^\mathcal E,V}&|V'\setminus V|\\ \hline w_1,w_2\in V, w_e\in V'&0&2&2&1\\ w_1\in V, w_2\notin V, w_e\in V'&1&1&1&1\\ w_1,w_2\notin V,w_e\in V'&2&0&0&1\\ w_1,w_2\in V, w_e\notin V'&2&2&2&0\\ w_1\in V,w_2\notin V, w_e\notin V'&1&1&1&0\\ w_1,w_2\notin V, w_e\notin V'&0&0&0&0 \end{array} \] This concludes the proof. \end{proof}
We continue our analysis of quasistable divisors in the case when $\Gamma$ is tree-like.
\begin{Prop} \label{prop:tree} Let $\Gamma$ be a tree-like graph, $v_0$ a vertex of $\Gamma$, and $\mu$ a degree-$d$ polarization on $\Gamma$. Then there is exactly one $(v_0,\mu)$-quasistable divisor of degree-$d$ on $\Gamma$. \end{Prop} \begin{proof}
First, we note that a divisor $D$ on $\Gamma$ is $(v_0,\mu)$-quasistable if and only if $\iota_*(D)$ is $(\iota(v_0),\iota_*(\mu))$-quasistable, where $\iota$ is the contraction of all loops of $\Gamma$. In fact, loops do not contribute to any of the values $\deg(D|_V)$, $\mu(V)$ and $\delta_V$ for $V\subset V(\Gamma)$. So, we can safely assume that $\Gamma$ is a tree.\par
We proceed by induction on the number of vertices. If $\Gamma$ has a only one vertex the result is clear. Assume that $\Gamma$ has at least two vertices and let $D_1$ and $D_2$ be $(v_0,\mu)$-quasistable divisors on $\Gamma$. Let $v$ be a vertex of $\Gamma$ of valence $1$ such that $v\neq v_0$. Hence \[ -\frac{1}{2}\leq D_i(v)-\mu(V)< \frac{1}{2}, \text{ for $i=1,2$} \] which means \[ D_i(v)= \left\lfloor\mu(V)+\frac{1}{2} \right \rfloor, \text{ for $i=1,2$}. \] Let $\iota\colon \Gamma\to \Gamma'$ be the contraction of the only edge $e$ incident to $v$. By the induction hypothesis, there exists a unique $(\iota(v_0),\iota_*(\mu))$-quasistable divisor on $\Gamma'$. In particular $\iota_*(D_1)=\iota_*(D_2)$, because, by Proposition \ref{prop:spec}, $\iota_*(D_1)$ and $\iota_*(D_2)$ are $(\iota(v_0),\iota_*(\mu))$-quasistable. This means that $D_1(w)=D_2(w)$ for every vertex $w$ not incident to $e$. Since $D_1(v)=D_2(v)$ and the degrees of $D_1$ and $D_2$ are equal, it follows that $D_1=D_2$.\par We note that the proof above actually gives a algorithm to find a $(v_0,\mu)$-quasistable divisor on $\Gamma$ hence, in particular, it also provides the existence. \end{proof}
\begin{Cor} \label{cor:unique} Let $\Gamma$ be a graph, $v_0$ a vertex of $\Gamma$, and $\mu$ a polarization on $\Gamma$. If $\mathcal E$ is the complement of a spanning tree of $\Gamma$, then there exists exactly one $(v_0,\mu^\mathcal E)$-quasistable divisor $D$ on $\Gamma^\mathcal E$ with $D(v')=-1$ for every $v'\in V(\Gamma^\mathcal E)\setminus V(\Gamma)$. \end{Cor} \begin{proof} Just combine Propositions \ref{prop:spec} and \ref{prop:tree}. \end{proof}
Let $\Gamma$ be a graph, $v_0$ a vertex of $\Gamma$, and $\mu$ a polarization on $\Gamma$.
The set $\mathcal{QD}_{v_0,\mu}(\Gamma)$ of $(v_0,\mu)$-quasistable pseudo-divisors on $\Gamma$ is a poset where the partial order is given by $(\mathcal E,D)\geq (\mathcal E',D')$ if $(\mathcal E,D)$ specializes to $(\mathcal E',D')$.\par
If $\iota\colon \Gamma\to \Gamma'$ is a graph specialization, by Proposition \ref{prop:spec} there is a natural map \begin{align} \label{eq:i*} \iota_*\colon \mathcal{QD}_{v_0,\mu}(\Gamma) \to& \mathcal{QD}_{\iota(v_0),\iota_*(\mu)}(\Gamma')\\
\nonumber (\mathcal E,D)\mapsto&\iota_*(\mathcal E,D). \end{align} This map is order preserving, hence it is continuous.\par
Given a nondisconnecting subset $\mathcal E$ of $E(\Gamma)$, again by Proposition \ref{prop:spec} we have an induced injection \begin{align*} \kappa_*\colon \mathcal{QD}_{v_0,\mu_\mathcal E}(\Gamma_\mathcal E) \to& \mathcal{QD}_{v_0,\mu}(\Gamma)\\
(\mathcal E',D')\mapsto&(\mathcal E'\cup \mathcal E, D) \end{align*} where $D(v)=D'(v)$ for every $v\in V(\Gamma)$. The map $\kappa$ is order preserving, hence it is continuous. Moreover, $\kappa$ is an open injection, since Proposition \ref{prop:spec} implies that $\text{Im}(\kappa)=\{(\mathcal E',D)\in \mathcal{QD}_{v_0,\mu}(\Gamma) ; \mathcal E\subset \mathcal E'\}$.\par
Given a specialization of graphs $\iota\colon\Gamma\rightarrow\Gamma'$ and inclusions $\mathcal E'\subset E(\Gamma')\subset E(\Gamma)$, we have an induced specialization $\iota_{\mathcal E'}\colon \Gamma_{\mathcal E'}\rightarrow \Gamma'_{\mathcal E'}$ which makes the following diagram a commutative diagram of topological spaces: \[ \begin{CD} \mathcal{QD}_{v_0,\mu_{\mathcal E'}}(\Gamma_{\mathcal E'}) @>\iota_{\mathcal E'*}>> \mathcal{QD}_{\iota(v_0),\iota_*(\mu_{\mathcal E'})}(\Gamma'_{\mathcal E'}) \\ @V\kappa_* VV @V\kappa'_*VV\\ \mathcal{QD}_{v_0,\mu}(\Gamma)@>\iota_*>> \mathcal{QD}_{\iota(v_0),\iota_*(\mu)}(\Gamma') \end{CD} \]
\begin{Exa} \label{exa:poset} Let $\Gamma$ be the graph with $2$ vertices $v_0$ and $v_1$ and $3$ edges $e_1,e_2,e_3$ connecting them. Let $\mu$ be the degree-$0$ polarization given by $\mu(v)=0$ for every $v\in V(\Gamma)$. The poset $ \mathcal{QD}_{v_0,\mu}(\Gamma)$ of pseudo-divisors of $\Gamma$ is as depicted in Figure \ref{fig:poset} (up to permutation of the edges). \end{Exa} \begin{figure}
\caption{The poset of pseudo-divisors.}
\label{fig:poset}
\end{figure}
Now, we prove a series of topological properties of the poset $\mathcal{QD}_{v_0,\mu}(\Gamma)$ (in Propositions \ref{prop:chain} and \ref{prop:cod1}) and of the map $\iota_*$ defined in Equation \eqref{eq:i*} (in Proposition \ref{prop:jclosed} and Corollary \ref{cor:sep}).
\begin{Prop} \label{prop:chain} Let $\Gamma$ be a graph, $v_0$ a vertex of $\Gamma$, and $\mu$ a polarization on $\Gamma$. The poset $\mathcal{QD}_{v_0,\mu}(\Gamma)$ is ranked of length $b_1(\Gamma)$. \end{Prop} \begin{proof}
Set $g=b_1(\Gamma)$. It is sufficient to prove that if $(\mathcal E,D)$ is a $(v_0,\mu)$-quasistable pseudo-divisor on $\Gamma$ such that $|\mathcal E|<g$, then we can find a $(v_0,\mu)$-quasistable pseudo-divisor $(\mathcal E',D')$ on $\Gamma$ with $\mathcal E\subset \mathcal E'$ and $|\mathcal E'|=|\mathcal E|+1$, and such that $(\mathcal E',D')$ specializes to $(\mathcal E,D)$. By Corollary \ref{cor:unique}, we can assume that $g\ge 1$. By Proposition \ref{prop:spec}, we have that $D_\mathcal E$ is a $(v_0,\mu_\mathcal E)$-quasistable divisor on $\Gamma_\mathcal E$, thus we can reduce to the case in which $\mathcal E=\emptyset$. In what follows, given $\mathcal E'\subset E(\Gamma)$, we will denote by $v_e$ the exceptional vertex of $\Gamma^{\mathcal E'}$ over $e$, for every $e\in \mathcal E'$.\par
If $\Gamma$ has a loop $e$ incident to a vertex $v_1$, we can take the pseudo-divisor $(\mathcal E',D')$, where $\mathcal E'=\{e\}$ and $D'(v_1)=D(v_1)+1$, $D'(v_e)=-1$ and $D'(v)=D(v)$ for every $v\in V(\Gamma^{\mathcal E'})\setminus\{v_1,v_e\}$. It is clear that $(\mathcal E',D')$ is $(v_0,\mu)$-quasistable and $(\mathcal E',D')$ specializes to $(\emptyset,D)$. Therefore, we can also assume that $\Gamma$ has no loops.\par
Fix an edge $e$ incident to $v_0$ and let $v_1\in V(\Gamma)$ be the other vertex incident to $e$. For $i=0,1$, we define the divisor $D_i$ on $\Gamma^{\{e\}}$ as \[ D_i(v):=\begin{cases} \begin{array}{ll} -1,&\text{if $v=v_e$};\\ D(v)+1,&\text{if $v=v_{1-i}$};\\ D(v),&\text{if $v\neq v_{1-i},v_e$.} \end{array} \end{cases} \]
If at least one between $D_0$ and $D_1$ is $(v_0,\mu^{\{e\}})$-quasistable, then we are done, since $(\{e\},D_i)$ specializes to $(\emptyset,D)$ for $i=0,1$. Otherwise, there are subsets $V_0', V_1'\subsetneq V(\Gamma^{\{e\}})$ such that, for $i=0,1$, we have \begin{equation} \label{eq:vi} \beta_{D_i}(V_i')\leq0,\text{ with strict inequality if $v_0\notin V_i'$}. \end{equation} Fix $i\in\{0,1\}$ and set $V_i:=V_i'\setminus\{v_e\}\subset V(\Gamma)$. We have \[
\beta_{D_i}(V_i')-\beta_D(V_i)=\deg(D_i|_{V_i'})-\deg(D|_{V_i})+\frac{\delta_{\Gamma^{\{e\}},V_i'}-\delta_{\Gamma,V_i}}{2}. \] By the table below, \[
\begin{array}{l|c|c}
\{v_e,v_0,v_1\}\cap V_i'& \deg(D_i|_{V_i'})-\deg(D|_{V_i}) & \delta_{\Gamma^{\{e\}},V_i'} -\delta_{\Gamma,V_i}\\ \hline \{v_e,v_0,v_1\}&\phantom{-}0&0\\ \{v_e,v_{1-i}\}&\phantom{-}0&0\\ \{v_e,v_{i}\}&-1&0\\ \{v_0,v_1\}&\phantom{-}1&2\\ \{v_e\}&-1&2\\ \{v_i\}&\phantom{-}0&0\\ \{v_{1-i}\}&\phantom{-}1&0\\ \emptyset&\phantom{-}0&0 \end{array} \] we deduce that, if $\{v_e,v_0,v_1\}\cap V_i'\ne\{v_e,v_i\}$, then $\beta_{D_i}(V_i')\geq\beta_D(V_i)$, and this contradicts Equation \eqref{eq:vi}. It follows that $\{v_e,v_0,v_1\}\cap V_i'=\{v_e,v_i\}$. This implies that $\beta_{D_i}(V_i')=\beta_D(V_i)-1$, which in turn implies that \begin{equation} \label{eq:vi1} \beta_D(V_i)\leq1,\text{ with strict inequality if $i=1$}. \end{equation}\par
If $V(\Gamma)=\{v_0, v_1\}$, then we have $V_i=\{v_i\}$, and hence $\beta_D(\{v_0\})+\beta_D(\{v_1\})<2$. However, by Lemma \ref{lem:cap}, we have that $\beta_D(\{v_0\})+\beta_D(\{v_1\})$ is the number of edges between $v_0$ and $v_1$ which is at least 2, since $g\ge 1$. This gives rise to a contradiction, so we are done when $\Gamma$ has exactly two vertices.\par
We now proceed by induction on the number of vertices of $\Gamma$. Let $\iota\colon\Gamma\to\Gamma'$ be the contraction of all edges in $E(v_0,v_1)$. By the induction hypothesis, there are an edge $e'\in E(\Gamma')\subset E(\Gamma)$ and a $(\iota(v_0),\iota_*(\mu))$-quasistable pseudo-divisor $(\{e'\},\widehat{D})$ on $\Gamma'$ such that $(\{e'\},\widehat{D})$ specializes to $(\emptyset,\iota_*(D))$. This means that if $w_0$ and $w_1$ are the vertices incident to $e'$, then $\widehat{D}(w_0)=\iota_*(D)(w_0)+1$, $\widehat{D}(v_{e'})=-1$ and $\widehat{D}(w)=\iota_*(D)(w)$ for every $w\in V(\Gamma')\setminus\{w_0,v_{e'}\}$. For $i=0,1$, if $\iota^{-1}(w_i)$ is a single vertex, we abuse notation and write $w_i$ for such vertex. Otherwise, we have that $\iota^{-1}(w_i)=\{v_0,v_1\}$, and again we abuse notation and write $w_i$ for the vertex in $\{v_0,v_1\}$ attached to $e'$. We define the pseudo-divisor $(\{e'\},D')$ on $\Gamma$ as \[ D'(v):=\begin{cases} \begin{array}{ll} -1&\text{if $v=v_{e'}$;}\\ D(v)+1&\text{if $v=w_0$;}\\ D(v)&\text{if $v\neq w_0,v_{e'}$.} \end{array} \end{cases} \] Then $\iota_*(\{e'\},D')=(\{e'\},\widehat{D})$ and there is a specialization of $(\{e'\},D') $ to $(\emptyset,D)$. All that is left to prove is that $(\{e'\},D')$ is $(v_0,\mu)$-quasistable. By contradiction, assume that there is a subset $V'\subsetneq V(\Gamma^{\{e'\}})$ such that \begin{equation} \label{eq:v'} \beta_{D'}(V')\leq 0,\text{ with strict inequality if $v_0\notin V'$}. \end{equation} By an argument similar to the one using the above table, we have $\{w_0,w_1, v_{e'}\}\cap V'=\{w_1,v_{e'}\}$ and hence $\beta_{D'}(V')=\beta_D(V)-1$, where $V=V'\setminus\{v_{e'}\}\subset V(\Gamma)$, Again, this implies that \begin{equation} \label{eq:v'1} \beta_{D}(V)\leq 1,\text{ with strict inequality if $v_0\notin V$}. \end{equation}
Moreover, if $|\{v_0,v_1\}\cap V'|\neq 1$, then $\beta_{D'}(V')=\beta_{\widehat{D}}(\iota^{\{e'\}}(V'))$, because $\iota_*(\{e'\},D')=(\{e'\},\widehat{D})$ and $V'$ is the inverse image of $\iota^{\{e'\}}(V')$ via $\iota^{\{e'\}}$. This contradicts Equation \eqref{eq:v'} and the fact that $\widehat D$ is $(\iota(v_0),\iota_*(\mu))$-quasistable. Therefore there exists $i\in\{0,1\}$ such that $v_{1-i}\in V$ and $v_{i}\notin V$.\par
By Lemma \ref{lem:cap} we have \begin{equation} \label{eq:2}
\beta_D(V\cup V_i)+\beta_D(V\cap V_i)+|E(V,V_i)|=\beta_D(V)+\beta_D(V_i)<2, \end{equation} where the inequality comes from Equations \eqref{eq:vi1} and \eqref{eq:v'1} (note that either $v_0\notin V$ or $i=1$, hence the inequality is indeed strict).\par Note that $\beta_D(V\cup V_i)\geq0$ and $\beta_D(V\cap V_i)\geq0$, since $(\emptyset,D)$ is $(v_0,\mu)$-quasistable. We have three cases to consider.
In the first case, $w_0\in V_i$ and $w_1\notin V_i$, hence $|E(V,V_i)|\geq2$, because the set $E(V,V_i)$ contains $e$ and $e'$. This contradicts Equation \eqref{eq:2}.\par
In the second case, $w_0\notin V_i$, hence $\beta_{D'}(V'\cup V_i)=\beta_D(V\cup V_i)-1$, because $\mu(V'\cup V_i)=\mu(V\cup V_i)$, $\deg(D'|_{V'\cup V_i})=\deg(D|_{V\cup V_i})+D'(v_{e'})$ and $\delta_{\Gamma^{\{e'\}},V'\cup V_i}=\delta_{\Gamma, V\cup V_i}$ (recall that $w_1\in V'$). However, since $v_0,v_1\in V'\cup V_i$, we have \[ \beta_{D'}(V'\cup V_i)=\beta_{\widehat{D}}(\iota^{\{e'\}}(V'\cup V_i))\geq 0, \]
hence $\beta_D(V\cup V_i)\geq1$. But the set $E(V,V_i)$ contains $e$, hence $|E(V,V_i)|\geq 1$, and this contradicts Equation \eqref{eq:2}.\par
In the third case, $w_0,w_1\in V_i$. The argument is similar to the one used in the second case. Indeed, we have $|E(V,V_i)|\geq 1$ and $\beta_{D'}(V'\cap (V_i\cup\{v_{e'}\}))=\beta_D(V\cap V_i)-1$. Since $v_0,v_1\notin V'\cap (V_i\cup\{v_{e'}\})$, we also have
\[ \beta_{D'}(V'\cap (V_i\cup\{v_{e'}\})=\beta_{\widehat{D}}(\iota^{\{e'\}}(V'\cap (V_i\cup\{v_{e'}\}))\geq 0, \] hence $\beta_D(V\cap V_i)\geq1$, contradicting again Equation \eqref{eq:2}. \end{proof}
\begin{Prop} \label{prop:jclosed} Let $\Gamma$ be a graph, $v_0$ a vertex of $\Gamma$, and $\mu$ be a polarization on $\Gamma$. If $\iota\colon\Gamma\rightarrow\Gamma'$ is a specialization of graphs, then the induced map $\iota_*\colon \mathcal{QD}_{v_0,\mu}(\Gamma)\to \mathcal{QD}_{\iota(v_0),\iota_*(\mu)}(\Gamma')$ is closed and surjective. \end{Prop} \begin{proof} We can assume that $\iota$ is the contraction of a single edge $e\in E(\Gamma)$, because a composition of closed and surjective maps is also closed and surjective.\par
We begin proving that $\iota_*$ is closed. It is enough to prove that if $(\mathcal E_1',D_1')\geq (\mathcal E_2',D_2')$ in $\mathcal {QD}_{\iota(v_0),\iota_*(\mu)}(\Gamma')$ and $(\mathcal E_1',D_1')=\iota_*(\mathcal E_1,D_1)$ for some $(\mathcal E_1,D_1)$ in $\mathcal{QD}_{v_0,\mu}(\Gamma)$ then there is $(\mathcal E_2,D_2)\in \mathcal{QD}_{v_0,\mu}(\Gamma)$ such that $(\mathcal E_1,D_1)\geq (\mathcal E_2,D_2)$ and $\iota_*(\mathcal E_2,D_2)=(\mathcal E_2',D_2')$. Since $(\mathcal E_1',D_1')\geq (\mathcal E_2',D_2')$, there are an inclusion $\mathcal E_2'\subset \mathcal E_1'$ and a compatible specialization $\Gamma'^{\mathcal E_1'}\rightarrow\Gamma'^{\mathcal E_2'}$ that takes $D_1'$ to $D_2'$. Since $\iota_*(\mathcal E_1,D_1)=(\mathcal E_1',D_1')$, we have that $\mathcal E_1'=\mathcal E_1\cap E(\Gamma')$ and the induced specialization $\Gamma^{\mathcal E_1}\rightarrow\Gamma'^{\mathcal E_1'}$ takes $D_1$ to $D_1'$. We set $\mathcal E_2:=\mathcal E_1\setminus E(\Gamma')$. Then we have a commutative diagram of specializations \[ \begin{CD} \Gamma^{\mathcal E_1}@>>> \Gamma^{\mathcal E_2}\\ @VVV @VVV\\ \Gamma'^{\mathcal E_1'}@>>> \Gamma'^{\mathcal E_2'} \end{CD} \] and the pushforward of $D_1$ to $\Gamma'^{\mathcal E_2'}$ must be $D_2'$. Define $D_2$ as the pushforward of $D_1$ to $\Gamma^{\mathcal E_2}$. Since the diagram is commutative, we have $\iota_*(\mathcal E_2,D_2)=(\mathcal E'_2,D'_2)$ which concludes the proof that $\iota_*$ is closed.
We now prove the surjectivity of $\iota_*$. Since $\iota_*$ is closed, it is enough to prove that $\iota_*^{-1}(\mathcal E',D')$ is nonempty for every maximal element $(\mathcal E',D')$ of $\mathcal{QD}_{\iota(v_0),\iota_*(\mu)}(\Gamma')$. By Proposition \ref{prop:chain}, a pseudo-divisor $(\mathcal E',D')$ is maximal if and only if $\mathcal E'$ is the complement of a spanning tree.
So let $\mathcal E'$ be the complement of a spanning tree on $\Gamma'$. Then, by Corollary \ref{cor:unique}, there exists a unique $(\iota(v_0),\iota_*(\mu))$-quasistable pseudo-divisor $(\mathcal E',D')$ on $\Gamma'$. Since $\mathcal E'$ is nondisconnecting on $\Gamma$, then $\Gamma_{\mathcal E'}$ is connected, which means that there exists a $(v_0,\mu_{\mathcal E'})$-quasistable divisor $D$ on $\Gamma_{\mathcal E'}$ (see Theorem \ref{thm:esteves}). We can use Proposition \ref{prop:spec} to get a $(v_0,\mu)$-quasistable divisor $(\mathcal E',D)$ on $\Gamma$. Since $\iota_*(\mathcal E',D)=(\mathcal E',\iota_{\mathcal E'*}(D))$ is $(\iota(v_0),\iota_*(\mu))$-quasistable by Proposition \ref{prop:spec}, the uniqueness of $(\mathcal E',D')$ implies that $\iota_*(\mathcal E',D)=(\mathcal E',D')$, and we are done. \end{proof} \begin{Cor} \label{cor:sep} Preserve the hypothesis of Proposition \ref{prop:jclosed}. If $\iota$ is a contraction of separating edges, then $\iota_*\colon \mathcal{QD}_{v_0,\mu}(\Gamma)\to \mathcal{QD}_{\iota(v_0),\iota_*(\mu)}(\Gamma')$ is a homeomorphism. \end{Cor} \begin{proof}
By Proposition \ref{prop:jclosed}, it is enough to prove that $\iota_*$ is injective. We can assume that $\iota$ is the contraction of a single edge $e\in E(\Gamma)$. Let $v_1$ and $v_2$ the vertices incident to $e$. Let $\Gamma_1$ be the connected component of $\Gamma_{\{e\}}$ containing $v_1$ and assume that $v_0\in\Gamma_1$.
Let $(\mathcal E_1,D_1)$ and $(\mathcal E_2,D_2)$ be $(v_0,\mu)$-quasistable divisors of $\Gamma$ such that $\iota_*(\mathcal E_1,D_1)=\iota_*(\mathcal E_2,D_2)$.
We have $e\notin \mathcal E_1,\mathcal E_2$, since $e$ is a separating edge (see Proposition \ref{prop:spec}). It follows that $\mathcal E_1=\mathcal E_2=:\mathcal E$. Therefore, $D_1$ and $D_2$ are divisor on $\Gamma^\mathcal E$, so we can assume that $\mathcal E=\emptyset$ (upon changing $\Gamma,\Gamma'$ with $\Gamma^\mathcal E,\Gamma'^\mathcal E$).\par
Since $\iota_*(D_1)=\iota_*(D_2)$, we have $D_1(v)=D_2(v)$ for every $v\notin\{v_1,v_2\}$. For $i\in\{1,2\}$ we have $0<\beta_{D_i}(V(\Gamma_1))\leq1$, which means that we have \[
\mu(V(\Gamma_1))-\frac{1}{2}<\deg(D_i|_{V(\Gamma_1)})\leq\mu(V(\Gamma_1))+\frac{1}{2}. \]
This implies that $\deg(D_1|_{V(\Gamma_1)})=\deg(D_2|_{V(\Gamma_1)})$, because both are integers. We deduce that $D_1(v_1)=D_2(v_1)$, and hence $D_1=D_2$. \end{proof}
\begin{Prop} \label{prop:cod1} Let $\Gamma$ be a graph, $v_0$ be a vertex of $\Gamma$, and $\mu$ be a polarization on $\Gamma$ of degree $d$. Then the poset $\mathcal{QD}_{v_0,\mu}(\Gamma)$ is connected in codimension 1. \end{Prop}
\begin{proof} Set $g:=b_1(\Gamma)$. We start proving the result for graphs with $g=1$. In this case the statement is equivalent to prove that $\mathcal{QD}_{v_0,\mu}(\Gamma)$ is connected. Since $g=1$, by Corollary \ref{cor:sep} we can reduce to the case in which $\Gamma$ is a cycle. \par We proceed by induction on the number of edges. The statement is clear if there is a single edge (which is a loop). Assume that $\Gamma$ has at least two edges and fix an edge $e\in E(\Gamma)$. By Corollary \ref{cor:unique}, there exists exactly one $(v_0,\{e\})$-quasistable pseudo-divisor $(\{e\},D)$ of degree-$d$ on $\Gamma$. Since $e$ is not a loop, there are exactly two divisors $D_1$ and $D_2$ on $\Gamma$ such that $(\{e\},D)\rightarrow (\emptyset,D_i)$ for $i=1,2$.
Let $\iota\colon\Gamma\to\Gamma/\{e\}$ be the contraction of the edge $e$. Then \[ (\emptyset,D'):=\iota_*(\{e\},D)=\iota_*(\emptyset, D_1)=\iota_*(\emptyset, D_2). \] We claim that the set of $(v_0,\mu)$-quasistable pseudo-divisors on $\Gamma$ specializing to the pseudo-divisor $(\emptyset,D')$ via $\iota$ is precisely $\{(\{e\},D),(\emptyset,D_1),(\emptyset,D_2)\}$. Indeed, the map $\iota_*\colon \mathcal{QD}_{v_0,\mu}(\Gamma)\to \mathcal{QD}_{\iota(v_0),\iota_*(\mu)}(\Gamma/\{e\})$ is surjective by Proposition \ref{prop:jclosed}. However \[
|\mathcal{QD}_{v_0,\mu}(\Gamma)|=2|E(\Gamma)|\quad\text{and}\quad| \mathcal{QD}_{\iota(v_0),\iota_*(\mu)}(\Gamma/\{e\})|=2|E(\Gamma')|, \] and hence \[
|\mathcal{QD}_{v_0,\mu}(\Gamma)|=|\mathcal{QD}_{\iota(v_0),\iota_*(\mu)}(\Gamma/\{e\})|+2. \]
Since $|\iota_*^{-1}(\emptyset, D')|\geq3$, the equality must hold, from which the claim follows.\par
If we consider $\mathcal{QD}_{v_0,\mu}(\Gamma)$ as a graph whose edges are its minimal elements and whose vertices are its remaining elements (which are maximal), then $\iota_*$ is simply the contraction of the two edges corresponding to $(\emptyset,D_1)$ and $(\emptyset,D_2)$, and meeting each other at the vertex corresponding to $(\{e\},D)$. Since, by the induction hypothesis, $\mathcal{QD}_{\iota(v_0),\iota_*(\mu)}(\Gamma/\{e\})$ is connected, we have that $\mathcal{QD}_{v_0,\mu}(\Gamma)$ is connected as well.\par
Now we prove the general case. By Lemma \ref{lem:tree}, it is sufficient to prove that if $\mathcal E_1$ and $\mathcal E_2$ are nondisconnecting subsets of $E(\Gamma)$ of cardinality $g$ such that $|\mathcal E_1\cap\mathcal E_2|=g-1$, then there is a path in codimension 1 in $\mathcal{QD}_{v_0,\mu}(\Gamma)$ connecting any two pseudo-divisors $(\mathcal E_1,D_1)$ and $(\mathcal E_2,D_2)$. Let $\Gamma':=\Gamma_{\mathcal E_1\cap \mathcal E_2}$. Note that $g_{\Gamma'}=1$. We have a open injection $\mathcal{QD}_{v_0,\mu_{\mathcal E_1\cap\mathcal E_2}}(\Gamma')\to \mathcal{QD}_{v_0,\mu}(\Gamma)$. Then it is sufficient to prove that there exists a path from $(\mathcal E_1\setminus \mathcal E_2,D_1)$ to $(\mathcal E_2\setminus \mathcal E_1,D_2)$ in $\mathcal{QD}_{v_0,\mu_{\mathcal E_1\cap\mathcal E_2}}(\Gamma')$. The existence of this path follows from the case $g_{\Gamma'}=1$ already proved.
\end{proof}
Let $d$ be an integer. We say that $\mu$ is a \emph{polarization (of degree $d$)} on $\mathbf{Graph}_{g,1}$ if $\mu$ is a collection of polarizations $\mu_\Gamma$ of degree $d$ for every genus-$g$ weighted stable graph with $1$ leg $\Gamma$, such that $\mu_{\Gamma'}=\iota_*(\mu_\Gamma)$ for every specialization $\iota\colon\Gamma\to\Gamma'$. In this case, we call $\mu$ a \emph{universal genus-$g$ polarization (of degree $d$)}. This polarization extends to every genus-$g$ semistable graph, since every genus-$g$ semistable graph is a subdivision of a stable graph. \begin{Exa} \label{exa:pol} Let $d$ be an integer. We give two examples of universal genus-$g$ polarizations of degree $d$. We have the \emph{canonical polarization}, given by \[ \mu_\Gamma(v)=\frac{d(2w(v)-2+\text{val}(v))}{2g-2}. \] We also have a polarization concentrated at the marked vertex $v_0$ of $\Gamma$, given by \[ \mu_\Gamma(v)=\begin{cases}
\begin{array}{ll}
0,&\text{ if $v\neq v_0$;}\\
d,&\text{ if $v=v_0$}.
\end{array}
\end{cases} \] Note that any linear combination of the above universal polarizations gives rise to a universal genus-$g$ polarization of degree $d$. These are the only universal polarizations one can consider on $\mathbf{Graph}_{g,1}$: this is proved in \cite[Section 5]{KP}. \end{Exa}
If $\mu$ is a universal genus-$g$ polarization, we define the category $\mathbf{QD}_{\mu,g}$ whose objects are triples $(\Gamma,\mathcal E,D)$ where $\Gamma$ is a genus-$g$ stable weighted graph with $1$ leg and $(\mathcal E,D)$ is a $(v_0,\mu_\Gamma)$-quasistable pseudo-divisor of $\Gamma$, and whose morphisms are given by the specializations. We define the poset $\mathcal{QD}_{\mu,g}$ associated to $\mathbf{QD}_{\mu,g}$ as \[ \mathcal{QD}_{\mu,g}:=\{(\Gamma,\mathcal E,D);\;(\Gamma,\mathcal E, D)\in \mathbf{QD}_{\mu,g}\}/\sim, \] where $(\Gamma,\mathcal E,D)\sim (\Gamma',\mathcal E',D')$ if there exists an isomorphism $\iota\colon\Gamma\rightarrow\Gamma'$ such that $(\mathcal E',D')=\iota_*(\mathcal E,D)$, and the ordering is given by specializations.
We end this section studying some topological properties of $\mathcal{QD}_{\mu,g}$.
\begin{Thm} \label{thm:maingraph} Let $\mu$ be a universal genus-$g$ polarization of degree $d$. Then the poset $\mathcal{QD}_{\mu,g}$ has pure dimension $4g-2$ and is connected in codimension $1$. Moreover the natural forgetful map $f\colon \mathcal{QD}_{\mu,g}\to \mathcal{SG}_{g,1}$ is continuous and for every genus-$g$ weighted graph $\Gamma$ with 1 leg, we have \[ f^{-1}([\Gamma])=\mathcal{QD}_{v_0,\mu_\Gamma}(\Gamma)/\textnormal{Aut}(\Gamma). \] \end{Thm} \begin{proof}
We prove that $\mathcal{QD}_{\mu,g}$ has pure dimension $4g-2$, which is equivalent to the fact that every maximal element of $\mathcal{QD}_{\mu,g}$ is of the form $(\Gamma,\mathcal E,D)$ where $\Gamma$ is a $3$-regular graph with $1$ leg and with zero weight function, and where $|\mathcal E|=g$. (Recall that a $3$-regular graph with $1$ leg is a graph such that every vertex $v$ satisfies $\text{val}(v)+\ell(v)=3$.) If $\Gamma$ is not a $3$-regular graph with $1$ leg and with zero weight function, then there exists a specialization $\Gamma'\rightarrow\Gamma$ with $\Gamma'$ a $3$-regular graph with $1$ leg and with zero weight function because $M_{g,1}^\text{trop}$ is pure of dimension $3g-2$. By Proposition \ref{prop:jclosed}, there is a specialization $(\Gamma',\mathcal E',D')\rightarrow (\Gamma,\mathcal E,D)$ of $(v_0,\mu)$-quasistable pseudo-divisors. If $|\mathcal E|<g$, by Proposition \ref{prop:chain} there is a specialization $(\Gamma, \mathcal E',D')\rightarrow (\Gamma,\mathcal E,D)$ with $|\mathcal E'|=g$.\par
We now prove that $\mathcal{QD}_{\mu,g}$ is connected in codimension $1$. Let $(\Gamma,\mathcal E,D)$ and $(\widehat{\Gamma},\widehat{\mathcal E},\widehat{D})$ be two maximal elements of $\mathcal{QD}_{\mu,g}$. By \cite[Theorem 3.2.5]{BMV} and \cite[Fact 4.12]{Caporaso}, there are two sequences: the first is a sequence $\Gamma=\Gamma_0,\Gamma_1,\ldots ,\Gamma_n=\widehat{\Gamma}$ of $3$-regular graphs with $1$ leg and the second is a sequence $\Gamma'_1 , \Gamma'_2 , \ldots , \Gamma'_n$ of codimension-$1$ graphs with $1$ leg; These sequences are endowed with specializations $\iota_k\colon \Gamma_k\rightarrow\Gamma'_{k+1}$ for $k=0,\dots,n-1$, and $\iota'_k\colon\Gamma_k\rightarrow\Gamma'_k$ for $k=1,\dots,n$, each one of which is the contraction of precisely one edge which is not a loop. This implies that the first Betti number of $\Gamma'_k$ is $b_1(\Gamma'_k)=g$ for every $k=1,\dots,n$; choose a $(v_0,\mu_{\Gamma'_k})$-quasistable pseudo-divisor $(\mathcal E'_k,D_k)$ on $\Gamma'_k$ with $|\mathcal E'_k|=g$ (this can be done by Corollary \ref{cor:unique}). By Proposition \ref{prop:jclosed} there exist pseudo-divisors $(\mathcal E_k,D_k)$ and $(\widehat{\mathcal E}_k,\widehat{D}_k)$ on $\Gamma_k$ for every $k=0,\ldots,n$ such that \[ \iota_{k,*}(\widehat{\mathcal E}_k,\widehat{D}_k)=(\mathcal E_{k+1}',D_{k+1}')\text{ for }k=0,\ldots,n-1,\text{ and }(\widehat{\mathcal E}_n,\widehat{D}_n)=(\widehat{\mathcal E},\widehat{D}), \] and such that \[ (\mathcal E_0,D_0)=(\mathcal E,D),\text{ and } \iota'_{k,*}(\mathcal E_k,D_k)=(\mathcal E_k',D_k')\text{ for }k=1,\ldots,n. \] So there is a path in codimension $1$ from $(\Gamma_k,\widehat{\mathcal E}_k,\widehat{D}_k)$ to $(\Gamma_{k+1},\mathcal E_{k+1},\widehat{D}_{k+1})$ for every $k=0,\dots,n-1$. However, Proposition \ref{prop:cod1} shows that $(\Gamma_k,\mathcal E_k,D_k)$ and $(\Gamma_k,\widehat{\mathcal E}_k,\widehat{D}_k)$ are connected in codimension $1$ for every $k=0,\dots,n$. This finishes the proof that $\mathcal{QD}_{\mu,g}$ is connected in codimension $1$.\par
The forgetful map $f$ is clearly order preserving. Let us prove that $f^{-1}([\Gamma])=\mathcal{QD}_{v_0,\mu}(\Gamma)/\textnormal{Aut}(\Gamma)$. There is a natural order-preserving map $h\colon \mathcal{QD}_{v_0,\mu_\Gamma}(\Gamma)\to \mathcal{QD}_{\mu,g}$, and we have that $f^{-1}([\Gamma])=\text{Im}(h)$. Moreover, $h(\mathcal E,D)=h(\mathcal E',D')$ if and only if there exists an automorphism $\iota\colon\Gamma\rightarrow\Gamma$ such that $\iota_*(\mathcal E,D)=(\mathcal E',D')$. This means that $\text{Im}(h)=\mathcal{QD}_{v_0,\mu_\Gamma}(\Gamma)/\textnormal{Aut}(\Gamma)$. \end{proof}
\section{The universal tropical Jacobian}\label{sec:univtropJ}
In this section we will extend the results of Section \ref{sec:quasigraph} to tropical curves. The analogues of the posets appearing in Section \ref{sec:quasigraph} will be polyhedral complexes. Moreover, we will introduce the Jacobian of a tropical curve by means of quasistable divisors, and prove that it is homeomorphic to the usual tropical Jacobian.
Let $X$ be a tropical curve. A degree-$d$ polarization on $X$ is a function $\mu\colon X\to\mathbb R$ such that $\mu(p)=0$ for all, but finitely many $p\in X$, and $\sum_{p\in X}\mu(p)=d$. We define the \emph{support} of $\mu$ as \[ \text{supp}(\mu):=\{p\in X;\mu(p)\neq 0\}. \]
Let $\mu$ be a degree-$d$ polarization on $X$. For every tropical subcurve $Y\subset X$, we define $\mu(Y):=\sum_{p\in Y}\mu(p)$. For any divisor $\mathcal D$ on $X$ and every tropical subcurve $Y\subset X$, we set \[
\beta_\mathcal D(Y):=\deg(\mathcal D|_Y)-\mu(Y)+\frac{\delta_Y}{2}. \]
We define the set $\Rel$ of \emph{relevant points of $X$} (with respect to $\mu$) and the set $\Rel_\mathcal D$ of \emph{$\mathcal D$-relevant points} as \[ \Rel:=V(X)\cup\text{supp}(\mu) \quad \text{ and } \quad \Rel_\mathcal D:=\Rel\cup\text{supp}(\mathcal D). \] Note that if $p$ is not $\mathcal D$-relevant, then $\beta_\mathcal D(p)=1$. Given a tropical subcurve $Y\subset X$, we also define \[
\rel_\mathcal D(Y)=|\Rel_\mathcal D\setminus Y|. \] We define the graphs $\Gamma_X$, $\Gamma_{X,\mathcal D}$, $\Gamma_{Y,\mathcal D}$, as the models of $X$ whose sets of vertices are \[ V(\Gamma_X)=\Rel, \quad\quad V(\Gamma_{X,\mathcal D})=\Rel_\mathcal D, \quad\quad V(\Gamma_{Y,\mathcal D})=V(Y)\cup \Rel_\mathcal D. \]
Note that a degree-$d$ polarization $\mu$ on $X$ induces a degree-$d$ polarization on $\Gamma_X$, $\Gamma_{X,\mathcal D}$ and $\Gamma_{Y,\mathcal D}$ which, abusing notation, we will denote by $\mu$.
\begin{Lem} \label{lem:beta} Let $X$ be a tropical curve and $Y,Z$ be tropical subcurves of $X$. Then \[ \beta_\mathcal D(Y\cap Z)+\beta_\mathcal D(Y\cup Z)=\beta_\mathcal D(Y)+\beta_\mathcal D(Z). \] In particular, if $Y\cap Z$ consists of a finite number of non $\mathcal D$-relevant points, then \[
\beta_\mathcal D(Y\cup Z)=\beta_\mathcal D(Y)+\beta_\mathcal D(Z)-|Y\cap Z|. \] \end{Lem} \begin{proof} The proof of the first equation follows simply observing that a point $p\in X$ contributes the same in each side of the equality. On the other hand, if $Y\cap Z$ consists of non $\mathcal D$-relevant points, then \[
\beta_\mathcal D(Y\cap Z)=\sum_{p\in Y\cap Z}\beta_\mathcal D(p)=|Y\cap Z|, \] and hence the second equality holds. \end{proof}
Given a tropical subcurve $Y\subset X$ and $\epsilon\in \mathbb{R}_{>0}$, we define the tropical subcurve \[ Y^{\epsilon}:=\overline{\bigcup_{p\in Y}B_\epsilon(p)}, \] where $B_\epsilon(p)$ is the ball in $X$ with radius $\epsilon$ and center $p$. Note that, for a sufficiently small $\epsilon\in \mathbb R_{>0}$, we have $\beta_\mathcal D(Y^\epsilon)=\beta_\mathcal D(Y)$. \par
\begin{Def}\label{def:quasistable} Let $X$ be a tropical curve. Let $\mu$ be a degree-$d$ polarization on $X$ and $\mathcal D$ be a divisor on $X$. We say that $\mathcal D$ is \emph{$\mu$-semistable} if for every tropical subcurve $Y\subset X$ we have $\beta_\mathcal D(Y)\geq0$. Given a point $p_0$ of $X$, we say that $\mathcal D$ is \emph{$(p_0,\mu)$-quasistable} if it is $\mu$-semistable and $\beta_\mathcal D(Y)>0$ for every proper subcurve $Y\subset X$ with $p_0\in Y$. \end{Def}
Note that, equivalently, $\mathcal D$ is $(p_0,\mu)$-quasistable if and only if for every tropical subcurve $Y\subset X$ we have $\beta_\mathcal D(Y)\leq\delta_{X,Y}$, with strict inequality if $p_0\notin Y$. Indeed, we can assume that $Y$ has no $\mathcal D$-relevant points in its border (just change $Y$ with some $Y^{\epsilon}$ for a sufficiently small $\epsilon$), interchange $Y$ with $\overline{X\setminus Y}$, and use Lemma \ref{lem:beta}.
The quasistability for a tropical curve and for one of its model are closely related, as it is illustrated by the next proposition and the subsequent corollary.
\begin{Prop} \label{prop:quasiquasi} Let $(X,p_0)$ be a pointed tropical curve and $\mu$ be a degree-$d$ polarization on $X$. A degree-$d$ divisor $\mathcal D$ on $X$ is $(p_0,\mu)$-quasistable if and only if $D$ is $(p_0,\mu)$-quasistable on $\Gamma_{X,\mathcal D}$, where $D$ is the divisor $\mathcal D$ seen as divisor on $\Gamma_{X,\mathcal D}$. \end{Prop} \begin{proof} Given a subset $V\subset V(\Gamma_{X,\mathcal D})$ there is an induced tropical subcurve $Y_V$ of $X$ defined as $Y_V:=V\cup\bigcup_{e\in E(V)} e$, where $E(V)$ are all the edges in $E(\Gamma)$ connecting two (possibly coincident) vertices in $V$. Then $\beta_D(V)=\beta_\mathcal D(Y_V)$, and this proves the ``only if'' implication.\par Conversely, let $Y$ be a subcurve of $X$ and define $V=Y\cap V(\Gamma_{{X,\mathcal D}})$. Let us prove that $\beta_\mathcal D(Y)\geq\beta_\mathcal D(Y_V)$.
First we show that, given an edge $e\in E(\Gamma_{X,\mathcal D})\setminus E(V)$, we have $\beta_\mathcal D(Y)\geq\beta_\mathcal D(Y_e)$, where $Y_e:=Y\setminus e^\circ$ and $e^\circ$ is the interior of $e$. By Lemma \ref{lem:beta}, we have \[ \beta_\mathcal D(Y)+\beta_\mathcal D(Y_e\cap e)=\beta_\mathcal D(Y_e)+\beta_\mathcal D(Y\cap e), \] then it is sufficient to prove that $\beta_\mathcal D(Y\cap e)\geq\beta_\mathcal D(Y_e\cap e)$.
We claim that $\delta_{Y\cap e}\geq \delta_{Y_e\cap e}$. Indeed, if $Y_e\cap e=\emptyset$ the result is trivial. So, we can assume that $Y_e\cap e=v$, where $v$ is a vertex of $\Gamma_{X,\mathcal D}$. Since the other vertex $v'$ incident to $e$ satisfies $v'\notin Y$, we have that $\delta_{Y\cap e}\geq\text{val}(v)$, which proves the claim.\par
So we get \begin{align*}
\beta_\mathcal D(Y\cap e)=&\deg(\mathcal D|_{Y\cap e})-\mu(Y\cap e)+\frac{\delta_{Y\cap e}}{2}\\
=&\deg(\mathcal D|_{Y_e\cap e})-\mu(Y_e\cap e)+\frac{\delta_{Y\cap e}}{2}\\
\geq&\deg(\mathcal D|_{Y_e\cap e})-\mu(Y_e\cap e)+\frac{\delta_{Y_e\cap e}}{2}\\
=&\beta_\mathcal D(Y_e\cap e). \end{align*}
The fact that $\beta_\mathcal D(Y)\geq\beta_\mathcal D(Y_e)$ for every $e\in E(\Gamma_{X,\mathcal D})\setminus E(V)$ means that we can assume that $Y$ does not contain any point in the interior of an edge outside $E(V)$.\par Now to prove that $\beta_\mathcal D(Y)\geq\beta_\mathcal D(Y_V)$, it suffices to show that, given an edge $e\in E(V)$, we have $\beta_\mathcal D(Y)\geq\beta_\mathcal D(Y^e)$, where $Y^e:=Y\cup e$. Again, by Lemma \ref{lem:beta}, we have \[ \beta_\mathcal D(Y^e)+\beta_\mathcal D(Y\cap e)=\beta_\mathcal D(Y)+\beta_\mathcal D(e). \] So it is enough that $\beta_\mathcal D(Y\cap e)\geq\beta_\mathcal D(e)$. Note that both $\mathcal D$ and $\mu$ are supported on $V(\Gamma_{X,\mathcal D})$, and $Y\cap e$ contains both vertices incident to $e$. Hence \begin{align*}
\beta_\mathcal D(Y\cap e)=&\deg(\mathcal D|_{Y\cap e})-\mu(Y\cap e)+\frac{\delta_{Y\cap e}}{2}\\
=&\deg(\mathcal D|_{e})-\mu(e)+\frac{\delta_{Y\cap e}}{2}\\
\geq&\deg(\mathcal D|_{e})-\mu(e)+\frac{\delta_{e}}{2}\\
=&\beta_\mathcal D(e). \end{align*} This concludes the proof. \end{proof} \begin{CorDef} \label{cor:quasiquasi} Preserve the notations of Proposition \ref{prop:quasiquasi} and let $\mathcal D$ be a $(p_0,\mu)$-quasistable degree-$d$ divisor on $X$. Then $\Gamma_{X,\mathcal D}$ is an $\mathcal E$-subdivision of $\Gamma_X$ for some $\mathcal E\subset E(\Gamma_X)$, and the pair $(\mathcal E,D)$ is a $(p_0,\mu)$-quasistable degree-$d$ pseudo-divisor on $\Gamma_X$, where $D$ is the divisor $\mathcal D$ seen as a divisor on $\Gamma_X^\mathcal E$. We call $(\mathcal E,D)$ the pseudo-divisor on $\Gamma_X$ induced by $\mathcal D$. \end{CorDef} \begin{proof} The fact that $\Gamma_{X,\mathcal D}$ is an $\mathcal E$-subdivision for some $\mathcal E\subset E(\Gamma_X)$ comes from Remark \ref{rem:subdivision}. The remaining statements are clear from Proposition \ref{prop:quasiquasi}. \end{proof}
Let $\iota\colon X\to Y$ be a specialization of tropical curves. Let $\mu$ be a degree-$d$ polarization on $X$ and $\mathcal D$ be a divisor on $X$ of degree $d$. We define a degree-$d$ polarization $\iota_*(\mu)$ on $Y$ and a degree-$d$ divisor $\iota_*(\mathcal D)$ on $Y$ as \[ \iota_*(\mu)(p')=\sum_{p\in \iota^{-1}(p')}\mu(p)\quad\text{ and }\quad\iota_*(\mathcal D)(p')=\sum_{p\in \iota^{-1}(p')}\mathcal D(p). \]
\begin{Lem}\label{lem:i*} Let $\iota\colon X\to Y$ be a specialization of tropical curves. Let $\mu$ be a degree-$d$ polarization of $X$ and $\mathcal D$ a $(p_0,\mu)$-quasistable divisor on $X$ of degree $d$, for some $p_0\in X$. Then $\iota_*(\mathcal D)$ is a $(\iota(p_0),\iota_*(\mu))$-quasistable divisor on $Y$ of degree $d$. \end{Lem} \begin{proof} For every proper tropical subcurve $Z$ of $Y$, we have \begin{align*}
\beta_{\iota_*(\mathcal D)}(Z)=&\deg(\iota_*(\mathcal D)|_Z)-\iota_*(\mu)(Z)+\frac{\delta_{Z}}{2}\\
=&\deg(\mathcal D|_{\iota^{-1}(Z)})-\mu(\iota^{-1}(Z))+\frac{\delta_{\iota^{-1}(Z)}}{2}\\
=&\beta_\mathcal D(\iota^{-1}(Z)), \end{align*} and the last term is always nonnegative, and it is positive if $p_0\in \iota^{-1}(Z)$ or, equivalently, if $\iota(p_0)\in Z$. This proves that $\iota_*(\mathcal D)$ is $(\iota(p_0),\iota_*(\mu))$-quasistable. \end{proof}
Next we have a key result stating that quasistable divisors can be chosen as canonical representatives for equivalence classes of divisors on a tropical curve.
\begin{Thm} \label{thm:quasistable} Let $(X,p_0)$ be a pointed tropical curve and $\mu$ a degree-$d$ polarization on $X$. Given a divisor $\mathcal D$ on $X$ of degree $d$, there exists a unique degree-$d$ divisor equivalent to $\mathcal D$ which is $(p_0,\mu)$-quasistable. \end{Thm}
\begin{proof} The set of points $p\in X$ such that $\{\mathcal D(p),\mu(p)\}\neq\{0\}$ is finite. Note that by Lemma \ref{lem:beta}, there exists a unique minimal subcurve $Y\subset X$ with $\beta_\mathcal D(Y)$ minimal, i.e., $\beta_\mathcal D(Z)\geq\beta_\mathcal D(Y)$ for every tropical subcurve $Z\subset X$ and the inequality is strict if $Z\subsetneq Y$. \par
Let $\ell_0$ be the minimum of the lengths of the edges in $\text{Out}(Y)$ (we use $\Gamma_{X,\mathcal D}$ as a model for $X$ to define $\text{Out}(Y)$). For each $e\in\text{Out}(Y)$ we give the orientation away from $Y$ and define the tropical subcurve $I_{e,\ell_0}:=\overline{p_{e,o}p_{e,\ell_0}}$ and we let $I_{e,\ell_0}^\circ$ be its interior. Let $\mathcal F:=\mathcal D_{Y,\ell_0}$ be the chip-firing divisor emanating from $Y$ with length $\ell_0$. Define $\mathcal D':=\mathcal D-\mathcal F$, and set \[ \Delta_Y:=Y\cap \overline{X\setminus Y} \quad \text{ and } \quad \Delta_{\mathcal F,Y}:=\text{supp}(\mathcal F)\setminus\Delta_Y. \] In other words, $\Delta_{\mathcal F,Y}$ are the points $p\in X$ where $\mathcal F(p)>0$. Note that $\mu(e^\circ)=0$ and $\text{supp}(\mathcal D)\cap e^\circ=\emptyset$ for every $e\in\text{Out}(Y)$.\par
In what follows, we will prove that, for every subcurve $Z$ of $X$, then $\beta_{\mathcal D'}(Z)\geq \beta_\mathcal D(Y)$, and if the equality holds, then $Z$ is strictly bigger then $Y$, in the sense that $\rel_{D'}(Z)<\rel_D(Y)$.\par
We begin proving some basic inequalities. Let $Z$ be a tropical subcurve of $X$ such that $Z\cap Y=\emptyset$ and $Z\cap I_{e,\ell_0}^\circ=\emptyset$ for every $e\in \text{Out}(Y)$. Define \[ W:=Z\cup Y\cup\underset{p_{e,\ell_0}\in Z}{\bigcup_{e\in\text{Out}(Y)}}I_{e,\ell_0}. \] We have \begin{align*}
\beta_\mathcal D(W)=&\deg(\mathcal D|_W)-\mu(W)+\frac{\delta_{W}}{2}\\
=&\deg(\mathcal D|_Y)+\deg(\mathcal D|_Z)-\mu(Y)-\mu(Z)+\frac{\delta_{Y}+\delta_{Z}}{2}\\&-|\{e\in\text{Out}(Y);p_{e,\ell_0}\in Z\}|\\
=&\beta_\mathcal D(Y)+\beta_\mathcal D(Z)-|\{e\in\text{Out}(Y);p_{e,\ell_0}\in Z\}|\\
=&\beta_\mathcal D(Y)+\beta_\mathcal D(Z)-\deg(\mathcal F|_{Z\cap\Delta_{\mathcal F,Y}}), \end{align*} where the last equality comes from Equation \eqref{eq:chip}. By the minimal property of $Y$ we have $\beta_\mathcal D(W)\geq\beta_\mathcal D(Y)$. We deduce that, for every tropical subcurve $Z$ of $X$ such that $Z\cap Y=\emptyset$ and $Z\cap I_{e,\ell_0}^\circ=\emptyset$ for every $e\in\text{Out}(Y)$, \begin{equation} \label{eq:betaF}
\beta_\mathcal D(Z)\geq\deg(\mathcal F|_{Z\cap\Delta_{\mathcal F,Y}}). \end{equation}
Now, let us prove that for every subcurve $Z$ of $X$ we have $\beta_{\mathcal D'}(Z)\geq\beta_D(Y)$. If $Z'$ is a connected component of $Z$ contained in $I_{e,\ell_0}^\circ$ for some $e\in$, then $\beta_{\mathcal D'}(Z')=1$. By Lemma \ref{lem:beta}, we have that $\beta_{\mathcal D'}(Z)=\beta_{\mathcal D'}(Z\setminus Z')+1$. Then, we can restrict to the case where $Z$ has no connected components contained in $I_{e,\ell_0}^\circ$ for every $e\in\text{Out}(Y)$. Define $Z_1:=Z\cap Y$ and \[ Z_2:=\overline{Z\setminus(Y\cup\bigcup_{e\in\text{Out}(Y)}I_{e,\ell_0})}. \] Then \begin{align*}
\beta_\mathcal D(Z)=&\deg(\mathcal D|_Z)-\mu(Z)+\frac{\delta_{Z}}{2}\\
=&\deg(\mathcal D|_{Z_1})+\deg(\mathcal D|_{Z_2})-\mu(Z_1)-\mu(Z_2)+\frac{\delta_{Z_1}+\delta_{Z_2}}{2}\\&-|\{e\in\text{Out}(Y);I_{e,\ell_0}\subset Z\}|\\
=&\beta_\mathcal D(Z_1)+\beta_\mathcal D(Z_2)-|\{e\in\text{Out}(Y);I_{e,\ell_0}\subset Z\}|, \end{align*} from which we get \begin{align*}
\beta_{\mathcal D'}(Z) =& \beta_\mathcal D(Z)-\deg(\mathcal F|_Z) \\
=&\beta_\mathcal D(Z)-\deg(\mathcal F|_{Z\cap \Delta_Y})-\deg(\mathcal F|_{Z\cap\Delta_{\mathcal F,Y}})\\
=&\beta_\mathcal D(Z_1)+(\beta_\mathcal D(Z_2)-\deg(\mathcal F|_{Z_2\cap\Delta_{\mathcal F,Y}}))\\&-(|\{e\in\text{Out}(Y);I_{e,\ell_0}\subset Z\}|+\deg(\mathcal F|_{Z_1\cap \Delta_Y})). \end{align*} However, by Equation \eqref{eq:chip}, we have \begin{equation} \label{eq:Z1}
|\{e\in\text{Out}(Y);I_{e,\ell_0}\subset Z\}|+\deg(\mathcal F|_{Z_1\cap \Delta_Y})\leq0 \end{equation} and, by Equation \eqref{eq:betaF}, and the fact that $Z\cap Y=\emptyset$ and $Z\cap I_{e,\ell_0}^\circ=\emptyset$ for every $e\in \text{Out}(Y)$, we have \[
\beta_\mathcal D(Z_2)-\deg(\mathcal F|_{Z_2\cap\Delta_{\mathcal F,Y}})\geq0, \]
so we deduce that $\beta_{\mathcal D'}(Z)\geq\beta_\mathcal D(Z_1)\geq\beta_\mathcal D(Y)$ (recall the minimal property of $Y$). Moreover, if $\beta_{\mathcal D'}(Z)=\beta_\mathcal D(Y)$, then $Z_1=Y$ and equality holds in Equation \eqref{eq:Z1}. In this case, $e\subset Z$ for every $e\in\text{Out}(Y)$ because $Z_1=Y$, from which we get $\deg(\mathcal F|_{Z_1\cap \Delta_Y})=\deg(\mathcal F|_{\Delta_Y})=|\text{Out}(Y)|$, and hence $\rel_{\mathcal D'}(Z)<\rel_\mathcal D(Y)$, because $Z$ contains the vertices incident to any edge $e_0\in \text{Out}(Y)$ of length $\ell_0$.\par
Repeating the process, we eventually arrive in the case where the minimal subcurve $Y\subset X$ with $\beta_\mathcal D(Y)$ minimal is empty. In this case, $\beta_{\mathcal D}(Y)=0$ and $\mathcal D$ is equivalent to a $\mu$-semistable divisor $\mathcal D'$. To prove that every $\mu$-semistable divisor $\mathcal D$ is equivalent to a $(p_0,\mu)$-quasistable divisor, just repeat the above process for the minimal curve $Y$ containing $p_0$ with $\beta_\mathcal D(Y)=0$.
Finally we prove the uniqueness in the statement. Assume that $\mathcal D_1$ and $\mathcal D_2$ are $(p_0,\mu)$-quasistable divisors on $X$ of degree $d$ such that $\mathcal D_1\sim \mathcal D_2$, then $\mathcal D_1=\mathcal D_2$.
Since $\mathcal D_1\sim \mathcal D_2$, there exists a rational function $f$ on $X$ such that $\mathcal D_1=\mathcal D_2+\textnormal{div}(f)$. By contradiction, assume that $f$ is not constant.\par Assume that $f$ is also nowhere constant. By Lemma \ref{lem:valP}, there exists a point $p\in X$ such that $\textnormal{ord}_p(f)\geq\delta_{X,p}$, then $\deg(\mathcal D_1|_p)\geq\deg(\mathcal D_2|_p)+\delta_{X,p}$. Using that $\mathcal D_1$ and $\mathcal D_2$ are $(p_0,\mu)$-quasistable, we get \[ \delta_{X,p}\geq\beta_{\mathcal D_1}(p)\geq\beta_{\mathcal D_2}(p)+\delta_{X,p}\geq\delta_{X,p}. \] Then $\beta_{\mathcal D_1}(p)=\delta_{X,p}$ and $\beta_{\mathcal D_2}(p)=0$. If $p\neq p_0$, the first equality is a contradiction, if $p= p_0$, the second equality is a contradiction.\par Assume now that there are segments of $X$ over which $f$ is constant. Let $\iota\colon X\rightarrow X'$ be the specialization of tropical curves obtained by contracting all the maximal segments of $X$ over which $f$ is constant. Note that $X'$ is not a point because $f$ is not constant. Then $\iota_*(\mathcal D_1)$ and $\iota_*(\mathcal D_2)$ are $(\iota(p_0),\iota_*(\mu))$-quasistable divisors on $X'$ by Lemma \ref{lem:i*}, and $f$ induces a rational function $f'$ on $X'$ which is nowhere constant, and such that $\iota_*(\mathcal D_1)=\iota_*(\mathcal D_2)+\textnormal{div}(f')$. By the first part of the proof, we have a contradiction. \end{proof}
Let $(X,p_0)$ be a pointed tropical curve with length function $\ell$, and let $\mu$ be a degree-$d$ polarization on $X$. Let $\Gamma_X$ be the model of $X$ whose vertices are the relevant points of $X$. For each $(p_0,\mu)$-quasistable pseudo-divisor $(\mathcal E,D)$ on $\Gamma_X$ we define polyhedra \begin{align*} \mathcal{P}_{(\mathcal E,D)}:=&\prod_{e\in\mathcal E}e=\prod_{e\in\mathcal E}[0,\ell(e)]\subset \mathbb R^{\mathcal E}\\ \mathcal{P}^\circ_{(\mathcal E,D)}:=&\prod_{e\in\mathcal E}e^\circ=\prod_{e\in\mathcal E}(0,\ell(e))\subset \mathbb R^{\mathcal E}, \end{align*} where $e^\circ$ denotes the interior of an edge $e$. Note that if $\mathcal E=\emptyset$, then $\mathcal{P}_{\mathcal E,D}$ is just a point. If $(\mathcal E,D)\rightarrow(\mathcal E',D')$ is a specialization of pseudo-divisors, then we have an induced face morphism of polyhedra $f\colon\mathcal{P}_{(\mathcal E',D')}\to\mathcal{P}_{(\mathcal E,D)}$. The polyhedron $\mathcal{P}^\circ_{(\mathcal E,D)}$ parametrizes $(p_0,\mu)$-quasistable divisors on $X$, whose induced pseudo-divisor on $\Gamma_X$ is $(\mathcal E,D)$.
\begin{Def} \label{def:jtrop} Let $(X, p_0)$ be a pointed tropical curve and $\mu$ be a degree-$d$ polarization on $X$. The \emph{Jacobian of $X$ with respect to $(p_0,\mu)$} is the polyhedral complex \[ J^\text{trop}_{p_0,\mu}(X)=\lim_{\longrightarrow}\mathcal{P}_{(\mathcal E,D)}, \] where the limit is taken over the poset $\mathcal{QD}_{p_0,\mu}(\Gamma_X)$. We have a set-theoretically decomposition \[ J^\text{trop}_{p_0,\mu}(X)=\coprod_{(\mathcal E,\mathcal D)} \mathcal{P}^\circ_{(\mathcal E,D)}, \] where the union is taken over $(\mathcal E,\mathcal D)\in\mathcal{QD}_{p_0,\mu}(\Gamma_X)$. \end{Def}
\begin{Exa} Let $X$ be the tropical curve associated to the graph with $2$ vertices and $3$ edges connecting such vertices, as in Figure \ref{fig:poset}, where all edges have length $1$. Let $\mu$ be the degree-$0$ polarization on $X$ given by $\mu(p)=0$ for every $p\in X$ and assume that $p_0$ is the leftmost vertex. Then the polyhedral complex $J^\text{trop}_{p_0,\mu}(X)$ is depicted in Figure \ref{fig:jac}. \begin{figure}
\caption{The Jacobian $J_{p_0,\mu}^{trop}(X)$.}
\label{fig:jac}
\end{figure}
By the description of the poset $\mathcal{QD}_{p_0,\mu}(\Gamma_X)$ in Example \ref{exa:poset}, we see that $J^\text{trop}_{p_0,\mu}(X)$ has $3$ cells of dimension 2, $6$ cells of dimension $1$, and $3$ cells of dimension $0$. Note that the outer edges are identified making $J^\text{trop}_{p_0,\mu}(X)$ a real torus. Also, the point associated to the zero divisor in $\Gamma_X$ is distinguished since it is contained in every cell of dimension 1, while the other cells of dimension 0 are contained in exactly $3$ cells of dimension 1. \end{Exa}
\begin{Rem} \label{rem:cont} There is a natural map $f\colon J^\text{trop}_{p_0,\mu}(X)\to \mathcal{QD}_{p_0,\mu}(\Gamma_X)$ and this map is continuous. Indeed, if $(\mathcal E,D)$ is a pseudo-divisor on $\Gamma_X$ and $T=\overline{\{(\mathcal E,D)\}}$ is the closure of $\{(\mathcal E,D)\}$ in $\mathcal{QD}_{p_0,\mu}(\Gamma_X)$, then $f^{-1}(T)=\mathcal{P}_{(\mathcal E,D)}$ which is closed. So $f$ is continuous, since every closed set in $\mathcal{QD}_{p_0,\mu}(\Gamma_X)$ is a finite union of such closures. \end{Rem}
Now we give some topological properties of the Jacobian of $X$ with respect to $(p_0,\mu)$, comparing it with the usual tropical Jacobian of $X$ (recall Equation \eqref{eq:Jtropdef}).
\begin{Thm} \label{thm:jX} Let $(X, p_0)$ be a pointed tropical curve and let $\mu$ be a degree-$d$ polarization on $X$. We have that $J^\text{trop}_{p_0,\mu}(X)$ has pure dimension $g$, it is connected in codimension $1$, and homeomorphic to $J^\text{trop}(X)$. \end{Thm}
\begin{proof} The fact that $J^\text{trop}_{p_0,\mu}(X)$ has pure dimension $g$ and connected in codimension $1$ follows from Propositions \ref{prop:chain} and \ref{prop:cod1}.\par
There exists a function $\alpha\colon J^\text{trop}_{p_0,\mu}(X)\to J^\text{trop}(X)$ that takes a $(p_0,\mu)$-quasistable divisor $\mathcal D$ to the class of $\mathcal D-dp_0$ in $J^\text{trop}(X)$. It follows from Theorem \ref{thm:quasistable} that $\alpha$ is a bijection. We now prove that $\alpha$ is a homeomorphism by showing that it is continuous (this is enough since $J^\text{trop}_{p_0,\mu}(X)$ is compact and $J^\text{trop}(X)$ is Hausdorff).\par Fix an orientation on $\Gamma_X$ and fix (not necessarily oriented) paths from $p_0$ to every vertex of $\Gamma_X$. Let $(\mathcal E,D)$ be a $(p_0,\mu)$-quasistable divisor on $\Gamma_X$ and define $D_0\in \text{Div}(\Gamma_X)$ by $D_0(v)=D(v)$ for every $v\in V(\Gamma_X)$. Let $\mathcal D_0$ be the divisor on $X$ induced by $D_0$. Given a divisor $\mathcal D$ on $X$ parametrized by a point in $\mathcal{P}_{(\mathcal E,D)}$, then there are real numbers $x_e\in[0,\ell(e)]$ such that $\mathcal D=\mathcal D_0-\sum_{e\in\mathcal E}p_{e,x_e}$.\par
For every $\mathcal D\in \mathcal{P}_{(\mathcal E,D)}$, we define the path $\gamma(\mathcal D)$ as \[ \gamma(\mathcal D):=\gamma(\mathcal D_0)+\sum_{e\in\mathcal E}\overrightarrow{p_{e,0}p_{e,x_e}}, \]
This induces a map \[ \alpha_{(\mathcal E,D)}\colon\mathcal{P}_{(\mathcal E,D)}\to\Omega(X)^\vee \] taking a divisor $\mathcal D$ to $\int_{\gamma(\mathcal D)}$. More precisely, the map $\alpha_{(\mathcal E,D)}$ is given by \begin{align*} \alpha_{(\mathcal E,D)}\colon\prod_{e\in\mathcal E}[0,\ell(e)]&\to \Omega(X)^\vee\\ (p_{e,x_e})_{e\in\mathcal E}&\mapsto \int_{\gamma(\mathcal D_0)}+\sum_{e\in\mathcal E}\frac{x_e}{\ell(e)}\int_{e}. \end{align*} This means that $\alpha_{(\mathcal E,D)}$ is an affine map, hence it is continuous. The composition of this map with the quotient map $\Omega(X)^\vee\rightarrow J^{trop}(X)$ is the restriction of $\alpha$ to $\mathcal{P}_{(\mathcal E,D)}$. Since $J^\text{trop}_{p_0,\mu}(X)$ is the direct limit of the polyhedra $\mathcal{P}_{(\mathcal E,D)}$, then $\alpha$ is continuous. \end{proof}
Now we move to the universal setting: we define a universal tropical Jacobian, give a modular description of the points that it parametrizes, and we prove the analogue of the properties stated in Theorem \ref{thm:maingraph}.
\begin{Def} Let $(\Gamma,v_0)$ be a graph with $1$ leg and $\mu$ be a degree-$d$ polarization on $\Gamma$. For each $(v_0,\mu)$-quasistable pseudo-divisor $(\mathcal E,D)$ on $\Gamma$, we define \[ \sigma_{(\Gamma,\mathcal E,D)}:=\mathbb{R}^{E(\Gamma^\mathcal E)}_{\geq0} \quad \text{and}\quad \sigma^\circ_{(\Gamma,\mathcal E,D)}:=\mathbb{R}^{E(\Gamma^\mathcal E)}_{>0}. \] Note that, if $\iota\colon(\Gamma,\mathcal E,D)\rightarrow(\Gamma',\mathcal E',D')$ is a specialization, then there exists a natural inclusion $\iota\colon \sigma_{(\Gamma',\mathcal E',D')}\to \sigma_{(\Gamma,\mathcal E,D)}$. Let $\mu$ be a universal genus-$g$ polarization. The \emph{universal Jacobian with respect to $\mu$} is defined as the generalized cone complex \[ J^\text{trop}_{\mu,g}:=\lim_{\longrightarrow} \sigma_{(\Gamma,\mathcal E,D)}=\coprod_{(\Gamma,\mathcal E,D)} \sigma_{(\Gamma,\mathcal E,D)}^\circ/\text{Aut}(\Gamma,\mathcal E,D) \] where the union is taken over all terns $(\Gamma,\mathcal E,D)$ running through all objects in the category $\mathbf{QD}_{\mu,g}^{op}$ (the opposite of the category $\mathbf{QD}_{\mu,g}$). \end{Def}
\begin{Prop} \label{prop:parameter} Let $\mu$ be a universal genus-$g$ polarization. The generalized cone complex $J^\text{trop}_{\mu,g}$ parametrizes equivalence classes $(X,\mathcal D)$, where $X$ is a stable pointed tropical curve of genus $g$ and $\mathcal D$ is a $(p_0,\mu)$-quasistable divisor on $X$, where $p_0$ is the marked point of $X$. \end{Prop}
\begin{proof} Let $(X,p_0)$ be a stable pointed tropical curve of genus $g$. The stable model $\Gamma:=\Gamma_{st}$ of $X$ is a genus-$g$ stable weighted graph with $1$ leg. The polarization $\mu_{\Gamma}$ induces a polarization on $X$, such that $\Gamma_X=\Gamma$ (recall that $\Gamma_X$ denotes the model of $X$ whose vertices are the relevant points).\par
It follows from Corollary \ref{cor:quasiquasi} that, if $\mathcal D$ is a $(p_0,\mu)$-quasistable divisor of $X$, then $\Gamma_{X,\mathcal D}=\Gamma^\mathcal E$ for some $\mathcal E\subset E(\Gamma_X)$. Note that $(\mathcal E,D)$ is a $(p_0,\mu)$-quasistable pseudo-divisor on $\Gamma$, where $D$ is the divisor on $\Gamma^\mathcal E$ induced by $\mathcal D$. Therefore $X$ corresponds to a point in $\sigma^\circ_{(\Gamma_X,\mathcal E,D)}=\mathbb R^{E(\Gamma^\mathcal E)}_{>0}$, and the pair $(X,\mathcal D)$ corresponds to a point in $J^\text{trop}_{\mu,g}$. If $(X,\mathcal D)$ and $(X',\mathcal D')$ are isomorphic, then the construction above will give rise to the same point in $J^\text{trop}_{\mu,g}$. On the other hand if $(X,\mathcal D)$ and $(X',\mathcal D')$ corresponds to the same point in $J^\text{trop}_{\mu,g}$ that belongs to a cell $\sigma^\circ_{(\Gamma,\mathcal E,D)}/\textnormal{Aut}(\Gamma,\mathcal E,D)$, then there is an isomorphism $\iota\colon\Gamma_X\rightarrow\Gamma_{X'}$ such that $\iota(\mathcal E)=\mathcal E'$ and $\iota_*(D)=D'$. Moreover, it follows that the metrics of $X$ and $X'$ are equal, hence $\iota$ induces an isomorphism of metric graphs between $X$ and $X'$ taking $\Gamma_{X,\mathcal D}$ to $\Gamma_{X',\mathcal D'}$, and hence $\mathcal D$ to $\mathcal D'$. This means that $(X,\mathcal D)$ and $(X',\mathcal D')$ are isomorphic.\par
Conversely, it is clear that, for every triple $(\Gamma,\mathcal E,D)$, every point in $\sigma^\circ_{(\Gamma,\mathcal E,D)}$ corresponds to a pair $(X,\mathcal D)$ with $X$ a stable pointed tropical curve of genus $g$ and $\mathcal D$ a $(p_0,\mu)$-quasistable divisor on $X$. \end{proof}
It follows from Proposition \ref{prop:parameter} that we have a natural forgetful map \[ \pi^{trop}\colon J^\text{trop}_{\mu,g}\to M^\text{trop}_{g,1}. \]
\begin{Rem} \label{rem:contmu} Note that the natural map $f\colon J^\text{trop}_{\mu,g}\to \mathcal{QD}_{\mu,g}$ is continuous. One can prove this fact as in Remark \ref{rem:cont}. \end{Rem}
\begin{Thm} \label{thm:maintrop} Let $\mu$ be a universal genus-$g$ polarization. The generalized cone complex $J^\text{trop}_{\mu,g}$ has pure dimension $4g-2$ and is connected in codimension $1$. The map $\pi^{trop}\colon J^\text{trop}_{\mu,g}\to M^\text{trop}_{g,1}$ is a map of generalized cone complexes. For every stable pointed tropical curve $X$ of genus $g$, we have a homeomorphism \[ (\pi^{trop})^{-1}([X])\simeq J^\text{trop}_{p_0,\mu}(X)/\textnormal{Aut}(X). \] \end{Thm} \begin{proof} The fact that $J^\text{trop}_{\mu,g}$ has pure dimension $4g-2$ and is connected in codimension $1$ follows from Theorem \ref{thm:maingraph} and Remark \ref{rem:contmu}.\par
For each cone $\sigma_{(\Gamma,\mathcal E,D)}$, we have that $\pi^{\text{trop}}$ induces a map $\sigma_{(\Gamma,\mathcal E,D)}\to M_{g,1}^\text{trop}$ that factors through a chain of maps \[ \sigma_{(\Gamma,\mathcal E,D)}=\mathbb R_{\geq0}^{E(\Gamma^\mathcal E)}\to \mathbb R_{\geq 0}^{E(\Gamma)}\rightarrow \mathbb R_{\geq 0}^{E(\Gamma)}/\textnormal{Aut}(\Gamma) \subset M_{g,1}^\text{trop}, \]
where the first one is defined in Equation \eqref{eq:hat} and the second one is the natural quotient map. Hence $\pi^{trop}$ is a morphism of generalized cone complexes.\par
There exists a natural map $h\colon J^\text{trop}_{p_0,\mu}(X)\to J^\text{trop}_{\mu,g}$ and we have $(\pi^{trop})^{-1}([X])=\text{Im}(h)$. Moreover, $h(\mathcal D)=h(\mathcal D')$ if and only if there exists an automorphism $\alpha\colon X\to X$ such that $\alpha_*(\mathcal D)=\mathcal D'$, which implies that $\text{Im}(h)=J^\text{trop}_{p_0,\mu}(X)/\textnormal{Aut}(X)$. \end{proof}
\begin{Rem}
It is easy to check that the maximal cells of $J^\text{trop}_{\mu,g}$ are of type $\sigma_{(\Gamma,\mathcal E,D)}/\textnormal{Aut}(\Gamma,\mathcal E,D)$ where $\Gamma$ is a 3-regular graph with weight function $0$ and $|\mathcal E|=g$. Moreover, the codimension $1$ cells of $J^\text{trop}_{\mu,g}$ are of the following type:
\begin{enumerate}
\item $\sigma_{(\Gamma,\mathcal E,D)}/\textnormal{Aut}(\Gamma,\mathcal E,D)$ where $\Gamma$ has weight function $0$ and has exactly one vertex of valence $4$ and all other vertices of valence $3$, and $|\mathcal E|=g$;
\item $\sigma_{(\Gamma,\mathcal E,D)}/\textnormal{Aut}(\Gamma,\mathcal E,D)$ where $\Gamma$ has weight function $0$ on all vertices but exactly one vertex of weight $1$ and valence $1$ and all other vertices of valence $3$, and $|\mathcal E|=g$;
\item $\sigma_{(\Gamma,\mathcal E,D)}/\textnormal{Aut}(\Gamma,\mathcal E,D)$ where $\Gamma$ is a $3$-regular graph of weight $0$, and $|\mathcal E|=g-1$.
\end{enumerate} \end{Rem}
We end this section introducing a compactification of $J^{\text{trop}}_{\mu,g}$. First of all, we set \[ \overline{\sigma}_{(\Gamma,\mathcal E,D)}:=\overline{\mathbb R}_{\geq0}^{E(\Gamma^\mathcal E)}\quad \text{ and }\quad \overline{\sigma}^\circ_{(\Gamma,\mathcal E,D)}:=\overline{\mathbb R}_{>0}^{E(\Gamma^\mathcal E)}, \] where $\overline{\mathbb R}=\mathbb R\cup\{\infty\}$. Then the compactification $\overline{J}^{\text{trop}}_{\mu,g}$ of $J^{\text{trop}}_{\mu,g}$ is defined as \[ \overline{J}^{\text{trop}}_{\mu,g}=\lim_{\longrightarrow}\overline{\sigma}_{(\Gamma,\mathcal E,D)} \] where $(\Gamma,\mathcal E,D)$ runs through all objects in the category $\mathbf{QD}_{\mu,g}^{op}$. We note that $\overline{J}^{\text{trop}}_{\mu,g}$ has a structure of extended generalized cone complexes as defined in \cite[Section 2]{ACP}. Moreover the map $\pi^\text{trop}\colon J^\text{trop}_{\mu,g}\to M^\text{trop}_{g,1}$ of generalized cone complexes extends to a map of extended generalized cone complexes \[ \overline{\pi}^\text{trop}\colon \overline{J}^\text{trop}_{\mu,g}\to \overline{M}^\text{trop}_{g,1} \] as defined in \cite[Section 2]{ACP}. The map $\overline{\pi}^\text{trop}$ when restricted to the cone $\overline{\mathbb R}_{\geq0}^{E(\Gamma^\mathcal E)}$ is the map $\overline{\mathbb R}_{\geq0}^{E(\Gamma^\mathcal E)}\to \overline{\mathbb R}_{\geq0}^{E(\Gamma)} $ defined in Equation \eqref{eq:hat} where $y_e=\infty$ if $x_{e'}=\infty$ for some edge $e'\in E(\Gamma^\mathcal E)$ over $e$. \par
\begin{Rem} The points of the boundary of $\overline{J}^\text{trop}_{\mu,g}$ parametrizes pairs $(X,\mathcal D)$ where $X$ is an extended tropical curve (in the sense of \cite[Section 3.3]{Caporaso}) and $\mathcal D$ a $(p_0,\mu)$-quasistable divisor on $X$. Here, the notion of $(p_0,\mu)$-quasistability for extended tropical curves can be given exactly as for tropical curves in Definition \ref{def:quasistable} (considering extended tropical subcurves). We do not know if there is an analogous of Theorem \ref{thm:quasistable} for extended tropical curves. \end{Rem}
\section{The skeleton of the Esteves' universal Jacobian}\label{sec:skeleton}
\subsection{The Esteves' universal compactified Jacobian}\label{sec:EstJac}
Let $(X,p_0)$ be a pointed nodal curve defined over an algebraically closed field $k$. Recall that a coherent sheaf $\mathcal I$ on $X$ is \emph{torsion-free} if it has no embedded components, \emph{rank-1} if it is invertible on a dense open subset of $C$, and \emph{simple} if $\text{Hom}(\mathcal I,\mathcal I)=k$. The \emph{degree} of $I$ is $\deg I=\chi(I)-\chi(\mathcal O_X)$.
Given an integer $d$, the \emph{degree-$d$ Jacobian} $\mathcal{J}_d(X)$ of $X$ is the scheme parametrizing the equivalence classes of invertible sheaves of degree $d$ on $X$. In general, $\mathcal{J}_d(X)$ is neither proper nor of finite type. A better behaved parameter space is obtained by resorting to torsion-free rank-$1$ sheaves and to stability conditions.\par
The scheme $\mathcal{J}_d(X)$ is an open dense subscheme of the scheme $\mathcal{S}pl_d(X)$ parametrizing simple torsion-free rank-$1$ sheaves of degree-$d$ on $X$. We refer to \cite{AK} and \cite{Es01} for the construction of $\mathcal{S}pl_d(X)$ and its properties. Recall that $\mathcal{S}pl_d(X)$ is universally closed over $k$ and connected but, in general, not separated and only locally of finite type. To deal with a manageable piece of it, we resort to polarizations.
Let $X_1,\ldots,X_m$ be the irreducible components of $X$. A \emph{degree-$d$ polarization on $X$} is any $m$-tuple of rational numbers $\mu=(\mu_1,\ldots,\mu_m)$ summing up to $d$. For every proper subcurve $Y$ of $X$, we set
\[
\mu(Y):=\sum_{X_i\subset Y} \mu_i.
\]
The notion of a degree-$d$ polarization on a curve can also be given by a vector bundle $\mathcal{F}$ on $X$ such that $\deg(\mathcal{F})=-d\cdot\text{rank}(\mathcal{F})$. In this case $\mu_i=-\deg(\mathcal{F}|_{X_i})/\text{rank}(\mathcal{F})$ for every $i=1,\ldots,m$ (see \cite[Section 1.2]{Es01} and \cite[Remark 4.6]{KP}).\par
Note that $\mu$ can be seen as a degree-$d$ polarization on the dual graph $(\Gamma_X, v_0)$ of the pointed curve $X$. Conversely, every polarization on the dual graph $(\Gamma_X, v_0)$ induces a polarization on $X$.\par
Let $I$ be a rank-$1$ degree-$d$ torsion-free sheaf on $X$. We can define a pseudo-divisor $(\mathcal E_I,D_I)$ on $\Gamma_X$ as follows. The set $\mathcal E_I\subset E(\Gamma_X)$ is precisely the set of edges corresponding to nodes where $I$ is not locally free. For every $v\in V(\Gamma^\mathcal E)$, we set \[ D_I(v)=\begin{cases}
\deg(I|_{X_v}),&\text{ if $v\in V(\Gamma_X)$};\\
-1,&\text{ if $v$ is exceptional},
\end{cases} \] where $X_v$ is the component of $X$ corresponding to $v\in V(\Gamma_X)$. We call $(\mathcal E_I,D_I)$ the \emph{multidegree} of $I$. We say that $I$ is $\mu$-semistable (respectively, \emph{$(p_0,\mu)$-quasistable}) if its multidegree $(\mathcal E_I,D_I)$ is a $\mu$-semistable (respectively, $(v_0,\mu)$-quasistable) pseudo-divisor on $\Gamma_X$. We call $(\Gamma_X,\mathcal E_I,D_I)$ the \emph{dual graph} of $(X,I)$. \par
Let $\mathcal{QD}_d(\Gamma_X)$ be the poset of all pseudo-divisors of degree-$d$ on $\Gamma_X$. The above construction gives rise to an anti-continuous function \begin{align*} \mathcal{S}pl_d(X)&\to \mathcal{QD}_d(\Gamma_X)\\
I&\mapsto (\mathcal E_I,D_I). \end{align*}
The above notions naturally extend to families.
Let $f\colon\mathcal X\to T$ be a family of pointed nodal curves with section $\sigma\colon T\to\mathcal X$. A polarization $\mu$ on $\mathcal X$ is the datum of polarizations on the fibers of $f$ that are compatible with specializations. We say that a sheaf $\mathcal I$ over $\mathcal X$ is \emph{$(\sigma,\mu)$-quasistable} if, for any closed point $t\in T$, the restriction of $\mathcal I$ to the fiber $f^{-1}(t)$ is a torsion-free rank-$1$ and $(\sigma(t),\mu)$-quasistable sheaf. There is an algebraic space $\overline{\mathcal{J}}_{\pi,\mu}$ parametrizing $(\sigma,\mu)$-quasistable sheaves over $\mathcal X$. This algebraic space is proper and of finite type (\cite[Theorems A and B]{Es01}) and it represents the contravariant functor $\mathbf{J}_{\pi,\mu}$ from the category of locally Noetherian $T$-schemes to sets, defined on a $T$-scheme $B$ by \[ \mathbf{J}_{\pi,\mu}(B):=\{(\sigma_B,\mu_B)\text{-quasistable sheaves over } \mathcal X\times_T B\stackrel{\pi_B}\longrightarrow B\}/\sim \] where $\sigma_B$ and $\mu_B$ are the pullback to $\mathcal X\times_T B$ of the section $\sigma$ and polarization $\mu$, and where $\sim$ is the equivalence relation given by $\mathcal I_1\sim \mathcal I_2$ if and only if there exists an invertible sheaf $\mathcal L$ on $B$ such that $\mathcal I_1\cong \mathcal I_2\otimes \pi_B^*\mathcal L$. \par
This construction can be extended to the universal setting. More precisely, let $\overline{\mathcal M}_{g,1}$ be the Deligne-Mumford stack parametrizing stable pointed genus-$g$ curves, and let $\mathcal M_{g,1}$ be its open locus. Let $\overline{\mathcal M}_{g,2}\to\overline{\mathcal M}_{g,1}$ be the universal family over $\overline{\mathcal M}_{g,1}$. Let $\mathcal{J}_{d,g,1}\rightarrow\mathcal M_{g,1}$ be the universal degree-$d$ Jacobian parametrizing invertible sheaves of degree-$d$ on smooth fibers of $\overline{\mathcal M}_{g,2}\to\overline{\mathcal M}_{g,1}$.
For each universal degree-$d$ polarization $\mu$ over $\overline{\mathcal M}_{g,2}\to\overline{\mathcal M}_{g,1}$, there is a proper and separated Deligne-Mumford stack $\overline{\mathcal{J}}_{\mu,g}$ over $\overline{\mathcal M}_{g,1}$ containing $\mathcal{J}_{d,g,1}$ as open dense subset. For every scheme $S$, we have \[ \overline{\mathcal{J}}_{\mu,g}(S)=\frac{\left\{(\pi,\sigma,\mathcal I);\begin{array}{l} \pi\colon \mathcal X\to S\text{ is a family of stable pointed genus-g curves,}\\ \mathcal I \text{ is a $(\sigma,\mu)$-quasistable torsion free rank-1 sheaf on $\mathcal X$}\end{array}\right\}}{\sim} \] where $(\pi_1,\sigma_1,\mathcal I_1)\sim(\pi_2,\sigma_2,\mathcal I_2)$ if there exist a $S$-isomorphism $f\colon \mathcal X_1\to \mathcal X_2$ and an invertible sheaf $\mathcal L$ on $S$, such that $\sigma_2=f\circ\sigma_1$ and $\mathcal I_1\cong f^*\mathcal I_2\otimes\pi_1^*\mathcal L$. We refer to \cite[Theorems A and B]{M15} and \cite[Corollary 4.4 and Remark 4.6]{KP} for more details on the stack $\overline{\mathcal{J}}_{\mu,g}$. In what follows we will consider the universal compactified Jacobian $\overline{\mathcal{J}}_{\mu,g}$ where $\mu$ is the canonical polarization (recall Example \ref{exa:pol}).
\subsection{The stratification of $\overline{\mathcal{J}}_{\mu,g}$}
In this section we study the stratification of the open embedding $\mathcal{J}_{d,g,1}\subset \overline{\mathcal{J}}_{\mu,g}$ and we collect some local properties of $\overline{\mathcal{J}}_{\mu,g}$.
Let $\mathcal{J}_{(\Gamma,\mathcal E,D)}$ be the substack of $\overline{\mathcal{J}}_{\mu,g}$ that parametrizes tuples $(\pi\colon \mathcal X\to S,\sigma,\mathcal I)$ where for each $s\in S$ we have that $(\mathcal X_s,\mathcal I|_{\mathcal X_s})$ have dual graph isomorphic to $(\Gamma,\mathcal E,D)$. Let $\mathcal{J}_{(\Gamma,\mathcal E,D)}\to \mathcal M_{\Gamma}$ be the forgetful map, and consider \[ \widetilde{\mathcal M}_\Gamma:=\prod_{v\in V(\Gamma)}\mathcal M_{w(v),\text{val}(v)+\ell(v)}. \] Let $\mathcal C_\Gamma\to \mathcal M_{\Gamma}$ be the universal family over $\mathcal M_\Gamma$ and define $\widetilde{\mathcal C}_\Gamma:=\mathcal C_\Gamma\times_{\mathcal M_\Gamma} \widetilde{\mathcal M}_\Gamma$. Consider the partial normalization $\widetilde{\mathcal C}_{\Gamma,\mathcal E}$ of $\widetilde{\mathcal C}_\Gamma$ over the nodes corresponding to edges in $\mathcal E$, and form the commutative diagram \[ \SelectTips{cm}{11} \begin{xy} <16pt,0pt>: \xymatrix{ \widetilde{\mathcal C}_{\Gamma,\mathcal E}\ar[dr]_{f}\ar[r] &\widetilde{\mathcal C}_\Gamma \ar[d]\ar[r]& \mathcal C_\Gamma\ar[d]\\
& \widetilde{\mathcal M}_\Gamma\ar[r] & \mathcal M_\Gamma } \end{xy} \] Let $F$ be the divisor on $\Gamma_\mathcal E$ such that $F(v)=\text{val}_\mathcal E(v)$ (recall that $\Gamma_\mathcal E$ is the graph obtained by removing the edges in $\mathcal E$ of $\Gamma$). Let $\widetilde{\mathcal{J}}_{(\Gamma,\mathcal E,D)}:=\mathcal{J}_{f, D_\mathcal E-F}$ be the relative Jacobian parametrizing invertible sheaves on $\widetilde{\mathcal C}_{\Gamma,\mathcal E}$ with multidegree $D_\mathcal E-F$. If $\mathcal L$ is the universal invertible sheaf over $\widetilde{\mathcal C}_{\Gamma,\mathcal E}\times_{\widetilde{\mathcal M}_\Gamma} \widetilde{\mathcal{J}}_{(\Gamma,\mathcal E,D)}$ then the pushforward of $\mathcal L$ to $\widetilde{\mathcal C}_\Gamma\times_{\widetilde{\mathcal M}_\Gamma}\widetilde{\mathcal{J}}_{(\Gamma,\mathcal E,D)}$ is a torsion free rank-$1$ sheaf with multidegree $(\mathcal E,D)$. Hence we have a map $\widetilde{\mathcal{J}}_{(\Gamma,\mathcal E,D)}\to \mathcal{J}_{(\Gamma,\mathcal E,D)}$ inducing a map \[ g\colon \widetilde{\mathcal{J}}_{(\Gamma,\mathcal E,D)}\to \mathcal{J}_{(\Gamma,\mathcal E,D)}\times_{\mathcal M_\Gamma}\widetilde{\mathcal M}_{\Gamma}. \] which is an isomorphism onto its image. We have a diagram \[ \SelectTips{cm}{11} \begin{xy} <16pt,0pt>: \xymatrix{ \widetilde{\mathcal{J}}_{(\Gamma,\mathcal E,D)}\ar[dr]_{h}\ar[r]^{g\;\quad\quad} &\mathcal{J}_{(\Gamma,\mathcal E,D)}\times_{\mathcal M_{\Gamma}} \widetilde{\mathcal M}_{\Gamma} \ar[d]\ar[r]& \widetilde{\mathcal M}_\Gamma\ar[d]\\
& \mathcal{J}_{(\Gamma,\mathcal E,D)}\ar[r] & \mathcal M_\Gamma } \end{xy} \] where $h$ is the composition of the projection and $g$. Recall that $\mathcal M_{\Gamma}=[\widetilde{\mathcal M}_\Gamma/\textnormal{Aut}(\Gamma)]$ (see \cite[Proposition 3.4.1]{ACP}); now we prove an analogous result for $\mathcal{J}_{\Gamma,\mathcal E,D}$.
\begin{Prop}\label{prop:stratades} We have an isomorphism \[ \mathcal{J}_{(\Gamma,\mathcal E,D)}\simeq\left[\frac{\widetilde{\mathcal{J}}_{(\Gamma,\mathcal E,D)}}{\textnormal{Aut}(\Gamma,\mathcal E,D)}\right] \]
and $h$ is the quotient map. \end{Prop}
\begin{proof} First, note that \[ \mathcal{J}_{(\Gamma,\mathcal E,D)}=\left[\frac{\mathcal{J}_{(\Gamma,\mathcal E,D)}\times_{\mathcal M_{\Gamma}}\widetilde{\mathcal M}_{\Gamma}}{\textnormal{Aut}(\Gamma)}\right], \] and we have a morphism \begin{equation} \label{eq:Jaut} \left[\frac{\widetilde{\mathcal{J}}_{(\Gamma,\mathcal E,D)}}{\textnormal{Aut}(\Gamma,\mathcal E,D)}\right]\to\left[\frac{\mathcal{J}_{(\Gamma,\mathcal E,D)}\times_{\mathcal M_{\Gamma}}\widetilde{\mathcal M}_{\Gamma}}{\textnormal{Aut}(\Gamma)}\right] \end{equation} induced by the equivariant morphism $g$. Let us construct the inverse of this morphism.\par
Let $\sigma$ be an automorphism of $\Gamma$. Define \[ g_\sigma\colon \widetilde{\mathcal{J}}_{(\Gamma,\sigma(\mathcal E),\sigma(D))}\to \mathcal{J}_{(\Gamma,\sigma(\mathcal E),\sigma(D))}\times_{\mathcal M_\Gamma}\widetilde{\mathcal M}_\Gamma \]
as we did for $g$. Note that $\mathcal{J}_{(\Gamma,\sigma(\mathcal E),\sigma(D))}=\mathcal{J}_{(\Gamma,\mathcal E,D)}$ in $\overline{\mathcal{J}}_{\mu, g}$ for every $\sigma\in\textnormal{Aut}(\Gamma)$.\par
Since $\mathcal M_\Gamma=[\widetilde{\mathcal M}_\Gamma/\textnormal{Aut}(\Gamma)]$, a point of $\mathcal{J}_{(\Gamma,\mathcal E,D)}\times_{\mathcal M_{\Gamma}}\widetilde{\mathcal M}_{\Gamma}$ parametrizes a triple $(X,I,\tau)$, where $[X]\in \mathcal M_\Gamma$ and $I$ is a torsion-free rank-1 sheaf on $X$, and where $\tau\colon\Gamma_X\to \Gamma$ is an isomorphism such that $\tau_*(\mathcal E_I,D_I)=(\sigma(\mathcal E),\sigma(D))$ for some $\sigma\in\textnormal{Aut}(\Gamma)$. Moreover, the triple $(X,I,\tau)$ is parametrized by a point in $g_\sigma(\widetilde{\mathcal{J}}_{\Gamma,\sigma(\mathcal E),\sigma(D)})$ if and only if $\tau_*(\mathcal E_I,D_I)=(\sigma(\mathcal E),\sigma(D))$. So if \[ g_\sigma(\widetilde{\mathcal{J}}_{(\Gamma,\sigma(\mathcal E),\sigma(D))})\cap g_{\sigma'}(\widetilde{\mathcal{J}}_{(\Gamma,\sigma'(\mathcal E),\sigma'(D))})\neq\emptyset \] then $(\sigma(\mathcal E),\sigma(D))=(\sigma'(\mathcal E),\sigma'(D))$, and hence $\sigma\textnormal{Aut}(\Gamma,\mathcal E,D)=\sigma'\textnormal{Aut}(\Gamma,\mathcal E,D)$. Conversely, if $\sigma\textnormal{Aut}(\Gamma,\mathcal E,D)=\sigma'\textnormal{Aut}(\Gamma,\mathcal E,D)$, then $(\sigma(\mathcal E),\sigma(D))=(\sigma'(\mathcal E),\sigma'(D))$, which implies $g_\sigma(\widetilde{\mathcal{J}}_{(\Gamma,\sigma(\mathcal E),\sigma(D))})=g_{\sigma'}(\widetilde{\mathcal{J}}_{(\Gamma,\sigma'(\mathcal E),\sigma'(D))})$.\par
Then, we have \[ \mathcal{J}_{(\Gamma,\mathcal E,D)}\times_{\mathcal M_{\Gamma}}\widetilde{\mathcal M}_{\Gamma}=\coprod_{i=1}^{N}g_{\sigma_i}(\widetilde{\mathcal{J}}_{(\Gamma,\sigma_i(\mathcal E),\sigma_i(D))}), \] where $N:=[\textnormal{Aut}(G)\colon\textnormal{Aut}(\Gamma,\mathcal E,D)]$ and $\sigma_i$ are chosen as representatives of the left cosets of $\textnormal{Aut}(\Gamma,\mathcal E,D)$ in $\textnormal{Aut}(\Gamma)$. Note that $\textnormal{Aut}(\Gamma)$ identifies the connected components of $\mathcal{J}_{(\Gamma,\mathcal E,D)}\times_{\mathcal M_{\Gamma}}\widetilde{\mathcal M}_{\Gamma}$, because $\sigma(g_{\sigma'}(\widetilde{\mathcal{J}}_{(\Gamma,\sigma'(\mathcal E),\sigma'(D))})=g_{\sigma\sigma'}(\widetilde{\mathcal{J}}_{(\Gamma,\sigma\sigma'(\mathcal E),\sigma\sigma'(D))})$. The chosen elements $\sigma_i$ can be used to define morphisms \[
\rho_i:=\sigma_i^{-1}|_{g_{\sigma_i}(\widetilde{\mathcal{J}}_{(\Gamma,\sigma_i(\mathcal E),\sigma_i(D))})}\colon g_{\sigma_i}(\widetilde{\mathcal{J}}_{(\Gamma,\sigma_i(\mathcal E),\sigma_i(D))})\to g(\widetilde{\mathcal{J}}_{(\Gamma,\mathcal E,D)})\cong \widetilde{\mathcal{J}}_{(\Gamma,\mathcal E,D)} \]
giving rise to an $\textnormal{Aut}(\Gamma)$-invariant morphism \[ \mathcal{J}_{(\Gamma,\mathcal E,D)}\times_{\mathcal M_{\Gamma}}\widetilde{\mathcal M}_{\Gamma}\stackrel{(\rho_i)_{1\le i\le N}}{\longrightarrow}\widetilde{\mathcal{J}}_{(\Gamma,\mathcal E,D)}\to\left[\frac{\widetilde{\mathcal{J}}_{(\Gamma,\mathcal E,D)}}{\textnormal{Aut}(\Gamma,\mathcal E,D)}\right], \] which in turn induces the inverse of the morphism in Equation \eqref{eq:Jaut}. \end{proof}
We need some results on the local geometry of $\overline{\mathcal{J}}_{\mu,g}$, where $\mu$ the canonical polarization. We will deduce them from known results about Caporaso-Pandharipande's compactification $\overline{\mathcal{J}}_{d,g,1}$ of $\mathcal{J}_{d,g,1}$. Let us introduce the moduli stack $\overline{\mathcal{J}}_{d,g,1}$.
Let $\mathcal{J}_{d,g}\rightarrow \mathcal M_{g}$ be the universal degree-$d$ Jacobian, parametrizing invertible sheaves on smooth curves. In \cite{C}, \cite{Pand} and \cite{C08}, Caporaso and Pandharipande introduced a moduli stack $\overline{\mathcal{J}}_{d,g}$ compactifying $\mathcal{J}_{d,g}$ over $\overline{\mathcal M}_g$. This stack can be viewed either as a moduli space of certain invertible sheaves, called \emph{balanced}, on semistable curves (this is done in \cite{C} and \cite{C08}), or as a moduli space of certain torsion-free rank-1 sheaves, called \emph{semistable}, on stable curves (this is done in \cite{Pand}). The two approaches give rise to isomorphic stacks (see \cite[Theorem 6.3]{EP}). For our purposes, it is better to follow the latter approach.
The setting can be extended to pointed curves. We let $\overline{\mathcal{J}}_{d,g,1}$ be the contravariant functor from the category of schemes to that of sets, taking a $k$-scheme $S$ to \[ \overline{\mathcal{J}}_{d,g,1}(S)=\frac{\left\{(\pi,\sigma,\mathcal I);\begin{array}{l} \pi\colon \mathcal X\to S\text{ is a family of stable pointed genus-g curves,}\\ \mathcal I \text{ is a $\mu$-semistable torsion free rank-1 sheaf on $\mathcal X$}\end{array}\right\}}{\sim} \] where $\sim$ is the same equivalence relation defined at the end of Section \ref{sec:EstJac}. Essentially the same proof of \cite[Theorem 6.3]{EP} shows that $\overline{\mathcal{J}}_{d,g,1}$ is isomorphic to the stack $\overline{\mathcal{P}}_{d,g,1}$ defined in \cite[Definition 4.1]{M11} by means of balanced invertible sheaves. Hence it follows from \cite[Theorem 4.2]{M11} that $\overline{\mathcal{J}}_{d,g,1}$ is a smooth and irreducible algebraic Artin stack of dimension $4g -2$ and it is universally closed over $\overline{\mathcal M}_{g,1}$.
\begin{Lem}\label{lem:openimm} There exists an open immersion of stacks $\overline{\mathcal{J}}_{\mu,g}\to \overline{\mathcal{J}}_{d,g,1}$. In particular for every point $[(X,I)]\in \overline{\mathcal{J}}_{\mu,g}$ we have \[ \widehat{\mathcal O}_{\overline{\mathcal{J}}_{\mu,g},[(X,I)]}\simeq \widehat{\mathcal O}_{ \overline{\mathcal{J}}_{d,g,1},[(X,I)]}. \] \end{Lem}
\begin{proof} Every $(p_0,\mu)$-quasistable torsion-free rank 1 sheaf on a stable pointed curves is $\mu$-semistable, hence there exists an injective map $\overline{\mathcal{J}}_{\mu,g}\to \overline{\mathcal{J}}_{d,g,1}$. It is an open immersion because quasistability is an open condition (see \cite[Proposition 34]{Es01}). \end{proof}
\begin{Rem} In general, the open immersion of Lemma \ref{lem:openimm} is not an isomorphism: there are $\mu$-semistable torsion free rank-$1$ sheaves that are not $(p_0,\mu)$-quasistable. \end{Rem}
Let $[(X,I)]$ be a point in $\overline{\mathcal{J}}_{\mu,g}$ and $(\Gamma,\mathcal E,D)$ be the dual graph of $(X,I)$. It follows from Lemma \ref{lem:openimm} and \cite[Equation 8.4]{CKV} that, up to choosing an orientation of $\Gamma$,
\begin{equation}
\label{eq:Jloc}
\widehat{\mathcal{O}}_{\overline{\mathcal{J}}_{\mu,g}[(C,I)]}=A\otimes k[[W_1,\ldots, W_{3g-2-|E(\Gamma)|},Z_1,\ldots, Z_{g-|\mathcal E|}]],
\end{equation} where \[ A:=\bigotimes_{e\in \mathcal E}\frac{k[[X_e,Y_e,T_e]]}{\langle X_eY_e-T_e\rangle}\otimes\left(\bigotimes_{e\in E(\Gamma)\setminus \mathcal E} k[[T_e]]\right). \]
The variables appearing above have the following modular interpretations. The variables $W$'s correspond to variations of the complex structure of the irreducible components of $X$, the variables $Z$'s correspond to deformations of $I$ where it is invertible, the variable $T_e$ corresponds to the smoothing of $X$ at the node associated to $e$, the variables $X_e$ and $Y_e$ correspond to the deformation of $I$ at the node associated to $e$. Moreover, $X_e$ and $Y_e$ correspond to the source and target of $e$, respectively; the construction is independent of the choice of the orientation.
We note that, although the results in \cite{CKV} hold for $\overline{\mathcal{J}}_{d,g}$, their arguments are based on deformation theory and can be naturally extended to the pointed case needed to analyze $\overline{\mathcal{J}}_{d,g,1}$.\par
The local ring at $[X]$ of $\overline{\mathcal M}_{g,1}$ is
\begin{equation}
\label{eq:Mloc}
\widehat{\mathcal O}_{\overline{\mathcal M}_{g,1},[X]}=\bigotimes_{e\in E(\Gamma)} k[[T_e]]\otimes k[[W_1,\ldots, W_{3g-2-|E(\Gamma)|}]],
\end{equation}
and the forgetful map $\pi\colon\overline{\mathcal{J}}_{\mu,g}\to \overline{\mathcal M}_{g,1}$ is induced locally at $[(X,I)]$ by the ring homomorphism $\pi^\#\colon \widehat{\mathcal O}_{\overline{\mathcal M}_{g,1},[X]}\rightarrow \widehat{\mathcal{O}}_{\overline{\mathcal{J}}_{\mu,g},[(X,I)]}$ given by $T_e\mapsto T_e$ and $W_i\mapsto W_i$ (see \cite[Equation 7.17]{CKV}).
Recall the notion of toroidal embedding of Deligne-Mumford stacks given in \cite[Definition 6.1.1]{ACP}.
\begin{Prop}\label{prop:stratification} The open immersion $\mathcal{J}_{d,g,1}\subset \overline{\mathcal{J}}_{\mu,g}$ is a toroidal embedding of Deligne-Mumford stacks and the forgetful map $\overline{\mathcal{J}}_{\mu,g}\to \overline{\mathcal M}_{g,1}$ is a toroidal morphism. Moreover, we have a partition
\begin{equation}\label{eq:strata}
\overline{\mathcal{J}}_{\mu,g}=\coprod_{(\Gamma,\mathcal E,D)\in \mathcal{QD}_{\mu,g}}\mathcal{J}_{(\Gamma,\mathcal E,D)},
\end{equation} where each $\mathcal{J}_{(\Gamma,\mathcal E,D)}$ is irreducible, and the following properties are equivalent for every $(\Gamma,\mathcal E,D),(\Gamma',\mathcal E',D')\in \mathcal{QD}_{\mu,g}$: \begin{enumerate}
\item
\label{thm-strata1} $\mathcal{J}_{(\Gamma,\mathcal E,D)}\cap \overline{\mathcal{J}}_{(\Gamma',\mathcal E',D')}\neq \emptyset$; \item
\label{thm-strata2}
$(\Gamma,\mathcal E,D)\geq (\Gamma',\mathcal E',D')$; \item
\label{thm-strata3} $\mathcal{J}_{(\Gamma,\mathcal E,D)}\subset \overline{\mathcal{J}}_{(\Gamma',\mathcal E',D')}$. \end{enumerate}
\end{Prop}
\begin{proof} The fact that $\mathcal{J}_{d,g,1}\subset \overline{\mathcal{J}}_{\mu,g}$ and the map $\overline{\mathcal{J}}_{\mu,g}\to \mathcal M_{g,1}$ are toroidal follows from Equations \eqref{eq:Jloc} and \eqref{eq:Mloc}, and from the local description of the map $\pi$.\par
The partition \eqref{eq:strata} follows from the fact that each pair $(X,I)$ has a unique dual graph, up to isomorphism. The locus $\mathcal{J}_{(\Gamma,\mathcal E,D)}$ is irreducible because the map $h\colon \widetilde{\mathcal{J}}_{(\Gamma,\mathcal E,D)}\rightarrow \mathcal{J}_{(\Gamma,\mathcal E,D)}$ is surjective by Proposition \ref{prop:stratades}, and $\widetilde{\mathcal{J}}_{(\Gamma,\mathcal E,D)}$ is irreducible.\par
To prove the equivalences we will use \cite[Proposition 3.4.1]{CC} and \cite[Proposition 3.4.2]{CC}. These propositions concern invertible sheaves. However, we have a canonical construction for passing from the setting of torsion-free rank-$1$ sheaves to the one of invertible sheaves. More precisely, if $\mathcal I$ is a torsion free rank-$1$ sheaf on a family of nodal curves $\mathcal X\to B$, we can construct a family of nodal curves $\widetilde{\mathcal X}\to B$ as $\widetilde{\mathcal X}:=\mathbb{P}(\mathcal I^\vee)=\text{Proj}(\text{Sym}(\mathcal I^\vee))$ which will add a $\mathbb{P}^1$ over each node where $\mathcal I$ is not locally free. Moreover, we have that $\mathcal I$ is the pushforward of $\mathcal O_{\widetilde{\mathcal X}}(-1)$ via the structural morphism $\widetilde{\mathcal X}\to \mathcal X$ (see \cite[Proposition 5.5]{EP}). \par
Let us prove that \eqref{thm-strata1} implies \eqref{thm-strata2}. If $[(X,I)]$ is a point in $\mathcal{J}_{(\Gamma,\mathcal E,D)}\cap \overline{\mathcal{J}}_{(\Gamma',\mathcal E',D')}$, then there exists a family of stable pointed curves $\mathcal X\to B$ with generic fiber in $\mathcal{J}_{(\Gamma',\mathcal E',D')}$ and special fiber $X$, and a $(\sigma,\mu)$-quasistable torsion-free rank-$1$ sheaf $\mathcal I$ on $\mathcal X$ that restricts to $I$ on $X$. If we take $\widetilde{\mathcal X}=\mathbb{P}(\mathcal I^\vee)$, then the dual graph of the generic fiber of $\widetilde{\mathcal X}$ will be $\Gamma'^{\mathcal E'}$, while the dual graph of the special fiber will be $\Gamma^\mathcal E$. As in \cite[Proposition 3.4.1]{CC}, we get a specialization $\iota\colon\Gamma^\mathcal E\to\Gamma'^{\mathcal E'}$ with $\iota_*(D)=D'$, hence $(\Gamma,\mathcal E,D)\geq (\Gamma',\mathcal E',D')$.\par
Let us prove that \eqref{thm-strata2} implies \eqref{thm-strata3}. Assume that there is a specialization $\iota\colon\Gamma^\mathcal E\to\Gamma'^{\mathcal E'}$ such that $\iota_*(D)=D'$; in particular $\mathcal E'$ is a subset of $\mathcal E$. Fix a point $[(X,I)]$ in $\mathcal{J}_{(\Gamma,\mathcal E,D)}$ and take $\widetilde{X}:=\mathbb{P}(I^\vee)$. Note that the dual graph of $(\widetilde{X},\mathcal O_{\widetilde{X}}(-1))$ is $(\Gamma^\mathcal E,\emptyset,D)$. Applying \cite[Proposition 3.4.2]{CC}, we find a family of nodal curves $\widetilde{\mathcal X}\to B$ and a line bundle $\mathcal L$ on $\widetilde{\mathcal X}$, such that the generic fiber of $(\widetilde{\mathcal X},\mathcal L)$ has dual graph $(\Gamma'^{\mathcal E'},\emptyset, D')$ and has special fiber $(\widetilde{X},\mathcal O_{\widetilde X}(-1))$. Let $f\colon\widetilde{\mathcal X}\to \mathcal X$ be the contraction of all rational curves in fibers corresponding to edges in $\mathcal E$ and set $\mathcal I:=f_*\mathcal L$. So $\mathcal X\to B$ is a family of stable pointed curves and $\mathcal I$ is a torsion-free rank-$1$ sheaf such
the generic fiber of $(\mathcal X,\mathcal I)$ has dual graph $(\Gamma',\mathcal E',D')$ and special fiber $(X,I)$ (see \cite[Propositions 5.4 and 5.5]{EP}). This shows that $\mathcal{J}_{(\Gamma,\mathcal E,D)}\subset\overline{\mathcal{J}}_{(\Gamma',\mathcal E',D')}$.\par
Finally, it is clear that \eqref{thm-strata3} implies \eqref{thm-strata1}. \end{proof}
\begin{Rem} Note that Proposition \ref{prop:stratification} tells us that the decomposition \eqref{eq:strata} is a stratification of $\overline{\mathcal{J}}_{\mu,g}$ by $\mathcal{QD}_{\mu,g}$, in the sense of \cite[Definition 1.3.2]{CC}. \end{Rem}
\subsection{The tropicalization of $\overline{\mathcal{J}}_{\mu,g}$} The goal of this section is to show that the skeleton of the Esteves' universal Jacobian $\overline{\mathcal{J}}_{\mu,g}$ is precisely $\overline{J}^{\text{trop}}_{\mu,g}$.
Let $\mathcal Y$ be a separated connected Deligne-Mumford stack over an algebraically closed field $k$. There is a Berkovich analytification $\mathcal Y^{an}$ of $\mathcal Y$ which is an analytic stack. We will work with the topological space underlying $\mathcal Y^{an}$, whose points are morphisms $\text{Spec}\,(K)\to \mathcal Y$, where $K$ is a non Archimedean valued field extension of the trivially valued field $k$, up to equivalence by further valued field extensions (see \cite{Be} and \cite[Section 3]{U17}). We abuse notation and use $\mathcal Y^{an}$ for both the stack and the topological space underlying it. Note that one can choose a representative $\text{Spec}\,(K)\to \mathcal Y$ with $K$ complete for each point in $\mathcal Y^{an}$. There is a distinguished subspace $\mathcal Y^{\beth}\subset \mathcal Y^{an}$ consisting of points $\text{Spec}\,(K)\to \mathcal Y$ that extends to $\text{Spec}\,(R)\to \mathcal Y$, where $R$ is the valuation ring of $K$. If $\mathcal Y$ is proper then $\mathcal Y^{\beth}=\mathcal Y^{an}$. \par
We recall the notion of monodromy associated to a toroidal embedding. Let $U\subset \mathcal Y$ be a toroidal embedding of Deligne-Mumford stacks. Define the sheaves $\textnormal{Mon}_{\mathcal Y}$ and $\textnormal{Eff}_{\mathcal Y}$ as the \'etale sheaves over $\mathcal Y$ such that, for every \'etale morphism $V\to \mathcal Y$ from a scheme $V$, we have that $\textnormal{Mon}_\mathcal Y(V)$ (respectively, $\textnormal{Eff}_\mathcal Y(V)$) is the group of Cartier divisors on $V$ (respectively, the submonoid of effective Cartier divisors on $V$) supported on $V\setminus U_V$, where $U_V=U\times_\mathcal Y V$. \par
For each stratum $W\subset \mathcal Y$ and a point $w\in W$, we have an action of the \'etale fundamental group $\pi_1^{et}(W,w)$ on the stalk $\textnormal{Mon}_{\mathcal Y,w}$ preserving $\textnormal{Eff}_{\mathcal Y,w}$. The \emph{monodromy group} $H_W$ is defined as the image of $\pi_1^{et}(W,w)$ in $\textnormal{Aut}(\textnormal{Mon}_{\mathcal Y,w})$.\par
For each stratum $W\subset \mathcal Y$, there is an associated extended cone \[ \overline{\sigma}_W:=\Hom_{\text{monoids}}(\textnormal{Eff}_{\mathcal Y,w},\overline{\mathbb R}_{\geq0}). \] One defines an extended generalized cone complex, called the \emph{skeleton} of $\mathcal Y$, as \[ \overline{\Sigma}(\mathcal Y):=\lim_{\longrightarrow}\overline{\sigma}_W, \] where the arrows $\overline{\sigma}_W\to\overline{\sigma}_{W'}$ are given by the inclusions $W'\subset \overline{W}$, where $\overline W$ is the closure of $W$ in $\mathcal Y$, and are also given, when $W=W'$, by the monodromy group $H_W$. For more details, see \cite{T} and \cite[Section 6]{ACP}.
There is a retraction map ${\bf p}_{\mathcal Y}\colon \mathcal Y^{\beth}\rightarrow \overline{\Sigma}(\mathcal Y)$ defined as follows. Let $\psi\colon \text{Spec}\,(R)\to \mathcal Y$ be a point in $\mathcal Y^{\beth}$, with $R$ complete. Let $w\in \mathcal Y$ be the image of the closed point in $\text{Spec}\,(R)$ and $W$ be the stratum of $\mathcal Y$ containing $w$. We have a chain of maps \[ \textnormal{Eff}_{\mathcal Y,w}\stackrel{\epsilon}{\longrightarrow} \widehat{\mathcal O}_{\mathcal Y,w}\stackrel{\psi^\#}{\longrightarrow} R\stackrel{\nu_R}{\longrightarrow} \overline{\mathbb R}_{>0}, \] where $\epsilon$ is the map that takes an effective divisor to its local equation and $\nu_R$ is the valuation of $R$. Note that the composition is a morphism of monoids: one defines ${\bf p}_{\mathcal Y}(\psi)\in \overline{\Sigma}(\mathcal Y)$ as the equivalence class of $\nu_R\circ\psi^\#\circ\epsilon\in\overline{\sigma}_{W}$. We refer to \cite[Section 6]{ACP} for details, in particular to \cite[Propositions 6.1.4. and 6.2.6]{ACP}.
The inclusions $\mathcal M_{g,n}\subset \overline{\mathcal M}_{g,n}$ and $\mathcal \mathcal{J}_{d,g,1}\subset \overline{\mathcal J}_{\mu,g}$ are embeddings of Deligne-Mumford stacks (see \cite[Section 3.3]{ACP} and Proposition \ref{prop:stratification}). We let $\overline{\Sigma}(\overline{\mathcal {M}}_{g,n})$ and $\overline{\Sigma}(\overline{\mathcal {J}}_{\mu,g})$ be the skeleta of $\overline{\mathcal M}_{g,n}$ and $\overline{\mathcal J}_{\mu,g}$, respectively.
The forgetful map $\pi\colon \overline{\mathcal J}_{\mu,g}\rightarrow \overline{\mathcal M}_{g,n}$ induces a natural map \[ \pi^{an}\colon\overline{\mathcal J}_{\mu,g}^{an} \rightarrow \overline{\mathcal M}_{g,n}^{an}. \]
\begin{Prop}\label{prop:funct} The map $\pi^{an}\colon\overline{\mathcal J}_{\mu,g}^{an} \rightarrow \overline{\mathcal M}_{g,n}^{an}$ restricts to a map of generalized extended cone complexes $\overline{\Sigma}(\pi)\colon \overline{\Sigma}(\overline{\mathcal{J}}_{\mu,g}) \rightarrow \overline{\Sigma}(\overline{\mathcal M}_{g,n})$. We have \[ {\bf p} _{\overline{\mathcal M}_{g,1}}\circ\pi^{an}=\overline{\Sigma}(\pi)\circ{\bf p} _{\overline{\mathcal{J}}_{\mu,g}}. \] \end{Prop}
\begin{proof} Since $\overline{\mathcal{J}}_{\mu,g}$ and $\overline{\mathcal M}_{g,n}$ is proper, then $\overline{\mathcal{J}}_{\mu,g}^{an}=\overline{\mathcal{J}}_{\mu,g}^\beth$ and $\overline{\mathcal M}_{g,n}^{an}=\overline{\mathcal M}_{g,b}^{\beth}$. The result follows just combining \cite[Proposition 6.1.8]{ACP} and Proposition \ref{prop:stratification}. \end{proof}
\begin{Def} \label{def:tropmap} The \emph{tropicalization map} \[ \text{trop}_{\overline{\mathcal{J}}_{\mu,g}}\colon \overline{\mathcal{J}}_{\mu,g}^{an}\to \overline{J}^{\text{trop}}_{\mu,g} \] is defined as follows. Fix a point $\psi\colon \text{Spec}\,(R)\to \overline{\mathcal{J}}_{\mu,g}$ with $R$ complete and let $[(X,I)]$ be the image of the closed point. We get a ring homomorphism \[ \psi^\#\colon \widehat{\mathcal O}_{\overline{\mathcal{J}}_{\mu,g},[(X,I)]}\to R. \] Let $(\Gamma,\mathcal E,D)$ be the dual graph of $[(X,I)]$. Recall the notation in Equation \eqref{eq:Jloc}. We define a length function $\ell\colon E(\Gamma^\mathcal E)\to \overline{\mathbb R}_{>0}$ given by $\ell(e)=\nu_R(\psi^\#(T_e))$ for $e\in E(\Gamma)\setminus \mathcal E$, while $\ell(e')=\nu_R(\psi^\#(X_e))$ and $\ell(e'')=\nu_R(\psi^\#(Y_e))$ for $e\in \mathcal E$, where $e'$ and $e''$ are the edges of $\Gamma^\mathcal E$ over $e\in E(\Gamma)$ corresponding to the source and target of $e$, respectively. Then, we define $\text{trop}(\psi)=[(X,\mathcal D)]$, where $X$ is the tropical curve $(\Gamma^\mathcal E,\ell)$ and $\mathcal D$ is the divisor on $X$ induced by $D$. \end{Def}
The previous definition is similar to the tropicalization map $\text{trop}_{\overline{\mathcal M}_{g,1}}\colon\overline{\mathcal M}_{g,1}^{an}\to\overline{\mathcal M}_{g,1}^\text{trop}$ in \cite[Lemma-Definition 2.4.1]{Viv13} and \cite[Section 1.1]{ACP}. As for $\text{trop}_{\overline{M}_{g,1}}$, we have that $\text{trop}_{\overline{\mathcal{J}}_{\mu,g}}$ does not depends on the choices of representative $\psi$ and local coordinates. This will be also consequence of Theorem \ref{thm:tropj}.
In the next proposition we compute the monodromy group $H_{\mathcal{J}_{(\Gamma,\mathcal E,D)}}$ of the stratum $\mathcal{J}_{(\Gamma,\mathcal E,D)}$ of the toroidal embedding $\mathcal{J}_{d,g,1}\subset \overline{\mathcal{J}}_{\mu,g}$.
\begin{Lem} \label{lem:mon} We have $H_{\mathcal{J}_{(\Gamma,\mathcal E,D)}}=\textnormal{Aut}(\Gamma,\mathcal E,D)$. \end{Lem} \begin{proof} Let $\Gamma_0$ be the graph with a single vertex of weight $g-1$ and a single loop. For $i=0,\dots,g-1$, let $\Gamma_i$ be the graph with a single edge connecting two vertices of weight $i$ and $g-i$, with the leg on the vertex with weight $i$. For $i=0,\dots,g-1$, let $D_i$ be the unique $(v_0,\mu)$-quasistable divisor on $\Gamma_i$ (the uniqueness is due to Proposition \ref{prop:tree}). By \cite[Theorem 3.2 and Corollary 3.3]{MV14} and the fact that $\overline{\mathcal{J}}_{\mu,g}\to \overline{\mathcal{J}}_{d,g}$ is an open immersion (see Lemma \ref{lem:openimm}), we have that the boundary of $\overline{\mathcal{J}}_{\mu,g}$ is given by the divisors $\Delta_{i,\mu}:=\overline{\mathcal{J}}_{\Gamma_i,\emptyset, D_i}$ for $i=0,\ldots, g-1$. (The difference with the description of the boundary of $\overline{\mathcal{J}}_{d,g}$ is that, using the notations of \cite[Corollary 3.3]{MV14}, the quasistability condition in $\overline{\mathcal{J}}_{\mu,g}$ selects just one between $\overline{\delta}_{i}^1$ and $\overline{\delta}_{i}^2$.) \par
We now follow essentially the same argument of \cite[Proposition 7.2.1]{ACP}. Let $w$ be a point in $\mathcal{J}_{(\Gamma,\mathcal E,D)}$. The set $E(\Gamma^\mathcal E)$ forms a group basis for $\textnormal{Mon}_{\overline{\mathcal{J}}_{\mu,g},w}$ and a monoid basis for $\textnormal{Eff}_{\overline{\mathcal{J}}_{\mu,g},w}$. Moreover, the locally constant sheaf of sets on $\mathcal{J}_{(\Gamma,\mathcal E,D)}$ whose stalk at every point is the set of edges $E(\Gamma^\mathcal E)$, is trivial when pulled back to $\widetilde{\mathcal{J}}_{(\Gamma,\mathcal E,D)}$. Hence the pull back of $\textnormal{Mon}_{\overline{\mathcal{J}}_{\mu,g}}$ and $\textnormal{Eff}_{\overline{\mathcal{J}}_{\mu,g}}$ to $\widetilde{\mathcal{J}}_{(\Gamma,\mathcal E,D)}$ are trivial. It follows from Proposition \ref{prop:stratades} that $\mathcal{J}_{(\Gamma,\mathcal E,D)}=[\widetilde{\mathcal{J}}_{(\Gamma,\mathcal E,D)}/\textnormal{Aut}(\Gamma,\mathcal E,D)]$, hence for $w\in \mathcal{J}_{(\Gamma,\mathcal E,D)}$, the action of $\pi_1^{et}(\mathcal{J}_{(\Gamma,\mathcal E,D)},w)$ on $\text{Mon}_{\mathcal{J}_{\mu,g},w}$ factors through its quotient $\textnormal{Aut}(\Gamma,\mathcal E,D)$. \end{proof}
We are ready to prove the result on the tropicalization of the Esteves' universal compactified Jacobian.
\begin{Thm} \label{thm:tropj} There is an isomorphism of extended generalized cone complexes \[ \Phi_{\overline{\mathcal{J}}_{\mu,g}}\colon \overline{\Sigma}(\overline{\mathcal{J}}_{\mu,g})\to \overline{J}^{\text{trop}}_{\mu,g}. \] Moreover, the following diagram is commutative \begin{eqnarray*} \SelectTips{cm}{11} \begin{xy} <16pt,0pt>: \xymatrix{ \overline{\mathcal{J}}_{\mu,g}^{an} \ar@/^2pc/[rr]^{\text{trop}_{\overline{\mathcal{J}}_{\mu,g}}} \ar[d]_{\pi^{an}} \ar[r]^{{\bf p}_{\overline{\mathcal{J}}_{\mu,g}}\;}
& \ar[r]^{{\Phi}_{\overline{\mathcal{J}}_{\mu,g}}} \ar[d]_{\overline{\Sigma}(\pi)} \overline{\Sigma}(\overline{\mathcal{J}}_{g,n}) & \overline{J}_{\mu,g}^{trop} \ar[d]_{\pi^{trop}} \\ \overline{\mathcal M}_{g,1}^{an} \ar@/_2pc/[rr]^{\text{trop}_{\overline{\mathcal M}_{g,1}}} \ar[r]^{{\bf p}_{\overline{ {\mathcal M}}_{g,1}}\;} & \ar[r]^{{\Phi}_{\overline{\mathcal M}_{g,1}}} \overline{\Sigma}(\overline{{\mathcal M}}_{g,1}) & \overline{M}_{g,1}^{trop}
} \end{xy} \end{eqnarray*} \end{Thm} \begin{proof}
Let $w$ be a point in a stratum $W=\mathcal{J}_{(\Gamma,\mathcal E,D)}$ of $\overline{\mathcal{J}}_{\mu,g}$. Then there is an isomorphism of monoids $\textnormal{Eff}_{\overline{\mathcal{J}}_{\mu,g},w}\to\mathbb{Z}_{\geq0}^{E(\Gamma^\mathcal E)}$ via Equation \eqref{eq:Jloc}, hence $\overline{\sigma}_W$ is isomorphic to $\overline{\sigma}_{(\Gamma,\mathcal E,D)}$. By \cite[Proposition 6.2.6]{ACP} and Lemma \ref{lem:mon}, we have \[ \overline{\Sigma}(\overline{\mathcal{J}}_{\mu,g})=\coprod_{(\Gamma,\mathcal E,D)} \overline{\sigma}^\circ_W/H_W\cong\coprod_{(\Gamma,\mathcal E,D)}\overline{\sigma}^\circ_{(\Gamma,\mathcal E,D)}/\textnormal{Aut}(\Gamma,\mathcal E,D). \] Moreover, by Proposition \ref{prop:stratification}, we have that $(\Gamma,\mathcal E,D)\geq (\Gamma',\mathcal E',D')$ in $\mathcal{QD}_{\mu,g}$ if and only if $\mathcal{J}_{\Gamma,\mathcal E,D}\subset \overline{\mathcal{J}}_{\Gamma',\mathcal E',D'}$ hence the extended generalized cone complex $\overline{\Sigma}(\mathcal{J}_{\mu,g})$ is isomorphic to $\overline{J}^\text{trop}_{\mu,g}$.\par
Let $\psi\colon \text{Spec}\,(R)\to \overline{\mathcal{J}}_{\mu,g}$ be a point in $\overline{\mathcal{J}}_{\mu,g}^{an}$, with $R$ complete, such that the image $w$ of the closed point in $\text{Spec}\,(R)$ lies in the stratum $\mathcal{J}_{(\Gamma,\mathcal E,D)}$ of $\overline{\mathcal{J}}_{\mu,g}$. The set $E(\Gamma^\mathcal E)$ can be seen as a monoid basis of the free monoid $\textnormal{Eff}_{\overline{\mathcal{J}}_{\mu,g},w}$. If $\text{trop}(\psi)=[(X,\mathcal D)]\in\overline{J}_{\mu,g}$ and $\ell\colon E(\Gamma^\mathcal E)\to \overline{\mathbb R}_{>0}$ is the length function of $X$, then $\ell$ factors through the composition \[ \ell\colon E(\Gamma^\mathcal E)\to \textnormal{Eff}_{\overline{\mathcal{J}}_{\mu,g},w}\stackrel{{\bf p}_{\overline{\mathcal{J}}_{\mu,g}}(\psi)}{\longrightarrow} \overline{\mathbb R}_{\geq0}. \] It follows that $\text{trop}_{\overline{\mathcal{J}}_{\mu,g}}=\Phi_{\overline{\mathcal{J}}_{\mu,g}}\circ{\bf p}_{\overline{\mathcal{J}}_{\mu,g}}$.
The fact that the square in the left hand side of the diagram in the statement is commutative follows from Proposition \ref{prop:funct}.
Finally, thanks to Equations \eqref{eq:Jloc} and \eqref{eq:Mloc}, we have that $\text{trop}_{\overline{\mathcal M}_{g,1}}\circ\pi^{an}(\psi)$ is the tropical curve $(\Gamma,\ell')$ where $\ell'(e)=\nu_R(\psi^\#T_{e})$. On the other hand, $\pi^\text{trop}\circ \text{trop}_{\overline{\mathcal{J}}_{\mu,g}}(\psi)$ is the tropical curve $(\Gamma^\mathcal E,\ell)$ where $\ell$ is as in Definition \ref{def:tropmap}. Since $\nu_R(\psi^\#T_e)=\nu_R(\psi^\#X_e)+\nu_R(\psi^\#Y_e)$ by Equation \eqref{eq:Jloc}, we deduce that $\ell'(e)=\ell(e')+\ell(e'')$ if $e\in \mathcal E$ and $e',e''$ are the edges of $\Gamma^\mathcal E$ over $e$, otherwise $\ell'(e)=\ell(e)$ if $e\notin \mathcal E$. Hence the tropical curves $(\Gamma,\ell')$ and $(\Gamma^\mathcal E,\ell)$ are isomorphic, and so \[ \text{trop}_{\overline{\mathcal M}_{g,1}}\circ\pi^{an}=\pi^\text{trop}\circ \text{trop}_{\overline{\mathcal{J}}_{\mu,g}}. \] This concludes the proof. \end{proof}
\section*{Acknowledgments}
We thank Lucia Caporaso, Renzo Cavalieri, Eduardo Esteves, Margarida Melo, Dhruv Ranganathan, and Martin Ulirsch for useful conversations and comments. The second author was supported by CNPq, processo 301314/2016-0. We thank the referees for carefully reading the paper and for the constructive suggestions.
\noindent{\smallsc Alex Abreu and Marco Pacini, Universidade Federal Fluminense, \\ Rua Prof. M. W. de Freitas, S/N, Istituto de Matem\'atica. Niter\'oi, Rio de Janeiro, Brazil. 24210-201. }\\ {\smallsl E-mail addresses: \small\[email protected]? \;\; and \;\; \small\[email protected]?}
\end{document} | arXiv |
If $y$ $=$ $\log_{e}{(\cos{\sqrt{x}})}$, then find $\dfrac{dy}{dx}$
The function $y$ is actually defined in terms of $x$. So, differentiate the function with respect to $x$ to get the derivative of $y$ in differential calculus.
Find the differentiation of the function
Differentiate both sides with respect to $x$ to find $\dfrac{dy}{dx}$ in mathematics.
$\dfrac{dy}{dx}$ $\,=\,$ $\dfrac{d}{dx} \log_{e}{(\cos{\sqrt{x}})}$
There is a direct formula in differential calculus to differentiate the natural logarithmic function.
$\dfrac{d}{dx} \log_{e}{x}$ $\,=\,$ $\dfrac{1}{x}$
Actually, the function of $y$ is in different form though natural logarithmic function is involved in defining this function. So, the above differentiation rule cannot be applied to the function $y$ in this case because the function is formed by the composition of three different functions $\log_{e}{x}$, $\cos{x}$ and $\sqrt{x}$.
In this type of cases, the chain rule is used to find the derivative of the function, formed by the composition of two or more functions.
Try Chain Rule for differentiating the function
The chain rule can be written in mathematical form as follows.
$\dfrac{d}{dx} {f[{g(x)}]} \,=\, {f'[{g(x)}]}.{g'{(x)}}$
Take $f[{g(x)}]$ $\,=\,$ $\log_{e}{(\cos{\sqrt{x}})}$ and $g{(x)}$ $\,=\,$ $\cos{\sqrt{x}}$. Firstly, find $f'[{g(x)}]$ and $g'{(x)}$, and then substitute them in chain rule.
Find $f'[{g(x)}]$
Differentiate the function $f[{g(x)}]$ with respect to $x$ to get $f'[{g(x)}]$.
$f'[{g(x)}]$ $\,=\,$ $\dfrac{d}{dx} {f[{g(x)}]}$ $\,=\,$ $\dfrac{d}{dx} \log_{e}{(\cos{\sqrt{x}})}$
Remember, just differentiate natural logarithmic function $\log_{e}{(\cos{\sqrt{x}})}$ by considering it as $\log_{e}{x}$ and don't differentiate the function $\cos{\sqrt{x}}$ because the meaning of function $f'[{g(x)}]$ is the differentiation of function $f$ in terms of $g{(x)}$.
$\implies f'[{g(x)}]$ $\,=\,$ $\dfrac{1}{\cos{\sqrt{x}}}$
Find $g'{(x)}$
Similarly, differentiate the function $g{(x)}$ with respect to $x$ to get $g'{(x)}$.
$g'{(x)} \,=\, \dfrac{d}{dx} g{(x)}$ $\,=\,$ $\dfrac{d}{dx} \cos{(\sqrt{x})}$
The differentiation of $\cos{x}$ can be done by derivative of cos rule but there is no formula for derivative of $\cos{(\sqrt{x})}$.
Substitute them in Chain Rule
Now, substitute the differentiated functions in chain rule formula.
$\dfrac{d}{dx} \log_{e}{(\cos{\sqrt{x}})}$ $\,=\,$ $\dfrac{1}{\cos{\sqrt{x}}}$ $\times$ $\dfrac{d}{dx} \cos{(\sqrt{x})}$
The differentiation of $\cos{(\sqrt{x})}$ is required to perform to complete the differentiation of the function $y$ but it cannot be performed directly because $\cos{(\sqrt{x})}$ is a composition of the functions $\cos{x}$ and $\sqrt{x}$. So, use chain rule one more time in order to find the derivative of function $\cos{(\sqrt{x})}$ with respect to $x$.
Use Chain Rule one more time to complete differentiation of function
$\dfrac{d}{dx} {f[{g(x)}]}$ $\,=\,$ $\dfrac{d}{dx} \cos{(\sqrt{x})}$
Now, take $f[{g(x)}]$ $\,=\,$ $\cos{(\sqrt{x})}$ and $g{(x)} \,=\, \sqrt{x}$, then find $f'[{g(x)}]$ and $g'{(x)}$. After that replace differentiated functions $f'[{g(x)}]$ and $g'{(x)}$ in chain rule.
$f'[{g(x)}]$ $\,=\,$ $\dfrac{d}{dx} {f[{g(x)}]}$ $\,=\,$ $\dfrac{d}{dx} \cos{\sqrt{x}}$
Differentiate the function $\cos{(\sqrt{x})}$ with respect to $x$ by considering it as $\cos{x}$.
$\implies f'[{g(x)}]$ $\,=\,$ $-\sin{\sqrt{x}}$
$g'{(x)} \,=\, \dfrac{d}{dx} g{(x)}$ $\,=\,$ $\dfrac{d}{dx} \sqrt{x}$
Differentiate the square root of $x$ with respect to $x$ by using derivative of square root of $x$ formula.
$\implies g'{(x)} \,=\, \dfrac{1}{2\sqrt{x}}$
Substitute the functions in chain rule formula to get the differentiation of $\cos{(\sqrt{x})}$.
$\dfrac{d}{dx} \cos{(\sqrt{x})}$ $\,=\,$ $-\sin{\sqrt{x}}$ $\times$ $\dfrac{1}{2\sqrt{x}}$
$\implies \dfrac{d}{dx} \cos{(\sqrt{x})}$ $\,=\,$ $\dfrac{-\sin{\sqrt{x}}}{2\sqrt{x}}$
Finding the Derivative of the function y
According to first time chain rule, the differentiation of the function $\log_{e}{(\cos{\sqrt{x}})}$ is calculated as follows.
But the differentiation of $\cos{(\sqrt{x})}$ is also calculated as follows.
$\dfrac{d}{dx} \cos{(\sqrt{x})}$ $\,=\,$ $\dfrac{-\sin{\sqrt{x}}}{2\sqrt{x}}$
Therefore, replace the differentiation of $\cos{(\sqrt{x})}$ in the derivative of $\log_{e}{(\cos{\sqrt{x}})}$ with respect to $x$.
$\implies \dfrac{d}{dx} \log_{e}{(\cos{\sqrt{x}})}$ $\,=\,$ $\dfrac{1}{\cos{\sqrt{x}}}$ $\times$ $\dfrac{-\sin{\sqrt{x}}}{2\sqrt{x}}$
Simplifying the differentiated function
The differentiation of function $y$ is successfully completed and the differentiated function can be further simplified by the trigonometry.
$\implies \dfrac{d}{dx} \log_{e}{(\cos{\sqrt{x}})}$ $\,=\,$ $\dfrac{1}{2\sqrt{x}}$ $\times$ $\dfrac{-\sin{\sqrt{x}}}{\cos{\sqrt{x}}}$
$\implies \dfrac{d}{dx} \log_{e}{(\cos{\sqrt{x}})}$ $\,=\,$ $\dfrac{1}{2\sqrt{x}}$ $\times$ $-\tan{\sqrt{x}}$
$\,\,\, \therefore \,\,\,\,\,\, \dfrac{dy}{dx}$ $\,=\,$ $\dfrac{d}{dx} \log_{e}{(\cos{\sqrt{x}})}$ $\,=\,$ $\dfrac{-\tan{\sqrt{x}}}{2\sqrt{x}}$ | CommonCrawl |
\begin{document}
\title{Conjugacy classes and characters for extensions of finite groups} \author[Tang]{Xiang Tang} \address{Department of Mathematics\\ Washington University\\ St. Louis\\ MO 63130\\ USA} \email{[email protected]}
\author[Tseng]{Hsian-Hua Tseng} \address{Department of Mathematics\\ Ohio State University\\ 100 Math Tower, 231 West 18th Ave. \\ Columbus \\ OH 43210\\ USA} \email{[email protected]}
\date{\today}
\begin{abstract} Let $H$ be an extension of a finite group $Q$ by a finite group $G$. Inspired by the results of duality theorems for \'etale gerbes on orbifolds, we describe the number of conjugacy classes of $H$ that maps to the same conjugacy class of $Q$. Furthermore, we prove a generalization of the orthogonality relation between characters of $G$. \end{abstract}
\maketitle
\section{Introduction}
Extensions of finite groups play an important role in the theory of finite groups. For example, the composition serious of a finite group $H$ consists of a sequence of subgroups $H_i$ \[ 1=H_0\lhd H_1\lhd H_2\lhd \cdots\lhd H_n=H, \] such that $H_i$ is a strict normal subgroup of $H_{i+1}$ with a simple quotient group $H_{i+1}/H_i$, for $i=0, \cdots, n-1$. Therefore, with the classification theorem of finite simple groups, the study of extensions of finite groups would describe and classify all finite groups.
The structure of extensions of finite groups has been studied for a long time, see \cite{Schreier}. In this paper, we look at extensions of finite groups from a geometric point of view. A finite group $G$ is a groupoid with one unit. In the language of stacks \cite{be-xu}, such a group(oid) corresponds to the classifying stack $BG$ of principal $G$-bundles. An extension of a finite group $Q$ by a finite group $G$ \[ 1\longrightarrow G\longrightarrow H\longrightarrow Q\longrightarrow 1 \] is equivalent to a $G$-gerbe \[ BH\longrightarrow BQ, \] a bundle of $BG$ over $BQ$, c.f. \cite{la-st-xu}.
Our study of extensions of finite groups is motivated by a conjecture in Mathematical physics \cite{hel-hen-pan-sh}. Let $\widehat{G}$ be the finite set of isomorphism classes of irreducible unitary representations of $G$. The above extension $H$ of $Q$ by $G$ gives a natural action of $Q$ on $\widehat{G}$. Consider the transformation groupoid $\widehat{G}\rtimes Q\rightrightarrows \widehat{G}$. There is a canonical class $c$ in $H^{2}(\widehat{G}\rtimes Q, U(1))$ associated to the extension $H$. The decomposition conjecture in \cite{hel-hen-pan-sh} suggests that the geometry of a $G$-gerbe associated to the extension $H$ is equivalent to the geometry of the orbifold associated to the groupoid $\widehat{G}\rtimes Q$ twisted by $c$. We studied this conjecture in \cite{TT} from the view point of noncommutative geometry. In particular, we proved that the group algebra of $H$ is Morita equivalent the $c$-twisted groupoid algebra of $\widehat{G}\rtimes Q$. The detail of this is reviewed in Section \ref{sec:groupalgebra}.
In this short note, we present two results from our analysis of the structure of ${\mathbb C} H$. One result concerns the relations between conjugacy classes of $H$ and $Q$, see Section \ref{subsec:conjugacy}. The other result concerns a generalized orthogonality relation between characters of $G$, see Section \ref{subsec:orthogonality}.
To the best of our knowledge, the results in this paper are new. We would like to thank I. M. Isaacs for discussions related to Question \ref{counting_question}. Tang's research is partially supported by NSF grant 0900985, and NSA grant H96230-13-1-02. Tseng's research is partially supported by by NSF grant DMS-0757722 and Simons Foundation collaboration grant.
\section{Group algebras of finite group extensions}\label{sec:groupalgebra} Consider an extension of finite groups as in \begin{equation}\label{eq:extension_reproduced} 1\longrightarrow G\stackrel{i}{\longrightarrow} H \stackrel{j}{\longrightarrow} Q\longrightarrow 1. \end{equation} As part of our study of gerbe duality, the structure of the group algebra ${\mathbb C} H$ is analyzed in \cite{TT}. We briefly recall the results.
Choose a section $s:Q\rightarrow H$ of $j:H\to Q$ above such that $j\circ s=id$, and $s(1)=1$. Since $G$ and $Q$ are finite groups, such a section $s$ always exists. For $q_1,q_2\in Q$ define $\tau(q_1, q_2):=s(q_1)s(q_2)s(q_1q_2)^{-1}$. It is easy to see that $\tau(q_1,q_2)\in \ker(j)=G$, so we obtain $$\tau:Q\times Q\rightarrow G.$$ Clearly $\tau$ is trivial (i.e. $\tau(-,-)=1$) if and only if $s: Q\to H$ is a group homomorphism, which in turn is equivalent to the extension (\ref{eq:extension_reproduced}) being a split extension.
The definition of $\tau$ may be written as \begin{equation} \label{eq:tau-def} s(q_1)s(q_2)=\tau(q_1,q_2)s(q_1q_2). \end{equation} By associativity, we have $(s(q_1)s(q_2))s(q_3)=s(q_1)(s(q_2)s(q_3))$. It follows that \begin{equation}\label{eq:tau-cocycle} \tau(q_1,q_2)\tau(q_1q_2,q_3)=s(q_1)\tau(q_2,q_3)s(q_1)^{-1}\tau(q_1,q_2q_3). \end{equation}
Given the section $s$, we can define a set-theoretic bijection between $H$ and $G\times Q$: $$\alpha: H\rightarrow G\times Q,\quad \alpha(h):=(hs(j(h))^{-1}, j(h)).$$ The inverse of $\alpha$ is $$G\times Q\to H, \quad (g,q) \mapsto i(g)s(q).$$ The group structure on $H$ induces a new group structure $\cdot$ on $G\times Q$ via $\alpha$. This group structure is given by \begin{equation}\label{eq:twisted-prod-gp} (g_1,q_1)\cdot (g_2,q_2)=(g_1\operatorname{Ad}_{s(q_1)}(g_2)\tau(q_1,q_2),q_1q_2). \end{equation} Here $\operatorname{Ad}_{h}(\cdot)$ denotes the conjugation action of an element $h\in H$ on $G$, which is an automorphism of $G$ because $G$ is normal in $H$. Denote by $$G\rtimes_{s,\tau}Q$$ the set $G\times Q$ with the group structure given by (\ref{eq:twisted-prod-gp}). The definition implies that $\alpha$ is a group isomorphism: $$\alpha: H \to G\rtimes_{s,\tau}Q.$$ It is easy to check that different choices of the section $s$ yield isomorphic groups $G\rtimes_{s,\tau}Q$.
The group isomorphism $\alpha$ naturally induces an isomorphism of group algebras $$\alpha: {\mathbb C} H \overset{\simeq}{\longrightarrow} {\mathbb C} (G\rtimes_{s,\tau}Q).$$ Given $s$ and $\tau$, we let an element $q\in Q$ act on ${\mathbb C} G$ by conjugation by $s(q)$. This does not give an action of $Q$ on ${\mathbb C} G$, and the failure of this to be an action is governed by $\tau$. In other words, this defines a $\tau$-twisted action of $Q$ on ${\mathbb C} G$. Hence the group algebra ${\mathbb C}(G\rtimes_{s,\tau}Q)$ can be written as a twisted crossed product algebra ${\mathbb C} G\rtimes _{s,\tau}Q$.
Let $\widehat{G}$ be the set of isomorphism classes of irreducible complex linear representations of $G$. Furthermore, for every element $[\rho]$ in $\widehat{G}$, we {\em choose} an irreducible representation in the class $[\rho]$ denoted by $$\rho:G\to \operatorname{End}(V_\rho),$$ where $V_\rho$ is a certain finite dimensional ${\mathbb C}$-vector space. The group algebra ${\mathbb C} G$ is isomorphic to a direct sum of matrix algebra $\oplus_{[\rho]\in \widehat{G}}\operatorname{End}(V_\rho)$: $$\beta: {\mathbb C} G\overset{\simeq}{\longrightarrow} \oplus_{[\rho]\in \widehat{G}}\operatorname{End}(V_\rho), \quad g\mapsto (\rho(g))_{[\rho]\in \widehat{G}}.$$ This is well-known, see e.g. \cite[Proposition 3.29]{fu-ha}.
Next we define an action of $Q$ on $\widehat{G}$. Let $\rho: G\to \operatorname{End}(V_\rho)$ be a ${\mathbb C}$-linear representation of $G$. Given $q\in Q$, we obtain another $G$ representation $\tilde{\rho}$ defined by $$G\ni g\mapsto \rho(\operatorname{Ad}_{s(q)}(g)).$$ It is easy to see that $\tilde{\rho}$ is irreducible if and only if $\rho$ is. If $s': Q\to H$ is another section of $j$, then we have $\rho\circ \operatorname{Ad}_{s(q)}=\rho\circ \operatorname{Ad}_{s'(q)}\operatorname{Ad}_{s'(q)^{-1}s(q)}$. Since $s'(q)^{-1}s(q)\in G$, $\operatorname{Ad}_{s'(q)^{-1}s(q)}$ is an inner automorphism of $G$. Hence $\rho\circ \operatorname{Ad}_{s(q)}$ and $\rho\circ \operatorname{Ad}_{s'(q)}$ are isomorphic $G$-representations. Therefore the assignment $(q, \rho)\mapsto \tilde{\rho}$ yields a right $Q$-action on $\widehat{G}$; namely, $q\in Q$ sends the class $[\rho]\in \widehat{G}$ to the class $[\tilde{\rho}]\in \widehat{G}$. For notational convenience, we write this right action as a left action. We denote the image of the isomorphism class $[\rho]\in \widehat{G}$ under the action by $q$ by $q([\rho])$. By abuse of notation, we denote the chosen irreducible $G$-representation that represents the class $q([\rho])$ also by $q([\rho]): G\to \operatorname{End}(V_{q([\rho])})$. Let $$\widehat{G}\rtimes Q:=(\widehat{G}\times Q \rightrightarrows\widehat{G})$$ be the groupoid associated to this $Q$-action on $\widehat{G}$.
By construction, the representation $q([\rho]): G\to \operatorname{End}(V_{q([\rho])})$ is equivalent to the representation $\tilde{\rho}: G\to \operatorname{End}(V_\rho)$ defined by $g\mapsto \rho(\operatorname{Ad}_{s(q)}(g))$. Therefore there exists a ${\mathbb C}$-linear isomorphism, $$T_q^{[\rho]}: V_{\rho}\to V_{q([\rho])},$$ that intertwines the two representations, namely $$\rho(\operatorname{Ad}_{s(q)}(g))={T^{[\rho]}_q}^{-1}\circ q([\rho])(g)\circ T^{[\rho]}_q.$$ We may choose $T^{[\rho]}_1$ to be the identity map on $V_\rho$. It can be shown that there are constants $c^{[\rho]}(q_1, q_2)$ such that $T^{q_1([\rho])}_{q_2}\circ T^{[\rho]}_{q_1}\circ \rho(\tau(q_1,q_2))\circ {T^{[\rho]}_{q_1q_2}}^{-1}$ is $c^{[\rho]}(q_1, q_2)$ times the identity map. In other words, \begin{equation} \label{eq:dfn-c}T^{q_1([\rho])}_{q_2}\circ T^{[\rho]}_{q_1}=c^{[\rho]}(q_1,q_2)T^{[\rho]}_{q_1q_2}\rho(\tau(q_1,q_2))^{-1}. \end{equation} Since the collection $\{\rho\}$ consists of unitary representations, the isomorphisms $T^{[\rho]}_q$ can also be chosen to be unitary. Therefore, $c^{[\rho]}(q_1,q_2)$ actually takes value in $U(1)$. By \cite[Proposition 3.1]{TT}, The function $$c:\widehat{G}\times Q\times Q\to U(1),\quad ([\rho], q_1,q_2)\mapsto c^{[\rho]}(q_1,q_2)$$ is a 2-cocycle on the groupoid $\widehat{G}\rtimes Q$ such that $c^{[\rho]}(1,q)=c^{[\rho]}(q,1)=1$ for any $[\rho]\in \widehat{G}, q\in Q$. The cohomology class defined by $c$ is independent of the choices of the section $s$ and the operator $T^{[\rho]}_q$.
Let $$C(\widehat{G}\rtimes Q,c),$$ be the {\em twisted groupoid algebra} associated to the cocycle $c$ on $\widehat{G}\rtimes Q$. We explain the definition of $C(\widehat{G}\rtimes Q, c)$ and refer the readers to \cite{tu-la-xu} for more details. By definition $C(\widehat{G}\rtimes Q, c)$ is the set of $C(\widehat{G})$-valued functions on $Q$, i.e., ${\mathbb C}$-valued functions on $\widehat{G}\times Q$. By abuse of notation, for $([\rho],q)\in \widehat{G}\times Q$ we also denote by $([\rho], q)$ the function on $\widehat{G}\times Q$ which takes value $1$ at $([\rho],q)$ and $0$ elsewhere. The collection $\{([\rho], q)\}$ of functions on $\widehat{G}\times Q$ forms an additive basis of $C(\widehat{G}\rtimes Q, c)$. The set $C(\widehat{G}\rtimes Q, c)$ is endowed with a product structure defined by \[ ([\rho],q)\circ([\rho'],q')=\left\{\begin{array}{ll}c^{[\rho]}(q,q')([\rho], qq')&\ \text{if\ }[\rho']=q([\rho])\\ 0&\ \text{otherwise}\end{array}\right. . \] The cocycle condition of $c$ implies that this product is associative.
Let $\oplus_{[\rho]\in \widehat{G}}\operatorname{End}(V_\rho)\otimes {\mathbb C} Q$ be the ${\mathbb C}$-vector space spanned by elements of the form $(x_\rho, q)$, where $x_\rho$ is an element in $\operatorname{End}(V_\rho)$ with $[\rho]\in \widehat{G}$ and $q\in Q$. We equip this space with a product $\circ$ defined as follows: \[ (x_{\rho_1},q_1)\circ (\tilde{x}_{\rho_2},q_2):= \left\{ \begin{array}{ll} (x_{\rho_1}{T^{[\rho_1]}_{q_1}}^{-1}\tilde{x}_{q_1([\rho_1])}T^{[\rho_1]}_{q_1}\rho_1(\tau(q_1,q_2)), q_1q_2),& \text{if}\ [\rho_2]=q_1([\rho_1]),\\ 0&\text{otherwise}. \end{array} \right. \] Let $$\oplus_{[\rho]}\operatorname{End}(V_{\rho})\rtimes_{T,\tau}Q$$ be the space $\oplus_{[\rho]\in \widehat{G}}\operatorname{End}(V_\rho)\otimes {\mathbb C} Q$ with the product $\circ$ defined above. We call this the twisted crossed product algebra. This algebra plays an important role in the following structure result on the group algebra ${\mathbb C} H$: \begin{prop}[\cite{TT}, Proposition 3.2]\label{prop:matrix-coeff-algebra} The map $$\kappa: G\times Q\ni (g,q)\mapsto \sum_{[\rho]\in \widehat{G}}(\rho(g),q)$$ defines an algebra isomorphism from the group algebra ${\mathbb C} G\rtimes_{s,\tau}Q$ to the twisted crossed product algebra $\oplus_{[\rho]}\operatorname{End}(V_{\rho})\rtimes_{T,\tau}Q$. Hence, $$\kappa\circ\alpha:{\mathbb C} H\to \oplus_{[\rho]}\operatorname{End}(V_{\rho})\rtimes_{T,\tau}Q$$ is an algebra isomorphism. \end{prop}
Proposition \ref{prop:matrix-coeff-algebra} is used in \cite[Section 3.2]{TT} to prove the following structure result of ${\mathbb C} H$: \begin{thm}[\cite{TT}, Theorem 3.1] \label{thm:local-mackey} The group algebra ${\mathbb C} H$ is Morita equivalent to the twisted groupoid algebra $C(\widehat{G}\rtimes Q, c)$. \end{thm}
We remark that the proof of Theorem \ref{thm:local-mackey} is done by explicitly constructing Morita equivalence bimodules between the two algebras.
Since $j:H\to Q$ is a surjective group homomorphism, $j$ induces a surjective homomorphism of algebras from ${\mathbb C} H$ to ${\mathbb C} Q$. It is well-known that the center of ${\mathbb C} Q$ has a canonical additive basis indexed by the conjugacy classes of $Q$. This decomposition of the center $Z({\mathbb C} Q)$ and the surjection ${\mathbb C} H\to {\mathbb C} Q$ implies that the center of ${\mathbb C} H$, as a vector space, decomposes into a direct sum of subspaces $Z({\mathbb C} H)_{\<q\right\rangle}$ indexed by conjugacy classes $\<q\right\rangle$ of $Q$, $$Z({\mathbb C} H)=\bigoplus_{\<q\right\rangle\subset Q} Z({\mathbb C} H)_{\<q\right\rangle}.$$ As shown in \cite[Section 3.2]{TT}, the center $Z(C(\widehat{G}\rtimes Q, c))$ decomposes into a direct sum of subspaces $Z(C(\widehat{G}\rtimes Q, c))_{\<q\right\rangle}$ indexed by conjugacy classes of $Q$, $$Z(C(\widehat{G}\rtimes Q, c))=\bigoplus_{\<q\right\rangle\subset Q}Z(C(\widehat{G}\rtimes Q, c))_{\<q\right\rangle}.$$
The explicit Morita equivalence bimodules in the proof of Theorem \ref{thm:local-mackey} yield an algebra isomorphism from the center of ${\mathbb C} H$ to the center of $C(\widehat{G}\rtimes Q, c)$, which we denote by $I$.
\begin{prop}[\cite{TT}, Proposition 3.4]\label{prop:conjugacy} The isomorphism $$I:Z({\mathbb C} H)\to Z(C(\widehat{G}\rtimes Q,c))$$ is compatible with the decompositions into subspaces indexed by conjugacy classes of $Q$, i.e., $I$ is an isomorphism from $Z({\mathbb C} H)_{\<q\right\rangle}$ to $Z(C(\widehat{G}\rtimes Q, c))_{\<q\right\rangle}$. \end{prop}
In the rest of this paper, we discuss some group-theoretic applications of our analysis of the group algebra $\mathbb{C} H$.
\section{Counting conjugacy classes in group extensions}\label{subsec:conjugacy}
Let $j: H\to Q$ be a surjective homomorphism of finite groups. Let $\<q\right\rangle\subset Q$ be a conjugacy class of $Q$. The pre-image $$j^{-1}(\<q\right\rangle)\subset H$$ may be partitioned into a disjoint union of conjugacy classes of $H$. It is natural to ask the following: \begin{question}\label{counting_question} How many conjugacy classes of $H$ are contained in $j^{-1}(\<q\right\rangle)$? \end{question} In this Section, we discuss an answer to this question.
Let $G$ be the kernel of $j: H\to Q$. Then we are in the situation of the exact sequence (\ref{eq:extension_reproduced}). The homomorphism $j: H\to Q$ induces a surjective homomorphism $\mathbf{j}: \mathbb{C} H\to \mathbb{C} Q$ between group algebras. This, in turn, induces a homomorphism $\mathbf{j} :Z(\mathbb{C} H)\to Z(\mathbb{C} Q)$ between centers. The centers $Z(\mathbb{C} H)$ and $Z(\mathbb{C} Q)$, viewed as vector spaces, admit natural bases, $\{1_{\<h\right\rangle}\}\subset Z(\mathbb{C} H)$ and $\{1_{\<q\right\rangle}\}\subset Z(\mathbb{C} Q)$, indexed by conjugacy classes. These bases satisfy the requirement that if $j(\<h\right\rangle)=\<q\right\rangle$, then $\mathbf{j}(1_{\<h\right\rangle})\in \mathbb{N} 1_{\<q\right\rangle}$. As $j(\<s(q)\right\rangle)=\<q\right\rangle$, the map $j:Z(\mathbb{C} H)\to Z(\mathbb{C} Q)$ is surjective. Let $$Z(\mathbb{C} H)_{\<q\right\rangle}:=\bigoplus_{\<h\right\rangle\subset j^{-1}(\<q\right\rangle)} \mathbb{C} 1_{\<h\right\rangle}.$$ By construction, the dimension $\text{dim}\, Z(\mathbb{C} H)_{\<q\right\rangle}$ is the number of conjugacy classes of $H$ that are contained in $j^{-1}(\<q\right\rangle)$. By Proposition \ref{prop:conjugacy}, the isomorphism $I: Z(\mathbb{C} H)\to Z(C(\widehat{G}\rtimes Q,c))$ restricts to an additive isomorphism $$Z(\mathbb{C} H)_{\<q\right\rangle}\simeq Z(C(\widehat{G}\rtimes Q, c))_{\<q\right\rangle}.$$ Clearly, the answer to Question \ref{counting_question} is the dimension $\text{dim}\, Z(C(\widehat{G}\rtimes Q, c))_{\<q\right\rangle}$, which we now compute.
Let $\widehat{G}^q\subset \widehat{G}$ be the subset consisting of elements fixed by $q\in Q$. Let $C(q)\subset Q$ be the centralizer subgroup of $q$. Then, by \cite{ru1}, we have that $Z(C(\widehat{G}\rtimes Q, c))_{\<q\right\rangle}$ is additively isomorphic to the $c$-twisted orbifold cohomology $H_{orb}^\bullet([\widehat{G}^q/C(q)], c)$. Decompose $\widehat{G}^q$ into a disjoint union of $C(q)$-orbits: \begin{equation}\label{orbit_decomp} \widehat{G}^q=\coprod_i O_i. \end{equation} For each $C(q)$-orbit $O_i$, pick a representative $[\rho_i]$ and denote by $Q_i:=\text{Stab}_{C(q)}([\rho_i])\subset C(q)$ the stabilizer subgroup of $[\rho_i]$. Consider the homomorphism $$\gamma_{-,q}^{[\rho_i]}: C(q)\to U(1), \quad C(q)\ni q_1\mapsto \gamma_{q_1,q}^{[\rho_i]}:=c^{[\rho_i]}(q_1,q)c^{[\rho_i]}(q,q_1)^{-1}.$$ Here, $c^{[\rho]}(-,-)$ is the cocycle defined in (\ref{eq:dfn-c}). It follows from (\ref{orbit_decomp}) that $$H_{orb}^\bullet([\widehat{G}^q/C(q)], c)\simeq \bigoplus H_{orb}^\bullet(BQ_i, c).$$ By \cite[Example 6.4]{ru1}, we have that $H_{orb}^\bullet(BQ_i, c)=\mathbb{C}$ if the following condition holds: \begin{equation}\label{condition:gamma=1} \gamma_{q_1,q}^{[\rho_i]}=1\,\, \text{ for all } q_1\in Q_i. \end{equation} Moreover, if (\ref{condition:gamma=1}) does not hold, then $H_{orb}^\bullet(BQ_i, c)=0$. It follows that $\text{dim}\, Z(C(\widehat{G}\rtimes Q, c))_{\<q\right\rangle}$ is equal to \[
\#\{O_i=C(q)\text{-orbit of }\widehat{G}^q| \text{ there exists } [\rho_i]\in O_i \text{ s.t. } \gamma_{q_1,q}^{[\rho_i]}=1\,\, \text{for all } q_1\in Q_i=\text{Stab}_{C(q)}([\rho_i])\}. \]
In summary, we have obtained the following theorem as an answer to Question \ref{counting_question}. \begin{theorem}\label{thm:conj-general} Let $H=G\rtimes_{s, \tau} Q$ be an extension of $Q$ by $G$. Consider the canonical quotient map $j:H\to Q$. For $q\in Q$, the number of conjugacy classes of $H$ that is mapped to the conjugacy class $\<q\right\rangle$ of $Q$ is equal to \begin{equation}\label{count_answer1}
\#\{O_i=C(q)\text{-orbit of }\widehat{G}^q| \text{ there exists } [\rho_i]\in O_i \text{ s.t. } \gamma_{q_1,q}^{[\rho_i]}=1\,\, \text{for all } q_1\in Q_i=\text{Stab}_{C(q)}([\rho_i])\}. \end{equation}
\end{theorem}
In the following, we discuss a few special cases of Theorem \ref{thm:conj-general}. \begin{example} If the group $G$ is abelian, then all irreducible representations of $G$ are $1$-dimensional, and all intertwiners
in (\ref{eq:dfn-c}) can be taken to be the identity. In this case, (\ref{count_answer1}) can be simplified to \begin{equation}\label{count_answer2} \begin{split}
\#\{O_i=C(q)\text{-orbit of }\widehat{G}^q| &\text{ there exists }[\rho_i]\in O_i, \\ &\text{s.t. }\rho_i(\tau(q_1,q)\tau(q,q_1)^{-1})=1\,\, \text{for all } q_1\in Q_i=\text{Stab}_{C(q)}([\rho_i])\}. \end{split} \end{equation} \end{example}
\begin{example} If the group $G$ is abelian and $H$ is a semi-direct product of $G$ and $Q$, then the cocycle $\tau(-,-)$ can be taken to be trivial. In this case, (\ref{count_answer1}) can be simplified to \begin{eqnarray}\label{count_answer2.5} &\#\{C(q)\text{-orbit of }\widehat{G}^q\}. \end{eqnarray} \end{example}
\begin{example} If the $Q$-action on $\widehat{G}$ is trivial\footnote{Equivalently, this means that the band of the gerbe $BH\to BQ$ is trivial.}, then $\widehat{G}^q=\widehat{G}$, and all intertwiners in (\ref{eq:dfn-c}) can be taken to be the identity. In this case, (\ref{count_answer1}) can be simplified to \begin{equation}\label{count_answer3} \begin{split} \#\{[\rho]&=\text{isomorphism class of irreducible }
G\text{-representations}|\\ &\hspace{4cm}\rho(\tau(q_1,q)\tau(q,q_1)^{-1})=1\,\, \text{for all }q_1\in C(q)\}. \end{split} \end{equation} \end{example}
\section{An orthogonality relation of characters}\label{subsec:orthogonality}
The material in this Section is inspired by the proof of the orthogonality relation given in \cite[Chapter 2, Section 12]{Be_Zh}. Using Proposition \ref{prop:matrix-coeff-algebra}, we prove a generalization of the orthogonality relation between characters of $G$. For $h\in H$, write the centralizer subgroup of $h$ by $C_H(h)$, and the number of elements in $C_H(h)$ by $|C_H(h)|$.
\begin{theorem}\label{thm:character} Let $H=G\rtimes_{s, \tau}Q$ be an extension of $Q$ by $G$. For $[\rho]\in \widehat{G}$, let $\chi^G_\rho$ be the character of the $G$-representation $V_\rho$. For $(g_1,\ g_2)\in G\times G$, \begin{equation}\label{generalized_orthogonal_relation} \sum_{[\rho]\in \widehat{G}}\sum_{q\in Q} \chi_\rho^G(g_1^{-1})\chi^G_{q([\rho])}(g_2)= \left\{ \begin{array}{ll}
|C_H(g_1)|,& \text{if } g_1 \text{ and } g_2 \text{ are conjugate {\em in }}H,\\ 0&\text{otherwise}. \end{array} \right. \end{equation}\end{theorem}
\begin{proof}
Consider (\ref{eq:extension_reproduced}) again. The group $H\times H$ acts naturally on the group algebra $\mathbb{C} H$ via $$(h_1, h_2)\cdot h=h_1^{-1}hh_2.$$ In this way, we may view $\mathbb{C} H$ as a representation of $H\times H$. Its character $\chi^{H\times H}_{\mathbb{C} H}$ can be calculated as follows: \begin{equation*} \begin{split}
\chi^{H\times H}_{\mathbb{C} H}((h_1, h_2))=&\#\{ h\in H| h_1^{-1}hh_2=h \}\\
=&\#\{ h\in H|hh_2h^{-1}=h_1 \}\\ =&\left\{ \begin{array}{ll}
|C_H(h_1)|,& \text{if } h_1 \text{ and } h_2 \text{ are conjugate in }H,\\ 0&\text{otherwise}. \end{array} \right. \end{split} \end{equation*}
We now consider $\mathbb{C} H$ as a representation of the subgroup $G\times G$. The above calculation gives the character of this representation: for $(g_1, g_2)\in G\times G$, \begin{equation}\label{character_answer1} \chi^{G\times G}_{\mathbb{C} H}((g_1,g_2))=\chi^{H\times H}_{\mathbb{C} H}((g_1,g_2))= \left\{ \begin{array}{ll}
|C_H(g_1)|,& \text{if } g_1 \text{ and } g_2 \text{ are conjugate in }H,\\ 0&\text{otherwise}. \end{array} \right. \end{equation} We calculate the character $\chi^{G\times G}_{\mathbb{C} H}$ by another method. By Proposition \ref{prop:matrix-coeff-algebra}, there is an isomorphism of algebras $$\mathbb{C} H\simeq\oplus_{[\rho]\in \widehat{G}}\operatorname{End}(V_{\rho})\rtimes_{T,\tau}Q.$$ Under this isomorphism, the $G\times G$ action on $\mathbb{C} H$ is identified with the following $G\times G$ action on $\oplus_{[\rho]\in \widehat{G}}\operatorname{End}(V_{\rho})\rtimes_{T,\tau}Q$: \begin{equation*} \begin{split} (g_1, g_2)\cdot (x_\rho, q):=&(\sum_{\rho_1}\rho_1(g_1^{-1}),1)\circ (x_\rho, q)\circ(\sum_{\rho_2}\rho_2(g_2),1)\\ =&(\rho(g_1^{-1})x_\rho {T_q^{[\rho]}}^{-1}q([\rho])(g_2)T_q^{[\rho]},q). \end{split} \end{equation*} Here, $\circ$ is the algebra structure on $\oplus_{[\rho]\in \widehat{G}}\operatorname{End}(V_{\rho})\rtimes_{T,\tau}Q$.
For each $\rho$, fix an isomorphism of $\operatorname{End}(V_\rho)$ with a matrix algebra, and let $e_{st}^\rho$ denote the standard basis of this matrix algebra. We use the symbol $(x_\rho)_{st}$ to denote the $s,t$-entry of $x_\rho\in \operatorname{End}(V_\rho)$. Then we have $(\rho(g_1^{-1})e_{st}^\rho {T_q^{[\rho]}}^{-1}q([\rho])(g_2)T_q^{[\rho]})_{st}=(\rho(g_1^{-1}))_{ss}({T_q^{[\rho]}}^{-1}q([\rho])(g_2)T_q^{[\rho]})_{tt}$. Therefore, \begin{equation*} \begin{split}
\text{tr}\,((g_1,g_2)|_{\operatorname{End}(V_\rho)\times \{q\}})=&\sum_{s,t} (\rho(g_1^{-1}))_{ss}({T_q^{[\rho]}}^{-1}q([\rho])(g_2)T_q^{[\rho]})_{tt}\\ =&\text{tr}\,(\rho(g_1^{-1}))\text{tr}\, ({T_q^{[\rho]}}^{-1}q([\rho])(g_2)T_q^{[\rho]})\\ =&\chi^G_\rho(g_1^{-1})\chi^G_{q([\rho])}(g_2), \end{split} \end{equation*} where $\chi^G_\rho$ and $\chi^G_{q([\rho])}$ denote the characters of the $G$-representations $\rho$ and $q([\rho])$. Summing over $[\rho]\in \widehat{G}$ and $q\in Q$, we find that \begin{equation}\label{character_answer2} \chi^{G\times G}_{\mathbb{C} H}((g_1,g_2))=\sum_{[\rho]\in \widehat{G}}\sum_{q\in Q} \chi_\rho(g_1^{-1})\chi_{q([\rho])}(g_2). \end{equation} Combining the above with (\ref{character_answer1}), we obtain the desired identity: \[ \sum_{[\rho]\in \widehat{G}}\sum_{q\in Q} \chi^G_\rho(g_1^{-1})\chi^G_{q([\rho])}(g_2)= \left\{ \begin{array}{ll}
|C_H(g_1)|,& \text{if } g_1 \text{ and } g_2 \text{ are conjugate {\em in }}H,\\ 0&\text{otherwise}. \end{array} \right. \] \end{proof}
\end{document} | arXiv |
\begin{definition}[Definition:Basic Universe Axioms]
Let $V$ be a basic universe.
Then $V$ is to satisfy the following axioms:
\end{definition} | ProofWiki |
R/simecol Simulation Model for the Battle of Iwo Jima
Posted on August 8, 2016 by José Romero in R bloggers | 0 Comments
[This article was first published on U.N.A. Matemáticas El Tigre, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)
It was about 71 years ago today that the Second World War on the Asian front ended. After the tragic Little Boy and Fat Man detonations over Hiroshima and Nagasaki on the 6th and 9th of August 1945, respectively, the Japanese surrendered in August 14th.
When I taught the Simulation and Modeling course for the Systems Engineering department at the UNEFA, we used to do an exercise in class on simulating the battle of Iwo Jima, one of the most decisive and sanguinary battles in the Pacific. This simulation was based on a System Dynamics model derived from the Lanchester Laws of combat and was implemented in R using the simecol/deSolve/FME packages. Prior to the class, we had covered the topic of Forrester diagrams and System Dynamic models in general1, the Lanchester Laws of combat and of course a bit of background on the use of the simecol and related packages for dynamic model fitting and simulation.
In this post, I will present a brief summary on the Lanchester Laws of Combat, some historical background on the battle of Iwo Jima and then go into the simulation model itself, detailing its implementation and data fitting in R through the said packages. Subsequently, I will contrast the model's output with the real historical data and derive some interpretations. To conclude the post, I will pose some "what if …" questions on alternative hypothetical scenarios and offer some thoughts about possible applications of simulation models like this one for teaching history in classrooms.
The Lanchester Models of Combat
Frederick Lanchester was a British engineer well-known for his inventions and contributions to the budding automotive industry at the end of the nineteenth century (he's the guy that invented the accelerator pedal, for example). Lanchester also made some contributions to the theory of aerodynamics and being concerned over the ways in which the aeronautical industry could change warfare (he was more of a pacifist, really), he set out to study the dynamics of warfare and develop mathematical models of battle through systems of differential equations, now known as the Lanchester Laws of Combat. These differential equation models are fairly abstract – they leave out many details that we commonly associate with battles and focus on the way in which number of combat units of two armies decrease over time.
Without loss of generality, we will represent in each time instant \(t\) the number of troops or combat units of the red army as \(R(t)\) or \(R\) and of the green army as \(G(t)\) or simply \(G\). At the start of the battle, at \(t=0\), \(R(0)\) and \(G(0)\) represent the number of combat units of each army. When \(R(t)\) or \(G(t)\) reach 0 at any given time, the corresponding army is left without troops and so the battle ends with the opposing army as victorious. Lanchester proposed two different scenarios for battle, according to the warfare technology used:
First Lanchester Law
In the first Lanchester Law, two armies confront each other in hand-to-hand combat. At each time instant, the number of casualties of each army is proportional to the number of confrontations between the combatant units and the effectiveness or lethality coefficient of the enemy army. This is represented mathematically by the following differential equations:
\[\begin{align*}\dfrac{dR}{dt} &= -g \cdot R \cdot G \\ \dfrac{dG}{dt} &= -r \cdot R \cdot G\end{align*} \]
Derivatives \(\tfrac{dR}{dt}\) and \(\tfrac{dG}{dt}\) indicate the way in which the battle unfolds and represent the rate of attrition of the red and green armies, respectively. The \(r\) y \(g\) parameters are the effectiveness coefficients of each army and represent the number of casualties on the enemy side inflicted during one time instant at each confrontation (represented by \(R \cdot G\)). This model of combat is also referred to as the the model of "unaimed fire", the word "fire" in this case not necessarily referring to artillery fire but rather to the directionality of discharge of the lethal armament used in combat. In fact, this model, being applied mainly to battle scenarios with no artillery can also be used for battles with low-precision artillery.
If we divide one equation by the other and isolate the red and green terms on each side, we obtain \(r \cdot dR = g \cdot dG\). In turn, if we integrate (with respect to \(t\)) this last equation we can see that at each time instant, the expression \(r\cdot R(t) – g \cdot G(t)\) remains constant. The terms \(r\cdot R\) and \(g \cdot G\) are called the fighting strengths of the reds and greens, respectively. The fighting strengths, which could also be considered as the number of troops eliminated the the red and green armies, determine result of the battle. Since \(r\cdot R(t) – g \cdot G(t)\) is constant throughout the whole time, the reds will win if \(r\cdot R(t) – g \cdot G(t) > 0\), the greens win if \(r\cdot R(t) – g \cdot G(t) < 0\) and if \(r\cdot R(t) - v \cdot G(t) = 0\), then both sides will be wiped out simultaneously. Since the fighting strengths are linear in \(R\) and \(G\), the first Lanchester Law is also known as the linear law of combat.
We can infer some characteristics of combat scenarios based on this linear law of combat:
An army's efficiency parameter and its number of troops are equally important factors to its fighting strength (ie. if you double the number of troops but decrease the efficiency to one half, the fighting strength will be the same).
If you manage to divide the enemy in two contingents and confront each contingent successively, this will not alter the result of the battle.
Second Lanchester Law
This law supposes a scenario where both sides use aimed artillery fire, hence the casualty infliction mechanism no longer depends on the number of hand-to-hand combat confrontations being produced, but rather on the number of troops firing upon the enemy:
\[\begin{align*} \dfrac{dR}{dt} &= -g \cdot G \\ \dfrac{dG}{dt} &= -r \cdot R\end{align*} \]
Again, dividing one equation by the other, isolating the terms corresponding to each army and integrating, we obtain the following constant expression: \(r\cdot R^2 -g \cdot G^2=0\). This time, the fighting strengths \(r\cdot R^2\) and \(g \cdot G^2\) are quadratic on \(R\) and \(G\), hence the alternative name of "square law of combat" being applied to Lanchester's Second Law. In two armies with equal number of troops, the more efficient one will prevail. If however, one army manages to divide the enemy in two or more contingents and confront each contingent successively, then it will certainly win over the other. This is what Von Clausewitz and Sun Tzu meant by advising generals not to divide their troops because each division would significantly reduce an army's fighting strength. In other words, if an army is numerically superior, then it must combat undivided against a more efficient enemy. As an example of this principle, Hannibal confronted a qualitatively superior enemy at Cannas by dividing him and prevailed. Conversely, the American strategy of flooding the island of troops at Iwo Jima proved effective in managing to establish a beach head.
Applications of the Lanchester combat models and further elaborations
Lanchester originally applied his model to the study of Nelson's naval tactics in the Battle of Traffalgar, that famous naval battle of the Napoleonic Wars. In more recent literature, Lanchester models were applied to the Kursk battle in 1943 involving panzers in the German and Soviet sides, the battle of Ardennes (again in World War II) and the naval conflict in the Atlantic in which the German U-Boats inflicted damage on the Allied convoys. In the Engel 1954 paper we find an application to the Battle of Iwo Jima, based on the data compiled by Cliiford and Moorehouse. On this last point, I must remark that the model fits to these battles has been partially successful due to the fact that under normal battle conditions, it is quite difficult to accurately compile day-to-day data of casualties on both sides.
The Lanchester models themselves can be further generalized to include reinforcement like so:
\[\begin{align*}\dfrac{dR}{dt} &= f_r(t) – g \cdot G \\ \dfrac{dG}{dt} &= f_g(t) -r \cdot R \end{align*} \]
Here, \(f_r(t)\) and \(f_g(t)\) represent the reinforcement rates of the red and green armies respectively. This is the model we will use for our Battle of Iwo Jima simulation, as the American side brought in reinforcements on several occasions, while, as we shall see, the Japanese had none.
If we are willing to relax our assumptions that the battle is based wholly on either an aimed or an unaimed fire combat model, then we get Bracken's generalized Lanchester Model:
\[\begin{align*} \dfrac{dR}{dt} &= -g \cdot R^q \cdot G^p \\ \dfrac{dG}{dt} &= -r \cdot R^p \cdot G^q\end{align*} \]
If we define \(\alpha=1+p-q\), we get \(gG^\alpha-rR^\alpha\) as the constant quantity and we can easily see how the first and second Lanchester models are particular cases of this model. However (see MacKay, 2005), when fitting this model to any particular battle there is an ambiguity in the estimates for \(p\) and \(q\), as several best fitting estimates could be obtained.
Some historical background on the Battle of Iwo Jima2
The island of Iwo Jima3 is a tiny island of some 8 square miles situated about 750 miles south from Tokyo in the Japanese mainland and halfway between Saipan and Tokyo. Were it not for this last fact, the place would not have been the stage of a major battle in the Second World War and would probably remain largely unknown today. By the summer of 1944, the Americans had gained control of Saipan and the Mariana islands, from where they staged bombing raids to mainland Japan with long range B-29 Superfortress bombers. However, the losses inflicted upon these bomber planes by Japanese fighter planes deploying from the two airfields in Iwo Jima were unsustainable to the Americans and furthermore, the island's radio station relayed early warning reports to the mainland to ready themselves for the bombing raids. For these reasons, it was of strategic importance to the Americans on their northward multi-pronged move towards Japan to seize control of Iwo Jima. And, naturally, the Japanese knew this.
For the Japanese, Iwo Jima was more than just strategically significant. Iwo Jima was sacred Japanese soil. On account of this, an eventual American invasion of Iwo Jima would be a heavy blow to the Japanese morale as for more than four thousand years, no foreign country had invaded Japan. Therefore, the Japanese were not ready to surrender the island without first inflicting heavy losses upon the American forces. Emperor Hirohito intrusted Lietenant General Tadamichi Kuribayashi4 with this mission.
Kuribayashi drew up his plan for the defense of the island upon the following premises:
The Japanese would engage in combat from underground, not on the surface of the island.
The battle would be defensive, the idea being to seriously erode the American forces.
No Japanese soldier would survive.
Every Japanese was to kill 10 Americans before being killed.
Kuribayashi's strategy stood apart from the usual Japanese tactics to which the Americans had become accustomed. He specifically forbade the famous "Banzai" suicide attacks to repel the American disembarkation on the island, which besides, were completely futile to prevent it, since the Japanese would not be receiving air or naval support, and hence no reinforcements. Instead, he began construction of a gallery of underground tunnels connecting a network of bunkers and pillboxes. By the time the Americans arrived, the Japanese had completed over 11 miles of tunnels and placed large artillery pieces protected by thick steel walls from within Mount Suribachi.
For their part, Americans were also preparing for what they designated as Operation Detachment (the invasion of Iwo Jima). According to their intelligence reports, there were some 14,000 Japanese troops on the island (there were really over 21,000). Before D-Day, and based on their experience at Saipan, Tarawa and Peleliu, the Marines requested for 10 days of previous bombings. The Navy finally acquiesced and on the 16th of February, 1945 began a 3-day bombing. Previously, though, B-29s had completed a 72 day bombing campaign.
The bombing campaign on Iwo Jima had been devastating … but only on the surface. Inexplicably, not only had the Japanese defensive fortifications not diminished, according to an American intelligence report, but they had grown in number from 450 to 750 3 months after the bombings began. For the unsuspecting American forces, the real surprise lay waiting for them underground…
On D-day (February 19th), the Americans disembarked 54,000 troops. During the first moments, many thought that no Japanese had survived the bombings and that the estimates on the number of Japanese troops were exaggerated. Then, as the troops made their way 300 yards into the beach and the spot became congested with troops and equipment (what Kuribayashi was waiting for), all hell broke loose. There was crossfire from Mount Suribachi and some hills to the north of the island. The beach offered no protection, as the fine and ashy volcanic sands made it impossible to dig trenches. As Japanese fire impacted upon the disembarkation boats, the amphibious vehicles got stuck in the fine sands. The beaches of Iwo Jima became an infernal chaos in the carnage that ensued.
During the first days of battle, it seemed like the Americans had made no progress at all. They were fighting an enemy they couldn't see – an enemy who fought from under the ground and usually at night. Upon burning underground burrows where Japanese were thought to be with flamethrowers, Americans were surprised to see Japanese troops re-emerge from burrows that were thought to be "cleansed" days ago. What the unsuspecting Americans didn't know is that these holes and burrows were part of a vast network of underground tunnels. In the meantime, from the 20th to the 23rd day of February, the Marines attempted to take Suribachi. A small group of Marines eventually managed to raise the American flag on the top of Mount Suribachi, a moment immortalized for posterity in the famous historical photograph by Joe Rosenthal5.
Fighting continued for 36 days after the start of the battle. Once Suribachi was taken, the Americans continued to slowly and painfully advance to the north of the island, encountering fierce pockets of Japanese resistance along the way. One can only surmise about the fierceness of the combat that took place on Iwo Jima by reading the names of certain places on a map of the island, such as "the meat grinder". During the whole of Operation Detachment, the Americans made two more disembarkation of troops: one of 6,000 men and another of 13,000 men on the third and sixth day of the battle, respectively. In total, the Americans sustained 28,000 casualties including about 8,000 dead. The cost of taking this small island was much more than they expected. As for the Japanese, over 20,000 died and about 1000 were taken prisoners. The last two holdouts6 did not surrender until 1951. It is said that Kuribayashi himself died in a suicide raid on the southern part of the island disguised as a common soldier without his rank of General insignia.
Data for the simulation study
The data for this simulation study is based on a report titled "The Iwo Jima Operation" written by Captain Clifford Moorehouse, an American officer. This report contains day-to-day records on the number of American casualties. Together with the data on the number of American troops put ashore on the three disembarkations, this data serves to reconstruct the number of active American troops on the given days of the 36-day battle. Of course, we must assume an estimate number of 21,500 Japanese troops at the start of the battle and 0 active Japanese troops at the end, in between which no estimates exist. The data for the number of active troops on each side is summarized in the following table, in which the "time" column indicates the day of the battle and the "ame" and "jap" columns indicate the active troops for the Americans and the Japanese, respectively:
0 0 21500 13 59549 NA 25 53347 NA
1 52839 NA 14 59345 NA 26 53072 NA
10 62339 NA 22 54796 NA 34 52155 NA
12 60667 NA 24 53938 NA 36 52140 0
The data on the three American disembarkations can be summarized by the following stepwise function definition:
\[f(t) = \left\{ \begin{array}{cl} 54000 & 0 \leq t \lt 1 \\ 0 & 1 \leq t \lt 2 \\ 6000 & 2 \leq t \lt 3 \\ 0 & 3 \leq t \lt 5 \\ 13000 & 5 \leq t \lt 6 \\ 0 & 6 \leq t \leq 36 \end{array} \right.\]
Engel (1954) estimated the effectiveness coefficients for the Americans and the Japanese to be 0.0106 and 0.0577 respectively. We can take these values to simulate the model and compare its fit to the actual observed number of active troops according to the Moorehouse data. Then we will fit these parameters using simecol's fitOdeModel function and see if we obtain an even better fit. I must remark that when using simecol, it is not necessary to know in advance the values of your model's parameters, as the fitOdeModel takes care of this problem for you. After all, estimating a model's parameters is one of the main reasons why we build simulation models in the first place.
R/simecol implementation for the simulation model of the Battle of Iwo Jima
We have come to the core of this blog entry: the implementation of the simulation model for the Battle of Iwo Jima using R and simecol. First off, let me briefly explain what simecol is. It is an R package for the simulation of ecological models through R's S4 object class system using a state-space representation. What this means (the S4 object class part) is that you can work with several model objects in memory having different parameter values, etc. and simulate them, fit their parameters separately and so on, using an object-oriented methodology that makes it easy for researchers without extensive knowledge of R programming to get their simulation models up and running quickly using a general purpose programming language like R, which provides them with flexibility in modeling while at the same time access to all the powerful statistical/analytical tools available through the base R and extensive package system.
The state-space representation part mentioned in the last paragraph means that the various types of models one can implement in simecol- 1) dynamic system models using differential equations (or simecol odeModel class objects. This is the type we will deal with here), 2) cellular automata models and 3) individual agent based models – are viewed as subtypes of a more general family of models in which you have: 1) a set of variables describing the state of the model at a given time, 2) rules governing the way these state variables change over time, 3) a set of model parameters 4) a specification for the time instants at which you wish to observe the state variables, 5) externalities (called inputs), which are not part of the state variables of the system interrelated amongst each other but which affect them and 6) a specification for a solver function, which updates the state variables based on the rules that describe their change over time.
Schematic diagram of simecol odeModel objects external inputs differential equations parameters state variables output (state variables) initial state (state variables) solver (rk4, lsoda, etc.) t ⇦ t + Δt
In the case of dynamic system models with differential equations, the rules describing the way the system changes over time are obviously characterized by a system of differential equations. These differential equations usually have parameters that further characterize the dynamic behavior of the system and which the researcher is usually interested in fitting from observed data. In this battle of Iwo Jima example, the parameters would be the effectiveness coefficients of the Americans and the Japanese. In our Iwo Jima model we also have the American reinforcements- those would be the inputs (or externalities) which are included in the differential equation describing how the number of American troops changes over time. Finally, the solver function is a specification of a particular numerical integrator that solves for the state variables using the differential equations, for example, 4th-order Runge-Kutta, Euler, LSODA, etc. In this sense, simecol serves as a wrapper for another package called deSolve, which includes the compiled FORTRAN numerical integration routines for fast execution in R.
Without further ado, here's the R script for the simulation of the Battle of Iwo Jima using the simecol library:
library(simecol)
iwojima_engel <- new("odeModel",
main = function (time, init, parms, ...) {
x <- init
p <- parms
f <- approxTime1(inputs, time, rule = 2)["reinforcements"]
dame <- f - p["j"] * x["jap"]
djap <- - p["a"] * x["ame"]
list(c(dame, djap))
parms = c(j=0.0577, a=0.0106),
times = c(from=0, to=36, by=0.01),
init = c(ame=0, jap=21500),
inputs = as.matrix(data.frame(
day = c(0, 0.999, 1, 1.999, 2, 2.999, 3, 4.999,
5, 5.999, 6, 36),
reinforcements = c(54000, 54000, 0, 0, 6000, 6000, 0, 0,
13000, 13000, 0, 0))),
solver = "lsoda"
#copy the iwojima Engel model onto another for fitting later
iwojima_fitode <- iwojima_engel
#now read the actual data...
obs <- read.csv2("iwo_jima.csv")
#the weightdf dataframe assigns equal weight to all observations,
#except for the NA values in the japanese column (those get 0 weight)
weightdf <- data.frame(ame=rep(1, nrow(obs)), jap = rep(0, nrow=(obs)))
weightdf[1,"jap"] <- 1
weightdf[nrow(obs),"jap"] <- 1
#return the ssq of the Engel model (a measure of goodness of fit)
ssqOdeModel(parms(iwojima_engel),
iwojima_engel,
obstime = obs$time,
yobs=obs[2:3],
weights=weightdf)
#and now we fit the model parameters with simecol's fitOdeModel
#including our weightdf as weights makes the PORT routine faster
result_fit <- fitOdeModel(iwojima_engel,
obstime=obs$time,
fn=ssqOdeModel,
weights=weightdf,
method="PORT",
scale=c(1/0.1, 1/0.01),
lower=c(j=0,a=0))
result_fit
#we update the new model parameters
parms(iwojima_fitode) <- result_fit$par
ssqOdeModel(parms(iwojima_fitode),
iwojima_fitode,
#finally we simulate and plot the graphs
iwojima_engel <- sim(iwojima_engel)
iwojima_fitode <- sim(iwojima_fitode)
svg("iwojima_engel.svg")
matplot([email protected],
main = "Model1 : Engel(1954)",
xlab = "Day",
ylab = "Number of active troops",
type = "l", lty = c("solid", "solid"),
col = c("blue","red"))
points(obs$time,obs$ame,col="blue",pch=19)
points(obs$time[c(1,36)],obs$jap[c(1,36)],col="red",pch=19)
graphics.off()
svg("iwojima_fitode.svg")
main = "Model2 : fitOdeModel",
Let's go over the code above. First, in line 1 we load the simecol library. In lines 2-20, we define an odeModel class object called "iwojima_engel". Notice how the "main" slot serves to specify the system of differential equations as a function returning the differentials on each state variable as a list. The other simecol methods will "know" that these differentials will correspond to the "ame" and "jap" variables because they are given as a list in the same order as we specify them in the "init" slot on line 13. Notice that the f function used to represent the American reinforcements is given as a function of the "inputs" slot defined in lines 14-18 (we will later explain what that approxTime1 function does). The "times" slot in line 12 is given in very small increments of 0.01 days so that the solver function (specified as "lsoda" in line 19) can do the numerical integration in small steps.
In line 24 we read in the Moorehouse data stored in a csv file (if you want to do this simulation exercise yourself, just take the Moorehouse data from the table above and put it in a csv file in the same directory as the R script (the separation character is a semicolon). Lines 27 to 29 simply create a data frame of the same dimensions as the "obs" data frame with 1 for valid values and 0 for NA values on that data frame. This simply serves to assign equal weights to each observed non-NA value so as to make the parameter fitting faster later on.
Lines 31-35 simply invoke the ssqOdeModel function on the iwojima_engel object to compute a measure of goodness of fit (consult the help page for ssqOdeModel to see how it is calculated). Lines 38-46 do the actual fitting of the iwojima_engel model object, producing a better-fitting set of parameters (remember, the American and Japanese effectiveness parameters). To do this, the fitOdeModel invokes a non-linear optimization routine using the PORT algorithm to obtain the set of parameter values that minimize the ssqOdeModel measure. In fact, this produces a better fit of the model (indicated by a lower ssqOdeModel value).
In line 48 we simply feed the new parameters into the iwojima_fitode, which was simply a copy of the iwojima_engel object and finally, in lines 55-76 we plot both model objects. Notice how the svg graphics device is used instead of pdf or png graphics. The svg graphics device produce better looking, non pixelating graphics that look great no matter how much you zoom in the page on your browser.
Model with parameters estimated by Engel (1954)
Model with parameters fitted by fitOdeModel
Finally, for comparison, we include a little table with the main results used to compare the different parameter values (Engel's estimations and the simecol estimations):
Engel (1954)
fitOdeModel
Japanese Effectiveness 0.0577000 0.05540003
American Effectiveness 0.0106000 0.01080927
ssqOdeModel 0.3167629 0.05121009
Interpretation of the simulation results and some what-if questions
Before I go on with the most fun part of modeling (playing with your models and interpreting them), I must clarify that simecol is a package originally intended for ecological modeling. It has also been used for engineering and economics modeling. However, because it is so versatile, I have used it for a military-related application such as simulating a battle using a Lanchester model, which is not an application for the package that its creators originally intended nor support. I do not mean by this that my view on this matter is different from the authors of simecol. In fact, I can see their wisdom in not supporting applications of simecol that they're afraid of.
What motivated me to present this model of the Battle of Iwo Jima are mainly educational reasons. First of all, the topic of historic and real battles is usually interesting for the young audiences to whom I've given classes on modeling and simulation. For me as a teacher, this serves as a sugar-coating to make the mathematical modeling and computational implementation topic-pills go down easier. Another important reason for me as a teacher is that these type of models lend themselves to interesting interpretations and discussions in class about politics and history. It gives us, as we shall see, a scientific framework for discussing about history and other social topics. To quote one of the authors on the bibliography I consulted:
Lanchester models provide an excellent example of the strengths (and weaknesses) of simple mathematical modeling. Further, as we have seen, the basic model leads to many other 'What if...?' questions which can be easily investigated. Many more such questions can be asked, and of course once one begins numerical simulations the possibilities are endless. To model warfare can seem more politically challenging (not to say incorrect) than to use ecology or epidemiology, but a little understanding of how military planners arrive at their tactical conclusions can also strip away mystique and (for this author, at least) expose some of the subject's limitations! (MacKay, 2005, p. 7)
Some possible 'what-if' questions or questions for interesting discussion:
Given the same effectiveness of the American and Japanese troops as those found in your model, with how many troops would the Japanese have won the Battle of Iwo Jima if the Americans had disembarked the same amount of troops on the island? Would it have been possible for the Japanese to amass this number of troops given the space and logistic limitations of Iwo Jima?
If the Japanese soldiers were 5 times more effective or lethal than the Americans, why did more Japanese die in Iwo Jima?
How many soldiers would the Japanese have needed to make the battle linger for, say, one year? How many American casualties would this have brought about? Would this have changed the outcome of the war?
Lanchester models are based on conventional warfare. Nowadays, this conventional warfare scenario is not as generalized as before. For example, it is said that in Venezuela there is a non-conventional war being waged, although there are no battles in the classical sense involving regular military forces. There are civilian casualties, such as those caused by one of the highest crime rates in the world. And now, there is an added element of extreme scarcity of foods and medical supplies which cause civilian casualties as well. How would you as a modern-day Lanchester propose a differential equation model for this war scenario? What state variables and parameters would you define?
As you can see, the subject lends itself to some very interesting discussions in class. I might add that mathematical modeling of this sort gives us a fresh new perspective on social issues and gives us the foundations for discussing about these issues in a more scientific and less dogmatic way. I think having this sort of discussion integrated into our educational system is much needed today, particularly in my country.
As a closing remark for the readers of this blog, let me say that I will post more about simecol and other related packages, such as FME (with more added functionalities for parameter fitting). And no, the other posts won't be about military/warfare models! Stay tuned...
I wish to thank, first and foremost, you, dear reader, for putting up with such a long post. My thanks go out to the creators of the simecol package. Dr. Thomas Petzoldt's kind help on the R-sig-dyn-mod mailing list has been invaluable in helping me come to grips with this powerful R modeling tool. I would also like to thank James Cagney for his historyanimated media on the Battle of Iwo Jima. It helped me to understand the military and historical background of the battle. Make sure to check out his historyanimated site! Finally, I would like to thank Professor Christine Lind for providing me with the Moorehouse data without which this post would not have been possible.
There are some excellent and thought provoking articles on the educational application of System Dynamics to school children without any Calculus background by the System Dynamics in Education Project, led by Jay Forrester himself. In a 1992 article, Forrester criticizes pre-college education as "poorly serving the needs of society", because it does not take into account the complex interactions between people and phenomena. The system dynamics view, Forrester argued, would empower schoolchildren to deal with complexity. Implementation of these ideas in the classroom entails the use of software like STELLA II or Vensim (both commercial packages with GUI's available both for Macintosh and Windows). It would be awesome if there was a such an application for R using the simecol and FME packages as backends for the actual simulation and fitting of SD models, but including a GUI with which schoolchildren could construct such models from Forrester diagrams... Oh! I'm daydreaming again.
There is an excellent series of animated presentations available at the Historyanimated site (see Cagney, 2008). I contacted the author himself, James Cagney, in 2011 for permission to use his material in class and for some class notes I was preparing and thinking about publishing on the web. He told me to add the following interesting note: "The animation for The Battle of Gettysburg (civilwaranimated) is used to train smoke jumpers in the western US (men who parachute from airplanes to fight forest fires) - they tell me that fighting a fire is just like fighting a battle.". So there it is, James!
Iwo Jima, or Iō-tō ("sulphur island" in japanese) was first visited by a Westerner in 1543, a Spaniard by the name of Bernardo de la Torre, who called the island "Isla de Sufre" or Sulphur Island (Sufre was the archaic Spanish spelling for azufre or sulphur). It is an appropriate name, as the southernmost tip of the island hosts a volcanic vent (hence the sulphur) that produces the island's most prominent geographic feature: Mount Suribachi. The fine volcanic ash sands of its beaches would prove a major nuisance for the americans as they tried to disembark amphibious vehicles during their invasion of the island. On account of this sulphur, no other scenario of any World War II battle could better resemble hell and in fact, ancient Japanese legend has it that the island was inhabited by demons.
A man descended from 5 generations of samurai, Koribayashi was also a veteran of the Manchurian and Chinese campaigns. He had previously served as military attaché in Washington during the 1920's, where he acquired extensive knowledge of Americans. About the latter, he wrote: "The U.S.A. is the last country in the world against whom the Japanese should wage war" (Anonymous, 2006).
At this point I would digress in class and talk to my students about how this Pulitzer-winning photograph was an artistic marvel. I would also mention to them that in Venezuela, we also have our own "Joe Rosenthal". His name was Héctor Rondon and his famous picture of a chaplain helping a wounded Venezuelan soldier in a street corner, amidst friendly and unfriendly crossfire, also won the Pulitzer Prize in 1963 and was featured in LIFE Magazine.
The events surrounding this "Pietá"-like picture were those of the Porteñazo, a military coup led by communist rebels backed by Cuba against the democratically elected president Romulo Betancourt in 1962. It is ironic that while today, the Chavista propaganda talks about US imperialism and intervention in our country before the so-called Socialist Revolution, the Cuban regime had tried on several ocassions to export their revolution to Venezuela (El Porteñazo was one such incident in which over 100 Venezuelan soldiers died). That was until Hugo Chavez came along and peacefully handed control of this country to the Cuban Castro regime without a single shot being fired. At any rate I leave you a small print of the photograph for your viewing.
On the subject of Japanese holdouts, there is the noteworthy case of Hiroo Onoda, who did not surrender in 1945 and spent 30 years in the jungles of the Philipines (if you didn't know about Onoda, look him up. It's one hell of a story). An english progressive rock band called Camel made a concept album called Nude inspired upon the true story of Hiroo Onoda (the name Nude derives from his family name Onoda).
Bibliographical References
ANONYMOUS. (2006). "Iwo Jima el Test Supremo". Post in online forum. Retrieved March 2011, from http://www.forosegundaguerra.com/viewtopic.php?p=18492
Battle of Iwo Jima. (2016, August 2). In Wikipedia, The Free Encyclopedia. Retrieved August 5, 2016, from https://en.wikipedia.org/wiki/Battle_of_Iwo_Jima
CAGNEY , J. (2008). PacificWarAnimated.com - Historical Animation of the Battle of Iwo Jima. Nonprofit History Animated Foundation. Oregon, US. Retrieved March 2011 from http://pacificwaranimated.com/Iwo.html
ENGEL, J. H. (1954). "A Verification of Lanchester's Law". Journal of the Operations Research Society of America, 2(2), pp. pp. 163–171.
Iwo Jima. (2016, July 31). In Wikipedia, The Free Encyclopedia. Retrieved August 5, 2016, from https://en.wikipedia.org/wiki/Iwo_Jima
LIND, C. (2009). Verifying Lanchester's Combat Model - Battle of Iwo Jima.
MACKAY, N. (2005). "Lanchester Combat Models". Retrieved March, 2011 from http://arxiv.org/pdf/math/0606300.
PETZOLDT, T. and RINKE , K. (2007). "simecol: An Object-Oriented Framework for Ecological Modeling in R". Journal of Statistical Software, 22(9), pp. 1–31. ISSN 1548-7660. http://www.jstatsoft.org/v22/i09.
R DEVELOPMENT CORE TEAM (2009). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. http://www.R-project.org.
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To leave a comment for the author, please follow the link and comment on their blog: U.N.A. Matemáticas El Tigre. | CommonCrawl |
Complex numbers $a,$ $b,$ $c$ form an equilateral triangle with side length 18 in the complex plane. If $|a + b + c| = 36,$ find $|ab + ac + bc|.$
Note that given complex numbers $a$ and $b$ in the plane, there are two complex numbers $c$ such that $a,$ $b,$ and $c$ form an equilateral triangle. They are shown as $c_1$ and $c_2$ below.
[asy]
unitsize(1 cm);
pair A, B;
pair[] C;
A = (2,-1);
B = (0,0);
C[1] = rotate(60,B)*(A);
C[2] = rotate(60,A)*(B);
draw(C[1]--A--C[2]--B--cycle);
draw(A--B);
label("$a$", A, SE);
label("$b$", B, NW);
label("$c_1$", C[1], NE);
label("$c_2$", C[2], SW);
[/asy]
Then for either position of $c,$
\[\frac{c - a}{b - a}\]is equal to $e^{\pm \pi i/6}.$ Note that both $z = e^{\pm \pi i/6}$ satisfy $z^2 - z + 1 = 0.$ Thus,
\[\left( \frac{c - a}{b - a} \right)^2 - \frac{c - a}{b - a} + 1 = 0.\]This simplifies to
\[a^2 + b^2 + c^2 = ab + ac + bc.\]Then
\[(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc = 3(ab + ac + bc).\]Hence,
\[|ab + ac + bc| = \frac{|a + b + c|^2}{3} = \frac{36^2}{3} = \boxed{432}.\] | Math Dataset |
\begin{document}
{\bf Comment on ``Sonoluminescence as Quantum Vacuum Radiation''}
We contest the recent claim that sonoluminescence may be explained in terms of quantum vacuum radiation \cite{CE1}. Due to fundamental physical limitations on bubble surface velocity, the predicted number of photons per flash is indeed much smaller than unity. Therefore, quantum vacuum radiation cannot be considered as an explanation of the observed sonoluminescence phenomenon.
As the starting point of our critical comment we will use the theoretical evaluation of radiation emitted into vacuum by a moving spherical interface between two dielectric media \cite{CE1,CE2}. In addition to the results presented in these papers we will calculate the number of photons emitted in each sonoluminescence cycle by interpreting it in terms of parametric emission. Each elementary radiation process corresponds to the emission of a pair of photons $\left( \omega ,\omega ^{\prime }\right) $ carrying an energy $\hbar (\omega +\omega ^{\prime })=\hbar \Omega $, where $\Omega $ is the frequency of the mechanical motion. Equation (10) of \cite{CE1} or equation (4.8) of \cite{CE2}, which give the energy radiated per pulsation of period $T$, are thus translated into the following expression of the number of radiated photons \begin{equation}
{\cal N}=\alpha \int_0^\infty {\rm d}\Omega \ \Omega ^5\ \left| \int_0^T{\rm
d}\tau \frac{R^2(\tau )}{c^2}e^{i\Omega \tau }\right| ^2 \label{NQR} \end{equation} $\alpha $ is a numerical factor ($\alpha \simeq 10^{-4}$ for a water-air interface), and $R(\tau )$ is the time-dependent radius.
An estimate of ${\cal N}$ may be derived from a simple lorentzian model (eq. (4.9) of \cite{CE2} with the same notations for $R_0$, $R_{\min }$ and $\gamma$) \begin{eqnarray} &&{\cal N}=\frac{15\pi ^2}{16}\alpha \beta ^4 \nonumber \\ &&\beta ^2=\frac{R_0^2-R_{\min }^2}{c^2\gamma ^2} \label{N} \end{eqnarray} The number of photons thus depends on a single parameter $\beta $ which is the typical velocity of bubble surface measured with respect to the speed of light. The time $\gamma $ characteristic of bubble collapse, which is assumed to be much shorter than $T$, determines the frequency spectrum of radiation but does not appear in the integrated photon number. A study of the law of variation of velocity then shows that the photon number (\ref{N}) may not exceed a value determined by the maximal velocity $\left( \frac{{\rm d}R}{{\rm d}t}\right) _{\max }$ of the bubble surface \begin{equation} {\cal N} \leq .1 \left( \frac{{\rm d}R}{c{\rm d}t}\right) _{\max }^4 \label{Nmax} \end{equation} {}For any maximal velocity smaller than the speed of light, the photon number (\ref{N}) therefore remains smaller than unity. In contrast, observed sonoluminescence corresponds to more than $10^5$ photons per flash (see for instance Fig.3 of \cite{sono1}).
Moreover, the characteristic velocity of the bubble surface corresponds to the sound velocity in air or water (see Fig.4 of \cite{sono2}) rather than to the speed of light. For a maximal velocity equal to sound velocity in water, ${\cal N}$ comes out as a very small number of the order of $10^{-23}$ photons. We may emphasize that, with a radius in the $\mu {\rm m}$ range (see Fig.3 of \cite{sono2}), a velocity of the order of $1{\rm km/s}$ leads to a typical time in the ${\rm ns}$ range, which fits experimental reports (see again Fig.3 of \cite{sono2}). A much shorter time scale is used in \cite{CE1} in order to explain the short duration of the flashes and the large width of the thermal-like spectrum. This very short time scale not only differs considerably from the measured values, but it also corresponds to supraluminal velocities.
The precise values of the numerical factors which play a role in the preceding discussion have been obtained from a specific model for the variation of the bubble radius. However, it can hardly be expected that a different variation law of bubble radius may fill the gap between $10^{-23}$ and $10^5$ photons. The predicted number of photons per flash remains much smaller than unity, which unavoidably leads to contest the claim that ``the theory of vacuum radiation seems to agree remarkably well with the experimental results on sonoluminescence'' \cite{CE1}.
\noindent Astrid Lambrecht\\ Max-Planck-Institut f\"{u}r Quantenoptik\\ D-85748 Garching, Germany\\
\noindent Marc-Thierry Jaekel\\ Laboratoire de Physique Th\'{e}orique de l'Ecole Nor\-ma\-le Su\-p\'{e}\-rieu\-re, F-75231 Paris Cedex 05, France\\
\noindent Serge Reynaud\\ Laboratoire Kastler Brossel, Universit\'{e} Pierre et Marie Curie, F-75252 Paris Cedex 05, France\\
\noindent LPTENS 96/40
\noindent PACS: 78.60.Mq, 03.70+k, 42.50.Lc
\begin{references} \bibitem{CE1} C. Eberlein, {\sl Physical Review Letters} {\bf 76} 3842 (1996)
\bibitem{CE2} C. Eberlein, {\sl Physical Review} {\bf A53} 2772 (1996)
\bibitem{sono1} R. Hiller, S.J. Putterman and B.P. Barber, {\sl Physical Review Letters} {\bf 69} 1182 (1992)
\bibitem{sono2} R. L\"{o}fstedt, B.P. Barber and S.J. Putterman, {\sl Physics of Fluids} {\bf A5} 2911 (1993) \end{references}
\end{document} | arXiv |
Phonon Conduction in Silicon Nanobeam Labyrinths
Far-field coherent thermal emission from polaritonic resonance in individual anisotropic nanoribbons
Sunmi Shin, Mahmoud Elzouka, … Renkun Chen
Selective excitation and imaging of ultraslow phonon polaritons in thin hexagonal boron nitride crystals
Antonio Ambrosio, Michele Tamagnone, … Federico Capasso
Controlled strong excitation of silicon as a step towards processing materials at sub-nanometer precision
Thanh-Hung Dinh, Nikita Medvedev, … Masaharu Nishikino
Sub-diffractional cavity modes of terahertz hyperbolic phonon polaritons in tin oxide
Flávio H. Feres, Rafael A. Mayer, … Ingrid D. Barcelos
Nanoscale mapping of optically inaccessible bound-states-in-the-continuum
Zhaogang Dong, Zackaria Mahfoud, … Joel K. W. Yang
Direct observation of highly confined phonon polaritons in suspended monolayer hexagonal boron nitride
Ning Li, Xiangdong Guo, … Peng Gao
Reshaping the phonon energy landscape of nanocrystals inside a terahertz plasmonic nanocavity
Xin Jin, Andrea Cerea, … Luca Razzari
Prominent radiative contributions from multiply-excited states in laser-produced tin plasma for nanolithography
F. Torretti, J. Sheil, … J. Colgan
Strain Engineering of Germanium Nanobeams by Electrostatic Actuation
Arman Ayan, Deniz Turkay, … Selcuk Yerci
Woosung Park1,
Giuseppe Romano2,
Ethan C. Ahn3,4,
Takashi Kodama1,
Joonsuk Park5,
Michael T. Barako ORCID: orcid.org/0000-0002-4745-45151,
Joon Sohn3,
Soo Jin Kim6,
Jungwan Cho1,7,
Amy M. Marconnet ORCID: orcid.org/0000-0001-7506-28888,
Mehdi Asheghi1,
Alexie M. Kolpak2 &
Kenneth E. Goodson1
Nanowires
Here we study single-crystalline silicon nanobeams having 470 nm width and 80 nm thickness cross section, where we produce tortuous thermal paths (i.e. labyrinths) by introducing slits to control the impact of the unobstructed "line-of-sight" (LOS) between the heat source and heat sink. The labyrinths range from straight nanobeams with a complete LOS along the entire length to nanobeams in which the LOS ranges from partially to entirely blocked by introducing slits, s = 95, 195, 245, 295 and 395 nm. The measured thermal conductivity of the samples decreases monotonically from ~47 W m−1 K−1 for straight beam to ~31 W m−1 K−1 for slit width of 395 nm. A model prediction through a combination of the Boltzmann transport equation and ab initio calculations shows an excellent agreement with the experimental data to within ~8%. The model prediction for the most tortuous path (s = 395 nm) is reduced by ~14% compared to a straight beam of equivalent cross section. This study suggests that LOS is an important metric for characterizing and interpreting phonon propagation in nanostructures.
Understanding phonon transport is crucial to engineering novel nanoscale systems, such as thermal energy storage1,2,3, heat-assisted memory4, 5, sensors6, 7, and integrated nanoelectronics8, 9. Previous studies of phonons in single crystalline silicon have established an increasingly comprehensive microscopic model of how phonons interact with introduced nanoscale features and boundaries10. In many nanostructures, e.g., nanobeams and nanowires, a key geometric feature is an unobstructed line-of-sight (LOS) for propagation through the medium. In these nanostructures, the boundaries are aligned along the dominant direction of heat flow and the LOS along the dominant direction of heat flow is typically much longer than the phonon mean free paths (MFPs)11,12,13,14,15. For the case of nanostructures with boundaries or interfaces that are oriented normal to the direction of heat flow, much progress has been made for superlattices and related nanostructures and has illustrated the impact of ballistic and even coherent transport16,17,18,19,20. However, the interplay of these two types of interfaces in practical structures remains both insufficiently understood and a barrier to the effective simulation of many practical and applied nanostructures.
In complex nanostructures, phonon scattering off longitudinal boundaries and obstructions (i.e. scattering sites oriented normal to the net heat flux vector) reduces the thermal conductivity. For example, thin films containing a high-density of nanoscale holes show a dramatic reduction in the in-plane thermal conductivity13, 21,22,23. These silicon structures contain a limited LOS path between adjacent holes in a given row in the direction of heat flow and a continuous LOS path between neighboring rows of holes in the direction of heat flow13, 21, 24. The contribution of the limited LOS path to thermal conductivity is not clearly understood, which makes it difficult to determine the impacts of the placement25,26,27, separation13 and size28 of holes on thermal transport. While previous work has explored the impact of forward boundary scattering29, the relevant physics is still elusive for phonons scattering off forward obstructions.
Here we demonstrate the impact of the limited LOS in silicon nanobeams that feature deliberately engineered forward obstructions along the direction of a net heat flow. We refer to these tortuous beams as labyrinths, which originates from the maze structure designed to hinder the escape of the mythological Minotaur. In nanostructured labyrinths, ballistic phonons are spatially confined by the boundaries. The baseline reference is a thin (thickness t = 80 nm) and narrow (width w = 470 nm) silicon beam that is 10 µm long, and phonons can travel unimpeded along the major axis of the beam. We introduce obstructions along this beam axis by adding offset slits to disrupt the thermal transport path (see Fig. 1). The slits are cut into the beam from the edge to increase the phonon scattering rate and further limit the maximum phonon axial-travel length. This effectively eliminates any continuous LOS at the point when the adjacent slits overlap each other. We fabricate six different samples, including a reference silicon nanobeam without slits and five nanobeams with slits that vary in width s from 95 nm to 395 nm, as shown in Fig. 1(a–f).
(a–f) Magnified view of nanostructured samples with various widths of slits up to 395 nm as marked on the images. (g) Schematic of a sample unit cell defining critical geometric dimensions. (h–m) Heat flux magnitude of the samples. The heat flux is obtained by solving the Boltzmann transport equation. The heat flux is normalized per sample and shown in the legend.
The experimental structures are patterned between hot and cold platforms, and are suspended for thermal characterization as shown in Fig. 2. Heat is generated in one island and conducts through the sample and the supporting beams, and temperature is measured in each platform to measure the thermal conductance of the sample30, 31. We extract the thermal conductivity of the samples by fitting the measured thermal conductance of the samples to the numerically calculated values using a finite element method (solved using COMSOL). The numerical simulation solves a three-dimensional heat equation that accounts for the volumetric change in thermal conductance between samples, assuming constant and homogeneous properties within the entire beams. We note that the finite element method thereby provides a reference, from the diffusive limit, for the impacts of geometry on transport, and the thermal conductivity allows us to examine departures from diffusive theory separately from geometry. All of the measurements are performed under vacuum to minimize convective heat losses from the beams, and the heat loss through radiation is considered negligible compared to the heat conduction through the samples at all temperatures in the study31. The thermal interfacial resistances between the sample and the platforms are negligible since the devices are monolithically fabricated, which minimizes contact resistances13, 22, 32. The uncertainty is primarily due to the measurement of the sample dimensions, and the uncertainty propagation is available in Supplementary Information.
Scanning electron microscope (SEM) images of the measurement structure with false-colored serpentine heater/thermometers. The SEM image is rotated by 45 degrees about the major beam axis. The residual shadow below the beam confirms that it is fully suspended over the substrate.
For data interpretation, it is helpful to categorically divide the samples into two categories, those having a continuous LOS (Fig. 1(a–c)) and those with a blocked LOS (Fig. 1(d–f)). In the continuous LOS category, an open channel exists through the center of the major beam axis between heat source and heat sink. In the blocked LOS category, the maximum phonon travel distance is dictated by the forward spacing between the boundaries, where the maximum phonon travel distance is comparable to phonon MFPs, (i.e. partially ballistic transport). The heat flux maps are obtained by solving the Boltzmann transport equation and clearly show the different LOS channels as seen Fig. 1(h–m). We also define the coordinate system in Fig. 1(g), where the x- and y-axes are defined with respect to the axial direction of the beam.
As slit width s increases, the phonon transport reduction mechanism changes from a constricted but continuous LOS to a forcibly tortuous heat flow and fully blocked LOS. The measured thermal conductivity of the samples decreases monotonically from ~47 W m−1 K−1 to ~31 W m−1 K−1 with increasing s as shown in Fig. 3. The model prediction described below shows a sharp change in thermal conductivity at transition from continuous LOS along the entire beam to blocked LOS by geometry (i.e., s = 235 nm). A combination of experimental results and the modeling prediction indicates that phonon suppression mechanisms between the continuous and blocked LOS categories are dissimilar. In the continuous LOS samples, the central open channel through the beams is the primary heat transport path as seen in Fig. 1(h–j). With decreasing width of the LOS, heat flux through the LOS channel is continuously reduced, and phonon traveling in x-direction is locally induced. Possible phonon trajectories can be grouped into two cases: phonons propagating through the central LOS and phonons experiencing additional boundary scattering between adjacent slits in y-axis. A fraction of these two cases is controlled with the slit width s, and this modulation results in monotonically reduced thermal conductivity with increasing s. In the blocked LOS samples, the LOS is limited by geometry, and the tortuous thermal path forces heat flow to change direction between x- and y-axis. The change of heat flow along the thermal path is shown in Fig. 1(k–m). In this regime, phonon propagation experience additional scattering from boundaries normal to the direction of predominant heat flow. The reduction in thermal transport is due to both spatially confined cross-section and the limited longitudinal dimension along dominant heat flow. The samples in the blocked LOS can be considered as a series of alternating thermal resistors, ones aligned in x direction and the other aligned in y direction. With decreasing slit width s, the cross-section of the resistors aligned with in y-axis is continuously reduced, and this transverse constriction further suppresses thermal conductivity in the blocked LOS. Also, each resistor has a limited LOS in a direction of heat flow, and this limited length scale impacts on phonon propagation through the beam.
Thermal conductivity k of the nanostructured silicon samples with varying slit width s. The blue solid line is a model prediction that is obtained by solving Boltzmann Transport equation.
We investigate the impact of limited LOS in a direction of the dominant heat flow by comparing the tortuous beam with s = 395 nm and a straight beam with an equivalent cross-section and length. The tortuous beam with s = 395 nm has a cross-sectional area of 75 nm × 80 nm cross-sectional area throughout the samples. The thermal conductivity for the straight beam with a cross-section of 75 nm × 80 nm is predicted to be 37.6 W m−1 K−1 using the model as described below. The tortuous beam with s = 395 nm are estimated to be 33.1 W m−1 K−1 and ~31 W m−1 K−1 from a model prediction and experiments results, respectively. Both estimations show ~14% and ~18%, respectively, smaller conductivity values than the prediction for the straight beam despite sharing the same cross-sectional area. This discrepancy of the thermal conductivity between these two geometries isolates the contribution from the limited LOS. Compared to the straight beam with w = 75 nm, the tortuous beam with s = 395 nm has 130 additional scattering corner sites. Following diffuse scattering from the boundary within the structures, these phonons scatter in the opposite direction of the dominant heat flow, resulting in a reduction of MFP22.
We model thermal transport in the nanostructured samples by solving the steady-state, phonon Boltzmann transport equation under the relaxation time approximation33. The thermal conductivity is described using a combination of bulk MFPs and the suppression function as given by34
$$k={\int }_{0}^{\infty }{\int }^{}{S}_{{\rm{\Lambda }}}({\rm{\Lambda }},{\rm{\Omega }})f({\rm{\Lambda }})d{\rm{\Omega }}d{\rm{\Lambda }}$$
where f is the differential MFP distributions for bulk silicon, S Λ is the directional phonon suppression function, Ω is a solid angle, and Λ is an integration variable for MFP of bulk silicon. The suppression function S Λ(Λ, Ω) describes the suppression of heat carried by phonons with MFP Λ and direction Ω with respect to Fourier's law prediction24. The directional suppression function is computed by the Boltzmann transport equation as described in Method section. The calculation in the present work uses MFPs for a bulk medium as input, where the bulk MFP spectra is obtained from ab initio calculations35. The calculation of the Boltzmann transport equation predicts thermal conductivity with varying s, and the prediction agrees with experimental results to within ~8%. This indicates that phonon suppression mechanisms are captured by the calculation of the Boltzmann transport equation, validating the use of diffuse scattering boundary conditions at room temperature36.
Based on the calculation of Boltzmann transport equation, we calculate the accumulated thermal conductivity with bulk MFPs to obtain a MFP-specific understanding of boundary scattering as a function of characteristic length scales. We compare three different cases: a thin film with thickness t = 80 nm, a straight beam with t = 80 nm and width w = 75 nm, and a tortuous beam with t = 80 nm, w = 75 nm, and s = 395 nm as shown in Fig. 4(a). The comparison among these cases shows the individual impact of respective dimensions: thickness t, width w, and the LOS. The accumulated thermal conductivity is
$$k({{\rm{\Lambda }}}_{bulk})={\int }_{0}^{{{\rm{\Lambda }}}_{bulk}}{\int }^{}{S}_{{\rm{\Lambda }}}({\rm{\Lambda }},{\rm{\Omega }})f({\rm{\Lambda }})d{\rm{\Omega }}d{\rm{\Lambda }}$$
where Λ bulk is phonon MFPs of bulk silicon. The function for thin film is calculated from the aforementioned model with a periodic boundary condition along x-axis. The accumulation functions show the impact of limiting dimensions, thickness t, width w, and the limited LOS in the dominant heat flow direction. The accumulation functions are reduced from bulk to the thin film, from the thin film to the straight beam, from the straight beam to the tortuous beam with s = 395 nm, and these corresponds to the impact of thickness, width, and the length of LOS, respectively. The accumulation functions for the thin film and the beam start to deviate from those for the bulk and the thin film, respectively, near t/3 = ~25 nm. The accumulation function for the sample with s = 395 nm shows suppression from a third longitudinal length, ~ 75 nm. The accumulation clearly shows the transition regime where boundary scattering becomes significant, and it is found that the limited LOS contributes to the reduction in thermal conductivity by suppressing long MFPs phonons.
(a) Schematics for simulations, which includes a thin film, a straight beam, and a tortuous beam. (b) Thermal conductivity accumulation with bulk MFP Λ bulk . The blue arrow indicates the MFPs, where a direction suppression function is shown at marked figures (c,d). Direction suppression function for (c) Λ bulk = ~9 nm, (d) Λ bulk = ~100 nm, and (e) Λ bulk = ~900 nm. Heat flows from ϕ = 90 to ϕ = 270. These suppression functions are normalized with the value of a thin film, and the relative magnitude is shared among (c–e).
To further investigate the impact of limited LOS, we use the directional suppression function, which provides angular information of suppressed heat flux for a given geometry. Since the samples are planar structures, we plot the directional suppression function on x-y plane, and the suppression function is defined as,
$${{\rm{S}}}_{{\rm{\Lambda }}}({\rm{\Lambda }},\varphi )=\frac{1}{2}{\int }_{0}^{\pi }{S}_{{\rm{\Lambda }}}({\rm{\Lambda }},\theta ,\varphi )sin(\theta )d\theta $$
where the function is integrated over the azimuthal angle θ. We note that a conventional suppression function SΛ is proportional to only the magnitude of heat flow, and the integration of SΛ(Λ, ϕ) over ϕ provides the ratio of the MFP dependent heat flow to the counterpart of the bulk. The directional suppression function SΛ is comprised of two symmetric lobes pointing toward +y and −y directions, and SΛ vanhishes along the x-axis since there is no net heat flux for ϕ = 0 and 180 degrees. The suppression functions are compared at MFP = ~9 nm, ~100 nm, and ~900 nm, as shown in Fig. 4(b–d). The MFPs are chosen to represent three cases, internal scattering process dominant region (diffusive regime), the transition from internal scattering to boundary scattering (quasi-ballistic regime), and boundary scattering dominant region (nearly ballistic regime). While the suppression functions for the case of MFP = ~9 nm are overlapped among all the geometries due to the relative dominance of internal scattering, the suppression functions for MFP = ~900 nm clearly shows the impact of each dimension. The suppression function is normalized with the case of thin films to show the individual impact of each limiting dimension. For the cases of MFP ~100 nm and ~900 nm, the lobe for the straight beam is narrowed in compared to that of thin films, and this change indicates the impact of the limited width, w = 75 nm. The longitudinal heat flow of the tortuous beam with s = 395 nm is further suppressed relative to the straight beams due to forward boundaries, and this impact is displayed in reduction of heat flux along y-axis for MFP ~100 nm and ~900 nm. The suppression function analysis suggests that the impact of limited LOS becomes relevant when the distance that phonons travel is comparable to the MFP. With diffuse boundary scattering, this length scale can be considered as a limiting dimension, such as the diameter of a nanowire. These findings suggest that controlling the longitudinal dimension along heat flow can manipulate phonon propagation, and the relative contribution of limited LOS to thermal conductivity is important because it reduces the length of longitudinal free path.
The combination of experiments and the Boltzmann transport equation calculations provides a quantitative description of the impact of conduction blocking the LOS channels along the direction of dominant heat flow. Tortuous thermal paths suppress the effective phonon MFP due to longitudinal features, and the suppression in heat flux is stronger than that from the cross-sectional confinement alone. This provides guidance to the design of nanostructures that achieve a lower limit to thermal conductivity. The suppression of phonon propagation using this limited LOS approach can be useful for thermoelectric applications. While previous studies of nanostructured thermoelectric materials show reduced thermal conductivity by increasing scattering events, such spatial confinement also causes simultaneous reduction of electrical conductivity, which degrades the figure or merit in applications such as thermoelectrics14, 21, 37. Engineering thermal paths enables manipulation of energy carriers by suppressing phonon thermal conductivity more efficiently with constant cross-sectional area, which aids in decoupling phonons and electrons into ballistic and diffusive regimes, respectively. This work improves the understanding of phonon conduction in modern devices where both ballistic and diffusive phonon transports are present since the length scales of all three spatial dimensions are comparable to the phonon MFP, such as in fin field effect transistors (FinFETs).
Sample Fabrication
We fabricate silicon nanobeams on silicon-on-insulator (SOI) wafers (Soitec Inc.) comprised of 340 nm-thick silicon device layer and a 1 µm-thick buried oxide (BOX) layer and patterned using electron beam lithography. First, the device layer is thinned down to ~80 nm via thermal oxidation and subsequent removal of the oxide using buffered oxide etchant (BOE). The reduced thickness of a device layer is measured using an optical ellipsometer and confirmed using transmission electron microscopy (TEM) imaging. We deposit ~25 nm-thick Al2O3 using an atomic layer deposition (ALD) for electrical passivation between the metal contacts and the device layer, and the 10 µm by 20 µm sample region window between the two islands is opened by etching Al2O3 using a combination of reactive ion etching (RIE) and wet etching using BOE. We pattern the nanobeams and slits using electron beam lithography, and RIE is used to remove the silicon to form the slits. The metal heater lines on the islands and the contact pads for electrical access are patterned using a combination of electron beam- and photo-lithography, and a lift-off process is followed after deposition of ~5 nm-thick Cr and ~40 nm-thick Pt using electron beam metal evaporation. The surrounding area of the islands and legs are etched using RIE, and they are suspended using gaseous hydrogen fluoride (HF) etching. All dimensions are measured using scanning electron microscopy (SEM) after fabrication and TEM is used to measure the thickness of silicon beam.
Boltzmann Transport Equation Calculation
The computational domain is a unit-cell of a beam as shown in Fig. 1(g), and the unit cell is a beam of width w = 470 nm, length l unitcell = 300 nm and thickness t = 80 nm. We solve the Boltzmann transport equation (BTE) with varying s from 0 to 395 nm as denoted in Fig. 1(a–f). Periodic boundary conditions are applied on the boundaries along the y-axis with a temperature difference of ΔT = 1 K, and diffuse boundary scattering is assumed for the other boundaries. The BTE solver uses the MFP spectra as only input38, which is computed by ab initio calculations35. The BTE is solved in space by the finite-volume method39 while the angular space is discretized by means of the discrete ordinate method. Specifically, we achieve convergence in the thermal conductivity when the BTE is solved for 100 MFPs, 96 polar angles (defined on the x-y plane) and 24 azimuthal angles. For the MFPs smaller than ~2 nm, we employ a diffusive equation, which enables accurate simulations and seamless integration with the BTE with a reasonable spatial discretization.
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The authors thank Keivan Esfarjani for sharing the MFP spectra, Aditya Sood for useful discussion, and Eric Pop for critical readings for the manuscript. This work is supported by the National Science Foundation under grant No. 1336734. Work at MIT was supported by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences (BES), Materials Sciences and Engineering Division under Award DE-SC0001299/DE-FG02-09ER46577. Sample preparation and characterization were performed at the Stanford Nano Shared Facilities (SNSF), supported by the National Science Foundation under award ECCS-1542152. M.T. B. appreciates support from the United States Department of Defense, Air Force Office of Scientific Research, National Defense Science and Engineering Graduate (NDSEG) Fellowship, 32 CFR 168a.
Department of Mechanical Engineering, Stanford University, Stanford, CA, 94305, USA
Woosung Park, Takashi Kodama, Michael T. Barako, Jungwan Cho, Mehdi Asheghi & Kenneth E. Goodson
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA
Giuseppe Romano & Alexie M. Kolpak
Department of Electrical Engineering, Stanford University, Stanford, CA, 94305, USA
Ethan C. Ahn & Joon Sohn
Department of Electrical and Computer Engineering, The University of Texas at San Antonio, San Antonio, TX, 78249, USA
Ethan C. Ahn
Department of Materials Science and Engineering, Stanford University, Stanford, CA, 94305, USA
Joonsuk Park
Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA, 94305, USA
Soo Jin Kim
Department of Mechanical Engineering, Kyung Hee University, Yongin-si, 446-701, South Korea
Jungwan Cho
School of Mechanical Engineering, Purdue University, West Lafayette, Indiana, 47907, USA
Amy M. Marconnet
Woosung Park
Giuseppe Romano
Takashi Kodama
Michael T. Barako
Joon Sohn
Mehdi Asheghi
Alexie M. Kolpak
Kenneth E. Goodson
W.P., A.M.M., and K.E.G. conceived the ideas for the project. W.P. designed and fabricated the experimental devices with help of E.C.A., T.K., J.S., and S.K. W.P. performed the thermal conductivity measurements. G.R. performed Boltzmann transport model calculations under supervision of A.M.K., W.P. and J.P. took scanning electron microscope. W.P. and M.T.B. wrote the main manuscript text. J.C., A.M.M., M.A., and K.E.G. commented on the manuscript. M.A. and K.E.G. guided the projects.
Correspondence to Kenneth E. Goodson.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Park, W., Romano, G., Ahn, E.C. et al. Phonon Conduction in Silicon Nanobeam Labyrinths. Sci Rep 7, 6233 (2017). https://doi.org/10.1038/s41598-017-06479-3
Received: 13 March 2017
Suppressed phonon conduction by geometrically induced evolution of transport characteristics from Brownian motion into Lévy flight
Yongjoon Kim
NPG Asia Materials (2022)
Structural optimization of silicon thin film for thermoelectric materials
Takuma Hori | CommonCrawl |
Global attractors of impulsive parabolic inclusions
Strong trajectory and global $\mathbf{W^{1,p}}$-attractors for the damped-driven Euler system in $\mathbb R^2$
July 2017, 22(5): 1857-1873. doi: 10.3934/dcdsb.2017110
Smooth attractors for weak solutions of the SQG equation with critical dissipation
Michele Coti Zelati 1,, and Piotr Kalita 2,
Department of Mathematics, University of Maryland, College Park, MD 20742, USA
Faculty of Mathematics and Computer Science, Jagiellonian University, 30-348 Kraków, Poland
* Corresponding author: Michele Coti Zelati
Received December 2015 Revised February 2016 Published March 2017
We consider the evolution of weak vanishing viscosity solutions to the critically dissipative surface quasi-geostrophic equation. Due to the possible non-uniqueness of solutions, we rephrase the problem as a set-valued dynamical system and prove the existence of a global attractor of optimal Sobolev regularity. To achieve this, we derive a new Sobolev estimate involving Hölder norms, which complement the existing estimates based on commutator analysis.
Keywords: Surface quasi-geostrophic equation, critical dissipation, global attractors, Sobolev regularity.
Mathematics Subject Classification: Primary: 35Q35, 35B41; Secondary: 35B4.
Citation: Michele Coti Zelati, Piotr Kalita. Smooth attractors for weak solutions of the SQG equation with critical dissipation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1857-1873. doi: 10.3934/dcdsb.2017110
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Michele Coti Zelati Piotr Kalita | CommonCrawl |
Error analysis in L p ⩽p⩽∞, for mixed finite element methods for linear and quasi-linear elliptic problems
Durán, Ricardo G.
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 22 (1988) no. 3, p. 371-387
MR 958875 | Zbl 0698.65060 | 3 citations in Numdam
@article{M2AN_1988__22_3_371_0,
author = {Dur\'an, Ricardo G.},
title = {Error analysis in $L^p \leqslant p \leqslant \infty $, for mixed finite element methods for linear and quasi-linear elliptic problems},
mrnumber = {958875},
language = {en},
url = {http://www.numdam.org/item/M2AN_1988__22_3_371_0}
Durán, Ricardo G. Error analysis in $L^p \leqslant p \leqslant \infty $, for mixed finite element methods for linear and quasi-linear elliptic problems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 22 (1988) no. 3, pp. 371-387. http://www.numdam.org/item/M2AN_1988__22_3_371_0/
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III. CONTINUOUS AUDIT INTELLIGENCE AS A SERVICE (CAIAAS) ARCHITECTURE
IV. AUDIT APP RECOMMENDATION8
V. TOWARD A CAI-BASED AUDIT PARADIGM10
VI. CHALLENGES
VII. CONCLUSION
editorial| December 28 2020
Continuous Audit Intelligence as a Service (CAIaaS) and Intelligent App Recommendations
Jun Dai;
Jun Dai
Miklos A. Vasarhelyi
Journal of Emerging Technologies in Accounting (2020) 17 (2): 1–15.
https://doi.org/10.2308/jeta-10751
Jun Dai, Miklos A. Vasarhelyi; Continuous Audit Intelligence as a Service (CAIaaS) and Intelligent App Recommendations. Journal of Emerging Technologies in Accounting 1 September 2020; 17 (2): 1–15. doi: https://doi.org/10.2308/jeta-10751
The audit profession is facing a major transition toward a tech-savvy environment, i.e., extensively employing technologies such as data analytics and continuous auditing in daily work. During this transition, one of the biggest challenges is the lack of skilled and experienced auditors who are able to use technologies effectively and efficiently. To solve the problem, this editorial proposes a new architecture, named Continuous Audit Intelligence as a Service (CAIaaS), to facilitate auditors to fully use technologies even with limited experience and knowledge. In the CAIaaS, auditors could capture and transmit their client data to a cloud, and then generate intelligent apps to accomplish specific tasks. Moreover, a recommender system could further suggest the most appropriate apps to use in a particular engagement. The CAIaaS platform and the recommender system, together with other intelligent audit aids, compose a CAI-based audit paradigm that enables semi-automatic app development and recommendations, and result analysis.
continuous auditing, CAIaaS, XaaS, audit apps, recommender systems
Recent advances in technology have increased the digitalization and the acquisition of intelligence of the auditing profession. Auditors are starting to explore IT applications designed for auditing purposes, such as continuous auditing (CA), audit data analytics (ADA), and robotic processing automation (RPA). The purpose is to enable risk identification and evidence collection from the entire population of company data as well as exogenous information, to automate repetitive audit procedures, and to change the frequency of performing audits from annually or quarterly to close to real time. Moreover, the recent vision of the next generation of auditing, named Audit 4.0, largely increases the degree of technology-based assurance (Alles and Gray 2019) by leveraging various emerging technologies that encompass Industry 4.01 upon audit processes (Dai and Vasarhelyi 2016). In Audit 4.0, auditors will "piggyback on technology promoted by Industry 4.0, especially the Internet of Things (IoT), Internet of Service (IoS), Cyber-Physical Systems (CPSs), and smart factories, to collect financial and operational information, as well as other audit-related data from an organization and its associated parties" (Dai and Vasarhelyi 2016). Audit 4.0 links the physical world to a "mirror" world by continuously transmitting the conditions, locations, surrounding environment, etc. of each object in an organization, as well as its business partners, to a virtual model of the value chain. Auditors could then rely on the information collected in a mirror world to build analytic models for anomaly identification and to automate a variety of audit processes such as remote inventory, cash balance evaluation, real-time faults, and irregularity detection.
Although auditors are increasingly aware of the value of intelligent technologies, surveys indicate that the adoption and use of technology are substantially below expectations (EY 2014; KPMG 2015; Li, Dai, Gershberg, and Vasarhelyi 2018). One reason is that the lack of skilled and experienced auditors impedes the transformation (Dai and Vasarhelyi 2016), especially for small audit firms or those in less-developed countries. Human IT skills, along with IT infrastructure and IT reconfigurability, are unique IT resources of a firm (Haseeb, Hussain, Ślusarczyk, and Jermsittiparsert 2019) and are difficult to acquire by small audit firms while being necessary to employ new technologies in audit tasks. To appropriately use technologies, auditors should use professional judgment to determine issues such as what tasks can be supported by those technologies, what data should be collected to build models, and what algorithms/tools are suitable to accomplish the tasks. Due to limited experience, auditors may only analyze portions of data and concentrate on issues they are familiar with or have experienced while ignoring other critical ones (O'Leary 2015). Also, auditors that are new to those data-driven technologies may not be able to create effective models, which could lead to failure of misstatement detection or overwhelming false alerts. Such an outcome may discourage auditors from exploring the beauty of those intelligent applications. Another reason is that large investments in IT infrastructure could hinder small audit firms from the adoption of new technologies (Munoko, Brown-Liburd, and Vasarhelyi 2020).
Solutions for the aforementioned challenges may be inspired by other industries. For example, an IT company offers customized image recognition models to video surveillance system developers by automatically training their images in a comprehensive cloud using built-in algorithms, which allows them to operate models at local devices.2 As a result, the developers without expertise in modeling can simply download the intelligent modules trained in the cloud and add to their video streams while protecting the data privacy and saving bandwidth costs because the incoming video streams are analyzed at their sites. A similar mechanism could be generated to help less-experienced auditors use intelligent applications. In the mechanism, a cloud could serve as the main host for the development and operation of a large number of intelligent apps and charge based on usage. Auditors could upload their clients' data to the cloud and use built-in functions to create intelligent apps and deploy them at clients' sites to perform CA/CM activities. Apps created in the cloud could be later used to help auditors that deal with similar clients.
As the number and variety of apps increase, it would be challenging to choose the right apps for each particular auditor and client, while choosing appropriate tools is crucial to perform effective analyses (Brown-Liburd, Issa, and Lombardi 2015). A potential solution is to rely on the recommender system technique to select the right apps. Recommender systems have been broadly used in ecommerce to predict user preferences on products. Such a technique could be employed to generate customized app suggestions based on client industry, auditor background knowledge, previous experience, familiarity with technology, etc.
This editorial explores the potential of a new architecture, named Continuous Audit Intelligence as a Service (CAIaaS), which could facilitate auditors fully using emerging technologies, even with limited experience and knowledge. In the CAIaaS, IoT is armed with the cloud as well as artificial intelligence, data mining, and machine learning to provide a comprehensive platform that allows auditors to capture and transmit their client data, automatically formulate CA/CM models and generate intelligent apps, and deploy at their own or clients' sites. If an auditor cannot perform model formulation or would like to explore new apps, a recommender system could further help to suggest the most appropriate apps to deploy in a particular engagement. Finally, a CAI-based audit paradigm is presented, which is composed of the CAIaaS platform, the recommender system, and other intelligent audit-aid systems.
The remainder of this editorial proceeds as follows: Section II provides the background of technologies that enable the CAIaaS. Section III presents the key elements and structure of CAIaaS architecture. Section IV improves the architecture by suggesting the most appropriate apps to auditors based on the recommender system technique. A CAI-based audit paradigm is demonstrated and discussed in Section V. The challenges facing the CAI-based audit paradigm adoption and implementation are discussed in Section VI. Section VII concludes and discusses potential research directions
Both academia and practice are exploring the potential use of emerging technologies in the field of auditing. Those technologies include among other topics the Internet of Things (IoT), cloud computing, and recommender systems. Also, several auditing-oriented IT applications, such as continuous auditing and monitoring, are gradually developed and used by audit firms and clients. This section summarizes the technologies and IT applications that are necessary to enable CAIaaS.
Big Data and the Internet of Things
In recent years, the term "Big Data" received increasing attention from the auditing profession because more data have been collected by organizations in the recent two years than in the previous 2,000 years (Syed, Gillela, and Venugopal 2013). One main resource that contributes to Big Data is the information captured or generated from the Internet of Things (IoT). IoT refers to millions of objects that are interconnected with each other via the internet (Xia, Yang, Wang, and Vinel 2012). The objects include sensors (collecting data from the physical world), actuators (receiving commands to take actions that impact the physical world), smartphones, home/work appliances, cars, and any other device or object that can be connected, monitored, or actuated (Biswas and Giaffreda 2014). The rapid development of IoT opens tremendous opportunities for a variety of fields such as living assistance, healthcare, transportation, city development, environment monitoring, agriculture, and manufactory (Wikipedia 2020).
Audit 4.0, the new auditing schemata proposed by Dai and Vasarhelyi (2016), explores various potential uses of IoT to facilitate auditors. By equipping goods with IoT devices, auditors could examine the locations and conditions of inventory items on their computers or smartphones and receive alerts when the sales data in financial statements do not match the actual movement of physical products. IoT could enhance the quality of inventory or assets examination, especially when the inventory or assets are less accessible using traditional audit methods. For example, a Chinese seafood company explained that the repeatedly dramatic changes in its financial statements were because a large number of scallops fled from its farms. It was later proved to be a major fraud of the company (Xinyue and Wei 2019). The examination of seafood inventories underwater is difficult. Such fraud could be instantly detected if IoT devices are placed in the farms monitoring the quantity and conditions of scallops.3 Another example is using IoT air quality monitors to continuously audit government officials' performance on air protection (Dai, He, and Yu 2019). Citizens, as well as other interested parties, can collect real data about their surrounding air quality using IoT monitors and report air pollution cases in real time and alert government auditors and officials to take appropriate actions.
The recent trend of combining IoT and artificial intelligence (AI), data mining (DM), and machine learning (ML) with cloud computing technology opens a new horizon for real-time data collection and analysis (Bacciu, Chessa, Gallicchio, and Micheli 2017; Biswas and Giaffreda 2014). The National Institute of Standards and Technology defines cloud computing as "a model for enabling ubiquitous, convenient, on-demand network access to a shared pool of configurable computing resources (e.g., networks, servers, storage, applications, and services) that can be rapidly provisioned and released with minimal management effort or service provider interaction" (Mell and Grance 2011). Cloud computing can provide elasticity and scalable computing resources, especially data storage and computing power, with minimal management effort by users, and can be accessed from anywhere anytime (Biswas and Giaffreda 2014). Audit firms can outsource the management of IT infrastructures to the cloud to save IT costs and labor. The cloud serves as a remote workstation with a large data center and a variety of auditing software, while it provides scalable IT solutions that allocate more computing and storage resources to auditors during peak time. The cloud plays an even more important role if auditors are armed with AI/DM/ML and IoT. For example, a recent dynamic audit solution (DAS) project4 launched by the AICPA and CPA.com encourages audit firms to develop data analytics and machine learning applications on a cloud. Operating AI/DM/ML algorithms upon an enormous amount of data collected from IoT devices and other sources, as well as providing fast discernment of risky or erroneous transactions from normal ones, requires substantial and specialized IT resources. The cloud can offer specialized hardware for AI/DM/ML operations such as GPU for intensive workloads, state-of-the-art analytical software, and a large space to store and analyze a very large amount of data.
Recommender Systems5
With the growth of the ecommerce market, companies seek a tool to accurately present customers with those products they are most likely to purchase. Recommender systems fulfill this task by predicting the preference a user would give to a certain item based on the features of the product or the user's social environment (Ricci and Shapira 2011). Recommender systems collect customers' preferences in the past, demographic information of users, or the attributes of items, and make suggestions based on those data. Such information can be obtained explicitly (by collecting user ratings, comments, etc.) or implicitly (by monitoring user behavior, such as websites visited and goods purchased) (Choi, Yoo, Kim, and Suh 2012; Lee, Cho, and Kim 2010).
Based on the type of underlying filtering algorithms, recommender systems generally can be divided into four categories: (1) demographic, (2) content-based, (3) collaborative, and (4) hybrid (Adomavicius and Tuzhilin 2005; Bobadilla, Ortega, Hernando, and Gutiérrez 2013). Demographic-filtering recommender systems (Krulwich 1997; Pazzani 1999) predict customers' preferences from the opinions of people who have similar demographic characteristics (such as gender, age, educational level, region, etc.). Content-based-filtering (CBF) recommender systems choose products for customers that are similar to those they preferred in the past (Lang 1995; Mooney and Roy 2000). They usually generate recommendations using the contents of the items such as category, production date, or even more complex information like textual descriptions. Collaborative-filtering (CF) recommender systems analyze customer ratings and suggest the items that are preferred by people with similar tastes. For example, the GroupLens recommender system (Resnick, Iacovou, Suchak, Bergstrom, and Riedl 1994) uses customer ratings to estimate their preferences and clusters users with similar preferences into groups. Hybrid recommendation systems combine the aforementioned methods to provide optimal performance (Burke 1999).
Continuous Auditing and Monitoring
The concept of CA was first proposed in academia, and its earliest application was developed for a corporate billing system (Vasarhelyi and Halper 1991) in 1991. Later, CA evolved into a much broader concept named "continuous assurance" (Vasarhelyi, Alles, and Williams 2010), which consists of three main components: continuous data assurance (CDA) (Kogan, Alles, Vasarhelyi, and Wu 2014), continuous controls monitoring (CCM) (Alles, Brennan, Kogan, and Vasarhelyi 2006), and continuous risk monitoring and assessment (CRMA) (Moon and Krahel 2020), providing assurance close to real time. CDA continuously and automatically executes transaction-level verification to provide timely assurance (Kogan et al. 2014). CCM monitors internal control activities for violations (Alles et al. 2006; Chan and Vasarhelyi 2011). CRMA focuses on identifying emerging and material risks and prioritizing audit and risk management control procedures (Moon and Krahel 2020). These components provide comprehensive, timely, and accurate assurance and preemptively address significant risks.
The use of continuous auditing (CA) and continuous monitoring (CM) in organizations is increasing with the recent advancements in technologies. For example, Curtis, Chui, and Pavur (2020) investigate the factors that influence management accountants' intentions to champion the adoption of CM in their organizations from the individual, organizational, and innovation-specific perspectives. Acar, Gal, Öztürk, and Usul (2020) demonstrate the use of a company's flowcharts to identify control points in business processes and therefore improve its continuous monitoring and auditing activities. Moon and Krahel (2020) propose a novel methodology that continuously monitors and assesses an organization's business risks using internal and external real-time data sources. O'Leary (2020) explores the use of signal theory to understand the characteristics of signals and their impacts on CA/CM systems. Codesso, Machado de Freitas, Wang, de Carvalho, and da Silva Filho (2020) illustrate a real-life implementation of a CA system for the tax processes at a large manufacturing and retail company. Eulerich, Georgi, and Schmidt (2020) provide empirical evidence showing factors that have significant impacts on the use of CA outcomes when internal auditors perform rule-based audit planning.
One of the main challenges in technology adoption by the auditing profession is the need for IT that supports very large data storage and fast analyses, as well as experts that can provide professional help on how to obtain optimal solutions. Auditors, especially those with less experience and knowledge in the use of technologies, will likely need assistance to build intelligent models to accomplish audit tasks. Audit firms, on the other hand, are not adopting because of the nature of standards, costly hardware and software, as well as investment in maintenance and IT labor.
The "as-a-service" business model could be a potential solution for part of the aforementioned challenge. "As-a-Service," also known as XaaS, is an emerging trend in the digital economy where "no one wants to sell you anything anymore, they just want to rent it to you" (Barker 2017, ¶1). XaaS allows consumers to pay only for the amount of computing and storage resources, as well as knowledge and labor, that they have used, which provides a cost-effective option for companies to operate their business (Perera, Zaslavsky, Christen, and Georgakopoulos 2014). Audit 4.0 also imagines a similar business model to provide flexible, low-cost, and high-quality audit services remotely using a large cloud (Dai and Vasarhelyi 2016). The cloud collects the requests from clients and matches them with the digitalized services offered by audit firms, and makes recommendations to the clients based on the timing and quality. Once an agreement is reached, the audit firm then offers the service remotely by analyzing the client's data in the cloud. The models, software, and hardware are paid per-use instead of purchasing, and auditors can obtain instant technical help from the cloud vendor. Such a subscription business model can largely reduce the upfront IT cost and later maintenance expenses for both audit firms and clients by outsourcing to a professional vendor. It can also improve market fairness, as smaller firms can afford advanced IT resources to provide high-quality audit services with limited initial investments.
To extend the service model in Audit 4.0, a Continuous Audit Intelligence as a Service (CAIaaS) architecture is proposed, which outsources the management of technologies and development of intelligent apps to professional providers. Continuous audit intelligence (CAI) refers to the use of AI/DM/ML algorithms upon data from a multiplicity of sources (such as IoT, ERPs, and websites) to identify risks and detect financial errors by promptly discerning problematic data elements from those tendencies from normal ones. In the CAIaaS architecture, data collected from IoT devices at the audit clients' sites, as well as ERP systems and other sources, are uploaded to a large secure cloud platform. Next, those data are prepared and standardized for formulating CA/CM models in the cloud. The models are then refined by auditors with audit or business logic and used to build intelligent apps. The apps are further deployed at auditors' computers to collect evidence or at clients' sites for monitoring transactions in real time. Those apps are also stored in a cloud marketplace for future use. As a result, burdens are largely alleviated from auditors, which allows them to focus on making auditing judgment based on the results from the intelligent apps.
Figure 1 demonstrates the CAIaaS architecture. This architecture mainly contains three stages: (1) data input, (2) app development, and (3) app deployment. In the first stage, both financial and nonfinancial data are collected from audit clients, not only through traditional approaches (bookkeeping), but also from IoT devices (e.g., cameras, sensors, GPS trackers), operational databases, and outside sources such as social media and news network, and uploaded to a cloud platform where data analyses and modeling are further processed. For example, GPS trackers and weighing sensors embedded in trucks could capture the real routes and weights of shipments, which could be further used to investigate the revenue of a logistics company. Cameras next to billboards could show solid evidence about whether an advertisement company claims fraudulent income by earning money from empty billboards. By using IoT devices, auditors can collect evidence, mainly from the physical world, anywhere, anytime.
The CAIaaS Architecture
The second stage is to develop apps that will be used for CA/CM purposes on a secure cloud-based platform. This platform includes three main processes: (1) data preparation and standardization, (2) model formulation, and (3) developing apps based on formulated and refined models. In the data preparation and standardization process, data are cleansed, integrated, and transformed to facilitate further analysis (Tan, Steinbach, and Kumar 2016). Moreover, data need to be standardized to a format that could be recognized by the AI/DM/ML algorithms provided by the platform. This could be a time-consuming process because organizations usually have their data models that need to be mapped to the one used in the platform. A potential solution could be that both companies and the platform provider follow industry standards (e.g., Auditing Data Standards issued by the AICPA)6 to build their data models, which will make the transmission of data from companies' IoT devices or ERP systems to the platform seamless (Dai 2017). Next, the prepared data are stored in a large data repository. The data repository also stores data that were previously uploaded from other clients. Auditors could choose to integrate anonymized and relevant data from the data repository with the current dataset, which could enrich the dataset to be used in the model formulation process and could enhance the accuracy of the data-driven models.
In the model formulation process, data prepared in the prior step are used to formulate CA/CM models that will facilitate auditors in detecting misstatements and fraud in a time-efficient manner. The platform is equipped with state-of-the-art software that can operate AI/DM/ML algorithms such as deep learning, Support Vector Machine (SVM), clustering, decision trees, regression, etc., which could be automatically or semi-automatically employed upon those data. For example, a regression could be applied to make predictions for future goods returns by analyzing their relationship with the conditions of inventory items and the surrounding environment (e.g., humidity, pressure, which can be obtained from IoT sensors). Such a prediction could be further used to monitor and detect abnormal return activities. Auditors may select appropriate parameters for models if they have sufficient knowledge and experience, but also may let the algorithms choose optimal ones if they are new to the technologies. Auditors then review the models and make them in compliance with audit standards or business policies. For example, auditors may disable certain data variables that provide redundant information and distort the results, assign a higher penalty cost to reduce a large number of false-positive cases, or set up a "materiality threshold" of transaction amounts to avoid overwhelming alerts. However, the refinement process could be sometimes limited, especially when AI algorithms with low interpretability (such as neural networks) are used to generate the models, as the internal logic of the models is difficult to understand. As a result, explainable AI (Samek, Wiegand, and Müller 2017) algorithms could be preferable in the development of models for audit purposes. Finally, the models are reformulated and ultimately achieve optimal performance.
In the third process, the refined models are built into intelligent apps by adding user interface, creating user manuals that define data input and output, and adding interfaces that allow performing CA/CM activities on auditors' and clients' computers, or their IoT devices. The platform then adds the newly developed apps to its app marketplace, which could benefit other auditors in the future. As a result, besides formulating models using their data, auditors could also find apps that can match their demands by searching the keyword of the semantic descriptions of apps on the market.
The third stage begins with auditors receiving apps and implementing these at clients' sites to monitor and detect risky or abnormal transactions in a continuous manner. RPAs (Huang and Vasarhelyi 2019) could be developed to routinize the data collection, monitoring, and alerting processes by linking data resources, apps, and email applications. Apps may also be deployed in auditors' computers to perform the end-of-period audit. Auditors and clients could operate the apps using the cloud if they have limited computing and storage resources. Apps may be deployed in the IoT devices at clients' sites so that the actuators7 could take instant actions to prevent fraudulent activities. For example, a smart locker could lock a warehouse when it detects that products without corresponding sales or shipping information are leaving the warehouse. Attached lights could turn red and flash if slow-moving inventory items are identified. After using apps in an engagement, auditors may provide feedback regarding performance and ratings to the apps for future improvement and recommendation.
The CAIaaS architecture provides auditors opportunities not only to generate intelligent apps using their clients' data, but also to choose apps from the marketplace. However, with the growth of the app marketplace, the increasing volume of apps is likely to diminish auditors' ability to manually seek the most appropriate ones. Therefore, the demand for a tool that can effectively choose apps, and even inspire auditors to explore new CA/CM models, will dramatically increase.
A recommender system could serve as a very valuable tool to identify the most appropriate apps to be used in a specific engagement and filter out irrelevant ones. This technology is superior to other information-filtering applications because of its ability to provide customized and meaningful recommendations (Zhou, Xu, Li, Josang, and Cox 2012). Unlike standard search engines that provide the same results for the same queries even though they are from different users, recommender systems can use personal characteristics and behaviors to provide personalized, relevant results to each user. Because of this advantage, recommender systems can suggest the right apps by analyzing the audit standards, audit clients, and auditors' historical preferences.
To help auditors select appropriate apps for a particular engagement, an app recommender system (ARS) is designed in this section.9 The framework of the ARS is shown in Figure 2. The proposed system makes app recommendations via three components: audit standards, audit clients, and auditors' preferences on apps. Recommendations based on audit standards are generated by creating a structure that categorizes apps by industry, business cycle, accounts, audit assertions, and audit objectives. These recommendations create a narrowed initial selection of apps that are then refined by audit clients and auditors' preferences. Recommendations based on audit clients are created to estimate the suitability of an app for a particular client. Recommendations based on auditors' preferences are also performed to predict the rating that a particular auditor will give to an app. The system creates a final score for each app by combining the results from these two filtrations, recommending apps with high scores to the auditor.
Design of the ARS
Recommendations Based on Audit Standards
In an engagement, auditors often follow five key steps to develop audit objectives: understanding objectives and responsibilities, dividing financial statements into cycles, knowing management assertions, knowing general audit objectives, and knowing specific audit objectives (Arens, Elder, and Mark 2012). The app selection process must follow this procedure, additionally controlling for the client's industry. The client industry is an important factor that would drive the choice of audit apps to use in particular engagements. Each industry has special business processes and risks in which auditors may need different models and data to collect evidence. For example, finance and insurance companies do not purchase or produce physical products, and thereby inventory-testing apps should be filtered out for those client types. Finance and insurance industry-specific apps should likewise be filtered out when dealing with retail clients. Similarly, water pollution is likely to be considered a significant risk for beverage companies but may have a moderate impact on other industries. The AICPA (2014) has provided guidance and delivered "how-to" advice for handling auditing issues in different industries. Within a specific industry, auditors must also identify business cycles (e.g., sales and collection, procurement and payment), individual accounts within those cycles, and associated management assertions. Auditors then test a specific audit objective based on such an assertion using apps.
A proper app recommender system must fit into this process. The proposed system is shown in Figure 3. Industry selection generates a list of industries that covers all possible industry categories. Business cycle selection links each industry to all possible business cycles for clients in that industry. Account selection associates each business cycle with all possible related accounts. Assertion selection links assertions with corresponding accounts. Objectives are linked to corresponding assertions during objectives selection. Finally, the system links all available audit apps with the audit objectives they can test. Each audit app may investigate one or more audit objectives, while each audit objective may also be linked to many audit apps since those audit apps cooperate to accomplish that audit objective. Using the system, an auditor could choose the client's industry and the relevant business cycle, account, assertion, and audit objective, and finally obtain a narrowed initial array of objective-appropriate audit apps. On very large multinationals or dynamically changing industries, or in heterogenous consolidated entities, much narrower recommender schemata may/should be used.
Recommendations Based on Audit Clients
It is possible that a narrowed initial selection still contains dozens of apps since new CA/CM models could be formulated and associated apps could be added to the marketplace every day. Therefore, the results of standards-based filtration should be further refined. Since audit problems usually have a large number of solutions and it is difficult to choose the best one, they often are solved using heuristics (rule-based) approaches (O'Leary and Watkins 1989). The ARS uses the heuristics approach. Specifically, if an app has been frequently used by auditors for similar clients (e.g., Adidas AG), that app is likely to be appropriate for the next such client (e.g., Nike, Inc.). A mechanism to identify such apps further refines and prioritizes recommendation results.
To generate client-based recommendations, a CF recommendation approach (derived from Zhang, Dai, P. Li, Q. Li, and Luo [2011]) is used to predict the suitability of an app to be used for an audit client. The underlying assumption is that the more the app has been used for similar audit clients in the past, the more suitable that app is for being used in a given instance. As shown in Figure 4, the approach has two clustering-based phases: (1) preparation, which groups audit clients based on an app usage matrix, and (2) recommendation, which makes predictions based on the nearest cluster to a given client.
Two-Phase Clustering-Based Recommendation
In the preparation phase, an app-usage matrix (shown in Table 1) is first created to record the usage frequency of each app for each audit client. This matrix is used as the basic data source to generate recommendations. Each row represents as audit client, and each column represents an app. Each cell, at the intersection of a row and a column, represents how many times a specific app has been used for a specific client in the past year. The reason to choose a short one-year window for calculating audit app usage is that in a dynamic environment, audit apps commonly gain or lose popularity due to updates or competition from other apps. One potential problem of the matrix is that it may be too sparse to be used for providing accurate recommendations. Apps, especially newly launched ones, could be known by only a few auditors, and the use of them in audit engagements would be even rare. To solve this problem, the app-usage matrix could be extended by adding clients' information such as firm size, risk profile, etc. Such information could reduce the sparseness of the matrix, and thereby improve the recommendation accuracy.
App-Usage Matrix
Based on the app-usage matrix, audit clients are then clustered into groups using classic clustering methods such as k-medoids (Han, Kamber, and Pei 2006). The main objective of the clustering is to accelerate the recommendation phase. Another benefit of the client clusters is to facilitate further mitigation of the sparseness problem. The ARS estimates the missing values in the app-usage matrix based on the information from the clients in the same cluster. This method is based on the assumption that the usage frequency of a certain app should be similar for the audit clients in the same clusters because those audit clients are similar to each other. Thus, it is reasonable to utilize the usage frequency of an app for audit clients to estimate the missing usage of the app for clients in the same cluster. For audit client i in cluster k, the missing usage for app j can be smoothed as:
\(\def\upalpha{\unicode[Times]{x3B1}}\)\(\def\upbeta{\unicode[Times]{x3B2}}\)\(\def\upgamma{\unicode[Times]{x3B3}}\)\(\def\updelta{\unicode[Times]{x3B4}}\)\(\def\upvarepsilon{\unicode[Times]{x3B5}}\)\(\def\upzeta{\unicode[Times]{x3B6}}\)\(\def\upeta{\unicode[Times]{x3B7}}\)\(\def\uptheta{\unicode[Times]{x3B8}}\)\(\def\upiota{\unicode[Times]{x3B9}}\)\(\def\upkappa{\unicode[Times]{x3BA}}\)\(\def\uplambda{\unicode[Times]{x3BB}}\)\(\def\upmu{\unicode[Times]{x3BC}}\)\(\def\upnu{\unicode[Times]{x3BD}}\)\(\def\upxi{\unicode[Times]{x3BE}}\)\(\def\upomicron{\unicode[Times]{x3BF}}\)\(\def\uppi{\unicode[Times]{x3C0}}\)\(\def\uprho{\unicode[Times]{x3C1}}\)\(\def\upsigma{\unicode[Times]{x3C3}}\)\(\def\uptau{\unicode[Times]{x3C4}}\)\(\def\upupsilon{\unicode[Times]{x3C5}}\)\(\def\upphi{\unicode[Times]{x3C6}}\)\(\def\upchi{\unicode[Times]{x3C7}}\)\(\def\uppsy{\unicode[Times]{x3C8}}\)\(\def\upomega{\unicode[Times]{x3C9}}\)\(\def\bialpha{\boldsymbol{\alpha}}\)\(\def\bibeta{\boldsymbol{\beta}}\)\(\def\bigamma{\boldsymbol{\gamma}}\)\(\def\bidelta{\boldsymbol{\delta}}\)\(\def\bivarepsilon{\boldsymbol{\varepsilon}}\)\(\def\bizeta{\boldsymbol{\zeta}}\)\(\def\bieta{\boldsymbol{\eta}}\)\(\def\bitheta{\boldsymbol{\theta}}\)\(\def\biiota{\boldsymbol{\iota}}\)\(\def\bikappa{\boldsymbol{\kappa}}\)\(\def\bilambda{\boldsymbol{\lambda}}\)\(\def\bimu{\boldsymbol{\mu}}\)\(\def\binu{\boldsymbol{\nu}}\)\(\def\bixi{\boldsymbol{\xi}}\)\(\def\biomicron{\boldsymbol{\micron}}\)\(\def\bipi{\boldsymbol{\pi}}\)\(\def\birho{\boldsymbol{\rho}}\)\(\def\bisigma{\boldsymbol{\sigma}}\)\(\def\bitau{\boldsymbol{\tau}}\)\(\def\biupsilon{\boldsymbol{\upsilon}}\)\(\def\biphi{\boldsymbol{\phi}}\)\(\def\bichi{\boldsymbol{\chi}}\)\(\def\bipsy{\boldsymbol{\psy}}\)\(\def\biomega{\boldsymbol{\omega}}\)\(\def\bupalpha{\bf{\alpha}}\)\(\def\bupbeta{\bf{\beta}}\)\(\def\bupgamma{\bf{\gamma}}\)\(\def\bupdelta{\bf{\delta}}\)\(\def\bupvarepsilon{\bf{\varepsilon}}\)\(\def\bupzeta{\bf{\zeta}}\)\(\def\bupeta{\bf{\eta}}\)\(\def\buptheta{\bf{\theta}}\)\(\def\bupiota{\bf{\iota}}\)\(\def\bupkappa{\bf{\kappa}}\)\(\def\buplambda{\bf{\lambda}}\)\(\def\bupmu{\bf{\mu}}\)\(\def\bupnu{\bf{\nu}}\)\(\def\bupxi{\bf{\xi}}\)\(\def\bupomicron{\bf{\micron}}\)\(\def\buppi{\bf{\pi}}\)\(\def\buprho{\bf{\rho}}\)\(\def\bupsigma{\bf{\sigma}}\)\(\def\buptau{\bf{\tau}}\)\(\def\bupupsilon{\bf{\upsilon}}\)\(\def\bupphi{\bf{\phi}}\)\(\def\bupchi{\bf{\chi}}\)\(\def\buppsy{\bf{\psy}}\)\(\def\bupomega{\bf{\omega}}\)\(\def\bGamma{\bf{\Gamma}}\)\(\def\bDelta{\bf{\Delta}}\)\(\def\bTheta{\bf{\Theta}}\)\(\def\bLambda{\bf{\Lambda}}\)\(\def\bXi{\bf{\Xi}}\)\(\def\bPi{\bf{\Pi}}\)\(\def\bSigma{\bf{\Sigma}}\)\(\def\bPhi{\bf{\Phi}}\)\(\def\bPsi{\bf{\Psi}}\)\(\def\bOmega{\bf{\Omega}}\)\begin{equation}\tag{1}{U_{i,j}} = \delta \left( {{{\bar U}_{k,j}}} \right) \end{equation}
where \({\bar U_{k,j}}\) is the average instances of the use of app j for all audit clients in cluster k, and δ is a coefficient that allows adjustment in the contribution of the data smoothing.
Next, the ARS re-clusters the clients using the smoothed app-usage matrix. After obtaining new audit client groups, the preparation phase ends. This step could be performed continuously without any human intervention.
When an auditor requests an app recommendation for a particular client, the recommendation phase begins. In this phase, the ARS predicts the usage frequency of an app for the target client using the average usage frequency for similar audit clients in the past. To speed up the selection of similar audit clients, the ARS first finds the top N similar client clusters for the target client and chooses the top M similar clients from those similar clusters.
In order to find the top N similar client clusters, the similarity between the target client and the centroid of each client cluster should be measured. The similarity could be calculated using the Pearson Correlation Coefficient (Sarwar, Karypis, Konstan, and Riedl 2001):
\begin{equation}\tag{2}s\left( {x,y} \right) = {{\sum\nolimits_{j = 1}^{\left| {x \cap y} \right|} {\left( {{U_{x,j}} - {{\bar U}_x}} \right)\left( {{U_{y,j}} - {{\bar U}_y}} \right)} } \over {\sqrt {\sum\nolimits_{j = 1}^{\left| {x \cap y} \right|} {{{\left( {{U_{x,j}} - {{\bar U}_x}} \right)}^2}\sqrt {\sum\nolimits_{j = 1}^{\left| {x \cap y} \right|} {{{\left( {{U_{y,j}} - {{\bar U}_y}} \right)}^2}} } } } }} \end{equation}
where \(s\left( {x,y} \right)\) denotes the similarity between clients x and y (in this case, x is the target client, and y is the central client in a cluster); \(\left| {x\!\cap\! y} \right|\) is the number of apps that have been used for both clients; \({\bar U_x}\) and \({\bar U_y}\) are the average app usage frequency for clients x and y; and Ux,j and Uy,j denote the usage frequency of app j for clients x and y.
Using the same formula, the similarity between the target client and each client in the top N similar clusters are calculated, which are then used to select the top M target clients. With the top M similar audit clients, the usage frequency of an app for the target client is predicted by taking the weighted average of deviations from the mean usage frequency of the app for similar audit clients. The weighted sum (Sarwar et al. 2001) could be used to predict the usage frequency of audit app j for audit client i:
\begin{equation}\tag{3}{P_{i,j}} = {{\sum\nolimits_{k = 1}^m {s\left( {i,k} \right) \times {U_{k,j}}} } \over {\sum\nolimits_{k = 1}^m {\left| {s\left( {i,k} \right)} \right|} }} \end{equation}
where Pi.j, represents the predicted usage frequency of app j for client i; m denotes the top M similar clients of the target client i; \(s\left( {i,k} \right)\) measures the similarity between client i and each similar client; and Uk,j represents the usage frequency of app j for client k (which is one of the similar clients). Using this formula, the potential usage frequency of each app for the target client is predicted by capturing how similar clients use the app.
Recommendations Based on Auditors' Preferences on Apps
Auditors' familiarity with the technique could also drive the choice of technique(s) to use in particular environments (Murthy and Groomer 2003); therefore, auditors' preferences can be used to further refine app recommendations. Auditors may have specific preferences regarding developers, versions, underlying CA/CM models, user interfaces, etc. Some auditors like apps developed by large firms more than those developed by small firms; some auditors prefer older, stable versions of apps, while others prefer the latest versions; some favor apps with a sophisticated user interface such as those allowing hand gestures, while others prefer more conventional operation. An effective recommendation system should incorporate these preferences to enhance result accuracy.
The ARS incorporates auditors' preferences into recommendations using a similar approach as used for the client-based recommendation. This approach is based on the assumption that auditors often choose apps that are consistent with their historical preferences, as well as the experiences and knowledge gained from a relatively large group of colleagues. Two auditors that have chosen the same apps in the past are likely to have similar preferences on apps in the future. The ratings of the first should influence the recommendations for the second. The preference-based approach also has two phases: preparation and recommendation. In the preparation phase, auditors are clustered based on preference similarity; in the recommendation phase, the system generates a list of apps for a specific auditor based on the app ratings from similar auditors.
In the preparation phase, an auditor-rating matrix (shown in Table 2) is created. Each row and column represents an auditor and an app, respectively. Each cell represents the rating that the auditor in the row has given to an app in the past. This matrix may have the same data sparseness problem as that in the app-usage matrix, as one auditor is likely to use and rate only a few apps. To weaken this problem, the matrix could also be extended by adding demographical information of auditors, such as their position levels, the accounting firms they work for, etc. Such information could facilitate the clustering of auditors and identifying those that have similar preferences on apps.
Auditor-Rating Matrix
Using the auditor-rating matrix, the ARS clusters similar auditors into groups, and smooths missing ratings using Formula (1). Then, auditors are re-clustered, and the preparation phase ends.
In the recommendation phase, the ARS predicts the rating of an audit app using the average of ratings that similar auditors have given to the app in the past. Specifically, the ARS first identifies the top N similar auditor clusters to the target auditor and then selects the top M similar auditors within those similar clusters. The similarity between the target auditor and the centroid of each auditor cluster could also be measured using Formula (2). After obtaining M most similar auditors, the ARS predicts the rating of an app given by the target auditor by using the weighted sum (Formula (3)) of the ratings that the similar auditors have given to the app.
Scores of Apps and Final Recommendation
The two predictions from the client-based and preference-based approaches are combined to generate a final, client- and auditor-specific recommendation score for an app using a weighted linear model. The final recommendation score for the app is calculated as:
\begin{equation}Score = \delta {P_u} + \left( {1 - \delta } \right){P_r}\end{equation}
where Pu represents the predicted usage frequency of the audit app for the target client, Pr represents the predicted rating that the target auditor will give to the app, and δ is the coefficient to adjust the contribution of each component. Finally, apps with high scores will be recommended to the auditor to perform CA/CM activities for the client.
Kozlowski (2016) describes an audit ecosystem in which various agents automatically execute functions such as importing client data into a standardized form, selecting appropriate audit apps to execute, and a feedback loop for unresolved results, etc. This section enriches the ecosystem by imaging a CAI-based paradigm that enables semi-automatic app development and recommendation, and also result analysis with the architecture and system proposed in this editorial as well as several other intelligent mechanisms.
The proposed CAI-based audit paradigm is shown in Figure 5. The paradigm is composed of a risk assessment module, a CAIaaS platform, an ARS, and a result analysis system with several intelligent modules, as well as the process of generating internal and external audit reports. The risk assessment module assists in locating the business cycles, accounts, and processes with high inherent risks. By deriving experienced auditor knowledge on information evaluation and judgment making concerning risks (Brown-Liburd, Mock, Rozario, and Vasarhelyi 2016), and integrating them into an expert system, the automatic risk assessment could be realized. Moreover, by incorporating the CRMA methodology (Moon and Krahel 2020), the module could identify emerging risks in a rapidly changing environment in addition to those that have been investigated by auditors in the past. Based on assessed business risks and auditors' judgment, auditors would gather relevant data that have been formalized by the Audit Data Standards, from IoT sensors, ERP systems, and other databases, as well as public websites. Next, auditors use the CAIaaS platform to formulate customized apps using the collected data. The ARS also suggests a couple of appropriate apps be used in the engagement according to auditors' preferences and the client's attributes. Later on, outcomes from the apps are further investigated by the result interpretation module, the exception prioritization module, and the exception investigation module. The result interpretation module explains the underlying mathematics or statistics of the results to facilitate auditors' decision making. Byrnes (2015) created a "super-app" to provide such knowledge for basic statistical models. The exception prioritization module ranks risky transactions by severity to avoid extremely heavy information load caused by Big Data. Methodologies proposed by Li, Chan, and Kogan (2016) and No, Lee, Huang, and Li (2019) for outlier prioritization could provide insights for the design of this module. The exception investigation module integrates auditors' knowledge and generates a list of exceptional exceptions that require further investigation. Issa (2013) devoted efforts in this area by designing a weighting system that utilizes experts' knowledge to identify irregularities. Finally, auditors use their professional judgment to determine whether sufficient evidence has been collected, and then either operate the process of generating final reports or retrieve to prior stages and perform more tests to collect useful evidence.
The CAI-Based Audit Paradigm
The corporate reporting environment is based on a wide set of statutory rules and regulations not designed for frequent reporting and not able to properly represent the dynamic environment of the current organization. The balance sheet and income statement, as an example, create a set of measurement adjustments such as owner's equity depreciation, goodwill, etc. that do not agree with real-life day-to-day operations. Articulating real-time databases with dashboards that represent the firm and call attention to discrepancies does not necessarily connect easily with traditional financial statements nor with the standard setters rules. Large economic actions inordinately affect monitoring schema, potentially creating false positives or hiding false negatives. These factors may lead to overreactions by management and misleading concerns of the auditors. Analytics and AI models, under current analytic technology, represent historical behavior and are better able to represent the linear part of operations, not extremes or fluctuations. For example, corporate behavior during a pandemic does not conform to normal standards, and exception models would naturally explode in alerts.
The idea of sharing intelligent apps could largely benefit auditors with little background and experience in using technologies. However, the capability of creating effective apps and using them appropriately to enhance the quality, efficiency, and value of an audit is a unique competitive advantage of auditors or audit firms; consequently, they may hesitate to share with others. Therefore, the app marketplace should set up solid policies to protect those intellectual properties and create a new business model to facilitate the trading of intelligent apps and providing CA/CM services in a cloud.
The third concern is regarding security and data ownership issues. The PCAOB requires auditors to maintain the confidentiality of client information (AU Section 339.11). With a very large amount of data captured and sent to the cloud for analysis, protecting the confidentiality of those data should be a critical issue. Besides data encryption, a hybrid analysis and storage mechanism could be performed. For example, auditors could send archived data to the cloud for model formulation purposes and download models to local servers and examine their effectiveness using clients' recent data. Furthermore, using one company's data to help build models for another organization may result in a potential breach of confidentiality. For example, rules learned from a client's past sales may reveal the sales patterns of the company. More regulations and detailed guidance regarding confidentiality protection and testing of apps for potential leakage are needed. Also, performing a comprehensive data anonymity mechanism, such as the one proposed by Kogan and Yin (2017), may facilitate learning empirical patterns while keeping them secret. Another issue is whether a cloud provider should use data provided by audit firms and the intelligent models formulated using those data to improve the apps used for other firms. The ownership of the data and models may be debatable. Some cloud providers treat them as their assets, while others only consider the cloud as a container. Regulations are necessary to stipulate the rights and obligations of cloud providers in the use of client data.
The quality control of apps is a critical and challenging task that may require the active engagement of standard-setting bodies. The PCAOB, for example, could perform accreditation to apps to ensure their compliance with existing standards. A "trial stage" may be set up for each newly created app that allows experienced auditors to evaluate its performance. Alternatively, regulators and professional agencies could move toward standardization on a collection of apps for each specific audit scenario. However, such standardization may harm the flexibility of using the state-of-the-art techniques/models to improve audit quality. Therefore, regulators need to find a balance between allowing auditors to explore emerging technologies and standardization of their use.
The inherent complexity of emerging technologies such as IoT and the cloud, and the anachronistic nature of accounting and audit standards, hinder technology adoption and full use by the auditing profession. Auditors with more or less knowledge and experience need the assistance of effective applications of technologies in their engagements. This editorial proposes a novel architecture, named CAIaaS, to fulfill this need. An app recommender system is further designed to provide customized suggestions for a particular auditor and client, enabling the selection of appropriate apps. Armed with the right tools, auditors would be able to efficiently perform audit activities and provide timely opinions. The CAIaaS, the app recommender system, as well as other intelligent mechanisms would together comprise a semiautomatic audit paradigm, which could facilitate the progressive transformation toward audit automation.
Future research should explore the implications that could explore methodologies and issues related to using information from one auditee in the analysis and modeling of other auditees and methods to protect confidentiality (Kogan and Yin 2017) while facilitating audit modeling and exception reporting.
The cloud computing model and capabilities bring a new set of capabilities to auditing as well as a new or changed set of security concerns to the assurance function. Furthermore, real-time reporting that is facilitated by this technology requires a substantial rethinking of corporate financial, predictive, and analytic reports into a modern form of informing stakeholders of the status of an organization.
Recommender models have taken a very important role in ecommerce marketing and advertising, but they have not been studied in the accounting/auditing literature. They bring together the formalization of many analytic technologies and can function on an evolving man × machine system, substantially improving assurance. Research is badly needed in this area.
A capital question that arises with the continuous assurance of organizational systems is the actual role of this process. It could be argued that close monitoring and detection of exceptions is a meta-control, not an assurance function. Research should reconceptualize the roles of measurement, control, and assurance in the systems that are emerging, managing data in real time and acting on prediction models.
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See, https://www.aicpa.org/interestareas/frc/assuranceadvisoryservices/auditdatastandards.html.
An actuator is an important component in IoT that receives commands from the internet to take actions that impact the physical world.
This section is based on "Three essays on audit technology: Audit 4.0, blockchain, and audit app" (Dai 2017, Chapter 4).
In addition to apps that can be found in the marketplace, many free routines (e.g., in Python) exist and may be considered.
The ideas, comments, and suggestions of Michael Alles, Soo Hyun Cho, Ivy Munoko, Qing Huang, and Chanyuan Zhang are very much appreciated.
Part of this editorial is based on "Three essays on audit technology: Audit 4.0, blockchain, and audit app" (Dai 2017).
Jun Dai, Michigan Technological University, College of Business, Department of Accounting, Houghton, MI, USA; Miklos A. Vasarhelyi, Rutgers, The State University of New Jersey, Rutgers Business School, Department of Accounting and Information Systems, Newark, NJ, USA.
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Subject: Continuous Audit Intelligence as a Service (CAIaaS) and Intelligent App Recommendations
Online Early | CommonCrawl |
Given that $a$ is an odd multiple of $7767$, find the greatest common divisor of $6a^2+49a+108$ and $2a+9$.
We can use the Euclidean Algorithm. The closest multiple of $2a+9$ that we can spot to $6a^2 + 49a + 108$ is $6a^2 + 49a + 99 = (2a+9)(3a+11),$ so we have
\begin{align*}
\text{gcd}\,(6a^2+49a+108,2a+9)
&=\text{gcd}\,(6a^2+49a+108-(2a+9)(3a+11),2a+9)\\
&=\text{gcd}\,(6a^2+49a+108-(6a^2+49a+99),2a+9)\\
&=\text{gcd}\,(9,2a+9).
\end{align*}Since $7767$ is a multiple of 9, both $2a$ and $9$ are multiples of $9$, $2a+9$ is also a multiple of $9$ so the greatest common divisor is $\boxed{9}$. | Math Dataset |
Surgically treated acromegaly patients have a similar quality of life whether controlled by surgery or requiring additional medical therapy (QuaLAT Study)
Muhammad Fahad Arshad ORCID: orcid.org/0000-0001-9932-09411,2,
Oluwafunto Ogunleye1,
Richard Ross2 &
Miguel Debono1,2
There is no consensus on quality of life (QOL) in patients with acromegaly requiring medical treatment after surgery compared with those achieving remission by surgery alone.
QuaLAT is a cross-sectional study comparing QOL in surgery-only treated acromegaly patients versus those requiring medical treatment post-surgery. Patients attending clinics were identified and divided into—Group 1: patients who had surgery only and were in biochemical remission, Group 2: all patients on medical treatment post-surgery, Group 3: patients from Group 2 with biochemical control. Participants were asked to fill three questionnaires; Acromegaly Quality of Life Questionnaire (ACROQOL), 36-Item Short Form Survey (SF36), and Fatigue Severity Scale (FSS).
There were 32 patients in Group 1 and 25 in Group 2. There was no difference in QOL scores between groups 1 and 2, as measured by ACROQOL (mean difference [MD] = − 2.5, 95% CI − 16.6 to 11.6; p = 0.72), SF36v2 [Physical component score (PCS) MD = − 4.9, 95% CI − 10.9 to 1.2; p = 0.12; mental component score MD = − 3.0, 95% CI − 10.5 to 4.4; p = 0.44], or FSS (MD = − 0.004, 95% CI − 1.14 to 1.33; p = 0.1). Comparison between groups 1 and 3 however showed that PCS (and 3 subdomains) was significantly better in group 3 (MD = − 8.3, 95% CI − 14.8 to -1.8; p = 0.01). All three QOL scores were lower when compared with healthy controls.
Medical treatment not only achieves a QOL comparable to surgery, it may also be associated with better QOL in physical subdomains. When compared with healthy controls, QOL remains worse in treated acromegaly patients compared to controls.
Acromegaly results from excessive secretion of growth hormone (GH) from tumours usually originating from pituitary somatotroph cells. The excess GH then stimulates the liver to produce insulin-like growth factor-1 (IGF-1), which causes most of the clinical manifestations of acromegaly [1]. Although the incidence of acromegaly has increased, it is a rare disease with an estimated prevalence of 8.6 cases per 100,000 [2]. For any chronic disease, improvement in the quality of life (QOL) is an important patient-related health outcome goal [3]. Different acromegaly treatment modalities have resulted in a clear reduction in mortality and morbidity [4] and it is well established that symptoms and QOL improve during acromegaly treatment [5], but in a relatively recent systematic review it was concluded that there is no overall consensus if any particular treatment is superior to the other in improving QOL [6].
The first-line treatment for acromegaly is trans-sphenoidal surgery (TSS), to remove or debulk the tumour [7]. However, if surgery is unable to achieve disease remission, medical treatment, such as somatostatin receptor ligands (SRL), dopamine agonists (DA), and pegvisomant are recommended to achieve biochemical control. Conventional radiotherapy or stereotactic radiosurgery (STRS) is considered if medical treatment is ineffective or intolerable and to treat residual tumour. While surgery can achieve remission in two-thirds of the cases [8], a significant proportion of patients will require some form of medical treatment afterwards to achieve hormonal control. Unfortunately irrespective of biochemical control health related quality of life remains impaired [6].
QoL reflects the subjective perception of health and effects of a disease. QOL in acromegaly has been a subject of interest in several published studies. Various validated questionnaires have been used to assess QOL in these patients. The most popular of these is Acromegaly Quality of Life Questionnaire (ACROQOL) developed by Webb et al. in 2002, which is a validated questionnaire for acromegaly patients [9]. Several other questionnaires have been used in various studies such as Patient-Assessed Acromegaly Symptom Questionnaire (PASQ) [10], and several other generic questionnaires such as 36-Item Short Form Survey (SF36) [11] and EuroQol-5 Dimensions (EQ-5D) [10]. Assessing the effects of symptoms and QOL is essential when looking at treatment outcomes [5].
When studies have assessed QOL in acromegaly the focus seems to be either on the effect of surgery [11, 12] or medications alone [13,14,15] or comparison of various medication combinations [16, 17]. It is still not clear if there is any difference in QOL among those who achieve remission by surgery only and those who require additional medical treatment post-surgery. Our literature review identified only one cross sectional study which looked at this question in the Japanese population [18]. They found that QOL is worse in medically treated patients, but this was a relatively small sized study (n = 26). We also found two older studies which directly compare QOL between those treated with surgery and those receiving medical treatment longitudinally both as first line treatment [19, 20]. These studies showed that the QOL was not different between the two groups, despite having improved from baseline. Another study from France showed that there was no difference in QOL among controlled patients whether treated with surgery ± medications or medications alone, however, among uncontrolled patients one subscale (psychological appearance) was better among surgical patients [21]. A significant proportion of patients will require some form of medical treatment post-surgery to control their IGF-1 levels. However, there is no clear consensus in the published literature on how the QOL in this group compares with those who achieve remission by surgery alone.
We therefore carried out a study to assess whether QOL between patients who have achieved remission post-surgery varies when compared to patients who receive medical treatment post-surgery.
QuaLAT (Quality of Life after Acromegaly Treatment) is a cross-sectional study carried out at the Department of Endocrinology, Sheffield Teaching Hospitals NHS Foundation Trust, a tertiary referral centre where all patients, aged 18 or above, with a histological diagnosis of a GH producing pituitary adenoma (diagnosed at least 6 months ago) were identified via our database. Exclusion criteria included having only non-surgical treatment for acromegaly, inability to provide informed consent, having a dire prognosis, or severe comorbidities unrelated to acromegaly which according to investigators may confound results. After exclusions, preliminary data for the remaining patients were collected to identify those who underwent trans-sphenoidal surgery (TSS) and went into disease remission biochemically (Group 1); that is achieved IGF1 levels within the age-and sex-specific reference range and a growth hormone suppressed level < 0.3 µg/L post OGTT and/or a growth hormone day curve average < 1.6 µg/L. Those who did not achieve biochemical remission after surgery and therefore required further medical treatment to control the disease were included in the other group (Group 2), irrespective of their biochemical status. There were some patients in Group 2 who had their last IGF-1 values and growth hormone levels outside the reference range. To avoid any potential confounding due to this factor, another group (Group 3) was crafted for the purpose of analysis, including only those patients from Group 2 who had controlled biochemistry.
The study was registered with Sheffield Teaching Hospitals research and development department (Reference STH20344). The study was approved by the Yorkshire and the Humber Research Ethics Committee (REC reference 19/YH/0373).
Study design and aims
In this study we aimed to compare the quality of life in those who were treated with surgery only vs those treated with surgery and medications for acromegaly, at a single tertiary care centre.
Eligible patients were approached via telephone and informed about the study. The willing participants were then posted the patient information sheet, a consent form, and the QOL questionnaires with two return envelopes to post back consent and questionnaires separately to maintain confidentiality. Phone numbers to contact the research team were provided in the patient information sheet, in case they had any additional questions or concerns.
Those who did not reply back within 28 days were given another phone call to check if they are still interested in participating in the study. If the interest was expressed once again, they were given another 28 days to send the documents. If the research team was unable to contact the patients, or the documents were not received within 28 days, it was deemed that the patients were no longer interested and therefore removed from the study.
Quality of life measurement
All the recruited patients were asked to fill three questionnaires to measure their QoL. One of these was a disease-specific questionnaire (ACROQOL) and the other two were generic questionnaires; SF36 and Fatigue severity scale (FSS). The permissions to use ACROQOL and SF36v2 were obtained prior to the study, while this was not required for FSS as it was freely available to use.
ACROQOL
ACROQOL consists of 22 questions, each having five options of responses scoring 1–5. The maximum score that can be reached is 110 (100%) which reflects the best QoL, with 22 (0%) being the lowest possible score. The score is calculated by the following equation [22]:
$$\frac{(X) - L}{{H - L}} \times 100$$
Where L is the lowest score of the subdomain of interest and H is the highest score of the same subdomain. There are two main categories for these questions: physical and psychological function, with the latter further subdivided into areas related to appearance and personal relationships.
Short form 36 (SF 36)
The SF36 is a questionnaire widely used for assessing health-related QoL. It is not disease-specific but has been used in acromegaly patients previously [11]. It consists of eight scales: physical functioning (PF), role-physical (RP), bodily pain (BP), general health (GH), vitality (VT), social functioning (SF), role-emotional (RE), and mental health (MH). The first four scales are related to physical health with the last four related to mental health. Like ACROQOL, final score is a percentage score with higher scores reflecting better QOL and vice versa. Calculation of the score is done in two steps: first pre-coded numeric values as assigned to the scores recorded by the participant according to the scoring key and second, the items in same scale are averaged together to create the eight scale scores [23].
Fatigue severity scale (FSS)
FSS is also a generic questionnaire, which consists of 9 questions with the patients score from 1 to 7 (1 means strong agreement and 7 means strong disagreement) assessing a specific acromegaly related symptom and its consequences. The questions evaluate the effects of fatigue on subject's motivation, exercise, physical functioning, interference in work, family and social life. The total score is calculated by either determining the mean of all scores. Unlike the first two questionnaires, higher score signifies worse QOL.
The demographic, treatment and biochemical data were collected for the successfully recruited patients. This was done by reviewing patients' electronic and paper case notes. In addition to age, gender, and ethnicity, the demographic data included the presence or absence of various co-morbidities often linked with acromegaly. The treatment data included the date of surgery, type of medications used to treat acromegaly (if any), use of conventional radiotherapy or stereotactic radiosurgery, and current hormone requirements. Latest IGF-1 values within the last 12 months of filling the questionnaires were recorded. All the IGF-1 values were measured using validated radioimmunoassay (Mediagnost GmBH, Reutlingen, Germany).
Sub analysis
To assess how our final results related to the normal healthy population we compared our results from all of the three questionnaires to those published in the literature. For this comparison, we selected studies which reported QOL in a large number of healthy participants and the population demographics were largely similar to our study groups and that reported the mean and standard deviation.
Our literature search identified only one study that reported the ACROQOL scores in a healthy non-diseased population as controls to validate their results [24]. The study participants in this study were Spanish and obese [body mass index (BMI) > 30 kg/m2], however the mean BMI was not provided. In case of SF-36, we selected the study reported by Jenkinson et al. [25] which reported scores from a very large group from the UK population (n = 8889) to test its reliability and consistency. Lastly, for FSS, the study reported by Ongre et al. [26] was chosen. This study reported FSS scores from 170 control participants from Norway and compared with scores from patients with Parkinson's disease.
All of the data were analysed using SPSS version 25 and GraphPad. Rates and percentages were calculated for categorical data. Comparison between continuous variables were summarised as mean difference (MD) and 95% confidence intervals (CI) of the differences as well as means and standard deviations (SD). Differences between the groups were tested by an independent sample T-test. A p-value of < 0.05 was considered significant for the study and two-tailed tests were used. The differences in quality of life scores between three groups were analysed by ANOVA followed by Fisher's least significance difference (LSD) test for post-hoc analysis. The differences in QOL scores between study cohort and healthy population was performed by independent sample T-test using GraphPad online calculator (https://www.graphpad.com/quickcalcs/ttest1.cfm).
The total number of acromegaly patients who were attending Sheffield Teaching Hospital clinics identified via our database was 142. Those who did not meet the inclusion criteria were excluded and the remaining (n = 107) were invited for the study. Of these, 67 patients were successfully approached and showed interest to participate. These patients were then posted the study documents including questionnaires. At the end of the study we received completed questionnaires from 57 participants, who were included in the final analysis. 32 participants had only surgical treatment and were in remission post-surgery (Group 1), while the remaining 25 patients had required some form of medical therapy after surgery (Group 2). Since there were some patients in this Group 2 who were still biochemically uncontrolled based on IGF1 levels above the normal range despite treatment, a third medically controlled only group was created for the purpose of analysis (Group 3) (Fig. 1).
Schematic representation of study illustrating how the final number of patients were reached in groups 1 (surgery only), 2 (surgery followed by medications), and 3 (surgery followed by medications and in biochemical remission)
Table 1 summarises the comparison of baseline demographic, treatment and biochemical data between the three groups. All of the patients in this study were Caucasians. There were no significant differences between Group 1 and 2 in age, duration of disease, gender distribution, use of conventional radiotherapy, or current requirement of hormonal treatments. Most of the patients in both groups were on single hormone replacement (12/14 in group 1 and 10/12 in group 2). The hormonal treatments were adequately replaced as patients are under regular endocrine follow up. More patients in the medical group underwent STRS, as one might expect. Also, there were a few uncontrolled patients in group 2 (n = 7), as mentioned earlier.
Table 1 Table showing baseline demographic, treatment and biochemical data comparison between the three groups
Groups 1 and 3 were similar with the exception of STRS. There were more males in the Group 3 (n = 12/18; p = 0.07). There was no significant difference in the incidence of other acromegaly related conditions [hypertension, T2DM, ischaemic heart disease, cardiomyopathy, heart failure, cardiac arrhythmias, obstructive sleep apnoea, mental health disorders such as anxiety or depression, dyslipidaemia (or current use of statins or fibrates), valvular heart disease (at least moderate aortic or mitral regurgitation), and bowel cancer] across all of the groups. The majority of patients in the medical group were on SRLs either alone (n = 16) or in combination with cabergoline (n = 1). Five patients were on cabergoline alone, and three patients were on pegvisomant treatment.
There was no difference in QOL scores between our main groups 1 and 2, as measured by all of three questionnaires (Table 2). Further comparison between subdomains and individual questions for all questionnaires (data not shown here) was also non-significant. However interestingly, when only medically controlled patients (Group 3) were compared with those in the surgical group, physical component score (PCS) of the SF36 questionnaire (and three of its subdomains: PF [MD = − 19.2, 95% CI − 37.2 to − 1.2; p = 0.04], RP [MD = − 22.8, 95% CI − 42.1 to − 3.5; p = 0.02], BP [MD = − 20.5, 95% CI − 36.3 to − 4.7; p = 0.01]) was significantly better in the medically controlled only patients (PCS MD = − 8.3, 95% CI − 14.8 to − 1.8; p = 0.014). This trend was also noted in the ACROQOL physical domain however the p value fell just short of significance (MD = − 15.4, 95% CI − 33.1 to 2.2; p = 0.09). A multivariate model including various variables (age, gender, duration of disease, hormone replacement, radiotherapy) also confirmed that while there was no difference between groups 1 and 2, the PCS of group 3 (and two of its subdomain [RP and BP] remained significantly better than group 1 [p = 0.04]).
Table 2 Comparison of QOL scores between surgical and medical group, calculated using ANOVA and Fisher's least significance difference (LSD) test for post-hoc analysis
QOL in acromegaly patients' vs healthy population
Figure 2 below shows the comparison of our ACROQOL scores with 157 healthy but obese controls from a Spanish population as reported by Webb et al. [24]. It can be seen that the difference in QOL is mainly in the psychological dimensions. Analogous comparisons of SF36 subdomains [25] and FSS scores [26] also mirrors the inferior QOL in all treated groups in our study (Figs. 3 and 4).
Comparison of mean ACROQL total and domain scores (with standard deviation) of acromegaly groups with healthy and obese Spanish population (asterisk symbol representing significant differences between groups; ***p < 0.001, **p < 0.01, *p < 0.05)
Comparison of mean SF36 subdomain scores (with standard deviation) of acromegaly groups with healthy controls (asterisk symbol representing significant differences between groups; ***p < 0.001, **p < 0.01, *p < 0.05)
Comparison of mean FSS scores (with standard deviation) of acromegaly groups with healthy controls (asterisk symbol representing significant differences between groups; ***p < 0.001, **p < 0.01, *p < 0.05)
Our study shows that patients who had medical treatment after surgery have a similar QOL when compared to patients with acromegaly who are in remission after surgery alone. Nonetheless, while treatment improves the QOL, it remains worse in acromegaly treated patients compared to the healthy population. Our findings highlight the important role medical treatment has in patients with uncontrolled acromegaly but also demonstrates the unmet need to develop more effective therapeutic strategies that, not only control the disease biochemically, but also improve patient well-being. In addition to ACROQOL, the only validated acromegaly questionnaire, we specifically assessed fatigue, an acromegaly related typical symptom, using FSS, a questionnaire that had never been studied previously in this patient group. The survey gives us insight on how a symptom could impact QOL, which was worse than in normal controls, and highlights the importance of assessing symptoms when studying treatment interventions in acromegaly. The third questionnaire we used was SF36, which is the most widely used QOL assessment tool globally and has been used frequently in acromegaly [11, 12].
QOL in patients with acromegaly is a concern. As first-line treatment to achieve remission in acromegaly, surgery is recommended as it can achieve higher remission rates compared with first line medical treatment (66% vs 45%) [10]. For those unable to achieve remission by surgery, long term medical treatment is the mainstay therapeutic strategy besides radiotherapy and a significant number of patients do achieve biochemical remission on maximal treatment, sometimes needing combination regimes [9]. Conversely long-term biochemical cure is not usually associated with normal life quality as highlighted by our findings. This has been shown by a number of studies but in view of the rarity of the disease most have used heterogenous groups and been cross-sectional as opposed to prospective [6]. Study populations vary within studies differing in treatment stage, disease control, length of remission and treatment received. This makes data less accurate and at times difficult to interpret. Unlike these studies we tried to create study groups which, apart from treatments received, were less heterogenous and studied sometime after surgery. Our groups did not differ in age, duration of disease from diagnosis, presence of hypopituitarism, presence of metabolic complications and mental health issues. Irrespective of this all our groups still achieved a QOL which is lower than the normal population.
Data from older longitudinal studies comparing QOL in surgical and medical treatment only (i.e., no surgery) have shown that QOL in both groups improves from baselines but is not different between the groups at 12 and 48 weeks [19, 20]. In our study, however, we excluded the few patients who required only medical treatment for acromegaly. Most of these patients had either inoperable tumours or were not medically fit enough to undergo surgery. These factors, therefore, would have confounded the QOL scores and besides in clinical practice, the vast majority of patients would first undergo surgery in line with the guidelines.
A Japanese study from 2014 also compared the QOL between surgical and medical patients [18]. This study was a cross-sectional study and showed that the QOL in surgical patients (n = 12) was better than those who were medically treated (n = 14). The authors also postulated that radiotherapy played an important role in lowering QOL scores in medical patients, as the difference between QOL scores was less significant once those with radiotherapy were excluded (n = 4) from the medical group. The results from our study are dissimilar to these findings however our study had more than twice the number of subjects. Besides, it is not clear from their published results, how many of the patients in the medical group also had surgery. This along with the differences in ethnicity between these two studies could possibly explain the dissimilarity in the results. In addition, we were also unable to confirm their finding with regards to radiotherapy, as removing radiotherapy patients from both groups in our study didn't alter the results, and the QOL scores remained comparable (supplementary table 1).
We have also found that medically controlled patients have better physical scores: SF36 PCS and possibly, ACROQOL physical subdomain. This could be related to the improvement in external features of the disease, better control of physical symptoms, or perhaps due to a sense of physical well-being from frequent monitoring and reassurance. Higher proportion of men in the medically controlled group, could have been a contributing factor since QOL scores in the general population tend to be worse in females [27] and in acromegaly QOL is affected more in females due to delay in diagnosis and added comorbidities [28] and more socioeconomic burden of the disease [29]. However, since the difference was still significant adjusting for gender, it was not a confounding factor in our study.
The reason for a low QOL in acromegaly patients despite treatment is debatable, but there are several contributing factors. Despite substantial improvements in healthcare in the recent past, there is still considerable delay in diagnosis of acromegaly, with some studies estimating the delay to be around 8–10 years from the onset of symptoms [30]. Owing to this delay severe irreversible damage such as changes in joint, soft tissues, voice, and physical appearance have already occurred by the time of diagnosis. Hence treatment, at best, can only slow down the progression of these changes, as the damage has already taken place [31]. There is an increased incidence of other chronic conditions in acromegaly which impair QOL such as depression [32], high BMI [33], type 2 diabetes [34]. Despite treatment, these conditions continue to progress in most cases, further deteriorating QOL. Some treatment options or resulting consequences could also be a contributing factor such as radiotherapy [35, 36] or hypopituitarism [36] both of which have been noted to reduce QOL in some studies. However, the systematic review by Geraedts et al. [6] didn't confirm the link between hypopituitarism and QOL. An interesting but unproven hypothesis is that current criteria for disease remission rely heavily on post treatment IGF-1 or GH levels. This may not reflect the true tissue exposure of GH and therefore may have a role to play in persistent low QOL in acromegaly despite treatment.
One of the main limitations of our study is the cross-sectional design and therefore absence of longitudinal data and control group. However, the latter was countered by including QOL data from large population cohorts. Secondly, our sample size was relatively small which may have decreased the statistical power. Also, the BMI data for the patients in this study was not available to us, which could be a potential confounding factor. Long-term, large prospective interventional studies looking at effects of different therapeutic strategies on symptoms and QOL are necessary to confirm our findings.
In conclusion, while surgery is superior in achieving biochemical remission in acromegaly, we have shown that medical treatment achieves similar QOL. In patients who do not achieve biochemical remission after surgery, medical treatment not only maintains QOL comparable to surgical patients in remission, but may also be associated with better QOL in physical subdomain. Secondly, compared to healthy controls, the QOL remains worse in acromegaly irrespective of the type of treatment. Strategies to diagnose acromegaly earlier and novel treatment modalities are required to improve this important patient-related health outcome.
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The study was funded by IPSEN Limited as a Medical Education Goods and Services (MEGS) grant.
Department of Endocrinology and Diabetes, Sheffield Teaching Hospitals, Glossop Road, Sheffield, S10 2 JF, UK
Muhammad Fahad Arshad, Oluwafunto Ogunleye & Miguel Debono
Department of Oncology and Metabolism, University of Sheffield, Sheffield, UK
Muhammad Fahad Arshad, Richard Ross & Miguel Debono
Muhammad Fahad Arshad
Oluwafunto Ogunleye
Richard Ross
Miguel Debono
Conception: MD, RR. Development of protocol: MD, MFA, OO, RR. Data collection tools: MFA. Data collection: MFA, OO. Data analysis: MFA, MD. Writing manuscript: MFA, MD, RR.
Correspondence to Muhammad Fahad Arshad.
None relevant to this study.
Yorkshire and the Humber Research Ethics Committee (REC reference 19/YH/0373).
Consent to participate
Supplementary file1 (DOCX 17 kb)
Arshad, M.F., Ogunleye, O., Ross, R. et al. Surgically treated acromegaly patients have a similar quality of life whether controlled by surgery or requiring additional medical therapy (QuaLAT Study). Pituitary 24, 768–777 (2021). https://doi.org/10.1007/s11102-021-01153-4
Acromegaly
Trans-sphenoidal surgery
Pituitary tumours | CommonCrawl |
Methodology to determine the parameters of historical earthquakes in China
Jian Wang1,
Guoliang Lin2 &
Zhe Zhang1
Geoscience Letters volume 4, Article number: 4 (2017) Cite this article
China is one of the countries with the longest cultural tradition. Meanwhile, China has been suffering very heavy earthquake disasters; so, there are abundant earthquake recordings. In this paper, we try to sketch out historical earthquake sources and research achievements in China. We will introduce some basic information about the collections of historical earthquake sources, establishing intensity scale and the editions of historical earthquake catalogues. Spatial–temporal and magnitude distributions of historical earthquake are analyzed briefly. Besides traditional methods, we also illustrate a new approach to amend the parameters of historical earthquakes or even identify candidate zones for large historical or palaeo-earthquakes. In the new method, a relationship between instrumentally recorded small earthquakes and strong historical earthquakes is built up. Abundant historical earthquake sources and the achievements of historical earthquake research in China are of valuable cultural heritage in the world.
Historical earthquake archives and field investigations
China has a long culture tradition to record natural phenomena. Since Shang (Yin) Dynasty (16–11 century B.C.), there had been officers appointed by governors to record natural disasters including earthquakes. Under Qin Dynasty (221–206 B.C.), the united country started to emerge. The regime consisted of the central and local governments. Meanwhile, there had been a reform and a standardization of the literal system. These elements had been making more literal recordings possible. From Western Han (206 B.C.–24 A.D.), earthquakes had been recorded as "catastrophes" in official historical archives. In Song Dynasty (960 A.D.–1279 A.D.), print technique was utilized widely, which enabled writing and propagation more convenient. In Yuan Dynasty (1271–1368 A.D.), the four-rank pyramid regime was formed, which included the central government in the highest rank followed by province, canton, and county hierarchically. From that time, the annals and various recordings of different rank governments became popular. After the middle of Ming Dynasty (about 1475 A.D.), there were more villages with growing population. Meanwhile, the society was stable and the administration was effective. In such situation, the number of earthquake recordings had been increasing greatly and more details of disasters were described gradually.
Two large-scale collections of historical earthquake archives
In general, the historical earthquake recordings scatter in different documents. The historical documents concerning earthquakes were extracted and compiled as books, which are known as "historical earthquake archives". In the long history of China, there were several books to pick out the historical sources of earthquakes. The first such book "Tai Ping Yu Lan" appeared in 977 A.D. A total of 45 items of earthquakes between the eleventh century B.C. and 618 A.D. were compiled by Li Fang. In Yuan Dynasty, a total of 268 items of earthquakes were compiled in a book "Wen Xian Tong Kao" by Ma Duanlin. In 1725 A.D., 654 items of earthquakes were compiled in a book "Gu Jin Tu Shu Ji Cheng" (Li et al. 1960).
At the end of Qing Dynasty, a Chinese named Huang Bolu, who was a French churchman that time, collected historical earthquake sources of China. About 3322 items of historical sources from ancient time to 1895 A.D. were compiled in a book (two volumes), written in French. The first volume was published in 1909 and the second in 1913.
In 1950s, the collection of historical earthquake sources participated by seismologists was carried out first time. During that time, there were about two hundred engineering projects which needed assessment of seismic intensity in the sites. For this purpose, historians and seismologists were organized by Academia Sinica to collect historical earthquake sources systematically. With more than 2 years of hard work together, the historians and seismologists searched through more than 8 thousand kinds of historical literature. About 15,000 items of historical earthquake sources were discovered and picked out. After checking and analyzing, they compiled these historical earthquake sources chronologically. The results were compiled in a two-volume book: "Chronological Archives of Historical Earthquakes in China" (Historical earthquake working group of Seismological Committee of Academia Sinica 1956).
The second large-scale collection of historical earthquake sources occurred in the 1970s. Since 1977, historians and seismologists have been involved in this work, organized by the State Seismological Bureau (SSB; since 1998, SSB has been renamed as CEA–China Earthquake Administration), China Academy of Social Sciences, and Academia Sinica. Seismologist Xie Yu-shou and historian Cai Mei-biao were the editors in chief. This time they explored historical earthquake sources in a wider scope. Based on "Chronological archives of Historical Earthquake in China," large amount of supplement and emendation were made. Some earthquake documents were discovered and translated from Manwen (one of official literatures in Qing Dynasty) and Zangwen (literature of Zangzu, i.e., Tibet) archives. After 5-year hard work, a five-volume "Compilation of Historical archives of Chinese Earthquakes" was accomplished and published successively in 8 books during 1983–1987 (Xie and Cai 1983). Meanwhile, most provinces and autonomous regions published their own regional compilation of historical earthquake archives.
Besides the two large-scale collections of historical earthquake recordings, there are some collections done by individuals, but in small scale and generally in local regions. The work of compiling historical earthquake archives from historical sources has never been ceased in China.
Although most disaster descriptions of historical earthquakes came from the historical earthquake archives, we can also get information through field investigation in some special situations. For example, the relics of ancient builds damaged by historical earthquakes remained. From these relics, the intensity can be judged more correctly. If lucky enough, we can even get new discovered sources of historical earthquake. Such information may be obtained from unearthed stone stelae, notes kept by individuals, genealogy of a large family, or local orally spread legends. Because the two large-scale collections were mainly focused on historical sources at the levels of central and provincial government, there might be some sources undiscovered at town level.
To evaluate the intensity of historical earthquake more correctly and accurately, investigation on local topography, soil layers, hydrogeology, building material and craft is very necessary. These elements are very important in intensity evaluation, which can help us understand real historical circumstances better. So the investigation on the relics of historical earthquake is emphasized, and such works have be done continuously in China (Shi et al. 1992; Wang et al. 1998a, 2010; Wang 2004).
Historical earthquake catalogues and spatial–temporal distribution
In the 1950s, the "Chronological Archives of Historical Earthquakes in China" was compiled and almost at the same time the "Catalogue of Chinese Earthquake" was also edited. Since that time, Chinese seismologists have endeavored unceasingly to study and revise the catalogues to make them more systematical and perfect. There are many earthquake catalogues in China. Here we mainly introduce the four formal editions of the catalogues.
Four formal editions of "The Catalogue of Chinese Earthquake"
First edition of "The Catalogue of Chinese Earthquakes (CCE)" was published in 1960 (Li et al. 1960). Li Shan-bang (Li S.B., also S.P., Lee) was the editor in chief. This catalogue has two volumes. Volume one is the catalogue of earthquakes with magnitude greater than 4.7, covering the time period from 1189 B.C. to 1955 A.D. Totally 1180 earthquakes were listed, including 585 historical earthquakes. Here the "historical earthquake" means the earthquake without instrumental recordings, which parameters were determined only by intensity data. In China, the end year of "historical earthquakes" was determined to be 1911 A.D (Table 1). In the first edition, the parameters of 585 historical earthquakes were determined mainly according to the information in "Chronological Data of Historical Earthquakes in China." Volume two of this edition is the catalogue in each county. In China, there are more than two thousand counties totally. About one thousand and six hundred counties have historical earthquake archives. Most of them were concentrated in the Eastern China.
The second edition of "The Catalogue of Chinese earthquakes" was published in 1971 (Seismic Working Group of Central government 1971). The chief editor was still Li Shan-bang actually, although named "working group." This four-volume catalogue covered the time period from 1177 B.C. to 1969 A.D. The total number of earthquakes listed with magnitude greater than 4.7 is 2257, including 589 historical earthquakes (Table 1).
The third edition of "The Catalogue of Chinese earthquakes" was published in 1983 (Gu et al. 1983). Gu Gong-xu was the chief editor. Lin Ting-huang and Shi Zhen-liang were vice editors. This two-volume catalogue contains information on 3187 earthquakes, including 619 historical earthquakes, which covered the time period from 1831 B.C. to 1969 A.D (Table 1).
The fourth edition of "The Catalogue of Chinese earthquakes" includes two parts. Part one "The Catalogue of Chinese Historical Strong Earthquakes" was published in 1995 (Min et al. 1995). This historical earthquake catalogue covered the time period from the twenty third century B.C. to 1911 A.D., with a time span of more than 4100 years. It contains 1034 historical earthquakes with magnitude greater than 4.7. Min Zi-qun was the chief editor. Part two "The Catalogue of Chinese Present Earthquakes" was published in 1999, the earthquake parameters were determined by investigations and instrumental recordings (Wang et al. 1999). This catalogue covered the time period from 1912 to 1990. A total of 4289 earthquakes were listed. Wang Su-yun, Wu Ge, and Shi Zhen-liang were the editors in chief (Table 1).
Table 1 Outline of the four editions of CCE
In order to compile the Seismic Intensity Zoning Map of China (1990), the concise catalogue of Chinese earthquakes (CCCE) was published (informally) in 1988, which covered the time period from 78 B.C. to 1986 A.D. Totally 5142 earthquakes with magnitude greater than 4.7 were listed. Min Zi-qun was the chief editor.
The four editions of CCE and a CCCE were all sponsored by the governments and recognized in whole country. There are some catalogues edited by individuals and local branches of State Seismological Bureau (SSB), which will not be introduced due to limited space.
Atlas of isoseismals
There are several atlases of isoseismals in China. Min Zi-qun compiled an atlas of isoseismals including 64 earthquakes (Min 1957). As an important basic data for seismic intensity zoning, the atlas of Chinese Earthquake Isoseismals was published in 1979 (Compilation Group of China Seismic Intensity Zoning Map SSB 1979). A total of 151 items of earthquake isoseismals were delineated in this atlas, the time period of which covered from 1125 A.D. to 1976 A.D.
The more complete and detailed works were done by cooperation between seismologists from Institute of Geophysics S.S.B. and historians from Institute of Chinese Historical Geography, Fudan University. Three-volume "Atlas of the historical earthquakes in China" was the cooperative results of seismologists, historians, and geographers (Institute of geophysics 1986, 1990a, b).
Spatial–temporal distribution of historical earthquakes
With the catalogue of the fourth edition (Min et al. 1995), we will analyze the spatial–temporal distribution of historical earthquakes in China.
Temporal distribution
The temporal distribution of historical earthquake in China is shown in Fig. 1. We can notice that the temporal distribution is not balanced. Although the first earthquake recording appeared in 2300 B.C., there were only 7 earthquakes recorded before Western Han Dynasty (started since 206 B.C.; for details, see "Appendix"). The number of recorded earthquakes increases gradually. From Western Han Dynasty to the end of Song Dynasty (206 B.C.–1279 A.D., totally 1485 years), there were 98 earthquakes recorded. There were 35 earthquakes recorded in Yuan Dynasty (1271 A.D.–1368 A.D.), although which covered a relatively short time span. Between the end of Yuan and the early years of Ming Dynasty, there were a lot of wars and society was not stable. There was an obvious gap in the earthquakes recorded in the corresponding time period. Remarkable changes appeared in the middle of Ming Dynasty (Since 1475 A. D). Almost 70% of recorded earthquakes occurred after that time. In the corresponding period, the society was stable and the administration was effective. So, the historical earthquake recordings can reflect the situation of the actual historical society to some extent from another aspect.
Temporal distribution of historical earthquakes in China
Spatial distribution is illustrated in Fig. 2, which is also not balanced. The most of recorded historical earthquakes were concentrated in the Eastern China.
Spatial distribution of historical earthquakes in China
Incremental frequency–magnitude distribution data are listed in Table 2. From Table 2, we can notice that the number of M4 earthquakes is even less than that of M5 earthquakes. It is obvious that the earthquakes below M5 are incomplete.
Table 2 Incremental frequency–magnitude distribution (2300 B.C.–1911 A.D.)
Intensity scale and traditional methods
The traditional methods to determine the parameters of historical earthquake rely heavily on the statistical relationship between intensity and the parameters. The relationship is a very important "bridge" in the traditional methods, which concerned the intensity scale. Firstly, we introduce the China intensity scale briefly.
China intensity scale
China intensity scale is a twelve-degree system. It was created by Li Shan-bang (Li 1954) and improved by Xie (1957). China Intensity Scale referred the intensity scales across the world, including the intensity scale of former Soviet Union in 1952 and Modified Mercalli Intensity Scale. In China Intensity Scale, seismic intensity was described according to four kinds of damage phenomena: house building; structure (including special ancient China buildings, such as memorial archway, pagoda, stele, rampart.); damage on the earth surface; and other phenomena. Glossary culled from historical records to describe the earthquake damage was subjoined to judge intensity from the historical documents easily (Li 1989). Chinese ancient words generally are different in font or meaning with nowadays. Such glossary would be helpful to those who are not familiar with ancient literature.
China Intensity Scale was supplemented and simplified by Liu (Liu 1978) according to new kinds of modern buildings destroyed by the recent years strong earthquakes both in China and across the world. China intensity scale was amended again in 1999 and became the National Standard then.
The procedures of tradition methods
Here we reconfirm the concept of "historical earthquake." In China, we define the "historical earthquake" as the earthquakes which parameters were only inferred from historical sources and not instrument recordings. Generally, the parameters of an earthquake include occurring time, epicenter, maximum intensity, and magnitude. If the data are enough, accuracy and focal depth may be inferred. The methods to determine the parameters of historical earthquakes include the following steps:
In a research region, historical recordings at all sites should be collected as completely as possible. The intensity of each site should be assessed according to the relevant destructive description.
Utilizing the events with instrumental parameters and intensities to work out the statistics or empirical relations between earthquake parameters and intensities. The relations will be extrapolated to infer the parameters of historical earthquakes.
In order to determine the magnitude of a historical earthquake, we should decide which statistical formula will be used according to the actual situation of intensity. For example, if a historical earthquake had enough recordings to draw out isoseismals, the earthquake magnitude can be inferred from the areas defined by the isoseismals (Wu 1989). The relation between magnitude and equivalent radius (R) of each intensity region was listed as follows:
$${\text{IV:}}\quad M = 2.16\lg R + 1.10$$
$${\text{V:}}\quad M = 2.06\lg R + 1.93$$
$${\text{VI:}}\quad M = 1.90\lg R + 3.06$$
$${\text{VII:}}\quad M = 1.85\lg R + 3.79$$
The area of each isoseismal in different magnitude bin had been statistics based on more data (Wang et al. 1998b). The results enrich the relations between magnitude and intensity.
If a historical earthquake only has the peak intensity, its parameters can be estimated by the relation between the parameters and the peak intensity.
If the peak intensity (I 0) is known, the formula (5) will be used to evaluate the magnitude of the historical earthquake (M):
$$M = 0. 5 8I_{0} + 1. 5$$
The formula (5) was used in the edition 1st, 2nd, and 3rd of "The Catalogue of Chinese Earthquakes" (Li et al. 1960; Seismic Working Group of Central government 1971; Gu et al. 1983). In "The catalogue of Chinese historical strong earthquakes" (Min et al. 1995), the relations were different in each zone of whole China.
$${\text{Eastern China:}}\quad M = 0. 5 7 9I_{0} + 1. 40 3$$
$${\text{Western China:}}\quad M = 0. 60 5I_{0} + 1. 3 7 6$$
$${\text{Taiwan island:}}\quad M = 0. 50 7I_{0} + 2. 10 8$$
If a historical earthquake recorded in a region with no damaging, only felt intensity data however, In this case, the relation between magnitude and radius of felt area (D) can be used:
$$M = 2. 2 1 {\text{lg}}D + 0. 6 3$$
Generally, the intensity of felt area may be intensity III or IV (Wu 1989).
The center of isoseismal or the equivalent center of several intensity sites might be taken as the epicenter generally. In some cases, the peak intensity site should be regarded as the epicenter.
No doubt, a suitable relation should be selected to evaluate the magnitude of a historical earthquake according to actual intensity data.
Anyway, the traditional methods have limitations, because they rely heavily on the historical earthquake archives. As we all know, there are many reasons that the historical earthquake archives might be incomplete. If a historical earthquake occurred in a place no persons living that time, there might be no disaster recorded at all. No doubt, a lot of historical earthquakes are missed which could not be sought by traditional methods.
In recent years, we tried a new approach to judge historical earthquake depending on modern instrumental data. In China, since 1960s, local earthquake observation networks have been built. With the accumulation of such small earthquake data, a very interesting phenomenon was discovered. There are many small earthquakes that occurred around the epicenters of strong historical earthquakes (Wang 1985). In order to analyze the phenomenon quantitatively, we suggested a method to deal with seismic pattern quantitatively and named it "calculation of seismic density" (Wang 1999, 2001).
Definition and calculation of the seismic density
Firstly, we introduce a metric known as the "seismic density" that quantifies the degree of clustering of the epicenter distribution. In a research region, we delineate a grid with interval Δ in longitude and latitude. For each grid node, we specify an outer circle centered on the node of maximum radius R. For the jth node, only those earthquakes with epicenters inside the circle will contribute to the index of seismic density. The index is then a mixture of physical and geometric measures of clustering of location, number, and magnitude (ratio to the size of the earthquake's fracture). For pragmatic reasons, we use metrics that are logarithmic (such as the magnitude), mainly to reduce the potential for large statistical sampling error that might occur using linear measures (such as energy or moment release). For consistency, we also use a logarithmic measure of the separation distance r ij . For the jth node at time t, the seismic density index is then defined by
$$I_{{j\text{, }t}} = \sum\limits_{{i = \text{1}}}^{n} {\frac{{\text{M}_{\text{i}} }}{{\Delta m\ln (r_{\text{ij}} )}}} ,\quad R_{ \hbox{min} } \le r_{\text{ij}} \le R$$
where Mi is the magnitude of ith earthquake. Δm is the difference between the threshold magnitude for complete reporting and the maximum magnitude, introduced as a rough normalization factor. The finite minimum distance R min avoids problems with the singularity in 1/(ln r ij) at r ij = 0. Once the calculation has been applied to every node, we can construct contours of the density index. The main ideas and steps are illustrated in Fig. 3.
A sketch map illustrating the seismic density index calculation. The circles on the left-hand side of the diagram represent the outer annulus of the middle diagram
The example in Beijing, China
We choose Beijing as an example to demonstrate the new method. Beijing is the capital of China, a metropolis which includes several administrational districts. The total area of Beijing is about 16,800 km2. Our research region is chosen as a quadrangle to cover the main part of administrational districts (115.5–117.5°E, 39.5–41.0°N). The earliest observation network has been established around Beijing since 1966, providing the best instrumentally recorded catalogue in China (Institute of Geophysics, CEA 2006). There are 926 earthquakes with M L ≥ 2 recorded from 1970 to 2014, which are estimated as a complete catalogue. There are 8 strong historical earthquakes (M ≥ 6), which parameters determined by extensive historical archives (in Fig. 4).
Plot of the Epicenters of events of different magnitude (blue circles) and the contour of the seismic density index defined in Eq. 10. The values of contours start from I j,t = 1 and increase in unit steps
The grid size Δ, search radius R, and resolution R min are determined by the accuracy of epicenter location. In our research region, the first-level accuracy is not larger than 5 km; so, we take the grid size Δ as 0.05 degree of longitude and latitude, roughly equal to 5 km. The R min represents the ultimate resolution of epicenter location. In China, the minimum distance that can be resolved by the networks is about 2~3 km (Institute of Geophysics, CEA 2006); therefore, we take R min = e (e≈2.71828) as a mathematically convenient number within this range. The maximum radius R should also depend on the grid size Δ and not contain too much overlapping data. Similarly it should be greater than or equal to (\(\Delta /\sqrt 2\)), so that events are not restricted to within the dimension of the elementary grid cell. Here we take R = 10 km, i.e., including all the data from the nearest and next-nearest neighbor nodes. The Δm = (M max–M min) is 3.0, given M min = 2.0 and M max = 5.0. The resulting contours of seismic density and the epicenter distribution for the magnitude thresholds M min = 2.0 are drawn in Fig. 4. We must point out that the metric is relative, which may be sensitive to the choice of parameters. Nevertheless the main characteristics of cluster identified remain robust (Wang 2001).
The highest seismic density is 50. The seismic density contours in Fig. 4 map out clear anomalies associated with the characteristics of the population of earthquake magnitudes and locations. Some strong historical earthquakes fall into seismic density zones, such as the M6 earthquake in 294 and M6¾ earthquake in 1484. One anomaly has two subsidiary structures, one seismic density center to the SE (near Sanhe) and another side lobe to the NE (near Huangsongyu). The epicenter of great M8 earthquake in 1679 is just located in the center near Sanhe. For the event in 1679, there were enough historical archives to draw out the isoseismals. The contour at intensity X and XI has a similar shape and orientation as those of seismic density (Fig. 5). Such elongated contours in the density anomaly are likely to be associated with the orientation of the causative rupture plane inferred from the major axis of the isoseismals. This complexity could also be due to a second smaller event that is not recorded, or to a major sub-event within the M8 earthquake, as seen for example in the Ms8 Wenchuan earthquake in 2008 (Zhang et al. 2008, 2010).
Comparison of seismic density with the isoseismals of the historical earthquakes M8 in 1679
Within each seismic density zone, the temporal distribution of earthquakes is analyzed quantitatively. According to the statistics, a regularity has been discovered that the temporal distribution of the seismic density zone varies with time very smoothly, which coincides with the epicenters of strong historical earthquakes. We can utilize this regularity to improve the strong historical earthquake locations with large error originally, and even find a new candidate zone of a missed strong historical earthquake (Wang et al. 2004a, 2010; Wang 2007, 2011).
Geophysics implication and application prospect
The modern-day seismicity can point out the locations of persistent weaknesses in the lithosphere. Such persistent but relatively stationary clustering is also observed in the other regions of the world such as in the New Madrid of the eastern US (Page and Hough 2014) and in Austria (Wang et al. 2008). This persistence has also been observed on a much smaller scale in the location of acoustic emissions (AE) during cyclic loading in laboratory rock deformation tests. Once a zone of weakness is created, subsequent loading in the next cycle appears to produce more AE events from the same volume (Sondergeld and Estey 1981). From the view of fracture mechanics, the occurrence of a strong earthquake means that the local crust medium has been experiencing a process from integrity to fragmentation. This process can be not backtracked usually.
We must point out that this method would only work as a method of identifying candidate zones for large historical or palaeo-earthquakes in cases where there is modern-day seismicity. However, this is not always the case. Many faults known to be active from palaeo-seismic observations in zones of shallow intra-plate continental deformation are not associated with current seismically active zones. This highlights the importance of combining studies of ours with alternate methods such as paleo-seismic and geodetic methods, as well as expanding the historical search for significant past earthquakes. Anyway, under some suitable circumstance, this method might help us to judge historical earthquake parameters and amend them, even find a new candidate site.
Combining historical earthquake and instrumental data, we can get the information of local crustal medium and inherent relationship between them. The results let us know more about the complex of seismicity, and meanwhile, we understand better about the geophysics behind them.
Retrospect and prospect
Since 1950s, huge numbers of investigation and research have been done on historical earthquakes for several generations in China. The resplendent accomplishments have been achieved. The investigations and researches can been concluded in the following aspects:
Comprehensive collection of historical earthquake archives;
Besides the textual archives, field investigations and excavations have been done on many important historical earthquake sites. Large evidences were found on ancient buildings, steles, and pagodas, and even tombs.
China seismic intensity scale has been established and improved.
Utilizing the earthquakes with both instrumental parameters and intensity data to get the statistics or empirical relations between them.
Compilations of historical earthquake catalogues and isoseismal atlases.
Probe a new approach to amend the parameters of strong historical earthquakes, which does not depend on the historical sources.
Application: historical earthquake data prolong the time and extend the space, which enable seismicity analysis more reasonably. The results are applied in long mid-term earthquake prediction, seismic hazard analysis, and seismic intensity zoning (Wang et al. 2004b).
Although a lot of work and research have been carried out, more efforts are still needed. Here, we just list out some possible further works:
Supplement collection of scattering historical sources.
After two large-scale collections, there are still historical earthquake documents that remain undiscovered. Because these documents are scattered, they are more difficult to collect.
Detailed field investigations and excavations on historical earthquake sites.
This kind of work is especially needed in the Western China, where population and historical documents are sparse. With less man-made demolishment and dry nature circumstance, the vestige of historical earthquakes might keep a longer time generally. Cooperation between seismologists and archaeologists on historical earthquakes research will bring more achievements.
It is important to reconstruct actual circumstance of a historical earthquake, such as considering building anti-earthquake ability in historical time period, effects of local hydrology, geology, topography, and physiognomy. This will help us comprehend the historical disaster documents synthetically.
Actual situation of historical sources is very complex. For example, some documents of a historical earthquake disaster were not original recordings, which might be already summarized. Generally, after a long time and undergoing number of wars, the original recordings had been missed. Because such summarized documents are easier to keep, they may have more chance to be retained till to nowadays. Common situation is that the disaster recordings were summarized at a county, but the actual peak intensity recordings at several villages might be missed. Unscrambling such summarized recordings depends on experience and knowledge of old experts to a large extent. There is an urgent task to make clear how the experts got their results, because some of experts have died and many experts are older than 70 years.
Seismic intensity data at sites derived from original disaster recordings are very valuable. We try our best to collect and unscramble historical sources to rebuild intensity data at sites. Seismic intensity database of electronic version should be built, which are more convenient to use intensity data to evaluate the parameters of historical earthquakes with quantitative methods (Lin and Wang 2012).
The seismic density method has been applied in some parts of the Eastern China. In the next step, we will expand it to the whole China and amend the Catalogue of Chinese Historical Earthquakes again. We are pursuing the wide international cooperation to test the seismic density method in different regions of the world. The final purpose should be to find out the general regularity behind complex seismicity (Wang et al. 2008).
Abundant historical earthquake archives and plentiful achievements of historical earthquake research in China are of valuable cultural heritage. We will continue to translate and introduce them in order for more experts to realize and utilize the treasure. It seems no possible to introduce Chinese historical earthquakes in all aspects in a paper. We just tried our best to outline the main points. There might be some omissions and misunderstandings.
SSB:
State Seismological Bureau (1973–1997)
CEA:
China Earthquake Administration (1998 to now)
CCE:
Catalogue of Chinese Earthquakes
CCCE:
Concise Catalogue of Chinese Earthquakes
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JW has made substantial contribution to conception and design, acquisition of data, and analysis and interpretation of data; GL has done some work concerned with collection of data and calculations, and took part in discussion; ZZ has done some work concerned with statistical analysis and drawing figures, and took part in discussion. All authors read and approved the final manuscript.
The partial work of this paper is supported by the NSFC of China and NERC/ESRC Newton of UK fund cooperation programme on Increasing Resilience to Natural Hazards in Earthquake-prone regions in China. We thank the two anonymous peers for their fair comments and helpful suggestions.
The partial work of this paper is supported by the NSFC of China with the project title: Probability and Uncertainty in Risk Estimation and Communication (PUREC-41661134014).
Institute of Geophysics, China Earthquake Administration, No 5. Minzu Daxue Nan Road, Haidian District, Beijing, 100081, China
Jian Wang
& Zhe Zhang
Yunnan Seismological Bureau, China Earthquake Administration, Kunming, China
Guoliang Lin
Search for Jian Wang in:
Search for Guoliang Lin in:
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Correspondence to Jian Wang.
Appendix: Table of Chinese Dynasties
Xia (21–16 century B.C.)
Shang (Yin) (16–11 century B.C.)
Western Zhou (11 century–771 B.C.)
Eastern Zhou (770–221 B.C.)
Spring and Autumn Period (770–475 B.C.)
Warring States Period (475–221 B.C.)
Qin (221–207 B.C.)
Western Han (206 B.C.–24 A.D.)
Eastern Han (24–220 A.D.)
Three Kingdoms (220–280)
Western Jin (265–316)
Eastern Jin (317–420)
Southern and Northern Dynasties (420–589)
Sui Dynasty (581–618)
Tang Dynasty (618–907)
Five Dynasties and Ten Kingdoms (907–979)
Song Dynasty (960–1279)
Yuan Dynasty (1271–1368)
Ming Dynasty (1368–1644)
Qing Dynasty (1644–1911)
The Republic of China (1911–1949)
The People's Republic of China (founded in 1949)
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Wang, J., Lin, G. & Zhang, Z. Methodology to determine the parameters of historical earthquakes in China. Geosci. Lett. 4, 4 (2017). https://doi.org/10.1186/s40562-017-0071-x
Historical earthquake
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Increased lignocellulosic inhibitor tolerance of Saccharomyces cerevisiae cell populations in early stationary phase
Venkatachalam Narayanan1,
Jenny Schelin1,
Marie Gorwa-Grauslund1,
Ed WJ van Niel1 &
Magnus Carlquist1
Production of second-generation bioethanol and other bulk chemicals by yeast fermentation requires cells that tolerate inhibitory lignocellulosic compounds at low pH. Saccharomyces cerevisiae displays high plasticity with regard to inhibitor tolerance, and adaptation of cell populations to process conditions is essential for reaching efficient and robust fermentations.
In this study, we assessed responses of isogenic yeast cell populations in different physiological states to combinations of acetic acid, vanillin and furfural at low pH. We found that cells in early stationary phase (ESP) exhibited significantly increased tolerance compared to cells in logarithmic phase, and had a similar ability to initiate growth in the presence of inhibitors as pre-adapted cells. The ESP cultures consisted of subpopulations with different buoyant cell densities which were isolated with flotation and analysed separately. These so-called quiescent (Q) and non-quiescent (NQ) cells were found to possess similar abilities to initiate growth in the presence of lignocellulosic inhibitors at pH 3.7, and had similar viabilities under static conditions. Therefore, differentiation into Q-cells was not the cause for increased tolerance of ESP cultures. Flow cytometry analysis of cell viability, intracellular pH and reactive oxygen species levels revealed that tolerant cell populations had a characteristic response upon inhibitor perturbations. Growth in the presence of a combination of inhibitors at low pH correlated with pre-cultures having a high frequency of cells with low pHi and low ROS levels. Furthermore, only a subpopulation of ESP cultures was able to tolerate lignocellulosic inhibitors at low pH, while pre-adapted cell populations displayed an almost uniform high tolerance to the adverse condition. This was in stark contrast to cell populations growing exponentially in non-inhibitory medium that were uniformly sensitive to the inhibitors at low pH.
ESP cultures of S. cerevisiae were found to have high tolerance to lignocellulosic inhibitors at low pH, and were able to initiate growth to the same degree as cells that were pre-adapted to inhibitors at a slightly acidic pH. Carbon starvation may thus be a potential strategy to prepare cell populations for adjacent stressful environments which may be beneficial from a process perspective for fermentation of non-detoxified lignocellulosic substrates at low pH. Furthermore, flow cytometry analysis of pHi and ROS level distributions in ESP cultures revealed responses that were characteristic for populations with high tolerance to lignocellulosic inhibitors. Measurement of population distribution responses as described herein may be applied to predict the outcome of environmental perturbations and thus can function as feedback for process control of yeast fitness during lignocellulosic fermentation.
Adverse impacts of climatic change and concerns over energy security could be abated by replacing petrochemicals with chemicals produced from lignocellulose, which is the most abundant renewable feedstock on the planet and is available from industrial and agricultural residues. Intense research and development for decades have led to the onset of commercial scale production of lignocellulosic bioethanol using the industrial workhorse Saccharomyces cerevisiae. Although many improvements could be done in biomass pretreatment [1, 2], enzymatic hydrolysis [3] and inhibitor detoxification [4], the robustness of S. cerevisiae towards adverse process conditions is still a key engineering target to increase productivity, avoid loss of fermentable sugars and therefore reduce production costs [5]. An important hurdle to overcome for maintaining high cell activity is the negative effect of lignocellulosic inhibitors produced by the most common pretreatment methods; these include furaldehydes such as furfural and hydroxymethylfurfural (HMF), phenolics such as vanillin and 4-hydroxybenzoic acid and weak organic acids such as acetic acid, formic acid and levulinic acid (see reviews [5,6,7]).
Cell tolerance to lignocellulosic inhibitors is a highly plastic phenotype and depends on the environment that the cell population has experienced before exposure. For example, pre-cultivation in lignocellulosic hydrolysate containing furfural and HMF leads to induced expression of genes coding for specific NADPH-dependent oxidoreductases, e.g. Adh6 [8], that reduce the aldehyde moiety into less inhibitory furfuryl alcohols resulting in a shortened latency phase in the fermentation [9]. Tolerance to vanillin is similarly correlated to increased reduction to the less toxic vanillyl alcohol [10]. Also the tolerance to acetic acid at low pH is increased by pre-cultivation in medium supplemented with acetic acid at slightly acidic pH [11]. The acid tolerance is partly caused by an induced expression of the HAA1 gene coding for a global transcription factor that activates multiple genes, including TPO1 and TPO2 coding for drug/H+-antiporters which export dissociated acetate from the cytoplasm [12, 13]. For these reasons, improved fermentation of lignocellulosic substrates can be reached through adapting cell populations by pre-exposure to moderate inhibitor levels in the pre-cultivation step [14, 15].
The level of cellular resistance to a specific stress is determined both by stress-specific and general mechanisms. For example, it was previously found that a slow growth rate correlates with increased tolerance towards a number of seemingly non-related stresses [16]. The extreme case are cells in stationary phase (SP), which are characterized by increased cell robustness to heat shock, osmotic stress, freeze–thaw stress and weak acid stress [17,18,19,20,21]. The higher robustness of SP-cells is often explained by activation of multiple cellular regulatory events upon nutrient starvation, including the environmental stress response (ESR), which leads to adjustment of cellular resources to promote survival in adjacent environments (see reviews [22, 23]). Based on this, it can be proposed that increased tolerance to lignocellulosic conditions may be reached without pre-exposure to inhibitors, for example, by allowing cells to reach SP by carbon starvation prior to the fermentation step.
The aim of the current study was to investigate correlations between the physiological state of yeast populations and their aptitude to tolerate combinations of lignocellulosic inhibitors (vanillin, furfuraldehyde and acetic acid) at low pH. In particular, physiological responses of cells in SP, including the previously described quiescent (Q) and non-quiescent (NQ) cells [24,25,26], were investigated in detail. Furthermore, flow cytometry (FCM) measurements of cell viability, the intracellular pH (pHi) and reactive oxygen species (ROS) levels were applied to compare the responses of cell populations in early stationary phase (ESP) to those of cells in logarithmic phase (LP), and cells pre-adapted to lignocellulosic inhibitors.
Strains and media
Saccharomyces cerevisiae strains used in this study are listed in Table 1. They were stored at −80 °C in Yeast Peptone (YP) medium containing 10 g L−1 yeast extract, 20 g L−1 peptone and 20 g L−1 glucose supplemented with 30% (v/v) glycerol and maintained on YP agar plates containing 10 g L−1 yeast extract, 20 g L−1 peptone, 20 g L−1 glucose and 15 g L−1 agar. A chemically defined medium [27] with 20 g L−1 glucose, buffered to pH 3.7, 5.0 or 6.5 with 50 mM potassium hydrogen phthalate [28] and supplemented with or without 6 g L−1 acetic acid, 0.75 g L−1 furfural and 0.2 g L−1 vanillin was used in all aerobic growth experiments. Vitamins, trace elements, furfural and vanillin were filter-sterilized to avoid changes in composition due to evaporation during autoclavation. Escherichia coli strain NEB5α (New England Biolabs) was recovered from 25% glycerol stock stored at −80 °C and used for subcloning of plasmid DNA and further propagation. Luria–Bertani broth (LB) (5 g L−1 yeast extract, 10 g L−1 tryptone and 5 g L−1 NaCl, pH 7.5) medium was used for culturing E. coli and 50 mg L−1 ampicillin was added to the LB medium when required. Media components were purchased from Sigma-Aldrich (Sweden), unless mentioned otherwise.
Table 1 S. cerevisiae strains used in this study
Construction of S. cerevisiae strain expressing pHluorin
Competent E. coli NEB5α cells were prepared using the RbCl method described in the subcloning notebook from Promega, which is adapted from the method described by [29]. The competent cells were transformed according to the supplier's instructions (New England Biolabs). Bacterial transformants were selected on solid LB plates (15 g L−1 agar), supplemented with ampicillin (50 mg L−1), for 16 h at 37 °C. Plasmid preparation from E. coli transformants was performed using GeneJet™ Plasmid Miniprep kit (Thermo Scientific, Waltham, USA). S. cerevisiae CEN.PK 113-5D was grown in liquid YPD medium for 14–16 h at 30 °C and 180 rpm in a rotary shake incubator (New Brunswick, Enfield, CT, USA) when preparing the strain for transformation. It was transformed with the URA3-based 2µ episomal plasmid pYES-pACT1-pHluorin [30] using the high-efficiency LiAc method [31], and the engineered yeast strain (henceforth mentioned as TMB3800) was selected on YNB-glucose plates (6.7 g L−1 Yeast Nitrogen Base without amino acids (Becton, Dickinson and Company, USA) supplemented with 20 g L−1 glucose and 15 g L−1 agar).
Aerobic batch cultivation in shake flasks
Cultures were grown in a rotary shake incubator at 180 rpm at 30 °C and cell concentrations were determined by optical density (OD) at 620 nm (Spectrophotometer U-1800, Hitachi, Berkshire, UK). Seed cultures were cultivated from single colonies of S. cerevisiae strains (from solid media) in 5 mL defined medium at pH 6.5 in a 50-mL conical tube until late exponential phase. Cells from the seed culture were harvested by centrifugation at 3056g for 5 min at 4 °C (Eppendorf centrifuge 5810 R, USA), washed with saline solution (0.9% NaCl) and subsequently used for inoculation of the pre-culture at an initial OD620 of 0.5. Cells from the pre-culture were harvested at different phases of cultivation and used to inoculate the final cultivation. Pre-cultivation and subsequent cultivations were performed in baffled shake flasks with the medium volume equivalent to 10% of baffled shake volume to maintain adequate aeration. All the experiments were carried out in biological replicates (n = 2 or 3) and measurements were carried out in technical triplicate.
TMB3500 was grown for 24 h until ESP, subsequently Q- and NQ-cells were separated by flotation [32]. For the flotation procedure, three different sterile colloidal solutions of non-toxic silanized silica particles in 0.9% NaCl, all formulated by QRAB (Alunda, Sweden) and produced by FertiPro N.V., Beernem, Belgium were used. The densities of the solutions were adjusted to previously reported buoyant cell densities for Q- (1.14 g mL−1) and NQ-cells (1.10 g mL−1) [24] as follows: BactXtractor-L (BX-L; density, 1.06 g mL−1), BactXtractor-M (BX-M; density, 1.12 g mL−1) and BactXtractor-H (BX-H; density, 1.29 g mL−1). The final densities of BX-L and BX-M were reached after dilution of BX-H with sterile 0.9% NaCl and measured using a DMA46 density metre (Instrument AB Lambda, Stockholm, Sweden). The flotation media were stored at 4 °C. Initially cells from an ESP culture of TMB3500 were harvested by centrifugation and homogenously re-suspended in 3 mL BX-H in a 15-mL Falcon Tube (Sarstedt, Nümbrecht, Germany). A discontinuous gradient was then created by the careful addition of 6 mL of BX-M followed by 2 mL of BX-L. The tube was centrifuged at 3056 g for 60 min at 4 °C using a swing-out centrifuge (Sigma 4-15C, Qiagen, Sweden) to separate Q- and NQ-cells. The two resulting cell fractions, due to differences in buoyant densities (Q-cells at the lower interphase between BX-H and BX-M layers, δ > 1.12; NQ-cells at the upper interphase between BX-M and BX-L layers, δ > 1.06), were collected using a syringe and needle. Each cell fraction was subsequently washed with sterile 0.9% NaCl and centrifuged at 3056g for 5 min at 4 °C. Cells from each fraction were visualized in the microscope (Nikon optiphot with Zeiss axiscam MRm, Sweden) and were characterized further in subsequent growth analyses.
Analysis of responses of unsorted ESP-, Q- and NQ-cells to lignocellulosic inhibitors
TMB3500 was pre-cultured for 24 h to reach ESP and cells were harvested by centrifugation at 3056 g for 5 min at 4 °C. Unsorted ESP-cells, Q-cells and NQ-cells were inoculated at an OD620 of 0.5 in 200 μL of 15 different media (Table 2) with varied concentrations of inhibitors (furfural, vanillin, acetic acid) at pH 3.7 in 96-well microtiter plates covered with a transparent plastic film (Breathe easy, Diversified biotech, USA) to prevent evaporation. Concentrations of the inhibitors were defined with a circumscribed central composite design equation (ccdesign) using Matlab (Release R2015a, The MathWorks, Inc., Natick, MA, USA). Growth was followed for 40 h by measuring OD620 with a multiscan ascent spectrophotometer (ThermoFisher Scientific, Sweden). Three-way ANOVAs (anovan) using Matlab were performed to see any effect of individual inhibitors on growth after 14 h with the different inocula (Unsorted ESP-, Q- and NQ-cells) (Additional file 1: Tables S1–3). A principle component analysis (PCA) of the whole dataset with ESP-, Q- and NQ-cells as loads and medium (M1-15) as scores was performed using the pca function in Matlab. Lag times and maximum specific growth rates were calculated by fitting the raw data to the modified Gompertz growth equation [33] using the Solver function to minimize sum of least squares in Excel (Microsoft, 2013, USA).
Table 2 Inhibitor compositions of the media as defined by circumscribed central composite design
Flow cytometry analysis
A BD Accuri™ C6 flow cytometer equipped with a Csampler (Becton–Dickinson, NJ, USA) was used to measure viability [34] and ROS [35], as described previously. Quality control of the instrument was made with 6 and 8 peak fluorescent calibration beads. The fluidics was set to medium flow rate (35 µL s−1), the threshold was set to 50,000 on the forward scatter channel, and 20,000 or 100,000 cells were collected at a rate between 3000 and 6000 events s−1. For determination of cell viability, 1 × 107 cells mL−1 in phosphate-buffered saline solution (PBS) (8 g L−1 NaCl, 0.2 g L−1 KCl, 1.42 g L−1 Na2HPO4 and 0.24 g L−1 KH2PO4, pH 7.4) were stained with propidium iodide (PI) (1 µg mL−1) and incubated in the dark for 10 min. A blue laser (488 nm) was used for the excitation, and PI emission was collected at 585/40 nm. For determination of ROS, 1 × 106 cells mL−1 in PBS solution were stained with dihydroethidium (DHE) (50 µg mL−1) and incubated in the dark for 20 min. DHE permeates into cells and gets oxidized to ethidium when exposed to superoxide in a dose-dependent manner. Ethidium then intercalates with DNA and emits red fluorescence proportional to intracellular ROS [35, 36]. A blue laser (488 nm) was used for the excitation, and DHE emission was collected at 585/40 nm. Cells in logarithmic phase grown in defined medium were used as live control and cells treated with 70% ethanol for 20 min were used as positive control for analysis of both viability and ROS. Autofluorescence was measured for unstained exponentially growing cells (CEN.PK 113-7D).
A Moflo XPD cell sorter (Becton–Dickinson, NJ, USA) with physically separated laser lines was used for ratiometric flow cytometry analysis of pHluorin fluorescence, as described previously [37]. Calibration of the instrument was made with fluorescent calibration beads (SPHERO Ultra rainbow fluorescent particles, 3.01 μM, Spherotech, USA). The threshold was set based on forward scatter obtained from blue laser (488 nm), and 100,000 cells were collected at a rate of approximately 5000 events/s. Excitation of pHluorin was made with a blue laser (488 nm) and a violet laser (405 nm), and the corresponding fluorescence emissions were collected with bandpass filters at 529/28 nm and 542/50 nm. The pH dependence of pHluorin was confirmed by measuring the ratio of fluorescence from the different excitation wavelengths, R405/488, in permeabilized cells in sodium phosphate buffer (0.2 M) at different pH ranging from 5.7 to 8. Permeabilization was made by incubating cells in PBS buffer (50 mM, pH 6.5) supplemented with digitonin (0.04 mM) for 15 min at room temperature on a turning table, as described previously [30].
Flow cytometry standard (FCS) data files were exported from the BD Accuri C6 software (BD Biosciences, USA) or saved directly from the Kaluza software (Beckman coulter, USA) and analysed with FlowJo v10.1 (FlowJo, LLC Ashland, OR, USA). For determination of percent viable cells, the gate was defined based on PI fluorescence (585/40 nm) of live and dead cell controls samples. The live control sample was exponentially growing cells in defined mineral medium, and the dead control was prepared by incubating cells in 70% ethanol for at least 30 min. An initial noise signal reduction was made for all samples by gating single cells based on the height and area of the forward scatter signal. pHluorin population was gated from autofluorescence (measured with a CEN.PK 113-7D strain without expression of pHluorin) using a bivariate plot of emissions at 542/50 nm and 529/28 nm, and excitation with the violet laser (405 nm) and the blue laser (488 nm), respectively. The pHluorin excitation ratio, R (F405/F488), for each cell was subsequently calculated using the derived parameter function in Flowjo. R (F405/488) for exponentially growing cells was used to differentiate cells with low and high pH. The cut-off was defined as the value (R = 0.400) dividing the population to two fractions: 1% lowest percentile (low pHi) and 99% highest percentile (high pHi) of exponentially growing cells. High and low ROS levels were defined similarly as the fluorescence intensity (height) at 585/50 nm dividing the LP-cell population in 1% lowest percentile (low ROS) and 99% highest percentiles (high ROS).
Tolerance to acetic acid is growth phase dependent
Acetic acid is one of the most important adversaries during fermentation of lignocellulosic substrates. Yeast cells display relatively high tolerance to acetic acid at low pH; however, the level of tolerance depends on several mechanisms that are differently regulated depending on additional extracellular conditions. To shed light on to whether the acetic acid tolerance phenotype can be induced by carbon starvation instead of pre-cultivation in the presence of acetic acid, cells were cultivated in aerobic batch mode in a defined mineral medium and harvested at different growth phases, i.e. at log phase (LP) (8–12 h), early stationary phase (ESP) (18–24 h) and late stationary phase (48 h). In parallel, cells were also pre-cultivated in a medium with 6 g L−1 acetic acid at pH 5.0, since this condition was previously shown to induce the desired acid tolerance [11]. Supplementation of acetic acid in the pre-cultivation medium inhibited growth already at pH 5.0, as reflected by an extended diauxic phase as well as a reduced growth rate during glucose assimilation compared to the reference cultivation (Fig. 1a). With a pKa for acetic acid of 4.76, the concentration of undissociated acetic acid experienced by the cells at pH 5.0 was approximately 2.2 g L−1. Although growth profiles differed substantially, the final biomass concentrations in the two pre-cultivations were the same with or without acetic acid in the medium. The cells of each of the pre-cultivations were re-inoculated in a new medium supplemented with 6 g L−1 acetic acid at pH 3.7.
Yeast response to acetic acid and low pH. a Pre-cultivation of the S. cerevisiae TMB3500 strain in aerobic batch mode in defined mineral medium at pH 5.0 without (round) or with 6 g L−1 acetic acid (square). b Cell dry weight (full lines) and % viability (dotted lines) after 24 h aerobic batch cultivation in defined mineral medium at pH 3.7 with 6 g L−1 acetic acid from pre-cultures at pH 5.0 with (blue) and without (grey) 6 g L−1 acetic acid. Error bars represent standard error of biological duplicates (n = 2)
The LP-cells not pre-exposed to acetic acid did not proliferate in the new medium (Fig. 1b). The poor tolerance to acetic acid at low pH was also clear from the poor viability after 24 h in the new medium (3–4% of the population). In contrast, pre-adapted cells that were pre-cultured the same length of time (8–12 h) were able to generate a cell biomass of ca 2.5 ± 0.1 gdw L−1 and cell viability was around 90% after 24 h (Fig. 1b).
The ESP-cells were able to initiate proliferation in the new medium, whether or not they were pre-grown in the presence of acetic acid at pH 5.0, resulting in a high biomass titre (2.4 gdw L−1) and viability (87%) after 24 h. This demonstrates that ESP-cells indeed had an induced acetic acid tolerance. In late SP (48 h pre-cultivation), cells displayed a reduced ability to grow, thus demonstrating a lower tolerance to acetic acid at low pH. However, for this long pre-cultivation time, the biomass titre was substantially lower if cells were pre-adapted than not (1.30 ± 0.65 gdw L−1 vs. 0.13 ± 0.01 gdw L−1). Survival was not correlated to growth for ESP- and late SP-cells, as shown by a high viability (>90%) even under static conditions (Fig. 1b).
Tolerance of ESP-cell subpopulations to combinations of lignocellulosic inhibitors
The conditional boundaries of ESP-cells to tolerate lignocellulosic inhibitors at pH 3.7 were further mapped by performing a series of cultivations in microtiter plates in media supplemented with different levels of acetic acid, vanillin and furfural according to a 23 factorial design (Table 2). In addition, we reasoned that the observed increase in tolerance to acetic acid of ESP-cells was due to their differentiation into Q-cells, which were previously shown to possess increased tolerance to heat shock and oxidative stress compared to NQ-cells [24]. To investigate this further, Q- and NQ-cell subpopulations were separated and both were analysed in parallel with the unsorted ESP-cells. Indeed, two cell fractions with different buoyant densities were separated using flotation (Fig. 2), whereas cells in exponential phase cells could not, which is in accordance with the previous studies [24, 25]. When analysed under the microscope, the Q-cells were round without buds in contrast to the NQ-cells that were a heterogeneous mixture of budding and non-budding cells (Fig. 2).
Separation of S. cerevisiae TMB3500 in ESP using flotation. a Unsorted SP-cells. b Discontinuous density gradient after centrifugation. c Q-cells d NQ-cells
The three cell populations (ESP-cells, Q-cells and NQ-cells) were each re-inoculated in 15 different medium compositions (M1-M15) and cell growth was monitored for 40 h. Concentration of glucose was low (20 g L−1) to avert osmotic stress to the yeast cell and to keep ethanol levels low, thereby removing their inhibitory effect as significant factors on the growth of the populations. Growth was observed in all media except M6, M8, M10 and M14, although the final biomass generated was substantially different (Fig. 3). To distinguish the individual and combinatory effect of inhibitors on growth, a 3-way analysis of variation (ANOVA) was used separately on each dataset generated for each test (Additional file 1: Tables S1–3). The first measured time point (14 h) (Fig. 3a) was used as input for the ANOVA, since the largest effect of these different inocula was observed in the latency phase. From the ANOVAs, it was seen that acetic acid, furfural and vanillin inhibited growth significantly (p < 0.001) for all inocula, although the effect of furfural was the smallest. The synergistic effect of two inhibitors (acetic acid and vanillin, acetic acid and furfural or vanillin and furfural) on growth was significant in all cases when using Q-cells as inoculum, indicating that this cell subpopulation required a longer latency period under the applied conditions (Additional file 1: Table S2). However, for unsorted ESP-cells and NQ-cells, a combinatorial effect was significant only for acetic acid and vanillin, and none of the combinations with furfural (p > 0.01) (Additional file 1: Tables S1 and S3).
Cell density after (a) 14 h and (b) 40 h cultivation in media M1-M15. Unsorted SP-cells (grey), Q-cells (blue) or NQ-cells (green) as inoculum. Error bars represent standard deviation of the mean for biological duplicates (n = 2) except for *M15 (n = 18)
A principal component analysis (PCA) of the three datasets combined was made to re-organize the information into new variables [principal components (PC)] that accounted for the majority of the variability in growth (Fig. 4). The input variables were set as yeast responses in terms of biomass formed after 14 h for the unsorted ESP-cells, Q-cells and NQ-cells. PC1 showed 98% variance with a high positive component loading for all subpopulations (Unsorted ESP-cells, 0.62; Q-cells, 0.44; NQ-cells, 0.65), demonstrating that growth of the three inocula was similar in the different media and was in agreement with the ANOVAs. A variation between the inocula was, however, seen in PC2 (2%), where Q-cells had a positive component loading (0.88), whereas the unsorted ESP-cells and NQ-cells were more similar to each other and had a slightly negative loading for biomass titre (unsorted ESP-cells, −0.42; NQ-cells, −0.22). From the score plot, it could be read that the largest influence of the different behaviour of Q-cells compared to ESP-cells and NQ-cells in both PC1 and PC2 was from growth in media M1, M3, M9 and M13, while the other media clustered together.
PCA analysis of factorial design experiment using unsorted SP-, Q- or NQ-cells as inoculum. Loading plot (squares) and Score plot (triangles)
To further analyse potential differences between the three inocula, lag times, maximum growth rates and the final biomass in an intermediate inhibitory medium (3.5 g L−1 acetic acid, 1 g L−1 furfural and 1 g L−1 vanillin) were estimated by fitting the modified Gompertz equation (Eq. 1) [33] to the experimental data
$$y = A\exp \left\{{- \exp \left[{\frac{{\mu_{\max} \times e}}{A}\left({\lambda - 1} \right) + 1} \right]} \right\}$$
where y is the logarithm of the relative population size [ln(N/N 0)], A is the asymptote [ln(N ∞/N 0)], µ max is the maximum specific growth rate (h−1), λ is the lag time (h), e is exp(1) and t is time (h).
Modelling of growth responses for unsorted ESP-, Q- and NQ-cells revealed a small difference in lag time (18 and 7% longer for Q-cells than for unsorted ESP-cells and NQ-cells, respectively, p < 0.001) (Table 3; Additional file 2: Figure S1). The maximum specific growth rate and the asymptote, A, were similar for the different inocula, meaning that once growth started the specific condition rather than the history of the population determined the growth rate and biomass yield.
Table 3 Fitting of the Gompertz equation to experimental data obtained for cultivation in medium 15 (n = 18)
Altogether the different analyses pointed towards that the shift into Q-cells was not determining the enhanced ability of ESP-cells to grow in the presence of inhibitors. In contrast, the trend was towards longer lag phases for Q-cells than for NQ-cells. On the other hand, since the lag time is determined by the number of viable cells at the start, it could also be that Q-cells were less tolerant to the inhibitors resulting in an initial drop in viability. However, FCM analysis after incubation for 24 h in static media demonstrated that cell viability was similar for unsorted ESP-cells, Q- and NQ-cells (Fig. 5). For medium M14 cell viability was substantially lower than for the other static media, demonstrating that this specific combination of inhibitors was most toxic.
Cell viability after 24 h incubation in medium supplemented with different amounts of acetic acid, vanillin and furfural at pH 3.7. SP- cells (grey), Q-cells (blue) and NQ-cells (green). Error bars represent standard error of biological duplicates (n = 2)
Intracellular pH distribution responses to lignocellulosic inhibitors at low external pH
Determining viability might not be an appropriate means to distinguish the ability to initiate growth in the presence of lignocellulosic inhibitors at low pH as viability of cells in static media remained also very high. Instead, the number of cells that are capable to maintain their pHi may be a better measure to predict the occurrence of growth when exposed to the harsh conditions. In a previous study, it was found that growth in the presence or absence of acetic acid at low extracellular pH correlated with the number of cells maintaining their pHi [38, 39]. To test this for ESP-cells, the response in pHi distribution in a CEN.PK strain over-expressing recombinant pHluorin was measured at lignocellulosic conditions by using ratiometric flow cytometry. Furthermore, measuring the response at the single-cell level could reveal any discrepancies in tolerance distribution within the different pre-culture populations.
The CEN.PK background was chosen because it is a well-established model strain for physiological studies within the yeast community [40], and it allowed for easy introduction of the pHluorin reporter system using the URA3 marker. Further, it was previously established that induction of acetic acid tolerance at low pH is displayed both in TMB3500 and in the laboratory strain CEN.PK 113-7D [11], and is thus strain-independent.
The CEN.PK strain was pre-cultured with or without acetic acid at pH 5.0, harvested in LP or in ESP and subsequently re-inoculated in defined mineral medium supplemented with 6 g L−1 acetic acid, 0.2 g L−1 vanillin and 0.75 g L−1 furfural at pH 4.5. Variation in pre-cultivation conditions indeed resulted in significant differences in growth profiles (Fig. 6, Additional file 3: Figure S2). LP-cells did not grow in the presence of inhibitors within the measured time interval of 24 h; it only grew in the control medium at pH 5.0 without supplementation of inhibitors (Additional file 4: Figure S3). Loss of viability could not explain the lack of growth for LP-cells since viability slightly reduced to about 85% during the first hour after which this level remained throughout the cultivation (Fig. 6). However, the number of LP-cells with kept physiological pHi had drastically reduced (Fig. 7a, e). Between the time of inoculation and before stable acquisition of cells in the FCM analysis (2 min), 95% of the population had a reduction in the fluorescence ratio from 0.54 to 0.25 (Fig. 7a, d, e). This demonstrates that nearly the total LP-cell population was sensitive to the inhibitory medium. Interestingly, a subpopulation (34 ± 27%) with a transiently higher ratio was observed after 30 min, but after 4 h the frequency of cells with high pHi was close to zero (Fig. 7a, d, e).
Effect of pre-cultivation on subsequent cultivation of yeast (CEN.PK 113-5D expressing pHluorin, TMB3800) in inhibitory medium. OD620 (full lines) and viability (dotted lines) as a function of cultivation time for the following inocula: LP-cells (grey), pre-adapted cells (blue) and ESP-cells (green). Error bars represent standard error of biological duplicates (n = 2)
Evolution of pHi in different yeast cell inocula (CEN.PK 113-5D expressing pHluorin, TMB3800) upon exposure to a mixture of lignocellulosic inhibitors. Representative histograms from the following inocula: a LP-cells, b cells pre-adapted in medium supplemented with the inhibitors at pH 5, and c ESP-cells. The vertical line displays cells with high and low ratio (F405/F488) and was defined as the number (=0.400) separating LP-cells (a, inoculum) into fractions of 1% lowest and 99% highest percentiles. d Mean ratio (F405/F488) and e frequency of cells with high ratio (>0.400) for the three different inocula (LP-cells, grey bars; pre-adapted cells, blue bars; ESP-cells, green bars) measured over time. Error bars represent standard error of biological duplicates (n = 2). Ratio (F405/F488) is the ratio of pHluorin fluorescence emission collected at 542/50 and 533/30 nm originating from excitation at 405 and 488 nm, respectively. Ratiometric flow cytometry measurement was performed with a Moflo cell sorter and data analysis was performed with Flowjo
Cells from the pre-adapted inoculum had a lower initial pHi than LP-cells in the medium without inhibitors (Fig. 7b, d, e). As cells were transferred to the medium with inhibitors at pH 4.5, a majority of the population (78 ± 1%) recovered to a higher pH and maintained it under the measured time period (Fig. 7b, d, e). The more uniform pHi response to the shift in environment and the low fraction of cells with low pHi indicates that inhibitor tolerance was relatively homogenously distributed within the pre-adapted cell population.
Finally, cell populations in ESP cultures displayed a high degree of heterogeneity with a high frequency of cells with low pHi prior to inhibitor exposure (Fig. 7c–e). Upon transfer to the inhibitory medium, a distinct subpopulation had an increase in ratio to a similar level as those from the pre-adapted cells, while a majority of cells kept a low pHi over the 4-h incubation period. Despite this, cells with low pH were still viable as measured with FCM analysis of PI-stained cells (Fig. 6). Altogether, this indicates that the ability of ESP populations to initiate growth under lignocellulosic conditions is only present in a fraction of the cells.
ROS level distribution responses to lignocellulosic inhibitors at low external pH
Low ROS levels have previously been associated with cell tolerance to multiple stress factors, e.g. exposure to furfural [41] or oxidative stress from hydrogen peroxide [42]. Therefore, we hypothesized that the ability to quench ROS contributes favourably to an acquired inhibitor tolerance observed herein for pre-adapted and ESP-cells. To verify this, population responses to exposure to the mixture of lignocellulosic inhibitors at low pH were analysed by FCM. It was found that all cultures consisted of cells with high or low ROS levels (Fig. 8), although the subpopulation distribution differed considerably (Fig. 8a–c, f). Mean ROS levels were ca. twofold higher for LP-cells than for pre-adapted cells and ESP-cells (Fig. 8d). As cells were transferred to the inhibitory environment, a subpopulation of the pre-adapted inoculum had a dramatic reduction (ca 3-fold) in ROS levels already within 2 min, whereas a slight increase was observed for LP-cells. ESP-cells displayed a higher degree of heterogeneity (Fig. 8e) and had a large cell fraction with low ROS levels (Fig. 8c, f) that was more stable than for the other two inocula over the measured time period. After 4 h, the differences between the inocula were levelled out, i.e. ROS levels for LP-cells were significantly reduced, while they were slightly increased for the pre-adapted cells (Fig. 8d).
Evolution of ROS in different yeast cell inocula (CEN.PK 113-7D) upon exposure to a mixture of lignocellulosic inhibitors. Representative histograms from the following inocula: a LP-cells, b cells pre-adapted in medium supplemented with the inhibitors at pH 5 and c ESP-cells. The vertical line displays cells with high and low ROS levels and was defined as the number (F585/40 = 52 × 104 channel number) separating the LP-cells (a, inoculum) into fractions of 1% lowest and 99% highest percentiles. d Mean F585/40 and e coefficient of variation (CV) of F585/40 for the three different inocula (LP-cells, grey; pre-adapted cells, blue; ESP-cells, green) measured over time. f Frequency of cells with low ROS levels over time. Error bars represent standard error of biological duplicates (n = 2). F585/40 is DHE fluorescence emission collected at 585/30 nm originating from excitation at 488 nm, respectively. Flow cytometry measurement was performed with a BD Accuri instrument and data analysis was performed with Flowjo. Histograms are from a representative experiment
ESP-cells displayed increased tolerance to lignocellulosic inhibitors at low pH
In this study, we demonstrated that early stationary phase cells have an increased tolerance to a combination of acetic acid, vanillin and furfural at low pH, as were reflected in immediate ability to initiate growth. The tolerance of ESP-cells was in range with the tolerance obtained by pre-adaptation in moderately inhibitory conditions. This was in stark contrast with exponentially growing cells that displayed a higher sensitivity towards all tested inhibitors. Allowing cells to enter ESP in the pre-cultivation step may therefore be beneficial from a process perspective as it will shorten the latency phase of the fermentation. It is well established from the previous studies that ESP-cells have high tolerance towards multiple stress factors [17, 19, 20]. Although the underlying mechanism behind tolerance is specific for each stress factor, cross-tolerance to different stressors is often observed and is generally ascribed to induction of the ESR upon nutrient depletion.
Inhibitor tolerance was similar for Q- and NQ-cells in ESP populations
SP-cells were previously found to consist of two subpopulations with distinct physiological state, i.e. the so-called Q- and NQ-cells [24]. The cell fractions can easily be separated from each other based on differences in buoyant cell density (δ Q = 1.14 g L−1 and δ NQ = 1.10 g L−1), for example, by density gradient centrifugation [24] or by flotation as described in the present study. Q-cells consists of young mother cells and unbudded daughter cells and was previously shown to have high mitochondrial activity, low ROS levels, high levels of reserve carbohydrates (glycogen and trehalose), and a high ability to re-enter the cell cycle upon nutrient-rich conditions. NQ-cells, on the other hand, are more heterogeneous and consist of cells with genomic instability, high levels of ROS, non-functional mitochondria and display apoptotic characteristics. With regard to stress tolerance, Q-cells possess higher ability to withstand heat shock and carbon starvation than NQ-cells [24, 26]. We therefore reasoned that the higher ability of ESP-cells to initiate growth in the presence of lignocellulosic inhibitors at low pH was due to differentiation into Q-cells. However, the ability of Q-cells to proliferate and remain viable was rather similar to that of NQ-cells under the whole range of tested conditions in our study. It follows that differentiation into Q-cells was not the underlying cause for improved tolerance of ESP-cells. Engineering of Q/NQ distributions in the pre-cultivation step may therefore not be a way forward to minimize the lag phase of a lignocellulosic fermentation process. Q-cells actually had a slight prolongation of the lag phase upon inoculation to new medium, which may be due to that they are arrested in the G0 phase, and once nutrients are provided they require time for metabolic and structural rearrangements before entering the mitotic cell cycle. NQ-cells on the other hand are arrested in different phases of the cell cycle without clear preference for a specific phase, and did not require the same time before initiating proliferation. Furthermore, any negative effect of long-term starvation of NQ-cells, as observed previously [26], was not observed in ESP. It can be deduced from our results that cells in late SP would behave differently compared to ESP-cells.
Vanillin was biocidal in combination with acetic acid at low pH
Yeast was sensitive to all tested inhibitors, although the degree of inhibition differed depending on the physiological state of the population. Acetic acid and vanillin were most detrimental for cell fitness, while furfural was inhibiting to a lesser extent at the concentration range used in the present study. The latter was probably due to an inherent high capacity of yeast to reduce furfural to the corresponding non-inhibitory alcohol, since pre-adapted cells behaved similarly to ESP-cells despite having a manifold higher specific furfural reductase activity (Additional file 5: Figure S3). It cannot be ruled out that the mechanisms behind the acquired inhibitor tolerance were different between pre-adapted cells and ESP-cells, and that furfural detoxification contributed favourably to the former. Inhibition was in most cases static rather than cidal, except for the combination of high amounts of vanillin (1.84 g L−1) and moderate levels of acetic acid (3.5 g L−1). At these concentrations, both vanillin and acetic acid were static on their own despite the low pH (3.7). The static effect of vanillin has been described previously [43], but herein we demonstrated that it is biocidal in combination with acetic acid at low pH. The underlying reasons for this is yet unknown, but it can be speculated that above a threshold concentration vanillin will inhibit carbon metabolism, resulting in an inability of the cell to efflux dissociated acetic acid and to keep pH homeostasis. Hence, the combination of vanillin and acetic acid has a synergistic effect on growth inhibition. Furthermore, our results indicated that furfural and acetic acid had a lower synergistic effect on growth inhibition. This is in contrast to a previous study on the combinatorial effect of different inhibitors (acetic acid, furfural and p-hydroxybenzaldehyde) in which it was found that furfural and acetic acid interacted synergistically on cell growth and ethanol production [44].
Tolerant cultures displayed characteristic population responses in pHi and ROS
Process control of yeast fitness distributions during fermentation requires identification of correlations between critical process parameters and cell properties that can be rapidly monitored at the single-cell level, such as with flow cytometry [45, 46]. Here we looked into pHi and ROS formation as two cell properties that correlate with inhibitor stress tolerance in yeast.
FCM measurement of intracellular pH
ESP-cells consist of two subpopulations with differences in intracellular pH, and addition of glucose in non-inhibitory medium results in recovery of physiological pH for the entire population [37]. By measuring responses in intracellular pH distribution as cells were transferred from the pre-culture to the inhibitory medium, we observed that only a subpopulation of ESP-cells was able to maintain their pHi. This implicates that inhibitor tolerance of ESP cultures lies at the level of a subpopulation. A similar behaviour was previously observed for Zygosaccharomyces bailii, which was shown to have a 1000-fold higher fraction of cells in ESP that was tolerant to weak organic acids, e.g. benzoic acid, sorbic acid and acetic acid, than exponentially growing cells [18]. Tolerance to acetic acid has previously been found to correlate to the cell pHi prior to exposure of the stress [47]. In our case, the average pHi of the ESP-cells was lower than for LP-cells, which is in agreement with the previous studies [30, 37]. Pre-adapted cells also had a lower pHi than LP-cells and a majority of the population was able to recover physiological pH within 2 min. Low pHi may be beneficial due to a lower amount of dissociated acid in the cytoplasm at the moment the cells are exposed to the harsh lignocellulosic conditions. It is unclear how cells with initially low pHi are able to rapidly restore intracellular pH, as was the case for pre-adapted cells and a subpopulation of the ESP-cells (Fig. 7). Maintaining physiological pH is a requirement for a functional glycolysis [48], enabling ATP-driven proton export to keep pH homeostasis under acidic condition (see review [49]). Rapid reduction in free inorganic phosphate due to the formation of fructose bisphosphate may play a role in bringing the pHi to neutral as was monitored in Lactococcus lactis (see review by [50]). Yet, there might be other specific mechanisms contributing to the reduction of acetic acid stress in combination with vanillin and furfural.
FCM measurement of ROS levels
A correlation between inhibitor tolerance and a high frequency of cells with low ROS levels was observed. This may be due to an increased capacity of inhibitor tolerant populations to quench ROS that was formed spontaneously or as a consequence of inhibitor exposure [41]. ROS are formed continuously in mitochondria under aerobic conditions, and are generally at higher level in exponentially growing cells than in SP-cells [51]. This may explain the observed discrepancies in inhibitor tolerance that were observed between pre-adapted cells, LP-cells and ESP-cells. The lower ROS levels of pre-adapted cells may be caused by inhibitor-specific mechanisms, for example, by an increased ability to reduce furfural by an induced furfural reductase activity. Another contributing factor may be an increased level of glutathione, which scavenges ROS by non-enzymatic oxidation, and is involved in maintaining redox homeostasis in the cell via NADPH-dependent glutathione reductase (GLR1) [52]. It was previously shown that by increasing intracellular glutathione levels by over-expression, the gene coding for γ-glutamylcysteine synthetase (GST1) resulted in improved growth in non-detoxified spruce hydrolysate [53]. An alternative way to reduce ROS levels is to introduce a biosynthetic pathway to L-ascorbic acid (vitamin C) that functions as a scavenger for oxygen radicals [54]. Our results give support to reduction of ROS levels as a suitable target for improving tolerance to inhibitors.
In this study, we demonstrate that cells in early stationary phase have increased tolerance to lignocellulosic inhibitors at low pH. Thus, allowing cells to enter ESP by carbon starvation during pre-cultivation may be a useful strategy to improve productivity in batch processes that are based on actively growing cells as biocatalysts for bioconversions that are limited by high amounts of inhibitors and low pH. Furthermore, flow cytometry as means to characterize population response profiles has demonstrated to be a sophisticated tool for prediction of yeast behaviour. Herein, we found that FCM-measured frequency of cells that recovered pHi and kept low ROS levels correlated with the ability of the yeast culture to initiate growth in harsh lignocellulosic conditions.
Log phase
ESP:
Early stationary phase
Stationary phase
NQ:
Non-quiescent
pHi:
Intracellular pH
ROS:
Environmental stress response
FCM:
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VN, MGG and EN participated in the design of the study. JS participated in the design of flotation experiments for the characterization of quiescent cells. VN and MC performed the experimental work and wrote the manuscript. MC conceived the study, and participated in the design and coordination of the study. All the authors read and approved the final manuscript.
We would like to thank Prof. Gertien Smits (University of Amsterdam, The Netherlands) for kindly providing us with the pHluorin plasmid. We would also like to thank Tim Berglund and Andreas Toytziaridis for technical assistance in strain cultivations, and Dr. Lisa Wasserstrom (Lund University) for construction of the pHluorin-expressing strain.
The datasets supporting the conclusions of this article are included within the article. Strains constructed in the current study are available from the corresponding author on request.
This work was financed by the Swedish Energy Agency (Energimyndigheten) (Grant Number P35350-1) and the Swedish Research Council FORMAS (Grant Number 229-2011-1052).
Division of Applied Microbiology, Department of Chemistry, Lund University, P.O. Box 124, SE 221 00, Lund, Sweden
Venkatachalam Narayanan, Jenny Schelin, Marie Gorwa-Grauslund, Ed WJ van Niel & Magnus Carlquist
Venkatachalam Narayanan
Jenny Schelin
Marie Gorwa-Grauslund
Ed WJ van Niel
Magnus Carlquist
Correspondence to Magnus Carlquist.
ANOVA:s of factorial design experiment using unsorted ESP-, Q- or NQ-cells as inoculum.
Fitting of the Gompertz equation to experimental data obtained for unsorted ESP-cells, Q-cells and NQ-cells.
Viability and growth of CEN.PK 113-7D (w/o pHluorin) in defined medium supplemented with lignocellulosic inhibitors.
Growth and fluorescence of CEN.PK 113-5D expressing pHluorin (TMB3800).
Specific furaldehyde reductase activity in crude cell extracts.
Narayanan, V., Schelin, J., Gorwa-Grauslund, M. et al. Increased lignocellulosic inhibitor tolerance of Saccharomyces cerevisiae cell populations in early stationary phase. Biotechnol Biofuels 10, 114 (2017). https://doi.org/10.1186/s13068-017-0794-0
Carbon starvation
Stress tolerance
Furfural
Quiescence
Population heterogeneity | CommonCrawl |
Positive current
In mathematics, more particularly in complex geometry, algebraic geometry and complex analysis, a positive current is a positive (n-p,n-p)-form over an n-dimensional complex manifold, taking values in distributions.
For a formal definition, consider a manifold M. Currents on M are (by definition) differential forms with coefficients in distributions; integrating over M, we may consider currents as "currents of integration", that is, functionals
$\eta \mapsto \int _{M}\eta \wedge \rho $
on smooth forms with compact support. This way, currents are considered as elements in the dual space to the space $\Lambda _{c}^{*}(M)$ of forms with compact support.
Now, let M be a complex manifold. The Hodge decomposition $\Lambda ^{i}(M)=\bigoplus _{p+q=i}\Lambda ^{p,q}(M)$ is defined on currents, in a natural way, the (p,q)-currents being functionals on $\Lambda _{c}^{p,q}(M)$.
A positive current is defined as a real current of Hodge type (p,p), taking non-negative values on all positive (p,p)-forms.
Characterization of Kähler manifolds
Using the Hahn–Banach theorem, Harvey and Lawson proved the following criterion of existence of Kähler metrics.[1]
Theorem: Let M be a compact complex manifold. Then M does not admit a Kähler structure if and only if M admits a non-zero positive (1,1)-current $\Theta $ which is a (1,1)-part of an exact 2-current.
Note that the de Rham differential maps 3-currents to 2-currents, hence $\Theta $ is a differential of a 3-current; if $\Theta $ is a current of integration of a complex curve, this means that this curve is a (1,1)-part of a boundary.
When M admits a surjective map $\pi :\;M\mapsto X$ :\;M\mapsto X} to a Kähler manifold with 1-dimensional fibers, this theorem leads to the following result of complex algebraic geometry.
Corollary: In this situation, M is non-Kähler if and only if the homology class of a generic fiber of $\pi $ is a (1,1)-part of a boundary.
Notes
1. R. Harvey and H. B. Lawson, "An intrinsic characterisation of Kahler manifolds," Invent. Math 74 (1983) 169-198.
References
• P. Griffiths and J. Harris (1978), Principles of Algebraic Geometry, Wiley. ISBN 0-471-32792-1
• J.-P. Demailly, $L^2$ vanishing theorems for positive line bundles and adjunction theory, Lecture Notes of a CIME course on "Transcendental Methods of Algebraic Geometry" (Cetraro, Italy, July 1994)
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average value, functions
Thread starter idontknow
University Math
idontknow
Find the average value of the functions below :
(1) \(\displaystyle f(x)=\lfloor x \rfloor ^x \; , \: x\in [1,2]\in \mathbb{R}\).
(2) \(\displaystyle a_n =\dfrac{1}{n} \; , \: n\in \mathbb{N} \).
Method required !
romsek
Well (1) is obviously 1 as the measure of the set where $\lfloor x \rfloor = 1, ~x \in [1,2]$ is $1$ and $1^x = 1~ \forall x$
The only other value $f(x)$ has on this interval is $f(2)=4$, but the interval $[2,2]$ has measure 0 so adds nothing to the average.
(2) $\bar{a} = \lim \limits_{N\to \infty} \sum \limits_{n=1}^N \dfrac{1}{n} =
\lim \limits_{N\to \infty} \dfrac 1 n H(n) \leq \lim \limits_{N\to \infty} \dfrac{\ln(N)+1}{N} = 0$
To be honest, I find the result in (2) troubling. The infinite sum of positive values produces an average of 0. That doesn't make sense to me, but such is the nature of infinity.
Last edited by a moderator: Jan 7, 2020
Reactions: idontknow
I agree that (2) seems non-sense to be 0 since \(\displaystyle \dfrac{1}{n}\) cannot be 0.
romsek said:
To be honest I find the result in (2) troubling. The infinite sum of positive values produces an average of 0. That doesn't make sense to me but such is the nature of infinity.
those equations above are a mess. They should read
$\bar{a} = \lim \limits_{N\to \infty} \dfrac 1 N \sum \limits_{n=1}^N \dfrac 1 n = \lim \limits_{N\to \infty} \dfrac 1 N H_N \leq \lim \limits_{N\to \infty}\dfrac{\ln(N)+1}{N} = 0$
mathman
idontknow said:
It seems almost logical. However the limit is less than any specified positive value, so it must be zero.
Maschke
The tail keeps pulling the average down. Sort of makes sense.
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Sensitivity of the DARWIN observatory to the neutrinoless double beta decay of \(^{136}\)Xe
Regular Article - Experimental Physics
F. Agostini1,
S. E. M. Ahmed Maouloud2,
L. Althueser3,
F. Amaro4,
B. Antunovic5,
E. Aprile6,
L. Baudis7,
D. Baur8,
Y. Biondi7,
A. Bismark7,8,
P. A. Breur9,
A. Brown7,
G. Bruno10,
R. Budnik11,
C. Capelli7,
J. Cardoso4,
D. Cichon12,
M. Clark13,
A. P. Colijn9,
J. J. Cuenca-García14,
J. P. Cussonneau15,
M. P. Decowski9,
A. Depoian13,
J. Dierle8,
P. Di Gangi1,
A. Di Giovanni10,
S. Diglio15,
J. M. F. dos Santos4,
G. Drexlin16,
K. Eitel14,
R. Engel14,
A. D. Ferella17,18,
H. Fischer8,
M. Galloway7,
F. Gao6,
F. Girard7,
F. Glück14,
L. Grandi19,
R. Größle14,
R. Gumbsheimer14,
S. Hansmann-Menzemer20,
F. Jörg12,
G. Khundzakishvili13,
A. Kopec13,
F. Kuger8,
L. M. Krauss21,
H. Landsman11,
R. F. Lang13,
S. Lindemann8,
M. Lindner12,
J. A. M. Lopes4,
A. Loya Villalpando9,
C. Macolino22,
A. Manfredini7,
T. Marrodán Undagoitia12,
J. Masbou15,
E. Masson22,
P. Meinhardt8,
S. Milutinovic5,
A. Molinario23,
C. M. B. Monteiro4,
M. Murra3,
U. G. Oberlack24,
M. Pandurovic5,
R. Peres7,
J. Pienaar19,
M. Pierre15,
V. Pizzella12,
J. Qin13,
D. Ramírez García8,
S. Reichard7,
N. Rupp12,
P. Sanchez-Lucas8,
G. Sartorelli1,
D. Schulte3,
M. Schumann8,
L. Scotto Lavina2,
M. Selvi1,
M. Silva4,
H. Simgen12,
M. Steidl14,
A. Terliuk20,
C. Therreau15,
D. Thers15,
K. Thieme7,
R. Trotta25,
C. D. Tunnell26,
K. Valerius14,
G. Volta7,
D. Vorkapic5,
C. Weinheimer3,
C. Wittweg3,
J. Wolf16,
J. P. Zopounidis2,
K. Zuber27 &
DARWIN Collaboration
The European Physical Journal C volume 80, Article number: 808 (2020) Cite this article
A preprint version of the article is available at arXiv.
The DARWIN observatory is a proposed next-generation experiment to search for particle dark matter and for the neutrinoless double beta decay of \(^{136}\)Xe. Out of its 50 t total natural xenon inventory, 40 t will be the active target of a time projection chamber which thus contains about 3.6 t of \(^{136}\)Xe. Here, we show that its projected half-life sensitivity is \(2.4\times {10}^{27}\,{\hbox {year}}\), using a fiducial volume of 5 t of natural xenon and 10 year of operation with a background rate of less than 0.2 events/(t \(\cdot \) year) in the energy region of interest. This sensitivity is based on a detailed Monte Carlo simulation study of the background and event topologies in the large, homogeneous target. DARWIN will be comparable in its science reach to dedicated double beta decay experiments using xenon enriched in \(^{136}\)Xe.
Neutrinos are the only known elementary particles that are Majorana fermion candidates, implying that they would be their own antiparticles. The most sensitive probe for the Majorana nature of neutrinos is an extremely rare nuclear decay process called neutrinoless double beta decay (\(0\nu \beta \beta \)), where a nucleus with mass number A and charge Z decays by emitting only two electrons and changes its charge by two units (A,Z)\(\longrightarrow \)(A,Z+2) + 2e\(^{-}\). The observation of this decay would mean that lepton number is violated by two units and, in the standard light Majorana neutrino exchange scenario, would yield information about the neutrino mass scale via the effective neutrino Majorana mass \(\langle m_{\beta \beta }\rangle = |\varSigma _i U^{2}_{ei}m_i|\). The sum is over the neutrino mass eigenstates, \(m_i\), and \(U_{ei}\), the corresponding entries in the lepton mixing matrix, which are complex numbers. The two-neutrino double beta decay mode (\(2\nu \beta \beta \)) is allowed in the Standard Model and has been observed in more than 10 nuclei [1]. In this case, the summed energy of the two electrons is a continuum, while for the \(0\nu \beta \beta \)-decay the distinct signature is a peak at the Q-value, the mass difference between the mother and daughter nuclei.
Experiments can observe a certain decay rate in a detector. The corresponding half-life is inversely proportional to \(\langle m_{\beta \beta }\rangle ^2\),
$$\begin{aligned} \frac{1}{T_{1/2}^{0\nu }} = \frac{\langle m_{\beta \beta }\rangle ^2}{m_e^2} G^{0\nu } |M^{0\nu }|^2, \end{aligned}$$
assuming that the decay is mediated by the exchange of a light Majorana neutrino. \(m_e\) is the mass of the electron, \(G^{0\nu }\) is the phase space factor, and \(M^{0\nu }\) is the nuclear matrix element. Recent experimental limits on \(T_{1/2}^{0\nu }\) and \(\langle m_{\beta \beta }\rangle \) are of the order \(T_{1/2}^{0\nu } \ge (\)10\(^{25}\)–10\(^{26}\)) year and \(\langle m_{\beta \beta }\rangle \le (0.06--0.17)\) eV, using a variety of nuclei and detector technologies [2, 3].
A particularly suitable isotope to search for the \(0\nu \beta \beta \)-decay with is \(^{136}\)Xe, with \(Q_{\beta \beta }\) = (2457.83±0.37) keV [4]. Current experiments use liquid xenon either in its pure form, EXO-200 [5], or xenon dissolved in liquid scintillator, KamLAND-Zen [6], and provide competitive constraints on the half-life. Future detectors that use xenon gas operated at high pressure, NEXT [7, 8] and PandaX-III [9], will add tracking capabilities for improved background rejection, while nEXO [10] proposes to operate a total of 5 t of isotopically enriched liquid xenon.
DARWIN [11] is a proposed observatory using 40 t of liquid natural xenon (LXe) in a time projection chamber (TPC) with the primary goal of searching for particle dark matter. Here, we demonstrate that DARWIN has a similar reach to dedicated future neutrinoless double beta decay experiments. This is due to its large, homogeneous target, and its ultra-low background, coupled to the capability of the TPC to simultaneously measure the location, energy, particle type and multiplicity of an event [12].
The paper is organized as follows: in Sect. 2 we provide a brief review of the baseline design of the DARWIN detector and describe the detector model utilized in our simulation study. Section 3 addresses the signal topology and how it is used to reject background events. In Sect. 4 we discuss the expected background sources, while the resulting background spectra and rates are presented in Sect. 5. We discuss DARWIN's sensitivity to \(0\nu \beta \beta \)-decay in Sect. 6 and give a summary and an outlook in Sect. 7.
The DARWIN observatory
DARWIN is a next-generation dark matter experiment that will operate a 40 t active (50 t total) liquid xenon TPC with the main goal to probe the entire experimentally accessible parameter space for weakly interacting massive particles (WIMPs) as dark matter candidates. Other physics goals include the search for the \(0\nu \beta \beta \)-decay, the real-time detection of solar pp neutrinos via electron scattering, the observation of supernova and solar \(^8\)B neutrinos via coherent neutrino nucleus scattering and the search for solar axions, galactic axion-like particles and dark photons.
The DARWIN detector is described in detail in [11]. In the baseline scenario, the detector is a cylindrical, two-phase (liquid and gas) xenon TPC with 2.6 m diameter and 2.6 m height. The TPC will be placed in a low-background, double-walled cryostat surrounded by an instrumented water tank to shield it from the environmental radioactivity and to record the passage of cosmic muons and their secondaries as well as for neutron thermalization.
Interactions in the TPC will give rise to a prompt signal (S1) from photons and a delayed, proportional scintillation signal (S2) from electrons transported by a homogeneous drift field and extracted into the gas phase. Both signals will be detected by photosensor arrays (made of photomultiplier tubes (PMTs), silicon photomultiplier (SiPM), or new types of sensors), providing the x-y-z-coordinates of an interaction, as well as its energy with < 1% \(1\,\sigma \) resolution for MeV energy depositions. Interactions separated by more than 15 mm are assumed to be individually identified in event reconstruction. This allows for separation between single scatters (as expected from \(0\nu \beta \beta \)-decays and dark matter particle interactions) and multiple scatters (as expected from many sources of backgrounds), as well as the definition of an inner (fiducial) volume with reduced background levels. The high density of the liquid xenon (\(\sim \)3 g/cm\(^3\)) ensures a short attenuation length for \(\gamma \)-rays.
The final location of the DARWIN experiment is yet to be decided. A good candidate is the Gran Sasso Underground Laboratory (LNGS) in Italy. We will use its overburden in this study.
Monte Carlo model of the detector
For the Monte Carlo event generation and particle propagation in geant4 we use a realistic model of the DARWIN detector. Its details are described in the following.
Drawing of DARWIN's double-walled cryostat and TPC, showing all components considered in the simulation
The TPC is enclosed within the outer and inner titanium cryostat (shown in Fig. 1), including torispherical domes, flanges and stiffening rings to minimize the amount of material. A dome-shaped pressurizable titanium vessel is placed on the inner cryostat floor to reduce the volume to be filled with liquid xenon while keeping the material budget low. A study based on previously-measured specific activities of cryostat materials [13, 14] showed that a cryostat made of titanium yields a lower background rate than a stainless steel cryostat of equal mechanical properties.
The inner cryostat contains the liquid xenon volume and the TPC. The TPC walls are formed by PTFE reflectors of 3 mm thickness with high reflectivity for the vacuum ultra-violet (VUV) scintillation light, surrounded by 92 cylindrical copper field shaping rings. The structure is reinforced with 24 PTFE support pillars. Titanium frames at the bottom and top of the TPC support the electrodes to establish drift and extraction fields. Two photosensor arrays are located at the top and bottom of the TPC cylinder, consisting of a structural copper support, a PTFE reflector disk, the VUV-sensitive photosensors and the sensors' cold electronics. Because the final sensor type is yet to be chosen for DARWIN and R&D on light sensor options [15,16,17,18] is ongoing, the top and bottom sensors have, for the majority of simulations, been simplified to two disks which properly account for the material budget and the associated activities of radioactive isotopes. This allows for a direct comparison between a baseline scenario with PMTs and an alternative based on SiPMs.
All the major components included in the simulations are listed in Table 1. The assumed radioactivity levels of the materials are discussed in Sect. 4 and listed in Table 2.
Table 1 List of detector components included in the geant4 geometry model of DARWIN stating their material composition and total mass
\(0\nu \beta \beta \) signal events in liquid xenon
In a \(0\nu \beta \beta \)-decay, the energy \(Q_{\beta \beta }\) is released mainly in the form of kinetic energy of the two electrons. In liquid xenon, the electrons thermalize within \(\mathcal {O}\)(mm) resulting in a single site (SS) signal topology, as shown in Fig. 2 (left). Bremsstrahlung photons emitted during electron thermalization travel some distance without energy deposition before scattering or being absorbed. Abundantly emitted low energy photons are likely to deposit their energy close to the decay position and remain unresolved in the DARWIN detector. Photons with energies above 300 keV have a mean free path of more than 15 mm and might travel larger distances before interacting. This can result in an energy deposition which is spatially separable and can cause a false identification as a multi site (MS) event, Fig. 2 (right).
Energy depositions are therefore spatially grouped using a density-based spatial clustering algorithm [19]. An energy deposition is considered as a new cluster if its distance to any previous energy deposition is larger than our selected separation threshold \(\epsilon \). Figure 3 shows the efficiency for signal acceptance and background rejection for photons and electrons with an energy of \(Q_{\beta \beta }\) as a function of \(\epsilon \).
Simulated energy deposition (color scale) of two different \(0\nu \beta \beta \)-events in the x-y-plane. Left: The two electrons thermalize along two non-resolvable back-to-back tracks. The emitted Bremsstrahlung photons yield detached energy depositions. Right: A \(\mathcal {O}\)(400 keV) photon Compton scatters 8 mm from the position of the decaying \(^{136}\)Xe nucleus and travels more than 2 cm without energy loss before absorption. The circles indicate the boundaries of individually resolvable clusters assuming a separation threshold \(\epsilon = {15}\,{\hbox {mm}}\)
The distribution of energy per electron and the angle between the two depend on the yet unknown decay mechanism. We assume a mass mixing mechanism and the most probable decay where the electrons are emitted back-to-back, each with a kinetic energy of \(Q_{\beta \beta }/2\). This assumption is compared in Fig. 3 to the predicted energy and angular distributions in the mass mixing (MM) model and a right-handed current (RHC) model presented in [20].
Efficiency of \(0\nu \beta \beta \) signal acceptance and background rejection as a function of the spatial separation threshold \(\epsilon \). The three signal lines (blue) compare different energy and angular distributions based on a back-to-back electron emission, a mass mixing (MM) mechanism and a right-handed current (RHC) model. The background rejection efficiency is shown for \(\gamma \)s (red) and electrons (green) with \(E = Q_{\beta \beta }\). The vertical line (grey) corresponds to the value of \(\epsilon \) assumed in this study. Bands indicate \(\pm 2\,\sigma \) uncertainties
We assume that a spatial separation between energy depositions of \(\epsilon = {15}\,{\hbox {mm}}\) can be resolved in the DARWIN TPC. This results in a signal acceptance of \(90.4\%\) (MM: \(88.7\%\), RHC: \(86.6\%\)) as SS events. Background events from electrons and photons with \(Q_{\beta \beta }\) energy are rejected as MS with an efficiency of \(17.7\%\) and \(85.1\%\), respectively. A smaller separation threshold \(\epsilon \) results in a larger fraction of misidentified \(0\nu \beta \beta \)-decays. Simultaneously, electrons from \(\beta \) decays and \(\gamma \)-ray events are more efficiently identified as MS. As we will discuss in Sect. 7, a lower spatial threshold can increase the sensitivity to \(0\nu \beta \beta \)-decays. The decrease in signal acceptance is overcompensated by the improved background rejection.
Background events in the \(0\nu \beta \beta \) energy range
We discuss all background sources which contribute events within the energy range of [2.3–2.7] MeV around \(Q_{\beta \beta }\). We consider intrinsic background events from radioactive decays (radiogenic) and those induced by cosmic neutrinos, muons and their secondaries (cosmogenic). Intrinsic events are homogeneously distributed in the liquid xenon. Likewise we study radiogenic background radiation from external sources emanating into the target.
Table 2 Assumed activity levels for the simulated materials and isotopes
Homogeneously distributed intrinsic background
The intrinsic background sources originate from noble gas isotopes or from interactions of cosmogenic particles with the xenon target:
\(^{8}\)B solar neutrinos are an irreducible background source. The expected rate of \(\nu \)-\(e^-\) scatterings is derived assuming a \(^{8}\)B-\(\nu \) flux of \(\phi = (5.46 \pm 0.66)\times 10^{6}\) cm\(^{-2}\) s\(^{-1}\) [21]. The calculation of scattering cross sections follows [22]. The electron neutrino survival probability is conservatively estimated to be \(P_{ee} = 0.50^{+5 \%}_{-30 \%}\) for neutrinos with \(E_\nu > Q_{\beta \beta }\).
\(^{137}\)Xe from cosmogenic activation: muon-induced neutrons produced in the liquid xenon can thermalize and be captured on a \(^{136}\)Xe nucleus, producing \(^{137}\)Xe, as measured by EXO-200 [23]. This isotope decays via a \(\beta ^{-}\) process with Q\(_\beta \) = 4.17 MeV and a half-life of 3.82 min. Assuming the depth of LNGS and previous simulations of the muon-induced neutron flux underground [24], we estimate the muon-induced \(^{137}\)Xe production rate in DARWIN to be \((6.9 \pm 0.4)\) atoms/(t\(\cdot \)year). Neutrons produced in the solid materials contribute about 5% of this rate. Activation of \(^{136}\)Xe due to radiogenic neutrons from the TPC materials has been found to be subdominant by more than two orders of magnitude. Activation of xenon in the non-shielded environment of the purification loop is non-negligible, but can be efficiently suppressed by a delayed re-feed of the LXe into the detector. Suppression by three orders of magnitude adds an additional 225 kg to the total xenon budget when cycling 1000 standard liter per minute.
The \(2\nu \beta \beta \) decay spectrum of \(^{136}\)Xe has been simulated assuming the measured half-life of \(T_{1/2}=(2.165\pm 0.061)\times 10^{21}\) year [25]. For the analytic spectrum we use the non-relativistic Primakoff-Rosen approximation for the interaction between nuclei and electrons in the parametrization discussed in [26]. This approximation is conservative as it overestimates the rate around the spectral end point.
\(^{222}\)Rn in LXe is assumed to be reduced by online cryogenic distillation [27] and stringent material selection to a concentration equivalent to 0.1 \(\mu \)Bq \(^{222}\)Rn activity per kg of xenon. Being crucial for the WIMP search, significant efforts are being undertaken to reach this design goal. The dominant intrinsic background contribution for the \(0\nu \beta \beta \) search originates from the \(\beta \)-decay of \(^{214}\)Bi (Q\(_\beta \) = 3.27 MeV). In 19.1\(\%\) of the cases it decays to the \(^{214}\)Po ground state without \(\gamma \)-emission, which renders the rejection based on spatial topology rather inefficient, as discussed in Sect. 3. The short half-life of the decay daughter \(^{214}\)Po (\(T_{1/2} = 164.3\,\mu \hbox {s}\)), however, allows for BiPo event tagging and suppression with more than 99.8 % efficiency [28].
External radiogenic background sources
Long-lived radionuclides are present in each detector material. Their decays, as well as the subsequent decays of their daughter isotopes, might introduce background in the target. Activity levels for all materials are listed in Table 2 and based on reports from previous or ongoing experiments [13, 14].
The natural decay chains of \(^{238}\)U, \(^{232}\)Th and \(^{235}\)U yield a background contribution primarily from \(\gamma \)-rays emitted by \(^{214}\)Bi- (E\(_\gamma \) = 2.45 MeV) and \(^{208}\)Tl-decays (E\(_\gamma \) = 2.61 MeV). The former two chains were split into their early and late component at \(^{226}\)Ra and \(^{228}\)Th, respectively, to account for radiogenic non-equilibrium.
\(^{60}\)Co \(\beta \)-decays dominantly (99.95%) via the two excited states of \(^{60}\)Ni. The de-excitation is temporally non-resolvable and spatial coincidences of the 1.17 MeV and the 1.33 MeV \(\gamma \)-events contribute to the background.
Among the radio-isotopes from cosmogenic material activation at sea level [29], \(^{44}\)Ti in the cryostat material is the most relevant, due to its long half-life (\(T_{1/2}\) = 59.1 year) and the subsequent decay of \(^{44}\)Sc with \(\gamma \)-emission at 2.66 MeV.
\(^{222}\)Rn contamination in the non-instrumented xenon surrounding the TPC can contribute to the \(^{214}\)Bi-induced \(\gamma \)-background. The rejection based on BiPo tagging described above cannot be applied since the subsequent alpha decays are not observed.
Analysis and background results
The background sources discussed in Sect. 4 are simulated with the geant4 particle physics simulation toolkit [30], using the detector model presented in Sect. 2.1. The equivalent of at least 100 years of DARWIN run time has been simulated for each material and isotope. In this section, we discuss the methods applied for event selection. The analytical background model, used for the profile-likelihood analysis in Sect. 6.2, is also described, and the background results are discussed.
Monte Carlo data processing and event selection
The energy depositions generated by geant4 per event undergo a density-based spatial clustering algorithm [19] to topologically distinguish signal-like single site (SS) from background-like multi site (MS) events, as discussed in Sect. 3. We assume a separation threshold \(\epsilon = {15}\,{\hbox {mm}}\) for the DARWIN TPC. This comparatively coarse clustering inevitably results in a fraction of \(\gamma \)-accompanied \(\beta \)-decays from background events, e.g., \(^{214}\)Bi decays which frequently occur with higher multiplicity, being falsely identified as SS and consequentially contributing to the background.
To account for the finite energy resolution of the detector, the combined energy deposited inside each cluster is smeared according to a resolution of
$$\begin{aligned} \frac{\sigma _E}{E} = \frac{a}{\sqrt{E[\mathrm {keV}]}} + b, \end{aligned}$$
with \( a = (0.3171 \pm 0.0065)\) and \(b=(0.0015 \pm 0.0002)\). At \(E = Q_{\beta \beta }\) this corresponds to \(\sigma _E / E= 0.8 \%\), as demonstrated in the XENON1T TPC [31]. The cluster position is smeared to account for the detector's spatial resolution which is conservatively assumed to be \(\sigma _{x,y}= \sigma _{z} = {10}\,{\hbox {mm}}\) above 2 MeV.
Constraining the target to a super-ellipsoidal-shaped fiducial volume (FV) allows us to exploit the excellent self-shielding capabilities of liquid xenon. To compensate for the reduced shielding power in the xenon gas phase, the FV is shifted slightly downwards from the center of the instrumented volume. The fiducial volume is optimized for each FV mass independently. We use the lifetime-weighted combined external background, after the selection of single site events, energy and spatial resolution smearing. Only events with an energy inside the \(0\nu \beta \beta \)-ROI of [2435–2481] keV, defined as the full width at half maximum (FWHM) range of the expected signal peak, are considered. The spatial distribution of external background events inside the active volume is shown in Fig. 4.
Spatial distribution of external background events inside the instrumented volume for 100 years of DARWIN run time. The colored lines indicate the contours of the optimized fiducial volumes containing different LXe target masses. The 5 t fiducial volume is used for the sensitivity estimate presented below
Background model and fiducial mass dependence
The selection of events within a fiducial volume removes all \(\alpha \)- and \(\beta \)-contributions originating from external sources. The \(\gamma \)-background is shown in Fig. 5 (bottom) for the 20 t fiducial volume. In the \(0\nu \beta \beta \)-ROI the background is composed of the absorption peak from \(^{214}\)Bi at \(E_{\mathrm {Bi}} = {2.45}\,{\hbox {MeV}}\) and Compton scattered photons, mainly from the \(^{208}\)Tl line (\(E_{\mathrm {Tl}} = {2.61}\,{\hbox {MeV}}\)). Compton scatterings inside the fiducial volume with the subsequent escape of the scattered lower energy \(\gamma \)-ray are strongly suppressed by fiducialization. The continuous background is dominated by photons that undergo an undetected Compton scatter outside the detector followed by their absorption in the fiducial volume.
Composition of the material-induced external background in the 20 t fiducial volume. Top: Relative contribution to the background in the \(0\nu \beta \beta \)-ROI by material and isotope. Bottom: Background spectra by isotope with the corresponding model fits. The relative contributions and spectral shapes are representative for smaller fiducial volumes
The continuous contribution from \(\gamma \)-rays emitted in \(^{44}\)Sc decays accounts for less than 1% of the external background. \(^{214}\)Bi decays with \(E_\gamma > Q_{\beta \beta }\) contribute with a similarly subdominant level. Spatial coincident absorption of both \(^{60}\)Co gammas accounts for only approximately \(10^{-3}\) of the total material background at \(E = {2.51}\,{\hbox {MeV}}\) in the 30 t fiducial volume. In the fiducial volume mass range of interest, it can be considered negligible. The largest background contribution in the ROI is induced by the absorption peak of 2.45 MeV \(\gamma \)-rays emitted by \(^{214}\)Bi decays in the detector materials. The contribution from \(^{214}\)Bi decays in the non-instrumented LXe around the TPC accounts for approximately 0.1% of the total material-induced background.
The relative contributions to the \(\gamma \)-background in the ROI are shown per material of origin in Fig. 5 (top). The similar contribution of cryostat-induced events from the walls and the combined PMT and electronics background originating from the top and bottom sensor array is a result of the optimization of the fiducial volume, which is properly balancing the r- and z-extent.
The spectral shape of the material-induced \(\gamma \)-back-ground is modelled with a Gaussian peak and an exponentially decreasing continuum for each line, as shown in Fig. 5 (bottom). We consider the 2.61 MeV \(^{208}\)Tl peak, the 2.66 MeV \(^{44}\)Sc peak and each contribution of \(^{214}\)Bi with \(E_\gamma > {2.0}\,{\hbox {MeV}}\). The ratio between the \(^{214}\)Bi and the \(^{44}\)Sc peaks to the \(^{208}\)Tl peak intensity is established using Monte Carlo data in fiducial volumes sufficiently large to provide high statistics. Similarly, each continuum contribution is tied to its corresponding peak intensity and a fixed relation between the three slope parameters is found. The only remaining free parameters of the combined model are the total intensity of the \(^{208}\)Tl peak and one common slope parameter. The model is tested and confirmed using a \(\chi ^2\) goodness-of-fit test on the combined external background in the fiducial mass range \(\le {20}\,{t}\).
The intrinsic background from \(^8\)B neutrinos is assumed to be flat. The spectra corresponding to \(^{137}\)Xe and \(^{222}\)Rn are approximated linearly falling in the [2.2–2.8 MeV] range. The slopes are obtained from Monte Carlo studies. The \(2\nu \beta \beta \) spectrum is convolved with the Gaussian energy resolution.
The suppression of the external background with decreasing fiducial mass is shown in Fig. 6, together with the target mass independent intrinsic contributions.
Background rate in the ROI versus fiducial mass. External contributions are combined. Fiducial volume independent intrinsic sources are shown per contribution. Bands indicate \(\pm 1\,\sigma \) uncertainties. At \({5}\,\hbox {t}\), the external sources contribute at the same level as the combined intrinsic background
Background rates in the \(0\nu \beta \beta \)-ROI
The fiducial volume is optimized for \(T^{0\nu }_{1/2}\) sensitivity, as discussed in detail in Sect. 6.1, and yields 5 t. The resulting background spectrum from intrinsic and external sources is shown for this fiducial mass in Fig. 7.
Table 3 Expected background index averaged in the \(0\nu \beta \beta \)-ROI of [2435–2481] keV, corresponding event rate in the 5 t FV and relative uncertainty by origin
The intrinsic background in the ROI is dominated by the gently falling \(\beta ^-\)-spectrum of \(^{137}\)Xe decay. Subdominant contributions are the electron scattering of solar \(^8\)B neutrinos and \(\beta ^-\)-events from \(^{214}\)Bi-decays which are not vetoed by BiPo tagging. The \(2\nu \beta \beta \) spectrum overlaps negligibly with the ROI, but dominates the background toward lower energies.
The model-estimated background indices for all contributions are summarized in Table 3. To validate the analytic model introduced in Sect. 5.2, we compare the background model estimate with the values derived by weighted event counting in the 5 t fiducial mass data from Monte Carlo. Both results are in agreement within the statistical errors. The model-derived uncertainty on the background, however, is a factor of 4 lower than the Poissonian statistics error in the simple counting approach. The uncertainties on intrinsic background sources account for statistical errors, the variation of the overlap with the \(0\nu \beta \beta \)-ROI based on the energy resolution and systematic uncertainties from (theory-driven) input parameters. The dominant contributions are the \(\nu _e\) survival probability and the neutrino flux (\(^8\)B \(\nu \)-\(e^-\) scattering), the \(^{136}\)Xe neutron capture cross-section (governing the \(^{137}\)Xe production rate) and the half-life of \(^{136}\)Xe (\(2\nu \beta \beta \) decay).
Predicted background spectrum around the \(0\nu \beta \beta \)-ROI for the 5 t fiducial volume. A hypothetical signal of 0.5 counts per year corresponding to \(T^{0\nu }_{1/2} \approx 2\times 10^{27}\,\hbox {year}\) is shown for comparison. Bands indicate \(\pm 1\,\sigma \) uncertainties
Sensitivity calculation
We use the background rates predicted in Sect. 5.3 to derive a limit on the half-life sensitivity at 90% confidence level (CL) as well as the \(3\,\sigma \) discovery potential for the \(0\nu \beta \beta \)-decay. The latter is defined as the minimal value of \(T_{1/2}^{0\nu }\) required to exclude the null hypothesis with a median significance of 99.7% CL.
Half-life sensitivity estimation
Based on the figure-of-merit estimator proposed in [32] we calculate the half-life sensitivity at 90% CL as:
$$\begin{aligned} T_{1/2}^{0\nu } = \ln {2} \frac{\epsilon \, f_{\text {ROI}}\, \alpha \, N_{A}}{1.64 \, M_{\mathrm {Xe}}} \frac{\sqrt{Mt}}{\sqrt{B\varDelta E}} , \end{aligned}$$
with \(\epsilon = 0.9\) being the detection efficiency of a single site \(0\nu \beta \beta \)-decay event, \(f_{\text {ROI}}=0.76\) the fraction of signal covered by the ROI, \(\alpha = 0.089 \) the abundance of \(^{136}\)Xe in natural xenon, \(N_{A}\) the Avogadro number in mol\(^{-1}\), \(M_{\textit{Xe}}\) the molar mass number of xenon in t/mol, M the fiducial mass in tons, t the exposure time in years, B the background index in t\(^{-1}\)year\(^{-1}\)keV\(^{-1}\), and \(\varDelta E\) the width of the ROI in keV. The value 1.64 is the number of standard deviations corresponding to a 90% CL.
Following Eq. (3) and using the background index for the 5 t fiducial mass (Table 3), we obtain a half-life sensitivity of \(2.0\times 10^{27}\,\hbox {year}\) (\(1.3\times 10^{27}\,{\hbox {year}}\)) after 10 (4) years of exposure.
This figure-of-merit estimation is an established tool to directly compare \(0\nu \beta \beta \) sensitivities of different experiments using common statistical methods and assumptions. It also allows for a straightforward assessment of the sensitivity as a function of different parameters, such as the fiducial mass. It does not, however, consider background uncertainties, but assumes perfect knowledge of the background rates.
Frequentist profile-likelihood analysis
To account for and effectively constrain the background uncertainties, we apply a profile-likelihood analysis based on the background model discussed in Sect. 5.2. The inserted signal is a Gaussian peak with \(Q_{\beta \beta }\) and \(\sigma _E(Q_{\beta \beta })\) according to Eq. (2), which is scaled by the \(^{136}\)Xe atoms in the target volume, an activity corresponding to \(T_{1/2}^{0\nu }\) and the detection efficiency, as shown in Fig. 7.
Background uncertainties from the model are treated as nuisance parameters with Gaussian constraining terms in the likelihood. For external background contributions, their variances are obtained either by the model fit on the spectrum corresponding to 5 t FV (\(^{208}\)Tl peak intensity and slope parameter) or extrapolation of the model parameters from larger fiducial volumes into the low fiducial mass range (\(^{214}\)Bi / \(^{208}\)Tl peak ratio, \(^{208}\)Tl continuum / \(^{208}\)Tl peak intensity). The uncertainty on the subdominant contribution from \(^{44}\)Sc has been neglected. For the intrinsic contributions, the variances correspond to the square of the errors listed in Table 3. The corresponding slope uncertainties are negligible.
We obtain a \(T_{1/2}^{0\nu }\) sensitivity limit of \(2.4\times 10^{27}\,\hbox {year}\) for a 10 year exposure with 5 t fiducial mass. The corresponding \(3\,\sigma \) discovery potential after 10 years exposure is \(1.1 \times 10^{27}\,\hbox {year}\).
The DARWIN observatory will reach a sensitivity to the neutrinoless double beta decay of \(^{136}\)Xe of \(2.4\times 10^27\) years \(T_{1/2}\) exclusion limit (90% CL) and a discovery sensitivity (\(3\,\sigma \)) of \(T_{1/2} = 1.1\times 10^27\) years after 10 years of exposure.
In the baseline scenario discussed above, the assumptions on radio-purity and detector performance are considered realistic or even conservative. In an optimistic scenario, the external background could be reduced by a factor of three or more. The required measures include the use of less radioactive PMTs (with reduced mass of ceramic [33]) and/or low radioactivity SiPMs, more stringent material selection to reach lower levels of radio-activity for PTFE [34], copper [35] and titanium, as well as more radio-pure electronics.
Intrinsic backgrounds, dominated by the muon-induced activation of \(^{136}\)Xe, are difficult to mitigate assuming the muon flux at 3500 meter water equivalent (mwe) depth of LNGS. A time- and spatial- muon veto might allow for suppression by up to a factor of two at an acceptable exposure loss. The \(^{137}\)Xe contribution would, however, become subdominant in a sufficiently deep laboratory. A total intrinsic background suppression by a factor of five or even eight could then be reached assuming a reduced BiPo tagging inefficiency of \(0.1\%\) and \(0.01\%\), respectively. Assuming a factor five reduction in external sources the latter scenario leads to a solar \(^8\)B neutrino dominated background.
The sensitivity could be increased by further exploitation of the SS/MS discrimination, discussed in Sect. 3. Despite increased signal rejection, the gain in background reduction dominates for spatial separation thresholds down to \(\epsilon = {3}\,{\hbox {mm}}\). The cluster separation in the x-y-plane would benefit from a higher granularity photosensor top array, featuring e.g. SiPMs. The z-position reconstruction is already more accurate and a combined three dimensional charge signal analysis will optimize the separation.
The largest sensitivity increase can be achieved with a combination of the above mentioned measures. Figure 8 shows the fiducial volume mass dependency (top) and time evolution (bottom) of the DARWIN half-life limit sensitivity (90% CL) calculated with the figure-of-merit estimator (see Sect. 6.1) for the baseline and different optimistic scenarios. The latter assume reduced spatial separation threshold \(\epsilon \), intrinsic and external background rates. Figure 9 translates the half-life limit sensitivity to the effective Majorana neutrino mass \(m_{\beta \beta }\) using Eq. (1), where the \(m_{\beta \beta }\) range corresponds to the range of published nuclear matrix elements [36]. Under the conservative baseline assumptions, DARWIN reaches a \(m_{\beta \beta }\) limit of [18–46 meV]. The neutrino dominated scenario yields a limit in the [11–28 meV] range. Future dedicated neutrinoless double beta decay experiments using either \(^{136}\)Xe or other isotopes are aiming for a similar science reach as DARWIN, as shown for comparison in Table 4 and in Fig. 8 (bottom).
DARWIN median \(T_{1/2}^{0\nu }\) sensitivity at 90% CL as a function of fiducial volume mass for 10 years of exposure (top) as well as of the exposure time for the optimized fiducial volume (bottom). The baseline design is compared with different optimistic scenarios. The latter assume a reduction of the external (ext.) and the intrinsic (int.) backgrounds and improved spatial separation threshold of 10 mm (red, blue) or 5 mm (green). Sensitivity projections for future \(^{136}\)Xe \(0\nu \beta \beta \) experiments are shown for comparison [8,9,10, 37]
Table 4 Comparison of \(T_{1/2}^{0\nu }\) and \(m_{\beta \beta }\) sensitivity limits (90% CL) between DARWIN and future \(0\nu \beta \beta \) experiments. For experiments using \(^{136}\)Xe the \(m_{\beta \beta }\) ranges are calculated with the nuclear matrix element ranges from [36], those using other isotopes are taken from [37]
The objective of detecting particle dark matter with a sensitivity down to the neutrino floor requires the DARWIN observatory to be an ultra-low background experiment. It additionally features a high \(^{136}\)Xe target mass, excellent energy resolution and single site discrimination capability. In the presented baseline scenario DARWIN will reach a sensitivity that approaches that of the tonne-scale proposed \(0\nu \beta \beta \) experiments. Under more optimistic assumptions, requiring adaptations to the baseline design, DARWIN will explore the full inverted hierarchy and will compete with the most ambitious proposed \(0\nu \beta \beta \) projects.
Effective Majorana neutrino mass vs. lightest neutrino mass. The sensitivity reach after 50 t\(\times \)year of exposure is shown for the baseline and the optimistic neutrino dominated scenario. The horizontal bands stem from the range of nuclear matrix elements [36]. Global sensitivity according to [38], oscillation parameters from [39, 40]
This manuscript has associated data in a data repository. [Authors' comment: Data is made available on reasonable request to the collaboration or the corresponding authors.]
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This work was supported by the Swiss National Science Foundation under Grants No 200020-162501 and No 200020-175863, by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreements No 674896, No 690575 and No 691164, by the European Research Council (ERC) grant agreements No 742789 (Xenoscope) and No 724320 (ULTIMATE), by the Max-Planck-Gesellschaft, by the Deutsche Forschungsgemeinschaft (DFG) under GRK-2149, by the US National Science Foundation (NSF) grants No 1719271 and No 1940209, by the Portuguese FCT, by the Netherlands Organisation for Scientific Research (NWO), by the Ministry of Education, Science and Technological Development of the Republic of Serbia and by grant ST/N000838/1 from Science and Technology Facilities Council (UK).
Department of Physics and Astronomy, University of Bologna and INFN-Bologna, 40126, Bologna, Italy
F. Agostini, P. Di Gangi, G. Sartorelli & M. Selvi
LPNHE, Université Pierre et Marie Curie, Université Paris Diderot, CNRS/IN2P3, 75252, Paris, France
S. E. M. Ahmed Maouloud, L. Scotto Lavina & J. P. Zopounidis
Institut für Kernphysik, Westfälische Wilhelms-Universität Münster, 48149, Münster, Germany
L. Althueser, M. Murra, D. Schulte, C. Weinheimer & C. Wittweg
LIBPhys, Department of Physics, University of Coimbra, 3004-516, Coimbra, Portugal
F. Amaro, J. Cardoso, J. M. F. dos Santos, J. A. M. Lopes, C. M. B. Monteiro & M. Silva
Vinca Institute of Nuclear Science, University of Belgrade, Mihajla Petrovica Alasa 12-14., Belgrade, Serbia
B. Antunovic, S. Milutinovic, M. Pandurovic & D. Vorkapic
Physics Department, Columbia University, New York, NY, 10027, USA
E. Aprile & F. Gao
Physik-Institut, University of Zurich, 8057, Zurich, Switzerland
L. Baudis, Y. Biondi, A. Bismark, A. Brown, C. Capelli, M. Galloway, F. Girard, A. Manfredini, R. Peres, S. Reichard, K. Thieme & G. Volta
Physikalisches Institut, Universität Freiburg, 79104, Freiburg, Germany
D. Baur, A. Bismark, J. Dierle, H. Fischer, F. Kuger, S. Lindemann, P. Meinhardt, D. Ramírez García, P. Sanchez-Lucas & M. Schumann
Nikhef and the University of Amsterdam, Science Park, 1098XG, Amsterdam, The Netherlands
P. A. Breur, A. P. Colijn, M. P. Decowski & A. Loya Villalpando
New York University Abu Dhabi, Abu Dhabi, United Arab Emirates
G. Bruno & A. Di Giovanni
Department of Particle Physics and Astrophysics, Weizmann Institute of Science, 7610001, Rehovot, Israel
R. Budnik & H. Landsman
Max-Planck-Institut für Kernphysik, 69117, Heidelberg, Germany
D. Cichon, F. Jörg, M. Lindner, T. Marrodán Undagoitia, V. Pizzella, N. Rupp & H. Simgen
Department of Physics and Astronomy, Purdue University, West Lafayette, IN, 47907, USA
M. Clark, A. Depoian, G. Khundzakishvili, A. Kopec, R. F. Lang & J. Qin
Institute for Nuclear Physics (IKP), Karlsruhe Institute of Technology (KIT), 76344, Eggenstein-Leopoldshafen, Germany
J. J. Cuenca-García, K. Eitel, R. Engel, F. Glück, R. Größle, R. Gumbsheimer, M. Steidl & K. Valerius
SUBATECH, IMT Atlantique, CNRS/IN2P3, Université de Nantes, 44307, Nantes, France
J. P. Cussonneau, S. Diglio, J. Masbou, M. Pierre, C. Therreau & D. Thers
Institute of Experimental Particle Physics (ETP), Karlsruhe Institute of Technology (KIT), 76344, Eggenstein-Leopoldshafen, Germany
G. Drexlin & J. Wolf
Department of Physics and Chemistry, University of L'Aquila, 67100, L'Aquila, Italy
A. D. Ferella
INFN-Laboratori Nazionali del Gran Sasso and Gran Sasso Science Institute, 67100, L'Aquila, Italy
Department of Physics & Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, 60637, USA
L. Grandi & J. Pienaar
Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany
S. Hansmann-Menzemer & A. Terliuk
The Origins Project Foundation, Phoenix, AZ, 85020, USA
L. M. Krauss
Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405, Orsay, France
C. Macolino & E. Masson
A. Molinario
Institut für Physik & Exzellenzcluster PRISMA+, Johannes Gutenberg-Universität Mainz, 55099, Mainz, Germany
U. G. Oberlack
Department of Physics, Imperial Centre for Inference and Cosmology, Imperial College London, London, SW7 2AZ, UK
R. Trotta
Department of Physics and Astronomy, Rice University, Houston, TX, 77005, USA
C. D. Tunnell
Institute for Nuclear and Particle Physics, TU Dresden, 01069, Dresden, Germany
K. Zuber
F. Agostini
S. E. M. Ahmed Maouloud
L. Althueser
F. Amaro
B. Antunovic
E. Aprile
L. Baudis
D. Baur
Y. Biondi
A. Bismark
P. A. Breur
A. Brown
R. Budnik
C. Capelli
J. Cardoso
D. Cichon
M. Clark
A. P. Colijn
J. J. Cuenca-García
J. P. Cussonneau
M. P. Decowski
A. Depoian
J. Dierle
P. Di Gangi
A. Di Giovanni
S. Diglio
J. M. F. dos Santos
G. Drexlin
K. Eitel
R. Engel
H. Fischer
M. Galloway
F. Gao
F. Girard
F. Glück
L. Grandi
R. Größle
R. Gumbsheimer
S. Hansmann-Menzemer
F. Jörg
G. Khundzakishvili
A. Kopec
F. Kuger
H. Landsman
R. F. Lang
M. Lindner
J. A. M. Lopes
A. Loya Villalpando
C. Macolino
A. Manfredini
T. Marrodán Undagoitia
J. Masbou
E. Masson
P. Meinhardt
S. Milutinovic
C. M. B. Monteiro
M. Murra
M. Pandurovic
R. Peres
J. Pienaar
M. Pierre
V. Pizzella
J. Qin
D. Ramírez García
S. Reichard
N. Rupp
P. Sanchez-Lucas
G. Sartorelli
D. Schulte
M. Schumann
L. Scotto Lavina
M. Selvi
M. Silva
H. Simgen
M. Steidl
A. Terliuk
C. Therreau
D. Thers
K. Thieme
K. Valerius
G. Volta
D. Vorkapic
C. Weinheimer
C. Wittweg
J. Wolf
J. P. Zopounidis
Correspondence to F. Kuger.
Funded by SCOAP3
Agostini, F., Maouloud, S.E.M.A., Althueser, L. et al. Sensitivity of the DARWIN observatory to the neutrinoless double beta decay of \(^{136}\)Xe. Eur. Phys. J. C 80, 808 (2020). https://doi.org/10.1140/epjc/s10052-020-8196-z
DOI: https://doi.org/10.1140/epjc/s10052-020-8196-z | CommonCrawl |
\begin{document}
\title{Globally Optimal Symbolic Regression}
\begin{abstract}
In this study we introduce a new technique for symbolic regression that guarantees global optimality.
This is achieved by formulating a mixed integer non-linear program (MINLP) whose solution is a symbolic mathematical expression of minimum complexity that explains the observations.
We demonstrate our approach by rediscovering Kepler's law on planetary motion using exoplanet data
and Galileo's pendulum periodicity equation using experimental data.
\end{abstract} \section{Introduction} Discovering mathematical models that explain the behavior of a system has a broad range of applications. Symbolic regression is an important field in machine learning whose goal is to find a symbolic mathematical expression that explains a dependent variable in terms of a number of independent variables for a given data set, most commonly as an explicit function of the independent variables. Unlike traditional (numerical) regression schemes, the functional form of the expression is not assumed to be known {\em a priori} \cite{kroll2017workflow}. The utility of the approach has been established for a broad range of applications \cite{connor1977scaling,willis1997genetic,davidson1999method,davidson2003symbolic}, including the discovery of equations \cite{todorovski1997declarative,langley1981data,schmidt2009distil,schmidt2010symbolic}.
Starting with \cite{koza}, symbolic regression problems have been typically solved with genetic programming \cite{banzhaf1998genetic,augusto2000symbolic}, an evolutionary {\it metaheuristic} related to genetic algorithms; another such technique is grammatical evolution \cite{ge}. Even before \cite{koza}, heuristics to find explicit functional relationships were developed as part of the BACON system, see \cite{langley1987}.
There has been much research into improving symbolic regression techniques. In \cite{Luo2017}, the separability of the desired functional form is exploited to speed up symbolic regression, whereas a set of candidate basis functions is used in \cite{ffx} for this purpose. Other recent work has focused on finding accurate constants in the derived symbolic mathematical expression \cite{dgp}. To discover meaningful functions in the context of physical systems,
the authors of \cite{schmidt2009distil} search for functions that not only match the data, but also have the property that partial derivatives also match the empirically computed partial derivatives. The same authors search for implicit functional relationships in \cite{schmidt2010symbolic}. Another approach populates a large hypothesis space of functional forms, on which sparse selection is applied \cite{brunton2016discovering}.
In this study we present a Mixed-Integer Non-Linear Programming (MINLP) formulation that produces the simplest symbolic mathematical expression
that satisfies a prescribed upper bound on the prediction error.
Our MINLP formulation can be solved to optimality using existing state-of-the-art global optimization solvers. The key advantage of our approach is that it produces a {\it globally optimal} mathematical expression while avoiding exhaustive search of the solution space.
Another advantage is that it produces correct real-valued constants (within a tolerance) directly; most other methods use specialized algorithms to refine constants \cite{dgp, topchy} and cannot guarantee global optimality. In addition, our formulation can, in principle, seamlessly incorporate additional constraints and objectives to capture domain knowledge and user preferences.
The process of creating a mathematical model of the relationship between the inputs and outputs of a system involves choices which affect model complexity and realism. One must trade-off competing objectives such as accuracy, generalizability, robustness, functional complexity, scalability (computational complexity) and interpretability. Mathematical modeling techniques range from a first-principles approach based on fundamental laws to empirically learned data-driven models in statistics and machine learning.
A first-principles approach was the historical norm before the computing era, and usually requires significant domain expertise. Recently, there has been a lot of progress in the automatic derivation of models in the form of data-driven approaches. Such approaches typically prescribe a functional form for the relationship between the inputs and outputs and then use the data to find the parameters that completely define the function. As an example, in linear regression one assumes a linear relationship and then one solves a least-squares problem to find the coefficients defining the linear relationship. Thus, prescribing a linear functional form also leads to a computationally tractable solution approach. Of course, if the output does not linearly depend on data, the model can be quite inaccurate. More general models (such as polynomials, which can yield an arbitrarily accurate model as they are dense in the set of continuous functions) usually require more data. Such models can be effective when (i) limited domain expertise is available, (ii) a lot of data is available, and (iii) the available data is representative of all system states.
Selecting a functional form a priori thus has many drawbacks: one with few parameters may be too inaccurate; one with too many parameters may yield an accurate model but may need a lot of data, limited generalizabilty, and yield limited interpretational insight.
\textcolor{red}{It is evident that each approach
offers unique advantages while also involving innate shortcomings. The challenge is therefore, to find a way to hybridize the two approaches.
This bring us to the fundamental modeling question as to whether one should derive, or let the data drive?
At some level the two modeling approaches
are very similar, where they primarily differ by the choice of functional form and the level of automation associated with the model formation. There is great synergy in combining the strengths of both. }
\section{A MINLP Formulation for Free-Form Model Discovery}
A symbolic regression scheme consists of a space of valid mathematical expressions together with a mechanism for its exploration.
A mathematical expression can be represented by a (rooted) expression tree where each node is labeled by one of the following entities: operators (such as $+, -$, $\times$, $\log$), variables (i.e., the independent variables), and constants.
Edges of the tree link these entities in a way that is consistent with a prescribed grammar.
For example, an expression tree for $2(w_1+w_2)^3+1$ is:
\begin{center}
{\small\begin{tikzpicture}[scale=1, sibling distance=1cm, grow=right, level distance = 2cm,
edge from parent path= {(\tikzparentnode.east) .. controls +(1,0) and +(-1,0) .. (\tikzchildnode.west)},
every node/.style = {shape=rectangle, rounded corners, draw, align=center, top color=white, bottom color=blue!20}]] \node[label=left:{root}] {$+$} child { node {1} } child { node {$\times$}
child { node {$\mbox{\scriptsize\textsf{pow}}(\cdot,\cdot)$}
child { node {3} }
child { node {$+$}child { node {$w_1$} }
child { node {$w_2$} } }
}
child { node {2} } }; \end{tikzpicture}
}\end{center} Here $+$, $\times$ and the power function are the operators, $w_1,w_2$ are the variables, and numbers $1$, $2$ and $3$ are the constants.
In our MINLP based approach we model the grammar of valid expression trees by a set of constraints. The discrete variables of the formulation are used to define the structure of the expression tree, and the continuous variables to evaluate the resulting symbolic expression for specific numerical values associated with the data. To choose among the many possible expressions that fit the data (within an error bound), we set the objective to minimize the description complexity of the expression. Consequently, the MINLP formulation has the form \begin{align} \min&&\mathcal{C}(f_{cqwz} )& \tag{Complexity}\label{eq:C}\\ \text{s.t.}&&{(c,q,w,z)} &\in \mathcal{T} \tag{Grammar}\label{eq:G}\\
&&v_i &=f_{cqwz}(x^i) ,~~\forall i\in I \tag{Prediction}\label{eq:M}\\
&& \mathcal{D}(v,y) &\le \epsilon \tag{Error}\label{eq:E} \end{align} where $c,q,w$ and $z$ are decision variables, $f_{cqwz}$ is the expression tree defined by these variables, $\mathcal{T}$ is the set of values of the decision variables that define valid expression trees of bounded size, $\mathcal{C}$ measures the description complexity of the expression tree, $\mathcal{D}$ measures error of the predicted values $v$, and $(x,y)$ are the observed data. In practice, MINLP solvers (e.g., BARON \cite{ts05}, COUENNE\cite{BeLeLiMaWa08}, SCIP\cite{MaherFischerGallyetal2017}) employ various convex relaxation schemes to obtain lower bounds on the objective function value and use these bounds in divide-and-conquer strategies to obtain {\it globally optimal solutions} \cite{ts05}. We note that our MINLP approach bears similarities to the one presented in \cite{horesh2016tech}, however unlike our model, the one presented in \cite{horesh2016tech} is computationally impractical.
\newcommand{\mathbb{R}}{\mathbb{R}} \newcommand{\{0,1\}}{\{0,1\}} \newcommand{\setminus}{\setminus} \newcommand{\etx}{expression tree}\newcommand{\et}{{\etx} }
\label{sec:problem} We next give the details of our formulation. The input to the formulation consists of a rooted binary tree, a set of candidate operators and an observed dataset. For each observation $i\in I$, we denote the value of the independent variables by $x^i\in\mathbb{R}^m$ and the dependent variable values by $y^i\in\mathbb{R}$. Let $D=\{1,\ldots,m\}$ denote the indices of the independent variables. Let $T=(N,E)$ be the input binary tree, let $s\in N$ denote root node and $L\subset N$ denote the leaf nodes. Each node $n\in N\setminus\{s\}$ in the tree has exactly one predecessor and each node $n\in N\setminus L$ has exactly two successors. Let $O=U\cup B$ denote the set of candidate operators where $U$ contains the unary operators (such as squareroot) and $B$ contains the binary operators (such as addition). Our formulation has two parts: The first part consists of constraints and decision variables that construct the \et and the second part is used to evaluate the difference between the estimated and actual dependent variable value of each observation using the constructed \etx.
{\bf \ref{eq:G}.} The first part of the formulation chooses a subtree of $T$ and assigns either an operator, a constant, or an input variable to each node of the chosen subtree in a consistent way. For each $n\in N$, we have a decision variable $u_n\in\{0,1\}$ to indicate if the node is used (active) in the \etx.
Decision variables $z_{n,o}\in\{0,1\}$ and $w_{n,d}\in\{0,1\}$ denote whether the operator $o\in O$
or the input variable $d\in D$ are assigned to node $n$ respectively. Finally we have a decision variable $q_n\in\{0,1\}$ if a constant value is assigned the node. The constraint \begin{equation}q_n +\sum_{d\in D} w_{n,d}+\sum_{o\in O} z_{n,o}= u_n, ~~ \forall n\in N,\label{eq:ops}\end{equation} enforces that each active node is assigned an operator, a constant, or one of the variables. Let $n\in N\setminus L$ be a non-leaf node with successors $l,r\in N$ where $l$ has a smaller index than $r$. Note that if node $n$ is assigned a binary operator then both nodes $l,r$ have to be active in the tree; and if $n$ is assigned a unary operator then exactly one of the nodes (say $l$) has to be active. Consequently, we have constraints $u_r= \sum_{o\in B} z_{n,o}$ and $u_l= \sum_{o\in B\cup U} z_{n,o}$. Notice that these constraints also enforce that if a node is inactive, or is assigned a constant or is a variable, then its successor nodes cannot be active. As the leaf nodes cannot be assigned operators, we also set $ z_{n,o}=0$ for all ${o\in O}, l\in L$. Finally, we define a continuous variable $c_n$
for $n\in N$ to denote the value of the constant when $q_n=1$. It is possible to show that any $c\in\mathbb{R}^{|N|}$ together with binary vectors $q,w,z$, and $u$ (of appropriate dimensions) that satisfy the constraints above define an \et and vice versa.
In addition to the basic model described above, we impose some additional constraints to improve computational performance (without compromising optimality). For example we make sure that if a node is assigned a binary operator, then both its successor nodes cannot be assigned constants at the same time. Similarly the left successor of a unary operator cannot be a constant. We also have several constraints to deal with symmetry as \etx s are inherently symmetric objects in the sense that the same function can be represented with different trees, for example by flipping two branches succeeding a commutative binary operator.
To avoid numerical problems, we also set bounds on the absolute value of the continuous variables.
{\bf \ref{eq:M}. } Using the first set of variables, the second part of the formulation computes the value of the function (defined by the \etx) for each observation. More precisely, we define a variable $v_{n,i}$ to denote the value of node $n\in N$ for observation $i\in I$. Value of a node clearly depends on the operator assigned to it and the values of its successor nodes. For example, if the addition operator $a\in B$ is assigned to node $n\in N$ that has successor nodes $r,l\in N$, then $v_{n,i}$ would be the sum of $v_{r,i}$ and $v_{l,i}$. Therefore we have a constraint: \begin{equation}v_{n,i}=q_nc_n+\sum_{d\in D}w_{n,d}x^i_d+\sum_{o\in B}z_{n,o}f_o(v_{r,i},v_{l,i})+\sum_{o\in U}z_{n,o}f_o(v_{l,i})~~~~\forall i\in I,~n\in N,\label{eq:val}\end{equation} where $f_o(\cdot)$ denotes the function on the argument(s) of the operator $o\in O$. Also note that $v_{n,i}\not=0$ only if $u_n=1$. Leaf nodes cannot be assigned operators, therefore the last two summations in \eqref{eq:val} are zero for these nodes.
{\bf \ref{eq:E}. }Recall that $s\in N$ denotes the root node of the \et and therefore $v_{s,i}$ corresponds to the estimated output variable for observation $i\in I$. We define the error for observation $i\in I$ as $(v_{s,i}-y^i)$ and relative error as $(v_{s,i}/y^i-1)$. Using these expressions, we can define a discrepancy function such as $\mathcal{D}(v_{s},y)=\sum_{i\in
I}(v_{s,i}-y^i)^2$. In our computational experiments, we control the relative error in our model by adding the constraint $\sum_{i\in I}(v_{s,i}/y^i-1)^2\le \epsilon $ for a given $\epsilon > 0$.
{\bf \ref{eq:C}. } We can define a function to measure description complexity of the model in various ways, for example $\mathcal{C}(f_{cqwz})=\sum_{n\in N}\sum_{o\in O}c_{o}z_{n,o},$ where $c_o\in \mathbb{R}$ is a complexity weight assigned to operator $o\in O$ by the user. In our computational experiments, we use the expression $\sum_{n\in N}u_n$ to minimize the total number of nodes in the \etx.
\section{Computational Experiments} We report our computational experience on the discovery of physical laws using real-world datasets. We solve the optimization problem described in Section \ref{sec:problem} with the MINLP solver BARON \cite{ts05} v17.8.9, using IBM ILOG CPLEX and IPOPT \cite{ipopt} as subsolvers. We can only use operators that are handled by BARON. Furthermore, when multiple globally optimal solution exists, we do not have control over which one is returned. Experiments are run on the cloud, so CPU speed cannot be precisely quantified.
{\bf Exoplanet Data.} The first set of experiments concerns the discovery of Kepler's law on planetary motion on a dataset taken from NASA \cite{NASAExoplanetArchive} reporting information on planetary systems. The dataset consists of tuples of the form (planet, star, $\tau, M, m, d$) where $\tau$ is the orbital period of the planet, $M$ the mass of the star, $m$ the mass of the planet, $d$ the major semi-axis of the orbit. Planet features are normalized to Earth's, star features to Sol's. $\tau$ is further normalized since its range is very large. We build three test problems from this dataset, including the systems listed in brackets: EP1 (Sol and Trappist), EP2 (HD 219134 and Kepler), EP3 (GJ 667C, HD 147018, HD 154857, HD 159243 and HD 159868). \begin{table}[tb]
\centering
\caption{Results for the exoplanet dataset. Time limit is 6
hours. Grey cells exceed the time limit.
$c$ represents constants (we do not report its value for brevity).} \small
\begin{tabular}{|l|*{6}{c|}}
\hline
& \multicolumn{6}{c|}{Maximum relative error} \\
\cline{2-7}
Dataset & 2\% & 5\% & 10\% & 20\% & 30\% & 50\% \\
\hline
EP1 & $\sqrt[3]{c\tau^2M}$ & $\sqrt[3]{c\tau^2M}$ & $\sqrt[3]{\tau^2}(M+c)$ & $\sqrt[3]{\tau}(\sqrt{\tau} + M)$ & $\sqrt{c\tau^2}$ & $\sqrt{\tau M}$ \\
EP2 & $\sqrt[3]{c\tau^2M}$ & $c\sqrt[3]{\tau^2}$ & $c\sqrt[3]{\tau^2}$ & $\sqrt[3]{c\tau^2}$ & $\sqrt{\tau}$ & $\sqrt{\tau}$ \\
EP3 & \cellcolor{gray!25} $\sqrt[3]{c\tau^2M}$ & $\sqrt[3]{c \tau^2M}$ & $\sqrt{\tau M} + \tau$ & \cellcolor{gray!25} $\sqrt[3]{c\tau^2M}$ & $\sqrt{c\tau} + c$ & $\sqrt{\tau}$ \\
\hline
\end{tabular}\label{tab:ep} \end{table}
We consider the set of operators $+,*,\sqrt{},\sqrt[3]{}$, and full binary expression trees with depth at most 3 (the root has depth 0) in which each operator can appear at most 3 times. We aim to predict $d$ in terms of the other input variables. Results are reported in Table \ref{tab:ep}, for different upper bounds applied to the relative model error. For small model errors, we always find a refined expression of Kepler's third law: $d = \sqrt[3]{c\tau^2M}$. Given the data, the more comprehensive formula would be $d = \sqrt[3]{c\tau^2(M+m)}$, but the dependency on $m$ is not picked up because $m$ is negligible compared to $M$. As model error upper bound increases, we find simpler formulas that do not accurately reflect Kepler's third law.
Most of the formulas returned are certified globally optimal in a relatively short period of time: on average, 50 minutes for the instances solved to global optimality (two instances hit the time limit and thus are not certified globally optimal within the time limit), standard deviation 75. The number of nodes explored by the Branch-and-Bound algorithm varies greatly: from 147 for the simplest formula $\sqrt{\tau}$, to over 600k for one instance that is not solved to optimality within the allotted runtime. The geometric mean is 19917. By contrast, the number of binary expression trees of depth $3$ with $4$ candidate operations, $3$ input variables, and numerical constants is $\approx 2 \cdot 10^7$ without accounting for symmetry. We remark that the number of nodes explored by Branch-and-Bound does not correspond to the number of candidate expression trees that have been examined: it is merely a measure of difficulty of the search.
{\bf Pendulum Data.} The second set of experiments concerns ten pendulums with different link lengths, and timestamps for the time at which each pendulum crosses the midpoint of its arc. The data is obtained from a high-resolution video recording of the pendulums, yielding hundreds of timestamps. From the raw data, we randomly extract ten tuples (one for each link length) of the form ($\ell, i, t_i, j, t_j$), where $\ell$ is the link length, $i$ and $j$ are integers, $t_i$ and $t_j$ are the timestamps for the $i$-th and $j$-th crossing of the midpoint, and $j > i$. We aim to predict $t_j$. \begin{table}[tb]
\centering
\caption{Results for the pendulum dataset. Time limit is 6 hours.
All formulas are certified optimal.}
\small
\begin{tabular}{|l|*{4}{c|}}
\hline
& \multicolumn{4}{c|}{Maximum relative error} \\
\cline{2-5}
Dataset & 0.5\% & 1\% & 2\% & 5\% \\
\hline
DS1 & $j(\ell - c) -\ell$ & $j\sqrt{\ell} + c$ & $j\sqrt{\ell}$ & $cj$ \\
DS2 & $j\sqrt{\ell}$ & $j\sqrt{\ell}$ & $cj$ & $cj$ \\
DS3 & $j\sqrt{\ell}$ & $j\sqrt{\ell}$ & $\ell - cj$ & $cj$ \\
DS4 & $j(\ell -c) -c$ & $j\sqrt{\ell}$ & $cj$ & $cj$ \\
\hline
\end{tabular}\label{tab:ds} \end{table}
The period of the pendulum is given by $\tau = 2\pi\sqrt{\ell/g}$. Hence, we aim to obtain the equation $t_j = \pi j\sqrt{\ell/g}$. The experimental setup is similar to the exoplanet dataset, but we add ``$-$'' to the list of operators. Table \ref{tab:ds} shows that many experiments return the correct formula except the multiplicative constant, since $\pi/\sqrt{g} \approx 1.002$ and its discrepancy from $1$ is too small to be significant, up to experimental error. With error bound $0.5\%$, we observe overfitting on DS1 and DS4: the simpler formula $j\sqrt{\ell}$ does not satisfy the error bound due to experimental error. Because the link lengths of the ten pendulums are close to each other (ranging from $0.28m$ to $0.307m$), for larger error bounds a simpler expression of the form $cj$ suffices. The average runtime is 2 minutes, standard deviation 2.3. The number of nodes ranges from 57 to 8134, geometric mean 299. Despite the presence of redundant variables, the algorithm quickly identifies the simplest formula explaining the data.
In this extended abstract we focused on small datasets and did not address scalability, which is a known limitation of Branch-and-Bound-based methods for MINLP. This is left for future research.
{\small
}
\end{document} | arXiv |
Forcing of late Pleistocene ice volume by spatially variable summer energy
Kristian Agasøster Haaga ORCID: orcid.org/0000-0001-6880-87251,2,3,
Jo Brendryen1,3,
David Diego1,2 &
Bjarte Hannisdal ORCID: orcid.org/0000-0002-7637-758X1,2,3
Scientific Reports volume 8, Article number: 11520 (2018) Cite this article
Cryospheric science
Palaeoclimate
Changes in Earth's orbit set the pace of glacial cycles, but the role of spatial variability in the insolation forcing of global ice volume remains unknown. Here, we leverage the intrinsic dynamical information in empirical records to show that ice volume responded to summer energy at high northern latitudes, as predicted by Milankovitch theory. However, the external forcing of ice volume encompasses insolation signals with a wide range of orbital frequency content, and cannot be fully accounted for by a unique time series. Southern mid-latitude insolation forcing coincides with the position of the subtropical front and the westerlies, which have been implicated in Quaternary climate changes. Dominant forcing modes at northern mid-latitudes are anti-phased with the canonical Milankovitch forcing, consistent with ice volume sensitivity to latitudinal insolation gradients.
Earth's climate and global ice volume oscillated in tune with changes in orbital geometry during the Quaternary period. Evidence of this relationship is derived from geochemical indicators in marine microfossils found in deep ocean sediment cores. Isotopic ratios in foraminiferal shells (tests) show that Northern Hemisphere ice sheets grew and receded periodically1. This periodicity is clearly visible in global sea level reconstructions2 and is also mirrored in other climate indices such as global sea surface temperature3. Because Earth's rotational axis and the shape of its orbit vary with similar frequencies as the glacial cycles, the orbital changes have been dubbed a pacemaker of Quaternary glacial-interglacial climate variability1. Hypotheses for the origin of the recurrent ice ages include obliquity, precession and combined orbital pacing of deglaciations by various mechanisms1,4,5,6,7,8,9,10,11,12. The extent to which latitudinal insolation acted as a dynamical forcing of late Pleistocene ice volume variability, however, has not been directly quantified.
In the Pleistocene, large paleo-ice sheets were located in the Northern Hemisphere and summer ablation is considered a key factor controlling ice sheet size. Milankovitch theory predicts that if insolation controls continental ice sheet dynamics, then northern latitude summer insolation plays a central role13. We test this prediction by quantifying the dynamical response of global ice volume to latitudinal insolation forcing. To achieve this, we target a recent reconstruction of global sea level (GSL)2 spanning the past 800,000 years. The advantage of using GSL over stacked benthic isotope records (e.g.14) is that it explicitly records global ice volume changes, and that it minimises bias due to temperature-driven fractionation and differences in oxygen isotopes between the Atlantic and Pacific basins2.
The GSL time series does not simply record a response to insolation forcing, but contains information on greenhouse gases, ocean circulation, and other processes in the climate system of which ice volume itself is an active component. However, dynamical systems theory shows that one can reconstruct the dynamics of a system of unknown complexity from an observed time series of a single variable15,16,17. Hence, if latitudinally varying insolation was a dynamical influence on changes in ice volume, then information about this insolation forcing should be recoverable from the GSL record of ice volume variations. To test this assertion, we use a model-free time series analysis method, convergent cross mapping (CCM)18,19. This method is based on state-space reconstruction from time delay embedding, and measures the extent to which a forcing time series can be predicted from a response time series. CCM can detect causal coupling in non-linear dynamical systems18, making the method well suited for studying climate dynamics, where non-linear interactions are ubiquitous. We use CCM to test if GSL can predict latitudinal insolation reconstructions. If GSL significantly predicts the insolation time series at a given latitude, then there is empirical support for local insolation at that latitude contributing to the dynamical forcing of ice volume.
Summer energy time series
Ice sheets are most sensitive to insolation during summer melting season. Choosing a meaningful insolation metric is thus crucial for investigating climate system responses to insolation forcing. Summer energy is defined as the sum of daily insolation over days of the year exceeding a specified insolation intensity threshold20 and varies with latitude and the choice of the threshold value21.
We use summer energy time series over all latitudes in 1° increments, generated at threshold values ranging from 0 Wm−2 to 500 Wm−2 in 25 Wm−2 increments. Integrating summer insolation over low thresholds yields time series representing longer summer periods, including the full annual insolation. Higher thresholds, on the other hand, yield time series representing peak summers (Fig. 1). Each summer energy time series corresponds to spatially separate physical forcing scenarios that span different portions of the year, and we seek to detect the dynamical influence of these local processes on global ice volume. We emphasize that we do not use these 1° increment insolation signals to force an ice sheet model. Instead, we use a model-free approach to quantify the dynamical contribution of local insolation at different latitudes on global ice volume.
Examples of summer energy reconstructions at different latitudes. (A) Yearly insolation intensity (solid lines) at 65°N and 35°N for the current orbital configuration. For low threshold values (stippled line), local summer energy is integrated over large portions of the year (shaded area). (B) For higher threshold values, fewer days are integrated, with summer energy representing the local insolation forcing during peak summer. (C) Corresponding summer energy forcing time series reconstructed for orbital configurations over the past 800 kyr. Latitudinal summer energy is computed using the code accompanying ref.20, which calculates daily insolation following ref.67 with orbital parameters from ref.68.
According to Milankovitch theory, the strongest forcing is expected to occur at latitudes where landmasses were ice-covered during glacial intervals, with weaker or no latitudinal forcing in southern and equatorial parts of the globe. In addition, peak summer is expected to dominate the insolation forcing. Our spatiotemporally explicit approach allows us to test both aspects of the theory without making mechanistic assumptions.
Significance assessment
We proceed by quantifying the extent to which GSL predicts each latitudinal summer energy time series. To determine whether the integrated insolation forcing at a given latitude is significant, we use the method of surrogate testing (see methods). A necessary condition for statistical detection of a causal response of ice volume to insolation forcing at a given latitude is that the prediction of summer energy time series from GSL is significant beyond their shared frequencies. This condition is tested by a null ensemble of amplitude-adjusted Fourier transform (AAFT) surrogates, which are constructed by randomising the data in a way that retains the autocorrelation function, or, equivalently, the periodogram of the original forcing time series. In order to resolve causal directionality, we compute CCM over a range of negative and positive time lags, and require that CCM skill must be stronger for negative lags (causal) than for positive lags (non causal; see methods). If both the surrogate test and the lag test pass, there is statistical evidence of dynamical forcing of sea level by insolation at that latitude.
Predicting summer energy from GSL on an orbitally independent age model
CCM analysis yields three connected regions of positive prediction skill in latitude-threshold space (Fig. 2). These three clusters represent insolation forcing occurring during different portions of the year in distinct latitudinal bands.
Predicting summer energy from global sea level records for the past 800,000 years. Each cell in the heat map indicates the CCM prediction skill when GSL is used to predict summer energy at that latitude and threshold. Low thresholds represent forcing by insolation over large portions of the year, whereas high thresholds represent peak summer insolation forcing. Non-zero skill implies that GSL contains information about summer energy beyond noise and shared frequencies, which, in the context of CCM, is interpreted as dynamical forcing of ice volume by summer energy at that latitude. The magnitude of CCM skill indicates the relative strength of summer energy forcing. Skill is set to zero for latitude-threshold specifications where the lagged causality test fails and/or the null hypotheses cannot be rejected at the 0.01 level (see methods). Contour lines indicate the fraction of obliquity (1/41 ± 1/150 kyr) to precession (1/21 ± 1/150 kyr) frequency band power for the corresponding summer energy time series, from obliquity dominated (dark red lines) to precession dominated (white lines).
In the latitudinal zone at 50–90°N, GSL predicts summer energy time series for a wide range of threshold values, primarily above 250 Wm−2 (cluster NH1 in Fig. 2). Prediction strengths in this latitudinal band are bi-modally distributed south and north of 70°N. North of 70°N, prediction is strongest when summer energy is defined by the 400–450 Wm−2 thresholds. South of 70°N, prediction strength peaks at the 300–350 Wm−2 thresholds. This prediction pattern coincides with the relative dominance of orbital frequencies in the forcing time series: prediction skill is strongest for time series with roughly equal contributions of obliquity and precession (contour lines in Fig. 2). From the maximum prediction strength at around 50/50 obliquity/precession, prediction skills decrease as the summer energy time series get more precession or obliquity dominated; this corresponds to insolation integrated over shorter and longer summer time windows, respectively.
Further, we detect significant prediction of summer energy by GSL in two continuous clusters in latitude-threshold space, one in each hemisphere (clusters NH2 and SH in Fig. 2), where GSL predicts summer energy, primarily for time series corresponding to thresholds of 50–400 Wm−2. The overall pattern for both clusters is that prediction is successful at lower latitudes for high threshold values, transitioning to increasingly higher latitudes as the threshold decreases. However, these clusters are hemispherically asymmetric. NH2 spans the latitudinal zone from 55°N to around the equator, while SH covers 30–65°S. GSL predicts time series with a wide range of obliquity-to-precession ratios for both clusters, but there are some cross-hemispheric differences (Fig. 3). SH corresponds to time series with frequency power distributed over the entire range of obliquity/precession ratios, while time series are more precession dominated for NH2. Annual integrated insolation at low threshold specifications (0–75 Wm−2) is also predicted by GSL. At these thresholds, successful prediction occurs in both hemispheres 45–55° (included in NH2 and SH).
Ratio of obliquity band (1/41 ± 1/150 kyr) to precession band (1/21 ± 1/150 kyr) variance in summer energy time series versus strength of coupling to ice volume indicated by CCM skill. Results are grouped according to the clusters in Fig. 2.
We have verified our results on three different age models for the GSL record. The original GSL age model is based on the LR04 stack, whose age model is tuned to an ice sheet model forced by insolation. This approach may introduce circularity between the orbital forcing and the putative ice volume response. Therefore, we have constructed two alternative age models for the sea level stack (Fig. 4; see methods). One is based on tuning by aligning bandpass-filtered sea level and speleothem records22, and one utilises the connection between North Atlantic sea surface temperatures and Asian monsoon intensity, linking North Atlantic benthic δ18O records with U-Th-dated Chinese speleothem records23,24. Both alternative age models are based on U-Th and ice core data, but the latter uses a tuning independent of both ice volume and orbital parameters. The two alternative models give overall similar non-lagged CCM results as the LR04 age model when predicting summer energy from GSL (Fig. 5). To avoid circularity, we here present results based on the age model that is independent of both sea level and orbital tuning.
Age models used in this study. Upper panel: the black line shows GSL on the original age model2, which is aligned with the LR04 stack14. The blue line shows GSL on the age model ('speleothem') constructed following the approach of ref.22 using band-pass filtering (see methods). The red line shows GSL on an orbitally independent age model ('speleoice') constructed by tuning a composite of North Atlantic SST proxy records to the speleothem record (see methods). Lower panel: age offsets between the original LR04-based GSL chronology and the two alternative age models. Dashed lines show the average offsets.
Comparison of non-lagged CCM results for the different age models considered in this study (see Fig. 4). Overall, CCM skills are higher for the LR04 age model, possibly reflecting stronger coupling induced by the tuning of the LR04 age model to the orbital forcing. All results discussed in the main text are based on the fully orbitally independent 'speleoice' age model.
Our model-free approach, which makes no a priori assumptions about the coupling between ice volume and insolation, neither through explicit mechanisms nor through age model construction, provides strong evidence of Milankovitch type forcing of ice volume. Northern Hemisphere ice sheets dominated the global ice volume signal during the past 800,000 years. The latitudinal zone from 50–80°N corresponds to the known range of land-based Northern Hemisphere ice sheets during the late Pleistocene, which reached as far south as 40°N during glacial maxima25. The successful prediction of summer energy by GSL in the NH1 cluster (Fig. 2) thus provides a data-driven confirmation that Northern Hemisphere summer energy acted as a dynamical forcing of Northern Hemisphere ice sheets.
We re-iterate that CCM does not make assumptions about properties or mechanistic behaviors of the ice sheets, or their interaction with other climate system components. Our approach only targets the intrinsic dynamical information in the observed GSL record, and estimates the strength of dynamical coupling over the entire 800 kyr interval covered by the GSL reconstruction. This way of looking at dynamical causality is fundamentally different from an event-based view26, in that an underlying dynamical coupling between insolation and the climate system might exist even if the relationship between components varies through time (e.g. lead-lag relationships between glacial terminations and insolation peaks). Thus, the detected couplings in our study capture not only direct, linear responses of ice sheets to specific insolation peaks, but also nonlinear, lagged effects which include modulation by other climatic factors such as greenhouse gases.
Causal pathways from local insolation to global ice volume vary with latitude. Different forcing scenarios overlap in their relative ratio of obliquity to precession variability (Figs 2 and 3) and might show strong covariance, but unique latitude-threshold combinations correspond to physically distinct causal chains leading from local insolation to ice volume variations. An example of this distinction is Antarctic climate, which, due to its phase coherence with selected Northern Hemisphere insolation time series, has been interpreted to be controlled by northern insolation27. However, Antarctic climate variability can also be explained by a local response to the duration of the Antarctic summer28. Both local and remote forcings might thus influence a given geographical region. Because of the uncertainty inherent in ice sheet reconstructions, the boundary separating regions of direct insolation influence from regions of indirect influence cannot be precisely resolved. The NH1 cluster represents mostly direct insolation forcing effects, but likely also some indirect effects for the southernmost forcing signals. Similarly, the northernmost forcing signals in NH2 may represent the direct effect of annual insolation on ice sheets at these latitudes.
Our analyses indicate that Northern Hemisphere ice sheets respond to local summer insolation, but that insolation in the SH cluster (Fig. 2) also contributes to ice volume variations. Although the dominant modes of summer energy forcing in the NH1 and SH clusters are redundant (Fig. 6; black and red lines), individual forcing time series are highly variable (Fig. 3). Our interpretation is that there are distinct physical processes occurring during different times of the year in different geographical regions, each having a unique causal pathway to ice volume, that may have worked in tandem to produce the observed global ice volume variations. Covariance between different local summer energy forcing time series might be strong, but local climate necessarily responds to the duration of summer and the magnitude of the integrated insolation intensity at that location, the effect of which may be magnified or suppressed by other climatic processes.
Comparing dominant latitudinal summer energy modes to the canonical Milankovitch forcing signal. The first principal component (PC1) time series of the three significant forcing clusters in Fig. 2 explain 94% (NH1), 77% (NH2), and 82% (SH) of the variance in each cluster. The conventional Milankovitch forcing is represented by summer energy at 65°N for the 350 Wm−2 threshold. For reference, the upper time series shows the sea level record (GSL) used to predict insolation curves (for details on the orbitally independent age model, see Fig. 4).
There are several climatic processes operating at southern mid-latitudes that might be significantly influenced by local insolation. For example, the Patagonian ice sheet resides at these latitudes, and it has been hypothesized that Patagonian ice sheet dynamics affect the flux of dust over the Southern Ocean and Antarctica29. Variation in the dust supply to surface waters drives natural iron fertilisation and regulates the intensity of the biological pump30. This process could influence ocean stratification and venting of CO2 from the deep oceans, resulting in global climatic impacts. We also note that the 40–60°S latitudinal band coincides with the position of the oceanic subtropical and sub-Antarctic front systems and the mid-latitude westerlies31. It has been proposed that the position of the subtropical front relative to the southern tip of Africa modulates glacial climate through regulation of heat and salt exchange between the Indian and Atlantic oceans by the Agulhas leakage, affecting the strength of the Atlantic meridional overturning circulation31,32,33. In addition, the position of the westerlies and Southern Ocean sea ice dynamics may regulate venting of deep water and the release of stored CO2 to the atmosphere, which in turn affects ice sheets in the Northern Hemisphere34.
Our analysis shows no statistically significant effect of southern high latitude summer energy on global sea level. The response to direct insolation forcing at high southern latitudes may be too small in comparison to that of northern hemisphere ice sheets for it to be detectable beyond our chosen null hypothesis. This result may be understood in terms of the difference between the Arctic and Antarctic ice sheets. Because the Antarctic ice sheet mostly loses mass along the continent edges28, it is less affected by albedo and elevation feedbacks35. Its influence on global sea level on the time scales considered here could thus be negligible compared to the Northern Hemisphere ice sheets, which show much larger variability over the same time span.
In addition to the direct effect of local northern insolation, Northern Hemisphere ice sheets were likely also influenced by local and global feedbacks involving basal sliding36, greenhouse gases34, changes in vegetation37, dust38 and glacial isostasy39. The explicit role of different mechanisms in the insolation-climate link, however, cannot be resolved with the global data sets used here. For regions in the Northern Hemisphere south of the maximal extent of the late Pleistocene ice sheets, the influence of local summer energy forcing on ice volume must necessarily be indirect. Overall, insolation forcing in the NH2 cluster is overall more precession dominated compared to further north in NH1 (Figs 2, 3 and 6). The sensitivity of local climatic processes to local insolation forcing at these latitudes was likely different from regions hosting large Northern Hemisphere ice sheets, where the detected insolation forcing is mostly restricted to the summer half-year. In the NH2 cluster, the dominant forcing mode is generally in anti-phase and inversely correlated with the direct Milankovitch type summer forcing of more northern latitudes (Fig. 6; blue line). This inverse relationship is consistent with a role of differential heating due to meridional insolation gradients in the Northern Hemisphere, which may regulate atmospheric fluxes of moisture and heat40,41.
From a linear perspective, we can distil the variance of clusters of summer energy time series in latitude-threshold space using the eigenvector of the covariance matrix of each cluster with the largest corresponding eigenvalue (Fig. 6). For both NH1 and SH clusters, their first principal component time series nearly perfectly covary with 65°N summer energy at the 350 Wm−2 threshold, which closely matches the caloric summer half-year insolation at 65°N that was considered by Milankovitch13. Using 65°N summer energy at 350 Wm−2 as a first approximation of regional insolation forcing of global ice volume on these time scales is thus supported by intrinsic dynamical evidence in the GSL record on an orbitally independent age model. In contrast, the first principal component time series for NH2 is more precession dominated and generally in antiphase with the NH1 and SH principal components. Hence, the dynamics of the summer energy time series in the NH2 cluster represents regional insolation forcing of global ice volume that is not captured by the canonical Milankovitch forcing. We emphasize that although the dominant insolation forcing modes for the NH1 and SH clusters are similar to the canonical Milankovitch forcing (Fig. 6), significant predictability occurs for a range of time series with very different characteristics within each cluster. The nature of our causality test ensures that any inferred dynamical coupling between global ice volume and summer energy on these time scales cannot be accounted for solely by the dominant modes of forcing or by orbital frequency content alone.
Both obliquity and precession components feature prominently in the dynamics of global ice volume. Our analysis shows that there is no clear answer to the question of which orbital parameter plays a greater overall dynamical role as a forcing of global ice volume (Figs 2 and 3). The spread of obliquity-to-precession frequency content in significantly predicted summer energy time series arises naturally as a consequence of spatiotemporal heterogeneity in the insolation forcing.
Convergent cross-mapping (CCM) analyses
If two variables belong to the same dynamical system, then there is a 1:1 mapping between the reconstructed state spaces of both variables16. CCM estimates to what extent such a mapping exists, using the amount of information from the "driving" variable that is encoded in the "response" variable, and vice versa18. To resolve causal directionality, we employed time-lagged analyses using the rEDM implementation42 of CCM. Rather than inferring causality whenever the optimal lag is negative19,43,44, we used a more stringent criterion to reduce the likelihood of false positives: the total significant CCM skill (ρccm) of negative lags (past affects future) had to exceed that of positive lags (future affects past):
$${\rho }_{ccm}^{causal}=\sum _{i=min(lag)}^{-1}\frac{{\rho }_{ccm}^{i}}{{n}_{lags}}-\sum _{i=1}^{max(lag)}\frac{{\rho }_{ccm}^{i}}{{n}_{lags}}$$
where results for zero lag were excluded to avoid bias in either direction. A directional causal forcing was detected if \({\rho }_{ccm}^{causal} > 0\). In contrast, \({\rho }_{ccm}^{causal}\le 0\) implied that there was no detectable causal effect. We chose this conservative approach to limit the likelihood of false positives; we did not infer strict causal delays, which might be biased45, but limited our interpretation to causal directionality. Our lagged causality test (Fig. 2) used a maximum lag of 12 kyr.
Choice of embedding dimensions
CCM estimates dynamical coupling using a time-delay reconstruction16,17,46 of the dynamics. We took the minimal embedding dimension (Emin) as the integer dimension strictly larger than twice the box-counting dimension17 of the reconstructed attractor, and required the false nearest neighbour (FNN) rate to be less than 0.01. We used the tseriesChaos47 and fractaldim48 R packages to estimate the minimal FNN (\({E}_{min}^{FNN < 0.01}\)) and box counting dimensions (\({E}_{min}^{box}\)). To ensure numerical stability when estimating the box-counting dimension, summer energy time series with >2% zero values were excluded from the analyses. Optimal embedding parameters Eopt were then selected by maximizing self-prediction using simplex projection49 over integer dimensions \(\{max\{{E}_{min}^{box},{E}_{min}^{FNN < 0.01}\},\ldots ,10\}\) with embedding lag 1, setting 10 as the maximum embedding dimension to limit computational cost. Optimal embedding parameters were estimated separately for each pair of GSL-insolation time series. For each cross mapping, we constructed embeddings using the optimal embedding dimension for the target variable (the presumed driver). Leave-k-out cross validation with an exclusion radius of 30 kyr was used on prediction libraries to limit autocorrelation bias.
Statistical acceptance criteria
CCM requires that the correlation between predicted and observed values increases with increasing library size18. We determined convergence by regression of ρccm on 20 different library sizes (L) in the range \((E+tau+max(lag,1),\mathrm{...},L)\) distributed among the smallest possible and largest possible library sizes. Analyses were labelled convergent if the value of the constant q in the expression \(\rho ={\rho }_{max}-{\rho }_{0}\cdot {e}^{-q(L-{L}_{0})}\) was positive; here, ρ is the median CCM predictive skill and L is the library size. The value of q was found through linear regression of the logarithmic transformation of the same equation, or \(q=[ln({\rho }_{0})-ln({\rho }_{max}-\rho )]/(L-{L}_{0})\). In addition, to be convergent, we tested whether CCM skills were higher at the largest library sizes compared to the lowest library sizes by the means of a Wilcoxon rank sum test, which had to reject the null at the 0.01 level. Non-convergent analyses were discarded from the calculation of \({\rho }_{ccm}^{causal}\).
The upper limit of the CCM skill for a given analysis is determined by the coupling strength between the variables, but also by process noise18. Therefore, to claim significant forcing, we establish a distribution in the form of an ensemble of amplitude-adjusted Fourier transform (AAFT) surrogate time series50. These surrogates are randomized realizations of the original time series that preserve both the histogram and the frequency power spectrum (i.e. autocorrelation) of the original insolation forcing time series. Rejecting the null hypothesis thus implies that the dynamical coupling between GSL and the insolation cannot be fully accounted for neither by noise properties nor by shared frequencies.
Rejection of the null hypothesis for each driver-response pair involved passing a one-sided rank-order test51 where the ρccm of the data had to exceed the 99th percentile ρccm of the surrogate ensemble. We used 400 surrogates, and verified the results at selected threshold-latitude configurations using 1,000 surrogates.
Orbitally independent chronology
We explored three different chronologies for the GSL record (Fig. 4). The original age model for the GSL record2 is aligned with the LR04 stack14, which is tuned to an orbitally forced ice sheet model. We constructed an alternative age model following the approach of ref.22, wherein a filtered GSL (band-pass filtered using a 22 ka Gaussian filter) was tuned to an equivalently filtered composite δ18O record from U/Th dated Chinese speleothems24. GSL was then aligned with the speleothem record by tiepoints determined from peaks and troughs in the band-pass filtered versions. The speleothem record goes back to 640 kyr; age control for the older parts of the GSL record was obtained by linear interpolation to a tie point on the LR04 stack at 787 kyr, close to the Brunhes-Matuyama boundary. In an effort to obtain an orbitally independent chronology, we constructed another GSL age model by tuning a North Atlantic SST composite proxy record to the speleothem record. The GSL record was then matched to the benthic δ18O records of the respective North Atlantic archives. This approach utilizes the close connection between millennial-scale North Atlantic climate and the intensity of the Asian Monsoon documented in several studies52,53,54. In the interval from 0 to 332 kyr we used the GICC05/NALPSpeleo chronology from ODP site 98323 with a benthic δ18O record from ref.55. In the interval from 332 to 553 kyr we used the SST record from IODP site U131356 and the corresponding benthic δ18O record56,57. In the interval older than 553 kyr, we used the abundance of the polar planktonic foraminifer Neogloboquadrina pachyderma sinistral and the benthic δ18O records from ODP site 98058. Age control beyond the reach of the speleothem composite was obtained by matching the ODP 980 data to the Epica Dome C methane record59 placed on a modified AICC2012 gas age chronology. The AICC201260 was modified by matching the methane record to the composite speleothem record from 554 to 627 kyr, and to two tie points at 783 and 792 kyr determined by Ar/Ar dated tephras and a δ18O record from an Italian lacustrine sediment sequence61, utilizing the close relationship between the atmospheric methane concentration, North Atlantic climate and Asian Monsoon intensity (e.g.62,63). Age models were constructed with Oxcal 4.364, using the P sequence65 and variable k66 options.
Code to reproduce all analyses and figures is available at https://github.com/kahaaga/Haaga_et_al_insolation.
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We thank Peter Huybers and George Sugihara for helpful comments on an earlier version of this manuscript. This work is funded by the Bergen Research Foundation (B.H.) and by the Norwegian Research Council grants nos. 231259 (B.H.) and 221999 (J.B.).
Department of Earth Science, University of Bergen, P.O. Box 7803, N-5020, Bergen, Norway
Kristian Agasøster Haaga, Jo Brendryen, David Diego & Bjarte Hannisdal
The K.G. Jebsen Centre for Deep Sea Research, P.O. Box 7803, N-5020, Bergen, Norway
Kristian Agasøster Haaga, David Diego & Bjarte Hannisdal
Bjerknes Centre for Climate Research (BCCR), Allégaten 70, N-5007, Bergen, Norway
Kristian Agasøster Haaga, Jo Brendryen & Bjarte Hannisdal
Kristian Agasøster Haaga
Jo Brendryen
David Diego
Bjarte Hannisdal
K.A.H., J.B., D.D. and B.H. designed research. K.A.H. and J.B. performed research. K.A.H. analysed data, and drafted the manuscript. All authors commented on and revised the final manuscript.
Correspondence to Kristian Agasøster Haaga.
The authors declare no competing interests.
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Haaga, K.A., Brendryen, J., Diego, D. et al. Forcing of late Pleistocene ice volume by spatially variable summer energy. Sci Rep 8, 11520 (2018). https://doi.org/10.1038/s41598-018-29916-3
Accepted: 20 July 2018
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Scientific Reports ISSN 2045-2322 (online) | CommonCrawl |
Amy, Betty, Cathy, and Daisy have $36$ apples that they plan to split among themselves in a ratio of $2:3:3:4$, respectively. How many apples will Betty and Cathy have together?
The four girls are dividing the $36$ apples into $2+3+3+4 = 12$ equal parts. Thus, there are $\frac{36}{12} = 3$ apples per part. Together, Betty and Cathy have $3+3 = 6$ parts, so they will have $3 \cdot 6 = \boxed{18}$ apples. | Math Dataset |
Adaptive coding and modulation using imperfect CSI in cognitive BIC-OFDM systems
Jeroen Van Hecke ORCID: orcid.org/0000-0002-1860-83431,
Paolo Del Fiorentino2,
Riccardo Andreotti2,
Vincenzo Lottici2,
Filippo Giannetti2,
Luc Vandendorpe3 &
Marc Moeneclaey1
EURASIP Journal on Wireless Communications and Networking volume 2016, Article number: 256 (2016) Cite this article
This work investigates adaptive coding and modulation (ACM) algorithms under the realistic assumption that the available channel state information (CSI) at the transmitter is imperfect due to estimation errors and/or feedback delays. First, we introduce an optimal performance metric for the secondary user (SU) bit-interleaved coded orthogonal frequency division multiplexing (BIC-OFDM) system, called the expected goodput (EGP). By using an accurate modeling approximation, we succeed in deriving a tractable and very accurate approximation for the EGP. This approximate EGP (AEGP) is then used for the derivation of several ACM algorithms which optimize the code rate and bit and energy allocation under a constraint on the interference caused to the PU network. In the numerical results, we show that the AEGP is far more accurate than previous attempts to model the GP in the presence of imperfect CSI. Further, we verify that, in spite of the imperfect nature of the available CSI, the derived ACM algorithms significantly increase the goodput of the SU network, compared to a non-adaptive selection of the transmission parameters.
To meet the demand of high data rates and the increasing amount of traffic, the current and next generation of wireless networks need spectrally efficient solutions such as multicarrier orthogonal frequency division multiplexing (OFDM) transmission, efficient channel coding techniques in the form of bit interleaved coded modulation (BICM) [1], and adaptive coding and modulation (ACM) [2]. To further increase the spectral efficiency, the idea of cognitive radio (CR) [3, 4] has been proposed. This technique allows unlicensed or secondary users (SUs) to transmit over sections of spectrum owned by licensed or primary users (PUs), on the condition that the former do not harm the quality of service (QoS) of the latter.
If channel state information (CSI) is available at the transmitter, ACM can significantly improve the performance of the network by adapting the transmission parameters, such as energy and bit allocation per subcarrier, constellation size, and code rate, to the actual state of the channel. However, in a wireless environment, the CSI at the transmitter, obtained from channel estimates fed back by the receiver, will be imperfect, due to channel estimation errors at the receiver and, in the case of a time-varying channel, the feedback delay on the return channel from the receiver to the transmitter. In [5], the authors show for a single user OFDM system that, even with CSI imperfections at the transmitter, the throughput of the system can be significantly increased by using adaptive modulation. The adaptation algorithms take the CSI imperfections into account, and their performance was shown to improve by having multiple estimates available at the transmitter. This means that, when multiple estimates are available, the network can tolerate larger channel estimation errors or longer delays, while still achieving an acceptable performance level. In [6], this scenario was extended to a multi-user OFDMA-system where the subcarriers are allocated to the user with the best signal-to-noise ratio (SNR) conditions and the number of bits per subcarrier are optimized by maximizing the average throughput. However, the results in [5, 6] were obtained for an uncoded OFDM system; this considerably simplifies the optimization problem (OP) because the probability of a bit error on a subcarrier only depends on the SNR of the considered subcarrier, but the results are of limited use in a practical scenario where channel coding is used.
In recent years, there have been several works [7–12] that studied resource allocation in cognitive underlay networks with imperfect CSI. However, these works used a more theoretic performance metric like the capacity metric or SNR and did not consider the difficult problem of implementing ACM in a practical coded multi-carrier transmission system. Because the bits are coded and the channel is frequency-selective, the throughput of the network depends upon a complicated function of the SNRs of all the subcarriers which are used for the transmission. A technique which allows to simplify the analytical expression for the performance metric is effective SNR mapping (ESM) [13]. This technique transforms the vector of subcarrier SNRs, which affect the codeword, into a scalar SNR. This effective SNR is the operating point at which an equivalent coded system, which uses the same modulation and coding scheme, operating over an additive white Gaussian noise (AWGN) channel, has the same performance as the system under consideration. A very promising mapping function, called the cumulant-generating function-based ESM (κESM), was introduced in [14]. This mapping function combines the simplicity of exponential ESM (EESM) with the accuracy of mutual information ESM (MIESM) [15]. Another advantage is that this mapping function can be used to optimize the coding rate together with the energy and bit allocation per subcarrier.
In [16], EESM has been applied to ACM in a multi-carrier system with feedback delays. The bit allocation per subcarrier and the code rate are selected such that the throughput gets maximized under a certain block error rate constraint. However, because the transmitter is unaware of the fact that the available CSI is delayed, the transmitter sometimes over- or underestimates the actual channel conditions which results in a loss of spectral efficiency. In [17], the throughput of a BIC-OFDM system is optimized under a target packet error rate (PER) constraint, where a packet can consist of multiple OFDM symbols. Also here, the considered adaptation algorithm at the transmitter does not account for CSI imperfections, which leads to a violation of the PER constraint when only delayed CSI is available.
Rationale and contributions. This paper deals with an ACM scheme for the SU link of a cognitive system based on a BIC-OFDM signaling with imperfect CSI at the SU transmitter, due to estimation errors or feedback delays. The performance metric we consider is the goodput (GP), which is similar to the throughput but considers only the number of information bits which are correctly received. The key idea behind the proposed method relies on optimizing the long-term average GP of the SU link, averaged over the realizations of both the actual channel and the available CSI at the SU transmitter, under the constraints of the total transmitted energy and the level of interference on the PU receivers. This can be achieved by optimizing the expected GP (EGP) metric.1 This optimal metric is the expected GP conditioned on the available CSI at the SU transmitter. In view of these features, our proposed scheme turns out to be more competitive, when compared to the current literature, as outlined in the sequel.
Instead of resorting to the often used information-theoretical capacity metric, a more practically relevant metric, i.e., the GP, is optimized, which gives the advantage of allowing the optimization of realistic modulation and coding formats.
Unlike the ad hoc approaches used in our previous work [19, 20], we now start from the optimal expression for the EGP. By using the statistical approximation for the effective SNR, which we introduced in [21], we now derive an analytical, tractable approximation for the EGP, which we call the approximate EGP (AEGP). In the numerical results, we show that the AEGP is a far more accurate approximation of the EGP, compared to the metrics used in [19, 20]. To the authors' knowledge, these works are the first ones which propose to use a practical metric, which takes care of the imperfect CSI, for the optimization of the transmission parameters.
In this work, we successfully combine the practical assumption of imperfect CSI with the accurate model of the effective SNR, which results in the AEGP metric. This AEGP metric, which takes care of the imperfect CSI, is proposed as the objective function of an OP to search for the optimal combination of the ACM parameters under the above mentioned transmit energy and interference constraints. By using the AEGP, packet errors or a loss in spectral efficiency by over- or underestimating the actual channel conditions are largely avoided. This differs from the approach taken in [16, 17], where the transmitter is unaware that its CSI is imperfect and only the impact of the imperfect CSI on the performance is investigated.
We derive several ACM solutions which optimize the code rate together with uniform or non-uniform bit allocation and uniform or non-uniform energy allocation. The performance of these algorithms is investigated for different types of CSI at the SU transmitter.
Although affected by imperfect CSI, extensive simulation runs show that the proposed ACM algorithms allow significant gains compared to non-adaptive ACM schemes. Further, depending on the quality level of the CSI, the resulting GP performance can be very close to that obtainable in scenarios where perfect CSI is employed.
Organization. In Section 2, we describe the cognitive BIC-OFDM system. In Section 3, we introduce the EGP metric and discuss the statistical approximation of the κESM. The ACM algorithms which select the code rate and the energy and bit allocations per subcarrier are derived in Section 4. The accuracy of the EGP metric and the performance of the ACM algorithms are validated in Section 5. The conclusions are presented in Section 6.
Notations. Expectation operator is E[ ·], [ ·]T is the transpose operator, [ ·]H is the Hermitian transpose operator, x∼(0,Σ) refers to a circular symmetric zero-mean Gaussian complex random vector with covariance matrix Σ, and the matrix I denotes the identity matrix. The ith column of the identity matrix is denoted by e i . The notation (X) i,j refers to the element on the ith row and jth column of the matrix X, while (x) i denotes the ith component of the vector x.
Cognitive BIC-OFDM system model
We consider a SU network, which consists of a point-to-point OFDM link, that occupies the same bandwidth as a PU network containing N PU PU receivers. Messages are transmitted by means of a packet-oriented BIC-OFDM communication system consisting of N subcarriers within a bandwidth B [14]. Each packet contains N p information bits and N CRC bits for the cyclic redundancy check (CRC), which leads to a total of N u=N p+N CRC bits per packet. These N u bits are first encoded by a convolutional encoder. Several convolutional codes are available at the transmitter; these are punctured versions of a rate 1/2 code, designated by their rate \(r\in {\mathcal {D}}_{\mathrm {r}}\). In the following step, these N u/r coded bits are randomly interleaved and Gray-mapped to N s unit-energy quadrature amplitude modulation (QAM) symbols. In the last step, we make use of OFDM, with N available subcarriers per OFDM symbol, to transmit the N s QAM symbols over a frequency-selective fading channel, which is assumed to be time-invariant for the whole packet transmission duration. The duration of an OFDM symbol will be denoted by T s. The SU receiver first performs a fast Fourier transform (FFT) on the received OFDM symbol; the kth OFDM subcarrier, with k∈{1,…,N} is observed at the corresponding FFT output as
$$ z_{k}{\overset{\bigtriangleup}{=}}\sqrt{E_{k}}H_{k}x_{k}+w_{k}, $$
where E k is the transmit energy on the kth subcarrier, H k is the corresponding channel coefficient, x k is the constellation symbol transmitted on subcarrier k containing \(m_{k}\in {\mathcal {D}}_{\mathrm {m}}\) coded bits and E[|x k |2]=1, and \(w_{k}\in {\mathcal {C}N}\left (0,\sigma _{\mathrm {w}}^{2}\right)\) is the additive noise contribution. The transmit energies are constrained by
$$ \sum_{k=1}^{N}E_{k}\leq E_{\text{max}} $$
where E max is the maximal transmit energy per OFDM symbol. Next, the SU receiver first performs soft demapping, and finally de-interleaves and decodes the packet; the CRC allows to verify whether the packet has been correctly decoded.
The received SNR associated with the kth subcarrier at the FFT output is defined as
$$ \gamma_{k}{\overset{\Delta}{=}}\frac{E_{k}|H_{k}|^{2}}{\sigma_{\mathrm{w}}^{2}}. $$
Let us arrange the received SNRs into a vector \(\boldsymbol {\Gamma }{\overset {\Delta }{=}} [\gamma _{1},\ldots,\gamma _{N}]\) for further use. We define the transmission mode (TM) \(\mathbf {\phi }{\overset {\Delta }{=}}\{\mathbf {m},r\}\in {\mathcal {D}}_{\mathrm {m}}^{N}\times {\mathcal {D}}_{\mathrm {r}}\), with \(\mathbf {m}{\overset {\Delta }{=}}[m_{1},\ldots,m_{N}]^{T}\). As not all N available subcarriers will necessarily be used for the transmission, we make a distinction between the set {1,…,N} of available subcarriers, and the set \(\mathcal {N\subseteq }\{1,\ldots,N\}\) of active subcarriers. When the kth subcarrier is not active (i.e., \(k\notin \mathcal {N}\)), we have E k =0 and m k =0.
Because of noise and/or feedback delays, the CSI available at the transmitter will often be imperfect. To make the description of our proposed approach quite general, we will denote the CSI, which is available at the transmitter about the actual channel realization \(\mathbf {H}{\overset {\Delta }{=}}[H_{1},\ldots,H_{N}]^{T}\), by the vector C S I. We make the assumption that H and C S I are jointly zero-mean circular symmetric Gaussian. It then follows that H conditioned on C S I is Gaussian, with expectation μ H|C S I =E H [H|C S I] and covariance matrix C H|C S I =E H [H H H|C S I]−μ H|C S I μ H|C S I H; note that μ H|C S I is the minimum mean-squared error (MMSE) estimate of H based on C S I. Some examples of C S I and the associated statistics are given in the Appendix section.
The signals transmitted in the SU network cause interference at the PU receivers, which should be constrained in order not to affect the PU QoS. Denoting by \(G_{k}^{(q)}\) the channel gain from the SU transmitter to the qth PU receiver, experienced by the kth subcarrier, the interference constraints can be expressed as \(\sum _{k\in {\mathcal {N}}}E_{k}|G_{k}^{(q)}|^{2}\leq {\mathcal {I}}_{q}\) for \(q\in {\mathcal {Q}}{\overset {\Delta }{=}}\{1,\ldots,N_{\text {PU}}\}\).
We denote by \(\mathbf {CSI}_{\text {PU}}=\{\mathbf {CSI}_{\text {PU}}^{(q)},q\in {\mathcal {Q}}\}\) the imperfect CSI available at the SU transmitter about its channels to the PU receivers. This CSI could be obtained from a band manager [22] or, assuming time-division duplexing in the PU network and channel reciprocity, this CSI could be extracted by the SU transmitter when the considered PU receiver has switched to a transmission mode. As only C S I PU and not the exact channel gains \(G_{k}^{(q)}\) are available at the SU transmitter, it can happen that the interference constraint at the PU receivers is violated. Therefore, alternative formulations of the interference constraints are needed that can be satisfied by the SU transmitter. A first possibility is to satisfy the interference constraints only on average, conditioned on the available \(\mathbf {CSI}_{\text {PU}}^{(q)}\). In this case, the interference constraint is replaced by
$$ \mathrm{E}_{\mathbf{G}^{(q)}}\left[\sum_{k\in{\mathcal{N}}}E_{k}|G_{k}^{(q)}|^{2}\,|\mathbf{CSI}_{\text{PU}}^{(q)}\right] \leq{\mathcal{I}}_{q},\quad\forall q\in{\mathcal{Q}}, $$
where \(\mathbf {G}^{(q)}{\overset {\Delta }{=}}[G_{1}^{(q)},\ldots,G_{N}^{(q)}]^{T}\). The expected value in (4) can be expressed as (\(\forall q\in {\mathcal {Q}}\))
$$\begin{array}{*{20}l} &\mathrm{E}_{\mathbf{G}^{(q)}}\left[\sum_{k\in{\mathcal{N}}}E_{k}\Big|G_{k}^{(q)}\Big|^{2}\,\Big|\mathbf{CSI}_{\text{PU}}^{(q)}\right] \\ &\qquad =\sum_{k\in{\mathcal{N}}}E_{k}\left(\left|\left(\boldsymbol{\mu}_{\mathbf{G}|\mathbf{CSI}_{\text{PU}}}^{(q)}\right)_{k}\right|^{2} + \left(\mathbf{C}_{\mathbf{G}|\mathbf{CSI}_{\text{PU}}}^{(q)}\right)_{k,k}\right), \end{array} $$
where we have assumed that the distribution of G (q) conditioned on \(\mathbf {CSI}_{\text {PU}}^{(q)}\) is Gaussian with mean μ G|C S I PU(q) and covariance matrix C G|C S I PU(q).
A second possibility is to define the interference constraint by means of uncertainty sets [23, 24]. By defining the uncertainty set \(\mathcal {S}_{k}^{(q)}\) as follows
$$ \mathcal{S}_{k}^{(q)}=\left\{ \hat{G}_{k}^{(q)}:\hat{G}_{k}^{(q)}=\left(\boldsymbol{\mu}_{\mathbf{G}|\mathbf{CSI}_{\text{PU}}}^{(q)}\right)_{k}+\alpha\epsilon,\|\epsilon\|\leq1\right\}, $$
the interference constraint is formulated as
$$ \sum_{k\in{\mathcal{N}}}E_{k}|\hat{G}_{k}^{(q)}|^{2}\leq{\mathcal{I}}_{q},\quad\forall q\in{\mathcal{Q}},\forall\hat{G}_{k}^{(q)}\in\mathcal{S}_{k}^{(q)} $$
where the complex scalar α defines the size of the uncertainty interval, which directly influences the minimum probability that the interference is below the interference threshold \({\mathcal {I}}_{q}\). The set of constraints in (7) can be reduced to a single constraint per PU receiver, by only considering the value of \(\hat {G}_{k}^{(q)}\) in \(\mathcal {S}_{k}^{(q)}\) which leads to the most restrictive constraint. Denoting this value by \(G_{k}^{*(q)}\), (7) is equivalent to
$$ \sum_{k\in{\mathcal{N}}}E_{k}|G_{k}^{*(q)}|^{2}\leq{\mathcal{I}}_{q},\quad\forall q\in{\mathcal{Q}}. $$
A third possibility, used in [12, 25, 26], is to neglect the statistical variation of \(G_{k}^{(q)}\) for given \(\mathbf {CSI}_{\text {PU}}^{(q)}\), and to use the following interference constraint
$$ \sum_{k\in{\mathcal{N}}}E_{k}|(\boldsymbol{\mu}_{\mathbf{G}|\mathbf{CSI}_{\text{PU}}}^{(q)})_{k}|^{2}\leq{\mathcal{I}}_{q},\quad\forall q\in{\mathcal{Q}}. $$
We note that these interference constraints can be linked to the concept of interference probability as defined in [12]. The interference probability (IP) for the qth PU receiver reads as
$$ \text{IP}_{q}=\text{Pr}\left(\sum_{k\in{\mathcal{N}}}E_{k}|G_{k}^{(q)}|^{2}>{\mathcal{I}}_{q}\right). $$
In the case that the dynamically allocated energy vector \(\mathbf {E}{\overset {\Delta }{=}}[E_{1},\ldots,E_{N}]^{T}\) leads to an intolerable IP, one can substitute \({\mathcal {I}}_{q}\) in the corresponding interference constraint by \(\kappa _{q}{\mathcal {I}}_{q}\). The scaling factor κ q is chosen such that IP q reaches an acceptable value, after finding a new dynamic allocation of the vector E which satisfies the new constraint.
Finally, it is clear that the constraints (5), (8), and (9) all have the same mathematical form. This means that our proposed algorithms are compatible with all these constraints. For the remainder of the paper however, we will consider the average interference constraint (5).
Goodput performance metric
The goodput (GP), being defined as the ratio of the number of correctly received information bits (associated with correctly decoded packets) and the actual transmission time, has a very clear practical interpretation. Normalizing the GP by dividing by the actual bandwidth N/T s, the GP corresponding to a given TM ϕ={m,r} and SNR vector Γ is expressed as
$$\begin{array}{*{20}l} \text{GP} =\frac{N_{\mathrm{p}}r}{NN_{\mathrm{u}}}\left(\sum_{k\in{\mathcal{N}}}m_{k}\right)\cdot(1-\text{\text{PER}}(\phi,\boldsymbol{\Gamma})), \end{array} $$
where PER(ϕ,Γ) is the packet error rate (PER) corresponding to the selected (ϕ,Γ). Note that the goodput (11) is a function of the actual channel realization H because of (3). As a performance measure of the SU network, we consider the long-term average of the goodput (11) over many channel realizations.
If perfect CSI were available at the transmitter (i.e., the transmitter knows the realizations of its channels to the SU receiver and PU receivers), the optimal way of selecting the transmission mode ϕ and the energy allocation vector E as a function of these realizations is to maximize (11) under the constraints on the SU transmit energy and the interference at the PU receivers, for the given realizations H and \(\{\mathbf {G}^{(q)},q\in \mathcal {Q}\)}. This selection obviously maximizes the long-term average goodput of the system, given by \(\text {GP}_{\text {avg}}=\mathrm {E}_{\mathbf {H},\left \{\mathbf {G}^{(q)},q\in \mathcal {Q}\right \}}\left [\text {GP}\right ]\).
However, when only imperfect CSI is available, the transmission parameters (ϕ,E) must be selected as functions of C S I and C S I PU, rather than H and \(\{\mathbf {G}^{(q)},q\in \mathcal {Q}\)}. Taking into account that for given ϕ and E, GP from (11) is a function of H and that the joint probability density function of H, C S I , and C S I PU can be factored as p(H,C S I,C S I PU)=p(H|C S I)p(C S I)p(C S I PU), the long-term average goodput can be written as
$$\begin{array}{*{20}l} {}\text{GP}_{\text{avg}} & \,=\,\mathrm{E}_{\mathbf{H},\mathbf{CSI},\mathbf{CSI}_{\text{PU}}}[\text{GP}] \\ &\,=\,\mathrm{E}_{\mathbf{CSI},\mathbf{CSI}_{\text{PU}}}\!\!\left[\!\!\frac{N_{\mathrm{p}}r}{NN_{\mathrm{u}}}\! \!\left(\sum_{k\in{\mathcal{N}}}\!\!m_{k}\!\!\right)\!\!\cdot\!\left(1\,-\,\mathrm{E}_{\mathbf{H}}\! \left[\text{\text{PER}}\!\left(\phi,\mathbf{\boldsymbol{\Gamma}}\right)\!|\mathbf{CSI}\right]\right)\!\!\right]\!. \end{array} $$
It follows from (12) that GPavg becomes maximum when for given (C S I,C S I PU) the transmission parameters (ϕ,E) maximize the expression between brackets in the second line of (12), under the constraints (2) and (4). This is equivalent to maximizing the expected goodput (EGP) metric, given by
$$\begin{array}{*{20}l} \text{EGP}&=\mathrm{E}_{\mathbf{H}}\left[\text{GP}|\mathbf{CSI}\right] \\ & =\frac{N_{\mathrm{p}}r}{NN_{\mathrm{u}}}\left(\sum_{k\in{\mathcal{N}}}m_{k}\right) \cdot\left(1-\mathrm{E}_{\mathbf{H}}\left[\text{\text{PER}}(\phi,\boldsymbol{\Gamma})|\mathbf{CSI}\right]\right). \end{array} $$
which is the conditional expectation of GP for given C S I and represents the optimal performance metric in terms of GPavg when only imperfect CSI is available at the transmitter.
The evaluation of PER(ϕ,Γ) is not an easy task. In [14], an accurate link performance evaluation model, referred to as κESM, has been proposed for the BIC-OFDM system. This model provides a closed-form expression for the effective SNR γ. The effective SNR γ has the important property that the PER of the considered BIC-OFDM system where the SNRs and transmission mode of the subcarriers are given by Γ and ϕ, respectively, is approximately equal to PERESM(r,γ), which denotes the PER of an equivalent BPSK system (i.e., using the same convolutional code with rate r) which operates over an AWGN channel with SNR equal to γ. The effective SNR is calculated as [14]
$$ \gamma{\overset{\Delta}{=}}-\beta\log(Y), $$
where β is a scaling coefficient which is optimized across all possible TMs [27]. Y is expressed as
$$ Y{\overset{\Delta}{=}}\frac{1}{\sum_{l\in{\mathcal{N}}}m_{l}}\sum_{k\in{\mathcal{N}}}\Omega_{k}, $$
and Ω k is given by
$$ \Omega_{k}{\overset{\Delta}{=}}\sum_{n=1}^{\frac{\sqrt{2^{m_{k}}}}{2}}\alpha_{k,n}e^{-\frac{\gamma_{k}n^{2}d_{k,\text{min}}^{2}}{4\beta}}, $$
where d k,min denotes the minimum Euclidean distance of the constellation used on the kth subcarrier, and α k,n is a known constant which depends on the chosen constellation.
The EGP from (13) can now be approximated by replacing PER(ϕ,Γ) by PERESM(r,γ), with γ given by (14). The reference curves PERESM(r,γ) can be stored in a lookup table for each code rate r from the set \({\mathcal {D}}_{\mathrm {r}}\). In order to compute the conditional expectation E H [PERESM(r,−β log(Y))|C S I], we need the distribution of Y conditioned on C S I. In [21], it was proposed to approximate Y conditioned on C S I by a random variable Z which follows a beta distribution with shaping parameters a and b, i.e., p Z (z)∝z a−1(1−z)b−1 for 0≤z≤1. The value of these shaping parameters is given by \(a=\frac {e(e-e^{2}-v)}{v}\) and \(b=\frac {(1-e)(e-e^{2}-v)}{v}\), where e=E H [Y|C S I] and v=Var H [Y|C S I]. For more details, we refer to [21], where closed-form expressions where derived for e and v. Note that the distribution of Z depends on the selected bit allocation through the variables α k,n , m k and d k,min. Using this approximating beta distribution, we obtain the approximate EGP (AEGP) given by
$$ {}\text{AEGP}=\!\frac{N_{\mathrm{p}}r}{NN_{\mathrm{u}}}\!\!\left(\sum_{k\in{\mathcal{N}}}m_{k}\right)\cdot\left(1\,-\,\mathrm{E}_{Z} \left[\text{PER}_{\text{ESM}}(r,-\beta\log(Z))\right]\right). $$
The expectation w.r.t. Z in (17) can be approximated by means of numerical integration.
Goodput optimization
In this section, we consider different algorithms the transmitter can employ to optimize the code rate r, the energy allocation E k and the bit allocation m k (\(\forall k\in {\mathcal {N}}\)) such that the AEGP from (17) is maximized, while satisfying the transmit energy constraint (2) and the interference constraints (4) at the PU receivers. These algorithms assume that only imperfect CSI is available at the transmitter.
Uniform energy and bit allocation
In this first subsection, we make the restriction that the bit and energy allocation is uniform and that all N available subcarriers are actually used, i.e., \(\mathcal {N}=\{1,\ldots,N\}\). For the bit and energy allocation, this means that
$$ m_{k}=m,\,E_{k}=E,\quad\forall k\in{\mathcal{N}}, $$
where \(m\in \mathcal {D}_{\mathrm {m}}\). Considering the constraints (2) and (4), the optimal uniform energy per subcarrier is given by
$$ E=\min\left(\min_{q\in\mathcal{Q}}\frac{\mathcal{I}_{q}}{\mathrm{E}_{\mathbf{G}^{(q)}} \left[\sum_{k\in{\mathcal{N}}}|G_{k}^{(q)}|^{2}\,|\mathbf{CSI}_{\text{PU}}^{(q)}\right]}, \frac{E_{\text{max}}}{|{\mathcal{N}}|}\right), $$
where the expected value can be found from (5) and \(|{\mathcal {N}}|\) denotes the number of active subcarriers. The transmitter will calculate the AEGP (17) for every TM ϕ={m,r}, and then selects the TM ϕ={m,r} which yields the largest AEGP. The pseudo-code of this optimization is outlined in Table 1.
Table 1 Uniform energy and bit allocation
Optimized energy and uniform bit allocation
In this subsection, we will adapt the previous algorithm such that the transmitter optimizes the energy per subcarrier, while the bit allocation remains uniform. As explained further, we will allow some of the subcarriers to be inactive, i.e., \(\mathcal {N}\subseteq \{1,2,\ldots,N\}\). We first have a closer look at the EGP from (13) where PER(ϕ,Γ) is replaced by PERESM(r,γ), i.e.,
$$\begin{array}{*{20}l} \text{EGP} \approx&\frac{N_{\mathrm{p}}r}{NN_{\mathrm{u}}}\left(\sum_{k\in{\mathcal{N}}}m_{k}\right) \\ & \cdot\left(1-\mathrm{E}_{\mathbf{H}}\left[\text{PER}_{\text{ESM}}\left(r,-\beta\log(Y(\mathbf{E}))\right)|\mathbf{CSI}\right]\right), \end{array} $$
where we have explicitly shown the dependence on the energy allocation vector E. Because the PER is a convoluted function of the individual subcarrier energies, an exact optimization of this metric will be very hard to obtain. Therefore, we suggest a more computationally efficient method, by optimizing the following simplification of the EGP
$$\begin{array}{*{20}l} \text{EGP} \approx&\frac{N_{\mathrm{p}}r}{NN_{\mathrm{u}}}\left(\sum_{k\in{\mathcal{N}}}m_{k}\right) \\ & \cdot\left(1-\text{PER}_{\text{ESM}}\left(r,-\beta\log\left(\mathrm{E}_{\mathbf{H}}\left[Y(\mathbf{E})|\mathbf{CSI}\right]\right)\right)\right), \end{array} $$
where the average is now taken inside the logarithm. As PERESM(r,γ) decreases with increasing γ, the maximization of (21) w.r.t. E is equivalent to the minimization of E H [Y(E)|C S I]. The latter function can be obtained analytically [21]:
$$ \mathrm{E}_{\mathbf{H}}\left[Y(\mathbf{E})|\mathbf{CSI}\right]=\frac{1}{\sum_{l\in{\mathcal{N}}}m_{l}}\,\sum_{k\in{\mathcal{N}}}\sum_{n=1}^{\frac{\sqrt{2^{m_{k}}}}{2}}g_{k,n}(E_{k}), $$
$$ g_{k,n}(E_{k})=\alpha_{k,n}\frac{e^{\frac{-\left|\left(\boldsymbol{\mu}_{\mathbf{H}|\mathbf{CSI}}\right)_{k}\right|^{2} \frac{E_{k}}{4\beta\sigma_{\mathrm{w}}^{2}}n^{2}d_{k,\text{min}}^{2}}{1+\frac{E_{k}}{4\beta\sigma_{\mathrm{w}}^{2}}n^{2}d_{k,\text{min}}^{2} \left(\mathbf{C}_{\mathbf{H}|\mathbf{CSI}}\right)_{k,k}}}}{1+\frac{E_{k}}{4\beta\sigma_{\mathrm{w}}^{2}}n^{2}d_{k,\text{min}}^{2} \left(\mathbf{C}_{\mathbf{H}|\mathbf{CSI}}\right)_{k,k}}\cdot $$
So the optimized energy allocation that maximizes the simplified EGP in (21) is found by solving the following OP
$$ \left\{ \begin{array}{ll} \mathbf{E}^{(\text{opt})} & =\arg\min_{\mathbf{E}}\sum_{k\in{\mathcal{N}}}\sum_{n=1}^{\frac{\sqrt{2^{m_{k}}}}{2}}g_{k,n}(E_{k})\\ s.t. & \sum_{k\in{\mathcal{N}}}E_{k}\leq E_{\text{max}}\\ & (4) \end{array}\right.. $$
According to [28], an OP is convex when both the constraints and the objective function are convex. From (24), it is clear that the constraints are convex, as they are linear in the components of E. Further, the convexity of the objective function follows from the fact that the second derivative of g k,n (E k ) with respect to E k can be shown to be non-negative; hence, each term of the objective function is convex, so that the entire objective function is convex as well. Therefore, the OP of (24) can be efficiently solved by using optimization tools such as CVX [29].
For the optimization of the EGP, we slightly adapt the algorithm outlined in Table 1. We start by considering all available subcarriers as active, i.e., \(\mathcal {N}=\{1,\ldots,N\}\). For every possible TM ϕ={m,r} the algorithm computes the approximation (17) of the EGP, using as energy allocation the solution of OP (24). Because the energy allocation now depends on the parameter m, it must now become part of the outer loop of the algorithm. For a given value of m, it might happen that for some k the optimized value of E k equals 0. In this case, the corresponding subcarriers are removed from the active set \({\mathcal {N}}\) by putting m k =0, which also removes the large terms with E k =0 (i.e., γ k =0) from (15) for the considered bit allocation. Finally, the algorithm selects the TM and the corresponding energy allocation yielding the largest value of the AEGP (17).
Uniform energy and greedy bit allocation
In this subsection we consider a uniform energy allocation according to (19) and an optimized bit allocation per subcarrier.
We first consider the simplified expression for the EGP (21):
$$\begin{array}{*{20}l} {}\text{EGP} \approx&\frac{N_{\mathrm{p}}r}{NN_{\mathrm{u}}}\left(\sum_{k\in{\mathcal{N}}}m_{k}\right) \\ &\cdot\left(1-\text{PER}_{\text{ESM}}\left(r,-\beta\log\left(\mathrm{E}_{\mathbf{H}} \left[Y\left(\mathbf{m},\mathbf{E}\right)|\mathbf{CSI}\right]\right)\right)\right), \end{array} $$
where now the dependence on the bit and energy allocation vectors m and E is explicitly shown. Considering (15), we notice that the simplified EGP from (25) only depends on the bit allocation through the quantity \(\sum _{k\in {\mathcal {N}}}\mathrm {E}_{\mathbf {H}}\left [\Omega _{k}(m_{k},E_{k})|\mathbf {CSI}\right ]\) and the sum \(\sum _{k\in {\mathcal {N}}}m_{k}{\overset {\Delta }{=}} M(\mathbf {m})\). Because the PER is a decreasing function of the effective SNR γ, the maximal value of the simplified EGP, for a fixed value of M(m), will be achieved for the bit allocation m and energy allocation E which minimizes
$$\begin{array}{*{20}l} & \arg\min_{\mathbf{E},\mathbf{m}}\sum_{k\in{\mathcal{N}}}\mathrm{E}_{\mathbf{H}}\left[\Omega_{k}(m_{k},E_{k})|\mathbf{CSI}\right] \\ &\qquad\quad=\arg\min_{\mathbf{E},\mathbf{m}}\sum_{k\in{\mathcal{N}}}\sum_{n=1}^{\frac{\sqrt{2^{m_{k}}}}{2}}g_{k,n}(m_{k},E_{k}). \end{array} $$
where g k,n (m k ,E k ) is given by (23), and the dependence on m k is shown explicitly. However, this represents a mixed integer programming problem, which is computationally very hard. In order to obtain a computationally efficient solution, we base our algorithm on the iterative suboptimal greedy algorithm described in [30].
In the current iteration, we modify the bit allocation from the previous iteration by adding 2 bits (because we restrict our attention to square QAM constellations, representing an even number of bits) to the subcarrier which leads to the smallest increase of \(\sum _{k\in {\mathcal {N}}}\mathrm {E}_{\mathbf {H}}\left [\Omega _{k}(m_{k},E_{k})|\mathbf {CSI}\right ]\). For the resulting bit and energy allocation, we determine the code rate r which leads to the highest AEGP (17). The iterative algorithm is initialized with m k =0 for all available subcarriers (yielding M(m)=0) and continues until all N available subcarriers have m max bits (yielding M(m)=m max N), where m max is the largest allowed number of bits in the constellation. At that point, we select the code rate r and the energy and bit allocation which correspond to the value of M(m) for which the AEGP (17) is maximal.
Now, we outline how the increase of \(\sum _{k\in {\mathcal {N}}}\mathrm {E}_{\mathbf {H}}\left [\Omega _{k}(m_{k},E_{k})|\mathbf {CSI}\right ]\) is evaluated. Let us denote by m the value of the bit allocation vector and by \(\mathcal {N}\) the set of active subcarriers, both referring to the previous iteration. We now introduce the quantity \(\delta _{k}^{(m_{k}+2)}(\mathbf {m})\) which is defined as the increase of (26) when the bit allocation on subcarrier k increases from m k to m k +2. If subcarrier k was not active in the previous iteration (i.e., m k =0), the set of active subcarriers increases from \(\mathcal {N}\) (previous iteration) to \(\mathcal {N}\cup \{k\}\) (current iteration), yielding the increase
$$\begin{array}{*{20}l} {}\delta_{k}^{(2)}\!(\mathbf{m})\! =&\mathrm{E}_{\mathbf{H}}\!\left[\!{\vphantom{\sum_{l\in{\mathcal{N}}}}}\Omega_{k}\!\left(2,E_{k}\!\left(\mathbf{m}+2\mathbf{e}_{k}\right)\right)\right. \\ &\left.+\!\!\sum_{l\in{\mathcal{N}}}\!\left(\Omega_{l}\!\left(m_{l},E_{l}\!\left(\mathbf{m}\,+\,2\mathbf{e}_{k}\right)\right) \,-\,\Omega_{l}\!\left(m_{l},E_{l}(\mathbf{m})\right)\right)\!|\mathbf{CSI}\!\right]\!, \end{array} $$
where E(m) and E(m+2e k ) denote the uniform energy allocations from (19) corresponding to the bit allocations m and m+2e k , respectively, related to the previous and the current iteration; because the corresponding set of active subcarriers has changed, E(m) and E(m+2e k ) are different, which makes in (27) the summation over l nonzero. If subcarrier k was already active in the previous iteration (i.e., m k >0), we obtain
$$\begin{array}{*{20}l} \delta_{k}^{m_{k}+2}(\mathbf{m}) & =\mathrm{E}_{\mathbf{H}}\left[\Omega_{k}\left(m_{k}+2,E_{k}\left(\mathbf{m}+2\mathbf{e}_{k}\right)\right)\right. \\ &\left. \quad-\Omega_{k}\left(m_{k},E_{k}(\mathbf{m})\right)|\mathbf{CSI}\right]. \end{array} $$
As in this case, the set of active subcarriers equals \(\mathcal {N}\) for both the previous and the current iteration, the uniform energy allocation from (19) satisfies E(m+2e k )=E(m). In the current iteration, the increments \(\delta _{k}^{m_{k}+2}(\mathbf {m})\) are computed for all k∈{1,…,N}; then, the subcarrier k which yields the lowest \(\delta _{k}^{m_{k}+2}(\mathbf {m})\) (k∈{1,…,N}) is selected, and the bit allocation for this subcarrier and M(m) are both increased by 2, compared to the previous iteration.
Suboptimal joint energy and bit allocation
The greedy bit allocation algorithm introduced in the previous subsection requires the reevaluation of the values of \(\delta _{k}^{m_{k}+2}(\mathbf {m})\) (∀k∈{1,…,N}) each time the set \(\mathcal {N}\) of active subcarriers is modified. The complexity would increase even further if we combined each step of the greedy bit allocation algorithm with the optimized energy allocation introduced in Section 4.2, which requires solving a convex optimization algorithm instead of a simple evaluation of Eq. (19).
To circumvent this complexity, we present a faster, less computationally intensive algorithm. We initialize the algorithm with the optimal uniform energy and bit allocation from Section 4.1. Then, as a first step we calculate for this specific uniform bit allocation the optimized energy allocation vector resulting from OP (24), for \(\mathcal {N}=\{1,\ldots,N\}\). In the second step, we optimize the bit allocation and code rate according to the greedy algorithm outlined in 4.3. Because during this step the energy allocation vector E is kept to its value resulting from the previous step, we can drop the dependency of \(\delta _{k}^{m_{k}+2}\) on m because \(\delta _{k}^{m_{k}+2}\) now depends only on m k for given k and, therefore, has to be evaluated only once for each m k (m k ≥0,∀k∈{1,…,N}). This considerably reduces the complexity. For more details, we refer to the pseudo-code of this algorithm shown in Table 2. As a final step, the optimized energy allocation vector E is recalculated according to Section 4.2, for the optimized TM resulting from the second step. The resulting values for the code rate r, energy allocation E, and bit allocation m are then used for the transmission.
Table 2 Suboptimal joint energy and bit allocation
Numerical results
We consider a communication system characterized by the parameters from Table 3, which uses orthogonal frequency-division multiple access (OFDMA) to support several users. Here, we concentrate on the performance of a user to which 48 data subcarriers are allocated, which is equal to one subchannel in the FUSC permutation mode of WiMax [31]. These subcarriers are considered to be evenly spaced across the available bandwidth. The channel impulse responses behave according to the ITU vehicular A model [32], with time variations according to Jakes' model [33]. We consider a single PU receiver (so we can drop the index q) and the channels between the different nodes are characterized by Tr(E[h h H])=1 and Tr(E[g g H])=10−3, where h and g denote the channel impulse responses corresponding to the channel frequency responses H and G, respectively; this yields E[|H k |2]=1 and E[|G k |2]=10−3 for k∈{1,…,N}. In this section, we will consider three types of CSI, i.e., estimated CSI, delayed CSI, and estimated and delayed CSI (see the Appendix section); we always assume that for both C S I and C S I PU, the same type of CSI is available at the transmitter. We note however that this is not a requirement for the proper functioning of our proposed algorithms.
Table 3 System parameters
The SNR is defined as
$$ \text{SNR}{\overset{\Delta}{=}}\frac{E_{\text{max}}}{N{\sigma_{w}^{2}}}. $$
As a performance indicator for the different resource allocation schemes, we will display (12), which denotes the average of the actual GP w.r.t. the joint probability density function of H, C S I, and C S I PU. This averaging involves the generation of realizations of C S I and C S I PU, from which the corresponding (m,E,r) are computed. For each such realization of (m,E,r), we generate realizations of H according to the conditional distribution p(H|C S I). For each such realization of H, we transmit and decode one packet using the transmission parameters (m,E,r) and verify whether a decoding error has occurred; averaging the indicator of a decoding error over the realizations of H yields E H [PER(ϕ,Γ)|C S I] corresponding to the considered realization of (m,E,r).
Accuracy of AEGP
In this subsection, we investigate how accurately the AEGP metric (17) approximates the EGP from (13). As a reference, we compare the accuracy with the predicted GP (PGP) introduced in [20] and the IC- κESM introduced in [19]. The PGP is obtained by neglecting the uncertainty on H given the actual CSI, and is calculated by substituting H by μ H|C S I in the expression (15) and using this deterministic value of Y to replace the random variable Z in (17). The IC- κESM is an approximation that only applies to delayed CSI. For this reason, we will compare the accuracy of these three metrics for the scenario where the transmitter only has delayed CSI available (see the "Delayed CSI" section in the Appendix). The following simulation parameters are used: SNR=10 dB, \({\mathcal {I}}_{q}/\sigma _{\mathrm {w}}^{2}=0\ \text {dB}\), and the value of f d τ d is equal to 0.05.
We generate 1000 realizations of C S I and C S I PU (see the Appendix section), and for each realization, the corresponding optimum uniform bit and energy allocation and code rate are obtained as described in Section 4.1. Then, for each realization of C S I, C S I PU, and the corresponding (m,E,r), we compute (i) the AEGP from (17); (ii) the PGP; (iii) the IC- κESM; (iv) the EGP from (13), where the average conditioned on C S I is replaced by an arithmetical average over 1000 realizations of H, generated according to the conditional distribution p(H|C S I) (see the Appendix section), and for each realization of H, it is verified whether the received packet is correctly decoded; and (v) the differences ε AEGP=|AEGP−EGP|, ε PGP=|PGP−EGP|, and ε IC−κESM=|IC−κESM−EGP|. Table 4 shows the average, the standard deviation, and the root mean-squared (rms) value of ε AEGP, ε PGP, and ε IC−κESM, resulting from the simulations; these numbers should be compared to the average of EGP over the CSI, which equals 1.42 bits/s/Hz. From Table 4, we observe that the AEGP is a very accurate estimate of the EGP, outperforming both the PGP and the IC- κESM by about one order of magnitude in terms of rms value. This result validates the accuracy of both the κESM and our approximation of Y by a beta-distributed random variable. The high accuracy of the AEGP metric makes it a very attractive objective function for the optimization of the SU transmission parameters. Further, we also note that being able to accurately describe the expected performance of a link will also have further benefits for more high level algorithms such as scheduling as the probability, of correctly allocating a user to a channel that satisfies its demands, will be increased.
Table 4 Accuracy of the AEGP, PGP and IC- κESM metric (SNR =10 dB, \({\mathcal {I}}_{q}/\sigma _{\mathrm {w}}^{2}=0\ \text {dB}\) and f d τ d=0.05)
The performance of the uniform energy and bit allocation algorithm described in Section 4.1 is investigated. As a reference, we will also show the performance in the case of perfect CSI and also for non-adaptive transmission.
In the case of perfect CSI, the optimal uniform energy allocation is given by
$$ E=\min\left(\min_{q\in\mathcal{Q}}\frac{\mathcal{I}_{q}}{\sum_{l\in{\mathcal{N}}}|G_{l}^{(q)}|^{2}},\frac{E_{\text{max}}}{|{\mathcal{N}}|}\right). $$
Using this uniform energy allocation, the GP metric (11) is computed for each possible TM { m,r} but with PER(ϕ,Γ) replaced by PERESM(r,γ). The TM which corresponds to the largest GP is then considered optimal.
In the case of non-adaptive transmission, the transmitter has no CSI available. This is equivalent to the case where the pdf of the channel gains conditioned on the CSI reduces to the unconditional pdf of the channel gains. Hence, the uniform energy allocation is obtained as
$$ E=\min\left(\min_{q\in\mathcal{Q}}\frac{\mathcal{I}_{q}}{\mathrm{E}_{\mathbf{G}^{(q)}} \left[\sum_{l\in{\mathcal{N}}}|G_{l}^{(q)}|^{2}\right]},\frac{E_{\text{max}}}{|{\mathcal{N}}|}\right). $$
For the above energy allocation, the transmitter selects, for the current value of SNR (29), the TM { m,r} which leads to the highest value of E H [GP], with GP given by (11).
Now, we will apply the algorithm described in Section 4.1. As a first example, we assume that the transmitter only has estimated CSI available (see the "Estimated CSI" section in the Appendix). The variance of the estimation error related to the PU and SU channels is equal to \(\sigma _{\mathrm {e}}^{2}=0,\ 10,\ 20,\ \text {and}\ 30~\text {dB}\). For the interference threshold, we consider \({\mathcal {I}}_{q}/\sigma _{\mathrm {w}}^{2}=0\ \text {dB}\). The results are shown in Fig. 1. We observe that the performance of the SU network clearly depends on the variance of the estimation error \(\sigma _{\mathrm {e}}^{2}\). For \(\sigma _{\mathrm {e}}^{2}=30~ \text {dB}\), there is almost no gain by exploiting CSI compared to a non-adaptive transmission algorithm, because the CSI is unreliable. However, when the value of \(\sigma _{\mathrm {e}}^{2}\) decreases, we consistently see a clear gain in performance by exploiting the CSI. When \(\sigma _{\mathrm {e}}^{2}=0\) dB, we notice there is a negligible difference between the algorithm using estimated CSI or perfect CSI. Further, we also note that there is almost no gain compared to non-adaptive transmission for small SNR.
GP using estimated CSI (\({\sigma }_{\mathrm {e}}^{2}=0,\ 10,\ 20,\ \text {and}\ 30\) dB)
In the following example, the transmitter only has access to delayed CSI (see the "Delayed CSI" section in the Appendix). The performance of the SU network is shown in Fig. 2 for a value of f d τ d equal to 0.01, 0.05, 0.1, and 0.2. It is clear from Fig. 2 that when f d τ d is equal to 0.2, there is almost no gain in performance compared to the non-adaptive transmission algorithm because the channel variations are too fast. However, for lower values of f d τ d, the GP of the SU network increases considerably. When f d τ d=0.01, the GP almost equals the performance of the algorithm which uses perfect CSI.
GP using delayed CSI (f d τ d=0.01, 0.05, 0.1, and 0.2)
In Fig. 3, we show the difference in performance between optimizing the AEGP, the PGP (as in [20]), and the IC- κESM (as in [19]). We show the performance for f d τ d equal to 0.05 and 0.2. For f d τ d=0.05, we can see a small performance benefit by optimizing the AEGP compared to the less accurate PGP and IC- κESM. When f d τ d=0.2, we notice that the performance improvement we get by using the AEGP or IC- κESM becomes significantly larger compared to using the PGP. In this case, the performance achieved by using the PGP drops even below the performance we would get by using the non-adaptive approach. This demonstrates that the PGP approximation is unable to accurately describe the expected goodput and is thus not suited as an objective function for the OPs, especially in the case of fast channel variations. While optimizing the IC- κESM is shown to achieve a similar performance as the optimization of the AEGP, the IC- κESM is far less general than the proposed AEGP as it can only be used in the scenario with delayed CSI described in the "Delayed CSI" section in the Appendix.
Comparison between AEGP, PGP, and IC- κESM using delayed CSI (f d τ d= 0.05 and 0.2)
In the last example, we combine the delayed CSI with the estimated CSI (see the "Estimated and delayed CSI" section in the Appendix). We choose f d τ d=0.2 and \(\sigma _{\mathrm {e}}^{2}=0\) dB. We investigate the performance for a different number (P) of available, delayed channel estimates, with corresponding delays τ d , 2τ d , …, P τ d . The performances are shown in Fig. 4 for P=1, 2, 3, and 4. We observe that the performance of the SU network can be significantly improved when the CSI consists of multiple delayed channel estimates. In this example, the GP increases by about 20 % when going from P=1 to P=4 for high SNRs. We note that it is not possible to reach the performance of an algorithm with perfect CSI, by increasing the number of estimates. As is clear from Fig. 4, there is no noticeable performance gain by going from P=3 to P=4.
GP using estimated and delayed CSI (\(\sigma _{\mathrm {e}}^{2}=0\) dB, f d τ d=0.2, P=1, 2, 3 and 4)
In Fig. 5, we investigate the impact of the interference threshold. We show the performance of the uniform bit and energy allocation algorithm when \({\mathcal {I}}_{q}/\sigma _{\mathrm {w}}^{2}=0,\ 5\ \text {and}\ 10\ \text {dB}.\) The resulting goodput is shown for the following simulation variables: f d τ d=0.2, \(\sigma _{\mathrm {e}}^{2}=0\) dB and P=3. We observe that the value of the interference threshold has a huge impact on the performance of the SU network. A too conservative value of the interference threshold will severely limit the achievable goodput of the SU network.
GP for different interference thresholds. (\(\sigma _{\mathrm {e}}^{2}=0\) dB, f d τ d=0.2, P=3, \({\mathcal {I}}_{q}/\sigma _{\mathrm {w}}^{2}=0,\ 5\ \text {and}\ 10\ \text {dB}\))
In this subsection, the optimized energy (OE) allocation from (24) and the uniform energy (UE) allocation are compared in terms of goodput. The following simulation parameters are chosen: \(\sigma _{\mathrm {e}}^{2}=0\) dB, f d τ d=0.2, P=3, and \({\mathcal {I}}_{q}/\sigma _{\mathrm {w}}^{2}=0\ \text {dB}\). Figure 6 shows the goodput resulting from the uniform energy and bit allocation described in Section 4.1, along with the goodput corresponding to the OE allocation for the same uniform bit (UB) allocation. We notice that for high SNR the OE allocation improves the goodput by about 8 % compared to UE allocation.
GP achieved by optimal and uniform energy allocation (\(\sigma _{\mathrm {e}}^{2}=0\) dB, f d τ d=0.2, P=3, \({\mathcal {I}}_{q}/\sigma _{\mathrm {w}}^{2}=0\ \text {dB}\))
Greedy bit allocation
Now, we investigate the performance of the SU network in the case where the SU transmitter optimizes the bit allocation per subcarrier. The simulation parameters are chosen as follows: \(\sigma _{\mathrm {e}}^{2}=0\) dB, f d τ d=0.2, P=3, and \({\mathcal {I}}_{q}/\sigma _{\mathrm {w}}^{2}=0\ \text {dB}\). We compare the performance of uniform bit and energy allocation (UB+UE), with our algorithm introduced in Section 4.3 which combines greedy bit allocation with uniform energy allocation (GB+UE). Further, we also consider the performance of the suboptimal algorithm introduced in Section 4.4 which combines the greedy bit allocation and optimized energy allocation (GB+OE). From Fig. 7, we notice that there is a considerable increase in GP when we apply GB instead of UB allocation. At low SNR, the transmitter is now capable of deactivating subcarriers with poor instantaneous channel gains, which considerably decreases the PER and improves GP. At higher SNR the transmitter can now better utilize the full capacity at each subcarrier by allocating a larger number of bits to a subcarrier with favorable channel gains. An even larger gain at higher SNR can be obtained by combining the GB with the OE allocation. In Fig. 7, we notice that the gain compared to uniform bit and energy allocation (UB+UE) amounts to 10 % for greedy bit and uniform energy allocation (GB+UE) and becomes nearly 20 % for greedy bit and optimized energy allocation (GB+OE). This additional gain is achieved by giving the transmitter the freedom of reallocating the energy over the subcarriers, which improves the performance in several ways: it can happen for example that subcarriers with less favorable channel gains now receive more energy, or that subcarriers causing strong interference at the PU are switched off to allow for a higher total transmit energy. We do notice however that at lower SNRs the GB+OE algorithm performs slightly worse than the GB+UE algorithm. This is a consequence of our suboptimal approach outlined in Section 4.4. However, the performance loss at low SNR is very small, and an optimal joint bit and energy allocation algorithm would require a much higher complexity.
Comparison of the goodput achieved by GB and UB allocation (\(\sigma _{\mathrm {e}}^{2}=0\) dB, f d τ d=0.2, P=3, \({\mathcal {I}}_{q}/\sigma _{\mathrm {w}}^{2}=0\ \text {dB}\))
To illustrate their complexity, we will compare the average computation times of the different resource allocation algorithms described in Section 4. The SNR is fixed at 20 dB and the simulation parameters are \(\sigma _{\mathrm {e}}^{2}=0\) dB, f d τ d=0.2, P=3 and \({\mathcal {I}}_{q}/\sigma _{\mathrm {w}}^{2}=0\ \text {dB}\). In Fig. 8, the computation time of the algorithms is shown as a function of the number of subcarriers N. We notice a slight increase in computation time for the optimized energy allocation (UB+OE) compared to the uniform energy allocation (UB+UE). However, a more significant increase in computation time occurs when implementing the greedy bit allocation. The greedy bit with uniform energy allocation (GB+UE) described in Section 4.3 clearly becomes unfeasible when the number of subcarriers becomes too high. Compared to GB+UE, the complexity is significantly reduced when using the suboptimal joint energy and bit allocation (GB+OE) described in Section 4.4, whose computation time increases much more slowly with N.
Comparison of the simulation time of the different bit and energy allocation algorithms (SNR=20 dB, \(\sigma _{\mathrm {e}}^{2}=0\) dB, f d τ d=0.2, P=3, \({\mathcal {I}}_{q}/\sigma _{\mathrm {w}}^{2}=0\ \text {dB}\))
In this paper, we have considered adaptive coding and modulation in a cognitive BIC-OFDM system, under the realistic assumption that only imperfect CSI is available. In order to tackle this problem, we introduced an optimum performance metric called the expected goodput (EGP), which is the expectation of the goodput, conditioned on the imperfect CSI.
A major advantage of this metric is that it allows the transmitter to account for the imperfections of the CSI by selecting its transmission parameters such that the best average goodput is achieved. To make the optimization of the code rate, bit and energy allocation tractable, we proposed a very accurate approximation of this performance metric, referred to as approximate EGP (AEGP). The numerical results clearly show that the ACM algorithms based on the AEGP have at least the same performance as the non-adaptive algorithms and, in most cases, clearly outperform them. Finally, we also show that, depending upon the quality of the available CSI, the proposed algorithms can come very close to the performance of algorithms with perfect CSI.
1 This EGP metric is different from the expected effective goodput metric proposed in [18]. The metric introduced in [18] takes into account the expected transmission time, which can vary because of the possibilities of retransmissions. It has however nothing to do with imperfect CSI which is the focus of the present paper.
2 Note that if we have a number of paths L<ν+1, only L diagonal elements of R h are strictly greater than 0.
Examples of different types of CSI at the transmitter
In the following, the impulse response of a generic channel between the SU transmitter and any receiver of the PU or SU network will be denoted by h(m,t), where the delay variable is represented by the discrete time index m associated with a sampling rate 1/T, and the time variability of the channel is indicated by a continuous time index t. Without any loss of generality, we can assume that h(m,t)=0 for m<0 and for m>ν, where ν is defined as the length of the cyclic prefix. For given t, the samples h(m,t) (0≤m≤ν) of the channel impulse vector \(\mathbf {h}(t){\overset {\Delta }{=}}[h(0,t),\ldots,h(\nu,t)]^{T}\) are assumed to be independent circular symmetric zero-mean Gaussian complex random variables; assuming stationarity w.r.t. the variable t, the covariance matrix of h(t) is given by2 \(\mathbf {R}_{h}\overset {\Delta }{=}\text {diag}({\sigma _{0}^{2}},\ldots,\sigma _{\nu }^{2})\). The time variations of the channel are described by Jakes' model [33], which gives \(\mathrm {E}\left [h(m,t+\tau _{\mathrm {d}})h^{*}(m,t)\right ]=\mathrm {J}_{0}(2\pi f_{\mathrm {d}}\tau _{\mathrm {d}}){\sigma _{m}^{2}}\), where J0(x) represents the zeroth-order Bessel function of the first kind, and f d denotes the Doppler spread.
Introducing the Fourier matrix \(\mathbf {F}\in \mathbb {C}^{N_{\text {car}}\times (\nu +1)}\) as
$$ {}\mathbf{F}_{k,l}{\overset{\Delta}{=}} e^{-j2\pi(k-1)(l-1)/N_{\text{car}}},\quad k=1,\ldots,N_{\text{car}};l=1,\ldots,\nu+1, $$
the time-varying frequency response of the channel can then be written as H(t)=F h(t) which has the covariance matrix R H =F R h F H. The kth component of H(t) denotes the channel gain which affects the kth subcarrier at time instant t.
In the following subsections, we consider a few possible examples of the type of CSI available at the transmitter. Each case leads to different expressions for the parameters μ H|C S I and C H|C S I , which completely describe the random variable H(t) conditioned on the available C S I as follows
$$ \mathbf{H}(t)=\boldsymbol{\mu}_{\mathbf{H}|\mathbf{CSI}}(t)+\mathbf{n}(t), $$
where \(\mathbf {n}(t)\sim {\mathcal {C}N}(0,\mathbf {C}_{\mathbf {H}|\mathbf {CSI}})\). The probability density function p(H(t)|C S I) is then given by \({\mathcal {C}N}(\boldsymbol {\mu }_{\mathbf {H}|\mathbf {CSI}}(t), \mathbf {C}_{\mathbf {H}|\mathbf {CSI}})\). If only N of the N car subcarriers are available at the transmitter, as is the case in the numerical section, we can define a smaller μ H|C S I and C H|C S I which only contain the elements corresponding to the available subcarriers.
Estimated CSI
In this subsection we determine the quantities μ H|C S I and C H|C S I in the case of channel estimation errors. The transmitter only has access to an estimated frequency response \(\tilde {\mathbf {H}}(t)\), which means that \(\mathbf {CSI}=\tilde {\mathbf {H}}(t)\). The estimated frequency response \(\tilde {\mathbf {H}}(t)\) is decomposed as
$$ \tilde{\mathbf{H}}(t)=\mathbf{H}(t)+\tilde{\mathbf{e}}(t), $$
where \(\tilde {\mathbf {e}}(t)\) and H(t) are statistically independent, \(\tilde {\mathbf {e}}(t)\sim {\mathcal {C}N}(0,\sigma ^{2}\mathbf {I}_{N_{\text {car}}})\). In Section 5, we will use the value of the normalized estimation error variance \(\sigma _{\mathrm {e}}^{2}\overset {\Delta }{=}\sigma ^{2}/\text {Tr}\left (\mathrm {E}\left [\mathbf {h}\mathbf {h}^{H}\right ]\right)\). It can be shown that
$$\begin{array}{*{20}l} \boldsymbol{\mu}_{\mathbf{H}|\mathbf{CSI}} & =\mathbf{R}_{\mathbf{H}}(\mathbf{R}_{\mathbf{H}}+\sigma^{2}\mathbf{I}_{N_{\text{car}}})^{-1}\tilde{\mathbf{H}}(t), \end{array} $$
$$\begin{array}{*{20}l} \mathbf{C}_{\mathbf{H}|\mathbf{CSI}} & =\mathbf{R}_{\mathbf{H}}-\mathbf{R}_{\mathbf{H}} \left(\mathbf{R}_{\mathbf{H}}+\sigma^{2}\mathbf{I}_{N_{\text{car}}}\right)^{-1}\mathbf{R}_{\mathbf{H}}. \end{array} $$
Note that in the case of perfect estimation (i.e., σ 2=0) we obtain perfect CSI, as (34), (35) and (36) reduce to \(\tilde {\mathbf {H}}(t)=\mathbf {H}(t)\), μ H|C S I =H(t) and C H|C S I =0.
Delayed CSI
Now we assume that the C S I is outdated, because of a delay in the feedback to the transmitter. At time instance t, the delayed CSI available at the transmitter is denoted by H(t−τ d), where τ d denotes the delay. In this case, it can be shown that
$$ \boldsymbol{\mu}_{\mathbf{H}|\mathbf{CSI}}=J_{0}\left(2\pi f_{\mathrm{d}}\tau_{\mathrm{d}}\right)\mathbf{H}(t-\tau_{\mathrm{d}}), $$
$$ \mathbf{C}_{\mathbf{H}|\mathbf{CSI}}=\left(1-J_{0}\left(2\pi f_{\mathrm{d}}\tau_{\mathrm{d}}\right)^{2}\right)\mathbf{R}_{\mathbf{H}}. $$
When τ d =0, we obtain perfect CSI, as (37) and (38) reduce to μ H|C S I =H(t) and C H|C S I =0.
Estimated and delayed CSI
In this section we assume that the CSI available at the transmitter is both delayed and estimated. We also consider the possibility that the transmitter has access to multiple delayed estimates. With P denoting the number of available estimates, the CSI which is available at the transmitter is given by
$$ \mathbf{CSI}=\left[\tilde{\mathbf{H}}\left(t-\tau_{\mathrm{d}}\right)^{T}\ldots\tilde{\mathbf{H}}\left(t-P\tau_{\mathrm{d}}\right)^{T}\right]^{T}, $$
where \(\tilde {\mathbf {H}}(t-k\tau _{\mathrm {d}})\) (∀k∈{1,…,P}) is defined as in (34). Defining the matrices
$$\begin{array}{*{20}l} \mathbf{X} & {\overset{\Delta}{=}}\left[J_{0}\left(2\pi f_{\mathrm{d}}\tau_{\mathrm{d}}\right),J_{0}\left(2\pi2f_{\mathrm{d}}\tau_{\mathrm{d}}\right)\ldots,J_{0}\left(2\pi Pf_{\mathrm{d}}\tau_{\mathrm{d}}\right)\right]\otimes\mathbf{R}_{\mathbf{H}}, \end{array} $$
$$\begin{array}{*{20}l} \mathbf{Y} & {\overset{\Delta}{=}}\mathbf{J}\otimes\mathbf{R}_{\mathbf{H}}+\mathbf{I}_{P}\otimes\sigma^{2}\mathbf{I}_{N_{\text{car}}}, \end{array} $$
where \(\mathbf {J}\in \mathbb {C}^{P\times P}\) with entries \(\mathbf {J}_{k,l}{\overset {\Delta }{=}} J_{0}\left (2\pi f_{\mathrm {d}}\tau _{\mathrm {d}}(k-l)\right)\), k=1,…,P;l=1,…,P, and ⊗ indicates the Kronecker product, it can be shown that
$$\begin{array}{*{20}l} \boldsymbol{\mu}_{\mathbf{H}|\mathbf{CSI}} & =\mathbf{X}\mathbf{Y}{}^{-1}\mathbf{CSI}, \end{array} $$
$$\begin{array}{*{20}l} \mathbf{C}_{\mathbf{H}|\mathbf{CSI}} & =\mathbf{R}_{\mathbf{H}}-\mathbf{X}\mathbf{Y}{}^{-1}\mathbf{X}^{H}. \end{array} $$
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J. Van Hecke is supported by a Ph. D. fellowship of the Research Foundation Flanders (FWO).
This work was supported by the European Commission in the framework of the FP7 Network of Excellence in Wireless COMmunications NEWCOM# (Grant agreement no. 318306), and the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office.
Ghent University, Department of Telecommunications and Information Processing, Ghent, 9000, Belgium
Jeroen Van Hecke
& Marc Moeneclaey
University of Pisa, Department of Information Engineering, Pisa, I-56122, Italy
Paolo Del Fiorentino
, Riccardo Andreotti
, Vincenzo Lottici
& Filippo Giannetti
Université Catholique de Louvain, Institute of Information and Communication Technologies, Electronics and Applied Mathematics (ICTEAM), Louvain-la-Neuve, 1348, Belgium
Luc Vandendorpe
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Correspondence to Jeroen Van Hecke.
Van Hecke, J., Del Fiorentino, P., Andreotti, R. et al. Adaptive coding and modulation using imperfect CSI in cognitive BIC-OFDM systems. J Wireless Com Network 2016, 256 (2016). https://doi.org/10.1186/s13638-016-0739-5
Effective SNR mapping (ESM)
Orthogonal frequency division multiplexing (OFDM)
Adaptive coding and modulation (ACM)
Imperfect channel state information
Goodput | CommonCrawl |
\begin{document}
\title{Universal Off-Policy Evaluation}
\begin{abstract}
When faced with sequential decision-making problems, it is often useful to be able to predict what would happen if decisions were made using a new policy. Those predictions must often be based on data collected under some previously used decision-making rule.
Many previous methods enable such \emph{off-policy} (or counterfactual) estimation of the \textit{expected} value of a performance measure called the \emph{return}.
In this paper, we take the first steps towards a \emph{\underline{un}iversal \underline{o}ff-policy estimator} (UnO)---one that provides off-policy estimates and high-confidence bounds for \emph{any} parameter of the return distribution.
We use UnO for estimating and simultaneously bounding the mean, variance, quantiles/median, inter-quantile range, CVaR, and the entire cumulative distribution of returns.
Finally, we also discuss UnO's applicability in various settings, including fully observable, partially observable (i.e., with unobserved confounders), Markovian, non-Markovian, stationary, smoothly non-stationary, and discrete distribution shifts.
\end{abstract}
\section{Introduction}
Problems requiring sequential decision-making are ubiquitous \citep{barto2017some}.
When online experimentation is costly or dangerous, it is essential to conduct off-policy evaluation before deploying a new policy; that is, one must leverage existing data collected using some policy $\beta$ (called a behavior policy) to evaluate a performance metric of another policy $\pi$ (called the evaluation policy).
For problems with high stakes, such as in terms of health \citep{liao2020off} or financial assets \citep{theocharous2015ad},
it is also crucial to provide high-confidence bounds on the desired performance metric to ensure reliability and safety.
Perhaps the most widely studied performance metric in the off-policy setting is the expected return \citep{SuttonBarto2}.
However, this metric can be limiting for many problems of interest.
Safety-critical applications, such as automated healthcare, require minimizing the chances of risk-prone outcomes, and so performance metrics such as value at risk (VaR) or conditional value at risk (CVaR) are more appropriate \citep{keramati2020being,brown2020bayesian}.
By contrast, applications like online recommendations are subject to noisy data and call for robust metrics like the median and other quantiles \citep{altschuler2019best}.
In order to improve user experiences, applications involving direct human-machine interaction, such as robotics and autonomous driving, focus on minimizing uncertainty in their outcomes and thus use metrics like variance and entropy \citep{kuindersma2013variable,tamar2016learning}.
Recent work in distributional reinforcement learning (RL) have also investigated estimating the cumulative distribution of returns \citep{bellemare2017distributional, dabney2020distributional} and its various statistical functionals \citep{rowland2019statistics}.
While it may even be beneficial to use all of these different metrics simultaneously to inform better decision-making, even individually estimating and bounding any performance metric, other than mean and variance, in the \textit{off-policy setting} has remained an open problem.
This raises the main question of interest: \textit{How do we develop a universal off-policy method---one that can estimate any desired performance metrics and can also provide finite-sample confidence bounds that hold simultaneously with high probability for those metrics?}
\textbf{Prior Work:}
Off-policy methods can be broadly categorized as model-based or model-free \citep{SuttonBarto2}.
Model-based methods typically require strong assumptions on the parametric model when statistical guarantees are needed.
Further, using model-based approaches to estimate parameters other than the mean can also require estimating the \textit{distribution} of rewards for \textit{every} state-action pair in order to obtain the complete return distribution for any policy.
By contrast, model-free methods are applicable to a wider variety of settings.
Unfortunately, the popular technique of using \textit{importance-weighted returns} \citep{precup2000eligibility} only corrects for the \textit{mean} under the off-policy distribution.
Recent work by \citet{chandak2021hcove} provides a specialized extension to only correct for the variance.
Outside RL, works in the econometrics and causal inference literature have also considered quantile treatments \citep{donald2014estimation,wang2018quantile} and inferences on counterfactual distributions \citep{dinardo1995labor,chernozhukov2013inference,firpo2016identification}, but these methods are not developed for sequential decisions and do not provide any high-confidence bounds with guaranteed coverage. Further, they often mandate stationarity, identically distributed data, and full observability (i.e., no confounding).
Existing frequentist high-confidence bounds are not only specifically designed for either the mean or variance, but also hold only \textit{individually} \citep{thomas2015highb, jiang2020minimax, chandak2021hcove}.
Instead of frequentist intervals, a Bayesian posterior distribution over the mean return and various statistics of that distribution can also be obtained \citep{yang2020offline}.
We are not aware of any method that provides off-policy bounds or even estimates for \emph{any} parameter of the return, while also handling different domain settings that are crucial for RL related tasks.
Therefore, a detailed discussion of existing work is deferred to Appendix \ref{apx:related}.
\textbf{Contributions:} We take the first steps towards a \textit{\underline{un}iversal \underline{o}ff-policy estimator} (UnO) that estimates and bounds the \textit{entire distribution} of returns, and then derives estimates and simultaneous bounds for all parameters of interest.
With UnO, we make the following contributions:
\textbf{A. } For \textit{any} distributional parameter (mean, variance, quantiles, entropy, CVaR, CDF, etc.), we provide an off-policy method to obtain \textbf{(A.1)} model-free estimators; \textbf{(A.2)} high-confidence bounds that have guaranteed coverage \textit{simultaneously} for all parameters and that, perhaps surprisingly, often nearly match or outperform prior bounds specifically designed for the mean and the variance; and
\textbf{(A.3)} approximate bounds using statistical bootstrapping that can often be significantly tighter.
\textbf{B. } The above advantages hold for \textbf{(B.1)} fully observable and partially observable (i.e., with unobserved confounders) settings,
\textbf{(B.2)} Markovian and non-Markovian settings, and \textbf{(B.3)} settings with stationary, smoothly non-stationary, and discrete distribution shifts in a policy's performance.
\textbf{Limitations:} \label{apx:limitations}
Our method uses importance sampling and thus \textbf{(1)} Requires knowledge of action probabilities under the behavior policy $\beta$, \textbf{(2)} Any outcome under the evaluation policy should have a sufficient probability of occurring under $\beta$, and \textbf{(3)} Variance of our estimators scales exponentially with the horizon length \citep{guo2017using,liu2018breaking}, which may be unavoidable in non-Markovian domains \citep{jiang2016doubly}.
\textbf{Notation: }
For brevity, we first restrict our focus to the stationary setting. In Section \ref{sec:confounding}, we discuss how to tackle non-stationarity and distribution shifts.
A \textit{partially observable Markov decision process} (POMDP) is a tuple $(\mathcal S, \mathcal O, \mathcal A, \mathcal P, \Omega, \mathcal R, \gamma, d_0)$, where $\mathcal S$ is the set of states, $\mathcal O$ is the set of observations, $\mathcal A$ is the set of actions, $\mathcal P$ is the transition function, $\Omega$ is the observation function, $\mathcal R$ is the reward function, $\gamma \in [0,1]$ is the discount factor, and $d_0$ is the starting state distribution.
Although our results extend to the continuous setting, for notational ease, we consider $\mathcal S, \mathcal A, \mathcal O$, and the set of rewards to be finite.
Since the true underlying states are only partially observable, the resulting rewards and transitions from one partially observed state to another are therefore also potentially non-Markovian \citep{singh1994learning}.
We write $S_t, O_t, A_t$, and $R_t$ to denote random variables for state, observation, action, and reward respectively at time $t$.
Let $\mathcal D$ be a data set $(H_i)_{i=1}^n$ collected using \textit{behavior} policies $(\beta_i)_{i=1}^n$, where each $H_i$ denotes the \textit{observed trajectory}
$( O_{0}, A_{0}, \beta(A_{0}|O_{0}), R_{0}, O_{1}, ...)$.
Notice that an observed trajectory contains $\beta(A_{t}|O_{t})$ and does not contain the states $S_t$, for all $t$.
Let $G_i \coloneqq \sum_{j=0}^T \gamma^j R_{j}$ be the \textit{return} of $H_i$, where $\forall i, \,\, G_{\min} < G_{i} < G_{\max}$ for some finite constants $G_{\min}$ and $G_{\max}$, and $T$ is a finite horizon length.
Let $G_\pi$ and $H_\pi$ be the random variables for returns and complete trajectories under any policy $\pi$, respectively.
Since the set of observations, actions, and rewards are finite, and $T$ is finite, the total number of possible trajectories is finite.
Let $\mathcal X$ be the finite set of returns corresponding to these trajectories.
Let $\mathscr H_\pi$ be the set of all possible trajectories for any policy $\pi$.
Sometimes, to make the dependence explicit, we write $g(h)$ to denote the return of trajectory $h$.
Further, to ensure that samples in $\mathcal D$ are informative,
we make a standard assumption that any outcome under $\pi$ has sufficient probability of occurring under $\beta$
(see Appendix \ref{apx:ass} for further discussion of assumptions in general),
\begin{ass} The set $\mathcal D$ contains independent (not necessarily identically distributed) observed trajectories generated using $(\beta_i)_{i=1}^n$, such that for some (unknown) $\varepsilon > 0$,
$(\beta_i(a|o)<\varepsilon)\implies(\pi(a| o) = 0)$, for all $o \in \mathcal O, a \in \mathcal A,$ and $i \in \{1,2,...,n\}$.
\thlabel{ass:support}
\end{ass}
\section{Idea Summary}
For the desired universal method, instead of considering each parameter individually, we suggest estimating the entire \textit{cumulative distribution function} (CDF) of returns first:
\begin{align}
\forall \nu \in \mathbb R, \quad\quad F_\pi(\nu) \coloneqq \Pr \Big(G_\pi \leq \nu \, \Big). \end{align}
Any distributional parameter, $\psi(F_\pi)$, can then be estimated from the estimate of $F_\pi$.
However, we only have off-policy data from a behavior policy $\beta$, and the typical use of importance sampling \citep{precup2000eligibility} only corrects for the mean return.
To overcome this, we propose an estimator $\hat F_n$ that uses importance sampling from the \textit{perspective of the CDF} to correct for the \textit{entire} distribution of returns.
The CDF estimate, $\hat F_n$, is then used to obtain a plug-in estimator $\psi(\hat F_n)$ for any distributional parameter $\psi(F_\pi)$.
Next, we show that this CDF-centric perspective provides the additional advantage that, if we can compute a $1-\delta$ \textit{confidence band} $\mathcal F: \mathbb R \rightarrow 2^{\mathbb R}$ such that
\begin{align}
\Pr \Big( \forall \nu \in \mathbb R, \,\, \Pr\big(G_\pi \leq \nu\big) \in \mathcal F(\nu) \Big) \geq 1 - \delta, \end{align}
then a $1-\delta$ upper (or lower) high-confidence bound on any parameter, $\psi(F_\pi)$, can be obtained by searching for a function $F$ that maximizes (or minimizes) $\psi(F)$ and $ \forall \nu \in \mathbb R$ has $F(\nu) \in \mathcal F(\nu)$.
\section{UnO: Universal Off-Policy Estimator}
In the\textit{ on-policy} setting, one approach for estimating any parameter of returns, $G_\pi$, might be to first estimate its \textit{cumulative distribution} $F_\pi$ and then use that to estimate its parameter $\psi(F_\pi)$.
However, doing this in the off-policy setting requires additional consideration as
the \textit{entire} distribution of the observed returns needs to be adjusted to estimate $F_\pi$ since the data is collected using behavior policies that can be different from the evaluation policy $\pi$.
We begin by observing that $\forall \nu \in \mathbb R, F_\pi(\nu)$ can be expanded using the fact that the probability that the return $G_\pi$ equals $x$ is the sum of the probabilities of the trajectories $H_\pi$ whose return equals $x$,
\begin{align}
F_\pi(\nu)
&= \Pr(G_\pi \leq \nu)
= \sum_{x\in\mathcal X, x \leq \nu} \Pr(G_\pi = x) = \sum_{x\in\mathcal X, x \leq \nu} \left( \sum_{h \in \mathscr H_\pi} \Pr(H_\pi = h) \mathds{1}_{\{g(h) = x\}} \right),\;\; \label{main:eqn:1} \end{align}
where $\mathds{1}_{A} = 1$ if $A$ is true and 0 otherwise.
Now, observing that the indicator function can be one for at most a single value less than $\nu$ as $g(h)$ is a deterministic scalar given $h$, \eqref{main:eqn:1} can be expressed as,
\begin{align}
F_\pi(\nu) &
= \sum_{h \in \mathscr H_\pi} \Pr(H_\pi = h) \sum_{x\in\mathcal X, x \leq \nu}\mathds{1}_{\{g(h) = x\}}
= \sum_{h \in \mathscr H_\pi} \Pr(H_\pi = h)\Big( \mathds{1}_{\{g(h) {\color{red}\leq} \nu\}}\Big), \label{main:eqn:2} \end{align}
where the red color is used to highlight changes.
Now, from \thref{ass:support} as $\forall \beta, \,\, \mathscr H_\pi \subseteq \mathscr H_\beta$,\footnote{Results can be extended to hybrid probability measures using Radon-Nikodym derivatives.}
\begin{align}
F_\pi(\nu) &= \sum_{h \in {\color{red}\mathscr H_\beta}} \Pr(H_\pi = h) \Big(\mathds{1}_{\{g(h) \leq \nu\}}\Big) = \sum_{h \in \mathscr H_\beta} \Pr(H_\beta = h) \frac{ \Pr(H_\pi = h)}{\Pr(H_\beta = h)} \Big(\mathds{1}_{\{g(h) \leq \nu\}} \Big). \label{main:eqn:4} \end{align}
The form of $F_\pi(\nu)$ in \eqref{main:eqn:4} is beneficial as it suggests a way to not only perform off-policy corrections for one specific parameter, as in prior works \citep{precup2000eligibility,chandak2021hcove}, but for the \textit{entire cumulative distribution function} (CDF) of return $G_\pi$.
Formally, let $\rho_i \coloneqq \prod_{j=0}^{T} \frac{\pi(A_{j}| O_{j})}{\beta_i(A_{j}|O_{j})}$ denote the importance ratio for $H_i$, which is equal to $\Pr(H_\pi=h)/\Pr(H_\beta=h)$ (see Appendix \ref{apx:proofs}).
Then, based on \eqref{main:eqn:4}, we propose the following non-parametric and model-free estimator for $F_\pi$.
\begin{align}
\forall \nu\in \mathbb{R}, \quad \hat F_n(\nu) \coloneqq \frac{1}{n} \sum_{i=1}^n \rho_i \mathds{1}_{\{G_i \leq \nu\}}.
\label{eqn:Festimator} \end{align}
Figure \ref{fig:IS} provides intuition for \eqref{eqn:Festimator}.
In the following theorem, we establish that this estimator, $\hat F_n$, is unbiased and not only pointwise consistent, but also a uniformly consistent estimator of $F_\pi$, even when the data $\mathcal D$ is collected using multiple behavior policies $(\beta_i)_{i=1}^n$.
The proof (deferred to Appendix \ref{apx:proofs}) also illustrates that by using knowledge of action probabilities under the behavior policies, no additional adjustments (e.g., front-door or backdoor \citep{pearl2009causality}) are required by $\hat F_n$ to estimate $F_\pi$, even when the domain is non-Markovian or has partial observability (confounders).
\begin{figure}
\caption{An illustration of return distributions for $\pi$ and $\beta$.
The CDF at any point $\nu$ corresponds to the area under the probability distribution up until $\nu$.
Having order statistics $(G_{(i)})_{i=1}^5$ of samples $(G_i)_{i=1}^5$ drawn using $\beta$, \eqref{eqn:Festimator} constructs an empirical estimate of the CDF
for $\pi$ (\textit{green} shaded region) by correcting for the probability of observing each $G_i$ using the \textit{importance-sampled counts} of $G_i \leq \nu$.
Additionally, weighted-IS (WIS) can be used as in \eqref{eqn:WISF} for a variance-reduced estimator for $F_\pi$.}
\label{fig:IS}
\end{figure}
\begin{thm}
\thlabel{thm:Funbiased} Under \thref{ass:support}, $\hat F_n$ is an unbiased and uniformly consistent estimator of $F_\pi$, \begin{align}
\forall \nu\in \mathbb{R}, \quad \mathbb{E}_{\mathcal D}\Big[\hat F_n(\nu)\Big] &= F_\pi(\nu), &
\underset{\nu \in \mathbb R}{\sup} \quad \Big|\hat F_n(\nu) - F_\pi(\nu) \Big| \overset{\text{a.s.}}{\longrightarrow} 0. \end{align} \end{thm}
\begin{rem} Notice that the value of $\hat F_n(\nu)$ can be more than one, even though $F_\pi(\nu)$ cannot have a value greater than one for any $\nu \in \mathbb R$.
This is an expected property of estimators based on importance sampling (IS).
For example, the IS estimates of expected return during off-policy mean estimation can be smaller or larger than the smallest and largest possible return when $\rho > 1$. \thlabel{rem:geq1} \end{rem}
Having an estimator $\hat F_n$ of $F_\pi$, any parameter $\psi(F_\pi)$ can now be estimated using $\psi(\hat F_n)$.
However, some parameters like the mean $\mu_\pi$, variance $\sigma^2_\pi$, and entropy ${\mathcal H_\pi}$, are naturally defined using the probability distribution $\text{d}F_\pi$ instead of the cumulative distribution $F_\pi$.
Similarly, parameters like the $\alpha$-quantile $ Q^\alpha_\pi$ and inter-quantile range (which provide tail-robust measures for the mean and deviation from the mean) and conditional value at risk $\text{CVaR}_\pi^\alpha$ (which is a tail-sensitive measure) are defined using the inverse CDF $F_\pi^{-1}(\alpha)$.
Therefore, let $(G_{(i)})_{i=1}^n$ be the \textit{order statistics} for samples $(G_i)_{i=1}^n$ and $G_{(0)} \coloneqq G_{\min}$.
Then, we define the off-policy estimator of the inverse CDF for all $\alpha \in [0,1]$, and the probability distribution estimator $\mathrm{d}\hat F_n$ as,
\begin{align}
\hat F^{-1}_n(\alpha) &\coloneqq \min \Big\{g \in (G_{(i)})_{i=1}^n \Big| \hat F_n(g) \geq \alpha \Big\}, \quad & \text{d}\hat F_n(G_{(i)}) \coloneqq \hat F_n(G_{(i)}) - \hat F_n(G_{(i-1)}), \label{eqn:inverseF} \end{align}
where $\text{d}\hat F_n(\nu)\coloneqq 0$ if $\nu \neq G_{(i)}$ for any $i \in (1,\dotsc,n)$.
Using \eqref{eqn:inverseF}, we now define off-policy estimators for parameters like the mean, variance, quantiles, and CVaR (see Appendix \ref{apx:UnOinverse} for more details on these).
This procedure can be generalized to any other parameter of $F_\pi$ for which a sample estimator $\psi(\hat F_n)$ can be directly created using $\hat F_n$ as a plug-in estimator for $F_\pi$.
\begin{align}
\mu_\pi(\hat F_n)&\coloneqq \sum_{i=1}^n \text{d}\hat F_n(G_{(i)}) G_{(i)},
&
\sigma^2_\pi(\hat F_n) &\coloneqq \sum_{i=1}^n \text{d} \hat F_n(G_{(i)}) \Big (G_{(i)} - \mu_\pi(\hat F_n) \Big)^2,
\\
{Q}^\alpha_\pi(\hat F_n) &\coloneqq \hat F_n^{-1}(\alpha),
& {\text{CVaR}}^\alpha_\pi(\hat F_n) &\coloneqq \frac{1}{\alpha}\sum_{i=1}^n \text{d}\hat F_n(G_{(i)}) G_{(i)} \mathds{1}_{\left\{G_{(i)} \leq {Q}^\alpha_\pi(\hat F_n)\right\}}. \end{align}
\begin{rem} Let $H_i$ be the observed trajectory for the $G_i$ that gets mapped to $G_{(i)}$ when computing the order statistics.
Note that $\text{d}\hat F_n(G_{(i)})$ equals $\rho_i/n$ for this $H_i$.
This implies that the estimator for the mean, $\mu_\pi(\hat F_n)$, reduces \textit{exactly} to the existing full-trajectory-based IS estimator \citep{precup2000eligibility}. \end{rem}
Notice that many parameters and their sample estimates discussed above are nonlinear in $F_\pi$ and $\hat F_n$, respectively (the mean is one exception).
Therefore, even though $\hat F_n$ is an unbiased estimator of $F_\pi$, the sample estimator, $\psi(\hat F_n)$, may be a biased estimator of $\psi(F_\pi)$.
This is expected behavior because even in the on-policy setting it is not possible to get unbiased estimates of some parameters (e.g., standard deviation), and UnO reduces to the on-policy setting when $\pi=\beta$.
However, perhaps surprisingly, we establish in the following section that even when $\psi(\hat F_n)$ is a biased estimator of $\psi(F_\pi)$, high-confidence upper and lower bounds can still be computed for both $F_\pi$ and $\psi(F_\pi)$.
\section{High-Confidence Bounds for UnO} \label{sec:Fbounds}
Off-policy estimators are typically prone to high variance, and when the domain can be non-Markovian, the curse of horizon might be unavoidable \citep{jiang2016doubly}. For critical applications, this might be troublesome \citep{thomas2019preventing} and thus
necessitates obtaining confidence intervals to determine how much our estimates can be trusted.
Therefore, in this section, we aim to construct a set of possible CDFs $\mathcal F: \mathbb R \rightarrow 2^{\mathbb R}$, called a \textit{confidence band}, such that the true $F_\pi(\nu)$ is within the set $\mathcal F(\nu)$ with high probability, i.e., $\Pr ( \forall \nu \in \mathbb R, \,\, F_\pi(\nu) \in \mathcal F(\nu) ) \geq 1 - \delta$, for any $\delta \in (0, 1]$.
Subsequently, we develop finite-sample bounds for any parameter $\psi(F_\pi)$ using $\mathcal F$.
In the on-policy setting, $\mathcal F$ can be constructed using the DKW inequality \citep{dvoretzky1956asymptotic} and its tight constants \citep{massart1990tight}.
However, its applicability to the off-policy setting is unclear as \textbf{(a)} unlike the on-policy CDF estimate, the ``steps'' of an off-policy CDF estimate are not of equal heights, \textbf{(b)} the ``steps'' do not sum to one (see Figure \ref{fig:Fband}) and the maximum height of the steps need not be known either, and \textbf{(c)} DKW assumes samples are identically distributed, however, off-policy data $\mathcal D$ might be collected using multiple different behavior policies.
This raises the question:\textit{ How do we obtain $\mathcal F$ in the off-policy setting?}
Before constructing a confidence band $\mathcal F$, let us first focus on obtaining bounds for a single point, $F_\pi(\kappa)$.
Let $X \coloneqq \rho ( \mathds{1}_{\{G \leq \kappa\}})$. Then, from \thref{thm:Funbiased}, we have that $\mathbb{E}_{\mathcal D}[X] = F_\pi(\kappa)$.
This implies that a confidence interval for the mean of $X$ provides a confidence interval for $F_\pi(\kappa)$.
Using this observation, existing confidence intervals for the mean of a bounded random variable can be directly applied to $X$ to obtain a confidence interval for $F_\pi(\kappa)$.
For example, \citet{thomas2015higha} present tight bounds for the mean of IS-based random variables by mitigating the variance resulting from the heavy tails associated with IS; we use their method on $\hat F_n(\kappa)$ to bound $F_\pi (\kappa)$.
Alternatively, recent work by \citet{kuzborskij2020confident} can potentially be used with a WIS-based $F_\pi$ estimate \eqref{eqn:WISF}.
Before moving further, we introduce some additional notation.
Let $(\kappa_i)_{i=1}^K$ be any $K$ ``key points'' and let $\texttt{CI}_-(\kappa_i, \delta_i)$ and $\texttt{CI}_+(\kappa_i, \delta_i)$ be the lower and the upper confidence bounds on $F_\pi(\kappa_i)$ constructed at each key point using the observation made in the previous paragraph, such that \begin{align} \forall i \in (1,...,K), \quad & \Pr\Big(\texttt{CI}_-(\kappa_i, \delta_i) \leq F_\pi(\kappa_i) \leq \texttt{CI}_+(\kappa_i, \delta_i)\Big) \geq 1 - \delta_i. \end{align}
We now use the following observation to obtain a band, $\mathcal F$, that contains $F_\pi$ with high confidence.
Because $F_\pi$ is a CDF, it is necessarily monotonically non-decreasing, and so if $F_\pi(\kappa_i) \geq \texttt{CI}_-(\kappa_i, \delta_i)$ then for any $ \nu \geq \kappa_i$, $F_\pi(\nu)$ must be no less than $\texttt{CI}_-(\kappa_i, \delta_i)$.
Similarly, if $F_\pi(\kappa_i) \leq \texttt{CI}_+(\kappa_i, \delta_i)$ then for any $ \nu \leq \kappa_i$, $F_\pi(\nu)$ must also be no greater than $\texttt{CI}_+(\kappa_i, \delta_i)$.
Let $\kappa_0 \coloneqq G_{\min}$, $\kappa_{K+1} \coloneqq G_{\max}$, $\texttt{CI}_-(\kappa_0, \delta_0) \coloneqq 0$, and $\texttt{CI}_+(\kappa_{K+1}, \delta_{K+1}) \coloneqq 1$; then, as illustrated in Figure \ref{fig:Fband}, we can construct a lower function $F_-$ and an upper function $F_+$ that encapsulate $F_\pi$ with high probability,
\begin{minipage}[c]{0.5\textwidth} \begin{align}
F_-(\nu) \coloneqq
\begin{cases}
1 & \text{if } \nu > G_{\max},
\\
\underset{\kappa_i \leq \nu}{\max} \,\,\, \texttt{CI}_-(\kappa_i,\delta_i) & \text{otherwise}.
\end{cases} \end{align}
\end{minipage}
\begin{minipage}[c]{0.5\textwidth}
\begin{align}
F_+(\nu) \coloneqq
\begin{cases}
0 & \text{if } \nu < G_{\min},
\\
\underset{\kappa_i \geq \nu}{\min} \,\,\, \texttt{CI}_+(\kappa_i,\delta_i) & \text{otherwise}.
\end{cases} \label{eqn:Fband2} \end{align} \end{minipage}
\begin{figure}
\caption{An illustration of $\hat F_n$ (in black) using five return samples and the confidence band $\mathcal F$ (red shaded region) computed using \eqref{eqn:Fband2} with confidence intervals (red lines) at three key points $(\kappa_i)_{i=1}^3$.
Notice that the vertical ``steps'' in $\hat F_n$ can be of different heights and their total can be greater than $1$ due to importance weighting. However, since we know that $F_\pi$ is never greater than $1$, $\mathcal F$ can be clipped at $1$. }
\label{fig:Fband}
\end{figure}
\begin{thm} Under \thref{ass:support}, for any $\delta \in (0, 1]$, if $\sum_{i=1}^K\delta_i \leq \delta$, then the confidence band defined by $F_-$ and $F_+$ provides guaranteed coverage for $F_\pi$.
That is, \begin{align}
\Pr \Big(\forall \nu \in \mathbb R, \,\, F_{-}(\nu) \leq F_\pi(\nu) \leq F_+(\nu) \Big) \geq 1 - \delta. \end{align}\thlabel{thm:Fguarantee1} \end{thm}
\begin{rem}
Notice that any choice of $(\kappa_i)_{i=1}^K$ results in a valid band $\mathcal F$.
However, $\mathcal F$ can be made tighter by optimizing over the choice of $(\kappa_i)_{i=1}^K$. In Appendix \ref{apx:UnOoptim}, we present one such method using cross-validation to minimize the area enclosed within $\mathcal F$. \end{rem}
Having obtained a high-confidence band for $F_\pi$, we now discuss how high-confidence bounds for any parameter $\psi(F_\pi)$ can be obtained using this band.
Formally, with a slight overload of notation let $\mathcal F$ be the set of all possible CDFs bounded between $F_-$ and $F_+$, that is, \begin{align}
\mathcal F \coloneqq \Big\{F \,\, \Big| \,\, \forall \nu \in \mathbb R, \,\, F_-(\nu) \leq F(\nu) \leq F_+(\nu)\Big\}. \end{align}
This band $\mathcal F$ contains many possible CDFs, one of which is $F_\pi$ with high probability.
Therefore, to get a lower or upper bound, $\psi_-$ or $\psi_+$, on $\psi(F_\pi)$, we propose deriving a CDF $F \in \mathcal F$ that minimizes or maximizes $\psi(F)$, respectively, and we show that these contain $\psi(F_\pi)$ with high probability:
\begin{align}
\psi_- &\coloneqq \underset{F \in \mathcal F}{\inf} \,\,\, \psi (F), \quad\quad
\psi_+ \coloneqq \underset{F \in \mathcal F}{\sup} \,\,\, \psi (F). \label{eqn:lub} \end{align}
\begin{thm} Under \thref{ass:support}, for any $1-\delta$ confidence band $\mathcal F$, the confidence interval defined by $\psi_-$ and $\psi_+$ provides guaranteed coverage for $\psi(F_\pi)$.
That is, \begin{align} \Pr \Big(\psi_- \leq \psi(F_\pi) \leq \psi_+ \Big) \geq 1 - \delta. \end{align} \end{thm}
While obtaining $\psi_-$ might not look straightforward, one can obtain closed-form expressions for many popular parameters of interest.
In other cases, simple algorithms exist for computing $\psi_-$ and $\psi_+$ \citep{romano2002explicit}.
Figure \ref{fig:geometry} provides geometric depictions of the closed-form expressions for some parameters.
\begin{rem}
Perhaps surprisingly, even though $\psi(\hat F_n)$ may be biased, we can obtain high-confidence bounds with guaranteed coverage on any $\psi(F_\pi)$ using the confidence band $\mathcal F$. In fact,
confidence bounds for \emph{all} parameters computed using \eqref{eqn:lub}
hold \emph{simultaneously} with probability at least $1-\delta$ as they are all derived from the same confidence band, $\mathcal F$. \end{rem}
\begin{figure*}
\caption{Given a confidence band $\mathcal F$, bounds for many parameters can be obtained using geometry.
\textbf{(Left)} For a lower bound on the mean, we would want a CDF $F \in \mathcal F$ that assigns as high a probability as possible on lower $G$ values, and $F_+$ is the CDF which does that.
To obtain the mean of $F_+$, we use the property that the mean of a distribution is the area above the CDF on the positive x-axis minus the area below the CDF on the negative x-axis \citep{anderson1969confidence}.
Hence, the mean of the distribution characterized by $F_+$ is the area of the shaded blue region minus the area of the shaded purple region, and this value is the high-confidence lower bound on the mean.
\textbf{(Middle)} Similarly,
within $\mathcal F$, $F_+$ characterizes the distribution with the smallest
$\alpha$-quantile.
\textbf{(Right)} Building upon the lower bounds for the mean and the quantile, \citet{thomas2019concentration} showed that the lower bound for $\alpha$-CVaR can be obtained using the area of the shaded blue region minus the area of the shaded purple region, normalized by $\alpha$.
To get the upper bounds on the mean, quantile, and CVaR, analogous arguments hold using the lower bound CDF $F_-$.
See Appendix \ref{apx:UnOoptim} for discussions of variance, inter-quantile, entropy, and other parameters.
}
\label{fig:geometry}
\end{figure*} \textbf{3.1. Statistical Bootstrapping:} An important advantage of having constructed an off-policy estimator of any $\psi(F_\pi)$ is that it opens up the possibility of using \textit{resampling}-based methods, like statistical bootstrapping \citep{efron1994introduction}, to obtain \textit{approximate} confidence intervals for $\psi(F_\pi)$.
In particular,
we can use the \textit{bias-corrected and accelerated} (BCa) bootstrap procedure to obtain $\psi_-$ and $\psi_+$ for $\psi(F_\pi)$.
This procedure is outlined in Algorithm \ref{alg:pboot} in Appendix \ref{apx:sec:boot}.
Unlike the bounds from \eqref{eqn:lub}, BCa-based bounds do not offer guaranteed coverage and need to be computed individually for each parameter $\psi$.
However, they can be combined with UnO to get significantly tighter bounds with less data, albeit without guaranteed coverage.
\section{Confounding, Distributional Shifts, and Smooth Non-Stationarities} \label{sec:confounding}
A particular advantage of UnO is the remarkable simplicity with which the estimates and bounds for $F_\pi$ or $\psi(F_\pi)$ can be extended to account for confounding, distributional shifts, and smooth non-stationarities that are prevalent in real-world applications \citep{dulac2019challenges}.
\textbf{Confounding / Partial Observability:}
Estimator $\hat F_n$ in \eqref{eqn:Festimator} accounts for partial observability when both $\pi$ and $\beta$ have the same observation set. However, in systems like automated loan approval \citep{thomas2019preventing}, data might have been collected using a behavior policy $\beta$ dependent on sensitive attributes like race and gender that may no longer be allowable under modern laws.
This can make the available observation, $\widetilde O$, for an evaluation policy $\pi$ different from the observations, $O$, for $\beta$, which may also have been a partial observation of the underlying true state $S$.
However, an advantage of many such automated systems (e.g., online recommendation, automated healthcare, robotics) is the direct availability of behavior probabilities $\beta_i(A|O)$.
In Appendix \ref{apx:proofs}, we provide generalized proofs for all the earlier results, showing that access to $\beta_i(A|O)$ allows UnO to handle various sources of confounding even when $\widetilde{O} \neq O$, without requiring any additional adjustments.
When $\beta_i(A|O)$ is not available, we allude to possible alternatives in Appendix \ref{apx:ass}.
\textbf{Distribution Shifts:} Many practical applications exhibit distribution shifts that might be discrete or abrupt.
One example is when a medical treatment developed for one demographic is applied to another \citep{gao2020deep}.
To tackle discrete distributional shifts, let $F^{(1)}_\pi$ and $F^{(2)}_\pi$ denote the CDFs of returns under policy $\pi$ in the first and the second domain, respectively.
To make the problem tractable, similar to prior work on characterizing distribution shifts \citep{berger2014kolmogorov}, we assume that the Kolmogorov-Smirnov distance between $F^{(1)}_\pi$ and $F^{(2)}_\pi$ is bounded.
\begin{ass} There exists $\epsilon \geq 0$, such that
$\underset{\nu \in \mathbb R}{\sup} \left | F^{(1)}_\pi(\nu) - F^{(2)}_\pi(\nu) \right| \leq \epsilon$.
\thlabel{ass:shift} \end{ass}
Given data $\mathcal D$ collected in the first domain, one can obtain the bounds $F^{(1)}_-$ and $F^{(1)}_+$ on $F^{(1)}_\pi$ as in Section \ref{sec:Fbounds}.
Now since $F^{(2)}_\pi$ can differ from $F^{(1)}_\pi$ by at most $\epsilon$ at any point, we propose the following bounds for $F^{(2)}_\pi$ for all $\nu \in \mathbb{R}$ and show that they readily provide guaranteed coverage for $F^{(2)}_\pi$:
\begin{align}
F^{(2)}_-(\nu) &\coloneqq \max(0,F^{(1)}_-(\nu) - \epsilon),
& F^{(2)}_+(\nu) \coloneqq \min(1,F^{(1)}_+(\nu) + \epsilon). \label{eqn:Fshiftbound} \end{align}
\begin{thm} Under \thref{ass:support,ass:shift}, $\forall \delta \in (0, 1]$, the confidence band defined by $F^{(2)}_-$ and $F^{(2)}_+$ provides guaranteed coverage for $F^{(2)}_\pi$.
That is, $\Pr (\forall \nu, \,\, F^{(2)}_{-}(\nu) \leq F^{(2)}_\pi(\nu) \leq F^{(2)}_+(\nu) ) \geq 1 - \delta.$
\thlabel{thm:Fguaranteeshift} \end{thm}
\textbf{Smooth Non-stationarity:} The stationarity assumption is unreasonable for applications like online tutoring or recommendation systems, which must deal with drifts of students' interests or seasonal fluctuations of customers' interests \citep{thomas2017predictive,theocharous2020reinforcement}.
In the worst case, however, even a small change in the transition dynamics can result in a large fluctuation of a policy's performance and make the problem intractable.
Therefore, similar to the work of \citet{chandak2020towards}, we assume that the distribution of returns for any $\pi$ changes smoothly over the past episodes $1$ to $L$, and the $\ell$ episodes in the future.
In particular, we assume that the trend of $F^{(i)}_\pi(\nu)$ for all $\nu$ can be modeled using least-squares regression using a nonlinear basis function $\phi : \mathbb R \rightarrow \mathbb R^d$ (e.g., the Fourier basis, which is popular for modeling non-stationary trends \citep{bloomfield2004fourier}).
\begin{ass}
For any $\nu$, $\exists w_\nu \in \mathbb{R}^d$, such that,
$\forall i \in [1,L+\ell], \,\,\,\, F^{(i)}_\pi(\nu) = \phi(i)^\top w_\nu.$
\thlabel{ass:ns} \end{ass}
Estimating $F_\pi^{(L+\ell)}$ can now be seen as a time-series forecasting problem.
Formally, for any key point $\kappa$, let $\hat F_n^{(i)}(\kappa)$ be the estimated CDF using $H_i$ observed in episode $i$.
From \thref{thm:Funbiased}, we know that $\hat F_n^{(i)}(\kappa)$ is an unbiased estimator of $F^{(i)}_\pi(\kappa)$;
therefore, $(\hat F_n^{(i)}(\kappa))_{i=1}^L$ is an unbiased estimate for the underlying time-varying sequence $(F_\pi^{(i)}(\kappa))_{i=1}^L$.
Now, using methods from time-series literature, the trend of $(\hat F_n^{(i)}(\kappa))_{i=1}^L$ can be analyzed to forecast $F^{(L+\ell)}_\pi(\kappa)$, along with its $\texttt{CI}$s.
In particular, we propose using \textit{wild bootstrap} \citep{mammen1993bootstrap,davidson2008wild}, which provides \textit{approximate} $\texttt{CI}$s with finite sample error of $O(L^{-1/2})$ while also handling
non-normality and heteroskedasticity, which would occur when dealing with IS-based estimates resulting from different behavior polices \citep{chandak2020towards}.
See Appendix \ref{apx:sec:NSboot} for more details.
Finally, using the bounds obtained using wild bootstrap at multiple key points, an entire confidence band can be obtained as discussed in Section \ref{sec:Fbounds}.
\section{Empirical Studies}
In this section, we provide empirical support for the established theoretical results for the proposed UnO estimator and high-confidence bounds.
To do so, we use the following domains: \textbf{(1)} An open source implementation \citep{xie2020deep} of the FDA-approved type-$1$ diabetes treatment simulator \citep{man2014uva}, \textbf{(2)} A stationary and a non-stationary recommender system domain, and \textbf{(3)} A continuous-state Gridworld with partial observability, where data is collected using multiple behavior policies.
Detailed description for domains and the procedures for obtaining $\pi$ and $\beta$ are provided in Appendix \ref{apx:empiricaldomain}; code is also publicly available \href{https://github.com/yashchandak/UnO}{here}.
In the following, we discuss four primary takeaway results.
\textbf{(A) Characteristics of the UnO estimator: } Figure \ref{fig:results} reinforces the universality of UnO.
As can be seen, UnO can accurately estimate the entire CDF and a wide range of its parameters: mean, variance, quantile, and CVaR.
\begin{figure*}
\caption{
Performance trend of the proposed estimators and bounds on three domains.
The black dashed line is the true value of $F_\pi$ or $\psi(F_\pi)$, green is our UnO estimator, red is our $\texttt{CI}$-based UnO bound, blue is the bootstrap version of our UnO bound, and yellow is the baseline bound for the mean \citep{thomas2015higha} or variance \citep{chandak2021hcove}.
Each bound has two lines (upper and lower); however, some are not visible due to overlaps.
The shaded regions are $\pm 2$ standard error,
computed using 30 trials.
The plots in the top row are for CDFs obtained using $3\times10^{4.5}$ samples.
The next four rows are for different parameters and share the same x-axis.
Bounds were obtained for a failure rate $\delta = 0.05$.
Since the UnO-Boot and Baseline-CI methods do not hold simultaneously for all the parameters, they were made to hold with failure rate of $\delta/4$ for a fair comparison (as there are 4 parameters in this plot).
}
\label{fig:results}
\end{figure*}
\textbf{(B) Comparison of UnO with prior work:} Recent works for bounding the mean \citep{jiang2020minimax,feng2021nonasymptotic} assume no confounding and Markovian structure. Therefore, for a fair comparison, we resort to the method of \citet{thomas2015higha} that can provide tight bounds even when the domain is non-Markovian or has confounding (partial observability).
Perhaps surprisingly, Figure \ref{fig:results} shows that the proposed guaranteed coverage bounds, termed \textit{UnO-CI} here, can be competitive with this existing specialized bound, termed \textit{Baseline-CI} here, for the mean.
In fact, UnO-CI can often require an order of magnitude less data compared to the specialized bounds for variance \citep{chandak2021hcove};
we refer readers to Appendix \ref{apx:empiricalS} for a discussion on potential reasons.
This suggests that the universality of UnO can be beneficial even when only one specific parameter is of interest.
\begin{figure}
\caption{ \textbf{(Top row)} True rewards (unknown to the RL agent) associated with each of the five items over the past $1000$ episodes for different \textit{speeds} of non-stationarity. Speed of $0$ indicates stationary setting and higher speeds indicates greater degrees of non-stationarity.
\textbf{(Bottom row)}. The black dashed line is the true value of the future distribution of returns under $\pi$: $F^{(L+\ell)}_\pi$, where $L=1000$ and $\ell = 1$. In red is our UnO bound that does not account for non-stationarity, and in blue is the wild-bootstrap version of our UnO bound that accounts for non-stationarity.
The shaded region corresponds to one standard error computed using 30 trials.
Bounds were obtained for a failure rate $\delta = 0.05$.
\textbf{(Left column)} In the stationary setting, both the variants of UnO bounds approximately contain the true future CDF $F_\pi^{(L+\ell)}$. In this setting, the UnO method designed only for stationary settings provides a tighter bound.
\textbf{(Middle \& Right columns)} As the domain becomes non-stationary, UnO bounds that do not account for non-stationarity fail to adequately bound the true future CDF $F_\pi^{(L+\ell)}$.
When the degree of non-stationarity is high, not accounting for non-stationarity can lead to significantly inaccurate bounds.
By comparison, UnO bounds that use wild bootstrap to tackle non-stationarity provide a more accurate bound throughout.
As expected, when the fluctuations due to non-stationarity increase, the width of the confidence band increases as well.
These results illustrate (a) the importance of accounting for non-stationarity, when applicable, and (b) the flexibility offered by our proposed universal off-policy estimator, UnO, to tackle such settings.
}
\label{fig:nsresults}
\end{figure}
\textbf{(C) Finite-sample confidence bounds for other parameters using UnO: }
Figure \ref{fig:results} demonstrates that UnO-CI also successfully addresses the open question of providing guaranteed coverage bounds for multiple parameters simultaneously without additional applications of the union bound.
As expected, bounds for parameters like variance and CVaR that depend heavily on the distribution tails take more samples to shrink than bounds on other parameters (like the median [quantile($0.5$)]).
Additional discussion on the observed trends for the bounds is provided in Appendix \ref{apx:empiricalS}.
The proposed UnO-Boot bounds, as discussed in Section 3.1, are approximate and might not always hold with the specified probability.
However, they stand out by providing \textit{significantly} tighter, and thus more practicable, confidence intervals.
\textbf{(D) Results for non-stationary settings: }
\label{apx:empiricalNS}
Results for this setting are presented in Figure \ref{fig:nsresults}.
As discussed earlier, online recommendation systems for tutorials, movies, advertisements and other products are ubiquitous.
However, the popular assumption of stationarity is seldom applicable to these systems.
In particular, personalizing for each user is challenging in such settings as interests of a user for different items among the recommendable products fluctuate over time.
For an example, in the context of online shopping, interests of customers can vary based on seasonality or other unknown factors.
To abstract such settings, in this domain the reward (interest of the user) associated with each item changes over time.
See Figure \ref{fig:nsresults} (top row) for visualization of the domain, for different ``speeds'' (degrees of non-stationarity).
In all the settings with different speeds, a uniformly random policy was used as a behavior policy $\beta$ to collect data for $1000$ episodes.
To test the efficacy of UnO, when the future domain can be different from the past domains, the evaluation policy was chosen to be a near-optimal policy for the future episode: $1000+1$.
\section{Conclusion}
We have taken the first steps towards developing a \emph{\underline{un}iversal \underline{o}ff-policy estimator} (UnO), closing
the open question of whether it is possible to estimate and provide finite-sample bounds (that hold with high probability) for \textit{any} parameter of the return distribution in the \textit{off-policy} setting, with minimal assumptions on the domain.
Now, without being restricted to the most common and basic parameters, researchers and practitioners can fully characterize the (potentially dangerous or costly) behavior of a policy without having to deploy it.
There are many new questions regarding how UnO can be improved for policy \textit{evaluation} by further reducing data requirements or weakening assumptions.
Using UnO for policy \textit{improvement} also remains an interesting future direction.
Subsequent to this work,
\citet{huang2021offpolicy} showed how models can be used to obtain UnO-style doubly robust estimators along with its convergence rates in the contextual bandit setting. This allows their method to also provide finite-sample uniform CDF bounds for a broad class of Lipschitz risk functionals.
\onecolumn \setcounter{thm}{0} \setcounter{ass}{0}
\appendix
\section{Notation}
\begin{table}[h]
\centering
\begin{tabular}{c|l c}
\hline \\
Symbol & Meaning \\
\hline \\
$\mathcal D$ & Data set of the observed trajectories\\
$n$ & Total number of observed trajectories in $\mathcal D$\\
$\pi$ & Evaluation policy\\
$\beta_i$ & Behavior policy for the $i^{\text{th}}$ trajectory\\
$\rho_i$ & Importance ratio for the observed trajectory $H_i$\\
$\mathcal S$ & State set\\
$\mathcal O$,
$\widetilde{\mathcal O}$ & Observation set for the behavior policy and the evaluation policy, respectively\\
$\mathcal A$ & Action set\\
$\mathcal P$ & Transition dynamics, $\mathcal P : \mathcal S \times \mathcal A \rightarrow \Delta(\mathcal S)$\\
$\mathcal R$ & Reward function, $\mathcal R: \mathcal S \times \mathcal A \rightarrow \Delta(\mathbb R)$\\
$\Omega$ & Observation function for behavior policy, $\Omega : \mathcal S \rightarrow \Delta(\mathcal O)$\\
$\Omega_2$ & Observation function for the evaluation policy, $\Omega_2: \mathcal S \times \mathcal O \rightarrow \Delta(\widetilde {\mathcal O})$\\
$\gamma$ & Discounting factor\\
$d_0$ & Starting state distribution\\
$T$ & Finite horizon length\\
$H_i$, $H_\pi$ & $i^\text{th}$ observed trajectory in the dataset and complete trajectory under policy $\pi$, respectively\\
$G_i$, $G_\pi$ & Return observed in the $i^\text{th}$ trajectory in the dataset and return under any policy $\pi$, respectively\\
$G_{\min}$, $G_{\max}$ & Minimum and maximum value of a return, respectively\\
$F_\pi, \mathrm{d}F_\pi$ & True CDF of returns under policy $\pi$ and its associated probability distribution, respectively\\
$\hat F_n$, $\bar F_n$ & Off-policy CDF estimator and weighted off-policy CDF estimator using $n$ samples, respectively\\
$F_-, F_+$ & Lower and upper bound on the CDF\\
$\mathcal F$ & The set of all CDFs between the upper bound and the lower bounds\\
$\kappa_i, K$ & $i^\text{th}$ key point and total number of key points, respectively\\
$\alpha$ & Value for defining inverse CDF-based statistics \\
$\psi$ & Generic functional for a distributional parameter/statistic\\
$\psi_-, \psi_+$ & Lower and upper bounds for $\psi(F_\pi)$ \\
$\delta$ & Failure rate for the bounds\\
$\mathcal D_\text{eval}, \mathcal D_\text{train}$ & Evaluation and training split of the dataset $\mathcal D$\\
$\texttt{CI}_-, \texttt{CI}_+$ & Lower and upper confidence bounds for a given random variable\\
$\theta$ & Parameters that are used to construct $\mathcal F$\\
$\mathscr A$ & Euclidean area enclosed within $\mathcal F$\\
$X_i^*$ & $i^\text{th}$ bootstrap resampled value for any random variable $X$\\
$\varepsilon$, $\epsilon$ & Some small value in \thref{ass:support} and \thref{ass:shift}, respectively\\
$w_\nu$, $\phi$ & Regression weights and basis function for the assumption on smooth non-stationarity\\
$L, \ell$ & Number of past and future episodes being considered in the smooth non-stationary setting
\end{tabular}
\caption{List of symbols used in the main paper and their associated meanings.}
\label{tab:my_label} \end{table}
\section{Broader Impact} \label{apx:broader}
While our estimators and bounds are both theoretically sound and intuitively simple, it is important for a broader audience to understand the limitations of our method, assumptions being made, and what can be done when these assumptions do not hold.
Understanding these assumptions can also help in mitigating any undesired biases in applications built around UnO and can thus avoid any potential negative societal impacts.
In the following, we briefly allude to possible alternatives when the required assumptions are violated.
\subsection{Discussion of Assumptions and Requirements of UnO} \label{apx:ass}
\paragraph{Knowledge of Subset Support:} Through \thref{ass:support}, UnO requires that all the behavior policies $(\beta_i)_{i=1}^n$ have sufficient support for actions that have non-zero probability under $\pi$.
Particularly, it requires that the $\beta(a|o)$ is bounded below by (an unknown) $\epsilon$ when $\pi(a|o) > 0$.
This ensures that importance ratios are bounded and thus simplifies analysis for UnO's consistency results and constructing confidence intervals.
This assumption is common both in the off-policy literature \citep{kallus2020double, xie2019towards, yang2020offline} and in real applications \citep{theocharous2015personalized}
The above assumption is also equivalent to assuming bounded exponentiated-Renyi-divergence (for $\alpha=\infty$) between the probability distributions of trajectories under the behavior and the evaluation policies \citep{metelli2020importance}.
As the UnO's bound for the CDF uses CIs for the mean as a sub-routine, the above assumption can be relaxed by using CIs for the mean that depend on Renyi-divergence for other values of $\alpha$ \citep{metelli2020importance}.
Similarly, consistency results for UnO rely upon finite variance, which can also be achieved by instead assuming that the Renyi-divergence is bounded for $\alpha=2$.
Alternatively, \thref{ass:support} can be relaxed to only absolute continuity by using methods that provide valid CIs for the mean by clipping the importance weights. (See the work by \citet[Theorem 1]{thomas2015higha} for removal of the upper bound on the importance weights when lower-bounding the mean, and the work by \citet[Theorem 5]{chandak2021hcove} for removal of the upper bound on the importance weights when upper-bounding the mean).
Furthermore, prior work has also shown how even the assumption of absolute continuity can in some cases be removed (See discussion around Eqn 8 in the appendix of the work by \citet{thomas2015higha}).
If the supports for the behavior and the evaluation policies are unequal, \citet{thomas2017importance} also present a technique to reduce variance resulting from IS.
Further, WIS might also be helpful in relaxing the assumptions on the IS ratios. Specifically, WIS-based mean bounds \citep{kuzborskij2020confident} can also be used along with the WIS-based UnO estimator \eqref{eqn:WISF} to get a valid confidence band for the entire CDF.
Using multi-importance sampling (MIS), the subset support requirement for \textit{all} $(\beta_i)_{i=1}^n$ can be relaxed to the requirement that the \textit{union of supports} under the behavior policies $(\beta_i)_{i=1}^n$ has sufficient support \citep{veach1995optimally,papini2019optimistic,metelli2020importance}.
MIS can also help in substantially reducing variance.
However, this relaxation requires an alternate assumption that a complete knowledge of all the behavior policies $(\beta_i)_{i=1}^n$, not just the probabilities of the action executed using them, is available.
\paragraph{Knowledge of Action Probabilities under Behavior Policies $(\beta_i)_{i=1}^n$:}
UnO requires access to the probability $\beta(a|o)$ (only the scalar probability value and not the entire policy $\beta$) of the actions available in the data set, $\mathcal D$, to compute the importance sampling ratios in \eqref{eqn:Festimator}.
Access to the probability $\beta(a|o)$ is often available when $\mathcal D$ is collected using an automated policy; however, it might not be available in some cases, such as when decisions were previously made by humans.
When the probability $\beta(a|o)$ is not available, one natural alternative is to estimate it from the data and use this estimate of $\beta(a|o)$ in the denominator of the importance ratios.
This technique is also known as regression importance sampling (RIS) and is known to provide biased but consistent estimates for the mean \citep{hanna2019importance, pavse2020reducing} in the Markov decision process setting (MDP) setting.
For UnO, $\hat F_n(\nu)$ is analogous to mean estimation of $X \coloneqq \rho \big(\mathds{1}_{\{G\leq \nu\}} \big)$, for any $\nu$.
Therefore, the findings of RIS can be directly extended to UnO in the MDP setting, where $\widetilde {O} = O = S$.
In the following, we provide a high-level discussion for the setting when $\beta(a|o)$ is \textit{not} available and the states are partially observed,
\begin{itemize}[leftmargin=*]
\item \textbf{Partial observability with $\widetilde O = O$:} In this setting, as $\beta(a|o) = \beta(a|\tilde o)$, one can use density estimation on the available data, $\mathcal D$, to construct an estimator $\hat \beta(a|o)$ of $\Pr(a|\tilde o) = \beta(a|\tilde o)$ and use RIS to get a biased but consistent estimator for $F_\pi$.
Here, bias results from the estimation error in $\hat \beta(a|o)$ but consistency follows as the true $\beta(a|o)$ can be recovered in the limit when $n \rightarrow \infty$.
In context of UnO, using $\hat \beta(a|o)$ instead of $\beta(a|o)$ violates the unbiased condition for $\hat F_n$, which was necessary to obtain the $\texttt{CI}$s and construct $\mathcal F$.
Therefore, high-confidence bounds with guaranteed coverage cannot be obtained using UnO in this setting.
However, point estimates and approximate bootstrap bounds can still be obtained.
\item\textbf{Partial observability with $\widetilde O \neq O$:} In this setting, using RIS will produce neither an unbiased nor a consistent estimator for $F_\pi$.
As $\mathcal D$ only has $\tilde o$ and not $o$, at best it is only possible to estimate $\Pr(a|\tilde o) = \sum_{x \in \mathcal O} \beta(a|x)\Pr(x|\tilde o)$ through density estimation using data $\mathcal D$.
However, in general, since $\beta(a|o) = \Pr(a|o) \neq \Pr(a|\tilde o)$ we cannot even consistently estimate the denominator for importance sampling unless some other stronger assumptions are made.
See work by \citet{namkoong2020off,tennenholtz2020off,bennett2020off} and \citet{kallus2020confounding} for possible alternative assumptions and approaches to tackle this setting.
\end{itemize}
\paragraph{Knowledge of $G_{\min}, G_{\max}$: } To construct the CDF band $\mathcal F$, UnO requires knowledge of $G_{\min}$ and $G_{\max}$ in \eqref{eqn:Fband2}.
Notice from Figure \ref{fig:Fband} that knowing $G_{\max}$ helps in clipping the \textit{\underline{l}ower \underline{b}ound} for the \textit{\underline{u}pper \underline{t}ail} (LBUT) of $\mathcal F$, which otherwise would have extended to $+\infty$.
Similarly, knowing $G_{\min}$ helps in clipping the \textit{{\underline{u}pper \underline{b}ound}} for the \textit{{\underline{l}ower \underline{t}ail}} (UBLT) of $\mathcal F$, which otherwise would have extended to $-\infty$.
Typically, even if $G_{\min}$ or $G_{\max}$ is not known, they can be obtained as $R_{\min}/(1-\gamma)$ or $R_{\max}/(1-\gamma)$, respectively, where $R_{\min}$ and $R_{\max}$ are known finite lower and upper bounds for any individual reward.
Otherwise, knowledge of $G_{\min}$ or $G_{\max}$ can be relaxed if the desired bound on $\psi$ does not depend on UBLT or LBUT, respectively.
For example, observe from Figure \ref{fig:geometry} that
(a) The lower bound for the mean or quantile does not depend on LBUT. Analogously, if only an upper bound for the mean or quantile is required, then UBLT is not needed.
(b) The lower bound on CVaR depends on UBLT, however, (for small values of $\alpha$) the upper bound on CVaR neither depends on LBUT nor UBLT.
(c) For an upper bound on variance, both LBUT and UBLT are required. However, for the variance's lower bound, neither LBUT nor UBLT are required. See Figure \ref{apx:fig:geometry} for intuition.
\paragraph{Knowledge of Function Class $\phi$: } For the smoothly non-stationary setting, through \thref{ass:ns}, UnO requires access to the basis functions $\phi$ that can be used with least-squares regression to analyze the trend in the distributions of returns $(F_\pi^{(i)}(\nu))_{i=1}^L$ for any $\nu \in \mathbb R$.
In practice, one can use sufficiently flexible basis functions to model time-series trends (e.g., Fourier basis \citep{bloomfield2004fourier}).
To avoid overfitting or underfitting, one could also use goodness-of-fit tests to select the functional class $\phi$ for the trend \citep{chen2003empirical}.
\paragraph{Knowledge of Bound $\epsilon$ on the Distribution Shift: } Unlike the smoothly non-stationary setting, if the underlying shift can be discrete and arbitrary, prior data may not contain any useful information towards characterizing the shift. Therefore, avoiding domain knowledge may be inevitable when setting the value for $\epsilon$ unless some other stronger assumptions are made.
\section{Extended Discussion on Related Work} \label{apx:related}
In the on-policy RL literature, parameters other than the mean have also been explored \citep{jaquette1973markov,sobel1982variance,chung1987discounted, white1988mean,dearden1998bayesian,lattimore2012pac,azar2013minimax}, and recent distributional RL methods extend this direction by estimating the entire distribution of returns \citep{morimura2010nonparametric,morimura2012parametric,bellemare2017distributional,dabney2018implicit,dabney2018distributional,dabney2020distributional,rowland2018analysis}.
Our work builds upon many of these ideas and extends them to the off-policy setting.
In the off-policy RL setup, there is a large body of literature that tackles the off-policy mean estimation problem \citep{precup2000eligibility,SuttonBarto2}.
Some works also aim at providing high-confidence off-policy mean estimation using concentration inequalities \citep{thomas2015higha,kuzborskij2020confident} or bootstrapping \citep{thomas2015highb, hanna2017bootstrapping,kostrikov2020statistical}.
Several recent approaches build upon a dual perspective for dynamic programming \citep{puterman1990markov,wang2007dual,nachum2020reinforcement} for both estimating and bounding the mean \citep{liu2018breaking, xie2019towards,jiang2020minimax,uehara2020minimax,dai2020coindice,feng2021nonasymptotic}.
However, these methods are restricted to domains with Markovian dynamics and full observability.
Some works have also focused on estimating the mean return in the setting where states are partially observed \citep{namkoong2020off,tennenholtz2020off,kallus2020confounding} or when there is non-stationarity \citep{chandak2020towards,chandak2020optimizing, khetarpal2020towards, padakandla2020survey}.
Recent work by \citet{chandak2021hcove} also looks at (high-confidence) off-policy variance estimation.
Our work extends these research directions by tackling these settings simultaneously, while also providing a general procedure to estimate and obtain high-confidence bounds for \textit{any} parameter of the distribution of returns.
Particularly, UnO is a single, unified, and universal procedure that can be used to mitigate the complexity associated with estimating different parameters for different domain settings.
A popular RL method that has similar name to UnO is the \textit{Universal value function approximator} (UVFA) by \citet{schaul2015universal}. However, UVFA is fundamentally different from UnO: UVFA estimates \textit{expected} return $\mathbb{E}[G_\pi]$ from a state given any desired goal.
By comparison, UnO estimates any parameter of the return $G_\pi$ for a single ``goal''.
Recent work by \citet{harb2020policy} and \citet{faccio2020parameter} propose using supervised learning to estimate parametric models that can map a \textit{representation} of a policy $\pi$ to the corresponding distribution of $G_\pi$.
By training over a given distribution of policies, new policies in the test set can be evaluated without using new data.
By comparison, UnO does not requires any parametric assumptions or any train-test distribution.
Further, UnO also provides high-confidence bounds for all the parameters of the return distribution.
\section{Proofs for Theoretical Results} \label{apx:proofs}
The main results in this paper are for the setting where both the evaluation and the behavior policies have the same observation set.
In the following, we present generalized results where the available observations, $\widetilde{O}$, for the evaluation policy can be different from the behavior policy's observations, $O$.
Further, for notational ease, in the main paper we had focused only on finite sets.
In the following, we present a more general setting where states, actions, observations, and rewards are all continuous.
Let $\Omega_2: \mathcal S \times \mathcal O \rightarrow \Delta(\widetilde {\mathcal O})$ be the distribution over $\widetilde{\mathcal O}$, conditioned on state $s \in \mathcal S$ and observation $o \in \mathcal O$, which determines how the observations $\widetilde O$ are generated.
Let $\mathcal D = (H_i)_{i=1}^n$ be the available observed trajectories, where each $H$ contains $(\widetilde O_{0}, A_{0}, \beta(A_{0}|O_{0}), R_{0}, \widetilde O_{1}, ...)$.
Note that when the random variables $\widetilde{O} = O = S$, we recover a standard fully observable MDP setting.
By comparison, $H_\pi$ is the random variable corresponding to the complete trajectory $(S_0, O_0, \widetilde O_{0}, A_{0}, R_{0}, S_1, O_1, \widetilde O_{1}, ...)$ under any policy $\pi$.
Of course, $H_\pi$ is unknown.
To make the dependence between a trajectory $h \in \mathscr H_\pi$ and its associated return $G$ and importance ratios $\rho$ explicit, we use the shorthand $g(h)$ and $\rho(h)$ to denote the return and importance ratios for the full trajectory $h$, respectively.
To tackle this generalized setting, we also generalize the support assumption introduced earlier,
\begin{ass}
The set $\mathcal D$ contains independent (not necessarily identically distributed) observed trajectories generated using $(\beta_i)_{i=1}^n$, such that for some (unknown) $\varepsilon > 0$,
$(\beta(a|o)<\varepsilon)\implies(\pi(a|\tilde o) = 0)$, for all $s \in \mathcal S, o \in \text{supp}(\Omega(s)), \tilde o \in \text{supp}(\Omega_2(s,o)), a \in \mathcal A,$ and $i \in \{1,\dotsc,n\}$.
\end{ass}
\begin{thm}
Under \thref{ass:support}, $\hat F_n$ is an unbiased and uniformly consistent estimator of $F_\pi$. That is, \begin{align}
\forall \nu\in \mathbb{R}, \quad \mathbb{E}_{\mathcal D}\Big[\hat F_n(\nu)\Big] &= F_\pi(\nu), &&
\underset{\nu \in \mathbb R}{\sup} \quad \Big|\hat F_n(\nu) - F_\pi(\nu) \Big| \overset{\text{a.s.}}{\longrightarrow} 0. \end{align} \end{thm}
\begin{proof}
This theorem has two results: unbiasedness and consistency of $\hat F_n$.
Therefore, we break the proof into two parts.
\paragraph{Part 1 (Unbiasedness). }
We begin by expanding $F_\pi$ for any $\nu \in \mathbb R$ using the definition of the CDF.
\begin{align}
F_\pi(\nu)
&= \Pr(G_\pi \leq \nu)
= \int_{-\infty}^{\nu} p(G_\pi = x) \mathrm{d}x = \int_{-\infty}^\nu \left( \int_{\mathscr H_\pi} p(H_\pi = h) \mathds{1}_{\{g(h) = x\}}\mathrm{d}h \right)\mathrm{d}x, \quad \label{eqn:1} \end{align}
where we used the fact that the probability density of the return $G_\pi$ being $x$ is the integral of the probability densities of the trajectories $h$ whose return equals $x$.
Therefore, as the integrands in \eqref{eqn:1} are finite and non-negative measurable functions, using Tonelli's theorem for interchanging the integrals, \eqref{eqn:1} can be expressed as,
\begin{align}
F_\pi(\nu) &= \int_{\mathscr H_\pi} p(H_\pi = h) \left( \int_{-\infty}^{\nu}\mathds{1}_{\{g(h) = x\}} \mathrm{d}x \right) \mathrm{d}h
= \int_{\mathscr H_\pi} p(H_\pi = h)\Big( \mathds{1}_{\{g(h) {\color{red}{\leq}} \nu\}}\Big) \mathrm{d}h, \label{eqn:2} \end{align}
where the last term follows because the output of $g(h)$ is a deterministic scalar given $h$ and thus the indicator function can be one for at most a single value less than $\nu$, and where the red color is used to highlight changes.
Next, using \thref{ass:support} to change the support of the distribution in \eqref{eqn:2} and using importance weights we obtain,
\begin{align}
F_\pi(\nu) &= \int_{\color{red}{\mathscr H_\beta}} p(H_\pi = h) \Big(\mathds{1}_{\{g(h) \leq \nu\}}\Big)\mathrm{d}h =
\int_{\mathscr H_\beta} p(H_\beta = h) \frac{ p(H_\pi = h)}{p(H_\beta = h)} \Big(\mathds{1}_{\{g(h) \leq \nu\}} \Big)\mathrm{d}h.\;\; \label{eqn:4} \end{align}
To simplify \eqref{eqn:4}, we recursively use the fact that $p(X, Y) = p(X)p(Y|X)$ and note that under a given policy $\pi$ the probability density of a trajectory with partial observations and non-Markovian structure is
\begin{align}
p(H_\pi = h) =& p(s_0)p(o_0|s_0) p(\tilde o_0 | o_0, s_0) p(a_0 | s_0, o_0, \tilde o_0; \pi)
\\
& \times \prod_{i=0}^{T-1} \Bigg(p(r_i | h_i) p(s_{i+1}|h_i) p(o_{i+1} | s_{i+1}, h_i) p(\tilde o_{i+1} | s_{i+1}, o_{i+1}, h_i)
\\
&\quad\quad \times p(a_{i+1} |s_{i+1}, o_{i+1}, \tilde o_{i+1}, h_i; \pi) \Bigg)p(r_T | h_T), \label{eqn:8}
\end{align}
where conditioning on $\pi$ emphasizes that each action is sampled using $\pi$, and $h_i$ represents the trajectory of all the states, partial observations, and actions up to time step $i$.
Therefore, using \eqref{eqn:8}, the ratio between $p(H_\pi = h)$ and $p(H_\beta = h)$ can be written as,
\begin{align}
\frac{ p(H_\pi = h)}{p(H_\beta = h)} &= \frac{p(a_0 | s_0, o_0, \tilde o_0; \pi)}{p(a_0 | s_0, o_0, \tilde o_0; \beta)}
\prod_{i=0}^{T-1} \frac{p(a_{i+1} |s_{i+1}, o_{i+1}, \tilde o_{i+1}, h_i; \pi)}{p(a_{i+1} |s_{i+1}, o_{i+1}, \tilde o_{i+1}, h_i; \beta)} \\
&=
\prod_{i=0}^T \frac{\pi(a_i|\widetilde o_i)}{\beta(a_i| o_i)}
\\
&= \rho(h). \label{eqn:5} \end{align}
Combining \eqref{eqn:4} and \eqref{eqn:5},
\begin{align}
F_\pi(\nu) &= \int_{\mathscr H_\beta} p(H_\beta = h)\rho(h) \Big(\mathds{1}_{\{g(h) \leq \nu\}} \Big) \mathrm{d}h. \label{eqn:6} \end{align}
Finally, it can be shown that our proposed estimator $\hat F_n$ is an unbiased estimator of $F_\pi$ by taking the expected value of $\hat F_n$,
\begin{align}
\mathbb{E}_{\mathcal D} \Big[\hat F_n (\nu) \Big] &= \mathbb{E}_{\mathcal D} \left[ \frac{1}{n} \sum_{i=1}^n \rho_i\Big( \mathds{1}_{\{G_i \leq \nu\}}\Big) \right]
\\
&= \frac{1}{n} \sum_{i=1}^n \mathbb{E}_{\mathcal D} \left[ \rho_i\Big( \mathds{1}_{\{G_i \leq \nu\}}\Big) \right]
\\
&= \frac{1}{n} \sum_{i=1}^n \int_{\mathscr{H}_{\beta_i}} p(H_{\beta_i} = h)\rho(h) \Big( \mathds{1}_{\{g(h) \leq \nu\}}\Big) \mathrm{d}h
\\
&\overset{(a)}{=} \frac{1}{n} \sum_{i=1}^n F_\pi(\nu)
\\
&= F_\pi(\nu), \label{eqn:9} \end{align}
where (a) follows from \eqref{eqn:6}, which holds for any behavior policy $\beta$ that satisfies \thref{ass:support}.
\textbf{Note:}
$H_\pi$ or $H_\beta$ were invoked only for the purposes of the proof.
Notice that the proposed estimator, $\hat F_n(\nu) = \frac{1}{n}\sum_{i=1}^n \rho_i\big( \mathds{1}_{\{G_i \leq \nu\}}\big)$, only depends on the quantities available in the observed trajectory $(H_i)_{i=1}^n$ from $\mathcal D$.
\paragraph{Part 2 (Uniform Consistency). }
For this part, we will first show pointwise consistency, i.e., for any $\nu$, $\hat F_n(\nu) \overset{\text{a.s.}}{\longrightarrow} F_\pi(\nu)$, and then we will use this to establish \textit{uniform} consistency, as required.
To do so, let \begin{align}
X_i \coloneqq \rho_i\Big( \mathds{1}_{\{G_i \leq \nu\}}\Big). \label{eqn:7} \end{align}
From \thref{ass:support}, we know that trajectories are independent and that $\beta(a|o) \geq \varepsilon$ when $\pi(a|\tilde o) > 0$.
This implies that the denominator in the IS ratio is bounded below when $\pi(a|\tilde o) \neq 0$, and hence the $X_i$'s are bounded above and have a finite variance.
Further, as established in \eqref{eqn:9}, the expected value of $X_i$ for all $i$ equals $F_\pi(\nu)$.
Therefore, using Kolmogorov's strong law of large numbers \citep[Theorem 2.3.10 with Proposition 2.3.10]{Sen1993},
\begin{align}
\hat F_n(\nu) = \frac{1}{n} \sum_{i=1}^n X_i \overset{\text{a.s.}}{\longrightarrow} \mathbb{E}_{\mathcal D} \left[ \frac{1}{n} \sum_{i=1}^n X_i\right] = F_\pi(\nu). \label{eqn:10} \end{align}
In the following, to obtain uniform consistency, we follow the proof for the Glivenko-Cantelli theorem \citep{glivenko1933sulla, cantelli1933sulla,gclemmanote1,gclemmanote2} using the pointwise consistency of the off-policy CDF estimator $\hat F_n$ established in \eqref{eqn:10}.
The proof relies upon the construction of $K$ key points such that the difference in $F_\pi$ at successive key points is bounded by a small $\epsilon_1$.
However, this would not be possible directly as there can be discontinuties/jumps in $F_\pi$ that are greater than $\epsilon_1$.
To tackle such discontinuties, we introduce some extra notation,
Formally, let, $\forall \nu \in \mathbb R$, \begin{align}
F_\pi(\nu^-) \coloneqq \Pr(G_\pi {\color{red}<} \nu) = F_\pi(\nu) - \Pr(G_\pi=\nu), && \hat F_n(\nu^-) \coloneqq \frac{1}{n} \sum_{i=1}^n \rho_i \Big(\mathds{1}_{\{G_i {\color{red} < }\nu\}} \Big). \label{eqn:20} \end{align}
Then, using arguments analogous to the ones used for \eqref{eqn:10}, it can be observed that \begin{align}
\hat F_n(\nu^-) \overset{\text{a.s}}{\longrightarrow} F_\pi(\nu^-). \label{eqn:12} \end{align}
Let $\epsilon_1 > 0$, and let $K$ be any value more than $1/\epsilon_1$.
Let $(\kappa_i)_{i=0}^K$ be $K$ key points, \begin{align}
G_{\min} = \kappa_0 < \kappa_1 \leq \kappa_2 .... \leq \kappa_{K-1} < \kappa_K = G_{\max}, \end{align} which create $K$ intervals such that for all $i \in (1, ..., K-1)$, \begin{align}
F_\pi(\kappa_i^-) \leq \frac{i}{K} \leq F_\pi(\kappa_i). \end{align}
Then by construction, if $\kappa_{i-1} < \kappa_i$, \begin{align}
F_\pi(\kappa_i^-) - F_\pi(\kappa_{i-1}) \leq \frac{i}{K} - \frac{i-1}{K} = \frac{1}{K} < \epsilon_1. \label{eqn:11} \end{align}
Intuitively, as $F_\pi$ is monotonically non-decreasing, \eqref{eqn:11} restricts the intermediate values for any $F_\pi(\nu)$, to be within an $\epsilon_1$ distance of the CDF values at its nearby key points.
Notice the role of $\kappa_i^-$ here: it would not have been possible to bound difference between $F_\pi(\kappa_i)$ and $F_\pi(\kappa_{i-1})$ by $\epsilon_1$ as there could have been `jumps' of value greater than $\epsilon_1$ in $F_\pi$.
However, $\kappa^-$ and $\kappa$ can be used to consider key points right before and after any jump in $F_\pi$, which ensures that we can always construct sequence of key points such that $F_\pi(\kappa_{i}^-) - F_\pi(\kappa_{i-1})$ is instead bounded by $\epsilon_1$.
For the CDF estimates at the key points, let,
\begin{align}
\Delta_n \coloneqq \max_{i \in (1...K-1)} \Big\{\left|\hat F_n(\kappa_i) - F_\pi(\kappa_i) \right|, \left|\hat F_n(\kappa_i^-) - F_\pi(\kappa_i^-) \right| \Big\}. \label{eqn:13} \end{align}
From \eqref{eqn:10} and \eqref{eqn:12}, as $\hat F_n(\nu)$ and $\hat F_n(\nu^-)$ are consistent estimators of $F_\pi(\nu)$ and $F_\pi(\nu^-)$, respectively, and since the maximum is over a finite set in \eqref{eqn:13}, it follows that as $n \rightarrow \infty$, \begin{align}
\Delta_n \overset{\text{a.s.}}{\longrightarrow} 0. \label{eqn:18} \end{align}
For any $\nu$, let $\kappa_{i-1}$ and $\kappa_i$ be such that $\kappa_{i-1} \leq \nu < \kappa_i$.
Then, \begin{align}
\hat F_n(\nu) - F_\pi(\nu) &\leq \hat F_n(\kappa_i^-) - F_\pi(\kappa_{i-1})
\\
&\leq \hat F_n(\kappa_i^-) - F_\pi(\kappa_{i}^-) + \epsilon_1, \label{eqn:14} \end{align} where the last step follows using \eqref{eqn:11}. Similarly, \begin{align}
\hat F_n(\nu) - F_\pi(\nu) &\geq \hat F_n(\kappa_{i-1}) - F_\pi(\kappa_{i}^-)
\\
&\geq \hat F_n(\kappa_{i-1}) - F_\pi(\kappa_{i-1}) - \epsilon_1. \label{eqn:15} \end{align} Then, using \eqref{eqn:14} and \eqref{eqn:15}, $\forall \nu \in \mathbb R$, \begin{align}
\hat F_n(\kappa_{i-1}) - F_\pi(\kappa_{i-1}) - \epsilon_1 \leq \hat F_n(\nu) - F_\pi(\nu) \leq \hat F_n(\kappa_i^-) - F_\pi(\kappa_{i}^-) + \epsilon_1, \label{eqn:16} \end{align} and thus using \eqref{eqn:13} and \eqref{eqn:16}, \begin{align}
\Big| \hat F_n(\nu) - F_\pi(\nu) \Big| \leq \Delta_n + \epsilon_1. \label{eqn:17} \end{align}
Using \eqref{eqn:18}, we obtain the following property of the upper bound in \eqref{eqn:17}: \begin{align}
\quad\quad \Delta_n + \epsilon_1 \overset{\text{a.s}}{\longrightarrow} \epsilon_1. \label{eqn:19} \end{align}
Finally, since \eqref{eqn:17} holds for $\forall \nu \in \mathbb R$ and \eqref{eqn:19} is valid for any $\epsilon_1 > 0$, making $\epsilon_1 \rightarrow 0$ gives the desired result,
\begin{align}
\underset{\nu \in \mathbb R}{\sup} \quad \Big|\hat F_n(\nu) - F_\pi(\nu) \Big| &\overset{\text{a.s.}}{\longrightarrow} 0. \label{eqn:21} \end{align} \end{proof}
\paragraph{Variance-reduced estimation:} \label{apx:sec:WIS}
It is known that importance-sampling-based estimators are subject to high variance, which can often be limiting in practice \citep{guo2017using}.
A popular approach to mitigate variance is to use \textit{weighted} importance sampling (WIS), which trades off variance for bias.
Leveraging this approach, we propose the following variance-reduced estimator, $\bar F_n$, of $F_\pi$,
\begin{align}
\forall \nu \in \mathbb R, \quad \bar F_n(\nu) &\coloneqq \frac{1}{\sum_{j=1}^n \rho_j} \left(\sum_{i=1}^n \rho_i \Big(\mathds{1}_{\{G_i \leq \nu\}}\Big) \right).
\label{eqn:WISF} \end{align}
In the following theorem, we show that $\bar F_n$ is a biased estimator of $F_\pi$, though it preserves consistency.
\begin{prop}
Under \thref{ass:support}, $\bar F_n$ may be biased but is a uniformly consistent estimator of $F_\pi$,
\begin{align}
\forall \nu\in \mathbb{R}, \quad \mathbb{E}_{\mathcal D}\Big[\bar F_n(\nu)\Big] &\neq F_\pi,
&
\underset{\nu \in \mathbb R}{\sup} \quad \Big|\bar F_n(\nu) - F_\pi(\nu) \Big| \overset{\text{a.s.}}{\longrightarrow} 0. \end{align} \thlabel{thm:WISFbiased} \end{prop}
\begin{proof} Similar to the proof for \thref{thm:Funbiased}, we break this proof in two parts, one to establish bias and the other to establish consistency of $\hat F_n$.
\paragraph{Part 1 (Biased):} We prove this using a counter-example. Let $n=1$ and $\pi \neq \beta_1$, so \begin{align}
\forall \nu\in \mathbb{R}, \quad \mathbb{E}_{\mathcal D}\Big[\bar F_n(\nu)\Big] &= \mathbb{E}_{\mathcal D}\left[ \frac{1}{\sum_{j=1}^1 \rho_j} \left(\sum_{i=1}^1 \rho_i \mathds{1}_{\{G_i \leq \nu\}} \right)\right]
\\
&= \mathbb{E}_{\mathcal D}\left[ \mathds{1}_{\{G_1 \leq \nu\}}\right]
\\
&\overset{(a)}{=} \int_{\mathscr H_{\beta_1}} p(H_{\beta_1} = h)\Big( \mathds{1}_{\{g(h) \leq \nu\}}\Big) \mathrm{d}h
\\
&= F_{\beta_1}(\nu)
\\
&\neq F_\pi(\nu), \end{align} where (a) follows analogously to \eqref{eqn:2}.
\paragraph{Part 2 (Uniform Consistency): }
First, we will establish pointwise consistency, i.e., for any $\nu$, $\bar F_n(\nu) \overset{\text{a.s.}}{\longrightarrow} F_\pi(\nu)$, and then we will use this to establish \textit{uniform} consistency, as required.
\begin{align}
\forall \nu \in \mathbb R, \quad \bar F_n(\nu) &= \frac{1}{\sum_{j=1}^1 \rho_j} \left(\sum_{i=1}^1 \rho_i \mathds{1}_{\{G_i \leq \nu\}} \right)
\\
&= \left(\frac{1}{n}\sum_{j=1}^n \rho_j\right)^{-1} \left(\frac{1}{n}\sum_{i=1}^n \rho_i \mathds{1}_{\{G_i \leq \nu\}} \right). \end{align}
Let $X_n \coloneqq \frac{1}{n}\sum_{j=1}^n \rho_j$ and $Y_n \coloneqq \frac{1}{n}\sum_{i=1}^n \rho_i \mathds{1}_{\{G_i \leq \nu\}}$.
Now, as $\bar F_n(\nu)$ is a continuous function of both $X_n$ and $Y_n$, if both $( \lim\limits_{n\rightarrow \infty} \, X_n)^{-1}$ and $(\lim\limits_{n \rightarrow \infty} \, Y_n)$ exist then using the continuous mapping theorem \citep[Theorem 2.3]{van2000asymptotic},
\begin{align}
\forall \nu \in \mathbb R, \quad \lim_{n \rightarrow \infty} \quad \bar F_n(\nu) &= \left(\lim_{n \rightarrow \infty} \, X_n\right)^{-1} \left(\lim_{n \rightarrow \infty} \, Y_n \right) \label{eqn:23}. \end{align}
Notice using Kolmogorov's strong law of large numbers \citep[Theorem 2.3.10 with Proposition 2.3.10]{Sen1993} that the term in the first parentheses will almost surely converge to the expected value of importance ratios, which equals one \citep{precup2000eligibility}.
Similarly, we know from \eqref{eqn:10} that the term in the second parentheses will converge to $F_\pi(\nu)$ almost surely. Therefore, both parenthetical terms of \eqref{eqn:23} exist, and thus \begin{align}
\forall \nu \in \mathbb R, \quad \bar F_n(\nu) \overset{\text{a.s.}}{\longrightarrow} (1)^{-1} (F_\pi(\nu)) = F_\pi(\nu) . \label{eqn:22} \end{align}
Now, similar to the proof for \thref{thm:Funbiased}, combining \eqref{eqn:22} with arguments from \eqref{eqn:20} to \eqref{eqn:21}, it can be observed that
\begin{align}
\underset{\nu \in \mathbb R}{\sup} \quad \Big|\bar F_n(\nu) - F_\pi(\nu) \Big| &\overset{\text{a.s.}}{\longrightarrow} 0. \end{align}
\end{proof}
\begin{thm} Under \thref{ass:support}, for any $\delta \in (0, 1]$, if $\sum_{i=1}^K\delta_i \leq \delta$, then the confidence band defined by $F_-$ and $F_+$ provides guaranteed coverage for $F_\pi$.
That is, \begin{align}
\Pr \Big(\forall \nu, \,\, F_{-}(\nu) \leq F_\pi(\nu) \leq F_+(\nu) \Big) \geq 1 - \delta. \end{align} \end{thm}
\begin{proof}
Let $A_i$ be the event that for the key point $\kappa_i$, $\texttt{CI}_-(\kappa_i, \delta_i) \leq F_\pi(\kappa_i) \leq \texttt{CI}_+(\kappa_i, \delta_i)$, for all $i \in (1,..., K)$. Let superscript $c$ denote a complementary event; then by the union bound, the total probability of the bounds holding at each key point simultaneously is
\begin{align}
\Pr\Big(\cap_{i=1}^K A_i\Big) &= 1 - \Pr\Big((\cap_{i=1}^K A_i)^c \Big)
= 1 - \Pr\Big(\cup_{i=1}^K A_i^c \Big)
\geq 1 - \sum_{i=1}^K\Pr\Big( A_i^c \Big)
\overset{(a)}{\geq} 1 - \delta,\;\;\;\; \label{eqn:24}
\end{align}
where $(a)$ holds because the conditions of the theorem assert that the sum of probabilities of the bounds failing at each key point is at most $\delta$.
Therefore, using \eqref{eqn:24},
\begin{align}
\Pr\left(\forall i \in (1,...,K), \,\, \texttt{CI}_-(\kappa_i, \delta_i) \leq F_\pi(\kappa_i) \leq \texttt{CI}_+(\kappa_i, \delta_i) \right) \geq 1 - \delta. \label{eqn:25}
\end{align}
Since by construction, at the key points $(\kappa_i)_{i=1}^K, F_-(\kappa_i) = \texttt{CI}_-(\kappa_i, \delta_i)$ and $F_+(\kappa_i) = \texttt{CI}_+(\kappa_i, \delta_i)$, it follows from \eqref{eqn:25} that
\begin{align}
\Pr\left(\forall i \in (1,...,K), \,\, F_-(\kappa_i) \leq F_\pi(\kappa_i) \leq F_+(\kappa_i) \right) \geq 1 - \delta. \label{eqn:26}
\end{align}
Using the monotonically non-decreasing property of a CDF, at any point $\nu \in \mathbb R$ such that $\kappa_i \leq \nu \leq \kappa_{i+1}$, we know that $ F_\pi(\kappa_{i}) \leq F_\pi(\nu) \leq F_\pi(\kappa_{i+1})$. Therefore, when the bounds at the key points hold, $F_\pi$ at the key points can also be upper and lower bounded: $ F_-(\kappa_{i}) \leq F_\pi(\nu) \leq F_+(\kappa_{i+1})$. Therefore, by \eqref{eqn:26} and the construct in \eqref{eqn:lub}, it immediately follows that
\begin{align}
\Pr \Big(\forall \nu, \,\, F_{-}(\nu) \leq F_\pi(\nu) \leq F_+(\nu) \Big) \geq 1 - \delta. \end{align} \end{proof}
\begin{thm} Under \thref{ass:support}, for any $1-\delta$ confidence band $\mathcal F$, the confidence interval defined by $\psi_-$ and $\psi_+$ provides guaranteed coverage for $\psi(F_\pi)$.
That is, \begin{align} \Pr \Big(\psi_- \leq \psi(F_\pi) \leq \psi_+ \Big) \geq 1 - \delta. \end{align} \end{thm}
\begin{proof} Recall that the confidence band $\mathcal F$ is a random variable dependent on the data $\mathcal D$.
Let $\mathbb{E}_{\mathcal F}[\cdot]$ represent expectation with respect to $\mathcal F$, then repeatedly using the law of total probability,
\begin{align}
\Pr\Big(\psi_- \leq \psi(F_\pi) \leq \psi_+ \Big)
&= \mathbb{E}_{\mathcal F}\left[\Pr\Big(\psi_- \leq \psi(F_\pi) \leq \psi_+ \Big | \mathcal F \Big) \right] \\
&= \mathbb{E}_{\mathcal F}\Big[\Pr\Big(\psi_- \leq \psi(F_\pi) \leq \psi_+ \Big| F_\pi \in \mathcal F, \mathcal F\Big) \Pr \Big(F_\pi \in \mathcal F \Big| \mathcal F\Big) \\ &\quad\quad\quad + \Pr\Big(\psi_- \leq \psi(F_\pi) \leq \psi_+ \Big| F_\pi \not \in \mathcal F, \mathcal F\Big) \Pr\Big(F_\pi \not \in \mathcal F \Big | \mathcal F\Big) \Big]
\\
&\geq \mathbb{E}_{\mathcal F}\left[\Pr\Big(\psi_- \leq \psi(F_\pi) \leq \psi_+ \Big| F_\pi \in \mathcal F, \mathcal F \Big) \Pr\Big(F_\pi \in \mathcal F\Big | \mathcal F\Big) \right]
\\
&\overset{(a)}{=} \mathbb{E}_{\mathcal F}\left[\Pr\Big(F_\pi \in \mathcal F \Big | \mathcal F\Big) \right]
\\
&= \Pr\Big(F_\pi \in \mathcal F \Big)
\\
&\overset{(b)}{\geq} 1 - \delta,
\end{align}
where $(a)$ follows from that fact that $F_\pi \in \mathcal F$ implies $\psi_- \leq \psi(F_\pi) \leq \psi_+ $. Step $(b)$ follows from \thref{thm:Fguarantee1}. \end{proof}
\begin{proof}[Proof (Alternate)] This proof is shorter but requires a theoretical construct of a \textit{set of sets of functions}.
That is, let $\mathbb F$ be any set of cumulative distribution functions and $\mathscr F$ be a set of such sets, such that \begin{align}
\mathscr F \coloneqq \Big\{ \mathbb F \,\, \Big| F_\pi \in \mathbb F \Big\}. \end{align}
In other words, $\mathbb F$ is the set of CDFs which contains the true CDF $F_\pi$, and $\mathscr F$ is the set of \textit{all} such sets $\mathbb F$. From \thref{thm:Fguarantee1}, we know that the confidence band $\mathcal F$ contains $F_\pi$ with probability at least $1-\delta$. Therefore, it also holds that
\begin{align}
\Pr(\mathcal F \in \mathscr F) \geq 1-\delta. \end{align}
However, the event $(\mathcal F \in \mathscr F)$ implies that $\psi_- \leq \psi(F_\pi) \leq \psi_+$ as $F_\pi$ is contained in this specific $\mathcal F$ used to construct $\psi_-$ and $\psi_+$.
Therefore, it also holds that \begin{align}
\Pr(\psi_- \leq \psi(F_\pi) \leq \psi_+) \geq 1 - \delta. \end{align}
\end{proof}
\begin{thm} Under \thref{ass:support,ass:shift}, for any $\delta \in (0, 1]$, the confidence band defined by $F^{(2)}_-$ and $F^{(2)}_+$ provides guaranteed coverage for $F^{(2)}_\pi$.
That is, \begin{align}
\Pr \Big(\forall \nu, \,\, F^{(2)}_{-}(\nu) \leq F^{(2)}_\pi(\nu) \leq F^{(2)}_+(\nu) \Big) \geq 1 - \delta. \end{align} \end{thm}
\begin{proof}
From \thref{ass:shift}, $\underset{\nu \in \mathbb R}{\sup} \left | F^{(1)}_\pi(\nu) - F^{(2)}_\pi(\nu) \right| \leq \epsilon$. Or equivalently, \begin{align}
\forall \nu \in \mathbb{R}, \quad F_\pi^{(1)}(\nu) - \epsilon \leq F_\pi^{(2)}(\nu) \leq F_\pi^{(1)}(\nu) + \epsilon. \label{eqn:27} \end{align}
Using \thref{thm:Fguarantee1} for the bound obtained on $F_\pi^{(1)}$ for the first domain, \begin{align}
\Pr \Big(\forall \nu, \,\, F_{-}^{(1)}(\nu) \leq F_\pi^{(1)}(\nu) \leq F_+^{(1)}(\nu) \Big) \geq 1 - \delta. \label{eqn:28} \end{align} Therefore, combining \eqref{eqn:27} and \eqref{eqn:28}, \begin{align}
\Pr \Big(\forall \nu, \,\, F_{-}^{(1)}(\nu) - \epsilon \leq F_\pi^{(2)}(\nu) \leq F_+^{(1)}(\nu) + \epsilon \Big) \geq 1 - \delta. \label{eqn:29} \end{align} Then by the construct in \eqref{eqn:Fshiftbound}, it follows from \eqref{eqn:29} that
\begin{align}
\Pr \Big(\forall \nu, \,\, F_{-}^{(2)}(\nu) \leq F_\pi^{(2)}(\nu) \leq F_+^{(2)}(\nu) \Big) \geq 1 - \delta. \end{align} \end{proof}
\section{Extended Discussion for UnO}
\subsection{Nuances for CDF Inverse and CVaR} \label{apx:UnOinverse}
For brevity, some nuances for $\hat F_n^{-1}(\alpha)$ and $\text{CVaR}_\pi^\alpha(\hat F_n)$ were excluded from the main paper.
We discuss them in this section.
As discussed earlier in \thref{rem:geq1}, it is possible that $\hat F_n(\nu) > 1$ for some $\nu \in \mathbb R$ due to the use of importance weighting.
Similarly, it is also possible that $\hat F_n(\nu) < 1$ for all $\nu \in \mathbb R$.
Specifically, if $\hat F_n(\nu) < \alpha$ for all $\nu$, then it raises the question: how can one obtain an estimate of $ F_\pi^{-1}(\alpha)$?
To resolve this issue, we use the following estimator of $ F_\pi^{-1}(\alpha)$ for UnO: \begin{align}
\hat F^{-1}_n(\alpha) \coloneqq \begin{cases}
\min \Big\{g \in (G_{(i)})_{i=1}^n \Big| \hat F_n(g) \geq \alpha \Big\}, & \text{if} \quad \exists \, g \,\,\text{s.t.}\,\, \hat F_n(g) \geq \alpha,
\\
\max (G_{(i)})_{i=1}^n & \text{otherwise.}
\end{cases}
\label{apx:eqn:inverseF} \end{align}
However, it is known from \thref{thm:Funbiased} that $\hat F_n$ is a uniformly consistent estimator of $F_\pi$.
Therefore, the edge case that $\hat F_n(\nu) < \alpha$ for all $\nu$ cannot occur in the limit as $n \rightarrow \infty$.
Resolving this is required mostly when the sample size is small.
Regarding CVaR, it is known \citep{acerbi2002coherence} that when the distribution of a random variable (which is $G_\pi$ for UnO) is continuous, then CVaR can be expressed as,
\begin{align}
\text{CVaR}^\alpha_\pi(F_\pi) &= \mathbb{E}\left[G_\pi \middle| G_\pi \leq F_\pi^{-1}(\alpha) \right], \label{eqn:apx:cvar1}
\end{align} and thus an off-policy sample estimator for \eqref{eqn:apx:cvar1} can be constructed as,
\begin{align}
{\text{CVaR}}^\alpha_\pi(\hat F_n) &\coloneqq \frac{1}{\alpha}\sum_{i=1}^n \text{d}\hat F_n(G_{(i)}) G_{(i)} \mathds{1}_{\left\{G_{(i)} \leq {Q}^\alpha_\pi(\hat F_n)\right\}}. \end{align}
However, for distributions that are not continuous, a more generic definition for CVaR is \citep{brown2007large},
\begin{align}
\text{CVaR}^\alpha_\pi(F_\pi) &= \inf_{g} \left\{g - \frac{1}{\alpha}\mathbb{E}\Big[\max\big(0,g - G_\pi \big) \Big] \right\}. \label{eqn:apx:cvar2}
\end{align}
We extend the sample estimator by \citet{brown2007large} for \eqref{eqn:apx:cvar2} and use the following off-policy estimator for UnO:
\begin{align}
\text{CVaR}^\alpha_\pi(\hat F_n) &\coloneqq \hat F_n^{-1}(\alpha) - \frac{1}{\alpha}\sum_{i=1}^{n} \text{d}\hat F_n(G_{(i)}) \left(\max\big(0, \hat F_n^{-1}(\alpha) - G_{(i)}\big) \right) \end{align}
\subsection{Optimizing Confidence Bands for Tighter Bounds: }
Constructing $\mathcal F$ requires selecting $K$ key points for which $\texttt{CI}$s are computed.
If too many key points are selected, then each $\delta_i$ has to be a very small positive value so that $\sum_{i=1}^K \delta_i \leq \delta$, as required by \thref{thm:Fguarantee1}.
This will make the confidence intervals wide at each key point.
In contrast, if too few key points are selected, then the confidence intervals at the $\kappa_i$'s will be relatively tighter, but this will not tighten the intervals \textit{between} the $\kappa_i$'s due to the way $F_-$ and $F_+$ are constructed in \eqref{eqn:Fband2}.
Further, the overall tightness of $\mathcal F$ is also affected by the location of each $\kappa_i$ and its respective failure rate $\delta_i$.
Therefore, to get a tight $\mathcal F$, we propose searching for a $\theta \coloneqq \left(K,(\kappa_i)_{i=1}^K, (\delta_i)_{i=1}^K\right)$ that minimizes the area enclosed in $\mathcal F$.
That is, let $\Delta_{i+1} \coloneqq \kappa_{i+1} - \kappa_i$, then the area enclosed in $\mathcal F$ is
\begin{align}
\mathscr A(\theta) \coloneqq \sum_{i=0}^{K} \left(\texttt{CI}_+(\kappa_{i+1}, \delta_{i+1})- \texttt{CI}_-(\kappa_{i}, \delta_{i}) \right) \Delta_{i+1}.
\end{align}
To avoid multiple comparisons \citep{benjamini1995controlling}, we first partition $\mathcal D$ into $\mathcal D_\text{train}$ and $\mathcal D_\text{eval}$. Subsequently, $\mathcal D_\text{train}$ is used to search for $\theta^*$ as follows, and then $\theta^*$ is used with $\mathcal D_\text{eval}$ to obtain $\mathcal F$.
\begin{align}
\theta^* \coloneqq \,\,& \underset{\theta}{ \argmin} \,\, \mathscr A(\theta) \label{eqn:Foptim}
\\
\text{s.t.} \quad\quad& G_{\min} < \kappa_i < G_{\max}, \quad \sum_{i=1}^K \delta_i \leq \delta, \quad \delta_i \geq 0, \quad & \forall i \in (1,...,K).
\end{align}
\begin{rem}
A global optimum of \eqref{eqn:Foptim} is not required---any feasible $\theta$ can be used with $\mathcal D_\text{eval}$ to obtain a confidence band $\mathcal F$. Optimization only helps by making the band tighter. \end{rem}
For our experimental results, when searching $\theta^*$ for \eqref{eqn:Foptim}, we keep the number of key points, $K$, fixed to $\log(n)$, where $n$ is the number of observed trajectory samples in $\mathcal D$. To search for the locations $(\kappa_i)_{i=1}^K$ and the failure rates $(\delta_i)_{i=1}^K$ at each key point, we use the BlackBoxOptim library\footnote{https://github.com/robertfeldt/BlackBoxOptim.jl} available in Julia \citep{bezanson2017julia}.
To perform this optimization, we construct $\mathcal D_\text{train}$ using $5\%$ of data from $\mathcal D$, and construct $\mathcal D_\text{eval}$ using the rest of the data.
Following the idea by \citet{thomas2015higha}, when searching for $\theta^*$ using $\mathcal D_\text{train}$, bounds for the key points $(\kappa_i)_{i=1}^K$ are obtained as if the number of samples are equal to the number of samples available in $\mathcal D_\text{eval}$
(see Equation $7$ in the work by \citet{thomas2015higha} for more discussion on this).
Instead of using a single split, one could potentially also leverage results by \citet{romano2019multiple} to use multiple splits; we leave this for future work.
\subsection{Bound Specialization} \label{apx:UnOspecial}
In \eqref{eqn:Foptim}, $\theta$ was searched to minimize the area $\mathscr A(\theta)$ enclosed within $\mathcal F(\theta)$, where $\mathcal F(\theta)$ represents the CDF band obtained using the parameter $\theta$.
This was done without any consideration of the downstream parameter $\psi$ for which the bounds would be constructed using $\mathcal F(\theta)$.
Therefore, the band $\mathcal F(\theta)$ is tight overall, but need not be the best possible if only a specific parameter $\psi$'s bounds are required using $\mathcal F(\theta)$.
For example, consider obtaining bounds for $\text{CVaR}_\pi^\alpha$.
As can be seen from the geometric insight in Figure \ref{fig:geometry}, bounds for CVaR are mostly dependent on the tightness of $\mathcal F(\theta)$ near the lower tail.
Therefore, if one can obtain $\mathcal F(\theta)$ that is tighter near the lower tail, albeit looser near the upper tail, that would provide a better bound for CVaR as opposed to a band $\mathcal F(\theta)$ that has uniform tightness throughout.
To get a tight $\mathcal F(\theta)$ in such cases where there is a single downstream parameter of interest, we propose searching for a $\theta \coloneqq \left(K,(\kappa_i)_{i=1}^K, (\delta_i)_{i=1}^K\right)$ that directly optimizes for the final parameter of interest instead of the area enclosed in $\mathcal F(\theta)$.
For example, if only the lower bound for $\psi(F_\pi)$ is required, then let \begin{align}
\mathscr \psi_-(\theta) &\coloneqq \underset{F \in \mathcal F(\theta)}{\inf} \,\,\, \psi (F). \end{align}
Next, the optimization using $\mathcal D_\text{train}$ can then be modeled as the following,
\begin{align}
\theta^* \coloneqq \,\,& \underset{\theta}{ \argmax} \,\, \mathscr \psi_-(\theta)
\\
\text{s.t.} \quad\quad& G_{\min} < \kappa_i < G_{\max}, \quad & \forall i \in (1,...,K),
\\
& \sum_{i=1}^K \delta_i \leq \delta, \quad \delta_i \geq 0, & \forall i \in (1,...,K), \end{align}
This would result in $\theta^*$ that when used with $\mathcal D_\text{eval}$ can be expected to provide the CDF band which will yield the highest lower bound for $\psi(F_\pi)$.
\subsection{Approximate Bounds for Any Parameter using Bootstrap} \label{apx:sec:boot}
In Algorithm \ref{alg:pboot}, we provide the pseudo code for obtaining bootstrap-based bounds for any parameter $\psi(F_\pi)$. In Line 1, $B$ datasets $(\mathcal D_i^*)_{i=1}^B$ are generated from $\mathcal D$ using resampling, and for each of these resampled data sets, $B$ (weighted IS-based) CDF estimates $(\bar F^{*}_{n,i})_{i=1}^B$ are obtained.
In Line 3, sample estimates $(\psi(\bar F^{*}_{n,i}))_{i=1}^B$ for the desired parameter $\psi(F_\pi)$ are constructed using the $B$ estimated CDFs.
In Line 4, these sample estimates for $\psi(F_\pi)$ can be subsequently passed to the bias-corrected and accelerated (BCa \citep{efron1994introduction}) bootstrap procedure to obtain approximate lower and upper bounds $(\psi_-, \psi_+)$.
\IncMargin{1em}
\begin{algorithm2e}[h]
\textbf{Input:} {Dataset $\mathcal D$}, Confidence level $1 - \delta$
\\
Bootstrap $B$ datasets $(\mathcal D_i^*)_{i=1}^B$ and create $(\bar F^{*}_{n,i})_{i=1}^B$
\\
Bootstrap estimates $(\psi(\bar F^{*}_{n,i}))_{i=1}^B$ using $(\bar F_{n,i}^*)_{i=1}^B$
\\
Compute $(\psi_-, \psi_+)$ using BCa($(\psi(\bar F^{*}_{n,i}))_{i=1}^B$, $\delta$)
\\
\textbf{Return} $(\psi_-, \psi_+)$
\caption{Bootstrap Bounds for $\psi(F_\pi)$}
\label{alg:pboot}
\end{algorithm2e}
\DecMargin{1em}
\subsection{Extended Discussion of High-Confidence Bounds for Any Parameter} \label{apx:UnOoptim}
Section \ref{sec:Fbounds} of the main paper discussed how high-confidence bounds $\psi_-$ and $\psi_+$ can be obtained for any parameter $\psi(F_\pi)$ using the confidence band $\mathcal F$.
Specifically, in Figure \ref{fig:geometry}, geometric insights for obtaining the analytical form of the bounds for the mean, quantile, and CVaR were discussed.
Extending that discussion, Figure \ref{apx:fig:geometry} provides geometric insights for bounding other parameters, namely variance, inter-quantile ranges, and entropy, in the off-policy setting.
An advantage of having the CDF band $\mathcal F$ is that it can permit bounding other novel parameters that might be of interest.
While analytical bounds using geometric insights, as discussed for a number of popular parameters, should also be the first attempt for the desired novel parameter, it may be the case that such geometric insight cannot be obtained.
In such cases, a CDF $F$ can be directly parameterized using a spline curve, or a piecewise non-decreasing function that is constrained to be within $\mathcal F$.
Depending on how rich this parameterization is, it may be feasible to use a black-box optimization routine and obtain a globally optimal $F$ that minimizes (maximizes) the desired parameter $\psi(F)$.
If not feasible, an approximate bound can be achieved by using the best found local optima.
\begin{figure*}
\caption{Similar to Figure \ref{fig:geometry}, given a confidence band $\mathcal F$, lower and upper bounds for several other parameters can also be obtained using simple geometric insights.
\textbf{(Left)} An upper bound for the variance can be obtained by observing that variance is maximized when the probability of events on either extreme are maximized. Therefore, the CDF $F \in \mathcal F$ for such a distribution will initially follow (from left to right) $F_+$ and then make a horizontal jump (at a specific jump point) to $F_-$, which it then follows until $1$. The variance of the distribution with this CDF, $F$, will give the desired upper bound. Analogously, the CDF that initially follows $F_-$ and then jumps vertically (at a specific jump point) to $F_+$, assigns highest probability to events near the mean and thus results in the lowest variance \citep{romano2002explicit}.
\textbf{(Middle)} An upper bound for the inter-quantile range can be obtained by maximizing the value of upper $\alpha_2$-quantile and subtracting the minimum value for the lower $\alpha_1$-quantile. This can be obtained by $F^{-1}_-(\alpha_2) - F^{-1}_+(\alpha_1)$.
Analogously, a lower bound can be obtained using $\max(0, F^{-1}_+(\alpha_2) - F^{-1}_-(\alpha_1))$.
\textbf{(Right)} An upper bound on the entropy can be obtained by what \citet{learned2008probabilistic} call a ``string-tightening'' algorithm. That is, if the ends of a tight string are held at the bottom-left and the upper-right corner of $\mathcal F$, and the entire string is constrained to be within $\mathcal F$, then the path of the string corresponds to the $F \in \mathcal F$ that has highest entropy. In our figure, such an $F$ corresponds to the CDF of the uniform distribution, which is known to have maximum entropy. Unless some stronger assumptions are made, the lower bound on differential entropy is typically $- \infty$ if there is any possibility of a point mass.}
\label{apx:fig:geometry}
\end{figure*}
\subsection{Tackling Smooth Non-stationarity using Wild Bootstrap}
\label{apx:sec:NSboot}
From \thref{thm:Funbiased}, it is known that the proposed estimator $\hat F_n(\kappa)$ provides unbiased estimates for $F_\pi(\kappa)$, even with a single observed trajectory.
In the non-stationary setting, let the true underlying CDF of returns for $\pi$ in the episode $i$ be $F^{(i)}_\pi(\kappa)$, and the estimate of $F^{(i)}_\pi(\kappa)$ using the trajectory observed during the episode $i$ be
\begin{align}
\hat F_n^{(i)}(\kappa) &\coloneqq \rho_{i} \mathds{1}_{\{G_i \leq \kappa\}} &\forall i \in \{1,2,...,L\}. \end{align}
Next, the trend of the sequence $\big(\hat F_n^{(i)}(\kappa) \big)_{i=1}^L$ can be analyzed to forecast $\hat F_n^{(L+\ell)}(\kappa)$ for the future episode $L + \ell$ when the policy $\pi$ will be executed.
Particularly, under \thref{ass:ns}, $\exists w_\kappa$, such that, $\forall i \in (1,...,L+\ell), \,\,\,\, F^{(i)}_\pi(\kappa) = \phi(i)^\top w_\kappa.$
Therefore, using the unbiased estimates $\big(\hat F_n^{(i)}(\kappa) \big)_{i=1}^L$ of $\big(F_\pi^{(i)}(\kappa) \big)_{i=1}^L$, we propose searching for $w_\kappa$ using least-squares regression. Let $X\coloneqq [1, 2, ...., L] $ be the episode numbers in the past, then the predicates $\Phi_\kappa$, the targets $Y_\kappa$, and the corresponding least-squares solution $w_\kappa$ can be obtained as,
\begin{align}
\Phi_\kappa &\coloneqq [\phi(X_1), \phi(X_2), ..., \phi(X_L)] & \in \mathbb{R}^{L \times d},\\
Y_\kappa &\coloneqq [\hat F_n^{(1)}(\kappa), \hat F_n^{(2)}(\kappa), ..., \hat F_n^{(L)}(\kappa)] & \in \mathbb{R}^{L\times 1},\\
w_\kappa &\coloneqq \left( \Phi_\kappa^\top \Phi_\kappa \right)^{-1}\Phi_\kappa^\top Y_\kappa & \in \mathbb{R}^{d \times 1}. \end{align}
Using $w_\kappa$, an unbiased estimate of $F_\pi^{(L+\ell)}(\kappa)$ can be obtained as,
\begin{align}
\hat F_n^{(L+\ell)}(\kappa) &\coloneqq \phi(L+\ell)^\top w_\kappa. \label{eqn:foreast} \end{align}
The point forecast $\hat F_n^{(L+\ell)}(\kappa)$ from \eqref{eqn:foreast} can then be combined with Algorithms 1 and 2 presented by \citet{chandak2020towards} to obtain wild-bootstrap-based confidence intervals for $F_\pi^{(L+\ell)}(\kappa)$. Once the confidence intervals are obtained at different key points, \eqref{eqn:Fband2} can be used to construct an entire confidence band for $F_\pi^{(L+\ell)}$.
\section{Empirical Details} \label{apx:empirical}
\subsection{Domain Details} \label{apx:empiricaldomain}
In this section, we discuss domain details and how $\pi$ and $\beta$ were selected for these domains.
The code for the domains, baselines \citep{thomas2015higha,chandak2021hcove}, and the proposed UnO estimator can be found at \href{https://github.com/yashchandak/UnO}{https://github.com/yashchandak/UnO}.
\paragraph{Recommender System: }
Systems for online recommendation of tutorials, movies, advertisements, etc., are ubiquitous \citep{theocharous2015ad,theocharous2020reinforcement}.
In these settings, it may be beneficial to fully characterize a customer's experience once the new system/policy is deployed.
To abstract such settings, we created a simulated domain where the user's interest for a finite set of items is represented using the corresponding item's reward.
Using an actor-critic algorithm \citep{SuttonBarto2}, we find a near-optimal policy $\pi$, which we use as the evaluation policy.
Let $\pi^\texttt{rand}$ be a random policy with uniform distribution over the actions (items).
Then for an $\alpha = 0.5$, we define the behavior policy $\beta(a|s) \coloneqq \alpha \pi(a|s) + (1-\alpha) \pi^\texttt{rand}(a|s)$ for all states and actions.
\paragraph{Gridworld: }
We also consider a standard continuous-state Gridworld with partial observability (which also makes the domain non-Markovian in the observations), stochastic transitions, and eight discrete actions corresponding to up, down, left, right, and the four diagonal movements.
The off-policy data was collected using two different behavior policies, $\beta_1$ and $\beta_2$, and the evaluation policies for this domain were obtained similarly as for the recommender system domain discussed above.
Particularly, using $\alpha = 0.5$, we define $\beta_1(a|o) \coloneqq \alpha \pi(a|0) + (1-\alpha) \pi^\texttt{rand}(a|o)$ for all states and actions. Similarly, $\beta_2$ was defined using $\alpha = 0.75$.
\paragraph{Diabetes Treatment: }
This domain is modeled using an open source implementation \citep{simglucose} of the U.S. Food and Drug Administration (FDA) approved Type-1 Diabetes Mellitus Simulator (T1DMS) \citep{man2014uva} for the treatment of type-1 diabetes.
An episode corresponds to a day, and each step of an episode corresponds to a minute in an \textit{in silico} patient's body and is governed by a continuous time nonlinear ordinary differential equation (ODE) \citep{man2014uva}.
In such potentially critical medical applications, it is important to go beyond just the expected performance and to characterize the risk associated with it, \textit{before deployment}.
To control the insulin injection, which is required for regulating the blood glucose level, we use a policy that controls the parameters of a \textit{basal-bolus controller}. This controller is based on the amount of insulin that a person with diabetes is instructed to inject prior to eating a meal \citep{bastani2014model}:
\begin{align}
\text{injection} = \frac{\text{current blood glucose} - \text{target blood glucose}}{CF} + \frac{\text{meal size}}{CR}, \end{align}
where ``current blood glucose'' is the estimate of the person's current blood glucose level, ``target blood glucose'' is the desired blood glucose, ``meal size'' is the estimate of the size of the meal the patient is about to eat, and $CR \in [CR_\texttt{min}, CR_\texttt{max}]$ and $CF \in [CF_\texttt{min}, CF_\texttt{max}]$ are two parameters of the controller that must be tuned based on the body parameters to make the treatment effective.
We designed an RL policy that acts on the discretized space of the parameters, $CR$ and $CF$, for the above basal-bolus controller.
Behavior and evaluation policies were selected similarly as discussed for the recommender system domain.
\subsection{Extended Discussion on Results for Stationary Settings} \label{apx:empiricalS}
The main results for the stationary setting are provided in Figure \ref{fig:results} of the main body.
In this section, we provide some additional discussion on the observed trends for the bounds.
Notice in Figure \ref{fig:results} that UnO-CI bounds for the variance can require up to an order of magnitude less data compared to the existing bound for the variance \citep{chandak2021hcove}.
This can be attributed to the fact that \citet{chandak2021hcove} construct the bounds using $\mathbb{E}[\rho G^2] - \mathbb{E}[\rho G]^2$, where it can be observed that the second term depends quadratically on $\rho$. This makes the variance of that term effectively ``doubly exponential'' in the horizon length. This does not happen in the CDF-based approach as the bounds at any key point $\kappa$ depend on $\mathbb{E}[\rho \mathds{1}_{G<\kappa}])$, which does not have any higher powers of $\rho$.
Another thing worth noting in Figure \ref{fig:results} is that not only the bounds for different parameters, but even the upper and lower bounds for the same parameter converge at different rates (especially for smaller values of $n$).
Therefore, there are two particular trends to observe: (a) how close the bounds are to the true value at the beginning, and (b) how quickly they improve.
Both of these depend on the direction for which clipping plays a major role and also how the bounds depend on the tails.
For example, for the mean, as the distributions are right skewed (because evaluating policy $\pi$ is a near-optimal policy), the bounds on the CDF are clipped more from the lower end (so that $F(\nu) >= 0$ always).
Therefore, since the upper bound on the mean depends on the lower CDF bound (see Figure \ref{fig:geometry}), it starts close to the estimate itself but the progress actually seems slow because shrinking CDFs bounds at any specific $F(\nu)$ from the lower end does not impact the bound until the point where clipping is not required anymore.
For variance, the upper bound depends on both the upper bound on the lower tail and the lower bound on the upper tail (see Figure \ref{apx:fig:geometry}), and these two benefit from clipping the least and also converge the slowest.
In contrast, the lower bound for variance depends on the upper bound on the upper tail and the lower bound on the lower tail, which are clipped immediately to be below 1 and above 0, respectively.
Appendix \ref{apx:ass} (knowledge of $G_{\min}$, $G_{\max}$) and Fig \ref{apx:fig:geometry} provide more intuition on this.
\end{document} | arXiv |
Uspekhi Mat. Nauk, 2000, Volume 55, Issue 2(332), Pages 3–94 (Mi umn267)
This article is cited in 39 scientific papers (total in 40 papers)
Decision problems for groups and semigroups
S. I. Adiana, V. G. Durnevb
a Steklov Mathematical Institute, Russian Academy of Sciences
b P. G. Demidov Yaroslavl State University
Abstract: The paper presents a detailed survey of results concerning the main decision problems of group theory and semigroup theory, including the word problem, the isomorphism problem, recognition problems, and other algorithmic questions related to them. The well-known theorems of Markov–Post, P. S. Novikov, Adian–Rabin, Higman, Magnus, and Lyndon are given with complete proofs. As a rule, the proofs presented in this survey are substantially simpler than those given in the original papers. For the sake of completeness, we first prove the insolubility of the halting problem for Turing machines, on which the insolubility of the word problem for semigroups is based. Specific attention is also paid to the simplest examples of semigroups with insoluble word problem. We give a detailed proof of the significant result of Lyndon that, in the class of groups presented by a system of defining relations for which the maximum mutual overlapping of any two relators is strictly less than one fifth of their lengths, the word problem is soluble, while insoluble word problems can occur when non-strict inequality is allowed. A proof of the corresponding result for finitely presented semigroups is also given, when the corresponding fraction is one half.
DOI: https://doi.org/10.4213/rm267
Full text: PDF file (665 kB)
References: PDF file HTML file
Russian Mathematical Surveys, 2000, 55:2, 207–296
Bibliographic databases:
UDC: 510.53+512.53+512.54
MSC: Primary 20F10, 20M05; Secondary 20E06, 03D40, 03D10
Received: 26.01.2000
Citation: S. I. Adian, V. G. Durnev, "Decision problems for groups and semigroups", Uspekhi Mat. Nauk, 55:2(332) (2000), 3–94; Russian Math. Surveys, 55:2 (2000), 207–296
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http://mi.mathnet.ru/eng/umn267
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\begin{document}
\title{Graph Neural Networks for Graph Drawing}
\author{Matteo~Tiezzi,
Gabriele~Ciravegna
and~Marco~Gori~\IEEEmembership{Fellow,~IEEE}
\IEEEcompsocitemizethanks{\IEEEcompsocthanksitem M. Tiezzi and M. Gori are with the Department of Information Engineering and Mathematics, University of Siena, 53100 Siena, Italy. M. Gori and G. Ciravegna are with MAASAI, Inria, I3S, CNRS, Université Côte d'Azur, Nice, France.\protect\\
Matteo Tiezzi is the corresponding author ([email protected]).
\IEEEcompsocthanksitem Accepted for publication in Transaction of Neural Networks and Learning Systems (TNNLS), Special Issue on Deep Neural Networks for Graphs: Theory, Models, Algorithms and Applications.
(DOI: \url{https://doi.org/10.1109/tnnls.2022.3184967}) } \thanks{2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.} }
\maketitle
\begin{abstract}
Graph Drawing techniques have been developed in the last few years with the purpose of producing aesthetically pleasing node-link layouts. Recently, the employment of differentiable loss functions has paved the road to the massive usage of Gradient Descent and related optimization algorithms. In this paper, we propose a novel framework for the development of {\em Graph Neural Drawers} (GND), machines that rely on neural computation for constructing efficient and complex maps. GNDs are Graph Neural Networks (GNNs) whose learning process can be driven by any provided loss function, such as the ones commonly employed in Graph Drawing. Moreover, we prove that this mechanism can be guided by loss functions computed by means of Feedforward Neural Networks, on the basis of supervision hints that express beauty properties, like the minimization of crossing edges. In this context, we show that GNNs can nicely be enriched by positional features to deal also with unlabelled vertexes. We provide a proof-of-concept by constructing a loss function for the edge-crossing and provide quantitative and qualitative comparisons among different GNN models working under the proposed framework. \end{abstract}
\begin{IEEEkeywords} Graph Drawing, Graph Representation Learning, Graph Neural Drawers, Graph Neural Networks \end{IEEEkeywords}
\IEEEpeerreviewmaketitle
\section{Introduction}
\IEEEPARstart{V}{isualizing} complex relations and interaction patterns among entities is a crucial task, given the increasing interest in structured data representations\cite{bronstein2021geometric}. The Graph Drawing \cite{battista1998graph} literature aims at developing algorithmic techniques to construct drawings of graphs -- i.e. mathematical structures capable to efficiently represent the aforementioned relational concepts with nodes and edges connecting them -- for example via the node-link paradigm~\cite{saket2014node, tamassia2013handbook, didimo2019survey}. The readability of graph layouts can be evaluated following some aesthetic criteria such as the number of crossing edges, minimum crossing angles, community preservation, edge length variance, etc. \cite{ahmed2020graph}. The final goal is to find suitable coordinates for the node positions, and this often requires to explicitly express and combine these criteria through complicated mathematical formulations ~\cite{cox2008multidimensional}. Moreover, effective approaches such as energy-based models~ \cite{jacomy2014forceatlas2, kamada1989algorithm} or spring-embedders~\cite{frick1994fast, brandes2008experimental} require hands-on expertise and trial and error processes to achieve certain desired visual properties. Additionally, such methods define loss or energy functions that must be optimized for each new graph to be drawn, often requiring to adapt algorithm-specific parameters. Lately, two interesting directions have emerged in the Graph Drawing community. The former one leverages the power of Gradient Descent to explore the manifold given by pre-defined loss functions or combinations of them. Stochastic Gradient Descent (SGD) can be used to move sub-samples of vertices couples in the direction of the gradient of spring-embedder losses~\cite{zheng2018graph} substituting complicated techniques such as Majorization~\cite{de1988convergence}. This approach has been extended to arbitrary optimization goals, or combinations of them, which can be optimized via Gradient Descent if the corresponding criterion can be expressed via smooth functions~\cite{ahmed2020graph}.
The latter novel direction consists in the exploitation of Deep Learning models. Indeed, the flexibility of neural networks and their approximation capability can come in handy also when dealing with the Graph Drawing scenario. For instance,
Neural networks are capable to learn the layout characteristics from plots produced by other graph drawing techniques~\cite{brandes2006eigensolver, wang2019deepdrawing}, as well as the underlying distribution of data~\cite{kwon2019deep}. Very recently, the node positions produced by graph drawing frameworks~\cite{brandes2006eigensolver} have been used as an input to Graph Neural Networks (GNNs)~\cite{DBLP:journals/tnn/ScarselliGTHM09, DBLP:journals/tnn/WuPCLZY21} to produce pleasing layout that minimize combinations of aesthetic losses~\cite{wang2021deepgd}.
We propose a framework, \textit{Graph Neural Drawers} (GND), which embraces both the aforementioned directions. We borrow the representational capability and computational efficiency of neural networks to prove that (1) differentiable loss functions guiding the common Graph Drawing pipeline can be provided directly by a neural network, a Neural Aesthete, \textcolor{black}{even when the required aesthetic criteria cannot be directly optimized.} In particular, we propose a proof-of-concept where we focus on the criteria of edge crossing, proving that a neural network can learn to identify if two arcs are crossing or not \textcolor{black}{and provide a differentiable loss function towards non-intersection}. \textcolor{black}{Otherwise, in fact, this simple aesthetic criterion cannot be achieved through direct optimization, because it is non-differentiable. Instead, the Neural Aesthete} provides a useful and flexible gradient direction that can be exploited by (Stochastic) Gradient Descent methods. Moreover, (2) we prove that \textcolor{black}{GNNs}, even in the non-attributed graph scenario if enriched with appropriate node positional features, can be used to process the topology of the input graph with the purpose of mapping the obtained node representation in a 2D layout. We compare various commonly used GNN models ~\cite{kipf2017semisupervised, velivckovic2018graph, xu2019powerful}, proving that the proposed framework is flexible enough to give these models the ability to learn a wide variety of solutions. In particular, GND is capable to draw graphs (\textit{i}) from supervised coordinates, i.e. emulating Graph Drawing Packages, (\textit{ii}) minimizing common aesthetic loss functions and, additionally, (\textit{iii}) by descending towards the gradient direction provided by the Neural Aesthete.
The paper is organized as follows. Section \ref{sec:related} introduces some basics on the Graph Drawing scenario as well as references on Gradient Descent approaches. Section \ref{sec:proposal_1} introduces the Neural Aesthete and provides a proof-of-concept on the edge crossing task. Section \ref{sec:proposal_2} describes the requirements of using GNNs to draw graphs in the non-attributed scenario, as well as the problem definition and the experimental evaluation. Finally, conclusions are drawn in Section \ref{sec:conclusion}.
\section{Related work} \label{sec:related}
There exists a large variety of methods in literature to improve graph readability. A straight-forward approach, that has been proved to be effective in improving the human understanding of the graph topology, consists in minimizing the number of crossing edges~\cite{purchase1997aesthetic}. However, the computational complexity of the problem is NP-hard, and several authors proposed complex solutions and algorithms to address this problem \cite{abrego2013rectilinear}. In~\cite{shabbeer2010optimal}, authors employ an Expectation-Maximization algorithm based on the decision boundary surface built by a Support Vector Machine. The underlying idea is that two edges do not cross if there is a line separating the node coordinates. Further aesthetic metrics have been explored, such as the minimization of node occlusions~\cite{chrobak1996convex}, neighborhood preservation, the maximization of crossing edges angular width~\cite{kaufmann2018heuristic} and many more~\cite{gibson2013survey, ahmed2020graph}. Given the graph drawing categorization depicted in surveys~\cite{gibson2013survey} (i.e. force-directed, dimension reduction, multi-level techniques), interesting and aesthetically pleasing layouts are produced by methods regarding a graph as a physical system, with forces acting on nodes with attraction and repulsion dynamics up to a stable equilibrium state~\cite{kamada1989algorithm}. Force-directed techniques inspired many subsequent works, from spring-embedders~\cite{fruchterman1991graph} to energy-based approaches~\cite{jacomy2014forceatlas2}. The main idea is to obtain the final layout of the graph minimizing the \textit{Stress} function (see Eq. \ref{eq:stress}). The forces characterizing this formulation can be thought of as springs connecting pairs of nodes. This very popular formulation, exploited for graph layout in the seminal work by Kamada and Kawai~\cite{kamada1989algorithm}, was optimized with the localized 2D Newton-Raphson method. Further studies employed various complicated optimization techniques, such as the \textit{Stress Majorization} approach which produces graph layout through an iterative resolution of simpler functions, as proposed by Gasner et al.~\cite{gansner2004graph}.
In this particular context, some recent contributions highlighted the advantages of using gradient-based methods to solve graph drawing tasks. The SGD method was successfully applied to efficiently minimize the Stress function in Zheng et al.~\cite{zheng2018graph}, displacing pairs of vertices following the direction of the gradient, computed in closed form.
A recent framework, $(GD)^2$, leverages Gradient Descent to optimize several readability criteria at once~\cite{ahmed2020graph}, as long as the criterion can be expressed by smooth functions. Indeed, thanks to the powerful auto-differentiation tools available in modern machine learning frameworks~\cite{paszke2017automatic}, several criteria such as ideal edge lengths, Stress Majorization, node occlusion, angular resolution and many others can be easily optimized. We build our first contribution upon these ideas, proving that neural networks can be used to learn decomposed single criteria (i.e., edge crossing) approximating smooth functions, with the purpose of providing a useful descent direction to optimize the graph layout.
Deep Learning has been successfully applied to data belonging to the non-Euclidean domain, e.g. graphs, in particular thanks to \textcolor{black}{GNNs}~\cite{DBLP:journals/spm/BronsteinBLSV17, DBLP:journals/tnn/WuPCLZY21}. The seminal work by Scarselli et al.~\cite{DBLP:journals/tnn/ScarselliGTHM09} proposes a model
based on an information diffusion process involving the whole graph, fostered by the iterative application of an \textit{aggregation function} among neighboring nodes up to an equilibrium point.
The simplification of this computationally expensive mechanism was the goal of several works which leverage alternative recurrent neural models~\cite{DBLP:journals/corr/LiTBZ15} or constrained fixed-point formulations. This problem was solved via Reinforcement Learning algorithms as done in \textcolor{black}{Stochastic Steady-state Embedding (SSE)}\cite{DBLP:conf/icml/DaiKDSS18} or cast to the Lagrangian framework and optimized by a gradient descent-ascent approach, like in \textcolor{black}{Lagrangian-Propagation GNNs (LP-GNNs)}~\cite{DBLP:conf/ecai/TiezziMMMG20}, even with the advantage of multiple layers of feature extraction~\cite{tiezzi2021deep}. The iterative nature of the aforementioned models inspired their classification into the umbrella of RecGNNs in recent surveys~\cite{DBLP:journals/tnn/WuPCLZY21, DBLP:journals/nn/BacciuEMP20}.
In addition to RecGNNs, several other flavours of GNN models have been proposed, such as the ConvGNNs~\cite{kipf2020deep} or Attentional GNNs~\cite{velivckovic2018graph, monti2017geometric, bresson2017residual}. All such models fit into the powerful concept of message exchange, the foundation on which is built the very general framework of Message Passing Neural Networks (MPNNs)\cite{gilmer2017neural, gilmer2020message}.
Recent works analyze the expressive capabilities of GNNs and their aggregation functions, following the seminal work on graph isomorphism by Xu et al. ~\cite{DBLP:journals/corr/abs-1810-00826}. The model proposed by the authors, Graph Isomorphism Network (GIN), leverage an injective aggregation function with the same representational power of the Weisfeiler-Leman (WL) Test~\cite{weisfeiler1968reduction}. Subsequent works (sometimes denoted with the term WL-GNNs) try to capture higher-order graph properties \cite{morris2019weisfeiler,maron2018invariant, corso2020principal, bodnar2021weisfeiler}. Bearing in mind that we deal with the non-attributed graph scenario, i.e., graphs lacking node features, we point out the importance of the nodal feature choice. Several recent works investigated this problem~\cite{cui2021positional, srinivasan2019equivalence, murphy2019relational}. We borrow the highly expressive Laplacian Eigenvector-based positional features described by Dwivedi et al. \cite{dwivedi2020benchmarking}.
There have been some early attempts in applying Deep Learning models and GNNs to the Graph Drawing scenario. Wang et al.~\cite{ wang2019deepdrawing} proposed a graph-based LSTM model able to learn and generalize the coordinates patterns produced by other graph drawing techniques. \textcolor{black}{ However, this approach is limited by the fact that the model drawing ability is highly dependent on the training data, such that processing different graph classes or layout styles requires re-collecting and re-training procedures. We prove that our approach is more general, given that we are able to learn both drawing styles from graph drawing techniques and to draw by minimizing aesthetic losses. Another very recent work, DeepGD \cite{wang2021deepgd}, consists in a message-passing GNN which process starting positions produced by graph drawing frameworks~\cite{brandes2006eigensolver}, to construct pleasing layouts that minimize combinations of aesthetic losses (Stress loss combined with others). Both DeepDraw and DeepGD share the common need of transforming the graph topology into a more complicated one: DeepDrawing~\cite{wang2019deepdrawing} introduces skip connections (\textit{fake edges}) among nodes in order to process the graph via a bidirectional-LSTM; DeepGD converts the input graph to a complete one, so that each node couples is directly connected, and requires to explicitly provide the shortest path between each node couple as an edge feature. The introduction of additional edges into the learning problem increase the computational complexity of the problem, hindering the model ability to scale to bigger graphs. More precisely, in the DeepGD framework the computational complexity grows quadratically in the number of nodes $O(N^2)$. Conversely, we show that the GNN are capable of producing aesthetically pleasing layouts without inserting additional edges, by simply leveraging powerful positional-structural features. Additionally, we introduce a novel neural-based mechanism, the \textit{Neural Aesthete}, capable to express differentiable aesthetic losses delivering flexible gradient direction also for non-differentiable goals. We show that this mechanism can be exploited by Gradient-descent based graph drawing techniques and by the proposed GND framework.
}
Finally, GNN-based Encoder-Decoder architectures can learn a generative model of the underlying distribution of data from a collection of graph layout examples~\cite{kwon2019deep}.
\section{Loss functions and Neural Aesthetes} \label{sec:proposal_1}
\subsection{ \textcolor{black}{Graph Drawing Algorithms}} Graph drawing algorithms typically optimize functions that somehow express a sort of beauty index, leveraging information visualization techniques, graph theory concepts, topology and geometry to derive a graphical visualization of the graph at hand in a bidimensional or tridimensional space~\cite{battista1998graph, di1994algorithms, yoghourdjian2018exploring}. Amongst others, typical beauty indexes are those of measuring the degree of edge crossings~\cite{purchase1997aesthetic}, the measurements to avoid small angles between adjacent or crossing edges, and measurements to express a degree of uniform allocation of the vertexes~\cite{gibson2013survey, ahmed2020graph}. All these requirements inherently assume that the graph drawing only consists of the allocation of the vertexes in the layout space, since the adjacent matrix of the graph can drive the drawing of the arcs as segments. However, we can also choose to link pairs of vertexes through a spline by involving some associated variables in the optimization process ~\cite{vismara2000experimental}.
Without loss of generality, in this work we restrict our objective to the vertex coordinates optimization, but the basis ideas can be extended also to the case of appropriate arc drawing.
As usual, we denote a graph by $G=(\mathcal{V},\mathcal{E})$, where $\mathcal{V}= \{v_1, \dots, v_N\}$ is a finite set of N {\em nodes} and $E \subseteq V \times V$ collects the {\em arcs} connecting them. The neighborhood of node $v_i$ is denoted by $\mathcal{N}_i$. We denote the coordinates of each vertex with $p_i : \mathcal{V} \mapsto {\rm I\!R^2} $, for a node $i$ mapped to a bi-dimensional space. We denote with $P \in {\rm I\!R^{N \times 2}}$ the matrix of the node coordinates.
One of the techniques that empirically proved to be very effective for an aesthetically pleasing node coordinates selection is the \textit{Stress} function~\cite{kamada1989algorithm}, \begin{equation}
\textsc{stress}(P) = \sum_{i < j} w_{ij}\big(||p_i -p_j|| - d_{ij}\big)^2
\label{eq:stress} \end{equation}
where \textcolor{black}{$p_i, p_j$} are the coordinates of vertices $i$ and $j$, respectively, $d_{ij}$ is the graph theoretic distance (or shortest-path) between node $i$ and $j$, and $w_{ij}$ is a weighting factor leveraged to balance the influence of certain pairs given their theoretical distance. Usually, it is defined as $w_{ij} = d_{ij}^{-\alpha}$ with $\alpha \in [0, 1,2]$. The optimization of this function is generally carried out leveraging complicated resolution methods (i.e., 2D Newton-Raphson method, Stress Majorization, etc.) that hinder its efficiency.
Recently, Gradient Descent methods were employed to produce graph layouts~\cite{zheng2018graph} by minimizing the Stress function, and noticeably, Ahmed et al.~\cite{ahmed2020graph} proposed a similar approach employing auto-differentiation tools. The advantage of this solution is that, as long as aesthetic criteria are characterized by smooth differentiable functions, it is possible to undergo an iterative optimization process\footnote{The variables of the optimization process are the node coordinates~$P$.} following, at each variable update step, the gradient of the criteria.
Clearly, the definition of aesthetic criteria as smooth functions could be hard to express.
For instance, while we can easily count the number of arc intersections, \textcolor{black}{devising a smooth function that may drive a continuous optimization of this problem is not trivial~\cite{shabbeer2010optimal, ahmed2020graph}. Indeed, finding the intersection of two lines,\footnote{\textcolor{black}{Obiously, in the case of arcs one should also check whether the intersection point, if exists, lies on one of the two segments.}} is as simple as solving the following equation system:} \begin{equation}
\begin{cases}
\textcolor{black}{
{a_1}x + {b_1}y + {c_1} = 0,} \\
\textcolor{black}{ {a_2}x + {b_2}y + {c_2} = 0
}
\end{cases}
\label{eq:inters} \end{equation} \textcolor{black}{By employing the classic Cramer's rule we can see that there is an intersection only in case of a non-negative determinant of the coefficient matrix $A, Det(A) = a_1b_2 - a_2b_1 \neq 0$. Clearly, the previous formula cannot be employed as a loss function in an optimization problem since it does not provide gradients.
To tackle this issue and provide a scoring function optimizable via gradient descent, we propose the \textit{Neural Aesthete}. }
\subsection{\textcolor{black}{The Neural Aesthete}} A major contribution of this paper is that of introducing the notion of \textit{Neural Aesthete}, which is in fact a neural network that learns beauty from examples with the perspective of generalizing to unseen data. The obtained modelled function that is expressed by the Neural Aesthete is smooth and differentiable by definition and offers a fundamental heuristic for graph drawing. As a proof-of-concept, we focus on edge crossing. In this case, we define the Neural Aesthete as a machine which processes two arcs as inputs and returns the information on whether or not they intersect one each other. Each arc is identified by the coordinates of the corresponding pair of vertices, $e_{u} = (p_i, p_j)$ for $e_{u} \in \mathcal{E}$. Hence, the Neural Aesthete $\nu(\cdot,\cdot,\cdot): {\cal E}^{2} \times \mathbb{R}^{m} \to \mathbb{R}$ operates on the concatenation of two arcs, $e_{u}$ and $e_{v}$ and returns \begin{equation}
y_{e_u, e_v} = \nu \big(\theta,e_{u}, e_{v})
\end{equation}
where $\theta \in \mathbb{R}^{m}$ is the vector which represents the weights of the neural network. The Neural Aesthete is learned by optimizing a cross-entropy loss function $L(y_{e_u, e_v}, \hat{y}_{e_u, e_v})$ over the arcs $(e_u, e_v) \in \mathcal{E}$, which is defined as: \begin{equation} \begin{split} L({y}_{e_u, e_v}, \hat{y}_{e_u, e_v}) = & - \big(\hat{y}_{e_u, e_v} \cdot log(y_{e_u, e_v}) \\ & \quad \ + (1 - \hat{y}_{e_u, e_v}) \cdot log(1 - y_{e_u, e_v})\big), \end{split} \end{equation}
where $\hat{y}_{e_u, e_v}$ is the target: \begin{equation}
\hat{y}_{e_u, e_v} = \begin{cases}
0, & \text{if } \quad (e_{u}, e_{v}) \text{ do not intersect} \\
1, & \text{otherwise}
\end{cases}
\label{eq:target} \end{equation} \textcolor{black}{ and the intersection of $e_u, e_v$ is automatically computed, e.g., by solving Eq. \ref{eq:inters}. }
Notice that the learning process from a finite set of supervised examples yields weights that allows us to estimate the probability of intersection of any two arcs. Basically, the learned output of the neural network can be regarded as a degree of intersection between any arc couple. Once learned, this characteristic of the Neural Aesthete comes in handy for the computation of the gradient of a loss function for Graph Drawing. In general, we want to move the extreme nodes defining the two arcs towards the direction of \textit{non-intersection}.
Hence, for the Graph Drawing task, the Neural Aesthete is able to process an unseen edge couple $(e_{u}, e_{v})$ randomly picked from the edge list $\mathcal{E}$, and to predict their degree of intersection $y_{e_u, e_v}$. We define the loss function $L(\cdot,\cdot)$ on this edge pair as the cross-entropy with respect to the target \textit{no-intersection}, $\hat{y}_{e_u{//}e_v} = 0$
\begin{equation}
H_{u, v} = L(y_{e_u, e_v}, \hat{y}_{e_u{//}e_v}) = - log(1 - y_{e_u, e_v})
\label{eq:dir_no_target} \end{equation}
This smooth and differential loss function foster the utilization of Gradient Descent methods to optimize the problem variables, i.e. the arc node coordinates $(e_u, e_v)$.
This same procedure can be replicated to all the graph edges,
\begin{equation}
H(P) = \sum_{(e_u, e_v) \in \mathcal{E}} L(y_{e_u, e_v}, \hat{y}_{e_u// e_v})
\label{eq:total_loss_neu} \end{equation}
Overall, a possible graph drawing scheme is the one which returns \begin{equation}
P^{\star} = \arg\min_{\cal P} H( P). \end{equation} This can be carried out by classic optimization methods. For instance, a viable solution is by gradient descent as follows: \[
P \leftarrow P - \eta \nabla_P H( P). \] where $\eta$ specifies the learning rate.
It is worth mentioning that, overall, this approach leverages the computational efficiency and parallelization capabilities of neural networks. Hence, the prediction of the edge-crossing degree can be carried out for many edge couples in parallel.
Moreover, this same approach can be conveniently combined with other aesthetic criteria, for instance coming from other Neural Aesthetes or from classical loss function (e.g., Stress). For example, we could consider \begin{align}
E = H(P) +\lambda_{A} A(\cdot)+ \lambda_{B} B(\cdot) \end{align} where $A(\cdot)$ and $B(\cdot)$ denotes other aesthetic criteria characterized by smooth differentiable functions.
\subsection{Example: Neural Aesthete on small-sized Random Graphs} \label{sec:prop1_ex} We provide a qualitative proof-of-concept example for the aforementioned Neural Aesthete for edge-crossing in Figure~\ref{fig:edge_cross}.
\begin{figure}
\caption{\textsc{Neural Aesthete for edge crossing}. Left-to right, Graph layouts with starting random node coordinates (\textsc{start}), optimized by minimizing stress function with Gradient Descent (\textsc{stress}), optimized by Gradient Descent applied on the Neural Aesthete for edge-crossing loss (\textsc{NA-cross}), optimized by alternating stress loss and Neural Aesthete loss in subsequent iterations (\textsc{combined}). We report the graph layouts generated in three random sparse graphs, one for each row.}
\label{fig:edge_cross}
\end{figure}
We built an artificial dataset composed of 100K entries to train the Neural Aesthete. Each entry of the dataset is formed by an input-target couple $(x, \hat{y})$. The input pattern $x$ corresponds to the Neural Aesthete arcs input positions\footnote{Which are defined as $x := [e_u, e_v] = [p_i, p_j, p_h, p_k]$, given two arcs $e_u = (p_i, p_j)$ and $e_v = (p_h, p_k)$.} as defined in Section \ref{sec:proposal_1}, whose node coordinates are randomly picked inside the interval $[0,1]$. The corresponding target $\hat{y}$ is defined as in Eq. \ref{eq:target}.
We balanced the dataset composition in order to have a comparable number of samples between the two classes (cross/no-cross). We trained a Neural Aesthete implemented as a Multi-Layer Perceptron (MLP) with two hidden layers of 100 nodes each and ReLu activation functions, minimizing the cross-entropy loss function with respect to the targets and leveraging the Adam optimizer \cite{kingma2014adam}. We tested the generalization capabilities of the learned model on a test dataset composed of 50K entries, achieving a test accuracy of $97\%$\footnote{\textcolor{black}{For comparison, a Decision Tree model, trained on the same dataset, only reaches a test accuracy of 78.7\%.}}. Hence, the learned model constitutes the Neural Aesthete for the task of edge crossing. Given an unseen input composed of a couple of arcs, the learned model outputs a probability distribution representing a degree of intersection. Following the common pipeline of Graph Drawing methods with Gradient Descent, the Neural Aesthete output represent a differentiable function that provides an admissible descent direction for the problem parameters $P$.
To test the capability of the proposed solution, we leverage an artificial dataset of random graphs with a limited number of nodes $(N \in [20, 40]$). We generated Erdős-Rényi graphs with the method presented in \cite{batagelj2005efficient} for efficiently creating sparse \footnote{The probability of edge creation has been set to $p=0.01$.} random networks, implemented in NetworkX \cite{hagberg2008exploring}. We selected the connected component of the generated graph having the biggest size (max node number).
Figure \ref{fig:edge_cross} reports a qualitative example of the proposed method in three graphs from the aforementioned dataset. To generate the graph layout, we carried on an optimization process on mini-batches of 10 arc-couples for an amount of 2K iterations (gradient steps). The first column depicts the starting random positions of the nodes; second column reports the graph layout obtained with an in-house implementation of the Stress function (see Eq. \ref{eq:stress}), optimized via Gradient Descent as done in \cite{ahmed2020graph}; third column contains the results obtained by optimizing the loss provided by the proposed Neural Aesthete for edge-crossing; fourth column reports the layouts obtained alternating the optimization of the Stress function and edge-crossing in subsequent update steps. It is noticeable to see how the solution provided by our approach is capable to avoid any arc intersection in these simple graphs. Moreover, the fact that the Neural Aesthete output represents a form of degree-of-intersection, seems to provide a good gradient direction that easily moves the arcs into a recognizable angle pattern, even when combined with other criteria (fourth column). The proposed proof-of-concept proves that Neural Aesthetes represent a feasible, general and efficient solution for Graph Drawing. In the following, we prove that this same approach can be used to guide the training process of different kind of Deep Neural Models.
\iffalse
{\color{red} \section{Marco old stuff}
We can go one step further and consider a large class of interaction fields $\varphi_{a}(\cdot)$ that are generated by a neural network. Hence, Eq.~(\ref{Kernel-based-if}) becomes \begin{equation}
\varphi_{a}(w,x) = \int_{0}^{1} d\alpha \ \eta(w,x- v_{a}(\alpha)). \label{NN-if} \end{equation} As already mentioned, in addition to the parametric representation given by Eq.~\ref{arc-seg}, we can assume that the more general dependency $
v_{a}(\alpha) = \nu(\pi_{a},\theta_{a},\alpha) $ where $\pi_{a}=(v_i,v_j)$ is the pair of vertexes corresponding with vertex $a$, $\theta \in \mathbb{R}^{d}$ is a vector of variables that are chosen in such a way to satisfy the condition on the vertexes. Clearly, in the case we use segments for connection vertexes the parameter $\theta_{a}$ is nil. As a consequence, after such a replacement, \begin{equation}
\varphi_{a}(w,x)=\varphi(\pi_{a},\theta,w,x). \label{GenIF} \end{equation}
Given the graph, the induced Harmony is \begin{equation}
H_{a}({\cal V})= \frac{1}{2}
\sum_{(\alpha,\beta) \in {\cal A}}\int_{\cal R}dx \ \varphi_{\alpha}(x) \varphi_{\beta}(x) \label{H-arc-def} \end{equation} We can promptly see that $H({\cal V})$ returns a positive number which expresses the degree of intersection of the arcs of the graph. In the case we choose the interaction field as dictated by Eq.~(\ref{GenIF}) then the induced Harmony, in addition to the dependence on the vertexes, also inherits the dependence on the weights $w$ of the neural net $\eta$ and the dependence on $\theta_{a}$, that is $H_{a}({\cal V}) = H({\cal V},\Theta,w)$, where
$\Theta = [\theta_1,\ldots,\theta_{|A|}]$.
The algorithmic expression of Harmony function $H_{a}$ can dramatically benefit form the sparsity of arc interaction. For any given ${\cal V}$ we can use the approximation
\begin{equation}
H_{a}^{\#}({\cal V}) = \frac{1}{2} \sum_{\kappa \in {\cal R}^{\#}}
\sum_{(i,j) \in {\cal A_{\kappa}}} \ \varphi_{\alpha_{\kappa,i}}(x_{\kappa})
\varphi_{\alpha_{\kappa,j}}(x_{\kappa}). \label{H-arc-def-app} \end{equation} Here ${\cal R}^{\#}$ is the set of indexes that denotes the partition of ${\cal R}$ is disjoint small boxes ${\cal U}_{\kappa}$, so as ${\cal R} = \bigcup_{\kappa \in {\cal R}^{\#}} {\cal U}_{\kappa}$ and $\forall i,j \in {\cal R}^{\#}, i \neq j: \ \ {\cal U}_{i} \bigcap {\cal U}_{j} = \emptyset$. The basic {\em sparsity assumption} here is that $A_{\kappa}$ typically contains just a few interacting segments. Of course, this number is determined by an appropriate thresholding criterion based on $\epsilon>0$, so as there is no interaction in window $\kappa \in {\cal R}^{\#}$ whenever $\varphi_{\kappa,h}<\epsilon$.\\ ~\\ \emph{\sc Boundary barriers}\\ The graph must be drawn in a given retina. For this reason we construct a barrier on the border $\partial {\cal R}$. In the simple case of a circular retina we can use the barrier function \begin{equation}
B({\cal V}) = \sum_{{\rm v} \in {\cal V}}\frac{1}{\rho^{2}+\big(r-v\big)^{2}} \end{equation} Finally we notice that most common expressions for harmonic drawings can be formalized within the same framework herein presented.\\ ~\\ \noindent{\sc Harmony minimization}\\ A possible graph drawing scheme is one which returns \begin{equation}
{\cal V}^{\star} = \arg\min_{\cal V} H_{a}({\cal V}). \end{equation} This can be carried out by classic optimization methods. For graphs the optimization can be carried out by gradient descent as follows \[
{\cal V} \leftarrow {\cal V} - \eta \nabla E({\cal V}). \] The generation process can become more sophistaced by enriching the optimization process thanks to the replacement of $H_{a}({\cal V})$ with $H({\cal V},\Theta,w)$. In this case we can exploit the computational complexity optimality of Backpropagation for the computation of $\nabla_{w} H({\cal V},\Theta,w)$. We assume that the overall optimzation process begins after having learned the weights of neural network $\eta$ on an appropriate $g_{\sigma}$. As already pointed out, different Harmony indexes can be used. For example, we could consider \begin{align}
E = H_{a} +\lambda_{c} H_{c}+ \lambda_{B} B \end{align} It is worth mentioning that, in addition to issues connected with the discovery of sub-optical solution, in general this optimization is remarkably expensive for some drawing requirements.
}
\fi
\section{GNNs for Graph Drawing} \label{sec:proposal_2}
The increasing adoption of \textcolor{black}{GNNs} in several research fields and practical tasks \cite{jumper2021highly, zhao2019t} opens the road to an even wider spectrum of problems. Clearly, Graph Drawing and GNNs seem inherently linked, even if the formalization of this learning process under the GNN framework is not trivial. As pointed out in Section \ref{sec:related}, some recent works leveraged GNNs-inspired models for Graph Drawing. DeepDrawing~\cite{ wang2019deepdrawing} employs a graph-based LSTM model to learn node layout styles from Graph Drawing frameworks.
DeepGD \cite{wang2021deepgd} is a concurrent work in which an MPNN processes starting node positions to develop pleasing layouts that minimize combinations of aesthetic losses (Stress loss combined with others). Starting node positions, however, needs to be initialized by standard graph drawing frameworks~\cite{brandes2006eigensolver}; in case a random initialization is employed, network performances deteriorate.
One of the drawbacks of both these approaches is the fact that they modify the graph topology, introducing additional connections that were not present in the original graph. This fact entails an increased computational burden for the model. Indeed, a complete graph requires many more message exchanges than a sparse one, being the computational complexity of the GNN propagation \textcolor{black}{linear in the number of edges} \cite{DBLP:journals/tnn/WuPCLZY21}. Moreover, the complete graph processed by DeepGD is enriched with edge features being the shortest path between the node connected to the edge. This solution gives big advantages in tasks closely connected with the Stress minimization but could prevent the network from generalizing to other tasks.
We propose an approach to Graph Drawing, GNDs, that leverages the computational efficiency of GNNs and, thanks to informative nodal features (Laplacian eigenvectors, see Sec. \ref{sec:pe}), is general enough to be applied to several learning tasks.
\subsection{Graph Neural Networks}
First and foremost, let us introduce some notation. We denote with $l_i$ all the input information (initial set of features) eventually attached to each node $i$ in a graph $G$. The same holds for an arc connecting two nodes $i$ and $j$, whose feature, if available, is denoted with $l_{(i,j)}$. Each node $i$ has an associated hidden representation (or state) $x_i\in \mathbb{R}^s$, which in recent models is initialized with the initial features, $x_i = l_i$ (but it is not necessarily the case in RecGNN models \cite{tiezzi2021deep}). Many GNN models can be efficiently described under the powerful umbrella of Message Passing Neural Networks (MPNNs) \cite{gilmer2017neural}, where the node state $x_i$ is iteratively updated at each iteration $t$, through an aggregation of the exchanged information among neighboring nodes $\mathcal{N}_i$, undergoing a message passing process. Formally, \begin{eqnarray}
\label{m1} x_{(i,j)}^{(t-1)} &=& \texttt{MSG}^{(t)}\left(x_i^{(t-1)}, x_j^{(t-1)}, l_{(i,j)}\right) \\
\label{m2} x_i^{(t)} &=& \texttt{AGG}^{(t)}\left(x_i^{(t-1)}, \sum_{j \in \mathcal{N}_i} x_{(i,j)}^{(t-1)}, l_i\right) \end{eqnarray}
where $x_{(i,j)}^{(t)}$ represent explicitly the message exchanged by two nodes, computed by a learnable map $\texttt{MSG}^t(\cdot)$.\footnote{Notice that in the case in which arc features $l_{(i,j)}$ are not available they are removed from the problem formulation.} Afterwards, $\texttt{AGG}^t(\cdot)$ aggregates the incoming messages from the neighborhood, eventually processing also local node information such as the node hidden state $x_i$ and its features $l_i$. The messaging and aggregation functions $\texttt{MSG}^{(t)}(\cdot)$, $\texttt{AGG}^{(t)}(\cdot)$ are typically implemented via Multi-Layer Perceptrons (MLPs) learned from data. Apart from RecGNN, other GNN models leverage a different set of learnable parameters for each iteration step. Hence, the propagation process of such models can be described as the outcome of a multi-layer model, in which, for example, the node hidden representation at layer $t$, $x_i^{(t)}$, is provided as input to the next layer, $t+1$. Therefore an $\ell$-step message passing scheme can be seen as an $\ell$-layered model.
This convenient framework is capable to describe several GNN models \cite{DBLP:journals/tnn/WuPCLZY21}. In this work, we focus our analysis on three commonly used GNN model from literature (i.e., GCN \cite{kipf2017semisupervised}, GAT \cite{velivckovic2018graph}, GIN \cite{DBLP:journals/corr/abs-1810-00826}) whose implementation is given in Table~\ref{tab:gnn}, characterized by different kinds of aggregation mechanisms (degree-norm, attention-based, injective/sum, respectively). Following the Table notation, in GCN $c_{u,v}$ denotes a normalization constant depending on node degrees; in GAT $\alpha^{(t-1)}_{u,v}$ is a learned attention coefficient which introduces anisotropy in the neighbor aggregation, $\sigma$ denotes a non-linearity and $W, W_0, W_1$ are learnable weight matrices; in GIN $\epsilon$ is a learnable parameter (which is usually set to zero).
\begin{table}
\caption{Common implementations of GNN aggregation mechanisms.
{
See the main text and the referenced papers for further details on the formulations.}
}
\centering
\begin{tabular}{lHc}
\toprule
Method: Funct.& Reference & Implementation \\
\midrule
GCN\cite{kipf2017semisupervised}: Mean & Kipf and Welling \cite{DBLP:conf/iclr/KipfW17} & \textcolor{black}{$\sigma \bigg( c_{v}W^{(t)} x_v{^{(t-1)}} + \sum_{u \in \mathcal{N}_v} c_{u,v} W^{(t)} x_u{^{(t-1)}}\bigg)$} \\
GAT\cite{velivckovic2018graph}: Att. & {Veličković et al. \cite{velivckovic2018graph} } & { $\sigma \big(\sum_{u \in \mathcal{N}_v} \alpha^{(t-1)}_{u,v} W^{(t)}x_u^{(t-1)}\big)$}\\
GIN\cite{DBLP:journals/corr/abs-1810-00826}: Sum & Xu et al. \cite{DBLP:journals/corr/abs-1810-00826} & $\text{MLP}{^{(t)}} \big((1 + \epsilon) x_v{^{(t-1)}} + \sum_{u \in \mathcal{N}_v} x_u{^{(t-1)}}$ \big) \\
\bottomrule\\
\end{tabular}
\label{tab:gnn} \end{table}
\subsection{Problem Formulation} \label{sec:pe}
Through the \textcolor{black}{GNDs} framework we propose to employ the representational and generalization capability of GNNs to learn to generate graph layouts. We formulate the problem as a \textit{node-focused} regression task, in which for each vertex belonging to the input graph we want to infer its coordinates in a bi-dimensional plane, conditioned on the graph topology and the target layout/loss function (see Section \ref{sec:dataset_crea}). Furthermore, in the GND framework, we propose to employ GNNs to learn to draw by themselves following the guidelines prescribed by Neural Aesthetes. (see Section \ref{sec:neu}).
In order to be able to properly solve the Graph Drawing task via GNDs, a crucial role is played by the expressive power of the GNN model and the nodal features which are used. In fact, in line with the aforementioned regression task, each node state must be uniquely identified to be afterwards mapped to a different 2D position in the graph layout. This problem is inherently connected with recent studies on the representational capabilities of GNNs (see Section \ref{sec:related} and \cite{you2019position}). Standard MP-GNNs have been proved to be less powerful than the 1-WL test \cite{morris2019weisfeiler}, both due to the lack of expressive power of the used aggregation mechanisms and to the existence of symmetries inside the graph. For instance, local isomorphic neighborhoods create indiscernible unfolding of the GNN computational structure. Hence, the GNN embeds isomorphic nodes to the same point in the high dimensional space of the states, hindering the Graph Drawing task. Some approaches address this problem proposing novel and more powerful architectures (WL-GNNs) that, however, tend to penalise the computational efficiency of the GNNs \cite{morris2019weisfeiler}. Moreover, given the fact that we focus on the task of drawing non-attributed graphs, it is even more important to enrich the nodes with powerful features able to identify both the position of nodes inside the graph (often referred to as Positional Encodings (PEs)\cite{dwivedi2020benchmarking}) and able to describe the neighboring structure.
\textcolor{black}{
Recently, it has been shown that the usage of random nodal features theoretically strengthens the representational capability of GNNs \cite{sato2021random,abboud2021surprising}. Indeed, setting random initial node embedding (i.e., different random values when processing the same input graph) enable GNNs to better distinguish local substructures, to learn distributed randomized algorithms and to solve matching problems with nearly optimal approximation ratios. Formally, the node features can be considered as random variables sampled from a probability distribution $\mu$ with support $D \subseteq \mathbb{R}^s$, \begin{equation}
l_i \sim \mu, \quad \forall i \in \mathcal{V} \end{equation} where $\mu$ can be instantiated as the Uniform distribution. The main intuition is that the underlying message passing process combines such high-dimensional and discriminative nodal features, fostering the detection of fixed substructures inside the graph \cite{sato2021random}. These approaches, which hereinafter we refer to as rGNNs, proved that classification tasks can be tackled in a novel way, with a paradigm shift from the importance of task-relevant information (the features values) to the relevance of the relationship among node values. However, the peculiar regression task addressed in this work requires both positional and structural knowledge, which is essential to identify and distinguish neighboring nodes. }
To address this issue, we keep standard GNN architectures and leverage Positional features defined as the Laplacian eigenvectors \cite{belkin2003laplacian} of the input graph, as introduced recently in GNNs \cite{dwivedi2020benchmarking}. Laplacian eigenvectors embed the graphs into the Euclidean space through a spectral technique, are unique and distance-preserving (far away nodes on the graph have large PE distance). Indeed, they can be considered hybrid positional-structural encodings, as they both define a local coordinate system and preserve the global graph structure.
Formally, they are defined via the factorization of the graph Laplacian matrix:
\begin{align} L =\textrm{I}-D^{-1/2}AD^{-1/2}=U^T\Lambda U,
\label{LapEig} \end{align} where $I$ is the $N \times N$ identity matrix, $D$ is the node degree matrix, $A$ is the adjacency matrix and $\Lambda$ and $U$ correspond respectively to the eigenvalues and eigenvectors. As proposed in \cite{dwivedi2020benchmarking}, we use the $k$ smallest non-trivial eigenvectors to generate a $k$-dimensional feature vector for each node, where $k$ is chosen by grid-search. \textcolor{black}{Noticeably, given that the smallest eigenvectors provide smooth encoding coordinates of neighboring nodes, during the message exchange each node receives and implicit feedback on its own positional-structural characteristics from all the nodes with which it is communicating. This process foster the regression task on the node coordinates, which receives useful information from their respective neighborhood. We believe that this is a crucial component of the model pipeline. }
\subsection{Experimental setup} \label{sec:dataset_crea}
We test the capabilities of the proposed framework comparing the performances of three commonly used GNN models (see Table \ref{tab:gnn}). In the following, we describe the learning tasks and the datasets employed for testing the different models. In Sections \ref{sec:super}, \ref{sec:stress} and \ref{sec:neu}, instead, we will give qualitative and quantitative evaluations for each learning problem, showing the generality of our approach.
Given the fact that the outputs of GND are node coordinates, we can impose on such predictions heterogeneous loss functions that can be optimized via BackPropagation. In the proposed experiments, we test the GND performances on the loss functions defined as the following: \begin{enumerate}[label=(\roman*)]
\item distance with respect to ground truth node coordinates belonging to certain layouts, produced by Graph Drawing packages (Section \ref{sec:super});
\item aesthetic loss functions (e.g. Stress) (Section \ref{sec:stress});
\item loss functions provided by Neural Aesthetes (Section \ref{sec:neu}). \end{enumerate} We assume to work with solely the graph topology, hence the node are not characterized by additional features.
We employed two different graph drawing datasets with different peculiarities. We chose to address small-size graphs ($\leq 100$ nodes) to assure the graph layout readability, since prior works highlighted node-link layouts are more suitable to small-size graphs \cite{wang2019deepdrawing, ghoniem2004comparison}. The former one is the \textsc{Rome} dataset{\footnote{http://www.graphdrawing.org/data.html}, a Graph Drawing benchmarking dataset containing 11534 undirected graphs with heterogeneous structures and connection patterns. We preprocessed the dataset removing three disconnected graphs\footnote{\textcolor{black}{Stress-based Graph drawing techniques cannot take into account disconnected graphs. However, one can easily draw each connected component separately and then plot them side by side.}}}. Each graph contains a number of nodes between 10 and 100. Some samples of the dataset are reported in the first column of Figures \ref{fig:sup_exp_kamada} and \ref{fig:sup_exp_spectral}, drawn with different layouts (see the following).
We built a second dataset, which we refer to as \textsc{Sparse}, with the same technique described in Section \ref{sec:prop1_ex}. We generated 10K Erdős-Rényi graphs following the method presented in \cite{batagelj2005efficient} for efficient sparse random networks and implemented in NetworkX \cite{hagberg2008exploring}. We randomly picked the probability of edge creation in the interval $(0.01, 0.05)$ and the number of nodes from 20 to 100. To improve the sparsity and readability, we discarded all the created graphs having both more than 60 nodes and more than 120 edges. Afterwards, we selected the connected component of the generated graph having the biggest size (max node number). We report in Figure \ref{fig:stats_dataset} a visual description of the datasets composition.
\begin{figure}
\caption{Datasets composition statistics. On the left, the histogram of the graph order (number of nodes for each graph $|\mathcal{V}|$) for both the analyzed datasets. On the right, the histogram of the graphs sizes (number of edges $|\mathcal{E}|$). The \textsc{Sparse} dataset is characterized by a sparse connection pattern.}
\label{fig:stats_dataset}
\end{figure}
In order to carry out the training process and afterwards evaluate the obtained performances, we split each of the datasets into three sets, (i.e. training, validation, test) with a ratio of (75\%, 10\%, 15\%).
\subsection{GNNs learn to draw from ground-truth examples} \label{sec:super}
The first experimental goal is focused on the task of learning to draw graph layouts given ground-truths node positions produced by Graph Drawing frameworks. Among several packages, we chose NetworkX \cite{hagberg2008exploring} for its completeness and ease of integration with other development tools. This framework provides several utilities to plot graph appearances. We choose two different classical layouts. The first is the \textsc{Kamada-Kawai} node layout \cite{kamada1989algorithm} computed by optimizing the Stress function. In few words, this force-directed method models the layout dynamic as springs between all pairs of vertices, with an ideal length equal to their graph-theoretic distance. The latter is the \textsc{Spectral} layout, which leverages the unnormalized Laplacian $\hat{L}$ and its eigenvalues to build cartesian coordinates for the nodes \cite{beckman1994theory}, \textcolor{black}{formally: \begin{equation}
\hat{L} = D - A = \hat{U}^T\hat{\Lambda} \hat{U},
\label{eq:spectral} \end{equation} where $\hat{\Lambda}$ and $\hat{U}$ correspond to the eigenvalues and eigenvectors, respectively, and using the first two non-trivial eigenvectors ($k=2$) as the actual node coordinates.\footnote{ \textcolor{black}{The NetworkX Spectral layout adds a rescaling of the node coordinates into the range $(-1,1)$ as a standard step.}} } \textcolor{black}{ We remark that Eq. \ref{LapEig} and \ref{eq:spectral} produce different outputs.} This layout tends to highlight clusters of nodes in the graph.\footnote{\textcolor{black}{See the referenced papers for further details on the layouts properties.}}
Each training graph is enriched by Positional Encodings defined as $k$-dimensional Laplacian Eigenvectors (see Section \ref{sec:pe}) and is processed by each of the tested GNN models to predict the node coordinates. Hence, we need a loss function capable to discern if the generated layout is similar to the corresponding ground truth. Furthermore, trained models should generalize the notion of graph layout beyond a simple one-to-one mapping. For these reasons, we leverage the Procrustes Statistic \cite{wang2019deepdrawing} as a loss function since it measures the shape difference among graph layouts independently of affine transformations such as translations, rotations and scaling. Given a graph composed of N nodes, the predicted node coordinates $P = (p_1, ..., p_N)$ and the ground-truth positions $\hat{P} = (\hat{p}_1, .., \hat{p}_N)$, the Procrustes Statistic similarity is defined as the squared sum of the distances between $P$ and $\hat{P}$ after a series of possible affine transformations \cite{wang2019deepdrawing}. Formally:
\begin{equation}
R^2 = 1 - \frac{\big(\mathrm{Tr}\hspace{1pt}(P^T\hat{P}\hat{P}^TP)^{\frac{1}{2}}\big)^2}{\mathrm{Tr}\hspace{1pt}(P^TP)\mathrm{Tr}\hspace{1pt}(\hat{P}^T\hat{P})}
\label{eq:procru} \end{equation}
where $\mathrm{Tr}(\cdot)$ denotes the trace operator and the obtained metric $R^2$ assumes values in the interval $[0,1]$, the lower the better. We will use the Procrustes Statistic-based similarity both as the loss function to guide the model training, and to evaluate its generalization capability on the test set.
We tested the proposed framework comparing the test performances obtained by the three different GNN models described in Table \ref{tab:gnn}, GCN, GAT, GIN.
All models are characterized by the ReLU non-linearity. The GAT model is composed by four attention heads. The $\epsilon$ variable in the GIN aggregation process is set to 0, as suggested in \cite{xu2019powerful}. We leverage the PyTorch implementation of the models provided by the Deep Graph Library (DGL)\footnote{https://www.dgl.ai/}.
We searched for the best hyper-parameters selecting the models with the lowest validation error obtained during training, in the following grid of values: size of node hidden states $x_i$ in $\{ 10, 25, 50\}$; learning rate $\eta$ in $\{10^{-4}, 10^{-3}, 10^{-2}\}$; the number of GNN layers in $\{ 2, 3, 5\}$; PE dimension $k$ in $\{5, 8\}$ (20 is added to the grid in the case of the \textsc{Sparse} dataset, given its greater node number lowerbound); drop-out rate in $\{ 0.0, 0.1\}$. We considered 100 epochs of training with an early stopping strategy given by a patience on the validation loss of 20 epochs. For each epoch, we sampled non-overlapping mini-batches composed by $\beta$ graphs, until all the training data were considered. We searched for the best mini-batch size $\beta$ in $\{ 32, 64, 128\}$. \textcolor{black}{We devised several competitors in order to asses the performances of the proposed approach. Given that Laplacian PEs available at node-level are powerful descriptors of the neighboring graph structure, we leverage a Multilayer Perceptron (MLP) as a baseline. This neural predictor learns a mapping to the node coordinates, solely exploiting the available local information. We compare the performances obtained by GNNs with Laplacian PEs against those achieved by the three corresponding variants of rGNNs, which we denote with rGCN, rGAT, rGIN. For a fair comparison, we searched in the same hyperparameter space for all the baseline and competitors. }
\begin{figure*}
\caption{\textsc{Kamada-Kawai} layout. Qualitative example of the predicted node coordinates for both the \textsc{Rome} dataset (first four column) and the \textsc{Sparse} dataset (subsequent four columns).
Each row depicts the Ground-Truth positions (GT), the graph layout produced by GCN, GAT, GIN model, left-to-right. We report the predictions on three different test graphs (rows).}
\label{fig:sup_exp_kamada}
\end{figure*}
\begin{table}[th!]
\centering
\caption{Procrustes Statistic similarity (defined in Eq. \ref{eq:procru}) on the test split of the \textsc{Rome } and \textsc{Sparse} dataset. We compare three GND models with two graph layouts generation, \textsc{Kamada-Kawai} and \textsc{Spectral}. We report the average values and standard deviations over three runs with different weights initialization.}
\resizebox{\columnwidth}{!}{
\begin{tabular}{llcccc}
\toprule
& \multirow{2}{*}{Model} & \multicolumn{2}{c}{\textsc{Rome}} & \multicolumn{2}{c} {\textsc{Sparse}} \\
\cmidrule(l){3-4} \cmidrule(l){5-6}
& & \textsc{Kamada} & \textsc{Spectral} & \textsc{Kamada} & \textsc{Spectral} \\
\midrule
\multirow{4}{*}{\rotatebox{90}{\textsc{Compet.}}} & \textcolor{black}{MLP} & 0.291 $\pm$ 0.000 & 0.144 $\pm$ 0.000 & 0.282 $\pm$ 0.001 & 0.131 $\pm$ 0.000 \\
& \textcolor{black}{rGCN} & 0.612 $\pm$ 0.009 & 0.527 $\pm$ 0.008 & 0.592 $\pm$ 0.006 & 0.532 $\pm$ 0.009 \\
&\textcolor{black}{rGAT} & 0.475 $\pm$ 0.008 & 0.437 $\pm$ 0.008 & 0.465 $\pm$ 0.015 & 0.425 $\pm$ 0.017 \\
& \textcolor{black}{rGIN} & 0.685 $\pm$ 0.001 & 0.590 $\pm$ 0.003 & 0.675 $\pm$ 0.010 & 0.603 $\pm$ 0.030 \\
\midrule
\multirow{3}{*}{\rotatebox{90}{\textsc{\textbf{GND}}}} & GCN & 0.241 $\pm$ 0.001 & $0.078 \pm 0.002$ & $0.228 \pm 0.000$ & $0.060 \pm 0.000$ \\
& GAT & \textbf{0.186} $\pm$ 0.000 & \textbf{0.057} $\pm$ 0.000 & \textbf{0.177} $\pm$ 0.001 & \textbf{0.045} $\pm$ 0.001 \\
& GIN & $0.240 \pm 0.002$ & $0.076 \pm 0.003$ & $0.229$ $\pm$ 0.003 & $0.059 \pm 0.001$ \\
\bottomrule
\end{tabular} }
\label{tab:sup_exp} \end{table}
\begin{figure*}\label{fig:sup_exp_spectral}
\end{figure*}
In Figures~\ref{fig:sup_exp_kamada} and \ref{fig:sup_exp_spectral}, we report a qualitative evaluation obtained by the best performing models for each different GNN architecture on three randomly picked graphs from the test set of each dataset. Figure~\ref{fig:sup_exp_kamada} shows the aforementioned evaluation in the case of the \textsc{Kamada-Kawai} layout supervision, both for the \textsc{Rome} dataset (first four columns, where the first one depicts the Ground Truth (GT)) and for the \textsc{Sparse} dataset. Figure~\ref{fig:sup_exp_spectral} shows the same analysis in the case of the \textsc{Spectral} layouts. The results show the good performances of the \textcolor{black}{GND} framework in generating two heterogeneous styles of graph layouts, learning from different ground truth node coordinates. In order to give a more comprehensive analysis, we report in Table \ref{tab:sup_exp} a quantitative comparison among the global Procrustes Statistic similarity values obtained on the test set by the best models, for both datasets. We report the average score and its standard deviation over three runs with different seeds for the weights random number generator.
\textcolor{black}{The strength of the Laplacian PE is validated by the decent performances yielded by the MLP baseline. Conversely, the random features characterizing the rGNNs are not sufficient to solve this node regression task. Some additional structural information is required in order to jointly represent the node position and its sorroundings. Indeed, all the models exploiting the proposed solution outperform the competitors. The improved performances with respect to MLP are due to the fact that nodes states receive implicit feedback on their own position during the message passing steps.} \textcolor{black}{The proposed GAT model with Laplacian PE} achieves the best performances in all the settings. We believe that the attention mechanism plays a crucial role in the task of distinguishing the right propagation patterns, alongside the fact that the multi-head attention mechanism provides a bigger number of learnable parameters with respect to the competitors.
In general, the \textsc{Spectral} layout is easier to be learned by the models. This can be due to the fact that the Laplacian PE represent an optimal feature for this task, given the common spectral approach. Even from a qualitative perspective generated layouts are almost identical to the Ground Truth. Vice versa, the \textsc{Kamada-Kawai} layout represents an harder task to be learned from ground-truth positions, especially in the case of the \textsc{Sparse} dataset. As pointed out in \cite{dwivedi2020benchmarking}, Laplacian-based PE have still some limitations given by natural symmetries such as the arbitrary sign of eigenvectors, that several recent works are trying to solve \cite{cui2021positional}.
\subsection{GNNs learn to draw minimizing aesthetic loss functions} \label{sec:stress}
In Section \ref{sec:super}, GNDs explicitly minimize the distances with respect to certain ground-truth node positions, hence learning to draw directly from data according to certain layouts. In this second experimental setting, instead, we want to build GNNs capable to draw at inference time respecting certain aesthetic criteria which are implicitly learnt during training. We defined our framework in such a way that powerful PE features are mapped to 2D coordinates. Given a smooth and differentiable loss function defined on such output, we can leverage the BP algorithm in order to learn to minimize heterogeneous criteria. We investigate the case in which the GNN models minimize the \textit{Stress function} (see Eq. \ref{eq:stress}) on the predicted node coordinates.
Only during the training phase, for each graph, we compute the shortest-path $d_{ij}$ among every node couple $(i,j)$. At inference time, the GND framework process the graph topology (the adjacency matrix) and the node features, directly predicting the node coordinates, without the need of any further information.
We use the same experimental setup, \textcolor{black}{competitors} and hyper-parameters selection grids of Section \ref{sec:super}. However, according to a preliminary run of the models which achieved poor performances, we varied the hidden state dimension grid to $\{ 100, 200, 300\}$. This means that this task need a bigger representational capability with respect to the previous one, which is coherent with the complex implicit nature of the learning problem. We set the Stress normalization factor to $w_{ij}= \frac{1}{d_{ij}}$ (hence, $\alpha=1$) and compute the averaged Stress function\footnote{We used the following \textit{average Stress} definition to avoid potential numerical issues:
$\textsc{stress}(P) = \frac{1}{D}\sum_{i < j} w_{ij}\big(||p_i -p_j|| - d_{ij}\big)^2 $
where $D$ is the number of considered node couples.}. For this experiment, we use the stress value obtained on the validation split as the metric to select the best performing model. \textcolor{black}{ For comparison, we report the stress loss values obtained by three State-of-the-art Graph Drawing methods. Neato \footnote{Implementation available through Graphviz, \url{https://graphviz.com}} leverage the stress majorization \cite{gansner2004graph} algorithm to effectively minimize the stress. PivotMDS \cite{brandes2006eigensolver} is a deterministic dimension reduction based approach. Finally, ForceAtlas2 \cite{jacomy2014forceatlas2} generates graph layouts through a force-directed method. }
\begin{table}[th!]
\centering
\caption{Average Stress loss value obtained on the Training set and Test set by the best selected models, for each dataset. We report the mean and standard deviation obtained over three runs initialized with different fixed seeds. \textcolor{black}{We do not report standard deviations for Neato and PivotMDS, being deterministic algorithms.}}
\resizebox{\columnwidth}{!}{
\begin{tabular}{@{}p{.2cm}p{1.cm}p{1.45cm}p{1.6cm}p{1.45cm}p{1.6cm}@{}}\toprule
& \multirow{2}{*}{Model} & \multicolumn{2}{c}{\textsc{Rome}} & \multicolumn{2}{c} {\textsc{Sparse}} \\
\cmidrule(l){3-4} \cmidrule(l){5-6}
& & \textsc{train loss} & \textsc{test loss} & \textsc{train loss} & \textsc{test loss} \\
\midrule
\multirow{3}{*}{\rotatebox{90}{\textsc{GD}}}& \textcolor{black}{ForceAtl.2} & 27.44 \scriptsize{$\pm$ 0.01} & 26.82 \scriptsize{$\pm$ 0.02} & 23.31 \scriptsize{$\pm$ 0.01} & 22.69 \scriptsize{$\pm$ 0.02} \\
& \textcolor{black}{Neato} & 4.35 & 4.34 & 5.61 & 5.66 \\
& \textcolor{black}{Piv.MDS} & 16.40 & 16.65 & 28.93 & 29.27 \\
\midrule
\multirow{4}{*}{\rotatebox{90}{\textsc{Compet.}}} & \textcolor{black}{MLP} & 1.07 \scriptsize{$\pm$ 0.01} & 1.06 \scriptsize{$\pm$ 0.03} & 1.16 \scriptsize{$\pm$ 0.02} & 1.18 \scriptsize{$\pm$ 0.00} \\
& \textcolor{black}{rGCN} & 1.25 \scriptsize{$\pm$ 0.01} & 1.24 \scriptsize{$\pm$ 0.04} & 1.70 \scriptsize{$\pm 0.02$} & 1.72 \scriptsize{$\pm 0.01$} \\
& \textcolor{black}{rGAT} & 0.98 \scriptsize{$\pm$ 0.02} & 0.97 \scriptsize{$\pm$ 0.00} & 1.34 \scriptsize{$\pm$ 0.01} & 1.36 \scriptsize{$\pm$ 0.01} \\
&\textcolor{black}{rGIN} & 1.11 \scriptsize{$\pm$ 0.03} & 1.49 \scriptsize{$\pm$ 0.04} & 1.49 \scriptsize{$\pm 0.07$} & 1.89 \scriptsize{$\pm$ 0.02} \\
\midrule
\multirow{3}{*}{\rotatebox{90}{\textsc{\textbf{GND}}}}& GCN & 0.51 \scriptsize{$\pm$ 0.02} & 0.53 \scriptsize{$\pm$ 0.03} & 0.56 \scriptsize{$\pm 0.05$} & 0.61 \scriptsize{$\pm 0.02$} \\
& GAT & {\bf 0.33} \scriptsize{$\pm$ 0.02} & {\bf 0.34} \scriptsize{$\pm$ {\bf 0.00}} & {\bf 0.30} \scriptsize{$\pm$ {\bf 0.02}} & {\bf 0.33} \scriptsize{$\pm$ {\bf 0.01}} \\
& GIN & 0.49 \scriptsize{$\pm$ 0.04} & 0.88 \scriptsize{$\pm$ 0.05} & 0.28 \scriptsize{$\pm 0.05$} & 0.81 \scriptsize{$\pm$ 0.01} \\
\bottomrule
\end{tabular} }
\label{tab:stress} \end{table}
\begin{figure}
\caption{\textsc{Stress minimization on Rome}. Qualitative example of the graph layout produced by three GNN models on the test graphs of the \textsc{Rome} dataset. Each row contains one of the same three graphs depicted in the first column of Figure \ref{fig:sup_exp_kamada} for comparison with the layout produced by Kamada-Kawai \cite{kamada1989algorithm}. }
\label{fig:stress_rome}
\end{figure}
We report in Figure \ref{fig:stress_rome} and \ref{fig:stress_random} some qualitative examples of the graph layouts produced by the best selected GNN models on test samples (the same graphs selected for Figure \ref{fig:sup_exp_kamada}) of the two datasets, following the aforementioned setting. Noticeably, all three models succeed in producing a layout that adheres to the typical characteristics of graphs obtained via Stress minimization. \textcolor{black}{In particular, for reference on the drawing style, the layouts of these same graphs generated via the Kamada-Kawai algorithm (that also minimize stress) are depicted in the first and fifth columns of Figure~\ref{fig:sup_exp_kamada}, for \textsc{Rome} and \textsc{Sparse} dataset respectively. Comparing the graph layout produced by the various GNN models and the aforementioned ones from Kamada-Kawai, also in this case it is easy to see from a qualitative analysis that the GAT model is the best performing one. } The peculiar characteristics of the \textsc{Sparse} dataset (sparse connection patterns, causing many symmetries and isomorphic nodes) entail a hardship in minimizing the stress loss function in some of the reported examples.
A quantitative comparison is reported in Table \ref{tab:stress}, with the stress values obtained by the best models for each competitor and dataset, both at training time and test time, averaged over three runs initialized with different seeds. Once again, GAT performs the best. The metrics obtained by the GIN model highlight an overfit of the training data, given the selected grid parameters. \textcolor{black}{GND models obtain better stress than all the SOTA Graph Drawing packages, with Neato being the best performing one in terms of stress minimization, as expected. Similar conclusion with respect to the previous experiment can be drawn regarding the results obtained by rGNNs and MLP. Indeed, these result show how learning to minimize stress requires both positional and structural knowledge, and that the message passing process foster the discriminative capability of the learned node states, with respect to solely exploiting local information. }
Summing up, the experimental campaign showed the generalization capabilities of the proposed framework even in the task of minimizing common aesthetic criteria imposed on the GNNs node-wise predictions, such as the \textit{stress function}, on unseen graphs. The GND framework is capable to predict node positions on unseen graphs respecting typical stress minimization layouts, without providing at inference time any explicit graph-theoretic/shortest-path information.
\begin{figure}
\caption{\textsc{Stress minimization on Sparse}. Same setting of Figure \ref{fig:stress_rome}.}
\label{fig:stress_random}
\end{figure}
\subsection{GNNs learn to Draw from Neural Aesthetes} \label{sec:neu}
In the previous Section, we showed that GNDs are capable to learn to minimize a differentiable smooth function that implicitly guides the node coordinates positioning. In a similar way, the Neural Aesthetes presented in Section \ref{sec:proposal_1} provide a smooth differentiable function that can be leveraged to find a good gradient descent direction for the learning parameters. In this Section, we mix the two proposals in order to build a Graph Neural Drawer that learns to generate graph layouts thanks to the gradients provided by the edge-crossing Neural Aesthete, and, eventually, to optimize the combination of several aesthetic losses.
At each learning epoch, GND minimizes the loss function $H(P)$ defined in Eq. \ref{eq:total_loss_neu}, over the whole edge list $\mathcal{E}$. The loss function can be computed as follows: the GNN model process the graph and predicts node-wise coordinates. Given such predicted node positions and the input graph adjacency matrix, the Neural Aesthete (which was trained beforehand as explained in Section \ref{sec:prop1_ex}) processes couples of arcs and output their degree-of-intersection. The overall loss function can then be composed by the contribution given by each of the considered arc-couples, as in Eq. \ref{eq:dir_no_target}.
We restrict our analysis \textcolor{black}{to the Rome dataset, exploiting } a GAT model with 2 hidden layers, an hidden size of node state of 25, PE dimension $k=10$, learning rate $ \eta= 10^{-2}$. We compare the graph layout generated by this model in three randomly picked test graphs, comparing three different loss function definitions: (\textit{i}) stress loss, (\textit{ii}) Neural-Aesthete edge-crossing based loss $H(P)$, (\textit{iii}) a combination of the two losses with a weighing factor $\lambda=0.5$ acting on the Neural Aesthete loss, in particular: \begin{equation}
\textsc{Loss(P)} = \textsc{Stress}(P) + \lambda H(P) . \end{equation}
We report in Figure \ref{fig:edge_cross_gnd} some qualitative results on three test graphs (one for each row). We compare the layout obtained optimizing the stress function (first column, see Section \ref{sec:stress}), the edge-crossing Neural Aesthete (second column) and the combination of the two losses.
The styles of the generated layout are recognizable with respect to the plain optimization of the Neural Aesthete with Gradient Descent (see Figure \ref{fig:edge_cross}), meaning that the GND framework is able to fit the loss provided by the Neural Aesthete and to generalize it to unseen graphs. Noticeable, the introduction of the combined loss functions (third column in Figure \ref{fig:edge_cross_gnd}) helps in better differentiating the nodes in the graph with respect to the case of solely optimizing stress. \textcolor{black}{The Neural Aesthete guided layouts (second and third column) tend to avoid edge intersections, as expected.} This opens the road to further studies in this direction, leveraging the generality of the Neural Aesthetes approach and the representation capability of GNNs.
\begin{figure}
\caption{\textsc{Learning from the Neural Aesthete}. We report the layouts obtained on three randomly picked test graphs \textcolor{black}{from the Rome dataset}, one for each row. Left-to-right: Graph layout generated by optimizing the stress loss function, the edge-crossing Neural Aesthete based loss (denoted with NA-Crossing), the combination of the two losses with a weighing factor $\lambda=0.5$. }
\label{fig:edge_cross_gnd}
\end{figure}
\subsection{Computational Complexity} \label{sec:comp}
The proposed framework leverages the same computational structure of the underlying GNN model, which we can generally describe, for each parameter update, as linear with respect to the edge number $\mathcal{O}\big(T(|\mathcal{V}|+ |\mathcal{E}|)\big)$, where $T$ is the number of iterations/layers, $|\mathcal{V}|$ the number of nodes and $|\mathcal{E }|$ the number of edges. Through our approach, there is not any increase in the computation related to the graph topology or the edge connection patterns. At inference time, the only additional requirement is the computation of the Laplacian PEs, requiring $\mathcal{O}(\mathcal{E}^{3/2})$, with $\mathcal{E}$ being the number of edges, that however can be improved with the Nystrom method \cite{fowlkes2004spectral, dwivedi2020benchmarking}.
\textcolor{black}{ \subsection{Scaling to bigger graphs} \label{sec:bigger}
Common Graph Drawing techniques based on multidimensional scaling \cite{cox2008multidimensional} or SGD \cite{zheng2018graph} require ad-hoc iterative optimization processes for each graph to be drawn. Additionally, dealing with large scale graphs -- both in terms of number of nodes and number of involved edges -- decreases the time efficiency of these approaches. Conversely, once a GND has been learned, the graph layout generation consist solely in the extraction of Laplacian PE followed by a forward pass on the chosen GNN backbone. In this Section, we prove the ability of GND to scale to real-world graphs, providing quantitative results in terms of computational times and a qualitative analysis on the obtained graph layouts, with respect to SOTA Graph Drawing techniques. We employed the best performing GAT model trained to minimize the stress loss on the Rome dataset ( Section \ref{sec:stress}). We test the model inference performances on bigger scale graphs from the SuiteSparse Matrix Collection.\footnote{\url{https://sparse.tamu.edu}} We report in Figure \ref{fig:timings} the computational times required by the different techniques to generate graph layouts of different scale, from the \texttt{dwt\_n} graph family. We analyze both the correlation on graph order (left -- varying number of nodes) and size (right -- varying number of edges). We compare the GND execution times against those of the NetworkX-GraphViz implementation of \texttt{neato} and \texttt{sfdp}, the latter being a multilevel force-directed algorithm that efficiently layouts large graphs. We also tested the Fruchterman-Reingold force-directed algorithm implemented in NetworkX (denoted with FR) and the PivotMDS implementation from the NetworKit \texttt{C++} framework \cite{staudt2016networkit}. The tests where performed in a Linux environment equipped with an Intel(R) Core(TM) i9-10900X CPU @ 3.70GHz, 128 GB of RAM and an NVIDIA GeForce RTX 3090 GPU (24 GB).
\begin{figure}\label{fig:timings}
\end{figure}
We report the average execution times over three runs (we omit the variances due to their negligible values). These results confirm the advantages of the proposed approach. While all the competitors require expensive optimization process that increase their impact with bigger graph scale, the fast inference step carried on by GNDs assures small timings even with big graphs. Computing Laplacian PE is scalable and does not hinder the time efficiency of the proposed method.
To asses the quality of the generated layouts, we report in Figure \ref{fig:tamu} a comparison among the ones yielded by GND the framework, \texttt{sfdp} and PivotMDS on several graphs from the SuiteSparse collection (we report the graph name, its order $|\mathcal{V}|$ and size $|\mathcal{E}|$). While we remark that in this experiment we exploited a GND model trained on a smaller scale dataset (i.e., Rome), the performances show a significant ability of the model to generalize the learned laws (e.g., the stress minimization in this case) to unseen graphs, even when dealing with diverging characteristics. However, we also remark that graphs having very diverse structures from the training distribution may be not correctly plotted. The causes of such performances drop are twofold. First, the intrinsic dependance of neural models on the inductive biases learned during the training process leads to an inability to generalize to unseen graph topologies. On the other hand, the limitations of Laplacian PE to discriminate certain graph simmetries or structures \cite{dwivedi2020benchmarking} may be further compounded with larger scale datasets, which is an active area of research \cite{cui2021positional}. }
\begin{figure}\label{fig:tamu}
\end{figure}
\section{Conclusion} \label{sec:conclusion}
Starting from some very interesting and promising results on the adoption of GNNs for graph drawing, which are mostly based on supervised learning, in this paper we proposed a general framework to emphasize the role of unsupervised learning schemes based on loss functions that enforce classic aesthetic measures. When working in such a framework, referred to as \textit{Graph Neural Drawers}, we open the doors towards the construction of a novel machine learning-based drawing scheme where the {\em Neural Aesthete} drives the learning of a GNN towards the optimization of beauty indexes. While we have adopted the {\em Neural Aesthetes} only from learning to minimize arc intersections, the same idea can be used for nearly any beauty index.
We show that our framework is effective also for drawing unlabelled graphs. In particular, we rely on the adoption of Laplacian Eigenvector-based positional features~\cite{dwivedi2020benchmarking} for attaching information to the vertexes, which leads to very promising results.
\section*{Acknowledgment} The authors would like to thank Giuseppe Di Battista for the insightful discussions and for useful suggestions on the Graph Drawing literature and methods. This work was partly supported by EU Horizon 2020 project AI4Media, under contract no. 951911 (https://ai4media.eu/).
\ifCLASSOPTIONcaptionsoff
\fi
\end{document} | arXiv |
Binomial process
A binomial process is a special point process in probability theory.
Definition
Let $P$ be a probability distribution and $n$ be a fixed natural number. Let $X_{1},X_{2},\dots ,X_{n}$ be i.i.d. random variables with distribution $P$, so $X_{i}\sim P$ for all $i\in \{1,2,\dots ,n\}$.
Then the binomial process based on n and P is the random measure
$\xi =\sum _{i=1}^{n}\delta _{X_{i}},$
where $\delta _{X_{i}(A)}={\begin{cases}1,&{\text{if }}X_{i}\in A,\\0,&{\text{otherwise}}.\end{cases}}$
Properties
Name
The name of a binomial process is derived from the fact that for all measurable sets $A$ the random variable $\xi (A)$ follows a binomial distribution with parameters $P(A)$ and $n$:
$\xi (A)\sim \operatorname {Bin} (n,P(A)).$
Laplace-transform
The Laplace transform of a binomial process is given by
${\mathcal {L}}_{P,n}(f)=\left[\int \exp(-f(x))\mathrm {P} (dx)\right]^{n}$
for all positive measurable functions $f$.
Intensity measure
The intensity measure $\operatorname {E} \xi $ of a binomial process $\xi $ is given by
$\operatorname {E} \xi =nP.$
Generalizations
A generalization of binomial processes are mixed binomial processes. In these point processes, the number of points is not deterministic like it is with binomial processes, but is determined by a random variable $K$. Therefore mixed binomial processes conditioned on $K=n$ are binomial process based on $n$ and $P$.
Literature
• Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
| Wikipedia |
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\title{Synthesis of arbitrary, two-mode, high visibility $N$-photon interference patterns} \author{Saroosh Shabbir} \affiliation{Department of Applied Physics, Royal Institute of Technology (KTH)\\ AlbaNova University Center, SE - 106 91 Stockholm, Sweden} \author{Marcin Swillo} \affiliation{Department of Applied Physics, Royal Institute of Technology (KTH)\\ AlbaNova University Center, SE - 106 91 Stockholm, Sweden} \author{Gunnar Bj\"{o}rk} \email[e-mail:]{[email protected]} \affiliation{Department of Applied Physics, Royal Institute of Technology (KTH)\\ AlbaNova University Center, SE - 106 91 Stockholm, Sweden}
\date{\today}
\begin{abstract}
Using coherent states and linear optics, we demonstrate the synthesis of arbitrary interference patterns and establish that neither the shape nor the visibility of $N$-photon interference patterns can be used as a quantum signature in general. Specific examples include saw-curve and rectangle curve interference patterns, and phase super-resolution with period shortening of up to 60 times compared to ordinary interference. The former two with a visibility close to 100\% and the latter with a visibility in excess of 57 \%. \end{abstract}
\pacs{42.25.Hz,42.50.St}
\maketitle The rapid development of experimental techniques has led to the demonstration of many remarkable quantum interference effects. A decade ago, it was shown that with specific $N$-photon quantum states, one could break the Rayleigh diffraction limit and make optical interference patterns with a smallest feature size $N$-times smaller than with ordinary light \cite{Boto}. This discovery laid the foundation for quantum lithography, and as the name conveys, it was thought that this was manifestly a quantum feature. However, superpositions and interference also manifest themselves in the classical world. It is therefore of interest to delineate what interference effects belong to the realm of the classical world, and which require quantum states.
In different contexts limits arise for how large a visibility one can obtain using classical and quantum light. For instance, in the two-photon Hong-Ou-Mandel experiment, one can in principle achieve 100 \% visibility with with both the input state $\ket{1}\otimes\ket{1}$ and the state $\ket{1}\otimes\ket{\alpha}$, where $\ket{\alpha}$ is a weak coherent state. However, two mutually phase randomized classical input states will never reach a visibility in excess of 50 \% \cite{Rarity}. Likewise, letting two classical states interfere in a Mach-Zehnder interferometer and measuring the probability of detecting $m$ and $N-m$ photons respectively in the two output ports will likewise never result in a visibility of the $\lambda/N$-period fringes $> 50$ \% \cite{Afek 2}. For three and four photon visibility experiments other limits to the obtainable visibility are 81.8 \% and 94.4 \%, respectively \cite{Ficek,Agafonov}. However, all these limits are derived with the reference to a specific setup and a specific detection method. Here, we consider the interference between two coherent states using a general $N$-photon projection measurement (realized with linear optics and coincidence detection). In this case we show that there is no difference between classical and quantum states neither regarding the visibility (that can be 100 \% in both cases) nor in the obtainable shape of the interference curve. Only by restricting the measurements to $\ket{m,N-m}\bra{m,N-m}$-projectors one will regain the results in \cite{Afek 2}. The distinguishing quantum signature is instead in success probability. Only with quantum states can one surpass limits to, e.g., phase sensitivity of classical light \cite{Nagata,Okamoto}. This is regardless of the detection method.
A general, two-mode, $N$-photon state can be written \begin{eqnarray} \ket{\psi} & = & \sum_{n=0}^{N} c_n \ket{n,N-n} \nonumber \\ && = \sum_{n=0}^{N} b_n (\hat{a}^{\dagger})^{n} (\hat{b}^{\dagger})^{N-n}\ket{0,0}, \label{Eq: Square wavefunction} \end{eqnarray} where $b_n = c_n/ \sqrt{n! (N-n)!}$. If we formally divide $\sum b_n (\hat{a}^{\dagger})^{n} (\hat{b}^{\dagger})^{N-n}$ with $(\hat{b}^{\dagger})^N$ and make the substitution $(\hat{a}^{\dagger}/\hat{b}^{\dagger})^n = z^n$ we get the complex polynomial \begin{equation} \sum_{n=0}^{N} b_n z^{n} = b_N(z-z_1)(z-z_2) \cdots (z-z_N), \label{Eq: Polynomial}\end{equation} where we have used the fact that any complex $N$-th degree polynomial has exactly $N$ complex roots. Hence, it is always possible to express any two-mode, $N$-photon state \begin{equation} \ket{\psi} = \frac{b_N}{\Pi_{n=1}^N {\cal N}_n}{\cal N}_1(\hat{a}^{\dagger}-z_1 \hat{b}^{\dagger})\cdots {\cal N}_N(\hat{a}^{\dagger}-z_N \hat{b}^{\dagger})\ket{0,0},
\label{Eq: Single projector expansion}\end{equation} where ${\cal N}_n$, are real numbers such that ${\cal N}_n^2(1+|z_n|^2)=1$. Thus, any two-mode, $N$-photon state can be written as a direct product of two-mode single photon states. Hofmann \cite{Hofmann} made the important observation that to make a probabilistic projection measurement onto $\ket{\psi}\bra{\psi}$, one could split the state to be measured into $N$ two-mode paths and using linear optics, unitarily transform the state in path $n$ as ${\cal N}_n(\hat{a}^{\dagger}-z_n \hat{b}^{\dagger})\ket{0,0} \rightarrow \hat{a}^{\dagger}_n \ket{0,0}$. Then, in the case one detects one photon in the $a_n$ mode of each path, one has in fact projected the input state onto $\ket{\psi}\bra{\psi}$. It is convenient to use polarization states and, e.g., take the two modes $a_n$ and $b_n$ to be horizontally and vertically polarized modes. Then any single photon polarization transformation can be achieved with two wave plates: The first compensating the relative phase between the $a_n$ and $b_n$ mode by $\theta_n=-\textrm{Arg}(z_n)$ so that the (in general elliptically polarized) state ${\cal N}_n(\hat{a}^{\dagger}_n-z_n \hat{b}^{\dagger}_n)\ket{0,0}$ is transformed to the linearly polarized state ${\cal N}_n(\hat{a}^{\dagger}_n-|z_n| \hat{b}^{\dagger}_n)\ket{0,0}$. The second being a polarizer rotated to the angle $\varrho_n = \arctan |z_n|$ so that the linearly polarized state passes it, but the orthogonal polarization is blocked.
This method was further developed in \cite{Guo} for measurement projection of NOON states, and demonstrated in \cite{RP,Kothe} to project out the NOON state components from coherent states to demonstrate $\lambda/N$-period interference. In \cite{Sun} the method was shown to provide a means of breaking the Heisenberg limit in interferometry when used with a $\ket{N/2,N/2}$ state input. A similar method was used in \cite{Zanthier,Oppel} to demonstrate resolution beyond the Rayleigh limit using independent light sources, such as thermal light. A scheme to synthesize arbitrary filter functions from $N$ cascaded polarization ``components,'' based on a similar factorization was suggested \cite{Harris} in 1967. However, in this proposal the individual components were frequency modes and not single photon states.
The stated factorization method can be taken further. Suppose the input state is a two-mode, linearly polarized coherent state
\begin{equation}\ket{\alpha,\alpha}=\exp(-|\alpha|^2)\sum_{m=0}^\infty \sum_{n=0}^\infty \frac{\alpha^{m+n}}{\sqrt{m! n!}}\ket{m,n},\end{equation} where for simplicity we take $\alpha$ to be real. We want to detect the interference between the phase-shifted input state and the projector $\ket{\psi}\bra{\psi}$ where $\ket{\psi}$ is given by Eq. (\ref{Eq: Square wavefunction}). The unitary, differential phase-shift operator is $\hat{U}(\phi) = \exp[-i \phi(\hat{a}^{\dagger} \hat{a} - \hat{b}^{\dagger}\hat{b})/2]$. The detected $N$-photon coincidence probability then becomes
\begin{eqnarray} P(\phi)& \propto & |\bra{\alpha,\alpha}\hat{U}^\dagger(\phi)\ket{\psi}|^2 \nonumber \\
& = & \alpha^{2N} \exp(-2 \alpha^2)\left |\sum_{n=0}^N b_n \exp(i \phi [2n-N]/2)\right |^2 \nonumber \\
& = & \alpha^{2N} \exp(-2 \alpha^2)\left |\sum_{n=0}^N b_n \exp(i \phi n)\right |^2.\label{Eq: Arbitrary expansion}\end{eqnarray} The sum within the absolute sign can be identified as a truncated Fourier series. Therefore, up to the highest ``frequency'' $N \phi/2$ in the parameter $\phi$, any function can be expanded in the exponential function basis. This means that we can mimic any $N$-photon state's projection onto any $N$-photon projector with a coherent state input!
\begin{figure}
\caption{Projection measurement setup. A weakly excited two-mode coherent state, linearly polarized at 45 degrees from the vertical, impinges from the left a birefringent wedge imparting the differential phase shift $\phi$. It is subsequently divided by a sequence of non-polarizing beam splitters into $N$ paths. In each path, a certain single photon state is projected out by preceding the detector with a polarizer set at $\varrho_n$ and a birefringent wedge set to the differential phase $\theta_n$. Finally the detection coincidences are recorded.}
\label{Fig: Setup}
\end{figure}
However, splitting a polarization, two-mode coherent state $\hat{U}(\phi)\ket{\alpha,\alpha}$ spatially into $N$ paths results in a product state of $N$ identical coherent states $\hat{U}(\phi)\ket{\alpha/\sqrt{N},\alpha/\sqrt{N}}$. The fact that this is a product state makes each state's projection probability onto a certain single photon projector, v.i.z.
\begin{equation} P_n(\phi) = |\bra{\alpha/\sqrt{N},\alpha/\sqrt{N}}\hat{U}^\dagger(\phi){\cal N}_n(\hat{a}^{\dagger}-z_n \hat{b}^{\dagger})\ket{0,0}|^2 \end{equation} statistically independent of any other coherent state's projection probability \cite{Glauber}. The total probability is simply the product $P(\phi) = \Pi_{n=0}^N P_n(\phi)$ of the $N$ individual projection probabilities. Thus, if one uses coherent input states it is not necessary to measure the ``clicks'' in each path in coincidence. Instead, each single photon projector $n$ can be measured separately using a weak coherent state. The result $P_n(\phi)$ is recorded, and the final probability $P(\phi)$ is subsequently obtained by multiplying all the individual probabilities as demonstrated for a NOON state in \cite{Kothe}. Hence, we need only to implement one of the arms in Fig. \ref{Fig: Setup} at a time, and reconfigure the bifrefringence $\theta_n$ and polarizer angle $\varrho_n$ between each run. Note that this is only possible for coherent states. If, e.g., quantum states or thermal states are used, as is the case in \cite{Sun} and \cite{Oppel}, respectively, there are correlations between the different single photon projection probabilities and a coincidence measurement is a must, whereas for coherent states either method works.
To demonstrate the generality of the method, we first synthesized a rectangle function interference pattern. That is, we would like to have
\begin{equation} |\bra{\alpha,\alpha} \hat{U}(\phi)\ket{\psi_s}|^2 \propto {\rm Rect}(\phi, \pi/2) = \left \{ \begin{array}{ll} 0 & |\phi| \leq \pi/2,\\ 1 & \mbox{otherwise.} \end{array} \right . \end{equation} To this end, for $N=30$ which gives a reasonable rectangle-like function, we compute the 31 lowest Fourier expansion coefficients of the Rect function and identify the coefficients with the expansion coefficients $b_n$ in Eq. (\ref{Eq: Arbitrary expansion}). Subsequently the expansion is factored in single photon projector terms by Eqs. (\ref{Eq: Polynomial}) and (\ref{Eq: Single projector expansion}) (see Supplementary Material). The thirty single photon projectors are then measured and multiplied. The result is displayed in Fig. \ref{Fig: Rect} \begin{figure}
\caption{Rectangular interference pattern. The solid (black) line (215 data points) shows raw data and the dashed (red) line shows the theoretically expected curve with the amplitude chosen for best fit. At selected data points, error bars show the $\pm \sigma$ statistical uncertainty.}
\label{Fig: Rect}
\end{figure} where the only fitting is the amplitude of the theoretically predicted coincidence probability. The experimental curve represents raw data, no background subtraction or any other data processing have been done. The maximum count rate employed in the experiments was around 5.2 MHz. A discussion about the error bars will follow below.
Next, we synthesized a Saw-curve interference pattern. The interference pattern being the square of the overlap between the input state and the projector, we want the ($2 \pi$-periodic) overlap between the input state and the measurement projector to be \begin{equation} |\bra{\alpha,\alpha}\hat{U}(\phi)\ket{\psi}|=|\phi/\pi|^{1/2} \end{equation} in the interval $\{-\pi,\pi\}$. Again finding the Fourier coefficients for $N=30$, plugging them into Eq. (\ref{Eq: Polynomial}), and factorizing the polynomial to obtain the single photon projectors (see Supplementary Material), we finally get the raw data displayed in Fig. \ref{Fig: Saw} where the only fitting is the amplitude of the theoretically predicted curve. This interference pattern has the particular feature that its derivative, which governs the phase sensitivity, is almost constant over large intervals. \begin{figure}
\caption{Saw-curve interference pattern. The solid (black) line shows raw data and the dashed (red) line shows the theoretically expected curve with the amplitude chosen for best fit. At selected data points, error bars show $\pm \sigma$ statistical uncertainty.}
\label{Fig: Saw}
\end{figure}
Finally, a NOON state \begin{equation} \ket{\psi_{\textrm{NOON}}} = \frac{1}{\sqrt{2}} \left ( \ket{N,0} - \ket{0,N}\right ), \end{equation} can give an interference pattern with the smallest feature size $\approx\lambda/(2 N)$, where $\lambda$ is the wavelength of the light \cite{Boto}. This feature size reduction can be explained in terms of maximizing a state's dynamical evolution speed \cite{Margolus,Sanders,Soderholm,Noon4} or in terms of $N$-photon quasi-particles having $N$ times the linear momentum of the photons making up the quasi particles, and thus having a de Broglie wavelength $N$ times smaller \cite{Jacobson,Fonseca,Edamatsu,WP}. Up to now, there have been many proposals and demonstrations \cite{Rarity,Fonseca,Edamatsu,WP,Angelo,Mitchell,RP, Kothe,Afek,Israel,Guo} of NOON-state interference with $N=2$ to $N=30$. The $N=6$ to $N=30$ experiments \cite{RP,Kothe}, were made with coherent light with the same method we now generalize. There has also been other demonstrations and proposals of multi-photon (i.e., using coincidence measurements) interference using non-entangled, classical states also showing a period shortening, but at the expense of a reduced interference visibility. In, e.g., \cite{Afek 2} periods down to 1/4 of the ``regular'' interference period were measured, but with a visibility of only about 29 \%. Using the method above, we obtained interference patterns for projection of a coherent state onto $N=10$, $15$, $30$, and $60$ NOON state projectors, shown in Fig. \ref{Fig: NOON}. We see that for $N=10$ we get an excellent fit with the predicted pattern. The poorest visibility, where the visibility of a fringe is defined by the adjacent local minimum and local maximum of the interference pattern is 99.5 \% in this case. Even for $N=60$ we get a reasonable fit with the theory and a minimum and maximum visibility of 57.5 \% and 86.2 \%.
\begin{figure}
\caption{NOON-state coincidence patterns. In the figures on the left, the solid (black) lines represent raw data points connected by straight lines. No background subtraction has been done. The dashed (red) curves are the expected $A\sin^2\left(N\phi/2 \right)$ with $A$ chosen for best fit. The figures on the right show a magnified part of the data points with error bars points and the fit curve. Error bars show $\pm \sigma$ statistical uncertainty. \textbf{a,} $N=10$. \textbf{b,} $N=15$. \textbf{c,} $N=30$. \textbf{d,} $N=60$. The boxed area shows the portion with highest visibility, with the maximum of 86.2 \% occurring at 1.75 $\pi$.}
\label{Fig: NOON}
\end{figure} The visibility is limited by five effects. The first one is the smallest step with which we can vary $\phi$. It is set by the combination of birefringent material (quartz in our case), the birefringent wedge angle which is 19 deg, and the linear motorized stage minimum step size 1 $\mu$m. This leads to a minimum phase difference resolution of 29 mrad which for $N=60$ is too poor (one period of the pattern, occupying $2 \pi/60$ radians, is probed by less than four measurement points). As seen in Fig \ref{Fig: NOON} (d), we simply ``miss'' many minima and maxima of the pattern, yielding a lower visibility than that set by other measurement errors and noise. For $N \leq 30$, the dominant source of error is the stochastically varying quantum efficiency of the detector. The measured coincidence probability is $P(\phi)= \Pi_{n=1}^N\eta_n P_n(\phi)$, where $\eta_n$ is the quantum detector efficiency for the single photon projector $n$. In our case, we use the same detector for each projector, and therefore $\eta_n \rightarrow \eta(n \tau)$, where $\tau = 480$ s is the time it took between measuring the point $\phi$ of one projector to the same point for the next projector. Assuming that the variation in quantum efficiency $\Delta \eta(n \tau) = \eta(n \tau)-\langle \eta(\tau)\rangle$ is uncorrelated between one projector to another, the expected standard deviation due to this error is \begin{equation} \langle (\Delta P(\phi))^2 \rangle^{1/2} = \sqrt{N} P(\phi) \frac{\langle \left [ \Delta \eta(n \tau)\right ]^2 \rangle^{1/2}}{\eta} .\end{equation} This equation shows two important features. Firstly, that due to the multiplication of detection events the error will grow with the square root of the projector's photon number. This sets an upper limit to the method's applicability. Secondly, the absolute error is proportional to the coincidence probability, or stated otherwise, the relative error is constant. This is clearly seen in our data as an excellent fit between experiment and theory as long as $P(\phi)$ is small. Another error source, but less important for our measurements, is small errors in the phases $\theta_n$. For a rapidly oscillating function, the exact position of each single photon projector's minimum has a substantial impact, as predicted in \cite{Kok}. In addition, for the NOON state interference, one should ideally be able to get all the light in and out of one linear polarizer. At best, we manage a perpendicular polarization suppression ratio of 46 dB, and typically the suppression rate is $> 40$ dB. Finally, the detector dark count also sets a limit. For our NOON state interference this error source is negligible as the dark count rate is $< 2$ Hz and the maximum count rate is 1.02 MHz. The detector efficiency (if it were constant) does not matter in the experiment as the detected state is a coherent state irrespective of $n$ and $\phi$. A low detector efficiency (about 20 \% for our detector) can easily be offset by a higher input intensity.
With the Hofmann-Guo method \cite{Hofmann,Guo}, it is possible to synthesize any interference pattern to, in principle, the precision given by $N$. The method does not limit the obtainable visibility, and we have shown that in practice, even $N=60$ NOON-state interference patterns can be recorded with $>50$ \% visibility. We have also demonstrated that unusual (non-sinusoidal) interference patterns can arise from multi-photon interference of coherent states. Hence, it is neither the shape nor the visibility of multi-photon interference patterns that delineate a general border between classical and quantum interference, although in special cases limits do apply.
However, with a coherent state input it is not possible to get quantum phase sensitivity \cite{Mitchell,Nagata,Okamoto} (i.e. break the standard quantum limit \cite{Caves,GL}). On the contrary, as discussed in some detail in \cite{Kothe}, for a given mean photon number, e.g., the NOON state interference curves have a lower phase sensitivity than ordinary (single fringe) interferometry with a coherent state.
It is possible to directly scale up the demonstrated method to the classical regime using ordinary photo detectors, and perhaps it is here the method will find applications. One reason we have stuck to working at the single photon detection level is to show that the coincidence method is very flexible in that allows the synthesis of \textit{any} two-mode, multi-photon projector, albeit at the expense of exponentially decreasing probability of coincidence events with increasing $N$. Another reason is to explore what practical limits (i.e., how the influence of various sources of errors) one encounters when one scales single-photon coincidence measurements to large numbers.
When using coherent state input, the switch from obtaining the interference curves by coincidence measurements to multiplication of single photon projection probabilities allows considerable savings in time and equipment. In our case, the maximum probability of detecting a photon in a single temporal mode, defined by the response time of the detector, is about 0.1 for each single photon projector. Hence, as the $N$ single photon projectors give statistically independent results, the probability of detecting, say, 10 photons in coincidence would be around $0.1^{10} = 10^{-10}$. With our detectors with a response time of 45 ns it would thus take at least $45\cdot 10^{-9} \cdot 10^{10}=450$ s to get a single coincidence click, on average. Detecting each (temporal) mode sequentially, instead of detecting $N$ (spatial) modes in parallel, it took about 900~s to measure the 60 projectors for a single phase shift $\phi$. Thus, taking 215 steps to cover the range of phase shifts from 0 to $2 \pi$ radians, a whole $N=60$ interference curve was obtained in about 8 h which is long, but feasible.
The authors thank C. Kothe, S. Takeuchi, K.~Edamatsu, and S. Harris for helpful discussions. The work was supported by the Swedish Research Council (VR) through grant 621-2011-4575 and through its support of the Linn\ae us Excellence Center ADOPT.
\section*{Supplementary material}\label{SM}
\subsection{Methods}
We have implemented a multi-photon experiment using a linearly polarized HeNe laser (whose polarization was further ``cleaned'' with a polarizer) with a power of 15 mW, neutral density filters, two birefringent wedges mounted on step-motor drive translators (with a precision of 1 $\mu$m) a polarizer mounted on a rotation stage (with a precision of 0.2 degrees), and a single photon sensitivity avalanche photo diode (APD), see Fig. \ref{Fig: Setup}. The intensity of the laser was adjusted by the neutral filters such that the mean time interval between two photons was 20 times longer ($\approx 1 \mu$s) than the dead time of the detector (45 ns). Thus, we ensure that essentially each recorded ``click'' stems from a single photon. The laser's polarization was carefully adjusted to be at 45 degrees from the horizontal, thus creating the desired input state $\ket{\alpha,\alpha}$, where $\alpha \ll 1$ in the temporal mode basis of the detector. The calibration of the wedges was found to give a $2 \pi$ phase shift with a relative translation of 0.215 mm.
Following the method in \cite{Kothe}, each arm of the interferometer was implemented sequentially and the results were recorded on the computer. For each arm $n$, the first birefringent wedge was set to impart the differential phase shift $\phi$ (from $0$ to $2 \pi$ for each projector) on the input state, the second birefringent wedge imparted the differential phase shift $\theta_n$ and the polarizer after the second wedge was rotated to the angle $\varrho_n$ to adjust the relative amplitude of the horizontal and vertical modes (see Fourier Expansion Coefficients). The state was subsequently focused on the APD and the number of ``clicks'' during one second was recorded as a function of $\phi$ in a computer memory. After all $N$ single photon projector counts [proportional to $P_n(\phi)$)] had been measured as a function of $\phi$, the counts at each setting of $\phi$ were multiplied together yielding a function proportional to $P(\phi)$.
All the displayed data is raw, that is, without the subtraction of dark counts and it was obtained with a counter gate time of 1 s.
\subsection{Fourier expansion coefficients}
Consider a $2 \pi$-periodic rectangle function defined by the two conditions \begin{equation} {\rm Rect}(\phi, \pi/2) = \left \{ \begin{array}{ll} 0 & |\phi| \leq \pi/2,\\ 1 & \mbox{otherwise.} \end{array} \right ., \end{equation} and ${\rm Rect}(\phi, \pi/2)={\rm Rect}(\phi + 2 l \pi, \pi/2)$ where $l$ is an arbitrary integer. Since the Rect function is even, it is advantageous (and natural) to use an even number of projectors to synthesize the function implying that $N$ should be chosen even. (The function can also be expanded for odd $N$ but with a less pleasing result). The Fourier expansion of this function can hence formally be written \begin{equation} {\rm Rect}(\phi, \pi/2) = \sum_{n=-N/2}^{N/2} b_{(2n+N)/2} \exp(i \phi n )\end{equation} where the expansion coefficients are \begin{eqnarray} b_{(2n+N)/2} & = & \frac{1}{2 \pi}\int_{- \pi}^{\pi} {\rm Rect}(\phi, \pi/2) \exp(-i \phi n ) d\phi \nonumber \\ & = & \left \{ \begin{array}{ll} 1/2 & n = 0, \\ -\frac{\sin(n \pi/2)}{n \pi} & n \neq 0, \end{array} \right .\\ \end{eqnarray} Using Eq. (\ref{Eq: Square wavefunction}), the associated polynomial for this state for, e.g., $N=10$ is hence \begin{equation} \frac{z^5}{2}-\frac{z^4 + z^6}{\pi}+\frac{z^2+z^8}{3 \pi} -\frac{1+z^{10}}{5\pi} = 0, \end{equation} with the roots \begin{eqnarray}
z_1 = z_2^* & = & 0.967612 + i 0.252442, \nonumber \\
z_3 = z_4^* & = & 0.723141 + i 0.6907, \nonumber \\
z_5 = z_6^* & = & 0.313207 + i 0.949685, \nonumber \\
z_7 = z_8^* & = & -0.463687 + i 0.29332, \nonumber \\
z_9 = z_{10}^* & = & -1.54027 + i 0.974347. \end{eqnarray}
The fact that the Rect function is an even function leads to the result that all the roots of the associated polynomial come in complex conjugate pairs. Inserting these roots into Eq. (\ref{Eq: Single projector expansion}) it is evident how to implement the ten single projectors. For example, the $n=1$ projector is implemented by introducing a birefringence of $-\textrm{Arg}(z_1)\approx -0.255$ rad $= -14.6$ degrees between the horizontal and vertical directions. This birefringence should be followed by a polarizer set at the angle $\arctan|z_1|=\pi/4$ rad. In Table \ref{Table 1} we list the settings of the birefringence and the polarizer angle for the 10 projectors in degrees.
\begin{table}[h]
\begin{tabular}{|c|c|c|}
\hline
Root $n$& $\varrho_n$ & $\theta_n$ \\ \hline
1 & 45.0 & -14.6 \\
2 & 45.0 & 14.6 \\
3 & 45.0 & -43.7 \\
4 & 45.0 & 43.7 \\
5 & 45.0 & -71.7 \\
6 & 45.0 & 71.7 \\
7 & 28.7 & -147.7 \\
8 & 28.7 & 147.7 \\
9 & 61.2 & -147.7 \\
10 & 61.2 & 147.7 \\
\hline \end{tabular} \caption{The experimental parameters for a $N=10$ Rect-function projectors.} \label{Table 1} \end{table} It is seen in Fig. \ref{Fig: Rect} that the resulting function is not a perfect rectangle, but Gibbs phenomenon makes the interference curve overshoot on the steep flanks. This effect can be reduced by the use of a Lanczos-Fourier expansion or a Ces\`{a}ro approximation of the Fourier series, at the expense of getting less steep flanks in both cases. Going to higher photon numbers one could in principle reduce the wiggles in the interval $\{ \pi/2, 3 \pi/2 \}$, but the overshoot height would remain the same, only the width of the overshooting peak could be decreased.
\begin{table}[t]
\begin{tabular}{|c|c|c|}
\hline
Root $n$& $\varrho_n$ & $\theta_n$ \\ \hline
1 & 56.2 & 0.0 \\
2 & 33.8 & 0.0 \\
3 & 66.6 & -74.0 \\
4 & 66.6 & 74.0 \\
5 & 23.4 & -74.0 \\
6 & 23.4 & 74.0 \\
7 & 21.7 & -141.7 \\
8 & 21.7 & 141.7 \\
9 & 68.3 & -141.7 \\
10 & 68.3 & 141.7 \\
\hline \end{tabular} \caption{The experimental parameters for a $N=10$ Saw-function projectors.} \label{Table 2} \end{table} In a similar manner, an $N=10$ expansion of the Saw$^{1/2}$ function \begin{equation} {\rm Saw}^{1/2}(\phi, 2 \pi) = \sum_{n=-N/2}^{N/2} b_{(2n+N)/2} \exp[-i \phi n ].\end{equation} Computing the expansion coefficients in the same manner as before, one arrives at \begin{equation} b_{(2n+N)/2} = \frac{
\sqrt{2 n}\sin(n \pi) -{\rm FS}(\sqrt{2 n})}{\sqrt{2} \pi n^{3/2}},\end{equation} where FS is the Fresnel sine function. Numerically, the coefficients are: \begin{eqnarray} b_5 & = & 2/3, \nonumber \\ b_4=b_6 & = & -0.1607, \nonumber \\ b_3=b_7 & = & -0.0273, \nonumber \\ b_2=b_8 & = & -0.0272, \nonumber \\ b_1=b_9 & = & -0.0109, \nonumber \\ b_0=b_{10} & = & -0.0121. \end{eqnarray}
The polynomial associated to the expansion is thus \begin{eqnarray} \frac{2 z^5}{3}- 0.1607 (z^4 + z^6)-0.0273(z^3 + z^7) && \nonumber \\- 0.0272(z^2 + z^8)-0.011 (z + z^9) && \nonumber \\ - 0.012(1 + z^{10}) & = & 0
\end{eqnarray} with the roots \begin{eqnarray} z_1 & = & 1.49215, \nonumber \\ z_2 & = & 0.670175, \nonumber \\ z_3 = y_4^* & = & 0.637336 + i 2.22359, \nonumber \\ z_5 = y_6^* & = & 0.119116 + i 0.415582, \nonumber \\ z_7 = y_8^* & = & -0.311654 + i 0.245907, \nonumber \\ z_9 = y_{10}^* & = & -1.97752+ i 1.56034. \end{eqnarray} The settings, in degrees for the birefringence and the polarizer angle are given in Table \ref{Table 2}.
\end{document} | arXiv |
Alfreds Meders
Alfreds Arnolds Adolfs Meders was a German-Latvian mathematician, and a student of Leopold Kronecker. He was a professor at the Riga Technical University until his mandatory repatriation to Germany in 1939.[2]
Alfreds Meders
Born(1873-09-19)September 19, 1873
Riga, Latvia
Died(1944-06-28)June 28, 1944
Poznań, Poland[1]
References
1. "Meder, Alfred Arnold Adolf". BBLD - Baltisches biografisches Lexikon digital. Retrieved 6 February 2016.
2. O'Connor, John J.; Robertson, Edmund F., "Alfreds Meders", MacTutor History of Mathematics Archive, University of St Andrews
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| Wikipedia |
\begin{definition}[Definition:Complex Vector Space]
Let $\C$ be the set of complex numbers.
Then the $\C$-module $\C^n$ is called the '''complex ($n$-dimensional) vector space'''.
\end{definition} | ProofWiki |
Biokinetics, dosimetry, and radiation risk in infants after 99mTc-MAG3 scans
J. Soares Machado ORCID: orcid.org/0000-0001-9475-21631,
J. Tran-Gia1,
S. Schlögl1,
A. K. Buck1 &
M. Lassmann1
Renal scans are among the most frequent exams performed on infants and toddlers. Due to the young age, this patient group can be classified as a high-risk group with a higher probability for developing stochastic radiation effects compared to adults. As there are only limited data on biokinetics and dosimetry in this patient group, the aim of this study was to reassess the dosimetry and the associated radiation risk for infants undergoing 99mTc-MAG3 renal scans based on a retrospective analysis of existing patient data.
Consecutive data were collected from 20 patients younger than 20 months (14 males; 6 females) with normal renal function undergoing 99mTc-MAG3 scans. To estimate the patient-specific organ activity, a retrospective calibration was performed based on a set of two 3D-printed infant kidneys filled with known activities. Both phantoms were scanned at different positions along the anteroposterior axis inside a water phantom, providing depth- and size-dependent attenuation correction factors for planar imaging. Time-activity curves were determined by drawing kidney, bladder, and whole-body regions-of-interest for each patient, and subsequently applying the calibration factor for conversion of counts to activity. Patient-specific time-integrated activity coefficients were obtained by integrating the organ-specific time-activity curves. Absorbed and effective dose coefficients for each patient were assessed with OLINDA/EXM for the provided newborn and 1-year-old model. The risk estimation was performed individually for each of the 20 patients with the NCI Radiation Risk Assessment Tool.
The mean age of the patients was 7.0 ± 4.5 months, with a weight between 5 and 12 kg and a body size between 60 and 89 cm. The injected activities ranged from 12 to 24 MBq of 99mTc-MAG3. The patients' organ-specific mean absorbed dose coefficients were 0.04 ± 0.03 mGy/MBq for the kidneys and 0.27 ± 0.24 mGy/MBq for the bladder. The mean effective dose coefficient was 0.02 ± 0.02 mSv/MBq. Based on the dosimetry results, an evaluation of the excess lifetime risk for the development of radiation-induced cancer showed that the group of newborns has a risk of 16.8 per 100,000 persons, which is about 12% higher in comparison with the 1-year-old group with 14.7 per 100,000 persons (all values are given as mean plus/minus one standard deviation except otherwise specified).
In this study, we retrospectively derived new data on biokinetics and dosimetry for infants with normal kidney function after undergoing renal scans with 99mTc-MAG3. In addition, we analyzed the associated age- and gender-specific excess lifetime risk due to ionizing radiation. The radiation-associated stochastic risk increases with the organ doses, taking age- and gender-specific influences into account. Overall, the lifetime radiation risk associated with the 99mTc-MAG3 scans is very low in comparison to the general population risk for developing cancer.
According to information from the American National Institute of Diabetes and Digestive and Kidney Diseases, there is a high incidence of kidney pathologies in infants [1]. Compared to adults, they have a ~20 times higher probability for developing defects in the urinary tract. Cases as urine reflux/blockage and infections are related with the infants' immature urinary system and/or malformation by birth [1]. Nuclear medicine renography for pediatric patients is one of the standard non-invasive diagnostic methods with advantages such as a potential detection of diseases in early stages and information on physiology with a high sensitivity [2, 3]. The standard tracer for examining the latter is 99mTc-MAG3 [2].
99mTc-MAG3 scans are often indicated for renal function examinations in infants because of the minimum recommended age (1 month), the short physical half-life of 99mTc (6.01 h), the high extraction rate of the radiopharmaceutical (60% in the first filtration), and the high kidney uptake (97%), providing a good image quality even for infants [2]. Therefore, renal scans with 99mTc-MAG3 are among the most frequent urinary tract exams performed on infants and toddlers [3].
The basic standard protocol for 99mTc-MAG3 renal scans performed for young children consists of a dynamic scan performed on a single-headed camera equipped with a low-energy collimator [2]. According to the European Association of Nuclear Medicine (EANM) dosage card, the recommended minimum injected activity is 15 MBq [4].
While there are no recommended absorbed dose limits, it is, based on the ALARA concept ("As Low As Reasonably Achievable"), highly recommended to always minimize the doses with regard to the image quality necessary for an accurate diagnosis [3]. It is also important to consider the risk from radiation exposure for this group of patients, as, due to the young age, this group could potentially be classified as a "high-risk group" with a higher probability for developing stochastic radiation effects compared to adults [3]. Cancer is a complex disease that depends on many factors such as age, gender, genetic predisposition, and lifestyle, and it can take years to develop [1, 3, 5, 6]. Therefore, an exposure to ionizing radiation at young ages will, most likely, increase the cancer risk [7]. According to information provided by BEIR VII (Committee on the Biological Effects of Ionizing Radiation) about risk estimation for exposure in childhood (studied cohort: the atomic bomb survivors and two case-control studies of thyroid cancer), the risk decreases with the age at the time of exposure [6, 7]. Based on these considerations, one can conclude that an accurate risk assessment is of particular importance for pediatric patients to minimize the risk of nuclear medicine imaging procedures [3]. However, there is few data on biokinetics, dosimetry, and risk estimation at low doses for pediatric patients. In a review by Eberlein et al., it was shown that, for 99mTc-MAG3 renal scans, the biokinetic and dosimetry data were published 26 years ago, with only four data sets specific for children [8].
One of the quantities proposed by the International Commission on Radiological Protection (ICRP) used for assessing radiation risk is the effective dose [5, 9]. According to ICRP, it can be applied in diagnostic exams to estimate the health detriment for a general group of exposed individuals, without considering ages and gender. Based on the effective dose values, the risk levels of different procedures can be compared and optimized whenever it seems reasonable or necessary [5, 9]. Therefore, the ICRP formalism is not applicable for performing individual risk estimation of radiation-induced effects.
A newly developed tool for individual patient risk estimation is the Radiation Risk Assessment Tool (RadRAT) developed by the National Cancer Institute's Division of Cancer Epidemiology and Genetics [10]. This online platform tool was developed based on BEIR VII from the National Academy concerning radiation health effects named Health Risks from Exposure to Low Levels of Ionizing Radiation. Utilizing the RadRAT calculator, it is possible to perform lifetime attributable risk (LAR) estimations of radiation-related cancer induction for low-level ionizing radiation with doses < 1 Gy for individuals based on their age, gender, year of exposure, uniform and/or non-uniform doses and organ-specific absorbed doses [10].
The aim of this study was, therefore, to reassess the biokinetics in infants undergoing 99mTc-MAG3 renal scans based on an image-based retrospective quantification and to derive organ absorbed doses as well as the associated risk for the related age group. The research comprised four steps: (1) phantom experiments for retrospective image quantification by depth- and size-dependent attenuation correction, (2) Estimation of patient-specific time-integrated activity coefficients (TIACs), (3) Calculation of the absorbed and effective doses, and (4) Risk assessment.
Patient data and measurement protocol
For this retrospective study, data from 20 consecutive patients with scintigraphy normal excretion with good wash-out were analyzed. Referral criteria for 99mTc-MAG3 scintigraphy included sonographic suspicion of either urinary tract dilation or obstructive uropathy. The preparation of the patients (incl. oral hydration with 10 ml/kg or by breastfeeding 30 min prior to injection) was performed according to the EANM guidelines for standard and diuretic renograms in children [2]. Furosemide (1 mg/kg i.v. in infants, 0.5 mg/kg in children above the age of 1 year) was injected following a F+20 protocol in all but two patients who showed almost complete tracer excretion 20 min after 99mTc-MAG3 administration. For a radiation-related absorbed dose and risk analysis, the patients were separated in two groups based on their age: 17 newborns (1.6–12.0 months) and 3 1-year-olds (13.0–20.0 months). As this study only included retrospectively analyzed data acquired within the clinical routine, our local ethics committee waived the need for further approval.
At our institution, 99mTc-MAG3 scans are typically performed on a single-head gamma camera (E.Cam Signature, Siemens Healthcare) equipped with a low-energy high-resolution (LEHR) collimator. The injected activities are patient-specifically calculated based on the Pediatric Dosage Card 2014 of the European Association of Nuclear Medicine [4]. The study protocol is a planar dynamic acquisition of 132 images centered at the patients' kidneys, started at the bolus injection, and lasting 35 min. The dynamic data are distributed in three phases (phase I: 40 images acquired over 1 min; phase II: 40 images acquired over 4 min; phase III: 52 images acquired over 30 min). The individual sizes and depths of each patient's organs were taken from previously acquired ultrasound data.
Determination of a depth- and size-dependent attenuation correction function
To estimate the patient-specific organ activities, a retrospective calibration was performed on the same gamma camera that had been used for the patient acquisitions (E.Cam Signature, Siemens Healthcare). The goal was to derive a calibration factor by taking into account the sizes and depths of the individual kidneys. To simulate different kidney sizes, two one-compartment kidney phantoms designed according to MIRD pamphlet 19 [11] and fabricated with a 3D printer as described by Tran-Gia et al. [12] (Fig. 1) were used (newborns: 8.6 mL, 1-year-old: 23.4 mL). Both kidneys were filled with 99mTc-solutions (newborn: 1.10 MBq/mL, 1-year-old: 0.98 MBq/mL). To simulate different kidney depths inside a patient, the phantoms were mounted in a body phantom (NEMA-NU2–2012, PTW-Freiburg) using a 3D-printed, depth-adjustable attachment system (distances of 8.2, 11.7, and 15.2 cm from the patient bed) which is also presented in [12] (Fig. 2a, b). After filling the phantom with water, static planar images (duration 600 s) were acquired for each depth position and kidney insert. Besides the acquisitions of the kidney phantom placed inside of the torso phantom, another acquisition was acquired with the kidney phantom placed directly on the patient bed to simulate a depth of 0 cm (i.e., approximately zero attenuation).
3D-printed 1-year-old and newborn kidney phantoms [12]
Phantom experiment. a Kidney insert (newborn) mounted on the torso-phantom. b Kidney insert mounted on the torso-phantom using the manufactured attachment system with three different depth positions: 8.2 cm, 11.7 cm, and 15.2 cm. c Exemplary ROI positioning for the phantom (dark blue) and the background (red)
All post processing was performed with vendor-specific software (E.Soft, Siemens Healthcare). For each measurement, regions-of-interest (ROIs) were drawn around the phantom insert and in the background (Fig. 2c). To estimate the scatter contribution of the phantom as accurately as possible, a background subtraction was performed prior to any further calculations. As the background and phantom ROIs were different in size (AreaPhantom ≠ AreaBgr), a ROI normalization had to be applied to the counts in the background ROI (CountsBgr):
$$ {\mathrm{Counts}}_{\mathrm{Bgr}\to \mathrm{Phantom}}(d)={\mathrm{Counts}}_{\mathrm{Bgr}}(d)\bullet \frac{{\mathrm{Area}}_{\mathrm{Phantom}}}{\ {\mathrm{Area}}_{\mathrm{Bgr}}} $$
Here, the parameter d represents the depth of the phantom (distance kidney ↔ patient bed). Based on the background counts normalized to the size of the phantom ROI (CountsBgr → Phantom), a depth-dependent calibration factor cfvolume (unit: cps/MBq or counts-per-second-per-MBq) was calculated as:
$$ {\mathrm{cf}}_{\mathrm{volume}}(d)=\frac{{\mathrm{Counts}}_{\mathrm{Phantom}}(d)-{\mathrm{Counts}}_{\mathrm{Bgr}\to \mathrm{Phantom}}(d)}{A\ast \Delta t} $$
where A represents the decay-corrected activity, and ∆t stands for the total acquisition duration.
Next, the depth-dependent calibration factors cfvolume(d) were divided by the calibration factor at depth zero cfvolume(d0) to obtain a unitless attenuation factor for a kidney-shaped organ of either age group (newborns and 1-year-olds):
$$ \mathrm{Attenuation}\ {\mathrm{Correction}\ \mathrm{Factor}}_{\mathrm{volume}}\ (d)=\frac{{\mathrm{cf}}_{\mathrm{volume}}(d)}{{\mathrm{cf}}_{\mathrm{volume}}\left({d}_0\right)} $$
To enable an attenuation correction for patient-specific organ depths, a depth-dependent attenuation correction function was approximated by a second-degree polynomial curve, which was separately fitted to the newborn and the 1-year-old data. Figure 4 shows these attenuation correction factors as a function of the depth d for newborns (blue) and 1-year-olds (orange). The calibration factor for any depth can be calculated based on these curves by inserting the patient-specific organ depth (Additional file 1: Table S1) into the fitted equations shown in the graphic.
Determination of time-activity curves in the patients
Another ROI analysis was performed to obtain the time-activity curves for different organs from the patient data. The ROIs were drawn around the kidneys and the bladder, with an additional ROI placed beside each organ to estimate the background (Fig. 3). The whole-body ROI (WB) covered the entire field-of-view. After background correction (subtraction of the counts in the background ROI normalized to the organ area from the organ ROI as described by Eqs. 1 and 2), the result was a temporal course of the counts in all examined organs.
ROI analysis of patients P20 (left, 20-month-old female) and P10 (right, 5-month-old female). a, b ROIs and background in kidneys (red: left kidney; green: right kidney) and bladder (blue) for multiple time points. The whole-body ROI covers the entire field-of-view and is not depicted. c, d Number of counts as a function of time for all ROIs
The conversion from counts to activity was performed based on pre-determined ultrasound-based patient data (kidney and bladder volumes, kidney depth): First, the kidney depth was inserted into both attenuation correction functions (Fig. 4) to obtain depth-corrected attenuation factors for the newborn and 1-year-old volumes. Subsequently, the calibration factor for the patient-specific kidney volume was linearly interpolated based on these volume-cf pairs (Fig. 4). Finally, kidney time-activity curves were obtained by dividing the number of counts in each temporal frame by the resulting factor and the acquisition time.
Depth-dependent attenuation correction factors obtained in the phantom experiment. Blue: newborns. Red:1-year-olds
The bladder time-activity curves were determined similarly, with the exception that no depth could be extracted from the ultrasound data. Instead, a depth of 5 cm was used for all bladders. The assumption that the kidney geometry is comparable to the approximately spherical geometry of the bladder is based on a study by Tran-Gia et al., where only a negligible difference occurred between the MIRD-based kidneys and a spherical model of similar volume [12].
The whole-body time-activity curves were obtained under the assumption that the total number of counts corresponded to the total administered activity. This typically holds for pediatric patients, as a large part of the body is included in the camera field-of-view. The coverage sufficiently represents the normalized whole-body biokinetics.
Patient-specific time-integrated activity coefficients
The patient-specific time-integrated activity coefficients (TIACs) were obtained by a time integration of the organ-specific time-activity curves. During the renal scans, 132 images were acquired starting at the bolus injection over a period between 0 and 35 min. While the organ-specific time-activity curves were integrated up to the last time point for each patient, only physical decay was assumed after the last time point. A combination of a trapezoidal integration for time points before the curve maximum and a bi-exponential function after the maximum was applied to all time-activity curves. While the bi-exponential functions were fitted in OriginPro 2016 (ADDITIVE GmbH), the integrals were calculated using Microsoft Office 365 Excel version 2016 (Microsoft Corporation).
Absorbed dose calculation
The absorbed dose coefficients (mGy/MBq) to the organs and the effective dose coefficients (mSv/MBq) for each patient were assessed by means of the newborn and the 1-year-old mathematical phantom provided by OLINDA/EXM [13].
Risk estimation
The risk estimation was performed individually for all 20 pediatric patients with RadRAT, published by the US National Cancer Institute. The following input data were used: gender; age; population group (U.S. 2000–2005); exposure year; organs like brain, breast (females), colon, gallbladder, liver, lungs, ovaries (females), kidneys, pancreas, red bone marrow, stomach, thyroid, urinary bladder, uterus (females); exposure rate (acute); dose distribution type (fixed value); organ-specific absorbed doses [10]. The result was the percentage risk in 100,000 persons for the development of stochastic radiation-induced effects for "lifetime attributable risk" and "future risk". The lifetime attributable risk (LAR) estimates the probability of cancer development and death by an individual arising from radiation exposure. The future risk is defined as the risk estimated for an individual from the present time until the end of the expected lifetime for developing cancer [6, 10].
Despite a high uncertainty in the individual risk estimation (90%), this information might assist in the establishment of more accurate recommendations for this high-risk group of pediatric patients for keeping the balance between sufficient imaging quality at the lowest possible patient radiation exposure [14].
All values will be given as mean plus/minus one standard deviation except otherwise specified.
Demographic data of the patient group
The Additional file 1: Table S2 shows the demographic data of the 20 patients (14 males; 6 females) classified into the two age groups (newborns and 1-year-olds). Ages are 1.6–20.0 months (mean 7.0 ± 4.6 months). Weight is 5–12 kg (mean 7.8 ± 1.9 kg). Body size is 60–89 cm (mean: 69.5 ± 7.8 cm). Injected activity is 12–24 MBq (mean 17.9 ± 2.6 MBq). Four patients had injected activities between 12 and 15 MBq, 14 patients between 16 and 20 MBq and 2 patients between 21 and 24 MBq. On average, the activities administered to our patients were 22% lower than the recommended injected activity values from EANM Dosage Card 2014 [4].
Depth- and size-dependent attenuation correction functions
The depth- and size-dependent attenuation correction curves are shown in Fig. 4. Separate second-degree polynomials were fitted separately for the newborn and the 1-year-old kidney for distances of 0, 8.2, 11.7, and 15.2 cm from the patient bed. As expected, the attenuation increases with the distance. While the attenuation is comparable for small depths < 5 cm (no difference for 0 cm), the attenuation of the newborn kidney phantom is higher than that of the 1-year-old phantom for larger depths (differences of 14% for 8.2 cm, 18% for 11.7 cm, and 22% for 15.2 cm).
The time-integrated activity coefficient (TIAC) values of the patient analysis are given in Table 1.
Table 1 Organ-specific time-integrated activity coefficients (TIACs) in hours for all patients (classified into age groups)
Comparing the patient age groups, the mean TIAC values for the newborn group were 47% higher for the kidneys, 50% lower for the bladder, and 80% higher for the whole-body. Although none of the patients had severe kidney function impairments, large inter-patient variations were observed. However, according to one-way ANOVA tests comparing the TIAC values between the newborn and the 1-year-old groups (p value 0.07), there was no significant difference (p > 0.05). In comparison with the ICRP 128 values for the 1-year-old group with normal renal function (kidneys 0.065 h; bladder 1.6 h; whole-body 0.23 h) [15], the differences of the mean TIAC values were 38% higher for the kidneys, 35% lower for the bladder, and 83% higher for the whole-body.
Dose calculation
Additional file 1: Table S3 shows the mean values of the organ-specific absorbed dose coefficients. For all patients, the kidneys absorbed dose coefficient values were between 0.004 and 0.131 mGy/MBq and the bladder absorbed dose coefficients were 0.01–0.93 mGy/MBq. The effective dose coefficients ranged from 0.001 to 0.063 mSv/MBq.
Table 2 shows the dose results clustered per age group. Comparing the two age groups, the newborns showed 66% higher absorbed doses for the kidneys. The 1-year-old group showed 6% higher bladder absorbed doses. The effective dose results were 24% higher for the 1-year-old group.
Table 2 Mean absorbed doses and effective doses for both age groups
In contrast to adults, excretion cannot be controlled or contained by newborns and toddlers. Therefore, it is possible to observe the influence of bladder voiding on the dosimetry. The patients who had bladder voiding before the last image (i.e., within ~ 30 min after the injection) showed lower absorbed dose values. The mean organ absorbed doses for the 13 patients with voiding were 0.5 ± 0.2 mGy for the kidneys, 3.3 ± 2.6 mGy for the bladder, and 0.1 ± 0.1 mGy for the whole-body. As expected, the mean organ absorbed doses for the 7 patients without voiding were 47% higher in the kidneys (0.9 ± 0.8 mGy), 57% higher in the bladder (7.7 ± 5.5 mGy), and 35% higher for the whole-body (0.2 ± 0.1 mGy).
The results of the excess lifetime risk estimation are given in Table 3. The mean excess lifetime risk, as well as the lower and upper bounds (limits) of the respective confidence intervals (CI) for the risk probability of all patients, classified per age and gender group, are listed in Table 3. The group of newborn patients has a mean risk value of 16.8 per 100,000 persons to develop cancer from radiation exposure, which is about 12% higher compared to that of the 1-year-old group (14.7 per 100,000). Comparing the mean excess lifetime risk values between different gender groups (Table 3), the female patients have a 29% higher risk than male patients. Related to the excess lifetime risk per cancer site (Additional file 1: Table S4), the main critical organs featuring higher risk values for the underlying patient group are the bladder, colon, and kidneys.
Table 3 Age- and gender-dependent mean excess lifetime risk (chances in 100,000 persons)
Figure 5 shows a comparison between the individual absorbed doses to the bladder and excess lifetime risk for the 20 patients. As expected, increased organ absorbed doses lead to a higher risk, independently of the age.
Absorbed doses to the bladder (mGy) and excess lifetime risk (chances in 100,000 persons) as a function of age (months). a Male group with 14 patients aged between 2 and 13 months. b Female group with 6 patients aged between 3 and 20 months
This retrospective study in infants with normal kidney function undergoing 99mTc-MAG3 is the first comprehensive study on biokinetics, dosimetry, and radiation-related risk in a larger group of patients. For retrospective image quantification, age-specific 3D-printed phantoms were manufactured and calibration measurements were performed.
For this study, we chose, for a direct comparison to the published data by Stabin et al. [16] and to the ICRP 128 data [15], to use the Cristy-Eckerman stylized phantoms provided by OLIDNA/EXM [13]. In addition, the effective doses provided by ICRP 128 are still calculated with the ICRP 60 tissue weighting factors [9]. Although new hybrid phantoms for pediatric patients have been developed by the University of Florida group and have been applied by Sgouros et al. in their study on DSMA absorbed doses [14], we believe that, for a retrospective organ dose assessment as performed in this study in a limited number of source organs (kidneys, bladder, whole-body), the accuracy of the dose calculation with OLINDA/EXM is sufficient as a basis for a risk estimate.
Compared to the pediatric patients' 99mTc-MAG3 data presented by Stabin et al. in [16], the absorbed dose coefficients observed in our study are lower for the newborns and higher for the 1-year-old patients. The kidney absorbed dose coefficients were 17% lower for the newborns and 25% higher for the 1-year-old group, and 22% lower for newborns and 76% higher for 1-year-olds in the bladder. Lastly, the remainder dose coefficient was 7% lower for newborns and 63% higher for 1-year-olds. This can be related to the difference of the number of patients (our study has 20 pediatric patients and Stabin et al. study has two pediatric patients in the age range considered) [16].
In an intravenous urography (IVU) study, Almen et al. showed an average absorbed dose per exposure of 0.68 mGy (range 0.48 to 1.10 mGy) for pediatric patients aged between 0 and 1 year [17]. In comparison, the 99mTc-MAG3 scans presented in this study resulted in a 9% lower kidney absorbed dose of 0.62 mGy average over 20 patients (range 0.10 to 2.62 mGy).
Overall, the mean effective dose per patient was less than 1 mSv, showing that the recommendations for administered activities based on the patient weight (EANM Pediatric Dosage Card 2014) keeps the absorbed doses low for pediatric patients undergoing renal 99mTc-MAG3 examinations [4].
The patients' mean excess lifetime risk was 16.5 per 100,000 people (lower boundary 6.6; upper boundary 32.6). Gender-wise, the male patient group showed a mean excess lifetime risk value of 14.7 per 100,000 persons compared to 20.7 per 100,000 for the female group (Table 3). These risk values are the lifetime overall risk for developing cancer.
Based on information from the US National Cancer Institute's Surveillance Epidemiology and End Results (SEER) of the American Cancer Society (database, 2010 to 2012), the risk for developing cancer is 42% in males and 38% in females [18]. The lifetime overall risk in male population is 286,000 times higher than the mean excess lifetime risk of our patients. Compared to the female population, the mean excess lifetime risk of our patients is approximately 182,000 times lower for all cancer types [18]. Similar results are shown for a comparison with the risk database (2012) from the Robert Koch Institute's (RKI) German Centre for Cancer Registry Data [19]. In Germany, the male population showed a lifetime overall risk of 50% for developing cancer [19], which is about 344,000 times higher than the mean excess risk for our male patients. The female population has a lifetime overall risk of 43% for developing cancer, which is approximately 208,000 times higher than the excess risk for our female patients [19]. According to these comparisons, the overall additional risk for our patient group can be considered as very low.
As expected, the excess lifetime risk of our pediatric patient group in comparison with adults undergoing the same exposure was higher for both genders. As an example, we estimated the excess lifetime risk of a 30-year-old adult by separately inserting the organ doses of male patient P18 and female patient P20 at the same exposure year in the RadRAT [10]. While the adult male showed a 61.2% lower risk, the adult female showed a 60.8% lower risk.
Based on a cohort of atomic bomb survivors, the study of Ozasa et al. [20] showed that the risk of a higher mortality caused by late effects of radiation exposure is increased during the lifespan. The rates of cancer deaths increased in proportion to age and dose of radiation [20]. In this cohort, the individuals who were exposed at younger ages presented a higher risk for different cancer sites [20]. In contrast, the risk decreases for those who were exposed at older ages [20,21,22].
Additional file 1: Table S4 presents the mean excess lifetime risk values clustered per cancer site for all patients. Except for the bladder, all other included organs show a maximum risk of 1 per 100,000 persons. The critical organs (highest risk values) were the bladder, colon, thyroid, lungs, kidneys, and bone marrow. In comparison, Ozasa et al. [20] presented similar results: besides the organs stated above, the breast (female), esophagus, gall bladder, and liver were reported as organs with the highest excess risk per cancer site. Conversely, the rectum, uterus (female), prostate (male), and kidneys (parenchyma) presented no significant excess risk [20].
The highest mean organ absorbed doses were observed in the bladder with values above 4 mGy. Thus, the estimated risk was higher for all ages and genders (Fig. 5). For male patients, the bladder absorbed doses were 28% lower than for female patients with the risk accordingly reduced by 25%.
Compared to our patient risk data, the lifetime overall risk for both genders of the general population for developing bladder cancer is above the mean excess lifetime risk, with values of approximately 33,000 (SEER) and 21,000 (RKI) time higher for males and 7293 (SEER) and 5118 (RKI) time higher for females [18, 19]. Bladder voiding influenced the risk; in comparison to the patient group without bladder voiding during the examination, the mean excess lifetime risk values of the patient group with voiding were 58% lower.
Our results show a tendency towards higher excess lifetime risks for female patients compared to males (Table 3) and gender-specific distinctions when comparing the organs' dose-risks between both genders (Additional file 1: Table S4) [6, 7, 20]. As an example, the excess lifetime risk values for the kidneys were higher for males than for females [20].
A one-way ANOVA test was performed to examine significant differences between the patient groups. The input data were the results for absorbed doses (mGy), excess lifetime risk (chances in 100,000), and excess lifetime risk per cancer site (chances in 100,000) for both age groups (newborns and 1-year-old groups) and both genders (male and female). According to the tests, no significant differences were found (p > 0.05). The p values for the age groups were 0.2 for kidneys absorbed dose (mGy), 0.9 for bladder absorbed dose (mGy), and 0.8 for excess lifetime risk (chances in 100,000). The p value for the gender groups was 0.4 for excess lifetime risk (chances in 100,000).
There are some shortcomings concerning the study; however, as it is a retrospective study with images taken at suboptimal time points for dosimetry, the doses reported might be overestimated due to the approximation of a physical decay after the last time point. The error associated with the calibration and the subsequent patient-specific correction adds to the uncertainty of the dose assessment. In this age group, however, the variability concerning morphology is rather low. For an estimate of the effective doses according to ICRP 103, the risk factors could not be applied as the data of the underlying voxel-based ICRP phantom are yet to be published [5].
Nevertheless, a risk-adapted, TIAC-based approach applied for organ-specific absorbed dose calculations, instead of reporting effective dose values obtained by multiplying the administered activities with constant values taken from the ICRP tables such as ICRP 128 [15], might lead to improvements of future recommendations for pediatric dosages in nuclear medicine diagnostics.
In this study, we retrospectively derived new data on biokinetics and dosimetry for infants with normal kidney function after undergoing renal 99mTc-MAG3 scans. In addition, we analyzed the associated age- and gender-specific excess lifetime risk due to ionizing radiation. The absorbed and effective doses were low when using the EANM pediatric dosage card for calculating the injected activities. The radiation-associated stochastic risk increases with the organ doses taking age- and gender-specific influences into account. In comparison with adults, the pediatric patient data show a slightly higher radiation-related risk (excess lifetime risk) for the same absorbed doses. Overall, however, the lifetime radiation risk associated with the 99mTc-MAG3 scans is very low when compared to the general population's risk for developing cancer.
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The authors would like to thank the department of radiology for the kind permission to use their ultrasound images and C. Lapa for helpful discussions of the 99mTc-MAG3 scan results.
Part of the work was financed by a scholarship of CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, development agency of the Brazilian Federal Government.
This publication was funded by the German Research Foundation (DFG) and the University of Wuerzburg in the funding Open-Access Publishing Programme.
Department of Nuclear Medicine, University Hospital Würzburg, Oberdürrbacher Str. 6, 97080, Würzburg, Germany
J. Soares Machado
, J. Tran-Gia
, S. Schlögl
, A. K. Buck
& M. Lassmann
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JTG and JSM contributed with the phantom design and printing. JSM, JTG, and SS performed the phantom experiments. JSM, JTG, and ML performed the data acquisition, data analysis, and calculations. JSM, ML, JTG, and AKB contributed to drafting the manuscript. All authors revised and approved the manuscript.
Correspondence to J. Soares Machado.
As this study is a retrospective analysis acquired within clinical routine, our local ethics committee waived the need for further approval.
Additional file 1: Table S1.
Patient-specific organ sizes classified per age groups (newborns and 1-year-olds). Table S2. Demographics clustered by age groups with patients' information of age, gender, weight, body size, and injected activity. Table S3. The values for Patient organ-specific mean absorbed dose coefficients. Table S4. Mean organ-specific absorbed doses and respective estimated excess lifetime risk per cancer site (chances in 100,000 persons) for newborns (1.6–11.0 months) and 1-year-olds (13.0–20.0 months) clustered per gender. (PDF 1213 kb)
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Inertial Tseng's extragradient method for solving variational inequality problems of pseudo-monotone and non-Lipschitz operators
Gang Cai 1,, , Yekini Shehu 2, and Olaniyi S. Iyiola 3,
School of Mathematics Science, Chongqing Normal University, Chongqing 401331, China
Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China
Department of Mathematics and Physical Sciences, , California University of Pennsylvania, PA, USA
* Corresponding author: G. Cai
Received September 2020 Revised January 2021 Early access May 2021
Fund Project: The first author is supported by the NSF of China (Grant No. 11771063), the Natural Science Foundation of Chongqing(cstc2020jcyj-msxmX0455), Science and Technology Project of Chongqing Education Committee (Grant No. KJZD-K201900504) and the Program of Chongqing Innovation Research Group Project in University (Grant no. CXQT19018)
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In this paper, we propose a new inertial Tseng's extragradient iterative algorithm for solving variational inequality problems of pseudo-monotone and non-Lipschitz operator in real Hilbert spaces. We prove that the sequence generated by proposed algorithm converges strongly to an element of solutions of variational inequality problem under some suitable assumptions imposed on the parameters. Finally, we give some numerical experiments for supporting our main results. The main results obtained in this paper extend and improve some related works in the literature.
Keywords: Tseng's extragradient method, variational inequality, pseudo-monotone operators, strong convergence.
Mathematics Subject Classification: Primary: 47H09, 47H10; Secondary: 47J20, 65K15.
Citation: Gang Cai, Yekini Shehu, Olaniyi S. Iyiola. Inertial Tseng's extragradient method for solving variational inequality problems of pseudo-monotone and non-Lipschitz operators. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021095
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Figure 1. Example 1: $ k = 20 $, $ N = 10 $
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Figure 9. Example 1: Different $ \gamma $ with $ (N, k) = (20, 10) $
Figure 10. Example 1: Different $ \gamma $ with $ (N, k) = (20, 21) $
Figure 17. Example 1: Different $ \mu $ with $ (N, k) = (20, 10) $
Figure 25. Example 2: Case I
Figure 26. Example 2: Case II
Figure 27. Example 2: Case III
Figure 28. Example 2: Case IV
Figure 29. Example 2: Case V
Figure 30. Example 2: Case VI
Figure 31. Example 2: Case I with different $ \gamma $
Figure 32. Example 2: Case II with different $ \gamma $
Figure 33. Example 2: Case III with different $ \gamma $
Figure 34. Example 2: Case IV with different $ \gamma $
Figure 35. Example 2: Case V with different $ \gamma $
Figure 36. Example 2: Case VI with different $ \gamma $
Figure 37. Example 2: Case I with different $ \mu $
Figure 38. Example 2: Case II with different $ \mu $
Figure 39. Example 2: Case III with different $ \mu $
Figure 40. Example 2: Case IV with different $ \mu $
Figure 41. Example 2: Case V with different $ \mu $
Figure 42. Example 2: Case VI with different $ \mu $
Figure 43. The value of error versus the iteration numbers for Example 3
Table 1. Methods Parameters Choice for Comparison
Proposed Alg. $ \epsilon_n = \frac{1}{n^2} $ $ \theta = 0.1 $ $ l=0.001 $ $ \beta_n = \frac{1}{n} $
$ \gamma = 0.99 $ $ \mu = 0.99 $
Thong Alg. (1) $ \epsilon_n = \frac{1}{(n + 1)^2} $ $ \theta = 0.1 $ $ \beta_n = \frac{1}{n + 1} $ $ \lambda = \frac{1}{1.01L} $
Thong Alg. (2) $ l=0.001 $ $ \gamma = 0.99 $ $ \mu = 0.99 $
Thong Alg. (3) $ \alpha_n = \frac{1}{n + 1} $ $ l=0.001 $ $ \gamma = 0.99 $ $ \mu = 0.99 $
Gibali Alg. $ \alpha_n = \frac{1}{n + 1} $ $ l=0.001 $ $ \gamma = 0.99 $ $ \mu = 0.99 $
Table 2. Example 1: Comparison among methods with different values of $ N $ and $ k $
$ N=10 $ $ N=20 $ $ N=30 $ $ N=40 $
$ k=20 $ Iter. Time Iter. Time Iter. Time Iter. Time
Proposed Alg. 3 1.3843 3 1.7672 3 1.7564 4 2.2017
Thong Alg. (1) 76 1.2902 139 2.7111 111 2.1715 232 37.7743
Thong Alg. (2) 2136 36.6812 1561 30.7776 1370 31.8672 1160 4.0453
Gibali Alg. 150 12.0085 235 20.3243 319 41.0421 315 4.2520
Thong Alg. (1) 72 1.1548 142 2.6436 136 2.888 207 4.4416
Thong Alg. (2) 1771 30.2921 1325 28.4023 1132 28.6053 920 26.5714
Thong Alg. (3) 101 1.5058 90 1.4923 156 2.9515 162 3.6149
Gibali Alg. 203 17.1568 255 30.849 282 31.6244 303 35.2953
Table 3. Example 1 Comparison: Proposed Alg. with different values $ \gamma $
$ (N, k) $ $ \gamma = 0.1 $ $ \gamma = 0.5 $ $ \gamma = 0.7 $ $ \gamma = 0.99 $
$ (20, 10) $ No. of Iterations 3 3 3 3
CPU (Time) 1.6337 1.4830 1.4773 1.3843
CPU (Time) 1.8664 0.85257 1.7597 1.7564
Table 4. Example 1 Comparison: Proposed Alg. with different values $ \mu $
$ (N, k) $ $ \mu = 0.1 $ $ \mu = 0.5 $ $ \mu = 0.7 $ $ \mu = 0.99 $
Proposed Alg. $ \epsilon_n = \frac{1}{(n + 1)^2} $ $ \theta = 0.5 $ $ l=0.01 $ $ \beta_n = \frac{1}{n + 1} $ $ \gamma = 0.99 $ $ \mu = 0.99 $
Gibali Alg. $ \alpha_n = \frac{1}{1 + n} $ $ l=0.01 $ $ \gamma = 0.99 $ $ \mu = 0.99 $
Table 6. Example 2: Prop. Alg. vs Gibali Alg. (Unaccel. Alg.)
No. of Iterations CPU Time
Prop. Alg. Gibali Alg. Prop. Alg. Gibali Alg.
Case I 17 1712 0.001243 0.1244
Case II 17 1708 0.001518 0.1248
Case III 17 1713 0.001261 0.1276
Case IV 17 1729 0.001202 0.1297
Case V 17 1715 0.001272 0.1258
Case VI 18 1835 0.001339 0.1564
$ \mu = 0.1 $ $ \mu = 0.5 $ $ \mu = 0.7 $ $ \mu = 0.99 $
Case I No. of Iterations 17 17 17 17
CPU (Time) 0.0011992 0.0012179 0.0013264 0.0012430
Case II No. of Iterations 17 17 17 17
Case III No. of Iterations 17 17 17 17
Case IV No. of Iterations 17 17 17 17
Case V No. of Iterations 17 17 17 17
Case VI No. of Iterations 18 18 18 18
$ \gamma = 0.1 $ $ \gamma = 0.5 $ $ \gamma = 0.7 $ $ \gamma = 0.99 $
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EPJ Nonlinear Biomedical Physics
Does changing Fitts' index of difficulty evoke transitions in movement dynamics?
Raoul Huys1,
Hester Knol1,
Rita Sleimen-Malkoun1,2,
Jean-Jacques Temprado2 &
Viktor K. Jirsa1
EPJ Nonlinear Biomedical Physics volume 3, Article number: 8 (2015) Cite this article
The inverse relationship between movement speed and accuracy in goal-directed aiming is mostly investigated using the classic Fitts' paradigm. According to Fitts' law, movement time scales linearly with a single quantity, the index of difficulty (ID), which quantifies task difficulty through the quotient of target width and distance. Fitts' law remains silent, however, on how ID affects the dynamic and kinematic patterns (i.e., perceptual-motor system's organization) in goal-directed aiming, a question that is still partially answered only.
Therefore, we here investigated the Fitts' task performed in a discrete as well as a cyclic task under seven IDs obtained either by scaling target width under constant amplitude or by scaling target distance under constant target width.
Under all experimental conditions Fitts' law approximately held. However, qualitative and quantitative dynamic as well as kinematic differences for a given ID were found in how the different task variants were performed. That is, while ID predicted movement time, its value in predicting movement organization appeared to be limited.
We conclude that a complete description of Fitts' law has yet to be achieved and speculate that the pertinence of the index of difficulty in studying the dynamics underlying goal-directed aiming may have to be reconsidered.
More than 60 years ago, Paul Fitts initiated a novel paradigm when he asked participants to cyclically move a stylus between two targets characterized by a width W and separated by a distance D [1]. By systematically varying D and W, he found that movement time MT related linearly to the ratio of D and W, MT = a + b × log2 (2D/W). This linear relation, now known as Fitts' law, was next found to hold also for discrete aiming [2]. In Fitts' law, the index of difficulty ID = log2 (2D/W) quantifies task difficulty as an informational quantity in bits [2, 3]. Over the years, several authors have voiced criticism as to the functional form of the scaling of MT with ID as well as on whether the ID as formulated by Fitts is the most appropriate one [4–8]. Regardless, few will debate that as a first approximation, MT scales linearly with the ID, which has been repeatedly shown in discrete and cyclical performances alike [9–13].
Fitts' law, however, is silent on how the movements' organization changes as the ID is scaled. In addressing this issue, one prominent class of models (sub-movement models) focuses on the presence and features of primary and corrective secondary (and sometimes more) sub-movements as a function of W and D [5, 6, 14]. The features associated with these movements, and those that deemed most relevant, are typically scalar variables (duration, [average] velocity, proportion of corrective movements, etc.).
Another prominent class of models (dynamical models) seeks to understand how the movements' kinematic and underlying dynamics change when D and/or W are systematically changed. In this case, the focus is geared towards trajectories in the Hooke's plane and/or phase space [11, 15, 16], and the identification of the dynamics as observable in the latter [17, 18]. In that regard, deterministic, autonomous, and time-continuous systems are unambiguously described by their flow in phase (or state) space (or vector field), i.e., the space spanned by the system's state variables [19]. For movements along a single physical direction, as in a (sliding) Fitts' task, it is commonplace to use the movement's position and its time-derivative velocity as the state variables [11, 20, 21] (but see [22, 23] for a critical discussion). The attractors that may live in such two-dimensional spaces are limited to (different kinds of) fixed points (i.e., points where velocity and acceleration are zero) and limit cycles (nonlinear closed orbits), which are associated with discrete and rhythmic movements, respectively [24, 25]. Changing the system's parameter(s) modifies the phase flow, and may evoke a bifurcation (i.e., a change in the system's solution(s)). If so, the parameter is referred to as a control or bifurcation parameter. Grounded in this latter perspective, the present study aimed to identify the dynamics, and further characterize the movements' kinematics, when changing the ID by varying W and D independently, in both discrete and cyclic versions of Fitts' task.
In that regard, for the cyclic Fitts' task version, it has been shown that gradually changing ID induces a gradual adjustment of the movement kinematics [11, 15], albeit less so when changing D than when changing W. In the latter case, the gradual adjustment may evoke an abrupt transition in the dynamics underlying the performance [17, 26] via a homoclinic bifurcation (i.e., from a limit cyclic dynamics to (two) fixed points, each having one stable and one unstable direction [i.e., saddles]); [17]). As hinted at, changing ID via target width W and distance D affects the aiming movement's velocity profiles differently [7, 11, 27]: increasing D mainly stretches the (bell-shaped) velocity profile, while increasing W renders it skewed (the deceleration phase lengthens relative to the acceleration phase). Thus, it is not clear if the scaling of ID per se induced the bifurcation in [17] or if effectively the manipulation of W did so. For discrete task performance, the pattern of kinematic changes as a function of ID (including the D and W differentiation) yields some similarities with those observed in the cyclic task version [12, 28]. The discrete and cyclic task, however, are fundamentally different in that, in the former but not the latter, movement velocity and acceleration must be zero before initiating the movement and upon ending it [10]. In this case, it seems self-evident to assume the existence of a fixed-point dynamics in the discrete task version. Various fixed-point dynamics scenarios, next to the above-mentioned connected saddles, are realizable, however. For instance, Schöner [25] has proposed that a fixed point (at location A) vanishes so as to temporally give way to a limit cycle—causing the movement, after which the limit cycle vanishes and the fixed point (at location B) re-occurs. Alternatively, a fixed point may be driven through phase space, more or less continuously changing the phase flow so that the system is 'dragged behind it" [29]. This scenario constitutes an interpretation of equilibrium point models [30] in the framework of dynamical systems [31]. In this case, the trajectories in the phase space can be expected to be 'wiggly' and reveal little local convergence (i.e., overlaid trajectories can be expected to have a similar thickness throughout). Yet another possible realization involves a competition in which an active fixed point at location A vanishes while simultaneously another at location B comes into existence [29]. Indeed, while the discrete Fitts' task must involve fixed points, what remains unknown is: i) whether they are similar under D and W induced ID scaling, ii) if ID changes evoke a transition between mechanisms, and iii) if the fixed points assumed in the discrete task are the same as those found for the (W induced ID scaling) cyclic task. In fact, for the cyclic task, it is not known either if a transition occurs if ID is altered via target distance D. Teasing apart the contributions of D and W to the scaling of the ID will allow us to investigate whether ID, which plays a primordial role in the Fitts' paradigm, acts as bifurcation parameter or if effectively either D or W or both do so.
Based on the above reasoning, in the discrete task we predicted to observe fixed-point dynamics under all conditions. Under the distance manipulation, for low ID, the (average) velocity can be expected to be very low. We therefore expected to find indications for the existence of a driven fixed-point scenario. For the cyclic task, in line with previous findings we predicted to observe a bifurcation from a limit cycle dynamics to a fixed-point dynamics with decreasing target width [17]. Finally, we expected to find evidence either for a limit cycle dynamics or the driven fixed-point at small target distances and time-invariant fixed points at large target distances.
We examined our predictions primarily by investigating the underlying Fitts' task performance under discrete and rhythmic task versions and identifying bifurcations (if existent). In addition, to further characterize task performance, we also extracted various kinematic features of the movements. Thereto, we designed a Fitts' task that was performed in the discrete and cyclic mode, and in which task difficulty was scaled via D and W separately in different sessions. We found that while ID predicted movement time, it did not uniquely predict the dynamics and kinematics associated with the task performances.
What participants do in a Fitts' task typically (slightly) deviates from the imposed task constraints. That is, the produced end-point variability and movement amplitude do not map one-to-one onto the task-defined target width W and distance D. As commonly done, we therefore computed the effective amplitude and the effective target width (see Methods ) and calculated the effective ID as ID e = log2(2A e /W e ). We report our results based on the ID e .
As a first step in our analysis, we examined how MT changed under the different experimental factors. MT was lower in the discrete task (mean ± SD = 0.75 ± 0.04) than in the cyclic task (mean ± SD = 0.80 ± 0.04; F(1,12) = 10.270, p < .01, η 2 = .461) and higher in the distance manipulation (mean ± SD = 0.82 ± 0.05) than in width manipulation (mean ± SD = 0.73 ± 0.03; F(1,12) = 23.975, p < .0001, η 2 = .666). The Task × Manipulation interaction (F(1,12) = 22.203, p < .005, η 2 = .649) showed, however, that the effect of Task only held for the distance manipulation (Fig. 1a). As expected, MT increased with ID e (F(1.449,17.383) = 148.827, p < .0001, η 2 = .925), but did so in a manner that interacted with Task (F(2.903,34.842) = 18.228, p < .0001, η 2 = .603; Fig. 1b), and Task and Manipulation (F(2.770,33.236) = 3.376, p < .05, η 2 = .220; Fig. 1c, d). For each task and manipulation combination, we linearly regressed MT against effective ID (for each participant), and investigated the regressions' slopes with a 2 (Task) × 2 (Manipulation) ANOVA. The average R 2 equalled .91 (±0.08). The slopes in the discrete task (mean ± SD = 0.19 ± 0.01) were smaller than in the cyclic task (mean ± SD = 0.24 ± 0.02; F(1,12) = 56.856, p < .0001, η 2 = .826), and those in distance conditions (mean ± SD = 0.18 ± 0.01) were smaller than those in the width conditions (mean ± SD = 0.24 ± 0.02; F(1,12) = 75.028, p < .0001, η 2 = .862). The significant Task × Manipulation interaction (F(1,12) = 6.929, p < .05, η 2 = .366) indicated that the effect of Manipulation was stronger in the cyclic task (mean ± SD = 0.20 ± 0.02 versus 0.28 ± 0.02 for distance and width scaling, respectively) than in the discrete task (mean ± SD = 0.16 ± 0.01 versus 0.19 ± 0.01 for distance and width scaling, respectively). Thus, both the task version (discrete, cyclic) and how ID was varied (via D or W) altered the rate of MT increase with ID e . At the same time, as a first approximation, the linear relation predicted by Fitts' law held in all Task × Manipulation conditions.
Average movement time. a MT is lower in the discrete than in the cyclic task in the distance but not in the width conditions. b At low ID e MT is higher in the discrete task than in the cyclic task, while at higher ID e this is inversed. c This effect owes largely to the width conditions. d For distance, the cyclic MT equals the discrete MT at low ID e but is (again) higher at high ID e
To investigate the dynamics associated with the movements in the various conditions, we computed the vector fields, and statistically analysed the maximal angle θ max [17]. In that regard, each vector k in a vector field has (up to) eight neighbouring vectors whose direction relative to vector k can be represented by an angle. θ max represents the maximum of the angles of vector k with its neighbouring vectors. For θ max > 90°, we consider that the movements in the corresponding trial pertained to a fixed-point dynamics. In almost all conditions, except for the cyclic-width condition at a low ID e and for a few trials in the cyclic-distance condition, indications for the existence of fixed points in the target regions were found (see Fig. 2 and Table 1). This observation was statistically corroborated by the ANOVA on θ max (Additional file 1), which indicated that θ max was higher in the discrete task (mean ± SD = 171° ± 0.6) than in the cyclic one (mean ± SD = 130° ± 3.5; F(1,12) = 123.893, p < .0001, η 2 = .912), and higher in the distance conditions (mean ± SD = 160° ± 2.3) relative to those of width (mean ± SD = 142° ± 1.4; F(1,12) = 99.556, p < .0001, η 2 = .892). As expected, θ max became larger as ID e increased (F(3.765,45.180) = 28.289, p < .0001, η 2 = .702). The distance versus width effect, however, was only observed for the cyclic task (Task × Manipulation, F(1,12) = 54.782, p < .0001, η 2 = .820). In addition, the increase of θ max with ID e occurred primarily in the width (Manipulation × ID e , F(3.010,36.126) = 18.737, p < .0001, η 2 = .610) and in the cyclic conditions (Task × ID e , F(3.552,42.267) = 43.048, p < .0001, η 2 = .782). Finally, the Task × Manipulation × ID e interaction (F(3.022,36.261) = 9.456, p < .0001, η 2 = .441) showed that θ max was high (>130°) and varied little only with ID e in all task–manipulation combinations except for that of cyclic–width. As can be seen in Table 1 (see also Additional file 2), the number of participants in which fixed points were identified (in correspondence with the criterion outlined above) always equalled the total number of participant (i.e., n = 13) in all discrete task conditions. In the cyclic-discrete task conditions, fixed points were always found for the higher ID e . However, all but one (2 ID e ) or 3 participants (1 ID e ) did not show a fixed-point dynamics at low ID e . Two of the participants that did not adhere to a fixed-point dynamics at ID e = 4.9 were 'stand-alone' incidences. In one participant no fixed points were found for the first three ID e' s, that is, this participant's behaviour in all likelihood showed a true transition. In the cyclic-width task conditions, all participants showed a transition from a limit cycle dynamics to a fixed points dynamics with increasing ID e , albeit at different ID e . Thus, across the board (with a single exception), a limit cycle dynamics was operative in the cyclic–width condition at ID e up to about 5.6, whereas a fixed-point dynamics governed all other conditions.
Angle diagrams. The upper versus lower two rows represent angle diagrams from a single participant of the discrete and cyclic mode, respectively. For both row pairs, the upper panels show the distance conditions while the lower ones display the width conditions. ID e increases form left to right. The colour coding (at the right) represents the maximal angle between neighbouring vectors. Red areas indicate locally opposing angles, implying the existence of a fixed point. Their absence suggests the existence of a limit cycle
Table 1 Number of participants for whom fixed points were identified per condition
Topologically, the vector fields of all conditions in which a fixed-point dynamics was identified were indistinguishable. For cyclic Fitts' task performance the gradual up-scaling of task difficulty, in particular through manipulation of target width, gradually increases the degree of the system's nonlinearity [11]. Fig. 2 suggests that this was also the case for the discrete task performances (see also [12]). The degree of nonlinearity does not uniquely map onto a system's topological organization. Its evolution as a function of ID e , however, informs about the quantitative changes in the movement dynamics. We summarize these changes via R AT/MT , which quantifies the degree of symmetry of the movement velocity profile. R AT/MT was lower in the discrete task (mean ± SE = 0.35 ± 0.02) than in the cyclic task (mean ± SD = 0.39 ± 0.04; F(1,12) = 28.982, p < .0001, η 2 = .707), but primarily so in the width conditions (Task × Manipulation (F(1,12) = 53.321, p < .0001, η 2 = .816; Fig. 3a). In line therewith, R AT/MT was lower in the distance manipulation (mean ± SD = 0.34 ± 0.02) than in that of the width manipulation (mean ± SD = 0.39 ± 0.01; F(1,12) = 36.564, p < .0001, η 2 = .753). At first glance, this latter finding appears to contradict established knowledge [7, 11, 12, 27]; it is important to recall therefore, that in the present distance manipulation, the fixed target width was always very small (0.31 cm). As expected, R AT/MT decreased as ID e increased (F(2.205,26.458) = 136.922, p < .0001, η 2 = .919). This decrease was stronger for discrete than for the cyclic conditions, and at high ID e the task mode differences vanished (Task × ID e ; F(2.908,34.899) = 14.305, p < .0001, η 2 = .544; Fig. 3b). Similarly, the interaction between Manipulation and ID e (F(2.068,24.822) = 42.956, p < .0001, η 2 = .782) indicated that the manipulation differences vanished with increasing ID e . Finally, the Task × Manipulation × ID e interaction (F(3.625,43.504) = 7.302, p < .0001, η 2 = .378) indicated that R AT/MT decreased faster in the cyclic than in the discrete task mode with increasing ID e for the width conditions but not so for the distance conditions (Fig. 3c, d). Thus, the velocity profiles became more skewed with increasing ID e (i.e., the nonlinearity increased). Whereas this effect was similar for both tasks in the distance manipulation, for the width manipulation, the profiles were more symmetric at low ID e in the cyclic task than in the discrete task. This latter difference vanished as in both task modes the profiles became more asymmetric (i.e., the movements became more nonlinear).
Ratio AT/MT. a The larger R AT/MT in the width condition relative to the distance conditions is most pronounced in the cyclic task mode. b At low ID e R AT/MT is lower in the discrete task than in the cyclic task, while at higher ID e R AT/MT is similar. c This interaction is due to the width conditions. d In the distance conditions R AT/MT is similar and decreasing with ID e for both task modes
As indicated in the Background section, we expected that under the distance manipulation at low ID a driven fixed point would govern the movements. In such a dynamic system, the phase space trajectories can be expected to be wiggly (and show little local convergence), and thus to be variable from one trial to the next. We therefore examined the trajectory variability using PCA (see also Additional file 3), and subjected the first eigenvalue to an ANOVA. Please note that the more variable the trajectories are, the smaller the first eigenvalue is. In fact, the variance that is not accounted for by the first principal component is orthogonal to it so that the first eigenvalue can be interpreted as reflecting the degree of convergence towards the trajectory associated with the first principal component. Neither the effect of Task nor that of ID e was significant (p > .1 and p > .05, respectively). In contrast, the trajectories were more variable (i.e., the 1st eigenvalue smaller) in the distance conditions (mean ± SE = .76 ± 0.01) than in those of width (mean ± SE = .84 ± 0.01; F(1,12) = 140.683, p < .0001, η 2 = .921). The Task × Manipulation interaction (F(1,12) = 17.132, p < .0005, η 2 = .588) indicated that the difference between task manipulations was larger in the cyclic task mode than in the discrete one. The significant interaction between Task and ID e (F(3.374,40.486) = 26.453, p < .0001, η 2 = .688), Manipulation and ID e (F(2.713,32.559) = 43.106, p < .0001, η 2 = .782), and Task, Manipulation and ID e (F(3.061,36.734) = 7.385, p < .005, η 2 = .381) are displayed in Fig. 4. In combination, these interactions showed that at low ID e , the trajectory variability in the discrete task was larger than that of the cyclic task, which inversed at high ID e (Fig. 4a). Further, the trajectories were most variable at low ID e in the distance manipulation, and the variability decreased as ID e increased (Fig. 4b), The inverse was observed for the width conditions. The effect of increasing ID via target distance was comparable for both tasks (Fig. 4c, d). Decreasing target width, however, hardly affected the trajectory variability in the discrete task, but it led to an increased variability in the cyclic task. For the latter, at low ID e , that is, when a limit cycle dynamics was invariantly present, the trajectories were the least variable in the entire dataset.
PCA: 1st eigenvalue. a Trajectory variability decreased with increasing ID e for the discrete task mode whereas it increased in the cyclic mode (from the 3rd ID e onwards). b Similarly, trajectory variability in the distance conditions decreased with increasing ID e whereas that in the width conditions increased. c, d This effect, however, was most pronounced for the cyclic task. In the discrete task trajectory variability in the width condition stayed about the same for all ID e
As for the movement organization under the different conditions, our results revealed identical topological organizations (fixed-point dynamics) under all but the cyclic-width task version at low ID e (and for one participant in the cyclic-distance task at low ID e ). At the same time, however, the discrete and cyclic task modes are set apart in terms of the degree of symmetry of movement velocities, and particularly, trajectory variability. By hypothesis, this distinction may imply that in the cyclic task mode, the dynamical organization (i.e., the phase flow) remains invariant throughout the entire trial, independent of whether a fixed-point or limit cycle dynamics is adhered to. For the discrete task mode, this will actually be the same. In this case, however, prior to and following each single aiming the movement-task organization will be (has to be) 'dismantled' (to return to the home position) and assembled for the execution of the next trial. Consequently, additional performance variability can be expected for the discrete task mode relative to the cyclic one as the trial-to-trial re-establishment of the movement organization adds a source of variability for the former task mode relative to the latter. In terms of Saltzman and Munhall's [32] wording, additional variability in repeated discrete aiming relative to continuous cyclic aiming is introduced in terms of parameter dynamics. We tested this hypothesis by focussing on the variability (through the coefficient of variation) of the variable central to Fitts' law, that is movement time. Consistent with the hypothesis, the movement time's coefficient of variation (CV MT ) was larger in the discrete task (mean ± SD = 1.18 ± 0.01) than in the cyclic one (mean ± SD = 1.12 ± 0.01; F(1,12) = 63.735, p < .0001, η 2 = .842), and also larger in the distance conditions (mean ± SD = 1.16 ± 0.01) than in the width conditions (mean ± SD = 1.13 ± 0.01; F(1,12) = 43.829, p < .0001, η 2 = .785). This latter effect was stronger in the cyclic than in the discrete task (Task × Manipulation; F(1,12) = 5.271, p < .05, η 2 = .305; Fig. 5a). Further, CV MT decreased with increasing ID e (F(4.306,51.667) = 6.812, p < .0001, η 2 = .362); this effect, however, was confined to the discrete task (Task × ID e ; F(3.201,38.417) = 7.810, p < .0001, η 2 = .394; Fig. 5b). Finally, the interaction between Manipulation and ID e (F(3.526,42.317) = 55.529, p < .0001, η 2 = .822) indicated that at low ID e , the CV MT under the distance manipulation, which decreased strongly as ID e increased, was almost twice as high as that under the width manipulation, which increased moderately as ID e increased (Fig. 5c).
CV movement time. a The distance-width CV MT difference is moderately stronger in the cyclic than in the discrete task mode. b In the discrete task CV MT increases with ID e whereas in the cyclic task it decreases. c In the distance conditions CV MT decreases with increasing ID e whereas in the width conditions it increases
The pattern of intra-participant R AT/MT variability (coefficient of variation) strongly resembled that of CV MT , and is reported in Additional file 4.
In the present study, we investigated both the dynamics and kinematics underlying Fitts' task performance during discrete and cyclic task modes when ID was varied through distance and width independently. Most importantly regarding the dynamics, consistent with previous observations [17], in the cyclic task setting a transition from a limit cycle to a fixed-point dynamics occurred when scaling the ID via target width. In contrast to the expectation voiced in that study, a similar transition was not observed here when varying ID via the distance separating the targets. Indeed, varying target distance did not affect the observed movement's topology, at least not in the presently studied range. In this case, fixed points were found throughout the entire ID range (except for a few cases, mainly one participant). This result suggests that, in general, the index of difficulty per se does not uniquely dictate the dynamical organization of rapid aiming movements, thereby disqualifying as a bifurcation parameter. That function, however, appeared to be fulfilled by target width, even though only so for the cyclic task mode. Indeed, varying target distance did not affect the movement's topology observed, at least not in the presently studied range. It cannot be ruled out, however, that the smallness of the target (0.31 cm) under the present distance manipulations hindered the occurrence of a limit cycle dynamics (or any other; see below) at specific target distances.
Furthermore, trajectory variability changed in opposing direction with decreasing target width for the discrete and cyclic task. This observation seems hard to reconcile with the idea that the kinematic re-organization as a function of (varying) target width for both task modes is the same. That is, even if varying target width drives the sensorimotor system through a bifurcation when operating in the discrete task mode, it is unlikely that the bifurcation type matches the one observed in the cyclic task. Confirmation (or not) of this hypothesis, however, will require further investigation.
Concerning the effects on the movement kinematics, we found that how ID was varied (i.e., via D or W) as well as the nature of the task (i.e., discrete or cyclic) had pronounced effects on the movement kinematics investigated. In that regard, it is often stated that scaling target distance versus its width 'simply' stretches the velocity profile or skews it, respectively [7, 11, 12]. Here, we found that although increasing the ID by scaling target width reduced the degree asymmetry of the movement's velocity profile more than under the distance scaling, the latter also reduced it (Fig. 3b). Again, this may to some extent be due to the smallness of the target under the present distance scaling. By comparison, we here used a target width smaller than the one used in [12] and [11] under a (modestly; relative to [10]) larger distance scaling. A closer look at these studies, however, shows that while, indeed, the degree of asymmetry increases markedly more under the width than distance scaling, categorically setting apart the effects of these variations in terms of skewing versus stretching the velocity profiles appears a simplification that does not do justice to the observations.
Further, the degree of asymmetry increased with decreasing target width. In that case, at low ID, the discrete movements were more asymmetric than the cyclic ones, while at high ID, this difference vanished as for both task modes the asymmetry increased. The initial difference, as well as the evolution, can be understood when considering that discrete movements always contain a zero velocity and acceleration start and end points, which emerge at higher ID only for cyclic movements. Indeed, under the distance scaling, the symmetry reduction was similar for the discrete and cyclic task mode (Fig. 3d), which was always governed by a fixed-point dynamics.
The effects of varying target distance versus width were not limited to the movement's velocity symmetry. Specifically, increasing target distance (under constant target width) reduced the movement's trajectory variability irrespective of task mode. As peak velocity also increases with increasing distance (Additional file 5), this effect contrasts the signal-dependent noise perspective [33]. Reasoning from a dynamical perspective, and assuming noise to be approximately constant, a reduction in trajectory variability may come about by an increased (more or less local) convergence of the phase flows underlying the movements (i.e., an increased tendency of the vectors pointing towards a manifold in the state space) and/or by an increased contribution of the deterministic dynamics relative to the stochastic dynamics (i.e., an increased length of the vectors). Under this perspective, increasing the ID via target distance is likely to result in an increased flow convergence for both task modes, which indeed occurs (reduced trajectory variability; see Fig. 2 and Additional file 3). The marked reduction in inter-aiming movement time variability (CV MT ; Fig. 5c) is consistent therewith. In contrast, varying ID via target width did not (globally) affect the trajectory variability in the discrete task mode, but resulted in a marked variability increase in the cyclic task mode: In these conditions, at low ID and governed by a limit cycle dynamics, trajectory variability was the lowest observed but it noticeably (but rather gradually) increased as the non-linearity increased (R AT/MT ; Fig. 3d) and a fixed point dynamics was created (Fig. 2).
We found support for the hypothesis that the variability across aiming movements is bigger in the discrete task mode than in the cyclic one. This might result from the dynamical organization (i.e., the phase flow) being more or less invariant throughout the entire trial, depending on the type of task: in the discrete task mode, the perceptual-motor system prepares the movement for each upcoming action (leading to more variability) while across repeated aiming movements (in the cyclic task mode), the dynamical organization is more invariant (less variable). To further investigate this interpretation, we calculated the Pearson correlation for the movement time variability (CV MT ) between all Task, Manipulation, and low and high ID condition pairs (NB: in order to obtain more data points, the lowest two ID conditions and the highest two ID conditions were taken together to form a 'low ID' and 'high ID' category). Our reasoning was that, if tasks share specific processes relevant for their (timed) behaviour, their variability ought to be correlated and, conversely, if not, no correlation is to be expected [34]. Accordingly, we expected that all correlations between pairs involving a discrete and cyclic condition would be non-significant, and that for the cyclic task the low ID – width condition would show no significant correlation with any of the others as the dynamics in the former condition (limit cycle) differed from the latter (fixed points).
As it can be appreciated in Fig. 6, these expectations turned out to be correct. Further, for the rhythmic task, all pairs except for those involving the low ID – width condition turned out significant, which fits the observation of similar dynamics (fixed points) and the proposition of being governed by an invariant dynamics across repetitive aiming movements. For the discrete task, however, the correlations were less straightforward: the CV MT of multiple pairs correlated, but not all did, and the 2 × 2 matrix was not symmetric. We therefore abstain from any further interpretations.
Pearson's correlation r for CV MT between conditions. Black arrows indicate significant correlations (p < .05; across correlations .41 < r < .70), the grey arrow indicates a marginal significant correlation (.1 < p < .05; r = .36). The absence of arrows indicates that the correlation was not significant. No correlations between any of the discrete and cyclic conditions were found; within the cyclic task mode, no correlation was found between the low ID – width condition and the others. For the discrete task mode, several significant correlations were identified
As discussed above, we found evidence that in a subset of these combinations limit cycle dynamics were observed, while in another subset, fixed points regimes were found. Some indications (not conclusive) were found that the nature of the fixed points might have been dissimilar in the later subset dependent on the experimental factors. Regardless, in all task and manipulation combinations, movement time increased as the ID increased. That is, this trade-off appeared independent of the dynamical organization underlying Fitts' task performance. The question then arises of what could be the origin of the increase of MT with ID? The different models available in the literature do not provide satisfying answers in this respect. For instance, the dynamical model proposed by Mottet and Bootsma [11] fails for discrete movements—for these an N-shape in the Hooke plane appears independent of ID. Similarly, models from the 'corrective (sub) movements class' ([5, 14]; see Background) fail to deal with performances in (at least a large part of) the limit cycle regime since no corrective sub-movements are made (acceleration is about highest around the targets, [11]) but MT still gets larger with increasing ID. That is, while both models have their merits in their respective domains, neither of them is able to explain the MT increase with increasing ID across the range of task conditions that is reported here, and in the literature more at large.
Consistent with the Fitts' law, movement time scaled (approximately) linearly with the index of difficulty ID under all task and manipulation conditions. However, the system's functional organization underlying task performance differed both qualitatively and quantitatively as a function of task mode (cyclic vs. discrete) and manipulation (D vs. W). Within the cyclic task mode, low IDs were associated with limit cycles or fixed points dependent on whether target width or distance was manipulated, respectively. In this respect, target width was the parameter causing a bifurcation at a critical value. Conversely, for the discrete task mode, we did not observe such a bifurcation. Both behavioural modes adhere to distinct functional organizations; for instance velocity and acceleration always have to vanish at the target in discrete task mode. Consistent herewith, analysis of movement time variability (CV MT ) set apart the discrete and cyclic task mode, even for IDs in which both task modes appeared governed by a fixed-point dynamics. We argue that their difference is due to the inherent need for the perceptual-motor system to instantiate every single aiming in the discrete but not cyclic task mode, thereby introducing variability at another level of the perceptual-motor organization (i.e., that of the parameter dynamics; [32]). In addition, it cannot be ruled out that the nature of the fixed-points, or the space within which they exist, is dissimilar across both task modes and ID-manipulations. While our present data do not allow us to either conclusively refute or confirm this hypothesis, the differential trajectory variability evolution as a function of the task modes and distance versus width manipulation provides a hint thereto.
Regardless, the functional organization underlying task performance at various IDs varied markedly as a function of task mode and manipulation. In fact, our results counter the idea that change in a single parameter (as a function of ID) of the dynamics and/or bifurcation structure can account for Fitts' law. Explanations in terms of correction strategies, however, as discussed above, also have their limitations. That is, explaining Fitts' law in terms of a single dynamical organization or movement strategy remains problematic. In fact, this may indicate that a full description of Fitts' law may require more than one control (or bifurcation) parameter. The remaining question, then, is which one? We found evidence that target width, rather than task difficulty, acts as a control parameter. No clear indication was found that target distance did so too (except for a single participant) even though changing target distance had the opposite effect (of width) on trajectory variability. This may be due to a differential effect of the degree of convergence of the phase flow for both parameters. This, however, is of yet an open question. Regardless, this discussion resonates with previously expressed doubts as to whether the notion of task difficulty as quantified through target distance divided by width is appropriate. For instance, Welford and colleagues proposed a definition incorporating two additive logarithmic D and W terms [8]. Further, task difficulty is insensitive to energetic cost [35], which is higher at the easy task difficulty spectrum. Also, anecdotal reports from our participants suggest that subjective difficulty does not map uniquely onto the index of difficulty (the low ID small width-small distance conditions were experienced as particularly difficult). That is, the explicit identification of the nature of the fixed points in the various task conditions as well as the control parameter(s) implicated seems of a particular interest for the Fitts' paradigm. If a second control parameter indeed exists, its identification may well alter the notion of task difficulty as currently understood.
Thirteen (self-declared) right-handed participants (7 females; age: 29.3 ± 3.8 years) performed aiming movements from a starting point to a target (discrete mode) or between two targets (cyclic mode) with a hand-held stylus (18 g, 156.5 × 14.9 mm, ~1 mm tip) across a digitizer tablet (Wacom Intuos XL, 1024 × 768 pixel resolution) under instructions stressing both speed and accuracy. Position time series were acquired from the tablet via custom-made software (sampled at 250 Hz). The targets were printed in red on white A3 paper that was positioned under the transparent sheet of the tablet. In the cyclic mode, for each condition two trial repetitions consisting of 50 horizontal reciprocal aiming movements each (i.e., 25 cycles) were performed in the transversal plane, once starting from the left target and once from the right target. In the discrete mode, four blocks consisting of 25 single aiming movements were performed twice; in two blocks movements were made toward a target positioned on the right side; in the two other blocks the direction was inversed. Cyclic trials with more than 6 errors and discrete blocks with more than 3 errors were redone (i.e., a 12% error rate was tolerated). In both task modes target distance and target width were manipulated independently, and chosen so as to allow for a large sampling of distance and width, respectively. In the width manipulation, target distance was set at 20 cm, and target width varied as follows: 4.20, 2.50, 1.49, 0.88, 0.53, 0.31, and 0.19 cm. In the distance manipulation, target width was set at 0.31 cm, and target distance was varied from 1.47, 2.48, 4.17, 7.01, 11.80, 19.84, and 33.37 cm. In both cases, this resulted in seven IDs (from 3.25 to 7.75, step size 0.75), which were administered randomly. Participants were familiarized with all task mode (discrete, cyclic) by manipulation (distance, width) conditions by performing 5 to 10 movements (until fast and successful performance) with ID = 3.25 and ID = 7.00. The familiarization ended when the participant reached a stable behaviour (i.e., moving fast and not missing the target). The width and distance manipulations were assessed in two experimental sessions lasting about 1½ hour each. Both the width and distance manipulations, as well as the cyclic and discrete tasks were counterbalanced across participants.
For the rhythmic movements, the peaks in the (horizontal) position time series were taken as movement initiations and terminations. The discrete movements' initiation and termination were defined as the moment its (absolute) velocity exceeded versus fell below 0.1 cm/s, respectively. For the termination, the additional criterion was used that the movement velocity had to remain below this velocity criterion for minimally 60 ms [5]. A secondary movement was deemed present if it lasted for minimally 100 ms, the velocity criterion was exceeded for at least half of the burst's time, and if the covered distance was minimally either 2 mm or ¼ of the target width. If present, the secondary movement's endpoint was taken as the movement's termination. (For movement time, we verified whether the inclusion of the secondary movement changed the patterns of results, which was not the case.) Movement time (MT) was defined as the average of the temporal differences between movement termination and onset. For each movement, the acceleration duration (AT) was defined as the moment of peak velocity minus movement initiation. The ratio AT/MT (R AT/MT ) measures of the degree of symmetry of the movement velocity's profile. Effective amplitude (A e ) was computed as the average distance traversed across repetitions, and effective target width (W e ) as 2 × 1.96 times the mean standard deviation at the movement terminations [36]. Next, effective ID was calculated as ID e = log2(2A e /W e ).
In order to reconstruct the vector field underlying the movements [37, 38], we computed the conditional probability distributions P(x,y,t|x 0 ,y 0 ,t 0 ) that indicated the probability to find the system at a state (x,y) at a time t given its state (x 0 , y 0 ) at an earlier time t 0 . These distributions were computed using all aiming movements in each condition using a grid size of 28 bins. Drift coefficients (i.e., the deterministic dynamics) were computed according to:
$$ \begin{array}{l}{D}_x\left(x,y\right)=\underset{\tau \to 0}{ \lim}\frac{1}{\tau }{\displaystyle \int \int \left(x\hbox{'}-x\right)P\left(x\hbox{'},y\hbox{'},t+\tau \left|x,y,t\right.\right)dx\hbox{'} dy}\hfill \\ {}{D}_y\left(x,y\right)=\underset{\tau \to 0}{ \lim}\frac{1}{\tau }{\displaystyle \int \int \left(y\hbox{'}-y\right)P\left(x\hbox{'},y\hbox{'}t+\tau \left|x,y,t\right.\right)dx\hbox{'} dy}\hfill \end{array} $$
These coefficients are the numerical representations of the x-, and y-component of the vector at each phase space position. From these coefficients, we computed the angle θ max for each bin between its corresponding vector and that of each of its neighbours (if existent), and extracted the corresponding maximal value in order to visualize the phase flows in terms of so-called angle diagrams [17, 39].
We performed a principal component analysis (PCA) to investigate trajectory variability (x,y). For each participant and condition all trajectories were resampled to 100 samples, and subjected to principal component analysis [17]. A PCA was done separately for the 50 left-to-right and 50 right-to-left aiming movements. The 1st eigenvalue λ 1 was next averaged.
The ANOVA on ID e showed multiple effects, we therefore created ID e block averages of MT, R AT/MT , θ max , and λ 1 that were subjected to a repeated measures ANOVA with Task (2), Manipulation (2), and ID e (7) (i.e., a total of 28 conditions) as within participant factors. The Greenhouse-Geisser correction was applied whenever necessary. Significant main effects (α = .05) were followed up by Bonferroni-corrected post hoc tests. (For completeness, the same analyses were performed for peak velocity, acceleration, and deceleration time; they are reported in Additional file 5, 6, 7, respectively).
The protocol was in agreement with the Declaration of Helsinki. Informed consent was obtained from all participants prior to the experiment.
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The research here reported was support by the French Centre National de la Recherche Scientifique.
Institut de Neurosciences des Systèmes, INSERM UMR1106, Aix-Marseille Université, Faculté de Médicine, la Timone, 27 Bd Jean Moulin, 13005, Marseille, France
Raoul Huys
, Hester Knol
, Rita Sleimen-Malkoun
& Viktor K. Jirsa
Aix-Marseille Université, CNRS, Institut des Sciences du Mouvement UMR 7287, 13288, 13009, Marseille, France
Rita Sleimen-Malkoun
& Jean-Jacques Temprado
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Correspondence to Raoul Huys.
All authors designed the study. HK and RSM conducted the experiment. RH, HK, and RSM analyzed the data. RH wrote the manuscript. All authors wrote, improved and approved the final manuscript.
Maximal vector field angles θmax.
Classification of dynamics for the cyclic task.
Phase plane trajectories and their variability.
Intra-participant RAT/MT variability CV RAT/MT.
Peak velocity (PV).
Acceleration time (AT).
Deceleration time (DT).
Huys, R., Knol, H., Sleimen-Malkoun, R. et al. Does changing Fitts' index of difficulty evoke transitions in movement dynamics?. EPJ Nonlinear Biomed Phys 3, 8 (2015) doi:10.1140/epjnbp/s40366-015-0022-4
DOI: https://doi.org/10.1140/epjnbp/s40366-015-0022-4
Fitts' law
Goal-directed aiming
PACS code
87.19.rs (movement)
MCS code | CommonCrawl |
\begin{document}
\begin{frontmatter}
\title{$L^2$ estimates and existence theorems for the $\overline{\partial}$ operators in infinite dimensions, II}
\author[0]{Zhouzhe Wang} \address[0]{School of Mathematics, Sichuan University, Chengdu, 610064, China. E-mail address: [email protected] }
\author[1]{Jiayang Yu} \address[1]{School of Mathematics, Sichuan University, Chengdu, 610064, China. E-mail address: [email protected].}
\author[2]{Xu Zhang} \address[2]{School of Mathematics, Sichuan University, Chengdu, 610064, China. E-mail address: zhang\[email protected].}
\begin{abstract} This paper is the second part of our series of works to establish $L^2$ estimates and existence theorems for the $\overline{\partial}$ operators in infinite dimensions. In this part, we consider the most difficult case, i.e., the underlying space is a general pseudo-convex domain. In order to solve this longstanding open problem, we introduce several new concepts and techniques, which have independent interest and pave the way for research that investigates some other issues in infinite-dimensional analysis. \end{abstract}
\begin{keyword} $\overline{\partial}$ equation \sep Gaussian measure \sep $L^2$ method \sep pseudo-convex domain \sep plurisubharmonic exhaustion function \MSC[2020] 46G20 \sep 26E15 \sep 46G05 \end{keyword}
\end{frontmatter}
\section{Introduction}
Quite many of rich, deep and beautiful results in modern mathematics are limited to finite dimensional spaces and their subspaces/submanifolds or function spaces defined on them. A fundamental reason for this is that, in finite dimensional spaces there exist Lebesgue measures (or more generally Haar measures on locally compact Hausdorff topological groups), which enjoy nearly perfect properties. Unfortunately, in the setting of infinite dimensions, though after the works of great mathematicians such as D.~Hilbert (\cite{Hil}), A.~N.~Kolmogorov (\cite{Kol56}), N.~Wiener (\cite{Wiener}), J.~von~Neumann (\cite{Neumann55}) and I.~M.~Gel'fand (\cite{Gel64}), so far people have NOT yet found an ideal substitute for Lebesaue/Haar measures so that in-depth studies of infinite-dimensional analysis are always possible. Roughly speaking, functional analysis is in some sense the part of linear algebra in infinite-dimensional spaces; while the part of calculus in infinite-dimensional spaces, i.e., infinite-dimensional analysis, to our opinion, very likely, requires the efforts of mathematicians of several generations or even many generations to be satisfactory. As a reference, the long process of classical calculus (on finite dimensions) from its birth to perfection, from I. Newton and G.~W.~Leibniz to H.~L.~Lebesgue, has lasted more than 200 years.
Infinite-dimensional analysis has a wide range of application backgrounds:
\begin{itemize} \item[{\rm (1)}] In probability theory, many useful random variables and stochastic processes are actually defined on infinite-dimensional spaces (e.g., \cite{Gel64, Kol56, Wiener}), including typically the classical Brownian motion defined on the space of all continuous functions on $[0, 1]$ which vanish at $0$, i.e. the Wiener space. Also, it is well-known that one of the most fundamental concepts in probability theory is the so-called ``independence", for which a proper measure for constructing nontrivial sequences of independent random variables is, simply to use the product measure of some given sequence of measures in finite dimensions;
\item[{\rm (2)}] In control theory, the value functions of optimal control problems for distributed parameter systems (including particularly controlled partial differential equations) are functions of infinitely many variables, and hence the corresponding Hamilton-Jacobi-Bellman equations are nonlinear partial differential equations with infinite number of variables (\cite{DaZ02, Li-Yong, Lions88}). On the other hand, quite naturally, functions of infinitely many variables are extensively used in the field of stochastic control, in both finite and infinite dimensions (\cite{Lu-Zhang, YZ99});
\item[{\rm (3)}] In physics, functions with infinite number of variables appear extensively in continuum mechanics, quantum mechanics, quantum field theory (\cite{Glimm, Neumann55}). The most relevant example in this respect is the Feynman path integrals (\cite{Feynman}). As pointed by N.~Wiener (\cite[p. 131]{Wiener}), ``the physicist is equally concerned with systems the dimensionality of which, if not infinite, is so large that it invites the use of limit-process in which it is treated as infinite. These systems are the systems of statistical mechanics...". \end{itemize}
In \cite[p. 14]{Atiyah}, M. Atiyah remarked that ``the 21st century might be the era of quantum mathematics or, if you like, of infinite-dimensional mathematics". Similarly to the role of analysis in mathematics, infinite-dimensional analysis is of fundamental importance in infinite-dimensional mathematics, especially when precise computation or estimate is indispensable. Similarly to the finite dimensional setting, infinite-dimensional analysis is the basis of infinite-dimensional mathematics.
Historically, infinite-dimensional analysis originated to a large extent from the study of probability theory and related fields (e.g., \cite{Kol56, Wiener} and so on). In the recent decades, due to the important and fundamental works by some leading probability experts including L. Gross (for abstract Wiener spaces, \cite{Gro65}), T. Hida (for white noise analysis, \cite{Hida}) and P. Malliavin (for Malliavin calculus, \cite{Mall}), significant progresses have been made in infinite-dimensional analysis. Nevertheless, as far as we know, due to the lack of nontrivial translation invariance measure and local compactness in the setting of infinite dimensions, there is still a long way to go before establishing an effective setting for infinite-dimensional analysis. In addition, most of these probabilistic experts' works consider infinite-dimensional analysis on quite general probability spaces (or even abstract probability spaces), whose results inevitably appear not powerful enough to solve some complicated problems in this respect.
In our opinion, infinite-dimensional analysis has the significance of being independent of probability theory, and should and can fully become an independent discipline and mathematical branch. In particular, ``randomness" should be removed to a certain extent (of course, probability theory will always be one of the most important application backgrounds of infinite-dimensional analysis), and ``(mathematical) analysis nature" should be returned to. As a reference, functional analysis originated (to a large extent in history) from the study of integral equations, but as an independent branch of mathematics, the great development of functional analysis is possible only after getting rid of the shackles of integral equations.
As the first step of returning to the above mentioned ``(mathematical) analysis nature", according to our understanding, one should first deeply study analytic functions in infinite-dimensional space, that is, infinite-dimensional complex analysis. This is because, similarly to the finite dimensions, analytic functions in infinite-dimensional spaces are these functions enjoying many elegant and elaborate (or, in some sense even the best) analysis properties on these spaces. Without a deep understanding of this class of functions, the in-depth study of infinite-dimensional analysis seems difficult to be carried out.
Since the last sixties, infinite-dimensional complex analysis has been a rapid developing field, in which one can find many works, for example, the early survey by L.~Nachbin (\cite{Nachbin}), the monographes by A. Bayoumi (\cite{Bayoumi}), S.~B.~Chae (\cite{Chae}), G.~Coeur\'e (\cite{Coeure}), J.~Colombeau (\cite{Col}), S.~Dineen (\cite{Din81, Din99}), M.~Herv\'e (\cite{Herve}), J.-M. Isidro and L. L. Stach\'o (\cite{IS85}), P.~Mazet (\cite{Maz}), J.~Mujica (\cite{Mujica}) and P.~Noverraz (\cite{Noverraz}), and the rich references cited therein. In particular, around 2000, L.~Lempert revisited this field and made significant progresses in a series of fundamentally important works (\cite{Lem98, Lem99, Lem00, Lem03}). On the other hand, infinite-dimensional complex analysis is also strongly linked with many other branches of mathematics (e.g., \cite{DG21, SS16}, just to mention a few).
Although great advance has already been made, to the best of our knowledge, compared with the situation of finite dimensions, infinite-dimensional complex analysis is still far from reaching the level of precision analysis. So far, a large number of results in infinite-dimensional complex analysis are still in the category of abstract analysis or soft analysis, which can be said to be very similarly to the situation of several complex analysis before the 1960s, as we shall explain below a little more.
As in the setting of several complex analysis, one of the most fundamental problems in infinite-dimensional complex analysis is the solvability of the following $\overline{\partial}$ equation: \begin{equation}\label{d-bar equation} \overline{\partial}u=f, \end{equation} where $f$ is an inhomogeneous term with $\overline{\partial}f=0$ and $u$ is an unknown, defined on some infinite-dimensional complex space or its suitable open subset. L. Lempert pointed out at \cite[p. 775]{Lem99} that, ``{\sl up to now not a single infinite dimensional Banach space and an open subset therein have been proposed where the equation $(\ref{d-bar equation})$ could be proved to be solvable under reasonably general conditions on $f$}". In our previous work \cite{YZ} (See \cite[Theorem 5.1, p. 542]{YZ} particularly), we provided an approach to solve (\ref{d-bar equation}) in the case of quite general forms $f$ on the infinite-dimensional space $\ell^p$ (for $ p \in [1, \infty)$). However, our approach therein applies only to the whole space of $\ell^p$, while the case of general pseudo-convex domains is much more difficult and interesting, which turns out to be the main concern of the present paper.
In finite dimensions, a powerful basic tool to solve the counterpart of (\ref{d-bar equation}) is the $L^2$ estimates, developed systematically by C.~B.~Morrey (\cite{Mor}), J.~J.~Kohn (\cite{Koh}) and particularly by L.~H\"omander in \cite{Hor65, Hor90} (See also two recent very interesting books by T.~Ohsawa (\cite{Ohsawa}) and by E.~J.~Straube (\cite{Straube})). Since the $L^2$ estimates found by L. H\"ormander are dimension-free, starting from the 1970s, very naturally, people (See P. Raboin \cite{Rab79} and the references therein) have been looking forward to extending it to infinite dimensions, but without complete success for more than 40 years. Indeed, it is a longstanding unsolved problem to establish the $L^2$ estimates in infinite dimensions, particularly for general pseudo-convex domains therein (See \cite[Theorem 3.1 at p. 533 and Corollary 5.2 at p. 544]{YZ} for some partial positive results in this respect for the whole space $\ell^p$).
Based on the above observations, as a key step for our research project on infinite-dimensional analysis, we shall focus first on the solvability of the $\overline{\partial}$ equations over infinite-dimensional pseudo-convex domains, and especially the $L^2$ estimates for them. We shall choose the Hilbert space $\ell^2$ as a basic space. In some sense, $\ell^2$ is the simplest infinite-dimensional space, closest to Euclidean spaces. Note that the essential difficulties for analysis problems on $\ell^2$ are by no means reduced too much though sometimes working in this space does provide some convenience. To see this, we note that, similarly to the proof of \cite[Theorem 3.1, p. 533]{YZ}, it is easy to derive an $L^2$ estimate for smooth functions/forms on any pseudo-convex domain in $\ell^2$. In order to solve the corresponding $\overline{\partial}$ equation, people need to extend such an estimate to all functions/forms with suitably weak smoothness only. In the case of finite dimensions, this extension was achieved by L.~H\"omander (See \cite[Proposition 2.1.1 at p. 100 and Theorem 2.1.4 at p. 104]{Hor65} and \cite[Lemma 4.1.3 at p. 80, and Lemma 4.2.1 and Theorem 4.2.2 at p. 84]{Hor90} for two different ways) with the classical Friedrichs mollifier technique which relies heavily upon an argument not available in any infinite-dimensional space (including $\ell^2$, of course), namely, the translation invariance of the Lebesgue measure. To overcome this difficulty, we have to construct some suitable working spaces (See $L^{2}_{\left( s,t\right) }\left( V,loc\right)$ and so on in Section \ref{secx3}) so that the involved problems in infinite dimensions can be reduced to that in finite dimensions, which is possible by introducing a simple but very useful reduction transformation in (\ref{Integration reduce dimension}) (See Proposition \ref{Reduce diemension} for its main properties). In some sense, such an idea to establish the desired $L^2$ estimates in infinite dimensions is quite natural but the full process to realize it is delicate. Indeed, as mentioned before, there are two ways to derive the corresponding $L^2$ estimates (for functions/forms with some weak smoothness only) in finite dimensions, both of which are possible to be generalized to the setting of infinite dimensions:
\begin{itemize} \item[{\rm (1)}] One way is that in \cite{Hor65} (See \cite[Proposition 2.1.1 at p. 100 and Theorem 2.1.4 at p. 104]{Hor65}), which is based on another key preliminary, i.e., \cite[Proposition 1.2.4, p. 96]{Hor65} to show the identity of weak and strong extensions of first order differential operators with some complicated boundary conditions for which one has to use the localization and flattening techniques (based respectively on partitions of unity and local coordinate systems) and in particular introduce some delicate modification of the Friedrichs mollifier technique (See \cite[Proof of Proposition 1.2.4, pp. 97--98]{Hor65}). We have tried a long time to extend \cite[Proposition 1.2.4, p. 96]{Hor65} to infinite dimensions but failed, for which the main difficulty is that the above mentioned techniques do not seem to be compatible with our reduction transformation (\ref{Integration reduce dimension}).
\item[{\rm (2)}] Another way is that in \cite{Hor90} (See \cite[Lemma 4.1.3 at p. 80, and Lemma 4.2.1 and Theorem 4.2.2 at p. 84]{Hor90}, which is based on a delicate choice of weight functions (\cite[(4.2.2) at p. 82 and Proof of Theorem 4.2.2 at p. 85]{Hor90}) so that for the corresponding $L^2$ estimate one only needs to consider those functions/forms with compact supports, and hence the boundary conditions for them become quite simple. Due to the lack of local compactness, when trying to extend such an argument to infinite dimensions, we have to modify essentially the usual spaces of smooth functions on Euclidean spaces. For this purpose, we introduce a new concept of uniform inclusion for two sets in $\ell^2$ (See Definition \ref{def of bounded contained}), and via which we may introduce the desired spaces $C^{\infty}_0(V)$ (and also $C^{\infty}_{F}(V)$ and so on) of smooth functions (in a certain new sense) on any nonempty open set V of $\ell^2$ (See Subsection \ref{definition of basic symbols} for the details). We may then define pseudo-convex domains in $\ell^2$ by means of plurisubharmonic exhaustion functions (See Definition \ref{pseudocovexdomain}), via which the desired weight functions can be constructed (See (\ref{weight function}) and (\ref{230503e1})). With these machines in hand, for the corresponding $L^2$ estimate in infinite dimensions we only need to consider those functions/forms which vanish on the boundaries, and hence our reduction transformation (\ref{Integration reduce dimension}) works well to reduce the original infinite-dimensional problems to a sequence of approximate ones in finite dimensions so that the Friedrichs mollifier technique can be employed (See the proof of Theorem \ref{general density4}). In this way, we succeed in establishing the desired $L^2$ estimates for the $\overline{\partial}$ operators in general pseudo-convex domains of $\ell^2$, and via which we obtain the solvability of the corresponding infinite-dimensional $\overline{\partial}$ equations. \end{itemize}
The strategy adopted in this work (i.e., choosing $\ell^2$ and/or its suitable subsets as typical underlying spaces) can be refined to propose a convenient framework for infinite-dimensional analysis (including real and complex analysis), in which differentiation (in some weak sense) and integration operations can be easily performed, integration by parts can be conveniently established under rather weak conditions (\cite{WYZ0}),
and especially some nice properties and consequences obtained by convolution (or even the Fourier transformation) in Euclidean spaces can be extended to infinite-dimensional spaces in some sense by taking the limit (See \cite{YZ-b} for more details). Compared to the existing tools in infinite-dimensional analysis (including abstract Wiener space, white noise analysis, Malliavin calculus and so on), our framework enjoys more convenient and clearer links with that of finite dimensions, and hence it is more convenient for computation and studying some analysis problems in infinite-dimensional spaces. Except for the $L^2$ estimates for the $\overline{\partial}$ operators (in general pseudo-convex domains of $\ell^2$) obtained in this paper, by modifying suitably the above strategy, we can also solve several other longstanding open problems, including Holmgren type theorem with infinite number of variables (\cite{YZ-a}) and the equivalence of Sobolev spaces over quite general domains of infinite-dimensional spaces (\cite{YZZ}).
It is well-known that the concepts of smooth functions in infinite-dimensional spaces are much more complicated than that in finite dimensions. The usual Fr\'echet and G\^ateaux derivatives, especially those of higher order, are actually quite inconvenient to use. In our framework, instead we use the partial derivatives (equivalent to some specific directional derivatives, which have lower requirements than all directional derivatives, i.e. the G\^ateaux derivative). The related regularity issues for solutions to the $\overline{\partial}$ equations considered in this paper are much more difficult than that in finite dimensions, which, as well as its applications in infinite-dimensional complex analysis, will be presented in our forthcoming papers.
The rest of this paper is organized as follows. Section \ref{secx2} is of preliminary nature, in which we introduce some basic notations and notions, including some important spaces of functions and particularly the definition of pseudo-convex domains of $\ell^2$. In Section \ref{secx3}, we define the $\overline{\partial}$ operators on our working spaces and present some properties of these operators. Section \ref{secx4} is devoted to establishing the key $L^2$ estimates for these operators, while Section \ref{sec5} is for solving the $\overline{\partial}$ equations on pseudo-convex domains of $\ell^{2}$.
\section{Preliminaries}\label{secx2}
\subsection{Notations and definitions}\label{definition of basic symbols} To begin with, we introduce some notations and definitions. Suppose that $G$ is a nonempty set and $(G,\cal J)$ is a topological space on $G$, then the smallest $\sigma$-algebra generated by all sets in $\cal J$ is called the Borel $\sigma$-algebra of $G$, denoted by $\mathscr{B}(G)$. For any $\sigma$-finite Borel measure $\nu$ on $G$ and $p\in [1,\infty)$, denote by $L^p(G,\nu)$ the Banach space of all
(equivalence classes of) complex-valued Borel-measurable functions $f$ on $G$ for which $\int_G|f|^pd\nu<\infty$ (Particularly, $L^2(G,\nu)$ is a Hilbert space with the canonical inner product). For any subset $A$ of $G$, denote by $\overline{A}$ the closure of $A$ in $G$, by $\partial A$ the boundary of $A$, and by $\chi_A(\cdot)$ the characteristic function of $A$ in $G$. Recall that $G$ is called a Lindel\"{o}f space if every open cover of $G$ has a countable subcover (e.g., \cite[p. 50]{Kel}).
Denote by $\mathbb{R}$ and $\mathbb{C}$ respectively the sets (or fields, or Euclidean spaces with the usual norms) of all real and complex numbers, and by $\mathbb{N}$ the set of all positive integers. Write $\mathbb{N}_0\triangleq \mathbb{N}\cup \{0\}$ and $\mathbb{C}^{\infty}=\prod\limits_{i=1}^{\infty}\mathbb{C}$, which is the countable Cartesian product of $\mathbb{C}$, i.e., the space of sequences of complex numbers (Nevertheless, in the sequel we shall write an element $\textbf{z}\in \mathbb{C}^{\infty }$ explicitly as $\textbf{z}=( z_{i})^{\infty }_{i=1}$ rather than $\textbf{z}=\{ z_{i}\}^{\infty }_{i=1}$, where $z_i\in\mathbb{C}$ for each $i\in \mathbb{N}$). Clearly, $\mathbb{C}^{\infty}$ can be identified with $(\mathbb{R}^2)^{\infty}=\prod\limits_{i=1}^{\infty}\mathbb{R}^2$. Set $$
\ell^{2}\triangleq\left\{( z_{i})^{\infty }_{i=1} \in \mathbb{C}^{\infty } :\; \sum\limits^{\infty }_{i=1} \left| z_{i}\right|^{2} <\infty \right\}. $$
Then, $\ell^2$ is a separable Hilbert space with the canonic inner product $\left( \cdot,\cdot\right)$ (and hence the canonic norm $\left| \left|\;\cdot\;\right| \right|$). From \cite[Theorem 15, p. 49]{Kel}, we see that every separable metric space is a Lindel\"{o}f space. Hence, in particular every open subset of $\ell^2$ is a Lindel\"{o}f space. For any $\textbf{a}\in \ell^{2}$ and $r\in(0,+\infty)$, set $$
B_{r}(\textbf{a})\triangleq \left\{ \textbf{z}\in \ell^{2} :\;\left| \left| \textbf{z}-\textbf{a}\right| \right|<r\right\} , $$ and we simply write $B_{r}$ for $B_{r}(0)$.
In the rest of this paper, unless other stated, we will fix a nonempty open set $V$ of $\ell^{2}$ and a sequence $\{a_i\}_{i=1}^{\infty}$ of positive numbers such that $$\sum\limits_{i=1}^{\infty} a_i <1. $$
Suppose that $f$ is a complex-valued function defined on $V$. As in \cite[p. 528]{YZ}, for each $\textbf{z}=(z_j)_{j=1}^{\infty}=(x_j+\sqrt{-1}y_j)_{j=1}^{\infty}\in V$ (here and henceforth $x_j,y_j\in \mathbb{R}$ for each $j\in \mathbb{N}$), we define the partial derivatives of $f(\cdot)$ (at $\textbf{z}$): $$ \begin{array}{ll} \displaystyle\frac{\partial f(\textbf{z})}{\partial x_j}\triangleq\lim_{\mathbb{R}\ni \tau\to 0}\frac{f(z_1,\cdots,z_{j-1},z_j+\tau,z_{j+1},\cdots)-f(\textbf{z})}{\tau},\\[3mm] \displaystyle\frac{\partial f(\textbf{z})}{\partial y_j}\triangleq\lim_{\mathbb{R}\ni \tau\to 0}\frac{f(z_1,\cdots,z_{j-1},z_j+\sqrt{-1}\tau,z_{j+1},\cdots)-f(\textbf{z})}{\tau}, \end{array} $$ provided that the above limits exist.
In the sequel, for each $k\in\mathbb{N}_0$, we denote by $C^{k} ( V)$ the set of all complex-valued functions $f$ on $V$ for which $f$ itself and all of its partial derivatives up to order $k$ are continuous on $V$. Write $$ C^{\infty} ( V)\triangleq \bigcap_{j=1}^{\infty}C^{j} ( V). $$ Note that throughout this work we use partial derivatives instead of the usual Fr\'echet derivatives. The following simple result reveals a big difference between these two notions of derivatives in infinite dimensions. \begin{proposition}\label{SMOOTH BUT NOT Frechet} There exists $f^0\in C^{\infty}(\ell^2)$ such that $f^0$ is not Fr\'{e}chet differentiable at some point $\textbf{z}_0\in\ell^2$. \end{proposition} \begin{proof} Let $$ f^0(\textbf{z})\triangleq \prod\limits_{j=1}^{\infty}(1+x_j^2\sin(2j^2\pi x_j))(1+y_j^2\sin(2j^2\pi y_j)),\quad \forall\;\textbf{z}=(x_j+\sqrt{-1}y_j)_{j=1}^{\infty}\in\ell^2. $$
Note that $|x_j^2\sin(2j^2\pi x_j)|+|y_j^2\sin(2j^2\pi y_j)|\leqslant x_j^2+y_j^2$ for each $j\in\mathbb{N}$ and $\sum\limits_{j=1}^{\infty}(x_j^2+ y_j^2)<\infty$, by \cite[Theorem 15.4, p. 299]{Rud87}, the infinite product $\prod\limits_{j=1}^{\infty}(1+x_j^2\sin(2j^2\pi x_j))(1+y_j^2\sin(2j^2\pi y_j))$ exists for each $\textbf{z}=(x_j+\sqrt{-1}y_j)_{j=1}^{\infty}\in\ell^2.$ It is easy to see that all order of partial derivatives of $f^0$ exist and $\frac{\partial f^0(\textbf{x})}{\partial x_i}=2(x_i\sin(2i^2\pi x_i)+i^2\pi x_i^2\cos(2i^2\pi x_i))\prod\limits_{j\neq i}(1+x_j^2\sin(2j^2\pi x_j))\prod\limits_{j=1}^{\infty}(1+y_j^2\sin(2j^2\pi y_j))$. Put $\textbf{z}_0\triangleq\left(\frac{1}{j}\right)_{j=1}^{\infty}\in\ell^2$. Then we have $\frac{\partial f^0(\textbf{z}_0)}{\partial x_i}=2\pi$ for each $i\in\mathbb{N}$. Let us use the contradiction argument to show that $f^0$ is not Fr\'{e}chet differentiable at $\textbf{z}_0$. If $f^0$ was Fr\'{e}chet differentiable at $\textbf{z}_0$, then it would hold that $\sum\limits_{i=1}^{\infty}\left(\left|\frac{\partial f^0(\textbf{z}_0)}{\partial x_i}\right|^2+\left|\frac{\partial f^0(\textbf{z}_0)}{\partial y_i}\right|^2\right)<\infty$, and hence $\lim\limits_{i\to\infty}\frac{\partial f^0(\textbf{z}_0)}{\partial x_i}=0$, which is a contradiction.
For simplicity, we only prove that $f^0\in C^0(\ell^2)$. For each $n\in\mathbb{N}$, let $$ f^0_n(\textbf{z})\triangleq \prod\limits_{j=1}^{n}(1+x_j^2\sin(2j^2\pi x_j))(1+y_j^2\sin(2j^2\pi y_j)),\quad \forall\;\textbf{z}=(x_j+\sqrt{-1}y_j)_{j=1}^{\infty}\in\ell^2. $$ Obviously, $\{f^0_n\}_{n=1}^{\infty}\subset C^0(\ell^2)$. By \cite[Proposition 2.1, p. 778]{Lem99}, we only need to prove that $f^0_n\to f^0$ uniformly on each compact subset $K$ of $\ell^2$. Firstly, since $K$ is compact and
$$|x_n^2\sin(2n^2\pi x_n)|+|y_n^2\sin(2n^2\pi y_n)|\leqslant x_n^2+y_n^2\leqslant ||\textbf{z}||^2,\quad \forall\;n\in\mathbb{N},
$$
we have $\sup\limits_{\textbf{z}\in K}||\textbf{z}||<\infty$ and hence both $\{x_n^2\sin(2n^2\pi x_n)\}_{n=1}^{\infty}$ and $\{y_n^2\sin(2n^2\pi y_n)\}_{n=1}^{\infty}$ are sequences of bounded functions on $K$. By \cite[Theorem 15.4, p. 299]{Rud87} again, we only need to prove that the series $\sum\limits_{j=1}^{\infty}(x_j^2+y_j^2)$ converges uniformly on $K$. Since $K$ is compact, for any given $\varepsilon>0$, one can find $k_1\in\mathbb{N}$ and $\textbf{z}_1=(x_j^1+\sqrt{-1}y_j^1)_{j=1}^{\infty}\in K,\cdots,\textbf{z}_{k_1}=(x_j^{k_1}+\sqrt{-1}y_j^{k_1})_{j=1}^{\infty}\in K$ such that for any $\textbf{z}=(x_j+\sqrt{-1}y_j)_{j=1}^{\infty}\in K$ there exists an integer $k$ with $1\leqslant k\leqslant k_1$ such that $||\textbf{z}_k-\textbf{z}||<\frac{\sqrt{\varepsilon}}{2}$. Note that there exists $N\in\mathbb{N}$ such that $\sum\limits_{j=N}^{\infty}\left((x_j^l)^2+(y_j^l)^2\right)<\frac{\varepsilon}{4}$ for all integers $l$ satisfying $1\leqslant l\leqslant k_1$. Therefore, by the triangle inequality, we have $$
\sum\limits_{j=N}^{\infty}(x_j^2+y_j^2)\leqslant \left(\sqrt{\sum\limits_{j=N}^{\infty}\left((x_j^k)^2+(y_j^k)^2\right)}+\sqrt{\sum\limits_{j=N}^{\infty}\left[(x_j-x_j^k)^2+(y_j-y_j^k)^2\right]}\right)^2
\leqslant \left(\frac{\sqrt{\varepsilon}}{2}+||\textbf{z}_k-\textbf{z}||\right)^2<\varepsilon $$ for all $\textbf{z}\in K$, which implies that the series $\sum\limits_{i=1}^{\infty}(x_i^2+y_i^2)$ converges uniformly on $K$. This completes the proof of Proposition \ref{SMOOTH BUT NOT Frechet}. \end{proof}
Recall that a continuous function on finite dimensional spaces is bounded on each bounded and closed subset of its domain. The following result shows that this does NOT hold anymore for continuous functions on $\ell^2$. \begin{proposition}\label{Frechet D but NOT bounded on boudned set} There exists a continuous function (on $\ell^2$) which is unbounded on some bounded, nonempty subset of $\ell^2$. \end{proposition} \begin{proof}
Let $\{\textbf{e}_k\}_{k=1}^{\infty}$ be an orthonormal basis on $\ell^2$. Then, $||\textbf{e}_k||=1$ for each $k\in\mathbb{N}$ and $||\textbf{e}_k-\textbf{e}_l||=2$ for all $k,l\in\mathbb{N}$ with $k\neq l$. Choose $\varphi\in C^{\infty}(\mathbb{R})$ such that $0\leqslant \varphi\leqslant 1$, $\varphi(x)=1$ for all $|x|\leqslant \frac{1}{256}$ and $\varphi(x)=0$ for all $|x|\geqslant \frac{1}{64}$. Let $$
f^0(\textbf{z})\triangleq \sum_{k=1}^{\infty}k\cdot \varphi(||\textbf{z}-\textbf{e}_k||^2),\quad \forall\;\textbf{z}\in\ell^2. $$
If $\textbf{z}\in B(\textbf{e}_{k_0},\frac{1}{4})$ for some $k_0\in\mathbb{N}$, then $\textbf{z}\notin \bigcup\limits_{k\neq k_0}B(\textbf{e}_{k},\frac{1}{8})$ and hence $f^0(\textbf{z})=k_0\cdot\varphi(||\textbf{z}-\textbf{e}_{k_0}||^2)$. If $\textbf{z}\in \ell^2\setminus \overline{\bigcup\limits_{k=1}^{\infty}B(\textbf{e}_{k},\frac{1}{8})}$, then $f^0(\textbf{z})=0$. Thus it is easy to see that $f^0$ is continuous at each point of $\ell^2$ and $f^0(\textbf{e}_k)=k$ for each $k\in\mathbb{N}$, which implies that $\sup\limits_{||\textbf{z}||\leqslant 1}|f^0(\textbf{z})|=\infty$. The proof of Proposition \ref{Frechet D but NOT bounded on boudned set} is completed. \end{proof}
\begin{remark}
For any infinite-dimensional Banach space $X$ with norm $||\cdot||_X$, by the Riesz Lemma (e.g., \cite[Lemma 2, p. 218]{Riesz}), there exists a sequence $\{h_k\}_{k=1}^{\infty}\subset X$ such that $||h_k||_X=1$ for each $k\in\mathbb{N}$ and $||h_k-h_l||_X>\frac{1}{2}$ for all $k,l\in\mathbb{N}$ with $k\neq l$. Then, similarly to the proof of Proposition \ref{Frechet D but NOT bounded on boudned set}, one can find a continuous function (on $X$) which is unbounded on some bounded, nonempty subset of $X$.
\end{remark}
Because of Proposition \ref{Frechet D but NOT bounded on boudned set}, we shall consider bounded and continuous functions on $\ell^2$ in this paper. More precisely, in the sequel, for each $k\in\mathbb{N}_0$, we denote by $C^{k}_{b}\left( V\right)$ the set of all complex-valued functions $f$ on $V$ for which $f$ itself and all of its partial derivatives up to order $k$ are bounded and continuous on $V$. Write $$C^{\infty}_b(V)\triangleq \bigcap_{j=1}^{\infty}C^{j}_b( V).
$$
The following notion will play a fundamental role in the sequel. \begin{definition}\label{def of bounded contained} A set $S\subset \ell^2$ is said to be uniformly included in $V$, denoted by $S\stackrel{\circ}{\subset} V$, if there exist $r,R\in(0,+\infty)$ such that $\cup_{\textbf{z}\in S}B_r(\textbf{z})\subset V$ and $S\subset B_R$. \end{definition} \begin{remark}
For any set $S\subset \ell^2$, it is easy to see that $S\stackrel{\circ}{\subset} V$ if and only if $\overline{S} \stackrel{\circ}{\subset} V$.
\end{remark}
For $f\in C^0(V)$, we call $\hbox{\rm supp$\,$} f\triangleq\overline{\{\textbf{z}\in V:\;f(\textbf{z})\neq 0\}}$ the support of $f$. For $k\in\mathbb{N}_0$, write $$ C^{k}_{0}\left( V\right) \triangleq \left\{ f\in C^{k}_{b}\left( V\right) :\;\hbox{\rm supp$\,$} f\stackrel{\circ}{\subset} V\right\},\quad C^{\infty}_0(V)\triangleq \bigcap_{j=1}^{\infty}C^{j}_0( V). $$ For each $n\in\mathbb{N}$, we denote by $C^{k}\left( \mathbb{C}^n\right)$ the set of all complex-valued functions $f$ on $\mathbb{C}^n$ for which $f$ itself and all of its partial derivatives up to order $k$ are continuous on $\mathbb{C}^n$. For any $f\in C^0\left( \mathbb{C}^n\right)$, we may define $\hbox{\rm supp$\,$} f$ similarly as above. Write $$ C^{k}_{c}\left( \mathbb{C}^n\right) \triangleq \left\{ f\in C^{k}\left( \mathbb{C}^n\right) :\;\hbox{\rm supp$\,$} f\hbox{ is compact in } \mathbb{C}^n\right\},\quad C^{\infty}_c\left(\mathbb{C}^n\right)\triangleq \bigcap_{j=1}^{\infty}C^{j}_c\left( \mathbb{C}^n\right). $$ In a similar way, we denote by $C^{\infty}_c\left(\mathbb{R}^n\right)$ the set of all complex-valued functions $f$ on $\mathbb{R}^n$ for which $f$ itself and all of its partial derivatives are continuous on $\mathbb{R}^n$, and $\hbox{\rm supp$\,$} f$ is compact in $\mathbb{R}^n$.
\begin{remark} Note that, since $V$ is assumed to be a nonempty open set of the infinite-dimensional Hilbert space $\ell^2$, an element $g$ in $C^{0}_{0}\left( V\right)$ does NOT need to have a compact support in $V$. Nevertheless, such a function $g$ is bounded and continuous on $V$ and $\hbox{\rm supp$\,$} g\stackrel{\circ}{\subset} V$, and therefore, as we shall see later, functions in $C^{0}_{0}\left( V\right)$ can be regarded as ideal (infinite-dimensional) counterparts of compactly supported continuous functions on nonempty open sets of Euclidean spaces. \end{remark}
For $f\in C^1(V)$ and $i\in\mathbb{N}$, write \begin{equation}\label{eq2301161} \begin{array}{ll} \displaystyle D_{x_{i}}f\triangleq\frac{\partial f}{\partial x_{i}},\quad D_{y_{i}}f\triangleq\frac{\partial f}{\partial y_{i}},\quad \partial_{i} f\triangleq\frac{1}{2}( D_{x_{i}}f-\sqrt{-1}D_{y_{i}}f),\\[3mm]
\displaystyle\overline{\partial }_{i} f\triangleq\frac{1}{2}( D_{x_{i}}f +\sqrt{-1}D_{y_{i}}f), \quad \delta_{i} f\triangleq \partial_{i} f-\frac{\overline{z_{i}} }{2a^{2}_{i}} \cdot f,\quad \overline{\delta_{i} } f\triangleq \overline{\partial_{i} } f-\frac{z_{i}}{2a^{2}_{i}}\cdot f. \end{array} \end{equation}
Denote by $\mathbb{N}_0^{(\mathbb{N})}$ the set of all finitely supported sequences of nonnegative integers, i.e., for each $\alpha =(\alpha_{j} )_{j=1}^\infty\in \mathbb{N}^{\left( \mathbb{N} \right) }_{0} $ with $\alpha_{m}\in\mathbb{N}_{0}$ for any $ m\in\mathbb{N}$, there exists $n\in\mathbb{N}$ such that $\alpha_j=0$ for all $j\geqslant n$. Set $k\triangleq \left| \alpha \right| \triangleq \sum\limits^{\infty }_{j=1} \alpha_{j}$. For $f\in C^k(V)$, write $$ \delta^{\alpha }f \triangleq \delta^{\alpha_{1} }_{1} \ \delta^{\alpha_{n} }_{2} \cdots \ \delta^{\alpha_{n} }_{n} f,\qquad \overline{\delta }^{\alpha } f\triangleq \overline{\delta }^{\alpha _{1} }_{1} \ \overline{\delta }^{\alpha _{2} }_{2} \ \cdots \ \overline{\delta }^{\alpha_{n} }_{n} f, $$ $$ \partial_{\alpha } f\triangleq \partial^{\alpha_{1} }_{1} \partial^{\alpha_{2} }_{2} \cdots \partial^{\alpha_{n} }_{n} f ,\qquad \overline{\partial}_{\alpha } f \triangleq \overline{\partial}_{1} ^{\alpha_{1}} \overline{\partial}_{2} ^{\alpha _{2} } \cdots \overline{\partial}_{n} ^{\alpha _{n} }f. $$
Put $$
C^{1}_{F}\left( V\right)\triangleq \left\{f\in C^{1}\left( V\right):\;\sup_{E} \left( \left| f\right| +\sum^{\infty }_{i=1} \bigg(\left| D_{x_i} f\right|^{2} +\left|D_{y_i} f\right|^{2}\bigg) \right) <\infty,\; \forall\; E\stackrel{\circ}{\subset}V\right\}. $$ Clearly, $C^{1}_{F}\left( V\right)\not\subset C^{1}_{b}\left( V\right)$ and $C^{1}_{b}\left( V\right)\not\subset C^{1}_{F}\left( V\right)$. For $k\in\mathbb{N}\cup \{\infty\}$, we write $$ \begin{array}{ll} \displaystyle C^{k}_{F}\left( V\right)\triangleq C^{k}\left( V\right)\cap C^{1}_{F}\left( V\right),\quad C^{k}_{b,F}\left( V\right)\triangleq C^{k}_{b}\left( V\right)\cap C^{1}_{F}\left( V\right),\quad C^{k}_{0,F}\left( V\right)\triangleq C^{k}_{0}\left( V\right)\cap C^{1}_{F}\left( V\right),\\[3mm]
\displaystyle C^{k}_{F^{k}}\left( V\right)\triangleq \left\{f\in C^{k}(V):\;\partial_{a} \overline{\partial_{\beta } } f\in C^{1}_{F}\left( V\right)\text{ for all }\alpha, \beta\in \mathbb{N}^{\left( \mathbb{N} \right) }_{0}\text{ with }|\alpha|+|\beta|<k\right\},\\[3mm] \displaystyle C^{k}_{b,F^{k}}\left( V\right)\triangleq C^{k}_{b}\left( V\right)\cap C^{k}_{F^{k}}\left( V\right),\quad C^{k}_{0,F^{k}}\left( V\right)\triangleq C^{k}_{0}\left( V\right)\cap C^{k}_{F^{k}}\left( V\right). \end{array} $$ It is obvious that $$ C^{k}_{F}\left( V\right)\supset C^{k}_{b,F}\left( V\right)\supset C^{k}_{0,F}\left( V\right),\quad C^{k}_{F^{k}}\left( V\right)\supset C^{k}_{b,F^{k}}\left( V\right)\supset C^{k}_{0,F^{k}}\left( V\right). $$
\subsection{Multiplicative System Theorems} The material of this short subsection is from \cite[pp. 111--112]{Dri} (with some modification).
Suppose that $\Omega$ is a nonempty set and $\mathbb{H}$ is a set of some bounded $\mathbb{R}$-valued (or $\mathbb{C}$-valued) functions on $\Omega$. We call $\{f_n\}_{n=1}^{\infty}\subset \mathbb{H}$ a bounded convergent sequence in $\mathbb{H}$ if
$\displaystyle\sup_{(n,\omega)\in\mathbb{N}\times\Omega}|f_n(\omega)|< \infty$, and $f(\omega)\triangleq\lim\limits_{n\to\infty}f_n(\omega)$ exists for every $\omega\in\Omega$. We say that $\mathbb{H}$ is closed under the bounded convergence if the limit of every bounded convergent sequence in $\mathbb{H}$ is still in $\mathbb{H}$.
A subset $\mathbb{M}$ of $\mathbb{H}$ is called a multiplicative system in $\mathbb{H}$ if for any $f,g\in\mathbb{M}$, it holds that $f\cdot g\in\mathbb{M}$. We denote by $\sigma(\mathbb{M})$ the smallest $\sigma$-algebra on $\Omega$ such that for any $f\in \mathbb{M}$, $f$ is a $\sigma(\mathbb{M})$-measurable function. It is easy to see that, for the case of $\mathbb{R}$-valued ({\it resp.} $\mathbb{C}$-valued) functions, $\sigma(\mathbb{M})$ is exactly the $\sigma$-algebra generated by all subsets (of $\Omega$) in the form $h^{-1}(A)$, where $h\in \mathbb{M}$ and $A\in \mathscr{B}(\mathbb{R})$ ({\it resp.} $A\in \mathscr{B}(\mathbb{C})$).
\begin{theorem}\label{Dynkin's Multiplicative System Theorem} \textbf{(Dynkin's Multiplicative System Theorem).} If $\mathbb{H}$ is a real linear space of some bounded $\mathbb{R}$-valued functions on $\Omega$, $1\in \mathbb{H}$, $\mathbb{H}$ is closed under the bounded convergence and $\mathbb{M}$ is a multiplicative system in $\mathbb{H}$ , then $\mathbb{H}$ contains all bounded $\mathbb{R}$-valued $\sigma(\mathbb{M})$-measurable functions. \end{theorem} The following is a complex version of Theorem \ref{Dynkin's Multiplicative System Theorem}. \begin{theorem}\label{Complex Multiplicative System Theorem} \textbf{(Complex Multiplicative System Theorem).} If $\mathbb{H}$ is a complex linear space of some bounded $\mathbb{C}$-valued functions on $\Omega$, $1\in \mathbb{H}$, $\mathbb{H}$ is closed under the bounded convergence, $\mathbb{M}$ is a multiplicative system in $\mathbb{H}$, and both $\mathbb{H}$ and $\mathbb{M}$ are closed under the complex conjugation, then $\mathbb{H}$ contains all bounded $\mathbb{C}$-valued $\sigma(\mathbb{M})$-measurable functions. \end{theorem}
\subsection{A Borel probability measure on $\ell^2$}\label{s2-3} For any given $a>0$, we define a probability measure $\mathcal{N}_a$ in $(\mathbb{C},\mathscr{B}(\mathbb{C}))$ by $$
\mathcal{N}_a(E)\triangleq \frac{1}{ 2\pi a^2 }\int_{E}e^{-\frac{x^2+y^2}{2a^2}}\,\mathrm{d}x\mathrm{d}y,\quad\,\forall\;E\in \mathscr{B}(\mathbb{C}). $$ We need to use a product measure on the space $\mathbb{C}^{\infty}$, endowed with the usual product topology. By the discussion in \cite[p. 9]{Da}, $\mathscr{B}(\mathbb{C}^{\infty})$ is precisely the product $\sigma$-algebra generated by the following sets $$ \bigcup_{k=1}^{\infty}\Bigg\{E_1\times E_2\times \cdots \times E_k\times \Bigg(\prod_{i=k+1}^{\infty}\mathbb{C}\Bigg):\;E_i\in \mathscr{B}(\mathbb{C}),\,1\leqslant i\leqslant k\Bigg\}. $$ Let $$
\mathcal{N}\triangleq\prod_{i=1}^{\infty}\mathcal{N}_{a_i} $$ be the product measure on $(\mathbb{C}^{\infty},\mathscr{B}(\mathbb{C}^{\infty}))$. Just as \cite[Exercise 1.10, p. 12]{Da}, one can show that every closed ball in $\ell^2$ lies in $\mathscr{B}(\mathbb{C}^{\infty})$ and $\mathscr{B}(\ell^2)=\{E\cap\ell^2:\; E\in \mathscr{B}(\mathbb{C}^{\infty})\}$. We shall need the following known simple but useful result (See \cite[Proposition 1.11, p. 12]{Da}).
\begin{proposition}\label{lem1} $\mathcal{N}(\ell^2)=1$. \end{proposition}
Thanks to Proposition \ref{lem1}, we obtain a Borel probability measure $P$ on $\ell^2$ by setting $$ P(E)\triangleq \mathcal{N}(E),\quad\forall\;E\in \mathscr{B}(\ell^2). $$ For any $n\in\mathbb{N}$, we denote by $C_c^{\infty}(\mathbb{C}^n)$ the set of all $\mathbb{R}$-valued, $C^{\infty}$-functions on $\mathbb{C}^n(\equiv \mathbb{R}^{2n})$ with compact supports. Clearly, each function in $C_c^{\infty}(\mathbb{C}^n)$ can also be viewed as a cylinder function on $\ell^2$. Set $$ \mathscr {C}_c^{\infty}(\mathbb{C}^n)\triangleq \left\{f+\sqrt{-1}g:\;f,g\in C_c^{\infty}(\mathbb{R}^{2n})\right\},\quad\mathscr {C}_c^{\infty}\triangleq \bigcup_{n=1}^{\infty}\mathscr {C}_c^{\infty}(\mathbb{C}^n). $$ By \cite[Proposition 2.4, p. 528]{YZ}, $\mathscr {C}_c^{\infty}$ is dense in $L^2(\ell^2,P)$.
We shall need the following result.
\begin{lemma}\label{strictly positive measure} For every nonempty open subset $O$ of $\ell^2$, it holds that $P(O)>0$. \end{lemma} \begin{proof} We use the contradiction argument. Suppose that $P(O)>0$ was not true. Then, we would have $P(O)=0$. Let $$ \mathscr{Q}\triangleq \bigcup\limits_{n=1}^{\infty}\left\{(r_1+\sqrt{-1}r_2,\cdots,r_{2n-1}+\sqrt{-1}r_{2n},0,\cdots,0,\cdots):\;r_1,\cdots,r_{2n}\hbox{ are real rational numbers}\right\}.
$$ One can show that $\mathscr{Q}$ is dense in $\ell^2$, and $\ell^2=\bigcup\limits_{\textbf{r}\in \mathscr{Q}}(\textbf{r}+O)$, where $\textbf{r}+O=\{\textbf{r}+\textbf{z}:\;\textbf{z}\in O\}$. By the conclusion (ii) of \cite[Theorem 2.8, p. 31]{Da} and $P(O)=0$, for each $\textbf{r}\in \mathscr{Q}$, we have $P(\textbf{r}+O)=0$. Note that $\mathscr{Q}$ is a countable set. Thus $1=P(\ell^2)\leqslant \sum\limits_{\textbf{r}\in \mathscr{Q}}P(\textbf{r}+O)=0$, which is a contradiction. This completes the proof of Lemma \ref{strictly positive measure}. \end{proof}
\begin{remark} As in \cite[p. 125]{Com Neg}, a probability measure with the property stated in Lemma \ref{strictly positive measure} is called a strictly positive measure. \end{remark}
As a consequence of Lemma \ref{strictly positive measure}, it is easy to see the following property. \begin{corollary}\label{property of strictly positive measure} Suppose that $E$ is a Borel subset of $\ell^2$ such that $P(E)=1$. Then $E$ is dense in $\ell^2$. \end{corollary}
For each $n\in\mathbb{N}$, we define a probability measure $\mathcal{N}^n$ in $(\mathbb{C}^n,\mathscr{B}(\mathbb{C}^n))$ by setting $$ \mathcal{N}^n\triangleq \prod_{i=1}^{n}\mathcal{N}_{a_i} $$ and a probability measure ${\widehat{ \mathcal{N}}^n}$ in $(\mathbb{C}^{\infty},\mathscr{B}(\mathbb{C}^\infty))$ by setting $$ \widehat{\mathcal{N}}^n\triangleq \prod_{i=n+1}^{\infty}\mathcal{N}_{a_i}. $$ Let \begin{equation}\label{220817e1zx} P_n(E)\triangleq \widehat{\mathcal{N}}^n(E),\quad\forall\;E\in \mathscr{B}(\ell^2). \end{equation} Then, by Proposition \ref{lem1} again, we obtain a Borel probability measure $P_n$ on $\ell^2$. Obviously, $$P=\mathcal{N}^n\times P_n,\quad\forall\; n\in \mathbb{N}.$$ Further, for any $f\in L^2(\ell^2, P)$, let \begin{equation}\label{Integration reduce dimension} f_n(\textbf{z}_n)\triangleq \int f(\textbf{z}_n,\textbf{z}^n)\,\mathrm{d}P_n(\textbf{z}^n), \end{equation} where $\textbf{z}_n=(x_{i} +\sqrt{-1}y_{i})_{i=1}^{n}\in \mathbb{C}^n$ and $\textbf{z}^n=(x_{i} +\sqrt{-1}y_{i})_{i=n+1}^\infty\in \ell^2$. The following result will play a crucial role in the sequel.
\begin{proposition}\label{Reduce diemension} For $f\in L^{2} ( \ell^{2} ,P )$ and $n\in\mathbb{N}$, the function $f_n$ given by (\ref{Integration reduce dimension}) can be viewed as a cylinder function on $\ell^2$ with the following properties: \begin{itemize} \item[{\rm (1)}]\label{z1}
\quad $\left| \left| f_{n}\right| \right|_{L^{2} ( \ell^{2} ,P ) } \leqslant \left| \left| f\right| \right|_{L^{2} ( \ell^{2} ,P ) } $; \item[{\rm (2)}] \label{z2}
\quad $\lim\limits_{n\rightarrow \infty } \left| \left| f_{n}-f\right| \right|_{L^{2} ( \ell^{2} ,P) } =0$; \item[{\rm (3)}] \label{z3}
\quad $\lim\limits_{n\rightarrow \infty } \int_{\ell^2}\big|| f_{n}|^{2} -| f|^{2}\big|\,\mathrm{d}P =0.$ \end{itemize} \end{proposition} \begin{proof} (1) Note that $$ \begin{array}{ll}
\displaystyle \int_{\ell^2}|f_{n}|^2\,\mathrm{d}P=\int_{\mathbb{C}^n}|f_n(\textbf{z}_n)|^2\,\mathrm{d}\mathcal{N}^n(\textbf{z}_n)\\[5mm]
\displaystyle =\int_{\mathbb{C}^n}\bigg|\int f(\textbf{z}_n,\textbf{z}^n)\cdot
\mathrm{d}P_n(\textbf{z}^n)\bigg|^2\mathrm{d}\mathcal{N}^n(\textbf{z}_n)\\[5mm]
\displaystyle \leqslant \int_{\mathbb{C}^n} \int |f(\textbf{z}_n,\textbf{z}^n)|^2\cdot \mathrm{d}P_n(\textbf{z}^n)\mathrm{d}\mathcal{N}^n(\textbf{z}_n)\\[5mm]
\displaystyle =\int_{\ell^2}|f|^2\,\mathrm{d}P, \end{array} $$ where the inequality follows from the Jensen inequality (See \cite[Theorem 3.3, p. 62]{Rud87}) and the first and the third equalities follows form the fact that $P=\mathcal{N}^n\times P_n$.
(2) Since $\mathscr {C}_c^{\infty}$ is dense in $L^2(\ell^2,P)$, for any $\varepsilon>0$, there exists $m\in\mathbb{N}$ and $g\in \mathscr{C}_c^{\infty}(\mathbb{C}^m)$ such that $$
\bigg(\int_{\ell^2} |f-g|^2\mathrm{d}P\bigg)^{\frac{1}{2}}<\frac{\varepsilon}{2}. $$ For any $n\geqslant m$, we define $g_n$ as that in (\ref{Integration reduce dimension}), i.e., $ g_n(\cdot)\triangleq \int g(\cdot,\textbf{z}^n)\,\mathrm{d}P_n(\textbf{z}^n)$. It is easy to check that $g_n=g$ and $$ \begin{array}{ll}
\displaystyle \bigg(\int_{\ell^2} |f_n-f|^2\mathrm{d}P\bigg)^{\frac{1}{2}}
=\bigg(\int_{\ell^2} |f_n-g_n+g-f|^2\mathrm{d}P\bigg)^{\frac{1}{2}}\\[3mm]
\displaystyle \leqslant\bigg(\int_{\ell^2} |f_n-g_n|^2\mathrm{d}P\bigg)^{\frac{1}{2}}+\bigg(\int |g-f|^2\mathrm{d}P\bigg)^{\frac{1}{2}}\\[3mm]
\displaystyle \leqslant 2\bigg(\int_{\ell^2} |g-f|^2\mathrm{d}P\bigg)^{\frac{1}{2}}<\varepsilon, \end{array} $$ where the second inequality follows from the conclusion (1).
(3) Note that $$ \begin{array}{ll} \displaystyle
\int_{\ell^2}\big|| f_{n}|^{2} -| f|^{2}\big|\,\mathrm{d}P =
\int_{\ell^2}\big|| f_{n}|-| f|\big|\cdot(| f_{n}|+| f|)\,\mathrm{d}P\\[3mm]
\displaystyle \leqslant \big|\big| |f_{n}|-| f|\big|\big|_{L^{2}\left( \ell^{2} ,P\right)}\cdot \big|\big| |f_{n}|+| f|\big|\big|_{L^{2}\left( \ell^{2} ,P\right)}\\[3mm]
\displaystyle \leqslant || f_{n}- f||_{L^{2}\left( \ell^{2} ,P\right)}\cdot(|| f_n||_{L^{2}\left( \ell^{2} ,P\right)}+ || f||_{L^{2}\left( \ell^{2} ,P\right)})\\[3mm]
\displaystyle \leqslant 2|| f_{n}- f||_{L^{2}\left( \ell^{2} ,P\right)}\cdot || f||_{L^{2}\left( \ell^{2} ,P\right)}, \end{array} $$ where the first inequality follows from the Cauchy-Schwarz inequality. The above inequality, combined with the conclusion (2), gives the desired conclusion (3). This completes the proof of Proposition \ref{Reduce diemension}. \end{proof}
We shall need the following infinite-dimensional Gauss-Green type Theorem.
\begin{theorem}\label{Gauss-Green Theorem}
If $f\in C^{1}_{0} ( \ell^2 )$, then for each $m\in\mathbb{N}$, $$ \int_{\ell^2}D_{x_m}f\,\mathrm{d}P =\int_{\ell^2}\frac{x_m }{a_m^2}\cdot f \,\mathrm{d}P. $$ \end{theorem} \begin{proof} For simplicity, we consider only $m=1$. Since $f\in C^{1}_{0} ( \ell^2 )$, there exists $R\in(0,+\infty)$ such that $\hbox{\rm supp$\,$} f\subset B_R$. Write $$
\ell^{2,1}\triangleq \left\{(\sqrt{-1}y_1,x_{2}+\sqrt{-1}y_{2},\cdots):\;y_{1},x_i,y_i\in\mathbb{R}\hbox{ for }i=2,3,\cdots,|y_{1}|^2+\sum_{i\not=1}(|x_i|^2+|y_i|^2)<\infty\right\}. $$ For any $\textbf{z}=(x_1+\sqrt{-1}y_1,x_{2}+\sqrt{-1}y_{2},\cdots)\in\ell^2$, where $x_i,y_i\in\mathbb{R}$ for each $i\in\mathbb{N}$, write $$ \textbf{z}^{x_1}\triangleq (\sqrt{-1}y_1,x_{2}+\sqrt{-1}y_{2},\cdots). $$ Clearly, $\textbf{z}^{x_{1}}\in\ell^{2,1}$ and $\ell^{2,1}$ is a real Hilbert space (with the canonical inner product). Let $$F(\textbf{z})\triangleq D_{x_1}\left(f(\textbf{z})\right)\cdot \frac{e^{-\frac{x_1^2}{2a_1^2}}}{\sqrt{2\pi a_1^2}}.$$ Then, $$ \int_{\ell^2}D_{x_1}f\,\mathrm{d}P =\int_{\mathbb{R}\times \ell^{2,1}}F(\textbf{z})\,\mathrm{d}x_1\mathrm{d}P^{\widehat{x_1}}(\textbf{z}^{x_1}), $$ where $P^{\widehat{x_{1}}}$ is the product measure $\prod_{i=1}^{\infty}\mathcal{N}_{a_i}$ without the $x_{1}$-component, i.e., it is the restriction of the product measure $\frac{1}{\sqrt{2\pi a_{1}^2}}\cdot e^{-\frac{ y_{1}^2}{2a_{1}^2}}\,\mathrm{d}y_{1}\times\Pi_{i\not= 1}\mathcal{N}_{a_i}$ on $\left(\ell^{2,1},\mathscr{B}\big(\ell^{2,1}\big)\right)$ (similarly to that in (\ref{220817e1zx})). Applying Fubini's theorem \cite[Theorem 8.8, p. 164]{Rud87}, we have $$ \int_{\mathbb{R}\times \ell^{2,1}}F(\textbf{z})\,\mathrm{d}x_1\mathrm{d}P^{\widehat{x_1}}(\textbf{z}^{x_1})=\int_{\ell^{2,1}}\left(\int_{\mathbb{R}}F_{\textbf{z}^{x_1}}(x_1)\,\mathrm{d}x_1\right)\mathrm{d}P^{\widehat{x_1}}(\textbf{z}^{x_1}), $$ where $F_{\textbf{z}^{x_1}}(x_1)\triangleq F((x_i+\sqrt{-1}y_{i})_{i=1}^\infty)$. By our assumption, it follows that $F_{\textbf{z}^{x_1}}\in C^1(\mathbb{R})$ and $\hbox{\rm supp$\,$} F_{\textbf{z}^{x_1}}\subset [-R,R]$ for each $\textbf{z}^{x_1}\in \ell^{2,1}$. Therefore, $$ \begin{array}{ll} \displaystyle \int_{\ell^{2,1}}\left(\int_{\mathbb{R}}F_{\textbf{z}^{x_1}}(x_1)\,\mathrm{d}x_1\right)\mathrm{d}P^{\widehat{x_1}}(\textbf{z}^{x_1}) =\int_{\ell^{2,1}}\left(\int_{-R}^{R}F_{\textbf{z}^{x_1}}(x_1)\,\mathrm{d}x_1\right)\mathrm{d}P^{\widehat{x_1}}(\textbf{z}^{x_1})\\[3mm] \displaystyle =\int_{\ell^{2,1}}\left(\int_{-R}^{R}D_{x_1}\left(f(\textbf{z})\right)\cdot \frac{e^{-\frac{x_1^2}{2a_1^2}}}{\sqrt{2\pi a_1^2}}\,\mathrm{d}x_1\right)\mathrm{d}P^{\widehat{x_1}}(\textbf{z}^{x_1}) =\int_{\ell^{2,1}}\left(\int_{-R}^{R} \frac{x_1}{a_1^2}\cdot f(\textbf{z}) \cdot \frac{e^{-\frac{x_1^2}{2a_1^2}}}{\sqrt{2\pi a_1^2}}\,\mathrm{d}x_1\right)\mathrm{d}P^{\widehat{x_1}}(\textbf{z}^{x_1})\\[3mm] \displaystyle =\int_{\ell^{2,1}}\left(\int_{\mathbb{R}} \frac{x_1}{a_1^2}\cdot f(\textbf{z}) \cdot \frac{e^{-\frac{x_1^2}{2a_1^2}}}{\sqrt{2\pi a_1^2}}\,\mathrm{d}x_1\right)\mathrm{d}P^{\widehat{x_1}}(\textbf{z}^{x_1}) =\int_{\ell^{2}} \frac{x_1}{a_1^2}\cdot f(\textbf{z}) \,\mathrm{d}P (\textbf{z}), \end{array} $$ where the third equality follows from the classical Newton-Leibniz formula. The proof of Theorem \ref{Gauss-Green Theorem} is completed. \end{proof}
Recall (\ref{eq2301161}) for the definition of $\overline{\partial}_{i} $ and $\delta_i$. As a consequence of Theorem \ref{Gauss-Green Theorem}, we have the following result: \begin{corollary}\label{integration by Parts or deltai} For any $f,g\in C^{1}_{0}\left( V\right)$, it holds that $$ \int_{V} \overline{\partial}_{i} f\cdot \bar{g} \,\mathrm{d}P=-\int_{V} f\cdot\overline{ \delta_{i} g} \;\mathrm{d}P,\quad \forall\; i\in\mathbb{N}. $$ \end{corollary} \begin{proof} Since $f,g\in C^{1}_{0}\left( V\right)$, there exists $r,R\in(0,+\infty)$ such that $\displaystyle\bigcup_{\textbf{z}\in \hbox{\rm supp$\,$} f\bigcup \hbox{\rm supp$\,$} g}B_r(\textbf{z})\subset V$ and $\hbox{\rm supp$\,$} f\cup\hbox{\rm supp$\,$} g\subset B_R$. Then both $f$ and $g$ can be viewed as elements in $ C^{1}_{0} ( \ell^2 )$ by extending their values to $\ell^2\setminus (\hbox{\rm supp$\,$} f\cup \hbox{\rm supp$\,$} g)$ by 0. Thus for each $i\in\mathbb{N}$, it holds that $$ \int_{V} \overline{\partial}_{i} f\cdot \overline{g} \,\mathrm{d}P =\int_{\hbox{\rm supp$\,$} f\bigcup \hbox{\rm supp$\,$} g} \overline{\partial}_{i} f\cdot \overline{g} \,\mathrm{d}P =\int_{\ell^2} \overline{\partial}_{i} f\cdot \overline{g} \,\mathrm{d}P =\int_{\ell^2}\frac{1}{2}( D_{x_{i}}f +\sqrt{-1}D_{y_{i}}f)\cdot \overline{g} \,\mathrm{d}P. $$ By Theorem \ref{Gauss-Green Theorem}, we have $$ \begin{array}{ll} \displaystyle \int_{V} \overline{\partial}_{i} f\cdot\overline{g} \,\mathrm{d}P\\[5mm] \displaystyle = -\int_{\ell^2}f\cdot \overline{ \frac{1}{2}( D_{x_{i}}g -\sqrt{-1}D_{y_{i}}g)} \,\mathrm{d}P +\int_{\ell^2}f\cdot \overline{ \frac{x_i-\sqrt{-1}y_i}{2a_i^2} g} \,\mathrm{d}P = \int_{\ell^2}f\cdot \overline{ \left(-\partial_{i}g +\frac{\overline{z_i}}{2a_i^2}\cdot g\right) }\,\mathrm{d}P\\[5mm] \displaystyle =-\int_{\ell^2}f\cdot \overline{\delta_i g}\,\mathrm{d}P=-\int_{\hbox{\rm supp$\,$} f\bigcup \hbox{\rm supp$\,$} g}f\cdot \overline{\delta_i g}\,\mathrm{d}P =-\int_{V}f\cdot \overline{\delta_i g}\,\mathrm{d}P. \end{array} $$ This completes the proof of Corollary \ref{integration by Parts or deltai}. \end{proof}
\begin{remark} We refer to \cite{WYZ0} for a more general infinite-dimensional Gauss-Green type Theorem. \end{remark}
Now, we fix a real-valued function $\varphi \in C^{2}_{F}\left( V\right)$. For any $\phi \in C^{1}(V)$ and $i\in\mathbb{N}$, write \begin{equation}\label{230419e11} \sigma_{i} \phi\triangleq \delta_{i} \phi-\phi \cdot\partial_{i}\varphi. \end{equation} Clearly, for all $h\in C^{2}(V)$ and $i,j\in\mathbb{N}$, \begin{equation}\label{commutaor formula} \left( \overline{\partial_i }\sigma_j-\sigma_j \overline{\partial_i }\right)h=-h\cdot (\overline{\partial_i } \partial_j \varphi) -\frac{ \overline{\partial_i}( \overline{z_j})}{2a_j^2}\cdot h. \end{equation} We have the following simple result: \begin{lemma}\label{integration by Parts} For $f,g\in C^{1}_{0}\left( V\right)$, $$ \int_{V} (\overline{\partial_{i} } f)\cdot \overline{g}\cdot e^{-\varphi }\,\mathrm{d}P=-\int_{V} f\cdot\overline{(\sigma_{i} g)}\cdot e^{-\varphi }\,\mathrm{d}P,\quad \forall\; i\in\mathbb{N}. $$ \end{lemma} \begin{proof} By Corollary \ref{integration by Parts or deltai}, it follows that $$ \begin{array}{ll} \displaystyle \int_{V} (\overline{\partial_{i} } f)\cdot \overline{g}\cdot e^{-\varphi }\,\mathrm{d}P =\int_{V} \overline{\partial_{i} } (f\cdot e^{-\varphi })\cdot \overline{g}\,\mathrm{d}P +\int_{V} (\overline{\partial_{i} }\varphi)\cdot f\cdot e^{-\varphi }\cdot \overline{g}\,\mathrm{d}P\\[3mm] \displaystyle =-\int_{V}f\cdot e^{-\varphi }\cdot \overline{\delta_i g}\,\mathrm{d}P +\int_{V} (\overline{\partial_{i} }\varphi)\cdot f\cdot e^{-\varphi }\cdot \overline{g}\,\mathrm{d}P =-\int_{V} f\cdot\overline{(\sigma_{i} g)}\cdot e^{-\varphi }\,\mathrm{d}P, \end{array} $$ which completes the proof of Lemma \ref{integration by Parts}. \end{proof}
\subsection{Pseudo-convex domains in $\ell^{2}$}
In this subsection, we shall introduce the concept of pseudo-convex domains in $\ell^{2}$ and present some of their elementary properties.
For any two subsets $E_1,E_2\subset \ell^2$, we define the distance $d(E_1,E_2)$ between $E_1$ and $E_2$ by
$$
d(E_1,E_2)\triangleq
\left\{
\begin{array}{ll}
\infty, &\hbox{if }E_1=\emptyset\hbox{ or }E_2=\emptyset,\\
\inf\big\{||\textbf{z}_1-\textbf{z}_2||:\;\textbf{z}_1\in E_1,\,\textbf{z}_2\in E_2\big\},\quad\;&\hbox{otherwise}.
\end{array}\right.
$$ For a nonempty subset $S$ of $V$, set $$
d_V(S)\triangleq \min\bigg\{d(S, \partial V),\;\frac{1}{\sup\limits_{\textbf{z}\in S}||\textbf{z}||}\bigg\}. $$ Here and henceforth, we agree that $\frac{1}{0}=\infty$ and $\frac{1}{\infty}=0$. Particularly, For $\textbf{z}\in V$, let $$
d_V(\textbf{z})\triangleq \min\bigg\{d(\{\textbf{z}\}, \partial V),\frac{1}{||\textbf{z}||}\bigg\}. $$ It is clear that $d_V(\textbf{z})\geqslant d_V(S)$ for each $\textbf{z}\in S$. The following simple result is a characterization of $S \stackrel{\circ}{\subset}V$.
\begin{lemma}\label{221018lem1} Suppose that $S$ is a nonempty subset of $V$. Then, $d_V(S)>0$ if and only if $S \stackrel{\circ}{\subset}V$. \end{lemma} \begin{proof}
If $S \stackrel{\circ}{\subset}V$, then by Definition \ref{def of bounded contained}, there exist $r,R\in(0,+\infty)$ such that $\cup_{\textbf{z}\in S}B_r(\textbf{z})\subset V$ and $S\subset B_R$. Thus $d(S, \partial V)\geqslant r$, $\frac{1}{\sup\limits_{\textbf{z}\in S}||\textbf{z}||}\geqslant \frac{1}{R}$ and hence $d_V(S)\geqslant \min\left\{r,\frac{1}{R}\right\}>0.$
Conversely, if $d_V(S)>0$, then for any $r\in \left(0,d_V(S)\right)$, it holds that $d(S, \partial V)> r$ and $\frac{1}{\sup\limits_{\textbf{z}\in S}||\textbf{z}||}> r$. Therefore, we have $\cup_{\textbf{z}\in S}B_r(\textbf{z})\subset V$ and $S\subset B_{\frac{1}{r}}$, and hence $S \stackrel{\circ}{\subset}V$. This completes the proof of Lemma \ref{221018lem1}. \end{proof}
The following result is useful to construct many subsets which are uniformly contained in $V$ (Recall Definition \ref{def of bounded contained}). \begin{lemma}\label{Exhaustion function lemma} Suppose that $\alpha$ is a real-valued function on $V$. Then, $\{ \textbf{v}\in V:\;\alpha (\textbf{v}) \leqslant \tau\} \stackrel{\circ}{\subset} V$ for all $ \tau\in \mathbb{R}$ if and only if $\lim\limits_{d_V(\textbf{z})\rightarrow 0 } \alpha (\textbf{z})=+\infty.$ \end{lemma}
\begin{proof} If $\lim\limits_{d_V(\textbf{z})\rightarrow 0 } \alpha (\textbf{z})=+\infty$, then for any given $\tau\in \mathbb{R}$, there exists $\delta>0$ such that $\alpha \left( \textbf{z}\right) >\tau+1$ holds for any $\textbf{z}\in V$ satisfying $d_V(\textbf{z})<\delta $. Thus $$ \left\{ \textbf{v}\in V:\;\alpha \left( \textbf{v}\right) \leqslant \tau\right\} \subset \left\{ \textbf{v}\in V:\;d_V(\textbf{v}) \geqslant \delta \right\}
\subset \{ \textbf{v}\in V:\;d(\{\textbf{v}\},\partial V) \geqslant \delta \}\cap \bigg\{ \textbf{v}\in V:\; ||\textbf{v}|| \leqslant \frac{1}{\delta} \bigg\} \stackrel{\circ}{\subset} V. $$
Conversely, suppose that $\left\{ \textbf{v}\in V:\;\alpha \left( \textbf{v}\right) \leqslant \tau\right\} \stackrel{\circ}{\subset} V$ for all $\tau\in \mathbb{R}$. For any given $M>0$, let $$ \delta \triangleq d_V ( \{ \textbf{v}\in V:\;\alpha \left( \textbf{v}\right) \leqslant M \}). $$ By Lemma \ref{221018lem1}, we have $\delta>0$. Then for any $\textbf{z}\in V$ with $d_V(\textbf{z}) <\delta$, we claim that \begin{equation}\label{230317e1} \textbf{z}\notin \{\textbf{v}\in V:\;\alpha \left(\textbf{v}\right) \leqslant M \}.
\end{equation} Otherwise, if $\textbf{z}\in \{\textbf{v}\in V:\;\alpha \left(\textbf{v}\right) \leqslant M \}$, then $d_V(\textbf{z})\geqslant d_V ( \{ \textbf{v}\in V:\;\alpha \left( \textbf{v}\right) \leqslant M \})=\delta$, which is a contradiction. Therefore, (\ref{230317e1}) holds, and hence $\alpha \left( \textbf{z}\right) >M$, which implies that $\lim\limits_{d_V(\textbf{z})\rightarrow 0} \alpha \left( \textbf{z}\right) =+\infty$. The proof of Lemma \ref{Exhaustion function lemma} is completed. \end{proof}
Based on Definition \ref{def of bounded contained} and motivated by the definition of exhaustion functions in finite dimensions, we introduce the following notion.
\begin{definition} A real-valued function $\alpha$ on $V$ is called an exhaustion function of $V$, if \begin{equation}\label{230317e2} \{\textbf{z}\in V:\;\alpha (\textbf{z}) \leqslant \tau\} \stackrel{\circ}{\subset} V, \quad\forall\;\tau\in \mathbb{R}. \end{equation} \end{definition}
\begin{remark} Note that in view of \cite[p. 16]{Ohsawa}, in the setting of finite dimensions the counterpart of (\ref{230317e2}) takes the form that $\{ \textbf{z}\in V:\;\alpha (\textbf{z}) \leqslant \tau\}$ is compact in $V$ for all $\tau\in \mathbb{R}$. As we shall see later, (\ref{230317e2}) is more convenient in infinite-dimensional spaces. \end{remark}
In this work, we use the following notion of pseudo-convex domains in $\ell^2$: \begin{definition}\label{pseudocovexdomain} $V$ is called a pseudo-convex domain in $\ell^2$, if there is an exhaustion function $\eta \in C^{\infty}_{F}\left( V\right)$ such that for every $n\in \mathbb{N}$, the following inequality holds on $V$, \begin{equation}\label{positivity condition}
\sum^{n}_{i=1} \sum^{n}_{j=1} ( \partial_i \overline{\partial_j} \eta )\cdot \zeta_{i}\cdot\overline{\zeta_{j}} \geqslant 0,\quad\forall\;(\zeta_{1},\cdots,\zeta_n)\in \mathbb{C}^n. \end{equation} The above $\eta$ is called a plurisubharmonic exhaustion function of $V$. \end{definition}
For the function $\eta$ in Definition \ref{pseudocovexdomain} and each $\tau\in\mathbb{R}$, write \begin{equation}\label{230117e1} V_{\tau}\triangleq \left\{ \textbf{z}\in V:\;\eta \left( \textbf{z}\right) \leqslant \tau\right\}, \qquad V^{o}_{\tau}\triangleq \big\{\textbf{z}\in V:\; B_{r}(\textbf{z})\subset V_\tau\hbox{ for some }r>0\big\}. \end{equation} Clearly, $V_{\tau}\stackrel{\circ}{\subset} V$ and $V^{o}_{\tau}$ is the interior of $V_\tau$ for all $\tau\in \mathbb{R}$. On the other hand, since $V_{0}\stackrel{\circ}{\subset} V$ and $\eta \in C^{\infty}_{F}\left( V\right)$, it is clear that $\inf\limits_{V_{0}} \eta >-\infty$ and $\inf\limits_{V} \eta=\min\left(\inf\limits_{V_{0}}\eta ,\;\inf\limits_{V\setminus V_{0}}\eta \right) >-\infty$.
\begin{remark} Definition \ref{pseudocovexdomain} is essentially motivated by \cite[Theorem 2.6.2, p. 44]{Hor90} and \cite[Theorem 2.6.7, p. 46]{Hor90}. There exist several other equivalent definitions of pseudo-convex domains in infinite-dimensional spaces, e.g., \cite[p. 361]{Lem03}, \cite[Definition 37.3, p. 274]{Mujica} and \cite[Definition 2.1.3, p. 41]{Noverraz}. Our Definition \ref{pseudocovexdomain} is equivalent to these definitions but the proof of this fact is quite complicated, and therefore we shall present it in a forthcoming paper \cite{WZ}.
Clearly, $\eta$ appeared in Definition \ref{pseudocovexdomain} is not unique. Since only second order partial derivatives are used in (\ref{positivity condition}), it seems more natural to require $\eta\in C^2_{F}\left( V\right)$ in Definition \ref{pseudocovexdomain}. Nevertheless, as we shall show in \cite{WZ} that, if one can find an exhaustion function $\eta\in C^2_{F}\left( V\right)$ satisfying (\ref{positivity condition}), then there exists another exhaustion function $\tilde\eta\in C^{\infty}_{F}\left( V\right)$ enjoying the same property. On the other hand, the $C^{\infty}_{F}\left( V\right)$-smoothness for plurisubharmonic exhaustion functions of $V$ will paly a key role in our main approximation results in the rest of this paper. \end{remark}
\begin{remark}\label{230422r1}
If necessary, we may replace the plurisubharmonic exhaustion function $\eta$ in Definition \ref{pseudocovexdomain} by $\eta + \left| \left|\cdot\right| \right|^{2}_{\ell^{2} } -\inf\limits_{V_{0}} \eta $. Then, we may assume that $\eta\geqslant 0$ and for any $n\in\mathbb{N}$, instead of (\ref{positivity condition}), the following inequality holds on $V$: \begin{equation}\label{regular condition for eta}
\sum_{1\leqslant i,j\leqslant n}\left( \partial_{i} \overline{\partial_{j} } \eta\right)\cdot \zeta_{i}\cdot\overline{\zeta_{j}} \geqslant \sum^{n}_{i=1} \left| \zeta_{i}\right|^{2} ,\quad\forall\;(\zeta_{1},\cdots,\zeta_n)\in \mathbb{C}^n . \end{equation} \end{remark}
One can find many pseudo-convex domains in $\ell^2$, as shown by the following result.
\begin{proposition}\label{pseudo-convex domains in l2}
{\rm (1)} Suppose that $V$ is a pseudo-convex domain in $\ell^2$. For any $\textbf{a}\in \ell^{2}$ and $\textbf{c}=(c_{i})_{i=1}^{\infty } \in \mathbb{C}^{\infty}$ with $ 0<\inf\limits_{i\geqslant 1} \left| c_{i}\right| \leqslant \sup\limits_{i\geqslant 1} \left| c_{i}\right| <\infty $, let
$$V+\textbf{a } \triangleq \left\{ \textbf{z}+\textbf{a}:\;\textbf{z}\in V\right\}, \quad\textbf{c}V \triangleq \left\{ \left( c_{i}z_{i}\right)^{\infty }_{i=1} :\;\textbf{z}=\left(z_{i}\right)^{\infty }_{i=1} \in V\right\}.
$$ Then both $V+\textbf{a }$ and $\textbf{c}V$ are pseudo-convex domains in $\ell^2$;
{\rm (2)} For any $\textbf{a}\in \ell^{2}$ and $r\in(0,+\infty)$, $B_{r}(\textbf{a})$ is a pseudo-convex domain in $\ell^2$;
{\rm (3)} If $S\subset \mathbb{C}^{m}$ (for some $m\in\mathbb{N}$) is a pseudo-convex domain in $\mathbb{C}^{m}$ (See \cite[Definition 2.6.8, p. 47]{Hor90}), then $V\triangleq \{(\textbf{z}_m,\textbf{z}^m):\;\textbf{z}_m\in S,\,\textbf{z}^m\in \ell^2\}$ is a pseudo-convex domain in $\ell^{2}$;
{\rm (4)} The Hilbert polydisc $\mathbb{D}_2^{\infty}\triangleq \big\{(z_i)_{i=1}^{\infty}\in\ell^2:\;|z_i|<1\text{ for all }i\in\mathbb{N} \big\} $ is a pseudo-convex domain in $\ell^{2}$. \end{proposition}
\begin{proof} (1) Suppose that $\eta$ is a plurisubharmonic exhaustion function of $V$. Let $\eta_{1} \left( \textbf{z}\right)\triangleq\eta \left( \textbf{z}-\textbf{a}\right)$ for $\textbf{z}\in V+\textbf{a}$ and $\eta_{2} \left( \textbf{z}\right)\triangleq\eta \left( \frac{z_{1}}{c_{1}} ,\frac{z_{2}}{c_{2}} ,\cdots \right)$ for $\textbf{z}=\left(z_{i}\right)^{\infty }_{i=1} \in \textbf{c}V$.
Clearly, $\eta_{1} \in C^{\infty}_{F}\left( V+\textbf{a}\right)$ and $\eta_{2} \in C^{\infty}_{F}\left( \textbf{c}V\right)$. Then $\eta_{1}$ and $\eta_{2}$ are plurisubharmonic exhaustion functions of $V+\textbf{a }$ and $\textbf{c}V$, respectively.
(2) Let $\eta \left( \textbf{z}\right)\triangleq-\ln \left( 1-\left| \left| \textbf{z}\right| \right|^{2} \right)$ for $\textbf{z}\in B_1.$ Then $\eta$ is a plurisubharmonic exhaustion function of $B_1$. By the conclusion (1), we see that $B_{r}(\textbf{a})$ is a pseudo-convex domain in $\ell^2$.
(3) By \cite[Theorem 2.6.11, p. 48]{Hor90}), there exists a strictly plurisubharmonic exhaustion function $\eta \in C^{\infty }\left(S \right)$ of $S$. Let $\tilde{\eta}(\textbf{z})\triangleq\eta(\textbf{z}_m)+\left| \left| \textbf{z}\right| \right|^{2}$ for $\textbf{z}=(\textbf{z}_m,\textbf{z}^m)\in V$. It is easy to check that $\tilde{\eta}$ is a plurisubharmonic exhaustion function of $V$.
(4) Let $\eta ( \textbf{z})\triangleq\prod^{\infty }_{i=1} \frac{1}{1-|z_{i}|^{2}}$ for $\textbf{z}=(z_i)_{i=1}^{\infty}\in\mathbb{D}_2^{\infty}$. Then $\eta$ is a plurisubharmonic exhaustion function of $\mathbb{D}_2^{\infty}$. This completes the proof of Proposition \ref{pseudo-convex domains in l2}. \end{proof}
We shall use the following assumption frequently. \begin{condition}\label{230424c1} Suppose that $V$ is a pseudo-convex domain in $\ell^{2}$ and $\eta$ is a plurisubharmonic exhaustion function of $V$. \end{condition}
The following result is about some elementary properties of pseudo-convex domains in $\ell^{2}$ and their plurisubharmonic exhaustion functions (Recall (\ref{230117e1}) for $V_{\tau}$ and $V^{o}_{\tau}$).
\begin{proposition}\label{properties on pseudo-convex domain} Let Condition \ref{230424c1} hold. Then, \begin{itemize} \item[$(1)$] For each $\tau\in\mathbb{R}$, there exists $C(\tau)\in(0,+\infty)$ such that \begin{equation}\label{230319e1}
\left| \eta \left(\textbf{z}\right) -\eta \left( \tilde{\textbf{z}} \right) \right| \leqslant C(\tau)\left| \left|\, \textbf{z}-\tilde{\textbf{z}}\,\right| \right|,\quad\forall\; \textbf{z},\tilde{\textbf{z}}\in V_{\tau}; \end{equation} \item[$(2)$] For any $\tau_1,\tau_2\in\mathbb{R}$ with $\tau_1<\tau_2$, it holds that $V_{\tau_1}\stackrel{\circ}{\subset} V^{o}_{\tau_2}$; \item[$(3)$] For any $S\stackrel{\circ}{\subset} V$, there exists $\tau\in\mathbb{R}$ such that $S\stackrel{\circ}{\subset} V^{o}_{\tau}$. \end{itemize} \end{proposition} \begin{proof} (1) Choose a real-valued function $\psi \in C^{\infty }\left( \mathbb{R} \right)$ such that $0\leqslant \psi \leqslant 1,\psi \left( x\right) =1$ for all $x\leqslant \tau$ and $\psi \left( x\right) =0$ for all $x>\tau+1$. Let $\Psi \left( \textbf{z}\right)\triangleq \eta \left( \textbf{z}\right) \psi \left( \eta \left( \textbf{z}\right) \right)$ for $\textbf{z}\in V$ and $\Psi \left( \textbf{z}\right)\triangleq 0$ for $\textbf{z}\in \ell^2\setminus V$. Then $\Psi\in C^{\infty}_{0,F}( \ell^2).$ For each $n\in\mathbb{N}$, set $$ \eta_n(\textbf{z}_n)\triangleq \int \Psi(\textbf{z}_n,\textbf{z}^n)\,\mathrm{d}P_n(\textbf{z}^n), $$ where $\textbf{z}_n=(x_{i} +\sqrt{-1}y_{i})_{i=1}^{n}\in \mathbb{C}^n$ and $\textbf{z}^n=(x_{i} +\sqrt{-1}y_{i})_{i=n+1}^\infty\in \ell^2$.
By H\"older's inequality, it follows that $$ \begin{array}{ll} \displaystyle
\sum^{n}_{j=1}\big(| D_{x_j} \eta_{n}|^{2}+| D_{y_j} \eta_{n}|^{2}\big)
= \sum^{n}_{j=1}\bigg(\bigg|\int D_{x_j} \Psi\,\mathrm{d}P_{n}\bigg|^{2}+\bigg|\int D_{y_j} \Psi\,\mathrm{d}P_{n}\bigg|^{2}\bigg)\\[3mm]
\displaystyle \leqslant \int \sum^{n}_{j=1}\bigg( |D_{x_j} \Psi|^{2}+|D_{y_j} \Psi|^{2}\bigg)\,\mathrm{d}P_{n}\\[3mm]
\displaystyle \leqslant \sup_{V} \sum^{\infty }_{j=1} \big(| D_{x_j}\Psi|^{2}+| D_{y_j} \Psi|^{2}\big)<\infty, \end{array} $$ where the last inequality follows from the fact that $\Psi\in C^{\infty}_{0,F}( \ell^2)$.
Write $C(\tau)\triangleq \max\left\{1,\sqrt{\sup\limits_{V} \sum\limits^{\infty }_{j=1} \big(| D_{x_j}\Psi|^{2}+| D_{y_j} \Psi|^{2}\big)}\right\}$. Then by the classical mean value theorem, we have $$
\left| \eta_{n}\left( \textbf{z}_n\right) -\eta_{n}\left(\tilde{\textbf{z}}_n\right) \right| \leqslant C(\tau)\left| \left| \textbf{z}_n -\tilde{\textbf{z}}_n \right| \right|_{\mathbb{C}^{n} } ,\quad \forall\;\textbf{z}_n ,\tilde{\textbf{z}}_n \in \mathbb{C}^{n}. $$ Viewing $\eta_n$ as a cylinder function on $\ell^2$, we have $$
| \eta_{n}( \textbf{z}) -\eta_{n}(\tilde{\textbf{z}}) |=| \eta_{n}( \textbf{z}_n) -\eta_{n}(\tilde{\textbf{z}}_n) | \leqslant C(\tau)\left| \left| \textbf{z}_n -\tilde{\textbf{z}}_n \right| \right|_{\mathbb{C}^{n} } \leqslant C(\tau)\left| \left| \textbf{z} -\tilde{\textbf{z}} \right| \right|,\quad \forall\; \textbf{z},\tilde{\textbf{z}}\in\ell^2, $$ where $\tilde{\textbf{z}}^n,\textbf{z}^n\in \ell^2,\,\, \tilde{\textbf{z}}_n,\textbf{z}_n\in \mathbb{C}^n$ and $\textbf{z}=(\textbf{z}_n,\textbf{z}^n)$, $\tilde{\textbf{z}}=(\tilde{\textbf{z}}_n,\tilde{\textbf{z}}^n)$. By the conclusion (2) of Proposition \ref{Reduce diemension}, there exists a subsequence $\{\eta_{n_{k}}\}_{k=1}^{\infty}$ and a Borel subset $E_0$ of $\ell^2$ with $P(E_0)=0$ such that $\lim\limits_{k\rightarrow \infty } \eta_{n_{k}}(\textbf{z})=\Psi(\textbf{z})$ for any $\textbf{z}\in \ell^2\setminus E_0$. For any $\textbf{z}, \tilde{\textbf{z}}\in\ell^2\setminus E_0$, letting $k\to\infty$, we have $$
| \Psi( \textbf{z}) -\Psi(\tilde{\textbf{z}}) | \leqslant C(\tau)\left| \left| \textbf{z} -\tilde{\textbf{z}} \right| \right|. $$ Combining the continuity of $\Psi$ with the above inequality and by Corollary \ref{property of strictly positive measure}, we see that $$
| \Psi( \textbf{z}) -\Psi(\tilde{\textbf{z}}) | \leqslant C(\tau)\left| \left| \textbf{z} -\tilde{\textbf{z}} \right| \right|,\quad \forall\;\textbf{z},\tilde{\textbf{z}}\in\ell^2. $$ Especially, we have the desired inequality (\ref{230319e1}).
(2) Obviously, $V_{\tau_1}\subset V^{o}_{\tau_2}$. Note that for $\textbf{z}\in V_{\tau_1}$ and $\tilde{\textbf{z}}\in \partial V^{o}_{\tau_2}$, we have $\eta(\textbf{z}) \leqslant \tau_1$ and $\eta(\tilde{\textbf{z}})=\tau_2$. By the above conclusion (1), we have $$
0<\tau_2-\tau_1<\left| \eta \left( \textbf{z} \right) -\eta \left( \tilde{\textbf{z}}\right) \right| \leqslant C\left( \tau_2\right) \left| \left|\textbf{z}-\tilde{\textbf{z}}\right| \right|, $$ which implies that $d(V_{\tau_1},\partial V^{o}_{\tau_2})\geqslant \frac{\tau_2-\tau_1}{C(\tau_2)}>0$ and $V_{\tau_1}$ is a bounded subset of $\ell^2$. Thus $V_{\tau_1}\stackrel{\circ}{\subset}V^{o}_{\tau_2}$.
(3) Firstly, we use the contradiction argument to prove that there exists $\tau_0\in\mathbb{R}$ such that $S\subset V_{\tau_0}$. Otherwise, for each $n\in\mathbb{N}$, there would exist $\textbf{z}_n\in S\backslash V_n$ and hence, by the definition of $V_\tau$ in (\ref{230117e1}), one has $\eta(\textbf{z}_n)>n$, which contradicts the fact that $\sup\limits_{\textbf{z}\in S}|\eta(\textbf{z})|<\infty$ (which follows from the assumption $\eta \in C^{\infty}_{F}\left( V\right)$ and the definition of $C^{\infty}_{F}\left( V\right)$). Then choosing $\tau>\tau_0$, by the above conclusion (2), we have $S\subset V_{\tau_0}\stackrel{\circ}{\subset}V^{o}_{\tau}$. The proof of Proposition \ref{properties on pseudo-convex domain} is thus completed. \end{proof}
\section{$\overline{\partial}$ operators on pseudo-convex domains}\label{secx3}
In the sequel, unless other stated, we fix $s,t\in \mathbb{N}_0$ with $s+t\geqslant 1$, a $\sigma$-finite Borel measure $\mu$ on $V$, a Borel subset $E$ of $V$ and a real-valued Borel-measurable function $w$ on $E$.
Write $$ (\ell^2)^{s+t}=\underbrace{\ell^2\times\ell^2\times\cdots\times\ell^2}_{s+t\hbox{ \tiny times}}. $$ For any strictly increasing multi-indices $I=(i_1,\cdots,i_s)$ and $J=(j_1,\cdots,j_t)$, i.e., $i_1,\cdots,i_s, j_1,\cdots,$ $j_t\in \mathbb{N}$, $i_1<\cdots<i_s$ and $j_1<\cdots<j_t$, we fix a number \begin{equation}\label{defnition of general st froms} c_{I,J}>0, \end{equation} and define $I\cup J\triangleq\{i_1,\cdots,i_s\}\cup \{j_1,\cdots,j_t\}$. For $j\in \mathbb{N}$, we define $J\cup\{j\}=\{j_1,\cdots,j_t\}\cup\{j\}$. As in \cite[p. 530]{YZ}, we define a complex-valued function $\mathrm{d}z^I\wedge \mathrm{d}\overline{z}^J$ on $(\ell^2)^{s+t}$ by $$ (\mathrm{d}z^I\wedge \mathrm{d}\overline{z}^J)(\textbf{z}^1,\cdots,\textbf{z}^{s+t})\triangleq \frac{1}{\sqrt{(s+t)!}}\sum_{\sigma\in S_{s+t}}(-1)^{s(\sigma)}\cdot\prod_{k=1}^{s}z_{i_k}^{\sigma_k}\cdot\prod_{l=1}^{t}\overline{z_{j_l}^{\sigma_{s+l}}}, $$ where $\textbf{z}^l=(z_j^l)_{j=1}^\infty\in\ell^2,\,\,l=1,\cdots, s+t$, $S_{s+t}$ is the permutation group of $\{1,\cdots,s+t\}$, $s(\sigma)$ is the sign of $\sigma=(\sigma_1,\cdots,\sigma_s,\sigma_{s+1},\cdots,\sigma_{s+t})$, and we agree that $0!=1$. We call the following (formal) summation an $(s,t)$-form on $E$:
\begin{equation}\label{230320e1}
\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| J\right| =t} f_{I,J}\,\mathrm{d}z_{I}\wedge \,\mathrm{d}\overline{z_{J}},
\end{equation}
where the sum $\sum\limits^{\prime }_{\left| I\right| =s} \sum\limits_{\left| J\right| =t}^{\prime } $ is taken only over strictly increasing multi-indices $I$ and $J$ with $\left| I\right| =s$ and $\left| J\right| =t$ (for which $\left| I\right|$ and $\left| J\right|$ stand for respectively the cardinalities of sets $\{i_1,\cdots,i_s\}$ and $\{j_1,\cdots,j_t\}$), and $f_{I,J}$ is a function on $E$ for any strictly increasing multi-indices $I$ and $J$. Clearly, each $f_{I,J}\,\mathrm{d}z_{I}\wedge \,\mathrm{d}\overline{z_{J}}$ can be viewed as a function on $E\times (\ell^2)^{s+t}$. Nevertheless, in the present setting of infinite dimensions (\ref{230320e1}) is an infinite series for which the convergence is usually not guaranteed, and therefore it is a formal summation (unless further conditions are imposed).
We need to introduce some working spaces which will play key roles in the sequel. Denote by $L^{2}\left(V,loc\right)$ the set of all Borel-measurable functions $f$ on $V$ for which $$
\int_{S} \left| f\right|^{2} \,\mathrm{d}P<\infty,\quad\forall\; S\in \mathcal{B}(V)\hbox{ with }S\stackrel{\circ}{\subset} V; $$
by $L^{2}_{\left( s,t\right) }\left( V,loc\right)$ the set of all $(s,t)$-forms $\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| J\right| =t} f_{I,J}\,\mathrm{d}z_{I}\wedge \,\mathrm{d}\overline{z_{J}}$ on $V$ for which $$
f_{I,J}\in L^{2}\left(V,loc\right)\hbox{ for each } I,J,\hbox{ and }\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| J\right| =t}c^{ }_{I,J} \int_{S} \left| f_{I,J}\right|^{2}\,\mathrm{d}P<\infty ,\;\forall\; S\in \mathcal{B}(V)\hbox{ with } S\stackrel{\circ}{\subset} V; $$
by $L^{2}_{\left( s,t\right) }\left( E,\mu \right)$ the set of all $(s,t)$-forms $\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| J\right| =t} f_{I,J}\,\mathrm{d}z_{I}\wedge \,\mathrm{d}\overline{z_{J}}$ on $E$ for which $$
f_{I,J}\in L^2(E,\mu)\hbox{ for each } I,J,\hbox{ and }\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| J\right| =t} c^{ }_{I,J}\int_{E} \left| f_{I,J}\right|^{2} \,\mathrm{d}\mu <\infty; $$
by $L^{2}_{\left( s,t\right)}\left(E,w\right)$ the set of all $(s,t)$-forms $\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| J\right| =t} f_{I,J}\,\mathrm{d}z_{I}\wedge \,\mathrm{d}\overline{z_{J}}$ on $E$ for which $$
f_{I,J}\in L^2(E,e^{-w}P)\hbox{ for each } I,J,\hbox{ and }\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| J\right| =t} c^{ }_{I,J}\int_{E} \left| f_{I,J}\right|^{2} e^{-w}dP<\infty . $$
\begin{remark} Clearly, both $L^{2}_{(s,t)}\left(E,w\right)$ and $L^{2}_{(s,t)}\left(E,\mu\right)$, endowed respectively with the following norms $$
\left|\left|f\right|\right|_{L^{2}_{\left( s,t\right) \left( E,\mu \right)}}=\sqrt{\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| J\right| =t} c^{ }_{I,J}\int_{E} \left| f_{I,J}\right|^{2} \,\mathrm{d}\mu},\quad\forall\;f=\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| J\right| =t} f_{I,J}\,\mathrm{d}z_{I}\wedge \,\mathrm{d}\overline{z_{J}}\in L^{2}_{\left( s,t\right) \left( E,\mu \right)} $$ and $$
\left|\left|f\right|\right|_{L^{2}_{\left( s,t\right) \left( E,w \right)}}=\sqrt{\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| J\right| =t} c^{ }_{I,J}\int_{E} \left| f_{I,J}\right|^{2} \,e^{-w}dP},\quad\forall\;f=\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| J\right| =t} f_{I,J}\,\mathrm{d}z_{I}\wedge \,\mathrm{d}\overline{z_{J}}\in L^{2}_{\left( s,t\right) \left( E,w \right)}, $$ are Hilbert spaces. On the other hand, $L^{2}_{\left( s,t\right) }\left( V,loc\right)$ can be viewed as the set of all locally square-integrable $(s,t)$-forms on $V$ (In particular, $L^{2}\left( V,loc\right)$ is the set of all locally square-integrable functions on $V$). \end{remark}
\begin{lemma}\label{density1} Assume that $\textbf{z}_0\in\ell^2$, $r\in(0,+\infty)$ and $\mu $ is a Borel measure on $B_r(\textbf{z}_0)$ such that $\mu \left( E\right)<\infty$ for all Borel subset $E\stackrel{\circ}{\subset} B_r(\textbf{z}_0)$. Then $C^{\infty }_{0,F^{\infty}}\left(B_r(\textbf{z}_0)\right)$ is dense in $L^{p}\left(B_r(\textbf{z}_0),\mu \right)$ for each $ p\in [1,\infty) $. \end{lemma}
\begin{proof}
Let $\eta \left( \textbf{z}\right)\triangleq-\ln \left( 1-\left| \left| \frac{\textbf{z}-\textbf{z}_0}{r}\right| \right|^{2} \right),\,\textbf{z}\in B_r(\textbf{z}_0)$ and $(B_r(\textbf{z}_0))_n^o\triangleq \{\textbf{z}'\in B_r(\textbf{z}_0):\;\eta(\textbf{z}')<n\},\,n\in\mathbb{N}$. Choose $\varphi_n\in C^{\infty}(\mathbb{R})$ such that $0\leqslant \varphi_n\leqslant 1$, $\varphi_n(x)=1$ for all $x\leqslant n-1$ and $\varphi_n(x)=0$ for all $x\geqslant n$. Let $\Phi_n(\cdot)\triangleq \varphi_n(\eta(\cdot))$ and $\mu_n(E)\triangleq\mu(E)$, $\forall\;E\in \mathscr{B}((B_r(\textbf{z}_0))_n^o)$. Then, $\Phi_n(\cdot)\in C^{\infty }_{0,F^{\infty}}\left(B_r(\textbf{z}_0)\right)$. For each $f\in L^{p}\left( B_r(\textbf{z}_0),\mu \right)$, one can view $\Phi_n f$ as an element of $L^p((B_r(\textbf{z}_0))_n^o,\mu_n)$. Moreover, by Lebesgue's Dominated Convergence Theorem, it holds that \begin{equation}\label{230123e1}
\lim_{n\to\infty}\int_{ B_r(\textbf{z}_0)}|\Phi_n f-f|^p\,\mathrm{d}\mu=0. \end{equation} For each $n\in\mathbb{N}$, we choose $\{\psi_m\}_{m=1}^{\infty}\subset C^{\infty}(\mathbb{R})$ such that $0\leqslant \psi_m\leqslant 1$, $\psi_m(x)=1$ for all $x<n-\frac{2}{m}$ and $\psi_m(x)=0$ for all $x\geqslant n-\frac{1}{m},\,m\in\mathbb{N}$. Then $\lim\limits_{m\to\infty}\psi_m(x)=1$ for all $x<n$ and $\lim\limits_{m\to\infty}\psi_m(x)=0$ for all $x\geqslant n$. Let $\Psi_m(\cdot)\triangleq \psi_m(\eta(\cdot))$. Then $\Psi_m(\cdot)\in C^{\infty }_{0,F^{\infty}}\left( (B_r(\textbf{z}_0))_n^o\right)$.
Write $$ \begin{array}{ll} \displaystyle \mathbb{M}_n \triangleq C^{\infty }_{0,F^{\infty}}\left( (B_r(\textbf{z}_0))_n^o\right),\\[3mm] \displaystyle \mathbb{N}_n\triangleq
\left\{f\in L^p((B_r(\textbf{z}_0))_n^o,\mu_n):\;\exists\; \{g_k\}_{k=1}^{\infty}\subset \mathbb{M}_n\text{ such that }\lim_{k\to\infty}\int_{(B_r(\textbf{z}_0))_n^o}|g_k-f|^p\,\mathrm{d}\mu_n=0\right\},\\[3mm] \displaystyle
\mathbb{H}_n\triangleq \left\{f\in \mathbb{N}_n:\; f\text{ is a bounded Borel-measurable function on }(B_r(\textbf{z}_0))_n^o \right\}. \end{array} $$ Then $\mathbb{H}_n$ and $\mathbb{M}_n$ are two family of bounded complex-valued functions on $(B_r(\textbf{z}_0))_n^o$ such that:
\begin{enumerate}
\item[(1)] $\mathbb{H}_n$ is a complex linear subspace which is closed under the complex conjugation and the bounded convergence;
\item[(2)] $\mathbb{M}_n\subset \mathbb{H}_n$ and $\mathbb{M}_n$ is a multiplicative system;
\item[(3)] $1\in \mathbb{H}_n$, which follows from the fact that $\{\Psi_m\}_{m=1}^{\infty}\subset \mathbb{M}_n$ and $\lim\limits_{m\to\infty}\Psi_m(\textbf{z})=1$ for all $\textbf{z}\in (B_r(\textbf{z}_0))_n^o$.
\end{enumerate}
\noindent By Theorem \ref{Complex Multiplicative System Theorem}, $\mathbb{H}_n$ contains all bounded complex valued $\sigma(\mathbb{M}_n)$-measurable functions.
We claim that \begin{equation}\label{230321e1} \sigma(\mathbb{M}_n)=\mathscr{B}((B_r(\textbf{z}_0))_n^o). \end{equation}
Indeed, it is easy to see that $\sigma(\mathbb{M}_n)\subset\mathscr{B}((B_r(\textbf{z}_0))_n^o)$. On the other hand, for each $\textbf{z}_0'\in (B_r(\textbf{z}_0))_n^o$ and $r_0>0$ so that the open ball $B_{r_0}(\textbf{z}_0')\subset (B_r(\textbf{z}_0))_n^o$, choose an integer $k_0>\frac{1}{r_0^2}$. For each integer $k\geqslant k_0$, choose $\varphi_k \in C_c^{\infty}(\mathbb{R})$ such that $0\leqslant \varphi_k\leqslant 1$, $\varphi_k(x)=1$ for all $x\in \mathbb{R}$ with $0\leqslant |x|\leqslant r_0^2-\frac{1}{k}$, and $\varphi_k(x)=0$ for all $x\in \mathbb{R}$ satisfying $|x|\geqslant r_0^2-\frac{1}{2k}$. Let $\Phi_k(\textbf{z})\triangleq \varphi_k(||\textbf{z}-\textbf{z}_0'||^2),\,\textbf{z}\in B_r(\textbf{z}_0),\,k\geqslant k_0$. It is easy to show that $\{\Phi_k\}_{k=k_0}^{\infty}\subset \mathbb{M}_n $, and hence $B_{r_0}(\textbf{z}_0')=\bigcup\limits_{k=k_0}^{\infty}\left\{\textbf{z}\in B_r(\textbf{z}_0):\Phi_k(\textbf{z})=1\right\}\in \sigma(\mathbb{M}_n)$. Since $(B_r(\textbf{z}_0))_n^o$ is a Lindel\"{o}f space, each non-empty open subset of $(B_r(\textbf{z}_0))_n^o$ is a countable union of open balls contained in $(B_r(\textbf{z}_0))_n^o$. Therefore, $\sigma(\mathbb{M}_n)$ contains all open subset of $(B_r(\textbf{z}_0))_n^o$, and hence (\ref{230321e1}) holds.
By (\ref{230321e1}), we conclude that $\mathbb{H}_n$ contains all bounded complex valued Borel-measurable functions. Combining the fact that the set of all bounded complex valued Borel-measurable functions is dense in $L^p((B_r(\textbf{z}_0))_n^o,\mu_n)$, we see that $L^p((B_r(\textbf{z}_0))_n^o,\mu_n)\subset \mathbb{N}_n$, and hence $L^p((B_r(\textbf{z}_0))_n^o,\mu_n)= \mathbb{N}_n$, which indicates that $ C^{\infty }_{0,F^{\infty}}\left( (B_r(\textbf{z}_0))_n^o\right)$ is dense in $L^p((B_r(\textbf{z}_0))_n^o,\mu_n)$. By (\ref{230123e1}) and noting that $\Phi_n f\in L^p((B_r(\textbf{z}_0))_n^o,\mu_n)$ for each $n$, we conclude that $C^{\infty }_{0,F^{\infty}}\left( B_r(\textbf{z}_0) \right)$ is dense in $L^{p}\left( B_r(\textbf{z}_0),\mu \right)$. The proof of Lemma \ref{density1} is completed. \end{proof}
As a consequence of Lemma \ref{density1}, we have the following result. \begin{corollary}\label{Test functions} Suppose that $ p\in(1,+\infty)$, $\mu$ is a Borel measure on $V$ such that $\mu \left( S\right)<\infty$ for all Borel subset $S \stackrel{\circ}{\subset} V$, and $f$ is a Borel-measurable function on $V$ so that for each $\textbf{z}_0\in V$, there exists $r\in(0,+\infty)$ for which $B_{r}(\textbf{z}_0)\stackrel{\circ}{\subset} V$, $\chi_{B_{r}(\textbf{z}_0)}\cdot f\in L^{p}\left(V,\mu \right)$ and \begin{equation}\label{jichuchou}
\int_{V} f\bar\phi d\mu =0,\quad\forall\;\phi\in C^{\infty }_{0,F^{\infty }}\left( B_{r}(\textbf{z}_0)\right). \end{equation} Then, $f=0$ almost everywhere with respect to $\mu$. \end{corollary} \begin{proof}
By Lemma \ref{density1}, $C^{\infty }_{0,F^{\infty }}\left( B_{r}(\textbf{z}_0)\right)$ is dense in $L^q(B_{r}(\textbf{z}_0),\mu\mid_{B_{r}(\textbf{z}_0)})$, where $q\triangleq\frac{1}{1-\frac{1}{p}}$ and $\mu\mid_{B_{r}(\textbf{z}_0)}(E)\triangleq \mu(E)$ for all $E\in \mathscr{B}\left(B_{r}(\textbf{z}_0)\right)$. Note that $\overline{\text{sgn}(f)}\cdot |f|^{p-1}\in L^q(B_{r}(\textbf{z}_0),\mu\mid_{B_{r}(\textbf{z}_0)})$ where sgn$(z)\triangleq \frac{z}{|z|}$ if $z\in\mathbb{C}\setminus\{0\}$ and sgn$(0)\triangleq 0$. By (\ref{jichuchou}), it holds that $$
\int_{B_{r}(\textbf{z}_0)}|f|^p\,\mathrm{d}\mu=0. $$ Since $V$ is a Lindel\"{o}f space, we may cover $V$ by countably many balls as above. Combining the Monotone Convergence Theorem, we conclude that $$
\int_{V}|f|^p\,\mathrm{d}\mu=0, $$ which implies that $f=0$ almost everywhere with respect to $\mu$. Hence we completed the proof of Corollary \ref{Test functions}. \end{proof}
As a further consequence of Lemma \ref{density1}, we have the following density result for the space of square-integrable functions on $V$. \begin{corollary}\label{density2} Assume that $\mu $ is a Borel measure on $V$ such that $\mu \left( S\right)<\infty$ for all Borel subset $S \stackrel{\circ}{\subset} V$. Then $C^{\infty }_{0,F^{\infty}}\left(V \right)$ is dense in $L^{2}\left( V,\mu \right)$. \end{corollary} \begin{proof} Note that $L^{2}\left( V,\mu \right)$ is a Hilbert space and obviously $C^{\infty }_{0,F^{\infty }}\left( V\right) \subset L^{2}\left( V,\mu \right)$. Write
$$
\left( C^{\infty }_{0,F^{\infty }}\left( V\right) \right)^{\bot }=\left\{f\in L^{2}\left( V,\mu \right):\;\int_Vf\bar gd\mu=0,\quad\forall\;g\in C^{\infty }_{0,F^{\infty }}\left( V\right)\right\}.
$$ For any $f\in \left( C^{\infty }_{0,F^{\infty }}\left( V\right) \right)^{\bot }$, we have $\int_{V} f\bar gd\mu=0$ for all $g\in C^{\infty }_{0,F^{\infty }}\left( V\right)$. By Corollary \ref{Test functions}, $f=0\in L^2\left( V,\mu \right)$, which implies that $C^{\infty }_{0,F^{\infty}}\left(V \right)$ is dense in $L^{2}\left( V,\mu \right)$. The proof of Corollary \ref{density2} is completed.
\end{proof}
Further, we have the following density result for the space of square-integrable $(s,t)$-forms on $V$. \begin{lemma}\label{0orderSobolev} Suppose that $w$ is a real-valued Borel-measurable function on $V$ such that $\int_Ee^{-w}\,\mathrm{d}P<\infty$ for all Borel subset $E\stackrel{\circ}{\subset} V$. Then $$
\bigcup^{\infty }_{n=1} \bigg\{\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t,\max \left\{ I\bigcup J\right\} \leqslant n} f_{I,J}dz_{I}\wedge d\overline{z_{J}} :\;f_{I,J}\in C^{\infty }_{0,F^{\infty}}\left( V\right)\text{ for all involved }I,J \bigg\} $$ is dense in $L^{2}_{(s,t) }( V,w)$. \end{lemma} \begin{proof} Obviously, $$
\bigcup^{\infty }_{n=1} \bigg\{\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t,\max \left\{ I\bigcup J\right\} \leqslant n} f_{I,J}dz_{I}\wedge d\overline{z_{J}} :\;f_{I,J}\in C^{\infty }_{0,F^{\infty}}\left( V\right)\text{ for all involved }I,J \bigg\}\subset L^{2}_{(s,t)}\left( V,w\right) . $$
For any $ f=\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t} f_{I,J}dz_{I}\wedge d\overline{z_{J}} \in L^{2}_{\left( s,t\right) }\left( V,w\right)$ and $\varepsilon >0$, there exists $n\in \mathbb{N}$ such that $$
\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t,\max \left\{ I\bigcup J\right\} >n} c_{I,J}\int_{V} \left| f_{I,J}\right|^{2} e^{-w}dP<\frac{\varepsilon }{2}. $$
By Corollary \ref{density2}, $C_{0,F^{\infty}}^{\infty}(V)$ is dense in $L^{2}(V,e^{-w}P)$. Then, for each strictly increasing multi-indices $I$ and $J$ with $\left| I\right| =s,\left| J\right| =t$ and $\max \left\{ I\bigcup J\right\} \leqslant n$, choosing $\varphi_{I,J} \in C^{\infty }_{0,F^{\infty}}\left( V\right)$ such that $$
\int_{V} \left| \varphi_{I,J} -f_{I,J}\right|^{2} e^{-w}dP<\frac{\varepsilon }{2n^{t+s}\max\limits_{\left| I\right| =s,\left| J\right| =t,\max \left\{ I\bigcup J\right\} \leqslant n} c_{I,J}} $$
and let $\varphi \triangleq\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t,\max \left\{ I\bigcup J\right\} \leqslant n} \varphi_{I,J} dz_{I}\wedge d\overline{z_{J}}.$ It is clear that $\left| \left| \varphi -f\right| \right|^{2}_{L^{2}_{(s,t)}\left( V,w\right) } <\varepsilon $. This completes the proof of Lemma \ref{0orderSobolev}. \end{proof}
Motivated by Corollary \ref{Test functions}, we have the following notions of supports of locally square-integrable functions/forms. \begin{definition}\label{230124def1} For any $f\in L^2(V,loc)$, the support of $f$, denoted by $\hbox{\rm supp$\,$} f$, is the set of all points $\textbf{z}_0$ in $V$ for which there exists no $r>0$ so that $B_{r}(\textbf{z}_0)\subset V$ and $\int_{B_{r}(\textbf{z}_0)} f\bar\phi dP=0$ for all $\phi \in C^{\infty }_{0,F^{\infty}}\left( B_{r}(\textbf{z}_0)\right).$
Generally, for any $$
f=\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| J\right| =t} f_{I,J}\,\mathrm{d}z_{I}\wedge \,\mathrm{d}\overline{z_{J}}\in L^{2}_{\left( s,t\right) }\left( V,loc\right), $$
the support of $f$, denoted by $\hbox{\rm supp$\,$} f$, is the set $\overline{\bigcup\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t} \hbox{\rm supp$\,$} f_{I,J}}$, where the union $\bigcup\limits^{\prime }$ is taken only over all strictly increasing multi-indices $I$ and $J$ with $\left| I\right| =s$ and $\left| J\right| =t$. \end{definition}
For each $n\in\mathbb{N}$, choose a real-valued function \begin{equation}\label{eq230131e1} \gamma_{n} \in C^{\infty }_{c}\left( \mathbb{C}^{n} \right) \end{equation} such that \begin{itemize} \item[(1)] $\gamma_{n} \geqslant 0$;
\item[(2)] $\int_{\mathbb{C}^{n} } \gamma_{n} \left( \textbf{z}_n\right) d\textbf{z}_n=1$, where $d\textbf{z}_n$ is the Lebesuge measure on $\mathbb{C}^n$;
\item[(3)]
$\hbox{\rm supp$\,$}\gamma_{n} \subset \{ \textbf{z}_n \in \mathbb{C}^{n} :\;\left| \left| \textbf{z}_n\right| \right|_{\mathbb{C}^{n} } \leqslant 1\} ;$
\item[(4)] $\gamma_{n} ( \textbf{z}_n) =\gamma_{n} ( \textbf{z}_n')$ for all $\textbf{z}_n,\textbf{z}_n'\in\mathbb{C}^{n}$ with $| | \textbf{z}_n | |_{\mathbb{C}^{n}} =\left| \left| \textbf{z}_n'\right| \right|_{\mathbb{C}^{n}} $. \end{itemize} For $\delta\in(0,+\infty)$, set $$\gamma_{n,\delta } \left( \textbf{z}_n\right) \triangleq \frac{1}{\delta^{2n} } \gamma_n \left( \frac{\textbf{z}_n }{\delta } \right),\quad\forall\;\textbf{z}_n \in \mathbb{C}^{n}. $$ Suppose that $f\in L^2(\ell^2,P)$. By the conclusion (1) of Proposition \ref{Reduce diemension}, we see that $f_n\in L^2(\mathbb{C}^n,\mathcal{N}^n)$, where $f_n$ is given by (\ref{Integration reduce dimension}). Suppose further that $\hbox{\rm supp$\,$} f\stackrel{\circ}{\subset} \ell^2$. Then, for any $\delta\in(0,+\infty)$, we may define the convolution $f_{n,\delta}(\cdot)$ of $f_n$ and $\gamma_{n,\delta}$ by \begin{equation}\label{Convolution after reduce dimension} f_{n,\delta}(\textbf{z}_n)\triangleq \int_{\mathbb{C}^n}f_n(\textbf{z}_n')\gamma_{n,\delta } ( \textbf{z}_n -\textbf{z}_n')\mathrm{d}\textbf{z}_n',\quad\;\textbf{z}_n \in \mathbb{C}^{n}. \end{equation} Then, $f_{n,\delta}\in C_c^{\infty}(\mathbb{C}^n)$. Set \begin{equation}\label{def for Gauss weight}
\varphi_n(\textbf{z}_n)\triangleq\prod_{i=1}^{n}\left(\frac{1}{2\pi a_i^2}\cdot e^{-\frac{|z_i|^2}{2a_i^2}}\right),\quad \; \textbf{z}_n=(z_1,\cdots,z_n)\in \mathbb{C}^n. \end{equation} We have the following result. \begin{proposition}\label{convolution properties} Suppose that $f\in L^2(\ell^2,P)$ with $\hbox{\rm supp$\,$} f\stackrel{\circ}{\subset} \ell^2$. Then, for each $n\in\mathbb{N}$, it holds that \begin{itemize}
\item[{\rm (1)}] $f_n\in L^2(\mathbb{C}^n, d\textbf{z}_n)\cap L^1(\mathbb{C}^n, d\textbf{z}_n)$, $f_{n,\delta}\in L^2(\ell^2,P)$ for any $\delta>0$ and $\lim\limits_{\delta\to 0+}||f_{n,\delta}-f_n||_{L^2(\ell^2,P)}=0$;
\item[{\rm (2)}] $\lim\limits_{\delta\rightarrow 0+} \int_{\ell^2}\big|| f_{n,\delta}|^{2} -| f_n|^{2}\big|\,\mathrm{d}P =0;$ \item[{\rm (3)}] For any $g\in L^2(\ell^2,P)$ with $\hbox{\rm supp$\,$} g\stackrel{\circ}{\subset} \ell^2$, $$ \int_{\ell^2}f_{n,\delta}g\,\mathrm{d}P=\int_{\ell^2}f\varphi_n^{-1}\cdot(g_{n}\varphi_n)_{n,\delta}\,\mathrm{d}P. $$ \end{itemize} \end{proposition} \begin{proof}
(1) Since $f_{n,\delta}\in C_c^{\infty}(\mathbb{C}^n)$, we have $f_{n,\delta}\in L^2(\ell^2,P)$. By our assumption there exists $r>0$ such that $\hbox{\rm supp$\,$} f\subset B_{r}$. By (\ref{Integration reduce dimension}) and (\ref{Convolution after reduce dimension}), we see that $\hbox{\rm supp$\,$} f_{n}\subset \left\{ \textbf{z}_n\in \mathbb{C}^{n} :\;\left| \left| \textbf{z}_n\right| \right|_{\mathbb{C}^{n} } \leqslant r\right\}$, and hence $\hbox{\rm supp$\,$} f_{n,\delta }\subset \left\{ \textbf{z}_n\in \mathbb{C}^{n} :\;\left| \left| \textbf{z}_n\right| \right|_{\mathbb{C}^{n} } \leqslant r+\delta \right\}$. By the conclusion (1) of Proposition \ref{Reduce diemension}, we have $f_{n}\in L^2(\mathbb{C}^n,d\textbf{z}_n)\cap L^1(\mathbb{C}^n,d\textbf{z}_n)$ and $$
\int_{\ell^2}|f_{n,\delta} -f_n|^{2}\,\mathrm{d}P
=\int_{\mathbb{C}^n} | f_{n,\delta} - f_n |^2\,\mathrm{d}\mathcal{N}^n
\leqslant \frac{1}{\left( 2\pi \right)^{n} a^{2}_{1}a^{2}_{2}\cdots a^{2}_{n}} \int_{\mathbb{C}^{n} } \left| f_{n,\delta }\left( \textbf{z}_n\right) -f_{n}\left( \textbf{z}_n\right) \right|^{2} \mathrm{d}\textbf{z}_n. $$
Since $\lim\limits_{\delta\to 0+}\int_{\mathbb{C}^{n} } \left| f_{n,\delta }\left( \textbf{z}_n\right) -f_{n}\left( \textbf{z}_n\right) \right|^{2} \mathrm{d}\textbf{z}_n=0$, we conclude that $\lim\limits_{\delta\to 0+}||f_{n,\delta}-f_n||_{L^2(\ell^2,P)}=0$.
(2) The proof is similar to that of the conclusion (3) of Proposition \ref{Reduce diemension}, and therefore we omit the details.
(3) Note that $$
\begin{array}{ll} \displaystyle
\int_{\ell^{2} } f_{n,\delta }g\mathrm{d}P=\int_{\ell^{2} } \left(\int_{\mathbb{C}^{n} } f_{n}\left( \textbf{z}_n'\right) \gamma_{n,\delta } \left( \textbf{z}_n-\textbf{z}_n'\right) \mathrm{d}\textbf{z}_n'\right)g\left( \textbf{z}_n,\textbf{z}^n\right) \mathrm{d}P\left( \textbf{z}_n,\textbf{z}^n\right) \\[3mm] \displaystyle =\int_{\ell^{2} } \left(\int_{\ell^{2} } f\left( \textbf{z}_n',\tilde{\textbf{z}}^n\right) \gamma_{n,\delta } \left( \textbf{z}_n-\textbf{z}_n'\right) \varphi^{-1}_{n} \left( \textbf{z}_n'\right) \mathrm{d}P\left( \textbf{z}_n',\tilde{\textbf{z}}^n\right) \right) g\left( \textbf{z}_n,\textbf{z}^n\right) \mathrm{d}P\left(\textbf{z}_n,\textbf{z}^n\right) \\[3mm] \displaystyle =\int_{\ell^{2} }f\left( \textbf{z}_n',\tilde{\textbf{z}}^n\right)\varphi^{-1}_{n} \left( \textbf{z}_n'\right)\left( \int_{\ell^{2} }g\left( \textbf{z}_n,\textbf{z}^n\right) \gamma_{n,\delta } \left( \textbf{z}_n-\textbf{z}_n'\right)\mathrm{d}P\left(\textbf{z}_n,\textbf{z}^n\right)\right)\mathrm{d}P\left( \textbf{z}_n',\tilde{\textbf{z}}^n\right) \\[3mm] \displaystyle =\int_{\ell^{2} }f\left( \textbf{z}_n',\tilde{\textbf{z}}^n\right)\varphi^{-1}_{n} \left( \textbf{z}_n'\right)\left( \int_{\mathbb{C}^{n} }g_n\left( \textbf{z}_n\right)\varphi_n(\textbf{z}_n) \gamma_{n,\delta } \left( \textbf{z}_n-\textbf{z}_n'\right)\mathrm{d}\textbf{z}_n\right)\mathrm{d}P\left( \textbf{z}_n',\tilde{\textbf{z}}^n\right) \\[3mm] \displaystyle =\int_{\ell^{2} } f\varphi^{-1}_{n} \cdot\left( g_{n}\varphi_{n} \right)_{n,\delta } \mathrm{d}P,
\end{array} $$ where the third equality follows from Fubini's theorem. This completes the proof of Proposition \ref{convolution properties}. \end{proof}
The following result provides a useful link between the forms on $V$ and that on finite dimensions. \begin{lemma}\label{approximation for st form}
Suppose that $\sup\limits_{\textbf{z}\in E}|w(\textbf{z})|<\infty$ for each $E\stackrel{\circ}{\subset} V$ and $f=\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t} f_{I,J}dz_{I}\wedge d\overline{z_{J}}\in L^{2}_{(s,t) }( V,w)$. Then, the following conclusions hold:
{\rm (1)} $L^{2}_{(s,t) }( V,w)\subset L^{2}_{(s,t) }( V,loc)$;
{\rm (2)} If $\hbox{\rm supp$\,$} f\stackrel{\circ}{\subset} V$ (in this case $f$ can be viewed as an element of $L^{2}_{(s,t) }( \ell^2,P)\left(\subset L^{2}_{\left( s,t\right) }\left( \ell^2,loc\right)\right)$ by extending the value of $f_{I,J}$ by 0 on $\ell^2\setminus V$ for each strictly increasing multi-indices $I$ and $J$, and $\hbox{\rm supp$\,$} f\stackrel{\circ}{\subset} \ell^2$), for each $n\in\mathbb{N}$ and $\delta\in(0,+\infty)$, letting $$
f_{n,\delta}\triangleq\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t,\max\{I\cup J\}\leqslant n} (f_{I,J})_{n,\delta}dz_{I}\wedge d\overline{z_{J}}, $$ where $(f_{I,J})_{n,\delta}$ is defined as that in (\ref{Convolution after reduce dimension}), then $$
\lim_{n\to\infty}\lim_{\delta\to 0+}||f_{n,\delta}-f||_{L^{2}_{(s,t) }( \ell^2,P)}=0. $$ \end{lemma} \begin{proof} The proof of the conclusion (1) is obvious, and hence we only prove the conclusion (2). For each $n\in\mathbb{N}$, write $$
f_{n}\triangleq\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t,\max\{I\cup J\}\leqslant n} (f_{I,J})_{n}dz_{I}\wedge d\overline{z_{J}}, $$ where $(f_{I,J})_{n}$ is defined similarly to that in (\ref{Integration reduce dimension}). By $f\in L^{2}_{(s,t) }( \ell^2,P)$, for any $\varepsilon>0$, there exists $n_0\in\mathbb{N}$ such that $$
\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t,\max\{I\cup J\}> n_0} c_{I,J}\int_{\ell^2}|f_{I,J}|^2\,\mathrm{d}P<\frac{\varepsilon}{8}. $$ By the conclusion (2) of Proposition \ref{Reduce diemension}, there exists $n_1\in\mathbb{N}$ such that $n_1\geqslant n_0$ and for all $n\geqslant n_1$, it holds that $$
\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t,\max\{I\cup J\}\leqslant n_0} c_{I,J}\int_{\ell^2}|(f_{I,J})_{n}-f_{I,J}|^2\,\mathrm{d}P<\frac{\varepsilon}{4}, $$ and by the conclusion (1) of Proposition \ref{Reduce diemension}, $$ \begin{array}{ll} \displaystyle
||f_{n}-f||_{L^{2}_{(s,t) }( \ell^2,P)}^2\\[3mm] \displaystyle =
\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t,\max\{I\cup J\}\leqslant n_0} c_{I,J}\int_{\ell^2}|(f_{I,J})_{n}-f_{I,J}|^2\,\mathrm{d}P+\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t,\max\{I\cup J\}> n} c_{I,J}\int_{\ell^2}|f_{I,J}|^2\,\mathrm{d}P\\[3mm]
\displaystyle \quad+\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t,n_0<\max\{I\cup J\}\leqslant n} c_{I,J}\int_{\ell^2}|(f_{I,J})_{n}-f_{I,J}|^2\,\mathrm{d}P\\[3mm]
\displaystyle \leqslant\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t,\max\{I\cup J\}\leqslant n_0} c_{I,J}\int_{\ell^2}|(f_{I,J})_{n}-f_{I,J}|^2\,\mathrm{d}P+
5\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t,\max\{I\cup J\}> n_0} c_{I,J}\int_{\ell^2}|f_{I,J}|^2\,\mathrm{d}P\\[3mm] \displaystyle <\varepsilon. \end{array} $$
Therefore, $\lim\limits_{n\to\infty}||f_{n}-f||_{L^{2}_{(s,t) }( \ell^2,P)}=0$. For each $n\in\mathbb{N}$ and $\delta\in(0,+\infty)$, $$
||f_{n,\delta}-f_n||_{L^{2}_{(s,t) }( \ell^2,P)}^2=
\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t,\max\{I\cup J\}\leqslant n} c_{I,J}\int_{\ell^2}|(f_{I,J})_{n,\delta}-(f_{I,J})_n|^2\,\mathrm{d}P. $$
Combining this with the conclusion (1) of Proposition \ref{convolution properties}, we see that $\lim\limits_{\delta\to 0+}||f_{n,\delta}-f_n||_{L^{2}_{(s,t) }( \ell^2,P)}=0$, which completes the proof of Lemma \ref{approximation for st form}. \end{proof}
As a consequence of Lemmas \ref{0orderSobolev} and \ref{approximation for st form}, we have the following density result. \begin{corollary}\label{cylinder function is dense}
Suppose that $\sup\limits_{\textbf{z}\in V} |w(\textbf{z})|<\infty$. Then $$
\bigcup^{\infty }_{n=1} \bigg\{\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t,\max \left\{ I\bigcup J\right\} \leqslant n} f_{I,J}dz_{I}\wedge d\overline{z_{J}} :\; f_{I,J}\in C^{\infty }_{c}\left(\mathbb{C}^n\right)\text{ for all involved }I,J \bigg\} $$ is dense in $L^{2}_{(s,t) }( V,w)$. \end{corollary}
We also need the following simple approximation result (which should be known but we do not find an exact reference). \begin{lemma}\label{F} Assume that $f\in C^{1 }_{b}( \ell^{2})$. Then there exists $\{h_{k }\}_{k=1}^{\infty}\subset C^{\infty }_{b,F^{\infty}}( \ell^{2})$ such that $\lim\limits_{k \rightarrow \infty } h_{k}=f$, $\lim\limits_{k \rightarrow \infty } D_{x_{j}}h_{k}=D_{x_{j}}f$ and $\lim\limits_{k \rightarrow \infty } D_{y_{j}}h_{k}=D_{y_{j}}f$ almost everywhere respect to $P$ for each $j\in\mathbb{N}$. Meanwhile, for each $j,k\in\mathbb{N}$, the following inequalities hold on $\ell^2$, \begin{equation}\label{6zqqqx} \left\{ \begin{array}{ll}
\displaystyle\left|h_{k }\right| \leqslant \sup_{\ell^{2} } \left| f\right| ,\ \left| \partial_{j} h_{k}\right| \leqslant \sup_{ \ell^{2} } \left| \partial_{j} f\right| ,\left| \overline{\partial_{j} } h_{k }\right| \leqslant \sup_{ \ell^{2} } \left| \overline{\partial_{j} } f\right|, \\[5mm]
\displaystyle\left| D_{x_{j}}h_{k }\right| \leqslant \sup_{\ell^{2} } \left| D_{x_{j}}f\right|\text{ and } \left| D_{y_{j}}h_{k}\right| \leqslant \sup_{\ell^{2} } \left| D_{y_{j}}f\right|. \end{array}\right. \end{equation} Furthermore, if $f\in C^{1}_{0,F}( \ell^{2})$, for each $k\in\mathbb{N}$, it also holds that \begin{equation}\label{6}
\sum^{\infty }_{j=1} \left| \partial_{j} h_{k}\right|^{2} \leqslant \sup_{\ell^{2} } \sum^{\infty }_{j=1} \left| \partial_{j} f\right|^{2} ,\quad\sum^{\infty }_{j=1} \left| \overline{\partial_{j} } h_{k}\right|^{2} \leqslant \sup_{\ell^{2} } \sum^{\infty }_{j=1} \left| \overline{\partial_{j} } f\right|^{2}. \end{equation} \end{lemma}
\begin{proof} For each $n\in\mathbb{N}$, let $$f_n(\textbf{z}_n)\triangleq \int f(\textbf{z}_n,\textbf{z}^n)\,\mathrm{d}P_n(\textbf{z}^n),
$$
where $\textbf{z}^n=(x_{i} +\sqrt{-1}y_{i})_{i=n+1}^\infty\in \ell^2$ and $\textbf{z}_n=(x_{i} +\sqrt{-1}y_{i})_{i=1}^{n}\in \mathbb{C}^n$. It is easy to see that $D_{x_j}f_n(\textbf{z}_n)=\int D_{x_j}f(\textbf{z}_n,\textbf{z}^n)\,\mathrm{d}P_n(\textbf{z}^n)$ for $j=1,\cdots, n$. By the conclusion (2) of Proposition \ref{Reduce diemension}, we have $\lim\limits_{n\rightarrow \infty } \int_{\ell^{2} } \left| f_{n}-f\right|^{2} dP=0$ and $\lim\limits_{n\rightarrow \infty } \int_{\ell^{2} } \left| D_{x_{j}} f_{n} -D_{x_{j}} f\right|^{2}dP=0.$ For the sequence of functions $\{\gamma_n\}_{n=1}^{\infty}$ given in (\ref{eq230131e1}) and each $\delta\in(0,+\infty)$, write $$ f_{n,\delta}(\textbf{z}_n)\triangleq \int_{\mathbb{C}^n}f_n(\textbf{z}_n')\gamma_{n,\delta } ( \textbf{z}_n -\textbf{z}_n')d\textbf{z}_n',\quad\textbf{z}_n \in \mathbb{C}^{n}. $$
Then, $f_{n,\delta}\in C^{\infty }_{b,F^{\infty}}( \ell^{2})$, $|f_{n,\delta}|\leqslant \sup\limits_{\mathbb{C}^n}|f_n|\leqslant \sup\limits_{\ell^2}|f|$, $|D_{x_j}f_{n,\delta}|\leqslant \sup\limits_{\mathbb{C}^n}|D_{x_j}f_n|\leqslant \sup\limits_{\ell^2}|D_{x_j}f|$, $|\partial_{j}f_{n,\delta}|\leqslant \sup\limits_{\mathbb{C}^n}|\partial_{j}f_n|\leqslant \sup\limits_{\ell^2}|\partial_{j}f|$, $$
\lim\limits_{\delta\rightarrow 0} \int_{\ell^{2} } \left| f_{n,\delta}-f_n\right|^{2} dP=0,\qquad
\lim\limits_{\delta\rightarrow 0} \int_{\ell^{2} } \left| D_{x_{j}} f_{n,\delta} -D_{x_{j}} f_n\right|^{2}dP=0 $$
for each fixed $j\in\mathbb{N}$. By the diagonal method we can pick a sequence of functions $\{g_k\}_{k=1}^{\infty}$ from $\{f_{n,\delta}:\;n\in\mathbb{N},\,\delta\in(0,+\infty)\}$ such that $\lim\limits_{k\rightarrow \infty } \int_{\ell^{2} } \left| g_{k}-f\right|^{2} dP=0$ and $\lim\limits_{k\rightarrow \infty } \int_{\ell^{2} } \left| D_{x_{j}} g_{k} -D_{x_{j}} f\right|^{2}dP=0$ for each $j\in\mathbb{N}$. By \cite[Theorem 6.3.1 (b) at pp. 171--172 and Section 6.5 at pp. 180--181]{Res} and using the diagonal method again, we can find a subsequence $\{h_k\}_{k=1}^{\infty}$ of $\{g_k\}_{k=1}^{\infty}$ such that $\lim\limits_{k \rightarrow \infty } h_{k}=f$ and $\lim\limits_{k \rightarrow \infty } D_{x_{j}}h_{k}=D_{x_{j}}f$ almost everywhere respect to $P$ for each $j\in\mathbb{N}$. Obviously, the sequence $\{h_k\}_{k=1}^{\infty}$ satisfies the conditions in (\ref{6zqqqx}). If $f\in C^{1}_{0,F}( \ell^{2})$, then by the Jensen inequality (See \cite[Theorem 3.3, p. 62]{Rud87}), $$ \begin{array}{ll} \displaystyle
\sum^{\infty }_{j=1} \left| \partial_{j} f_{n,\delta}( \textbf{z}_n)\right|^{2}
= \sum^{n}_{j=1} \left| \partial_{j} f_{n,\delta}( \textbf{z}_n )\right|^{2}
\leqslant\int_{\mathbb{C}^n}\sum^{n}_{j=1} \left| \partial_{j} f_{n }(\textbf{z}_n')\right|^{2}\gamma_{n,\delta } ( \textbf{z}_n -\textbf{z}_n')d\textbf{z}_n'\\[3mm]
\displaystyle \leqslant \sup_{\textbf{z}_n'\in\mathbb{C}^n}\sum^{n}_{j=1} \left| \partial_{j} f_{n }(\textbf{z}_n')\right|^{2}
= \sup_{\textbf{z}_n'\in\mathbb{C}^n}\sum^{n}_{j=1} \left| \int\partial_{j} f (\textbf{z}_n',\textbf{z}^n)\,\mathrm{d}P_n(\textbf{z}^n)\right|^{2}\\[3mm]
\displaystyle \leqslant \sup_{\textbf{z}_n'\in\mathbb{C}^n} \int\sum^{n}_{j=1} \left|\partial_{j} f (\textbf{z}_n',\textbf{z}^n)\right|^{2}\,\mathrm{d}P_n(\textbf{z}^n)
\leqslant \sup_{\textbf{z}\in\ell^2} \sum^{n}_{j=1} \left|\partial_{j} f (\textbf{z})\right|^{2}\leqslant \sup_{\textbf{z}\in\ell^2} \sum^{\infty}_{j=1} \left|\partial_{j} f (\textbf{z})\right|^{2}, \end{array} $$ and hence we have proved \eqref{6}. This completes the proof of Lemma \ref{F}. \end{proof}
For $i\in\mathbb{N} $ and strictly increasing multi-indices $L=(l_1,\cdots,l_t)$ and $K$ with $\left| L\right| =t$ and $\left| K\right| =t+1$,
we set \begin{equation}\label{230204e3} \varepsilon_{iL}^{K}\triangleq\left\{ \begin{array}{ll}0, &\hbox{ if }K\neq \{i\}\cup L,\\ \hbox{the sign of the permutation }\left(\begin{array}{ll}i,l_1,\cdots,l_t\\ \qquad K\end{array}\right), &\hbox{ if }K= \{i\}\cup L. \end{array}\right. \end{equation}
\begin{definition}\label{def for db}
We say that $f=\sum\limits^{\prime }_{\left| I\right| =s} \sum\limits^{\prime }_{\left| J\right| =t} f_{I,J}\,\mathrm{d}z_{I}\wedge \,\mathrm{d}\overline{z_{J}}\in L^{2}_{(s,t)}\left( V,loc\right)$ is in the domain $ D_{\overline{\partial } }$ of $\overline{\partial }$ if there exists $g=\sum\limits^{\prime }_{\left| I\right| =s} \sum\limits^{\prime }_{\left| K\right| =t+1} g_{I,K}\,\mathrm{d}z_{I}\wedge \,\mathrm{d}\overline{z_{K}}\in L^{2}_{(s,t+1)}\left( V,loc\right)$ such that for all strictly increasing multi-indices $I$ and $K$ with $\left| I\right| =s$ and $\left| K\right| =t+1$, it holds that \begin{equation}\label{230204e5}
\left( -1\right)^{s+1} \int_{V} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t} \varepsilon^{K}_{iJ} f_{I,J}\overline{\delta_{i} \phi }\,\mathrm{d}P=\int_{V} g_{I,K}\bar{\phi }\,\mathrm{d}P,\quad \forall \;\phi \in C^{\infty }_{0,F}\left( V\right). \end{equation} In this case, we define $\overline{\partial } f\triangleq g$. \end{definition}
\begin{remark}\label{230327r1}
Note that, for any fixed strictly increasing multi-index $K$ with $\left| K\right|=t+1$, by the definition of $\varepsilon^{K}_{iJ}$ in (\ref{230204e3}), there exist at most finitely many non-zero terms in the series $\sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t} \varepsilon^{K}_{iJ} f_{I,J}\overline{\delta_{i} \phi }$ in (\ref{230204e5}). \end{remark}
\begin{remark}\label{230407r1}
If $f=\sum\limits^{\prime }_{\left| I\right| =s} \sum\limits^{\prime }_{\left| J\right| =t} f_{I,J}\,\mathrm{d}z_{I}\wedge \,\mathrm{d}\overline{z_{J}}\in L^{2}_{(s,t)}\left( V,loc\right)$ satisfies $ f_{I,J}\in C_0^1(V)$ for each strictly increasing multi-indices $I$ and $J$ with $\left| I\right| =s$ and $\left| J\right| =t$, then, by Corollaries \ref{integration by Parts or deltai} and \ref{density2}, $ f\in D_{\overline{\partial } }$, and $$
\overline{\partial } f=\left( -1\right)^{s} \sum\limits^{\prime }_{\left| I\right| =s} \sum\limits^{\prime }_{\left| K\right| =t+1} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t} \varepsilon^{K}_{iJ} \overline{\partial}_{i} f_{I,J}\,\mathrm{d}z_{I}\wedge \,\mathrm{d}\overline{z_{K}}. $$ \end{remark}
We have the following result. \begin{proposition}\label{weak equality for more test functions}
Under Condition \ref{230424c1}, for any $f=\sum\limits^{\prime }_{\left| I\right| =s} \sum\limits^{\prime }_{\left| J\right| =t} f_{I,J}\,\mathrm{d}z_{I}\wedge \,\mathrm{d}\overline{z_{J}}\in L^{2}_{(s,t)}\left( V,loc\right)$, the following assertions hold:
{\rm (1)} If $f\in D_{\overline{\partial } }$, then for all strictly increasing multi-indices $I$ and $K$ with $\left| I\right|=s$ and $\left| K\right|=t+1$, \begin{equation}\label{230203e1}
\left( -1\right)^{s+1} \int_{V} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t} \varepsilon^{K}_{iJ} f_{I,J}\overline{\delta_{i} \phi }\,\mathrm{d}P=\int_{V} \left( \overline{\partial } f\right)_{I,K} \bar{\phi }\,\mathrm{d}P,\quad \forall\; \phi \in C^{1 }_{0}\left( V\right); \end{equation}
{\rm (2)}
If $f\in D_{\overline{\partial } }$ and $\hbox{\rm supp$\,$} f \stackrel{\circ}{\subset}V$, then for all strictly increasing multi-indices $I$ and $K$ with $\left| I\right|=s$ and $\left| K\right|=t+1$, \begin{equation}\label{230204e2}
\left( -1\right)^{s+1} \int_{V} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t} \varepsilon^{K}_{iJ} f_{I,J}\overline{\delta_{i} \phi }\,\mathrm{d}P=\int_{V} \left( \overline{\partial } f\right)_{I,K} \bar{\phi }\,\mathrm{d}P,\quad \forall\; \phi \in C^{1 }_{b}\left( V \right); \end{equation}
{\rm (3)}
If $\eta\in C^{\infty}_{F^{\infty}}\left( V\right)$, $g=\sum\limits^{\prime }_{\left| I\right| =s} \sum\limits^{\prime }_{\left| K\right| =t+1} g_{I,K}\,\mathrm{d}z_{I}\wedge \,\mathrm{d}\overline{z_{K}}\in L^{2}_{(s,t+1)}\left( V,loc\right)$, and the following equality $$
\left( -1\right)^{s+1} \int_{V} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t} \varepsilon^{K}_{iJ} f_{I,J}\overline{\delta_{i} \phi }\,\mathrm{d}P=\int_{V} g_{I,K}\bar{\phi } \,\mathrm{d}P,\quad \forall\; \phi \in C^{\infty }_{0,F^{\infty }}\left( V\right) $$
holds for any strictly increasing multi-indices $I$ and $K$ with $\left| I\right|=s$ and $\left| K\right|=t+1$, then $f\in D_{\overline{\partial } }$ and $\overline{\partial } f\triangleq g$. \end{proposition}
\begin{proof} (1) Choose $\{\varphi_{k}\}_{k=1}^{\infty} \subset C^{\infty }\left( \mathbb{R}\right)$ such that for each $k\in\mathbb{N}$, $0\leqslant\varphi_{k} \leqslant1$, $\varphi_{k} \left( x\right) =1$ for all $x\leqslant k$ and $\varphi_{k} \left( x\right) =0$ for all $x>k+1$. Recall Condition \ref{230424c1} for $\eta$. Let \begin{equation}\label{230204e1} \Phi_{k} \triangleq\varphi_{k}( \eta ). \end{equation} Then $\Phi_{k} \in C^{\infty }_{0,F}( V).$ For any fixed $\phi \in C^{1 }_{0}\left( V\right)$, by Proposition \ref{properties on pseudo-convex domain}, there exists $r\in(0,+\infty)$ such that $\hbox{\rm supp$\,$} \phi \subset V_{r}$ (Recall (\ref{230117e1}) for the notation $V_r$). We may view $\phi$ as an element of $C^{1 }_{0}( \ell^2)\subset C^{1 }_{b}( \ell^2)$ by the zero extension. By Lemma \ref{F}, there exists $\{\phi_{k}\}_{k=1}^{\infty}\subset C^{\infty }_{b,F^\infty}( \ell^{2})$ such that the conditions corresponding to (\ref{6zqqqx}) are satisfied (for which $f$ and $h_k$ therein are replaced by $\phi$ and $\phi_k$, respectively). For any $k,l\in\mathbb{N}$ with $k>r$, we have $\phi_{l}\Phi_{k} \in C^{\infty }_{0,F}\left( V\right)$, and hence, by Definition \ref{def for db}, \begin{equation}\label{230204e4}
\left( -1\right)^{s+1} \int_{V} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t} \varepsilon^{K}_{iJ} f_{I,J}\overline{\delta_{i} \left( \phi_{l } \Phi_{k} \right) }\,\mathrm{d}P=\int_{V} \left( \overline{\partial } f\right)_{I,K} \overline{\phi_{l } \Phi_{k} }\,\mathrm{d}P. \end{equation} Letting $l\rightarrow \infty $ in (\ref{230204e4}) and noting Remark \ref{230327r1}, we have \begin{equation}\label{230204e25}
\left( -1\right)^{s+1} \int_{V} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t} \varepsilon^{K}_{iJ} f_{I,J}\overline{\delta_{i} \left( \phi \Phi_{k} \right) }\,\mathrm{d}P=\int_{V} \left( \overline{\partial } f\right)_{I,K} \overline{\phi \Phi_{k} }\,\mathrm{d}P. \end{equation} Since $\hbox{\rm supp$\,$} \phi\subset V_k$ and $\Phi_k=1$ on $V_k$, by (\ref{230204e25}) we obtain the desired (\ref{230203e1}).
(2) Since $\hbox{\rm supp$\,$} f \stackrel{\circ}{\subset}V$, by Proposition \ref{properties on pseudo-convex domain} again, there exists $r\in(0,+\infty)$ such that $\hbox{\rm supp$\,$} f \subset V_{r}.$ Then for any $k\in\mathbb{N}$ with $k>r$ and $\phi \in C^{1 }_{b}\left( V \right)$, $\phi \Phi_{k}\in C^{1 }_{0}( V)$ (where $\Phi_{k}$ is given by (\ref{230204e1})). By the above conclusion (1), we see that (\ref{230204e25}) holds for the present $\phi \Phi_{k}$. Hence, by $\hbox{\rm supp$\,$} f\subset V_k$ and $\Phi_k=1$ on $V_k$, we obtain the desired (\ref{230204e2}).
(3) By our assumption, $\{\Phi_{k}\}_{k=1}^{\infty} \subset C^{\infty }_{0,F^{\infty}}( V) $ (See (\ref{230204e1}) for $\Phi_{k}$). For any $\phi \in C^{\infty}_{0, F}( V)$, we choose $\{\phi_{k}\}_{k=1}^{\infty}\subset C^{\infty }_{b,F^{\infty}}( \ell^{2})$ as in the proof of the conclusion (1). For any $k,l\in\mathbb{N}$ such that $k>r$, we have $\phi_{l}\Phi_{k} \in C^{\infty }_{0,F^{\infty}}\left( V\right)$ and hence $$
\left( -1\right)^{s+1} \int_{V} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t} \varepsilon^{K}_{iJ} f_{I,J}\overline{\delta_{i} \left( \phi_{l} \Phi_{k} \right) }\,\mathrm{d}P=\int_{V}g_{I,K}\overline{\phi_{l } \Phi_{k} }\,\mathrm{d}P. $$ By the same argument as that in the proof of the conclusion (1), from the above equality we obtain the desired (\ref{230204e5}). The proof of Proposition \ref{weak equality for more test functions} is completed. \end{proof}
\begin{definition}\label{definition of T S} For any given real-valued functions $w_{1},w_{2},w_{3}\in C^{2}_{F}\left( V\right)$, we define two linear unbounded operators $T:\;L^{2}_{(s,t)}\left( V,w_{1}\right) \to L^{2}_{(s,t+1)}\left( V,w_{2}\right)$ and $S:\;L^{2}_{(s,t+1)}\left( V,w_{2}\right)\to L^{2}_{(s,t+2)}\left( V,w_{3}\right)$ as follows: $$ \left\{ \begin{array}{ll} D_{T}\triangleq \left\{ u\in L^{2}_{(s,t)}\left( V,w_{1}\right) :\;u\in D_{\overline{\partial }}\;\text{ and }\;\overline{\partial } u\in L^{2}_{(s,t+1)}\left( V,w_{2}\right) \right\},\\[3mm] Tu\triangleq \overline{\partial } u,\quad\forall\;u\in D_T, \end{array}\right. $$ $$ \quad\;\left\{ \begin{array}{ll} D_{S}\triangleq \left\{ f\in L^{2}_{\left( s,t+1\right) }\left( V,w_{2}\right) :\;f\in D_{\overline{\partial }}\;\text{ and }\;\overline{\partial } f\in L^{2}_{(s,t+2)}\left( V,w_{3}\right) \right\},\\[3mm] Sf\triangleq \overline{\partial } f,\quad\forall\;f\in D_S, \end{array}\right. $$ and call $ R_{T}\triangleq \left\{ Tu :\;u\in D_{T} \right\}$ and $ N_{S}\triangleq \left\{ f :\;f\in D_{S}\text{ and }Sf=0 \right\}$ the range of $T$ and the kernel of $S$, respectively. \end{definition}
In the rest of this section, to simplify the presentation, unless other stated, we fix $w_{1},w_{2},w_{3}$, $T$ and $S$ as that in Definition \ref{definition of T S}. \begin{lemma}\label{densly defined closed} The operator $T$ (resp. $S$) is densely defined and closed from $L^{2}_{(s,t)}\left( V,w_{1}\right)$ (resp. $L^{2}_{(s,t+1)}\left( V,w_{2}\right)$) into $L^{2}_{(s,t+1)}\left( V,w_{2}\right)$ (resp. $L^{2}_{(s,t+2)}\left( V,w_{3}\right)$). \end{lemma} \begin{proof} It is easy to see that $$
\bigcup^{\infty }_{n=1} \bigg\{ \sum^{\prime }_{\left| I\right| =s,\left| J\right| =t,\max \left\{ I\bigcup J\right\} \leqslant n} f_{I,J}dz_{I}\wedge d\overline{z_{J}} :\;f_{I,J}\in C^{\infty }_{0,F}\left( V\right)\text{ for all }I,J \bigg\}\subset D_T $$ and $$
\bigcup^{\infty }_{n=1} \bigg\{ \sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\max \left\{ I\bigcup J\right\} \leqslant n} f_{I,J}dz_{I}\wedge d\overline{z_{J}} :\;f_{I,J}\in C^{\infty }_{0,F}\left( V\right)\text{ for all }I,J \bigg\}\subset D_S. $$ Combining Lemma \ref{0orderSobolev} and the fact that $ C^{\infty }_{0,F^{\infty}}\left( V\right)\subset C^{\infty }_{0,F}\left( V\right)$, we see that both $T$ and $S$ are densely defined operators. Similarly to the proof of \cite[Lemma 2.5, p. 529]{YZ}, we can prove that both $T$ and $S$ are closed operators. The proof of Lemma \ref{densly defined closed} is completed. \end{proof}
In the sequel, we shall denote by $T^*$ and $S^*$ respectively the adjoint operators of $T$ and $S$ (e.g., \cite[Definition 4.2.2, p. 177]{Kra}). Meanwhile, we denote by $D_{T^*}$ and $D_{S^*}$ respectively the domains of $T^*$ and $S^*$.
We have the following simple result. \begin{proposition}\label{Pre-exactness} For each $u\in D_T$, it holds that $Tu\in D_S$ and $S(Tu)=0$, i.e., $R_T\subset N_S$. \end{proposition}
\begin{proof} The proof is very similar to that of \cite[Lemma 2.6, p. 531]{YZ}, and therefore we omit the details. \end{proof}
For any fixed $f=\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} f_{I,J}dz_{I}\wedge d\overline{z_{J}}\in L^{2}_{(s,t+1)}\left(V,loc\right)$, strictly increasing multi-indices $K$ and $L$ with $\left| K\right| =s$ and $\left| L\right| =t$, $i\in\mathbb{N} $, we write (Recall (\ref{230204e3}) and (\ref{defnition of general st froms}) respectively for $\varepsilon^{J}_{iL}$ and $c_{K,J}$ for which $J$ is any strictly increasing multi-index with $\left| J\right| =t+1$):
\begin{equation}\label{gener111}
f_{K,iL}\triangleq \sum\limits^{\prime }_{\left| J\right| =t+1}\varepsilon^{J}_{iL}\cdot f_{K,J},\qquad c_{K,iL}\triangleq \sum\limits^{\prime }_{\left| J\right| =t+1}|\varepsilon^{J}_{iL}|\cdot c_{K,J}.
\end{equation}
\begin{remark}\label{230329r1}
By (\ref{230204e3}), it is easy to see that, for any given $i\in\mathbb{N} $ and strictly increasing multi-indices $K$ and $L$ with $\left| K\right| =s$ and $\left| L\right| =t$, there is at most one non-zero term in each of the series $\sum\limits^{\prime }_{\left| J\right| =t+1}\varepsilon^{J}_{iL}\cdot f_{K,J}$ and $\sum\limits^{\prime }_{\left| J\right| =t+1}|\varepsilon^{J}_{iL}|\cdot c_{K,J}$ in (\ref{gener111}).
\end{remark}
We need the following technical result. \begin{lemma}\label{T*formula}
Assume that $ f=\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} f_{I,J}dz_{I}\wedge d\overline{z_{J}} \in D_{T^{\ast }}$, $\phi \in C^{1}_0\left( V\right)$, and \begin{equation}\label{phi begin to D condition}
\int_{V} \left| \phi \right|^{2} e^{-w_{1}}\,\mathrm{d}P<\infty \ \ \text{ and } \ \ \sum^{\infty }_{j=1} c_{K,jL}\int_{V} \left| \partial_{j} \phi \right|^{2} e^{-w_{2}}\,\mathrm{d}P<\infty \end{equation}
for some given strictly increasing multi-indices $K$ and $L$ with $\left| K\right|=s$ and $\left| L\right|=t$. Then, $$ c_{K,L}\int_{V} \left( T^{\ast }f\right)_{K,L} \phi e^{-w_{1}}\,\mathrm{d}P=\left( -1\right)^{s} \sum_{j=1}^{\infty} c_{K,jL} \int_{V}\partial_{j} \phi \cdot f_{K,jL}e^{-w_{2}}\,\mathrm{d}P. $$
\end{lemma} \begin{proof}
Let $g\triangleq \sum\limits^{\prime }_{\left| I_2\right| =s,\left| L_2\right| =t} g_{I_2,L_2}dz_{I_2}\wedge d\overline{z_{L_2}}$, where $g_{K,L}\triangleq \overline{\phi}$ and $g_{I_2,L_2}\triangleq 0$ for any other strictly increasing multi-indices $I_2$ and $L_2$ with $\left| I_2\right|=s$ and $\left| L_2\right|=t$. Fix any $ \varphi \in C^{\infty }_{0,F}\left( V\right)$ and strictly increasing multi-indices $I_3$ and $J_3$ with $\left| I_3\right|=s$ and $\left| J_3\right|=t+1$. If $I_3\neq K$ or there does not exist $j_0\in\mathbb{N}$ such that $\{j_0\}\cup L=J_3$, then we have $$
\left( -1\right)^{s+1} \int_{V} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| L_3\right| =t} \varepsilon^{J_3}_{iL_3} g_{I_3,L_3}\overline{\delta_{i} \varphi }\,\mathrm{d}P=0. $$ If $I_3=K$ and there exists $j_0\in\mathbb{N}$ such that $\{j_0\}\cup L=J_3$, then we have $$ \begin{array}{ll}
\displaystyle \left( -1\right)^{s+1} \int_{V} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| L_3\right| =t} \varepsilon^{J_3}_{iL_3} g_{K,L_3}\overline{\delta_{i} \varphi }\,\mathrm{d}P\\[3mm] \displaystyle =\left( -1\right)^{s+1} \int_{V} \varepsilon^{J_3}_{j_0L}\cdot \overline{\phi}\cdot\overline{\delta_{j_0} \varphi }\,\mathrm{d}P=\left( -1\right)^{s} \int_{V} \varepsilon^{J_3}_{j_0L}\cdot \left(\overline{\partial_{j_0}}\overline{\phi}\right)\cdot\overline{ \varphi }\,\mathrm{d}P=\left( -1\right)^{s} \int_{V}\sum_{j=1}^{\infty} \varepsilon^{J_3}_{jL}\cdot \left(\overline{\partial_{j}}\overline{\phi}\right)\cdot\overline{ \varphi }\,\mathrm{d}P, \end{array} $$ where the second equality follows from Corollary \ref{integration by Parts or deltai}. Hence, by the condition \eqref{phi begin to D condition}, we see that $g\in L^{2}_{(s,t)}\left( V,w_{1}\right)$ and $$
(-1)^s\sum^{\prime }_{\left| J\right| =t+1} \sum_{j=1}^{\infty}\varepsilon^{J}_{jL}\overline{\partial_{j} }\overline{\phi} dz_{K}\wedge d\,\overline{z_{J}}\in L^{2}_{(s,t+1)}\left( V,w_{2}\right). $$ By Definition \ref{def for db}, $g\in D_T$ and $
Tg = (-1)^s\sum^{\prime }_{\left| J\right| =t+1} \sum_{j=1}^{\infty}\varepsilon^{J}_{jL}\overline{\partial_{j} }\overline{\phi} dz_{K}\wedge d\,\overline{z_{J}}$. By $(T^{\ast }f,g)_{L^2_{(s,t)}(V,w_1)}=
(f,Tg)_{L^2_{(s,t+1)}(V,w_2)}$, using the fact that $\varepsilon^{J}_{jL}=\varepsilon^{J}_{jL}|\varepsilon^{J}_{jL}|$ and noting Remark \ref{230329r1}, we have $$ \begin{array}{ll} \displaystyle c_{K,L}\int_{V} \left( T^{\ast }f\right)_{K,L} \phi e^{-w_{1}}\,\mathrm{d}P
=\left( -1\right)^{s} \sum^{\prime }_{\left| J\right| =t+1} \sum_{j=1}^{\infty} c_{K,J}\varepsilon^{J}_{jL}\int_{V}f_{K,J}\cdot\overline{\overline{ \partial_{j}} \overline{\phi}} e^{-w_{2}}\,\mathrm{d}P\\[3mm]
\displaystyle =\left( -1\right)^{s} \sum^{\prime }_{\left| J\right| =t+1} \sum_{j=1}^{\infty} c_{K,J}\varepsilon^{J}_{jL}\int_{V}f_{K,J}\cdot\partial_{j} \phi e^{-w_{2}}\,\mathrm{d}P =\left( -1\right)^{s} \sum_{j=1}^{\infty} c_{K,jL}\int_{V}f_{K,jL}\cdot\partial_{j}\phi e^{-w_{2}}\,\mathrm{d}P, \end{array} $$ which completes the proof of Lemma \ref{T*formula}. \end{proof}
Now we derive an explicit expression of $T^*$ (Recall (\ref{gener111}) for $f_{I,iL}$): \begin{proposition}\label{general formula of T*}
Suppose that $f=\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} f_{I,J}dz_{I}\wedge d\overline{z_{J}}\in L^{2}_{(s,t+1)}\left( V,w_{2}\right)$, where $f_{I,J}=0$ for all strictly increasing multi-indices $I$ and $J$ with $\max \left\{ I\bigcup J\right\} >n_{0}$ for some $n_{0}\in \mathbb{N}$ and $f_{I,J}\in C^{1}_{0,F}\left( V\right)$ for all strictly increasing multi-indices $I$ and $J$ satisfying $\max \left\{ I\bigcup J\right\} \leqslant n_{0}$. Then, $f\in D_{T^{\ast }}$ and $$
T^{\ast }f= \left( -1\right)^{s+1} e^{w_{1}-w_{2}} \sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n_{0}}\sum^{n_{0}}_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot (\delta_{i} f_{I,iL}-f_{I,iL}\cdot \partial_{i} w_{2})dz_{I}\wedge d\overline{z_{L}}. $$ \end{proposition} \begin{proof} Let $$
g\triangleq \left( -1\right)^{s+1} e^{w_{1}-w_{2}} \sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n_{0}}\sum^{n_{0}}_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot(\delta_{i} f_{I,iL}-f_{I,iL}\cdot\partial_{i} w_{2})dz_{I}\wedge d\overline{z_{L}}. $$
Then, $g\in L^{2}_{(s,t)}\left( V,w_{1}\right)$. We will prove that $\left( Tu,f\right)_{L^{2}_{(s,t+1)}\left( V,w_{2}\right) } =\left( u,g\right)_{L^{2}_{(s,t)}\left( V,w_{1}\right) } $ for all $u\in D_{T}$, which implies that $f\in D_{T^*}$ and $T^*f=g$. Since $f_{I,J}e^{-w_{2}}\in C_0^1(V)$ for all strictly increasing multi-indices $I$ and $J$, by the conclusion (1) of Proposition \ref{weak equality for more test functions}, using the equality $\varepsilon^{J}_{jL}=\varepsilon^{J}_{jL}|\varepsilon^{J}_{jL}|$ and recalling Remark \ref{230329r1}, we obtain that $$ \begin{array}{ll} \displaystyle
\left( Tu,f\right)_{L^{2}_{(s,t+1)}\left( V,w_{2}\right) } =\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| J\right| =t+1} c_{I,J}\int_{V} \left( Tu\right)_{I,J} \overline{f_{I,J}} e^{-w_{2}}\,\mathrm{d}P\\[3mm]
\displaystyle =\left( -1\right)^{s+1} \sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} \sum^{\prime }_{\left| L\right| =t} \sum^{\infty }_{i=1} c_{I,J}\varepsilon^{J}_{iL} \int_{V} u_{I,L}\overline{\delta_{i} (f_{I,J}e^{-w_{2}})} \,\mathrm{d}P\\[3mm]
\displaystyle =\left( -1\right)^{s+1} \sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} \sum^{\prime }_{\left| L\right| =t} \sum^{\infty }_{i=1} c_{I,J}\varepsilon^{J}_{iL} \int_{V} u_{I,L}e^{w_{1}-w_{2}}(\overline{\delta_{i} f_{I,J}-f_{I,J}\partial_{i} w_{2}}) e^{-w_{1}}\,\mathrm{d}P\\[3mm]
\displaystyle =\left( -1\right)^{s+1} \sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} c_{I,L}\int_{V} u_{I,L}\sum^{\prime }_{\left| J\right| =t+1} \sum^{\infty }_{i=1}\varepsilon^{J}_{iL}\cdot \frac{c_{I,J}}{c_{I,L}}\cdot e^{w_{1}-w_{2}}(\overline{\delta_{i} f_{I,J}-f_{I,J}\partial_{i} w_{2}} )e^{-w_{1}}\,\mathrm{d}P\\[3mm]
\displaystyle =\left( -1\right)^{s+1} \sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} c_{I,L}\int_{V} u_{I,L} \sum^{\infty }_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot e^{w_{1}-w_{2}}(\overline{\delta_{i} f_{I,iL}-f_{I,iL}\partial_{i} w_{2}} )e^{-w_{1}}\,\mathrm{d}P. \end{array} $$ Since $f_{I,J}=0$ for all strictly increasing multi-indices $I$ and $J$ with $\max \left\{ I\cup J\right\} >n_{0}$, and noting the definition of $f_{I,iL}$ in (\ref{gener111}), we obtain that $$ \begin{array}{ll}
\displaystyle \left( -1\right)^{s+1} \sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} c_{I,L}\int_{V} u_{I,L} \sum^{\infty }_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot e^{w_{1}-w_{2}}(\overline{\delta_{i} f_{I,iL}-f_{I,iL}\partial_{i} w_{2}} )e^{-w_{1}}\,\mathrm{d}P\\[3mm]
\displaystyle =\left( -1\right)^{s+1} \sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n_{0}} c_{I,L}\int_{V} u_{I,L} \sum^{n_0 }_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot e^{w_{1}-w_{2}}(\overline{\delta_{i} f_{I,iL}-f_{I,iL}\partial_{i} w_{2}} )e^{-w_{1}}\,\mathrm{d}P\\[3mm] \displaystyle =\left( u,g\right)_{L^{2}_{(s,t)}\left( V,w_{1}\right) }. \end{array} $$ The proof of Proposition \ref{general formula of T*} is completed. \end{proof}
The following proposition will be useful in the sequel. \begin{proposition}\label{support argument} Under Condition \ref{230424c1}, the following results hold.
\begin{itemize}
\item[$(1)$] Suppose that $f\in D_{T^{\ast}}$, $\hbox{\rm supp$\,$} f \stackrel{\circ}{\subset} V_{r}^{o}$ for some $r\in(0,+\infty)$, and $\sup\limits_{i\in\mathbb{N}}c_{I,iL}<\infty$ for each strictly increasing multi-indices $I$ and $L$ with $\left| I\right|=s$ and $\left| L\right|=t$ (See (\ref{gener111}) for $ c_{I,iL}$). Then $\hbox{\rm supp$\,$} (T^{\ast}f) \stackrel{\circ}{\subset} V_{r}^{o}$;
\item[$(2)$] If $f\in D_{T}$ and $\hbox{\rm supp$\,$} f \stackrel{\circ}{\subset} V_{r}^{o}$ for some $r\in(0,+\infty)$, then $\hbox{\rm supp$\,$} (Tf) \stackrel{\circ}{\subset} V_{r}^{o}$.
\end{itemize} \end{proposition} \begin{proof} (1) Without loss of generality, we assume that $f\not\equiv0$. Hence $\hbox{\rm supp$\,$} f\not=\emptyset$. By Lemma \ref{221018lem1}, $d\left( \hbox{\rm supp$\,$} f,\partial V^{o}_{r}\right) >0$. Let $U\triangleq\left\{ \textbf{z}\in V^{o}_{r}:\;d\left( \textbf{z},\hbox{\rm supp$\,$} f\right) <\frac{1}{2} d\left( \hbox{\rm supp$\,$} f,\partial V^{o}_{r}\right) \right\}$. We claim that $d\left( U,\partial V^{o}_{r}\right) \not=0$. Otherwise, we could find $\{\textbf{z}^{1 }_{n}\}_{n=1}^{\infty}\subset U$ and $\{\textbf{z}^{2 }_{n}\}_{n=1}^{\infty}\subset \partial V^{o}_{r} $ so that $\lim\limits_{n\rightarrow \infty } d\left( \textbf{z}^{1 }_{n},\textbf{z}^{2 }_{n}\right) =0.$ Then, there exists $n_0\in\mathbb{N}$ such that for all $n\geqslant n_0$, it holds that $d\left( \textbf{z}^{1 }_{n},\textbf{z}^{2 }_{n}\right)<\frac{1}{2} d\left( \hbox{\rm supp$\,$} f,\partial V^{o}_{r}\right)$, and hence $$
d\left( \textbf{z}^{1 }_{n},\hbox{\rm supp$\,$} f\right) \geqslant d\left( \textbf{z}^{2 }_{n},\hbox{\rm supp$\,$} f\right) -d\left( \textbf{z}^{1 }_{n},\textbf{z}^{2 }_{n}\right)
\geqslant d\left( \partial V^{o}_{r},\hbox{\rm supp$\,$} f\right) -d\left( \textbf{z}^{1}_{n},\textbf{z}^{2 }_{n}\right) >\frac{1}{2} d\left(\hbox{\rm supp$\,$} f,\partial V^{o}_{r}\right), $$
which is a contradiction. Thus $ d\left( U,\partial V^{o}_{r}\right) >0$. Hence $ d\left( \overline{U},\partial V^{o}_{r}\right) >0$, and by Lemma \ref{221018lem1} again, $\overline{U}\stackrel{\circ}\subset V^{o}_{r}$. Further, for all strictly increasing multi-indices $I$ and $L$ with $\left| I\right| =s$ and $\left| L\right| =t$ and $ \phi \in C^{\infty }_{0,F}\left( V\backslash \overline{U}\right)$ (which can be viewed as a subset of $C^{1 }_{0,F}\left( V\right)$ by the zero extension to $\overline{U}$), noting that $$
\int_{V} \left| \phi \right|^{2} e^{-w_{1}}\,\mathrm{d}P<\infty \text{ and } \sum^{\infty }_{j=1} c_{I,jL}\int_{V} \left| \partial_{j} \phi \right|^{2} e^{-w_{2}}\,\mathrm{d}P\leqslant \sup_{i\in\mathbb{N}}c_{I,iL} \sum^{\infty }_{j=1}\int_{V} \left| \partial_{j} \phi \right|^{2} e^{-w_{2}}\,\mathrm{d}P<\infty , $$ by Lemma \ref{T*formula}, we have (See (\ref{gener111}) for $ f_{I,jL}$)
$$
c_{I,L}\int_{V\backslash\overline{U} } \left( T^{\ast }f\right)_{I,L} \phi e^{-w_{1}}\,\mathrm{d}P=\left( -1\right)^{s} \sum^{\infty }_{j=1} \int_{V} c_{I,jL}f_{I,jL}\partial_{j} \phi e^{-w_{2}}\,\mathrm{d}P=0.
$$ By Corollary \ref{density2}, we have $\left( T^{\ast }f\right)_{I,L} =0$ almost everywhere on $V/\overline{U}$ with respect to $P$. Therefore, $\hbox{\rm supp$\,$} T^{\ast}f \subset \overline{U}\stackrel{\circ}{\subset} V_{r}^{o}$.
(2) Similarly to the above and the proof of the conclusion (1) of Proposition \ref{weak equality for more test functions}, we can show the conclusion (2) of Proposition \ref{support argument}. \end{proof}
Combining Lemma \ref{T*formula} and Proposition \ref{support argument}, we have the following variant of Lemma \ref{T*formula}. \begin{corollary}\label{weak T*formula}
Assume that Condition \ref{230424c1} holds, $ f=\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} f_{I,J}dz_{I}\wedge d\overline{z_{J}} \in D_{T^{\ast }}$ with $\hbox{\rm supp$\,$} f \stackrel{\circ}{\subset} V$, $\sup\limits_{i\in\mathbb{N}}c_{I,iL}<\infty$, $\phi \in C^{1}_0\left( V\right)$, and for all $E\in \mathscr{B}(V)$ with $E \stackrel{\circ}{\subset} V$, \begin{equation}\label{phi begin to D condition'}
\int_{E} \left| \phi \right|^{2} e^{-w_{1}}\,\mathrm{d}P<\infty \text{ and } \sum^{\infty }_{j=1} c_{I,jL}\int_{E} \left| \partial_{j} \phi \right|^{2} e^{-w_{2}}\,\mathrm{d}P<\infty \end{equation}
hold for some strictly increasing multi-indices $I$ and $L$ with $\left| I\right|=s$ and $\left| L\right|=t$. Then, \begin{equation}\label{230221e1} c_{I,L}\int_{V} \left( T^{\ast }f\right)_{I,L} \phi e^{-w_{1}}\,\mathrm{d}P=\left( -1\right)^{s} \sum_{j=1}^{\infty} c_{I,jL} \int_{V}\partial_{j} \phi \cdot f_{I,jL}e^{-w_{2}}\,\mathrm{d}P. \end{equation} \end{corollary} \begin{proof} Recall that $V$ is a pseudo-convex domain in $\ell^{2}$ and $\eta(\in C_F^\infty(U))$ is a plurisubharmonic exhaustion function of $V$ (See Condition \ref{230424c1}). By the conclusion (3) of Proposition \ref{properties on pseudo-convex domain}, we may find $r\in(0,+\infty)$ such that $\hbox{\rm supp$\,$} f \stackrel{\circ}{\subset} V_r^o$. Choose $h\in C^{\infty}(\mathbb{R})$ such that $0\leqslant h(\cdot)\leqslant 1$, $h(x)=0$ for all $x\geqslant r+1$ and $h(x)=1$ for all $x\leqslant r$. Let $\Phi\triangleq h(\eta)$. Then $\Phi\in C_{0,F}^{\infty}(V)$. By (\ref{phi begin to D condition'}), we see that $$
\int_{V} \left| \phi \cdot\Phi \right|^{2} e^{-w_{1}}\,\mathrm{d}P=\int_{V_{r+1}^o} \left| \phi \cdot\Phi \right|^{2} e^{-w_{1}}\,\mathrm{d}P
\leqslant \int_{V_{r+1}^o} \left| \phi \right|^{2} e^{-w_{1}}\,\mathrm{d}P<\infty $$ and $$ \begin{array}{ll}
\displaystyle \sum^{\infty }_{j=1} c_{I,jL}\int_{V} \left| \partial_{j} (\phi \cdot \Phi )\right|^{2} e^{-w_{2}}\,\mathrm{d}P \\[3mm]
\displaystyle =\sum^{\infty }_{j=1} c_{I,jL}\int_{V} \left| \Phi\cdot\partial_{j} \phi +\phi \cdot \partial_{j} \Phi \right|^{2} e^{-w_{2}}\,\mathrm{d}P \\[3mm]
\displaystyle \leqslant 2\sum^{\infty }_{j=1} c_{I,jL}\int_{V} \left| \Phi\cdot\partial_{j} \phi \right|^{2} e^{-w_{2}}\,\mathrm{d}P+2\sum^{\infty }_{j=1} c_{I,jL}\int_{V} \left| \phi \cdot \partial_{j} \Phi \right|^{2} e^{-w_{2}}\,\mathrm{d}P\\[3mm]
\displaystyle =2\sum^{\infty }_{j=1} c_{I,jL}\int_{V_{r+1}^o} \left| \Phi\cdot\partial_{j} \phi \right|^{2} e^{-w_{2}}\,\mathrm{d}P+2\sum^{\infty }_{j=1} c_{I,jL}\int_{V_{r+1}^o} \left| \phi \cdot \partial_{j} \Phi \right|^{2} e^{-w_{2}}\,\mathrm{d}P\\[3mm]
\displaystyle \leqslant 2\sum^{\infty }_{j=1} c_{I,jL}\int_{V_{r+1}^o} \left|\partial_{j} \phi \right|^{2} e^{-w_{2}}\,\mathrm{d}P+2 \cdot \left(\sup_{j\in\mathbb{N}} c_{I,jL} \right)\cdot \left(\sup_{V_{r+1}^o}\sum^{\infty }_{j=1}\left|\partial_{j} \Phi \right|^{2}\right)\cdot\int_{V_{r+1}^o} \left| \phi \right|^{2} e^{-w_{2}}\,\mathrm{d}P<\infty. \end{array} $$ Thus $\phi\cdot \Phi$ satisfies the condition \eqref{phi begin to D condition} in Lemma \ref{T*formula}, and hence $$ c_{I,L}\int_{V} \left( T^{\ast }f\right)_{I,L} \phi\cdot \Phi \cdot e^{-w_{1}}\,\mathrm{d}P=\left( -1\right)^{s} \sum_{j=1}^{\infty} c_{I,jL} \int_{V}\partial_{j}( \phi\cdot \Phi)\cdot f_{I,jL}e^{-w_{2}}\,\mathrm{d}P. $$ Since $\hbox{\rm supp$\,$} f \stackrel{\circ}{\subset} V_r^o$ and $\sup\limits_{i\in\mathbb{N}}c_{I,iL}<\infty$, by the conclusion (1) of Proposition \ref{support argument}, we see that $\hbox{\rm supp$\,$} T^*f \stackrel{\circ}{\subset} V_r^o$. Then, $$ \begin{array}{ll} \displaystyle c_{I,L}\int_{V} \left( T^{\ast }f\right)_{I,L} \phi e^{-w_{1}}\,\mathrm{d}P = c_{I,L}\int_{V_r^o} \left( T^{\ast }f\right)_{I,L} \phi e^{-w_{1}}\,\mathrm{d}P\\[3mm] \displaystyle = c_{I,L}\int_{V_r^o} \left( T^{\ast }f\right)_{I,L} \phi\cdot \Phi \cdot e^{-w_{1}}\,\mathrm{d}P= c_{I,L}\int_{V} \left( T^{\ast }f\right)_{I,L} \phi\cdot \Phi \cdot e^{-w_{1}}\,\mathrm{d}P, \end{array} $$ and similarly $\left( -1\right)^{s} \sum\limits_{j=1}^{\infty} c_{I,jL} \int_{V}\partial_{j}( \phi\cdot \Phi)\cdot f_{I,jL}e^{-w_{2}}\,\mathrm{d}P = \left( -1\right)^{s} \sum\limits_{j=1}^{\infty} c_{I,jL} \int_{V}(\partial_{j} \phi )\cdot f_{I,jL}e^{-w_{2}}\,\mathrm{d}P$. Thus we obtain the desired equality (\ref{230221e1}). This completes the proof of Corollary \ref{weak T*formula}. \end{proof}
To end this section, we put \begin{equation}\label{230419e21}
\mathscr{M}\triangleq\left\{ \mathfrak{m}\in C^{1}_b\left( V\right) :\;\sup_{V} \bigg(\sum^{\infty }_{i=1} (|\partial_{i} \mathfrak{m} |^{2}+ | \overline{\partial_{i} } \mathfrak{m}|^{2} )\cdot e^{w_{1}-w_{2}} \bigg) <\infty \right\}. \end{equation} Also, we introduce the following assumption (Recall (\ref{gener111}) for $c_{I,iJ}$). \begin{condition}\label{230423ass1}
Suppose that $\displaystyle c_1^{s,t}\triangleq \sup_{|I|=s,|J|=t,i\in\mathbb{N}}\frac{ c_{I,iJ} }{c_{I,J}}<\infty$ for all $s,t\in\mathbb{N}_0$. \end{condition}
The following result will be useful later. \begin{proposition}\label{general multiplitier} Under Conditions \ref{230424c1} and \ref{230423ass1}, for any $\mathfrak{m}\in \mathscr{M}$, it holds that \begin{itemize}
\item[$(1)$] If $f\in D_{T}$, then $\mathfrak{m}\cdot f\in D_{T}$ and $T\left( \mathfrak{m}\cdot f\right) =\mathfrak{m}\cdot (Tf)+(T\mathfrak{m}) \wedge f,$ where $$
(T\mathfrak{m})\wedge f\triangleq \left( -1\right)^{s} \sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| K\right| =t+1} \left(\sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t} \varepsilon^{K}_{iJ} f_{I,J}\overline{\partial_{i} } \mathfrak{m} \right)dz_{I}\wedge d\overline{z_{K}}; $$
\item[$(2)$] If $g\in D_{T^{\ast }}$, then $\mathfrak{m} \cdot g\in D_{{T}^{\ast }}$. \end{itemize} \end{proposition}
\begin{proof}
(1) Note that for any $\phi \in C^{\infty }_{0,F}\left( V\right)$, $\mathfrak{m}\phi\in C_0^1(V)$. Hence, for any strictly increasing multi-indices $I$ and $K$ with $\left| I\right| =s$ and $\left| K\right| =t+1$, by the conclusion (1) of Proposition \ref{weak equality for more test functions}, we have $$
\begin{array}{ll} \displaystyle
\left( -1\right)^{s+1} \int_{V} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t} \varepsilon^{K}_{iJ} \mathfrak{m} f_{I,J}\overline{\delta_{i} \phi } \,\mathrm{d}P
=\left( -1\right)^{s+1} \int_{V} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t} \varepsilon^{K}_{iJ} f_{I,J}\overline{(\delta_{i} \left( \overline{\mathfrak{m} } \phi \right) -\partial_{i} \overline{\mathfrak{m} } \cdot \phi )}\,\mathrm{d}P\\[3mm]
\displaystyle =\int_{V} \mathfrak{m} (T f)_{I,K}\overline{\phi } \,\mathrm{d}P+\left( -1\right)^{s} \int_{V} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t} \varepsilon^{K}_{iJ} f_{I,J}\overline{\partial_{i} } \mathfrak{m} \cdot \overline{\phi } \,\mathrm{d}P\\[3mm] \displaystyle =\int_{V} \mathfrak{m} (T f)_{I,K}\overline{\phi }\,\mathrm{d}P+ \int_{V}((T\mathfrak{m} )\wedge f)_{I,K} \cdot \overline{\phi } \,\mathrm{d}P.
\end{array} $$ By (\ref{gener111}), it follows that (Recall Condition \ref{230423ass1} for $c_1^{s,t}$) \begin{equation}\label{general inequality for Teta wedege f} \begin{array}{ll}
\displaystyle\left| \left|(T\mathfrak{m}) \wedge f\right| \right|_{L^{2}_{\left( s,t+1\right) }\left( V,w_{2}\right) }^2 \\[3mm]\displaystyle
\leqslant \left( t+1\right) \sum^{\prime }_{\left| I\right| =s,\left| K\right| =t+1} c_{I,K}\int_{V} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t} |\varepsilon^{K}_{iJ} f_{I,J}\overline{\partial_{i} } \mathfrak{m} |^{2}e^{-w_{2}}\,\mathrm{d}P \\[3mm]\displaystyle=\left( t+1\right) \sum^{\prime }_{\left| I\right| =s,\left| K\right| =t+1}\int_{V} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t}\frac{ c_{I,K}\cdot|\varepsilon^{K}_{iJ}|}{c_{I,J}}|\overline{\partial_{i} } \mathfrak{m}|^2\cdot c_{I,J}\cdot|f_{I,J}|^{2}e^{-w_{2}}\,\mathrm{d}P \\[3mm]\displaystyle= \left( t+1\right) \sum^{\prime }_{\left| I\right| =s }\int_{V} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t}\frac{ c_{I,iJ} }{c_{I,J}}|\overline{\partial_{i} } \mathfrak{m}|^2\cdot c_{I,J}\cdot|f_{I,J}|^{2}e^{-w_{2}}\,\mathrm{d}P \\[3mm]\displaystyle\leqslant \left( t+1\right) \cdot c_1^{s,t}\cdot\int_{V} \left(\sum^{\infty }_{i=1} |\overline{\partial_{i} } \mathfrak{m} |^{2}\cdot e^{w_{1}-w_{2}}\right)\cdot \left(\sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| J\right| =t} c_{I,J}\cdot\left| f_{I,J}\right|^{2}\right) e^{-w_{1}}\,\mathrm{d}P\\[3mm]\displaystyle\leqslant \left( t+1\right) \cdot c_1^{s,t}\cdot\sup_{V} \sum^{\infty }_{i=1} (|\overline{\partial_{i} } \mathfrak{m} |^{2}e^{w_{1}-w_{2}})\cdot \left| \left| f\right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V,w_{1}\right) } <\infty. \end{array} \end{equation}
Also, $||\mathfrak{m}\cdot(Tf)||_{L^{2}_{\left( s,t+1\right) }\left( V,w_{2}\right) } \leqslant \sup\limits_{V}|\mathfrak{m}|\cdot ||Tf||_{L^{2}_{\left( s,t+1\right) }\left( V,w_{2}\right) }$. Therefore, $\mathfrak{m} \cdot f \in D_{T}$ and $T ( \mathfrak{m}\cdot f ) =\mathfrak{m}\cdot (Tf)+(T\mathfrak{m}) \wedge f.$
(2) Clearly, $\overline{\mathfrak{m}}\in \mathscr{M}$. For any $u\in D_{T}$, by conclusion (1), it holds that $\overline{\mathfrak{m}}\cdot u\in D_T$ and $T\left( \overline{\mathfrak{m}}\cdot u\right) =\overline{\mathfrak{m}}\cdot (Tu)+(T\overline{\mathfrak{m}}) \wedge u$ and hence $$ \begin{array}{ll} \displaystyle
\left( Tu,\mathfrak{m} \cdot g\right)_{L^{2}_{(s,t+1)}\left( V,w_{2}\right) } =\left( \overline{\mathfrak{m} }\cdot (Tu),g\right)_{L^{2}_{(s,t+1)}\left( V,w_{2}\right) } \\[3mm] \displaystyle =\left(T\left( \overline{\mathfrak{m}}\cdot u\right),g\right)_{L^{2}_{\left( s,t+1\right) }\left( V,w_{2}\right) } -\left( (T\overline{\mathfrak{m}}) \wedge u,g\right)_{L^{2}_{\left( s,t+1\right) }\left( V,w_{2}\right) } \\[3mm] \displaystyle =\left( u,\mathfrak{m}\cdot T^{\ast }g\right)_{L^{2}_{\left( s,t\right) }\left( V,w_{1}\right) } -\left( (T\overline{\mathfrak{m}}) \wedge u,g\right)_{L^{2}_{\left( s,t+1\right) }\left( V,w_{2}\right) }.
\end{array} $$ By \eqref{general inequality for Teta wedege f}, we have $$ \begin{array}{ll} \displaystyle
| ( Tu,\mathfrak{m}\cdot g )_{L^{2}_{(s,t+1)}\left( V,w_{2}\right) } |
\leqslant |( u,\mathfrak{m}\cdot T^{\ast }g)_{L^{2}_{\left( s,t\right) }\left( V,w_{1}\right) }| +|( (T\overline{\mathfrak{m}}) \wedge u,g)_{L^{2}_{\left( s,t+1\right) }\left( V,w_{2}\right) }|\\[3mm]
\displaystyle \leqslant \left( || \mathfrak{m}\cdot T^{\ast }g||_{L^{2}_{\left( s,t\right) }\left( V,w_{1}\right) } +C\cdot\left| \left| g\right| \right|_{L^{2}_{\left( s,t+1\right) }\left( V,w_{2}\right) } \right) \cdot \left| \left| u\right| \right|_{L^{2}_{\left( s,t\right) }\left( V,w_{1}\right) }\\[3mm]
\displaystyle \leqslant \left( \sup_V|\mathfrak{m}|\cdot || T^{\ast }g ||_{L^{2}_{\left( s,t\right) }\left( V,w_{1}\right) } +C \cdot\left| \left| g\right| \right|_{L^{2}_{\left( s,t+1\right) }\left( V,w_{2}\right) } \right) \cdot \left| \left| u\right| \right|_{L^{2}_{\left( s,t\right) }\left( V,w_{1}\right) }, \end{array} $$
where $C\triangleq \sqrt{\left( t+1\right) \cdot c_1^{s,t}\cdot \sup\limits_{V} \sum\limits^{\infty }_{i=1} (\left| \partial_{i} \mathfrak{m}\right|^{2} e^{w_{1}-w_{2}})}$. By the definition of $T^*$ (e.g., \cite[Definition 4.2.2, p. 177]{Kra}), we have $\mathfrak{m}\cdot g\in D_{T^{\ast }}$ and this completes the proof of Proposition \ref{general multiplitier}. \end{proof}
\section{$L^2$ estimates for the $\overline{\partial}$ equations on pseudo-convex domains in $\ell^{2}$}\label{secx4}
The purpose of this section is to establish the key $L^2$ estimates for the $\overline{\partial}$ equations (on pseudo-convex domains in $\ell^{2}$) by means of some suitable approximations.
From now on, we fix a pseudo-convex domain $V$ in $\ell^{2}$ and a plurisubharmonic exhaustion function $\eta(\in C_F^\infty(V))$ of $V$. We need to choose suitably the weight functions $w_{1},w_{2}$ and $w_{3}$ in Definition \ref{definition of T S}. For this purpose, we need the following notion. \begin{definition}
A complex valued function $f$ on a nonempty open subset $U$ of $\ell^2$ is called a locally bounded function on $U$ if $\sup\limits_{E} \left| f\right| <\infty$ for all $E\stackrel{\circ}{\subset}U$. \end{definition}
We also need the following simple result. \begin{lemma}\label{majority funtion}
Suppose that Condition \ref{230424c1} holds and $f$ is a locally bounded function on $V$. Then, there exists $g\in C^{\infty}_{F}(V)$ such that $|f(\textbf{z})|\leqslant g(\textbf{z})$ for all $\textbf{z}\in V$. \end{lemma} \begin{proof}
For each $j\in \mathbb{N}$, define $V_j$ as that in (\ref{230117e1}). Then $V=\bigcup\limits^{\infty }_{j=1} V_{j}=V_1\bigcup \bigcup\limits^{\infty }_{j=1} \left(V_{j+1}\setminus V_j\right)$. Since $f$ is a locally bounded function on $V$, we have $c_j\triangleq \sup\limits_{\textbf{z}\in V_{j}}|f(\textbf{z})|<\infty$ for each $j\in\mathbb{N}$. Choose $h \in C^{\infty }\left( \mathbb{R} \right)$ such that $h(x)\geqslant c_{j+1}$ for any $x\in( j,j+1],\,j\in\mathbb{N}$ and $h(x)=c_1$ for any $x\leqslant 1$. Let $g\triangleq h (\eta)$. It is obvious that $g\in C^{\infty}_{F}(V)$ such that $|f(\textbf{z})|\leqslant g(\textbf{z})$ for all $\textbf{z}\in V$ which completes the proof of Lemma \ref{majority funtion}. \end{proof}
For every $k\in\mathbb{N}$, let $$ h_{k} \left( t\right) \triangleq\begin{cases}0,&t>k +1,\\ \left( t-k -1\right)^{2} \left( 2t+1-2k \right),&k \leqslant t\leqslant k +1,\\ 1,&t<k. \end{cases} $$
Then, $h_{k }\in C^{1 }\left( \mathbb{R}\right)$, $0\leqslant h_{k}\leqslant 1$, $h_{k}(x)=1$ for all $x<k$, $h_{k } \left( x\right) =0$ for all $x>k+1$ and $\left| h^{\prime }_{k} \right| \leqslant \frac{3}{2}$. We construct a sequence of cut-off functions $\{X_{k}\}$ on $V$ as follows. \begin{equation}\label{section2}
X_{k}\triangleq h_{k }( \eta) \in C^{1}_{0,F}\left( V\right). \end{equation} Note that for each $k\in\mathbb{N}$, $$
\ln \left( 1+\sum\limits^{\infty }_{i=1} |\overline{\partial_{i} } X_{k}|^{2}\right)
=\ln \left( 1+\sum\limits^{\infty }_{i=1} |h_k^{'}(\eta)|^2\cdot|\overline{\partial_{i} }\eta|^{2}\right)
\leqslant \ln \left( 1+\frac{9}{4}\cdot\sum\limits^{\infty }_{i=1}|\overline{\partial_{i} }\eta|^{2}\right). $$
Since $\ln \left( 1+\frac{9}{4}\cdot\sum\limits^{\infty }_{i=1}|\overline{\partial_{i} }\eta|^{2}\right)$ is a locally bounded function on $V$, by Lemma \ref{majority funtion}, there exists $\psi \in C^{\infty }_{F}\left( V\right)$ such that $\ln \left( 1+\frac{9}{4}\cdot\sum\limits^{\infty }_{i=1}|\overline{\partial_{i} }\eta|^{2}\right) \leqslant \psi $. Therefore, \begin{equation}\label{majority function}
\sum^{\infty }_{i=1} |\overline{\partial_{i} } X_{k}|^{2}\leqslant e^{\psi },\quad \forall\; k\in\mathbb{N}. \end{equation} In the rest of this paper, unless other stated, we shall choose the weight functions in Definition \ref{definition of T S} as follows: \begin{equation}\label{weight function}
w_{1} =\varphi -2\psi ,\quad w_{2} =\varphi -\psi ,\quad w_{3} =\varphi , \end{equation} where $\varphi \in C^{2}_{F}\left( V\right)$ is a given real-valued function (satisfying some suitable condition to be given later).
Recall that $T$ and $S$ are given as that in Definition \ref{definition of T S}. We need to approximate suitably the elements in $ D_{S}\bigcap D_{T^{\ast }}$, for which the whole process is divided into two parts. The following proposition is the first part of our approximation process (Recall \eqref{weight function} for $w_1,w_2$ and $w_3$ and \eqref{section2} for $\{X_{k}\}_{k=1}^{\infty}$). \begin{proposition}\label{general cut-off density3}
Under Conditions \ref{230424c1} and \ref{230423ass1}, for each $f\in D_{S}\bigcap D_{T^{\ast }}$, it holds that $X_k\cdot f\in D_{S}\bigcap D_{T^{\ast }}$ for any $k\in\mathbb{N}$ and $$
\lim_{k \rightarrow \infty }\Big( \left| \left| S\left( X_{k}\cdot f\right) -X_{k}\cdot (Sf)\right| \right|_{w_{3}} +\left| \left| T^{\ast }\left( X_{k}\cdot f\right) -X_{k }\cdot (T^{\ast }f)\right| \right|_{w_{1}} +\left| \left| X_{k}\cdot f-f\right| \right|_{w_{2}} \Big) =0, $$
where we simply denote $ \left| \left| \cdot \right| \right|_{L^{2}_{\left( s,t+i-1\right) }\left( V,w_{i}\right) }$ by $\left| \left| \cdot \right| \right|_{w_{i}}$ for each $i=1,2,3$. \end{proposition} \begin{proof} By \eqref{section2}, we see that $\{X_{k}\}_{k=1}^{\infty}\subset \mathscr{M}$ (Recall (\ref{230419e21}) for $\mathscr{M}$). Hence, similarly to the proof of the conclusion (1) of Proposition \ref{general multiplitier}, one can show that $X_{k}\cdot f\in D_{S}$ for each $k\in\mathbb{N}$ and $S\left( X_{k}\cdot f\right) =X_{k}\cdot (Sf)+(SX_{k}) \wedge f,$ where $$
(SX_{k})\wedge f\triangleq \left( -1\right)^{s} \sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| K\right| =t+2} \left(\sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t+1} \varepsilon^{K}_{iJ} f_{I,J}\overline{\partial_{i} } X_{k} \right)dz_{I}\wedge d\overline{z_{K}}. $$ By the conclusion (2) of Proposition \ref{general multiplitier}, we have $X_k\cdot f\in D_{T^{\ast }}$, and hence $X_k\cdot f\in D_{S}\bigcap D_{T^{\ast }}$, and (Recall Condition \ref{230423ass1} for $c_1^{s,t+1}$) \begin{eqnarray*}
&&\left| \left| S\left( X_{k}\cdot f\right) -X_{k}\cdot(Sf)\right| \right|^{2}_{w_{3} }
=\sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| K\right| =t+2} c_{I,K}\int_{V} \left|\sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t+1} \varepsilon^{K}_{i,J} f_{I,J}\overline{\partial_{i} } X_{k }\right|^{2}e^{-w_{3}}\,\mathrm{d}P\\
&&\leqslant (t+2)\sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| K\right| =t+2} c_{I,K}\int_{V}\sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t+1} \left|\varepsilon^{K}_{i,J} f_{I,J}\overline{\partial_{i} } X_{k }\right|^{2}e^{-w_{3}}\,\mathrm{d}P\\
&&=(t+2)\sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| K\right| =t+2}\int_{V}\sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t+1} \frac{c_{I,K}\cdot|\varepsilon^{K}_{i,J}|}{c_{I,J}} \cdot c_{I,J}\cdot |f_{I,J}|^2\cdot|\overline{\partial_{i} }X_{k } |^{2}e^{-w_{3}}\,\mathrm{d}P\\
&&=(t+2)\sum^{\prime }_{\left| I\right| =s}\int_{V}\sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t+1} \frac{c_{I,iJ}}{c_{I,J}} \cdot c_{I,J}\cdot |f_{I,J}|^2\cdot|\overline{\partial_{i} }X_{k } |^{2}e^{-w_{3}}\,\mathrm{d}P\\
&&\leqslant (t+2)\cdot c_1^{s,t+1}\cdot\sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| J\right| =t+1}c_{I,J}\cdot\int_{V} |f_{I,J}|^2\cdot\sum^{\infty }_{i=1} |\overline{\partial_{i} }X_{k } |^{2}\cdot e^{-w_{3}}\,\mathrm{d}P. \end{eqnarray*}
Recalling \eqref{majority function} and \eqref{weight function}, we have $\sum\limits^{\infty }_{i=1} |\overline{\partial_{i} } X_{k}|^{2}\leqslant e^{\psi }=e^{w_{3}-w_{2}}$ for all $k\in\mathbb{N}.$ Thus $$
\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J} \left| f_{I,J}\right|^{2} \cdot\sum^{\infty }_{i=1} |\overline{\partial_{i} } X_{k }|^{2} e^{-w_{3}} \leqslant\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J} \left| f_{I,J}\right|^{2} e^{-w_{2}} . $$
Note that for each $\textbf{z}\in V$, if $k>\eta(\textbf{z})$, then $|\overline{\partial_{i} } X_{k}\left( \textbf{z}\right) |^{2}=0$ for all $i\in\mathbb{N}$. Hence, $$
\lim_{k \rightarrow \infty } \sum^{\infty }_{i=1} |\overline{\partial_{i} } X_{k }(\textbf{z})|^{2}=0,\quad \forall\; \textbf{z}\in V.\ $$ By Lebesgue's Dominated Convergence Theorem, it follows that $$
\lim_{k\rightarrow \infty }\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} \sum^{\infty }_{i=1} |\overline{\partial_{i} } X_{k}|^{2}\left| f_{I,J}\right|^{2} e^{-w_{3}}\,\mathrm{d}P=0,
$$
and hence $\lim\limits_{k\rightarrow \infty } \left| \left| S\left( X_{k}\cdot f\right) -X_{k}\cdot(Sf)\right| \right|^{2}_{w_{3}} =0.$
On the other hand, for every $u\in D_{T}$, by Proposition \ref{general multiplitier}, and noting \eqref{majority function} and \eqref{weight function} again, we obtain that \begin{eqnarray*}
&&\left|\left( T^{\ast }\left( X_{k}\cdot f\right) -X_{k}\cdot(T^{\ast }f),u\right)_{L^{2}_{\left( s,t\right) }\left( V,w_{1}\right) } \right|
=\left|\left(f, X_{k}\cdot(Tu)\right)_{L^{2}_{\left( s,t+1\right) }\left( V,w_{2}\right) } -\left( f,T(X_{k}\cdot u)\right)_{L^{2}_{\left( s,t+1\right) }\left( V,w_{2}\right) } \right|\\
&&=\left|\left( f,(T X_{k})\wedge u\right)_{L^{2}_{\left( s,t+1\right) }\left( V,w_{2}\right) } \right|=\left| \sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| J\right| =t+1} c_{I,J}\int_{V} f_{I,J}\overline{((TX_{k})\wedge u)_{I,J}} e^{-w_{2}}\,\mathrm{d}P\right| \\
&&\leqslant \sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| J\right| =t+1} c_{I,J}\int_{V} \left|f_{I,J}e^{-\frac{w_{2}}{2} }\right|\cdot \left|((TX_{k})\wedge u)_{I,J}e^{-\frac{w_{2}}{2} }\right|\,\mathrm{d}P\\
&&\leqslant \int_{V} \left(\sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| J\right| =t+1}c_{I,J} |f_{I,J}|^{2}e^{-w_{2}}\right)^{\frac{1}{2} }\cdot \left(\sum^{\prime }_{\left| I\right| =s}\sum_{\left| J\right| =t+1}^{\prime } c_{I,J}|((TX_{k})\wedge u)_{I,J}|^{2}e^{-w_{2}}\right)^{\frac{1}{2} }\,\mathrm{d}P\\
&&\leqslant \sqrt{(t+1)\cdot c_1^{s,t}} \cdot\int_{V\setminus V_{k-1}} \left(\sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| J\right| =t+1} c_{I,J} |f_{I,J}|^{2}e^{-w_{2}}\right)^{\frac{1}{2} }\cdot \left(\sum^{\prime }_{\left| I\right| =s}\sum_{\left| L\right| =t}^{\prime } c_{I,J}|u_{I,L}|^{2}e^{-w_{1}}\right)^{\frac{1}{2} }\,\mathrm{d}P\\
&&\leqslant \sqrt{(t+1)\cdot c_1^{s,t}} \cdot\left(\int_{V\setminus V_{k-1}} \sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| J\right| =t+1} c_{I,J}|f_{I,J}|^{2}e^{-w_{2}}\,\mathrm{d}P\right)^{\frac{1}{2} }\cdot \left| \left| u\right| \right|_{L^{2}_{\left( s,t\right) }\left( V,w_{1}\right) } , \end{eqnarray*} where the third inequality follows from the facts that $((TX_{k})\wedge u)_{I,J}=0$ on $V_{k-1}$ for all $I,J$, \begin{eqnarray*}
\sum^{\infty }_{i=1} |\overline{\partial_{i} } X_{k}|^{2}\leqslant e^{\psi }=e^{w_{2}-w_{1}} \end{eqnarray*} and $$
\begin{array}{ll} \displaystyle
\sum_{\left| I\right| =s}\sum^{\prime }_{\left| J\right| =t+1}c_{I,J} |((T X_{k })\wedge u)_{I,J}|^{2}e^{-w_{2}}=\sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| J\right| =t+1} c_{I,J}\left|\sum^{\infty }_{i=1} \sum^{\prime }_{\left| L\right| =t} \varepsilon^{J}_{iL} u_{I,L}\overline{\partial_{i} } X_{k}\right|^{2}e^{-w_{2}}\\[3mm]
\displaystyle \leqslant \left( t+1\right)\cdot \sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| J\right| =t+1}c_{I,J} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| L\right| =t} |\varepsilon^{J}_{iL} u_{I,L}\overline{\partial_{i} } X_{k}|^{2}e^{-w_{2}}\\[3mm]
\displaystyle \leqslant \left( t+1\right)\cdot \sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| J\right| =t+1}\sum^{\infty }_{i=1} \sum^{\prime }_{\left| L\right| =t} \frac{c_{I,J} |\varepsilon^{J}_{iL}|}{c_{I,L}}\cdot c_{I,L} \cdot|u_{I,L}|^2\cdot|\overline{\partial_{i} } X_{k}|^{2}e^{-w_{2}}\\[3mm]
\displaystyle \leqslant \left( t+1\right) \cdot \sum^{\prime }_{\left| I\right| =s} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| L\right| =t} \frac{c_{I,iL} }{c_{I,L}}\cdot c_{I,L} \cdot|u_{I,L}|^2\cdot|\overline{\partial_{i} } X_{k}|^{2}e^{-w_{2}}\\[3mm]
\displaystyle \leqslant \left( t+1\right)\cdot c_1^{s,t}\cdot \sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} c_{I,L}|u_{I,L}|^2\sum^{\infty }_{i=1}|\overline{\partial_{i} } X_{k}|^{2}e^{-w_{2}}\\[3mm]
\displaystyle \leqslant \left( t+1\right)\cdot c_1^{s,t}\cdot \sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| L\right| =t} c_{I,L}|u_{I,L}|^{2}e^{-w_{1}}.
\end{array} $$ Since $D_T$ is dense in $L^{2}_{\left( s,t\right) }\left( V,w_{1}\right)$, we deduce that $$
\left| \left| T^{\ast }\left( X_{k}\cdot f\right) -X_{k}\cdot( T^{\ast }f)\right| \right|_{L^{2}_{(s,t)}\left( V,w_{1}\right) }\leqslant \sqrt{(t+1)\cdot c_1^{s,t}} \left(\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| J\right| =t+1} c_{I,J}\int_{V\setminus V_{k-1}} |f_{I,J}|^{2}e^{-w_{2}}\,\mathrm{d}P\right)^{\frac{1}{2} }. $$
Note that the right side of the above inequality tends to zero as $k\to\infty$, which implies that $\lim\limits_{k\rightarrow \infty } \left| \left| T^{\ast }\left( X_{k}\cdot f\right) -X_{k} \cdot(T^{\ast }f)\right| \right|_{L^{2}_{(s,t)}\left( V,w_{1}\right) } =0.$
Since $|X_{k}|\leqslant 1$, we have $\sum\limits^{\prime }_{\left| I\right| =s}\sum\limits^{\prime }_{\left| J\right| =t+1} c_{I,J}|\left( X_{k}-1\right) f_{I,J}|^{2}e^{-w_{2}}\leqslant 4 \sum\limits^{\prime }_{\left| I\right| =s}\sum\limits^{\prime }_{\left| J\right| =t+1} c_{I,J}\left| f_{I,J}\right|^{2} e^{-w_{2}}$. We also note that for each $\textbf{z}\in V$, if $k>\eta(\textbf{z})$, then $X_{k}\left( \textbf{z}\right)=1$, and hence $$ \lim_{k \rightarrow \infty } X_{k }(\textbf{z}) =1,\ \forall\; \textbf{z}\in V.\ $$ By Lebesgue's Dominated Convergence Theorem, we arrive at $$
\begin{array}{ll} \displaystyle
\lim_{k\rightarrow \infty } \left| \left| X_{k} \cdot f-f\right| \right|^{2}_{L^{2}_{(s,t+1)}\left( V,w_{2}\right) }
=\lim_{k \rightarrow \infty } \int_{V} \sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| J\right| =t+1} c_{I,J}|\left( X_{k}-1\right) f_{I,J}|^{2}e^{-w_{2}}\,\mathrm{d}P \\[3mm]
\displaystyle = \int_{V} \lim_{k\rightarrow \infty }\sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| J\right| =t+1} c_{I,J}|\left( X_{k}-1\right) f_{I,J}|^{2}e^{-w_{2}}\,\mathrm{d}P=0 .
\end{array} $$ The proof of Proposition \ref{general cut-off density3} is completed. \end{proof}
The following technical result will be useful later. \begin{lemma}\label{general I,P}
Let Conditions \ref{230424c1} and \ref{230423ass1} hold. Then, for any $g=\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t+1 } g_{I,J}dz_{I}\wedge d\overline{z_{J}}\in D_{T^{\ast}}$ with $\hbox{\rm supp$\,$} g\stackrel{\circ}{\subset} V$, the following conclusions hold:
\begin{itemize}
\item[$(1)$] For each strictly increasing multi-indices $I$ and $L$ with $\left| I\right| =s$ and $\left| L\right| =t$, the series $$\sum\limits^{\infty }_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot g_{I,iL}\cdot\partial_{i} w_2$$ converges almost everywhere (with respect to $P$) on $V$, where $g_{I,iL}\triangleq \sum\limits^{\prime }_{\left| J\right| =t+1}\varepsilon^{J}_{iL}g_{I,J}$ for $i\in\mathbb{N}$;
\item[$(2)$] The following two forms \begin{equation}\label{general def of I and P}
\mathbb{I}g\triangleq \sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} \left(\sum^{\infty }_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot g_{I,iL}\cdot\partial_{i} w_{2}\right)dz_{I}\wedge d\overline{z_{L}} ,\quad \mathbb{P}g\triangleq \mathbb{I}g-\left( -1\right)^{s}\cdot e^{w_{2}-w_{1}}\cdot (T^{\ast }g) \end{equation}
satisfy that $\mathbb{I}g\in L^{2}_{(s,t)}\left( V,w_{1}\right)$ and $\mathbb{P}g\in L^{2}_{(s,t)}\left( V,w_{1}\right)$. Furthermore, if there exists $n_{0}\in \mathbb{N}$ such that $g_{I,J}=0$ for all strictly increasing multi-indices $I$ and $J$ with $\left| I\right| =s$, $\left| J\right| =t+1$ and $\max \left\{ I\cup J\right\} >n_{0}$. Then,
$\left( \mathbb{I}g\right)_{I,L} =0$ and $\left( \mathbb{P}g\right)_{I,L} =0$ for all strictly increasing multi-indices $I$ and $L$ with $\left| I\right| =s$, $\left| L\right| =t$ and $\max \left\{ I\cup L\right\} >n_{0}.$ \end{itemize}
\end{lemma}
\begin{proof}
(1) Since $T^*g\in L^{2}_{(s,t)}\left( V,w_{1}\right)$, by Proposition \ref{support argument}, we have $\hbox{\rm supp$\,$} (T^*g)\stackrel{\circ}{\subset}V$. Since $\hbox{\rm supp$\,$} g\stackrel{\circ}{\subset}V$, by the conclusion (3) of Proposition \ref{properties on pseudo-convex domain}, there exists $r\in(0,+\infty)$ such that $\hbox{\rm supp$\,$} g\subset V_r^o\subset V_r$. Then for each strictly increasing multi-indices $I$ and $L$ with $\left| I\right| =s$ and $\left| L\right| =t$, by the Cauchy-Schwarz inequality, we have \begin{eqnarray*}
&&\int_{V} \sum^{\infty }_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot |g_{I,iL}\cdot \partial_{i}w_{2}|\,\mathrm{d}P\\
&&=\sum^{\infty }_{i=1}\int_{V}\frac{c_{I,iL}}{c_{I,L}}\cdot |g_{I,iL}|\cdot e^{-\frac{w_{2}}{2}}\cdot e^{\frac{w_{2}}{2}}\cdot|\partial_{i}w_{2}|\,\mathrm{d}P\\
&&\leqslant \left(\sum^{\infty }_{i=1} \left(\frac{c_{I,iL}}{c_{I,L}}\right)^2\cdot\int_{V} \left| g_{I,iL}\right|^{2} e^{-w_{2}}\,\mathrm{d}P\right)^{\frac{1}{2} }\cdot \left(\sum^{\infty }_{i=1} \int_{V_{r}} \left| \partial_{i}w_{2}\right|^{2} e^{w_{2}}\,\mathrm{d}P\right)^{\frac{1}{2} }\\
&&\leqslant \sqrt{\frac{1}{c_{I,L}}\cdot\left(\sup_{i\in\mathbb{N}}\frac{c_{I,iL}}{c_{I,L}}\right)}\cdot\left(\sum^{\infty }_{i=1} c_{I,iL} \cdot\int_{V} \left| g_{I,iL}\right|^{2} e^{-w_{2}}\,\mathrm{d}P\right)^{\frac{1}{2} }\cdot \left(\sum^{\infty }_{i=1} \int_{V_{r}} \left| \partial_{i}w_{2}\right|^{2} e^{w_{2}}\,\mathrm{d}P\right)^{\frac{1}{2} }. \end{eqnarray*} Also, $$
\begin{array}{ll}
\displaystyle \sum^{\infty }_{i=1} c_{I,iL} \cdot\int_{V} \left| g_{I,iL}\right|^{2}\cdot e^{-w_{2}}\,\mathrm{d}P=\sum^{\infty }_{i=1}\int_{V} c_{I,iL} \cdot g_{I,iL}\cdot\overline{g_{I,iL}} \cdot e^{-w_{2}}\,\mathrm{d}P\\[3mm]
\displaystyle =\sum^{\infty }_{i=1} \int_{V}\sum^{\prime }_{\left| J_1\right| =t+1,\left| J_2\right| =t+1,\left| J_3\right| =t+1}\left|\varepsilon^{J_1}_{iL}\right|\cdot\varepsilon^{J_2}_{iL}\cdot \varepsilon^{J_3}_{iL}\cdot c_{I,J_1}\cdot g_{I,J_2}\cdot \overline{g_{I,J_3}}\cdot e^{-w_{2}}\,\mathrm{d}P\\[3mm]
\displaystyle =\sum^{\infty }_{i=1} \int_{V}\sum^{\prime }_{\left| J\right| =t+1}\left|\varepsilon^{J}_{iL}\right|^3\cdot c_{I,J}\cdot \left| g_{I,J}\right|^2\cdot e^{-w_{2}}\,\mathrm{d}P, \end{array} $$ which gives \begin{equation}\label{230414e3}
\begin{array}{ll}
\displaystyle \sum^{\prime }_{\left| I\right| =s,\left| L\right| =t}\sum^{\infty }_{i=1} c_{I,iL} \cdot\int_{V} \left| g_{I,iL}\right|^{2}\cdot e^{-w_{2}}\,\mathrm{d}P\\[3mm]
\displaystyle =\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1}\int_{V}\sum^{\infty }_{i=1} \sum^{\prime }_{\left| L\right| =t}\left|\varepsilon^{J}_{iL}\right|^3\cdot c_{I,J}\cdot \left| g_{I,J}\right|^2\cdot e^{-w_{2}}\,\mathrm{d}P\\[3mm]
\displaystyle =(t+1)\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1}c_{I,J}\cdot \int_{V}\left| g_{I,J}\right|^2\cdot e^{-w_{2}}\,\mathrm{d}P. \end{array} \end{equation} Hence,
$$\int_{V} \sum^{\infty }_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot |g_{I,iL}\cdot \partial_{i}w_{2}|\,\mathrm{d}P<\infty, $$ which implies that the series $\sum\limits^{\infty }_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot g_{I,iL}\partial_{i} w_{2}$ converges almost everywhere (with respect to $P$) on $V$.
(2) \ \ By (\ref{230414e3}), it follows that (Recall Condition \ref{230423ass1} for $c_1^{s,t}$) \begin{eqnarray*}
&&\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t}c_{I,L}\cdot\int_{V} \left|\sum^{\infty }_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot g_{I,iL}\cdot\left(\partial_{i} w_{2}\right)\right|^{2}\cdot e^{-w_1}\,\mathrm{d}P\\
&&=\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t}c_{I,L}\cdot\int_{V_r} \left|\sum^{\infty }_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot g_{I,iL}\cdot\left(\partial_{i} w_{2}\right)\right|^{2} e^{-w_1}\,\mathrm{d}P\\
& &\leqslant \sum^{\prime }_{\left| I\right| =s,\left| L\right| =t}c_{I,L}\cdot\int_{V_{r}} \left(\sum^{\infty }_{i=1} \left(\frac{c_{I,iL}}{c_{I,L}}\right)^2\cdot\left| g_{I,iL}\right|^{2}\right)\cdot e^{-w_2}\cdot \left( \sum^{\infty }_{i=1} \left| \partial_{i}w_{2}\right|^{2} \right)\cdot e^{w_2-w_1} \,\mathrm{d}P\\
& &\leqslant \sup_{V_{r}}\left( \sum^{\infty }_{i=1} \left|\partial_{i}w_{2}\right|^{2}\cdot e^{w_2-w_1} \right)\cdot c_1^{s,t}\cdot\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t}\int_{V} \left(\sum^{\infty }_{i=1}c_{I,iL} \cdot\left| g_{I,iL}\right|^{2}\right)\cdot e^{-w_2} \,\mathrm{d}P\\
&&= \sup_{V_{r}}\left( \sum^{\infty }_{i=1} \left|\partial_{i}w_{2}\right|^{2}\cdot e^{w_2-w_1} \right)\cdot c_1^{s,t}\cdot(t+1)\cdot \sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1}c_{I,J}\cdot\int_{V}\left| g_{I,J}\right|^{2}\cdot e^{-w_2} \,\mathrm{d}P\\ &&<\infty, \end{eqnarray*} and hence $\mathbb{I}g\in L^{2}_{(s,t)}\left( V,w_{1}\right)$.
By the definition of $\mathbb{I}g$, it is easy to see that $\left( \mathbb{I}g\right)_{I,L} =0$ for all strictly increasing multi-indices $I$ and $L$ with $\left| I\right| =s,\left| L\right| =t$ and $\max \left\{ I\cup L\right\} >n_{0}$. For any $\phi\in C^{\infty }_{0,F}\left( V\right)$ and strictly increasing multi-indices $I$ and $L$ with $\left| I\right| =s,\left| L\right| =t$ and $\max \left\{ I\cup L\right\} >n_{0}$, noting that \begin{eqnarray*}
\sum^{\infty }_{j=1} |c_{I,jL}|\int_{V} \left| \partial_{j} \phi \right|^{2}\cdot e^{-w_{2}}\,\mathrm{d}P
\leqslant c_{I,L}\cdot \left(\sup_{i\in\mathbb{N}} \frac{c_{I,iL}}{c_{I,L}}\right)\cdot \sum^{\infty }_{j=1}\int_{V} \left| \partial_{j} \phi \right|^{2} \cdot e^{-w_{2}}\,\mathrm{d}P<\infty , \end{eqnarray*} using Lemma \ref{T*formula}, we obtain that \begin{eqnarray*}
\int_{V} \left( T^{\ast }g\right)_{I,L} \cdot \phi \cdot e^{-w_{1}}dP=\frac{1}{c_{I,L}}\left( -1\right)^{s} \sum_{j=1}^{\infty} c_{I,jL} \int_{V}\partial_{j} \phi \cdot g_{I,jL}\cdot e^{-w_{2}}\,\mathrm{d}P=0. \end{eqnarray*} By Corollary \ref{density2}, $C^{\infty }_{0,F}\left( V\right)$ is dense in $L^2(V,e^{-w_1}P)$, and hence $\left( T^{\ast }g\right)_{I,L}=0$ and $\left( \mathbb{P}g\right)_{I,L}=0$ for above indices $I$ and $L$. This completes the proof of Lemma \ref{general I,P}. \end{proof}
The following theorem is the second part of our approximations (Recall Definition \ref{definition of T S} for the operators $T$ and $S$). \begin{theorem}\label{general density4}
Under Conditions \ref{230424c1} and \ref{230423ass1}, for any $f\in D_{S}\bigcap D_{T^{\ast }}$ with $\hbox{\rm supp$\,$} f\stackrel{\circ}{\subset} V_{r}^{o}$ for some $r\in(0,+\infty)$, there exists $\{h_{k}\}_{k=1}^{\infty}\subset D_{S}\cap D_{T^{\ast }} $ and $\{n_k\}_{k=1}^{\infty}\subset\mathbb{N}$ such that for each $k\in\mathbb{N}$, $\left( h_{k }\right)_{I,J} =0$ for all strictly increasing multi-indices $I$ and $J$ with $\left| I\right| =s,\left| J\right| =t+1$ and $\max \left\{ I\bigcup J\right\} >n_{k}$ and $\left( h_{k }\right)_{I,J} \in C^{\infty}_{0,F}(V)$ for all strictly increasing multi-indices $I$ and $J$ with $\left| I\right| =s,\left| J\right| =t+1$ and $\max \left\{ I\bigcup J\right\}\leqslant n_{k}$. Moreover, $$
\lim_{k \rightarrow \infty } \left(\left| \left| T^{\ast } h_k -T^{\ast }f\right| \right|_{L^{2}_{(s,t)}\left( V,w_{1}\right) } +\left| \left| h_k-f\right| \right|_{L^{2}_{(s,t+1)}\left( V,w_{2}\right) } +\left| \left| S h_k -Sf\right| \right|_{L^{2}_{(s,t+2)}\left( V,w_{3}\right) } \right) =0. $$ \end{theorem} \begin{proof}
Note that for each strictly increasing multi-indices $I,J$ and $K$ with $\left| I\right|=s,\left| J\right| =t+1$ and $\left| K\right|=t+2$, $\hbox{\rm supp$\,$} f_{I,J}\stackrel{\circ}{\subset} V_{r}^{o}$ and by Proposition \ref{support argument}, $\hbox{\rm supp$\,$} (Sf)_{I,K}\stackrel{\circ}{\subset} V_{r}^{o}$. Since $\sup\limits_{\textbf{z}\in V_{r}^{o}}|w_2(\textbf{z})|<\infty$ and $\sup\limits_{\textbf{z}\in V_{r}^{o}}|w_3(\textbf{z})|<\infty$, we have $f_{I,J},(Sf)_{I,K}\in L^2(V,P)$. For $n\in\mathbb{N}$ and $\delta\in(0,+\infty)$, let \begin{eqnarray*}
f_{n}&\triangleq &\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\max \left\{ I\cup J\right\} \leqslant n} (f_{I,J})_{n}dz_{I}\wedge d\overline{z_{J}}, \\
f_{n,\delta }&\triangleq &\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\max \left\{ I\cup J\right\} \leqslant n} (f_{I,J})_{n,\delta }dz_{I}\wedge d\overline{z_{J}}, \\
\left( Sf\right)_{n,\delta } & \triangleq &\sum^{\prime }_{\left| I\right| =s,\left| K\right| =t+2,\max \left\{ I\cup K\right\} \leqslant n} (\left( Sf\right)_{I,K})_{n,\delta } dz_{I}\wedge d\overline{z_{K}}, \end{eqnarray*} where $(f_{I,J})_{n}$ is defined as that in (\ref{Integration reduce dimension}), and $(f_{I,J})_{n,\delta }$ and $(\left( Sf\right)_{I,K})_{n,\delta }$ are defined as that in (\ref{Convolution after reduce dimension}). Clearly, $f_{n}$ and $f_{n,\delta }$ (resp. $\left( Sf\right)_{n,\delta }$) may be viewed $(s,t+1)$-forms (resp. an $(s,t+2)$-form) on $V$ (and also on $\ell^2$), and \begin{equation}\label{230409e1} \begin{aligned}
&\left(f_{n}\right)_{I,J}=\begin{cases} \left(f_{I,J}\right)_{n},&\max \left\{ I\cup J\right\} \leqslant n,\\ 0,&\max \left\{ I\cup J\right\} > n, \end{cases}\qquad
\left(f_{n,\delta}\right)_{I,J}=\begin{cases} \left(f_{I,J}\right)_{n,\delta},&\max \left\{ I\cup J\right\} \leqslant n,\\ 0,&\max \left\{ I\cup J\right\} > n, \end{cases}\\[5mm]
&
\left(\left(Sf\right)_{n,\delta}\right)_{I,K}=\begin{cases} \left(\left(Sf\right)_{I,K}\right)_{n,\delta},&\max \left\{ I\cup K\right\} \leqslant n,\\ 0,&\max \left\{ I\cup K\right\} > n. \end{cases} \end{aligned} \end{equation}
Let $$ h \left( x\right)\triangleq \begin{cases} 1,&x\in (-\infty ,0],\\ \left( e^{\frac{1}{x-1} }-1\right) e^{-\frac{e^{\frac{1}{x-1} }}{x} }+1,&x\in \left( 0,1\right), \\ 0,&x\in [1,+\infty ). \end{cases} $$
For each $\rho>r$ and $x\in\mathbb{R}$, set $h_{\rho}(x)\triangleq h(x-\rho)$ and $\eta_{\rho }\triangleq h_{\rho } \left( \eta \right)$. Then $\eta_{\rho }\in C^{\infty }_{0,F}\left( V\right)$, $h,h_{\rho}\in C^{\infty }\left( \mathbb{R}\right)$, $0\leqslant h\leqslant 1$, $0\leqslant h_{\rho}\leqslant 1$, $\ h_{\rho } \left( x\right) =1$ for all $x<\rho ,\ h_{\rho } \left( x\right) =0$ for all $x>\rho +1$ and $\left| h_{\rho}^{\prime } \right| \leqslant C$ where $C\triangleq \sup\limits_{\mathbb{R}}| h^{\prime } |<\infty.$
Note that $(\eta_{\rho }\cdot f_{n,\delta })_{I,J}\in C^{\infty }_{0,F}\left( V\right)$ for each strictly increasing multi-indices $I$ and $J$ with $\left| I\right| =s,\left| J\right| =t+1$ and $\max\{I\cup J\}\leqslant n$, and $(\eta_{\rho }\cdot f_{n,\delta })_{I,J}=0$ for each strictly increasing multi-indices $I$ and $J$ with $\left| I\right| =s,\left| J\right| =t+1$ and $\max\{I\cup J\}>n$. Since $\hbox{\rm supp$\,$} \eta_{\rho}\subset V_{\rho+1}$, by the conclusion (2) of Proposition \ref{properties on pseudo-convex domain}, for each $\rho'>\rho+1$, it holds that $\hbox{\rm supp$\,$} \eta_{\rho } \stackrel{\circ}{\subset} V_{\rho^{\prime } }^{o}$. By Proposition \ref{general formula of T*} and the proof of Lemma \ref{densly defined closed}, we see that $\eta_{\rho }\cdot f_{n,\delta }\in D_{T^{\ast }}\cap D_S$. We will prove that \begin{equation}\label{230418e03} \begin{array}{ll}
\displaystyle \lim\limits_{n\rightarrow \infty } \lim\limits_{\delta \rightarrow 0^{+}}\left(\left| \left| T^{\ast }\left( \eta_{\rho }\cdot f_{n,\delta }\right) -T^{\ast }f\right| \right|_{L^{2}_{(s,t)}\left( V,w_{1}\right) } +\left| \left| \eta_{\rho }\cdot f_{n,\delta }-f\right| \right|_{L^{2}_{(s,t+1)}\left( V,w_{2}\right) }\right.\\[3mm]
\displaystyle \left.\qquad\qquad +\left| \left| S\left( \eta_{\rho }\cdot f_{n,\delta }\right) -Sf\right| \right|_{L^{2}_{(s,t+2)}\left( V,w_{3}\right) }\right)\\[3mm] \displaystyle = 0. \end{array} \end{equation} As a consequence of (\ref{230418e03}), the desired sequence $\{h_k\}_{k=1}^{\infty}$ in Theorem \ref{general density4} can be chosen from $\{\eta_{\rho }\cdot f_{n,\delta }:\;\rho>r,\,n\in\mathbb{N},\,\delta\in(0,+\infty)\}$. The proof of the equality (\ref{230418e03}) is quite long, and therefore we divide it into several steps.
\textbf{Step 1:} We shall prove that \begin{equation}\label{230406e1}
\lim\limits_{n\rightarrow \infty } \lim\limits_{\delta \rightarrow 0^{+}} \left| \left| \eta_{\rho }\cdot f_{n,\delta }-f\right| \right|_{L^{2}_{(s,t+1)}\left( V_{\rho^{\prime } },w_{2}\right) } =0. \end{equation} Since $\hbox{\rm supp$\,$} f \stackrel{\circ}{\subset} V^{o}_{r}$, $\rho>r$ and $\eta_{\rho }=1$ on $V^{o}_{r}$, we have $\eta_{\rho }\cdot f=f$. Note also that \begin{eqnarray*}
&& \left| \left| \eta_{\rho }\cdot f_{n,\delta }-f\right| \right|_{L^{2}_{(s,t+1)}\left( V_{\rho^{\prime } },w_{2}\right) } \\
&&\leqslant \left| \left| \eta_{\rho }\cdot f_{n,\delta }-\eta_{\rho }\cdot f_{n}\right| \right|_{L^{2}_{(s,t+1)}\left( V_{\rho^{\prime } },w_{2}\right) } +\left| \left| \eta_{\rho }\cdot f_{n}-\eta_{\rho } \cdot f\right| \right|_{L^{2}_{(s,t+1)}\left( V_{\rho^{\prime } },w_{2}\right) } +\left| \left| \eta_{\rho }\cdot f-f\right| \right|_{L^{2}_{(s,t+1)}\left( V_{r},w_{2}\right) }\\
&&\leqslant \left| \left| f_{n,\delta }- f_{n}\right| \right|_{L^{2}_{(s,t+1)}\left( V_{\rho^{\prime } },w_{2}\right) } +\left| \left| f_{n}- f\right| \right|_{L^{2}_{(s,t+1)}\left( V_{\rho^{\prime } },w_{2}\right) } \\
&&\leqslant \sqrt{\sup_{V_{\rho^{\prime } }} e^{-w_2}}\cdot\left| \left| f_{n,\delta }- f_{n}\right| \right|_{L^{2}_{(s,t+1)}\left( V_{\rho^{\prime } },P\right) } +\sqrt{\sup_{V_{\rho^{\prime } }} e^{-w_2}}\cdot\left| \left| f_{n}- f\right| \right|_{L^{2}_{(s,t+1)}\left( V_{\rho^{\prime } },P\right) }. \end{eqnarray*} Since $\sup\limits_{V_{\rho^{\prime } }} e^{-w_2}<\infty$ and by the proof of Lemma \ref{approximation for st form}, we obtain the desired equality (\ref{230406e1}).
\textbf{Step 2:} In this step, we shall show that \begin{equation}\label{230406e2}
\lim\limits_{n\rightarrow \infty } \lim\limits_{\delta \rightarrow 0^{+}} \left| \left| S(\eta_{\rho }\cdot f_{n,\delta })-Sf\right| \right|_{L^{2}_{(s,t+2)}\left( V_{\rho^{\prime } },w_{3}\right) } =0. \end{equation} Since $\eta_{\rho }\cdot (Sf)=Sf$ and \begin{eqnarray*}
&&\left| \left| S\left( \eta_{\rho }\cdot f_{n,\delta }\right) -Sf\right| \right|_{L^{2}_{(s,t+2)}\left( V_{\rho^{\prime } },w_{3}\right) }\\
&&\leqslant \left| \left| S\left( \eta_{\rho }\cdot f_{n,\delta }\right) -\eta_{\rho }\cdot \left( Sf\right)_{n,\delta } \right| \right|_{L^{2}_{(s,t+2)}\left( V_{\rho^{\prime } },w_{3}\right) } +\left| \left| \eta_{\rho }\cdot \left( Sf\right)_{n,\delta } -\eta_{\rho }\cdot (Sf)\right| \right|_{L^{2}_{(s,t+2)}\left( V_{\rho^{\prime } },w_{3}\right) }\\
&&\quad+\left| \left| \eta_{\rho }\cdot (Sf)-Sf\right| \right|_{L^{2}_{(s,t+2)}\left( V_{r},w_{3}\right) }\\
&&=\left| \left| S\left( \eta_{\rho }\cdot f_{n,\delta }\right) -\eta_{\rho }\cdot \left( Sf\right)_{n,\delta } \right| \right|_{L^{2}_{(s,t+2)}\left( V_{\rho^{\prime } },w_{3}\right) } +\left| \left| \eta_{\rho }\cdot \left( Sf\right)_{n,\delta } -\eta_{\rho }\cdot (Sf)\right| \right|_{L^{2}_{(s,t+2)}\left( V_{\rho^{\prime } },w_{3}\right) }, \end{eqnarray*} it suffices to prove that
\begin{equation}\label{230407e1}
\lim\limits_{n\rightarrow \infty } \lim\limits_{\delta \rightarrow 0^{{}_{+}}} \left| \left| \eta_{\rho }\cdot \left( Sf\right)_{n,\delta } -\eta_{\rho }\cdot (Sf)\right| \right|_{L^{2}_{(s,t+2)}\left( V_{\rho^{\prime } },w_{3}\right) } =0
\end{equation}
and
\begin{equation}\label{230407e2}
\lim\limits_{n\rightarrow \infty } \lim\limits_{\delta \rightarrow 0^{+}}\left| \left| S\left( \eta_{\rho }\cdot f_{n,\delta }\right) -\eta_{\rho } \cdot\left( Sf\right)_{n,\delta } \right| \right|_{L^{2}_{(s,t+2)}\left( V_{\rho^{\prime } },w_{3}\right) }=0. \end{equation}
By the conclusion (2) of Proposition \ref{support argument}, $\hbox{\rm supp$\,$} Sf \stackrel{\circ}{\subset} V^{o}_{r}$. Since $0\leqslant \eta_{\rho } \leqslant 1$ and $\sup\limits_{V_{\rho^{\prime } }}e^{-w_3}<\infty$, we have $$
\left| \left| \eta_{\rho } \cdot\left( Sf\right)_{n,\delta } -\eta_{\rho }\cdot (Sf)\right| \right|_{L^{2}_{(s,t+2)}\left( V_{\rho^{\prime } },w_{3}\right) } ^2
\leqslant \left(\sup_{V_{\rho^{\prime } }}e^{-w_3}\right)\cdot \left| \left|\left( Sf\right)_{n,\delta } - Sf\right| \right|_{L^{2}_{(s,t+2)}\left( V,P\right) } ^2. $$ By the conclusion (2) of Lemma \ref{approximation for st form}, we obtain (\ref{230407e1}).
To show (\ref{230407e2}), we claim that \begin{equation}\label{230407e9} S( \eta_{\rho }\cdot f_{n,\delta }) =\eta_{\rho }\cdot \left( Sf\right)_{n,\delta } +(S\eta_{\rho }) \wedge f_{n,\delta },
\end{equation} where $$
(S\eta_{\rho }) \wedge f_{n,\delta }\triangleq \left( -1\right)^{s} \sum^{\prime }_{\left| I\right| =s,\max\{I\}\leqslant n} \sum^{\prime }_{\left| K\right| =t+2} \left(\sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t+1,\max\{J\}\leqslant n} \varepsilon^{K}_{iJ} (f_{n,\delta })_{I,J}\overline{\partial_{i} } (\eta_{\rho }) \right)dz_{I}\wedge d\overline{z_{K}}. $$
Indeed, for each strictly increasing multi-indices $I$ and $K$ with $\left| I\right| =s$ and $\left| K\right| =t+2$, we show first that \begin{equation}\label{230407e5}
(-1)^s\sum^{\prime }_{\left| J\right| =t+1,\max\{J\}\leqslant n} \sum^{n}_{i=1} \varepsilon^{K}_{iJ} \overline{\partial_{i} } (f_{n,\delta })_{I,J}=(( Sf )_{n,\delta })_{I,K}. \end{equation} For this purpose, we consider several cases. The first case is $\max\{I\cup K\}\le n$. In this case, by (\ref{230409e1}), we have $$ \begin{array}{ll}
\displaystyle (-1)^s\sum^{\prime }_{\left| J\right| =t+1,\max\{J\}\leqslant n} \sum^{n}_{i=1} \varepsilon^{K}_{iJ} \overline{\partial_{i} } (f_{n,\delta })_{I,J}(\textbf{z}_n)=(-1)^s\sum^{\prime }_{\left| J\right| =t+1} \sum^{\infty}_{i=1} \varepsilon^{K}_{iJ} \overline{\partial_{i} } (f_{n,\delta })_{I,J}(\textbf{z}_n)\\[3mm]
\displaystyle =(-1)^s\sum^{\prime }_{\left| J\right| =t+1} \sum^{\infty}_{i=1} \varepsilon^{K}_{iJ} \int_{\mathbb{C}^{n} } (f_{I,J})_{n} ( \textbf{z}_n^{'}) \overline{\partial_{i} } \gamma_{n,\delta } (\textbf{z}_n-\textbf{z}_n^{'}) \,\mathrm{d}\textbf{z}_n^{'}\\[3mm]
\displaystyle =(-1)^s\sum^{\prime }_{\left| J\right| =t+1} \sum^{\infty}_{i=1} \varepsilon^{K}_{iJ} \int_{\ell^{2}} f_{I,J} ( \textbf{z}_n',{\textbf{z}^n}' ) \overline{\partial_{i} } \gamma_{n,\delta } (\textbf{z}_n- \textbf{z}_n' ) \varphi^{-1}_{n} (\textbf{z}_n') \,\mathrm{d}P( \textbf{z}_n',{\textbf{z}^n}' )\\[3mm]
\displaystyle =(-1)^{s+1}\sum^{\prime }_{\left| J\right| =t+1} \sum^{\infty}_{i=1} \varepsilon^{K}_{iJ} \int_{\ell^{2}} f_{I,J} \overline{\delta_{i} \left( \gamma_{n,\delta }(\textbf{z}_n-\cdot) \varphi^{-1}_{n} \right) } \,\mathrm{d}P\\[3mm] \displaystyle =\int_{\ell^{2}} \left( Sf\right)_{I,K} \gamma_{n,\delta }(\textbf{z}_n-\cdot) \varphi^{-1}_{n} \,\mathrm{d}P=(( Sf )_{n,\delta })_{I,K}(\textbf{z}_n), \end{array} $$ where $\textbf{z}_n\in\mathbb{C}^n$, $\varphi_{n} $ is given by (\ref{def for Gauss weight}) and the fifth equality follows from the fact that $ \gamma_{n,\delta }(\textbf{z}_n-\cdot) \varphi^{-1}_{n}\in C_b^1(V)$ and the conclusion (2) of Proposition \ref{weak equality for more test functions}. The second case is $\max\{I\}> n$, for which, by (\ref{230409e1}) again, each side of (\ref{230407e5}) is equal to $0$. The third case is $\max\{K\}> n$. In this case, by the definition of $\varepsilon^{K}_{iJ} $, the left side of (\ref{230407e5}) is equal to $0$; while, by (\ref{230409e1}) once more, so is its right side.
In view of Remark \ref{230407r1}, it follows that \begin{equation}\label{230410e1}
S( \eta_{\rho }\cdot f_{n,\delta }) =\left( -1\right)^{s} \sum\limits^{\prime }_{\left| I\right| =s} \sum\limits^{\prime }_{\left| K\right| =t+2} \sum^{\prime }_{\left| J\right| =t+1} \sum^{\infty}_{i=1} \varepsilon^{K}_{iJ}\cdot \overline{\partial_{i} } \left(\eta_{\rho }\cdot(f_{n,\delta })_{I,J}\right)\,\mathrm{d}z_{I}\wedge \,\mathrm{d}\overline{z_{K}}. \end{equation}
Also, by the conclusion (1) of Proposition \ref{general multiplitier}, for each strictly increasing multi-indices $I$ and $K$ with $\left| I\right| =s$ and $\left| K\right| =t+2$, noting (\ref{230407e5}), we have \begin{eqnarray*}
&&(-1)^s\sum^{\prime }_{\left| J\right| =t+1} \sum^{\infty}_{i=1} \varepsilon^{K}_{iJ}\cdot \overline{\partial_{i} } \left(\eta_{\rho }\cdot(f_{n,\delta })_{I,J}\right)=(-1)^s\sum^{\prime }_{\left| J\right| =t+1,\max\{J\}\leqslant n} \sum^{\infty}_{i=1} \varepsilon^{K}_{iJ}\cdot \overline{\partial_{i} } \left(\eta_{\rho }\cdot(f_{n,\delta })_{I,J}\right)\\
&&=\eta_{\rho }\cdot(-1)^s\sum^{\prime }_{\left| J\right| =t+1,\max\{J\}\leqslant n} \sum^{\infty}_{i=1} \varepsilon^{K}_{iJ} \cdot \overline{\partial_{i} } (f_{n,\delta })_{I,J}
+(-1)^s\sum^{\prime }_{\left| J\right| =t+1,\max\{J\}\leqslant n} \sum^{\infty}_{i=1} \varepsilon^{K}_{iJ}\cdot \overline{\partial_{i} } (\eta_{\rho })\cdot(f_{n,\delta })_{I,J}\\
&&=\eta_{\rho }\cdot(-1)^s\sum^{\prime }_{\left| J\right| =t+1,\max\{J\}\leqslant n} \sum^{n}_{i=1} \varepsilon^{K}_{iJ}\cdot \overline{\partial_{i} } (f_{n,\delta })_{I,J}
+(-1)^s\sum^{\prime }_{\left| J\right| =t+1,\max\{J\}\leqslant n} \sum^{\infty}_{i=1} \varepsilon^{K}_{iJ}\cdot \overline{\partial_{i} } (\eta_{\rho })\cdot(f_{n,\delta })_{I,J}\\ &&=\eta_{\rho }\cdot(( Sf )_{n,\delta })_{I,K}+\left((S\eta_{\rho }) \wedge f_{n,\delta }\right)_{I,K}, \end{eqnarray*} which, combined with (\ref{230410e1}), yields (\ref{230407e9}).
By (\ref{230407e9}), noting that $\varepsilon_{iJ}^K=\left|\varepsilon_{iJ}^K\right|\varepsilon_{iJ}^K$ for each $i\in\mathbb{N}$ and strictly increasing multi-indices $J$ and $K$ with $\left| J\right| =t+1$ and $\left| K\right| =t+2$, and recalling (\ref{gener111}) for $c_{I,iJ}$, we obtain that \begin{equation}\label{230407e6} \begin{aligned}
&\left| \left| S\left( \eta_{\rho } \cdot f_{n,\delta }\right) -\eta_{\rho }\cdot \left( Sf\right)_{n,\delta } \right| \right|^{2}_{L^{2}_{(s,t+2)}\left( V_{\rho^{\prime } },w_{3}\right) }\\
&=\left| \left|(S\eta_{\rho }) \wedge f_{n,\delta }\right| \right|^{2}_{L^{2}_{t+2}\left( V_{\rho^{\prime } },w_{3}\right) } \\
&= \sum^{\prime }_{\left| I\right| =s,\left| K\right| =t+2,\max \left\{ I \right\} \leqslant n} c_{I,K}\int_{V_{\rho^{\prime } }}\left|\sum^{\prime }_{\left| J\right| =t+1,\max\{J\}\leqslant n} \sum^{\infty}_{i=1} \varepsilon^{K}_{iJ}\cdot \overline{\partial_{i} } (\eta_{\rho })\cdot(f_{I,J})_{n,\delta }\right|^2e^{-w_{3}}\,\mathrm{d}P\\
&\leqslant \sum^{\prime }_{\left| I\right| =s,\left| K\right| =t+2,\max \left\{ I \right\} \leqslant n} c_{I,K}\int_{V_{\rho^{\prime } }}\left(\sum^{\prime }_{\left| J\right| =t+1,\max\{J\}\leqslant n} \sum^{\infty}_{i=1} \left|\varepsilon^{K}_{iJ}\right|^2\right)\\
&\qquad\qquad\qquad\qquad\qquad\quad\cdot\left(\sum^{\prime }_{\left| J\right| =t+1,\max\{J\}\leqslant n} \sum^{\infty}_{i=1} \left|\varepsilon^{K}_{iJ}\cdot \overline{\partial_{i} } (\eta_{\rho })\cdot(f_{I,J})_{n,\delta }\right|^2\right)e^{-w_{3}}\,\mathrm{d}P\\
&=(t+2)\sum^{\prime }_{\left| I\right| =s,\left| K\right| =t+2,\max \left\{ I \right\} \leqslant n} c_{I,K}\int_{V_{\rho^{\prime } }}\sum^{\prime }_{\left| J\right| =t+1,\max\{J\}\leqslant n} \sum^{\infty}_{i=1} \left|\varepsilon^{K}_{iJ}\cdot \overline{\partial_{i} } (\eta_{\rho })\cdot(f_{I,J})_{n,\delta }\right|^2e^{-w_{3}}\,\mathrm{d}P\\
&= (t+2)\int_{V_{\rho^{\prime } }} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\max \left\{ I\bigcup J\right\} \leqslant n} c_{I,iJ}\cdot\left| \overline{\partial_{i} } \eta_{\rho } \right|^{2} \cdot\left| (f_{I,J})_{n,\delta }\right|^{2} \cdot e^{-w_{3}}\,\mathrm{d}P\\
&\leqslant C_1\int_{V_{\rho^{\prime } }} \left| h^{\prime }_{\rho } \left( \eta \right) \right|^{2} \sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\max \left\{ I\bigcup J\right\} \leqslant n} c_{I,J}\left| (f_{I,J})_{n,\delta }\right|^{2}\,\mathrm{d}P, \end{aligned} \end{equation}
where $C_1\triangleq (t+2)\cdot \sup\limits_{V_{\rho^{\prime } }}\left(\sum\limits^{\infty }_{i=1} \left| \overline{\partial_{i} } \eta \right|^{2}\cdot e^{-w_3}\right)\cdot c_1^{s,t+1}<\infty$ (Recall Condition \ref{230423ass1} for $c_1^{s,t+1}$).
Since $f_{I,J}\in L^2(V,P)$ for each strictly increasing multi-indices $I$ and $J$ with $\left| I\right| =s$ and $\left| J\right| =t+1$, by the conclusion (2) of Proposition \ref{convolution properties}, for each $n\in\mathbb{N}$, it holds that $$
\lim_{\delta \rightarrow 0^{+}} \int_{V_{\rho^{\prime } }} \left| | (f_{I,J})_{n,\delta } |^{2} - | (f_{I,J})_{n}|^{2} \right|dP=0. $$ Therefore, $$
\lim_{\delta \rightarrow 0^{+}} \int_{V_{\rho^{\prime } }} \left|\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\max \left\{ I\bigcup J\right\} \leqslant n} c_{I,J} | (f_{I,J})_{n,\delta } |^{2} -\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\max \left\{ I\bigcup J\right\} \leqslant n} c_{I,J} | (f_{I,J})_{n}|^{2} \right|dP=0. $$ Note that \begin{eqnarray*}
\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J} \int_{V_{\rho^{\prime } }} | f_{I,J}|^{2}\,\mathrm{d}P
\leqslant \left(\sup_{V_{\rho^{\prime } }}e^{w_2}\right)\cdot\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J} \int_{V_{\rho^{\prime } }} | f_{I,J}|^{2}e^{-w_2}\,\mathrm{d}P<\infty. \end{eqnarray*} For any given $\varepsilon>0$, there exists $n_0\in\mathbb{N}$ such that \begin{eqnarray*}
\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\,\max\{I\cup J\}>n_0} c_{I,J} \int_{V_{\rho^{\prime } }} | f_{I,J}|^{2}\,\mathrm{d}P <\frac{\varepsilon}{4}. \end{eqnarray*} By the conclusion (3) of Proposition \ref{Reduce diemension}, there exits an integer $n_1>n_0$ such that for all $n\geqslant n_1$, it holds that \begin{eqnarray*}
\int_{V_{\rho^{\prime } }} \sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\,\max\{I\cup J\}\leqslant n_0} c_{I,J}\left| | (f_{I,J})_n|^{2}- | f_{I,J}|^{2}\right|\,\mathrm{d}P <\frac{\varepsilon}{2}. \end{eqnarray*} Then for all $n\geqslant n_1$, noting that $\hbox{\rm supp$\,$} f\subset V_{\rho'}$, we have \begin{eqnarray*}
&&\int_{V_{\rho^{\prime } }} \left|\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\max \left\{ I\bigcup J\right\} \leqslant n} c_{I,J}\left| (f_{I,J})_{n}\right|^{2} -\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\left| f_{I,J}\right|^{2} \right|\,\mathrm{d}P\\
&&\leqslant \sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\,\max\{I\cup J\}\leqslant n_0} c_{I,J}\int_{V_{\rho^{\prime } }} \left|\left| (f_{I,J})_{n}\right|^{2} -\left| f_{I,J}\right|^{2} \right|\,\mathrm{d}P\\
&&\quad+\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\,n_0<\max\{I\cup J\}\leqslant n } c_{I,J}\int_{V_{\rho^{\prime } }} \left|\left| (f_{I,J})_{n}\right|^{2} -\left| f_{I,J}\right|^{2} \right|\,\mathrm{d}P\\
&&\quad+\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\max \left\{ I\bigcup J\right\} >n} c_{I,J}\int_{V_{\rho^{\prime } }} \left| f_{I,J}\right|^{2} \,\mathrm{d}P\\
&&<\frac{\varepsilon}{2}+2\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\max \left\{ I\bigcup J\right\} >n_0} c_{I,J} \int_{V_{\rho^{\prime } }} | f_{I,J}|^{2}\,\mathrm{d}P<\varepsilon, \end{eqnarray*} where the second inequality follows from the conclusion (1) of Proposition \ref{Reduce diemension}. Thus $$
\lim_{n\rightarrow \infty } \int_{V_{\rho^{\prime } }} \left|\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\max \left\{ I\bigcup J\right\} \leqslant n} c_{I,J}\left| (f_{I,J})_{n}\right|^{2} -\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\left| f_{I,J}\right|^{2} \right|\,\mathrm{d}P=0, $$ and hence $$
\lim_{n\rightarrow \infty } \lim_{\delta \rightarrow 0^{+}} \int_{V_{\rho^{\prime } }} \left|\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\max \left\{ I\bigcup J\right\} \leqslant n} c_{I,J}\left| (f_{I,J})_{n,\delta }\right|^{2} -\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\left| f_{I,J}\right|^{2} \right|\,\mathrm{d}P=0. $$
Since $\sup_{x\in\mathbb{R}}\left| h^{\prime }_{\rho }(x) \right| =C<\infty$, we have \begin{equation}\label{230407e7}
\lim_{n\rightarrow \infty } \lim_{\delta \rightarrow 0^{+}} \int_{V_{\rho^{\prime } }} \left| h^{\prime }_{\rho } \left( \eta \right) \right|^{2} \cdot\left|\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\max \left\{ I\bigcup J\right\} \leqslant n} c_{I,J}\left| (f_{I,J})_{n,\delta }\right|^{2} -\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\left| f_{I,J}\right|^{2} \right|\,\mathrm{d}P=0. \end{equation}
Combining (\ref{230407e6}) and (\ref{230407e7}), we arrive at \begin{equation}\label{end of STEP 2} \begin{array}{ll}
\displaystyle\lim_{n\rightarrow \infty } \lim_{\delta \rightarrow 0^{+}}\left| \left| S\left( \eta_{\rho }\cdot f_{n,\delta }\right) -\eta_{\rho } \cdot\left( Sf\right)_{n,\delta } \right| \right|^{2}_{L^{2}_{(s,t+2)}\left( V_{\rho^{\prime } },w_{3}\right) }\\[5mm]\displaystyle
\leqslant C_1\lim_{n\rightarrow \infty } \lim_{\delta \rightarrow 0^{+}}\int_{V_{\rho^{\prime } }} \left| h^{\prime }_{\rho } \left( \eta \right) \right|^{2} \sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\max \left\{ I\bigcup J\right\} \leqslant n} c_{I,J}\left| (f_{I,J})_{n,\delta }\right|^{2}\,\mathrm{d}P\\[5mm]\displaystyle
= C_1\int_{V_{\rho^{\prime } }} \left| h^{\prime }_{\rho } \left( \eta \right) \right|^{2} \sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\left| f_{I,J} \right|^{2}\,\mathrm{d}P
= C_1\int_{V_{r}} \left| h^{\prime }_{\rho } \left( \eta \right) \right|^{2} \sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\left| f_{I,J} \right|^{2}\,\mathrm{d}P=0, \end{array} \end{equation} which gives (\ref{230407e2}).
\textbf{Step 3:} In this step, we shall prove that $\lim\limits_{n\rightarrow \infty } \lim\limits_{\delta \rightarrow 0^{+}} \left| \left| \mathbb{P}\left( \eta_{\rho } \cdot f_{n,\delta }\right) -\mathbb{P}f\right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) } =0$.\\ Since $\eta_{\rho } \cdot(\mathbb{P}f)=\mathbb{P}f$ and \begin{eqnarray*}
&&\left| \left| \mathbb{P}\left( \eta_{\rho }\cdot f_{n,\delta }\right) -\mathbb{P}f\right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) }\\
&&\leqslant \left| \left|\mathbb{P}\left( \eta_{\rho }\cdot f_{n,\delta }\right) -\eta_{\rho } \cdot\left(\mathbb{P}f\right)_{n,\delta } \right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) }+\left| \left| \eta_{\rho }\cdot \left(\mathbb{P}f\right)_{n,\delta } -\eta_{\rho }\cdot (\mathbb{P}f)\right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) }\\
&& \quad +\left| \left| \eta_{\rho }\cdot (\mathbb{P}f)-\mathbb{P}f\right| \right|_{L^{2}_{(s,t)}\left( V_{r},w_{1}\right) }\\
&&=\left| \left|\mathbb{P}\left( \eta_{\rho }\cdot f_{n,\delta }\right) -\eta_{\rho } \cdot\left(\mathbb{P}f\right)_{n,\delta } \right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) }+\left| \left| \eta_{\rho }\cdot \left(\mathbb{P}f\right)_{n,\delta } -\eta_{\rho }\cdot (\mathbb{P}f)\right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) }, \end{eqnarray*} it suffices to prove that
\begin{equation}\label{230407e4}\lim\limits_{n\to\infty}\lim\limits_{\delta\to 0+}\left| \left|\mathbb{P}\left( \eta_{\rho }\cdot f_{n,\delta }\right) -\eta_{\rho } \cdot\left( \mathbb{P}f\right)_{n,\delta } \right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) }=0. \end{equation} and
\begin{equation}\label{230407e3}\lim\limits_{n\to\infty}\lim\limits_{\delta\to 0+}\left| \left| \eta_{\rho }\cdot \left(\mathbb{P}f\right)_{n,\delta } -\eta_{\rho }\cdot (\mathbb{P}f)\right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) }=0 \end{equation}
The proof of (\ref{230407e3}) is similarly to that of (\ref{230407e1}), and therefore we omit the details. In the rest of this step, we shall prove (\ref{230407e4}).
By Proposition \ref{general formula of T*} and Lemma \ref{general I,P}, we have $\eta_{\rho }\cdot f_{n,\delta }\in D_{T^*},\mathbb{P}f,\mathbb{P}\left( \eta_{\rho } \cdot f_{n,\delta }\right) ,\mathbb{I}f,\mathbb{I}\left( \eta_{\rho }\cdot f_{n,\delta }\right) \in L^{2}_{\left( s,t\right) }\left( V,w_{1}\right) $ (Recall (\ref{general def of I and P}) for the definition of $\mathbb{I}$ and $\mathbb{P}$). For every $\phi \in C^{\infty }_{0,F}\left(V_{\rho^{\prime} }^{o}\right) $ and sufficiently small $\delta>0$, each strictly increasing multi-indices $I$ and $L$ with $\left| I\right| =s,\left| L\right| =t ,\max\{I\cup L\}\leqslant n$, noting that $
\left(f_{n,\delta }\right)_{I,iL}=\sum_{|J|=t+1}'\varepsilon_{iL}^J \left(f_{n,\delta }\right)_{I,J}=\sum_{|J|=t+1,\max\{J\}\leqslant n}'\varepsilon_{iL}^J \left(f_{n,\delta }\right)_{I,J}$ for each $n,i\in\mathbb{N}$, we have $$
\begin{array}{ll} \displaystyle \int_{V} \left(\mathbb{P} ( \eta_{\rho }\cdot f_{n,\delta } ) \right)_{I,L}\cdot\phi \,\mathrm{d}P =\int_{V} \left(\mathbb{ I}(\eta_{\rho}\cdot f_{n,\delta }) \right)_{I,L}\cdot \phi \,\mathrm{d}P-\left( -1\right)^{s} \int_{V} \left( T^{\ast } ( \eta_{\rho}\cdot f_{n,\delta } ) \right)_{I,L}\cdot e^{w_{2}}\cdot\phi\cdot e^{-w_{1}}\,\mathrm{d}P\\[3mm] \displaystyle =\int_{V} \sum^{n}_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot\eta_{\rho }\cdot (f_{n,\delta })_{I,iL}\cdot(\partial_{i}w_{2})\cdot\phi \,\mathrm{d}P-\sum^{n}_{i=1} \int_{V}\frac{c_{I,iL}}{c_{I,L}}\cdot \eta_{\rho }\cdot (f_{n,\delta })_{I,iL}\cdot\partial_{i}(e^{w_{2}}\cdot\phi )\cdot e^{-w_{2}}\,\mathrm{d}P\\[3mm] \displaystyle =-\sum^{n}_{i=1} \int_{V} \frac{c_{I,iL}}{c_{I,L}}\cdot\eta_{\rho }\cdot (f_{n,\delta })_{I,iL}\cdot(\partial_{i}\phi )\,\mathrm{d}P\\[3mm] \displaystyle =\sum^{n}_{i=1} \int_{V}\frac{c_{I,iL}}{c_{I,L}}\cdot \eta_{\rho } \cdot \delta_{i} (f_{n,\delta })_{I,iL}\cdot\phi \,\mathrm{d}P+\sum^{n}_{i=1} \int_{V}\frac{c_{I,iL}}{c_{I,L}}\cdot \partial_{i}(\eta_{\rho }) \cdot (f_{n,\delta })_{I,iL}\cdot\phi\,\mathrm{d}P,
\end{array} $$ where the second equality follows from Lemma \ref{T*formula} and the final one follows from Corollary \ref{integration by Parts or deltai}. By (\ref{general def of I and P}) and the conclusion (3) of Proposition \ref{convolution properties}, we have $$ \begin{aligned} &\int_{V} \left( (\mathbb{P}f)_{I,L}\right)_{n,\delta }\cdot \phi \,\mathrm{d}P =\int_{V} \left( (\mathbb{I}f)_{I,L}\right)_{n,\delta }\cdot \phi \,\mathrm{d}P-\left( -1\right)^{s} \int_{V} \left( e^{w_{2}-w_{1}}\cdot (T^{\ast }f)_{I,L}\right)_{n,\delta }\cdot\phi \,\mathrm{d}P\\ &=\int_{V} \left( \mathbb{I}f\right)_{I,L}\cdot \varphi^{-1}_{n} \cdot\left( \phi_{n} \cdot\varphi_{n} \right)_{n,\delta }\,\mathrm{d}P-\left( -1\right)^{s} \int_{V} \left( T^{\ast }f\right)_{I,L}\cdot \varphi^{-1}_{n} \cdot e^{w_{2}-w_{1}}\cdot \left( \phi_{n} \cdot\varphi_{n} \right)_{n,\delta } \,\mathrm{d}P\\ &=\int_{V} \left( \sum^{\infty }_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot f_{I,iL}\cdot\partial_{i}w_{2}\right) \cdot \varphi^{-1}_{n} \cdot\left( \phi_{n}\cdot \varphi_{n} \right)_{n,\delta } \,\mathrm{d}P\\ &\quad-\left( -1\right)^{s} \int_{V} \left( T^{\ast }f\right)_{I,L}\cdot \varphi^{-1}_{n}\cdot e^{w_{2}}\cdot\left( \phi_{n}\cdot \varphi_{n} \right)_{n,\delta } \cdot e^{-w_{1}}\,\mathrm{d}P. \end{aligned} $$ Hence, noting that $\hbox{\rm supp$\,$} f\stackrel{\circ}{\subset} V_{r}^{o}$ and the function $\varphi^{-1}_{n}\cdot e^{w_{2}}\cdot \eta_\rho\cdot\left( \phi_{n} \cdot\varphi_{n} \right)_{n,\delta }$ satisfies the assumption \eqref{phi begin to D condition'}, by Corollary \ref{weak T*formula}, $ \left( T^{\ast }f\right)_{I,L}= \left( T^{\ast }f\right)_{I,L}\cdot \eta_\rho$, and $\partial_{i} \phi_{n} =\left(\partial_{i} \phi\right)_{n} $, $ f_{I,iL}\cdot \eta_\rho= f_{I,iL}$ and $ f_{I,iL}\cdot (\partial_{i} \eta_\rho)= 0$ for $i=1,2,\cdots,n$, we obtain that $$ \begin{aligned} &\int_{V} \left( (\mathbb{P}f)_{I,L}\right)_{n,\delta } \cdot \phi \,\mathrm{d}P\\ &=\int_{V} \left( \sum^{\infty }_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot f_{I,iL}\cdot\partial_{i}w_{2}\right)\cdot \varphi^{-1}_{n}\cdot \left( \phi_{n} \cdot \varphi_{n} \right)_{n,\delta }\,\mathrm{d}P\\ &\quad-\sum^{\infty }_{i=1} \int_{V} \frac{c_{I,iL}}{c_{I,L}}\cdot f_{I,iL}\cdot \partial_{i}\left(\varphi^{-1}_{n}\cdot e^{w_{2}}\cdot \eta_\rho\cdot \left( \phi_{n}\cdot \varphi_{n} \right)_{n,\delta } \right)\cdot e^{-w_{2}}\,\mathrm{d}P\\ &=\int_{V} \left( \sum^{\infty }_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot f_{I,iL}\cdot \partial_{i}w_{2}\right)\cdot \varphi^{-1}_{n}\cdot \left( \phi_{n} \cdot \varphi_{n} \right)_{n,\delta }\,\mathrm{d}P\\ &\quad-\sum^{\infty }_{i=1} \int_{V}\frac{c_{I,iL}}{c_{I,L}}\cdot f_{I,iL}\cdot \partial_{i}\left(\varphi^{-1}_{n}\cdot e^{w_{2}}\cdot \eta_\rho\right)\cdot \left( \phi_{n}\cdot \varphi_{n} \right)_{n,\delta } \cdot e^{-w_{2}}\,\mathrm{d}P\\ &\quad-\sum^{\infty }_{i=1} \int_{V}\frac{c_{I,iL}}{c_{I,L}}\cdot f_{I,iL}\cdot \varphi^{-1}_{n}\cdot \partial_{i}\left(\left( \phi_{n} \cdot \varphi_{n} \right)_{n,\delta } \right)\,\mathrm{d}P\\ &=\int_{V} \left( \sum^{\infty }_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot f_{I,iL}\cdot \partial_{i}w_{2}\right) \cdot \varphi^{-1}_{n} \cdot \left( \phi_{n} \cdot \varphi_{n} \right)_{n,\delta } \,\mathrm{d}P\\ &\quad-\sum^{\infty }_{i=1} \int_{V} \frac{c_{I,iL}}{c_{I,L}}\cdot f_{I,iL}\cdot \left(\partial_{i}\varphi^{-1}_{n} +\left(\partial_{i}w_{2}+\partial_{i}\eta_\rho\right)\cdot \varphi^{-1}_{n} \right)\cdot \left( \phi_{n} \cdot \varphi_{n} \right)_{n,\delta } \,\mathrm{d}P\\ &\quad-\sum^{n}_{i=1} \int_{V}\frac{c_{I,iL}}{c_{I,L}}\cdot f_{I,iL}\cdot \varphi^{-1}_{n}\cdot \left((\partial_{i} \phi_{n} )\cdot \varphi_{n} \right)_{n,\delta } \,\mathrm{d}P-\sum^{n}_{i=1} \int_{V} \frac{c_{I,iL}}{c_{I,L}}\cdot f_{I,iL}\cdot \varphi^{-1}_{n}\cdot \left( \phi_{n} \cdot \partial_{i}\varphi_{n} \right)_{n,\delta }\,\mathrm{d}P\\ &=\int_{V} \left( \sum^{n}_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot f_{I,iL}\cdot \partial_{i} w_{2}\right) \cdot \varphi^{-1}_{n} \left( \phi_{n} \cdot \varphi_{n} \right)_{n,\delta } \,\mathrm{d}P\\ &\quad -\sum^{n}_{i=1} \int_{V} \frac{c_{I,iL}}{c_{I,L}}\cdot f_{I,iL}\cdot \left(\frac{\overline{z_{i}} }{2a^{2}_{i}} \cdot \varphi^{-1}_{n} +\left(\partial_{i} w_{2} \right)\cdot \varphi^{-1}_{n}\right)\cdot \left( \phi_{n} \cdot \varphi_{n} \right)_{n,\delta } \,\mathrm{d}P\\ &\quad-\sum^{n}_{i=1} \int_{V} \frac{c_{I,iL}}{c_{I,L}}\cdot f_{I,iL}\cdot \varphi^{-1}_{n}\cdot \left( (\partial_{i} \phi_{n} ) \cdot \varphi_{n} \right)_{n,\delta } \,\mathrm{d}P+\sum^{n}_{i=1} \int_{V}\frac{c_{I,iL}}{c_{I,L}}\cdot f_{I,iL}\cdot \varphi^{-1}_{n} \cdot \left( \phi_{n} \cdot \frac{\overline{z_{i}} }{2a^{2}_{i}}\cdot \varphi_{n} \right)_{n,\delta } \,\mathrm{d}P\\ &=-\sum^{n}_{i=1} \int_{V} \frac{c_{I,iL}}{c_{I,L}}\cdot f_{I,iL}\cdot \frac{\overline{z_{i}} }{2a^{2}_{i}} \cdot \varphi^{-1}_{n}\cdot \left( \phi_{n} \cdot \varphi_{n} \right)_{n,\delta } \,\mathrm{d}P\\ &\quad-\sum^{n}_{i=1} \int_{V}\frac{c_{I,iL}}{c_{I,L}}\cdot f_{I,iL}\cdot \varphi^{-1}_{n} \cdot \left( (\partial_{i} \phi)_{n} \cdot \varphi_{n} \right)_{n,\delta }\,\mathrm{d}P+\sum^{n}_{i=1} \int_{V}\frac{c_{I,iL}}{c_{I,L}}\cdot f_{I,iL}\cdot \varphi^{-1}_{n} \cdot \left( \phi_{n} \cdot \frac{\overline{z_{i}} }{2a^{2}_{i}} \cdot \varphi_{n} \right)_{n,\delta } \,\mathrm{d}P. \end{aligned} $$ Hence, by the conclusion (3) of Proposition \ref{convolution properties} again and noting that $\partial_{i}\phi=\eta_{\rho'+1}\cdot \left(\partial_{i}\phi\right)$ for $i=1,2,\cdots,n$ , we have \begin{eqnarray*} &&\int_{V} \left( (\mathbb{P}f)_{I,L}\right)_{n,\delta }\cdot \phi \,\mathrm{d}P\\ &&=-\sum^{n}_{i=1} \int_{V}\frac{c_{I,iL}}{c_{I,L}}\cdot \left(f_{I,iL}\cdot\frac{\overline{z_{i}} }{2a^{2}_{i}} \right)_{n,\delta }\cdot\phi \,\mathrm{d}P-\sum^{n}_{i=1} \int_{V}\frac{c_{I,iL}}{c_{I,L}}\cdot (f_{I,iL})_{n,\delta }\cdot \eta_{\rho'+1}\cdot\partial_{i} \phi \,\mathrm{d}P\\ &&\quad+\sum^{n}_{i=1} \int_{V}\frac{c_{I,iL}}{c_{I,L}}\cdot (f_{I,iL})_{n,\delta }\cdot\frac{\overline{z_{i}} }{2a^{2}_{i}}\cdot \phi \,\mathrm{d}P\\ &&=-\sum^{n}_{i=1} \int_{V}\frac{c_{I,iL}}{c_{I,L}}\cdot \left(f_{I,iL}\cdot\frac{\overline{z_{i}} }{2a^{2}_{i}}\right)_{n,\delta }\cdot\phi \,\mathrm{d}P+\sum^{n}_{i=1} \int_{V}\frac{c_{I,iL}}{c_{I,L}}\cdot \delta_{i} \left((f_{I,iL})_{n,\delta }\cdot\eta_{\rho'+1}\right) \cdot\phi \,\mathrm{d}P\\ &&\quad+\sum^{n}_{i=1} \int_{V}\frac{c_{I,iL}}{c_{I,L}}\cdot (f_{I,iL})_{n,\delta }\cdot\frac{\overline{z_{i}} }{2a^{2}_{i}} \cdot\phi \,\mathrm{d}P,\\ &&=-\sum^{n}_{i=1} \int_{V}\frac{c_{I,iL}}{c_{I,L}}\cdot \left(f_{I,iL}\cdot\frac{\overline{z_{i}} }{2a^{2}_{i}} \right)_{n,\delta }\cdot\phi \,\mathrm{d}P+\sum^{n}_{i=1} \int_{V}\frac{c_{I,iL}}{c_{I,L}}\cdot \delta_{i} (f_{I,iL})_{n,\delta }\cdot\phi \,\mathrm{d}P\\ &&\quad+\sum^{n}_{i=1} \int_{V}\frac{c_{I,iL}}{c_{I,L}}\cdot (f_{I,iL})_{n,\delta }\cdot\frac{\overline{z_{i}} }{2a^{2}_{i}} \cdot\phi \,\mathrm{d}P, \end{eqnarray*} where the second equality follows from Corollary \ref{integration by Parts or deltai}. By Corollary \ref{density2}, we have \begin{equation}\label{Pfndelta} \begin{aligned}
&\left( \mathbb{P}f\right)_{n,\delta } =\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} \sum^{n}_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot\left( -\left(f_{I,iL}\cdot\frac{ \overline{z_{i}}}{2a^{2}_{i}} \right)_{n,\delta }+ \delta_{i} (f_{I,iL})_{n,\delta }+ (f_{I,iL})_{n,\delta }\cdot\frac{ \overline{z_{i}} }{2a^{2}_{i}} \right) dz_{I}\wedge d\overline{z_{L}}. \end{aligned} \end{equation}
On the other hand, by Proposition \ref{general formula of T*} and the definition of $\mathbb{P}$ in (\ref{general def of I and P}), it follows that \begin{eqnarray}\label{Petarhofndelta}
\mathbb{P}\left( \eta_{\rho }\cdot f_{n,\delta }\right) =\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \{ I\bigcup L\} \leqslant n} \left(\sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot\left( \eta_{\rho } \cdot\delta_{i} (f_{I,iL})_{n,\delta }+(\partial_{i} \eta_{\rho }) \cdot (f_{I,iL})_{n,\delta }\right)\right)dz_{I}\wedge d\overline{z_{L}} . \end{eqnarray}
By \eqref{Pfndelta} and \eqref{Petarhofndelta}, we have $$ \begin{aligned}
&\left| \left| \mathbb{P}\left( \eta_{\rho }\cdot f_{n,\delta }\right) -\eta_{\rho }\cdot (\mathbb{P}f)_{n,\delta } \right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) } \\
&\leqslant\left|\left|\eta_{\rho }\cdot \sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} \sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot \left( - \left(f_{I,iL}\cdot \frac{\overline{z_{i}} }{2a^{2}_{i}}\right)_{n,\delta }+ (f_{I,iL})_{n,\delta }\cdot \frac{ \overline{z_{i}}}{2a^{2}_{i}} \right) dz_{I}\wedge d\overline{z_{L}} \right|\right|_{L^{2}_{\left( s,t\right) }\left( V_{\rho^{\prime } },w_{1}\right) }\\
&\quad +\left| \left| \sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \{ I\bigcup L\} \leqslant n} \left(\sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot(\partial_{i} \eta_{\rho }) \cdot (f_{I,iL})_{n,\delta }\right)dz_{I}\wedge d\overline{z_{L}} \right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) }\\
&\leqslant\sqrt{\sup_{V_{\rho^{\prime } }}e^{-w_1}}\cdot\left[\left|\left|\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} \sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot\left( (f_{I,iL})_{n,\delta }\cdot \frac{ \overline{z_{i}}}{2a^{2}_{i}}\right.-\left(f_{I,iL}\cdot \frac{\overline{z_{i}} }{2a^{2}_{i}}\right)_{n,\delta }\right) dz_{I}\wedge d\overline{z_{L}} \right|\right|_{L^{2}_{\left( s,t\right) }\left( V_{\rho^{\prime } },P\right) }\\
&\quad +\left.\left| \left| \sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \{ I\bigcup L\} \leqslant n} \left(\sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot(\partial_{i} \eta_{\rho }) \cdot (f_{I,iL})_{n,\delta }\right)dz_{I}\wedge d\overline{z_{L}} \right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },P\right) }\right]. \end{aligned} $$ Note that \begin{eqnarray*}
&&\left|\left|\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} \sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot\left( (f_{I,iL})_{n,\delta }\cdot \frac{ \overline{z_{i}}}{2a^{2}_{i}} -\left(f_{I,iL}\cdot \frac{\overline{z_{i}} }{2a^{2}_{i}}\right)_{n,\delta }\right) dz_{I}\wedge d\overline{z_{L}} \right|\right|^2_{L^{2}_{\left( s,t\right) }\left( V_{\rho^{\prime } },P\right) }\\
&&=\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} c_{I,L}\cdot\int_{V_{\rho^{\prime } }} \left|\sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot\left( (f_{I,iL})_{n,\delta }\cdot \frac{ \overline{z_{i}}}{2a^{2}_{i}} -\left(f_{I,iL}\cdot \frac{\overline{z_{i}} }{2a^{2}_{i}}\right)_{n,\delta }\right)\right|^2\,\mathrm{d}P\\
&&\leqslant n\cdot\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} c_{I,L}\cdot\sum^{n}_{i=1}\left(\frac{c_{I,iL}}{c_{I,L}}\right)^2\cdot\int_{V_{\rho^{\prime } }}\left| (f_{I,iL})_{n,\delta }\cdot \frac{ \overline{z_{i}}}{2a^{2}_{i}} -\left(f_{I,iL}\cdot \frac{\overline{z_{i}} }{2a^{2}_{i}}\right)_{n,\delta }\right|^2\,\mathrm{d}P, \end{eqnarray*} $(f_{I,iL})_{n}\cdot \overline{z_{i}} =(f_{I,iL}\cdot \overline{z_{i}} )_{n}$ for all $1\leqslant i\leqslant n$ and \begin{eqnarray*}
&&\left(\int_{V_{\rho^{\prime } }}\left| (f_{I,iL})_{n,\delta }\cdot \frac{ \overline{z_{i}}}{2a^{2}_{i}} -\left(f_{I,iL}\cdot \frac{\overline{z_{i}} }{2a^{2}_{i}}\right)_{n,\delta }\right|^2\,\mathrm{d}P\right)^{\frac{1}{2}}\\
&&\leqslant\left(\int_{V_{\rho^{\prime } }}\left| (f_{I,iL})_{n,\delta }\cdot \frac{ \overline{z_{i}}}{2a^{2}_{i}} -(f_{I,iL})_{n}\cdot \frac{\overline{z_{i}}}{2a_i^2}\right|^2\,\mathrm{d}P\right)^{\frac{1}{2}}+\left(\int_{V_{\rho^{\prime } }}\left| \left(f_{I,iL}\cdot \frac{\overline{z_{i}}}{2a_i^2} \right)_{n}-\left(f_{I,iL}\cdot \frac{\overline{z_{i}} }{2a^{2}_{i}}\right)_{n,\delta }\right|^2\,\mathrm{d}P\right)^{\frac{1}{2}}\\
&&\leqslant\sup_{V_{\rho^{\prime } }}\left|\frac{ \overline{z_{i}}}{2a^{2}_{i}}\right|\cdot\left(\int_{V_{\rho^{\prime } }}\left| (f_{I,iL})_{n,\delta } -(f_{I,iL})_{n}\right|^2\,\mathrm{d}P\right)^{\frac{1}{2}}+\left(\int_{V_{\rho^{\prime } }}\left| \left(f_{I,iL}\cdot \frac{\overline{z_{i}}}{2a_i^2} \right)_{n}-\left(f_{I,iL}\cdot \frac{\overline{z_{i}} }{2a^{2}_{i}}\right)_{n,\delta }\right|^2\,\mathrm{d}P\right)^{\frac{1}{2}}, \end{eqnarray*} which tends zero as $\delta\to 0+$ (by the conclusion (1) of Proposition \ref{convolution properties}). Thus, for all $n\in\mathbb{N}$, \begin{eqnarray*}
\lim_{\delta\to 0+}\left|\left|\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} \sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot\left( (f_{I,iL})_{n,\delta }\cdot \frac{ \overline{z_{i}}}{2a^{2}_{i}} -\left(f_{I,iL}\cdot \frac{\overline{z_{i}} }{2a^{2}_{i}}\right)_{n,\delta }\right) dz_{I}\wedge d\overline{z_{L}} \right|\right|^2_{L^{2}_{\left( s,t\right) }\left( V_{\rho^{\prime } },P\right) }=0. \end{eqnarray*} We also note that \begin{eqnarray*}
&& \lim_{\delta\to 0+}\left| \left| \sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \{ I\bigcup L\} \leqslant n} \left(\sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot(\partial_{i} \eta_{\rho }) \cdot (f_{I,iL})_{n,\delta }\right)dz_{I}\wedge d\overline{z_{L}} \right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },P\right) }^2\\
&&= \lim_{\delta\to 0+}\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \{ I\bigcup L\} \leqslant n} c_{I,L}\cdot\int_{V_{\rho^{\prime } }}\left|\sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot(\partial_{i} \eta_{\rho }) \cdot (f_{I,iL})_{n,\delta }\right|^2\,\mathrm{d}P\\
&&=\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \{ I\bigcup L\} \leqslant n} c_{I,L}\cdot\int_{V_{\rho^{\prime } }}\left|\sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot(\partial_{i} \eta_{\rho }) \cdot (f_{I,iL})_{n }\right|^2\,\mathrm{d}P\\
&&\leqslant\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \{ I\bigcup L\} \leqslant n} c_{I,L}\cdot\int_{V_{\rho^{\prime } }}\left(\sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot|\partial_{i} \eta_{\rho }|^2\right) \cdot \left(\sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot(f_{I,iL})_{n }|^2\right)\,\mathrm{d}P\\
&&\leqslant C_2\cdot\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \{ I\bigcup L\} \leqslant n} c_{I,L}\cdot\int_{V_{\rho^{\prime } }}|h_{\rho }'(\eta)|^2 \cdot \left(\sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot|(f_{I,iL})_{n }|^2\right)\,\mathrm{d}P\\
&&=C_2\cdot(t+1)\cdot\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\max \{ I\bigcup J\} \leqslant n} c_{I,J}\cdot\int_{V_{\rho^{\prime } }}|h_{\rho }'(\eta)|^2 \cdot |(f_{I,J})_{n }|^2\,\mathrm{d}P, \end{eqnarray*}
where $C_2\triangleq c_1^{s,t}\cdot\sup\limits_{V_{\rho^{\prime } }}\left(\sum\limits^{\infty }_{i=1} \left| \overline{\partial_{i} } \eta \right|^{2}\right)<\infty$. Similarly to \eqref{end of STEP 2}, we have \begin{eqnarray*}
&&\lim_{n\to\infty}\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\max \{ I\bigcup L\} \leqslant n} c_{I,J}\cdot\int_{V_{\rho^{\prime } }}|h_{\rho }'(\eta)|^2 \cdot |(f_{I,J})_{n }|^2\,\mathrm{d}P\\
&&=\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\cdot\int_{V_{\rho^{\prime } }}|h_{\rho }'(\eta)|^2 \cdot |f_{I,J}|^2\,\mathrm{d}P=0, \end{eqnarray*} and hence $$
\lim_{n\to\infty}\lim_{\delta\to 0+}\left| \left| \sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \{ I\bigcup L\} \leqslant n} \left(\sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot(\partial_{i} \eta_{\rho }) \cdot (f_{I,iL})_{n,\delta }\right)dz_{I}\wedge d\overline{z_{L}} \right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },P\right) }=0. $$ Therefore, we obtain the desired equality (\ref{230407e3}).
\textbf{Step 4:} In this step, we prove that $\lim\limits_{n\rightarrow \infty } \lim\limits_{\delta \rightarrow 0^{+}} \left| \left| \mathbb{I}\left( \eta_{\rho }\cdot f_{n,\delta }\right) -\mathbb{I}f\right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) } =0$. Since $\hbox{\rm supp$\,$} (\mathbb{I}f)\stackrel{\circ}{\subset} V_{r}^{o}$, $\eta_{\rho } \cdot(\mathbb{I}f)=\mathbb{I}f$ and \begin{eqnarray*}
&&\left| \left| \mathbb{I}\left( \eta_{\rho }\cdot f_{n,\delta }\right) -\mathbb{I}f\right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) }\leqslant \left| \left| \mathbb{I}\left( \eta_{\rho }\cdot f_{n,\delta }\right) -\eta_{\rho }\cdot (\mathbb{I}f)\right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) } +\left| \left| \eta_{\rho }\cdot (\mathbb{I}f)-\mathbb{I}f\right| \right|_{L^{2}_{(s,t)}\left( V_{r},w_{1}\right) }, \end{eqnarray*} it suffices to prove that \begin{equation}\label{230417e1}
\lim\limits_{n\to\infty}\lim\limits_{\delta\to 0+}\left| \left| \mathbb{I}\left( \eta_{\rho }\cdot f_{n,\delta }\right) -\eta_{\rho } \cdot\left(\mathbb{I}f\right) \right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) }=0. \end{equation}
By the definition of $\mathbb{I}$ in (\ref{general def of I and P}), it follows that $$
\mathbb{I}\left( \eta_{\rho }\cdot f_{n,\delta }\right) =\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} \left(\sum^{n}_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot\eta_{\rho }\cdot (f_{I,iL})_{n,\delta }\cdot\partial_{i} w_{2}\right)dz_{I}\wedge d\overline{z_{L}} . $$
By the conclusion (2) of Proposition \ref{convolution properties} and noting that $\sup\limits_{V_{\rho^{\prime } }}|\eta_{\rho }\cdot (\partial_{i} w_{2})\cdot e^{-w_1}|<\infty$ for each $i=1,2,\cdots,n$, we deduce that \begin{equation}\label{230418e01}
\lim_{\delta \rightarrow 0^{+}} \left| \left| \mathbb{I}\left( \eta_{\rho }\cdot f_{n,\delta }\right) -\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} \left(\sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot \eta_{\rho }\cdot (f_{I,iL})_n\cdot\partial_{i} w_{2}\right )dz_{I}\wedge d\overline{z_{L}}\right| \right|_{L^{2}_{\left( s,t\right) }\left( V_{\rho^{\prime } },w_{1}\right) } =0. \end{equation}
Let $\theta \triangleq h_{\rho^{\prime }+1 } (\eta )$. Then $\theta\in C^{\infty }_{0,F}\left( V\right)$. For any $g=\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right|=t+1}g_{I,J}dz_{I}\wedge d\overline{z_{L}}\in L^{2}_{(s,t+1)}\left( \ell^{2} ,P\right)$, similarly to the proof of (\ref{230414e3}), we have \begin{equation}\label{general commutator estimation} \begin{aligned}
&\Bigg|\Bigg|\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t} \left(\sum^{\infty }_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot g_{I,iL}\cdot\partial_{i} \left( \theta\cdot w_{2}\right) \right)dz_{I}\wedge d\overline{z_{L}}\\
&\quad-\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n}\left(\sum^{n}_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot (g_{I,iL})_n\cdot\partial_{i} \left( \theta\cdot w_{2}\right) \right)dz_{I}\wedge d\overline{z_{L}} \Bigg|\Bigg|^{2}_{L^{2}_{\left( s,t\right) }\left( V_{\rho^{\prime } },w_{1}\right) }\\
&\leqslant 2\left| \left| \sum^{\prime }_{\left| I\right| =s,\left| L\right| =t} \left(\sum^{\infty }_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot g_{I,iL}\cdot\partial_{i} \left( \theta\cdot w_{2}\right) \right)dz_{I}\wedge d\overline{z_{L}} \right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V_{\rho^{\prime } },w_{1}\right) } \\
&\quad +2\left| \left| \sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} \left(\sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot (g_{I,iL})_{n}\cdot\partial_{i} \left( \theta\cdot w_{2}\right) \right)dz_{I}\wedge d\overline{z_{L}} \right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V_{\rho^{\prime } },w_{1}\right) } \\
&= 2\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t} c_{I,L}\int_{V_{\rho^{\prime } }} \left| \sum^{\infty }_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot g_{I,iL}\cdot\partial_{i} \left( \theta\cdot w_{2}\right) \right|^{2}e^{-w_1}\,\mathrm{d}P\\
&\quad+2\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} c_{I,L}\int_{V_{\rho^{\prime } }} \left| \sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot (g_{I,iL})_{n}\cdot\partial_{i} \left( \theta\cdot w_{2}\right) \right|^{2}e^{-w_1}\,\mathrm{d}P\\
&\leqslant 2C_3\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t} c_{I,L}\int_{V_{\rho^{\prime } }} \sum^{\infty }_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot |g_{I,iL}|^{2}\,\mathrm{d}P
+2C_3\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} c_{I,L}\int_{V_{\rho^{\prime } }} \sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot |(g_{I,iL})_{n} |^{2}\,\mathrm{d}P\\
&=2C_3\cdot(t+1)\cdot\left(\sum^{\prime }_{\left| I\right| =s,\left| J\right|=t+1} c_{I,J}\int_{V_{\rho^{\prime } }} \sum^{\infty }_{i=1} |g_{I,J}|^{2}\,\mathrm{d}P
+ \sum^{\prime }_{\left| I\right| =s,\left| J\right|=t+1,\max \left\{ I\bigcup L\right\} \leqslant n} c_{I,J}\int_{V_{\rho^{\prime } }} \sum^{n}_{i=1} |(g_{I,J})_{n} |^{2}\,\mathrm{d}P\right)\\
&\leqslant 4C_3\cdot(t+1)\cdot\left(\sum^{\prime }_{\left| I\right| =s,\left| J\right|=t+1} c_{I,J}\int_{\ell^2} \sum^{\infty }_{i=1} |g_{I,J}|^{2}\,\mathrm{d}P\right)=4C_3\cdot(t+1)\cdot ||g||^2_{ L^{2}_{(s,t+1)}\left( \ell^{2} ,P\right) }, \end{aligned} \end{equation}
where $C_3\triangleq \left(\sup\limits_{V_{\rho^{\prime } }}\sum\limits^{\infty }_{i=1} \left| \partial_{i} \left( \theta\cdot w_{2}\right) \right|^{2}\right)\cdot c_1^{s,t}<\infty$ and the third inequality follows from the conclusion (1) of Proposition \ref{Reduce diemension}.
Write $$
\mathscr{C}_{(s,t+1)}\triangleq\bigcup^{\infty }_{m=1} \left\{ \sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\max \left\{ I\bigcup J\right\} \leqslant m} h_{I,J}dz_{I}\wedge d\overline{z_{J}}:\;h_{I,J}\in C^{\infty }_{c}(\mathbb{C}^m) \right\}. $$
Combining the sequence of cut-off functions given in \eqref{section2}, the final part of the proof of Proposition \ref{general cut-off density3} and Lemma \ref{approximation for st form}, we see that $\mathscr{C}_{(s,t+1)}$ is dense in $L^{2}_{(s,t+1)}\left( \ell^{2} ,P\right)$. As a consequence, for every $\varepsilon>0$, there exists $m\in\mathbb{N}$ and $g=\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\max \left\{ I\bigcup J\right\} \leqslant m} g_{I,J}dz_{I}\wedge d\overline{z_{J}}\in \mathscr{C}_{(s,t+1)}$ such that $\left| \left| f-g\right| \right|^{2}_{L^{2}_{(s,t+1)}\left( \ell^{2} ,P\right) } \leqslant \frac{\varepsilon}{8C_3(t+1)+1} $. Then for any $n>m$, noting that $\theta(\textbf{z})=1$ for all $\textbf{z}\in V_{\rho^{\prime } }$, we have \begin{eqnarray*}
&&\left| \left| \mathbb{I}f-\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t, \max \left\{I\bigcup L\right\} \leqslant n} \left(\sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot (f_{I,iL})_{n}\cdot\partial_{i} w_{2}\right)dz_{I}\wedge d\overline{z_{L}}\right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V_{\rho^{\prime } },w_{1}\right) } \\
&&=\Bigg| \Bigg| \sum^{\prime }_{\left| I\right| =s,\left| L\right| =t} \left(\sum^{\infty }_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot f_{I,iL}\cdot\partial_{i} w_{2}\right)dz_{I}\wedge d\overline{z_{L}}\\
&&\quad-\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t, \max \left\{I\bigcup L\right\} \leqslant n} \left(\sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot (f_{I,iL})_{n}\cdot\partial_{i} w_{2}\right)dz_{I}\wedge d\overline{z_{L}}\Bigg| \Bigg|^{2}_{L^{2}_{\left( s,t\right) }\left( V_{\rho^{\prime } },w_{1}\right) } \\
&&=\Bigg| \Bigg|\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t} \left(\sum^{\infty }_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot f_{I,iL}\cdot\partial_{i} \left( \theta\cdot w_{2}\right) \right)dz_{I}\wedge d\overline{z_{L}} \\
&&\quad-\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} \left(\sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot (f_{I,iL})_{n}\cdot\partial_{i} \left( \theta\cdot w_{2}\right) \right)dz_{I}\wedge d\overline{z_{L}}\Bigg| \Bigg|^{2}_{L^{2}_{\left( s,t\right) }\left( V_{\rho^{\prime } },w_{1}\right) } \\
&&\leqslant 2\Bigg| \Bigg| \sum^{\prime }_{\left| I\right| =s,\left| L\right| =t} \left(\sum^{\infty }_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot (f_{I,iL}-g_{I,iL})\cdot\partial_{i} \left( \theta\cdot w_{2}\right)\right)dz_{I}\wedge d\overline{z_{L}}\\
&&\quad-\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} \left(\sum^{n}_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot ((f_{I,iL})_{n}-(g_{I,iL})_{n})\cdot\partial_{i} \left( \theta\cdot w_{2}\right)\right)dz_{I}\wedge d\overline{z_{L}} \Bigg| \Bigg|^{2}_{L^{2}_{\left( s,t\right) }\left( V_{\rho^{\prime } },w_{1}\right) } \\
&&\quad+2\Bigg| \Bigg| \sum^{\prime }_{\left| I\right| =s,\left| L\right| =t} \left(\sum^{\infty }_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot g_{I,iL}\cdot\partial_{i} \left( \theta\cdot w_{2}\right)\right)dz_{I}\wedge d\overline{z_{L}}\\
&&\quad-\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} \left(\sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot (g_{I,iL})_{n}\cdot\partial_{i} \left( \theta\cdot w_{2}\right) \right)dz_{I}\wedge d\overline{z_{L}} \Bigg| \Bigg|^{2}_{L^{2}_{\left( s,t\right) }\left( V_{\rho^{\prime } },w_{1}\right) }\\
&& \leqslant 8C_3\cdot(t+1)\cdot\left| \left| f-g\right| \right|^{2}_{L^{2}_{\left( s,t+1\right) }\left( \ell^{2} ,P\right) } \leqslant \varepsilon, \end{eqnarray*}
where the second inequality follows from \eqref{general commutator estimation} and the fact that $g=\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t+1,\max \left\{ I\bigcup J\right\} \leqslant m} g_{I,J}dz_{I}\wedge d\overline{z_{J}}\in \mathscr{C}_{(s,t+1)}$, and hence $$
\begin{array}{ll}
\displaystyle \sum^{\prime }_{\left| I\right| =s,\left| L\right| =t} \left(\sum^{\infty }_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot g_{I,iL}\cdot\partial_{i} \left( \theta\cdot w_{2}\right) \right)dz_{I}\wedge d\overline{z_{L}} \\[3mm]
\displaystyle \quad-\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} \left(\sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot (g_{I,iL})_{n}\cdot\partial_{i} \left( \theta\cdot w_{2}\right) \right)dz_{I}\wedge d\overline{z_{L}} \\[3mm]
\displaystyle =\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant m}\left(\sum^{m}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot g_{I,iL}\cdot\partial_{i} \left( \theta\cdot w_{2}\right) \right)dz_{I}\wedge d\overline{z_{L}}\\[3mm]
\displaystyle \quad-\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant m} \left(\sum^{m}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot (g_{I,iL})_{n}\cdot\partial_{i} \left( \theta\cdot w_{2}\right) \right)dz_{I}\wedge d\overline{z_{L}} \\[3mm]
\displaystyle =\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant m} \left(\sum^{m}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot g_{I,iL}\cdot\partial_{i} \left( \theta\cdot w_{2}\right) \right)dz_{I}\wedge d\overline{z_{L}}\\[3mm]
\displaystyle \quad-\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant m}\left(\sum^{m}_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot g_{I,iL}\cdot\partial_{i} \left( \theta\cdot w_{2}\right) \right)dz_{I}\wedge d\overline{z_{L}} \\[3mm] \displaystyle =0.
\end{array} $$ By the arbitrariness of $\varepsilon$, we have \begin{equation}\label{230418e02}
\lim_{n\rightarrow \infty } \left| \left| \mathbb{I}f-\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} \left(\sum^{n}_{i=1}\frac{c_{I,iL}}{c_{I,L}}\cdot (f_{I,iL})_{n}\cdot\partial_{i} w_{2}\right)dz_{I}\wedge d\overline{z_{L}} \right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) } =0. \end{equation} Since \begin{eqnarray*}
&&\left| \left| \eta_{\rho } \cdot(\mathbb{I}f)-\mathbb{I}\left( \eta_{\rho }\cdot f_{n,\delta }\right) \right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) } \\
&&\leqslant \left| \left| \eta_{\rho }\cdot (\mathbb{I}f)-\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} \left(\sum^{n}_{i=1} \eta_{\rho }\cdot\frac{c_{I,iL}}{c_{I,L}}\cdot (f_{I,iL})_{n}\cdot\partial_{i} w_{2}\right)dz_{I}\wedge d\overline{z_{L}} \right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) }\\
&&\quad+\left| \left| \mathbb{I}\left( \eta_{\rho }\cdot f_{n,\delta }\right) -\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} \left(\sum^{n}_{i=1} \eta_{\rho }\cdot\frac{c_{I,iL}}{c_{I,L}}\cdot (f_{I,iL})_n\cdot\partial_{i} w_{2}\right )dz_{I}\wedge d\overline{z_{L}}\right| \right|_{L^{2}_{\left( s,t\right) }\left( V_{\rho^{\prime } },w_{1}\right) }\\
&&\leqslant \left| \left| \mathbb{I}f-\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} \left(\sum^{n}_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot (f_{I,iL})_{n}\cdot\partial_{i} w_{2}\right)dz_{I}\wedge d\overline{z_{L}} \right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) }\\
&&\quad+\left| \left| \mathbb{I}\left( \eta_{\rho }\cdot f_{n,\delta }\right) -\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t,\max \left\{ I\bigcup L\right\} \leqslant n} \left(\sum^{n}_{i=1} \eta_{\rho }\cdot \frac{c_{I,iL}}{c_{I,L}}\cdot (f_{I,iL})_n\cdot\partial_{i} w_{2}\right )dz_{I}\wedge d\overline{z_{L}}\right| \right|_{L^{2}_{\left( s,t\right) }\left( V_{\rho^{\prime } },w_{1}\right) }, \end{eqnarray*} by (\ref{230418e01}) and (\ref{230418e02}), we obtain the desired equality (\ref{230417e1}).
\textbf{Step 5:} Finally, we come to show that $$ \begin{aligned}
&
\lim\limits_{n\rightarrow \infty } \lim\limits_{\delta \rightarrow 0^{+}}\left( \left| \left| T^{\ast }\left( \eta_{\rho }\cdot f_{n,\delta }\right) -T^{\ast }f\right| \right|_{L^{2}_{(s,t)}\left( V,w_{1}\right) } +\left| \left| \eta_{\rho }\cdot f_{n,\delta }-f\right| \right|_{L^{2}_{(s,t+1)}\left( V,w_{2}\right) } +\left| \left| S\left( \eta_{\rho }\cdot f_{n,\delta }\right) -Sf\right| \right|_{L^{2}_{(s,t+2)}\left( V,w_{3}\right) }\right) \\ & = 0. \end{aligned} $$ Combining Steps 3 and 4, and noting $T^*\left( \eta_{\rho }\cdot f_{n,\delta }\right)=(-1)^{s}e^{w_1-w_2}\cdot\left(\mathbb{I}\left( \eta_{\rho }\cdot f_{n,\delta }\right)-\mathbb{P}\left( \eta_{\rho }\cdot f_{n,\delta }\right)\right)$, $T^*f=(-1)^{s}e^{w_1-w_2}\cdot(\mathbb{I}f-\mathbb{P}f)$ and $\sup\limits_{V_{\rho^{\prime } }} e^{w_{1}-w_{2}}<\infty$, we have \begin{equation} \begin{array}{ll}
\displaystyle \lim_{n\rightarrow \infty } \lim_{\delta \rightarrow 0^{+}} \left| \left| T^{\ast }\left( \eta_{\rho }\cdot f_{n,\delta }\right) -T^{\ast }f\right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) } \\[3mm]
\displaystyle =\lim_{n\rightarrow \infty } \lim_{\delta \rightarrow 0^{+}} \left| \left| \left( -1\right)^{s} e^{w_{1}-w_{2}}\cdot\left(\mathbb{I}\left( \eta_{\rho }\cdot f_{n,\delta }\right) - \mathbb{I}f \right)-\left( -1\right)^{s} e^{w_{1}-w_{2}}\cdot\left(\mathbb{P}\left( \eta_{\rho }\cdot f_{n,\delta }\right)-\mathbb{P}f\right)\right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) }\\[3mm]
\displaystyle \leqslant \left(\sup\limits_{V_{\rho^{\prime } }} e^{w_{1}-w_{2}}\right)\cdot\left(\lim_{n\rightarrow \infty } \lim_{\delta \rightarrow 0^{+}} \left| \left| \mathbb{I}\left( \eta_{\rho }\cdot f_{n,\delta }\right) - \mathbb{I}f \right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) }+\lim_{n\rightarrow \infty } \lim_{\delta \rightarrow 0^{+}}\left| \left| \mathbb{P}\left( \eta_{\rho }\cdot f_{n,\delta }\right)-\mathbb{P}f\right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) }\right)\\[3mm] \displaystyle =0.\label{general 11111} \end{array} \end{equation} By Steps 1 and 2, and noting $\eqref{general 11111}$, we have \begin{equation}\label{230418e5} \begin{aligned}
&\lim\limits_{n\rightarrow \infty } \lim\limits_{\delta \rightarrow 0^{+}}\left( \left| \left| T^{\ast }\left( \eta_{\rho }\cdot f_{n,\delta }\right) -T^{\ast }f\right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) } +\left| \left| \eta_{\rho }\cdot f_{n,\delta }-f\right| \right|_{L^{2}_{(s,t+1)}\left( V_{\rho^{\prime } },w_{2}\right) } \right.\\
&\qquad\qquad\left.+\left| \left| S\left( \eta_{\rho }\cdot f_{n,\delta }\right) -Sf\right| \right|_{L^{2}_{(s,t+2)}\left( V_{\rho^{\prime } },w_{3}\right) }\right)\\ &= 0. \end{aligned} \end{equation} By the fact that $\hbox{\rm supp$\,$} \eta_{\rho }\subset V^{o}_{\rho^{\prime } }$ and Proposition \ref{support argument}, we have \begin{eqnarray*} &&\hbox{\rm supp$\,$} (\eta_{\rho }\cdot f_{n,\delta }) \stackrel{\circ}{\subset} V^{o}_{\rho^{\prime } }, \quad\hbox{\rm supp$\,$} \left(S\left( \eta_{\rho }\cdot f_{n,\delta }\right)\right) \stackrel{\circ}{\subset} V^{o}_{\rho^{\prime } },\quad \hbox{\rm supp$\,$} \left(T^{\ast }\left( \eta_{\rho }\cdot f_{n,\delta }\right) \right) \stackrel{\circ}{\subset} V^{o}_{\rho^{\prime } },\\ &&\hbox{\rm supp$\,$} f \stackrel{\circ}{\subset} V^{o}_{\rho^{\prime } }, \quad\hbox{\rm supp$\,$} (Sf) \stackrel{\circ}{\subset} V^{o}_{\rho^{\prime } }, \quad\hbox{\rm supp$\,$} (T^{\ast }f) \stackrel{\circ}{\subset} V^{o}_{\rho^{\prime } }. \end{eqnarray*} Hence $$
\begin{aligned}
&\left| \left| T^{\ast }\left( \eta_{\rho }\cdot f_{n,\delta }\right) -T^{\ast }f\right| \right|_{L^{2}_{(s,t)}\left( V,w_{1}\right) } +\left| \left| \eta_{\rho }\cdot f_{n,\delta }-f\right| \right|_{L^{2}_{(s,t+1)}\left( V,w_{2}\right) } +\left| \left| S\left( \eta_{\rho }\cdot f_{n,\delta }\right) -Sf\right| \right|_{L^{2}_{(s,t+2)}\left( V,w_{3}\right) } \\&=\left| \left| T^{\ast }\left( \eta_{\rho }\cdot f_{n,\delta }\right) -T^{\ast }f\right| \right|_{L^{2}_{(s,t)}\left( V_{\rho^{\prime } },w_{1}\right) } +\left| \left| \eta_{\rho }\cdot f_{n,\delta }-f\right| \right|_{L^{2}_{(s,t+1)}\left( V_{\rho^{\prime } },w_{2}\right) } +\left| \left| S\left( \eta_{\rho }\cdot f_{n,\delta }\right) -Sf\right| \right|_{L^{2}_{(s,t+2)}\left( V_{\rho^{\prime } },w_{3}\right) } .
\end{aligned} $$ Combining this with (\ref{230418e5}), we obtain the desired equality (\ref{230418e03}), which completes the proof of Theorem \ref{general density4}. \end{proof}
\begin{remark} We should note that $f_{n,\delta }$ may be not in $D_{T^{\ast }}\cap D_S$, but $\eta_{\rho }\cdot f_{n,\delta }\in D_{T^{\ast }}\cap D_S$. \end{remark}
In the sequel, we need the following two assumptions the coefficients in (\ref{defnition of general st froms}) (Recall (\ref{gener111}) for $c_{I,iJ}$). \begin{condition}\label{230423ass2}
Suppose that $\displaystyle c_0^{s,t}\triangleq \inf_{|I|=s,|J|=t,i\in\mathbb{N}}\frac{ c_{I,iJ} }{c_{I,J}}>0$. \end{condition}
\begin{condition}\label{230423ass3} Suppose that the coefficients in (\ref{defnition of general st froms}) satisfy the following condition \begin{eqnarray}\label{multiplictive condition for coefficient} c_{I,J}\cdot c_{I,J'}=c_{I,L}\cdot c_{I,K} \end{eqnarray}
for all strictly increasing multi-indices $I,J,J',L$ and $K$ with $|I|=s,|J|=t+1,|J'|=t+1,|L|=t,|K|=t+2, J\cup J'=K$ and $J\cap J'=L$. \end{condition}
We also need the following assumption:
\begin{condition}\label{230423ass4} Suppose that the real-valued function $\varphi$ in \eqref{weight function} satisfies that for each $n\in\mathbb{N}$, the following inequality holds on $V$, \begin{eqnarray}\label{230419e12} \sum_{1\leqslant i,j\leqslant n} (\partial_{i} \overline{\partial_{j} } \varphi) \cdot\zeta_{i}\cdot\overline{\zeta_{j}}
\geqslant \left( 2\sum\limits^{n}_{i=1} |\partial_{i} \psi |^{2}+2e^{\psi }-\frac{1}{2} \right)\cdot\left( \sum^{n}_{i=1} \left| \zeta_{i}\right|^{2}\right),\quad \forall\; (\zeta_{1},\cdots,\zeta_{n})\in\mathbb{C}^n. \end{eqnarray} \end{condition}
We are in a position to establish an infinite-dimensional version of \cite[Lemma 4.2.1, p. 84]{Hor90} as follows (Recall Condition \ref{230423ass2} for $c_0^{s,t}$): \begin{lemma}\label{general estimation similar to Lemma 4.2.1} Under Conditions \ref{230424c1}, \ref{230423ass1}, \ref{230423ass2}, \ref{230423ass3} and \ref{230423ass4}, it holds that \begin{eqnarray}\label{230419e13}
\left| \left| T^{\ast }f\right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V,w_{1}\right) } +\left| \left| Sf\right| \right|^{2}_{L^{2}_{\left( s,t+2\right) }\left( V,w_{3}\right) }\geqslant c_0^{s,t}\cdot\left| \left| f\right| \right|^{2}_{L^{2}_{\left( s,t+1\right) }\left( V,w_{2}\right) } \end{eqnarray}
for any $f=\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t+1 } f_{I,J}dz_{I}\wedge d\overline{z_{J}}\in D_{S}\bigcap D_{T^{\ast }}$ satisfying that there exists $m\in\mathbb{N}$ such that $f_{I,J}=0$ for all strictly increasing multi-indices $I$ and $J$ with $\max{\left\{ I\bigcup J\right\} }>m$ and $f_{I,J}\in C^{2}_{0,F}\left( V\right)$ for all strictly increasing multi-indices $I$ and $J$ with $\max{\left\{ I\bigcup J\right\} }\leqslant m$. \end{lemma}
\begin{proof} We proceed as in the proof of \cite[Lemma 4.2.1, p. 84]{Hor90}. Since $w_3=\varphi$, \begin{eqnarray*}
\left| \left| Sf\right| \right|^{2}_{L^{2}_{\left( s,t+2\right) }\left( V,w_{3}\right) }
&=&\sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| K\right| =t+2} c_{I,K}\int_{V} \left| \sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t+1} \varepsilon^{K}_{i,J}\cdot \overline{\partial_{i} } f_{I,J }\right|^{2} e^{-w_{3}}\,\mathrm{d}P\\
&=&\sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| K\right| =t+2} \int_{V} \sum_{1\leqslant i,j<\infty } \sum^{\prime }_{\left| J\right|=t+1 ,\left| J^{\prime }\right| =t+1} c_{I,K}\cdot\varepsilon^{K}_{i,J}\cdot \varepsilon^{K}_{j,J^{\prime }}\cdot \overline{\partial_{j} } f_{I,J^{\prime }}\cdot\partial_{i} \overline{f_{I,J}}\cdot e^{-w_{3}}\,\mathrm{d}P \\
&=&\sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| J\right|=t+1 ,\left| J^{\prime }\right| =t+1}\sum_{1\leqslant i,j<\infty }\int_{V} c_{I,iJ}\cdot\varepsilon^{j,J^{\prime }}_{i,J}\cdot \overline{\partial_{j} } f_{I,J^{\prime }}\cdot\partial_{i} \overline{f_{I,J}}\cdot e^{-w_{3}}\,\mathrm{d}P, \end{eqnarray*}
where $c_{I,iJ}$ is defined as that in (\ref{gener111}), and we have used the fact that $\varepsilon^{K}_{i,J}\cdot \varepsilon^{K}_{j,J^{\prime }}\not=0$ if and only if $i\not\in J$, $j\not\in J^{\prime }$ and $K=\{i\}\cup J=\{j\}\cup J^{\prime }$ (in this case $\varepsilon^{K}_{i,J}\cdot \varepsilon^{K}_{j,J^{\prime }}=\varepsilon^{j,J^{\prime }}_{i,J}$, and $\varepsilon^{j,J^{\prime }}_{i,J}$ is the sign of the permutation from $j J^{\prime }$ to $i J$). We shall rearrange the terms in the above sum. Note that if $i=j$, then $\varepsilon^{j,J^{\prime }}_{i,J}\neq 0$ if and only if $j\notin J$ and $J^{\prime }=J$. If $i\neq j$, then $\varepsilon^{j,J^{\prime }}_{i,J}\neq 0$ if and only if there exists a strictly increasing multi-index $L$ with $|L|=t$ so that $i\not\in L$, $j\not\in L$, $J^{\prime }=L\cup \{i\}$ and $J=L\cup \{j\}$, in which case, \begin{eqnarray*} \varepsilon^{j,J^{\prime }}_{i,J}=\varepsilon^{j,J^{\prime }}_{j,i,L}\cdot\varepsilon^{j,i,L}_{i,j,L}\cdot\varepsilon^{i,j,L}_{i,J}=-\varepsilon^{j,J^{\prime }}_{j,i,L}\cdot\varepsilon^{i,j,L}_{i,J}=-\varepsilon^{J^{\prime }}_{i,L}\cdot\varepsilon^{j,L}_{J}. \end{eqnarray*} Combining \eqref{multiplictive condition for coefficient} in Condition \ref{230423ass3}, and noting that by (\ref{gener111}), $c_{I,iL}=0$ for $i\in L$, $c_{I,jL}=0$ for $j\in L$ and $c_{I,iJ}=0$ for $i\in J$, we then have \begin{eqnarray*}
\left| \left| Sf\right| \right|^{2}_{L^{2}_{\left( s,t+2\right) }\left( V,w_{3}\right) } &=&\sum^{\prime }_{\left| I\right| =s}\sum_{i\notin J} \sum^{\prime }_{\left| J\right| =t+1}c_{I,iJ}\cdot \int_{V} |\overline{\partial_{i} } f_{I,J}|^{2}\cdot e^{-\varphi }\,\mathrm{d}P\\
&&-\sum^{\prime }_{\left| I\right| =s}\sum_{i\neq j} \sum^{\prime }_{\left| J\right| ,\left| J^{\prime }\right| =t+1} \sum^{\prime }_{\left| L\right| =t} \frac{c_{I,iL}\cdot c_{I,jL}}{c_{I, L}}\int_{V}\varepsilon^{J^{\prime }}_{i,L}\cdot\varepsilon^{j,L}_{J}\cdot \partial_{i} \overline{f_{I,J}} \cdot \overline{\partial_{j} } f_{I,J^{\prime }}\cdot e^{-\varphi }\,\mathrm{d}P\\
&=&\sum^{\prime }_{\left| I\right| =s}\sum_{i\notin J} \sum^{\prime }_{\left| J\right| =t+1} c_{I,iJ}\int_{V} |\overline{\partial_{i} } f_{I,J}|^{2}\cdot e^{-\varphi }\,\mathrm{d}P\\
&&-\sum^{\prime }_{\left| I\right| =s}\sum_{i\neq j} \sum^{\prime }_{\left| L\right| =t} \frac{c_{I,iL}\cdot c_{I,jL}}{c_{I, L}}\int_{V} \partial_{i} \overline{f_{I,jL}}\cdot\overline{\partial_{j} } f_{I,iL}\cdot e^{-\varphi }\,\mathrm{d}P\\
&\geqslant&\sum^{\prime }_{\left| I\right| =s}\sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t+1}c_{I,iJ} \int_{V} |\overline{\partial_{i} } f_{I,J}|^{2}\cdot e^{-\varphi }\,\mathrm{d}P\\
&&-\sum^{\prime }_{\left| I\right| =s}\sum_{1\leqslant i,j<\infty } \sum^{\prime }_{\left| L\right| =t}\frac{c_{I,iL}\cdot c_{I,jL}}{c_{I, L}} \int_{V} \partial_{i} \overline{f_{I,jL}}\cdot \overline{\partial_{j} } f_{I,iL}\cdot e^{-\varphi }\,\mathrm{d}P, \end{eqnarray*} where the last inequality follows from the fact that \begin{eqnarray*}
\sum^{\prime }_{\left| I\right| =s}\sum_{ i=1}^{\infty } \sum^{\prime }_{\left| L\right| =t}\frac{c_{I,iL}^2}{c_{I, L}} \int_{V} |\overline{\partial_{i} } f_{I,iL}|^2\cdot e^{-\varphi }\,\mathrm{d}P\geqslant 0. \end{eqnarray*} Recalling (\ref{230419e11}) for the operator $\sigma_{i}$, by Lemma \ref{integration by Parts}, we obtain that \begin{equation}\label{middle equality} \begin{aligned}
&\sum^{\prime }_{\left| I\right| =s} \sum_{1\leqslant i,j<\infty } \sum^{\prime }_{\left| L\right| =t} \frac{c_{I,iL}\cdot c_{I,jL}}{c_{I, L}}\int_{V} \left((\sigma_{i} f_{I,iL})\cdot\overline{(\sigma_{j} f_{I,jL})}-\overline{(\overline{\partial_{i}}f_{I,jL})}\cdot(\overline{\partial_{j}}f_{I,iL})\right)\cdot e^{-\varphi }\,\mathrm{d}P\\
&=-\sum^{\prime }_{\left| I\right| =s} \sum_{1\leqslant i,j<\infty } \sum^{\prime }_{\left| L\right| =t}\frac{c_{I,iL}\cdot c_{I,jL}}{c_{I, L}} \int_{V} f_{I,iL} \cdot \left(\overline{\overline{\partial_{i} } (\sigma_{j} f_{I,jL})- \sigma_{j} (\overline{\partial_{i} } f_{I,jL})}\right)\cdot e^{-\varphi }\,\mathrm{d}P\\
&=\sum^{\prime }_{\left| I\right| =s} \sum_{1\leqslant i,j<\infty } \sum^{\prime }_{\left| L\right| =t} \frac{c_{I,iL}\cdot c_{I,jL}}{c_{I, L}}\int_{V} f_{I,iL}\cdot \overline{f_{I,jL}}\cdot \left( \partial_{i} \overline{\partial_{j} } \varphi +\frac{ \overline{\partial_i }(\overline{z_{j}})}{2a^{2}_{j}} \right) \cdot e^{-\varphi }\,\mathrm{d}P, \end{aligned} \end{equation} where the last equality follows from \eqref{commutaor formula}.
By Proposition \ref{general formula of T*} , we have $T^{\ast }f =e^{-\psi }\cdot X+e^{-\psi }\cdot Y $, where $$ \begin{aligned}
&X\triangleq(-1)^{s+1}\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} \sum^{\infty }_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot \sigma_{i} f_{I,iL}dz_{I}\wedge d\overline{z_{L}} ,\\
&Y\triangleq(-1)^{s+1}\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} \sum^{\infty }_{i=1} \frac{c_{I,iL}}{c_{I,L}}\cdot f_{I,iL}\cdot\partial_{i}\psi dz_{I}\wedge d\overline{z_{L}}. \end{aligned} $$ Then, $$
\left| \left| e^{-\psi }\cdot X\right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V,w_{1}\right) } \leqslant 2\left| \left| e^{-\psi }\cdot Y\right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V,w_{1}\right) } +2\left| \left| T^{\ast }f\right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V,w_{1}\right) }, $$ $$
\begin{aligned}
\left| \left| e^{-\psi }\cdot X\right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V,w_{1}\right) } &=\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} \int_{V} \sum_{1\leqslant i,j<\infty }\frac{c_{I,iL}\cdot c_{I,jL}}{c_{I,L}}\cdot \overline{ (\sigma_{i} f_{I,iL})}\cdot(\sigma_{j} f_{I,jL})\cdot e^{-w_{1}-2\psi}\,\mathrm{d}P\\
&=\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} \int_{V} \sum_{1\leqslant i,j<\infty } \frac{c_{I,iL}\cdot c_{I,jL}}{c_{I,L}}\cdot \overline{(\sigma_{i} f_{I,iL})}\cdot(\sigma_{j} f_{I,jL})\cdot e^{-\varphi}\,\mathrm{d}P
\end{aligned} $$ and $$
\begin{aligned}
\left| \left| e^{-\psi }\cdot Y\right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V,w_{1}\right) } &=\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} \frac{1}{c_{I,L}}\int_{V} \bigg|\sum^{\infty }_{i=1} c_{I,iL}\cdot f_{I,iL}\cdot\partial_{i}\psi \bigg|^{2}\cdot e^{-w_{1}-2\psi }\,\mathrm{d}P\\
&\leqslant \sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t}\int_{V} \left(\sum^{m}_{i=1}\frac{|c_{I,iL}|^2}{c_{I,L}}\cdot \left| f_{I,iL}\right|^{2} \right) \cdot \left(\sum^{m }_{i=1} |\partial_{i}\psi|^{2}\right)e^{-\varphi }\,\mathrm{d}P,
\end{aligned} $$
where the inequality follows from Cauchy-Schwarz inequality and the assumption that $f_{I,J}=0$ for all strictly increasing multi-indices $I$ and $J$ with $\left| I\right| =s$, $\left| L\right| =t$ and $\max{\left\{ I\bigcup J\right\} }>m$. Combining the above inequalities and equalities we have $$ \begin{aligned}
& 2\left| \left| T^{\ast }f\right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V,w_{1}\right) } +\left| \left| Sf\right| \right|^{2}_{L^{2}_{\left( s,t+2\right) }\left( V,w_{3}\right) }\\
&\geqslant \sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} \sum_{1\leqslant i,j<\infty }\frac{c_{I,iL}\cdot c_{I,jL}}{c_{I,L}}\cdot \int_{V}\overline{(\sigma_{i} f_{I,iL})}\cdot(\sigma_{j} f_{I,jL})\cdot e^{-\varphi}\,\mathrm{d}P\\
&\quad-2\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} \int_{V} \left(\sum^{m}_{i=1} \frac{|c_{I,iL}|^2}{c_{I,L}}\cdot \left| f_{I,iL}\right|^{2} \right) \cdot \left(\sum^{m }_{i=1} |\partial_{i}\psi |^{2}\right)\cdot e^{-\varphi }\,\mathrm{d}P\\
&\quad+\sum^{\prime }_{\left| I\right| =s} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t+1}c_{I,iJ} \int_{V} |\overline{\partial_{i} } f_{I,J}|^{2}\cdot e^{-\varphi }\,\mathrm{d}P-\sum^{\prime }_{\left| I\right| =s} \sum_{1\leqslant i,j<\infty } \sum^{\prime }_{\left| L\right| =t} \frac{c_{I,iL}\cdot c_{I,jL}}{c_{I, L}}\int_{V} \partial_{i} \overline{f_{I,jL}}\cdot \overline{\partial_{j} } f_{I,iL}\cdot e^{-\varphi }\,\mathrm{d}P\\
&=\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} \int_{V} \sum_{1\leqslant i,j<\infty }\frac{c_{I,iL}\cdot c_{I,jL}}{c_{I, L}}\left( \overline{(\sigma_{i} f_{I,iL})}\cdot(\sigma_{j} f_{I,jL})-\partial_{i} \overline{f_{I,jL}}\cdot \overline{\partial_{j} } f_{I,iL}\right)\cdot e^{-\varphi}\,\mathrm{d}P \\
&\quad+\sum^{\prime }_{\left| I\right| =s}\sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t+1} c_{I,iJ}\int_{V} |\overline{\partial_{i} } f_{I,J}|^{2}\cdot e^{-\varphi }\,\mathrm{d}P- 2\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} \int_{V} \left(\sum^{m}_{i=1} \frac{|c_{I,iL}|^2}{c_{I,L}}\cdot\left| f_{I,iL}\right|^{2}\right) \cdot \left(\sum^{m }_{i=1} |\partial_{i}\psi |^{2}\right)\cdot e^{-\varphi }\,\mathrm{d}P\\
&=\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t}\frac{c_{I,iL}\cdot c_{I,jL}}{c_{I, L}}\int_{V} \sum_{1\leqslant i,j<\infty }f_{I,iL}\cdot\overline{f_{I,jL}}\cdot \left( \partial_{i} \overline{\partial_{j} } \varphi +\frac{\overline{\partial_i} (\overline{z_j})}{2a^{2}_{j}} \right)\cdot e^{-\varphi}\,\mathrm{d}P \\
&\quad+\sum^{\prime }_{\left| I\right| =s}\sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t+1} c_{I,iJ}\int_{V} |\overline{\partial_{i} } f_{I,J}|^{2}\cdot e^{-\varphi }\,\mathrm{d}P- 2\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} \int_{V} \left(\sum^{m}_{i=1}\frac{|c_{I,iL}|^2}{c_{I,L}}\cdot \left| f_{I,iL}\right|^{2} \right) \cdot \left(\sum^{m }_{i=1} |\partial_{i}\psi |^{2}\right)\cdot e^{-\varphi }\,\mathrm{d}P\\
&=\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} \int_{V} \sum_{1\leqslant i,j\leqslant m }\frac{c_{I,iL}\cdot c_{I,jL}}{c_{I, L}}\cdot f_{I,iL}\cdot\overline{f_{I,jL}}\cdot \left( \partial_{i} \overline{\partial_{j} } \varphi +\frac{\overline{\partial_i} (\overline{z_j})}{2a^{2}_{j}} \right)\cdot e^{-\varphi}\,\mathrm{d}P \\
&\quad+\sum^{\prime }_{\left| I\right| =s} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t+1} c_{I,iJ}\int_{V} |\overline{\partial_{i} } f_{I,J}|^{2}\cdot e^{-\varphi }\,\mathrm{d}P- 2\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} \int_{V} \left(\sum^{m}_{i=1} \frac{|c_{I,iL}|^2}{c_{I,L}}\cdot \left| f_{I,iL}\right|^{2} \right) \cdot\left( \sum^{m }_{i=1} |\partial_{i}\psi |^{2}\right)\cdot e^{-\varphi }\,\mathrm{d}P, \end{aligned} $$ where the second equality follows from \eqref{middle equality}. Thus \begin{equation}
\begin{aligned}
&2\left| \left| T^{\ast }f\right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V,w_{1}\right) }+\left| \left| Sf\right| \right|^{2}_{L^{2}_{\left( s,t+2\right) }\left( V,w_{3}\right) }+2\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} \int_{V} \left(\sum^{m}_{i=1} \frac{|c_{I,iL}|^2}{c_{I,L}}\cdot \left| f_{I,iL}\right|^{2} \right) \cdot \left(\sum^{m }_{i=1} |\partial_{i}\psi |^{2}\right)\cdot e^{-\varphi }\,\mathrm{d}P \\
&\geqslant \sum^{\prime }_{\left| I\right| =s}\sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t+1} c_{I,iJ}\int_{V} |\overline{\partial_{i} } f_{I,J}|^{2}\cdot e^{-\varphi }\,\mathrm{d}P\\
&\quad+\sum^{\prime }_{\left| I\right| =s} \sum_{1\leqslant i,j\leqslant m } \sum^{\prime }_{\left| L\right| =t}\frac{c_{I,iL}\cdot c_{I,jL}}{c_{I, L}} \int_{V} f_{I,iL}\cdot\overline{f_{I,jL}} \cdot\left( \partial_{i} \overline{\partial_{j} } \varphi +\frac{\overline{\partial_i} (\overline{z_j})}{2a^{2}_{j}} \right)\cdot e^{-\varphi }\,\mathrm{d}P\\
&= \sum^{\prime }_{\left| I\right| =s}\sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t+1} c_{I,iJ}\int_{V} |\overline{\partial_{i} } f_{I,J}|^{2}\cdot e^{-\varphi }\,\mathrm{d}P+\sum^{\prime }_{\left| I\right| =s} \sum_{1\leqslant i,j\leqslant m } \sum^{\prime }_{\left| L\right| =t}\frac{c_{I,iL}\cdot c_{I,jL}}{c_{I, L}} \int_{V} f_{I,iL}\cdot\overline{f_{I,jL}} \cdot\left( \partial_{i} \overline{\partial_{j} } \varphi \right)\cdot e^{-\varphi }\,\mathrm{d}P\\
&\quad+\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} \int_{V}\left( \sum_{i=1}^{m}\frac{|c_{I,iL}|^2}{c_{I, L}}\cdot|f_{I,iL}|^2\cdot\frac{1}{2a^{2}_{i}}\right)\cdot e^{-\varphi }\,\mathrm{d}P.\label{general inequality for smooth functions}
\end{aligned} \end{equation} By \eqref{general inequality for smooth functions} and the assumption (\ref{230419e12}) in Condition \ref{230423ass4}, we have (Recall Condition \ref{230423ass2} for $c_0^{s,t}$) $$ \begin{aligned}
&2\left| \left| T^{\ast }f\right| \right|^{2}_{L^{2}_{\left( s,t+1\right) }\left( V,w_{1}\right) } +\left| \left| Sf\right| \right|^{2}_{L^{2}_{\left( s,t+2\right) }\left( V,w_{3}\right) }\\
&\geqslant \sum^{\prime }_{\left| I\right| =s}\sum^{\infty }_{i=1} \sum^{\prime }_{\left| J\right| =t+1} c_{I,iJ}\int_{V} |\overline{\partial_{i} } f_{I,J}|^{2}\cdot e^{-\varphi }\,\mathrm{d}P+
\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} \sum^{m}_{i=1} \frac{|c_{I,iL}|^2}{c_{I,L}} \int_{V} \left| f_{I,iL}\right|^{2}\cdot\left( 2e^{\psi }+\frac{1}{2a_i^2}-\frac{1}{2} \right)\cdot e^{-\varphi } \,\mathrm{d}P \\
&\geqslant 2\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} \sum^{m}_{i=1} \frac{|c_{I,iL}|^2}{c_{I,L}}\cdot\int_{V} \left| f_{I,iL}\right|^{2}\cdot e^{\psi-\varphi } \,\mathrm{d}P\\
&\geqslant 2 c_0^{s,t}\cdot\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| L\right| =t} \sum^{m}_{i=1} |c_{I,iL}| \cdot\int_{V} \left| f_{I,iL}\right|^{2}\cdot e^{-w_2} \,\mathrm{d}P\\
&= 2c_0^{s,t}\cdot(t+1)\cdot\left| \left| f\right| \right|^{2}_{L^{2}_{\left( s,t+1\right) }\left( V,w_{2}\right) }, \end{aligned} $$ which gives the desired inequality (\ref{230419e13}). This completes the proof of Lemma \ref{general estimation similar to Lemma 4.2.1}. \end{proof}
As an immediate consequence of Proposition \ref{general cut-off density3}, Theorem \ref{general density4} and Lemma \ref{general estimation similar to Lemma 4.2.1}, we have the following main result in this section. \begin{theorem}\label{general key inequality} Under Conditions \ref{230424c1}, \ref{230423ass1}, \ref{230423ass2}, \ref{230423ass3} and \ref{230423ass4}, it holds that $$
\left| \left| T^{\ast }f\right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V,w_{1}\right) } +\left| \left| Sf\right| \right|^{2}_{L^{2}_{\left( s,t+2\right) }\left( V,w_{3}\right) }
\geqslant c_0^{s,t}\cdot \left| \left| f\right| \right|^{2}_{L^{2}_{\left( s,t+1\right) }\left( V,w_{2}\right) },\quad \forall \;f\in D_{S}\bigcap D_{T^{\ast }}. $$ \end{theorem} \begin{remark} Recall that for any strictly increasing multi-indices $I$ and $J$, it was chosen in \cite{YZ}
$$C_{I,J}=2^{|I|+|J|}\cdot \left(\prod\limits_{i\in I}a_i^2\right)\cdot \left(\prod\limits_{j\in J}a_j^2\right)$$ for the working space (See \cite[p. 530]{YZ}). In that case, $$
\inf\limits_{|I|=s,\,|L|=t,\,i\in \mathbb{N}}\frac{c_{I,iL} }{c_{I,L}}=2\inf\limits_{i\in \mathbb{N}}a_i^2=0. $$ Therefore, if we choose the working space as that in \cite{YZ}, then the result in Theorem \ref{general key inequality} becomes trivial. \end{remark}
\section{Solving the $\overline{\partial}$ equations on pseudo-convex domains in $\ell^{2}$}\label{sec5}
In this section, we shall solve the $\overline{\partial}$ equations on pseudo-convex domains in $\ell^{2}$.
Combining Proposition \ref{Pre-exactness}, Theorem \ref{general key inequality} and \cite[Corollary 2.1, p. 522]{YZ}, we have the following result immediately.
\begin{proposition} \label{general Solving d-bar on V}
Under Conditions \ref{230424c1}, \ref{230423ass1}, \ref{230423ass2}, \ref{230423ass3} and \ref{230423ass4}, it holds that $R_T=N_S$, and for each $f\in N_S$ there exists $g\in D_T$ so that $Tg=f$ and $ \sqrt{c_0^{s,t}}\cdot \left| \left| g\right| \right|_{L^{2}_{(s,t)}\left( V,w_{1}\right) } \leqslant \left| \left| f\right| \right|_{L^{2}_{(s,t+1)}\left( V,w_{2}\right) }$. \end{proposition} In order to solve the $\overline{\partial}$ equation on a more general working space, we need the following result from \cite[Lemma 7.2.10, p. 179]{Tu} (Such a result was implicitly used in \cite[Proof of Theorem 4.2.2 at p. 85]{Hor90}). \begin{lemma}\label{smooth convext increasing function} Suppose that $g_0$ is an increasing nonnegative function on $[0,+\infty )$. Then there exists a convex, increasing, and real analytic function $g$ on $[0,+\infty )$ such that $$ g^{\prime \prime } (x)\geqslant g^{\prime } (x) \geqslant g (x)\geqslant g_0 (x),\quad\forall\; x\in[0,+\infty). $$ \end{lemma} \begin{proof} Since we have not found an exact reference (in English) for Lemma \ref{smooth convext increasing function}, for the readers' convenience, following \cite[Lemma 7.2.10, p. 179]{Tu}, we shall give below a detailed proof. Without loss of generality, we assume that $g_0\geqslant1$. Let $N_{1}=1$, for $k\geq 2$, we assume that $N_ {1},N_ {2},\cdots, N_ {k-1}\in\mathbb{N}$ have been chosen such that $$ N_{l}> \max\left\{ \frac{\ln(g_0(l+1))}{\ln\frac{l}{l-1}},N_{l-1}\right\},\quad l=2,\cdots,k-1. $$ Choose $N_k\in\mathbb{N}$ such that $$ N_{k}> \max\left\{ \frac{\ln(g_0(k+1))}{\ln\frac{k}{k-1}},N_{k-1}\right\}, $$ then, $$ \frac{1}{k}(g_0(k+1))^{\frac{1}{N_{k}}}\in \bigg[\frac{1}{k},\frac{1}{k-1}\bigg). $$ By induction, we obtain a sequence $\{N_{k}\}_{k=1}^{\infty}$ of positive integers. Let $$ a_{0}\triangleq g_0(2)\cdot e,\quad a_{l}\triangleq\frac{1}{k}(g_0(k+1))^{\frac{1}{N_{k}}}\cdot e^{\frac{1}{\sqrt{l}}} \;\hbox{ for }\; l=N_{k},\cdots, N_{k+1}-1,\quad\forall\; k,l\in\mathbb{N}. $$
Note that $\{a_{l}\}_{l=1}^{\infty} $ is a decreasing sequence and for $l=N_{k},\cdots, N_{k+1}-1$, we have $$ \frac{1}{l}\leqslant\frac{1}{N_k}\leqslant\frac{1}{k}\leqslant\frac{e^{\frac{1}{\sqrt{N_{k+1}}}}}{k}\leqslant a_l\leqslant \frac{e^{\frac{1}{\sqrt{N_{k}}}}}{k-1}\leqslant \frac{e^{\frac{1}{\sqrt{k}}}}{k-1} $$ which implies that $\lim\limits_{l\rightarrow \infty } a_{l}=0.$
Hence, $\sum\limits_{n=0}^{\infty }\bigg(\prod\limits_{i=0}^{n}a_{i}\bigg)\cdot|x|^{n}<\infty$ for all $x\in\mathbb{R}$. Let $$ g(x)\triangleq \sum_{n=0}^{\infty }\bigg(\prod_{i=0}^{n}a_{i}\bigg)\cdot x^{n},\quad\forall\; x\in\mathbb{R}. $$ Then $g$ is an increasing and real analytic function on $\mathbb{R}$.
Since $a_{l}\geqslant \frac{1}{l} $ for all $l\in\mathbb{N}$, for each $x\geqslant 0$, we have \begin{eqnarray*} g^{\prime \prime } \left( x\right) &=&\sum_{n=2}^{\infty }a_n\cdot n\cdot a_{n-1}\cdot(n-1)\cdot\bigg(\prod_{i=0}^{n-2}a_{i}\bigg)\cdot x^{n-2}\\ &\geqslant &\sum_{n=2}^{\infty } a_{n-1}\cdot(n-1)\cdot\bigg(\prod_{i=0}^{n-2}a_{i}\bigg)\cdot x^{n-2} =\sum_{n=1}^{\infty } a_{n}\cdot n\cdot\bigg(\prod_{i=0}^{n-1}a_{i}\bigg)\cdot x^{n-1} =g^{\prime } \left(x\right)\\ &\geqslant&\sum_{n=1}^{\infty } \bigg(\prod_{i=0}^{n-1}a_{i}\bigg)\cdot x^{n-1} =\sum_{n=0}^{\infty } \bigg(\prod_{i=0}^{n}a_{i}\bigg)\cdot x^{n} =g(x)>0. \end{eqnarray*} Thus $g$ is a convex function on $[0,+\infty)$.
For $x\in[0,2]$, it follows that $$ g \left(x\right) \geqslant a_{0}=g_0\left( 2\right)\cdot e\geqslant g_0\left( 2\right) \geqslant g_0 \left( x\right). $$ For $x\in \left( 2,+\infty \right) $, we have $$ \begin{array}{ll} \displaystyle g\left( x\right) \!\!\!&\geqslant g\left( \left[ x\right] \right) \geqslant \prod^{N_{\left[ x\right] }}_{i=0} a_{i}\cdot \left[ x\right]^{N_{\left[ x\right] }} \geqslant \left( a_{N_{\left[x\right] }}\cdot \left[x\right] \right)^{N_{\left[x\right] }} =\left( \left(g_0\left( 1+\left[x\right] \right)\right)^{\frac{1}{N_{\left[x\right] }} }\cdot e^{\frac{1}{\sqrt{N_{\left[x\right] }} } }\right)^{N_{\left[x\right] }} \\ &\displaystyle \geqslant g_0\left( 1+\left[x\right] \right) \geqslant g_0\left(x\right), \end{array} $$ where $\left[x\right] $ represents the maximum integer that does not exceed $x$. Thus $g(x)\geqslant g_0(x)$ for all $x\in[0,+\infty)$. This completes the proof of Lemma \ref{smooth convext increasing function}. \end{proof}
The following theorem is an infinite-dimensional version of \cite[Theorem 4.2.2, p. 84]{Hor90}. \begin{theorem}\label{general LL}
Under Conditions \ref{230424c1}, \ref{230423ass1}, \ref{230423ass2} and \ref{230423ass3}, for any $f=\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t+1 } f_{I,J}dz_{I}\wedge d\overline{z_{J}}\in L^{2}_{(s,t+1)}\left( V,loc \right)$ with $\overline{\partial}f=0$, there exists $u\in L^{2}_{(s,t)}\left( V,loc\right)$ such that $\overline{\partial}u=f$. \end{theorem} \begin{proof} By Remark \ref{230422r1}, we may assume that the plurisubharmonic exhaustion function $\eta$ satisfies $\eta\geqslant 0$ and (\ref{regular condition for eta}). Since $V_{j+1}\setminus V_{j}\subset V_{j+1}\stackrel{\circ}{\subset}V$ (Recall (\ref{230117e1}) for $V_j$, where $j\in \mathbb{N}$), we have $$
0\leqslant\int_{V_{j+1}\setminus V_{j}} \sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\cdot\left| f_{I,J}\right|^{2}\,\mathrm{d}P<\infty,\quad \forall\;j\in\mathbb{N}. $$ Choosing a sequence $\{ b_{j}\}^{\infty }_{j=1}$ of positive numbers such that $$
\sum^{\infty }_{j=1}b_{j} \int_{V_{j+1}\setminus V_{j}} \sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\cdot\left| f_{I,J}\right|^{2}\,\mathrm{d}P<\infty . $$ Let $$
h\left( x\right) \triangleq \begin{cases}\sum\limits^{j}_{i=1} \left| \ln \frac{1}{b_{i} } +\sup\limits_{V_{i+1}\setminus V_{i}} \psi \right| , &j<x\leqslant j+1,j\in\mathbb{N}, \\ 0,&0\leqslant x\leqslant 1,\end{cases} $$ where the function $\psi\in C^{\infty}_{F}(V)$ is given between \eqref{section2} and \eqref{majority function}. Set $$
g_0\left( x\right) \triangleq1+h\left( x\right) +\sup_{V_{x}} \left( 2\sum\limits^{\infty }_{i=1} \left| \partial_{i}\psi \right|^{2} +2e^{\psi }\right),\quad\forall\; x\in[0,+\infty) . $$ Obviously, $g_0$ is an increasing nonnegative function on $[0,+\infty )$. By Lemma \ref{smooth convext increasing function}, there exists a convex, increasing, and real analytic function $g$ on $[0,+\infty )$ such that $$ g^{\prime \prime} \left( x\right) \geqslant g^{\prime } \left( x\right) \geqslant g \left( x\right) \geqslant g_0\left( x\right),\quad\forall\; x\in [0,+\infty). $$ Let \begin{equation}\label{230503e1} \varphi \triangleq g \left( \eta \right).
\end{equation} By the properties of $g$ and $\eta$, noting (\ref{230503e1}), it is easy to see that $\varphi\in C^{2}_{F}\left( V\right) $. Let $w_{1} =\varphi -2\psi $, $w_{2} =\varphi -\psi$ and $w_{3} =\varphi$ as that in \eqref{weight function}. Note that \begin{eqnarray}\label{condtion for D g eta}
g^{\prime } \left( \eta \left( \textbf{z}\right) \right) \geqslant g_0\left( \eta \left( \textbf{z}\right) \right) \geqslant \sup_{V_{\eta \left( \textbf{z}\right) }} \left( 2\sum\limits^{\infty }_{i=1} \left| \partial_{i}\psi \right|^{2} +2e^{\psi }\right) \geqslant 2\sum\limits^{\infty }_{i=1} \left|\partial_{i}\psi \left( \textbf{z}\right) \right|^{2} +2e^{\psi \left( \textbf{z}\right) } ,\quad\forall\;\textbf{z}\in V, \end{eqnarray} and $$ g \left( \eta \left( \textbf{z}\right) \right) \geqslant h\left( \eta \left( \textbf{z}\right) \right) \geqslant \ln \frac{1}{b_{j} } +\sup_{ V_{j+1}\setminus V_{j}} \psi \geqslant \ln \frac{1}{b_{j} } +\psi \left( \textbf{z}\right),\quad \forall\; \textbf{z}\in V_{j+1}\setminus V_{j}, $$ and hence $$ \sup_{ V_{j+1}\setminus V_{j}} e^{-\left( \varphi -\psi \right) }\leqslant b_{j}. $$ Then, \begin{eqnarray*}
&&\sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| J\right| =t+1}c_{I,J} \cdot\int_{V} \left| f_{I,J}\right|^{2} e^{-w_{2}}\,\mathrm{d}P\\
&&=\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| J\right| =t+1} c_{I,J} \cdot\int_{V_{1}} \left| f_{I,J}\right|^{2} \cdot e^{-\left( \varphi -\psi \right) }\,\mathrm{d}P+\sum^{\infty }_{j=1} \sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| J\right| =t+1} c_{I,J} \cdot\int_{V_{j+1}\setminus V_{j}} \left| f_{I,J}\right|^{2}\cdot e^{-\left( \varphi -\psi \right) }\,\mathrm{d}P\\
&&\leqslant \left(\sup_{V_1}e^{-\left( \varphi -\psi \right) }\right)\cdot\sum^{\prime }_{\left| I\right| =s} \sum^{\prime }_{\left| J\right| =t+1}c_{I,J} \cdot \int_{V_{1}} \left| f_{I,J}\right|^{2} \,\mathrm{d}P+\sum^{\infty }_{j=1} b_{j} \int_{V_{j+1}\setminus V_{j}} \sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J} \cdot\left| f_{I,J}\right|^{2}\,\mathrm{d}P<\infty, \end{eqnarray*} which implies that $f\in L^{2}_{(s,t+1)}\left( V,w_{2}\right)$.
On the other hand, for each $n\in\mathbb{N}$, by \eqref{regular condition for eta} and \eqref{condtion for D g eta}, we have $$
\begin{aligned}
\sum_{1\leqslant i,j\leqslant n} \left(\partial_{i} \overline{\partial_{j} }\left( g\left( \eta \right) \right) \right)\cdot \zeta_{i}\cdot\overline{\zeta_{j}}&=g^{\prime \prime } \left( \eta \right) \cdot\left| \sum^{n}_{i=1}\zeta_{i}\cdot \partial_{i}\eta\right|^{2} +g^{\prime } \left( \eta \right)\cdot \sum_{1\leqslant i,j\leqslant n}\left( \partial_{i} \overline{\partial_{j} }\eta\right)\cdot \zeta_{i}\cdot\overline{\zeta_{j}}\\ &\geqslant g^{\prime } \left( \eta \right) \cdot \sum_{1\leqslant i,j\leqslant n}\left(\partial_{i} \overline{\partial_{j} }\eta \right)\cdot\zeta_{i}\cdot\overline{\zeta_{j}}\\
&\geqslant \left( 2\sum\limits^{\infty }_{i=1} \left| \partial_{i} \psi \right|^{2} +2e^{\psi }\right) \cdot\left( \sum^{n}_{i=1} \left| \zeta_{i} \right|^{2}\right) ,\quad\forall\;(\zeta_{1},\cdots,\zeta_n)\in \mathbb{C}^n .
\end{aligned} $$ Recall Definition \ref{definition of T S} for the operators $T$ and $S$, and recall Condition \ref{230423ass2} for $c_0^{s,t}$. Since $\overline{\partial}f=0$, we have $Sf=0$. Hence by Proposition \ref{general Solving d-bar on V}, there exists $u\in L^{2}_{(s,t)}\left( V,w_{1}\right)$ such that $Tu=f$ and \begin{eqnarray}\label{L2 estimation for u}
\sqrt{c_0^{s,t}}\cdot \left| \left| u\right| \right|_{L^{2}_{(s,t)}\left( V,w_{1}\right) } \leqslant \left| \left| f\right| \right|_{L^{2}_{(s,t+1)}\left( V,w_{2}\right) }. \end{eqnarray} Note that for any $r>0$, \begin{eqnarray*}
\sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| J\right| =t}c_{I,L} \cdot\int_{V_r} \left|u_{I,L}\right|^{2} \,\mathrm{d}P
\!\!&\leqslant&\!\!\sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| J\right| =t}c_{I,L} \cdot\int_{V_r} \left|u_{I,L}\right|^{2} \cdot e^{-w_{1}}\cdot e^{w_{1}}\,\mathrm{d}P\\
&\leqslant&\!\!\left(\sup_{V_r}e^{w_{1}}\right)\cdot\left(\sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| J\right| =t}c_{I,L} \cdot\int_{V_r} \left|u_{I,L}\right|^{2} \cdot e^{-w_{1}}\,\mathrm{d}P\right)\\ &<&\!\!\infty, \end{eqnarray*} which implies that $u\in L^{2}_{(s,t)}\left( V_{r},P\right)$. By the conclusion (3) of Proposition \ref{properties on pseudo-convex domain} and the definition of $L^{2}_{(s,t)}\left( V,loc\right)$, we have $u\in L^{2}_{(s,t)}\left( V,loc\right)$. Recalling again the definition of the operator $T$ in Definition \ref{definition of T S}, we conclude that $\overline{\partial}u=f$. The proof of Theorem \ref{general LL} is completed. \end{proof}
In order to extend the \emph{a priori} estimate in \eqref{L2 estimation for u}, we need the following technical lemma which is motivated by \cite[Lemma 41.5, p. 304]{Mujica}.
\begin{lemma}\label{calculus lemma} If $0<x_1<x_2<+\infty$ and $g$ is a non-decreasing real-valued function on $[0,+\infty)$ so that $g(x)=0$ for all $x\in[0,x_2]$, then there exists $G\in C^2(\mathbb{R})$ such that $G(x)=0$ for all $x\leqslant x_1$, $G^{''}(x)\geqslant 0$ for all $x\in[0,+\infty)$, $G(x)\geqslant g(x)$ and $G'(x)\geqslant g(x)$ for all $x\in[0,+\infty)$. \end{lemma} \begin{proof}
Choose $r_0\triangleq0<r_1\triangleq x_1<r_2\triangleq\frac{x_1+x_2}{2}<r_3\triangleq x_2<\cdots<r_j<r_{j+1}<\cdots$ for $j\in\{4,5,\cdots,\}$ so that $\lim\limits_{j\to\infty}r_j=\infty$. For each $j\in\mathbb{N}$, let $\lambda_j\triangleq \sup\limits_{x\in[0,r_j]}g(x)$. Meanwhile, we choose $\{\varphi_j\}_{j=1}^{\infty}\subset C^2(\mathbb{R})$ such that $0\leqslant\varphi_j\leqslant 1$, $\varphi_j'(x)\geqslant 0$ for all $x\in\mathbb{R}$, $\varphi_j(x)= 0$ for all $x \leqslant r_{j-1}$, and $\varphi_j(x)= 1$ for all $x \geqslant r_{j}$. Then $\lambda_1=\lambda_2=\lambda_3=0$. Write \begin{eqnarray*} \Phi \triangleq \lambda_1+\sum_{j=1}^{\infty}(\lambda_{j+1}-\lambda_j)\varphi_j=\sum_{j=3}^{\infty}(\lambda_{j+1}-\lambda_j)\varphi_j. \end{eqnarray*} Then $\Phi(x)=0$ for all $x\leqslant r_2$. If $x\in[r_j,r_{j+1}]$ for some $j\in\mathbb{N}$, it holds that $\varphi_1(x)=\cdots=\varphi_j(x)=1$, $0=\varphi_{j+2}(x)=\cdots $ and hence \begin{eqnarray*} \Phi(x)=\lambda_1+\sum_{k=1}^{j+1}(\lambda_{k+1}-\lambda_k)\varphi_k(x)\geqslant \lambda_1+\sum_{k=1}^{j}(\lambda_{k+1}-\lambda_k)=\lambda_{j+1}\geqslant g(x). \end{eqnarray*} Obviously, $\Phi\in C^2(\mathbb{R})$, $\Phi'(x)\geqslant 0$ and $\Phi(x)\geqslant g(x)$ for all $x\in[0,+\infty)$. Set $d_j\triangleq \sup\limits_{[0,r_j]}\max\{\Phi',g\}$ for each $j\in\mathbb{N}$ and \begin{eqnarray*} h\triangleq d_1+\sum_{j=1}^{\infty}(d_{j+1}-d_j)\varphi_j . \end{eqnarray*} Similarly to the above, we can show that $h\in C^2(\mathbb{R})$, $h(x)\geqslant \max\{\Phi'(x),g(x)\}$ and $h'(x)\geqslant 0$ for all $x\in[0,+\infty)$, $d_1=d_2=0$, and $h(x)=0$ for all $x\leqslant x_1$. Let \begin{eqnarray*} G(x)\triangleq \int_{-\infty}^{x}h(\tau)\,\mathrm{d}\tau,\quad \forall\; x\in \mathbb{R}. \end{eqnarray*} Then, $G\in C^2(\mathbb{R})$, $G(x)\geqslant \int_0^x \Phi'(\tau)\,\mathrm{d}\tau=\Phi(x)\geqslant g(x)$, $G'(x)=h(x)\geqslant g(x),\,G^{''}(x)=h'(x)\geqslant 0$ for all $x\in[0,+\infty)$ and $G(x)=0$ for all $x\leqslant x_1$. The proof of Lemma \ref{calculus lemma} is completed. \end{proof} The following theorem is an infinite-dimensional version of \cite[Lemma 4.4.1, p. 92]{Hor90} (Recall Condition \ref{230423ass2} for $c_0^{s,t}$). \begin{theorem}\label{general L} Suppose that Conditions \ref{230424c1}, \ref{230423ass1}, \ref{230423ass2} and \ref{230423ass3} hold, and the real-valued function $\varphi$ in \eqref{weight function} is chosen so that for each $n\in\mathbb{N}$,
$$
\sum_{1\leqslant i,j\leqslant n}\left( \partial_{i} \overline{\partial_{j} } \varphi\left(\textbf{z}\right) \right)\cdot\zeta_{i}\cdot\overline{\zeta_{j}} \geqslant c \cdot\sum_{1\leqslant i\leqslant n} \left| \zeta_{i}\right|^{2},\quad\forall\; (\textbf{z},\zeta_{1},\cdots,\zeta_n)\in V\times\mathbb{C}^n, $$
where $c$ is a positive, continuous function on $V$ such that $\sup\limits_{ E} c <\infty$ for all $E\stackrel{\circ}{\subset} V $. Then, for any $f=\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t+1 } f_{I,J}dz_{I}\wedge d\overline{z_{J}}\in L^{2}_{(s,t+1)}\left( V,\varphi \right) $ with $\overline{\partial}f=0$ and $\sum^{\prime }\limits_{\left| I\right| =s}\sum^{\prime }\limits_{\left| J\right| =t+1} c_{I,J}\int_{V} \frac{\left| f_{I,J}\right|^{2} }{c} \cdot e^{-\varphi }\,\mathrm{d}P<\infty$, there exists $u\in L^{2}_{(s,t)}\left( V,\varphi \right)$ such that $\overline{\partial}u=f$ and \begin{equation}\label{230423e1}
\sum^{\prime }_{\left| I\right| =s}\sum^{\prime }_{\left| L\right| =t} c_{I,J}\int_{V} \left| u_{I,L}\right|^{2}\cdot e^{-\varphi }\,\mathrm{d}P\leqslant \frac{ 2\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} \frac{\left| f_{I,J}\right|^{2} }{c} \cdot e^{-\varphi }\,\mathrm{d}P }{c_0^{s,t}\cdot(t+1)}. \end{equation} \end{theorem} \begin{proof} As in the proof of Theorem \ref{general LL}, by Remark \ref{230422r1}, we may assume that the plurisubharmonic exhaustion function $\eta$ satisfies $\eta\geqslant 0$ and (\ref{regular condition for eta}).
Similarly to the construction of $h_{\rho}(\cdot)$ at the beginning of the proof of Theorem \ref{general density4}, one can find a sequence of functions $\{h_k\}_{k=1}^{\infty}\subset C^{\infty }\left( \mathbb{R}\right)$ such that for each $k\in\mathbb{N}$, $0\leqslant h_k\leqslant 1$, $h_{k} \left( x\right) =1$ for $ x<k $ and $h_{k } \left( x\right) =0$ for $ x>k+1$, and $\left| h^{\prime }_{k} \right| \leqslant C$ for a positive number $C$ which is independent of $k$.
Let $X_{k}\triangleq h_{k}(\eta)$ for $k\in\mathbb{N}$. Then $X_{k}\in C^{\infty}_{0,F}\left( V\right)$, and $$
\sup_{k\in\mathbb{N}} \;\ln \bigg(1+\sum^{\infty }_{j=1} \left| \overline{\partial_{j} } X_{k }\right|^{2} \bigg) =\sup_{k\in\mathbb{N}} \; \ln \left(1+|h_{k}^{'} (\eta)|^2\cdot\left(\sum^{\infty }_{j=1} \left| \overline{\partial_{j} } \eta\right|^{2}\right)\right) $$ is a locally bounded function on $V$. By Lemma \ref{majority funtion}, there exists $\Psi \in C^{\infty }_{F}\left( V\right)$ such that $$
\ln \left(1+\sum\limits^{\infty }_{j=1} \left| \overline{\partial_{j} } X_{k }\right|^{2} \right)\leqslant \Psi,\quad\forall\; k\in\mathbb{N} $$
and hence $\sum\limits^{\infty }_{j=1} \left| \overline{\partial_{j} } X_{k}\right|^{2} \leqslant e^{\Psi }$ for all $k\in\mathbb{N}$.
For each $a>0$, choose $a_2>a_1>a$ and $\varphi_{a}\in C^{\infty }\left( \mathbb{R} \right)$ such that $0\leqslant \varphi_{a}\leqslant 1 $, $\varphi_{a} \left( x\right) =0$ for $x<a_1$ and $\varphi_{a} \left( x\right) =1$ for $x>a_2.$ Let $\eta_{a} \triangleq\varphi_{a} \left( \eta \right)$ and $\psi_a\triangleq \eta_{a} \cdot \Psi $. It is easy to see that $\psi_a\in C^{2}_{F}\left( V\right)$. For every $k >a_2$, we have $X_{k}\left( \textbf{z}\right) =1$ for all $\textbf{z}\in V_{a_2}$ and $\psi_a \left( \textbf{z}\right) =0$ for all $\textbf{z}\in V_{a_1}$. When $\eta(\textbf{z})\geqslant a_2$, we have $\eta_a(\textbf{z})=1$ and hence $e^{\psi_a (\textbf{z})}=e^{\Psi (\textbf{z})}\geqslant\sum\limits^{\infty }_{j=1} \left| \overline{\partial_{j} } X_{k }(\textbf{z})\right|^{2}$. When $\eta(\textbf{z})< a_2$, we have $e^{\psi_a (\textbf{z})}\geqslant 0=\sum\limits^{\infty }_{j=1} \left| \overline{\partial_{j} } X_{k }(\textbf{z})\right|^{2}$. Hence, for each $k >a_2$,
$$e^{\psi_a (\textbf{z})}\geqslant\sum\limits^{\infty }_{j=1} \left| \overline{\partial_{j} } X_{k }(\textbf{z})\right|^{2},\quad \forall\;\textbf{z}\in V, $$ and $\psi_a$ has the same property as $\psi$ in \eqref{majority function}.
Let $$
h\left( x\right) \triangleq 2\sup_{V_{x}} \psi_a +2\sup_{V_{x}}\left( \sum^{\infty }_{i=1} \left| \partial_{i}\psi_a \right|^{2}\right),\quad x\in[0,+\infty). $$ Then $h\left( x\right) =0$ for all $0\leqslant x\leqslant a_1$ and $h$ is a non-decreasing nonnegative function on $[0,+\infty )$. By Lemma \ref{calculus lemma}, there exists $H\in C^2(\mathbb{R})$ such that $H(x)=0$ for all $x\leqslant a$, $H^{''}(x)\geqslant 0$ for all $x\in[0,+\infty)$, and $H(x)\geqslant h(x)$ and $H'(x)\geqslant h(x)$ for all $x\in[0,+\infty)$. Note that for each $\textbf{z}\in V$, we have \begin{eqnarray*}
H \left( \eta \left( \textbf{z}\right) \right) \geqslant 2\sup_{V_{\eta \left( \textbf{z}\right) }} \psi_a \geqslant 2\psi_a \left( \textbf{z} \right),\quad H^{\prime } \left( \eta \left( \textbf{z}\right) \right) \geqslant 2\sup_{V_{\eta \left( \textbf{z}\right) }} \left(\sum^{\infty }_{i=1} \left| \partial_{i}\psi_a \right|^{2}\right) \geqslant 2\sum^{\infty }_{i=1} \left| \partial_{i}\psi_a \left( \textbf{z}\right) \right|^{2}, \end{eqnarray*} and hence, by (\ref{regular condition for eta}), for each $n\in\mathbb{N}$, the following inequality holds on $V$, $$
H^{\prime } \left( \eta \right) \cdot \sum_{1\leqslant i,j\leqslant n}\left( \partial_{i}\overline{\partial_{j}}\eta\right)\cdot \zeta_{i}\cdot\overline{\zeta_{j}}\geqslant 2\left(\sum^{\infty }_{i=1} \left| \partial_{i}\psi_a \right|^{2} \right)\cdot\left( \sum^{n}_{i=1} \left| \zeta_{i}\right|^{2}\right),\quad \forall\;(\zeta_1,\cdots,\zeta_n)\in\mathbb{C}^n. $$ Then $H(\eta)\in C^2_F(V)$ and $\varphi +H\left( \eta \right)$ enjoys the same property as $\varphi$. Moreover, for each $n\in\mathbb{N}$, the following inequality holds on $V$, \begin{equation}\label{230424e1} \begin{aligned} &\sum_{1\leqslant i,j\leqslant n} \left(\partial_{i} \overline{\partial_{j} } (\varphi+H\left( \eta \right)) \right)\cdot\zeta_{i}\cdot\overline{\zeta_{j}}\\ &=\sum_{1\leqslant i,j\leqslant n} \left(\partial_{i} \overline{\partial_{j} } \varphi\right)\cdot\zeta_{i}\cdot\overline{\zeta_{j}} +H^{\prime \prime }\left( \eta \right) \cdot\sum_{1\leqslant i,j\leqslant n}\partial_{i} \eta \cdot\overline{\partial_{j} } \eta\cdot \zeta_{i}\cdot\overline{\zeta_{j}} +H^{\prime }\left( \eta\right) \cdot \sum_{1\leqslant i,j\leqslant n} \left(\partial_{i}\overline{\partial_{j}}\eta \right)\cdot\zeta_{i}\cdot\overline{\zeta_{j}}\\
&\geqslant c\cdot\sum^{n}_{i=1} \left| \zeta_{i}\right|^{2}+ 2\cdot\left(\sum^{\infty }_{i=1} \left| \partial_{i}\psi_a \right|^{2} \right)\cdot\left(\sum^{n}_{i=1} \left| \zeta_{i}\right|^{2}\right),\quad \forall\;(\zeta_1,\cdots,\zeta_n)\in\mathbb{C}^n. \end{aligned} \end{equation}
Let $$ \widetilde{w_{1}}\triangleq\varphi +H\left( \eta \right) -2\psi_a,\quad\widetilde{w_{2}}\triangleq\varphi +H\left( \eta \right) -\psi_a,\quad\widetilde{w_{3}}\triangleq\varphi +H\left( \eta \right). $$ Suppose that $\widetilde{T}$ and $\widetilde{S}$ are the corresponding operators in Definition \ref{definition of T S} with $w_1, w_2$ and $w_3$ replaced respectively by $\widetilde{w_1},\widetilde{w_2}$ and $\widetilde{w_3}$. Then all properties for $T$ and $S$ also hold for $\widetilde{T}$ and $\widetilde{S}$. Now, in view of \eqref{general inequality for smooth functions} (and noting that for the proof of \eqref{general inequality for smooth functions} the condition (\ref{230419e12}) is NOT used) and using (\ref{230424e1}), we obtain that \begin{equation}\label{general formula 14}
c_0^{s,t}\cdot(t+1)\cdot\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} c\cdot\left| g_{I,J}\right|^{2} \cdot e^{-\left( \varphi +H\left( \eta \right) \right) }dP\leqslant 2\left| \left| \widetilde{T}^{\ast }g\right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V,\widetilde{w_{1}}\right) } +\left| \left| \widetilde{S}g\right| \right|^{2}_{L^{2}_{\left( s,t+2\right) }\left( V,\widetilde{w_{3}}\right) }
\end{equation}
for every $g=\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t+1 }g_{I,J}dz_{I}\wedge d\overline{z_{J}}\in D_{\widetilde{T}^{\ast}}\bigcap D_{\widetilde{S}} $ satisfying that there exists $k\in \mathbb{N}$ such that $g_{I,J}=0$ for all strictly increasing multi-indices $I$ and $J$ with $\left| I\right| =s,\left| J\right| =t+1$ and $\max \left\{ I\bigcup J\right\} >k$, and $g_{I,J}\in C^{2}_{0,F}\left( V\right)$ for all strictly increasing multi-indices $I$ and $J$ with $\left| I\right| =s,\left| J\right| =t+1$ and $\max \left\{ I\bigcup J\right\} \leqslant k$. By Proposition \ref{general cut-off density3} and Theorem \ref{general density4}, for every $g\in D_{\widetilde{S}}\cap D_{\widetilde{T}^{\ast }}$, there exists $$ \{g_{n}\}_{n=1}^{\infty}\subset D_{\widetilde{T}^{\ast }}\cap D_{\widetilde{S}} $$
satisfying that, for each $n\in\mathbb{N}$, there exists $k_{n}\in \mathbb{N}$ such that $(g_{n})_{I,J}=0$ for all strictly increasing multi-indices $I$ and $J$ with $\left| I\right| =s,\left| J\right| =t+1$ and $\max \left\{ I\bigcup J\right\} >k_{n}$, $(g_{n})_{I,J}\in C^{2}_{0,F}\left( V\right)$ for all strictly increasing multi-indices $I$ and $J$ with $\left| I\right| =s,\left| J\right| =t+1,\max \left\{ I\bigcup J\right\}\leqslant k_{n}$, and $$ \begin{aligned}
&\lim_{n\rightarrow \infty }\left( \left| \left| \sqrt{c}\cdot e^{-\frac{\psi_a }{2} }\cdot g_{n}-\sqrt{c}\cdot e^{-\frac{\psi_a }{2} }\cdot g\right| \right|_{L^{2}_{\left( s,t+1\right) }\left( V_{r},\widetilde{w_{2}}\right) } +\left| \left| \widetilde{T}^{\ast }g_{n}-\widetilde{T}^{\ast }g\right| \right|_{L^{2}_{\left( s,t\right) }\left( V_{r},\widetilde{w_{1}}\right)} +\left| \left| \widetilde{S}g_{n}-\widetilde{S}g\right| \right|_{L^{2}_{\left( s,t+2\right) }\left( V_{r},\widetilde{w_{3}}\right)}\right) \\ & =0,\qquad \forall\;r>0. \end{aligned}
$$ By \eqref{general formula 14}, for each $n\in\mathbb{N}$, we have $$
c_0^{s,t}\cdot(t+1)\cdot\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V_r} c\cdot\left| (g_n)_{I,J}\right|^{2} \cdot e^{-\left( \varphi +H\left( \eta \right) \right) }\,\mathrm{d}P\leqslant 2\left| \left| \widetilde{T}^{\ast }g_n\right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V,\widetilde{w_{1}}\right) } +\left| \left| \widetilde{S}g_n\right| \right|^{2}_{L^{2}_{\left( s,t+2\right) }\left( V,\widetilde{w_{3}}\right) }. $$ Letting $n\to\infty$ in the above inequality, we obtain that $$ \begin{aligned}
&c_0^{s,t}\cdot(t+1)\cdot\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V_{r}} c\cdot\left| g_{I,J}\right|^{2} \cdot e^{-\left( \varphi +H \left( \eta \right) \right) }\,\mathrm{d}P\\
&\leqslant 2\left| \left|\widetilde{T}^{\ast }g\right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V,\widetilde{w_{1}}\right) } +\left| \left| \widetilde{S}g\right| \right|^{2}_{L^{2}_{\left( s,t+2\right) }\left( V,\widetilde{w_{3}}\right) },\quad \forall\; g\in D_{\widetilde{S}}\cap D_{\widetilde{T}^{\ast }}. \end{aligned} $$ By the arbitrariness of $r$, we then have \begin{equation}\label{a priori estimate} \begin{aligned}
&c_0^{s,t}\cdot(t+1)\cdot\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} c\cdot\left| g_{I,J}\right|^{2} \cdot e^{-\left( \varphi +H \left( \eta \right) \right) }\,\mathrm{d}P\\
&\leqslant 2\left| \left|\widetilde{T}^{\ast }g\right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V,\widetilde{w_{1}}\right) } +\left| \left|\widetilde{S}g\right| \right|^{2}_{L^{2}_{\left( s,t+2\right) }\left( V,\widetilde{w_{3}}\right) },\quad \forall\; g\in D_{\widetilde{S}}\cap D_{\widetilde{T}^{\ast }}. \end{aligned} \end{equation}
By $H \left( \eta \right) \geqslant 2\psi_a$, we have $e^{-\varphi -H \left( \eta \right) +\psi_a }\leqslant e^{-\varphi -\frac{H\left( \eta \right) }{2} }$. Since $\widetilde{w_{2}}=\varphi+H(\eta)-\psi_a\geqslant \varphi$ and $f\in L^{2}_{(s,t+1)}\left( V,\varphi \right)$ with $\overline{\partial}f=0$, we have $f\in L^{2}_{(s,t+1)}\left( V,\widetilde{w_{2}} \right)$ and $f\in N_{\widetilde{S}}$. Hence, for every $g=\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t+1 }g_{I,J}dz_{I}\wedge d\overline{z_{J}}\in D_{\widetilde{S}}\bigcap D_{\widetilde{T}^{\ast }}$, it holds that
\begin{equation}
\begin{aligned}
&\left| \left( g,f\right)_{L^{2}_{(s,t+1)}\left( V,\widetilde{w_{2}}\right) } \right|^{2} =\bigg|\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} f_{I,J}\cdot\overline{g_{I,J}}\cdot e^{-\widetilde{w_{2}}}\,\mathrm{d}P\bigg|^{2}\\
&\leqslant \bigg|\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} |f_{I,J}\cdot\overline{g_{I,J}}| \cdot e^{-\varphi -H \left( \eta \right) +\psi_a }\,\mathrm{d}P\bigg|^{2}\\
&\leqslant \bigg|\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} |f_{I,J}\cdot\overline{g_{I,J}}|\cdot e^{-\varphi -\frac{H\left( \eta \right) }{2} }\,\mathrm{d}P\bigg|^{2}\\
&\leqslant \left(\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} \frac{\left| f_{I,J}\right|^{2} }{c}\cdot e^{-\varphi }\,\mathrm{d}P\right)\cdot \left(\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} c\cdot\left| g_{I,J}\right|^{2}\cdot e^{-\left( \varphi +H\left( \eta \right) \right) }\,\mathrm{d}P\right)\\
&\leqslant \left(2\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} \frac{\left| f_{I,J}\right|^{2} }{c} \cdot e^{-\varphi }\,\mathrm{d}P\right)\cdot \frac{\left(\left| \left|\widetilde{T}^{\ast }g\right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V,\widetilde{w_{1}}\right) } +\left| \left|\widetilde{S}g\right| \right|^{2}_{L^{2}_{\left( s,t+2\right) }\left( V,\widetilde{w_{3}}\right) }\right)}{c_0^{s,t}\cdot(t+1)},\label{general meidiation inequality}
\end{aligned} \end{equation} where the third inequality follows from Cauchy-Schwarz inequality and the fourth inequality follows from \eqref{a priori estimate}.
If $g\in N^{\bot }_{\widetilde{S}}$, by $R_{\widetilde{T}}\subset N_{\widetilde{S}}$, we have $ N_{\widetilde{S}}^{\perp}\subset R_{\widetilde{T}}^{\perp}=N_{\widetilde{T}^{\ast }}$ (where the equality $R_{\widetilde{T}}^{\perp}=N_{\widetilde{T}^{\ast }}$ follows from \cite[Exercise 2, p. 178]{Kra}), and hence $g\in N_{\widetilde{T}^{\ast }}$, which gives $\widetilde{T}^{\ast }g=0$. Recall that $f\in N_{\widetilde{S}}$, which, combined with $g\in N^{\bot }_{\widetilde{S}}$, gives $\left( g,f\right)_{L^{2}_{(s,t+1)}\left( V,\widetilde{w_{2}}\right) }=0$.
If $g\in N_{\widetilde{S}} \bigcap D_{\widetilde{T}^{\ast }}$, by \eqref{general meidiation inequality}, we have \begin{eqnarray}
\left| \left( g,f\right)_{L^{2}_{(s,t+1)}\left( V,\widetilde{w_{2}}\right) } \right|^{2} \leqslant \left(2\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} \frac{\left| f_{I,J}\right|^{2} }{c}\cdot e^{-\varphi }\,\mathrm{d}P\right)\cdot \frac{\left| \left|\widetilde{T}^{\ast }g\right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V,\widetilde{w_{1}}\right) }}{c_0^{s,t}\cdot(t+1)}.\label{general second inequality} \end{eqnarray}
For any $g\in D_{\widetilde{T}^{\ast }}$, write $g=g_{1}+g_{2}$, where $ g_{1}\in N^{\bot }_{\widetilde{S}}$ and $g_{2}\in N_{\widetilde{S}}$. Since $N^{\bot }_{\widetilde{S}}\subset R_{\widetilde{T}}^{\perp}=N_{{\widetilde{T}}^{\ast }}\subset D_{{\widetilde{T}}^*}$ and $g\in D_{{\widetilde{T}}^{\ast }}$, we have $g_{1},g_{2}\in D_{{\widetilde{T}}^{\ast }}$ and hence $$
\begin{aligned}
\left| \left( g,f\right)_{L^{2}_{(s,t+1)}\left( V,\widetilde{w_{2}}\right) } \right|^{2} &=\left| \left( g_{2},f\right)_{L^{2}_{(s,t+1)}\left( V,\widetilde{w_{2}}\right) } \right|^{2} \\&\leqslant \left(2\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} \frac{\left| f_{I,J}\right|^{2} }{c}\cdot e^{-\varphi }\,\mathrm{d}P\right)\cdot \frac{\left| \left| \widetilde{T}^{\ast }g_2\right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V,\widetilde{w_{1}}\right)} }{c_0^{s,t}\cdot(t+1)} \\
&=\left(2\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} \frac{\left| f_{I,J}\right|^{2} }{c}\cdot e^{-\varphi }\,\mathrm{d}P\right)\cdot \frac{\left| \left| \widetilde{T}^{\ast }g-\widetilde{T}^{\ast }g_1\right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V,\widetilde{w_{1}}\right) }}{c_0^{s,t}\cdot(t+1)} \\
&=\left(2\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} \frac{\left| f_{I,J}\right|^{2} }{c}\cdot e^{-\varphi }\,\mathrm{d}P\right)\cdot \frac{\left| \left| \widetilde{T}^{\ast }g\right| \right|^{2}_{L^{2}_{\left( s,t\right) }\left( V,\widetilde{w_{1}}\right) }}{c_0^{s,t}\cdot(t+1)} ,
\end{aligned} $$
where the first inequality follows from \eqref{general second inequality} and the last equality follows from the fact that $\widetilde{T}^{\ast }g_{1}=0$. If $\sum^{\prime }\limits_{\left| I\right| =s}\sum^{\prime }\limits_{\left| J\right| =t+1}c_{I,J} \int_{V} \frac{\left| f_{I,J}\right|^{2} }{c}\cdot e^{-\varphi }\,\mathrm{d}P<\infty$, by the Hahn-Banach theorem, there exists $u_a\in L^{2}_{(s,t)}\left( V,\widetilde{w_{1}}\right)$ such that $\left( g,f\right)_{L^{2}_{(s,t+1)}\left( V,\widetilde{w_{2}}\right) } =\left( \widetilde{T}^*g,u_a\right)_{L^{2}_{(s,t)}\left( V,\widetilde{w_{1}}\right) } $ for all $g\in D_{\widetilde{T}^*}$ and $$
\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t} c_{I,L}\int_{V} \left| (u_a)_{I,L} \right|^{2} \cdot e^{-\widetilde{w_{1}}}\,\mathrm{d}P\leqslant \frac{ 2\sum^{\prime }\limits_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} \frac{\left| f_{I,J}\right|^{2} }{c}\cdot e^{-\varphi }\,\mathrm{d}P }{c_0^{s,t}\cdot(t+1)}. $$ Since $\widetilde{T}^{**}=\widetilde{T}$ (e.g., \cite[Exercise 2, p. 178]{Kra}), we conclude that $\overline{\partial}u_a=\widetilde{T}u_a=f$. Since $\widetilde{w_{1}}(\textbf{z})=\varphi(\textbf{z})$ for all $\textbf{z}\in V_{a}$, we have $$
\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t} c_{I,L}\int_{V_{a}} \left| \left( u_{a}\right)_{I,L} \right|^{2} \cdot e^{-\varphi }\,\mathrm{d}P\leqslant \frac{ 2\sum^{\prime }\limits_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} \frac{\left| f_{I,J}\right|^{2} }{c} \cdot e^{-\varphi }\,\mathrm{d}P }{c_0^{s,t}\cdot(t+1)} . $$ For every $b\geqslant a$, by a similar procedure, we can find $ u_{b}\in L^{2}_{(s,t)}\left( V,\widehat{w_{1}} \right)$ (where $\widehat{w_{1}}$ is the corresponding weight function for $b$) such that $\overline{\partial}u_{b}=f$ and \begin{eqnarray}\label{general estimation for ub}
\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t} c_{I,L}\int_{V_{b}} \left| \left( u_{b}\right)_{I,L} \right|^{2} \cdot e^{-\varphi }\,\mathrm{d}P\leqslant \frac{ 2\sum^{\prime }\limits_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} \frac{\left| f_{I,J}\right|^{2} }{c} \cdot e^{-\varphi }\,\mathrm{d}P }{c_0^{s,t}\cdot(t+1)}. \end{eqnarray} Note that for each $b\in[a,+\infty)$, by Corollary \ref{cylinder function is dense}, $L^ {2}_ {(s, t)} \left (V_{b}, \varphi \right) $ is a separable Hilbert space.
By the Banach-Alaoglu Theorem (e.g., \cite[Theorem 3.15 at p. 66 and Theorem 3.17 at p. 68]{Rud91}), there exists $\{ b_{j}\}^{\infty }_{j=1} \subset \left( a,+\infty \right)$ and $\tilde{u}_{a}\in L^{2}_{(s,t)}\left( V_{a},\varphi \right)$ such that $\lim\limits_{j\rightarrow \infty } b_{j}=\infty $ and $\lim\limits_{j\rightarrow \infty } u_{b_{j}}\rightharpoonup \tilde{u}_{a} $ weakly in $L^{2}_{(s,t)}\left( V_{a},\varphi \right)$. For $a+1$, by the same argument, there exists a subsequence $\{b_{j_i}\}_{i=1}^{\infty}$ of $\{ b_{j}\}^{\infty }_{j=1}$ and $\tilde{u}_{a+1}\in L^{2}_{(s,t)}\left( V_{a+1},\varphi \right)$ such that $\lim\limits_{i\rightarrow \infty }u_{b_{j_i}}\rightharpoonup \tilde{u}_{a+1} $ weakly in $L^{2}_{(s,t)}\left( V_{a+1},\varphi \right)$. By induction and the diagonal process, there exists $\{\tilde{b}_j\}_{j=1}^{\infty}\subset (a,+\infty)$ and $\tilde{u}_{a+k}\in L^{2}_{(s,t)}\left( V_{a+k},\varphi \right)$ for each nonnegative integer $k$ such that \begin{itemize} \item[(1)]$\lim\limits_{j\rightarrow \infty } \tilde{b}_{j}=\infty $; \item[(2)]$\lim\limits_{j\rightarrow \infty }u_{\tilde{b}_{j}}= \tilde{u}_{a+k} $ weakly in $L^{2}_{(s,t)}\left( V_{a+k},\varphi \right)$ for each nonnegative integer $k$;
\item[(3)]$(\tilde{u}_{a+k_1})_{I,L}(\textbf{z}) =(\tilde{u}_{a+k_2})_{I,L}(\textbf{z})$ for all $\textbf{z}\in V_{a+k_2}$, positive integers $k_1$ and $k_2$ with $k_1\geqslant k_2$, and strictly increasing multi-indices $I$ and $L$ with $\left| I\right| =s$ and $\left| L\right| =t$. \end{itemize}
Define $u_{I,J}(\textbf{z})\triangleq (\tilde{u}_{a+k})_{I,L}(\textbf{z})$ for $\textbf{z}\in V_{a+k}$, $k\in\mathbb{N}$ and strictly increasing multi-indices $I$ and $L$ with $\left| I\right| =s$ and $\left| L\right| =t$. Set $u=\sum\limits^{\prime }_{\left| I\right| =s,\left| L\right| =t }u_{I,L}dz_{I}\wedge d\overline{z_{L}}$. Then $u\in L^{2}_{(s,t)}\left( V,loc\right)$ and we shall prove that $\overline{\partial}u=f$.
Since $\overline{\partial}u_{\tilde{b}_{j}}=f$ for each $j\in\mathbb{N}$, for every $\phi \in C^{\infty }_{0,F}\left( V\right)$ and strictly increasing multi-indices $I$ and $J$ with $\left| I\right| =s$ and $\left| J\right| =t+1$, we have $$
\left( -1\right)^{s+1} \int_{V} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| L\right| =t} \varepsilon^{J}_{iL}\cdot ( u_{\tilde{b}_{j}} )_{I,L} \cdot\overline{\delta_{i} \phi }\,\mathrm{d}P=\int_{V} f_{I,J}\cdot\overline{\phi }\,\mathrm{d}P. $$ Since $\hbox{\rm supp$\,$} \phi \stackrel{\circ}{\subset} V$, by the conclusion (3) of Proposition \ref{properties on pseudo-convex domain}, there exists $j_{0}\in\mathbb{N}$ such that $\hbox{\rm supp$\,$} \phi \subset V_{a+j_{0}}$. Hence, for $j>j_{0}$, we have $$
\int_{V} f_{I,J}\cdot\overline{\phi }\,\mathrm{d}P=\left( -1\right)^{s+1} \int_{V_{a+j_{0}}} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| L\right| =t} \varepsilon^{J}_{iL}\cdot ( u_{\tilde{b}_{j}})_{I,L}\cdot \overline{\delta_{i} \phi } \,\mathrm{d}P. $$ Letting $j\rightarrow \infty$ in the above equality, we arrive at $$
\int_{V} f_{I,J}\cdot\overline{\phi }\,\mathrm{d}P=\left( -1\right)^{s+1} \int_{V_{a+j_{0}}} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| L\right| =t} \varepsilon^{J}_{iL}\cdot u_{I,L}\cdot\overline{\delta_{i} \phi } \,\mathrm{d}P=\left( -1\right)^{s+1} \int_{V} \sum^{\infty }_{i=1} \sum^{\prime }_{\left| L\right| =t} \varepsilon^{J}_{iL}\cdot u_{I,L}\cdot\overline{\delta_{i} \phi } \,\mathrm{d}P, $$ which implies that $\overline{\partial}u=f$. Since for each $k\in\mathbb{N}$, $\lim\limits_{j\rightarrow \infty }u_{\tilde{b}_{j}}= \tilde{u}_{a+k} $ weakly in $L^{2}_{(s,t)}\left( V_{a+k},\varphi \right)$, by \eqref{general estimation for ub}, we conclude that $$
\left| \left| u\right| \right|^{2}_{L^{2}_{(s,t)}\left( V_{a+k},\varphi \right) } \leqslant \varliminf_{j\rightarrow \infty } || u_{\tilde{b}_{j}}||^{2}_{L^{2}_{(s,t)}\left( V_{a+k},\varphi \right) } \leqslant\frac{ 2\sum^{\prime }\limits_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} \frac{\left| f_{I,J}\right|^{2} }{c}\cdot e^{-\varphi }\,\mathrm{d}P }{c_0^{s,t}\cdot(t+1)}. $$ Letting $k\to\infty$ in the above, we obtain the desired estimate (\ref{230423e1}). The proof of Theorem \ref{general L} is completed. \end{proof}
Finally, we show the following infinite-dimensional analogue of \cite[Theorem 4.4.2, p. 94]{Hor90}. \begin{corollary}\label{general Theorem 4.4.2Hor90} Suppose that Conditions \ref{230424c1}, \ref{230423ass1}, \ref{230423ass2} and \ref{230423ass3} hold, and the real-valued function $\varphi$ in \eqref{weight function} is chosen so that for each $n\in\mathbb{N}$,
$$ \sum_{1\leqslant i,j\leqslant n}\left( \partial_{i} \overline{\partial_{j} } \varphi\left(\textbf{z}\right)\right)\cdot\zeta_{i}\cdot\overline{\zeta_{j}} \geqslant 0,\quad\forall\; (\textbf{z},\zeta_{1},\cdots,\zeta_n)\in V\times\mathbb{C}^n. $$
Then, for any $f=\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t+1 }f_{I,J}dz_{I}\wedge d\overline{z_{J}}\in L^{2}_{(s,t+1)}\left( V,\varphi \right)$ with $\overline{\partial}f=0$, there exists $u=\sum\limits^{\prime }_{\left| I\right| =s,\left| L\right| =t }u_{I,L}dz_{I}\wedge d\overline{z_{L}}$ $\in L^{2}_{(s,t)}\left( V,\varphi \right)$ such that $\overline{\partial}u=f$ and \begin{equation}\label{230423e2}
\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t} c_{I,J}\int_{V} \left( 1+\left| \left| \textbf{z}\right| \right|^{2}_{\ell^{2} } \right)^{-2} \cdot \left| u_{I,L}\right|^{2}\cdot e^{-\varphi }\,\mathrm{d}P\leqslant \frac{\sum^{\prime }\limits_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} \left| f_{I,J}\right|^{2} \cdot e^{-\varphi }\,\mathrm{d}P}{c_0^{s,t}\cdot(t+1)}. \end{equation} Furthermore, if $V$ is bounded, then \begin{equation}\label{230423e31}
\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t} c_{I}\int_{V} \left| u_{I,L}\right|^{2}\cdot e^{-\varphi }\,\mathrm{d}P\leqslant \left( 1+\sup\limits_{\textbf{z}\in V} \left| \left| \textbf{z}\right| \right|^{2}_{\ell^{2} } \right)^{2} \cdot \frac{\sum^{\prime }\limits_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} \left| f_{I,J}\right|^{2} \cdot e^{-\varphi }\,\mathrm{d}P}{c_0^{s,t}\cdot(t+1)}. \end{equation} \end{corollary} \begin{proof}
Let $\tilde{\varphi } \left( \textbf{z}\right)\triangleq\varphi \left(\textbf{z}\right) +2\ln \left( 1+\left| \left| \textbf{z}\right| \right|^{2}_{\ell^{2} } \right)$ for $\textbf{z}\in V$. Then $\tilde{\varphi }\in C^{2}_{F}\left( V\right)$ and for each $n\in\mathbb{N}$, it hold that $$
\begin{aligned}
\sum_{1\leqslant i,j\leqslant n}\left( \partial_{i}\overline{\partial_{j}}\tilde{\varphi }\left( \textbf{z}\right) \right)\cdot \zeta_{i}\cdot\overline{\zeta_{j}}&\geqslant 2\sum_{1\leqslant i,j\leqslant n} \left(\partial_{i} \overline{\partial_{j} } \ln \left( 1+\left| \left| \textbf{z}\right| \right|^{2}_{\ell^{2} } \right) \right)\cdot\zeta_{i}\cdot\overline{\zeta_{j}} \\
&= 2\sum_{1\leqslant i,j\leqslant n} \frac{z_{j}\cdot\overline{z_{i}} }{\left( 1+\left| \left| \textbf{z}\right| \right|^{2}_{\ell^{2} } \right)^{2} } \cdot\zeta_{i}\cdot\overline{\zeta_{j}} + \frac{2}{1+\left| \left| \textbf{z}\right| \right|^{2}_{\ell^{2} } } \cdot\left(\sum_{i=1}^{n}\left| \zeta_{i}\right|^{2} \right) \\
&\geqslant \frac{2}{\left( 1+\left| \left| \textbf{z}\right| \right|^{2}_{\ell^{2}} \right)^{2} }\cdot\left( \sum^{n}_{i=1} \left| \zeta_{i}\right|^{2}\right),\quad\forall\; (\zeta_{1},\cdots,\zeta_n)\in \mathbb{C}^n,\,\textbf{z}=(z_i)_{i=1}^{\infty}\in V,
\end{aligned} $$ and \begin{eqnarray*}
\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} \left| f_{I,J}\right|^{2} \cdot e^{-\tilde{\varphi } }\,\mathrm{d}P
&\leqslant &\sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} \left( 1+\left| \left| \textbf{z}\right| \right|^{2}_{\ell^2} \right)^{2} \cdot \left| f_{I,J}\right|^{2} \cdot e^{-\tilde{\varphi } }\,\mathrm{d}P\\
&=& \sum^{\prime }_{\left| I\right| =s,\left| J\right| =t+1} c_{I,J}\int_{V} \left| f_{I,J}\right|^{2} \cdot e^{-\varphi }\,\mathrm{d}P\\ &<&\infty. \end{eqnarray*}
Thus $f=\sum\limits^{\prime }_{\left| I\right| =s,\left| J\right| =t+1 }f_{I,J}dz_{I}\wedge d\overline{z_{J}}\in L^{2}_{(s,t+1)}\left( V,\tilde{\varphi } \right) $. By Theorem \ref{general L}, there exists $u=\sum\limits^{\prime }_{\left| I\right| =s,\left| L\right| =t }u_{I,L}dz_{I}\wedge d\overline{z_{L}}\in L^{2}_{(s,t)}\left( V,\tilde{\varphi} \right)\subset L^{2}_{(s,t)}\left( V,loc\right)$ such that $\overline{\partial}u=f$ and $$
\sum^{\prime }_{\left| I\right| =s,\left| L\right| =t} c_{I,L}\int_{V} \left| u_{I,L}\right|^{2}\cdot e^{-\tilde{\varphi } }\,\mathrm{d}P\leqslant \frac{\sum^{\prime }\limits_{\left| I\right| =s,\left| J\right| =t+1} c_{I,L}\int_{V} \left( 1+\left| \left| \textbf{z}\right| \right|^{2}_{\ell^2} \right)^{2} \cdot \left| f_{I,J}\right|^{2}\cdot e^{-\tilde{\varphi } }\,\mathrm{d}P}{c_0^{s,t}\cdot(t+1)}<\infty, $$ which gives (\ref{230423e2}).
Clearly, if $V$ is bounded, then we have the estimate (\ref{230423e31}). This completes the proof of Corollary \ref{general Theorem 4.4.2Hor90}. \end{proof}
\section*{References}
\end{document} | arXiv |
will be true if $K\to\infty$.
The right hand we know is NOT the right tri-nomial develop, so this is the absurdum we are looking for.
The reason why it will works for $n=2$ and not for $n>2$ can be summarized as: it's due to mixed terms of higher degree (from 2).
And a short cut will be: first derivate = curve.
It was a very hard work for me (8 years long)... I hope it's close to the full stop (for many reasons).
So it's clear that the right hand is NOT the right tri-nomial develop, so this is the absurdum we are looking for.
I already did another trip, just doble longer, where will be clear that assuming $A^=C^n-B^n$ for true in the integers there is no continuty if we assume, and computate in this way from right (+1/K) or from the left side (-1/K), since the two limits, once reduced in this way are different.
I will check the computation to avoid errors, and I've also to check witch of the possible cooling, or warming, is the shortest one to show the absurdum.
So if all computation is right I'm alowd to say that this prove that the initial equation must be wrong ? | CommonCrawl |
\begin{document}
\pagestyle{headings}
\title{Consensus-Based Distributed Feature Selection for Inductive Logic Programming Applications}
\author{Haimonti Dutta and Ashwin Srinivasan} \date{27 April, 2014}
\maketitle
\section*{Information About this Paper}
\begin{description} \item[Main contribution of the paper.]
In this paper, we describe a consensus-based learning algorithm for the discovery of features in ILP applications using a network of computational units. Our algorithm assumes that nodes in a distributed environment can independently sample features from a large feature space, learn local models and estimate stochastic gradients of their loss functions. They share information with neighbours to update their view of the global model. For a category of models (those with convex loss functions), we prove that the nodes converge to a consensus model. Controlled experiments on synthetic data suggest that the consensus model that results from the distributed approach is comparable to that obtained with a centralised approach that requires substantially more time and computational effort. Some experimental results with real-world data are also provided. These show performances that are comparable to the best reported in the literature, suggesting that for problems that have been considered by ILP, the models we construct are not overly restrictive. Finally, we provide arguments supporting the conjecture that the consensus-based approach will always converge to model that is within some small error bound of the model from a centralised approach with the same number of features. \item[What evidence is provided.]
Evidence is provided in the form a theoretical proof of convergence
and empirical results on synthetic and real-world data. \item[Related work.] The paper has an extensive discussion of related
work.
\end{description}
\end{document} | arXiv |
\begin{document}
\thispagestyle{empty}
\setlength{\headsep}{2cm} {{\centering
{\huge{{\textbf{Limiting Distributions of Scaled}}}
{\huge{{\textbf{Eigensections in a GIT-Setting}}}}
{\LARGE{\scshape{Daniel Berger}}}
}}
\begin{abstract} Let $\mathrm{\mathbf{L}\rightarrow \mathbf{X}}$ be a base point free $\mathrm{\mathbb{T}=T^{\mathbb{C}}}$-linearized hermitian line bundle over a compact variety $\mathrm{\mathbf{X}}$ where $\mathrm{T=\left(S^{1}\right)^{m}}$ is a real torus. The main focus of this paper is to describe the asymptotic behavior of a certain class of sequences $\mathrm{\left(s_{n}\right)_{n}}$ of $\mathrm{\mathbb{T}}$-eigensections $\mathrm{s_{n}\in H^{0}\left(\mathbf{X},\mathbf{L}^{n}\right)}$ as $\mathrm{n\rightarrow \infty}$, introduced by {\scshape{Shiffman, Tate}} and {\scshape{Zelditch}}, and its connection to the geometry of the Hilbert quotient $\mathrm{\pi\!:\!\bf{X}^{ss}_{\xi}\rightarrow \mathbf{X}^{ss}_{\xi}/\!\!/\mathbb{T}}$ where $\mathrm{\xi\in \mathfrak{t}^{*}}$.
Using these sequences $\mathrm{\left(s_{n}\right)_{n}}$ we will first define a naturally associated sequence of probability measures $\mathrm{\left(\bm{\nu}_{n}\left(y\right)\right)_{n}}$ on each fiber $\mathrm{\pi^{-1}\left(y\right)}$ of the Hilbert quotient where $\mathrm{y}$ varies in a Zariski dense subset $\mathrm{\mathbf{Y}_{0}}$ of the base $\mathrm{\mathbf{X}^{ss}_{\xi}/\!\!/\mathbb{T}}$. In the main part of this paper we will then show that $\mathrm{\left(\bm{\nu}_{n}\left(y\right)\right)_{n}}$ converges uniformly over $\mathrm{\mathbf{Y}_{0}}$ to a Dirac fiber measure whose support is completely determined by the hermitian bundle metric $\mathrm{h}$ and the asymptotic geometry of the rescaled weight vectors $\mathrm{\left(n^{-1}\xi_{n}\right)_{n}}$ given by the initial sequence $\mathrm{\left(s_{n}\right)_{n}}$.
The essential step of the proof is based on the work of {\scshape{D.\! Barlet}} whose results provide us with the construction of an equivariant, dimensional-theoretical {\itshape{flattening}} $\mathrm{\Pi\!:\!\widetilde{\bf{\,X\,}}\rightarrow \widetilde{\bf{\,Y\,}}}$ of the Hilbert quotient $\mathrm{\pi\!:\!\bf{X}^{ss}_{\xi}\rightarrow \mathbf{X}^{ss}_{\xi}/\!\!/\mathbb{T}}$ which turns out to be crucial in order to guarantee uniform estimates over all of $\mathrm{\mathbf{Y}_{0}}$.
\end{abstract}
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\setcounter{page}{1} \section{Introduction and Statement of Results} The aim of this thesis is to examine the asymptotic geometry of a certain class of sequences of eigensections of a line bundle by describing the convergence properties of a naturally associated measure sequence. In this discussion, we let $\mathrm{\mathbf{X}}$ be a projective, normal, purely m-dimensional variety and $\mathrm{\mathbb{T}=T^{\mathbb{C}}}$ a complex torus with the unique maximal compact subgroup $\mathrm{T=\big(S^{1}\big)^{m}}$. We equip $\mathrm{\mathbf{X}}$ with an algebraic action $\mathrm{\phi\!:\!\mathbb{T}\times \mathbf{X}\rightarrow \mathbf{X}}$ of the complex torus $\mathrm{\mathbb{T}}$ and assume $\mathrm{\phi}$ to be compatible with the holomorphic structure of $\mathrm{\mathbf{X}}$. Furthermore, we will fix a based point free line bundle $\mathrm{p\!:\!\mathbf{L}\rightarrow \mathbf{X}}$ over $\mathrm{\mathbf{X}}$ and an algebraic $\mathrm{\mathbb{T}}$-action $\mathrm{\widehat{\phi}\!:\!\mathbb{T}\times \mathbf{L}\rightarrow \mathbf{L}}$ which projects down to $\mathrm{\phi}$ so that the corresponding morphisms of the fibers $\mathrm{\mathbf{L}_{\vert x}}$ are linear transformations. In the sequel, we will refer to $\mathrm{\mathbf{L}}$ as a $\mathrm{\mathbb{T}}$-{{linearized}} line bundle. Moreover, let $\mathrm{h}$ \label{Hermitian Bundle Metric} be a $\mathrm{T}$-invariant smooth, hermitian, positive bundle metric on $\mathrm{\mathbf{L}}$ and let $\mathrm{\mathfrak{t}}$ be the Lie algebra of the compact form $\mathrm{T\subset \mathbb{T}^{\mathbb{C}}}$. In this context, there exists ({cf.\! }\cite{Gu-St}) a naturally associated moment map $\mathrm{\mu\!:\!\mathbf{X}\rightarrow \mathfrak{t}^{*}}$ \label{Notation Moment Map} with respect to the K\"ahler form $\mathrm{p^{*}\omega=-\frac{i}{2}\partial \overline{\partial} \vert \cdot\vert_{h}^{2}}$ \label{Notation Kaehler form} given by the formula $$\mathrm{p^{*}\mu^{\xi}=-\frac{1}{4}d^{c}\mathbf{log}\,\vert \cdot\vert_{h}^{2}\,\widehat{X}_{\xi}}$$ where $\mathrm{\widehat{X}_{\xi}\label{Notation Fundamental Vector Field on L}}$ denotes the fundamental vector field of $\mathrm{\widehat{\phi}}$ on the total space $\mathrm{\mathbf{L}}$ (in the sequel we will use the notation $\mathrm{\vert \cdot \vert^{2}_{h}=\vert \cdot\vert^{2}}$\label{Notation Norm of Section with Respect to H}).
If $\mathrm{\xi\in Im\left(\mu\right)}$, it is possible to define an equivalence relation on the Zariski open, $\mathrm{\mathbb{T}}$-invariant set of semistable points $$\mathrm{\mathbf{X}^{ss}_{\xi}=\left\{x\in \mathbf{X}\!:\!\,cl\left(\mathbb{T}.x\right)\cap \mu^{-1}\left(\xi\right)\neq \varnothing\right\}\label{Notation Set of Semistable Points}}$$ given by $$\mathrm{z_{0}\sim z_{1}:\Leftrightarrow cl\left(\mathbb{T}.z_{0}\right)\cap cl\left(\mathbb{T}.z_{1}\right)\neq \varnothing.}$$ Having defined $\mathrm{\sim}$, it is possible (cf. \cite{He-Hu2}, {pp.\ }{\bf{\oldstylenums{310}-\oldstylenums{349}}}) to equip the induced quotient $\mathrm{\mathbf{X}^{ss}_{\xi}/\!\sim}$ with a unique, holomorphic structure of a complex space denoted by $\mathrm{\mathbf{X}^{ss}_{\xi}/\!\!/\mathbb{T}\label{Notation Hilbert Quotient}}$ so that the quotient map $\mathrm{\pi\!:\!\bf{X}^{ss}_{\xi}\rightarrow \mathbf{X}^{ss}_{\xi}/\!\!/\mathbb{T}}$ \label{Notation Hilbert Quotient Map} is holomorphic and has the following two characteristic properties:
\begin{enumerate} \item If $\mathrm{\mathcal{O}_{\mathbf{X}}}$ denotes the sheaf of holomorphic functions on $\mathrm{\mathbf{X}}$ and $\mathrm{\mathcal{O}_{\mathbf{X}^{ss}/\!\!/\mathbb{T}}}$ and is the sheaf of holomorphic functions on $\mathrm{\mathbf{X}^{ss}_{\xi}/\!\!/\mathbb{T}}$, then $\mathrm{(\pi_{*}\mathcal{O}_{\mathbf{X}^{ss}_{\xi}})^{\mathbb{T}}=\mathcal{O}_{\mathbf{X}^{ss}_{\xi}/\!\!/\mathbb{T}}}$.
\item We have $\mathrm{cl\,\left(\mathbb{T}.x\right)\cap cl\,\left(\mathbb{T}.y\right)\neq\varnothing}$ if and only if $\mathrm{\pi\left(x\right)=\pi\left(y\right)}$. \end{enumerate} As a generalization of the work of {\scshape{Shiffman, Tate}} and {\scshape{Zelditch}} (cf.\ \cite{S-T-Z}) and the results of {\scshape{Huckleberry}}, {\scshape{Sebert}} (cf.\ \cite{Hu-Se}), we link the geometry of the sequence of $\mathrm{\mathbb{T}}$-representation spaces, given by $\mathrm{H^{0}\left(\mathbf{X},\mathbf{L}^{n}\right)}$, as $\mathrm{n\rightarrow \infty}$ to the geometry of the quotient $\mathrm{\pi\!:\!\mathbf{X}^{ss}_{\xi}\rightarrow \mathbf{X}^{ss}_{\xi}/\!\!/\mathbb{T}\eqqcolon\mathbf{Y}\label{Notation Abbreviation for Hilbert Quotient}}$. To be more precise, we show that for each choice of $\mathrm{\xi\in Im\left(\mu\right)}$, it is possible to construct a convergent measure sequence which localizes along $\mathrm{\mu^{-1}\left(\xi\right)}$ by using sequences of $\mathrm{\mathbb{T}}$-eigensections $\mathrm{s_{n}\in H^{0}\left(\mathbf{X},\mathbf{L}^{n}\right)}$, i.e.\ $$\mathrm{{\bf{exp}}\left(\eta\right).s_{n}=e^{2\,\pi\,\sqrt{-1}\xi_{n}\left(\eta\right)}\,s_{n}\text{ where }\eta\in \mathfrak{t},\,\xi_{n}\in\mathfrak{t^{*}_{\mathbb{Z}}}}$$ whose rescaled weights $\mathrm{\frac{\xi_{n}}{n}}$ asymptotically approximate the chosen $\mathrm{\xi\in \mathfrak{t}^{*}}$ appropriately as $\mathrm{n\rightarrow \infty}$.
As a starting point, we show (cf.\ {\bf{Theorem 1}}) that, given $\mathrm{\xi\in Im\left(\mu\right)}$, there exists a finite cover $\mathrm{\left\{\mathbf{X}^{i}\right\}_{i\in I}\text{ of }\mathbf{X}^{ss}_{\xi}}$ consisting of open, $\mathrm{\pi}$-saturated subsets and a finite collection $$\mathrm{\left\{\left( s_{n}^{i} \right)_{n}\right\}_{i\in I},\text{ where }s_{n}^{i}\in H^{0}(\mathbf{X},\mathbf{L}^{n})}$$ of sequences consisting of $\mathrm{\mathbb{T}}$-eigensections such that the following properties are fulfilled: \begin{enumerate} \item $\mathrm{\frac{\xi^{i}_{n}}{n}\rightarrow \xi}$ where $\mathrm{\vert \xi^{i}_{n}-n\,\xi\vert\in \mathcal{O}\left(1\right)}$ for each $\mathrm{i}$.
\item $\mathrm{\mathbf{X}^{i}\subset \mathbf{X}(s_{n}^{i})\coloneqq \left\{x\in \mathbf{X}:\, s_{n}^{i}\left(x\right)\neq 0\right\}\label{Notation GTZU}}$ for all $\mathrm{n}$ big enough.
\item $\mathrm{\mathbf{X}^{i}\cap \mathbb{T}.\mu^{-1}\left(\xi\right)=\mathbf{X}^{i}\cap \mathbb{T}.\mu^{-1}\left(n^{-1}\xi^{i}_{n}\right)}$ for all $\mathrm{n\geq N_{0}}$. \end{enumerate} The construction of such a {\itshape{tame collection}} will be the first step of the present work and involves the combinatorial analysis of the sets $\mathrm{\mu\left(cl\left(\mathbb{T}.x\right)\right)}$, $\mathrm{x\in \mathbf{X}^{ss}_{\xi}}$, which are known to be convex polytopes in $\mathrm{\mathfrak{t}^{*}}$ ({cf.\ }\cite{Ati}). The crucial step of the existence proof is to control the dependence of the geometry of $\mathrm{\mu\left(cl\left(\mathbb{T}.x\right)\right)}$ as $\mathrm{x}$ varies in $\mathrm{\mathbf{X}^{ss}_{\xi}}$. Even in the special case $\mathrm{\mathbf{X}^{s}_{\xi}=\mathbf{X}^{ss}_{\xi}}$, i.e.\ where every $\mathrm{\pi}$-fiber is given by a $\mathrm{\mathbb{T}}$-orbit, the shape and position of $\mathrm{\mu\left(cl\left(\mathbb{T}.x\right)\right)}$ can in general vary considerably.
After having proven the existence of $\mathrm{\left\{\left(s_{n}^{i}\right)_{n}\right\}_{i}}$ with the aforementioned properties, it is possible to define for each finite, open cover $\mathrm{\mathfrak{U}=\left\{\mathbf{U}_{i}\right\}_{i}}$ of $\mathrm{\mathbf{Y}}$ subordinate to $\mathrm{\left\{\pi\left(\mathbf{X}^{i}\right)\right\}_{i}}$, i.e.\ $\mathrm{\mathbf{U}_{i}\subset \pi\left(\mathbf{X}^{i}\right)}$, a finite collection $\mathrm{\left\{\bm{\nu}^{\mathfrak{U}}_{n}\right\}_{n}=\left\{\bm{\nu}^{i}_{n}\right\}_{i}}$ of sequences of $\mathrm{\pi}$-fiber probability measures on Zariski open subsets of $\mathrm{\mathbf{X}^{ss}_{\xi}}$ by using the corresponding norm functions $\mathrm{\vert s^{i}_{n}\vert^{2}}$ with respect to the hermitian bundle metric $\mathrm{h}$. The precise construction of the collection $\mathrm{\left\{\bm{\nu}^{\mathfrak{U}}_{n}\right\}_{n}}$ is based on the observation that the ambiguity of the norm sequence $\mathrm{\left(\vert s_{n}^{i}\vert^{2}\right)_{n}}$, which is only well-determined up to scalar multiplication, can be abolished if one considers the normalized sequence given by $\mathrm{\Vert s_{n}^{i}\Vert^{-2}\vert s_{n}^{i}\vert^{2}}$. Here $\mathrm{\Vert s_{n}^{i}\Vert^{2}}$ denotes the fiber integral of the function $\mathrm{\vert s_{n}^{i}\vert^{2}}$ over $\mathrm{\pi_{y}\coloneqq \pi^{-1}\left(y\right)}$. The collection $\mathrm{\left\{\bm{\nu}^{\mathfrak{U}}_{n}\right\}_{n}}$ associated to the tame collection $\mathrm{\left\{s_{n}^{i}\right\}_{i}}$ is then given by $$\mathrm{\bm{\nu}_{n}^{i}\left(y\right)\left(\mathbf{A}\right)\coloneqq \int_{\mathbf{A}}\Vert s_{n}^{i}\Vert^{-2}\vert s_{n}^{i}\vert^{2}\,d\,[\pi_{y}]\text{ for }\mathbf{A}\text{ measurable and }y\in \mathbf{U}_{i}\cap \mathbf{Y}_{0}}$$ where $\mathrm{\int_{\mathbf{A}}d\,[\pi_{y}]}$ is defined to be the integral over $\mathrm{\mathbf{A}\subset \pi^{-1}\left(y\right)}$ of the restriction $\mathrm{\omega^{dim_{\mathbb{C}}\,\pi_{y}}\vert\pi_{y}}$ with a certain multiplicity\footnote{All relevant details of the theory of fiber integration can be found in \cite{Kin}.}. Since the dimension $\mathrm{k_{y}=dim_{\mathbb{C}}\,\pi^{-1}\left(y\right)}$ of a fiber $\mathrm{\pi^{-1}\left(y\right)}$ for $\mathrm{y\in \mathbf{Y}}$ can change as $\mathrm{y}$ moves in $\mathrm{\mathbf{Y}}$ and since the construction of each $\mathrm{\bm{\nu}_{n}^{i}\left(y\right)}$ involves $\mathrm{k_{y}}$, one can not expect that $\mathrm{\bm{\nu}_{n}^{\mathfrak{U}}}$ defines a uniform object over the full base $\mathrm{\mathbf{Y}}$. However, it is possible to find a Zariski dense subset $\mathrm{\mathbf{Y}_{0}\subset \mathbf{Y}}$, so that $\mathrm{\pi^{-1}\left(y\right)}$ for $\mathrm{y\in \mathbf{Y}_{0}}$ are purely $\mathrm{k}$-dimensional complex varieties (set $\mathrm{\mathbf{X}_{0}\coloneqq \pi^{-1}\left(\mathbf{Y}_{0}\right)}$). Over this set, it is reasonable to examine the convergence properties of the measure sequence $\mathrm{\bm{\nu}_{n}^{i}\left(\cdot\right)}$ for each $\mathrm{i\in I}$. More precisely, if $\mathrm{f\in\mathcal{C}^{0}\!\left(\mathbf{X}\right)}$ is a continuous function and if $\mathrm{f_{red}}$ denotes the {\itshape{reduced function}} on the base $\mathrm{\mathbf{Y}}$ given by the restriction $\mathrm{\overline{f}\vert \mu^{-1}\left(\xi\right)}$ of the averaged function $\mathrm{\overline{f}\left(x\right)=\int_{T}f\left(t.x\right)\,d\nu_{T}}$ \label{Notation Averaged Function} with respect to the {\scshape{Haar}} measure $\mathrm{\nu_{T}}$, we prove the following theorem.
\begin{theo_3*}\label{Theorem Uniform Convergence of the Tame Measure Sequence} \textnormal{[\scshape{Uniform Convergence of the Tame Measure Sequence}]}
\\ For for every tame collection $\mathrm{\left\{s_{n}^{i}\right\}_{i}}$ there exists a finite cover $\mathrm{\mathfrak{U}}$ of $\mathrm{\mathbf{Y}}$ with $\mathrm{\mathbf{U}_{i}\subset}$ $\mathrm{\pi\left(\mathbf{X}^{i}\right)}$ so that the collection of fiber probability measures $\mathrm{\left\{\bm{\nu}^{\mathfrak{U}}_{n}\right\}}$ associated to $\mathrm{\left\{s^{i}_{n}\right\}_{i}}$ converges uniformly on $\mathrm{\mathbf{Y}_{0}}$ to the fiber Dirac measure of $\mathrm{\mu^{-1}\left(\xi\right)\cap \mathbf{X}_{0}}$, i.e.\ for every $\mathrm{i\in I}$ and every $\mathrm{f\in \mathcal{C}^{0}\left(\mathbf{X}\right)}$, we have $$\begin{xy}
\xymatrix{\mathrm{\left(y\mapsto \int_{\pi^{-1}\left(y\right)}f\, d\bm{\nu}_{n}^{i}\left(y\right)\right)}\ar[rr]^{\mathrm{\,\,\,\,\,}}&&\mathrm{f_{red}\text{ uniformly on }\mathbf{U}_{i}\cap \mathbf{Y}_{0}.\label{Notation Reduced Function}}
} \end{xy}$$ \end{theo_3*}
Furthermore, if $\mathrm{\left\{D^{\mathfrak{U}}_{n}\right\}=\left\{D_{n}^{i}\right\}_{i}}$ denotes the corresponding collection of cumulative fiber probability densities given by $$\mathrm{D^{i}_{n}\left(y,\cdot \right)\!:\!t\mapsto D^{i}_{n}\left(y,t \right)\coloneqq \int_{\left\{ \frac{\vert s_{n}^{i}\vert^{2}}{\left\Vert s_{n}^{i}\right\Vert ^{2}}\geq t\right\} \cap\pi^{-1}\left(y\right)}d\,[\pi_{y}]\text{ for }y\in\mathbf{ U}_{i}\cap \mathbf{Y}_{0}}$$ the following convergence result can be proved. \begin{theo_4*}\label{Theorem Uniform Convergence of the Tame Distribution Sequence} \textnormal{[\scshape{Uniform Convergence of the Tame Distribution Sequence}]}
\\ For every $\mathrm{t\in \mathbb{R}}$ and every tame collection $\mathrm{\left\{s_{n}^{i}\right\}_{i}}$ there exists a finite cover $\mathrm{\mathfrak{U}}$ of $\mathrm{\mathbf{Y}}$ with $\mathrm{\mathbf{U}_{i}\subset}$ $\mathrm{\pi\left(\mathbf{X}^{i}\right)}$ so that the collection of cumulative fiber probability densities $\mathrm{\left\{D^{\mathfrak{U}}_{n}\left(\cdot,t\right)\right\}}$ associated to $\mathrm{\left\{s^{i}_{n}\right\}_{i}}$ converges uniformly on $\mathrm{\mathbf{Y}_{0}}$ to the zero function on $\mathrm{\mathbf{Y}_{0}}$, i.e.\ for every $\mathrm{i\in I}$ we have $$\begin{xy}
\xymatrix{\mathrm{\left(y\mapsto D_{n}^{i}\left(y,t\right)\right)}\ar[rr]^{\mathrm{\,\,\,\,\,}}&&\mathrm{0\text{ uniformly on }\mathbf{U}_{i}\cap \mathbf{Y}_{0}\subset \pi\left(\mathbf{X}^{i}\right).}
} \end{xy}$$ \end{theo_4*}
In this sense, we have shown that each tame sequence of eigensections $\mathrm{\left(s_{n}^{i}\right)_{n}}$, attached to a prescribed weight $\mathrm{\xi}$, gives rise to a sequence of fiber measures over $\mathrm{\mathbf{Y}_{0}}$, which independently of the choice of $\mathrm{\left(s_{n}^{i}\right)_{n}}$, localizes uniformly along the critical $\mathrm{\mu^{-1}\left(\xi\right)}$. The localization property of the measure sequence $\mathrm{\bm{\nu}_{n}^{\mathfrak{U}}\left(\cdot\right)}$ attached to the tame collection $\mathrm{\left\{\left(s^{i}_{n}\right)\right\}_{i}}$ is a consequence of the fact that the corresponding sequences of strictly plurisubharmonic functions $\mathrm{\varrho^{i}_{n}\!:\!\mathbf{X}^{i}\rightarrow \mathbb{R}}$ given by $$\mathrm{\varrho^{i}_{n}\coloneqq -\frac{1}{n}\mathbf{log}\,\vert s_{n}^{i}\vert^{2}}$$ converge (along with all derivatives) uniformly on compact sets to a strictly plurisubharmonic function $\mathrm{\varrho^{i}}$. It is crucial to note that the restriction $\mathrm{\varrho^{i}\vert \pi^{-1}\left(y\right)}$ of $\mathrm{\varrho^{i}}$ to each fiber of the projection $\mathrm{\pi\!:\!\mathbf{X}^{ss}_{\xi}\rightarrow \mathbf{Y}}$ takes on its $\mathrm{T}$-invariant minimum along the uniquely defined $\mathrm{T}$-orbit $\mathrm{T.x_{y}\subset \pi^{-1}\left(y\right)}$ given by $$\mathrm{T.x_{y}=\mu^{-1}\left(\xi\right)\cap \pi^{-1}\left(y\right).}$$ Using this observation, it is then possible to deduce estimates of the magnitude of $\mathrm{\varrho^{i}_{n}}$ and hence of $\mathrm{e^{-n\,\varrho^{i}_{n}}=\vert s_{n}^{i}\vert^{2}}$ outside a $\mathrm{T}$-invariant, relatively compact tube of $\mathrm{\mu^{-1}\left(\xi\right)}$ as $\mathrm{n}$ tends to infinity (cf.\ {\bf{Theorem 2}}).
Apart form the determination of the asymptotic behavior of $\mathrm{\varrho^{i}_{n}}$, which plays an essential role in the proof of {\bf{Theorem 3, 4}}, it is also necessary to deal with the following issue: The fact that $\mathrm{\bm{\nu}_{n}^{\mathfrak{U}}\left(\cdot\right)}$ is defined over a non-compact base makes a direct application of the standard convergence theorems of measure theory considerably more difficult. Therefore, a good portion of the proof of the above convergence theorem will be devoted to resolving this issue by constructing a new quotient $\mathrm{\Pi\!:\!\widetilde{\bf{\,X\,}}\rightarrow \widetilde{\bf{\,Y\,}}}$ which extends the restricted quotient $\mathrm{\pi\!:\!\mathbf{X}_{0}\rightarrow \mathbf{Y}_{0}\label{Notation Inverse Image of Y_0 Under The Projection Map}}$, so that the following diagram commutes
$$\begin{xy} \xymatrix{ \mathrm{\mathbf{X}_{0}}\ar@{^{(}->}[rrr]\ar[d]_{\mathrm{\pi\vert \mathbf{X}_{0}}} &&& \mathrm{\widetilde{\bf{\,X\,}}}\ar[d]^{\mathrm{\Pi}}\\ \mathrm{\mathbf{Y}_{0}}\ar@{^{(}->}[rrr]&&& \mathrm{\widetilde{\bf{\,Y\,}}}\\ } \end{xy} $$ and so that all fibers of $\mathrm{\Pi}$ are compact and of pure dimension $\mathrm{k}$.
The construction of this equivariant, dimensional-theoretical {\itshape{flattening}} $\mathrm{\Pi\!:\!\widetilde{\bf{\,X\,}}\rightarrow \widetilde{\bf{\,Y\,}}}$, which is based on results of {\scshape{D.\! Barlet}}, allows us to realize the fiber measure sequence $\mathrm{\bm{\nu}_{n}^{\mathfrak{U}}\left(\cdot\right)}$ as a restriction of a measure sequence defined on $\mathcal{\widetilde{\bf{\,X\,}}}$. Since $\mathrm{\widetilde{\bf{\,X\,}}}$ and all its fibers are compact, it is then possible to show the above mentioned convergence properties of $\mathrm{\left\{\bm{\nu}_{n}^{\mathfrak{U}}\right\}_{n}}$ by applying results concerning the continuity of fiber integration.
In the last part of this work, we return to the initial sequence of $\mathrm{\mathbb{T}}$-eigensections $\mathrm{\left(s_{n}\right)_{n}}$ and examine the convergence properties of the fiber measure sequence induced by $\mathrm{\vert s_{n}\vert^{2}\Vert s_{n}\Vert^{-2}}$. Unlike in the tame case, we first have to face the problem that the function $\mathrm{\Vert s_{n}\Vert^{-2}}$ is only well defined for all $\mathrm{y\in \mathbf{Y}}$ with the property $\mathrm{s_{n}\vert\pi^{-1}\left(y\right)\not\equiv0}$. The task of defining a maximal, $\mathrm{n}$-stable set in $\mathrm{\mathbf{Y}}$, on which we can consistently write down a measure sequence $\mathrm{\left(\bm{\nu}_{n}\right)_{n}}$ for all $\mathrm{n\in \mathbb{N}}$ big enough attached to $\mathrm{\left(s_{n}\right)_{n}}$, naturally leads to the notion of a {\itshape{removable singularity}}. Having available this concept, which is based on the idea that certain singularities can be "divided out" by multiplying $\mathrm{s_{n}}$ with locally defined, invariant holomorphic functions, we are able to uniquely extend the measure sequence beyond its original set of definition for all $\mathrm{n\geq N_{0}}$ on an open subset of $\mathrm{\mathbf{Y}_{0}}$ which is given by $\mathrm{\mathbf{R}_{N_{0}}\cap \mathbf{Y}_{0}}$. Here, $\mathrm{\mathbf{R}_{N_{0}}}$ denotes the open subset of all singularities $\mathrm{y\in \mathbf{Y}}$ which are removable for all $\mathrm{n\geq N_{0}}$. Once the maximal set of definition given by $\mathrm{\mathbf{R}_{N_{0}}\cap \mathbf{Y}_{0}}$ is found, we continue our discussion by analyzing the convergence properties of the measure sequence $\mathrm{\left(\bm{\nu}_{n}\right)_{n}}$ over $\mathrm{\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$ and obtain the following two results.
\begin{theo_5.b*}\label{theo_5.b*}\textnormal{[\scshape{Uniform Convergence of the Initial Distribution Sequence}]}
\\ For fixed $\mathrm{t\in \mathbb{R}}$ the sequence $\mathrm{\left(D_{n}\left(\cdot,t\right)\right)_{n}}$ converges uniformly on $\mathrm{\mathbf{Y}_{0}\cap\mathbf{R}_{N_{0}}}$ to the zero function. \end{theo_5.b*}
\begin{theo_6.b*}\label{theo_6.b*}\textnormal{[\scshape{Uniform Convergence of the Initial Measure Sequence}]}
\\ For $\mathrm{f\in \mathcal{C}^{0}\!\left(\mathbf{X}\right)}$ the sequence $$\mathrm{\left(y\mapsto \int_{\pi^{-1}\left(y\right)}f\,d\bm{\nu}_{n}\left(y\right)\right)_{n}}$$ converges uniformly over $\mathrm{\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$ to the reduced function $\mathrm{f_{red}}$. \end{theo_6.b*} Since the deviation of the initial measure sequence $\mathrm{\left(\bm{\nu}_{n}\right)}$ induced by $\mathrm{\left(s_{n}\right)_{n}}$ from a tame, locally defined measure sequence $\mathrm{\left(\bm{\nu}^{i}_{n}\right)_{n}}$ is completely described by the locally defined sequence of functions $\mathrm{\left(\triangle^{i}_{n}\right)_{n}}$ given by $\mathrm{s_{n}=\triangle^{i}_{n}\cdot s^{i}_{n}}$, it is not surprising that the proof of both theorems is based on technics we already used before when proving {\bf{Theorem 3, 4}}. These are combined with certain analytic facts about the growth properties of $\mathrm{\triangle^{i}_{n}}$ as $\mathrm{\varrho^{i}\rightarrow \infty}$ on $\mathrm{\pi^{-1}\left(y\right)}$ and as $\mathrm{n\rightarrow \infty}$.
{\setlength{\parindent}{0cm}{\bf{Acknowledgements.}} The author would like to express his gratitude to his advisor Professor {\scshape{Alan Huckleberry}} who initiated this project and provided permanent support and advise whenever needed from its very beginning.}
Furthermore, the author is indebted to the constructive remarks of Prof.\ Dr.\ {\scshape{Heinzner}}, Prof.\ Dr.\ {\scshape{Winkelmann}}, Prof.\ Dr.\ {\scshape{Wurzbacher}} and Dr.\ {\scshape{Tsanov}}.
Finally, the author would like to thank the {\scshape{Studienstiftung des Deutschen Volkes}} for financial support which included a scholarship for the author's years as a student and PhD candidate at {\scshape{Ruhr-Universit{\"a}t Bochum}} and for his time as a visiting student at {\scshape{Massachusetts Institute of Technology}}. Also in this regard, the author owes special thanks to Prof.\ {\scshape{Vogan}} who generously supported the research period at MIT.
\section{Existence of Tame Sequences}\label{Existence of Tame Sequences}
The aim of this section is to prove the following theorem.
\begin{theo_1*} \textnormal{[\scshape{Existence of Tame Sequences}]}
\\ If $\mathrm{\mathbf{L}\rightarrow \mathbf{X}}$ is as above and $\mathrm{\xi\in Im\left(\mu\right)}$, then we can find a finite cover $$\mathrm{\left\{\mathbf{X}^{i}\right\}_{i\in I}\text{ of }\mathbf{X}^{ss}_{\xi}}$$ consisting of open, $\mathrm{\pi}$-saturated subsets and a finite collection $$\mathrm{\left\{\left( s_{n}^{i} \right)_{n}\right\}_{i\in I},\text{ where }s_{n}^{i}\in H^{0}(\mathbf{X},\mathbf{L}^{n})\label{Notation i-th tame sequence}}$$ of sequences consisting of $\mathrm{\mathbb{T}}$-eigensections such that the following properties are fulfilled: \begin{enumerate} \item $\mathrm{\frac{\xi^{i}_{n}}{n}\rightarrow \xi}$ where $\mathrm{\vert \xi^{i}_{n}-n\,\xi\vert\in \mathcal{O}\left(1\right)}$ for each $\mathrm{i}$\label{Notation Approximating Sequence of Weight Vectors}.
\item $\mathrm{\mathbf{X}^{i}\subset \mathbf{X}(s_{n}^{i})}$\label{Notation i-th set} for all $\mathrm{n}$ big enough.
\item $\mathrm{\mathbf{X}^{i}\cap \mathbb{T}.\mu^{-1}\left(\xi\right)=\mathbf{X}^{i}\cap \mathbb{T}.\mu^{-1}\left(\frac{\xi^{i}_{n}}{n}\right)}$ for all $\mathrm{n\geq N_{0}}$. \end{enumerate} \end{theo_1*} In the sequel, we will refer to a collection $\mathrm{\left\{\left(s_{n}^{i}\right)\right\}_{i}}$ of sequences of $\mathrm{\mathbb{T}}$-eigensections $\mathrm{s_{n}^{i}\in H^{0}}$ $\mathrm{\left(\mathbf{X},\mathbf{L}^{n}\right)}$ with the aforementioned properties as {\itshape{tame}}. Before we proceed with the proof of {\bf{Theorem 1}}, let us consider an example of a tame collection. \begin{example}\label{Example First Example} \textnormal{Let $\mathrm{\mathbf{X}=\mathbb{C}\mathbb{P}^{1}\times \mathbb{C}\mathbb{P}^{1}}$ with K\"aher form $\mathrm{\omega=p_{1}^{*}\omega_{FS}+p_{2}^{*}\omega_{FS}}$ where $\mathrm{p_{i}\!:\!}$ $\mathrm{\mathbf{X}}$ $\mathrm{\rightarrow}$ $\mathrm{\mathbb{C}\mathbb{P}^{1}}$ denotes the $\mathrm{i}$-th projection, $\mathrm{i\in\left\{0,1\right\}}$. Equip $\mathrm{\mathbf{X}}$ with the $\mathrm{\mathbb{T}\cong\mathbb{C}^{*}}$-action given by $$\mathrm{t.\left(\left[z_{0}\!:\!z_{1}\right],\left[\zeta_{0}\!:\!\zeta_{1}\right]\right)=\left(\left[t\, z_{0}\!:\!z_{1}\right],\left[t\, \zeta_{0}\!:\!\zeta_{1}\right]\right)}$$ and consider the $\mathrm{\mathbb{T}}$-linearization on $\mathrm{\mathbf{L}=p_{0}^{*}\mathcal{O}_{\mathbb{C}\mathbb{P}^{1}}\left(1\right)\otimes p_{1}^{*}\mathcal{O}_{\mathbb{C}\mathbb{P}^{1}}\left(1\right)\rightarrow \mathbf{X}}$ induced by the $\mathrm{\mathbb{T}}$-action on $\mathrm{\mathbb{C}\mathbb{P}^{4}}$ given by $$\mathrm{t.[u_{0}\!:\!u_{1}\!:\!u_{2}\!:\!u_{3}\!:\!v]=[t\,u_{0}\!:\!u_{1}\!:\!u_{2}\!:\!t^{-1}\,u_{3}\!:\!v].}$$ Here we view $\mathrm{\mathbf{L}}$ embedded in $\mathrm{\mathcal{O}_{\mathbb{C}\mathbb{P}^{3}}\left(1\right)\cong\mathbb{C}\mathbb{P}^{4}\setminus \{[0\!:\!0\!:\!0\!:\!0\!:\!1]\}}$ realized as the cone over $$\mathrm{\mathbf{X}\cong\left\{[u]\in \mathbb{C}\mathbb{P}^{4}\!:\!\, u_{0}u_{3}-u_{1}u_{2}=0, v=0\right\} \subset \left\{[u]\in \mathbb{C}\mathbb{P}^{4}\!:\!\,v=0\right\}\cong \mathbb{C}\mathbb{P}^{3}.}$$ Furthermore, let $\mathrm{H}$ denote the standard hermitian metric on $\mathrm{\mathcal{O}_{\mathbb{C}\mathbb{P}^{3}}\left(1\right)}$ which is given by $\mathrm{H([u\!:\!v])=\Vert u\Vert^{-2}\vert v\vert^{2}}$ for $\mathrm{u=\left(u_{0},u_{1},u_{2},u_{3}\right)\neq 0}$ and which is $\mathrm{T}$-invariant with respect to the above action. Define $\mathrm{h\coloneqq H\vert \mathbf{X}}$.}
\textnormal{If $\mathrm{\xi=0}$ and $\mathrm{\mu}$ denotes the moment map induced by the hermitian bundle metric $\mathrm{h}$, it follows that $$\mathrm{\mu^{-1}\left(0\right)=\left\{\left([z_{0}\!:\!z_{1}],[\zeta_{0}\!:\!\zeta_{1}]\right)\!:\!\,z_{0}\zeta_{0}-z_{1}\zeta_{1}=0\right\}\subset \mathbb{C}\mathbb{P}^{1}\times \mathbb{C}\mathbb{P}^{1}.}$$ A calculation shows that the only eigensections $\mathrm{s_{n}\in H^{0}\left(\mathbf{X},\mathbf{L}^{n}\right)}$ which can be part of a tame sequences in the sense of {\bf{Theorem 1}} are given by the collection $$\mathrm{\left\{ s_{n,k}=z_{0}^{n-k}z_{1}^{k}\zeta_{0}^{k}\zeta_{1}^{n-k},\,0\leq k\leq n\right\} \subset H^{0}\left(\mathbf{X},\mathbf{L}^{n}\right).}$$ A tame collection is for example given by $$\mathrm{\Big\{\!\left(z_{0}^{n}\zeta_{1}^{n}\right)_{n},\left(z_{1}^{n}\zeta_{0}^{n}\right)_{n}\!\Big\}.\,\, \boldsymbol{\Box}}$$} \end{example}
For the proof of the existence of tame sequences, we will first restate an important fact about the geometry of an arbitrary $\mathrm{\mathbb{T}}$-orbit closure $\mathrm{cl\left(\mathbb{T}.x\right)}$ \label{Notation T-Orbit Closure} where $\mathrm{x\in\mathbf{X}}$:
Every image $\mathrm{\mu(cl(\mathbb{T}.x))\subset \mathfrak{t}^{*}}$ of an arbitrary orbit closure $\mathrm{ cl(\mathbb{T}.x)\subset \mathbf{X}}$ is the convex hull of the image of finitely many fixed points $\mathrm{x_{i}\in {\bf{Fix}}^{\mathbb{T}}\coloneqq \{x\in \mathbf{X}\!:\!\,t.x}$ \label{Notation Fix Point Set Of T-Action} $\mathrm{=x}$ $\mathrm{\text{for all }t\in \mathbb{T}\}}$ ({cf.\ }\cite{Ati}), i.e.\: \begin{equation}\label{Equation Convex Hull} \mathrm{\mu(cl(\mathbb{T}.x))=\mathfrak{Conv}\,\left\{\mu({\bf{S}}_{x})\right\}} \end{equation} where $$\mathrm{{\bf{S}}_{x}=\left\{\sigma_{x,j_{x}}\in cl\left(\mathbb{T}.x\right)\cap {\bf{Fix}}^{\mathbb{T}},\, 1\leq j_{x}\leq m_{x}\right\}\subset {\bf{Fix}}^{\mathbb{T}}}$$ \label{Notation T-Fixed Points Contained In The Closure Of T.x} is a finite set and $\mathrm{\mathfrak{Conv}\,\left\{\mu\left({\bf{S}}_{x}\right)\right\}}$ \label{Notation Convex Hull Of A Subset} denotes the convex hull of the corresponding image $\mathrm{\mu\left({\bf{S}}_{x}\right)}$.
Consider the decomposition of the $\mathrm{\mathbb{T}}$-representation space $\mathrm{H^{0}(\mathbf{X},\mathbf{L})}$ in its eigenspaces $\mathrm{\mathbb{C}s_{k}}$ where $\mathrm{1\leq k\leq m\coloneqq dim_{\mathbb{C}}\,H^{0}(\mathbf{X},\mathbf{L})}$. For any non-empty $\mathrm{J\subset \left\{1,\dots, m\right\}}$ let $\mathrm{\mathcal{S}_{J}\coloneqq \left\{s_{j}\!:\!\,j\in J\right\}}$ $\mathrm{\subset H^{0}\left(\mathbf{X},\mathbf{L}\right)}$ \label{Notation Set Of T-Eigensections Indexed By J} and let $\mathrm{{\bm{\mathfrak{S}}}_{J}\subset \mathfrak{t}^{*}}$ \label{Notation Set Of All Characters Corresponding To S_x} be the set of all characters corresponding to the set $\mathrm{\mathcal{S}_{J}}$.
We introduce the following notation: For a non-empty $\mathrm{J\subset \{1,\dots,m\}}$, let $$\mathrm{\mathbf{M}_{J}\coloneqq\left\{x\in \mu^{-1}\left(\xi\right)\!:\!\, s_{j}\left(x\right)\neq 0\,\text{ for all } j\in J, s_{j}\left(x\right)= 0\,\text{ for all } j\in\complement\, J\right\}. \label{Notation Definition of M_J}}$$ Note that $\mathrm{\mathbf{M}_{J}\subset \mu^{-1}\left(\xi\right)}$ is a $\mathrm{T}$-invariant, open subset which might be empty for certain non-empty subindices $\mathrm{J\subset \{1,}$ $\mathrm{\dots,m\}}$. Furthermore, the collection $\mathrm{\left\{\mathbf{M}_{J}\right\}_{J\neq \varnothing}}$ is finite and cover $\mathrm{\mu^{-1}\left(\xi\right)}$. The first claim follows by the fact that $\mathrm{dim_{\mathbb{C}}\,H^{0}\left(\mathbf{X},\mathbf{L}\right)<\infty}$ and the second claim is a direct consequence of the assumption that $\mathrm{\mathbf{L}}$ is base point free.
The next lemma establishes a connection between the geometry of the image of $\mathrm{\mathbb{T}.x}$ under the moment map $\mathrm{\mu}$ where $\mathrm{x\in \mathbf{M}_{J}}$ and the convex set $\mathrm{\mathfrak{Conv}\,{\bm{\mathfrak{S}}}_{J}}$.
\begin{lemma}\label{Lemma Convex Hull Moment Map} Let $\mathrm{x\in \mathbf{M}_{J}}$, then it follows that $$\mathrm{\mu(cl(\mathbb{T}.x))=\mathfrak{Conv}\,{\bm{\mathfrak{S}}}_{J}}$$ and, if $\mathrm{x}$ is not a $\mathrm{\mathbb{T}}$-fixed point, $$\mathrm{\xi\in\mu\left(\mathbb{T}.x\right)=Relint\,\mathfrak{Conv}\,{\bm{\mathfrak{S}}}_{J}}$$ where $\mathrm{Relint\,\mathfrak{Conv}\,{\bm{\mathfrak{S}}}_{J}}$ \label{Notation Relative Interior} denotes the relative interior of $\mathrm{{\bm{\mathfrak{S}}}_{J}\subset \mathfrak{t}^{*}}$. \end{lemma} \begin {proof} First of all note that since the image of $\mathrm{cl\left(\mathbb{T}.x\right)}$ is known to be the convex hull of the image $\mathrm{\mu\left({\bf{S}}_{x}\right)}$ where $\mathrm{{\bf{S}}_{x}}$ are the $\mathrm{\mathbb{T}}$-fixed points in $\mathrm{cl\left(\mathbb{T}.x\right)}$, the first claim follows as soon as we have shown that $\mathrm{\mu\left(\sigma_{x,j_{x}}\right)\in {\bm{\mathfrak{S}}}_{J}}$ for all $\mathrm{\sigma_{x,j_{x}}\in {\bf{S}}_{x}}$.
Let $\mathrm{\sigma_{x,j_{x}}\in {\bf{S}}_{x}}$. Since $\mathrm{\mathbf{L}}$ is assumed to be base point free there exists at least one $\mathrm{\mathbb{T}}$-eigensection $\mathrm{s}$ which does not vanish at $\mathrm{\sigma_{x,j_{x}}}$. As $\mathrm{x\in \mathbf{M}_{J}}$, such an $\mathrm{s}$ is necessarily given by $\mathrm{s=s_{j}}$ for $\mathrm{j\in J}$. If $\mathrm{\xi_{j}}$ denotes its corresponding character, we deduce $\mathrm{\mu\left(\sigma_{x,j_{j}}\right)=\xi_{j}}$ which is a direct consequence of the following reasoning: Let $\mathrm{D\!:\!\mathcal{A}^{0}\left(\mathbf{L}\right)\rightarrow\mathcal{A}^{1}\left(\mathbf{L}\right)}$ be the uniquely defined, hermitian connection associated to $\mathrm{h}$ and recall that we have the formula ({cf.\! }\cite{Gu-St}) \begin{equation}\label{Equation Formula Connection Moment Map} \mathrm{D_{\mathbf{X}_{\eta}}s+2\,\sqrt{-1}\mu^{\eta}s=\frac{d}{dt}_{t=0}{\bf{exp}}(\sqrt{-1}\,t\,\eta).s.\text{ for all }\eta\in \mathfrak{t}.} \end{equation} If we apply formula \ref{Equation Formula Connection Moment Map} to $\mathrm{s=s_{j}}$ at the fixed point $\mathrm{\sigma_{x,j_{x}}}$, we deduce $\mathrm{\mu^{\eta}(\sigma_{x,j_{x}})=\xi_{j}(\eta)}$ for all $\mathrm{\eta \in \mathfrak{t}}$. So it follows $\mathrm{\mu(\sigma_{x,j_{x}})=\xi_{j}}$ and hence $$\mathrm{\mu\left(cl\left(\mathbb{T}.x\right)\right)= \mathfrak{Conv}\,{\bm{\mathfrak{S}}}_{J}}$$ as claimed.
The second claim follows from the fact that $\mathrm{\xi=\mu\left(x\right)}$ for $\mathrm{x\in \mathbf{M}_{J}\subset \mu^{-1}\left(\xi\right)}$ and the general fact that $\mathrm{cl\left(f\left(\mathbf{A}\right)\right)=f\left(cl\left(\mathbf{A}\right)\right)}$ for a continuous map $\mathrm{f\!:\!\mathbf{X}\rightarrow \mathbf{Y}}$ where $\mathrm{\mathbf{X}}$ is compact and $\mathrm{\mathbf{A}\subset \mathbf{X}}$. \end{proof}
For the proof of the next proposition, which will be crucial for the existence of tame sequences, we need the following technical lemma.
\begin{lemma}\label{Lemma Convex Polytope} Let $\mathrm{{\bm{\mathfrak{P}}}=\mathfrak{Conv}\,\left\{(q_{1},\dots,q_{m})\right\}}$, $\mathrm{q_{1},\cdots,q_{m}\in \mathbb{R}^{n}}$ and let $$\mathrm{{\bm{\mathfrak{P}}}_{n^{-1}\mathbb{N}^{0}}\coloneqq \mathfrak{Conv}_{n^{-1}\mathbb{N}^{0}}\left\{q_{1},\dots,q_{m}\right\}\coloneqq \Bigg\{ p=\sum_{j}\nu_{j}q_{j}\!:\!\,\sum_{j}\nu_{j}=1,\,\nu_{j}\in n^{-1}\mathbb{N}^{0}\Bigg\}.}$$ If $\mathrm{\xi\in{\bm{\mathfrak{P}}}}$, then there exists a sequence $\mathrm{\xi_{n}=\sum_{j}\nu_{j,n}q_{j}}$ so that the following conditions are fulfilled: \begin{enumerate} \item $\mathrm{\xi_{n}\rightarrow \xi}$\\[-0.6 cm]
\item $\mathrm{\mathrm{\left|\xi_{n}-n\,\xi\right|\in\mathcal{O}\left(1\right)}}$\\[-0.6 cm] \item $\mathrm{\xi_{n}\subset {\bm{\mathfrak{P}}}_{n^{-1}\mathbb{N}^{0}}}$ for all $\mathrm{n}$ big enough.\\[-0.6 cm] \item There exists $\mathrm{N_{0}\in \mathbb{N}}$ and a partition $\mathrm{J=J_{0}\cup J_{1}}$, $\mathrm{J_{1}\neq \varnothing}$ of $\mathrm{J=\left\{1,\dots,m\right\}}$ so that $\mathrm{\nu_{j,n}=0}$ for all $\mathrm{j\in J_{0}}$, $\mathrm{n\geq N_{0}}$ and $\mathrm{\nu_{j,n}>0}$ for all $\mathrm{j\in J_{1}}$, $\mathrm{n\geq N_{0}}$.\\[-0.6 cm] \item The sequences $\mathrm{\left(\nu_{j,n}\right)_{n}}$ are convergent so that the limits are strictly positive for all $\mathrm{j\in J_{1}}$. \end{enumerate} \end{lemma} \begin{proof} Let $\mathrm{{\bm{\mathfrak{P}}}_{0}\subset \mathbb{R}^{m}}$ denote the convex hull of the standard basis $\mathrm{\{e_{1},\dots,e_{m}\}}$ in $\mathrm{\mathbb{R}^{m}}$, then \begin{equation}\label{Equation Image Given By Matrix} \mathrm{{\bm{\mathfrak{P}}}\coloneqq \mathfrak{Conv}\,\left\{q_{1},\dots,q_{m}\right\}={\bf{M}}({\bm{\mathfrak{P}}}_{0})} \end{equation} for the matrix $\mathrm{\bf{M}}$ whose columns are given by $\mathrm{(q_{1},\dots,q_{m})}$. By \ref{Equation Image Given By Matrix} we find $\mathrm{\nu\in {\bm{\mathfrak{P}}}_{0}}$ with $\mathrm{\xi={\bf{M}}\left(\nu\right)}$. Without restriction of generality, we can assume that \begin{equation}\label{Equation In Order To Guarantee} \mathrm{\nu\in Relint\,\mathfrak{Conv}\,\{e_{\ell},\dots,e_{m}\}}, \end{equation} where $\mathrm{1\leq \ell\leq m-1}$ because otherwise, $\mathrm{\xi=q_{j_{0}}}$ for one $\mathrm{1\leq j_{0}\leq m}$ and the claim follows immediately by choosing $\mathrm{\nu_{j_{0},n}=1}$ and $\mathrm{\nu_{j,n}=0}$ for all $\mathrm{j\neq j_{0}}$.
Now, it is direct to see that we can choose a sequence $\mathrm{\left( \nu_{n}\right)_{n}}$ in $\mathrm{\mathfrak{Conv}_{n^{-1}\mathbb{N}^{0}}\,\{e_{\ell},}$ $\mathrm{\dots,e_{m}\}}$ so that $\mathrm{\nu_{n}\rightarrow \nu}$ where all limits are strictly positive. Furthermore, we can always guarantee that $\mathrm{\left|\nu_{n}-n\,\nu\right|}$ $\mathrm{\in\mathcal{O}\left(1\right)}$. Set $$\mathrm{\left(\xi_{n}\right)_{n}=\left( {\bf{M}}(\nu_{n})\right)_{n}\subset {\bm{\mathfrak{P}}}_{n^{-1}\mathbb{N}^{0}}.}$$
The first claim follows by $\mathrm{\xi_{n}=\mathbf{M}\left(\nu_{n}\right)\rightarrow \mathbf{M}\left(\nu\right)=\xi}$, the second claim is a direct consequence of $\mathrm{\left|\xi_{n}-n\xi\right|\leq\left\Vert \mathbf{M}\right\Vert _{\infty}\left|\nu_{n}-n\nu\right|\in\mathcal{O}\left(1\right)}$ and the third, resp.\ fifth claim follows by construction. The fourth claim results from equation \ref{Equation In Order To Guarantee}:\ We have $\mathrm{\nu_{j,n}=0}$ for all $\mathrm{j}$ with $\mathrm{1\leq j \leq \ell-1}$ and $\mathrm{\nu_{j,n}>0}$ for all $\mathrm{j}$ with $\mathrm{\ell\leq j\leq m}$ and $\mathrm{n}$ big enough which follows by $\mathrm{\nu_{n}\rightarrow \nu \in Relint\, \mathfrak{Conv}}$ $\mathrm{\{e_{\ell},\dots,e_{m}\}}$. Hence, we have $\mathrm{J_{0}=\left\{1,\dots,\ell-1\right\}}$ and $\mathrm{J_{1}=\left\{\ell,\dots,m\right\}}$. \end{proof}
The following proposition will be the essential step in order to prove {\bf{Theorem 1}}.
\begin{prop}\label{Proposition Existence of Tame Sequences} If $\mathrm{\mathbf{L}\rightarrow \mathbf{X}}$ is as above and $\mathrm{\xi\in Im(\mu)}$, then for each $\mathrm{x \in \mathbf{M}_{J}}$ there exists a sequence $$\mathrm{\left(s^{J}_{n}\right)_{n}, s^{J}_{n}\in H^{0}(\mathbf{X},\mathbf{L}^{n})}$$ of $\mathrm{\xi^{J}_{n}}$-eigensections with the following characteristic properties:
\begin{enumerate} \item $\mathrm{\frac{\xi^{J}_{n}}{n}\rightarrow \xi}$ where $\mathrm{\vert\xi^{J}_{n}-n\,\xi\vert\in \mathcal{O}\left(1\right)}$\\[-0.6 cm] \item The set $\mathrm{\mathbf{X}(s^{J}_{n}) \coloneqq\{x\in \mathbf{X}\!:\! s_{n}^{J}(x)\neq 0\}}$ is independent of $\mathrm{n}$ for $\mathrm{n}$ big enough. \\[-0.6 cm] \item $\mathrm{\mathbf{X}^{J}\coloneqq\pi^{-1}\left(\pi\left(\mathbf{M}_{J}\right)\right)\subset \mathbf{X}(s^{J}_{n})}$.\\[-0.6 cm] \item $\mathrm{\mathbf{X}^{J} \cap \mathbb{T}.\mu^{-1}\left(\xi\right)=\mathbf{X}^{J} \cap \mathbb{T}.\mu^{-1}\left(\frac{\xi^{J}_{n}}{n}\right)}$ for all $\mathrm{n}$ big enough.
\end{enumerate} \end{prop} \begin{proof} By {\bf{Lemma \ref{Lemma Convex Hull Moment Map}}} we know that $\mathrm{\mu\left(cl\left(\mathbb{T}.x\right)\right)=\mathfrak{Conv}\, {\bm{\mathfrak{S}}}_{J}}$. By applying {\bf{Lemma \ref{Lemma Convex Polytope}}} we find a sequence $$\mathrm{\left(\xi_{n}^{J,\prime}\right)_{n}\subset\mathfrak{Conv}\,{\bm{\mathfrak{S}}}_{J}}$$ so that $$\mathrm{\xi^{J,\prime}_{n}=\sum_{j=1}^{Card\,J}\nu_{j,n}^{J}\xi_{j}}$$ where $\mathrm{n\cdot\nu_{j,n}^{J}\in \mathbb{N}\cup\left\{0\right\}}$ for all $\mathrm{n\in \mathbb{N}}$ and $\mathrm{\left(\xi^{\prime}_{n}\right)_{n}}$ converges to $\mathrm{\xi}$ so that $\mathrm{\vert \xi^{J,\prime}_{n}-n\,\xi\vert\in \mathcal{O}\left(1\right)}$. Note that we have $\mathrm{n\cdot\sum_{j=1}^{Card J}\nu_{j,n}^{J}=n}$ and that $$\mathrm{\xi^{J,\prime}_{n}=\sum_{j=1}^{Card\,J}\nu_{j,n}^{J}\xi_{j}=\sum_{j\in J_{1}}\nu_{j,n}^{J}\xi_{j}}$$ (recall that, according to {\bf{Lemma \ref{Lemma Convex Polytope}}}, we have a partition $\mathrm{J=J_{0}\cup J_{1}}$, $\mathrm{J_{0}\neq\varnothing}$, where $\mathrm{\nu_{j,n}^{J}=0}$ for $\mathrm{j\in J_{0}}$). Consider $$\mathrm{s_{n}^{J}\coloneqq\prod_{j=1}^{Card\, J}s_{j}^{n\cdot\nu_{j,n}^{J}}\in H^{0}(\mathbf{X},\mathbf{L}^{n\cdot\sum_{j=1}^{Card J}\nu_{j,n}^{J}})=H^{0}(\mathbf{X},\mathbf{L}^{n}),}$$ which defines a sequence of eigensections whose associated weight vectors $\mathrm{\xi^{J}_{n}}$ approximate $\mathrm{\xi}$: $$\mathrm{\frac{\xi_{n}^{J}}{n}=\sum_{j=1}^{Card\,J}\nu_{j,n}^{J}\xi_{j}\rightarrow \xi.}$$ Furthermore, by the fourth claim of {\bf{Lemma \ref{Lemma Convex Polytope}}}, we know that $\mathrm{\nu_{j,n}=0}$ for all $\mathrm{j\in J_{0}}$ and $\mathrm{\nu_{j,n}>0}$ for all $\mathrm{j\in J_{1}}$ for all $\mathrm{n}$ big enough. Therefore we deduce $$\mathrm{\mathbf{X}\left(s^{J}_{n}\right)=\complement\bigcup_{j\in J_{1}}\left\{s_{j}=0\right\}}$$ for all $\mathrm{n}$ big enough which proves the second claim.
The third property can be proved as follows: If $\mathrm{z\in \pi^{-1}\left(\pi\left(\mathbf{M}_{J}\right)\right)}$, then we have $\mathrm{cl\left(\mathbb{T}.z\right)}$ $\mathrm{\cap \mathbb{T}.\mathbf{M}_{J}\neq \varnothing}$. Since $\mathrm{cl\left(\mathbb{T}.z\right)}$ is $\mathrm{\mathbb{T}}$-invariant, it follows that $\mathrm{\mathbb{T}.\mathbf{M}_{J}\subset cl\left(\mathbb{T}.z\right)}$. As $\mathrm{s_{n}^{J}\vert \mathbb{T}.\mathbf{M}_{J}\not\equiv 0}$ for all $\mathrm{n}$ big enough, it follows that $\mathrm{s_{n}^{J}\vert cl\left(\mathbb{T}.z\right)\not\equiv 0}$ and hence $\mathrm{s_{n}^{J}\left(z\right)\neq 0}$ for all $\mathrm{n}$ big enough.
It remains to verify the fourth claim. For this, let $\mathrm{x\in \mathbf{X}^{J}\cap \mathbb{T}.\mu^{-1}\left(\xi\right)}$. Note that we can assume that $\mathrm{x}$ is not a $\mathbb{T}$ fixed point: If $\mathrm{x}$ is a $\mathbb{T}$ fixed point, then it follows $\mathrm{x\in \mathbf{M}_{J}\subset \mu^{-1}\left(\xi\right)}$ and hence $\mathrm{\xi_{j}=\xi}$ for all $\mathrm{j\in J}$. In particular, we find $\mathrm{\xi=n^{-1}\xi^{J}_{n}}$ for all $\mathrm{n}$ big enough, so the fourth claim is immediate. In the sequel, let $\mathrm{x\in \mathbf{X}^{J}\cap \mathbb{T}.\mu^{-1}\left(\xi\right)}$ be not a $\mathrm{\mathbb{T}}$ fixed point. Observe that $\mathrm{x\in \mathbb{T}.\mathbf{M}_{J}}$ and hence $\mathrm{\xi\in\mu\left(\mathbb{T}.x\right)}$ or $$\mathrm{\xi\in Relint\,\mu\left(cl\left(\mathbb{T}.x\right)\right)=Relint\,\mathfrak{Conv}\,{\bm{\mathfrak{S}}}_{J}}$$ by {\bf{Lemma \ref{Lemma Convex Hull Moment Map}}}. Since $\mathrm{n^{-1}\xi^{J,\prime}_{n}\rightarrow \xi}$ and $\mathrm{\xi\in Relint\,\mathfrak{Conv}\,{\bm{\mathfrak{S}}}_{J}}$, we have $$\mathrm{n^{-1}\xi^{J,\prime}_{n}\in Relint\,\mu\left(cl\left(\mathbb{T}.x\right)\right)=Relint\,\mathfrak{Conv}\,{\bm{\mathfrak{S}}}_{J}}$$ for all $\mathrm{n}$ big enough as well. Therefore we deduce $$\mathrm{x\in \mathbb{T}.\mu^{-1}\left(n^{-1}\xi^{J,\prime}_{n}\right)}$$ for all $\mathrm{n}$ big enough, which proves the inclusion $\mathrm{"\!\!\subset\!"}$.
Now, let $\mathrm{x\in \mathbf{X}^{J} \cap \mathbb{T}.\mu^{-1}\left(n^{-1}\xi^{J}_{n}\right)}$. Note that, as in the previous case, we can assume that $\mathrm{x}$ is not a $\mathrm{\mathbb{T}}$-fixed point. Moreover, let $\mathrm{\mathbb{T}.z_{x}}$ be the unique closed orbit in $\mathrm{\pi^{-1}\left(\pi\left(x\right)\right)}$ i.e.\ $$\mathrm{\pi^{-1}\left(\pi\left(x\right)\right)\cap \mathbb{T}.\mu^{-1}\left(\xi\right)=\mathbb{T}.z_{x}.}$$ Note that $\mathrm{z_{x}\in \mathbb{T}.\mathbf{M}_{J}}$. If the claim were false, i.e.\ if $\mathrm{\mathbb{T}.x\neq\mathbb{T}.z_{x}}$, then we would deduce $$\mathrm{\xi\in \mathfrak{Conv}\,{\bm{\mathfrak{S}}}_{J}= \mu\left(cl\left(\mathbb{T}.z_{x}\right)\right)\subset bd\,\mu\left(cl\left(\mathbb{T}.x\right)\right)}$$ where $\mathrm{n^{-1}\xi^{J,\prime}_{n}\in \mu\left(cl\left(\mathbb{T}.z_{x}\right)\right)}$ for all $\mathrm{n}$. In particular, by the above inclusion, it then follows that $\mathrm{n^{-1}\xi^{J,\prime}_{n}\notin \mu\left(\mathbb{T}.x\right)}$ for all $\mathrm{n}$ in contradiction to $\mathrm{x\in \mathbf{X}{J} \cap \mathbb{T}.\mu^{-1}\left(n^{-1}\xi^{J}_{n}\right)}$ for all $\mathrm{n}$ big enough. Therefore, the assumption is false and it follows that $\mathrm{x\in \mathbb{T}.\mathbf{M}_{J}}$ and hence $\mathrm{\xi\in\mu\left(\mathbb{T}.x\right)}$ or $\mathrm{x\in \mathbb{T}.\mu^{-1}\left(\xi\right)}$ which proves $\mathrm{"\!\!\supset\!"}$. \end{proof}
After having shown {\bf{Proposition \ref{Proposition Existence of Tame Sequences}}} we can prove that the set of semistable points $\mathrm{\mathbf{X}^{ss}_{\xi}}$ can be covered by the $\mathrm{n}$-stable complements of finitely many sequences $\mathrm{\left(s_{n}^{i}\right)_{n}}$ of eigensections whose associated weight sequences $\mathrm{\left(\xi^{i}_{n}\right)_{n}}$ approximate the ray $\mathrm{\mathbb{R}^{\geq 0}\xi}$.
\begin{theo_1*} \textnormal{[\scshape{Existence of Tame Sequences}]}
\\ If $\mathrm{\mathbf{L}\rightarrow \mathbf{X}}$ is as above and $\mathrm{\xi\in Im\left(\mu\right)}$, then we can find a finite cover $$\mathrm{\left\{\mathbf{X}^{i}\right\}_{i\in I}\text{ of }\mathbf{X}^{ss}_{\xi}}$$ consisting of open, $\mathrm{\pi}$-saturated subsets and a finite collection $$\mathrm{\left\{\left( s_{n}^{i} \right)_{n}\right\}_{i\in I},\text{ where }s_{n}^{i}\in H^{0}(\mathbf{X},\mathbf{L}^{n})}$$ of sequences consisting of $\mathrm{\mathbb{T}}$-eigensections such that the following properties are fulfilled: \begin{enumerate} \item $\mathrm{\frac{\xi^{i}_{n}}{n}\rightarrow \xi}$ where $\mathrm{\vert \xi^{i}_{n}-n\,\xi\vert\in \mathcal{O}\left(1\right)}$ for each $\mathrm{i}$.
\item $\mathrm{\mathbf{X}^{i}\subset \mathbf{X}(s_{n}^{i})}$ for all $\mathrm{n}$ big enough.
\item $\mathrm{\mathbf{X}^{i}\cap \mathbb{T}.\mu^{-1}\left(\xi\right)=\mathbf{X}^{i}\cap \mathbb{T}.\mu^{-1}\left(\frac{\xi^{i}_{n}}{n}\right)}$ for all $\mathrm{n\geq N_{0}}$. \end{enumerate} \end{theo_1*} \begin{proof} First of all choose an indexing $\mathrm{I}$ of all non-empty $\mathrm{\mathbf{M}_{J}}$ and recall that the collection $\mathrm{\left\{\mathbf{M}_{i}\right\}_{i\in I}}$ defines a finite cover $\mathrm{\mu^{-1}\left(\xi\right)}$. Hence, the corresponding collection $\mathrm{\left\{\mathbf{X}^{i}\right\}_{i\in I}}$ of open, $\mathrm{\pi}$-saturated subset $\mathrm{\mathbf{X}^{i}=\pi^{-1}\left(\pi\left(\mathbf{M}_{i}\right)\right)}$ defines a finite cover of $\mathrm{\mathbf{X}^{ss}_{\xi}}$ and the claim is a direct consequence of {\bf{Proposition \ref{Proposition Existence of Tame Sequences}}}. \end{proof}
Before we proceed with the proof of {\bf{Proposition \ref{Proposition Uniform Convergence Potential Functions in the Abelian Case}}}, we consider the following example of {\bf{Theorem 1}}.
\begin{example} \textnormal{Let $\mathrm{\mathbf{X}=\mathbf{\Sigma}_{m}}$, $\mathrm{m\in \mathbb{N}}$, be the $\mathrm{m}$-th {\scshape{Hirzebruch}}-Surface (cf. \cite{Hir}) which is defined as the projectivization $\mathrm{\mathbb{P}\left(\mathcal{O}_{\mathbb{C}\mathbb{P}^{1}}\left(1\right)\oplus \mathcal{O}_{\mathbb{C}\mathbb{P}^{1}}\left(m\right)\right)}$ and which is isomorphic to the hypersurface $\mathrm{\{z_{0}^{m}\zeta_{1}}$ $\mathrm{-z_{1}^{m}\zeta_{2}=0\}\subset \mathbb{C}\mathbb{P}^{1}\times \mathbb{C}\mathbb{P}^{2}}$.}
\textnormal{Consider the $\mathrm{\mathbb{T}=\mathbb{C}^{*}}$-action on $\mathrm{\mathbb{C}\mathbb{P}^{5}}$ given by $\mathrm{t.[u]=[u_{0}\!:\!t\,u_{1}\!:\!t\,u_{2}\!:\!u_{3}\!:\!t\,u_{4}\!:\!t\,u_{5}]}$ which pulls back to the $\mathrm{\mathbb{C}^{*}}$-action on $\mathrm{\mathbb{C}\mathbb{P}^{1}\times \mathbb{C}\mathbb{P}^{2}}$ given by $$\mathrm{t.\left(\left[z_{0}\!:\!z_{1}\right],\left[\zeta_{0}\!:\!\zeta_{1}\!:\!\zeta_{2}\right]\right)=([z_{0}\!:\!z_{1}],[\zeta_{0}\!:\!t\,\zeta_{1}\!:\!t\,\zeta_{2}])}$$ via the {\scshape{Segre}}-embedding $\mathrm{\sigma_{1,2}\!:\!\mathbb{C}\mathbb{P}^{1}\times \mathbb{C}\mathbb{P}^{2}\hookrightarrow \mathbb{C}\mathbb{P}^{5}}$. Moreover, fix the $\mathrm{\mathbb{C}^{*}}$-linearization on $\mathrm{\mathcal{O}_{\mathbb{C}\mathbb{P}^{5}}\left(1\right)}$ $\mathrm{\rightarrow \mathbb{C}\mathbb{P}^{5}}$ given by $\mathrm{t.[u,\zeta]=[t.u\!:\!\zeta]}$ where we have used the identification $$\mathrm{\mathcal{O}_{\mathbb{C}\mathbb{P}^{5}}\left(1\right)\cong \mathbb{C}\mathbb{P}^{6}\setminus\left\{[0\!:\!{\dots}\!:\!0\!:\!1]\right\}.}$$ It is direct to check that this $\mathrm{\mathbb{C}^{*}}$-linearization can be pulled back to a $\mathrm{\mathbb{C}^{*}}$-linearization on $\mathrm{\mathbf{L}\coloneqq \left(\sigma_{1,2}^{*}\mathcal{O}_{\mathbb{C}\mathbb{P}^{5}}\left(1\right)\right)\vert \mathbf{\Sigma}_{m}}$. Furthermore, the moment map of the $\mathrm{\mathbb{C}^{*}}$-linearization on $\mathrm{\mathcal{O}_{\mathbb{C}\mathbb{P}^{5}}\left(1\right)}$ $\mathrm{\rightarrow \mathbb{C}\mathbb{P}^{5}}$ associated to the standard hermitian metric $\mathrm{h}$ on $\mathrm{\mathcal{O}_{\mathbb{C}\mathbb{P}^{5}}\left(1\right)}$ defined by $$\mathrm{h\left(\left(u_{0},\zeta_{0}\right),\left(u_{0},\zeta_{0}\right)\right)=\left\Vert u\right\Vert ^{-2}\zeta_{0}\overline{\zeta}_{1}}$$ yields a moment map $\mathrm{\mu\!:\!\mathbf{X}\rightarrow \mathfrak{t}^{*}}$ on $\mathrm{X}$ which is given by $$\mathrm{\mu\left(\left[z_{0}\!:\!z_{1}\right],\left[\zeta_{0}\!:\!\zeta_{1}\!:\!\zeta_{2}\right]\right)=\frac{1}{2}\frac{\left|z_{0}\zeta_{1}\right|^{2}+\left|z_{0}\zeta_{2}\right|^{2}+\left|z_{1}\zeta_{1}\right|^{2}+\left|z_{1}\zeta_{2}\right|^{2}}{\left|z_{0}\zeta_{0}\right|^{2}+\left|z_{0}\zeta_{1}\right|^{2}+\left|z_{0}\zeta_{2}\right|^{2}+\left|z_{1}\zeta_{0}\right|^{2}+\left|z_{1}\zeta_{1}\right|^{2}+\left|z_{1}\zeta_{2}\right|^{2}}}.$$}
\textnormal{If $\mathrm{\xi=\frac{1}{2\,\sqrt{2}}\in Im\,\mu=[0,\frac{1}{2}]}$, then $$\mathrm{\mu^{-1}\left(\xi\right)=\left\{\left(\sqrt{2}-1\right)\left(\left|z_{0}\zeta_{1}\right|^{2}+\left|z_{0}\zeta_{2}\right|^{2}+\left|z_{1}\zeta_{1}\right|^{2}+\left|z_{1}\zeta_{2}\right|^{2}\right)-\left|z_{0}\zeta_{0}\right|^{2}-\left|z_{1}\zeta_{2}\right|^{2}=0\right\}\cap \mathbf{\Sigma}_{m}}$$ and $$\mathrm{\mathbf{X}^{ss}_{\xi}=\mathbf{X}\setminus\left(\left\{ \zeta_{0}=0\,\vee\,\zeta_{1}=\zeta_{2}=0\right\}\right) \subset\mathbb{C}\mathbb{P}^{1}\times \mathbb{C}\mathbb{P}^{2}}$$ where $\mathrm{\pi\!:\!\mathbf{X}^{ss}_{\xi}\rightarrow \mathbf{Y}\cong \mathbb{C}\mathbb{P}^{1}}$ can be identified with the restriction $\mathrm{p_{\mathbb{C}\mathbb{P}^{1}}\vert \mathbf{\Sigma}_{m}=\pi}$ of the projection map $\mathrm{p_{\mathbb{C}\mathbb{P}^{1}}\!:\!\mathbb{C}\mathbb{P}^{1}\times \mathbb{C}\mathbb{P}^{2}\rightarrow \mathbb{C}\mathbb{P}^{1}}$. A further analysis of the example shows that $\mathrm{6=dim_{\mathbb{C}}}$ $\mathrm{H^{0}\left(\mathbf{\Sigma}_{m},\mathbf{L}\right)}$ where $$\mathrm{\begin{array}{rclrclrclrclrclrcl} \mathrm{s_{1}} & \mathrm{=} & \mathrm{z_{0}\zeta_{0},} & \mathrm{s_{2}} & \mathrm{=} & \mathrm{z_{0}\zeta_{1},} & \mathrm{s_{3}} & \mathrm{=} & \mathrm{z_{0}\zeta_{2},} & \mathrm{s_{4}} & \mathrm{=} & \mathrm{z_{1}\zeta_{0},} & \mathrm{s_{5}} & \mathrm{=} & \mathrm{z_{1}\zeta_{1},} & \mathrm{s_{6}} & \mathrm{=} & \mathrm{z_{1}\zeta_{2}}\\ \mathrm{\xi_{1}} & \mathrm{=} & \mathrm{0,} & \mathrm{\xi}_{2} & \mathrm{=} & \mathrm{1,} & \mathrm{\xi_{3}} & \mathrm{=} & \mathrm{1,} & \mathrm{\xi_{4}} & \mathrm{=} & \mathrm{0}, & \mathrm{\xi_{5}} & \mathrm{=} & \mathrm{1} & \mathrm{\xi_{6}} & \mathrm{=} & \mathrm{1}.\end{array}}$$ Moreover, it is direct to check that $\mathrm{\mathbf{M}_{J}\neq \varnothing}$ if and only if $\mathrm{J\in\left\{\left\{ 1,3\right\} ,\left\{ 4,5\right\},J\right\}}$. A tame collection is for example given by $$\mathrm{\left\{\mathbf{X}^{i}\right\}_{i=1,2}=\left\{\pi^{-1}\left(\mathbb{C}\mathbb{P}^{1}\setminus\left\{[1\!:\!0]\right\}\right),\,\pi^{-1}\left(\mathbb{C}\mathbb{P}^{1}\setminus\left\{[0\!:\!1]\right\}\right)\right\}}$$ where $$\mathrm{\left\{\left(s^{i}_{n}\right)_{n}\right\}_{i=1,2}=\left\{\Big(s_{1}^{n-\left\lfloor \frac{n}{2}\right\rfloor }s_{3}^{\left\lfloor \frac{n}{2}\right\rfloor }\Big)_{n},\,\Big(s_{4}^{n-\left\lfloor \frac{n}{2}\right\rfloor }s_{5}^{\left\lfloor \frac{n}{2}\right\rfloor }\Big)_{n}\right\}. \,\boldsymbol{\Box}}$$} \end{example}
We complete this section by proving the following proposition. \begin{prop}\label{Proposition Uniform Convergence Potential Functions in the Abelian Case} If $\mathrm{\left(s^{i}_{n}\right)_{n}}$ is a tame collection, then the associated collection of sequences of strictly plurisubharmonic potentials $\mathrm{\left(\varrho^{i}_{n}\right)_{n}}$ given by $$\mathrm{\varrho_{n}^{i}=-\frac{1}{n}\mathbf{log}\,\vert s_{n}^{i}\vert^{2}\label{Notation Sequence of S.p.s.h. Functions Associated to the Tame Sequence}}$$ converges uniformly on every compact set of $\mathrm{\mathbf{X}^{i}}$ to a smooth strictly plurisubharmonic function $\mathrm{\varrho^{i}\!:\!\mathbf{X}^{i}\rightarrow \mathbb{R}}$\label{Notation S.p.s.h. Limit Function Associated to the Tame Sequence}. Moreover, the same is true for all its derivatives. \end{prop}
\begin{proof} First of all recall (cf.\ proof of {\bf{Proposition \ref{Proposition Existence of Tame Sequences}}}) that each sequence $\mathrm{\left(s^{i}_{n}\right)_{n}}$ is given by $$\mathrm{s_{n}^{J}\coloneqq\prod_{j=1}^{Card J}s_{j}^{n\cdot\nu_{j,n}^{J}},}$$ where $\mathrm{J\subset \left\{1,\dots,m\right\}}$ is a finite, suitable index set and $\mathrm{\big(\nu_{j,n}^{J}\big)_{n}}$ a sequence of integers such that $$\mathrm{\sum_{j=1}^{Card\,J}\nu^{J}_{j,n}\xi _{j}=\sum_{j\in J_{1}}\nu_{j,n}^{J}\xi_{j}\rightarrow \xi}$$ where $\mathrm{J=J_{0}\cup J_{1}}$, $\mathrm{J_{1}\neq \varnothing}$, $\mathrm{\nu_{j}=0}$ for all $\mathrm{j\in J_{0}}$ and all $\mathrm{n}$ big enough, resp.\ $\mathrm{\nu_{j}>0}$ for all $\mathrm{j\in J_{1}}$ and all $\mathrm{n}$ big enough. Hence, it follows that $$\mathrm{\varrho^{i}_{n}=-\sum_{j\in J_{1}}\nu^{J}_{j,n}\mathbf{log}\,\vert s_{j}\vert^{2}, \nu_{j,n}^{J}>0\text{ for }j\in J_{1}}$$ for all $\mathrm{n}$ big enough. Recall that $\mathrm{\mathbf{X}\left(s^{J}_{n}\right)=\complement\bigcup_{j\in J_{1}}\left\{s_{j}=0\right\}}$ for all $\mathrm{n}$ big enough. As the sequences $\mathrm{\big(\nu_{j,n}^{J}\big)_{n}}$ are convergent with strictly positive limits for all $\mathrm{j\in J_{1}}$, it follows that $\mathrm{\varrho^{i}_{n}}$ converges uniformly on compact subsets of $\mathrm{\mathbf{X}^{i}}$ in all derivatives to the smooth s.p.s.h. ($\mathrm{\coloneqq}${\itshape{s}}trictly {\itshape{p}}luri{\itshape{s}}ub{\itshape{h}}armonic)\ function $\mathrm{\varrho^{i}\!:\!\mathbf{X}^{i}\rightarrow \mathbb{R}}$ given by $$\mathrm{\varrho^{i}=-\sum_{j\in J_{1}}\,\underset{n\rightarrow\infty}{lim}\nu_{j,n}^{J}\,\mathbf{log}\,\left|s_{j}\right|^{2}.}$$ \end{proof}
\section{Uniform Localization Proposition}\label{Uniform Localization Proposition}
In this section fix a tame sequence $\mathrm{\left(s_{n}^{i}\right)_{n}}$ and let $\mathrm{\left(\varrho_{n}^{i}\right)_{n}}$ be the associated sequence of strictly plurisubharmonic functions. Moreover, recall that by {\bf{Theorem 1}}, the subset $\mathrm{\mathbf{X}^{i}}$ is $\mathrm{\pi}$-saturated and contained in the complement of the zero set $\mathrm{\mathbf{X}\left(s_{n}^{i}\right)}$ of $\mathrm{s_{n}^{i}}$ for all $\mathrm{n}$ big enough.
We start this section by recalling the following basic fact (cf.?\cite{He-Hu2}, {\bf{{pp.\ }\oldstylenums{310}-\oldstylenums{349}}}).
\begin{lemma} Let $\mathrm{\varrho^{i}\!:\!\mathbf{X}^{i}\rightarrow \mathbb{R}^{\geq 0}}$ and let $\mathrm{\mathbf{Y}^{i}=\pi\left(\mathbf{X}^{i}\right)\label{Notation Image of X_i Under Projection Map}}$ be as above, then for each $\mathrm{y\in \mathbf{Y}^{i}}$ there exists a $\mathrm{\pi}$-saturated, open subset $\mathrm{\mathbf{V}^{i}=\pi^{-1}\left(\mathbf{W}^{i}\right)}$\label{Notation V^i}\label{Notation W^i} of $\mathrm{\mathbf{X}^{ss}_{\xi}}$ where $\mathrm{\mathbf{W}^{i}\subset \mathbf{Y}^{i}}$ is open so that $\mathrm{\left(\varrho^{i}\times \pi\right)\vert \mathbf{V}^{i}}$ is proper. \end{lemma}
Note that it is always possible to assume that $\mathrm{\mathbf{W}^{i}\subset \mathbf{Y}^{i}}$ is a compact neighborhood which we will do form now on. Moreover, since $\mathrm{\mathbf{Y}}$ is compact finitely many of those compact neighborhoods $\mathrm{\mathbf{W}^{i}}$ will cover $\mathrm{\mathbf{Y}}$.
In the sequel, we will work with the normalized s.p.s.h function $\mathrm{\hat{\varrho}^{i}}$, resp. $\mathrm{\hat{\varrho}^{i}_{n}}$ on $\mathrm{\mathbf{X}^{i}}$ defined by $$\mathrm{\hat{\varrho}^{i}\coloneqq \varrho^{i}-\pi^{*}\varrho^{i}_{red}\text{ resp.\! }\hat{\varrho}^{i}_{n}\coloneqq \varrho^{i}_{n}-\pi^{*}\varrho^{i}_{n,red}.}$$ Since $\mathrm{\pi^{*}\varrho^{i}}$ is continuous and since $\mathrm{\mathbf{W}^{i}\subset \mathbf{Y}^{i}}$ was chosen to be a compact neighborhood, the above lemma remains valid for $\mathrm{\hat{\varrho}^{i}}$, resp{.\! }for $\mathrm{\hat{\varrho}^{i}_{n}}$. In the sequel, we will set $\mathrm{\varrho^{i}=\hat{\varrho}^{i}}$ and $\mathrm{\hat{\varrho}^{i}_{n}=\varrho^{i}_{n}}$.
We define $$\mathrm{T\left(\epsilon,\mathbf{W}^{i}\right)\coloneqq \left(\varrho^{i}\times \pi\right)^{-1}\left([0,\epsilon]\times \mathbf{W}^{i}\right)\label{Notation Compact Neighborhood Tube}.}$$ By the above lemma, $\mathrm{T\left(\epsilon,\mathbf{W}^{i}\right)}$ is a relatively compact subset of $\mathrm{\mathbf{X}^{i}\subset \mathbf{X}^{ss}_{\xi}}$ which is $\mathrm{T}$-invariant by its definition. Furthermore, define $$\mathrm{T^{c}\left(\epsilon,\mathbf{W}^{i}\right)\coloneqq \left(\varrho^{i}\times \pi\right)^{-1}\left([\epsilon,\infty)\times \mathbf{W}^{i}\right)\label{Notation Complement of Compact Neighborhood Tube}.}$$
\begin{remark} \label{Preparation Remark} Note that there exists $\mathrm{N_{0}\left(\epsilon\right)\in\mathbb{N}}$ so that $$\mathrm{\mu^{-1}\left(n^{-1}\xi^{i}_{n}\right)\cap\pi^{-1}\left(\mathbf{W}^{i}\right)\subset T\left(\epsilon,\mathbf{W}^{i}\right)\text{ for all }n\geq N_{0}\left(\epsilon\right).}$$ Otherwise there would exist a sequence $\mathrm{\left(x_{n}\right)_{n}}$ in $\mathrm{\mu^{-1}\left(n^{-1}\xi^{i}_{n}\right)\cap\pi^{-1}\left(\mathbf{W}^{i}\right)}$ so that $\mathrm{x_{n}}$ $\mathrm{\notin}$ $\mathrm{T(\epsilon,}$ $\mathrm{\mathbf{W}^{i})}$ for all $\mathrm{n\in \mathbb{N}}$ big enough. Since $\mathrm{\mathbf{X}}$ and $\mathrm{\mathbf{W}^{i}}$ are compact, we can find a convergent subsequence $\mathrm{\left(x_{n_{j}}\right)_{j}}$ so that $\mathrm{x_{n_{j}}\rightarrow x_{0}}$ where $\mathrm{\pi\left(x_{0}\right)\in \mathbf{W}^{i}}$ and $\mathrm{x_{0}\in \mu^{-1}\left(\xi\right)}$ which follows by $\mathrm{x_{n_{j}}\in \mu^{-1}(n^{-1}\xi^{i}_{n_{j}})}$ and $\mathrm{{n_{j}}^{-1}\xi^{i}_{n_{j}}\rightarrow \xi}$. Hence, we have $\mathrm{x_{0}\in \mu^{-1}\left(\xi\right)\cap \pi^{-1}\left(\mathbf{W}^{i}\right)}$. On the other hand, we have assumed that $\mathrm{x_{n_{j}}\notin T\left(\epsilon,\mathbf{W}^{i}\right)}$ for all $\mathrm{n}$ big enough. However, since $\mathrm{T\left(\epsilon,\mathbf{W}^{i}\right)}$ is an open neighborhood of $\mathrm{\mu^{-1}\left(\xi\right)\cap \pi^{-1}\left(\mathbf{W}^{i}\right)}$ in $\mathrm{\pi^{-1}\left(\mathbf{W}^{i}\right)}$ this yields a contradiction to $\mathrm{x_{0}\in \mu^{-1}\left(\xi\right)\cap \pi^{-1}\left(\mathbf{W}^{i}\right)}$. \end{remark}
Before we prove {\bf{Theorem 2}}, we need the following preparation.
\begin{lemma} \label{Preparation Lemma} Let $\mathrm{\left(s_{n}^{i}\right)_{n}}$ be a tame sequence as above, $\mathrm{\pi^{-1}\left(y\right)\subset \mathbf{X}^{i}}$ and let $\mathrm{\mathbb{T}.z_{y}}$ be the unique closed orbit in $\mathrm{\pi^{-1}\left(y\right)}$ then it follows that $\mathrm{\varrho^{i}_{n}\vert \pi^{-1}\left(y\right)\cap \mathbb{T}.z_{y}}$ takes on a unique minimum along the set $$\mathrm{T.\mu^{-1}\left(n^{-1}\xi^{i}_{n}\right)\cap \pi^{-1}\left(y\right)}$$ which is contained in $\mathrm{\mathbb{T}.\mu^{-1}\left(\xi\right)\cap \pi^{-1}\left(y\right)}$. \end{lemma} \begin{proof} Note that since $\mathrm{\left(s_{n}^{i}\right)_{n}}$ is a tame sequence we have $$\mathrm{\mathbb{T}.\mu^{-1}\left(\xi\right)\cap \mathbf{X}^{i}=\mathbb{T}.\mu^{-1}\left(n^{-1}\xi^{i}\right)\cap \mathbf{X}^{i}}$$ by the third claim of {\bf{Theorem 1}}. This shows that the set $$\mathrm{T.\mu^{-1}\left(n^{-1}\xi^{i}_{n}\right)\cap \pi^{-1}\left(y\right)}$$ is contained in $\mathrm{\mathbb{T}.\mu^{-1}\left(\xi\right)\cap \pi^{-1}\left(y\right)}$ so the second claim is proved.
The first claim is a direct consequence of the fact that the unique minimum of $\mathrm{\varrho^{i}_{n}}$ on $$\mathrm{\mathbb{T}.\mu^{-1}\left(n^{-1}\xi^{i}_{n}\right)\cap \pi^{-1}\left(y\right)}$$ is known to be equal to $\mathrm{T.\mu^{-1}\left(n^{-1}\xi^{i}_{n}\right)\cap \pi^{-1}\left(y\right)}$. \end{proof}
We can now prove the uniform Localization Proposition.
\begin{theo_2} \label{Theorem Localization of the Sequence of Potential Functions} \textnormal{[\scshape{Uniform Localization of the Potential Functions}]}
\\ Let $\mathrm{\left(s_{n}^{i}\right)_{n}}$ be a tame sequence and $\mathrm{\left(\varrho_{n}^{i}\right)_{n}}$ the associated sequence of strictly plurisubharmonic functions. Let $\mathrm{\mathbf{W}^{i}\subset\pi\left(\mathbf{X}^{i}\right)}$ and $\mathrm{\epsilon>0}$ be as above and let $\mathrm{\delta>0}$ be given. Then there exists $\mathrm{N_{0}\in\mathbb{N}}$ so that $$\mathrm{\varrho_{n}^{i}\left(x\right)\geq\epsilon-\delta}$$ for all $\mathrm{x\in T^{c}\left(\epsilon,\mathbf{W}^{i}\right)}$ and all $\mathrm{n\geq N_{0}}$. \end{theo_2} \begin{proof} First of all, by {\bf{Remark \ref{Preparation Remark}}}, we can assume that \begin{equation} \mathrm{\mu^{-1}\left(n^{-1}\xi^{i}_{n}\right)\cap\pi^{-1}\left(\mathbf{W}^{i}\right)\subset T\left(\epsilon,\mathbf{W}^{i}\right)\text{ for all }n\geq N_{0}\left(\epsilon\right).}\label{Critical Subsets Contained in Tube} \end{equation} Moreover, note that $\mathrm{\varrho^{i}_{n}}$ converges uniformly on relatively compact sets and hence on $\mathrm{T\left(\epsilon,\mathbf{W}^{i}\right)}$. Therefore, we can find an $\mathrm{N_{0}\in \mathbb{N}}$ so that \begin{equation} \mathrm{\varrho^{i}_{n}\left(x\right)\geq\epsilon-\delta}\label{Convergence on Boundary} \end{equation} for all $\mathrm{x}$ in the compact subset $\mathrm{\left(\varrho^{i}\times \pi\right)^{-1}\left(\left\{\epsilon\right\}\times \mathbf{W}^{i}\right)}$ and all $\mathrm{n\geq N_{0}}$. We continue the proof by considering the following two cases:
1) First of all let, $\mathrm{x\in T^{c}\left(\epsilon,\mathbf{W}^{i}\right)\cap \mathbb{T}.z_{x}}$ where $\mathrm{\mathbb{T}.z_{x}}$ is the unique closed orbit in the fiber $\mathrm{{\pi^{-1}\left(\pi\left(x\right)\right)}}$. We have to show that $$\mathrm{\varrho^{i}_{n}\vert T^{c}\left(\epsilon,\mathbf{W}^{i}\right)\cap \mathbb{T}.z_{x}>\epsilon-\delta.}$$
Applying {\bf{Lemma \ref{Preparation Lemma}}}, we know that the restriction $\mathrm{\varrho^{i}_{n}\vert \mathbb{T}.z_{x}}$ of the strictly plurisubharmonic function $\mathrm{\varrho^{i}_{n}}$ on the unique closed orbit $\mathrm{\mathbb{T}.z_{x}}$ in the fiber $\mathrm{\pi^{-1}}$ $\mathrm{\left(\pi\left(x\right)\right)}$ $\mathrm{\subset \mathbf{X}^{i}}$ takes on its minimum along $$\mathrm{\mu^{-1}\left(n^{-1}\xi^{i}\right)\cap \pi^{-1}\left(\pi\left(x\right)\right)\subset \mathbb{T}.z_{x}.}$$
If $\mathrm{\mathfrak{m}_{x}\coloneqq \mathfrak{\bm{Ann}}\,\mathfrak{t}_{x}\subset \mathfrak{t}^{*}}$ denotes the annihilator $\mathrm{\left\{ \eta\in\mathfrak{t}^{*}:\,\left\langle \eta,\xi\right\rangle =0\text{ for all }\xi\in\mathfrak{t}_{x}\right\} }$ of the isotropy $\mathrm{\mathfrak{t}_{x}}$, then $\mathrm{\mathbb{T}.z_{x}}$ is isomorphic to the homogenous vector bundle with typical fiber $\mathrm{\mathfrak{m}_{x}}$: $$\mathrm{\mathbb{T}.z_{x}\cong T\times^{T_{x_{z}}}\mathfrak{m}_{z}}$$ Note that the zero section in $\mathrm{T\times^{T_{x_{z}}}\mathfrak{m}_{z}}$ is mapped under the above identification to the $\mathrm{T}$-orbit $$\mathrm{T.\mu^{-1}\left(\xi\right)\cap \mathbb{T}.z_{x}.}$$ Moreover, using this identification, it follows that the restriction of $\mathrm{\varrho^{i}_{n}}$ on $\mathrm{\mathbb{T}.z_{x}}$ yields a smooth function on $\mathrm{T\times^{T_{x_{z}}}\mathfrak{m}_{z}}$ whose restriction on each fiber $\mathrm{[\left\{t\right\}\times \mathfrak{m}_{z}]}$, $\mathrm{t\in T}$ is a strictly convex function with a unique minimum given by $\mathrm{[\left\{t\right\}\times \left\{m_{n}\right\}]}$ for $\mathrm{m_{n}\in \mathfrak{m}_{z}}$. Note that the tube $\mathrm{T\left(\epsilon,\mathbf{W}^{i}\right)\cap \mathbb{T}.z_{x}}$ is isomorphic to a tube of the zero section in $\mathrm{T\times^{T_{x_{0}}}\mathfrak{m}_{x}}$ and also note that this tube contains $\mathrm{[T\times \left\{m_{n}\right\}]}$ for all $\mathrm{n}$ big enough by the remark at the beginning of this proof which is based on {\bf{Remark \ref{Preparation Remark}}} .
We continue the proof by connecting $\mathrm{x=\left[\left\{ t\right\} \times\left\{ \eta_{x}\right\} \right]}$, for $\mathrm{\eta_{x}\in \mathfrak{m}_{z}}$ suitable, and the unique minimum $\mathrm{m_{n}=[\left\{t\right\}\times \left\{m_{n}\right\}]}$ of $\mathrm{\varrho^{i}_{n}}$ with a line $\mathrm{\Lambda\!:\!\mathbb{R}\rightarrow [\left\{t\right\}\times \mathfrak{m}_{x}]}$ in the vector space $\mathrm{[\left\{t\right\}\times \mathfrak{m}_{x}]}$ so that $\mathrm{\Lambda\left(0\right)=m_{n}}$ and $\mathrm{\Lambda\left(1\right)=x}$. Note that this line intersects $\mathrm{\varrho^{i,-1}\left(\epsilon\right)}$ in, say $\mathrm{\Lambda\left(\tau_{n}\right)=y_{n}}$, where $\mathrm{0<\tau_{n}<x}$, because the minimum of $\mathrm{\varrho^{i}_{n}}$ is contained in the tube $\mathrm{T\left(\epsilon,\mathbf{W}^{i}\right)}$ (for all $\mathrm{n}$ big enough) whereas $\mathrm{x}$ is not by our assumption. To sum up, we have a convex function $\mathrm{\Lambda^{*}\varrho^{i}_{n}}$ on $\mathbb{R}$ with a unique minimum at $\mathrm{0}$ so that $\mathrm{\Lambda^{*}\varrho^{i}_{n}\left(\tau_{n}\right)\geq \epsilon-\delta}$ for all $\mathrm{n}$ big enough by equation \ref{Convergence on Boundary}. Hence, it follows that $\mathrm{\varrho^{i}_{n}\left(x\right)\geq \epsilon-\delta}$ as well for all $\mathrm{n}$ big enough.
2) The next step is to show that the inequality $\mathrm{\varrho^{i}_{n}\left(x\right)\geq\epsilon-\delta }$ also holds for all $\mathrm{x\in T^{c}(\epsilon,}$ $\mathrm{\mathbf{W}^{i})}$ so that $\mathrm{x}$ is not contained in the unique closed orbit $\mathrm{\mathbb{T}.z_{x}}$ of the fiber $\mathrm{\pi^{-1}\left(\pi\left(x\right)\right)}$.
Let us assume that this is not true. By the {\scshape{Hilbert}} Lemma ({cf.\ }\cite{Kra}), we can find a one parameter group $\mathrm{\gamma\!:\!\mathbb{C}\rightarrow \mathbb{T}}$ so that $$\mathrm{\mathrm{\underset{\mathrm{t\rightarrow0}}{\mathrm{lim}}\,\gamma\left(t\right).x\in \mathbb{T}.z_{x}}.}$$
Note that neither the pull back $\mathrm{\gamma^{*}\varrho^{i}}$ nor the pull back $\mathrm{\gamma^{*}\varrho^{i}_{n}}$ attains its $\mathrm{S^{1}}$-invariant minimum on $\mathrm{\mathbb{C}^{*}}$ since otherwise it would follow that $\mathrm{\varrho^{i}\vert Im\left(\gamma\vert \mathbb{C}^{*}\right)}$ and $\mathrm{\varrho^{i}_{n}\vert Im\left(\gamma\vert \mathbb{C}^{*}\right)}$ attain their $\mathrm{S^{1}}$-invariant minimum on $\mathrm{\mathbb{T}.x}$. However, this would yield a contradiction to the assumption $\mathrm{\mathbb{T}.x\neq \mathbb{T}.z_{x}}$ and the claim of {\bf{Lemma} \ref{Preparation Lemma}}. Hence, it follows that $\mathrm{t\mapsto \gamma^{*}\varrho^{i}_{n}\left(t\right)}$ and $\mathrm{t\mapsto \gamma^{*}\varrho^{i}\left(t\right)}$ where $\mathrm{t\in \mathbb{C}^{*}}$ are strictly monotone decreasing when $\mathrm{\vert t\vert\rightarrow 0}$ (they can not be strictly increasing since $\mathrm{s^{i}_{n}}$ does not vanish on $\mathrm{\mathbb{T}.z_{x}}$). The present case can be subdivided into the following cases:
2.a) Assume that $$\mathrm{x_{0}=\mathrm{\underset{\mathrm{t\rightarrow0}}{\mathrm{lim}}\,\gamma\left(t\right).x\in \mathbb{T}.z_{x}\cap T^{c}\left(\epsilon,\mathbf{W}^{i}\right)},}$$ then we have $$\mathrm{\varrho^{i}_{n}\left(x_{0}\right)\geq\epsilon-\delta}$$ for all $\mathrm{n\geq N_{0}}$ by case 1). Since $\mathrm{\gamma^{*}\varrho^{i}_{n}}$ is monotone decreasing for $\mathrm{\vert t\vert\rightarrow 0}$ we deduce that $$\mathrm{\delta-\epsilon\geq\gamma^{*}\varrho^{i}_{n}\left(1\right)=\varrho^{i}_{n}\left(x\right)}$$ for all $\mathrm{n\geq N_{0}}$ as claimed.
2.b) Assume that $$\mathrm{x_{0}=\mathrm{\underset{\mathrm{t\rightarrow0}}{\mathrm{lim}}\,\gamma\left(t\right).x\in \mathbb{T}.z_{x}\cap T\left(\epsilon,\mathbf{W}^{i}\right)},}$$ then there exists $\mathrm{t\in \mathbb{C}^{*}}$ so that $\mathrm{\gamma\left(t\right).x\in \varrho^{i,-1}\left(\epsilon\right)}$ and hence $\mathrm{\varrho^{i}_{n}\left(\gamma\left(t\right).x\right)=\epsilon-\delta}$ for all $\mathrm{n\geq N_{0}}$ by \ref{Convergence on Boundary}. As in case 2.a), it then follows that $\mathrm{\varrho^{i}_{n}\left(x\right)\geq\delta-\epsilon}$ for all $\mathrm{n\geq N_{0}}$ as claimed. \end{proof}
We close this section with the following corollary which slightly generalizes the claim of {\bf{Theorem \ref{Theorem Localization of the Sequence of Potential Functions}}}.
In the context of {\bf{Theorem \ref{Theorem Localization of the Sequence of Potential Functions}}}, fix $\mathrm{m_{0}\in \mathbb{N}}$ and consider the sequence $\mathrm{\big(s^{i}_{n}\cdot s^{i,-1}_{m_{0}}\big)_{n}}$ of meromorphic $\mathrm{\eta^{i,m_{0}}_{n}\coloneqq \xi^{i}_{n}-\xi^{i}_{m_{0}}}$-eigensections. Since we have $\mathrm{s^{i}_{n}\left(x\right)}$ $\mathrm{\neq 0}$ for all $\mathrm{x\in \mathbf{X}^{i}}$ and all $\mathrm{n\in \mathbb{N}}$ and hence in particular for $\mathrm{n=m_{0}}$, we can define $$\mathrm{\varrho^{i,m_{0}}_{n}\coloneqq -\frac{1}{n}\mathbf{log}\,\vert s^{i}_{n}\cdot s^{i,-1}_{m_{0}}\vert^{2}.}$$ It is direct to check that the above definition yields a sequence of smooth, strictly plurisubharmonic, $\mathrm{T}$-invariant functions for all $\mathrm{n\geq m_{0}}$ on the $\mathrm{\pi}$-saturated open set $\mathrm{\mathbf{X}^{i}}$. Furthermore, it is known that $\mathrm{\varrho^{i,m_{0}}_{n}\vert \pi^{-1}\left(y\right)}$ for $\mathrm{y\in \pi\left(\mathbf{X}^{i}\right)}$ takes on its unique minimum along the set $$\mathrm{\mu^{-1}\left(n^{-1}\eta^{i,m_{0}}_{n}\right)\cap \pi^{-1}\left(y\right).}$$ Therefore, after having applied the argumentation of {\bf{Remark \ref{Preparation Remark}}}, we can assume, as at the beginning of the proof of {\bf{Theorem \ref{Theorem Localization of the Sequence of Potential Functions}}}, that $$\mathrm{\mu^{-1}\left(n^{-1}\eta^{i,m_{0}}_{n}\right)\cap\pi^{-1}\left(\mathbf{W}^{i}\right)\subset T\left(\epsilon,\mathbf{W}^{i}\right)\text{ for all }n\text{ big enough.}}$$ The proof of {\bf{Theorem \ref{Theorem Localization of the Sequence of Potential Functions}}} now translates verbatim to the sequence $\mathrm{\left(\varrho^{m_{0}}_{n}\right)_{n}}$ and yields the following corollary.
\begin{cor}\label{Remark Convergence Theorem For The Shifted Sequence} If $\mathrm{\delta>0}$, $\mathrm{m_{0}\in \mathbb{N}}$ fixed, then it follows $$\mathrm{\varrho^{i,m_{0}}_{n}\left(x\right)\geq \epsilon-\delta}$$ for all $\mathrm{x\in T^{c}\left(\epsilon,\mathbf{W}^{i}\right)}$ and all $\mathrm{n}$ big enough. \end{cor}
\section{Fiber Probability Measure Sequence}
\subsection{Definition of $\mathrm{\widehat{\pi}\!:\!\widehat{\mathbf{\,X\,}}\rightarrow \mathbf{Y}}$ and $\mathrm{\mathbf{Y}_{0}}$}\label{Definition of Y_{0}}
As before, let $\mathrm{\mathbf{X}}$ be a purely $\mathrm{m}$-dimensional, normal $\mathrm{\mathbb{T}}$-variety with K\"ahler structure $\mathrm{\omega}$ and let $\mathrm{\mathbf{Y}=\mathbf{X}^{ss}_{\xi}/\!\!/\mathbb{T}}$ be the associated Hilbert quotient. Note that we can always assume $\mathrm{\mathbf{Y}}$ to be purely dimensional, {i.e.\ }$\mathrm{n=}$ $\mathrm{dim_{\mathbb{C}}\,\mathbf{Y}}$: Since $\mathrm{\mathbf{X}}$ is normal by our assumption, it follows (cf.\ \cite{He-Hu1}, {p.\ }{\bf{\oldstylenums{124}}}) that $\mathrm{\mathbf{Y}}$ is normal as well. In particular, it follows that $\mathrm{\mathbf{Y}}$ is locally of pure dimension (cf.\ \cite{Gr-Re}, {p.\ }{\bf{\oldstylenums{125}}}) and by considering the connected components of $\mathrm{\mathbf{Y}}$, which are finite in number, we can confine ourselves to the case where $\mathrm{\mathbf{Y}}$ is of pure dimension $\mathrm{n=}$ $\mathrm{dim_{\mathbb{C}}\,\mathbf{Y}}$.
Now, recall that $\mathrm{\mathbf{X}^{ss}_{\xi}}$ is Zariski open and Zariski dense in $\mathrm{\mathbf{X}}$. Consider the compact variety $\mathrm{\widehat{\mathbf{\,X\,}}}$ \label{Notation X Hat} defined by the normalization of $\mathrm{cl\,{\bf{\Gamma_{\pi}}}}$, where $\mathrm{{\bf{\Gamma_{\pi}}}}$ denotes the graph of $\mathrm{\pi}$, i.e. $$\mathrm{{\bf{\Gamma}}_{\pi}\coloneqq \left\{\left(x,y\right)\in \mathbf{X}^{ss}_{\xi}\times \mathbf{Y}\!:\!\,\pi\left(x\right)=y\right\}\label{Notation Graph of Quotient Map}.}$$ Furthermore, we define the algebraic map $\mathrm{\widehat{\pi}\!:\!\widehat{\mathbf{\,X\,}}\rightarrow \mathbf{Y}}$ \label{Notation Regular Map between cal X and Y} by $\mathrm{\widehat{\pi}\coloneqq p_{\mathbf{Y}}\vert cl\,\mathbf{\Gamma}_{\pi}\circ\zeta}$, where $\mathrm{\zeta\!:\!\widehat{\,\mathbf{X}\,}=}$ $\mathrm{\left(cl\,\mathbf{\Gamma}_{\pi}\right)^{nor}\rightarrow cl\,\mathbf{\Gamma}_{\pi}}$ \label{Notation Normalization Map Zeta} denotes the normalization map and $\mathrm{p_{\mathbf{Y}}}$ the projection map $\mathrm{p_{\mathbf{Y}}\!:\!\mathbf{X}}$ $\mathrm{\times \mathbf{Y}\rightarrow}$ $\mathrm{\mathbf{Y}}$. Moreover, endow $\mathrm{\widehat{\mathbf{\,X\,}}}$ with the $\mathrm{\mathbb{T}}$-action induced by the lift (for its existence see \cite{Gr-Re}, {pp.\! }{\bf{\oldstylenums{164} f.}}) of the $\mathrm{\mathbb{T}}$-action on $\mathrm{cl\,\mathbf{\Gamma}_{\pi}}$, defined by $\mathrm{t.\left(x,y\right)=\left(t.x,y\right)}$ and equip $\mathrm{\widehat{\mathbf{\,X\,}}}$ with a smooth $\mathrm{\left(2,2\right)}$-form $\mathrm{\omega^{\prime}}$ \label{Notation Smooth (2,2,)-form} given by $\mathrm{\omega^{\prime}\coloneqq \left(p_{\mathbf{X}}\vert cl\left({\mathbf{\Gamma}}_{\pi}\right)\circ\zeta\right)^{*}\omega}$. For the sake of completeness, we note the following remark.
\begin{remark} \label{Remark T-Invariance of Fibers} The graph $\mathrm{{\bf{\Gamma}}_{\pi}\mathrm{\subset \mathbf{X}\times \mathbf{Y}}}$ is $\mathrm{\mathbb{T}}$-invariant with respect to the action on $\mathrm{\mathbf{X}\times \mathbf{Y}}$ given by $\mathrm{t.\left(x,y\right)=\left(t.x,y\right)}$. Since $\mathrm{{\bf{\Gamma}}_{\pi}}$ is a Zariski open and Zariski dense subset of $\mathrm{\mathbf{X}}$, it follows that $\mathrm{cl\,{\bf{\Gamma}}_{\pi}}$ is $\mathrm{\mathbb{T}}$-invariant. Moreover, as $\mathrm{\zeta}$ is $\mathrm{\mathbb{T}}$-equivariant, all fibers $\mathrm{\widehat{\pi}^{-1}\left(y\right)}$, $\mathrm{y\in \mathbf{Y}}$ are $\mathrm{\mathbb{T}}$-invariant as well. \end{remark}
Note that since $\mathrm{\mathbf{X}}$ is a assumed to be of pure dimension $\mathrm{m}$, it follows that $\mathrm{cl\,\mathbf{\Gamma}_{\pi}}$ is likewise a purely $\mathrm{m}$-dimensional subvariety of $\mathrm{\mathbf{X}\times \mathbf{Y}}$. As $\mathrm{\zeta}$ is finite, we deduce that $\mathrm{\widehat{\mathbf{\,X\,}}}$ is of pure dimension $\mathrm{m}$ too.
The next step is to find a Zariski open subset $\mathrm{\mathbf{Y}_{0}\subset \mathbf{Y}}$ so that the fibers of the restricted projection $\mathrm{\widehat{\pi}\vert \widehat{\pi}^{-1}\left(\mathbf{Y}_{0}\right)\!:\! \widehat{\pi}^{-1}\left(\mathbf{Y}_{0}\right)\rightarrow \mathbf{Y}_{0}}$ are all purely $\mathrm{k}$-dimensional varieties. The existence of $\mathrm{{\bf{Y}}_{0}}$ is a direct consequence of known facts in complex analysis and algebraic geometry: By a theorem of {\scshape{Cartan}} and {\scshape{Remmert}} ({cf.\! }\cite{Loj}, {p.\ }{\bf{\oldstylenums{271} f.}}), it follows that the subset $\mathrm{{\bf{E}}\subset \widehat{\mathbf{\,X\,}}}$ defined by $$\mathrm{{\bf{E}}\coloneqq \left\{x\in\widehat{\mathbf{\,X\,}}\!:\, k<dim_{\mathbb{C},x}\,\widehat{\pi}^{-1}\left(\widehat{\pi}\left(x\right)\right)\right\}\subset \widehat{\mathbf{\,X\,}}}$$ is a proper analytic subset of $\mathrm{\widehat{\,\mathbf{X}\,}}$ where $\mathrm{k=m-n=dim_{\mathbb{C}}\widehat{\mathbf{\,X\,}}-dim_{\mathbb{C}}\,\mathbf{Y}}$. Hence, by {\scshape{Chow}}'s Theorem it follows that $\mathrm{\mathbf{E}}$ is a proper algebraic subset. Applying the Direct Image Theorem (cf.\ \cite{Gr-Re}, {p.\ }{\bf{\oldstylenums{207}}}), one deduces that the image $\mathrm{\widehat{\pi}\left(\mathbf{E}\right)}$ is a proper analytic subset of $\mathrm{\mathbf{Y}}$. In particular it is a proper algebraic subvariety of $\mathrm{\mathbf{Y}}$. Now, set $\mathrm{\mathbf{Y}_{0}\coloneqq \complement\, \widehat{\pi}\left(\mathbf{E}\right)}$ \label{Notation Subset Y_0} and note that all fibers of $\mathrm{\widehat{\pi}}$ over $\mathrm{\mathbf{Y}_{0}}$ are purely $\mathrm{k}$-dimensional by construction.
For later use, we introduce the following notation: Let $\mathrm{\mathbf{X},\mathbf{Y}}$ be purely dimensional complex spaces ($\mathrm{m}$ $\mathrm{=dim_{\mathbb{C}}\,\mathbf{X}}$, $\mathrm{n}$ $\mathrm{=dim_{\mathbb{C}}\,\mathbf{Y}}$) where $\mathrm{\mathbf{Y}}$ is assumed to be normal and let $\mathrm{F\!:\!\mathbf{X}\rightarrow \mathbf{Y}}$ be a holomorphic map so that all non-empty fibers $\mathrm{F^{-1}\left(y\right)}$ are of pure dimension $\mathrm{k=m-n}$. Then $\mathrm{F}$ is called a $\mathrm{k}$-{\itshape{fibering}}. Note that $\mathrm{\pi\vert \pi^{-1}\left(\mathbf{Y}_{0}\right): \pi^{-1}\left(\mathbf{Y}_{0}\right)\rightarrow \mathbf{Y}_{0}}$ is a $\mathrm{k}$-fibering.
An example for the construction of $\mathrm{\widehat{\,\mathbf{X}\,}}$ as described above is given by the next example.
\begin{example}\label{Example Proper Inclusion} \textnormal{Let $\mathrm{\mathbf{X}=\mathbb{C}\mathbb{P}^{1}\times \mathbb{C}\mathbb{P}^{1}}$ with the K\"aher form $\mathrm{\omega=p_{1}^{*}\omega_{FS}+p_{2}^{*}\omega_{FS}}$ where $\mathrm{p_{i}\!:\!}$ $\mathrm{\mathbf{X}}$ $\mathrm{\rightarrow}$ $\mathrm{\mathbb{C}\mathbb{P}^{1}}$ denotes the $\mathrm{i}$-th projection, $\mathrm{i\in\left\{0,1\right\}}$ equipped with the $\mathrm{\mathbb{T}\cong\mathbb{C}^{*}}$-action given by $$\mathrm{t.\left(\left[z_{0}z_{1}\right],\left[\zeta_{0}\!:\!\zeta_{1}\right]\right)=\left(\left[t.z_{0}\!:\!z_{1}\right],\left[t.\zeta_{0}\!:\!\zeta_{1}\right]\right)}$$ and consider the moment map given by (cf.\ {\bf{Example \ref{Example First Example}}}) $$\mathrm{\mu\left(\left[z_{0}\!:\!z_{1}\right],\left[\zeta_{0}\!:\!\zeta_{1}\right]\right)=\frac{\vert z_{1}\vert^{2}}{\Vert z\Vert^{2}}-\frac{\vert\zeta_{0}\vert^{2}}{\Vert\zeta\Vert^{2}}.}$$ In particular, we have $$\mathrm{\mathbf{X}_{\xi=\frac{1}{2}}^{ss}=\mathbf{X}\setminus\left\{ \left\{ z_{1}=0\right\} \cup\left\{ \zeta_{1}=0\right\} \cup\left\{ \left(\left[0\!:\!1\right],\left[0\!:\!1\right]\right)\right\} \right\} }$$ where $\mathrm{\pi\!:\!\mathbf{X}^{ss}_{0}\rightarrow }$ $\mathrm{\mathbf{X}^{ss}_{0}/\!\!/\mathbb{T}=\mathbf{Y}\cong \mathbb{C}\mathbb{P}^{1}}$ is given by the map $\mathrm{\pi\!:\!\left(\left[z_{0}\!:\!z_{1}\right],\left[\zeta_{0}\!:\!\zeta_{1}\right]\right)\rightarrow[z_{0}}$ $\mathrm{\zeta_{1}\!:\!z_{1}\zeta_{0}]}$. Furthermore, a calculation shows that $$\mathrm{\widehat{\mathbf{\,X\,}}=cl\,\mathbf{\Gamma}_{\pi}=\left\{ z_{0}\zeta_{1}\xi_{1}-z_{1}\zeta_{0}\xi_{0}=0\right\} \subset \mathbf{X}\times\mathbb{C}\mathbb{P}^{1}}$$ if $\mathrm{\left[\xi_{0},\xi_{1}\right]}$ are the homogeneous coordinates of $\mathrm{\mathbb{C}\mathbb{P}^{1}}$. A further analysis of the geometry of $\mathrm{\widehat{\mathbf{\,X\,}}}$ and $\mathrm{\mathbf{Y}}$ reveals that $\mathrm{\mathbf{Y}_{0}=\mathbf{Y}}$. Note that we have $$\mathrm{p_{\mathbf{X}}\left(\widehat{\pi}^{-1}\left(\left[\zeta_{0},\zeta_{1}\right]\right)\right)=cl\left({\pi}^{-1}\left(\left[\zeta_{0},\zeta_{1}\right]\right)\right)\text{ for all }\left[\zeta_{0},\zeta_{1}\right]\text{ such that }\zeta_{0,1}\neq0.}$$ For $\mathrm{[\zeta_{0}]=[1\!:\!0]}$ and $\mathrm{\zeta_{1}=[0\!:\!1]}$, we have a proper inclusion $$\mathrm{p_{\mathbf{X}}\left(\widehat{\pi}^{-1}\left(\left[\zeta_{i}\right]\right)\right)\supsetneqq cl\left(\pi^{-1}\left(\left[\zeta_{i}\right]\right)\right)}$$ for all $\mathrm{i}$ $\mathrm{\in\left\{0,1\right\}}$.\,$\boldsymbol{\Box}$} \end{example}
\subsection{The Fiber Probability Measure Sequence (Tame Case)}\label{Definition of the Tame Fiber Probability Measure Sequence}
Let $\mathrm{\mathfrak{U}=\left\{\mathbf{U}_{j}\right\}_{j}}$ be a finite cover of $\mathrm{\mathbf{Y}}$ then, after having chosen a finite refinement $\mathrm{\mathfrak{U}^{\prime}}$ of $\mathrm{\mathfrak{U}}$ (for convenience set $\mathrm{\mathfrak{U}=\mathfrak{U}^{\prime}}$), it follows by {\bf{Section \ref{Existence of Tame Sequences}}} that there exists a tame collection $\mathrm{\left\{\left(s_{n}^{i}\right)\right\}_{i}}$ so that $\mathrm{\mathbf{U}_{j_{i}}\subset\pi\left(\mathbf{X}^{i}\right)=\mathbf{Y}^{i}}$. By changing the index set $\mathrm{J}$ of the finite cover $\mathrm{\mathfrak{U}}$ we can always assume that $\mathrm{\mathbf{U}_{j_{i}}=\mathbf{U}_{i}}$.
Let $\mathrm{\mathbf{Y}_{0}}$ be as in {\bf{Section} \ref{Definition of Y_{0}}} and set $$\mathrm{\Vert s_{n}^{i}\Vert^{2}\!:\!\mathbf{X}^{i}\cap \mathbf{Y}_{0}\rightarrow \mathbb{R}^{\geq 0},\,\Vert s_{n}^{i}\Vert^{2}\left(x\right)\coloneqq \int_{\pi\left(\pi^{-1}\left(x\right)\right)}\vert s_{n}^{i}\vert^{2}\,d\,[\pi_{y}]\label{Notation Fiber Integral Initial Sequence}}$$ where $\mathrm{\int_{\pi^{-1}\left(\pi\left(x\right)\right)}d\,[\pi_{y}]}$ denotes the fiber integral of $\mathrm{\omega^{k}}$ with respect to the $\mathrm{k}$-fibering $\mathrm{\pi\!:\!\mathbf{X}_{0}}$ $\mathrm{\rightarrow \mathbf{Y}_{0}}$ as defined in the work of {\scshape{J. King}} ({cf.\! }\cite{Kin}).
\begin{remark} Since we have $$\mathrm{\int_{\pi^{-1}\left(y\right)}\vert s_{n}^{i}\vert^{2}\left(\omega\vert\pi^{-1}\left(y\right)\right)^{k}\leq\int_{\widehat{\pi}^{-1}\left(y\right)}\left(\zeta\circ p_{\mathbf{X}}\right)^{*}\vert s_{n}^{i}\vert^{2}\left(\omega^{\prime}\vert\widehat{\pi}^{-1}\left(y\right)\right)^{k}}$$ it follows by the compactness of $\mathrm{\widehat{\pi}^{-1}\left(y\right)}$ where $\mathrm{y\in \mathbf{Y}}$, that $$\mathrm{\Vert s_{n}^{i}\Vert^{2}<\infty \text{ for all }y\in \mathbf{Y}_{0}\subset\mathbf{Y}.}$$ \end{remark}
The next step is to define a sequence of collections of fiber probability densities attached to $\mathrm{\mathfrak{U}=\left\{\mathbf{U}_{i}\right\}_{i}}$ as follows.
\begin{definition} \label{Notation Sequence of Collections of Fiber Distribution Densities}Let $\mathrm{\mathfrak{U}=\left\{\mathbf{U}_{i}\right\}_{i}}$ and $\mathrm{\left\{\left(s_{n}^{i}\right)\right\}_{i}}$ be as above, i.e.\ $\mathrm{\mathbf{U}_{i}\subset\pi\left(\mathbf{X}^{i}\right)}$, then define a sequence of collections of fiber distribution densities on $\mathrm{\pi^{-1}\left(\mathbf{U}_{i}\right)\cap \mathbf{X}_{0}}$ by $$\mathrm{\phi^{i}_{n}\!:\!x\mapsto \phi^{i}_{n}\left(x\right)=\Vert s_{n}^{i}\Vert^{-2}\left(x\right)\vert s_{n}^{i}\vert^{2}\left(x\right).}$$ In the sequel, the terminology $\mathrm{\left\{\phi^{\mathfrak{U}}_{n}\right\}=\left\{\phi^{i}_{n}\right\}_{i}}$ will be used. Furthermore, $\mathrm{\left\{\phi^{\mathfrak{U}}_{n}\right\}}$ will be referred to as the collection of fiber distribution densities associated to $\mathrm{\left\{s_{n}^{i}\right\}_{i}}$. \end{definition}
Having defined $\mathrm{\left\{\phi^{\mathfrak{U}}_{n}\right\}}$, it is self-evident to introduce
\begin{definition}\label{Definition Fiber Probability Measure Tame Case} Let $\mathrm{\left\{\phi^{\mathfrak{U}}_{n}\right\}}$ be a sequence of collections of fiber distribution densities associated to a tame collection $\mathrm{\left\{s_{n}^{i}\right\}_{i}}$, then define a sequence of collections of fiber probability measures over $\mathrm{\mathbf{Y}_{0}}$ by $$\mathrm{\bm{\nu}_{n}^{i}\left(y\right)\!:\!\mathbf{A}\mapsto\bm{\nu}_{n}^{i}\left(y\right)\left(\mathbf{A}\right)\coloneqq\int_{\mathbf{A}}d{\bm{\nu}}^{i}_{n}\coloneqq \int_{\mathbf{A}}\phi^{i}_{n}\,d\,[\pi_{y}]\label{Notation Sequence Of Fiber Probability Measures}}$$ where $\mathrm{y\in \mathbf{U}_{i}\cap \mathbf{Y}_{0}}$ and $\mathrm{\mathbf{A}\subset\pi^{-1}\left(y\right)}$ measurable.
As in {\bf{Definition \ref{Notation Sequence of Collections of Fiber Distribution Densities}}}, set $\mathrm{\left\{\bm{\nu}^{\mathfrak{U}}_{n}\right\}=\left\{\bm{\nu}^{i}_{n}\right\}_{i}}$ \label{Notation Collection Of Fiber Probability Measures} and refer to $\mathrm{\left\{\bm{\nu}^{\mathfrak{U}}_{n}\right\}}$ as the collection of fiber probability measures associated to $\mathrm{\left\{s_{n}^{i}\right\}_{i}}$. \end{definition}
We complete our definitions with
\begin{definition}\label{Definition Sequence Of Collections Of Fiber Probability Densities} Let $\mathrm{\left\{\phi^{\mathfrak{U}}_{n}\right\}}$ be a sequence of collections of fiber distribution densities associated to a tame collection $\mathrm{\left\{s_{n}^{i}\right\}_{i}}$, then define a sequence of collections of cumulative fiber probability densities over $\mathrm{\mathbf{Y}_{0}}$ by $$\mathrm{D_{n}^{i}\left(y,\cdot\right)\!:\!t\mapsto D_{n}^{i}\left(y,t\right)\coloneqq \int_{\left\{\phi^{i}_{n}\geq t\right\}\cap \pi^{-1}\left(\pi\left(x\right)\right)}d\,[\pi_{y}]\label{Notation Cumulative Distribution Densities}}$$ where $\mathrm{y\in \mathbf{U}_{i}\cap \mathbf{Y}_{0}}$.
As in the aforementioned definitions, we set $\mathrm{\left\{D^{\mathfrak{U}}_{n}\right\}=\left\{D^{i}_{n}\right\}_{i}}$ \label{Notation Collection Of Cumulative Fiber Probability Densities} and refer to $\mathrm{\left\{D^{\mathfrak{U}}_{n}\right\}}$ as the collections of cumulative fiber probability densities associated to $\mathrm{\left\{s_{n}^{i}\right\}_{i}}$. \end{definition}
In {\bf{Section \ref{Section Uniform Convergence Theorems In The Tame Case}}} we will give a prove of the following two convergence results.
\begin{theo_3*}\label{Theorem Uniform Convergence of the Tame Measure Sequence} \textnormal{[\scshape{Uniform Convergence of the Tame Measure Sequence}]}
\\ For for every tame collection $\mathrm{\left\{s_{n}^{i}\right\}_{i}}$ there exists a finite cover $\mathrm{\mathfrak{U}}$ of $\mathrm{\mathbf{Y}}$ with $\mathrm{\mathbf{U}_{i}\subset}$ $\mathrm{\pi\left(\mathbf{X}^{i}\right)}$ so that the collection of fiber probability measures $\mathrm{\left\{\bm{\nu}^{\mathfrak{U}}_{n}\right\}}$ associated to $\mathrm{\left\{s^{i}_{n}\right\}_{i}}$ converges uniformly on $\mathrm{\mathbf{Y}_{0}}$ to the fiber Dirac measure of $\mathrm{\mu^{-1}\left(\xi\right)\cap \mathbf{X}_{0}}$, i.e.\ for every $\mathrm{i\in I}$ and every $\mathrm{f\in \mathcal{C}^{0}\left(\mathbf{X}\right)}$, we have $$\begin{xy}
\xymatrix{\mathrm{\left(y\mapsto \int_{\pi^{-1}\left(y\right)}f\, d\bm{\nu}_{n}^{i}\left(y\right)\right)}\ar[rr]^{\mathrm{\,\,\,\,\,}}&&\mathrm{f_{red}\text{ uniformly on }\mathbf{U}_{i}\cap \mathbf{Y}_{0}.\label{Notation Reduced Function}}
} \end{xy}$$ \end{theo_3*}
\begin{theo_4*}\label{Theorem Uniform Convergence of the Tame Distribution Sequence} \textnormal{[\scshape{Uniform Convergence of the Tame Distribution Sequence}]}
\\ For every $\mathrm{t\in \mathbb{R}}$ and every tame collection $\mathrm{\left\{s_{n}^{i}\right\}_{i}}$ there exists a finite cover $\mathrm{\mathfrak{U}}$ of $\mathrm{\mathbf{Y}}$ with $\mathrm{\mathbf{U}_{i}\subset}$ $\mathrm{\pi\left(\mathbf{X}^{i}\right)}$ so that the collection of cumulative fiber probability densities $\mathrm{\left\{D^{\mathfrak{U}}_{n}\left(\cdot,t\right)\right\}}$ associated to $\mathrm{\left\{s^{i}_{n}\right\}_{i}}$ converges uniformly on $\mathrm{\mathbf{Y}_{0}}$ to the zero function on $\mathrm{\mathbf{Y}_{0}}$, i.e.\ for every $\mathrm{i\in I}$ we have $$\begin{xy}
\xymatrix{\mathrm{\left(y\mapsto D_{n}^{i}\left(y,t\right)\right)}\ar[rr]^{\mathrm{\,\,\,\,\,}}&&\mathrm{0\text{ uniformly on }\mathbf{U}_{i}\cap \mathbf{Y}_{0}\subset \pi\left(\mathbf{X}^{i}\right).}
} \end{xy}$$ \end{theo_4*}
\subsection{The Fiber Probability Measure Sequence (Non-Tame Case)}\label{Definition of the Fiber Probability Measure Sequence (Non-Tame Case)} Let $\mathrm{\left(s_{n}\right)_{n}}$ be a sequence of $\mathrm{\mathbb{T}}$-eigensections so that $\mathrm{\vert \xi_{n}-n\,\xi\vert\in \mathcal{O}\left(1\right)}$. Moreover, let $\mathrm{\pi\!:\!\mathbf{X}^{ss}_{\xi}\rightarrow \mathbf{Y}=\mathbf{X}^{ss}_{\xi}/\!\!/\mathbb{T}}$ be the projection map attached to the Hilbert quotient associated to the level subset $\mathrm{\mu^{-1}\left(\xi\right)}$ and $\mathrm{\pi_{n}\!:\!\mathbf{X}^{ss}_{n^{-1}\xi_{n}}\rightarrow \mathbf{Y}_{n}=\mathbf{X}^{ss}_{n^{-1}\xi_{n}}/\!\!/\mathbb{T}}$ the corresponding Hilbert quotient associated to the level subset $\mathrm{\mu^{-1}\left(n^{-1}\xi_{n}\right)}$.
The aim of this subsection is to define a sequence $\mathrm{\left(\bm{\nu}_{n}\right)_{n}}$ of fiber probability measures over the base $\mathrm{\mathbf{Y}_{0}}$ by $$\mathrm{\bm{\nu}_{n}\left(y\right)\!:\!\mathbf{A}\mapsto\bm{\nu}_{n}\left(y\right)\left(\mathbf{A}\right)\coloneqq\int_{\mathbf{A}}\frac{\vert s_{n}\vert^{2}}{\Vert s_{n}\Vert^{2}}\, d\,[ \pi_{y}].}$$ However, since $\mathrm{\mathbf{X}\left(s_{n}\right)=\left\{x\in \mathbf{X}\!:\!\,?s_{n}\left(x\right)\neq 0\right\}}$ moves as $\mathrm{n\rightarrow \infty}$, the above measure is not well defined on all of $\mathrm{\bf{Y}}$:\ For example if $\mathrm{y\in \mathbf{Y}}$ is so that $\mathrm{\pi^{-1}\left(y\right)\subset \left\{s_{n}=0\right\}}$, then the above measure is not defined for $\mathrm{n}$ at $\mathrm{y}$. In some cases, it is however possible to circumvent this problem. For this we will first introduce the following definition.
\begin{definition}\label{Definition Removable Singularity} \textnormal{[\scshape{Removable Singularity}]} \\A point $\mathrm{y\in \mathbf{Y}}$ is defined to be a removable singularity of order $\mathrm{N_{0}}$ for the fiber measure $\mathrm{\bm{\nu}_{n}}$ if there exists an open neighborhood $\mathrm{\mathbf{U}_{y}\subset \mathbf{Y}}$ of $\mathrm{y}$, a sequence $\mathrm{\left(f_{y,n}\right)_{n}}$ of local holomorphic functions $\mathrm{f_{n}\in \mathcal{O}\left(\mathbf{U}_{y}\right)}$ and $\mathrm{N_{0}\in \mathbb{N}}$ so that $$\mathrm{\widehat{s}_{f_{y,n}}\coloneqq s_{n}\cdot\pi^{*}f_{y,n}^{-1}\label{Notation Local Extension Of S_{n}}}$$ defines a local holomorphic section on $\mathrm{\pi^{-1}\left(\mathbf{U}_{y}\right)}$ for all $\mathrm{n\geq N_{0}}$ which does not vanish identically on $\mathrm{\pi^{-1}\left(y^{\prime}\right)}$ for all $\mathrm{y^{\prime}\in \mathbf{U}_{y}}$.
In the sequel, we will denote the set of all removable singularities of order $\mathrm{N_{0}}$ by $\mathrm{\mathbf{R}_{N_{0}}}$\label{Notation Removable Singularity Of Order N_0}. \end{definition} \begin{remark} The local extension $\mathrm{\widehat{s}_{f_{y,n}}}$ is again a $\mathrm{\xi_{n}}$-eigensection. \end{remark}
As a corollary of the definition, we deduce
\begin{cor}\label{Corollary Independence Of Extension} If $\mathrm{y\in\mathbf{Y}_{0}}$ is a removable singularity for $\mathrm{\bm{\nu}_{n}}$ of order $\mathrm{N_{0}\in\mathbb{N}}$, then after having shrunken $\mathrm{\mathbf{U}_{y}}$ appropriately, the quotient $\mathrm{\vert \widehat{s}_{f_{y,n}}\vert^{2}}$ $\mathrm{\Vert \widehat{s}_{f_{y,n}}\Vert^{-2}}$ is independent of the local scaling functions $\mathrm{\left(f_{y,n}\right)_{n}}$, i.e.\ we have $$\mathrm{\frac{\vert \widehat{s}_{f^{0}_{y,n}}\vert^{2}}{\Vert \widehat{s}_{f^{0}_{y,n}}\Vert^{2}}=\frac{\vert \widehat{s}_{f^{1}_{y,n}}\vert^{2}}{\Vert \widehat{s}_{f^{1}_{y,n}}\Vert^{2}}}$$ over $\mathrm{\mathbf{U}_{y}\subset \mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$. \end{cor} \begin{proof} First of all, since $\mathrm{y}$ is contained in the open set $\mathrm{\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$, we can assume that $\mathrm{\mathbf{U}_{y}\subset \mathbf{Y}_{0}}$.
Let $\mathrm{\left(f^{i}_{y,n}\right)_{n}}$, $\mathrm{f^{i}_{y,n}\in \mathcal{O}\left(\mathbf{U}_{y}\right)}$ where $\mathrm{i\in\left\{0,1\right\}}$. As in {\bf{Definition \ref{Definition Removable Singularity}}}, we have $\mathrm{\widehat{s}_{f^{i}_{y,n}}}$ $\mathrm{=s_{n}\cdot}$ $\mathrm{\pi^{*}f^{i,-1}_{y,n}}$. Throughout this proof, we use the abbreviation $\mathrm{f^{i}_{y,n}=f_{i,n}}$ for $\mathrm{i\in\left\{0,1\right\}}$. Using this notation, it follows that \begin{equation}\label{Equation Quotient Of Holomorphic Functions} \mathrm{\widehat{s}^{\phantom{-1}}_{f_{0,n}}=\pi^{*}(f_{1,n}^{\phantom{-1}}f_{0,n}^{-1})\cdot \widehat{s}_{f_{1,n}}} \end{equation} where $\mathrm{h_{0,1,n}\coloneqq f_{1,n}^{\phantom{-1}}f_{0,n}^{-1}}$ yields a sequence of meromorphic function defined on $\mathrm{\mathbf{U}_{y}}$. We claim that after having shrunken $\mathrm{\mathbf{U}_{y}}$, each $\mathrm{h_{0,1,n}}$ is bounded from above and below away form zero on $\mathrm{\mathbf{U}_{y}}$.
For this note the following: Since $\mathrm{\widehat{s}_{f_{1},n}\vert \pi^{-1}\left(y\right)\not\equiv 0}$ there exists one $\mathrm{x\in \pi^{-1}\left(y\right)}$ so that $\mathrm{\widehat{s}_{f_{1},n}\left(x\right)\neq 0}$ and hence $\mathrm{\vert \widehat{s}_{f_{1},n}\vert^{2}\left(x\right)>0}$. In particular, we find an open neighborhood $\mathrm{\mathbf{V}\subset \mathbf{X}_{0}}$ of $\mathrm{x}$ so that \begin{equation}\label{Equation Simplification Of This Proof} \mathrm{\vert \widehat{s}_{f_{1},n}\vert^{2}(x^{\prime})\geq c>0} \end{equation} for all $\mathrm{x^{\prime}\in \mathbf{V}}$. By continuity we also have \begin{equation}\label{Equation Simplification Of This Proof2} \mathrm{\vert \widehat{s}_{f_{0},n}\vert^{2}(x^{\prime})\leq C<\infty} \end{equation} for all $\mathrm{x^{\prime}\in \mathbf{V}}$.
Since $\mathrm{\mathbf{V}\subset \mathbf{X}_{0}}$ and since $\mathrm{\pi\vert \mathbf{X}_{0}\!:\!\mathbf{X}_{0}\rightarrow \mathbf{Y}_{0}}$ is a $\mathrm{k}$-fibering and hence an open map ({cf.\! }\cite{Loj}, {p.\ }{\bf{\oldstylenums{297} f.}}), we can assume (after having shrunken ${\mathbf{V}}$ and $\mathrm{\mathbf{U}_{y}}$ appropriately) that $\mathrm{\pi\left(\mathbf{V}\right)=\mathbf{U}_{y}}$. Combining \ref{Equation Simplification Of This Proof}, \ref{Equation Simplification Of This Proof2} and \ref{Equation Quotient Of Holomorphic Functions} we deduce $\mathrm{\vert h_{0,1,n}\vert^{2}\left(y^{\prime}\right)\leq C^{\prime}<\infty}$ for all $\mathrm{y^{\prime}\in \mathbf{U}_{y}}$. Reversing the roles of $\mathrm{f_{0}}$ and $\mathrm{f_{1}}$ shows (after having shrunken $\mathrm{\mathbf{U}_{y}}$ again) that $\mathrm{\vert h_{0,1,n}\vert^{2}\left(y^{\prime}\right)\geq c^{\prime}>0}$. Hence, the meromorphic function $\mathrm{h_{0,1,n}= f_{1,n}^{\phantom{-1}}f_{0,n}^{-1}}$ is bounded from above and below away from zero on $\mathrm{\mathbf{U}_{y}}$. As the base $\mathrm{\mathbf{Y}}$ is assumed to be normal, we can apply {\scshape{Riemann}}'s Extension Theorem (cf.\ \cite{Gr-Re}, {p.\ }{\bf{\oldstylenums{144}}}) in order to deduce that $\mathrm{h_{0,1,n}}$ yields a non-vanishing holomorphic function on $\mathrm{\mathbf{U}_{y}}$. All in all we, deduce $$\mathrm{\frac{\vert \widehat{s}_{f^{0}_{y,n}}\vert^{2}}{\Vert \widehat{s}_{f^{0}_{y,n}}\Vert^{2}}\cdot \frac{\pi^{*}\vert h_{0,1,n}\vert^{2}}{\pi^{*}\vert h_{0,1,n}\vert^{2}}=\frac{\vert \widehat{s}_{f^{1}_{y,n}}\vert^{2}}{\Vert \widehat{s}_{f^{1}_{y,n}}\Vert^{2}}}$$ over $\mathrm{\mathbf{U}_{y}}$ as claimed.
\end{proof} In this sense, the quotient $\mathrm{\vert s_{n}\vert^{2}\Vert s_{n}\Vert^{-2}}$ can be uniquely extended onto $\mathrm{\mathbf{R}_{N_{0}}\cap \mathbf{Y}_{0}}$. Assume now that $\mathrm{\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}\neq \varnothing}$. Using the independence of $\mathrm{\vert \widehat{s}_{f_{y,n}}\vert^{2}\Vert \widehat{s}_{f_{y,n}}\Vert^{-2}}$ of the chosen sequence $\mathrm{\left(f_{y,n}\right)_{n}}$, $\mathrm{f_{y,n}\in \mathcal{O}\left(\mathbf{U}_{y}\right)}$ shown in {\bf{Corollary \ref{Corollary Independence Of Extension}}}, we can define
\begin{definition} \label{Notation Sequence of Collections of Fiber Distribution Densities Initial Sequence}Let $\mathrm{\left(s_{n}\right)_{n}}$ be sequence of $\mathrm{\xi_{n}}$-eigensections whose rescaled weights approximate $\mathrm{\xi}$, then define a sequence of fiber distribution densities over $\mathrm{\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$ by $$\mathrm{\phi_{n}\!:\!x\mapsto \phi_{n}\left(x\right)\coloneqq \Vert \widehat{s}_{f_{y,n}}\Vert^{-2}\left(x\right)\vert \widehat{s}_{f_{y,n}}\vert^{2}\left(x\right)}$$ where $\mathrm{\left(f_{y,n}\right)_{n}}$ is a sequence of holomorphic functions $\mathrm{f_{y,n}\in \mathcal{O}\left(\mathbf{U}_{y}\right)}$ as in {\bf{Definition \ref{Definition Removable Singularity}}}. \end{definition}
As in the tame case we introduce the following two definitions.
\begin{definition}\label{Definition Fiber Probability Measure Non Tame Case} Let $\mathrm{\left(\phi_{n}\right)_{n}}$ be the sequence of fiber distribution densities associated to $\mathrm{\left(s_{n}\right)_{n}}$ as defined in {\bf{Definition \ref{Notation Sequence of Collections of Fiber Distribution Densities Initial Sequence}}}, then we define a sequence of fiber probability measures parametrized over $\mathrm{\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$ by $$\mathrm{\bm{\nu}_{n}\left(y\right)\!:\!\mathbf{A}\mapsto\bm{\nu}_{n}\left(y\right)\left(\mathbf{A}\right)\coloneqq\int_{\mathbf{A}}d{\bm{\nu}}_{n}\coloneqq \int_{\mathbf{A}}\phi_{n}\,d\,[\pi_{y}]}$$ for $\mathrm{\mathbf{A}\subset\pi^{-1}\left(y\right)}$ measurable where $\mathrm{y\in \mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$. \end{definition}
\begin{definition} Let $\mathrm{\left(s_{n}\right)_{n}}$ be sequence of $\mathrm{\xi_{n}}$-eigensections whose rescaled weights approximate $\mathrm{\xi}$, then we define a sequence of cumulative fiber probability densities over $\mathrm{\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$ by $$\mathrm{D_{n}\left(y,\cdot\right)\!:\!t\mapsto D_{n}\left(y,t\right)\coloneqq \int_{\left\{\phi_{n}\geq t\right\}\cap \pi^{-1}\left(\pi\left(x\right)\right)}d\,[\pi_{y}]}$$ where $\mathrm{y\in \mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$. \end{definition}
The aim of {\bf{Section \ref{Section Uniform Convergence Theorems In The Non-Tame Case}}} is to give a proof of the following two convergence results.
\begin{theo_5.b*}\label{theo_5.b*}\textnormal{[\scshape{Uniform Convergence of the Initial Distribution Sequence}]}
\\ For fixed $\mathrm{t\in \mathbb{R}}$ the sequence $\mathrm{\left(D_{n}\left(\cdot,t\right)\right)_{n}}$ converges uniformly on $\mathrm{\mathbf{Y}_{0}\cap\mathbf{R}_{N_{0}}}$ to the zero function. \end{theo_5.b*}
\begin{theo_6.b*}\label{theo_6.b*}\textnormal{[\scshape{Uniform Convergence of the Initial Measure Sequence}]}
\\ Let $\mathrm{f\in \mathcal{C}^{0}\!\left(\mathbf{X}\right)}$ then the sequence $$\mathrm{\left(y\mapsto \int_{\pi^{-1}\left(y\right)}f\,d\bm{\nu}_{n}\left(y\right)\right)_{n}}$$ converges uniformly over $\mathrm{\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$ to the reduced function $\mathrm{f_{red}}$. \end{theo_6.b*}
We close this section by showing that in general there exists no $\mathrm{N_{0}\in \mathbb{N}}$ so that $\mathrm{\mathbf{Y}=\mathbf{R}_{N_{0}}}$. Furthermore, the example shows that even the extreme case $\mathrm{\mathbf{R}_{N_{0}}=\varnothing}$ for all $\mathrm{N_{0}\in \mathbb{N}}$ is possible.
\begin{example}\label{Example Existence Of Removable Singularities} \textnormal{Let $\mathrm{\mathbf{X}\subset \mathbb{C}\mathbb{P}^{k}\times \mathbb{C}\mathbb{P}^{k}}$ equipped with the $\mathrm{\mathbb{T}=\mathbb{C}^{*}}$ action given by $$\mathrm{t.\left([\zeta_{0}\!:\!{\dots}\!:\!\zeta_{k}],[z_{0}\!:\!{\dots}\!:\!z_{k}]\right)=\left([\zeta_{0}\!:\!{\dots}\!:\!\zeta_{k}],[t^{-1}z_{0}\!:\!{\dots}\!:\!t^{-1}z_{k-1}\!:\!z_{k}]\right).}$$ In the sequel, we will consider the $\mathrm{\mathbb{T}}$-linearization $\mathrm{\mathbf{L}=p_{0}^{*}\mathcal{O}_{\mathbb{C}\mathbb{P}^{k}}\left(1\right)\otimes p_{1}^{*}\mathcal{O}_{\mathbb{C}\mathbb{P}^{k}}\left(1\right)}$ of the $\mathrm{\mathbb{T}}$-action on $\mathrm{\mathbf{X}}$ induced by the trivial $\mathrm{\mathbb{T}}$-action on the first factor $\mathrm{\mathcal{O}\left(1\right)_{\mathbb{C}\mathbb{P}^{k}}\rightarrow \mathbb{C}\mathbb{P}^{k}}$ given by $$\mathrm{t.[\zeta_{0}\!:\!{\dots}\!:\!\zeta_{k}\!:\!\zeta]=[\zeta_{0}\!:\!{\dots}\!:\!\zeta_{k}\!:\!\zeta]}$$ where we have used the identification $\mathrm{\mathcal{O}\left(1\right)_{\mathbb{C}\mathbb{P}^{k}}\cong \mathbb{C}\mathbb{P}^{k+1}\setminus\left\{[0\!:\!\dots\!:\!0\!:\!1]\right\}}$ and the $\mathrm{\mathbb{T}}$-action on the second factor $\mathrm{\mathcal{O}_{\mathbb{C}\mathbb{P}^{k}}\left(1\right)\rightarrow \mathbb{C}\mathbb{P}^{k}}$ given by $$\mathrm{t.[z_{0}\!:\!{\dots}\!:\!z_{k}\!:\!\zeta]=[t^{-1}z_{0}\!:\!{\dots}\!:\!t^{-1}z_{k-1}\!:\!z_{k}\!:\!\zeta].}$$ A calculation shows that the $\mathrm{\xi=0}$-level of the associated moment map is given by the set $$\mathrm{\mu^{-1}\left(0\right)=\mathbb{C}\mathbb{P}^{k}\times \left\{[0\!:\!{\dots}\!:\!0\!:\!1]\right\}}$$ and it is also direct to verify that one can identify $\mathrm{\mathbf{Y}=\mathbf{X}^{ss}_{\xi=0}/\!\!/\mathbb{T}\cong \mathbb{C}\mathbb{P}^{k}}$ where $\mathrm{\mathbf{X}^{ss}_{0}\cong}$ $\mathrm{\mathbb{C}\mathbb{P}^{k}\times \mathbb{C}^{k}}$. Furthermore, each fiber of the quotient map $\mathrm{\pi}$ is isomorphic to $\mathrm{\mathbb{C}^{k}}$ equipped with the inverse diagonal action.}
\textnormal{We will now consider the sequence $\mathrm{\left(s_{n}\right)_{n}}$ whose rescaled weights converge to $\mathrm{\xi=0}$ defined by $$\mathrm{s_{n}=\sum_{i=0}^{k-1}\zeta^{n}_{i}\,z_{i}\,z_{k}^{n-1}\in H^{0}\left(\mathbf{X},\mathbf{L}^{n}\right).}$$ Note that we have $$\mathrm{s_{n}\vert \pi^{-1}\left([0\!:\!{\dots}\!:\!0\!:\!1]\right)\equiv 0 \text{ for all }n\in \mathbb{N}.}$$}
\textnormal{Using the homogenous standard coordinates $\mathrm{\zeta^{\prime}_{i}=\frac{\zeta_{i}}{\zeta_{k}}}$, $\mathrm{z^{\prime}_{i}=\frac{z_{i}}{z_{k}}}$, $\mathrm{0\leq i\leq k-1}$ on the open subset $$\mathrm{\mathbf{U}_{k,k}=\left\{\left([z],[\zeta]\right)\!:\!\,\zeta_{k}\neq 0,\,z_{k}\neq 0\right\}\subset \mathbf{X}^{ss}_{0}\subset \mathbb{C}\mathbb{P}^{k}\times \mathbb{C}\mathbb{P}^{k},}$$ the restriction $\mathrm{\pi\vert \mathbf{U}_{k,k}\rightarrow \mathbf{V}_{k}\cong \mathbb{C}^{k}}$ of the quotient map $\mathrm{\pi\!:\!\mathbf{X}^{ss}_{0}\rightarrow \mathbb{C}\mathbb{P}^{k}}$ is given by the projection map $\mathrm{\pi\vert \mathbf{U}_{k,k}\!:\!\mathbf{U}_{k,k}\cong\mathbb{C}^{k}\times \mathbb{C}^{k}\rightarrow \mathbb{C}^{k}}$. With respect to this trivialization the sequence $\mathrm{s_{n}\vert \mathbf{U}_{k,k}}$ is given by $$\mathrm{s_{n}\vert {\mathbf{U}_{k,k}}=\sum_{i=0}^{k-1}\zeta^{\prime,n}_{i}\,z^{\prime}_{i}}$$ where $\mathrm{\pi^{-1}\left([0\!:\!{\dots}\!:\!0\!:\!1]\right)=\left\{0\right\}\times \mathbb{C}^{k}}$. To shorten notation, we will just write $\mathrm{s_{n}\vert \mathbf{U}_{k,k}}$ $\mathrm{=s_{n}}$ throughout the rest of this example and set $\mathrm{\zeta^{\prime}_{i}=\zeta_{i}}$, $\mathrm{z^{\prime}_{i}=z_{i}}$ for all $\mathrm{0\leq i\leq k-1}$.}
\textnormal{Assume now that $\mathrm{\zeta_{0}=[0\!:\!{\dots}\!:\!0\!:\!1]}$ is a removable singularity for the fiber measure $\mathrm{\bm{\nu}_{n}}$ (we will fix $\mathrm{n}$ henceforth), i.e.\ there exists a non-vanishing holomorphic function $\mathrm{f\in \mathcal{O}\left(\mathbf{U}\right)}$ defined on an open neighborhood $\mathrm{\mathbf{U}\subset \mathbb{C}^{k}}$ of $\mathrm{\zeta_{0}}$ so that $\mathrm{\widehat{s}_{n}\coloneqq s_{n}\cdot\pi^{*}f^{-1}_{n}}$ is holomorphic and does not vanish identically on $\mathrm{\pi^{-1}\left(\zeta^{\prime}\right)}$ for all $\mathrm{y^{\prime}\in \mathbf{U}}$. Note that, after having shrunken $\mathrm{\mathbf{U}}$, we can assume that $\mathrm{\zeta_{0}}$ is an isolated zero of the function $\mathrm{f}$. In particular, the restriction $\mathrm{\widehat{s}_{n}\vert \pi^{-1}\left(\zeta^{\prime}\right)}$ defines a non-vanishing linear one form on $\mathrm{\pi^{-1}\left(\zeta^{\prime}\right)\cong\mathbb{C}^{k}}$ for all $\mathrm{\zeta^{\prime}\in \mathbf{U}}$. Using this, it follows that for each sequence $\mathrm{\left(\zeta_{m}\right)_{m}}$ in $\mathrm{\mathbf{U}}$ converging to $\mathrm{\zeta_{0}}$, the sequence of one-codimensional subspaces in $\mathrm{\mathbb{C}^{k}\cong \pi^{-1}\left(\zeta_{m}\right)}$ given by $\mathrm{{\bm{\mathfrak{H}}}\left(\zeta_{m}\right)=\left\{x\in\mathbb{C}^{k}\!:\!\,\widehat{s}_{n}\left(\zeta_{m}\right)=0\right\} }$ must converge to a uniquely defined one-codimensional subspace which is independent of the choice of $\mathrm{\left(\zeta_{m}\right)_{m}}$. However, this is a contradiction to the equation $\mathrm{\widehat{s}_{n}=s_{n}\cdot }$ $\mathrm{\pi^{*}f^{-1}}$ and the fact that $\mathrm{f}$ is non-zero on $\mathrm{\mathbf{U}\setminus\left\{\zeta_{0}\right\}}$. For example, consider the collection of sequences given by $$\mathrm{\left\{ \zeta_{m}^{i}\right\} _{i}=\left\{ \left(0,\dots,\mathrm{m^{-1}},\dots,0\right)_{m}\right\}_{i},}$$ then we have $$\mathrm{{\bm{\mathfrak{H}}}\left(\zeta_{m}^{i}\right)=\left\{ x\in\mathbb{C}^{k}\!:\!\,\widehat{s}_{n}\left(\zeta_{m}^{i}\right)=0\right\} \rightarrow\left\{ x\in\mathbb{C}^{k}\!:\!\, z_{i}=0\right\}}$$ so the limit is not independent of the chosen sequence. Hence, we deduce a contradiction and it follows that $\mathrm{\zeta_{0}}$ is a not a removable singularity for any $\mathrm{N_{0}\in \mathbb{N}}$ in the sense of {\bf{Definition \ref{Definition Removable Singularity}}}, i.e we have $\mathrm{\zeta_{0}\notin \mathbf{R}_{N_{0}}}$ for all $\mathrm{N_{0}}$.}
\textnormal{Furthermore, by slightly changing the above sequence $\mathrm{\left(s_{n}\right)_{n}}$, we can show that $\mathrm{\mathbf{R}_{N_{0}}=\varnothing}$ for each $\mathrm{N_{0}}$:\ Choose a dense sequence $\mathrm{\left(\zeta_{n}\right)_{n}}$ in the quotient $\mathrm{\mathbf{Y}\cong \mathbb{C}\mathbb{P}^{k}}$ and let $\mathrm{\left(\Phi_{n}\right)_{n}}$, $\mathrm{\Phi_{n}\in}$ $\mathrm{Aut\left(\mathbf{Y}\right)}$ be a sequence of projective transformations so that $\mathrm{\Phi_{n}\left(\zeta_{0}\right)=\zeta_{n}}$. Define a new sequence of eigensections by $$\mathrm{s^{\prime}_{n}\left([\zeta],[z]\right)\coloneqq s_{1}\left([\Phi_{n}\left(\zeta\right)],[z]\right)}$$ and consider the sequence given by $$\mathrm{s_{n}\coloneqq \prod_{i=1}^{n}s^{\prime}_{i}.}$$ It is direct to see that $\mathrm{\mathbf{R}_{N_{0}}=\varnothing}$ because for each open neighborhood $\mathrm{\mathbf{U}}$ of any point $\mathrm{y\in \mathbf{Y}}$, the subset $\mathrm{\mathbf{U}\cap \left\{\zeta_{n}\right\}}$ is dense by construction and $\mathrm{\zeta_{n}}$ is non-removable. $\boldsymbol{\Box}$} \end{example}
As a consequence of this example we conclude
\begin{remark} There are examples of approximating sequences $\mathrm{\left(s_{n}\right)_{n}}$ so that $\mathrm{\mathbf{R}_{N_{0}}=\varnothing}$ for all $\mathrm{N_{0}\in \mathbb{N}_{0}}$. \end{remark}
As a further-reaching question, one could ask whether the set of all $\mathrm{\xi}$-approximating sequences of $\mathrm{\xi_{n}}$-eigensections $\mathrm{\left(s_{n}\right)_{n}}$, with the property that $\mathrm{\mathbf{R}_{N_{0}}=\varnothing}$ for all $\mathrm{N_{0}}$, is "thin" as a subset of $\mathrm{\bigoplus_{n=0}^{\infty}H^{0}\left(\mathbf{X},\mathbf{L}^{n}\right)}$.
It turns out that there is no definitive answer to this question: In the context of {\bf{Example \ref{Example First Example}}}, one can show that each singularity of $\mathrm{s\in H^{0}\left(\mathbf{X},\mathbf{L}^{n}\right)}$ is removable and hence $\mathrm{\mathbf{Y}=\mathbf{R}_{N_{0}}}$ for all $\mathrm{N_{0}\in \mathbb{N}}$ and any choice of $\mathrm{\left(s_{n}\right)_{n}}$. On the other hand, if $\mathrm{k=2}$ in {\bf{Example \ref{Example Existence Of Removable Singularities}}}, it turns out that a $\mathrm{\xi_{n}}$-eigensection $\mathrm{s_{n}\in H^{0}\left(\mathbf{X},\mathbf{L}^{n}\right)}$, which has been randomly chosen with respect to a choice of a Lebesgue measure on $\mathrm{H^{0}\left(\mathbf{X},\mathbf{L}^{n}\right)}$ (induced by a choice of a basis), has almost surely at least one non-removable singularity $\mathrm{y_{n}\in \mathbf{Y}}$. Moreover, if one randomly chooses a $\mathrm{\xi}$-approximating sequence of $\mathrm{\xi_{n}}$-eigensections in this setting, it turns out that the set $\mathrm{\left\{y_{n}\right\}_{n\in \mathbb{N}}}$ is almost surely dense in $\mathrm{\mathbf{Y}}$.
\section{The $\mathrm{k}$-Fibering $\mathrm{\Pi\!:\!\widetilde{\bf{\,X\,}}\rightarrow \widetilde{\bf{\,Y\,}}}$}
\subsection{Construction of the $\mathrm{k}$-Fibering $\mathrm{\Pi\!:\!\widetilde{\bf{\,X\,}}\rightarrow \widetilde{\bf{\,Y\,}}}$}
Let $\mathrm{\widehat{\pi}\!:\!\widehat{\mathbf{\,X\,}}\rightarrow \mathbf{Y}}$ be the holomorphic map between the purely dimensional varieties $\mathrm{\widehat{\mathbf{\,X\,}}}$ and $\mathrm{\mathbf{Y}}$ as defined in {\bf{Section \ref{Definition of Y_{0}}}} and recall that there exits a Zariski-dense subset $\mathrm{\mathbf{Y}_{0}\subset \mathbf{Y}}$ so that all fibers $\mathrm{\widehat{\pi}^{-1}\left(y\right)}$ for $\mathrm{y\in \mathbf{Y}_{0}}$ are purely $\mathrm{k}$-dimensional, compact subvarieties not necessarily irreducible. Recall that we have assumed $\mathrm{\mathbf{X}}$ to be normal and hence, it follows (cf.\ \cite{He-Hu1}, {p.\ }{\bf{\oldstylenums{124}}}) that $\mathrm{\mathbf{Y}}$ is normal and therefore, in particular, the open subset $\mathrm{\mathbf{Y}_{0}}$ as well.
By \cite{Bar1} the following is known in the above context: There exists a holomorphic map $\mathrm{\varphi^{\widehat{\pi}}:\mathbf{Y}_{0}\rightarrow\mathcal{C}^{k}(\mathbf{\widehat{\,\bm{X}\,}})}$ \label{Notation Regular Map Universal Property Chow Scheme} into the cycle space $\mathrm{\mathcal{C}^{k}(\mathbf{\widehat{\,\bm{X}\,}})}$ \label{Notation Cycle Space Of All k-Dim Cycles} of all compact $\mathrm{k}$-dimensional cycles $$\mathrm{{\bm{\mathfrak{C}}}=\sum_{i\in I}n_{i}\mathbf{C}_{i}, n_{i}\in \mathbb{N}, \mathbf{C}_{i}\subset \widehat{\bf{\,X\,}} \text{ globally irreducible subspaces of }\widehat{\bf{\,X\,}}\label{Notation Cycle Of Dimension K}}$$ of dimension $\mathrm{k}$ so that the support\footnote{The support $\mathrm{\vert{\bm{\mathfrak{C}}}\vert}$ of a cycle $\mathrm{{\bm{\mathfrak{C}}}=\sum_{i\in I}n_{i}\mathbf{C}_{i}}$ is defined by $\mathrm{\vert{\bm{\mathfrak{C}}}\vert=\bigcup_{i\in I}\mathbf{C}_{i}}$\label{Notation Support Of A Cycle}.} $\mathrm{\vert \varphi^{\widehat{\pi}}\left(y\right)\vert}$ of the cycle $\mathrm{{\bm{\mathfrak{C}}}_{y}\coloneqq \varphi^{\widehat{\pi}}\left(y\right)}$ \label{Notation Cycle Of Dimension K Associated To The Fiber Of} for $\mathrm{y\in {\bf{Y}}_{0}}$ is equal to the set theoretic fiber $\mathrm{\widehat{\pi}^{-1}\left(y\right)}$, i.e.\ we have \begin{equation}\label{Equation Equality Cycles Barlet Construction} \mathrm{\vert {\bm{\mathfrak{C}}}_{y}\vert=\widehat{\pi}^{-1}\left(y\right)} \text{ for all }\mathrm{y\in \mathbf{Y}_{0}.} \end{equation} Furthermore, since $\mathrm{\widehat{\bf{\,X\,}}}$ and $\mathrm{\mathbf{Y}}$ are compact, there exists (cf.\ \cite{Bar1}) a proper modification $\mathrm{\sigma\!:\!\widetilde{\,\bf{Y}\,}\rightarrow \mathbf{Y}}$ with center $\mathrm{\mathbf{Y}\setminus \mathbf{Y}_{0}}$, a proper modification $\mathrm{\Sigma\!:\!\widetilde{\,\mathbf{X}\,}\rightarrow \widehat{\,\bf{X}\,}}$\label{Notation Blow Up Space} \label{Notation Blow Up Map onto X} with center $\mathrm{\widehat{\pi}^{-1}\left(\mathbf{Y}\setminus \mathbf{Y}_{0}\right)}$ and a surjective holomorphic map $\mathrm{\Pi\!:\!\widetilde{\,\mathbf{X}\,}\rightarrow \widetilde{\,\mathbf{Y}\,}}$ \label{Notation Blown Up Projection Map} so that the following diagram commutes: $$\begin{xy}\label{Commutating Diagram} \xymatrix{ \mathrm{\widehat{\mathbf{\,X\,}}}\ar[d]^{\mathrm{\widehat{\pi}}} &&& \mathrm{\widetilde{\bf{\,X\,}}}\ar[d]^{\mathrm{\Pi}}\ar[lll]_{\mathrm{\Sigma}}\\ \mathrm{\mathbf{Y}}&&& \mathrm{\widetilde{\bf{\,Y\,}}}\ar[lll]_{\mathrm{\sigma}}\\ } \end{xy} $$ The compact, complex space $\mathrm{\widetilde{\bf{\,Y\,}}}$ is given by \begin{equation} \mathrm{\widetilde{\,\bf{Y}\,}\coloneqq cl\,\left\{\left(y,{\bm{\mathfrak{C}}}\right)\in {\bf{Y}}_{0}\times\mathcal{C}^{k}(\widehat{\,\bf{X}\,})\!:\!\, \varphi^{\widehat{\pi}}\left(y\right)={\bm{\mathfrak{C}}}\right\} \subset \mathbf{Y}\times\mathcal{C}^{k}(\widehat{\,\bf{X}\,})}\label{Notation Blown Up Space Associated to Y} \end{equation} and the holomorphic map $\mathrm{\sigma\!:\!\widetilde{\,\bf{Y}\,}\rightarrow \mathbf{Y}}$ \label{Notation Blow Up Map onto Y} is defined by $\mathrm{\sigma\coloneqq p_{\mathbf{Y}}\vert\widetilde{\,\bf{Y}\,}}$. Moreover, if $\mathrm{{\bm{\mathfrak{X}}}\subset \mathcal{C}^{k}(\widehat{\,\bf{X}\,})\times}$ $\mathrm{\widehat{\,\bf{X}\,}}$\label{Notation Universal Space} denotes the universal space defined by $\mathrm{{\bm{\mathfrak{X}}}\coloneqq \left\{\big({\bm{\mathfrak{C}}},x\right)\in \mathcal{C}^{k}(\widehat{\,\bf{X}\,})\times \widehat{\,\bf{X}\,}: x\in \vert {\bm{\mathfrak{C}}}\vert\big\}}$, then $\mathrm{\widetilde{\bf{\,X\,}}}$ is compact and given by $$\mathrm{\widetilde{\bf{\,X\,}}=\left(\mathbf{Y}\times {\bm{\mathfrak{X}}}\right)\cap \big(\widetilde{\bf{\,Y\,}}\times\widehat{\bf{\,X\,}}\big)}$$ where $\mathrm{\Sigma}$ is the restriction of the projection $\mathrm{p_{}\!:\!\mathbf{Y}\times\mathcal{C}^{k}(\widehat{\,\bf{X}\,})\times\widehat{\bf{\,X\,}}\rightarrow \widehat{\bf{\,X\,}}}$ to $\mathrm{\widetilde{\bf{\,X\,}}}$. Note that the above construction, whose details can be found in \cite{Bar1}, implies that the fiber $\mathrm{\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}\right)\in}$ $\mathrm{ \widetilde{\bf{\,X\,}}}$ identifies with $\mathrm{\vert{\bm{\mathfrak{C}}}\vert\subset \widehat{\,\bf{X}\,}}$. Using this identification, we will simply write $\mathrm{\Pi^{-1}\left(y,\bm{\mathfrak{C}}\right)=\vert \bm{\mathfrak{{C}}}\vert}$ henceforth.
\begin{remark} Note that $\mathrm{\Pi\!:\!\widetilde{\bf{\,X\,}}\rightarrow \widetilde{\bf{\,Y\,}}}$ is a holomorphic map whose fibers are purely $\mathrm{k}$-dimensional by construction. Furthermore, $\mathrm{\widetilde{\bf{\,X\,}}}$ and $\widetilde{\bf{\,Y\,}}$ are both purely dimensional where $\mathrm{dim_{\mathbb{C}}\widetilde{\bf{\,X\,}}}$ $\mathrm{=dim_{\mathbb{C}}\bf{\,X\,}}$ and $\mathrm{dim_{\mathbb{C}}\widetilde{\bf{\,Y\,}}=dim_{\mathbb{C}}\bf{\,Y\,}}$ (which is a well known fact of the theory of proper modifications, cf.\ {\textnormal{\cite{Gr-Re}}}, {p.\ }{\bf{\oldstylenums{214}}}). \end{remark}
Moreover, we have the following lemma.
\begin{lemma}\label{Invariance of Cycles} The support $\mathrm{\Pi^{-1}\left(y,\bm{\mathfrak{C}}\right)=\vert \bm{\mathfrak{{C}}}\vert}$ is $\mathrm{\mathbb{T}}$-invariant subset of $\mathrm{\widehat{\pi}^{-1}\left(y\right)}$. \end{lemma} \begin{proof} By the commutativity of the previous diagram, we deduce that $\mathrm{\vert{\bm{\mathfrak{C}}} \vert\subset \widehat{\pi}^{-1}\left(y\right)}$, hence it remains to verify that $\mathrm{\vert{\bm{\mathfrak{C}}} \vert}$ is $\mathrm{\mathbb{T}}$-invariant. In order to prove this, we can proceed as follows: Since the set $$\mathrm{\left\{\left(y,{\bm{\mathfrak{C}}}\right)\in {{\bf{Y}}}_{0}\times\mathcal{C}^{k}(\widehat{\,\bf{X}\,})\!:\!\, \varphi^{\widehat{\pi}}\left(y\right)={\bm{\mathfrak{C}}}\right\}}$$ is Euclidean dense in $\mathrm{\widetilde{\bf{\,Y\,}}}$, we can choose a sequence $\mathrm{\left(y_{n}\right)_{n}}$\label{Notation y,frak C_y} in $\mathrm{\mathbf{Y}_{0}}$ so that $\mathrm{\left(y_{n},{\bm{\mathfrak{C}}}_{y_{n}}\right)\rightarrow}$ $\mathrm{\left(y,{\bm{\mathfrak{C}}}\right)}$ where $\mathrm{{\bm{\mathfrak{C}}}_{y_{n}}}$ are the cycles whose underlying sets are equal to $\mathrm{\widehat{\pi}^{-1}\left(y_{n}\right)}$ by property \ref{Equation Equality Cycles Barlet Construction}. The $\mathrm{\mathbb{T}}$-invariance of the limit cycle $\mathrm{{\bm{\mathfrak{C}}}}$ follows by the following reasoning: Let $\mathrm{x=t.x_{0}\in }$ $\mathrm{\mathbb{T}.\mathfrak {C}}$ where $\mathrm{x_{0}\in{\bm{\mathfrak{C}}} }$. Then, since $\mathrm{{\bm{\mathfrak{C}}}_{n}\rightarrow {\bm{\mathfrak{C}}}}$ means convergence in the Hausdorff topology of the underlying support, there exists a sequence $\mathrm{x_{n}\in {\bm{\mathfrak{C}}}_{n}}$ so that $\mathrm{x_{n}\rightarrow x_{0}}$. Since $\mathrm{\vert {\bm{\mathfrak{C}}}_{y}\vert=\widehat{\pi}^{-1}\left(y\right)}$ for all $\mathrm{y\in \mathbf{Y}_{0}}$, it follows that $\mathrm{\vert {\bm{\mathfrak{C}}}_{y_{n}}\vert=\widehat{\pi}^{-1}\left(y_{n}\right)}$ for all $\mathrm{n\in \mathbb{N}}$. Using the $\mathrm{\mathbb{T}}$-invariance of $\mathrm{\widehat{\pi}^{-1}\left(y\right)}$ (cf.\ {\bf{Remark \ref{Remark T-Invariance of Fibers}}}), we deduce that $\mathrm{t.x_{n}\in\vert {\bm{\mathfrak{C}}}_{y_{n}}\vert}$. By the continuity of the action it follows that $\mathrm{t.x_{n}\rightarrow t.x_{0}}$. So $\mathrm{\left(t.x_{n}\right)_{n}}$ is a convergent sequence with limit $\mathrm{t.x_{0}}$ where $\mathrm{t.x_{n}\in {\bm{\mathfrak{C}}}_{n}}$ and $\mathrm{{\bm{\mathfrak{C}}}_{n}\rightarrow {\bm{\mathfrak{C}}}}$. By the definition of the Hausdorff topology, it then follows that $\mathrm{t.x_{0}\in{\bm{\mathfrak{C}}}}$ and hence $\mathrm{\mathbb{T}.{\bm{\mathfrak{C}}}\subset {\bm{\mathfrak{C}}}}$ which proves the claim $\mathrm{\mathbb{T}.{\bm{\mathfrak{C}}}={\bm{\mathfrak{C}}}}$. \end{proof}
We close this section with the following remark and example.
\begin{remark} In general ({cf.\! }{\bf{Example \ref{Example Blowing Up Example}}}) $\mathrm{\vert \bm{\mathfrak{{C}}}\vert}$ is a proper subset of $\mathrm{\widehat{\pi}^{-1}\left(y\right)}$. \end{remark}
\begin{example}\label{Example Blowing Up Example} \textnormal{Let $\mathrm{\mathbf{X}=\mathbb{C}\mathbb{P}^{3}}$ equipped with the $\mathrm{\mathbb{T}=\mathbb{C}^{*}}$ action given by $$\mathrm{t.[z_{0}\!:\!z_{1}\!:\!z_{2}\!:\!z_{3}]=[t^{-1}z_{0}\!:\!t\,z_{1}\!:\!t\,z_{2}\!:\!z_{3}]}$$ and consider the Hilbert quotient $$\mathrm{\pi\!:\!\mathbf{X}^{ss}_{0}=\mathbb{C}\mathbb{P}^{3}\setminus\left(\left\{[1\!:\!0\!:\!0\!:\!0]\right\}\cup\left\{z_{0}=z_{3}=0\right\}\right)\rightarrow \mathbf{Y}\cong\mathbb{C}\mathbb{P}^{2}}$$ associated to the $\mathrm{0=\xi}$-level set of the moment map $$\mathrm{\mu\!:\![z]\mapsto\Vert z\Vert^{-2}\left(-\vert z_{0}\vert^{2}+\vert z_{1}\vert^{2}+\vert z_{2}\vert^{2}\right).}$$ The corresponding projection map $\mathrm{\pi}$ is given by $\mathrm{\pi\!:\![z]\mapsto[\zeta]=[z_{0}z_{1}\!:\!z_{0}z_{2}\!:\!z_{3}^2]}$ where $\mathrm{[\zeta]\in }$ $\mathrm{\mathbf{Y}\cong \mathbb{C}\mathbb{P}^{2}}$. Note that all fibers $\mathrm{\widehat{\pi}^{-1}\left([\zeta]\right)}$ over $\mathrm{\mathbf{Y}_{0}=\mathbb{C}\mathbb{P}^{2}\setminus \left\{[0\!:\!0\!:\!1]\right\}}$ are of pure dimension one and of degree two. Moreover, it is direct to verify that these fibers can be parameterized by $$\mathrm{\gamma_{[\zeta]}\!:\!t\mapsto [\zeta_{2}t_{1}^{2}\!:\!\zeta_{0}t_{0}^{2}\!:\!\zeta_{1}t_{0}^{2}\!:\!\zeta_{2}t_{0}t_{1}]\text{ for }\zeta\in \mathbf{Y}_{0}}$$ and that they are given as the zero set of the following system of equations\!:\! $$\mathrm{\zeta_{2}z_{0}z_{1}-\zeta_{0}z^{2}_{3}=0,\,\zeta_{2}z_{0}z_{2}-\zeta_{1}z_{3}^{2}=0\text{ where }\zeta\in \mathbf{Y}_{0}.}$$ Let $\mathrm{\mathbf{U}_{2}=\left\{[\zeta]\in \mathbb{C}\mathbb{P}^{2}\!:\!\,\zeta_{2}\neq 0\right\}}$ and set $\mathrm{c_{0}\coloneqq \zeta_{2}^{-1}\zeta_{0}}$, $\mathrm{c_{1}\coloneqq \zeta_{2}^{-1}\zeta_{1}}$ and consider $\mathrm{\mathbf{U}_{2}^{*}\coloneqq}$ $\mathrm{\mathbf{Y}_{0}\cap \mathbf{U}_{2}}$ which we can identify with $\mathrm{\mathbb{C}^{2}\setminus \left\{0\right\}}$. As mentioned before, there exists a holomorphic map $\mathrm{\varphi^{\widehat{\pi}}\!:\!\mathbf{U}^{*}_{2}\hookrightarrow \mathcal{C}^{1}(\widehat{\,\mathbf{X}\,})}$. It turns out that all fibers of $\mathrm{\widehat{\pi}}$ are compact subvarieties of degree $\mathrm{2}$ in $\mathrm{\mathbb{C}\mathbb{P}^{3}}$. Hence, it follows that the image of $\mathrm{\varphi^{\widehat{\pi}}}$ is contained in the cycle space component which can be identified with the compact connected {\scshape{Chow}} Variety $\mathrm{\mathcal{C}_{1,2}\left(\mathbb{C}\mathbb{P}^{3}\right)}$ of all $\mathrm{1}$-dimensional cycles in $\mathrm{\mathbb{C}\mathbb{P}^{3}}$ of degree $\mathrm{2}$ which itself is realized as closed variety in the projective space $\mathrm{\mathbb{C}\mathbb{P}^{\nu_{3,1,2}}}$ (for a rigorous definition cf.\ \cite{Sha}).}
\textnormal{Recall that the Chow coordinates of a cycle $\mathrm{{\bm{\mathfrak{C}}}}$ in $\mathrm{\mathbf{X}\subset \mathbb{C}\mathbb{P}^{m}}$ of degree $\mathrm{d}$ and dimension $\mathrm{k}$ are given by the coefficients of the Chow form $\mathrm{\mathfrak{F}_{{\bm{\mathfrak{C}}},\mathbb{C}\mathbb{P}^{m}}}$ \label{Notation Chow Form}, i.e.\ by the coefficients of a polynomial homogenous in $\mathrm{k+1}$ groups $\mathrm{\xi_{0}^{(i)},\dots,\xi^{(i)}_{m}}$, $\mathrm{i\in \left\{0,\dots,k\right\}}$ of $\mathrm{m+1}$ indeterminates of degree $\mathrm{d}$ modulo multiplication with a non-vanishing complex number $\mathrm{\lambda\in \mathbb{C}^{*}}$ (cf.\ \cite{Sha}). In the sequel, let $\mathrm{{\bm{\mathfrak{C}}}_{c}}$ for $\mathrm{c\in \mathbb{C}^{2}\setminus \left\{0\right\}}$ and let $\mathrm{\mathfrak{F}_{{\bm{\mathfrak{C}}}_{c},\mathbb{C}\mathbb{P}^{3}}}$ be the corresponding Chow form. A calculation shows that $$\mathrm{\begin{array}{rl}
& \mathrm{\mathfrak{F}_{{\bm{\mathfrak{C}}}_{c},\mathbb{C}\mathbb{P}^{3}}\left(\xi_{0}^{\left(0\right)},\xi_{1}^{\left(0\right)},\xi_{2}^{\left(0\right)},\xi_{3}^{\left(0\right)},\xi_{0}^{\left(1\right)},\xi_{1}^{\left(1\right)},\xi_{2}^{\left(1\right)},\xi_{3}^{\left(1\right)}\right)}\\[0.4 cm] =& \mathrm{c_{0}^{2}\,{\xi_{1}^{(0)}}^{2}{\xi_{0}^{(1)}}^{2}+c_{1}^{2}\,{\xi_{2}^{\left(0\right)}}^{2}{\xi_{0}^{\left(1\right)}}^{2}+c_{0}^{2}\,{\xi_{0}^{\left(0\right)}}^{2}{\xi_{1}^{\left(1\right)}}^{2}+c_{1}^{2}\,{\xi_{0}^{\left(0\right)}}^{2}{\xi_{2}^{\left(1\right)}}^{2}+2\, c_{0}\,c_{1}\,{\xi_{0}^{\left(0\right)}}^{2}\xi_{1}^{\left(1\right)}\xi_{2}^{\left(1\right)}}\\[0.4 cm] &\mathrm{+c_{0}\,\xi_{0}^{\left(0\right)}\xi_{1}^{\left(0\right)}{\xi_{3}^{\left(1\right)}}^{2}+c_{1}\,\xi_{0}^{\left(0\right)}\xi_{2}^{\left(0\right)}{\xi_{3}^{\left(1\right)}}^{2}+c_{0}\,{\xi_{3}^{\left(0\right)}}^{2}\xi_{0}^{\left(1\right)}\xi_{1}^{\left(1\right)}+c_{1}\,{\xi_{3}^{\left(0\right)}}^{2}\xi_{0}^{\left(1\right)}\xi_{2}^{\left(1\right)}}\\[0.4 cm]
&\mathrm{+2\,c_{0}\,c_{1}\,\xi_{1}^{\left(0\right)}\xi_{2}^{\left(0\right)}{\xi_{0}^{\left(1\right)}}^{2}-c_{0}\,\xi_{0}^{\left(0\right)}\xi_{3}^{\left(0\right)}\xi_{1}^{\left(1\right)}\xi_{3}^{\left(1\right)}-c_{1}\,\xi_{0}^{\left(0\right)}\xi_{3}^{\left(0\right)}\xi_{2}^{\left(1\right)}\xi_{3}^{\left(1\right)}-c_{0}\,\xi_{1}^{\left(0\right)}\xi_{3}^{\left(0\right)}\xi_{0}^{\left(1\right)}\xi_{3}^{\left(1\right)}}\\[0.4 cm]
&\mathrm{-c_{1}\,\xi_{2}^{\left(0\right)}\xi_{3}^{\left(0\right)}\xi_{0}^{\left(1\right)}\xi_{3}^{\left(1\right)}-2\,c_{0}^{2}\,\xi_{0}^{\left(0\right)}\xi_{1}^{\left(0\right)}\xi_{0}^{\left(1\right)}\xi_{1}^{\left(1\right)}-2\,c_{0}\,c_{1}\,\xi_{0}^{\left(0\right)}\xi_{1}^{\left(0\right)}\xi_{0}^{\left(1\right)}\xi_{2}^{\left(1\right)}}\\[0.4 cm]
&\mathrm{-2\,c_{0}\,c_{1}\,\xi_{0}^{\left(0\right)}\xi_{2}^{\left(0\right)}\xi_{0}^{\left(1\right)}\xi_{1}^{\left(1\right)}-2\,c_{1}^{2}\,\xi_{0}^{\left(0\right)}\xi_{2}^{\left(0\right)}\xi_{0}^{\left(1\right)}\xi_{2}^{\left(1\right)}}\\ \end{array}}$$ so the map $\mathrm{\varphi^{\widehat{\pi}}\!:\!\mathbb{C}^{2}\setminus \left\{0\right\}\rightarrow \mathcal{C}_{1,2}\left(\mathbb{C}\mathbb{P}^{3}\right)}$ is given by $$\mathrm{\begin{array}{rcl} \mathrm{\varphi^{\widehat{\pi}}\!:\!\mathbb{C}^{2}\setminus \left\{0\right\}\ni\left(c_{1},c_{2}\right)} & \mapsto & \mathrm{[c_{0}^{2}\!:\!c_{0}^{2}\!:\!\!-2\, c_{0}^{2}\!:\!c_{1}^{2}\!:\!c_{1}^{2}\!:\!\!-2\, c_{1}^{2}\!:\!2\, c_{0}c_{1}\!:\!2\, c_{0}c_{1}\!:\!\,-2\, c_{0}c_{1}\!:\!}\\[0,2 cm]
& & \mathrm{-2\, c_{0}c_{1}\!:\!-c_{0}\!:\!\!-c_{0}\!:\!c_{0}\!:\!c_{0}\!:\!\!-c_{1}\!:\!\!-c_{1}\!:\!c_{1}\!:\!c_{1}\!:\!0\!:\!...\!:\!0].}\end{array}}$$ The closure of the graph $$\mathrm{\left\{\left(c,{\bm{\mathfrak{C}}}\right)\!:\!\,c\in\mathbb{C}^{2}\setminus \left\{0\right\},\,\varphi^{\widehat{\pi}}\left(c\right)={\bm{\mathfrak{C}}}\right\}\subset \mathbb{C}^{2}\times \mathcal{C}_{1,2}\left(\mathbb{C}\mathbb{P}^{3}\right)}$$ of the map $\mathrm{\varphi^{\widehat{\pi}}}$ turns out to be isomorphic to the blow-up $\mathrm{ {\bm{\mathfrak{Bl}}}\left(0,\mathbb{C}^{2}\right)}$ of the origin $\mathrm{0\in \mathbb{C}^{2}}$. This can be seen in the following way: Let $$\mathrm{\varphi\!:\!\mathbb{C}^{2}\setminus\left\{0\right\}\ni\left(c_{0},c_{1}\right)\mapsto \left(\left(c_{0},c_{1}\right),[c_{0}\!:\!c_{1}]\right) \in {\bm{ {\bm{\mathfrak{Bl}}}}}\left(0,\mathbb{C}^{2}\right)}$$ where $\mathrm{ {\bm{\mathfrak{Bl}}}\left(0,\mathbb{C}^{2}\right)=\left\{\left(\left(c_{0},c_{1}\right),[c^{\prime}_{0}\!:\!c^{\prime}_{1}]\right)\!:\!\,c_{0}c^{\prime}_{1}-c_{1}c^{\prime}_{0}=0\right\}\subset \mathbb{C}^{2}\times \mathbb{C}\mathbb{P}^{1}}$. We have $$\mathrm{cl\left(\varphi\left(\mathbb{C}^{2}\setminus\{0\}\right)\right)= {\bm{\mathfrak{Bl}}}\left(0,\mathbb{C}^{2}\right).}$$ We can embed $\mathrm{ {\bm{\mathfrak{Bl}}}\left(0,\mathbb{C}^{2}\right)\subset \mathbb{C}^{2}\times \mathbb{C}\mathbb{P}^{1}\subset \mathbb{C}\mathbb{P}^{2}\times \mathbb{C}\mathbb{P}^{1}}$ into $\mathrm{\mathbb{C}\mathbb{P}^{5}}$ using the {\scshape{Segre}} map. The composition of $\mathrm{\varphi}$ with the {\scshape{Segre}} embedding yields an embedding $\mathrm{\widehat{\varphi}\!:\!\mathbb{C}^{2}\setminus}$ $\mathrm{\left\{0\right\}\hookrightarrow \mathbb{C}\mathbb{P}^{5}}$ given by $$\mathrm{\widehat{\varphi}\!:\!\mathbb{C}^{2}\setminus\left\{0\right\}\mapsto [c_{0}\!:\!c_{1}\!:\!c_{0}^2\!:\!c_{0}c_{1}\!:\!c_{0}c_{1}\!:\!c_{1}^{2}].}$$ It is direct to see that there exists a projective transformation $\mathrm{\Phi}$ of $\mathrm{\mathbb{C}\mathbb{P}^{\nu_{3,1,2}}}$ so that the following diagram commutes: $$\begin{xy} \xymatrix{ \mathrm{\mathbb{C}^{2}\setminus\left\{0\right\}} \ar@{^{(}->}[d] \ar@{^{(}->}[rrrrr]^{\zeta_{\mathbb{C}^{2}\setminus\left\{0\right\}}} &&&&&\mathrm{\mathcal{C}_{1,2}\left(\mathbb{C}\mathbb{P}^{3}\right)\subset}\hspace{-0.9 cm}& \mathrm{\mathbb{C}\mathbb{P}^{\nu_{3,1,2}}}\\ \mathrm{ {\bm{\mathfrak{Bl}}}\left(0,\mathbb{C}^{2}\right)}&**[r] \hspace{-1cm}\subset \mathbb{C}\mathbb{P}^{2}\times \mathbb{C}\mathbb{P}^{1}\ar@{^{(}->}[rrrr]^-{\text{{\scshape{Segre}} map}}&&& &\mathrm{\mathbb{C}\mathbb{P}^{5}\subset }\hspace{ -1.85 cm} & \mathrm{\mathbb{C}\mathbb{P}^{\nu_{3,1,2}}}\ar[u]_{\Phi} \\ } \end{xy} $$ Let $\mathrm{\left(\left(c_{n},{\bm{\mathfrak{C}}}_{c_{n}}\right)\right)_{n}}$ be the sequence in $\mathrm{\widetilde{\bf{\,Y\,}}\cap\sigma^{-1}\left(\mathbb{C}^{2}\setminus{0}\right)}$ given by $\mathrm{c_{n}=n^{-1}\left(c^{\prime}_{0},c_{1}^{\prime}\right)}$. The above formulas show that $\mathrm{\left({\bm{\mathfrak{C}}}_{c_{n}}\right)_{n}}$ converges to $$\mathrm{{\bm{\mathfrak{C}}}_{c^{\prime}}=[0\!:\!0\!:\!0\!:\!0\!:\!0\!:\!0\!:\!0\!:\!0\!:\!-c^{\prime}_{0}\!:\!-c^{\prime}_{0}\!:\!c^{\prime}_{0}\!:\!c^{\prime}_{0}\!:\!-c^{\prime}_{1}\!:\!-c^{\prime}_{1}\!:\!c^{\prime}_{1}\!:\!c^{\prime}_{1}\!:\!0\!:\!...\!:\!0]}$$ which corresponds to the point $\mathrm{\left(\left(0,0\right),[c^{\prime}_{1}\!:\!c^{\prime}_{2}]\right)\in {\bm{\mathfrak{Bl}}}\left(0,\mathbb{C}^{2}\right)}$ under the above identification. In particular we have $\mathrm{\sigma\left(0,{\bm{\mathfrak{C}}}_{c^{\prime}}\right)=[0\!:\!0\!:\!1]\in \mathbb{C}^{2}}$. In order to determine the limit cycle $$\mathrm{{\bm{\mathfrak{C}}}_{c^{\prime}}\in \widetilde{\bf{\,Y\,}}\cap\sigma^{-1}\left(\mathbf{U}_{2}\right)}$$ we consider the Chow form $\mathrm{\mathfrak{F}_{{\bm{\mathfrak{C}}}_{c^{\prime}},\mathbb{C}\mathbb{P}^{3}}}$ $$\mathrm{\begin{array}{rl} & \mathrm{\mathfrak{F}_{{\bm{\mathfrak{C}}}_{c^{\prime}},\mathbb{C}\mathbb{P}^{3}}\left(\xi_{0}^{\left(0\right)},\xi_{1}^{\left(0\right)},\xi_{2}^{\left(0\right)},\xi_{3}^{\left(0\right)},\xi_{0}^{\left(1\right)},\xi_{1}^{\left(1\right)},\xi_{2}^{\left(1\right)},\xi_{3}^{\left(1\right)}\right)}\\[0.4 cm] =& \mathrm{c^{\prime}_{0}\,\xi_{0}^{\left(0\right)}\xi_{1}^{\left(0\right)}{\xi_{3}^{\left(1\right)}}^{2}+c^{\prime}_{1}\,\xi_{0}^{\left(0\right)}\xi_{2}^{\left(0\right)}{\xi_{3}^{\left(1\right)}}^{2}+c^{\prime}_{0}\,{\xi_{3}^{\left(0\right)}}^{2}\xi_{0}^{\left(1\right)}\xi_{1}^{\left(1\right)}+c^{\prime}_{1}\,{\xi_{3}^{\left(0\right)}}^{2}\xi_{0}^{\left(1\right)}\xi_{2}^{\left(1\right)}-c^{\prime}_{0}\,\xi_{0}^{\left(0\right)}\xi_{3}^{\left(0\right)}\xi_{1}^{\left(1\right)}\xi_{3}^{\left(1\right)}}\\[0.4 cm] &\mathrm{-c^{\prime}_{1}\,\xi_{0}^{\left(0\right)}\xi_{3}^{\left(0\right)}\xi_{2}^{\left(1\right)}\xi_{3}^{\left(1\right)}-c^{\prime}_{0}\,\xi_{1}^{\left(0\right)}\xi_{3}^{\left(0\right)}\xi_{0}^{\left(1\right)}\xi_{3}^{\left(1\right)}-c^{\prime}_{1}\,\xi_{2}^{\left(0\right)}\xi_{3}^{\left(0\right)}\xi_{0}^{\left(1\right)}\xi_{3}^{\left(1\right)}}\end{array}}$$ which turns out to be reducible:}
\textnormal{$$\mathrm{\begin{array}{rl} & \mathrm{\mathfrak{F}_{{\bm{\mathfrak{C}}}_{c^{\prime}},\mathbb{C}\mathbb{P}^{3}}\left(\xi_{0}^{\left(0\right)},\xi_{1}^{\left(0\right)},\xi_{2}^{\left(0\right)},\xi_{3}^{\left(0\right)},\xi_{0}^{\left(1\right)},\xi_{1}^{\left(1\right)},\xi_{2}^{\left(1\right)},\xi_{3}^{\left(1\right)}\right)}\\[0.4 cm] =&\mathrm{\mathrm{\overset{\mathfrak{F}_{\widetilde{{\bm{\mathfrak{C}}}}_{c^{\prime}},\mathbb{C}\mathbb{P}^{3}}}{\overbrace{\left(\mathrm{c^{\prime}_{0}\,\xi_{1}^{\left(0\right)}\xi_{3}^{\left(1\right)}+c_{1}^{\prime}\,\xi_{2}^{\left(0\right)}\xi_{3}^{\left(1\right)}-c^{\prime}_{0}\,\xi_{3}^{\left(0\right)}\xi_{1}^{\left(1\right)}-c^{\prime}_{1}\,\xi_{3}^{\left(0\right)}\xi_{2}^{\left(1\right)}}\right)}}}\cdot\overset{\mathfrak{F}_{{\bm{\mathfrak{C}}}_{0},\mathbb{C}\mathbb{P}^{3}}}{\overbrace{\left(\mathrm{\xi_{0}^{\left(0\right)}\xi_{3}^{\left(1\right)}-\xi_{3}^{\left(0\right)}\xi_{0}^{\left(1\right)}}\right)}}}\\ \end{array}}$$ A direct computation shows that $\mathrm{\mathfrak{F}_{\widetilde{{\bm{\mathfrak{C}}}}_{c^{\prime}},\mathbb{C}\mathbb{P}^{3}}}$ is the Chow form associated to the line $$\mathrm{\widetilde{{\bm{\mathfrak{C}}}}_{c^{\prime}}=\left\{z_{0}=0,\,z_{4}=0, \,c^{\prime}_{2}z_{1}-c_{1}^{\prime}z_{2}=0\right\}}$$ and $\mathrm{\mathfrak{F}_{{\bm{\mathfrak{C}}}_{0},\mathbb{C}\mathbb{P}^{3}}}$ is the corresponding Chow form of the line $$\mathrm{{\bm{\mathfrak{C}}}_{0}=\left\{z_{1}=0,\,z_{2}=0,\,z_{3}=0\right\}}$$ where $\mathrm{{\bm{\mathfrak{C}}}_{c^{\prime}}=\widetilde{{\bm{\mathfrak{C}}}}_{c^{\prime}}+ {\bm{\mathfrak{C}}}_{0}}$. In particular, note that $\mathrm{{\bm{\mathfrak{C}}}_{c^{\prime}}\neq \widehat{\pi}^{-1}\left([0\!:\!0\!:\!1]\right)}$ where $$\mathrm{\widehat{\pi}^{-1}\left([0\!:\!0\!:\!1]\right)=\left\{z_{0}=0\right\}\cup\left\{z_{1}=z_{2}=z_{3}=0\right\}.\,\boldsymbol{\Box}}$$} \end{example}
\subsection{Fiber Integral Properties of the $\mathrm{k}$-Fibering $\mathrm{\Pi\!:\!\widetilde{\bf{\,X\,}}\rightarrow \widetilde{\bf{\,Y\,}}}$}
As in {\bf{Section \ref{Uniform Localization Proposition}}}, let $\mathrm{\mathbf{X}^{i}}$ be the $\mathrm{\mathbb{T}}$-invariant Zariski open subset of $\mathrm{\mathbf{X}\left(s_{n}^{i}\right)\subset\mathbf{X}^{ss}_{\xi}}$ of {\bf{Theorem 1}}, where $\mathrm{\mathbf{X}\left(s_{n}^{i}\right)}$ is the $\mathrm{n}$-stable complement of the zero set of the tame sequence $\mathrm{\left(s^{i}_{n}\right)_{n}}$. As before, let $\mathrm{\varrho^{i}\!:\!\mathbf{X}^{i}}$ $\mathrm{\rightarrow \mathbb{R}}$ be the normalized s.p.s.h.\ limit function and recall the definition of the compact tube $\mathrm{T\left(\epsilon,\mathbf{W}^{i}\right)}$ $\mathrm{\subset \mathbf{X}^{i}}$ for $\mathrm{\mathbf{W}^{i}\subset \pi\left(\mathbf{X}^{i}\right)}$ a compact neighborhood and $\mathrm{\epsilon>0}$ given in {\bf{Section \ref{Uniform Localization Proposition}}}: $$\mathrm{T\left(\epsilon,\mathbf{W}^{i}\right)= \left(\varrho^{i}\times \pi\right)^{-1}\left([0,\epsilon]\times \mathbf{W}^{i}\right).}$$ Define the corresponding tube $\mathrm{\widetilde{\,T\,}\left(\epsilon,\mathbf{W}^{i}\right)}$ \label{Notation Compact Corresponding Neighborhood Tube} in $\mathrm{\widetilde{\bf{\,X\,}}}$ by $$\mathrm{\widetilde{\,T\,}\left(\epsilon,\mathbf{W}^{i}\right)\coloneqq \Sigma^{-1}\left(T\left(\epsilon,\mathbf{W}^{i}\right)\right)}$$ where we have used the fact that $\mathrm{\mathbf{X}^{ss}_{\xi}\supset T\left(\epsilon,\mathbf{W}^{i}\right)}$ is naturally embedded in $\mathrm{\widehat{\mathbf{\,\bm{X}\,}}}$ via $\mathrm{\zeta^{-1}\vert \mathbf{X}^{ss}_{\xi}}$ (recall that $\mathrm{\zeta\vert \zeta^{-1}\big(\mathbf{X}^{ss}_{\xi}\big)}$ is biholomorphic because $\mathrm{\mathbf{X}^{ss}_{\xi}}$ is assumed to be normal). Note that $\mathrm{\widetilde{\,T\,}\left(\epsilon,\mathbf{W}^{i}\right)}$ projects down via $\mathrm{\Pi}$ onto the compact neighborhood $\mathrm{\widetilde{\bf{W}}^{i}\coloneqq\sigma^{-1}\left(\mathbf{W}^{i}\right)}$. \label{Notation cal W^i}
The first aim of this section is to show that the fiber integral $\mathrm{vol\left(\pi^{-1}\left(y\right)\right)=\int_{\pi^{-1}\left(y\right)}d\,[\pi_{y}]}$ \label{Notation Volume of a Fiber Over Y_0} is bounded as $\mathrm{y}$ varies in $\mathrm{\mathbf{Y}_{0}}$. \begin{lemma}\label{Lemma Equality of Volume} For all $\mathrm{y\in \mathbf{W}^{i}\cap \mathbf{Y}_{0}}$ we have $$\mathrm{vol\left(T\left(\epsilon,{\mathbf{W}}^{i}\right)\cap\pi^{-1}\left(y\right)\right)=vol\left(\widetilde{\,T\,}\left(\epsilon,{\bf{W}}^{i}\right)\cap\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}_{y}\right)\right)}$$ where the right hand side is the fiber integral of the projection taken with respect to $\mathrm{\Omega^{k}}$ where $\mathrm{\Omega\coloneqq\Sigma^{*}\omega^{\prime}}$ \label{Notation Capital Omega}.
Furthermore, we have $$\mathrm{vol\left(\pi^{-1}\left(y\right)\right)\leq vol\left(\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}_{y}\right)\right)}$$ for all $\mathrm{y \in\mathbf{Y}_{0}}$. \end{lemma} \begin{proof} This is a direct consequence of the following reasoning: Recall that $\mathrm{\Sigma}$ is a modification with center $\mathrm{\widehat{\pi}^{-1}\left(\mathbf{Y}\setminus \mathbf{Y}_{0}\right)}$ and $\mathrm{\zeta\vert\zeta^{-1}\big(\mathbf{X}^{ss}_{\xi}\big)}$ is an isomorphism (as before we consider $\mathrm{\mathbf{X}^{ss}_{\xi}}$ embedded in $\mathrm{\mathbf{\Gamma}_{\pi}}$). Hence, it follows that the open subset $$\mathrm{\Pi^{-1}\left(\zeta^{-1}\left(\mathbf{X}_{0}\right)\right), \text{ where }\mathbf{X}_{0}=\pi^{-1}\left(\mathbf{Y}_{0}\right)}$$ in $\mathrm{\widehat{\,\mathbf{X}\,}}$ is mapped isomorphically on $\mathrm{\pi^{-1}\left(\mathbf{Y}_{0}\right)}$ by $\mathrm{\zeta\circ\Sigma}$. Therefore, if $\mathrm{y\in \mathbf{Y}_{0}}$, we deduce that $\mathrm{\pi^{-1}\left(y\right)}$ is biholomorphic to $$\mathrm{\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}_{y}\right)\cap \Sigma^{-1}\big(\zeta^{-1}\left(\mathbf{X}^{ss}_{\xi}\right)\big)}$$ via $\mathrm{\left(\zeta\circ \Sigma\right)^{-1}}$. Using the fact that $\mathrm{\widetilde{\,T\,}\left(\epsilon,{\bf{W}}^{i}\right)}$ is defined as the pull back of $\mathrm{T\left(\epsilon,{\bf{W}}^{i}\right)}$ and that $\mathrm{\Omega=(\zeta\circ}$ $\mathrm{\Sigma)^{*}\omega}$, it follows that the volume of $\mathrm{T\left(\epsilon,{\mathbf{W}}^{i}\right)\cap\pi^{-1}\left(y\right)}$ with respect to $\mathrm{\omega}$ is equal to the volume of $\mathrm{\widetilde{\,T\,}\left(\epsilon,{\bf{W}}^{i}\right)\cap\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}_{y}\right)}$ with respect to $\mathrm{\Omega}$, which proves the first claim.
The second claim is an immediate consequence of the above argumentation: Via $\mathrm{\left(\zeta\circ \Sigma\right)^{-1}}$, the fiber $\mathrm{\pi^{-1}\left(y\right)}$, where $\mathrm{y\in \mathbf{Y}_{0}}$, is biholomorphic to $$\mathrm{\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}_{y}\right)\cap \Sigma^{-1}\big(\zeta^{-1}\left(\mathbf{X}^{ss}_{\xi}\right)\big).}$$ In particular, it can be seen as a subset of $\mathrm{\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}_{y}\right)}$ realized by $\mathrm{\left(\zeta\circ \Sigma\right)^{-1}}$. \end{proof}
\begin{remark} In general, the inequality in {\bf{Lemma \ref{Lemma Equality of Volume}}} is a strict inequality. This is exhibited in {\bf{Example \ref{Example Proper Inclusion}}}, where $$\mathrm{cl\left(\pi^{-1}\left(\left[1\!:\!0\right]\right)\right)=\left\{ z_{1}=0\right\} \text{ and }cl\left(\pi^{-1}\left(\left[0\!:\!1\right]\right)\right)=\left\{ \zeta_{1}=0\right\}}$$ on the one hand and $$\mathrm{p_{\mathbf{X}}\left(\widehat{\pi}^{-1}\left(\left[1\!:\!0\right]\right)\right)=\left\{ z_{1}=0\right\} \cup\left\{ \zeta_{0}=0\right\} \text{ resp. }}$$ $$\mathrm{p_{\mathbf{X}}\left(\widehat{\pi}^{-1}\left(\left[0\!:\!1\right]\right)\right)=\left\{ \zeta_{1}=0\right\} \cup\left\{ z_{0}=0\right\} }$$ on the other hand. Hence, $\mathrm{cl\left(\pi^{-1}\left(\left[\zeta_{i}\right]\right)\right)}$ $\mathrm{i\in\left\{0,1\right\}}$ is properly contained as an irreducible component in $\mathrm{p_{\mathbf{X}}\left(\widehat{\pi}^{-1}\left([\zeta_{i}]\right)\right)}$. \end{remark}
As a direct consequence of {\bf{Lemma \ref{Lemma Equality of Volume}}}, we deduce the following two corollaries.
\begin{cor}\label{Boundedness of Fiber Integral} Let $\mathrm{\mathbf{Y}_{0}}$ be as in {\bf{Section \ref{Definition of Y_{0}}}}, then there exists a constant $\mathrm{C>0}$ so that $$\mathrm{vol\left(\pi^{-1}\left(y\right)\right)=\int_{\pi^{-1}\left(y\right)}\,d\,[\pi_{y}]\leq C}$$ for all $\mathrm{y\in \mathbf{Y}_{0}}$. \end{cor} \begin{proof} By the second claim of {\bf{Lemma \ref{Lemma Equality of Volume}}}, we have $$\mathrm{vol\left(\pi^{-1}\left(y\right)\right)\leq vol\left(\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}_{y}\right)\right)\text{ for all }y\in\mathbf{Y}_{0}.}$$ Since the projection of $\mathrm{\widetilde{\bf{\,Y\,}}}$ on $\mathrm{\mathcal{C}^{k}\big(\widehat{\bf{\,X\,}}\big)}$ is a compact subset, the claim then follows by the fact that the volumes of all cycles which are contained in a compact subset of $\mathrm{\mathcal{C}^{k}\big(\widehat{\bf{\,X\,}}\big)}$ are uniformly bounded from above ({cf.\! }\cite{Bar2}). \end{proof}
\begin{lemma}\label{Lemma Intersection Zero Measure} Let $\mathrm{\left(y,{\bm{\mathfrak{C}}}\right)\in\widetilde{\bf{\,Y\,}}}$, then $\mathrm{\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}_{y}\right)\cap \widetilde{\,T\,}\left(0,\mathbf{W}^{i}\right)}$ is of measure zero concerning the measure induced by $\mathrm{\Omega}$.
Moreover, the restriction of the form $\mathrm{\Omega}$ on $\mathrm{\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}\right)\cap \widetilde{\,T\,}\left(\epsilon,\mathbf{W}^{i}\right)}$ where $\mathrm{\epsilon>0}$ is non-zero. \end{lemma} \begin{proof} First of all note, that we can view $\mathrm{\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}\right)\cap \widetilde{\,T\,}\left(\epsilon,\mathbf{W}^{i}\right)}$ as a $\mathrm{\mathbb{T}}$-invariant, closed, $\mathrm{k}$-dimensional complex subspace in $\mathrm{\pi^{-1}\left(y\right)\cap T\left(\epsilon,\mathbf{W}^{i}\right)}$. In fact, each fiber $\mathrm{\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}\right)}$ is given by the $\mathrm{\mathbb{T}}$-invariant, $\mathrm{k}$-dimensional subvariety $\mathrm{\vert{\bm{\mathfrak{C}}}\vert\subset \widehat{\,\bf{X}\,}}$. The identification is then induced by $\mathrm{p_{\mathbf{X}}\vert cl\left({\mathbf{\Gamma}}_{\pi}\right)\circ\zeta}$ which is biholomorphic over $\mathrm{\mathbf{X}^{ss}_{\xi}\supset T\left(\epsilon,{\mathbf{W}}^{i}\right)}$. Now, the second claim of the lemma is an immediate consequence of this fact combined with $\mathrm{\omega^{\prime}=\left(p_{\mathbf{X}}\vert cl\left({\mathbf{\Gamma}}_{\pi}\right)\circ\zeta\right)^{*}\omega}$.
The first claim follows from the fact that the minimal closed, $\mathrm{\mathbb{T}}$-invariant complex space of $\mathrm{\pi^{-1}\left(y\right)\cap T\left(\epsilon,\mathbf{W}^{i}\right)}$ containing $$\mathrm{\pi^{-1}\left(y\right)\cap T\left(0,\mathbf{W}^{i}\right)=\pi^{-1}\left(y\right)\cap \mu^{-1}\left(\xi\right)}$$ is given by the unique closed orbit $\mathrm{\mathbb{T}.z_{y}}$ which contains $\mathrm{\pi^{-1}\left(y\right)\cap T\left(0,\mathbf{W}^{i}\right)}$ as a total real submanifold. \end{proof}
The rest of this section is devoted to the proof of the existence of uniform estimates concerning the fiber volume.
\begin{prop}\label{Continuity of Fiber Integral} Let $\mathrm{\Delta>0}$ and $\mathrm{\mathbf{W}^{i}\subset \pi\left(\mathbf{X}^{i}\right)}$ be as above, then there exists $\mathrm{\epsilon_{\Delta}>0}$ so that $$\mathrm{vol\left(\pi^{-1}\left(y\right)\cap T\left(\epsilon_{\Delta},\mathbf{W}^{i}\right) \right)\leq\Delta}$$ for all $\mathrm{y\in \mathbf{W}^{i}\cap \mathbf{Y}_{0}}$.
Moreover, if $\mathrm{\epsilon>0}$, then there exists $\mathrm{\delta>0}$ so that $$\mathrm{\delta\leq vol\left(\pi^{-1}\left(y\right)\cap T\left(\epsilon,\mathbf{W}^{i}\right) \right)}$$ for all $\mathrm{y\in \mathbf{W}^{i}\cap \mathbf{Y}_{0}}$. \end{prop} \begin{proof} Choose a sequence $\mathrm{\epsilon_{n}\rightarrow 0}$ and a sequence $\mathrm{\left(\psi_{n}\right)_{n}}$ of smooth cut-off functions on $\mathrm{\widetilde{\bf{\,X\,}}}$ so that $$\mathrm{\psi_{n}\vert \widetilde{\,T\,}\left(\epsilon_{n},\mathbf{W}^{i}\right)\equiv 1\text{ and }supp\,\psi_{n}\cap \widetilde{\,T\,}^{c}\left(\epsilon_{n+1},\mathbf{W}^{i}\right)=\varnothing}$$ which is possible since $\mathrm{\widetilde{\,T\,}\left(\epsilon_{n},\mathbf{W}^{i}\right)}$ is relatively compact. In particular, we have $$\mathrm{\Pi^{-1}\big(\widetilde{\bf{W}}^{i}\big)\cap supp\,\psi_{n}\subset \widetilde{\,T\,}\left(\epsilon_{n+1},\mathbf{W}^{i}\right)}$$ and therefore $$\mathrm{\Pi^{-1}\big(\widetilde{\bf{W}}^{i}\big)\cap supp\,\psi_{n}\downarrow \widetilde{\,T\,}\left(0,\mathbf{W}^{i}\right) \text{ as }n\rightarrow \infty.}$$ Hence, by the first claim of {\bf{Lemma \ref{Lemma Intersection Zero Measure}}}, the intersection of the set $\mathrm{supp\,\psi_{n}}$ and a fiber of $\mathrm{\Pi}$ over $\mathrm{\widetilde{\bf{W}}^{i}}$ converges monotonically decreasing to a set of measure zero with respect to the measure induced by $\mathrm{\Omega}$. Set $\mathrm{\Omega_{n}\coloneqq \psi_{n}\Omega^{k}}$.
To sum up, $\mathrm{\left(\Omega_{n}\right)_{n}}$ is a sequence of smooth $\mathrm{\left(k,k\right)}$-forms on $\mathrm{\widetilde{\bf{\,X\,}}}$ with compact support where $\mathrm{\Pi\!:\!\widetilde{\bf{\,X\,}}\rightarrow \widetilde{\bf{\,Y\,}}}$ is a holomorphic map between purely dimensional compact complex spaces so that each fiber $\mathrm{\Pi^{-1}\left(y\right)}$ is purely $\mathrm{k}$-dimensional. Furthermore, for each $\mathrm{\left(y,{\bm{\mathfrak{C}}}\right)\in\widetilde{\bf{W}}^{i}\subset \widetilde{\bf{\,Y\,}}}$ we know that $$\mathrm{\int_{\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}\right)}\Omega_{n}\downarrow 0.}$$
In order to finish the proof of {\bf{Proposition \ref{Continuity of Fiber Integral}}}, we need the following lemma, which we will proved at the end of this section.
\begin{lemma}\label{Lemma Fiber Integration} Let $\mathrm{\Pi\!:\!\widetilde{\bf{\,X\,}}\rightarrow \widetilde{\bf{\,Y\,}}}$ be a $\mathrm{k}$-fibering between compact spaces and let $\mathrm{{\widetilde{\bf{W}}}\subset \widetilde{\bf{\,Y\,}}}$ be a closed subset.
\begin{enumerate} \item Let $\mathrm{\left(\Omega_{n}\right)_{n}}$ be a sequence of smooth, positive $\mathrm{\left(k,k\right)}$-forms on $\mathrm{\widetilde{\bf{\,X\,}}}$ and assume that $$\mathrm{\int_{\Pi^{-1}\left(y\right)}\Omega_{n}\downarrow 0}$$ for each $\mathrm{y\in{\widetilde{\bf{W}}}\subset \widetilde{\bf{\,X\,}}}$ and let $\mathrm{\Delta>0}$. Then there exists $\mathrm{n_{\Delta}\in \mathbb{N}}$ so that $$\mathrm{\mathrm{\int_{\Pi^{-1}\left(y\right)}\Omega_{n}\leq\Delta}\text{ for all }y\in {\widetilde{\bf{W}}}\text{ and all }n\geq n_{\Delta}.}$$ \item If $\mathrm{\Omega}$ is a smooth, positive $\mathrm{\left(k,k\right)}$-form on $\mathrm{\widetilde{\bf{\,X\,}}}$ so that $$\mathrm{\int_{\Pi^{-1}\left(y\right)}\Omega>0}$$ for all $\mathrm{y\in {\widetilde{\bf{W}}}}$. Then there exists $\mathrm{\delta>0}$ so that $$\mathrm{\delta\leq\int_{\Pi^{-1}\left(y\right)}\Omega\text{ for all }y\in {\widetilde{\bf{W}}}.}$$ \end{enumerate} \end{lemma}
Applying the first statement of the above lemma to the fiber integral yields the existence of $\mathrm{n_{\Delta}\in \mathbb{N}}$ so that $$\mathrm{\left(y\mapsto\int_{\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}\right)}\Omega_{n}\right)\leq\Delta\text{ for all }n\geq n_{\Delta}\text{ and all }\left(y,{\bm{\mathfrak{C}}}\right)\in {\widetilde{\bf{W}}}^{i}.}$$ By the choice $\mathrm{\psi_{n}\vert\widetilde{\,T\,}\left(\epsilon_{n},\mathbf{W}^{i}\right)\equiv1}$, we deduce \begin{equation*} \begin{split} \mathrm{\left(y,{\bm{\mathfrak{C}}}\right)\mapsto \int_{\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}\right)}\Omega_{n}}&=\mathrm{\int_{\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}\right)}\psi_{n}\Omega^{k}}\\[0,2 cm] &\geq\mathrm{vol\left(\widetilde{\,T\,}\left(\epsilon_{n},\mathbf{W}^{i}\right)\cap \Pi^{-1}\left(y,{\bm{\mathfrak{C}}}_{y}\right)\right)}\\ \end{split} \end{equation*} for all $\mathrm{\left(y,{\bm{\mathfrak{C}}}\right)\in {\widetilde{\bf{W}}}^{i}}$. Now, set $\mathrm{n=n_{\triangle}}$, resp.\ $\mathrm{\epsilon=\epsilon_{n_{\triangle}}}$ and note that for all $\mathrm{y\in \mathbf{W}^{i}\cap \mathbf{Y}_{0}}$ we have $$\mathrm{vol\left(T\left(\epsilon,\mathbf{W}^{i}\right)\cap\pi^{-1}\left(y\right)\right)=vol\left(\widetilde{\,T\,}\left(\epsilon,\mathbf{W}^{i}\right)\cap\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}_{y}\right)\right)}$$ by {\bf{Lemma \ref{Lemma Equality of Volume}}}. Hence, it follows that $$\mathrm{vol\left(T\left(\epsilon,\mathbf{W}^{i}\right)\cap\pi^{-1}\left(y\right)\right)\leq\Delta\text{ for all }y\in \mathbf{W}^{i}\cap \mathbf{Y}_{0}}$$ as claimed.
The second claim is a direct consequence of the second claim of the above lemma applied to the smooth form $\mathrm{\Omega_{n_{0}}}$, where $\mathrm{n_{0}\in \mathbb{N}}$ is chosen so that $\mathrm{\epsilon_{n_{0}+1}<\epsilon}$. In fact, we have \begin{equation}\label{Equation wisa fwwq} \begin{split} \mathrm{\left(y,{\bm{\mathfrak{C}}}\right)\mapsto \int_{\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}\right)}\Omega_{n_{0}}}&=\mathrm{\int_{\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}\right)}\psi_{n_{0}}\Omega^{k}}\\[0,2 cm] &\leq\mathrm{vol\left(\widetilde{\,T\,}\left(\epsilon,\mathbf{W}^{i}\right)\cap \Pi^{-1}\left(y,{\bm{\mathfrak{C}}}_{y}\right)\right)}\\ \end{split} \end{equation} for all $\mathrm{\left(y,{\bm{\mathfrak{C}}}\right)\in {\widetilde{\bf{W}}}^{i}}$ which follows by $\mathrm{\Pi^{-1}\big(\widetilde{\bf{W}}^{i}\big)\cap supp\,\psi_{n}\subset \widetilde{\,T\,}\left(\epsilon_{n+1},\mathbf{W}^{i}\right)}$. Note that $\mathrm{\Omega_{n_{0}}}$ has compact support and that the left hand side of \ref{Equation wisa fwwq} is non-zero for all $\mathrm{\left(y,{\bm{\mathfrak{C}}}\right)\in {\widetilde{\bf{W}}}^{i}}$ which follows by the second claim of {\bf{Lemma \ref{Lemma Intersection Zero Measure}}} applied to $\mathrm{\epsilon=\epsilon_{n_{0}}}$: $$\mathrm{0<vol\left(\widetilde{\,T\,}\left(\epsilon_{n_{0}},\mathbf{W}^{i}\right)\cap \Pi^{-1}\left(y,{\bm{\mathfrak{C}}}_{y}\right)\right)\leq\int_{\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}\right)}\Omega_{n_{0}}\text{ for all }\left(y,{\bm{\mathfrak{C}}}\right)\in {\widetilde{\bf{W}}}^{i}.}$$ Applying the equality $$\mathrm{vol\left(T\left(\epsilon,\mathbf{W}^{i}\right)\cap\pi^{-1}\left(y\right)\right)=vol\left(\widetilde{\,T\,}\left(\epsilon,\mathbf{W}^{i}\right)\cap\Pi^{-1}\left(y,{\bm{\mathfrak{C}}}_{y}\right)\right)}$$ for all $\mathrm{y\in \mathbf{W}^{i}\cap \mathbf{Y}_{0}}$ and using the second claim of {\bf{Lemma \ref{Lemma Fiber Integration}}} completes the proof. \end{proof}
It remains to prove {\bf{Lemma \ref{Lemma Fiber Integration}}}.
\begin{proof} (of {\bf{Lemma \ref{Lemma Fiber Integration}}}) Let $\mathrm{\xi\!:\!\widetilde{\bf{\,Y\,}}^{nor}\rightarrow \widetilde{\bf{\,Y\,}}}$ \label{Notation Normalization Map Xi} be the normalization of $\mathrm{\widetilde{\bf{\,Y\,}}}$ and consider the pull back space $\mathrm{\xi^{*}\widetilde{\bf{\,X\,}}}$ of $\mathrm{\widetilde{\bf{\,X\,}}}$ which is defined to be the complex space given by $$\mathrm{\xi^{*}\widetilde{\bf{\,X\,}}\coloneqq\left\{ \left(x,\widehat{y}\right)\!:\!\,\Pi\left(x\right)=\xi\left(\widehat{y}\right)\right\} \subset\widetilde{\bf{\,X\,}}\times\widetilde{\bf{\,Y\,}}^{nor}.}$$ Note that we have a projection map $\mathrm{\xi^{*}\Pi\!:\!\xi^{*}\widetilde{\bf{\,X\,}}\rightarrow \widetilde{\bf{\,Y\,}}^{nor}}$ which is given by the map $\mathrm{\Pi\times }$ $\mathrm{Id_{\widetilde{\bf{\,Y\,}}^{nor}}}$ and whose fibers are purely $\mathrm{k}$-dimensional since they are exactly given by the fibers of the projection associated to the universal space $\mathrm{\widetilde{\bf{\,X\,}}}$. Moreover, note that each smooth $\mathrm{\left(k,k\right)}$-form $\mathrm{\Omega}$ defined on $\mathrm{\widetilde{\bf{\,X\,}}}$ induces a smooth $\mathrm{\left(k,k\right)}$-form on $\mathrm{\xi^{*}\widetilde{\bf{\,X\,}}}$ via its pull back with respect to the projection map $\mathrm{p_{\widetilde{\bf{\,X\,}}}\vert \xi^{*}\widetilde{\bf{\,X\,}}\!:\!\xi^{*}\widetilde{\bf{\,X\,}}\rightarrow \widetilde{\bf{\,X\,}}}$. In the sequel, we will label this form by $\mathrm{\Omega^{nor}}$. Since $\mathrm{\xi^{*}\widetilde{\bf{\,X\,}}}$ is compact as a closed subset of the compact space $\mathrm{\widetilde{\bf{\,X\,}}\times\widetilde{\bf{\,Y\,}}^{nor}}$, it follows that $\mathrm{\Omega^{nor}}$ has compact support. Furthermore, the inverse image $\mathrm{\widetilde{\bf{W}}^{nor}\coloneqq\xi^{-1}\big(\widetilde{\bf{W}}\big)}$ of the compact subset $\mathrm{\bf{W}}$ is likewise compact for the same reason.
We will now prove the first claim of the lemma. For this, note that \begin{equation}\label{Equation for This Proof} \mathrm{\int_{\Pi^{-1}\left(y\right)}\Omega_{n}=\int_{\Pi^{nor,-1}\left(\widehat{y}\right)}\Omega_{n}^{nor} \text{ for all }\widehat{y}\text{ with }\xi\left(\widehat{y}\right)=y.} \end{equation}
Since $\mathrm{\Omega^{nor}_{n}}$ defines a sequence of smooth $\mathrm{\left(k,k\right)}$-forms with compact support and since the map fulfills all requirements of theorem in \cite{Kin}, {pp.\ }{\bf{\oldstylenums{185}-\oldstylenums{220}}}, we deduce that the fiber integral $$\mathrm{\widehat{y}\mapsto\int_{\Pi^{nor,-1}\left(\widehat{y}\right)}\Omega_{n}^{nor}}$$ defines a continuous function over the base $\mathrm{\widetilde{\bf{\,Y\,}}^{nor}}$. So the sequence given by $$\mathrm{\left(\widehat{y}\mapsto\int_{\Pi^{nor,-1}\left(\widehat{y}\right)}\Omega_{n}^{nor}\right)_{n}}$$ defines a sequence of continuous functions over $\mathrm{\widetilde{\bf{\,Y\,}}^{nor}}$ which will converge for each point by the assumption of the lemma to the zero function. By {\scshape{Dini}}'s convergence theorem of strictly decreasing sequence of continuous functions, it follows that this sequence of functions converges uniformly over the compact subset $\mathrm{\widetilde{\bf{W}}^{nor}}$ to the zero function. Hence, for each $\mathrm{\Delta>0}$ we can find $\mathrm{n_{\Delta}\in \mathbb{N}}$ so that $$\mathrm{\int_{\Pi^{nor,-1}\left(\widehat{y}\right)}\Omega_{n}^{nor}\leq\Delta\text{ for all }\widehat{y}\in \widetilde{\bf{W}}^{nor}\text{ and all }n\geq n_{\Delta}.}$$ By equation \ref{Equation for This Proof} and the fact that $\mathrm{\widetilde{\bf{W}}^{nor}=\xi^{-1}\big(\widetilde{\bf{W}}\big)}$ the first claim is shown.
The second claim is a direct consequence of the fact that $\mathrm{\Omega^{nor}}$ has compact support and by the assumption of the lemma which is given by $$\mathrm{0<\int_{\Pi^{-1}\left(y\right)}\Omega\text{ for all }y\in \widetilde{\bf{W}}.}$$ Hence, we deduce that $$\mathrm{0<\int_{\Pi^{nor,-1}\left(\widehat{y}\right)}\Omega^{nor}\text{ for all }\widehat{y}\text{ so that }\xi\left(\widehat{y}\right)=y\in \widetilde{\bf{W}}}$$ by \ref{Equation for This Proof}. Note that $\mathrm{\widehat{y}\mapsto\int_{\Pi^{nor,-1}\left(\widehat{y}\right)}\Omega^{nor}}$ is a continuous function which does not vanish over $\mathrm{\widetilde{\bf{\,W\,}}^{nor}}$ by the above inequality. Hence, there exists $\mathrm{\delta>0}$ so that $\mathrm{\delta\leq }$ $\mathrm{\int_{\Pi^{nor,-1}\left(\widehat{y}\right)}\Omega^{nor}}$ for all $\mathrm{\widehat{y}}$ in the compact subset $\mathrm{\widetilde{\bf{W}}^{nor}}$. This proves the claim by applying \ref{Equation for This Proof} again. \end{proof}
For the sake of completeness, we will close this section by stating the theorem of {\scshape{J. King}}, which we have used in the proof of the preceding proposition.
\begin{theo} \textnormal{(cf.\ \cite{Kin}, {pp.\ }{\bf{\oldstylenums{185}-\oldstylenums{220}}})}\label{Theorem King}
\\ Let $\mathrm{F\!:\!\mathbf{X}\rightarrow \mathbf{Y}}$ be a $\mathrm{k}$-fibering between complex purely dimensional spaces $\mathrm{\mathbf{X},\mathbf{Y}}$ where $\mathrm{m=}$ $\mathrm{dim_{\mathbb{C}}\,\mathbf{X}}$, $\mathrm{n=dim_{\mathbb{C}}\,\mathbf{Y}}$ and assume that $\mathrm{\mathbf{Y}}$ is normal. If $\mathrm{\Omega}$ is a continuous, complex valued $\mathrm{\left(k,k\right)}$-form on $\mathrm{\mathbf{X}}$ with compact support, then the fiber integral $$\mathrm{y\mapsto \int_{F^{-1}\left(y\right)}\Omega\,d\,[F_{y}]\coloneqq \int_{F^{-1}\left(y\right)}\nu_{F}\,\Omega\vert F^{-1}\left(y\right)}$$ \label{Notation Fiber Integration with Respect to a k-Fibering} defines a continuous function on $\mathrm{\mathbf{Y}}$ where $\mathrm{\nu_{F}}$ denotes the order\label{Notation Order Of F At Point}\,\footnote{For a rigorous definition of the order of a $\mathrm{k}$-fibering in the point $\mathrm{x\in \mathbf{X}}$ {cf.\! }\cite{Kin}.} of the $\mathrm{k}$-fibering $\mathrm{F\!:\!\mathbf{X}\rightarrow \mathbf{Y}}$. \end{theo}
\section{Uniform Convergence in the Tame Case}\label{Section Uniform Convergence Theorems In The Tame Case} \subsection{Uniform Convergence of the Fiber Probability Measures}
The aim of this section is to prove {\bf{Theorem 3}}. For this recall, that for each $\mathrm{x\in \mu^{-1}\left(\xi\right)}$ there exists a $\mathrm{\mathbb{T}}$-invariant Zariski open subset $\mathrm{\mathbf{X}^{i}\subset \mathbf{X}^{ss}_{\xi}}$ which is contained in the $\mathrm{n}$-stable complement of the zero set of the tame sequence $\mathrm{\left(s^{i}_{n}\right)_{n}}$ given by $\mathrm{\mathbf{X}\left(s_{n}^{i}\right)\subset\mathbf{X}^{ss}_{\xi}}$ ({cf.\! }{\bf{Theorem 1}}). Moreover, let $\mathrm{\varrho^{i}\!:\!\mathbf{X}^{i}\rightarrow \mathbb{R}}$ be the normalized s.p.s.h.\ limit function as defined at the beginning of {\bf{Section \ref{Uniform Localization Proposition}}} and recall that there exists a compact neighborhood $\mathrm{\mathbf{W}^{i}\subset \pi\left(\mathbf{X}^{i}\right)}$ so that $\mathrm{T\left(\epsilon,\mathbf{W}^{i}\right)\subset \mathbf{X}^{i}}$ given by $$\mathrm{T\left(\epsilon,\mathbf{W}^{i}\right)=\left(\varrho^{i}\times \pi\right)^{-1}\left([0,\epsilon]\times \mathbf{W}^{i}\right)\text{ (cf.\ {\bf{Section \ref{Uniform Localization Proposition}}})}}$$ defining a compact $\mathrm{T}$-invariant tube for all $\mathrm{\epsilon\geq 0}$.
We begin the proof with the following lemma.
\begin{lemma}\label{Lemma Reduced Function} Let $\mathrm{f\in \mathcal{C}^{0}\!\left(\mathbf{X}\right)}$ be a $\mathrm{T}$-invariant, continuous function. Given $\mathrm{\sigma>0}$, there exists $\mathrm{\epsilon_{\sigma}>0}$ so that $$\mathrm{\left|f\vert\left(\pi^{-1}\left(y\right)\cap T\left(\epsilon_{\sigma},\mathbf{W}^{i}\right)\right)-f_{red}\left(y\right)\right|\leq\sigma}$$ for all $\mathrm{y\in \mathbf{W}^{i}\subset \mathbf{Y}^{i}}$. \end{lemma} \begin{proof} Let us assume that $\mathrm{\epsilon_{\sigma}>0}$ with the above property does not exists. Then we can find a sequence $\mathrm{\left(\epsilon_{\sigma,n}\right)_{n}}$ of positive numbers with $\mathrm{\epsilon_{\sigma,n}\rightarrow 0}$, a sequence $\mathrm{\left(y_{n}\right)_{n}}$ in $\mathrm{\mathbf{W}^{i}}$ and a lifted sequence $\mathrm{\left(x_{n}\right)_{n}}$ in $\mathrm{\pi^{-1}\left(\mathbf{W}^{i}\right)}$ (i.e.\ $\mathrm{\pi\left(x_{n}\right)=y_{n}}$) where $$\mathrm{x_{n}\in T\left(\epsilon_{\sigma,n},\mathbf{W}^{i}\right)}$$ and so that $$\mathrm{\left\vert f\left(x_{n}\right)-f_{red}\left(y_{n}\right) \right\vert\geq \epsilon>0}$$ for all $\mathrm{n}$.
By the compactness of $\mathrm{\mathbf{W}^{i}}$, we can assume that $\mathrm{y_{n}\rightarrow y\in \mathbf{W}^{i}}$. Furthermore, by the compactness of $\mathrm{T\left(\epsilon_{\sigma,n},\mathbf{W}^{i}\right)}$, we have can assume that the lifted sequence $\mathrm{\left(x_{n}\right)_{n}}$, which is contained in the sequence of the compact nested subsets given by $\mathrm{T\left(\epsilon_{\sigma,n},\mathbf{W}^{i}\right)}$, is convergent as well, i.e.\ $\mathrm{x_{n}\rightarrow x}$. Since $$\mathrm{T\left(\epsilon_{\sigma,n},\mathbf{W}^{i}\right)\downarrow \mu^{-1}\left(\xi\right)\cap \pi^{-1}\left(\mathbf{W}^{i}\right)}$$ we have $\mathrm{x_{n}\rightarrow x\in \mu^{-1}\left(\xi\right)\cap \pi^{-1}\left(\mathbf{W}^{i}\right)}$.
So, all in all, we have found a convergent sequence $\mathrm{\left(x_{n}\right)_{n}}$ in $\mathrm{\pi^{-1}\left(\mathbf{W}^{i}\right)}$ so that $$\mathrm{x_{n}\rightarrow x\in \mu^{-1}\left(\xi\right)\cap \pi^{-1}\left(\mathbf{W}^{i}\right) }$$ and $$\mathrm{\vert f\left(x_{n}\right)-f_{red}\left(\pi\left(x_{n}\right)\right)\vert=\left\vert f\left(x_{n}\right)-f_{red}\left(y_{n}\right) \right\vert\geq \epsilon>0}$$ for all $\mathrm{n}$. From the continuity of $\mathrm{\pi, f}$ and the fact that $\mathrm{f_{red}\left(\pi\left(x\right)\right)=f_{red}\left(y\right)}$ $\mathrm{=f\left(x\right)}$ we deduce a contradiction. \end{proof}
After this preparation, we can prove the uniform convergence of the measure sequence with respect to the weak topology. For this recall that $\mathrm{\overline{f}}$ for $\mathrm{f\in \mathcal{C}^{0}\!\left(\mathbf{X}\right)}$ is defined to be the averaged function given by $\mathrm{\overline{f}\left(x\right)=\int_{T}f\left(t.x\right)\,d\nu_{T}}$ where denotes the {\scshape{Haar}} measure. Moreover, recall the definition of the sequence $\mathrm{\left({\bm{\nu}_{n}^{i}}\right)_{n}}$ of fiber probability measures attached to the tame sequence $\mathrm{\left(s_{n}^{i}\right)_{n}}$ (cf.\,?{\bf{Definition \ref{Definition Fiber Probability Measure Tame Case}}}).
\begin{prop}\label{Proposition Uniform Convergence} Let $\mathrm{f\in \mathcal{C}^{0}\!\left(\mathbf{X}\right)}$ and $\mathrm{\mathbf{W}^{i}\subset}$ $\mathrm{\mathbf{Y}^{i}=}$ $\mathrm{\pi\left(\mathbf{X}^{i}\right)}$ as before. Then the sequence of functions on $\mathrm{\mathbf{W}^{i}\cap \mathbf{Y}_{0}}$ given by $$\mathrm{y\mapsto \int_{\pi^{-1}\left(y\right)}f\,d\bm{\nu}_{n}^{i}\left(y\right)}$$ converges uniformly on $\mathrm{\mathbf{W}^{i}\cap \mathbf{Y}_{0}}$ to the reduced function $\mathrm{f_{red}\vert \mathbf{W}^{i}\cap \mathbf{Y}_{0}\!:\!\mathbf{W}^{i}\cap \mathbf{Y}_{0} \rightarrow \mathbb{R}.}$ \end{prop} \begin{proof} First, let us show that it is enough to prove the claim for continuous $\mathrm{T}$-invariant functions: Let us assume that the claim is valid for all continuous functions which are $\mathrm{T}$-invariant and assume that $\mathrm{f\in \mathcal{C}^{0}\!\left(\mathbf{X}\right)}$ is arbitrary. Using the $\mathrm{T}$-invariance of $\mathrm{\bm{\nu}_{n}}$ and $\mathrm{\nu_{T}}$ and the fact $\mathrm{\int_{T}d\nu_{T}=1}$ it follows that: $$\mathrm{\int_{t\in T}\left[\int_{\pi^{-1}\left(y\right)}t^{*}f\,d\bm{\nu}_{n}\left(y\right)\right]\,d\nu_{T}=\int_{\pi^{-1}\left(y\right)}f\,d\bm{\nu}_{n}\left(y\right).}$$ Interchanging the order of integration yields \begin{equation}\label{Equation Interchanging Integration} \mathrm{\int_{\pi^{-1}\left(y\right)}f\,d\bm{\nu}_{n}\left(y\right)=\int_{\pi^{-1}\left(y\right)}\left[\int_{t\in T}t^{*}f\,d\nu_{T}\right]\,d\bm{\nu}_{n}\left(y\right)=\int_{\pi^{-1}\left(y\right)}\overline{f}\,d\bm{\nu}_{n}\left(y\right).} \end{equation} Since $\mathrm{\overline{f}}$ is $\mathrm{T}$-invariant and continuous and since we have assumed that the claim is true for those functions, we deduce by \ref{Equation Interchanging Integration} that $$\mathrm{\left(\int_{\pi^{-1}\left(y\right)}f\,d\bm{\nu}_{n}\left(y\right)\right)=\left(y\mapsto \int_{\pi^{-1}\left(y\right)}\overline{f}\,d\bm{\nu}_{n}\left(y\right)\right)}$$ converges uniformly on $\mathrm{\mathbf{W}^{i}\cap \mathbf{Y}_{0}}$ to the reduced function $\mathrm{f_{red}=\vert \mathbf{W}^{i}\cap \mathbf{Y}_{0}\!:\!\mathbf{W}^{i}\cap \mathbf{Y}_{0} \rightarrow \mathbb{R}}$. Hence, it remains to verify the claim for continuous, $\mathrm{T}$-invariant functions.
If $\mathrm{\sigma>0}$ and $\mathrm{f\in\mathcal{C}^{0}\!\left(\mathbf{X}\right)}$, then by {\bf{Lemma \ref{Lemma Reduced Function}}}, there exists $\mathrm{\epsilon_{\sigma}>0}$ so that $$\mathrm{\left|f\vert\left(\pi^{-1}\left(y\right)\cap T\left(\epsilon_{\sigma},\mathbf{W}^{i}\right)\right)-f_{red}\left(y\right)\right|\leq\frac{\sigma}{2}}$$ for all $\mathrm{y\in \mathbf{W}^{i}\subset \mathbf{Y}^{i}}$. By {\bf{Theorem 2}} we can find $\mathrm{N_{0}\in \mathbb{N}}$ so that for all $\mathrm{n\geq N_{0}}$ we have $$\mathrm{\varrho^{i}_{n}\left(x\right)\geq \frac{\epsilon_{\sigma}}{2}}$$ for all $\mathrm{x\in T^{c}\left(\epsilon_{\sigma},\mathbf{W}^{i}\right)}$. Therefore, we deduce that \begin{equation*} \begin{split}
\mathrm{\left\vert \int_{\pi^{-1}\left(y\right)}f\,d\bm{\nu}^{i}_{n}\left(y\right)-f_{red}\left(y\right)\right\vert} &\leq \mathrm{\frac{\sigma}{2}\frac{\int_{\pi^{-1}\left(y\right)\cap T\left(\epsilon_{\sigma},\mathbf{W}^{i}\right)}\left|s_{n}^{i}\right|^{2}\,d\,[\pi_{y}]}{\left\Vert s_{n}^{i}\right\Vert ^{2}\left(y\right)}}\\[0,2 cm] &+ \mathrm{e^{-n\,\frac{\epsilon_{\sigma}}{2}}\,C\left(f\right)\,\frac{\int_{\pi^{-1}\left(y\right)\cap T^{c}\left(\epsilon_{\sigma},\mathbf{W}^{i}\right)}\,d\,[\pi_{y}]}{\left\Vert s_{n}^{i}\right\Vert ^{2}\left(y\right)}}\end{split}
\end{equation*} for all $\mathrm{y\in \mathbf{W}^{i}\cap \mathbf{Y}_{0}}$ where $\mathrm{C\left(f\right)\coloneqq \underset{x\in \mathbf{X}}{max}\left|f-\pi^{*}f_{red}\right|<\infty}$.
By {\bf{Proposition \ref{Proposition Uniform Convergence Potential Functions in the Abelian Case}}} the sequence $\mathrm{\varrho^{i}_{n}}$ converges uniformly on compact subsets of $\mathrm{\pi^{-1}\left(\mathbf{Y}^{i}\right)=\mathbf{X}^{i}}$ to $\mathrm{\varrho^{i}}$ and hence, for each $\mathrm{\delta>0}$, there exists $\mathrm{N^{\prime}_{0}\in \mathbb{N}}$ so that $$\mathrm{\varrho^{i}_{n}\leq \epsilon^{\prime}+\delta}$$ for all $\mathrm{x\in T\left(\epsilon^{\prime},\mathbf{W}^{i}\right)}$. Here we will choose $\mathrm{\delta>0}$ and $\mathrm{\epsilon^{\prime}>0}$ so that $\mathrm{\epsilon^{\prime}+\delta<\frac{\epsilon_{\sigma}}{2}}$. It then follows for all $\mathrm{y\in \mathbf{W}^{i}\cap \mathbf{Y}_{0}}$ \begin{equation*} \begin{split}
\mathrm{\left\Vert s_{n}^{i}\right\Vert^{2}\left(y\right)} &\geq \mathrm{\int_{\pi^{-1}\left(y\right)\cap T\left(\epsilon^{\prime},\mathbf{W}^{i}\right)}\left|s_{n}^{i}\right|^{2}\,d\,[\pi_{y}]}\\[0,2 cm] &\geq \mathrm{e^{-n\,\left(\epsilon^{\prime}+\delta\right)}\,\int_{\pi^{-1}\left(y\right)\cap T\left(\epsilon^{\prime},\mathbf{W}^{i}\right)}\,d\,[\pi_{y}].}\end{split} \end{equation*} So we deduce \begin{equation*} \begin{split}
\mathrm{\left\vert \int_{\pi^{-1}\left(y\right)}f\,d\bm{\nu}^{i}_{n}\left(y\right)-f_{red}\left(y\right)\right\vert} &\leq \mathrm{\frac{\sigma}{2}\frac{\int_{\pi^{-1}\left(y\right)\cap T\left(\epsilon_{\sigma},\mathbf{W}^{i}\right)}\left|s_{n}^{i}\right|^{2}\,d\,[\pi_{y}]}{\left\Vert s_{n}^{i}\right\Vert ^{2}\left(y\right)}}\\[0,2 cm] &+ \mathrm{e^{-n\,\left(\frac{\epsilon_{\sigma}}{2}-\epsilon^{\prime}-\delta\right)}\,C\left(f\right)\,\frac{\int_{\pi^{-1}\left(y\right)\cap T^{c}\left(\epsilon_{\sigma},\mathbf{W}^{i}\right)}\,d\,[\pi_{y}]}{\int_{\pi^{-1}\left(y\right)\cap T\left(\epsilon^{\prime},\mathbf{W}^{i}\right)}\,d\,[\pi_{y}]}}\end{split} \end{equation*} for all $\mathrm{y\in \mathbf{W}^{i}\cap \mathbf{Y}_{0}}$. Using {\bf{Corollary \ref{Boundedness of Fiber Integral}}}, we know that the dominator of the last term is bounded for all $\mathrm{y\in \mathbf{Y}_{0}}$ and by the second claim of {\bf{Proposition \ref{Continuity of Fiber Integral}}}, we know that the denominator is bounded away form zero as $\mathrm{y}$ varies in $\mathrm{\mathbf{W}^{i}\cap \mathbf{Y}_{0}}$. Therefore, we find a constant $\mathrm{\Gamma<\infty}$ so that \begin{equation*} \begin{split}
\mathrm{\left\vert \int_{\pi^{-1}\left(y\right)}f\,d\bm{\nu}^{i}_{n}\left(y\right)-f_{red}\left(y\right)\right\vert} &\leq \mathrm{\frac{\sigma}{2}\frac{\int_{\pi^{-1}\left(y\right)\cap T\left(\epsilon_{\sigma},\mathbf{W}^{i}\right)}\left|s_{n}^{i}\right|^{2}\,d\,[\pi_{y}]}{\left\Vert s_{n}^{i}\right\Vert ^{2}\left(y\right)}}\\[0,2 cm] &+ \mathrm{e^{-n\,\left(\frac{\epsilon_{\sigma}}{2}-\epsilon^{\prime}-\delta\right)}\,C\left(f\right)\,\Gamma.}\end{split}
\end{equation*} Since $$\mathrm{\frac{\int_{\pi^{-1}\left(y\right)\cap T\left(\epsilon_{\sigma},\mathbf{W}^{i}\right)}\left|s_{n}^{i}\right|^{2}\,d\,[\pi_{y}]}{\left\Vert s_{n}^{i}\right\Vert ^{2}\left(y\right)}\leq 1}$$ for all $\mathrm{n\in \mathbb{N}}$ and all $\mathrm{y\in \mathbf{W}^{i}\cap \mathbf{Y}_{0}}$, we find $$\mathrm{\left\vert \int_{\pi^{-1}\left(y\right)}f\,d\bm{\nu}^{i}_{n}\left(y\right)-f_{red}\left(y\right)\right\vert \leq \frac{\sigma}{2}+e^{-n\,\left(\frac{\epsilon_{\sigma}}{2}-\epsilon^{\prime}-\delta\right)}\,C\left(f\right)\,\Gamma.}$$ Since $\mathrm{\frac{\epsilon_{\sigma}}{2}-\epsilon^{\prime}-\delta>0}$ there exists $\mathrm{N_{0}^{\prime\prime}\geq N^{\prime}_{0}}$ so that $$\mathrm{\left\vert \int_{\pi^{-1}\left(y\right)}f\,d\bm{\nu}^{i}_{n}\left(y\right)-f_{red}\left(y\right)\right\vert \leq \sigma}$$ for all $\mathrm{n\geq N^{\prime\prime}_{0}}$ as claimed. \end{proof}
We sum up the results in
\begin{theo_3*}\label{Theorem Uniform Convergence of the Tame Measure Sequence} \textnormal{[\scshape{Uniform Convergence of the Tame Measure Sequence}]}
\\ For for every tame collection $\mathrm{\left\{s_{n}^{i}\right\}_{i}}$ there exists a finite cover $\mathrm{\mathfrak{U}}$ of $\mathrm{\mathbf{Y}}$ with $\mathrm{\mathbf{U}_{i}\subset}$ $\mathrm{\pi\left(\mathbf{X}^{i}\right)}$ so that the collection of fiber probability measures $\mathrm{\left\{\bm{\nu}^{\mathfrak{U}}_{n}\right\}}$ associated to $\mathrm{\left\{s^{i}_{n}\right\}_{i}}$ converges uniformly on $\mathrm{\mathbf{Y}_{0}}$ to the fiber Dirac measure of $\mathrm{\mu^{-1}\left(\xi\right)\cap \mathbf{X}_{0}}$, i.e.\ for every $\mathrm{i\in I}$ and every $\mathrm{f\in \mathcal{C}^{0}\left(\mathbf{X}\right)}$, we have $$\begin{xy}
\xymatrix{\mathrm{\left(y\mapsto \int_{\pi^{-1}\left(y\right)}f\, d\bm{\nu}_{n}^{i}\left(y\right)\right)}\ar[rr]^{\mathrm{\,\,\,\,\,}}&&\mathrm{f_{red}\text{ uniformly on }\mathbf{U}_{i}\cap \mathbf{Y}_{0}.\label{Notation Reduced Function}}
} \end{xy}$$ \end{theo_3*} \begin{proof} Let $\mathrm{\left\{s^{i}_{n}\right\}_{i}}$ be a tame collection and $\mathrm{\mathfrak{U}}$ be a finite open cover of $\mathrm{\mathbf{Y}}$ so that $$\mathrm{\mathbf{U}^{i} \subset \mathbf{W}^{i}\subset \mathbf{Y}^{i}=\pi\left(\mathbf{X}^{i}\right)}$$ where $\mathrm{\mathbf{W}^{i}}$ is a compact neighborhood as defined at the beginning of {\bf{Section \ref{Uniform Localization Proposition}}}.
By {\bf{Proposition \ref{Proposition Uniform Convergence}}} there exists $\mathrm{N^{i}_{0}\in \mathbb{N}}$ so that $$\mathrm{\left|\int_{\pi^{-1}\left(y\right)}f\, d\bm{\nu}_{n}^{i}\left(y\right)-f_{red}\left(y\right)\right|\leq\epsilon}$$ for all $\mathrm{y\in \mathbf{Y}_{0}\cap \mathbf{W}^{i}}$ and all $\mathrm{n\geq N^{i}_{0}}$ which proves the claim. \end{proof}
\subsection{Uniform Convergence of the Fiber Distribution Densities}
In this section we will give a proof of {\bf{Theorem 4}}. For this, let $\mathrm{\left(D_{n}^{i}\left(\cdot,t\right)\right)_{n}}$ be the sequence of cumulative fiber distribution functions on $\mathrm{\mathbf{W}^{i}\cap \mathbf{Y}_{0}}$ as defined in {\bf{Definition \ref{Definition Sequence Of Collections Of Fiber Probability Densities}}} associated to a tame sequence $\mathrm{\left(s_{n}^{i}\right)_{n}}$ and let $\mathrm{\mathbf{W}^{i}\subset \pi\left(\mathbf{X}^{i}\right)}$ be a compact neighborhood so that the $\mathrm{T}$-invariant tube $\mathrm{T\left(\epsilon,\mathbf{W}^{i}\right)\subset \mathbf{X}^{i}}$ as defined in {\bf{Section \ref{Uniform Localization Proposition}}} is compact.
\begin{prop}\label{Proposition Uniform Convergence of The Distribution Sequence} The sequence $\mathrm{\left(D_{n}^{i}\left(\cdot,t\right)\right)_{n}}$ of distribution functions on $\mathrm{\mathbf{W}^{i}\cap \mathbf{Y}_{0}}$ converges uniformly on $\mathrm{\mathbf{W}^{i}\cap \mathbf{Y}_{0}}$ to the zero function for all $\mathrm{t\geq 0}$. \end{prop} \begin{proof} First of all fix $\mathrm{\sigma>0}$. Then by the first part of {\bf{Proposition \ref {Continuity of Fiber Integral}}} there exists $\mathrm{\epsilon_{\sigma}>0}$ so that $$\mathrm{vol\left(\pi^{-1}\left(y\right)\cap T\left(\epsilon_{\sigma},\mathbf{W}^{i}\right)\right)\leq \sigma}$$ for all $\mathrm{y\in \mathbf{W}^{i}\cap \mathbf{Y}_{0}}$. Hence, we are finished as soon as we have proved that there exists $\mathrm{N_{0}\in \mathbb{N}}$ so that $$\mathrm{\left\{x\in \pi^{-1}\left(\mathbf{W}^{i}\cap \mathbf{Y}_{0}\right)\!:\!\,\frac{\vert s_{n}\vert^{2}}{\Vert s_{n}\Vert^{2}}\left(x\right)\geq t\right\}\subset T\left(\epsilon_{\sigma},\mathbf{W}^{i}\right)}$$ for all $\mathrm{n\geq N_{0}}$.
By {\bf{Theorem 2}} we find $\mathrm{N_{0}\in \mathbb{N}}$, so that $\mathrm{\varrho_{n}^{i}\left(x\right)\geq \frac{\epsilon_{\sigma}}{2}}$ for all $\mathrm{x\in T^{c}(\epsilon_{\sigma},\mathbf{W}^{i})}$ and all $\mathrm{n\geq N_{0}}$. Therefore, we deduce \begin{equation}\label{Equation Proof}
\mathrm{\frac{\left|s_{n}\right|^{2}}{\left\Vert s_{n}\right\Vert ^{2}}\leq\frac{e^{-n\,\frac{\epsilon_{\sigma}}{2}}}{\left\Vert s_{n}\right\Vert ^{2}}} \end{equation}
for all $\mathrm{n\geq N_{0}}$ and all all $\mathrm{x\in T^{c}\left(\epsilon_{\sigma},\mathbf{W}^{i}\right)\cap \pi^{-1}\left(\mathbf{Y}_{0}\right)}$. Moreover, since $\mathrm{\varrho^{i}_{n}}$ converges uniformly to $\mathrm{\varrho^{i}}$ on the compact subsets like $\mathrm{T\left(\frac{\epsilon_{\sigma}}{5},\mathbf{W}^{i}\right)}$ ({cf.\! }{\bf{Proposition \ref{Proposition Uniform Convergence Potential Functions in the Abelian Case}}}), there exists $\mathrm{N^{\prime}_{0}\geq N_{0}}$ so that $$\mathrm{\varrho^{i}_{n}\left(z\right)\leq \varrho^{i}\left(z\right)+\frac{\epsilon_{\sigma}}{5}}$$ for all $\mathrm{z\in T\left(\frac{\epsilon_{\sigma}}{5},\mathbf{W}^{i}\right)}$ and all $\mathrm{n\geq N_{0}^{\prime}}$, i.e.\ $$\mathrm{\varrho^{i}_{n}\left(z\right)\leq \frac{2}{5} \,\epsilon_{\sigma}\text{ on }T\left(\frac{\epsilon_{\sigma}}{5},\mathbf{W}^{i}\right).}$$ So we deduce $$\mathrm{\Vert s_{n}\Vert^{2}\left(y\right)\geq e^{-n\,\frac{2}{5}\,\epsilon_{\sigma}} \int_{\pi^{-1}\left(y\right)\cap T\left(\frac{\epsilon_{\sigma}}{5},\mathbf{W}^{i}\right)}\,d\,[\pi_{y}]}$$ for all $\mathrm{y\in \mathbf{W}^{i}\cap \mathbf{Y}_{0}}$. If we substitute this into equation \ref{Equation Proof}, it follows that $$\mathrm{\mathrm{\frac{\left|s_{n}\right|^{2}}{\left\Vert s_{n}\right\Vert ^{2}}\left(x\right)\leq\frac{e^{-n\,\frac{1}{10}\,\epsilon_{\sigma}}}{\int_{\pi^{-1}\left(\pi\left(x\right)\right)\cap T\left(\frac{\epsilon_{\sigma}}{5},\mathbf{W}^{i}\right)}\,d\,[\pi_{y}]}}}$$ for all $\mathrm{x\in T^{c}\left(\epsilon_{\sigma},\mathbf{W}^{i}\right)\cap \pi^{-1}\left(\mathbf{Y}_{0}\right)}$ and all $\mathrm{n\geq N_{0}^{\prime}}$. By the second part of {\bf{Proposition \ref{Continuity of Fiber Integral}}}, we know that the denominator is bounded away form zero as $\mathrm{x}$ varies in $\mathrm{\pi^{-1}\left(\mathbf{W}^{i}\cap \mathbf{Y}_{0}\right)}$ and therefore we find $\mathrm{N_{0}\in \mathbb{N}}$, $\mathrm{N_{0}\geq N^{\prime}_{0}}$ so that $$\mathrm{\mathrm{\frac{\left|s_{n}\right|^{2}}{\left\Vert s_{n}\right\Vert ^{2}}\left(x\right)\leq t}}$$ for all $\mathrm{x\in T^{c}\left(\epsilon_{\sigma},\mathbf{W}^{i}\right)\cap \pi^{-1}\left(\mathbf{Y}_{0}\right)}$ and all $\mathrm{n\geq N_{0}}$. So it follows that $$\mathrm{\left\{x\in \pi^{-1}\left(\mathbf{W}^{i}\cap \mathbf{Y}_{0}\right)\!:\!\,\frac{\vert s_{n}\vert^{2}}{\Vert s_{n}\Vert^{2}}\left(x\right)\geq t\right\}\subset T\left(\epsilon_{\sigma},\mathbf{W}^{i}\right)}$$ for all $\mathrm{n\geq N_{0}}$ and therefore $$\mathrm{D_{n}\left(y,t\right)\leq vol\left\{\pi^{-1}\left(y\right)\cap T\left(\epsilon_{\sigma},\mathbf{W}^{i}\right)\right\} \leq \sigma .}$$ for all $\mathrm{y\in \mathbf{W}^{i}\cap \mathbf{Y}_{0}}$ and all $\mathrm{n\geq N_{0}}$. \end{proof}
To sum up, we have proved
\begin{theo_4*}\label{Theorem Uniform Convergence of the Tame Distribution Sequence} \textnormal{[\scshape{Uniform Convergence of the Tame Distribution Sequence}]}
\\ For every $\mathrm{t\in \mathbb{R}}$ and every tame collection $\mathrm{\left\{s_{n}^{i}\right\}_{i}}$ there exists a finite cover $\mathrm{\mathfrak{U}}$ of $\mathrm{\mathbf{Y}}$ with $\mathrm{\mathbf{U}_{i}\subset}$ $\mathrm{\pi\left(\mathbf{X}^{i}\right)}$ so that the collection of cumulative fiber probability densities $\mathrm{\left\{D^{\mathfrak{U}}_{n}\left(\cdot,t\right)\right\}}$ associated to $\mathrm{\left\{s^{i}_{n}\right\}_{i}}$ converges uniformly on $\mathrm{\mathbf{Y}_{0}}$ to the zero function on $\mathrm{\mathbf{Y}_{0}}$, i.e.\ for every $\mathrm{i\in I}$ we have $$\begin{xy}
\xymatrix{\mathrm{\left(y\mapsto D_{n}^{i}\left(y,t\right)\right)}\ar[rr]^{\mathrm{\,\,\,\,\,}}&&\mathrm{0\text{ uniformly on }\mathbf{U}_{i}\cap \mathbf{Y}_{0}\subset \pi\left(\mathbf{X}^{i}\right).}
} \end{xy}$$ \end{theo_4*} \begin{proof} As in the proof of {\bf{Theorem 3}}, using the compactness of $\mathrm{\mathbf{Y}}$, we can find by means of {\bf{Theorem 1}} a tame collection $\mathrm{\left\{s^{i}_{n}\right\}_{i}}$ and an finite open cover $\mathrm{\mathfrak{U}}$ of $\mathrm{\mathbf{Y}}$ so that $$\mathrm{\mathbf{U}^{i} \subset \mathbf{W}^{i}\subset \mathbf{Y}^{i}=\pi\left(\mathbf{X}^{i}\right)}$$ where $\mathrm{\mathbf{W}^{i}}$ is a compact neighborhood as defined at the beginning of {\bf{Section \ref{Uniform Localization Proposition}}}. The claim then follows by applying {\bf{Proposition \ref{Proposition Uniform Convergence}}} to the sequence $\mathrm{\left(D^{i}_{n}\left(\cdot,t\right)\right)_{n}}$. \end{proof}
\section{Uniform Convergence in the Non-Tame Case}\label{Section Uniform Convergence Theorems In The Non-Tame Case}
Throughout this section we make the following assumption. \begin{agree}\label{General Agreement} There exists $\mathrm{N_{0}\in \mathbb{N}}$ so that $\mathrm{\mathbf{R}_{N_{0}}\subset \mathbf{Y}}$ is non-empty. \end{agree}
Recall that $\mathrm{\mathbf{R}_{N_{0}}\cap \mathbf{Y}_{0}}$ is open. Furthermore, if $\mathrm{y\in\mathbf{R}_{N_{0}}\cap \mathbf{Y}_{0}}$, there exists an open neighborhood $\mathrm{{\mathbf{U}}_{y}\subset\mathbf{R}_{N_{0}}\cap \mathbf{Y}_{0}}$ and a sequence of local, holomorphic $\mathrm{\xi_{n}}$-eigensections $\mathrm{\widehat{s}_{f_{y},n}}$ defined on the open $\mathrm{\pi}$-saturated subset $\mathrm{\pi^{-1}\left(\mathbf{U}_{y}\right)}$ so that $\mathrm{\widehat{s}_{f_{y},n}\vert \pi^{-1}\left(y^{\prime}\right)\not \equiv0}$ for all $\mathrm{n\geq N_{0}}$ and all $\mathrm{y^{\prime}\in \mathbf{U}_{y}}$. Here $\mathrm{\widehat{s}_{f_{y,n}}}$ is given by (cf.\ {\bf{Section \ref{Definition of the Fiber Probability Measure Sequence (Non-Tame Case)}}}) $$\mathrm{\widehat{s}_{f_{y,n}}=s_{n}\cdot\pi^{*}f^{-1}_{y,n}\text{ for all }y^{\prime}\in \mathbf{U}_{y}\subset \mathbf{Y}}$$ where $\mathrm{\left(f_{y,n}\right)_{n}}$ is a sequence of holomorphic functions $\mathrm{f_{y,n}\in \mathcal{O}\left(\mathbf{U}_{y}\right)}$.
Using {\bf{Theorem 2}}, there exists a tame sequence $\mathrm{\left(s^{i}_{n}\right)_{n}}$ so that $\mathrm{y\in \mathbf{Y}^{i}=\pi\left(\mathbf{X}^{i}\right)}$. Note that after having shrunken $\mathrm{\mathbf{U}_{y}}$, we can always assume that $\mathrm{\mathbf{U}_{y}\subset \mathbf{Y}^{i}}$. We now define a sequence $\mathrm{\big(\triangle_{f_{y,n}}^{i}\big)_{n}}$ of holomorphic $\mathrm{\xi_{n}-\xi^{i}_{n}}$-eigenfunctions on $\mathrm{\pi^{-1}\left(\mathbf{U}_{y}\right)}$ by $$\mathrm{s^{i}_{n}\cdot\triangle_{f_{y,n}}^{i}=\widehat{s}_{f_{y,n}}\label{Notation Difference Function}.}$$ Note that $\mathrm{\triangle^{i}_{f_{y,n}}\vert \pi^{-1}\left(y^{\prime}\right)\not \equiv0}$ for all $\mathrm{n\geq N_{0}}$ and all $\mathrm{y^{\prime}\in \mathbf{U}_{y}}$.
Moreover, recall that the sequence $$\mathrm{\phi_{n}\!:\!x\mapsto \phi_{n}\left(x\right)\coloneqq \Vert \widehat{s}_{f_{y,n}}\Vert^{-2}\left(x\right)\vert \widehat{s}_{f_{y,n}}\vert^{2}\left(x\right)}$$ is independent of the choice of $\mathrm{\left(f_{y,n}\right)_{n}}$, $\mathrm{f_{y,n}\in \mathcal{O}\left(\mathbf{U}_{y}\right)}$, over $\mathrm{\mathbf{U}_{y}\subset \mathbf{R}_{N_{0}}\cap \mathbf{Y}_{0}}$ ({cf.\! }{\bf{Corollary \ref{Corollary Independence Of Extension}}}). Therefore, we will not longer specify $\mathrm{\left(f_{y,n}\right)_{n}}$ and just write $\mathrm{\triangle^{i}_{f_{y,n}}=\triangle^{i}_{y,n}}$ and $\mathrm{s_{n}=\widehat{s}_{f_{y,n}}}$.
\subsection{Analysis of $\mathrm{\triangle^{i}_{n}\vert \pi^{-1}\left(y\right)}$}
The decomposition $\mathrm{s_{n}=s^{i}_{n}\cdot \triangle^{i}_{y,n}}$ introduced above will play a crucial role in the proof of $\mathrm{\mathbf{Theorem\, 5.a,\, b}}$ and $\mathrm{\mathbf{Theorem\, 6.a,\,b}}$. Since the sequence of local functions given by $\mathrm{\triangle^{i}_{y,n}}$ can be seen as measure of the difference between the initial sequence of eigensections $\mathrm{s_{n}}$ and the tame sequence $\mathrm{\left(s^{i}_{n}\right)_{n}}$, it is desirable to control the growth of $\mathrm{\triangle^{i}_{y,n}}$. However, in general the restrictions of the sequence of functions given by $\mathrm{\triangle^{i}_{y,n}}$ to the fibers of the quotient map $\mathrm{\pi}$ turns out to be unbounded. The aim of this section is to prove (cf. {\bf{Proposition \ref{Theorem Estimates Tales of Triangle}}}) that there always exists $\mathrm{m_{0}\in \mathbb{N}}$ so that $\mathrm{s^{i}_{m_{0}}\cdot\triangle^{i}_{y,n}}$ is uniformly bounded over the quotient and takes on its maximum on the fibers in a given neighborhood of $\mathrm{\mu^{-1}\left(\xi\right)}$ for all $\mathrm{n}$ big enough.
\begin{prop} \label{Theorem Estimates Tales of Triangle} Let $\mathrm{\mathbf{W}\subset\mathbf{Y}}$ be a compact neighborhood (we can assume that $\mathrm{\mathbf{W}\subset \mathbf{Y}^{i}}$), then there exists $\mathrm{m_{0}\in \mathbb{N}}$ so that the restriction of $\mathrm{\vert s^{i}_{m_{0}}\cdot\triangle_{y,n}^{i}\vert^{2}}$ on $\mathrm{\pi^{-1}\left(y\right)}$ takes on its maximum in $\mathrm{\pi^{-1}\left(y\right)\cap}$ $\mathrm{ T\left(\epsilon,\mathbf{W}\right)}$ for all $\mathrm{n}$ big enough and all $\mathrm{y\in \mathbf{W}\cap \mathbf{R}_{N_{0}}}$. \end{prop}
Before we prove the above claim, we first have to show the following lemma.
\begin{lemma} \label{Lemma Fibers at Stage n Are Completely Contained In} If $\mathrm{\left(\eta_{m}\right)_{m}}$ is an arbitrary sequence in $\mathrm{\mathfrak{t}^{*}}$ so that $\mathrm{\frac{1}{m}\eta_{m}\rightarrow \xi}$, then there exists $\mathrm{N_{0}\in \mathbb{N}}$ so that $$\mathrm{\mathbf{X}^{ss}_{m^{-1}\eta_{m}}\subset \mathbf{X}^{ss}_{\xi},}$$ for all $\mathrm{m\geq N_{0}}$.
Furthermore, for all $\mathrm{m\geq N_{0}}$, each fiber of $\mathrm{\pi_{m}}$ is entirely contained in a fiber of $\mathrm{\pi}$. \end{lemma} \begin{proof} Since $\mathrm{m^{-1}\eta_{m}\rightarrow \xi}$ it follows that the sequence of compact sets given by $\mathrm{\mu^{-1}(m^{-1}}$ $\mathrm{\eta_{m})}$ converges to $\mathrm{\mu^{-1}\left(\xi\right)}$. From the fact $\mathrm{\mathbf{X}^{ss}_{\xi}}$ is an open neighborhood of $\mathrm{\mu^{-1}\left(\xi\right)}$ we deduce the existence of $\mathrm{N_{0}\in \mathbb{N}}$ so that $\mathrm{\mu^{-1}\left(m^{-1}\eta_{m}\right)\subset \mathbf{X}^{ss}_{\xi}}$ for all $\mathrm{m\geq N_{0}}$. As a consequence, we conclude $$\mathrm{\mathbf{X}^{ss}_{m^{-1}\eta_{m}}\subset \mathbf{X}^{ss}_{\xi}}$$ for all $\mathrm{m\geq N_{0}}$: Let $\mathrm{x\in \mathbf{X}^{ss}_{m^{-1}\eta_{m}}}$ (in the sequel fix $\mathrm{m\geq N_{0}}$). Then by the definition of set of semistable points, we can find $\mathrm{y}$ so that $$\mathrm{y\in cl\left(\mathbb{T}.x\right)\cap \mu^{-1}\left(m^{-1}\eta_{m}\right).}$$ Since $\mathrm{y\in \mu^{-1}\left(m^{-1}\eta_{m}\right)\subset {\mathbf{X}}^{ss}_{\xi}}$, it follows that $$\mathrm{cl\left(\mathbb{T}.y\right)\cap \mu^{-1}\left(\xi\right)\neq \varnothing.}$$ However, since $\mathrm{cl\left(\mathbb{T}.y\right)\subset cl\left(\mathbb{T}.x\right)}$, it follows that $\mathrm{cl\left(\mathbb{T}.x\right)\cap \mu^{-1}\left(\xi\right)\neq \varnothing}$ which proves that $\mathrm{x\in \mathbf{X}^{ss}_{\xi}}$ and hence $\mathrm{\mathbf{X}^{ss}_{m^{-1}\eta_{m}}\subset \mathbf{X}^{ss}_{\xi}}$ as claimed.
Using the fact that $\mathrm{\mathbf{X}^{ss}_{m^{-1}\eta_{m}}\subset \mathbf{X}^{ss}_{\xi}}$ and the universality of the Hilbert quotient map $\mathrm{\pi_{m}}$, there exists an algebraic map $\mathrm{\varphi_{m}\!:\!\mathbf{Y}_{m}\coloneqq \mathbf{X}^{ss}_{m^{-1}\eta_{m}}/\!\!/\mathbb{T}\rightarrow \mathbf{Y}}$ for each $\mathrm{m\geq N_{0}}$ so that the following diagram commutes: $$\begin{xy} \xymatrix{ \mathrm{\mathbf{X}^{ss}_{m^{-1}\eta_{m}}} \ar@{^{(}->}[d] \ar[r]^{\mathrm{\pi_{m}}} &\mathrm{\mathbf{Y}_{m}}\ar[dd]^{\mathrm{\varphi_{m}}}\\ \mathrm{\mathbf{X}^{ss}_{\xi}}\ar[rd]^{\mathrm{\pi}}& & \\ &\mathrm{\mathbf{Y}}\\ } \end{xy} $$ This proves the second claim. \end{proof}
Before we proceed with the proof of {\bf{Proposition \ref{Theorem Estimates Tales of Triangle}}}, we note the following remark.
\begin{remark}\label{Remark Minimum Of Potential Functions} Let $\mathrm{\mathbf{U}\subset \mathbf{Y}_{m}}$ be an open subset, $\mathrm{\mathbf{V}\coloneqq \pi_{m}^{-1}\left(\mathbf{U}\right)}$ and let $$\mathrm{\sigma\in H^{0}\left(\mathbf{V},\mathbf{L}^{m}\vert \mathbf{V}\right)}$$ be a local $\mathrm{\eta_{m}}$-eigensection over $\mathrm{\mathbf{V}}$. In this situation, it follows for $\mathrm{y\in{\bf{U}}}$ that the restriction of $\mathrm{\vert \sigma\vert^{2}}$ to $\mathrm{\pi^{-1}_{m}\left(y\right)}$ takes on its maximum on $\mathrm{\mu^{-1}\left(m^{-1}\eta_{m}\right)\cap \pi^{-1}_{m}\left(y\right)}$. To see this, recall that by the construction of the algebraic Hilbert quotient, we can always find a global $\mathrm{N\cdot\eta_{m}}$-eigensection $\mathrm{\sigma^{\prime}\in H^{0}\left(\mathbf{X},\mathbf{L}^{N\cdot m}\right)}$ for $\mathrm{N}$ big enough so that $\mathrm{\sigma^{\prime}\left(x\right)\neq 0}$ for all $\mathrm{x\in \pi_{m}^{-1}\left(y\right)}$. Therefore, $\mathrm{\sigma^{\prime,-1} \cdot\sigma^{N}}$ defines a holomorphic $\mathrm{\mathbb{T}}$-invariant function on $\mathrm{\pi^{-1}_{m}\left(y\right)}$.
The next step is to consider two possibilities: If $\mathrm{\sigma^{\prime,-1}\cdot \sigma^{N}\equiv 0}$ then it follows that $\mathrm{\sigma\equiv 0}$ and hence the claim is true. Otherwise, it follows that $\mathrm{\sigma\left(x\right)\neq 0}$ for all $\mathrm{x\in \pi_{m}^{-1}\left(y\right)}$. In this case, the claim follows by the theory of the Hilbert quotient because $\mathrm{-\mathbf{log}\,\vert \sigma\vert^{2}}$ defines a smooth plurisubharmonic potential on $\mathrm{\pi^{-1}\left(y\right)}$ of the shifted moment map data and in this case the claim is known. \end{remark}
\begin{proof} (of {\bf{Proposition \ref{Theorem Estimates Tales of Triangle}}}) Let $\mathrm{y\in \mathbf{W}\cap \mathbf{R}_{N_{0}}}$. First of all, note that $\mathrm{s^{i}_{m}\cdot \triangle^{i}_{y,n}}$ defines a local holomorphic $\mathrm{\eta_{m,n}\coloneqq \xi^{i}_{m}+\left(\xi_{n}-\xi^{i}_{n}\right)}$ eigensection over $\mathrm{\pi^{-1}\left(\mathbf{U}_{y}\right)\supset\pi^{-1}\left(y\right)}$ for all $\mathrm{n}$. As $\mathrm{\vert \xi_{n}-\xi^{i}_{n}\vert\in \mathcal{O}\left(1\right)}$, it follows that the set $\mathrm{\left\{\xi_{n}-\xi_{n}^{i}\right\}_{n}\subset \mathfrak{t}^{*}_{\mathbb{Z}}}$ is finite. Hence, it is enough to prove the claim under the assumption that $\mathrm{\xi_{n}-\xi^{i}_{n}}$ is a constant weight $\mathrm{\xi_{0}\in \mathfrak{t}^{*}}$. In the sequel, set $\mathrm{\eta_{m}\coloneqq \xi^{i}_{m}+\xi_{0}}$. Since $\mathrm{m^{-1}\eta_{m}\rightarrow \xi}$ we can apply {\bf{Lemma \ref{Lemma Fibers at Stage n Are Completely Contained In}}}, in order to find $\mathrm{m_{0}\in \mathbb{N}}$ so that $\mathrm{\mathbf{X}^{ss}_{m_{0}^{-1}\eta_{m_{0}}}\subset \mathbf{X}^{ss}_{\xi}}$. Set $$\mathrm{\triangle^{i}_{y,m_{0},n}\coloneqq s^{i}_{m_{0}}\cdot \triangle^{i}_{y,n}}$$ and note that $\mathrm{\triangle^{i}_{y,m_{0},n}}$ induces a local holomorphic $\mathrm{\xi^{i}_{m}+\xi_{0}}$-eigensection over the open, $\mathrm{\pi_{m_{0}}}$-saturated subset $$\mathrm{\mathbf{V}_{m_{0},y}\coloneqq \left(\varphi_{m_{0}}\circ\pi_{m_{0}}\right)^{-1}\left(\mathbf{U}_{y}\right)\subset \mathbf{X}^{ss}_{m_{0}^{-1}\eta_{m_{0}}}}$$ for all $\mathrm{n}$ big enough. In fact: By the above assumption, $\mathrm{\triangle^{i}_{y,n}}$ is of fixed weight $\mathrm{\xi_{0}}$ for all $\mathrm{n\in \mathbb{N}}$.
Note that by {\bf{Remark \ref{Remark Minimum Of Potential Functions}}}, it follows that the strictly plurisubharmonic function given by $\mathrm{-\mathbf{log}\,\vert \triangle^{i}_{y,m_{0},n}\vert^{2}}$ takes on its uniquely defined minimum on $\mathrm{\pi^{-1}_{m_{0}}\left(\widetilde{y}\right)\cap \mu^{-1}\left(m_{0}^{-1}\eta_{m_{0}}\right)}$ for all $\mathrm{n}$ big enough and all $\mathrm{\widetilde{y}\in \varphi_{m_{0}}^{-1}\left(\mathbf{U}_{y}\right)}$. Equivalently, the restriction of $\mathrm{\vert \triangle^{i}_{y,m_{0},n}\vert^{2}}$ to on $\mathrm{\pi^{-1}_{m_{0}}\left(\widetilde{y}\right)}$ takes on its uniquely defined maximum on $\mathrm{\pi^{-1}_{m_{0}}\left(\widetilde{y}\right)\cap \mu^{-1}\left(m_{0}^{-1}\eta_{m_{0}}\right)}$ for all $\mathrm{n}$ big enough and all $\mathrm{\widetilde{y}\in \varphi_{m_{0}}^{-1}\left(\mathbf{U}_{y}\right)}$.
By the commutative diagram of {\bf{Lemma \ref{Lemma Fibers at Stage n Are Completely Contained In}}}, it then follows that the restriction of $\mathrm{\vert \triangle^{i}_{y,m_{0},n}\vert^{2}}$ to $\mathrm{\pi^{-1}\left(y^{\prime}\right)\cap \mathbf{X}^{ss}_{m_{0}^{-1}\eta_{m_{0}}}}$ takes on its maximum in $\mathrm{\mu^{-1}\left(m_{0}^{-1}\eta_{m_{0}}\right)\cap \pi^{-1}\left(y^{\prime}\right)}$ $\mathrm{\cap \mathbf{X}^{ss}_{m_{0}^{-1}\eta_{m_{0}}}}$ for all $\mathrm{y^{\prime}\in \mathbf{U}_{y}}$ and all $\mathrm{n}$ big enough. Since $\mathrm{\mathbf{X}^{ss}_{m_{0}^{-1}\eta_{m_{0}}}}$ is Zariski dense in $\mathrm{\mathbf{X}^{ss}_{\xi}}$ it follows by continuity that the restriction of $\mathrm{\vert \triangle^{i}_{y,m_{0},n}\vert^{2}}$ to $\mathrm{\pi^{-1}\left(y^{\prime}\right)}$ takes on its maximum on $$\mathrm{\mu^{-1}\left(m_{0}^{-1}\eta_{m_{0}}\right)\cap \pi^{-1}\left(y^{\prime}\right)}$$ for all $\mathrm{y^{\prime}\in \mathbf{U}_{y}}$ and all $\mathrm{n}$ big enough - in particular this holds for $\mathrm{y\in \mathbf{U}_{y}}$ itself. The claim then follows by the fact that we can always assume that $$\mathrm{\mu^{-1}\left(m_{0}^{-1}\eta_{m_{0}}\right)\cap \pi^{-1}\left(\mathbf{W}^{i}\right)\subset T\left(\epsilon,\mathbf{W}^{i}\right)}$$ for $\mathrm{m_{0}}$ big enough. \end{proof}
We close this section with the proof of
\begin{lemma}\label{Lemma The Maximum Of Triangle Is Fiberwisely Not Zero} Let $\mathrm{\mathbf{W}\subset\mathbf{Y}}$ be a compact neighborhood of $\mathrm{y_{0}\in \mathbf{Y}}$ (we can assume that $\mathrm{\mathbf{W}\subset \mathbf{Y}^{i}}$), and $\mathrm{\epsilon>0}$, then there exists $\mathrm{N_{0}\in \mathbb{N}}$ so that $$\mathrm{\underset{x\in\pi^{-1}\left(y\right)\cap T\left(\epsilon,\mathbf{W}\right)}{max}\,\left|\triangle_{y,n}^{i}\right|^{2}\left(x\right)>0}$$ for all $\mathrm{y\in \mathbf{W}\cap \mathbf{R}_{N_{0}}}$ and all $\mathrm{n}$ big enough.
In particular, it follows that $$\mathrm{\underset{x\in\pi^{-1}\left(y\right)\cap T\left(\epsilon,\mathbf{W}\right)}{max}\,\left|s^{i}_{m_{0}}\cdot\triangle_{y,n}^{i}\right|^{2}\left(x\right)>0}$$ for all $\mathrm{y\in \mathbf{W}\cap \mathbf{R}_{N_{0}}}$ and all $\mathrm{n}$ big enough. \end{lemma} \begin{proof} First of all note that the second claim is a direct consequence of the first because $\mathrm{\vert s^{i}_{m_{0}}\vert^{2}\left(x\right)>0}$ for all $\mathrm{x\in \mathbf{X}^{i}}$ and all $\mathrm{m_{0}\in \mathbb{N}}$.
We already know that $\mathrm{\triangle^{i}_{y,n}\vert \pi^{-1}\left(y\right)\not\equiv 0}$ for all $\mathrm{n\in \mathbb{N}}$ and all $\mathrm{y\in \mathbf{W}\cap \mathbf{R}_{N_{0}}}$. Throughout the proof we will fix $\mathrm{y}$ and let $$\mathrm{\pi^{-1}\left(y\right)=\bigcup_{j}\mathbf{C}_{j,y}}$$ be the decomposition of $\mathrm{\pi^{-1}\left(y\right)}$ in its global irreducible components $\mathrm{\mathbf{C}_{j,y}}$. It is direct to see that these components are $\mathrm{\mathbb{T}}$-invariant and hence each $\mathrm{\mathbf{C}_{j,y}}$ intersects $\mathrm{\mu^{-1}\left(\xi\right)}$ non-trivially:\ In fact, let $\mathrm{z\in \mathbf{C}_{j,y}}$ and choose a one parameter subgroup $\mathrm{\gamma\!:\!\mathbb{C}^{*}\rightarrow \mathbb{T}}$ so that $$\mathrm{z_{0}=\underset{t\rightarrow0}{lim}\,\gamma\left(t\right).z\in \mathbb{T}.x_{y}}$$ where $\mathrm{\mathbb{T}.x_{y}}$ is the unique closed orbit in the fiber $\mathrm{\pi^{-1}\left(y\right)}$. Since $\mathrm{\mathbf{C}_{j,y}}$ is closed and $\mathrm{\mathbb{T}}$ invariant it follows that $\mathrm{z_{0}\in \mathbb{T}.x_{y}\cap \mathbf{C}_{j,y}}$. Again by the invariance of $\mathrm{\mathbf{C}_{j,y}}$ and the fact that $\mathrm{\mu^{-1}\left(\xi\right)\cap \mathbb{T}.x_{y}=\mu^{-1}\left(\xi\right)\cap \pi^{-1}\left(y\right)}$ it follows that $\mathrm{\mu^{-1}\left(\xi\right)\cap \mathbf{C}_{j,y}\neq \varnothing}$. In particular, the open set $$\mathrm{T\left(\epsilon,\mathbf{W}\right)\cap \mathbf{C}_{j,y}}$$ is always non-empty in $\mathrm{\mathbf{C}_{j,y}}$. So if $$\mathrm{\underset{x\in\pi^{-1}\left(y\right)\cap T\left(\epsilon,\mathbf{W}\right)}{max}\,\left|\triangle_{y,n}^{i}\right|^{2}\left(x\right)=0}$$ for $\mathrm{y\in \mathbf{W}\cap\mathbf{R}_{N_{0}}}$, it would follow that $\mathrm{\triangle^{i}_{y,n}}$ would vanish identically on each non-empty open subset $\mathrm{T\left(\epsilon,\mathbf{W}\right)\cap \mathbf{C}_{j,y}}$ of the irreducible component $\mathrm{\mathbf{C}_{j,y}}$. Hence, it would follow by the Identity Principle that $\mathrm{\triangle^{i}_{y,n}\vert \mathbf{C}_{j,y}\equiv 0}$ for all $\mathrm{j}$ and therefore $\mathrm{\triangle^{i}_{y,n}\vert \pi^{-1}\left(y\right)\equiv 0}$ in contradiction to $\mathrm{\triangle^{i}_{y,n}\vert \pi^{-1}\left(y\right)\not\equiv 0}$. \end{proof}
\subsection{A Local Proposition Concerning Fiber Integration}\label{A Local Proposition Concerning Fiber Integration} In this section we will prove a technical proposition ({cf.\ }{\bf{Proposition \ref{Proposition Local Version Of Proposition}}}) which will be of crucial importance when proving {\bf{Theorem 5.a,\ b}} and {\bf{Theorem 6.a,\ b}}.
A first step towards {\bf{Proposition \ref{Proposition Local Version Of Proposition}}} is the following technical lemma.
\begin{lemma}\label{Lemma Bound Of Norm Squares Of Roots} Let $$\mathrm{z^{d}+\alpha_{d-1}z^{d-1}+\dots+\alpha_{0},\,\text{ where }\alpha_{i}\in \mathbb{C}\text{ for }0\leq i\leq d-1}$$ be a monic polynomial of degree $\mathrm{d}$ and let $\mathrm{\zeta_{i}}$, $\mathrm{1\leq i\leq d}$ the corresponding roots, then there exists a constant $\mathrm{c_{d}>0}$ which only depends of the degree $\mathrm{d}$, so that $$\mathrm{\sum_{i=1}^{d}\vert \zeta_{i}\vert^{2}\geq c_{d}\left(\sum_{i=0}^{d-1}\vert \alpha_{i}\vert^{2}\right)^{\frac{1}{d}}.}$$ \end{lemma} \begin{proof} Consider the holomorphic map $\mathrm{F\!:\!\mathbb{C}^{d}\rightarrow \mathbb{C}^{d}}$ defined by $$\mathrm{F\!:\!\left(\zeta_{1},\dots,\zeta_{d}\right)\mapsto \left(\mathfrak{P}_{0}\left(\zeta_{1},\dots,\zeta_{d}\right),\dots,\mathfrak{P}_{d-1}\left(\zeta_{1},\dots,\zeta_{d}\right)\right)}$$ where $\mathrm{\left\{\mathfrak{P}_{\ell}\right\}_{0\leq \ell\leq d-1}}$ is the set of all elementary symmetric polynomials, i.e.\ $$\mathrm{\begin{array}{rcl} \mathrm{\mathfrak{P}_{d-1}\left(\zeta_{1},\dots,\zeta_{d}\right)} & \mathrm{=} & \mathrm{\left(-1\right)\,\sum_{1\leq j\leq d}\zeta_{j}}\\[0.2 cm] \mathrm{\mathfrak{P}_{d-2}\left(\zeta_{1},\dots,\zeta_{d}\right)} & \mathrm{=} & \mathrm{\left(+1\right)\,\sum_{1\leq j_{1}<j_{2}\leq d}\zeta_{j_{1}}\zeta_{j_{2}}}\\[0.2 cm]
& \mathrm{\vdots}\\[0.2 cm] \mathrm{\mathfrak{P}_{d-\ell}\left(\zeta_{1},\dots,\zeta_{d}\right)} & \mathrm{=} & \mathrm{\left(-1\right)^{\ell}\sum_{1\leq j_{1}<\dots<j_{\ell}\leq d}\zeta_{j_{1}\leq d}\dots\zeta_{j_{\ell}}}\\[0.2 cm]
& \mathrm{\vdots}\\[0.2 cm] \mathrm{\mathfrak{P}_{0}\left(\zeta_{1},\dots,\zeta_{d}\right)} & = & \mathrm{\left(-1\right)^{d}\zeta_{1}\dots\zeta_{m}}\end{array}}$$ If $\mathrm{\alpha=\left(\alpha_{0},\dots,\alpha_{d-1}\right)\in \mathbb{C}^{d}}$, then by {\scshape{Vieta}}'s Formula we deduce that $$\mathrm{F^{-1}\left(\alpha\right)=\left\{\zeta\in \mathbb{C}\!:\!\, \zeta^{d}+\alpha_{d-1}\,\zeta^{d-1}+\alpha_{d-2}\,\zeta^{d-2}+\dots +\alpha_{0}=0 \right\}}$$ so the inverse image of $\mathrm{\alpha\in \mathbb{C}^{d}}$ contains exactly all roots of the polynomial with coefficients $\mathrm{\alpha_{\ell}}$ for $\mathrm{0\leq\ell\leq d-1}$. In particular, it follows that $\mathrm{F^{-1}\left(0\right)=\left\{0\right\}}$ and by continuity there exists $\mathrm{c_{d}> 0}$ so that $\mathrm{{\bm{\Delta}}_{c_{d}}\subset Int\,F^{-1}\left({\bm{\Delta}}_{1}\right)}$ where $\mathrm{{\bm{\Delta}}_{\lambda}\subset \mathbb{C}^{d}}$ is the closed ball of radius $\mathrm{\lambda>0}$ in $\mathrm{\mathbb{C}^{d}}$. It is direct to check that $$\mathrm{F\left(\lambda\,\zeta_{1},\dots,\lambda\,\zeta_{d}\right)=\left(\lambda^{d}\,\mathfrak{P}_{0}\left(\zeta_{1},\dots,\zeta_{d}\right),\dots,\lambda\,\mathfrak{P}_{d-1}\left(\zeta_{1},\dots,\zeta_{d}\right)\right)\text{ for all }\lambda\in \mathbb{C}}$$ and hence, we deduce that $$\mathrm{{\bm{\Delta}}_{\lambda^{\frac{1}{d}}\,c_{d}}\subset Int\,F^{-1}\left({\bm{\Delta}}_{\lambda}\right)}$$ for all $\mathrm{\lambda\geq 0}$. This can be reformulated as follows: If $\mathrm{z^{d}+\alpha_{d-1}z^{d-1}+\dots+\alpha_{0}}$ is a monic polynomial of degree $\mathrm{d}$ so that $\mathrm{\sum_{i=0}^{d-1}\vert \alpha_{i}\vert^{2}=\lambda}$, {i.e.\! }$\mathrm{\alpha\in bd\, {\bm{\Delta}}_{\lambda}}$, then $\mathrm{F^{-1}\left(\alpha\right)\notin{\bm{\Delta}}_{\lambda^{\frac{1}{d}}\,c_{d}}}$, i.e. $$\mathrm{\sum_{i=1}^{d}\vert \zeta_{i}\vert^{2}\geq \lambda^{\frac{1}{d}}\cdot c_{d}}$$ where $\mathrm{F^{-1}\left(\alpha\right)=\left\{\zeta_{i}\right\}_{1\leq j\leq d}}$ is the set of the corresponding roots and therefore $$\mathrm{\sum_{i=1}^{d}\vert \zeta_{i}\vert^{2}\geq c_{d}\left(\sum_{i=0}^{d-1}\vert \alpha_{i}\vert^{2}\right)^{\frac{1}{d}}}$$ as claimed. \end{proof}
Let $\mathrm{F\!:\!\mathbf{X}\rightarrow \mathbf{Y}}$\label{Notation D-Sheeted Covering} be $\mathrm{k}$-fibering, i.e.\ a holomorphic map between purely dimensional complex spaces where $\mathrm{m=dim_{\mathbb{C}}\,\mathbf{X}}$ and $\mathrm{n=dim_{\mathbb{C}}\,\mathbf{Y}}$ so that $\mathrm{F^{-1}\left(y\right)}$ is a purely dimensional complex space of dimension $\mathrm{k=m-n}$ for all $\mathrm{y\in \mathbf{Y}}$. The relevant examples of $\mathrm{k}$-fiberings, which we have in mind, are given by $\mathrm{\widehat{\pi}\!:\!\widehat{\mathbf{\,X\,}}\rightarrow \mathbf{Y}}$, $\mathrm{\pi\vert \mathbf{X}_{0}\!:\!\mathbf{X}_{0}\rightarrow \mathbf{Y}_{0}}$ where $\mathrm{\mathbf{X}_{0}=\pi^{-1}\left(\mathbf{Y}_{0}\right)}$ and $\mathrm{\Pi\!:\!\widetilde{\bf{\,X\,}}\rightarrow \widetilde{\bf{\,Y\,}}}$.
If $\mathrm{x_{0}\in \mathbf{X}}$ and $\mathrm{F\!:\!\mathbf{X}\rightarrow \mathbf{Y}}$ a $\mathrm{k}$-fibering, then there exists ({cf.\ }\cite{Kin}, {p.\ }{\bf{\oldstylenums{205}}}) an open neighborhood $\mathrm{\mathbf{U}\subset \mathbf{X}}$ of $\mathrm{x_{0}}$ which can be realized via an isomorphism $\mathrm{\Phi}$ as a closed analytic subset $\mathrm{\mathbf{Z}\subset\mathbf{Q}= \mathbf{Q}_{0}\times \mathbf{Q}_{1}\subset \mathbb{C}^{\upkappa}\times \mathbb{C}^{k}}$ of a relatively compact, open product set $\mathrm{\mathbf{Q}_{0}\times \mathbf{Q}_{1}}$ and an open neighborhood $\mathrm{\mathbf{B}\subset \mathbf{Y}}$ of $\mathrm{y_{0}=F\left(x_{0}\right)\in \mathbf{Y}}$ so that the following holds: If $$\mathrm{z^{\left(0\right)}=\big(z^{\left(0\right)}_{1},\dots,z^{\left(0\right)}_{\upkappa}\big),\text{ resp. }z^{\left(1\right)}=\big(z^{\left(1\right)}_{1},\dots,z^{\left(1\right)}_{k}\big).}$$ are the standard coordinates on $\mathrm{\mathbb{C}^{\upkappa}}$, resp.\ on $\mathrm{\mathbb{C}^{k}}$, then the restriction of the projection map $\mathrm{p\!:\!\mathbb{C}^{\upkappa}\times \mathbb{C}^{k}\rightarrow}$ $\mathrm{\mathbb{C}^{k}}$ to the closed $\mathrm{k}$-dimensional space $\mathrm{\mathbf{Z}_{y}\coloneqq \Phi\left(F^{-1}\left(y\right)\cap \mathbf{U}\right)}$ of $\mathrm{\mathbf{Q}=\mathbf{Q}_{0}\times \mathbf{Q}_{1}}$ induces a $\mathrm{d}$-sheeted covering map $\mathrm{p\vert\mathbf{\mathbf{Z}}_{y} \!:\!\mathbf{\mathbf{Z}}_{y}\rightarrow \mathbf{Q}_{1}}$ onto $\mathrm{\mathbf{Q}_{1}}$ for all $\mathrm{y\in \mathbf{B}\subset \mathbf{Y}}$. Moreover, by the theory of finite $\mathrm{d}$-sheeted coverings the following is known ({cf.\ }\cite{Gr-Re}, {pp.\ }{\bf{\oldstylenums{133}-\oldstylenums{146}}}): For each $\mathrm{d}$-sheeted covering map $\mathrm{p\vert \mathbf{Z}_{y}\rightarrow \mathbf{Q}_{1}}$ where $\mathrm{y\in \mathbf{B}}$, the inclusion $$\mathrm{\mathcal{O}\left(\mathbf{Q}_{1}\right)\subset \left(p\vert \mathbf{Z}_{y}\right)_{*}\mathcal{O}\left(\mathbf{Z}_{y}\right)}$$ is a finite, integral ring extension so that for all for each $\mathrm{f\in \mathcal{O}\left(\mathbf{Z}_{y}\right)}$, there exists $\mathrm{\alpha_{f,j}\in \mathcal{O}\left(\mathbf{Q}_{1}\right)}$, $\mathrm{0\leq j\leq d-1}$ with \begin{equation}\label{Equation Polynomial Equation Induces By Finite Covers} \mathrm{f^{d}+\left(p\vert \mathbf{Z}_{y}\right)^{*}\alpha_{f,d-1}\,f^{d-1}+\dots+\left(p\vert \mathbf{Z}_{y}\right)^{*}\alpha_{f,0}=0} \end{equation} on $\mathrm{\mathbf{Z}_{y}}$.
\begin{example} \textnormal{Let $\mathrm{\widehat{\mathbf{\,X\,}}=\left\{ z_{0}\zeta_{1}\xi_{1}-z_{1}\zeta_{0}\xi_{0}=0\right\} \subset \mathbf{X}\times\mathbb{C}\mathbb{P}^{1}}$ be as in example {\bf{Example \ref{Example Proper Inclusion}}} where $\mathrm{\widehat{\mathbf{\,Y\,}}\cong \mathbf{Y}\cong \mathbb{C}\mathbb{P}^{1}}$ and $\mathrm{\Pi=\widehat{\pi}=p_{\mathbb{C}\mathbb{P}^{1}}}$ and consider the open neighborhood $\mathrm{\mathbf{U}\subset\widehat{\mathbf{\,X\,}}}$ of $\mathrm{\left([1\!:\!0],[0\!:\!1],[1\!:\!0]\right)}$ which is isomorphic to the affine variety $\mathrm{\left\{\xi-z\zeta=0\right\}\subset \mathbb{C}^{3}}$ where $\mathrm{z=z_{0}^{-1}z_{1}}$, $\mathrm{\zeta=\zeta_{1}^{-1}\zeta_{0}}$, $\mathrm{\xi =\xi_{0}^{-1}\xi_{1}}$. After a linear change of coordinates it follows that $\mathrm{\widehat{\mathbf{\,X\,}}\cap \mathbf{U}\cong\left\{\xi-z^{\prime,2}-\zeta^{\prime,2}=0\right\}}$ and we can choose $\mathrm{\mathbb{C}^{2}\times \mathbb{C}=\mathbf{B}_{0}\times \mathbf{B}_{1}}$ where $\mathrm{\zeta^{\prime}, \xi}$ are the coordinates of the first factor and $\mathrm{z^{\prime}}$ of the second one in order to deduce a two sheeted, global projection $\mathrm{p\vert \mathbf{Z}_{\xi}:\mathbf{Z}_{\xi}\rightarrow \mathbf{Q}_{1}}$ onto $\mathrm{\mathbf{Q}_{1}=\mathbb{C}}$ for all $\mathrm{\xi\in \mathbf{Y}\setminus \left\{[0\!:\!1]\right\}}$.}
\textnormal{If $\mathrm{f\in \mathcal{O}\left(\mathbf{Z}_{\xi}\right)}$ is given by the restriction of the polynomial $\mathrm{p\left(z^{\prime}\right)=\sum_{j=0}^{m}c_{j}z^{\prime,j}}$ to $\mathrm{\mathbf{Z}_{\xi}}$ one can verify that $$\mathrm{\alpha_{f,1}\left(z^{\prime}\right)=-2\sum_{\substack{\textnormal{j=0}\\[0.05 cm] \textnormal{even}}}^{m}c_{j}\left(z^{\prime,2}-\xi\right)^{\frac{j}{2}}}$$ and $$\mathrm{\alpha_{f,0}\left(z^{\prime}\right)=\sum_{j=0}^{m}\left(-1\right)^{j}c_{j}^{2}\left(z^{\prime,2}-\xi\right)^{j}+2\sum^{m}_{\substack{\textnormal{0=j}<\textnormal{k}\\[0.05cm] \textnormal{j,k even}}}c_{j}c_{k}\left(z^{\prime,2}-\xi\right)^{\frac{1}{2}\left(j+k\right)}. \,\boldsymbol{\Box}}$$} \end{example}
Now let $\mathrm{\mathbf{V}\subset \mathbb{C}^{\upkappa}\times \mathbb{C}^{k}}$ be an open neighborhood of $\mathrm{x=0\in \mathbb{C}^{\upkappa}\times \mathbb{C}^{k}}$ containing $\mathrm{\mathbf{Q}}$ and equip $\mathrm{\mathbf{V}}$ with a smooth K\"ahler form $\mathrm{\omega}$.
With the help of {\bf{Lemma \ref{Lemma Bound Of Norm Squares Of Roots}}} and the existence of \ref{Equation Polynomial Equation Induces By Finite Covers}, we deduce the following lemma.
\begin{lemma} \label{Lemma Lower Bound Of Integral} Let $\mathrm{y\in\mathbf{B}}$ and $\mathrm{f\in \mathcal{O}\left(\mathbf{Z}_{y}\right)}$ where $\mathrm{\alpha_{f,j}}$ $\mathrm{\in \mathcal{O}\left(\mathbf{Q}_{1}\right)}$, $\mathrm{1\leq j\leq d-1}$ are as above. Then it follows $$\mathrm{\int_{\mathbf{Z}_{y}}\vert f\vert^{2} \,d\,[\mathbf{Z}_{y}]\geq d\cdot c_{k}\int_{\mathbf{Q}_{1}}\left(\sum_{j=0}^{d-1}\vert \alpha_{f,j}\vert^{2}\right)^{\frac{1}{d}}\,\omega_{0}^{q}}$$ where $\mathrm{\omega_{0}}$ denotes the standard K\"ahler form, $\mathrm{c_{d}}$ the constant of {\bf{Lemma \ref{Lemma Bound Of Norm Squares Of Roots}}} and $\mathrm{d}$ the degree of the covering map. \end{lemma} \begin{proof} It is known (cf.\ \cite{Kin}, {pp.\ }{\bf{\oldstylenums{185}-\oldstylenums{220}}}) that the right hand side of the above inequality can be be bounded from below by $$\mathrm{d\,\int_{\mathbf{Q}_{1}}\left(z^{\left(1\right)}\mapsto \sum_{j=1}^{d}\left\vert \zeta_{f,j}\big(z^{\left(1\right)}\big)\right\vert^{2}\right)\,\omega^{q}_{0}}$$ where $\mathrm{\zeta_{f,j}\left(z^{\left(1\right)}\right)}$ are the $\mathrm{d}$ roots of the polynomial equation $$\mathrm{z^{d}+\alpha_{f,d-1}\big(z^{\left(1\right)}\big)\,?z^{d-1}+\dots+\alpha_{f,0}\big(z^{\left(1\right)}\big)=0.}$$ The claim then follows by using the inequality proved in {\bf{Lemma \ref{Lemma Bound Of Norm Squares Of Roots}}}. \end{proof}
After this preparation, we can now prove the announced proposition.
\begin{prop} \label{Proposition Local Version Of Proposition} Let $\mathrm{\left(y_{n}\right)_{n}}$ be a sequence in $\mathrm{\mathbf{B}}$ converging to $\mathrm{y_{0}\in \mathbf{B}}$ and let $\mathrm{\left(f_{n}\right)_{n}}$, $\mathrm{f_{n}\in \mathcal{O}\left(\mathbf{Z}_{y_{n}}\right)}$ be a sequence of uniformly bounded holomorphic functions so that $$\mathrm{\int_{\mathbf{Z}_{n}}\vert f_{n}\vert^{2}\,d\,[\mathbf{Z}_{n}]\rightarrow 0.}$$
Then it follows that for each compact subset $\mathrm{\mathbf{K}\subset \mathbf{Q}_{1}}$ and each $\mathrm{\epsilon>0}$ there exists $\mathrm{N_{\epsilon}\left(\mathbf{K}\right)}$ $\mathrm{\in\mathbb{N}}$ so that $$\mathrm{\vert f_{n}\vert^{2}\leq \epsilon\text{ on }p^{-1}\left(\mathbf{K}\right)\cap \mathbf{Z}_{n}\text{ for all }n\geq N_{\epsilon}\left(\mathbf{K}\right).}$$ \end{prop} \begin{proof} First of all set $\mathrm{\alpha_{n,j}\coloneqq\alpha_{f_{n},j}\in \mathcal{O}\left(\mathbf{Z}_{y_{n}}\right)}$ for $\mathrm{0\leq j\leq d-1}$. By the assumption combined with {\bf{Lemma \ref{Lemma Lower Bound Of Integral}}} we deduce that \begin{equation}\label{Equation Sum Converges Against Zero} \mathrm{d\cdot c_{d}\int_{\mathbf{Q}_{1}}\left(\sum_{j=0}^{d-1}\vert \alpha_{n,j}\vert^{2}\right)^{\frac{1}{d}}\,\omega_{0}^{q}\rightarrow 0} \end{equation} where \begin{equation}\label{Equation F As Zero Of Polynomial} \mathrm{f_{n}^{d}+\left(p\vert \mathbf{Z}_{n}\right)^{*}\alpha_{n,d-1}\,f_{n}^{d-1}+\dots+\left(p\vert \mathbf{Z}_{n}\right)^{*}\alpha_{n,0}=0.} \end{equation} Since the sequence $\mathrm{\left(f_{n}\right)_{n}}$ is uniformly bounded, i.e.\ there exists $\mathrm{C>0}$ so that $\mathrm{\vert f_{n}\vert^{2}\leq C}$ on $\mathrm{\mathbf{Z}_{n}}$ for all $\mathrm{n}$, it follows by equation \ref{Equation F As Zero Of Polynomial} in combination with {\bf{Lemma \ref{Lemma Bound Of Norm Squares Of Roots}}} that $$\mathrm{d\cdot C\geq c_{d}\,\left(\sum_{j=0}^{d-1}\vert \alpha_{n,j}\vert^{2}\right)^{\frac{1}{d}}\text{ on }\mathbf{Q}_{1}\text{ for all }n.}$$ Hence, the sequences $\mathrm{\left(\alpha_{n,j}\right)_{n}}$, where $\mathrm{\alpha_{n,j}\in\mathcal{O}\left(\mathbf{Q}_{1}\right)}$, are uniformly bounded as well for all $\mathrm{1\leq j\leq d-1}$. By the theorem of {\scshape{Montel}} (after having chosen a subsequence) it follows that $$\mathrm{\alpha_{n,j}\rightarrow \alpha_{j}\in \mathcal{O}\left(\mathbf{K}\right)\text{ uniformly on the compact subset }\mathbf{K}\subset \mathbf{Q}_{1}.}$$ The next step is to show that $\mathrm{\alpha_{j}\equiv 0}$ for all $\mathrm{j}$. Let us assume that this is false, i.e.\ there exist at least one $\mathrm{\alpha_{\ell}\not\equiv 0}$ for $\mathrm{0\leq \ell\leq d-1}$. Since $\mathrm{\alpha_{n,j}\rightarrow \alpha_{j}}$ converges uniformly on $\mathrm{\mathbf{K}}$, it follows by \ref{Equation Sum Converges Against Zero} that $$\mathrm{0=\underset{n\rightarrow\infty}{lim}\,c_{d}\int_{\mathbf{K}}\left(\sum_{j=0}^{d-1}\vert \alpha_{n,j}\vert^{2}\right)^{\frac{1}{d}}\,\omega_{0}^{q}=c_{d}\int_{\mathbf{K}}\left(\sum_{j=0}^{d-1}\vert \alpha_{j}\vert^{2}\right)^{\frac{1}{d}}\,\omega_{0}^{q}}$$ which yields a contradiction to the assumption that $\mathrm{\alpha_{\ell}\not\equiv 0}$ on $\mathrm{\mathbf{K}}$ for at least one $\mathrm{\ell}$. Hence, we deduce that $\mathrm{\alpha_{j}\equiv 0}$ on $\mathrm{\mathbf{K}}$ for all $\mathrm{0\leq j\leq d-1}$. As the convergence is uniform, we find for an arbitrary $\mathrm{\Gamma>0}$ an integer $\mathrm{N\left(\Gamma\right)\in \mathbb{N}}$ so that $\mathrm{\vert \alpha_{n,j}\vert^{2}\leq \Gamma}$ for all $\mathrm{n\geq N\left(\Gamma\right)}$ on $\mathrm{\mathbf{K}}$. By equation \ref{Equation F As Zero Of Polynomial} combined with the general fact that if $\mathrm{\zeta\in \mathbb{C}}$ is a root of $\mathrm{z^{d}+\alpha_{d-1}\,z^{d-1}+\dots+}$ $\mathrm{\alpha_{0}=0}$ then $\mathrm{\vert \zeta\vert\leq 2\,\underset{1\leq j\leq d}{max}\,\vert \alpha_{d-j}\vert^{\frac{1}{j}} }$ (cf.\ \cite{Dem}), we deduce $$\mathrm{\vert f_{n}\vert\leq 2\,\underset{1\leq j\leq d}{max}\,\Gamma^{\frac{1}{j}} \text{ on }p^{-1}\left(\mathbf{K}\right)\cap \mathbf{Z}_{n}\text{ for all }n\geq N\left(\Gamma\right)}$$ which proves the claim:\ Choose $\mathrm{\Gamma_{\epsilon}>0}$ so that $\mathrm{2\,\underset{1\leq j\leq d}{max}\,\Gamma_{\epsilon}^{\frac{1}{j}}=\epsilon}$ and set $\mathrm{N_{\epsilon}\left(\mathbf{K}\right)\coloneqq N(\Gamma_{\epsilon})}$. \end{proof}
\subsection{Local Uniform Convergence on $\mathrm{\mathbf{R}_{N_{0}}\cap\mathbf{Y}_{0}}$}
Recall that the set of all removable singularities $\mathrm{\mathbf{R}_{N_{0}}}$ of the measure sequence $\mathrm{\left(\bm{\nu}_{n}\right)_{n}}$ induced by the $\mathrm{\xi}$-approximating sequence $\mathrm{\left(s_{n}\right)_{n}}$ of $\mathrm{\xi_{n}}$-eigensections $\mathrm{s_{n}\in H^{0}\left(\mathbf{X},\mathbf{L}^{n}\right)}$ is assumed to be non-empty throughout this section (cf.\ {\bf{General Agreement}} at the beginning of page \pageref{General Agreement}). The first step towards the proof of {\bf{Theorem 5.a}} and {\bf{Theorem 6.a}} is the following proposition.
\begin{prop}\label{Proposition Tales Are Uniformly Bounded} Let $\mathrm{\mathbf{W}\subset\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$ be a compact neighborhood of $\mathrm{y_{0}\in \mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$ (we can assume that $\mathrm{\mathbf{W}\subset \mathbf{Y}^{i}}$) and $\mathrm{\epsilon>0}$, then after having shrunken $\mathrm{\mathbf{W}}$, there exists a constant $\mathrm{C>0}$ and $\mathrm{m_{0}\in\mathbb{N}}$ so that $$\mathrm{\underset{x\in\pi^{-1}\left(y\right)}{max}\,\,\,\frac{\left|s_{m_{0}}^{i}\cdot\triangle_{y,n}^{i}\right|^{2}}{\int_{\pi^{-1}\left(y\right)\cap T\left(\epsilon,\mathbf{W}\right)}\left|s_{m_{0}}^{i}\cdot\triangle_{y,n}^{i}\right|^{2}\,d\,[\pi_{y}]}<C}$$ for all $\mathrm{n\geq m_{0}}$ and all $\mathrm{y\in{\mathbf{W}}}$. \end{prop} \begin{proof} First of all by {\bf{Proposition \ref{Theorem Estimates Tales of Triangle}}} we know that there exists $\mathrm{m_{0}}$ so that \begin{equation}\label{Equation Maximum Is Contained In}
\mathrm{\underset{x\in\pi^{-1}\left(y\right)}{max}\,\left|s_{m_{0}}^{i}\cdot\triangle_{y,n}^{i}\right|^{2}=\underset{x\in\pi^{-1}\left(y\right)\cap T\left(\frac{\epsilon}{2},\mathbf{W}^{i}\right)}{max}\,\left|s_{m_{0}}^{i}\cdot\triangle_{y,n}^{i}\right|^{2}}
\end{equation} for all $\mathrm{n}$ big enough and all $\mathrm{y\in \mathbf{W}\cap \mathbf{R}_{N_{0}}=\mathbf{W}}$ (by the above assumption, $\mathrm{\mathbf{W}}$ is contained in $\mathrm{\mathbf{R}_{N_{0}}\cap \mathbf{Y}_{0}}$). Furthermore, by lemma {\bf{Lemma \ref{Lemma The Maximum Of Triangle Is Fiberwisely Not Zero}}} we deduce that $$\mathrm{\underset{x\in\pi^{-1}\left(y\right)\cap T\left(\epsilon,\mathbf{W}^{i}\right)}{max}\,\left|s_{m_{0}}^{i}\cdot\triangle_{y,n}^{i}\right|^{2}>0}$$ for all $\mathrm{n}$ big enough and all $\mathrm{y\in \mathbf{W}\cap \mathbf{R}_{N_{0}}=\mathbf{W}}$. Hence, $$\mathrm{\int_{\pi^{-1}\left(y\right)\cap T\left(\epsilon,\mathbf{W}\right)}\frac{\left|s_{m_{0}}^{i}\cdot\triangle_{y,n}^{i}\right|^{2}}{\underset{x\in\pi^{-1}\left(y\right)\cap T\left(\epsilon,\mathbf{W}\right)}{max}\,\left|s_{m_{0}}^{i}\cdot\triangle_{y,n}^{i}\right|^{2}}\,d\,[\pi_{y}]}$$ is well defined for all $\mathrm{n}$ big enough and all $\mathrm{y\in \mathbf{W}}$. Note that the claim is shown, as soon as we have shown that, after having shrunken $\mathrm{\mathbf{W}}$, there exists $\mathrm{c>0}$ (set $\mathrm{C\coloneqq c^{-1}}$) so that $$\mathrm{\int_{\pi^{-1}\left(y\right)\cap T\left(\epsilon,\mathbf{W}\right)}\frac{\left|s_{m_{0}}^{i}\cdot\triangle_{y,n}^{i}\right|^{2}}{\underset{x\in\pi^{-1}\left(y\right)\cap T\left(\epsilon\mathbf{W}\right)}{max}\,\left|s_{m_{0}}^{i}\cdot\triangle_{y,n}^{i}\right|^{2}}\,d\,[\pi_{y}]>c}$$ for all $\mathrm{n}$ big enough and all $\mathrm{y\in \mathbf{W}}$.
Let us assume that this is not the case, then there exists a sequence $\mathrm{y_{n}\in \mathbf{W}}$ converging to $\mathrm{y_{0}}$ so that \begin{equation}\label{Equation Integral Converges To Zero}
\mathrm{\int_{\pi^{-1}\left(y_{n}\right)\cap T\left(\epsilon,\mathbf{W}\right)}\frac{\left|s_{m_{0}}^{i}\cdot\triangle_{y_{n},n}^{i}\right|^{2}}{\underset{x\in\pi^{-1}\left(y_{n}\right)\cap T\left(\epsilon,\mathbf{W}\right)}{max}\,\left|s_{m_{0}}^{i}\cdot\triangle_{y_{n},n}^{i}\right|^{2}}\,d\,[\pi_{y_{n}}]\rightarrow 0.}
\end{equation} To shorten notation we will set $$\mathrm{f_{n}\coloneqq \frac{s^{i}_{m_{0}}\cdot \triangle^{i}_{y_{n},n}}{\underset{x\in\pi^{-1}\left(y_{n}\right)\cap T\left(\epsilon,\mathbf{W}\right)}{max}\,\left|s_{m_{0}}^{i}\cdot\triangle_{y_{n},n}^{i}\right|}\Bigg\vert\pi^{-1}\left(y_{n}\right)}$$ throughout the rest of this proof where $\mathrm{f_{n}\in H^{0}\left(\pi^{-1}\left(y_{n}\right),\mathbf{L}^{m_{0}}\vert \pi^{-1}\left(y_{n}\right)\right)}$. By equation \ref{Equation Maximum Is Contained In} it follows that $$\mathrm{\underset{x\in\pi^{-1}\left(y_{n}\right)}{max}\,\left|f_{n}\right|^{2}=\underset{x\in\pi^{-1}\left(y_{n}\right)\cap T\left(\frac{\epsilon}{2},\mathbf{W}\right)}{max}\,\left|f_{n}\right|^{2}=1}$$ for all $\mathrm{n}$ big enough where $\mathrm{y_{n}\rightarrow y_{0}\in Int\,\mathbf{W}}$. Therefore we can find a lift of $\mathrm{\left(y_{n}\right)_{n}}$, i.e.\ a sequence $\mathrm{\left(x_{n}\right)_{n}}$ in $\mathrm{T\left(\frac{\epsilon}{2},\mathbf{W}\right)}$ so that $\mathrm{\pi\left(x_{n}\right)=y_{n}}$ with the property \begin{equation}\label{Equation Norm Is Equal To One}
\mathrm{\vert f_{n}\vert^{2}\left(x_{n}\right)=1=\underset{x\in\pi^{-1}\left(y_{n}\right)}{max}\,\left|f_{n}\right|^{2}}. \end{equation} As $\mathrm{x_{n}\in T\left(\frac{\epsilon}{2},\mathbf{W}\right)}$ where $\mathrm{y_{n}=\pi\left(x_{n}\right)\rightarrow y_{0}\in Int\,\mathbf{W}}$, we can assume using the compactness of $\mathrm{T\left(\frac{\epsilon}{2},\mathbf{W}\right)}$ (after having chosen a subsequence) that $\mathrm{x_{n}\rightarrow x_{0}\in }$ $\mathrm{Int\,T\left(\epsilon,\mathbf{W}\right)}$. Hence, there exists an open neighborhood $\mathrm{\mathbf{U}}$ of $\mathrm{x_{0}}$ which is contained in the interior of $\mathrm{T\left(\epsilon,\mathbf{W}\right)}$ so that $\mathrm{x_{n}\in \mathbf{U}}$ for all $\mathrm{n}$ big enough.
Since we have $\mathrm{x_{0} \in \pi^{-1}\left(y_{0}\right)\subset\pi^{-1}\left(\mathbf{Y}_{0}\right)}$, we can assume that the open neighborhood $\mathrm{\mathbf{U}}$ (after having shrunken it) is isomorphic via an holomorphic embedding $\mathrm{\Phi\!:\!\mathbf{U}\rightarrow \mathbb{C}^{\upkappa}\times \mathbb{C}^{k}}$ (let $\mathrm{\Phi\left(x_{0}\right)=0=\left(0,0\right)\in \mathbf{Q}_{0}\times \mathbf{Q}_{1}}$) to a closed analytic subset in a relatively compact product neighborhood $\mathrm{\mathbf{Q}=\mathbf{Q}_{0}\times \mathbf{Q}_{1}\subset}$ $\mathrm{\mathbb{C}^{\upkappa}\times \mathbb{C}^{k}}$ so that each $\mathrm{\Phi\left(\pi^{-1}\left(y\right)\cap \mathbf{U}\right)=\mathbf{Z}_{y}}$, where $\mathrm{y}$ varies in an open neighborhood $\mathrm{\mathbf{B}\subset \mathbf{Y}_{0}}$ of $\mathrm{\pi\left(x_{0}\right)}$, is a closed analytic subset which yields a surjective, $\mathrm{d}$-sheeted covering of $\mathrm{\mathbf{Q}_{1}}$ given by $\mathrm{p\vert \mathbf{Z}_{y}\rightarrow \mathbf{Q}_{1}}$. The existence of such a neighborhood $\mathrm{\mathbf{U}}$ has been treated in {\bf{Section \ref{A Local Proposition Concerning Fiber Integration}}}, page {\bf{\oldstylenums{\pageref{Notation D-Sheeted Covering}}}}.
After having shrunken $\mathrm{\mathbf{U}}$ again, we can also assume that $\mathrm{\mathbf{L}^{m_{0}}\vert \mathbf{U} \cong \mathbf{U}\times \mathbb{C}}$. In particular, the norm $\mathrm{\vert s_{m_{0}}^{i}\vert^{2}}$ of the holomorphic sections $\mathrm{s_{m_{0}}^{i}}$ over $\mathrm{\mathbf{U}}$ is then given by a smooth, strictly positive function $\mathrm{\alpha\in\mathcal{C}^{\infty}\left(\mathbf{U}\right)}$, independent of $\mathrm{n\in\mathbb{N}}$, so that $\mathrm{\vert f_{n} \vert^{2}=\alpha\cdot \vert \triangle^{i}_{y_{n},n}\vert^{2}}$. It is important to note that $\mathrm{\vert f_{n} \vert^{2}}$ denotes the norm of the restricted local section $\mathrm{f_{n}}$ with respect to the hermitian bundle metric $\mathrm{h}$, whereas $\mathrm{\vert \triangle^{i}_{y_{n},n}\vert^{2}}$ is the norm induced by the absolute value of the complex numbers. In the sequel, we will use the abbreviation $\mathrm{g_{n}\coloneqq \triangle^{i}_{y_{n},n}\in \mathcal{O}\left(\mathbf{Z}_{y_{n}}\right)}$, i.e.\ we have $\mathrm{\vert f_{n}\vert^{2}=\alpha\cdot \vert g_{n}\vert^{2}}$ on $\mathrm{{\bf{Z}}_{y_{n}}}$. Note that we have $$\mathrm{\underset{x\in \mathbf{Z}_{y_{n}}}{max}\vert f_{n}\vert \mathbf{Z}_{y_{n}}\vert^{2}=\vert f_{n}\left(x_{n}\right)\vert^{2}=1}$$ because $\mathrm{x_{n}\in \mathbf{U}}$ for all $\mathrm{n}$ big enough where $\mathrm{\Phi\left(\pi^{-1}\left(y_{n}\right)\cap \mathbf{U}\right)=\mathbf{Z}_{y_{n}}}$. In other words, if $\mathrm{A>0}$, resp.\ $\mathrm{a>0}$ denotes the maximum, resp.\ the minimum of $\mathrm{\alpha}$ on $\mathrm{\mathbf{U}}$ we deduce that \begin{equation}\label{Equation Bla bla bla bli blub} \mathrm{a^{-1}\geq \underset{x\in \mathbf{Z}_{y_{n}}}{max}\vert g_{n}\vert \mathbf{Z}_{y_{n}}\vert^{2}\text{ and } \vert g_{n}\vert^{2}\left(x_{n}\right)\geq A^{-1}>0} \end{equation} for all $\mathrm{n}$ big enough. Furthermore, by assumption \ref{Equation Integral Converges To Zero}, it follows that $$\mathrm{\int_{\mathbf{Z}_{n}}\vert f_{n}\vert^{2}d\,[\mathbf{Z}_{n}]\geq a\,\int_{\mathbf{Z}_{n}}\vert g_{n}\vert ^{2}d\,[\mathbf{Z}_{y_{n}}]\rightarrow 0\text{ and hence }\int_{\mathbf{Z}_{n}}\vert g_{n}\vert ^{2}d\,[\mathbf{Z}_{y_{n}}]\rightarrow 0.}$$
We are now in the situation of {\bf{Proposition \ref{Proposition Local Version Of Proposition}}}: We have a sequence $\mathrm{\mathbf{Z}_{n}=\mathbf{Z}_{y_{n}}}$ and a sequence $\mathrm{\left(g_{n}\right)_{n}}$ of uniformly bounded holomorphic functions on $\mathrm{\mathbf{Z}_{n}}$ given by $\mathrm{g_{n}}$ so that $$\mathrm{\int_{\mathbf{Z}_{n}}\vert g_{n}\vert^{2}\,d\,[\mathbf{Z}_{n}]\rightarrow 0.}$$ In particular, if $\mathrm{\mathbf{K}\subset \mathbf{Q}_{1}}$ is a compact neighborhood of $\mathrm{0\in \mathbf{K}}$ and if $\mathrm{\epsilon=\frac{1}{2}A^{-1}}$ we deduce, by {\bf{Proposition \ref{Proposition Local Version Of Proposition}}}, that there exists $\mathrm{N_{\frac{1}{2}A^{-1}}\left(\mathbf{K}\right)}$ so that \begin{equation}\label{Equation sd<vh dsb} \mathrm{\vert g_{n}\vert^{2}\leq \frac{1}{2}A^{-1}\text{ on }p^{-1}\left(\mathbf{K}\right)\cap \mathbf{Z}_{n}\text{ for all }n\geq N_{\frac{1}{2}A^{-1}}\left(\mathbf{K}\right).} \end{equation} However, since $\mathrm{x_{n}\rightarrow x_{0}=0\in p^{-1}\left(\mathbf{K}\right)}$, it follows that $\mathrm{x_{n}\in p^{-1}\left(\mathbf{K}\right)\cap \mathbf{Z}_{n}}$ for all $\mathrm{n}$ big enough. According to the second inequality of \ref{Equation Bla bla bla bli blub}, we have $\mathrm{\vert g_{n}\vert^{2}}$ $\mathrm{\left(x_{n} \right)\geq A^{-1}}$ and hence a contradiction to \ref{Equation sd<vh dsb}. Therefore, the assumption \ref{Equation Integral Converges To Zero} is false and the claim is proven. \end{proof}
\begin{theo_5.a*}\label{theo_5.a*}\textnormal{[\scshape{Locally Uniform Convergence of the Initial Distribution Sequence}]}
\\ Let $\mathrm{y\in \mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$, $\mathrm{t\in \mathbb{R}}$ and let $\mathrm{\mathbf{W}\subset \mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$ be a compact neighborhood\,\footnote{From now on, we will always assume that $\mathrm{\mathbf{W}\subset \mathbf{Y}^{i}}$ which is possible without restriction of generality.}of $\mathrm{y_{0}}$. Then after having shrunken $\mathrm{\mathbf{W}}$, the sequence $\mathrm{\left(D_{n}\left(\cdot,t\right)\right)_{n}}$ converges uniformly to the zero function over $\mathrm{\mathbf{W}}$. \end{theo_5.a*} \begin{proof} Let $\mathrm{\epsilon>0}$, then by the first part of {\bf{Proposition \ref {Continuity of Fiber Integral}}} there exists $\mathrm{\sigma_{\epsilon}>0}$ so that \begin{equation}\label{Equation 6} \mathrm{vol\left(\pi^{-1}\left(y\right)\cap T\left(\sigma_{\epsilon},\mathbf{W}\right)\right)\leq \epsilon} \end{equation} for all $\mathrm{y\in \mathbf{W}\cap \mathbf{Y}_{0}=\mathbf{W}}$. Hence, it is enough to show that \begin{equation}\label{Equation Distribution Functions Is Small Outside Local Case} \mathrm{\phi_{n}=\frac{\vert s_{n}\vert^{2}}{\Vert s_{n}\Vert^{2}}\leq t} \end{equation} on $\mathrm{T^{c}\left(\sigma_{\epsilon},\mathbf{W}\right)}$ for all $\mathrm{n}$ big enough. Recall that $\mathrm{\vert s_{n}\vert^{2}\Vert s_{n}\Vert^{-2}}$ is an abbreviation for the local description given by $\mathrm{\vert s_{y,n}\vert^{2}\Vert s_{y,n}\Vert^{-2}}$ on $\mathrm{\pi^{-1}\left(\mathbf{U}_{y}\right)}$.
The first step is to write $$\mathrm{\phi_{n}=\frac{\vert s_{n}^{i}\cdot s_{m}^{i,-1}\vert^{2}\cdot \vert s_{m}^{i}\cdot \triangle^{i}_{y,n}\vert^{2}}{\int_{\pi^{-1}\left(y\right)}\vert s_{n}^{i}\cdot s_{m}^{i,-1}\vert^{2}\cdot \vert s_{m}^{i}\cdot \triangle^{i}_{y,n}\vert^{2}\,d\,[\pi_{y}]}}$$ for arbitrary $\mathrm{m\in \mathbb{N}}$ which is possible because $\mathrm{s^{i}_{m}\left(x\right)\neq 0}$ for all $\mathrm{x\in \mathbf{X}^{i}}$ and all $\mathrm{m\in \mathbb{N}}$. Using {\bf{Lemma \ref{Lemma The Maximum Of Triangle Is Fiberwisely Not Zero}}} we deduce that \begin{equation}\label{Equation Intergal In The Denominator Will Not Be Zero}
\mathrm{\underset{x\in\pi^{-1}\left(y\right)\cap T\left(\epsilon_{\frac{\sigma}{2}},\mathbf{W}\right)}{max}\,\left|s_{m}^{i}\cdot\triangle_{y,n}^{i}\right|^{2}>0} \end{equation} for all $\mathrm{n}$ big enough and all $\mathrm{y\in \mathbf{W}}$ and hence we can estimate \begin{equation}\label{Equation 2} \mathrm{\phi_{n}\leq\frac{\vert s_{n}^{i}\cdot s_{m}^{i,-1}\vert^{2}\cdot \vert s_{m}^{i}\cdot \triangle^{i}_{y,n}\vert^{2}}{\int_{\pi^{-1}\left(y\right)\cap T\left(\frac{\sigma_{\epsilon}}{2},\mathbf{W}\right)}\vert s_{n}^{i}\cdot s_{m}^{i,-1}\vert^{2}\cdot \vert s_{m}^{i}\cdot \triangle^{i}_{y,n}\vert^{2}\,d\,[\pi_{y}]}}\end{equation} for all $\mathrm{n,m\in \mathbb{N}}$ and all $\mathrm{x\in\pi^{-1}\left( \mathbf{W}\right)}$.
Note that $\mathrm{\varrho^{m}_{n}\coloneqq -\frac{1}{n}\mathbf{log}\,\vert s^{i}_{n}\cdot s^{i,-1}_{m} \vert^{2}=\varrho^{i}_{n}-\frac{1}{n}\mathbf{log}\,\vert s^{i,-1}_{m}\vert^{2}}$ defines for a fixed $\mathrm{m}$ and for all $\mathrm{n\geq m }$ a strictly plurisubharmonic function on $\mathrm{\pi^{-1}\left(\mathbf{W}\right)}$ which converges uniformly on compact subsets to $\mathrm{\varrho^{i}}$. Therefore, for each $\mathrm{m}$, there exists $\mathrm{N_{m}\in \mathbb{N}}$ so that $\mathrm{\varrho^{m}_{n}\left(x\right)\leq \frac{5}{8}\sigma_{\epsilon}}$ for all $\mathrm{x\in T\left(\frac{\sigma_{\epsilon}}{2},\mathbf{W}\right)}$ and for all $\mathrm{n\geq N_{m}}$ or equivalently \begin{equation}\label{Equation 1} \mathrm{\vert s^{i}_{n}\cdot s^{i,-1}_{m}\vert^{2}\left(x\right)\geq e^{-n\,\frac{5}{8}\,\sigma_{\epsilon}}} \end{equation} for all $\mathrm{x\in T\left(\frac{\sigma_{\epsilon}}{2},\mathbf{W}\right)}$ and for all $\mathrm{n\geq N_{m}}$. Using inequality \ref{Equation 1} and \ref{Equation 2}, we deduce \begin{equation}\label{Equation 3} \mathrm{\phi\leq e^{n\,\frac{5}{8}\,\sigma_{\epsilon}}\,\frac{\vert s_{n}^{i}\cdot s_{m}^{i,-1}\vert^{2}\cdot \vert s_{m}^{i}\cdot \triangle^{i}_{y,n}\vert^{2}}{\int_{\pi^{-1}\left(y\right)\cap T\left(\frac{\sigma_{\epsilon}}{2},\mathbf{W}\right)}\vert s^{i}_{m}\cdot\triangle_{y,n}\vert^{2}\,d\,[\pi_{y}]}}\end{equation} for all $\mathrm{n\geq N_{m}}$ an all $\mathrm{x\in\pi^{-1}\left( \mathbf{W}\right)}$.
Note that by {\bf{Corollary \ref{Remark Convergence Theorem For The Shifted Sequence}}} we know that $\mathrm{\varrho^{m}_{n}\left(x\right)\geq \frac{7}{8}\, \sigma_{\epsilon}}$ or equivalently \begin{equation}\label{Equation 4} \mathrm{\vert s^{i}_{n}\cdot s^{i,-1}_{m}\vert^{2}\left(x\right)\leq e^{-\frac{7}{8}\,n\,\sigma_{\epsilon}}} \end{equation} for all $\mathrm{x\in T^{c}\left(\sigma_{\epsilon},\mathbf{W}\right)}$ and all $\mathrm{n\geq N_{m}^{\prime}}$. So if we combine \ref{Equation 4} and \ref{Equation 3} we deduce \begin{equation}\label{Equation 5}
\mathrm{\phi\leq e^{-n\,\frac{1}{4}\,\sigma_{\epsilon}}\,\frac{\vert s_{m}^{i}\cdot \triangle^{i}_{y,n}\vert^{2}}{\int_{\pi^{-1}\left(y\right)\cap T\left(\frac{\sigma_{\epsilon}}{2},\mathbf{W}\right)}\vert s^{i}_{m}\cdot\triangle^{i}_{y,n}\vert^{2}\,d\,[\pi_{y}]}}\end{equation} for all $\mathrm{x\in T^{c}\left(\sigma_{\epsilon},\mathbf{W}\right)}$ and all $\mathrm{n\geq max\,\left\{N^{\phantom{\prime}}_{m},\, N_{m}^{\prime}\right\}}$. After having shrunken $\mathrm{\mathbf{W}}$ there exists (cf.\ {\bf{Proposition \ref{Proposition Tales Are Uniformly Bounded}}}) $\mathrm{C>0}$ and $\mathrm{m_{0}\in \mathbb{N}}$ so that $$\mathrm{\underset{x\in\pi^{-1}\left(y\right)}{max}\,\,\,\frac{\left|s_{m_{0}}^{i}\cdot\triangle_{y,n}^{i}\right|^{2}}{\int_{\pi^{-1}\left(y\right)\cap T\left(\frac{\sigma_{\epsilon}}{2},\mathbf{W}\right)}\left|s_{m_{0}}^{i}\cdot\triangle_{y,n}^{i}\right|^{2}\,d\,[\pi_{y}]}<C}$$ for all $\mathrm{n}$ big enough and all $\mathrm{y\in \mathbf{W}}$. In particular combining this with inequality \ref{Equation 5} we deduce \begin{equation}\label{Equation Which Is Necessary} \mathrm{\phi_{n}\leq e^{-n\,\frac{1}{4}\,\sigma_{\epsilon}}\cdot C} \end{equation} for all $\mathrm{n}$ big enough (i.e.\ at least $\mathrm{n\geq max\,\left\{N^{\phantom{\prime}}_{m_{0}},\, N_{m_{0}}^{\prime}\right\}}$) and all $$\mathrm{x\in T^{c}\left(\sigma_{\epsilon},\mathbf{W}\right)\cap \pi^{-1}(\mathbf{W})=T^{c}(\sigma_{\epsilon},\mathbf{W}).}$$ In other words we have \begin{equation}\label{Equation 6} \mathrm{\phi_{n}\leq t} \end{equation} on $\mathrm{T^{c}(\sigma_{\epsilon},\mathbf{W})}$ for all $\mathrm{n}$ big enough which proves the claim because of equation \ref{Equation Distribution Functions Is Small Outside Local Case}. \end{proof}
As a direct consequence of the above theorem we deduce {\bf{Theorem 6.a}}. \begin{theo_6.a*}\label{theo_6.a*}\textnormal{[\scshape{Locally Uniform Convergence of the Initial Measure Sequence}]}
\\ Let $\mathrm{y_{0}\in \mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$, $\mathrm{t\in \mathbb{R}}$ and let $\mathrm{\mathbf{W}\subset \mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$ be a compact neighborhood of $\mathrm{y_{0}}$. Moreover, let $\mathrm{f\in \mathcal{C}^{0}\!\left(\mathbf{X}\right)}$. Then after having shrunken $\mathrm{ \mathbf{W}}$, the sequence $$\mathrm{\left(y\mapsto \int_{\pi^{-1}\left(y\right)}f\,d\bm{\nu}_{n}\left(y\right)\right)_{n}}$$ converges uniformly on $\mathrm{\mathbf{W}}$ to the reduced function $\mathrm{f_{red}}$. \end{theo_6.a*} \begin{proof} By the same argumentation as in the proof of {\bf{Proposition \ref{Proposition Uniform Convergence}}}, it is enough to prove the claim for all $\mathrm{f\in\mathcal{C}^{0}\!\left(\mathbf{X}\right)}$ which are $\mathrm{T}$-invariant. Hence, in the sequel let $\mathrm{f}$ be continuous, $\mathrm{T}$-invariant function on $\mathrm{\mathbf{X}}$ and let $\mathrm{\epsilon>0}$. Then by {\bf{Lemma \ref{Lemma Reduced Function}}} there exists $\mathrm{\sigma_{\epsilon}>0}$ so that \begin{equation}\label{Equation Integral Of F With Respect To}
\mathrm{\left|f\vert\left(\pi^{-1}\left(y\right)\cap T\left(\sigma_{\epsilon},\mathbf{W}\right)\right)-f_{red}\left(y\right)\right|\leq\frac{\epsilon}{2}} \end{equation} for all $\mathrm{y\in \mathbf{W}}$. Furthermore, note that we have \begin{equation*} \begin{split} &\mathrm{\int_{\pi^{-1}\left(y\right)}\left(f-\pi^{*}f_{red}\right)\,d\bm{\nu}_{n}\left(y\right)}\\[0,2 cm] =&\mathrm{\int_{\pi^{-1}\left(y\right)\cap T\left(\sigma_{\epsilon},\mathbf{W}\right)}\left(f-\pi^{*}f_{red}\right)\,\phi_{n}\,d\,[\pi_{y}]}\\[0,2 cm] +&\mathrm{\int_{\pi^{-1}\left(y\right)\cap T^{c}\left(\sigma_{\epsilon},\mathbf{W}\right)}\left(f-\pi^{*}f_{red}\right)\,\phi_{n}\,d\,[\pi_{y}].}\\ \end{split} \end{equation*} Since $\mathrm{f}$ is bounded as continuous function on a compact space, we find $\mathrm{\Gamma\coloneqq max\, \{ f-}$ $\mathrm{\pi^{*}f_{red}\}<\infty}$. Moreover, we can assume that $\mathrm{f-\pi^{*}f_{red}\not\equiv 0}$ and therefore $\mathrm{\Gamma^{-1}<\infty}$ is defined. Furthermore, using {\bf{Corollary \ref{Boundedness of Fiber Integral}}}, there exists a constant $\mathrm{C>0}$ so that $$\mathrm{\int_{\pi^{-1}\left(y\right)\cap T^{c}\left(\sigma_{\epsilon},\mathbf{W}\right)}\,d\,[\pi_{y}]\leq C}$$ for all $\mathrm{y\in \mathbf{W}\subset \mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$. According to the proof of {\bf{Theorem 5.a}}, we know by \ref{Equation Which Is Necessary} that, after having replaced $\mathrm{\mathbf{W}}$ by a smaller compact neighborhood, $$\mathrm{\phi_{n}\leq C^{-1}\cdot\Gamma^{-1}\cdot\frac{\epsilon}{4}\text{ on }T^{c}(\sigma_{\epsilon},\mathbf{W})\text{ for all }n\text{ big enough.}}$$ Hence, it follows that
\begin{equation*} \begin{split} &\mathrm{\left\vert\int_{\pi^{-1}\left(y\right)}\left(f-\pi^{*}f_{red}\right)\,d\bm{\nu}_{n}\left(y\right)\right\vert\leq \int_{\pi^{-1}\left(y\right)}\vert f-\pi^{*}f_{red}\vert\,d\bm{\nu}_{n}\left(y\right)}\\[0,2 cm] \leq&\mathrm{\int_{\pi^{-1}\left(y\right)\cap T\left(\sigma_{\epsilon},\mathbf{W}\right)}\left\vert f-\pi^{*}f_{red}\right\vert\,\phi_{n}\,d\,[\pi_{y}]+\frac{\epsilon}{4}}\\ \end{split}
\end{equation*} for all $\mathrm{n}$ big enough and all $\mathrm{y\in \mathbf{W}}$. Using inequality \ref{Equation Integral Of F With Respect To}, we deduce $$\mathrm{\left|\int_{\pi^{-1}\left(y\right)}f\, d\bm{\nu}_{n}\left(y\right)-f_{red}\left(y\right)\right|\leq\frac{\epsilon}{2}\int_{\pi^{-1}\left(y\right)\cap T\left(\sigma_{\epsilon},\mathbf{W}\right)}\phi\,d\,[\pi_{y}]+\frac{\epsilon}{4}}$$ for all $\mathrm{y\in \mathbf{W}}$ and all $\mathrm{n}$ big enough. Since $\mathrm{\int_{\pi^{-1}\left(y\right)\cap T\left(\sigma_{\epsilon},\mathbf{W}\right)}\phi_{n}\leq 1}$ for all $\mathrm{n}$ we find $$\mathrm{\left|\int_{\pi^{-1}\left(y\right)}f\, d\bm{\nu}_{n}\left(y\right)-f_{red}\left(y\right)\right|\leq\frac{3}{4}\,\epsilon}$$ for all $\mathrm{y\in\mathbf{W}}$ and all $\mathrm{n}$ big enough and therefore the theorem is proven. \end{proof}
\subsection{Global Uniform Convergence on $\mathrm{\mathbf{R}_{N_{0}}\cap\mathbf{Y}_{0}}$}
The strategy of the proof of the globally uniform convergence theorems on $\mathrm{\mathbf{R}_{N_{0}}\cap\mathbf{Y}_{0}}$ runs along the following lines: If $\mathrm{y_{0}\in cl\,\left(\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}\right)}$\footnote{Recall that we have assumed that $\mathrm{\mathbf{R}_{n_{0}}\neq \varnothing}$; in particular $\mathrm{\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$ is non-empty and euclidean open.}, then we find an open neighborhood $\mathrm{\mathbf{W}\subset \mathbf{Y}}$ of $\mathrm{y_{0}}$ so that we have uniform convergence over $\mathrm{\mathbf{W}\cap \mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$. Since $\mathrm{cl\left(\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}\right)}$ is compact, we can cover $\mathrm{\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$ by finitely many of such compact neighborhoods and the claim follows.
We begin this section by proving an extended version {\bf{Proposition \ref{Proposition Tales Are Uniformly Bounded}}}.
\begin{prop}\label{Proposition Tales Are Uniformly Bounded On Y} Let $\mathrm{\mathbf{W}\subset\mathbf{Y}}$ be a compact neighborhood of $\mathrm{y_{0}\in cl\,\left(\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}\right)}$ (recall: we have assumed that $\mathrm{\mathbf{W}\subset \mathbf{Y}^{i}}$) and $\mathrm{\epsilon>0}$, then after having shrunken $\mathrm{\mathbf{W}}$, there exists a constant $\mathrm{C>0}$ and $\mathrm{m_{0}\in\mathbb{N}}$ so that $$\mathrm{\underset{x\in\pi^{-1}\left(y\right)}{max}\,\,\,\frac{\left|s_{m_{0}}^{i}\cdot\triangle_{y,n}^{i}\right|^{2}}{\int_{\pi^{-1}\left(y\right)\cap T\left(\epsilon,\mathbf{W}\right)}\left|s_{m_{0}}^{i}\cdot\triangle_{y,n}^{i}\right|^{2}\,d\,[\pi_{y}]}<C}$$ for all $\mathrm{n}$ big enough and all $\mathrm{y\in \mathbf{W}\cap \mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$. \end{prop} \begin{proof} Let us assume that this is not the case, then as in the proof of {\bf{Proposition \ref{Proposition Tales Are Uniformly Bounded}}}, we can find a sequence $\mathrm{\left(y_{n}\right)_{n}}$ in $\mathrm{y_{n}\in \mathbf{Y}_{0}\cap {\mathbf{R}}_{N_{0}}}$ converging to $\mathrm{y_{0}\in cl\left(\mathbf{Y}_{0}\cap {\bf{R}}_{N_{0}}\right)}$ so that \begin{equation}\label{Equation Integral Converges To Zero Global Case}
\mathrm{\int_{\pi^{-1}\left(y_{n}\right)\cap T\left(\epsilon,\mathbf{W}\right)}\frac{\left|s_{m_{0}}^{i}\cdot\triangle_{y_{n},n}^{i}\right|^{2}}{\underset{x\in\pi^{-1}\left(y_{n}\right)\cap T\left(\epsilon,\mathbf{W}\right)}{max}\,\left|s_{m_{0}}^{i}\cdot\triangle_{y_{n},n}^{i}\right|^{2}}\,d\,[\pi_{y_{n}}]\rightarrow 0.} \end{equation} Using the same notation as in the proof of {\bf{Proposition \ref{Proposition Tales Are Uniformly Bounded}}}, we can write \begin{equation}\label{Equation Integral Converges To Zero Global Case} \mathrm{\int_{\pi^{-1}\left(y_{n}\right)\cap T\left(\epsilon,\mathbf{W}\right)}\vert f_{n}\vert^{2}\,d\,[\pi_{y_{n}}]\rightarrow 0.}
\end{equation} Furthermore, as as in the proof of {\bf{Proposition \ref{Proposition Tales Are Uniformly Bounded}}}, we can assume that $$\mathrm{\underset{x\in\pi^{-1}\left(y_{n}\right)}{max}\,\left|f_{n}\right|^{2}=\underset{x\in\pi^{-1}\left(y_{n}\right)\cap T\left(\frac{\epsilon}{2},\mathbf{W}\right)}{max}\,\left|f_{n}\right|^{2}=1}$$ for all $\mathrm{n}$ big enough where $\mathrm{y_{n}\rightarrow y_{0}\in Int\,\mathbf{W}}$. Moreover, as before we find a lift $\mathrm{\left(x_{n}\right)_{n}}$ of the sequence $\mathrm{\left(y_{n}\right)_{n}}$, i.e.\ a sequence $\mathrm{\left(x_{n}\right)_{n}}$ in $\mathrm{T\left(\frac{\epsilon}{2},\mathbf{W}\right)\cap \pi^{-1}\left(\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}\right)}$ so that $\mathrm{\pi\left(x_{n}\right)=y_{n}}$ where \begin{equation}\label{Equation Norm Is Equal To One}
\mathrm{\vert f_{n}\vert^{2}\left(x_{n}\right)=1=\underset{x\in\pi^{-1}\left(y_{n}\right)}{max}\,\left|f_{n}\right|^{2}=\underset{x\in\pi^{-1}\left(y_{n}\right)\cap T\left(\epsilon,\mathbf{W}\right)}{max}\,\left|f_{n}\right|^{2}}. \end{equation}
Recall that we have the following commutative diagram:
$$\begin{xy} \xymatrix{ \mathrm{\mathbf{X}^{ss}_{\xi}\subset \mathbf{X}}&&\mathrm{\widehat{\mathbf{\,X\,}}}\ar[d]_{\mathrm{\widehat{\pi}}}\ar[ll]^{\mathrm{p_{\mathbf{X}}\vert cl{\mathbf{\Gamma}}_{\pi}\circ \zeta}} &&& \mathrm{\widetilde{\bf{\,X\,}}}\ar[d]^{\mathrm{\Pi}}\ar[lll]_{\mathrm{\Sigma}}\\ &&\mathrm{\mathbf{Y}}&&& \mathrm{\widetilde{\bf{\,Y\,}}}\ar[lll]_{\mathrm{\sigma}}\\ } \end{xy} $$ Set $\mathrm{P\coloneqq p_{\mathbf{X}}\vert cl {\mathbf{\Gamma}}_{\pi}\circ\zeta\circ\Sigma}$ and define $\mathrm{\widetilde{\mathbf{L}}\coloneqq P^{*}\mathbf{L}}$. Since $\mathrm{\Sigma}$ is surjective and the above diagram commutes, we can find a lift $\mathrm{\left(\widetilde{x}_{n}\right)_{n}}$ in $\mathrm{\widetilde{\bf{\,X\,}}}$ of $\mathrm{\left(x_{n}\right)_{n}}$ under $\mathrm{P}$, i.e.\ we have $\mathrm{P\left(\widetilde{x}_{n}\right)=x_{n}}$. In the sequel, let $\mathrm{\left(\widetilde{y}_{n}\right)_{n}}$ be the sequence in $\mathrm{\widetilde{\bf{\,Y\,}}}$ given by $\mathrm{\widetilde{y}_{n}=\Pi\left(\widetilde{x}_{n}\right)}$ and note that $\mathrm{\sigma\left(\widetilde{y}_{n}\right)=y_{n}}$. Moreover, we have $\mathrm{\widetilde{x}_{n}\in \widetilde{\,T\,}\big(\frac{\epsilon}{2},\bf{W}\big) \subset \widetilde{\,T\,}\big(\epsilon,\bf{W}\big)}$ (recall:\ $\mathrm{\widetilde{\bf{W}}\coloneqq \sigma^{-1}\left(\mathbf{W}\right)}$) for all $\mathrm{n}$. Arguing as before, since $\mathrm{\widetilde{x}_{n}\in \widetilde{\,T\,}\big(\frac{\epsilon}{2},\bf{W}\big)}$ for all $\mathrm{n}$, we can assume that (after having chosen a subsequence) $$\mathrm{\widetilde{x}_{n}\rightarrow\widetilde{x}_{0}\in Int\,\widetilde{\,T\,}\big(\epsilon,\bf{W}\big).}$$ Hence, there exists an open neighborhood $\mathrm{\mathbf{U} \subset \widetilde{\,T\,}\big(\epsilon,\bf{W}\big)}$ of $\mathrm{\widetilde{x}_{0}}$ so that $\mathrm{\widetilde{x}_{n}\in \mathbf{U}}$ for all $\mathrm{n}$ big enough. In the sequel, let $\mathrm{\widetilde{y}_{0}\coloneqq \Pi\left(\widetilde{x}_{0}\right)\in \widetilde{\bf{\,Y\,}}}$.
We will now define $\mathrm{\widetilde{f}_{n}}$ on $\mathrm{\widetilde{\,T\,}\big(\epsilon,\bf{W}\big)\cap}$ $\mathrm{\Pi^{-1}\left(\widetilde{y}_{n}\right)}$ by the pull back of the restriction of $\mathrm{f_{n}}$ to $\mathrm{T\left(\epsilon,\mathbf{W}\right)\cap \pi^{-1}\left(y_{n}\right)}$. Note that we have $\mathrm{\vert \widetilde{f}_{n}?\vert^{2}\left(\widetilde{x}_{n}\right)=1}$ for all $\mathrm{n}$ and more precisely \begin{equation}\label{Equation 12} \mathrm{\underset{x\in\widetilde{\,T\,}(\epsilon,{\bf{W}})\cap\Pi^{-1}\left(\widetilde{y}_{n}\right)}{max}\vert \widetilde{f}_{n}\vert^{2}=\vert \widetilde{f}_{n}\vert^{2}\left(x_{n}\right)=1} \end{equation} which is a direct consequence of \ref{Equation Norm Is Equal To One}.
Since $\mathrm{\Pi\!:\!\widetilde{\bf{\,X\,}}\rightarrow \widetilde{\bf{\,Y\,}}}$ is a $\mathrm{k}$-fibering, we can proceed as in the proof of {\bf{Proposition \ref{Proposition Tales Are Uniformly Bounded}}}: After having shrunken $\mathrm{\mathbf{U}}$ we can assume that the open neighborhood $\mathrm{\mathbf{U}}$ is isomorphic to a closed analytic subset in a relatively compact product neighborhood $\mathrm{\mathbf{Q}=\mathbf{Q}_{0}\times \mathbf{Q}_{1}\subset \mathbb{C}^{\upkappa}\times}$ $\mathrm{\mathbb{C}^{k}}$, i.e.\ there exists an isomorphism $\mathrm{\Phi\!:\!\mathbf{U}\rightarrow \mathbb{C}^{\upkappa}\times \mathbb{C}^{k}}$ (let $\mathrm{\Phi\left(\widetilde{x}_{0}\right)=0}$) so that for each $\mathrm{\widetilde{y}}$ in an open neighborhood $\mathrm{\mathbf{B}\subset \widetilde{\bf{\,Y\,}}}$ of $\mathrm{\Pi\left(\widetilde{x}_{0}\right)=\widetilde{y}_{0}}$ each $\mathrm{\Phi\left(\Pi^{-1}(\widetilde{y}\right)}$ $\mathrm{\cap \mathbf{U})=\mathbf{Z}_{\widetilde{y}}}$ is a closed analytic subset which yields a $\mathrm{d}$-sheeted covering onto $\mathrm{\mathbf{Q}_{1}}$ given by the restriction $\mathrm{p\vert \mathbf{Z}_{\widetilde{y}_{n}}\rightarrow \mathbf{Q}_{1}}$. We can now finish the proof exactly like the proof of {\bf{Proposition \ref{Proposition Tales Are Uniformly Bounded}}}. First of all after having shrunken $\mathrm{\mathbf{U}}$ we can assume that $\mathrm{\widetilde{f}_{n}}$ is represented by $\mathrm{\widetilde{f}_{n}=\widetilde{\alpha} \cdot \widetilde{g}_{n}}$ where $\mathrm{\widetilde{\alpha}\in \mathcal{C}^{\infty}\left(\mathbf{U}\right)}$ and $\mathrm{\widetilde{g}_{n}\in \mathcal{O}\left(\mathbf{Z}_{y_{n}}\right)}$. Furthermore, we also have \begin{equation}\label{Equation Bla bla bla bli blub asf} \mathrm{a^{-1}\geq \underset{x\in \mathbf{Z}_{\widetilde{y}_{n}}}{max}\vert \widetilde{g}_{n}\vert \mathbf{Z}_{\widetilde{y}_{n}}\vert^{2}\text{ and }\vert \widetilde{g}_{n}\vert^{2}\left(\widetilde{x}_{n}\right)\geq A^{-1}>0} \end{equation} for all $\mathrm{n}$ big enough, where $\mathrm{A>0}$, resp.\ $\mathrm{a>0}$ denotes the maximum, resp.\ the minimum of $\mathrm{\widetilde{\alpha}}$ on $\mathrm{\mathbf{U}}$. We can now apply {\bf{Proposition \ref{Proposition Local Version Of Proposition}}} to the sequence $\mathrm{\widetilde{g}_{n}}$: Each $\mathrm{\widetilde{g}_{n}}$ is a holomorphic function on $\mathrm{\widetilde{\mathbf{Z}}_{n}\coloneqq\mathbf{Z}_{\widetilde{y}_{n}}}$. Furthermore, this sequence is uniformly bounded by $\mathrm{a^{-1}}$ and by assumption \ref{Equation Integral Converges To Zero Global Case}, we have $$\mathrm{\int_{\widetilde{\mathbf{Z}}_{n}}\vert \widetilde{g}_{n}\vert^{2}\,d\,[\widetilde{\mathbf{Z}}_{n}]\rightarrow 0.}$$ Using {\bf{Proposition \ref{Proposition Local Version Of Proposition}}}, it follows that for a compact subset $\mathbf{K}\subset \mathbf{Q}_{1}$ and for $\mathrm{\epsilon=\frac{1}{2}A^{-1}}$ we have \begin{equation}\label{Equation wlisac?kns} \mathrm{\vert \widetilde{g}_{n}\vert^{2}\leq \frac{1}{2}A^{-1}\text{ on }p^{-1}\left(\mathbf{K}\right)\cap \widetilde{\mathbf{Z}}_{n}\text{ for all }n\geq N_{\frac{1}{2}A^{-1}}\left(\mathbf{K}\right).} \end{equation} However, as $\mathrm{\widetilde{x}_{n}\rightarrow x_{0}=0\in p^{-1}\left(\mathbf{K}\right)}$ we know that $\mathrm{\widetilde{x}_{n}\in p^{-1}\left(\mathbf{K}\right)\cap \widetilde{\mathbf{Z}}_{n}}$ for all $\mathrm{n}$ big enough. By the second inequality of \ref{Equation Bla bla bla bli blub asf}, we know that $\mathrm{\vert g_{n}\vert^{2}}$ $\mathrm{\left(x_{n} \right)\geq A^{-1}}$ which yields a contradiction to \ref{Equation wlisac?kns}. Consequently, the assumption \ref{Equation Integral Converges To Zero Global Case} is false and the claim of the proposition holds. \end{proof}
As a consequence of {\bf{Proposition \ref{Proposition Tales Are Uniformly Bounded On Y}}} we deduce a generalization of {\bf{Theorem 5.a}}. \begin{theo_5.b*}\label{theo_5.b*}\textnormal{[\scshape{Uniform Convergence of the Initial Distribution Sequence}]}
\\ For fixed $\mathrm{t\in \mathbb{R}}$ the sequence $\mathrm{\left(D_{n}\left(\cdot,t\right)\right)_{n}}$ converges uniformly on $\mathrm{\mathbf{Y}_{0}\cap\mathbf{R}_{N_{0}}}$ to the zero function. \end{theo_5.b*} \begin{proof} Note that by the compactness of $\mathrm{cl\left(\mathbf{Y}_{0}\cap\mathbf{R}_{N_{0}}\right)}$ the claim follows as soon as we have shown that for each $\mathrm{y_{0}\in cl\left(\mathbf{Y}_{0}\cap\mathbf{R}_{N_{0}}\right)}$, there exists a compact neighborhood $\mathrm{\mathbf{W}\subset \mathbf{Y}}$ of $\mathrm{y_{0}}$ (after having shrunken $\mathrm{\mathbf{W}}$ we can assume that $\mathrm{\mathbf{W}\subset \mathbf{Y}^{i}}$) so that $\mathrm{D_{n}\left(\cdot,t\right)}$ converges uniformly on $\mathrm{\mathbf{W}\cap \mathbf{Y}_{0}\cap\mathbf{R}_{N_{0}}}$ to the zero function. The proof of the latter claim is similar to the proof of {\bf{Theorem 5.a}}: First of all fix $\mathrm{y_{0}\in cl\left(\mathbf{Y}_{0}\cap\mathbf{R}_{N_{0}}\right)}$ and $\mathrm{\epsilon>0}$. As in the proof of {\bf{Theorem 5.a}}, we apply {\bf{Proposition \ref {Continuity of Fiber Integral}}} in order to find $\mathrm{\sigma_{\epsilon}>0}$ so that \begin{equation}\label{Equation 6} \mathrm{vol\left(\pi^{-1}\left(y\right)\cap T\left(\sigma_{\epsilon},\mathbf{W}\right)\right)\leq \epsilon} \end{equation} for all $\mathrm{y\in \mathbf{W}\cap \mathbf{Y}_{0}}$. Hence, it is enough to show that after having shrunken $\mathrm{\mathbf{W}}$ we have \begin{equation}\label{Equation Distribution Functions Is Small Outside Global Case} \mathrm{\phi_{n}=\frac{\vert s_{n}\vert^{2}}{\Vert s_{n}\Vert^{2}}\leq t} \end{equation} on $\mathrm{T^{c}\big(\sigma_{\epsilon},\mathbf{W}\big)\cap \pi^{-1}\left(\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}\right)}$ for all $\mathrm{n}$ big enough. Note that we can now perform the same steps as in the proof {\bf{Theorem 5.a}} over the set $\mathrm{\mathbf{W}\cap \mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$.
First of all, we write $$\mathrm{\phi_{n}=\frac{\vert s_{n}^{i}\cdot s_{m}^{i,-1}\vert^{2}\cdot \vert s_{m}^{i}\cdot \triangle^{i}_{y,n}\vert^{2}}{\int_{\pi^{-1}\left(y\right)}\vert s_{n}^{i}\cdot s_{m}^{i,-1}\vert^{2}\cdot \vert s_{m}^{i}\cdot \triangle^{i}_{y,n}\vert^{2}\,d\,[\pi_{y}]}}$$ where $\mathrm{m\in \mathbb{N}}$ is arbitrary and by {\bf{Lemma \ref{Lemma The Maximum Of Triangle Is Fiberwisely Not Zero}}}, we know that \begin{equation}\label{Equation Intergal In The Denominator Will Not Be Zero Global Case}
\mathrm{\underset{x\in\pi^{-1}\left(y\right)\cap T\left({\frac{\sigma_{\epsilon}}{2}},\mathbf{W}\right)}{max}\,\left|s_{m}^{i}\cdot\triangle_{y,n}^{i}\right|^{2}>0} \end{equation} for all $\mathrm{n}$ big enough and all $\mathrm{y\in \mathbf{W}\cap \mathbf{R}_{N_{0}}}$. Therefore we deduce the estimate \begin{equation}\label{Equation 2 Global Case} \mathrm{\phi_{n}\leq\frac{\vert s_{n}^{i}\cdot s_{m}^{i,-1}\vert^{2}\cdot \vert s_{m}^{i}\cdot \triangle^{i}_{y,n}\vert^{2}}{\int_{\pi^{-1}\left(y\right)\cap T\left(\frac{\sigma_{\epsilon}}{2},\mathbf{W}\right)}\vert s_{n}^{i}\cdot s_{m}^{i,-1}\vert^{2}\cdot \vert s_{m}^{i}\cdot \triangle^{i}_{y,n}\vert^{2}\,d\,[\pi_{y}]}}\end{equation} for all $\mathrm{n,m\in \mathbb{N}}$ and all $\mathrm{x\in\pi^{-1}\left( \mathbf{W\cap \mathbf{R}_{N_{0}}}\cap \mathbf{Y}_{0}\right)}$. Note that the right hand side is well defined because the denominator of the term on the right hand side of the above inequality is not zero according to equation \ref{Equation Intergal In The Denominator Will Not Be Zero Global Case}.
As before we know that $\mathrm{\varrho^{m}_{n}\coloneqq -\frac{1}{n}\mathbf{log}\,\vert s^{i}_{n}\cdot s^{i,-1}_{m} \vert^{2}=\varrho^{i}_{n}-\frac{1}{n}\mathbf{log}\,\vert s^{i,-1}_{m}\vert^{2}}$ is a strictly plurisubharmonic function on $\mathrm{\pi^{-1}\left(\mathbf{W}\right)}$ for $\mathrm{m}$ fixed and for all $\mathrm{n\geq m }$ which converges uniformly on compact subsets to $\mathrm{\varrho^{i}}$. In particular we deduce that $\mathrm{\varrho^{m}_{n}\left(x\right)\leq \frac{5}{8}\sigma_{\epsilon}}$ for all $\mathrm{x\in T\left(\frac{\sigma_{\epsilon}}{2},\mathbf{W}\right)}$ and all $\mathrm{n\geq N_{m}\in \mathbb{N}}$ big enough where $\mathrm{m}$ is fixed. Combined this with inequality \ref{Equation 2 Global Case}, it follows that \begin{equation}\label{Equation 3 Global Case} \mathrm{\phi_{n}\leq e^{n\,\frac{5}{8}\,\sigma_{\epsilon}}\,\frac{\vert s_{n}^{i}\cdot s_{m}^{i,-1}\vert^{2}\cdot \vert s_{m}^{i}\cdot \triangle^{i}_{y,n}\vert^{2}}{\int_{\pi^{-1}\left(y\right)\cap T\left(\frac{\sigma_{\epsilon}}{2},\mathbf{W}\right)}\vert s^{i}_{m}\cdot\triangle_{y,n}\vert^{2}\,d\,[\pi_{y}]}}\end{equation} on $\mathrm{\pi^{-1}\left(\mathbf{R}_{N_{0}}\cap \mathbf{Y}_{0}\cap \mathbf{W}\right)}$ for all $\mathrm{n\geq N_{m}}$.
On the other hand, using {\bf{Corollary \ref{Remark Convergence Theorem For The Shifted Sequence}}}, we have $\mathrm{\varrho^{m}_{n}\left(x\right)\geq \frac{7}{8}\, \sigma_{\epsilon}}$ or equivalently \begin{equation}\label{Equation 4 Global Case} \mathrm{\vert s^{i}_{n}\cdot s^{i,-1}_{m}\vert^{2}\left(x\right)\leq e^{-\frac{7}{8}\,n\,\sigma_{\epsilon}}} \end{equation} for all $\mathrm{x\in T^{c}\left(\sigma_{\epsilon},\mathbf{W}\right)}$ and all $\mathrm{n\geq N_{m}^{\prime}}$. Combining \ref{Equation 4 Global Case} and \ref{Equation 3 Global Case} we deduce \begin{equation}\label{Equation 5 Global Case}
\mathrm{\phi_{n}\leq e^{-n\,\frac{1}{4}\,\sigma_{\epsilon}}\,\frac{\vert s_{m}^{i}\cdot \triangle^{i}_{y,n}\vert^{2}}{\int_{\pi^{-1}\left(y\right)\cap T\left(\frac{\sigma_{\epsilon}}{2},\mathbf{W}\right)}\vert s^{i}_{m}\cdot\triangle^{i}_{y,n}\vert^{2}\,d\,[\pi_{y}]}}\end{equation} for all $\mathrm{x\in T^{c}\left(\sigma_{\epsilon},\mathbf{W}\right)\cap \pi^{-1}\left(\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}\right)}$ and all $\mathrm{n\geq max\,\left\{N_{m},\, N_{m}^{\prime}\right\}}$. Since $\mathrm{y_{0}\in cl\,(\mathbf{Y}_{0}\cap }$ $\mathrm{\mathbf{R}_{N_{0}})}$, we can apply {\bf{Proposition \ref{Proposition Tales Are Uniformly Bounded On Y}}} in order to deduce, after having shrunken $\mathrm{\mathbf{W}}$, the existence of $\mathrm{C>0}$ and $\mathrm{m_{0}\in \mathbb{N}}$ so that $$\mathrm{\underset{x\in\pi^{-1}\left(y\right)}{max}\,\,\,\frac{\left|s_{m_{0}}^{i}\cdot\triangle_{y,n}^{i}\right|^{2}}{\int_{\pi^{-1}\left(y\right)\cap T\left(\frac{\sigma_{\epsilon}}{2},\mathbf{W}\right)}\left|s_{m_{0}}^{i}\cdot\triangle_{y,n}^{i}\right|^{2}\,d\,[\pi_{y}]}<C}$$ for all $\mathrm{n}$ big enough and all $\mathrm{y\in \mathbf{W}\cap \mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$. In combination with inequality \ref{Equation 5 Global Case}, it then follows \begin{equation}\label{Equation Which Is Necessary Global Case} \mathrm{\phi_{n}\leq e^{-n\,\frac{1}{4}\,\sigma_{\epsilon}}\cdot C} \end{equation} for all $\mathrm{n}$ big enough (i.e.\ at least $\mathrm{n\geq max\,\left\{N_{m},\, N_{m}^{\prime}\right\}}$) and all $$\mathrm{x\in T^{c}\left(\sigma_{\epsilon},\mathbf{W}\right)\cap \pi^{-1}(\mathbf{W}\cap \mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}})=T^{c}(\sigma_{\epsilon},\mathbf{W})\cap \pi^{-1}\left(\mathbf{Y}_{0}\cap\mathbf{R}_{N_{0}}\right).}$$ To sum up we have found that \begin{equation}\label{Equation 6} \mathrm{\phi_{n}\leq t} \end{equation} on $\mathrm{T^{c}(\sigma_{\epsilon},\mathbf{W})\cap \pi^{-1}\left(\mathbf{Y}_{0}\cap\mathbf{R}_{N_{0}}\right)}$ for all $\mathrm{n}$ big enough which proves the claim because of equation \ref{Equation Distribution Functions Is Small Outside Global Case}. \end{proof}
As a direct consequence of the proof of {\bf{Theorem 5.b}} we can prove {\bf{Theorem 6.b}}.
\begin{theo_6.b*}\label{theo_6.b*}\textnormal{[\scshape{Uniform Convergence of the Initial Measure Sequence}]}
\\ Let $\mathrm{f\in \mathcal{C}^{0}\!\left(\mathbf{X}\right)}$ then sequence $$\mathrm{\left(y\mapsto \int_{\pi^{-1}\left(y\right)}f\,d\bm{\nu}_{n}\left(y\right)\right)_{n}}$$ converges uniformly over $\mathrm{\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$ to the reduced function $\mathrm{f_{red}}$. \end{theo_6.b*} \begin{proof} As before (cf.\ proof of {\bf{Theorem 6.a}}) it is enough to consider the case where $\mathrm{f\in\mathcal{C}^{0}\!\left(\mathbf{X}\right)}$ is $\mathrm{T}$-invariant.
Let $\mathrm{\epsilon>0}$. Again, by the compactness of $\mathrm{cl\left(\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}\right)}$, the proof of the claim reduces to the following statement: If $\mathrm{y_{0}\in cl\left(\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}\right)}$, then there exists a compact neighborhood $\mathrm{\mathbf{W}}$ so that the claim is true over the set $\mathrm{\mathbf{W}\cap \mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$. In order to show this, we proceed similar as in the proof of {\bf{Theorem 6.a}}: First of all, by {\bf{Lemma \ref{Lemma Reduced Function}}} there exists $\mathrm{\sigma_{\epsilon}>0}$ so that \begin{equation}\label{Equation Integral Of F With Respect To Global Version}
\mathrm{\left|f\vert\left(\pi^{-1}\left(y\right)\cap T\left(\sigma_{\epsilon},\mathbf{W}\right)\right)-f_{red}\left(y\right)\right|\leq\frac{\epsilon}{2}} \end{equation} for all $\mathrm{y\in \mathbf{W}}$. Furthermore, as before we have the decomposition \begin{equation*} \begin{split} &\mathrm{\int_{\pi^{-1}\left(y\right)}\left(f-\pi^{*}f_{red}\right)\,d\bm{\nu}_{n}\left(y\right)}\\[0,2 cm] =&\mathrm{\int_{\pi^{-1}\left(y\right)\cap T\left(\sigma_{\epsilon},\mathbf{W}\right)}\left(f-\pi^{*}f_{red}\right)\,\phi_{n}\,d\,[\pi_{y}]}\\[0,2 cm] +&\mathrm{\int_{\pi^{-1}\left(y\right)\cap T^{c}\left(\sigma_{\epsilon},\mathbf{W}\right)}\left(f-\pi^{*}f_{red}\right)\,\phi_{n}\,d\,[\pi_{y}].}\\ \end{split} \end{equation*} Again we have $\mathrm{\Gamma\coloneqq max\,\left\{ f-\pi^{*}f_{red}\right\}<\infty}$ because $\mathrm{f}$ is continuous and without restriction of generality we can assume that $\mathrm{f-\pi^{*}f_{red}\not\equiv 0}$ so that $\mathrm{\Gamma^{-1}<\infty}$ exists. Moreover, as in the proof of {\bf{Theorem 5.b}}, using {\bf{Corollary \ref{Boundedness of Fiber Integral}}}, there exists $\mathrm{C>0}$ so that $$\mathrm{\int_{\pi^{-1}\left(y\right)\cap T^{c}\left(\sigma_{\epsilon},\mathbf{W}\right)}\,d\,[\pi_{y}]\leq C}$$ for all $\mathrm{y\in \mathbf{W}\cap \mathbf{Y}_{0}}$. By inequality \ref{Equation Which Is Necessary Global Case} in the proof of {\bf{Theorem 5.b}} we know, after having shrunken $\mathrm{\mathbf{W}}$, that $\mathrm{\phi_{n}\leq C^{-1}\cdot\Gamma^{-1}\cdot\frac{\epsilon}{4}\text{ on } T^{c}\big(\sigma_{\epsilon},\mathbf{W}\big)\cap\pi^{-1}(\mathbf{Y}_{0}\cap\mathbf{R}_{N_{0}})}$ for all $\mathrm{n\in \mathbb{N}}$ big enough and hence
\begin{equation*} \begin{split} &\mathrm{\left\vert\int_{\pi^{-1}\left(y\right)}\left(f-\pi^{*}f_{red}\right)\,d\bm{\nu}_{n}\left(y\right)\right\vert\leq \int_{\pi^{-1}\left(y\right)}\vert f-\pi^{*}f_{red}\vert\,d\bm{\nu}_{n}\left(y\right)}\\[0,2 cm] \leq&\mathrm{\int_{\pi^{-1}\left(y\right)\cap T\left(\sigma_{\epsilon},\mathbf{W}\right)}\left\vert f-\pi^{*}f_{red}\right\vert\,\phi_{n}\,d\,[\pi_{y}]+\frac{\epsilon}{4}}\\ \end{split}
\end{equation*} for all $\mathrm{n}$ big enough and all $\mathrm{y\in \mathbf{W}\cap \mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$. By \ref{Equation Integral Of F With Respect To Global Version} we have $$\mathrm{\left|\int_{\pi^{-1}\left(y\right)}f\, d\bm{\nu}_{n}\left(y\right)-f_{red}\left(y\right)\right|\leq\frac{\epsilon}{2}\int_{\pi^{-1}\left(y\right)\cap T\left(\sigma_{\epsilon},\mathbf{W}\right)}\phi\,d\,[\pi_{y}]+\frac{\epsilon}{4}}$$ for all $\mathrm{y\in \mathbf{W}\cap \mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$ and all $\mathrm{n}$ big enough. As $\mathrm{\int_{\pi^{-1}\left(y\right)\cap T\left(\sigma_{\epsilon},\mathbf{W}\right)}\phi_{n}\leq 1}$ for all $\mathrm{n}$ we deduce $$\mathrm{\left|\int_{\pi^{-1}\left(y\right)}f\, d\bm{\nu}_{n}\left(y\right)-f_{red}\left(y\right)\right|\leq\frac{3}{4}\,\epsilon}$$ for all $\mathrm{y\in \mathbf{W}\cap \mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$ and all $\mathrm{n}$ big enough as claimed. So with $\mathrm{\mathbf{W}\subset \mathbf{Y}}$ we have found a compact neighborhood of $\mathrm{y_{0}\in cl\left(\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}\right)}$ so that the claim is true over $\mathrm{\mathbf{W}\cap \mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}}$ and hence the claim follows by the introducing statement of this proof. \end{proof}
We close this section by giving an alternative formulation of our results in the language of operators. For this let $\mathrm{f\in \mathcal{C}^{0}\!\left(\mathbf{X}\right)}$ and consider the continuous, $\mathrm{T}$-invariant function $\mathrm{\overline{f}}$ on $\mathrm{\mathbf{X}}$ defined by $$\mathrm{\overline{f}\left(x\right)\coloneqq \int_{T}f\left(t.x\right)d\nu_{T}}$$ where $\mathrm{\nu_{T}}$ denotes the {\scshape{Haar}} measure on $\mathrm{T}$. Furthermore, we introduce $$\mathrm{\Phi_{n}\left(f\right)\left(y\right)\coloneqq \int_{\pi^{-1}\left(y\right)}\overline{f}\,d\bm{\nu}_{n}\left(y\right)\text{for all }f\in \mathcal{C}^{0}\!\left(\mathbf{X}\right)\text{ and all }y\in \mathbf{R}_{N_{0}}\cap \mathbf{Y}_{0}.\label{Notation Phi_n}}$$ If $\mathrm{\mathcal{C}^{0}_{bd}\!\left(\mathbf{R}_{N_{0}}\cap \mathbf{Y}_{0}\right)}$ denotes the space of all bounded continuous functions on $\mathrm{\mathbf{R}_{N_{0}}\cap \mathbf{Y}_{0}}$ then we have the following lemma.
\begin{lemma} Each $\mathrm{\Phi_{n}}$ (for $\mathrm{n}$ big enough) defines an operator from $\mathrm{\mathcal{C}^{0}\left(\mathbf{X}\right)}$ to $\mathrm{\mathcal{C}^{0}_{bd}(\mathbf{R}_{N_{0}}}$ $\mathrm{\cap \mathbf{Y}_{0})}$ for all $\mathrm{n}$ big enough. \end{lemma} \begin{proof} Let $\mathrm{f\in\mathcal{C}^{0}\!\left(\mathbf{X}\right)}$, $\mathrm{K\coloneqq max\, \left\{\vert f\vert\right\}=max\, \left\{\vert \overline{f}\vert\right\}}$ and $\mathrm{y\in \mathbf{R}_{N_{0}}\cap \mathbf{Y}_{0}}$. Choose a compact neighborhood $\mathrm{\mathbf{W}\subset \mathbf{R}_{N_{0}}\cap \mathbf{Y}_{0}}$ of $\mathrm{y}$ and let $\mathrm{\epsilon_{m}\in \mathbb{R}^{\geq 0}}$ be a strictly increasing sequence converging to $\mathrm{\infty}$. Furthermore, choose a sequence $\mathrm{\left(\psi_{m}\right)_{m}}$ of smooth, $\mathrm{T}$-invariant cut-off functions defined on $\mathrm{\mathbf{X}}$ so that $$\mathrm{\psi_{m}\vert T\left(\epsilon_{m},\mathbf{W}\right)\equiv 1,\text{ } supp\,\psi_{m}\cap T^{c}\left(\epsilon_{m+1},\mathbf{W}\right)=\varnothing\text{ and } supp\,\psi_{m}\subset \pi^{-1}\left(\mathbf{Y}_{0}\cap \mathbf{R}_{N_{0}}\right).}$$ Since $\mathrm{\pi\vert \mathbf{X}_{0}\!:\!\mathbf{X}_{0}\rightarrow \mathbf{Y}_{0}}$ is a $\mathrm{k}$-fibering, it follows by {\bf{Theorem \ref{Theorem King}}} that $\mathrm{\Phi_{n}\left(\psi_{m}\,f\right)\in \mathcal{C}^{0}(\mathbf{W})}$ for all $\mathrm{m\in \mathbb{N}}$. We calculate \begin{equation*} \begin{split}
\mathrm{\vert \Phi_{n}\left(\psi_{m}\,f\right)-\Phi_{n}\left(f\right)\vert\left(y\right)}=&\mathrm{\left|\int_{\pi^{-1}\left(y\right)}\left(\psi_{m}\,\overline{f}-\overline{f}\right)\,d\bm{\nu}_{n}\left(y\right)\right|}\\[0,2 cm]
\leq&\mathrm{2\,K\int_{\pi^{-1}\left(y\right)\cap T^{c}\left(\epsilon_{m},\mathbf{W}\right)}\left|\psi_{m}-1\right|\,\phi_{n}\,d\,[\pi_{y}]}\\[0,2 cm] \leq&\mathrm{2\,K\int_{\pi^{-1}\left(y\right)\cap T^{c}\left(\epsilon_{m},\mathbf{W}\right)}\phi_{n}\,d\,[\pi_{y}].}\\ \end{split} \end{equation*} Using equation \ref{Equation Which Is Necessary} (for $\mathrm{\sigma_{\epsilon}=\sigma_{m}}$) in the proof of {\bf{Theorem 5.a}}, we deduce that $$\mathrm{\vert \Phi_{n}\left(\psi_{m}\,f\right)-\Phi_{n}\left(f\right)\vert \leq 2\cdot e^{-n\,\frac{1}{4}\,\epsilon_{m}}\cdot C\cdot K}$$ for all $\mathrm{n}$ big enough and all $\mathrm{y\in \mathbf{W}}$. Hence, for a fixed $\mathrm{n\in \mathbb{N}}$ big enough, the sequence of continuous functions on $\mathrm{\mathbf{W}}$ given by $\mathrm{\left(\Phi_{n}\left(\psi_{m}\,f\right)\vert \mathbf{W}\right)_{m}}$ converges uniformly to $\mathrm{\Phi_{n}\left(f\right)\vert \mathbf{W}}$ as $\mathrm{m\rightarrow \infty}$ and consequently $\mathrm{\Phi_{n}\left(f\right)\vert \mathbf{W}\in \mathcal{C}^{0}\!\left(\mathbf{W}\right)}$. Since $\mathrm{y\in \mathbf{R}_{N_{0}}\cap \mathbf{Y}_{0}}$ was chosen arbitrarily, it follows that $\mathrm{\Phi_{n}\left(f\right)\in \mathcal{C}^{0}\!\left(\mathbf{R}_{N_{0}}\cap \mathbf{Y}_{0}\right)}$ for all $\mathrm{f\in \mathcal{C}^{0}\!\left(\mathbf{X}\right)}$ and all $?\mathrm{n\in \mathbb{N}}$ big enough as claimed. \end{proof}
\begin{remark} Even if $\mathrm{\complement\left(\mathbf{R}_{N_{0}}\cap \mathbf{Y}_{0}\right)}$ is a proper analytic subset of $\mathrm{\mathbf{Y}}$, it is in general not possible to extend $\mathrm{\Phi_{n}\!:\!\mathcal{C}^{0}\!\left(\mathbf{X}\right)\rightarrow \mathcal{C}^{0}\!\left(\mathbf{R}_{N_{0}}\cap \mathbf{Y}\right)}$ to an operator $\mathrm{\widehat{\Phi}_{n}\!:\!\mathcal{C}^{0}\!\left(\mathbf{X}\right)\rightarrow }$ $\mathrm{\mathcal{C}^{0}\!\left(\mathbf{Y}\right)}$: Using {\bf{Example \ref{Example Existence Of Removable Singularities}}}, one can find a continuous function $\mathrm{f\in\mathcal{C}^{0}\!\left(\mathbf{X}\right)}$ so that $\mathrm{\Phi_{n}\left(f\right)}$ has no continuous extension in $\mathrm{[0\!:\!{\dots}\!:\!0\!:\!1]\in \mathbf{Y}=\mathbb{C}\mathbb{P}^{k}}$. \end{remark}
If $\mathrm{\mathfrak{Red}\!:\!\mathcal{C}^{0}\!\left(\mathbf{X}\right)\rightarrow \mathcal{C}^{0}_{bd}\!\left(\mathbf{R}_{N_{0}}\cap \mathbf{Y}_{0}\right)}$ denotes the operator given by $\mathrm{\mathfrak{Red}\left(f\right)\coloneqq \overline{f}_{red}\vert \mathbf{R}_{N_{0}}\cap \mathbf{Y}_{0}}$, then we deduce the following corollary.
\begin{cor} The operator sequence $\mathrm{\left(\Phi_{n}\right)_{n}}$ converges to $\mathrm{\mathfrak{Red}}$ with respect to the topology induced by the supremum norm on $\mathrm{\mathcal{C}^{0}\!\left(\mathbf{X}\right)}$ and $\mathrm{\mathcal{C}^{0}_{bd}\!\left(\mathbf{R}_{N_{0}}\cap \mathbf{Y}\right)}$. \end{cor} \begin{proof} Apply {\bf{Theorem 6.b}}. \end{proof}
\addcontentsline{toc}{section}{Index of Notation} \hspace{-0.6 cm}{\Large{\bf{Index of Notation}}}
\noindent\begin{tabular}{@{}p{2.5 cm}p{12.35cm}@{}} $\mathrm{\mathcal{C}^{k}(\widehat{\,\mathbf{X}\,})}$ & The complex space of all $\mathrm{k}$-dimensional cycles $\mathrm{{\bm{\mathfrak{C}}}}$ in $\mathrm{\widehat{\,\mathbf{X}\,}}$, \textbf{S. \oldstylenums{\pageref{Notation Cycle Space Of All k-Dim Cycles}}}\\[0.05 cm]
$\mathrm{{\bm{\mathfrak{C}}}}$ & $\mathrm{k}$-dimensional cycle $\mathrm{\sum_{i}n_{i}\mathbf{C}_{i}}$, $\mathrm{n_{i}\in \mathbb{N}}$, $\mathrm{dim_{\mathbb{C}}\,\mathbf{C}_{i}=k}$ in $\mathrm{\widehat{\,\mathbf{X}\,}}$, \textbf{S. \oldstylenums{\pageref{Notation Cycle Of Dimension K}}}\\[0.05 cm]
$\mathrm{{\bm{\mathfrak{C}}}_{y}}$ & $\mathrm{k}$-dimensional cycle associated to the fiber $\mathrm{\widehat{\pi}^{-1}\left(y\right)}$ for $\mathrm{y\in \mathbf{Y}_{0}}$, \textbf{S. \oldstylenums{\pageref{Notation Cycle Of Dimension K Associated To The Fiber Of}}}\\[0.05 cm]
$\mathrm{\vert {\bm{\mathfrak{C}}}\vert}$&Support $\mathrm{\vert{\bm{\mathfrak{C}}}\vert=\bigcup_{i\in I}\mathbf{C}_{i}}$ of the cycle $\mathrm{{\bm{\mathfrak{C}}}=\sum_{i}n_{i}\mathbf{C}_{i}}$, \textbf{S. \oldstylenums{\pageref{Notation Support Of A Cycle}}}\\[0.05 cm]
$\mathrm{cl\left(\mathbb{T}.x\right)}$&Zariski closure of the $\mathrm{\mathbb{T}}$-orbit $\mathrm{\mathbb{T}.x}$ where $\mathrm{x\in \mathbf{X}}$, \textbf{S. \oldstylenums{\pageref{Notation T-Orbit Closure}}}\\[0.05 cm]
$\mathrm{\mathfrak{Conv}\,\mathbf{A}}$&Convex hull of a subset $\mathrm{\mathbf{A}\subset \mathfrak{t}^{*}}$, \textbf{S. \oldstylenums{\pageref{Notation Convex Hull Of A Subset}}}\\[0.05 cm]
$\mathrm{D_{n}^{i}\left(\cdot,t\right)}$& Sequence of cumulative distribution densities associated to the tame sequence $\mathrm{\left(s^{i}_{n}\right)_{n}}$, \textbf{S. \oldstylenums{\pageref{Notation Cumulative Distribution Densities}}}\\[0.05 cm]
$\mathrm{D_{n}^{\mathfrak{U}}\left(\cdot,t\right)}$& Sequence of distribution functions associated to the open cover $\mathrm{\mathfrak{U}}$, \textbf{S. \oldstylenums{\pageref{Notation Collection Of Cumulative Fiber Probability Densities}}}\\[0.05 cm]
$\mathrm{\triangle^{i}_{f_{y,n}}\left(=\triangle^{i}_{y,n}\right)}$& Holomorphic $\mathrm{\xi_{n}-\xi^{i}_{n}}$-eigenfunction on $\mathrm{\pi^{-1}\left(\mathbf{U}_{y}\right)}$ where $\mathrm{y\in \mathbf{R}_{N_{0}}}$ defined by $\mathrm{s^{i}_{n}\cdot\triangle_{f_{y,n}}^{i}=\widehat{s}_{f_{y,n}}}$, \textbf{S. \oldstylenums{\pageref{Notation Difference Function}}}\\[0.05 cm]
$\mathrm{{\bf{Fix}}^{\mathbb{T}}}$& Set of all $\mathrm{\mathbb{T}}$-fixed points in $\mathrm{\mathbf{X}}$, \textbf{S. \oldstylenums{\pageref{Notation Fix Point Set Of T-Action}}}\\[0.05 cm]
$\mathrm{\overline{f}}$& Averaged function defined by $\mathrm{\overline{f}\left(x\right)=\int_{T}f\left(t.x\right)\,d\nu_{T}}$, \textbf{S. \oldstylenums{\pageref{Notation Averaged Function}}}\\[0.05 cm]
$\mathrm{f_{red}}$& Function on the quotient $\mathrm{\mathbf{Y}=\mathbf{X}^{ss}_{\xi}/\!\!/\mathbb{T}}$ induced by the restriction $\mathrm{\overline{f}\vert \mu^{-1}\left(\xi\right)}$ of the averaged function $\mathrm{\overline{f}}$, \textbf{S. \oldstylenums{\pageref{Notation Reduced Function}}}\\[0.05 cm]
$\mathrm{\mathfrak{F}_{{\bm{\mathfrak{C}}},\mathbb{C}\mathbb{P}^{m}}}$& Chow form associated to a cycle $\mathrm{{\bm{\mathfrak{C}}}}$ in $\mathrm{\mathbb{C}\mathbb{P}^{m}}$, \textbf{S. \oldstylenums{\pageref{Notation Chow Form}}}\\[0.05 cm]
${\bm{\Gamma}_{\pi}}$& Graph of the quotient map $\mathrm{\pi\!:\!\mathbf{X}^{ss}_{\xi}\rightarrow \mathbf{X}^{ss}_{\xi}/\!\!/\mathbb{T}=\mathbf{Y}}$ in $\mathrm{\mathbf{X}\times \mathbf{Y}}$, \textbf{S. \oldstylenums{\pageref{Notation Graph of Quotient Map}}}\\[0.05 cm]
$\mathrm{h}$& Hermitian, positive, $\mathrm{T}$-invariant bundle metric on $\mathrm{\mathbf{L}}$, \textbf{S. \oldstylenums{\pageref{Hermitian Bundle Metric}}}\\[0.05 cm]
$\mathrm{\int_{F^{-1}\left(y\right)}f\,d\,[F_y]}$& Fiber integral of $\mathrm{f}$ with respect to a $\mathrm{k}$-fibering $\mathrm{F\!:\!\mathbf{X}\rightarrow \mathbf{Y}}$, \textbf{S. \oldstylenums{\pageref{Notation Fiber Integration with Respect to a k-Fibering}}}\\[0.05 cm]
$\mathrm{\mathbf{M}_{J}}$& Subset of all points $\mathrm{x\in\mu^{-1}\left(\xi\right)}$ so that $\mathrm{s_{j}\left(x\right)\neq 0}$ for all $\mathrm{j\in J}$ and $\mathrm{s_{j}\left(x\right)\neq 0}$ for all $\mathrm{j\in J^{c}}$, \textbf{S. \pageref{Notation Definition of M_J}}\\[0.05 cm]
$\mathrm{\mu}$& Moment map associated to the hermitian, positive, $\mathrm{T}$-invariant bundle metric $\mathrm{h}$, \textbf{S. \oldstylenums{\pageref{Notation Moment Map}}}\\[0.05 cm]
$\mathrm{\bm{\nu}^{i}_{n}}$& Sequence of fiber measures associated to the tame sequence $\mathrm{\left(s^{i}_{n}\right)_{n}}$, \textbf{S. \oldstylenums{\pageref{Notation Sequence Of Fiber Probability Measures}}}\\[0.05 cm]
$\mathrm{\bm{\nu}^{\mathfrak{U}}_{n}}$& Sequence of fiber measures associated to the open cover $\mathrm{\mathfrak{U}}$, \textbf{S. \oldstylenums{\pageref{Notation Collection Of Fiber Probability Measures}}}\\[0.05 cm]
$\mathrm{\nu_{F}\left(x\right)}$& Order of a $\mathrm{k}$-fibering $\mathrm{F\!:\!\mathbf{X}\rightarrow \mathbf{Y}}$ at a point $\mathrm{x\in \mathbf{X}}$, \textbf{S. \oldstylenums{\pageref{Notation Order Of F At Point}}}\\[0.05 cm]
$\mathrm{\omega}$& Naturally associated K{\"a}hler form given by $\mathrm{\omega=-\frac{\sqrt{-1}}{\,\,2}\partial\overline{\partial}\,\mathbf{log}\,\vert \cdot\vert_{h}^{2}}$, \textbf{S. \oldstylenums{\pageref{Notation Kaehler form}}}\\[0.05 cm]
$\mathrm{\omega^{\prime}}$ & Smooth (2,2,)-form on $\mathrm{\widehat{\mathbf{\,X\,}}}$ given by $\mathrm{\omega^{\prime}= \left(p_{\mathbf{X}}\vert cl\left({\mathbf{\Gamma}}_{\pi}\right)\circ\zeta\right)^{*}\omega}$, \textbf{S. \oldstylenums{\pageref{Notation Smooth (2,2,)-form}}}\\[0.05 cm]
$\mathrm{\Omega}$ & Smooth (2,2,)-form on the compact variety $\mathrm{\widetilde{\bf{\,X\,}}}$ given by $\mathrm{\Omega\coloneqq\Sigma^{*}\omega^{\prime}}$, \textbf{S. \oldstylenums{\pageref{Notation Capital Omega}}}\\[0.05 cm]
$\mathrm{\pi}$& Algebraic projection map $\mathrm{\pi\!:\!\mathbf{X}^{ss}_{\xi}\rightarrow \mathbf{X}^{ss}_{\xi}/\!\!/\mathbb{T}}$ of the Hilbert Quotient, \textbf{S. \oldstylenums{\pageref{Notation Hilbert Quotient Map}}}\\[0.05 cm]
$\mathrm{\widehat{\pi}}$& Algebraic projection map from the compact variety $\mathrm{\widehat{\bf{\,X\,}}}$ to $\mathrm{\mathbf{Y}}$ given by $\mathrm{\widehat{\pi}= p_{\mathbf{Y}}\vert cl\,\mathbf{\Gamma}_{\pi}\circ \zeta}$, \textbf{S. \oldstylenums{\pageref{Notation Regular Map between cal X and Y}}}\\[0.05 cm]
$\mathrm{\Pi}$& Surjective, holomorphic $\mathrm{k}$-fibering from $\mathrm{\mathbf{\widetilde{\,\bf{X}\,}}}$ to $\mathrm{\widetilde{\,\bf{Y}\,}}$, \textbf{S. \oldstylenums{\pageref{Notation Blown Up Projection Map}}}\\[0.05 cm]
$\mathrm{\varphi^{\widehat{\pi}}}$& Holomorphic map from $\mathrm{\mathbf{Y}_{0}}$ into the cycle space $\mathrm{\mathcal{C}^{k}(\widehat{\,\mathbf{X}\,})}$ induced by the $\mathrm{k}$-fibering $\mathrm{\widehat{\pi}\vert \widehat{\pi}^{-1}\left(\mathbf{Y}_{0}\right)\rightarrow \mathbf{Y}_{0}}$, \textbf{S. \oldstylenums{\pageref{Notation Regular Map Universal Property Chow Scheme}}}\\[0.05 cm] \end{tabular}
\noindent\begin{tabular}{@{}p{2.5 cm}p{12.35cm}@{}}
$\mathrm{\Phi_{n}}$& Continuous operator from $\mathrm{\mathcal{C}^{0}\left(\mathbf{X}\right)}$ to $\mathrm{\mathcal{C}^{0}_{bd}\left(\mathbf{R}_{N_{0}}\cap \mathbf{Y}_{0}\right)}$ induced by $\mathrm{\bf{\nu}_{n}}$, \textbf{S. \oldstylenums{\pageref{Notation Phi_n}}}\\[0.05 cm]
$\mathrm{Relint\,\mathbf{A}}$& Relative interior of a convex subset $\mathrm{\mathbf{A}\subset \mathfrak{t}^{*}}$, \textbf{S. \oldstylenums{\pageref{Notation Relative Interior}}}\\[0.05 cm]
$\mathrm{\mathbf{R}_{N_{0}}}$& The set of all removable singularities of order $\mathrm{N_{0}\in \mathbb{N}}$ of the fiber measure sequence $\mathrm{\left(\bm{\nu}_{n}\right)_{n}}$, \textbf{S. \oldstylenums{\pageref{Notation Removable Singularity Of Order N_0}}}\\[0.05 cm]
$\mathrm{\varrho^{i}}$& S.p.s.h limit function associated to the tame sequence $\mathrm{\left(s^{i}_{n}\right)_{n}}$, \textbf{S. \oldstylenums{\pageref{Notation S.p.s.h. Limit Function Associated to the Tame Sequence}}}\\[0.05 cm]
$\mathrm{\varrho^{i}_{n}}$& Sequence of s.p.s.h functions associated to the tame sequence $\mathrm{\left(s^{i}_{n}\right)_{n}}$, \textbf{S. \oldstylenums{\pageref{Notation Sequence of S.p.s.h. Functions Associated to the Tame Sequence}}}\\[0.05 cm]
$\mathrm{s^{i}_{n}}$& $\mathrm{i}$-th tame sequence associated to the ray $\mathrm{\mathbb{R}^{\geq 0}\xi}$ where $\mathrm{\xi\in \mathfrak{t}^{*}}$, \textbf{S. \oldstylenums{\pageref{Notation i-th tame sequence}}}\\[0.05 cm]
$\mathrm{\widehat{s}_{f_{y,n}}}$& Extension of $\mathrm{s_{n}}$ given by $\mathrm{\widehat{s}_{f_{y,n}}=s_{n}\cdot\pi^{*}f^{-1}_{y,n}}$ for all $\mathrm{n\geq N_{0}\in \mathbb{N}}$ where $\mathrm{f_{y,n}\in \mathcal{O}\left(\mathbf{U}_{y}\right)}$ and $\mathrm{y\in \mathbf{R}_{N_{0}}}$, \textbf{S. \oldstylenums{\pageref{Notation Local Extension Of S_{n}}}}\\[0.05 cm]
$\mathrm{\sigma}$& Proper, surjective holomorphic map from $\mathrm{\widetilde{\bf{\,Y\,}}}$ to $\mathrm{\mathbf{Y}}$ given by $\mathrm{p_{\mathbf{Y}}\vert \widetilde{\bf{\,Y\,}}}$, \textbf{S. \oldstylenums{\pageref{Notation Blow Up Map onto Y}}}\\[0.05 cm]
$\mathrm{\Sigma}$& Proper, surjective holomorphic map from $\mathrm{\widetilde{\,\bf{X}\,}}$ to $\mathrm{\widehat{\,\bf{X}\,}}$, \textbf{S. \oldstylenums{\pageref{Notation Blow Up Map onto X}}}\\[0.05 cm]
$\mathrm{\vert s\vert^{2}}$& Norm function of a section $\mathrm{s\in H^{0}\left(\mathbf{X},\mathbf{L}\right)}$ with respect to the bundle metric $\mathrm{h}$, \textbf{S. \oldstylenums{\pageref{Notation Norm of Section with Respect to H}}}\\[0.05 cm]
$\mathrm{\Vert s^{i}_{n}\Vert^{2}}$& Fiber integral of $\mathrm{\vert s^{i}_{n}\vert ^{2}}$ over the subset $\mathrm{\mathbf{Y}_{0}\subset \mathbf{Y}}$ with respect to $\mathrm{\pi\vert \pi^{-1}\left(\mathbf{Y}_{0}\right)}$, \textbf{S. \oldstylenums{\pageref{Notation Fiber Integral Initial Sequence}}}\\[0.05 cm]
$\mathrm{{\bf{S}}_{x}}$& Set of all $\mathrm{\mathbb{T}}$-fixed points contained in the Zariski closure $\mathrm{\mathbb{T}.x}$ where $\mathrm{x\in\mathbf{X}}$, \textbf{S. \oldstylenums{\pageref{Notation T-Fixed Points Contained In The Closure Of T.x}}}\\[0.05 cm]
$\mathrm{\mathcal{S}_{J}}$& Set of all $\mathrm{\mathbb{T}}$-eigensections indexed by $\mathrm{J\subset \left\{1,\dots,m\right\}}$ where $\mathrm{m=dim_{\mathbb{C}}}$ $\mathrm{H^{0}\left(\mathbf{X},\mathbf{L}\right)}$, \textbf{S. \oldstylenums{\pageref{Notation Set Of T-Eigensections Indexed By J}}}\\[0.05 cm]
$\mathrm{{\bm{\mathfrak{S}}}_{J}}$& Set of all characters $\mathrm{\xi_{j}\in \mathfrak{t}^{*}_{\mathbb{Z}}}$ attached to the collection of all $\mathrm{\mathbb{T}}$-eigensections given by $\mathrm{\mathcal{S}_{J}}$, \textbf{S. \oldstylenums{\pageref{Notation Set Of All Characters Corresponding To S_x}}}\\[0.05 cm]
$\mathrm{T\left(\epsilon,\mathbf{W}^{i}\right)}$& Compact $\mathrm{\epsilon}$-neighborhood tube around $\mathrm{\mu^{-1}\left(\xi\right)}$ over $\mathrm{\mathbf{W}^{i}\subset \mathbf{Y}}$, \textbf{S. \oldstylenums{\pageref{Notation Compact Neighborhood Tube}}}\\[0.05 cm]
$\mathrm{T^{c}\left(\epsilon,\mathbf{W}^{i}\right)}$& Complement of $\mathrm{T\left(\epsilon,\mathbf{W}^{i}\right)}$ in $\mathrm{\pi^{-1}\left(\mathbf{W}^{i}\right)}$, \textbf{S. \oldstylenums{\pageref{Notation Complement of Compact Neighborhood Tube}}}\\[0.05 cm]
$\mathrm{\widetilde{\,T\,}\left(\epsilon,\mathbf{W}^{i}\right)}$& Compact $\mathrm{\epsilon}$-neighborhood tube in $\mathrm{\widetilde{\bf{\,X\,}}}$ induced by $\mathrm{T\left(\epsilon,\mathbf{W}^{i}\right)}$, \textbf{S. \oldstylenums{\pageref{Notation Compact Corresponding Neighborhood Tube}}}\\[0.05 cm]
$\mathrm{\mathbf{V}^{i}}$& Inverse image of the compact neighborhood $\mathrm{\mathbf{W}^{i}\subset \mathbf{Y}}$ under the projection $\mathrm{\pi}$, \textbf{S. \oldstylenums{\pageref{Notation V^i}}}\\[0.05 cm]
$\mathrm{vol\left(\pi^{-1}\left(y\right)\right)}$& Volume of the fiber $\mathrm{\pi^{-1}\left(y\right)}$ for $\mathrm{y\in \mathbf{Y}_{0}}$ with respect to the form $\mathrm{\omega^{k}\vert \pi^{-1}\left(y\right)}$ counted with multiplicities, \textbf{S. \oldstylenums{\pageref{Notation Volume of a Fiber Over Y_0}}}\\[0.05 cm]
$\mathrm{\mathbf{W}^{i}}$& Compact neighborhood contained in $\mathrm{\mathbf{Y}^{i}}$, \textbf{S. \oldstylenums{\pageref{Notation W^i}}}\\[0.05 cm]
$\mathrm{\widetilde{\bf{W}}^{i}}$& Compact neighborhood in $\mathrm{\widetilde{\bf{\,Y\,}}}$ induced by $\mathrm{\mathbf{W}^{i}}$ via $\mathrm{\widetilde{\bf{W}}^{i}=\sigma^{-1}\left(\mathbf{W}^{i}\right)}$, \textbf{S. \oldstylenums{\pageref{Notation cal W^i}}}\\[0.05 cm]
$\mathrm{\mathbf{\widehat{\,X\,}}}$& Compact variety defined as the normalization of the compact variety subvariety $\mathrm{cl\,\bm{\Gamma}_{\pi}\subset\mathbf{X}\times \mathbf{Y}}$, \textbf{S. \oldstylenums{\pageref{Notation X Hat}}}\\[0.05 cm]
$\mathrm{\widetilde{\bf{\,X\,}}}$& Compact variety given by $\mathrm{\widetilde{\bf{\,X\,}}=\left(\mathbf{Y}\times {\bm{\mathfrak{X}}}\right)\cap \big(\widetilde{\bf{\,Y\,}}\times\widehat{\bf{\,X\,}}\big)}$, \textbf{S. \oldstylenums{\pageref{Notation Blow Up Space}}}\\[0.05 cm]
$\mathrm{\mathbf{X}_{0}}$& Inverse image of $\mathrm{\mathbf{Y}_{0}}$ with respect to the projection map $\mathrm{\pi\!:\!\mathbf{X}^{ss}_{\xi}\rightarrow \mathbf{Y}}$, \textbf{S. \oldstylenums{\pageref{Notation Inverse Image of Y_0 Under The Projection Map}}}\\[0.05 cm]
$\mathrm{\mathbf{X}^{i}}$& Open, $\mathrm{\pi}$-saturated subset of $\mathrm{\mathbf{X}^{ss}_{\xi}\cap \mathbf{X}(s^{i}_{n})}$ given by $\mathrm{\mathbf{X}^{i}\coloneqq\pi^{-1}\left(\pi\left(\mathbf{M}_{i}\right)\right)}$, \textbf{S. \oldstylenums{\pageref{Notation i-th set}}}\\[0.05 cm]
$\mathrm{\mathbf{X}\left(s_{n}^{i}\right)}$& n-stable, Zariski open subset of $\mathrm{\mathbf{X}}$ given by $\mathrm{\left\{x\in \mathbf{X}:\, s_{n}^{i}\left(x\right)\neq 0\right\}}$, \textbf{S. \oldstylenums{\pageref{Notation GTZU}}}\\[0.05 cm]
$\mathrm{\mathbf{X}^{ss}_{\xi}}$& Set of semistable points with respect to the level subset $\mathrm{\mu^{-1}\left(\xi\right)}$, \textbf{S. \oldstylenums{\pageref{Notation Set of Semistable Points}}}\\[0.05 cm]
\end{tabular}
\noindent\begin{tabular}{@{}p{2.5 cm}p{12.35cm}@{}} $\mathrm{\mathbf{X}^{ss}_{\xi}/\!\!/\mathbb{T}}$& Hilbert Quotient with respect to the level subset $\mathrm{\mu^{-1}\left(\xi\right)}$, \textbf{S. \oldstylenums{\pageref{Notation Hilbert Quotient}}}\\[0.05 cm]
$\mathrm{\widehat{X}_{\xi}}$& Fundamental vector field on $\mathrm{\mathbf{L}}$ generated by the flow induced by $\mathrm{\xi\in \mathfrak{k}}$, \textbf{S. \oldstylenums{\pageref{Notation Fundamental Vector Field on L}}}\\[0.05 cm]
$\mathrm{{\bm{\mathfrak{X}}}}$& Universal space defined by $\mathrm{{\bm{\mathfrak{X}}}\coloneqq \left\{\big({\bm{\mathfrak{C}}},x\right)\in \mathcal{C}^{k}(\widehat{\,\bf{X}\,})\times \widehat{\,\bf{X}\,}: x\in \vert {\bm{\mathfrak{C}}}\vert\big\}}$, \textbf{S. \oldstylenums{\pageref{Notation Universal Space}}}\\[0.05 cm]
${\xi}$& Normalization map $\mathrm{\xi\!:\!\widetilde{\bf{\,Y\,}}^{nor}\rightarrow \widetilde{\bf{\,Y\,}}}$ of $\mathrm{\widetilde{\bf{\,Y\,}}}$, \textbf{S. \oldstylenums{\pageref{Notation Normalization Map Xi}}}\\[0.05 cm]
$\mathrm{\xi^{i}_{n}}$& Sequence of weights associated to a tame sequence $\mathrm{\left(s^{i}_{n}\right)_{n}}$, \textbf{S. \oldstylenums{\pageref{Notation Approximating Sequence of Weight Vectors}}}\\[0.05 cm]
$\mathrm{\left(y,{\bm{\mathfrak{C}}}_{y}\right)}$& Point in $\mathrm{\sigma^{-1}\left(\mathbf{Y}_{0}\right)\subset \widetilde{\bf{\,Y\,}}\subset \mathbf{Y}\times \mathcal{C}^{k}(\widehat{\,\bf{X}\,})}$ where $\mathrm{\varphi^{\widehat{\pi}}\left(y\right)={\bm{\mathfrak{C}}}_{y}}$ for $\mathrm{y\in \mathbf{Y}_{0}}$, \textbf{S. \oldstylenums{\pageref{Notation y,frak C_y}}}\\[0.05 cm]
$\mathrm{\mathbf{Y}}$& Abbreviation for the Hilbert Quotient $\mathrm{\mathbf{X}^{ss}_{\xi}/\!\!/\mathbb{T}}$, \textbf{S. \oldstylenums{\pageref{Notation Abbreviation for Hilbert Quotient}}}\\[0.05 cm]
$\mathrm{\mathbf{Y}^{i}}$& Image of $\mathrm{\mathbf{X}^{i}}$ under the projection map $\mathrm{\pi\!:\!\mathbf{X}^{ss}_{\xi}\rightarrow \mathbf{Y}}$, \textbf{S. \oldstylenums{\pageref{Notation Image of X_i Under Projection Map}}}\\[0.05 cm]
$\mathrm{\mathbf{Y}_{0}}$& Subset of all points in $\mathrm{\mathbf{Y}}$ so that $\mathrm{\widehat{\pi}^{-1}\left(y\right)}$ is a $\mathrm{k}$-dimensional subvariety of $\mathrm{\mathbf{\widehat{\,X\,}}}$, \textbf{S. \oldstylenums{\pageref{Notation Subset Y_0}}}\\[0.05 cm]
$\widetilde{\bf{\,Y\,}}$& Complex subspace of $\mathrm{\mathbf{Y}\times \mathcal{C}^{k}(\widehat{\,\mathbf{X}\,})}$ given by the closure of the graph of $\mathrm{\varphi^{\widehat{\pi}}\!:\!\mathbf{Y}_{0}\rightarrow \mathcal{C}^{k}(\widehat{\,\mathbf{X}\,})}$, \textbf{S. \oldstylenums{\pageref{Notation Blown Up Space Associated to Y}}}\\[0.05 cm]
${\zeta}$& Normalization map $\mathrm{\zeta\!:\!\widehat{\,\mathbf{X}\,}=\left(cl\,\mathbf{\Gamma}_{\pi}\right)^{nor}\rightarrow cl\,\mathbf{\Gamma}_{\pi}}$ of the compact variety $\mathrm{cl\,\mathbf{\Gamma}_{\pi}\subset}$ $\mathrm{ \mathbf{X}\times \mathbf{Y}}$, \textbf{S. \oldstylenums{\pageref{Notation Normalization Map Zeta}}}
\end{tabular}
\addcontentsline{toc}{section}{References}
\end{spacing}
\end{document} | arXiv |
gumbel vs normal distribution
@Jess That's better, because demonstrating an alternative approach was the motivation to write this answer. In Monopoly, if your Community Chest card reads "Go back to ...." , do you move forward or backward? Are min$(X_1,\ldots,X_n)$ and min$(X_1Y_1,\ldots,X_nY_n)$ independent for $n$ to infinity? These distributions differ in their location and scale parameters: the mean ("average") of the distribution defines its location, and the standard deviation ("variability") defines the scale. So you took $F$ to be the standard normal CDF. The Gumbel-Softmax distribution is a continuous distribution that approximates samples from a categorical distribution and also works with backpropagation. To study the shapes of these distributions, we can shift each one back to the left by some amount $b_n$ and rescale it by $a_n$ to make them comparable. And so how might the associated series be obtained? When F is a Normal distribution, the particular limiting extreme value distribution is a reversed Gumbel, up to location and scale. Finding the mean of the max order statistic drawn from standard normal, Extreme Value Theory - Normalizing constants for Generalized Extreme Value distribution, Using extreme value theory to estimate bounds, How to find the $(a_n,b_n)$ for extreme value theory, Limiting distribution of maximum of i.i.d. Let $0 \lt q \lt 1$. Also, de Haan examines the sufficient condition already differentiated. The case where μ = 0 and β = 1 is called the standard Gumbel distribution. The case where μ = 0 and β = 1 is called the standard Gumbel distribution. Can it be justified that an economic contraction of 11.3% is "the largest fall for more than 300 years"? $\xi_a = F^{-1}(a)$. How to solve this puzzle of Martin Gardner? Yes, that's true, I realized this shortly after I posted my comment so I deleted it immediately. there is a lower bound of zero) then the Weibull distribution should be used in preference to the Gumbel. Can I run my 40 Amp Range Stove partially on a 30 Amp generator. MathJax reference. 10.5 of the book H.A. @renrenthehamster I have added relevant material. Each of the previous graphs has been shifted to place its median at $0$ and to make its interquartile range of unit length. In probability theory and statistics, the generalized extreme value (GEV) distribution is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. I'm not quite sure I understood your solution. (This general approach should succeed in finding $a_n$ and $b_n$ for any continuous distribution. Let the scale parameter be $\beta$ and the location parameter be $\alpha$. Gumbel Distribution There are essentially three types of Fisher-Tippett extreme value distributions. This is from ch. What LEGO piece is this arc with ball joint? Recalling the definition of $F_n(x) = F^n(x)$, the solution is, $$b_n = x_{1/2;n},\ a_n = x_{3/4;n} - x_{1/4;n};\ G_n(x) = F_n(a_n x + b_n).$$, Because, by construction, the median of $G_n$ is $0$ and its IQR is $1$, the median of the limiting value of $G_n$ (which is some version of a reversed Gumbel) must be $0$ and its IQR must be $1$. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. An indirect way, is as follows: It behaves like $\left(2 \log(n) - \log(2\pi)\right)^{-1/2}$ for large $n$. Please note, to clarify some assertions appearing elsewhere in this thread, that. Then, if, $$\lim_{x\rightarrow F^{-1}(1)}\left (\frac d{dx}\frac {(1-F(x))}{f(x)}\right) =0 \Rightarrow X_{(n)} \xrightarrow{d} G(x)$$, Using the usual notation for the standard normal and calculating the derivative, we have, $$\frac d{dx}\frac {(1-\Phi(x))}{\phi(x)} = \frac {-\phi(x)^2-\phi'(x)(1-\Phi(x))}{\phi(x)^2} = \frac {-\phi'(x)}{\phi(x)}\frac {(1-\Phi(x))}{\phi(x)}-1$$, Note that $\frac {-\phi'(x)}{\phi(x)} =x$. The second appears to be more difficult; that is the issue addressed here. This is why Gumbel generally applies to e.g. Asking for help, clarification, or responding to other answers. Can you solve it or find it in literature? Gumbel Distribution The Gumbel distribution is used to model the largest value from a relatively large set of independent elements from distributions whose tails decay relatively fast, such as a normal or exponential distribution. Extreme Value Theory - Show: Normal to Gumbel. random variables, and $f(x)$ their common density. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. "Question closed" notifications experiment results and graduation, MAINTENANCE WARNING: Possible downtime early morning Dec 2/4/9 UTC (8:30PM…, Extreme value distribution for univariate normal: Derive parameters of the Gumbel, Examples of convergence in distribution using CDF directly, Variance of maximum of Gaussian random variables, Normalization to non-degenerate distribution. FTG asserts that sequences (a n) and (b n) can be chosen so that these distribution functions converge pointwise at every x to some extreme value distribution, up to scale and location. There appear to be different conventions concerning the Gumbel distribution. Properties The Gumbel distribution is a continuous probability distribution. How does linux retain control of the CPU on a single-core machine? Making statements based on opinion; back them up with references or personal experience. Translated by Norman Johnson. What makes cross input signature aggregation complicated to implement? But beware because some of the notation has different content in de Haan -for example in the book $f(t)$ is the probability density function, while in de Haan $f(t)$ means the function $w(t)$ of the book (i.e. will work fine (and are as simple as possible). The Maximum of $X_1,\dots,X_n. By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy. Naturally, low values are restricted in the sense that 0 is the absolute minimum. I followed through and agree that the sufficient condition is satisfied. OOP implementation of Rock Paper Scissors game logic in Java. \sim$ i.i.d. When $F$ is a Normal distribution, the particular limiting extreme value distribution is a reversed Gumbel, up to location and scale. Gumbel Distribution represents the distribution of extreme values either maximum or minimum of samples used in various distributions. Why were there only 531 electoral votes in the US Presidential Election 2016? I will adopt the convention that the CDF of a reversed Gumbel distribution is, up to scale and location, given by $1-\exp(-\exp(x))$. For absolutely continuous distributions, Richard von Mises (in a 1936 paper "La distribution de la plus grande de n valeurs", which appears to have been reproduced -in English?- in a 1964 edition with selected papers of his), has provided the following sufficient condition for the maximum of a sample to converge to the standard Gumbel, $G(x)$: Let $F(x)$ be the common distribution function of $n$ i.i.d. rev 2020.11.24.38066, The best answers are voted up and rise to the top, Cross Validated works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. When the $X_i$ are iid with common distribution function $F$, the distribution of the maximum $X_{(n)}$ is, $$F_n(x) = \Pr(X_{(n)}\le x) = \Pr(X_1 \le x)\Pr(X_2 \le x) \cdots \Pr(X_n \le x) = F^n(x).$$. What is this part which is mounted on the wing of Embraer ERJ-145? Thank you! The most common is the type I distribution, which are sometimes referred to as Gumbel types or just Gumbel distributions. There are essentially three types of Fisher-Tippett extreme value distributions. The Gumbel distribution gives the asymptotic distribution of the minimum value in a sample from a distribution such as the normal distribution. What is this part of an aircraft (looks like a long thick pole sticking out of the back)? Statistica Neerlandica, 30(4), 161-172." Where should small utility programs store their preferences? The main difference between the normal distribution and the logistic distribution lies in the tails and in the behavior of the failure rate function. So we have to evaluate the limit, $$\lim_{x\rightarrow \infty}\left (x\frac {(1-\Phi(x))}{\phi(x)}-1\right) $$, But $\frac {(1-\Phi(x))}{\phi(x)}$ is Mill's ratio, and we know that the Mill's ratio for the standard normal tends to $1/x$ as $x$ grows. The question asks two things: (1) how to show that the maximum $X_{(n)}$ converges, in the sense that $(X_{(n)}-b_n)/a_n$ converges (in distribution) for suitably chosen sequences $(a_n)$ and $(b_n)$, to the Standard Gumbel distribution and (2) how to find such sequences.
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gumbel vs normal distribution 2020 | CommonCrawl |
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\title{ Lucas Numbers with the Lehmer property }
\author{ {\sc Bernadette~Faye}\and {\sc Florian~Luca} } \address{ Ecole Doctorale de Mathematiques et d'Informatique \newline Universit\'e Cheikh Anta Diop de Dakar \newline BP 5005, Dakar Fann, Senegal and\newline School of Mathematics, University of the Witwatersrand \newline Private Bag X3, Wits 2050, Johannesburg, South Africa } \email{[email protected]}
\address{ School of Mathematics, University of the Witwatersrand \newline Private Bag X3, Wits 2050, Johannesburg, South Africa \newline } \email{[email protected]}
\maketitle
\begin{abstract} A composite positive integer $n$ is \textit{Lehmer} if $\phi(n)$ divides $n-1$, where $\phi(n)$ is the Euler's totient function. No Lehmer number is known, nor has it been proved that they don't exist. In 2007, the second author \cite{FL1} proved that there is no Lehmer number
in the Fibonacci sequence. In this paper, we adapt the method from \cite{FL1} to show that there is no Lehmer number in the companion Lucas sequence of the Fibonacci sequence $(L_n)_{n\ge 0}$ given by $L_0=2, L_1=1$ and $L_{n+2}=L_{n+1}+L_n$ for all $n\geq0$. \end{abstract}
\section{Introduction} Let $\phi(n)$ be the Euler function of a positive integer $n$. Recall that if $n$ has the prime factorization $$ n=p_1^{\alpha_1} p_2^{\alpha_2}\cdots p_k^{\alpha_k}, $$ then $$ \phi(n)=(p_1-1)p_1^{\alpha_1-1} (p_2-1)p_2^{\alpha_2-1}\cdots (p_k-1)p_k^{\alpha_k-1}. $$ Lehmer \cite{Leh} conjectured that if $\phi(n)\mid n-1$ then $n$ is a prime. To this day, the conjecture remains open. Counterexamples to Lehmer's conjecture have been dubbed {\it Lehmer numbers}. Several people worked on getting larger and larger lower bounds on a potential Lehmer number. For a positive integer $m$, we write $\omega(m)$ for the number of distinct prime factors of $m$. Lehmer himself proved that if $N$ is Lehmer, then $\omega(N)\geq 7$. This has been improved by Cohen and Hagis \cite{coh} to $\omega(N)\geq 14.$ The current record $\omega(N)\ge 15$ is due to Renze \cite{joh}. If additionally $3\mid N$, then $\omega(N)\geq 40\cdot 10^{6}$ and $N>10^{36\cdot 10^{7}}.$
Not succeeding in proving that there are no Lehmer numbers, some researchers have settled for the more modest goal of proving that there are no Lehmer numbers in certain interesting subsequences of positive integers. For example, in \cite{FL1}, Luca proved that there is no Fibonacci number which is Lehmer. In \cite{GL}, it is shown that there is no Lehmer number in the sequence of Cullen numbers $\{C_n\}_{n\ge 1}$ of general term $C_n=n2^n+1$, while in \cite{Dajune} the same conclusion is shown to hold for generalized Cullen numbers. In \cite{L1}, it is shown that there is no Lehmer number of the form $(g^n-1)/(g-1)$ for any $n\ge 1$ and integer $g \in [2,1000]$.
Here, we apply the same argument as in \cite{FL1}, to the Lucas sequence companion of the Fibonacci sequence given by $L_0=2, L_1=1$ and $L_{n+2}=L_{n+1}+L_n$ for all $n\geq0$. Putting $(\alpha,\beta)=((1+{\sqrt{5}})/2,(1-{\sqrt{5}})/2)$ for the two roots of the characteristic equation $x^2-x-1=0$ of the Lucas sequence, the Binet formula \begin{equation} \label{eq:1} L_n=\alpha^n +\beta^n\qquad {\text{\rm holds~for~all}}\qquad n\ge 0. \end{equation} There are several relations among Fibonacci and Lucas numbers which are well-known and can be proved using the Binet formula \eqref{eq:1} for the Lucas numbers and its analog $$ F_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}\qquad {\text{\rm for~all}}\qquad n\ge 0 $$ for the Fibonacci numbers. Some of them which are useful for us are \begin{equation} \label{r1} L_n^{2}-5F_n^{2}=4(-1)^n, \end{equation} \begin{equation} \label{eq:2} L_n=L_{n/2}^2-2(-1)^{n/2}\quad {\text{\rm valid~for~ all~ even}}\quad n, \end{equation} whereas for odd $n$ \begin{equation} \label{eq:3} L_n-1=\left\{ \begin{matrix} 5F_{(n+1)/2} F_{(n-1)/2} & {\text{\rm if}} & n\equiv 1\pmod 4;\\ L_{(n+1)/2} L_{(n-1)/2} & {\text{\rm if}} & n\equiv 3\pmod 4. \end{matrix}\right. \end{equation} Our result is the following: \begin{theorem} There is no Lehmer number in the Lucas sequence. \end{theorem}
\section{Proof} Assume that $L_n$ is Lehmer for some $n$. Clearly, $L_n$ is odd and $\omega(L_n)\ge 15$ by the main result from \cite{joh}. The product of the first $15$ odd primes exceeds $1.6\times 10^{19}$, so $n\ge 92$. Furthermore,
\begin{equation} \label{eq:15} 2^{15}\mid 2^{\omega(L_n)}\mid \phi(L_n)\mid L_n-1. \end{equation}
If $n$ is even, formula \eqref{eq:2} shows that $L_n-1=L_{n/2}^2+1$ or $L_{n/2}^2-3$ and numbers of the form $m^2+1$ or $m^2-3$ for some integer $m$ are never multiples of $4$, so divisibility \eqref{eq:15} is impossible. If $n\equiv 3\pmod 8$, relations \eqref{eq:3} and \eqref{eq:15} show that $2^{15}\mid L_{(n+1)/2}L_{(n-1)/2}$. This is also impossible since no member of the Lucas sequence is a multiple of $8$, fact which can be easily proved by listing its first $14$ members modulo $8$: $$ 2,~1,~3,~4,~7,~3,~2,~5,~7,~4,~3,~7,~2,~1, $$ and noting that we have already covered the full period of $\{L_m\}_{m\ge 0}$ modulo $8$ (of length $12$) without having reached any zero.
So, we are left with the case when $n\equiv 1\pmod{4}.$
Let us write $$ n=p_1^{\alpha_1}\cdots p_k^{\alpha_k}, $$ with $p_1<\cdots<p_k$ odd primes and $\alpha_1,\ldots,\alpha_k$ positive integers. If $p_1=3$, then $L_n$ is even, which is not the case. Thus, $p_1\geq 5$.
Here, we use the argument from \cite{FL1} to bound $p_1$. Since most of the details are similar, we only sketch the argument. Let $p$ be any prime factor of $L_n$. Reducing formula \eqref{eq:1} modulo $p$ we get that $-5F_n^2\equiv -4\pmod p$. In particular, $5$ is a quadratic residue modulo $p$, so by Quadratic Reciprocity also $p$ is a quadratic residue modulo $5$. Now let $d$ be any divisor of $n$ which is a multiple of $p_1$. By Carmichael's Primitive Divisor Theorem for the Lucas numbers (see \cite{RD}), there exists a primitive prime $p_d\mid L_d$, such that $p_d\nmid L_{d_1}$ for all positive $d_1<d.$ Since $n$ is odd and $d\mid n$, we have $L_d\mid L_n$, therefore $p_d\mid L_n$. Since $p_d$ is primitive for $L_d$ and a quadratic residue modulo $5$, we have $p_d\equiv 1\pmod d$ (if $p$ were not a quadratic residue modulo $5$, then we would have had that $p_d\equiv -1\pmod 5$, which is less useful for our problem). In particular, \begin{equation} \label{eq:p1} p_1\mid d\mid p_d-1\mid \phi(L_n). \end{equation} Collecting the above divisibilities \eqref{eq:p1} over all divisors $d$ of $n$ which are multiples of $p_1$ and using \eqref{eq:3}, we have \begin{equation} \label{eq:5} p_1^{\tau(n/p_1)} \mid \phi(L_n)\mid L_n-1\mid 5 F_{(n-1)/2}F_{(n+1)/2}. \end{equation} In the above, $\tau(m)$ is the number of divisors of $m$. If $p_1=5$, then $5\mid n$, therefore $5\nmid F_{(n\pm 1)/2}$ because a Fibonacci number $F_m$ is a multiple of $5$ if and only if its index $m$ is a multiple of $5$. Thus, $\tau(n/p_1)=1$, so $n=p_1$, which is impossible since $n>92$.
Assume now that $p_1>5$. Since $$ \gcd(F_{(n+1)/2}, F_{(n-1)/2})=F_{\gcd((n+1)/2,(n-1)/2)}=F_1=1, $$ divisibility relation \eqref{eq:5} shows that $p_1^{\tau(n/p_1)}$ divides $F_{(n+\varepsilon)/2}$ for some $\varepsilon\in \{\pm 1\}$. Let $z(p_1)$ be the order of appearance of $p_1$ in the Fibonacci sequence, which is the minimal positive integer $\ell$ such that $p_1\mid F_{\ell}$. Write \begin{equation} \label{eq:Wall} F_{z(p_1)}=p_1^{e_{p_1}} m_{p_1}, \end{equation} where $m_{p_1}$ is coprime to $p_1$. It is known that $p_1\mid F_k$ if and only if $z(p_1)\mid k$. Furthermore, if $p_1^t\mid F_k$ for some $t>e_{p_1}$, then necessarily $p_1\mid k$. Since for us $(n+\varepsilon)/2$ is not a multiple of $p_1$ (because $n$ is a multiple of $p_1$), we get that $\tau(n/p_1)\le e_{p_1}$. In particular, if $p_1=7$, then $e_{p_1}=1$, so $n=p_1$, which is false since $n>92$. So, $p_1\ge 11$. We now follow along the argument from \cite{FL1} to get that \begin{equation} \label{eq:30} \tau(n)\leq 2\tau(n/p_1)\leq \frac{(p_1+1)\log\alpha}{\log p_1}. \end{equation} Further, since $(L_n-1)/\phi(L_n)$ is an integer larger than $1$, we have \begin{equation} \label{r:1} 2<\frac{L_n}{\phi(L_n)}\leq \prod_{p\mid L_n}\(1+\frac{1}{p-1}\)<\exp\(\sum_{p\mid L_n}\frac{1}{p-1}\), \end{equation} or \begin{equation} \label{eq:4} \log 2 \leq \sum_{p \mid L_n}\frac{1}{p-1}. \end{equation} Letting for a divisor $d$ of $n$ the notation ${\mathcal P_d}$ stand for the set of primitive prime factors of $L_d$, the argument from \cite{FL1} gives \begin{equation} \label{eq:imp} \sum_{p \in \mathcal{P}_d}\frac{1}{p-1} \leq \frac{0.9}{d} + \frac{2.2\log\log d}{d}. \end{equation} Since the function $x\mapsto (\log\log x)/x$ is decreasing for $x>10$ and all divisors $d>1$ of $n$ satisfy $d>10$, we have, using \eqref{eq:30}, that \begin{eqnarray} \label{eq:8} \sum_{p\mid L_n}\frac{1}{p-1}&=& \sum_{d\mid n}\sum_{p\in \mathcal{P}_d}\frac{1}{p-1} \leq \sum_{\substack{d\mid n\\ d>1}}\(\frac{0.9}{d} + \frac{2.2\log\log d}{d}\)\\ &\leq & \(\frac{0.9}{p_1} + \frac{2.2\log\log p_1}{p_1}\)\tau(n) \nonumber \\ &\leq & (\log\alpha)\frac{(p_1+1)}{\log p_1}\cdot \(\frac{0.9}{p_1} + \frac{2.2\log\log p_1}{p_1}\),\nonumber \end{eqnarray} which together with inequality \eqref{eq:4} leads to \begin{equation} \label{eq:9} \log p_1\leq \frac{(\log\alpha)}{\log 2}\left(\frac{p_1+1}{p_1}\right)(0.9 + 2.2\log\log p_1). \end{equation} The above inequality \eqref{eq:9} implies $p_1<1800$. Since $p_1<10^{14}$, a calculation of McIntosh and Roettger \cite{MC} shows that $e_{p_1}=1$. Thus, $\tau(n/p_1)=1$, therefore $n=p_1.$ Since $n\ge 92$, we have $p_1\ge 97$. Going back to the inequalities \eqref{eq:4} and \eqref{eq:imp}, we get $$ \log 2<\frac{0.9}{p_1}+\frac{2.2 \log\log p_1}{p_1}, $$ which is false for $p_1\ge 97$. The theorem is proved.
\section*{Acknowledgments} B. F. thanks OWSD and Sida (Swedish International Development Cooperation Agency) for a scholarship during her Ph.D. studies at Wits.
\end{document} | arXiv |
Graham Higman
Graham Higman FRS[1] (19 January 1917 – 8 April 2008) was a prominent English mathematician known for his contributions to group theory.
Graham Higman
Born
Graham Higman
(1917-01-19)19 January 1917
Louth, Lincolnshire, England
Died8 April 2008(2008-04-08) (aged 91)
Oxford, England
CitizenshipUnited Kingdom
Alma materBalliol College, Oxford
Known forHigman group
Higman's embedding theorem
Higman's lemma
HNN extension
Higman–Sims group
Hall–Higman theorem
AwardsSenior Berwick Prize (1962)
LMS De Morgan Medal (1974)
Sylvester Medal (1979)
Scientific career
FieldsMathematics, Group theory
InstitutionsUniversity of Oxford
Doctoral advisorJ. H. C. Whitehead
Doctoral students
• Jonathan Lazare Alperin
• Rosemary A. Bailey
• Marston Conder
• John Mackintosh Howie
• Peter M. Neumann
• Sheila Oates Williams
Biography
Higman was born in Louth, Lincolnshire, and attended Sutton High School, Plymouth, winning a scholarship to Balliol College, Oxford.[2] In 1939 he co-founded The Invariant Society, the student mathematics society,[3] and earned his DPhil from the University of Oxford in 1941. His thesis, The units of group-rings, was written under the direction of J. H. C. Whitehead. From 1960 to 1984 he was the Waynflete Professor of Pure Mathematics at Magdalen College, Oxford.
Higman was awarded the Senior Berwick Prize in 1962 and the De Morgan Medal of the London Mathematical Society in 1974. He was the founder of the Journal of Algebra and its editor from 1964 to 1984. Higman had 51 D.Phil. students, including Jonathan Lazare Alperin, Rosemary A. Bailey, Marston Conder, John Mackintosh Howie, and Peter M. Neumann.
He was also a local preacher in the Oxford Circuit of the Methodist Church. During the Second World War he was a conscientious objector, working at the Meteorological Office in Northern Ireland and Gibraltar.
He died in Oxford.[2]
Publications
• Higman, Graham (1940). "The units of group-rings". Proceedings of the London Mathematical Society. (2). 46: 231–248. doi:10.1112/plms/s2-46.1.231.
• Feit, Walter; Higman, Graham (1964). "The nonexistence of certain generalized polygons". Journal of Algebra. 1 (2): 114–131. doi:10.1016/0021-8693(64)90028-6.
• Graham Higman (1966) Odd characterisations of finite simple groups, U. of Michigan Press
• *Graham Higman (1974), Finitely presented infinite simple groups, Notes on Pure Mathematics, vol. 8, Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, ISBN 978-0-7081-0300-5, MR 0376874
• Graham Higman and Elizabeth Scott (1988), Existentially closed groups, LMS Monographs, Clarendon Press, Oxford[4]
See also
• Higman–Sims group, named after Donald G. Higman, but studied also by Graham Higman.
• Higman's embedding theorem
• Feit-Higman theorem
• Higman group
• Higman's lemma
• HNN extension
• Hall–Higman theorem
Notes
1. Conder, Marston D. E. (2022). "Graham Higman. 19 January 1917—8 April 2008". Biographical Memoirs of Fellows of the Royal Society. 73.
2. Collins, Michael (8 May 2008). "Professor Graham Higman: Leading group theorist". Obituaries. The Independent. Retrieved 14 October 2008.
3. The Early History of the Invariant Society by Robin Wilson, printed in The Invariant (2010), Ben Hoskin
4. Hickin, Kenneth (1990). "Review: Existentially closed groups by Graham Higman and Elizabeth Scott" (PDF). Bull. Amer. Math. Soc. (N.S.). 23 (1): 242–249. doi:10.1090/s0273-0979-1990-15943-9.
References
• Interview on YouTube
• Death notice, Oxford University Gazette, 17 April 2008 Archived 24 April 2008 at the Wayback Machine
External links
• O'Connor, John J.; Robertson, Edmund F., "Graham Higman", MacTutor History of Mathematics Archive, University of St Andrews
• Graham Higman at the Mathematics Genealogy Project
De Morgan Medallists
• Arthur Cayley (1884)
• James Joseph Sylvester (1887)
• Lord Rayleigh (1890)
• Felix Klein (1893)
• S. Roberts (1896)
• William Burnside (1899)
• A. G. Greenhill (1902)
• H. F. Baker (1905)
• J. W. L. Glaisher (1908)
• Horace Lamb (1911)
• J. Larmor (1914)
• W. H. Young (1917)
• E. W. Hobson (1920)
• P. A. MacMahon (1923)
• A. E. H. Love (1926)
• Godfrey Harold Hardy (1929)
• Bertrand Russell (1932)
• E. T. Whittaker (1935)
• J. E. Littlewood (1938)
• Louis Mordell (1941)
• Sydney Chapman (1944)
• George Neville Watson (1947)
• A. S. Besicovitch (1950)
• E. C. Titchmarsh (1953)
• G. I. Taylor (1956)
• W. V. D. Hodge (1959)
• Max Newman (1962)
• Philip Hall (1965)
• Mary Cartwright (1968)
• Kurt Mahler (1971)
• Graham Higman (1974)
• C. Ambrose Rogers (1977)
• Michael Atiyah (1980)
• K. F. Roth (1983)
• J. W. S. Cassels (1986)
• D. G. Kendall (1989)
• Albrecht Fröhlich (1992)
• W. K. Hayman (1995)
• R. A. Rankin (1998)
• J. A. Green (2001)
• Roger Penrose (2004)
• Bryan John Birch (2007)
• Keith William Morton (2010)
• John Griggs Thompson (2013)
• Timothy Gowers (2016)
• Andrew Wiles (2019)
Authority control
International
• ISNI
• VIAF
National
• Germany
• Israel
• United States
• Czech Republic
• Netherlands
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
People
• Deutsche Biographie
Other
• IdRef
| Wikipedia |
\begin{document}
\pagestyle{plain} \title{Learning Lyapunov Functions for Hybrid Systems}
\begin{abstract}
We propose a sampling-based approach to learn Lyapunov functions for a class of discrete-time autonomous hybrid systems that admit a mixed-integer representation. Such systems include autonomous piecewise affine systems, closed-loop dynamics of linear systems with model predictive controllers, piecewise affine/linear complementarity/mixed-logical dynamical system in feedback with a ReLU neural network controller, etc. The proposed method comprises an alternation between a learner and a verifier to find a valid Lyapunov function inside a convex set of Lyapunov function candidates. In each iteration, the learner uses a collection of state samples to select a Lyapunov function candidate through a convex program in the parameter space. The verifier then solves a mixed-integer quadratic program in the state space to either validate the proposed Lyapunov function candidate or reject it with a counterexample, i.e., a state where the Lyapunov condition fails. This counterexample is then added to the sample set of the learner to refine the set of Lyapunov function candidates. By designing the learner and the verifier according to the analytic center cutting-plane method from convex optimization, we show that when the set of Lyapunov functions is full-dimensional in the parameter space, our method finds a Lyapunov function in a finite number of steps. We demonstrate our stability analysis method on closed-loop MPC dynamical systems and a ReLU neural network controlled PWA system.
\end{abstract}
\section{Introduction}
Hybrid systems have become widespread within the systems and control community in the last decades thanks to their flexibility in modeling the interaction of continuous and discrete dynamical systems that frequently arise in cyber-physical systems (CPS) \cite{bemporad1999control,borrelli2017predictive}. As today's cyber-physical systems are getting more complex, developing new methods for design, analysis, and control of hybrid systems is increasingly important.
Analysis and control design for general hybrid systems is challenging and therefore, various methods have been proposed to tackle special classes of hybrid systems such as linear complementarity (LC) systems~\cite{heemels2000linear, van1998complementarity}, mixed logical dynamical (MLD) systems~\cite{bemporad1999control}, and piecewise affine (PWA) systems~\cite{sontag1981nonlinear}. While these classes are mathematically equivalent~\cite{heemels2001equivalence}, their representation could have a high impact on their numerical tractability. When it comes to stability analysis, various methods have been proposed for PWA systems~\cite{biswas2005survey, lin2009stability, sun2010stability} while methods that directly deal with MLD and LC systems are relatively scarce. Nevertheless, stability analysis tools for PWA systems are applicable to MLD and LC systems as they can be transformed into PWA systems~\cite{heemels2001equivalence}.
Transforming various types of hybrid systems into a PWA representation for stability analysis may not always be efficient. For example, for a PWA system in feedback with a ReLU neural network controller, although the closed-loop dynamics is PWA, identifying the PWA representation may be tedious and the stability analysis task may become very challenging since the number of partitions generated by the ReLU network can be very large~\cite{pascanu2013number}.
In this paper, we propose a learning-based approach to stability analysis of hybrid systems that admit a mixed-integer formulation. These systems include PWA, MLD, LC systems and ReLU networks. Our method comprises a learner and a verifier, which iteratively search for a Lyapunov function from a target class $\mathcal{F}$ of Lyapunov functions (e.g., quadratic or piecewise quadratic). In each iteration, the learner uses a set of samples of the hybrid system to localize $\mathcal{F}$ by a convex set $\tilde{\mathcal{F}} \supseteq \mathcal{F}$ and then solves a semidefinite program (SDP) to select a Lyapunov function candidate from $\tilde{\mathcal{F}}$. The verifier then solves a mixed-integer program in the state space to either validate the Lyapunov function candidate or reject it with a counterexample, i.e., a state where the Lyapunov condition fails. This counterexample is then added to the sample set of the learner to refine the set $\tilde{\mathcal{F}}$ of Lyapunov function candidates. By designing the alternation between the learner and the verifier according to the analytic center cutting-plane method (ACCPM), we show that when the set $\mathcal{F}$ of Lyapunov functions is full-dimensional and contains a norm ball with radius $\epsilon > 0$ in the parameter space, our method is guaranteed to find a Lyapunov function in $\mathcal{O}({n}^3/\epsilon^2)$ steps, where $n$ is the ambient dimension.
\subsection{Related work} \emph{LMI-based stability analysis of PWA systems}: Among various stability analysis methods for PWA dynamical systems~\cite{lin2009stability, sun2010stability}, linear matrix inequality (LMI)-based approaches are relatively prominent. These methods construct an SDP whose solution gives a valid Lyapunov function. For continuous-time PWA systems, LMI-based approaches to synthesize piecewise affine \cite{johansson2003piecewise}, piecewise quadratic (PWQ) \cite{johansson1997computation} and piecewise polynomial Lyapunov functions~\cite{prajna2003analysis} have been proposed. The adaptation of these Lyapunov function synthesis methods to handle discrete-time PWA systems is summarized in~\cite{biswas2005survey}. For discrete-time PWA systems, a common feature of the LMI-based methods is computing the transition map between all pairs of modes. This step may become time-consuming when the number of modes is large.
\noindent \emph{Sampling-based Synthesis Methods}: The iterative approach of alternating between a learning module and a verification module to synthesize a certificate for control systems is known as the Counter-Example Guided Inductive Synthesis (CEGIS) framework proposed by~\cite{solar2006combinatorial, solar2008program} in the verification community. The application of CEGIS to Lyapunov function synthesis for continuous-time nonlinear autonomous systems can be found in~\cite{ahmed2020automated, abate2020formal,kapinski2014simulation} using Satisfiability Modulo Theory (SMT) solvers for verification. In general, the termination of the iterative procedures in these works is not guaranteed. Notably, Ravanbakhsh et al.~\cite{ravanbakhsh2019learning} apply the CEGIS framework to synthesize control Lyapunov functions for nonlinear continuous-time systems and provide finite-step termination guarantees for the iterative algorithm through careful design of the learner which essentially implements the maximum volume ellipsoid cutting-plane method~\cite{Tarasov1988}. Our work differs from~\cite{ravanbakhsh2019learning} in the algorithm design and the application on hybrid systems. Other than the CEGIS framework, a learning-based approach to synthesize control barrier functions for hybrid systems is proposed in~\cite{lindemann2020learning}.
\subsection{Notations} We denote the set of real numbers by $\mathbb{R}$, the set of integers by $\mathbb{Z}$, the $n$-dimensional real vector space by $\mathbb{R}^n$, and the set of $n \times m$-dimensional real matrices by $\mathbb{R}^{n \times m}$. The standard inner product between two matrices $A, B \in \mathbb{R}^{n \times m}$ is given by $\langle A, B \rangle = \text{tr}(A^\top B)$ and the Frobenius norm of a matrix $A \in \mathbb{R}^{n \times m}$ is given by $\lVert A \rVert_F = (\text{tr}(A^\top A))^{1/2}$. Denote $\mathbb{S}^n$ the set of $n \times n$-dimensional symmetric matrices, and $\mathbb{S}^n_{+}$ ($\mathbb{S}^n_{++}$) the set of $n \times n$-dimensional positive semidefinite (definite) matrices. Given a set $\mathcal{S} \subseteq \mathbb{R}^{n_x + n_y}, \text{Proj}_x (\mathcal{S}) = \{ x \in \mathbb{R}^{n_x} \vert \exists y \in \mathbb{R}^{n_y} \text{ s.t. } (x, y) \in \mathcal{S}\}$ denotes the orthogonal projection of $\mathcal{S}$ onto the subspace $\mathbb{R}^{n_x}$. We denote $\text{int}(\mathcal{S})$ the set of all interior points in $\mathcal{S}$.
\section{Mixed-integer formulation of hybrid systems}
Consider a discrete-time autonomous hybrid system \begin{equation}\label{eq:nonlinear_sys} x_+ = f(x) \end{equation} where $x \in \mathbb{R}^{n_x}$ is the state and $f:\mathbb{R}^{n_x} \mapsto \mathbb{R}^{n_x}$ is a continuous function. Without loss of generality, assume system~\eqref{eq:nonlinear_sys} has an equilibrium at the origin, i.e., $0 = f(0)$. Let $\mathcal{R} \subset \mathbb{R}^{n_x}$ be the domain of the system and $\mathcal{R}$ is a compact set which contains the origin in its interior. Denote $x_k$ the state of system~\eqref{eq:nonlinear_sys} at time $k$ and $x_0$ the initial state. The nonlinear dynamics $x_+ = f(x)$ with domain $\mathcal{R}$ can be equivalently described through its graph defined as \begin{equation}\label{eq:graph} \text{gr}(f) = \{(x, y) \in \mathbb{R}^{n_x} \times \mathbb{R}^{n_x} \vert x \in \mathcal{R}, y = f(x) \}. \end{equation}
In this paper, we study the Lyapunov stability of the origin of~\eqref{eq:nonlinear_sys} for a class of hybrid systems that admit a mixed-integer formulation.
\begin{definition}[Mixed-integer formulation of a set~\cite{marcucci2019mixed}] \label{def:MIF}
For a set $\mathcal{Q} \subset \mathbb{R}^{n_z}$, consider the set $\mathcal{L}_\mathcal{Q} \subseteq \mathbb{R}^{n_z} \times \mathbb{R}^{ n_\lambda} \times \mathbb{Z}^{n_\mu}$ in a lifted space given by
\begin{equation}
\mathcal{L}_\mathcal{Q} = \{ (z \in \mathbb{R}^{n_z}, \lambda \in \mathbb{R}^{n_\lambda}, \mu \in \mathbb{Z}^{n_\mu}) \vert \ell(z, \lambda, \mu) \leq v\},
\end{equation}
with a function $\ell: \mathbb{R}^{n_z} \times \mathbb{R}^{ n_\lambda} \times \mathbb{Z}^{n_\mu} \rightarrow \mathbb{R}^{n_\ell}$ and a vector $v \in \mathbb{R}^{n_\ell}$. The set $\mathcal{L}_\mathcal{Q}$ is a mixed-integer formulation of $\mathcal{Q}$ if $\text{Proj}_z(\mathcal{L}_\mathcal{Q}) = \mathcal{Q}$. If the function $\ell$ is linear, we call the related formulation mixed-integer linear (MIL). \end{definition} In the next subsections, we show how to find mixed-integer formulations for PWA, MLD, LC systems and ReLU networks.
\subsection{Piecewise affine systems} \label{sec:PWA_representation} Consider a discrete-time piecewise affine system with control inputs \begin{equation} \label{eq:pwa_dynamics} x_+ = \psi_i(x, u) = A_i x + B_i u + c_i, \forall (x, u) \in \mathcal{R}_i, \end{equation} where $\mathcal{R}_i = \{(x, u) \in \mathbb{R}^{n_x} \times \mathbb{R}^{n_u} \vert F_i x + G_i u \leq h_i\}$ for $i \in \mathcal{I} = \{1, 2, \allowdisplaybreaks \cdots, N_{mode}\}$ are polyhedral partitions of the state-input space $\mathcal{R} = \bigcup_{i \in \mathcal{I}} \mathcal{R}_i$. We assume that the partitions $\mathcal{R}_i$ are bounded for all $i \in \mathcal{I}$. To make the PWA system~\eqref{eq:pwa_dynamics} well-posed, we assume that $\text{int}(\mathcal{R}_i) \cap \text{int}(\mathcal{R}_j) = \emptyset, \forall i \neq j$ and $f_i(x,u) = f_j(x,u), \forall (x,u) \in \mathcal{R}_i \cap \mathcal{R}_j$ if the intersection is not an empty set.
We denote the PWA dynamics~\eqref{eq:pwa_dynamics} collectively as $x_+ = \psi(x, u)$. The graph of $\psi(x,u)$ is given by
$\text{gr}(\psi) = \bigcup_{i\in \mathcal{I}} \text{gr} (\psi_i)$,
where each graph $\text{gr}(\psi_i)$ is defined as \begin{equation} \text{gr}(\psi_i) = \{(x, u, x_+) \vert Q_i [x^\top \ u^\top \ x_+^\top]^\top \leq q_i \}, \end{equation} with \begin{equation} Q_i = \begin{bmatrix} A_i & B_i & -I \\ -A_i & -B_i & I \\ F_i & G_i & 0 \end{bmatrix}, \quad q_i = \begin{bmatrix} -c_i \\ c_i \\h_i \end{bmatrix}. \end{equation}
In this paper, we apply a disjunctive programming-based formulation~\cite{marcucci2019mixed} which states that $x_+ = \psi(x,u)$ is equivalent to the following set of constraints~\cite{marcucci2019mixed} \begin{equation} \label{eq:pwa_MIL_constr} \begin{aligned} & F_i x_i + G_i u_i \leq \mu_i h_i, \ \mu_i \in \{0, 1\}, \ \forall i \in \mathcal{I}, \\ & 1 = \sum_{i \in \mathcal{I}} \mu_i, \ x = \sum_{i \in \mathcal{I}} x_i, \ u =\sum_{i\in \mathcal{I}} u_i \\ & x_+ = \sum_{i\in \mathcal{I}} (A_i x_i + B_i u_i + \mu_i c_i). \end{aligned} \end{equation} In the disjunctive programming formulation~\eqref{eq:pwa_MIL_constr}, we have $N_{mode}$ binary variables $\mu_i$ and $2N_{mode}$ auxiliary continuous variables $\{x_i, u_i\}$. The binary variable $\mu_i$ can be interpreted as the indicator of the mode where the state-input pair $(x, u)$ lives in. Note that $\mu_i = 1$ imposes $\mu_j = 0, \forall j \neq i$. By the boundedness of the partitions $\mathcal{R}_j$, we have $F_j x_j + G_j u_j \leq \mu_j h_j = 0 \Rightarrow x_j = 0, u_j = 0$, and correspondingly $x = \sum_{k \in \mathcal{I}} x_k = x_i, u = \sum_{k \in \mathcal{I}} u_k = u_i, x_+ = \sum_{k \in \mathcal{I}} (A_k x_k + B_k u_k + \mu_k c_k) = A_i x_i + B_i u_i + c_i$. We can also obtain a mixed-integer formulation of the PWA dynamics~\eqref{eq:pwa_MIL_constr} through the big-M method~\cite{vielma2015mixed}.
When the PWA system~\eqref{eq:pwa_dynamics} is interconnected with a controller which also has a mixed-integer formulation, we can describe the closed-loop dynamics through mixed-integer constraints. When an autonomous PWA system is considered, we obtain its mixed-integer formulation by removing the control input related variables in~\eqref{eq:pwa_MIL_constr}.
\subsection{Linear complementarity systems} \label{sec:LCS_representation} Consider a discrete-time linear complementarity system~\cite{heemels2001equivalence, heemels2000linear} \begin{subequations} \label{eq:LCS}
\begin{align}
\begin{split}
&x_+ = Ax + B_1 u + B_2 w
\end{split}\\
\begin{split}
&v = E_1 x + E_2 u + E_3 w + g_4
\end{split}\\
\begin{split} \label{eq:LC_constr}
&0 \leq v \perp w \geq 0
\end{split}
\end{align} \end{subequations} where $v, w \in \mathbb{R}^s$ and $\perp$ denotes that $v^\top w = 0$. The complementarity constraint~\eqref{eq:LC_constr} can be equivalently formulated as a set of mixed-integer linear constraints through the big-M method~\cite{vielma2015mixed} as \begin{equation} \label{eq:MIL_LCS} \begin{aligned} v_i w_i = 0 \Leftrightarrow 0 \leq v_i \leq \mu_i M_{1,i}, \ 0 \leq w_i \leq (1 - \mu_i) M_{2,i}, \end{aligned} \end{equation} where $\mu_i \in \{0, 1\}$ and the subscript $i$ denotes the $i$-th entry of the variable $v$ and $w$. The binary variable $\mu_i$ either forces $v_i$ to be zero ($\mu_i = 0$) or forces $w_i$ to be zero ($\mu_i = 1$). In general, selecting the big-M values $M_{1, i}$ and $M_{2, i}$ is no simple task~\cite{Kleinert2020lunch}. In practice, when the big-M values are hard to obtain, we can alternatively use the special-ordered set constraint in Gurobi~\cite{gurobi} which forces constraint~\eqref{eq:LC_constr} through a branching rule instead of specifying the big-M's explicitly.
One interesting application of the formulations in~\eqref{eq:LCS} and~\eqref{eq:MIL_LCS} is in the description of the closed-loop MPC systems. We refer the readers to~\cite{simon2016stability} for the details of this formulation.
\subsection{Mixed-logical dynamical systems} The mixed-logical dynamical system~\cite{bemporad1999control} can be written as \begin{align} &x_+ = A x + B_1 u + B_2 \delta + B_3 z \\ &E_1 x + E_2 u + E_3 \delta + E_4 z \leq g_5 \end{align} where $x = [x_r^\top \ x_b^\top]^\top$ with $x_r \in \mathbb{R}^{n_r}$ and $x_b \in \{0,1\}^{n_b}$ ($u$ has a similar structure), $z \in \mathbb{R}^{n_z}$ and $\delta \in \{0, 1\}^{n_\delta}$ are auxiliary variables. The MLD system is explicitly constructed through mixed-integer linear constraints.
\subsection{ReLU neural networks} Consider a PWA system in feedback with an $L$-layer ReLU neural network controller $u = \pi(x)$, where $\pi: \mathbb{R}^{n_x} \rightarrow \mathbb{R}^{n_u}$ is given by \begin{equation} \label{eq:nn_controller} \begin{aligned} z_0 &= x\\ z_{\ell+1} &= \max(W_\ell z_\ell + b_\ell, 0), \quad \ell = 0, \cdots, L -1 \\ \pi(x) &= W_{L} z_{L} + b_L. \end{aligned} \end{equation} Here $z_0 = x \in \mathbb{R}^{n_0} \ (n_0 = n_x)$ is the input to the neural network, $z_{\ell+1} \in \mathbb{R}^{n_{\ell+1}}$ is the vector representing the output of the $(\ell+1)$-th hidden layer with $n_{\ell+1}$ neurons, $\pi(x) \in \mathbb{R}^{n_{L+1}} \ (n_{L+1} = n_u)$ is the output of the neural network, and $W_\ell \in \mathbb{R}^{n_{\ell + 1} \times n_\ell}, b_\ell \in \mathbb{R}^{n_{\ell + 1}}$ are the weight matrix and the bias vector of the $(\ell+1)$-th hidden layer, respectively.
Consider a scalar ReLU function $y=\max(0,x)$ where $\underline{x} \leq x \leq \bar{x}$. Then it can be shown that the ReLU function admits the following mixed-integer representation~\cite{tjeng2017evaluating}, \begin{equation} \begin{aligned} &y=\max(0,x), \ \underline{x} \leq x \leq \bar{x} \iff \\ &y \geq 0, \ y \geq x, y \leq x - \underline{x} (1-t), \ y \leq \bar{x} t, \ t \in \{0,1\}, \end{aligned} \end{equation} where the binary variable $t \in \{0,1\}$ is an indicator of the activation function being active ($y=x$) or inactive ($y=0$). For a ReLU network described by the equations in \eqref{eq:nn_controller}, let $\underline{m}^{\ell}$ and $\bar{m}^{\ell}$ be the element-wise lower and upper bounds on the input to the $(\ell+1)$-th activation layer, i.e., $\underline{m}_{\ell} \leq W_{\ell} z_{\ell} +b_{\ell}\leq \bar{m}_{\ell}$. Then the neural network equations are equivalent to a set of mixed-integer constraints:
\begin{equation} \begin{aligned} \label{eq:MIL_NN} & z_{\ell+1} = \max(W_\ell z_\ell + b_\ell,0) \iff \\ & \begin{cases} z_{\ell+1} \geq W_\ell z_\ell + b_\ell \\ z_{\ell+1} \leq W_\ell z_\ell + b_\ell - \mathrm{diag}(\underline{m}_\ell) (\mathbf{1}-t_\ell) \\ z_{\ell+1} \geq 0 \\ z_{\ell+1} \leq \mathrm{diag}(\bar{m}_\ell) t_\ell, \end{cases} \end{aligned} \end{equation}
where $t_{\ell} \in \{0,1\}^{n_{\ell+1}}$ is a vector of binary variables for the $(\ell+1)$-th activation layer. We note that the element-wise pre-activation bounds $\{\underline{m}_{\ell} , \bar{m}_{\ell}\}$ can be found by, for example, interval bound propagation or linear programming assuming known bounds on the input of the neural network \cite{weng2018towards,hein2017formal,wong2018provable}. Combined with the MIL formulation of PWA systems shown in Section~\ref{sec:PWA_representation}, the closed-loop dynamics of a ReLU neural network controlled PWA system admits a mixed-integer formulation.
\section{Stability Analysis via Lyapunov functions}
The convergence behavior of system~\eqref{eq:nonlinear_sys} around its equilibrium points can be studied by Lyapunov stability.
\begin{definition} \normalfont{(Lyapunov stability \cite[Chapter~13]{haddad2011nonlinear})} \label{def:Lyap_stability}
The equilibrium point $x = 0$ of the autonomous system~\eqref{eq:nonlinear_sys} is
\begin{itemize}
\item \textbf{Lyapunov stable} if for each $\epsilon >0$, there exists $\delta = \delta(\epsilon)$ such that if
$\lVert x_0 \rVert < \delta$, then $\lVert x_k \rVert < \epsilon, \forall k \geq 0$.
\item \textbf{asymptotically stable} if it is Lyapunov stable and there exists $\delta > 0$ such that if
$\lVert x_0 \rVert < \delta$, then $\lim_{k\rightarrow \infty} \|x_k\| = 0$.
\end{itemize} \end{definition}
Since the hybrid dynamics~\eqref{eq:nonlinear_sys} is nonlinear, Lyapunov stability is often a local property and it is of interest to estimate its region of attraction defined as \begin{definition}[Region of attraction]
The region of attraction $\mathcal{O}$ of the nonlinear system~\eqref{eq:nonlinear_sys} is the set of states from which the trajectory of system~\eqref{eq:nonlinear_sys} converges to the origin, i.e., $\mathcal{O} = \{ x_0 \in \mathcal{R} \vert \lim_{t \rightarrow \infty} x_t = 0\}$. \end{definition}
In this work, we do not assume the domain $\mathcal{R}$ to be positive invariant. Instead, we introduce the region of interest (ROI) $\mathcal{X} \subseteq \mathcal{R}$ which is a polytopic set given by \begin{equation} \mathcal{X} := \{ x \in \mathbb{R}^{n_x} \vert F_\mathcal{X} x \leq h_\mathcal{X}\}, \end{equation} to guide our search for an inner approximation $\tilde{\mathcal{O}}$ of the ROA $\mathcal{O}$. We can certify the asymptotic stability of system~\eqref{eq:nonlinear_sys} and find an $\tilde{\mathcal{O}}$ by constructing Lyapunov functions defined in the following theorem:
\begin{theorem} \label{thm:Lyapunov} \normalfont{\cite[Chapter~13]{haddad2011nonlinear}}
Consider the discrete-time nonlinear hybrid system~\eqref{eq:nonlinear_sys}. If there is a continuous function $V(x): \mathcal{R} \mapsto \mathbb{R}$ with domain $\mathcal{R}$ such that
\begin{subequations}
\label{eq:Lyap_conditions}
\begin{align}
\begin{split} \label{eq:Lyap_pos_constr}
& V(0) = 0 \text{ and } V(x) > 0, \forall x \in \mathcal{X} \setminus \{0\}
\end{split} \\
\begin{split} \label{eq:general_Lyap_cond_3}
& V(f(x)) - V(x) \leq 0, \forall x \in \mathcal{X},
\end{split}
\end{align}
\end{subequations}
where the set $\mathcal{X}$ is the region of interest (ROI) satisfying $\mathcal{X} \subseteq \mathcal{R}$ and $0 \in \text{int}(\mathcal{X})$, then the origin is Lyapunov stable. If, in addition,
\begin{equation}\label{eq:Lyap_diff_constr}
V(f(x)) - V(x) < 0, \forall x \in \mathcal{X} \setminus \{ 0 \},
\end{equation}
then the origin is asymptotically stable. \end{theorem} We call any $V(\cdot)$ satisfying~\eqref{eq:Lyap_pos_constr} a Lyapunov function \emph{candidate}. If additionally, $V(\cdot)$ satisfies the condition~\eqref{eq:general_Lyap_cond_3} or~\eqref{eq:Lyap_diff_constr}, then $V(\cdot)$ is called a \emph{valid} Lyapunov function candidate, or simply, a Lyapunov function. Since asymptotic stability is our primary focus in this paper, Lyapunov functions refer to any $V(\cdot)$ satisfying constraints~\eqref{eq:Lyap_pos_constr} and~\eqref{eq:Lyap_diff_constr} unless specified otherwise. Once a Lyapunov function $V(x)$ is obtained, an inner estimate of the ROA is given by
$\mathcal{\tilde{O}} = \{ x \vert V(x) \leq \tau\}$,
where $\tau = \inf_{x \in \mathcal{R} \setminus \mathcal{X}} V(x)$. In other words, $\mathcal{\tilde{O}} \subset \mathcal{X}$ is the largest sublevel set of $V(x)$ that is contained in $\mathcal{X}$.
\begin{remark} The ROI $\mathcal{X}$ can be set according to prior knowledge or from simulation of the nonlinear hybrid dynamics~\eqref{eq:nonlinear_sys}. See Section~\ref{sec:simulation} for examples of choosing the ROI. \end{remark}
\subsection{Lyapunov function parameterization} Searching for a Lyapunov function in the function space is intractable since the problem is infinite-dimensional in this space. Instead, we reduce our search space to the class of Lyapunov functions defined by \begin{equation} \label{eq: multi-step Lyapunov} V^{(k)}(x;P) = {z^{(k)}}^\top P z^{(k)}, \end{equation} where $P \in \mathbb{S}^{(k+1)n_x}_{++}$, and $z^{(k)}$ is given by \begin{equation} \label{eq:basis} z^{(k)} = [x^\top \quad f(x)^{(1),\top} \quad f^{(2), \top}(x) \quad \cdots \quad f^{(k), \top}(x)]^\top \end{equation} Here we use the notation $f^{(0)}(x) = x, f^{(k+1)} = f(f^{(k)}(x)), \forall k \geq 0$. We call $V^{(k)}(x;P)$ the Lyapunov function candidate of order $k$, which is a quadratic function in composition with the system dynamics $f(x)$ evolved for $k$ steps. Indeed, we can increase the complexity of the function class monotonically by increasing the order $k$.
Since $P$ is positive definite, searching for a valid Lyapunov function of the form \eqref{eq: multi-step Lyapunov} reduces to find a matrix $P$ that satisfies the Lyapunov difference condition ~\eqref{eq:Lyap_diff_constr}. Explicitly, we can characterize the space of matrices $P$ that admit a valid Lyapunov function candidate as
\begin{equation} \label{eq:target_set} \mathcal{F} \!= \!\{P \in \mathbb{S}^{(k+1)n_x} \vert \alpha I \! \preceq P \! \preceq \!\beta I, \Delta V^{(k)}(x, P) < 0, \forall x \in \mathcal{X} \setminus \{0\} \}. \end{equation} where $0 \leq \alpha < \beta$, and $\Delta V^{(k)}(x, P)$ is the Lyapunov difference given by \begin{equation} \Delta V^{(k)}(x, P) := V^{(k)}(f(x);P) - V^{(k)}(x;P). \end{equation}
The constraint $\alpha I \preceq P$ guarantees the condition~\eqref{eq:Lyap_pos_constr}, while the constraint $P \preceq \beta I$ is imposed to make $\mathcal{F}$ bounded without loss of generality since we can always scale $P$ while satisfying~\eqref{eq:Lyap_diff_constr}.
We call $\mathcal{F}$ the target set. It follows that $\mathcal{F}$ is convex since it is defined by semidefinite as well as linear constraints on $P$. As $\alpha I \preceq P \preceq \beta I$ imposes an additional constraint on the condition number of $P$, in practice we choose $\beta/\alpha$ large or simply set $\alpha = 0$~\footnote{As will be shown next, the proposed method always finds a feasible solution in the interior of $\mathcal{F}$. Therefore, choosing $\alpha = 0$ does not affect the positive definiteness of the solution.}. Then finding a Lyapunov function in a given function class $V^{(k)}(x;P)$ is stated as the following problem:
\begin{problem}\label{prob:target_set}
For each parameterized function class $V^{(k)}(x;P)$ and the target set $\mathcal{F}$ defined in~\eqref{eq:target_set}, find a feasible point in $\mathcal{F}$ or certify that $\mathcal{F}$ is empty. \end{problem}
Although the target set $\mathcal{F}$ is convex, the fact that it is characterized by infinitely many linear constraints (i.e., the constraints $\Delta V^{(k)}(x, P)$ $<0, \forall x \in \mathcal{X} \setminus \{0\}$) poses computational challenges to solving Problem~\ref{prob:target_set}. In this work, we propose a learning-based approach to address this challenge by iteratively drawing state samples to refine our over-approximation of the target set $\mathcal{F}$. By designing the learning strategy based on ACCPM from convex optimization, we show that when the target set $\mathcal{F}$ is full-dimensional in the parameter space of $P$, our method is guaranteed to find a feasible point in $\mathcal{F}$ in a finite number of steps.
\begin{remark}
The parameterization of $V^{(k)}(x;P)$ is inspired by the finite-step Lyapunov function~\cite{aeyels1998new, bobiti2016sampling} and the non-monotonic Lyapunov functions~\cite{ahmadi2008non} which also construct Lyapunov function candidates using the system states several steps ahead. $V^{(k)}(x;P)$ allows us to parameterize complex function classes with a relatively small number of parameters. For example, when $f(x)$ is a PWA function with $N_{mode}$ (possibly large) partitions, $V^{(1)}(x;P)$ is a PWQ function with the same partitions in the state space. \end{remark}
\section{Learning Lyapunov functions from counterexamples}
\label{sec:ACCPM}
We recall from the previous section that finding a feasible point of the convex set $\mathcal{F}$ is computationally intractable since the condition $\Delta V^{(k)}(x;P) < 0$ must hold for all $x \in \mathcal{X} \setminus \{0\}$. To overcome this intractability, we adopt a learning-based approach, in which we first select a set of finite samples $\mathcal{S} = \{x^1, x^2, \cdots, x^N\} \subset \mathcal{X}$ and then enforce the linear constraint $\Delta V^{(k)}(x;P) < 0$ to hold only for $x \in \mathcal{S}$. This results in an over-approximation of $\mathcal{F}$ given by
\begin{equation} \label{eq:localization}
\tilde{\mathcal{F}} = \{ P \in \mathbb{S}^{(k+1)n_x} \vert \alpha I \preceq P \preceq \beta I, \Delta V^{(k)}(x, P) \leq 0, \forall x \in \mathcal{S}\}. \end{equation} We call $\tilde{\mathcal{F}}$ the localization set. Finding a feasible point in $\tilde{\mathcal{F}}$ now becomes a tractable convex feasibility problem. However, there is no guarantee that a $P \in \tilde{\mathcal{F}}$ would correspond to a valid Lyapunov function, even if the number of samples approaches infinity. As one of our contributions, we propose an efficient learning strategy based on the analytic center cutting-plane method (ACCPM) to iteratively grow the sample set and refine the set $\tilde{\mathcal{F}}$ until we find a feasible point in $\mathcal{F}$. In the next subsections, we describe the proposed approach and provide finite-step termination guarantees when $\mathcal{F}$ is non-empty and satisfies the following assumption:
\begin{assumption} \label{assump:nonempty} The target set $\mathcal{F}$ defined in~\eqref{eq:target_set} is full-dimensional and there exists $P_{center}\in \mathbb{S}^{(k+1)n_x}$ such that $\{P \in \mathbb{S}^{(k+1)n_x} \vert \lVert P - P_{center} \rVert_F \leq \epsilon \} \subset \mathcal{F}$ where $\lVert \cdot \rVert_F$ is the Frobenius norm. \end{assumption}
\subsection{Analytic center cutting-plane method} Cutting-plane methods~\cite{atkinson1995cutting, elzinga1975central,boyd2007localization} are iterative algorithms to find a point in a target convex set $\mathcal{F}$ or to determine whether $\mathcal{F}$ is empty. In these methods, we have no information on $\mathcal{F}$ except for a ``cutting-plane oracle'', which can verify whether $P \in \mathcal{F}$ for a given $P$. Let $\tilde{\mathcal{F}}$ be a localization set defined by a finite set of inequalities that over approximates the target set, i.e., $\mathcal{F} \subseteq \tilde{\mathcal{F}}$. If $\tilde{\mathcal{F}}$ is empty, then we have proof that the target set $\mathcal{F}$ is also empty. Otherwise, we query the oracle at a point $P \in \tilde{\mathcal{F}}$. If $P \in \mathcal{F}$, the oracle returns `yes' and the algorithm terminates; if $P \notin \mathcal{F}$, it returns `no' together with a separating hyperplane that separates $P$ and $\mathcal{F}$. In the latter case, the cutting-plane method updates the localization set by $\tilde{\mathcal{F}} \leftarrow \tilde{\mathcal{F}} \cap \{\text{half space defined by the separating hyperplane}\}$ as shown in Fig.~\ref{fig:cutting_plane_demo}. This process continues until either a point in the target set is found or the target set is certified to be empty.
Based on how the query point is chosen, different cutting-plane methods have been proposed including the center of gravity method \cite{Lev65}, the maximum volume ellipsoid (MVE) cutting-plane method \cite{Tarasov1988}, the Chebyshev center cutting-plane method \cite{elzinga1975central}, the ellipsoid method \cite{khachiyan1980polynomial, ellipsoidYudin}, and the analytic center cutting-plane method \cite{goffin1993computation, nesterov1995cutting, atkinson1995cutting}. In this paper, we use the analytic center cutting-plane method since it allows the localization set $\tilde{\mathcal{F}}$ to be described by linear matrix inequalities. In the ACCPM, the query point $P$ is chosen as the analytic center of the localization set $\tilde{\mathcal{F}}$.
\begin{figure}\label{fig:cutting_plane_demo}
\end{figure}
\begin{definition}[Analytic center~\cite{boyd2004convex}] \label{def:analytic_center}
The analytic center $x_{ac}$ of a set of convex inequalities and linear equalities
$h_i(x) \leq 0, i = 1,\! \cdots\!, m, \ Fx = g$,
is defined as the solution of the convex problem
\begin{equation} \label{eq:analytic_center}
\begin{aligned}
\underset{x}{\mathrm{minimize}} & \quad - \sum_{i=1}^m \log(-h_i(x)) \quad \text{subject to} \quad Fx = g.
\end{aligned}
\end{equation} \end{definition}
We design the learning strategy according to the ACCPM by constructing a learner, which proposes Lyapunov function candidates based on a set of samples, and a verifier, which serves as a cutting-plane oracle and updates the sample set with counterexamples.
\subsection{The learner} Let $\mathcal{S} = \{x^1, x^2, \cdots, x^N \} \subset \mathcal{X}$ be a collection of samples from $\mathcal{X}$. The localization set $\tilde{\mathcal{F}}$ in the space of $P$ is given by~\eqref{eq:localization},
which represents the learner's knowledge about $\mathcal{F}$ by observing the $k$-step trajectories of the dynamical system~\eqref{eq:nonlinear_sys} starting from $\mathcal{S}$. According to the ACCPM, the learner proposes a Lyapunov function candidate $V^{(k)}(x; P_{ac})$ with $P_{ac}$ as the analytic center of $\tilde{\mathcal{F}}$:
\begin{equation}\label{eq:ac_optimization} \begin{aligned} P_{ac}: = \underset{P}{\text{argmin}} & \quad -\sum_{x\in \mathcal{S}} \log( -\Delta V^{(k)}(x,P) ) \\ &\quad \quad \quad - \log\det (\beta I - P) - \log \det (P - \alpha I). \end{aligned} \end{equation} This is a convex program that can be solved efficiently through off-the-shell convex optimization solvers. We denote the objective function in~\eqref{eq:ac_optimization} as $\phi(\tilde{\mathcal{F}})$ and call it the potential function on the set $\tilde{\mathcal{F}}$. If \eqref{eq:ac_optimization} is infeasible, then we have a proof that no Lyapunov function exists in the function class of order $k$. Otherwise, the learner proposes $V^{(k)}(x; P_{ac}) = {z^{(k)}}^\top P_{ac} z^{(k)}$ as a Lyapunov function candidate. Due to the log-barrier function in~\eqref{eq:ac_optimization}, $P_{ac}$ is in the interior of $\tilde{\mathcal{F}}$. When the sample set $\mathcal{S}$ is empty, the potential function simply becomes $\phi(\tilde{\mathcal{F}}) = -\log \det(\beta I - P) - \log \det (P - \alpha I)$ with $P_{ac} = \frac{\alpha + \beta}{2}I$ as the optimal solution.
\subsection{The verifier} Suppose the learner proposes the Lyapunov function candidate $V^{(k)}(x; P)$ by solving~\eqref{eq:ac_optimization}. Given $V^{(k)}(x; P)$, the verifier either ensures that this function satisfies the constraints in~\eqref{eq:Lyap_pos_constr} and~\eqref{eq:Lyap_diff_constr}, or returns a state where constraint~\eqref{eq:Lyap_pos_constr} or~\eqref{eq:Lyap_diff_constr} is violated as a counterexample. Since the log-barrier function in~\eqref{eq:ac_optimization} guarantees $P \succ 0$, constraint~\eqref{eq:Lyap_pos_constr} is readily satisfied and the verifier must check the violation of constraint~\eqref{eq:Lyap_diff_constr}. This can be done by solving the optimization problem
\begin{alignat}{2} \label{eq:verifier_formulation} &\underset{x \in \mathcal{X} \setminus \{0\}}{\mathrm{maximize}} \quad && \Delta V^{(k)}(x,P). \end{alignat}
When $f(x)$ has a mixed-integer formulation \begin{equation*}
\mathcal{L}_{\text{gr}(f)} = \{(x, x_+, \lambda, \mu) \in \mathbb{R}^{n_x} \times \mathbb{R}^{n_x} \times \mathbb{R}^{n_\lambda}\times \mathbb{Z}^{n_\mu} \vert \ell(x, x_+, \lambda, \mu) \leq v \}, \end{equation*} for Lyapunov function candidates $V^{(k)}(x;P)$ of order $k$, we write problem~\eqref{eq:verifier_formulation} explicitly as a mixed-integer quadratic program (MIQP):
\begin{subequations} \label{eq:MIQP_f}
\begin{align}
\begin{split}\label{eq:MIQP_f_obj}
\underset{ \{x_i\}, \{\lambda_i\}, \{\mu_i\} }{\text{maximize}} & \quad \begin{bmatrix}
x_1 \\ x_2 \\ \vdots \\ x_{k+1}
\end{bmatrix}^\top P \begin{bmatrix}
x_1 \\ x_2 \\ \vdots \\ x_{k+1}
\end{bmatrix} -
\begin{bmatrix}
x_0 \\ x_1 \\ \vdots \\ x_{k}
\end{bmatrix}^\top P \begin{bmatrix}
x_0 \\ x_1 \\ \vdots \\ x_{k}
\end{bmatrix}
\end{split} \\
\begin{split} \label{eq:MIQP_f_ROI}
\text{subject to} & \quad x_0 \in \mathcal{X}
\end{split} \\
\begin{split} \label{eq:MIQP_f_exclusion}
& \quad \lVert x_0 \rVert_\infty \geq \epsilon
\end{split}\\
\begin{split} \label{eq:MIQP_f_MIL}
& \quad \ell(x_i, x_{i+1}, \lambda_i, \mu_i) \leq v_i, i = 0, 1, \cdots, k
\end{split}
\end{align} \end{subequations} where the objective function~\eqref{eq:MIQP_f_obj} is equal to the Lyapunov difference $\Delta V^{(k)}(x_0,P)$, constraint~\eqref{eq:MIQP_f_ROI} restricts the search of counterexamples inside the ROI $\mathcal{X}$, constraint~\eqref{eq:MIQP_f_exclusion} excludes a small $\ell_\infty$ norm ball centered at the origin $B_\epsilon = \{ x \vert \lVert x \rVert_\infty < \epsilon \}$ from the search space, and constraints~\eqref{eq:MIQP_f_MIL} enforce the dynamical constraint $(x_i, x_{i+1}) \in \text{gr}(f)$, or equivalently $x_{i+1}= f(x_i)$ for $i =0, \cdots, k$.
Denote $p^*$ the optimal value and $x_0^*$ the $x_0$-component of the optimal solution of~\eqref{eq:MIQP_f}. When $p^* \geq 0$, $x_0^*$ is a counterexample for the Lyapunov function candidate $V^{(k)}(x;P)$ since $\Delta V^{(k)}(x_0^*, P) \geq 0$. Then, the separating hyperplane induced by the counterexample $x_0^*$ is given as $\Delta V^{(k)}(x_0^*, P) = 0$ where $\Delta V^{(k)}(x_0^*, P)$ is interpreted as a linear function in the matrix variable $P$. When $p^* < 0 $, we certify the convergence of system trajectories to a neighborhood of the origin as shown in the following corollary. \begin{corollary} \label{coro:convergence}
Let $\Omega = \{ x \vert \allowdisplaybreaks V^{(k)}(x;P) \leq \allowdisplaybreaks \gamma\}$ be the largest sublevel set of $V^{(k)}(x;P)$ inside $\mathcal{X}$ with $\gamma = \inf_{x \in \mathcal{R}\setminus \mathcal{X}} V^{(k)}(x;P)$. Define the successor set of $B_\epsilon$ as $\text{suc}(B_\epsilon):=\{y\vert y = f(x), x \in B_\epsilon \}$. Assume $B_\epsilon \subset \Omega$, $\text{suc}(B_\epsilon) \subset \Omega$, and $\Omega_{loc}$ is the smallest sublevel set of $V^{(k)}(x;P)$ such that $\text{suc}(B_\epsilon) \subseteq \Omega_{loc}$. If $p^* < 0$, then we have $\lim_{t \rightarrow \infty} x_t \in \Omega_{loc}$ for all $x_0 \in \Omega$. \end{corollary}
\begin{proof}
First, note that all trajectories starting from $x_0 \in \Omega$ and $x_0 \notin B_\epsilon$ reach $B_\epsilon$ in a finite number of steps since $V^{(k)}(x_{t+1};P) - V^{(k)}(x_t; P) \leq p^* < 0$ as long as $x_t \notin B_\epsilon$ and $V^{(k)}(x_t; P)$ is lower-bounded by $0$ for all $x_t$. If $x_t$ never reaches $B_\epsilon$, we will have a contradiction that $V^{(k)}(x_N; P) < 0$ for some $N$. After the trajectories reach $B_\epsilon$, the subsequent states will remain inside $\Omega_{loc}$ by construction. Therefore, we have $\lim_{t \rightarrow \infty} x_t \in \Omega_{loc}$ for all $x_0 \in \Omega$. \end{proof}
The alternation between the learner and the verifier is summarized in Algorithm~\ref{alg:ACCPM}. In the next subsection, we show that when the target set $\mathcal{F}$ satisfies Assumption~\ref{assump:nonempty}, our proposed algorithm is guaranteed to find a feasible point in $\mathcal{F}$ in a finite number of steps.
\begin{algorithm}[htb] \caption{Learning-based Lyapunov function synthesis}\label{alg:ACCPM}
\begin{algorithmic}[1]
\Procedure{LearningLyapunov}{}
\State $\mathcal{S}_1 = \emptyset$ \Comment{initialize the sample set}
\State $i=1$
\While {True}
\State Generate $\tilde{\mathcal{F}}_i$ from sample set $\mathcal{S}_i$ by~\eqref{eq:localization}.
\If {$\tilde{\mathcal{F}}_{i} = \emptyset$}
\State Return: Status = Infeasible, $P_* = \emptyset$.
\EndIf
\State $P^{(i)} = \arg \min \eqref{eq:ac_optimization} \text{ with } $$\mathcal{S}_i$ \Comment{find analytic center}
\State solve~\eqref{eq:MIQP_f} with $P^{(i)}$ \Comment{query the verifier}
\If {$\max \ $\eqref{eq:MIQP_f} < 0}
\State Return: Status = Feasible, $P_* = P^{(i)}$.
\Else
\State $x_*^{(i)} = \arg \min_{x_0} \eqref{eq:MIQP_f}$ \Comment{find a counterexample}
\State $\mathcal{S}_{i+1} = \mathcal{S}_i \cup \{x^{(i)}_*\}$
\EndIf
\State $i = i+1$
\EndWhile
\EndProcedure
\end{algorithmic} \end{algorithm}
\begin{remark} In this paper, the verifier is constructed as a global optimization problem~\eqref{eq:verifier_formulation}. Alternative constructions of the verifier do exist, e.g., by using SMT solvers as shown in~\cite{ahmed2020automated, abate2020formal}. The termination results in the next subsection will always hold no matter how the verifier is formulated. \end{remark}
\subsection{Convergence Analysis} The convergence and complexity of the ACCPM have been studied in~\cite{atkinson1995cutting,nesterov1995cutting, ye1992potential, luo2000polynomial,goffin1996complexity, sun2002analytic} under various assumptions on the localization set, the form of the separating hyperplane, whether multiple cuts are applied, etc. Directly related to Algorithm~\ref{alg:ACCPM} and the search for $V^{(k)}(x;P)$ is~\cite{sun2002analytic} which analyzes the complexity of the ACCPM with a matrix variable and semidefiniteness constraints. Notably, it provides an upper bound on the number of iterations that Algorithm~\ref{alg:ACCPM} can run before termination when the target set is non-empty. In~\cite{sun2002analytic}, it is assumed that \begin{itemize}
\item A1: $\mathcal{F}$ is a convex subset of $\mathbb{S}^{n}$.
\item A2: See Assumption~\ref{assump:nonempty}.
\item A3: $\mathcal{F} \subset \{P \in \mathbb{S}^{n} \vert 0 \preceq P \preceq I \}$. \end{itemize} For the Lyapunov function candidate class of order $k$, we have $n = (k+1)n_x$. Let the ACCPM start with the localization set $\tilde{\mathcal{F}}_1 = \{ P \vert 0 \preceq P \preceq I\}$ and initialize the first query point $P^{(1)} = \frac{1}{2}I$ correspondingly. If at iteration $i$ a query point $P^{(i)}$ is rejected by the oracle, a separating hyperplane of the form $\langle D_i, P - P^{(i)} \rangle = 0$ with $\lVert D_i \rVert_F = 1$ is given by the verifier. By induction, we have that at iteration $i > 1$, the localization set $\tilde{\mathcal{F}_{i}}$ is \begin{equation*} \tilde{\mathcal{F}_{i}} = \{ P \vert 0 \preceq P \preceq I, \langle D_j, P \rangle \leq c_j, j = 1, \cdots, i-1 \}. \end{equation*} with $D_j$ defining the separating hyperplane and $c_j = \langle D_j, P^{(j)} \rangle$. The query point at iteration $i$ is given by $P^{(i)} = \arg\min \phi(\tilde{\mathcal{F}_{i}})$ with the potential function \begin{equation*} \phi(\tilde{\mathcal{F}_{i}}) = -\sum_{j = 1}^{i-1} \log(c_j - \langle D_j, P\rangle) - \log \det (I -P) - \log \det(P). \end{equation*}
\begin{theorem} \label{thm:ACCPM_termination} Under Assumption~\ref{assump:nonempty} on the target set $\mathcal{F}$, Algorithm~\ref{alg:ACCPM} finds a feasible point in~$\mathcal{F}$ in at most $O(((k+1)n_x)^3/\epsilon^2)$ iterations. \end{theorem}
\begin{proof} The proof follows from~\cite{sun2002analytic} which states for a general target set $\mathcal{F}\subset \mathbb{S}^{n}$ under assumptions A$1$ to A$3$, the analytic center cutting-plane method with separating hyperplanes of the form $\langle D_i, P \rangle \leq c_i$ is shown to find a feasible point in at most $O(n^3/\epsilon^2)$ iterations. For the sequence of localization sets $\tilde{\mathcal{F}}_i$, \cite{sun2002analytic} computes an upper bound on the potential function $\phi(\tilde{\mathcal{F}}_i)$ which is approximately $i \log( \frac{1}{\epsilon})$, and a lower bound on $\phi(\tilde{\mathcal{F}}_i)$ which is proportional to $\frac{i}{2}\log(\frac{i}{n^3})$. Since the ACCPM must terminate before the lower bound exceeds the upper bound of $\phi(\tilde{\mathcal{F}}_i)$, we obtain the $O(n^3/\epsilon^2)$ upper bound on the number of iterations.
To show that the result in~\cite{sun2002analytic} applies to Algorithm~\ref{alg:ACCPM}, note that for each counterexample $x_*^{(i)}$ found by the verifier in iteration $i$, the separating hyperplane can be constructed as $\Delta V^{(k)}(x_*^{(i)}, P) \leq \Delta V^{(k)}(x_*^{(i)},P^{(i)})$ since $\Delta V^{(k)}(x_*^{(i)},P^{(i)}) \geq 0$. We can rewrite this cutting plane in the form $\langle D_i, P \rangle \leq \langle D_i, P^{(i)} \rangle$ by setting $\hat{D}_i = z_{k,*}^{+, (i)} z_{k,*}^{+,(i), \top} - z_{k,*}^{(i)} z_{k,*}^{(i), \top}, D_i = \hat{D}_i / \lVert \hat{D}_i \rVert_F$, where $z_{k,*}^{(i)}$ denotes the basis~\eqref{eq:basis} with $x = x_*^{(i)}$, and $z_{k,*}^{+, (i)}$ denotes the basis~\eqref{eq:basis} after setting $x = f(x_*^{(i)})$. Then with $\alpha = 0, \beta = 1$ in~\eqref{eq:target_set}, Algorithm~\ref{alg:ACCPM} satisfies assumptions A$1$ to A$3$ and it terminates in at most $O(((k+1)n_x)^3/\epsilon^2)$ iterations according to~\cite{sun2002analytic}. \end{proof}
Theorem~\ref{thm:ACCPM_termination} provides a finite-step termination guarantee for Algorithm~\ref{alg:ACCPM} when the target set satisfies Assumption~\ref{assump:nonempty}. However, when $\mathcal{F}$ is empty, we do not have such a guarantee. Certification of the non-existence of Lyapunov functions in $V^{(k)}(x;P)$ relies on detecting that an over-approximation $\tilde{\mathcal{F}}$ is empty. Hence, a quick expansion of the sample set $\mathcal{S}$ as shown in Section~\ref{sec:early_termination} is preferred.
\subsection{Implementation}
\subsubsection{Complexity of the learner} For a sample set $\mathcal{S}$ with $N$ samples, the localization set $\tilde{\mathcal{F}}$ in ~\eqref{eq:localization} is described by $N$ linear as well as two semidefinite constraints on the variable $P \in \mathbb{S}^{(k+1)n_x}$. We first decide if $\tilde{\mathcal{F}}$ is empty by solving an SDP feasibility problem. If the SDP is infeasible, then $\tilde{\mathcal{F}} = \emptyset$ and so is $\mathcal{F}$. If $\mathcal{\mathcal{F}}$ is non-empty, we move on to the analytic center problem~\eqref{eq:ac_optimization} which can be solved, e.g., through an infeasible start Newton's method~\cite{boyd2004convex}.
\subsubsection{Complexity of the MIQP} Since MIQP is well-known to be NP-hard, we use the number of binary variables, which we denote by $N_{\mu}$ in~\eqref{eq:MIQP_f}, as a rough measure of the complexity of~\eqref{eq:MIQP_f}. When $f(x)$ is a PWA function with $N_{mode}$ modes, we have $N_{mode}$ binary variables in the MIL formulation of $f(x)$. For the LC system~\eqref{eq:LCS}, $N_\mu$ equals the dimension of the orthogonal variables $v$ and $w$. $N_\mu$ is explicitly given in the MLD system and is equal to the number of neurons in the mixed-integer formulation of the ReLU networks. It follows that for the LC systems and ReLU networks, it is possible to use a small number of integer variables to encode a PWA system with many more modes. Since we need to evaluate the hybrid dynamics $f(x)$ for $k$ times when $V^{(k)}(x;P)$ is applied, the number of binary variables $N_\mu$ in~\eqref{eq:MIQP_f} is largely linear in the order $k$.
The actual solving time of the MIQP has a complex dependence not only on the number of variables and constraints but also on how the constraints are formulated. The exploration of the numerical performance of the proposed algorithm is left for future research.
\subsubsection{Solvability of the MIQP}
The optimization problem~\eqref{eq:MIQP_f} is a nonconvex MIQP since the quadratic objective function is indefinite. Therefore, the relaxation of the problem after removing the integrality constraints would result in a nonconvex quadratic program. Nonconvex MIQP can be solved to global optimality through Gurobi v$9.0$~\cite{gurobi} by transforming the nonconvex quadratic expression into a bilinear form and applying spatial branching~\cite{belotti2013mixed}. More information on solving nonconvex mixed-integer nonlinear programming can be found in~\cite{vigerske2013decomposition, tawarmalani2013convexification, belotti2009branching}. In this paper, we rely on Gurobi to solve the nonconvex MIQP~\eqref{eq:MIQP_f} automatically.
\subsubsection{Exclusion of the origin} In constraint~\eqref{eq:MIQP_f_exclusion}, we add a guard $B_\epsilon$ at the origin to approximate the exact constraint $x\in \mathcal{X} \setminus \{0\}$. Since constraint~\eqref{eq:MIQP_f_exclusion} allows a mixed-integer linear formulation through the big-M method, problem~\eqref{eq:MIQP_f} is an MIQP. Adding $B_\epsilon$ bounds $p^*$ off from $0$ if $V_k(x;P)$ is in fact a Lyapunov function and we can decide the negativity of $p^*$ by checking if $p^* < -\epsilon_{tol}$ for some tolerance $\epsilon_{tol} > 0$ to handle round-off errors in computation.
When the dynamics $x_+ = f(x)$ is linear inside $B_\epsilon$, i.e., $x_+ = Ax$ for $x \in B_\epsilon$, we can show convergence to the origin of the system trajectories starting inside $B_\epsilon$ by checking the magnitude of the eigenvalues of $A$. Combined with Corollary~\ref{coro:convergence}, asymptotic convergence inside the sublevel set $\Omega$ can be established.
\subsubsection{Early termination of the MIQP} \label{sec:early_termination} We can terminate the Branch $\&$ Bound~\cite{wolsey1999integer} algorithm in solving~\eqref{eq:MIQP_f} once a feasible solution is found with non-negative objective function since in this case we already have a counterexample.
\section{Numerical examples}
\label{sec:simulation} We demonstrate our method through two examples: closed-loop MPC systems and ReLU neural network controlled PWA systems. In particular, we compare the performances of our method with the LMI-based Lyapunov function synthesis methods~\cite{biswas2005survey} on the MPC example. Algorithm~\ref{alg:ACCPM} is implemented in Python $3.7$ with Gurobi v9.0~\cite{gurobi} and the LMI-based method is implemented in the MPT3 toolbox~\cite{MPT3} with Mosek~\cite{mosek} in Matlab. All the simulation is implemented on an Intel i7-6700K CPU with $32$ GB of RAM.
Throughout the numerical experiments in this section, Algorithm~\ref{alg:ACCPM} is run with $\alpha = 0.0, \beta = 1.0$ and $\epsilon_{tol} = 10^{-8}$ to decide negativity of the optimal value of the MIQP. The Gurobi solver is set with feasibility tolerance $10^{-9}$, integer tolerance $10^{-9}$, and optimality tolerance $10^{-9}$. In addition, we terminate the MIQP once it finds a feasible solution that generates an objective $\geq 10^{-4}$ which means a counterexample is already found.
\subsection{A 2-dimensional closed-loop MPC system} \label{sec:2d_example} For an unstable linear system $x_+ = Ax + Bu$ with \begin{equation} \label{eq:2d_dyn} A = \begin{bmatrix} 1.2 & 1.2 \\ 0 & 1.2 \end{bmatrix},\quad B = \begin{bmatrix} 1 \\ 0.5 \end{bmatrix}, \end{equation} we design an MPC controller with horizon $T = 10$, state constraint $[-5 \ -5]^\top \leq x \leq [5 \ 5]^\top$, control input constraint $-1 \leq u \leq 1$, stage cost $p(x, u) = x^\top Q x + u^\top R u$ with $Q = 10 I, R = 1$, and terminal cost $q(x) = x^\top P_\infty x$ where $P_\infty = \text{DARE}(A, B, Q, R)$ is the solution to the discrete algebraic Riccati equation defined by $(A, B, Q, R)$. The terminal set is chosen as the maximum positive invariant set~\cite[Chapter 10]{borrelli2017predictive} of the closed-loop system $x_+ = (A + B K_\infty) x$ where $K_\infty = -(B^\top P_\infty B + R)^{-1}B^\top P_\infty A$.
\begin{figure}
\caption{Domain $\mathcal{R}$ and the $211$ modes of the closed-loop MPC system of~\eqref{eq:2d_dyn}.}
\label{fig:MPC_2d_domain}
\end{figure}
With the above setup, we obtain the explicit MPC controller, which is a PWA function of the state $x$ with $211$ partitions, through the MPT3 toolbox in Matlab. The domain $\mathcal{R}$ of the explicit MPC is a polytope shown in Fig.~\ref{fig:MPC_2d_domain} together with its partitions. After obtaining $\mathcal{R}$, we verify that it is positive invariant for the closed-loop dynamics. Through the LMI-based method, we are able to synthesize a discontinuous PWQ Lyapunov function in MPT3 which verifies $\mathcal{R}$ is the ROA for the closed-loop MPC system. The total running time is $38.890$ seconds, with $0.25$ spent in solving the constructed SDP and the rest in computing the transition map.
\subsubsection{Algorithm~\ref{alg:ACCPM} with PWA representation} We import the PWA representation of the closed-loop MPC dynamics which we denote as $x_+ = f_{cl}(x)$ from MPT3 and run Algorithm~\ref{alg:ACCPM} with the mixed-integer formulation of $f_{cl}(x)$ as shown in Section~\ref{sec:PWA_representation}. Since the domain $\mathcal{R}$ is positively invariant, we set $\mathcal{X} = \mathcal{R}$. For Lyapunov function candidates of order $k = 0, 1$, Algorithm~\ref{alg:ACCPM} certifies the non-existence of Lyapunov functions after $1.709$ seconds with $4$ iterations and $28.333$ seconds with $7$ iterations, respectively. With $V^{(k)}(x;P)$ of order $k= 2$, Algorithm~\ref{alg:ACCPM} terminates in $14$ iterations with a valid Lyapunov function candidate and the total running time is $4379.716$ seconds. We plot the counterexamples found in each iteration in Fig.~\ref{fig:MPC_2d_counterexamples}. The accumulated running time of Algorithm~\ref{alg:ACCPM} in each iteration is shown in Fig.~\ref{fig:time_compare}.
\begin{figure}
\caption{Sequence of counterexamples found in Algorithm~\ref{alg:ACCPM}.}
\caption{Accumulated running time of Algorithm~\ref{alg:ACCPM}. }
\caption{(a) Sequence of counterexample states found by the verifier in Algorithm~\ref{alg:ACCPM} with the PWA representation of the closed-loop MPC system. (b) Accumulated running time of Algorithm~\ref{alg:ACCPM} in each iteration with mixed-integer formulations induced by the PWA (blue) and LC (orange) representations of the closed-loop MPC system. }
\label{fig:MPC_2d_counterexamples}
\label{fig:time_compare}
\end{figure}
\subsubsection{Algorithm~\ref{alg:ACCPM} with LCS representation} As shown in Section~\ref{sec:LCS_representation} and~\cite{simon2016stability}, we can obtain a mixed-integer formulation of the closed-loop MPC dynamics through its LC system representation instead of the PWA one. Based on this mixed-integer formulation, we run Algorithm~\ref{alg:ACCPM} with Lyapunov function candidates $V^{(k)}(x;P)$ of order $k = 2$. The algorithm terminates in $368.618$ seconds with $20$ iterations and returns a valid Lyapunov function which certifies that the domain $\mathcal{R}$ is the ROA. The accumulated running time at each iteration is summarized in Fig.~\ref{fig:time_compare}.
\subsubsection{Discussion} Fig.~\ref{fig:time_compare} shows that Algorithm~\ref{alg:ACCPM} depends on the mixed-integer representation of the system. In this example, we need $211$ binary variables to describe the map $x_+ = f_{cl}(x)$ using the PWA representation, while with the LCS representation, we only need to use $64$ binary variables to describe the map $u = \pi_{MPC}(x)$, and hence the closed-loop dynamics. However, when compared with the LMI-based method which synthesizes a PWQ discontinuous Lyapunov functions in $38.890$ seconds, Algorithm~\ref{alg:ACCPM} is rather inefficient. This is partly due to that the continuous Lyapunov function candidate class $V^{(k)}(x;P)$ is more conservative than the discontinuous one~\cite{biswas2005survey}. To compensate for the conservatism, we need to apply high order $V^{(k)}(x;P)$ which increases the complexity of the MIQP. In the next subsection, we show that Algorithm~\ref{alg:ACCPM} can synthesize a Lyapunov function when the LMI-based method fails.
\subsection{A 4-dimensional closed-loop MPC system} Consider model predictive control of a randomly generated $4$-dimensional open-loop unstable linear system given by \begin{equation*} A = \begin{bmatrix} 0.4346 & -0.2313& -0.6404& 0.3405 \\ -0.6731& 0.1045& -0.0613 & 0.3400 \\ -0.0568& 0.7065& -0.0861& 0.0159\\ 0.3511 & 0.1404 & 0.2980& 1.0416 \end{bmatrix}, \ B = \begin{bmatrix} 0 \\ -0.0065 \\ -0.5238 \\ 0.4605 \end{bmatrix}. \end{equation*} With the state constraint $\lVert x \rVert_\infty \leq 5$ and the input constraint $-1 \leq u \leq 1$, we design the MPC controller by choosing horizon $T = 10$, stage cost with $Q = 10I, R = 1$, terminal cost with $P_\infty$ and the terminal set as in Section~\ref{sec:2d_example}. The explicit MPC controller is constructed through MPT3 and has $193$ partitions in its domain $\mathcal{R}$, which is validated to be positive invariant under the closed-loop dynamics. Although it took only $29.9$ seconds to compute the transition map, the constructed SDP is ill-posed and Mosek failed to find a feasible solution.
Then we apply Algorithm~\ref{alg:ACCPM} to synthesize a Lyapunov function $V^{(k)}(x;P)$ with $k = 1$ and the LCS representation of the closed-loop MPC system. The ROI is chosen as $\mathcal{X} = \mathcal{R}$. Algorithm~\ref{alg:ACCPM} terminates in $9$ iterations with a valid Lyapunov function candidate and therefore proves that the closed-loop system is asymptotically stable in the domain $\mathcal{R}$. The total running time is $680.515$ seconds and the accumulated running time in each iteration is plotted in Fig.~\ref{fig:MPC_4d_time}.
\begin{figure}
\caption{Accumulated solver time of Algorithm~\ref{alg:ACCPM} in each iteration for the $4$-dimensional closed-loop MPC system.}
\label{fig:MPC_4d_time}
\end{figure}
\subsection{Neural network controlled PWA system} We use a hybrid system example from~\cite{marcucci2017approximate} and consider the inverted pendulum shown in Fig.~\ref{fig:pendulum_and_controller} with parameters $m = 1, \ell = 1, g = 10, k = 100, d = 0.1$. We denote the angle and the angular velocity of the pendulum by $q$ and $\dot{q}$, respectively, and define the system state as $x = (q, \dot{q})$. By linearizing the dynamics of the inverted pendulum around $x = 0$, we obtain a hybrid system which has two modes: not in contact with the elastic wall (mode $1$) and in contact with the elastic wall (mode $2$). After discretizing the model using the explicit Euler scheme with a sampling time $h = 0.01$, a PWA model $x_+ = \psi(x, u)$ of the form~\eqref{eq:pwa_dynamics} is obtained with the following parameters \begin{equation} \label{eq:pendulum_dynamics} \begin{aligned} & A_1 = \begin{bmatrix} 1 & 0.01 \\ 0.1 & 1 \end{bmatrix}, B_1 = \begin{bmatrix} 0 \\ 0.01 \end{bmatrix}, c_1 = \begin{bmatrix} 0 \\ 0 \end{bmatrix}, \\ & \mathcal{R}_1 = \{x \vert [-0.2 \ -1.5]^\top \leq x \leq [0.1 \ 1.5]^\top \}. \\ & A_2 = \begin{bmatrix} 1 & 0.01\\ -0.9 & 1 \end{bmatrix}, B_2 = \begin{bmatrix} 0 \\ 0.01 \end{bmatrix}, c_2 = \begin{bmatrix} 0 \\ 0.1 \end{bmatrix}, \\ & \mathcal{R}_2 = \{x \vert [0.1 \ -1.5]^\top \leq x \leq [0.2 \ 1.5]^\top\}. \end{aligned} \end{equation}
\begin{figure}
\caption{A ReLU neural network (right) that approximates a hybrid MPC controller is applied on the inverted pendulum system with an elastic wall (left).}
\label{fig:pendulum_and_controller}
\end{figure}
We then synthesize a hybrid MPC controller $\pi_{MPC}(x)$ for the PWA system~\cite{marcucci2019mixed} where the control input constraints are given by $-4 \leq u \leq 4$ and the horizon of MPC is set as $T = 10$. The stage and terminal costs are given by $Q = I, R = 1$, and $P_\infty = \text{DARE}(A_1, B_1, Q, R)$. We evaluate $\pi_{MPC}(x)$ on a uniform $40 \times 40$ grid samples from the state space and let the reference ROI $\mathcal{X}_0$ be the convex hull of all the feasible state samples. Then the ROI $\mathcal{X} = \gamma \mathcal{X}_0$ with $0 < \gamma \leq 1$ is applied to guide the search for an estimate of ROA.
A total number of $1354$ feasible samples of state and control input pairs $(x, \pi_{MPC}(x))$ are generated to train a ReLU neural network $\pi(x)$ in Keras~\cite{chollet2015keras} to approximate the MPC controller. The neural network has $2$ hidden layers with $20$ neurons in each layer and its output layer bias term is modified after training to guarantee $\pi(0) = 0$. We plot the neural network controller in Fig.~\ref{fig:pendulum_and_controller} and set $\epsilon = 0.0158$ in~\eqref{eq:MIQP_f} since the closed-loop dynamics is linear and asymptotically stable inside $B_\epsilon$.
For Lyapunov function candidates of order $k=0$ and $k=1$, we run Algorithm~\ref{alg:ACCPM} with ROI $\mathcal{X} = \gamma \mathcal{X}_0$ of varying values of $\gamma$. For the quadratic function class $V^{(0)}(x;P)$, the largest ROI is given by $\mathcal{X} = 0.86 \mathcal{X}_0$ through bisection with which Algorithm~\ref{alg:ACCPM} terminates in $16$ iterations with a total running time of $13.687$ seconds. The corresponding estimate of ROA is shown in Fig.~\ref{fig:IP_quad_ROA}. For the PWQ function class $V^{(1)}(x;P)$, the largest ROI is given by $\mathcal{X} = 1.0 \mathcal{X}_0$ in which case Algorithm~\ref{alg:ACCPM} terminates in $11$ iterations with a total running time of $90.917$ seconds. The estimate of ROA obtained by the found PWQ Lyapunov function candidate is shown in Fig.~\ref{fig:IP__ROA}. It shows that the synthesized Lyapunov function in $V^{(1)}(x;P)$ is less conservative compared with the one in $V^{(0)}(x;P)$ and Algorithm~\ref{alg:ACCPM} can obtain non-trivial estimates of ROA for the neural network controlled hybrid systems.
\begin{figure}
\caption{Estimate of ROA of the neural network controlled system by a quadratic Lyapunov function in $V^{(0)}(x;P)$. }
\label{fig:IP_quad_ROA}
\caption{Estimate of ROA of the neural network controlled system by a PWQ Lyapunov function in $V^{(1)}(x;P)$. }
\label{fig:IP__ROA}
\caption{Estimates of ROA found by Algorithm~\ref{alg:ACCPM} with Lyapunov function candidates in $V^{(0)}(x;P)$ and $V^{(1)}(x;P)$. The partitions of the state space are marked by the black boxes. Simulated closed-loop trajectories with the neural network controller are plotted for a grid of initial conditions.}
\label{fig:IP_ROA}
\end{figure}
\section{Conclusion}
We have proposed a learning-based method to learn Lyapunov functions for autonomous hybrid systems that have a mixed-integer formulation, including piecewise affine, linear complementarity, mixed logical dynamical systems and ReLU neural networks. By designing the method according to the analytic center cutting-plane method, we show that the proposed algorithm is guaranteed to find a Lyapunov function in a finite number of steps when the set of Lyapunov functions is full-dimensional in the parameter space. Our method is an alternative to the LMI-based Lyapunov function synthesis approach which relies on the piecewise affine representation of hybrid systems.
\small
\end{document} | arXiv |
\begin{document}
\title{Irreducible modules for equivariant map superalgebras and their extensions}
\author{Lucas Calixto} \address{Departamento de Matem\'atica\\
Instituto de Ci\^encias Exatas\\
UFMG\\
Belo Horizonte, Minas Gerais, Brazil, 30.123-970} \email{[email protected]}
\author{Tiago Macedo} \address{Department of Mathematics and Statistics\\
University of Ottawa\\
Ottawa, ON K1N 6N5\\
\ and \
Universidade Federal de S\~ao Paulo\\
S\~ao Jos\'e dos Campos, S\~ao Paulo, Brazil, 12.247-014} \email{[email protected]}
\begin{abstract} Let $\Gamma$ be a group acting on a scheme $X$ and on a Lie superalgebra $\text{$\mathfrak{g}$}$, both defined over an algebraically closed field of characteristic zero $\text{$\Bbbk$}$. The corresponding equivariant map superalgebra $M(\text{$\mathfrak{g}$}, X)^\Gamma$ is the Lie superalgebra of equivariant regular maps from $X$ to $\text{$\mathfrak{g}$}$. In this paper we complete the classification of finite-dimensional irreducible $M(\text{$\mathfrak{g}$}, X)^\Gamma$-modules when $\text{$\mathfrak{g}$}$ is a finite-dimensional simple Lie superalgebra, $X$ is of finite type and $\Gamma$ is a finite abelian group acting freely on the rational points of $X$, by classifying these $M(\text{$\mathfrak{g}$},X)^\Gamma$-modules in the case where $\lie g$ is a periplectic Lie superalgebra. We also describe extensions between irreducible modules in terms of homomorphisms and extensions between modules for certain finite-dimensional Lie superalgebras. \end{abstract}
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\section{Introduction}
Lie superalgebras have a wide range of applications in many areas of physics and mathematics, such as supersymmetry, string theory, conformal field theory, and number theory. In the study of symmetry, for instance, while Lie algebras describe bosonic degrees of freedom, Lie superalgebras also allow fermionic degrees of freedom \cite{Var04}. In number theory, affine Kac-Moody superalgebras and their representations can be used to study problems related to sums of squares and sums of triangular numbers \cite{KW94}. For more examples, see, for instance, \cite{FL85, Ser85, GLS01, FK02}.
It is usually the case that the representation theory of Lie superalgebras is more complicated than that of their Lie algebra counterparts. For instance, the category of finite-dimensional modules for a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero is always semisimple, while the category of finite-dimensional modules for a finite-dimensional simple Lie \textit{super}algebra is not necessarily so. It is thus important to describe extensions between their irreducible modules. Despite being a subject of intense study since the birth of supersymmetry, these extensions are not known in general, and, in contrast to extensions for Lie algebras, their study is often done case by case. See, for instance, \cite{FL84, fuks86, Pol88, SZ98, SZ99, Gru00, Gru03, BKN10, Bag12} for some of these results.
The main goal of the current paper is to develop the representation theory of certain Lie superalgebras known as equivariant map superalgebras. Equivariant map superalgebras generalize, on the one hand, simple Lie superalgebras, and on the other hand, current and loop Lie algebras. They are constructed in the following way. Consider a scheme $X$, a Lie superalgebra $\text{$\mathfrak{g}$}$, both defined over an algebraically closed field of characteristic zero $\text{$\Bbbk$}$, and a group $\Gamma$ that acts on $X$ and $\text{$\mathfrak{g}$}$ by automorphisms. The corresponding equivariant map superalgebra $M(X, \text{$\mathfrak{g}$})^\Gamma$ is the Lie superalgebra of $\Gamma$-equivariant regular maps from $X$ to $\text{$\mathfrak{g}$}$. If one denotes by $A$ the coordinate ring of $X$, then $M(X, \lie g)^\Gamma$ can be identified with the Lie subsuperalgebra $(\text{$\mathfrak{g}$} \otimes_\text{$\Bbbk$} A)^\Gamma$ of $\text{$\mathfrak{g}$} \otimes_\text{$\Bbbk$} A$ that consists of its $\Gamma$-fixed points under the diagonal action.
In the particular case where \lie g is a Lie algebra, $M(X, \lie g)^\Gamma$ is called equivariant map algebra. Representations of these Lie algebras have been a subject of intense research for the last thirty years (see, for instance, the survey \cite{NS13}). One reason is that representations of $M(\text{$\Bbbk$}^\times, \lie g)^\Gamma$, known as twisted loop algebras, and $M(\text{$\Bbbk$}, \lie g)^\Gamma$, known as twisted current algebras, are closely related to those of affine Kac-Moody Lie algebras. In fact, when $\text{$\mathfrak{g}$}$ is a finite-dimensional simple Lie algebra and $\Gamma$ is the group of automorphisms of the Dynkin diagram of $\text{$\mathfrak{g}$}$, the twisted current algebra is a parabolic subalgebra of the affine Kac-Moody Lie algebra associated to $\lie g$ and $\Gamma$, and the twisted loop algebra is its centerless derived subalgebra (see \cite[Section 13.1]{kumar02}).
Finite-dimensional irreducible representations of equivariant map algebras were classified by Neher, Savage and Senesi \cite{NSS12} in the case where $\text{$\mathfrak{g}$}$ is a finite-dimensional Lie algebra and $\Gamma$ is a finite group. Finite-dimensional simple Lie superalgebras over an algebraically closed field of characteristic zero and finite-dimensional irreducible representations of the so-called basic classical Lie superalgebras were classified by Kac in \cite{kac77} and \cite{kac78}. In \cite{savage14}, Savage classified irreducible finite-dimensional representations of equivariant map superalgebras in the case where $\lie g$ is a basic classical Lie superalgebra (or $\lie{sl}(n,n)$, $n\geq 1$, if $\Gamma$ is trivial), $X$ has a finitely-generated coordinate ring and $\Gamma$ is a finite abelian group acting freely on the rational points of $X$. Moving beyond basic classical Lie superalgebras, the first author, Moura and Savage classified finite-dimensional irreducible representations of equivariant map queer Lie superalgebras in \cite{CMS15}. While in the basic classical setting those irreducible representations were isomorphic to tensor products of generalized evaluation representations, in the queer case they are irreducible products of evaluation representations (see Section \ref{ss:map.superalgebras} for details). In \cite{Bag15}, Bagci extended this classification to equivariant map superalgebras where $\lie g$ is of Cartan type.
In the current paper, we complete the classification of finite-dimensional irreducible representations of equivariant map superalgebras by describing these modules for $M(\text{$\mathfrak{g}$}, X)^\Gamma$ when $\lie g={\lie p}(n)$, a periplectic Lie superalgebra, $X$ has a finitely-generated coordinate ring, $\Gamma$ is a finite abelian group acting on $\text{$\mathfrak{g}$}$ and $X$, and such that the induced action of $\Gamma$ on the rational points of $X$ is free. In Theorem~\ref{thm:main.p(n)} we prove that, similarly to the case where $\text{$\mathfrak{g}$}$ is basic classical of type II, all irreducible finite-dimensional $M(\text{$\mathfrak{g}$}, X)^\Gamma$-modules are evaluation modules. The technique used to prove this result is similar to the one used in \cite{savage14}. Behind this technique are the facts that one can choose a Cartan subalgebra of ${\lie p}(n)$ that is purely even and abelian, that the root space decomposition of $\lie p(n)$ with respect to such a subalgebra is relatively similar to that of the basic case, and that ${\lie p}(n)_{\bar 0}$ is a semisimple Lie algebra.
Equipped with a complete classification of irreducible finite-dimensional $M(\text{$\mathfrak{g}$}, X)^\Gamma$-modules, one can inquire about their extensions. This is the second problem that we address in the current paper. In Theorem~\ref{thm:main} we describe extensions between finite-dimensional irreducible $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-modules in terms of homomorphisms and extensions between finite-dimensional indecomposable (irreducible, in some cases) modules for a finite-dimensional Lie superalgebra of the form $\text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}^n$, where ${\text{$\mathsf{m}$}}$ is a maximal ideal of $A$ and $n$ is a positive integer. The technique used here is similar to the one used in \cite{NS13} for the non-super case. The main difference in this super setting is that one needs to describe the kernel of certain transgression maps, which we do in Proposition~\ref{prop:ker.transgression}.
\subsection*{Organization of the paper}
In Section~\ref{section:notation} we fix the notation and state some results that will be used throughout the paper. In Section~\ref{sec:periplectic} we describe the structure of periplectic Lie superalgebras, construct and classify the finite-dimensional irreducible $M(\text{$\mathfrak{g}$}, X)^\Gamma$-modules when $\text{$\mathfrak{g}$}$ is of periplectic type, $X$ has a finitely-generated coordinate ring, $\Gamma$ is a finite abelian group acting on $\text{$\mathfrak{g}$}$ and $X$, and such that the action of $\Gamma$ on the rational points of $X$ is free (see Section~\ref{ss:class.pn} for the main result of this section). In Section~\ref{s:transgression} we review the construction of Lyndon-Hochschild-Serre spectral sequences in order to describe the kernel of certain transgression maps that are relevant to the computation of extensions between finite-dimensional irreducible $M(\text{$\mathfrak{g}$}, X)^\Gamma$-modules (see Proposition~\ref{prop:ker.transgression}). Using results of previous sections, in Section~\ref{Exts} we develop a general technique to describe $p$-extensions between finite-dimensional irreducible $M(\text{$\mathfrak{g}$}, X)^\Gamma$-modules, and describe $1$-extensions between these modules in terms of homomorphisms and extensions between finite-dimensional indecomposable (irreducible, in some cases) modules for a finite-dimensional Lie superalgebra of the form $\text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}^n$, where ${\text{$\mathsf{m}$}}$ is a maximal ideal of $A$ and $n$ is a positive integer (see Theorem~\ref{thm:main} for the main result of this section). We finish the paper with some examples and applications.
\subsection*{Acknowledgments}
The authors would like to thank E. Neher and A. Savage for helpful discussions and for their comments on earlier versions of this paper. The authors would also like to thank an anonymous referee for their thorough reading and suggestions. The first author was supported by FAPESP grant 2013/08430-4 and PRPq grant ADRC-05/2016. The second author was supported by CNPq grant 232462/2014-3.
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\section{Notation and Preliminaries} \label{section:notation}
\subsection{Notation}
Let $\text{$\Bbbk$}$ denote an algebraically closed field of characteristic zero, $\bb Z_2$ denote the finite field with two elements $\{\bar0, \bar1\}$, and $\bb Z_{>0}$ denote the set of positive integers. All vector spaces, algebras, and tensor products will be considered over the field $\text{$\Bbbk$}$ (unless otherwise specified). A vector space $V$ is said to be a super space if it is $\bb Z_2$-graded; that is, there exist subspaces $V_{\bar0}, V_{\bar1} \subseteq V$ such that $V = V_{\bar0} \oplus V_{\bar1}$. We denote by $| \cdot |$ the degree of a homogeneous element in a super space $V$; that is, $|v| = z$ for all $v \in V_z$ and $z \in \bb Z_2$.
\subsection{Finite-dimensional Lie superalgebras} \label{ss:lie.sup}
In this subsection we follow \cite{Mus12, gav14}.
A Lie superalgebra is a $\bb Z_2$-graded vector space $\lie g = \lie g_{\bar0} \oplus \lie g_{\bar1}$ with a $\bb Z_2$-graded linear transformation $[\cdot , \cdot] \colon \text{$\mathfrak{g}$} \otimes \text{$\mathfrak{g}$} \to \text{$\mathfrak{g}$}$ that satisfies $\bb Z_2$-graded versions of anticommutativity and Jacobi identity.
For any Lie superalgebra $\text{$\mathfrak{g}$} = \lie g_{\bar0} \oplus \lie g_{\bar1}$, the subspace $\text{$\mathfrak{g}$}_{\bar0}$ inherits the structure of a Lie algebra and the subspace $\text{$\mathfrak{g}$}_{\bar1}$ inherits the structure of a $\text{$\mathfrak{g}$}_{\bar 0}$-module. A finite-dimensional simple Lie superalgebra $\lie g$ is said to be classical if the $\lie g_{\bar0}$-module $\lie g_{\bar1}$ is completely reducible. Otherwise it is said to be of Cartan type. When $\text{$\mathfrak{g}$}$ is a classical Lie superalgebra, the $\lie g_{\bar0}$-module $\lie g_{\bar1}$ is either irreducible or a direct sum of two irreducible modules. In the first case, $\lie g$ is said to be of type II, and in the second case, $\lie g$ is said to be of type I. If a classical Lie superalgebra admits an even nondegenerate invariant bilinear form, then it is said to be basic; otherwise, it is said to be strange. Table \ref{table} summarizes the classification of finite-dimensional simple Lie superalgebras with nonzero odd parts proved by V.~Kac in \cite[Theorem~5]{kac77}.
\begin{table} \begin{tabular}{l l} \toprule Lie superalgebra & Classification \\ \midrule $A(m,n)$, \ $m > n \ge 0$ & Basic, type I \\ $A(n,n)$, \ $n \ge 1$ & Basic, type I \\ $B(m,n)$, \ $m \ge 0$, $n \ge 1$ & Basic, type II \\ $C(n+1)$, \ $n \ge 1$ & Basic, type I \\ $D(m,n)$, \ $m \ge 2$, $n \ge 1$ & Basic, type II \\ $D(2,1;\alpha)$, \ $\alpha \ne 0,-1$ & Basic, type II \\ $F(4)$ & Basic, type II \\ $G(3)$ & Basic, type II \\ $\lie p(n)$, \ $n \ge 2$ & Strange, type I \\ $\lie q(n)$, \ $n \ge 2$ & Strange, type II \\ $H(n)$, \ $n \ge 4$ & Cartan type \\ $S(n)$, \ $n \ge 3$ & Cartan type \\ $\tilde S(n)$, \ $n = 2m, m \ge 2$ & Cartan type \\ $W(n)$, \ $n \ge 2$ & Cartan type \\ \bottomrule \end{tabular}
\caption{Classification of finite-dimensional simple Lie superalgebras with nonzero odd parts.} \label{table} \end{table}
When $\lie g$ is a classical Lie superalgebra, a Cartan subalgebra of $\lie g$ is defined to be a Cartan subalgebra of the Lie algebra $\lie g_{\bar 0}$. When $\text{$\mathfrak{g}$}$ is a Lie superalgebra of Cartan type, it admits a $\bb Z$-grading $\text{$\mathfrak{g}$} = \bigoplus_{-1 \le i} \text{$\mathfrak{g}$}_i$ that is compatible with the $\bb Z_2$-grading; that is, $\text{$\mathfrak{g}$}_{\bar 0} = \bigoplus_{i \in 2 \bb Z}\text{$\mathfrak{g}$}_i$ and $\text{$\mathfrak{g}$}_{\bar 1}=\bigoplus_{i \in 2\bb Z}\text{$\mathfrak{g}$}_{i+1}$, and such that $\text{$\mathfrak{g}$}_0$ is a reductive Lie algebra. In this case, a Cartan subalgebra of $\lie g$ is defined to be a Cartan subalgebra of the Lie algebra $\lie g_0$. (It is worth noting that when $\text{$\mathfrak{g}$}$ is of type $\tilde S$, this is only a $\bb Z$-grading as a vector space, not as Lie superalgebras.)
Let $\text{$\mathfrak{g}$}$ be a finite-dimensional simple Lie superalgebra, and let $\lie h$ be a Cartan subalgebra of $\lie g$. The action of $\lie h$ on $\text{$\mathfrak{g}$}$ is diagonalizable and we have a root space decomposition \[ \text{$\mathfrak{g}$}=\lie h\oplus \bigoplus_{\alpha\in \lie h^* \setminus \{0\}}\text{$\mathfrak{g}$}_{\alpha}, \quad \textup{where} \quad \text{$\mathfrak{g}$}_\alpha = \{ x \in \text{$\mathfrak{g}$} \mid [h,x] = \alpha(h) x \textup{ for all } h \in \lie h\}. \] Let $\Delta = \{ \alpha \in \lie h^* \setminus \{ 0 \} \mid \lie g_\alpha \neq 0 \}$ denote the set of roots of \lie g and $Q$ denote the subgroup of $\lie h^*$ generated by $\Delta$.
Let \lie g be a finite-dimensional simple Lie superalgebra. Then the set of isomorphism classes of finite-dimensional irreducible \lie g-modules is in bijection with a subset $\Lambda^+$ of $\lie h^*$ (see \cite[Theorem~8]{kac77}). We denote by $\Lambda$ the subgroup of $\lie h^*$ generated by $\Lambda^+$, and by $V(\lambda)$ the irreducible $\lie g$-module corresponding to $\lambda \in \Lambda^+$, namely the unique finite-dimensional irreducible $\text{$\mathfrak{g}$}$-module of highest weight $\lambda$. Let $\lambda^* \in \Lambda^+$ denote the highest weight of the dual \lie g-module $V(\lambda)^*$.
For a Lie superalgebra $\text{$\mathfrak{g}$}$ and a $\text{$\mathfrak{g}$}$-module $V$, consider the space of linear endomorphisms of the vector space $V$, $\End(V)$, and define \begin{gather*} \End_{\lie g}(V) = \{\varphi\in \End (V)\mid \varphi(x v)=x\varphi(v), \text{ for all }x\in \text{$\mathfrak{g}$},\ v\in V\}, \\ \End(V)_z = \{ \phi \in \End (V) \mid \phi (v) \in V_{z+z'} \textup{ for all } v \in V_{z'}, z' \in \bb Z_2 \}, \\ \End_{\lie g}(V)_{\bar 0}=\End_{\lie g} (V)\cap \End (V)_{\bar 0} \text{ and } \End_{\lie g}(V)_{\bar 1}=\End_{\lie g} (V)\cap \End (V)_{\bar 1}. \end{gather*} Let $\lie g^1$ and $\lie g^2$ be two finite-dimensional Lie superalgebras, and $V_1$ and $V_2$ be irreducible finite-dimensional modules for $\text{$\mathfrak{g}$}^1$ and $\text{$\mathfrak{g}$}^2$ respectively. The ($\lie g^1 \oplus \lie g^2$)-module $V_1 \otimes V_2$ is reducible only if $\End_{\lie g^1} (V_1)_{\bar 1} \neq 0$ and $\End_{\lie g^2} (V_2)_{\bar 1} \neq 0$ (see \cite[Proposition~8.4]{Che95}). In this case, by Schur's Lemma for Lie superalgebras, $\End_{\lie g^i} (V_i)_{\bar 1} = \text{$\Bbbk$} \varphi_i$ for some $\varphi_i^2 = 1$, $i \in \{1,2\}$, and \[ \widehat{V} = \left\{ v \in V_1 \otimes V_2 \mid \left( \sqrt{-1} \varphi_1 \otimes \varphi_2 \right) v = v \right\} \] is an irreducible ($\lie g^1\oplus\lie g^2$)-submodule satisfying $V_1 \otimes V_2 \cong \widehat{V}\oplus \widehat{V}$ (see \cite[p.~27]{Che95}). Define the irreducible product $V_1 \htimes V_2$ to be \[ V_1 \htimes V_2 = \begin{cases} V_1\otimes V_2, & \textup{if $V_1 \otimes V_2$ is irreducible}, \\ \widehat V, & \text{otherwise}. \end{cases} \] If $\lie g^1$ and $\lie g^2$ are finite-dimensional simple Lie superalgebras not of type $\lie q$, then the irreducible product is always equal to the tensor product.
Given $\ell>1$, finite-dimensional Lie superalgebras $\lie g^1, \dotsc, \lie g^\ell$, and for each $i \in \{1, \dotsc, \ell\}$, an irreducible finite-dimensional $\lie g^i$-module $V_i$, define the $(\lie g^1 \oplus \dotsb \oplus \lie g^\ell)$-module $V_1 \htimes \dotsb \htimes V_\ell$ to be \[ V_1 \htimes \dotsb \htimes V_\ell = (V_1 \htimes \dotsb \htimes V_{\ell-1}) \htimes V_\ell. \] Up to isomorphism, $\htimes$ is associative and, when $\lie g^1 = \dots = \lie g^\ell$, $\htimes$ is also commutative (see \cite[Lemma~6.2]{CMS15}). Also, define \begin{equation} \label{eq:kappa.mods} \kappa(V_1, \ldots, V_{\ell}) = \sum_{i=2}^{\ell} \dim \End_{\left( \text{$\mathfrak{g}$}^1 \oplus \dotsb \oplus \text{$\mathfrak{g}$}^{i-1} \right)} \left( V_1 \htimes \dotsb \htimes V_{i-1} \right)_{\bar 1} \dim \End_{\lie g^i} (V_i)_{\bar1}. \end{equation} One can prove by induction that $\kappa (V_1, \dotsc, V_\ell) \le \ell-1$ and that \[ V_1 \otimes \dotsb \otimes V_\ell \cong (V_1 \htimes \dotsb \htimes V_\ell)^{\oplus 2^{\kappa (V_1, \dotsc, V_\ell)}}. \]
\subsection{Equivariant map superalgebras} \label{ss:map.superalgebras}
In this subsection, we review results proved in \cite{savage14, Bag15, CMS15}.
Let $\lie g$ be a finite-dimensional simple Lie superalgebra and $A$ be an associative, commutative, finitely-generated $\text{$\Bbbk$}$-algebra with unit. The map superalgebra $\text{$\mathfrak{g} \otimes A$}$ is the Lie superalgebra with underlying vector space $\lie g \otimes_\text{$\Bbbk$} A$, with $\bb Z_2$-grading given by $(\lie g \otimes A)_z = \lie g_z \otimes A$, $z \in \bb Z_2$, and with Lie superbracket extending bilinearly \[ [x \otimes a, y \otimes b] = [x,y] \otimes ab, \quad \textup{for all } x,y \in \text{$\mathfrak{g}$} \textup{ and } a,b \in A. \] Let $\Gamma$ be a group acting on $\text{$\mathfrak{g}$}$ and $A$ by automorphisms. One can induce an action of $\Gamma$ on $\text{$\mathfrak{g} \otimes A$}$ by extending linearly \[ \gamma (x \otimes a) = \gamma(x) \otimes \gamma (a), \quad \textup{for all } \gamma \in \Gamma, x \in \text{$\mathfrak{g}$} \textup{ and } a \in A. \] The equivariant map superalgebra $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$ is the Lie subsuperalgebra of $\text{$\mathfrak{g} \otimes A$}$ consisting of its $\Gamma$-fixed points: \[ \text{$(\mathfrak{g} \otimes A)^\Gamma$} = \{ x \in \text{$\mathfrak{g} \otimes A$} \mid \gamma x = x \textup{ for all } \gamma \in \Gamma \}. \]
Let $\maxspec (A)$ denote the set of maximal ideals of $A$. Notice that the action of $\Gamma$ on $A$ induces an action of $\Gamma$ on $\maxspec(A)$ that is explicitly given by $\gamma {\text{$\mathsf{m}$}} = \{ \gamma a \mid a \in {\text{$\mathsf{m}$}} \} \in \maxspec(A)$ for all ${\text{$\mathsf{m}$}} \in \maxspec(A)$ and $\gamma \in \Gamma$. For most finite-dimensional simple Lie superalgebras $\text{$\mathfrak{g}$}$, if $\Gamma$ is a finite abelian group acting freely on $\maxspec (A)$, then every finite-dimensional irreducible $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-module can be described in terms of generalized evaluation modules. These generalized evaluation modules are defined in the following way. Given ${\text{$\mathsf{m}$}} \in \maxspec(A)$ and $n \in \bb Z_{>0}$, define ${\rm ev}_{{\text{$\mathsf{m}$}}^n}$ to be the homomorphism of Lie superalgebras given by the composition \[ {\rm ev}_{{\text{$\mathsf{m}$}}^n} \colon \text{$\mathfrak{g} \otimes A$} \to \text{$\mathfrak{g} \otimes A$} / \lie g \otimes {\text{$\mathsf{m}$}}^n \iso \text{$\mathfrak{g} \otimes A$} / {\text{$\mathsf{m}$}}^n. \]
\begin{center}\textit{ For the rest of this subsection, assume that $\Gamma$ is a finite group acting freely on $\maxspec (A)$. }\end{center}
In this case, the restriction of ${\rm ev}_{{\text{$\mathsf{m}$}}^n}$ to $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$ is surjective (see \cite[Lemma~5.6]{savage14}), and induces a surjective homomorphism of Lie superalgebras \[ {\rm ev}_{{\text{$\mathsf{m}$}}^n}^\Gamma \colon \text{$(\mathfrak{g} \otimes A)^\Gamma$} \to \text{$\mathfrak{g} \otimes A$} / {\text{$\mathsf{m}$}}^n. \] Given a $\text{$\mathfrak{g} \otimes A$} / {\text{$\mathsf{m}$}}^n$-module $V$ with associated representation $\rho \colon \text{$\mathfrak{g} \otimes A$} / {\text{$\mathsf{m}$}}^n \to \lie{gl}(V)$, define $\Gev{{\text{$\mathsf{m}$}}^n} (V)$ to be the $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-module with associated representation given by the pull-back of $\rho$ along ${\rm ev}_{{\text{$\mathsf{m}$}}^n}^\Gamma$, \[ \Gev{{\text{$\mathsf{m}$}}^n}(\rho) \colon \text{$(\mathfrak{g} \otimes A)^\Gamma$} \pra[{\rm ev}_{{\text{$\mathsf{m}$}}^n}^\Gamma] \text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}^n \pra[\ \rho \ ] \lie{gl} (V). \]
Given ${\text{$\mathsf{m}$}} \in \maxspec(A)$ and $n \in \bb Z_{>0}$, denote by ${\rm Irred}(\text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}^n)$ the set of isomorphism classes of finite-dimensional irreducible $\text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}^n$-modules, and denote by $\cal R$ the disjoint union \[ \cal R = \bigsqcup_{\atop{n \in \bb Z_{>0}}{{\text{$\mathsf{m}$}} \in \maxspec(A)}} {\rm Irred}(\text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}^n). \] Notice that the action of $\Gamma$ on $\text{$\mathfrak{g} \otimes A$}$ induces an action of $\Gamma$ on $\cal R$. Namely, let $[V]$ be an element in $\cal R$, and let $V$ be a representative of the class $[V]$ with associated representation $\rho \colon \text{$\mathfrak{g} \otimes A$} / {\text{$\mathsf{m}$}}^n \to \lie{gl}(V)$. For each $\gamma \in \Gamma$, define $\gamma\cdot [V]$ in $\cal R$ to be the isomorphism class of the $\text{$\mathfrak{g} \otimes A$} / (\gamma{\text{$\mathsf{m}$}})^n$-module $V^\gamma$, whose associated representation $\rho^\gamma \colon \text{$\mathfrak{g} \otimes A$} / (\gamma{\text{$\mathsf{m}$}})^n \to \lie{gl}(V)$ is given by $\rho^\gamma (x) = \rho (\gamma^{-1} x)$, for all $x \in \text{$\mathfrak{g} \otimes A$} / (\gamma{\text{$\mathsf{m}$}})^n$.
Denote by $\cal P$ the set of $\Gamma$-equivariant functions $\pi \colon \maxspec(A) \to \cal R$ satisfying the following conditions: \begin{enumerate}[$\bullet$] \item For each ${\text{$\mathsf{m}$}} \in \maxspec(A)$, $\pi ({\text{$\mathsf{m}$}}) \in {\rm Irred}(\text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}^n)$ for some $n > 0$;
\item $\pi({\text{$\mathsf{m}$}})$ is the isomorphism class of the trivial (one-dimensional irreducible) module for all but finitely many ${\text{$\mathsf{m}$}} \in \maxspec(A)$. \end{enumerate} Notice that for any two representatives $V$ and $V'$ of $\pi({\text{$\mathsf{m}$}})\in {\rm Irred} (\text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}^n)$, there is an isomorphism of $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-modules $\Gev{{\text{$\mathsf{m}$}}^n} V \cong \Gev{{\text{$\mathsf{m}$}}^n} V'$. Thus we will abuse notation and for each maximal ideal ${\text{$\mathsf{m}$}} \subseteq A$, we will denote by $\pi({\text{$\mathsf{m}$}})$ an arbitrary but fixed choice of $(\text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}^n)$-module representative of $\pi({\text{$\mathsf{m}$}})$.
Given $\pi \in \cal P$, define its support to be $\Supp(\pi) = \{ {\text{$\mathsf{m}$}} \in \maxspec(A) \mid \pi({\text{$\mathsf{m}$}}) \textup{ is nontrivial} \}$, and let $\Supp_*(\pi)$ be a subset of $\maxspec(A)$ which contains exactly one element of each $\Gamma$-orbit in $\Supp(\pi)$. Since every $\pi \in \cal P$ is $\Gamma$-equivariant, up to isomorphism, the $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-module $\widehat{\bigotimes}_{{\text{$\mathsf{m}$}} \in \Supp_* (\pi)} \Gev{{\text{$\mathsf{m}$}}^n} \pi({\text{$\mathsf{m}$}})$ is independent of the choice of $\Supp_* (\pi)$ (see \cite[Lemma~5.9]{savage14}). Thus for every $\pi \in \cal P$, we fix an arbitrary subset $\Supp_*(\pi)$ as above and define $\cal V(\pi)$ to be the $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-module \[ \cal V(\pi) = \widehat{\bigotimes_{{\text{$\mathsf{m}$}} \in \Supp_*(\pi)}} \Gev{{\text{$\mathsf{m}$}}^n}\pi({\text{$\mathsf{m}$}}). \]
For most finite-dimensional simple Lie superalgebras, it is known that every finite-dimensional irreducible $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-module is isomorphic to $\cal V(\pi)$ for a unique $\pi \in \cal P$. This was proved by Savage when $\text{$\mathfrak{g}$}$ is a basic Lie superalgebra (see \cite[\textsection 7]{savage14}), by Bagci when $\text{$\mathfrak{g}$}$ is a Lie superalgebra of Cartan type (see \cite[Theorem 4.3]{Bag15}), and by the first author, Moura and Savage when $\text{$\mathfrak{g}$}$ is a queer Lie superalgebra (see \cite[Theorem~7.1]{CMS15}). In Section \ref{sec:periplectic}, we will show that, if $\text{$\mathfrak{g}$}$ is of type $\lie p$, then every finite-dimensional irreducible $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-module is also isomorphic to $\cal V(\pi)$ for a unique $\pi \in \cal P$, and that $n = 1$ for all ${\text{$\mathsf{m}$}} \in \Supp_*(\pi)$. This completes the classification of finite-dimensional irreducible modules for equivariant map superalgebras associated to finite-dimensional simple Lie algebras.
Moreover, if $\lie g$ is of type II, $H$, $S$ or $\tilde S$, then every finite-dimensional irreducible $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-module is isomorphic to $\cal V(\pi)$ for a unique $\pi \in \cal P$ with $n = 1$ for all ${\text{$\mathsf{m}$}} \in \Supp_*(\pi)$ (these modules are called evaluation modules by Savage, see \cite[Definition~5.2]{savage14}). It is important to remark that if $\lie g$ is of type I, then there exist finite-dimensional irreducible $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-modules which are isomorphic to generalized evaluation modules but not to evaluation modules (see \cite[\textsection 4]{rao13}).
\begin{remark} \label{rmk:gather.gen.ev} Let $n, n' \in \bb Z_{> 0}$ with $n' \le n$, let ${\text{$\mathsf{m}$}}$ be a maximal ideal in $A$, and let $V$ be a $\text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}^{n'}$-module with corresponding representation $\rho \colon \text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}^{n'} \to \lie{gl} (V)$. Since $n' \le n$, then ${\text{$\mathsf{m}$}}^n \subseteq {\text{$\mathsf{m}$}}^{n'}$, and therefore we have a canonical projection $\pi \colon \text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}^n \twoheadrightarrow \text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}^{n'}$. We can thus regard $V$ as a $\text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}^n$-module with representation given by $\rho \circ \pi$.
Notice that as representations of $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$, $\Gev{{\text{$\mathsf{m}$}}^{n'}} (\rho \circ \pi)$ and $\Gev{{\text{$\mathsf{m}$}}^{n}} (\rho)$ are the same. Hence, the $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-modules $\Gev{{\text{$\mathsf{m}$}}^{n'}} V$ and $\Gev{{\text{$\mathsf{m}$}}^{n}} V$ are isomorphic. Moreover, if $\rho$ is an irreducible representation of $\text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}^{n'}$, then $\rho \circ \pi$ is an irreducible representation of $\text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}^{n}$. As a consequence, given any two finite-dimensional irreducible $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-modules $V$ and $V'$, one loses no generality in assuming that there exist integers $\ell, n \in \bb Z_{>0}$, maximal ideals ${\text{$\mathsf{m}$}}_1, \dotsc, {\text{$\mathsf{m}$}}_\ell \in \maxspec(A)$ in distinct $\Gamma$-orbits, and for each $i \in \{ 1, \dotsc, \ell \}$, irreducible $\text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}_i^n$-modules $V_i, V'_i$, such that $V \cong \widehat{\bigotimes}_{i=1}^\ell \Gev{{\text{$\mathsf{m}$}}_i^n} V_i$ and $V' \cong \widehat{\bigotimes}_{i=1}^\ell \Gev{{\text{$\mathsf{m}$}}_i^n} V'_i$. \end{remark}
Given $\pi \in \cal P$, let $\ell, n_1, \dotsc, n_\ell \in \bb Z_{>0}$, ${\text{$\mathsf{m}$}}_1, \dotsc, {\text{$\mathsf{m}$}}_\ell$ be maximal ideals of $A$ in distinct $\Gamma$-orbits, and for each $i \in \{ 1, \dotsc, \ell \}$, let $V_i$ be an irreducible $\text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}_i^{n_i}$-module such that $\cal V (\pi) \cong \widehat{\bigotimes}_{i=1}^\ell \Gev{{\text{$\mathsf{m}$}}_i^{n_i}} V_i$. Recall from \eqref{eq:kappa.mods} the definition of $\kappa(V_1, \dotsc, V_\ell)$, and define $\kappa (\pi)$ to be \begin{equation}\label{eq:kappa.function} \kappa(\pi) = \kappa (V_1, \dotsc, V_\ell), \end{equation} and notice that $\cal V(\pi)^{\oplus 2^{\kappa(\pi)}} \cong \bigotimes_{{\text{$\mathsf{m}$}} \in \Supp(\pi)} \Gev{{\text{$\mathsf{m}$}}^n} \pi ({\text{$\mathsf{m}$}})$ for all $\pi \in \cal P$.
\subsection{Ideals}\label{subsec:ideals}
Throughout this subsection, let $\lie g$ be a finite-dimensional simple Lie superalgebra and $A$ be an associative, commutative, finitely-generated $\text{$\Bbbk$}$-algebra with unit. Define the support of an ideal $I \subseteq A$ to be \[ \Supp (I) = \{ {\text{$\mathsf{m}$}} \in \maxspec A \mid I \subseteq {\text{$\mathsf{m}$}}\}. \]
\begin{lemma} \label{lem:assoc-alg-facts} Let $I$ and $J$ be ideals of $A$. \begin{enumerate}[$(a)$] \item \label{lem-item:support-power} For any $n>0$, we have $\Supp(I)=\Supp(I^n)$.
\item \label{lem-item:finiteCod-and-finiteSupp} If $A$ is finitely generated, then $\Supp(I)$ is finite if and only if $I$ has finite codimension in $A$.
\item \label{lem-item:product-intersection} If $\Supp(I) \cap \Supp(J) = \varnothing$, then $I+J=A$ and $IJ = I \cap J$. Moreover, $I^m + J^n = A$ and $I^m J^n = I^m \cap J^n$ for any $m, n > 0$.
\item \label{lem-item:Noetherian-power-radical} If $A$ is Noetherian, then every ideal $I \subseteq A$ contains a power of its radical. In particular, $\rad I = \prod_{{\text{$\mathsf{m}$}} \in \Supp (I)} {\text{$\mathsf{m}$}}$. \end{enumerate} \end{lemma}
\begin{proof} To prove part \eqref{lem-item:support-power}, fix $n > 0$. It is clear that $\Supp(I) \subseteq \Supp (I^n)$. The reverse inclusion follows from the fact that maximal ideals are also prime. Indeed, suppose ${\text{$\mathsf{m}$}}$ is a maximal ideal containing $I^n$. Then for all $a \in I$, we have $a^n \in {\text{$\mathsf{m}$}}$. Since maximal ideals are prime ideals, this implies that $a \in {\text{$\mathsf{m}$}}$, showing that $I \subseteq {\text{$\mathsf{m}$}}$. The proofs of parts \eqref{lem-item:finiteCod-and-finiteSupp}, \eqref{lem-item:product-intersection} and \eqref{lem-item:Noetherian-power-radical} can be found in \cite[\S 2.1]{savage14}. \end{proof}
The next result is a generalization of \cite[Corollary~4.17]{savage14} and will be used in the proof of Lemma~\ref{lem:h1(ga,M)}.
\begin{proposition} \label{prop:ann.fin.dim} Let $\Gamma$ be a finite group acting freely on $\maxspec (A)$ and $M$ be a finite-di\-men\-sion\-al $(\text{$\mathfrak{g} \otimes A$})^\Gamma$-module. Then there exist $\ell, n \in \bb Z_{>0}$ and ${\text{$\mathsf{m}$}}_1, \dots, {\text{$\mathsf{m}$}}_\ell \in \maxspec(A)$, such that ${\text{$\mathsf{m}$}}_1^n \cdots {\text{$\mathsf{m}$}}_\ell^n$ is a $\Gamma$-invariant finite-codimensional ideal of $A$ and $(\lie g \otimes {\text{$\mathsf{m}$}}_1^n \cdots {\text{$\mathsf{m}$}}_\ell^n)^\Gamma M = 0$. \end{proposition}
\begin{proof} We begin by proving the case where $\Gamma$ is trivial. Let $\rho \colon \lie g \otimes A \to \lie{gl} (M)$ be the representation of $\text{$\mathfrak{g} \otimes A$}$ corresponding to $M$. Notice that $\ker\rho$ is a finite-codimensional ideal of $\text{$\mathfrak{g} \otimes A$}$, since $M$ is finite dimensional. Thus, by \cite[Proposition~8.1]{savage14}, $\ker\rho$ must be of the form $\text{$\mathfrak{g}$}\otimes I$, for some finite-codimensional ideal $I \subseteq A$. Now, by Lemma~\ref{lem:assoc-alg-facts}\eqref{lem-item:finiteCod-and-finiteSupp}, the fact that $I$ is finite-codimensional implies that $I$ has finite support; that is, $\Supp(I) = \{ {\text{$\mathsf{m}$}}_1, \ldots, {\text{$\mathsf{m}$}}_\ell \}$ for some $\ell > 0$. Since these maximal ideals are pairwise distinct, the radical of $I$ is given by $\rad I = {\text{$\mathsf{m}$}}_1 \cdots {\text{$\mathsf{m}$}}_\ell$. Moreover, since $A$ is assumed to be finitely generated, by Lemma~\ref{lem:assoc-alg-facts}\eqref{lem-item:Noetherian-power-radical}, there exists $n > 0$, such that $(\rad I)^n \subseteq I$; that is, such that ${\text{$\mathsf{m}$}}_1^n \cdots {\text{$\mathsf{m}$}}_\ell^n \subseteq I$. We thus conclude that there exist $n, \ell \in \bb Z_{>0}$ and ${\text{$\mathsf{m}$}}_1, \dots, {\text{$\mathsf{m}$}}_\ell \in \maxspec(A)$ such that $(\lie g \otimes {\text{$\mathsf{m}$}}_1^n \cdots {\text{$\mathsf{m}$}}_\ell^n) M = 0$.
For the case where $\Gamma$ acts non trivially on M, we recall that, by \cite[Proposition~8.5]{savage14}, $M$ is the restriction of a finite-dimensional $\text{$\mathfrak{g} \otimes A$}$-module $M'$. From the case where $\Gamma$ is trivial, since $M'$ is finite dimensional, there exist $n, k \in \bb Z_{>0}$ and maximal ideals ${\text{$\mathsf{m}$}}_1, \dots, {\text{$\mathsf{m}$}}_k \subseteq A$, such that $(\lie g \otimes {\text{$\mathsf{m}$}}_1^n \cdots {\text{$\mathsf{m}$}}_k^n) M = 0$. Consider all ideals of the form $\gamma {\text{$\mathsf{m}$}}_i$, with $\gamma \in \Gamma$ and $i \in \{1, \dotsc, k\}$. Since $\Gamma$ is a finite group, we can enumerate the elements of $\{ \gamma{\text{$\mathsf{m}$}}_i \mid \gamma \in \Gamma, i \in \{1, \dotsc, k\} \}$ as ${\text{$\mathsf{m}$}}_1, \dotsc, {\text{$\mathsf{m}$}}_\ell$, with $\ell \le |\Gamma| k$. (Notice that $\ell \le |\Gamma| k$ only if ${\text{$\mathsf{m}$}}_i \notin \Gamma {\text{$\mathsf{m}$}}_j$, for $1 \le i \ne j \le k$.) Moreover, notice that ${\text{$\mathsf{m}$}}_1^n \cdots {\text{$\mathsf{m}$}}_\ell^n$ is a $\Gamma$-invariant finite-codimensional ideal and \[ (\text{$\mathfrak{g}$}\otimes{\text{$\mathsf{m}$}}_1^n \cdots {\text{$\mathsf{m}$}}_\ell^n)^\Gamma M = (\text{$\mathfrak{g}$}\otimes{\text{$\mathsf{m}$}}_1^n \cdots {\text{$\mathsf{m}$}}_k^n)^\Gamma M' \subseteq (\text{$\mathfrak{g}$}\otimes{\text{$\mathsf{m}$}}_1^n \cdots {\text{$\mathsf{m}$}}_k^n)M' =0. \qedhere \] \end{proof}
\subsection{Cohomology of Lie superalgebras} \label{ss:homology.cohomology}
Let $\lie g = \lie g_{\bar0} \oplus \lie g_{\bar1}$ be a Lie superalgebra and $V, U$ be $\lie g$-modules. As usual, the $n$-th extension group $\Ext^n_{\lie g}(V, U)$ denotes the $n$-th right derived functor of $\hom_{U(\lie g)}(V, -)$ applied to $U$. In particular, the cohomology of $\lie g$ with coefficients in $U$, is $\h^\bullet (\text{$\mathfrak{g}$}, U) = \Ext^\bullet_\lie g (\text{$\Bbbk$}, U)$.
In order to compute $\Ext^n_{U(\lie g)}(V,U)$, first notice that $\Lambda^n \lie g \otimes V$ and $\hom_\text{$\Bbbk$} (\Lambda^n \lie g, V)$ are naturally $\bb Z_2$-graded. Namely, for all $n \ge 0$ and $z \in \bb Z_2$, \begin{gather*}
(\Lambda^n \lie g \otimes V)_z = \{ x_1 \wedge \dotsb \wedge x_n \otimes v \mid |x_1| + \dotsb + |x_n| + |v| = z \}, \\ \hom_\text{$\Bbbk$} (\Lambda^n \lie g, V)_z = \{ f \colon \Lambda^n \lie g \to V \mid f(\lambda) \in V_{{w+z}}, \ \lambda \in (\Lambda^n\lie g)_{w}, w \in \bb Z_2 \}. \end{gather*} Then consider $\hom_\text{$\Bbbk$} (V, U)$ as a $\text{$\mathfrak{g}$}$-module, where, for homogeneous elements $f \in \hom_\text{$\Bbbk$} (V, U)$ and $x\in \text{$\mathfrak{g}$}$, we have
\[
(xf)(v)=x (f(v))-(-1)^{|x||f|} f(x v) \quad \text{ for all } v \in V.
\] Finally, identify $\Ext^\bullet_\lie g (V,U)$ with the cohomology of the cocomplex \begin{equation} \label{chevalley.cocomplex} 0 \pra \hom_\text{$\Bbbk$} (V, U) \pra[\partial^0] \hom_\text{$\Bbbk$} (\text{$\mathfrak{g}$}, \hom_\text{$\Bbbk$}(V, U)) \pra[\partial^1] \hom_{\text{$\Bbbk$}} (\Lambda^2 \text{$\mathfrak{g}$} , \hom_\text{$\Bbbk$}(V, U)) \pra[\partial^2] \dotsb , \end{equation} where, for all $f \in \hom_\text{$\Bbbk$} (\Lambda^n \text{$\mathfrak{g}$}, \hom_\text{$\Bbbk$}(V, U))$, its image under the differential $\partial^n$ extends linearly \begin{align} \label{eq:codiffs} \partial^nf & (x_0 \wedge \dots \wedge x_n) \notag \\
&= \sum_{0 \le i \le n} (-1)^{i+|x_i|(|f|+|x_0|+ \cdots + |x_{i-1}|)}x_i f(x_0 \wedge \dots \wedge \widehat{x_i} \wedge \dots \wedge x_n) \\
&{\quad}+ \sum_{0 \le i < j \le n} (-1)^{j+|x_j|(|x_{i+1}|+ \cdots + |x_{j-1}|)} f(x_0 \wedge \dots x_{i-1}\wedge [x_i, x_j] \wedge x_{i+1}\wedge\dots \wedge \widehat{x_j} \wedge \dots \wedge x_n), \notag \end{align} for all homogeneous elements $x_0, \dotsc, x_n \in \text{$\mathfrak{g}$}$, $n \ge 0$. Notice that, the differentials $\partial^\bullet$ respect the $\bb Z_2$-grading on $\hom_\text{$\Bbbk$} (\Lambda^\bullet \text{$\mathfrak{g}$}, \hom_\text{$\Bbbk$}(V, U))$, thus it induces a $\bb Z_2$-grading on $\Ext^\bullet_\text{$\mathfrak{g}$} (V, U)$. We denote $\h^\bullet (\text{$\mathfrak{g}$}, U)_z$ by $\h^\bullet_z (\text{$\mathfrak{g}$}, U)$ for all $z \in \bb Z_2$.
We now state some homological techniques that will be used in this paper. This first lemma reduces the computation of extensions between finite-dimensional modules to the computation of cohomologies.
\begin{lemma} \label{lem:isos} If $\lie s$ is a Lie superalgebra and $U$, $V$ and $W$ are finite-dimensional $\lie s$-modules, then we have the following isomorphisms of $\lie s$-modules: \begin{gather*} U \otimes V \cong V \otimes U, \quad (U\otimes V)^* \cong U^*\otimes V^*, \\ \hom_\text{$\Bbbk$} (U \otimes V, W) \cong \hom_\text{$\Bbbk$} (U, V^*\otimes W), \textup{ and} \\ \Ext^n_{\lie s} (V,U) \cong \h^n (\lie s, V^* \otimes U), \ \textup{for all } n>0. \end{gather*} \end{lemma} \begin{proof} The proofs are similar to those in \cite[Lemma 3.1.13]{kumar02}. \end{proof}
The following lemma reduces the computation of the cohomology of any trivial $\lie s$-module; that is, any $\lie s$-module $V$ such that $\lie s \cdot V=0$, to that of the trivial $\lie s$-module $\text{$\Bbbk$}$.
\begin{lemma} \label{lem:triv.mods} If $\lie s$ is a Lie superalgebra and $V$ is a finite-dimensional trivial $\lie s$-module, then \[ \h^\bullet (\lie s, V) \cong \h^\bullet(\lie s, \text{$\Bbbk$}) \otimes V. \] \end{lemma} \begin{proof} This proof follows directly from the definition of the differentials \eqref{eq:codiffs}. \end{proof}
The following proposition is a special case of the well known K\"unneth formula.
\begin{proposition} \label{prop:kunneth} If $\lie s^1, \lie s^2$ are Lie superalgebras, $U_1, V_1$ are $\lie s^1$-modules and $U_2, V_2$ are $\lie s^2$-modules, then \[ \Ext^n_{\lie s^1 \oplus \lie s^2} (U_1 \otimes U_2, V_1 \otimes V_2) \cong \bigoplus_{p+q=n} \Ext^p_{\lie s^1} (U_1, V_1) \otimes \Ext^q_{\lie s^2} (U_2, V_2), \quad n \ge 0. \] \end{proposition} \begin{proof} The proof follows from \cite[Theorem 3.6.3]{weibel94} using standard arguments. \end{proof}
The following result is a graded version of Lyndon-Hochschild-Serre spectral sequence. In the superalgebra setting, a filtration by powers of an ideal turns out to be $\bb Z_2$-graded, thus yielding two spectral sequences that converge respectively to even and odd cohomologies (see Section~\ref{s:transgression} for further details).
\begin{proposition} \label{prop:lhsss} If $\lie s$ is a Lie superalgebra, $V$ is a $\lie s$-module and $\lie i \subseteq \lie s$ is an ideal, then there exist first-quadrant cohomology convergent spectral sequences \[ E_{2}^{p,q} \cong \h^p_z (\lie s/\lie i, \h^q_z (\lie i, V)) \Rightarrow \h^{p+q}_z (\lie s, V), \quad z \in \bb Z_2. \] \end{proposition} \begin{proof} This proof is given in \cite[Chapter 1, \textsection 6.5]{fuks86} and \cite[Theorem~16.6.6]{Mus12}. \end{proof}
The following result is a superalgebra generalization of a well-known result for Lie algebra cohomology. We record it as it will be used to prove Lemma~\ref{lem:com.even}.
\begin{lemma} \label{lem:h1aC} For any Lie superalgebra $\lie s$, we have \[ \h^1_{\bar 0} (\lie s, \text{$\Bbbk$}) \cong (\lie s_{\bar 0} / ([\lie s_{\bar 0}, \lie s_{\bar 0}] + [\lie s_{\bar 1}, \lie s_{\bar 1}]))^* \quad \textup{and} \quad \h^1_{\bar 1} (\lie s, \text{$\Bbbk$}) \cong \h^0 (\lie s_{\bar 0}, \lie s_{\bar 1}^*) \cong (\lie s_{\bar 1} / [\lie s_{\bar 0}, \lie s_{\bar 1}])^*. \] \end{lemma} \begin{proof} Recall that the cocomplex $\Lambda^\bullet \lie s^*$ is $\bb Z_2$-graded, that the differential $\partial^\bullet$ preserves this grading, and that it induces a $\bb Z_2$-grading on $\h^\bullet (\lie s, \text{$\Bbbk$})$. We will compute each graded component of $\h^1 (\lie s, \text{$\Bbbk$})$. First notice that the restriction of $\partial^1$ to the even part is $\partial^1_{\bar 0} \colon \lie s_{\bar 0}^* \to \Lambda^2 \lie s_{\bar 0}^* \oplus S^2 \lie s_{\bar 1}^*$, and that the restriction of $\partial^1$ to the odd part is $\partial^1_{\bar 1} \colon \lie s_{\bar 1}^* \to \lie s_{\bar 0}^* \otimes \lie s_{\bar 1}^*$.
By definition, $\h^1 (\lie s, \text{$\Bbbk$}) = \ker (\partial^1) / \im (\partial^0)$, where $\partial^0 \colon \text{$\Bbbk$} \to \lie s^*$ is given by $\partial^0 (\lambda) (x) = x \cdot \lambda = 0$ for all $\lambda \in \text{$\Bbbk$}, x \in \lie s$, and $\partial^1 \colon \lie s^* \to \Lambda^2 \lie s^*$ is given by $\partial^1 (\varphi)(x \wedge y) = - \varphi ([x,y])$ for all $\varphi \in \lie s^*, x,y \in \lie s$. So, in order to compute $\h^1 (\lie s, \text{$\Bbbk$})$ it is enough to determine $\ker (\partial^1)$.
From the formula of $\partial^1$, it follows that $\partial^1_{\bar 0} (\varphi) = 0$ if and only if $\varphi ([\lie s_{\bar 0}, \lie s_{\bar 0}] + [\lie s_{\bar 1}, \lie s_{\bar 1}]) = 0$. Thus $\h^1_{\bar 0} (\lie s, \text{$\Bbbk$}) \cong (\lie s_{\bar 0} / [\lie s_{\bar 0}, \lie s_{\bar 0}] + [\lie s_{\bar 1}, \lie s_{\bar 1}])^*$. Also from the formula of $\partial^1$, one can see that $\ker(\partial^1_{\bar 1})$ is the kernel of $\partial^0$ in the cocomplex for computing the cohomology of the Lie algebra $\lie s_{\bar 0}$ with coefficients in $\lie s_{\bar 1}^*$ (see Section \ref{ss:homology.cohomology}). Thus $\h^1_{\bar 1} (\lie s, \text{$\Bbbk$}) \cong \h^0 (\lie s_{\bar 0}, \lie s_{\bar 1}^*)$. \end{proof}
The next result follows from the previous one. It will be used in the proof of Lemma~\ref{lem:h1(ga,M)}.
\begin{lemma} \label{lem:com.even} Let $\lie g$ be a finite-dimensional simple Lie superalgebra, $A$ be an associative, commutative algebra with unit, $\Gamma$ be an abelian group acting on $\text{$\mathfrak{g}$}$ and $A$ by automorphisms, and $I$ be a $\Gamma$-invariant ideal of $A$. If $\lie g_{\bar 1}$ is nonzero, then \[ \h_{\bar0}^{1} ( (\lie g \otimes I)^\Gamma, \text{$\Bbbk$} ) \cong \left(\left(\lie g_{\bar 0} \otimes I/I^2\right)^\Gamma \right)^* \quad \textup{and} \quad \h_{\bar1}^{1} ( (\lie g \otimes I)^\Gamma, \text{$\Bbbk$}) \cong \left(\left(\lie g_{\bar 1} \otimes I/I^2\right)^\Gamma\right)^*. \] \end{lemma} \begin{proof} This proof follows from the fact that $\lie g$ is a simple Lie algebra and Lemma~\ref{lem:h1aC}. \details{ First notice that $\lie g_{\bar 0}\oplus [\lie g_{\bar 0}, \lie g_{\bar 1}]$ and $[\lie g_{\bar 1}, \lie g_{\bar 1}]\oplus \lie g_{\bar 1}$ are ideals of $\lie g$. When $\text{$\mathfrak{g}$}$ is simple and $\text{$\mathfrak{g}$}_{\bar1}$ is nonzero, this implies that $[\text{$\mathfrak{g}$}_{\bar0}, \text{$\mathfrak{g}$}_{\bar1}] = \text{$\mathfrak{g}$}_{\bar1}$ and $[\text{$\mathfrak{g}$}_{\bar1}, \text{$\mathfrak{g}$}_{\bar1}] = \text{$\mathfrak{g}$}_{\bar0}$. Hence: \[ \left[ (\lie g_z \otimes I)^\Gamma, (\lie g_{\bar1} \otimes I)^\Gamma \right] = \left[ \lie g_z \otimes I, \lie g_{\bar1} \otimes I \right]^\Gamma = \left( [\lie g_z, \lie g_{\bar1}] \otimes I^2 \right)^\Gamma = (\lie g_{z+\bar1} \otimes I^2)^\Gamma, \quad z \in \bb Z_2. \] The result follows from Lemma~\ref{lem:h1aC}. } \end{proof}
The next corollary follows from Lemma~\ref{lem:com.even}.
\begin{corollary} \label{cor:h1(ga,C)=0} Let $\lie g$ be a finite-dimensional simple Lie superalgebra with $\lie g_{\bar 1} \neq 0$, $A$ be an associative, commutative algebra with unit, and $\Gamma$ be an abelian group acting on $\text{$\mathfrak{g}$}$ and $A$ by automorphisms. If $M$ and $N$ are finite-dimensional trivial $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-modules, then $\Ext^1_\text{$(\mathfrak{g} \otimes A)^\Gamma$} (M, N) = 0$. In particular, $\h^1 (\text{$(\mathfrak{g} \otimes A)^\Gamma$}, \text{$\Bbbk$}) = 0$. \end{corollary} \begin{proof} This proof follows directly from Lemmas~\ref{lem:isos}, \ref{lem:triv.mods} and \ref{lem:com.even}. \details{ By Lemma~\ref{lem:isos}, $\Ext^1_\text{$(\mathfrak{g} \otimes A)^\Gamma$} (M, N) \cong \h^1 (\text{$(\mathfrak{g} \otimes A)^\Gamma$}, M^* \otimes N)$. Since $M$ and $N$ are finite-dimensional trivial $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-modules, by Lemma~\ref{lem:triv.mods}, \[ \h^1 (\text{$(\mathfrak{g} \otimes A)^\Gamma$}, M^* \otimes N) \cong \h^1 (\text{$(\mathfrak{g} \otimes A)^\Gamma$}, \text{$\Bbbk$}) \otimes (M^* \otimes N). \] Since $A$ is an associative, commutative algebra with unit, then $A^2 = A$. Thus, by Lemma~\ref{lem:com.even}, $\h^1 (\text{$(\mathfrak{g} \otimes A)^\Gamma$}, \text{$\Bbbk$}) = 0$. The result follows. } \end{proof}
\section{Equivariant periplectic map Lie superalgebras} \label{sec:periplectic}
\subsection{Structure of periplectic Lie superalgebras}
Given $n\geq 2$, the periplectic Lie superalgebra $\lie p(n)$ is the Lie subalgebra of $\lie{gl}(n+1, n+1)$ whose elements are matrices of the form \begin{equation}\label{eq:matrixper}
M=\left(\begin{array}{r|r}
A & B \\
\hline
C & -A^t \\
\end{array}\right), \end{equation} where $A \in \lie{sl}_{n+1}$, $B=B^t$ and $C^t=-C$.
\begin{center}\textit{ Throughout this section, we will denote $\lie p(n)$ by $\text{$\mathfrak{g}$}$. }\end{center}
The even part, $\text{$\mathfrak{g}$}_{\bar0}$, is isomorphic to the Lie algebra $\lie{sl}_{n+1}$. The structure of $\text{$\mathfrak{g}$}_{\bar 1}$ is the following. Let $S^2 (\text{$\Bbbk$}^{n+1})$ (resp. $\Lambda^2 (\text{$\Bbbk$}^{n+1})^*$) denote the second symmetric (resp. exterior) power of $\text{$\Bbbk$}^{n+1}$ (resp. $(\text{$\Bbbk$}^{n+1})^*$), with the action of $\lie{sl}_{n+1}$ induced by matrix multiplication, and let $\text{$\mathfrak{g}$}_{\bar 1}^+$ (resp. $\text{$\mathfrak{g}$}_{\bar 1}^-$) be the set of all matrices of the form \eqref{eq:matrixper} such that $A=C=0$ (resp. $A=B=0$). As a $\text{$\mathfrak{g}$}_{\bar 0}$-module, we have: $\text{$\mathfrak{g}$}_{\bar 1} \cong \text{$\mathfrak{g}$}_{\bar 1}^+ \oplus \text{$\mathfrak{g}$}_{\bar 1}^-$, $\text{$\mathfrak{g}$}_{\bar 1}^+ \cong S^2 (\text{$\Bbbk$}^{n+1})$ and $\text{$\mathfrak{g}$}_{\bar 1}^- \cong \Lambda^2 (\text{$\Bbbk$}^{n+1})^*$.
If we set $\text{$\mathfrak{g}$}_{-1} = \text{$\mathfrak{g}$}_{\bar 1}^-$, $\text{$\mathfrak{g}$}_0 = \text{$\mathfrak{g}$}_{\bar 0}$ and $\text{$\mathfrak{g}$}_1 = \text{$\mathfrak{g}$}_{\bar 1}^+$, then $\text{$\mathfrak{g}$}= \text{$\mathfrak{g}$}_{-1} \oplus \text{$\mathfrak{g}$}_0 \oplus \text{$\mathfrak{g}$}_1$ is a $\bb Z$-grading of $\text{$\mathfrak{g}$}$ which is compatible with the $\bb Z_2$-grading; that is, $\text{$\mathfrak{g}$}_{\bar 0} = \text{$\mathfrak{g}$}_0$ and $\text{$\mathfrak{g}$}_{\bar 1} = \text{$\mathfrak{g}$}_{-1} \oplus \text{$\mathfrak{g}$}_1$. Let $\lie h \subseteq \text{$\mathfrak{g}$}_0$ be the Cartan subalgebra of $\text{$\mathfrak{g}$}_0$ formed by diagonal matrices. Since $\text{$\mathfrak{g}$}_0$ is isomorphic to $\lie{sl}_{n+1}$, we can identify $\lie h$ with its dual via the bilinear, nondegenerate, $\text{$\mathfrak{g}$}_0$-invariant form $(A_1, A_2) = {\rm tr} (A_1 A_2)$. For $i \in \{ 1, \dotsc, n+1 \}$, let $\varepsilon_i$ be the unique element in $\lie h^*$ satisfying \[ \varepsilon_i (E_{j,j} + E_{k,k} - E_{n+j+1, n+j+1} - E_{n+k+1, n+k+1}) = \delta_{i,j} + \delta_{i,k} \qquad \textup{for all $1 \le j \ne k \le n+1$}. \]
The roots of $\text{$\mathfrak{g}$}$ are described as follows: \begin{enumerate}[$\bullet$] \item Roots of $\text{$\mathfrak{g}$}_{-1}$: $-\varepsilon_i-\varepsilon_j$, where $1 \leq i < j\le n+1$.
\item Roots of $\text{$\mathfrak{g}$}_0$: $\varepsilon_i-\varepsilon_j$, where $1 \le i \ne j \le n+1$.
\item Roots of $\text{$\mathfrak{g}$}_1$: $\varepsilon_i+\varepsilon_j$, where $1 \le i \le j \le n+1$. \end{enumerate}
Choose a triangular decomposition $\lie n_0^-\oplus \lie h\oplus\lie n_0^+$ of the Lie algebra $\text{$\mathfrak{g}$}_0$ such that the positive roots are: $\varepsilon_i-\varepsilon_j$, with $i<j$. We fix a triangular decomposition $\lie n^- \oplus \lie h \oplus \lie n^+$ of $\lie p(n)$, where $\lie n^\pm = \text{$\mathfrak{g}$}_{\pm 1} \oplus \lie n_0^\pm$, and denote by $\lie b = \lie h \oplus \lie n^+$ a Borel subalgebra of $\lie p(n)$. Notice that all the roots of $\text{$\mathfrak{g}$}_1$ are positive.
\subsection{Construction of finite-dimensional irreducible modules}
Recall that throughout this section we are denoting $\text{$\mathfrak{g}$} = \lie p(n)$, $n \ge 2$. Let $A$ be an associative commutative algebra with unit. In this subsection we will construct some finite-dimensional irreducible $\lie p(n) \otimes A$-modules which will be used in the next subsection.
For each $\psi\in (\lie h\otimes A)^*$, denote by $\text{$\Bbbk$}_\psi$ the one-dimensional $\lie b \otimes A$-module where the action of $\lie n^+\otimes A$ is trivial and the action of $\lie h\otimes A$ is given by $\psi$. Define a Verma type module $M(\psi)$ to be \[ M(\psi) = U(\text{$\mathfrak{g}$}\otimes A) \otimes_{U(\lie b \otimes A)} \text{$\Bbbk$}_\psi. \] Since a submodule of $M(\psi)$ is proper if and only if its intersection with $\text{$\Bbbk$}_\psi$ is trivial, $M(\psi)$ admits a unique maximal non-proper submodule. Let $V(\psi)$ denote the unique irreducible quotient of $M(\psi)$ by such a submodule.
\begin{proposition} Every finite-dimensional irreducible $\text{$\mathfrak{g} \otimes A$}$-module is isomorphic to $V(\psi)$, for some $\psi \in (\lie h \otimes A)^*$. \end{proposition}
\begin{proof} The proof of \cite[Proposition~4.5]{savage14} only requires the existence of a nonzero weight vector $v \in V$. Since $\lie h \otimes A$ is abelian, such a vector always exists. \end{proof}
The annihilator $\Ann_A (V)$ of a $\text{$\mathfrak{g} \otimes A$}$-module $V$ is, by definition, the sum of all ideals $I$ of $A$ such that $(\text{$\mathfrak{g}$} \otimes I)V=0$. The support of $V$ is defined to be \[ \Supp (V) = \Supp (\Ann_A(V)). \]
\begin{proposition}\label{prop:irred.tensor} The tensor product of two irreducible finite-dimensional $\text{$\mathfrak{g} \otimes A$}$-modules with disjoint supports is irreducible as well. \end{proposition}
\begin{proof} This proof is similar to that of \cite[Proposition~4.12]{savage14}.
\details{ Assume that $V_1$, $V_2$ are irreducible finite-dimensional $\text{$\mathfrak{g} \otimes A$}$-modules with disjoint supports, and let $\rho_1$ and $\rho_2$ be the representations associated to $V_1$ and $V_2$, respectively. If $I_1$ and $I_2$ denote their annihilators, then the representation $\rho_1\otimes \rho_2$ associated to the action of $\text{$\mathfrak{g} \otimes A$}$ on $V_1\otimes V_2$ factors through the composition \begin{equation} \label{eq:tensor-product-factor-map} \text{$\mathfrak{g} \otimes A$} \hookrightarrow (\text{$\mathfrak{g} \otimes A$}) \oplus (\text{$\mathfrak{g} \otimes A$}) \twoheadrightarrow (\text{$\mathfrak{g} \otimes A$}/I_1) \oplus (\text{$\mathfrak{g} \otimes A$}/I_2), \end{equation} where the injective homomorphism on the left is the diagonal map and the surjective homomorphism on the right is induced by the quotient maps. Notice that for $i=1,2$, $V_i$ is an irreducible finite-dimensional $\text{$\mathfrak{g} \otimes A$}/I_i$-module, and its highest weight space has dimension one. Since homomorphisms of modules preserve weight spaces, we have $\End_\text{$\mathfrak{g} \otimes A$} (V_1) \cong \End_\text{$\mathfrak{g} \otimes A$} (V_2) \cong \text{$\Bbbk$}$. Therefore, it follows from \cite[Proposition~8.4]{Che95} that $V_1\otimes V_2$ is an irreducible $(\text{$\mathfrak{g} \otimes A$}/I_1) \oplus (\text{$\mathfrak{g} \otimes A$}/I_2)$-module.
Now, by the fact that the supports of $I_1$ and $I_2$ are disjoints, we have that $I_1\cap I_2 = I_1 I_2$ and then $A=I_1+I_2$. Therefore $\text{$\mathfrak{g}$}\otimes (A/I_1I_2)\cong (\text{$\mathfrak{g} \otimes A$}/I_1)\oplus (\text{$\mathfrak{g} \otimes A$}/I_2)$, and we have the following commutative diagram: \[ \xymatrix@R=6ex@C=4ex{ \text{$\mathfrak{g} \otimes A$} \ar[r]^-{\Delta} \ar@{->>}[d] & (\text{$\mathfrak{g} \otimes A$}) \oplus (\text{$\mathfrak{g} \otimes A$}) \ar@{->>}[d] \\ \text{$\mathfrak{g} \otimes A$}/I_1I_2 \ar[r]^-{\cong} & (\text{$\mathfrak{g} \otimes A$}/I_1) \oplus (\text{$\mathfrak{g} \otimes A$}/I_2). } \] This proves that \eqref{eq:tensor-product-factor-map} is surjective, and therefore our result follows. \qedhere } \end{proof}
This proof of the next result follows ideas contained in \cite[Theorem~4.16]{savage14} (see also \cite[Lemma~2.3]{Bag15} and \cite[Proposition~4.3]{BLL15}).
\begin{proposition}\label{prop:quasi-fin.and.ann} Let $\psi\in (\lie h\otimes A)^*$. The weight spaces of $V(\psi)$ are finite dimensional if and only if there exists an ideal $I$ of $A$ of finite codimension such that $(\text{$\mathfrak{g}$}\otimes I)V(\psi)=0$. \end{proposition}
\begin{proof} Suppose all the weight spaces of $V(\psi)$ are finite dimensional and let $v$ be a highest weight vector of $V(\psi)$. Let $\Delta^-$ denote the set $\{ -\varepsilon_i \pm \varepsilon_j \mid 1 \le i < j \le n \}$. For each $\alpha \in \Delta^-$, define $I_\alpha$ to be $\{ a \in A \mid (\text{$\mathfrak{g}$}_{\alpha}\otimes a)v=0\}$. Since each weight space of $V(\psi)$ is finite dimensional, $I_\alpha$ is an ideal of $A$ of finite codimension. Let $I$ be $\bigcap_{\alpha \in \Delta^-} I_\alpha$. Since $\text{$\mathfrak{g}$}$ is finite dimensional, $I$ is an intersection of finitely many finite-codimensional ideals. In particular, $I$ is also a finite-codimensional ideal of $A$. We claim that $(\text{$\mathfrak{g}$}\otimes I)V(\psi)=0$. Indeed, the fact that $(\lie n^+\otimes A)v=0$ follows from the fact that $v$ is a highest weight vector. The fact that $(\lie n^-\otimes I)v=0$ follows from the construction of $I$. Finally, notice that $\lie h\subseteq [\lie n^-,\lie n^+]$, and so $(\lie h\otimes I)v\subseteq [\lie n^-\otimes I,\lie n^+\otimes A]v=0$. Thus $(\text{$\mathfrak{g}$}\otimes I)v=0$ and hence $W=\{w\in V(\psi) \mid (\text{$\mathfrak{g}$}\otimes I)v=0\}$ is a non-trivial submodule of $V(\psi)$. Since $V(\psi)$ is irreducible, we conclude that $W=V(\psi)$.
Suppose now there exists a finite-codimensional ideal $I$ of $A$ such that $(\text{$\mathfrak{g}$}\otimes I)V(\psi)$. Then the action of $\text{$\mathfrak{g} \otimes A$}$ on $V(\psi)$ factors through an action of the finite-dimensional Lie superalgebra $\text{$\mathfrak{g} \otimes A$}/I$. Thus, by standards arguments using the PBW Theorem, all the weight spaces of $V(\psi)$ are finite dimensional. \end{proof}
\begin{center}\textit{ For the remainder of this section, we will assume that $A$ is finitely generated. }\end{center}
\begin{proposition} Let $\psi\in (\lie h\otimes A)^*$. The support of $V(\psi)$ is finite if and only if there exists a finite-codimensional ideal $I$ of $A$ such that $(\text{$\mathfrak{g}$}\otimes I)V(\psi)=0$. \end{proposition}
\begin{proof} Recall that $\Supp V(\psi) = \Supp \Ann_A(V(\psi))$. So $\Ann_A(V(\psi))$ has finite codimension if and only if there exists a finite-codimensional ideal $I$ of $A$ such that $(\text{$\mathfrak{g}$}\otimes I)V(\psi)=0$. Since $A$ is finitely generated, $\Ann_A(V(\psi))$ has finite codimension if and only if its support is finite. \end{proof}
Before stating the next result, recall that $\text{$\mathfrak{g}$}_{\bar 0}$ is a finite-dimensional simple Lie algebra.
\begin{proposition}\label{prop:radical.ann} If $V$ is an irreducible finite-dimensional $\text{$\mathfrak{g} \otimes A$}$-module, then $(\text{$\mathfrak{g}$}\otimes J)V=0$ for some radical ideal $J$ of $A$ of finite codimension. \end{proposition}
\begin{proof} Since $V$ is irreducible and has finite dimension, Proposition~\ref{prop:quasi-fin.and.ann} implies that $(\text{$\mathfrak{g}$}\otimes I)V=0$ for some finite-codimensional ideal $I$ of $A$. Let $J=\sqrt I$ be the radical of $I$. We will show that $(\text{$\mathfrak{g}$}\otimes J)V=0$. Since we are assuming that $A$ is finitely generated (and in particular, Noetherian), there exists some power of $J$ that is contained in $I$. Then $\text{$\mathfrak{g}$}\otimes (I/J)$ is a solvable Lie superalgebra satisfying the following property: \[ \left[ (\text{$\mathfrak{g}$}\otimes (J/I))_{\bar 1},(\text{$\mathfrak{g}$}\otimes (J/I))_{\bar 1}\right] = \left[ \text{$\mathfrak{g}$}_{\bar 1}, \text{$\mathfrak{g}$}_{\bar 1} \right] \otimes (J^2/I) \subseteq \text{$\mathfrak{g}$}_{\bar 0}\otimes (J^2/I) = \left[ (\text{$\mathfrak{g}$}\otimes (J/I))_{\bar 0}, (\text{$\mathfrak{g}$}\otimes (J/I))_{\bar 0} \right], \] where the last equality follows from the fact that $\text{$\mathfrak{g}$}_{\bar 0}$ is a simple Lie algebra. Therefore, it follows from \cite[Proposition~5.2.4]{kac77}, that any irreducible finite-dimensional representation of $\text{$\mathfrak{g}$}\otimes J$ is one-dimensional. Then there exists a nonzero vector $w\in V$, and an element $\mu \in (\text{$\mathfrak{g}$}\otimes J)^*$, such that $xw=\mu(x)w$ for all $x\in \text{$\mathfrak{g}$}\otimes J$. We claim that $\mu=0$. Since $V$ is finite dimensional, for any $z\in \lie n^\pm\otimes J$, there exists $m\geq 0$ such that $z^mw=\mu(z)^m w=0$. In other words, $\mu(\lie n^\pm\otimes J)=0$, and hence $(\lie n^\pm\otimes J)w=0$. Let $\mu'$ denote the restriction of $\mu$ to $\text{$\mathfrak{g}$}_{\bar 0}\otimes J$. Since $\text{$\mathfrak{g}$}_{\bar 0}$ is a simple Lie algebra, it follows that the kernel of $\mu'$ must be $\text{$\mathfrak{g}$}_{\bar 0}\otimes J$. In particular, $\mu'(\lie h \otimes J)=0$, since $\lie h\subseteq \text{$\mathfrak{g}$}_{\bar 0}$. We have thus proved that $(\text{$\mathfrak{g}$}\otimes J)w=0$. Now the result follows from the irreducibility of $V$ along with the fact that $W=\{v\in V \mid (\text{$\mathfrak{g}$}\otimes J)v=0\}$ is a nonzero submodule of $V$. \end{proof}
\subsection{Classification of finite-dimensional irreducible modules} \label{ss:class.pn}
Recall that we are assuming that $\text{$\mathfrak{g}$} = \lie p (n)$, $n \ge 2$, and that $A$ is an associative, commutative, finitely-generated algebra with unit. In this subsection we will classify all the finite-dimensional irreducible $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-modules in the case where $\Gamma$ is a finite abelian group acting on $\text{$\mathfrak{g}$}$ and $A$ by automorphisms and such that the induced action of $\Gamma$ on $\maxspec (A)$ is free.
Recall from Section~\ref{ss:map.superalgebras} that for every ${\text{$\mathsf{m}$}} \in \maxspec(A)$ one may consider the composition \[ {\rm ev}_{{\text{$\mathsf{m}$}}} \colon \text{$\mathfrak{g} \otimes A$} \to \text{$\mathfrak{g} \otimes A$} / \lie g \otimes {\text{$\mathsf{m}$}} \iso \text{$\mathfrak{g}$}. \] Furthermore for a $\text{$\mathfrak{g}$}$-module $V$ with associated representation $\rho \colon \text{$\mathfrak{g}$} \to \lie{gl}(V)$, the module $\ev{{\text{$\mathsf{m}$}}} (V)$ is defined to be the $\text{$\mathfrak{g} \otimes A$}$-module with associated representation given by the pull-back of $\rho$ along ${\rm ev}_{{\text{$\mathsf{m}$}}}$: \[ \ev{{\text{$\mathsf{m}$}}}(\rho) \colon \text{$\mathfrak{g} \otimes A$} \pra[{\rm ev}_{{\text{$\mathsf{m}$}}}] \text{$\mathfrak{g}$} \pra[\ \rho \ ] \lie{gl} (V). \] The $\text{$\mathfrak{g} \otimes A$}$-module $\ev{{\text{$\mathsf{m}$}}} (V)$ is called an evaluation module and its associated representation $\ev{{\text{$\mathsf{m}$}}}(\rho)$ is called an evaluation representation. Define $\Gev{{\text{$\mathsf{m}$}}}(\rho)$ to be the restriction of $\ev{{\text{$\mathsf{m}$}}}(\rho)$ to $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$.
\begin{theorem}\label{thm:irred.classification} Every irreducible finite-dimensional $\text{$\mathfrak{g}$}\otimes A$-module $V$ is isomorphic to a tensor product of evaluation modules. \end{theorem}
\begin{proof} Since $V$ is finite dimensional, by Proposition~\ref{prop:radical.ann}, there exists a radical ideal $I$ of $A$ of finite codimension such that $(\text{$\mathfrak{g}$}\otimes I)V=0$. Since $A$ is finitely generated and $I$ has finite codimension, Lemma~\ref{lem:assoc-alg-facts}~\eqref{lem-item:finiteCod-and-finiteSupp} implies that the support of $I$ is finite. If $\Supp (I) = \{{\text{$\mathsf{m}$}}_1,\ldots, {\text{$\mathsf{m}$}}_n\} \subseteq \maxspec(A)$, then $I=\sqrt{I}={\text{$\mathsf{m}$}}_1\cdots {\text{$\mathsf{m}$}}_n$. Therefore the action of $\text{$\mathfrak{g}$}\otimes A$ on $V$ factors through the map \begin{equation} \label{eq:quotient} \text{$\mathfrak{g}$}\otimes A \twoheadrightarrow \text{$\mathfrak{g}$}\otimes A/I \iso \bigoplus_{i=1}^n \text{$\mathfrak{g}$}\otimes A/{\text{$\mathsf{m}$}}_i \iso \text{$\mathfrak{g}$}^{\oplus n}. \end{equation} In other words, $V$ is isomorphic to the pull-back of an irreducible finite-dimensional $\text{$\mathfrak{g}$}^{\oplus n}$-module $W$ along \eqref{eq:quotient}. Notice that $W$ is also an irreducible finite-dimensional module for $U\left( \text{$\mathfrak{g}$}^{\oplus n}\right) \cong U (\text{$\mathfrak{g}$})^{\otimes n}$. By \cite[Proposition~8.4]{Che95}, there exist irreducible finite-dimensional modules $V_1, \dotsc, V_n$ for $U(\text{$\mathfrak{g}$})$ such that $W$ is either isomorphic to $V_1 \otimes \dotsm \otimes V_n$ or to a proper submodule of $V_1\otimes \dotsm \otimes V_n$. Since $V_1 \otimes \dotsm \otimes V_n$ is irreducible by Proposition~\ref{prop:irred.tensor}, $W \cong V_1 \otimes \dotsm \otimes V_n$, and $V$ is isomorphic to a tensor product of evaluation modules. \end{proof}
\begin{center}\textit{ \begin{minipage}{.82\textwidth} From now on we will assume that $\Gamma$ is a finite abelian group acting on $\text{$\mathfrak{g}$}$ and $A$ by automorphisms and such that the induced action of $\Gamma$ on $\maxspec (A)$ is free. \end{minipage} } \end{center}
Let $\irred(\text{$\mathfrak{g}$})$ (resp. $\irred\text{$(\mathfrak{g} \otimes A)^\Gamma$}$) be the set of isomorphism classes of irreducible finite-di\-men\-sion\-al modules for $\text{$\mathfrak{g}$}$ (resp. $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$). Let $[V] \in \irred (\text{$\mathfrak{g}$})$ denote the isomorphism class of a $\text{$\mathfrak{g}$}$-module $V$. Notice that, if $V$ and $V'$ are isomorphic \text{$\mathfrak{g}$}-modules, then $\Gev{{\text{$\mathsf{m}$}}}(V)$ and $\Gev{{\text{$\mathsf{m}$}}} (V')$ are isomorphic \text{$(\mathfrak{g} \otimes A)^\Gamma$}-modules. Therefore, for each $[V] \in \irred(\text{$\mathfrak{g}$})$, we define $\Gev{{\text{$\mathsf{m}$}}}[V]$ to be the isomorphism class of $\Gev{{\text{$\mathsf{m}$}}}(V)$ in $\irred\text{$(\mathfrak{g} \otimes A)^\Gamma$}$.
Also recall that the action of $\Gamma$ on $\text{$\mathfrak{g}$}$ induces an action of $\Gamma$ on $\irred(\text{$\mathfrak{g}$})$. Namely, if $V$ is a \text{$\mathfrak{g}$}-module representative of $[V] \in \irred(\text{$\mathfrak{g}$})$ with associated representation $\rho \colon \text{$\mathfrak{g}$} \to \lie{gl} (V)$, then $\gamma [V] = [V^\gamma]$, where $V^\gamma$ is a \text{$\mathfrak{g}$}-module with underlying vector space $V$ and associated representation $\rho' \colon \text{$\mathfrak{g}$} \to \lie{gl}(V)$ given by $\rho'(x) = \rho (\gamma^{-1} x)$ for all $x \in \text{$\mathfrak{g}$}$.
Let $\cal P$ be the set of $\Gamma$-equivariant functions $\pi \colon \maxspec(A) \to \irred(\text{$\mathfrak{g}$})$ such that $\pi({\text{$\mathsf{m}$}})=[\text{$\Bbbk$}]$ for all but finitely many distinct ${\text{$\mathsf{m}$}} \in \maxspec(A)$. Given $\pi \in \cal P$, recall that its support is defined to be $\Supp(\pi) = \{ {\text{$\mathsf{m}$}} \in \maxspec(A) \mid \pi({\text{$\mathsf{m}$}}) \textup{ is nontrivial} \}$. Let $X_*$ be denote the set of all finite subsets ${\text{$\mathsf{M}$}} \subseteq \maxspec(A)$ satisfying the following property: if ${\text{$\mathsf{m}$}}$ and ${\text{$\mathsf{m}$}}'$ are distinct elements in ${\text{$\mathsf{M}$}}$, then ${\text{$\mathsf{m}$}}\notin\Gamma{\text{$\mathsf{m}$}}'$. As in Section~\ref{ss:map.superalgebras}, for each $\pi \in \cal P$, fix an element $\Supp_ *(\pi)$ in $X_*$ containing one element of each $\Gamma$-orbit in $\Supp(\pi)$, and define $\cal V(\pi)$ to be the $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-module \[ \cal V(\pi) = \bigotimes_{{\text{$\mathsf{m}$}} \in \Supp_*(\pi)} \Gev{{\text{$\mathsf{m}$}}}\pi({\text{$\mathsf{m}$}}). \]
\begin{lemma}\label{lem:properties_map} With the above notation, the following hold: \begin{enumerate}[(a)] \item \label{lem:ev.well.def} For every $\pi\in \cal P$, the isomorphism class of $\cal V(\pi)$ does not depend on the choice of $\Supp_*(\pi)$.
\item \label{lem:ev.irred} For every $\pi\in \cal P$, the $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-module $\cal V(\pi)$ is irreducible.
\item \label{lem:properties_map.injection} The map $\cal P \to \irred \text{$(\mathfrak{g} \otimes A)^\Gamma$}$ given by $\pi \mapsto \cal V(\pi)$ is injective. \end{enumerate} \end{lemma}
\begin{proof} Part~\eqref{lem:ev.well.def} follows from \cite[Lemma~5.9]{savage14}. Part~\eqref{lem:ev.irred} follows from Proposition~\ref{prop:irred.tensor} along with the fact that the map $ \Gev{{\text{$\mathsf{m}$}}}$ is surjective for all ${\text{$\mathsf{m}$}} \in \maxspec(A)$. Part~\eqref{lem:properties_map.injection} follows from \cite[Proposition~5.11]{savage14}. Notice that the condition of $\text{$\mathfrak{g}$}$ being basic is not used in the proofs of the results cited from \cite{savage14}. \end{proof}
\begin{proposition}\label{prop:restriction.property} Every finite-dimensional $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-module $V$ is isomorphic to the restriction of a finite-di\-men\-sion\-al $\text{$\mathfrak{g} \otimes A$}$-module $V'$ whose support is in $X_*$. Moreover, $V$ is irreducible if and only if $V'$ is. \end{proposition}
\begin{proof} The proof of this fact for any finite-dimensional simple Lie superalgebra is the same as the proof of \cite[Proposition~8.5]{savage14}. Notice that, although the fact that $\Supp (V')$ is an element of $X_*$ is not stated, it is also proved there. \end{proof}
\begin{theorem}\label{thm:main.p(n)} Let $A$ be an associative, commutative, finitely-generated algebra with unit, $\Gamma$ be a finite abelian group acting on $A$ and $\text{$\mathfrak{g}$}$ by automorphisms, and such that the induced action of $\Gamma$ on $\maxspec(A)$ is free. The map $\cal P \to \irred \text{$(\mathfrak{g} \otimes A)^\Gamma$}$ given by $\pi \mapsto \cal V(\pi)$ is a bijection. \end{theorem}
\begin{proof} Recall from Lemma~\ref{lem:properties_map}~\eqref{lem:properties_map.injection} that the map $\cal P \to \irred \text{$(\mathfrak{g} \otimes A)^\Gamma$}$ given by $\pi \mapsto \cal V(\pi)$ is injective. Let $V$ be an irreducible finite-dimensional $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-module. By Proposition~\ref{prop:restriction.property}, $V$ is isomorphic to the restriction of an irreducible finite-dimensional $\text{$\mathfrak{g} \otimes A$}$-module $V'$, whose support is in $X_*$. Hence, by Theorem~\ref{thm:irred.classification}, $V' \cong \bigotimes_{i=1}^n \ev{{\text{$\mathsf{m}$}}_i}(V_i)$ for some $n \ge 0$, $\{ {\text{$\mathsf{m}$}}_1,\dotsc, {\text{$\mathsf{m}$}}_n \}\in X_*$ and irreducible finite-dimensional \text{$\mathfrak{g}$}-modules $V_1, \dotsc, V_n$. Thus, $V$ is isomorphic to $\cal V (\pi)$, where $\pi ({\text{$\mathsf{m}$}}_i)=[V_i]$ for all $i \in \{1, \dotsc, n\}$, and $\pi ({\text{$\mathsf{m}$}})=[\text{$\Bbbk$}]$ for all ${\text{$\mathsf{m}$}} \not\in \Supp(V')$. \end{proof}
\section{Transgression maps} \label{s:transgression}
Let $\mathfrak{s}$ be a Lie superalgebra, and consider its exterior algebra $\Lambda^\bullet \mathfrak{s}$ with the $\mathfrak{s}$-module structure induced from the adjoint representation. Given a proper ideal $\mathfrak{i} \subset \mathfrak{s}$, define an increasing filtration $0 = \Lambda^\bullet_0 \subsetneq \Lambda^\bullet_1 \subsetneq \Lambda^\bullet_2 \subsetneq \dots \subsetneq \Lambda^\bullet = \Lambda^\bullet \mathfrak{s}$ by: \begin{gather*} \Lambda^n_p := \textup{span}_\C \{ x_1 \wedge \dots \wedge x_n \mid x_{i_1}, \dotsc, x_{i_k} \in {\mathfrak{i}}\textup{ for some } k > n-p, \ 1 \le i_1 < \dotsb < i_k \le n \}, \ 0 < p \le n, \\ \Lambda^n_p := \Lambda^n \mathfrak{s} \quad \textup{for all } p > n. \end{gather*} Now, given a $\mathfrak{s}$-module $M$, define a decreasing filtration $\dots \subsetneq C^\bullet_2 \subsetneq C^\bullet_1 \subsetneq C^\bullet_0 = \hom_\text{$\Bbbk$} \left( \Lambda^\bullet \mathfrak{s}, M \right)$ by \[ C^\bullet_p := \left\{ f \colon \Lambda^\bullet \to M \mid f(\Lambda^\bullet_p) = 0 \right\}.
\]
Recall that $\h^\bullet (\mathfrak{s}, M)$, the cohomology of $\mathfrak{s}$ with coefficients in $M$, is the cohomology of the cocomplex $\left( \hom_\text{$\Bbbk$} (\Lambda^\bullet \mathfrak{s}, M), \partial^\bullet \right)$, where $\partial^\bullet$ is given in \eqref{eq:codiffs}. Moreover, notice that $\partial^n (C^n_p) \subseteq C^{n+1}_p$ for all $p, n \ge 0$.
In this section, we will review the construction of the Lyndon-Hochschild-Serre spectral sequence that converges to $\h^\bullet (\mathfrak{s}, M)$ in the case where $\mathfrak{i} \cdot M = 0$. Even though most of this material is not new (see \cite{HS53} and \cite[\S16.6]{Mus12}), we will use it to describe the transgression map in a few cases that will be important in the proof of Theorem~\ref{thm:main} (see Proposition~\ref{prop:ker.transgression}).
\subsection{$E_0$-page} \label{ss:E0.lhs}
Let $E_0^{p,q} := C^{p+q}_p / C^{p+q}_{p+1}$ for all $p, q \ge 0$. Recall that $C^n_p$ consists of linear maps in $\hom_\text{$\Bbbk$} \left( \Lambda^n, M \right)$ that vanish on $\Lambda^n_p$ (and similarly for $C^n_{p+1}$). Hence, we have the following isomorphisms of vector spaces \begin{equation} \label{eq:E0.iso} E_0^{p,q} \cong \frac{\hom_\text{$\Bbbk$} (\Lambda^{p+q} / \Lambda^{p+q}_{p}, M)}{\hom_\text{$\Bbbk$} (\Lambda^{p+q} / \Lambda^{p+q}_{p+1}, M)} \cong \hom_\text{$\Bbbk$} (\Lambda^{p+q}_{p+1} / \Lambda^{p+q}_p, M) \qquad \textup{ for all } p, q \ge 0, \end{equation} where the second isomorphism is given by the First Isomorphism Theorem and the restriction of linear maps in $\hom_\text{$\Bbbk$} (\Lambda^{p+q} / \Lambda^{p+q}_{p}, M)$ to $\Lambda^{p+q}_{p+1}/\Lambda^{p+q}_p$.
Now recall that, as vector spaces, $\mathfrak{s} \cong {\mathfrak{i}}\oplus (\mathfrak{s}/{\mathfrak{i}})$ and thus, $\Lambda^n \mathfrak{s} \cong \bigoplus_{j=0}^n \Lambda^{n-j}(\mathfrak{i}) \otimes \Lambda^j (\mathfrak{s}/{\mathfrak{i}})$. Using these isomorphisms and the definition of $\Lambda^n_p$, we see that, as vector spaces, \begin{equation} \label{eq:Lambda.iso} \Lambda^n_p \cong \bigoplus_{j=0}^{p-1} \Lambda^{n-j}(\mathfrak{i}) \otimes \Lambda^j (\mathfrak{s}/{\mathfrak{i}}) \quad \textup{ and } \quad \Lambda^{p+q}_{p+1} / \Lambda^{p+q}_p \cong \Lambda^q(\mathfrak{i}) \otimes \Lambda^p (\mathfrak{s}/{\mathfrak{i}}) \quad \textup{ for all } \ p, q, n \ge 0. \end{equation} Coupling these isomorphisms with the isomorphisms \eqref{eq:E0.iso}, we obtain the following isomorphisms of vector spaces: \[ E_0^{p,q} \cong \hom_\text{$\Bbbk$} (\Lambda^q(\mathfrak{i}) \otimes \Lambda^p (\mathfrak{s}/{\mathfrak{i}}), M) \quad \textup{ for all } \ p, q \ge 0. \] Explicitly, a linear map $f \in \hom_\text{$\Bbbk$}(\Lambda^q(\mathfrak{i}) \otimes \Lambda^p (\mathfrak{s}/{\mathfrak{i}}), M)$ corresponds to the restriction of a linear map $F \in \hom_\text{$\Bbbk$}(\Lambda^{p+q}, M)$ (that vanishes on $\Lambda^{p+q}_{p}$) to $\Lambda^{p+q}_{p+1}$, and the elements in $\mathfrak{s}/\mathfrak{i}$ are identified with elements in $\mathfrak{s}$ via a vector space splitting of the canonical projection $\mathfrak{s} \twoheadrightarrow \mathfrak{s}/\mathfrak{i}$.
Consider the map $D_0^{p,q} \colon E_0^{p,q} \to E_0^{p, q+1}$, explicitly given by $D_0^{p,q}(F + C^{p+q}_{p+1}) = \partial^{p+q}F + C^{p+q+1}_{p+1}$. Using the explicit formula of $\partial^{p+q}F$ \eqref{eq:codiffs} and the isomorphisms \eqref{eq:E0.iso}, \eqref{eq:Lambda.iso}, $D_0^{p,q}$ induces a map $d_0^{p,q} : \hom_\text{$\Bbbk$} (\Lambda^q(\mathfrak{i}) \otimes \Lambda^p (\mathfrak{s}/{\mathfrak{i}}), M) \to \hom_\text{$\Bbbk$} (\Lambda^{q+1}(\mathfrak{i}) \otimes \Lambda^p (\mathfrak{s}/{\mathfrak{i}}), M)$, which can be explicitly written as \begin{align} \label{d0.lhs} d_0^{p,q}f {}&{}(i_0 \wedge \dotsb \wedge i_q \otimes x_1 \wedge \dotsb \wedge x_p) \notag \\
{}&{}= \sum_{0 \le \ell \le q} (-1)^{\ell+|i_\ell|(|F|+|i_0| + \dotsb + |i_{\ell-1}|)} i_\ell \, F(i_0 \wedge \dotsb \wedge \widehat{i_\ell} \wedge \dotsb \wedge i_q \wedge {\widetilde x}_1 \wedge \dotsb \wedge {\widetilde x}_p) \\ &{\quad}+ \sum_{0 \le j < k \le q} (-1)^{\sigma(j,k)} F(i_0 \wedge \dotsb i_{j-1} \wedge [i_j, i_k] \wedge i_{j+1} \wedge \dotsb \wedge \widehat{i_k} \wedge \dotsb \wedge i_q \wedge {\widetilde x}_1 \wedge \dotsb \wedge {\widetilde x}_p), \notag \end{align}
where $\sigma(j,k) = k+|i_k|(|i_{j+1}|+ \dotsb + |i_{k-1}|)$, $i_0, \dotsc, i_q \in {\mathfrak{i}}$, $x_1, \dotsc, x_p \in \mathfrak{s}/\mathfrak{i}$, and ${\widetilde x}_1, \dotsc, {\widetilde x}_p$ are their corresponding representatives in $\mathfrak{s}$. \details{ Notice that \eqref{d0.lhs} does not, in fact, depend on the choice of these representatives, as any other choice would differ from this one by extra terms in ${\mathfrak{i}}$, and $f$ vanishes in $\Lambda^{p+q}_{p+1}$, that is, when more than $q$ of its arguments are in ${\mathfrak{i}}$. }
Now recall that, for each $p,q \ge 0$, the tensor-hom adjunction induces an isomorphism of vector spaces $E_0^{p,q} \cong \hom_\text{$\Bbbk$} \left( \Lambda^q (\mathfrak{i}), \hom_\text{$\Bbbk$} (\Lambda^p (\mathfrak{s}/{\mathfrak{i}}), M) \right)$. Using this isomorphism and \eqref{d0.lhs}, one can check that, for each $p, q \ge 0$, the map $D_0^{p,q}$ corresponds to the $q$-th differential of the cocomplex $\left( \hom_\text{$\Bbbk$} \left( \Lambda^\bullet (\mathfrak{i}), \hom_\text{$\Bbbk$} (\Lambda^p (\mathfrak{s}/{\mathfrak{i}}), M) \right),\, \partial^\bullet_{\mathfrak{i}}\right)$, used to compute $\h^\bullet (\mathfrak{i}, \hom_\text{$\Bbbk$} (\Lambda^p (\mathfrak{s}/{\mathfrak{i}}), M))$.
\details{
In fact, $\hom_\text{$\Bbbk$} (\Lambda^p (\mathfrak{s}/{\mathfrak{i}}), M)$ admits a $\mathfrak{s}$-module structure explicitly given by $(sf)(x) = s(f(x)) -(-1)^{|s||f|} f(s \cdot x)$ for all homogeneous $s \in \mathfrak{s}$, $f \in \hom_\text{$\Bbbk$} (\Lambda^p (\mathfrak{s}/{\mathfrak{i}}), M)$, and $x \in \Lambda^p (\mathfrak{s}/{\mathfrak{i}})$. Since $[{\mathfrak{i}},\mathfrak{s}]\subseteq {\mathfrak{i}}$, it follows that $i \cdot x = 0$ for all $i \in \mathfrak{i}$ and $x \in \Lambda^p (\mathfrak{s}/{\mathfrak{i}})$. Hence, the $\mathfrak{i}$-module structure on $\hom_\text{$\Bbbk$} (\Lambda^p (\mathfrak{s}/{\mathfrak{i}}), M)$ is given by $(if) (x) = i (f(x))$ for all $i \in {\mathfrak{i}}$, $x \in \Lambda^p (\mathfrak{s}/{\mathfrak{i}})$ and $f \in \hom_\text{$\Bbbk$} (\Lambda^p (\mathfrak{s}/{\mathfrak{i}}), M)$. The claim follows from the tensor-hom adjunction, \eqref{eq:codiffs} and \eqref{d0.lhs}. }
\subsection{$E_1$-page} \label{ss:E1.lhs}
Let $E_1^{p, q} := \h^q \left( E_0^{p, \bullet}, D_0^{p, \bullet} \right)$ for all $p, q \ge 0$. As a direct consequence of the arguments of Section~\ref{ss:E0.lhs}, there are isomorphisms of vector spaces: \[ E_1^{p,q} \cong \h^q ({\mathfrak{i}}, \hom_\text{$\Bbbk$} (\Lambda^p (\mathfrak{s}/{\mathfrak{i}}), M)) \quad \textup{ for all } \ p,q \ge 0. \]
Moreover, for each $p, q \ge 0$, there is an isomorphism of vector spaces \[ \h^q ({\mathfrak{i}}, \hom_\text{$\Bbbk$} (\Lambda^p (\mathfrak{s}/{\mathfrak{i}}), M)) \cong \hom_\text{$\Bbbk$} (\Lambda^p (\mathfrak{s}/{\mathfrak{i}}), \h^q ({\mathfrak{i}}, M)). \] \details{ By definition, every element in $\h^q ({\mathfrak{i}}, \hom_\text{$\Bbbk$} (\Lambda^p (\mathfrak{s}/{\mathfrak{i}}), M))$ is an equivalence class of linear maps in $\hom_\text{$\Bbbk$} (\Lambda^q(\mathfrak{i}), \hom_\text{$\Bbbk$}(\Lambda^p(\mathfrak{s}/\mathfrak{i}), M))$. By the tensor-hom adjunction, there are isomorphisms $\hom_\text{$\Bbbk$} (\Lambda^q(\mathfrak{i}), \hom_\text{$\Bbbk$} (\Lambda^p (\mathfrak{s}/{\mathfrak{i}}), M)) \cong \hom_\text{$\Bbbk$} (\Lambda^q(\mathfrak{i}) \otimes \Lambda^p (\mathfrak{s}/{\mathfrak{i}}), M) \cong \hom_\text{$\Bbbk$} (\Lambda^p (\mathfrak{s}/{\mathfrak{i}}), \hom_\text{$\Bbbk$}(\Lambda^q(\mathfrak{i}), M))$. Under these isomorphisms, the differential $\partial^q_{\mathfrak{i}}$ of the cocomplex $\hom_\text{$\Bbbk$} (\Lambda^\bullet(\mathfrak{i}), \hom_\text{$\Bbbk$}(\Lambda^p(\mathfrak{s}/\mathfrak{i}), M))$ corresponds to the differential $d_0^{p,q} : \hom_\text{$\Bbbk$} (\Lambda^q(\mathfrak{i}) \otimes \Lambda^p (\mathfrak{s}/{\mathfrak{i}}), M) \to \hom_\text{$\Bbbk$} (\Lambda^{q+1}(\mathfrak{i}) \otimes \Lambda^p (\mathfrak{s}/{\mathfrak{i}}), M)$, which corresponds, on the cocomplex $\hom_\text{$\Bbbk$}(\Lambda^p(\mathfrak{s}/\mathfrak{i}), \hom_\text{$\Bbbk$}(\Lambda^\bullet(\mathfrak{i}), M))$, to the post-composition with the differential $\partial^q_{\mathfrak{i}, M}$. This implies that $\h^q(\mathfrak{i}, \hom_\text{$\Bbbk$}(\Lambda^p(\mathfrak{s}/\mathfrak{i}), M))$ is isomorphic to $E_1^{p,q}$, which is isomorphic to $\hom_\text{$\Bbbk$} (\Lambda^p(\mathfrak{s}/\mathfrak{i}), \h^q(\mathfrak{i}, M))$.
} Now, recall that $\h^q ({\mathfrak{i}}, M)$ is a $\mathfrak{s}/{\mathfrak{i}}$-module (that is, a $\mathfrak{s}$-module with trivial ${\mathfrak{i}}$-action) with its $\mathfrak{s}$-module structure induced from $\hom_\text{$\Bbbk$} (\Lambda^q(\mathfrak{i}), M)$. \details{
Explicitly, $(sf) (x) = s(f(x)) -(-1)^{|s||f|} f(s \cdot x)$ for all homogeneous $s \in \mathfrak{s}$, $f \in \hom_\text{$\Bbbk$} (\Lambda^q(\mathfrak{i}), M)$, and $x \in \Lambda^q(\mathfrak{i})$. The fact that $\mathfrak{i}$ acts trivially on $\h^q(\mathfrak{i}, M)$ follows from the fact that the elements in $\h^q (\mathfrak{i}, M)$ are equivalence classes (modulo $\im \partial^{q-1}_{\mathfrak{i}, M}$) of linear maps $h \in \hom_\text{$\Bbbk$} (\Lambda^q(\mathfrak{i}), M)$ such that $\partial^q_{\mathfrak{i}, M}(h) = 0$. } Thus, one can consider $\h^\bullet (\mathfrak{s}/\mathfrak{i}, \h^q(\mathfrak{i}, M))$, the cohomology of the cocomplex $\hom_\text{$\Bbbk$} (\Lambda^\bullet (\mathfrak{s}/{\mathfrak{i}}), \h^q({\mathfrak{i}}, M))$, whose differential will be denoted by $\partial_{\mathfrak{s}/{\mathfrak{i}}}^\bullet$ and is given by \eqref{eq:codiffs}.
Since every element in $E_1^{p,q}$ is an equivalence class (modulo $\im D_0^{p, q-1}$) of elements in $E_0^{p,q}$, and every element in $E_0^{p,q}$ is the restriction (to $\Lambda^{p+q}_{p+1}$) of a linear map in $\hom_\text{$\Bbbk$}(\Lambda^{p+q}, M)$, we can define a map $D_1^{p,q} \colon E_1^{p,q} \to E_1^{p+1, q}$ by $D_1^{p,q}(F + \im D_0^{p, q-1}) = \partial^{p+q}F + \im D_0^{p+1, q-1}$. Using the explicit formula for $\partial^{p+q}F$ \eqref{eq:codiffs}, the tensor-hom adjunction and \eqref{d0.lhs}, we obtain that \begin{equation} \label{d1.lhs} \partial^{p+q}F ({\widetilde x}_0 \wedge \dotsb \wedge {\widetilde x}_p \wedge i_1 \wedge \dotsb \wedge i_q) = (\partial^p_{\mathfrak{s}/{\mathfrak{i}}}F ({\widetilde x}_0 \wedge \dots \wedge {\widetilde x}_p)) (i_1 \wedge \dots \wedge i_q) + \im D_0^{p+1,q-1}, \end{equation} for all homogeneous $i_1, \dotsc, i_{q-1} \in {\mathfrak{i}}$, $x_0, \dotsc, x_{p+1} \in \mathfrak{s} / {\mathfrak{i}}$, and their corresponding representatives ${\widetilde x}_0, \dotsc, {\widetilde x}_{p+1} \in \mathfrak{s}$. This implies that $D_1^{p,q} : E_1^{p,q} \to E_1^{p+1,q}$ induces a map \[ d_1^{p,q} : \hom_\text{$\Bbbk$} (\Lambda^p(\mathfrak{s}/\mathfrak{i}), \h^q(\mathfrak{i}, M)) \to \hom_\text{$\Bbbk$} (\Lambda^{p+1}(\mathfrak{s}/\mathfrak{i}), \h^q(\mathfrak{i}, M)), \] which is the $p$-th differential of the cocomplex $( \hom_\text{$\Bbbk$} \left( \Lambda^\bullet (\mathfrak{s}/{\mathfrak{i}}), \h^q({\mathfrak{i}}, M) \right),\, \partial_{\mathfrak{s}/{\mathfrak{i}}}^\bullet )$.
\subsection{$E_2$-page and the transgression map} \label{ss:E2+trans}
Let $E_2^{p,q} := \h^p \left( E_1^{\bullet, q}, D_1^{\bullet, q} \right)$ for all $p, q \ge 0$. As a direct consequence of the arguments of Section~\ref{ss:E1.lhs}, there are isomorphisms \begin{equation} \label{eq:E2.iso} E_2^{p,q} \cong \h^p \left( \mathfrak{s}/{\mathfrak{i}}, \h^q ({\mathfrak{i}}, M) \right) \quad \textup{ for all } \ p, q \ge 0. \end{equation} It is known (see, for instance, \cite[Chapter 1, \textsection 6.5]{fuks86}) that the spectral sequence thus obtained converges to the cohomology of $\mathfrak{s}$ with coefficients in $M$; that is, \begin{equation} \label{eq:LHSss} E_2^{p,q} \cong \h^p \left( \mathfrak{s}/{\mathfrak{i}}, \h^q ({\mathfrak{i}}, M) \right) \Rightarrow \h^{p+q} (\mathfrak{s}, M). \end{equation}
Since every element in $E_2^{p,q}$ is an equivalence class (modulo $\im D_1^{p-1, q}$) of elements in $E_1^{p,q}$, every element in $E_1^{p,q}$ is an equivalence class (modulo $\im D_0^{p, q-1}$) of elements in $E_0^{p,q}$, and every element in $E_0^{p,q}$ is the restriction (to $\Lambda^{p+q}_{p+1}$) of a linear map in $\hom_\text{$\Bbbk$}(\Lambda^{p+q}, M)$, we can define a map $D_2^{p,q} \colon E_2^{p,q} \to E_2^{p+2, q-1}$ by $D_2^{p,q} (F + \im D_1^{p-1,q}) = \partial^{p+q}F + \im D_1^{p+1, q-1}$. Using the explicit formula for $\partial^{p+q}F$ \eqref{eq:codiffs} and \eqref{d1.lhs}, we can obtain that \begin{align} \label{d2.lhs} &\partial^{p+q}F({\widetilde x}_0 \wedge \dotsb \wedge {\widetilde x}_{p+1} \wedge i_1 \wedge \dotsb \wedge i_{q-1}) \\ &{\ }= \sum_{0 \le j < k \le p+1} (-1)^{\sigma(j,k)} F({\widetilde x}_0 \wedge \dotsb \wedge {\widetilde x}_{j-1} \wedge [{\widetilde x}_j, {\widetilde x}_k] \wedge {\widetilde x}_{j+1} \wedge \dotsb \wedge \widehat{{\widetilde x}_k} \wedge \dotsb \wedge {\widetilde x}_{p+1} \wedge i_1 \wedge \dotsb \wedge i_{q-1}) \notag \\ &{\qquad}+ \im D_1^{p+1, q-1}, \notag \end{align}
where $\sigma(j,k) = k+|x_k|(|x_{j+1}|+ \cdots + |x_{k-1}|)$, for all homogeneous $i_1, \dotsc, i_{q-1} \in {\mathfrak{i}}$, $x_0, \dotsc, x_{p+1} \in \mathfrak{s}/\mathfrak{i}$, and their corresponding representatives ${\widetilde x}_0, \dotsc, {\widetilde x}_{p+1} \in \mathfrak{s}$. For each $p,q \ge 0$, denote by \[ d_2^{p,q} : \h^p(\mathfrak{s}/\mathfrak{i}, \h^q(\mathfrak{i}, M)) \to \h^{p+2} (\mathfrak{s}/\mathfrak{i}, \h^{q-1}(\mathfrak{i}, M)) \] the map induced by $D_2^{p,q}$ via the isomorphism \eqref{eq:E2.iso}.
\begin{definition} \label{defn:transgression} The transgression map of the Lyndon-Hochschild-Serre spectral sequence \eqref{eq:LHSss} is defined to be \[ d_2^{0,1} \colon \h^0 (\mathfrak{s}/\mathfrak{i}, \h^1(\mathfrak{i}, M)) \to \h^2(\mathfrak{s}/\mathfrak{i}, \h^0(\mathfrak{i}, M)). \] \end{definition}
In the next result, we describe the kernel of this transgression map in two particular cases that are important to the proof of Theorem~\ref{thm:main}.
\begin{proposition} \label{prop:ker.transgression} Let $\mathfrak{s}$ be a Lie superalgebra and $\mathfrak{i} \subsetneq \mathfrak{s}$ be an ideal. \begin{enumerate}[(a)] \item \label{item:ker.transgression.ev.mod} If $\mathfrak{s}/{\mathfrak{i}}$ is a Lie subalgebra of $\mathfrak{s}$, that is, $[{\widetilde x}_j, {\widetilde x}_k] \in \mathfrak{s} \setminus {\mathfrak{i}}$ for all ${\widetilde x}_j, {\widetilde x}_k \in \mathfrak{s} \setminus {\mathfrak{i}}$, then $d_2^{0,1} = 0$. In particular, if $M$ is a finite-dimensional $\text{$\mathfrak{g}$}$-module, $\mathfrak{s} = \text{$(\mathfrak{g} \otimes A)^\Gamma$}$ and $\mathfrak{i} = \text{$\mathfrak{g}$} \otimes I$, $I = \prod_{\gamma \in \Gamma} \gamma{\text{$\mathsf{m}$}}$, ${\text{$\mathsf{m}$}} \in \maxspec(A)$, then the kernel of the transgression map \[ d_2^{0,1} : \hom_{(\lie g \otimes A/I)^\Gamma} \left( (\lie g \otimes I / I^2)^\Gamma, \Gev{{\text{$\mathsf{m}$}}} M \right) \to \h^2 \left( (\text{$\mathfrak{g} \otimes A$}/I)^\Gamma, \Gev{{\text{$\mathsf{m}$}}} M \right) \] is isomorphic to $\hom_\text{$\mathfrak{g}$} \left( \text{$\mathfrak{g}$}, M \right)^{\oplus \dim_{A/{\text{$\mathsf{m}$}}} {\text{$\mathsf{m}$}} / {\text{$\mathsf{m}$}}^2}$.
\item \label{item:ker.transgression.gen.ev.mod} If $[\mathfrak{s}/{\mathfrak{i}}, \mathfrak{s}/{\mathfrak{i}}] = {\mathfrak{i}}$, then $d_2^{0,1} f ({\widetilde x}_0 \wedge {\widetilde x}_1) = 0$ for all ${\widetilde x}_0, {\widetilde x}_1$, if and only if $f = 0$. In particular, if $\mathfrak{s} = \text{$(\mathfrak{g} \otimes A)^\Gamma$}$ and $\mathfrak{i} = \text{$\mathfrak{g}$} \otimes I$, $I = \prod_{\gamma \in \Gamma} (\gamma{\text{$\mathsf{m}$}})^n$, ${\text{$\mathsf{m}$}} \in \maxspec(A)$, $n > 1$, then the kernel of the transgression map $d_2^{0,1} : \hom_{(\lie g \otimes A/I)^\Gamma} \left( (\lie g \otimes I / I^2)^\Gamma, \Gev{{\text{$\mathsf{m}$}}^n} M \right) \to \h^2 \left( (\text{$\mathfrak{g} \otimes A$}/I)^\Gamma, \Gev{{\text{$\mathsf{m}$}}^n} M \right)$ is $0$. \end{enumerate} \end{proposition}
\begin{proof}
Part~\eqref{item:ker.transgression.ev.mod} follows from \eqref{d2.lhs}, the fact that $(\text{$\mathfrak{g} \otimes A$}/I)^\Gamma \cong \text{$\mathfrak{g}$}$, and the fact that $(\text{$\mathfrak{g}$} \otimes I/I^2)^\Gamma \cong \text{$\mathfrak{g}$} \otimes {\text{$\mathsf{m}$}} / {\text{$\mathsf{m}$}}^2 \cong \text{$\mathfrak{g}$}^{\oplus \dim_{A/{\text{$\mathsf{m}$}}} {\text{$\mathsf{m}$}} / {\text{$\mathsf{m}$}}^2}$. To prove part~\eqref{item:ker.transgression.gen.ev.mod}, first notice that, by \eqref{d2.lhs}, we have that $\partial^1 f ({\widetilde x}_0 \wedge {\widetilde x}_1) = (-1)^{1+|x_1| |x_0|} f([{\widetilde x}_0, {\widetilde x}_1])$, for all representatives ${\widetilde x}_0, {\widetilde x}_1 \in \mathfrak{s}$ of corresponding elements $x_0, x_1 \in \mathfrak{s}/{\mathfrak{i}}$. Since $[\mathfrak{s}/{\mathfrak{i}}, \mathfrak{s}/{\mathfrak{i}}] = {\mathfrak{i}}$ by hypothesis, it follows that $\partial^1 f ({\widetilde x}_0 \wedge {\widetilde x}_1) = 0$ for all ${\widetilde x}_0, {\widetilde x}_1$ if and only if $f(\mathfrak{i}) = 0$. \end{proof}
\section{Extensions} \label{Exts}
Throughout this section, we will assume that $\lie g$ is a finite-dimensional simple Lie superalgebra, that $A$ is an associative, commutative, finitely-generated algebra with unit, that $\Gamma$ is a finite abelian group acting on $\text{$\mathfrak{g}$}$ and $A$ by automorphisms, and that the action of $\Gamma$ on $\maxspec (A)$ is free.
We begin by using Lemma \ref{lem:isos} to reduce the problem of describing $p$-extensions between finite-dimensional irreducible $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-modules to that of describing the $p$-th cohomology of $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$ with coefficients in indecomposable modules.
\begin{proposition} \label{prop:ext=h} Let $\pi, \pi' \in \cal P$. There exist $n, \ell, q_1, \dotsc, q_\ell \in \bb Z_{>0}$, maximal ideals ${\text{$\mathsf{m}$}}_1, \dotsc, {\text{$\mathsf{m}$}}_\ell \subseteq A$ in distinct $\Gamma$-orbits, and, for each $i \in \{ 1 , \dotsc, \ell \}$, $j \in \{ 1, \dotsc, q_i \}$, a finite-dimensional indecomposable $\text{$\mathfrak{g} \otimes A$} / {\text{$\mathsf{m}$}}_i^n$-module $M_{i, j}$, such that \begin{gather*} \hom_\text{$\Bbbk$} ( \cal V(\pi), \cal V(\pi') )^{\oplus 2^{\kappa(\pi)+\kappa(\pi')}} \cong \bigoplus_{\atop{1 \le j_i \le q_i}{1 \le i \le \ell}} \Gev{{\text{$\mathsf{m}$}}_1^n} M_{1, j_1} \otimes \dotsb \otimes \Gev{{\text{$\mathsf{m}$}}_\ell^n} M_{\ell, j_\ell} \quad \textup{and} \\ \Ext^p_\text{$(\mathfrak{g} \otimes A)^\Gamma$} ( \cal V(\pi), \cal V(\pi') )^{\oplus 2^{\kappa(\pi)+\kappa(\pi')}} \cong \bigoplus_{\atop{1 \le j_i \le q_i}{1 \le i \le \ell}} \h^p \left( \text{$(\mathfrak{g} \otimes A)^\Gamma$}, \, \Gev{{\text{$\mathsf{m}$}}_1^n} M_{1, j_1} \otimes \dotsb \otimes \Gev{{\text{$\mathsf{m}$}}_\ell^n} M_{\ell, j_\ell} \right), \quad p>0. \end{gather*} \end{proposition}
\begin{proof} From Remark~\ref{rmk:gather.gen.ev}, there exist $n, \ell \in \bb Z_{>0}$, maximal ideals ${\text{$\mathsf{m}$}}_1, \dotsc, {\text{$\mathsf{m}$}}_\ell \subseteq A$ in distinct $\Gamma$-orbits, and finite-dimensional irreducible $\text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}_i^n$-modules $V_i$ and $V'_i$ such that \[ \cal V (\pi) \cong \widehat{\bigotimes}_{i=1}^\ell \Gev{{\text{$\mathsf{m}$}}_i^n} V_i \quad \textup{and} \quad \cal V (\pi') \cong \widehat{\bigotimes}_{i=1}^\ell \Gev{{\text{$\mathsf{m}$}}_i^n} V'_i. \] Thus, there are isomorphisms of $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-modules \begin{align*} \hom_\text{$\Bbbk$} \left( \cal V (\pi), \cal V (\pi') \right)^{\oplus 2^{\kappa(\pi)+\kappa(\pi')}} &\cong \hom_\text{$\Bbbk$} \left( \cal V (\pi)^{\oplus 2^{\kappa(\pi)}}, \cal V (\pi')^{\oplus 2^{\kappa(\pi')}} \right) \\ &\cong \left( \bigotimes_{i=1}^\ell \Gev{{\text{$\mathsf{m}$}}_i^n} V_i^* \right) \otimes \left( \bigotimes_{i=1}^\ell \Gev{{\text{$\mathsf{m}$}}_i^n} V'_i \right) \\ &\cong \bigotimes_{i=1}^\ell \Gev{{\text{$\mathsf{m}$}}_i^n} \left( V_i^* \otimes V'_i \right). \end{align*} For each $i \in \{1, \dotsc, \ell\}$, since $V^*_i$ and $V'_i$ are finite dimensional, there exist $q_i > 0$ and indecomposable $\lie g\otimes A/{\text{$\mathsf{m}$}}_i^n$-modules, $M_{i,1}, \dotsc, M_{i, q_i}$, such that $V^*_i \otimes V'_i \cong \bigoplus_{j=1}^{q_i} M_{i,j}$. Thus there exist isomorphisms of $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-modules \[ \bigotimes_{i=1}^\ell \Gev{{\text{$\mathsf{m}$}}_i^n} \left( V_i^* \otimes V'_i \right) \cong \bigotimes_{i = 1}^\ell \left( \bigoplus_{j_i=1}^{q_i} \Gev{{\text{$\mathsf{m}$}}_i^n} \left( M_{i, j_i} \right) \right) \cong \bigoplus_{\atop{1 \le j_i \le q_i}{1 \le i \le \ell}} \Gev{{\text{$\mathsf{m}$}}_1^n} M_{1, j_1} \otimes \dotsb \otimes \Gev{{\text{$\mathsf{m}$}}_\ell^n} M_{\ell, j_\ell}. \] This proves the first statement. The second statement follows from the first one and Lemma~\ref{lem:isos}. \end{proof}
This next result is a particular case of Proposition~\ref{prop:ext=h} that will be used in to prove Corollary~\ref{cor:Ext1=0}.
\begin{corollary} \label{cor:ext=h.disjoint.supp} Let $V$ and $V'$ be finite-dimensional irreducible \text{$(\mathfrak{g} \otimes A)^\Gamma$}-modules. If the supports of $V$ and $V'$ are disjoint, then $V^* \htimes V'$ is irreducible and, for all $p>0$, \[ \Ext_\text{$(\mathfrak{g} \otimes A)^\Gamma$}^p (V, V') \cong \begin{cases} \h^p (\text{$(\mathfrak{g} \otimes A)^\Gamma$}, V^* \htimes V'), & \textup{ if } V^*\otimes V'\textup{ is irreducible},\\ \h^p (\text{$(\mathfrak{g} \otimes A)^\Gamma$}, V^* \htimes V')^{\oplus 2}, & \textup{ otherwise}. \end{cases} \] \end{corollary}
Our next goal is to reduce the problem of determining 1-extensions between finite-dimensional $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-modules to that of determining homomorphisms and extensions between finite-dimensional $\text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}^n$-modules, where ${\text{$\mathsf{m}$}}$ is a maximal ideal of $A$ and $n$ is a positive integer. We start with a general result regarding first cohomology. Recall the definition of transgression map (Definition~\ref{defn:transgression}).
\begin{lemma} \label{lem:h1(ga,M)} If $M$ is a finite-dimensional $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-module, then there exists a finite-codi\-men\-sion\-al $\Gamma$-invariant ideal $I \subseteq A$, such that \[ \h^1 \left( \text{$(\mathfrak{g} \otimes A)^\Gamma$}, M \right) \cong \h^1 \left( (\lie g \otimes A / I)^\Gamma, M \right) \oplus K, \] where $K$ is the kernel of the transgression map \[ t_M\colon \hom_{(\lie g \otimes A/I)^\Gamma} \left( (\lie g \otimes I / I^2)^\Gamma, M \right) \to \h^2 \left( (\text{$\mathfrak{g} \otimes A$}/I)^\Gamma, M \right). \] \end{lemma}
\begin{proof} Since $M$ is a finite-dimensional $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-module, by Proposition~\ref{prop:ann.fin.dim}, there exists a $\Gamma$-invariant finite-codimensional ideal $I \subseteq A$ such that $(\text{$\mathfrak{g}$} \otimes I)^\Gamma M = 0$. By Proposition~\ref{prop:lhsss}, there exists a first-quadrant cohomology spectral sequence associated to $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$ and $(\lie g \otimes I)^\Gamma$, namely \begin{equation} \label{eq:LHSss1} E_2^{p,q} \cong \h^p \left( (\lie g \otimes A/I)^\Gamma, \h^q \left( (\lie g \otimes I)^\Gamma, M \right) \right) \Rightarrow \h^{p+q} \left( \text{$(\mathfrak{g} \otimes A)^\Gamma$}, M \right). \end{equation} Since \eqref{eq:LHSss1} is a first-quadrant cohomology spectral sequence, we have an isomorphism of vector spaces $\h^1 (\text{$(\mathfrak{g} \otimes A)^\Gamma$}, M) \cong E_\infty^{1,0} \oplus E_\infty^{0,1}$. Moreover, \[ E_\infty^{1,0} = E_2^{1,0} \cong \h^1 \left( (\lie g \otimes A/I)^\Gamma, M \right) \quad \textup{ and } \quad E_\infty^{0,1} = E_3^{0,1} \cong \ker (d_2^{0,1} \colon E_2^{0,1} \to E_2^{2,0}), \] where $d_2^{0,1}$ is the transgression map $t_M$.
In order to finish the proof, we only need to describe $E_2^{0,1}$ and $E_2^{2,0}$. By \eqref{eq:LHSss1}, \[ E_2^{0,1} \cong \h^0 \left( (\lie g \otimes A/I)^\Gamma, \h^1 \left( (\lie g \otimes I)^\Gamma, M \right) \right) \quad \textup{and} \quad E_2^{2,0} \cong \h^2 ((\lie g \otimes A/I)^\Gamma, M). \] Since $(\lie g \otimes I)^\Gamma$ acts trivially on $M$, by Lemma~\ref{lem:triv.mods}, there is an isomorphism of $(\lie g \otimes A/I)^\Gamma$-modules $\h^\bullet \left( (\lie g \otimes I)^\Gamma, M \right) \cong \h^\bullet \left( (\lie g \otimes I)^\Gamma, \text{$\Bbbk$} \right) \otimes M$. Moreover, by Lemma~\ref{lem:com.even}, $\h^1 \left( (\lie g \otimes I)^\Gamma, \text{$\Bbbk$} \right)$ is isomorphic to $((\lie g \otimes I/I^2)^\Gamma)^*$ as a $(\lie g \otimes A/I)^\Gamma$-module. Thus $E_2^{0,1} \cong \hom_{(\lie g \otimes A/I)^\Gamma} \left( (\lie g \otimes I/I^2)^\Gamma, M \right)$. \end{proof}
As a consequence of Lemmas~\ref{lem:isos} and \ref{lem:h1(ga,M)}, we obtain the following result.
\begin{corollary} \label{cor:extp.fd} If $V, V'$ are finite-dimensional irreducible $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-modules, then $\Ext^1_\text{$(\mathfrak{g} \otimes A)^\Gamma$} (V, V')$ is finite dimensional. \end{corollary}
\begin{proof} Since $V$ and $V'$ are finite-dimensional modules, by Lemma~\ref{lem:isos}, $\Ext^1_{\text{$(\mathfrak{g} \otimes A)^\Gamma$}} (V, V')$ is isomorphic to $\h^1 \left( \text{$(\mathfrak{g} \otimes A)^\Gamma$}, V^* \otimes V' \right)$. By Lemma~\ref{lem:h1(ga,M)}, $\h^1 \left( \text{$(\mathfrak{g} \otimes A)^\Gamma$}, V^* \otimes V' \right)$ is isomorphic to a subspace of \begin{equation} \label{eq:h1.sub} \h^1 \left( (\lie g \otimes A / I)^\Gamma, V^* \otimes V' \right) \oplus \hom_{(\lie g \otimes A/I)^\Gamma} \left( (\lie g \otimes I/I^2)^\Gamma, V^* \otimes V' \right), \end{equation} where $I$ is a finite-codimensional $\Gamma$-invariant ideal of $A$. Since $\text{$\mathfrak{g}$}$, $A/I$, $I/I^2$, and $V^* \otimes V'$ are finite dimensional, both terms in \eqref{eq:h1.sub} are finite dimensional. This proves that $\h^1 \left( \text{$(\mathfrak{g} \otimes A)^\Gamma$}, V^* \otimes V' \right)$ is finite dimensional, and finishes the proof. \end{proof}
We emphasize the relevance of Corollary~\ref{cor:extp.fd} by contrasting it with a case where $\Ext^1$ is not finite dimensional. Namely, let $\Gamma$ be a group acting by automorphisms on an abelian Lie superalgebra $\lie a$ and on an associative commutative algebra with unit $B$. For any finite-dimensional trivial $(\lie a \otimes B)^\Gamma$-modules $M$ and $M'$, \[ \Ext^1_{(\lie a \otimes B)^\Gamma} (M, M') \cong \hom_\text{$\Bbbk$} \left( (\lie a \otimes B)^\Gamma, M^* \otimes M' \right) \] is finite dimensional if and only if $(\lie a \otimes B)^\Gamma$ is finite dimensional. In particular, when $\Gamma$ is trivial and $B$ is infinite dimensional, $\Ext^1_{(\lie a \otimes B)^\Gamma} (M, M')$ is infinite dimensional.
Recall from Corollary~\ref{cor:h1(ga,C)=0} that $\h^1 (\text{$(\mathfrak{g} \otimes A)^\Gamma$}, \text{$\Bbbk$}) = 0$. The next result gives a vanishing condition for $\h^1 (\text{$(\mathfrak{g} \otimes A)^\Gamma$}, M)$ when $M$ is a finite-dimensional $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-module of the form $\bigotimes_{i=1}^\ell \Gev{{\text{$\mathsf{m}$}}_i^{n_i}} M_i$, generalizes \cite[Theorem 3.6]{kodera10} and \cite[Theorem 3.7]{NS15}.
\begin{proposition} \label{prop:h1(ga,M)} Let $\ell, n_1, \dotsc, n_\ell \in \bb Z_{>0}$, ${\text{$\mathsf{m}$}}_1, \dotsc, {\text{$\mathsf{m}$}}_\ell \subseteq A$ be maximal ideals in distinct $\Gamma$-orbits, for each $i \in \{ 1, \dotsc, \ell \}$, let $M_i$ be a finite-dimensional $\lie g \otimes A/{\text{$\mathsf{m}$}}_i^{n_i}$-module, and $M = \bigotimes_{i=1}^\ell \Gev{{\text{$\mathsf{m}$}}_i^{n_i}} M_i$.
\begin{enumerate}[(a)] \item \label{prop:h1(ga,M)a} If $\hom_{\lie g \otimes A/{\text{$\mathsf{m}$}}_i^{n_i}} (\text{$\Bbbk$}, M_i) = 0$ for more than one index $i$, then $\h^1 \left( \text{$(\mathfrak{g} \otimes A)^\Gamma$}, M \right) = 0$.
\item \label{prop:h1(ga,M)b} If $\hom_{\lie g \otimes A/{\text{$\mathsf{m}$}}_i^{n_i}} (\text{$\Bbbk$}, M_i) = 0$ for exactly one index $i$, then \[ \h^1 \left( \text{$(\mathfrak{g} \otimes A)^\Gamma$}, M \right) \cong \h^1 \left( \text{$(\mathfrak{g} \otimes A)^\Gamma$}, \Gev{{\text{$\mathsf{m}$}}_i^{n_i}} M_i \right) \otimes \bigotimes_{j \ne i} \hom_{\lie g \otimes A/{\text{$\mathsf{m}$}}_j^{n_j}} (\text{$\Bbbk$}, M_j). \]
\item \label{prop:h1(ga,M)c} If $\hom_{\lie g \otimes A/{\text{$\mathsf{m}$}}_i^{n_i}} (\text{$\Bbbk$}, M_i) \ne 0$ for all indices $i$, then \[ \h^1 \left( \text{$(\mathfrak{g} \otimes A)^\Gamma$}, M \right) \cong \bigoplus_{i=1}^\ell \left( \h^1 \left( \text{$(\mathfrak{g} \otimes A)^\Gamma$}, \Gev{{\text{$\mathsf{m}$}}_i^{n_i}} M_i \right) \otimes \bigotimes_{j \ne i} \hom_{\lie g \otimes A/{\text{$\mathsf{m}$}}_j^{n_j}} (\text{$\Bbbk$}, M_j) \right). \] \end{enumerate} \end{proposition}
\begin{proof} By Proposition \ref{prop:kunneth}, we have \[ \h^1 \left( \text{$(\mathfrak{g} \otimes A)^\Gamma$}, M \right) \cong \bigoplus_{i=1}^\ell \left( \h^1 \left( \text{$(\mathfrak{g} \otimes A)^\Gamma$}, \Gev{{\text{$\mathsf{m}$}}_i^{n_i}} M_i \right) \otimes \bigotimes_{j\neq i} \hom_\text{$(\mathfrak{g} \otimes A)^\Gamma$} \left( \text{$\Bbbk$}, \Gev{{\text{$\mathsf{m}$}}_j^{n_j}} M_j \right) \right). \] Now, notice that $\hom_\text{$(\mathfrak{g} \otimes A)^\Gamma$} \left( \text{$\Bbbk$}, \Gev{{\text{$\mathsf{m}$}}_k^{n_k}} M_k \right) \cong \hom_{\text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}_k^{n_k}} \left( \text{$\Bbbk$}, M_k \right)$ for all $k \in \{1, \dotsc, \ell\}$. This proves part~\eqref{prop:h1(ga,M)c}. If $\hom_{\lie g \otimes A/{\text{$\mathsf{m}$}}_k^{n_k}} (\text{$\Bbbk$}, M_k) = 0$ for more than one index $k$, then for each $i$, there exists $j \neq i$ such that $\hom_{\lie g \otimes A/{\text{$\mathsf{m}$}}_j^{n_j}} (\text{$\Bbbk$}, M_j) = 0$. This proves part~\eqref{prop:h1(ga,M)a}. If $\hom_{\lie g \otimes A/{\text{$\mathsf{m}$}}_i^{n_i}} (\text{$\Bbbk$}, M_i) = 0$ for exactly one index $i$, then $\bigotimes_{j \neq k} \hom_{\lie g \otimes A/{\text{$\mathsf{m}$}}_j^{n_j}} (\text{$\Bbbk$}, M_j) = 0$ for all $k \ne i$. This proves part~\eqref{prop:h1(ga,M)b}. \end{proof}
The next result generalizes \cite[Lemma 3.3]{kodera10} and \cite[Proposition~3.6]{NS15}.
\begin{corollary} \label{cor:Ext1=0} Let $V$ and $V'$ be nontrivial, finite-dimensional, irreducible $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-modules. If the supports of $V$ and $V'$ are disjoint, then $\Ext^1_{\text{$(\mathfrak{g} \otimes A)^\Gamma$}} (V, V') = 0$. \end{corollary} \begin{proof} This proof follows directly from Corollary~\ref{cor:ext=h.disjoint.supp}, the classification of finite-dimensional irreducible $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-modules given in Section~\ref{sec:periplectic} and Proposition~\ref{prop:h1(ga,M)}\eqref{prop:h1(ga,M)a}. \end{proof}
Recall that we are assuming that $\lie g$ is a finite-dimensional simple Lie superalgebra, $A$ is an associative, commutative, finitely-generated algebra with unit, and $\Gamma$ is a finite abelian group acting on $\text{$\mathfrak{g}$}$ and $A$ by automorphisms, such that the induced action of $\Gamma$ on $\maxspec (A)$ is free. Now we state, and prove, the main result of this section. It describes 1-extensions between finite-dimensional irreducible $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-modules in terms of homomorphisms and extensions between finite-dimensional $\text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}^n$-modules.
\begin{theorem} \label{thm:main} Let $\pi, \pi' \in \cal P$, $\ell, n_1, \dotsc, n_\ell \in \bb Z_{>0}$, ${\text{$\mathsf{m}$}}_1, \dotsc, {\text{$\mathsf{m}$}}_\ell \subseteq A$ be maximal ideals in distinct $\Gamma$-orbits, and $V_i, V'_i$ be finite-dimensional irreducible $\lie g \otimes A/{\text{$\mathsf{m}$}}_i^{n_i}$-modules such that $\cal V (\pi) = \widehat{\bigotimes}_{i=1}^\ell \Gev{{\text{$\mathsf{m}$}}_i^{n_i}} V_i$ and $\cal V (\pi') = \widehat{\bigotimes}_{i=1}^\ell \Gev{{\text{$\mathsf{m}$}}_i^{n_i}} V_i'$. For each $i \in \{1, \dotsc, \ell\}$, let $d_i$ denote $\delta_{1, n_i} \dim_{A/{\text{$\mathsf{m}$}}_i} {\text{$\mathsf{m}$}}_i/{\text{$\mathsf{m}$}}_i^2$.
\begin{enumerate}[$(a)$] \item \label{thm:main.a} If $V_i$ is not isomorphic to $V_i'$ for two or more indices $i$, then $\Ext_{\text{$(\mathfrak{g} \otimes A)^\Gamma$}}^1(\cal V (\pi), \cal V (\pi')) = 0$.
\item \label{thm:main.b} If $V_i$ is isomorphic to $V_i'$ for all but one index $i$, then \[ \Ext_{\text{$(\mathfrak{g} \otimes A)^\Gamma$}}^1 (\cal V (\pi), \cal V (\pi'))^{\oplus 2^{\kappa(\pi) + \kappa(\pi')}} \cong \Ext^1_{\lie g \otimes A/{\text{$\mathsf{m}$}}_i^{n_i}} (V_i, V'_i) \oplus \hom_\text{$\mathfrak{g}$} \left( \text{$\mathfrak{g}$} \otimes V_i, V_i' \right)^{\oplus d_i}. \]
\item \label{thm:main.c} If $V_i$ is isomorphic to $V_i'$ for all $i \in \{1, \dotsc, \ell\}$, then \[ \Ext_{\text{$(\mathfrak{g} \otimes A)^\Gamma$}}^1 (\cal V (\pi), \cal V (\pi'))^{\oplus 2^{\kappa(\pi) + \kappa(\pi')}} \cong \bigoplus_{i=1}^\ell \left( \Ext^1_{\lie g \otimes A/{\text{$\mathsf{m}$}}_i^{n_i}} (V_i, V'_i) \oplus \hom_\text{$\mathfrak{g}$} \left( \text{$\mathfrak{g}$} \otimes V_i, V_i' \right)^{\oplus d_i} \right). \] \end{enumerate} \end{theorem}
\begin{proof} Denote $\left( \bigotimes_{i=1}^\ell \Gev{{\text{$\mathsf{m}$}}_i^{n_i}} V_i \right)$ by $V$, $\left( \bigotimes_{i=1}^\ell \Gev{{\text{$\mathsf{m}$}}_i^{n_i}} V'_i \right)$ by $V'$, and recall from Section \ref{ss:map.superalgebras} that $V \cong \cal V (\pi)^{\oplus 2^{\kappa(\pi)}}$ and $V' \cong \cal V(\pi')^{\oplus 2^{\kappa(\pi')}}$. Thus, by Lemma~\ref{lem:isos}, we have \[ \Ext^1_\text{$(\mathfrak{g} \otimes A)^\Gamma$} (\cal V (\pi), \cal V (\pi'))^{\oplus 2^{\kappa(\pi) + \kappa(\pi')}} \cong \h^1 \left( \text{$(\mathfrak{g} \otimes A)^\Gamma$}, V^* \otimes V' \right). \] Now, notice that $V^* \otimes V' \cong \bigotimes_{i=1}^\ell \Gev{{\text{$\mathsf{m}$}}_i^{n_i}} (V_i^* \otimes V'_i)$, that $V_i^* \otimes V_i'$ are finite dimensional, and that \[ \hom_{\text{$\mathfrak{g} \otimes A$} / {\text{$\mathsf{m}$}}_i^{n_i}} (\text{$\Bbbk$}, V_i^* \otimes V'_i) \cong \hom_{\text{$\mathfrak{g} \otimes A$} / {\text{$\mathsf{m}$}}_i^{n_i}} (V_i, V'_i) \cong \begin{cases} 0, & \textup{if $V_i \not\cong V'_i$,} \\ \text{$\Bbbk$}, & \textup{if $V_i \cong V'_i$.} \end{cases} \] \begin{enumerate}[(a)]\itemsep1ex \item To prove part~\eqref{thm:main.a}, notice that, if $V_i \not\cong V_i'$ for two or more indices $i$, then by Proposition~\ref{prop:h1(ga,M)}\eqref{prop:h1(ga,M)a}, \[ \Ext^1_\text{$(\mathfrak{g} \otimes A)^\Gamma$} (\cal V (\pi), \cal V (\pi')) = 0. \]
\item If $V_i \not\cong V_i'$ for exactly one index $i$, then by Proposition~\ref{prop:h1(ga,M)}\eqref{prop:h1(ga,M)b}, we have \[ \Ext^1_\text{$(\mathfrak{g} \otimes A)^\Gamma$} (\cal V (\pi), \cal V (\pi'))^{\oplus 2^{\kappa(\pi) + \kappa(\pi')}} \cong \h^1 \left( \text{$(\mathfrak{g} \otimes A)^\Gamma$}, \Gev{{\text{$\mathsf{m}$}}_i^{n_i}} (V_i^* \otimes V_i') \right). \] Now, let $I = \prod_{\gamma \in \Gamma} (\gamma {\text{$\mathsf{m}$}}_i)^{n_i}$. By Lemma~\ref{lem:h1(ga,M)}, we have \[ \Ext^1_\text{$(\mathfrak{g} \otimes A)^\Gamma$} (\cal V (\pi), \cal V (\pi'))^{\oplus 2^{\kappa(\pi) + \kappa(\pi')}} \cong \h^1 \left( \text{$\mathfrak{g} \otimes A$}/{\text{$\mathsf{m}$}}_i^{n_i}, (V_i^* \otimes V_i') \right) \oplus K_i, \] where $K_i$ is the kernel of the transgression map \[ t_{_{V_i^* \otimes V_i}} \colon \hom_{(\lie g \otimes A/I)^\Gamma} \left( (\lie g \otimes I / I^2)^\Gamma, \Gev{{\text{$\mathsf{m}$}}_i^{n_i}} (V_i^* \otimes V_i') \right) \to \h^2 \left( (\text{$\mathfrak{g} \otimes A$}/I)^\Gamma, \Gev{{\text{$\mathsf{m}$}}_i^{n_i}} (V_i^* \otimes V_i') \right). \] To finish the proof of part~\eqref{thm:main.b}, notice that, by Proposition~\ref{prop:ker.transgression}, $K_i \cong \hom_\text{$\mathfrak{g}$} \left( \text{$\mathfrak{g}$} \otimes V_i, V_i' \right)^{\oplus d_i}$.
\item If $V_i \cong V_i'$ for all $i \in \{1, \dots, \ell\}$, then by Proposition~\ref{prop:h1(ga,M)}\eqref{prop:h1(ga,M)c}, we have \[ \Ext^1_\text{$(\mathfrak{g} \otimes A)^\Gamma$} (\cal V (\pi), \cal V (\pi'))^{\oplus 2^{\kappa(\pi) + \kappa(\pi')}} \cong \bigoplus_{i=1}^\ell \h^1 \left( \text{$(\mathfrak{g} \otimes A)^\Gamma$}, \Gev{{\text{$\mathsf{m}$}}_i^{n_i}} (V_i^* \otimes V_i') \right). \] The rest of the proof of part~\eqref{thm:main.c} follows from Lemma~\ref{lem:h1(ga,M)} and Proposition~\ref{prop:ker.transgression} using, for each $i \in \{1, \dotsc, \ell\}$, the same arguments that we used to prove part~\eqref{thm:main.b}. \qedhere \end{enumerate} \end{proof}
Theorem~\ref{thm:main} generalizes \cite[Theorem~3.7]{NS13} to the super setting. The particular case where the irreducible module is an evaluation one is treated in the following example.
\begin{example}[Evaluation modules] \label{eg:blocks.ev} Let $\cal V(\pi) = \widehat{\bigotimes}_{i=1}^\ell \Gev{{\text{$\mathsf{m}$}}_i} V (\lambda_i)$ and $\cal V(\pi') = \widehat{\bigotimes}_{i=1}^\ell \Gev{{\text{$\mathsf{m}$}}_i} V(\mu_i)$ be irreducible finite-dimensional $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-modules. In this case, $d_i = \dim_{A/{\text{$\mathsf{m}$}}_i} {\text{$\mathsf{m}$}}_i / {\text{$\mathsf{m}$}}_i^2$ and Theorem~\ref{thm:main} yields: \begin{enumerate}[$(i)$] \item $\Ext_{\text{$(\mathfrak{g} \otimes A)^\Gamma$}}^1(\cal V (\pi), \cal V (\pi')) = 0$, when $\lambda_i \ne \mu_i$ for two or more indices $i$.
\item $\Ext_{\text{$(\mathfrak{g} \otimes A)^\Gamma$}}^1 (\cal V (\pi), \cal V (\pi'))^{\oplus 2^{\kappa(\pi) + \kappa(\pi')}} \cong \Ext^1_\text{$\mathfrak{g}$} (V(\lambda_i), V(\mu_i)) \oplus \hom_\text{$\mathfrak{g}$} \left( \text{$\mathfrak{g}$} \otimes V(\lambda_i), V(\mu_i) \right)^{\oplus d_i}$, when $\lambda_i = \mu_i$ for all but one index $i$.
\item $\Ext_{\text{$(\mathfrak{g} \otimes A)^\Gamma$}}^1 (\cal V (\pi), \cal V (\pi'))^{\oplus 2^{\kappa(\pi) + \kappa(\pi')}} \cong \bigoplus_{i=1}^\ell \Ext^1_\text{$\mathfrak{g}$} (V(\lambda_i), V(\mu_i)) \oplus \hom_\text{$\mathfrak{g}$} \left( \text{$\mathfrak{g}$} \otimes V(\lambda_i), V(\mu_i) \right)^{\oplus d_i}$, when $\lambda_i = \mu_i$ for all $i \in \{1, \dotsc, \ell\}$. \end{enumerate} Thus, $\cal V(\pi)$ and $\cal V(\pi')$ are in the same block if and only if, for each $i \in \{1, \dots, \ell\}$, either: $\lambda_i - \mu_i \in Q$, or $V(\lambda_i)$ and $V(\mu_i)$ are in the same block in the category of finite-dimensional $\text{$\mathfrak{g}$}$-modules. In particular, if $\text{$\mathfrak{g}$}$ is of type II, $\lie p(n)$, $\lie q(n)$, $S(n)$, $\tilde S(n)$ or $H(n)$, since all finite-dimensional irreducible $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-modules are evaluation modules, this gives a description of the block decomposition of the category of finite-dimensional $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-modules.
Moreover, if either $\lambda$ or $\mu$ are \emph{typical}, then $\Ext^1_\text{$\mathfrak{g}$} (V(\lambda), V(\mu)) = 0$ (see \cite[Theorem 1]{kac78}). Thus, if either $\lambda_i$ or $\mu_i$ is \emph{typical} for each $i \in \{ 1, \dotsc, \ell \}$, then $\Ext_{\text{$(\mathfrak{g} \otimes A)^\Gamma$}}^1 (\cal V (\pi), \cal V (\pi'))$ depends only on $\hom_\text{$\mathfrak{g}$} \left( \text{$\mathfrak{g}$} \otimes V(\lambda_i), V(\mu_i) \right)$. In particular, if $\text{$\mathfrak{g}$}$ is of type $B(0,n)$, since all the weights are typical, the blocks of the category of finite-dimensional $\text{$(\mathfrak{g} \otimes A)^\Gamma$}$-modules are parametrized by the so-called spectral character (compare with \cite[Proposition 4.5]{kodera10} and \cite[Theorem 5.19]{NS15}).
Now, let $\text{$\mathfrak{g}$}$ be a Lie superalgebra of type $A$ or $C$ (type I), $\text{$\Bbbk$}$ be the trivial $\text{$\mathfrak{g}$}$-module and $V(\lambda)$ be an irreducible finite-dimensional $\text{$\mathfrak{g}$}$-module of highest weight $\lambda\in \lie h^*$. Let $\rho_1$ be the half sum of the odd positive roots of $\text{$\mathfrak{g}$}$, $\alpha_{\max}$ (resp. $\alpha_{\min}$) be the maximal (resp. minimal) root of $\text{$\mathfrak{g}$}$, and $\mu^{(j)}$ be as in \cite[Theorem~1.1]{SZ07}. For any maximal ideal ${\text{$\mathsf{m}$}}\in\maxspec A$, we have
\[ \Ext_{\text{$(\mathfrak{g} \otimes A)^\Gamma$}}^1 (\Gev{{\text{$\mathsf{m}$}}} \text{$\Bbbk$}, \Gev{{\text{$\mathsf{m}$}}} V(\lambda)) \cong \Ext^1_{\lie g} (\text{$\Bbbk$}, V(\lambda)) \oplus \hom_\text{$\mathfrak{g}$} \left( \text{$\mathfrak{g}$} , V(\lambda) \right)^{\oplus d},
\] where $d=\dim_{A/{\text{$\mathsf{m}$}}} {\text{$\mathsf{m}$}}/{\text{$\mathsf{m}$}}^2$. Thus $\Ext_{\text{$(\mathfrak{g} \otimes A)^\Gamma$}}^1 (\Gev{{\text{$\mathsf{m}$}}} \text{$\Bbbk$}, \Gev{{\text{$\mathsf{m}$}}} V(\lambda))\neq 0$ if and only if $\Ext^1_{\lie g} (\text{$\Bbbk$}, V(\lambda)) \ne 0$ or $\hom_\text{$\mathfrak{g}$} \left( \text{$\mathfrak{g}$} , V(\lambda) \right) \ne 0$. From \cite[Theorem~1.1]{SZ07}, we obtain that: \begin{enumerate}[$\bullet$] \item in type $A$, $\Ext_{\text{$(\mathfrak{g} \otimes A)^\Gamma$}}^1 (\Gev{{\text{$\mathsf{m}$}}} \text{$\Bbbk$}, \Gev{{\text{$\mathsf{m}$}}} V(\lambda))\neq 0$ if and only if $\lambda \in \{-\alpha_{\min}, \alpha_{\max}, \mu^{(0)}, \dotsc, \mu^{(n-1)} \}$, \item in type $C$, $\Ext_{\text{$(\mathfrak{g} \otimes A)^\Gamma$}}^1 (\Gev{{\text{$\mathsf{m}$}}} \text{$\Bbbk$}, \Gev{{\text{$\mathsf{m}$}}} V(\lambda))\neq 0$ if and only if $\lambda \in \{-\alpha_{\min},\ 2\rho_1,\ \alpha_{\max}\}$. \end{enumerate} \end{example}
\begin{remark} In the published version of this paper, Example~5.9 was incorrect (see \cite[Remark~3.6]{CM}). \end{remark}
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29 results in 81Txx
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Published online by Cambridge University Press: 10 June 2022, pp. 990-1004
Print publication: December 2022
We relate the quantum cohomology of minuscule flag manifolds to the tt*-Toda equations, a special case of the topological–antitopological fusion equations which were introduced by Cecotti and Vafa in their study of supersymmetric quantum field theories. To do this, we combine the Lie-theoretic treatment of the tt*-Toda equations of Guest–Ho with the Lie-theoretic description of the quantum cohomology of minuscule flag manifolds from Chaput–Manivel–Perrin and Golyshev–Manivel.
TAMARKIN–TSYGAN CALCULUS AND CHIRAL POISSON COHOMOLOGY
Lie algebras and Lie superalgebras
Algebraic geometry: Foundations
EMILE BOUAZIZ
Published online by Cambridge University Press: 28 February 2022, pp. 751-765
We construct and study some vertex theoretic invariants associated with Poisson varieties, specializing in the conformal weight $0$ case to the familiar package of Poisson homology and cohomology. In order to do this conceptually, we sketch a version of the calculus, in the sense of [12], adapted to the context of vertex algebras. We obtain the standard theorems of Poisson (co)homology in this chiral context. This is part of a larger project related to promoting noncommutative geometric structures to chiral versions of such.
Combinatorics of the geometry of Wilson loop diagrams II: Grassmann necklaces, dimensions, and denominators
Susama Agarwala, Siân Fryer, Karen Yeats
Journal: Canadian Journal of Mathematics / Volume 74 / Issue 6 / December 2022
Published online by Cambridge University Press: 23 July 2021, pp. 1625-1672
Wilson loop diagrams are an important tool in studying scattering amplitudes of SYM $N=4$ theory and are known by previous work to be associated to positroids. In this paper, we study the structure of the associated positroids, as well as the structure of the denominator of the integrand defined by each diagram. We give an algorithm to derive the Grassmann necklace of the associated positroid directly from the Wilson loop diagram, and a recursive proof that the dimension of these cells is thrice the number of propagators in the diagram. We also show that the ideal generated by the denominator in the integrand is the radical of the ideal generated by the product of Grassmann necklace minors.
Large N behaviour of the two-dimensional Yang–Mills partition function
Abstract harmonic analysis
Thibaut Lemoine
Journal: Combinatorics, Probability and Computing / Volume 31 / Issue 1 / January 2022
Published online by Cambridge University Press: 30 June 2021, pp. 144-165
We compute the large N limit of the partition function of the Euclidean Yang–Mills measure on orientable compact surfaces with genus $g\geqslant 1$ and non-orientable compact surfaces with genus $g\geqslant 2$ , with structure group the unitary group ${\mathrm U}(N)$ or special unitary group ${\mathrm{SU}}(N)$ . Our proofs are based on asymptotic representation theory: more specifically, we control the dimension and Casimir number of irreducible representations of ${\mathrm U}(N)$ and ${\mathrm{SU}}(N)$ when N tends to infinity. Our main technical tool, involving 'almost flat' Young diagram, makes rigorous the arguments used by Gross and Taylor (1993, Nuclear Phys. B400(1–3) 181–208) in the setting of QCD, and in some cases, we recover formulae given by Douglas (1995, Quantum Field Theory and String Theory (Cargèse, 1993), Vol. 328 of NATO Advanced Science Institutes Series B: Physics, Plenum, New York, pp. 119–135) and Rusakov (1993, Phys. Lett. B303(1) 95–98).
GRADIENT FLOWS OF HIGHER ORDER YANG–MILLS–HIGGS FUNCTIONALS
PAN ZHANG
Journal: Journal of the Australian Mathematical Society / Volume 113 / Issue 2 / October 2022
Published online by Cambridge University Press: 31 May 2021, pp. 257-287
Print publication: October 2022
In this paper, we define a family of functionals generalizing the Yang–Mills–Higgs functionals on a closed Riemannian manifold. Then we prove the short-time existence of the corresponding gradient flow by a gauge-fixing technique. The lack of a maximum principle for the higher order operator brings us a lot of inconvenience during the estimates for the Higgs field. We observe that the $L^2$ -bound of the Higgs field is enough for energy estimates in four dimensions and we show that, provided the order of derivatives appearing in the higher order Yang–Mills–Higgs functionals is strictly greater than one, solutions to the gradient flow do not hit any finite-time singularities. As for the Yang–Mills–Higgs k-functional with Higgs self-interaction, we show that, provided $\dim (M)<2(k+1)$ , for every smooth initial data the associated gradient flow admits long-time existence. The proof depends on local $L^2$ -derivative estimates, energy estimates and blow-up analysis.
Combinatorics of the geometry of Wilson loop diagrams I: equivalence classes via matroids and polytopes
Journal: Canadian Journal of Mathematics / Volume 74 / Issue 4 / August 2022
Published online by Cambridge University Press: 26 February 2021, pp. 1177-1208
Print publication: August 2022
Wilson loop diagrams are an important tool in studying scattering amplitudes of SYM $N=4$ theory and are known by previous work to be associated to positroids. We characterize the conditions under which two Wilson loop diagrams give the same positroid, prove that an important subclass of subdiagrams (exact subdiagrams) corresponds to uniform matroids, and enumerate the number of different Wilson loop diagrams that correspond to each positroid cell. We also give a correspondence between those positroids which can arise from Wilson loop diagrams and directions in associahedra.
Twisted Donaldson invariants
Infinite-dimensional manifolds
Higher algebraic $K$-theory
TSUYOSHI KATO, HIROFUMI SASAHIRA, HANG WANG
Journal: Mathematical Proceedings of the Cambridge Philosophical Society / Volume 171 / Issue 3 / November 2021
Print publication: November 2021
Fundamental group of a manifold gives a deep effect on its underlying smooth structure. In this paper we introduce a new variant of the Donaldson invariant in Yang–Mills gauge theory from twisting by the Picard group of a 4-manifold in the case when the fundamental group is free abelian. We then generalise it to the general case of fundamental groups by use of the framework of non commutative geometry. We also verify that our invariant distinguishes smooth structures between some homeomorphic 4-manifolds.
A CLASS OF NONHOLOMORPHIC MODULAR FORMS II: EQUIVARIANT ITERATED EISENSTEIN INTEGRALS
Zeta and $L$-functions: analytic theory
Discontinuous groups and automorphic forms
FRANCIS BROWN
Journal: Forum of Mathematics, Sigma / Volume 8 / 2020
Published online by Cambridge University Press: 28 May 2020, e31
We introduce a new family of real-analytic modular forms on the upper-half plane. They are arguably the simplest class of 'mixed' versions of modular forms of level one and are constructed out of real and imaginary parts of iterated integrals of holomorphic Eisenstein series. They form an algebra of functions satisfying many properties analogous to classical holomorphic modular forms. In particular, they admit expansions in $q,\overline{q}$ and $\log |q|$ involving only rational numbers and single-valued multiple zeta values. The first nontrivial functions in this class are real-analytic Eisenstein series.
Simple Formulas for Constellations and Bipartite Maps with Prescribed Degrees
Geometric probability and stochastic geometry
Baptiste Louf
Journal: Canadian Journal of Mathematics / Volume 73 / Issue 1 / February 2021
Published online by Cambridge University Press: 12 November 2019, pp. 160-176
Print publication: February 2021
We obtain simple quadratic recurrence formulas counting bipartite maps on surfaces with prescribed degrees (in particular, $2k$-angulations) and constellations. These formulas are the fastest known way of computing these numbers.
Our work is a natural extension of previous works on integrable hierarchies (2-Toda and KP), namely, the Pandharipande recursion for Hurwitz numbers (proved by Okounkov and simplified by Dubrovin–Yang–Zagier), as well as formulas for several models of maps (Goulden–Jackson, Carrell–Chapuy, Kazarian–Zograf). As for those formulas, a bijective interpretation is still to be found. We also include a formula for monotone simple Hurwitz numbers derived in the same fashion.
These formulas also play a key role in subsequent work of the author with T. Budzinski establishing the hyperbolic local limit of random bipartite maps of large genus.
ON BORWEIN'S CONJECTURES FOR PLANAR UNIFORM RANDOM WALKS
YAJUN ZHOU
Journal: Journal of the Australian Mathematical Society / Volume 107 / Issue 3 / December 2019
Published online by Cambridge University Press: 09 October 2019, pp. 392-411
Let $p_{n}(x)=\int _{0}^{\infty }J_{0}(xt)[J_{0}(t)]^{n}xt\,dt$ be Kluyver's probability density for $n$-step uniform random walks in the Euclidean plane. Through connection to a similar problem in two-dimensional quantum field theory, we evaluate the third-order derivative $p_{5}^{\prime \prime \prime }(0^{+})$ in closed form, thereby giving a new proof for a conjecture of J. M. Borwein. By further analogies to Feynman diagrams in quantum field theory, we demonstrate that $p_{n}(x),0\leq x\leq 1$ admits a uniformly convergent Maclaurin expansion for all odd integers $n\geq 5$, thus settling another conjecture of Borwein.
The Batalin–Vilkovisky Algebra in the String Topology of Classifying Spaces
Homotopy theory
Katsuhiko Kuribayashi, Luc Menichi
For almost any compact connected Lie group $G$ and any field $\mathbb{F}_{p}$, we compute the Batalin–Vilkovisky algebra $H^{\star +\text{dim}\,G}(\text{LBG};\mathbb{F}_{p})$ on the loop cohomology of the classifying space introduced by Chataur and the second author. In particular, if $p$ is odd or $p=0$, this Batalin–Vilkovisky algebra is isomorphic to the Hochschild cohomology $HH^{\star }(H_{\star }(G),H_{\star }(G))$. Over $\mathbb{F}_{2}$ , such an isomorphism of Batalin–Vilkovisky algebras does not hold when $G=\text{SO}(3)$ or $G=G_{2}$. Our elaborate considerations on the signs in string topology of the classifying spaces give rise to a general theorem on graded homological conformal field theory.
A NOTE ON GUNNINGHAM'S FORMULA
Projective and enumerative geometry
JUNHO LEE
Journal: Bulletin of the Australian Mathematical Society / Volume 98 / Issue 3 / December 2018
Gunningham ['Spin Hurwitz numbers and topological quantum field theory', Geom. Topol.20(4) (2016), 1859–1907] constructed an extended topological quantum field theory (TQFT) to obtain a closed formula for all spin Hurwitz numbers. In this note, we use a gluing theorem for spin Hurwitz numbers to re-prove Gunningham's formula. We also describe a TQFT formalism naturally induced by the gluing theorem.
CLE PERCOLATIONS
Equilibrium statistical mechanics
Markov processes
JASON MILLER, SCOTT SHEFFIELD, WENDELIN WERNER
Journal: Forum of Mathematics, Pi / Volume 5 / 2017
Published online by Cambridge University Press: 03 October 2017, e4
Conformal loop ensembles (CLEs) are random collections of loops in a simply connected domain, whose laws are characterized by a natural conformal invariance property. The set of points not surrounded by any loop is a canonical random connected fractal set — a random and conformally invariant analog of the Sierpinski carpet or gasket.
In the present paper, we derive a direct relationship between the CLEs with simple loops ( $\text{CLE}_{\unicode[STIX]{x1D705}}$ for $\unicode[STIX]{x1D705}\in (8/3,4)$, whose loops are Schramm's $\text{SLE}_{\unicode[STIX]{x1D705}}$-type curves) and the corresponding CLEs with nonsimple loops ( $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ with $\unicode[STIX]{x1D705}^{\prime }:=16/\unicode[STIX]{x1D705}\in (4,6)$, whose loops are $\text{SLE}_{\unicode[STIX]{x1D705}^{\prime }}$-type curves). This correspondence is the continuum analog of the Edwards–Sokal coupling between the $q$-state Potts model and the associated FK random cluster model, and its generalization to noninteger $q$.
Like its discrete analog, our continuum correspondence has two directions. First, we show that for each $\unicode[STIX]{x1D705}\in (8/3,4)$, one can construct a variant of $\text{CLE}_{\unicode[STIX]{x1D705}}$ as follows: start with an instance of $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$, then use a biased coin to independently color each $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ loop in one of two colors, and then consider the outer boundaries of the clusters of loops of a given color. Second, we show how to interpret $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ loops as interfaces of a continuum analog of critical Bernoulli percolation within $\text{CLE}_{\unicode[STIX]{x1D705}}$ carpets — this is the first construction of continuum percolation on a fractal planar domain. It extends and generalizes the continuum percolation on open domains defined by $\text{SLE}_{6}$ and $\text{CLE}_{6}$.
These constructions allow us to prove several conjectures made by the second author and provide new and perhaps surprising interpretations of the relationship between CLEs and the Gaussian free field. Along the way, we obtain new results about generalized $\text{SLE}_{\unicode[STIX]{x1D705}}(\unicode[STIX]{x1D70C})$ curves for $\unicode[STIX]{x1D70C}<-2$, such as their decomposition into collections of $\text{SLE}_{\unicode[STIX]{x1D705}}$-type 'loops' hanging off of $\text{SLE}_{\unicode[STIX]{x1D705}^{\prime }}$-type 'trunks', and vice versa (exchanging $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D705}^{\prime }$). We also define a continuous family of natural $\text{CLE}$ variants called boundary conformal loop ensembles (BCLEs) that share some (but not all) of the conformal symmetries that characterize $\text{CLE}$s, and that should be scaling limits of critical models with special boundary conditions. We extend the $\text{CLE}_{\unicode[STIX]{x1D705}}$/ $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ correspondence to a $\text{BCLE}_{\unicode[STIX]{x1D705}}$/ $\text{BCLE}_{\unicode[STIX]{x1D705}^{\prime }}$ correspondence that makes sense for the wider range $\unicode[STIX]{x1D705}\in (2,4]$ and $\unicode[STIX]{x1D705}^{\prime }\in [4,8)$.
PyCFTBoot: A Flexible Interface for the Conformal Bootstrap
Algorithms - Computer Science
Connor Behan
Journal: Communications in Computational Physics / Volume 22 / Issue 1 / July 2017
Published online by Cambridge University Press: 03 May 2017, pp. 1-38
Print publication: July 2017
We introduce PyCFTBoot, a wrapper designed to reduce the barrier to entry in conformal bootstrap calculations that require semidefinite programming. Symengine and SDPB are used for the most intensive symbolic and numerical steps respectively. After reviewing the built-in algorithms for conformal blocks, we explain how to use the code through a number of examples that verify past results. As an application, we show that the multi-correlator bootstrap still appears to single out the Wilson-Fisher fixed points as special theories in dimensions between 3 and 4 despite the recent proof that they violate unitarity.
DUBROVIN'S SUPERPOTENTIAL AS A GLOBAL SPECTRAL CURVE
Deformations of analytic structures
P. Dunin-Barkowski, P. Norbury, N. Orantin, A. Popolitov, S. Shadrin
Journal: Journal of the Institute of Mathematics of Jussieu / Volume 18 / Issue 3 / May 2019
Published online by Cambridge University Press: 17 April 2017, pp. 449-497
Print publication: May 2019
We apply the spectral curve topological recursion to Dubrovin's universal Landau–Ginzburg superpotential associated to a semi-simple point of any conformal Frobenius manifold. We show that under some conditions the expansion of the correlation differentials reproduces the cohomological field theory associated with the same point of the initial Frobenius manifold.
Lattice Boltzmann Simulation of Steady Flow in a Semi-Elliptical Cavity
Basic methods in fluid mechanics
Incompressible viscous fluids
Junjie Ren, Ping Guo
Journal: Communications in Computational Physics / Volume 21 / Issue 3 / March 2017
Print publication: March 2017
The lattice Boltzmann method is employed to simulate the steady flow in a two-dimensional lid-driven semi-elliptical cavity. Reynolds number (Re) and vertical-to-horizontal semi-axis ratio (D) are in the range of 500-5000 and 0.1-4, respectively. The effects of Re and D on the vortex structure and pressure field are investigated, and the evolutionary features of the vortex structure with Re and D are analyzed in detail. Simulation results show that the vortex structure and its evolutionary features significantly depend on Re and D. The steady flow is characterized by one vortex in the semi-elliptical cavity when both Re and D are small. As Re increases, the appearance of the vortex structure becomes more complex. When D is less than 1, increasing D makes the large vortexes more round, and the evolution of the vortexes with D becomes more complex with increasing Re. When D is greater than 1, the steady flow consists of a series of large vortexes which superimpose on each other. As Re and D increase, the number of the large vortexes increases. Additionally, a small vortex in the upper-left corner of the semi-elliptical cavity appears at a large Re and its size increases slowly as Re increases. The highest pressures appear in the upper-right corner and the pressure changes drastically in the upper-right region of the cavity. The total pressure differences in the semi-elliptical cavity with a fixed D decrease with increasing Re. In the region of themain vortex, the pressure contours nearly coincide with the streamlines, especially for the cavity flow with a large Re.
Homogeneity of cohomology classes associated with Koszul matrix factorizations
Local theory
Homological methods
Alexander Polishchuk
Journal: Compositio Mathematica / Volume 152 / Issue 10 / October 2016
In this work we prove the so-called dimension property for the cohomological field theory associated with a homogeneous polynomial $W$ with an isolated singularity, in the algebraic framework of [A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, J. Reine Angew. Math. 714 (2016), 1–122]. This amounts to showing that some cohomology classes on the Deligne–Mumford moduli spaces of stable curves, constructed using Fourier–Mukai-type functors associated with matrix factorizations, live in prescribed dimension. The proof is based on a homogeneity result established in [A. Polishchuk and A. Vaintrob, Algebraic construction of Witten's top Chern class, in Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) (American Mathematical Society, Providence, RI, 2001), 229–249] for certain characteristic classes of Koszul matrix factorizations of $0$. To reduce to this result, we use the theory of Fourier–Mukai-type functors involving matrix factorizations and the natural rational lattices in the relevant Hochschild homology spaces, as well as a version of Hodge–Riemann bilinear relations for Hochschild homology of matrix factorizations. Our approach also gives a proof of the dimension property for the cohomological field theories associated with some quasihomogeneous polynomials with an isolated singularity.
A Feynman integral via higher normal functions
$K$-theory in number theory
Surfaces and higher-dimensional varieties
Families, fibrations
Spencer Bloch, Matt Kerr, Pierre Vanhove
Journal: Compositio Mathematica / Volume 151 / Issue 12 / December 2015
Published online by Cambridge University Press: 06 August 2015, pp. 2329-2375
We study the Feynman integral for the three-banana graph defined as the scalar two-point self-energy at three-loop order. The Feynman integral is evaluated for all identical internal masses in two space-time dimensions. Two calculations are given for the Feynman integral: one based on an interpretation of the integral as an inhomogeneous solution of a classical Picard–Fuchs differential equation, and the other using arithmetic algebraic geometry, motivic cohomology, and Eisenstein series. Both methods use the rather special fact that the Feynman integral is a family of regulator periods associated to a family of $K3$ surfaces. We show that the integral is given by a sum of elliptic trilogarithms evaluated at sixth roots of unity. This elliptic trilogarithm value is related to the regulator of a class in the motivic cohomology of the $K3$ family. We prove a conjecture by David Broadhurst which states that at a special kinematical point the Feynman integral is given by a critical value of the Hasse–Weil $L$-function of the $K3$ surface. This result is shown to be a particular case of Deligne's conjectures relating values of $L$-functions inside the critical strip to periods. | CommonCrawl |
Edwin E. Floyd
Edwin Earl Floyd (8 May 1924, in Eufaula, Alabama – 9 December 1990) was an American mathematician, specializing in topology (especially cobordism theory).
Education and career
Floyd studied received in 1943 his bachelor's degree from the University of Alabama and in 1948 his Ph.D. from the University of Virginia under Gordon Whyburn with thesis The extension of homeomorphisms.[1] He was in the academic year 1948–1949 an instructor at Princeton University and became in 1949 a member of the faculty of the University of Virginia, where in the 1960s he collaborated with Pierre Conner in research on cobordism theory. At the University of Virginia, he was the chair of the department of mathematics from 1966 to 1969 and since 1966 the Robert C. Taylor Professor of Mathematics. In 1974 he became the dean of the Faculty of Arts and Sciences and in 1981 the vice-president and provost of the university. In the academic years 1958/59 and 1963/64 he was a visiting scholar at the Institute for Advanced Study.[2]
From 1960 to 1964 he was a Sloan Fellow. In 1962 he was an invited speaker at the International Congress of Mathematicians in Stockholm and gave a talk Some connections between cobordism and transformation groups. In 1964 he was the Hedrick Lecturer of the Mathematical Association of America. In 1981 he received the Thomas Jefferson Award of the University of Virginia. His burial was in University Cemetery, Charlottesville, Virginia.[3]
Personal life
Floyd had a daughter, Sally.[4]
Selected publications
Articles
• Floyd, E. E. (1949). "A nonhomogeneous minimal set". Bull. Amer. Math. Soc. 55 (10): 957–960. doi:10.1090/s0002-9904-1949-09318-7. MR 0033535.
• Floyd, E. E. (1952). "On periodic maps and the Euler characteristics of associated spaces". Trans. Amer. Math. Soc. 72 (1): 138–147. doi:10.2307/1990658. JSTOR 1990658.
• Floyd, E. E. (1955). "Real-valued mappings of spheres". Proc. Amer. Math. Soc. 6 (6): 957–959. doi:10.1090/s0002-9939-1955-0073978-9. MR 0073978.
• Floyd, E. E. (1957). "Fixed point sets of compact abelian Lie groups of transformations". Annals of Mathematics. 66 (1): 30–35. doi:10.2307/1970115. JSTOR 1970115.
• with R. W. Richardson: Floyd, E. E.; Richardson, R. W. (1959). "An action of a finite group on an 𝑛-cell without stationary points". Bull. Amer. Math. Soc. 65 (2): 73–76. doi:10.1090/s0002-9904-1959-10282-2. MR 0100848.
• with Pierre E. Conner: Conner, P. E.; Floyd, E. E. (1960). "Fixed point free involutions and equivariant maps". Bull. Amer. Math. Soc. 66 (6): 416–441. doi:10.1090/s0002-9904-1960-10492-2. MR 0163310.
• with P. E. Conner: Conner, P. E.; Floyd, E. E. (1966). "Maps of odd period". Annals of Mathematics. 84 (2): 132–156. doi:10.2307/1970515. JSTOR 1970515.
• with P. E. Conner: Conner, P. E.; Floyd, E. E. (1959). "On the construction of periodic maps without fixed points". Proc. Amer. Math. Soc. 10 (3): 354–360. doi:10.1090/s0002-9939-1959-0105115-x. hdl:2027/mdp.39015095249366. MR 0105115.
• with P. E. Conner: Conner, P. E.; Floyd, E. E. (1962). "Differential periodic maps". Bull. Amer. Math. Soc. 68 (2): 76–86. doi:10.1090/s0002-9904-1962-10730-7. MR 0133834.
• Floyd, E. E. (1971). "Stiefel-Whitney numbers of quaternionic and related manifolds". Trans. Amer. Math. Soc. 155: 77–94. doi:10.1090/s0002-9947-1971-0273632-8. MR 0273632.
Books
• with Pierre E. Conner: Conner, P. E.; Floyd, E. E. (1964). Differentiable periodic maps. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer. doi:10.1007/978-3-662-41633-4. ISBN 978-3-662-41635-8. 2nd edn. 1979
• with P. E. Conner: Conner, P. E.; Floyd, E. E. (1966). The relation of cobordism to K-theories. Lecture Notes in Mathematics, vol. 28. Vol. 28. doi:10.1007/BFb0071091. ISBN 978-3-540-03610-4.
• with P. E. Conner: Conner, Pierre E.; Floyd, E. E. (1966). Torsion in 𝑆𝑈-bordism. No. 60. American Mathematical Society. ISBN 9780821812600.
References
1. Edwin E. Floyd at the Mathematics Genealogy Project
2. Floyd, Edwin E. | Institute for Advanced Study
3. Thomas E. Spencer (1998). Where they're buried: a directory containing more than twenty thousand names of notable persons buried in American cemeteries. Genealogical Publishing Com. pp. 443. ISBN 9780806348230.
4. Hafner, Katie (September 8, 2019). "Sally Floyd, Who Helped Things Run Smoothly Online, Dies at 69". The New York Times. Retrieved 4 August 2022.
Authority control
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| Wikipedia |
Hierarchical combinatorial deep learning architecture for pancreas segmentation of medical computed tomography cancer images
Min Fu1,2,
Wenming Wu3,
Xiafei Hong3,
Qiuhua Liu1,2,
Jialin Jiang3,
Yaobin Ou1,
Yupei Zhao3 &
Xinqi Gong2
BMC Systems Biology volume 12, Article number: 56 (2018) Cite this article
Efficient computational recognition and segmentation of target organ from medical images are foundational in diagnosis and treatment, especially about pancreas cancer. In practice, the diversity in appearance of pancreas and organs in abdomen, makes detailed texture information of objects important in segmentation algorithm. According to our observations, however, the structures of previous networks, such as the Richer Feature Convolutional Network (RCF), are too coarse to segment the object (pancreas) accurately, especially the edge.
In this paper, we extend the RCF, proposed to the field of edge detection, for the challenging pancreas segmentation, and put forward a novel pancreas segmentation network. By employing multi-layer up-sampling structure replacing the simple up-sampling operation in all stages, the proposed network fully considers the multi-scale detailed contexture information of object (pancreas) to perform per-pixel segmentation. Additionally, using the CT scans, we supply and train our network, thus get an effective pipeline.
Working with our pipeline with multi-layer up-sampling model, we achieve better performance than RCF in the task of single object (pancreas) segmentation. Besides, combining with multi scale input, we achieve the 76.36% DSC (Dice Similarity Coefficient) value in testing data.
The results of our experiments show that our advanced model works better than previous networks in our dataset. On the other words, it has better ability in catching detailed contexture information. Therefore, our new single object segmentation model has practical meaning in computational automatic diagnosis.
Recently, due to the great development in deep neural network and increasing medical needs, Computer-Aided Diagnosis (CAD) system has become a new fashion. The high morbidity of pancreas cancers leads to great interest in developing useful CAD methods for diagnosis and treatment, in which accurate pancreas segmentation is fundamentally important. Therefore, developing an advanced pancreas segmentation method is necessary.
Nowadays, pancreas segmentation from Computed Tomography (CT) images is still an open challenge. The accuracy of pancreas segmentation in CT scans is still limit to 73% Dice Similarity Coefficient (DAC) on the patients without pancreatic cancer lesion [1,2,3,4,5,6], since the pancreas with cancer lesion are more challenging to be segmented. Previous efforts in pancreas segmentation are all referred as MALF (Multi-Atlas Registration & Label Fusion), a top-down model fitting method [1,2,3,4]. To optimize the per-pixel organ labeling process, they are all based on applying volumetric multiple atlas registration [7,8,9] and robust label fusion approach [10,11,12].
Recently, a new bottom-up pancreas segmentation method [5] has been reported, based on probability maps, which are aggregated to classify image regions, or super-pixels [13,14,15], into pancreas or non-pancreas label. By leveraging mid-level visual representations of image, this method aims to enhance the segmentation accuracy of highly deformable organs, such as the pancreas segmentation. Furtherly, this work has been improved [6] by using a set of multi-scale and multi-level deep Convolutional Neural Networks (CNN) to confront the high complexity of pancreas appearance in CT images.
In the past few years, deep CNN has become popular in the computer vision community, owing to its ability to accomplish various state-of-the-art tasks, such as image classification [16,17,18], semantic segmentation [19, 20] and object detection [21,22,23,24]. And there is a recent trend of applying it in edge detection, object segmentation and object detection [25] in medical imaging, and a series of deep learning based approaches have been invented. Fully Convolution Network (FCN) [20] adopts a skip architecture combining information from a deep layer and a shallow layer, which could produce accurate and detailed segmentations. Besides, the network could take input in arbitrary size and produce correspondingly-sized output. Holistically-nested edge detection (HED) [26] has been developed to perform image-to-image training and prediction. This deep learning model leverages fully convolutional neural networks and deeply-supervised nets, and accomplishes the task of object boundary detection by automatically learning rich hierarchical representations [17]. In the observation that only adopting the features from the last convolutional stage would cause losing some useful richer hierarchical features when classifying pixels to edge or non-edge class, richer convolutional features network (RCF) has been developed. Combining the multistage outputs, it accomplishes the task of edge detection better.
However, when it comes to the single object segmentation (pancreas segmentation), the RCF does not achieve great performance as in edge detection, because the detailed texture information of the object caught by the network is not accurate enough. To overcome this difficulty, we propose a novel multi-stage up-sampling structure into the network, to accomplish the task of single object segmentation (pancreas segmentation) more perfectly. In the following method section, we will explain our dataset, the detail of the multi-layer up-sampling structure,the loss function we used, the whole workflow, and the evalution criteria.Besides, the experiment result will be shown in the results section.
Our dataset are the real pancreas cancer CT images from the General Surgery Department of Peking Union Medical College Hospital. There are totally 59 patients, including 15 patients with non-pancreas diseases, and 44 with pancreas-related diseases, with a sum of 236 image slices. With the informed consent, patients' information, including name, gender, age, are confidential. At the slice level, one patient has 4 abdomen CT images in different phases, such as non-enhanced phase, arterial phase, portal phase, delayed phase. Additionally, the five sorts of pancreas-related diseases included in the dataset are: PDAC (Pancreatic Ductal Adenocarcinoma), PNET (Pancreatic Neuroendocrine Tumors), IPMN (Intraductal Papillary Mucinous Neoplasia), SCA (Serous CystAdenoma of the pancreas), and SPT (Solid Pseudopapillary Tumour of the pancreas) (Fig. 1).
Examples of the six types (including non-disease) of abdomen CT image for (a) Healthy, (b) PNET, (c) PDAC, (d) IPMN, (e) SCA, (f) SPT. From row1 to row4 are non-enhanced phase, arterial phase, portal phase, delayed phase
Multi-layer up-sampling structure
Inspired by the previous work on deep convolutional neural network [17, 26], we design our network by modifying the RCF network [27]. Based on Holistically-nested Edge Detection (HED) network, it is an edge detection architecture aiming to extract visually salient edges and object boundaries from natural images [27].
The whole network contains a feature extraction network and 5 feature fusing layers with up-sampling layers. The feature extraction network contains 13 conv layers and 4 pooling layers [27], which are divided into 5 stages (shown in Fig. 2). Different from the traditional classification network, there is no fully connected layer in the network. Besides, to get richer interior information and improve the overall performance, the RCF network combines the hierarchical features extracted from the 5 stages of the convolutional layers.
Architecture of our network. Part (a) shows the main structure of our network. In the feature extract network, each color box stands for a conv layer, and the conv layers are divided into 5 different stages in different colors. Furtherly, each stage is connected to a features fusing layer. After that, an up-sampling structure is used to de-convolute the extracted features to the initial size. Part (b) and (c) separately show the up-sampling structure of the RCF network and ours
Each stage combines a feature fusing layer, i.e., each convolutional layer in each stage is connected to a convolutional layer with kernel size 1*1 and channel depth 21 and then the resulting is accumulated using an element-wise layer to attain hybrid features [26], and a 1*1–1 convolutional layer follows them. After the feature fusing layer, an up-sampling structure (also called de-convolution) is used to up-sample the feature map to the input image size. Beneficial from the non-full-connection layers and up-sampling structures, the network can duel with input images in arbitrary size and output the response-size probability map.
In the up-sampling process, the images outputted by the last layer has to be resized as the input images, thus more detailed texture information is added into the images. The starting point of our network design lies in the construction of this detailed texture information.
The novel network proposed by us is shown in the part (a) of Fig. 2. Compared with RCF, our modifications can be described as following: We adopt the multi-layer up-sampling structures to replace the four de-convolutional layers. Then on the stage 2 to 5, the 1*1–1 conv layer is connected by the multi-layer up-sampling structure, and the output images of them are combined in the fusion stage.
Our novel structure consists of several up-sampling layers that include diverse convolutional kernels. We initialize them with bilinear interpolation. Then in the training process, the convolutional kernels in the layers continuously learn and adjust the parameters during iteration and repeated optimization.
Compared with the task of edge detection, single object segmentation requires the model containing far more accurate detailed texture information. In the previous RCF network, the de-convolutional layer could produce the loss pixels and resize the images, but resulting from the simple bilinear interpolation, the information added is too coarse to segment the object. As we all know, in an image, there are strong relationships between the neighbor pixels, and it is an ideal method to produce a missing pixel by using its nearest neighbors. However, adopting only one step of up-sampling may lead to produce a pixel by comparably far ones since too much pixels are missed in the images. In contrast, a multi-layer up-sampling structure ensures that a missing pixel is produced by its neighbors by multi-step up-sampling, and furtherly guarantees higher quality of output on each stage. Additionally, different from simple bilinear interpolation, the pattern, that the convolutional kernels adjust the parameters during the training process, assures the up-sampling operation and the whole model fit the local dataset better by producing a set of optimized parameters. The comparison of up-sampling structure in the RCF network and ours is shown in the part (b) and (c) in the Fig. 2.
Hence, we acquire multi-stage outputs with more accurate detailed texture information helpful to single object segmentation. We show the intermediate results from each stage in Fig. 3. Compared with the five outputs of RCF, they are obviously in higher quality. And the quantized advantages are shown in section 3.
Example of multistage output. The first column is the original input from our datasets. And from row 1 to row 6 are the six classes of pancreas disease, namely healthy, PNET, PDAC, IPMN, SCA, SPT. From the column 2 to column 6 are the output of stage 1 to 5 from our model
To train and optimize our segmentation model, we adopt per pixel loss function [26], and thus necessary to have the ground-truth maps. Each CT scan has been labeled by an annotator with medical knowledge. The ground-truth maps show the edge possibility of each pixel. 0 means that the annotator does not label at this pixel, and 1 means that the annotator labels at this pixel. Additionally, the negative sample consists of pixels with possibility value equal to 0, and the positive sample consists of other pixels.
$$ L(W)=\kern0.5em {\sum}_{i=1}^{\mid I\mid}\left({\sum}_{k=1}^Kl\left({X}_i^{(k)};W\right)+l\left({X}_i^{fuse};W\right)\right), $$
K means the number of stages making output. As shown in the Equation1, the loss value of each image is the addition of loss value of each pixel, which is made of loss value of each stage-out and fusion stage.\( l\left({X}_i^{(k)};W\right) \) denotes the loss value of a pixel in the k-th stage. Similarly, \( l\left({X}_i^{fuse};W\right) \) denotes the loss value of a pixel in the fusion stage. X i is the activation value (feature vector) at pixel i. W is all the parameters in our network. |I| is the number of all pixel in an image.
$$ l\left({X}_i;\mathrm{W}\right)=\left\{\begin{array}{c}\alpha \ast \log \left(1-P\left({X}_i;W\right)\right)\ if\ {y}_i=0\\ {}\beta \ast \log P\left({X}_i;W\right)\kern2.5em otherwise\end{array}\right. $$
P(X i ; W) is the edge possibility value at pixel i. P denotes the standard sigmoid function.
$$ \left\{\begin{array}{c}\upalpha =\uplambda \ast \frac{\left|{Y}^{+}\right|}{\left|{Y}^{+}\right|+\left|{Y}^{-}\right|}\\ {}\upbeta =\frac{\mid {Y}^{+}\mid }{\mid {Y}^{+}\mid +\mid {Y}^{-}\mid}\end{array}\right. $$
To balance the negative and positive sample, we adopt the hyper-parameter λ (λ is set as 1.1 when training). Y+ denotes the positive sample of an image, and Y− denotes the negative sample of an image.
Workflow of our segmentation
We implement a deep learning framework based on our new multi-layer up-sampling neural network for pancreas segmentation (Fig. 4). The segmentation pipeline consists of two modules, model training and optimization (Fig. 4).
Workflow of the segmentation process. The data with manually label are used to training and optimization. When the whole architecture is trained, the architecture receives the input CT images and directly output the pancreas segmentation result
In the model training module, firstly we preprocess both the original CT images and the ground truth images. The original images are in different size about 400 pixels*500 pixels. We resize the images' height to 256 pixels and keep the ratio of each image's height and width. Reducing the size of the images can not only speed the model training, but also retain more information of the original data.
After resizing the image size, to enlarge the training dataset and prevent the deep learning model over-fitting, we do the data augmentation basing on [28], such as translation transform and scale transform. After that, we trained our multi-layer up-sampling neural network based on Convolution Neural Network (CNN). Since the dataset is still small, we adopt transfer learning, i.e., fine-tuning our CNN models pre-trained from BSDS500 dataset [26] (a natural dataset for edge detection) to our medical CT image tasks, which [29] has examine why transfer learning from pre-trained natural dataset is useful in medical image tasks. After pre-training, the model gets an original set of parameters, and then was fine-tuned in our dataset, so that the network could easily converge in our dataset with a higher speed.
Our advanced model outputs a probability map of each training data. The probability map is in response-size with the input image, whose pixels are the probability of the corresponding pixel's belonging to pancreas. Besides, to highlight the pancreas, we rescale the probability map from the grey [0, 1] to [0,255] and do the gray value inversion, so in the probability map, darker region has higher probability to be pancreas.
The optimization module is divided into 3 steps: fusing, maximum connected area and threshold filter. In the fusing step, a set of probability maps belonging to the same input image is fused into a new image. To predict a specific pixel, we simply count the probability maps with its probability larger than 0. Then the specific pixel of a fuse image is made up of the mean of true positive pixel. In the maximum connected area step, after transforming the fuse image to binary image, we search the fused image's pixels to find the non-zeros neighbors of current pixel, and obtain one or several connected areas. Then we select the region with maximum area. In the filter step, we simply get a mask showing the maximum connected area, and use it to segment the pancreas from the original input image.
Here, P is the prediction image, G is the ground-truth image, and S is the area of foreground in certain image. Then we have the following criteria:
Precision (also called positive predictive value), is the fraction of correctly predicted foreground area among that in prediction
$$ Precision=\frac{S\left(P\bigcap G\right)}{S(P)} $$
where S(P ⋂ G) is the interaction area in foreground of P and G.
Recall (also known as sensitivity), is the fraction of correctly predicted foreground area over that in ground-truth.
$$ Recall=\frac{S\left(P\bigcap G\right)}{S(G)} $$
Dice Similarity Coefficient (DSC), measures the similarity of prediction image and ground-truth image. The definition of DSC is the same as F1 score. Here we also give its relationship with precision and recall.
$$ {\displaystyle \begin{array}{l} DSC\left(P,G\right)=\frac{2^{\ast }S\left(P\cap G\right)}{S(P)+S(G)}=\kern0.5em \frac{2}{\frac{S(P)}{S\left(P\cap G\right)}+\frac{S(G)}{S\left(P\cap G\right)}}\\ {}\kern8.5em =\frac{2}{\frac{1}{precision}+\frac{1}{recall}}=\frac{2\ast precision\ast recall}{precision+ recall}\end{array}} $$
Jaccard similarity coefficient, also known as Intersection over Union (originally coined coefficient de communauté by Paul Jaccard), is a statistic used for comparing the similarity and diversity of prediction image and ground-truth image. It is defined as the size of the intersection area divided by the size of the union area:
$$ \mathrm{Jaccard}\left(\mathrm{P},\mathrm{G}\right)=\frac{S\left(P\bigcap G\right)}{S\left(P\bigcup G\right)}=\frac{S\left(P\bigcap G\right)}{S(P)+S(G)-S\left(P\bigcap G\right)} $$
All of the criterias ranges from 0 to 1, with best value at 1 and worst at 0.
In our experiment, we randomly split the dataset of 59 patients into 5-folds, training and testing folds, with 10, 10, 10, 10 and 9 for each one. Then we do data augmentation, such as zooming in, flipping, rotating for each training data and enlarge the data into 128 times, and the whole dataset up to 30,208 images.
Besides, our CNN model is pre-trained in BSDS500 dataset and fine-tuned in our dataset with stochastic gradient descent (SGD) algorithm and step-wise learning schedule to optimize. The model is implemented by a deep learning framework CAFFE [30] and run over one NVIDIA QUADRO M4000 GPU.
Using 5-fold cross-validation, we could achieve a mean of precision of 76.83%, a mean of recall of 78.74%, a mean of DSC of 75.92%, and the mean of JACCARD of 63.29%. Apart from the recall one, all of them are higher than the RCF network. At the same time, our method with multi-scale input (OURS-MS) reaches 77.36%, 79.12%, 76.36%, 63.72% in mean of precision, recall, DSC and Jaccard. Table 1 show the detailed performance of three models.
Table 1 Compare the three segmentation models' performance in four measurements: precision, recall, DSC and Jaccard index
In the pancreas segmentation task, the number of positive samples is much less than that of negative samples, which means that the Precision-Recall (PR) curve can better reflect the performance of the prediction [31]. Figure 5 shows the Recall value can reach more than 90% while the Precision value is still more than 60%, which means that we could attain excellent reservation of the pancreas organ area in a decent precision.
The Precision-Recall curve. The blue, orange and green curves stand for the performance of RCF, our model and OURS-MS.
Our model's performances in different types of pancreas cancer are shown in Table 2. We can see that the values of four measurements are comparably high, and the standard deviations are not too big, which means that our model is robust in different types of pancreas cancer.
Table 2 Model's performance in different types of pancreas cancer (with healthy type)
Our model's performances in different phases are shown in Table 3. We can see that the values of four measurements are comparably high, and the standard deviations are not too big, which means that our model is robust in different phases.
Table 3 Model's performance in different phases. The Phase1 to Phase4 are non-enhanced phase, arterial phase, portal phase, delayed phase
Figure 6 shows some examples of the pancreas segmentation result, a comparison of ground-truth and output of our model. The red curve is ground-truth annotation, and the green curve highlights the output. We can easily find that the two curves of four images share high similarity, and high accuracy has been gained in our model. Images in row1 get the best performance, where the DSC values are around 94%, images in row2 get the DSC value on quartile2, around 79%, and those in row3 reach the DSC values around 70%, which is on the quartile1.
Some examples of pancreas segmentation result. Red curve shows the ground truth while green for the predicted. Row1 are in the best performance, row2 are on the quartile2 and row3 on the quartile1
We summarize our contributions as follow. In this paper, we design an automatically pancreas segmentation architecture based on deep learning model, and get a 76.36% DSC value.
We extend the Richer Convolutional Feature network to pancreas segmentation and improve the RCF network with multi-layer up-sampling structure and get over 1% better performance in pancreas segmentation. Besides, we find that, in experiment, testing with multi-scale input and training with data augmentation, especially rotation, can improve the performance of the network.
Significantly, our model is robust in different types of pancreas cancer and different phases of CT images.
Computer-Aided Diagnosis
DSC:
Dice Similarity Coefficient
FCN:
Fully Convolution Network
HED:
Holistically-nested edge detection
IPMN:
Intraductal Papillary Mucinous Neoplasia
MALF:
Multi-Atlas Registration & Label Fusion
PNET:
Pancreatic Neuroendocrine Tumors
Precision-Recall
RCF:
Richer Convolutional Feature network
SCA:
Serous CystAdenoma of the pancreas
SGD:
stochastic gradient descent
SPT:
Solid Pseudopapillary Tumour of the pancreas
Std:
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This research was supported by National Natural Science Fundation of China (31670725, 91730301) to Xinqi Gong.
The publication cost of this article was funded by National Natural Science Fundation of China (91730301).
All the data are provided by the General Surgery Department of Peking Union Medical College Hospital. All the patient have signed the informed consent.
About this supplement
This article has been published as part of BMC Systems Biology Volume 12 Supplement 4, 2018: Selected papers from the 11th International Conference on Systems Biology (ISB 2017). The full contents of the supplement are available online at https://bmcsystbiol.biomedcentral.com/articles/supplements/volume-12-supplement-4.
Mathematics Department, School of Information, Renmin University of China, Beijing, China
Min Fu, Qiuhua Liu & Yaobin Ou
Mathematical Intelligence Application Lab, Institute for Mathematical Sciences, Renmin University of China, Beijing, China
Min Fu, Qiuhua Liu & Xinqi Gong
Department of General Surgery, Peking Union Medical College Hospital, Chinese Academy of Medical Sciences and Peking Union Medical College, Beijing, China
Wenming Wu, Xiafei Hong, Jialin Jiang & Yupei Zhao
Min Fu
Wenming Wu
Xiafei Hong
Qiuhua Liu
Jialin Jiang
Yaobin Ou
Yupei Zhao
Xinqi Gong
X.Q.G supervised the project and designed the ideas. M. F, W.M.W, X.F.H, Q.H. L and J.L.J. did the experiments and drafted the initial manuscript. Y.P.Z and Y.B.O participated in supervision. All authors discussed the results and commented on the manuscript. All authors read and approved the final manuscript.
Correspondence to Yaobin Ou, Yupei Zhao or Xinqi Gong.
Fu, M., Wu, W., Hong, X. et al. Hierarchical combinatorial deep learning architecture for pancreas segmentation of medical computed tomography cancer images. BMC Syst Biol 12, 56 (2018). https://doi.org/10.1186/s12918-018-0572-z
Pancreas segmentation
Single object segmentation | CommonCrawl |
For the sake of organizing the review, we have divided the literature according to the general type of cognitive process being studied, with sections devoted to learning and to various kinds of executive function. Executive function is a broad and, some might say, vague concept that encompasses the processes by which individual perceptual, motoric, and mnemonic abilities are coordinated to enable appropriate, flexible task performance, especially in the face of distracting stimuli or alternative competing responses. Two major aspects of executive function are working memory and cognitive control, responsible for the maintenance of information in a short-term active state for guiding task performance and responsible for inhibition of irrelevant information or responses, respectively. A large enough literature exists on the effects of stimulants on these two executive abilities that separate sections are devoted to each. In addition, a final section includes studies of miscellaneous executive abilities including planning, fluency, and reasoning that have also been the subjects of published studies.
l-theanine (Examine.com) is occasionally mentioned on Reddit or Imminst or LessWrong32 but is rarely a top-level post or article; this is probably because theanine was discovered a very long time ago (>61 years ago), and it's a pretty straightforward substance. It's a weak relaxant/anxiolytic (Google Scholar) which is possibly responsible for a few of the health benefits of tea, and which works synergistically with caffeine (and is probably why caffeine delivered through coffee feels different from the same amount consumed in tea - in one study, separate caffeine and theanine were a mixed bag, but the combination beat placebo on all measurements). The half-life in humans seems to be pretty short, with van der Pijl 2010 putting it ~60 minutes. This suggests to me that regular tea consumption over a day is best, or at least that one should lower caffeine use - combining caffeine and theanine into a single-dose pill has the problem of caffeine's half-life being much longer so the caffeine will be acting after the theanine has been largely eliminated. The problem with getting it via tea is that teas can vary widely in their theanine levels and the variations don't seem to be consistent either, nor is it clear how to estimate them. (If you take a large dose in theanine like 400mg in water, you can taste the sweetness, but it's subtle enough I doubt anyone can actually distinguish the theanine levels of tea; incidentally, r-theanine - the useless racemic other version - anecdotally tastes weaker and less sweet than l-theanine.)
Amongst the brain focus supplements that are currently available in the nootropic drug market, Modafinil is probably the most common focus drug or one of the best focus pills used by people, and it's praised to be the best nootropic available today. It is a powerful cognitive enhancer that is great for boosting your overall alertness with least side effects. However, to get your hands on this drug, you would require a prescription.
Stayed up with the purpose of finishing my work for a contest. This time, instead of taking the pill as a single large dose (I feel that after 3 times, I understand what it's like), I will take 4 doses over the new day. I took the first quarter at 1 AM, when I was starting to feel a little foggy but not majorly impaired. Second dose, 5:30 AM; feeling a little impaired. 8:20 AM, third dose; as usual, I feel physically a bit off and mentally tired - but still mentally sharp when I actually do something. Early on, my heart rate seemed a bit high and my limbs trembling, but it's pretty clear now that that was the caffeine or piracetam. It may be that the other day, it was the caffeine's fault as I suspected. The final dose was around noon. The afternoon crash wasn't so pronounced this time, although motivation remains a problem. I put everything into finishing up the spaced repetition literature review, and didn't do any n-backing until 11:30 PM: 32/34/31/54/40%.
Serotonin, or 5-hydroxytryptamine (5-HTP), is another primary neurotransmitter and controls major features of the mental landscape including mood, sleep and appetite. Serotonin is produced within the body by exposure, which is one reason that the folk-remedy of "getting some sun" to fight depression is scientifically credible. Many foods contain natural serotonergic (serotonin-promoting or releasing) compounds, including the well-known chemical L-Tryptophan found in turkey, which can promote sleep after big Thanksgiving dinners.
Another common working memory task is the n-back task, which requires the subject to view a series of items (usually letters) and decide whether the current item is identical to the one presented n items back. This task taxes working memory because the previous items must be held in working memory to be compared with the current item. The easiest version of this is a 1-back task, which is also called a double continuous performance task (CPT) because the subject is continuously monitoring for a repeat or double. Three studies examined the effects of MPH on working memory ability as measured by the 1-back task, and all found enhancement of performance in the form of reduced errors of omission (Cooper et al., 2005; Klorman et al., 1984; Strauss et al., 1984). Fleming et al. (1995) tested the effects of d-AMP on a 5-min CPT and found a decrease in reaction time, but did not specify which version of the CPT was used.
The title question, whether prescription stimulants are smart pills, does not find a unanimous answer in the literature. The preponderance of evidence is consistent with enhanced consolidation of long-term declarative memory. For executive function, the overall pattern of evidence is much less clear. Over a third of the findings show no effect on the cognitive processes of healthy nonelderly adults. Of the rest, most show enhancement, although impairment has been reported (e.g., Rogers et al., 1999), and certain subsets of participants may experience impairment (e.g., higher performing participants and/or those homozygous for the met allele of the COMT gene performed worse on drug than placebo; Mattay et al., 2000, 2003). Whereas the overall trend is toward enhancement of executive function, the literature contains many exceptions to this trend. Furthermore, publication bias may lead to underreporting of these exceptions.
This calculation - reaping only \frac{7}{9} of the naive expectation - gives one pause. How serious is the sleep rebound? In another article, I point to a mice study that sleep deficits can take 28 days to repay. What if the gain from modafinil is entirely wiped out by repayment and all it did was defer sleep? Would that render modafinil a waste of money? Perhaps. Thinking on it, I believe deferring sleep is of some value, but I cannot decide whether it is a net profit.
Barbaresi WJ, Katusic SK, Colligan RC, Weaver AL, Jacobsen SJ. Modifiers of long-term school outcomes for children with attention-deficit/hyperactivity disorder: Does treatment with stimulant medication make a difference? Results from a population-based study. Journal of Developmental and Behavioral Pediatrics. 2007;28:274–287. doi: 10.1097/DBP.0b013e3180cabc28. [PubMed] [CrossRef]
As with any thesis, there are exceptions to this general practice. For example, theanine for dogs is sold under the brand Anxitane is sold at almost a dollar a pill, and apparently a month's supply costs $50+ vs $13 for human-branded theanine; on the other hand, this thesis predicts downgrading if the market priced pet versions higher than human versions, and that Reddit poster appears to be doing just that with her dog.↩
Analyzing the results is a little tricky because I was simultaneously running the first magnesium citrate self-experiment, which turned out to cause a quite complex result which looks like a gradually-accumulating overdose negating an initial benefit for net harm, and also toying with LLLT, which turned out to have a strong correlation with benefits. So for the potential small Noopept effect to not be swamped, I need to include those in the analysis. I designed the experiment to try to find the best dose level, so I want to look at an average Noopept effect but also the estimated effect at each dose size in case some are negative (especially in the case of 5-pills/60mg); I included the pilot experiment data as 10mg doses since they were also blind & randomized. Finally, missingness affects analysis: because not every variable is recorded for each date (what was the value of the variable for the blind randomized magnesium citrate before and after I finished that experiment? what value do you assign the Magtein variable before I bought it and after I used it all up?), just running a linear regression may not work exactly as one expects as various days get omitted because part of the data was missing.
Adderall is a mix of 4 amphetamine salts (FDA adverse events), and not much better than the others (but perhaps less addictive); as such, like caffeine or methamphetamine, it is not strictly a nootropic but a cognitive enhancer and can be tricky to use right (for how one should use stimulants, see How To Take Ritalin Correctly). I ordered 10x10mg Adderall IR off Silk Road (Wikipedia). On the 4th day after confirmation from seller, the package arrived. It was a harmless looking little padded mailer. Adderall as promised: 10 blue pills with markings, in a double ziplock baggy (reasonable, it's not cocaine or anything). They matched pretty much exactly the descriptions of the generic I had found online. (Surprisingly, apparently both the brand name and the generic are manufactured by the same pharmacorp.)
I have also tried to get in contact with senior executives who have experience with these drugs (either themselves or in their firms), but without success. I have to wonder: Are they completely unaware of the drugs' existence? Or are they actively suppressing the issue? For now, companies can ignore the use of smart drugs. And executives can pretend as if these drugs don't exist in their workplaces. But they can't do it forever.
Too much caffeine may be bad for bone health because it can deplete calcium. Overdoing the caffeine also may affect the vitamin D in your body, which plays a critical role in your body's bone metabolism. However, the roles of vitamin D as well as caffeine in the development of osteoporosis continue to be a source of debate. Significance: Caffeine may interfere with your body's metabolism of vitamin D, according to a 2007 Journal of Steroid Biochemistry & Molecular Biology study. You have vitamin D receptors, or VDRs, in your osteoblast cells. These large cells are responsible for the mineralization and synthesis of bone in your body. They create a sheet on the surface of your bones. The D receptors are nuclear hormone receptors that control the action of vitamin D-3 by controlling hormone-sensitive gene expression. These receptors are critical to good bone health. For example, a vitamin D metabolism disorder in which these receptors don't work properly causes rickets.
No. There are mission essential jobs that require you to live on base sometimes. Or a first term person that is required to live on base. Or if you have proven to not be as responsible with rent off base as you should be so your commander requires you to live on base. Or you're at an installation that requires you to live on base during your stay. Or the only affordable housing off base puts you an hour away from where you work. It isn't simple. The fact that you think it is tells me you are one of the "dumb@$$es" you are referring to above.
More recently, the drug modafinil (brand name: Provigil) has become the brain-booster of choice for a growing number of Americans. According to the FDA, modafinil is intended to bolster "wakefulness" in people with narcolepsy, obstructive sleep apnea or shift work disorder. But when people without those conditions take it, it has been linked with improvements in alertness, energy, focus and decision-making. A 2017 study found evidence that modafinil may enhance some aspects of brain connectivity, which could explain these benefits.
It is not because of the few thousand francs which would have to be spent to put a roof [!] over the third-class carriages or to upholster the third-class seats that some company or other has open carriages with wooden benches. What the company is trying to do is to prevent the passengers who can pay the second class fare from traveling third class; it hits the poor, not because it wants to hurt them, but to frighten the rich. And it is again for the same reason that the companies, having proved almost cruel to the third-class passengers and mean to the second-class ones, become lavish in dealing with first-class passengers. Having refused the poor what is necessary, they give the rich what is superfluous.
One often-cited study published in the British Journal of Pharmacology looked at cognitive function in the elderly and showed that racetam helped to improve their brain function.19 Another study, which was published in Psychopharmacology, looked at adult volunteers (including those who are generally healthy) and found that piracetam helped improve their memory.20
One symptom of Alzheimer's disease is a reduced brain level of the neurotransmitter called acetylcholine. It is thought that an effective treatment for Alzheimer's disease might be to increase brain levels of acetylcholine. Another possible treatment would be to slow the death of neurons that contain acetylcholine. Two drugs, Tacrine and Donepezil, are both inhibitors of the enzyme (acetylcholinesterase) that breaks down acetylcholine. These drugs are approved in the US for treatment of Alzheimer's disease.
And as before, around 9 AM I began to feel the peculiar feeling that I was mentally able and apathetic (in a sort of aboulia way); so I decided to try what helped last time, a short nap. But this time, though I took a full hour, I slept not a wink and my Zeo recorded only 2 transient episodes of light sleep! A back-handed sort of proof of alertness, I suppose. I didn't bother trying again. The rest of the day was mediocre, and I wound up spending much of it on chores and whatnot out of my control. Mentally, I felt better past 3 PM.
Intrigued by old scientific results & many positive anecdotes since, I experimented with microdosing LSD - taking doses ~10μg, far below the level at which it causes its famous effects. At this level, the anecdotes claim the usual broad spectrum of positive effects on mood, depression, ability to do work, etc. After researching the matter a bit, I discovered that as far as I could tell, since the original experiment in the 1960s, no one had ever done a blind or even a randomized self-experiment on it.
Looking at the prices, the overwhelming expense is for modafinil. It's a powerful stimulant - possibly the single most effective ingredient in the list - but dang expensive. Worse, there's anecdotal evidence that one can develop tolerance to modafinil, so we might be wasting a great deal of money on it. (And for me, modafinil isn't even very useful in the daytime: I can't even notice it.) If we drop it, the cost drops by a full $800 from $1761 to $961 (almost halving) and to $0.96 per day. A remarkable difference, and if one were genetically insensitive to modafinil, one would definitely want to remove it.
Sounds too good to be true? Welcome to the world of 'Nootropics' popularly known as 'Smart Drugs' that can help boost your brain's power. Do you recall the scene from the movie Limitless, where Bradley Cooper's character uses a smart drug that makes him brilliant? Yes! The effect of Nootropics on your brain is such that the results come as a no-brainer.
The price is not as good as multivitamins or melatonin. The studies showing effects generally use pretty high dosages, 1-4g daily. I took 4 capsules a day for roughly 4g of omega acids. The jar of 400 is 100 days' worth, and costs ~$17, or around 17¢ a day. The general health benefits push me over the edge of favoring its indefinite use, but looking to economize. Usually, small amounts of packaged substances are more expensive than bulk unprocessed, so I looked at fish oil fluid products; and unsurprisingly, liquid is more cost-effective than pills (but like with the powders, straight fish oil isn't very appetizing) in lieu of membership somewhere or some other price-break. I bought 4 bottles (16 fluid ounces each) for $53.31 total (thanks to coupons & sales), and each bottle lasts around a month and a half for perhaps half a year, or ~$100 for a year's supply. (As it turned out, the 4 bottles lasted from 4 December 2010 to 17 June 2011, or 195 days.) My next batch lasted 19 August 2011-20 February 2012, and cost $58.27. Since I needed to buy empty 00 capsules (for my lithium experiment) and a book (Stanovich 2010, for SIAI work) from Amazon, I bought 4 more bottles of 16fl oz Nature's Answer (lemon-lime) at $48.44, which I began using 27 February 2012. So call it ~$70 a year.
But there are some potential side effects, including headaches, anxiety and insomnia. Part of the way modafinil works is by shifting the brain's levels of norepinephrine, dopamine, serotonin and other neurotransmitters; it's not clear what effects these shifts may have on a person's health in the long run, and some research on young people who use modafinil has found changes in brain plasticity that are associated with poorer cognitive function.
Following up on the promising but unrandomized pilot, I began randomizing my LLLT usage since I worried that more productive days were causing use rather than vice-versa. I began on 2 August 2014, and the last day was 3 March 2015 (n=167); this was twice the sample size I thought I needed, and I stopped, as before, as part of cleaning up (I wanted to know whether to get rid of it or not). The procedure was simple: by noon, I flipped a bit and either did or did not use my LED device; if I was distracted or didn't get around to randomization by noon, I skipped the day. This was an unblinded experiment because finding a randomized on/off switch is tricky/expensive and it was easier to just start the experiment already. The question is simple too: controlling for the simultaneous blind magnesium experiment & my rare nicotine use (I did not use modafinil during this period or anything else I expect to have major influence), is the pilot correlation of d=0.455 on my daily self-ratings borne out by the experiment?
An entirely different set of questions concerns cognitive enhancement in younger students, including elementary school and even preschool children. Some children can function adequately in school without stimulants but perform better with them; medicating such children could be considered a form of cognitive enhancement. How often does this occur? What are the roles and motives of parents, teachers, and pediatricians in these cases? These questions have been discussed elsewhere and deserve continued attention (Diller, 1996; Singh & Keller, 2010).
If you could take a drug to boost your brainpower, would you? This question, faced by Bradley Cooper's character in the big-budget movie Limitless, is now facing students who are frantically revising for exams. Although they are nowhere near the strength of the drug shown in the film, mind-enhancing drugs are already on the pharmacy shelves, and many people are finding the promise of sharper thinking through chemistry highly seductive.
L-Alpha glycerylphosphorylcholine or choline alfoscerate, also known as Alpha GPC is a natural nootropic which works both on its own and also in combination with other nootropics. It can be found in the human body naturally in small amounts. It's also present in some dairy products, wheat germ, and in organic meats. However, these dietary sources contain small quantities of GPC, which is why people prefer taking it through supplements.
Kratom (Erowid, Reddit) is a tree leaf from Southeast Asia; it's addictive to some degree (like caffeine and nicotine), and so it is regulated/banned in Thailand, Malaysia, Myanmar, and Bhutan among others - but not the USA. (One might think that kratom's common use there indicates how very addictive it must be, except it literally grows on trees so it can't be too hard to get.) Kratom is not particularly well-studied (and what has been studied is not necessarily relevant - I'm not addicted to any opiates!), and it suffers the usual herbal problem of being an endlessly variable food product and not a specific chemical with the fun risks of perhaps being poisonous, but in my reading it doesn't seem to be particularly dangerous or have serious side-effects.
Scientists found that the drug can disrupt the way memories are stored. This ability could be invaluable in treating trauma victims to prevent associated stress disorders. The research has also triggered suggestions that licensing these memory-blocking drugs may lead to healthy people using them to erase memories of awkward conversations, embarrassing blunders and any feelings for that devious ex-girlfriend.
With something like creatine, you'd know if it helps you pump out another rep at the gym on a sustainable basis. With nootropics, you can easily trick yourself into believing they help your mindset. The ideal is to do a trial on yourself. Take identical looking nootropic pills and placebo pills for a couple weeks each, then see what the difference is. With only a third party knowing the difference, of course.
…The Fate of Nicotine in the Body also describes Battelle's animal work on nicotine absorption. Using C14-labeled nicotine in rabbits, the Battelle scientists compared gastric absorption with pulmonary absorption. Gastric absorption was slow, and first pass removal of nicotine by the liver (which transforms nicotine into inactive metabolites) was demonstrated following gastric administration, with consequently low systemic nicotine levels. In contrast, absorption from the lungs was rapid and led to widespread distribution. These results show that nicotine absorbed from the stomach is largely metabolized by the liver before it has a chance to get to the brain. That is why tobacco products have to be puffed, smoked or sucked on, or absorbed directly into the bloodstream (i.e., via a nicotine patch). A nicotine pill would not work because the nicotine would be inactivated before it reached the brain.
There is no shortage of nootropics available for purchase online that can be shipped to you nearly anywhere in the world. Yet, many of these supplements and drugs have very little studies, particularly human studies, confirming their results. While this lack of research may not scare away more adventurous neurohackers, many people would prefer to […]
Nature magazine conducted a poll asking its readers about their cognitive-enhancement practices and their attitudes toward cognitive enhancement. Hundreds of college faculty and other professionals responded, and approximately one fifth reported using drugs for cognitive enhancement, with Ritalin being the most frequently named (Maher, 2008). However, the nature of the sample—readers choosing to answer a poll on cognitive enhancement—is not representative of the academic or general population, making the results of the poll difficult to interpret. By analogy, a poll on Vermont vacations, asking whether people vacation in Vermont, what they think about Vermont, and what they do if and when they visit, would undoubtedly not yield an accurate estimate of the fraction of the population that takes its vacations in Vermont.
A television advertisement goes: "It's time to let Focus Factor be your memory-fog lifter." But is this supplement up to task? Focus Factor wastes no time, whether paid airtime or free online presence: it claims to be America's #1 selling brain health supplement with more than 4 million bottles sold and millions across the country actively caring for their brain health. It deems itself instrumental in helping anyone stay focused and on top of his game at home, work, or school. Learn More...
In most cases, cognitive enhancers have been used to treat people with neurological or mental disorders, but there is a growing number of healthy, "normal" people who use these substances in hopes of getting smarter. Although there are many companies that make "smart" drinks, smart power bars and diet supplements containing certain "smart" chemicals, there is little evidence to suggest that these products really work. Results from different laboratories show mixed results; some labs show positive effects on memory and learning; other labs show no effects. There are very few well-designed studies using normal healthy people.
Smart pills have huge potential and several important applications, particularly in diagnosis. Smart pills are growing as a highly effective method of endoscopy, particularly for gastrointestinal diseases. Urbanization and rapid lifestyle changes leaning toward unhealthy diets and poor eating habits have led to distinctive increasing lifestyle disorders such as gastroesophageal reflux disease (GERD), obesity, and gastric ulcers.
Fish oil (Examine.com, buyer's guide) provides benefits relating to general mood (eg. inflammation & anxiety; see later on anxiety) and anti-schizophrenia; it is one of the better supplements one can take. (The known risks are a higher rate of prostate cancer and internal bleeding, but are outweighed by the cardiac benefits - assuming those benefits exist, anyway, which may not be true.) The benefits of omega acids are well-researched.
But there would also be significant downsides. Amphetamines are structurally similar to crystal meth – a potent, highly addictive recreational drug which has ruined countless lives and can be fatal. Both Adderall and Ritalin are known to be addictive, and there are already numerous reports of workers who struggled to give them up. There are also side effects, such as nervousness, anxiety, insomnia, stomach pains, and even hair loss, among others.
Despite decades of study, a full picture has yet to emerge of the cognitive effects of the classic psychostimulants and modafinil. Part of the problem is that getting rats, or indeed students, to do puzzles in laboratories may not be a reliable guide to the drugs' effects in the wider world. Drugs have complicated effects on individuals living complicated lives. Determining that methylphenidate enhances cognition in rats by acting on their prefrontal cortex doesn't tell you the potential impact that its effects on mood or motivation may have on human cognition.
However, when I didn't stack it with Choline, I would get what users call "racetam headaches." Choline, as Patel explains, is not a true nootropic, but it's still a pro-cognitive compound that many take with other nootropics in a stack. It's an essential nutrient that humans need for functions like memory and muscle control, but we can't produce it, and many Americans don't get enough of it. The headaches I got weren't terribly painful, but they were uncomfortable enough that I stopped taking Piracetam on its own. Even without the headache, though, I didn't really like the level of focus Piracetam gave me. I didn't feel present when I used it, even when I tried to mix in caffeine and L-theanine. And while it seemed like I could focus and do my work faster, I was making more small mistakes in my writing, like skipping words. Essentially, it felt like my brain was moving faster than I could.
Another important epidemiological question about the use of prescription stimulants for cognitive enhancement concerns the risk of dependence. MPH and d-AMP both have high potential for abuse and addiction related to their effects on brain systems involved in motivation. On the basis of their reanalysis of NSDUH data sets from 2000 to 2002, Kroutil and colleagues (2006) estimated that almost one in 20 nonmedical users of prescription ADHD medications meets criteria for dependence or abuse. This sobering estimate is based on a survey of all nonmedical users. The immediate and long-term risks to individuals seeking cognitive enhancement remain unknown.
A provisional conclusion about the effects of stimulants on learning is that they do help with the consolidation of declarative learning, with effect sizes varying widely from small to large depending on the task and individual study. Indeed, as a practical matter, stimulants may be more helpful than many of the laboratory tasks indicate, given the apparent dependence of enhancement on length of delay before testing. Although, as a matter of convenience, experimenters tend to test memory for learned material soon after the learning, this method has not generally demonstrated stimulant-enhanced learning. However, when longer periods intervene between learning and test, a more robust enhancement effect can be seen. Note that the persistence of the enhancement effect well past the time of drug action implies that state-dependent learning is not responsible. In general, long-term effects on learning are of greater practical value to people. Even students cramming for exams need to retain information for more than an hour or two. We therefore conclude that stimulant medication does enhance learning in ways that may be useful in the real world.
^ Sattler, Sebastian; Mehlkop, Guido; Graeff, Peter; Sauer, Carsten (February 1, 2014). "Evaluating the drivers of and obstacles to the willingness to use cognitive enhancement drugs: the influence of drug characteristics, social environment, and personal characteristics". Substance Abuse Treatment, Prevention, and Policy. 9 (1): 8. doi:10.1186/1747-597X-9-8. ISSN 1747-597X. PMC 3928621. PMID 24484640. | CommonCrawl |
sequences with a fractal dimension
This is inspired by the self-similarity of the celebrated Golay-Rudin-Shapiro sequence, more exactly, of its alternating partial sums. (This latter one is oeis 020990). The pictures show the 550 first terms, then the 9000 first terms.
It makes sense to define a certain fractal $F$ as the "limit" of the graph $\Gamma=\{(n,a_n)\}_{n\ge0}$.
More precisely:
Fix a rectangle $R\subset\mathbb R^2$, e.g. the unit square. Take the part $\Gamma_k$ of $\Gamma$ between $n=2^{2k-1}$ and $n=2^{2k+1}-1$ and rescale it to a graph $\Gamma^0_k$ that fits $R$ best.
Then because of the geometrical (almost-) similarity of the $\Gamma^0_k$, the limit $F:=\lim\limits_{k\to\infty}\Gamma^0_k\subset\mathbb R^2$ is well-defined. Note that its Hausdorff dimension is $d=3/2$.
the sequence oeis 004074 that defines likewise the Blancmange curve, dimension $d=1$
sequences linked to the Gray code, like 003188 or 006068, both with $d=1$
Stern's diatomic series (a.k.a. Stern-Brocot sequence or $fusc$ function) yields a fractal with dimension $d=\frac{\ln 3}{\ln 2}$
it makes sense to relate (if not to identify) the Cantor set with the sequence $1,0,1,0,0,0,1,0,1,... $ where $a_n=1$ iff the ternary representation of $n$ has only 0's and 2's (equivalently, the cellular automaton where at each step $1 \mapsto 101$ and $0 \mapsto000$), and to say this sequence has dimension $\frac{\ln2}{\ln3}$. Likewise for the "fat Cantor set" iterating 11100111 (dimension $\frac{\ln5}{\ln8}$) and all other sorts of Cantor dust.
the devil's staircase, obtained by "integrating" the Cantor set, corresponds to this sequence, and a "mirrored" version of it can be found here.
Other sequences of Toothpick and Cellular Automata type
Like for most other sequences of this kind, the ressemblance is best seen when looking at a range from either $1$ to $2^n$, or (for some, like the Blancmange curve) from $2^{n-1}$ to $2^n$.
Note that it is not at all straightforward or even possible to define a fractal for every self-similar integer sequence $a=(a_n)_{n\ge0}$ (self-similar meaning as usual that there is a $k\ge2$ and $\lambda$ such that $a_n=\lambda a_{kn}$). On the other hand, there are also sequences with a fractal-like appearance without being self-similar in the above sense.
Has the idea of the "fractal dimension" of certain sequences been investigated before?
ds.dynamical-systems fractals integer-sequences
WolfgangWolfgang
Interestingly, fractal dimensions of the human DNA sequence have been analyzed and different embeddings considered: http://biocomplexity.indiana.edu/jglazier/docs/papers/20_DNA_Analysis.pdf.
Andreas RüdingerAndreas Rüdinger
$\begingroup$ "...indicating the presence of significant information content in DNA sequences not explained by base or dimer frequencies". Interesting indeed! $\endgroup$ – Wolfgang Jan 7 '13 at 8:53
Taking a truncated integer sequence is essentially the same as defining the "heights" of the line endpoints of a 2-dimensional curve, similar to http://www.gameprogrammer.com/fractal.html my guess is that this is not really new, but rather a cumbersome way to define self-similar curves.
Making a reasonable scaling will in some examples, as above, give a well-defined limit curve. Maybe it is possible to compute the Hausdorff dimension by doing a discrete Fourier transform of the sequence. The spectrum should be quite interesting, I think.
Per AlexanderssonPer Alexandersson
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On the boundary of the twindragon
Self-Similar Graphs | CommonCrawl |
\begin{definition}[Definition:Hyperbolic Function]
There are six basic '''hyperbolic functions''', as follows:
\end{definition} | ProofWiki |
Improving magnetic resonance imaging with smart and thin metasurfaces
Endri Stoja1,
Simon Konstandin2,
Dennis Philipp2,
Robin N. Wilke2,
Diego Betancourt1,
Thomas Bertuch1,
Jürgen Jenne2,3,
Reiner Umathum2,3 &
Matthias Günther2,4
Over almost five decades of development and improvement, Magnetic Resonance Imaging (MRI) has become a rich and powerful, non-invasive technique in medical imaging, yet not reaching its physical limits. Technical and physiological restrictions constrain physically feasible developments. A common solution to improve imaging speed and resolution is to use higher field strengths, which also has subtle and potentially harmful implications. However, patient safety is to be considered utterly important at all stages of research and clinical routine. Here we show that dynamic metamaterials are a promising solution to expand the potential of MRI and to overcome some limitations. A thin, smart, non-linear metamaterial is presented that enhances the imaging performance and increases the signal-to-noise ratio in 3T MRI significantly (up to eightfold), whilst the transmit field is not affected due to self-detuning and, thus, patient safety is also assured. This self-detuning works without introducing any additional overhead related to MRI-compatible electronic control components or active (de-)tuning mechanisms. The design paradigm, simulation results, on-bench characterization, and MRI experiments using homogeneous and structural phantoms are described. The suggested single-layer metasurface paves the way for conformal and patient-specific manufacturing, which was not possible before due to typically bulky and rigid metamaterial structures.
Magnetic Resonance Imaging (MRI)1,2 is the most versatile and powerful imaging modality available for clinical use nowadays. For almost five decades, the technology was improved, extended, and continues to evolve, still not reaching the physical limits. Technical and physiological limitations are hampering the advancement and constrain physically feasible developments, making it increasingly challenging to innovate.
While in clinical applications static magnetic field strengths of 1.5 and 3 Tesla are most common, research scanners have been developed with field strengths of \(7\,\)T, \(9.4\,\)T, and even higher to benefit from the increase in signal-to-noise ratio (SNR)3,4. However, working at higher field strengths becomes progressively difficult due to the direct proportionality to the Larmor frequency, which defines the nuclear resonance frequency of (hydrogen) atoms. A key limiting factor is the specific absorption rate (SAR) of the deposited radio frequency (RF) power that increases almost quadratically with frequency, setting practical limitations due to tissue heating5. Besides general challenges related to strong magnetic fields, electromagnetic wave phenomena become more relevant due to the shorter wavelength of the RF field. Huge efforts have to be taken to tackle issues arising from the fact that the RF wavelength is on the order of the imaged object's dimensions, eventually creating interference patterns and standing waves6. As a consequence, MRI is sensitive to motion artifacts, which should be circumvented. At higher field strengths, the longitudinal relaxation time \(T_1\) of tissue increases, leading to an intrinsic limitation on the achievable imaging speed for some applications7.
A main technical (and physiological) obstacle in MRI is also the use of gradient magnetic fields for localization, which substantially limit the technically possible imaging speed8,9. Despite recent advances in fast imaging approaches10 such as parallel imaging11,12 and compressed sensing13, image acquisition can be considered comparably slow. To improve the imaging performance, the patient-specific design of MRI equipment certainly is a possible approach14 but usually related to high costs as well as complicated design and manufacturing when it comes to, e.g., tailored receive coil arrays.
The best solution strategy to enhance the MRI performance and proceed to the next level of medical imaging should offer increased imaging efficiency (a metric of SNR, contrast, and speed) over large volumes of interest without the immediate need of higher background field strengths. So-called metamaterials (MTMs) are an up-and-coming solution in this respect. Here we show that a smart metasurface - a dynamic two-dimensional MTM - yields a significant SNR improvement for \(3\,\)T MRI without affecting the transmit field. Electromagnetic MTMs, as outlined by the seminal works of Vesselago, Pendry, Schurig, Leonhardt, and others15,16,17,18,19 are artificially constructed structures, consisting of a usually periodic arrangement of dielectric or conducting unit elements (small L-C circuits). Prominent application examples include MTM cloaks, perfect lenses, and ultimately MRI applications20. MTMs offer to manipulate the amplitude (focusing effects) as well as the phase of incident radiation21,22 and also polarization-independent modulation can be achieved23. The unit cells can be considered as single meta-atoms on a sub-wavelength scale. Therefore, an incident electromagnetic field is subject to a macroscopic influence induced by the interactions of all meta-atoms. Hence, w.r.t. RF field interaction the MTM can be viewed as a homogeneous material slab, which is effectively described by (anisotropic and dispersive) permeability and permittivity. In contrast to naturally occurring materials, MTMs can be designed to have arbitrary positive and negative values for both parameters. This leads to, e.g., field enhancement or focusing, phase changes, and tailored reflection and transmission properties. Usually, metasurfaces yield desired effects only for a specific target frequency with a very narrow bandwidth. This property makes them ideal tools for MRI, which similarly builds on narrow bandwidth RF radiation. It was shown previously that MTMs can be used to significantly improve the SNR without the need of stronger background fields24,25,26,27,28,29,30,31.
However, passive structures as those presented in the available literature will of course influence both, the transmit (Tx) as well as the receive (Rx) field. The influence during Tx corresponds to a local increase of power transmitted, posing a potential threat of tissue heating and high SAR values. Moreover, obviously the Tx-field will be disturbed leading to a range of unwanted effects and subtle consequences. Thus, it is utterly important to carefully design, test, and use MTMs in MRI applications as patient safety must be assured at all times in research as well as clinical routine. To do so, dynamical MTMs can be developed, which are (de-)tuned depending on the imaging phase, i.e., discrete or continuous resonance states are to be introduced. This is indeed not trivial due to the MRI scanner's strong magnetic field environment, which limits electromagnetic signal communication close to the machine and causes some known control mechanisms for MTMs not to work. So far, only very few tunable MRI MTMs have been presented in general, and there are even less proposals that achieve the tuning depending on the imaging phase without the need of active intervention32,33. Non-linear MTMs32 are introduced as a promising solution since they can be sensitive to the imaging phase but also MRI-compatible additional battery-powered electronic sensing circuits33 can be used to take care of the state-switching. However, in all known cases the suggested MTMs are bulky structures (such as helix-shaped configurations or 3d arrangements of conducting bars), putting patient-specific design, patient comfort, and flexible or conformal manufacturing benefits beyond reach. The thickness of several centimeters also limits possible imaging applications. To overcome these limitations, we developed smart and ultra-thin metasurface enhancement plates (EPs) for MRI. Here, "smart" refers to automatic self-detuning in the Tx phase of the imaging. The functionality was designed and tested via simulations and verified in on-bench experiments and MRI scans at \(3\,\)T. Depending on the imaging sequence parameters, the SNR is increased up to eightfold (by a conservative measure) in slices close to the metasurface. A generalization to different field strengths is easily possible, such that EPs can be constructed for, e.g., \(1.5\,\)T or \(7\,\)T scanners.
Note that the MTMs (in general and those presented here in particular) do not act as MRI coils. Rather their effect is to locally redistribute (focus) the Rx field (or Tx field in other applications) such that in combination with the scanner's Tx and Rx coils the SNR can be enhanced, leading to increased imaging efficiency. The smart metasurfaces here are designed to be used in combination with the MRI scanner's integrated body coil. Thus, they allow to build a kind of effective, local, universal (referred to different imaging applications with the same metasurface), and wireless Rx coil, which is more cost-effective and simpler w.r.t. dedicated Rx coils. MTMs which work in combination with other (local) Rx coils can also be designed but care needs to be taken concerning the interaction of close resonant structures. However, as resonant, conducting structures are involved in the construction, the design of MTMs for use in MRI applications may adopt and modify conventional Rx/Tx coil methodologies and well-known geometries.
Metasurface design paradigm and (de-)tuning mechanism
In the following, we present the final results of our iterative metamaterial design loop for smart metasurfaces. To avoid any additional control circuitry components and batteries, we follow a design paradigm for non-linear MTMs32, such that each of our smart metasurface EPs consists of a non-linear system of two inductively coupled and resonant substructures. One of these is a linear 2d MTM (linear metasurface) and the other one is a non-linear, single-loop tuning resonator; see Figs. 1d, 2a and the supplementary material for a visual impression. In such an arrangement, resonant hybrid modes34 are determined by the properties of each subsystem and the coupling mechanism. The fundamental mode is of primary interest for MRI applications due to the largest region of interest (ROI) that can be covered. Our design has the advantage that instead of manipulating every unit cell of the linear MTM separately, only the tuning resonator needs to be controlled to vary the resonance of the full EP. Details on a theoretical description of a similar arrangement via the coupled mode theory have been published before32 and we outline the most important steps in the methods section; see also the supplementary material and figures therein. Details on the design parameters for the EPs are outlined in the methods section as well.
On-bench- characterization of the metasurfaces. (a) Simulation of the first two hybrid eigenmodes (normalized H-field component orthogonal to the metasurface in arbitrary units). (b) Measurement of the effective S21 parameter (magnitude) on the central axis as a function of distance and input power at 123.5 MHz with a 3-axis scanning stage; see the supplementary material. (c) Enhancement factor curves for selected input power levels from (b). (d) Symmetric on-bench setup with untuned sniffer coils that leads to the scattering parameters S11 and S21 shown in (e) and (f) as a function of incident excitation power.
Orientation of the metasurface in MRI scans. (a) 3D view of the setup, the photo shows the arrangement in the MRI scanning room on the patient table. (b) Setup as seen from an axial plane. The SNR monitoring line and the two regions of interest are indicated. Between the metasurface and the phantom, a Rohacell layer of thickness d = 1.5 cm is used. (c) Sagittal view of the orientation and recorded slices. The phantom was positioned in the isocenter. Note that for structural images with the Kiwi fruit, the slices are parallel to the enhancement plate.
Following the design methodology, each of our EPs is conceptually composed of (i) a linear metasurface resonator consisting of capacitively-loaded and coupled flat-wire unit cells, and (ii) an outer single-loop tuning resonator, which is milled on the same dielectric substrate and tightly covers the inner structure. The substrate is less than \(0.6\,\)mm thick with only \(17\,\mu\)m of copper cladding, see the methods section for details. The tuning resonator is loaded with semi-conductor elements that are sensitive to the incident power level via the induced current in the loop. In the Tx phase of the MRI sequence, substantially higher power is incident as compared to the Rx phase, in which the signal is caused by the relaxation of the excited magnetization. Hence, the resonance behavior of the tuning resonator is sensitive to the imaging phase and, thus, also the hybrid modes of the full EP are influenced by the incident power. The result is a smart, non-linear metasurface of which the resonance properties change between Tx and Rx without any additional need for user influence, control circuitry, batteries, or active (de-)tuning.
The resonance at low incident power is made to coincide with the MRI scanner's Larmor frequency (123.5 MHz here, see the methods section) by the design of the metasurface such that the EP is active and resonant in the Rx phase while it is sufficiently detuned (almost invisible) in Tx. For the first version (EP1), varactor diodes are soldered in series into the tuning resonator, while in the second case (EP2) an MRI compatible limiter diode is used, see Fig. 3e. To provide the possibility of manual fine-tuning of the resonance frequency at low incident power levels, a small trimmer capacitor was soldered into the tuning resonator in either case. It allows to tune and characterize the smart metasurface on-bench in presence of a phantom before using it in the MRI scanner. For the inner metasurface, the wire-resonator unit cells are loaded with capacitors, which are implemented as simple rectangular patches on a ground plane (on the back layer, see the figures in the supplementary material). In this way, the intensive use of lumped elements can be avoided to increase the Q-factor. In a next step, the manufacturing on flexible substrates for patient-specific and/or conformal design for many different applications becomes possible due to the compactness of the metasurfaces, the structural design that avoids lumped elements, and laser milling on thin sub-millimeter substrates. This is a big advantage over existing (bulky) solutions that use helix-structures or 3d arrangements of wire-shaped conducting bars with a thickness of several centimeters in total. Note that our metasurface resonator itself resembles properties of an open low-pass birdcage coil with the leg capacitors being a series of the two structural parallel-plate ones at the wire ends. The metasurface design was successively iterated based on simulations and measurement results, which are presented in the next sections.
MRI results for homogeneous phantom scans. The SNR in slice# 4 is shown for a flip angle of \(70\,\)deg (approx. Ernst angle) with Rx by (a) the SLC, (b) the body coil, (c) body coil + EP1, and (d) body coil + EP2. The SNR in the two ROIs is indicated, respectively. (e) Photo of the two EPs. The insets show the elements that are responsible for detuning in Tx. f) The normalized SNR enhancement (w.r.t. the body coil) in slice# 4 on a central vertical line, see Fig. 2. At a certain distance, the EPs outperform the SLC. Either EP leads to a sixfold increase in SNR close to the surface, which falls off almost linearly in contrast to the SLC's exponential behavior.
Simulation, and on-bench characterization results
For the \(B_1\) field enhancement in MRI, we are interested in the lowest-order resonant hybrid mode of the EP. Eigenmode simulations were conducted to study the resonant modes of the substructures at first separately and then as a coupled system. The lowest-order coupled modes involving the fundamental mode of the inner, linear metasurface are two: one in which the fields generated by the substructures are in phase (mode of interest), whereas in the other one they are out of phase, see Fig. 1a. In the next step, a plane wave excitation was used for full-wave simulations of the entire structure in presence of a cylindrical phantom to assess the characteristics and to feedback the design loop.
After successive design cycles, prototypes are manufactured for on-bench measurements of (i) the resonance frequency as a function of the incident power, and (ii) the quality factor to assure the desired functionality. A typical setup composed of shielded sniffer coils connected to a vector network analyzer was used, see Fig. 1d and the supplementary material. For S11 measurements at different incident power levels, the sniffer coil is positioned some distance above the center of the EP. The self-detuning of the smart metasurface is clearly observable at increasing incident power, see Fig. 1e. The same setup is used to fine-tune the EP at the target frequency at low incident power by variation of the trimmable capacitor. For the quality factor we find \({Q = 110}\) for EP1 and \({Q = 103}\) for EP2. In Fig. 1f, the detuning of the EP as a function of the incident power is shown in terms of S21 measurements with the sniffer coils positioned at a distance of \(300\,\)mm from each-other and the EP in the center. This detuning can be observed as a shift towards lower frequencies of the maximum of S21 magnitude or as the drop of the S21 magnitude at a fixed target frequency. In addition, a custom-made 3-axis scanning stage was used to further characterize the spatial dependence of the S-parameter response as a function of incident power as shown in Fig. 1b,c; see also the supplementary material for the setup. In the non-detuning regime (below about \(-15\,\)dBm input power), we observe an almost exponential decay of the field enhancement in the direction orthogonal to the EP. Depending on the input power, enhancement factors of about 10 can be observed close to the surface in free space, see Fig. 1.
Magnetic resonance images and validation
MRI measurements prove the expected functionality of the smart metasurfaces. A gradient echo sequence is used with the following parameters: \({TR = 100\,}\)ms, \({TE = 5\,}\)ms, field-of-view = \(\{ 64\times 64, 128 \times 128 \}\,\) \(\hbox {mm}^2\), matrix size = \({128\times 128}\), bandwidth = \(250\,\)Hz/Px, eight slices with thickness of \(5\,\)mm and distance of \(15\,\)mm. The imaging is done with a homogeneous cylindrical phantom (with both, EP1 and EP2) and with a Kiwi fruit for structural images (with EP1). For high SNR images of the Kiwi, also measurements with increased \({TR = 1\,}\)s are performed. The scanner's body coil is used for Tx throughout all experiments and also for Rx in combination with the smart metasurfaces EP1 and EP2. For comparison, imaging is also done with a local single loop coil (SLC) for Rx while the EP is removed. Besides the integrated body coil of the MRI scanner, a SLC might be considered a standard to which a new MTM device should be compared in a first step.
All measurements are performed with a single sequence implemented into the vendor-independent gammaSTAR framework (Fraunhofer MEVIS)35,36 with dedicated pauses between measurements to avoid frequency adjustments between successive experiments with/without the metasurfaces, see also the methods section. To investigate the de-tuning behavior and to separate effects on the Tx and Rx fields, 10 measurements with equidistant nominal flip angles between 0 and 90 degrees are recorded to determine the Ernst angle, i.e., the flip angle leading to a maximal SNR. This approach allows to genuinely characterize MTMs in MRI applications and to clearly differentiate between Tx and Rx modifications of the \(B_1\) field, see the methods section, but this method is often missing in existing literature. Reference measurements are performed without any metasurface in either case using only the body coil. The geometrical setup containing the phantom and the smart metasurface is shown in Fig. 2.
The noise for all SNR evaluations is taken from the respective \(0\,\)deg flip angle measurement, which is a better and more conservative measure than taking the noise from apparently "signal-free" areas in the actual images. The results in Figs. 3 and 4 prove that the detuning mechanism works well for both EPs. The Ernst angle is the same for measurements with and without the smart metasurfaces, see Fig. 4d and the supplementary material. Hence, in the Tx phase of the imaging sequence, the EPs remain silent and do not affect the \(B_1\) field. In the Rx phase, however, they lead to a sixfold increase in SNR for homogeneous phantom measurements with \({TR = 100\,}\)ms and to an almost eightfold SNR increase for structural measurements with \({TR = 1\,}\)s, see Figs. 3, 4, 5, and 6. The homogeneous phantom results in Fig. 3 show that both EPs work approximately equally well. From Fig. 3f, one can see that after a certain penetration depth, roughly the diameter of the SLC, the metasurfaces yield larger SNR improvements and outperform the local coil. Moreover, the profile for the SLC decreases almost exponentially, whereas the enhancement factor due to the smart metasurfaces seems to decrease only linearly. This behavior is confirmed in simulations and offers yet another advantage compared to local coils and existing MTM solutions.
MRI results for structural images with EP1 using a kiwi fruit and \({TR = 100\,}\)ms. The SNR is shown for slice# 5 and Rx by (a) the SLC, (b) the body coil, and (c) body coil + smart metasurface (EP1). Note that the slices are parallel to the metasurface. The SNR in the ROI is indicated, respectively. (d) SNR in the ROI as a function of the flip angle in slice# 5. The Ernst angle is not shifted in presence of the metasurface. Hence, the SNR increase is only due to influence on the Rx field. (e) The SNR as a function of distance from the SLC / EP for the measurement with \({TR = 1\,}\)s and flip angle of \(75\,\)deg (approx. Ernst angle). At a certain distance, the metasurface outperforms the SLC. Compared to the body coil, the metasurface leads to an eightfold increase in the SNR close to the surface.
Additional MRI results for structural images with the Kiwi fruit and \({TR = 100\,}\)ms. The plots show the comparison of results for the body coil (top) and the body coil + smart metasurface (bottom) in each panel. The flip angle is the approx. Ernst angle in this configuration. The respective SNR in the ROI is indicated in the subfigures.
Additional MRI results for structural images with the Kiwi fruit and \({TR = 1\,}\)s. The plots show the comparison of results for the body coil (top) and the body coil + smart metasurface (bottom) in each panel. The flip angle is the approx. Ernst angle in this configuration. The respective SNR in the ROI is indicated in the subfigures.
The metasurfaces introduce local focussing effects that may be compared in some sense to effects of a lens. However, note that our metasurface design differs from what is usually called "metamaterial lens". The smart metasurface's functionality might be best described in terms of a local resonant structure (composed of an array of wire-shaped unit cells) exited by the Rx field, which inductively couples to the body coil resonator of the MRI-scanner. In this case, we have near-field coupling in contrast to far-field coupling by, e.g., a lens. Then, the incident field induces a current in the single unit cells (Faraday's law of induction) and, in turn, the currents flowing in the unit cells induce magnetic fields superimposed on the incident one. The respective currents depend on the material, the mutual coupling and the geometry - these parts we can influence by the metasurface design. The full enhancement plate has multiple resonant modes, of which the ground mode is of major interest here. It also depends, as shown by the theoretical description, on the coupling to the non-linear outer tuning loop. Compared to the existing solutions for detunable MRI MTMs32,33, our smart metasurfaces exhibit a larger penetration depth and slower decrease of the SNR enhancement factor while maintaining a similar or even better performance. It is a subtle problem though to make a quantitative comparison as it depends on, e.g., the size of the respective MTMs, the imaged object, and the MRI sequence. From looking at SNR enhancement only, other authors claim an improvement factor of about 1532. However, for this result a small \(2\times 1\) configuration with (just) two helix-shaped unit cells is used. Of course, this smaller configuration can yield higher local SNR effects very close to the metamaterial but these are rapidly dropping as distance increases. Thus, the enhancement is at the expense of a larger field-of-view. As the field-of-view in MRI (for most applications) is larger than this \(2\times 1\) configuration, a larger MTM structure - such as the one presented here - is needed accordingly. In the supplementary material of the paper, the authors show results for a \(4\times 4\) configuration of their MTM, which is pretty bulky in size; see Figure S3 in their supplementary material. For this configuration, which compares roughly to our thin metasurface in the planar dimensions, a SNR enhancement of about 6–7 is reported but also a different repetition time of TR = \(3\,\)s is used for the MRI sequence. Note that our smart metasurfaces yield a similar or even better enhancement with TR = \(\{100,1000\}\,\)ms.
Our results clearly show that the SNR in MRI can be substantially improved with smart yet thin MTMs. Safety concerns are taken care of by the automatic Tx detuning and the sub-millimeter structures set the foundation for the possibility of flexible and conformal manufacturing. In some cases, the metasurfaces in combination with the scanner's body coil even outperform a local SLC. Future work devoted to the improvement and construction of even better and advanced MTMs will also address a detailed comparison to local Rx coils and the potential use of MTMs as their (partial) replacement in imaging applications. If the same image quality can be achieved without local coils, which are wired to the scanner's interface, metasurfaces have the potential to improve patient comfort and function effectively as "universal wireless receive coils".
Simulation, metasurface design, and theoretical modeling
All numerical electromagnetic simulations were performed using CST Microwave Studio (Dassault Systèmes, France) with a frequency domain solver. Two different types of simulations, eigenmode analysis and full plane wave excitations, are employed. The overall design requirement is to have the full smart metasurface resonant in the Rx phase of the imaging process at the MRI scanner's Larmor frequency, which is 123.5 MHz in our case (\(3\,\)T Magnetom Skyra, Siemens Healthineers, Germany). To achieve this goal, the wire-resonator metasurface can be varied in its overall dimension and the length of substructures as well as the number of wire resonators can be adjusted. Changing the size of single wire resonators, the MTM unit cells, allows to shift the resonance frequency. Furthermore, the variation of the end parts, the parallel plate capacitors, allows to vary the resonance and coupling of the unit cells. Since the full structure gives rise to hybrid eigenmodes, also the coupling and the properties of the outer, non-linear tuning resonator change the overall resonance frequency. Hence, parameter studies and optimization were performed to find the best possible combination. For the two manufactured EPs, we use \({N = 14}\) wire-resonator unit cells, which are \(10\,\)mm wide each, with printed wires of \(1\,\)mm width. The capacitive end patches have the same size for the two EPs, \({9\times 3\,}\) \(\hbox {mm}^2\), see Fig. 1 and the supplementary material. The length of an individual unit cell is \(180\,\)mm. Each EP consists of two subsystems: the inner, linear metasurface and the outer, non-linear tuning resonator. These two subsystems were designed to individually resonate at slightly higher frequencies, respectively, so that when inductively coupled together they would resonate at the target frequency of the MRI system.
For the simulations, semiconductor components (varactor diodes for EP1 and the limiter diode for EP2), are modeled as fixed-value capacitors (lumped elements) with different discrete values for Tx and Rx, respectively. Full wave simulations are performed with open boundary conditions in all directions and plane wave excitations with circular polarization. The propagation direction of the plane wave is in the plane of the metasurface along the symmetry axis orthogonal to the wire resonators. A homogeneous phantom of which the properties such as density, permittivity, and conductivity resemble average values for tissue (\(\epsilon _r = 78\), \(\sigma = 0.7 \,\)S/m) was included to simulate the loading. Evaluations of the numerical results show that the H-field at the target frequency inside the phantom is stronger in the resonant case (corresponding to the Rx phase of MRI) and almost unaffected in the detuned case (corresponding to the Tx phase of the MRI sequence). Eigenmode simulations indicate that two different hybrid modes exist that involve the fundamental mode of the linear metasurface, one of which has the two subsystems in phase, see Fig. 1. For the other mode, the two are exactly out of phase but the homogeneity is improved in the central part. For MRI, the first mode is of primary interest due to the larger region of interest covered.
An approximate theoretical description of the smart metasurfaces can be obtained using the coupled mode theory as outlined in the available literature32,37. For the linear part of the system, the inner metasurface structure, the mode amplitude \(y_1\) depends on an excitation x according to
$$\begin{aligned} {\dot{y}}_1 = a_1(\omega _1) \, y_1 + b_1 \, x \, , \quad a_1(\omega _1) = i \omega _1 - 1/\tau _r - 1/\tau _i \, , \quad b_1 = \sqrt{2/\tau _r} \, . \end{aligned}$$
Here, \(\omega _1\) is the resonance frequency, \(\tau _r\) is the decay time due to radiation losses, and \(\tau _i\) is the decay time associated with intrinsic losses. The overdot denotes the time derivative. A similar equation can be constructed for the tuning resonator,
$$\begin{aligned} {\dot{y}}_2 = a_2(\omega _2) \, y_2 + b_2 \, x \, , \end{aligned}$$
but now the resonance frequency \(\omega _2\) depends on the resonator's amplitude as well, \(\omega _2 = \omega _2(y_2)\). We may assume a linear relation such that \(\omega _2 = \Omega _2 + \kappa |y_2|\) with a constant \(\kappa\) that depends on the properties of the chosen non-linear element and a small amplitude limit \(\Omega _2\). Finally, the full metasurface is described by the coupled, non-linear system of differential equations in the time domain,
$$\begin{aligned} {\dot{y}}_1&= \big ( i \omega _1 - 1/\tau _{r,1} - 1/\tau _{i,1} \big ) y_1 + \sqrt{2/\tau _{r,1}} \, x + i k y_2 \, , \end{aligned}$$
(3a)
$$\begin{aligned} {\dot{y}}_2&= \big ( i \Omega _2 + i\kappa |y_2| - 1/\tau _{r,2} - 1/\tau _{i,2} \big ) y_2 + \sqrt{2/\tau _{r,2}} \, x + i k y_1 \, , \end{aligned}$$
(3b)
Which includes the coupling strength k. In the supplementary material details on the solution are outlined and it is shown how to qualitatively compare the reflection of the system to the S11-measurements we performed with an untuned sniffer coil, see Fig. 1e.
Manufacturing, on-bench measurements and MRI scans
The metasurfaces are manufactured at the Fraunhofer Institute for High Frequency Physics and Radar Techniques (FHR) by laser milling on Rogers 4003c substrate with a \(17\, \mu\)m copper layer on both sides. The tuning resonator loop used for automatic detuning during the Tx phase is milled on the same layer as the capacitively-loaded wires. The substrate's thickness is \(0.508\,\)mm. For EP1, the varactor diode-loaded tuning loop has three readily available Skyworks SMV2020 varactor diodes in series to have it resonate at a suitable frequency while keeping the geometrical dimensions adjusted to the inner substructure. The safe use of these varactor diodes in a 3 T experiment was previously reported32. However, varactor diodes connected in series give rise to (unwanted) non-linear effects38,39. For EP2, the limiter diode-loaded loop is of the same size as the varactor-loaded loop of EP1 but uses the UMX9989AP diode (Microsemi, USA). For low incident power, this limiter diode has a typical capacitance of about \(4\,\)pF. For high incident power, the limiter diode conducts and, thus, on/off states are realized as extreme cases depending on the incident power level, i.e., depending on the MRI imaging phase. In the two extreme states, the resonance frequency of the metasurface is different, with the low incident power state's resonance frequency matched to the MRI scanner's Larmor frequency. On the back layer of the EPs, the ground stripes for the capacitively coupled wire resonators of the inner metasurface are milled, see the supplementary material for an overview of front and back layers of the two manufactured prototypes.
The characterization of the prototypes is performed on-bench with a symmetric coil setup and with a dedicated 3-axis scanning stage, respectively, see Fig. 1 and the supplementary material. Fine-tuning is achieved via the trimmable capacitor in the tuning resonator, which has a range of \(2\,\)pF to \(6\,\)pF (Sprague-Goodman SGC3S060). The on-bench measurements are performed with two untuned \(60\,\)mm diameter sniffer coils and a vector network analyzer (N5242A, Keysight Technologies, USA), see Fig. 1 and the supplementary material. The scattering S-parameters are measured as functions of the incident power to verify the desired de-tuning effect for high field strengths. The Q-factor of the metasurfaces was determined from S21 measurements via the width at the \(3\,\)dB decay from the maximum. For EP1 it is \({Q = 110}\), and for EP2 we have \({Q = 103}\). Hence, EP1 performs slightly better in on-bench performance tests. However, the detuning for high incident power is smoother for EP2. Note that the UMX diodes have been specifically designed for MRI applications. Slight differences in the performance can also be due to non-perfect fine-tuning and positioning accuracy but it has to be noted that both designs fulfill their purpose perfectly as they detune with increasing incident power. The spatial dependence of the S-parameter distribution, as depicted in Fig. 1b and c, is determined with a workshop-built setup including a high-precision 3-axis scanning stage (SF600, GAMPT, Germany), see the supplementary material. This setup consists of a low-bandpass, custom-build, trimmable Tx coil with a diameter of \(30\,\)cm and a small Rx coil with a diameter of \(2.5\,\)cm, both of which are connected to a 2-Port vector network analyzer (E5061B ENA Series Network Analyzer, Keysight Technologies, USA). In this setup, the metasurface is mounted on a pylon of polystyrene at a distance of about \(60\,\)cm to the Tx coil. The effective scattering parameter S21, as depicted in Fig. 1b, is measured on the central axis behind the metasurface and in planes parallel to the surface to characterize the enhancement factor vs. distance and the field homogeneity, respectively. The effective values (magnitudes) are obtained by subtracting (on a log scale) the background signal, i.e., the data from measurements without any metamaterial. The deduced enhancement factor is on the order of magnitude of the enhancement seen in the MRI scans.
For the MRI scans, the scanner's body coil is used for Tx and also for Rx in combination with the smart metasurfaces, respectively. For comparison, we also use a \(70\,\)mm SLC for Rx in absence of any EP. Such a local coil constitutes a Rx standard to which MRI-compatible MTMs should be compared in a first step beyond the scanner's body coil. However, such a comparison is often missing in the existing literature. When the SLC is used, it is placed at the position of the removed EP to allow for a fair comparison. All experiments are performed with a single sequence that offers sufficient time to place, exchange, or remove the metasurfaces between successive acquisitions without allowing the scanner to do any new adjustments in the meantime. In this way the genuine comparison and characterization is possible. We use gradient echo sequences with the following parameters (if not stated otherwise in figure captions): \(TR = \{100 ms,\, 1 s\}\), \(TE = 5\,\)ms, field-of-view = \(\{ 64\times 64, 128 \times 128 \}\,\) \(\hbox {mm}^2\), matrix size = \(128 \times 128\), bandwidth = \(250\,\)Hz/Px, eight slices with thickness of \(5\,\)mm and distance of \(15\,\)mm. These sequences are implemented into the unique, vendor-independent gammaSTAR framework (Fraunhofer MEVIS)35,36, which offers exceptional flexibility in sequence design and is optimally suited for MTM characterization as shown here. It also allows, in principle, to repeat our analysis with MRI systems from different vendors. Reducing the scanner's internal image scaling factor is important. Even with the huge SNR enhancement factor, due to the presence of the smart metasurface, artificial image saturation should be avoided. We performed measurements with a homogeneous cylindrical phantom, see Figs. 2 and 3, and also with a Kiwi fruit for structural images, see Figs. 4 and 5. The supplementary material contains additional MRI results.
To prove that the smart detuning works in the Tx phase, imaging was performed for 10 different flip angles with equidistant spacing, ranging from \(0\,\)deg to \(90\,\)deg. In the supplementary material we show additional results (to those in Figs. 4 and 5), which relate the SNR enhancement in different ROIs for the homogenous and structural phantom measurements to the nominal flip angle. As can be seen in all these plots, the presence of a smart metasurface does not affect the position of the maximal SNR, i.e., the position of the flip angle that leads to a maximal SNR (Ernst angle). Hence, the SNR improvement due to the smart metasurface is related to effects on the Rx field only. If the Ernst angle for any MTM in MRI measurements was shifted (as compared to body coil or SLC measurements), the MTM influences also the Tx field, which poses a potential threat to the patient. Uncontrolled modifications of the Tx field by MTMs are to be avoided since also all scanner-integrated safety mechanisms are not adapted to this situation.
The definition of the noise obviously influences the SNR enhancement that the metasurfaces yield in comparison to the body coil measurements. To characterize its effect, we evaluated the SNR performance for five different noise definitions/regions. The most reliable and conceptually acceptable noise measure is to calculate the standard deviation of the 'signal' in a large region of the \(0\,\)deg flip angle measurements. This also proves to be the most conservative measure in our consideration. For comparison, we also calculated the noise as the standard deviation in the apparently signal-free areas in the edges of each image. The results are presented in the supplementary material.
For the structural images with the Kiwi fruit and \({TR = 1\,}\)s, the SNR enhancement factor due to the smart metasurface is about eight in nearby slices, see Fig. 4 and the supplementary material. For the measurements with \({TR = 100\,}\)ms, the enhancement factor is a bit less, see Figs. 4 and 5. However, especially in this case the influence of the metasurface is most prominent as the fruit can barely be recognized in the plots without its presence but becomes clearly visible with internal structure with a smart metasurface being used.
Note that EP2 performs slightly better in MRI experiments, see Fig. 3. The performance difference can be attributed to the Tx-detuning strategy via varactor and limiter diodes, respectively, but also positioning accuracy after exchange of the metasurface introduces possible differences. The UMX limiter diodes have been specifically designed for MRI experiments but they are more expensive though.
As can be seen in Fig. 4, the SNR enhancement due to the metasurface tends to fall off almost linearly as compared to the faster drop-off behavior of the SLC. This effect is also supported by simulation data, which shows that close to the surface, the enhancement can be best fitted by a linear decay. However, the measurements with the 3-axis scanning stage show an almost exponential decay of the enhancement vs. distance in the normal direction to the surface, see Fig. 1. This difference can be explained by (i) the absence of a phantom in the lab measurements and (ii) the different normalization of the enhancement. Whilst for the MRI scans, the enhancement is calculated w.r.t. the body coil SNR, which is almost constant for the homogeneous phantom scans, the lab measurements yield the enhancement normalized to an empty scan, which itself has a distance-dependent behavior related to the H-field of the Tx coil.
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This work was supported by the Fraunhofer MAVO project MetaRF (Grant No. MAVO 142-600555). The authors want to thank Peter Erhard for discussions and comments and Michael Jäger for support in the production of prototypes.
Open Access funding enabled and organized by Projekt DEAL.
Fraunhofer FHR, Fraunhoferstraße 20, 53343, Wachtberg, Germany
Endri Stoja, Diego Betancourt & Thomas Bertuch
Fraunhofer MEVIS, Max-von-Laue-Str. 2, 28359, Bremen, Germany
Simon Konstandin, Dennis Philipp, Robin N. Wilke, Jürgen Jenne, Reiner Umathum & Matthias Günther
Division of Medical Physics in Radiology, German Cancer Research Center DKFZ, Im Neuenheimer Feld 280, 69120, Heidelberg, Germany
Jürgen Jenne & Reiner Umathum
MR-Imaging and Spectroscopy, Faculty 01, University of Bremen, Otto-Hahn-Allee 1, 28359, Bremen, Germany
Matthias Günther
Endri Stoja
Simon Konstandin
Dennis Philipp
Robin N. Wilke
Diego Betancourt
Thomas Bertuch
Jürgen Jenne
Reiner Umathum
E.S. was responsible for the design and simulation of the metasurface EPs and performed on-bench characterizations. T.B., R.U. and J.J. suggested components and advised the design process. D.B. assisted with simulations and on-bench measurements. S.K. and M.G. drafted and performed the MRI measurements. S.K. developed and implemented the sequence into the gammaSTAR framework and evaluated measurement data. R.N.W. performed on-bench S-parameter measurements and developed the 3-axis S-parameter measurement setup. D.P. assisted with simulations and metasurface design, evaluated the MRI measurements, drafted the paper, and produced the figures. M.G. and T.B. initialized the research project. All authors were involved in the discussion and interpretation of the results presented and all contributed to writing the final paper.
Correspondence to Dennis Philipp.
Supplementary material 1 (pdf 6642 KB)
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Stoja, E., Konstandin, S., Philipp, D. et al. Improving magnetic resonance imaging with smart and thin metasurfaces. Sci Rep 11, 16179 (2021). https://doi.org/10.1038/s41598-021-95420-w
DOI: https://doi.org/10.1038/s41598-021-95420-w | CommonCrawl |
\begin{document}
\title{Stability and Rate of Convergence of\\ the Steiner Symmetrization} \date{} \author{D.I. Florentin, A. Segal} \maketitle \begin{abstract} We present a direct analytic method towards an estimate for the rate of convergence (to the Euclidean Ball) of Steiner symmetrizations. To this end we present a modified version of a known stability property of the Steiner symmetrization. \end{abstract}
\section{Introduction and results}\label{Sec_Intro}
Let $(\mathbb R^n, \iprod{\cdot}{\cdot})$ be some fixed Euclidean structure, and let $\mathcal{K}^n$ be the class of all compact convex sets in $\mathbb R^n$. Denote by $D_n$ the Euclidean unit ball, by $S^{n-1}$ its boundary and by $\kappa_n=|D_n|$ its Lebesgue measure. Fix a direction $u \in S^{n-1}$ and denote its orthogonal hyperplane by $H=\{x\in \mathbb R^n: \iprod{x}{u}=0\}$. Obviously, each point $x \in \mathbb R^n$ can be uniquely decomposed as $x=y + tu$ where $y \in H$ and $t \in \mathbb R$. The Steiner symmetral of a set $K$ with respect to $u$ is defined to be \[
S_u(K) = \left\{(y,t)\,:\, K \cap (y+\mathbb R u) \ne \emptyset,\quad |t|\le
\frac{|K \cap (y + \mathbb R u)|}{2}\right\}. \] The Steiner symmetrization has several important properties. For one, it reduces the surface area while preserving volume. Clearly, this process makes the set more ``round'' in some sense, so one would expect that applying mutliple Steiner symmetrizations is a process that converges to the Euclidean ball - the only fixed point of this operation. It was shown by Gross \cite{Gross} that for each convex set there exists a sequence of symmetrizations that converges in the Hausdorff metric to a ball with the same volume. This result was improved by Mani-Levitska \cite{Mani} where it was shown that a random sequence of Steiner symmetrizations applied to a convex set, converges almost surely to a ball. However, these proofs do not provide results regarding the rate of convergence. The first estimate of the rate is due to Hadwiger \cite{Hadwiger}, who showed that $\left(c\frac{\sqrt{n}}{\varepsilon^2}\right)^n$ symmetrizations are enough to transform a convex set to a new set with Hausdorff distance at most $\varepsilon$ from the Euclidean ball. Later, Bourgain, Lindenstrauss and Milman \cite{BLM} proved an isomorphic result, stating that in order to reach some fixed distance from the Euclidean Ball, roughly $n\log n$ symmetrizations suffice. In recent years this bound was reduced to $3n$ by Klartag and Milman \cite{KM_Isomorph-Stein}. Klartag \cite{K_Isomet-Mink+Stein} also improved the isometric result of Hadwiger, showing that the rate of convergene is almost exponential. More precisely: \begin{thm}[Klartag]\label{thm-Klartag-Convergence}
Let $K\in\mathcal{K}^n$ be a convex body with $|K|=|D_n|$, and let $\varepsilon \in (0,\frac{1}{2})$. There exist $Cn^4(\log\varepsilon)^2$ Steiner symmetrization transforming $K$ into a body $K'$ satisfying \[ (1-\varepsilon)D_n \subset K \subset (1+\varepsilon)D_n. \] \end{thm}
In \cite{K_Isomet-Mink+Stein} Klartag first provided a bound of
$Cn|\log \varepsilon |$ steps on the convergence rate when applying the {\em Minkowski symmetrization} $M_u K$, a linear operation on the support function, by means of controlling the decay of the non-constant spherical harmonics of the support function. The proof of Theorem \ref{thm-Klartag-Convergence} consists mainly of the bound for Minkowski symmetrizations, together with the inclusion $S_u K\subseteq M_u K$. A byproduct of this approximation is that the bound for Steiner symmetrizations is polynomial in the dimension $n$ rather than linear. It is conjectured that the correct dependence is indeed linear, as in the case of Minkowski symmetrizations. The goal of this paper is to provide a direct estimate for the convergence rate of Steiner symmetrizations in the Nikodym pseudo metric, defined in Section \ref{Sec-Stability}. It may be formulated as follows, where $A\Delta B$ is the {\em symmetric difference} of the sets $A$ and $B$. \begin{thm}\label{Thm_Main}
Let $K\in \mathcal{K}^n$ be a convex body with $|K|=|D_n|$ and let $\varepsilon \in(0,1)$. There exist $c\frac{n^{13}\log^3 n}{\varepsilon^\gamma}$
Steiner symmetrizations transforming $K$ into a body $K'$ satisfying \[
\frac{|K' \Delta D_n|}{|D_n|} < \varepsilon, \] where $\gamma={4+\frac{2}{\log n}}$. \end{thm} Obviously, Theorem \ref{Thm_Main} provides a non optimal bound (for example, by equivalence of the Hausdorff and Nikodym metrics, one can derive a better bound from Theorem \ref{thm-Klartag-Convergence}). However, the polynomial bound presented in this proof is obtained using a self contained, direct analysis of Steiner symmetrization, which may lead to similar results in the case of non convex sets, where there are no estimates analogous to Theorem \ref{thm-Klartag-Convergence}. The main ingredient of our proof is a quantitative estimate regarding the change in surface area under a Steiner symmetrization. It is a well known fact that surface area decreases under a Steiner symmetrization. However, a quantitative version of this statement was only recently provided, by Barchiesi, Cagnetti and Fusco \cite{BCF}. Their statement contains factors which are exponential in the dimension and have a direct effect on the estimate of the convergence rate. In Section \ref{Sec-Stability} we provide a slightly different version with an improved dependence on the dimension. To this end we require a Poincar\'{e} type inequality for convex domains, which we obtain in the following section.
\section{Poincar\'{e} type inequalities for convex domains}\label{Sec_Poincare} We wish to establish a weighted Poincar\'{e} type inequality for convex domains. We denote by $\rho: K\to \mathbb R^+$ the distance to the boundary of $K$, that is \[
\rho(x)=\min_{y\in \partial K} \{ |x-y| \}. \]
Our main result in this section is the following theorem: \begin{thm}\label{Thm-OurPoincare} Let $n\ge 2$ and let $K \in \mathcal{K}^n$ be such that $rD_n \subseteq K \subseteq R D_n$. If $b: K \to \mathbb R$ is a bounded function with mean zero with respect to $\rho$ (i.e. $\int_K b\rho=0$), then for every $\lambda\in (2,\infty)$ one has \[
\int_K |b| \le C
\left(\frac{||b||_\infty|K|}{\beta} \right)^{1-\frac{1}{\lambda}}
\cdot \left(n\frac{R}{r}\int_K |{\nabla} b|\rho\right)^{\frac{1}{\lambda}}, \] where $\beta=\frac{\lambda-2}{\lambda-1}\in (0,1)$. \end{thm}
We collect a few technical lemmas before proving Theorem \ref{Thm-OurPoincare}. \begin{lem}\label{Lem_Switch-Vol-Surf.Area} Let $n\ge 1$ and let $K\in\mathcal{K}^n$ be such that $rD_n \subseteq K \subseteq R D_n$. Then \[ \frac{n}{R} \le
\frac{|\partial K|}{|K|} \le \frac{n}{r} .\] \end{lem} \begin{proof} By the definition we have: \[
|\partial K| =
\lim_{t\to 0} \frac{ |K + t D_n| - |K| }{t} \le
\lim_{t\to 0} \frac{ |K + \frac{t}{r} K| - |K| }{t} =
\frac{n |K|}{r} .\] Replacing $D_n$ with $K/R$ in the above limit yields the other direction. \end{proof}
\begin{lem}\label{Lem_Bound-1-over-Rho} Let $n\ge 1$, $K\in\mathcal{K}^n$. For every $\beta\in (0,1)$ we have \[ I_{\beta} = \int_K \frac{1}{\rho^{1-\beta}} <
\frac{C n^{1-\beta}|K|}{\beta r^{1-\beta}},\] where $r$ is the inner radius of $K$, and $C$ is some positive constant. \end{lem}
\begin{proof} First, recall the {\em Beta function} defined for positive $x$ and $y$ by \[ B(x,y) = \int_0^1 t^{x-1}(1-t)^{y-1}dt.\]
The function $\rho$ is bounded (from above and below) by the $K$-distance-to-the-boundary function $\rho_K(x)=\min_{y\in \partial K}
\{ ||x-y||_K \}$ $= 1 - ||x||_K$, whose corresponding integral is easily estimated. Indeed, if $rD_n \subseteq K \subseteq RD_n$ then \begin{equation*}
\frac{1}{R}|x| \le ||x||_K \le \frac{1}{r}|x|. \end{equation*} In particular we get a lower bound on $\rho$: \begin{equation}\label{eq-rho_bound}
\rho(x) = \min_{y\in \partial K}|x-y| \ge r \min_{y\in\partial K}||x-y||_K = r \rho_K(x) . \end{equation} By Fubini's theorem: \begin{eqnarray*} \int_K \frac{1}{\rho_K(x)^{1-\beta}} &=&
\int_0^\infty \left|\left\{x \in K : \rho_K^{-(1-\beta)}(x) > t\right\}\right|dt \\ \\ &=&
\int_0^1 |K| dt + \int_1^\infty \left|\left\{x \in K : \rho_K^{-(1-\beta)}(x) > t\right\}\right|dt \\ \\ &=&
|K|\left(1 + \int_1^\infty\left(1 - (1-t^{-\frac{1}{1-\beta}})^n \right) dt\right) \\ \\ &=&
|K|\left(1 + \int_0^1 (1-s^n)(1-s)^{\beta-2}(1-\beta)ds\right) \\ \\ &=&
n|K|\int_0^1(1-s)^{\beta-1}s^{n-1}ds = n|K|B(n,\beta) \\ \\ &=&
n|K|\frac{n+\beta}{\beta}B(n, 1+\beta) \\ \\ &=&
n|K|\left(\frac{n+\beta}{\beta}\right)\frac{\Gamma(n)\Gamma(1+\beta)}{\Gamma(1+n+\beta)} \\ \\
&<&\frac{2 n^2 |K| \Gamma(n)}{\beta \Gamma(1+n+\beta)} <
\frac{C n^{1-\beta} |K|}{\beta}, \end{eqnarray*} for some $C > 0$, since $\Gamma(n)n^{1+\beta}< C_1\Gamma(1+n+\beta)$. Therefore, by \eqref{eq-rho_bound} we conclude that \begin{eqnarray*} I_{\beta} &=& \int_K \frac{1}{\rho^{1-\beta}} \leq \frac{1}{r^{1-\beta}}\int_K \frac{1}{\rho_K(x)^{1-\beta}}
< \frac{C n^{1-\beta}|K|}{\beta r^{1-\beta}}. \end{eqnarray*} \end{proof}
The last tool we require is the following weighted Poincar\'{e} type inequality, due to Chua and Wheeden (in fact, in \cite{ChW} they prove a more general result). \begin{thm}[Chua, Wheeden]\label{thm-PoincareChua} Let $K \in \mathcal{K}^n$ and let $f$ be a Lipschitz function. If \[ \int_K f\rho = 0, \] then \[
\int_K |f|\rho \leq C \text{diam}(K) \int_K |{\nabla} f|\rho, \] where $C>0$ is some universal constant. \end{thm}
We turn now to prove the main result of this section. \begin{proof}[{\bf Proof of Theorem \ref{Thm-OurPoincare}.}] Let $\lambda > 2$. By the H\"{o}lder inequality we have \begin{equation} \label{eq-Reverse_Holder}
\int_K|b|^{\frac{1}{\lambda}}\le
\left(\int_K |b|\rho\right)^{\frac{1}{\lambda}} \cdot \left(\int_K \rho^{\frac{1}{1-\lambda}}\right)^{1-\frac{1}{\lambda}}
\end{equation} We write $\lambda = \frac{2-\beta}{1-\beta}$ for $\beta \in(0,1)$, so that $(\lambda-1)(1-\beta)=1$. By Lemma \ref{Lem_Bound-1-over-Rho}, \begin{eqnarray*} \left(\int_K \rho^{\frac{1}{1-\lambda}}\right)^{1-\frac{1}{\lambda}} &=& \left(\int_K \frac{1}{\rho^{1-\beta}}\right)^{1-\frac{1}{\lambda}} <
\left(\frac{C n^{1-\beta}|K|}{\beta r^{1-\beta}}\right)^{1-\frac{1}{\lambda}} \\ \\ &=& \left(\frac{n}{r}\right)^{\frac{1}{\lambda}}\cdot
\left(\frac{C|K|}{\beta} \right)^{1-\frac{1}{\lambda}} \end{eqnarray*} Combining the two estimates we get \[
\int_K |b| \le
||b||_\infty^{1-\frac{1}{\lambda}}\int_K|b|^{\frac{1}{\lambda}}\le
\left(\frac{C||b||_\infty|K|}{\beta} \right)^{1-\frac{1}{\lambda}}
\left(\frac{n}{r}\int_K |b|\rho\right)^{\frac{1}{\lambda}}. \] Since $\int_K b\rho = 0$, and $diam(K) \le 2R$, we may apply Theorem \ref{thm-PoincareChua} to obtain: \[
\int_K |b| \le
\left(\frac{C||b||_\infty|K|}{\beta} \right)^{1-\frac{1}{\lambda}}
\left(\frac{nR}{r}\int_K |{\nabla} b|\rho\right)^{\frac{1}{\lambda}}. \] \end{proof}
\section{Stability for the Steiner symmetrization}\label{Sec-Stability}
Let $K \in \mathcal{K}^n$ and $u \in S^{n-1}$. It is well known that the surface area $|\partial K|$ decreases under a Steiner symmetrization, but until recently this phenomenon was not quantified. Barchiesi, Cagnetti and Fusco showed in \cite{BCF} that for a convex body $K$ satisfying
$rD_n \subseteq K \subseteq RD_n$, the following holds \begin{equation} \label{eq-FuscoStability} A(K, S_u K) \leq n4^{n+1}\left(\frac{R}{r}\right)^{2n}\sqrt{\delta_u(K)}, \end{equation} where the {\em surface area deficit} $\delta_u$ is defined by \[
\delta_u(K) = 1 - \frac{|\partial S_u K|}{|\partial K|}, \] and the {\em Nikodym pseudo metric} $A$ is defined by \[
A(K, T) = \inf_{x_0 \in \mathbb R^n}\frac{|(rK) \Delta (x_0 + T))|}{|T|}, \]
for $r^n=\frac{|T|}{|K|}$. In this section we show that the dependence on the dimension and the quantity $\frac{R}{r}$ in (\ref{eq-FuscoStability}) can be reduced to polynomial at the cost of slightly worsening the exponent of $\delta_u(K)$ (i.e. decreasing it below $1/2$). More precisely:
\begin{thm}\label{thm-SteinerStability} Let $n\ge 2$ and let $K \in \mathcal{K}^n$ such that $rD_n \subseteq K \subseteq R D_n$. Then for every $\lambda\in (2,\infty)$ and $u\in S^{n-1}$ we have: \[ A(K,S_u K)\le C \left(\frac{1}{\beta} \right)^{1-\frac{1}{\lambda}}
\left( n\frac{R}{r} \right)^{1+\frac{1}{\lambda}}
\delta_u(K)^{\frac{1}{2\lambda}}, \] where $\beta=\frac{\lambda-2}{\lambda-1}\in (0,1)$, and $C$ is some constant. \end{thm}
The proof of Theorem \ref{thm-SteinerStability} follows the methods of \cite{BCF}, combined with Theorem \ref{Thm-OurPoincare}. Denote the orthogonal projection of $K$ to $u^\perp$ by $P = Proj_{u^\perp}(K)$. For each $x \in P$, we consider the ``fiber above $x$ in $K$'', namely
$K \cap (x+\mathbb R u)$. We denote its length by $L(x) = |K \cap (x+\mathbb R u)|$ and its barycenter by $b(x)$.
By Brunn's principle, $L$ is concave. The following lemma gives a local upper bound for the gradient of a concave function, in terms of the distance from the boundary of its domain.
\begin{lem}\label{Lem_Bound_grad_L} Let $P \in \mathcal{K}^n$, and denote by $\rho(x)=dist(x, \partial P)$ the distance to $\partial P$. If $L:P\to\mathbb R$ is a concave function with oscillation $\Delta L:=\sup\{L\}-\inf\{L\}$, then \[
|{\nabla} L(y)| \leq \frac{\Delta L}{\rho(y)}. \] \end{lem} \begin{proof} Let $y\in P$, and consider $x = y - \rho(y)v \in P$, where
$v = \frac{{\nabla} L(y)}{|{\nabla} L(y)|}$. Then \[ L(y) - L(x) = \int_0^{\rho(y)} \frac{\partial L}{\partial v}(x + tv)dt \ge \int_0^{\rho(y)}\frac{\partial L}{\partial v}(y)dt =
\rho(y)|{\nabla} L(y)|, \]
by concavity. Therefore $|{\nabla} L(y)|\le \frac{\Delta L}{\rho(y)}$, as required. \end{proof}
\begin{proof}[{\bf Proof of Theorem \ref{thm-SteinerStability}.}] As before, denote $P = Proj_{u^\perp}(K)= Proj_{u^\perp}(S_u K)$ and $\rho(x) = dist(x, \partial P)$. The fiber $K \cap (x+\mathbb R u)$ has endpoints with heights $b\pm \frac{L}{2}$, thus: \begin{eqnarray*}
|\partial K| - |\partial S_u K| &=&
\int_P \left(\sqrt{1+\left|{\nabla} b + \frac{{\nabla} L}{2}\right|^2} + \sqrt{1+\left|{\nabla} b - \frac{{\nabla} L}{2}\right|^2} - 2\sqrt{1 + \left|\frac{{\nabla} L}{2}\right|^2} \right) \\ \\ &=& \int_P \frac{N}{D}\ge \left(\int_P N^\frac{1}{2}\right)^2 \left(\int_P D\right)^{-1}, \end{eqnarray*} where \begin{eqnarray*} N &=&
\frac{1}{2}\left(\sqrt{1+\left|{\nabla} b + \frac{{\nabla} L}{2}\right|^2} + \sqrt{1+\left|{\nabla} b - \frac{{\nabla} L}{2}\right|^2}\right)^2 -
\frac{1}{2}\left(2\sqrt{1 + \left|\frac{{\nabla} L}{2}\right|^2} \right)^2 \\ \\ &=&
\sqrt{\left(1 + \frac{1}{4}|{\nabla} L|^2 + |{\nabla} b|^2 \right)^2 - \iprod{{\nabla} L}{{\nabla} b}^2} - \left(1 + \frac{1}{4}|{\nabla} L|^2 - |{\nabla} b|^2\right) \\ \\ &=&
\frac{ 4\left(1+\frac{1}{4}|{\nabla} L|^2 \right)|{\nabla} b|^2 - \iprod{{\nabla} b}{{\nabla} L}^2 } {\sqrt{\left(1 + \frac{1}{4}|{\nabla} L|^2 + |{\nabla} b|^2 \right)^2 - \iprod{{\nabla} L}{{\nabla} b}^2} + \left(1 + \frac{1}{4}|{\nabla} L|^2 - |{\nabla} b|^2\right) } \\ \\
&\geq& \frac{4|{\nabla} b|^2}{\sqrt{\left(1 + \frac{1}{4}|{\nabla} L|^2 + |{\nabla} b|^2 \right)^2 - \iprod{{\nabla} L}{{\nabla} b}^2} + \left(1 + \frac{1}{4}|{\nabla} L|^2 - |{\nabla} b|^2\right)}
\equiv \frac{4|{\nabla} b|^2}{Q}, \end{eqnarray*} and \[ D = \frac{1}{2}\left(
\sqrt{1+\left|{\nabla} b + \frac{1}{2}{\nabla} L\right|^2} + \sqrt{1+\left|{\nabla} b - \frac{1}{2}{\nabla} L\right|^2} \right)
+ \sqrt{1 + \frac{1}{4}|{\nabla} L|^2}, \]
so that $\int_P D = \frac{|\partial K| + |\partial S_u K|}{2} \le
|\partial K|$. Thus: \begin{equation}\label{eq-surface_deficit_est}
\delta_u(K) = \frac{|\partial K| - |\partial S_u K|}{|\partial K|}\ge
\left(\frac{1}{|\partial K|} \int_P \sqrt{N}\right)^2. \end{equation} Next, we bound $N$ from below. Since $\sqrt{a^2-x^2} \leq a-\frac{x^2}{2a}$ we have
\begin{eqnarray*}
Q&=&\sqrt{\left(1 + \frac{1}{4}|{\nabla} L|^2 + |{\nabla} b|^2 \right)^2 - \iprod{{\nabla} L}{{\nabla} b}^2} + \left(1 + \frac{1}{4}|{\nabla} L|^2 - |{\nabla} b|^2\right) \\ \\
&\leq & 2 + \frac{1}{2}|{\nabla} L|^2 - \frac{\iprod{{\nabla} b}{{\nabla} L}^2}{2 + \frac{1}{2}|{\nabla} L|^2 + 2|{\nabla} b|^2} \leq 2 + \frac{1}{2}|{\nabla} L|^2 \\ \\
&\leq& 2\frac{R^2}{\rho^2} + \frac{1}{2}|{\nabla} L|^2 \leq \frac{4R^2}{\rho^2}, \end{eqnarray*} where the last inequality is due to Lemma \ref{Lem_Bound_grad_L} (here $\Delta L\le 2 R$). Therefore \[
\sqrt{N} \ge \frac{|{\nabla} b|\rho}{R}. \] Plugging this back to \eqref{eq-surface_deficit_est}, we may bound the surface area deficit: \begin{equation}\label{Eq_Surface-Deficit} \sqrt{\delta_u(K)} \ge
\frac{1}{R|\partial K|} \int_P |{\nabla} b| \rho. \end{equation}
In order to bound the Nikodym pseudo metric by the integral of the barycenter, we first note that since $K\subseteq R D_n$, we have
$|b|\le R$. Moreover, $K$ may be shifted parallel to $u$, so without loss of generality we may assume $\int_P b\rho=0$. This shift cannot exceed $R$, thus the new barycenter is bounded by $2R$. We have: \begin{equation}\label{Eq_Volume-Deficit}
A(K, S_u K) \le \frac{|K \Delta S_u K|}{|K|} \le \frac{1}{|K|}\int_P |b|, \end{equation} where the second inequality is due to the fact that
$| [-\frac{L}{2},\frac{L}{2}] \Delta
[b-\frac{L}{2},b+\frac{L}{2}] |\le |b|$. Since we assumed that $\int_P b\rho=0$, we may apply Theorem \ref{Thm-OurPoincare} to get
(recall $||b||_\infty\le 2R$):
\begin{eqnarray*} A(K, S_u K) &\le&
\frac{1}{|K|}
\left(\frac{C||b||_\infty|P|}{\beta} \right)^{1-\frac{1}{\lambda}}
\left(\frac{nR}{r}\int_P |{\nabla} b|\rho\right)^{\frac{1}{\lambda}}
\\ \\ &\le&
\frac{1}{|K|}
\left(\frac{2CR|P|}{\beta} \right)^{1-\frac{1}{\lambda}}
\left(\frac{nR^2|\partial K|}{r} \sqrt{\delta_u(K)}\right)^{\frac{1}{\lambda}}
\\ \\ &=&
\frac{|\partial K|^{\frac{1}{\lambda}}}{|K|}
\left(\frac{2C|P|}{\beta} \right)^{1-\frac{1}{\lambda}}
\frac{n^\frac{1}{\lambda}R^{1+\frac{1}{\lambda}}}{r^\frac{1}{\lambda}}
\delta_u(K)^{\frac{1}{2\lambda}}
\\ \\ &\le&
\frac{|\partial K|}{|K|}
\left(\frac{C}{\beta} \right)^{1-\frac{1}{\lambda}}
\frac{n^\frac{1}{\lambda}R^{1+\frac{1}{\lambda}}}{r^\frac{1}{\lambda}}
\delta_u(K)^{\frac{1}{2\lambda}}
\\ \\ &\le&
\left(\frac{C}{\beta} \right)^{1-\frac{1}{\lambda}}
\left( n\frac{R}{r} \right)^{1+\frac{1}{\lambda}}
\delta_u(K)^{\frac{1}{2\lambda}}.
\end{eqnarray*}
The last two inequalities hold since $P$ is a $n-1$ dimensional set contained in $S_u K$, thus $2|P| \le |\partial S_u K| \le
|\partial K|$. Moreover, by Lemma \ref{Lem_Switch-Vol-Surf.Area}, $\frac{|\partial K|}{|K|}\le \frac{n}{r}$. \end{proof}
\section{Rate of convergence}
In this section we prove Theorem \ref{Thm_Main}. The proof is based on the following idea. Assume that $|K|=|D_n|$. Due to Theorem \ref{thm-SteinerStability}, as long as one can find a direction $u$
for which $A(K, R_uK)$ is not very small, there exists a Steiner symmetrization which reduces the surface area of $K$ by a factor. Since the surface area cannot drop below $n|D_n|$ (isoperimetric inequality), the number of such operations is bounded. Next, one has to show that if $A(K, R_uK)$ is small in every direction, then so is $A(K,D_n)$. Let us formulate this last statemnt precisely before proving the main theorem.
\begin{lem}\label{lem-CloseToReflections} Let $K\subset \mathbb R^n$ be a compact star shaped body and let $\varepsilon >0$. Denote by $R_u$ the reflection with respect to $u^\perp$. If $A(K, R_uK)<\varepsilon$ for all $u\in S^{n-1}$, then $A(K, D_n) < 4n\varepsilon$. \end{lem} \begin{proof} First note that $A(K, R_{u_m}\dots R_{u_1}K) < m\varepsilon$ for any $m \le n$. The proof goes by induction, where the case $m=1$ is assumed to hold. For $m \ge 2$ one has \begin{eqnarray*} A(K, R_{u_m}\ldots R_{u_1}K) &=& A(R_{u_m}K, R_{u_{m-1}}\ldots R_{u_1}K) \\
&\le& A(R_{u_m}K, K) + A(K, R_{u_{m-1}}\ldots R_{u_1}K) \\ &<& \varepsilon + (m-1)\varepsilon = m\varepsilon. \end{eqnarray*} Every isometry $u\in O(n)$ is generated by at most $n$ reflections, thus $A(K, uK) < n\varepsilon$. This may be written as follows, in terms of the radial function $\rho$ of $K$: \begin{equation}\label{Eq_SymmDiffIsometry}
A(K, uK) = \frac{|K \Delta uK|}{|K|} = \frac{|D_n|}{|K|}
\int_{S^{n-1}} |\rho(x)^n - \rho(ux)^n| d\sigma(x) < n\varepsilon, \end{equation}
where $\sigma$ is the normalized Haar measure on the sphere. Without loss of generality, assume from now on that $|K|=|D_n|$. Note that if $u$ is selected at random with respect to the Haar measure on $SO(n)$, then for every $x\in S^{n-1}$, the point $ux$ is distributed uniformly on $S^{n-1}$. Thus averaging (\ref{Eq_SymmDiffIsometry}) over $u\in SO(n)$ yields: \begin{equation}\label{Eq_SymmDiffBall}
\int_{S^{n-1}}
\int_{S^{n-1}} |\rho(x)^n - \rho(y)^n| d\sigma(x) d\sigma(y) < n\varepsilon. \end{equation} Consider the sets $A=\{x \in S^{n-1} : \rho(x) \ge 1\}$ and
$B=\{x \in S^{n-1} : \rho(x) \le 1\}$. Since $|K \setminus D_n| =
|D_n \setminus K|$, we have \begin{eqnarray*} \frac{1}{2}A(K, D_n) &=& \frac{1}{2}
\int_{S^{n-1}} |\rho(x)^n - 1| d\sigma(x) =
\int_A |\rho(x)^n - 1| d\sigma(x)\\ \\ &=& \frac{1}{\sigma(B)} \int_B \int_A
|\rho(x)^n - 1| d\sigma(x)d\sigma(y)\\ \\&\le& \frac{1}{\sigma(B)} \int_B \int_A
|\rho(x)^n - \rho(y)^n| d\sigma(x) d\sigma(y)\\ \\&\le& \frac{1}{\sigma(B)} \int_{S^{n-1}} \int_{S^{n-1}}
|\rho(x)^n-\rho(y)^n|d\sigma(x)d\sigma(y). \end{eqnarray*} This implies $A(K,D_n) < \frac{2n\varepsilon}{\sigma(B)}$ by \eqref{Eq_SymmDiffBall}, and similarly one has $A(K,D_n) < \frac{2n\varepsilon}{\sigma(A)}$, so combining the two we get \[ A(K, D_n) < 4n\varepsilon. \] \end{proof}
\begin{proof}[{\bf Proof of Theorem \ref{Thm_Main}.}]
Assume without loss of generality that $|K| = |D_n|$. Apply $n$ Steiner symmetrizations to $K$ with respect to some orthogonal basis to obtain a new convex body $K_0$ which is unconditional, and in particular centrally symmetric. By John's theorem, there exists an ellipsoid $\mathcal{E}$ such that \[ \mathcal{E} \subset K_0 \subset \sqrt{n}\mathcal{E}. \] There exist $n$ Steiner symmetrizations which transform $\mathcal{E}$ to an Euclidean ball (see \cite{KM_Isomorph-Stein}, Lemma 2.6). Applying these symmetrizations to $K_0$, we obtain a body $K_1$ satisfying \[ r_1D_n \subset K_1 \subset \sqrt{n}r_1D_n, \] for some $r_1 > 0$. Thus the inner and outer radii of $K_1$ satisfy
$\frac{R}{r} \le \sqrt{n}$. Note that $|D_n| = |K_1| \le |\sqrt{n}
rD_n|$ which implies that $\frac{1}{r}\le \sqrt{n}$. Hence, by Lemma \ref{Lem_Switch-Vol-Surf.Area} \[
|\partial K_1| \leq \frac{n}{r}|K_1| \le n^{3/2}|D_n|. \] Fix $\varepsilon_0 > 0$. If there exists $u_1 \in S^{n-1}$ with $A(K_1, S_{u_1}K_1) > \varepsilon_0$, denote $K_2 = S_{u_1} K_1$. By Theorem \ref{thm-SteinerStability}, combined with the bound $\frac{R}{r} \le \sqrt{n}$, we get \begin{eqnarray*} \label{eq-SurfAreaReduction}
n|D_n| = |\partial D_n| \le |\partial K_2| &\le&
|\partial K_1| \left(1 - \frac{A(K_1, S_{u_1} K_1)^{2\lambda}}{n^{3(\lambda+1)} }\left(\frac{\beta}{C}\right)^{2(\lambda-1)}
\right) \\ \\ &\le &
|\partial K_1| \left( 1 - \frac{\varepsilon_0^{2\lambda}}{n^{3(\lambda+1)} } \left(\frac{\beta}{C}\right)^{2(\lambda-1)} \right). \end{eqnarray*} If there exists $u_2\in S^{n-1}$ with $A(K_2, S_{u_2}K_2) > \varepsilon_0$, denote by $K_3 = S_{u_2}K_2$. Continue this process for $m$ steps. Then $K_{m+1}$ satisfies \begin{eqnarray*}
n|D_n| \le |\partial K_{m+1}| &\le& |\partial K_1|\left( 1 - \left(\frac{\beta}{C}\right)^{2(\lambda-1)} \frac{\varepsilon_0^{2\lambda}}{n^{3(\lambda+1)}} \right)^m \\ \\ &\le &
n^{3/2}|D_n|\left( 1 - \left(\frac{\beta}{C}\right)^{2(\lambda-1)} \frac{\varepsilon_0^{2\lambda}}{n^{3(\lambda+1)} } \right)^m. \end{eqnarray*} Thus, \[ 0 \leq \frac{1}{2}\log n + m \log \left( 1 - \left(\frac{\beta}{C}\right)^{2(\lambda-1)} \frac{\varepsilon_0^{2\lambda}}{n^{3(\lambda+1)} } \right). \] Set $\lambda = 2 + \frac{1}{\log n}$, so $\beta=\frac{1}{1 +\log n}$. Hence, the number of such steps is bounded by \begin{equation}\label{Eq_mBound} m \le \left(C(1 +\log n)\right)^{2 + \frac{2}{\log n}}\log n \left( \frac{n^{9 + \frac{3}{\log n}}}{\varepsilon_0^{4 + \frac{2}{\log n}}} \right) < c \left( \frac{ n^9 \log^3n}{\varepsilon_0^{4 + \frac{2}{\log n}}} \right), \end{equation} for some $c>0$. The resulting body $K'$ thus satisfies $A(K', S_u K') < \varepsilon_0$ for all $u \in S^{n-1}$, which in turn implies that $A(K', R_u K') < 2\varepsilon_0$ for all $u \in S^{n-1}$, where $R_u K_m$ is the reflection of $K_m$ with respect to $u^\perp$. By Lemma \ref{lem-CloseToReflections} we conclude that \[ A(K_m, D_n) < 8n\varepsilon_0. \] Let $\varepsilon>0$. Plugging $\varepsilon_0 = \varepsilon/(8n)$ into \eqref{Eq_mBound} completes the proof. \end{proof}
\begin{rem}{\rm The dependence in the dimension $n$ in Theorem \ref{Thm_Main} is clearly not optimal (as mentioned before, the sharp bound is believed to be linear). For example, the bound for the ratio $R/r$ may be reduced to a constant, rather than $\sqrt{n}$, which results in decreasing the power $13$ to $10$. This may be done by one of the isomrphic results mentioned in the introduction. }\end{rem}
\noindent Dan Itzhak Florentin, [email protected]\\ \noindent Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel.\\
\noindent Alexander Segal, [email protected]\\ \noindent School of Mathematical Science, Tel Aviv University, Tel Aviv, Israel.
\end{document} | arXiv |
\begin{document}
\title{The lifespan of classical solutions
of one dimensional wave equations
with semilinear terms\
of the spatial derivative}
\begin{abstract} This paper is devoted to the lifespan estimates of small classical solutions of the initial value problems for one dimensional wave equations with semilinear terms of the spatial derivative of the unknown function. It is natural that the result is same as the one for semilinear terms of the time-derivative. But there are so many differences among their proofs. Moreover, it is meaningful to study this problem in the sense that it may help us to investigate its blow-up boundary in the near future. \end{abstract}
\section{Introduction} In this paper, we consider the initial value problems; \begin{equation} \label{initial_value_problem} \left\{ \begin{array}{ll}
u_{tt}-u_{xx}=|u_x|^p
&\mbox{in}\quad \mathbb R\times(0,T),\\
u(x,0)=\varepsilon f(x),\ u_t(x,0)=\varepsilon g(x),
& x\in\mathbb R, \end{array} \right. \end{equation} where $p>1$, and $T>0$. We assume that $f$ and $g$ are given smooth functions of compact support and a parameter $\varepsilon>0$ is \lq\lq small enough". We are interested in the lifespan $T(\varepsilon)$, the maximal existence time, of classical solutions of (\ref{initial_value_problem}). Our result is that there exists positive constants $C_1,C_2$ independent of $\varepsilon$ such that $T(\varepsilon)$ satisfies \begin{equation} \label{result} C_1\varepsilon^{-(p-1)}\le T(\varepsilon) \le C_2\varepsilon^{-(p-1)}. \end{equation}
We note that, even if $|u_x|^p$ is replaced with $|u_t|^p$, (\ref{result}) still holds. Such a result is due to Zhou \cite{Zhou01} for the upper bound of $T(\varepsilon)$, and Kitamura, Morisawa and Takamura \cite{KMT23} for the lower bound of $T(\varepsilon)$.
\par As model equations to ensure the optimality of the general theory for nonlinear wave equations by Li, Yu and Zhou \cite{LYZ91, LYZ92}, the nonlinear term $|u_t|^p$ is sufficient to be studied except for the \lq\lq combined effect" case. See Morisawa, Sasaki and Takamura \cite{MST} and Kido, Sasaki, Takamatsu and Takamura \cite{KSTT}
for this direction with a possibility to improve the general theory. But it is quite meaningful to deal with also $|u_x|^p$
because their proofs are technically different from each others. Moreover, there is no result on its blow-up boundary due to lack of the monotonicity of the solution, while the one for $|u_t|^p$ is well-studied by Sasaki \cite{Sasaki18,Sasaki21}, and Ishiwata and Sasaki \cite{IS20a,IS20b}. See Remark \ref{rem:non-definite} below. It is also remarkable that it can be studied if the nonlinear term has a special form of both $u_t$ and $u_x$. See Sasaki \cite{Sasaki22} for this direction. Our research may help us to study the blow-up boundary for the equation in (\ref{initial_value_problem}) near future. \par This paper is organized as follows. In the next section, the preliminaries are introduced. Moreover, (\ref{result}) are divided into two theorems. Section 3 is devoted to the proof of the existence part, the lower bound of $T(\varepsilon)$, of (\ref{result}). The main strategy is the iteration method for the system of integral equations for $(u,u_x)$, which is essentially due to Kitamura, Morisawa and Takamura \cite{KMT23}. They employed it for the system of integral equations for $(u,u_t)$
to construct a classical solution of the wave equation with nonlinear term $|u_t|^p$, which is originally introduced by John \cite{John79}. In the section 4, following Rammaha \cite{Rammaha95, Rammaha97}. we prove the blow-up part, the upper bound of $T(\varepsilon)$, of (\ref{result}). We note that the method to reduced to $u$-closed integral inequality by Zhou \cite{Zhou01} for the nonlinear term $|u_t|^p$ cannot be applicable to (\ref{initial_value_problem}) because a time delay appears in the reduced ordinary differential inequality. Rammaha \cite{Rammaha95, Rammaha97} overcomes this difficulty by employing weighted functionals along with the characteristic direction in studyng two or three dimensional wave equations with nonlinear terms of spatial derivatives.
\section{Preliminaries and main results} Throughout this paper, we assume that the initial data $(f,g)\in C_0^2(\mathbb R)\times C^1_0(\mathbb R)$ satisfies \begin{equation} \label{supp_initial}
\mbox{\rm supp }f,\ \mbox{supp }g\subset\{x\in\mathbb R: |x|\le R\},\quad R\geq1. \end{equation} Let $u$ be a classical solution of (\ref{initial_value_problem}) in the time interval $[0,T]$. Then the support condition of the initial data, (\ref{supp_initial}), implies that \begin{equation} \label{support_sol}
\mbox{supp}\ u(x,t)\subset\{(x,t)\in\mathbb R\times[0,T]:|x|\leq t+R\}. \end{equation} For example, see Appendix of John \cite{John_book} for this fact.
\par It is well-known that $u$ satisfies the integral equation; \begin{equation} \label{u}
u(x,t)=\varepsilon u^0(x,t)+L(|u_x|^p)(x,t), \end{equation} where $u^0$ is a solution of the free wave equation with the same initial data, \begin{equation} \label{u^0} u^0(x,t):=\frac{1}{2}\{f(x+t)+f(x-t)\}+\frac{1}{2}\int_{x-t}^{x+t}g(y)dy, \end{equation} and a linear integral operator $L$ for a function $v=v(x,t)$ in Duhamel's term is defined by \begin{equation} \label{nonlinear} L(v)(x,t):=\frac{1}{2}\int_0^tds\int_{x-t+s}^{x+t-s}v(y,s)dy. \end{equation} Then, one can apply the time-derivative to (\ref{u}) to obtain \begin{equation} \label{u_t}
u_t(x,t)=\varepsilon u_t^0(x,t)+L'(|u_x|^p)(x,t) \end{equation} and \begin{equation} \label{u^0_t} u_t^0(x,t)=\frac{1}{2}\{f'(x+t)-f'(x-t)+g(x+t)+g(x-t)\}, \end{equation} where $L'$ for a function $v=v(x,t)$ is defined by \begin{equation} \label{nonlinear_derivative} L'(v)(x,t):=\frac{1}{2}\int_0^t\{v(x+t-s,s)+v(x-t+s,s)\}ds. \end{equation} Therefore, $u_t$ is expressed by $u_x$. On the other hand, applying the space-derivative to (\ref{u}), we have \begin{equation} \label{u_x}
u_x(x,t)=\varepsilon u_x^0(x,t)+\overline{L'}(|u_x|^p)(x,t) \end{equation} and \begin{equation} \label{u^0_x} u_x^0(x,t)=\frac{1}{2}\{f'(x+t)+f'(x-t)+g(x+t)-g(x-t)\}, \end{equation} where $ \overline{L'} $ for a function $v=v(x,t)$ is defined by \begin{equation} \label{nonlinear_derivative_conjugate} \overline{L'}(v)(x,t):= \frac{1}{2}\int_0^t\{v(x+t-s,s)-v(x-t+s,s)\}ds. \end{equation}
\begin{rem} \label{rem:non-definite} In view of (\ref{u_x}), it is almost impossible to obtain a point-wise positivity of $u_x$. This fact prevents us from studying its blow-up boundary as stated in Introduction. \end{rem}
Moreover, applying one more time-derivative to (\ref{u_x}) yields that \begin{equation} \label{u_xt}
u_{xt}(x,t)=\varepsilon u_{xt}^0(x,t)+{L'}(p|u_x|^{p-2}u_xu_{xt})(x,t) \end{equation} and \begin{equation} \label{u^0_xt} u_{xt}^0(x,t)=\frac{1}{2}\{f''(x+t)-f''(x-t)+g'(x+t)+g'(x-t)\}. \end{equation} Similarly, we have that \[
u_{tt}(x,t)=\varepsilon u^0_{tt} + |u_x|^p(x,t) + \overline{L'}(p|u_x|^{p-2}u_xu_{xt})(x,t). \] Therefore, $u_{tt}$ is expressed by $u_x,u_{xt}$ and so is $u_{xx}$.
\par First, we note the following fact.
\begin{prop} \label{prop:system} Assume that $(f,g)\in C_0^2(\mathbb R)\times C_0^1(\mathbb R)$. Let $w$ be a $C^1$ solution of (\ref{u_x}) in which $u_x$ is replaced with $w$. Then, \[ u(x,t):= \int^x_{-\infty}w(y,t)dy \] is a classical solution of (\ref{initial_value_problem}) in $\mathbb R\times[0,T]$. \end{prop} \par\noindent {\bf Proof.} This is easy along with the computations above in this section.
$\Box$
\vskip10pt \par Our results are divided into the following two theorems.
\begin{thm} \label{thm:lower-bound} Assume (\ref{supp_initial}). Then, there exists a positive constant $\varepsilon_1=\varepsilon_1(f,g,p,R)>0$ such that a classical solution $u\in C^2(\mathbb R\times[0,T])$ of (\ref{initial_value_problem}) exists as far as $T$ satisfies \begin{equation} \label{lower-bound} T \leq C_1\varepsilon^{-(p-1)} \end{equation} where $0<\varepsilon\leq\varepsilon_1$, and $C_1$ is a positive constant independent of $\varepsilon$. \end{thm}
\begin{thm} \label{thm:upper-bound} Assume (\ref{supp_initial}) and \begin{equation} \label{asummption1} f(x), g(x)\geq0,\ \mbox{and}\ f(x)\not\equiv0. \end{equation} Then, there exists a positive constant $\varepsilon_2=\varepsilon_2(f,p,R)>0$ such that any classical solution of (\ref{initial_value_problem}) in the time interval $[0,T]$ cannot exist as far as $T$ satisfies \begin{equation} \label{upperr-bound} T>C_2\varepsilon^{-(p-1)} \end{equation} where $0<\varepsilon\leq\varepsilon_2$, and $C_2$ is a positive constant independent of $\varepsilon$. \end{thm}
The proofs of above theorems are given in following sections.
\section{Proof of Theorem \ref{thm:lower-bound}} \par According to Proposition \ref{prop:system}, we shall construct a $C^1$ solution of (\ref{u_x}) in which $u_x=w$ is the unknown function. Let $\{w_j\}_{j\in \mathbb N}$ be a sequence of $C^1(\mathbb R\times[0,T])$ defined by \begin{equation} \label{w_j} \left\{ \begin{array}{l}
w_{j+1}=\varepsilon u_x^0+\overline{L'}(|w_j|^p), \\ w_1=\varepsilon u_x^0. \end{array} \right. \end{equation} Then, in view of (\ref{u_xt}), $(w_j)_t$ has to satisfy \begin{equation} \label{w_j_t} \left\{ \begin{array}{l}
(w_{j+1})_t=\varepsilon u_{xt}^0+{L'}(p|w_j|^{p-2}w_j(w_j)_t), \\ (w_1)_t=\varepsilon u_{xt}^0, \end{array} \right. \end{equation} so that the functional space in which $\{w_j\}$ converges is \[
X:=\{w\in C^1(\mathbb R\times[0,T]): \|w\|_X<\infty, \mbox{ supp }w \subset \{(x,t)\in\mathbb R\times[0,T] : |x|\leq t+R\}\} \] which is equipped with a norm \[
\|w\|_X:=\|w\|+\|w_t\| \] where \[
\|w\|:= \sup_{(x,t)\in\mathbb R\times[0,T]}|w(x,t)|. \] We note that (\ref{u_x}) implies that \[ \begin{array}{l}
\text{ supp } w_j\subset\{(x,t)\in\mathbb R\times[0,T] : |x|\leq t+R\}\\
\Longrightarrow\text{ supp } w_{j+1}\subset\{(x,t)\in\mathbb R\times[0,T] : |x|\leq t+R\}. \end{array} \] \par The following lemma provides us a priori estimate. \begin{prop} \label{prop:apriori}
Let $w\in C(\mathbb R\times[0,T])$ and supp\ $w\subset\{(x,t)\in\mathbb R\times[0,T]:|x|\leq t+R\}$. Then \begin{equation} \label{apriori}
\|L'(|w|^p)\|\leq C\|w\|^p(T+R) \end{equation} where $C$ is a positive constant independent of $T$ and $\varepsilon$. \end{prop}
\par\noindent {\bf Proof.} The proof of Proposition \ref{prop:apriori} is completely same as the one of Proposition 3.1 in Morisawa, Sasaki and Takamura \cite{MST}.
$\Box$
\vskip5pt \par Let us continue to prove Theorem \ref{thm:lower-bound}. Set \[
M:=\sum_{\alpha=0}^2\|f^{(\alpha)}\|_{L^\infty(\mathbb R)} + \sum_{\beta=0}^1\|g^{(\beta)}\|_{L^\infty(\mathbb R)}. \]
\vskip10pt \par\noindent {\bf The convergence of the sequence $\v{\{w_j\}}$.}
\par First we note that $\|w_1\| \leq M\varepsilon$ by (\ref{u^0_x}). (\ref{w_j}) and (\ref{apriori}) yield that \[
\|w_{j+1} \| \leq M \varepsilon + C \|w_j\|^p(T+R) \] because it is trivial that \[
|\overline{L'}(v)| \leq L'(|v|). \] Therefore the boundedness of $\{w_j\}$, i.e. \begin{equation} \label{bound_w}
\|w_j\| \leq 2M\varepsilon \quad(j\in\mathbb N), \end{equation} follows from \begin{equation} \label{condi1} C(2M\varepsilon)^p (T+R)\leq M\varepsilon. \end{equation}
Assuming (\ref{condi1}), one can estimate $\|w_{j+1}-w_j\|$ as follows. \[\begin{split}
\|w_{j+1}-w_j\|
& =\|\overline{L'}(|w_{j}|^p-|w_{j-1}|^p ) \| \leq \|{L'}(||w_{j}|^p-|w_{j-1}|^p |) \| \\
&\leq 2^{p-1}p\|{L'}((|w_{j}|^{p-1}+|w_{j-1}|^{p-1} )|w_{j}-w_{j-1}|) \| \\
&\leq 2^{p-1}pC(\|w_{j}\|^{p-1}+\|w_{j-1}\|^{p-1})(T+R)\|w_{j}-w_{j-1}\| \\
&\leq 2^{p}pC(2M\varepsilon)^{p-1}(T+R)\|w_{j}-w_{j-1}\|. \end{split}\] Therefore the convergence of $\{w_j\}$ follows from \[
\|w_{j+1}-w_j\| \leq \frac{1}{2}\|w_j-w_{j-1}\| \] provided (\ref{condi1}) and \begin{equation} \label{condi2} 2^{p}pC(2M\varepsilon)^{p-1}(T+R) \leq \frac{1}{2} \end{equation} are fulfilled.
\vskip10pt \par\noindent {\bf The convergence of the sequence $\v{\{(w_j)_t\}}$.}
\par First we note that $\|(w_1)_t\| \leq M\varepsilon$ by (\ref{u^0_xt}). Assume that (\ref{condi1}) and (\ref{condi2}) are fulfilled. Since (\ref{w_j_t}) and (\ref{apriori}) yield that \[\begin{split}
\|(w_{j+1})_t\|
&\leq M\varepsilon+ \|{L'}(p|w_j|^{p-2}w_j(w_j)_t)\| \\
&\leq M\varepsilon + \|{L'}(p|w_j|^{p-1}|(w_j)_t|)\| \\
&\leq M\varepsilon + pC\|w_j\|^{p-1}(T+R)\|(w_j)_t\| \\
&\leq M\varepsilon + pC(2M\varepsilon)^{p-1}(T+R)\|(w_j)_t\|, \\ \end{split}\] the boundness of $\{(w_j)_t\}$, i.e. \[
\|(w_j)_t\|\leq 2M\varepsilon, \] follows as long as it is fulfilled that \begin{equation} \label{condi3} pC(2M\varepsilon)^{p-1} (T+R)\leq 1. \end{equation} Assuming (\ref{condi3}), one can estimate $\{(w_{j+1})_t-(w_j)_t\}$ as follows. Noting that \[ \begin{split}
&||w_j|^{p-2}w_j-|w_{j-1}|^{p-2}w_{j-1}| \\ &\leq \left\{ \begin{array}{ll}
(p-1)2^{p-2}(|w_j|^{p-2}+|w_{j-1}|^{p-2})|w_j-w_{j-1}| & \mbox{for}\ p\ge2,\\
2|w_j-w_{j-1}|^{p-1} & \mbox{for}\ 1<p<2, \end{array} \right. \end{split} \] we have \[ \begin{array}{l}
\|(w_{j+1})_t-(w_j)_t\|
= \|L'(p|w_j|^{p-2}w_j(w_j)_t -p|w_{j-1}|^{p-2}w_{j-1}(w_{j-1})_t) \| \\
\leq p \|L'(|w_j|^{p-1} | (w_j)_t -(w_{j-1})_t |) \|
+ p\|L'(| |w_j|^{p-2}w_j -|w_{j-1}|^{p-2}w_{j-1}||(w_{j-1})_t | ) \| \\
\leq pC \|w_j\|^{p-1}(T+R) \|(w_j)_t -(w_{j-1})_t \| \\ \quad + \left\{ \begin{array}{ll}
L'(p(p-1)2^{p-2}(|w_j|^{p-2}+|w_{j-1}|^{p-2})|w_j-w_{j-1}||(w_{j-1})_t | & \mbox{for}\ p\geq2,\\
L'(2p|w_j-w_{j-1}|^{p-1}|(w_{j-1})_t |) & \mbox{for}\ 1<p<2 \end{array} \right. \\
\leq pC \|w_j\|^{p-1} (T+R) \|(w_j)_t -(w_{j-1})_t \| \\ \quad + \left\{ \begin{array}{ll}
p(p-1)2^{p-2}C(\|w_j\|^{p-2}+\|w_{j-1}\|^{p-2})\|w_j-w_{j-1}\|\|(w_{j-1})_t \|& \mbox{for}\ p\geq2,\\
2pC\|w_j-w_{j-1}\|^{p-1}\|(w_{j-1})_t \| & \mbox{for}\ 1<p<2 \end{array} \right. \\
\leq pC (2M\varepsilon)^{p-1} (T+R) \|(w_j)_t -(w_{j-1})_t \| + O\left(\displaystyle\frac{1}{2^{j\min(p-1,1)}}\right). \end{array} \] Therefore, we obtain the convergence of $\{(w_j)_t\}$ provided \begin{equation} \label{condi4} pC(2M\varepsilon)^{p-1} (T+R)\leq \frac{1}{2}. \end{equation}
\vskip10pt \par\noindent {\bf Continuation of the proof.} \par The convergence of the sequence $\{w_j\}$ to $w$ in the closed subspace of $X$
satisfying $\|w\|, \|w_t\| \leq 2M\varepsilon$ is established by (\ref{condi1}), (\ref{condi2}), (\ref{condi3}), and (\ref{condi4}), which follow from \[ 2^{p+1}pC(2M)^{p-1}\varepsilon^{p-1}(T+R) \leq 1. \] Therefore the statement of Theorem \ref{thm:lower-bound} is established with \[ \varepsilon_1:=(2^{p+2}pC(2M)^{p-1}R)^{-1/(p-1)},\quad C_1:= 2^{p+1}pC(2M)^{p-1} \] because $R\leq (2C_1)^{-1}\varepsilon^{-(p-1)}$ holds for $0<\varepsilon\leq\varepsilon_1$.
$\Box$
\section{Proof of Theorem \ref{thm:upper-bound}} \par Following Rammaha \cite{Rammaha95}, set \[ H(t):=\int^t_0(t-s)ds\int^{s+R}_{s+R_0}u(x,s)dx \] where $R_0$ is some fixed point with $0<R_0<R$. We may assume that there exists a point $x_0\in(R_0,R)$ such that $f(x_0)>0$ because of the assumption (\ref{asummption1}) and of a possible shift of $x$-variable.
\par Then it follows that \begin{equation} \label{H''} \begin{array}{ll} H''(t) &\displaystyle=\int^{t+R}_{t+R_0}u(x,s)dx\\ &\displaystyle=\frac{\varepsilon}{2}\int^{t+R}_{t+R_0} \left\{f(x+t)+f(x-t)+\frac{1}{2}\int_{x-t}^{x+t}g(y)dy\right\}dx+\frac{1}{2}F(t), \end{array} \end{equation} where \[
F(t):=\int^{t+R}_{t+R_0}dx\int_0^tds\int_{x-t+s}^{x+t-s}|u_x(y,s)|^pdy. \] By virtue of (\ref{asummption1}) and (\ref{H''}), we have that \[ H''(t)\geq \frac{\varepsilon}{2}\int^{t+R}_{t+R_0}f(x-t)dx \geq2C_f\varepsilon \] where \[ C_f:=\frac{1}{4}\int^R_{R_0}f(y)dy>0. \] Integrating this inequality in [$0,t$] twicely and noting that $H'(0)=H(0)=0$, we have \begin{equation} \label{esti_H} H(t)\geq C_f\varepsilon t^2. \end{equation}
\par On the other hand, $F(t)$ can be rewritten as \[
F(t)=\int_0^tds\int_{t+R_0}^{t+R}dx\int_{x-t+s}^{x+t-s}|u_x(y,s)|^pdy. \] From now on, we assume that \begin{equation} \label{R_1} t \geq R_1:=\frac{R-R_0}{2}>0. \end{equation} Then, inverting the order on $(y,x)$-integral, for $0 \leq s \leq t-R_1$, we have that \[ \begin{split}
&\int_{t+R_0}^{t+R}dx\int_{x-t+s}^{x+t-s}|u_x(y,s)|^pdy\\ &=\left(\int_{s+R_0}^{s+R}\int_{t+R_0}^{y+t-s} +\int_{s+R}^{2t-s+R_0}\int_{t+R_0}^{t+R}
+\int_{2t-s+R_0}^{2t-s+R}\int_{y-t+s}^{t+R}\right)|u_x(y,s)|^pdxdy\\
&\geq \int_{s+R_0}^{s+R}dy\int_{t+R_0}^{y+t-s}|u_x(y,s)|^pdx. \end{split} \] Similarly, for $t-R_1\leq s \leq t$, we also have that \[ \begin{split}
&\int_{t+R_0}^{t+R}dx\int_{x-t+s}^{x+t-s}|u_x(y,s)|^pdy\\ &=\left(\int_{s+R_0}^{2t-s+R_0}\int_{t+R_0}^{y+t-s} +\int_{2t-s+R_0}^{s+R}\int_{y-t+s}^{y+t-s}
+\int_{s+R}^{2t-s+R}\int_{y-t+s}^{t+R}\right)|u_x(y,s)|^pdxdy\\
&\geq \int_{s+R_0}^{2t-s+R_0}dy\int_{t+R_0}^{y+t-s}|u_x(y,s)|^pdx
+\int_{2t-s+R_0}^{s+R}dy\int_{y-t+s}^{y+t-s}|u_x(y,s)|^pdx. \end{split} \] Hence we obtain that \[ \begin{split} F(t)
&\geq \int_0^{t-R_1}ds\int_{s+R_0}^{s+R}(y-s-R_0)|u_x(y,s)|^pdy\\
&\ +\int_{t-R_1}^tds\int_{s+R_0}^{2t-s+R_0}(y-s-R_0)|u_x(y,s)|^pdy\\
&\ +\int_{t-R_1}^tds\int_{2t-s+R_0}^{s+R}2(t-s)|u_x(y,s)|^pdy. \end{split} \] Therefore it follows from (\ref{R_1}) and \[ 1=\frac{y-s-R_0}{y-s-R_0} \geq \frac{y-s-R_0}{R-R_0} \geq \frac{y-s-R_0}{2t} \] that \[ \begin{split} F(t)
&\geq \int_0^{t-R_1}\frac{t-s}{t}ds\int_{s+R_0}^{s+R}(y-s-R_0)|u_x(y,s)|^pdy\\
&\ +\int_{t-R_1}^t\frac{t-s}{t}ds\int_{s+R_0}^{2t-s+R_0}(y-s-R_0)|u_x(y,s)|^pdy\\
&\ +\int_{t-R_1}^t2(t-s)ds\int_{2t-s+R_0}^{s+R}\frac{y-s-R_0}{2t}|u_x(y,s)|^pdy\\
&= \frac{1}{t}\int_0^t(t-s)ds\int_{s+R_0}^{s+R}(y-s-R_0)|u_x(y,s)|^pdy.\\ \end{split} \] In this way, (\ref{asummption1}), (\ref{H''}) and the estimate of $F(t)$ above yield that \[
H''(t)\geq \frac{1}{2}F(t)\geq \frac{1}{2t}\int_0^t(t-s)ds\int_{s+R_0}^{s+R}(y-s-R_0)|u_x(y,s)|^pdy \quad\mbox{for}\ t\geq R_1. \] Moreover, it follows from (\ref{support_sol}), integration by parts and H\"older's inequality that \[ \begin{split}
|H(t)|
&=\left|\int_0^t(t-s)ds\int_{s+R_0}^{s+R}\partial_y(y-s-R_0)u(y,s)dy\right|\\
&=\left|\int_0^t(t-s)ds\int_{s+R_0}^{s+R}(y-s-R_0)u_x(y,s)dy\right|\\
&\leq \int_0^t(t-s)ds\int_{s+R_0}^{s+R}(y-s-R_0)|u_x(y,s)|dy\\
&\leq \left(\int_0^t(t-s)ds\int_{s+R_0}^{s+R}(y-s-R_0)|u_x(y,s)|^p dy\right)^{1/p}I(t)^{1-1/p},\\ \end{split} \] where \[ I(t):=\int_0^t(t-s)ds\int_{s+R_0}^{s+R}(y-s-R_0)dy=\frac{1}{4}t^2(R-R_0)^2=t^2R_1^2. \]
\par Hence we obtain that \begin{equation} \label{esti_H''}
H''(t)\geq \frac{1}{2}R_1^{-2(p-1)}t^{1-2p}|H(t)|^p\quad\mbox{for}\ t\geq R_1. \end{equation} Therefore, the argument in Rammaha \cite{Rammaha95} can be applied to (\ref{esti_H}) and (\ref{esti_H''}) to ensure that
there exist positive constants $\varepsilon_2=\varepsilon_2(f,p,R)$ and $C_2$
independent of $\varepsilon$ such that a contradiction appears provided \[ T>C_2\varepsilon^{-(p-1)} \] holds for $0<\varepsilon \leq \varepsilon_2$. The proof is now completed.
$\Box$
\end{document} | arXiv |
Optimization of electromagnetic wave propagation through a liquid crystal layer
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April 2015, 8(2): 313-321. doi: 10.3934/dcdss.2015.8.313
Conley's theorem for dispersive systems
Hahng-Yun Chu 1, , Se-Hyun Ku 1, and Jong-Suh Park 1,
Department of Mathematics, Chungnam National University, 79, Daehak-ro, Yuseong-gu, Daejeon 305-764, South Korea, South Korea, South Korea
Received April 2013 Revised November 2013 Published July 2014
In this article, we study Conley's theorem about the chain recurrence in dynamical systems, that is, the chain recurrent set of continuous map $f$ is the complement of union of $B_{U}(A)-A$, where $A$ is an attractor and $B_{U}(A)$ is a basin of $A$. In this paper, we generalize this theorem to dispersive systems on noncompact spaces.
Keywords: Chain recurrence, basin of attractor, compact-valued dispersive systems., attractor, dispersive systems.
Mathematics Subject Classification: Primary: 37B20; Secondary: 37B25, 37B3.
Citation: Hahng-Yun Chu, Se-Hyun Ku, Jong-Suh Park. Conley's theorem for dispersive systems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 313-321. doi: 10.3934/dcdss.2015.8.313
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Hahng-Yun Chu Se-Hyun Ku Jong-Suh Park | CommonCrawl |
\begin{definition}[Definition:Rational Decision-Maker]
Let $G$ be a game.
Let $P$ be a player of $G$.
Let $A$ be the set of moves available to $P$.
Let $C$ be the set $C$ of consequences of the moves available to $P$.
Let $g: A \to C$ be the consequence function on $A$.
Let $B \subseteq A$ be a the set of moves which are feasible.
$P$ is a '''rational decision-maker''' {{iff}} $P$ chooses an element $a^* \in B$ such that:
:$\forall a \in a^*: \map g {a^*} \succsim \map g a$
where $\succsim$ is the preference relation on $C$.
That is, that $P$ solves the problem:
:$\ds \max_{a \mathop \in B} \map U {\map g a}$
where $U$ is the utility function on $C$.
It is assumed that $P$ uses the same preference relation for all $B \subseteq A$.
\end{definition} | ProofWiki |
What happens when the definition of disability changes? The case of obesity
Jennifer Bennett Shinall ORCID: orcid.org/0000-0001-6848-11521
This paper examines how Congress's 2008 expansion of who is disabled under the Americans with Disabilities Act (ADA) impacts the labor market outcomes of newly covered individuals. Focusing on obese individuals, I exploit variation in coverage of obesity before and after the 2008 expansion to identify effects of the legal change, but I find no improvement in the labor market outcomes of the obese. Although the 2008 expansion was intended to remedy the unintended consequences of the ADA and improve labor market outcomes of the disabled, these early estimates suggest that the expansion has not yet achieved Congress's stated goals.
JEL codes: J14, J21, J78, K31
When Congress first passed the Americans with Disabilities Act (ADA) on July 26, 1990, its goals were lofty: it aimed "to invoke the sweep of congressional authority…in order to address the major areas of discrimination faced day-to-day by people with disabilities" (42 U.S.C. § 12101(b)(4)). Congress expressly intended "to provide a clear and comprehensive national mandate for the elimination of discrimination against individuals with disabilities" throughout their lives, including in employment, and thus to improve the outcomes of the disabled in the labor market and beyond (42 U.S.C. § 12101(b)(1)). Specifically, Title I of the ADA sought to further Congress's goals in the labor market by prohibiting discrimination against the disabled with regard to the "terms, conditions, and privileges of employment" and by requiring employers to provide "reasonable accommodation" to disabled employees who are capable of "perform[ing] the essential functions of the employment position" (42 U.S.C. §§ 12111(8), 12112(a)).
Yet empirical studies suggest that, over the past two decades, Title I of the ADA has fallen far short of Congress's intentions. Two well-known papers by DeLeire (2000) and Acemoglu and Angrist (2001) demonstrate that, instead of improving the position of the disabled in the labor market, employment has actually declined for the disabled since Title I went into effect on July 26, 1992. Subsequent papers such as Hotchkiss (2004) and Bound and Waidmann (2002) have questioned the methodology of earlier estimates, but even after re-estimating the wage and employment outcomes of the disabled after the ADA, these papers admit that labor market outcomes for the disabled, at best, remained the same after the effective date of Title I.
Although multiple issues may have contributed to the ineffectiveness of Title I, certainly one of the greatest obstacles encountered by ADA plaintiffs was the fact that Congress's definition of disability in the Act was anything but clear. Congress defined an ADA disability as "(A) a physical or mental impairment that substantially limits one or more of the major life activities of such individual; (B) a record of such an impairment; or (C) being regarded as having such an impairment." (ADA 1990 §3(2)). But Congress failed to define what the terms "impairment," "substantially limits," "major life activities," and "regarded as" precisely meant. Nor did Congress provide any rules of construction for the undefined terms in the ADA.
As a result, years of litigation ensued over the meaning of these terms and, more broadly, over who was disabled for the purposes of the ADA. Coverage under the ADA's definition may have been relatively certain for individuals confined to a wheelchair, but coverage was much less certain for individuals affected by a host of other debilitating conditions, including cancer, diabetes, and heart disease. Among the conditions for which coverage was uncertain under the 1990 ADA was obesity.Footnote 1 The negative effects of obesity on wage and employment outcomes are well documented (Cawley 2004; Pagan and Davila 1997; Averett and Korenman 1996). Similar to the labor market penalty experienced by individuals with most other types of disabilities, the obesity penalty may consist of both a penalty related to reduced productivity, suggested by the long list of comorbidities associated with obesity, including musculoskeletal conditions (Hergenroeder et al. 2011; Alley & Chang 2007), and a penalty related to taste-based discrimination, suggested by the persistence of the negative wage and employment effects for women, but not for men, after controlling for individual characteristics (Shinall 2016; Cawley 2004).Footnote 2 Still, the question with which most courts wrestled regarding obesity was whether it was a covered "impairment" under the 1990 ADA (as opposed to an uncovered physical condition). With the exception of a single federal appellate court, federal courts generally answered this question in the negative.
Fortunately for the obese (and many other individuals seeking disability status under the ADA), the legal environment improved in 2008. In response to decisions by the US Supreme Court and federal appellate courts that had severely limited coverage under the ADA, Congress passed the Americans with Disabilities Act Amendments Act (ADAAA). The ADAAA clarified many of the definitional ambiguities of the 1990 Act, including definitions of the crucial terms "substantially limits," "major life activities," and "regarded as" within its provisions. Moreover, Congress provided rules of construction in the amended Act that admonished federal courts to construe the term "disability…in favor of broad coverage of individuals…to the maximum extent permitted" (42 U.S.C. § 12102(4)(A)).
In expanding the coverage of the ADA in 2008, Congress did not specifically identify obesity or any other medical condition as a covered disability.Footnote 3 Nonetheless, Congress's broad command has led the US Equal Employment Opportunity Commission (EEOC) to revise its official ADA guidance substantially. With respect to obesity, the prior guidance stated that obesity would only be a covered disability under the ADA in "rare circumstances," but the current guidance now states that "severe obesity, which has been defined as body weight more than 100 % over the norm…is clearly an impairment" (EEOC 2009). The EEOC's definition of severe obesity roughly equates to the medical definition of morbid obesity, which is a body mass index (BMI) of 40 or more (the BMI of a normal-weight person ranges from 18.5 to less than 25). Congress's broad command also appears to have convinced federal courts to view the claims of morbidly obese individuals seeking the ADA's protections more favorably.
As a result of the ADAAA, obese individuals—and particularly morbidly obese individuals—are positioned to see improvements in labor market outcomes now more than ever before. Because the legal treatment of obesity has shifted since the 2008 Amendments, the possibility of seeing a large and substantial improvement in nationwide employment outcomes of the heaviest individuals in the labor market seems quite plausible. Consequently, the principal focus of this paper will be to examine how morbidly obese individuals have fared in terms of employment and in terms of presence in the labor market since the passage of the ADAAA. Using 2004 through 2013 data from the Behavioral Risk Factor Surveillance System (BRFSS), I estimate the 2008 Amendments' labor market effects. In spite of the potential for the amended ADA to improve the labor market outcomes of obese individuals, I find little evidence that it has actually improved their outcomes. These estimates suggest that, so far, the ADAAA in action has not been much more effective in the labor market than its predecessor.
My investigation of how expanding the definition of disability under the ADA has impacted the heaviest individuals in the labor market will proceed as follows. Section 2 discusses the legal treatment of obese individuals under the 1990 version of the ADA, and Section 3 examines how this legal treatment has changed since the 2008 Amendments. Section 4 elaborates on my data and empirical methodology, and Section 5 presents my results. Section 6 discusses the implications of these results both for the obese and for the disabled generally, considering why the ADA in action may diverge from the ADA in print.
The debate surrounding the effectiveness of the 1990 ADA
In passing the original 1990 Act and the 2008 Amendments, the unambiguous intent of Congress was to improve labor market outcomes for the disabled. Yet most prior studies of the original version of the ADA indicate that it has failed to accomplish Congress's stated intent, which immediately raises questions about why and how a law with such a clear purpose could fail to accomplish its purpose. Prior work has primarily blamed the Act's imposition of all costs associated with employing a disabled worker on the employer. As part of the employer's duty not to discriminate in the hiring, compensation, promotion, or firing processes, employers are required to provide "reasonable accommodations" to disabled individuals unless such accommodations would "impose an undue hardship" on the employer (42 U.S.C. § 12112). The ADA requires employers to bear the entire cost of reasonably accommodating the disabled worker; the ADA's equal pay provision prohibits employers from taking the cost of reasonable accommodation out of a disabled worker's paycheck. In addition, if the employer had previously refused to hire the disabled because of personal distaste for the disabled or because of the perceived distaste of customers, clients, and other employees, the employer must fully bear the costs of distaste in order to comply with the ADA.
To see why the ADA may not help, and even may harm, labor market outcomes of the disabled on balance, consider the following example. Suppose the employer is considering taking a tangible employment action—such as hiring, promoting, or retaining—with respect to a disabled employee. The employer will consider the expected profits the disabled worker would generate, π d, minus the expected cost of complying with the ADA, a, which includes the cost of reasonable accommodation the employer would face if he decided to hire, promote, or retain the disabled worker. If the employer (or the employer's customers) harbors any animus or taste-based preferences against the disabled, such costs would also be included in a. Against this consideration, the employer will weigh the expected profits minus the expected costs of making the contrary employment decision—here, not hiring, not promoting, or discharging the disabled worker. In this case, the expected profits, π n, would be generated by the alternative worker hired, promoted, or retained in the disabled worker's place; the expected costs would include the costs of non-compliance with the ADA, including a potential lawsuit brought by the disabled worker. As long as
$$ {\pi}_{\mathrm{d}}-a<{\pi}_{\mathrm{n}}-c, $$
the employer will decide not to hire, promote, or retain the disabled worker. From Eq. (1), it follows that as long as the expected cost of compliance is sufficiently larger than the expected cost of non-compliance, the employer will decline to take a positive employment action with respect to the disabled worker.Footnote 4
In line with the above model, economists have focused on the high compliance costs the Act imposed on employers, particularly the compliance costs imposed by the reasonable accommodation provision, to explain their findings that employment of the disabled declined after the ADA. DeLeire (2000), for instance, concluded that the increased costs to firms arising from the ADA's accommodation mandate was responsible for a 7.2 percentage point decline in employment of disabled men after 1990. Acemoglu and Angrist (2001) similarly blamed accommodation costs (and costs arising from disputes over what constitutes a reasonable accommodationFootnote 5) for their finding that disabled men and women worked between 1 and 4 fewer weeks per year after passage of the 1990 ADA. Yet the evidence available on the costs of reasonable accommodation casts some doubt on the argument that compliance costs are solely responsible for the ineffectiveness of the original ADA. According to a 2013 employer interview study by the Job Accommodation Network (2013), 58 % of disability accommodations cost the employer nothing. Of the 42 % of accommodations that are costly, the median employer expenditure is only $500.
Since previous authors have principally focused on compliance costs, they have paid less attention to the potential importance of non-compliance costs. Yet according to Eq. (1), relatively low non-compliance costs (when compared to employer compliance costs) could have the same negative consequences for wage and employment outcomes of the disabled population. Non-compliance costs may be low for many reasons, including (1) if a disabled worker is unlikely to sue an employer who improperly denies his/her coverage under the ADA, (2) if a defendant employer has a good chance of winning an ADA lawsuit in which the case is close or in which the employer has improperly denied coverage, and (3) if the damages available to successful ADA plaintiffs are too low to deter employers from improperly denying coverage to disabled workers.
Evidence from legal scholars indicates that at least (1) and (2) may pose a problem for potential ADA plaintiffs. Clermont and Schwab (2004), for example, showed that in all the ADA employment cases filed in federal court between 1998 and 2001, 1371 resulted in wins for the defendant, but only 117 resulted in wins for the plaintiff. These raw numbers suggested to the authors that employers who chose to litigate disability disputes had a good chance of prevailing in federal court. The prospects of ADA plaintiffs in federal court, on the other hand, appeared quite discouraging to the authors. Moreover, follow-up research by Clermont and Schwab (2009) demonstrated that new ADA employment case filings in federal court declined sharply between 2002 and 2007. Given their prior findings, the authors worried that potential ADA plaintiffs might be discouraged by high costs or low probability of success from filing lawsuits against employers.Footnote 6
Although the low plaintiff success rate and the decline in plaintiff filing rate under the ADA were part of larger trends in all employment discrimination litigation (Clermont and Schwab 2004, 2009), the definitional ambiguity of the 1990 Act may at least partially explain these trends in the case of the ADA. Recall from Section 1 that Congress failed to define several key terms and failed to provide rules of construction in the original Act. This definitional ambiguity could have led to inconsistent decisions among federal courts, which over time could have convinced employers that they had a good chance of winning close ADA cases (and simultaneously discouraged potential ADA plaintiffs about their prospects in federal court).
Such inconsistent decisions under the original Act are apparent in the case of obesity. In Cook v. Department of Mental Health (1993), the First Circuit upheld a jury award of $100,000 to a morbidly obese job applicant, Bonnie Cook, after the Rhode Island Department of Mental Health refused to rehire her as an institutional attendant because she was morbidly obese. Cook had held this position twice previously, voluntarily leaving both times with a clean employee record, and she had always been morbidly obese. In reaching its decision, the First Circuit was persuaded that Cook could not just "simply lose weight and rid herself of any concomitant disability"; the evidence demonstrated that Cook would have to deal with a dysfunctional metabolism for the rest of her life no matter how much weight she lost. Moreover, the court placed particular importance on the fact that Cook's obesity arose from an underlying physiological condition.
The Cook decision had the potential to revolutionize the treatment of obese workers—especially morbidly obese workers—under the 1990 version of the ADA. And yet, in the years following the First Circuit's decision, Cook proved to be the exception, not the rule. Since the 1993 case, the Second, Sixth, and Eleventh Circuits have all distinguished or disagreed with Cook. In Andrews v. State of Ohio, 104 F.3d 803 (6th Cir. 1997), the Sixth Circuit refused to grant relief to Ohio State Highway Patrol officers who had been disciplined at work after failing to meet the weight limits set by the Highway Patrol Fitness Program. The court noted that the appendix to the relevant rule in the Code of Federal Regulations, 29 C.F.R. § 1630.2(h), held that the "definition of the term 'impairment' does not include physical characteristics such as eye color, hair color, left-handedness, or height, weight or muscle tone that are within 'normal' range and are not the result of a physiological disorder." Thus, to hold that "a mere physical characteristic, without more, equal[s] a physiological disorder," the court concluded, "would debase [the] high purpose [of] the statutory protections available to those truly handicapped." The Sixth Circuit distinguished its holding from the Cook case by noting that the plaintiff in Cook had presented expert testimony that her morbid obesity arose from a physiological impairment of the metabolism. Almost a decade later, the Sixth Circuit reaffirmed this decision in E.E.O.C. v. Watkins Motor Lines.Footnote 7
The Second and Eleventh Circuits were also less generous than the First Circuit to the obese. In Francis v. City of Meriden, 129 F.3d 281 (2d Cir. 1997), the Second Circuit declined to grant relief to a firefighter who had been suspended without pay after failing to meet the department weight standard and refusing to take a body fat or fitness test. The Second Circuit agreed with the Sixth Circuit that physical characteristics not arising from a physiological condition were not impairments for the purposes of the ADA. In Greenberg v. Bellsouth Telecommunications, Inc., 498 F.3d 1258 (11th Cir. 2007), the Eleventh Circuit declined to grant ADA protection to an obese worker who actually did suffer from diabetes, hypertension, hypothyroidism, and other physiological conditions because the worker failed to show that he was "unable to work in a broad class of jobs." Furthermore, even though Cook remains good law in the First Circuit today, never again has the First Circuit found an obese individual disabled for the purposes of the ADA in a reported decision.Footnote 8
Federal courts' later treatment of obesity under the 1990 version of the ADA suggests that the Cook decision was an outlier. Given the general lack of success of obese plaintiffs bringing ADA claims subsequent to Cook, the case, at best, stands for the proposition that morbid obesity could be a covered disability under the original ADA in rare circumstances and only when it arose from an underlying physiological condition. In light of the case law that followed Cook, it seems unlikely that the decision convinced many employers to reform their treatment of all obese applicants and employees. At most, Cook might have encouraged some employers in the First Circuit to take more care in their treatment of the morbidly obese.
Obesity and the 2008 Americans with Disabilities Act Amendments Act
As noted in Section 1, the legal environment substantially improved for obese workers, and disabled workers generally, with the passage of the ADAAA in 2008. The ADA Amendments were Congress's response to four US Supreme Court cases (ADAAA § 2). The first three, nicknamed the "Sutton trilogy," consisted of Sutton v. United Air Lines (1999), Murphy v. United Parcel Service, Inc. (1999), and Albertson's, Inc. v. Kirkingburg (1999). Decided together on June 22, 1999, the three cases severely limited the definition of disability under the original version of the ADA by holding that an individual was not disabled if corrective measures could ameliorate the individual's condition. The fourth case, Toyota Motor Manufacturing, Ky., Inc. v. Williams (2002), decided in 2002, even more severely limited the definition of disability under the original ADA by holding that "the central inquiry must be whether the claimant is unable to perform the variety of tasks central to most people's daily lives, not whether the claimant is unable to perform the tasks associated with her specific job." Thus, even if an ADA plaintiff's impairment substantially limited her ability to do her job, she would not have been considered disabled under the ADA after Toyota unless her impairment also substantially limited her ability to function in daily life.
The Amendments specifically reversed these decisions, with the explicit goal of reducing the number of ADA suits in which employers could dispute whether the plaintiff was disabled for the purposes of the Act. Thinking about the effect of the ADAAA in terms of the model presented in Eq. (1), the ADAAA should have raised the per worker cost of employer non-compliance. If the ADAAA successfully reduced the number of ADA cases in which the employer could contest the existence of a disability, the ADAAA should have reduced the chance that an employer will win an ADA case when the case is close or when the employer has improperly denied an employee ADA coverage. The ADAAA may have also encouraged previously discouraged, disabled employees to seek remedies from employers for discrimination. On the other hand, the ADAAA does not require employers to provide additional accommodations to disabled workers; the standard for employer compliance remains to provide disabled workers with reasonable accommodations that do not result in an undue hardship. Thus, the ADAAA should not have raised the per worker cost of compliance.
Not only has coverage of disabilities generally become more certain since the 2008 Amendments—thus raising the cost of employer non-compliance—but also coverage of obesity in particular has become more certain. In addition to revising its compliance guidelines to reflect the view that severe obesity is now a covered disability under the revised Act, the EEOC has filed two ADA lawsuits involving morbidly obese plaintiffs following the passage of the ADAAA. In September 2010, the EEOC filed its first public interest suit, EEOC v. Resources for Human Development, Inc., which involved a morbidly obese woman in New Orleans. Lisa Harrison had been terminated from her job at a residential treatment facility despite an excellent performance record. Even though Harrison had already weighed over 400 lb at the time of her hiring, she weighed 527 lb at the time of her termination. The district court denied the employer's motion for summary judgment, finding that Harrison's "severe obesity…[wa]s clearly an impairment," and Harrison need not prove the underlying basis of her obesity in order to gain protection under the ADA. Before the case could go to trial, Harrison's employer settled with the EEOC for $125,000.Footnote 9 In September 2011, The EEOC filed a second public interest suit involving a morbidly obese Houston man. The EEOC v. BAE Systems, Inc. suit arose after an employer, BAE Systems, fired a plant employee, Ronald Kratz, because of his weight. Throughout his employment, Kratz was able to perform the essential functions of his job, despite weighing approximately 450 lb at his hiring and 680 lb at his termination. Before the district court could rule on any motions, BAE Systems settled the suit with Kratz for $55,000. (Equal Employment Opportunity Commission, 2012b).
Moreover, favorable treatment of ADA claims made by morbidly obese individuals has extended beyond the two EEOC suits. In two cases filed by private, morbidly obese litigants with right-to-sue letters from the EEOC, federal district courts located in the Fifth Circuit have acknowledged that severe obesity limiting a major life activity can now be considered a disability under the expansive terms of the ADAAA.Footnote 10 A Mississippi district court in Lowe v. American Eurocopter, LLC, held, "Based on the substantial expansion of the ADA by the ADAAA, defendant's assertion that plaintiff's weight cannot be considered a disability is misplaced."Footnote 11 Another district court in Louisiana similarly found that a plaintiff's obesity was a disability because it substantially limited her breathing, which was a major life activity.Footnote 12 And a very recent case out of a federal district court in Missouri (located in the Eighth Circuit) has similarly agreed that obesity claims under the ADA will fare much better in the post-ADAAA regime. The court noted that the pre-ADAAA case law requiring "obesity [to be] related to an underlying physiological disorder or condition…[was] based on the more restrictive approach that was applied before Congress passed the Americans with Disabilities Amendments Act of 2008."Footnote 13 Furthermore, the Missouri district court found the employer's argument that obesity could qualify as a disability under the ADA only in "rare circumstances…misplaced as that language has been omitted [from the EEOC Compliance Guidelines] following the ADAAA."Footnote 14
Indeed, only one district court so far has been dismissive of a morbidly obese plaintiff's ADA claim under the 2008 Amendments. Yet in Powell v. Gentiva Health Services (2014), much of the Alabama district court's dismissive language was traceable to the unfavorable facts of the case.Footnote 15 The plaintiff, a former account executive for a hospice provider, only pled the first type of ADA liability in her complaint, claiming that she was limited in a major life activity under 42 U.S.C. § 12101(1)(A). (She did not claim that her former employer regarded her as substantially limited or that she had a record of a limiting impairment under subparts (B) and (C).) Contrary to her claims, however, the plaintiff referred to herself in the deposition as merely "overweight"; she also testified that she exercised regularly and that she did not have any health conditions or limitations as a result of her obesity. The final nail in the coffin came from the plaintiff's deposition testimony that her obesity "[a]bsolutely [did] not" impact her ability to perform her former account executive position.
Beyond the reported cases and settlements, the ADAAA has evidently led to a substantial number of obese plaintiffs filing suit under the amended ADA; a 2012 estimate reported that 48 obesity-related ADA cases had already been filed since the Amendments went into effect (Gordon 2012). Although only a handful of these cases have resulted in publicly available decisions or settlements, a perceptible shift has occurred in how attorneys view obesity claims under the ADA since the 2008 Amendments. Employer defense firms across the country have published numerous articles warning employers of the regime change with regard to obesity since the Amendments.Footnote 16 Moreover, despite the very small number of obesity cases under the ADAAA that have been decided by district courts—and despite the fact that no federal court of appeals has weighed in on the issue—employment defense attorneys' advice is largely the same: "Employers should assume that, post-ADAAA, obese employees are protected, and focus on providing reasonable accommodations" (Montgomery 2012).
This perceptible shift not only in how courts view obesity-related ADA claims but also in how attorneys view obesity-related ADA claims makes obesity a ripe area for study in the post-ADAAA regime. If employers are listening to courts—or at least listening to their attorneys—they have good reason to take obesity claims under the ADA more seriously. They have good reason to believe that under the amended ADA, courts will view less favorably any improper treatment of a severely obese employee or applicant whose weight substantially limits a major life activity.
Empirical methodology and data
To evaluate whether this expansion in the definition of disability under the ADAAA has impacted the labor market outcomes of the obese, I take a similar empirical approach as Carpenter (2006), who previously examined a related question. This paper asks whether the 2008 ADAAA has improved labor market outcomes of the obese; Carpenter asked whether the 1990 ADA improved labor market outcomes of the obese. Specifically, Carpenter examined whether employment outcomes of the obese improved after Cook, the sole positive federal court decision under the 1990 ADA. Carpenter found that employment of the obese, but not the morbidly obese, increased after 1993 (the year of the Cook decision) by two percentage points for men and four percentage points for women nationwide.
Carpenter attributed these increases to Cook, yet this attribution is problematic in light of the case law discussed in Section 2. Cook was an outlier decision with respect to coverage of obesity under the 1990 ADA. At most, Cook might have improved employment outcomes of the morbidly obese in the First Circuit.Footnote 17 But Carpenter found no improvements in nationwide employment for the morbidly obese; he only found improvements in nationwide employment for the regularly obese. Moreover, Carpenter only examined nationwide employment outcomes, not outcomes specific to the First Circuit. To determine whether employment outcomes of the morbidly obese improved in the First Circuit after Cook, Appendix Table 9 replicates Carpenter's results, restricting his nationwide data to observations from the First Circuit only. According to these estimates, employment of the morbidly obese may have actually declined for men and women in the First Circuit after Cook (although the point estimates are not statistically significant). In sum, whatever was driving the increases in employment outside the First Circuit for the regularly obese in the mid-1990s, it does not appear to be the Cook decision.
Cook and the 1990 version of the ADA may not have improved employment outcomes of morbidly obese workers. Yet as discussed in Section 3, the 2008 amended version of the ADA is positioned to remedy the original Act's prior shortcomings. Increased support from the EEOC as well as consistently positive case law in favor of covering morbidly obese workers under the ADA should encourage employers to rethink how they treat weight in the workplace. Certainly, post-ADAAA legal developments have convinced defense attorneys to rethink their advice to employers regarding weight in the workplace. If employers are in fact responding to these legal developments, then they should be more discouraged from taking adverse employment actions against workers on the basis of weight now than under the previous legal regime, and employment of the morbidly obese should increase in the post-ADAAA period.
To test whether employment of the morbidly obese actually increased after the ADAAA, I use a difference-in-differences approach, the same methodology used by Carpenter (2006):
$$ \Pr \left({Y}_{ict}=1\right)={X}_{ict}\beta +{O}_{ict}{\gamma}_1+{P}_t{\gamma}_2+\left({O}_{ict}\ast {P}_t\right){\gamma}_3+{C}_c{\sigma}_1+{T}_t{\sigma}_2+\left({C}_c\ast {T}_t\right){\sigma}_3+{\varepsilon}_{ict} $$
Here, Y ist is the labor market outcome of interest for individual i, who lives in federal circuit c at time t. This analysis will investigate two principal outcomes: first, whether the individual is employed and, second, whether the individual is in the labor market (since improvement in legal protections may encourage disabled individuals to enter the labor market).Footnote 18 X is a vector of individual demographic characteristics that includes age, race, ethnicity, education, and marital status, and O is a dummy variable indicating whether the individual is obese.Footnote 19 P is a post-2008 indicator variable, representing the post-ADAAA period. C is a vector of circuit dummy variables for all 12 geographically defined federal circuits, and T is a set of year dummy variables. The coefficient of interest, γ 3, is the double-difference estimate. That is, γ 3 represents the change in the employment of obese workers after the ADAAA, differencing out (1) changes in employment of the obese before the ADAAA and (2) changes in employment of the non-obese both before and after the ADAAA. Because the dependent variable is a non-continuous dummy variable (equal to 1 if the respondent is employed/in the labor market), I will estimate Eq. (2) using a linear probability model to avoid the concerns raised by Ai and Norton (2003) regarding the reliability of difference-in-differences estimates with probit estimation.
To estimate the effects of the ADAAA on labor market outcomes of the obese, I use data from the BRFSS, the same data used by Carpenter (2006) to estimate the effect of the original ADA on labor market outcomes of obese individuals. The BRFSS is an annual health survey dataset administered by the Centers for Disease Control (CDC) since 1984; data are collected from respondents continuously throughout the year. Although only 15 states participated in the first year of the BRFSS, most states participated by 1990, and all states and territories have participated from 1994 until the most recent year of availability, 2013. Besides being the data that Carpenter used for his previous, related study of obesity and the ADA, the BRFSS has both a size and a time advantage for this particular study over the few other labor market datasets that contain weight and height information.
The BRFSS's size advantage is that, unlike other datasets used to study obesity and the labor market (such as the National Longitudinal Survey of Youth, or NLSY), the BRFSS has contained at least 50,000 respondents every year since its inception. The large number of respondents allows this study to distinguish between the obese and the morbidly obese. Such a distinction is important given that all of the positive developments in ADA litigation have involved morbidly obese (not regularly obese) individuals. Similarly, the EEOC guidance says that even under the expanded ADAAA regime, only severe obesity (double normal body weightFootnote 20), which roughly parallels the medical definition of morbid obesity (double normal BMI), is a covered disability under the amended ADA. Even though 25 to 33 % of the country is obese, only about 6.6 % of the country is morbidly obese (Sturm and Hattori 2013). Given the relatively small number of morbidly obese individuals, a large amount of observations is critical for this study.
The BRFSS's time advantage is that it contains 5 years of post-ADAAA observations, which should allow the present study to identify any meaningful changes in the labor market outcomes of the obese—and particularly, in the labor market outcomes of the morbidly obese—since the 2008 Amendments. Other datasets that contain both labor market and BMI information on their respondents have few, if any, post-2008 observations. The Eating and Health Module, for example, ends in 2008 and so contains no post-ADAAA observations. The NLSY 1979 contains just two post-ADAAA observations, and even though the NLSY 1997 contains several post-ADAAA observations, the oldest respondents in that dataset are only 32.
Because the focus of the BRFSS is health status and behaviors, the survey asks respondents to self-report their weight and height,Footnote 21 from which I calculate their BMIs. Because pregnancy can dramatically, but temporarily, alter a person's weight, I drop pregnant women from the sample. I also drop individuals from the sample who have BMIs greater than 100.Footnote 22 Since the focus of this study is employment, I will also focus on individuals between the typical working ages of 18 and 65, inclusive. In addition to health-related questions, the BRFSS asks respondents about their demographic characteristics and labor market status, although these questions are more limited. Among the demographic characteristics available in the BRFSS are age, gender, race, ethnicity, education, marital status, and presence of children. The BRFSS also asks respondents whether they are employed, to which they can answer that they are employed for wages, self-employed, out of work for less than 1 year, out of work for more than 1 year, a homemaker, a student, retired, or unable to work.
To construct my first dependent variable of interest, employment, I define employed as employed for wages, define unemployed as out of work (either for less or more than 1 year), and drop all other respondents. Throughout the paper, I use this definition of employment for two reasons. First, this definition of employment offers the best match to the Bureau of Labor Statistics (BLS) definition, which defines employed as doing any work for pay or profit during a given week and defines unemployed as not having a job but actively looking for one during the past 4 weeks. Second, although the self-employed would normally be counted as employed under the BLS definition, I drop self-employed respondents for consistency with prior disability literature, including Baldwin (2006) and Beegle and Stock (2003).
To construct my second dependent variable of interest, in the labor market, I define workers who are employed for wages, self-employed, or out of work as being in the labor market and workers who are a homemaker, a student, retired, or unable to work as being out of the labor market. Plausibly, disabled workers who may have been discouraged by lack of success in the original ADA regime may re-enter the labor market in the post-ADAAA period, hoping that their fortunes in the labor market will improve. For this reason, it is important to monitor movement into (or out of) the labor market during this time period, which this second dependent variable of interest will allow me to do in the next section.
Summary statistics for the BRFSS data are presented by BMI classification and gender in Table 1. These data, which include observations from 2004 to 2013, contain 1,168,553 observations of men and 1,642,813 observations of women ages 18 to 65 (inclusive). For men, the means of the underweight, overweight, obese, and morbidly obese are statistically different from the means of the normal weight for virtually every characteristic. The summary statistics for underweight and morbidly obese men have much in common: compared to normal-weight men, both are more likely to be a member of a minority group, less likely to be employed or in the labor market, less likely to have a child, and less likely to have graduated from college. In contrast, overweight men are actually more successful in the labor market than normal-weight men, and overweight men are also the most likely of all BMI groups to have graduated from college. Moreover, heavier men are more likely to be married than normal-weight men, and they tend to be older than normal-weight men.
Table 1 Summary statistics for 2004–2013 BRFSS respondents in the labor market, ages 18 to 65
For women, clear weight-based patterns in the data are apparent as BMI classification increases beyond normal weight. As a woman gets heavier and rises above the normal-weight BMI classification, she becomes increasingly less likely to be employed or in the labor market, less likely to be married, less likely to have a child, and less likely to have graduated from college. Moreover, black women, Hispanic women, and older women are relatively overrepresented in the overweight, obese, and morbidly obese groups. On the labor market, marriage, and education fronts, normal-weight women appear to be the most successful of all BMI classifications. The correlations between BMI, demographic, education, and labor market outcomes seen in the BRFSS are very much in line with the correlations seen previously in the NLSY, National Health and Nutrition Examination Survey, and the Eating and Health Module.Footnote 23
To estimate the effect of the ADAAA on the employment and labor market outcomes of the obese, I use the 2004–2013 BRFSS data and the double-difference model set out in Eq. (2). The results for men and women are reported in Table 2. Because the ADAAA went into effect on January 1, 2009, the post dummy variable in the double-difference regressions is equal to 1 in the post-2008 period. An important concern with studying the post-2008 period, of course, is that my results will pick up the effects of the Great Recession and financial collapse of 2008. For that reason, I include a control for the national unemployment rate (by sex) in addition to the usual demographic controls (detailed in Section 5) in the reported estimates.Footnote 24 In all four columns of Table 2, the regression estimates include federal circuit fixed effects, year fixed effects,Footnote 25 and circuit*year fixed effects.Footnote 26 All standard errors are heteroscedasticity-robust and clustered by federal judicial circuit.
Table 2 The effect of the ADAAA on obese individuals (post-2008 effect)
Turning first to the estimates for men, the coefficient of interest is the post*morbidlyobese variable. Given the fact that all positive post-ADAAA developments (the EEOC guidance, the EEOC litigation, and the private litigation) have all involved morbidly obese (not regularly obese) plaintiffs, any effects of the ADAAA should be greatest for, if not limited to, the morbidly obese. The post-2008 interaction effects for the overweight and the obese are presented for comparison. In sharp contrast to Carpenter's estimates of the original ADA's effects, the estimates of the ADAAA's effects are unpromising for morbidly obese workers who seek legal protection. The point estimate for the effect of the ADAAA on morbidly obese men's employment since 2008 is negative and statistically significant, indicating that employment for morbidly obese men declined by 2.8 percentage points in the post-2008 period. For the labor market regressions in column 2, the point estimate is small and positive but nowhere close to statistically significant. The results in Table 2 suggest that after the ADAAA went into effect on January 1, 2009, morbidly obese men have seen a decline in employment without an offsetting rise in labor market participation.
The estimates for the post-2008 ADAAA effect on women do not appear any more promising for morbidly obese women who seek legal protection. In columns 3 and 4 of Table 2, the point estimates of the coefficient on post*morbidlyobese are negative, although statistically insignificant, in both the employment regression and the labor market regression. These results suggest that, at best, morbidly obese women have not experienced any improvement in employment outcomes since the ADAAA, nor have they been sufficiently encouraged by the more favorable legal regime to increase their labor market participation.
In spite of the ADAAA's promise—particularly for groups like the morbidly obese, who have seen quite a few favorable legal developments in the post-ADAAA period—the results in Table 2 indicate that not much has changed in terms of employment or labor market participation. If anything, outcomes have declined for the morbidly obese since most of the point estimates are negative. Nonetheless, looking back at the history of the pro-obesity developments under the ADAAA raises the question of whether 2008 is the appropriate year around which to test. Although the ADAAA went into effect at the beginning of 2009, courts did not begin hearing ADAAA cases until much later. In fact, the EEOC did not file its first obesity-related lawsuit under the ADAAA until September 2010, and its second suit came a year later in 2011. Both cases settled in 2012. Because the best publicized obesity-related developments under the ADAAA did not begin until late 2010, arguably, the status of obesity under the amended ADA might have remained uncertain to employers (and their attorneys) until then. Thus, perhaps the more appropriate year around which to test is 2010, not 2008, in order to see the ADAAA's effect on labor market outcomes of the obese.Footnote 27
To test this hypothesis, Table 3 repeats the estimates in Table 2 using 2008 through 2013 BRFSS data and 2010 as the operative year. Even with these changes in the analysis, the results in Table 3 do not look much different than the results in Table 2. With respect to the men's results in Table 3, the point estimate of the effect of the post-2010 period on morbidly obese men's employment is negative, although statistically insignificant. The point estimate of the effect on morbidly obese men's labor market participation after 2010 is similarly negative and insignificant. Nor are the post-2010 effects of the ADAAA positive for morbidly obese women. According to column 3 of Table 3, morbidly obese women saw a 1.9 percentage point decline in employment in the post-2010 period. This finding is particularly striking given that the observations from the pre-comparison period (2008, 2009, and 2010) fall largely during the Great Recession. Furthermore, morbidly obese women did not increase their labor market participation during this period. The point estimate on the post*morbidlyobese coefficient in the labor market regression (column 4) is negative, although statistically insignificant.
No matter the period tested—post-2008 or post-2010—the labor market outcomes of morbidly obese individuals do not appear to have improved as a result of the ADAAA and may have actually declined. Because this finding is incongruous with the positive developments in obesity-related ADA case law, additional robustness checks are warranted. Appendix Table 10 reports the results of placebo estimates of the employment and labor market participation effects of the post-2007, post-2009, and post-2011 periods on the morbidly obese; the point estimates are generally negative.Footnote 28 Table 4 reports a re-estimate of the post*morbidlyobese coefficient using eight different specifications for both the post-2008 period and the post-2010 period. Row 1 reports the baseline estimates from Tables 2 and 3 for comparison. Row 2 adds interaction terms for year of interview times BMI classification (with the normal-weight category omitted); row 3 additionally includes interaction terms for federal judicial circuit times BMI classification (with the normal-weight category omitted). Row 4 explores whether the effects of the ADAAA may be more visible for younger respondents and restricts the sample to respondents ages 18 to 45 (inclusive), which is the same age group studied by Carpenter (2006). Row 5 examines whether expanding the morbidly obese category to include individuals with a BMI of greater than or equal to 35 meaningfully changes the results.Footnote 29 Row 6 substitutes state fixed effects for federal circuit fixed effects and clusters the standard errors at the state level instead of the federal circuit level. None of these robustness checks meaningfully change the results; in fact, many of the negative point estimates that were statistically insignificant in the baseline regressions become statistically significant (and more negative) in these alternative specifications.
Table 4 Robustness checks: estimate of post*morbidlyobese interaction term
Row 7 of Table 4 re-estimates the baseline regressions adding an indicator variable and interaction effectsFootnote 30 for the ten jurisdictions across the USA that prohibit discrimination on the basis of weight and/or personal appearance. The ten jurisdictions are Michigan; Washington, DC; San Francisco, CA; Santa Cruz, CA; Madison, WI; Urbana, IL; Binghamton, NY; Howard County, MD; Harford County, MD; and Prince George's County, MD. Obese individuals working in these jurisdictions who experience labor market discrimination have an alternative remedy to the ADA, which makes controlling for the availability of such laws potentially important. The only caveat to controlling for these laws is that in 2013, the BRFSS stopped identifying observations by county for privacy reasons. Thus, controlling for these laws requires me to cut a year off on each end of the time periods, so that the post-2008 estimate uses 2005–2012 data and the post-2010 estimate uses 2009–2012 data. Despite all these changes, the results are remarkably similar to the baseline regressions. The point estimates are of similar magnitudes, and the already negative point estimate of the post2010*morbidlyobese coefficient becomes statistically significant as a result of the additional controls and time restrictions.
The next robustness check in row 8 of Table 4 is a bit more complicated. The Cook case was the law of the First Circuit before the ADAAA, and it remains good law there today. It was never explicitly agreed with by another circuit and in fact was distinguished or disagreed with by other circuits during the pre-ADAAA period. Thus, it is possible that the better specification is one that compares outcomes of the morbidly obese outside the First Circuit (where the law more dramatically changed after the ADAAA) to outcomes of the morbidly obese inside the First Circuit (where the ADAAA was less revolutionary) in the pre- and post-ADAAA periods. Carrying out such a comparison requires a triple-difference estimate, where the coefficient of interest is notinthefirstcircuit*post*morbidlyobese.Footnote 31 Even in this specification, the results are not substantially different from the baseline estimates. Apparently, the employment of morbidly obese men dropped much more dramatically outside the First Circuit than within the First Circuit during the post-2008 period (hence, the point estimate in column 1 triples to −0.065). The labor market participation rates for women also dropped more dramatically outside the First Circuit relative to within the First Circuit during the post-2008 period. In general, the other results are statistically insignificant, with the exception of a very large, positive point estimate indicating an increase in morbidly obese women's labor market participation outside the First Circuit during the post-2010 period. Given that this result is somewhat of an anomaly compared to the other estimates and is so large (7.1 percentage points), it is difficult to conclude that the increase is solely the result of EEOC enforcement of the ADAAA after 2010. At best, this result suggests that EEOC enforcement of the ADAAA may have encouraged some morbidly obese women outside the First Circuit to enter the labor market after 2010.
A final robustness check is presented in row 9 of Table 4 for the post-2010 estimates. Because the BRFSS data identifies the month and the year of interview, these estimates take advantage of the month data and test the effect of the post-September 2010 period (the EEOC filed its first ADAAA lawsuit on behalf a morbidly obese individual in that month) instead of the post-2010 period. The results are again substantially similar to the baseline estimates, although the estimated coefficient on the post*morbidlyobese interaction term for women's employment actually becomes more negative.
Table 5 considers whether the effects of the ADAAA have been heterogeneous on certain sub-populations and, thus, re-estimates the baseline results in Tables 2 and 3 for whites, blacks, Hispanics, individuals with low levels of education (defined as a high school diploma or less), and individuals with high levels of education (defined as at least some college). Even though minority status and socio-economic status are not legally relevant to a federal disability claim (that is, being a member of a minority group does not make it easier to prove a disability discrimination claim in court), several compelling reasons exist for considering possible heterogeneity in the ADAAA's effects. First, as seen in the BRFSS summary statistics (Table 1) and other prior work on obesity, obesity disproportionately affects individuals who have low levels of education and who self-identify as a minority. Second, Cawley (2004) presented evidence of heterogeneity in the effect of weight on wages, finding the most negative impact for white women and the least negative impact for black men.
Table 5 Heterogeneity of post*morbidlyobese interaction term estimates across sub-populations
Table 5 reports a re-estimate of the post*morbidlyobese coefficient on these sub-populations for both the post-2008 period and the post-2010 period. The point estimates for each sub-population, like the point estimates for the combined populations, are almost entirely negative. Indeed, considering these sub-populations separately often increases the estimated magnitude of the negative coefficient on the post*morbidlyobese interaction term. Perhaps the most notable result is for whites in the post-2010 period. These estimates suggest that employment declined by 2.2 percentage points for morbidly obese white women and 3.4 percentage points for morbidly obese white men since EEOC enforcement of the ADAAA began in late 2010. Labor market participation of morbidly obese white women also decreased by 1.9 percentage points. Cawley's (2004) results indicated that morbidly obese white women were in the most need of legal protection in the labor market, and yet, the results presented in Table 5 suggest that they may have been most harmed by the ADAAA.
A final, alternative specification is presented in Table 6. Because morbidly obese plaintiffs have been disproportionately successful in district courts within the Fifth Circuit—both EEOC enforcement suits were filed there, and two other district courts have stated in summary judgment orders that severe obesity can now be a disability under the ADAAA—it is possible that any positive effects of the ADAAA may be strongest there. The EEOC lawsuits received quite a bit of local press,Footnote 32 so attorney awareness of the EEOC's position on obesity is likely to have been greater within this region than nationwide. A greater awareness of the EEOC's position—and later, the EEOC's success in convincing a federal court—that severe obesity is a disability under the ADA may have translated into plaintiffs' lawyers seeking out or taking on more clients with an obesity claim under the ADA. A greater awareness may have also translated into more defense lawyers advising their employer clients to take greater care with their obese employees and applicants. This hypothesis is tested in Table 6 with a triple-difference estimate, comparing labor market outcomes of morbidly obese men and women inside the Fifth Circuit to the outcomes of morbidly obese men and women outside the Fifth Circuit, before and after the EEOC lawsuits began in late 2010.
Table 6 The effect of EEOC enforcement post-2010 in the Fifth Circuit
The results in Table 6 are more positive than the prior estimates, although still somewhat mixed. After 2010, morbidly obese men's employment rates increased by 5.4 percentage points in the Fifth Circuit compared to the rest of the country, but their labor market participation rates declined by almost an equal amount (5.1 percentage points). The data on morbidly obese women in the Fifth Circuit after 2010 demonstrate the opposite pattern: employment rates declined by 2.1 percentage points for these women after 2010 compared to women in the rest of the country, but this decline is offset by a greater increase (3.3 percentage points) in the labor force participation rate of Fifth Circuit morbidly obese women during the post-2010 period.
Tables 7 and 8 repeat similar robustness and heterogeneity checks for the Fifth Circuit triple-difference estimates as already presented for the nationwide double-difference estimates (see Tables 4 and 5). For the most part, the sign and magnitude of the post * fifth * morbidlyobese coefficients in the Table 7 robustness checks remain similar to the estimated baseline coefficients in Table 6: the results indicate that ADAAA enforcement in the Fifth Circuit had a negative effect on morbidly obese men's labor market participation and morbidly obese women's employment, but a positive effect on morbidly obese men's employment and morbidly obese women's labor market participation. In most estimates, the positive and negative effects are almost perfectly offsetting. The heterogeneity checks in Table 8 reveal that any dominant, positive effects of Fifth Circuit enforcement are largely concentrated among Hispanic respondents and low-education respondents.
Table 7 Robustness checks: estimate of post*fifth*morbidlyobese interaction term
Table 8 Heterogeneity of post*fifth*morbidlyobese interaction term estimates across sub-populations
In sum, if the ADAAA has improved labor market outcomes for anyone, its effects have been isolated to individuals living in the area of the country that has most actively enforced the new federal disability discrimination regime. Caution should be taken in interpreting the Fifth Circuit results since the employment and labor market participation results are often offsetting.Footnote 33 Still, even if employment and labor market participation results in the Fifth Circuit cancel each other out, the post-2010 Fifth Circuit results are notable because, unlike the nationwide results, they are not entirely negative. Overall, the estimates presented in this section provide a cautionary tale to policymakers, suggesting that in the absence of enforcement, new anti-discrimination laws may not help—and may even hurt—their intended beneficiaries.
By more clearly defining what it meant to be disabled under the ADA, and by explicitly overturning four previous judicial interpretations that had severely restricted what it meant to be disabled under the ADA, Congress had dual purposes in passing the 2008 Amendments to the ADA. First, Congress hoped to reduce the uncertainty over who was disabled under the ADA, and second, Congress hoped to do so by greatly expanding the coverage of the ADA "to the maximum extent permitted" (42 U.S.C. § 12102(4)(A)). According to EEOC Legal Counsel Peggy Mastroianni, the agency's ultimate aspiration for the ADAAA has been to create a system in which "'employers are expeditiously determining whether individuals have disabilities and then moving on to the heart of the matter: judging fitness based on individual abilities and qualifications, and providing accommodation where needed to do the job.' Over time, that should result in a reduction in ADA litigation" (Dorrian 2014).
Thus, the EEOC and Congress have been united in their vision for the 2008 Amendments to virtually eliminate the question of who is disabled for the purposes of the ADA. In terms of the model presented in Eq. (1), their hope has been that the Amendments would raise the costs of non-compliance for employers. Yet this paper suggests that the lofty goals of the EEOC and Congress have so far remained unattained, particularly in areas of the country that have not seen much enforcement of the expanded federal disability regime. Taking as its subject a group that has experienced a legal sea change as a result of the Amendments, this paper concludes that the morbidly obese have not seen systematic improvement in labor market outcomes since the 2008 Act. The post-ADAAA case law indicates that at least a few morbidly obese individuals have benefited from the increased protections of the ADAAA, but the data suggest that these improved labor market outcomes may not extend beyond the few morbidly obese workers who have successfully brought a lawsuit.
So why hasn't the ADAAA worked thus far for the morbidly obese—in spite of strong support by the EEOC and favorable district court decisions? One possibility is that the law in action may not exactly match the law on paper. For instance, if enough morbidly obese workers, employers, and/or employment lawyers remain largely unaware of the legal change, then enough morbidly obese workers may not be coming forward to enforce their newfound rights under the ADAAA. In fact, some evidence exists to support this lack-of-awareness hypothesis. According to EEOC Legal Counsel Mastroianni, not only do some morbidly obese individuals appear to be unaware of the 2008 Amendments but also "some courts are mistakenly citing and applying pre-ADAAA case law" (Dorrian 2014).
Even assuming that most courts, lawyers, and morbidly obese workers are aware of the positive changes brought about by the ADAAA, social and economic factors may discourage morbidly obese workers from coming forward to enforce their rights. Because the ADAAA was passed on the brink of a recession, and unemployment rates have remained relatively high during the post-ADAAA period, morbidly obese workers may be reluctant to challenge employers out of fear of losing their jobs in a tight labor market. Or perhaps morbidly obese workers are scared that coming forward will damage their reputation among potential future employers and will make finding a new job impossible.
Whatever the explanation behind the lack of improvement in the labor market outcomes of the morbidly obese since 2008, Resources for Human Development, BAE Systems, and the other pro-obesity ADAAA decisions would not be the first legal cases that failed to stimulate meaningful, on-the-ground change. Even landmark cases such as Brown v. Board of Education have been criticized for failing to have any systematic, on-the-ground impact, in spite of their symbolic impact (Rosenberg 2008; Bell 2004). Indeed, the findings presented here are in line with recent findings from an interview-based study conducted by George et al. (2011) regarding the on-the-ground impact of another employment discrimination case, Jespersen v. Harrah's Operating Co (9th Cir. 2006).Footnote 34 This 2006 Ninth Circuit decision, which centered on a casino bartender in Reno challenging her former employer's sex-based grooming standards, announced that "[i]f a grooming standard imposed on either sex amounts to impermissible sex stereotyping…a plaintiff of either sex may challenge that requirement" under Title VII. The case effectively opened the door to appearance-based, sex-stereotyping claims in the Ninth Circuit.
In spite of the widespread media coverage of Jespersen—and the widespread discussion of the case among lawyers and legal academics—the authors found that Las Vegas casino workers, who stood to gain the most from the decision, were completely unaware of it. And when the authors made them aware of the favorable legal decision, the casino workers expressed unwillingness to come forward and enforce their newfound rights. Among the excuses the workers gave included fears of losing their current hard-to-get, high-paying jobs and of being blacklisted from similar, future jobs (George et al. 2011).
This study suggests that the ADAAA has not been any more effective on the ground for morbidly obese workers than Jespersen has been for Ninth Circuit casino workers. Determining whether the 2008 Amendments will ultimately be a success or a failure will require more time and more study of other disabled populations. But these early estimates suggest that the ADAAA is following the same disappointing trajectory as the 1990 version of the Act.
Throughout this paper, weight categories are defined according to body mass index (BMI), which is calculated using the following equation: \( \mathrm{B}\mathrm{M}\mathrm{I}=\frac{\mathrm{weight}\left(\mathrm{lb}\right)\times 703}{{\left(\mathrm{height}\left(\mathrm{in}\right)\right)}^2} \). Using BMI, individuals are then classified as underweight if their BMI is less than 18.5, normal weight if their BMI is greater than or equal to 18.5 but less than 25.0, overweight if their BMI is greater than or equal to 25.0 but less than 30.0, obese if their BMI is greater than or equal to 30.0 but less than 40.0, and morbidly obese if their BMI is greater than or equal to 40.0.
Shinall (2016) suggests that, after comparing the occupational characteristics of the obese to the non-obese, the taste-based discrimination component is the dominant component of the obesity labor market penalty, particularly for women.
Whether an individual's disability is covered is always determined on a case-by-case basis. Note, however, that 42 U.S.C. § 12102(2) provides a non-exhaustive list of actions that Congress considers to be major life activities.
In order for the employer to pass over the disabled worker, the expected cost of accommodation must exceed the expected cost of non-compliance plus the difference in the expected profits of the workers (a > c + (π d − π nd)). The difference in expected profits of the two workers should not be very large as long as the non-disabled worker is the next best option after the disabled worker. Although this example has been stated in terms of the employer considering a positive employment action with respect to the disabled worker, the example could easily be reworked to situations in which the employer is considering taking a negative employment action against a disabled worker.
Terming such costs "firing costs," Acemoglu and Angrist (2001) believed that employers avoided hiring the disabled altogether in order to avoid the additional costs associated with their employment.
In fact, Clermont and Schwab (2004, 2009) argued, based on the notable decline in federal court case filings, that potential ADA plaintiffs (and employment discrimination plaintiffs generally) might be discouraged by high costs and low probability of success from filing lawsuits against employers.
E.E.O.C. v. Watkins Motor Lines, 463 F.3d 436 (6th Cir 2006) held that "to constitute an ADA impairment, a person's obesity, even morbid obesity, must be the result of a physiological condition."
In fact, one district court in the First Circuit that has subsequently discussed the case concluded that "Cook is instructive, yet not dispositive, on the issue of morbid obesity as an 'impairment.'" Nedder v. Rivier College, 908 F. Supp. 66, 75 n.8 (D.N.H. 1995). See also Katz v. City Metal Co., 87 F.3d 26, 31 (1st Cir. 1996); Mandel v. Boston Phoenix, Inc., 456 F.3d 198, 208 (1st Cir. 2006) (citing Cook only in its discussions of evidentiary burdens).
E.E.O.C. v. Resources for Human Development, Inc., 827 F. Supp. 2d 688, 694 (E.D. La. 2011); Equal Employment Opportunity Commission (2012a).
Melson v. Chetofield, 2009 WL 537457 (E.D. La. Mar. 4, 2009); Lowe v. American Eurocopter, LLC, 2010 WL 5232523 (N.D. Miss. Dec. 16, 2010).
2010 WL 5232523, at *8 (N.D. Miss. Dec. 16, 2010).
2009 WL 537457, at *3 (E.D. La. Mar. 4, 2009).
Whittaker v. America's Car Mart, Inc., 2014 WL 1648816, at *2 (E.D. Mo. April 24, 2014).
The Southern District of Alabama, where Powell was decided, is located within the Eleventh Circuit.
For examples of such articles, see Hager and Schwedler (2013) and Egan (2013).
If a national company with a presence in the First Circuit sets its human resources policy based on the most restrictive federal circuit, then it would be possible for Cook to have an impact outside the First Circuit. Nonetheless, any effects from the decision would always be strongest in the First Circuit.
A perennial issue in the disability literature is the role that changes in labor force participation have played in any observed changes in employment outcomes, which is why it is important to identify movement in and out of the labor market for the present study. For a richer discussion of changes in labor force participation rates after the passage of a disability law, see Hotchkiss (2004) (discussing the ADA) and Beegle and Stock (2003) (discussing state disability laws).
Note that in Eq. (2), I assume for simplicity that only two BMI groups, obese and non-obese, exist. In the actual empirical analyses, I will separately consider all five medical classifications of BMI: underweight (BMI < 18.5), overweight (25.0 ≤ BMI < 30.0), obese (30.0 ≤ BMI < 40.0), and morbidly obese (BMI ≥ 40), with normal weight (18.5 ≤ BMI < 25.0), as the omitted category.
The definition of "severe obesity" in the EEOC guidance seems to imply a meaning of double normal body weight given an individual's height, since the medical definition of normal body weight varies based on a person's height. Moreover, the fact that the EEOC has only taken on cases involving morbidly obese (not regularly obese) plaintiffs further indicates that the EEOC's use of the term "severe obesity" corresponds to morbid obesity.
Using self-reported weight and height data may raise concerns about systematic measurement error (in particular, systematic under-reporting of weight and/or systematic over-reporting of height), which could bias the results. Cawley (2004) developed a correction for self-reporting measurement error that uses NHANES data, which contains both self-reported weight and height and measured weight and height. Cawley (2004), Cawley et al. (2007), Lakdawalla and Philipson (2007), and Baum and Chou (2011) have all implemented the correction, but none of these authors have found that the correction changes their results in a meaningful way. These papers all studied the effect of obesity on wages and other labor market outcomes. A likely reason that the correction does not make much of a difference is that the specification used by these authors (as well as the specification used by the present paper) is only sensitive to large errors in measurement of weight and height. By using BMI categories (as opposed to a continuous BMI variable), measurement error will only affect the estimation if a person has so underreported her weight (or overreported her height) that she moves from one BMI category to another (i.e., from morbidly obese to obese, from overweight to normal weight, etc.).
A woman of average height (5 ft, 4 in.) would have to weigh almost 600 lb to have a BMI of greater than 100; a man of average height (5 ft, 10 in.) would have to weigh almost 700 lb to have a BMI of greater than 100. Keeping the few individuals in the sample with BMIs of over 100 does not meaningfully change the results.
For summary statistics using these datasets, see Cawley (2004), Ogden et al. (2010), and Shinall (2016).
Although the coefficient on the unemployment rate is highly statistically significant, in fact, it has little effect on the estimated coefficients for the variable of interest, post*morbidlyobese.
Controlling for year fixed effects should quell any remaining concerns about picking up the effects of the 2008 recession.
Excluding the circuit*year fixed effects has negligible effects on the estimated coefficients for the variable of interest, post*morbidlyobese.
Since the first EEOC lawsuit came in late 2010, and BRFSS observations are taken throughout the year, any effects of EEOC enforcement should be seen most clearly in the data during the post-2010 period, not the post-2009 period. A robustness check in Table 4 utilizes the recorded month of the BRFSS interview for each respondent and tests the post-September 2010 effect (instead of the post-2010 effect) and finds very similar results.
For a visual comparison of pre-trends in the data (which look quite similar, regardless of BMI classification), Appendix Figures 1 and 2 graph employment and labor market participation by BMI classification and gender. Appendix Table 10 also reports the estimated effects of the post-2008 period on two groups examined in post-1993 robustness checks by Carpenter (2006): smokers and diabetics. Carpenter argued that the 1993 Cook decision should not have had an effect on either group, and in fact did not find a post-1993 effect for either group, and used the robustness checks of these two groups to bolster his positive, statistically significant results for obesity. For the purposes of the present analysis, the ADAAA should not have affected smokers. American disability laws (and the surrounding case law) have never protected smokers, and in fact, the American legal regime has increasingly marginalized smokers in recent years. Thus, it is not surprising that the estimated post-2008 effect on smokers is negative and statistically significant in Appendix Table 10. Diabetics, in contrast, might have benefited from the expanded disability protections provided by the ADAAA. Although diabetes has not been as litigated as much as obesity in the post-ADAAA regime, the EEOC has similarly added diabetes to its list of potentially covered disabilities in the agency's revised ADAAA compliance guidelines. Like the estimated effects for obesity, the estimated effects of the post-2008 period on the labor market outcomes of diabetics are mostly zero, with the exception of a small, positive increase in the labor market participation of diabetic men since 2008.
The baseline specification defines morbidly obese individuals as anyone with a BMI of 40 or greater (instead of 35 or greater) for two reasons. First, the EEOC's definition of severe obesity (which the agency asserts is a disability for the purposes of the ADA) is double normal body weight, and a BMI of 35 is less than double normal body weight. Second, physicians classify all individuals with a BMI of 40 or greater as morbidly obese; physicians only classify individuals with a BMI between 35 and 40 as morbidly obese if they have significant comorbidities associated with their weight.
The additional controls are local law (a dummy variable equal to 1 for jurisdictions with a weight/personal appearance law), local law*underweight, local law*overweight, local law*obese, and local law*morbidlyobese.
An alternative to get rid of the potentially confounding effects of the First Circuit is to simply drop the First Circuit observations from the double-difference baseline regressions. Dropping the First Circuit produces very similar results to the baseline estimates.
For examples, see Waller (2011), Sixel (2011a), Sixel (2011b), and Sixel (2012). The charge-filing party in the Resources for Human Development case, Lisa Harrison, died of complications of morbid obesity shortly before the EEOC filed its suit against her former employer. As a result, the BAE Systems case generally received more media attention because reporters were able to interview the charge-filing party, Ronald Kratz.
Appendix Table 11 provides another reason to be cautious about the Fifth Circuit results: the post-2009 and post-2011 placebo tests produce some positive, statistically significant estimates.
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The author extends a special thanks to Joni Hersch, W. Kip Viscusi, R. Lawrence Van Horn, James Blumstein, Kevin Stack, and Richard Burkhauser for the helpful feedback throughout the course of this project.
Responsible editor: Pierre Cahuc
Vanderbilt University Law School, 131 21st Avenue South, Nashville, Tennessee, 37203, USA
Jennifer Bennett Shinall
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Table 9 Carpenter (2006) Nationwide BRFSS estimates compared to the First Circuit BRFSS estimates
Table 10 Estimated value of post-interaction term in placebo estimates
Table 11 Estimated value of Fifth Circuit post-interaction term in placebo estimates
2003–2013 men's and women's employment rates in the BRFSS data, by BMI classification. Notes: See Tables 2 and 3 for the definition of employment used throughout this paper
2003–2013 men's and women's labor market participation rates in the BRFSS data, by BMI classification. Notes: See Tables 2 and 3 for the definition of labor market participation used throughout this paper
Shinall, J.B. What happens when the definition of disability changes? The case of obesity. IZA J Labor Econ 5, 2 (2016). https://doi.org/10.1186/s40172-016-0041-0 | CommonCrawl |
Hair cortisol concentrations correlate negatively with survival in a wild primate population
Josué H. Rakotoniaina ORCID: orcid.org/0000-0003-1052-48201,2,
Peter M. Kappeler1,2,
Eva Kaesler2,
Anni M. Hämäläinen3,
Clemens Kirschbaum4 &
Cornelia Kraus1
BMC Ecology volume 17, Article number: 30 (2017) Cite this article
Glucocorticoid hormones are known to play a key role in mediating a cascade of physiological responses to social and ecological stressors and can therefore influence animals' behaviour and ultimately fitness. Yet, how glucocorticoid levels are associated with reproductive success or survival in a natural setting has received little empirical attention so far. Here, we examined links between survival and levels of glucocorticoid in a small, short-lived primate, the grey mouse lemur (Microcebus murinus), using for the first time an indicator of long-term stress load (hair cortisol concentration). Using a capture-mark-recapture modelling approach, we assessed the effect of stress on survival in a broad context (semi-annual rates), but also under a specific period of high energetic demands during the reproductive season. We further assessed the power of other commonly used health indicators (body condition and parasitism) in predicting survival outcomes relative to the effect of long-term stress.
We found that high levels of hair cortisol were associated with reduced survival probabilities both at the semi-annual scale and over the reproductive season. Additionally, very good body condition (measured as scaled mass index) was related to increased survival at the semi-annual scale, but not during the breeding season. In contrast, variation in parasitism failed to predict survival.
Altogether, our results indicate that long-term increased glucocorticoid levels can be related to survival and hence population dynamics, and suggest differential strength of selection acting on glucocorticoids, body condition, and parasite infection.
Identifying the links between physiological traits and fitness is vital for understanding the proximate mechanisms of selection that regulate natural populations. Glucocorticoid (GC) hormones are commonly employed as a biomarker of health or relative condition, both at the individual and the population level, since they mediate an array of physiological processes that can directly or indirectly impact fitness [1, 2]. As part of the hypothalamic-pituitary-adrenal (HPA) axis activity, GCs play a key role in the reallocation of resources in response to actual or perceived ecological challenges, such as inclement climatic conditions or predation pressure, that are associated with increased energetic needs [3,4,5]. Whereas the increase in GCs during acute stress is generally adaptive, chronic elevation of GC levels can compromise reproduction, immune function, and ultimately survival, thus reducing fitness [6,7,8,9,10]. For instance, individuals of various taxa have been found to potentially suffer fitness consequences of high GC concentrations through their increased susceptibility to parasite infection because of the immunosuppressive effects of GC [11,12,13,14,15]. The negative relationship between GC levels and fitness is at the core of the "Cort-fitness hypothesis", which posits that high levels of baseline GCs indicate poor individual or population condition [16, 17].
However, to date, tests of the Cort-fitness hypothesis in wild populations are rare [9, 16], and in some of the few studies that attempted to do so, this hypothesis failed to receive empirical support [reviewed in 16]. Plausible explanations of this failure include the lack of repeatability of both the cort-fitness relationship and GC measures. Indeed, the cort-fitness relationship has been frequently described to be context-dependent, and factors such as sex and reproductive strategy may influence the nature of this correlation [18, 19]. Furthermore, GCs are known to show strong fluctuations over time [5, 20] and within-individual variability can potentially mask the effect of GC variation on fitness among individuals [20]. The majority of studies that have examined this correlation were based on a single sampling of GCs, using established biomarkers of acute stress (i.e., serum, saliva or urine), potentially biasing estimates of overall individual condition. Hence, when multiple sampling is not possible, the use of a biomarker of chronic stress might better illuminate this relationship.
Recently, the measurement of hair cortisol concentration (HCC) has emerged as a promising new tool for monitoring long‐term HPA axis activity [21,22,23]. Free circulating GCs are thought to be incorporated into the hair shaft throughout its growth [24, 25]. Therefore, HCC is unaffected by potential short-term fluctuation in cortisol secretion and allows an assessment of accumulated cortisol levels over a wider time window compared to traditionally used matrices. Moreover, cortisol levels in hair were proven to be highly stable and consistent within individuals [26,27,28]. While HCC has been applied to monitor individuals' response to adverse conditions in several species (e.g., humans: [29]; chimpanzees: [30]; vervet monkeys: [31, 32]; marmosets: [33]; squirrels: [34]; cows: [35]; wolves: [36]), to our knowledge, no attempt was previously made to connect this indicator to measures of fitness in a wild population.
Studies of small, short-lived species can be advantageous when assessing the relationship between GCs and fitness in a wild setting, because in such study systems, physiological state can be measured repeatedly throughout the individual lifespan. The grey mouse lemur (Microcebus murinus), a small-bodied nocturnal primate (Lemuriformes: Cheirogaleidae), presents several key features allowing investigations of the potential relationship between individual condition and fitness. First, its average lifespan in the wild is 2–3 years, with a high annual turnover of around 50% [37, 38]. Second, in its highly seasonal habitat, strong fluctuation of water and food availability affects the feeding behaviour of M. murinus [39], but also several health indicators such as body mass, parasitism and levels of faecal GC metabolites [37, 40, 41]. Condition-dependent mortality is suggested to occur in mouse lemurs [37], and risky behaviour significantly influences male mortality during the breeding season [38]. While predation has been invoked as the probable leading cause of mortality [42], the proximate physiological mechanisms accompanying this non-random mortality remain unknown.
In this study, we test the hypothesis that high HCC (as a measure of long-term stress level) is related to individual survival in a wild population of M. murinus. Additionally, we assess the power of two other health indicators (size-adjusted body mass and patterns of parasitism) to predict survival. Specifically, we evaluate the potential link between these three health indicators and survival at two different levels. First, in order to define a general pattern, we test whether they can predict survival by assessing their relationship on a semi-annual basis, following the seasonal fluctuation of environmental conditions [43]. Second, we estimate the significance of individual condition under a specific context of high energetic demands by focusing on survival rates at the end of the dry season. The short mating season occurs during this period [44, 45], and it is associated with the lowest body mass and highest faecal GC metabolite levels [37, 40, 46]. We therefore predicted that individuals with high HCC suffer from lower survival. Furthermore, as individuals in poor health should be more vulnerable to ecologically adverse conditions, we expect survival to be positively correlated with general body condition, while individuals that exhibit a high degree of parasitism should face higher mortality.
Study site and population
The study was conducted in Kirindy Forest, which is a concession operated by the CNFEREF (Centre National de Formation, d´Etudes et de Recherche en Environnement et Foresterie), located approximately 60 km north of Morondava and a part of a dry deciduous forest in central western Madagascar [43]. We focused on a population of M. murinus from a 25 ha area (500 × 500 m) locally known as N5. This population has been continuously monitored since 2002.
Capture-mark-recapture
As a part of the long-term live capturing protocol, we conducted monthly capture sessions during the mouse lemurs' active period (from September to April) between 2012 and 2014. Trapping sessions consisted of three consecutive nights of capturing, using Sherman live traps baited with banana. Traps were set at 25 m intervals, at the intersections of a grid system of foot trails, in the late afternoon at 40–200 cm height, and checked in the early morning the next day. After being anesthetized with 0.02 ml ketamine (Ketavet®, Pfizer, Germany), captured animals were individually marked (or only identified without anesthetization if recaptured) with a subcutaneous transponder (Trovan EURO ID, Germany) and sexed, and standard field measurements, such as head width and body mass, were recorded. Hair samples for cortisol analysis and faecal samples for parasitology were also collected during capture sessions. While hair samples and morphometric measurements were collected semi-annually (in September–October and March–April), faecal samples were obtained opportunistically at a monthly rate.
To assess the relationship between HCC and semi-annual survival probabilities, we used the results of capture sessions held in October 2012, 2013, 2014, April 2013, and March 2014, during which a total of 171 individuals (74 females, 97 males) were captured. The same dataset, except the October 2014 session, was used to assess the effect of body condition on semi-annual survival probabilities, for a total of 149 individuals (63 females, 86 males). The link between survival probabilities and the health indicators (HCC, body condition, and parasitism) over the reproductive period was assessed by using data collected during monthly trapping sessions between September 2012 and April 2013. This dataset included 48 individuals (16 females and 32 males).
Sample collection and analysis
Assessment of hair cortisol concentration
In order to avoid potential variation of hair cortisol concentration (HCC) from different body regions [47, 48], we collected hair samples consistently from the animals' dorso-caudal region, where pelage coloration was reported to vary little across individuals and sexes [49], using a pet grooming clipper (Aesculap Isis GT 420). The detailed protocol for washing and extracting hair cortisol is described by Gao et al. [50], and all laboratory analyses were conducted at the University of Dresden (Germany). As a minor modification to the original protocol, because the hair structure of M. murinus prohibited the measurement of individual hair, we extracted cortisol using 7.5 mg of sample after washing (twice in 3 ml isopropanol for 3 min) and drying. The sample was further incubated with 40 μl internal standard and 2.4 ml methanol for 18 h at room temperature in a glass vial. After centrifugation at 10,000 rpm for 3 min, 1.6 ml of the clear supernatant was dried at 50 °C under a constant stream of nitrogen. The dry residue was re-suspended using 175 μl double-distilled water, 100 μl of which was used for cortisol concentration determination with liquid chromatography tandem mass spectrometry (LC–MS/MS). This assessment was performed using a Shimadzu HPLC-tandem mass spectrometry system (Shimadzu, Canby, Oregon) coupled to an ABSciex API 5000 Turbo-ion-spray triple quadrupole tandem mass spectrometer (AB Sciex, Foster City, California) with purification by on-line solid-phase extraction [50].
Although the mode (synchronous or asynchronous) and rate of hair growth are unknown for mouse lemurs, we are confident that the amount of hair we used for the HCC analysis is sufficient to accurately reflect a mean baseline cortisol concentration for this species. Indeed, as M. murinus has very dense fur, consisting of very thin hair. The number of hairs in a 7.5 mg sample (containing whole strands) is largely above 50–100 hairs, the number recommended by Fourie et al. [51] when studying medium-sized primates. Also, we observed that hair did not fully regrow after a month, motivating us to sample hair only at a semi-annual rate.
Assessment of general body condition
Instead of using body mass (BM) per se, we computed the scaled mass index (SMI) to assess body condition, thus controlling for the allometric relationship between body mass and body size [52]. This index yields an individual value of body mass standardized to the mean body size of all individuals present in the population. We used head width (HW) as a body size measurement due to its strong positive correlation with body mass in this species [see also 53]. The scaled mass index for every individual i was calculated as follows:
$$\mathop {SMI}\nolimits_{i} = \mathop {BM}\nolimits_{i} \left[ {\frac{{\mathop {HW}\nolimits_{0} }}{{\mathop {HW}\nolimits_{i} }}} \right]^{{\mathop b\nolimits_{SMA} }}$$
where HW0 (=21.92 mm) is the arithmetic mean of HW for the population and b SMA (=3.888) the slope of the standardized major axis (SMA) regression of ln (BM) on ln (HW). We used the software RMA [54] to calculate the value of b SMA .
Assessment of parasitism pattern
Fresh faeces collected opportunistically from handling bags or traps were weighted, directly homogenized with 10% formaldehyde and stored in 2 ml screw cap Sarstedt tubes. Parasite eggs and oocysts were identified under microscopic examination following a slightly modified Ritchie's ether sedimentation method [55]. Parasites were further classified up to the genus level based on egg or oocyst shape, size and internal structure [56,57,58], and prevalence, morphotype richness and occurrence of multiple morphotype infections were used to characterize the pattern of parasitism, as detailed in Rakotoniaina et al. [59].
To control for potential observer bias, we used blind observation by coding samples prior to laboratory analysis of hair cortisol levels and faecal parasites.
Modelling outline and candidate set of models
Semi-annual survival
In order to statistically estimate the link between HCC and SMI, as well as semi-annual survival (Φ), we used multistate capture-mark-recapture models [60,61,62,63,64] implemented in the program MARK version 8.0 [65], which account for recapture (p) and state-transition (ψ) probabilities. For each capture session, each individual was first assigned to a high or low HCC and SMI state, using the population median HCC or SMI value of the considered session as a cut-off point. Afterwards, in order to check if the correlation with survival is stronger at high ends of the health indicator values, we explored models where the categorization cut-off was based on the third quartile of HCC and SMI values. We could not use actual HCC and SMI values because modelling individually time-varying covariates (a different covariate value per individual at each recapture event) in MARK requires including a value of the covariates at each capture event even for missing (not recaptured) animals. Therefore, the multistate approach (using HCC/SMI categories) allows us to incorporate a variable value of the covariates between capture events (by accounting for ψ) while controlling for missing individuals (by accounting for p). Unlike HCC and SMI, the effect of parasitism on survival could not be modelled using this approach since we could not obtain faecal samples for every single individual at every single capture event. Therefore, indices of parasite infection were only considered for the assessment of reproductive season survival (see below).
Following Burnham and Anderson [66], we constructed a priori a candidate set of biologically plausible models (see Additional file 1). We assessed the goodness-of-fit of the global models and obtained an estimation of the variance inflation factor ĉ with the median-ĉ approach implemented in MARK. This method suggested that our data were slightly overdispersed [models with categorization set using the median: ĉ(HCC) = 1.204, ĉ(SMI) = 1.432; models with categorization set using the third quartile: ĉ(HCC) = 1.213, ĉ(SMI) = 1.432], thus model selection statistics were adjusted accordingly. Owing to the rather small sample size, we based our model selection on AICc (or QAICc in the presence of overdispersion) [67], which is an adjusted variant of the Akaike's information criterion (AIC). The difference (Δ i ) between the AICc of the most parsimonious model and a given model i and the normalized Akaike weights (w i ) were used to interpret the results of model selection. Hence, models with Δ i ≤ 2 were considered to have a strong support while models with 4 < Δ i < 7 have intermediate support and models with Δ i > 10 have negligible support [66]. The relatively low values of the Akaike weights of our top models (<0.9) for the HCC and the SMI datasets indicated model selection uncertainty and therefore, we adopted multi-model inference techniques over a confidence subset of models (all models with relative likelihood >0.05; see Additional files 2, 3). Thus, the importance of a variable [given as w + (variable)] was determined by summing the Akaike weights of models containing the variable of interest. Parameter estimates and their unconditional standard errors were calculated by averaging over all models in our confidence subset of models [66, 68].
We established all our candidate model sets by including factors known to influence mouse lemurs' survival [38] and consequently considered the factors sex (s) and time (t) in addition to our measure of the animal condition (c; high/low HCC or SMI). As a global model, we used \(\varPhi \left( {c*s + t} \right)\;p\left( {s + t} \right) \, \psi \left( {c*t} \right)\) (*: interactive effect, +: additive effect). Subsequently, all possible additive combinations of c, s and t and their single effects were used to model survival probability (Φ). Recapture probability (p) was additionally considered to be time dependant or constant over time. The condition index (c) was not included to model recapture probability since previous studies have reported a lack of a link between stress responses and previous capture experience [69], but also an increasing recapture probability ("trap happiness") of most individuals in this mouse lemur population [38], suggesting no long-term physiological cost of capture activities. Finally, we further modelled state-transition probability (ψ) to depend only on c. We fitted a total of 54 models for each condition index by considering all combinations of parameterization used for Φ, p and ψ.
Breeding season survival
We further estimated the potential association between our health indicators (HCC, SMI and parasitism) and survival (Φ) and recapture probabilities (p) over the breeding season by using the "Cormack-Jolly-Seber" model for open populations [70,71,72] implemented in MARK. In order to get an accurate estimation of (Φ) and (p) over the reproductive season, we used data from monthly trapping sessions conducted between the end of the dry season (September 2012) and the end of the rainy season (April 2013). However, as data for January and February 2013 were missing, we controlled for the bias that this gap may have induced to the estimation of (Φ) and (p) by manually adjusting the time interval of trapping sessions between December and March (3 months instead of one) in our models. Thereafter, we proceeded in two steps. First, we established a starting set of models by using \(\varPhi \left( {s* t} \right)\;p\left( {s + t} \right)\) as the global model and further comparing all possible permutation of models with an effect of \(\left( {s + t} \right)\), s and t on survival probability and s and t on recapture probability along with a constant Φ and p (see Additional file 4). Then, our health parameter values (HCC, SMI, parasite morphotype richness, overall prevalence and multiple species infection) were successively included as an individual covariate (for the first month only) to the most parsimonious model (basic model) among the starting set to assess if their inclusion improved the fit of the model. Additionally, we checked for potentially normalizing selection that might favour optimal HCC and body mass values and therefore tested for a quadratic effect of HCC and SMI on survival probabilities. We also fitted models including natural log-transformed HCC and SMI values to our monthly capture data. As above, the goodness-of-fit of the global model was assessed using the median-ĉ approach (all ĉ were <1) and model selection was based on the information theoretical approach [66].
Additionally, we tested for potential intercorrelations among HCC and the other health indicators. If existing, those correlations might mask or interfere with the assessment of their independent links to survival. Yet, no association was detected; neither between HCC and SMI (r 2 = 0.006, df = 227, p = 0.247) nor between HCC and parasitism pattern (species richness: r 2 = 0.053, df = 46, p = 0.112; overall prevalence: r 2 = 0.068, df = 46, p = 0.073; multiple infection: r 2 = 0.062, df = 46, p = 0.087).
Semi-annual survival relative to HCC values
Multistate models applied on HCC revealed that survival is lower for mouse lemurs with elevated levels of hair cortisol. The gap in survival probability between low and high HCC individuals is larger at high ends of HCC values (Fig. 1a, b). Individuals with low HCC had on average (based on geometric mean across years regardless of sex) a 9.8% higher chance to survive than those with high HCC when the categorization was set using the median HCC value. This gap increased to up to 13.9% when the categories were defined using the third quartile value (Fig. 1a, b). In both cases, in addition to the HCC effect, the best-supported models (Δ i < 2) also suggested a sex difference in survival (Table 1). Females survived relatively better than males (Fig. 1a, b; geometric means over time where the median is used as categorization cut-off: Φ low HCC F = 0.758, Φ low HCC M = 0.724, Φ high HCC F = 0.664, Φ high HCC M = 0.622; geometric means over time where the third quartile is used as categorization cut-off: Φ low HCC F = 0.729, Φ low HCC M = 0.694, Φ high HCC F = 0.594, Φ high HCC M = 0.550). Yet, multi-model inference emphasized that the relative importance of the effect of HCC on survival was higher than the effect of sex independently of the method used to set the categories [median cut-off: w + (HCC) = 0.701; w + (sex) = 0.504; third quartile cut-off: w + (HCC) = 0.756; w + (sex) = 0.487], further highlighting the strong support for a lowered survival of individuals with high HCC values.
Semi-annual survival probabilities of M. murinus. Estimates are relative to: hair cortisol concentration (a and b), where the categorization cut-off is the median (a), or the third quartile (b); and the scaled mass index (c and d), where the categorization cut-off is the median (c) or the third quartile (d). Presented are model-averaged maximum likelihood estimates and unconditional standard errors (Filled symbols/low low value of the condition index, Open symbols/high high value of the condition index, Circles/F females, Squares/M males). Estimates are averaged (geometric mean) over capture sessions
Table 1 Model selection statistics (multistate approach) for semi-annual survival (Φ), recapture (p) and state-transition (ψ) probabilities of M. murinus depending on hair cortisol concentration and general body condition (measured as scaled mass index)
In both approaches, our candidate set of models showed limited support for between-season variation in survival [median cut-off: w + (t) = 0.170; third quartile cut-off: w + (t) = 0.108) but instead a strong variability of recapture probabilities through time [median cut-off: w + (t) = 0.805; third quartile cut-off: w + (t) = 0.756]. All models in the confidence sets supported that a transition to a given state (high or low HCC) depended mainly on the current state of the individual (Table 1).
Semi-annual survival relative to SMI values
We found only weak support for an effect of SMI on survival in comparison to the effect of sex when categories were established according to the median SMI value [w + (sex) = 0.625; w + (SMI) = 0.441]. On average (based on the geometric mean across seasons), females had around 6% higher chance to survive to the next season than males but the difference of survival probability between conditions was negligible (Fig. 1c). However, our results also suggested that M. murinus in very good body condition (categories based on the third quartile regardless of sexes) survive on average 13.7% better than low condition animals (Fig. 1d), and strong support for a positive effect of SMI on survival was obtained with those models [w + (SMI) = 0.744; w + (sex) = 0.513].
Additionally, a time varying recapture probability structure was strongly supported by our confidence set of models in both scenarios [median cut-off: w + (t) = 0.895; third quartile cut-off: w + (t) = 0.884].
Health indicators and survival over the breeding season
The most parsimonious model (basic model) among the starting set contained time varying survival and recapture probabilities \(\left( {\varPhi \left( t \right)\;p\left( t \right)} \right)\). All models including HCC as a predictor of survival had a better fit than the basic model (Table 2). The top model (with natural log-transformed HCC) was more than four times better supported than the basic model [w (ln(HCC)) = 0.366, w (t) = 0.0.83; 0.366/0.083 = 4.404]. In the two best-supported models (Δ i < 2) we found a negative relationship between HCC and survival [ln (HCC), Fig. 2a; HCC, Fig. 2b]. In contrast, we found no evidence of a link between survival and SMI or parasitemia, as the basic model performed better than all the models containing the other health indicators as covariates (SMI, multiple parasite species richness, parasite morphotype richness and overall prevalence; Table 2).
Table 2 Model selection statistics for monthly survival (Φ) and recapture (p) probabilities of M. murinus
Monthly survival probabilities of M. murinus relative to the hair cortisol concentration. Considered is linear effect on the natural log-transformed HCC (values presented here are back transformed to the original scale). Presented are main effect (solid line) and 95% CI (dashed lines). Histograms represent the sample size for each category
In this study, we tested the hypothesis that high HCC (as a measure of long-term activation of the HPA axis) translates into reduced survival. Using wild mouse lemurs previously known to face condition-dependent mortality [37], we further tested the correlation of two other health indicators (body condition and patterns of parasitism) with survival in order to compare their predictive power for fitness outcomes. As predicted by the Cort-fitness hypothesis [16], both semi-annual survival and survival over the reproductive period were negatively associated with the level of accumulated hair cortisol. The first approach revealed that the relationship between HCC and survival is particularly strong at the high end of HCC values. Furthermore, our result suggested that individuals in extremely good condition enjoy higher survival probabilities than the ones with mid to low SMI values. In contrast, there was little support for the effects of SMI and measures of parasitism (multiple parasite species richness, parasite morphotype richness and overall prevalence) on survival over the reproductive period.
The relationship between stress and survival in a wild primate population
Our results provide general support for the Cort-fitness hypothesis which posits that elevated GCs are associated with a decline in one component of fitness [16]. Under this hypothesis, individuals in poor quality are assumed to perceive their environment as challenging and therefore secrete higher levels of GC than good quality individuals. However, similar to what has been reported by several other studies [73, 74], our data suggest that the range of GC concentration can be wide and thus, emphasize the need for caution when interpreting differences in GC concentrations without a proper assessment of their biological significance [75]. Individual significant differences in GC concentrations do not necessarily translate into significant diverging biological effects and conversely, slight changes in hormonal levels could be important. For instance, Pride [8] found that GC can be a sensitive indicator of survival probabilities but especially at very high values. Although natural selection can strongly operate on GC regulation, the high inter-individual variability for this trait seen in the wild could be maintained if the divergent stress responses offer alternative strategies with differing payoffs depending on environmental conditions [76, 77].
Furthermore, the assessment of how HCC correlates with survival of M. murinus over the reproductive season supported the notion that the relationship between GC and fitness can be context-dependent [9, 78]. Indeed, the benefit of having a relatively low stress load seems to be maximized prior to entering the breeding season as previously observed for various bird species [17, 79]. Individuals that are already strongly affected by challenging conditions during the dry season might not be able to cope with the additional costs related to the mating season. This brief period is particularly challenging for male mouse lemurs, which show a drastic increase in mortality coupled with significant body condition deterioration over the mating season [38, 46] that might be proximately mediated by the adverse physiological consequences of the stress accumulated over the dry season. Unlike females, which hibernate for several months, male mouse lemurs stay active and only undergo daily torpor during the dry winter [46, 80]. Staying active and the need to be energetically prepared to face the mating season [81] seems to physiologically affect males, which showed a higher HCC than females at the end of the dry season [59]. However, we could not detect this sex difference in the present data set, probably due to the limited sample size. This limitation emphasizes the importance of chronologically isolating specific processes in the life cycle in order to comprehend the proximate mechanisms that impact the survival of a given population.
High mortality rate as a cost of high reproductive success was described for several study systems [82, 83], and GC hormones were suggested to be central in mediating this trade-off [84]. Lee and Cockburn [85] proposed that during the mating period, animals (especially males) may exhibit an adaptive stress response, which can compromise their survival but promotes reproductive fitness by permitting a redirection of energy to reproduction. For instance, such terminal investment was detected in male arctic ground squirrels [86], and it was evoked that it may occur in mammalian species with similar life history traits characterized by a single annual breeding opportunity per year coupled with high between-year mortality [84]. This physiological adaptation might occur in male grey mouse lemurs facing strong intrasexual competition over access to receptive females [45] and thus, it could severely affect individuals that are showing already signs of high stress load at the beginning of the mating season. An estimation of the reproductive success of animals showing high GC levels will help to test this hypothesis.
Several reasons could lead to the high mortality of chronically stressed individuals, including impaired immune and inflammatory responses leading to impaired resistance to diseases [87], or a maladaptive adrenocortical response to additional unpredictable stressors that might impair the animals' coping ability [88]. However, our results seem to argue against the hypothesis of increased mortality due to increased parasitism. All models that included an indicator of parasitism failed to support a monthly survival trend and suggested that multiple parasite species infection, parasite morphotype richness, and overall prevalence were poor predictors of individual survival. Acquired immunity against helminth infections [41] might further explain the relatively low selective pressure on parasitism. As parasite virulence and host tolerance might also be highly variable, these findings highlight the degree of uncertainty associated with the use of basic measures of parasitism as biomarker of health without information on parasite pathogenicity.
Additionally, chronically stressed individuals could fail to mount an adaptive HPA activity response to an acute stressor such as predation which might increase their vulnerability during the mating season. For instance, the grey mouse lemur is known to be preyed upon by several predators such as snakes, owls or another lemur (Mirza coquereli), and although they face a continuously high predation risk [42, 89, 90], this threat might be maximal at the peak of the activity period of both predators and prey. In this case, an inappropriate physiological response to the presence of a predator could be translated into reduced reaction time or escaping ability of the high-stress individuals. Several studies have reported that GC responses to acute stressors were down-regulated in animals exposed to chronic stress [3, 91]. For instance, in lemurs, Tecot [92] found that Eulemur rubriventer showed an attenuated GC response to known seasonal environmental challenges in altered habitats. While this response could be aimed at reducing the detrimental effects of chronic elevation of GC levels, it may negatively affect the capacity of an animal to face acute life-threatening stressors.
Age is a factor that could influence the GC-survival relationship, but it could unfortunately not be addressed in this study. Previous studies suggested that both survival and GC profile are age-dependent in grey mouse lemurs. For instance, Kraus et al. [38] reported lower survival of juveniles over the dry winter but no significant difference between juvenile and adult survival probabilities over the summer. Also, while older individuals were found to have higher faecal GC metabolites during the breeding season [40], Rakotoniaina et al. [59] showed that HCC was higher in juvenile M. murinus. The contradicting results found in those studies might have arisen from the different matrices used to assess physiological stress but also from the definition of age: while the first study used estimates of individual age, the latter applied age categorization. This age-GC link was also detected in various study systems [93,94,95,96,97] and is mainly assumed to be associated with the impaired ability of aged individuals to cope with challenges [98]. Considering age is therefore recommended in future studies examining the link between GC and fitness.
Body condition, stress and survival
As suggested previously [37], we found that mortality is condition-dependent in M. murinus. In fact, very good condition (measured as scaled mass index) was associated with a high semi-annual survival probability. Poor quality individuals can be more vulnerable to diseases [99] and predation [100] but also have a lower capacity to face competition [101]. While body mass has been reported to consistently decline in chronically stressed individuals [102], it is still necessary to disentangle and define the causal effect of body condition and stress hormones on the survival output at high values of these health indicators. As body condition and GC levels were not correlated in our study population, selection on either of these traits should not influence the detection of survival selection on the other one [74].
Also, body condition was not correlated with monthly survival, which may indicate that different selective pressures acting on this trait might have opposite effects during the mating season. For instance, if individuals in good condition are more active than weak individuals in this period, this may increase their probability to encounter predators [103,104,105]. The benefits of being in better condition (e.g. low susceptibility to disease, high success in resource competition) could therefore be curtailed by increased predation risk. Yet, if body condition does not correlate with survival from low to mid values, as suggested by the multistate analysis, this failure of body condition to explain monthly survival could also arise from the limited sample size being used, which does not allow us to detect this trend. Overall, our results indicate the possibility that physiological traits are under stronger selection in terms of survival consequences than body condition. Additional studies of the heritability and effects of these health indicators on reproductive success would be needed, however, to confirm the overall selective potential for these traits.
Hair cortisol concentration as a reliable health indicator
The gold standard of validating a biological indicator of health is to show that it correlates with fitness. Here, we demonstrate for the first time that HCC exhibited such a correlation in a wild population. Although the exact mechanism of incorporation of cortisol in the hair shaft is not yet well understood [106,107,108], it is mostly assumed that cortisol contained in hair is representative of free systemic concentration [For a review see 21, 22 but see 109, 110]. Similar to our study, Patterson et al. [74] found that free GC hormone levels may be more relevant than morphological traits as a predictor of survival in white-crowned sparrows.
The fundamental advantage of using hair as a matrix to assess cortisol levels is the broadness of the time window reflected by HCC. While HCC is assumed to account for up to several months of stress load, traditionally used biomarkers of stress (blood, faeces, urine, saliva) are point estimators and could fail to describe the true individual condition. Under natural conditions where animals cannot be continuously observed and sampled, it is difficult to obtain a reliable measure of the baseline GC level with such biomarkers. However, most studies that investigated this aspect in the wild rely solely on limited sampling of indicators of short-term stress responses that are likely affected, for instance, by daily level fluctuations or individual stressful events experienced prior to sampling [e.g. 73, 111]. Overall, this might explain the large inconsistency in the results so far reported from studies that attempted to link stress and fitness [16]. At present, the lack of precise information on hair growth rate in M. murinus limits our estimation about the period of accumulation of cortisol recorded with the hair samples. However, when an animal was captured and sampled early in September, we observed that the hair had fully regrown after a subsequent recapture of the same individual in December. Thus, we are confident that HCC in our study reflected the mean cortisol load over, at least, a substantial part of each period between sampling sessions.
Moreover, HCC has been shown to be heritable and reported to represent an individual trait that is affected by genes and environment [31]. Since cortisol is known to be closely linked to a series of other phenotypic traits [76, 112], levels of hair cortisol may indicate individual quality, where low quality individuals that might perceive their environment as more challenging secrete higher cortisol levels [113]. For instance in our study population, considering that HCC reflects an accumulation of cortisol over an extended period of time, and that individuals are assumed to face similar extrinsic pressures, it is very likely that individual differences in HCC reflect true differences in condition rather than a potential difference in exposure to various stressors. However, consistent individual behavioural responses to external stimuli, also refered to as "personality" [sensu 114], could interfere with this individual quality-GC profile relationship. Indeed, several studies have demonstrated that specific personality traits such as boldness can strongly correlate with HPA axis responses [115,116,117,118]. An investigation of such a relationship by conducting personality tests combined with HCC measurements might enrich the interpretation of information obtained from hair cortisol levels.
This study provides support for the Cort-fitness hypothesis by demonstrating that survival is negatively associated with levels of hair cortisol concentrations in a wild grey mouse lemur population. This study therefore provides a first confirmation of the predictive power of HCC variation on individual fitness in a wild setting. Moreover, we demonstrate that, while GC, body condition and parasite resistance could all influence individual survival, their effects might differ in strength. Thus, we emphasize that care must be taken when interpreting such indices without prior knowledge of their effect on fitness. Although our approach is correlational and does not permit the identification of the exact causes of mortality, it suggests that variation in GC hormone concentrations alone may underlie demographic fluctuations of natural populations. Thus, these results highlight the need to consider environmental pressures that can affect GC levels as potential threats to survival. Since population decline is often hard to measure, the assessment of an individual health indicator such as long-term stress levels could, therefore, provide an easier alternative for detecting issues emerging at the population level and ultimately predicting wild populations' responses to environmental challenges.
HCC:
hair cortisol concentration
HPA axis:
hypothalamic–pituitary–adrenal axis
BM:
body mass
SMI:
scaled mass index
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CKraus and PK originally formulated the idea. CKirschbaum developed and implemented the hormone analyses. JR, EK, and AH collected field data. JR analyzed the data and wrote the manuscript with input from PK, AH, EK, and CKraus. All authors read and approved the final manuscript.
We thank Bruno Tsiverimana and l´Equipe Kirindy as well as Pauline Thomas for their valuable help in collecting data, Léonard Razafimanantsoa and Rodin Rasoloarison for administrative and logistic support, Vololomboahangy Andrianaja and Pascaline Ravoniarimbinina for making the collaboration with the Institut Pasteur of Madagascar possible, Juliane Graß, Christina Glaschke, research technicians in Institut Pasteur de Madagascar and Technische Universität Dresden for helping in laboratory works, Henning Lahmann for helping with the long-term database. We thank two anonymous reviewers for their constructive comments that improved an earlier version of this manuscript. We acknowledge the collaboration of the Département de Biologie Animale of the University of Antananarivo. The Ministère de l´Environnement, de l´Ecologie, de la Mer et des Forêts and the CNFEREF Morondava authorized research in Kirindy.
The dataset supporting the results of this article is available in the Figshare repository, https://dx.doi.org/10.6084/m9.figshare.5259415 [119].
All research activities conducted in Madagascar got an official approval from the Ministère de l´Environnement, de l´Ecologie, de la Mer et des Forêts and comply with animal care national legislations of Madagascar.
This work was supported by the "Deutscher Akademischer Austausch Dienst" [Awarded to JRakotoniaina, A/12/90426] and the "Deutsche Forschungsgemeinschaft" [Awarded to CKraus, KR3834/4-1].
Department of Sociobiology/Anthropology, Georg-August University of Göttingen, Kellnerweg 6, 37077, Göttingen, Germany
Josué H. Rakotoniaina, Peter M. Kappeler & Cornelia Kraus
Behavioral Ecology and Sociobiology Unit, Deutsches Primatenzentrum, Kellnerweg 4, 37077, Göttingen, Germany
Josué H. Rakotoniaina, Peter M. Kappeler & Eva Kaesler
Department of Biological Sciences, University of Alberta, Edmonton, AB, T6G 2E9, Canada
Anni M. Hämäläinen
Department of Psychology, TU Dresden, Andreas-Schubert-Bau, Zellescher Weg 19, 01069, Dresden, Germany
Clemens Kirschbaum
Josué H. Rakotoniaina
Peter M. Kappeler
Eva Kaesler
Cornelia Kraus
Correspondence to Josué H. Rakotoniaina.
Additional file 1.
Full set of starting candidate models. This additional file presents the full list of candidate set of biologically plausible models that were constructed a priori and used for the semi-annual survival estimation.
Confidence set of models for semi-annual survival of M. murinus depending on hair cortisol concentration. This additional file presents the full list of models with relative likelihood > 0.05 obtained from the semi-annual survival estimation of M. murinus depending on hair cortisol concentration.
Confidence set of models for semi-annual survival of M. murinus depending on scaled mass index. This additional file presents the full list of models with relative likelihood > 0.05 obtained from the semi-annual survival estimation of M. murinus depending on scaled mass index.
Starting set of models for monthly survival assessment of M. murinus. This additional file presents the full list of starting set of models used in order to identify the basic model for the monthly survival estimation of M. murinus depending on hair cortisol concentration and scaled mass index.
Rakotoniaina, J.H., Kappeler, P.M., Kaesler, E. et al. Hair cortisol concentrations correlate negatively with survival in a wild primate population. BMC Ecol 17, 30 (2017). https://doi.org/10.1186/s12898-017-0140-1
Received: 15 March 2017
Accepted: 03 August 2017
Cort-fitness hypothesis
Body condition
Microcebus murinus | CommonCrawl |
APPOL 2 Workshop
University of Bologna Residential Center
Bertinoro (Forlì), Italy
[ What the Meeting is About
| Seminar Schedule
| Important Dates
| Location
| How to Reach Bertinoro
| List of Participants
| Abstracts
| Organization and Sponsorship
| Local Weather Forecast]
What the Meeting is About
Appol 2 is a thematic network of the European Union, focussing on on-line and approximation algorithms. The Appol network consists of 15 universities.
The format of this meeting will be about two, two and a half days of lectures, followed by a few days during which research problems will be discussed in a congenial atmosphere.
Tentative Schedule
Thomas Erlebach Frits Spieksma
Alexander Hall Leen Stougie
Martin Skutella Sai Anand
Fernandez Dela Vega Guochuan Zhang
Gerold Jäger Rene Sitters
Stefano Leonardi
Arrival: Saturday 22 March, 2003
Departures: 28-29 March, 2003
Last day of talks:
Tuesday 25 March, 2003
End of workshop:
The meeting will be held in the small medieval hilltop town of Bertinoro. This town is in Emilia Romagna about 50km east of Bologna at an elevation of about 230m. Here is a map putting it in context. It is easily reached by train and taxi from Bologna and is close to many splendid Italian locations such as Ravenna, a treasure trove of byzantine art and history, and the Republic of San Marino (all within 35km) as well as some less well-known locations like the thermal springs of Fratta Terme and the castle and monastic gardens of Monte Maggio. Bertinoro can also be a base for visiting some of the better-known Italian locations such as Padua, Ferrara, Vicenza, Venice, Florence and Siena.
Bertinoro itself is picturesque, with many narrow streets and walkways winding around the central peak. The meeting will be held in a redoubtable ex-Episcopal fortress that has been converted by the University of Bologna into a modern conference center with computing facilities and Internet access. From the fortress you can enjoy a beautiful the vista that stretches from the Tuscan Apennines to the Adriatic coast.
How to Reach Bertinoro
List of confirmed participants so far. More can join later.
Susanne Albers, Universität Freiburg
Sai Anand, ETH Zürich
Georg Baier, Technische Universität Berlin
Euripides Bampis, Universitè D'Evry
Debora Donato, Università La Sapienza di Roma
Fabian Ennecke , ETH Zürich
Thomas Erlebach, ETH Zürich
Wenceslas Fernandez De La Vega, Universitè Paris-Sud
Amos Fiat, University of Tel Aviv
Alexander Hall, ETH Zürich
Gerold Jäger, Universität Kiel
Klaus Jansen, Universität Kiel
Miklos Kresz, University of Szeged
Stefano Leonardi, Università La Sapienza di Roma
Alberto Marchetti Spaccamela, Università La Sapienza di Roma
Ioannis Milis, Athens University of Economics and Business
Marc Nunkesser, ETH Zürich
Alessandro Panconesi, Università La Sapienza di Roma
Rene Sitters, Technische Universiteit Eindhoven
Martin Skutella, Technische Universität Berlin
Ines Spenke, Technische Universität Berlin
Frits Spieksma, Katholieke Universiteit Leuven
Leen Stougie, Technische Universiteit Eindhoven
Gabor Szabo, ETH Zürich
Joris van de Klundert, Universiteit Maastricht
Peter Widmayer, ETH Zürich
Guochuan Zhang, Universität Kiel
Organization and Sponsorship
Stefano Leonardi Università La Sapienza di Roma
Alessandro Panconesi Università La Sapienza di Roma
Local Organization
Andrea Bandini, Elena Della Godenza, Centro Congressi di Bertinoro
Sponsored by BICI Bertinoro International Center for Informatics
On Paging with Locality of Reference
by Susanne Albers, Universität Freiburg
Motivated by the fact that competitive analysis yields too pessimistic results when applied to the paging problem, there has been considerable research interest in refining competitive analysis and in developing alternative models for studying online paging. The goal is to devise models in which theoretical results capture phenomena observed in practice. In this talk we propose a new, simple model for studying paging with locality of reference. The model is closely related to Denning's working set concept and directly reflects the amount of locality that request sequences exhibit. We demonstrate that our model is reasonable from a practical point of view. We use the page fault rate to evaluate the quality of paging algorithms, which is the performance measure used in practice. We develop tight or nearly tight bounds on the fault rates achieved by popular paging algorithms such as LRU, FIFO, deterministic Marking strategies and LFD. It shows that LRU is an optimal online algorithm, whereas FIFO and Marking strategies are not optimal in general. We present an experimental study comparing the page fault rates proven in our analyses to the page fault rates observed in practice. This is the first such study for an alternative/refined paging model.
(Joint work with Lene Favrholdt and Oliver Giel.)
A Linear Bound on the Diameter of the Transportation Polytope
by Leen Stougie, Technische Universiteit Eindhoven
We prove that the combinatorial diameter of the skeleton of the polytope of feasible solutions of any $m \times n$ transportation problem is less than $8 \, (m+n)$.
A competitive algorithm for the general two server-problem
by Rene Sitters, Technische Universiteit Eindhoven
We consider the general on-line two server problem in which at each step both servers receive a request, which is a point in a metric space. One of the servers has to be moved to its request. The special case where the requests are points on the real line is known as the CNN-problem. It has been a well-known open question if an algorithm with a constant competitive ratio exists for this problem. We answer this question in the affirmative sense by providing an algorithm that is 23808-competitive on any metric space. The constant is huge and the algorithm is rather ugly but our main goal was indeed to settle the question. We believe that our result gives new insight in the problem and will lead to more and much better algorithms for the general $k$-server problem in the near future.
Joint work with Leen Stougie and Willem de Paepe.
Polynomial time approximation schemes for metric MIN-BISECTION
by Fernandez De La Vega, Universitè Paris Sud
We design the first polynomial time approximation schemes (PTASs) for the problem of Metric MIN-BISECTION: given a finite metric space, divide the points into two halves so as to minimize the sum of distances across that partition. Our approximation schemes depend on biased sampling and on a new application of linearized quadratic programs with randomized rounding of non-uniformly bounded numbers.
Joint work with Marek Karpinski and Claire Kenyon
Smoothed Competitive Analysis of the Multi-Level Feedback Algorithm
by Stefano Leonardi, Università di Roma La Sapienza
Spielman and Teng invented the concept of smoothed analysis to explain the success of algorithms that are known to work well in practice while presenting poor worst case performance. Spielman and Teng [STOC 01] proved that the simplex algorithm runs in expected polynomial time if the input instance is smoothened with a normal distribution. We extend this notion to analyze online algorithms. In particular we introduce smoothed competitive analysis to study the Multi-Level Feedback (MLF) algorithm, at the basis of the scheduling policies of Unix and Windows NT, when the processing times of jobs released over time are only known at time of completion.
We show that, if the k least significant of K bits describing the processing time of a job are randomly changed, MLF achieves a tight smoothed competitive ratio of O(2K-k) to minimize the average flow time. A direct consequence is a first constant approximation for this problem in the average case under a quite general class of probability distributions.
Joint work with L. Becchetti, A. Marchetti-Spaccamela, G. Schäfer, and T. Vredeveld.
Approximating minimum cuts for valid paths in the Internet
by Thomas Erlebach, ETH Zürich
A recently proposed model of the Internet at the level of autnomous systems (ASs) is a graph in which the edges are classified according to the economic relationships between the ASs that they connect: Edges can be customer-provider edges or peer-to-peer edges. It is assumed that a path in the network is only valid if it consists of a sequence of customer-provider edges, followed by zero or one peer-to-peer edges, followed by a sequence of provider-customer edges. Motivated by robustness considerations, we consider the problem of computing a minimum vertex cut that separates two given vertices (meaning that no valid path between them remains after removing the vertices of the cut from the network). We show that the problem is NP-hard and present a 2-approximation algorithm based on linear programming techniques.
(joint work with Danica Vukadinovic)
Flows over time - approximation and complexity
by Martin Skutella, Technische Universität Berlin
The intention of the talk is to give an introduction into the area of "flows over time" or "dynamic flows". Flows over time have been introduced about forty years ago by Ford and Fulkerson and have many real-world applications such as, for example, traffic control, evacuation plans, production systems, communication networks, and financial flows. Flows over time are modeled in networks with capacities and transit times on the arcs. The transit time of an arc specifies the amount of time it takes for flow to travel from the tail to the head of that arc. In contrast to the classical case of static flows, a flow over time specifies a flow rate entering an arc for each point in time and the capacity of an arc limits the rate of flow into the arc at each point in time. We discuss recent approximation and hardness results for flows over time with multiple commodities and costs. The talk is based on joint work with Lisa Fleischer, Alex Hall, and Steffen Hippler.
Routing and Call Control Algorithms in Ring Networks
by Sai Anand, ETH Zürich
FPTAS for Flows Over Time with Inflow-Dependent Transit Times
by Alexander Hall, ETH Zürich
Motivated by applications in road traffic control, we study flows in networks featuring special characteristics. In contrast to classical static flow problems, time plays an important role. Firstly, there are transit times on the arcs of the network which specify the amount of time it takes for flow to travel through an arc; in particular, flow values on arcs may change over time. Secondly, the transit time of an arc varies with the current amount of flow using this arc. The latter feature is crucial for various real-life applications of flows over time.
K\"ohler et al. (Proceedings of ESA 2002, LNCS 2461, pp. 599-611) study the single commodity case and give a (2+$\varepsilon$)-approximation based on the notion of temporarily repeated flows, where flow is pushed at constant rates into (possibly) several paths which do not change over time. We are able to generalize this result to the multicommodity case, by modifying the underlying network and the way flow is pushed into the corresponding paths. Applying the same technique to condensed time-expanded networks allows variation of flow paths over time. This approach even yields a fully polynomial time approximation scheme for the multicommodity case, which is known to be NP-hard. Furthermore we show that under a certain restriction of valid flows, the single commodity case is NP-hard as well.
Approximation of a retrieval problem for parallel disks
by Frits Spieksma, Katholieke Universiteit Leuven
When handling large collections of data, the communication between fast internal memory and slow external memory (e.g. disks) can be a performance bottleneck. We study the problem of maximizing the throughput (that is, the number of requests served per time-unit) for parallel disks. This retrieval problem can be formulated as assigning a maximum number of size 1 and size 2 jobs to machines of limited capacity. We show that the LP-relaxation of an integer programming formulation is half-integral. Further, we sketch 2/3-approximation algorithms for this problem.
Joint work with Joep Aerts and Jan Korst.
Improved Approximation Algorithms for Maximum Graph Partition Problems
by Gerold Jäger, Universität Kiel
Joint work with Anand Srivastav
We consider a weighted, directed graph with n vertices and the problem of dividing the set of vertices in two parts of sizes k and n-k, so that the edges of some of four regions (edges on the two subgraphs and edges between the two subgraphs in both directions) becomes maximum. For example, MAX-k-DENSE-SUBGRAPH is the problem of determining a subset S of size k, so that the sum of the edge weights of the subgraph induced by S becomes maximum. There are six nontrivial problems of this kind.
Halperin and Zwick have introduced an algorithm based on semidefinite programming, which for the case k = n/2 and these six maximum graph partition problems delivers the best known approximation ratios. We show improvements of their techniques and generalize the algorithm for arbitrary k.
On weighted rectangle packing
by Guochuan Zhang, Universität Kiel
Joint work with Klaus Jansen.
Approximation algorithms for max-min resource sharing, with an application to fractional weighted graph coloring
by Klaus Jansen, Universität Kiel
We generalize a method by Grigoriadis et al. to compute an approximate solution of the max-min resource sharing (and fractional covering) problem with $M$ nonnegative concave (linear) constraints $f_m$ on a convex set $B$ to the case with general approximate block solvers (i.e. with only constant, logarithmic, or even worse approximation ratios). The algorithm is based on a Lagrangian decomposition which uses a modified logarithmic potential function and on several other ideas (scaling phase strategy, two stopping rules in a phase, eliminating functions $f_m$ larger than a threshold value $T$, reducing the step length and taking a convex combination among different iterates in a phase). We show that the algorithm runs in $O(M \epsilon^{-2} \ln (M \epsilon^{-1}))$ iterations (or block optimization steps), a data and approximation ratio independent bound. Furthermore we show how to use this framework for the fractional weighted graph coloring problem.
Maintained by Alessandro Panconesi | CommonCrawl |
Pinwheel tiling
In geometry, pinwheel tilings are non-periodic tilings defined by Charles Radin and based on a construction due to John Conway. They are the first known non-periodic tilings to each have the property that their tiles appear in infinitely many orientations.
Wikimedia Commons has media related to Pinwheel tiling.
Conway's tessellation
Let $T$ be the right triangle with side length $1$, $2$ and ${\sqrt {5}}$. Conway noticed that $T$ can be divided in five isometric copies of its image by the dilation of factor $1/{\sqrt {5}}$.
By suitably rescaling and translating/rotating, this operation can be iterated to obtain an infinite increasing sequence of growing triangles all made of isometric copies of $T$. The union of all these triangles yields a tiling of the whole plane by isometric copies of $T$.
In this tiling, isometric copies of $T$ appear in infinitely many orientations (this is due to the angles $\arctan(1/2)$ and $\arctan(2)$ of $T$ each being algebraically independent to $\pi $ over the reals.). Despite this, all the vertices have rational coordinates.
The pinwheel tilings
Radin relied on the above construction of Conway to define pinwheel tilings. Formally, the pinwheel tilings are the tilings whose tiles are isometric copies of $T$, in which a tile may intersect another tile only either on a whole side or on half the length $2$ side, and such that the following property holds. Given any pinwheel tiling $P$, there is a pinwheel tiling $P'$ which, once each tile is divided in five following the Conway construction and the result is dilated by a factor ${\sqrt {5}}$, is equal to $P$. In other words, the tiles of any pinwheel tilings can be grouped in sets of five into homothetic tiles, so that these homothetic tiles form (up to rescaling) a new pinwheel tiling.
The tiling constructed by Conway is a pinwheel tiling, but there are uncountably many other different pinwheel tilings. They are all locally undistinguishable (i.e., they have the same finite patches). They all share with the Conway tiling the property that tiles appear in infinitely many orientations (and vertices have rational coordinates).
The main result proven by Radin is that there is a finite (though very large) set of so-called prototiles, with each being obtained by coloring the sides of $T$, so that the pinwheel tilings are exactly the tilings of the plane by isometric copies of these prototiles, with the condition that whenever two copies intersect in a point, they have the same color in this point.[1] In terms of symbolic dynamics, this means that the pinwheel tilings form a sofic subshift.
Generalizations
Radin and Conway proposed a three-dimensional analogue which was dubbed the quaquaversal tiling.[2] There are other variants and generalizations of the original idea.[3]
One gets a fractal by iteratively dividing $T$ in five isometric copies, following the Conway construction, and discarding the middle triangle (ad infinitum). This "pinwheel fractal" has Hausdorff dimension $d={\frac {\ln 4}{\ln {\sqrt {5}}}}=\log _{5}(16)\approx 1.7227$.
Use in architecture
Federation Square, a building complex in Melbourne, Australia, features the pinwheel tiling. In the project, the tiling pattern is used to create the structural sub-framing for the facades, allowing for the facades to be fabricated off-site, in a factory and later erected to form the facades. The pinwheel tiling system was based on the single triangular element, composed of zinc, perforated zinc, sandstone or glass (known as a tile), which was joined to 4 other similar tiles on an aluminum frame, to form a "panel". Five panels were affixed to a galvanized steel frame, forming a "mega-panel", which were then hoisted onto support frames for the facade. The rotational positioning of the tiles gives the facades a more random, uncertain compositional quality, even though the process of its construction is based on pre-fabrication and repetition. The same pinwheel tiling system is used in the development of the structural frame and glazing for the "Atrium" at Federation Square, although in this instance, the pin-wheel grid has been made "3-dimensional" to form a portal frame structure.
References
1. Radin, C. (May 1994). "The Pinwheel Tilings of the Plane". Annals of Mathematics. 139 (3): 661–702. CiteSeerX 10.1.1.44.9723. doi:10.2307/2118575. JSTOR 2118575.
2. Radin, C., Conway, J., Quaquaversal tiling and rotations, preprint, Princeton University Press, 1995
3. Sadun, L. (January 1998). "Some Generalizations of the Pinwheel Tiling". Discrete and Computational Geometry. 20 (1): 79–110. arXiv:math/9712263. CiteSeerX 10.1.1.241.1917. doi:10.1007/pl00009379. S2CID 6890001.
External links
• Pinwheel at the Tilings Encyclopedia
• Dynamic Pinwheel made in GeoGebra
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
\begin{definition}[Definition:True State]
Let $G$ be a game whose outcome is determined by the realization of a random variable $X$.
The particular state of $G$ which $X$ actually takes is known as the '''true state (of the world)'''.
\end{definition} | ProofWiki |
What is the largest value of $n$ less than 100,000 for which the expression $8(n-2)^5-n^2+14n-24$ is a multiple of 5?
By the Binomial Theorem, \begin{align*}
(n - 2)^5 &= n^5 - \binom{5}{1} \cdot 2n^4 + \binom{5}{2} \cdot 2^2 n^3 - \binom{5}{3} \cdot 2^3 n^2 \\
&\qquad + \binom{5}{4} \cdot 2^4 n - 2^5 \\
&= n^5 - 10n^4 + 40n^3 - 80n^2 + 80n - 32.
\end{align*} Note that this reduces to $n^5 - 32 \equiv n^5 + 3 \pmod{5}$. Therefore, \begin{align*}
8(n - 2)^5 - n^2 + 14n - 24 &\equiv 8(n^5 + 3) - n^2 + 14n - 24 \\
&\equiv 8n^5 + 24 - n^2 + 14n - 24 \\
&\equiv 3n^5 - n^2 - n \pmod{5}.
\end{align*}
If $n \equiv 0 \pmod{5}$, then \[3n^5 - n^2 - n \equiv 3 \cdot 0^5 - 0^2 - 0 \equiv 0 \pmod{5}.\] If $n \equiv 1 \pmod{5}$, then \[3n^5 - n^2 - n \equiv 3 \cdot 1^5 - 1^2 - 1 \equiv 1 \pmod{5}.\] If $n \equiv 2 \pmod{5}$, then \[3n^5 - n^2 - n \equiv 3 \cdot 2^5 - 2^2 - 2 \equiv 90 \equiv 0 \pmod{5}.\] If $n \equiv 3 \pmod{5}$, then \[3n^5 - n^2 - n \equiv 3 \cdot 3^5 - 3^2 - 3 \equiv 717 \equiv 2 \pmod{5}.\] If $n \equiv 4 \pmod{5}$, then \[3n^5 - n^2 - n \equiv 3 \cdot 4^5 - 4^2 - 4 \equiv 3052 \equiv 2 \pmod{5}.\]
Therefore, the given expression is a multiple of 5 if and only if $n \equiv 0$ or $n \equiv 2 \pmod{5}$.
The largest value of $n$ less than 100000 that is congruent to 0 or 2 modulo 5 is $\boxed{99997}$. | Math Dataset |
\begin{document}
\title{ Remarks on surfaces with $c_1^2 =2\chi -1$ \\ having non-trivial\footnotetext{2000 Mathematics Subject Classification. Primary 14J29; Secondary 13J10, 32G05} $2$-torsion\footnotetext{Key Words and Phrases. surfaces of general type, torsion group, moduli space}} \author{Masaaki MURAKAMI
} \date{} \maketitle \begin{center} On the occasion of $60$-th birthday of Prof.\,Fabrizio Catanese \end{center} \begin{abstract} We shall show that any complex minimal surface of general type with $c_1^2 = 2\chi -1$ having non-trivial $2$-torsion divisors, where $c_1^2$ and $\chi$ are the first Chern number of a surface and the Euler characteristic of the structure sheaf respectively, has the Euler characteristic $\chi$ not exceeding $4$. Moreover, we shall give a complete description for the surfaces of the case $\chi =4$, and prove that the coarse moduli space for surfaces of this case is a unirational variety of dimension $29$.
Using the description, we shall also prove that our surfaces of the case $\chi = 4$ have non-birational bicanonical maps and no pencil of curves of genus $2$, hence being of so called non-standard case for the non-birationality of the bicanonical maps.
\end{abstract}
\section{Introduction} \label{sectn:introduction}
In classification of regular surfaces of general type, the torsion parts of the Picard groups (the torsion groups for short) sometimes play an important role. One of the reasons for this lies in variety of topological types under single values of numerical invariants, which is common especially in cases of small geometric genus; the torsion group of a regular surface, isomorphic to the first homology group with integral coefficients, carries information that the numerical invariants $c_1^2$ and $\chi$ do not.
Studies on surfaces of general type done using the torsion groups are well-known for cases of vanishing geometric genus (see, e.g., Barth-Peters-Van de Ven \cite[p.\ 237]{complexsurf}). In those studies, they tried to determine the structures of surfaces with given isomorphism classes of the torsion groups. There are, however, some other cases of numerical invariants for which similar studies have been successfully developed. Consider the case $c_1^2 = 2\chi -2$. In this case, by Ciliberto-Mendes Lopes \cite{on2chi-2}, the orders of the torsion groups do not exceed $2$, and the Euler characteristics $\chi$'s for the cases of non-trivial torsion do not exceed $5$. Complete descriptions for the surfaces with non-trivial torsion with $\chi =2$, $3$, $4$, and $5$ are given in Catanese-Debarre \cite{pg1k2=2'}, Ciliberto-Mendes Lopes \cite{on2chi-2}, Bartalesi-Catanese \cite{withtorsion}, and Ciliberto-Mendes Lopes \cite{on2chi-2} respectively. We remark that even in cases of vanishing geometric genus, complete descriptions are known only for a small number of classes.
In the present paper, we study minimal surfaces with $c_1^2= 2 \chi -1$ having non-trivial $2$-torsion divisors. Note that if $X$ is a minimal surface with $c_1^2 = 2 \chi -1$, then $X$ has vanishing irregularity, hence geometric genus $p_g = \chi -1$. We shall prove the bound $\chi \leq 4$ for the Euler characteristics $\chi$'s (Theorem \ref{thm:maintheorem}), describe the surfaces of the case $\chi = 4$ (Theorem \ref{thm:completedescription}, Remark \ref{rem:meaning}), and study the moduli space for surfaces of this case (Theorem \ref{thm:moduli}). By the main theorem of \cite{bound'''}, the order of the torsion group of a minimal surface with $c_1^2 = 2\chi -1$ is at most $3$ if $\chi =2$, and at most $2$ if $\chi \geq 3$. Thus for our surfaces with $\chi \geq 2$, two conditions $\mathbb{Z}/2 \subset \mathrm{Tors}$ and $\mathrm{Tors} \simeq \mathbb{Z}/2$ are equivalent, where $\mathrm{Tors}$ denotes the torsion group. The case $\chi =1$ on this line is that of the numerical Godeaux surfaces (i.e., minimal surfaces of general type with $c_1^2=1$ and $p_g= 0$).
Surfaces with $c_1^2 = 2 \chi -1$, $\chi = 4$, and $\mathrm{Tors} \simeq \mathbb{Z} / 2$ are known to exist and can be found in \cite{nonstandardpg3}. In \cite{nonstandardpg3}, Ciliberto and Mendes Lopes completely classified regular surfaces with $p_g = 3$ having non-birational bicanonical maps and without genus $2$ pencils, i.e., regular surfaces with $p_g = 3$ and of non-standard case for the non-birationality of the bicanonical maps. Among their results, they showed that any regular surface of non-standard case with $c_1^2 = 7$ and $p_g =3$ is obtained by performing a certain operation on what is known as Du Val's ancestor with $c_1^2=8$ and $p_g = 4$. Since these surfaces have non-trivial $2$-torsion divisors, as has been shown in \cite{nonstandardpg3}, these are examples of our surfaces for the case $\chi =4$. In fact, our structure theorem for surfaces with $c_1^2 = 2 \chi -1$, $\chi = 4$, and $\mathrm{Tors} \simeq \mathbb{Z} / 2$ shows that although we start from the different assumption, the resulting surfaces are exactly those seen in the paper \cite{nonstandardpg3}.
Our complete description for the surfaces with $\chi =4$ asserts that any such surface $X$ is obtained roughly as a free quotient by $\mathbb{Z} /2$ of a double cover of the Hirzebruch surface $\varSigma_d = \mathbb{P} (\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1} (d))$ ($d=0$ or $2$). We shall describe the branch divisor of the double cover, and determine the free action by $\mathbb{Z} /2$ (Theorem \ref{thm:completedescription}, Remark \ref{rem:involutiondescription}). The branch divisor of the double cover turns out to be a member of the quadruple anticanonical system having exactly two $[3,3]$-points. The action by $\mathbb{Z} /2$ turns out to be a lifting of that on the Hirzebruch surface $\varSigma_d$.
This description induces another description of our surfaces of the case $\chi =4$ (Proposition \ref{prop:anotherdescription}), which is almost the same as a description appearing in Ciliberto-Mendes Lopes \cite{nonstandardpg3}. Using our descriptions, we shall show that our surfaces of the case $\chi = 4$ has non-birational bicanonical maps and no pencil of curves of genus $2$ (Proposition \ref{prop:bicanonical}), hence completely coinciding with those seen in \cite{nonstandardpg3} (see also Remark \ref{rem:notecmdescr}).
The coincidence of the resulting surfaces certainly implies possibility of another proof of our complete description, i.e., of a proof, like one for the case $c_1^2 = 2 \chi -2$ in Ciliberto-Mendes Lopes \cite{on2chi-2}, by showing that our surfaces with $\chi =4$ are of non-standard case for the non-birationality of the bicanonical maps. We however do not chose this way. We remark that our method has an advantage in the sense that we can show the irreducibility of the moduli space in a very explicit and elementary way.
The present paper is organized as follows. In order to show our main theorem, we follow Miyaoka \cite{MiyaokaTri} and Reid \cite{Sfpgk2'}, and take the unramified double cover $Y \to X$ corresponding to a torsion divisor. We study its canonical map $\varPhi_{K_Y}$ using the action by the Galois group of $Y$ over $X$. In Section \ref{scn:statement}, we state our main results and show, on the assumption $\chi \geq 4$, that we have $\deg \varPhi_{K_Y} =1$ or $2$, and that $\deg \varPhi_{K_Y} =1$ implies $\chi =4$. Note here that to obtain our main theorem, we only need to study the case $\chi \geq 4$. In Section \ref{scn:deg=2}, we study the case $\deg \varPhi_{K_Y} =2$. We divide this case into three according to the degree of the canonical image $Z = \varPhi_{K_Y} (Y) \subset \mathbb{P}^n$: the case $\deg Z = n+1$, the case $\deg Z =n$, and the case $\deg Z = n-1$. We shall classify non-degenerate surfaces in $\mathbb{P}^n$ of degree $n+1$ of which minimal desingularizations have vanishing irregularities (Proposition \ref{prop:deg=n+1}), and use this classification to study the case $\deg Z = n+1 $.
In Section \ref{scn:deg=1}, we study the case $\deg \varPhi_{K_Y} =1$ and $\chi =4$, and then prove Theorems \ref{thm:maintheorem} and \ref{thm:completedescription}. In the case $\deg \varPhi_{K_Y} =1$ and $\chi =4$, the surface $Y$ has the first Chern number $14$, geometric genus $7$, and irregularity $0$. Hence the surface $Y$ in this case is a canonical surface whose invariant lies on the Castelnuovo line. We use results given in Ashikaga-Konno \cite{3pg-7}
to exclude this case. Finally in Section \ref{scn:onmodulispace}, we study the coarse moduli space for the surfaces of the case $\chi =4$, and prove Theorem \ref{thm:moduli}. To prove the unirationality of the moduli space and the uniqueness of the deformation type, we describe our surfaces of the case $\chi =4$ as double planes, which is almost the same as the description in Ciliberto-Mendes Lopes \cite{nonstandardpg3} for the surfaces of the non-standard case (see also Ciliberto-Francia-Mendes Lopes \cite{onbicanonicalmaps}).
Using the two descriptions of our surfaces, we show that our surfaces of the case $\chi =4$ in fact are of the non-standard case for the non-birationality of bicanonical maps.
{\sc Acknowledgment}
The author performed the first half of the computations included in this article during his stay at Max-Planck-Institut f\"{u}r Mathematik in Bonn from October 2004 until March 2005. He expresses his deepest gratitude to the institute for the comfortable environment and the financial support which he received during his stay. During the final preparation of the manuscript, the author was supported by the research grant fellowship by the University of Padua, and then later by DFG Forschergruppe 790 ``Classification of algebraic surfaces and compact complex manifolds''. The author is thankful to the referee, who suggested him the proof of Proposition \ref{prop:bicanonical}, shorter than the proof in the earlier version. The author is thankful also to Prof.\,Kazuhiro Konno for letting him know the paper \cite{evenI}, to Prof.\,Margarida Mendes Lopes for her kind comments on the first manuscript, and to Prof.\,Fabrizio Catanese and Prof.\,Ingrid Bauer for giving him comfortable environment for the study at Bayreuth, where he performed the final revision of the manuscript.
{\sc Notation and Terminology}
Let $S$ be a compact complex manifold of dimension $2$. We denote by $c_1 (S)$, $p_g(S)$, and $q(S)$ the first Chern class, the geometric genus, and the irregularity of $S$ respectively. The torsion group of $S$, denoted by $\mathrm{Tors} (S)$, is the torsion part of the Picard group of $S$. If $V$ is a complex manifold, $K_V$ is a canonical divisor of $V$. For a coherent sheaf $\mathcal{F}$ on $V$, we denote by $H^i (\mathcal{F})$, $h^i (\mathcal{F})$, and $\chi (\mathcal{F})$ the $i$-th cohomology group, its dimension $\dim_{\mathbb{C}} H^i (\mathcal{F})$, and the Euler characteristic $\sum (-1)^i h^i (\mathcal{F})$ respectively. Let $f: V \to W$ be a morphism to a complex manifold $W$, and $D$, a divisor on $W$. Then $f^* (D)$ and $f^{-1}_* (D)$ denote the total transform and the strict transform respectively of $D$. The symbol $\sim$ means the linear equivalence of divisors. We denote by $\varSigma_d \to \mathbb{P}^1$ the Hirzebruch surface of degree $d$. The divisors $\varDelta_0$ and $\varGamma$ are its minimal section and its fiber respectively. Let $C$ be a curve on $S$. We denote by $\mathrm{mult}_x\, C$ the multiplicity of $C$ at a point $x \in S$. Let $x$ be a triple point of a reduced curve $C$ on $S$, and $S^{\prime} \to S$, the blowing-up at $x$. Assume that the strict transform $C^{\prime}$ of $C$ has an infinitely near triple point $x^{\prime}$. Then the point $x$ is called a $[3, 3]$-point of $C$, if the strict transform $C^{\prime \prime}$ to $S^{\prime \prime}$, where $S^{\prime \prime} \to S^{\prime}$ is the blowing-up at $x^{\prime}$, has at most negligible singularities on the exceptional locus of $S^{\prime \prime} \to S$.
\section{Statement of the main theorem} \label{scn:statement}
In \cite{bound'''}, we obtained a bound for the orders of the torsion groups of minimal surfaces with $c_1^2 = 2\chi -1$ and $\chi \geq 2$. In the present paper, we study the case of $2$-torsion divisors, and sharpen the bound. Our goals are a bound for the Euler characteristic $\chi$, a complete description for the surfaces of the case of maximal $\chi$, and the unirationality of the moduli space for surfaces of this case. The following three are the main theorems:
\begin{theorem} \label{thm:maintheorem} Let $X$ be a minimal surface of general type with $c_1^2 =2\chi -1$ and torsion group $\mathrm{Tors} (X) \simeq \mathbb{Z} /2$. Then the Euler characteristic $\chi$ of the structure sheaf does not exceed $4$. \end{theorem}
\begin{theorem} \label{thm:completedescription} Let $X$ be a minimal surface with $c_1^2 = 2\chi -1$, $\chi =4$, and torsion group $\mathrm{Tors} (X) \simeq \mathbb{Z} /2$. Then the unramified double cover $Y$ of $X$ admits a generically two-to-one morphism $f$ onto the Hirzebruch surface $\varSigma_d$ of degree $d=0$ or $2$ satisfying the following conditions$:$
$\mathrm{i}$ $)$ the action by the Galois group
$G = \mathrm{Gal} (Y/X) \simeq \mathbb{Z} /2$
of $Y$ over $X$ induces one on $\varSigma_d$,
of which fixed locus is a set of four points on $\varSigma_d$ $;$
$\mathrm{ii}$ $)$ the branch divisor $B$ of $f$ is a member of the
linear system $|-4K_{\varSigma_d}|$ passing no fixed
point of the action by $G$ $;$
$\mathrm{iii}$
$)$ the branch divisor $B \in |-4K_{\varSigma_d}|$
has exactly two $[3,3]$-points, and all other singularities,
if any, are negligible ones. \end{theorem}
\begin{theorem} \label{thm:moduli} Any two minimal surfaces with $c_1^2 = 2 \chi -1$, $\chi =4$, and $\mathrm{Tors} \simeq \mathbb{Z}/2$ are equivalent under deformation of complex structures. The coarse moduli space for minimal surfaces with these invariants is a unirational variety of dimension $29$. \end{theorem}
Theorem \ref{thm:maintheorem} sharpens the bound given in \cite{bound'''} into the following:
\begin{theorem} \label{thm:modifiedbound} Let $X$ be a minimal algebraic surface with $c_1^2=2\chi -1$. Then the following hold$:$
$\mathrm{i}$ $)$ if $\chi =2$, then $\sharp \mathrm{Tors} (X) \leq 3$ $;$
$\mathrm{ii}$ $)$ if $\chi \geq 3$, then $\sharp \mathrm{Tors} (X) \leq 2$ $;$
$\mathrm{iii}$ $)$ if $\chi \geq 5$, then $\sharp \mathrm{Tors} (X) =1$. \end{theorem}
\begin{remark} \label{rem:involutiondescription} In Theorem \ref{thm:completedescription}, we can describe the action by $G$ on $\varSigma_d$ more concretely: if an involution of the Hirzebruch surface $\varSigma_d$ has exactly four fixed points ($d$: even), then there exists an open cover $\{ U_i \}_{i=0, 1}$ of $\varSigma_d$ satisfying $U_i = \{ (u_i, (t_i : 1))\} = \mathbb{C} \times \mathbb{P}^1$, $u_0 = 1/ u_1$, and $t_0 = u_1^d t_1$, such that this involution is given by \begin{equation} \label{eq:involution}
(u_0, t_0) \mapsto (-u_0, -t_0). \end{equation}
\end{remark}
\begin{remark} \label{rem:meaning} Theorem \ref{thm:completedescription} asserts that any minimal surface $X$ with $c_1^2 =2\chi -1$, $\chi =4$, and $\mathrm{Tors} (X) \simeq \mathbb{Z} /2$ is obtained by the following procedure: $1$) set $d=0$ or $2$; the involution (\ref{eq:involution}) defines an action by $G= \mathbb{Z} /2$ on the Hirzebruch surface $\varSigma_d$;
$2$) take a reduced member $B \in |-4K_{\varSigma_d}|$ stable under this action that satisfies the conditions ii) and iii) in Theorem \ref{thm:completedescription}; $3$) take the double cover of $\varSigma_d$ branched along $B$, and denote by $Y$ its minimal desingulraization; there exists a unique free lifting to $Y$ of the action by $G$ on $\varSigma_d$; $4$) take the quotient of $Y$ by this free action.
It is not difficult to check that this procedure in fact gives surfaces of the case $\chi =4$ for sufficiently general $B$. \end{remark}
\begin{remark} Let $\varSigma_d$ be the Hirzebruch surface which appears in Theorem \ref{thm:completedescription}. It is obvious from Rmarks \ref{rem:involutiondescription} and \ref{rem:meaning} that the fibration $\varSigma_d \to \mathbb{P}^1$ induces a hyperelliptic fibration $Y \to \mathbb{P}^1$ of genus $3$ and that the divisor class of a fiber of this fibration is stable under the action by the Galois group $G = \mathrm{Gal} (Y / X)$. So we obtain a hyperelliptic fibration $X = Y/G \to \mathbb{P}^1 / G$ of genus $3$ with two multiple fibers $2 A_1$ and $2 A_2$ corresponding to the fixed points of the action by $G$ on $\mathbb{P}^1$. As is explained also in \cite[p.\,$85$]{nonstandardpg3}, the difference $A_1 - A_2$ gives a non-trivial $2$-torsion divisor of our surface $X$. \end{remark}
In what follows, $X$ is a minimal surface with $c_1^2 = 2\chi -1$, $\chi = \lambda \geq 4$, and $\mathrm{Tors} (X) \simeq \mathbb{Z} /2$. We denote by $\pi : Y \to X$ the unramimfied double cover corresponding to the torsion group $\mathrm{Tors} (X)$. Note that we have assumed $\lambda \geq 4$. The following lemma follows from
the unbranched covering trick.
\begin{lemma} $K_Y^2 = 2(2\lambda -1)$, $p_g(Y) = 2\lambda -1$, and $q(Y) =0$. \end{lemma}
In order to show Theorems \ref{thm:maintheorem} and \ref{thm:completedescription}, we study the canonical map $\varPhi_{K_Y} : Y \to \mathbb{P}^n$ of $Y$, where $n= 2\lambda -2$. We denote by $Z = \varPhi_{K_Y} (Y)$ the canonical image of the surface $Y$.
\begin{proposition} \label{prop:degphiKY} The canonical image $Z$ is a surface. The equality $\deg \varPhi_{K_Y} =1$ or $2$ holds. Moreover, if $\deg \varPhi_{K_Y} =1$, then $\lambda =4$. \end{proposition}
Proof. Since we have assumed $\lambda \geq 4$, we have \[
K_Y^2 -3p_g(Y) = -(2\lambda -1) \leq -7 . \] By this together with $q(Y) =0$ and \cite[Theorem 1.1]{smallc1-3}, we see that
$|K_Y|$ is not composite with a pencil. Thus we have \[
\deg \varPhi_{K_Y} \leq \frac{K_Y^2}{\deg Z}
\leq \frac{2(n+1)}{n-1} = 2 + \frac{4}{n-1} \leq 2 + \frac{4}{5}, \] hence $\deg \varPhi_{K_Y} \leq 2$. The second assertion follows from Castelnuovo's inequality. \qed
If $\lambda =4$, then the Chern invariant of $Y$ is on the Castelnuovo line. Thus we can use results given in \cite{3pg-7} to study the case $\deg \varPhi_{K_Y} =1$.
\section{The case $\deg \varPhi_{K_Y} =2$} \label{scn:deg=2}
In this section, we study the case $\deg \varPhi_{K_Y} =2$. We begin with the study of the base locus of the canonical
system $|K_Y|$.
Let $|M|$ and $F$ be the variable part and the fixed part of the
linear system $|K_Y|$. We take the shortest composite $p: \tilde{Y} \to Y$ of
quadric transformations such that the variable part $|L|$ of
$p^{*} |M|$ is free from base points, and denote by $E$ the
fixed part of $p^{*} |M|$.
Then we have $p^*|K_Y| = |L| + E + p^*F$ and \begin{equation} \label{eq:KY^2}
K_Y^2 = L^2 + LE + MF + K_Y F, \end{equation} where each term of the right hand is a non-negative integer. Note that the eigenvectors of the natural action by $G = \mathrm{Gal} (Y/X)$ span the the space of global section $H^0 (\mathcal{O}_Y (K_Y))$. This implies that the linear systems
$|K_Y|$, $|M|$, and $F$ are spanned by the pull-backs of divisors on $X$. Hence, for example, we have $MF \equiv 0$ $\mathrm{mod}$ $2$, since $\pi : Y \to X$ is of mapping degree $2$. In the same way, we obtain \begin{equation} \label{eq:mod2}
L^2 \equiv LE = -E^2 \equiv MF \equiv K_YF \equiv 0
\quad \mathrm{mod} \quad 2 \end{equation} (for the detail, see \cite[Section 3]{3tors'}).
\begin{proposition} \label{prop:L^2} Let $M$, $F$, $L$, and $E$ be divisors as above. Then one of the following holds$:$
$1$
$)$ $|K_Y|=|L|$ $:$
the canonical system $|K_Y|$ is free from base points$;$
$2$ $)$ $L^2 = K_Y^2 -2$, $F=0$, and $LE =2$ $;$
$3$-$1$ $)$ $L^2 = K_Y^2 -4$, $F=0$, and $LE =4$ $;$
$3$-$2$
$)$ $L^2 = K_Y^2 -4$, $|L|=|M|$, $K_Y F=0$, and $F^2 = -4$.
\end{proposition} Proof. First, note that we have $L^2 = K_Y^2$, $K_Y^2 -2$, or $K_Y^2 -4$. This follows from (\ref{eq:mod2}) and \cite[Lemma 2]{quintic}. Second, note that \begin{equation} \label{eq:MFmod4}
MF \equiv 0 \quad \mathrm{mod} \quad 4. \end{equation} This follows from the Riemann--Roch theorem, since we have $MF = M(M + K_Y) - 2M^2 = M(M + \pi^* K_X) - 2 M^2$, $\deg \pi = 2$, and $M \sim \pi^*M^{\prime}$ for a certain divisor $M^{\prime}$ on $X$. Then the assertion follows from (\ref{eq:KY^2}), (\ref{eq:MFmod4}), (\ref{eq:mod2}), and Hodge's index theorem. \qed
In case $3$-$1$), the number of the base points of $|M|$ cannot be $1$, since the action by $G$ on $Y$ has no fixed point. Thus in this case, the morphism $p : \tilde{Y} \to Y$ is a composite of four quadric transformations. In the same way, we see that, in case $2$), the morphism $p : \tilde{Y} \to Y$ is a blowing-up of $Y$ at two distinct points. In case $3$-$2$), the divisor $F$ is a sum of two fundamental cycles of rational double points.
We denote by $\varPhi_L : \tilde{Y} \to Z \subset \mathbb{P}^n$
the morphism associated with the linear system $|L|$. The action by $G$ on $Y$ induces one on $\tilde{Y}$. We study the morphism $\varPhi_L$ using this action.
\subsection{The case $|K_Y|=|L|$}
Let us first exclude case $1$) in Proposition \ref{prop:L^2}.
In what follows, we assume $|K_Y| = |L|$. Thus we have $\deg Z = n+1$. We shall prove the following proposition in Appendix.
\begin{proposition} \label{prop:deg=n+1} Let $n \geq 4$ be an integer, $Z$, a non-degenerate surface in $\mathbb{P}^n$ of degree $n+1$, and $Z^{\prime} \to Z$, its minimal desingularization. Assume that the morphism $Z^{\prime} \to Z$ is given by
a complete linear system $|D^{\prime}|$ and that $q(Z^{\prime}) = 0$ holds. Then $n$ does not exceed $11$. Further, there exist an integer $0 \leq d \leq 3$ and a blowing-up $r : Z^{\prime} \to \varSigma_d$ at $($possibly infinitely near$)$ $11 - n$ points such that the equivalence $D^{\prime} \sim - K_{Z^{\prime}} + r^* \varGamma $ holds. Here, the divisor $\varGamma$ is a fiber of the Hirzebruch surface $\varSigma_d \to \mathbb{P}^1$. \end{proposition}
In our case, we have $n = 2\lambda -2$, $\lambda \geq 4$, and $q (Y) =0$. Moreover $Z$ is the canonical image of $Y$. Thus our surface $Z = \varPhi_{K_Y} (Y)$ satisfies all the conditions in the proposition above. It follows that there exist an integer $0 \leq d \leq 3$ and a blowing-up $r : Z^{\prime} \to \varSigma_d$ at $11-n$ points such that the morphism $\varPhi_{D^{\prime}} : Z^{\prime} \to Z$, where $\varPhi_{D^{\prime}}$ is a morphism corresponding to the complete linear system
$|D^{\prime}| = | -K_{Z^{\prime}} + r^* \varGamma|$, gives the minimal desingularization of $Z$.
\begin{proposition} \label{prop:branchdivisor} The canonical map $\varPhi_{K_Y} : Y \to Z$ lifts to a morphism $f^{\prime} : Y \to Z^{\prime}$. The branch divisor $B^{\prime}$ of $f^{\prime}$ is a member of
the linear system $|2(2D^{\prime} - r^* \varGamma)|$ having at most negligible singularities. \end{proposition}
Proof. Let us first show the liftability of the canonical map $\varPhi_{K_Y}$. Let $p^{\prime} : Y^{\prime} \to Y$ be the shortest composite of quadric transformations such that the morphism $\varPhi_{K_Y} \circ p^{\prime}$ factors through $\varPhi_{D^{\prime}} : Z^{\prime} \to Z$. We denote by $f^{\prime} : Y^{\prime} \to Z^{\prime}$ the unique morphism satisfying $\varPhi_{K_Y} \circ p^{\prime} = \varPhi_{D^{\prime}} \circ f^{\prime}$. Then we have $K_{Y^{\prime}} \sim {p^{\prime}}^* K_Y + \eta$ for a certain effective divisor $\eta$ on $Y^{\prime}$.
If $f^{\prime}_* \eta =0$, then $p^{\prime} : Y^{\prime} \to Y$ is an isomorphism
. Thus we only need to show $f^{\prime}_* \eta =0$.
So we prove the equality above. Let $R^{\prime}$ be the ramification divisor of $f^{\prime}$, and $B^{\prime} = f^{\prime}_* R^{\prime}$, its direct image. Then from $ R^{\prime} \sim K_{Y^{\prime}} - {f^{\prime}}^* K_{Z^{\prime}}
\sim {f^{\prime}}^* (2D^{\prime} - r^*\varGamma) + \eta $, we infer \begin{equation} \label{eq:equivBprime}
B^{\prime} \sim 2(2D^{\prime} -r^*\varGamma + \alpha), \end{equation} where $\alpha$ is a divisor satisfying $2\alpha \sim f^{\prime}_* \eta$. We denote by $Y^{\prime \prime} \to Z^{\prime}$ the double cover branched along $B^{\prime}$, and by $Y^{\sharp} \to Y^{\prime \prime}$ its canonical resolution.
To show the equality $f^{\prime}_* \eta =0$, we compute the Euler characteristic $\chi (\mathcal{O}_{Y^{\sharp}})$ in two ways and compare them. Note that $\dim (\varPhi_{K_Y} \circ p^{\prime}) (\eta) =0$,
and that any general member of $|r^*\varGamma|$ is a $0$-curve. It follows
$D^{\prime} \alpha = {D^{\prime} f^{\prime}_* \eta }/2 =0$ and $D^{\prime} (r^* \varGamma) = -K_{Z^{\prime}} (r^* \varGamma) =2$. Thus by (\ref{eq:equivBprime}) and \cite[Lemma 6]{quintic}, we obtain
\begin{align}
\chi (\mathcal{O}_{Y_{\sharp}})
&= 2 + \frac{1}{2} (2D^{\prime} -r^*\varGamma + \alpha)
((2D^{\prime} -r^*\varGamma + \alpha) + K_{Z^{\prime}})
- \beta \notag \\
&= 2 + \frac{1}{2} (2D^{\prime} -r^*\varGamma + \alpha)
(D^{\prime} + \alpha )
- \beta \notag \\
&= {D^{\prime}}^2 + 1
- \frac{1}{4} (r^*\varGamma) (f^{\prime}_* \eta )
+ \frac{1}{8}(f^{\prime}_* \eta)^2
- \beta ,
\label{eq:chiOYsharp} \end{align}
where $\beta$ is a term coming from essential singularities of the branch divisor $B^{\prime}$. Here, we have three inequalities \begin{equation}
- \frac{1}{4} (r^*\varGamma) (f^{\prime}_* \eta ) \leq 0 , \quad
\frac{1}{8}(f^{\prime}_* \eta)^2 \leq 0 , \quad
\text{and} \quad
- \beta \leq 0. \label{eq:twoinequalities} \end{equation}
The first one follows from the absence of base points of $|r^*\varGamma|$, the second one from ${D^{\prime}}^2 > 0$ and $D^{\prime} f^{\prime}_* \eta = 0$, and the last one from the definition of $\beta$. Meanwhile we have $\chi (\mathcal{O}_{Y^{\sharp}}) =
\chi (\mathcal{O}_Y) = n +2 = {D^{\prime}}^2 +1$. Thus by (\ref{eq:chiOYsharp}) and (\ref{eq:twoinequalities}), we obtain $(f^{\prime}_* \eta)^2 =0$, from which together with Hodge's index theorem, we infer $f^{\prime}_* \eta =0$. Hence the canonical map $\varPhi_{K_Y}$ lifts.
The remaining assertion easily follows from the proof above. \qed
Note that the action by $G = \mathrm{Gal} (Y/X)$ on $Y$ induces one on $Z^{\prime}$. We can verify it as follows. Since $Z$ is the canonical image of our surface $Y$, the action on $Y$ induces one on $Z$. Meanwhile the surface $Z^{\prime}$ is the minimal desingularization of our surface $Z$. Thus this action on $Z$ induces one on $Z^{\prime}$.
\begin{lemma} \label{lm:onedimcomp} The induced action by $G$ on $Z^{\prime}$ is non-trivial. The fixed locus of this action has a one-dimensional irreducible component $C_0^{\prime}$ satisfying ${C_0^{\prime}}^2 \equiv 1 $ $\mathrm{mod}$ $2$. \end{lemma}
Proof. The first assertion is trivial, since the action on $Y$ has no fixed point. Let us show the second assertion. Let $\{ z_1, \ldots , z_b \}$ be the set of isolated fixed points of the action on $Z^{\prime}$, and $r^{\prime \prime} : Z^{\prime \prime} \to Z^{\prime}$ the blowing-up at these $b$ points. We denote by $C_i^{\prime \prime}$ the $(-1)$-curve lying over $z_i$. Let $\{ C_1^{\prime}, \ldots , C_a^{\prime} \}$ be the set of $1$-dimensional irreducible components of the fixed locus of the action on $Z^{\prime}$. We use the same symbol $C_i^{\prime}$ for the total transform to $Z^{\prime \prime}$ of the divisor $C_i^{\prime}$. Note that the divisor $\sum_{i=1}^{a} C_i^{\prime} + \sum_{i=1}^{b} C_i^{\prime \prime}$ has no singularity, since we have $G \simeq \mathbb{Z}/2$. It follows that the quotient $Z^{\prime \prime} /G$ is smooth, where the action by $G$ is the lifting of that on $Z^{\prime}$. We denote by $\bar{C_i^{\prime}}$ and $\bar{C_i^{\prime \prime}}$ the image to $Z^{\prime \prime} /G$ of the divisor $C_i^{\prime}$ and that of the divisor $C_i^{\prime \prime}$, respectively. Then since the branch divisor $\sum_{i=1}^{a} \bar{C_i^{\prime}} + \sum_{i=1}^{b} \bar{C_i^{\prime \prime}}$ is linearly equivalent to twice a divisor on $Z^{\prime \prime} /G$, we have \[
\sum_{i=1}^{a}{C_i^{\prime}}^2 - b =
(\sum_{i=1}^{a} C_i^{\prime} + \sum_{i=1}^{b} C_i^{\prime \prime})^2 =
(\sum_{i=1}^{a} \bar{C_i^{\prime}} + \sum_{i=1}^{b} \bar{C_i^{\prime \prime}})^2 /2
\equiv 0 \quad \mathrm{mod} \quad 2. \] Meanwhile, since $K_{Z^{\prime \prime}}$ is linearly equivalent to a pull-back of a divisor on $Z^{\prime \prime} /G$, we have $K_{Z^{\prime \prime}}^2 = K_{Z^{\prime}}^2 -b =n-3-b \equiv 0 $ $\mathrm{mod}$ $2$, hence $b \equiv 1$ $\mathrm{mod}$ $2$. Thus we infer $\sum_{i=1}^{a}{C_i^{\prime}}^2 \equiv 1$ $\mathrm{mod}$ $2$, which implies the second assertion. \qed
\begin{lemma} \label{lm:bcneqzero} Let $C_0^{\prime}$ be an irreducible curve as in Lemma $\ref{lm:onedimcomp}$. Then $B^{\prime}C_0^{\prime} \neq 0$ holds. \end{lemma}
Proof. We derive a contradiction by assuming $B^{\prime}C_0^{\prime} =0$. Assume that $B^{\prime}C_0^{\prime} =0$ holds. Then by Proposition \ref{prop:branchdivisor}, we have \[
(2D^{\prime} - r^* \varGamma ) C_0^{\prime} =
(-2K_{Z^{\prime}} + r^* \varGamma ) C_0^{\prime} =0. \] If $(r^* \varGamma) C_0^{\prime} =0$, then by the equality above, we obtain $K_{Z^{\prime}} C_0^{\prime} =0$, which contradicts ${C_0^{\prime}}^2 \equiv 1$ $\mathrm{mod}$ $2$. Thus we have $(r^* \varGamma) C_0^{\prime} > 0$, hence $-2K_{Z^{\prime}} C_0^{\prime} = -(r^* \varGamma) C_0^{\prime} < 0$. It follows $C_0^{\prime}$ is a fixed component of
the anti-canonical system $|-K_{Z^{\prime}}|$. Then since $-K_{\varSigma_d} \sim 2\varDelta_0 + (2+ d) \varGamma$, we obtain $(r^* \varGamma) C_0^{\prime} \leq 2$, hence $(r^* \varGamma) C_0^{\prime} = 2K_{Z^{\prime}} C_0^{\prime} =2$. Thus $r_* C_0^{\prime} \sim 2\varDelta_0 + c \varGamma$ holds for a certain integer $c \geq 1$. Meanwhile since $0 \leq d \leq 3$, we have $h^0(\mathcal{O}_{Z^{\prime}}(-K_{Z^{\prime}})) \geq h^0(\mathcal{O}_{\varSigma_d}(-K_{\varSigma_d})) -(11 -n) = n-2$. Thus we obtain \begin{multline} n-2 \leq h^0(\mathcal{O}_{Z^{\prime}}(-K_{Z^{\prime}}))
= h^0(\mathcal{O}_{Z^{\prime}}(-K_{Z^{\prime}} - C_0^{\prime})) \\
\leq h^0(\mathcal{O}_{\varSigma_d}
(-K_{\varSigma_d} - r_* C_0^{\prime}))
= 3 + d -c, \notag \end{multline} hence $ c -d \leq 5 - n <0$. It follows $ (r_* C_0^{\prime}) \varDelta_0 = (c -d) -d <0$, which contradicts the irreducibility of $C_0^{\prime}$. Hence $B^{\prime} C_0^{\prime} \neq 0$ holds. \qed
Now let us exclude case $1$) in Proposition \ref{prop:L^2}.
\begin{proposition} \label{prop:case1exclusion} Case $1$ $)$ in Proposition $\ref{prop:L^2}$ does not occur. \end{proposition}
Take an irreducible curve $C_0^{\prime}$ as in Lemma \ref{lm:onedimcomp}. Then by Lemma \ref{lm:bcneqzero}, we have $B^{\prime} \cap C_0^{\prime} \neq \emptyset$. So let us take a point $x \in B^{\prime} \cap C_0^{\prime}$. Then the preimage ${f^{\prime}}^{-1} (x) \subset Y$ is stable under the action by $G$ on $Y$. By Proposition \ref{prop:branchdivisor}, however, the set ${f^{\prime}}^{-1} (x)$ is either a point or a base space of the fundamental cycle of a rational double point. This implies that the action by $G$ on ${f^{\prime}}^{-1} (x)$ has a fixed point, which contradicts the definition of $\pi : Y \to X$. Thus we have the assertion. \qed
\subsection{The case $L^2 = K_Y^2 -4$} Next we exclude cases $3$-$1$) and $3$-$2$) in Proposition \ref{prop:L^2}. In these two cases, we have $L^2 = 2(n-1)$; hence the canonical image $Z$ is a non-degenerate surface in $\mathbb{P}^n$ of minimal degree $n-1$. Thus from the well-known classification, it follows that our $Z$ is a image of the Hirzebruch surface $Z^{\prime} = \varSigma_d$ by the morphism associated with the complete linear system
$|D^{\prime}| = |\varDelta_0 + \frac{n-1+d}{2}\varGamma|$, where $0 \leq d \leq n-1$ and $d \equiv n-1$ $\mathrm{mod}$ $2$ (see \cite{ratsfI}or \cite[Lemma 1.2]{smallc1-1}). Let us denote this morphism by $\varPhi_{D^{\prime}}: Z^{\prime} \to Z \subset \mathbb{P}^n$. Then $\varPhi_{D^{\prime}}$ is an embedding if $d < n-1$, and is the contraction of $\varDelta_0$ if $d = n-1$.
Note that in the later case, our $Z$ is a cone over a rational curve embedded in $\mathbb{P}^{n-1}$ by $\mathcal{O}_{\mathbb{P}^1} (n-1)$.
For the case $d < n-1$, the lemma below is trivial. For the case $d = n-1$, we can give a proof by the same method as in \cite[Lemma 1.5]{smallc1-1}.
\begin{lemma} The morphism $\varPhi_L : \tilde{Y} \to Z$ lifts to a morphism $f^{\prime} : \tilde{Y} \to Z^{\prime}$. \end{lemma}
By the same argument as in the exclusion of case $1$), we see that the action by $G$ on $Y$ induces one on $Z^{\prime}$.
Let us recall the morphism $p: \tilde{Y} \to Y$ and the
base locus of $|K_Y|$. In case $3$-$1$) in Proposition \ref{prop:L^2}, the morphism $p$ is the blowing-up at (possibly infinitely near) four points, which we shall call $y_1, \ldots , y_4$. Let $E_i$ denote the total transform to $\tilde{Y}$ of the $(-1)$-curve corresponding to $y_i$. Then we have $E = \sum_{i=1}^4 E_i$ and $L E_i =1$ ($1 \leq i \leq 4$).
Since the action by $G$ on the set of base points of $|M|$ has no fixed point, we have only two cases: i) the case where $y_1, \ldots , y_4$ are four distinct points on $\tilde{Y}$, and ii) the case where $y_1$ and $y_2$ are distinct points on $\tilde{Y}$, and $y_{i +2}$ is infinitely near to $y_i$ for $i =1$, $2$. In the later case, the divisor $E_i^{\prime} = E_i - E_{i +2}$ is a $(-2)$-curve satisfying $L E_i^{\prime} =0$.
Meanwhile in case $3$-$2$), the morphism $p : \tilde{Y} \to Y$ is an isomorphism. Hence we may assume $\tilde{Y} = Y$.
We have $|M|=|L|$ and $F = \sum_{i=1,2} F_i$, where $F_i$ is a fundamental cycle of a rational double point. Since the action on $Y$ has no fixed point, we have $F_1 \cap F_2 = \emptyset$; hence the generator of $G$ maps $F_1$ onto $F_2$. It follows $L F_1 = L F_2 =2$.
In what follows, we put $T = 2E$ for case $3$-$1$), and $T =F$ for case $3$-$2$). Then we have \[
K_{\tilde{Y}} \sim L +T. \]
\begin{lemma} \label{lm:Tmod4} Let $T$ be the divisor above. Then $\varGamma (f^{\prime}_* T) \equiv 2$ $\mathrm{mod}$ $4$ holds. \end{lemma}
Proof. Since $d \equiv n-1 \equiv 1$ $\mathrm{mod}$ $2$, we have $d \neq 0$. Thus the action by $G$ on $Z^{\prime} =\varSigma_d$ induces one on $\mathbb{P}^1$ via the natural fibration $\varSigma_d \to \mathbb{P}^1$ of the Hirzebruch surface.
It follows there exists a member $\varGamma_0 \in |\varGamma|$ stable under the action by $G$. Let us take a blowing-up $\tilde{X} \to X$ such that $\tilde{Y} = \tilde{X} \times_X Y$ holds. The base change $\tilde{\pi} : \tilde{Y} \to \tilde{X}$ is an unramified double cover satisfying $\mathrm{Gal} (\tilde{Y}/\tilde{X}) \simeq \mathrm{Gal} (Y/X)$. Then since ${f^{\prime}}^* \varGamma_0$ is stable under the action by $G$ on $\tilde{Y}$, the divisor ${f^{\prime}}^* \varGamma_0$ is a pull-back by $\tilde{\pi}$ of a certain divisor on $\tilde{X}$. Thus from $\tilde{\pi}^* K_{\tilde{X}} \sim K_{\tilde{Y}}$ and the Riemann--Roch theorem, we infer \[
({f^{\prime}}^* \varGamma_0)^2
+ ({f^{\prime}}^* \varGamma_0) K_{\tilde{Y}}
= ({f^{\prime}}^* \varGamma_0) (L + T)
= 2 + ({f^{\prime}}^* \varGamma_0) T
\equiv 0 \quad \mathrm{mod} \quad 4. \] Hence we have the assertion. \qed
\begin{lemma} \label{lm:-1and-2curves} The morphism $f^{\prime} : \tilde{Y} \to Z^{\prime}$ contracts no $(-1)$-curve on $\tilde{Y}$. Further, the following hold$:$
$\mathrm{i}$ $)$ if $C$ is a $(-1)$-curve on $\tilde{Y}$ satisfying $LC=1$,
then $f^{\prime}_* C \sim \varGamma$ $;$
$\mathrm{ii}$ $)$ if $C$ is a $(-2)$-curve on $\tilde{Y}$ satisfying $LC=0$,
then $f^{\prime}$ contracts $C$. \end{lemma}
Proof. The first assertion trivially follows from the definition of $p : \tilde{Y} \to Y$. In order to prove i) and ii), we put $f^{\prime}_* C \sim a \varDelta_0 + b \varGamma$. We denote by $\theta$ the involution of $\tilde{Y}$ over $Z^{\prime}$. This involution exists, since $f^{\prime}$ contracts no $(-1)$-curve.
First, let us prove the assertion i). Assume that $C$ is a $(-1)$-curve on $\tilde{Y}$ satisfying $LC=1$. Then since $L \sim {f^{\prime}}^* D^{\prime}$, we have \begin{equation} \label{eq:fprimec}
(\varDelta_0 + d \varGamma) f^{\prime}_* C
+ \frac{n-1-d}{2} \varGamma f^{\prime}_* C =1, \end{equation} where each term of the left hand is a non-negative integer. Thus we obtain $(\varDelta_0 + d \varGamma) f^{\prime}_* C =0$ or $1$.
Assume that $(\varDelta_0 + d \varGamma) f^{\prime}_* C =1$. Then we have $f^{\prime}_* C \sim a \varDelta_0 + \varGamma$ and $(\frac{n-1-d}{2})a =0$. Thus, in this case, we only have to show $a=0$, which is trivial if $n-1-d \neq 0$. If $n-1-d=0$, then by the irreducibility of $C$, we have $\varDelta_0 f^{\prime}_* C = 1 -a (n-1) \geq 0$, hence $a=0$. Assume next that $(\varDelta_0 + d \varGamma) f^{\prime}_* C =0$. Then by (\ref{eq:fprimec}), we obtain $f^{\prime}_* C = \varDelta_0 $ and $d = n-3$. We exclude this case as follows. We have ${f^{\prime}}^* \varDelta_0 = C + \theta (C) + \xi$ for a certain effective divisor $\xi$ exceptional with respect to $f^{\prime}$. It follows \[ ({f^{\prime}}^* \varDelta_0)^2 = (C + \theta (C) + \xi)(C + \theta (C)) \geq C^2 + {\theta (C)}^2 + 2 C \theta (C) \geq -4, \] hence $-2(n-3) \geq -4$. This contradicts $\lambda \geq 4$. Thus we have $(\varDelta_0 + d \varGamma) f^{\prime}_* C \neq 0$, which completes the proof of the assertion i).
Next, let us prove the assertion ii). Assume that $f^{\prime}(C)$ is a curve. Then since $\varPhi_{D^{\prime}}$ contracts $f^{\prime}(C)$, we have $d = n-1$ and $f^{\prime}(C) = \varDelta_0$. Note that we have $f^{\prime}_*C = \varDelta_0$ or $2\varDelta_0$, since $\deg f^{\prime} =2$. Assume that $f^{\prime}_*C = \varDelta_0$. Then we have ${f^{\prime}}^* \varDelta_0 = C + \theta (C) + \xi$ for a certain effective divisor $\xi$ exceptional with respect to $f^{\prime}$. Then by the same method as in the proof of i), we obtain $-2(n-1) = ({f^{\prime}}^* \varDelta_0)^2 \geq -8$, which contradicts $\lambda \geq 4$.
Assume next that $f^{\prime}_*C = 2\varDelta_0$. Then we have ${f^{\prime}}^* \varDelta_0 = C + \xi$ for a certain effective divisor $\xi$ exceptional with respect to $f^{\prime}$. Then again by the same method, we obtain $-2(n-1) \geq -2$, which contradicts $\lambda \geq 4$. Thus we have the assertion ii). \qed
If our $Y$ is of case $3$-$1$) in Proposition \ref{prop:L^2}, then by the lemma above we have $f^{\prime}_* T = 2f^{\prime}_* E \sim 8 \varGamma$, which contradicts Lemma \ref{lm:Tmod4}. Thus we have the following: \begin{proposition} \label{prop:case3-1exclusion} Case $3$-$1$ $)$ in Proposition $\ref{prop:L^2}$ does not occur. \end{proposition}
So in what follows, we assume that our $Y$ is of case $3$-$2$) in Proposition \ref{prop:L^2}.
\begin{lemma} \label{lm:-2curves} Let $C$ be an irreducible component of $F_1$ satisfying $D^{\prime} f^{\prime}_* C > 0$. Then one of the following holds$:$
$\mathrm{i}$ $)$ $D^{\prime} f^{\prime}_* C =1$ and $f^{\prime}_* C \sim \varGamma$ $;$
$\mathrm{ii}$ $)$ $D^{\prime} f^{\prime}_* C =2$ and $f^{\prime}_* C \sim 2 \varGamma$ $;$
$\mathrm{iii}$ $)$ $D^{\prime} f^{\prime}_* C =2$, $f^{\prime}_* C = \varDelta_0$,
and $d = n - 5 =1$. \end{lemma}
Proof. First, note that if $f^{\prime}(C) = \varDelta_0$, then we have $C \neq \theta (C)$, where $\theta$ is the involution of $\tilde{Y}=Y$ over $Z^{\prime}$. We can verify this as follows. Let $\iota$ be the generator of the Galois group $G$, and
$\iota |_{Z^{\prime}}$, the corresponding automorphism of $Z^{\prime}$. Then since $d \neq 0$, we have $f^{\prime} (\iota (C))
= \iota |_{Z^{\prime}} (f^{\prime}(C)) = \varDelta_0 = f^{\prime} (C)$. This means $C \neq \theta(C) = \iota (C)$, since we have $\iota (C) \subset F_2$ and $F_1 \cap F_2 = \emptyset$. Next, note that $C$ is a $(-2)$ curve satisfying $0 < D^{\prime} f^{\prime}_* C \leq D^{\prime} f^{\prime}_* F_1 = 2$. Then we can prove the assertion by the same method as in the proof of Lemma \ref{lm:-1and-2curves}. \qed
By $D^{\prime} f^{\prime}_* F_1 =2$ together with Lemmas \ref{lm:-1and-2curves} and \ref{lm:-2curves}, we see that either of the following holds:
a) $f^{\prime}_* F_1 = f^{\prime}_* (\iota (F_2)) \sim 2\varGamma$;
b) $f^{\prime}_* F_1 = f^{\prime}_* (\iota (F_2)) = \varDelta_0$,
and $d = n-5 =1$,
\noindent where $\iota$ is the generator of the Galois group of $G$. Case a) above, however, contradicts the assertion in Lemma \ref{lm:Tmod4}. Thus we have the following:
\begin{lemma} \label{lm:F1F2image} $f^{\prime}_* F_1 = f^{\prime}_* F_2 = \varDelta_0$ and $d =n-5 =1$. \end{lemma}
Now let us study the morphism $f^{\prime} : \tilde{Y} =Y \to Z^{\prime} = \varSigma_1$. Let $R^{\prime}$ be the ramification divisor of $f^{\prime}$, and $B^{\prime} = f^{\prime}_* R^{\prime}$, the branch divisor. Then by the lemma above we obtain \begin{equation} \label{eq:ramificationdivisor} R^{\prime}
\sim {f^{\prime}}^*(3\varDelta_0 + 6\varGamma)
+ \sum_{i =1, 2} F_i \qquad \text{and} \qquad B^{\prime} \sim 2(4\varDelta_0 + 6\varGamma) .
\end{equation} We take the double cover of $Z^{\prime}$ with branch divisor $B^{\prime}$, and denote by $Y^{\sharp}$ its canonical resolution. Let us recall how to obtain the canonical resolution. Set $Z^{\prime}_0 = Z^{\prime}$ and $B^{\prime}_0 = B^{\prime}$. We define $Z^{\prime}_i$ and $B^{\prime}_i$ inductively as follows. Choose a singularity $z_i$, if any, of $B^{\prime}_{i-1}$, and take the blowing-up $q^{\prime}_i : Z^{\prime}_i \to Z^{\prime}_{i-1}$ at this point. We denote by $\varepsilon_i$ the $(-1)$-curve corresponding to $z_i$. Let $m_i$ be the multiplicity of $B^{\prime}_{i-1}$ at $z_i$, and $[\frac{m_i}{2}]$, the largest integer not exceeding $\frac{m_i}{2}$. Then we define $B^{\prime}_i$ by $B^{\prime}_i = {q^{\prime}_i}^* B^{\prime}_{i-1} - 2[\frac{m_i}{2}] \varepsilon_i$. For a certain $s \geq 0$, the divisor $B^{\prime}_s$ is non-singular. So take the double cover $f^{\sharp} : \tilde{Y_s} \to Z^{\sharp} = Z^{\prime}_s$ with branch divisor $B^{\sharp} = B^{\prime}_s$. Then this $\tilde{Y_s}$ is our canonical resolution $Y^{\sharp}$. Put $q^{\prime} = (q^{\prime}_1 \circ q^{\prime}_2 \circ \cdots \circ q^{\prime}_s): Z^{\sharp} \to Z^{\prime}$. There exists a natural birational morphism $p^{\sharp} : Y^{\sharp} \to \tilde{Y}$ satisfying $q^{\prime} \circ f^{\sharp} = f^{\prime} \circ p^{\sharp}$. We use the same symbol $\varepsilon_i$ for the total transform to $Z^{\sharp}$ of the $(-1)$-curve $\varepsilon_i \subset Z^{\prime}_i$. Note, for our case, the action by the Galois group $G = \mathrm{Gal} (Y/X)$ on $\tilde{Y}$ induces one on $Z^{\sharp}$ and one on $Y^{\sharp}$. This action on $Y^{\sharp}$ is free.
By the same method as in \cite[Section 2]{quintic}, we obtain the following:
\begin{proposition} \label{prop:i1i2} There exist $i_1$ and $i_2$ $(i_1 < i_2)$ satisfying $[\frac{m_{i_1}}{2}] = [\frac{m_{i_2}}{2}] = 2$. For any $i \neq i_1$, $i_2$, the equality $[\frac{m_i}{2}] = 1$ holds. The morphism $p^{\sharp} : Y^{\sharp} \to \tilde{Y} = Y$ is a composite of two quadric transformations. \end{proposition}
Thus the branch divisor $B^{\prime}$ has an essential singularity. By the proposition above, we obtain \begin{equation} \label{eq:KYsharp}
K_{Y^{\sharp}} \sim {f^{\sharp}}^*
({q^{\prime}}^*(2\varDelta_0 + 3\varGamma)
- \varepsilon_{i_1} - \varepsilon_{i_2}) . \end{equation}
\begin{lemma} \label{lm:bprimeessentialsing} Every essential singularity of $B^{\prime}$ lies on $\varDelta_0$. \end{lemma}
Proof. Since $f^{\prime}$ contracts no $(-1)$-curve, ${f^{\prime}}^* B^{\prime} - 2 R^{\prime} = 2 \zeta^{\prime}$ holds for a certain effective divisor $\zeta^{\prime}$ on $\tilde{Y}$. This $\zeta^{\prime}$ satisfies \begin{equation} \label{eq:zetaprime}
2 \zeta^{\prime} \sim 2 ({f^{\prime}}^*(\varDelta_0) - \sum_{i =1, 2} F_i), \end{equation} since we have (\ref{eq:ramificationdivisor}). Let $\zeta^{\prime} = \sum \zeta^{\prime}_i$ be the decomposition into connected components. Note that $f^{\prime}$ maps each $\zeta^{\prime}_i$ to a point on $Z^{\prime}$. Then, for any $i$ satisfying $f^{\prime} (\zeta^{\prime}_i) \notin \varDelta_0$, we infer from (\ref{eq:zetaprime}) that ${\zeta^{\prime}_i}^2 = \zeta^{\prime}_i \zeta^{\prime} =0$, hence $\zeta^{\prime}_i =0$, which implies the assertion. \qed
\begin{lemma} \label{lm:fixedpartKYsharp} Let $\eta^{\sharp} \sim K_{Y^{\sharp}} - {p^{\sharp}}^* K_{\tilde{Y}}$ be the exceptional divisor corresponding to $p^{\sharp} : Y^{\sharp} \to \tilde{Y}$.
Then the fixed part of $|K_{Y^{\sharp}}|$ is given by $\sum_{i=1,2} {p^{\sharp}}^* F_i + \eta^{\sharp}$. Further, the linear equivalence $\sum_{i=1,2} {p^{\sharp}}^* F_i + \eta^{\sharp} \sim {f^{\sharp}}^*({q^{\prime}}^*\varDelta_0 -\varepsilon_{i_1} -\varepsilon_{i_2})$ holds, where $i_1$ and $i_2$ are integers given in Proposition $\ref{prop:i1i2}$. \end{lemma}
Proof. The first assertion follows from
$|K_{Y^{\sharp}}|= |K_{\tilde{Y}}| + \eta^{\sharp}$,
since $|L|$ has no base point. The second assertion follows from (\ref{eq:KYsharp}) and $\sum {p^{\sharp}}^* F_i + \eta^{\sharp} \sim K_{Y^{\sharp}} - {p^{\sharp}}^* L \sim K_{Y^{\sharp}} - {p^{\sharp}}^* {f^{\prime}}^* D^{\prime}$. \qed
\begin{lemma} \label{lm:gamma1}
There exists a member $\varGamma_1 \in |\varGamma|$ contained in the fixed locus of the action by $G$ on $Z^{\prime} = \varSigma_1$. \end{lemma}
Proof. The action by $G$ on $Z^{\prime} = \varSigma_1$ induces one on $\mathbb{P}^1$ via the natural fibration $Z^{\prime} = \varSigma_1 \to \mathbb{P}^1$ of the Hirzebruch surface. Let us show that this induced action on $\mathbb{P}^1$ is non-trivial.
There exists a member $\varDelta_1 \in |\varDelta_0 + \varGamma|$ stable under the action by G satisfying $\varDelta_1 \cap \varDelta_0 = \emptyset$. Assume that the induced action on $\mathbb{P}^1$ is trivial. Then this $\varDelta_1$ is contained in the fixed locus of the action by $G$ on $Z^{\prime}$. From this together with $B^{\prime} \varDelta_1 = 12$ and Lemma \ref{lm:bprimeessentialsing}, it follows that $B^{\prime}$ has a smooth point or a negligible singularity that is stable under the action by $G$. This, however, leads us to a contradiction by the same argument as in the proof of Proposition \ref{prop:case1exclusion}. Thus the induced action on $\mathbb{P}^1$ is non-trivial. Now take two fibers of $Z^{\prime} \to \mathbb{P}^1$ that lie over the fixed points of the action on $\mathbb{P}^1$. Since $Z^{\prime} = \varSigma_1$, one of these two fibers are contained in the fixed locus of the action by $G$. \qed
Let us exclude case $3$-$2$) in Proposition \ref{prop:L^2}. \begin{proposition} \label{prop:case3-2exclusion} Case $3$-$2$ $)$ in Proposition $\ref{prop:L^2}$ does not occur. \end{proposition}
Proof.
Let $\varGamma_1 \in |\varGamma|$ be the member as in Lemma \ref{lm:gamma1}. By (\ref{eq:ramificationdivisor}), we have $B^{\prime} \varGamma_1 =8$, hence $B^{\prime} \cap \varGamma_1 \neq \emptyset$. If a smooth point or a negligible singularity of $B^{\prime}$ lies on $B^{\prime} \cap \varGamma_1$, we can derive a contradiction by the same argument as in the proof of Proposition \ref{prop:case1exclusion}. Thus by Lemma \ref{lm:bprimeessentialsing}, we see that $B^{\prime} \cap \varGamma_1 = \varDelta_0 \cap \varGamma_1$ and that this point is an essential singularity of $B^{\prime}$. So we put $\varDelta_0 \cap \varGamma_1 = \{ z_1\}$, where the point $z_1$ is the center of the first blowing-up $q^{\prime}_1: Z^{\prime}_1 \to Z^{\prime}_0 = Z^{\prime}$ in the procedure to obtain the canonical resolution $Y^{\sharp}$. Then, by Proposition \ref{prop:i1i2}, we have $3 \leq m_1 \leq 5$. If $m_1$ is odd, then the strict transform $\varepsilon_1^{\sharp} \simeq \mathbb{P}^1 \subset Z^{\sharp}$ of the exceptional curve $\varepsilon_1 \subset Z_1^{\prime}$ is a component of $B^{\sharp}$ stable under the action by $G$. This, however, leads us to a contradiction, since the action by $G$ on $Y^{\sharp}$ is free. It follows $m_1 \equiv 0$ $\mathrm{mod}$ $2$, hence $m_1 =4$. Thus we have $B_1^{\prime} = {q_1^{\prime}}^* B^{\prime} - 4 \varepsilon_1$ and $B_1^{\prime} {q_1^{\prime}}^{-1}_* (\varGamma_1) =4$, where the divisor ${q_1^{\prime}}^{-1}_* (\varGamma_1)$ is the strict transform of $\varGamma_1$ by $q_1^{\prime} : Z_1^{\prime} \to Z^{\prime}$. Note that the action by $G$ on $Z^{\prime}$ induces one on $Z_1^{\prime}$, and that the strict transform ${q_1^{\prime}}^{-1}_* (\varGamma_1)$ is contained in the fixed locus of this induced action. By the same argument as that on $\varGamma_1$ above, we see that the point $B_1^{\prime} \cap {q_1^{\prime}}^{-1}_* (\varGamma_1) = \varepsilon_1 \cap {q_1^{\prime}}^{-1}_* (\varGamma_1)$ is an essential singularity of $B_1^{\prime}$, that we can set $\varepsilon_1 \cap {q_1^{\prime}}^{-1}_* (\varGamma_1) = \{ z_2 \}$, where the point $z_2$ is the center of the second blowing-up $q_2^{\prime} : Z_2^{\prime} \to Z_1^{\prime}$, and that $m_2= 4$, where $m_2$ is the multiplicity of $B_1^{\prime}$ at $z_2$. Thus we have $i_1 =1$ and $i_2 =2$, where $i_1$ and $i_2$ are the integers given in Proposition \ref{prop:i1i2}.
Now we derive a contradiction. Let $\varGamma_1^{\sharp}$ be the strict transform to $Z^{\sharp}$ of of the divisor $\varGamma_1$. Note that we have $z_1 \in \varGamma_1$ and $z_2 \in {q_1^{\prime}}^{-1}_* (\varGamma_1)$. Thus by Lemma \ref{lm:fixedpartKYsharp}, we obtain \[
f^{\sharp}_* (\sum {p^{\sharp}}^* F_i + \eta^{\sharp})
\varGamma_1^{\sharp}
= 2( \varDelta_0 \varGamma + {\varepsilon_1}^2 + {\varepsilon_2}^2 )
= -2 < 0. \] From this together with Lemma \ref{lm:F1F2image}, we infer that the divisor $\varGamma_1^{\sharp}$ is the image by $f^{\sharp}$ of an irreducible component of $\eta^{\sharp}$, which contradicts the equality $\dim (q^{\prime} \circ f^{\sharp}) (\eta^{\sharp})= \dim (f^{\prime} \circ p^{\sharp}) (\eta^{\sharp}) = 0$. Hence we have the assertion. \qed
\subsection{The case $L^2 = K_Y^2 -2$}
Finally, we study case $2$) in Proposition \ref{prop:L^2}. It will turn out that $\lambda =4$ in this case, and that the surfaces of this case have the structure as in the statement of Theorem \ref{thm:completedescription}. In what follows, we assume that our $Y$ is of case $2$) in Proposition \ref{prop:L^2}, hence $ \deg Z = L^2 /2 = n$. Note that in this case, the morphism $p : \tilde{Y} \to Y$ is a blowing-up at two distinct points on $Y$. Let $E_1$ and $E_2$ denote the $(-1)$-curves corresponding to the centers of this blowing-up. Then we have
$p^*|K_Y| = |L| + \sum_{i=1, 2} E_i$ and $ L E_1 = L E_2 = 1 $. The Galois group $G = \mathrm{Gal} (Y/X)$ acts transitively on the set $\{ E_1, E_2 \}$. We denote by $Z^{\prime}$ the minimal desingularization of $Z$.
\begin{lemma} \label{lm:L^2-2resoution} There exists a blowing-up $r : Z^{\prime} \to \mathbb{P}^2$ at $($possibly infinitely near$)$ $9-n$ points such that the anticanonical morphism $Z^{\prime} \to Z \subset \mathbb{P}^n$ of $Z^{\prime}$ gives the minimal desingularization of $Z$. \end{lemma}
Proof. Note that our $Z = \varPhi_{K_Y} (Y)$ is a non-degenerate surface in $\mathbb{P}^n$ of degree $n$. Hence our $Z$ is one of the following (see \cite{ratsfI} or \cite[Section 3]{smallc1-4}):
i) a projection of a surface of degree $n$ in $\mathbb{P}^{n + 1}$ from a point outside the surface;
ii) the Veronese embedding into $\mathbb{P}^8$ of a quadric in $\mathbb{P}^3$ ($n=8$);
iii) the anticanonical image of $\mathbb{P}^2$ blown up at $9-n$ points;
iv) a cone over an elliptic curve in $\mathbb{P}^{n-1}$ of degree $n$.
Since $Z^{\prime} \to Z$ is given by a complete linear system, case i) above is impossible for our case. Since $q (Y) = 0$, case iv) also is impossible. Thus it suffices to exclude case ii). In case ii), however, the divisor $L$ is linearly equivalent to twice a divisor on $\tilde{Y}$, which contradicts the equality $LE_i =1$. Hence we have the assertion. \qed
In what follows, we put $D^{\prime} = - K_{Z^{\prime}}$ and denote by $\varPhi_{D^{\prime}} : Z^{\prime} \to Z \subset \mathbb{P}^n$ the anticanonical map of $Z^{\prime}$. Note that the action by $G = \mathrm{Gal} (Y/X)$ on $\tilde{Y}$ induces one on $Z^{\prime}$.
\begin{lemma} \label{lm:lifalilitytcriterion} If the surface $Z^{\prime}$ has no $(-2)$-curve, or if every $(-2)$-curve on $Z^{\prime}$ is stable under the action by $G$ on $Z^{\prime}$, then $\varPhi_L : \tilde{Y} \to Z \subset \mathbb{P}^n$ lifts to a morphism $f^{\prime} : \tilde{Y} \to Z^{\prime}$. \end{lemma}
Proof. Take the shortest composite $p^{\prime} : Y^{\prime} \to \tilde{Y}$ of quadric transformations such that $Y^{\prime}$ admits a morphism $f^{\prime} : Y^{\prime} \to Z^{\prime}$ satisfying $\varPhi_L \circ p^{\prime} = \varPhi_{D^{\prime}} \circ f^{\prime}$. Then the action by $G$ on $\tilde{Y}$ induces one on $Y^{\prime}$.
Note that $f^{\prime}$ contracts no $(-1)$-curve. This follows from $LE_i =1$ and the definition of $p^{\prime}$, since the surface $Y$ is of general type. To obtain the assertion, we only need to show that $p^{\prime} : Y^{\prime} \to \tilde{Y}$ is an isomorphism. Assume that $p^{\prime} : Y^{\prime} \to \tilde{Y}$ is not an isomorphism. Then there exists a $(-1)$-curve $C$ on $Y^{\prime}$ exceptional with respect to $p^{\prime}$. Since the anticanonical map $\varPhi_{D^{\prime}} : Z^{\prime} \to Z \subset \mathbb{P}^n$ contracts $f^{\prime} (C)$ to a point, the curve $f^{\prime} (C)$ is a $(-2)$-curve on $Z^{\prime}$, hence, by the assumption in the statement, stable under the action by $G$ on $Z^{\prime}$. Meanwhile by the same method as in Lemma \ref{lm:-1and-2curves}, we see that $f^{\prime}_* C = f^{\prime} (C)$ or $2 f^{\prime} (C)$, and that if $f^{\prime}_* C = f^{\prime} (C)$, then $C$ is a component of the ramification divisor of $f^{\prime}$. It follows that $C \simeq \mathbb{P}^1$ is stable under the action by $G$ on $Y^{\prime}$, which implies the existence of fixed points of this action.
This, however, contradicts the definition of $\pi : Y \to X$. Thus we have the assertion. \qed
\begin{lemma} \label{lm:fprimeE_i} Assume that $\varPhi_L : \tilde{Y} \to Z$ lifts to a morphism $f^{\prime} : \tilde{Y} \to Z^{\prime}$. Then $f^{\prime} (E_1)$ and $f^{\prime} (E_2)$ are $(-1)$-curves on $Z^{\prime}$. Further, the following hold$:$
$\mathrm{i}$ $)$ $f^{\prime}_* E_i = f^{\prime} (E_i)$ for $i =1$, $2$ $;$
$\mathrm{ii}$ $)$ the ramification divisor $R^{\prime}$ of $f^{\prime}$ satisfies $R^{\prime} \sim {f^{\prime}}^*(-2K_{Z^{\prime}}) + 2 \sum_{i =1, 2} E_i $ $;$
$\mathrm{iii}$ $)$ the branch divisor $B^{\prime}$ of $f^{\prime}$ satisfies $B^{\prime} \sim -4K_{Z^{\prime}} + 2 \sum_{i=1, 2} f^{\prime} (E_i)$ $;$
$\mathrm{iv}$ $)$ $f^{\prime} (E_1)$ and $f^{\prime} (E_2)$ are distinct components of the branch divisor $B^{\prime}$. \end{lemma}
Proof. The first assertion and the assertion i) follow from $E_i L = E_i {f^{\prime}}^* D^{\prime} =1$, which implies $\varPhi_L (E_i)$ is a line in $\mathbb{P}^n$. The assertions ii) and iii) follow from $D^{\prime} \sim -K_{Z^{\prime}}$ and the assertion i). So it suffices to prove the assertion iv). Let us prove the assertion iv). Let $\theta$ be the involution of $\tilde{Y}$ over $Z^{\prime}$. Since $Y$ is of general type, the divisors $E_1$ and $E_2$ are the only $(-1)$-curves on $\tilde{Y}$. It follows that if $f^{\prime}(E_1) \neq f^{\prime}(E_2)$, then $\theta (E_i) = E_i$ holds for $i=1$, $2$. Thus we only need to show $f^{\prime}(E_1) \neq f^{\prime}(E_2)$. Assume that $f^{\prime}(E_1) = f^{\prime}(E_2)$. Then ${f^{\prime}}^*(f^{\prime}(E_1)) =
{f^{\prime}}^*(f^{\prime}(E_2)) = E_1 + E_2 + \xi$ holds for a certain effective divisor $\xi$ exceptional with respect to $f^{\prime}$. Since we have $E_1 \cap E_2 = \emptyset$, we see, by the same method as in the proof of Lemma \ref{lm:-1and-2curves}, that $\xi ^2 = -(E_1 + E_2) \xi = 0$, hence $\xi =0$. It follows ${f^{\prime}}^*(f^{\prime}(E_1)) =
{f^{\prime}}^*(f^{\prime}(E_2)) = E_1 + E_2$. From this together with the assertions ii) and iii), we infer ${f^{\prime}}^* B^{\prime} - 2R^{\prime} =0$, which implies that the branch divisor $B^{\prime}$ has at most negligible singularities. Thus by \cite[Lemma 6]{quintic}, we obtain
\[
\chi (\mathcal{O}_{\tilde{Y}}) =
2\chi (\mathcal{O}_Z^{\prime})
+ \frac{1}{2} (-2K_{Z^\prime} + \sum f^{\prime} (E_i))
( -K_{Z^\prime} + \sum f^{\prime} (E_i))
= n + 3, \] which contradicts $\chi (\mathcal{O}_Y) = n + 2$.
Thus we have $f^{\prime} (E_1) \neq f^{\prime} (E_2)$, which completes the proof of the assertion iv). \qed
\begin{lemma} If the surface $Y$ is of case $2$ $)$ in Proposition $\ref{prop:L^2}$, then $\lambda = 4$. \end{lemma}
Proof. By Lemma \ref{lm:L^2-2resoution}, we have $n = 2 \lambda - 2 \leq 9$, hence $\lambda \leq 5$. Thus we only need to exclude the case $\lambda =5$. Assume $\lambda =5$. Then $r : Z^{\prime} \to \mathbb{P}^2$ is a blowing-up at one pint, hence $Z^{\prime} = \varSigma_1$. Thus by Lemmas \ref{lm:lifalilitytcriterion} and \ref{lm:fprimeE_i}, we see that $\varPhi_L : \tilde{Y} \to Z$ lifts to a morphism $f^{\prime} : \tilde{Y} \to Z^{\prime}$, and that $f^{\prime} (E_i)$'s are $(-1)$-curves. The minimal section $\varDelta_0$, however, is the unique $(-1)$-curve on the Hirzebruch surface $\varSigma_1$. Thus we have $f^{\prime} (E_1) = f^{\prime} (E_2) =\varDelta_0$, which contradicts Lemma \ref{lm:fprimeE_i}. Hence we have the assertion. \qed
Thus we only need to study the case $\lambda =4$. In what follows we assume $\lambda =4$, hence $n=6$. In this case, the morphism $r : Z^{\prime} \to \mathbb{P}^2$ is a blowing-up at three points.
\begin{lemma} \label{lm:case2structuer} Assume that $\varPhi_L : \tilde{Y} \to Z$ lifts to a morphism $f^{\prime} : \tilde{Y} \to Z^{\prime}$, and that $f^{\prime} (E_1) \cap f^{\prime} (E_2) = \emptyset$ holds. Let $r^{\prime} : Z^{\prime} \to W$ denote the blowing-down of the two $(-1)$-curves $f^{\prime} (E_1)$ and $f^{\prime} (E_2)$. Then the branch divisor $B$ of the morphism $r^{\prime} \circ f^{\prime} : \tilde{Y} \to W$ is a
member of the linear system $|-4 K_W|$ having $[3, 3]$-points at $r^{\prime} (f^{\prime} (E_1))$ and $r^{\prime} (f^{\prime} (E_2))$. Except for these two $[3,3]$-points, the branch divisor $B$ has at most negligible singularities. Further, the surface $Y$ gives the minimal desingularization of the double cover $($of the surface $W$ $)$ with branch divisor $B$. \end{lemma}
Proof. Note that $f^{\prime}$ contracts no $(-1)$-curve. Thus the divisor ${f^{\prime}}^* B^{\prime} - 2R^{\prime}$, linearly equivalent to $2(\sum {f^{\prime}}^* (f^{\prime}(E_i)) - 2 \sum E_i)$ by Lemma \ref{lm:fprimeE_i}, is twice a certain effective divisor $\zeta$ on $\tilde{Y}$. We have $\zeta E_j = (\sum {f^{\prime}}^* (f^{\prime}(E_i)) - 2 \sum E_i) E_j =1$, hence $\sharp (\zeta \cap E_j) = 1$ for $j =1$, $2$. So we put $\{ z_j \} = f^{\prime} (\zeta \cap E_j)$. Then the point $z_j \in f^{\prime} (E_j)$, where $1 \leq j \leq 2$, is an essential singularity of the branch divisor $B^{\prime}$. Meanwhile, by Lemma \ref{lm:fprimeE_i}, we see that the divisor $B^{\prime} - \sum f^{\prime} (E_i)$ is effective, and that $(B^{\prime} - \sum f^{\prime} (E_i)) f^{\prime}(E_j) = 3$ for each $j=1$, $2$, from which we infer $\mathrm{mult}_{z_j} (B^{\prime} - \sum f^{\prime} (E_i)) \leq 3$. If, moreover, we have $\mathrm{mult}_{z_j} (B^{\prime} - \sum f^{\prime} (E_i)) \leq 2$, then $z_j$ is a negligible singularity of the branch divisor $B^{\prime}$; the singularity $z_j$ of $B^{\prime}$ decomposes into a sum of points of multiplicity at most $2$ by the blowing-up at $z_j$. Thus we obtain $\mathrm{mult}_{z_j} (B^{\prime} - \sum f^{\prime} (E_i)) = 3$, hence $(B^{\prime} - \sum f^{\prime} (E_i)) \cap f^{\prime} (E_j) = \{ z_j\}$ and $\mathrm{mult}_{z_j} B^{\prime} = 4$. Let $q_1 \circ q_2 : Z_2^{\prime} \to Z^{\prime}$ be the blowing-up at $z_1$ and $z_2$, and $\varepsilon_j = (q_1 \circ q_2)^{-1}(z_j) $, the $(-1)$-curve corresponding to $z_j$. Then by the same method as in \cite[Section 2]{quintic}, we infer that the divisor $B_2^{\prime} = (q_1 \circ q_2)^* B^{\prime} - 4 \sum_{i=1,2} \varepsilon_i$ has at most negligible singularities, and that the surface $\tilde{Y}$ gives the canonical resolution of the double cover with branch divisor $B^{\prime}$. It follows that $B = r^{\prime}_* B^{\prime}$
has $[3,3]$-points at $r^{\prime} (f^{\prime}(E_1))$ and $r^{\prime} (f^{\prime}(E_2))$, that the divisor $B$ has no essential singularity except for these two $[3,3]$-points, and that the surface $Y$ gives the minimal desingularization of the double cover with branch divisor $B$. Now all we have left is the linear equivalence $B \sim - 4K_W$, which, however, is trivial by iii) in Lemma \ref{lm:fprimeE_i}. \qed
\begin{lemma} \label{lm:case2action} Let $r^{\prime} : Z^{\prime} \to W$ be the blowing-down given in Lemma $\ref{lm:case2structuer}$. Then the surface $W$ is the Hirzebruch surface $\varSigma_d$ of degree $d=0$ or $2$. The action by $G = \mathrm{Gal} (Y/X)$ on $Z^{\prime}$ induces one on $W$, of which fixed locus is a set of four isolated points. Further, none of these four fixed points lies on the branch divisor $B$. \end{lemma}
Proof. The action by $G$ on $Z^{\prime}$ induces one on $W$, since the divisor $f^{\prime} (E_1) + f^{\prime} (E_2)$ is stable under the action by $G$.
Note that the anticanonical system $|-K_{Z^{\prime}}|$ has no fixed component. From this together with $K_W^2 = K_Z^2 +2 = 8$, we see that $W = \varSigma_d$ for a certain integer $0 \leq d \leq 2$.
Let us show that the class of $\varGamma$, a fiber of the Hirzebruch surface $W = \varSigma_d \to \mathbb{P}^1$, is stable under the action by $G$ on $W$. If the class of $\varGamma$ is not stable, then we see that $d =0$ and that the generator of $G$ maps $\varGamma$ to a
member of the linear system $|\varDelta_0|$. It follows that there exists an irreducible member
$\varDelta \in |\varDelta_0 + \varGamma|$ contained in the fixed locus of the action by $G$ on $W$. We have $\varDelta \cap B \neq \emptyset$, since $\varDelta$ is a $2$-curve. Meanwhile since the Galois group $G$ acts transitively on the set $\{ r^{\prime} (f^{\prime} (E_1)), r^{\prime} (f^{\prime} (E_2)) \}$, neither $r^{\prime} (f^{\prime} (E_1))$ nor $r^{\prime} (f^{\prime} (E_2))$ lies on $\varDelta$. Thus by Lemma \ref{lm:case2structuer}, every point in $\varDelta \cap B$ is at most a negligible singularity of $B$. Then the same argument as in the proof of Proposition \ref{prop:case1exclusion} leads us to a contradiction. Hence the class of $\varGamma$ is stable.
Now let us show the assertions. The argument above shows that the action by $G$ on $W$ induces one on $\mathbb{P}^1$ via the natural fibration of the Hirzebruch surface $W = \varSigma_d \to \mathbb{P}^1$. Note that if this induced action on $\mathbb{P}^1$ is trivial, then there exists an irreducible member
$\varDelta_1 \in |\varDelta_0 + d \varGamma|$ contained in the fixed locus of the action by $G$ on $W$, which, together with the same argument as in the case of $\varDelta$ above, leads us to a contradiction. Thus the induced action on $\mathbb{P}^1$ is non-trivial.
It follows that $|\varGamma|$ has exactly two members stable under the action on $W$, which we shall call $\varGamma_1$ and $\varGamma_2$. The same argument as in the case of $\varDelta$ above shows that the induced action on $\varGamma_i$ is non-trivial for each $i =1$, $2$. Thus we see that $d \neq 1$, and that if $d=0$ or $2$, then the fixed locus of the induced action on $W$ is a set of four isolated points. The absence of the fixed points lying on $B$ follows from the same argument as in the case of $\varDelta$ above. \qed
By Lemmas \ref{lm:case2structuer} and \ref{lm:case2action}, we see that if $\varPhi_L$ lifts to $f^{\prime} : \tilde{Y} \to Z^{\prime}$, and if $f^{\prime} (E_1) \cap f^{\prime} (E_2) = \emptyset$, then our surface $X$ has the structure as in Theorem \ref{thm:completedescription}. Let us check that these two conditions are in fact satisfied. To do this, we study the arrangement of $(-1)$-curves and $(-2)$-curves on $Z^{\prime}$, and use Lemmas \ref{lm:lifalilitytcriterion} and \ref{lm:fprimeE_i}.
Let $r_i : Z^{\prime}_i \to Z^{\prime}_{i-1}$, where $-2 \leq i \leq 0$, be the blowing-up such that $r = (r_{-2} \circ r_{-1} \circ r_{0}) : Z^{\prime}_0 = Z^{\prime} \to Z^{\prime}_{-3} = \mathbb{P}^2 $ holds. We denote by $z_i \in Z^{\prime}_{i-1}$ and $\varepsilon_i = r_i^{-1} (z_i)$ the center of the blowing-up $r_i$ and its corresponding $(-1)$-curve, respectively. For each $-2 \leq i \leq 0$, we denote by $\varepsilon_i^{\prime}$ the strict transform to $Z^{\prime}$ of the exceptional curve $\varepsilon_i$. For the total transform to $Z^{\prime}$ of $\varepsilon_i$, we use the same symbol $\varepsilon_i$.
\begin{lemma} Let $m \leq 2$ be a non-negative integer, and $C$, a $(-m)$-curve on $Z^{\prime}$ not exceptional with respect to $r: Z^{\prime} \to \mathbb{P}^2$. Then $C$ is a strict transform to $Z^{\prime}$ of a line on $\mathbb{P}^2$ passing exactly $m+1$ of the tree points $z_{i}$'s $($ $-2 \leq i \leq 0$ $)$. \end{lemma}
Proof. Let $l$ be a line on $\mathbb{P}^2$. Then we have $C \sim m_0 r^* (l) - \sum_{i = -2}^0 n_i \varepsilon_i$ for certain integers $m_0 \geq 1$ and $n_i \geq 0$'s. Note that $C^2 = -m$ and $-K_{Z^{\prime}}C = 2 -m$, since $C$ is a $(-m)$-curve. Thus we have \begin{equation} \label{eq:CK} m_0^2 - \sum_{i= -2}^0 n_i^2 = -m, \quad \quad 3 m_0 - \sum_{i= -2}^0 n_i = -m + 2. \end{equation} From these equalities, we infer \[
5 \sum_{i= -2}^0 n_i^2
+ \sum_{-2 \leq i < j \leq 0} (n_i - n_j)^2
+ \sum_{i= -2}^0 (n_i + m-2)^2
= 9m + 4 (m-2)^2 \leq 18, \] hence $\sum_{i= -2}^0 n_i^2 \leq 3$. Thus we have $n_i^2 = n_i$ for any $-2 \leq i \leq 0$. By this together with the equalities (\ref{eq:CK}), we obtain $m_0 = 1$ and $\sum_{i= -2}^0 n_i = m +1 $. Thus we have the assertion. \qed
We study the arrangement of $(-1)$-curves and $(-2)$-curves on $Z^{\prime}$ according to the configuration of the centers $z_i$'s of the blowing-up $r: Z^{\prime} \to \mathbb{P}^2$. First, we study the case where no two of the centers $z_{-2}$, $z_{-1}$, and $z_{0}$ are infinitely near. This case is divided into two cases: case $2$-$1$-$1$) and case $2$-$1$-$2$).
Case $2$-$1$-$1$): the case where the centers $z_{-2}$, $z_{-1}$, and $z_0$ are not collinear. In this case, the surface $Z^{\prime}$ has no $(-2)$-curve. Thus $\varPhi_L$ lifts to a morphism $f^{\prime} : \tilde{Y} \to Z^{\prime}$. There exist exactly six $(-1)$-curves: $\varepsilon_{-2}$, $\varepsilon_{-1}$, $\varepsilon_{0}$, $r^{-1}_*(l_{-2,-1})$, $r^{-1}_*(l_{-1,0})$, and $r^{-1}_*(l_{-2,0})$, where $l_{i,j}$ denotes the line on $\mathbb{P}^2$ passing $z_i$ and $z_j$. Let $(X_0 : X_1: X_2)$ be homogeneous coordinates of $\mathbb{P}^2$ satisfying $z_{-2} = (1:0:0)$, $z_{-1} = (0:1:0)$, and $z_0 = (0:0:1)$. For each $(a,b) \in \mathbb{C}^{\times} \times \mathbb{C}^{\times}$, we denote by $\varphi_{(a,b)}$ the automorphism of $Z^{\prime}$ corresponding to the projective transformation $(X_0 : X_1: X_2) \mapsto (X_0 : a X_1: b X_2)$.
Let us study the induced action by $G$ on $Z^{\prime}$. Let $\mathrm{Aut} (Z^{\prime})$ be the group of analytic automorphisms of the surface $Z^{\prime}$, and $D_6$, the dihedral group of degree $6$. Then we have a short exact sequence \[
0 \to \mathbb{C}^{\times} \times \mathbb{C}^{\times}
\to \mathrm{Aut}(Z^{\prime})
\to D_6 \to 0, \] where the morphism $\mathbb{C}^{\times} \times \mathbb{C}^{\times} \to \mathrm{Aut}(Z^{\prime}) $ is given by $(a,b) \mapsto \varphi_{(a,b)}$, and the morphism $\mathrm{Aut}(Z^{\prime}) \to D_6$ corresponds to the transitions of six $(-1)$-curves on $Z^{\prime}$. Let $\varphi_{\sigma}$ and $\varphi_{\tau}$ be the automorphisms of $Z^{\prime}$ corresponding to the Cremona transformation $(X_0 : X_1: X_2) \mapsto (X_2 X_0: X_0 X_1: X_1 X_2)$ and the morphism $(X_0 : X_1: X_2) \mapsto (X_0 : X_2: X_1)$, respectively. Then we have \[
(\varphi_{\sigma})^6 = \mathrm{id}_{Z^{\prime}} \qquad
(\varphi_{\tau})^2 = \mathrm{id}_{Z^{\prime}} \qquad
\varphi_{\sigma} \circ \varphi_{\tau}
\circ \varphi_{\sigma} \circ \varphi_{\tau} = \mathrm{id}_{Z^{\prime}}. \] Thus the short exact sequence above splits. We denote by $\sigma$ and $\tau$ the image by $\mathrm{Aut} (Z^{\prime}) \to D_6$ of $\varphi_{\sigma}$ and $\varphi_{\tau}$, respectively. We have a group homomorphism $G \to \mathrm{Aut} (Z^{\prime})$ corresponding to the action by $G$ on $Z^{\prime}$. Composing this homomorphism with $\mathrm{Aut} (Z^{\prime}) \to D_6$, we obtain a group homomorphism $\alpha : G \to D_6$. Note that by Lemma \ref{lm:fprimeE_i}, the morphism $\alpha$ is an injection of $G$ into $D_6$. Hence the image $\alpha (G)$ is conjugate to $\langle \tau \rangle$, $ \langle \sigma^3 \tau \rangle$, or $\langle \sigma^3 \rangle$ in $D_6$.
Assume that the image $\alpha (G)$ is conjugate to $\langle \tau \rangle$ in $D_6$. Replacing the morphism $r : Z^{\prime} \to \mathbb{P}^2$ if necessary, we may assume that $\alpha (G) = \langle \tau \rangle$. Then since the Galois group $G$ acts transitively on the set $\{ f^{\prime} (E_1), f^{\prime} (E_2) \}$, we have $\{ f^{\prime} (E_1), f^{\prime} (E_2) \} = \{ r^{-1}_* (l_{-2, -1}), r^{-1}_* (l_{-2, 0}) \}$ or $\{ \varepsilon_{-1}, \varepsilon_0 \}$, hence $f^{\prime} (E_1) \cap f^{\prime} (E_2) = \emptyset$. It follows that the surface $W$, where $r^{\prime} : Z^{\prime} \to W$ is the blowing-down of the two $(-1)$-curves $f^{\prime} (E_1)$ and $f^{\prime} (E_2)$, is isomorphic to the Hirzebruch surface $\varSigma_1$, which contradicts Lemma \ref{lm:case2action}. Thus $\alpha (G)$ is not conjugate to $\langle \tau \rangle$.
Assume that the image $\alpha (G)$ is conjugate to $\langle \sigma^3 \tau \rangle$ in $D_6$. Replacing the morphism $r : Z^{\prime} \to \mathbb{P}^2$ if necessary, we may assume that $\alpha (G) = \langle \sigma^3 \tau \rangle$. Then the blowing-down of the two $(-1)$-curves $\varepsilon_{-2}$ and $r^{-1}_* (l_{-1, 0})$ gives a birational morphism $r^{\prime \prime} : Z^{\prime} \to \varSigma_0 = \mathbb{P}^1 \times \mathbb{P}^1 $ satisfying $r^{\prime \prime}_* (\varepsilon_0) \sim
r^{\prime \prime}_* (r^{-1}_* (l_{-2, -1})) \sim \varDelta_0$ and $r^{\prime \prime}_* (\varepsilon_{-1}) \sim
r^{\prime \prime}_* (r^{-1}_* (l_{-2, 0})) \sim \varGamma$. Note that the action by $G$ on $Z^{\prime}$ induces one on $\varSigma_0 = \mathbb{P}^1 \times \mathbb{P}^1$. We take homogeneous coordinates $((\xi_0 : \xi_1), (\eta_0 : \eta_1))$ of $\varSigma_0 = \mathbb{P}^1 \times \mathbb{P}^1$ in such a way that $r^{\prime \prime} (\varepsilon_{-2}) = ((1 : 0), (1 : 0))$ and $r^{\prime \prime} (r^{-1}_* (l_{-1, 0})) = ((0 : 1), (0 : 1))$ hold, and that the automorphism of $\varSigma_0$ corresponding to the generator $\iota$ of $G$ is given by $((\xi_0 : \xi_1), (\eta_0 : \eta_1)) \mapsto ((\eta_1 : \eta_0), (\xi_1 : \xi_0))$. Since we have $-K_{Z^{\prime}} \sim {r^{\prime \prime}}^* (-K_{\varSigma_0}) - \varepsilon_{-2} - r^{-1}_* (l_{-1, 0})$, the space $H^0(\mathcal{O}_{Z^{\prime}} (-2K_{Z^{\prime}}))$ corresponds to a certain subspace $V$ of $H^0(\mathcal{O}_{\varSigma_0} (-2K_{\varSigma_0}))$. Every element in $V$ is a homogeneous polynomial $\psi (\xi_0, \xi_1, \eta_0, \eta_1)$ of bidegree $(4, 4)$ vanishing with multiplicity at least $2$ at $((1 : 0), (1 : 0))$ and $((0 : 1), (0 : 1))$.
Note that we have a natural inclusion $V \hookrightarrow H^0(\mathcal{O}_Y (2K_Y))$, since we have $L \sim {f^{\prime}}^* D^{\prime}$. We denote by $\phi$ the restriction to $V$ of the natural action by $G$ on $H^0_Y (\mathcal{O}(2K_Y))$. Let $\phi^{\prime} (\iota)$ be the automorphism of $V$ given by $\psi (\xi_0, \xi_1, \eta_0, \eta_1) \mapsto \psi (\eta_1, \eta_0, \xi_1, \xi_0)$. Then $\iota \mapsto \phi^{\prime} (\iota)$, where $\iota$ is the generator of the Galois group $G$, gives another action $\phi^{\prime}$ by $G$ on $V$. Note that for any $g \in G$ and $\psi \in V$, the two elements $\phi (g) \psi$ and $\phi^{\prime} (g) \psi$ defines the same divisor on $\varSigma_0$. From this we infer that $\phi = c \phi^{\prime}$ for a certain character $c \in \mathrm{Char} (G)$.
Now let $V^+$ be the set of all elements in $V$ stable under the action $\phi^{\prime}$. Then by $\phi = c \phi^{\prime}$, we see that $V^+ \subset H^0(\mathcal{O}_X (2K_X - T_c))$ for a torsion divisor $T_c \in \mathrm{Pic} (X)$ corresponding to the character $c$. Meanwhile, by the Riemann--Roch theorem, we have $h^0(\mathcal{O}_X (2K_X - T_c)) = \chi + K_X^2 = 11$. The space $V^+$, however, has a base consisting of twelve elements: \[
\xi_0^i \xi_1^{4-i} \eta_0^j \eta_1^{4-j}
+ \xi_0^{4-j} \xi_1^{j} \eta_0^{4-i}\eta_1^{i}
\qquad
( 0 \leq i, \quad 0 \leq j, \quad 2 \leq i+j \leq 4). \] This contradicts the inequality $\dim V^+ \leq h^0(\mathcal{O}_X (2K_X - T_c))$. Hence, the image $\alpha (G)$ is not conjugate to $\langle \sigma^3 \tau \rangle$ in $D_6$.
Thus we have $\alpha (G) = \langle \sigma^3 \rangle$. Hence, replacing $r : Z^{\prime} \to \mathbb{P}^2$ if necessary, we may assume that $\{ f^{\prime} (E_1), f^{\prime} (E_2) \} = \{ \varepsilon_{-2}, r^{-1}_* (l_{-1, 0}) \}$. Then the surface $W$ as in Lemma \ref{lm:case2structuer}, obtained by blowing down the two $(-1)$-curves $f^{\prime} (E_1)$ and $f^{\prime} (E_2)$ of $Z^{\prime}$, is isomorphic to the Hirzebruch surface $\varSigma_0$. Thus by Lemmas \ref{lm:case2structuer} and \ref{lm:case2action}, our surface $X$, in case $2$-$1$-$1$), has the structure as in the case $d=0$ in Theorem \ref{thm:completedescription}.
Case $2$-$1$-$2$): the case where three points $z_{-2}$, $z_{-1}$, and $z_0$ are collinear. Let $l_{-2, -1}$ be the line on $\mathbb{P}^2$ passing the tree points $z_{i}$'s above. Then the strict transform $r^{-1}_* (l_{-2, -1})$ is the unique $(-2)$-curve on $Z^{\prime}$. Hence by Lemma \ref{lm:lifalilitytcriterion}, the morphism $\varPhi_L : \tilde{Y} \to Z$ lifts to $f^{\prime} : \tilde{Y} \to Z^{\prime}$. Meanwhile the surface $Z^{\prime}$ has exactly three $(-1)$-curves: $\varepsilon_{-2}$, $\varepsilon_{-1}$, and $\varepsilon_0$. Replacing $r : Z^{\prime} \to \mathbb{P}^2$ if necessary, we may assume $\{ f^{\prime} (E_1), f^{\prime} (E_2)\} = \{ \varepsilon_{-2}, \varepsilon_{-1} \}$ by Lemma \ref{lm:fprimeE_i}. Let $r^{\prime} : Z^{\prime} \to W$ be the blowing-down as in Lemma \ref{lm:case2structuer} of the two $(-1)$-curves $f^{\prime} (E_1)$ and $f^{\prime} (E_2)$. Then we have $W = \varSigma_1$, $r^{\prime}_* (\varepsilon_0) = \varDelta_0$, and $r^{\prime}_* (r^{-1}_*(l_{-2, -1})) \sim \varGamma$, which contradicts Lemma \ref{lm:case2action}. Thus case $2$-$1$-$2$) does not occur.
Next, we study the case where $z_{-2}$ and $z_{-1}$ are distinct points on $\mathbb{P}^2$, and $z_0$ is infinitely near to $z_{-1}$. We denote by $l_{-2, -1}$ the unique line on $\mathbb{P}^2$ passing $z_{-2}$ and $z_{-1}$. This case is divided into two cases: case $2$-$2$-$1$) and case $2$-$2$-$2$).
Case $2$-$2$-$1$): the case where $z_0$ does not lie on the strict transform $(r_{-2} \circ r_{-1})^{-1}_* (l_{-2, -1})$ of $l_{-2, -1}$ by $r_{-2} \circ r_{-1}$. Let $l_{-1, 0}$ be the unique line on $\mathbb{P}^2$ whose strict transform $(r_{-2} \circ r_{-1})^{-1}_* (l_{-1, 0})$ by $r_{-2} \circ r_{-1}$ passes $z_0$. Then the surface $Z^{\prime}$ has a unique $(-2)$-curve $\varepsilon_{-1}^{\prime}$, and exactly four $(-1)$-curves $\varepsilon_{-2}$, $\varepsilon_0$, $r^{-1}_* (l_{-2, -1})$, and $r^{-1}_* (l_{-1, 0})$. Hence, by Lemma \ref{lm:lifalilitytcriterion}, the morphism $\varPhi_L : \tilde{Y} \to Z$ lifts to $f^{\prime} : \tilde{Y} \to Z^{\prime}$. Note that $\{ \varepsilon_0 , r^{-1}_* (l_{-2, -1})\}$ is the set of all $(-1)$-curves intersecting the unique $(-2)$-curve $\varepsilon_{-1}^{\prime}$. Thus we have $\{ f^{\prime} (E_1), f^{\prime} (E_2)\} = \{ \varepsilon_0 , r^{-1}_* (l_{-2, -1}) \}$ or $\{ \varepsilon_{-2} , r^{-1}_* (l_{-1, 0}) \}$, hence, in particular, $f^{\prime} (E_1) \cap f^{\prime} (E_2) = \emptyset$. We denote by $r^{\prime} : Z^{\prime} \to W$ the blowing-down as in Lemma \ref{lm:case2structuer} of the two $(-1)$-curves $f^{\prime} (E_1)$ and $f^{\prime} (E_2)$. If $\{ f^{\prime} (E_1), f^{\prime} (E_2)\} = \{ \varepsilon_0 , r^{-1}_* (l_{-2, -1}) \}$, then we have $W = \varSigma_0$, $r^{\prime}_* (\varepsilon_{-1}^{\prime}) \sim \varDelta_0$ and $r^{\prime}_* (\varepsilon_{-2}) \sim
r^{\prime}_* (r^{-1}_* (l_{-1, 0})) \sim \varGamma$. If on the other hand $\{ f^{\prime} (E_1), f^{\prime} (E_2)\} = \{ \varepsilon_{-2} , r^{-1}_* (l_{-1, 0}) \}$, then we have $W = \varSigma_2$, $r^{\prime}_* (\varepsilon_{-1}^{\prime}) = \varDelta_0$, and $r^{\prime}_* (\varepsilon_0) \sim
r^{\prime}_* (r^{-1}_* (l_{-2, -1})) \sim \varGamma$. Thus by lemmas \ref{lm:case2structuer} and \ref{lm:case2action}, our surface $X$, in case $2$-$2$-$1$), has the structure as in the case $d=0$ or the case $d=2$ in Theorem \ref{thm:completedescription}, according as $\{ f^{\prime} (E_1), f^{\prime} (E_2)\} = \{ \varepsilon_0 , r^{-1}_* (l_{-2, -1}) \}$ or $\{ f^{\prime} (E_1), f^{\prime} (E_2)\} = \{ \varepsilon_{-2} , r^{-1}_* (l_{-1, 0}) \}$ respectively.
Case $2$-$2$-$2$): the case where $z_0$ lies on the strict transform $(r_{-2} \circ r_{-1})^{-1}_* (l_{-2, -1})$. In this case, the surface $Z^{\prime}$ has exactly two $(-2)$-curves $\varepsilon_{-1}^{\prime}$ and $r^{-1}_* (l_{-2, -1})$, and exactly two $(-1)$-curves $\varepsilon_{-2}$ and $\varepsilon_0$. Note that every $(-2)$-curve on $Z^{\prime}$ is stable under the action by $G$ on $Z^{\prime}$; the divisor $r^{-1}_* (l_{-2, -1})$ is the unique $(-2)$-curve intersecting all $(-1)$-curves on $Z^{\prime}$. Thus by Lemma \ref{lm:lifalilitytcriterion}, the morphism $\varPhi_L : \tilde{Y} \to Z$ lifts to $f^{\prime} : \tilde{Y} \to Z^{\prime}$ . Then it follows from Lemma \ref{lm:fprimeE_i} that $\{ f^{\prime} (E_1), f^{\prime} (E_2)\} = \{ \varepsilon_{-2} , \varepsilon_0 \}$. This, however, contradicts the transitivity of the action by $G$ on $\{ f^{\prime} (E_1), f^{\prime} (E_2)\}$, since $\varepsilon_0$ is the unique $(-1)$-curve intersecting all $(-2)$-curves on $Z^{\prime}$. Thus case $2$-$2$-$2$) does not occur.
Finally, we study the case where $z_{-1}$ is infinitely near to $z_{-2}$, and $z_0$ is infinitely near to $z_{-1}$. We denote by $l_{-2, -1}$ the unique line on $\mathbb{P}^2$ whose strict transform $(r_{-2})^{-1}_* (l_{-2, -1})$ passes $z_{-1}$. Note that $Z^{\prime}$ has no $(-3)$-curve, since
the linear system $|-K_{Z^{\prime}}|$ has no fixed component. Thus the point $z_0$ does not lie on the strict transform $(r_{-1})^{-1}_* (\varepsilon_{-2})$. This case is divided into two cases: case $2$-$3$-$1$) and case $2$-$3$-$2$).
Case $2$-$3$-$1$): the case where $z_0$ does not lie on the strict transform $(r_{-2} \circ r_{-1})^{-1}_* (l_{-2, -1})$. In this case, the surface $Z^{\prime}$ has exactly two $(-2)$-curves $\varepsilon_{-2}^{\prime}$ and $\varepsilon_{-1}^{\prime}$, and exactly two $(-1)$-curves $\varepsilon_0$ and $r^{-1}_* (l_{-2, -1})$. Since $\varepsilon_{-2}^{\prime}$ is the unique $(-2)$-curve intersecting no $(-1)$-curve on $Z^{\prime}$, every $(-2)$-curve is stable under the action by $G$ on $Z^{\prime}$. Thus by Lemma \ref{lm:lifalilitytcriterion}, the morphism $\varPhi_L : \tilde{Y} \to Z$ lifts to $f^{\prime} : \tilde{Y} \to Z^{\prime}$. Then it follows from Lemma \ref{lm:fprimeE_i} that $\{ f^{\prime} (E_1), f^{\prime} (E_2)\} = \{ \varepsilon_0 , r^{-1}_* (l_{-2, -1})\}$, hence $f^{\prime} (E_1) \cap f^{\prime} (E_2) = \emptyset $. Let $r^{\prime} : Z^{\prime} \to W$ be the blowing-down as in Lemma \ref{lm:case2structuer} of the two $(-1)$-curves $f^{\prime} (E_1)$ and $f^{\prime} (E_2)$. Then we have $W = \varSigma_2$, $r^{\prime}_* (\varepsilon_{-2}^{\prime}) = \varDelta_0$, and $r^{\prime}_* (\varepsilon_{-1}^{\prime}) \sim \varGamma$. Thus by Lemmas \ref{lm:case2structuer} and \ref{lm:case2action}, our surface $X$, in case $2$-$3$-$1$), has the structure as in the case $d=2$ in Theorem \ref{thm:completedescription}.
Case $2$-$3$-$2$): the case where $z_0$ lies on the strict transform $(r_{-2} \circ r_{-1})^{-1}_* (l_{-2, -1})$. In this case, the surface $Z^{\prime}$ has exactly three $(-2)$-curves $\varepsilon_{-2}^{\prime}$, $\varepsilon_{-1}^{\prime}$, and $r^{-1}_* (l_{-2, -1})$, and a unique $(-1)$-curve $\varepsilon_0$. Note that $\varepsilon_{-2}^{\prime}$ is the unique $(-2)$-curve intersecting no $(-1)$-curve on $Z^{\prime}$, and that $\varepsilon_{-1}^{\prime}$ is the unique $(-2)$-curve intersecting $\varepsilon_{-2}^{\prime}$. Thus every $(-2)$-curve on $Z^{\prime}$ is stable under the action by $G$ on $Z^{\prime}$. Thus by Lemma \ref{lm:lifalilitytcriterion}, the morphism $\varPhi_L : \tilde{Y} \to Z$ lifts to $f^{\prime} : \tilde{Y} \to Z^{\prime}$. This, however, contradicts Lemma \ref{lm:fprimeE_i}, since $\varepsilon_0$ is the unique $(-1)$-curve on $Z^{\prime}$. Hence, case $2$-$3$-$2$) does not occur.
Thus we have the following:
\begin{proposition} \label{prop:case2conclusion} Assume that the surface $Y$ is of case $2$ $)$ in Proposition $\ref{prop:L^2}$. Then $\lambda = 4$. Further, the surface $X$ in this case has the structure as in Theorem $\ref{thm:completedescription}$. \end{proposition}
\section{The case $\deg \varPhi_{K_Y} = 1$} \label{scn:deg=1}
In this section, we exclude the case $\deg \varPhi_{K_Y} =1$ and give a proof for Theorems \ref{thm:maintheorem} and \ref{thm:completedescription}. In what follows, we assume that $\deg \varPhi_{K_Y} =1$. Note that by Proposition \ref{prop:degphiKY}, we have $\lambda =4$, hence $K_Y^2 = 14$, $p_g (Y) =7$, and $q (Y) =0$. Thus our $Y$ is a canonical surface whose invariant lies on the Castelnuovo line. By \cite[Lemma 1.1]{3pg-7},
the canonical system $|K_Y|$ is free from base points; hence the canonical map $\varPhi_{K_Y} : Y \to \mathbb{P}^n$ is a morphism, where $n = 2\lambda -2 =6$. In what follows, we frequently use results given in \cite{3pg-7}.
Let $\mathcal{Q} \subset \mathbb{P}^n$ be the intersection of all quadrics containing the canonical image $Z = \varPhi_{K_Y} (Y)$. By \cite[Section $1$]{3pg-7}, we obtain the following:
\begin{proposition} \label{prop:intersectionofquadrics} Let $\mathcal{Q}$ be the variety defined above. Then either of the following holds$:$
$1$ $)$ $\mathcal{Q}$ is the image by $\varPhi_T$ of the variety $\mathcal{Q}^{\prime} = \mathbb{P} (\mathcal{O}_{\mathbb{P}^2} \oplus \mathcal{O}_{\mathbb{P}^2} (2))$, where $\varPhi_T$ is the morphism associated with a tautological divisor $T$ of the $\mathbb{P}^1$-bundle $\mathrm{pr}_{\mathcal{Q}^{\prime}} : \mathbb{P} (\mathcal{O}_{\mathbb{P}^2} \oplus \mathcal{O}_{\mathbb{P}^2} (2)) \to \mathbb{P}^2$ $;$
$2$ $)$ $\mathcal{Q}$ is the image by $\varPhi_T$ of the variety $\mathcal{Q}^{\prime} = \mathbb{P} (\bigoplus_{i=0}^2 \mathcal{O}_{\mathbb{P}^1} (a_i))$, where $\varPhi_T$ is the morphism associated with a tautological divisor $T$ of the $\mathbb{P}^2$-bundle $\mathrm{pr}_{\mathcal{Q}^{\prime}} : \mathbb{P} (\bigoplus_{i=0}^2 \mathcal{O}_{\mathbb{P}^1} (a_i)) \to \mathbb{P}^1$, and $0 \leq a_0 \leq a_1 \leq a_2$ and $\sum_{i=0}^2 a_i = n-2$. \end{proposition}
First, we exclude case $1$) in the proposition above.
\begin{proposition} \label{prop:deg1case1exclusion} Case $1$ $)$ in Proposition $\ref{prop:intersectionofquadrics}$ does not occur. \end{proposition}
Proof. Assume that our $\mathcal{Q}$ is as in case $1$) in Proposition \ref{prop:intersectionofquadrics}. Then $\mathcal{Q}$ is a cone over the Veronese surface. Let $p_0$ be the vertex of $\mathcal{Q}$, and $\varLambda$, the linear system consisting of pull-backs by $\varPhi_{K_Y}$ of all hyperplanes in $\mathbb{P}^n$ passing $p_0$. We denote by $\varLambda_0$ and $G_0$ its variable part and fixed part respectively. By \cite[Proof of Claim I]{3pg-7}, the linear system $\varLambda_0$ is free from base points and induces $\varPhi_{\varLambda_0} : Y \to \mathbb{P}^{n-1}$, a morpshism of mapping degree $3$ onto its image. The image $\varPhi_{\varLambda_0} (Y)$ is the Veronese surface, i.e., the projective plane $\mathbb{P}^2$ embedded in $\mathbb{P}^5$ by $\mathcal{O}_{\mathbb{P}^2} (2)$. Note that by the definition of $\mathcal{Q}$, the varity $\mathcal{Q}$ and its vertex $p_0$ are stable under the action by $G = \mathrm{Gal} (Y/X)$ on $\mathbb{P}^n$. This implies that the subspace of $H^0 (\mathcal{O}_Y (K_Y))$ corresponding to $\varLambda$ is stable under the action by $G$ on $H^0 (\mathcal{O}_Y (K_Y))$. Thus the action by $G$ on $Y$ induces one on $\varPhi_{\varLambda_0} (Y) = \mathbb{P}^2$. Now let us derrive a contradiction. Since $G \simeq \mathbb{Z} / 2$, the fixed locus of this induced action contains a line $l_0$ on $\mathbb{P}^2$. Then the divsor $\varPhi_{\varLambda_0}^* (l_0)$, stable under the action by $G$, is a pull-back by $\pi : Y \to X$ of that on $X$. We however have $\varPhi_{\varLambda_0}^* (l_0)^2 = \deg \varPhi_{\varLambda_0} =3$, which contradicts $\deg \pi =2$. Thus we have the assertion. \qed
Next, we exclude case $2$) in Proposition \ref{prop:intersectionofquadrics}.
\begin{lemma} \label{lm:a0=0} If the variety $\mathcal{Q}$ is as in case $2$ $)$ of Proposition \ref{prop:intersectionofquadrics}, then $a_0 =0$. \end{lemma}
Proof. Assume that our variety $\mathcal{Q}$ is as in case $2$) in Proposintion \ref{prop:intersectionofquadrics} and that $a_0 > 0$. Then $\varPhi_T : \mathcal{Q}^{\prime} \to \mathbb{P}^n$ is an embedding. We identify $\mathcal{Q}$ and $\mathcal{Q}^{\prime}$ by $\varPhi_T$. By the same arguement as in the proof of Proposintion \ref{prop:deg1case1exclusion}, we see that the variety $\mathcal{Q}$ is stable under the action by $G$ on $\mathbb{P}^n$.
Let $P$ be a fiber of the $\mathbb{P}^2$-bundle $\mathrm{pr}_{\mathcal{Q}^{\prime}} : \mathcal{Q}=\mathcal{Q}^{\prime} \to \mathbb{P}^1$. Then $P$ and $T$ generate the Picard group of $\mathcal{Q}$. Using this, we see easily that if a divisor $P^{\prime}$ on $\mathcal{Q}$ satisfies ${P^{\prime}}^3 = K_{\mathcal{Q}}{P^{\prime}}^2 =0$ and $h^0 (\mathcal{O}_{\mathcal{Q}} (P^{\prime})) =2$, then $P^{\prime} \sim P$. Thus the class of $P$ is stable under the action by $G$ on $\mathcal{Q}$. It follows that this action induces one on $\mathbb{P}^1$ via the projection $\mathrm{pr}_{\mathcal{Q}^{\prime}} : \mathcal{Q}=\mathcal{Q}^{\prime} \to \mathbb{P}^1$,
and that there exsits a member $P_0 \in |P|$ stable under the action on $\mathcal{Q}$. Now let us derrive a contradiction. Since $G \simeq \mathbb{Z} / 2$, the fixed locus of the action by $G$ on $P_0 = \mathbb{P}^2$ contains a line $l_0$. Hence the action on $Z$ has a fixed points. By \cite[Theorem 1.5]{3pg-7}, however, the surface $Z$ has at most rational double points as its singularities. Thus, by the same arguement as in the proof of Proposition \ref{prop:case1exclusion}, we infer that the action on $Y$ has fixed points, which contradicts the definition of $\pi : Y \to X$. \qed
\begin{proposition} \label{prop:prop:deg1case2exclusion} Case $2$ $)$ in Proposition $\ref{prop:intersectionofquadrics}$ does not occur. \end{proposition}
Proof. Assume that our $\mathcal{Q}$ is as in case $2$) in Proposition \ref{prop:intersectionofquadrics}. Then by Lemma \ref{lm:a0=0} and \cite[Claim II]{3pg-7}, we have $a_0 = 0$ and $a_1 > 0$. It follows that our $\mathcal{Q}$ is a cone over the Hirzebruch surface $\varSigma_{a_2 - a_1}$ embedded in $\mathbb{P}^{n-1}$ by
$|\varDelta_0 + a_2 \varGamma|$. Let $p_0$ be the vertex of $\mathcal{Q}$, and $\varLambda$, the linear system consisting of the pull-backs by $\varPhi_{K_Y}$ of all hyperplanes in $\mathbb{P}^n$ passing $p_0$. We denote by $\varLambda_0$ and $G_0$ the variable part and the fixed part of $\varLambda$ respectively. By \cite[Proof of Claim II]{3pg-7}, the linear system $\varLambda_0$ is free from base points and induces $\varPhi_{\varLambda_0} : Y \to \mathbb{P}^{n-1}$, a morphism of degree $3$ onto its image. The image $\varPhi_{\varLambda_0} (Y)$ is the Hirzebruch surface $\varSigma_{a_2 - a_1}$ embedded in $\mathbb{P}^{n-1}$ by
$|\varDelta_0 + a_2 \varGamma|$. By the same arguement as in the proof of Proposition \ref{prop:deg1case1exclusion}, we see that the action by $G$ on $Y$ induces one on $\varSigma_{a_2 - a_1}$. The class of $\varDelta_0 + a_2 \varGamma$ and that of $ -K_{\varSigma_{a_2 - a_1}}$ are stable under this induced action on $\varSigma_{a_2 - a_1}$; hence so are the class of $\varDelta_0$ and that of $\varGamma$.
Thus there exist members $\varDelta_1 \in |\varDelta_0|$
and $\varGamma_1 \in |\varGamma|$ stable under the action on $\varSigma_{a_2 - a_1}$. Then from $\varPhi_{\varLambda_0}^* (\varDelta_1)
\varPhi_{\varLambda_0}^* (\varGamma_1)
= \deg \varPhi_{\varLambda_0} =3$, we derrive a contradiction by the same arguement as in the proof of Proposition \ref{prop:deg1case1exclusion}. Thus we have the assertion. \qed
Now we are ready to prove Theorems \ref{thm:maintheorem} and \ref{thm:completedescription}.
{\sc Proof of Theorems \ref{thm:maintheorem} and \ref{thm:completedescription}}.
By Propositions \ref{prop:degphiKY},
\ref{prop:intersectionofquadrics},
\ref{prop:deg1case1exclusion},
and \ref{prop:prop:deg1case2exclusion}, we have $\deg \varPhi_{K_Y} = 2$. Thus Theorems \ref{thm:maintheorem} and \ref{thm:completedescription} follow from Propsitions \ref{prop:L^2},
\ref{prop:case1exclusion},
\ref{prop:case3-1exclusion},
\ref{prop:case3-2exclusion},
and \ref{prop:case2conclusion}. \qed
\begin{remark} \label{rem:onthedescription} Let $X_{(1)}$ and $X_{(2)}$ be two minimal complex surfaces as in Theorem \ref{thm:completedescription}, $\pi_{(i)} : Y_{(i)} \to X_{(i)}$ ($i =1$, $2$), the unramified double cover corresponding to the torsion group, $f_{(i)} : Y_{(i)} \to W_{(i)} = \varSigma_{d_{(i)}}$, the generically two-to-one morphism as in Theorem \ref{thm:completedescription}, and $B_{(i)}$, the branch divisor of $f_{(i)}$. Then if $X_{(1)}$ and $X_{(2)}$ are isomorphic to each other, so are the triplets
$(W_{(1)}, \iota |_{W_{(1)}}, B_{(1)})$
and $(W_{(2)}, \iota |_{W_{(2)}}, B_{(2)})$,
where $\iota |_{W_{(i)}}$ denotes the involution of $W_{(i)}$ corresponding to the generator of the Galoirs group of $\pi_{(i)}$. This is verified as follows. Let $p_{(i)} : \tilde{Y}_{(i)} \to Y_{(i)}$ be the shortest composite of quadric transformations such that the variable part of
$p_{(i)}^* |K_{Y_{(i)}}|$ is free from base points, and $r^{\prime}_{(i)} : Z_{(i)}^{\prime} \to W_{(i)} = \varSigma_{d_{(i)}}$, the blowing-up at two $[3,3]$-points of the branch divisor $B_{(i)}$. Then $f_{(i)}$ induces a morphims $\tilde{f}_{(i)} : \tilde{Y}_{(i)} \to Z_{(i)}^{\prime}$. The projection $r_{(i)}^{\prime}$ is the blowing-down of the image by $\tilde{f}_{(i)}$ of the exceptional divisor of $p_{(i)} : \tilde{Y}_{(i)} \to Y_{(i)}$. Since $Z_{(i)}^{\prime}$ is the minimal desingularization of the canonical image of $Y_{(i)}$, we have the assertion. \end{remark}
\section{The moduli space for the case $\chi=4$} \label{scn:onmodulispace}
In this section, we shall study the moduli space for surfaces as in Theorem \ref{thm:completedescription}, and give a proof for Theorem \ref{thm:moduli}. For this purpose, we shall first study the explicit description of our surfaces in more detail.
Let $X$ be a minimal algebraic surface with $c_1^2= 2\chi -1$, $\chi = 4$, and $\mathrm{Tors} \simeq \mathbb{Z} /2$. We denote by $\pi : Y \to X$ the unramified double cover corresponding to the torsion group, and by $p: \tilde{Y} \to Y$, the shortest composite of quadric transformations such that the variable part of
$p^*|K_Y|$ is free from base points. Then there exist an even integer $0 \leq d \leq 2$ and a generically two-to-one morphism $f: Y \to W = \varSigma_d$ satisfying the three conditions given in Theorem \ref{thm:completedescription}.
In what follows, we denote by $\iota |_{W}$ the involution of $W$ corresponding to the generator of the Galois group $G = \mathrm{Gal} (Y/X)$.
Let $r^{\prime} : Z^{\prime} \to W$ be the blowing-up at two $[3,3]$-points, which we shall call $w_1$ and $w_2$, of the branch divisor $B$ of $f$. Then $f_W= f \circ p : \tilde{Y} \to W$ lifts to a morphism $f^{\prime} : \tilde{Y} \to Z^{\prime}$. We denote by $e_i = {r^{\prime}}^{-1} (w_i)$ the exceptional divisor of $r^{\prime}$ lying over $w_i$. Let $\tilde{r} : \tilde{Z} \to Z^{\prime}$ be the blowing-up at two quadraple points, which we shall call $w_1^{\prime} \in e_1$ and $w_2^{\prime} \in e_2$, of the branch divisor of $f^{\prime}$. Then $f^{\prime}$ lifts to a morphism $\tilde{f} : \tilde{Y} \to \tilde{Z}$. We denote by $e_i^{\prime} = \tilde{r}^{-1} (w_i^{\prime})$ the exceptional divisor of $\tilde{r}$ lying over $w_i^{\prime}$. Let us use the same symbol $e_i$ for the total transform to $\tilde{Z}$ of the divisor $e_i \subset Z^{\prime}$. Then there exists a reduced member
$\tilde{B}_0 \in |(r^{\prime} \circ \tilde{r})^* (-4K_W)
- 3 \sum e_i - 3 \sum e_i^{\prime}|$ satisfying $\tilde{B}_0 \cap \tilde{r}^{-1}_* (e_1) = \tilde{B}_0 \cap \tilde{r}^{-1}_* (e_2) = \emptyset$ such that the branch divisor of $\tilde{f}$ is given by $\tilde{B}_0 + \sum \tilde{r}^{-1}_* (e_i)$. Note that the divisor $\tilde{B}_0$ has at most negligible singularities. In what follows, $\varDelta_0$ and $\varGamma$ denote the minimal section and a fiber respectively of the Hirzebruch surface $W = \varSigma_d \to \mathbb{P}^1$.
\begin{lemma} \label{lm:configw1w1prime}
Let $\iota |_{Z^{\prime}}$ be the involution of $Z^{\prime}$ induced
by the involution $\iota |_W $ of $W$. Then the configuration of
the four points $w_1$, $w_2 = \iota |_W (w_1)$, $w_1^{\prime}$,
and $w_2^{\prime} =\iota |_{Z^{\prime}} (w_1^{\prime})$ satisfies the following three conditions$:$
i$)$ if $d=2$, then $w_1 \notin \varDelta_0$\ $;$
ii$)$ if the two points $w_1$ and $w_2$ lie on one and the same member
of the linear system $|\varGamma|$, then for each $i =1$, $2$, the point $w_i^{\prime}$ does not lie on the strict transform to $Z^{\prime}$ of this member$;$
iii$)$ if $d$ equals $0$, and the two points $w_1$ and $w_2$ lie on
one and the same member of the linear system $|\varDelta_0|$, then for each $i =1$, $2$, the point $w_i^{\prime}$ does not lie on the strict transform to $Z^{\prime}$ of this member. \end{lemma}
Proof. i). Assume that $d=2$ and $w_1 \in \varDelta_0$. Then since $\varDelta_0$ is stable under the action by $G$ on $W$, we have $w_2 \in \varDelta_0$. Thus ${r^{\prime}}^{-1}_* (\varDelta_0)$ is a $(-4)$-curve on $Z^{\prime}$, hence ${r^{\prime}}^{-1}_* (\varDelta_0) (-K_{Z^{\prime}}) < 0$. It follows that ${r^{\prime}}^{-1}_* (\varDelta_0)$ is a
fixed component of the linear systme $|-K_{Z^{\prime}}|$. This is impossible, since by the proof of our complete descripiton
the pull-back ${f^{\prime}}^* |-K_{Z^{\prime}}|$ is the variable part of
$|K_{\tilde{Y}}|$. Thus we have $w_1 \notin \varDelta_0$ for the case $d=2$.
ii). Assume that $w_1$ and $w_2$ lie on one and the same member
$\varGamma_0 \in |\varGamma|$.
Then since $w_2 = \iota|_W (w_1)$, the member $\varGamma_0$ is stable under the action by $G$. It follwos that $\varGamma_0$
passes exactly two of the fixed points of the involution $\iota |_W$. Moreover if $w_1^{\prime} \in {r^{\prime}}^{-1}_* (\varGamma_0)$, then we obtain $w_2^{\prime} \in {r^{\prime}}^{-1}_* (\varGamma_0)$, $(r^{\prime} \circ \tilde{r})^{-1}_* (\varGamma_0) \sim (r^{\prime} \circ \tilde{r})^* (\varGamma) - \sum e_i - \sum e_i^{\prime}$, and $\tilde{B}_0 ((r^{\prime} \circ \tilde{r})^{-1}_* (\varGamma_0)) = -4 < 0$. The last inequality implies that $\varGamma_0$ is an irreducible component of the branch divisor $B$. This however is impossible, since, by the condition in Theorem \ref{thm:completedescription}, the branch divisor $B$ cannot pass
any fixed points of the involution $\iota |_W$. Thus we have $w_1^{\prime} \notin {r^{\prime}}^{-1}_* (\varGamma_0)$.
iii). By the same arguement as in the proof of ii), we can prove iii). \qed
\begin{remark} \label{rem:gamma1gamma2} As shown in the proof of Lemma above, if the two points $w_1$ and $w_2$ lie on one and the same member
$\varGamma_0 \in | \varGamma |$, then this $\varGamma_0$ is stable under the action by $G$ on $W$. There exist exactly two
members of $|\varGamma|$ stable under the action by $G$. In what follows, we denote by $\varGamma_1$ and $\varGamma_2$ these two members. For each $i=1$, $2$, exactly two fixed points of the action by $G$ lie on $\varGamma_i$. \end{remark}
Next let us show that if conversely the configuration of four points $w_i$'s and $w_i^{\prime}$'s satisfies the three conditions in Lemma \ref{lm:configw1w1prime}, then the proceedure implied by our structure theorem in fact produces a minimal surface with the desired invariants. Some of the results below will be used later, in our proof of the uniqueness of the deformation type. Let $W = \varSigma_d$ be the Hirzebruch surface of degree $d=0$ or $2$,
and $\iota |_W$, the involution (\ref{eq:involution}) given in Remark \ref{rem:involutiondescription}.
Take a point $w_1 \in W$ outside the fixed locus of $\iota |_W$. We denote by $r^{\prime} : Z^{\prime} \to W$ the blowing-up at
two points $w_1$ and $w_2 = \iota |_W (w_1)$, and by $e_i = {r^{\prime}}^{-1} (w_i)$, the exceptional curve lying over $w_i$.
Let $\iota |_{Z^{\prime}}$ be the involution of $Z^{\prime}$
induced by $\iota |_W$. Take a point $w_1^{\prime} \in e_1 \subset Z^{\prime}$. We donote by $\tilde{r} : \tilde{Z} \to Z^{\prime}$ the blowing-up at two points $w_1^{\prime}$ and
$w_2^{\prime} = \iota |_{Z^{\prime}} (w_1^{\prime})$, and by $e_i^{\prime} = \tilde{r}^{-1} (w_i^{\prime})$, the exceptional curve lying over $w_i^{\prime}$. We use the same symbol $e_i$ for the total transform to $\tilde{Z}$ of the divisor $e_i$ on $Z^{\prime}$. We assume that the configuration of $w_i$'s and $w_i^{\prime}$'s satisfies the three conditions i), ii), and iii) in Lemma \ref{lm:configw1w1prime}.
Let $\varGamma_1$ and $\varGamma_2$ be two distinct members of
$|\varGamma|$ stable under
the natural action by $G = \langle \iota |_W\rangle$ on $W$ (see Remark \ref{rem:gamma1gamma2}). We take the minimal section $\varDelta_0$ and an irreducible member
$\varDelta_{\infty} \in |\varDelta_0 + d \varGamma |$ such that both are stable under the action by $G$, and $\varDelta_0 \cap \varDelta_{\infty} = \emptyset$ holds. Let $m$ be a positive integer. Since the divisor $m(\varDelta_0 + \frac{d+2}{2} \varGamma_1) $ is stable under the action by $G$, we obtain a natural action on $H^0 (\mathcal{O}_W (m(\varDelta_0 + \frac{d+2}{2} \varGamma) ))$ by identifying this space with that of meromorphic functions with poles at most $m(\varDelta_0 + \frac{d+2}{2} \varGamma_1 )$.
We put $ \varLambda_m = |m(\varDelta_0 + \frac{d+2}{2} \varGamma)|$, and denote by $\varLambda_m^+$ and $\varLambda_m^-$ the subsystems of $\varLambda_m$ corresponding to the eigenspaces of eigenvalues $+1$ and $-1$ repectively
with respect to ${\iota |_W}^*$. Moreover, for an effective divisor $C$ on $\tilde{Z}$, we put \begin{align} \varLambda_m (C) &= \{ D \in \varLambda_m ; (r^{\prime} \circ \tilde{r})^*(D) -C \succeq 0 \} \quad
&\tilde{\varLambda}_m (C) = (r^{\prime} \circ \tilde{r})^* \varLambda_m (C) - C \notag \\
\varLambda_m^+ (C) &= \{ D \in \varLambda_m^+ ; (r^{\prime} \circ \tilde{r})^*(D) -C \succeq 0 \} \quad
&\tilde{\varLambda}_m^+ (C) = (r^{\prime} \circ \tilde{r})^* \varLambda_m^+ (C) - C \notag \\
\varLambda_m^- (C) &= \{ D \in \varLambda_m^- ; (r^{\prime} \circ \tilde{r})^*(D) -C \succeq 0 \} \quad
&\tilde{\varLambda}_m^- (C) = (r^{\prime} \circ \tilde{r})^* \varLambda_m^- (C) - C , \notag
\end{align}
where the symbol $\succeq 0$ means effectiveness of a divisor. We abbreviate $\tilde{\varLambda}_m (0)$, $\tilde{\varLambda}_m^+ (0)$, and $\tilde{\varLambda}_m^- (0)$ to $\tilde{\varLambda}_m$, $\tilde{\varLambda}_m^+$, and $\tilde{\varLambda}_m^-$ respectively. Note that if $\tilde{f} : \tilde{Y} \to \tilde{Z}$ is the generically two-to-one morphism obtained as in the begining of this section from our structure theorem, then we have $\tilde{B}_0 \in \tilde{\varLambda}_8^+ ( 3 \sum e_i + 3 \sum e_i^{\prime} )$, where $\tilde{B}_0 + \sum \tilde{r}^{-1}_* (e_i)$ gives the branch divisor of $\tilde{f} : \tilde{Y} \to \tilde{Z}$.
\begin{lemma} \label{lm:baseptfreesystems}
1$)$ The linear system $\tilde{\varLambda}_2^+ $ has no base point.
2$)$ The linear system $\tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime} )$ has no base point.
3$)$ The linear system $\tilde{\varLambda}_8^+ (3\sum e_i + 3\sum e_i^{\prime} )$ has no base point.
4$)$ The linear systems $|-K_{Z^{\prime}}|$ and $|-K_{\tilde{Z}}|$ have no base point.
\end{lemma}
Proof. Since we have $
\tilde{\varLambda}_2^+ + 3 \tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime} ) \subset
\tilde{\varLambda}_8^+ (3\sum e_i + 3\sum e_i^{\prime} ) $, the assertion 3) follows from the assertions 1) and 2).
Assume that we have the assertions 1) and 2). Then by $ \tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime} ) \subset
|- K_{\tilde{Z}}| $, we see that
the linear system $|- K_{\tilde{Z}}|$ has no base point. Moreover, by this together with the Riemann--Roch theorem and the vanishing theorem, we obtain $h^0 ( \mathcal{O}_{\tilde{Z}} (-K_{\tilde{Z}})) = \chi ( \mathcal{O}_{\tilde{Z}} ) + K_{\tilde{Z}}^2 = 5$. Meanwhile, since $r^{\prime} : Z^{\prime} \to W$ is the blowing-up at two points $w_i^{\prime}$'s, we have $h^0 ( \mathcal{O}_{Z^{\prime}} (-K_{Z^{\prime}})) \geq h^0 ( \mathcal{O}_W (-K_{W})) -2 = 7$. Thus we obtain $ h^0 ( \mathcal{O}_{Z^{\prime}} (-K_{Z^{\prime}})) - h^0 ( \mathcal{O}_{\tilde{Z}} (-K_{\tilde{Z}})) \geq 2$, which implies that neither of the two points $w_i^{\prime}$'s
is a base point of $|-K_{Z^{\prime}}|$.
From this, we infer that $|-K_{Z^{\prime}}|$ has no base point. So the assertion 4) also follows from the assertions 1) and 2).
Thus we only need to show the assertions 1) and 2). First, let us show the assertion 1).
Let $C_0$ be a general member of $| \varGamma |$. Then since the divisor
$2\varDelta_0 + \frac{d+2}{2} (C_0 + \iota |_W (C_0)) \in \varLambda_2$ is stable under the action by $G$, and the divisor \[
(2\varDelta_0 + \frac{d+2}{2} (C_0 + \iota |_W (C_0)))
- 2(\varDelta_0 + \frac{d+2}{2} \varGamma_1) \] has no support at $\varDelta_{\infty} \cap \varGamma_2$, the divisor
$2\varDelta_0 + \frac{d+2}{2} (C_0 + \iota |_W (C_0))$ is a member of $\varLambda_2^+$. Thus the base locus of $\varLambda_2^+$ is contained in $\varDelta_0$. Using a similar argument, we can show that
$2\varDelta_{\infty} + \frac{2- d}{2} (C_0 + \iota |_W (C_0)) \in \varLambda_2^+$, so that the base locus of $\varLambda_2^+$ is contained in $\varDelta_{\infty}$. Thus since $\varDelta_0 \cap \varDelta_{\infty} = \emptyset$, the linear system $\tilde{\varLambda}_2^+$ has no base point. Hence we have the assertion 1).
Next, let us show the assertion 2). We shall show it by dividing our situation into several cases. In what follows, for each $i = 1$, $2$, we denote by $\varGamma_{(i)}$
the unique member of $| \varGamma |$ passing $w_i$.
Case $1$-$1$: the case where $d=0$ holds, and the two points $w_1$ and $w_2$ lie neither on one and the same member
of $| \varGamma |$ nor on that of $|\varDelta_0 |$. In this case, for each $i=1$, $2$, we denote by $\varDelta_{(i)}$
the unique member of $|\varDelta_0|$ passing $w_i$. This case is divided into two subcases: case $1$-$1$-$1$ and case $1$-$1$-$2$.
Case $1$-$1$-$1$; the subcase of case $1$-$1$ where $w_1^{\prime} \notin {r^{\prime}}^{-1}_* (\varGamma_{(1)})$ and $w_1^{\prime} \notin {r^{\prime}}^{-1}_* (\varDelta_{(1)})$. In this case, take global coordinates $(s_1^{\prime}, \xi_1^{\prime})$ of $W \setminus (\varGamma_{(2)} \cup \varDelta_{(2)}) \simeq \mathbb{A}^2$ such that $\varGamma_{(1)}$ is given by $s_1^{\prime} =0$, $\varGamma_{(2)}$ by $s_1^{\prime} =\infty$, $\varDelta_{(1)}$ by $\xi_1^{\prime} =0$, and $\varDelta_{(2)}$ by $\xi_1^{\prime} =\infty$.
Then the involution $\iota |_W$ is given by $(s_1^{\prime}, \xi_1^{\prime}) \mapsto (1 / s_1^{\prime} , 1 / \xi_1^{\prime} )$, and the linear system $\varLambda_2^+$ is spanned by the five elements ${s_1^{\prime}}^l {\xi_1^{\prime}}^m + { s_1^{\prime}}^{2-l} {\xi_1^{\prime}}^{2-m}$ ($0 \leq l \leq2 $, $ 0 \leq m \leq2$). Thus the linear system $\varLambda_2^+ (\sum e_i + \sum e_i^{\prime}) $ is spanned by the three elements \[
a_0 (s_1^{\prime} + s_1^{\prime} {\xi_1^{\prime}}^2) + b_0 (\xi_1^{\prime} + { s_1^{\prime}}^2 \xi_1^{\prime} ), \qquad \quad s_1^{\prime} \xi_1^{\prime}, \qquad \quad {s_1^{\prime}}^2 + {\xi_1^{\prime}}^2 , \] where $a_0 \neq 0$ and $b_0 \neq 0$ are certain non-zero complex numbers. From this, we infer that the set $\{ w_1 , w_2 \}$ forms the base locus of $\varLambda_2^+ (\sum e_i + \sum e_i^{\prime}) $, and that any general member of this linear system is smooth. By this together with $\sum \varGamma_{(i)} + \sum \varDelta_{(i)} \in \varLambda_2^+ (\sum e_i + \sum e_i^{\prime})$, we see that the linear system $\tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime})$ has no base point.
Case $1$-$1$-$2$: the subcase of case $1$-$1$ where $w_1^{\prime} \in {r^{\prime}}^{-1}_* (\varGamma_{(1)})$ or $w_1^{\prime} \in {r^{\prime}}^{-1}_* (\varDelta_{(1)})$. Since the proof is the same, we only give a proof for the case $w_1^{\prime} \in {r^{\prime}}^{-1}_* (\varGamma_{(1)})$. Assume that $w_1^{\prime} \in {r^{\prime}}^{-1}_* (\varGamma_{(1)})$. Since we have
$C_0 + \iota |_W (C_0) + \sum \varGamma_{(i)} \in \varLambda_2^+ (\sum e_i + \sum e_i^{\prime})$
for any general member $C_0$ of $| \varDelta_0 |$, the base locus of $\tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime})$ is contained in $\sum (r^{\prime} \circ \tilde{r})^{-1}_* (\varGamma_{(i)})$. Meanwhile, since we have
$C_1 + \iota |_W (C_1) \in \varLambda_2^+ (\sum e_i + \sum e_i^{\prime})$ for any general member $C_1$ of $\varLambda_1 (\sum e_i)$, the base locus of $\tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime})$ is contained in $\sum \tilde{r}^{-1}_* (e_i)$. Since we have $(\sum (r^{\prime} \circ \tilde{r})^{-1}_* (\varGamma_{(i)})) \cap (\sum \tilde{r}^{-1}_* (e_i) ) = \emptyset $, we see that the linear system $\tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime})$ has no base point.
Case $1$-$2$: the case where $d=0$ holds, and the two points $w_1$ and $w_2$ lie on one and the same member of
$| \varGamma |$ or $| \varDelta_0 |$. In this case, for each $i =1$, $2$, we denote by
$\varDelta_{(i)}$ the unique member of $| \varDelta_0 |$ passing $w_i$. By exchanging $\varDelta_0$ and $\varGamma$ if necessary, we may assume that the two points $w_1$ and $w_2$ lie on one
and the member $\varGamma_0 \in | \varGamma |$. Moreover, by Remark \ref{rem:gamma1gamma2}, by exchanging $\varGamma_1$ and $\varGamma_2$ if necessary, we may assume that $\varGamma_0 = \varGamma_{(1)} = \varGamma_{(2)} = \varGamma_1$. Then this case is divided into two subcases: case $1$-$2$-$1$ and case $1$-$2$-$2$.
Case $1$-$2$-$1$: the subcase of case $1$-$2$ where $w_1^{\prime} \notin {r^{\prime}}^{-1}_* ( \varDelta_{(1)})$. Note that we have assumed the condition ii) of Lemma \ref{lm:configw1w1prime} for our configuration, so that we have $w_1^{\prime}$, $w_2^{\prime} \notin {r^{\prime}}^{-1}_* (\varGamma_1)$.
For any general member $C_0 \in | \varDelta_0 |$, we have
$C_0 + \iota |_W (C_0) + 2 \varGamma_1 \in \varLambda_2^+ (\sum e_i + \sum e_i^{\prime})$. Thus the base locus of $\tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime})$ is contained in $2 (r^{\prime} \circ \tilde{r})^{-1}_* (\varGamma_1) + \sum \tilde{r}^{-1}_* (e_i)$. Meanwhile for any general $C_1$ ($ \neq \varDelta_{(1)} + \varGamma_1$) $\in \varLambda_1 (e_1 + e_1^{\prime})$, we have
$C_1 + \iota |_W (C_1) \in \varLambda_2^+ (\sum e_i + \sum e_i^{\prime})$ and the irreducibility and smoothness at $w_1$ of $C_1$. Thus the base locus of $\tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime})$ is contained in
$(r^{\prime} \circ \tilde{r})^{-1}_* (C_1 + \iota |_W (C_1))$. Since $2 (r^{\prime} \circ \tilde{r})^{-1}_* (\varGamma_1) + \sum \tilde{r}^{-1}_* (e_i)$ and
$(r^{\prime} \circ \tilde{r})^{-1}_* (C_1 + \iota |_W (C_1))$ do not intersect each other, we see that the linear system $\tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime})$ has no base point.
Case $1$-$2$-$2$: the subcase of case $1$-$2$ where $w_1 \in {r^{\prime}}^{-1}_* (\varDelta_{(1)})$. By the same argument as one given in the proof for case $1$-$2$-$1$, we see that the base locus of $\tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime})$ is contained in $2 (r^{\prime} \circ \tilde{r})^{-1}_* (\varGamma_1) + \sum \tilde{r}^{-1}_* (e_i)$.
Meanwhile, for any general member $C_1 \in | \varGamma |$, we have
$C_1 + \iota |_W (C_1) + \sum \varDelta_{(i)} \in \varLambda_2^+ ( \sum e_i + \sum e_i^{\prime})$. Thus the base locus of $\tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime})$ is contained in $\sum (r^{\prime} \circ \tilde{r})^{-1}_* (\varDelta_{(i)})$. Since $2 (r^{\prime} \circ \tilde{r})^{-1}_* (\varGamma_1) + \sum \tilde{r}^{-1}_* (e_i)$ and $\sum (r^{\prime} \circ \tilde{r})^{-1}_* (\varDelta_{(i)})$ do not intersect each other, we see that the linear system $\tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime})$ has no base point.
Case $2$-$1$: the case where $d=2$ holds, and the two points $w_1$ and $w_2$ do not lie on one and the same
member of $| \varGamma |$. Note that since we have assumed the condition i) in Lemma \ref{lm:configw1w1prime}, we have $w_1 \notin \varDelta_0$. Note also that for this case, or more generally for case $d=2$, we have $\dim \varLambda_1^- =1$, and any general member of this linear system is an irreducible curve stable under the action by $G$ that passes two points $\varDelta_{\infty} \cap \varGamma_1$ and $\varDelta_{\infty} \cap \varGamma_2$. We denote by $\varDelta_1$ the unique member of $\varLambda_1^-$ that passes the two points $w_1$ and $w_2$. Then this case is divided into three subcases: case $2$-$1$-$1$, case $2$-$1$-$2$, and case $2$-$1$-$3$.
Case $2$-$1$-$1$: the subcase of case $2$-$1$ where $w_1^{\prime} \notin {r^{\prime}}^{-1}_* (\varGamma_{(1)})$ and $w_1^{\prime} \notin {r^{\prime}}^{-1}_* (\varDelta_{1})$. Since the divisor $\varDelta_0 + 2 \varGamma_{(1)}$ is the unique reducible member of $\varLambda_1 (e_1 + e_1^{\prime})$, and we have $h^0 (\mathcal{O}_W (\varDelta_0 + 2 \varGamma)) =4$, any general member of $\varLambda_1 (e_1 + e_1^{\prime})$ is irreducible and non-singular. By this together with $\varDelta_0 + 2 \varGamma_{(1)} \in \varLambda_1 (e_1 + e_1^{\prime})$, we see that $\tilde{\varLambda}_1 (e_1 + e_1^{\prime})$ has no base point,
and $(r^{\prime} \circ \tilde{r})^{-1}_* (C_0 + \iota |_W (C_0)) \in \tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime})$ for any general member $C_0 \in \varLambda_1 (e_1 + e_1^{\prime})$. Thus we see that $\tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime})$ has no base point.
Case $2$-$1$-$2$: the subcase of case $2$-$1$ where $w_1^{\prime} \in {r^{\prime}}^{-1}_* (\varGamma_{(1)})$. Since we have $2 \varDelta_0 + \sum \varGamma_{(i)}
+ C_0 + \iota |_W (C_0) \in \varLambda_2^+ (\sum e_i + \sum e_i^{\prime})$
for any general member $C_0 \in | \varGamma |$, the base locus of $\tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime})$ is contained in $2 (r^{\prime} \circ \tilde{r})^* (\varDelta_0) + \sum (r^{\prime} \circ \tilde{r})^{-1}_* (\varGamma_{(i)})$. By this together with $2 \varDelta_1 \in \varLambda_2^+ (\sum e_i + \sum e_i^{\prime})$, we see that the linear system $\tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime})$ has no base point.
Case $2$-$1$-$3$: the subcase of case $2$-$1$ where $w_1^{\prime} \in {r^{\prime}}^{-1}_* (\varDelta_1)$. Since we have $C_0 + \varDelta_1 \in \varLambda_2^+ (\sum e_i + \sum e_i^{\prime})$ for any general member $C_0 \in \varLambda_1^-$, the base locus of $\tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime})$ is contained in $(r^{\prime} \circ \tilde{r})^{-1}_* (\varDelta_1)$. By this together with $2 (\varDelta_0 + \sum \varGamma_{(i)}) \in \varLambda_2^+ (\sum e_i + \sum e_i^{\prime})$, we see that the linear system $\tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime})$ has no base point.
Case $2$-$2$: the case where $d=2$ holds, and the two points $w_1$ and $w_2$
lie on one and the same member of $| \varGamma |$. By Remark \ref{rem:gamma1gamma2}, we may assume $w_1$, $w_2 \in \varGamma_1$. Note that we have assumed the conditions i) and ii) of Lemma \ref{lm:configw1w1prime} for our configuration, so that we have $w_1 \notin \varDelta_0$ and $w_1^{\prime} \notin {r^{\prime}}^{-1}_* (\varGamma_1)$. Since we have
$2 \varDelta_0 + 2 \varGamma_1 + C_0 + \iota |_W (C_0) \in \varLambda_2^+ (\sum e_i + \sum e_i^{\prime})$
for any general member $C_0 \in | \varGamma |$, the base locus of $\tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime})$ is contained in $2 (r^{\prime} \circ \tilde{r})^* (\varDelta_0) + 2 (r^{\prime} \circ \tilde{r})^{-1}_* (\varGamma_1) + \sum \tilde{r}^{-1}_* (e_i)$. Meanwhile since we have $h^0 (\mathcal{O}_W (\varDelta_0 + 2 \varGamma)) = 4$, we have
$C_1 + \iota |_{W} (C_1) \in \varLambda_2^+ (\sum e_i + \sum e_i^{\prime})$ for an irreducible member $C_1 \in \varLambda_1 (e_1 + e_1^{\prime})$. Since $2 (r^{\prime} \circ \tilde{r})^* (\varDelta_0) + 2 (r^{\prime} \circ \tilde{r})^{-1}_* (\varGamma_1) + \sum \tilde{r}^{-1}_* (e_i)$ and $(r^{\prime} \circ \tilde{r})^{-1}_*
(C_1 + \iota |_{W} (C_1))$ do not intersect each other, we see that the linear system $\tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime})$ has no base point.
Now that we have shown the absence of base points of $\tilde{\varLambda}_2^+ (\sum e_i + \sum e_i^{\prime})$ for all the eight cases $1$-$1$-$1$, \ldots, $2$-$2$, we have the assertion 2). \qed
Let $\tilde{B}_0$ be a reduced member of $\tilde{\varLambda}_8^+ (3 \sum e_i + 3 \sum e_i^{\prime})$ that has at most negligible singularities, satisfies $\tilde{B}_0 \cap \sum \tilde{r}^{-1}_* (e_i) = \emptyset $, and passes no fixed point of the action by $G$ on $\tilde{Z}$. Existence of such $\tilde{B}_0$ is ensured by Lemma \ref{lm:baseptfreesystems}.
Let $\tilde{Y}$ be the canonical resolution of the double cover of $\tilde{Z}$ branched along $\tilde{B}_0 + \sum \tilde{r}^{-1}_* (e_i)$, and $\tilde{f} : \tilde{Y} \to \tilde{Z}$, the natural projection.
We have $ \tilde{f}^* ({\tilde{r}}^{-1}_* (e_i)) = 2 E_i$ for a $(-1)$-curve $E_i$ on $\tilde{Y}$. Let $p : \tilde{Y} \to Y$ be the blowing-down of $E_1$ and $E_2$.
Then we have $|K_{\tilde{Y}}| =
(\tilde{r} \circ \tilde{f})^* |-K_{Z^{\prime}}| + 2 \sum E_i$.
Since $|-K_{Z^{\prime}}|$ has no base point by Lemma \ref{lm:baseptfreesystems}, we see that $Y$ is a minimal surface with $c_1^ 2 = 14$ and $\chi =8$. By \cite[Lemma 3.1]{ngsffermat} and \[
(r^{\prime} \circ \tilde{r}) (\tilde{B}_0) \varGamma_1 \equiv
(r^{\prime} \circ \tilde{r}) (\tilde{B}_0) \varGamma_2 \equiv
(r^{\prime} \circ \tilde{r}) (\tilde{B}_0) \varDelta_0 \equiv
0 \mod 4, \] there exists a unique free lifting to $\tilde{Y}$ of the action by $G$ on $W$. Let $X = Y/G$ be the quotient of $Y$ by the induced free action by $G$ on $Y$. Then by \cite[Theorem 1]{bound'''} or \cite[(ii) in Theorem A]{on2chi-2}, the surface $X$ is a minimal surface with $c_1 ^2 = 2 \chi -1$, $\chi =4$, and $\mathrm{Tors} (X) \simeq \mathbb{Z} /2$. Thus we have the following:
\begin{proposition} \label{prop:sufficientprocedure} Let $W = \varSigma_d$ be the Hirzebruch surface of degree $d=0$ or $2$. Let $r^{\prime} : Z^{\prime} \to W$ be the blowing-up at
two points $w_1$ and $w_2 = \iota |_W (w_1)$,
where $\iota |_W$ is the involution of $W$ given in Remark \ref{rem:involutiondescription},
and $w_1$, a point outside the fixed locus of $\iota |_W$. Let $ \tilde{r} : \tilde{Z} \to Z^{\prime}$ be the blowing-up at two points $w_1^{\prime}$ and $w_2^{\prime}
= \iota |_{Z^{\prime}} (w_1^{\prime})$,
where $\iota |_{Z^{\prime}}$ is the induced involution of $Z^{\prime}$, and $w_1^{\prime}$, a point infinitely near to $w_1$. Put $e_i = {r^{\prime}}^{-1} (w_i)$ and $e_i^{\prime} = \tilde{r}^{-1}(w_i^{\prime})$ for each $i=1$, $2$, and assume that the configuration of $w_i$'s and $w_i^{\prime}$'s satisfies all the three conditions in Lemma \ref{lm:configw1w1prime}. Let $\tilde{B}_0$ be a reduced member of $\tilde{\varLambda}_8^+ ( 3 \sum e_i + 3 \sum e_i^{\prime})$ that has at most negligible singularities, satisfies $\tilde{B}_0 \cap \sum \tilde{r}^{-1}_* (e_i) = \emptyset$, and passes no fixed point of the induced action on $\tilde{Z}$ by
$G = \langle \iota |_W \rangle$.
Let $\tilde{Y}$ be the canonical resolution of the double cover of $\tilde{Z}$ branched along $\tilde{B}_0 + \sum \tilde{r}^{-1}_* (e_i)$, and $\tilde{f} : \tilde{Y} \to \tilde{Z}$, the natural projection. Let $p : \tilde{Y} \to Y$ be the blowing-down of two $(-1)$-curves $E_1 = {\tilde{f}}^{-1} ({\tilde{r}}^{-1}_* (e_1))$ and $E_2 = {\tilde{f}}^{-1} ({\tilde{r}}^{-1}_* (e_2))$.
Then there exists a unique free lifting to $\tilde{Y}$ of the action by $G$ on $\tilde{Z}$, and the quotient $Y/G$ of $Y$ by the induced free action is a minimal surface with $c_1^2 = 2 \chi -1$, $\chi =4$, and $\mathrm{Tors} \simeq \mathbb{Z} / 2$.
\end{proposition}
Our Theorem \ref{thm:completedescription} together with Remark \ref{rem:involutiondescription} and Lemma \ref{lm:configw1w1prime} says that all minimal surfaces with $c_1^2 = 2 \chi -1$, $\chi =4$, and $\mathrm{Tors} \simeq \mathbb{Z} / 2$ are obtained by the procedure as in the proposition above. We use the following lemma in order to show the uniqueness of the deformation type.
\begin{lemma} \label{lm:b0tildenonsing} Let $r^{\prime} : Z^{\prime} \to W$ and $\tilde{r} : \tilde{Z} \to Z^{\prime}$, $w_i \in W$ and $w_i^{\prime} \in Z^{\prime}$ for $i=1$, $2$, and $e_i = {r^{\prime}}^{-1} (w_i)$ and $e_i^{\prime} = \tilde{r}^{-1} (w_i^{\prime})$ for $i =1$, $2$ be the morphisms, points, and divisors respectively as in Proposition \ref{prop:sufficientprocedure}. Then any general member $\tilde{B}_0$ of $\tilde{\varLambda}_8^+ ( 3 \sum e_i + 3 \sum e_i^{\prime})$ is non-singular and reduced. Further $h^i ( \mathcal{O}_{\tilde{Z}} (\tilde{B}_0)) =0$ holds for any positive integer $i > 0$. \end{lemma}
Proof. The first assertion follows from 3) in Lemma \ref{lm:baseptfreesystems}. The second assertion follows from 3) and 4) in Lemma \ref{lm:baseptfreesystems} and the vanishing theorem. \qed
Now let us show the uniqueness of the deformation type and the unirationality of the moduli space. For this purpose, we shall give another description of our surface $X$. Let $r^{\prime} : Z^{\prime} \to W$, $\tilde{r} : \tilde{Z} \to Z^{\prime}$, $w_i$, $w_i^{\prime}$, $e_i$, and $e_i^{\prime}$ be as in Proposition \ref{prop:sufficientprocedure}. Let $\varGamma_1$ and $\varGamma_2$ be as in Remark \ref{rem:gamma1gamma2}. We take the minimal section $\varDelta_0$ and an irreducible member
$\varDelta_{\infty} \in |\varDelta_0 + d \varGamma|$ satisfying $\varDelta_0 \cap \varDelta_{\infty} = \emptyset$ such that both are stable under the action by $G$. Note that if $d = 2$, such $\varDelta_{\infty}$'s form a one-dimensional family.
The fixed locus of the action by $G$ on $\tilde{Z}$ is a set of four isolated points: $\{ (r^{\prime} \circ \tilde{r})^{-1} (\varGamma_i \cap \varDelta_j)\}_{i=1,2, \ j= 0, \infty}$. Let $\bar{r} : \bar{Z} \to \tilde{Z}$ be the blowing-up at these four points. For $i =1$, $2$ and $j=0$, $\infty$, we define the divisors $J_{i j}$ on $\bar{Z}$ as follows:
if $d = 0$, then $J_{i j} = (r^{\prime} \circ \tilde{r} \circ \bar{r})^{-1} (\varGamma_i \cap \varDelta_j)$ for any $i=1$, $2$ and $j=0$, $\infty$;
if $d = 2$, then $J_{1 0} = (r^{\prime} \circ \tilde{r} \circ \bar{r})^{-1} (\varGamma_1 \cap \varDelta_{\infty})$, $J_{1 \infty} = (r^{\prime} \circ \tilde{r} \circ \bar{r})^{-1} (\varGamma_1 \cap \varDelta_0)$, and $J_{2 j} = (r^{\prime} \circ \tilde{r} \circ \bar{r})^{-1} (\varGamma_2 \cap \varDelta_j)$ for any $j = 0$, $\infty$.
\noindent Moreover for $i=1$, $2$ and $j= 0$, $\infty$, we define the divisors $\bar{\varGamma}_i$, $\bar{\varDelta}_j$, $\bar{e}_i$, and ${\bar{e}}_i^{\prime}$ on $\bar{Z}$ by $\bar{\varGamma}_i = (r^{\prime} \circ \tilde{r} \circ \bar{r})^{-1}_* (\varGamma_i)$, $\bar{\varDelta}_j = (r^{\prime} \circ \tilde{r} \circ \bar{r})^{-1}_* (\varDelta_j)$, $\bar{e}_i = \bar{r}^* ( \tilde{r}^{-1}_* (e_i))$, and ${\bar{e}}_i^{\prime} = \bar{r}^* (e_i^{\prime})$. The four divisors $J_{i j}$'s form the set of all irreducible exceptional curves of $\bar{r} : \bar{Z} \to \tilde{Z}$. The action by $G$ on $\tilde{Z}$ lifts to one on $\bar{Z}$. Note that $\sum J_{i j}$ gives the fixed locus of the induced action by $G$ on $\bar{Z}$.
Now let $\bar{V} = \bar{Z} / G$ be the quotient of $\bar{Z}$ by the induced action by $G$, and $\bar{\wp} : \bar{Z} \to \bar{V}$, the natural projection. Then $\bar{V}$ is smooth, and $\sum J_{i j}$ gives the ramification divisor of $\bar{\wp} : \bar{Z} \to \bar{V}$. For $i=1$, $2$ and $j=0$, $\infty$, we define the divisors $\bar{I}_{i j}$, $\bar{G}_i$, and $\bar{D}_j$ on $\bar{V}$ by $\bar{I}_{i j} = \bar{\wp} (J_{i j})$, $\bar{G}_i = \bar{\wp} (\bar{\varGamma}_i)$, and $\bar{D}_j = \bar{\wp} (\bar{\varDelta}_j)$. Moreover we define the divisors $\bar{\lambda}$ and ${\bar{\lambda}}^{\prime}$ on $\bar{V}$ by $\bar{\lambda} = \bar{\wp} (\bar{e}_1) = \bar{\wp} (\bar{e}_2)$ and ${\bar{\lambda}}^{\prime} = \bar{\wp} (\bar{e}_1^{\prime}) = \bar{\wp} (\bar{e}_2^{\prime})$. The divisors $\bar{I}_{i j}$'s are non-singular rational curves with selfintersection ${\bar{I}_{i j}}^2 = -2$. Note that $\sum \bar{I}_{i j}$ gives the branch divisor of $\bar{\wp} : \bar{Z} \to \bar{V}$.
Let $\bar{\nu} : \bar{V} \to \tilde{V}$ be the blowing-down of the $(-1)$-curve $\bar{\lambda}^{\prime}$. For $i =1$, $2$ and $j=0$, $\infty$, we define the divisor $\tilde{I}_{i j}$ on $\tilde{V}$ by $\tilde{I}_{i j} = \bar{\nu} (\bar{I}_{i j})$. Moreover we define the divisor $\tilde{\lambda}$ on $\tilde{V}$ by $\tilde{\lambda} = \bar{\nu} (\bar{\lambda})$. The divisors $\tilde{I}_{i j}$ and $\tilde{\lambda}$ are non-singular rational curves with $\tilde{I}_{i j}^2 = -2$ and $\tilde{\lambda}^2 = -1$ respectively.
Let $\tilde{\nu} : \tilde{V} \to V^{\prime}$ be the blowing-down of the $(-1)$-curve $\tilde{\lambda}$. For $i =1$, $2$ and $j=0$, $\infty$, we define the divisors $I_{i j}^{\prime}$, $G_i^{\prime}$, and $D_j^{\prime}$ on $V^{\prime}$ by $I_{i j}^{\prime} = (\tilde{\nu} \circ \bar{\nu})_* (\bar{I}_{i j})$, $G_i^{\prime} = (\tilde{\nu} \circ \bar{\nu})_* (\bar{G}_i)$, and $D_j^{\prime} = (\tilde{\nu} \circ \bar{\nu})_* (\bar{D}_j)$. The divisors $I_{i j}^{\prime}$, $G_i^{\prime}$, $D_0^{\prime}$, and $D_{\infty}^{\prime}$ are non-singular rational curves with ${I_{i j}^{\prime}}^2 = -2$, ${G_i^{\prime}}^2 = -1$, ${D_0^{\prime}}^2 = - (d+2)/ 2$, and ${D_{\infty}^{\prime}}^2 = (d-2)/ 2$ respectively.
Let $\nu^{\prime} : V^{\prime} \to V^{\prime \prime}$ be the blowing-down of the two $(-1)$-curves $G_1^{\prime}$ and $G_2^{\prime}$. For $i=1$, $2$ and $j =0$, $\infty$, we define the divisors $I_{i j}^{\prime \prime}$ and $D_j^{\prime \prime}$ on $V^{\prime \prime}$ by $I_{i j}^{\prime \prime} = \nu^{\prime} (I_{ij}^{\prime})$ and $D_j^{\prime \prime} = \nu^{\prime} (D_j^{\prime})$. The divisors $I_{i j}^{\prime \prime}$, $D_0^{\prime \prime}$, and $D_{\infty}^{\prime \prime}$ are non-singular rational curves with ${I_{i j}^{\prime \prime}}^2 = -1$, ${D_0^{\prime \prime}}^2 = -(d+2)/2$ and ${D_{\infty}^{\prime \prime}}^2 = (d - 2)/2$ respectively.
Let $\nu^{\prime \prime} : V^{\prime \prime} \to V^{\prime \prime \prime}$ be the blowing-down of the two $(-1)$-curves $I_{1 \infty}^{\prime \prime}$ and $I_{2 \infty}^{\prime \prime}$. For $i=1$, $2$ and $j=0$, $\infty$, we define the divisors $I_{i 0}^{\prime \prime \prime}$ and $D_j^{\prime \prime \prime}$ on $V^{\prime \prime \prime}$ by $I_{i 0}^{\prime \prime \prime} = \nu^{\prime \prime} (I_{i 0 }^{\prime \prime})$ and $D_j^{\prime \prime \prime} = \nu^{\prime \prime} (D_j^{\prime \prime})$. The divisors $I_{i 0}^{\prime \prime \prime}$, $D_0^{\prime \prime \prime}$, and $D_{\infty}^{\prime \prime \prime}$ are non-singular rational curves with ${I_{i 0}^{\prime \prime \prime}}^2 = 0$, ${D_0^{\prime \prime \prime}}^2 = -1$, and ${D_{\infty}^{\prime \prime \prime}}^2 = 1$. By $K_{V^{\prime \prime \prime}}^2 = 8$, we see easily that $V^{\prime \prime \prime}$ is isomorphic to the Hirzebruch surface $\varSigma_1$ of degree $1$, where $D_0^{\prime \prime \prime}$ and $I_{1 0}^{\prime \prime \prime} \sim I_{2 0}^{\prime \prime \prime}$ give the minimal section and the fiber class respectively. We define the point $v_1^{\prime \prime \prime} \in V^{\prime \prime \prime}$ by $v_1^{\prime \prime \prime} = \nu^{\prime \prime} (I_{1 \infty}^{\prime \prime })$. Note that we have $v_1^{\prime \prime \prime} \notin D_0^{\prime \prime \prime}$ if $d =0$, and $v_1^{\prime \prime \prime} \in D_0^{\prime \prime \prime}$ if $d=2$.
We put $\nu = ( \nu^{\prime \prime} \circ \nu^{\prime} \circ \tilde{\nu} \circ \bar{\nu}) : \bar{V} \to V^{\prime \prime \prime} \simeq \varSigma_1$, and use the same symbols $I_{i \infty}^{\prime \prime}$, $G_i^{\prime}$, and $\tilde{\lambda}$ for the total transforms to $\bar{V}$ of the divisors $I_{i \infty}^{\prime \prime} \subset V^{\prime \prime}$, $G_i^{\prime} \subset V^{\prime}$, and $\tilde{\lambda} \subset \tilde{V}$ respectively. Note that the morphism $\nu : \bar{V} \to V^{\prime \prime \prime} \simeq \varSigma_1$ is a blowing-up at six points some of which are infinitely near.
\begin{proposition} \label{prop:anotherdescription}
The linear system $|-4K_{\bar{V}} + \tilde{\lambda} + \bar{\lambda}^{\prime}|
= |\nu^* (-4 K_{V^{\prime \prime \prime}}) - 4\sum I_{i \infty}^{\prime \prime} - 4\sum G_i^{\prime}
- 3\tilde{\lambda} -3 \bar{\lambda}^{\prime}|$ has no base point. Let $\bar{A}_0$ be a reduced member of
$|-4K_{\bar{V}} + \tilde{\lambda} + \bar{\lambda}^{\prime}|$ that has at most negligible singularities, and satisfies $\bar{A}_0 \cap \bar{\lambda} = \emptyset$ and $\bar{A}_0 \cap \sum \bar{I}_{i j} = \emptyset$.
Let $\bar{X}$ be the canonical resolution of the double cover of $\bar{V}$ branched along $\bar{A}_0 + \bar{\lambda} + \sum \bar{I}_{i j}$, and $\bar{h} : \bar{X} \to \bar{V}$, the natural projection. Let $\bar{X} \to X$ be the blowing-down of five $(-1)$-curves $\bar{h}^{-1} (\bar{\lambda})$, $\bar{h}^{-1} (\bar{I}_{1 0})$, $\bar{h}^{-1} (\bar{I}_{2 0})$, $\bar{h}^{-1} (\bar{I}_{1 \infty})$, and $\bar{h}^{-1} (\bar{I}_{2 \infty})$.
Then $X$ is a minimal surface with $c_1^2 = 2 \chi -1$, $\chi =4$, and $\mathrm{Tors} \simeq \mathbb{Z} / 2$. Conversely, for any minimal surface $X_{(1)}$ with these invariants, there exist configuration of $w_1$ and $w_1^{\prime}$ and a reduced member $\bar{A}_0$ as above such that the surface $X$ constructed by this procedure is isomorphic to $X_{(1)}$. \end{proposition}
Proof. Since $K_{\bar{Z}} \sim \bar{\wp}^* (K_{\bar{V}}) + \sum J_{i j}$, $2J_{i j} = \bar{\wp}^* ({\bar{I}}_{i j})$, $\bar{\wp}^* (\bar{\lambda}) = \sum \bar{e}_i$, $\bar{\wp}^* (\bar{\lambda}^{\prime}) = \sum \bar{e}_i^{\prime}$, and $\tilde{\lambda} \sim \bar{\lambda} + \bar{\lambda}^{\prime}$, we have \[
\bar{\wp}^* (-4 K_{\bar{V}} + \tilde{\lambda} + \bar{\lambda}^{\prime})
\sim
\bar{r}^* ( (r^{\prime} \circ \tilde{r})^* ( - 4K_W)
- 3 \sum e_i -3 \sum e_i^{\prime}). \] By this together with \begin{equation} \label{eq:halfofsumIij} \sum \bar{I}_{i j} = \nu^* (\sum I_{i 0}^{\prime \prime \prime}) - 2 \sum G_i^{\prime} \sim 2 (I_{1 0}^{\prime \prime \prime} - \sum G_i^{\prime}), \end{equation} we obtain \begin{align} &\bar{r}^* \tilde{\Lambda}_8^+ (3 \sum e_i + 3 \sum e_i^{\prime}) =
\bar{\wp}^* |- 4 K_{\bar{V}} + \tilde{\lambda} + \bar{\lambda}^{\prime}|, \notag \\ &\bar{r}^* \tilde{\Lambda}_8^- (3 \sum e_i + 3 \sum e_i^{\prime}) =
\bar{\wp}^* |- 4 K_{\bar{V}} + \tilde{\lambda} + \bar{\lambda}^{\prime}
- (I_{1 0}^{\prime \prime \prime} - \sum G_i^{\prime})| + \sum J_{i j}. \notag \end{align} Thus the absence of base points of
$|- 4 K_{\bar{V}} + \tilde{\lambda} + \bar{\lambda}^{\prime}|$ follows from 3) of Lemma \ref{lm:baseptfreesystems}. Let $\bar{A}_0 \in
|- 4 K_{\bar{V}} + \tilde{\lambda} + \bar{\lambda}^{\prime}|$, $\bar{h} : \bar{ X} \to \bar{V}$, and $X$ be a reduced member, the induced morphism, and the obtained surface respectively as in the statement. Then $\tilde{B}_0 = \bar{r} (\bar{\wp}^*(\bar{A}_0))$ satisfies all the conditions given in Proposition \ref{prop:sufficientprocedure}. Thus for this $\tilde{B}_0$, we obtain morphisms $\tilde{f} : \tilde{Y} \to \tilde{Z}$ and $p : \tilde{Y} \to Y$ as in Proposition \ref{prop:sufficientprocedure} and a minimal surface $Y/G$ with $c_1^2 = 2\chi -1$, $\chi =4$, and $\mathrm{Tors} \simeq \mathbb{Z} /2$. Note that the preimage by $\tilde{f}$ of the set $\{ (r^{\prime} \circ \tilde{r})^{-1} (\varGamma_i \cap \varDelta_j) \}_{i=1, 2, \ j=0, \infty}$ is composed of exactly eight points. We denote by $\bar{Y} \to \tilde{Y}$ the blowing-up at these eight points. Then the morphism $\tilde{f} : \tilde{Y} \to \tilde{Z}$ induces a generically two-to-one morphism $\bar{f} : \bar{Y} \to \bar{Z}$. Moreover,
the natural free action by $G = \langle \iota |_W \rangle$ on $\tilde{Y}$ lifts to one on $\bar{Y}$ that is compatible with the induced action by $G$ on $\bar{Z}$. Thus $\bar{f} : \bar{Y} \to \bar{Z}$ induces a natural morphism $\bar{Y} /G \to \bar{V} = \bar{Z} /G$. Since the branch divisor of $\bar{Y} /G \to \bar{V} = \bar{Z} /G$ is $\bar{A}_0 + \bar{\lambda} + \sum \bar{I}_{i j}$, and $\bar{V}$ has no non-trivial torsion divisor, the morphism $\bar{Y} / G \to \bar{V}$ coincides with $\bar{h}: \bar{X} \to \bar{V}$. Thus by Proposition \ref{prop:sufficientprocedure}, $X \simeq Y/G$ is a minimal surface with $c_1^2 = 2 \chi -1$, $\chi =4$, and $\mathrm{Tors} \simeq \mathbb{Z} / 2$. The final assertion follows from Theorem \ref{thm:completedescription} and Lemma \ref{lm:configw1w1prime}. \qed
\begin{remark} \label{rem:notecmdescr} The description above for our surfaces of the case $\chi =4$ is almost the same as the description in Ciliberto-Mendes Lopes \cite[Section 1]{nonstandardpg3} of the surfaces of the non-standard case for the non-birationality of bicanonical maps (see also \cite[(b) in Theorem 3.1]{onbicanonicalmaps}). We emphasize here that in the present paper we have put neither the assumption of the non-birationality of bicanonical maps nor the assumption of the absence of pencils of curves of genus $2$.
By the description above, it is almost clear that our surfaces coincide with those found in \cite{nonstandardpg3}. To be precise, however, by showing the following proposition, we shall prove that they in fact coincide. \end{remark}
\begin{proposition} \label{prop:bicanonical} Any minimal surface $X$ with $c_1^2 = 2 \chi -1$, $\chi =4$, and $\mathrm{Tors} \simeq \mathbb{Z} /2$ has non-birational bicanonical map. Moreover, it has no pencil of curves of genus $2$. \end{proposition}
Proof. Let $X$ be a minimal surface with $c_1^2 = 2 \chi -1$, $\chi =4$, and $\mathrm{Tors} \simeq \mathbb{Z} / 2$, and assume that $X$ has a pencil of curves of genus $2$. This pencil is rational, since $c_1^2 \geq 2$. Let $\vartheta$ be a non-trivial $2$-torsion divisor, and $C$, a general member of the pencil. Then by the same method as in the proof of (i), Lemma $1. 2$ of \cite{on2chi-2},
we see that $| K_X + \vartheta| = |(\chi - 1) C| + D$ for a certain effective divisor $D$ satisfying $K_X D = 1$, $CD = 2$, $D^2 = 3 - 2\chi$, and $\mathcal{O}_C (D) \not\simeq \mathcal{O}_C (K_C)$. Since $K_X D = 1$, the divisor $D$ contains an irreducible curve $D_1$ satisfying $K_X D_1 = 1$, and all other components of $D$ are $(-2)$-curves. Then since $-3 \leq D_1^2 + D_1 (D - D_1) = D_1 D = K_X D_1 - (\chi -1) C D_1 = 1 - (\chi -1) C D_1 $, we obtain $0 \leq C D_1 \leq 1$. Assume that $CD_1 = 1$. Then $D - D_1$ contains a $(-2)$-curve $D_2$ such that $C D_2 =1$, and so $-2 \leq D_2^2 + D_2 (D - D_2) = D D_2 \leq - (\chi - 1) CD_2$, hence a contradiction. Thus we have $C D_1 = 0$. In this case, if we let $D = D^{\prime} + D^{\prime \prime}$ be the decomposition of $D$ such that $D^{\prime}$ is the sum of all the irreducible components meeting $C$ with positive intersection number, and $D^{\prime \prime}$ is the sum of all the irreducible components meeting $C$ with intersection number $0$, then we have $D_1 \subset D^{\prime \prime}$ and $C D^{\prime}$ = 2. Since $D^{\prime}$ has at most two irreducible components and any irreducible component of $D^{\prime}$ is a $(-2)$-curve, by using exactly the same argument as in the proof of Lemma $2. 1$ of \cite{on2chi-2}, we obtain a contradiction. Hence the surface $X$ has no pencil of curves of genus $2$.
Thus in order to prove Proposition \ref{prop:bicanonical}, it only remains to prove that $X$ has non-birational bicanonical map. For this purpose, let us use the notation in Proposition \ref{prop:anotherdescription}. We have $H^0 (\bar{h}_* \mathcal{O}_{\bar{X}} (2K_{\bar{X}})) = H^0(\mathcal{O}_{\bar{V}}(2(K_{\bar{V}} + \varrho))) \oplus H^0(\mathcal{O}_{\bar{V}}(2K_{\bar{V}} + \varrho )) $ for a certain divisor $\varrho$ with $\bar{A}_0 + \bar{\lambda} + \sum \bar{I}_{i,j} \sim 2 \varrho$. We however have \[ 2 K_{\bar{V}} + \varrho \quad \sim \quad \nu^* (I_{1 0}^{\prime \prime \prime}) -\sum G_i^{\prime} + \tilde{\lambda}, \] hence $h^0 (\mathcal{O}_{\bar{V}}(2K_{\bar{V}} + \varrho )) = 0$. This implies that the bicanonical map of $\bar{X}$ factors through the rational map of $\bar{V}$ associated to the linear system
$|2 (K_{\bar{V}} + \varrho)|$. Hence we have the assertion. \qed
To give a proof for Theorem \ref{thm:moduli}, we also need the following lemma:
\begin{lemma} \label{lm:a0barcoh} Let $\bar{V}$ be the smooth surface as in Proposition \ref{prop:anotherdescription}. Then for a member $\bar{A}_0 \in
|- 4 K_{\bar{V}} + \tilde{\lambda} + \bar{\lambda}^{\prime}|$, the following equalities hold$:$
\[
h^0 (\mathcal{O}_{\bar{V}} (\bar{A}_0)) = 29, \qquad
h^1 (\mathcal{O}_{\bar{V}} (\bar{A}_0)) = 0 , \qquad
h^2 (\mathcal{O}_{\bar{V}} (\bar{A}_0)) = 0. \] \end{lemma}
Proof. Since we have $\bar{\wp}^* (\bar{A}_0) \in \bar{r}^* \tilde{\varLambda}_8^+ (3\sum e_i + 3\sum e_i^{\prime})$, the equality $h^i (\mathcal{O}_{\bar{V}} (\bar{A}_0)) = 0$ for any $i >0$ follows from Lemma \ref{lm:b0tildenonsing}. From this together with the Riemann--Roch theorem, we infer $h^0 (\mathcal{O}_{\bar{V}} (\bar{A}_0)) = 29$. \qed
As the first part of our proof for Theorem \ref{thm:moduli}, we shall show the following:
\begin{lemma} \label{lm:irredmoduli} Any two minimal algebraic surfaces with $c_1^2 = 2 \chi -1$, $\chi = 4$, and $\mathrm{Tors} \simeq \mathbb{Z} /2$ are equivalent under deformation of complex structures. The coarse moduli space $\mathcal{M}$ for surfaces with these invariants is irreducible. \end{lemma}
Proof. Let $\mathcal{M}$ be the coarse moduli space for minimal surfaces $X$'s with $c_1^2 = 2 \chi -1$, $\chi = 4$, and $\mathrm{Tors} \simeq \mathbb{Z} /2$. In what follows, for a surface $X$ with these invariants, we denote by $[X]$ the point in $\mathcal{M}$ corresponding to the isomorphism class of $X$. As a reference point of $\mathcal{M}$, let us fix a surface $X_{(1)}$ with these invariants for which $d =0$ holds, the two points $w_1$ and $w_2$ lie neither on one and the same member
of $| \varGamma |$ nor on that of $|\varDelta_0 |$, and $\bar{A}_0$ is smooth. Below, we shall give an irreducible component $\mathcal{M}_{(1)}$ of $\mathcal{M}$ containing $[X_{(1)}]$, and show that for any $X$ with these invariants we have $[X] \in \mathcal{M}_{(1)}$ and $X$ has the same deformation type as that of the reference surface $X_{(1)}$. We divide our situation into several cases according to $d$, the configuration of $w_i$'s and $w_i^{\prime}$'s, and smoothness of $\bar{A}_0$ for our $X$. In what follows, $\epsilon$ and $\epsilon_0$ will denote positive real numbers small enough. We shall replace these numbers with smaller ones without mentioning it explicitly.
Case $1$-$1$: the case where $d=0$ holds, the two points $w_1$ and $w_2$ lie neither on one and the same member
of $| \varGamma |$ nor on that of $|\varDelta_0 |$. This case splits into two subcases: case $1$-$1$-$1$ and case $1$-$1$-$2$.
Case $1$-$1$-$1$: the subcace of case $1$-$1$ where $\bar{A}_0$ is smooth. From the point of view of description as in Proposition \ref{prop:anotherdescription}, this case corresponds to the case where $v_1^{\prime \prime \prime} \notin D_0^{\prime \prime \prime}$, $v_0^{\prime} \notin \sum I_{i j}^{\prime} + \sum G_i^{\prime} + \sum D_j^{\prime}$, and moreover $\bar{A}_0$ is smooth, where we put $v_1^{\prime \prime \prime} = \nu^{\prime \prime} (I_{1 \infty}^{\prime \prime})$ and $v_0^{\prime} = \tilde{\nu} (\tilde{\lambda})$. Note that for all $X$'s of this case $\tilde{V}$'s have one and the same isomorphism class. Let $\mathrm{pr}_{\tilde{V} \times \tilde{\lambda}} : \tilde{V} \times \tilde{\lambda} \to \tilde{\lambda} \simeq \mathbb{P}^1$ be the trivial family. Let $\mathrm{pr}_{\tilde{V} \times \tilde{\lambda}, \tilde{V}}: \tilde{V} \times \tilde{\lambda} \to \tilde{V}$ be the first projection.
Then we can easily construct an analytic family $\mathrm{pr}_{\bar{\mathcal{V}}} : \bar{\mathcal{V}} \to \tilde{\lambda} \simeq \mathbb{P}^1$ together with a projection $\mathrm{pr}_{\bar{\mathcal{V}}, \tilde{V} \times \tilde{\lambda}} : \bar{\mathcal{V}} \to \tilde{V} \times \tilde{\lambda}$ satisfying the following condition: for each $t \in \tilde{\lambda}$, the natural projection $\bar{V}_t = {\mathrm{pr}_{\bar{\mathcal{V}}}}^{-1} (t) \to \tilde{V} = {\mathrm{pr}_{\tilde{V} \times \tilde{\lambda}}}^{-1} (t)$ is the blowing-up at $t \in \tilde{\lambda} \subset \tilde{V}$ with exceptional divisor $\bar{\lambda}^{\prime}_t $. Let us denote by $\bar{\lambda}_t$ and $\tilde{\lambda}_t$ the strict transform and the total transform by $\bar{V}_t = {\mathrm{pr}_{\bar{\mathcal{V}}}}^{-1} (t) \to \tilde{V} = {\mathrm{pr}_{\tilde{V} \times \tilde{\lambda}}}^{-1} (t)$ of the divisor $\tilde{\lambda}$, respectively. We denote by $\mathrm{pr}_{\bar{\mathcal{V}}, \tilde{V}} : \bar{\mathcal{V}} \to \tilde{V}$ the composite of two projections $\mathrm{pr}_{\bar{\mathcal{V}}, \tilde{V} \times \tilde{\lambda}}$ and $\mathrm{pr}_{\tilde{V} \times \tilde{\lambda}, \tilde{V}}$.
Consider the divisor $-4K_{\bar{\mathcal{V}}} + {\mathrm{pr}_{\bar{\mathcal{V}}, \tilde{V}}}^* (\tilde{\lambda}) + \cup_t \bar{\lambda}_t^{\prime}$ on $\bar{\mathcal{V}}$. The restriction to $\bar{V}_t$ of this divisor is linearly equivalent to $- 4K_{\bar{V}_t} + \tilde{\lambda}_t + \bar{\lambda}_t^{\prime}$. Since we have $h^1 (\mathcal{O}_{\bar{V}_t} (- 4K_{\bar{V}_t} + \tilde{\lambda}_t + \bar{\lambda}_t^{\prime})) =0$ and $h^0 (\mathcal{O}_{\bar{V}_t} (- 4K_{\bar{V}_t} + \tilde{\lambda}_t + \bar{\lambda}_t^{\prime})) = 29$ by Lemma \ref{lm:a0barcoh}, it follows that the direct image \[
\mathcal{F}_0 = \mathrm{{pr}_{\bar{\mathcal{V}}}}_* \mathcal{O}_{\bar{\mathcal{V}}} (-4K_{\bar{\mathcal{V}}} + {\mathrm{pr}_{\bar{\mathcal{V}}, \tilde{V}}}^* (\tilde{\lambda}) + \cup_t \bar{\lambda}_t^{\prime}) \] is a locally free sheaf on $\tilde{\lambda} \simeq \mathbb{P}^1$ of rank $29$. We denote by $\mathcal{F}_0^{\vee}$ the dual sheaf of $\mathcal{F}_0$ on $\tilde{\lambda}$.
Let $\mathrm{pr}_{\mathbb{P}} : \mathbb{P} = \mathbb{P} (\mathcal{F}_0^{\vee}) \to \tilde{\lambda}$ be the $\mathbb{P}^{28}$-bundle over $\tilde{\lambda}$ associated with $\mathcal{F}_0^{\vee}$. Then $\mathbb{P}$ is the projectivised total space of vector bundle $\mathcal{F}_0$. We consider the Cartesian diagram \[ \begin{CD} \bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P} @>\text{$ $}>> \mathbb{P} \\ @V\text{$ $}VV @VV\text{$\mathrm{pr}_{\mathbb{P}}$}V \\ \bar{\mathcal{V}} @>\text{$\mathrm{pr}_{\bar{\mathcal{V}}}$}>> \tilde{\lambda}, \end{CD} \] and denote by $\mathrm{pr}_{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}, \, \bar{\mathcal{V}}} : \bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P} \to \bar{\mathcal{V}}$, $\mathrm{pr}_{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}, \, \mathbb{P}}: \bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P} \to \mathbb{P}$, and $\mathrm{pr}_{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}}: \bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P} \to \tilde{\lambda}$ the first projection, the second projection, and the induced natural projection respectively.
Let $\mathcal{O}_{\mathbb{P}} (1)$ be the tautological bundle of $ \mathrm{pr}_{\mathbb{P}} : \mathbb{P} = \mathbb{P} (\mathcal{F}_0^{\vee}) \to \tilde{\lambda}$. Then there exists a natural non-zero global section \[
\varPsi_0 \in
H^0 ({\mathrm{pr}_{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}, \, \mathbb{P}}}^* \mathcal{O}_{\mathbb{P}} (1) \otimes {\mathrm{pr}_{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}, \, \bar{\mathcal{V}}}}^* \mathcal{O}_{\bar{\mathcal{V}}} ( -4 K_{\bar{\mathcal{V}}} + {\mathrm{pr}_{\bar{\mathcal{V}}, \, \tilde{V}}}^* (\tilde{\lambda}) + \cup_t \bar{\lambda}_t^{\prime})) \] on $\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}$ satisfying the following condition: for each open set $U \subset \tilde{\lambda}$ such that
the restriction $\mathcal{F}_0 |_U$ is trivial, the restriction $\varPsi_0
|_{{\mathrm{pr}_{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}}}^{-1} (U)}$ of $\varPsi_0$ to ${\mathrm{pr}_{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}}}^{-1} (U)$ is given by $\varPsi_0
|_{{\mathrm{pr}_{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}}}^{-1} (U)} = \sum_{i=1}^{29} a_i \psi_i$, where $\{ \psi_1, \ldots , \psi_{29} \}$ and $\{ a_1, \ldots , a_{29} \}$ are
a base of $H^0 (\mathcal{F}_0 |_{U})$ and its dual base respectively (note here that we have the natural isomorphism ${\mathrm{pr}_{\mathbb{P}}}_* \mathcal{O}_{\mathbb{P}} (1) \simeq \mathcal{F}_0^{\vee}$). We denote by $\bar{\alpha}_0 = (\varPsi_0)$ the divisor on $\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}$ defined by the section $\varPsi_0$.
For each $u \in \mathbb{P}$, we put $t(u) = \mathrm{pr}_{\mathbb{P}} (u) \in \tilde{\lambda}$. Then we have the natural isomorphism ${\mathrm{pr}_{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}, \, \mathbb{P} }}^{-1} (u) \simeq \bar{V}_{t(u)}$. Moreover, via this identification, the restriction $\bar{A}_{0 \, u} = \bar{\alpha}_0
|_{{\mathrm{pr}_{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}, \, \mathbb{P} }}^{-1} (u) } $ $\in
| -4 K_{\bar{V}_{t(u)}} + \tilde{\lambda}_{t(u)}
+ \bar{\lambda}^{\prime}_{t(u)}| $ is a divisor on $\bar{V}_{t(u)}$ given by the local defining function
$\sum_{i=1}^{29} a_i (u) \psi_i |_{\bar{V}_{t(u)}}$. Let $\mathbb{P}_0 \subset \mathbb{P}$ be the set of all $u$'s such that ${\bar{A}_{0 \, u}}$ is a reduced smooth divisor satisfying $\bar{A}_{0 \, u} \cap (\bar{\lambda}_{t(u)} + \sum_{i=1, 2, \ j=0, \infty} \bar{I}_{ij \, t(u)}) = \emptyset$, where $\bar{I}_{ij \, t}$ ($t \in \tilde{\lambda}$) is the restriction to $\bar{V}_t$ of the divisor ${\mathrm{pr}_{\bar{\mathcal{V}}, \tilde{V}}}^* (\tilde{I}_{ij})$. Then $\mathbb{P}_0$ is a non-empty Zariski open subset of $\mathbb{P} = \mathbb{P} (\mathcal{F}_0^{\vee})$. Since $\mathbb{P} \to \tilde{\lambda}$ is a $\mathbb{P}^{28}$-bundle over a non-singular rational curve $\tilde{\lambda} \simeq \mathbb{P}^1$, there exists a covering $\{ U_{\mu}^{\vee}\}_{\mu}$ of $\mathbb{P}$ by a finite number of Zariski open subsets $U_{\mu}^{\vee}$'s satisfying the following condition: for any $\mu$, the restriction
$\mathcal{O}_{\mathbb{P}} (1) |_{U_{\mu}^{\vee}}$ is trivial, and $U_{\mu}^{\vee}$ is isomorphic to the $29$-dimensional linear space $\mathbb{A}^{29}$. We fix one such cover $\{ U_{\mu}^{\vee}\}_{\mu}$, and put $U_{\mu}^0 = U_{\mu}^{\vee} \cap \mathbb{P}_0$ for each $\mu$.
Let $\mathrm{pr}_{\bar{\mathcal{V}}, V^{\prime \prime \prime}} : \bar{\mathcal{V}} \to V^{\prime \prime \prime}$ and $\mathrm{pr}_{\bar{\mathcal{V}}, V^{\prime}} : \bar{\mathcal{V}} \to V^{\prime}$ be the natural projections $\nu^{\prime \prime} \circ \nu^{\prime} \circ \tilde{\nu} \circ \mathrm{pr}_{\bar{\mathcal{V}}, \tilde{V}}$ and $\tilde{\nu} \circ \mathrm{pr}_{\bar{\mathcal{V}}, \tilde{V}}$ respectively. Then since the restriction to ${\mathrm{pr}_{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}, \, \mathbb{P}}}^{-1} (U_{\mu}^0)$ of ${\mathrm{pr}_{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}, \, \mathbb{P}}}^* \mathcal{O}_{\mathbb{P}}(1) $ is trivial, it follows from (\ref{eq:halfofsumIij}) that the restriction to ${\mathrm{pr}_{{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}, \, \mathbb{P}}}}^{-1} (U_{\mu}^0)$ of the divisor \begin{equation} \label{eq:familyofbranchdivisors} \bar{\alpha}_0 + {\mathrm{pr}_{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}, \, \bar{\mathcal{V}}} }^* (\cup_t \bar{\lambda}_t + {\mathrm{pr}_{{\bar{\mathcal{V}} , \tilde{V}}}}^* (\sum \tilde{I}_{i j})) \end{equation} is linearly equivalent to twice the restriction to
${\mathrm{pr}_{{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}, \, \mathbb{P}}}}^{-1} (U_{\mu}^0)$ of the divisor \[ {\mathrm{pr}_{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}, \, \bar{\mathcal{V}}} }^* (-2 K_{\bar{\mathcal{V}}} + {\mathrm{pr}_{\bar{\mathcal{V}}, \tilde{V}}}^* (\tilde{\lambda}) + {\mathrm{pr}_{\bar{\mathcal{V}}, V^{\prime \prime \prime}}}^*
(I_{1 0}^{\prime \prime \prime}) - {\mathrm{pr}_{\bar{\mathcal{V}}, V^{\prime}}}^*
(\sum G_i^{\prime})). \] So let $\bar{\mathcal{X}}^{(\mu)} \to {\mathrm{pr}_{{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}, \, \mathbb{P}}}}^{-1} (U_{\mu}^0)$ be the double cover branched along the restriction to ${\mathrm{pr}_{{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}, \, \mathbb{P}}}}^{-1} (U_{\mu}^0)$ of the divisor (\ref{eq:familyofbranchdivisors}). Composing this morphism with the projection $\mathrm{pr}_{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}, \, \mathbb{P}} : \bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P} \to \mathbb{P}$, we obtain an analytic family $\mathrm{pr}_{\bar{\mathcal{X}}^{(\mu)}} : \bar{\mathcal{X}}^{(\mu)} \to U_{\mu}^0$. For each $u \in U_{\mu}^0$, we put $\bar{X}^{(\mu)}_u = {\mathrm{pr}_{\bar{\mathcal{X}}^{(\mu)}}}^{-1} (u)$.
The inverse image by $\bar{\mathcal{X}}^{(\mu)} \to {\mathrm{pr}_{{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}, \, \mathbb{P}}}}^{-1} (U_{\mu}^0)$ of ${\mathrm{pr}_{\bar{\mathcal{V}} \times_{\tilde{\lambda}} \mathbb{P}, \, \bar{\mathcal{V}}} }^* (\cup_t \bar{\lambda}_t + {\mathrm{pr}_{{\bar{\mathcal{V}} , \tilde{V}}}}^* (\sum \tilde{I}_{i j}))$ gives a family over $U_{\mu}^0$ whose fiber over each $u \in U_{\mu}^0$ is a sum of five disjoint $(-1)$-curves on $\bar{X}^{(\mu)}_u$. Blowing down this family of disjoint five $(-1)$-curves relatively to $\mathrm{pr}_{\bar{\mathcal{X}}^{(\mu)}} : \bar{\mathcal{X}}^{(\mu)} \to U_{\mu}^0$, we obtain an analytic family $\mathrm{pr}_{\mathcal{X}^{(\mu)}} : \mathcal{X}^{(\mu)} \to U_{\mu}^0$. For each $u \in U_{\mu}^0$, we put $X^{(\mu)}_u = {\mathrm{pr}_{\mathcal{X}^{(\mu)}}}^{-1} (u)$. Then by Proposition \ref{prop:anotherdescription}, for each $u \in U_{\mu}^0$, the fiber $X^{(\mu)}_u$ is a minimal surface with $c_1^2 = 2\chi -1$, $\chi =4$, and $\mathrm{Tors} \simeq \mathbb{Z} / 2$.
Let $U_{\mu}^0 \to \mathcal{M}$ be the natural morphism induced from the family $\mathrm{pr}_{\mathcal{X}^{(\mu)}} : \mathcal{X}^{(\mu)} \to U_{\mu}^0$, i.e., the morphism given by $u \mapsto [X^{(\mu)}_u]$, where $[X^{(\mu)}_u]$ is a point in $\mathcal{M}$ corresponding to the isomorphism class of the fiber $X^{(\mu)}_u$. The two morphisms $U_{\mu_1}^0 \to \mathcal{M}$ and $U_{\mu_2}^0 \to \mathcal{M}$ coincide on $U_{\mu_1}^0 \cap U_{\mu_2}^0$. Thus gluing $U_{\mu}^0 \to \mathcal{M}$'s, we obtain a morphism $\mathbb{P}_0 \to \mathcal{M}$ given locally by $u \mapsto [X^{(\mu)}_u]$. Since $\mathbb{P}_0$ is irreducible, the image of this $\mathbb{P}_0 \to \mathcal{M}$ lies on an irreducible component of the moduli space $\mathcal{M}$. We fix one such irreducible component and denote it by $\mathcal{M}_{(1)}$.
Now let $X$ be a minimal surface with $c_1^2 = 2\chi -1$, $\chi =4$, and $\mathrm{Tors} \simeq \mathbb{Z} /2$ for which $d=0$ holds, the two points $w_1$ and $w_2$ lie neither on
one and the same member of $|\varDelta_0|$
nor on that of $|\varGamma|$, and $\bar{A}_0$ is smooth. Then by Proposition \ref{prop:anotherdescription} and the construction of $\mathrm{pr}_{\mathcal{X}^{(\mu)}} : \mathcal{X}^{(\mu)} \to U_{\mu}^0$ above, there exist a $\mu$ and a $u \in U_{\mu}^0$ such that $X \simeq X^{(\mu)}_u$ holds. Since $\mathbb{P}_0$ is connected, we infer that $X$ has the same deformation type as that of the reference surface $X_{(1)}$. Moreover, we infer that the corresponding point $[X]$ lies on the irreducible component $\mathcal{M}_{(1)}$
.
Case $1$-$1$-$2$: the subcase of case $1$-$1$ where $\bar{A}_0$ is singular. Let $X$ be a minimal surface with $c_1^2 = 2\chi -1$, $\chi =4$, and $\mathrm{Tors} \simeq \mathbb{Z} /2$ of this case. In this case, $\bar{A}_0$ has at most negligible singularities, and by Proposition \ref{prop:anotherdescription},
the linear system $|\bar{A}_0|$ has no base point. Thus by the same method as in \cite[Proof of Theorem $4$]{quintic}, we obtain an analytic family $\mathrm{pr}_{\mathcal{X}} : \mathcal{X} \to N
= \{ u \in \mathbb{C} : |u| < \epsilon \}$ of minimal surfaces with $c_1^2 = 2 \chi -1$, $\chi =4$, and $\mathrm{Tors} \simeq \mathbb{Z} / 2$ such that $X_u = {\mathrm{pr}_{\mathcal{X}}}^{-1} (u)$ is of case $1$-$1$-$1$ for each $u \neq 0 \in N$, and $X_0 = {\mathrm{pr}_{\mathcal{X}}}^{-1} (0) \simeq X$.
From this together with the results for case $1$-$1$-$1$, we infer that $X$ has the same deformation type as that of the reference surface $X_{(1)}$, and that the point $[X]$ lies on the irreducible component $\mathcal{M}_{(1)}$ in the proof for case $1$-$1$-$1$.
Case $1$-$2$: the case where $d=0$ holds, and the two points $w_1$ and $w_2$ lie on one and the same
member of $| \varGamma |$ or $| \varDelta_0 |$. In this case, by Remark \ref{rem:gamma1gamma2}, we may assume that the two points $w_1$ and $w_2$ lie
on the member $\varGamma_1 \in |\varGamma|$. Then by Lemma \ref{lm:configw1w1prime}, we have $w_1^{\prime} \notin {r^{\prime}}^{-1}_* (\varGamma_1)$. This case is divided into two subcases: case $1$-$2$-$1$ and case $1$-$2$-$2$.
Case $1$-$2$-$1$: the subcase of case $1$-$2$ where $\bar{A}_0$ is smooth. From the point of view of description as in Proposition \ref{prop:anotherdescription}, this case corresponds to the case where $v_1^{\prime \prime \prime} \notin D_0^{\prime \prime \prime}$, $v_0^{\prime} \in G_1^{\prime} \setminus (\sum_j I_{1 j}^{\prime})$, and moreover $\bar{A}_0$ is smooth, where we put $v_1^{\prime \prime \prime} = \nu^{\prime \prime} (I_{1 \infty}^{\prime \prime})$ and $v_0^{\prime} = \tilde{\nu} (\tilde{\lambda})$. Let $X$ be a minimal surface with $c_1^2 = 2\chi -1$, $\chi =4$, and $\mathrm{Tors} \simeq \mathbb{Z} /2$ of this case. Let $\epsilon$ be a positive real number small enough.
We put $N = \{ t \in \mathbb{C} : |t| < \epsilon \}$, and denote by $\mathrm{pr}_{V^{\prime} \times N} : V^{\prime} \times N \to N $ the trivial family. Let $\mathrm{pr}_{ V^{\prime} \times N, \, V^{\prime}} : V^{\prime} \times N \to V^{\prime}$ be the first projection.
Then we can easily construct anatitic families $\mathrm{pr}_{\tilde{\mathcal{V}}} : \tilde{\mathcal{V}} \to N$, $\mathrm{pr}_{\bar{\mathcal{V}}} : \bar{\mathcal{V}} \to N$ together with projections $\mathrm{pr}_{\tilde{\mathcal{V}}, \, V^{\prime} \times N} : \tilde{\mathcal{V}} \to V^{\prime} \times N$, $\mathrm{pr}_{\bar{\mathcal{V}}, \tilde{\mathcal{V}}} : \bar{\mathcal{V}} \to \tilde{\mathcal{V}}$ satisfying the following conditions: for each $t \in N$, the projection $\tilde{V}_t = {\mathrm{pr}_{\tilde{\mathcal{V}}}}^{-1} (t) \to V^{\prime} = {\mathrm{pr}_{V^{\prime} \times N}}^{-1} (t)$ is the blowing-up at $v^{\prime} (t)$ with exceptional divisor $\tilde{\lambda}_t$, where $v^{\prime}: N \to V^{\prime} \times N$ is a holomorphic section of the analytic family $\mathrm{pr}_{V^{\prime} \times N} : V^{\prime} \times N \to N$ ; for each $t \in N$, the projection $\bar{V}_t = {\mathrm{pr}_{\bar{\mathcal{V}}}}^{-1} (t) \to \tilde{V}_t = {\mathrm{pr}_{\tilde{\mathcal{V}}}}^{-1} (t) $ is the blowing-up at $\tilde{v} (t)$ with exceptional divisor $\bar{\lambda}^{\prime}_t$, where $\tilde{v} : N \to \tilde{\mathcal{V}}$ is a holomorphic section of the analytic family $\mathrm{pr}_{\tilde{\mathcal{V}}} : \tilde{\mathcal{V}} \to N$ ; $v^{\prime} (0) = v^{\prime}_0 (= \tilde{\nu} (\tilde{\lambda}))$ holds, and $\mathrm{pr}_{ V^{\prime} \times N, \, V^{\prime}} (v^{\prime} (t)) \in \sum G_i^{\prime} + \sum D_j^{\prime} + \sum I_{ij}^{\prime}$ if and only if $t =0$ ; $\tilde{v} (0) = \tilde{v}_0 (= \bar{\nu} (\bar{\lambda}^{\prime}) )$ holds, and $\tilde{v} (t) \in \tilde{\lambda}_t$ for any $t \in N$. Note that from the conditions above, we have in particular $\tilde{V}_0 = \tilde{V}$ and $\bar{V}_0 = \bar{V}$. Let us denote by $\bar{\lambda}_{t}$ the strict transform of $\tilde{\lambda}_t$ by $\bar{V}_t = {\mathrm{pr}_{\bar{\mathcal{V}}}}^{-1} (t) \to \tilde{V}_t = {\mathrm{pr}_{\tilde{\mathcal{V}}}}^{-1} (t) $.
Consider the divisor $-4 K_{\bar{\mathcal{V}}} + {\mathrm{pr}_{\bar{\mathcal{V}}, \tilde{\mathcal{V}}}}^* (\cup_t \tilde{\lambda}_t) + \cup_t \bar{\lambda}^{\prime}_t$ on $\bar{\mathcal{V}}$. The restriction to $\bar{V}_0 = \bar{V}$ of this divisor is linearly equivalent to $-4K_{\bar{V}} + \tilde{\lambda} + \bar{\lambda}^{\prime}$. Since we have $h^1 (\mathcal{O}_{\bar{V}} (-4K_{\bar{V}} + \tilde{\lambda} + \bar{\lambda}^{\prime})) =0$ by Lemma \ref{lm:a0barcoh}, there exists a non-zero global section $\varPsi \in H^0 (\mathcal{O}_{\bar{\mathcal{V}}} (-4 K_{\bar{\mathcal{V}}} + {\mathrm{pr}_{\bar{\mathcal{V}}, \tilde{\mathcal{V}}}}^* (\cup_t \tilde{\lambda}_t) + \cup_t \bar{\lambda}^{\prime}_t))$ on $\bar{\mathcal{V}}$ satisfying the following conditions:
the restriction $(\varPsi) |_{\bar{V}_0}$ to $\bar{V}_0$ coincides with $\bar{A}_0$, where $( \varPsi )$ denotes the divisor on $\bar{\mathcal{V}}$ defined by the global section $\varPsi$; for any $t \in N$, the restriction
$(\varPsi) |_{\bar{V}_t}$ to $\bar{V}_t$ is a reduced non-singular divisor on $\bar{V}_t$; the divisor $(\varPsi)$ does not intersects $\cup_t \bar{\lambda}_t + {\mathrm{pr}_{\bar{\mathcal{V}}, V^{\prime}}}^* (\sum I_{i j}^{\prime})$, where $\mathrm{pr}_{\bar{\mathcal{V}}, V^{\prime}} : \bar{\mathcal{V}} \to V^{\prime}$ is the composite of three projections $\mathrm{pr}_{\bar{\mathcal{V}}, \tilde{\mathcal{V}}}$, $\mathrm{pr}_{\tilde{\mathcal{V}}, \, V^{\prime} \times N}$, and $\mathrm{pr}_{V^{\prime} \times N, \, V^{\prime}}$.
Let $\bar{\mathcal{X}} \to \bar{\mathcal{V}}$ be the double cover branched along $( \varPsi ) + \cup_t \bar{\lambda}_t + {\mathrm{pr}_{\bar{\mathcal{V}}, V^{\prime}}}^* (\sum I_{i j}^{\prime})$. Then composing this morphism with the projection $\mathrm{pr}_{\bar{\mathcal{V}}} : \bar{\mathcal{V}} \to N$, we obtain an analytic family $\mathrm{pr}_{\bar{\mathcal{X}}} : \bar{\mathcal{X}} \to N$. For each $t \in N$, we put $\bar{X}_t = {\mathrm{pr}_{\bar{\mathcal{X}}}}^{-1} (t)$. The inverse image by $\bar{\mathcal{X}} \to \bar{\mathcal{V}}$ of $\cup_t \bar{\lambda}_t + {\mathrm{pr}_{\bar{\mathcal{V}}, V^{\prime}}}^* (\sum I_{i j}^{\prime})$ gives a family over $N$ whose fiber over each $t \in N$ is a disjoint union of five $(-1)$-curves on $\bar{X}_t$. Blowing down this family of five $(-1)$-curves relatively to $\mathrm{pr}_{\bar{\mathcal{X}}} : \bar{\mathcal{X}} \to N$, we obtain an analytic family $\mathrm{pr}_{\mathcal{X}} : \mathcal{X} \to N$. Then by the construction of $\mathrm{pr}_{\mathcal{X}} : \mathcal{X} \to N$ above, we have $X_0 = {\mathrm{pr}_{\mathcal{X}}}^{-1} (0) \simeq X$, and for each $t \neq 0 \in N$, the fiber $X_t = {\mathrm{pr}_{\mathcal{X}}}^{-1} (t)$ is a minimal surface with $c_1^2 = 2\chi -1$, $\chi =4$, and $\mathrm{Tors} \simeq \mathbb{Z} /2$ of case $1$-$1$-$1$.
From this together with the results for case $1$-$1$-$1$, we infer that $X$ has the same deformation type as that of the reference surface $X_{(1)}$, and that the point $[X]$ lies on the irreducible component $\mathcal{M}_{(1)}$ in the proof for case $1$-$1$-$1$.
Case $1$-$2$-$2$: the subcase of case $1$-$2$ where $\bar{A}_0$ is singular.
Let $X$ be a minimal surface with $c_1^2 = 2\chi -1$, $\chi =4$, and $\mathrm{Tors} \simeq \mathbb{Z} /2$ of this case.
Then using the same argument as in case $1$-$1$-$2$, we infer from the results for case $1$-$2$-$1$ that $X$ has the same deformation type as that of the reference surface $X_{(1)}$, and that the point $[X]$ lies on the irreducible component $\mathcal{M}_{(1)}$ in the proof for case $1$-$1$-$1$.
Case $2$-$1$: the case where $d=2$ holds, and the two points
$w_1$ and $w_2$ do not lie on one and the same member of $| \varGamma |$. Note that in this case we have $v_0^{ \prime } \notin D_0^{\prime} $ by Lemma \ref{lm:configw1w1prime}, where we put $v_0^{ \prime } = \tilde{\nu} (\tilde{\lambda})$. This case splits into two subcases: case $2$-$1$-$1$ and case $2$-$1$-$2$.
Case $2$-$1$-$1$: the subcase of case $2$-$1$ where $\bar{A}_0$ is smooth. From the point of view of description as in Proposition \ref{prop:anotherdescription}, this case corresponds to the case where $v_1^{\prime \prime \prime} \in D_0^{\prime \prime \prime}$, $v_0^{\prime} \notin D_0^{\prime} + \sum G_i^{\prime} + \sum I_{ij}^{\prime}$, and moreover $\bar{A}_0$ is smooth, where we put $v_1^{\prime \prime \prime} = \nu^{\prime \prime} ( I_{1 \infty}^{\prime \prime})$ and $v_0^{\prime} = \tilde{\nu} (\tilde{\lambda})$. Let $X$ be a minimal surface with $c_1^2 = 2\chi -1$, $\chi =4$, and $\mathbb{Z} /2$ of this case. Let $\epsilon$ be a positive real number small enough.
We put $N = \{ t \in \mathbb{C} : |t| < \epsilon \}$, and denote by $\mathrm{pr}_{V^{\prime \prime \prime} \times N } : V^{\prime \prime \prime} \times N \to N $ the trivial family. Let $\mathrm{pr}_{V^{\prime \prime \prime} \times N, \, V^{\prime \prime \prime}} : V^{\prime \prime \prime} \times N \to V^{\prime \prime \prime}$ be the first projection.
Let us take a holomorphic section $v_{(1)}^{\prime \prime \prime} : N \to V^{\prime \prime \prime} \times N$ satisfying the following conditions: $\mathrm{pr}_{V^{\prime \prime \prime} \times N, \, V^{\prime \prime \prime}} (v_{(1)}^{\prime \prime \prime} (0)) = v_1^{\prime \prime \prime} $ ($= \nu^{\prime \prime} (I_{1 \infty}^{\prime \prime})$) holds; $\mathrm{pr}_{V^{\prime \prime \prime} \times N, \, V^{\prime \prime \prime}} (v_{(1)}^{\prime \prime \prime} (t)) \in I_{1 0}^{\prime \prime \prime}$ for any $t \in N$; $\mathrm{pr}_{V^{\prime \prime \prime} \times N, \, V^{\prime \prime \prime}} (v_{(1)}^{\prime \prime \prime} (t)) = v_1^{\prime \prime \prime}$ if and only if $t=0$.
Recall that the configuration corresponding to Case $1$-$1$-$1$ was $v_1^{\prime \prime \prime} \notin D_0^{\prime \prime \prime}$, $v_0^{\prime} \notin \sum I_{i j}^{\prime} + \sum G_i^{\prime} + \sum D_j^{\prime}$. Thus using the holomorphic section $v_{(1)}^{\prime \prime \prime} : N \to V^{\prime \prime \prime} \times N$ above, and by the same arguement as in the proof for case $1$-$2$-$1$, we obtain an analytic family $\mathrm{pr}_{\mathcal{X}} : \mathcal{X}
\to N= \{ t \in \mathbb{C} : |t| < \epsilon \}$ satisfying the following conditions: $X_0 = {\mathrm{pr}_{\mathcal{X}}}^{-1} (0) \simeq X$; and for each $t \neq 0 \in N$, the fiber $X_t = {\mathrm{pr}_{\mathcal{X}}}^{-1} (t)$ is a minimal surface with $c_1^2 = 2\chi -1$, $\chi =4$, and $\mathrm{Tors} \simeq \mathbb{Z} /2$ of case $1$-$1$-$1$. From this together with the results for case $1$-$1$-$1$, we infer that $X$ has the same deformation type as that of the reference surface $X_{(1)}$, and that the point $[X]$ lies on the irreducible component $\mathcal{M}_{(1)}$ in the proof for case $1$-$1$-$1$.
Case $2$-$1$-$2$: the subcase of case $2$-$1$ where $\bar{A}_0$ is singular.
Let $X$ be a minimal surface with $c_1^2 = 2\chi -1$, $\chi =4$, and $\mathrm{Tors} \simeq \mathbb{Z} /2$ of this case.
Then using the same argument as in case $1$-$1$-$2$, we infer from the results for case $2$-$1$-$1$ that $X$ has the same deformation type as that of the reference surface $X_{(1)}$, and that the point $[X]$ lies on the irreducible component $\mathcal{M}_{(1)}$
in the proof for case $1$-$1$-$1$.
Case $2$-$2$: the case where $d=2$ holds, the two points $w_1$ and $w_2$ lie on one and the same
member of $| \varGamma |$. Note that in this case we have $v_0^{\prime} \notin D_0^{\prime}$ by Lemma \ref{lm:configw1w1prime}, where we put $v_0^{\prime} = \tilde{\nu} (\tilde{\lambda})$. In this case, by Remark \ref{rem:gamma1gamma2}, we may assume that the two points $w_1$ and $w_2$ lie
on the member $\varGamma_1 \in |\varGamma|$. Then by Lemma \ref{lm:configw1w1prime}, we have $w_1^{\prime} \notin {r^{\prime}}^{-1}_* (\varGamma_1)$. This case is divided into two subcases: case $2$-$2$-$1$ and case $2$-$2$-$2$.
Case $2$-$2$-$1$: the subcase of case $2$-$2$ where $\bar{A}_0$ is smooth. From the point of view of description as in Proposition \ref{prop:anotherdescription}, this case corresponds to the case where $v_1^{\prime \prime \prime} \in D_0^{\prime \prime \prime}$, $v_0^{\prime} \in G_1^{\prime} \setminus (\sum_j I_{1 j}^{\prime})$, and moreover $\bar{A}_0$ is smooth, where we put $v_1^{\prime \prime \prime} = \nu^{\prime \prime} (I_{1 \infty}^{\prime \prime})$ and $v_0^{\prime} = \tilde{\nu} (\tilde{\lambda})$. Note that by Lemma \ref{lm:configw1w1prime}, we have $\tilde{v}_0 \notin {\tilde{\nu}}^{-1}_* (G_1^{\prime}) $, where we put $\tilde{v}_0 = \bar{\nu} (\bar{\lambda}^{\prime})$. Let $X$ be a minimal surface with $c_1^2 = 2\chi -1$, $\chi =4$, and $\mathrm{Tors} \simeq \mathbb{Z} /2$ of this case. Let $\epsilon$ be a positive real number small enough.
We put $N = \{ t \in \mathbb{C} : |t| < \epsilon \}$, and denote by $\mathrm{pr}_{V^{\prime} \times N} : V^{\prime} \times N \to N $ the trivial family. Let $\mathrm{pr}_{ V^{\prime} \times N, \, V^{\prime}} : V^{\prime} \times N \to V^{\prime}$ be the first projection. Let us take a holomorphic section $v^{\prime} : N \to V^{\prime} \times N$ satisfying the following conditions: $\mathrm{pr}_{ V^{\prime} \times N, \, V^{\prime}} (v^{\prime}(0)) = v_0^{\prime} (= \tilde{\nu} (\tilde{\lambda}))$ holds; $\mathrm{pr}_{ V^{\prime} \times N, \, V^{\prime}} (v^{\prime}(t)) \in D_0^{\prime} + \sum G_i^{\prime} + \sum I_{ij}^{\prime}$ if and only if $t =0$.
Recall that the configuration corresponding to case $2$-$1$-$1$ was $v_1^{\prime \prime \prime} \in D_0^{\prime \prime \prime}$, $v_0^{\prime} \notin D_0^{\prime} + \sum G_i^{\prime} + \sum I_{ij}^{\prime}$. Thus using the holomorphic section $v^{\prime} : N \to V^{\prime} \times N$ above, and by the same arguement as in the proof for case $1$-$2$-$1$, we obtain an analytic family $\mathrm{pr}_{\mathcal{X}} : \mathcal{X}
\to N= \{ t \in \mathbb{C} : |t| < \epsilon \}$ satisfying the following conditions: $X_0 = {\mathrm{pr}_{\mathcal{X}}}^{-1} (0) \simeq X$; for each $t \neq 0 \in N$, the fiber $X_t = {\mathrm{pr}_{\mathcal{X}}}^{-1} (t)$ is a minimal surface with $c_1^2 = 2\chi -1$, $\chi =4$, and $\mathrm{Tors} \simeq \mathbb{Z} /2$ of case $2$-$1$-$1$. From this together with the results for case $2$-$1$-$1$, we infer that $X$ has the same deformation type as that of the reference surface $X_{(1)}$, and that the point $[X]$ lies on the irreducible component $\mathcal{M}_{(1)}$
in the proof for case $1$-$1$-$1$.
Case $2$-$2$-$2$: the subcase of case $2$-$2$ where $\bar{A}_0$ is singular.
Let $X$ be a minimal surface with $c_1^2 = 2\chi -1$, $\chi =4$, and $\mathrm{Tors} \simeq \mathbb{Z} /2$ of this case.
Then using the same argument as in case $1$-$1$-$2$, we infer from the results for case $2$-$2$-$1$ that $X$ has the same deformation type as that of the reference surface $X_{(1)}$, and that the point $[X]$ lies on the irreducible component $\mathcal{M}_{(1)}$
in the proof for case $1$-$1$-$1$.
Now that we have the results for all the eight cases $1$-$1$-$1$, \ldots , $2$-$2$-$2$, we have the assertion. \qed
Note that from the proof above, we see that the morphism $\mathbb{P}_0 \to \mathcal{M}$ in the proof for case $1$-$1$-$1$ is dominant.
Now let us prove Theorem \ref{thm:moduli}.
{\sc Proof of Theorem \ref{thm:moduli}}
Let $\mathbb{P}_0 \to \mathcal{M}$ be the morphism $u \mapsto [X^{(\mu)}_u]$ given in the proof (for case $1$-$1$-$1$) of Lemma \ref{lm:irredmoduli}. Recall that we have $X^{(\mu_1)}_u \simeq X^{(\mu_2)}_u$ if $u \in U^0_{\mu_1} \cap U^0_{\mu_2}$. So in what follows, we abbreviate $X^{(\mu)}_u$ to $X_u$. Since $\mathbb{P}_0 \to \mathcal{M}$ is a dominant morphism from the $29$-dimensional variety $\mathbb{P}_0$, we only need to show that for each $u_0 \in \mathbb{P}_0$, there exist at most eight $u \in \mathbb{P}_0$'s satisfying $[X_u] = [X_{u_0}]$. Recall also that for all $X$'s of case $1$-$1$-$1$ in the proof of Lemma \ref{lm:irredmoduli}, $\tilde{V}$'s have one and the same isomorphism class. In what follows, we assume that $W$, $Z^{\prime}$, $\tilde{V}$, and the configuration of $w_i$'s are those for $X$'s of case $1$-$1$-$1$.
Let $\mathrm{Aut} (W)$ be the group of analytic automorphisms
of $W \simeq \varSigma_0$, and $\iota |_W$, the involution of $W$ as in Proposition \ref{prop:sufficientprocedure}.
Let $\mathrm{Aut} (W, \iota |_W, \{ w_i \} )$ be the subgroup of $\mathrm{Aut} (W)$ consisting of all $\sigma \in \mathrm{Aut} (W)$'s satisfying
$(\iota |_W) \circ \sigma = \sigma \circ (\iota |_W)$ and $\sigma (\{ w_i\}_{i=1,2}) = \{ w_i\}_{i=1,2}$.
Since $\mathrm{Aut} (W, \iota |_W, \{ w_i \} )$ acts naturally on the sets $\{ w_i \}_{i=1, 2}$ and $\{\varDelta_0, \varDelta_{\infty}, \varGamma_1, \varGamma_2 \}$, we have corresponding group homomorphisms
$\mathrm{Aut} (W, \iota |_W, \{ w_i \} ) \to \frak{S}_2$ and
$\mathrm{Aut} (W, \iota |_W, \{ w_i \} ) \to D_4$, where $\frak{S}_2$ and $D_4$ denote the symmetric group of degree $2$ and the dihedral group of degree $4$ respectively. It is easy to see that the product
$\mathrm{Aut} (W, \iota |_W, \{ w_i \} ) \to \frak{S}_2 \times D_4$ of these two morphisms is an isomorphism.
Let $Z^{\prime} /G$ be the quotient of the surface $Z^{\prime}$
by the natural action by the group $G = \langle \iota |_W \rangle$. Then the quotient $Z^{\prime} /G$ has four nodes, and the natural morphism $\tilde{V} \to Z^{\prime} /G$ gives the minimal desingularization of $Z^{\prime} /G$. Thus via the diagram $\tilde{V} \to Z^{\prime}/G \leftarrow Z^{\prime} \to W$,
the action by $\mathrm{Aut} (W, \iota |_W, \{ w_i \} )$ on the surface $W$ induces one on the pair $(\tilde{V}, \tilde{\lambda})$. Let
$\frak{S}_2 = \langle \iota |_W \rangle
\to \mathrm{Aut} (W, \iota |_W, \{ w_i \} )$ be the natural inclusion. Since $\frak{S}_2$ acts trivially on $\tilde{V}$ via this inclusion, we obtain a natural action
by $\mathrm{Aut} (W, \iota |_W, \{ w_i \} ) / \frak{S}_2 \simeq D_4$ on the pair $(\tilde{V}, \tilde{\lambda})$. This action on $(\tilde{V}, \tilde{\lambda})$ induces one on $\mathbb{P}_0$. Remark \ref{rem:onthedescription} however implies that we have $X_{u_1} \simeq X_{u_2}$ if and only if two points $u_1 \in \mathbb{P}_0$ and $u_2 \in \mathbb{P}_0$ belong to the same orbit
of the action by $\mathrm{Aut} (W, \iota |_W, \{ w_i \} ) / \frak{S}_2$ on $\mathbb{P}_0$. Thus by $\sharp D_4 =8$, we see that for any $u_0 \in \mathbb{P}_0$ there exist at most eight $u \in \mathbb{P}_0$'s satisfying $X_{u} \simeq X_{u_0}$. Hence we have the assertion. \qed
\section{Appendix}
{\sc Proof of Proposition \ref{prop:deg=n+1}}. Let us prove Proposition \ref{prop:deg=n+1}. The method we employ here is the same as the one used in \cite[Proof of Lemma 4.5]{evenI}, to which we refer the readers for details of the following argument. Let $Z \subset \mathbb{P}^n$, where $n \geq 4$, be a non-degenerate surface satisfying the assumptions in Proposition \ref{prop:deg=n+1}, and $Z^{\prime} \to Z$, its minimal desingularization. Since we have $\deg Z < 2n -2$, and
$Z^{\prime} \to Z$ is given by a complete linear system $|D^{\prime}|$, the surface $Z^{\prime}$ is a rational surface not isomorphic to $\mathbb{P}^2$. Thus, for an integer $d$, the surface $Z^{\prime}$ admits a birational morphism $r : Z^{\prime} \to \varSigma_d = Z^{\prime}_0$ onto the Hirzebruch surface $\varSigma_d$ of degree $d$. Let $D^{\prime}_0$ be a general member of the linear system
$r_* |D^{\prime}|$, and $\varepsilon^{\prime}_i$'s, the total transforms to $Z^{\prime}$ of the $(-1)$-curves appearing at the blowings up in $Z^{\prime} \to Z$. Then we have $D^{\prime} \sim r^* D^{\prime}_0 - \sum_{i=1}^s m_i\varepsilon^{\prime}_i$, where $m_i$'s, $s \in \mathbb{Z}$.
\begin{lemma} There exists an $r: Z^{\prime} \to Z^{\prime}_0$ as above such that for any $i$'s, the equality $m_i =1$ holds. \end{lemma}
Proof of Lemma. Note that the general member $D^{\prime}$ is a non-singular irreducible curve on $Z^{\prime}$. If $h^1 (\mathcal{O}_{D^{\prime}}(D^{\prime})) > 0$, then by Clifford's theorem on special divisors, we have ${D^{\prime}}^2 \geq 2 (h^0 (\mathcal{O}_{D^{\prime}}(D^{\prime}))-1)$, which contradicts $n \geq 4$. Thus we have $h^1 (\mathcal{O}_{D^{\prime}}(D^{\prime})) = 0$. From this together with the natural short exact sequence $0 \to \mathcal{O}_{Z^{\prime}} \to \mathcal{O}_{Z^{\prime}}(D^{\prime}) \to \mathcal{O}_{D^{\prime}}(D^{\prime}) \to 0$ and the Riemann--Roch theorem, we infer \[
\chi (\mathcal{O}_{Z^{\prime}}(D^{\prime}) ) = n+1, \] $D^{\prime} K_{Z^{\prime}} = {D^{\prime}}^2 + 2 (1 - \chi (\mathcal{O}_{Z^{\prime}}(D^{\prime}) )) = 1-n$, and $(K_{Z^{\prime}} + D^{\prime}/2 )D^{\prime}= (3-n)/2 <0$. Thus by Cone Theorem, we find that if $Z^{\prime}$ is not the Hirzebruch surface, then there exists a $(-1)$-curve $\varepsilon^{\prime}$ on $Z^{\prime}$ satisfying $(K_{Z^{\prime}} + D^{\prime}/2 ) \varepsilon^{\prime} <0$. Since $Z^{\prime} \to Z$ contracts no $(-1)$-curve, we obtain $D^{\prime} \varepsilon^{\prime} =1$. Let $r^{\prime} : Z^{\prime} \to Z^{\prime \prime}$ be the blowing-down of $\varepsilon^{\prime}$. We put $D^{\prime \prime} = r^{\prime}_* (D^{\prime})$. If $Z^{\prime \prime}$ is not the Hirzebruch surface, then the same argument as above ensures the existence of a $(-1)$-curve $\varepsilon^{\prime \prime}$ on $Z^{\prime \prime}$ satisfying $D^{\prime \prime} \varepsilon^{\prime \prime} =1$ (for the detail, see \cite[Lemma 4.4]{evenI}). We can repeat the same steps until we obtain the Hirzebruch surface. \qed
In what follows, we assume our $r$ satisfies the condition in the lemma above, hence $D^{\prime} \sim r^* D^{\prime}_0 - \sum_{i=1}^s \varepsilon^{\prime}_i$. We put $D^{\prime}_0 \sim a \varDelta_0 + b \varGamma$, where if $d=0$, we chose $\varDelta_0$ and $\varGamma$ in such a way that $b \geq a$. Then by $\chi (\mathcal{O}_{Z^{\prime}}(D^{\prime}) ) = n+1$ and ${D^{\prime}}^2 = n+1$, we obtain the following three equalities: \begin{align} n+1 &= D^{\prime}( D^{\prime} - K_{Z^{\prime}})/2 +1
= (a+1)(b- ad/2) + a -s +1, \notag \\ n+1 &= 2a (b - ad/2) -s, \label{eql:chidprime} \\ 0 &= {D^{\prime}}^2 - \chi (\mathcal{O}_{Z^{\prime}}(D^{\prime}) )
= (a-1)(b- ad/2) - (a+1). \label{eql:dprimesquare-chi} \end{align} Note that we have $b- ad/2 \geq a$ if $d \neq 1$, and that $b - ad/2 \geq a/2$ if $d =1$. Thus by (\ref{eql:chidprime}) and (\ref{eql:dprimesquare-chi}), we find $a =2$, $b =d +3$, and $s= 11-n$, hence $D^{\prime} \sim - K_{Z^{\prime}} + r^* \varGamma$.
Since $|D^{\prime}_0|$ has no fixed component, we obtain $d \leq 3$. \qed
\begin{flushright} \begin{minipage}{25em} Masaaki Murakami \\ University of Bayreuth, Lehrstuhl Mathematik VIII \\ Universitaetsstrasse 30, D-95447 Bayreuth, Germany\\ \texttt{[email protected]} \end{minipage} \end{flushright}
\end{document} | arXiv |
\begin{definition}[Definition:Segment of Circle/Angle in Segment]
{{EuclidSaid}}
:''{{:Definition:Euclid's Definitions - Book III/8 - Angle in Segment}}''
{{EuclidDefRefNocat|III|8|Angle in Segment}}
:''{{:Definition:Euclid's Definitions - Book III/9 - Stand on Circumference}}''
{{EuclidDefRefNocat|III|9|Stand on Circumference}}
:320px
Such a segment is said to '''admit''' the angle specified.
Category:Definitions/Circles
\end{definition} | ProofWiki |
Find the value of $\sqrt{12-\!\sqrt{12-\!\sqrt{12-\cdots}}}$.
Letting $x = \sqrt{12-\!\sqrt{12-\!\sqrt{12-\cdots}}}$, we have $x = \sqrt{12 - x}$. Therefore, $x^2 = 12 -x$, so $x^2 + x - 12=0$, or $(x+4)(x-3) = 0$. Clearly, $x$ must be positive, so $x = \boxed{3}$. | Math Dataset |
Shift rule
The shift rule is a mathematical rule for sequences and series.
Here $n$ and $N$ are natural numbers.
For sequences, the rule states that if $(a_{n})$ is a sequence, then it converges if and only if $(a_{n+N})$ also converges, and in this case both sequences always converge to the same number.[1]
For series, the rule states that the series $\sum \limits _{n=1}^{\infty }a_{n}$ converges to a number if and only if $\sum \limits _{n=1}^{\infty }a_{n+N}$ converges.[2]
References
1. Ueltschi, Daniel (2011), Analysis –MA131 (PDF), University of Warwick, p. 31.
2. Alcock, Lara (2014), How to Think About Analysis, Oxford University Press, p. 102, ISBN 9780191035371.
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Latin American Economic Review
Comparison of different regimes
Product competition and R&D investment under spillovers within full or partial collusion games
Kai Zhao†1Email author
Latin American Economic Review201524:4
Received: 22 March 2014
Accepted: 8 April 2015
The paper investigates firms' behavior and outcomes (levels of cost-reducing R&D, output, profit and welfare in equilibrium) in a differentiated duopoly with process innovation. One of the important features in this paper is that spillovers operate in the R&D stage and are tied to the degree of product substitutability as well as the extent of technological proximity/alienation of the research paths leading to cost reduction. Using this feature, the paper tries to explore and compare four separate organization setups (Full Competition, Semi-collusion in Production, Semi-collusion in R&D and Full Collusion). It is found that under technological proximity, competitions at the upstream stage depress R&D investment, and firms colluding in R&D regardless of their production strategy always yield more profit and generate higher social welfare than firms colluding in output; under technological alienation, R&D cooperation may reduce firms' interest to invest in R&D, and it is possible that firms in the Full Collusion regime produce most and generate the highest level of social welfare.
Spillover
Semi-collusion
Product differentiation
Horizontal merger
Nowadays, economies in Latin America are becoming more and more knowledge based. Innovation becomes essential to spur economic growth and to raise living standards. At the firm level, either competition or collusion could reward innovation by providing strong incentives for firms to be more efficient than their rivals. This paper aims to study the extent to which innovation incentives in a duopoly change according to the extent of product substitutability and the "technological distance" of firms. We draw particular attention to firms' (full/partial) collusive behavior and attempt to address the following questions: What type of collusion (partial, full, none) should firms choose, and which one is more conducive to technological advancement and a firm's growth? How do firms choose different types of collusion, and how do these affect market outcomes? Can the collusive strategy improve the consumer surplus and the social welfare, and which one serves best?
Innovation through R&D investment leads to more efficient use of resources, creating sustainable competitive advantages. The most important aspect of R&D investment is the externality (spillovers) which has been studied through the divergence between the social and private returns of production process. The public goods feature of knowledge generates spillovers which allow others to use the owner's innovation free of charge. Due to the spillover effect, the rate of return from an innovation is lesser and as a result, the incentives for carrying out R&D are reduced. The individual firm fears that competitors use its internal research results and thus probably increase their profits without having to bear the expenses. Therefore, the researching firm will only have limited incentive to invest in R&D. However, from the collective viewpoint, spillovers strengthen the dissemination of new knowledge available for the whole society, and improve the social welfare (Amir 2000).
Within a game where firms are first engaged in costly research efforts to adopt a lower-cost technology and then compete in a Cournot fashion with homogeneous products, (D'Aspremont and Jacquemin 1988) (henceforth "AJ") show that firms invest more under R&D cooperation than under R&D competition for sufficiently high spillover effects (full competition versus full cooperation). Kamien et al. (1992) (henceforth "KMZ") extend the AJ model to a more general framework with product differentiation and allow firms to participate in a research joint venture (RJV). They show that firms should be encouraged to form a RJV only if they coordinate their R&D decisions while maintaining competition for sales. Concerning the welfare effects of cooperative R&D with spillovers, cooperation raises social welfare when the spillover is high (Suzumura 1992).
Compared to aforementioned works, this paper emphasizes the "close relationship" between product differentiation and R&D spillovers. The key feature is to consider that the extent of product differentiation determines the ability of a firm to appropriate its rival's R&D effort. In addition, this ability is influenced by the sensibility of spillovers relative to product differentiation, in other words, technological distance. Several explanations can be provided to justify this "close relationship". First, when products are close substitutes, R&D efforts are less firm specific and a firm can more easily benefit from the discovery of a more efficient production technique resulting from rival's R&D effort. Second, the exchange of technological information between engineers of competing firms is recognized as an important source of R&D spillovers (Severinov 2001). Spillovers are believed to be higher between technological neighbors. According to this view, the ability to make productive use of another firm's knowledge depends on the degree of technological distance between firms. Every technology has a somewhat unique set of applications and language. Researchers in similar technological fields will interact in professional organizations, publish in commonly read journals, and, increasingly, browse a common set of web pages. It is natural to consider that the dissemination of technological knowledge across competing firms is strong when firms' technologies are similar. Furthermore, the above-mentioned "close relationship" is divided into two categories: concave relationship (technological proximity) where firms adopt similar technologies (i.e., the similar smart phones produced by Apple, Blackberry, Nokia ...), convex relationship (technological alienation) where firms adopt different technologies (i.e., electricity can be produced by different technologies). To be more concrete, we take the electricity production, for example, electric power companies are differentiated by voltages, a commercial consumer may need a voltage level of 11 kV or 440 V while a residential consumer needs power at level of 240 V, this difference of voltages refers to product differentiation. The electricity can be produced by different technologies (i.e., solar panels, wind turbines, nuclear energy), this refers to the extent of technological distance. The R&D flow between companies employing the same output (voltage) and the same technique is obviously greater.
In location models, the distance between firms determines the degree of product differentiation. By considering that R&D spillover depends negatively on firms' product location, it is shown that R&D effort is positively associated with the differentiation of products1 (Piga and Poyago-Theotoky 2005). However, they do not address the important issue of cooperative behavior between firms in their models.
In this paper, we consider a two-stage game where firms with heterogeneous products competing in a Cournot fashion engage in upstream R&D and downstream production. At each stage, the competing firms can either coordinate their decisions or adopt non-cooperative strategy. This assumption allows us to compare the Sub-game Perfect Nash Equilibrium (henceforth "SPNE") emerging in the four separate scenarios : full competition, semi-collusion in Production2, Semi-collusion in R&D3 and Full Collusion4. Compared to Kamien et al. (1992) which claim that the R&D investment by firms engaged in Semi-collusion in R&D is unambiguously greater than that in the Full Competition regime irrespective of spillovers, we demonstrate in fact which regime generates more R&D effort in equilibrium depends upon both the degree of product differentiation and the extent of technological distance. If we restrict our attention to the concave relationship, Full Collusion participants spend most on R&D, and Semi-collusion participants spend more than firms in the Full Competition regime. This ranking of R&D efforts is unalterable and independent of the product differentiation, and the competition at the upstream stage depresses R&D investment. Firms colluding in R&D regardless of their production strategy always yield more profit and generate higher social welfare than firms colluding in output independently of R&D strategy. When products are close substitutes, the synergy effects prevail over the anti-competitive effects due to the high spillovers, Full Collusion becomes a welfare-enhancing regime. Focusing on the convex relationship, R&D cooperation may reduce firms' interest to invest in R&D, and it is possible that firms in the Full Collusion regime produce most and generate the highest level of social welfare. Furthermore, horizontal mergers might be interpreted as a Full Collusion where the participants coordinate their decisions with respect to all of strategic variables. Thus, we launch the discussion about antitrust policy, and shed light on the leniency of the total welfare standard and the restrictiveness of the consumer welfare standard.
The rest of this paper is organized as follows. Section 2 presents the model and solves the SPNE in the four alternative regimes. We compare R&D effort, profit, consumer surplus and social welfare according to firms' behavior (competitive or collusive) in Sect. 3. Section 4 concludes this paper.
2 The model
2.1 Hypothesis
Consider an industry with two firms producing imperfectly substitutable goods. The representative consumer has a quasi-linear utility function
$$U(q_i,q_j)=a(q_i+q_j)-\frac{1}{2}(q_i^2+q_j^2+2 \gamma q_i q_j)$$
where "\(q_i\)" is the output of firm \(i\); "\(a\)" is a constant which is assumed to be sufficiently large so that all firms product positive amounts in equilibrium; "\(\gamma\)" measures the substitutability5 between the products and \(\gamma \in [0,1)\). The utility function generates the following inverse demand function faced by firm \(i\):
$$p_i(q_i,q_j)=a-q_i-\gamma q_j$$
The production technology exhibits a constant marginal cost "\(c\)" which can be reduced by investing in R&D. Due to spillovers (\(\beta\)), the R&D effort not only leads to a decrease in its own marginal cost, but also reduces the marginal cost of the rival firm. Given the R&D effort \(x_j\) of firm \(j\) (\(j=1,2\) and \(i \ne j\)), firm \(i\)'s effective marginal cost is
$$C_i(x_i,x_j)=c-x_i-\beta x_j$$
The R&D cost is assumed to be quadratic (\(\frac{1}{2} x_i^2\)), which reflects the decreasing returns to R&D effort.
The individual profit of firm \(i\) is defined by
$$\pi _i=\left( p_i( q_i, q_j) - C_i(x_i, x_j) \right) q_i-\frac{1}{2}x_i^2 \quad {\text{with}} \; i\ne j; i,j=1,2$$
The social welfare is the sum of producer surplus (denoted by \(PS\)) and consumer surplus (denoted by \(CS\)):
$$W=PS+CS \quad {\text {with}} \; PS =\pi _i+\pi _j, \, CS = U-p_i q_i-p_j q_j$$
The key feature of the model is to consider that the extent of product substitutability (\(\gamma\)) determines the ability of a firm to appropriate its rival' R&D effort. When products are less differentiated, competing firms share closer technological spaces, and one firm can benefit more from the rival's effort. We assume that the relationship between the spillover parameter (\(\beta\)) and the degree of product differentiation (\(\gamma\)) is described by:
$$\beta (\gamma ,h)=\gamma ^h \quad {\text {with}} \; h \in \left[ \frac{1}{2}, \frac{3}{2}\right] , \, \gamma \in [0, 1)$$
where the parameter "\(h\)" determines both the sensibility of the R&D spillover to the degree of product differentiation, in other words, the measure of technological distance 6, and the level of spillovers for a given value of differentiation (see Fig. 1). The assumption \(h \in [\frac{1}{2}, \frac{3}{2}]\) is necessary to guarantee the equilibrium existence in the four alternative scenarios. The range of \(h\) permits us to touch upon the issue of concavity (technological proximity, \(h<1\)) and convexity (technological alienation, \(h>1\)). As \(\frac{\partial \beta }{\partial \gamma }>0\) and \(\frac{\partial \beta }{\partial h}<0\), we incur that, for any given value of \(\gamma\), the concave relationship implies a more important spillover effect than the convex relationship. From the perspective of technological distance, the concavity refers to the situations where firms adopt similar technologies. Under concavity condition, the more differentiated are the products (close to 0), R&D spillovers are more sensitive to \(\gamma\). One can imagine that the concavity (\(h < 1\)) corresponds to industries that are geographically concentrated and that rely upon sources of basic scientific knowledge or general purpose technologies (GPT) in the cluster7 benefit most from the exchange of knowledge and technology. By contrast, under convexity condition, the less differentiated are products (close to 1), the more sensitive R&D spillovers with respect to \(\gamma\), and the convexity delineates the situations where firms adopt different technologies.
R&D spillovers and product differentiation. Source own graphic
We consider a two-stage game where firms act simultaneously at each stage. Firms select a strategic action (R&D effort) at the first stage anticipating correctly its impact at the second stage. The two competing firms can either coordinate their decisions or adopt non-cooperative strategy at each stage. When firms collude in one dimension (R&D or production) and compete in another one, such behavior is called semi-collusion (Fershtman and Gandal 1994). We compare the SPNE emerging in the four alternative scenarios (Table 1) such as Full Competition, Semi-collusion in Production, Semi-collusion in R&D and Full Collusion.
Four alternative scenarios. Source own table
Four alternative scenarios
First stage (R&D)
Seconde stage (production)
Full competition (regime F)
Firms compete in R&D; each firm decides its own R&D level given R&D efforts of the other firm
Firms compete; each firm decides its own output to maximize the individual profit
Semi-collusion in Production (Production Cartel) (regime P)
Firms coordinate their production activities to maximize the joint profit
Semi-collusion in R&D (R&D Cartel) (regime R)
Firms coordinate their R&D activities to maximize the joint profit; cooperative behavior in R&D does not change the level of spillovers
Full collusion (Horizontal Merger) (regime M)
2.2 Sub-game equilibrium in the four regimes
2.2.1 Full competition
We begin with regime \(F\), where there is no cooperation in any of the stages. The SPNE is obtained by backward induction. Firm \(i\) chooses output \(q_i\) to maximize individual profit \(\pi _i\), and the firm \(i\)'s output as a function of R&D efforts is given by:
$$\begin{aligned} q_i^F\left( x_i^F,x_j^F\right) =\frac{A(2-\gamma )+(2-\gamma ^{h+1})x_i^F+(2 \gamma ^h-\gamma )x_j^F}{4-\gamma ^2} \quad {\text{with}} \, A=a-c>0 \end{aligned}$$
The sign of the derivative \(\frac{\partial q_i^F(x_i^F,x_j^F)}{\partial x_i^F }\) is unambiguously positive, it demonstrates that the output of firm \(i\) increases with its own R&D effort. By contrast, concerning the sign of \(\frac{\partial q_i^F(x_i^F,x_j^F)}{\partial x_j^F }\), we have
\(\frac{\partial q_i^F\left( x_i^F,x_j^F\right) }{\partial x_j^F } < 0\), if \(h> 1+ \frac{\log (\frac{1}{2})}{\log \gamma }\)
\(\frac{\partial q_i^F\left( x_i^F,x_j^F\right) }{\partial x_j^F } > 0\), otherwise
When the technological distance is large enough (\(h> 1+ \frac{\log (\frac{1}{2})}{\log \gamma }\)), the technologies adopted by firms are very different, one firm's production will be negatively affected by its rival's R&D investment. By substituting Eq. (7) into the profit function Eq. (4), we can rewrite the profit function as \(\pi _i^F(x_i^F,x_j^F)\). In the first stage, each firm chooses R&D effort independently to maximize the individual profit. The SPNE of per-firm R&D effort, output, profit and social welfare is given by:
$$\begin{aligned} x^F= \frac{2A(2-\gamma ^{h+1})}{\Psi _F}, \quad q^F= \frac{A(2-\gamma )(2+\gamma )}{\Psi _F} \end{aligned}$$
$$\begin{aligned} \pi ^F= \frac{A^2 \Xi _F}{\Psi _F^2}, \quad W^F = \frac{A^2 \Omega _F}{\Psi _F^2} \end{aligned}$$
$$\begin{aligned} \Psi _F&= (4\gamma +8-\gamma ^3-2\gamma ^2)+2(\gamma ^{2h+1}+\gamma ^{h+1}-2\gamma ^h-2) > 0 \\ \Xi _F&= (\gamma ^2-4)^2-2(\gamma ^{h+1}-2)^2 > 0\\ \Omega _F&= (48+16\gamma -24\gamma ^2-8\gamma ^3+3\gamma ^4+\gamma ^5)-4(\gamma ^{h+1}-2)^2 > 0 \end{aligned}$$
2.2.2 Semi-collusion in production
Semi-collusion in Production is denoted by \(P\), firms choose their R&D efforts non-cooperatively, but select their outputs cooperatively. Firm \(i\)'s output, as a function of R&D effort, can be expressed as:
$$\begin{aligned} q_i^P(x_i^P,x_j^P)=\frac{A(1-\gamma )+(1-\gamma ^{h+1})x_i^P+(\gamma ^h-\gamma )x_j^P}{2(1-\gamma ^2)} \end{aligned}$$
The derivative \(\frac{\partial q_i^P(x_i^P,x_j^P)}{\partial x_i^P }\) is always positive, and \(\frac{\partial q_i^P(x_i^P,x_j^P)}{\partial x_j^P }\) is positive when \(h<1\) (concave relationship); negative while \(h>1\)(convex relationship).
The SPNE:
$$\begin{aligned} x^P= \frac{A(2-\gamma ^{h+1}-\gamma )}{\Psi _P},\quad q^P = \frac{2A(1-\gamma )}{\Psi _P} \end{aligned}$$
$$\begin{aligned} \pi ^P= \frac{A^2 \Xi _P}{2\Psi _P^2}, \quad W^P = \frac{A^2 \Omega _P}{\Psi _P^2} \end{aligned}$$
$$\begin{aligned} \Psi _P&= 4 (1-\gamma ^2)+\gamma ^h(2\gamma +\gamma ^{h+1}-2)+\gamma -2 > 0 \\ \Xi _P&= 8(\gamma ^3-\gamma ^2-\gamma +1)-(\gamma ^{h+1}-2)^2+4\gamma -\gamma ^2-2\gamma ^{h+2} > 0\\ \Omega _P&= 12(\gamma ^3-\gamma ^2-\gamma +1)-(\gamma ^{h+1}-2)^2+4\gamma -\gamma ^2-2\gamma ^{h+2} > 0 \end{aligned}$$
2.2.3 Semi-collusion in R&D
Firms coordinate their R&D investment in the R&D stage, and then maintain competition in the production stage. This regime is abbreviated by \(R\)
$$\begin{aligned} x^R= \frac{2A(1+\gamma ^h)}{\Psi _R}, \quad q^R = \frac{A(2+\gamma )}{\Psi _R} \end{aligned}$$
$$\begin{aligned} \pi ^R= \frac{A^2 }{\Psi _R}, \quad W^R = \frac{A^2 \Omega _R}{\Psi _R^3} \end{aligned}$$
$$\begin{aligned} \Psi _R&= (\gamma +2)^2-2(\gamma ^h+1)^2 >0 \\ \Omega _R& {}= (\gamma ^5+11\gamma ^4+46\gamma ^3+86\gamma ^2+64\gamma +16) +8 (\gamma ^{4h}+4\gamma ^{3h}+6\gamma ^{2h}+4\gamma ^h) \\&\quad-(40\gamma ^{2h}+80\gamma ^h+2\gamma ^{2h+3}+96\gamma ^{h+1}+48\gamma ^{2h+1}+36\gamma ^{h+2}+18\gamma ^{2h+2}+4\gamma ^{h+3}) \\& \quad > 0 \end{aligned}$$
2.2.4 Full collusion (horizontal merger)
Despite the ostensibly widespread use of Full Collusion to exploit the complementarities in firm's R&D process, the formal literature on R&D has almost focus exclusively on research joint venture, whereby firms share out technological knowledge (\(\beta =1\)) while continuing to compete against each other in product market (see Kamien et al. 1992).8 Here, we regard this scenario as the framework of multi-dimensional coordination in which firms cooperate in both R&D and production stages. Since, the products are imperfectly substitutable, Full Collusion9 means that the firms maximize their joint profit in each stage.
The SPNE of R&D effort, output, profit and welfare is given by
$$\begin{aligned} x^M= \frac{A(1+\gamma ^h)}{\Psi _M}, \quad q^M = \frac{A}{\Psi _M} \end{aligned}$$
$$\begin{aligned} \pi ^M= \frac{A^2}{2\Psi _M^2}, \quad W^M = \frac{A^2 \Omega _M}{\Psi _M^2} \end{aligned}$$
$$\begin{aligned} \Psi _M& = 2 (1+\gamma )-(\gamma ^h+1)^2 > 0 \\ \Omega _M&= 3 (1+\gamma )-(\gamma ^h+1)^2 = \Psi _M+(1+\gamma ) > 0 \end{aligned}$$
In the following section, we will compare these four aforementioned regimes in terms of significative relevance such as R&D investment, profit, consumer surplus and social welfare.
3 Comparison of different regimes
3.1 R&D effort
We start with the comparison of R&D investment level and address the question: which regime generates the highest level of R&D effort in equilibrium? To compare individual levels of R&D under different regimes, let us define the functions \(f_k(\gamma ,h)\), \(g_k(\gamma ,h)\) and \(j_F(\gamma ,h)\)
$$\begin{aligned} \left\{ \begin{array}{ll} f_k(\gamma ,h) = x^M(\gamma ,h)-x^k(\gamma ,h) \quad {\text {with}} \; k = \{F, P, R \} \\ g_k(\gamma ,h) = x^R(\gamma ,h)-x^k(\gamma ,h) \quad {\text {with}} \; k = \{F, P \} \\ j_F(\gamma ,h) = x^P(\gamma ,h)-x^F(\gamma ,h) & \\ \end{array}\right. \end{aligned}$$
We plot the curves \(f_k(\gamma ,h)=0\), \(g_k(\gamma ,h)=0\), \(j_F(\gamma ,h)=0\) in \(\gamma\) and \(h\) space and this pattern implies the ranking of R&D efforts into five zones (Fig. 2).
R&D investment ranking. Source own graphic
When firms have same behavior in the upstream stage, the downstream cooperation can incite firms to exert more R&D investment.
When firms adopt different technologies and produce differentiated goods (cf. Fig. 2, green area), the firms colluding in production will invest most in R&D.
Under technological proximity, firms with two-stage cooperation have most incentive to invest in R&D without ambiguity.
Based on Fig. 2, the R&D efforts (equilibrium) in the different regimes are arranged in the following form
\(x^P > x^F > x^M > x^R\) (zone I)
\(x^P > x^M > x^F > x^R\) (zone II)
\(x^P > x^M > x^R > x^F\) (zone III)
\(x^M > x^P > x^R > x^F\) (zone IV)
\(x^M > x^R > x^P > x^F\) (zone V)
\(\square\)
First of all, we find when firms have same behavior (cooperation or competition) in the upstream R&D stage, firms allowed to cooperate in the product market always exert more R&D efforts in equilibrium, compared to firms competing in the downstream stage (\(x^M > x^R\) and \(x^P > x^F\) \(\forall \ \gamma , h\)). As we know, R&D efforts reduce the marginal cost and indirectly lead to a decrease of the product price. When firms can collude in the downstream stage, they restrict their outputs for a given R&D effort and as a consequence, the negative impact of R&D efforts on the product price is alleviated. Conversely, an intense product competition dissipates the benefits of R&D effort and, therefore, shrinks the incentive to invest in R&D. The output cooperation has a positive impact on R&D investment and then induces firms to undertake more R&D than they would under competition in the downstream stage.
The output cooperation reinforces the R&D effort for a given behavior at upstream stage. However, when the behavior at downstream stage is given, the R&D cooperation does not unambiguously increase research efforts. If we compare the regime \(F\) with the regime \(R\) (corresponding, respectively, to the lowest level in terms of R&D effort), it is found that R&D cooperation could be detrimental to R&D effort in zone I and zone II. This finding is in sharp contrast with the existing literature, for instance, Kamien et al. (1992) show that \(x^R\) is unambiguously greater than \(x^F\) without taking into account the close relationship emphasized in this paper.
The striking outcome we find here is that R&D investment under regime \(P\) can be the largest (cf. Fig. 2, green area). It is different from the conventional wisdom that merged (two-stage cooperation) firms have more incentive to invest in R&D, because they appropriate all of the R&D efforts. The spillover effect (in zones I,II and III) constitutes a positive, but very small externality. When firms cooperate in the upstream stage (regimes M, R), on the one hand this small externality is internalized, on the other hand the R&D cooperation cannot promote the spending on common research of firms due to technological alienation (convexity). However, Semi-collusion in Production can intensify the R&D competition by production cooperation, and incites firms to invest more in R&D. Therefore, the regime P leads to the highest level of R&D effort in green area. Moreover, if we restrict our attention to the case where the relationship between product differentiation and R&D spillover is concave (red area), the ranking of R&D efforts (\(x^M > x^R > x^P > x^F\)) does not alter, and it is independent of the product differentiation. It means that the Full Collusion participants spend more on R&D than Semi-collusion ones, under concave relationship (technological proximity).
From the aggregate surplus point of view, the welfare performance of R&D investment in the different scenarios can be gauged, and we compare them with the First-Best welfare criterion (Suzumura 1992). "Appendix 1" provides the proof of the expression \(x^{\mathrm{FB}}.\)
$$\begin{aligned} x^{\mathrm{FB}}=\frac{A(1+\gamma ^h)}{(1+\gamma )-(1+\gamma ^h)^2} \end{aligned}$$
Obviously, \(x^{\mathrm{FB}}\) is the significant standard accessing whether the R&D investment is efficient, when the denominator \((1+\gamma )-(1+\gamma ^h)^2\) is positive.
Socially first-best R&D. Source own graphic
In Fig. 3, we plot the curves in \(\gamma \in [0,\frac{1}{4}]\) and \(h \in [1, \frac{3}{2}]\) space to zoom and emphasize the area \(x^{\mathrm{FB}}>\max \{x^F, x^R, x^P, x^M\}\). The smooth curve \(x^{\mathrm{FB}}=0\) divides the pattern into two parts and the left one represents \(x^{\mathrm{FB}}>0\). The intersection area between the smooth curve and the zigzag curve (\(x^{\mathrm{FB}}=x^P\)) defines combinations of \(\gamma\) and \(h\) where \(x^P > x^{\mathrm{FB}}\). D'Aspremont and Jacquemin (1988) and Henriques (1990) show that the social optimum R&D effort was unambiguously greater than the level of R&D investment in equilibrium under the fully cooperative or non-cooperative or mixed10 game. Compared to them, we find the similar result when firms produce sufficiently heterogeneous goods. Furthermore, it is worthwhile to note \(x^P\) can be higher than \(x^{\mathrm{FB}}\) in an infinitesimal area where a higher level of R&D effort corresponds to a wasteful duplication.
3.2 Output and consumer surplus
Due to symmetric equilibria, output is considered as an index of consumer surplus \(({\text{CS}}^k=(1+\gamma )(q^k)^2\) with \(k=\{ F, P, R, M \})\). We trace out the meaningful areas by plotting the following curves:
$$\begin{aligned} \left\{ \begin{array}{ll} R_k(\gamma ,h) = q^M(\gamma ,h)-q^k(\gamma ,h) \quad {\text {with}} \; k = \{F, P, R \} \\ V_k(\gamma ,h) = q^R(\gamma ,h)-q^k(\gamma ,h) \quad {\text {with}} \; k = \{F, P \} \\ Z_F(\gamma ,h) = q^P(\gamma ,h)-q^F(\gamma ,h) \end{array}\right. \end{aligned}$$
The output (consumer surplus) ranking. Source own graphic
The level of output and consumer welfare in fully cooperative scenario can be higher (cf. Fig. 4, green area) than that in partially cooperative or fully non-cooperative situations.
When firms adopt similar technologies (concavity), R&D cooperation (regimes M and R) encourages firms to produce more, and leads to fierce output competition.
Firms under Full Competition can produce most and achieve the highest level of consumer welfare (red area), if and only if they use very different technologies and produce highly differentiated goods.
Based on Fig. 4, the individual output equilibrium in the different regimes is arranged in the following form
\(q^F > q^R > q^P > q^M\) (zone I)
\(q^R > q^F > q^P > q^M\) (zone II)
\(q^R > q^F > q^M > q^P\) (zone III)
\(q^R > q^M > q^F > q^P\) (zone IV)
\(q^R > q^M > q^P > q^F\) (zone V)
\(q^M > q^R > q^F > q^P\) (zone VI)
\(q^M > q^R > q^P > q^F\) (zone VII)
Apart from Semi-collusion in Production, each regime can yield the highest level of output (consumer surplus) for plausible parameter combinations. When firms produce sufficiently similar goods, the Full Collusion regime ensures the highest level (green area). This finding is in contrast with the traditional literature "the firms under Full Competition always produce more than the firms under Full Collusion scenarios"11. The reason behind this is the substitutability–spillover relationship: the low level of differentiation on the one hand generates the high level of spillovers, on the other hand, it induces firms under Full Collusion to spend more on R&D (Result 1), accordingly the marginal cost of Full Collusion participants is sufficiently reduced, firms under Full Collusion have interest to expand their output. We also find that the output level is the highest in the regime \(R\) when the goods are sufficiently differentiated (zones II,III,IV,V). Furthermore, if the sensibility parameter \(h\) is comparatively large (technological alienation), Full Competition generates the highest output level (zone I). The reason of the instable relationship between \(q^R\) and \(q^M\) arises from the sensibility of output to R&D effort: in the symmetric equilibria, the sensibility under regime \(R\) and \(M\) is, respectively, given by
$$\begin{aligned} \frac{\partial q^R}{\partial x^R}= \frac{2-\gamma ^{h+1}+2\gamma ^h-\gamma }{4-\gamma ^2} > 0 \end{aligned}$$
$$\begin{aligned} \frac{\partial q^M}{\partial x^M}= \frac{1-\gamma ^{h+1}+\gamma ^h-\gamma }{2(1-\gamma ^2)} > 0 \end{aligned}$$
It is found that \(\frac{\partial q^R}{\partial x^R} > \frac{\partial q^M}{\partial x^M}\), this inequality discloses that the output in the regime \(R\) is more sensitive to R&D effort compared to the one in the regime \(M\). In addition, \(x^M > x^R\) holds true at all time (Result 1). Indeed, \(q^M\) can be greater than \(q^R\) in some zones (VI and VII).
There is no stable hierarchy because the impact of R&D effort is complicated and exerts two conflicting effects on the output of rival firm. On the one hand, R&D effort is managed to induce the firm to expand output at expense of its rival by cutting down its own production cost. It is considered as the substitutability effect (an increase in its own output leads to a decrease in rival's output) which is greater, the more substitutable the products are. On the other hand, the R&D effort can reduce the rival firm's cost, thereby increase its rival firm's output. It is regarded as the spillover effect (boosting rival's output) which is greater the larger the spillover is. Since the spillover depends positively on the degree of product differentiation, when products are quasi homogeneous, both substitutability effect and spillover effect enlarge. Whether the output (consumer surplus) increases depends on the interplay of these two conflicting effects. If the spillover effect prevails over the substitutability effect, firms are motivated to expand output; otherwise, they prefer to shrink output.
According to Fig. 4, it is clear that firms colluding in R&D produce more than firms competing on R&D (\(q^R, q^M > q^P, q^F\)) when the relationship between substitutability and spillover is concave (\(h<1\)). This result holds always true regardless of product differentiation. Under the circumstance that the leakage of know-how is relatively strong (concave relationship), firms cooperating on R&D are willing to spend more on R&D efforts (Result 1), the marginal costs of both firms are reduced so much that the spillover effect prevails over the substitutability effect, and firms are motivated to expand output. The curve \(V_F=0\) is a watershed of the relationship between \(q^R\) and \(q^F\) which is consistent with the corollary shown in Kamien et al. (1992)12.
The relationship \(q^R > q^P\) holds true for all \(\gamma\) and \(h\). The intuition behind this stems from the variation of competition intensity13. Under regime \(R\), upstream collusion leads to much more fierce rivalry in non-cooperative output stage. Furthermore, since firms collude in output under regime \(P\), the market becomes looser, and the firms have more incentives to increase the price by reducing output.
We find also that the firms colluding in output produce less than the firms competing in production market when the goods are sufficiently differentiated (zones I, II, III). First, the downstream output cooperation induces firms to increase the price and decrease the output; second, as the low value for \(\gamma\) generates the small spillovers, the R&D efforts exerted by firm \(i\) cannot sufficiently reduce its rival 's marginal cost, this spillover effect is not strong enough to compensate the decrease in output due to production cooperation, therefore, firms have to shrink output.
3.3 Profit
According to Brod and Shivakumar (1999)14 (henceforth, "BS"), the profit under Full Competition could be greater than under Semi-collusion in Production in some cases. When there is the "close relationship" between product differentiation and R&D spillovers, we have the following result:
The firms in Full Collusion are most profitable while the firms in Full Competition are least profitable.
When firms adopt similar technologies (concavity), they prefer taking part in R&D Cartel to joining in Production Cartel (cf. Fig. 5, red area).
When firms adopt different technologies (convexity), the firms in Production Cartel could generate more profit than that in R&D Cartel (cf. green area \(\pi ^P > \pi ^R\) and white area \(\pi ^P < \pi ^R\)).
The equilibrium individual firm's profits are arranged:
$$\begin{aligned} \pi ^M &> \max {[\pi ^P,\pi ^R]} > \min {[\pi ^P,\pi ^R]} > \pi ^F \, \, \, \forall \, \gamma , h \\ \pi ^M &> \pi ^R > \pi ^P > \pi ^F \, \, \, \forall \, \gamma \quad {\text {if}} \, \, h < 1 \end{aligned}$$
When the spillover is relative to the product differentiation, the profit of the firms in the regime \(P\) always prevails over the one in the regime \(F\). This result is in contrast with Brod and Shivakumar (1999) which shows that the profit under regime \(F\) could be higher than under regime \(P\). Furthermore, in line with semi-collusion literature (Matsui 1989; Fershtman and Gandal 1994), we establish the possibility that R&D Cartel is less profitable than Production Cartel.
We find that the profit of firms with fully cooperative behavior prevails over one-dimension cooperation profit which is higher than the profit earned by the firm in Full Competition. It is only that the relationship between two types of semi-delegation can be altered. The alluring question is which type of semi-collusion (Production Cartel or R&D Cartel) will be more beneficial for firms.
Consider \(\Delta\) as the difference of profits in two semi-collusion scenarios:
$$\begin{aligned} \Delta =\pi ^P - \pi ^R \end{aligned}$$
We examine the profit ranking with the same method used in the previous subsection. The result is illustrated in Fig. 5. The interesting conclusion which emerges from this figure is that both Semi-collusion in R&D and Semi-collusion in Production can yield more profit.
Two types of semi-collusion. Source own graphic
Under concave relationship (technological proximity), firms colluding in R&D generate always more profit than firms colluding in output. The intuition of this result is the following: compared to the regime \(P\), the distinctive advantage of the regime \(R\) is that firms invest more in R&D under concave relationship (See Result 1), thereby, firms are more competitive due to cost-saving by R&D investment; furthermore, according to Result 2, firms in the regime \(R\) produce more than firms in the regime \(P\). Despite the fact that R&D investment is expensive, the profit of the firms in the regime \(R\) is still higher than that in the regime \(P\) when \(h < 1\).
The inverse outcome \(\pi ^P > \pi ^R\) can take place for some plausible \(\gamma\) under convexity condition (technological alienation). In particular, when \(h\) is approximately greater than the critical value which is equal to \(1.12\), \(\pi ^P>\pi ^R\) holds always true.
3.4 Social welfare
In general, the welfare is damaged by collusion: in one-stage game, the collusion always harms the welfare; whereas in two-stage game where firms first choose R&D efforts, collusion reduces welfare if it occurs in each of the two stages15. We determine which regime is the most relevant with regard to aggregate surplus (Fig. 6).
The welfare ranking. Source own graphic
Full Collusion can generate the highest level in social welfare, in particular when firms produce the similar goods (cf. Fig. 6, green area).
When firms produce the differentiated goods (red area), Semi-collusion in R&D enhances most the social welfare.
Based on Fig. 6, the social welfare ranking will be:
\(W^F > W^R > W^P > W^M\) (zone I)
\(W^R > W^F > W^P > W^M\) (zone II)
\(W^R > W^F > W^M > W^P\) (zone III)
\(W^R > W^M > W^F > W^P\) (zone IV)
\(W^R > W^M > W^P > W^F\) (zone V)
\(W^M > W^R > W^F > W^P\) (zone VI)
\(W^M > W^R > W^P > W^F\) (zone VII)
We highlight that the collusive behavior in both stages could enhance the welfare (zones VI, VII). If we consider the social welfare equilibrium level in the Full Competition regime as the criterion value, not only the Full Collusion regime but also Semi-collusion can improve the welfare. For example, the regime \(R\) is the welfare dominant regime when products are sufficiently differentiated. We find also under concavity condition, firms colluding in R&D regardless of their production strategy always enhance more social welfare than firms colluding in output independently of R&D strategy. Semi-collusion in Production can lead to a decrease in social welfare under convexity condition (zones I, II, III, IV).
Although the hierarchies in terms of welfare are the same as the ones concerning consumer surplus (output) which are depicted in Sect. 3.2 (Result 2), it is clear that there are some points of dissimilarity, such as the location of the different zones and the size of zones. In virtue of this dissimilarity, the discussion on antitrust policy is unsealed. In what follows, we focus on the difference between consumer welfare standard and total welfare standard.
3.5 Merger control: consumer welfare standard Vs total welfare standard
On the basis of Result 4, we conclude that society can benefit from not only the cooperative behavior in one dimension (Semi-collusion in R&D or in Production) but also from the horizontal merger (Full Collusion). Therefore, all regimes can yield the highest level of welfare for plausible parameter combinations.
Nowadays, most countries have laws or regulations that require competition authorities to scrutinize horizontal mergers. These authorities normally do not examine whether a particular merger is likely to affect welfare because it substantially lessens competition (USA) or significantly impedes effective competition (European Union). The US or EU applies a consumer welfare criteria to mergers. Canada, Australia and New Zealand, however, consider a merger's effects on aggregate surplus and had a very explicit aggregate surplus standard (Motta 2004).
Consequently, we make use of both total welfare standard and consumer welfare standard within our framework, to analyze the difference between two above-mentioned criteria, to examine whether the merger prohibited under aggregate welfare standard can be authorized under consumer welfare standard or inversely.
From the perspective of competition policy, the regimes Full Competition and Semi-collusion in R&D are considered as benchmarks. The competition authorities authorize the merger satisfying the following condition using total welfare standard:
$$\begin{aligned} W^M > {\mathrm{max}} \{ W^F, W^R \} \end{aligned}$$
using consumer welfare standard:
$$\begin{aligned} {\mathrm{CS}}^M > {\mathrm{max}} \{ {\mathrm{CS}}^F, {\mathrm{CS}}^R \} \end{aligned}$$
Total welfare standard Vs consumer welfare standard. Source own graphic
In Fig. 7, on the right side of curve Consumer Welfare Standard, the horizontal merger is accepted by consumer welfare standard. Total welfare standard authorizes the merger when the beach of parameter combination locates to the right of the curve named Total Welfare Standard. It is straightforward that there is the gap (dashed area) between two mentioned curves which sheds light on the looseness of the total welfare standard and the preciseness of the consumer surplus standard. Due to the prohibition by competition authorities, in the left side, the firms have to lean to the less attracting regimes which yield less profit compared to merger one. Therefore, the firms prefer the Semi-collusion in R&D (semi-collusion16) in the prohibited merger zone (\(\pi ^R > \pi ^F\)).
4 Concluding remarks
In the traditional one-dimensional framework, collusion increases producer profits, but damages consumer welfare without ambiguity (Textbook17 view). However, this argument ignores the effects of other non-production activities, such as R&D. Recently, as shown in Revisionist18 view, within two-dimensional game, semi-collusion may be profitable and efficient (Brod and Shivakumar 1999) under some circumstances, while it can be unprofitable and inefficient. Previous works have shown whether producers and consumers would be better off under product market cooperation depends particularly on product differentiation and R&D spillovers.
This paper emphasizes the "close relationship" between product differentiation and spillovers, and studies the significative relevance in the scenarios where firms can either coordinate their decisions or adopt non-cooperative strategy (Full Competition, Full Collusion and Semi-collusion regimes) at each stage. Kamien et al. (1992) claim that the investment by firms engaged in the regime \(R\) is unambiguously greater than that in the regime \(F\) irrespective of spillovers. We demonstrate in fact which regime generates more R&D effort in equilibrium depends upon both the degree of product differentiation and the technological distance. If we restrict our attention to the concave relationship, the ranking of R&D efforts is unalterable and independent of the product differentiation, competitions at the upstream stage depress R&D investment. Firms colluding in R&D regardless of their production strategy always yield more profit and generate higher social welfare than firms colluding in output independently of R&D strategy. When products are close substitutes, Full Collusion is a welfare-enhancing regime.
In addition, a discussion about antitrust policy is carried out. By focusing upon the distinctness of different antitrust criteria, this framework sheds light on the looseness of the total welfare standard and the preciseness of the consumer welfare standard. This outcome will be verified, in future work, by considering the interaction between Competition Authorities and firms, in a context of asymmetric information19.
There are some possible extensions of this framework: first, we will check the robustness of the result obtained in this paper, when there would be more than two firms in the market; second, we will investigate whether our model can get the similar results within a dynamic20 duopoly game, by supposing the R&D investments for cost-reducing innovation over continuous time; third, the parameter of spillover depends only on the degree of product differentiation in this model, however, the government can control the parameter of spillover using the intellectual property right policy, and it is an important extension of this model to enrich the policy implication; fourth, the degree of product differentiation is exogenously given in our model, however, firms have strategic incentives to control to maximize their profit, and it is better to consider the case that the degree of product differentiation is determined endogenously.
The greater the distance between firms, the more differentiated the firms' products, the less the R&D spillover.
It is also called "Production Cartel", see Brod and Shivakumar (1999).
R&D Cartel.
The Full Collusion regime could also be considered as horizontal merger.
If \(\gamma =0\), firms' products are not substitutable and each firm acts as a monopolist. Note that, when products are perfect substitutes, the spillover obviously equals to 1 and the game cannot be solved. See D'Aspremont and Jacquemin (1988).
From the perspective of technological distance, it is straightforward that the more technologies are similar, the greater are spillovers, for a given level of product differentiation.
See more in Audretsch and Feldman (1996), Baptista and Swann (1998).
Kamien et al. (1992) provide a thorough analysis of RJV, contrasting the case of RJV Competition where firms pool R&D results, but behave non-cooperatively at both stages, and RJV Cartelization (the pooling of R&D results with cooperative determination of R&D investment, but competition in subsequent product market stage). Suzumura (1992) and Suzumura and Yanagawa (1993) contain a closely related analysis. D'Aspremont and Jacquemin (1988) do allow for merger under which firms pool R&D results and cooperate in both stage of the game. It is worth noting that there are the analysis of the converse case to RJV, where all firms compete in R&D stage, but then collude in outputs, see Fershtman and Gandal (1994) and Brod and Shivakumar (1999).
The Full Collusion regime could be considered as horizontal merger.
Firms cooperate in R&D, but remain non-cooperative in output. This game corresponds to the Semi-collusion in R&D within our framework.
D'Aspremont and Jacquemin (1988) and Henriques (1990) demonstrate the level of output in non-cooperative two-stage case is always higher than that in fully cooperative situation. In addition, they claim that the mixed game can generate more output than non-cooperative two-stage game for large spillovers. These models based on the assumption of homogenous goods.
They demonstrate the price (output) in R&D cartelization is less (more) than the price in R&D competition if and only if \(\gamma \le 2 \beta\).
See Fershtman and Gandal (1994).
In an one-stage game, cartels increase industry profits and exacerbate the consumer surplus. In a model where firms collude in production, but compete in R&D, the cartel members may be worse off and consumers better off due to over-investment by firms eager to improve their position in the cartel. Brod and Shivakumar (1999) analyze a two-stage model and examine the effect of semi-collusion when the non-production activity is R&D. Firms choose their R&D effort in a first stage and output in a second stage. They shed light on the fact that in the presence of spillovers, firms and consumers could be both better off, peradventure both worse off, by a semi-collusive production cartel. We are attired by this fascinating outcome. Thereupon, we try to approach the in-depth analysis and understand the driving forces of this result. We find, however, that the findings of Brod and Shivakumar (1999) are disputable. The incorrect SPNE values of per-firm R&D effort, output and profit due to improper handling result in the inaccuracy of their main propositions. When the goods are sufficiently substitutable, the proposition 1 does not hold. In other words, there is no absolute predominance of production cartel in terms of R&D effort. Since the optimum equilibrium of cartel at the production stage could be negative for certain combination parameters (the degree of product differentiation and the level of spillovers), we find the region D depicted as "Consumers prefer Production Cartel; firms prefer Competition" could not always satisfy the conditions mentioned in proposition 2. In "Appendix 2", we focus upon their calculative errors, and show what the correct solution can be.
See D'Aspremont and Jacquemin (1988), Suzumura (1992).
Note that in reality, the Production Cartel is prohibited. Thus, we exclude it in antitrust control analysis.
The textbook view: while the firms benefit from product market collusion, consumer welfare is higher under non-cooperation in the product market. See more in Jacquemin and Slade (1989).
The revisionist view: if the firms have the options for non-production activities, such as R&D, before production, producers can be worse off and consumers can be better off. See more in Matsui (1989), Mitchell (1993) and Fershtman and Gandal (1994).
See more in Besanko and Spulber (1989), Pénard and Souam (2002).
See more in Cellini and Lambertini (2009).
\(\Phi -\Phi _{BS}=-4b\delta (1-\gamma ^2)\gamma < 0\).
I am grateful to Nicolas Le Pape, Thierry Penard and Bernard Franck for thoughtful comments and suggestions. I also thank Said Souam and Jean-Pascal Gayant. We have received helpful comments on earlier drafts of this article from participants at the ESEM, AFSE and seminar participants at GAINS. This work is supported by the Research Funds of Huaqiao University (HQHRZD2014-03). I am grateful for the very painstaking efforts made by the editor and the referee in providing me with very valuable suggestions and comments. All remaining errors are mine.
The author declares that he has no competing interests.
Open AccessThis 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.
Appendix 1: First-best
The social optimum R&D effort derived from the first-best function welfare:
$$\begin{aligned} W(x_i,x_j,q_i,q_j)= & {} \sum _{i=1}^{2} \pi _i(x_i,x_j,q_i,q_j) + u(x_i,x_j,q_i,q_j) \\&-\sum _{i=1}^{2} p_i(x_i,x_j,q_i,q_j) \ q_i(x_i,x_j,q_i,q_j) \end{aligned}$$
By backward induction, \(q^{\mathrm{FB}}(x_i,x_j)\) is the socially First-Best output profile corresponding to \(x_i\) and \(x_j\). It is achieved by:
$$\begin{aligned} q^{\mathrm{FB}}(x_i,x_j) \equiv {\rm argmax} _{q>0} W(x_i,x_j,q_i,q_j) \end{aligned}$$
Then, the first-best welfare function \(W^{\mathrm{FB}}\) is defined by:
$$\begin{aligned} W^{\mathrm{FB}}(x_i,x_j) \equiv W^{\mathrm{FB}}\big (x_i,x_j,q^{\mathrm{FB}}(x_i,x_j)\big ) \end{aligned}$$
$$\begin{aligned} x^{\mathrm{FB}}&\equiv {\mathrm{argmax}}_{x>0} W^{\mathrm{FB}}(x_i,x_j) \\&= \frac{A(1+\gamma ^h)}{(1+\gamma )-(1+\gamma ^h)^2} \end{aligned}$$
Appendix 2: Review of Brod and Shivakumar (1999)
There are two regimes: the one is Competition where firms compete in both the R&D and the output markets; the other one is Production Cartel where the firms compete in the R&D market, but collude in output market. The superscript "C" stands for Competition and "P" signifies Production Cartel.
The game is solved by backward induction and we characterize the equilibrium outcomes of this game.
The SPNE values of per-firm R&D effort, output and profit are given by:
$$\begin{aligned} x^C&= \frac{2A}{\theta }(2-\beta \gamma ) \\ q^C&= \frac{\delta A}{\theta }(2-\gamma )(2+\gamma )\\ \pi ^C&= \frac{\delta A^2 \Delta }{\theta ^2} \end{aligned}$$
where \(A=a-c\), \(\theta =(2-\gamma )(2+\gamma )^2b\delta -2(1+\beta )(2-\beta \gamma ) > 0\) and \(\Delta =(2-\gamma )^2(2+\gamma )^2b \delta -2(2-\beta \gamma )^2 > 0.\)
In the paper of Brod and Shivakumar (1999), the expression of \(\Delta\) displayed in page 225 is, however, \(\Delta _{\rm BS}=(2-\gamma )^2(2+\gamma )^2b\delta -2(1+\beta )(2-\beta \gamma )^2 > 0\). We have \(\Delta -\Delta _{\rm BS}=2 \beta (2-\beta \gamma )^2 >0\) that generates the underestimate of the real profit.
Production cartel
The symmetric equilibrium of R&D effort, output and profit corresponds to the following solutions:
$$\begin{aligned} x^P&= \frac{A}{\Phi }(2-(1+\beta )\gamma ) \\ q^P&= \frac{2 \delta A}{\Phi }(1-\gamma ) \\ \pi ^P&= \frac{\delta A^2 \Gamma }{2\Phi ^2} \end{aligned}$$
where \(\Phi =\gamma +\beta ^2 \gamma + 4 b \delta (1-\gamma ^2)-2\beta (1-\gamma )-2\) and \(\Gamma =-4+8b\delta +8b\delta \gamma ^3+4\gamma (1+\beta -2b\delta )-\gamma ^2(1+2\beta +\beta ^2+8b\delta )\). As mentioned in BS, the product \(b \delta\) can be expressed in the same units as output, they assume \(b \delta =1\) to simplify expressions. We find whether these two expressions(\(\Phi\),\(\Gamma\)) are positive or not depends on the combination of parameters \(\gamma\) and \(\beta\).
Whereas, BS consider that \(\Phi _{\mathrm{BS}}=4(1-\gamma )(1+\gamma )^2b\delta -(1+\beta )(2-(1+\beta ) \gamma )>0\) and \(\Gamma _{\mathrm{BS}}=8(1-\gamma )^2b\delta -(2-(1+\beta )\gamma )>0\). Compared to our results, we have \(\Phi -\Phi _{\mathrm{BS}}=-4b\delta (1-\gamma ^2)\gamma < 0\). It is clear that there is the underestimate on R&D effort and output. These errors due to improper handling generate the distinctive change in the following analysis. Furthermore, BS regard mistakenly \(\Phi _{\mathrm{BS}}\) and \(\Gamma _{\mathrm{BS}}\) as the positive terms. Taking \(\Phi _{\mathrm{BS}}\) as an example, we illustrate here \(\Phi _{\mathrm{BS}}\) is negative when
\(\gamma \in (0.927441 , 0.927886]\) and \(\beta \in (\tilde{\beta _1}, \tilde{\beta _2} )\)
\(\gamma \in [0.927886, 1]\) and \(\beta \in (0, \tilde{\beta _2})\)
with \(\tilde{\beta _1}=\frac{1-\gamma }{\gamma }-\sqrt{ \frac{1-4\gamma -4\gamma ^2+4\gamma ^3+4\gamma ^4}{\gamma ^2} }\) and \(\tilde{\beta _2}=\frac{1-\gamma }{\gamma }+\sqrt{ \frac{1-4\gamma -4\gamma ^2+4\gamma ^3+4\gamma ^4}{\gamma ^2} }.\)
A reappraisal of the main propositions in Brod and Shivakumar (1999).
Since \(\Phi _{BS}>0\), BS claimed the R&D effort in regime Production Cartel is always significant, the firms colluding in output spared no effort to invest in R&D for \(0\le \beta \le 1\) and for all \(0 \le \gamma <1\). In fact, their finding is not true, the crux of the matter is that the \(\Phi\) could be negative 21 in certain circumscription where the optimum equilibrium R&D effort is meaningless. We find that the member firm of cartel could have no interest in R&D processes when the goods are sufficiently homogenous, precisely \(\gamma \in (\hat{\gamma } , 1]\) with \(\hat{\gamma }=\frac{(1+\beta )^2+\sqrt{33-28\beta +6\beta ^2+4\beta ^3+\beta ^4}}{8}\). In this instance, the \(x^P\) will be inferior to \(x^C\), then the proposition 1 is not always true.
In addition, Brod and Shivakumar (1999) claimed that "it is easy to show that as \(\beta\) rises, the difference \(x^P-x^C\) declines" in page 226. As a matter of fact, the \(\frac{\partial (x^P-x^C)}{\partial \beta }\) could be positive. Whether this gap enlarges or shrinks depends upon the combination of two parameters \(\beta\) and \(\gamma\). To be more legible and intuitionistic, we illustrate this outcome with the following graphic.
The effect of \(\beta\) on the difference \(x^P-x^C\). Source own graphic
On the basis of Fig. 8, apart from the dashed zone which represents the flaw of their proposition 1, we have not only the region, corresponding to the finding of BS, in which the relative valuation of R&D is reduced as spillovers increase, but also the region where the gap enlarges following the rise of spillovers. The primary reason of omitting this positive aspect of \(\beta\) stems from the underestimate of R&D effort in regime P.
Brod and Shivakumar (1999) try to compare two mentioned regimes in terms of both individual and collective incentive. They consider output as an index of consumer surplus.
$$\begin{aligned} q^P-q^C= & {} \frac{2 \delta A}{\Phi }(1-\gamma )-\frac{\delta A}{\theta }(2-\gamma )(2+\gamma ) \\= & {} \frac{A\delta \left( 2(1-\gamma )\theta -(2-\gamma )(2+\gamma )\Phi \right) }{\Phi \theta } \end{aligned}$$
It is straightforward, \(q^P-q^C\) has the same sign as the following expression:
$$\begin{aligned} f(\gamma ,\beta )=\frac{2(1-\gamma )\theta -(2-\gamma )(2+\gamma )\Phi }{\Phi \theta }=\frac{f_{BS}(\gamma ,\beta )}{\Phi \theta } \end{aligned}$$
Due to improper handling and error of judgement about \(\Phi\), it is mistakenly deemed that the difference \(q^P-q^C\) has the same sign as the expression \(f_{BS}(\gamma ,\beta )=2(1-\gamma )\theta -(2-\gamma )(2+\gamma )\Phi =-2\gamma ^4+(\beta ^2+2\beta +3) \gamma ^3-2\gamma ^2(2\beta ^2+3\beta -3)-4\gamma (1-\beta )\) displayed in page 227. As the case stands, the difference \(q^P-q^C\) is also influenced by the denominator \(\Phi \theta\).
Concerning the difference of profit \(\pi ^P-\pi ^C\),
$$\begin{aligned} \pi ^P-\pi ^C= & {} \frac{\delta A^2 \Gamma }{2\Phi ^2}-\frac{\delta A^2 \Delta }{\theta ^2} \\= & {} \frac{A^2\delta (\Gamma \theta ^2-2\Delta \Phi ^2)}{2\Phi ^2\theta ^2} \\\ne & {} \frac{A^2\delta (\Gamma _{BS} \theta ^2-2\Delta _{BS}\Phi _{BS}^2)}{2\Phi _{BS}^2\theta ^2} \end{aligned}$$
it is straightforward that \(\pi ^P-\pi ^C\) has the same sign as
$$\begin{aligned} g(\gamma ,\beta )=\Gamma \theta ^2-2\Delta \Phi ^2 \ne \Gamma _{BS} \theta ^2-2\Delta _{BS}\Phi _{BS}^2 \end{aligned}$$
According to Fig. 2 in Brod and Shivakumar (1999) page 228, there are always \(q_\mathrm{BS}^P > q_\mathrm{BS}^C\) and \(\pi _\mathrm{BS}^P < \pi _\mathrm{BS}^C\) in region D. Practically, we can find the inverse outcome \(q^P < q^C\) even \(\pi ^P > \pi ^C\) in this region.
Institute for quantitative economics, Research center for the applied statistics and big data, Huaqiao University, 668, Jimei Avenue, 361021 Xiamen, China
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December 2019, 24(12): 6725-6743. doi: 10.3934/dcdsb.2019164
Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations
Zhuan Ye ,
Department of Mathematics and Statistics, Jiangsu Normal University, 101 Shanghai Road, Xuzhou 221116, Jiangsu, China
* Corresponding author: Zhuan Ye
Received July 2018 Revised February 2019 Published July 2019
Fund Project: The author is supported by the National Natural Science Foundation of China (No. 11701232) and the Natural Science Foundation of Jiangsu Province (No. BK20170224)
This paper addresses the Cauchy problem of the three-dimensional inhomogeneous incompressible micropolar equations. We prove the global existence and exponential decay-in-time of strong solution with vacuum over the whole space $ \mathbb{R}^{3} $ provided that the initial data are sufficiently small. The initial vacuum is allowed.
Keywords: Inhomogeneous micropolar equations, vacuum, global strong solution.
Mathematics Subject Classification: Primary: 35Q35, 35B65, 76N10; Secondary: 76D05.
Citation: Zhuan Ye. Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6725-6743. doi: 10.3934/dcdsb.2019164
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\begin{document}
\RestyleAlgo{ruled} \title{Variational Quantum Pulse Learning\\
}
\newcommand{\HW}[1]{\textcolor{blue}{[HW: #1]}}
\author{\IEEEauthorblockN{ Zhiding Liang*\textsuperscript{1} \ \ Hanrui Wang*\textsuperscript{2} \ \ Jinglei Cheng\textsuperscript{3} \ \ Yongshan Ding\textsuperscript{4} \ \ Hang Ren\textsuperscript{5} \ \ Zhengqi Gao\textsuperscript{2}\\ Zhirui Hu\textsuperscript{6} \ \ Duane S. Boning\textsuperscript{2} \ \ Xuehai Qian\textsuperscript{3} \ \ Song Han\textsuperscript{2} \ \ Weiwen Jiang\textsuperscript{6} \ \ Yiyu Shi\textsuperscript{1}} \IEEEauthorblockA{ \textsuperscript{1}University of Notre Dame, IN, USA. \textsuperscript{2}Massachusetts Institute of Technology, MA, USA.\\ \textsuperscript{3}University of Southern California, CA, USA. \textsuperscript{4} Yale University, CT, USA.\\ \textsuperscript{5}University of California, Berkeley, CA, USA. \textsuperscript{6}George Mason University, VA, USA.\\ *These authors contributed to the work equally and should be regarded as co-first authors.\\ Corresponding authors: [email protected], [email protected]
} }
\maketitle
\begin{abstract} Quantum computing is among the most promising emerging techniques to solve problems that are computationally intractable on classical hardware. A large body of existing works focus on using variational quantum algorithms on the gate level for machine learning tasks, such as the variational quantum circuit (VQC). However, VQC has limited flexibility and expressibility due to limited number of parameters, e.g. only one parameter can be trained in one rotation gate. On the other hand, we observe that quantum pulses are lower than quantum gates in the stack of quantum computing and offers more control parameters. Inspired by the promising performance of VQC, in this paper we propose variational quantum pulses (VQP), a novel paradigm to \textit{directly train quantum pulses} for learning tasks. The proposed method manipulates variational quantum pulses by pulling and pushing the amplitudes of pulses in an optimization framework. Similar to variational quantum algorithms, our framework to train pulses maintains the robustness to noise on Noisy Intermediate-Scale Quantum (NISQ) computers. In an example task of binary classification, VQP learning achieves up to 11\% and 9\% higher accuracy compared with VQC learning on the qiskit pulse simulators (with system model from real machine) and ibmq-jarkata, respectively, demonstrating its effectiveness and feasibility. Stability for VQP to obtain reliable results has also been verified in the presence of noise.
\end{abstract}
\begin{IEEEkeywords} Variational Quantum Circuit, Quantum Computing, Quantum Machine Learning, Variational Quantum Pulse, Quantum Optimal Control \end{IEEEkeywords}
\section{Introduction} \label{sec1} Quantum computing has the properties of superposition and entanglement, which grants quantum speedup for certain problems \cite{alexeev2021quantum, ding2020quantum, gokhale2020optimization, niu2020hardware}. Compared with classical computing, exponential speedup has been demonstrated in areas such as quantum chemistry \cite{peruzzo2014variational,cao2019quantum}, finance \cite{ghosh2018identifying}, and machine learning \cite{jiang2021co, wang2021exploration, jiang2021machine}. Therefore, quantum computing has been considered among the best candidates for solving some complex computational problems that cannot be solved by classical computers. In recent decades, technologies such as superconducting transmon that support quantum computers have evolved substantially. Google released Sycamore \cite{arute2019quantum} in 2019 and claimed to achieve quantum supremacy, which sampled a circuit with 53 quantum bits (qubits), and the distribution is difficult for a classical computer to simulate in principle. IBM also released a 127-qubit quantum computer at the end of 2021. It is probable that quantum computers with around 1,000 qubits can be manufactured in the coming decade.
The workflow to run programs on quantum hardware can be divided into two subsequent steps. Firstly, the quantum programs are synthesized and compiled into quantum gates. When the programs are sent to the real quantum hardware, the quantum gates will be transformed into quantum pulses with a look-up table. For currently available superconducting quantum computers, the final control signals are in the form of microwave pulses. A major focus of quantum programs today is the variational quantum algorithm, especially for machine learning tasks, such as RobustQNN \cite{wang2021roqnn} and QuantumFlow \cite{jiang2021co}, which uses a classical optimizer to train a parametrized quantum circuit.
Most of the variational quantum algorithms manipulate the parameters at the gate level. For example, quantum neural networks (QNN) encode the input data \cite{wang2021quantumnas, tacchino2020variational, wang2022chip} and perform machine learning tasks on a quantum computer by building and training parametric quantum gate networks. Some state-of-the-art QNNs obtain high accuracy on several classical datasets to demonstrate potential advantage \cite{wang2021quantumnas}. Robust QNNs have also been proposed to mitigate the high noise in NISQ quantum computer \cite{liang2021can}. Decent accuracy has been demonstrated even in presence of noise \cite{wang2021roqnn}. Some recent work tackles the quantum problem with machine learning. For example, reinforcement learning is adopted to find optimal parameters at the circuit level \cite{ostaszewski2021reinforcement}.
However, variational quantum gates have limited flexibility. A controlled rotation gate comes with only one parameter that can be trained. As pointed out by \cite{sim2019expressibility}, the expressibility and entangling capability of parameterized quantum circuits are mainly affected by the number of parameters. Pulses-level operations, being lower than gate level in the stack of quantum computing, can provide finer controls and thus granting more flexible manipulation of internal parameters. Therefore, we contemplate that the increased parameters at the pulse level will give us advantages during the training process while maintaining the circuit latency. We term such a novel paradigm as {\em variational quantum pulse (VQP) learning}, which though intuitively appealing is in the uncharted territory. Note that a seemingly relevant study is on algorithms to generate pulses to manipulate the qubits \cite{leung2017speedup, khaneja2005optimal, caneva2011chopped, peng2021deep, porotti2022gradient, sivak2021model}. Our goal is fundamentally different. We are not interested in the effective ways to generate pulses, but rather making various pulse parameters learnable for machine learning tasks.
The architecture of VQP learning can be divided into three parts. The first part is to split the traditional QNN into the encoding circuit and trainable circuit (VQC). The encoding circuit is converted into pulse schedule and saved for later use. Then, the trainable circuit is also converted into pulse schedule. But the amplitudes are extracted from the pulse schedule. These amplitudes are the parameters that will be updated in the training process. The second part is to use an optimization framework to iteratively update the amplitudes with the target to minimize the error of classification tasks. The third part is to reconstruct the pulses with group of amplitudes obtained from optimization. Then, the trained VQP can be used for inference tasks. The major contributions of this work can be described as follow:
\begin{itemize}
\item This is the first work that verifies the trainability of VQP for machine learning tasks. We train VQP for binary classification tasks and obtain stable results in noisy environments.
\item Comparing the results of VQC learning and VQP learning for the same tasks demonstrates the advantages of VQP and sheds light on why VQP learning are more promising. \end{itemize}
The paper is organized as follows. Section \ref{sec1} briefly introduces the background for VQP. Section \ref{sec2} introduces the motivation of this work, gives a definition of VQP, and discusses the possible advantages of VQP learning. Section \ref{sec3} proposes a framework for VQP learning, describes the components of the framework and the optimization methods. Section \ref{sec4} unfolds the experimental results that validate the idea of VQP. Section \ref{sec5} discusses the purpose of extracting pulse amplitudes from the pulse, Section \ref{sec6} concludes the paper with a perspective of the future of VQP learning.
\begin{figure}
\caption{Conceptual illustration of VQP for QML tasks.}
\label{fig:VQP}
\end{figure} \section{Background and Motivation} \label{sec2}
\begin{figure}
\caption{An example QNN that uses VQC for QML tasks.}
\label{fig:VQC}
\end{figure}
Our idea mainly comes from the usage of the VQC for the machine learning tasks. VQC enables quantum gates to be parameterized \cite{kandala2017hardware}, such as assigning a value to the angle of a rotation gate, thus making the quantum circuit trainable. The training parameter is actually the angle of the rotation gate, which can also be understood as phase shift training. More specifically, a variational quantum state $|\psi(x, \theta)\rangle = \Phi(x, \theta) |0......0\rangle$ is prepared by parameterizing the quantum circuit, where $x$ is the input data, $\theta$ is a set of free variables for training, and $\Phi()$ is parameterized quantum circuit. Usually, the VQC is trained by quantum-classical co-optimization to find the optimal set of parameters for the circuit. Quantum neural network (QNN) is a quantum machine learning model that uses VQC to encode features on the input data and then performs complex-valued linear transformations. For image classification tasks, it is necessary to first form a quantum encoding circuit by encoding pixels to several rotation gates, and then use VQC to compute and process the information passed to it. After the computation, we measure the qubits based on the Z-axis and get expectation values, and then calculate the corresponding probability, which is already a digital number because the measurement carries out the transformation from quantum state to classical data. VQC and QNN have great potential for applications in quantum machine learning (QML) \cite{biamonte2017quantum, ahsan2022quantum}, quantum simulation and optimization \cite{moll2018quantum, cheng2020accqoc}. \subsection{Variational Quantum Pulse}
Inspired by the VQC, we propose the concept of the VQP. First of all, a general understanding is needed that the workflow of quantum computing at the software level is from the logic quantum circuits of high-level programming. Then the quantum circuits are mapped to the physical qubits after "transpiler". The quantum gates will further be "translated" into quantum pulses through a lookup table. These pulses are what quantum machine really corresponds to and processes with. In this paper, we define a VQP as a set of trainable quantum pulses that are parameterized, where a trainable pulse is defined as a pulse described by specific parameterization such as frequency, duration, amplitude, shape (e.g. Gaussian, square wave, etc.), etc.
\subsection{Variational Quantum Pulse Learning} Also inspired by the fact that VQC can be trained, we believe that VQP has the potential to be trainable. However, the wide variety of parameters of VQP can make the search space for training too large, so that the whole training is not doable. In this case, we narrow down the parameters of the attempted training to the amplitude item. As mentioned in the work \cite{wright2022deep}, a physical neural network can be constructed within a physical system in nature as long as parameters and an executable simulator exist, which also means that the parameters are trainable. And from the result of work \cite{meitei2021gate}, we can also make a guess that pulse should have the trainability.
Moreover, the training of amplitude is actually pulling and pushing the set of pulses in the pulse model without changing the shape of the pulse (the nature of quantum gates is mainly determined by the shape, e.g. the function of entanglement of CX gates is not affected by modifying amplitudes in a limited range). With the support of Qiskit's OpenPulse\cite{mckay2018qiskit}, We can take the amplitude both in the drive channel and control channel into VQP training to train the parameters that cannot be considered in the parameterized quantum circuit. The control channels are device-dependent. The role of these channels is described by the Hamiltonian returned by the device, with enough details to allow its operation. Based on this property, making amplitudes in control channels learnable may give the ability for tolerant noise in each specific device. \begin{figure}
\caption{Comparison between VQC and VQP on a binary classification task from mini-MNIST datasets on qiskit pulse simulator with system model from ibmq\_quito.}
\label{fig:motivation}
\end{figure}
Furthermore, given the limited variety of quantum gates available in quantum circuits, we should not limit our vision to the existing quantum gates. The training of VQP can be viewed as a black-box optimization problem in which the quantum gates corresponding to the pulses in the optimized black box are unknown, but these unknown gates may have better results for our QML tasks. Figure \ref{fig:VQP} illustrates the concept to use VQP for classification tasks. The process is quite similar to QNN that uses VQC. We first use encoding circuit to encode the image pixels by putting them as angles in the rotation gate and transforming them to pulses format. Then we build the trainable pulses either can be transformed from a circuit (e.g. VQC) or put parameters in pulse builder, do the VQP training to process pulses, which can be in a pulse simulator or a real quantum machine. After that, we measure all the qubits and store the information in the acquire channel for calculating the expectation values by using Softmax. After that, we can obtain the probabilities based on expectation values.
We have set up a simple experiment with MNIST two-class classification to see if VQP learning can show some advantages over VQC learning. We use qiskit pulse simulator with system model from ibmq\_quito as the backend to process the quantum circuit and pulse schedule. Due to the limited resource on the quantum computer, we configured the experiments using 10, 20, 100 images for training and the same number of images for testing respectively. The baseline VQC is randomly generated by interleaving U3 and CU3 as shown in Figure \ref{fig:VQC}. The results for the three settings are shown in Figure \ref{fig:motivation}. It is obvious that in all the cases, VQP learning can achieve much better accuracy than VQC learning.
Apprantely, the performance of VQP learning, like all other learning framerworks, heavily depends on how well the training is carried out. Yet a major and unique challenge for VQP learning is that back-propagation is not possible to be calculated from the backend of qiskit OpenPulse. Therefore, a non-gradient-based optimization framework to enable VQP learning is needed. This will be discussed in the following sections. \begin{table}[] \centering \renewcommand*{\arraystretch}{1} \setlength{\tabcolsep}{5.5pt} \footnotesize \begin{tabular}{cccc} \toprule \midrule \multirow{3}{*}{Form of CX gate} & \multicolumn{3}{c}{Time Duration} \\
& Pulse simulator & Pulse simulator & Pulse simulator \\
& (Quito) & (Belem) & (Jakarta)\\\midrule CRX($\pi$) gate & 26832.0dt & 32016.0dt & 26832.0dt \\ \midrule CX gate & 25136.0dt & 27728.0dt & 25136.0dt \\ \midrule \bottomrule \end{tabular}
\caption{Time duration for CX gate in different forms on pulse simulator with system models of ibmq\_quito, ibmq\_belem, and ibmq\_jakarta.}
\label{CXgate}
\end{table}
\subsection{Advantages of VQP learning} For quantum computers in the NISQ era, noise is a very big problem. VQC training architectures always need tens or even hundreds of gates to support learning. For VQP learning, pulse has more parameters, so the number of gates needed is greatly reduced. The reduction in the number of gates will directly lead to a decrease in latency, which means that the decoherence error can be reduced significantly. In this case, the process of VQP learning can be more robust to the environment of noise than VQC learning.
Moreover, the U gate is composed of some RZ gates as well as two fixed RX gates, RX($-\pi/2$) and RX($\pi/2$). This means that for circuit-level training of the U gate, the process of angle changes is only addressed on the Virtual Z and does not have a real impact on the physical amplitude. On the other hand, VQP learning, with amplitude as a parameter, actually changes the physical amplitudes of pulses.
Finally, VQP can also reduce the time duration for some special quantum gates. For example, in the gate level training, a CX gate needs to be treated as a special CRX($\pi$) gate, the time duration of which is shown in Table \ref{CXgate} for pulse simulator with system model of ibmq\_quito, ibmq\_belem, and ibmq\_jakarta. While at the pulse level, we can train directly on the pulse of the CX gate, and the time duration is shorter on the same machine.
\begin{figure}
\caption{Overview of the VQP learning framework.}
\label{fig:workflo}
\end{figure}
\section{VQP Learning Framework} \subsection{Overview} \label{sec3}
In this section, we present an overview of the VQP learning framework. As shown in Figure \ref{fig:workflo}, we use optimizer continues communicate with the environment. We first get a parameterized quantum circuit as a baseline and transform it into a pulse schedule. Then we put this set of pulses into the pulse optimization framework and give the amplitude list of pulses and the corresponding accuracy list of pulses to the optimizer as the initial parameters. After finding the optimized amplitude list with the lowest error rate, we need to reconstruct the pulse based on this amplitude list. The reconstructed pulses are finally deployed and executed on the pulse simulator in the qiskit test mock or on a real quantum device. Subsection B below describes the circuit to pulse transform process. Subsection C proposes the optimization framework for VQP learning. Subsection D illustrates the process of pulse reconstruction based on the optimized amplitudes.
\subsection{Quantum Circuit \& Pulse Transform} The baseline for this work is a VQC that can also be used for QML tasks. An example is shown in Figure \ref{fig:VQC}. Since there is no existing method for pulse encoding, we choose to continue to use the circuit encoding approach to encode classical data into quantum states. Then we transform the encoded circuit part, which already contains the input data and tasks, into a pulse schedule. This process can be achieved in qiskit's OpenPulse. After successfully encoding the data, we transform the trainable circuit part into pulse schedule by the same method in order to ensure the fairness of the comparison in the experiments. After the transformation, we get two parts of the pulse schedule: encoding pulses and variational quantum pulses, which can also be considered as trainable pulses. Then we need to fix these encoding pulses, and only variational quantum pulses are considered in the following training process. \subsection{Pulse Optimization Framework} \begin{figure}
\caption{The schematic of pulse optimization framework.}
\label{fig:BO}
\end{figure}
A wide range of algorithms can be possible candidates for this framework, such as RL evolutionary algorithms, genetic algorithms~\cite{kennedy1995particle,Ingber1993SA}, Bayesian optimization~\cite{gao2022bayesopt,gao2019bayesopt2,shahriari2016bayesopt}, etc. For this optimization framework, we extract the amplitudes of the pulses that are obtained as described in the previous subsection. At this point, the amplitudes are complex numbers, which means that the pulse optimization framework cannot handle it. But from the amplitude, we can associate the magnitudes and angles of these complex numbers. Then we combine the computed magnitudes and angles into a single array, which can be trained and optimized by the pulse optimization framework. It is worth noting that we need to revert this array back to the form of the complex numbers in the objective function to obtain the amplitude and then reconstruct pulses, because this objective function should be evaluated on a qiskit pulse simulator or a real quantum machine. Doing so allows our pulse optimization framework to tolerate random noise, which we believe can gain benefits in NISQ era quantum devices.
Here, we use Bayesian optimization (BO)~\cite{gao2022bayesopt,gao2019bayesopt2,shahriari2016bayesopt} to explain the optimization framework: combining the property of pulses, we propose a pulse BO framework. As shown in Figure \ref{fig:BO}, we first take the amplitude list extracted from pulse as the initial input for the BO. After the quantum machine executes, we can get an error rate, denoted as the initial objective values, also part of input for the BO. At the same time, we need to set the search space for BO. For an executable pulse, its norm needs to be no larger than one. We build a Gaussian Process (GP) regression model with RBF kernel based on the initial amplitude list, yielding a surrogate model for the real simulation. The model calculates the mean and variance of the function values at each point. With the GP model accessible, we constructs and optimize an acquisition function, which is used to decide the location to sample points. We choose the lower confidence bound (LCB) as the acquistion function in this work with the aim of finding the minimum error rate. After finding the minimum error rate, we can know the current searched best list, denormalize these data, and then recalculate the magnitude and angle of the current searched best list into amplitude, after which we get the optimized amplitude list. \begin{algorithm}[t] \caption{Variational Pulse BO Learning}\label{alg:one} \KwData{$\rho$, $\chi$, $M$, $D$} // $\rho$ is the amplitude list, $\chi$ is the search bound, $D$ consists of $x_i$ and $y_i$, $M$ is the Gaussian Process Regression model. \\ $D \gets $InitPulses$(\rho, \chi)$\;
\For{$i \gets |D|$ to $N\_total$}{ // iterative optimization
$p(y|x,D) \gets FitModel(M, D)$\;
// Acquisition function actively searches for the next optimized amplitudes.\\
$x_i$ $\gets$ $argmin_{x \in \chi}S(x, p(y|x,D))$\;
// Calculate corresponding error rate by processing in quantum machine.\\
$y_i$ $\gets$ $f(x_i)$\;
$D$ $\gets$ $D \bigcup (x_i, y_i)$\; } \end{algorithm}
Algorithm \ref{alg:one} presents the simplified pseudo code of the proposed variational pulse BO learning framework. $\rho$ is the amplitude list from initial pulses for optimization, $\chi$ is the search bound, constraint the norm of amplitude of pulses need to be less or equal to one, $D$ is consists of $x_i$ and $y_i$, $x_i$ is the hyper-parameter and is amplitude list during optimization in this case, and $y_i$ is the error rate that results by executing $x_i$ by quantum machines, $M$ is the Gaussian Process Regression model. By using $\rho$ inference on quantum device backend, we can get $(x_i, y_i)$ pairs, $y_i$ is the error rate in this case. And we set $N_{total}$ as the iteration times for optimization, we set the Gaussian process regression model, set the kernel method with normalized $y_i$, and fit the normalized $x_i$ to the Gaussian form. Then, $S$ is proposed as an acquisition function to select parameters by constraints, aiming at minimize $y$. Finally we compute the updated $y_i$ with updated $x_i$.
\subsection{Pulse Reconstruction} After obtaining the optimized amplitude list, we design a waveform reconstruction function. This function first stores the optimized amplitude in the modified list. Then it extracts the amplitude values from the initial pulses and overwrites the initial amplitude values one by one by iteratively storing the optimized amplitude in the modified list. The other parameters in the initial pulses are kept unchanged, so as to build new pulses with only the amplitude change.
\section{Experiments} \label{sec4} \subsection{Experiments Setup} \textbf{Dataset:} We evaluate the proposed method using the binary classification task, which is the same as that used in \cite{jiang2021co}. Specifically, it is a binary classification. In generating the dataset, we will associate 1 out of 2 classes to two input values ($x$ and $y$). For example, we associate class 0 to inputs of x = 0.2 and y = 0.6; and class 1 to x = 0.8 and y = 0.8. On top of the created dataset, we will divide them to to the train set and test set. And we also use the MNIST dataset, we do center-cropped images to 24 × 24, and then down-sample to 4×4 for two-class. We also setup a method to make sure the number of images we get from different classes keeping same.
\textbf{VQP Framework and Baseline:} We apply the gate based QNN as the baseline, which is implemented by TorchQuantum, an existing library for quantum machine learning. Figure \ref{fig:VQC} shows the detailed QNN structure of the baseline.
In VQP framework, we obtain the initial pulse by converting the baseline QNN, and then, we will learn the parameters in the pulse to iteratively
tuning pulses
on the pulse simulator/real quantum devices. The number of iterations is set as 30. With a trained VQP, we can stream the test set to get the accuracy. In the evaluation, we apply both gradient method and the same BO framework for VQC training. The optimization objective is to minimize the error rate (i.e., $1 - accuracy$). Limited by the property of optimizer, for training parameter in high dimensional space, the optimization results is unstable, so we run five seeds for every benchmark and calculate the average as the result we reporting in this paper.
\begin{figure}
\caption{The coupling map of all quantum devices we use.}
\label{fig:coupling}
\end{figure}
\textbf{Measurement of VQP Learning: }After run the pulse schedule in quantum machine, we can obtain the quantum state $\phi$ from computational qubit $|q\rangle$.
By setting the shots as $x$, say $x=256$, we can obtain the probabilities of results to be state $|0\rangle$ (corresponding to output of +1) and $|1\rangle$ (corresponding to output of -1).
For example if we get 156 $|0\rangle$ and 100 $|1\rangle$, then we can calculate the probability of -1 as $(156 * 1 + 100 * (-1))/256 = 0.21875$.
\textbf{Quantum Hardware and Backend Setting: }We use the pulse simulator and quantum computer provided by IBM Quantum. First, we employ pulse simulators as pulse simulator (Quito), pulse model (Belem), and pulse simulator (Lima), these simulators can be imported from the qiskit test mock, which are the simulators that pull the real data from the real machine. These settings are due to pulse simulator do not consider the physical error from qubits, but it will occur algorithmic error during Hamiltonian estimating. And as we want to keep circuit learning and pulse learning in a fair environment, we run both in pulse simulator, pulse level we do training on pulse and processed in pulse simulator in every optimization iteration, and for circuit level, we do training on circuit and transform the optimized circuit to pulse then processed in pulse simulator in every optimization iteration. Second, we also conduct the evaluation on ibmq\_jakarta, a real quantum machine that can process pulse schedule.
Kindly note that noise is considered in both pulse simulator (algorithmic error) and real machine (physical error from qubits), indicating we evaluate the proposed
method in a noisy environment. \begin{table}[t] \centering \renewcommand*{\arraystretch}{1} \setlength{\tabcolsep}{5pt} \footnotesize \begin{tabular}{ccccccc} \toprule \hline \multirow{3}{*}{Task Data} & \multicolumn{3}{c}{Accuracy} \\
& Pulse simulator & Pulse simulator & Pulse simulator \\
&(Quito) & (Belem) & (Lima) \\ \midrule
Initial 20 & 0.45 & 0.45 & 0.5 \\ \textbf{+ VQP learning} & \textbf{0.65} & \textbf{0.6} & \textbf{0.55}\\ \midrule Initial 100 & 0.51 & 0.5 & 0.5 \\ \textbf{+ VQP learning} & \textbf{0.57} & \textbf{0.63} & \textbf{0.61} \\ \midrule Initial MNIST 20 & 0.5 & 0.5 & 0.45 \\ \textbf{+ VQP learning} & \textbf{0.55} & \textbf{0.66} & \textbf{0.6} \\ \midrule Initial MNIST 100 & 0.5 & 0.5 & 0.51 \\ \textbf{+ VQP learning} & \textbf{0.57} & \textbf{0.61} & \textbf{0.61} \\ \midrule
\bottomrule \end{tabular}
\caption{Binary classification task results on different dataset size and run on pulse simulators with system models from ibmq\_quito, ibmq\_belem, ibmq\_lima, separately.}
\label{result1}
\end{table}
\begin{table}[t] \centering \renewcommand*{\arraystretch}{1} \setlength{\tabcolsep}{6.5pt} \footnotesize \begin{tabular}{ccccl} \toprule \midrule \multirow{2}{*}{Model} & \multicolumn{2}{c}{Accuracy} \\
& Pulse simulator (Belem) & ibmq\_jakarta \\ \midrule VQC learning 20 & 0.57 & 0.58 \\ \textbf{VQP learning 20} & \textbf{0.6} & \textbf{0.69} \\ \midrule VQC learning 100 & 0.61 & 0.59 \\ \textbf{VQP learning 100} & \textbf{0.63} & \textbf{0.64} \\ \midrule VQC learning MNIST 20 & 0.6 & 0.56 \\ \textbf{VQP learning MNIST 20} & \textbf{0.66} & \textbf{0.62} \\ \midrule VQC learning MNIST 100 & 0.57 & 0.62 \\ \textbf{VQP learning MNIST 100} & \textbf{0.61} & \textbf{0.71} \\ \midrule \bottomrule \end{tabular} \caption{Classification tasks results by VQP learning VS VQC learning with same BO framework on pulse simulator with system model from ibmq\_Belem and ibmq\_jakarta.}
\label{result2}
\end{table} \begin{table}[t] \centering \renewcommand*{\arraystretch}{1} \setlength{\tabcolsep}{25.5pt} \footnotesize \begin{tabular}{ccc} \toprule \midrule Model & \# of Gates & Accuracy \\ \midrule
VQC\_base & 9 & 0.62 \\ \midrule
\textbf{VQP} & \textbf{9} & \textbf{0.71} \\ \midrule VQC* & 12 & 0.68 \\ \midrule \bottomrule \end{tabular} \caption{Comparison the MNIST two-class classification task with 100 images and result on ibmq\_jakarta between VQC\_base, VQP, VQC* with different number of gates and all by same BO framework.} \label{result3} \end{table} \begin{table}[t] \centering \renewcommand*{\arraystretch}{1} \setlength{\tabcolsep}{20pt} \footnotesize \begin{tabular}{ccc} \toprule \midrule Model & \# of Gates & Accuracy \\ \midrule \textbf{VQP} & \textbf{9} & \textbf{0.71} \\ \midrule VQC with gradient & 9 & 0.73 \\ \midrule VQC* with gradient & 12 & 0.77 \\ \midrule \bottomrule \end{tabular} \caption{Put VQC with different gates in gradient based method and VQP in the BO framework, test on ibmq\_jakarta and get results.}
\label{WithBO}
\end{table}
\begin{table}[t] \centering \renewcommand*{\arraystretch}{1} \setlength{\tabcolsep}{5pt} \footnotesize \begin{tabular}{cccc} \toprule \midrule \multirow{2}{*}{Model} & \multirow{2}{*}{\# of Gate} & \multicolumn{2}{c}{Time Duration} \\
& & ibmq\_jakarta & Pulse simulator (Belem) \\ \midrule \textbf{VQP} & \textbf{9} & \textbf{40816.0dt} & \textbf{45168.0dt} \\ \midrule VQC* & 12 & 58896.0dt & 58768.0dt \\ \midrule \textbf{VQP\_transpiled} & \textbf{11} & \textbf{32368.0dt} & \textbf{32816.0dt} \\ \midrule VQC*\_transpiled & 17 & 53008.0dt & 46192.0dt \\ \midrule \bottomrule \end{tabular} \caption{Time duration for VQP, VQC*, VQP\_transpiled, and VQC*\_transpiled based on ibmq\_jakarta and pulse simulator with system model of Belem.}
\label{models}
\end{table}
\begin{table}[t] \centering \renewcommand*{\arraystretch}{1} \setlength{\tabcolsep}{2.6pt} \footnotesize \begin{tabular}{lccc} \toprule \midrule \multicolumn{1}{r}{\small \underline{Use system model of $\rightarrow$}} & \multirow{2}{*}{Pulse simulator (Lima)} & \multirow{2}{*}{Pulse simulator (Quito)} \\
{\small \underline{Inference on $\downarrow$}} & & \\
\midrule
Pulse simulator (Lima) & \cellcolor{blue!20} \textbf{0.65} & \cellcolor{red!20} 0.4 \\
Pulse simulator (Quito) & \cellcolor{red!20} 0.35 & \cellcolor{blue!20} \textbf{0.55} \\ \midrule \bottomrule \end{tabular} \caption{Run and test accuracy result on different models.}
\label{dependence}
\end{table} \subsection{Main Results}
\textbf{VQP Learning Result: }
We take the initial pulses inference on quantum machine's result as an initial result, and we set the different size of tasks: (1) 20 'train' data and 20 'test' data, which is described as 'Initial 20'; (2) 'Initial 100' for 100 'train' data and 100 'test' data; (3) 'Initial MNIST 20' for 20 'train' image and 20 'test' image in MNIST two-class; (4) 'Initial MNIST 100' for 100 'train' image and 100 'test' image in MNIST two-class.
Table \ref{result1} report the experimental results. VQP learning can improve accuracy for all settings.
Specifically, compared with 'Initial 100', VQP learning can get an average improvement of 10\% accuracy. The improvement of VQP is 12\% over 'Initial MNIST 20'.
\textbf{Comparison between VQP and VQC: }We also compared the results of VQC and VQP on the same task. VQC uses the nine-gate learnable circuit.
For a fair comparison, we performed 30 iteration same BO on two benchmarks, pulse simulator with system model of ibmq\_belem, and real quantum machine ibmq\_Jakarta, executing 256 shots each.
As can be seen from the Table \ref{result2}, the results of VQP learning achieved 60\% and 69\% accuracy for the 20-data. VQC learning, on the other hand, obtained 57\% and 58\% accuracy in pulse simulator (Belem) and ibmq\_jakarta. VQP learning show better performance in the 8 groups tasks, and achieve up to 71\% accuracy on the MNIST two-class 100 image classification task, meanwhile, VQC learning also gain the best performance on this task but with 62\% accuracy.
An observation is that VQP learning obtain up to 94\% accuracy under noise environment in real quantum machine, but sometimes only push performance to gain a little bit beneifts. This is because of non-gradient based optimizer sometimes cannot handle the hyperparameters in high dimensional space. Thus, a differentiable simulator is highly demand for the pulse learning. However, from another aspects that we can say 94\% is at least a second optimal solution that can be achieve by VQP learning. \begin{figure*}
\caption{Pulse visualization of CX gate before and after amplitude tuning in pulse simulator (Belem).}
\label{fig:vis}
\end{figure*} \begin{figure}
\caption{Schematic of varying the amplitude of pulse.}
\label{fig:vary}
\end{figure}
Overall, from the above results, VQP learning optimized for parameter amplitudes at pulse level can show advantage over VQC learning trained on parameter angles at circuit level under the same conditions.
\textbf{Further Experiment on VQP and VQC: }To verify the conjecture of VQP learning advantage in the Section \ref{sec2}. We conduct another set of experiment to compare VQP and VQC learning.
As can be seen from the Table \ref{models}, VQC* with 12 gates achieves 68\% accuracy on MNIST two-class classification, while VQC\_base with nine gates achieves only 62\% accuracy. This indicates that the training results get improved when the number of gates of VQC increases (i.e., more parameters); however, VQP learning with only 9 gates still outperform VQC learning with 12 gates. This observation illustrate
that VQP learning has more trainable parameters than the gate based VQC for better learning.
In the Table \ref{WithBO}, we report the results of VQC learning in gradient based method on ibmq\_jakarta. It is obviously that training VQC in gradient based method can gain the benefits. And now VQC learning with 9 gates already has the better performance than VQP leanring in terms of accuracy.
\textbf{Device Dependence of VQP: }VQP is device-dependent. As shown in Table \ref{dependence}, if we train on pulse simulator (Lima) and test on pulse simulator (Lima), we can obtain 65\% accuracy, but if we train on pulse simulator (Lima) and test on pulse simulator (Quito), the accuracy is only 35\%, which shows that device-specific is important. This is because the pulse for the same gate can vary from model to model. And from the work \cite{proctor2022measuring}, we think in the quantum machine the device dependence of VQP can also result in the noise is structured-dependent.
\section{Discussion} \label{sec5} We further discuss the purpose of extracting pulse amplitudes from the pulse schedules. Here, we provide the visualization on the change of variational quantum pulses in the process of optimization. Based on Equation \ref{eq1}, the Hamiltonian of the control pulses will be updated accordingly in the training process. Moreover, the latency of 12 gates is necessarily larger than that of 9 gates. On ibmq\_jakarta and pulse simulator (Belem), we tested the time duration of 9 gates VQP against 12 gates VQC*, which have similar accuracy for both using BO framework.
From the table, we can see that the time duration of VQP for 9 gates is much shorter than that of VQC* for 12 gates in all different cases. From this result, we can intuitively see the advantage of VQP learning over VQC learning in terms of latency, which means that the robustness of VQP learning on noise not only comes from the optimization framework but also the property that its decoherence error is smaller.
\begin{equation} \begin{aligned} H = \sum_{i=0}^{1} (U_i(t)+D_i(t)) \sigma_i^{X} + \sum_{i=0}^{1} 2\pi \nu_i (1-\sigma_i^{Z})/2 \\+ \omega_B a_B a^{\dagger}_B + \sum_{i=0}^{1} g_i \sigma_i^{X} (a_B + a_B^{\dagger}) \label{eq1} \end{aligned} \end{equation}
\textbf{Vary of Amplitudes of Pulses: } As shown in Figure \ref{fig:vary}, the process of VQP learning is equivalent to pulling or pushing initial pulses. In the process of iterative optimization, multiple attempts are made to vary amplitudes until the error rate is minimized. Figure \ref{fig:vis} show the visualization of the pulse of CX gate on pulse simulator (Belem). The plot on the left is the pulse of the CX gate before amplitdue tuning, while the plot on the right is the new pulse after amplitdue tuning. This shows how the proposed framework to directly tune the physical amplitudes.
\textbf{Analytical Understanding of Amplitude Tuning: }In the process of VQP learning, we can analyze the physical quantities affected by amplitude tuning according to Equation \ref{eq1}. This equation describes the drive Hamiltonian, where $D_i(t)$ is mixed by the signal on drive channel for qubit $i$ and local oscillator (LO) at frequency corresponding to the signal. $U_i(t)$ is mixed by the signal on control channel for qubit $i$ and some combinations of qubit LO's that specified by device. $\sigma_X, \sigma_Y$ and $\sigma_Z$ are Pauli operators. $\nu_i$ is the estimated frequency of qubits in the qubit i, $g_i$ is the coupling strength between qubits, $\omega_B$ is the frequency of buses, $a_B$ and $a^{\dagger}_B$ are the ladder operator for buses. The vectors actually effected by amplitudes tuning are $D_i(t)$ and $U_i(t)$. $D_i(t)$ and $U_i(t)$ can be obtained by the Equation \ref{eq2}:
\begin{equation} \begin{aligned} D_i(t) = Re (d_i(t)e^{iw_{d_i}t})\\ U_i(t) = Re [u_i(t)e^{i(w_{d_i} - w_{d_j}) t})] \label{eq2} \end{aligned} \end{equation} where $d_i(t)$ and $u_i(t)$ are the signal of qubit $i$ on drive channel and control channel, respectively. Amplitude is the intensity of signal, which means when we do amplitudes tuning, we change the intensity of signal, so that affect on the vary of $d_i(t)$ and $u_i(t)$. It is known from Equation \ref{eq2} that the changes in $d_i(t)$ and $u_i(t)$ affect the $D_i(t)$ and $U_i(t)$ in Equation \ref{eq1}.
\section{Conclusion and Prospective} \label{sec6} For QNN, its potential advantage over classical neural networks is that the search space of the unitary matrix can be increased with the number of qubits, such that the neural network can learn more and performs better. For VQP learning, the pulse has more parameters than the circuit, which means pulse learning may obtain better expressibility and entangling capability. The focus of this work is to propose a novel paradigm to use VQP for quantum learning. We demonstrate the advantages of VQP learning over VQC learning for ML tasks. Also, the reduced decoherence error due to the reduced latency from the small demand for gate number in VQP learning is really important for quantum machines in the NISQ era, since noise is one of the major problems in the NISQ era.
The potential of VQP is huge. It can be more flexible to tune and process the parameters inside the circuit. As we discussed in the experimental section, it currently achieves results only based on an unoptimized VQC architecture, and we expect to see better results if we apply it on an optimized circuit-level architecture. The current VQP learning has pitfalls, one being the limited resources of the real quantum machine and the other being that we do not have an effective simulator for pulses. The simulator that can support pulses in the OpenPulse interface provided by Qiskit is slow to execute, which makes it difficult to experiment with larger and more complex tasks at the moment. As well, this pulse simulator is not differentiable, which also leaves us at the moment to use non gradient based optimizer in VQP learning, which tend to be more stochastic in parametric tasks in high dimensional spaces. Therefore an efficient and differentiable pulse simulator is urgently needed.
In addition, training and optimization methods for VQP are worth investigating, and we plan to implement a more efficient optimization and training framework in the future. Gradient-based machine learning for VQP learning may still be possible with advances in the supporting platforms.
\section*{Acknowledgment} We thanks Thomas Alexander for patient guided on qiskit OpenPulse, also thanks and Dr. Xiangliang Zhang for valuable discussion about the optimization framework. We acknowledge the use of IBM Quantum services for this work.
\printbibliography
\end{document} | arXiv |
List of Posts written during Feb 2020
This is a list of of posts written during the month Feb 2020
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This is the complete list of categories of posts written during Feb 2020. They have been ordered by the publish date, with the most recent first.
(solved)Question 1 SSC-CGL 2018 June 4 Shift 1
Published: 06-Feb-2020
In a triangle ABC, the points F and E respectively on AB and AC sides are such that FE || BC and FE divide the triangle into two parts with equal area. If AD ⊥ BC and AD intersect FE at point G, then GD: AG =?
The average of twelve numbers is 42. The last five numbers have an average of 40 and the first four numbers have an average of 44. The sixth number is 6 less than the fifth number and 5 less than the seventh number. What will be the average of the 5th and 7th numbers?
Trigonometric: $\displaystyle \frac{2 + \tan^2 \theta + \cot^2 \theta}{\sec \theta \cosec \theta} = ?$
Four years ago, the ratio of the ages of A and B was 4: 5. Eight years from now, the ratio of the ages of A and B will be 11: 13. What is the sum of the present age of both of them?
A truck covers a distance of 384 km at a certain speed. If the speed is reduced by 16 km / h, it will take two hours more to cover the same distance. What is the 75% of the original speed (in km / h)?
Someone sold an item at a loss of 15%. If he sold it for Rs 30.60 more then he would get 9% profit. In order to get 10% profit, he has to sell the item in what amount?
If a number of 9 digits is 985x3678y, the number is divisible by 72, then the value of (4x - 3y) will be:
An amount becomes 8,028 in 3 years at a fixed percentage interest rate and 12,042 in 6 years, when the interest is compounded annually. What is the actual amount?
The ratio of efficiencies of A, B and C is 2 : 5 : 3. On working together, all three of them can complete work in 27 days. In how many days will both B and C together complete the 4/9th part of that work?
(solved)Question 10 SSC-CGL 2018 June 4 Shift 1
If 120 is reduced by x%, the same result will be obtained if 40 is increased by x%. Then x% of 210 will be what percentage less than (x + 20) % of 180?
After giving two successive discounts each of x% on the marked price of an item, total discount is Rs. 259.20. If the face value of the object is Rs. 720, what will be the value of x?
The radius of a circle with O center is 10 cm, PQ and PR are the chords of 12 cm. PO cuts the chord QR at point S. What is the length of OS?
A circle is drawn inside a triangle ABC. The circle touches the sides AB, BC and AC at the points R, P and Q respectively. If AQ = 4.5 cm, PC = 5.5 cm and BR = 6 cm, then the perimeter of triangle ABC is?
In ∆ ABC, the bisectors of ∠B and ∠C meet at point O inside the triangle. If ∠BOC = 122 degree , what will be the measure of ∠A?
If $\displaystyle \sin \theta = \frac{p^2 - 1}{p^2 + 1}$, then $\displaystyle \cos \theta = ?$
If $\displaystyle x^{4} + x^{-4} = 194, x \gt 0$, then what would be the value of $\displaystyle (x - 2)^2 ?$
If $\displaystyle \frac{5\sqrt 5 x^3 - 81\sqrt 3 y^3}{\sqrt 5 x - 3\sqrt 3 y} = $ $\displaystyle Ax^2 + By^2 +Cxy, $ then $\displaystyle 6A + B - \sqrt {15}C = ?$
If $\displaystyle x + y + z = 19$, $\displaystyle x^2 + y^2 + z^2 = 133$ and $\displaystyle xz = y^2$, then the difference between $\displaystyle x$ and $\displaystyle z$ is?
If $\displaystyle 4 - 2\sin^2 \theta - 5\cos \theta = 0$, with $\displaystyle 0\degree \lt \theta \lt 90 \degree$, then the value of $\displaystyle \sin \theta + \tan \theta$ is?
A solid cube with a volume of 13824 cu. cm is cut into eight cubes of the same volume. The ratio of the surface area of the original cube and the total sum of the surface area of three smaller cubes will be?
$\displaystyle \bigg(\frac{\sin \theta - 2\sin^3 \theta}{2\cos^3 \theta - \cos \theta}\bigg)^2 + 1 = ?$
Raju's income is 20% more than his expenditure. If his income increases by 60% and his expenditure increase by 70%, then what percentage of the savings will increase/decrease?
The ratio of competencies A, B and C is 4: 5: 3. On working together, all three of them complete work in 25 days. In how many days will both A and C together complete 35% of the work?
Renu bought an item for 1,240 and sold it at a loss of 25%. With that amount, she bought another item and sold it for 40% profit. What percentage of profit did Renu get?
The nominal value of an article is 315. It is sold in 288. If this leads to a loss of 4%, then what percentage of the item was marked more than the cost?
C/C++ Program to draw a circle with stars without graphics.h
The following program first creates an empty 2-dimensional array, then stores star('*') character into all the elements that would fall on or near the circumference of the circle.
C/C++ Program to draw an angle with stars
The following program draws two arms of a given angle. The horizontal arm is fairly simple. But the inclined arm is drawn by using a 2-D array by filling stars in those elements that satisfy the equation y = x tanθ.
A takes 30 minutes longer than B to cover a distance of 15 km at a certain speed. However, if A doubles his speed, he covers the same distance in an hour less time than B. What is the speed of B (in km / h)?
C/C++ Program to make a digital clock with stars
This program creates a digital clock that displays the current system time as in the familiar dot matrix numeric clocks. The time keeps updating every 15 seconds. The display character can be set to any ASCII character.
Simple C, C++ Program to download a file from internet URL
This is a ten line C, C++ program that connects to the internet and downloads the html source of a web page. The given program supports both http and https requests and works even if there is a url redirect from one protocol to the other, ie., from http-to-https or from https-to-http.
The average weight of some students in a class is 68.5 kg. If four new students of 72.2 kg, 70.8 kg, 70.3 kg, 66.7 kg are enrolled in the class, the average weight of students increases by 300 g. Initially, how many students were there in the class?
If x is subtracted from each of 23, 39, 32 and 56, then the numbers obtained in this sequence are in proportion. What will be the mean proportional between (x + 4) and (3x+ 1)?
C, C++ Program to extract URLs(hyperlinks) from a web page of a website
This program connects to the internet and downloads the html source of a web page. After that it extracts links (urls) from the downloaded string, and displays them on the console. The program can be easily altered to extract urls from a string or a text file also.
XML RSS News Feed Reader with bare C++
A native C++ program that connects to an rss feed site and reads the XML title and description tags in a loop. We have used a bbc rss-xml feed in this progam to create a news ticker.
A circle inside a triangle ABC, whose center O is created. On increasing AO, it meets the circle [or triangle] on K and AD ⊥BC. If ∠B = 80° and ∠C = 64°, then the measure of ∠DAK is?
☞ (C, C++ Minor Project)Management of Tasks and Appointments with a Calendar (with Project Report and Source Code)
This is a minor project for summer training /internship of engineering and school students. This project contains source code and also complete project report. The features of this project are that it allows a student to check leap year, view colored calendar, and add appointments, and view them alongside.
In ΔABC, AD ⊥ BC and BE ⊥ AC. AD and BE cut each other at F. If BF = AC, what will be the measure of ∠ABD?
Circles of radius 10 cm and 8 cm cut each other at points P and Q. PQ = 12 cm and the distance between the centers of the circle is x cm, then the value of x (correct to 1 decimal place)?
ΔABC is similar to ΔDEF. The area of ∆ABC is 100 sq. cm and the area of ∆DEF is 49 sq. cm. If ΔABC's altitude is 5 cm, then ΔDEF will have corresponding altitude?
If θ is acute and cos^2 θ = 3 (cot^2 θ - cos^2 θ), then the value of (trigonometric expression) is?
Take Picture with USB Web Camera with 2 lines of C, C++ Program
Following is a pure native C program to take picture from your attached web camera. The best thing is that this is a really short, 2 line program. We have tested with logitech USB camera.
Chemistry XI Ch 2 (29)
Physics XI Ch 2 (24)
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Scheduled talks
Speaker: Nora Salone (Studium Doktoranckie NCBJ)
Title: Electromagnetic transition form factors and Dalitz decays of hyperons
Abstract: This project aims to gain information about the hyperon structure through the study of Dalitz decays of a hyperon resonance to a ground-state hyperon and an electron-positron pair. The usual framework of fixed target experiments, albeit very suitable for nucleons, is not as effective for hyperon resonances. One should consequently change the explored kinematical region, from space-like to time-like $q^2$, with the aid of crossing symmetry.
After parametrizing the corresponding baryon-photon-baryon vertex through the use of electromagnetic transition form factors, we formulate double differential decay rates for different spin-parity combinations of the initial state resonance ($J^P = \frac{1}{2}^\pm, \frac{3}{2}^\pm$) transitioning to a ground-state hyperon ($J^P = \frac{1}{2}^+$). Such decay rates are then computed at $q^2=0$ ("QED-type" approximation) and compared to the original quantities where a "radius" structure has been implemented through a low-energy approximation of the form factors. This parallelism can give a rough estimate for the measurement accuracy needed to distinguish between a structure-less and a composite hyperon, namely the minimum requirements for the hyperon internal structure to be "seen".
Further information on electromagnetic transition form factors can be acquired through the self-analyzing weak decay of the ground-state hyperon: computing the respective multi-differential four-body decay width results in an additional term containing a relative phase between combinations of the original form factors.
Speaker: Yashwanth Prabhu (Studium Doktoranckie NCBJ)
Title: On the Determination of 𝛿CP with Accelerator Neutrinos
Abstract: One of the most important open questions in particle physics is whether the CP symmetry is violated in the leptonic sector- more specifically in the neutrino sector. It is well known the CP symmetry is violated in the quark sector. Discovery of CP violation in the neutrino sector will have implications on the observed matter and antimatter asymmetry in the Universe. The leptonic CP violation arises through the phase 𝛿CP which is a parameter in the neutrino mixing matrix. If 𝛿CP takes a non-conserving value, it will result in CP violation.
In my thesis, I studied the effect of 𝛿CP on neutrino and anti-neutrino oscillation probabilities and extended the same to the study of event rates at long baseline neutrino experiments. In this talk, I will discuss the results obtained from my analysis of accelerator neutrinos that travel 1,300 km before detection.
Speaker: Michał Jędrzejczyk (Studium Doktoranckie NCBJ)
Title: Establishment of reasonable model to simulate emergency passive coolant system in HTTR reactor
Abstract: International Atomic Energy Agency (IAEA) Coordinated Research Program (CRP) on "Heat Transport and Afterheat Removal for Gas-cooled Reactors under Accident Conditions" started in November 1993. In this program, benchmark tasks were proposed for the analysis of passive afterheat removal from gas-cooled reactors (GCR) under accident conditions. The specific objective of the benchmark program is to capture the essential heat transfer features of reactor-to-reactor vessel cooling system (VCS) and provide useful information applicable to a wide variety of designs, operating conditions and model parameters.
In the present study, a 1/6 scale model of VCS for High-Temperature Test Reactor (HTTR) was used to develop a reasonably accurate thermal-hydraulics model of a passive cooling system. HTTR is a graphite-moderated gas-cooled research reactor in Oarai, Ibaraki, Japan, operated by the Japan Atomic Energy Agency (JAEA). The reasonable 2D model was established by using ANSYS Fluent software. In the study temperature profiles of the outside of the scaled reactor vessel for three experiment configurations were obtained numerically and compared with experimental results. The numerical results showed good agreement with the experimental ones. Moreover, a simplified numerical approach has been proposed and new heat transfer coefficients were determined. The approach allows for a two-fluid system simulation with significantly reduced computational costs.
Speaker: Luis Eduardo Suelves (Studium Doktoranckie NCBJ)
Title: Anisotropic multiplicative bias in weak lensing shear estimates
Abstract: Gravitational weak lensing, the weak regime of gravitational lensing phenomena, arises in the sky as a slight shape distortion of observed galaxies, quantified as a change in their ellipticity. As the only information of a galaxy's ellipticity that we have is in the observation itself, the gravitational effect has to be extracted statistically. Simply speaking, the intrinsic shape of a set of galaxies at an astronomical frame would stack to that of a circle, therefore, any deviation from this stacked shape would be produced by some mass distribution.
This Master Thesis project was dedicated to characterize the systematic effects that can bias the measurements in weak lensing and cosmic shear surveys of the shear, a main quantity to characterize weak lensing studies. We produced highly simplified simulations of astronomical frames, with a uniform shear applied to the galaxies, using the Stuff/SkyMaker package, and then applied the KSB shear estimation method. The formalism used to calibrate the measurement systematics, called calibration bias, differs from the one found usually in the literature by the introduction of extra cross-components.
Speaker: Mateusz Kmieć (Studium Doktoranckie NCBJ)
Title: Feasibility Studies of CPT Violation Measurement in Flavour Oscillations of the Neutral D Meson
Abstract: Mesons are bound quark-antiquark pairs. Flavoured neutral mesons are defined as mesons with no electric charge and non-zero strangeness, charm or beauty content. The weak interactions mix neutral-mesons with their antiparticles
leading to spontaneous transitions between meson and antimeson quantum states, which can serve as a sensitive interferometer facilitating precision testing of CPT invariance.
The main objective of my master's thesis was to perform feasibility studies of the CPT violation (CPTV) measurement in the system of the neutral D meson. CPT symmetry is one of the fundamental symmetries of the Standard Model (SM). The measurement of CPTV would mean that there exists physics beyond the SM. My goal was to probe the level of sensitivity of testing CPTV in the system of the neutral D meson. For this purpose, I created a Monte Carlo (MC) generator of neutral meson decays, where CPTV was controlled by a complex phenomenological parameter z. The MC generator was used to simulate the CPT violation effect at the level of z=0.1 for an ensemble of 100 pseudo-experiments. Each experiment consisted of N=6.5*107 of MC generated events corresponding to the number of D0 -> K+pi– decays collected by the LHCb (2011-2012). For such statistics, the CPT violation effect would be seen at seven standard deviations level. This can be contrasted with the best experimental limit for the parameter z of order O(1) provided by the FOCUS collaboration.
Speaker: Victor Martínez-Fernández (Studium Doktoranckie NCBJ)
Title: CP violation in the Minimal Linear sigma Model
Abstract: In this seminar we review the generalities of composite Higgs (CH) models that aim to solve the Standard Model hierarchy problem with the introduction of the Higgs as a Nambu-Goldstone boson as well as a new strong sector with new heavy particles. In particular we work with a renormalizable CH model, the Minimal Linear sigma model (MLsM). The phenomenology of this model is extended with the study of the electron electric dipole moment (eEDM) in accordance with the experimental constraints furnished by the ACME Collaboration in order to set limits on the MLsM CP-violating phases.
Our interest in the eEDM stems from the fact that a non-zero value implies CP violation. Since it is a low-energy observable, we perform an integration-out of the heavy fields, obtaining an effective field theory that at 2 loop describes an eEDM (Barr-Zee diagram).
Speaker: Maitrayee Mandal (Studium Doktoranckie NCBJ)
Title: Improving the Tau Appearance Study in Atmospheric Neutrinos with Neutron Capture Information at the Super-Kamiokande Experiment
Abstract: The Super-Kamiokande (SK) experiment is dedicated to the detection and understanding of neutrino physics. Currently, few experiments constrain the tau neutrino sector and therefore, improving the detection of the appearance of tau neutrinos in atmospheric neutrino flux at SK is an interesting problem. To identify the tau signal from the background, a neural network is utilised at SK. The predominant background consists of neutral current interactions of neutrinos of all flavors. Lesser neutron captures per event are expected in case of the tau signal than in the predominant background, however the present neural network does not include an input of neutron capture information. The recent SK-Gd upgrade will result in 90% of the neutrons produced in the detector being detected and recognised. In the presented study, we show that adding a new input corresponding to the number of neutron captures per event allows for better classification of the tau-signal. We also observe a positive correlation of initial kinetic energy of the event with the separation of signal and background due to neutron captures.
Speaker: Paritosh Verma (Studium Doktoranckie NCBJ)
Title: Searching for gravitational waves from pulsars in Jordan Brans Dicke theory
Abstract: I shall talk about gravitational waves in Jordan Brans Dicke (JBD) theory. There are two tensor polarization states in the General theory of relativity (GR) but there can also be vector and scalar polarization states in alternative theories of gravity. The JBD theory is one of the attempts to modify the general theory of relativity by varying gravitational constant G and it has three polarization states. The first two states are the same as in GR and the third one is the scalar polarization. We have extracted these three polarizations for a particular case of a rotating neutron star with a mountain and then calculated the F-statistic. Finally, we have developed a simulation to estimate the amplitudes from quadrupole as well as dipole emission. | CommonCrawl |
Synthesis and in vitro antitumor activity of (1E,4E)-1-aryl-5-(2-((quinazolin-4-yl)oxy)phenyl)-1,4-pentadien-3-one derivatives
Hui Luo1,3,4,
Shengjie Yang2,3,4,
Da Hong2,
Wei Xue3,4 &
Pu Xie1
Chemistry Central Journal volume 11, Article number: 23 (2017) Cite this article
Cancer is one of the leading causes of death and only second to heart diseases. Recently, preclinical studies have demonstrated that curcumin had a number of anticancer properties. Thus, we planned to synthesize a series of curcumin analogs to assess their antiproliferation efficacy.
A series of (1E,4E)-1-aryl-5-(2-((quinazolin-4-yl)oxy)phenyl)-1,4-pentadien-3-one derivatives (curcumin analogs) were synthesized and characterized by IR, NMR, and elemental analysis techniques. All of the prepared compounds were screened for antitumor activities against MGC-803, PC3, and Bcap-37 cancer cell lines. A significant inhibition for cancer cells were observed with compound 5f and also less toxic on NIH3T3 normal cells. The mechanism of cell death induced by compound 5f was further investigated by acridine orange/ethidium bromide staining, Hoechst 33,258 staining, TUNEL assay, and flow cytometry cytometry, which revealed that the compound can induce cell apoptosis in MGC-803 cells.
This study suggests that most of the derivatives could inhibit the growth of human cancer cell lines. In addition, compound 5f could induce apoptosis of cancer cells, and it should be subjected to further investigation as a potential anticancer drug candidate.
Cancer is one of the leading causes of death and only second to heart diseases [1, 2]. The efficacy of current chemotherapeutics is low and undesirable side effects are still unacceptably high [3–5]. Hence, the development of novel, and less toxic and anti-cancer agents remains an important and challenging goal of medicinal chemist worldwide, and much attention has recently been paid to the discovery and development of new, more selective anticancer agents [3, 6–8].
Natural products have become a leading category of compounds in improving the rational drug design for novel anti-cancer therapeutics [9, 10]. Curcumin is a natural phenolic compound originally isolated from turmeric, a rhizome used in India for centuries as a spice and medicinal agent [11]. A literature survey reveals that curcumin, and its derivatives (analogs) have various pharmacological activities and medicinal applications such as antioxidant [12, 13], anti-inflammatory [12, 14], anti-HIV [15, 16], anti-angiogenesis and so on [12]. Recently, preclinical studies have demonstrated that curcumin had a number of anticancer properties, such as growth inhibition and induction of apoptosis in a variety of cancer cell lines [17–19]. Its mechanisms of action include inhibition of transcriptional factor NF-jB, HSP90 and epigenetic modulation related to direct inhibition of the catalytic site of DNMT-1 [20]. Moreover, the latest research shows that curcumin can effectively suppress NF-kB activity and COX-2 expression, as well as cell proliferation/survival in the setting of NSCLC [21]. Consequently, analogues of curcumin with similar safety profiles but increased anticancer activity have been developed in recent years [22]. Chandru et al. synthesized four novel dienone cyclopropoxy curcumin analogs by nucleophilic substitution reaction with cyclopropyl bromide, and found that the tumor growth inhibitory effects of synthetic dienone cyclopropoxy curcumin analogs could be mediated by promoting apoptosis and inhibiting tumor angiogenesis [23]. New 1,5-diaryl-1,4-pentadien-3-one derivatives (curcumin analogs), which can effectively inhibit proliferation of cancer cells at very low concentrations, were synthesized [24, 25], and we also found that curcumin analogs exhibited promising ex vivo antiviral bioactivities against tobacco mosaic virus and cucumber mosaic virus [26].
In order to discover more potent and selective anticancer agents based on curcumin scafforld, we have synthesized a series of (1E,4E)-1-aryl-5-(2-((quinazolin-4-yl)oxy)phenyl)-1,4-pentadien-3-one derivatives (eleven novel compounds 5a, 5b, 5d, 5f–5h, and 5j–5n) (Fig. 1). In our present study, all the target compounds were evaluated for their activity against MGC-803, PC3, and Bcap-37 cancer cell lines. Furthermore, the possible mechanism of MGC-803 cell growth inhibition by compound 5f was also investigated in this paper.
Design of the target compounds
Target compounds 5a–5n were synthesized as shown in Scheme 1. The starting material 2-aminobenzoic acid was conveniently cyclized to intermediate 1 by heating it with formamide at 140–145 °C as described in the literature. Upon refluxing with freshly distilled phosphorus oxychloride and pentachlorophosphorane, intermediate 1 yielded the corresponding 4-chloro derivative 2. Treatment of salicylaldehyde with acetone in the presence of sodium hydride at room temperature got intermediate 3. The key intermediates 4 were synthesized by reacting intermediate 3 with substituted 4-chloroquiazoline 2 in the present of K2CO3 in CH3CN at 30–50 °C for 6 h. And then, the target compounds 5a–5n were synthesized by reacting the substituted aldehydes with 4 in the present of anhydrous alcohol in acetone at room temperature. The structures of the final products were confirmed by their IR, 1H NMR, 13C NMR, and elemental analysis techniques.
Synthetic pathway to target compounds 5a–5n
Evaluation of anti-tumor bioactivity of synthetic compounds
The in vitro antitumor activity of the newly synthesized compounds 5a–5n were evaluated against a panel of three human cancer cell lines, including human gastric cancer cell line MGC-803, human prostate cancer cell line PC3, and human breast cancer cell line Bcap-37, and one normal cell line NIH3T3 (mouse embryo fibroblast cell line) by MTT method. Adriamycin (ADM) was chosen as a reference drug due to its availability and widespread use. Each experiment was repeated at least three times. The results are presented in Table 1.
Table 1 Effect of title compounds against cell viability of different cell lines
As depicted in Table 1, the title compounds suppressed proliferation of the above three cancer cell lines in different extents (IC50 values of 0.85–15.64 μM), and exhibited broad spectrum antitumor activity. Among these studied compounds, the inhibitory ratios of 5d–5g, and 5m against MGC-803 cells at 10 μM were 87.5, 87.0, 90.7, 85.9, and 81.1%, respectively, and their IC50 values were 1.72, 1.89, 0.85, 2.02, and 2.05 μM, respectively, similar to that of ADM (0.74 μM). Compounds 5d, 5f, 5g, and 5m displayed higher inhibitory activities against PC3 cells at 10 μM than that of the rest compounds, with inhibitory ratios of 86.3, 93.0, 83.1, and 81.2%, respectively, which were similar to or higher than that of ADM (91.2%). The inhibitory ratios of 5f and 5g against Bcap-37 cells at 10 μM, were 76.5 and 74.9% (IC50 values of 4.98 and 5.61 μM), respectively, which were higher than that of the rest compounds. Also noteworthy is that the potency of the compounds was generally more pronounced against the MGC-803 cells than against PC3 and Bcap-37 cells. Moreover, the antiproliferation activities of the title compounds against NIH3T3 normal cell line were also evaluated. Most of the title compounds showed stronger antiproliferative activities against the cancer cell lines than NIH3T3 lines. Compound 5f, which showed excellent levels of inhibition against MGC-803, PC3, and Bcap-37 cancer cells, have no significant activity against NIH3T3 cells, with inhibitory ratio of 21.5% at 10 μM. That is to say that the compound was less toxic on normal fibroblasts than on the investigated cancer cell lines and more selective to cancer cells.
Subsequently structure–activity relationships (SAR) studies were performed to determine how the substituents affected the anticancer activity. To examine SAR, different substituent groups were introduced into R1 and R2 in the quiazoline ring. Based on the activity values indicated in Table 1, the relationships of the activities with different R1 and R2 (type, position, and number of substituents) were deduced. Two main conclusions were drawn. On the one hand, compared with the same substituents on quiazoline, the corresponding molecules containing a 6-methyl group always had higher inhibitory rates than the compound containing a 8-methyl group. For example, the IC50 values of 5f (R1: 6-methyl, R2: 2,6-dichlorophenyl) and 5m (R1: 8-methyl, R2: 2,6-dichlorophenyl) on MGC-803 cells were 0.85 and 2.05 μM, respectively. By contrast, the inhibition rates of 5c (R1: 6-methyl, R2: p-chlorophenyl) and 5j (R1: 8-methyl, R2: p-chlorophenyl) at 10 μM were 79.2 and 76.3% on MGC-803 cells, 76.5 and 71.9% on PC3 cells, and 54.2 and 45.4% on Bcap-37 cells, respectively. On the other hand, when R2 was o-flurophenyl-fixed, the compounds always showed weak activity. For example, the inhibition rates of 5a (R1: 6-methyl, R2: o-flurophenyl) at 10 μM were 71.9, 68.5, and 44.1% on the three cancer cells, respectively, which suggested the weaker activity than that of the rest compounds.
Apoptosis is one of the major pathways that lead to the process of cell death [27]. Most cancer cells retain their sensitivity to some apoptotic stimuli from chemotherapeutic agent [28]. In the present study, compound 5f was selected and its mechanism of growth inhibition of MGC-803 cells was evaluated. To determine whether antiproliferation and cell death are associated with apoptosis, MGC-803 cells were stained with acridine orange (AO)/ethidium bromide (EB) staining and Hoechst 33,258 staining after exposure to compound 5f and observed under fluorescence microscopy.
It is well known that AO can pass through cell membranes, but EB cannot. Under the fluorescence microscope, living cells appear green. Necrotic cells stain red but have a nuclear morphology resembling that of viable cells. Apoptotic cells appear green, and morphological changes such as cell blebbing and formation of apoptotic bodies will be observed [29].
Representative images of the cells treated with 10 μM of HCPT (used as positive control) and 1, 5, 10 μM of compound 5f for 12 h are shown in Fig. 2a. While treatment of cells with HCPT and compound 5f, the apoptotic cells with typical apoptotic features, such as staining brightly, condense chromatin and fragment nuclei were observed. These results suggested that the proliferative inhibition and the death of target cells upon treatment with compound 5f were consequent to the induction of apoptosis.
Apoptosis induction studies of compound 5f. a AO\EB staining. b Hoechst 33,258 staining
Membrane-permeable Hoechst 33,258 was a blue fluorescent dye and stained the cell nucleus. When cells were treated with Hoechst 33,258, live cells with uniformly light blue nuclei were observed under fluorescence microscope, while apoptotic cells exhibited bright blue because of karyopyknosis and chromatin condensation, and the nuclei of dead cells could not be stained [30]. MGC-803 cells treated with compound 5f at concentrations of 1, 5, and 10 μM for 12 h were stained with Hoechst 33,258, with HCPT as positive control at 10 μM for 12 h. The results are illustrated in Fig. 2b.
Figure 2b shows that MGC-803 cells treated with the negative control DMSO were normally blue. Compared with the negative control, a part of cells with smaller nuclei and condensed staining appeared in the positive control group. After treated with compound 5f, the cells exhibited strong blue fluorescence and revealed typical apoptotic morphology. These findings demonstrate that compound 5f induced apoptosis against MGC-803 cell lines, consistent with the results for AO/EB double staining.
To further verify AO/EB and Hoechst 33,258 staining results, TUNEL assay was also carried out. TUNEL (Terminal deoxynucleotidyl Transferase Biotin-dUTP Nick End Labeling) is a very popular assay for identifying apoptotic cells. The assay identifies apoptotic cells in situ by using terminal deoxynucleotidyl transferase (TdT) to transfer biotin-dUTP to these strand breaks of cleaved DNA. The biotin-labeled cleavage sites are then detected by reaction with HRP conjugated streptavidin and visualized by DAB showing brown color [24]. MGC-803 cells treated with compound 5f at 5 μM for 6, 12, and 18 h were stained with TUNEL, with HCPT as positive control at 5 μM for 18 h. As shown in Fig. 3, cells in control group (DMSO treatment) did not appear as brown precipitates. However, the cells treated with compound 5f and HCPT appeared as brown precipitate. We further concluded that compound 5f induced apoptosis against MGC-803.
Apoptosis was assayed with TUNEL after treatment of MGC-803 cells with 5 μM 5f, and observed under light microscopy
In addition, the apoptosis ratios induced by compound 5f in MGC-803 cells were determined by flow cytometry, using Annexin V/PI double staining. Flow cytometry was performed on the total cell population (including both adherent and detached cells) and apoptosis detection was carried out as mentioned above. This double staining procedure discriminated necrotic cells (Q1, Annexin−/PI+), late apoptotic cells (Q2, Annexin+/PI+), intact cells (Q3, Annexin−/PI−) and early apoptotic cells (Q4, Annexin+/PI−) [31, 32]. As shown in Fig. 4, compound 5f could induce apoptosis of MGC-803 cells, and the highest apoptosis ratio (26.4%) was obtained after 24 h of treatment at a concentration of 10 μM. For the positive control HCPT, the apoptosis ratio was only 22.3% after 24 h of treatment at a concentration of 10 μM. In addition, as shown in Fig. 5, the apoptosis of MGC-803 cells treated with compound 5f gradually increased in a time-dependent manner.
The apoptosis ratios of MGC-803 cells treated with compound 5f and HCPT
Annexin V/PI dual staining of MGC-803 cell lines. a Negative control; b treated with HCPT at 10 μM for 24 h; c–e treated with compound 5f at 10 μM for 6, 12, and 24 h, respectively
As a development of our previous studies, we have synthesized and evaluated in vitro a series of (1E,4E)-1-aryl-5-(2-((quinazolin-4-yl)oxy)phenyl)-1,4-pentadien-3-one derivatives as potential antitumor agents. Most of the derivatives exhibited equivalent inhibitory activities against MGC-803, PC3, and Bcap-37 cancer cells. Compound 5f appeared to be more effective than other compounds against the three cells, with IC50 values of 0.85, 1.37, and 4.98 μM, respectively. And compounds 5f was found to exhibit a good degree of selectivity towards cancer cells than normal cells. In addition, the apoptosis-inducing activity of compound 5f in MGC-803 cells was investigated by AO/EB staining, Hoechst 33,258 staining, TUNEL assay, and flow cytometry. The results revealed that the compound may inhibit cell growth by inducing apoptosis, with apoptosis ratio of 26.4% at 10 μM for 24 h, which was higher than that of HCPT (22.3% at 10 μM for 24 h). Further studies on the specific mechanisms of compound 5f in MGC-803 cells are currently underway.
Melting points were determined by using an XT-4 binocular microscope (Beijing Tech Instrument Co., China) without correction. IR spectra were recorded on a Bruker VECTOR 22 spectrometer. NMR spectra were recorded in a CDCl3 solvent using a JEOL-ECX 500 NMR spectrometer operating at 500 MHz for 1H, and at 125 MHz for 13C by using TMS as internal standard. Elemental analysis was performed on an Elementar Vario-III CHN analyzer. Silica gel (200–300 mesh) and TLC plates (Qingdao Marine Chemistry Co., Qingdao, China) were used for chromatography. All solvents (Yuda Chemistry Co., Guiyang, China) were analytical grade, and used without further purification unless otherwise noted.
Synthetic procedures
6-methyl-quinazolin-4(3H)-one, 8-methyl-quinazolin-4(3H)-one, 6-methyl-4-chloroquiazoline, and 8-methyl-4-chloroquiazoline were prepared according to a previously described method [33]. Intermediate (E)-4-(2-hydroxyphenyl)-3-butylene-2-one was prepared according to a previously reported [34].
General synthetic procedures for the preparation of compounds 5a–5n
Compounds 2 (10 mmol), 3 (10 mmol) and K2CO3 (70 mmol) in 20 mL of acetonitrile was stirred at 30–40 °C for 3.5 h. The reaction mixture was concentrated and allowed to cool. The solid product obtained was filtered, and recrystallized with ethanol to afford the desired solid compound 4a or 4b, respectively. To the mixture of compound 4a or 4b (0.5 mmol) and sodium hydroxide (1%) in 20 mL of 75 vol% ethanol/water solution was added substituted aldehydes (0.5 mmol). The reaction mixture was stirred at room temperature overnight. The reaction mixture was concentrated and suspended in water (20 mL), adjusted with 5% HCl to pH 7, and filtered. Recrystallization with ethanol afforded the desired solid compounds 5a–5n.
(1E,4E)-1-(2-fluorophenyl)-5-(2-((6-methylquinazolin-4-yl)oxy)phenyl)penta-1,4-dien-3-one (5a)
Yield: 52.6%; yellow powder; mp: 121–123 °C; IR (KBr, cm−1) ν: 3442, 1657, 1622, 1596, 1465, 1398, 1356, 1221, 983; 1H NMR (CDCl3, 500 MHz) δ: 8.70 (s, 1H, Qu-2-H), 8.23 (d, J = 12.00 Hz, 1H, F–Ar–CH=), 7.93 (d, J = 8.6 Hz, 1H, Ar–CH=), 7.78–7.85 (m, 3H, Qu-5,7,8-H), 7.47–7.50 (m, 3H, F–Ar-4,6-H, Ar-3-H), 7.30–7.39 (m, 5H, F–Ar-3,5-H, Ar-4,5-H, F–Ar–C=CH), 7.10 (d, J = 16.0 Hz, 1H, Ar–C=CH), 6.81 (d, J = 14.8 Hz, 1H, Ar-6-H), 2.61 (s, 3H, CH3); 13C NMR (CDCl3, 125 MHz) δ: 188.8, 166.4, 153.4, 153.4, 151.7, 150.4, 136.9, 136.7, 136.5, 131.7, 129.4, 128.2, 127.9, 127.7, 127.2, 127.1, 126.6, 126.5, 123.6, 123.5, 122.3, 116.4, 21.9; Anal. Calcd for C25H19FN2O2: C 76.08; H 4.67; N 6.83; Found: C 76.42; H 4.78; N 6.80.
(1E,4E)-1-(2-chlorophenyl)-5-(2-((6-methylquinazolin-4-yl)oxy)phenyl)penta-1,4-dien-3-one (5b)
Yield: 46.3%; yellow powder; mp: 152–154 °C; IR (KBr, cm−1) ν: 3445, 1653, 1618, 1584, 1481, 1400, 1359, 1223, 986; 1H NMR (CDCl3, 500 MHz) δ: 8.69 (s, 1H, Qu-2-H), 8.22 (d, J = 8.0 Hz, 1H, Cl–Ar–CH=), 7.76–7.95 (m, 4H, Ar–CH=, Qu-5,7,8-H), 7.38–7.53 (m, 3H, Cl–Ar-3,6-H, Ar-3-H), 7.23–7.31 (m, 5H, Cl–Ar-4,5-H, Ar-5-H, Ar–C=CH, Cl–Ar–C=CH), 7.21 (m, 1H, Ar-4-H), 6.82 (d, J = 14.8 Hz, 1H, Ar-6-H), 2.62 (s, 3H, CH3); 13C NMR (CDCl3, 125 MHz) δ: 188.6, 167.1, 154.3, 153.1, 151.0, 142.6, 142.1, 136.5, 134.5, 133.3, 132.6, 130.0, 129.6, 129.4, 127.4, 125.8, 125.6, 122,9, 122.7, 121.2, 116.3, 17.7; Anal. Calcd for C26H19ClN2O2: C 73.2; H 4.50; N 6.56; Found: C 73.27; H 4.56; N 6.42.
(1E,4E)-1-(4-chlorophenyl)-5-(2-((6-methylquinazolin-4-yl)oxy)phenyl)penta-1,4-dien-3-one (5c)
Yield: 55.8%; yellow powder; mp: 173–176 °C; IR (KBr, cm−1) ν: 3445, 1653, 1622, 1558, 1489, 1373, 1229, 986; 1H NMR (CDCl3, 500 MHz) δ: 8.70 (s, 1H, Qu-2-H), 8.23 (d, J = 12.0 Hz, 1H, Cl–Ar–CH=), 7.93 (d, J = 8.6 Hz, 1H, Ar–CH=), 7.78–7.85 (m, 3H, Qu-5,7,8-H), 7.47–7.50 (m, 3H, Cl–Ar-2,6-H, Ar-3-H), 7.30–7.39 (m, 5H, Cl–Ar-3,5-H, Ar-4,5-H, Cl–Ar–C=CH), 7.10 (d, J = 16.0 Hz, 1H, Ar–C=CH), 6.81 (d, J = 14.8 Hz, 1H, Ar-6-H), 2.61 (s, 3H, CH3); 13C NMR (CDCl3, 125 MHz) δ: 185.7, 167.4, 154.3, 153.1, 148.1, 147.4, 134.5, 134.1, 133.5, 132.3,131.3, 130.0, 129.8, 129.2, 128.9, 127.3, 127.1, 122.8, 122.6, 121.1, 116.3, 17.8; Anal. Calcd for C26H19ClN2O2: C 73.15; H 4.49; N 6.56; Found: C 72.43; H 4.12; N 6.79.
(1E,4E)-1-(2-chloro-5-nitrophenyl)-5-(2-((6-methylquinazolin-4-yl)oxy)phenyl)penta-1,4-dien-3-one (5d)
Yield: 58.2%; yellow powder; mp: 176–178 °C; IR (KBr, cm−1) ν: 3445, 1653, 1622, 1576, 1522, 1458, 1348, 1277, 1221, 983; 1H NMR (CDCl3, 500 MHz) δ: 8.68 (s, 1H, Qu-2-H), 8.40 (s, 1H, Cl–Ar-6-H), 8.21 (d, J = 15.0 Hz, 1H, Cl–Ar-4-H), 8.10–8.12 (d, J = 10.0 Hz, 1H, Qu-8-H), 7.73–7.92 (m, 5H, Cl–Ar–CH=, Qu-5,7-H, Cl–Ar-3-H, Ar–CH=), 7.54–7.57 (m, 2H, Cl–Ar–C=CH, Ar-3-H), 6.91–7.41 (m, 4H, Ar-4,5,6-H, Ar–C=CH), 2.61 (s, 3H, CH3); 13C NMR (CDCl3, 125 MHz) δ: 187.8, 179.6, 158.5, 153.3, 151.9, 146.8, 138.1, 136.7, 136.4, 134.6, 132.1, 131.3, 130.7, 130.2, 128.4, 127.8, 126.7, 125.1, 123.6, 122.9, 122.5, 122.2, 116.7, 21.9; Anal. Calcd for C26H18N3O4: C 66.18; H 3.84; N 8.90; Found: C 65.81; H 3.66; N 9.30.
(1E,4E)-1-(2,4-dichlorophenyl)-5-(2-((6-methylquinazolin-4-yl)oxy)phenyl)penta-1,4-dien-3-one (5e)
Yield: 60.5%; yellow powder; mp: 211–214 °C; IR (KBr, cm−1) ν: 3443, 1655, 1618, 1582, 1499, 1371, 1225, 986; 1H NMR (CDCl3, 500 MHz) δ: 8.68 (s, 1H, Qu-2-H), 8.21 (s, 1H, Qu-5-H), 7.60–7.93 (m, 4H, Qu-7,8-H, Cl–Ar–CH=, Ar–CH=), 7.38–7.43 (m, 4H, Cl–Ar-3-H, Ar-3-H, Cl–Ar-5,6-H), 7.26–7.31 (m, 3H, Ar-4,5-H, Cl–Ar–C=CH), 7.12 (d, J = 16.5 Hz, 1H, Ar–C=CH), 6.80 (d, J = 16.1 Hz, 1H, Ar-6-H), 2.61 (s, 3H, CH3); 13C NMR (CDCl3, 125 MHz) δ: 188.5, 167.1, 153.4, 153.1, 151.4, 142.7, 137.9, 136.6, 136.5, 134.5, 132.3, 131.6, 130.2, 130.1, 128.7, 128.3, 127.6, 127.4, 125.0, 122.7, 121.2, 116.2, 17.7; Anal. Calcd for C26H18Cl2N2O2: C 67.69; H 3.93; N 6.07; N 7.07; Found: C 67.56; H 3.45; N 5.65.
(1E,4E)-1-(2,6-dichlorophenyl)-5-(2-((6-methylquinazolin-4-yl)oxy)phenyl)penta-1,4-dien-3-one (5f)
Yield: 55.2%; yellow powder; mp: 187–189 °C; IR (KBr, cm−1) ν: 3443, 1655, 1618, 1582, 1499, 1333, 1225, 986; 1H NMR (CDCl3, 500 MHz) δ: 8.68 (s, 1H, Qu-2-H), 8.20 (s, 1H, Qu-5-H), 7.89 (d, J = 8.5 Hz, 1H, Qu-8-H), 7.80–7.85 (m, 2H, Ar–CH=, Cl–Ar–CH=), 7.73 (d, J = 8.8 Hz, 1H, Qu-7-H), 7.52–7.61 (m, 2H, Cl–Ar-3,5-H), 7.39 (m, 1H, Cl–Ar-4-H), 7.24–7.30 (m, 3H, Ar-3,5-H, Ar–C=CH), 7.15 (m, 1H, Ar-4-H), 7.06 (d, J = 16.0 Hz, 1H, Cl–Ar–C=CH), 7.00 (d, J = 16.5 Hz, 1H, Ar-6-H), 2.60 (s, 3H, CH3); 13C NMR (CDCl3, 125 MHz) δ: 188.9, 166.4, 153.4, 151.7, 150.4, 138.4, 137.5, 136.7, 136.5, 135.2, 132.9, 132.3, 131.8, 129.9, 128.9, 128.2, 128.0, 127.9, 127.5, 126.7, 123.6, 122.2, 116.0, 21.9; Anal. Calcd for C26H18Cl2N2O2: C 67.69; H 3.93; N 6.07; Found: C 68.06; H 4.14; N 6.11.
1E,4E)-1-(2,5-dimethoxyphenyl)-5-(2-((6-methylquinazolin-4-yl)oxy)phenyl)penta-1,4-dien-3-one (5g)
Yield: 49.6%; yellow powder; mp: 122–123 °C; IR (KBr, cm−1) ν: 3443, 1653, 1618, 1576, 1497, 1458, 1360, 1223, 1114, 1045; 1H NMR (CDCl3, 500 MHz) δ: 8.68 (s, 1H, Qu-2-H), 8.22 (s, 1H, Qu-5-H), 7.81–7.92 (m, 5H, Qu-7,8-H, Ar–CH=, CH3O–Ar–CH=, Ar-3-H), 7.75 (d, J = 8.6 Hz, 1H, CH3O–Ar–C=CH), 7.51 (m, 1H, Ar-5-H), 7.38 (m, 1H, Ar-4-H), 7.17 (d, J = 16.0 Hz, 1H, Ar–C=CH), 6.99 (d, J = 2.8 Hz, 1H, Ar-6-H), 6.89–6.94 (m, 2H, CH3O–Ar-3,6-H), 6.81 (d, J = 2.8 Hz, 1H, CH3O–Ar-4-H), 3.76 (s, 6H, 2-OCH3), 2.57 (s, 3H, CH3); 13C NMR (CDCl3, 125 MHz) δ: 189.3, 166.5, 153.5, 153.4, 153.2, 151.6, 150.4, 138.8, 138.4, 136.6, 136.2, 131.5, 128.4, 128.1, 127.8, 127.1, 126.7, 126.6, 123.5, 122.4, 117.6, 113.2, 112.5, 56.1, 55.8, 21.9; Anal. Calcd for C28H24N2O4: C 74.3; H 5.35; N 6.19; Found: C 74.3; H 5.48; N 5.95.
(1E,4E)-1-(2-fluorophenyl)-5-(2-((8-methylquinazolin-4-yl)oxy)phenyl)penta-1,4-dien-3-one (5h)
Yield: 50.4%; yellow powder; mp: 155–157 °C; IR (KBr, cm−1) ν: 3445, 1653, 1620, 1582, 1506, 1481, 1398, 1223, 984; 1H NMR (CDCl3, 500 MHz) δ: 8.79 (s, 1H, Qu-2-H), 8.31 (d, J = 8.0 Hz, 1H, F–Ar–CH=), 7.77–7.85 (m, 3H, Qu-5,7-H, Ar–CH=), 7.67 (d, J = 16.5 Hz, 1H, F–Ar-6-H), 7.59 (m, 1H, Qu-6-H), 7.53 (m, 1H, F–Ar-4-H), 7.29–7.43 (m, 4H, Ar-3,5-H, F–Ar-3,5-H), 7.05–7.14 (m, 3H, Ar-4-H, F–Ar–C=CH, Ar–C=CH), 6.95 (d, J = 16.5 Hz, 1H, Ar-6-H), 2.76 (s, 3H, CH3); 13C NMR (CDCl3, 125 MHz) δ: 188.8, 167.1, 153.2, 151.7, 151.1, 136.9, 136.6, 136.0, 134.5, 131.9, 131.9, 129.3, 128.4, 128.1, 127.8, 127.8, 127.6, 126.6, 124.5, 123.6, 121.1, 116.4, 17.8; Anal. Calcd for C25H19FN2O2: C 76.08; H 4.67; N 6.83; Found: C 75.81; H 4.53; N 7.04.
(1E,4E)-1-(2-chlorophenyl)-5-(2-((8-methylquinazolin-4-yl)oxy)phenyl)penta-1,4-dien-3-one (5i)
Yield: 41.8%; yellow powder; mp: 152–154 °C; IR (KBr, cm−1) ν: 3443, 1655, 1616, 1595, 1481, 1406, 1358, 1229, 979; 1H NMR (CDCl3, 500 MHz) δ: 8.79 (s, 1H, Qu-2-H), 8.30 (d, J = 8.5 Hz, 1H, Cl–Ar–CH=), 7.96 (d, J = 16.5 Hz, 1H, Ar–CH=), 7.76–7.85 (m, 3H, Qu-5,6,7-H), 7.50-7.59 (m, 3H, Ar-3-H, Cl–Ar-3,6-H), 7.38-7.40 (m, 2H, Cl–Ar-4,5-H), 7.29–7.39 (m, 2H, Cl–Ar–C=CH, Ar–C=CH), 7.14–7.25 (m, 2H, Ar-4,5-H), 6.81 (d, J = 16.0 Hz, 1H, Ar-6-H), 2.77 (s, 3H, CH3); 13C NMR (CDCl3, 125 MHz) δ: 188.7, 167.1, 153.2, 151.7, 151.1, 139.2, 137.2, 136.6, 135.4, 134.6, 131.7, 131.3, 130.3, 128.4, 128.3, 128.1, 127.7, 127.6, 127.1, 126.6, 123.6, 121.1, 116.1, 17.8; Anal. Calcd for C26H19ClN2O2: C 73.15; H 4.49; N 6.56; Found: C 73.04; H 4.74; N 6.76%.
(1E,4E)-1-(4-chlorophenyl)-5-(2-((8-methylquinazolin-4-yl)oxy)phenyl)penta-1,4-dien-3-one (5j)
Yield: 58.6%; yellow powder; mp: 161–163 °C; IR (KBr, cm−1) ν: 3445, 1647, 1616, 1576, 1481, 1406, 1358, 1227, 937; 1H NMR (CDCl3, 500 MHz) δ: 8.79 (s, 1H, Qu-2-H), 8.20–8.34 (m, 3H, Qu-5,6,7-H), 7.72–7.86 (m, 4H, Ar–CH=, Ar-3-H, Cl–Ar–C=CH, Cl–Ar=CH), 7.52–7.64 (m, 4H, Cl–Ar-2,3,5,6-H), 7.41-7.42 (m, 1H, Ar-5-H), 7.30–7.32 (m, 1H, Ar-4-H), 7.11–7.14 (d, J = 15.0 Hz, 1H, Ar-6-H), 6.93–6.96 (d, J = 15.0 Hz, 1H, Ar–C=CH), 2.77 (s, 3H, CH3); 13C NMR (CDCl3, 125 MHz) δ: 188.6, 167.1, 153.2, 153.1, 151.7, 151.1, 141.9, 136.9, 136.7, 134.6, 131.7, 129.5, 129.2, 128.4, 128.1, 127.6, 127.2, 126.7, 125.8, 123.6, 121.1, 17.8; Anal. Calcd for C26H19ClN2O2: C 73.15; H 4.49; N 6.56; Found: C 73.36; H 4.65; N 6.86.
(1E,4E)-1-(2-chloro-5-nitrophenyl)-5-(2-((8-methylquinazolin-4-yl)oxy)phenyl)penta-1,4-dien-3-one (5k)
Yield: 54.5%; yellow powder; mp: 198–200 °C; IR (KBr, cm−1) ν: 3420, 1676, 1626, 1560, 1522, 1479, 1402, 1348, 1221, 980; 1H NMR (CDCl3, 500 MHz) δ: 8.78 (s, 1H, Qu-2-H), 8.39 (s, 1H, Cl–Ar-6-H), 8.32 (d, J = 8.0 Hz, 1H, Cl–Ar-4-H), 8.13 (d, J = 8.3 Hz, 1H, Qu-5-H), 7.76-7.88 (m, 4H, Ar–CH=, Cl–Ar–CH=, Qu-6, 7-H), 7.45–7.59 (m, 3H, Ar-3-H, Cl–Ar-3-H, Cl–Ar–C=CH), 7.23–7.40 (m, 2H, Ar-4,5-H), 7.10 (d, J = 12.5 Hz, 1H, Ar–C=CH), 6.93 (d, J = 16.0 Hz, 1H, Ar-6-H), 2.75 (s, 3H, CH3);13C NMR (CDCl3, 125 MHz) δ: 187.8, 167.1, 153.1, 151.8, 151.0, 146.7, 141.6, 138.2, 136.7, 136.6, 134.6, 132.0, 131.3, 130.1, 128.6, 127.7, 126.9, 126.7, 125.1, 123.7, 122.5, 120.9, 116.1, 17.7; Anal. Calcd for C26H18ClN3O4: C 66.18; H 3.84; N 8.90; Found: C 66.30; H 3.84; N 8.86.
(1E,4E)-1-(2,4-dichlorophenyl)-5-(2-((8-methylquinazolin-4-yl)oxy)phenyl)penta-1,4-dien-3-one (5l)
Yield: 58.6%; yellow powder; mp: 175–178 °C; IR (KBr, cm−1) ν: 3445, 1653, 1618, 1576, 1481, 1408, 1358, 1229, 984; 1H NMR (CDCl3, 500 MHz) δ: 8.79 (s, 1H, Qu-2-H), 8.30 (d, J = 8.0 Hz, 1H, Cl–Ar–CH=), 7.76–7.94 (m, 3H, Ar–CH=, Qu-5,7-H), 7.53–7.57 (m, 2H, Qu-6-H, Cl–Ar-3-H), 7.38–7.47 (m, 3H, Ar-3-H, Cl–Ar-5,6-H), 7.29–7.31 (m, 2H, Cl–Ar-4-H), 7.38–7.41 (m, 2H, Cl–Ar–C=CH, Ar-5-H), 7.15–7.17 (m, 2H, Ar–C=CH, Ar-4H), 6.78 (d, J = 16.5 Hz, 1H, Ar-6-H), 2.77 (s, 3H, CH3); 13C NMR (CDCl3, 125 MHz) δ: 188.5, 167.1, 153.2, 151.8, 151.1, 139.1, 137.5, 136.7, 135.4, 134.6, 134.1, 133.3, 131.8, 131.7, 129.2, 128.4, 128.0, 127.6, 127.4, 126.7, 125.8, 123.6, 121.0, 116.1, 17.7; Anal. Calcd for C26H18Cl2N2O2: C 67.69; H 3.93; N 6.07; Found: C 67.27; H 4.03; N 5.96%.
(1E,4E)-1-(2,6-dichlorophenyl)-5-(2-((8-methylquinazolin-4-yl)oxy)phenyl)penta-1,4-dien-3-one (5m)
Yield: 56.1%; yellow powder; mp: 161–163 °C; IR (KBr, cm−1) ν: 3421, 1676, 1620, 1587, 1481, 1400, 1359, 1225, 984; 1H NMR (CDCl3, 500 MHz) δ: 8.76 (s, 1H, Qu-2-H), 8.28 (d, J = 8.5 Hz, 1H, Ar–CH=), 7.73–7.85 (m, 3H, Cl–Ar–CH=, Qu-5,7-H), 7.52–7.59 (m, 3H, Cl–Ar-3,5-H, Qu- 6-H), 7.29–7.41 (m, 4H, Ar-3, 5-H, Cl–Ar-4-H, Cl–Ar–C=CH), 7.16 (m, 1H, Ar-4-H), 7.07 (d, J = 16.0 Hz, 1H, Ar–C=CH), 6.98 (d, J = 17.0 Hz, 1H, Ar-6-H), 2.75 (s, 3H, CH3); 13C NMR (CDCl3, 125 MHz) δ: 188.96, 167.10, 153.15, 151.70, 150.99, 137.59, 136.75, 135.68, 135.17, 134.57, 132.97, 131.85, 129.88, 128.85, 128.36, 127.88, 127.65,127.45, 126.72, 123.60, 121.04, 116.02, 17.79; Anal. Calcd for C26H18Cl2N2O2 (461): C, 67.69; H, 3.93; N, 6.07; N, 7.07%. Found: 67.36; H, 3.96; N, 5.84%.
(1E,4E)-1-(2,5-dimethoxyphenyl)-5-(2-((8-methylquinazolin-4-yl)oxy)phenyl)penta-1,4-dien-3-one (5n)
Yield: 43.6%; yellow powder; mp: 176–178 °C; IR (KBr, cm−1) ν: 3445, 1647, 1616, 1570, 1491, 1373, 1211, 984; 1H NMR (CDCl3, 500 MHz) δ: 8.79 (s, 1H, Qu-2-H), 8.30 (d, J = 8.6 Hz, 1H, CH3O–Ar–CH=), 7.75–7.92 (m, 4H, Ar–CH=, Qu-5,6,7-H), 7.50–7.59 (m, 2H, Ar-3,5-H), 7.39 (m, 1H, Ar-4-H), 7.15–7.29 (m, 2H, CH3O–Ar–C=CH, Ar–C=CH), 6.98 (s, 1H, CH3O–Ar-6-H), 6.89–6.93 (m, 2H, Ar-6-H, CH3O–Ar-3-H), 6.81 (d, J = 8.6 Hz, 1H, CH3O–Ar-4-H), 3.77 (s, 6H, 2CH3O), 2.76 (s, 3H, CH3); 13C NMR (CDCl3, 125 MHz) δ: 189.25, 167.14, 153.57, 153.18, 151.66, 151.03, 138.78, 136.58, 136.25, 134.51, 131.45, 128.41, 128.23, 127.55, 127.24, 126.58, 124.21, 123.54, 121.16, 120.94, 117.61, 116.16, 113.22, 112.47, 56.08, 55.85, 17.75. Anal. Calcd for C28H24N2O4 (453): C, 74.32; H, 5.35; N, 6.19; %. Found: C, 74.55; H, 5.68; N, 5.95%.
Human gastric cancer cell line MGC-803, human prostate cancer cell line PC3, and human breast cancer cell line Bcap-37 and one normal cell line NIH3T3 were obtained from Cell Bank of Type Culture Collection of Chinese Academy of Sciences (Shanghai, China). NIH3T3 was routinely maintained in a DMEM medium, while all the other cell lines were cultured in a 1640 medium. All the cells were grown in the medium supplemented with 10% FBS at 37 °C with 5% CO2.
MTT assay
The growth-inhibitory effects of the test compounds were determined on MGC-803, PC3, Bcap-37, and NIH3T3 cells. All cell types were seeded into 96-well plates at a density of 2 × 103 cells/well 100 μL of the proper culture medium and incubated with increasing concentrations of the compounds at 37 °C under cell culturing conditions. An MTT assay (Roche Molecular Biochemicals, 1465-007) was performed 72 h later according to the instructions provided by Roche. The precipitated formazan crystals were dissolved in SDS, and the absorbance was read at 595 nm with a microplate reader (BIO-RAD, model 680), which is directly proportional to the number of living cells in culture. The experiment was performed in triplicate. The percentage cytotoxicity was calculated using the formula.
$$\% {\text{Cytotoxicity}} = \left[ {\left( {{\text{Control}}_{\text{abs}} - {\text{Blank}}_{\text{abs}} } \right) - \left( {{\text{Test}}_{\text{abs}} - {\text{Blank}}_{\text{abs}} } \right)} \right]/\left( {{\text{Control}}_{\text{abs}} - {\text{Blank}}_{\text{abs}} } \right)\; \times \; 100$$
AO/EB staining
Cells were seeded in 6-well culture plates at a density of 5 × 104 cells/mL in 0.6 mL of medium and allowed to adhere to the plates overnight. The cells were incubated with different concentrations of compounds or vehicle solution (0.1% DMSO) in a medium containing 10% FBS for 12 h. After the treatment, the cover slip with monolayer cells was inverted on the glass slide with 20 μL of AO/EB stain (100 μg/mL), and finally analyzed for morphological characteristics of cell apoptosis under a fluorescence microscope (Olympus Co., Japan).
Hoechst 33,258 staining
Cells were seeded in 6-well culture plates at a density of 5 × 104 cells/mL in 0.6 mL of medium and allowed to adhere to the plates overnight. The cells were incubated with different concentrations of compounds or vehicle solution (0.1% DMSO) in a medium containing 10% FBS for 12 h. After the treatment, the cells were fixed with 4% paraformaldehyde for 10 min, followed by incubation with Hoechst 33,258 staining solution (Beyotime) for 5 min and finally analyzed for morphological characteristics of cell apoptosis under a fluorescence microscope (Olympus Co., Japan).
Flow cytometry analysis
To further quantitative analysis of apoptosis, the cells were washed with PBS, stained with annexinV-FITC and propidium iodide (PI) using the AnnexinV-FITC kit (KeyGEN BioTECH). The cells were then subjected to flow cytometry according to manufacturer's instructions and the stained cells were analyzed by FACS can flow cytometer (Becton–Dickinson, CA, USA).
All statistical analysis was performed with SPSS Version 19.0. Data was analyzed by one-way ANOVA. Mean separations were performed using the least significant difference method. Each experiment was replicated thrice, and all experiments yielded similar results. Measurements from all the replicates were combined, and treatment effects were analyzed.
ADM:
adriamycin
AO/EB:
acridine orange/ethidium bromide
13C NMR:
13C nuclear magnetic resonance
DMSO:
FCM:
HCPT:
10-hydroxyl camptothecine
1H NMR:
proton nuclear magnetic resonance
IR:
infra-red
MTT:
3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide
TUNEL:
terminal deoxynucleotidyl transferase biotin-dUTP nick end labeling
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HL and SY synthesized the compounds and carried out most of the bioassay experiments. DH took part in the compound structural elucidation and bioassay experiments. WX carried out some structure elucidation experiments. PX assisted in structural elucidation experiments. All authors read and approved the final manuscript.
The authors wish to thank the Scientific Research of Guizhou (No. 20126006) for the financial support.
Guizhou Fruit Institute, Guizhou Academy of Agricultural Sciences, Guiyang, 550006, P. R. China
Hui Luo & Pu Xie
R&D Center, Sinphar Tian-Li Pharmaceutical Co., Ltd, Hangzhou, 311100, P. R. China
Shengjie Yang & Da Hong
State Key Laboratory Breeding Base of Green Pesticide and Agricultural Bioengineering, Key Laboratory of Green Pesticide and Agricultural Bioengineering, Ministry of Education, Guizhou University, Guiyang, 550025, P. R. China
Hui Luo, Shengjie Yang & Wei Xue
Ctr for R&D of Fine Chemicals, Guizhou University, Guiyang, 550025, P. R. China
Hui Luo
Shengjie Yang
Da Hong
Wei Xue
Pu Xie
Correspondence to Hui Luo.
Hui Luo and Shengjie Yang contributed equally to this work
Luo, H., Yang, S., Hong, D. et al. Synthesis and in vitro antitumor activity of (1E,4E)-1-aryl-5-(2-((quinazolin-4-yl)oxy)phenyl)-1,4-pentadien-3-one derivatives. Chemistry Central Journal 11, 23 (2017). https://doi.org/10.1186/s13065-017-0253-9
Asymmetric curcumin analogs
Quinazoline derivatives of curcumin
Antitumor activity
MGC-803 | CommonCrawl |
Dario Graffi
Dario Graffi (10 January 1905 – 28 December 1990) was an influential Italian mathematical physicist, known for his researches on the electromagnetic field, particularly for a mathematical explanation of the Luxemburg effect,[3] for proving an important uniqueness theorem for the solutions of a class of fluid dynamics equations including the Navier-Stokes equation,[4] for his researches in continuum mechanics and for his contribution to oscillation theory.[5]
Dario Graffi
Born(1905-01-10)10 January 1905
Rovigo
Died28 December 1990(1990-12-28) (aged 85)
Bologna
NationalityItalian
Awards
• Golden medal "Benemeriti della Scuola, della Cultura, dell'Arte" (1964)[1]
• Prize of the President of the Italian Republic (1965)[2]
Scientific career
Fields
• Continuum mechanics
• Theory of electromagnetism
• Fluid dynamics
• Oscillation theory
Academic advisors
• Pietro Burgatti
• Quirino Majorana
• Emanuele Foà
Notable studentsLamberto Cesari
Life and academic career
Dario Graffi was born in Rovigo, the son of Michele, a yarn wholesale trader and of Amalia Tedeschi.[6] He attended the Istituto tecnico in his home town, specializing in physics and mathematics, but got his diploma in Bologna in 1921, where his family had moved a year before.[6]
He graduated from the University of Bologna in Physics in 1925,[7] when he was 20,[8] and in mathematics in 1927,[7] when he was 22:[8] both the degrees were awarded cum laude,[9]
Honors
He was awarded the Golden medal "Benemeriti della Scuola, della Cultura, dell'Arte" in 1964, and a year later, the Accademia Nazionale dei Lincei awarded him the Prize of the President of the Italian Republic.[2]
Work
Research activity
Graffi is known for his researches on the electromagnetic field, particularly for a mathematical explanation of the Luxemburg effect, for proving an important uniqueness theorem for the solutions of a class of fluid dynamics equations including the Navier-Stokes equation,[10] for his researches in continuum mechanics and for his contribution to oscillation theory.
Selected publications
Graffi published 181 works.[11] lists of his publications are included in references (Cercignani 1992, pp. 108–114) and in the biographical section of his "Selected works" (1999, pp. XX–XXVI): however, the set of lecture notes (Graffi 1981) is not listed in any of his publication lists.
Scientific papers
• Graffi, Dario (1936), "Una teoria ereditaria dell'effetto Lussemburgo" [An hereditary theory of the Luxemburg effect], Rendiconti del Seminario Matematico della Università di Padova (in Italian), 7: 36–54, JFM 62.0546.02, Zbl 0015.13705. In this paper, written only few years after the discovery of the effect itself, Dario Graffi proposes a theory of the Luxemburg effect based on Volterra's theory of hereditary phenomena.
• Graffi, Dario (1949) [14 April 1942], "Sul modello del Rocard per le oscillazioni di rilassamento" [On Rocard's model for relaxation oscillations], Atti della Accademia delle Scienze di Ferrara (in Italian), 21–22 – Anni Accademici (1938–1943): 255–257
• Graffi, Dario (1953), "Il Teorema di Unicita nella Dinamica dei Fluidi Compressibili" [The uniqueness theorem in the dynamics of compressible fluids], Journal of Rational Mechanics and Analysis (in Italian), 2 (1): 99–106, doi:10.1512/iumj.1953.2.52004, ISSN 0022-2518, MR 0052270, Zbl 0050.19604. In this paper, Graffi extends to compressible viscous fluids a uniqueness theorem for the solutions to Navier-Stokes equation in bounded domains, previously proved only for incompressible fluids by Emanuele Foà and rediscovered by David Dolidze.[12]
• Graffi, Dario (1955), "Il teorema di unicità per i fluidi incompressibili, perfetti, eterogenei" [The uniqueness theorem for incompressible, perfect, heterogeneous fluids] (PDF), Revista de la Unión Matemática Argentina y de la Asociación Física Argentina (in Italian), XVII: 73–77, ISSN 0041-6932, MR 0082829, Zbl 0074.20206.
• Graffi, Dario (2 November 1959), "Sur un théorème d'unicité pour le mouvement d'un fluide visqueux dans un domaine illimité" [On a uniqueness theorem for the motion of a viscous fluid on an unbounded domain], Comptes rendus hebdomadaires des séances de l'Académie des sciences (in French), 249 (2): 1741–1743, MR 0109550, Zbl 0088.41201, available at Gallica. A short research note announcing the results of the author on the uniqueness of solutions of the Navier-Stokes equations on unbounded domains under the hypothesis of constant fluid velocity at infinity.
• Graffi, Dario (1960), "Sul teorema di unicità nella dinamica dei fluidi" [On the uniqueness theorem in fluid mechanics], Annali di Matematica Pura ed Applicata, IV Serie (in Italian), 50: 379–387, doi:10.1007/BF02414524, ISSN 0373-3114, MR 0122198, S2CID 122474925, Zbl 0102.41103 (online version ISSN 1618-1891). In this paper, Graffi extends his uniqueness theorem for the solutions of Navier-Stokes equations on unbounded domains relaxing previously assumed hypotheses on the behaviour of the velocity at infinity.
• Graffi, Dario (1962), "Sui teoremi di unicita nella dinamica dei fluidi" [On uniqueness theorems in fluid dynamics], Rendiconti del Seminario Matematico e Fisico di Milano (in Italian), 32 (1): 80–91, doi:10.1007/BF02925666, MR 0148304, S2CID 117119086, Zbl 0128.43503. The published text of a conference held at the Seminario Matematico e Fisico di Milano, exposing mainly his researches on the uniqueness of the solutions to the Navier-Stokes equations.
• Graffi, Dario (December 1974), "Sull'espressione dell'energia libera nei materiali viscoelastici lineari" [On the formula of free energy in linear viscoelastic materials], Annali di Matematica Pura ed Applicata, IV Serie (in Italian), 98: 273–279, doi:10.1007/BF02414027, ISSN 0373-3114, MR 0345497, S2CID 120967727, Zbl 0281.73027 (online version ISSN 1618-1891). In this paper, Graffi introduces the free energy now called Graffi–Volterra free energy after him.
• Graffi, Dario; Fabrizio, Mauro (1989), "Sulla nozione di stato per materiali viscoelastici di tipo "rate"" [On the notion of state for "rate" type viscoelastic materials], Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali, Serie VIII (in Italian), 83 (1): 201–208, MR 1142459, Zbl 0732.73023.
• Graffi, Dario; Fabrizio, Mauro (1989), "Non unicità dell'energia libera per materiali viscoelastici" [Non uniqueness of free energy for viscoelastic materials], Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali, Serie VIII (in Italian), 83 (1): 209–214, MR 1142460, Zbl 0732.73024.
Books
• Graffi, Dario (1980), Nonlinear partial differential equations in physical problems, Research Notes in Mathematics, vol. 42, Boston–London–Melbourne: Pitman Advanced Publishing Program, pp. IV+105, ISBN 978-0-273-08474-7, ISSN 0269-3674, MR 0580946, Zbl 0453.35001, reviewed by Ablowitz, Mark J. (1984), "Nonlinear Partial Differential Equations in Physical Problems. by D. Graffi", SIAM Review, 26 (1): 125, doi:10.1137/1026014, ISSN 0036-1445, JSTOR 2029690, by Konhauser, Joseph (April 1982), "Telegraphic Reviews. Differential Equations, P*. Nonlinear Partial Differential Equations in Physical Problems. D. Graffi", The American Mathematical Monthly, 89 (4): C25–C36, ISSN 0002-9890, JSTOR 2320227 (also available with online ISSN 1930-0972), and by Marseglia, E. (1981), "Book reviews. Nonlinear Partial Differential Equations in Physical Problems. By D. Graffi", Philosophical Magazine B, 49 (1): 184, Bibcode:1981PMagB..43..183., doi:10.1080/01418638108225813, ISSN 1364-2812 (also available with online ISSN 1463-6417).
• Graffi, Dario (1981), Questioni sull'elettromagnetismo [Issues on electromagnetism] (in Italian), Napoli: Liguori editore, p. 247, ISBN 978-88-207-1136-8. A set of lecture notes of a course held by Graffi in the years 1977–1978.
• Graffi, Dario (November 1999), Fabrizio, M.; Grioli, G.; Renno, P. (eds.), Opere scelte [Selected works] (in Italian, English, and French), Bologna: C.N.R. – Gruppo Nazionale per la Fisica Matematica, pp. XXXII+458. Dario Graffi's "Selected works", containing a choice of his research papers reprinted in their original typographical form.
Historical, commemorative and survey works
• Graffi, Dario (1949–1950), "Emanuele Foà", Rendiconto delle Sessioni dell'Accademia delle Scienze dell'Istituto di Bologna, Classe di Scienze Fisiche, nuova serie (in Italian), LIV: 46–54. An obituary, with a list of his publications.
• Graffi, Dario (1975), "L'Elettromagnetismo in Levi-Civita", in Segre, Beniamino; Cattaneo, Carlo; Bompiani, Enrico; Colombo, Giuseppe; Finzi, Bruno; Graffi, Dario; Radicati di Brozolo, Luigi; Tricomi, Francesco Giacomo (eds.), Tullio Levi-Civita. Convegno internazionale celebrativo del centenario della nascita (Roma, 17–19 dicembre 1973) [Tullio Levi-Civita. International congress for the celebration of the centenary of his birth (Rome, 17–19 December 1973)], Atti dei Convegni Lincei (in Italian), vol. 8, Roma: Accademia Nazionale dei Lincei, pp. 171–177, ISSN 0391-805X. "Electromagnetism in the work of Levi-Civita" (English translation of the contribution title) is a survey of some of the works of Levi-Civita on the theory of electromagnetism.
• Graffi, Dario (1982), "Un grande accademico di Modena: Agostino Cauchy" [A great academician of Modena: Augustin Cauchy], Accademia Nazionale di Scienze Lettere e Arti di Modena. Atti, C. S., Serie VI (in Italian), XXIV: 345–355. The published text of the prolusion to the opening of the 1976 academic year, commemorating Augustin Louis Cauchy and describing his relationships with the Accademia Nazionale di Scienze Lettere e Arti di Modena, the first one in Italy having elected him member.
• Graffi, Dario (1992), "L'opera di Vito Volterra sui fenomeni ereditari e alcune sue conseguenze", in Amaldi, E.; Amerio, L.; Fichera, G.; Gregory, T.; Grioli, G.; Martinelli, E.; Montalenti, G.; Pignedoli, A.; Salvini, Giorgio; Scorza Dragoni, Giuseppe (eds.), Convegno internazionale in memoria di Vito Volterra (8–11 ottobre 1990) [International congress in memory of Vito Volterra (8–11 October 1990)], Atti dei Convegni Lincei (in Italian), vol. 92, Roma: Accademia Nazionale dei Lincei, pp. 39–76, ISSN 0391-805X, MR 1783028, Zbl 0977.01022, archived from the original on 2017-01-07, retrieved 2015-07-26. "The work of Vito Volterra on hereditary phenomena and some of their consequences" is an ample technical survey paper on the research work of Vito Volterra on hereditary phenomena in mathematical physics.
See also
• Gaetano Fichera
• Heat equation
• Ordinary differential equation
• Wave propagation
Notes
1. Cercignani (1992, p. 104),(Ridolfi 1976, p. 357).
2. Cercignani (1992, p. 104), (Ridolfi 1976, p. 357).
3. (Morro 1993, p. 43).
4. See for example (Serrin 1963, §5), (Morro 1993, p. 42), (Fabrizio 2012, p. 184).
5. (Fabrizio 2012, pp. 184–185).
6. See reference (Consiglio Scientifico del G.N.F.M. 1999, p. XI) or its English translation in (Fabrizio 2012, p. 189).
7. (Ridolfi 1976, p. 357), (Cercignani 1992, p. 101, 1992a, p. III)
8. (Morro 1993, p. 42).
9. (Ridolfi 1976, p. 357), (Cercignani 1992, p. 101, 1992a, p. III): Morro (1993, p. 42) seems to state only that he graduated cum laude in physics.
10. See for example (Serrin 1963, §5).
11. According to Cercignani (1992, p. 104): Morro (1993, p. 42) states that Graffi published 181 scientific works.
12. See (Serrin 1959, p. 251, footnote 1).
References
Biographical references
• Accademia delle Scienze di Torino (2013), Dario GRAFFI, Accademia delle Scienze di Torino, retrieved 22 June 2014.
• Accademia Nazionale dei Lincei (2012), Annuario dell'Accademia Nazionale dei Lincei 2012 – CDX dalla Sua Fondazione (PDF) (in Italian), Roma: Accademia Nazionale dei Lincei, p. 734, archived from the original (PDF) on 2016-03-04, retrieved 2014-06-22. The "Yearbook" of the renowned Italian scientific institution, including an historical sketch of its history, the list of all past and present members as well as a wealth of informations about its academic and scientific activities.
• Barbieri, Francesco; Taddei, Ferdinando (2006), L'Accademia di Scienze, Lettere e Arti di Modena dalle origini (1683) al 2005. Tomo I – La storia e i soci [The Academy of Sciences, Letters and Arts of Modena from its origins (1963) to 2005. Tome I – History and members] (PDF) (in Italian), Modena: Mucchi Editore, p. 359, ISBN 978-88-7000-419-9, archived from the original (PDF) on 2015-11-06, retrieved 2014-06-22. The first part ("Tomo") of an extensive work on the "Accademia di Scienze, Lettere e Arti di Modena", reporting the history of the academy and biographies of members up to the year 2006.
• Cagiano De Azevedo, Paola; Gerardi, Elvira, eds. (2005), Reale Accademia d'Italia. Inventario dell'archivio [Royal Academy of Italy. Inventory of the archive] (PDF), Pubblicazioni degli Archivi di Stato - Strumenti (in Italian), vol. CLXVII, Roma: Ministero per i Beni Culturali e Ambientali – Dipartimento per i Beni Archivistici e Librari – Direzione Generale per gli Archivi, pp. lxxxiv+492, ISBN 978-88-7125-264-3, freely available from the Ministero per i Beni Culturali e Ambientali - Dipartimento per i Beni Archivistici e Librari - Direzione Generale per gli Archivi. The complete inventory of the Reale Accademia d'Italia, which incorporated the Accademia Nazionale dei Lincei between 1939 and 1944.
• Cercignani, Carlo (1992a), "Dario Graffi", Annali di Matematica Pura ed Applicata (in Italian), 161 (1): III, doi:10.1007/BF01759629, eISSN 1618-1891, ISSN 0373-3114, S2CID 186230528.
• Papini, Pier Luigi (1991), "Dario Graffi", Notiziario dell'Unione Matematica Italiana (in Italian), anno XVII (1–2): 53, ISSN 0393-0998.
• Ridolfi, Roberto, ed. (1976), "Dario Graffi", Biografie e bibliografie degli Accademici Lincei [Biographies and bibliographies of the Lincean Academicians] (in Italian), Roma: Accademia Nazionale dei Lincei, pp. 357–363. The biographical and bibliographical entry (updated up to 1976) on Dario Graffi, published under the auspices of the Accademia dei Lincei in a book collecting many profiles of its living members up to 1976.
• Zanobetti, Dino (2002), "Giulio Supino e Emanuele Foà", in Mirri, Domenico; Arienti, Stefano (eds.), La cattedra negata. Dal giuramento di fedeltà al fascismo alle leggi razziali nell'Università di Bologna [The denied professorship. From the allegiance oath to fascism to the racial laws in the University of Bologna] (in Italian), Bologna: CLUEB, pp. 85–94, doi:10.1400/34421, ISBN 978-88-491-1848-3. Recollections of Giulio Supino and Emanuele Foà by Dino Zanobetti, professor emeritus of Electrical engineering and one of their former students, with some notices on the first years of the academic career of Dario Graffi.
General references
• Cercignani, Carlo (1992), (with a contribution of Gaetano Fichera), "Dario Graffi", Atti della Accademia Nazionale dei Lincei. Classe delle Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Supplemento (in Italian), 3: 101–114, ISSN 1121-3094, Zbl 0785.01015, with publication list.
• Consiglio Scientifico del G.N.F.M. (November 1999), "Dario Graffi", in Fabrizio, M.; Grioli, G.; Renno, P. (eds.), Opere scelte [Selected works] (in Italian), Bologna: C.N.R. – Gruppo Nazionale per la Fisica Matematica, pp. XI–XXXII, translated as:
• Fabrizio, Mauro (2012), "Dario Graffi in a Complex Historical Period", in Coen, Salvatore (ed.), Mathematicians in Bologna 1861–1960, translated by Bosello, Carlo Alberto, Basel: Birkhäuser, pp. 179–195, doi:10.1007/978-3-0348-0227-7_7, ISBN 978-3-0348-0226-0, MR 2934582, with publication list.
• Morro, Angelo (1993), "Dario Graffi", Atti dell'Accademia Ligure di Scienze e Lettere (in Italian), Genova, XLIX – Annata 1992: 42–44.
• Nardini, Renato; Caprioli, Luigi (1992), "Dario Graffi", Bollettino dell'Unione Matematica Italiana, Sezione A, Serie VII (in Italian), 6 (2): 297–307, ISSN 0392-4033, MR 1177932, Zbl 0755.01027, with publication list.
Scientific references
• Fabrizio, Mauro (1994), "Su alcune importanti ricerche di Dario Graffi" [On some important researches of Dario Graffi], Atti e Memorie della Accademia Nazionale di Scienze Lettere e Arti di Modena, Serie VII (in Italian), X (1992–93): 23–27. A short commemoration surveying Graffi's contribution to continuum physics, specifically to elasticity, electromagnetism and the mathematical theory of hereditary phenomena.
• Franchi, Franca; Straughan, Brian (2002), "Convection, diffusion and pollution: the model of Dario Graffi", in Capitani, Ovidio; Capriz, Gianfranco; Caputo, Michele; Fabrizio, Mauro; Graffi, Sandro; Grioli, Giuseppe; Podio Guidugli, Paolo; Renno, Pasquale (eds.), Nuovi progressi nella fisica matematica dall'eredità di Dario Graffi – Convegno Internazionale (Bologna, 24–27 maggio 2000) [New advances in mathematical physics from Dario Graffi's heritage], Atti dei Convegni Lincei, vol. 177, Roma: Accademia Nazionale dei Lincei, pp. 257–265, ISBN 978-88-218-0868-5, ISSN 0391-805X.
• Nardini, Renato (1994), "Una sintesi dell'opera scientifica di Dario Graffi" [A synthesis of the scientific work of Dario Graffi], Atti e Memorie della Accademia Nazionale di Scienze Lettere e Arti di Modena, Serie VII (in Italian), X (1992–93): 19–22. A a short commemorative survey of Graffi's scientific contributions.
• Serrin, James (1959), "Mathematical principles of classical fluid mechanics", in Flügge, Siegfried; Truesdell, Clifford A. (eds.), Fluid Dynamics I/Strömungsmechanik I, Handbuch der Physik (Encyclopedia of Physics), vol. VIII/1, Berlin–Heidelberg–New York: Springer-Verlag, pp. 125–263, Bibcode:1959HDP.....8..125S, doi:10.1007/978-3-642-45914-6_2, ISBN 978-3-642-45916-0, MR 0108116, Zbl 0102.40503.
• Serrin, James (1963), "The initial Value problem for the Navier-Stokes equations", in Langer, Rudolph E. (ed.), Nonlinear problems. Proceedings of a symposium conducted by the Mathematics Research Center, United States Army, at the University of Wisconsin, Madison, April 30-May 2, 1962., Madison: The University of Wisconsin Press, pp. 69–98, hdl:2027/uc1.b3836930, MR 0150444, Zbl 0115.08502.
• Straughan, Brian (2004) [1992], The energy method, stability, and nonlinear convection, Applied Mathematical Sciences, vol. 91 (2nd revised ed.), New York: Springer-Verlag, pp. xii+447, doi:10.1007/978-0-387-21740-6, ISBN 978-0-387-00453-2, MR 2003826, Zbl 1032.76001
• Straughan, Brian (2008), Stability and wave motion in porous media, Applied Mathematical Sciences, vol. 165, New York: Springer-Verlag, pp. xiv+437, doi:10.1007/978-0-387-76543-3, ISBN 978-0-387-76541-9, MR 2433781, Zbl 1149.76002
• Straughan, Brian (2011), Heat waves, Applied Mathematical Sciences, vol. 177, New York: Springer-Verlag, pp. xii+318, doi:10.1007/978-1-4614-0493-4, ISBN 978-1-4614-0492-7, MR 2663899, Zbl 1232.80001
Publications dedicated to him
• AA. VV. (December 1976), "Issue dedicated to Dario Graffi", Annali di Matematica (in French, English, and Italian), 108 (1): 1–389, ISSN 0373-3114 (e–ISSN 1618-1891) (subscription required).
• Capitani, Ovidio; Capriz, Gianfranco; Caputo, Michele; Fabrizio, Mauro; Graffi, Sandro; Grioli, Giuseppe; Podio Guidugli, Paolo; Renno, Pasquale, eds. (2002), Nuovi progressi nella fisica matematica dall'eredità di Dario Graffi – Convegno Internazionale (Bologna, 24–27 maggio 2000) [New advances in mathematical physics from Dario Graffi's heritage – International Congress (Bologna, 24–27 May 2000)], Atti dei Convegni Lincei (in English, French, and Italian), vol. 177, Roma: Accademia Nazionale dei Lincei, p. 304, ISBN 978-88-218-0868-5, ISSN 0391-805X. The proceedings of the international congress, held in Bologna on May 24–27, 2000, at the Accademia delle Scienze dell'Istituto di Bologna, under the auspices of the Accademia Nazionale dei Lincei.
External links
• Morando, Adriano (2002), "GRAFFI, Dario", Enciclopedia Treccani, Dizionario Biografico degli Italiani (in Italian), vol. LVIII, retrieved December 29, 2015. The biographical entry about Dario Graffi in the "Dizionario Biografico degli Italiani (Biographical Dictionary of Italians)" section of the Enciclopedia Treccani.
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| Wikipedia |
\begin{document}
\title{$M$-vector analogue for the $\cv\dv$-index}
\author{Kalle Karu}
\address{Department of Mathematics\\ University of British Columbia \\
1984 Mathematics Road\\
Vancouver, B.C. Canada V6T 1Z2}
\email{[email protected]}
\begin{abstract} A well-known conjecture of McMullen, proved by Billera, Lee and Stanley, describes the face numbers of simple polytopes. The necessary and sufficient condition is that the toric $g$-vector of the polytope is an $M$-vector, that is, the vector of dimensions of graded pieces of a standard graded algebra $A$. Recent work by Murai, Nevo and Yanagawa suggests a similar condition for the coefficients of the $\cv\dv$-index of a poset $P$. The coefficients of the $\cv\dv$-index are conjectured to be the dimensions of graded pieces in a standard multigraded algebra $A$. We prove the conjecture for simplicial spheres and we give numerical evidence for general shellable spheres. In the simplicial case we construct the multi-graded algebra $A$ explicitly using lattice paths.
\end{abstract}
\maketitle
\section{Introduction}
\setcounter{equation}{0}
We start by recalling the conjecture of McMullen \cite{McMullen}, proved by Billera, Lee and Stanley \cite{BilleraLee, Stanley}, that describes all possible face numbers of simple polytopes. A polytope $P$ is simple if its normal fan is simplicial. Let $f_i$ be the number of $i$-dimensional cones in the normal fan, $i=0,\ldots,n=\dim P$. One encodes these numbers in the toric $h$-vector $(h_0,\ldots,h_n)$, using the equality of polynomials \[ \sum_{i=0}^n h_i t^i = \sum_{i=0}^n f_i(t-1)^{n-i}.\] The $h$-numbers satisfy the Euler's equation $h_0=1$ and the Dehn-Sommerville equations $h_i = h_{n-i}$. Define the toric $g$-vector $(g_0,\ldots,g_{\lfloor n/2 \rfloor})$ by \[ g_0=1,\quad g_i = h_i-h_{i-1}, \text{ for } i>0.\] Then all linear relations among the face numbers $f_i$ are given by $g_0=1$, $g_i\geq 0$. However, there are additional non-linear relations. The precise condition is that there exists a standard graded $k$-algebra $A = \oplus_i A_i$, with $A_0=k$ the ground field and $A$ generated in degree one, such that \[ g_i = \dim A_i, \quad \text{ for } i \geq 0.\] Such vectors $(g_i)$ that are given by dimensions of graded pieces of a standard graded algebra $A$ are called M-vectors (after Macaualy) and they can be described by nonlinear inequalities.
Murai and Nevo \cite{MuraiNevo} conjectured that similar nonlinear relations hold among coefficients of the $\cv\dv$-index of a poset $P$. Further evidence for this conjecture was given by Murai and Yanagawa \cite{MuraiYanagawa}. The conjecture is that there exists a standard multigraded algebra $A$, such that the dimensions of the graded pieces of $A$ give the coefficients of the $\cv\dv$-index.
Let us recall the definition of the $\cv\dv$-index so that we can state the precise conjecture. Let $P$ be a graded poset of rank $n+1$, with rank function $\rho$. We consider chains in $P$ \[ \hat{0} < x_1 < x_2 < \cdots < x_m < \hat{1}. \] The type of this chain is $S = \{\rho(x_1),\ldots,\rho(x_m)\} \subset \{1,\ldots,n\}$. Let $f_S$ be the number of chains of type $S$ in $P$. We encode these flag numbers in the numbers $h_S$ by the formula \[ \Psi_P(t_1,\ldots,t_n) := \sum_{S} h_S t^S = \sum_S f_S (t-1)^{\overline{S}},\] where \[ t^S = \prod_{i\in S} t_i,\quad (t-1)^T = \prod_{i\in T} (t_i-1), \quad \overline{S} = \{1,\ldots,n\}{\smallsetminus} S.\] When the poset $P$ is Eulerian, then the numbers $h_S$ satisfy the generalized Dehn-Sommerville equations given by Bayer and Billera \cite{BayerBillera}, for example $h_S = h_{\overline{S}}$. The complete set of these linear relations is described by writing $\Psi_P$ as a polynomial in two noncommuting variables ${\bf c}$ and ${\bf d}$ of degree $1$ and $2$, respectively. A $\cv\dv$-monomial corresponds to a polynomial in the variables $t_i$ by replacing ${\bf c}$ with $t_i+1$ and ${\bf d}$ with $t_i+t_{i+1}$ as illustrated in the example: \[ {\bf c}{\bf d}{\bf d}{\bf c} = (t_1+1)(t_2+t_3)(t_4+t_5)(t_6+1).\] Then by Bayer and Klapper \cite{BayerKlapper}, the polynomial $\Psi_P$ for an Eulerian poset $P$ of rank $n+1$ can be expressed as a homogeneous polynomial of degree $n$ in the variables ${\bf c}$ and ${\bf d}$. The polynomial $\Psi_P({\bf c},{\bf d})$ is called the $\cv\dv$-index of $P$. When $P$ is Gorenstein* (this means, the simplicial complex formed by chains in $P$ is a purely $(n-1)$-dimensional homology sphere), then the $\cv\dv$-index has non-negative coefficients \cite{Stanley2, Karu}. Moreover, the $\cv\dv$-index is the most efficient encoding of the flag numbers $f_S$ in the sense that there are no more linear relations among the coefficients of $\Psi_P({\bf c},{\bf d})$ that hold for all Gorenstein* posets $P$, other than the coefficients being nonnegative and the coefficient of ${\bf c}^n$ being equal to one \cite{Stanley2}.
Define the multidegree of a degree $n$ $\cv\dv$-monomial $M$ as a zero-one vector in ${\mathbb{Z}}^n$ obtained by replacing each ${\bf c}$ with $0$ and each ${\bf d}$ with $10$. For example, \[ \operatorname{mdeg}({\bf c}{\bf d}{\bf d}{\bf c}) = (0,1,0,1,0,0).\]
For a $\cv\dv$-monomial $M$ with multidegree $v=\operatorname{mdeg}(M)$, write $\Psi_{P,v}$ for the coefficient of $M$ in $\Psi_P({\bf c},{\bf d})$. Let $\Psi_{P,v}=0$ for vectors $v$ that are not multidegrees of $\cv\dv$-monomials.
We say that a ${\mathbb{Z}}^n$-graded $k$-algebra $A$ (associative, commutative, with $1$) is standard multigraded if $A_0 = k$ and $A$ is generated in degrees $e_i = (0,\ldots,0,1,0,\ldots 0)$ for $i=0,\ldots,n$.
The following conjecture is the main subject of this article.
\begin{conjecture}[Murai-Nevo] \label{conj-main} Let $P$ be a Gorenstein* poset of rank $n+1$. Then there exists a standard ${\mathbb{Z}}^n$-graded $k$-algebra $A = \oplus_v A_v$, such that for any $v\in{\mathbb{Z}}^n$ \[ \Psi_{P,v} = \dim A_v. \] \end{conjecture}
In \cite{MuraiNevo} the conjecture was given in a different but equivalent form. Conjecture~4.3 in \cite{MuraiNevo} states that the numbers $\Psi_{P,v}$ should be the flag numbers of an $(n-1)$-coloured simplicial complex. Stated differently, Murai and Nevo conjectured that the algebra $A$ in Conjecture~\ref{conj-main} should be a polynomial ring modulo monomial relations. An arbitrary multigraded algebra $A$ can be degenerated to this form using a Gr\"obner basis.
Murai and Yanagawa \cite{MuraiYanagawa} proved that for any Gorenstein* poset $P$ and any set $S=\{i_1,\ldots,i_m\}$, \[ \Psi_{P, e_{i_1}+\cdots+e_{i_m}} \leq \prod_{j=1}^m \Psi_{P,e_{i_j}}.\] The case $m=2$ of this result was proved earlier in \cite{MuraiNevo}. If Conjecture~\ref{conj-main} is true, then this inequality follows from the surjectivity of the multiplication map \[ A_{e_{i_1}}\otimes A_{e_{i_2}}\otimes\cdots \otimes A_{e_{i_m}} \to A_{e_{i_1}+\cdots+e_{i_m}}.\]
In the shellable case we generalize this result to:
\begin{theorem} \label{thm-ineq} Let $P$ be a shellable Gorenstein* poset of rank $n+1$. Then for any $v,w\in {\mathbb{Z}}^n_{\geq 0}$ \[ \Psi_{P,v+w} \leq \Psi_{P,v} \cdot \Psi_{P,w}.\] \end{theorem}
The exact definition of shellability that we use here is given in Section~2 below. Shellable posets include all face posets of convex polytopes.
The inequalities given in Theorem~\ref{thm-ineq} unfortunately are not enough for the existence of an algebra $A$ as in Conjecture~\ref{conj-main}. For example, consider the degree $6$ homogeneous $\cv\dv$-polynomial \[ {\bf c}^6 +2({\bf d}{\bf c}^4+{\bf c}^2{\bf d}{\bf c}^2+{\bf c}^4{\bf d})+({\bf c}^2{\bf d}^2+{\bf d}{\bf c}^2{\bf d}+{\bf d}^2{\bf c}^2) +2{\bf d}^3.\] This polynomial satisfies all the inequalities of the theorem, but it is not hard to check that there cannot exist a standard multigraded $k$-algebra $A$ as in the conjecture. I do not know if this polynomial is the $\cv\dv$-index of any Gorenstein* poset.
The $\cv\dv$-index of a simplicial complex $P$ is much easier to understand. In fact the $\cv\dv$-index is determined by the face numbers $f_i$ of $P$. For simplicial complexes we prove:
\begin{theorem} \label{thm-simpl} Let $P$ be the poset of a Gorenstein* simplicial complex. Then Conjecture~\ref{conj-main} holds for $P$. \end{theorem}
We give an explicit construction of the algebra $A$ in Section~4 below.
In the special case where $P$ is the Boolean lattice $B_{n+1}$, the algebra $A$ can be described in terms of lattice paths as follows.
\begin{definition} \label{def-admfun}
Let $M$ be a $\cv\dv$-monomial of multidegree \[ \operatorname{mdeg}(M) = v = (v_1,\ldots,v_n).\] A function \[ f: \{0,1,\ldots,n\} \to {\mathbb{Z}} \] is called {\em admissible} for $M$ if the following is satisfied \begin{itemize} \item (Range of $f$) $f(0)=0$, $f(n)=n$, $0\leq f(i) \leq i$ for $i=1,\ldots,n-1$. \item (Strict ascent for $0$) If $v_i=0$, then $f(i-1)< f(i)$. \item (Weak descent for $1$) If $v_i=1$, then $f(i-1)\geq f(i)$. \item (Bound on descent) If $v_i=1$, $i>1$, then $f(i-1)-f(i)\leq f(i-2)+1$. \end{itemize} \end{definition}
Admissible functions give the coefficients of the $\cv\dv$-index of $B_{n+1}$:
\begin{theorem} \label{thm-Bn}
The coefficient of a $\cv\dv$-monomial $M$ in $\Psi_{B_{n+1}}$ is the number of admissible functions for $M$. \end{theorem}
This theorem is due to Fan and He \cite{FanHe}. Our notation is a bit different from their's and we also give a different proof using a shelling of $B_{n+1}$. There are other combinatorial interpretations of the coefficients of $\Psi_{B_{n+1}}$, for example in terms of Andr\'e permutations \cite{Purtill, Stanley2}.
\begin{figure}
\caption{Admissible functions for ${\bf c}{\bf d}{\bf d}{\bf c}$ and ${\bf d}{\bf d}$.}
\end{figure}
To visualize admissible functions $f$, we extend them to piecewise linear functions $f: [0,n]\to{\mathbb{R}}$. Figure~1(a) shows an example of an admissible function for $M={\bf c}{\bf d}{\bf d}{\bf c}$. Figure~1(b) shows all $4$ admissible functions for $M={\bf d}{\bf d}$. Note that the function $f=(0,0,2,0,4)$ is not admissible for ${\bf d}{\bf d}$ because it does not satisfy the ``bound on descent'' condition. The full $\cv\dv$-index of $B_{5}$ is \[ \Psi_{B_5} = {\bf c}^4 + 3 {\bf d} {\bf c}^2 + 5 {\bf c}{\bf d}{\bf c} + 3{\bf c}^2 {\bf d} + 4{\bf d}^2.\]
We define the algebra $A$ for $B_{n+1}$ by taking the admissible functions for all $\cv\dv$-monomials of degree $n$ as the $k$-basis. The product of two admissible functions is defined to be either zero or an admissible function. More precisely, let $M_1$, $M_2$, $M_3$ be $\cv\dv$-monomials of degree $n$, so that \[ \operatorname{mdeg}(M_1) + \operatorname{mdeg}(M_2) = \operatorname{mdeg}(M_3).\] Let $f_1$ and $f_2$ be two functions admissible for $M_1$ and $M_2$, respectively. Define \[ f_1 \cdot f_2 = \begin{cases}
\operatorname{min}(f_1,f_2) & \text{if $\operatorname{supp}(f_1)\cap\operatorname{supp}(f_2) = \emptyset$ and $\operatorname{min}(f_1,f_2)$ is admissible for $M_3$,} \\
0 & \text{otherwise.}
\end{cases} \] Here $\operatorname{min}(f_1,f_2)$ is computed pointwise and the support of $f$ is
\[ \operatorname{supp}(f) = \{i\in\{0,\ldots,n\}| f(i)<i\}.\] We prove in Section~3 that this multiplication turns $A$ into a standard multigraded $k$-algebra, hence $B_{n+1}$ satisfies Conjecture~\ref{conj-main}. The algebra $A$ for a general Gorenstein* simplicial complex is constructed similarly using lattice paths.
It is not difficult to see that an admissible function $f$ in the algebra $A$ constructed above is a product of a unique set of admissible functions $f_{i_j}$ of multidegree $e_{i_j}$. This means that the algebra $A$ is monomial and one can construct from it an $n$-coloured simplicial complex as conjectured by Murai and Nevo. Each admissible function of degree $e_i$ gives a vertex coloured with $i$ and a set of these vertices span a simplex if the product of the corresponding functions is nonzero.
{\bf Acknowledgement.} I thank Satoshi Murai for interesting conversation about the main conjecture and for explaining the equivalence of Conjecture~\ref{conj-main} and Conjecture~4.3 in \cite{MuraiNevo}.
\section{Shellable fans}
We will use the terminology of fans as in the theory of toric varieties \cite{Fulton}. Since we are only interested in the poset of cones in a fan $\Delta$, everything below also applies to the posets of regular $CW$-complexes. Thus, our fans are not fans over polyhedral complexes, but fans over $CW$-complexes. This implies for example that the intersection of two cones in a fan $\Delta$ is not necessarily a cone but a subfan of the boundary of each cone. One can generalize this further to Gorenstein* posets, but since we are interested in shellable posets, we do not get anything beyond $CW$-complexes.
\begin{definition}
An $n$-dimensional fan $\Delta$ is a fan over a regular $(n-1)$-dimensional $CW$-complex $C$, where $C$ is homeomorphic to the sphere $S^{n-1}$ or to the disk $D^{n-1}$. \end{definition}
In particular, a fan is Gorenstein* if and only if it is a fan over a regular $CW$-sphere.
For an $n$-dimensional fan $\Delta$, let $\Delta^{\leq m}$ be its $m$-skeleton, consisting of all cones of dimension at most $m$. Also let $\partial\Delta$ be the boundary, which is a subfan of $\Delta^{\leq n-1}$ and let $\operatorname{Int}(\Delta) = \Delta{\smallsetminus} \partial\Delta$ (as a set of cones). Similarly, let \[ (\operatorname{Int} \Delta)^{\leq n-1} = \Delta^{\leq n-1}{\smallsetminus} \partial\Delta.\]
We define shellability inductively on dimension.
\begin{definition}\label{def-shell} The unique $0$-dimensional fan is shellable. For $n>0$, an $n$-dimensional fan $\Delta$ is shellable if $\partial\Delta$ is shellable and there exists a decomposition \[ (\operatorname{Int} \Delta)^{\leq n-1} = \bigsqcup_i \operatorname{Int} \Delta_i,\] where each $\Delta_i$ is an $n-1$ dimensional shellable fan (over a sphere or a disks). \end{definition}
\begin{remark} The usual shelling of an $n$-dimensional fan consists of ordering maximal cones $\sigma_1,\ldots,\sigma_N$, so that if $\Sigma_j$ is the fan generated by $\sigma_1,\ldots,\sigma_j$, then $\sigma_{j+1}\cap \Sigma_j$ is a purely $(n-1)$-dimensional fan (over a sphere or a disk). Taking $\Delta_j = \sigma_{j+1}\cap \Sigma_j$ gives a shelling as in Definition~\ref{def-shell}, provided that $\partial \Delta$ and $\Delta_j$ are again shellable. It is known that projective fans, i.e., fans over the faces of a convex polytope, are shellable. \end{remark}
There are two natural operations that take shellable fans to disjoint unions of shellable fans. Define: \begin{align*} \partial: \Delta &\mapsto \partial\Delta, \\ C: \Delta &\mapsto \bigsqcup_i \Delta_i, \end{align*} where $\Delta_i$ are as in Definition~\ref{def-shell}. We extend the operations from disjoint unions to disjoint unions of shellable fans. In particular, we can compose the operations. When applying an $n$-fold composition of the operations to an $n$-dimensional fan, the result is a disjoint union of $m$ $0$-dimensional fans. We identify this disjoint union with the integer $m$.
Note that $\partial\circ\partial (\Delta) = \emptyset$. Thus, to get interesting compositions of $\partial$ and $C$, we have to precede each $\partial$ with a $C$, unless $\partial$ is the first operation in the composition. Define a third operation: \[ D: \Delta \mapsto \partial\circ C(\Delta).\] Then the interesting compositions that yield non-empty results are of the form \[ M(C,D), \quad M(C,D)\circ\partial,\] where $M(C,D)$ is a monomial in $C$ and $D$.
The following theorem is a special case of results proved in \cite{Karu}.
\begin{theorem}\label{thm-old} Let $\Delta$ be an $n$-dimensional shellable fan. Then the polynomial $\Psi_\Delta$ has the form \[ \Psi_\Delta = f_n({\bf c},{\bf d}) + g_{n-1}({\bf c},{\bf d}) t_n\] where $f_n$ and $g_{n-1}$ are homogeneous $\cv\dv$-polynomials of degree $n$ and $n-1$, respectively. Moreover \begin{enumerate} \item If $M(c,d)$ is a $\cv\dv$-monomial of degree $n$, then the coefficient of $M$ in $f_n$ is $M(C,D)(\Delta)$. \item If $M(c,d)$ is a $\cv\dv$-monomial of degree $n-1$, then the coefficient of $M$ in $g_{n-1}$ is $M(C,D)\circ\partial(\Delta)$.
\end{enumerate}
\end{theorem}
\begin{remark} The theorem implies that $g_{n-1} = \Psi_{\partial \Delta}$ and that the $\cv\dv$-polynomials $f_n$ and $g_{n-1}$ have nonnegative coefficients. The operation $C$ depends on the choice of the decomposition in the definition of shelling, but the $n$-fold composition of $\partial$ an $C$ applied to $\Delta$ is independent of the choices.
Another implication of the theorem is that compositions of the operations $\partial$ and $C$ determine all flag numbers $f_S$ of $\Delta$. To see in a different way why this is true, note that if $\Delta$ is a fan over a sphere, then the flag numbers of $\Delta^{\leq n-1}$ determine the flag numbers of $\Delta$. If $\Delta$ is a fan over a disk, then we need the flag numbers of both $\partial\Delta$ and $\Delta^{\leq n-1}$ to determine the flag numbers of $\Delta$. Cutting $\operatorname{Int} \Delta$ into disjoint pieces simply expresses the set of chains in $\operatorname{Int} \Delta$ as a disjoint union (indexed by $\Delta_i$ containing the smallest element in the chain). Thus, the recursive construction of shelling explains why the flag numbers of $\Delta$ are determined by the operations $C$ and $\partial$. \end{remark}
Given a monomial $M(\partial,C)$, we define its multidegree by replacing each $C$ with $0$ and each $\partial$ with $1$. This in particular defines the multidegree of a monomial $M(C,D)$, which agrees with the previously defined multidegree of $M({\bf c},{\bf d})$.
The following theorem together with Theorem~\ref{thm-old} implies Theorem~\ref{thm-ineq}. Note that by a Gorenstein* shellable poset we simply mean a shellable fan over a sphere.
\begin{theorem} \label{thm-ineq2} Let $M_1, M_2, M_3$ be be $n$-fold compositions of the operations $\partial$ and $C$, such that \[ \operatorname{mdeg} M_1 + \operatorname{mdeg} M_2 = \operatorname{mdeg} M_3.\] Then for any shellable $n$-dimensional fan $\Delta$ \[ M_3(\Delta) \leq M_1(\Delta) \cdot M_2(\Delta).\] \end{theorem}
\begin{remark} Theorem~\ref{thm-ineq2} is more general than Theorem~\ref{thm-ineq} because $\Delta$ need not be a fan over a sphere and the monomials $M_i$ may end with $\partial$.
\end{remark}
We need the following lemma.
\begin{lemma} Let $\Delta$ be an $n$-dimensional shellable fan and $\Pi\subset \Delta$ an $m$-dimensional shellable subfan, where $\Pi$ is a fan over the $(m-1)$-sphere, $m<n$. If $M(\partial,C)$ is any $m$-fold composition of $\partial$ and $C$, then \[ M(\Pi) \leq M\circ C^{n-m}(\Delta).\] \end{lemma}
\begin{proof} I only know how to prove this using sheaves on the fan $\Delta$. Using notation from \cite{Karu}, we replace the fans by the constant sheaves on them to get an exact sequence: \[ 0\to {\mathbb{R}}_\Pi \longrightarrow {\mathbb{R}}_{\Delta^{\leq m}} \longrightarrow {\mathbb{R}}_{\Delta^{\leq m}{\smallsetminus} \Pi} \to 0.\] The sheaves here are Cohen-Macaulay sheaves on $\Delta^{\leq m}$. The operations $C$ and $D$ can be applied to any Cohen-Macaulay sheaf and the result agrees with our definition of the operators $C$ and $D$. (Since by assumption $\partial\Pi = \emptyset$, we may assume that $M(\partial,C)$ is in fact a monomial in $C$ and $D$.) From \cite{Karu} the operations $C$ and $D$ are additive on exact sequences, hence \[ M\circ C^{n-m}({\mathbb{R}}_\Delta) = M({\mathbb{R}}_{\Delta^{\leq m}}) = M({\mathbb{R}}_\Pi) + M({\mathbb{R}}_{\Delta^{\leq m}{\smallsetminus} \Pi}).\] Since all summands are nonnegative, the result follows. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm-ineq}]
Switching $M_1$ and $M_2$ if necessary, we may assume that \begin{align*} M_1 &= M_1' \partial C^j,\\ M_2 &= M_2' C C^j,\\ M_3 &= M_3' \partial C^j, \end{align*} where $j\geq 0$ and \[ \operatorname{mdeg} M_3' = \operatorname{mdeg} M_1' + \operatorname{mdeg} M_2'.\] Since $\Delta$ is shellable, we get \[ \partial C^j(\Delta) = \bigsqcup_i \Pi_i,\] where $\Pi_i$ are $(n-j-1)$-dimensional fans over spheres. Moreover, $\Pi_i$ are subfans of $\Delta$, hence the previous lemma applies. Now \begin{align*}
M_3(\Delta) = \sum_i M_3'(\Pi_i) &\leq \sum_i M_1'(\Pi_i) \cdot M_2'(\Pi_i)\\
&\leq \sum_i M_1'(\Pi_i) \cdot M_2'C^{j+1}(\Delta)\\
&= (\sum_i M_1'(\Pi_i)) \cdot M_2'C^{j+1}(\Delta) = M_1(\Delta) \cdot M_2(\Delta). \end{align*} Here in the first inequality we used induction on dimension of the fans and in the second inequality we applied the lemma to the fans $\Pi_i\subset \Delta$ and monomial $M_2'$. \end{proof}
\section{Boolean lattices}
Let $\Pi_n$ be the $n$-dimensional fan over the boundary of an $(n+1)$-simplex. The poset of cones in $\Pi_n$ is the Boolean lattice $B_{n+1}$. Geometrically, $\Pi_n$ is the fan of the toric variety ${\mathbb{P}}^n$.
For $\Pi_n$ we can write down an explicit shelling. Let $\sigma_n$ be the fan consisting of an $n$-dimensional simplicial cone and all its faces. In the shelling of $\Pi_n$ we will encounter fans of the type $\Pi_k \times \Pi_l$ and $\Pi_k\times \sigma_l$. These include the special cases $\Pi_k = \Pi_k\times \sigma_0$ and $\sigma_l = \Pi_0\times\sigma_l$. One can now easily verify: \begin{align*}
\partial (\Pi_k\times\Pi_l) &= \emptyset,\\ \partial (\Pi_k\times \sigma_l) &= \Pi_k\times \partial \sigma_l = \Pi_k\times \Pi_{l-1},\\ C (\Pi_k\times \sigma_l) &= C (\Pi_k) \times \sigma_l = \bigsqcup_{i=1,\ldots,k} \Pi_{i-1}\times \sigma_{k-i} \times \sigma_l = \bigsqcup_{i=1,\ldots,k} \Pi_{i-1}\times \sigma_{k+l-i},\\ C (\Pi_k\times\Pi_l) &= \bigsqcup_{ {i=0,\ldots, k \atop j=0,\ldots,l} \atop (i,j) \neq (0,0)} \Pi_{i+j-1} \times \sigma_{k+l-i-j}. \end{align*}
We want to count the number of 0-dimensional fans obtained by applying an $n$-fold composition of $\partial$ and $C$ to $\Pi_n$. Each such $0$-dimensional fan is the result of making a choice at every step we apply $C$: choose either $i$ or $(i,j)$. We encode these choices in a sequence of $n+1$ integers, one for each fan, as follows. Applying $C$ to $\Pi_k\times \sigma_l$, the numbers are: \[ \begin{tabu}{cccc}
C: & \Pi_k\times\sigma_l & \xmapsto{\text{choose $1\leq i\leq k$}} & \Pi_{i-1}\times\sigma_l \\ &k &\xmapsto{\hspace*{2.2cm}} & i-1 \end{tabu} \] Applying $\partial$ followed by $C$: \[ \begin{tabu}{ccccc}
\Pi_k \times \sigma_{l+1} & \stackrel{\partial}{\longmapsto} & \Pi_k\times\Pi_l &\xmapsto{\text{C: choose $i,j$}}& \Pi_{i+j-1}\times\sigma_{k+l-i-j} \\ k & \longmapsto & k+j & \xmapsto{\hspace*{1.7cm}} & i+j-1 \end{tabu} \] If $\partial$ is the last operation: \[ \begin{tabu}{ccc}
\Pi_0 \times \sigma_{1} & \stackrel{\partial}{\longmapsto} & \Pi_0\times\Pi_0 \\ 0 & \longmapsto & 0 \end{tabu} \]
Applying a degree $n$ monomial $M(\partial,C)$ to $\Pi_n$ and fixing a choice at each step, we get a sequence of $n+1$ fans and the associated sequence of $n+1$ integers: \[ n=i_n \longmapsto i_{n-1} \longmapsto \cdots\longmapsto i_0 = 0.\] We define the function $f: \{0,\ldots, n\} \to {\mathbb{Z}}$ by $f(j) = i_j$.
\begin{example} Consider the monomial ${\bf c}{\bf d}{\bf d}{\bf c} = C\partial C\partial C C$ applied to $\Pi_6$: \[ \begin{tabu}{ccccccccccccc} \Pi_6 &\stackrel{C}{\mapsto} & \Pi_3\times \sigma_2 &\stackrel{C}{\mapsto} & \Pi_2\times \sigma_2 &\stackrel{\partial}{\mapsto} & \Pi_2\times \Pi_1 &\stackrel{C: (1,0)}{\mapsto} & \Pi_0\times \sigma_2 &\stackrel{\partial}{\mapsto} & \Pi_0\times \Pi_1 &\stackrel{C: (0,1)}{\mapsto}& \Pi_0\times \Pi_0 \\ 6 &\mapsto & 3 &\mapsto & 2 &\mapsto & 2 &\mapsto & 0 &\mapsto & 1 &\mapsto &0 \end{tabu} \] Note that when applying $C$ to $\Pi_k\times\Pi_l$, we need to specify the choice of $(i,j)$ as this is not determined by the resulting fan. The function constructed from this sequence: \[ (f(0), f(1), f(2), f(3), f(4), f(5), f(6)) = (0,1,0,2,2,3,6)\] is the admissible function for ${\bf c}{\bf d}{\bf d}{\bf c}$ shown in Figure~1(a). \end{example}
We claim that the functions constructed as above for a ${\bf c}{\bf d}$-monomial $M({\bf c},{\bf d})$ are precisely the admissible functions for $M$. This implies that the number of admissible functions for $M$ is the coefficient of $M$ in $\Psi_{\Pi_n}$.
\begin{lemma}
A function $f$ is admissible for a ${\bf c}{\bf d}$-monomial $M$ if and only if it is obtained by the construction above. \end{lemma}
\begin{proof} Let us first check that a function constructed above is admissible. First notice that the integers assigned to each fan are nonnegative and no bigger than the dimension of the fan. The first three conditions of admissibility are now easy to check from the definition. The ``bound on descent'' applies to the sequence \[ k \longmapsto k+j \longmapsto i+j-1.\] We need to check that \[ (k+j)-k \leq (i+j-1)+1,\] which follows from $i\geq 0$.
Conversely, given an admissible function, that means, a sequence of numbers $i_n\mapsto i_{n-1} \mapsto \cdots \mapsto i_0$, we construct the sequence of fans. Most numbers $i_j$ correspond to fans $\Pi_{i_j}\times \sigma_{j-i_j}$. The exceptions are the targets of the operation $\partial$, where the fans are \[ \begin{tabu}{ccc} i_k & \longmapsto & i_{k-1}=i_k+j \\
\Pi_{i_k} \times \sigma_{k-i_k} & \stackrel{\partial}{\longmapsto} & \Pi_{i_k}\times\Pi_{k-i_k-1} \end{tabu} \] If $\partial$ is followed by $C$, then from the sequence of numbers \[ i_k \longmapsto i_{k-1}=i_k+j \longmapsto i_{k-2} = i+j -1 \] we get the choice $(i,j) = (i_{k-2}-i_{k-1}+i_k +1,i_{k-1}-i_k)$ when applying the operation $C$. The fact that $0\leq i \leq i_k$, $0\leq j \leq k-i_k-1$ and $(i,j)\neq (0,0)$ follows from the admissibility of the function. \end{proof}
Recall the algebra $A$ defined in the introduction. It has a basis consisting of admissible functions for all ${\bf c}{\bf d}$-monomials of degree $n$, and multiplication is defined so that the product of two basis elements is again a basis element or zero.
\begin{lemma} $A$ is a standard multigraded algebra. \end{lemma}
\begin{proof} The multiplication is clearly commutative. The function $f(i)=i$ acts as $1$. We need to prove associativity of the multiplication. The operation of taking the minimum is associative. We only need to rule out the case that $f_1\cdot f_2=0$, but $f_1\cdot(f_2\cdot f_3)\neq 0$. In this case the supports of $f_1, f_2, f_3$ are disjoint and $g=\operatorname{min}(f_1,f_2)$ is not admissible, it does not satisfy the ``bound on descent'' condition. Let us show that then $\operatorname{min}(g,f_3)$ also does not satisfy this condition.
Suppose $g$ has a descent at $i$: $g(i) \leq g(i-1) > g(i-2)$ and the descent is too big, $g(i-1)-g(i) > g(i-2)+1$. These conditions imply that $i, i-2 \in \operatorname{supp}(g)$. The support condition implies that $f_3(i-2)= i-2$, and since $f_3$ must have an ascent at $i-1$, $f_3(i-2)<f_3(i-1)=i-1$. Thus, none of $i-2, i-1, i$ lie in $\operatorname{supp}(f_3)$, hence $\operatorname{min}(g,f_3)$ takes the same values as $g$ at $i-2,i-1, i$, and the ``bound on descent" condition is also violated for $\operatorname{min}(g,f_3)$.
It remains to prove that $A$ is generated in degrees $e_i$, $i=1,\ldots,n$. Let $f$ be an admissible function that has descents at $l_1,l_2,\ldots,l_m$, $f(l_j-1)\geq f(l_j)$. Define for $j=0,\ldots,m-1$ \[ f_j(i) = \begin{cases}
f(i) & \text{if $i=l_j,\ldots,l_{j+1}-1$,}\\ i &\text{otherwise,}
\end{cases}
\] and \[ f_m(i) = \begin{cases}
f(i) & \text{if $i=l_m,\ldots,n$,}\\ i &\text{otherwise.}
\end{cases}
\] Then $f_j$ is an admissible function lying in degree $e_{l_j}$ and $f=f_1\cdot f_2\cdots f_m$. \end{proof}
This finishes the proof of Theorem~\ref{thm-simpl} for the posets $B_{n+1}$.
\section{General simplicial spheres}
Consider now an $n$-dimensional fan $\Delta$ over a Gorenstein* simplicial complex (note that the complex may not be homeomorphic to a sphere, it is only a homology sphere).
First assume that $\Delta$ is shellable in the usual sense: there exists an ordering of the maximal cones $\delta_1,\delta_2,\ldots,\delta_N$, such that if $\Sigma_j$ is the fan generated by $\delta_1,\ldots,\delta_j$, then $\delta_{j+1}\cap\Sigma_j$ is a purely $n-1$-dimensional subfan of $\partial \delta_{j+1}$. This subfan is then necessarily of the form $\Pi_k\times \sigma_{n-k-1}$ for some $k=0,\ldots,n-1$. In fact, the number of fans $\Pi_k\times \sigma_{n-k-1}$ we get by shelling $\Delta$ is equal to the number $h_{k+1}$ of $\Delta$. Thus, the shelling of $\Delta$ differs from the shelling of $\Pi_n$ only at the first step; while $\Pi_n$ yields one copy of $\Pi_k\times \sigma_{n-k-1}$ for each $k=0,\ldots,n-1$, a general $\Delta$ yields $h_{k+1}$ copies of the same fan. We encode this in the admissible functions $f:\{0,\ldots,n\}\to {\mathbb{Z}}$ as follows. We draw $h_{k+1}$ edges from $(n-1,k)$ to $(n,n)$. Now an admissible function is a piecewise linear function $f:[0,n]\to{\mathbb{R}}$ as before, except that if $f(n-1)=k$, then the graph of $f$ on $[n-1,n]$ must be any one of the $h_{k+1}$ edges. (See Figre 2.) The number of such admissible functions again gives the coefficients of the ${\bf c}{\bf d}$-index of $\Delta$ and the algebra $A$ is constructed the same way as for the fan $\Pi_n$.
\begin{figure}
\caption{Graphs of admissible function.}
\end{figure}
Now let $\Delta$ be a general Gorenstein* simplicial fan, not necessarily shellable. Let $h_k$ be the $h$-numbers of $\Delta$ and construct the algebra $A$ using the numbers $h_k$ as in the shellable case. This algebra has the correct dimensions of graded pieces (the coefficients of the ${\bf c}{\bf d}$-index of $\Delta$) because the numbers $h_k$ determine the face numbers $f_i$ of $\Delta$, and these determine the flag numbers $f_S$, hence also the ${\bf c}{\bf d}$-index of $\Delta$. This proves Theorem~\ref{thm-simpl}.
\end{document} | arXiv |
Why is the reciprocal of the Hubble constant equal to the age of the universe?
I understand that the Hubble constant is the gradient of the line of best fit when we plot Redshift against distance. I understand why the reciprocal of the gradient would give a value for time. But why do we know (or assume) that this value of time is equal to the age of the universe? How do we know it isn't equal to something else, or it isn't just an arbitrary value?
cosmology time space-expansion
$\begingroup$ See physics.stackexchange.com/questions/254744/… $\endgroup$ – J.G. Jan 3 '18 at 16:21
$\begingroup$ If $H_0$ is the Hubble constant of today, I do not believe cosmology claims $\frac{1}{H_0}$ is the exact age of the universe. But according to models, it gives a good rough estimate of the age. But it is not obvious how good it will be. $\endgroup$ – Jeppe Stig Nielsen Jan 3 '18 at 16:28
$\begingroup$ en.wikipedia.org/wiki/Hubble%27s_law#Hubble_time $\endgroup$ – Ben Crowell Jan 3 '18 at 16:54
Hubble's law shows that the redshift velocity of an object is proportional to the distance to the object:
$$v=H \cdot D $$
The redshift velocity being the velocity that would give the observed redshift. At low velocities the amount of redshift is proportional to the redshift velocity.
If you assume the universe to expand linearly, i.e., that the distance between comoving objects grows linearly in time, then the apparent velocity at which a given object seems to move away due to the expansion of space remains constant over time. Assuming the object moved at this same velocity since the big bang, you can calculate how long ago the distance to the object was zero. This time is given by:
$$t=\frac{D}{v}=\frac{1}{H}$$
Note that a consequence of this assumption is that Hubble's constant is actually not a constant, but changes over time: $H=\frac{1}{t}$
Note also that it is far from trivial that this assumption would be legitimate. It depends on the amount of matter and energy in the universe, whether the rate of expansion is constant or not.
For more info see Hubble's law
CrimsonCrimson
Not the answer you're looking for? Browse other questions tagged cosmology time space-expansion or ask your own question.
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Age of the universe $ \equiv 1/H$?
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How does calculating Hubble time work? | CommonCrawl |
\begin{document}
\title {Last-Hitting Times and Williams' Decomposition \\of the Bessel Process of Dimension 3 \\at its Ultimate Minimum} \date{} \author{F. Thomas Bruss and Marc Yor \\Universit\'e Libre de Bruxelles and Universit\'e de Paris VI} \maketitle
\begin{abstract}\noindent In this note we shortly recall the importance of last-hitting times in theory and applications of optimal stopping. As a small contribution to this domain we then propose a concise proof of David Williams' decomposition of the Bessel Process of dimension 3 (BES(3)), starting from $r>0$ at its ultimate minimum. This discussion is strongly motivated by our interest in properties of last hitting times in general, and here specifically, directly linked with the forthcoming reading guide of Nikeghbali and Platen on this subject.
\noindent \noindent {\bf Keywords:} Brownian motion, Bessel processes, stopping times, measurability,\\ best choice problems, last arrival problem, compassionate-use clinical trials.
\noindent {\bf AMS subject classification:} {60 H 30}; secondary {60 G 40}.
\noindent {\bf Running title:} Last Hitting Times and Williams' Decomposition \end{abstract}
\section{Introduction}\label{intro}
Only in trivial cases, last hitting times are at the same time stopping times because the "current- measurability" requirement is usually not satisfied. Hence it is typically harder to deal with last hitting times than with stopping times for which we know quite an impressive collection of Theorems and tools. As Chung (see citation of Nikeghbali and Platen(2012)) among others conclude, last hitting times must therefore be {\it avoided at all costs.}
This is one way to see things, but one should admit that, often enough, reality looks somewhat different. Indeed, ironically, many interesting problems in the theory of optimal stopping require us to deal with last hitting times, and not with stopping times. And so, the attitude has changed, and the work of Jeulin (1980) and others had quite an influence on this development.
In their recent paper, Nikeghbali and Platen cite several interesting examples from the domain of Mathematical Finance. In order to add to the motivation, we would like to slightly broaden the horizon and look at a few other examples.
{\bf Examples and Motivation}
Best-choice problems, or secretary problems, are typical representatives of a last-hitting time problem, namely problems of stopping on the last "improvement" of a stochastic process. In some of these problems, the difficulty stemming from the last-hitting time character disappears. To give a very simple example, suppose we observe sequentially variables $X_1, X_2, \cdots$ and would like to maximize, for a given objective function $f,$ the expected total return, that is we seek $$ \arg \max_\tau f(X_1, X_2, \cdots , X_\tau).$$ Suppose now that the optimal payoff for stopping {\it after} time $t$ does not depend on ${\cal F}_t,$ where $({\cal F}_s)$ denotes the natural filtration. Then
$$\sup_{\tau \ge t} \mathrm E\left(f(X_1, X_2, \cdots, X_\tau |{\cal F}_t)\right)=\sup_{\tau \ge t}\mathrm E(f(X_{t+1}, X_{t+2}, \cdots, X_\tau))$$ so that RHS as well as $X_1, X_2, \cdots, X_t$ are both ${\cal F}_t$-measurable. Hence it suffices to compare at each time $t$ the value $f(X_1, X_2, \cdots, X_t)$ with the RHS supremum in order to take the optimal decision.
{\bf Harder last-hitting time problems}
In more difficult problems the ${\cal F}_t$-independence is typically no longer satisfied. However, external information about the underlying process may help us to change nevertheless the last-hitting time problem into a tractable stopping problem. See for example the {\it last-arrival} problem (Bruss and Yor (2012)) which is a continuous time problem, and where the relevant external information about the underlying process is implied by a related martingale. There may be other examples of such an approach.
There are also certain other problems where the {\it last} hitting time objective is hiding behind other objectives, as for example the objective to discover the {\it first} time a random subset of a given set becomes complete. We give one specific example of this in the important field of clinical trials, more precisely, in so-called compassionate-use clinical trials.
In such trials, a sequence of patients is treated with a drug (sometimes without FDA-approval) which may have serious side effects, the only justification being that it may be the last hope. (These trials require a special written consent of patients.) Treatments are typically sequential so that the physician or statistician may learn form preceding observations.
A little reflection shows that these trials pose a difficult ethical problem. The conscientious physician should try to save all lives which can be saved, and, at the same time avoid all unnecessary sufferings caused by the treatment. Since he or she is no prophet, the goal must be to stop (in a given sequence of patients within a fixed horizon) with maximum probability with the first patient completing the random subset of successes that is stop with the last success. Indeed, then {\it all} successes are covered, whereas the remaining patients (de facto not savable by the drug) do not have to suffer unnecessarily. If the success probability for each patient is known beforehand , then the optimal strategy follows immediately from the odds-algorithm (see Bruss (2000)). However, the physician may have almost no information about the respective success probabilities, and then the general solution of the optimal stopping problem is an open problem.
These examples second the motivation of the studies of Nikeghbali and Platen, as well as ours in the present note. Last-hitting times are often important. Admittedly, no model in these papers seems tractable enough to deal for instance with the {\it general} compassionate-use stopping problem described above. However, this indicates that it may be worth trying to view last-hitting times from any possible angle, and this is what we try in this paper.
Our goal is to add to the understanding by looking at what can be done with enlargements of filtrations (in this case together with Girsanov's Theorem), even though we confine our interest to special processes.
{\bf The BES(3)-process and Williams' Theorem}
\noindent {\bf (1.1)}~~In their survey about last passage times, the Nikeghbali and Platen (2012) illustrate some of their formulae with the following example:
Let $(R_t)_{t\ge 0}$ be a BES(3)-process on $\mathbb R_+$ starting from $r>0.$ Denote by $({\cal F}_t)_{t \ge 0}$ its natural filtration, and let $I_t$ denote the current infimum of the process $R$ at time $t$, that is, \begin{align*} I_t=\inf_{s \le t}R_s.\end{align*} The following results can be found in Nikeghbali and Platen around Corollary 4.10:
(a) $I_{\infty}$ follows the same distribution as the random variable $rU,$ where $U$ is uniform on $[0,1].$
(b) The Az\'ema-supermartingale associated with the random time $g$ at which the process $(R)$ reaches $I_\infty$ is given by \begin{align*}Z_t \equiv P(g>t|{\cal F}_t)=\frac{I_t}{R_t}.\end{align*}
(c) The Laplace transform of the law of $g$ is \begin{align*} \mathrm E \left(e^{-\lambda g}\right)= \frac{1}{\sqrt{2 \lambda} r} \left(1-e^{-{\sqrt{2 \lambda} r}}\right). \end{align*}
(d) The density of $g$ denoted by $p(t)$ equals \begin{align*} p(t)=\frac{1}{\sqrt{2 \pi t}\, r} \left(1-e^{-(r^2/2t)}\right). \end{align*}
Our aim in the remaining part of this note is to show Williams' decomposition of a BES(3)-process at its ultimate minimum, and how this decomposition is closely connected with (a)-(b)-(c)-(d).
\noindent {\bf (1.2)}~~Recall that if $(B_t)_{t\ge 0}$ is a Brownian motion starting from $0$ and $a$ is a real constant, then the law of the first hitting time of $a$ by $(B_t),$ denoted by $T_a^{(B)}$ is given by \begin{align}
P\left(T_a^{(B)}\in dt \right)= \frac{dt}{\sqrt{2 \pi t^3}}\, |\,a\,|\, {\rm exp} \left(-\frac{a^2}{2t}\right). \end{align}This well-known fact allows us to rewrite the statements (c) and (d) above as \begin{align} g \overset{ \cal L}{=} T_{r U}^{(B)}, \end{align} where $U$ is independent of $(B)$ and uniform on $[0,1],$ and where $ \overset{ \cal L}{=}$ denotes identity in law. This can be verified using (c) and (d). In fact, (2) may be understood via the classical decomposition of the process $(R)$ before and after time $g,$ due to Williams (1974).
\section{Williams' decomposition of $(R)$, before and after $g$, via progressive enlargement}
\noindent {\bf (2.1)}~~Figure 1 describes well the decomposition of a BES(3)-process. \begin{figure}
\caption{ This graph presents a simulation of a Bessel process of dimension 3 based on three independent simulations of $U[-1/2,1/2]$-random walks $S^x_k, S^y_k,S^z_k,$ where $k$ runs from $1$ to $1200.$ The starting point is chosen $S^x_0=4, S^y_0=4,S^z_0=2,$ so that the starting level of the simulated process is $B_0=r=6.$ The minimum height is indicated by the supporting horizontal line. In this simulation it equals $1.41963.$ Note that this is $I_{1200}$ and not $I_\infty$, of course. Its level would be uniformly distributed on $[0,B_0]=[0,6]$ if the horizon were infinite. }
\end{figure}
\noindent Note that this figure is nothing else but a (simulated finite-horizon) version of the Figure 5 in Revuz-Yor (1999) (see Proposition 3.10 and Theorem 3.11 in Ch. 6, Sect. 3) where the BES(3)-process is considered starting from level $c:=r.$
\noindent {\bf (2.2)}~~ We now state precisely Williams' Decomposition Theorem before and after time $g.$
\noindent {\bf Theorem 1} (Williams (1974))
\noindent Consider the following three independent random objects:
(i)\,\, ~a Brownian motion $(B'_t)_{t\ge 0}$ with $B'_0=r>0;$
(ii)\, ~a uniform random variable $U$ on $[0,1];$
(iii) ~a BES(3)-process $(\tilde R_t)_{t\ge 0}$ with $\tilde R_0=0;$
\noindent Then the process $(R)$ defined by
\begin{align} R_t=\begin{cases} B'_t &\mbox{,~if~} t \le g\\ r U+\tilde R_{t-g} & \mbox{,~if~} t \ge g\end{cases}
\end{align} with $g=\inf\{u \ge 0: B'_u=rU\}$ is a BES(3)-process starting from $r>0.$
\noindent We note that the pre-$g$-Browian-motion found in (3) explains the result (2). Indeed, if $B'_t=r-B_t^{(0)}$ then $$ g=\inf\left\{u\ge 0: B_u^{(0)}=r(1-U)\right\},\eqno(2')$$ which implies (2).
\noindent
{\bf 2.3}~We now proceed to the proof of the Theorem via the enlargement formula which describes the additive decomposition of the BES(3)-process $(R_t)$ in the filtration $\left({\cal F}_t^g)\right)$ containing
the filtration $({\cal F}_t),$ and making $g$ a stopping time.
Firstly, we have $$R_t=r+B_t+\int_0^t \frac{ds}{R_s}, \eqno(4)$$ where $(B_t)$ is a Brownian motion with respect to the filtration $({\cal F}_t).$
Secondly, the enlargement formula (see e.g. Jeulin (1980)) yields $$r+B_t=B'_t+\int_0^{g\wedge t} \frac{d<B,Z>_u}{Z_u}+\int_g^t \frac{d<B,1-Z>_u}{1-Z_u} \eqno(5) $$ with $(B'_t)$ being a Brownian motion with respect to $({\cal F}_t^{g}).$
Thirdly, we deduce from (b) the two identities $$\frac{d<B,Z>_u}{Z_u}=-\frac{du}{R_u}, \mbox{~for~} u\le g \eqno(6)$$ and $$\frac{d<B,1-Z>_u}{1-Z_u}=\frac{I_\infty du}{R_u(R_u-I_\infty)} \mbox{,~for~} u>g .\eqno(7)$$ These two identities imply (using (4) and (6)) , and also $$\frac{1}{R_u}+\frac{I_\infty}{R_u(R_u-I_\infty)} =\frac{1}{R_u-I_\infty}$$ the form of the pre-$g$-process and the form of the post-$g$-process.
Finally, for the proof of (3) to be complete, it remains to prove that the process $(B')$ is independent of the random variable $I_\infty \overset{\cal L}=\,rU,$ or more precisely, that, given $I_\infty = a$, the pre-$g$-process is just the process $(B'_u)_{u\le T'_a}$ with obvious notation. This is asserted in the following proposition:
{\bf Proposition}: Let $(\Phi_u)_{u\ge 0}$ be a non-negative predictable process on path-space. Further, let $P_r$ denote the law of the process $(R)$ starting from $r$ and let $P'_r$ denote the law of the Brownian motion $(B')$ starting from $r.$ Then, for $a<r,$
\begin{align*}\mathrm E_r\left[\Phi_g | I_\infty=a\right]&=\mathrm E_r\left[\Phi_{T_a}|T_a <\infty\right]~~~~~~~~(8.1)\\&\equiv \mathrm E_r \left[\Phi_{T_a}|I_\infty<a\right]~~~~~~~~~(8.2)\\&=\mathrm E'_r\left[\Phi(B'_u; u \le T'_a )\right]~~~~~~(8.3)\end{align*}
{\bf Proof}: The equality between the RHS of (8.1) and (8.3) follows, as we will show, from Doob's absolute continuity relationship, namely$$P_r/ {\cal F}_t=\left(\frac{X_{t \wedge T_0}}{r}\right)P'_r / {\cal F}_t,$$ on the canonical path-space $C([0, \infty],\mathbb R),$ where $(X_t)$ denotes the coordinate process on path-space. Indeed, this equality may be extended when replacing the time $t$ by a stopping time. Restricting ${\cal F}_{T_a}$ on the set $\{T_a<\infty\}$ we get then in particular $$P_r/({\cal F}_{T_a}\cap \{T_a<\infty\})=\left(\frac{a}{r}\right) P'_r/{\cal F}_{T_a}, ~0<a<r,$$ which yields the desired result.
\noindent identity (8.2) is obvious, since the equality $\{T_a<\infty\}=\{I_\infty<a\}$ holds $P_r$-almost surely.
The proof of the equality (8.1) is slightly more subtle. We start with the identity $$\mathrm E\left[{\bf 1}_{\{g \le t\}}\varphi(I_\infty)\right]= E\left[(1-Z_t)\,\int_0^t\varphi(I_s)d(1-Z_s)\right]\eqno(9)$$ which holds for any Borel-measurable function $\varphi:[0, \infty[ \to \mathbb R_+.$ To see this, note that $$\mathrm E\left[{\bf 1}_{\{g\le t\}}\varphi(I_t)\right]=\mathrm E\left[(1-Z_t)\varphi(I_t)\right].$$ Assuming $\varphi \in {\cal C}^1$ the latter becomes by partial integration $$\mathrm E\left[\int_0^t\varphi'(I_s)dI_s(1-Z_s)\right]+\mathrm E\left[\int_0^t\varphi(I_s)d(1-Z_s)\right].$$ We note that the expectation involving $\varphi'$ vanishes, since $1-Z_s$ vanishes $dI_s$ almost everywhere. Thus a monotone class argument implies that (9) holds for every non-negative Borel-measurable function $\varphi.$
Next, from the additive decomposition of $(1-Z_s)$, we obtain $$\mathrm E\left[{\bf 1}_{\{g\le t\}}\varphi(I_\infty)\right]=\mathrm E\left[\int_0^t\varphi(I_s)\left(-\frac{dI_s}{I_s}\right)\right]=\mathrm E\left[\int_{I_t}^r\varphi(a) \frac{da}{a}\right].$$ Since $\{I_t\le a\}=\{t \ge T_a\}$ the latter can also be written as $$\mathrm E\left(\int_0^r \varphi(a)\frac{da}{a}{\bf 1}_{\{T_a\le t\}}\right),$$ so that from (9) $$\mathrm E\left[{\bf 1}_{\{g\le t\}}\varphi(I_\infty)\right]=\mathrm E\left(\int_0^r \varphi(a)\frac{da}{a}{\bf 1}_{\{ T_a \le t\}}\right) \eqno(10).$$ Now note that the identity (10) still holds if we replace $t$ by a generic stopping time. Applying again the monotone class theorem gives us then $$\mathrm E\left[\Phi_g\varphi(I_\infty)\right]=\mathrm E\left[ \int_0^r \varphi(a) \frac{da}{a}\Phi_{T_a}{\bf 1}_{\{T_a < \infty\}}\right]. \eqno (11)$$ Finally, using $I_\infty \overset{{\cal L}}= rU$ with $U$ being uniform on $[0,1]$ under $P_r$ (see (2)), we see that identity (11) implies identity (8.1).
\qed \section{Concluding remarks} The statement of the theorem invites for a proof chosing between, on the one hand, initial enlargement with $I_\infty,$ and, on the other hand, progressive enlargement with $g.$ However we have not exactly proceeded like this; the Proposition plays the role of the initial enlargement method and relies on a classical Girsanov relationship between $P_r$ and $P'_r$.
In conclusion we find it interesting to present the above as an example of the potential of enlargement techniques, and here specifically, of a melange of enlargement techniques and Girsanov's theorem. Having said so, we know that this approach can in priciple be done for higher dimensions; however, the corresponding Theorem 1 would look more complicated.
{\bf Acknowledgement}
The authors are grateful to Monique Jeanblanc for providing them with a preprint of Nikeghbali and Platen (to appear).
{\bf References}:
F.T. Bruss: {\it Sum the odds to one and stop}, Ann. Probab., Vol. 28, No 3, 1384-1391, (2000)
F.T. Bruss and M. Yor: {\it Stochastic processes with proportional increments and the last-arrival problem}, Stoch. Proc. and Th. Applic., Vol. 122 (9), 3239-3261, (2012)
Th. Jeulin: {\it Semi-martingales et grossissement d'une filtration}, LNM Springer 833, (1980).
A. Nikeghbali and E. Platen: {\it A reading guide for last passage times with financial applications in view}. To appear in Finance and Stochastics (2012)
D. Revuz and M. Yor: {\it Continuous martingales and Brownian motion.}3rd Edition, Springer (1999).
D. Williams: {\it Path decomposition and continuity of local times for one-dimensional diffusions, I}, Proceed. of London Math. Soc. (3), 28, 738-768, (1974).
\noindent Authors' adresses:
\noindent Universit\'e Libre de Bruxelles\\Facult\'e des sciences\\D\'epartement de Math\'ematique, Campus Plaine CP 210\\ B-1050 Brussels, Belgium.
\noindent
Universit\'e Pierre et Marie Curie\\ Laboratoire des Probabilit\'es\\4, place Jusieu, Tour 56, \{\cal F}-75252 Paris Cedex 05, France
\end{document} | arXiv |
\begin{document}
\title{On sequentiality and well-bracketing in the $\pi$-calculus}
\begin{abstract} The $\pi$-calculus is used as
a model for programming languages. Its contexts exhibit arbitrary concurrency, making them very discriminating. This may prevent validating desirable behavioural equivalences in cases when more disciplined contexts are expected.
In this paper we focus on two such common disciplines: sequentiality, meaning that at
any time there is a single thread of computation, and well-bracketing, meaning that calls to external services obey a stack-like discipline. We formalise the disciplines by means of type systems. The main focus of the paper is on studying the consequence of the disciplines on behavioural
equivalence. We define and study labelled bisimilarities for sequentiality and well-bracketing. These relations are coarser than ordinary bisimilarity.
We prove that they are sound for the respective (contextual) barbed equivalence, and also
complete
under a certain technical condition.
We show the usefulness of our techniques on a number of examples, that have mainly to do with the representation of functions and store.
\end{abstract}
\section{Introduction}\label{s:intro}
The $\pi$-calculus has been advocated as a model to give semantics to, and reason about, various forms of programming languages, including those
with higher-order features. Strengths of the $\pi$-calculus are its rich algebraic theory and its wide spectrum of proof techniques. Concurrency is at the heart of the $\pi$-calculus: computation is interaction between concurrent processes. The operators of the calculus are simple (parallelism, input, output, restriction being the main ones) and unconstrained. This yields
an amazing expressive power~--- the calculus
can model a variety of programming idioms~\cite{SanWal}. However, this also makes the contexts of the calculus very discriminating; as a consequence, behavioural equivalences, which are supposed to be preserved by all the contexts of the calculus,
are rather demanding relations.
Higher-level languages may be syntactically quite different from a language for pure concurrency such as the $\pi$-calculus. For instance, the paradigmatic higher-order programming language,
the $\lambda$-calculus, is a pure calculus of functions and, in both its call-by-name and call-by-value variants, is
sequential~--- it is even deterministic. A variety of extensions of it have been considered; examples of additional features are references, control operators, non-determinism, (constrained) forms of concurrency. The specific set of syntactic features chosen for the language determines the ways in which the contexts of the language may interact with the terms. In any case, the patterns of interaction are usually more disciplined than those that arise in $\pi$-calculus representations of those terms.
Thus
there are $\lambda$-terms that are indistinguishable within the (pure) $\lambda$-calculus whose $\pi$-calculus images can be separated by appropriate $\pi$-contexts.
A well-known way of imposing a discipline to the $\pi$-calculus is to equip it with a type system. Such systems are intended to capture communication patterns that occur frequently when programming in the $\pi$-calculus. A number of type systems have been considered: e.g., capability types (formalising
the intended I/O usage of names that are exchanged among processes), linearity (formalising the property that certain names may be used at most once), session types (formalising the communication protocols in the dialogues between two or more processes), and so on~\cite{DBLP:journals/mscs/PierceS96,DBLP:journals/toplas/KobayashiPT99,DBLP:conf/esop/HondaVK98,AnconaBB0CDGGGH16}. Type systems have also been designed to capture specific properties of processes, such as termination, deadlock-freedom, lock-freedom \cite{DBLP:journals/toplas/Kobayashi98,DBLP:journals/iandc/Kobayashi02,DBLP:journals/iandc/DengS06,DBLP:journals/iandc/YoshidaBH04,DBLP:journals/toplas/KobayashiS10}. Types impose constraints on the set of legal contexts in which well-typed terms are supposed to be used; this can make behavioural equivalences usefully coarser.
A further step is then to tune the proof techniques of the $\pi$-calculus to such type systems, so to be able to actually prove the behavioural equalities that only hold in presence of types. Typically this is investigated in the coinductive setting of bisimilarity, and achieved by
refining and/or modifying the standard bisimilarity clauses so to take the usage of types into account. The resulting bisimilarity should be sound with respect to contextually-defined forms of bisimilarity such as \emph{barbed equivalence} (or \emph{congruence}); ideally, it should also be complete.
In barbed \equivalence, the bisimulation game is played only on internal actions, and certain success signals, the barbs, are used to monitor the computation. In the standard barbed \equivalence, an arbitrary context may be added, once (at the beginning), on top of the tested processes. In \emph{reduction-closed} barbed \equivalence \cite{HoYo95,SanWal}, the context may be dynamically updated, by adding further components during the computation. Reduction-closed barbed \equivalence usually allows simpler proofs of completeness, and
does not require any hypothesis of image-finiteness on the state space of the tested processes. In contrast, standard barbed \equivalence is more robust~--- reduction-closed barbed \equivalence may sometimes be over-discriminating~\cite{SaWa01}.
In this paper we focus on the $\pi$-calculus representation of \emph{sequentiality} and \emph{well-bracketing}. `Sequentiality' intuitively indicates the existence of a
single thread of computation. 'Well-bracketing' is a terminology
borrowed from
game semantics, and
used to refer to
a language without control operators, in which the call-return interaction behaviour
between a term and its context follows a stack discipline. Our main objectives are to define bisimilarity-based proof techniques for type systems in the $\pi$-calculus that formalise the sequentiality and well-bracketing notions. We actually work with the \emph{asynchronous} $\pi$-calculus, \Api, as this is the calculus that is usually adopted in the literature for modelling higher-order languages.
In \Api,
sequentiality is the property that, at any time, at most one process \emph{is active}, or \emph{carries the thread}; that is, the process has the control on the computation and decides what the next computation step can be. In other words, we
never find two sub-components of a system both of which contain an \emph{interaction redex} (a pair of an input and an output processes at the same name).
In the (standard) encodings of the $\lambda$-calculus \cite{DBLP:journals/mscs/Milner92,DBLP:journals/iandc/Sangiorgi94}, a process is active, i.e., it carries the thread, when it contains an unguarded output particle. Indeed, the $\pi$-calculus terms obtained from the encodings give rise to computations in which, syntactically, at any time there is at most one unguarded output particle.
An input process that consumes that output will in turn become active.
Our type system is more general, in that we allow also input processes to carry the thread. The type system specifies whether a name may carry the thread in output or in input; we call these names \emph{output-controlled} and \emph{input-controlled}. While the output-controlled are the most important ones (for instance, they play a central role in the modelling of functions), input-controlled names may be
useful too, for instance, in the representation of references or locks. A reference $\ell$ that contains the value $n$ is represented in \Api
by an output particle $\outpi \ell n$; and a process accessing the reference will do so by performing an input at $\ell$. Thus an input at
$\ell$ indicates ownership of the current computation thread.
As remarked above, sequentiality implies absence of parallel computation threads.
Sequentiality however does not exclude non-determinism. An output particle $\outpi ab$ that owns the thread may have the possibility of interacting with different input processes at $a$ (and symmetrically for input processes owning the thread).
Indeed we also admit internal non-determinism (i.e, processes such as $\tau.P + \tau.Q$ that
may chose to reduce either to $P$ or to $Q$ without interactions with the environment), both in active and in inactive processes.
The type system for well-bracketing is a refinement of that for sequentiality, in which a stack of \emph{continuation names} keeps track of the structure of calls and returns among the processes. These stacks are similar to those used in the implementation of compilers for languages (or fragments of languages) adopting well-bracketing, or used in well-bracketed forms of game semantics.
Finding proof techniques to reason about sequentiality and well-bracketing
presents a number of caveats, that have mainly to do with the soundness and completeness of the resulting bisimilarity with respect to barbed \equivalence. We briefly discuss below a couple of issues concerning completeness.
In the proof of completeness one has to show that the contexts of the language are at least as discriminating as the labelled bisimilarity. In standard proofs, one defines special contexts that interact with the tested processes and, at the same time, emit certain signals to the outside so to provide information on the kind of interactions that have occurred with the processes. Such behaviour of the testing contexts is however inherently concurrent~--- the context has to interact with the tested processes and, at the same time, emit signals to the outside~--- and is therefore liable to break the typing discipline for sequentiality (and hence also well-bracketing).
Further problems arise in proofs about reduction-closed barbed \equivalence. The reason why completeness proofs for reduction-closed barbed \equivalence may be simpler than with standard barbed \equivalence is that the testing context may be incrementally adjusted, after every interaction step with the tested processes. This however requires the existence of special components in the
contexts to handle the fresh names generated by
the tested processes. Specifically, the task of these components is to ensure that
new pieces of contexts, added later, will be able to access such fresh names. Again, these components represent parallel
threads, and break the sequentiality and well-bracketing
disciplines. For this reason in the paper we cannot appeal to reduction-closed forms of barbed \equivalence, remaining within the standard
notions and therefore requiring an image-finiteness condition.
In the case of well-bracketing the problems above are enhanced by the presence of continuation names. These names
are \emph{linear}~\cite{DBLP:journals/toplas/KobayashiPT99} (they may only be used once),
\emph{input receptive} \cite{DBLP:journals/tcs/Sangiorgi99} (the input-end of the name should always be available), and output-controlled. This places further constraints on the use of such names within contexts that test the processes.
For the above reasons, the completeness proofs for sequentiality and well-bracketing present significant technical differences, both between them and from completeness proofs in the literature.
In the paper we propose labelled bisimilarities that allow us to reason about processes following the sequentiality or well-bracketing disciplines. We prove that the bisimilarities are sound with respect to barbed equivalence. We also establish completeness, on processes with only output-controlled names. We do not know whether completeness holds in the general case, with also input-controlled names. \ds{so we say somewhere why the case of input-controlled names is
problematic? } \daniel{the following sentence is just after the proof sketch for
completeness-seq:\\
\textit{The reasoning sketched above does not apply if input-controlled names are allowed, intuitively because this introduces processes that exhibit no barbs while controlling the thread, such as, e.g., $u.\out x$. }} We also study some refinements of the bisimilarities: one is obtained by injecting ideas from bisimilarities for calculi with references \cite{DBLP:conf/concur/HirschkoffPS20}; other refinements are forms of `up-to techniques'. We illustrate applications of our techniques on a number of examples, most of which have to do with the representation of functions and references. Usually the examples are about equalities that only hold under the sequentiality or well-bracketing disciplines;
other examples show that sequentiality and well-bracketing may make equalities simpler to prove because there are fewer observables to take into account.
\emph{Paper outline.} We introduce some background in Section~\ref{s:api}. We study sequentiality in Section~\ref{s:seq}, and well-bracketing in Section~\ref{s:wb}: in each case, we present our type system, define an appropriate notion of bisimilarity, and show some examples or laws that we can derive. Related and future works are discussed in Section~\ref{s:ccl}. For lack of space, some technical definitions and proofs are given in~\cite{HPS:lics21:long}.
\section{Background: the (asynchronous) $\pi$-calculus}\label{s:api} We recall here the standard syntax of the asynchronous $\pi$-calculus, \Api, from~\cite{DBLP:journals/tcs/AmadioCS98}: $$ \begin{array}{rcl}
P,Q& ::=& \outpi{a}{\many b} \ensuremath{~\big|~} !\inpi{a}{\many b}.P \ensuremath{~\big|~} P|Q \ensuremath{~\big|~} \respi{a}P \ensuremath{~\big|~} G
\\[.1em]
G,G'& ::=& \ensuremath{\boldsymbol{0}} \ensuremath{~\big|~} \inpi{a}{\many b}.P \ensuremath{~\big|~} \tau.P \ensuremath{~\big|~} [a=b]G \ensuremath{~\big|~} G+G' \end{array} $$
Names are ranged over by $a,b,...$. In prefixes $\outpi{a}{\many b}$ and $ \inpi{a}{\many b}.P $, name $a$ is the \emph{subject} and $\til b$ are the \emph{objects}. We use a tilde, like in
\many b, for (possibly empty) tuples of names; similarly
$\respi{\many{a}} P$ stands for a sequence of restrictions.
As usual, we write $a.P$ and $\out a$ when the object of a prefix is the empty tuple.
We use $\sum_{i\in I} G_i$ (resp. $\prod_{i\in I} P_i$) for $G_{i_1}
+ \dots + G_{i_n}$ (resp. $P_{i_1} | \dots | P_{i_n}$) where $I =
\set{i_1,\dots,i_n}$.
We write $P\sub a b$ for the result of replacing name $b$ with $a$ in
$P$ in a capture-avoiding way.
Contexts, $C$, are processes containing a single occurrence of a
special constant, the
hole (written $\contexthole$).
The {\em static\/} contexts, ranged over by $E$, have the form $\res{\many a}(P | \contexthole)$.
In examples, for readability we sometimes use basic data values such as integers
and booleans.
The definition of structural congruence, written $\equiv$, and of the strong and weak labelled
transitions, written
$\strans{\mu}$, $\wtrans{}$, and $\wtrans{\hat\mu}$, are standard and are given in~\cite{HPS:lics21:long}.
We note $\fnames{P}$ (resp. $\fnames{\mu}$) the set of free names of $P$ (resp. $\mu$). We sometimes abbreviate reductions $P \arr\tau P' $ as $P \longrightarrow P'$.
The calculi in the paper will be typed. For simplicity we define our type systems as refinements of the most basic type system for $\pi$-calculus, namely
Milner's {\em sorting} \cite{Mil91}, in which names are partitioned into a collection of {\em types} (or sorts), and a sorting function maps types onto types. If a name type $S$ is mapped onto a type $T$, this means that names in $S$ may
only carry names in $T$. We assume that there is a sorting system under which all processes we manipulate are well-typed.
We write $\typedOK \Delta P$ when process $P$ is well-typed under
$\Delta$, and similarly for other objects, such as contexts.
The reference behavioural equivalence for us will be the context-closure of barbed bisimulation. We focus on barbed equivalence (as opposed to barbed congruence) because it is simpler (notably, we do not need to consider issues of closure of the labelled bisimulations under name substitutions). The definition of barbed bisimulation uses the reduction relation $\Longrightarrow $ along with an observation predicate $\Dwa_a$ for each name $a$, which detects the possibility of performing an output to the external environment along $a$. Moreover, since we work in a typed setting, such an output should be allowed by the typing of the tested processes.
Thus, we write $\typedOKp \Delta {P \Dwa_a}$ if $\Delta$ is a typing for $P$ (i.e., $\typedOK \Delta P$ holds), there is an output $\mu$ with subject $a$
s.t.\ $P \Arr\mu P'$, and such a transition is observable under the typing $\Delta$. The meaning of 'observable under a typing' will depend on the specific type system adopted; in the case of the plain sorting, all transitions are observable.
Having typed processes, in the definition of barbed equivalence we may only test processes with
contexts that respect the typing of the processes.
\begin{definition} \label{d:stCtT}
$\qct$ is a \emph{$\conTT \Gamma {\Delta }$ context} if $\typedOK{ \Gamma }{\qct}$ holds, using the typing for the processes plus the
rule $
\inferrule{ }{\typedOK{ \Delta}{\contexthole}} $ for the hole. \end{definition}
Similarly, $P$ is a \emph{$\Delta$-process} if $\typedOK \Delta P$. We assume (as in usual Subject-Reduction properties for type systems) that typing is invariant under reduction.
\iflong This is the case for the type systems we present in this paper. \fi \begin{definition}[Barbed bisimulation, equivalence, and congruence] \label{d:bb} {\em Barbed $ \Delta$-bisimulation} is the largest symmetric relation $\bbis \Delta$
on $\Delta$-processes s.t.\
$P \bbis \Delta Q$ implies:
\begin{enumerate} \item whenever $P \longrightarrow P'$
then there exists $Q'$ such that
$Q \Longrightarrow Q'$ and $P' \bbis \Delta Q'$;
\item for each name $a$,
$\typedOKp \Delta {P \Downarrow_a}$ iff $ \typedOKp \Delta {Q \Downarrow_a}$.
\end{enumerate} Two $\Delta$-processes $P$ and $Q$ are {\em barbed equivalent at $ \Delta$}, written $P \beq \Delta Q$, if for each $\conTT \Gamma {\Delta }$ static context $E$ it holds that $E[ P] \bbis\Gamma E[ Q]$. \emph{Barbed congruence at $\Delta$}, $\bcong \Delta$, is defined in the same way but employing all $\conTT \Gamma {\Delta }$ contexts (rather than only the static ones). \end{definition}
Barbed equivalence in the plain (untyped)
\Api, $\beq{}$, can be proved to coincide with the ordinary labelled early asynchronous bisimilarity,
on
image-finite processes,
exploiting the $n$-approximants of the labelled equivalences.
We recall that the class of {\em image-finite processes} is the largest subset ${\cal I}$ of
processes that is derivation closed and s.t.\ $P \in {\cal I}$ implies that, for all $\mu$, the set $\{P' \; \st\; P \Arr{\mu } P'\}$, quotiented by alpha conversion, is finite. In the remainder of the paper, we omit the
adjectives `early' and `asynchronous' in all bisimilarities.
\iflong Similarly, the relation we present below is usually called \emph{asynchronous bisimulation}, due to the second clause for input transitions; we simply call it \emph{bisimulation}, and do so for the coinductive equivalences we consider in the paper, which are all asynchronous.
\fi
\begin{definition}[Bisimulation] \label{d:bisa}
A relation $\ensuremath{\mathcal{R}}$ on processes is a \emph{bisimulation} if whenever $P\ensuremath{\mathrel{\mathcal{R}}} Q$
and $P\strans{\mu}P'$, then one of these
two clauses holds:
\begin{enumerate} \item
there is $Q'$ such that $Q\wtrans{\hat\mu}Q'$ and $P'\ensuremath{\mathrel{\mathcal{R}}} Q'$;
\item $\mu = \earlyinpi{a}{\many{b}}$ and there is $Q'$ such that $Q|\outpi{a}{\many{b}}\wtrans{}Q'$ and $P'\ensuremath{\mathrel{\mathcal{R}}} Q'$.
\end{enumerate} Moreover the converse holds too, on the transitions from $Q$.
\emph{Bisimilarity}, $\ensuremath{\approx}$, is
the largest
bisimulation.
\end{definition}
\begin{theorem}[\cite{DBLP:journals/tcs/AmadioCS98}]
\label{t:bc} On image-finite processes, relations $ \beq{}$ and $\ensuremath{\approx}$ coincide. \end{theorem}
\section{Sequentiality}\label{s:seq}
In this section we study sequentiality. We first formalise it by means of a type system, and then we examine its impact on behavioural equivalence.
\subsection{Type system} As mentioned in Section~\ref{s:intro}, intuitively, sequentiality ensures us that at any time
at most one interaction can occur in a system; i.e., there is a single computation
thread. A process that holds the thread decides what the next interaction can
be. It does so by offering a single particle (input or output) that controls the thread. The process may offer multiple particles, but only one of them may control the thread. The control on the thread attached to a particle is determined by the subject name of that particle. A given name may exercise the control on the thread either in output or in input; in the former case we say that the name is \emph{output-controlled},
in the latter case the name is \emph{input-controlled}. For instance, suppose that $x,y,z$ are output-controlled and $u,v$ are input-controlled. Then the following process correctly manages the thread and will indeed be typable in our type system: \[
P \defi u. (\outC x | y.\out{x}) | z. \outC y | \outC v \] The initial particles in $P$ are $u, z, \outC v$; however only $u$ controls the thread, as $z$ is output-controlled and $v$ is input-controlled. When the input at $u$ is consumed, the new particles $ \outC x, y$ are available, where $ \outC x$ now controls the thread, as both names $x,y$ are output-controlled. An external process that consumes the particle $\outC x$ will acquire the control over the thread.
For instance, a process such as $Q \defi \outC u | \inpC x . Q' $ initially does not hold the thread;
in the parallel composition $P |Q$, after the two interactions at $u$ and $x$, the control on the thread will be acquired by $Q'$:
\[ P | Q \longrightarrow \longrightarrow (y.\out{x} | z. \outC y | \outC v) | Q' \] Now $Q'$ will decide on the next interaction; for instance, it may offer an output at $y$ or $z$, or an input at $v$. It may only offer one of these, though it may offer other particles that do not control the thread.
\noindent {\bf Notation.} {\em
In the remainder, $x,y,z$ range over output-controlled names,
$u,v,w$ over input-controlled names;
we recall that $a,b,c$ range over the set of all names. }
The name used is therefore an indication of its type. For instance,
in $\respi xP$, $x$ is output-controlled, and can be only
alpha-converted using another output-controlled name.
The type system for sequentiality is presented in Figure~\ref{f:typ:rule:seq1}. Judgements are of the form $\typs{\eta}{P}$, for $\eta \in \{0,1 \}$. A judgement
$\typs{1}{P}$ indicates that $P$ owns the thread, i.e., $P$ is \emph{active}, and
$\typs{0}{P}$ otherwise, i.e., $P$ is \emph{inactive}.
We recall that we only present the additional typing constraints given by sequentiality,
assuming the existence of a sorting under which all processes are well-typed (thus the fully-fledged typing judgements would be the form $ \Delta ; \typs{\eta}{P}$, rather than $ \typs{\eta}{P}$).
Some remarks on the rules in Figure~\ref{f:typ:rule:seq1}: a rule with a double conclusion is an
abbreviation for more rules with the same premises but separate
conclusions. The continuation of an input always owns control on the thread; the input itself may or may not have the control (rules \trans{I-Act} and \trans{I-Ina}). A $\tau$-prefix is neutral w.r.t.\ the thread. The rule for parallel composition makes sure that the control on the thread is granted to only one of the components; in contrast, in the rule for sum, the control is maintained for both summands. Operators $\ensuremath{\boldsymbol{0}}$ and match cannot own the thread; this makes sure that the thread control is always exercised.
\begin{figure*}
\caption{The typing rules for sequentiality}
\label{f:typ:rule:seq1}
\end{figure*}
We present some behavioural properties that highlight the meaning of sequentiality. A reduction $P\arr\tau P'$ is an
\emph{interaction} if it has been obtained from a communication between an
input and an output (formally, its derivation in the LTS
of~\cite{HPS:lics21:long}
uses rule \trans{AComm}).
\iflong
A pair of an unguarded input and an unguarded output at the
same name
form an \emph{interaction redex}. \fi In a sequential system, one may not find two disjoint interactions. \iflong interaction redexes. TOFIX\fi
\begin{prop}
Whenever $\typs{\ensuremath{\eta}}{P}$, there exists no $P_1,P_2,
\many{a}$ such that $P \equiv
\respi{\many{a}}(P_1|P_2)$ with $P_1 \strans\tau P_1'$ and
$P_2\strans\tau P_2'$, and both these transitions are interactions.
\end{prop}
An inactive process may not perform interactions.
\begin{prop} If $\typs{0}{P}$, then \iflong no interactions may occur in $P$; i.e., \fi there is no $P'$ with $P \arr\tau P'$ and this transition is an interaction. \end{prop}
An inactive process may however perform $\tau$-reductions, notably to resolve internal choices. In other words such internal choices represent internal matters for a process, orthogonal with respect to the overall interaction thread. The possibility for inactive processes to accommodate internal choices will be important in our completeness proof.
However, an inactive process may only perform a finite number of $\tau$-reductions. A process $P$ \emph{is divergent} if it can perform an infinite sequence of reductions, i.e., there are $P_1, P_2, \ldots,$ with $P \longrightarrow P_1 \longrightarrow P_2 \ldots P_n \longrightarrow \ldots$.
\begin{prop} If $\typs{0}{P}$ then $P$ is not divergent. \end{prop}
In contrast, an active process may be divergent, through sequences of reductions containing infinitely many interactions.
Sequentiality imposes constraints on the interactions that a `legal' (i.e., well-typed) context may undertake with a process. For the definition of barbed bisimulation and equivalence we must therefore define the meaning of observability.
The following definition of \emph{type-allowed transitions} shows what such `legal' interactions can be.
\begin{definition}[Type-allowed transitions]\label{d:tat}
We write $\sttrans{\ensuremath{\eta}}{P}{\mu}{P'}$ if $\typs{\ensuremath{\eta}}{P}$, and $P \strans{\mu} P'$, and
one of the following clauses holds:
\begin{enumerate}
\item $\ensuremath{\eta} = 0$
\item $\mu = \tau$
\item $\ensuremath{\eta} = 1$ and $\mu = \earlyinpi{u}{\many{a}}$ for some $u,\many{a}$ or
$\mu = \respi{\many{a}}\outpi{x}{\many{b}}$ for some
$\many{a},x,\many{b}$.
\end{enumerate} \end{definition} Clause (1) says that all interactions between an inactive process and the context are possible;
this holds because the context is active and may therefore decide on the next interaction with the process. Clause (2) says that internal reductions may always be performed. Clause (3) says that the only visible actions observable in
active processes are those carrying the thread; this holds because the observer is inactive, and it is therefore up to the process to decide on the next interaction.
We now examine how typing evolves under legal actions. We recall that $x$ stands
for an output-controlled name.
\begin{definition}
We write $\rconf{\ensuremath{\eta}}{P}\strans{\mu}\rconf{\ensuremath{\eta}'}{P'}$ when
$\sttrans{\ensuremath{\eta}}{P}{\mu}{P'}$ and:
\begin{enumerate}
\item if $\mu = \earlyinpi{x}{\many{a}}$, then $\ensuremath{\eta}' = 1$.
\item if $\mu = \respi{\many{a}}\outpi{x}{\many{b}}$, then $\ensuremath{\eta}' = 0$.
\item otherwise $\ensuremath{\eta}' = \ensuremath{\eta}$.
\end{enumerate} \end{definition}
\begin{theorem}[Subject Reduction] \label{l:subred:seq1}
If $\typs{\ensuremath{\eta}}{P}$ and
$\rconf{\ensuremath{\eta}}{P}\strans{\mu}\rconf{\ensuremath{\eta}'}{P'}$ then $\typs{\ensuremath{\eta}'}{P'}$. \end{theorem} \iflong In the third clause above, in the case of an interaction, we necessarily have $\ensuremath{\eta}=1$.
\fi Weak type-allowed transition are defined as expected,
exploiting the
invariance of typing under reductions: $\wttrans{\ensuremath{\eta}}{P}{\mu}{P'}$ holds if there are $P_0,P_1$ with $P \Longrightarrow P_0 $, $\sttrans{\ensuremath{\eta}}{P_0}{\mu}{P_1}$ and $P_1\Longrightarrow P'$.
\subsection{Behavioural equivalence} To tune Definition~\ref{d:bb} of barbed bisimulation and equivalence to the setting of sequentiality, we have to specify the meaning of observables. An observable $\typps \eta {P \Dwa_a}$ holds if there are $P'$ and an output action $\mu$ such that
$\wttrans{\ensuremath{\eta}}{P}{\mu}{P'}$ and the subject of $\mu$ is $a$. Following Definition~\ref{d:stCtT}, in barbed equivalence, the legal contexts are the $\conTT \eta{\eta'}$ static contexts. \iflong
To remind ourselves of the sequentiality constraint, we \else We \fi
write barbed equivalence at $\eta$ as $\beq \eta$. \iflong , and sometimes call it \emph{sequential barbed equivalence at $\ensuremath{\eta}$}. \fi Thus $P \beq \eta
Q$ holds if
$\typs{\ensuremath{\eta}}{P,Q}$ and $E[P] \bbis{\eta'} E[Q]$,
for any $\eta'$ and
any $\conTT{\eta'} \eta $ static context $E$.
We are now ready to define the labelled bisimilarity to be used on sequential processes, which is our main proof technique for barbed equivalence. A \emph{typed process relation} is a set of triplets $(\ensuremath{\eta},P,Q)$ with $\typs{\ensuremath{\eta}}{P,Q}$.
\begin{definition}[Sequential Bisimulation]\label{d:sb}
A typed process relation $\ensuremath{\mathcal{R}}$ is a \emph{sequential bisimulation} if whenever
$(\ensuremath{\eta},P,Q)\in\ensuremath{\mathcal{R}}$ and $\rconf{\ensuremath{\eta}}{P}\strans{\mu}\rconf{\ensuremath{\eta}'}{P'}$, then one of the two following clauses holds:
\begin{enumerate} \item there is $Q'$ such that $Q \wtrans{\hat\mu}Q'$ and $(\ensuremath{\eta}',P',Q')\in\ensuremath{\mathcal{R}}$;
\item $\mu = \earlyinpi{a}{\many{b}}$ and there is $Q'$ such that $Q|\outpi{a}{\many{b}} \wtrans{}Q'$ with $(\ensuremath{\eta}',P',Q')\in\ensuremath{\mathcal{R}}$. \end{enumerate} Moreover, the converse of (1) and (2) holds on the transitions from $Q$. Processes
$P$ and $Q$ are \emph{sequentially bisimilar at $\ensuremath{\eta}$}, written $P \wbiss{\ensuremath{\eta}} Q$, if
$(\ensuremath{\eta}, P, Q)\in \ensuremath{\mathcal{R}}$ for some sequential bisimulation $\ensuremath{\mathcal{R}}$. \end{definition}
In clause (2), $Q
| \outpi a b$ is well-typed, be $a$ an input- or output-controlled name. Clauses (1) and (2) are the same as for ordinary bisimilarity $\ensuremath{\approx}$ (Definition~\ref{d:bisa}); typing however prevents certain transitions to be considered as challenge transitions in the bisimulation game. Thus the resulting bisimilarity becomes coarser.
Ordinary bisimilarity is included in the sequential one (the inclusion is strict, see Section~\ref{ss:exSeq}). \iflong
also a proof technique for barbed equivalence. Indeed, in any bisimulation, the subset of well-typed processes
yields a sequential bisimulation. \fi \begin{proposition} \label{p:asy_seq} For $\typs{\eta}{P,Q}$, if
$P \ensuremath{\approx} Q $ then also
$P \wbiss{\ensuremath{\eta}} Q$. \end{proposition}
\iflong
all the examples in the next section fail for $\ensuremath{\approx}$. \fi
\begin{theorem}[Soundness] \label{t:sound_seq}
If $P\wbiss{\ensuremath{\eta}} Q$, then $P \beq{\ensuremath{\eta}} Q$. \end{theorem}
As usual, the proof of Theorem~\ref{t:sound_seq} relies on the preservation of \wbiss\ensuremath{\eta}{} under parallel composition, which requires some care in order to enforce sequentiality. This is ensured by typability.
Theorem~\ref{t:sound_seq} allows us to use the labelled bisimilarity \wbiss{\ensuremath{\eta}} as a proof technique for typed barbed equivalence.
This proof technique is also complete,
assuming only output-controlled names (i.e., the thread may only be exercised by output particles, not by the input ones).
\begin{theorem}[Completeness on output-controlled names]\label{t:completeness}
For all image-finite processes $P,Q$ that only use
output-controlled names, and for all \ensuremath{\eta}, if $P \beq{\ensuremath{\eta}} Q$ then $P \wbiss{\ensuremath{\eta}} Q$. \end{theorem}
The completeness proof can be found in~\cite{HPS:lics21:long}.
While the overall structure of the proof is standard, the technical details are specific to sequentiality.
As usual, we rely on a stratification of bisimilarity and approximants
$\wbissn{\ensuremath{\eta}}{n}$, \iflong
for the $n$th of \wbiss\ensuremath{\eta}{} (the definition is standard, and is given in Appendix). As usual, \wbiss\ensuremath{\eta}{} coincides with the intersection of its approximants). \fi
and reason by contradiction to show that if $\typs{\ensuremath{\eta}}{P,Q}$ and $P \not\wbissn{\ensuremath{\eta}}n Q$, then there is a $\conTT{\ensuremath{\eta}'}{\ensuremath{\eta}}$ static context $E$ such that $E[P] \not\bbis{\ensuremath{\eta}'} E[Q]$. \iflong Because of typing, the definition of such $E$ requires some care. \fi The case $\ensuremath{\eta}=0$ (tested processes are inactive) is rather standard: the context
$E$ is of the form $\respi{\many{x}}(\contexthole | \out{z} | z.R)$, for some fresh $z$, and some ``tester process'' $R$. The barb at $z$ allows us to detect when the tested process interacts with $R$.
The delicate case is when $\ensuremath{\eta}=1$ (tested processes are active): the context must be inactive and hence
cannot have an unguarded output at $z$.
We use in this case a \iflong $\conTT{1}{1}$ \fi context of the form
$E \defi \respi{\many{x}}(\contexthole | G_R+G)$.
Process $G_R$ is the tester process, and $G$ is
$\sum_{y\in S} \inpi y{\many{y'}}.\out{z}$, defined for some fresh $z$ and some set $S$
containing $\fnames P\cup\fnames Q$.
$G$ satisfies the following
property: for any $P_0$ and for any $x$,
if $\typps{1}{P_0\wbarb{\out{x}}}$ then $\typps{1}{P_0 | G
\wbarb{\out{z}}}$.
Thus, as soon as $P_0$ exhibits some barb, we have
$\typps{1}{E[P_0] \wbarb{\out{z}}}$, and $P_0$ cannot interact with $R$
without removing the barb at $z$, which allows us to reason as in the case $\ensuremath{\eta}=0$.
The proof schema above does not apply if input-controlled names are allowed, intuitively because in this case the processes being tested may be active and perform an input (at an input-controlled name), thus maintaining the thread; both before and after the transition the testing context is passive and hence unable to signal, with appropriate barbs, which interaction occurred.
\subsection{Examples} \label{ss:exSeq}
With respect to ordinary bisimilarity, in sequential bisimilarity (\wbiss{\ensuremath{\eta}}) fewer challenges are allowed. This may both mean that certain processes, otherwise distinguishable, become equal, and that certain equalities are simpler to prove because
the state space of the processes to be examined is reduced.
We
present some equalities of the first kind (valid for \wbiss\ensuremath{\eta}{}
only). In Section~\ref{s:ex:ref}, we also show a refinement of \wbiss{\ensuremath{\eta}} useful for reasoning about references.
\subsubsection{Basic examples}
In the type system, $\ensuremath{\boldsymbol{0}}$ is inactive~---- without the thread. We write $\ensuremath{\boldsymbol{0}}_1 $ to abbreviate $ \respi{x}(\out{x})$ (an active process without transitions).
\begin{example} \label{exa:singu} While a component of a system is active, other components cannot be observed. Thus, if the active component keeps the thread, the existence of other components is irrelevant. Indeed we have, for any $R, Q$ inactive:
$$R | \ensuremath{\boldsymbol{0}}_1 \wbiss{1} R | \respi{x}(\out{x} | !x.\out{x}) \wbiss{1} Q |
\respi{x}(\out{x} | !x.\out{x}) \wbiss{1} \ensuremath{\boldsymbol{0}}_1\iflong,\fi$$
\iflong More generally, say that a process $P$ is \emph{singular} if it never releases the thread; that is, the set of singular processes is the largest set ${{\cal T}}$ of processes such that for all $P \in {\cal T}$ we have $\typs{1}{P}$ and whenever $\sttrans{1}{P}{\mu}{P'}$ then $\mu$ is not an output and also $P' \in {\cal T}$. Then for all $P \in {\cal T}$, and for all processes $Q_1,Q_2$ with $\typs{0}{Q_1,Q_2}$ we have \[
Q_1 | P \wbiss{1} Q_2 | P
. \] The relation containing all such pairs
$ (Q_1 | P, Q_2 | P)$ of processes is a sequential bisimulation. \fi
\end{example}
\begin{example} \label{ex:exp} An unguarded occurrence of an input at an input-controlled name becomes the only observation that can be made in a process. This yields the following equalities
\begin{mathpar} \begin{array}{rcl}
u.P | x.Q &\wbiss{1}& u.(P | x.Q) \\
u.P | \out{v} &\wbiss{1}& u.(P | \out{v}) \hskip 1cm \text{ for }u\neq v \end{array}
\end{mathpar} \iflong For the first equality, one shows that
$$\{ (1, u.P | x.Q , u.(P | x.Q)) \} \cup \Id \, , $$ where $\Id$ is the (typed) identity relation, is a sequential bisimulation. One proves the second equality similarly.
\fi \end{example}
\iflong
By sequentiality, it is also not possible to access parts of
processes that would require a simultaneous/parallel
reception. The following example illustrates this.
\fi
\begin{example}
Consider the process
$$P \defi \respi{y',z'}(!x.(z'.\out{z}|\out{y'}) | !y.\out{z'}).$$
The output at $z$ becomes observable if both an input at $x$ and an input at $y$ are consumed, so that the internal reduction at $z'$ can take place. However the input at $x$ acquires the thread, preventing a further immediate input at $y$; similarly for the input at $y$. Indeed we have
$ P
\wbiss{0} x.\ensuremath{\boldsymbol{0}}_1 + y.\ensuremath{\boldsymbol{0}}_1 \iflong\defi Q \fi $. \iflong To prove the equality, we can use $ \{P, Q \} \cup \wbiss{1}$, which is easily proved to be a sequential
bisimulation. The derivatives of $P$ and $Q$ are singular (and stable, i.e., unable to reduce further) processes; therefore they are in \wbiss{1}, as discussed in Example~\ref{exa:singu}.
Under the ordinary
bisimilarity,
$P$ and $Q$ are distinguished because the sequence of transitions $P \strans{x}\strans{y}\strans{\out{z_3}}$ cannot be matched by $Q$. \fi \end{example}
\iflong \begin{example} This example informally discusses why sequentiality can help reducing the number of pairs of processes to examine in a bisimulation proof. Suppose we wish to prove
the equality between the two processes \[ \begin{array}{rcl}
P_1 &\defi & \outC x |
\inpC y. R | \inpC u .Q \\
P_2 &\defi & \outC x |
\inpC u.Q | \inpC y .R \end{array}
\] and such processes are typable under the sequentiality system. The difference between $P_1$ and $P_2$ comes from a commutativity of parallel composition. We may therefore use ordinary bisimilarity, as this is a sound proof technique for typed barbed equivalence.
One may prove, more generally, that parallel composition is commutative and derive the equality. However suppose we wish to use the bisimulation method, concretely, on $P$ and $Q$. The two processes have $3$ initial transitions (with labels $\outC x, y$, and $u$), and the subtrees of transitions emanating from the such derivatives have similar size. Under ordinary bisimulation, we have to examine all the states in the $3$ subtrees.
Under sequential bisimulation, however, only the input at $y$ is observable (it is the only one carrying the thread), thus removing $2$ of the $3$ initial subtrees. Further pruning may be possible later on, again exploiting the fact that under sequentiality only certain transitions are observable. \end{example} \fi \subsubsection{Examples with references}\label{s:ex:ref}
We now consider a few examples involving references. For this, we use the standard encoding of references into \Api, and we enhance the bisimilarity for sequentiality so to take references into account.
We use $n,m,...$ to range over the entities stored in references (which can be names or values belonging to a first-order data type like booleans and integers) and placeholders for them. Name $\ell$ is used to represent a reference.
In \Api, a
reference $\ell$ holding a value $n$ is represented as an output particle \outpi{\ell}n. A process that contains a reference $\ell$ should have, at any time, exactly one unguarded output at $\ell$, meaning that at any time the reference has a unique value. We say that in this case $\ell$ \emph{is accessible}. The read and write operations on $\ell$ are written as follows: \[ \begin{array}{rcl}
\rreadL{m}.R & \defi & \inpi{\ell}{m}.(\outpi\ell m | R) \\
\rwriteL{n}.R & \defi & \inpi\ell{m'}.(\outpi\ell n | R)~\mbox{ for }m'\notin\fnames R \end{array} \]
Thus a name $\ell$ used to encode a reference is input-controlled, as an action on a reference is represented by an input at $\ell$ | we use $\ell$ rather than $u,v,...$ to stress the fact that names used to represent references obey constraints that go beyond input-control.
Proof techniques for the representation of references in \Api have been studied in \cite{DBLP:conf/concur/HirschkoffPS20}. Adopting them requires enhancing our type system with information about references, which simply consists in declaring which names represent references.
In the definition of barbed equivalence, the main constraint is that the tested context should make sure that all existing reference names are accessible. To reason about references, several definitions of labelled bisimilarity are presented in \cite{DBLP:conf/concur/HirschkoffPS20}, varying on the forms of constraints imposed on transitions. Here we only import the simplest such constraint: it forbids observations of input transitions $P \arr{\earlyinpi \ell n}P'$ at a reference name $\ell$ when $\ell$ is accessible in
$P$ (i.e., an unguarded output at $\ell$ occurs in $P$). Such a constraint represents the fact that an observer may not pretend to own a
reference when the reference is accessible in the process.
Formally, with the addition of references, judgements in the type system become of the form $\typsR S \eta P$, where $S$ is a finite set of reference names, meaning that $\typs \eta P$ holds and that $S$ is the set of accessible reference names in $P$.
The definition of type-allowed transitions, $\sttransR S {\ensuremath{\eta}}{P}{\mu}{P'}$, is the same as before (Definition~\ref{d:tat}) with the addition, in clause (3),
of the constraint \begin{center} $ $
if $\mu$ is an input $ \earlyinpi{\ell}{n}$ at a reference name $\ell$ then $\ell\not\in
S$.
$(*) $ \end{center}
Finally the definition of \emph{sequential bisimilarity with references at $(S, \ensuremath{\eta})$}, written $\wbRS{S}{\ensuremath{\eta}}$ is the same as that of sequential bisimilarity (Definition~\ref{d:sb}), just using $\typsR S \ensuremath{\eta} {P,Q}$ and $\sttransR S {\ensuremath{\eta}}{P}{\mu}{P'}$ in place of $\typs{\ensuremath{\eta}}{P,Q}$ and
$\sttrans{\ensuremath{\eta}}{P}{\mu}{P'}$.
It is straightforward to extend the soundness proof for sequential bisimilarity w.r.t.\ barbed equivalence (Theorem~\ref{t:sound_seq}) to the case of sequential bisimilarity with references. \iflong We refer to \cite{DBLP:conf/concur/HirschkoffPS20} for further details on proof techniques for references in \Api. \fi
\begin{example}\label{e:rewr} This example shows that reading or writing on a global reference is not subject to interferences from the outside, as these operations require the thread:
$$ \begin{array}{rcl}
\outpi{\ell}{n} | \rreadL{m}.R &\wbRS{\ell}{1}& \outpi{\ell}{n} | R\sub n m\\
\outpi{\ell}{n} | \rwriteL{m}.R &\wbRS{\ell}{1}& \outpi{\ell}{m} | R \end{array}$$
Indeed, in each law, if $P$ (resp. $Q$) is the process on the left-hand (resp. right-hand) side, then the relation $\{((\ell ; 1),P,Q)\}\cup \Id$ is a sequential bisimulation, when taking
the constraint $(*)$ for references into account. \end{example}
\begin{example}[Fetch-and-add, swap] We consider fetch-and-add and swap operations, often found in operating systems. The first, written
$\rfaaL{n}{m}$
atomically increments by $n$ the content of the reference $\ell$,
and returns the original value as $m$; the second, written
$\rswapL{n}{m}$, atomically sets the content
of
$\ell$
to $n$ and returns the original value as $m$: \[ \begin{array}{rcl}
\rfaaL{n}{m}.R & \defi & \inpi{\ell}{m}.(\outpi{\ell}{m+n} | R) \\
\rswapL{n}{m}.R & \defi & \inpi{\ell}{m}.(\outpi{\ell}{n} | R) \end{array} \] These operations may be mimicked by a combination of read and write operations (we take $m'\notin\fnames R$):
\[\begin{array}{rcl}
\rfaaD{n}{m}.R & \defi & \rreadL{m}.\rwriteL{m+n}. R \\ & = &
\inpi{\ell}{m}.(\outpi{\ell}{m} |
\inpi\ell {m'}.(\outpi{\ell}{m+n} | R))
\\
\rswapD{n}{m}.R & \defi & \rreadL{m}.\rwriteL{n}. R \\ & = &
\inpi{\ell}{m}.(\outpi{\ell}{m} |
\inpi\ell {m'}.(\outpi{\ell}{n} | R)) \end{array}\]
For this mimicking to be correct, sequentiality is necessary. To see this, consider the simple case when $R \defi \outpi c m$. In the ordinary \Api, processes $ \rswapL{n}{m}.R$ and $ \rswapD{n}{m}.R$ are distinguished, intuitively because the observer is capable of counting the two inputs and the two outputs at $\ell$ in $ \rswapD{n}{m}.R$ (against only one in $ \rswapL{n}{m}.R$) and/or is capable of detecting the output $\outpi{\ell}{m}$ in $ \rswapD{n}{m}.R$.
The processes are also distinguished with the proof techniques for references in~\cite{DBLP:conf/concur/HirschkoffPS20}, intuitively because, after the initial input $\inpi{\ell}{m}$ (whereby the processes read the content of the reference),
an observer may interact with the derivative of $ \rswapD{n}{m}.R$ and use its output $\outpi{\ell}{m}$
so to know the value that had been read. Such an observation is not possible with $ \rswapL{n}{m}.R$.
In contrast, the two processes are equal if we take sequentiality into account. That is, we have: $$ \rswapL{n}{m}.R \wbRS{\emptyset}{1}
\rswapD{n}{m}.R$$ This is proved by showing that the relation
$$ \begin{array}{l}
\cup_{m'} \{((\ell;1), \outpi{\ell}{n} | R\sub{m'} m,~ \outpi{\ell}{m'} | \rwriteL{n}.R \sub{m'} m)
\\[3pt] \cup \; \Id\; \cup \; \{((\emptyset;1),\; \: \rswapL{n}{m}.R, \; \: \rswapD{n}{m}.R)\}
\end{array}
$$
is a sequential bisimulation.
The equivalence between $ \rfaaL{n}{m}.R$ and $ \rfaaD{n}{m}.R$ is established using a similar relation.
\end{example}
\begin{example}[Optimised access]\label{e:optim:access}
Two consecutive read and/or write operations can be
transformed into an equivalent single operation. \[ \begin{array}{rcl}
\rwriteL{n}.\rwriteL{m}.R & \wbRS{\emptyset}{1} & \rwriteL{m}.R \\
\rwriteL{n}.\rreadL{m}.R & \wbRS{\emptyset}{1}& \rwriteL{n}.R\sub n m \\
\rreadL{m}.\rreadL{m'}.R & \wbRS{\emptyset}{1}& \rreadL{m}.R\sub{m}{m'} \end{array} \] For the first equality, one shows that \begin{align*}
\Id\; \cup \;&\{((\emptyset;1), \rwriteL{n}.\rwriteL{m}.R,\; \: \rwriteL{m}.R)\}\\
\cup \; &\{((\ell;1), \outpi{\ell}{n} | \rwriteL{m}.R,\; \: \outpi{\ell}{m} | R)\} \end{align*} is a sequential bisimulation. The second law is treated similarly. In both cases, the relation exhibited is finite.
For the third equality, one defines $\ensuremath{\mathcal{R}}$ as
$$
\cup_{n}\set{((\ell;1), \outpi{\ell}{n}|\rreadL{m'}.R\sub{n}{m}, \outpi{\ell}{n} | R\sub{n,n}{m,m'}
)}$$
Then $\{((\emptyset;1), \rreadL{m}.\rreadL{m'}.R,\;
\rreadL{m}.R\sub{m}{m'})\}$ $ \cup\;\ensuremath{\mathcal{R}} \; \cup \; \Id$
is a sequential bisimulation.
\end{example}
\section{Well-bracketing}\label{s:wb}
\subsection{Type System}
We now go beyond sequentiality, so to handle well-bracketing. In
languages without control operators, this means that return-call interactions among terms follow a stack-based discipline.
Intuitively, a well-bracketed system is a sequential system offering services. When interrogated, a service, say $A$, acquires the thread and is supposed to return a final result (unless the computation diverges) thus releasing the thread. During its computation, $A$ may however interrogate another service, say $B$, which, upon completion of its computation, will return the result to $A$. In a similar manner, the service $B$,
during its computation, may call yet another service $C$, and will wait for the return
from $C$ before resuming its computation. $B$ may also delegate to $C$ the task of
returning a result to $A$. In any case, the `return' obligation may not be thrown away or
duplicated.
The implementation of this policy requires
\emph{continuation names}. For instance, when calling $B$, process $A$ transmits a fresh name, say $p$, that will be used by $B$ (or other processes delegated by $B$) to return the result to $A$. Moreover, $A$ waits for such a result, via an input at $p$. Therefore continuation names are \emph{linear}~\cite{DBLP:journals/toplas/KobayashiPT99}~--- they may only be used once~--- and \emph{input receptive} \cite{DBLP:journals/tcs/Sangiorgi99}~--- the input-end of the name must be made available as soon as the name is created; and they are output-controlled: they carry the thread in output.
In short, the `well-bracketing' type system defined in this section
refines the type discipline for sequentiality by adding linear-receptive names and enforcing a stack discipline on the usage of such names. Proof techniques for well-bracketing will be studied in Section~\ref{s:wb:bis}.
\begin{figure*}
\caption{Type system for well-bracketing}
\label{f:typ:rule:int}
\end{figure*}
Thus, with well-bracketing, we have three kinds of names: output-controlled names (ranged over by $x,y,z,...$) and input-controlled names (ranged over by $u,v,w...$), as in the previous section; and continuation names, ranged over by $p,q,r...$. As before, names $a,b,c...$ range over the union of output- an input-controlled names.
Continuation names may only be sent at output-controlled names. Indeed, any output at an output-controlled name must carry exactly one continuation name. Without this constraint the type system for well-bracketing would be more complex, and
it is unclear whether it would be useful in practice.
By convention, we assume that, in a tuple of names transmitted over an output-controlled name, the last name is a continuation name. We write $\many a,p$ for such a tuple of names.
The type system is presented in Figure~\ref{f:typ:rule:int}.
Judgements are of the form \[ \typr \sigma P \] where $ \sigma $ is a \emph{stack}, namely
a sequence of \emph{input-} and \emph{output-tagged}
continuation names, in which the input and output tags alternate, always terminating with an output tag unless the sequence is
empty: \[
\begin{array}{c}
\sigma ~ ::= ~ \sigmaO \midd \sigmaI
\\
\sigmaO ~ ::= ~ \Tag p \OO, \sigmaI
\qquad\qquad
\sigmaI ~ ::= ~ \Tag p \II, \sigmaO \midd \emptyset
\end{array} \]
Moreover: a name may appear at most once with a given tag; and, if a name appears with both tags, then the input occurrence should immediately follow the output occurrence, as for $p$ in $\Tag{p'}\II, \Tag p \OO, \Tag p \II, \sigma$.
We
write $p\in\sigma$ if name $p$ appears in $\sigma$, and $\sz{\sigma}$ for the length of the sequence $\sigma$.
Intuitively, a stack expresses the expected usage of the free continuation names in a process. For instance, if \[ \typr{\Tag {p_1} \OO, \Tag {p_2} \II, \Tag {p_3} \OO, \Tag {p_3} \II, \Tag {p_4} \OO} P
\] then $p_1,..,p_4$ are the free continuation names in $P$; among these, $p_1$ will be used first, in an output ($p_1$ may be the subject or an object of the output); then $p_2$ will be used, in an input interaction with the environment. $P$ possesses both the output and the input capability on $p_3$, and may use both capabilities by performing a reduction at $p_3$; or $P$ may transmit the output capability and then use the input one; the computation for $P$ terminates with an output at $p_4$.
This behaviour however concerns only the free continuation names of $P$: at any time
when an output usage is expected, $P$ may decide to create a new continuation name and
send it out, maintaining its input end. The Subject Reduction Theorem~\ref{t:SR:wb} will formalise the behaviour concerning continuations names in stacks.
As simple examples of typing, we can derive $$\typr{\Tag p\OO}{\outpi pa} \quad \mbox{and}\quad \typr{\Tag p\OO}{\inpi ua.\outpi pa}$$
In the latter typing, by rule \trans{wb-Inp3}, an input at an input-controlled name has the thread, and does not affect the stack because $u$ is not a continuation name.
The same stack can be used to type a process that invokes a service at $x$ before sending the result at $p$, as in
$$\typr{\Tag p\OO}{\respi q(\outpi x{b,q} | \inpi qc.\outpi pc)}$$
where $q$ is a fresh continuation name created when calling $x$. To type the process without the
restriction at $q$, the stack should mention the input and output
capabilities for
$q$:
$$\typr{\Tag q\OO,\Tag q\II,\Tag p\OO}{\outpi x{b,q} | \inpi qc.\outpi
pc}$$
For another example, the process $$P_0 \defi \inpi pa.\outpi {p'}a | \inpi qb.\outpi {q'}b$$ can be typed using two stacks: we have both \typr{\Tag p\II,\Tag {p'}\OO,\Tag
q\II,\Tag{q'}\OO}{P_0} and
\typr{\Tag q\II,\Tag {q'}\OO,\Tag
p\II,\Tag{p'}\OO}{P_0}. The choice of the stack depends on whether
the call answered at $p$ has been made before or after the call answered at
$q$.
We comment on the rules of the type system. In \trans{wb-Out1} and \trans{wb-Out2} the obligation in the stack is fulfilled (the output capability on the only name in the stack is used in \trans{wb-Out1} and
transmitted in \trans{wb-Out2}).
\iflong
Note, in \trans{wb-Out2}, that an output at an output-controlled name $x$, as in the system with sequentiality, passes the thread to another process (i.e., it is like calling a service $x$). \\COMMENT: not clear that \trans{wb-Out2} actually shows that.\\ \fi
As explained above, the last name in the tuple transmitted at $x$ is a continuation name, and the only one being transmitted (to enforce the stack discipline).
In contrast, in \trans{wb-Out3} an output at an input-controlled name does not own the thread and therefore may not carry continuation names. In \trans{wb-Inp1} the input-tagged name on top of the stack is used.
Rule \trans{wb-Inp2} is the complement of \trans{wb-Out2}. In \trans{wb-Inp3}, an input at an input-controlled name maintains the thread. In all rules for input and $\tau$ prefixes,
the stack in the premise of the rules
may not contain input-tagged continuation names because
their input capability must be unguarded (as they are receptive names). The same occurs in rule \trans{wb-Sum}, following~\cite{DBLP:journals/tcs/Sangiorgi99} where
choice on inputs at receptive names is disallowed (though the constraint could be relaxed). Matching is allowed on plain names, but not on continuation names; this is typical of type systems where the input and output capabilities on names are separate \cite{SanWal}; moreover, no continuation name may appear in the process underneath, to make sure that the obligations on continuation names are not eschewed. In \trans{wb-Res1} a continuation name is created, and then its output and input capabilities are inserted into the stack. In the rule, $\xi,\sigma$ is a decomposition of the stack for $\res p P$ where $ \sigma $ is a stack beginning with an output tag; hence $\xi$ is either empty or it is of the form $\sigma', \Tag p \II$, (i.e. $\xi$ is an initial prefix of the stack, either empty or ending with an input tag).
\iflong \ds{thanks Enguerrand for your example: very nice. I'd consider adding it into the
explanations, in fact. It is commented here below.
We should just explain this, and that rule Res1 goes beyond ordinary stack, putting Enguerrand's example}\fi
Rule \trans{wb-Res2} is for continuation names that do not appear in the body of the restriction (this form of rule is common in type systems for linearity, to simplify the assertion of Subject Reduction). In rule \trans{wb-Par}, the typing stack is split to type the two process components $P_1$ and $P_2$; splitting of the typing is usual in type systems with linearity. Here, however, the split must respect the order of the names. That is, the stack in the conclusion should be an interleaving of the two stacks in the premises, as by the following definition.
\begin{definition}[Interleaving]
We write $\inter{\sigma_1}{\sigma_2}{\sigma_3}$ if (i) $\sigma_1$ is a stack, and (ii) $ \sigma_1 $ is an interleaving of $\sigma_2$ and $\sigma_3$ as by the following inductive rules:
\begin{enumerate}
\item $\inter{\emptyset}{\emptyset}{\emptyset}$
\item\label{inter:deux} $\inter{\Tag{p}{\OO},\sigma_1}{\sigma_2}
{\Tag{p}{\OO},\sigma_3}$ if $\inter{\sigma_1}{\sigma_2}{\sigma_3}$
\item\label{inter:trois} $\inter{\Tag{p}{\OO},\sigma_1}{\Tag{p}{\OO},\sigma_2}{\sigma_3}$ if $\inter{\sigma_1}{\sigma_2}{\sigma_3}$
\item the same as~\rref{inter:deux} and~\rref{inter:trois}
with $\Tag{p}{\II}$ instead of $\Tag p\OO$ \end{enumerate}
\end{definition}
If a name appears both in $\sigma_2$ and in $\sigma_3$ with the same tag, then $\theinter{\sigma_2}{\sigma_3}$ may not contain any stack.
\iflong\daniel{I commented here the discussion about examples of processes
that cannot be typed}\fi
Being stack-ordered means that $p.\out{q} | q.\out{p}$ cannot be typed. Indeed, the left process would require $p$ before $q$ in the stack, whereas the right process needs the opposite.
In rule \trans{Res1}, having the possibility to add names in the middle of the stack is mandatory to preserve typability after reduction. Consider for instance:
$\respi{q}(\respi{p}(\outpi{b}{p} | p.\out{q}) | q.\out{p'})
\strans{\respi{p}\outpi{b}{p}} \respi{q}(p.\out{q} | q.\out{p'})$ To type the derivative of the transition above, we have to use rule \trans{Res1} with $\xi = \Tag{p}{\II}$ and $\sigma' = \Tag{p'}{\OO}$.
Typability in the type system of Figure~\ref{f:typ:rule:int} implies typability in the type system for sequentiality. Indeed, when $\typr \sigma P$, if the first name in $\sigma $ is output-tagged then $P$ is active, otherwise $P$ is inactive. We write $\qseqF $ for the function that `forgets' the well-bracketing information in a stack, therefore $\seqF \sigma = 1$ if $\sigma = \Tag p \OO, \sigma'$, for some $p$ and $ \sigma'$, and $\seqF \sigma =0 $ otherwise.
\begin{proposition} \label{p:erase} If $\typr \sigma P$ then also
$\typs{\seqF \sigma} P$. \end{proposition}
In Definition~\ref{d:typr}, we extend type-allowed transitions to processes with continuation names. As previously, we must ensure that the process is typed, and that the transition is allowed by sequentiality (clauses (1) and (2) below). Clause (3) says that the first continuation name observed must be on top of the stack, and that the input or output capability on a continuation name may not be exercised by the environment when both capabilities are owned by the process.
\begin{definition} \label{d:typr} We write $\typpr{\sigma}{P\strans{\mu}P'}$ when \begin{enumerate}
\item $\typr{\sigma}{P}$
\item $\sttrans{\seqF{\sigma}}{P}{\mu}{P'}$ and
\item if $p\in \fnames\mu$ and $p \in \sigma$, then either
$\sigma = \Tag{p}{\OO},\sigma'$ or $\sigma =
\Tag{p}{\II},\sigma'$ for some $\sigma'$;
moreover,
if $p\in{\sigma'}$, then $p$ is not the
subject of $\mu$.
\end{enumerate}
\end{definition} We exploit type-allowed transitions to define transitions with stacks, which make explicit the evolution of the stack. \begin{definition}
We note $\rconf{\sigma}{P} \strans{\mu} \rconf{\sigma'}{P'}$
when
$\typpr{\sigma}{P\strans{\mu}P'}$ and
\begin{enumerate}
\item\label{sr:out:p} if $\mu = \respi{\many{b}}\outpi{p}{\many{a}}$, then $\sigma =
\Tag{p}{\OO},\sigma'$
\item\label{sr:inp:p} if $\mu = \earlyinpi{p}{\many{a}}$, then $\sigma = \Tag{p}{\II},\sigma'$
\item\label{sr:out:extr} if $\mu = \respi{\many{c},p}\outpi{a}{\many{b},p}$, then
$\sigma'=\Tag{p}{\II},\sigma$
\item\label{sr:out:std} if $\mu = \respi{\many{c}}\outpi{a}{\many{b},p}$, then $\sigma =
\Tag{p}{\OO},\sigma'$
\item\label{sr:inp:oc} if $\mu = \earlyinpi{a}{\many{b},p}$, then $\sigma' = \Tag{p}{\OO},\sigma$
\item\label{sr:tau} if $\mu = \tau$, then for $\sigma = \Tag{p}{\OO},\Tag{p}{\II},\sigma''$ and
$p \notin \fnames{P'}$, we have $\sigma' = \sigma''$,
otherwise $\sigma' = \sigma$.
\end{enumerate} \end{definition} In cases~\rref{sr:out:p}, \rref{sr:out:std} (resp.\ \rref{sr:inp:p}), we must have $\sigma = \Tag{p}{\OO},\sigma''$ (resp.\ $\Tag{p}{\II},\sigma''$) by definition of type-allowed transitions.
In clauses~\rref{sr:out:p} and~\rref{sr:inp:p}, the action is an input or an output at a continuation name that must be on top of $\sigma$, and is then removed.
In clause~\rref{sr:out:extr}, the action extrudes a continuation name, and then, following the stack discipline, the process waits for an answer on that name.
In clause~\rref{sr:out:std}, emitting a free continuation name amounts to passing the output capability on that name to the environment.
Dually, in clause~\rref{sr:inp:oc}, receiving a continuation name imposes to use it in output.
Finally, in clause~\rref{sr:tau}, a $\tau$ transition may come from an interaction at a continuation name, in which case $\sigma$ is modified. It can also come from an interaction at a restricted name
or from an internal choice; in such cases, $\sigma$ is unchanged.
\begin{theorem}[Subject Reduction]
\label{t:SR:wb}
If $\typr{\sigma}{P}$ and
$\rconf{\sigma}{P}\strans{\mu}\rconf{\sigma'}{P'}$ then $\typr{\sigma'}{P'}$. \end{theorem}
\iflong\daniel{I commented here the remark about ``trajectory'' of
continuation names}\fi
If a process owns both the input and the output capability on a continuation name $p$, then the environment may not use $p$. Semantically, this is the same as having a restriction on $p$ in the process. It is therefore safe, in the definition of barbed bisimulation and observability,
to assume that all such restrictions are syntactically present, i.e.,
there is a single occurrence of any free continuation name. We call \emph{clean} such processes.
\begin{definition} \label{d:clean} A stack $ \sigma $ is \emph{clean} if no name appears in $ \sigma $ both output- and input-tagged.
A process $P$ is \emph{clean} if $\typr \sigma P$ for some clean $\sigma$. \end{definition}
On clean processes, typing is preserved by reduction.
\begin{proposition} \label{p:clean_red} If $\typr \sigma {P }$ for $\sigma$ clean, then $P \longrightarrow P'$ implies
$\typr \sigma {P'}$. \end{proposition}
Defining barbed bisimulation on clean processes, we can use the $\qseqF$ function above to recast observability in the well-bracketing system from that in the sequentiality system: thus, for $\sigma $ clean, we have $\typpr \sigma {P \Dwa_a}$ (resp.\ $\typpr \sigma {P \Dwa_p}$) if $\typps {\seqF\sigma} {P \Dwa_a}$ (resp.\ $\typps {\seqF\sigma} {P \Dwa_p}$).
In the definition of barbed equivalence, the contexts testing the processes must be clean. Writing $\beq \sigma$ for barbed equivalence at $\sigma$, we have
$P \beq \sigma
Q$ if
$\typr{ \sigma }{P,Q}$, and for any clean $\sigma'$ and
any $\conTT{\sigma'} \sigma $ static context $E$, it holds that $E[P] \bbis{\sigma'} E[Q]$ (note that $\sigma$ itself need not be clean).
\subsection{Discreet Processes}
In this section we put forward the subclass of
\emph{discreet} processes, in which all continuation names that are exported must be private, and show how to transform any process into a discreet one. Then, on discreet processes: \begin{itemize} \item[(1)] we express
a behavioural property
that formalises the stack-like discipline on the usage of continuation names; \item[(2)]
we
develop
proof techniques,
in form of labelled bisimilarities, to reason about the behaviour of
well-typed processes. \end{itemize}
(Concerning (2), while the technical details are quite different, we follow the approach of proof techniques for receptive names in \cite{DBLP:journals/tcs/Sangiorgi99}, where the techniques are first defined on processes where only fresh names may be sent.)
\begin{definition}[Discreet processes]
A process $P$ is \emph{discreet} if any free continuation name
$p\in\fnames{P}$ may not appear in the object of an output, and,
in any sub-process $\inpi{x}{\many a,q}.Q$, the same holds for $q$ in $Q$.
The definition is extended to contexts, yielding
\emph{discreet contexts}. \end{definition}
If $E$ and $P$ are discreet, then so is $E[P]$.
We can transform all well-typed processes into discreet processes using the law in Lemma~\ref{l:gl} below. The law transforms the output of a \emph{global} continuation name $p$ into the output of a \emph{local} name $q$. In general, all outputs of continuation names in a process $P$
are local, as a global output
corresponds to $P$
delegating a stack-like obligation to
another process. In other words, in general the transformation of a non-discreet process into a discreet one will modify only a few outputs of the initial process.
The law in Lemma~\ref{l:gl}
is valid for barbed congruence, not just barbed equivalence, and may therefore be applied to any component of a given process.
\begin{lemma} \label{l:gl}
$\outpi{x}{\many{a},p} \bcong{\Tag{p}{\OO}} \respi{q}(\outpi{x}{\many{a},q} | \inpi{q}{\many{b}}.\outpi{p}{\many{b}})$.
\end{lemma}
Thus, \iflong TODO FIX In the light of Lemma~\ref{l:gl}, \fi in the definition of barbed equivalence (and congruence) it is sufficient to consider discreet contexts.
A discreet process may only export private continuation names. Dually,
the process may only receive fresh continuation names from a discreet context.
We call \emph{discreet} the transitions that satisfy this property.
\begin{definition}[Discreet transitions] A typed transition
$\typpr{\sigma}{P\strans{\mu}P'}$ is \emph{discreet} if
any continuation name in the object of $\mu$ is not free in $\sigma$ (and hence
also in $P$).
\end{definition}
\begin{lemma} If $P$ is discreet, and $\typpr{\sigma}{P\strans{\mu}P'}$ is discreet,
then $P'$ is discreet.
If, moreover,
$P$ is clean, then so is $P'$. \end{lemma}
\subsection{The Well-bracketing property on traces} Following game semantics \cite{DBLP:journals/iandc/HylandO00}, we formalise well-bracketing, that is, the stack-like behaviour of continuation names for well-typed processes, using traces of actions.
In this section, all processes are discreet and clean. A trace for such a process is obtained from a sequence of discreet transitions emanating from the process, with the expected freshness conditions to avoid ambiguity among names.
\begin{definition}[Trace]\label{d:trace} A sequence of actions $\mu_1, \ldots, \mu_n$ is a \emph{trace for a (discreet and clean) process $P_0$ and a stack $\sigma_0$} if there are
$ \sigma_1,\dots,\sigma_n, P_1,\dots,P_n$ such that for all $0\leq j < n$ we have
$\rconf{\sigma_j}{P_j}\strans{\mu_{j+1}}\rconf{\sigma_{j+1}}{P_{j+1}}$, where the transition is discreet, and moreover all continuation names appearing as object
in $\mu_{j+1}$ are fresh (i.e., the names may not appear in any
$\mu_i$ for $i\leq j$). \end{definition} The notion of discreet transition already imposes that continuation names in object position do not appear free in the process. The final condition in Definition~\ref{d:trace} on continuation names ensures us that for actions like $\respi{p}\outpi{x}{\many{a},p}$, name $p$ is fresh, and that after an action $\outpi{p}{\many{a}}$ (thus the only allowed interaction at $p$ has been played), name $p$ cannot be reintroduced, e.g., in an action $\earlyinpi{x}{\many a,p}$.
We simply say that $\mu_1, \ldots, \mu_n$ is a trace, or is a trace for $P$, when the stack or the process are clear from the context. \iflong \ds{not good $\mu$ for traces, because used usually for
actions. Anyhow, the lemma has to be reformulated, as also it is not clear what $\sigma_0 , \sigma_n$ are (wrt $\mu_i$). Introduce the lemma first and explain what it wants to say (eg the stack-like behaviour); maybe possible to simplify it talking about "suffix" of a stack"? } \begin{lemma}
For all traces $\mu_i$, if $\sigma_0 = \sigma_n$, then for some $\sigma_i'$
(resp. $\xi_i'$), $\sigma_i = \sigma_i', \sigma_0$ (resp. $\xi_i',\sigma_0$)
for all $i$. \end{lemma} \fi
The well-bracketing property is best described with the notion of questions and answers. \begin{definition} For a trace $\mu_1, \ldots, \mu_n$, we set $\mu_i \curvearrowright \mu_j$ if $i < j$ and:
\begin{enumerate}
\item either $\mu_i = \respi{\many{c},p}\outpi{a}{\many{b},p}$ and $\mu_j =
\earlyinpi{p}{\many{a'}}$,
\item or $\mu_i = \earlyinpi{a}{\many{b},p}$ and $\mu_j = \respi{\many{c}}
\outpi{p}{\many{a'}}$.
\end{enumerate}
Actions $\mu_i$ (with a continuation name in object position) are called \emph{questions}, while
actions $\mu_j$ (with a continuation name in subject position) are called \emph{answers}. \end{definition} A discreet transition is either an internal transition, or a question, or an answer. A question mentioning a continuation name $p$
is matched by an answer at \iflong the same name\fi $p$. When
questions and answers are seen as delimiters (`$[_p$',`$]_p$', different for each continuation name), a well-bracketed trace is a substring of a Dyck word. \iflong It can be expressed as follows. \fi
\begin{remark}
For a discreet transition $\rconf{\sigma}{P}\strans{\mu}\rconf{\sigma'}{P'}$, the value
$\sz{\sigma'}-\sz{\sigma}$ is 1 for a question, 0 for an internal action, and
$-1$ for an answer. \end{remark}
\begin{lemma}[Uniqueness]
Given a trace $\mu_1,\dots,\mu_n$,
if $\mu_i \curvearrowright \mu_j$ and $\mu_{i'} \curvearrowright \mu_{j'}$, then we have ($i=i'$ iff $j=j'$). \end{lemma}
\begin{definition}[Well-bracketing]\label{d:wb}
A trace $\mu_1,\dots,\mu_n$ is
\emph{well-bracketed} if for all $i < j$, if $\mu_i$ is a question and $\mu_j$
is an answer with
$\mu_i\not\curvearrowright \mu_k$
and
$\mu_k \not\curvearrowright \mu_j$
for all $i < k < j$, then $\mu_i \curvearrowright \mu_j$. \end{definition}
To prove that all traces are well-bracketed, \iflong(Proposition~\ref{p:wb}), \fi we need the following property relating questions and answers to stacks.
\begin{lemma} Let $\mu_1, \ldots, \mu_n$ be a trace, and $ \sigma_0, \ldots, \sigma_n$ be the corresponding stacks, as in Definition~\ref{d:trace}. Suppose $\sigma_0 = \sigma_n$, and for all $i$, $\sz{\sigma_i} > \sz{\sigma_0}$. Then \iflong $\mu_1$ is a question, $\mu_n$ is an answer, and \fi $\mu_1 \curvearrowright \mu_n$. \end{lemma}
\begin{prop}\label{p:wb}
Any trace (as by Definition~\ref{d:trace}) is well-bracketed. \end{prop}
\subsection{Bisimulation and Full Abstraction}\label{s:wb:bis}
As in Section~\ref{s:seq},
a \emph{wb-typed relation on processes} is a set of triplets $(\sigma,P,Q)$ with $\typr{\sigma}{P,Q}$. \begin{definition}[WB-Bisimulation]
A wb-typed relation $\ensuremath{\mathcal{R}}$ on discreet processes is a \emph{wb-bisimulation} if
whenever $(\sigma,P,Q)\in \ensuremath{\mathcal{R}}$ and $\rconf{\sigma}{P}\strans{\mu}\rconf{\sigma'}{P'}$
is discreet, then one of the three following clauses holds:
\begin{enumerate}
\item there is $Q'$ with $Q\wtrans{\hat\mu}Q'$
and $(\sigma', P', Q')\in\ensuremath{\mathcal{R}}$
\item $\mu = \earlyinpi{x}{\many a,p}$ and for some fresh $q$,
there is $Q'$ with
$Q | \respi{q}(\outpi{x}{\many a,q} |
\inpi{q}{\many b}.\outpi{p}{\many b}) \wtrans{} Q'$
and $(\sigma',P',Q')\in\ensuremath{\mathcal{R}}$
\item $\mu = \earlyinpi{u}{\many{a}}$ and there is $Q'$
with $Q | \outpi{u}{\many{a}} \wtrans{} Q'$
and $(\sigma',P',Q')\in\ensuremath{\mathcal{R}}$,
\end{enumerate}
and symmetrically for the transitions from $Q$.
Processes $P$ and $Q$ are \emph{wb-bisimilar at $\sigma$}, noted $P \ensuremath{\approx_{\rm{wb}}}^\sigma Q$, if
$(\sigma,P,Q)\in\ensuremath{\mathcal{R}}$ for some wb-bisimulation $\ensuremath{\mathcal{R}}$. \end{definition}
Compared to Definition~\ref{d:sb}, the clause for input actions is here split into two clauses. In clause (2), we apply Lemma~\ref{l:gl} to obtain a discreet process.
WB-bisimulation is sound with respect to barbed equivalence for all discreet processes. The main result concerns preservation by parallel composition: \begin{lemma}[Parallel composition]
If $P \ensuremath{\approx_{\rm{wb}}}^\sigma Q$, then for any discreet process $R$ and stacks
$\sigma',\sigma''$ such that $\typr{\sigma'}{R}$ and
$\inter{\sigma''}{\sigma}{\sigma'}$, we have $P|R \ensuremath{\approx_{\rm{wb}}}^{\sigma''} Q|R$. \end{lemma}
Note that even if $P,R$ are clean, $P|R$ needs not be so.
\begin{theorem}[Soundness]
$\ensuremath{\approx_{\rm{wb}}}^\sigma \protect{\subseteq} \beq{\sigma}$. \end{theorem}
To prove soundness, we show that $\ensuremath{\approx_{\rm{wb}}}^\sigma$ is preserved by all discreet static contexts. By Lemma~\ref{l:gl}, we can then replace any non-discreet context with a discreet one. \iflong
without changing its behaviour. \fi
We further refine the coinductive technique given by $\ensuremath{\approx_{\rm{wb}}}^\sigma$ by introducing some up-to techniques, which make it possible to work with smaller relations.
We write $P \strans{}_{\textrm{d}} P'$ when the reduction is \emph{deterministic},
meaning that whenever $P\strans{\mu}P''$, then $\mu = \tau$ and $P' \equiv P''$. Similarly, we write $P\wtrans{}_{\mathrm{d}} P'$ if all reduction steps are deterministic.
Moreover, for a relation $\ensuremath{\mathcal{R}}$, we write $(\sigma,P,Q)\in\ensuremath{{\wredd\RR^{\rm{C}}}}$ when there exists a stack $\sigma'$, a
$\conTT{\sigma}{\sigma'}$ context $E$, and processes $P',Q'$ such that $Q \equiv E[Q']$,
$P \wtrans{}_{\mathrm{d}} E[P']$ and $(\sigma', P', Q')\in\ensuremath{\mathcal{R}}$.
\begin{definition}[Up-to static contexts and up-to deterministic reductions] \label{d:upto}
A wb-typed relation $\ensuremath{\mathcal{R}}$ on discreet processes is a \emph{wb-bisimulation up-to
static contexts
and up-to deterministic reductions} if whenever $(\sigma,P,Q)\in \ensuremath{\mathcal{R}}$, for any discreet transition
$\rconf{\sigma}{P}\strans{\mu}\rconf{\sigma'}{P'}$, one of the
following clauses holds:
\begin{enumerate}
\item there is $Q'$ with $Q\wtrans{\hat\mu}Q'$ and $(\sigma',P',Q')\in\ensuremath{{\wredd\RR^{\rm{C}}}}$,
\item $\mu = \earlyinpi{x}{\many{a},p}$ and for some fresh $q$, there
is $Q'$ with
$Q | \respi{q}(\outpi{x}{\many{a},q} | q.\out{p}) \wtrans{} Q'$
and $(\sigma',P',Q')\in\ensuremath{{\wredd\RR^{\rm{C}}}}$,
\item $\mu = \earlyinpi{u}{\many{a}}$ and, there
is $Q'$ with $Q | \outpi{u}{\many{a}} \wtrans{} Q'$
and $(\sigma',P',Q')\in\ensuremath{{\wredd\RR^{\rm{C}}}}$,
\end{enumerate} and symmetrically for the transitions from $Q$. \end{definition}
\iflong \engue{Davide est ok pour cette remarque dans une version longue:
Note: on ne peut pas utiliser d'up-to deterministic $\tau$ sur des communications de noms "illégalement" libres (i.e $\out{p} | p.Q \strans{\tau} Q$) vu que ça change le typage.} \fi
\begin{lemma}
If $\ensuremath{\mathcal{R}}$ is a wb-bisimulation up-to static contexts and up-to
deterministic reductions,
then $(\sigma,P,Q)\in\ensuremath{\mathcal{R}}$ implies $P\ensuremath{\approx_{\rm{wb}}}^\sigma Q$. \end{lemma}
\subsubsection{Completeness}\label{s:compl:wb}
As in Section~\ref{s:seq}, we prove completeness for processes that
only use output-controlled names.
\begin{theorem}[Completeness]\label{t:completeness:wb}
For all image-finite, discreet and clean processes $P,Q$ that only use
output-controlled names, and for all $\sigma$, if $P \beq{\sigma} Q$ then $P \ensuremath{\approx_{\rm{wb}}}^\sigma Q$. \end{theorem} As for Theorem~\ref{t:completeness}, the crux of the proof is defining the discriminating static contexts. The additional difficulty is related to receptiveness of continuation names: we cannot use $z.R$ or $G_R + T$, as in Section~\ref{s:seq}, when the tester process, $R$ or $G_R$, contains an input at a free continuation name.
Suppose $\typr{\sigma}{P}$, for $P$
discreet and clean. We decompose $\sigma$ as $\xi, \Tag{p_1}{\OO}, \Tag{q_1}{\II}, \dots, \Tag{p_{n-1}}{\OO}, \Tag{q_{n-1}}{\II}, \Tag{p_n}{\OO}$ for $\xi=\emptyset$ or $\xi=\Tag{q}\II$, and
then define, for fresh
$q_n$ and $\many{x_i}$,
$$E_{\sigma}^{\many{x_i}} \,\defi\, \respi{\many{p_i},\many{q_i}}(\contexthole | \prod_{i \leq n} \inpi{p_i}{\many{y}}.\outpi{x_i}{\many{y},q_i}).$$
\vskip -.2cm \noindent We have $\typr{\xi, \Tag{q_n}{\OO}}{E_{\sigma}^{\many{x_i}}[P]}$.
Intuitively, $E_{\sigma}^{\many{x_i}}$ forwards information from the $p_i$'s (which are in $\sigma$) to the $x_i$'s. Accordingly, the tester process can use names in \many{x_i} (rather than
in \many{p_i}), and can use them in input.
Let $\ofnames{- }$ denote the set of free output-controlled names. We distinguish two cases.
If $\xi = \emptyset$, then $P$ is active. To follow the reasoning in the proof of Theorem~\ref{t:completeness}, we work with $E$ of the form
$$\respi{\many{x}}(E_{\sigma}^{\many{x_i}} | R + \prod_{y\in S} \inpi{y}{\many{y'},p}.\outpi{z}{p}),$$ for some set \iflong TOFIX $S$ of names containing $\ofnames P\cup\ofnames Q$.\fi $S\supseteq\ofnames P\cup\ofnames Q \cup \many{x_i}$ \iflong\engue{alternatively
$\ofnames{E^{\many{x_i}}_\sigma[P],E^{\many{x_i}}_\sigma[Q]}
\subseteq S$} \fi and fresh $z$.
If $\xi = \Tag{q}{\II}$, then $P$ is inactive.
We reason with $E$ of the form
$\respi{\many{x},q}(E_{\sigma}^{\many{x_i}} | \outpi{z}{q} | \inpi{z}{q'}.R)$. By typing, only the continuation name $q'$ received at $z$ may appear free in $R$. Such $q'$ will be instantiated with $q$ and then $R$ will use it to test the input at $q$ from the tested processes.
(A restriction on $q$ is needed, as
the overall process has to be clean.)
In both cases, the resulting $E$ is a $\conTT{\Tag{q_n}{\OO}}{\sigma}$ static context where $q_n$ is a fresh continuation name. More details on the proof can be found in~\cite{HPS:lics21:long}.
\subsubsection{An Example}\label{s:awkward} \label{ss:exWB}
We explain how the techniques we have introduced allow us to reason about a well-known example, the \textit{well-bracketed state change} (sometimes called `awkward', or `very awkward', example)~\cite{DBLP:conf/popl/AhmedDR09,DBLP:journals/jfp/DreyerNB12,DBLP:journals/pacmpl/Jaber20}. It is usually presented in ML thus:
\vskip .1cm $ $ \hskip -.8cm \begin{tabular}{rcl} $M_1$ & $\defi$ & \smalltexttt{let $\ell$ = ref 0 in fun y ->}\\
&& \smalltexttt{ ($\ell$ := 0; y() ; $\ell$ := 1; y() ; !$\ell$) } \\
$M_2 $ & $ \defi$ & \smalltexttt{fun y -> (y() ; y() ; 1)} \end{tabular}
\vskip .2cm \noindent Function $M_2$ makes two calls to an external function \smalltexttt{y} and returns $1$. The other term, $M_1$, between the two calls, modifies a local reference \smalltexttt{$\ell$}, which is then used to return the final result. Intuitively, equivalence between the two functions holds because: (i) the reference \smalltexttt{$\ell$} in $M_1$ represents a local state, not accessible from an external function; (ii)
computation respects well-bracketing (e.g., the language does not have control operators like call/cc). \iflong
(with such an operator, it
would be possible to reinstall the value \texttt{0} at \texttt{$\ell$}, and
obtain $0$ as final result). \fi
Below are the translations of $M_1$ and $M_2$, following a standard encoding of functions and references in \Api, and using the notations for references from Section~\ref{s:ex:ref}: \vskip -.4cm \begin{align*}
\psapp{\enco{M_1}}{p'} &\defi \respi{x,\ell}(\outpi{\ell}{0} | Q)
\quad\mbox{with}\\
Q &\defi \outpi{p'}{x} | !\inpi{x}{y,p}.\rwriteL{0}.\respi{q}(\outpi{y}{q} |\\
&\qquad\qquad
\; q.\rwriteL{1}.\respi{r}(\outpi{y}{r} | r.\rreadL{n}.\outpi{p}{n}))\\
\psapp{\enco{M_2}}{p'} &\defi \respi{x}(\outpi{p'}{x} | !\inpi{x}{y,p}.\respi{q}(\outpi{y}{q} |\\
&\qquad\qquad\qquad \qquad\qquad q.\respi{r}(\outpi{y}{r} | r.\outpi{p}{1}))) \end{align*}
\vskip -.2cm \noindent $\psapp{\enco{M_1}}{p'}$ has a unique transition, $\psapp{\enco{M_1}}{p'}\strans{\respi{x}\outpi{p'}{x}} P_1$. Similarly, let $P_2$ be the unique derivative from $\psapp{\enco{M_2}}{p'}$.
The equivalence between $\psapp{\enco{M_1}}{p'}$ and $\psapp{\enco{M_2}}{p'}$ follows immediately from
$P_1\ensuremath{\approx_{\rm{wb}}}^{\emptyset}P_2$.
To prove the latter,
we exhibit a relation $\ensuremath{\mathcal{R}}$ containing the triple $(\emptyset, P_1, P_2)$ and show that $\ensuremath{\mathcal{R}}$ is a wb-bisimulation up-to deterministic reductions and static context. \iflong \ds{Intuitively, $\ensuremath{\mathcal{R}}$ contains the triples such that ...\\ see if there is a simple explanation above, following how Seguei does. We can also do without} \fi To see the importance of well-bracketing, $\ensuremath{\mathcal{R}}$ contains the triple
$$\begin{array}{l}
\big(\,\Tag{q_2}{\II}, \Tag{p_2}{\OO}, \Tag{r_1}{\II},
\Tag{p_1}{\OO},\\
~\respi{\ell}(\outpi{\ell}{0} | Q | q_2.\rwriteL{1}.\respi{r_2}(\outpi{y}{r_2}\\
\qquad\qquad
| r_2.\rreadL{n}.\outpi{p_2}{n})) | r_1.\rreadL{n}.\outpi{p_1}{n}),\\[.1em]
~P_2 | q_2.\respi{r_2}(\outpi{y}{r_2} | r_2.\outpi{p_2}{1}) | r_1.\outpi{p_1}{1}
\,\big) \end{array}$$
\noindent Without the well-bracketing constraint, the first process in the triple could perform an input at $r_1$, an internal transition, and finally an output {\outpi{p_1}0}. The second process cannot emit $0$, which would allow us to distinguish $P_1$ and $P_2$. With well-bracketing, since $r_1$ is not on top of the stack in the triple, the initial transition on $r_1$ is ruled out.
The details of the definition of $\ensuremath{\mathcal{R}}$ can be found in~\cite{HPS:lics21:long}.
In the same Appendix, we also
discuss a simplified example, which exposes the main difficulties. The primary simplification consists in using linear functions\iflong, so that the \Api terms do not need replication and can be finite\fi. Some twisting in the ML terms is necessary, as $M_1$ and $M_2$ become equivalent~--- even dropping well-bracketing~--- if they can be used at most once.
\vskip -.3cm \section{Related work and conclusions} \label{s:ccl}
\label{ss:rela}
\vskip -.2cm
Sequentiality is a form of linearity, hence \iflong the rules for \fi our type system has similarity with, and borrow ideas from, \iflong type \fi
systems with linear types, in languages for concurrency or functional languages, including types for managing locks as in~\cite{DBLP:journals/iandc/Kobayashi02}.
The type system in~\cite{DBLP:conf/popl/LeviS00} ensures one that terms of the Ambient calculus are single-threaded, a notion similar
to the sequentiality for \Api examined in this paper. The type system in~\cite{DBLP:conf/tlca/BergerHY01} has been designed so to make
the encoding of PCF into the $\pi$-calculus fully abstract. The system therefore goes beyond sequentiality as described in our paper. For instance, the system presents a form of duality on types and
ensures that computations are stateless, hence also deterministic. Indeed, the only behaviours inhabited by the types are those in the image of the PCF terms. Types ensure the uniqueness of the computation thread, and such a thread is carried by outputs (the thread cannot be carried by input processes, as in our system). The system \cite{DBLP:conf/tlca/BergerHY01} has been further refined in~\cite{DBLP:journals/iandc/YoshidaBH04}, adding causality information and acyclicity constraints, so to ensure strong normalisation of well-typed processes. The issue of finding labelled bisimilarity characterisations of barbed equivalence or reduction-closed barbed equivalence is extensively discussed in \cite{SanWal}; see also \cite{DBLP:journals/mscs/HennessyR04} for an example involving types. \iflong SEE IF WE KEEP THE FOLLOWING SENTENCE We are not aware of bisimulation-based proof techniques for more advanced type systems such as those in \cite{DBLP:conf/popl/LeviS00,DBLP:conf/tlca/BergerHY01} mentioned above. \fi
Type systems for linearity and receptiveness in the $\pi$-calculus have been introduced in \cite{DBLP:journals/toplas/KobayashiPT99,DBLP:journals/iandc/IgarashiK00,DBLP:journals/tcs/Sangiorgi99}.
The way we formulate well-bracketing (Definition~\ref{d:wb}) is inspired by
`well-bracketed strategies' in game semantics~\cite{DBLP:journals/iandc/HylandO00,DBLP:conf/lics/Laird97},
used in functional programming languages and extensions thereof (they have in turn inspired type systems for $\pi$-calculi with stack-like information and input/output alternation, e.g., \cite{DBLP:conf/tlca/BergerHY01,DBLP:journals/entcs/Honda02}).
The notion of \emph{well-bracketed control flow} is studied in the field of secure compilation, for a wider class of languages. In works like~\cite{DBLP:journals/pacmpl/SkorstengaardDB19,DBLP:conf/csfw/PatrignaniDP16}, the technique of fully abstract compilation guarantees control flow correctness (and, in particular, well-bracketing) against low-level attacks.
Several methods have been proposed to establish contextual equivalence of sequential programs that include higher-order and stateful computation, including the above-mentioned game semantics, (step-indexed Kripke) logical relations \cite{DBLP:conf/popl/AhmedDR09,DBLP:journals/jfp/DreyerNB12}, dedicated forms of bisimulations designed on top of an operational semantics of the languages~\cite{DBLP:conf/popl/KoutavasW06,DBLP:conf/lics/SangiorgiKS07,DBLP:journals/entcs/KoutavasLS11,DBLP:conf/concur/MadiotPS14,DBLP:conf/fscd/BiernackiLP20}. Works like~\cite{DBLP:journals/pacmpl/Jaber20} or algorithmic game semantics~\cite{DBLP:journals/fmsd/MurawskiT18}, aim at automatically establishing contextual equivalences, by relying on model-checking techniques.
The main goal of this paper was to tailor some of the most prominent proof techniques in the $\pi$-calculus~--- those based on labelled bisimilarity~--- to the sequentiality and well-bracketing disciplines. This is instrumental to the use of the $\pi$-calculus as a model of programming languages, as sequentiality and well-bracketing are often found in programming languages or subsets of them. We have shown the usefulness of our techniques on a number of examples, that have mainly to do with the representation of functions and store~--- none of the equalities in the examples is valid in the ordinary bisimilarity of the calculus.
In Section~\ref{ss:exSeq} we have combined our proof technique for sequentiality with techniques concerning the representation of references in $\pi$-calculus from \cite{DBLP:conf/concur/HirschkoffPS20}. The resulting technique allows us in some cases to reason about programs with store without an explicit representation of the store (as usually required in the techniques in the literature, recalled above). This avoids universal quantifications on the possible values contained in the store, thus reducing the size of the relation to consider, sometimes making them finite. Further possibilities of reducing the size of relations may be possible by defining `up-to techniques' for our bisimilarities,
as exemplified by the up-to technique considered in Definition~\ref{d:upto} and applied in Section~\ref{ss:exWB}.
Our treatment of sequentiality raises a few technical questions that deserve
further investigation. We would like to see whether our proof of completeness (Theorem~\ref{t:completeness}) could be extended to handle input-controlled names. Similarly, we do not know whether the result still holds if internal choice is disallowed in inactive processes. The usual encoding of an internal choice $\tau.P + \tau.Q$ in terms of parallel composition as
$\respi
c\,(\out c|c{}.P|c{}.Q)$, for some fresh $c$, is not applicable because
the latter process is active (for instance, the encoding is not valid within a context
testing active processes). Indeed, if the result still holds, the current \iflong
completeness \fi proof \iflong
would probably \else might \fi
require some significant modifications.
For similar reasons, it is unclear if and how our completeness proof could be tuned to handle reduction-closed variants of barbed equivalence~\cite{HoYo95,SanWal}.
In the \emph{asynchronous} $\pi$-calculus considered in this paper,
an interaction involves only one prefixed process (the input). Therefore, in the type
systems, this process always
acquires the control on the thread after the interaction. In a \emph{synchronous} setting, in contrast, an interaction involves also an output prefix. Hence the type systems could be richer, specifying, for each name, where the control on the thread goes after an interaction at that name. The representation of references in Section~\ref{ss:exSeq}, however, might have to be revisited as it relies on the asynchronous model.
We have studied proof techniques for sequentiality and well-bracketing in the $\pi$-calculus based on labelled bisimilarities. We would like to examine also the impact of the disciplines on algebraic theory and modal logics.
\section*{Acknowledgments} Prebet acknowledges support from the Universit{\'e} Franco-Italienne, programme Vinci 2020.
Sangiorgi acknowledges support from the MIUR-PRIN project `Analysis of Program Analyses' (ASPRA, ID: \texttt{201784YSZ5\_004}).
\end{document}
\appendices
\section{The Asynchronous $\pi$-calculus: Operational Semantics}\label{a:def:api}
We give the definition of structural congruence in Figure~\ref{f:str:cong}.
\begin{figure}
\caption{Structural congruence in \Api}
\label{f:str:cong}
\end{figure}
The (early) Labelled Transition System for \Api{} is presented in Figure~\ref{fig:sos:pil:procs}, actions being defined by the following grammar:
$$\mu ::= \tau \ensuremath{~\big|~} \earlyinpi{a}{\many{b}} \ensuremath{~\big|~} \respi{\many{c}}\outpi{a}{\many{b}}$$
We have $\fnames{\earlyinpi{a}{\many{b}}}=\set a\cup\many b$, and $\fnames{\respi{\many{c}}\outpi{a}{\many{b}}}=(\set a\cup\many b)\setminus\many c$.
\begin{figure*}
\caption{Labelled Transition Semantics for \Api}
\label{fig:sos:pil:procs}
\end{figure*}
\section{Completeness for sequential bisimilarity}\label{a:compl:seq}
\begin{definition}[Approximants of sequential bisimilarity] We define a sequence $(\wbissn{\ensuremath{\eta}}{n})_{n\geq 0}$ of
typed relations:
\begin{enumerate}
\item $P\wbissn{\ensuremath{\eta}}{0} Q$ if $\typs{\ensuremath{\eta}}{P,Q}$.
\item For $1 \leq i$, relation $\wbissn{\ensuremath{\eta}}{i}$ is defined by:
$P\wbissn{\ensuremath{\eta}}{i} Q$ if whenever $\rconf{\ensuremath{\eta}}{P}\strans{\mu}\rconf{\ensuremath{\eta}'}{P'}$,
one of these two clauses holds:
\begin{description}
\item[--] there is $Q'$ such that $Q\wtrans{\hat\mu}Q'$ and $P' \wbissn{\ensuremath{\eta}'}{i-1} Q'$;
\item[--] $\mu = \earlyinpi{a}{b}$ and there is $Q'$ such that $Q|\outpi{a}{b}\wtrans{}Q'$ and $P' \wbissn{\ensuremath{\eta}'}{i-1} Q'$,
\end{description}
and symmetrically for the transitions of $Q$.
\item Then $P \wbissn{\ensuremath{\eta}}{\omega} Q$ if $P \wbissn{\ensuremath{\eta}}{i} Q$ for all $i$.
\end{enumerate} \end{definition} Notice that $\wbissn{\ensuremath{\eta}}{0} \supseteq \wbissn{\ensuremath{\eta}}{1} \supseteq \dots \supseteq \wbissn{\ensuremath{\eta}}{n} \supseteq \dots \supseteq \wbissn{\ensuremath{\eta}}{\omega} \supseteq \wbiss{\ensuremath{\eta}}$.
\begin{lemma}\label{l:img:eq}
If $P,Q$ are image-finite, $P\wbissn{\ensuremath{\eta}}{\omega} Q$ iff $P \wbiss{\ensuremath{\eta}} Q$. \end{lemma}
We say that a process $P$ is \emph{singular} if it never releases the thread; that is, the set of singular processes is the largest set ${{\cal T}}$ of processes such that for all $P \in {\cal T}$ we have $\typs{1}{P}$ and whenever $\sttrans{1}{P}{\mu}{P'}$ then
$P' \in {\cal T}$.
Singular processes are sequentially equivalent to $\ensuremath{\boldsymbol{0}}_1$ (an active process without transitions).
When only output-controlled names are used, we can give the following characterisation of singular processes: \begin{lemma}\label{l:singular}
Suppose $P$ is active, and uses only output-controlled names.
Then $P$ is singular iff there is no $z$ such that $P\Downarrow_{\out{z}}$.
\end{lemma} This is not true for processes with input-controlled names: for instance, $u.\out{x}$ is not singular but does not have a visible barb (it needs to interact on $u$ first).
The following property is the main technical ingredient to derive completeness.
\begin{prop}\label{p:compl:seq}
For all $n\geq 0$, if $\typs{\ensuremath{\eta}}{P,Q}$, and $P \not\wbissn{\ensuremath{\eta}}{n} Q$ and both $P$ and
$Q$ only use
output-controlled names and are image-finite, then there exists $R$
such that for any fresh name $z$, any $\many{x},S$ with
$\many{x}\subseteq\fnames{P}\cup\fnames{Q}\subseteq S$,
we define
$$R_0 \defi\left\{\begin{array}{l l}
\out{z} | z.R & \mbox{ when }\typs{0}{P,Q}\\
R + \sum_{y\in S} \inpi{y}{\many{y}'}.\out{z} & \mbox{ when }\typs{1}{P,Q}\end{array}\right.$$
and we have:
\begin{enumerate}
\item either $\respi{\many x}(P' | R_0) \not\bbis{1}
\respi{\many x}(Q | R_0)$ for all $P'$ such that $P \wtrans{} P'$
\item or $\respi{\many x}(P | R_0) \not\bbis{1}
\respi{\many x}(Q' | R_0)$ for all $Q'$ such that $Q \wtrans{} Q'$.
\end{enumerate} \end{prop} In the second case for the definition of $R_0$, we define $R$ as a guarded process, in order for the sum to make sense.
\begin{proof}
We reason by induction on $n$. For $n = 0$, there is nothing to prove.
For $n > 0$, suppose that $P \not\wbissn{\ensuremath{\eta}}{n} Q$. Thus, there exists $\mu$
such that $\rconf{\ensuremath{\eta}}{P}\strans{\mu}\rconf{\ensuremath{\eta}'}{P'}$ and
for all $Q'$ with $Q \wtrans{\hat\mu} Q'$
(or when $\mu = \earlyinpi{x}{\many{y}}$, all $Q'$ with
$Q|\outpi{x}{\many{y}} \wtrans{} Q'$),
we have $P' \not\wbissn{\ensuremath{\eta}'}{n-1} Q'$.
We note $\set{Q_i}$ for $i\in I$ the set of all such
$Q'$. This set is finite by
hypothesis. We also write $S'_I = \bigcup_{i\in I}\fnames{Q_i} \cup
\fnames{P'}$, which is also a finite set.
We show that clause 1) holds. For that, we distinguish two
cases, according to whether $P'$ is singular or not.
\noindent\textbf{First case: $P'$ is singular}
Since $P'$ is singular, $\ensuremath{\eta}' = 1$. For all $i$, as
$P' \not\wbiss{1} Q_i$, we have $Q_i\wbarb{\out{z_i}}$ for
some $z_i$.
By definition, $S'_I$ contains all such $z_i$'s.
By Theorem~\ref{l:subred:seq1} (Subject Reduction), there are two possible cases:
\begin{itemize}
\item when $\mu = \tau$, we can take $R \defi \ensuremath{\boldsymbol{0}}$.
We then have to show that for any $z,\many{x},S, Q'$ with $Q \wtrans{} Q'$,
we have $A \not\bbis{1} B$ where
\begin{mathpar}
A\defi \respi{\many x}(P | \ensuremath{\boldsymbol{0}} + \sum_{y\in S} \inpi{y}{\many{y}'}.\out{z})
\and
B \defi \respi{\many x}(Q' | \ensuremath{\boldsymbol{0}} + \sum_{y\in S} \inpi{y}{\many{y}'}.\out{z})
\end{mathpar}
We reason by contradiction. We have, since $\mu=\tau$,
$A \strans{} \respi{\many x}(P' | \sum_{y\in S}
\inpi{y}{\many{y}'}.\out{z}) \defi A'$ which is also
singular.
Thus there is $B'$ such that $B \wtrans{} B'$ and \linebreak $A'
\bbis{1} B'$.
In such a situation, $B'$ can be of 2 forms:
\begin{itemize}
\item $\respi{\many x}(Q_i | \sum_{y\in S} \inpi{y}{\many{y}'}.\out{z})$ for some $i\in I$ (with $Q' \wtrans{} Q_i$)
\item $\respi{\many x}(Q'' | \out{z})$
\end{itemize}
The second process clearly is not barbed bisimilar to $A'$.
For the first one, notice that $\fnames{Q_i} \subseteq \fnames{Q}$ and
$\fnames{P'} \subseteq \fnames{P}$. Thus, $S'_I \subseteq S$. As $Q_i
\wbarb{\out{z_i}}$ and $z_i \in S'_I$;
this means that $B'\wbarb{\out{z}}$ and thus this process not barbed bisimilar to $A'$.
\item when $\mu = \earlyinpi{x'}{\many{y}'}$, given a fresh name $z'$, we define $R \defi \outpi{x'}{\many{y}'} | \sum_{y\in S'_I} \inpi{y}{\many{y}'}.\out{z'}$.
We then have to show that for any $z,\many{x},S, Q'$ with $Q \wtrans{} Q'$
$$A\defi \respi{\many x}(P | z.R | \out{z}) \not\bbis{1}
\respi{\many x}(Q' | z.R | \out{z}) \defi B$$
We reason by contradiction. \\
We have $A \strans{}\strans{} \respi{\many x}(P' |
\sum_{y\in S'_I} \inpi{y}{\many{y}'}.\out{z'}) \defi A'$
which is also singular.
Thus there should exist $B'$ such that $B \wtrans{} B'$ and $A' \bbis{1} B'$.
We observe that $B'$ can be of 3 forms:
\begin{itemize}
\item $\respi{\many x}(Q_i | \sum_{y\in S} \inpi{y}{\many{y}'}.\out{z'})$ for some $i\in I$ (with $Q'|\outpi{x'}{\many{y}'} \wtrans{} Q_i$)
\item $\respi{\many x}(Q'' | z.R | \out{z})$
\item $\respi{\many x}(Q'' | \out{z'})$
\end{itemize}
The latter two are clearly not barbed bisimilar to $A'$.
For the first case, $Q_i \wbarb{\out{z_i}}$ and $z_i \in S'_I$,
this means that $B'\wbarb{\out{z'}}$, and thus $B'$ is not barbed bisimilar to $A'$.
\end{itemize}
\noindent\textbf{Second case: $P'$ is not singular.}
We know in this case that
either $P'$ is inactive,
or $P'\wbarb{\out{z}}$ for some $z$ (by Lemma~\ref{l:singular}).
We note $R_i$ for the process that we obtain, by induction,
for each pair $(P',Q_i)$.
\begin{itemize}
\item when $\mu = \tau$ and $\typs{1}{P'}$, given fresh names $(z_i)_{i\in I}$, we pose $R \defi \sum_{i\in I} \tau.(R_i + \sum_{y\in S'_I}\inpi{y}{\many{y}'}.\out{z_i})$.
We reason by contradiction. \\
We have
$A \strans{} \respi{\many x}(P' | R + \sum_{y\in S} \inpi{y}{\many{y}'}.\out{z}) \defi A'$
Thus there should exist $B'$ such that $B \wtrans{} B'$ and $A' \bbis{1} B'$.
As $P'$ is not singular, we have by
Lemma~\ref{l:singular} $P'\wbarb{\out{z'}}$ for some
$z'\in S'_I$ ($P'$ is active because \typs{1}{P'}).
So $A' \wbarb{\out{z_i}}$ for all $i\in I$
Therefore $B'$ must be of the form
$\respi{\many x}(Q_i | R + \sum_{y\in S} \inpi{y}{\many{y}'}.\out{z})$ for some
$i\in I$ (with $Q' \wtrans{} Q_i$)
Now, we have two cases depending on the clause of the
proposition that holds, by induction, for $(P',Q_i)$:
\begin{itemize}
\item if clause 1) holds, then $B' \strans{} B'_i$ with $$B'_i \defi \respi{\many{x}}(Q_i | R_i + \sum_{y\in S'_I}\inpi{y}{\many{y}'}.\out{z_i})$$
This means there is $A'_i$ with $A' \wtrans{} B'_i$ and $A'_i \bbis{1} B'_i$.
As we do not have $B'_i\wbarb{z_j}$ for $j\neq i$, we must have that
\begin{itemize}
\item either $A'_i \equiv \respi{\many{x}}(P'' |
R_i + \sum_{y\in S'_I}\inpi{y}{\many{y}'}.\out{z_i})$ with $P' \wtrans{} P''$
\item or $A'_i \equiv \respi{\many{x}}(P'' |
\out{z_i})$ for some $P''$
\end{itemize}
However the second case is not possible as $B'_i \strans{} \not\wbarb{\out{z_i}}$.
Using structural congruence, we can suppose
that in $A'_i$ and $B'_i$, $\many{x}$
contains only names from $\fnames{P',Q_i}$.
We are in a situation where we can apply the induction hypothesis, and
this is in contradiction with $A'_i \bbis{1} B'_i$.
\item if clause 2) holds, then $A' \strans{} A'_i$ with
$$A'_i \defi \respi{\many{x}}(P' | R_i + \sum_{y\in S'_I}\inpi{y}{\many{y}'}.\out{z_i})$$
This means there is $B'_i$ with $B' \wtrans{} B'_i$ and $A'_i \bbis{1} B'_i$.
As we do not have $A'_i\wbarb{z_j}$ for $j\neq i$, we must have that
\begin{itemize}
\item either $B'_i \equiv \respi{\many{x}}(Q_i' |
R_i + \sum_{y\in S'_I}\inpi{y}{\many{y}'}.\out{z_i})$ with $Q_i \wtrans{} Q_i'$
\item or $B'_i \equiv \respi{\many{x}}(Q'' |
\out{z_i})$ for some $Q''$
\end{itemize}
However the second case is not possible as $A'_i \strans{} \not\wbarb{\out{z_i}}$.
Using structural congruence, we can suppose
that in $A'_i$ and $B'_i$, $\many{x}$
contains only names from $\fnames{P',Q_i}$.
We are in a situation where we can apply the induction hypothesis, and
this is in contradiction with $A'_i \bbis{1} B'_i$.
\end{itemize}
\item when $\mu = \earlyinpi{x'}{\many{y}'}$, given
fresh names $(z_i)_{i\in I}$, we set $R \defi
\outpi{x'}{\many{y}'} | \sum_{i\in I} \tau.(R_i +
\sum_{y\in S'_I}\inpi{y}{\many{y}'}.\out{z_i})$
and we can conclude.
\item when $\mu = \tau$ and $\typs{0}{P'}$, given
fresh names $(z_i)_{i\in I}$, we pose $R \defi
\sum_{i\in I} \tau.(\out{z_i} | z_i.R_i)$
and we can conclude.
\item when $\mu = \respi{\many{y}^2}\outpi{x'}{\many{y}'}$, given fresh names
$z',z'', (z_i)_{i\in I}$,
we note $\many{y}^2 = y^2_1, \dots, y^2_n$ and $\many{y}'=\many{y}^2, y^1_1,
\dots y^1_m$.
Then we set
\begin{align*}
R \defi&~ \inpi{x'}{\many{x}^2,\many{x}^1}.\big(\,\out{z'}
\\&
| \sum_{j\leq n}\sum_{y''\in\fnames{P,Q}} [x^2_j =
y'']z'.\out{z''}
\\[.1em]&
+ \sum_{i\in I}[x^1_1 = y^1_1]
\dots[x^1_m = y^1_m]z'.(\out{z_i} | z_i.R_i)\,\big)
\end{align*}
and we can conclude.
\end{itemize} \end{proof}
\begin{proof}[Proof of Theorem~\ref{t:completeness}] We suppose $P\beq{\ensuremath{\eta}}Q$ and $P,Q$ are image-finite.
We reason by contradiction:
If $P\not\wbiss{\ensuremath{\eta}} Q$, then by Lemma~\ref{l:img:eq}, $P\not\wbissn{\ensuremath{\eta}}{\omega} Q$.
Thus there exists $n$ such that $P\not\wbissn{\ensuremath{\eta}}{n} Q$.
By Proposition~\ref{p:compl:seq}, there exists some static context $E$ such that $E[P] \not\bbis{\ensuremath{\eta}} E[Q]$.
So $P\not\beq{\ensuremath{\eta}}Q$, which is absurd. \end{proof}
\section{Completeness for WB-bisimulation}\label{a:compl:wb}
The proof of completeness for $\ensuremath{\approx_{\rm{wb}}}^\sigma$ w.r.t.\ $\beq\sigma$ has the same overall structure as the proof for sequential bisimulation. Technically, however, because of the presence of continuation names, the details are different.
The following lemma is a direct consequence of the usage discipline of continuation names.
\begin{lemma}\label{l:lin}
If $\typr{\sigma}{P|Q}$ then
$$\respi{p}(\outpi{p}{\many{a}} | \inpi{p}{\many{b}}.P | Q) \wbiss{\sigma}
P\sub{\many{a}}{\many{b}} | Q$$ \end{lemma} By soundness, this equivalence also holds for barbed equivalence at $\sigma$.
Suppose $\typr{\sigma}{P}$. We recall the definition of $E^{\many{x_i}}_\sigma$, given in Section~\ref{s:compl:wb}:
With $\sigma=\xi, \Tag{p_1}{\OO}, \Tag{q_1}{\II}, \dots, \Tag{p_{n-1}}{\OO}, \Tag{q_{n-1}}{\II}, \Tag{p_n}{\OO}$ and $\xi=\emptyset$ or $\xi=\Tag{q}\II$, we define, for fresh $q_n$ and $\many{x_i}$,
$$E_{\sigma}^{\many{x_i}} \,\defi\, \respi{\many{p_i},\many{q_i}}(\contexthole | \prod_{i \leq n} \inpi{p_i}{\many{y}}.\outpi{x_i}{\many{y},q_i}).$$
When $\rconf{\sigma}{P}\strans{\respi{\many{x}}\outpi{p_1}{\many{y}}}\rconf{\sigma'}{P'}$, then with $\many{x_i} = x_1,\many{x_i}'$ $$\typpr{\Tag{q_n}{\OO}}{E^{\many{x_i}}_\sigma[P] \strans{\tau}
\respi{q_1,\many{x}}(E^{\many{x_i}'}_{\sigma'}[P'] | \outpi{x_1}{\many{y},q_1})}$$ By Lemma~\ref{l:lin}, this internal transition does not change the behaviour of the process. Thus, in the proof of Proposition~\ref{p:compl:wb}, we reason modulo this kind of reductions.
As in Appendix~\ref{a:compl:seq}, forbidding input-controlled names allows us to gain information on the behaviour of singular processes. \begin{lemma}[Wb-typability and singular processes]
Suppose a clean active process $P$ does not use input-controlled
names, and suppose $\typr{\Tag{p}{\OO},\sigma}{P}$. Then $P$ is not
singular iff $P\Downarrow{\out{p}}$ or $P\Downarrow_{\out{z}}$ for some $z$.
Additionally, if $P$ is singular, so is $E[P]$ for any $\rho$ and any
$\conTT{\rho}{\Tag{p}{\OO},\sigma}$ static context $E$. \end{lemma} Under the same hypotheses, this lemma implies that $P$ is not singular iff $E_\sigma^{\many{x_i}}[P]\wbarb{\out{z}}$ for some $z$.
The following proposition is the counterpart, for WB-bisimulation, of Proposition~\ref{p:compl:seq}. It refers to the stratification of WB-bisimulation, which is defined like the stratification of sequential bisimulation.
\begin{prop}\label{p:compl:wb}
For all $n\geq 0$, $\sigma = \xi, \sigma''$ and $\sz{\xi} \leq 1$, if $\typr{\sigma}{P,Q}$ and $P \not\ensuremath{\approx_{\rm{wb}}}^{\sigma,n} Q$, then for any fresh names
$\many{x_i}$ there exists $R$ such that for any fresh $z$ and any $\many{x},S$
with $\many{x}\subseteq \ofnames{P,Q}$ and $\ofnames{P,Q}\cup\many{x_i} \subseteq S$,
we have:
\begin{itemize}
\item When $\xi = \Tag{p}{\II}$ for some $p$:\\
We note $R_0 \defi \outpi{z}{p} | \inpi{z}{p}.R$. One of the
following holds:
\begin{enumerate}
\item either for all $P'$
such that $P \wtrans{} P'$
$$\respi{\many x, p}(E^{\many{x_i}}_{\sigma}[P'] |
R_0) \not\bbis{\Tag{q_n}{\OO}}
\respi{\many x, p}(E^{\many{x_i}}_{\sigma}[Q] |
R_0)$$
\item or for all $Q'$ such that $Q \wtrans{} Q'$
$$\respi{\many x, p}(E^{\many{x_i}}_{\sigma}[P] | R_0)
\not\bbis{\Tag{q_n}{\OO}} \respi{\many x, p}(E^{\many{x_i}}_{\sigma}[Q'] | R_0)$$
\end{enumerate}
\item When $\xi = \emptyset$, we note $R_0\defi R + \sum_{y\in S}
\inpi{y}{\many{y}',p}.\outpi{z}{p}$. One of the following holds:
\begin{enumerate}
\item either
$\respi{\many x}(E^{\many{x_i}}_{\sigma}[P'] | R_0) \not\bbis{\Tag{q_n}{\OO}}
\respi{\many x}(E^{\many{x_i}}_{\sigma}[Q] | R_0)$ for all $P'$ such that $P \wtrans{} P'$,
\item or
$\respi{\many x}(E^{\many{x_i}}_{\sigma}[P] | R_0) \not\bbis{\Tag{q_n}{\OO}}
\respi{\many x}(E^{\many{x_i}}_{\sigma}[Q'] | R_0)$ for all $Q'$ such that $Q \wtrans{} Q'$.
\end{enumerate}
\end{itemize} \end{prop}
The proof of this proposition is structured like the proof of Proposition~\ref{p:compl:seq}. We therefore do not provide all details, but instead highlight the technical points that are specific.
\begin{proof}
We reason by induction on $n$. For $n = 0$, there is nothing to prove.
For $n > 0$, suppose that $P \not\wbissn{\sigma}{n} Q$. Thus, there exists $\mu$
such that $\rconf{\sigma}{P}\strans{\mu}\rconf{\sigma'}{P'}$ and
for all $Q'$ with $Q \wtrans{\hat\mu} Q'$
(or when $\mu = \earlyinpi{x}{\many{y},p}$, all $Q'$ with
\linebreak $Q|\respi{q}(\outpi{x}{\many{y},q} | \inpi{q}{\many{y}'}.\outpi{p}{\many{y}'}) \wtrans{} Q'$ for any fresh $q$),
we have $P' \not\wbissn{\sigma'}{n-1} Q'$.
We note $\set{Q_i}$ for $i\in I$ the set of all such $Q'$. This set is finite by
hypothesis. We also write \linebreak $S'_I = \bigcup_{i\in I}\ofnames{Q_i} \cup \ofnames{P'} \cup\many{x_i}$, which is also a finite set.
We show that clause 1) holds.
We distinguish two cases, according to whether $P'$ is singular or not.
\noindent\textbf{First case: $P'$ is singular.}
Since $P'$ is singular, $\sigma' = \Tag{p_j}{\OO},\sigma_0$
for some $p_j, \sigma_0$. For all $i$, as
$P' \not\wbiss{1} Q_i$, we have $E^{\many{x_i}}_{\sigma'}[Q_i]\wbarb{\out{z_i}}$
for some $z_i \in S'_I$.
By Theorem~\ref{t:SR:wb}, there are two possible cases:
\begin{itemize}
\item when $\mu = \tau$ and $\sigma' = \sigma = \Tag{p_1}{\OO},\sigma''$ for
some $p_1, \sigma''$. We set $R \defi \ensuremath{\boldsymbol{0}}$.
We then have to show that for any $z,\many{x},S, Q'$ with $Q \wtrans{} Q'$,
we have $A \not\bbis{\Tag{q_n}{\OO}} B$ where
\begin{mathpar}
A\defi \respi{\many x}(E^{\many{x_i}}_\sigma[P] | \ensuremath{\boldsymbol{0}} + \sum_{y\in S} \inpi{y}{\many{y}',q}.\outpi{z}{q})
\and
B \defi \respi{\many x}(E^{\many{x_i}}_\sigma[Q'] | \ensuremath{\boldsymbol{0}} + \sum_{y\in S} \inpi{y}{\many{y}',q}.\outpi{z}{q})
\end{mathpar}
We reason by contradiction.
We have
$A \strans{} \respi{\many x}\big(\,E^{\many{x_i}}_\sigma[P'] | \sum_{y\in S} \inpi{y}{\many{y}',q}.\outpi{z}{q}\,\big) \defi A'$ which is also singular.
Thus there should exist $B'$ such that \linebreak $B
\wtrans{} B'$ and $A' \bbis{\Tag{q_n}{\OO}} B'$.
We observe then that $B'$ can be of 3 forms:
\begin{itemize}
\item $\respi{\many x}(E^{\many{x_i}}_\sigma[Q_i] | \sum_{y\in S} \inpi{y}{\many{y}',q}.\outpi{z}{q})$ for some $i\in I$ (with $Q' \wtrans{} Q_i$)
\item $\respi{\many{x},q_1}(Q'' | \outpi{x_1}{\many{y},q_1} | \sum_{y\in S} \inpi{y}{\many{y}',q}.\outpi{z}{q})$
\item $\respi{\many x,q}(Q'' | \outpi{z}{q})$
\end{itemize}
The latter two are clearly not barbed bisimilar to $A'$.
For the first case, we have that $S'_I\subseteq S$ and
$E^{\many{x_i}}_\sigma[Q_i] \wbarb{\out{z_i}}$
for some $z_i \in S'_I$, so $B'\wbarb{\out{z}}$.
Thus, $B'$ is not barbed bisimilar to $A'$, because $A'$ is singular
so we have not $A'\wbarb{\out{z}}$.
\item when $\mu = \earlyinpi{x'}{\many{y}',p_0}$, then $\sigma = \Tag{q_0}{\II},\sigma''$ for some $q_0,\sigma''$ and $\sigma' =
\Tag{p_0}{\OO},\sigma$.
Given a fresh name $z'$, we define
\begin{align*}
R \defi & \respi{p_0}(\outpi{x'}{\many{y}',p_0} | \inpi{p_0}{\many{y}'}.\outpi{x_0}{\many{y}',q_0})\\
& | \sum_{y\in S'_I} \inpi{y}{\many{y}',q}.\outpi{z'}{q}
\end{align*}
We note $R'$ the process obtained by applying the law of Lemma~\ref{l:gl} to
$\outpi{x'}{\many{y}',p_0}$ in $R$. (Thus $R \bcong{\Tag{q_0}{\OO}} R'$)
We then have to show that for any $z,\many{x},S, Q'$ with \linebreak
$Q \wtrans{} Q'$,
we have $A\not\bbis{\Tag{q_n}{\OO}} B$ where
\begin{mathpar}
A\defi \respi{\many x, q_0}(E^{\many{x_i}}_\sigma[P] | \inpi{z}{q_0}.R | \outpi{z}{q_0})
\and
B\defi \respi{\many x, q_0}(E^{\many{x_i}}_\sigma[Q'] | \inpi{z}{q_0}.R' | \outpi{z}{q_0})
\end{mathpar}
We reason by contradiction. We have \\
$A \strans{}\strans{} \respi{\many x}(E^{x_0,\many{x_i}}_{\sigma'}[P'] |
\sum_{y\in S'_I} \inpi{y}{\many{y}',q}.\outpi{z'}{q}) \defi A'$ which is
also singular.
Thus there should exist $B'$ such that $B \wtrans{} B'$ and $A' \bbis{\Tag{q_n}{\OO}} B'$.
We observe that $B'$ can be of 3 forms:
\begin{itemize}
\item $\respi{\many x}(E^{x_0,\many{x_i}}_{\sigma'}[Q_i] | \sum_{y\in S'_I}
\inpi{y}{\many{y}',q}.\outpi{z'}{q})$ for some $i\in I$ \\
(with
$Q'|\respi{r_0}(\outpi{x'}{\many{y}',r_0} |
\inpi{r_0}{\many{y}'}.\outpi{p_0}{\many{y}'})
\wtrans{} Q_i$)
\item $\respi{\many x}(Q'' | \inpi{z}{q_0}.R' | \outpi{z}{q_0})$
\item $\respi{\many x}(Q'' | \outpi{z'}{q_0})$
\end{itemize}
The latter two are clearly not barbed bisimilar to $A'$.
For the first case, either $E^{x_0,\many{x_i}}_{\sigma'}[Q_i] \wbarb{\out{z_i}}$ for some $z_i \in S'_I$, so we have that $B'\wbarb{\out{z'}}$.
Thus, $B'$ is not barbed bisimilar to $A'$, because $A'$ is singular
so we have not $A'\wbarb{\out{z'}}$.
\end{itemize}
\textbf{Second case: if $P'$ is not singular.}
We examine the different possibilities for the transition along $\mu$.
\begin{itemize}
\item $\mu = \earlyinpi{x'}{\many{y}',p_0}$, then $\xi = \Tag{q_0}{\II}$ and
$\sigma' = \Tag{p_0}{\OO},\sigma$.
Given fresh names $(z_i)_{i\in I}$, we define
\begin{align*}
R \defi & \respi{p_0}
(\outpi{x'}{\many{y}',p_0} | \inpi{p_0}{\many{y}'}.\outpi{x_0}{\many{y}',q_0})\\
& | \sum_{i\in I}\tau.(R_i + \sum_{y\in S'_I}\inpi{y}{\many{y}',q}.\outpi{z_i}{q})
\end{align*}
and call $R'$ the process obtained by applying the law of Lemma~\ref{l:gl} to $\outpi{x_0}{\many{y}',q_0}$ in $R$. (Thus $R \bcong{\Tag{q_0}{\OO}} R'$)
We then have to show that for any $z,\many{x},S,Q'$ with \linebreak $Q \wtrans{} Q'$, we have
$A \not\bbis{\Tag{q_n}{\OO}} B$ where
\begin{mathpar}
A \defi \respi{\many{x},q_0}(E^{\many{x_i}}_\sigma[P] | \inpi{z}{q_0}.R | \outpi{z}{q_0})
\and
B \defi \respi{\many{x},q_0}(E^{\many{x_i}}_\sigma[Q'] | \inpi{z}{q_0}.R' | \outpi{z}{q_0})
\end{mathpar}
We reason by contradiction.
We have $A \strans{}\strans{} A'$ with
$$A'\defi \respi{\many{x}}(E^{x_0,\many{x_i}}_{\sigma'}[P'] | \sum_{i\in I}\tau.(R_i + \sum_{y\in S'_I}\inpi{y}{\many{y}',q}.\outpi{z_i}{q}))$$
Thus, there exists $B'$ such that $B \wtrans{} B'$ and \linebreak $A' \bbis{\Tag{q_n}{\OO}} B'$.
As $P'$ is not singular, $E^{x_0,\many{x_i}}_{\sigma'}[P']\wbarb{\out{z'}}$ for some $z'\in S'_I$,
so $A'\wbarb{\out{z_i}}$ for all $z_i$.
This implies that $B'$ is of the form $$\respi{x}(E^{x_0,\many{x_i}}_{\sigma'}[Q_i] | \sum_{i\in I}\tau.(R_i + \sum_{y\in S'_I}\inpi{y}{\many{y}',q}.\outpi{z_i}{q}))$$
with $Q'|\respi{r_0}(\outpi{x'}{\many{y}',r_0} |
\inpi{r_0}{\many{y}'}.\outpi{p_0}{\many{y}'})
\wtrans{} Q_i$.
We conclude this case by reasoning like in the proof
of Theorem~\ref{t:completeness}.
\iflong
\engue{C'est-à-dire qu'on regarde la clause qui est vraie pour $(P',Q_i)$. Je triche juste sur le fait qu'il peut y avoir une transition en trop qu'on annule avec le Lemma~\ref{l:lin}}
\fi
\item $\mu = \respi{\many{y}^2}\outpi{p_1}{\many{y}'}$. We note $\many{y}^2 = y^2_1,
\dots,y^2_n$ and $\many{y}' = \many{y}^2, y^1_1,\dots,y^1_m$. Given fresh
names $z',z'',(z_i)_{i\in I}$, we define
\begin{align*}
\hspace{-1em}R & \defi \inpi{x_1}{\many{x}^2,\many{x}^1,q_1}.(\outpi{z'}{q_1} | R_1)\quad\mbox{with}\\
\hspace{-1em}R_1 & \defi \sum_{j\leq n}\sum_{y''\in\ofnames{P,Q}}[x^2_j = y'']\inpi{z'}{q}.\outpi{z''}{q}\\
& +
\sum_{i\in I}[x^1_1 = y^1_1]\dots[x^1_m = y^1_m]\inpi{z}{q}.(\outpi{z_i}{q} |
\inpi{z_i}{q'}.R_i)
\end{align*}
We then have to show that for any $z,\many{x},S,Q'$ with \linebreak $Q \wtrans{} Q'$, we have
$A \not\bbis{\Tag{q_n}{\OO}} B$ where
\begin{mathpar}
A \defi \respi{\many{x}}(E^{\many{x_i}}_\sigma[P] | R + \sum_{y\in S} \inpi{y}{\many{y}',q}.\outpi{z}{q})
\and
B \defi \respi{\many{x}}(E^{\many{x_i}}_\sigma[Q'] | R + \sum_{y\in S} \inpi{y}{\many{y}',q}.\outpi{z}{q})
\end{mathpar}
We reason by contradiction. We let $\many{x_i}'$ stand
for $\many{x_i} = x_1,\many{x_i}'$.
We have $A \wbarb{z}$ and $A \strans{} \strans{} A'$ with
$$A' \defi \respi{\many{x},\many{y}^2, q_1}(E^{\many{x_i}'}_{\sigma'}[P'] | R_1).$$
So there should exist $B'$ such that $B \wtrans{} B'$ and $A' \bbis{\Tag{q_n}{\OO}} B'$.
As $A'\wbarb{z'}$ and $A'\wbarb{z_i}$ for all $i\in I$, but not $A'\wbarb{z''}$ nor $A'\wbarb{z}$, we have
that $B'$ can only be of the form $\respi{\many{x},\many{y}^2, q_1}(E^{\many{x_i}'}_{\sigma'}[Q_i] | R_0)$ for some $i\in I$ (with \linebreak$Q' \wtrans{\hat\mu} Q_i$)
We conclude this case by reasoning like in the proof
of Theorem~\ref{t:completeness}. \iflong \engue{C'est-à-dire qu'on regarde la clause qui est vraie pour $(P',Q_i)$. Je triche juste sur le fait qu'il peut y avoir une transition en trop qu'on annule avec le Lemma~\ref{l:lin}} \fi
\item $\mu = \tau$ and $\xi = \emptyset$. Given fresh names $(z_i)_{i\in I}$,
we define $R \defi \sum_{i\in I} \tau.(R_i + \sum_{y\in S'_I} \inpi{y}{\many{y}',q}.
\outpi{z_i}{q})$, and we can conclude.
\item $\mu = \tau$ and $\xi = \Tag{p}{\II}$. Given fresh names $(z_i)_{i\in I}$,
we define $R \defi \sum_{i\in I} \tau.(\outpi{z_i}{p} | \inpi{z_i}{p}.R_i)$,
and we can conclude.
\item $\mu = \earlyinpi{p}{\many{y}'}$. Given fresh names $(z_i)_{i\in I}$,
we define $$R \defi \outpi{p}{\many{y}'}
| \sum_{i\in I}\tau.(R_i + \sum_{y\in S'_I}\inpi{y}{\many{y}',q}.\outpi{z_i}{q})$$ \iflong \engue{Même triche à la fin avec le Lemma~\ref{l:lin}}\fi
and we can conclude.
\item $\mu = \respi{\many{y}^2,q}\outpi{x'}{\many{y}',q}$. We note $\many{y}^2 = y^2_1,
\dots,y^2_n$ and $\many{y}' = \many{y}^2, y^1_1,\dots,y^1_m$. Given fresh
names $z',z'',(z_i)_{i\in I}$, we define
\begin{align*}
\hspace{-1em}R &\defi \inpi{x'}{\many{x}^2,\many{x}^1,q_1}.(\outpi{z'}{q_1} | R_1)\\
\hspace{-1em}R_1 &\defi \sum_{j\leq n}\sum_{y''\in\ofnames{P,Q}}[x^2_j = y'']\inpi{z'}{q}.\outpi{z''}{q}\\
& +
\sum_{i\in I}[x^1_1 = y^1_1]\dots[x^1_m = y^1_m]\inpi{z}{q}.(\outpi{z_i}{q} |
\inpi{z_i}{q'}.R_i)
\end{align*}
We conclude this case by reasoning like in the case for $\mu = \respi{\many{y}^2}\outpi{p_1}{\many{y}'}$.
\end{itemize} \end{proof}
Once this proposition is proved, we reason as in the case of Sequential Bisimilarity to deduce completeness.
\section{Examples with WB Bisimulation (Section~\ref{s:awkward})}\label{a:awkward}
\subsection{A Simplified Example}
We discuss below a simplified example, which exposes the main difficulties that arise when studying the well-bracketed state change example. The primary simplification consists in using single-use functions, i.e., without replication. As a consequence, the two calls to the external function
are split into two separate functions. In ML, this corresponds to the terms $N$ and $L $ below: \begin{center} \begin{tabular}{rcl} $N$ & $ \defi$ & \smalltexttt{let x = ref 0 in} $(N_1, N_2)$ \\ $N_1$ & $ \defi$ & \smalltexttt{fun f -> x := 1; f (); !x} \\ $N_2$ & $ \defi$ & \smalltexttt{fun f -> x := 0; f (); x := 1} \\ \ \\ $L$ & $ \defi$ & $(L_1, L_2)$ \\ $L_1$ & $ \defi$ &\smalltexttt{fun f -> f (); 1} \\ $L_2$ & $ \defi$ &\smalltexttt{fun f -> f (); ()} \end{tabular}
\end{center} with the constraint that each term may only be called once. (Such a simplification would not work on the well-bracketed state change example, as $M_1$ and $M_2$ become equivalent even dropping well-bracketing if used at most once.)
Intuitively, \begin{center} \smalltexttt{let} $(h_1, h_2) = N$ \smalltexttt{in fun f -> } $h_2$ \smalltexttt{f} ; $h_1$ \smalltexttt{f} \end{center}
is similar to a single use of
$M_1$, and the same for $L$ and $M_2$.
The \Api translation of the above terms is as follows:
\begin{mathpar}
\begin{array}{rcl} \psapp{\enco{N}}{p_0} & \defi &
\respi{\ell,x,y}(\outpi{\ell}{0} | \outpi{p_0}{x,y} | P_1 | P_2)\\
P_1 & \defi &\inpi{x}{z, p}.\rwriteL{1}.\respi{p'}(\outpi{z}{\star, p'} | p'.\rreadL{n}.\outpi{p}{n})
\\
P_2 & \defi& \inpi{y}{z,q}.\rwriteL{0}.\respi{q'}(\outpi{z}{\star,q'} | q'.\rwriteL{1}.\out{q}) \\
\psapp{\enco{L}}{p_0} & \defi & \respi{x,y}(\outpi{p_0}{x,y}
| Q_1 | Q_2)\\
Q_1
&\defi &\inpi{x}{z, p}.\respi{p'}(\outpi{z}{\star, p'} | p'.\outpi{p}{1})
\\
Q_2
&\defi& \inpi{y}{z, q}.\respi{q'}(\outpi{z}{\star, q'} | q'.\out{q})
\end{array}
\end{mathpar} Indeed, the translation of an ML program is parametrised upon a continuation name (here $p_0$).
We introduce a slight abuse of notation, and write $\psapp{\enco
N}{p_0}\strans{\respi{x,y}\outpi{p_0}{x,y}} \enco{N}$, and similarly for notation $\enco L$.
Below we show that ${\enco N}$ and ${\enco L}$ are equivalent. First however we note that well-bracketing is necessary for this. Sequentiality alone is not sufficient: under the type system for sequentiality we can observe the following trace from $\enco N$:
$$
\enco N
\strans{\earlyinpi{x}{\star,p}}
\wtrans{\respi{p'}\outpi{z}{\star,p'}}\strans{\earlyinpi{y}{\star,q}}
\wtrans{\outpi{z}{\star, q'}}\strans{p'}\wtrans{\outpi{p}{0}}$$
This trace is not well-bracketed: function $x$ is called
first, but the continuation $p$ is returned before $q$. Process $\enco L$ may not produce such a trace~--- it may only emit $1$. However, the above non-well-bracketed trace is the only trace from $\enco N$ that may produce $0$.
We prove $\enco N
\ensuremath{\approx_{\rm{wb}}}^{\emptyset} \enco L$ by defining a relation, and relying on up-to techniques for
WB-bisimulation.
For that, we use the abbreviations \[ \begin{array}{rclrcl}
P_1' & \defi& p'.\rreadL{n}.\outpi{p}{n} , &
Q_1' & \defi& p'.\outpi{p}{1} , \\
P_2' & \defi& q'.\rwriteL{1}.\out{q}&
Q_2' & \defi& q'.\out{q} \end{array} \]
This allows us to define the following relation, called $\ensuremath{\mathcal{R}}$: $$\arraycolsep=1.4pt\begin{array}{l l l}
\{ (\emptyset,&\respi{\ell}(\outpi{\ell}{0}|P_1|P_2),& Q_1|Q_2),\\
((\Tag{p'}{\II},\Tag{p}{\OO}),& \respi{\ell}(\outpi{\ell}{1} | P_1' | P_2),&
Q_1' | Q_2),\\
((\Tag{q'}{\II},\Tag{q}{\OO}),& \respi{\ell}(\outpi{\ell}{0} | P_1| P_2'),&
Q_1 | Q_2'),\\
((\Tag{q'}{\II},\Tag{q}{\OO},\Tag{p'}{\II},\Tag{p}{\OO}),&
\respi{\ell}(\outpi{\ell}{0} | P_1'| P_2'),&
Q_1' | Q_2'),\\
(\emptyset,& \respi{\ell}(\outpi{\ell}{1}|P_2),& Q_2),\\
(\emptyset,& \respi{\ell}(\outpi{\ell}{1}|P_1),& Q_1),\\
((\Tag{p'}{\II},\Tag{p}{\OO},\Tag{q'}{\II},\Tag{q}{\OO}),&
\respi{\ell}(\outpi{\ell}{1} | P_1'| P_2'),&
Q_1' | Q_2'),\\
((\Tag{p'}{\II},\Tag{p}{\OO}),& \respi{\ell}(\outpi{\ell}{1} | P_1'),& Q_1'),\\
((\Tag{q'}{\II},\Tag{q}{\OO}),& \respi{\ell}(\outpi{\ell}{0} | P_2'),& Q_2'),\\
((\Tag{q'}{\II},\Tag{q}{\OO}),& \respi{\ell}(\outpi{\ell}{1} | P_2'),& Q_2'),\\
(\emptyset,& \respi{\ell}\outpi{\ell}{1},& \ensuremath{\boldsymbol{0}})\} \end{array}$$
$\ensuremath{\mathcal{R}}$ is
a wb-bisimulation up-to deterministic reduction and up-to static
context.
\iflong \ds{above: it would be useful to comment on what are the delicate pairs above; ie the
pairs in which the constraints on well-bracketing matter}\fi
\noindent\textbf{Without up-to techniques.}
We define some additional relations, which allow us to
show the improvement brought by up-to techniques.
For this, we introduce the following notations for processes
related to $P_1$ and $Q_1$ respectively:
\begin{mathpar}
P_1^a \defi \rwriteL{1}.P_1^b
\and
P_1^b \defi \respi{p'}(\outpi{z}{\star,p'} | P_1^c)
\and
P_1^c \defi p'.P_1^d
\and
P_1^d \defi \rreadL{n}.\outpi{p}{n}
\\
Q_1^c \defi p'.\outpi{p}{1}
\and
Q_1^b \defi \respi{p'}(\outpi{z}{\star,p'} | Q_1^c)
\end{mathpar}
and similarly for processes $P_2$ and $Q_2$.
We can remark that $P_i^c$ is the same as $P_i'$ introduced above
to define $\ensuremath{\mathcal{R}}$.
\begin{align*}
\ensuremath{\mathcal{R}}' \defi\{& (\Tag{p}{\OO}, \respi{\ell}(\outpi{\ell}{0} | P_1^a | P_2),
Q_1^a | Q_2),\\
& (\Tag{p}{\OO}, \respi{\ell}(\outpi{\ell}{0} | P_1 | P_2^a), Q_1 | Q_2^b),\\
& ((\Tag{q}{\OO},\Tag{p'}{\II},\Tag{p}{\OO}), \respi{\ell}(\outpi{\ell}{1} | P_1^c
| P_2^a), Q_1^c | Q_2^b)\\
& (\Tag{p}{\OO}, \respi{\ell}(\outpi{\ell}{1} | P_1^d | P_2), \outpi{p}{1} | Q_2)\\
& (\Tag{q}{\OO}, \respi{\ell}(\outpi{\ell}{0} | P_1 | P_2^d), Q_1 | \out{q})\\
& ((\Tag{p}{\OO},\Tag{q'}{\II},\Tag{q}{\OO}), \respi{\ell}(\outpi{\ell}{0} | P_1^a
| P_2^c), Q_1^b | Q_2^c)\\
& ((\Tag{q}{\OO},\Tag{p'}{\II},\Tag{p}{\OO}), \respi{\ell}(\outpi{\ell}{0} | P_1^c
| P_2^d), Q_1^c | \out{q})\\
& (\Tag{q}{\OO}, \respi{\ell}(\outpi{\ell}{1} | P_2^a), Q_2^b)\\
& (\Tag{p}{\OO}, \respi{\ell}(\outpi{\ell}{1} | P_1^a), Q_1^b)\\
& ((\Tag{p}{\OO},\Tag{q'}{\II},\Tag{q}{\OO}), \respi{\ell}(\outpi{\ell}{1} | P_1^d
| P_2^c), \outpi{p}{1} | Q_2^c)\\
& (\Tag{q}{\OO}, \respi{\ell}(\outpi{\ell}{0} | P_2^d), \out{q})\\
& (\Tag{q}{\OO}, \respi{\ell}(\outpi{\ell}{1} | P_2^d), \out{q})\\
& (\Tag{p}{\OO}, \respi{\ell}(\outpi{\ell}{1} | P_1^d), \outpi{p}{1})\\
\}
\end{align*}
\begin{align*}
\!\ensuremath{\mathcal{R}}'' \defi\{& (\Tag{p}{\OO}, \respi{\ell}(\outpi{\ell}{1} | P_1^b | P_2),
Q_1^b | Q_2),\\
& (\Tag{p}{\OO}, \respi{\ell}(\outpi{\ell}{0} | P_1 | P_2^b), Q_1 | Q_2^b),\\
& ((\Tag{q}{\OO},\Tag{p'}{\II},\Tag{p}{\OO}), \respi{\ell}(\outpi{\ell}{0} | P_1^c
| P_2^b), Q_1^c | Q_2^b)\\
& (\Tag{p}{\OO}, \respi{\ell}(\outpi{\ell}{1} | \outpi{p}{1} | P_2), \outpi{p}{1} | Q_2)\\
& (\Tag{q}{\OO}, \respi{\ell}(\outpi{\ell}{1} | P_1 | \out{q}), Q_1 | \out{q})\\
& ((\Tag{p}{\OO},\Tag{q'}{\II},\Tag{q}{\OO}), \respi{\ell}(\outpi{\ell}{1} | P_1^b
| P_2^c), Q_1^b | Q_2^c)\\
& ((\Tag{q}{\OO},\Tag{p'}{\II},\Tag{p}{\OO}), \respi{\ell}(\outpi{\ell}{1} | P_1^c
| \out{q}), Q_1^c | \out{q})\\
& (\Tag{q}{\OO}, \respi{\ell}(\outpi{\ell}{0} | P_2^b), Q_2^b)\\
& (\Tag{p}{\OO}, \respi{\ell}(\outpi{\ell}{1} | P_1^b), Q_1^b)\\
& ((\Tag{p}{\OO},\Tag{q'}{\II},\Tag{q}{\OO}), \respi{\ell}(\outpi{\ell}{1} |
\outpi{p}{1} | P_2^c), \outpi{p}{1} | Q_2^c)\\
& (\Tag{q}{\OO}, \respi{\ell}(\outpi{\ell}{1} | \out{q}), \out{q})\\
& (\Tag{p}{\OO}, \respi{\ell}(\outpi{\ell}{1} | \outpi{p}{1}), \outpi{p}{1})\\
\}
\end{align*}
We see that $\ensuremath{\mathcal{R}}$, $\ensuremath{\mathcal{R}}'$ and $\ensuremath{\mathcal{R}}''$ are comparable in size.
We have that:
\begin{itemize}
\item $\ensuremath{\mathcal{R}}\uplus\ensuremath{\mathcal{R}}'$ is a wb-bisimulation up-to static context.
\item
$\ensuremath{\mathcal{R}}\uplus\ensuremath{\mathcal{R}}''$ is a wb-bisimulation up-to deterministic
$\tau$.
\item
$\ensuremath{\mathcal{R}}\uplus\ensuremath{\mathcal{R}}'\uplus\ensuremath{\mathcal{R}}''$ is a wb-bisimulation.
\end{itemize}
\subsection{Well-bracketed state change}
We present the proof for the example described in Section~\ref{s:awkward}.
We need to introduce some notations in order to reason about the sub-processes which are generated by calls to the functions in the encodings of $M_1$ and $M_2$.
To simplify notations, we assume that names extruded from $\respi{q}$ (resp. $\respi{r}$) are in the set $\set{q_i \ensuremath{~\big|~} i \in I}$ (resp. $\set{r_j
\ensuremath{~\big|~} j \in J}$) for some set $I$ (resp. $J$). Continuation names received on $x$ will be noted $p_i$ or $p_j$ depending on which prefix they are underneath.
\begin{mathpar}
P^1_i = q_i.\respi{r}(\outpi{y}{r} | r.\outpi{p_i}{1})
\and
P^2_j = r_j.\outpi{p_j}{1}
\and
Q_0 = !\inpi{x}{y,p}.\rwriteL{0}.\respi{q}(\outpi{y}{q} | q.\rwriteL{1}.\respi{r}(\outpi{y}{r} | r.\rreadL{n}.\outpi{p}{n}))
\and
Q^1_i = q_i.\rwriteL{1}.\respi{r}(\outpi{y}{r} | r.\rreadL{n}.\outpi{p_i}{n})
\and
Q^2_j = r_j.\rreadL{n}.\outpi{p_j}{n} \end{mathpar}
Given an ordered list $s$ of the indices in $I\uplus J$, we
write $\sigma(s)$ for the stack defined inductively as follows:
\begin{mathpar}
\sigma(\emptyset) = \emptyset
\and
\sigma(i;s) = \Tag{q_i}{\II},\Tag{p_i}{\OO}, \sigma(s)
\and
\sigma(j;s) = \Tag{r_j}{\II},\Tag{p_j}{\OO}, \sigma(s) \end{mathpar}
We can then define the relation: \begin{align*}
\ensuremath{\mathcal{R}} \defi \{& (\emptyset, P, Q),\quad (\emptyset, P, \respi{\ell}(\outpi{\ell}{1} | Q_0)),\\
&(\sigma(i;s), P|\prod_{i\in I} P^1_i | \prod_{j\in J} P^2_j,\\
& \respi{\ell}(\outpi{\ell}{0} | Q_0 | \prod_{i\in I} Q^1_i
| \prod_{j\in J} Q^2_j)),\\
&(\sigma(i;s), P|\prod_{i\in I} P^1_i | \prod_{j\in J} P^2_j,\\
& \respi{\ell}(\outpi{\ell}{1} | Q_0 | \prod_{i\in I} Q^1_i
| \prod_{j\in J} Q^2_j)),\\
&(\sigma(j;s), P|\prod_{i\in I} P^1_i | \prod_{j\in J} P^2_j,\\
& \respi{\ell}(\outpi{\ell}{1} | Q_0 | \prod_{i\in I} Q^1_i
| \prod_{j\in J} Q^2_j))\\
\} \end{align*}
$\ensuremath{\mathcal{R}}$ is a wb-bisimulation up-to deterministic $\tau$ and up-to static context.
\end{document} | arXiv |
Preprint CASE REPORT | doi:10.20944/preprints202101.0249.v1
How Fear of COVID-19 Can Affect Treatment Choices for Anaplastic Large Cell Lymphomas Alk+ Therapy: A Case Report
Antonello Sica, Caterina Sagnelli, Beniamoni Casale, Gino Svaneta, Massimiliano Creta, Armando Calogero, Renato Franco, Evangelista Sagnelli, Andrea Ronchi
Subject: Medicine & Pharmacology, Allergology Keywords: Crizotinib; Anaplastic Large Cell Lymphomas ALK+; bridge therapy in NHL ALK+; ALK+ patients; anticancer therapy
Background: The t (2; 5) chromosomal rearrangement and resulting nucleophosmin (NPM1) -ALK fusion was first observed in 1994 in anaplastic large cell lymphoma (ALCL), a T-cell lymphoma responsive to cyclophosphamide, abriblastine, vincristine and prednisone in approximately 80% of cases; refractory cases usually respond favorably to brentuximab-vedotin. These treatments are regarded as a bridge to allogeneic hematopoietic stem cell transplantation (allo-SCT). Nowadays, transplant procedures and monitoring of chemotherapy patients proceed very slowly because the SARS-CoV-2 pandemic has heavily clogged the hospitals in all countries. Results: A 40-year-old Caucasian woman was first seen at our clinical center in June 2020. She had ALCL ALK +, a history of failure to two previous therapeutic lines and was in complete remission after 12 courses of Brentuximab, still pending allo-SCT after two failed donor selection. Facing of a new therapeutic failure, we requested the Italian drug regulatory agency, and obtained the authorization, to administer 250 mg twice a day of Crizotinib, a drug incomprehensibly not registered for ALCL ALK +. Conclusions: The response to Crizotinib was optimal, since no adverse event occurred, and CT-PET persisted negative; this drug has proved to be a valid bridge to allo-SCT
Investigation of Seismic Behavior of V-Shaped Bridge with Memory Alloy Separator in Earthquake Orientation -Affected Joints
Saleh Salehi fereidouni, Qiang Han, Xiuli Du
Subject: Engineering, Civil Engineering Keywords: v-shaped bridge columns; bridge junctions; shape memory alloy; seismic separator; seismic performance
In recent decades, widespread damage to structures due to destructive earthquakes has encouraged researchers to use seismic control systems. One of the most important structures is bridges, which should maintain their service after severe earthquakes. One of the ways to control the bridge is to isolation the deck from the substructure, which results in high displacements at different points in the bridge and the residual displacements within the separator itself, thus requiring repair and replacement of these systems after severe earthquakes. Therefore, the use of new seismic separation systems on bridges has always been important. The purpose of this project is to utilize a new memory-shaped seismic separator system to investigate the seismic performance of v-shaped column bridges , in which case the effect of earthquake orientation and its effects on responses the earthquake will also be assessed. Due to the ever-increasing advances in various intelligent materials, such as shape memory alloys, and their unique capacities, such as back-centering, and high fatigue and corrosion resistance, in this study, these materials have been used as a complementary component along with lead core separators. The shape memory alloy has been used as 4 vertical rods that connect two separating parts to each other and in this research the diameter and location of these rods will be calculated using optimization algorithm in Matlab software.
Nikfar Domination in Neutrosophic Graphs
Mohammadesmail Nikfar
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: Neutrosophic graph, bridge, tree, e ective edge, nikfar domination.
Many various using of this new-born fuzzy model for solving real-world problems and urgent requirements involve introducing new concept for analyzing the situations which leads to solve them by proper, quick and e cient method based on statistical data. This gap between the model and its solution cause that we introduce nikfar domination in neutrosophic graphs as creative and e ective tool for studying a few selective vertices of this model instead of all ones by using special edges. Being special selection of these edges a ect to achieve quick and proper solution to these problems. Domination hasn't ever been introduced. So we don't have any comparison with another de nitions. The most used graphs which have properties of being complete, empty, bipartite, tree and like stu and they also achieve the names for themselves, are studied as fuzzy models for getting nikfar dominating set or at least becoming so close to it. We also get the relations between this special edge which plays main role in doing dominating with other special types of edges of graph like bridges. Finally, the relation between this number with other special numbers and characteristic of graph like order are discussed.
Efficiency of Artificial Neural Networks in Determining Scour Depth at Composite Bridge Piers
Ata Amini, Shahriar Hamidi, Ataollah Shirzadi, Javad Behmanesh, Shatirah Akib
Subject: Engineering, Civil Engineering Keywords: Local Scour; Sediment; Bridge Design; Pier Geometry; ANN
Scouring is the most common cause of bridge failure. This study was conducted to evaluate the efficiency of the Artificial Neural Networks (ANN) in determining scour depth around composite bridge piers. The experimental data, attained in different conditions and various pile cap locations, were used to obtain the ANN model and to compare the results of the model with most well-known empirical, HEC-18 and FDOT, methods. The data were divided into training and evaluation sets. The ANN models were trained using the experimental data, and their efficiency was evaluated using statistical test. The results showed that to estimate scour at the composite piers, feedforward propagation network with three neurons in the hidden layer and hyperbolic sigmoid tangent transfer function was with the highest accuracy. The results also indicated a better estimation of the scour depth by the proposed ANN than the empirical methods.
Presentation a New Method for Determining of Bridge Condition Index by Using Analytical Hierarchy Process
Saeid Darban, Hosein Ghasemzadeh Tehrani, Nader Karballaeezadeh
Subject: Engineering, Civil Engineering Keywords: transportation infrastructure; bridge management system; concrete bridges; bridge condition index; analytical hierarchy process; expert system
This paper proposes a method for determining the bridge condition index (BCI) in concrete bridges, which is based on the views of bridge experts. First, eight indices were defined for a concrete bridge including structure, hydrology, safety, load impact, geotechnical and seismicity, strategic importance, facilities, and finally traffic and pavement. Each index consists of several sub-indices. Next, a series of questionnaires about the relative importance of indices and their sub-indices were prepared and distributed among bridge experts. Experts' views were analyzed by Expert Choice software and the relative importance (weight) of each index and each sub-index was determined using the analytical hierarchy process (AHP). Then, based on experts' views, an average score was assigned to each sub-index for any condition. Now the bridge inspectors can examine the bridge and determine the scores of sub-indices. Each index's score is the sum of the weighted score assigned to its' sub-indices and BCI is the sum of weighted scores assigned to indices. Higher values of BCI indicate a better condition. Therefore, bridges with lower BCI take priority in maintenance activities. To apply the proposed method, five bridges were selected in Semnan province, Iran, and BCI calculation of these bridges were conducted.
Nikfar Domination Versus Others: Restriction, Extension Theorems and Monstrous Examples
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: fuzzy graph, fuzzy bridge, α-strong edge, nikfar domination, dynamic networks.
The aim of this expository article is to present recent developments in the centuries-old discussion on the interrelations between several types of domination in graphs. However, the novelty even more prominent in the newly discovered simplified presentations of several older results. Domination can be seen as arising from real-world application and extracting classical results as first described by this article.The main part of this article, concerning a new domination and older one, is presented in a narrative that answers two classical questions: (i) To what extend must closing set be dominating? (ii) How strong is the assumption of domination of a closing set? In a addition, we give an overview of the results concerning domination. The problem asks how small can a subset of vertices be and contain no edges or, more generally how can small a subset of vertices be and contain other ones. Our work was as elegant as it was unexpected being a departure from the tried and true methods of this theory that had dominated the field for one fifth a century. This expository article covers all previous definitions. The inability of previous definitions in solving even one case of real-world problems due to the lack of simultaneous attentions to the worthy both of vertices and edges causing us to make the new one. The concept of domination in a variety of graphs models such as crisp, weighted and fuzzy, has been in a spotlight. We turn our attention to sets of vertices in a fuzzy graph that are so close to all vertices, in a variety of ways, and study minimum such sets and their cardinality. A natural way to introduce and motivate our subject is to view it as a real-world problem. In its most elementary form, we consider the problem of reducing waste of time in transport planning. Our goal here is to first describe the previous definitions and the results, and then to provide an overview of the flows ideas in their articles. The final outcome of this article is twofold: (i) Solving the problem of reducing waste of time in transport planning at static state; (ii) Solving and having a gentle discussions on problem of reducing waste of time in transport planning at dynamic state. Finally, we discuss the results concerning holding domination that are independent of fuzzy graphs. We close with a list of currently open problems related to this subject. Most of our exposition assumes only familiarity with basic linear algebra, polynomials, fuzzy graph theory and graph theory.
Potential Hazards at the New York City Bridges, 1982 – 2006
Bojidar Yanev
Subject: Engineering, Automotive Engineering Keywords: bridge; condition; flag; forecast; management; sustainability
New York State Department of Transportation designates potentially hazardous conditions on bridges as flags. From 1982 until 2006 the flags issued for the bridges owned by New York City underwent all phases typical of crises, including a gradual increase, an exponential expansion, an extended peak, a gradual decline, and a convergence to a higher but manageable level. The attempts to forecast the flag pattern as it was developing are reviewed for possible relevance to management of the transportation infrastructure and in general.
Myosin Cross-bridge Behaviour in Contracting Muscle – The T1 Curve of Huxley & Simmons (1971) Revisited
Carlo Knupp, John M. Squire
Subject: Life Sciences, Biophysics Keywords: myosin filament stiffness; actin filament stiffness; myosin cross-bridge stiffness; muscle transients; weak binding heads; contractile mechanism; cross-bridge cycle; rigor muscle
The stiffness of the myosin cross-bridges is a key factor in analysing possible scenarios to explain myosin head changes during force generation in active muscles. The seminal study of Huxley and Simmons (1971: Nature 233: 533) suggested that most of the observed half-sarcomere instantaneous compliance (=1/stiffness) resides in the myosin heads. They showed with a so-called T1 plot that, after a very fast release, the half-sarcomere tension reduced to zero after a step size of about 60Å (later with improved experiments reduced to 40Å). However, later X-ray diffraction studies showed that myosin and actin filaments themselves stretch slightly under tension, which means that most (at least two-thirds) of the half sarcomere compliance comes from the filaments and not from cross-bridges. Here we have used a different approach, namely to model the compliances in a virtual half sarcomere structure in silico. We confirm that the T1 curve comes almost entirely from length changes in the myosin and actin filaments, because the calculated cross-bridge stiffness (probably greater than 0.4 pN/Å) is higher than previous studies have suggested. In the light of this, we present a plausible modified scenario to describe aspects of the myosin cross-bridge cycle in active muscle. In particular, we suggest that, apart from the filament compliances, most of the cross-bridge contribution to the instantaneous T1 response comes from weakly-bound myosin heads, not myosin heads in strongly attached states. The strongly attached heads would still contribute to the T1 curve, but only in a very minor way, with a stiffness that we postulate could be around 0.1 pN/Å, a value which would generate a working stroke close to 100 Å from the hydrolysis of one ATP molecule. The new program can serve as a tool to calculate sarcomere elastic properties for any vertebrate striated muscle once various parameters have been determined (e.g. tension, T1 intercept, temperature, X-ray diffraction spacing results).
Vertex Domination in Fuzzy Graphs
Subject: Mathematics & Computer Science, Other Keywords: fuzzy graph; fuzzy bridge; fuzzy tree; $\alpha$-strong arc; vertex domination
We introduce a new variation on the domination theme which we call vertex domination as reducing waste of time in transportation planning and optimization of transport routes. We determine the vertex domination number $\gamma_v$ for several classes of fuzzy graphs. The bounds is obtained for it. In fuzzy graphs, monotone decreasing property and monotone increasing property are introduced. We prove both of the vizing's conjecture and the Grarier-Khelladi's conjecture are monotone decreasing fuzzy graph property for vertex domination. We obtain Nordhaus-Gaddum (NG) type results for these parameters. The relationship between several classes of operations on fuzzy graphs with the vertex domination number of them is studied. Finally, we discuss about vertex dominating set of a fuzzy tree by using the bridges and $\alpha$-strong edges equivalence.
Optimizing Efficiency and Motility of A Polyvalent Molecular Motor
Mark Rempel, Eldon Emberly
Subject: Physical Sciences, Acoustics Keywords: molecular motor; burnt bridge ratchet; computational model
Molecular motors play a vital role in the transport of material within the cell. A family of motors of growing interest are burnt bridge ratchets (BBRs). BBRs rectify spatial fluctuations into directed motion by creating and destroying motor-substrate bonds. It has been shown that the motility of a BBR can be optimized as a function of the system parameters. However, the amount of energy input required to generate such motion and the resulting efficiency has been less well characterized. Here, using a deterministic model, we calculate the efficiency of a particular type of BBR, namely a polyvalent hub interacting with a surface of substrate. We find that there is an optimal burn rate and substrate concentration that leads to optimal efficiency. Additionally, the substrate turnover rate has important implications on motor efficiency. We also consider the effects of force-dependent unbinding on the efficiency and find that under certain conditions the motor works more efficiently when bond breaking is included. Our results provide guidance for how to optimize the efficiency of BBRs.
Improvement Measures for Bridge Inspection Efficiency Using Spatial Information Technology
Jae Kang LEE, Min Jun Kim, Jung Ok Kim, Jin Soo Kim
Subject: Engineering, Civil Engineering Keywords: bridge maintenance and inspection; UAVs; machine vision
The economic development and infrastructure of a nation are closely interrelated. In addition, public trust in national infrastructure facilities is closely linked to the preservation of the advantages provided by these facilities to the public. Since the 1970s, Korea has achieved exponential economic growth over a short period of time and the number of infrastructure facilities has increased correspondingly. This compressed economic development has been underpinned by the national infrastructure, whose safety and usability have been excluded from the scope of the development. However, after around 30 years, structural deterioration coupled with general insensitivity to safety in today's society has considerably reduced public trust in using the infrastructure. Realistically, policies that mainly focus on developing new technologies related to infrastructure construction have led to practical limitations that discourage the development of technologies for maintenance or inspection. Furthermore, current maintenance works face certain limitations caused by various reasons: insufficient budget, increasing number of infrastructure facilities requiring maintenance, shortage of manpower, and rapidly increasing number of aging infrastructure facilities. To overcome these limitations, a new approach is required that is different from general inspection methods under the existing rules and regulations. In this context, this study aimed to explore the efficiency of bridge inspection and maintenance by unmanned aerial vehicles (UAVs) that could observe inaccessible areas, could be conveniently and easily controlled, and could offer high economic benefits. To this end, various tests were performed on elevated bridges, and suitable UAV images were obtained. The obtained UAV images were inspected by using machine vision technology, thereby excluding subjective evaluations by humans. Methods for enhancing the objectivity of the inspection were also discussed. The test results showed that both the efficiency and objectivity of the proposed method were better than those of the existing bridge maintenance and inspection methods.
A Moving 3D Laser Scanner for Automated Underbridge Inspection
Marco Tarabini, Hermes Giberti, Silvio Giancola, Matteo Sgrenzaroli, Remo Sala, Federico Cheli
Subject: Engineering, Civil Engineering Keywords: laser; construction monitoring; measurements; uncertainty; bridge inspection
Recent researches proved that the underbridge geometry can be reconstructed by mounting a 3D laser scanner on a motorized cart travelling on a walkway located under the bridge. The walkway is moved by a truck and the accuracy of the bridge model depends on the accuracy of the trajectory of the scanning head with respect to a fixed reference system. In this paper, we describe the metrological characterization of a method that uses non-contact systems to identify the relative motion of the cart with respect to the walkway; the orientation of the walkway with respect to the bridge is determined using inclinometers and optical rails, while the position of the truck with respect to the bridge is measured using a conventional odometer. The measurement uncertainty of the proposed system was initially evaluated by numerical simulations and successively verified by experiments in laboratory conditions. The complete system has then been tested in operative conditions; the validity of the proposed approach has been demonstrated by comparing the geometry of buildings reconstructed with the proposed system with the geometry obtained with a static scan. Results evidenced that the errors are approximately 6 mm.
Thermal Bridge Modeling and a Dynamic Analysis Method Using the Analogy of a Steady-State Thermal Bridge Analysis and System Identification Process for Building Energy Simulation: Methodology and Validation
Heegang Kim, Myoungsouk Yeo
Subject: Arts & Humanities, Architecture And Design Keywords: thermal bridge; modeling and dynamic analysis; system identification
It is challenging to apply heat flow through a thermal bridge, which requires the analysis of 2D or 3D heat transfer to building energy simulation(BES). Research on the dynamic analysis of thermal bridges has been underway for many years, but their utilization remains low in BESs. This paper proposes a thermal bridge modeling and a dynamic analysis method that can be easily applied to BESs. The main idea begins with an analogy of the steady-state analysis of thermal bridges. As with steady-state analysis, the proposed method first divides the thermal bridge into a clear wall, where the heat flow is uniform, and the sections that are not the clear wall (the thermal bridge part). For the clear wall part, the method used in existing BESs is applied and analyzed. The thermal bridge part (TB part) is modeled with the linear time-invariant system (LTI system) and the system identification process is performed to find the transfer function. Then, the heat flow is obtained via a linear combination of the two parts. This method is validated by comparing the step, sinusoidal and annual outdoor temperature response of the finite differential method(FDM) simulation. When the thermal bridge was modeled as a third-order model, the root mean square error(RMSE) of annual heat flow with the FDM solution of heat flow through the entire wall was about 0.1W.
New Cassane Diterpenoids from Caesalpinia sappan and Their Antiplasmodial Activity
Nai-Liang Zhu, Zhong-Hao Sun, Mei-Geng Hu, Tong-Yu Wu, Jing-Quan Yuan, Hai-Feng Wu, Yu Tian, Peng-Fei Li, Jun-Shan Yang, Guo-Xu Ma, Xu-Dong Xu
Subject: Chemistry, Medicinal Chemistry Keywords: Caesalpinia sappan; cassane diterpenes; N bridge; antimalarial activity
One new cassane diterpene possessing an unusual N bridge between C-19 and C-20 named caesalsappanin R (1), as well as another new diterpene caesalsappanin S (2), were isolated from the seeds of Caesalpinia sappan with methanol extract. Their structures were determined by spectroscopic analysis and examined alongside existing data from prior studies. Their biological activities were profiled by their antiplasmodial activity.
Photoinduced Electron Transfer in Organized Assemblies. Case Studies
Antonio Santoro, Giovanni Bella, Ambra Maria Cancelliere, Scolastica Serroni, Giuliana Lazzaro, Sebastiano Campagna
Subject: Chemistry, Physical Chemistry Keywords: Electron transfer; artificial photosynthesis; supramolecular assemblies; Donor-bridge-acceptor system; transient absorption spectroscopy
In this review, photoinduced electron transfer processes in specifically designed assembled ar-chitectures have been investigated on the basis of recent results reported from our laboratories. A convenient and useful way to study these kinds of systems has been described in order to under-stand the rules that drive to a light induced charge separation and its subsequent decay to the ground state, also with the aim of offering a tutorial for young researchers. Assembled systems of covalent or supramolecular nature have been presented and some functional multicomponent systems for the conversion of light energy into chemical energy have been discussed.
Investigation of Dynamic Analysis of Corrosion Effect under to Chloride Induced in Reinforced Concrete Bridge Columns
Abdolreza Tangtakabi, Mohammad Hasan Ramesht, Ali Golsoorat Pahlaviani, Towhid Pourostam
Subject: Engineering, Civil Engineering Keywords: reinforced concrete bridge (RCB); chloride corrosion; seismic performance; plastic hinge; overload analysis
One of the influential factors in estimating the service life of reinforced concrete bridges (RCB) is determining the long-term seismic performance of these structures. Corrosion due to chloride ion diffusion leads to the destruction of critical members of the RCB during the useful life of the structure. So, the long-term seismic performance of the bridge deteriorates as a result. It is essential to study the effect of corrosion deterioration on the long-term seismic performance of bridges in the southern regions of Iran, near the coasts of the Persian Gulf and the Oman Sea, because of the seismicity of the region and high corrosion rate of reinforced concrete (RC) members is the result of environmental conditions. In order to investigate this issue, considering studies about environmental conditions of southern Iran, the onset time of corrosion in the columns, as seismic critical members of the bridge, was determined. Based on that, the corrosion's effect on characteristics of RC at specific time points during the bridge's useful life (0, 15, 30, 45, 60, 75 and 90 years) have been calculated. The effects of corrosion include deterioration of the core and cover concrete properties, steel bar and the connection between concrete and steel bar. In the next step, at each time point, according to the modified stress-strain relationships, the moment-curvature analysis of the bridge pier was done, and the properties of the plastic hinge were determined. Finally, based on the obtained data about plastic hinge characteristics at each time point, overload analysis of the bridge was performed in both longitudinal and transverse directions. Then the capacity curves of RCB were compared at the mentioned time-points. The results show that the capacity of the bridge deteriorates over time due to corrosion. Therefore, a proposal to increase the value of base shear design has been made to ensure the long-term seismic performance of RCB in corrosive environments.
Glial Bridge Ecology: Cellular Mechanisms that Drive Spinal Cord Regeneration in Zebrafish
Corbin J. Schuster, Robert M. Kao
Subject: Life Sciences, Cell & Developmental Biology Keywords: glial bridge; ctgfa; Fgf signaling; MAPK signaling; shh; slit2/3; Wnt signaling; genetic compensation; glial bridge cycle; spinal cord regeneration; termination signal; central nervous system; peripheral nervous system; zebrafish
Zebrafish have been found to be the premier model organism in biological and biomedical research, specifically offering many advantages in developmental biology and genetics. This unique aquatic species has been found to have the capacity to regenerate their spinal cord after injury. However, the complete molecular and cellular mechanisms behind glial bridge formation in the central and peripheral nervous systems upon glial cell injury remains unclear. This review paper focuses on the molecular mechanisms and cellular processes that underlie spinal cord regeneration in four initial phases: proliferation and initial migration; migration and differentiation; glial bridge formation; and remodeling. We propose that within these four phases the cellular mechanisms that underlie spinal cord regeneration each express a terminating signal that aborts one step of the process and initiates the next. Specifically, future studies would be devoted to investigate transmitting signals in the spinal cord injury micro-environment in hope to contribute to the understanding of underlying cellular mechanisms by connecting each process of spinal cord regeneration in zebrafish.
Low Concentration Response Hydrogen Sensors Based on Wheatstone Bridge
Hongchuan Jiang, Xiaoyu Tian, Xinwu Deng, Xiaohui Zhao, Luying Zhang, Wanli Zhang, Jianfeng Zhang, Yifan Huang
Subject: Materials Science, Other Keywords: hydrogen sensors; PdNi thin films; Wheatstone bridge; low concentration
MEMThe PdNi film hydrogen sensors with Wheatstone bridge structure were designed and fabricated by the micro-electro-mechanical system (MEMS) technology. The integrated sensors consisted of four PdNi alloy film resistors. The interval two of them were shielded with silicon nitride film and used as reference resistance, while the others were used for hydrogen sensing. The PdNi alloy films and SiN films were deposited by magnetron sputtering. The morphology and microstructure of the PdNi films were characterized with X-ray diffraction (XRD). The output resistance signal was converted to millivolt output voltage signal for easy data acquisition. Hydrogen (H2) sensing properties of PdNi film hydrogen sensor with Wheatstone bridge structure was investigated under different temperatures (30℃, 50℃ and 70℃) and H2 concentrations (from 10 ppm to 0.4%). The hydrogen sensor demonstrated good response at different hydrogen concentrations and high repeatability in cycle testing under 0.4% H2 concentration. Under 10ppm hydrogen, the PdNi film hydrogen sensor had evident and collectable output voltage of 600 μV.
New Switched-Dual-Source Multilevel Inverter for Symmetrical and Asymmetrical Operation
Kennedy Aganah, Cristopher Luciano, Mandoye Ndoye, Gregory Murphy
Subject: Keywords: multilevel; inverter; single phase; reduced switch-count; h-bridge
The past two decades has seen a growing demand for high-power, high-voltage utility scale inverters mostly fueled by the integration of large solar PV and wind farms. Multilevel inverters have emerged as the industry choice for these megawatt range inverters because their reduced voltage stress, capable of generating an almost sinusoidal voltage, in-built redundancy, among others. This paper present a new Switched-Source Multilevel Inverter (SS MLI) architecture. The new inverter show superior over existing topologies. It has reduced voltage stress on the semiconductor, uses less number of switches –reduced size/weight/cost and increased efficiency. The new SSMLI is comprised of two voltage sources (V1, V2) and 6 switches. It is capable of generating 5-level output voltage in symmetric modes (i.e., V1 = V2), and 7-level output voltage in asymmetric modes (i.e., V1 ≠ V2). To demonstrate the validity of the proposed inverter, simulations results using MATLAB® /Simulink® for 5- and 7-level output voltages are presented . The simulations are also verified experimentally using a laboratory prototype.
Vertex Domination in t-Norm Fuzzy Graphs
Subject: Mathematics & Computer Science, Other Keywords: t-norm Fuzzy graph; t-norm; fuzzy tree; bridge; α-strong edges; vertex domination
For the first time, We do fuzzification the concept of domination in crisp graph on a generalization of fuzzy graph by using membership values of vertices, α-strong edges and edges. In this paper, we introduce the first variation on the domination theme which we call vertex domination. We determine the vertex domination number γv for several classes of t-norm fuzzy graphs which include complete t-norm fuzzy graph, complete bipartite t-norm fuzzy graph, star t-norm fuzzy graph and empty t-norm fuzzy graph. The relationship between effective edges and α-strong edges is obtained. Finally, we discuss about vertex dominating set of a fuzzy tree with respect to a t-norm ⨂ by using the bridges and α-strong edges equivalence.
The Transient Mechanics of Muscle Require Only a Single Force-Producing Crossbridge State and a 100 Å Working Stroke
Subject: Biology, Anatomy & Morphology Keywords: muscle transients; myosin cross-bridge cycle; isotonic shortening; length steps
An informative probe of myosin crossbridge behaviour in active muscle is a mechanical transient experiment where, for example, a fully active muscle initially held at constant length is suddenly shortened to a new fixed length giving a force transient, or has its load suddenly reduced giving a length transient. We describe the simplest crossbridge mechanical cycle we could find to model these transients. We show using the statistical mechanics of 50,000 crossbridges that a simple cycle with two actin-attached cross-bridge states, one producing no force and the other producing force, will explain much of what has been observed experimentally and we discuss the implications of this modelling for our understanding of how muscle works. We show that this same simple model will explain reasonably well the isotonic mechanical and X-ray transients under different loads observed by Reconditi et al (2004, Nature 428, 578) and that there is no need to invoke different crossbridge step sizes under these different conditions; a step size of 100 Å works well for all loads. We do not claim that this model provides a total mechanical explanation of how muscle works. But we do suggest that only if there are other observations that cannot be explained by this simple model should something more complicated be considered.
Physical Modelling of Bridge Pier Scour and Its Effect on Dynamic Response: Towards Real-Time Monitoring of Bridges
Fatemeh Rahimi, Alireza Keshavarzy, Mohsen Askari
Subject: Engineering, Civil Engineering Keywords: bridge stability; scour; physical modelling; structural health monitoring; modal analysis
The generation of vortices around bridge piers can lead to removal of riverbed materials from around piers, especially during flood events and heavy rainfalls, which can compromise the stability of the bridge and consequently its failure, if not properly detected and mitigated. Bridge failures pose serious threats to the local socio-economic and public safety and can cost lives. As such, real-time monitoring and early warning systems for scour-induced bridge failure can serve as a vital tool to protect the community and civil infrastructure against disastrous events. Vibration-base monitoring of bridge scour is an attractive option due to its low cost and relatively easy installation without the need to block the road and close the bridge to traffic. While limited number of previous studies have shown the capabilities of acceleration-based monitoring techniques in this area of research, they generally lack a rigorous framework for data analysis within and relating those to evaluate scour. This paper is an attempt to provide such framework that would enable a fast and low-cost analysis of vibration data within a physical modelling study on a simplified bridge pier. To achieve this, three experiments were conducted on an ideal single-pier scaled model bridge in a hydraulic flume, where water flows at a velocity near the critical velocity for the sand bed which in turn generates a scour hole around the pier. Then, the vibration data recorded using two mounted wireless accelerometers, were used to conduct an operational modal analysis through which the natural frequencies are extracted. The extracted natural frequencies and measured scour depths are then used to provide a chart that relates these two parameters. The results of this study showed the promising capability of the vibration-based data analysis in finding this relationship, indicating an up to around 30-50% reduction in natural frequencies as a result of around 50% scour ratio (ratio of maximum scour depth to buried depth), and beyond 50% scour ratio the natural frequencies remain constant. While the data presented in this paper are preliminary, they clearly show a promising potential for application in real-time monitoring of bridge stability under the effect of local scour and further works are underway to enrich the experimental data and empower the proposed methodologies at the laboratory scale.
Assessment of Mechanical Properties of Corroded Prestressing Strands
Chi-Ho Jeon, Cuong Duy Nguyen, Chang-Su Shim
Subject: Engineering, Civil Engineering Keywords: corrosion; prestressed concrete bridge; prestressing steel; section loss; strength; ductility
The corrosion of prestressing steel in prestressed concrete bridges is a critical issue for bridge maintenance. To assess structures with corroded strands, it is necessary to define the mechanical properties of the strands and their influence on the structural behavior. In this study, corroded strands are taken from external tendons in existing bridges and tested to define the effects of corrosion on the tensile properties of the strand. Empirical equations for the tensile strength and ductility of the corroded strand are proposed using test results. The most corroded wire governs the mechanical properties of the strand. Experiments on prestressed concrete beams with a single corroded strand are conducted to investigate the structural behavior. A reduction in the flexural strength and maximum deformation is observed from the experiment. According to the section loss of a wire in a strand and its location in a beam, the flexural capacity can be evaluated using the proposed equation. The reduced ultimate strain of the corroded strand can be the governing factor of the flexural strength.
Can Polyolefin Fibre Reinforced Concrete Improve the Sustainability of a Flyover Bridge?
Alejandro Enfedaque, Marcos G. Alberti, Jaime C. Galvez, Marino Rivero, Jose Simon-Talero
Subject: Engineering, Civil Engineering Keywords: concrete sustainable evaluations; flyover bridge; reinforced concrete slab; polyolefin fibres.
The use of polyolefin fibre reinforced concrete (PFRC) as an alternative for reducing or even eliminating the reinforcing steel bars employed in reinforced concrete has become real in the past years. This contribution analyses the improvements in sustainability that a change in the aforementioned reinforcement configuration might provide in a flyover bridge. Economic, environmental and social parameters of both possibilities were studied by means of the integrated value model for sustainable assessment use (Modelo Integrado de Valor para una Evaluación Sostenible, MIVES) used in Spain, which is a multi-criteria decision-making method based on the value function concept and the seminars delivered by experts. The results of the MIVES method showed that the use of PFRC in combination with reinforced concrete (RC) has a sustainability index 22% higher. An analysis of the parameters that form this evaluation shows that there are no remarkable differences in the financial costs between the two possibilities studied. Nevertheless, social and environmental aspects provide with a better qualification the option of building a bridge by using PFRC combined with RC.
Application of Analytical Hierarchy Process for Structural Health Monitoring and Prioritizing of Concrete Bridges in Iran
Saeid Darban, Hosein Ghasemzadeh Tehrani, Nader Karballaeezadeh, Amir Mosavi
Subject: Engineering, Civil Engineering Keywords: transportation infrastructure; concrete bridges; structural health monitoring; bridge condition index; analytical hierarchy process; prioritizing
This paper proposes a method for monitoring the structural health of concrete bridges in Iran. In this method, the bridge condition index (BCI) of bridges is determined by the analytical hierarchy process. BCI constitutes eight indices that are scored based on the experts' views, including structural, hydrology and climate, safety, load impact, geotechnical and seismicity, strategic importance, facilities, and traffic and pavement. Experts' views were analyzed by Expert Choice software, and the relative importance (weight) of indices were determined using the analytical hierarchy process (AHP). Then, the gave scores of experts were assigned to indices for various conditions. Bridge inspectors can examine the bridge, determine the scores of indices, and compute BCI. Higher values of BCI indicate better conditions. Therefore, bridges with lower BCI take priority in maintenance activities. Five bridges in Iran, Semnan province, were selected as the case studies, and BCI calculation of these bridges was conducted.
Power Balancing Control for Grid Energy Storage System in PV Applications—Real Time Digital Simulation Implementation
Sridhar Vavilapalli, Sanjeevikumar Padmanaban, Umashankar Subramaniam, Lucian MIHET-POPA
Subject: Engineering, Electrical & Electronic Engineering Keywords: active power control; battery charging; dual active bridge; energy storage system; hardware-in-loop
Grid energy storage system for PV Applications is connected with three different power sources i.e. PV Array, Battery and the Grid. It is advisable to have Isolation between these three different sources to provide safety for the equipment. The configuration proposed in this paper provides the complete isolation between the three sources. A Power Balancing Control (PBC) for this configuration is proposed to operate the system in three different modes of operation. Control of a dual active bridge (DAB) based battery charger which provides a galvanic isolation between batteries and other sources is explained briefly. Various modes of operation of a Grid energy storage system are also presented in this paper. Hardware-In-Loop (HIL) Simulation is carried out to check the performance of the system and the PBC algorithm. Power circuit (comprises of inverter, dual active bridge based battery charger, grid, PV cell, batteries, contactors and switches) is simulated and the controller hardware and user interface panel are connected as HIL with the simulated power circuit through Real Time Digital Simulator (RTDS). HIL simulation results are presented to explain the control operation, steady state performance in different modes of operation and the dynamic response of the system.
Applicability of Large-Span Structures for Presentations of Archaeological Sites
Natasa C Zivaljevic-Luxor, Hartmut Pasternak
Subject: Engineering, Automotive Engineering Keywords: archaeological site, roof, shelter, bridge, sustainability, aesthetics, heritage presentation, decision making
The purpose of the study was to provide support in the decision-making process for architects and engineers regarding large-spans structures for the presentation of archaeological sites in situ - construction of roofs, shelters, and bridges. We examined existing practice and analyzed their engineering classifications looking for a pattern in their application regarding sustainability and relation between type of cultural heritage site and type of applied large span structure. Contemporary engineering structures at built heritage sites create a sharp contrast between old and new. A presentation of cultural heritage in situ requires an understanding of heritage theory and internationally accepted doctrine which exceeds common engineering education. Nevertheless, application of large-span structures, which often take advantages of state-of-art construction-technologies nowadays, is also an aesthetical statement that affects the appearance of the site. Therefore, we gave an overview of the theoretical background of aesthetical issues and the overall ethics of the decision-making process in such sensitive cases. Within the framework of heritage-presentation, engineering and architectural issues, and selected case studies, we concluded in favor of the application of large-span structures under certain conditions.
Effect of Fiber Reinforced Polymer Tubes Filled with Recycled Materials and Concrete on Structural Capacity of Pile Foundations
Visar Farhangi, Moses Karakouzian
Subject: Engineering, Civil Engineering Keywords: Pile design; Fiber Reinforced Polymer; GFRP; FRP; Composite Piles; Bridge design
This paper deals with analyzing the structural responses of glass-fiber-reinforced polymer (GFRP) tubes filled with recycled and concrete material for developing composite piles, as an alternative to traditional steel reinforced piles in bridge foundations. The Full-scale GFRP composite piles included three inner and outer layers, using a fiber-oriented material that was inclined longitudinally, almost 40 degrees from the horizontal axis of the pile. The segment between these two layers was inclined 80 degrees from the longitudinal axis of the tube. The behavior of the filled GFRP tubes was semi-linear, and resulted in increasing the total ductility and strength of the piles. Adjusting the material's properties, such as the EAxial, EHoop, and Poisson ratios optimized the results. The lateral strength of the GFRP composite pile and pre-stressed piles are comparable in both axial and lateral loading conditions.
Form-Finding Analysis on the Rail Cable Shifting System of the Long Span Suspension Bridges
Pan Quan, Yan Donghuang, Yi Zhuangpeng
Subject: Engineering, Civil Engineering Keywords: Suspension bridge; Girder construction; RCS process; Form-finding analysis; Model test
The determination of the non-loading condition of the rail cable shifting (RCS) system, which consists of main cables, hangers and rail cables, is the premise of the girder erection for the long-span suspension bridges. An analytical form-finding analysis model of shifting system is established according to the basic assumptions of flexible cable structures. Herein, the rail cable is discretized into segmental linear cable elements and the main cable is discretized into segmental catenary elements. Moreover, the calculation and analysis equation of each member and their iterative solutions are derived by taking the elastic elongation of the sling into account. In addition, by taking the girder construction of Aizhai suspension bridge as engineering background, a global scale model of the RCS system is designed and manufactured; also the test system and working conditions are established. The comparison between the test results and analytical results shows the presented analytical method is correct and effective. The process is simplified in the analytical method, and the computational results and precision can satisfy the practical engineering requirements. In addition, the proposed method is suitable to apply to the computation analysis of similar structures.
Preprint HYPOTHESIS | doi:10.20944/preprints202012.0148.v1
Somatic Rhythmic Motion Effective on Peristaltic Circulation of Cerebrospinal Fluid: Hypothesis for Music- and Sport-Based Interventions
Huibing Tan, Torin Chiles, Yinhua Li, Tianyi Zhang, Hangqi Liu, Yunge Jia, Wei Hou, Xinghang Wang, Chenxu Rao, Zichun Wei, Ximeng Xu, Xiaoxin Wen, Siyu Tian
Subject: Keywords: Music-making; Cerebrospinal fluid; Myodural bridge; Somatic rhythmic motion; CSF-static compartment
Cerebrospinal fluid (CSF)-contacting neurons (CSF-N) located in the surface of both brain ventricles and the central canal (cc) in the spinal cord. The cc and CSF maintain a proliferative niche for neural progenitor cells and play a vital role in development of the brain. The CSF circulates in the ventricles and the subarachnoid spaces with the CSF rhythmic flow: cardiac pulsation and respiratory fluctuation. A new concept of CSF motion may be contrary to the classical one that the direction of CSF motion may vary in direction and may be dynamic in its location. The CSF pressure may also depend on the body position. Moderate music-making has been considered a potential approach for rehabilitative and restorative therapy of brain dysfunctions. Recently, we find that the CSF-Ns are present in both the interior CFS in the cc and also exterior CSF around the surface of the spinal cord. We hypothesize that CSF-N as mechanical sensors in the spinal cord could sense motion of the spinal cord. The myodural bridge is a ligament connecting a pair of deep, upper-neck muscles to the dura mater, which envelops the arachnoid mater and contains the CSF surrounding the brain and the spinal cord. We figure out the term "CSF-static compartment" and classify CSF storage location as rostral pool and caudal pool to demonstrate our hypothesis. We presume that the somatic body movement with music-making and rehabilitation-based interventions would orchestrate the CSF motion with head movement, myodural bridge stretching and puling as well as spinal bending.
New Design of Phase-Shifted Full-Bridge Power Converter for Photovoltaic Application
Wasan Phetphimoon, Krischonme Bhumkittipich
Subject: Engineering, Electrical & Electronic Engineering Keywords: phase-shifted full-bridge; resonant converter; photovoltaics; zero voltage switching; power loss
This paper presents the design of a high frequency zero voltage switching (ZVS) full-bridge converter with a phase-shifted driving signal for photovoltaic applications. The resonant power converter can provide high-power capacity under high-frequency operation. The proposed power converter can also reduce the size of the transformer under the same power rating. The high-frequency transformer was developed by using the resonant and switching frequencies of the power converter to reduce the switching loss and to improve the system efficiency. Phase-shifted modulation was selected to drive the switches of a full-bridge power converter based on the switching loss minimization method. The desired output voltage was controlled using a closed-loop controller under a loop gain stability margin. The simulation results showed that the output voltage can be controlled to the desired constant when the input voltage changes from 30 VDC to 60 VDC. The desired output voltage of power converter is constant at 400 VDC. The power converter can transfer the DC supply to a 220 VAC household via grid-connected inverter. Therefore, the proposed study showed the effectiveness of the phase-shift ZVS full-bridge power converter with high-frequency transformer. This power converter can control the operation of the desired voltage system and has a small sizing of power converter system, low switching loss, and high system efficiency.
Comparative Analysis of the Stability of Prosthetic Screws under Cyclic Loading in Implant Prosthodontics: An In Vitro Study
Santo Catapano, Mattia Ferrari, Nicola Mobilio, Marco Montanari, Massimo Corsalini, Francesco Grande
Subject: Medicine & Pharmacology, Allergology Keywords: preload loss; conical abutment screw; Multi-Unit-Abutment; OT-Bridge; prosthetic connection; implant-supported prosthesis; loosening torque; tightening torque
Background: To compare the loss of preload in absence of loading and after a fixed number of ideal masticatory cycles in two different connection systems using all-on-four prosthetic model. Methods: Two equal models of an edentulous mandible rehabilitated with all-on-four technique with two types of abutment system (MUA and OT-Bridge) supporting a hybrid prosthesis, were used. Initial torque values of the prosthetic fixing screw, after ten minutes from initial screw tightening and after 400000 masticatory cycles were registered using a mechanical torque gauge. Differences between initial and final torque values were reported for each anchoring system and the two systems were finally compared. Results: No statistically significant differences regarding the loss of preload between MUA and OT-Bridge system were found after 400000 masticatory cycles; however, in MUA system it was found between anterior and posterior implant screws. A significant difference in preload loss was found only for MUA system comparing the initial screw torque to that measured after 10 minutes from the tightening in absence of cyclic loadings. Conclusions: MUA and OT-Bridge are reliable prosthetic anchoring systems able to tolerate repeated masticatory cycles also on distal cantilever in all-on-four rehabilitation model without any significant loss of preload in screw tightening
Proposed O-GalNAc/Gal Gycosylation Pathways in Blood Group O and Non-O Blood Group Phenotypes During Plasmodium falciparum Infections Driving Evolution
Peter Arend
Subject: Life Sciences, Biochemistry Keywords: trans-species O-glycosylation; trans-species functional bridge; phenotype-specific plasma glycosylation; glycosidic exclusion; ontogenetic Tn formation
The coevolution of species drives diversity in animals and plants and contributes to natural selection, whereas in host–parasite coevolution, a parasite may complete an incomplete evolutionary/developmental function by utilizing the host cell's machinery. Analysis of related older data suggests that Plasmodium falciparum (P. falciparum), the pathogen of malaria tropica, cannot survive outside its human host because it is unable to perform the evolutionarily first protein glycosylation of serologically A-like, O-GalNAcα1-Ser/Thr-R, Tn antigen ("T nouvelle") formation, owing to its inability for synthesizing the amino sugar N-acetyl-d-galactosamine (GalNAc). This parasite breaks the species barrier via hijacking the host's physiological A-like/Tn formation through abundantly expressing serine residues and creating hybrid A-like/Tn structures, which in the human blood group O(H) are attacked by the germline-encoded nonimmune polyreactive immunoglobulin M (IgM), exerting the highly anti-A/B/H-aggressive isoagglutinin activities. These activities physiologically undergo the ABO(H) blood group phenotype formation, occurring on the surfaces of red blood cells (RBC), epithelial and endothelial cells and on plasma proteins by identical glycosylation, performed by the ABO(H)-allelic glycotransferases, phenotypically downregulating the anti-A/B/H-reactive IgM (isoagglutinin) activities in the non-O blood groups. ABO(H) phenotype diversity, this way glycosidically linked and molecularly connected to humoral immunity, becomes exposed to the evolution.
Reconfigurable, Multi-channel and Modular Bioimpedance Spectroscopy System on Field Programmable Gate Arrays
Anis Nurashikin Nordin, Ahmed Al-Hashimi, Amelia Wong Azman
Subject: Engineering, Electrical & Electronic Engineering Keywords: Bioimpedance Spectroscopy; Field Programmable Gate Array; Digital Auto Balance Bridge; Multichannel data acquisition;
This paper presents the design and implementation of a multichannel bio-impedance spectroscopy system on field programmable gate arrays (FPGA). The proposed system is capable of acquiring multiple signals from multiple bio-impedance sensors, process the data on the FPGA and store the final data in the on-board Memory. The system employs the Digital Automatic Balance Bridge (DABB) method to acquire data from biosensors. The DABB measures initial data of a known impedance to extrapolate the value of the impedance for the device under test. This method offers a simpler design because the balancing of the circuit is done digitally in the FPGA rather than using an external circuit. Calculations of the impedance values for the device under test were done in the processor. The final data is sent to an onboard Flash Memory to be stored for later access. The control unit handles the interfacing and the scheduling between these different modules (Processor, Flash Memory) as well as interfacing to multiple Balance Bridge and multiple biosensors. The system has been simulated successfully and has comparable performance to other FPGA based solutions. The system has a robust design that is capable of handling and interfacing input from multiple biosensors. Data processing and storage is also performed with minimal resources on the FPGA.
Static and Dynamic Testing of Sei Dareh Cable-Stayed Bridge West Sumatera, Indonesia
Sumargo Sumargo, Afdhal Lazuardiansyah Ramdhani
Subject: Engineering, Automotive Engineering Keywords: Structural health monitoring; bridge load testing; dynamic testing; operational modal analysis; experimental modal analysis
The Sei Dareh Bridge is a cable-stayed bridge located in West Sumatra Province, Indonesia. The bridge, has a main span of 123 meters length and 9 meters wide, crosses the Batanghari River. Traffic load is transmitted through 4 prestressed cables to a 42.4 meter high pylon made of concrete. Bridge deck and traffic loads are directly supported by steel box girders as main beams that are reinforced laterally with cross beams IWF 800.300.16.24 and stringers IWF 350.350.12.19. This paper discusses static and dynamic testing on the bridge which aims to assess the feasibility before it is opened for public. Based on the test, it was concluded that the 73% static load could not be achieved because the deflection that occurred was beyond the allowable deflection. This is exacerbated by the sound of a loud clanging sound on the ST2-X1 prestressed cable when loading to 240 tons or 58% of the targeted load. In addition, this bridge is included in the "lazy bridge" category because it takes 24 hours to return to an undeformed condition after loading. As a recommendation for this bridge, it is necessary to carry out a structural health monitoring system (SHMS) regularly on the vehicle floor and cables.
New Damage Evolution Law for Epoxy Asphalt Concrete in Long-Span Steel Bridge Considering Wheel Load and Temperature Variation
Xun Qian Xu, Xiao Yang, Wei Huang, Hong Liang Xiang, Wei Yang
Subject: Engineering, Civil Engineering Keywords: long-span steel bridges; steel bridge deck pavement (sbdp); epoxy asphalt (ea); fatigue damage evolution law; micro properties; fatigue test
Epoxy asphalt (EA) concrete is widely used in constructing long-span steel bridge pavements (SBDPs). This study aims to derive a fatigue damage evolution law, conducting an experimental investigation of SBDP. First, a general theoretical form of the fatigue damage evolution law of materials is established based on the thermal motion of atoms. Then, fatigue experiments demonstrate that this evolution law well represents the known damage–life relationships of SBDP. Taking into account the experimental relationships between damage and fatigue life under symmetrical cyclic loadings with different overload amplitudes and temperature variations, a detailed damage evolution law is deduced. Finally, the role of damage accumulation is discussed on the basis of the proposed damage evolution law for the extreme situation of heavy overload and severe environments. The results show that both heavy loading and falling temperatures increase the fatigue damage of SBDP considerably; therefore, SBDP should avoid heavy loading combined with winter temperatures. EA shows a fatigue life two to three times longer than that of modified matrix asphalt (SMA) or guss asphalt (GA). For the same thickness, EA pavement is demonstrated to be more suitable for an anti-fatigue design of large-span SBDP under high traffic flows and low temperatures.
Durability Assessment Method of Hollow Thin-Walled Bridge Piers under Rockfall Impact Based on Damage Response Surface
Fei Li, Yikang Liu, Jian Yang
Subject: Engineering, Civil Engineering Keywords: rockfall impact; impact resistance; hollow thin-walled bridge pier; response surface model; dura-bility assessment
Continuous rigid frame bridges across valleys are often at the risk of rockfalls caused by heavy rainfalls, earthquakes and debris flows in a mountainous country. Hollow thin-walled bridge piers (HTWBP) in valleys are exposed to the threat of the impact of accidental rockfalls. In the current research, ANSYS/LS-DYNA is used to establish a high-precision rockfall-HTWBP model. The rockfall-HTWBP model is verified against a scaled impact test of a previous research. A mesh independence test is also performed to obtained an appropriate mesh size. Based on the rockfall-HTWBP model, the impact force, damage and dynamic response characteristics of HTWBP under the rockfall impact are studied. In addition, a damage assessment criteria is proposed based on the response surface model combined with Central Composite Design method and Box-Behnken Design method. The main conclusions are as follows: 1)The impact force of rockfall has a substantial impulse characteristic, and the duration of the impulse load is approximately 0.01s. 2)The impacted surface of the pier is dominated by the final elliptic damage with the conical and strip damage areas as the symmetry axis. The cross-sectional damage mode is compression failure in the impact area and shear failure at the corner. 3)The maximum displacement occurs in the middle height of the pier. The maximum displacement increases with impact height, impact velocity and rockfall diameter and decreases with the uniaxial compressive strength of the concrete. 4) The initial impact velocity and diameter of the rockfall are the most significant parameters affecting the damage indices. In addition, a damage assessment method with a damage zoning diagram based on the response surface method is established for the fast assessment of the damage level of impacted HTWBP.
Gray Public Space Under Urban Bridge - A Case Study of Public Arts Space Micro-Transformation Focusing on "Regional Culture" and "Art for All"
Ting Liu Liu, Wu Cao, Yuyi Liu
Subject: Arts & Humanities, Theory Of Art Keywords: Urban grey space; Space under bridge; Public Art; Micro-transformation; Regional culture; Art for all
Since the 21st century, China's urbanization process has been rapid development, the concept and function of urban public space in the city has been gradually paid attention to. In order to guarantee life and water, most urban construction relies on rivers, and Bridges are the most important way to communicate between urban areas. The main functional part of the bridge is the span structure, that is, the bearing structure of the bridge, and the lower part of the "gray" space formed by the bridge structure. Considering the social level, with the economic growth and urbanization development, people have brought a better living environment and quality of life, and also improved the requirements for urban public environment. In the increasingly tense urban space, how to use and transform the space under the bridge is a problem that needs to be considered and solved. In view of this problem, in this study, we try to solve the micro-transformation of space under Bridges in cities through public art from the perspective of "regional culture" and "art for all". This paper analyzes the micro-transformation of space art under Bridges in two large cities of Shanghai and Foshan, namely, the space under Bridges under Songhong Road in Shanghai, the space under Bridges under Central Of Suzhou River and the space under Bridges under Pingsheng Bridge in Foshan. This paper discusses the cultural intervention of "regional culture" in the micro-transformation of the space under the bridge, and the influence and effect of "art for all" on the public art space under the bridge after the transformation to the community and the public.
Application of the Incremental Modal Analysis for Bridges (IMPAb) Subjected to Near-Fault Ground Motions
Alessandro Vittorio Bergami, Gabriele Fiorentino, Davide Lavorato, Bruno Briseghella, Camillo Nuti
Subject: Engineering, Civil Engineering Keywords: near field; pulse like ground motions; bridge, non-linear static analysis; non-linear dynamic analysis
Near-fault ground motions can cause severe damage to civil structures, including bridges. Safety assessment of these structures for near fault ground motion is usually performed through Non-Linear Dynamic Analyses, while faster methods are often used. IMPAb (Incremental Modal Pushover Analysis for Bridges) permits to investigate the seismic response of a bridge by considering the effects of higher modes, which are often relevant for bridges. In this work, IMPAb is applied to a bridge case study considering near-fault pulse-like ground motion records. The records were analyzed and selected from the European Strong Motion Database and the pulse parameters were evaluated. In the paper results from standard pushover procedures and IMPAb are compared with nonlinear Response-History Analysis (NRHA), considering also the vertical component of the motion, as benchmark solutions and incremental dynamic analysis (IDA). Results from the case study demonstrate that the vertical seismic action has a minor influence on the structural response of the bridge. Therefore IMPAb, which can be applied considering vertical motion, remains very effective conserving the original formulation of the procedure, and can be considered a well performing procedure also for near-fault events.
Design and Optimization of an Efficient (96.1%) and Compact (2 kW/dm3) Bidirectional Isolated Single-Phase Dual Active Bridge Ac–Dc Converter
Jordi Everts
Subject: Engineering, Electrical & Electronic Engineering Keywords: ac–dc power converters; battery chargers; dual active bridge; DAB; optimal design; power MOSFETs; single-stage
The growing attention for plug-in electric vehicles, and the associated high-performance demands, have initiated a development trend towards highly efficient and compact on-board battery chargers. These isolated ac-dc converters are most commonly realized using two conversion stages, combining a non-isolated power factor correction (PFC) rectifier with an isolated dc-dc converter. This, however, involves two loss stages and a relatively high component count, limiting the achievable efficiency and power density and resulting in high costs. In this paper a single-stage converter approach is analyzed to realize a single-phase ac-dc converter, combining all functionalities into one conversion stage and thus enabling a cost-effective efficiency and power density increase. The converter topology consists of a quasi-lossless synchronous rectifier followed by an isolated dual active bridge (DAB) dc-dc converter, putting a small filter capacitor in between. To show the performance potential of this bidirectional, isolated ac-dc converter, a comprehensive design procedure and multi-objective optimization with respect to efficiency and power density is presented, using detailed loss and volume models. The models and procedures are verified by a 3.7 kW hardware demonstrator, interfacing a 400 V dc-bus with the single-phase 230 V, 50 Hz utility grid. Measurement results indicate a state-of-the-art efficiency of 96.1% and power density of 2.2 kW/dm3, confirming the competitiveness of the investigated single-stage DAB ac-dc converter.
A Two-step FE Model Updating Approach for System and Damage Identification of Full-Scale Prestressed Bridge Girders
Niloofar Malekghaini, S. Farid Ghahari, Hamed Ebrahimian, Matthew Bowers, Eric Ahlberg, Ertugrul Taciroglu
Subject: Engineering, Civil Engineering Keywords: Modal-based model updating, Bayesian model updating, System identification, Damage identification, Operational health monitoring, I-girder, Bridge, Aging.
The average age of in-service bridges has increased in recent years in the United States. To address this issue, structural health monitoring and damage identification approaches can be employed to prioritize maintenance/replacement of aging bridges. Among the damage identification and operational health monitoring approaches, finite element (FE) model updating methods can offer a solution to evaluate the mechanics-based characteristics of bridges. However, in a real-world setting, unidentifiability and mutual dependency between model parameters, modeling errors, especially due to boundary conditions, as well as ill-conditioning of updating algorithms can pose challenges to the application of FE model updating methods. To address these challenges, this study presents a two-step FE model updating approach. In the first step, modal-based model updating is used to estimate linear model parameters mainly related to the stiffness of boundary conditions and material properties. In the second step, in order to refine parameter estimation accounting for nonlinear response behavior of the bridge, a time-domain model updating is carried out. In this step, boundary conditions are fixed at their final estimates using modal-based model updating. To prevent the convergence of updating algorithm to local solutions, the initial estimates for nonlinear material properties are selected based on their corresponding final estimates in the modal-based model updating. To validate the applicability of the two-step FE model updating approach, a series of forced-vibration experiments are designed and carried out on a pair of decommissioned and deteriorated prestressed bridge I-girders. After carrying out the two-step FE model updating, the final estimates of concrete compressive strength are shown to provide reasonable assessment of the damage extent in the girders.
Numerical Deformation Analysis of Reinforced Lightweight Aggregate Concrete Flexural Members
Darius Bacinskas, Deividas Rumsys, Gintaris Kaklauskas
Subject: Engineering, Civil Engineering Keywords: lightweight aggregate concrete; reinforced concrete; slab; bridge girder; curvature; short-term loading; tension stiffening; constitutive model; numerical modelling.
In the modern construction industry, lightweight aggregate concrete (LWAC) is often used in the production of load-bearing structural members. LWAC can be up to 40% lighter by volume in comparison to normal strength concrete. On the other hand, the lack of adequate numerical models often limits the practical application of innovative building materials, such as lightweight concrete, in real projects. This trend is due to the uncertainties in design standard methods and calculation errors, the level of which is generally unacceptable to civil engineers in terms of safety and reliability. In the present paper, a comparative numerical deformation analysis of a full-scale bridge deck slab and girder has been carried out. Using the physical model proposed by the authors and the finite element software ATENA, the deformations of full–scale lightweight and traditional reinforced concrete elements under short-term effects of permanent and variable loads was compared. Depending on the safety and serviceability limit requirements, it was found that the amount of longitudinal reinforcement in lightweight reinforced concrete elements can be reduced compared to normal reinforced concrete elements with the same parameters. The results of the numerical analysis show that the deformation analysis model proposed by the authors can be a reliable tool for the design of lightweight concrete flexural members by selecting the optimum geometrical and reinforcement parameters limited by the stiffness condition.
Thermal Bridge Modeling According to Time-varying Indoor Temperature for Dynamic Building Energy Simulation Using System Identification
Heegang Kim, Jihye Kim, Myoungsouk Yeo
Subject: Engineering, Other Keywords: thermal bridge; data-driven system modeling; system identification; time-varying indoor temperature; dynamic analysis; building energy simulation; building envelope
It is not easy to dynamically analyze thermal bridges that require multidimensional analysis in building energy simulations, which are mostly one-dimensional platforms. To solve this problem, many studies have been conducted and, recently, a study was conducted to model the thermal bridge based on the data by approaching this in a similar way to steady-state analysis, showing high accuracy. This was an early-stage study, which is only applicable when the indoor temperature is constant. By extending this study, a thermal bridge model that can be applied even when the indoor temperature changes over time is proposed and validated. Since the governing equation, the heat diffusion equation, is linear, the key idea is to create and apply two thermal bridge transfer function models by expressing the heat flow entering the room as a linear combination of the transfer function for indoor temperature and the transfer function for outdoor temperature. For the proposed thermal bridge model, the NRMSE of the model itself showed a high accuracy of 99.9%, and in the verification through annual simulation using the model, the NRMSE showed an accuracy of 88.8%.
Why Blood Group A Individuals Are at Risk Whereas Blood Group O Individuals Might Be Protected from SARS-CoV-2 (COVID-19) Infection: A Hypothesis Regarding How the Virus Invades the Human Body via Abo(H) Blood Group-Determining Carbohydrates
Subject: Keywords: COVID-19; SARS-CoV-2–human carbohydrate interaction; trans-species glycosylation; A-like/Tn structure; trans-species glycan bridge
While the angiotensin converting enzyme 2 (ACE2) protein is defined as the primary severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) receptor, the viral serine molecule might be mobilized by the host's transmembrane protease serine subtype 2 (TMPRSS2) enzyme from the viral spike (S) protein and hijack the host's N-acetyl-D-galactosamine (GalNAc) metabolism. The resulting hybrid, serologically A-like/Tn (T-nouvelle) structure potentially acts as a host–pathogen functional molecular bridge. In humans, this intermediate structure will hypothetically be replaced by ABO(H) blood group-specific, mucin-type structures, in the case of infection hybrid epitopes, implicating the phenotypically glycosidic accommodation of plasma proteins. The virus may, by mimicking the synthetic pathways of the ABO(H) blood groups, bind to the cell surfaces of the blood group O(H) by formation of a hybrid H-type antigen as the potential precursor of hybrid non-O blood groups, which does not affect the highly anti-glycan aggressive anti-A and anti-B isoagglutinin activities, exerted by the germline-encoded nonimmune immunoglobulin M (IgM). In the non-O blood groups, which have developed from the H-type antigen, these IgM activities are downregulated by phenotypic glycosylation, while adaptive immunoglobulins might arise in response to the hybrid A and B blood group structures, bonds between autologous carbohydrates and foreign peptides, suggesting the exertion of autoreactivity. The non-O blood groups thus become a preferred target for the virus, whereas blood group O(H) individuals, lacking the A/B phenotype-determining enzymes and binding the virus alone by hybrid H-type antigen formation, have the least molecular contact with the virus and maintain the critical anti-A and anti-B isoagglutinin activities, exerted by the ancestral IgM, which is considered the humoral spearhead of innate immunity.
Fatigue Assessment of Prestressed Concrete Slab-between-Girder Bridges
Eva O.L. Lantsoght, Rutger Koekkoek, Cor van der Veen, Henk Sliedrecht
Subject: Engineering, Civil Engineering Keywords: assessment; bridge evaluation; compressive membrane action; concrete bridges; fatigue; fatigue assessment; live loads; prestressed concrete; punching shear; scale model
In the Netherlands, the assessment of existing prestressed concrete slab-between-girder bridges showed that the thin, transversely prestressed slabs may be critical for static and fatigue punching when evaluated using the recently introduced Eurocodes. On the other hand, compressive membrane action increases the capacity of these slabs and changes the failure mode from bending to punching shear. To improve the assessment of the existing prestressed slab-between-girder bridges in the Netherlands, two 1:2 scale models of an existing bridge, the Van Brienenoord Bridge, were built in the laboratory and tested monotonically as well as under cycles of loading. The result of these experiments is: 1) the static strength of the decks, showing that compressive membrane action significantly enhances the punching capacity, and 2) the Wöhler curve of the decks, showing that compressive membrane action remains under fatigue loading. The experimental results can then be used for the assessment of the most critical existing slab-between-girder bridge. The outcome is that the bridge has sufficient punching capacity for static and fatigue loads, and thus that the existing slab-between-girder bridges in the Netherlands fulfil the code requirements for static and fatigue punching.
Blackcurrant Leaf Chlorosis Associated Virus: Next-Generation Sequencing Reveals an Extraordinary Virus with Multiple Genomic Components Including Evidence of Circular RNA
Delano James, James Phelan, Daniel Sanderson
Subject: Biology, Plant Sciences Keywords: Idaeovirus; Blackcurrant leaf chlorosis associated virus; next-generation sequencing (NGS); bridge reads; abutting primers; RNase R digestion; circular RNA; concatenated RNA
Blackcurrant leaf chlorosis associated virus (BCLCaV) was detected recently by next-generation sequencing (NGS) and proposed as a new and distinct species in the genus Idaeovirus. Genomic components of BCLCaV that were detected and confirmed include: 1) RNA-1 that is monocistronic and encodes the replicase complex; 2) a bicistronic RNA-2 that encodes a movement protein (MP) and the coat protein (CP) of the virus, with open reading frames (ORF) that overlap by a single adenine (A) nucleotide (nt) representing the third position of an opal stop codon of the MP ORF2a and the first position of the start codon of the CP ORF2b; 3) a subgenomic form of RNA-2 (RNA-3) that contains ORF2b; and 4) a concatenated form of RNA-2 that consists of a complementary and inverted RNA-3 conjoined to the full-length RNA-2. Analysis of NGS-derived paired-end reads revealed the existence of bridge reads encompassing the 3'-terminus and 5'-terminus of RNA-2 or RNA-3 of BCLCaV. The full RNA-2 or RNA-3 could be amplified using outward facing or abutting primers; also RNA-2/RNA-3 could be detected even after three consecutive RNase R enzyme treatments with denaturation at 95 oC preceding each digestion. Evidence was obtained indicating that there are circular forms of BCLCaV RNA-2 and RNA-3.
Design and Shape Optimization of Strain Gauge Load Cell for Axial Force Measurement for Test Benches
Omar Sabah Al-Dahiree, Mohammad Osman Tokhi, Nabil Hassan Hadi, Nassar Rasheid Hmoad, Raja Ariffin Raja Ghazilla, Hwa Jen Yap, Emad Abdullah Albaadani
Subject: Engineering, Mechanical Engineering Keywords: Strain gauge load cell; machine design; axial force measurement; force test bench; shape optimiza-tion; finite element method (FEM); Wheatstone bridge; amplifier circuit; calibration test
The load cell is an indispensable component of many engineering machinery and industrial automation for measuring and sensing force and torque. This paper describes the design and analysis of the strain gauge load cell, from the conceptional design stage to shape optimization (based on the finite element method (FEM) technique) and calibration, providing ample load capacity with low-cost material (aluminum 6061) and highly accurate force measurement. The amplifier circuit of the half Wheatstone bridge configuration with two strain gauges has been implemented experimentally with an actual load cell prototype. The calibration test was conducted to evaluate the load cell characteristics and derive the governing equation for sensing the unknown load depending on the measured output voltage. The measured sensitivity of the load cell is approximately 15 mV/N and 446.8 µV/V at a maximum applied load of 30 kg. The findings are supported by FEM results and experiments with an acceptable percentage of errors, which revealed an overall error of 6% in the worst situation. Therefore, the proposed load cell meets the design considerations for axial force measurement for the laboratory test bench, which has a light weight of 20 g and a maximum axial force capacity of 300 N with good sensor characteristics. | CommonCrawl |
Compute $\dbinom{15}{2}$.
\begin{align*}
\dbinom{15}{2} &= \dfrac{15!}{13!2!} \\
&= \dfrac{15\times 14}{2\times 1} \\
&= 15 \times \dfrac{14}{2} \\
&= 15 \times 7 \\
&= \boxed{105}.
\end{align*} | Math Dataset |
\begin{document}
\maketitle
\begin{abstract}
In this paper we obtain new characterizations of the $q$-uniformly convex and smooth Banach spaces by using Carleson measures.
These measures are defined by Poisson integral associated with Bessel operators and Banach valued $BMO$-functions. By the way we describe
$q$-uniformly convexity and smoothness of a Banach space in terms of the mapping properties of the Lusin integral defined by the Poisson
semigroup for Bessel operators.
\end{abstract}
\section{Introduction} \label{sec:intro}
It is well-known that vector valued harmonic analysis and geometry of Banach spaces are closely connected. Some geometric properties of a Banach
space $\mathbb{B}$ are characterized by the boundedness in $\mathbb{B}$-valued $L^p$ or $BMO$ spaces of some harmonic analysis operators
(Riesz transforms, imaginary powers, Littlewood-Paley $g$-functions, \dots). These properties have also a description by using
martingales transforms. The celebrated papers of Bourgain \cite{Bou} and Burkholder \cite{Bu} concerning to $UMD$ (\emph{Unconditional Martingale Difference}) spaces contain the first main
results of this theory. In the last years this area has a great activity. In \cite{Xu} Xu studied the one side Littlewood-Paley theory for
Banach valued functions and he obtained new characterizations for the uniformly convex and smooth Banach spaces. The results in \cite{Xu}
were generalized by Martínez, Torrea and Xu \cite{MTX} to the diffusion semigroup setting. Harmonic analysis operators associated with
Bessel, Hermite, Laguerre and Ornstein-Uhlenbeck operators allow also to characterize $UMD$, convexity and smoothness properties of
Banach spaces (see \cite{AMST}, \cite{AST}, \cite{BFMT}, \cite{BFR}, \cite{BFRST1}, \cite{HTV}, amongst others).\\
Recently, Ouyang and Xu \cite{OX} have studied the relationship between vector valued $BMO$ functions and the Carleson measures defined by their
Poisson integrals. They obtained new characterizations for those Banach spaces admitting an equivalent norm which is $q$-uniformly convex or
$q$-uniformly smooth.\\
In this paper we use the Poisson integrals associated with Bessel operators to define Carleson measures that allow us to characterize
(modulus renorming) $q$-uniformly convex and smooth Banach space. We consider the Banach valued odd $BMO$ functions on $\mathbb{R}$.
In \cite{BCFR2} the scalar space of odd $BMO$ functions was described by using Carleson measures.\\
Assume that $\mathbb{B}$ is a Banach space. We say that a locally integrable function $f: \mathbb{R} \longrightarrow \mathbb{B}$ has bounded mean oscillation,
written $f \in BMO(\mathbb{R},\mathbb{B})$, when
$$\|f\|_{BMO(\mathbb{R},\mathbb{B})} = \sup_{I \subset \mathbb{R}} \frac{1}{|I|} \int_I \|f(x)-f_I\|_\mathbb{B} dx,$$
where the supremum is taken over all bounded intervals $I$ in $\mathbb{R}$. Here $f_I = \frac{1}{|I|} \int_I f(x)dx$, where the integral is understood in the Bochner sense, and $|I|$ denotes the length of $I$, for every bounded interval $I$ in $\mathbb{R}$. By $BMO_o(\mathbb{R},\mathbb{B})$ we represent the space of the odd functions in $BMO(\mathbb{R},\mathbb{B})$. According to the well-known John-Nirenberg property we can see that a $\mathbb{B}$-valued locally integrable and odd function $f$ on $\mathbb{R}$ is in $BMO_o(\mathbb{R},\mathbb{B})$ if, and only if, for some (equivalently, for any)
$1 \leq p < \infty$, there exists $C>0$ such that
\begin{equation} \label{1.1}
\left( \frac{1}{|I|} \int_I \|f(x)-f_I\|_\mathbb{B}^p dx \right)^{1/p}
\leq C,
\end{equation}
for every interval $I=(a,b)$, $0<a<b<\infty$, and
\begin{equation}\label{1.2}
\left( \frac{1}{|I|} \int_I \|f(x)\|_\mathbb{B}^p dx \right)^{1/p}
\leq C,
\end{equation}
for each interval $I=(0,b)$, $0<b<\infty$. Moreover, for every $f \in BMO_o(\mathbb{R},\mathbb{B})$ and $1\leq p < \infty$,
$\|f\|_{BMO(\mathbb{R},\mathbb{B})} \simeq \inf \{ C>0, \text{\eqref{1.1} and \eqref{1.2} hold}\}$.
As in the classical case if $f \in BMO(\mathbb{R},\mathbb{B})$, then $\int_0^\infty \|f(x)\|_\mathbb{B} / (1+x^2)dx<\infty$.\\
In \cite{MS} Muckenhoupt and Stein developed the harmonic analysis theory in the ultraspherical and Bessel settings. Taking as a starting point
the ideas in \cite{MS} in the last years several authors have investigated boundedness properties of harmonic analysis operators associated
with Bessel operators (\cite{BBFMT}, \cite{BCC1}, \cite{BCN}, \cite{BFS}, \cite{BS}, \cite{NS}).\\
We consider, for every $\lambda>0$, the Bessel operator $\Delta_\lambda = -x^{-\lambda} \frac{d}{dx}x^{2\lambda}\frac{d}{dx}x^{-\lambda}$,
$x \in (0,\infty)$, and the Hankel transformation $h_\lambda$ defined by
$$h_\lambda(f)(x)=\int_0^\infty \sqrt{xy} J_{\lambda-1/2}(xy) f(y) dy, \quad x \in (0,\infty),$$
for every $f \in L^1(0,\infty) \cap L^2(0,\infty)$. Here, $J_\nu$ denotes the Bessel function of the first kind and order $\nu$. $h_\lambda$ can be extended to $L^2(0,\infty)$ as an isometry of $L^2(0,\infty)$ where $h_\lambda^{-1}=h_\lambda$. If $f \in C_c^\infty(0,\infty)$, the space of
smooth functions with compact support, we have that
$$h_\lambda(\Delta_\lambda f)(y)=y^2 h_\lambda(f)(y), \quad y \in (0,\infty).$$
We define the operator $\tilde{\Delta}_\lambda$ as follows,
$$\tilde{\Delta}_\lambda f = h_\lambda(y^2 h_\lambda (f)), \quad f \in D(\tilde{\Delta}_\lambda),$$
where the domain $D(\tilde{\Delta}_\lambda)$ of $\tilde{\Delta}_\lambda$ is
$$D(\tilde{\Delta}_\lambda) = \{f \in L^2(0,\infty) : y^2 h_\lambda (f) \in L^2(0,\infty)\}.$$
$\tilde{\Delta}_\lambda$ is a closed and positive operator. Note that $\tilde{\Delta}_\lambda f = \Delta_\lambda f$, $f \in C_c^\infty(0,\infty)$.
In the sequel we refer to $\tilde{\Delta}_\lambda$ also by $\Delta_\lambda$.\\
By $\{P_t^\lambda\}_{t>0}$ we represent the Poisson semigroup associated with $\Delta_\lambda$, or, in other words, the semigroup of operators generated by
$-\sqrt{\Delta_\lambda}$. According to \cite[(16.4)]{MS} we can write, for every $f \in L^p(0,\infty)$, $1 \leq p \leq \infty$,
$$P_t^\lambda(f)(x)=\int_0^\infty P_t^\lambda(x,y)f(y)dy, \quad x,t \in (0,\infty),$$
where
\begin{equation}\label{Bessel_kernel}
P_t^\lambda(x,y)=\frac{2\lambda (xy)^\lambda t}{\pi} \int_0^\pi \frac{(\sin \theta)^{2\lambda-1}}{[(x-y)^2+t^2+2xy(1-\cos \theta)]^{\lambda+1}} d\theta, \quad x,y,t \in (0,\infty).
\end{equation}
$\{P_t^\lambda\}_{t>0}$ is a contractive semigroup in $L^p(0,\infty)$, $1 \leq p \leq \infty$. Since the kernel function $P_t^\lambda(x,y)\geq 0$,
$x,y,t \in (0,\infty)$, the operator $P_t^\lambda$ is also a contraction in the Lebesgue-Bochner space $L^p((0,\infty),\mathbb{B})$, for every $1 \leq p \leq \infty$
and $t>0$.\\
We say that a positive measure $\mu$ on $(0,\infty) \times (0,\infty)$ is Carleson when there exists $C>0$ satisfying
$$\frac{\mu(I \times (0,|I|))}{|I|} \leq C,$$
for every bounded interval $I$ in $(0,\infty)$. It is well-known that the functions of bounded mean oscillation on $\mathbb{R}^n$ can be characterized by using
Carleson measures. In \cite{BCFR2} the following result was established.
\begin{Th2}[{\cite[Theorem 1.1]{BCFR2}}]
Let $\lambda>0$. Assume that $f$ is a locally integrable function in $[0,\infty)$. If we define $f_o$ as the odd extension of $f$ to $\mathbb{R}$, then
$f_o \in BMO_o(\mathbb{R})$ if and only if $(1+x^2)^{-1}f \in L^1(0,\infty)$ and the measure $\gamma_f$ given by
$$d\gamma_f(x,t) = \left| t \partial_t P_t^\lambda(f)(x) \right|^2 \frac{dxdt}{t}$$
is Carleson on $(0,\infty) \times (0,\infty)$.
\end{Th2}
We now recall the definitions of convexity and smoothness for a Banach space $\mathbb{B}$. The modulus of convexity $\delta_\mathbb{B}$ and of smoothness $\rho_\mathbb{B}$ are defined by
$$\delta_\mathbb{B}(\varepsilon)=\inf\{1-\| \frac{a+b}{2} \|_\mathbb{B} : a,b \in \mathbb{B}, \|a\|_\mathbb{B}=\|b\|_\mathbb{B}=1, \|a-b\|_\mathbb{B}=\varepsilon \}, \quad 0< \varepsilon < 2,$$
and
$$\rho_\mathbb{B}(t)=\sup\{ \frac{\|a+tb\|_\mathbb{B} + \|a-tb\|_\mathbb{B}}{2} : a,b \in \mathbb{B}, \|a\|_\mathbb{B}=\|b\|_\mathbb{B}=1 \}, \quad t>0.$$
We say that $\mathbb{B}$ is uniformly convex (respectively, uniformly smooth) when $\delta_\mathbb{B}(\varepsilon)>0$ (respectively, $\lim_{t \to 0} \rho_\mathbb{B}(t)/t=0$).
Also, $\mathbb{B}$ is called $q$-uniformly convex, $q \geq 2$ (respectively, $q$-uniformly smooth, $1<q\leq 2$) when there exist $C>0$ such that $\delta_\mathbb{B}(\varepsilon) \geq C \varepsilon^q$,
$0<\varepsilon<2$ (respectively, $\rho_\mathbb{B}(t) \leq C t^q$, $t>0$).\\
Pisier \cite{Pi} proved that $\mathbb{B}$ has an equivalent norm that is $q$-uniformly convex (respectively, $q$-uniformly smooth) if and only if $\mathbb{B}$ has martingale
cotype $q$ (respectively, martingale type $q$). Xu \cite{Xu} established the corresponding characterization when the martingale type and cotype is replaced by the
Lusin type and cotype associated with the Poisson semigroup for the torus. The result of Xu was extended to the diffusion semigroup setting in \cite{MTX}.
Similar properties have been obtained in the Bessel (\cite{BFMT}) and Laguerre (\cite{BFRST1}) contexts.
Recently, Ouyang and Xu \cite{OX} characterized those Banach spaces having an equivalent norm that is $q$-uniformly convex or smooth by using Carleson measures and $\mathbb{B}$-valued $BMO$ functions, and lately Jiao \cite{Ji} gave the martingale version of this result. \\
In this paper we obtain new characterizations for $q$-uniformly convexity and smoothness of a Banach space by using Carleson measures associated with the Bessel
Poisson integrals $P_t^\lambda(f)$ of $f$ belonging to $BMO_o(\mathbb{R},\mathbb{B})$.\\
The main results of this paper are the following ones.
\begin{Th}\label{Th_3.1_OX}
Let $\mathbb{B}$ be a Banach space, $\lambda>1$ and $2 \leq q < \infty$. Then, the following statements are equivalent.
\begin{itemize}
\item[$(i)$] There exists $C>0$ such that, for every $f \in BMO_o(\mathbb{R},\mathbb{B})$, the measure $d\mu_f$ defined by
$$d\mu_f(x,t)=\|t \partial_t P_t^\lambda (f)(x)\|_\mathbb{B}^q \frac{dxdt}{t}$$
is Carleson on $(0,\infty)\times (0,\infty)$ and
$$ \sup_{I} \frac{1}{|I|} \int_0^{|I|} \int_I \|t \partial_t P_t^\lambda(f)(x) \|^q_\mathbb{B} \frac{dx dt}{t}
\leq C \|f\|^q_{BMO_o(\mathbb{R},\mathbb{B})},$$
where the supremum is taken over all bounded intervals $I$ in $(0,\infty)$. \\ \quad
\item[$(ii)$] $\mathbb{B}$ has an equivalent norm which is $q$-uniformly convex.
\end{itemize}
\end{Th}
\begin{Th}\label{Th_4.1_OX}
Let $\mathbb{B}$ be a Banach space, $\lambda>1$ and $1 < q \leq 2$. Then, the following assertions are equivalent.
\begin{itemize}
\item[$(i)$] There exists $C>0$ such that, for every odd
$\mathbb{B}$-valued function $f$, satisfying that $(1+x^2)^{-1}f \in L^1(\mathbb{R},\mathbb{B})$,
$$ \|f\|^q_{BMO_o(\mathbb{R},\mathbb{B})}
\leq C \sup_{I} \frac{1}{|I|}\int_0^{|I|} \int_I \|t \partial_t P_t^\lambda(f)(x) \|^q_\mathbb{B} \frac{dx dt}{t},$$
where the supremum is taken over all bounded intervals $I$ in $(0,\infty)$. \\ \quad
\item[$(ii)$] $\mathbb{B}$ has an equivalent $q$-uniformly smooth norm.
\end{itemize}
\end{Th}
Theorems~\ref{Th_3.1_OX} and \ref{Th_4.1_OX} can be seen as versions of \cite[Theorems 3.1 and 4.1]{OX}, respectively.\\
In order to prove our theorems we need to obtain characterizations of the convexity and smoothness for a Banach space by using certain area
integrals involving the Poisson semigroup $\{P_t^\lambda\}_{t>0}$.\\
We define the following sets
$$\Gamma(x)=\{(y,t) \in \mathbb{R}\times (0,\infty) : |y-x|<t\}, \quad x \in \mathbb{R},$$
and
$$\Gamma_+(x)=\{(y,t) \in (0,\infty)\times (0,\infty) : |y-x|<t\}, \quad x \in (0,\infty).$$
We extend the definition of the Poisson kernel $P_t^\lambda(x,y)$ given in \eqref{Bessel_kernel} to $\mathbb{R} \times \mathbb{R}$,
for every $t>0$, as follows
\begin{equation*}
P_t^\lambda(x,y)=\frac{2\lambda |xy|^\lambda t}{\pi} \int_0^\pi \frac{(\sin \theta)^{2\lambda-1}}{[(x-y)^2+t^2+2xy(1-\cos \theta)]^{\lambda+1}} d\theta.
\end{equation*}
Note that, for every $t>0$,
\begin{equation}\label{(*)}
P_t^\lambda(x,y)=P_t^\lambda(-x,y)=P_t^\lambda(x,-y), \quad x,y \in \mathbb{R}.
\end{equation}
We consider the Lusin integrals associated with the Poisson semigroup $\{P_t^\lambda\}_{t>0}$ defined by
\begin{equation*}
S_{\lambda}^q(f)(x)= \left( \int_{\Gamma(x)} \left\| t \partial_t P_t^\lambda(f)(y) \right\|_\mathbb{B}^q \frac{dt dy}{t^2} \right)^{1/q}, \quad x \in \mathbb{R},
\end{equation*}
and
\begin{equation*}
S_{\lambda,+}^q(f)(x)= \left( \int_{\Gamma_+(x)} \left\| t \partial_t P_t^\lambda(f)(y) \right\|_\mathbb{B}^q \frac{dt dy}{t^2} \right)^{1/q}, \quad x \in (0,\infty),
\end{equation*}
where $q>1$ and $f$ is a strongly $\mathbb{B}$-valued measurable function defined on $(0,\infty)$ such that
$$\int_0^\infty \frac{\|f(y)\|_{\mathbb{B}}}{1+y^2}dy<\infty.$$
It is not hard to see that
\begin{equation}\label{equivalentes}
S^q_{\lambda,+}(f)(x) \leq S^q_{\lambda}(f)(x) \leq 2^{1/q} S^q_{\lambda,+}(f)(x), \quad x \in (0,\infty).
\end{equation}
We denote by $H^1(\mathbb{R},\mathbb{B})$ the $\mathbb{B}$-valued Hardy space. $H^1_o(\mathbb{R},\mathbb{B})$ is the subspace of $H^1(\mathbb{R},\mathbb{B})$ constituted by all
those odd functions in $H^1(\mathbb{R},\mathbb{B})$. A strongly measurable $\mathbb{B}$-valued function $a$ defined on $\mathbb{R}$ such that
$\int_\mathbb{R} a(x)dx=0$ is called an $\infty$-atom (respectively, a $2$-atom) when there exists a bounded interval $I$ in $\mathbb{R}$
such that $\supp(a)\subset I$ and $\|a\|_{L^\infty(\mathbb{R},\mathbb{B})} \leq 1/|I|$ (respectively, $\|a\|_{L^2(\mathbb{R},\mathbb{B})} \leq 1/|I|^{1/2}$).
According to well-known atomic representations of the elements of $H^1(\mathbb{R},\mathbb{B})$ (\cite{Hy}, \cite[p. 34, 40]{J}) we can see that a strongly measurable $\mathbb{B}$-valued
odd function $f$ defined on $\mathbb{R}$ is in $H_o^1(\mathbb{R},\mathbb{B})$ if, and only if, $f=\sum_{j=1}^\infty \lambda_j a_j$ in $L^1((0,\infty),\mathbb{B})$, where $\{\lambda_j\}_{j=1}^\infty \subset \mathbb{C}$ satisfies that $\sum_{j=1}^\infty |\lambda_j|<\infty$ and $\{a_j\}_{j=1}^\infty$
is a sequence of strongly measurable $\mathbb{B}$-valued functions defined on $(0,\infty)$ such that, for every $j \in \mathbb{N}$, $a_j$ is an
$\infty$-atom supported on $(0,\infty)$ or there exists $\beta>0$ for which $\supp(a_j) \subset [0,\beta]$ and
$\|a\|_{L^\infty((0,\infty),\mathbb{B})} \leq 1/\beta$. Also, we can characterize the elements of $H_o^1(\mathbb{R},\mathbb{B})$ similarly by using $2$-atoms. By proceeding as in the proof of \cite[Theorem 2.1]{Fr} we can show that a strongly measurable $\mathbb{B}$-valued
odd function $f$ defined on $\mathbb{R}$ is in $H_o^1(\mathbb{R},\mathbb{B})$ if, and only if, $f=\sum_{j=1}^\infty \lambda_j a_j$ in $L^1((0,\infty),\mathbb{B})$, where $\{\lambda_j\}_{j=1}^\infty \subset \mathbb{C}$ is such that $\sum_{j=1}^\infty |\lambda_j|<\infty$, and $\{a_j\}_{j=1}^\infty$
is a sequence of strongly measurable $\mathbb{B}$-valued functions defined on $(0,\infty)$ such that, for every $j \in \mathbb{N}$, $a_j$ is an
$\infty$-atom supported on $(0,\infty)$ or $a_j=b_j \chi_{(0,\delta_j)}/\delta_j$, for a certain $b_j \in \mathbb{B}$, being $\|b_j\|_\mathbb{B}=1$
and $\delta_j>0$. Here, $\chi_{(0,\delta)}$ denotes the characteristic function of $(0,\delta)$, for every $\delta>0$. The topology of $H^1_o(\mathbb{R},\mathbb{B})$ is defined by the norms associated in the usual way with the above atomic representations.\\
In \cite{BFMT} the martingale type and cotype of a Banach space is characterized by using Littlewood-Paley $g$-functions associated with the Poisson
semigroup $\{P_t^\lambda\}_{t>0}$. In the next result we establish the corresponding properties involving Lusin area integrals $S^q_{\lambda,+}$.
This proposition has interest in itself and it is useful in the proof of Theorems~\ref{Th_3.1_OX} and \ref{Th_4.1_OX}.
\begin{Prop}\label{Lem_principal}
Let $\mathbb{B}$ be a Banach space, $\lambda>0$ and $2 \leq q < \infty$. Then, the following assertions are equivalent.
\begin{itemize}
\item[$(i)$] For some $1<p<\infty$, there exists $C>0$ such that
$$\|S^q_{\lambda,+}(f)\|_{L^p(0,\infty)} \leq C \|f\|_{L^p((0,\infty),\mathbb{B})}, \quad f \in L^p((0,\infty),\mathbb{B}).$$
\item[$(ii)$] For every $1<p<\infty$, there exists $C>0$ such that
$$\|S^q_{\lambda,+}(f)\|_{L^p(0,\infty)} \leq C \|f\|_{L^p((0,\infty),\mathbb{B})}, \quad f \in L^p((0,\infty),\mathbb{B}).$$
\item[$(iii)$] There exists $C>0$ for which
$$ \|S^q_{\lambda,+}(f)\|_{L^1(0,\infty)} \leq C \|f\|_{H^1_o(\mathbb{R},\mathbb{B})}, \quad f \in H^1_o(\mathbb{R},\mathbb{B}).$$
\item[$(iv)$] $\mathbb{B}$ has an equivalent $q$-uniformly convex norm. \\ \quad
\item[$(v)$] $\mathbb{B}^*$, the dual space of $\mathbb{B}$, has an equivalent $q'$-uniformly smooth norm, where $q'=q/(q-1)$.
\end{itemize}
\end{Prop}
This paper is organized as follows. In Section~\ref{sec:propo} we prove Proposition~\ref{Lem_principal}. Proofs of Theorems~\ref{Th_3.1_OX}
and \ref{Th_4.1_OX} are presented in Sections~\ref{sec:proof1} and \ref{sec:proof2}, respectively. In order to make
Sections~\ref{sec:proof1} and \ref{sec:proof2} more legible we include in Section~\ref{sec:appendix} (Appendix) the proofs of some auxiliary results
that we need to show Theorems~\ref{Th_3.1_OX} and \ref{Th_4.1_OX}.\\
Throughout this paper by $C$ we always denote a positive constant that is not necessarily the same in each occurrence. The duality pairing between
a Banach space $\mathbb{B}$ and its dual $\mathbb{B}^*$ will be represented by $\langle \cdot , \cdot \rangle_{\mathbb{B} \times \mathbb{B}^*}$ or simply $\langle \cdot , \cdot \rangle$.\\
\textbf{Acknowledgements}. The authors would like to thank Professor José Luis Torrea our always helpful discussions with him about vector
valued harmonic analysis.
\section{Proof of Proposition~\ref{Lem_principal}}\label{sec:propo}
In order to proof Proposition~\ref{Lem_principal} we use \cite[Lemma~4.2]{OX} where the convexity and smoothness of a Banach space $\mathbb{B}$
is described in terms of the boundedness properties of the Lusin area integral associated with the classical Poisson integral.\\
If $f$ is a strongly measurable $\mathbb{B}$-valued function defined in $\mathbb{R}$ such that $\int_\mathbb{R} \|f(x)\|_\mathbb{B}/(1+x^2)dx<\infty$,
and $q>1$ the $q$-Lusin area integral $S^q(f)$ is defined by
$$S^q(f)(x)= \left( \int_{\Gamma(x)} \left\| t \partial_t P_t(f)(y) \right\|_\mathbb{B}^q \frac{dt dy}{t^2} \right)^{1/q}, \quad x \in \mathbb{R},$$
where $P_t(f)$ represents the Poisson integral of $f$ given as follows
$$P_t(f)(y)=\int_\mathbb{R} P_t(y-z)f(z)dz, \quad y \in \mathbb{R}, \ t>0.$$
As usual, we denote the Poisson kernel by
$$P_t(z)=\frac{1}{\pi} \frac{t}{t^2+z^2}, \quad z \in \mathbb{R}, \ t>0.$$
We also consider the following partial Lusin integrals
$$S^q_+(f)(x)= \left( \int_{\Gamma_+(x)} \left\| t \partial_t P_t(f)(y) \right\|_\mathbb{B}^q \frac{dt dy}{t^2} \right)^{1/q}, \quad x \in (0,\infty),$$
and
$$ S_{+,loc}^q(f)(x)= \left( \int_{\Gamma_+(x)} \left\| t \partial_t P_t(f\chi_{(x/2,2x)})(y) \right\|_\mathbb{B}^q \frac{dt dy}{t^2} \right)^{1/q}, \quad x \in (0,\infty),$$
with $q>1$.\\
We prove Proposition~\ref{Lem_principal} in two steps. Firstly, we establish that the $L^p$-boundedness of $S^q$ is equivalent to the
$L^p$-boundedness of $S^{q}_{\lambda,+}$, for every $1<p<\infty$.
\begin{Lem}\label{Lem_Lp}
Let $\mathbb{B}$ be a Banach space, $\lambda>0$, $2 \leq q < \infty$, and $1<p<\infty$. Then, the following assertions are equivalent.
\begin{itemize}
\item[$(i)$] $S^q$ is bounded from $L^p(\mathbb{R},\mathbb{B})$ into $L^p(\mathbb{R})$.\\
\item[$(ii)$] $S^q_+$ is bounded from $L^p(\mathbb{R},\mathbb{B})$ into $L^p(0,\infty)$.\\
\item[$(iii)$] $S^q_{+,loc}$ is bounded from $L^p((0,\infty),\mathbb{B})$ into $L^p(0,\infty)$.\\
\item[$(iv)$] $S^q_{\lambda,+}$ is bounded from $L^p((0,\infty),\mathbb{B})$ into $L^p(0,\infty)$.\\
\end{itemize}
\end{Lem}
\begin{proof}
$(i) \Rightarrow (ii)$. It is sufficient to note that $S_+^q(f)(x) \leq S^q(f)(x)$, $x \in (0,\infty)$, for every $f \in L^p(\mathbb{R},\mathbb{B})$.\\
$(ii) \Rightarrow (i)$. Let $f \in L^p(\mathbb{R},\mathbb{B})$. We decompose $f$ as follows, $f=f_o + f_e$,
where $f_o(x)=(f(x)-f(-x))/2$ and $f_e(x)=(f(x)+f(-x))/2$, $x \in \mathbb{R}$. We have that
$$P_t(f_o)(y) = \int_0^\infty [P_t(y-z)-P_t(y+z)]f_o(z)dz, \quad y \in \mathbb{R}, \ t>0,$$
and
$$P_t(f_e)(y) = \int_0^\infty [P_t(y-z)+P_t(y+z)]f_e(z)dz, \quad y \in \mathbb{R}, \ t>0.$$
For every $t>0$, the function $\|t\partial_t P_t(f_o)(y)\|_\mathbb{B}$ is even. Then, $S^q(f_o)$ is also an even function and
\begin{equation*}\label{Sq-Sq+}
S^q(f_o)(x) \leq 2^{1/q} S_+^q(f_o)(|x|), \quad x \in \mathbb{R}.
\end{equation*}
Hence, we get
$$\|S^q(f_o)\|_{L^p(\mathbb{R})} \leq 2^{1/p+1/q} \|S^q_+(f_o)\|_{L^p(0,\infty)}.$$
In a similar way we obtain
$$\|S^q(f_e)\|_{L^p(\mathbb{R})} \leq 2^{1/p+1/q} \|S^q_+(f_e)\|_{L^p(0,\infty)}.$$
The above inequalities allow us to show that $(ii)$ implies $(i)$.\\
$(ii) \Leftrightarrow (iii)$. We are going to see that there exists $C>0$ such that
\begin{equation}\label{2.1}
\|S^q_+(f) - S^q_{+,loc}(f)\|_{L^p(0,\infty)} \leq C \|f\|_{L^p(\mathbb{R},\mathbb{B})}, \quad f \in L^p(\mathbb{R},\mathbb{B}).
\end{equation}
Let $f \in L^p(\mathbb{R},\mathbb{B})$. We can write
$$P_t(f)(y)
= \int_0^\infty P_t(y-z) f(z)dz + \int_0^\infty P_t(y+z) f(-z)dz, \quad y,t \in (0,\infty).$$
By applying Minkowski's inequality we get
\begin{align*}
|S^q_+(f)(x) - S^q_{+,loc}(f)(x)|
\leq & \left( \int_{\Gamma_+(x)} \left\| t \partial_t \left(
\int_{\mathbb{R}} P_t(y-z)f(z)dz - \int_{x/2}^{2x} P_t(y-z)f(z)dz
\right) \right\|_\mathbb{B}^q \frac{dtdy}{t^2} \right)^{1/q} \\
\leq & \mathcal{J}_1(\|f\|_\mathbb{B})(x) + \mathcal{J}_2(\|\tilde{f}\|_\mathbb{B})(x), \quad x \in (0,\infty),
\end{align*}
where $\tilde{f}(x)=f(-x)$, $x \in (0,\infty)$, and the operators $\mathcal{J}_i$, $i=1,2$, are defined by
$$\mathcal{J}_1(g)(x) = \int_{(0,x/2) \cup (2x,\infty)} \left\| t \partial_t P_t(y-z) \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)} g(z) dz, \quad x \in (0,\infty),$$
and
$$\mathcal{J}_2(g)(x) = \int_0^\infty \left\| t \partial_t P_t(y+z) \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)} g(z) dz, \quad x \in (0,\infty).$$
Our objective \eqref{2.1} will be established when we prove that the operators $\mathcal{J}_1$ and $\mathcal{J}_2$
are bounded from $L^p(0,\infty)$ into itself.\\
First, we observe that
\begin{align} \label{bound1}
& \left\| t \partial_t P_t(y-z) \right\|^q_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)}
= \int_{\Gamma_+(x)} \left| t \partial_t P_t(y-z) \right|^q \frac{dt dy}{t^2}
\leq C \int_0^\infty \int_{|x-y|}^\infty \frac{t^{q-2}}{(|y-z|+t)^{2q}} dt dy \nonumber \\
& \quad \leq C \int_0^\infty \int_{|x-y|}^\infty \frac{dt}{(|y-z|+t)^{q+2}} dy
\leq C \left(\int_{I_{x,z}} \frac{dy}{(|y-z|+|x-y|)^{q+1}} + \int_{\mathbb{R} \setminus I_{x,z}} \frac{dy}{(|y-z|+|x-y|)^{q+1}} \right) \nonumber\\
& \quad \leq \frac{C}{|x-z|^q}, \quad x,z \in (0,\infty), \ x \neq z.
\end{align}
Here, $I_{x,z}$ represents the interval $(\min\{x,z\},\max\{x,z\})$. We also get
\begin{align}\label{bound2}
\left\| t \partial_t P_t(y+z) \right\|^q_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)}
& \leq C \left(\int_0^x + \int_x^\infty \right) \frac{dy}{(y+z+|x-y|)^{q+1}}
\leq \frac{C}{(x+z)^q}, \quad x,z \in (0,\infty).
\end{align}
These estimates lead to, for $i=1,2$,
$$|\mathcal{J}_i(g)| \leq C \left(H_0(|g|) + H_\infty(|g|) \right),$$
where $H_0$ and $H_\infty$ denote the Hardy type operators defined by
$$H_0(g)(x)=\frac{1}{x}\int_0^x g(y) dy, \quad H_\infty(g)(x)=\int_x^\infty \frac{g(y)}{y} dy, \quad x \in (0,\infty).$$
Since $H_0$ and $H_\infty$ are bounded from $L^p(0,\infty)$ into itself (see \cite{Mu}), we conclude that $\mathcal{J}_1$ and
$\mathcal{J}_2$ are bounded from $L^p(0,\infty)$ into itself. Now the desired equivalence follows from \eqref{2.1}.\\
$(iii) \Leftrightarrow (iv)$. This property will be proved when we show that there exists $C>0$ such that
\begin{equation}\label{7.1}
\|S^q_{+,loc}(f) - S^q_{\lambda,+}(f)\|_{L^p(0,\infty)} \leq C \|f\|_{L^p((0,\infty),\mathbb{B})}, \quad f \in L^p((0,\infty),\mathbb{B}).
\end{equation}
We decompose the Bessel Poisson kernel as follows
\begin{align}\label{7.2}
P_t^\lambda(x,y)
= & \frac{2\lambda (xy)^\lambda t}{\pi} \int_0^{\pi/2} \frac{(\sin \theta)^{2\lambda-1}}{[(x-y)^2+t^2+2xy(1-\cos \theta)]^{\lambda+1}}d\theta \nonumber \\
& + \frac{2\lambda (xy)^\lambda t}{\pi} \int_{\pi/2}^\pi \frac{(\sin \theta)^{2\lambda-1}}{[(x-y)^2+t^2+2xy(1-\cos \theta)]^{\lambda+1}} d\theta \nonumber \\
= & P_{t,1}^\lambda(x,y)+P_{t,2}^\lambda(x,y), \quad x,y,t \in (0,\infty).
\end{align}
By applying Minkowski's inequality we get
\begin{equation}\label{2.3a}
|S^q_{+,loc}(f) - S^q_{\lambda,+}(f)| \leq \sum_{j=1}^3 \mathcal{K}_j(\|f\|_\mathbb{B}),
\end{equation}
where
\begin{equation*}
\mathcal{K}_1(g)(x) = \int_{(0,x/2) \cup (2x,\infty)} \left\| t \partial_t P_{t,1}^\lambda(y,z) \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)} g(z) dz, \quad x \in (0,\infty),
\end{equation*}
\begin{equation*}
\mathcal{K}_2(g)(x) = \int_0^\infty \left\| t \partial_t P_{t,2}^\lambda(y,z) \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)} g(z) dz, \quad x \in (0,\infty),
\end{equation*}
and
\begin{equation*}
\mathcal{K}_3(g)(x) = \int_{x/2}^{2x} \left\| t \partial_t [P_{t,1}^\lambda(y,z)-P_t(y-z)] \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)} g(z) dz, \quad x \in (0,\infty).
\end{equation*}
Our objective is to see that $\mathcal{K}_j$ is a bounded operator in $L^p(0,\infty)$, for $j=1,2,3$.\\
According to \cite[p. 481--482]{BCFR2} we have that
\begin{equation}\label{Pt1}
|\partial_t P_{t,1}^\lambda(y,z)| \leq C \frac{z^\lambda}{(|z-y|+t)^{\lambda+2}}, \quad y,z,t \in (0,\infty).
\end{equation}
Also, we get
\begin{equation}\label{Pt2}
|\partial_t P_{t,2}^\lambda(y,z)| \leq C \frac{z^\lambda}{(z+y+t)^{\lambda+2}}, \quad y,z,t \in (0,\infty).
\end{equation}
By proceeding as in \eqref{bound1} and \eqref{bound2} we can see that
\begin{equation}\label{||Pt1||}
\left\| t \partial_t P_{t,1}^\lambda(y,z) \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)}
\leq C \frac{z^\lambda}{|x-z|^{\lambda+1}}, \quad x,z \in (0,\infty), \ x \neq z,
\end{equation}
and
\begin{equation}\label{||Pt2||}
\left\| t \partial_t P_{t,2}^\lambda(y,z) \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)}
\leq C \frac{z^\lambda}{(x+z)^{\lambda+1}}, \quad x,z \in (0,\infty).
\end{equation}
Then, we get that
$$|\mathcal{K}_1(g)| + |\mathcal{K}_2(g)| \leq C (H_0(|g|)+H_\infty(|g|)).$$
Therefore, $\mathcal{K}_1$ and $\mathcal{K}_2$ are bounded from $L^p(0,\infty)$ into itself.\\
Next, we deal with the most involved operator $\mathcal{K}_3$. In order to do this we introduce the new kernels
\begin{equation*}
P_{t,1,1}^\lambda(y,z)=\frac{2\lambda (yz)^\lambda t}{\pi} \int_0^{\pi/2} \frac{\theta^{2\lambda-1}}{[(y-z)^2+t^2+2yz(1-\cos \theta)]^{\lambda+1}} d\theta, \quad y,z,t \in (0,\infty),
\end{equation*}
and
\begin{equation*}
P_{t,1,2}^\lambda(y,z)=\frac{2\lambda (yz)^\lambda t}{\pi} \int_0^{\pi/2} \frac{\theta^{2\lambda-1}}{[(y-z)^2+t^2+yz\theta^2]^{\lambda+1}} d\theta, \quad y,z,t \in (0,\infty).
\end{equation*}
We can write
\begin{align*}
|\mathcal{K}_3(g)(x)|
\leq & \int_{x/2}^{2x} \left\| t \partial_t [P_{t,1}^\lambda(y,z)-P_{t,1,1}^\lambda(y,z)] \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)} |g(z)| dz \\
& + \int_{x/2}^{2x} \left\| t \partial_t [P_{t,1,1}^\lambda(y,z)-P_{t,1,2}^\lambda(y,z)] \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)} |g(z)| dz \\
& + \int_{x/2}^{2x} \left\| t \partial_t [P_{t,1,2}^\lambda(y,z)-P_t(y-z)] \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)} |g(z)| dz,
\quad x \in (0,\infty).
\end{align*}
By arguing like in \cite[p. 483--487]{BCFR2}, we deduce that, for each $x \in (0,\infty)$ and $x/2<z<2x$,
\begin{itemize}
\item $\displaystyle \left\| t \partial_t [P_{t,1}^\lambda(y,z)-P_{t,1,1}^\lambda(y,z)] \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)}
\leq \frac{C}{z} \left( 1 + \log_+\frac{z}{|x-z|} \right),$ \\
\item $\displaystyle \left\| t \partial_t [P_{t,1,1}^\lambda(y,z)-P_{t,1,2}^\lambda(y,z)] \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)}
\leq \frac{C}{z} \left( 1 + \log_+\frac{z}{|x-z|} \right),$\\
\item $\displaystyle \left\| t \partial_t [P_{t,1,2}^\lambda(y,z)-P_t(y-z)] \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)}
\leq \frac{C}{z}.$ \\ \quad
\end{itemize}
There (in \cite[p. 483--487]{BCFR2}), the case $q=2$ is considered, but the same arguments are still valid for $2 \leq q < \infty$.\\
Hence, we have that
\begin{equation}\label{2.8}
|\mathcal{K}_3(g)(x)|
\leq C \int_{x/2}^{2x} \frac{1}{z} \left( 1 + \log_+\frac{z}{|x-z|} \right) |g(z)| dz, \quad x \in (0,\infty).
\end{equation}
We denote $\displaystyle C_0= \int_{1/2}^2 \frac{1}{u}\left( 1 + \log_+\frac{u}{|1-u|} \right) du$. By using Jensen's inequality we get
\begin{align*}
|\mathcal{K}_3(g)(x)|^p
\leq & C \left( \int_{1/2}^1 \frac{1}{u}\left( 1 + \log_+\frac{u}{|1-u|} \right) |g(xu)|du \right)^p \\
\leq & C C_0^{p-1} \int_{1/2}^1 \frac{1}{u}\left( 1 + \log_+\frac{u}{|1-u|} \right) |g(xu)|^p du, \quad x \in (0,\infty).
\end{align*}
Then,
$$\|\mathcal{K}_3(g)\|_{L^p(0,\infty)}
\leq C \|g\|_{L^p(0,\infty)}, \quad g \in L^p(0,\infty).$$
Putting together the above estimations we obtain \eqref{7.1}, and the proof of $(iii) \Leftrightarrow (iv)$ is finished.
\end{proof}
In the following lemma we establish the endpoint result $p=1$.
\begin{Lem}\label{Lem_L1}
Let $\mathbb{B}$ be a Banach space, $\lambda>0$ and $2 \leq q < \infty$. Then, the following statements are equivalent.
\begin{itemize}
\item[$(i)$] $\displaystyle \|S^q(f)\|_{L^1(\mathbb{R})} \leq C \|f\|_{H^1(\mathbb{R},\mathbb{B})}, \quad f \in H^{1}(\mathbb{R},\mathbb{B}),$ \\
\item[$(ii)$] $\displaystyle \|S^q_{\lambda,+}(f)\|_{L^1(0,\infty)} \leq C \|f\|_{H^1_o(\mathbb{R},\mathbb{B})}, \quad f \in H^1_o(\mathbb{R},\mathbb{B}).$ \\ \quad
\end{itemize}
\end{Lem}
\begin{proof}
$(i) \Rightarrow (ii)$. We claim that the properties below are equivalent. \\
\begin{itemize}
\item[$(a)$] $\displaystyle \|S^q(f)\|_{L^1(\mathbb{R})} \leq C \|f\|_{H^1_o(\mathbb{R},\mathbb{B})}, \quad f \in H^1_o(\mathbb{R},\mathbb{B}).$ \\
\item[$(b)$] $\displaystyle \|S^q_+(f)\|_{L^1(0,\infty)} \leq C \|f\|_{H^1_o(\mathbb{R},\mathbb{B})}, \quad f \in H^1_o(\mathbb{R},\mathbb{B}).$ \\
\item[$(c)$] $\displaystyle \|S^q_{+,loc}(f)\|_{L^1(0,\infty)} \leq C \|f\|_{H^1_o(\mathbb{R},\mathbb{B})}, \quad f \in H^1_o(\mathbb{R},\mathbb{B}).$ \\
\item[$(d)$] $\displaystyle \|S^q_{\lambda,+}(f)\|_{L^1(0,\infty)} \leq C \|f\|_{H^1_o(\mathbb{R},\mathbb{B})}, \quad f \in H^1_o(\mathbb{R},\mathbb{B}).$ \\ \quad
\end{itemize}
This implies what we are looking for.\\
$(a) \Leftrightarrow (b)$. This equivalence can be proved by proceeding as in the proof of
$(i) \Leftrightarrow (ii)$ of Lemma~\ref{Lem_Lp}.\\
$(b) \Leftrightarrow (c)$. Let $f \in H^1_o(\mathbb{R},\mathbb{B})$. Since $f$ is odd, Minkowski's inequality implies that
\begin{align*}
|S^q_+(f)(x)-S^q_{+,loc}(f)(x)|
\leq \mathcal{H}_1(\|f\|_\mathbb{B})(x) + \mathcal{H}_2(\|f\|_\mathbb{B})(x), \quad x \in (0,\infty),
\end{align*}
where
\begin{equation*}
\mathcal{H}_1(g)(x) = \int_{(0,x/2) \cup (2x,\infty)} \left\| t \partial_t [P_t(y-z)-P_t(y+z)] \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)} g(z) dz, \quad x \in (0,\infty),
\end{equation*}
and
\begin{equation*}
\mathcal{H}_2(g)(x) = \int_{x/2}^{2x} \left\| t \partial_t P_t(y+z) \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)} g(z) dz, \quad x \in (0,\infty).
\end{equation*}
A straightforward manipulation leads to
\begin{align*}
\left| \partial_t[P_t(z-y)-P_t(z+y)] \right|
\leq & C \left( \frac{zy}{((z-y)^2+t^2)((z+y)^2+t^2)} + \frac{zyt^2 (z^2+y^2+t^2)}{((z-y)^2+t^2)^2((z+y)^2+t^2)^2} \right) \\
\leq & C \frac{zy}{((z-y)^2+t^2)((z+y)^2+t^2)}
\leq C \frac{\sqrt{z}}{(|z-y|+t)^2\sqrt{z+y+t}} \\
\leq & C \frac{\sqrt{z}}{(|z-y|+t)^{5/2}}, \quad y,z,t \in (0,\infty).
\end{align*}
Hence, by proceeding as in \eqref{bound1} we obtain
\begin{align*}
\int_{\Gamma_+(x)} \left|t \partial_t[P_t(z-y)-P_t(z+y)] \right|^q \frac{dtdy}{t^2}
\leq & C \int_{\Gamma_+(x)} \frac{z^{q/2}t^{q-2}}{(|z-y|+t)^{5q/2}} dt dy \\
\leq & C \frac{z^{q/2}}{|x-z|^{3q/2}}, \quad x,z \in (0,\infty), \ x \neq z.
\end{align*}
Then, we get
$$\left\| t \partial_t [P_t(y-z)-P_t(y+z)] \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)}
\leq C \frac{\sqrt{z}}{|x-z|^{3/2}}, \quad x,z \in (0,\infty), \ x \neq z.$$
Therefore,
$$\| \mathcal{H}_1(g) \|_{L^1(0,\infty)}
\leq C \|g\|_{L^1(0,\infty)}, \quad g \in L^1(0,\infty).$$
Moreover, according to \eqref{bound2}, we have that
$$|\mathcal{H}_2(g)(x)|
\leq C \int_{x/2}^{2x} \frac{|g(z)|}{z} dz, \quad x \in (0,\infty),$$
and it follows that
$$\| \mathcal{H}_2(g) \|_{L^1(0,\infty)}
\leq C \|g\|_{L^1(0,\infty)}, \quad g \in L^1(0,\infty).$$
Hence, we conclude that
$$ \| S^q_+(f)-S^q_{+,loc}(f) \|_{L^1(0,\infty)}
\leq C \|f\|_{L^1((0,\infty),\mathbb{B})}
\leq C \|f\|_{H^1_o(\mathbb{R},\mathbb{B})}.$$
\quad
$(c) \Leftrightarrow (d)$. Let $f \in H^1_o(\mathbb{R},\mathbb{B})$. By \eqref{2.3a} we have to analyze the operators $\mathcal{K}_j$, $j=1,2,3$.
From \eqref{||Pt1||} it follows that
\begin{align*}
\int_0^\infty \mathcal{K}_1(\|f\|_\mathbb{B})&(x) dx
\leq C \int_0^\infty \int_{(0,x/2) \cup (2x,\infty)} \frac{z^\lambda}{|x-z|^{\lambda+1}} \|f(z)\|_\mathbb{B} dz dx \\
\leq & C \int_0^\infty \|f(z)\|_\mathbb{B} \left(\int_0^{z/2} \frac{z^\lambda}{|x-z|^{\lambda+1}} dx + \int_{2z}^\infty \frac{z^\lambda}{|x-z|^{\lambda+1}} dx\right) dz
\leq C \int_0^\infty \|f(z)\|_\mathbb{B} dz,
\end{align*}
and from \eqref{||Pt2||} we deduce, in a similar way, that
$$\int_0^\infty \mathcal{K}_2(\|f\|_\mathbb{B})(x) dx
\leq C \int_0^\infty \|f(z)\|_\mathbb{B} dz.$$
Finally, \eqref{2.8} implies that
\begin{align*}
\int_0^\infty \mathcal{K}_3(\|f\|_\mathbb{B})(x) dx
\leq & C \int_0^\infty \int_{x/2}^{2x} \frac{1}{z} \left( 1 + \log_+\frac{z}{|x-z|} \right) \|f(z)\|_\mathbb{B} dz dx \\
\leq & C \int_0^\infty \|f(z)\|_\mathbb{B} \int_{z/2}^{2z} \frac{1}{z} \left( 1 + \log_+\frac{z}{|x-z|} \right) dx dz
\leq C \int_0^\infty \|f(z)\|_\mathbb{B} dz.
\end{align*}
By combining the above estimates we get
$$\|S^q_{+,loc}(f) - S^q_{\lambda,+}(f)\|_{L^1(0,\infty)}
\leq C \|f\|_{L^1((0,\infty),\mathbb{B})}
\leq C \|f\|_{H^1_o(\mathbb{R},\mathbb{B})}.$$
Thus $(i) \Rightarrow (ii)$ is established.\\
$(ii) \Rightarrow (i)$. Let $f \in H^1(\mathbb{R},\mathbb{B})$. We write $f=f_o+f_e$, where
$$f_o(x)=\frac{f(x)-f(-x)}{2}, \quad f_e(x)=\frac{f(x)+f(-x)}{2}, \quad x \in \mathbb{R}.$$
It is clear that $f_o$, $f_e \in H^1(\mathbb{R},\mathbb{B})$. Moreover
$\|f_o\|_{H^1(\mathbb{R},\mathbb{B})} \leq \|f\|_{H^1(\mathbb{R},\mathbb{B})}$ and $\|f_e\|_{H^1(\mathbb{R},\mathbb{B})} \leq \|f\|_{H^1(\mathbb{R},\mathbb{B})}$.
Assume that $(ii)$ holds. Since $f_o \in H^1_o(\mathbb{R},\mathbb{B})$, by using that $(d) \Leftrightarrow (a)$, we get
\begin{equation}\label{2.9}
\|S^q(f_o)\|_{L^1(\mathbb{R})}
\leq C \|f_o\|_{H^1(\mathbb{R},\mathbb{B})}.
\end{equation}
We define $H^1_e(\mathbb{R},\mathbb{B})$ the space constituted by all those even functions in $H^1(\mathbb{R},\mathbb{B})$. We consider the properties
$(a')$, $(b')$, $(c')$ and $(d')$ that are analogous to $(a)$, $(b)$, $(c)$ and $(d)$, respectively, but replacing $H^1_o(\mathbb{R},\mathbb{B})$
by $H^1_e(\mathbb{R},\mathbb{B})$. By proceeding as in the odd case we can see that $(a') \Leftrightarrow (b')$ and
$(c') \Leftrightarrow (d')$. We are going to see that $(b') \Leftrightarrow (c')$.\\
Suppose that $h \in H^1_e(\mathbb{R},\mathbb{B})$ and that $h = \sum_{j=1}^\infty \lambda_j a_j$, where, for each $j \in \mathbb{N}$, $a_j$ is a
$H^1(\mathbb{R},\mathbb{B})$-atom, and $\{\lambda_j\}_{j=1}^\infty \subset \mathbb{C}$ is such that $\sum_{j=1}^\infty |\lambda_j|<\infty$. Then,
$h=\sum_{j=1}^\infty \lambda_j (a_j+\tilde{a}_j)/2$, being $\tilde{a}_j(x)=a_j(-x)$, $x \in \mathbb{R}$, and $j \in \mathbb{N}$. We define
$b_j$ and $\gamma_j$, $j \in \mathbb{N}$, as follows
\begin{itemize}
\item $b_j=a_j$ and $\gamma_j=\lambda_j/2$, provided that $\supp(a_j) \subset [0,\infty),$
\item $b_j=\tilde{a}_j$ and $\gamma_j=\lambda_j/2$, when $\supp(a_j) \subset (-\infty,0],$
\item if $\supp(a_j) \cap (0,\infty) \neq \varnothing$ and $\supp(a_j) \cap (-\infty,0) \neq \varnothing$, then
$b_j=\chi_{(0,\infty)}(a_j+\tilde{a}_j)/2$ and $\gamma_j=\lambda_j$.
\end{itemize}
Thus, $b_j$ is a $H^1(\mathbb{R},\mathbb{B})$-atom. Indeed, in the two first cases it is clear. Assume now that $b_j=\chi_{(0,\infty)}(a_j+\tilde{a}_j)/2$
where $\supp(a_j) \cap (0,\infty) \neq \varnothing$, $\supp(a_j) \cap (-\infty,0) \neq \varnothing$, $\supp(a_j) \subset [-\alpha,\beta]$
being $0<\alpha<\beta$ (similarly, $0<\beta<\alpha$), and $\|a_j\|_{L^\infty(\mathbb{R},\mathbb{B})} \leq 1/(\beta+\alpha)$. Then, $\supp(b_j) \subset [0,\beta]$,
$\|b_j\|_{L^\infty(\mathbb{R},\mathbb{B})} \leq \|a_j\|_{L^\infty(\mathbb{R},\mathbb{B})} \leq 1/(\beta+\alpha) \leq 1/\beta$, and
$$\int_{\mathbb{R}} b_j(x) dx
= \frac{1}{2}\int_{-\alpha}^\beta a_j(x) dx =0.$$
From now on we write $h = \sum_{j=1}^\infty 2\gamma_j g_j$, where $g_j(x)=b_j(|x|)/2$, $x \in \mathbb{R}$ and $j \in \mathbb{N}$, and $b_j$
and $\gamma_j$, $j \in \mathbb{N}$, are those ones we have just defined.\\
We can write
\begin{align*}
& |S^q_+(h)(x) - S^q_{+,loc}(h)(x)|
\leq \left( \int_{\Gamma_+(x)} \left\| t\partial_t \left(
\int_{\mathbb{R}} P_t(y-z) h(z) dz - \int_{x/2}^{2x} P_t(y-z) h(z) dz
\right)\right\|^q_\mathbb{B} \frac{dtdy}{t^2} \right)^{1/q} \\
& \quad \leq 2\sum_{j=1}^\infty |\gamma_j| \left( \int_{\Gamma_+(x)} \left\| t\partial_t \left(
\int_{\mathbb{R}} P_t(y-z) g_j(z) dz - \int_{x/2}^{2x} P_t(y-z) g_j(z) dz
\right)\right\|^q_\mathbb{B} \frac{dtdy}{t^2} \right)^{1/q}, \ x \in (0,\infty).
\end{align*}
Assume that $g$ is an even $H^1(\mathbb{R},\mathbb{B})$-atom such that $\supp(g) \subset [-\beta,\beta]$ with $\beta>0$ and $\|g\|_{L^\infty(\mathbb{R},\mathbb{B})}\leq 1/2\beta$.
We have that
$$\int_{\mathbb{R}} P_t(y-z) g(z) dz
= \int_0^\beta [P_t(y-z) + P_t(y+z)]g(z) dz, \quad y \in \mathbb{R}.$$
For every $x \in (0,\infty)$, by using Minkowski's inequality we get
\begin{align*}
& \left( \int_{\Gamma_+(x)} \left\| t\partial_t \left( \int_{\mathbb{R}} P_t(y-z) g(z) dz - \int_{x/2}^{2x} P_t(y-z) g(z) dz
\right)\right\|^q_\mathbb{B} \frac{dtdy}{t^2} \right)^{1/q}\\
& \qquad = \left( \int_{\Gamma_+(x)} \left\| t\partial_t \left( \int_{(0,x/2)\cup(2x,\infty)} P_t(y-z) g(z) dz + \int_{0}^\infty P_t(y+z) g(z) dz
\right)\right\|^q_\mathbb{B} \frac{dtdy}{t^2} \right)^{1/q} \\
& \qquad\leq \chi_{(0,2\beta)}(x) \int_0^{x/2} \left\| t \partial_t P_t(y-z) \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)} \|g(z)\|_\mathbb{B} dz \\
& \qquad \qquad + \chi_{(2\beta, \infty)}(x) \int_0^{\beta} \left\|t \partial_t [P_t(y-z)-P_t(y+\beta)] \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)} \|g(z)\|_\mathbb{B} dz\\
& \qquad \qquad + \int_{2x}^\infty \left\| t \partial_t P_t(y-z) \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)} \|g(z)\|_\mathbb{B} dz \\
& \qquad \qquad + \chi_{(0,2\beta)}(x) \int_0^{x/2} \left\| t \partial_t P_t(y+z) \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)} \|g(z)\|_\mathbb{B} dz \\
& \qquad \qquad + \chi_{(2\beta, \infty)}(x) \int_0^{\beta} \left\|t \partial_t [P_t(y+z)-P_t(y+\beta)] \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)} \|g(z)\|_\mathbb{B} dz\\
& \qquad \qquad + \int_{x/2}^\infty \left\| t \partial_t P_t(y+z) \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)} \|g(z)\|_\mathbb{B} dz \\
&\qquad = \sum_{i=1}^6 \mathpzc{H}_{i}(\|g\|_\mathbb{B})(x).
\end{align*}
Note that it is possible to introduce the factor $t \partial_t P_t(y+\beta)$, because $g$ is even and has zero mean.\\
Our goal is to see that, for a certain $C>0$ independent of $g$,
\begin{equation*}
\| \mathpzc{H}_{i}(\|g\|_\mathbb{B}) \|_{L^1(0,\infty)}
\leq C, \quad i=1, \dots, 6.
\end{equation*}
According to \eqref{bound1} it follows that
\begin{align*}
\mathpzc{H}_{1}(\|g\|_\mathbb{B})(x)
\leq & C \chi_{(0,2\beta)}(x) \int_0^{x/2} \frac{\|g(z)\|_\mathbb{B}}{x-z} dz
\leq C \frac{\chi_{(0,2\beta)}(x)}{\beta}, \quad x \in (0,\infty),
\end{align*}
and then
\begin{equation}\label{2.10}
\| \mathpzc{H}_{1}(\|g\|_\mathbb{B}) \|_{L^1(0,\infty)}
\leq C.
\end{equation}
In a similar way \eqref{bound2} leads to
\begin{equation}\label{2.11}
\| \mathpzc{H}_{4}(\|g\|_\mathbb{B}) \|_{L^1(0,\infty)}
\leq C.
\end{equation}
By using again \eqref{bound1} and \eqref{bound2} we obtain
\begin{align*}
\mathpzc{H}_{3}(\|g\|_\mathbb{B})(x) + \mathpzc{H}_{6}(\|g\|_\mathbb{B})(x)
\leq & C \int_{x/2}^\infty \frac{\|g(z)\|_\mathbb{B}}{z} dz, \quad x \in (0,\infty).
\end{align*}
Since the Hardy operator $H_\infty$ is bounded from $L^1(0,\infty)$ into itself, we conclude that
\begin{equation}\label{2.12}
\| \mathpzc{H}_{3}(\|g\|_\mathbb{B}) \|_{L^1(0,\infty)} + \| \mathpzc{H}_{6}(\|g\|_\mathbb{B}) \|_{L^1(0,\infty)}
\leq C \int_0^\beta \|g(z)\|_\mathbb{B} dz
\leq C.
\end{equation}
In order to analyze $\mathpzc{H}_{j}(\|g\|_\mathbb{B})$, $j=2,5$, we claim that
\begin{equation}\label{y+beta}
\left\|t \partial_t [P_t(y \pm z)-P_t(y+\beta)] \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)}
\leq C \frac{\beta}{|x-z|^2}, \quad x \in (0,\infty), \ 0<z<\beta \text{ and } x \neq z.
\end{equation}
If \eqref{y+beta} holds we obtain
\begin{align}\label{2.14}
\| \mathpzc{H}_{2}(\|g\|_\mathbb{B}) \|_{L^1(0,\infty)} + \| \mathpzc{H}_{5}(\|g\|_\mathbb{B}) \|_{L^1(0,\infty)}
\leq & C \int_0^\beta \|g(z)\|_\mathbb{B} \int_{2\beta}^\infty \frac{\beta}{|x-z|^2} dx dz \nonumber \\
\leq & C \int_0^\beta \|g(z)\|_\mathbb{B} \frac{\beta}{2\beta-z} dz
\leq C.
\end{align}
Note that the constants $C>0$ in \eqref{2.10}-\eqref{2.14} do not depend on $g$.\\
To justify \eqref{y+beta} we observe that
\begin{align*}
\partial_t [P_t(y - z)-P_t(y+\beta)]
= & \frac{\beta^2-z^2+2y(\beta+z)}{\pi} \left( \frac{1}{(t^2+|y-z|^2)(t^2+(y+\beta)^2)} \right. \\
& \left. - 2t^2 \frac{2t^2+(y+\beta)^2+|y-z|^2}{(t^2+|y-z|^2)^2(t^2+(y+\beta)^2)^2} \right), \quad y,z \in \mathbb{R}, \ t>0.
\end{align*}
Moreover, if $0<y<\infty$ and $0<z<\beta$, $|y-z| \leq y + \beta$, and
\begin{align*}
|\partial_t [P_t(y - z)-P_t(y+\beta)]|
\leq C & \frac{\beta(y+\beta)}{(t+|y-z|)^2(t+y+\beta)^2}
\leq \frac{\beta}{(t+|y-z|)^3}.
\end{align*}
In a similar way we can see, for each $0<y<\infty$ and $0<z<\beta$,
\begin{align*}
|\partial_t [P_t(y + z)-P_t(y+\beta)]|
\leq C \frac{\beta}{(t+y+z)^2(t+y+\beta)}
\leq \frac{\beta}{(t+|y-z|)^3}.
\end{align*}
By proceeding as in \eqref{bound1} we obtain \eqref{y+beta}.\\
Putting together all the above estimations we conclude that
\begin{align*}
\|S^q_+(h) - S^q_{+,loc}(h)\|_{L^1(0,\infty)}
\leq & C \sum_{j=1}^\infty |\gamma_j|
\leq C \sum_{j=1}^\infty |\lambda_j|.
\end{align*}
Hence,
\begin{align*}
\|S^q_+(h) - S^q_{+,loc}(h)\|_{L^1(0,\infty)}
\leq & C \|h\|_{H^1(\mathbb{R},\mathbb{B})}.
\end{align*}
Thus, $(b') \Leftrightarrow (c')$ is established.\\
Assume again that $h \in H^1_e(\mathbb{R},\mathbb{B})$. We define $H$ as the odd extension of $h_{|_{[0,\infty)}}$ to $\mathbb{R}$. It is clear
that $H \in H^1_o(\mathbb{R},\mathbb{B})$ and $\|H\|_{H^1(\mathbb{R},\mathbb{B})} \leq C \|h\|_{H^1(\mathbb{R},\mathbb{B})}$. Then, according to $(ii)$ we get
\begin{align*}
\|S^q_{\lambda,+}(h)\|_{L^1(0,\infty)}
= & \|S^q_{\lambda,+}(H)\|_{L^1(0,\infty)}
\leq C \|H\|_{H^1(\mathbb{R},\mathbb{B})}
\leq C \|h\|_{H^1(\mathbb{R},\mathbb{B})}.
\end{align*}
Hence, we have that
\begin{equation}\label{2.15}
\|S^q(f_e)\|_{L^1(\mathbb{R})}
\leq C \|f_e\|_{H^1(\mathbb{R},\mathbb{B})}.
\end{equation}
By combining \eqref{2.9} and \eqref{2.15} we conclude that
\begin{equation*}
\|S^q(f)\|_{L^1(\mathbb{R})}
\leq C \|f\|_{H^1(\mathbb{R},\mathbb{B})}.
\end{equation*}
Thus the proof of this lemma is completed.
\end{proof}
The proof of Proposition~\ref{Lem_principal} is now consequence of Lemmas~\ref{Lem_Lp} and \ref{Lem_L1} and \cite[Lemma 4.2]{OX}.
\section{Proof of Theorem~\ref{Th_3.1_OX}} \label{sec:proof1}
\subsection{Proof of $(ii) \Rightarrow (i)$} \label{subsec:2->1}
Assume that $\mathbb{B}$ has an equivalent norm which is $q$-uniformly convex.\\
Let $f \in BMO_o(\mathbb{R},\mathbb{B})$ and take $I=(a,b)$ such that $0 \leq a<b<\infty$. We denote by $2I$
the interval $(0,\infty) \cap (x_I-|I|,x_I+|I|)$ where $x_I=(a+b)/2$. We decompose $f$ as follows:
$$f \chi_{(0,\infty)}=(f-f_{2I})\chi_{2I} + (f-f_{2I}) \chi_{(0,\infty)\setminus 2I} + f_{2I}=f_1 + f_2 + f_3.$$
We are going to show, for $i=1,2,3,$
\begin{equation}\label{objetivo2}
\left( \frac{1}{|I|}\int_0^{|I|} \int_I \|t \partial_t P_t^\lambda(f_i)(x) \|^q_\mathbb{B} \frac{dx dt}{t} \right)^{1/q}
\leq C \|f\|_{BMO_o(\mathbb{R},\mathbb{B})},
\end{equation}
where the constant $C>0$ depends neither on $I$ nor on $f$.\\
Firstly, we prove \eqref{objetivo2} for $i=1$. Note that $|I \cap (x-t,x+t)| \sim t$, when $x \in I$ and $0<t<|I|$. We have that
\begin{align} \label{3.1.a}
\int_0^{|I|} \int_I \|t \partial_t P_t^\lambda(f_1)(x) \|^q_\mathbb{B} \frac{dx dt}{t}
\leq & C \int_0^{|I|} \int_I \|t \partial_t P_t^\lambda(f_1)(x) \|^q_\mathbb{B} \frac{|I \cap (x-t,x+t)|}{t^2} dx dt \nonumber\\
= & C \int_0^{|I|} \int_I \int_{I \cap (x-t,x+t)} \|t \partial_t P_t^\lambda(f_1)(x) \|^q_\mathbb{B} dz \frac{dx dt}{t^2} \nonumber \\
\leq & C \int_I \int_{\Gamma_+(z)} \|t \partial_t P_t^\lambda(f_1)(x) \|^q_\mathbb{B} \frac{dt dx }{t^2}dz
\leq C \|S^q_{\lambda,+}(f_1)\|_{L^q(0,\infty)}^q.
\end{align}
By using Proposition~\ref{Lem_principal} and John-Nirenberg's inequality we get
\begin{align*}
\left( \frac{1}{|I|}\int_0^{|I|} \int_I \|t \partial_t P_t^\lambda(f_1)(x) \|^q_\mathbb{B} \frac{dx dt}{t} \right)^{1/q}
\leq & C \left( \frac{1}{|I|} \|S^q_{\lambda,+}(f_1)\|_{L^q(0,\infty)}^q \right)^{1/q} \\
\leq & C \left( \frac{1}{|I|} \int_{2I} \| f(x) - f_{2I}\|_\mathbb{B}^q dx\right)^{1/q}
\leq C \|f\|_{BMO_o(\mathbb{R},\mathbb{B})}.
\end{align*}
On the other hand, from \cite[(b), p. 86]{MS} we deduce that
\begin{align}\label{3.2a}
|t\partial_t P_t^\lambda(x,y)|
\leq & C \left( P_t^\lambda(x,y) + t^3 (xy)^\lambda \int_0^\pi \frac{(\sin \theta)^{2\lambda-1}}{(x^2 + y^2 + t^2 - 2xy \cos \theta)^{\lambda+2}} d\theta \right) \nonumber \\
\leq & C P_t^\lambda(x,y)
\leq C \frac{t}{t^2+(x-y)^2}, \quad x,y,t \in (0,\infty).
\end{align}
Then, we can write, for every $x \in I$ and $t>0$,
\begin{align*}
\| t\partial_t P_t^\lambda(f_2)(x) \|_\mathbb{B}
\leq & C \int_{(0,\infty) \setminus 2I} \frac{t}{t^2+(x-y)^2} \|f(y)-f_{2I}\|_\mathbb{B} dy \\
\leq & C \int_{(0,\infty) \setminus 2I} \frac{t}{t^2+(x_I-y)^2} \|f(y)-f_{2I}\|_\mathbb{B} dy \\
\leq & C \frac{t}{|I|} \sum_{k=1}^\infty \frac{1}{2^k} \left( \frac{1}{2^k |I|} \int_{2^{k+1}I \cap (0,\infty)} \|f(y)-f_{2^{k+1}I}\|_\mathbb{B} dy + \|f_{2^{k+1}I} - f_{2I}\|_\mathbb{B} \right) \\
\leq & C \frac{t}{|I|} \sum_{k=1}^\infty \frac{k}{2^k} \|f\|_{BMO_o(\mathbb{R},\mathbb{B})}
\leq C \frac{t}{|I|} \|f\|_{BMO_o(\mathbb{R},\mathbb{B})}.
\end{align*}
Hence,
\begin{align*}
\left( \frac{1}{|I|}\int_0^{|I|} \int_I \|t \partial_t P_t^\lambda(f_2)(x) \|^q_\mathbb{B} \frac{dx dt}{t} \right)^{1/q}
\leq & C \left( \frac{1}{|I|^{q+1}}\int_0^{|I|} t^{q-1} dt \int_I dx \right)^{1/q} \|f\|_{BMO_o(\mathbb{R},\mathbb{B})} \\
\leq & C \|f\|_{BMO_o(\mathbb{R},\mathbb{B})}.
\end{align*}
Finally, we show \eqref{objetivo2} for $i=3$. Observe that in the classical case (see \cite{OX}) this term does not appear because the
corresponding integral vanishes. First of all, we notice that
$$\|f_{2I}\|_\mathbb{B} \leq \frac{1}{2|I|} \int_0^{x_I+|I|} \|f(x)\|_\mathbb{B} dx \leq \frac{x_I+|I|}{2|I|} \|f\|_{BMO_o(\mathbb{R},\mathbb{B})}.$$
Then, in order to establish \eqref{objetivo2} for $i=3$ it is sufficient to show that
\begin{equation}\label{objetivo3}
\frac{(x_I+|I|)^q}{|I|^{q+1}}\int_0^{|I|} \int_I |t\partial_t P_t^\lambda(1)(x)|^q \frac{dx dt}{t}
\leq C,
\end{equation}
where $C>0$ does not depend on $I$.\\
By taking into account that $|t \partial_t P_t^\lambda(x,y)| \leq C P_t^\lambda(x,y)$, $x,y,t \in (0,\infty)$,
(see \eqref{3.2a}) we can write
\begin{align*}
|t\partial_t P_t^\lambda(1)(x)|
\leq & C \left( \int_0^{x/2} P_t^\lambda(x,y) dy + \left| t\partial_t \int_{x/2}^{3x/2} P_t^\lambda(x,y) dy \right| + \int_{3x/2}^\infty P_t^\lambda(x,y) dy \right)
= C \sum_{j=1}^3 J_j(x,t).
\end{align*}
According to \cite[(b), p. 86]{MS} we have that,
\begin{equation}\label{J1}
J_1(x,t)
\leq C \int_0^{x/2} \frac{(xy)^\lambda t}{[(x-y)^2+t^2]^{\lambda+1}} dy
\leq C \frac{x^{2\lambda+1} t}{(x^2+t^2)^{\lambda+1}}
\leq C \frac{t}{t+x}, \quad x,t \in (0,\infty),
\end{equation}
\begin{equation}\label{J2}
J_2(x,t)
\leq C \int_{x/2}^{3x/2} \frac{(xy)^\lambda t}{[(x-y)^2+t^2]^{\lambda+1}} dy
\leq C \frac{x^{2\lambda+1}}{t^{2\lambda+1}}, \quad x,t \in (0,\infty),
\end{equation}
and
\begin{align}\label{J3}
J_3(x,t)
\leq & C \int_{3x/2}^\infty \frac{(xy)^\lambda t}{[(x-y)^2+t^2]^{\lambda+1}} dy
\leq C x^\lambda t \int_{3x/2}^\infty \frac{y^\lambda }{(y^2+t^2)^{\lambda+1}} dy \nonumber \\
\leq & C x^\lambda t \int_{3x/2}^\infty \frac{1 }{(y+t)^{\lambda+2}} dy
\leq C \frac{x^\lambda t}{(x+t)^{\lambda+1}}
\leq C \frac{t}{t+x}, \quad x,t \in (0,\infty).
\end{align}
We need also to estimate $J_2$ in a different way. The classical Poisson kernel is introduced as follows
\begin{align*}
J_2(x,t)
\leq & \left|\int_{x/2}^{3x/2} t\partial_t [P_t^\lambda(x,y) - P_t(x-y)] dy \right| + \left| t\partial_t\int_{x/2}^{3x/2} P_t(x-y) dy \right|
= \sum_{j=1}^2 J_{2,j}(x,t), \ x,t \in (0,\infty).
\end{align*}
The function under the integral sign in $J_{2,1}$ is decomposed as follows
\begin{align}\label{3.5}
t\partial_t & [P_t^\lambda(x,y) - P_t(x-y)]
= P_t^\lambda(x,y)- P_t(x-y) \nonumber \\
& - \left( \frac{4(xy)^\lambda t^3 \lambda (\lambda+1)}{\pi} \int_0^\pi \frac{(\sin \theta)^{2\lambda-1}}{((x-y)^2 + t^2 + 2xy(1- \cos \theta))^{\lambda+2}} d\theta
- \frac{2t^3}{\pi} \frac{1}{((x-y)^2+t^2)^2}\right) \nonumber \\
= & (P_{t,1}^\lambda(x,y)- P_t(x-y)) + P_{t,2}^\lambda(x,y) \nonumber \\
& - \left( \frac{4(xy)^\lambda t^3 \lambda (\lambda+1)}{\pi} \int_0^{\pi/2} \frac{(\sin \theta)^{2\lambda-1}}{((x-y)^2 + t^2 + 2xy(1- \cos \theta))^{\lambda+2}} d\theta
- \frac{2t^3}{\pi} \frac{1}{((x-y)^2+t^2)^2}\right) \nonumber \\
& - \frac{4(xy)^\lambda t^3 \lambda (\lambda+1)}{\pi} \int_{\pi/2}^\pi \frac{(\sin \theta)^{2\lambda-1}}{(x^2+y^2 + t^2 - 2xy\cos \theta)^{\lambda+2}} d\theta,
\quad x,y,t \in (0,\infty),
\end{align}
where as above $P_{t,1}^\lambda$ and $P_{t,2}^\lambda$ is defined in \eqref{7.2}.\\
Firstly note that
\begin{align}\label{3.6}
\Big| (xy)^\lambda t^3 & \int_{\pi/2}^\pi \frac{(\sin \theta)^{2\lambda-1}}{(x^2 + y^2 + t^2 - 2xy \cos \theta)^{\lambda+2}} d\theta \Big|
\leq C P_{t,2}^\lambda(x,y) \nonumber \\
\leq & C \frac{(xy)^\lambda t}{(x+y+t)^{2\lambda+2}} \int_{\pi/2}^\pi (\sin \theta)^{2\lambda-1} d\theta
\leq C \frac{t}{xy}, \quad x,y,t \in (0,\infty).
\end{align}
Since $\sin \theta \sim \theta$ and $2(1-\cos \theta) \sim \theta^2$, $\theta \in [0,\pi/2]$, by using the mean value theorem we obtain
\begin{align}\label{3.7}
& \left| \frac{2(xy)^\lambda t^3 \lambda (\lambda+1)}{\pi} \int_0^{\pi/2} \frac{(\sin \theta)^{2\lambda-1}}{((x-y)^2 + t^2 + 2xy(1- \cos \theta))^{\lambda+2}} d\theta
- \frac{t^3}{\pi} \frac{1}{((x-y)^2+t^2)^2}\right| \nonumber \\
& \quad \leq \left| \frac{2(xy)^\lambda t^3 \lambda (\lambda+1)}{\pi} \int_0^{\pi/2} \frac{(\sin \theta)^{2\lambda-1}-\theta^{2\lambda-1}}{((x-y)^2 + t^2 + 2xy(1- \cos \theta))^{\lambda+2}} d\theta \right| \nonumber \\
& \qquad + \left| \frac{2(xy)^\lambda t^3 \lambda (\lambda+1)}{\pi} \int_0^{\pi/2}\theta^{2\lambda-1}
\left( \frac{1}{((x-y)^2 + t^2 + 2xy(1- \cos \theta))^{\lambda+2}} - \frac{1}{((x-y)^2 + t^2 + xy \theta^2)^{\lambda+2}}\right) d\theta \right| \nonumber \\
& \qquad + \left| \frac{2(xy)^\lambda t^3 \lambda (\lambda+1)}{\pi} \int_0^{\pi/2} \frac{\theta^{2\lambda-1}}{((x-y)^2 + t^2 + xy \theta^2)^{\lambda+2}} d\theta
- \frac{t^3}{\pi} \frac{1}{((x-y)^2+t^2)^2}\right| \nonumber \\
& \quad \leq C \left( (xy)^\lambda t^3 \int_0^{\pi/2} \frac{\theta^{2\lambda+1}}{((x-y)^2 + t^2 + xy\theta^2)^{\lambda+2}} d\theta
+ (xy)^{\lambda+1} t^3 \int_0^{\pi/2} \frac{\theta^{2\lambda+3}}{((x-y)^2 + t^2 + xy\theta^2)^{\lambda+3}} d\theta \right) \nonumber\\
& \qquad + \frac{t^3}{\pi}\left| \frac{2 \lambda (\lambda+1)}{((x-y)^2+t^2)^2} \int_0^{\frac{\pi}{2}\sqrt{\frac{xy}{(x-y)^2+t^2}}} \frac{u^{2\lambda-1}}{(1+u^2)^{\lambda+2}} du
- \frac{1}{((x-y)^2+t^2)^2}\right| \nonumber
\end{align}
\begin{align}
& \quad \leq C t^3 \left( \frac{1}{xy((x-y)^2+t^2)} \int_0^{\frac{\pi}{2}\sqrt{\frac{xy}{(x-y)^2+t^2}}} \frac{u^{2\lambda+1}}{(1+u^2)^{\lambda+2}} du
+ \frac{1}{((x-y)^2+t^2)^2} \int_{\frac{\pi}{2}\sqrt{\frac{xy}{(x-y)^2+t^2}}}^\infty \frac{u^{2\lambda-1}}{(1+u^2)^{\lambda+2}} du\right) \nonumber \\
& \quad \leq C t^3 \left( \frac{1}{xy((x-y)^2+t^2)} \int_0^\infty \frac{u^{2\lambda+1}}{(1+u^2)^{\lambda+2}} du
+ \frac{1}{((x-y)^2+t^2)^2} \frac{1}{1+\frac{xy}{(x-y)^2+t^2}}\int_{0}^\infty \frac{u^{2\lambda-1}}{(1+u^2)^{\lambda+1}} du\right) \nonumber\\
& \quad \leq C \frac{t}{xy}, \quad x,y,t \in (0,\infty).
\end{align}
We have used that $ \int_0^\infty u^{2\lambda-1}/(1+u^2)^{\lambda+2} du = 1/(2\lambda(\lambda+1))$.\\
Finally, by proceeding in a similar way and using that $ \int_0^\infty u^{2\lambda-1}/(1+u^2)^{\lambda+1} du = 1/2\lambda$,
we get
\begin{align}\label{3.8}
& |P_{t,1}^\lambda(x,y)- P_t(x-y)|
= \left| \frac{2(xy)^\lambda t \lambda }{\pi} \int_0^{\pi/2} \frac{(\sin \theta)^{2\lambda-1}}{((x-y)^2 + t^2 + 2xy(1- \cos \theta))^{\lambda+1}} d\theta
- \frac{t}{\pi} \frac{1}{(x-y)^2+t^2}\right| \nonumber \\
& \quad \leq \left| \frac{2(xy)^\lambda t\lambda }{\pi} \int_0^{\pi/2} \frac{(\sin \theta)^{2\lambda-1}-\theta^{2\lambda-1}}{((x-y)^2 + t^2 + 2xy(1- \cos \theta))^{\lambda+1}} d\theta \right| \nonumber \\
& \qquad + \left| \frac{2(xy)^\lambda t \lambda }{\pi} \int_0^{\pi/2}\theta^{2\lambda-1}
\left( \frac{1}{((x-y)^2 + t^2 + 2xy(1- \cos \theta))^{\lambda+1}} - \frac{1}{((x-y)^2 + t^2 + xy \theta^2)^{\lambda+1}}\right) d\theta \right| \nonumber \\
& \qquad + \left| \frac{2(xy)^\lambda t \lambda }{\pi} \int_0^{\pi/2} \frac{\theta^{2\lambda-1}}{((x-y)^2 + t^2 + xy \theta^2)^{\lambda+1}} d\theta
- \frac{t}{\pi} \frac{1}{(x-y)^2+t^2}\right| \nonumber \\
& \quad \leq C \left( (xy)^\lambda t \int_0^{\pi/2} \frac{\theta^{2\lambda+1}}{((x-y)^2 + t^2 + xy\theta^2)^{\lambda+1}} d\theta
+ (xy)^{\lambda+1} t \int_0^{\pi/2} \frac{\theta^{2\lambda+3}}{((x-y)^2 + t^2 + xy\theta^2)^{\lambda+2}} d\theta \right. \nonumber\\
& \qquad \left. + t \left| \frac{2 \lambda }{(x-y)^2+t^2} \int_0^{\frac{\pi}{2}\sqrt{\frac{xy}{(x-y)^2+t^2}}} \frac{u^{2\lambda-1}}{(1+u^2)^{\lambda+1}} du
- \frac{1}{(x-y)^2+t^2}\right| \right) \nonumber\\
& \quad \leq C t \left( \int_0^{\pi/2} \frac{\theta}{(x-y)^2 + t^2 + xy\theta^2} d\theta +
\frac{1}{(x-y)^2+t^2}\int_{\frac{\pi}{2}\sqrt{\frac{xy}{(x-y)^2+t^2}}}^\infty \frac{u^{2\lambda-1}}{(1+u^2)^{\lambda+1}} du\right) \nonumber\\
& \quad \leq C \frac{t}{xy} \left(1+ \log \left( 1 + \frac{xy}{(x-y)^2} \right) \right)
, \quad x,y,t \in (0,\infty), \quad x \neq y.
\end{align}
Putting together \eqref{3.5}-\eqref{3.8} we obtain
$$J_{2,1}(x,t)
\leq C \frac{t}{x} \int_{x/2}^{3x/2} \frac{1}{y} \left( 1+\log \left( 1 + \frac{x^2}{(x-y)^2} \right) \right) dy
\leq C \frac{t}{x}, \quad x,t \in (0,\infty).$$
Moreover, we have that
\begin{align*}
J_{2,2}(x,t)
&= \left |t\partial_t \int_{x/2}^{3x/2} P_t(x-y) dy \right|
= \frac{2}{\pi} \left |t\partial_t \int_{0}^{x/2} \frac{t}{t^2+u^2} du \right|
= \frac{2}{\pi} \left |t\partial_t \left( \frac{\pi}{2} - \int_{x/2}^\infty \frac{t}{t^2+u^2} du \right) \right| \nonumber \\
& \leq C t \int_{x/2}^\infty \frac{du}{u^2} \leq C \frac{t}{x}, \quad x,t \in (0,\infty).
\end{align*}
Hence, it follows that
\begin{equation}\label{3.9}
J_2(x,t)
\leq C \frac{t}{x}, \quad x,t \in (0,\infty).
\end{equation}
We now prove \eqref{objetivo3}. Suppose firstly $|I| \leq x_I$. Then, since $q \geq 2$,
\begin{align*}
\frac{(x_I+|I|)^q}{|I|^{q+1}}\int_0^{|I|} \int_I |t\partial_t P_t^\lambda(1)(x)|^q \frac{dx dt}{t}
\leq &C \frac{x_I^q}{|I|^{q+1}}\int_0^{|I|} t^{q-1} dt \int_{x_I-|I|/2}^{x_I+|I|/2} \frac{dx}{x^q}\\
\leq &C \frac{x_I^q}{(x_I-|I|/2)^{q}}
\leq C,
\end{align*}
because $|t\partial_t P_t^\lambda(1)(x)| \leq C t/x$, $x,t \in (0,\infty)$ (see \eqref{J1}, \eqref{J3} and \eqref{3.9}).\\
Assume now that $|I|>x_I$. From \eqref{J1} and \eqref{J3} we deduce
\begin{align*}
\frac{(x_I+|I|)^q}{|I|^{q+1}}\int_0^{|I|} \int_I J_i(x,t)^q \frac{dx dt}{t}
\leq & \frac{C}{|I|} \int_0^{|I|} t^{q-1} \int_0^{x_I+|I|} \frac{dx}{(t+x)^q} dt
\leq C, \quad i=1,3.
\end{align*}
Finally, from \eqref{J2} and \eqref{3.9} it follows that
\begin{align*}
\frac{(x_I+|I|)^q}{|I|^{q+1}}\int_0^{|I|} \int_I J_2(x,t)^q \frac{dx dt}{t}
\leq & \frac{C}{|I|} \int_I \left( \int_0^x \left( \frac{t}{x} \right)^q \frac{dt}{t} + \int_x^\infty \left( \frac{x}{t} \right)^{(2\lambda+1)q} \frac{dt}{t}\right) dx
\leq C.
\end{align*}
Hence, \eqref{objetivo3} is established and the proof of $(ii) \Rightarrow (i)$ is finished.
\begin{flushright}
\qed
\end{flushright}
\subsection{Proof of $(i) \Rightarrow (ii)$} \label{subsec:1->2}
Assume that $(i)$ holds. According to Proposition~\ref{Lem_principal}, in order to see that $\mathbb{B}$ has an equivalent $q$-uniformly convex norm
it is enough to prove that there exists $C>0$ such that
\begin{equation}\label{16.1}
\| S^q_{\lambda,+}(f) \|_{L^q(0,\infty)}
\leq C \|f\|_{L^q((0,\infty),\mathbb{B})}, \quad f \in L^q((0,\infty),\mathbb{B}).
\end{equation}
Note firstly that \eqref{16.1} is a finite dimensional inequality in the following sense: if \eqref{16.1} holds when $\mathbb{B}$ is replaced
by $\mathbb{E}$, where $\mathbb{E}$ is a finite dimensional subspace of $\mathbb{B}$, with a constant $C>0$ independent of $\mathbb{E}$, then \eqref{16.1}
is also true for every $f \in L^q((0,\infty),\mathbb{B})$ with the same constant $C>0$. This fact is a consequence of the density of
$L^q(0,\infty) \otimes \mathbb{B}$ into $L^q((0,\infty),\mathbb{B})$. Recall that every $f \in L^q(0,\infty) \otimes \mathbb{B}$ can be written as $f=\sum_{j=1}^n b_j f_j$, where $b_j \in \mathbb{B}$, $f_j \in L^q(0,\infty)$, $j=1, \dots, n$, and $n \in \mathbb{N}$.\\
Let $\mathbb{E}$ be a subspace of $\mathbb{B}$ such that $\dim \mathbb{E}<\infty$. Applying again Proposition~\ref{Lem_principal}, instead of proving \eqref{16.1}
for functions taking values in $\mathbb{E}$, it is sufficient to show that
\begin{equation}\label{3.10}
\| S^q_{\lambda,+}(f) \|_{L^1(0,\infty)}
\leq C \|f\|_{H^1_o(\mathbb{R},\mathbb{E})}, \quad f \in H^1_o(\mathbb{R},\mathbb{E}),
\end{equation}
being $C>0$ a constant independent of $\mathbb{E}$.
Moreover, \eqref{3.10} holds provided that, for a certain $C>0$,
\begin{equation}\label{3.11}
\| S^q_{\lambda}(f) \|_{BMO(0,\infty)}
\leq C \|f\|_{L^\infty((0,\infty),\mathbb{E})}, \quad f \in L^\infty_c((0,\infty),\mathbb{E}),
\end{equation}
where $L^\infty_c((0,\infty),\mathbb{E})$ denotes the space of functions in $L^\infty((0,\infty),\mathbb{E})$ that have compact support. To make easier the reading of this part, the proof of that \eqref{3.11} implies \eqref{3.10} will be included in Section~\ref{sec:appendix} (see Proposition~\ref{Prop_BMO_H1}).\\
Observe that \eqref{3.11} can be written as follows
\begin{equation}\label{objetivo4}
\left\| t \partial_t P_t^\lambda(f)(x+y) \right\|_{BMO\left((0,\infty),L^q\left(\Gamma(0),\frac{dtdy}{t^2},\mathbb{E}\right)\right)}
\leq C \|f\|_{L^\infty((0,\infty),\mathbb{E})}, \quad f \in L^\infty_{c}((0,\infty),\mathbb{E}).
\end{equation}
Inequality \eqref{objetivo4} will be proved by using duality. Our objective is to show that, there exists $C>0$ such that for every
$f \in L^\infty_{c}((0,\infty),\mathbb{E})$ and $h \in H^1\left((0,\infty), L^{q'}\left(\Gamma(0),\frac{dtdy}{t^2}, \mathbb{E}^*\right) \right)$,
\begin{equation}\label{3.13}
|\langle t \partial_t P_t^\lambda(f)(x+y), h(x,y,t) \rangle|
\leq C \|f\|_{L^\infty((0,\infty),\mathbb{E})} \|h\|_{H^1\left((0,\infty), L^{q'}\left(\Gamma(0),\frac{dtdy}{t^2}, \mathbb{E}^*\right) \right)}.
\end{equation}
Recalling the atomic definition of $H^1\left((0,\infty), L^{q'}\left(\Gamma(0),\frac{dtdy}{t^2}, \mathbb{E}^*\right) \right)$,
by density arguments it is sufficient to prove \eqref{3.13} for every $f \in L^\infty_{c}((0,\infty),\mathbb{E})$ and
$h \in L^\infty_{c}\left((0,\infty), L^{q'}\left(\Gamma(0),\frac{dtdy}{t^2},\mathbb{E}^*\right) \right)
\cap H^1\left((0,\infty), L^{q'}\left(\Gamma(0),\frac{dtdy}{t^2},\mathbb{E}^*\right) \right) $.\\
Let $f \in L^\infty_{c}((0,\infty),\mathbb{E})$ and $h \in L^\infty_{c}\left((0,\infty), L^{q'}\left(\Gamma(0),\frac{dtdy}{t^2},\mathbb{E}^*\right) \right)$.
We can write,
\begin{align*}
\langle t \partial_t P_t^\lambda(f)(x+y), h(x,y,t) \rangle
=& \int_0^\infty \int_{\Gamma(0)} \langle t \partial_t P_t^\lambda(f)(x+y),h(x,y,t) \rangle \frac{dtdy}{t^2} dx\\
=& \lim_{N \to \infty} \int_0^\infty \int_{\Gamma_N(0)} \langle t \partial_t P_t^\lambda(f)(x+y),h(x,y,t)\rangle \frac{dtdy}{t^2} dx,
\end{align*}
where, for every $N \in \mathbb{N}$, the truncated cone $\Gamma_N(0)$ is defined by
\begin{equation}\label{conoN}
\Gamma_N(0)=\{(y,t) \in \Gamma(0) : 1/N < t < N\}.
\end{equation}
Note that the above limit exists because the integral is absolutely convergent. Indeed, for every $x \in (0,\infty)$,
$S^q_\lambda(f)(x) \leq 2^{1/q} S_{\lambda,+}^q(f)(x).$ Then, according to Proposition~\ref{Lem_principal},
since $\dim \mathbb{E}<\infty$, $S_\lambda^q$ is bounded from $L^2((0,\infty),\mathbb{E})$ into $L^2(0,\infty)$. By applying Hölder's inequality and by
taking into account that $f$ and $h$ have compact support we deduce that the integral under analysis is absolutely convergent.\\
Interchanging the order of integration we get
\begin{equation}\label{3.14}
\langle t \partial_t P_t^\lambda(f)(x+y), h(x,y,t) \rangle
= \lim_{N \to \infty} \int_0^\infty \langle f(z) , \Psi_N(h)(z) \rangle dz,
\end{equation}
where, for each $N \in \mathbb{N}$,
\begin{equation*}
\Psi_N(h)(z) = \int_{\Gamma_N(0)} \int_0^\infty t \partial_t P_t^\lambda(x+y,z)h(x,y,t) dx \frac{dtdy}{t^2}, \quad z \in (0,\infty).
\end{equation*}
The interchange in the order of integration is justified because by using Hölder's inequality we obtain
\begin{align}\label{finito}
\int_0^\infty \int_0^\infty \int_{\Gamma_N(0)} & |t \partial_t P_t^\lambda(x+y,z)| \|h(x,y,t)\|_{\mathbb{E}^*} \|f(z)\|_{\mathbb{E}} \frac{dtdy}{t^2} dx dz \nonumber \\
\leq & C \|f\|_{L^\infty((0,\infty),\mathbb{E})} \int_{\supp(f)} \int_{\supp(h)}
\left(\int_{\Gamma_N(0)} \|h(x,y,t)\|_{\mathbb{E}^*}^{q'} \frac{dtdy}{t^2} \right)^{1/q'} \nonumber \\
& \times \left(\int_{\Gamma_N(0)} |t \partial_t P_t^\lambda(x+y,z)|^{q} \frac{dtdy}{t^2} \right)^{1/q} dx dz \nonumber \\
\leq & C \|f\|_{L^\infty((0,\infty),\mathbb{E})} \|h\|_{L^\infty\left((0,\infty), L^{q'}\left(\Gamma(0),\frac{dtdy}{t^2},\mathbb{E}^*\right) \right)} \nonumber \\
& \times \int_{\supp(f)} \int_{\supp(h)} \left(\int_{\Gamma_N(x)} |t \partial_t P_t^\lambda(y,z)|^{q} \frac{dtdy}{t^2} \right)^{1/q} dx dz, \quad N \in \mathbb{N}.
\end{align}
Then, since $\supp(f)$ and $\supp(h)$ are compact, by using \eqref{Pt1} and \eqref{Pt2} we conclude that
$$\int_0^\infty \int_0^\infty \int_{\Gamma_N(0)} |t \partial_t P_t^\lambda(x+y,z)| \|h(x,y,t)\|_{\mathbb{E}^*} \|f(z)\|_{\mathbb{E}} \frac{dtdy}{t^2} dx dz
< \infty, \quad N \in \mathbb{N}.$$
For the incoming reasoning it is convenient to consider the operator
$$g \longmapsto t \partial_t P_t^\lambda(\Psi_N(g))(x+y).$$
In order to make this proof more legible the main properties of this operator will be shown in Section~\ref{sec:appendix} (Appendix 2).\\
By interchanging the order of integration we can write, for each $x,t \in (0,\infty)$, $y \in \mathbb{R}$, and $N \in \mathbb{N}$,
\begin{align}\label{(*b)}
t \partial_t P_t^\lambda(\Psi_N(h))(x+y)
= & \int_0^\infty \int_{\Gamma_N(0)} \int_0^\infty t \partial_t P_t^\lambda(x+y,z) s \partial_s P_s^\lambda(v+u,z)h(v,u,s) dv \frac{dsdu}{s^2} dz \nonumber\\
= & \int_0^\infty \int_{\Gamma_N(0)} k^\lambda_{s,t}(x,y;u,v) h(v,u,s)\frac{dsdu}{s^2} dv
= \Phi_N(h)(x,y,t),
\end{align}
where the kernel $k^{\lambda}_{s,t}$ is given by
$$ k^\lambda_{s,t}(x,y;u,v) = \int_0^\infty t \partial_t P_t^\lambda(x+y,z) s \partial_s P_s^\lambda(v+u,z) dz,
\quad v,x,s,t \in (0,\infty), \ u,y \in \mathbb{R}.$$
The interchange in the order of integration is justified because the integral is absolutely convergent.
Indeed, according to \eqref{(*)}, \eqref{Pt1} and \eqref{Pt2} we have that
\begin{align*}
\int_0^\infty \int_0^\infty \int_{\Gamma_N(0)} & |t \partial_t P_t^\lambda(x+y,z)| \ |s \partial_s P_s^\lambda(v+u,z)| \ \|h(v,u,s)\|_{\mathbb{E}^*} \frac{dsdu}{s^2} dv dz \\
\leq & C \int_0^\infty \frac{t z^\lambda}{(||x+y|-z|+t)^{\lambda+2}} \int_{\supp(h)} \left(\int_{\Gamma_N(0)} \|h(v,u,s)\|_{\mathbb{E}^*}^{q'} \frac{dsdu}{s^2} \right)^{1/q'} \\
& \times \left(\int_{\Gamma_N(0)} |s \partial_s P_s^\lambda(v+u,z)|^{q} \frac{dsdu}{s^2} \right)^{1/q} dv dz <\infty, \
x,t \in (0,\infty), \ y \in \mathbb{R}, \ N \in \mathbb{N}.
\end{align*}
In Section~\ref{sec:appendix}, Proposition~\ref{Pro5.2}, we establish that the sequence
of operators $\{\Phi_N\}_{N \in \mathbb{N}}$ is uniformly bounded from $H^1\left((0,\infty),L^{q'}\left(\Gamma(0),\frac{dtdy}{t^2},\mathbb{E}^*\right)\right)$
into $L^1\left((0,\infty),L^{q'}\left(\Gamma(0),\frac{dtdy}{t^2},\mathbb{E}^*\right)\right)$.\\
We now return back to \eqref{3.14}. Let $N \in \mathbb{N}$. We can write
\begin{equation}\label{3.17}
\int_0^\infty \langle f(z) ,\Psi_N(h)(z) \rangle dz
= 4 \int_0^\infty \int_0^\infty \langle t\partial_tP_t^\lambda(f)(y) , t\partial_tP_t^\lambda(\Psi_N(h))(y) \rangle \frac{dydt}{t}.
\end{equation}
This equality can be shown by proceeding as in the proof of \cite[Proposition 4.4]{BCFR2} and by taking into account the following facts:
\begin{itemize}
\item $f \in L^\infty_c((0,\infty),\mathbb{E})$.
\item $(1+z^2)^{-1}\Psi_N(h) \in L^1((0,\infty),\mathbb{E}^*)$. Indeed, arguing as in \eqref{finito} it can be proved that
$\Psi_N(h) \in L^\infty((0,\infty),\mathbb{E}^*)$.
\item Since condition $(i)$ is assumed, if we define
$$C_\lambda^q(f)(x)
=\left( \sup_{I \ni x} \frac{1}{|I|}\int_0^{|I|} \int_I \|t \partial_t P_t^\lambda(f)(y)\|_\mathbb{E}^q \frac{dydt}{t} \right)^{1/q}, \quad x \in (0,\infty),$$
where the supremum is taken over all bounded intervals $I \subset (0,\infty)$ such that $x \in I$, then $C_\lambda^q(f) \in L^\infty(0,\infty)$.
\item $\Phi_N(h) \in L^1\left((0,\infty),L^{q'}\left(\Gamma(0),\frac{dtdy}{t^2},\mathbb{E}^*\right)\right)$ because
$h \in H^1\left((0,\infty),L^{q'}\left(\Gamma(0),\frac{dtdy}{t^2},\mathbb{E}^*\right)\right)$ (Proposition~\ref{Pro5.2}).
\end{itemize}
By using Hölder's inequality and \eqref{3.17} if follows that (see \cite[Proposition 4.3]{BCFR2})
\begin{align*}
\left|\int_0^\infty \langle f(z) ,\Psi_N(h)(z) \rangle dz \right|
\leq & C \int_0^\infty C_\lambda^q(f)(x) S_{\lambda,+}^{q'}(\Psi_N(h))(x)dx \\
\leq & C \|C_\lambda^q(f)\|_{L^\infty(0,\infty)} \|S_{\lambda,+}^{q'}(\Psi_N(h))\|_{L^1(0,\infty)}.
\end{align*}
Finally, since $(i)$ holds we get
\begin{align*}
|\langle t \partial_t P_t^\lambda(f)(x+y), h(x,y,t) \rangle|
\leq C & \|C_\lambda^q(f)\|_{L^\infty(0,\infty)} \limsup_{N \to \infty} \|\Phi_N(h)\|_{L^1\left((0,\infty),L^{q'}\left(\Gamma(0),\frac{dtdy}{t^2},\mathbb{E}^*\right)\right)} \\
\leq C & \|f_o\|_{BMO_o(\mathbb{R},\mathbb{E})} \|h\|_{H^1\left((0,\infty),L^{q'}\left(\Gamma(0),\frac{dtdy}{t^2},\mathbb{E}^*\right)\right)},
\end{align*}
being $f_o$ the odd extension of $f$ to $\mathbb{R}$.\\
Thus the proof of $(ii)$ is completed.
\begin{flushright}
\qed
\end{flushright}
\section{Proof of Theorem~\ref{Th_4.1_OX}} \label{sec:proof2}
\subsection{Proof of $(ii) \Rightarrow (i)$} \label{subsec2:1->2}
Assume that $(ii)$ holds. Let $f$ be an odd $\mathbb{B}$-valued function such that $\int_0^\infty \|f(z)\|_\mathbb{B}/(1+z^2) dz <\infty$.
If
$$\sup_{I} \frac{1}{|I|} \int_0^{|I|} \int_I \|t \partial_t P_t^\lambda(f)(y)\|_\mathbb{B}^q \frac{dydt}{t}=\infty,$$
where the supremum is taken over all the bounded intervals $I \subset (0,\infty)$, we have nothing to prove. Assume that
$$\sup_{I} \frac{1}{|I|} \int_0^{|I|} \int_I \|t \partial_t P_t^\lambda(f)(y)\|_\mathbb{B}^q \frac{dydt}{t}<\infty.$$
According to \cite[Propositions 4.3 and 4.4]{BCFR2}, for every $g \in L^\infty_c(0,\infty) \otimes \mathbb{B}^*$,
\begin{align}\label{18.1}
\left| \int_0^\infty \langle f(x) , g(x) \rangle dx \right|
= & 4 \left| \int_0^\infty \int_0^\infty \langle t\partial_t P_t^\lambda(f)(y) , t\partial_t P_t^\lambda(g)(y) \rangle \frac{dy dt}{t} \right| \nonumber \\
\leq &C \int_0^\infty C_\lambda^q(f)(x) S_{\lambda,+}^{q'}(g)(x) dx
\leq C \| C_\lambda^q(f) \|_{L^\infty(0,\infty)} \|S_{\lambda,+}^{q'}(g)\|_{L^1(0,\infty)},
\end{align}
being
$$C_\lambda^q(f)(x)=\left( \sup_{I \ni x} \frac{1}{|I|} \int_I \int_0^{|I|} \|t \partial_t P_t^\lambda(f)(y)\|^q_{\mathbb{B}} \frac{dtdy}{t} \right)^{1/q}, \quad x \in (0,\infty),$$
and
$$ S_{\lambda,+}^{q'}(g)(x)
= \left( \int_{\Gamma_+(x)} \| t \partial_t P_t^\lambda(g)(y) \|^{q'}_{\mathbb{B}^*} \frac{dtdy}{t^2}\right)^{1/q'}, \quad x \in (0,\infty).$$
Moreover, by using \cite[Colloraries 2.6 and 3.2]{Xu} and Proposition~\ref{Lem_principal} we deduce that
\begin{equation} \label{18.2}
\|S_{\lambda,+}^{q'}(g)\|_{L^1(0,\infty)} \leq C \|g\|_{H^1_o(\mathbb{R},\mathbb{B}^*)}, \quad g \in H^1_o(\mathbb{R},\mathbb{B}^*).
\end{equation}
Let $I = (a,b)$, being $0<a<b<\infty$. By applying \cite[Lemma 2.3]{GLY} and taking into account that $\overline{C_c(I) \otimes \mathbb{B}^*}=L^2(I,\mathbb{B}^*)$
and $(f-f_I)_I=0$, we obtain,
\begin{align*}
\left(\frac{1}{|I|}\int_I \|f(x)-f_I\|_{\mathbb{B}}^2 dx \right)^{1/2}
= & \frac{1}{|I|^{1/2}}
\sup_{\substack{g \in L^2(I,\mathbb{B}^*) \\ \|g\|_{L^2(I,\mathbb{B}^*)} \leq 1}}
\left| \int_I \langle f(x) - f_I , g(x) \rangle dx \right| \\
= & \sup_{\substack{g \in C_c(I) \otimes \mathbb{B}^* \\ \|g\|_{L^2(I,\mathbb{B}^*)} \leq 1}}
\left| \int_I \langle f(x), \frac{g(x) - g_I}{|I|^{1/2}} \rangle dx \right|.
\end{align*}
If $g \in C_c(I)\otimes \mathbb{B}^*$ it is clear that $h=(g-g_I)\chi_I/2|I|^{1/2}$ is a $2$-atom in $(0,\infty)$ for $H^1_o(\mathbb{R},\mathbb{B}^*)$, because $\supp(h) \subset I$, $h_I=0$ and
\begin{align*}
\|h\|_{L^2((0,\infty),\mathbb{B}^*)}
\leq & \frac{\sqrt{2}}{2|I|^{1/2}} \left( \int_I \|g(x)\|^2_{\mathbb{B}^*}dx + \frac{1}{|I|} \left(\int_I \|g(x)\|_{\mathbb{B}^*}dx\right)^2 \right)^{1/2}
\leq \frac{1}{|I|^{1/2}}.
\end{align*}
Hence, by applying \eqref{18.1} and \eqref{18.2}, we deduce
\begin{align*}
\left(\frac{1}{|I|}\int_I \|f(x)-f_I\|_{\mathbb{B}}^2 dx \right)^{1/2}
\leq & C \| C_\lambda^q(f) \|_{L^\infty(0,\infty)} \sup_{\substack{g \in C_c(I) \otimes \mathbb{B}^* \\ \|g\|_{L^2(I,\mathbb{B}^*)} \leq 1}}
\left\| \frac{g - g_I}{2|I|^{1/2}} \right\|_{H^1_o(\mathbb{R},\mathbb{B}^*)}
\leq C \| C_\lambda^q(f) \|_{L^\infty(0,\infty)}.
\end{align*}
Suppose now that $I=(0,\beta)$, for some $\beta>0$. By proceeding as before we get
\begin{align*}
\left(\frac{1}{\beta}\int_0^\beta \|f(x)\|_{\mathbb{B}}^2 dx \right)^{1/2}
= & \sup_{\substack{g \in C_c(0,\beta) \otimes \mathbb{B}^* \\ \|g\|_{L^2((0,\beta),\mathbb{B}^*)} \leq 1}}
\left| \int_0^\beta \langle f(x), \frac{g(x)}{\beta^{1/2}} \rangle dx \right|
\end{align*}
and the same conclusion follows because $h=g\chi_{(0,\beta)}/\beta^{1/2}$ satisfies $\supp(h) \subset [0,\beta]$ and $\|h\|_{L^2((0,\infty),\mathbb{B}^*)} \leq 1/\beta^{1/2}$.\\
Thus $(i)$ is established.
\begin{flushright}
\qed
\end{flushright}
\subsection{Proof of $(i) \Rightarrow (ii)$} \label{subsec2:1->2}
Suppose that $(i)$ holds. In order to see $(ii)$, according to Proposition~\ref{Lem_principal},
\cite[Colloraries 2.6 and 3.2]{Xu} and \eqref{equivalentes}, we prove that, for some $C>0$,
\begin{equation}\label{19.1}
\| S_{\lambda}^{q'} (g) \|_{L^{q'}(0,\infty)} \leq C \|g\|_{L^{q'}((0,\infty),\mathbb{B}^*)}, \quad g \in L^{q'}((0,\infty),\mathbb{B}^*).
\end{equation}
Moreover, it is sufficient to see that, there exists $C>0$ such that
\begin{equation}\label{19.2}
\| S_\lambda^{q'} (g) \|_{L^{q'}(0,\infty)} \leq C \|g\|_{L^{q'}((0,\infty),\mathbb{E}^*)}, \quad g \in L^{q'}((0,\infty),\mathbb{E}^*),
\end{equation}
for every subspace $\mathbb{E}$ of $\mathbb{B}$, being $\dim \mathbb{E} < \infty$.
Indeed, assume that \eqref{19.2} holds and take $g \in L^{q'}((0,\infty),\mathbb{B}^*)$. By \cite[Lemma 2.3]{GLY}, we can write
\begin{align*}
\| S_{\lambda}^{q'} (g) \|_{L^{q'}(0,\infty)}
& = \|t\partial_t P_t^\lambda(g)(x+y)\|_{L^{q'}\left((0,\infty) \times \Gamma(0),\frac{dtdy}{t^2}dx,\mathbb{B}^*\right) } \\
& = \sup_{\substack{G \in L^{q}\left((0,\infty) \times \Gamma(0),\frac{dtdy}{t^2}dx,\mathbb{B}\right) \\
\|G\|_{L^{q}\left((0,\infty) \times \Gamma(0),\frac{dtdy}{t^2}dx,\mathbb{B}\right) }\leq 1}}
\left|\int_0^\infty \int_{\Gamma(0)} \langle t \partial_t P_t^\lambda(g)(x+y), G(x,y,t) \rangle \frac{dtdy}{t^2} dx \right| \\
& = \sup_{\substack{G \in L^{q}\left((0,\infty)\times\Gamma(0),\frac{dtdy}{t^2}dx \right)\otimes \mathbb{B} \\
\|G\|_{L^{q}\left((0,\infty) \times \Gamma(0),\frac{dtdy}{t^2}dx,\mathbb{B}\right) }\leq 1}}
\left|\int_0^\infty \int_{\Gamma(0)} \langle t \partial_t P_t^\lambda(g)(x+y), G(x,y,t) \rangle \frac{dtdy}{t^2} dx \right|.
\end{align*}
Observe that in the last equality we have applied that $L^{q}\left((0,\infty)\times\Gamma(0),\frac{dtdy}{t^2}dx \right)\otimes \mathbb{B}$
is a dense subspace of $L^{q}\left((0,\infty) \times \Gamma(0),\frac{dtdy}{t^2}dx,\mathbb{B}\right)$. Fix $\varepsilon>0$. There exists
$G \in L^{q}\left((0,\infty) \times \Gamma(0),\frac{dtdy}{t^2}dx \right)\otimes \mathbb{B}$, such that
$\|G\|_{L^{q}\left((0,\infty) \times \Gamma(0),\frac{dtdy}{t^2}dx,\mathbb{B}\right) }\leq 1$ and
\begin{align*}
\| S_{\lambda}^{q'} (g) \|_{L^{q'}(0,\infty)}
\leq \left|\int_0^\infty \int_{\Gamma(0)} \langle t \partial_t P_t^\lambda(g)(x+y), G(x,y,t) \rangle \frac{dtdy}{t^2} dx \right| + \varepsilon,
\end{align*}
being $G = \sum_{j=1}^n a_j G_j$, $a_j \in \mathbb{B}$, $G_j \in L^{q}\left((0,\infty) \times \Gamma(0),\frac{dtdy}{t^2}dx \right) $, $j=1, \dots, n$
and $n \in \mathbb{N}$. If we define $\mathbb{E}=\spann\{a_j\}_{j=1}^n$, it is clear that
\begin{align*}
\langle t \partial_t P_t^\lambda(g)(x+y), G(x,y,t) \rangle_{\mathbb{B}^* \times \mathbb{B}}
= & \langle t \partial_t P_t^\lambda(g)(x+y), G(x,y,t) \rangle_{\mathbb{E}^* \times \mathbb{E}}, \quad x\in (0,\infty), \ (y,t) \in \Gamma(0),
\end{align*}
because every element of $\mathbb{B}^*$ can be seen by restriction as an element of $\mathbb{E}^*$. Hence, by Hölder's inequality and \eqref{19.2} we conclude that
\begin{align*}
\| S_{\lambda}^{q'} (g) \|_{L^{q'}(0,\infty)}
& \leq \left|\int_0^\infty \int_{\Gamma(0)} \langle t \partial_t P_t^\lambda(g)(x+y), G(x,y,t) \rangle_{\mathbb{E}^* \times \mathbb{E}} \frac{dtdy}{t^2} dx \right| + \varepsilon \\
& \leq \|t \partial_t P_t^\lambda(g)(x+y)\|_{L^{q'}\left((0,\infty) \times \Gamma(0),\frac{dtdy}{t^2}dx,\mathbb{E}^*\right) }
\|G\|_{L^{q}\left((0,\infty) \times \Gamma(0),\frac{dtdy}{t^2}dx,\mathbb{E}\right) } + \varepsilon \\
& \leq C\|g\|_{L^{q'}\left((0,\infty), \mathbb{E}^*\right)} + \varepsilon
\leq C\|g\|_{L^{q'}\left((0,\infty), \mathbb{B}^*\right)} + \varepsilon,
\end{align*}
and this gives \eqref{19.1}.\\
Let $\mathbb{E}$ be a finite dimensional subspace of $\mathbb{B}$. In order to prove \eqref{19.2}, by the equivalences shown in Proposition~\ref{Lem_principal},
we are going to see that there exists $C>0$, independent of $\mathbb{E}$, such that
\begin{equation*}\label{4.3XU}
\| S_\lambda^{q'} (g) \|_{L^1(0,\infty)} \leq C \|g\|_{H^1_o(\mathbb{R},\mathbb{E}^*)}, \quad g \in H^{1}_o(\mathbb{R},\mathbb{E}^*).
\end{equation*}
Fix $g \in H^{1}_o(\mathbb{R},\mathbb{E}^*)$.
We denote, for every $N \in \mathbb{N}$, $\mathbb{E}_N=L^{q}\left( \Gamma_N(0),\dfrac{dtdy}{t^2}, \mathbb{E} \right)$, where the truncated cone $\Gamma_N(0)$
is defined in \eqref{conoN}. By invoking \cite[Corollary III.2.13]{DU} we have that
$\mathbb{E}_N^*=L^{q'}\left( \Gamma(0),\dfrac{dtdy}{t^2}, \mathbb{E}^* \right)$ and
$\left(L^1((0,\infty),\mathbb{E}_N^*)\right)^* = L^\infty((0,\infty),\mathbb{E}_N)$.\\
It is clear that
$$S_\lambda^{q'}(g)(x)
= \lim_{N \to \infty} \left( \int_{\Gamma_N(0)} \|t \partial_t P_t^\lambda(g)(x+y) \|_{\mathbb{E}^*}^{q'} \frac{dtdy}{t^2} \right)^{1/q'}, \quad x \in (0,\infty).$$
Let $m \in \mathbb{N}$ and $N \in \mathbb{N}$. Assume that $G \in L^\infty_c([0,\infty),\mathbb{E}^*)$. Estimations \eqref{(*)}, \eqref{Pt1} and \eqref{Pt2} lead to
\begin{align*}
\int_0^m & \left( \int_{\Gamma_N(0)} \| t \partial_t P_t^\lambda (G)(x+y)\|_{\mathbb{E}^*}^{q'} \frac{dtdy}{t^2} \right)^{1/q'} dx \\
\leq & C \int_0^m \left( \int_{-N}^{N} \int_{1/N}^N
\left( \int_{\supp(G)} \frac{tz^\lambda}{(||x+y|-z|+t)^{\lambda+2}} \|G(z)\|_{E^*} dz \right)^{q'} \frac{dtdy}{t^2} \right)^{1/q'} dx \\
\leq & C \int_0^m \left( \int_{-N}^{N} \int_{1/N}^N
\left( \left(\int_{\supp(G) \cap [0,2(m+N)]} + \int_{\supp(G)\cap [2(m+N),\infty)} \right) \right. \right.\\
& \left. \left. \times \frac{tz^\lambda}{(||x+y|-z|+t)^{\lambda+2}} \|G(z)\|_{E^*} dz \right)^{q'} \frac{dtdy}{t^2} \right)^{1/q'} dx \\
\leq & C \|G\|_{L^\infty((0,\infty),\mathbb{E}^*)} \int_0^m \left( \int_{-N}^{N} \int_{1/N}^N
\left( \left( \frac{(2(m+N))^\lambda}{t^{\lambda+1}} + \frac{1}{t} \right) \int_{\supp(G)} dz \right)^{q'} \frac{dtdy}{t^2} \right)^{1/q'} dx\\
\leq & C \|G\|_{L^\infty((0,\infty),\mathbb{E}^*)} |\supp(G)|,
\end{align*}
where $C>0$ does not depend on $G$.\\
Hence, recalling the atomic representation of the elements of $H^1_o(\mathbb{R},\mathbb{E}^*)$ we deduce that
$t\partial_t P_t^\lambda(g) \in L^1\left((0,m),L^{q'}\left(\Gamma_N(0),\dfrac{dtdy}{t^2},\mathbb{E}^*\right)\right)$.\\
According to \cite[Lemma 2.3]{GLY} we have that
\begin{align*}
& \left\| \left( \int_{\Gamma_N(0)} \| t \partial_t P_t^\lambda (g)(x+y)\|_{\mathbb{E}^*}^{q'} \frac{dtdy}{t^2} \right)^{1/q'} \right\|_{L^1(0,m)} \\
& \qquad \qquad = \sup_{\substack{ h \in L^\infty((0,m),\mathbb{E}_N) \\ \|h\|_{L^\infty((0,m),\mathbb{E}_N)} \leq 1}}
\left| \int_0^m \int_{\Gamma_N(0)} \langle t \partial_t P_t^\lambda (g)(x+y) , h(x,y,t) \rangle \frac{dtdy}{t^2} dx \right|.
\end{align*}
Let $h \in L^\infty((0,m),\mathbb{E}_N)$ such that $\|h\|_{L^\infty((0,m),\mathbb{E}_N)} \leq 1$. By using Hölder's inequality and repeating the above
manipulations we can see that
$$\int_0^m \int_{\Gamma_N(0)} \int_0^\infty \left| \langle t \partial_t P_t^\lambda (x+y,z)g(z) , h(x,y,t) \rangle \right| dz \frac{dtdy}{t^2} dx
< \infty.$$
Then, we can interchange the order of integration to write
\begin{align*}
& \int_0^m \int_{\Gamma_N(0)} \langle t \partial_t P_t^\lambda(g)(x+y) , h(x,y,t) \rangle \frac{dtdy}{t^2} dx
= \int_0^\infty \langle g(z) , \Psi_N( \chi_{(0,m)}(x) h)(z) \rangle dz,
\end{align*}
where, as above,
$$\Psi_N(H)(z)
= \int_{\Gamma_N(0)} \int_0^\infty t \partial_t P_t^\lambda(x+y,z) H(x,y,t) dx \frac{dtdy}{t^2}, \quad z \in (0,\infty).$$
According to \eqref{(*)} and \eqref{3.2a} we get
\begin{align*}
\| \Psi_N( \chi_{(0,m)}(x) h)(z) \|_{\mathbb{E}_N}
& \leq \int_0^m \int_{\Gamma_N(0)} |t \partial_t P_t^\lambda(x+y,z)| \ \|h(x,y,t)\|_\mathbb{E} \frac{dtdy}{t^2} dx\\
& \leq C \int_0^m \int_{\Gamma_N(0)} \frac{t}{(|x+y|-z)^2+t^2} \ \|h(x,y,t)\|_\mathbb{E} \frac{dtdy}{t^2} dx\\
& \leq C \int_0^m \left( \int_{-N}^{N} \int_{1/N}^N \left(\frac{t}{(|x+y|-z)^2+t^2}\right)^{q'} \frac{dtdy}{t^2} \right)^{1/q'} dx
\leq C, \ z \in (0,\infty),
\end{align*}
where $C>0$ depends on $m$ and $N$. Thus, this function is locally integrable.\\
Suppose for a moment that there exists $C>0$ independent of $\mathbb{E}$, $m$ and $N \in \mathbb{N}$ such that
\begin{equation}\label{objetivo6}
\sup_{I} \frac{1}{|I|} \int_0^{|I|} \int_I \|t \partial_t P_t^\lambda(\Psi_N(\chi_{(0,m)}h))(x)\|_\mathbb{E}^q \frac{dxdt}{t}
\leq C,
\end{equation}
where the supremum is taken over all the bounded intervals $I \subset (0,\infty)$. Since $(i)$ holds, by using duality,
\eqref{objetivo6} leads to
\begin{align*}
\left| \int_0^\infty \langle g(z) , \Psi_N( \chi_{(0,m)}(x) h)(z) \rangle dz \right|
& \leq C \|g\|_{H^1_o(\mathbb{R},\mathbb{E}^*)} \| \left(\Psi_N(\chi_{(0,m)}(x) h)\right)_o\|_{BMO_o(\mathbb{R},\mathbb{E})} \\
& \leq C \|g\|_{H^1_o(\mathbb{R},\mathbb{E}^*)},
\end{align*}
where $C$ depends neither on $m,N \in \mathbb{N}$ nor on $\mathbb{E}$.\\
We conclude, by taking $m \to \infty$, that
$$\left\| \left( \int_{\Gamma_N(0)} \| t \partial_t P_t^\lambda (g)(x+y)\|_{\mathbb{E}^*}^{q'} \frac{dtdy}{t^2} \right)^{1/q'} \right\|_{L^1(0,\infty)}
\leq C \|g\|_{H^1_o(\mathbb{R},\mathbb{E}^*)},$$
where $C>0$ is independent of $N \in \mathbb{N}$, and by taking $N \to \infty$, it follows that
\begin{align*}
\|S_\lambda^{q'}(g)\|_{L^1(0,\infty)}
\leq & C \|g\|_{H^1_o(\mathbb{R},\mathbb{E}^*)}.
\end{align*}
We now prove \eqref{objetivo6}. Fix $m,N \in \mathbb{N}$ and a bounded interval $I \subset (0,\infty)$. We decompose
$H_{m}=\chi_{(0,m)}h$ as follows
$$H_{m}
= H_{m} \chi_{2I} + H_{m} \chi_{(0,\infty) \setminus 2I}
= H_{m}^1 + H_{m}^2.$$
By proceeding as in \eqref{3.1.a} and by using Proposition~\ref{Pro5.2} we get
\begin{align}\label{4.4}
\frac{1}{|I|} \int_0^{|I|} & \int_I \|t \partial_t P_t^\lambda(\Psi_N(H_{m}^1))(x)\|_\mathbb{E}^q \frac{dxdt}{t}
= \frac{1}{|I|} \int_0^{|I|} \int_I \|\Phi_N(H_{m}^1)(x,0,t)\|_\mathbb{E}^q \frac{dxdt}{t} \nonumber \\
\leq & C \frac{1}{|I|} \|\Phi_N(H_{m}^1)\|^q_{L^q\left((0,\infty),L^q\left( \Gamma(0),\frac{dtdy}{t^2},\mathbb{E} \right)\right)}
\leq C \frac{1}{|I|} \int_{2I} \|H_{m}(z,x,t)\|^q_{\mathbb{E}_N} dz
\leq C \|h\|^q_{L^\infty((0,m),\mathbb{E}_N)}
\leq C,
\end{align}
where $C>0$ does not depend on $m$ and $N$. Here $\Phi_N$ is defined as in \eqref{(*b)}.\\
On the other hand, for each $x,t \in (0,\infty)$,
\begin{align*}
\|t \partial_t P_t^\lambda(\Psi_N(H_{m}^2))(x)\|_\mathbb{E}
\leq & \int_{(0,m) \setminus 2I} \int_{\Gamma_N(0)} |k_{s,t}^\lambda(x,0;u,v)| \|h(v,u,s)\|_\mathbb{E} \frac{dsdu}{s^2} dv.
\end{align*}
We claim that
\begin{equation}\label{kst}
|k^\lambda_{s,t}(x,y;u,v)|
\leq C \frac{st}{(||x+y|-|u+v||+s+t)^3}, \quad v,x,s,t \in (0,\infty), \ u,y \in \mathbb{R}.
\end{equation}
Indeed, let $v,x,s,t \in (0,\infty)$ and $u,y \in \mathbb{R}$.
Since $\{P_t^\lambda\}_{t \geq 0}$ is a semigroup of operators we have that
$$\int_0^\infty P_t^\lambda(x,z)P_s^\lambda(z,y)dz=P_{t+s}^\lambda(x,y).$$
Then, we can write
\begin{align*}
k^\lambda_{s,t}(x,y;u,v)
= & t s \partial_t \partial_s \int_0^\infty P_t^\lambda(x+y,z) P_s^\lambda(z,u+v) dz
= t s \partial_t \partial_s P_{t+s}^\lambda(x+y,u+v) \nonumber \\
= & t s \partial_r^2 P_r^\lambda(x+y,u+v)_{|{r=t+s}}.
\end{align*}
By \cite[in the bottom of the p. 280]{BFMT} it follows that \eqref{kst} holds.\\
From \eqref{kst} we deduce that, for every $x,v,t \in (0,\infty)$,
\begin{align*}
\int_{\Gamma_N(0)}& |k_{s,t}^\lambda(x,0;u,v)| \|h(v,u,s)\|_\mathbb{E} \frac{dsdu}{s^2}
\leq \|h\|_{L^\infty((0,m),\mathbb{E}_N)} \left( \int_{\Gamma_N(0)} |k_{s,t}^\lambda(x,0;u,v)|^{q'} \frac{dsdu}{s^2} \right)^{1/q'} \\
\leq & C t \left( \int_{\mathbb{R}} \int_{|u|}^\infty \frac{dsdu}{(|x-|u+v||+s+t)^{2q'+2}} \right)^{1/q'}
\leq C t \left( \int_{\mathbb{R}} \frac{du}{(|x-|u+v||+|u|+t)^{2q'+1}} \right)^{1/q'}.
\end{align*}
In order to estimate the last integral we distinguish several cases. We have that
\begin{align*}
\int_0^\infty \frac{du}{(|x-u-v|+u+t)^{2q'+1}}
\leq & \int_0^{|x-v|} \frac{du}{(|x-v|+t)^{2q'+1}} + \int_{|x-v|}^\infty \frac{du}{(-|x-v|+2u+t)^{2q'+1}} \nonumber \\
\leq & \frac{C}{(|x-v|+t)^{2q'}}, \quad x,v,t \in (0,\infty),
\end{align*}
and
\begin{align*}
\int_0^\infty & \frac{du}{(|x-|u-v||+u+t)^{2q'+1}}
\leq \int_0^{\max\{\min\{v,x-v\},0\}} \frac{du}{(x-v+t)^{2q'+1}} \nonumber \\
& + \int_{\max\{\min\{v,x-v\},0\}}^v \frac{du}{(2u+v-x+t)^{2q'+1}}
+ \int_v^{x+v} \frac{du}{(x+v+t)^{2q'+1}}
+ \int_{x+v}^\infty \frac{du}{(2u-x-v+t)^{2q'+1}} \nonumber \\
\leq & \frac{C}{(|x-v|+t)^{2q'}}, \quad x,v,t \in (0,\infty).
\end{align*}
Hence, we obtain
$$ \int_{\Gamma_N(0)} |k_{s,t}^\lambda(x,0;u,v)| \|h(v,u,s)\|_\mathbb{E} \frac{dsdu}{s^2}
\leq C \frac{t}{(|x-v|+t)^{2}}, \quad x,v,t \in (0,\infty).$$
Finally, we deduce
\begin{align}\label{45.1}
\frac{1}{|I|} \int_0^{|I|} \int_I \|t \partial_t P_t^\lambda(\Psi_N(H_{m}^2))(x)\|_\mathbb{E}^q \frac{dxdt}{t}
\leq C & \frac{1}{|I|} \int_0^{|I|} \int_I t^{q-1} \left(\int_{(0,\infty) \setminus 2I} \frac{dv}{|x-v|^2} \right)^q dxdt
\leq C,
\end{align}
where $C>0$ depends neither on $m,N$ nor on $I$.\\
By combining \eqref{4.4} and \eqref{45.1} we establish \eqref{objetivo6}.\\
Thus the proof of this theorem is completed.
\begin{flushright}
\qed
\end{flushright}
\section{Appendices} \label{sec:appendix}
In this section we show two results that have been very useful in the proof of Theorems~\ref{Th_3.1_OX} and \ref{Th_4.1_OX}.
\subsection{Appendix 1}\label{subsec:app1}
The following property was used in Subsection~\ref{subsec:1->2}.
\begin{Prop}\label{Prop_BMO_H1}
Let $\mathbb{B}$ be a Banach space, $\lambda>0$ and $2 \leq q < \infty$. Suppose that
\begin{equation}\label{BMO-L^inf}
\|S^q_\lambda(f)\|_{BMO(0,\infty)} \leq C \|f\|_{L^\infty((0,\infty),\mathbb{B})}, \quad f \in L^\infty_{c}((0,\infty),\mathbb{B}).
\end{equation}
Then,
\begin{equation}\label{H1-L1}
\|S^q_{\lambda,+}(f)\|_{L^1(0,\infty)} \leq C \|f\|_{H^1_o(\mathbb{R},\mathbb{B})}, \quad f \in H^1_o(\mathbb{R},\mathbb{B}).
\end{equation}
\end{Prop}
\begin{proof}
According to \eqref{equivalentes} to show \eqref{H1-L1} it is sufficient to see that, for every $f \in H^1_o(\mathbb{R},\mathbb{B})$,
\begin{equation}\label{5.3}
\|S^q_{\lambda}(f)\|_{L^1(0,\infty)} \leq C \|f\|_{H^1_o(\mathbb{R},\mathbb{B})}.
\end{equation}
Let $f \in H^1_o(\mathbb{R},\mathbb{B})$. We write $f=\sum_{j=1}^\infty \lambda_j a_j$ on $(0,\infty)$ where, for every $j \in \mathbb{N}$,
$a_j$ is an $o$-atom and $\lambda_j \in \mathbb{C}$, being $\sum_{j=1}^\infty |\lambda_j|<\infty$. Here by $o$-atom we mean the class
of atoms defined in the introduction, as follows: $a$ is an
$\infty$-atom supported on $(0,\infty)$ or $a=b \chi_{(0,\delta)}/\delta$, for a certain $b \in \mathbb{B}$, being $\|b\|_\mathbb{B}=1$
and $\delta>0$. \\
We have that
$$\partial_t \int_0^\infty P_t^\lambda(y,z) f(z)dz
= \sum_{j=1}^\infty \lambda_j \int_0^\infty \partial_t P_t^\lambda(y,z) a_j(z)dz , \quad t\in (0,\infty), \ y \in \mathbb{R}.$$
This equality is justified because the serie
$$\sum_{j=1}^\infty |\lambda_j| \int_0^\infty |\partial_t P_t^\lambda(y,z)| \|a_j(z)\|_\mathbb{B} dz$$
is uniformly convergent in $y \in \mathbb{R}$ ant $t \in K$, for every compact subset $K \subset (0,\infty)$. Indeed, let $K$ be a compact subset of
$(0,\infty)$. By \eqref{(*)} and \eqref{3.2a} we get
$$|\partial_t P_t^\lambda(y,z)|
\leq C, \quad t \in K, \ y \in \mathbb{R}, \ z \in (0,\infty).$$
Then, since $a_j$ is an $o$-atom, for every $j \in \mathbb{N}$, it follows that
\begin{align*}
\sum_{j=1}^\infty |\lambda_j| \int_0^\infty |\partial_t P_t^\lambda(y,z)| \|a_j(z)\|_\mathbb{B} dz
\leq C \sum_{j=1}^\infty |\lambda_j|
< \infty, \quad t \in K, \ y \in \mathbb{R}.
\end{align*}
Hence, we can write
\begin{align*}
S_\lambda^q(f)(x)
= & \left( \int_{\Gamma(x)} \left\| t \partial_t P_t^\lambda\left( \sum_{j=1}^\infty \lambda_j a_j \right) (y) \right\|^q_\mathbb{B} \frac{dtdy}{t^2}\right)^{1/q} \nonumber \\
= & \left( \int_{\Gamma(x)} \left\|\sum_{j=1}^\infty \lambda_j t \partial_t P_t^\lambda\left( a_j \right) (y) \right\|^q_\mathbb{B} \frac{dtdy}{t^2}\right)^{1/q}
\leq \sum_{j=1}^\infty |\lambda_j| S_\lambda^q(a_j)(x), \quad x \in (0,\infty).
\end{align*}
In order to see \eqref{5.3} it is sufficient to show that there exists $C>0$ such that
\begin{equation}\label{L1-atom}
\|S_\lambda^q(a)\|_{L^1(0,\infty)} \leq C,
\end{equation}
for every $o$-atom $a$. \\
To prove \eqref{L1-atom} we use a procedure employed by Journé (\cite[p. 49-51]{J}).
Let $a$ be an $o$-atom supported in the interval $I=(z_0-|I|/2,z_0+|I|/2) \subset (0,\infty)$ and
$\|a\|_{L^\infty((0,\infty),\mathbb{B})} \leq 1/|I|$. We denote again $2I=(z_0-|I|,z_0+|I|) \cap (0,\infty)$ and
by $J$ we represent the interval $(z_0+|I|,z_0+|I|+|2I|)$.\\
By \cite[Lemma 1.1 (b), p. 217]{G} and \eqref{BMO-L^inf} we get
\begin{align*}
\frac{1}{|2I|} \int_{2I} S_\lambda^q(a)(x)dx
\leq & \frac{1}{|2I|} \int_{2I} |S_\lambda^q(a)(x) - S_\lambda^q(a)_{2I} | dx + |S_\lambda^q(a)_{2I} - S_\lambda^q(a)_{J}| + S_\lambda^q(a)_{J} \\
\leq & C \|S_\lambda^q(a)\|_{BMO(0,\infty)} + S_\lambda^q(a)_{J}
\leq C \|a\|_{L^\infty((0,\infty),\mathbb{B})} + S_\lambda^q(a)_{J}
\leq \frac{C}{|I|} + S_\lambda^q(a)_{J}.
\end{align*}
Hence, since $J \subset (0,\infty) \setminus 2I$ and $|J|=|2I|$,
\begin{align*}
\|S_\lambda^q(a)\|_{L^1(0,\infty)}
= & \int_{2I} S_\lambda^q(a)(x)dx + \int_{(0,\infty) \setminus 2I} S_\lambda^q(a)(x)dx
\leq C + |2I| S_\lambda^q(a)_{J} + \int_{(0,\infty) \setminus 2I} S_\lambda^q(a)(x)dx \\
\leq & C \left(1 + \int_{(0,\infty) \setminus 2I} S_\lambda^q(a)(x)dx \right).
\end{align*}
By writing $P_t^\lambda(y,z)=P_{t,1}^\lambda(y,z) + P_{t,2}^\lambda(y,z)$, $t,z \in (0,\infty)$, $y \in \mathbb{R}$, where
$$P_{t,1}^\lambda(y,z)
= \frac{2\lambda (|y|z)^\lambda t}{\pi} \int_0^{\pi/2} \frac{(\sin \theta)^{2\lambda-1}}{[(|y|-z)^2+t^2+2|y|z(1-\cos \theta)]^{\lambda+1}}d\theta, $$
it follows that
\begin{align}\label{objetivo5}
\int_{(0,\infty) \setminus 2I} S_\lambda^q(a)(x)dx
\leq & \int_{(0,\infty) \setminus 2I} \left\| \int_0^\infty t \partial_t P_{t,1}^\lambda(y,z)a(z)dz \right\|_{L^q\left(\Gamma(x),\frac{dtdy}{t^2},\mathbb{B}\right)} dx \nonumber \\
& + \int_{(0,\infty) \setminus 2I} \left\| \int_0^\infty t \partial_t P_{t,2}^\lambda(y,z)a(z)dz \right\|_{L^q\left(\Gamma(x),\frac{dtdy}{t^2},\mathbb{B}\right)} dx .
\end{align}
According to \eqref{(*)} and \eqref{||Pt2||} and using Minkowski's inequality we have that
\begin{align*}
\int_{(0,\infty) \setminus 2I} & \left\| \int_0^\infty t \partial_t P_{t,2}^\lambda(y,z)a(z)dz \right\|_{L^q\left(\Gamma(x),\frac{dtdy}{t^2},\mathbb{B}\right)} dx \\
\leq & 2 \int_{(0,\infty) \setminus 2I} \int_0^\infty \left\| t \partial_t P_{t,2}^\lambda(y,z) \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)} \|a(z)\|_\mathbb{B} dz dx\\
\leq & C \int_{(0,\infty) \setminus 2I} \int_I \frac{z^\lambda}{(x+z)^{\lambda+1}} \|a(z)\|_\mathbb{B} dz dx
\leq \frac{C}{|I|} \int_I z^\lambda \int_{(0,\infty) \setminus 2I} \frac{1}{(x+z)^{\lambda+1}} dx dz \\
\leq & \frac{C}{|I|} \int_I z^\lambda \left( \frac{1}{(z_0+|I|+z)^{\lambda}} + c(I) \left(\frac{1}{z^\lambda} + \frac{1}{(z_0-|I|+z)^{\lambda}} \right)\right) dz
\leq C,
\end{align*}
where $C>0$ does not depend on $a$. Here $c(I)=0$, when $z_0 \leq |I|$, and $c(I)=1$, provided that $z_0>|I|$.\\
We now deal with the integral involving $P_{t,1}^\lambda$ in \eqref{objetivo5}.
Assume first that $a=b\chi_{(0,\delta)}/\delta$, where $\delta>0$ and $b \in \mathbb{B}$ such that $\|b\|_\mathbb{B}=1$.
By using Minkowski's inequality and \eqref{||Pt1||} it follows that
\begin{align*}
\int_{(0,\infty) \setminus 2I} & \left\| \int_0^\infty t \partial_t P_{t,1}^\lambda(y,z)a(z)dz \right\|_{L^q\left(\Gamma(x),\frac{dtdy}{t^2},\mathbb{B}\right)} dx
\leq \frac{\|b\|_\mathbb{B}}{\delta}\int_{2\delta}^\infty \left\| \int_0^\delta t \partial_t P_{t,1}^\lambda(y,z)dz \right\|_{L^q\left(\Gamma(x),\frac{dtdy}{t^2}\right)} dx \\
\leq & \frac{2}{\delta}\int_{2\delta}^\infty \int_0^\delta \left\| t \partial_t P_{t,1}^\lambda(y,z) \right\|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)} dz dx
\leq \frac{C}{\delta} \int_0^\delta \delta^\lambda \int_{2\delta}^\infty \frac{dx}{|x-z|^{\lambda+1}} dz
\leq C,
\end{align*}
where $C>0$ is independent of $a$.\\
Suppose now that $\int_I a(z)dz=0$. By the fundamental theorem of calculus and Minkowski's inequality we can write
\begin{align*}
\int_{(0,\infty) \setminus 2I} & \left\| \int_I t\partial_t P_{t,1}^\lambda(y,z) a(z)dz \right\|_{L^q\left(\Gamma(x),\frac{dtdy}{t^2},\mathbb{B}\right)} dx \\
= & \int_{(0,\infty) \setminus 2I} \left\| \int_I \left[t\partial_t P_{t,1}^\lambda(y,z) - t\partial_t P_{t,1}^\lambda(y,z_0)\right] a(z)dz \right\|_{L^q\left(\Gamma(x),\frac{dtdy}{t^2},\mathbb{B}\right)} dx\\
\leq & \int_{(0,\infty) \setminus 2I} \int_I \|a(z)\|_B \|t\partial_t P_{t,1}^\lambda(y,z) - t\partial_t P_{t,1}^\lambda(y,z_0)\|_{L^q\left(\Gamma(x),\frac{dtdy}{t^2}\right)} dz dx \\
\leq & \frac{1}{|I|} \int_{(0,\infty) \setminus 2I} \int_I \left| \int_{z_0}^z \|t\partial_t \partial_u P_{t,1}^\lambda(y,u)\|_{L^q\left(\Gamma(x),\frac{dtdy}{t^2}\right)} du \right| dz dx.
\end{align*}
We are going to see that
\begin{equation}\label{5.6}
\frac{1}{|I|} \int_{(0,\infty) \setminus 2I} \int_I \left| \int_{z_0}^z \|t\partial_t \partial_u P_{t,1}^\lambda(y,u)\|_{L^q\left(\Gamma(x),\frac{dtdy}{t^2}\right)} du \right| dz dx
\leq C,
\end{equation}
where $C>0$ does not depend on $I$.\\
We have that, for every $u,t \in (0,\infty)$ and $y \in \mathbb{R}$,
\begin{align*}
\partial_t \partial_u P_{t,1}^\lambda(y,u)
= & \frac{2\lambda^2 |y|^\lambda u^{\lambda-1}}{\pi} \int_0^{\pi/2} \frac{(\sin \theta)^{2\lambda-1}}{[(|y|-u)^2+t^2+2|y|u(1-\cos \theta)]^{\lambda+1}} d\theta \\
& - \frac{4\lambda^2(\lambda+1) |y|^\lambda u^{\lambda-1}t^2}{\pi} \int_0^{\pi/2} \frac{(\sin \theta)^{2\lambda-1}}{[(|y|-u)^2+t^2+2|y|u(1-\cos \theta)]^{\lambda+2}} d\theta \\
& - \frac{4\lambda(\lambda+1) (|y|u)^\lambda }{\pi} \int_0^{\pi/2} \frac{(\sin \theta)^{2\lambda-1}[(u-|y|) +|y|(1-\cos \theta)]}{[(|y|-u)^2+t^2+2|y|u(1-\cos \theta)]^{\lambda+2}} d\theta\\
& + \frac{8\lambda(\lambda+1)(\lambda+2) (|y|u)^\lambda t^2}{\pi} \int_0^{\pi/2} \frac{(\sin \theta)^{2\lambda-1}[(u-|y|) +|y|(1-\cos \theta)]}{[(|y|-u)^2+t^2+2|y|u(1-\cos \theta)]^{\lambda+3}} d\theta.
\end{align*}
Since $\sin \theta \sim \theta$ and $2(1-\cos \theta) \sim \theta^2$, when $\theta \in [0,\pi/2]$, it follows that
\begin{align*}
|\partial_t \partial_u P_{t,1}^\lambda(y,u)|
\leq &C \left(u^{\lambda-1} \int_0^{\pi/2} \frac{|y|^\lambda \theta^{2\lambda-1}}{[(|y|-u)^2+t^2+|y|u\theta^2]^{\lambda+1}} d\theta
+ u^{\lambda} \int_0^{\pi/2} \frac{|y|^\lambda \theta^{2\lambda-1}}{[(|y|-u)^2+t^2+|y|u\theta^2]^{\lambda+3/2}} d\theta \right)\\
= & C \left( A_t^\lambda(y,u) + B_t^\lambda(y,u) \right), \quad u,t \in (0,\infty), \ y \in \mathbb{R}.
\end{align*}
We analyze firstly $A_t^\lambda(y,u)$. Assume $0 < \lambda \leq 1$. We get
\begin{align*}
\frac{|y|^\lambda \theta^{\lambda}}{[(|y|-u)^2+t^2+|y|u\theta^2]^{\lambda+1}}
\leq C \frac{(|y|u\theta^2)^{\lambda/2} }{[(|y|-u)^2+t^2+|y|u\theta^2]^{\lambda+1}}
\leq \frac{C}{(||y|-u|+t)^{\lambda+2}}, \quad 0 < |y| < 2u,
\end{align*}
and
\begin{align*}
\frac{|y|^\lambda }{[(|y|-u)^2+t^2+|y|u\theta^2]^{\lambda+1}}
\leq C \frac{|y|^{\lambda} }{[(|y|-u)^2+t^2]^{\lambda+1}}
\leq \frac{C}{(||y|-u|+t)^{\lambda+2}}, \quad 0 < 2u < |y|.
\end{align*}
Hence,
\begin{align*}
A_t^\lambda(y,u)
\leq C \frac{u^{\lambda-1}}{(||y|-u|+t)^{\lambda+2}}, \quad u,t \in (0,\infty), \ y \in \mathbb{R}.
\end{align*}
By proceeding as in \eqref{bound1} we obtain
$$\|t A_t^\lambda(y,u) \|_{L^q\left(\Gamma(x),\frac{dtdy}{t^2}\right)}
\leq C \|t A_t^\lambda(y,u) \|_{L^q\left(\Gamma_+(x),\frac{dtdy}{t^2}\right)}
\leq C \frac{u^{\lambda-1}}{|x-u|^{\lambda+1}}, \quad u,x \in (0,\infty).$$
Also notice that
\begin{align*}
\int_{(0,\infty) \setminus 2I} \frac{dx}{|x-z_0|^{\lambda+1}}
\leq \frac{C}{|I|^\lambda},
\end{align*}
and that $|x-u| \geq |x-z_0|/2$ provided that $u \in I$ and $x \in (0,\infty)\setminus 2I$. By combining these facts we conclude
\begin{align} \label{5.7}
\frac{1}{|I|} \int_{(0,\infty) \setminus 2I} &\int_I \left| \int_{z_0}^z \|t A_t^\lambda(y,u) \|_{L^q\left(\Gamma(x),\frac{dtdy}{t^2}\right)} du \right| dz dx
\leq \frac{C}{|I|} \int_{(0,\infty) \setminus 2I} \frac{dx}{|x-z_0|^{\lambda+1}} \int_I \left| \int_{z_0}^z u^{\lambda-1} du \right| dz \nonumber \\
\leq & \frac{C}{|I|^{\lambda+1}}\int_I \left| z^\lambda - z_0^\lambda \right| dz
= \frac{C}{|I|^{\lambda+1}} \left(\int_0^{|I|/2} [(z_0+w)^\lambda - z_0^\lambda] dw + \int_{-|I|/2}^0 [z_0^\lambda - (z_0+w)^\lambda] dw \right) \nonumber\\
\leq & \frac{C}{|I|^{\lambda+1}} \int_0^{|I|/2} w^\lambda dw
\leq C.
\end{align}
We have used that $(a+b)^\alpha \leq a^\alpha + b^\alpha$, when $a,b>0$ and $0 < \alpha \leq 1$.\\
Suppose now $\lambda>1$. We have that
\begin{align*}
A_t^\lambda(y,u)
= & \int_0^{\pi/2} \left(\frac{|y|u\theta^2}{(|y|-u)^2+t^2+|y|u\theta^2} \right)^{\lambda-1}
\frac{|y|\theta}{(|y|-u)^2+t^2+|y|u\theta^2)^2} d\theta \\
\leq C & \int_0^{\pi/2} \frac{|y|\theta}{[(|y|-u)^2+t^2+|y|u\theta^2]^{2}}d\theta
=C A_t^1(y,u) , \quad u,t \in (0,\infty), \ y \in \mathbb{R}.
\end{align*}
Then, by using what we have proved in the above case we get
\begin{align}\label{5.8}
\frac{1}{|I|} \int_{(0,\infty) \setminus 2I} &\int_I \left| \int_{z_0}^z \|t A_t^\lambda(y,u) \|_{L^q\left(\Gamma(x),\frac{dtdy}{t^2}\right)} du \right| dz dx
\leq C.
\end{align}
Finally, to treat the term $B_t^\lambda(y,u)$ we make the change of variables $\theta = \phi \sqrt{(|y|-u)^2 + t^2}/\sqrt{|y|u}$ and we obtain
\begin{align*}
B_t^\lambda(y,u)
\leq & \frac{C}{(||y|-u|+t)^3} \int_0^\infty \frac{\phi^{2\lambda-1}}{(1+\phi^2)^{\lambda+1/2}} d\phi
\leq \frac{C}{(||y|-u|+t)^3}, \quad u,t \in (0,\infty), \ y \in \mathbb{R}.
\end{align*}
As above it follows that
\begin{align}\label{5.9}
\frac{1}{|I|} \int_{(0,\infty) \setminus 2I} &\int_I \left| \int_{z_0}^z \|t B_t^\lambda(y,u) \|_{L^q\left(\Gamma(x),\frac{dtdy}{t^2}\right)} du \right| dz dx
\leq C.
\end{align}
Note that the constants $C$ in \eqref{5.7}, \eqref{5.8} and \eqref{5.9} do not depend on $I$. Thus \eqref{5.6} is shown.\\
Putting together all the estimations that we have just obtained, \eqref{L1-atom} is proved and the proof of this proposition is finished.
\end{proof}
\subsection{Appendix 2}\label{subsec:app2}
In this part we study in detail the operator $\Phi_N$, $N \in \mathbb{N}$, which appears in Subsections~\ref{subsec:1->2} and \ref{subsec2:1->2}.
We prove that the sequence $\{\Phi_N\}_{N \in \mathbb{N}}$ can be seen as a uniform (in a suitable sense) family of vector
valued Calderón-Zygmund operators. Consequently, the mapping properties that we need for $\Phi_N$, $N \in \mathbb{N}$, follow from the general theory
(\cite{RbRuT}).\\
Let $\mathbb{B}$ be a Banach space, $1 < q < \infty$ and $N \in \mathbb{N}$. For every $h \in L^\infty_c\left((0,\infty),L^q\left(\Gamma_N(0), \dfrac{dydt}{t^2},\mathbb{B}\right)\right)$
we define
$$ \Phi_N(h)(x,y,t)
= \int_0^\infty \int_{\Gamma_N(0)} k^\lambda_{s,t}(x,y;u,v) h(v,u,s)\frac{dsdu}{s^2} dv,
\quad x,t \in (0,\infty), \ y \in \mathbb{R},$$
where
$$ k^\lambda_{s,t}(x,y;u,v) = \int_0^\infty t \partial_t P_t^\lambda(x+y,z) s \partial_s P_s^\lambda(v+u,z) dz,
\quad v,x,s,t \in (0,\infty), \ u,y \in \mathbb{R}.$$
In Subsection~\ref{subsec:1->2} it was proved that the integral defining $ \Phi_N(h)(x,y,t)$ is absolutely convergent for every
$h \in L^\infty_c\left((0,\infty),L^q\left(\Gamma_N(0), \dfrac{dydt}{t^2},\mathbb{B}\right)\right)$, $x,t \in (0,\infty)$, $y \in \mathbb{R}$
and $2 \leq q < \infty$. Notice that this property is also true for $1<q<2$.\\
To simplify the notation we write in the sequel $\mathbb{F}^q=L^q\left(\Gamma(0),\dfrac{dt dy}{y^2},\mathbb{B}\right)$.
\begin{Lem}\label{Lem5.1}
Let $\mathbb{B}$ be a Banach space, $\lambda>0$ and $1<q<\infty$. Then, for every $N \in \mathbb{N}$, the operator $\Phi_N$ is bounded from
$L^q((0,\infty),\mathbb{F}^q)$ into itself. Moreover, there exists $C>0$ such that, for every $N \in \mathbb{N}$,
$$\|\Phi_N(g)\|_{L^q((0,\infty),\mathbb{F}^q)}
\leq C \|g\|_{L^q((0,\infty),\mathbb{F}^q)}, \quad g \in L^q((0,\infty),\mathbb{F}^q).$$
\end{Lem}
\begin{proof}
Let $N \in \mathbb{N}$ and $g \in L^q((0,\infty),\mathbb{F}^q)$. Hölder's inequality implies that
\begin{align*}
|\Phi_N(g)(x,y,t)|
\leq & \left(\int_0^\infty \int_{\Gamma_N(0)} |k^\lambda_{s,t}(x,y;u,v)| \frac{dsdu}{s^2} dv \right)^{1/q'} \\
& \times \left(\int_0^\infty \int_{\Gamma_N(0)} |k^\lambda_{s,t}(x,y;u,v)| \ \|g(v,u,s)\|_\mathbb{B}^q \frac{dsdu}{s^2} dv \right)^{1/q},
\quad x,t \in (0,\infty), \ y \in \mathbb{R}.
\end{align*}
By \eqref{kst} it follows that
\begin{align}\label{int_kst}
\int_0^\infty \int_{\Gamma_N(0)} |k^\lambda_{s,t}(x,y;u,v)| \frac{dsdu}{s^2} dv
\leq & C \int_{\Gamma(0)}\int_0^\infty \frac{st}{(z+s+t)^3} dz \frac{dsdu}{s^2}
\leq C \int_0^\infty \frac{t}{(s+t)^2}ds
\leq C.
\end{align}
Note that $C$ does not depend on $N$.
Now \eqref{int_kst} leads to
\begin{align*}
\|\Phi_N(g)\|_{L^q((0,\infty),\mathbb{F}^q)}^q
\leq C & \int_{\Gamma_N(0)} \int_0^\infty \left( \int_0^\infty \int_{\Gamma(0)} |k^\lambda_{s,t}(x,y;u,v)| \frac{dtdy}{t^2} dx \right) \|g(v,u,s)\|_\mathbb{B}^q \frac{dsdu}{s^2} dv \\
\leq C & \|g\|^q_{L^q((0,\infty),\mathbb{F}^q)}.
\end{align*}
\end{proof}
In the next lemma we introduce, for every $N \in \mathbb{N}$, a family $\{K_N^\lambda(x,v)\}_{x,v \in (0,\infty), \ x \neq v}$ of bounded
operators in $\mathbb{F}^q$. We prove that $K_N^\lambda(x,v)$, $x,v \in (0,\infty)$, $x \neq v$, satisfies the standard Calderón-Zygmund conditions
uniformly in $N \in \mathbb{N}$.
\begin{Lem}\label{Lem5.2}
Let $\mathbb{B}$ be a Banach space, $\lambda>1$ and $1<q<\infty$. For every $N \in \mathbb{N}$, and $x,v \in (0,\infty)$, $x \neq v$, we define
$$K_N^\lambda(x,v)(h)(y,t)
= \int_{\Gamma_N(0)} k^\lambda_{s,t}(x,y;u,v) h(u,s)\frac{dsdu}{s^2}, \quad h \in \mathbb{F}^q.$$
Then,
\begin{itemize}
\item[$(a)$] $K_N^\lambda(x,v)$, $N \in \mathbb{N}$, is bounded from $\mathbb{F}^q$ into itself and there exists $C>0$ such that for every $N \in \mathbb{N}$
and $x,v \in (0,\infty)$, $x \neq v$,
$$ \|K_N^\lambda(x,v)(h)\|_{\mathbb{F}^q} \leq \frac{C}{|x-v|} \|h\|_{\mathbb{F}^q}, \quad h \in \mathbb{F}^q.$$
\item[$(b)$] There exits $C>0$ such that, for every $N \in \mathbb{N}$,
$$ \|(K_N^\lambda(x_1,v) - K_N^\lambda(x_2,v))(h)\|_{\mathbb{F}^q} \leq C \frac{|x_1-x_2|}{|x_1-v|^2}\|h\|_{\mathbb{F}^q},$$
being $h \in \mathbb{F}^q$ and $|x_1-v|>2|x_1-x_2|$, $x_1,x_2,v \in (0,\infty)$.
\item[$(c)$] There exits $C>0$ such that, for every $N \in \mathbb{N}$,
$$ \|(K_N^\lambda(x,v_1) - K_N^\lambda(x,v_2))(h)\|_{\mathbb{F}^q} \leq C \frac{|v_1-v_2|}{|v_1-x|^2}\|h\|_{\mathbb{F}^q},$$
being $h \in \mathbb{F}^q$ and $|x-v_1|>2|v_1-v_2|$, $x,v_1,v_2 \in (0,\infty)$.
\end{itemize}
\end{Lem}
\begin{proof}
$(a)$ Note firstly that if $h \in \mathbb{F}^q$ we have that, for every $N \in \mathbb{N}$ and $v,x \in (0,\infty)$, with $x \neq v$,
\begin{align*}
\|K_N^\lambda(x,v)(h)\|_{\mathbb{F}^q}
\leq & \left( \int_{\Gamma(0)}
\int_{\Gamma_N(0)} \|h(u,s)\|_\mathbb{B}^q \frac{dsdu}{s^2}
\left(\int_{\Gamma_N(0)} |k^\lambda_{s,t}(x,y;u,v)|^{q'} \frac{dsdu}{s^2} \right)^{q/q'}
\frac{dtdy}{t^2} \right)^{1/q} \\
\leq & C \left( \int_{\Gamma_+(x)}
\left(\int_{\Gamma_+(v)} |k^\lambda_{s,t}(0,y;u,0)|^{q'} \frac{dsdu}{s^2} \right)^{q/q'}
\frac{dtdy}{t^2} \right)^{1/q} \|h\|_{\mathbb{F}^q},
\end{align*}
being $C>0$ independent of $N$. Then, $(a)$ is established when we prove that for a certain $C>0$
\begin{equation}\label{5.11}
\left( \int_{\Gamma_+(x)}\left(\int_{\Gamma_+(v)} |k^\lambda_{s,t}(0,y;u,0)|^{q'} \frac{dsdu}{s^2} \right)^{q/q'} \frac{dtdy}{t^2} \right)^{1/q}
\leq \frac{C}{|x-v|}, \quad x,v \in (0,\infty), \ x \neq v.
\end{equation}
We write
$$k^\lambda_{s,t}(0,y;u,0)
= ts \partial_r^2 P_{r,1}^\lambda(y,u)_{|_{r=t+s}} + ts \partial_r^2 P_{r,2}^\lambda(y,u)_{|_{r=t+s}},
\quad y,u,t,s \in (0,\infty),$$
where $P_{r,1}^\lambda(y,u)$ and $P_{r,2}^\lambda(y,u)$ are given by \eqref{7.2}.
We have that
\begin{align*}
\partial_r^2 P_{r,1}^\lambda(y,u)
= & -\frac{12\lambda(\lambda+1) (yu)^\lambda r}{\pi} \int_0^{\pi/2} \frac{(\sin \theta)^{2\lambda-1}}{[(y-u)^2+r^2+2yu(1-\cos \theta)]^{\lambda+2}} d\theta \\
& + \frac{8\lambda(\lambda+1)(\lambda+2) (yu)^\lambda r^3}{\pi} \int_0^{\pi/2} \frac{(\sin \theta)^{2\lambda-1}}{[(y-u)^2+r^2+2yu(1-\cos \theta)]^{\lambda+3}} d\theta \\
= & \mathcal{L}_{r,1}^{\lambda}(y,u) + \mathcal{L}_{r,2}^{\lambda}(y,u), \quad r,y,u \in (0,\infty).
\end{align*}
It is clear that $ |\mathcal{L}_{r,2}^{\lambda}(y,u)| \leq C |\mathcal{L}_{r,1}^{\lambda}(y,u)|$, $r,y,u \in (0,\infty)$.
Moreover, by taking into account that $\sin \theta \sim \theta$ and $2(1-\cos \theta) \sim \theta^2$, when $\theta \in [0,\pi/2]$,
and making the change of variables $\theta = \sqrt{|y-u|^2+r^2}\phi/\sqrt{uy}$, we get
$$ |\mathcal{L}_{r,1}^{\lambda}(y,u)|
\leq C r (yu)^\lambda \int_0^{\pi/2} \frac{\theta^{2\lambda-1}}{((y-u)^2+r^2+yu\theta^2)^{\lambda+2}} d\theta
\leq \frac{C}{(|y-u|+r)^3}, \quad r,y,u \in (0,\infty).$$
Then, since $|y-u| + t+ s \sim |x-v|+t+s$, when $(y,t) \in \Gamma_+(x)$ and $(u,s) \in \Gamma_+(v)$, we obtain
\begin{align*}
& \left( \int_{\Gamma_+(x)}\left(\int_{\Gamma_+(v)} |st \partial_r^2 P_{r,1}^\lambda(y,u)_{|r=s+t}|^{q'} \frac{dsdu}{s^2} \right)^{q/q'}\frac{dtdy}{t^2} \right)^{1/q} \\
& \qquad \qquad \leq C \left( \int_{\Gamma_+(x)}t^{q-2}\left(\int_{\Gamma_+(v)} \frac{s^{q'-2}}{(|x-v|+s+t)^{3q'}} dsdu \right)^{q/q'}dtdy \right)^{1/q} \\
& \qquad \qquad \leq C \left( \int_{\Gamma_+(x)}t^{q-2}\left(\int_0^\infty \frac{s^{q'-1}}{(|x-v|+s+t)^{3q'}} ds \right)^{q/q'}dtdy \right)^{1/q} \\
& \qquad \qquad \leq C \left( \int_{\Gamma_+(x)}t^{q-2}\left(\int_0^\infty \frac{ds}{(|x-v|+s+t)^{2q'+1}} ds \right)^{q/q'}dtdy \right)^{1/q} \\
& \qquad \qquad \leq C \left( \int_0^\infty \int_{|y-x|\leq t}\frac{t^{q-2}}{(|x-v|+t)^{2q}} dtdy \right)^{1/q}
\leq \frac{C}{|x-v|}, \quad x,v \in (0,\infty), \ x \neq v.
\end{align*}
In a similar way we can get that
$$\left( \int_{\Gamma_+(x)}\left(\int_{\Gamma_+(v)} |st \partial_r^2 P_{r,2}^\lambda(y,u)_{|r=s+t}|^{q'} \frac{dsdu}{s^2} \right)^{q/q'}\frac{dtdy}{t^2} \right)^{1/q}
\leq \frac{C}{|x-v|}, \quad x,v \in (0,\infty), \ x \neq v.$$
Hence \eqref{5.11} is established and $(a)$ is proved.\\
$(b)$ By proceeding as above and using Minkowski's inequality we can see that
\begin{align*}
\|K_N^\lambda(x_1,v)(h) & - K_N^\lambda(x_2,v)(h)\|_{\mathbb{F}^q} \\
\leq & \left( \int_{\Gamma(0)} \left(\int_{\Gamma(0)} |k^\lambda_{s,t}(x_1,y;u,v) - k^\lambda_{s,t}(x_2,y;u,v)|^{q'} \frac{dsdu}{s^2} \right)^{q/q'}
\frac{dtdy}{t^2} \right)^{1/q} \|h\|_{\mathbb{F}^q}\\
\leq & \left( \int_{\Gamma(0)} \left(\int_{\Gamma(0)} \left| \int_{x_1}^{x_2} \left| \partial_z k^\lambda_{s,t}(z,y;u,v) \right| dz \right|^{q'} \frac{dsdu}{s^2} \right)^{q/q'}
\frac{dtdy}{t^2} \right)^{1/q} \|h\|_{\mathbb{F}^q} \\
\leq & \left| \int_{x_1}^{x_2} \left( \int_{\Gamma(0)} \left(\int_{\Gamma(0)} \left| \partial_z k^\lambda_{s,t}(z,y;u,v) \right|^{q'} \frac{dsdu}{s^2} \right)^{q/q'}
\frac{dtdy}{t^2} \right)^{1/q}dz \right| \|h\|_{\mathbb{F}^q}.
\end{align*}
Hence, $(b)$ is shown when we prove that
\begin{equation}\label{5.12}
\left| \int_{x_1}^{x_2} \left( \int_{\Gamma(0)} \left(\int_{\Gamma(0)} \left| \partial_z k^\lambda_{s,t}(z,y;u,v) \right|^{q'} \frac{dsdu}{s^2} \right)^{q/q'}
\frac{dtdy}{t^2} \right)^{1/q}dz \right|
\leq C \frac{|x_1-x_2|}{|x_1-v|^2},
\end{equation}
for every $x_1,x_2,v \in (0,\infty)$ such that $|x_1-v|>2|x_1-x_2|$.\\
From now on, we take into account that $\lambda>1$. Suppose that $x_1,x_2,v \in (0,\infty)$ such that $|x_1-v|>2|x_1-x_2|$. We can write
\begin{align}\label{5.13}
& \left| \int_{x_1}^{x_2} \left( \int_{\Gamma(0)} \left(\int_{\Gamma(0)} \left| \partial_z k^\lambda_{s,t}(z,y;u,v) \right|^{q'} \frac{dsdu}{s^2} \right)^{q/q'}
\frac{dtdy}{t^2} \right)^{1/q}dz \right| \nonumber \\
& \qquad \leq C \left| \int_{x_1}^{x_2} \left( \int_{\Gamma_+(z)} \left(\int_{\Gamma_+(v)} \left| \partial_y k^\lambda_{s,t}(0,y;u,0) \right|^{q'} \frac{dsdu}{s^2} \right)^{q/q'}
\frac{dtdy}{t^2} \right)^{1/q}dz \right|.
\end{align}
By keeping the notation in the proof of $(a)$, straightforward manipulations lead to
\begin{align*}
\partial_y \partial_r^2 P_{r,1}^\lambda(y,u)
= & -\frac{12\lambda^2(\lambda+1) (yu)^{\lambda-1}u r}{\pi} \int_0^{\pi/2} \frac{(\sin \theta)^{2\lambda-1}}{[(y-u)^2+r^2+2yu(1-\cos \theta)]^{\lambda+2}} d\theta \\
& +\frac{24\lambda(\lambda+1)(\lambda+2) (yu)^\lambda r}{\pi} \int_0^{\pi/2} \frac{(\sin \theta)^{2\lambda-1}((y-u)+u(1-\cos \theta))}{[(y-u)^2+r^2+2yu(1-\cos \theta)]^{\lambda+3}} d\theta \\
& +\frac{8\lambda^2(\lambda+1)(\lambda+2) (yu)^{\lambda-1}u r^3}{\pi} \int_0^{\pi/2} \frac{(\sin \theta)^{2\lambda-1}}{[(y-u)^2+r^2+2yu(1-\cos \theta)]^{\lambda+3}} d\theta\\
& - \frac{16\lambda(\lambda+1)(\lambda+2)(\lambda+3) (yu)^\lambda r^3}{\pi} \int_0^{\pi/2} \frac{(\sin \theta)^{2\lambda-1}((y-u)+u(1-\cos \theta))}{[(y-u)^2+r^2+2yu(1-\cos \theta)]^{\lambda+4}} d\theta \\
= & \mathcal{L}_{r,1,1}^{\lambda}(y,u)+ \mathcal{L}_{r,1,2}^{\lambda}(y,u)
+ \mathcal{L}_{r,2,1}^{\lambda}(y,u) + \mathcal{L}_{r,2,2}^{\lambda}(y,u) , \quad r,y,u \in (0,\infty).
\end{align*}
If we define
$$\mathcal{L}_{r,3}^{\lambda}(y,u)
= (yu)^\lambda \int_0^{\pi/2} \frac{(\sin \theta)^{2\lambda-1}}{[(y-u)^2+r^2+2yu(1-\cos \theta)]^{\lambda+2}} d\theta, \quad r,y,u \in (0,\infty),$$
we have the following relations,
\begin{itemize}
\item $\mathcal{L}_{r,1,2}^{\lambda}(y,u) \leq C \left(\mathcal{L}_{r,1,1}^{\lambda}(y,u) + \mathcal{L}_{r,3}^{\lambda}(y,u)\right), \quad r,y,u \in (0,\infty),$
\item $\mathcal{L}_{r,2,1}^{\lambda}(y,u) \leq C \mathcal{L}_{r,1,1}^{\lambda}(y,u) , \quad r,y,u \in (0,\infty),$
\item $\mathcal{L}_{r,2,2}^{\lambda}(y,u) \leq C \left(\mathcal{L}_{r,1,1}^{\lambda}(y,u) + \mathcal{L}_{r,3}^{\lambda}(y,u)\right), \quad r,y,u \in (0,\infty).$
\end{itemize}
Therefore, it is sufficient to analyze $\mathcal{L}_{r,1,1}^{\lambda}(y,u)$ and $\mathcal{L}_{r,3}^{\lambda}(y,u)$, $r,y,u \in (0,\infty)$.\\
Now we can see
$$\mathcal{L}_{s+t,3}^{\lambda}(y,u)
\leq \dfrac{C}{(|y-u|+s+t)^4}
\leq \dfrac{C}{(|z-v|+s+t)^4}, \quad (y,t) \in \Gamma_+(z), \ (u,s) \in \Gamma_+(v).$$
Moreover, we have that
$$\mathcal{L}_{s+t,1,1}^{\lambda}(y,u)
\leq C \dfrac{1}{|z-v|(|z-v|+s+t)^3}, \quad y \geq |z-v|, \ (y,t) \in \Gamma_+(z), \ (u,s) \in \Gamma_+(v),$$
and, since $\lambda>1$,
\begin{align*}
\mathcal{L}_{s+t,1,1}^{\lambda}(y,u)
\leq & C (yu)^{\lambda-1}u (s+t) \int_0^{\pi/2} \frac{\theta^{2(\lambda-1)}}{[(y-u)^2+(s+t)^2+yu\theta^2]^{\lambda+2}} d\theta \\
\leq & C \frac{u(s+t)}{(|y-u|+s+t)^6}
\leq C \frac{|u-y|+y}{(|y-u|+s+t)^5}
\leq \frac{C}{(|z-v|+s+t)^4} \\
\leq & \frac{C}{|z-v|(|z-v|+s+t)^3}, \quad y \leq |z-v|, \ (y,t) \in \Gamma_+(z), \ (u,s) \in \Gamma_+(v).
\end{align*}
The same computations made in the proof of $(a)$ give us
$$\left( \int_{\Gamma_+(z)}
\left(\int_{\Gamma_+(v)} |st \partial_y \partial_r^2 P_{r,1}^\lambda(y,u)_{|r=s+t} |^{q'} \frac{dsdu}{s^2} \right)^{q/q'}
\frac{dtdy}{t^2} \right)^{1/q}
\leq \frac{C}{|z-v|^2}, \quad v,z \in (0,\infty).$$
Similarly we can obtain
$$\left( \int_{\Gamma_+(z)}
\left(\int_{\Gamma_+(v)} |st \partial_y \partial_r^2 P_{r,2}^\lambda(y,u)_{|r=s+t} |^{q'} \frac{dsdu}{s^2} \right)^{q/q'}
\frac{dtdy}{t^2} \right)^{1/q}
\leq \frac{C}{|z-v|^2}, \quad v,z \in (0,\infty).$$
Hence, we conclude that
$$\left( \int_{\Gamma_+(z)}
\left(\int_{\Gamma_+(v)} |\partial_y k^\lambda_{s,t}(0,y;u,0) |^{q'} \frac{dsdu}{s^2} \right)^{q/q'}
\frac{dtdy}{t^2} \right)^{1/q}
\leq \frac{C}{|z-v|^2}, \quad v,z \in (0,\infty).$$
From \eqref{5.13} it follows that
\begin{align*}
& \left| \int_{x_1}^{x_2} \left( \int_{\Gamma(0)} \left(\int_{\Gamma(0)} \left| \partial_z k^\lambda_{s,t}(z,y;u,v) \right|^{q'} \frac{dsdu}{s^2} \right)^{q/q'}
\frac{dtdy}{t^2} \right)^{1/q}dz \right|
\leq C \left| \int_{x_1}^{x_2} \frac{1}{|z-v|^2} dz \right|
\leq C \frac{|x_1-x_2|}{|v-x_1|^2},
\end{align*}
for each $x_1,x_2,v \in (0,\infty)$ such that $|x_1-v|>2|x_1-x_2|.$\\
Thus, \eqref{5.12} is shown and the proof of $(b)$ is completed.\\
$(c)$ The proof of $(c)$ is essentially the same one of $(b)$.
\end{proof}
We now obtain a representation of the operator $\Phi_N$ as a vector valued integral operator, for every $N \in \mathbb{N}$.
\begin{Lem}\label{Lem5.3}
Let $\mathbb{B}$ be a Banach space, $\lambda>0$, $1<q<\infty$ and $N \in \mathbb{N}$. We denote by $K_N^\lambda(x,v)$, $x,v \in (0,\infty)$, $x \neq v$,
the operator introduced in Lemma~\ref{Lem5.2}. Then,
\begin{equation}\label{5.14}
\Phi_N(g)(x)
= \int_0^\infty K_N^\lambda(x,v)(g(v)) dv, \quad \text{a.e. } x \notin \supp(g), \
g \in L^\infty_{c}(0,\infty) \otimes \left( L^q(\Gamma(0),\frac{dtdy}{t^2}) \otimes \mathbb{B} \right),
\end{equation}
where if $g \in L^\infty_{c}((0,\infty),\mathbb{F}^q)$ we represent
\begin{itemize}
\item for every $v \in (0,\infty)$, $g(v)(y,t)=g(v,y,t)$, $(y,t) \in \Gamma(0)$,
\item for every $x \in (0,\infty)$, $\Phi_N(g)(x)(u,s)=\Phi_N(g)(x,u,s)$, $(u,s) \in \Gamma(0)$.
\end{itemize}
The integral in \eqref{5.14} is understood in the $\mathbb{F}^q$-Bochner sense.
\end{Lem}
\begin{proof}
It is sufficient to show the result when $\mathbb{B}$ has finite dimension.\\
Let $g \in L^\infty_{c}((0,\infty),\mathbb{F}^q)$. We are going to see that, for almost all $x \notin \supp(g)$,
\begin{equation}\label{5.15}
\int_0^\infty \int_{\Gamma_N(0)} k^\lambda_{s,t}(x,y;u,v) g(v,u,s)\frac{dsdu}{s^2} dv
= \left(\int_0^\infty K_N^\lambda(x,v)(g(v)) dv\right)(x,y,t),
\end{equation}
in the sense of equality in $\mathbb{F}^q$. Note that the $\mathbb{F}^q$-Bochner integral in the right hand side is absolutely convergent
for every $x \notin \supp(g)$. Indeed, according to Lemma~\ref{Lem5.2}, $(a)$, we get
\begin{align*}
\int_0^\infty \|K_N^\lambda(x,v)(g(v))\|_{\mathbb{F}^q} dv
\leq &C \int_{\supp(g)} \frac{\|g(v)\|_{\mathbb{F}^q}}{|x-v|}dv \\
\leq &C \|g\|_{L^\infty((0,\infty),\mathbb{F}^q)} \int_{\supp(g)} \frac{dv}{|x-v|}
< \infty, \quad x \notin \supp(g).
\end{align*}
In order to show \eqref{5.15} it is enough to see that, for every $H \in \left( L^q((0,\infty),\mathbb{F}^q)\right)^*$ and $x \notin \supp(g)$
\begin{align}\label{dualidad}
\langle H(x,y,t), \int_0^\infty \int_{\Gamma_N(0)} k^\lambda_{s,t}(x,y;u,v) g(v,u,s)\frac{dsdu}{s^2} dv\rangle
= \langle H, \int_0^\infty K_N^\lambda(x,v)(g(v)) dv \rangle.
\end{align}
Let $H \in \left( L^q((0,\infty),\mathbb{F}^q)\right)^*$ and $x \notin \supp(g)$. By \cite[Corollary III.2.13]{DU},
$\left( L^q((0,\infty),\mathbb{F}^q)\right)^*=L^{q'}((0,\infty),(\mathbb{F}^q)^*)$, where $(\mathbb{F}^q)^*=L^{q'}\left(\Gamma(0),\frac{dt dy}{t^2}, \mathbb{B}^*\right)$.
Hence, there exists $h \in L^{q'}((0,\infty),(\mathbb{F}^q)^*)$ such that
$$\langle H, G \rangle
= \int_0^\infty \langle h(x),G(x) \rangle_{(\mathbb{F}^{q})^* \times \mathbb{F}^q } dx, \quad G \in L^q((0,\infty),\mathbb{F}^q).$$
Hence, we can write
\begin{align*}
\langle H(x,y,t), \int_0^\infty \int_{\Gamma_N(0)} & k^\lambda_{s,t}(x,y;u,v) g(v,u,s)\frac{dsdu}{s^2} dv\rangle \\
= & \int_0^\infty \int_{\Gamma(0)} \int_0^\infty \int_{\Gamma_N(0)} k^\lambda_{s,t}(x,y;u,v) \langle h(x,y,t), g(v,u,s) \rangle \frac{dsdu}{s^2} dv \frac{dtdy}{t^2} dx.
\end{align*}
Moreover, well-known properties of the Bochner integrals lead us to
\begin{align*}
\langle H, \int_0^\infty K_N^\lambda(x&,v)(g(v)) dv \rangle
= \int_0^\infty \langle H , K_N^\lambda(x,v)(g(v)) \rangle dv \\
= & \int_0^\infty \int_0^\infty \int_{\Gamma(0)} \int_{\Gamma_N(0)} k^\lambda_{s,t}(x,y;u,v) \langle h(x,y,t), g(v,u,s) \rangle \frac{dsdu}{s^2} \frac{dtdy}{t^2}dx dv.
\end{align*}
To obtain \eqref{dualidad} we only need to show that the last integral is absolutely convergent.
For this purpose we apply Hölder's inequality and \eqref{int_kst} as follows
\begin{align*}
\int_0^\infty & \int_0^\infty \int_{\Gamma(0)} \int_{\Gamma_N(0)} |k^\lambda_{s,t}(x,y;u,v)| \ \|g(v,u,s)\|_\mathbb{B} \frac{dsdu}{s^2} \|h(x,y,t)\|_{\mathbb{B}^*} \frac{dtdy}{t^2}dx dv \\
\leq & \left(\int_0^\infty \int_{\Gamma(0)} \int_0^\infty \int_{\Gamma_N(0)} |k^\lambda_{s,t}(x,y;u,v)| \|h(x,y,t)\|_{\mathbb{B}^*}^{q'} \frac{dsdu}{s^2}dv \frac{dtdy}{t^2}dx \right)^{1/q'}\\
& \times \left( \int_0^\infty \int_{\Gamma(0)} \int_0^\infty \int_{\Gamma_N(0)} |k^\lambda_{s,t}(x,y;u,v)| \|g(v,u,s)\|_{\mathbb{B}}^q \frac{dsdu}{s^2}dv \frac{dtdy}{t^2}dx \right)^{1/q} \\
\leq & C \|h\|_{L^{q'}((0,\infty),(\mathbb{F}^q)^*)} \|g\|_{L^q((0,\infty),\mathbb{F}^q)}
< \infty.
\end{align*}
The proof is finished.
\end{proof}
By using Lemmas~\ref{Lem5.1}, \ref{Lem5.2} and \ref{Lem5.3} and as a consequence of the theory of vector valued Calderón-Zygmund
operators (see \cite{RbRuT})
we obtain the following result that we used in the proof of Theorems~\ref{Th_3.1_OX} and \ref{Th_4.1_OX}.
\begin{Prop}\label{Pro5.2}
Let $\mathbb{B}$ be a Banach space, $\lambda>1$, $1<q<\infty$. Then, for each $N \in \mathbb{N}$, the operator $\Phi_N$ can be extended
\begin{itemize}
\item[$(a)$] to $L^p((0,\infty), \mathbb{F}^q)$ as a bounded operator from $L^p((0,\infty), \mathbb{F}^q)$ into itself, for every $1<p<\infty$;
\item[$(b)$] to $H^1((0,\infty), \mathbb{F}^q)$ as a bounded operator from $H^1((0,\infty), \mathbb{F}^q)$ into $L^1((0,\infty), \mathbb{F}^q)$.
\end{itemize}
Moreover, for every $1<p<\infty$ there exists $C_p>0$ such that
$$\|\Phi_N(g)\|_{L^p((0,\infty), \mathbb{F}^q)}
\leq C_p \|g\|_{L^p((0,\infty), \mathbb{F}^q)}, \quad g \in L^p((0,\infty), \mathbb{F}^q),$$
and there exists $C_1>0$ such that
$$\|\Phi_N(g)\|_{L^1((0,\infty), \mathbb{F}^q)}
\leq C_1 \|g\|_{H^1((0,\infty), \mathbb{F}^q)}, \quad g \in H^1((0,\infty), \mathbb{F}^q),$$
for every $N \in \mathbb{N}$.
\end{Prop}
\end{document} | arXiv |
Population Health Metrics
The cross-sectional average length of healthy life (HCAL): a measure that summarizes the history of cohort health and mortality
Markus Sauerberg ORCID: orcid.org/0000-0001-9524-446X1,2,
Michel Guillot3,4 &
Marc Luy1,2
Population Health Metrics volume 18, Article number: 21 (2020) Cite this article
Healthy life years have superseded life expectancy (LE) as the most important indicator for population health. The most common approach to separate the total number of life years into those spent in good and poor health is the Sullivan method which incorporates the health dimension to the classic period life table, thus transforming the LE indicator into the health expectancy (HE) indicator. However, life years derived from a period life table and health prevalence derived from survey data are based on different conceptual frameworks.
We modify the Sullivan method by combining the health prevalence data with the conceptually better fitting cross-sectional average length of life (CAL). We refer to this alternative HE indicator as the "cross-sectional average length of healthy life" (HCAL). We compare results from this alternative indicator with the conventional Sullivan approach for nine European countries. The analyses are based on EU-SILC data in three empirical applications, including the absolute and relative level of healthy life years, changes between 2008 and 2014, and the extent of the gender gap.
HCAL and conventional HE differ in each of these empirical applications. In general, HCAL provides larger gains in healthy life years in recent years, but at the same time greater declines in the proportion of healthy life years. Regarding the gender gap, HCAL provides a more favourable picture for women compared to conventional HE. Nonetheless, the extent of these differences between the indicators is only of minor extent.
Albeit the differences between HE and HCAL are small, we found some empirical examples in which the two indicators led to different conclusions. It is important to note, however, that the measurement of health and the data quality are much more important for the healthy life years indicator than the choice of the variant of the Sullivan method. Nonetheless, we suggest to use HCAL in addition to HE whenever possible because it widens the spectrum of empirical analyses and serves for verification of results based on the highly sensitive HE indicator.
Healthy life years have superseded life expectancy as the most important indicator for population health. It enables researchers to investigate, e.g. the proportion of life years spent in good/poor health, trends in life years spent in good respective poor health (the "compression-expansion-debate"), and differences between women and men [1,2,3,4]. In order to estimate the quality dimension of life years, the health expectancy indicator (HE) has been developed, which combines mortality and morbidity in a single indicator by incorporating the health dimension into the life table [5]. Even though several methods have been proposed for this purpose, the approach developed by Sullivan [6] is the most prominent one up to now [7]. It uses age-specific prevalence (proportions) of the population in the (un)healthy state, usually obtained from cross-sectional survey data, to apportion the life table person-years lived between the states of good and poor health [8].
In the application of the Sullivan method, it is frequently overlooked that life years derived from a period life table and health prevalence derived from survey data are based on different conceptual frameworks. Whereas the former reflects the life span of a hypothetical population constructed on the basis of current age-specific death rates, the latter reflects the actual health condition of real individuals [9, 10]. To overcome this conceptual mismatch between health and mortality information, it was suggested to base the Sullivan method on the "Cross-Sectional Average Length of Life" (CAL) instead of conventional period LE [11,12,13]. To our knowledge, this approach has not been applied empirically so far. We aim at closing this research gap by using this variant of the Sullivan approach, to which we refer as "Cross-Sectional Average Length of Healthy Life" (HCAL). Our central research question is to what extent the underlying mortality indicator, i.e. LE vs. CAL, affects the resulting estimates for healthy life years, i.e. HE vs. HCAL.
The paper is structured as follows: We start with a conceptual description of CAL in comparison to period and cohort LE to demonstrate that CAL is a combination of these two approaches. Then, we construct the HCAL indicator and discuss the difference between the mortality information in CAL and conventional LE with respect to its applicability to the Sullivan method. The empirical section starts with a description of data, followed by the presentation of our results. Here, we compare HCAL and conventional HE for nine European populations with regard to the absolute and relative level of healthy years, changes between 2008 and 2014, and differences between women and men. Finally, we discuss the advantages and disadvantages of HCAL as an alternative to HE.
Life expectancy and cross-sectional average length of life
Longevity measures usually follow a period or cohort concept. In a cohort life table, the observed age-specific survival probabilities define the survivorship function for a particular cohort born in time t. Integrating across all ages yields cohort LE at birth (\( {e}_0^c \)), i.e. the mean age at death for this particular cohort. Formally, \( {e}_0^c \) can be written as
$$ {e}_0^c(t)={\int}_0^{\infty }{p}_c\left(x,t\right) dx $$
with pc(x, t) being the probability for individuals born in time t to survive until age x. Because \( {e}_0^c \) can only be calculated for extinct cohorts, and thus reflects past mortality conditions, period LE is a more convenient summary measure for current mortality levels and for tracking recent mortality trends. In this concept, the age-specific survival probabilities do not correspond to one particular birth cohort but to one particular period, i.e. constructed from the observed age-specific death rates of this calendar year. Integrating the resulting period-specific survival probabilities over all ages leads to period LE at birth (\( {e}_0^p \)) for year t
$$ {e}_0^p(t)={\int}_0^{\infty }p\left(x,t\right) dx $$
with p(x, t) being the probability for individuals to survive until age x if they had been exposed to the survival probabilities prevailing at time t throughout their lives from birth to age x. Consequently, period LE reflects the average age at death of a hypothetical cohort under the assumption that the period-specific death rates remain unchanged over their entire life course.
The CAL concept combines the two classic concepts in the sense that it (1) refers to actual cohort mortality (i.e. it is based on longitudinal survival probabilities) and (2) corresponds to all cohorts alive in a given period (resulting in a cross-sectional summary measure of mortality experiences). CAL was originally introduced by Brouard [14] and further elaborated by Guillot [15] and Canudas-Romo and Guillot [16]. In the literature, this mixed period cohort concept has been labelled the "wedge-period perspective" [17], "cross-sectional cohort average" [18], or the "cross-sectional cohort mortality index" [19]. CAL is based on cohort survival probabilities (proportion of survivors) from birth until the last age reached at time t. Integrating this function across all ages yields CAL(t) as
$$ \mathrm{CAL}(t)={\int}_0^{\infty }{p}_c\left(x,t-x\right) dx $$
with pc(x, t − x) being the probability that a member of the cohort born at time t − x survives until age x. CAL(t) can be interpreted as a period longevity measure in the sense that it "[...] refers to a particular period t, but takes into account the actual mortality conditions to which cohorts present in the population at time t have been subject" ([15], p. 42).
Figure 1 illustrates the three demographic concepts using the example of French males in 2015, all constructed with data of the Human Mortality Database [20]. The upper panel shows the basic concepts in a Lexis surface. While the classic life table concept summarizes the mortality experiences of one single cohort (real or hypothetical) over its life course, CAL includes all mortality experiences experienced by cohorts alive in a given period. The lower panel shows the empirical survivorship functions corresponding to the three concepts. The areas under the curves yield cohort LE, period LE, and CAL, respectively, being 53.13 years for the cohort born in 1915, 79.02 years for the calendar year 2015, and 73.85 years for CAL in 2015. The 1915 birth cohort experienced relatively high mortality over all ages. Periods LE and CAL show similar survivorship patterns up to age 30. Then, the p(x, t) function of period LE is more rectangular compared to the pc(x, t − x) function of CAL. This is because the pc(x, t − x) function corresponds to actual cohorts of which many experienced higher mortality than current conditions which are reflected in period LE. The specific construction of CAL is also the reason why pc(x, t − x) is not a monotonically decreasing function. Whenever a cohort has been exposed to higher mortality conditions compared to the mortality experience of the previous (older) cohort, pc(x, t − x) will increase. This can be seen in Fig. 1 for French males born in 1945 (who reached age 70 in 2015). The proportion of cohort survivors of the 1944 birth cohort (who reached age 71 in 2015) is higher compared to the 1945 birth cohort even though they were born earlier, and therefore were longer exposed to the risk of dying.
Three demographic concepts for the measurement of life years. The upper panel shows the period perspective, the cohort perspective, and the cross-sectional cohort average concept in a Lexis surface. The lower panel shows the empirical survivorship functions corresponding to the three concepts using data for French males in 2015. Period LE, cohort LE, and CAL are defined as the area under the curves. The three measures are given by integrating the corresponding survivorship function over all ages (depicted next to the curves)
Whether an indicator should be based on the cohort, period, or cross-sectional cohort average perspective depends on the purpose of its use. The synthetic cohort approach is a powerful tool if the aim is to examine period mortality, e.g. to investigate changes in period death rates. However, the actual survival trajectory of individuals is usually poorly captured in the hypothetical cohort scenario. For that reason, the cohort perspective is essential to analyse the real mortality experience of people in the framework of age and calendar time. However, a population at a certain time is composed of a number of cohorts. Therefore, the experience of one cohort is not representative for the entire population. This is the central advantage of the cross-sectional cohort average concept. It refers to the population as a whole by taking into account the complete mortality history of cohorts, i.e. how the actual population alive results from the cohorts' past mortality experiences. Nonetheless, the empirical application of CAL is rare, mainly because of its high demand on the data. It has been used to study the impact of mortality on population size and growth [15], to evaluate population momentum [21], and to compare populations in terms of their mortality history [16].
Derivation of HE and HCAL with the Sullivan method
The Sullivan method divides the total number of life years into those spent in good and in poor health. Using the life table notation, life years are expressed as person-years lived (nLx). The nLx function allows to define eop(t) as the sum of all age-specific person-years lived (divided by the period life table radix l0p):
$$ {e}_0^p(t)=\frac{1}{l_0^{\mathrm{p}}}{\sum \limits_{x=0}^{\infty}}_n{L}_x^p(t) $$
with \( {}_n{L}_x^p \) being the number of person-years lived between age x and x + n in a life table for the period t and \( {l}_0^p \) the corresponding number of people alive at age 0 (i.e. the number of newborns). CAL can be constructed from person-years lived as well. In fact, CAL is the sum of the age- and cohort-specific person-years lived divided by the cohort life table radix \( {l}_0^c \), i.e. the number of newborns to which all cohorts are standardized:
$$ \mathrm{CAL}(t)=\frac{1}{l_0^{\mathrm{c}}}{\sum \limits_{x=0}^{\infty}}_n{L}_x^c\left(t-x-n,t-x\right) $$
with \( {}_n{L}_x^c \) being the number of person-years lived between age x and x + n in the life table for the cohort born between (t – x − n) and (t − x) and \( {l}_0^c \) the corresponding number of people alive at age 0.
The Sullivan method is based on the idea of applying the age-specific prevalence (proportions) of the population in an (un)healthy state to the age-specific person-years lived. In this way, the total life years in each age interval can be divided into those spent in good and in poor health. Summing up only the healthy person-years lived across all ages gives HE and HCAL, respectively, from:
$$ \mathrm{HE}(t)=\frac{1}{l_0^p}\sum \limits_{x=0}^{\infty}\left(1{-}_n{\pi}_x(t)\right)\cdot {}_n{L}_x^p(t) $$
$$ \mathrm{HCAL}(t)=\frac{1}{l_0^c}\sum \limits_{x=0}^{\infty}\left(1{-}_n{\pi}_x(t)\right)\cdot {}_n{L}_x^c\left(t-x-n,t-x\right) $$
with nπx being the age-specific prevalence (proportion) of poor health in the age interval x to x + n at time t. The proportion of healthy life years on total life years is given by the ratios HE/LE and HCAL/CAL, respectively. Alternatively, HE and HCAL can be derived directly from the corresponding survivorship functions by weighting the survival probabilities with the population proportions of individuals being in good health. Integrating the derived functions across all ages yields HE and HCAL in continuous time from
$$ \mathrm{HE}(t)={\int}_0^{\infty }p\left(x,t\right)\cdotp \left(1-\pi \left(x,t\right)\right) dx $$
$$ \mathrm{HCAL}(t)={\int}_0^{\infty }{p}_c\left(x,t-x\right)\cdotp \left(1-\pi \left(x,t\right)\right) dx $$
Equations 8 and 9 demonstrate that HE and HCAL solely differ in terms of the underlying survivorship function (p(x, t) vs. pc(x, t − x)), while the π(x, t) function remains the same for both measures. The combination of pc(x, t − x) with π(x, t), i.e. HCAL, is illustrated in Fig. 2 with data for French males in 2015. Each of the vertical lines in the right panel corresponds to a proportion of cohort survivors. The left shows the proportion of poor health according to the EU-SILC data [22]. For example, about 80% of the 1955 birth cohort survived up to 2015 (i.e. reached age 60) and approximately 30% of the same birth cohort reported to be mildly or strongly limited in 2015. Combining these two quantities gives the probability of being both healthy and alive in 2015: 0.8 · (1 – 0.3) = 0.56. The age-specific survival in good health (i.e. free of limitations) is shaded in dark grey.
Combining cohort survivorship with proportions of individuals being in the healthy state. The vertical lines on the left side give the share of unhealthy individuals in each cohort in 2015 on the basis of data for French males, i.e. the π(x, t) function. The right side shows the corresponding proportions of cohort survivors pc(x, t − x). Combining these two quantities gives the proportion of being both, healthy and alive in 2015, defined as pc(x, t − x) · (1 − π(x, t)). Accordingly, only the dark grey shaded vertical lines refer to cohort survivors being in good health. The remaining light-shaded lines give the unhealthy share of cohort survivors
Since both the pc(x, t − x) and the π(x, t) function correspond to the same group of individuals, HCAL creates a consistent combination of mortality and health quantities. By contrast, combining the p(x, t) function with the π(x, t) function, the conventional Sullivan method procedure (Eq. 8) combines survival probabilities corresponding to one hypothetical cohort with the health state-specific prevalence of several real cohorts. Strictly speaking, this results in a probability, which is neither reflecting healthy survival in a synthetic cohort fashion nor in a real cohort perspective. A detailed description of the particular implications for HE and HCAL resulting from the different conceptual approaches can be found in the Appendix.
Data sources for single age-specific mortality and prevalence
We estimate HE and HCAL with data for Denmark, Finland, France, Germany, Italy, the Netherlands, Norway, Sweden, and the UK for the years 2008 to 2014. To test whether the indicators provide different results, we compare the corresponding country-specific estimates with regard to the total level of healthy life years, genders differences, and changes over time. The HE and HCAL estimation requires single age-, period-, and cohort-specific death rates and single age-specific proportions of the (un)healthy population observed in the given country and year.
Age-specific death rates for estimating LE and CAL were taken from the Human Mortality Database (HMD). For the UK, HMD data is available from 1922 onwards. However, calculating (H)CAL in 2005 requires data beginning in 1905 (defining age 100 as the highest age). To obtain mortality rates before the year 1922, we combined HMD data for the UK with HMD data for England and Wales which is available from 1841 onwards. For Germany, we used cohort mortality data from Destatis [23] because HMD provides German mortality data only from 1956 onwards. While Destatis publishes cohort life tables, the HMD provides cohort death rates only for cohorts that have lived at least 30 years (from age 0 to 29). Yet, period life tables are available also for more recent years. Therefore, we reconstructed the cohort survivorship for the (H)CAL calculation for all countries but Germany by combining the age-specific death rates of the period life tables longitudinally along the cohorts' life course (see supplementary material). This technique has already been used in previous empirical estimations of CAL [16].
Age-specific prevalence data was taken from the European Union Statistics on Income and Living Conditions (EU-SILC) [22]. We defined "being healthy" on the basis of the "Global Activity Limitation Indicator" (GALI). GALI has been developed for providing a harmonized health indicator for monitoring population health in Europe [24] and refers to the question: "For at least the past six months, to what extent have you been limited because of a health problem in activities people usually do?" with the three answer categories "strongly limited", "limited, but not strong", and "not limited". We defined being in the healthy state if respondents reported to be "not limited". Unfortunately, the harmonization of GALI is still imperfect, hampering the comparison of HE estimates between countries and over time [25]. Previous research found that health indicators are sensitive with respect to the mode of data collection [26], the choice of the survey [27], and the wording of the health survey question [28]. Our selection of countries and time span was therefore driven by avoiding any substantial breaks in the time series and by choosing countries with the required mortality data available in the HMD (besides Germany for which data was taken from Destatis).
The country-specific sample sizes and the prevalence of being unhealthy using GALI are presented in Table 1 (separated by gender). Some data problems become apparent in these figures. For example, the prevalence of being unhealthy decreased strongly in Sweden between 2013 and 2014, and we, therefore, excluded Sweden from the time trend analysis. In Norway and Finland, we find extreme outliers in the prevalence values in 2011 and 2013, respectively, while for Italy, no EU-SILC data is available in 2010. These breaks and outliers are also mentioned in the Eurostat database [29], indicating that we cannot analyse the full time span 2008 to 2014, but at least we can compare the years 2008 and 2014. Because the health data is highly fluctuating between single age groups, we applied the R package "MortalitySmooth". The package has been developed for smoothing count data, which can be assumed to be Poisson-distributed [30] and provides two smoothing functions: "Mort1Dsmooth" assumes smoothness in a one-dimensional (over age) way and "Mort2Dsmooth" for a two-dimensional setting (over ages and years). We applied "Mort2Dsmooth" to the data for countries without a break or an outlier between 2008 and 2014 (France, the UK, Denmark, the Netherlands, and Germany). The remaining countries (Finland, Norway, Italy, and Sweden) were smoothed in a one-dimensional way. Single age-specific proportions of being unhealthy were derived from the smoothed health data. In order to take into account also the uncertainty from the survey sample size, we approximated single age-specific standard errors by using the approximation formula ([8], p. 27), i.e. applying the smoothed proportions of being unhealthy to the observed number of persons in the corresponding age intervals. These standard errors were used to approximate 95% confidence intervals for HE and HCAL estimates which are presented in Tables 4 and 5. As to be expected, the uncertainty in single age-specific prevalence data is substantial, and we do not find statistically significant differences between HE and HCAL in an any of our empirical analyses. Therefore, we compare the two indicators for healthy life years without confidence intervals in the following section. We come back to this issue at the end of the paper when we discuss the properties of HE and HCAL.
Table 1 Total sample size N (unweighted) and total prevalence of being unhealthy 휋 (weighted) for nine European countries from 2008 to 2014
Level of healthy life years estimated with HCAL and conventional HE
Table 2 shows the estimates LE, HE, CAL, and HCAL for the nine European countries, separately for females and males. As expected, CAL is lower than LE in each country and for each gender group. This results from the fact that CAL includes also (higher) historical death rates, whereas LE is solely build up from recently observed (relatively lower) death rates. Interestingly, the differences between HE and HCAL is smaller than the differences between LE and CAL. This relationship reverses in relative terms, however. The ratio HCAL/CAL is slightly higher than the ratio HE/LE in all nine countries (and for both genders). This is due to the relative difference between the p(x, t) function and the pc(x, t − x) function. In relative terms, the pc(x, t − x) function is higher at young ages and lower at older ages compared to the p(x, t) function. In other words, the relative number of deaths is higher at young ages and lower at older ages on the basis of CAL. Since the prevalence of individuals in the unhealthy state is usually low in younger ages but increases with age, HCAL provides higher proportions of healthy life years than conventional HE (see Appendix for more details). Especially, Italian, French, and German males show a (comparatively) large gap between the ratios HE/LE and HCAL/CAL because these populations experienced high mortality in the past, particularly during the World War II. For example, Italian males spend 81.06% of their total life years in good health on the basis of HCAL, whereas the proportion of healthy life years is only 78.44%. Including the mortality history of cohorts results also in a different country ranking on the basis of CAL. While LE ranks Italy (for both genders) relatively high, CAL favours Sweden and Norway. However, HE and HCAL rank the nine analysed countries similar for females and males, the ranking changes only slightly. This is because the prevalence of activity limitations varies substantially between countries, compensating most mortality differences between LE and CAL.
Table 2 LE, HE, CAL, and HCAL in absolute and relative terms for nine European countries in 2014
Changes in healthy life years over time: compression vs. expansion of morbidity
Figure 3 shows changes in LE, HE, CAL, and HCAL from 2008 to 2014 for France, the Netherlands, Denmark, and Germany, separated by gender. While CAL increases in a more or less linear fashion, LE shows some fluctuations over time. The robust trend in CAL results from the fact that it is based on a large number of age-specific death rates, and thus, it is not much affected by short-term fluctuations in period mortality (for more details see [15]). Nonetheless, the trend in HCAL is not as linearly increasing as the trend in CAL. Instead, it follows the trend in HE, indicating that prevalence data is the driving force in the corresponding time trend in healthy life years and that the choice of the basic survival function does not matter significantly.
Time trend in LE, HE, CAL, and HCAL from 2008 to 2014. The figure shows how LE (black solid line), HE (black dashed line), CAL (grey solid line), and HCAL (grey dashed line) changed between 2008 and 2014. The time trends are presented for France, the Netherlands, Denmark, and Germany (males on the left side of the figure and females on the right side)
Moreover, CAL is increasing faster than LE over time. While LE for French males increases between 2008 and 2014 by 1.67 years, CAL rises by about 2 years in the same period (see Table 3). Consequently, the increase in HCAL is also higher than the increase in HE (0.52 years vs. 0.29 years).
Table 3 Difference in the estimates of LE, HE, CAL, and HCAL in absolute and relative terms for eight European countries from 2008 to 2014
Note that differences in the increase in the total number of life years according to CAL and LE affect also the trend in the proportion of healthy life years, i.e. the ratio HCAL/CAL and the ratio HE/LE. In general, the increase in CAL exceeds the increase in LE between 2008 and 2014, resulting in higher decreases in the HCAL/CAL ratio. Denmark appears as a special case because both males and females show higher gains in LE compared to CAL. Accordingly, the reduction in the proportion of healthy life years is more pronounced on the basis of conventional HE.
Nonetheless, the proportions HCAL/CAL and HE/LE largely agree on the direction of the trend in healthy life years, i.e. whether we observe an expansion or compression of morbidity. The only exception is Italian females. Whereas the HE/LE ratio indicates a relative increase in healthy life years, i.e. relative compression of morbidity, the HCAL/CAL ratio suggests a slight trend in the direction of relative morbidity expansion.
Gender differences in healthy life years
Both mortality indicators (LE and CAL) show a female advantage in the total number of life years for all analysed populations (see Fig. 4). This gender gap is larger for CAL, indicating that the difference between male and female mortality was higher in the past compared to recent years. As a consequence, the gender gap in healthy life years according to HCAL is weighted stronger in the direction of females, i.e. HCAL provides either a larger female advantage or a smaller female disadvantage compared to HE. The differences between HCAL and HE in the extent of the gender gap differ, therefore, depending on whether females or males have a higher number of healthy life years. In Finland, Norway, Sweden, Denmark, and the Netherlands, males show a lower age-specific prevalence of activity limitation compared to women. Consequently, the gender gap is larger based on conventional HE in these populations. Yet, in countries where the age-specific prevalence of activity limitation is lower among females (Italy, the UK, and France), the gender gap is larger according to HCAL. In Germany, we find a specific situation in which HE and HCAL appear as contradictive in terms of the direction of the gender gap. The female survival advantage in CAL is large enough to compensate their higher age-specific prevalence of activity limitations, resulting in more healthy life years for females in HCAL, whereas conventional HE gives more healthy life years for males.
Gender gap in LE, HE, CAL, and HCAL in 2014. The figure shows the difference between males and females in LE (dark grey bars), HE (less-hatched bars), CAL (light grey bars), and HCAL (more-hatched bars) for Finland, Italy, Norway, Sweden, the UK, Denmark, France, Germany, and the Netherlands in 2014
In relative terms, the proportion of healthy life years on total life years is higher for males in all nine countries. This male advantage is larger according to HCAL than according to HE (see Table 2). As mentioned above, differences between HE and HCAL in relative terms result from the relative difference between the p(x, t) function and the pc(x, t − x) function, leading to an (un)favourable age-specific weighting scheme. The comparatively high mortality for males measured with CAL promotes a situation in which higher weights are assigned to the young ages with low prevalence of being unhealthy (see Fig. 6).
We started this paper with a description of three different concepts for measuring longevity in a population. We have concluded that only cohort LE is an appropriate choice if the aim is to examine the real-life course experience of people in the framework of age and calendar time. However, the concepts of period LE and CAL are more convenient for monitoring health and mortality on the population level as they summarize the mortality information of an entire population, instead of focusing exclusively on a specific group of individuals. In addition, CAL and period LE are more timely than cohort LE because they can be estimated also for recent periods. However, CAL includes a high proportion of historic death rates, while period LE solely reflects recent mortality rates (see [18] for more details).
Whereas period LE reflects the life course of one hypothetical cohort by linking together a set of age-specific death rates observed in a given period, CAL summarizes the complete mortality experience of all actual living cohorts from theirs birth until the current period. A specific feature of CAL is that it takes into account the natural process of survival, i.e., the overall mortality in a certain year or period is conceptualized as the product of past exposures and health behaviour that have accumulated over the entire life span of the currently living cohorts [31]. Therefore, an observed increase in CAL between two points in time corresponds to the factual longevity gains experienced by individuals present in the given population. By contrast, trends in period LE are more difficult to interpret and might be distorted by several effects such as cohort and tempo effects or heterogeneity [19, 32]. For example, the stagnation and rise seen in Danish women's period LE has been attributed to specific cohorts rather than to changes in period mortality conditions [33], and public health researchers are currently investigating to which extent the recent observed stalling in period LE in the UK and Europe represents a "real" deterioration of population health [34].
Period LE and CAL can be extended to HE and HCAL by applying the Sullivan method. While HCAL links the proportions of healthy individuals observed in a given population to the corresponding proportions of cohort survivors, conventional HE combines the health information of real cohorts with the survival trajectory of a hypothetical cohort. These features make CAL a more appropriate basis for the estimation of healthy life years with the Sullivan method. However, CAL is less affected by changes in recent death rates and increases also in years where period LE decreases. Therefore, one could argue that conventional HE estimates are more timely than HCAL estimates.
It is important to note that the Sullivan method itself has been criticized for producing misleading results regarding monitoring changes in population health [35,36,37]. The main argument of these critiques focused on using prevalence instead of incidence data. In terms of health, prevalence reflects the proportion of individuals in the unhealthy state at a given point in time. This includes individuals who transitioned from healthy to unhealthy in the observation period as well as those who experienced this transition already in the past. The incidence of being unhealthy, however, refers exclusively to individuals who experienced transitions during the given calendar year (or period). As a consequence, health indicators estimated with the Sullivan method cannot capture a sudden short-term change in population health regardless of the choice of the mortality information [38]. Nevertheless, prevalence-based indicators such as HE and HCAL are convenient for measuring the current health composition of a population, i.e. the actual proportion of healthy/unhealthy individuals in a population [39].
The use of health prevalence data makes HCAL the conceptually more coherent indicator for monitoring population health. The data demands are, however, somewhat higher for HCAL than for HE. This leads to the question, whether results between HCAL and HE differ to an extent that justifies this higher data demand. Therefore, we compared the HCAL indicator to conventional HE in three empirical applications for nine European countries. In general, HCAL is lower in absolute terms and slightly larger in relative terms due to incorporating (higher) historical death rates. Our examination of the gender gap in health and mortality shows that HCAL provides a larger female advantage in healthy life years than conventional HE. This finding illustrates the implication of using period mortality (in LE) instead of cohort mortality information (in CAL). The gender-specific differences in the prevalence of being unhealthy can be (partly) attributed to the different health risks and exposures, i.e. the mortality, experienced by women and women over their life span [40, 41]. In the case of HCAL, prevalence is related to the corresponding mortality history, which had considerable higher levels for men compared to women. Conventional HE, on the other hand, relates the prevalence to current period death rates, which show lower gender differences in mortality than actual cohorts have been experienced over their life courses.
The probably most discussed question in health research is, whether gains in longevity are spent primarily in good or poor health, in the context of the so-called "expansion vs. compression of morbidity debate". The ratio of LE/HE respective CAL/HCAL is particularly relevant in this context because it shows the relative share of healthy life years on total life years. In general, we found larger gains in CAL compared to period LE between 2008 and 2014, resulting in a slightly faster decreasing HCAL/CAL ratio compared to the LE/HE ratio. This can be interpreted as a stronger relative expansion of morbidity on the basis of HCAL, i.e. gains in longevity are mostly spent in poor health.
Nonetheless, the overall trend in healthy life years is very similar for both measures, indicating that the prevalence is the driving force in HCAL as well as in conventional HE. Albeit we observed gains in total life years for all analysed countries between 2008 and 2014, regardless if measured with period LE or CAL. Many populations still experienced declines in healthy life years in the same period. Naturally, these declines are attributed to increases in the age-specific prevalence of being in the unhealthy state. This finding demonstrates once more the great relevance for accurate health data for analysing healthy life years. Therefore, our results suggest that the health data incorporates large problems as we have shown by unexplainable jumps in the data or outliers. One example for this is the relatively large decrease of about 4 years in conventional HE for Finish females between 2008 and 2014 which is more attributed to random fluctuations in the health data than to real health deteriorations in the Finish population [42]. Last but not the least, the statistical insecurity related to the health data is so high that confidence intervals are not helpful for the analysis. This can be seen from the data presented in Tables 4 and 5. Taking into account these 95% confidence intervals, there are no statistically significant differences between HCAL and conventional HE as well as in the changes in healthy life years over time. This uncertainty results almost entirely from the health data rather than from the mortality data. For this reason, the lack of statistical difference does not imply that the two conceptual approaches are indistinguishable.
We used the GALI health indicator for our empirical applications which refers to a self-reported survey question about longstanding limitations in daily activities. In general, GALI has been validated positively, i.e. GALI is strongly associated with limitations in activities of daily living (such as washing, getting dressed or out of the bed), intermediately associated with limitations in instrumental activity of daily living (such as the need of assistance in doing light housework or managing medication), and somewhat lower association with physical limitations (biting, chewing, or kneeling) in most European countries [43]. Nevertheless, self-reported survey questions might still be influenced by age, culture, and social background of the respondent [44]. For these reasons, the presented trends should be interpreted with caution. In this paper, they served primarily the purpose of demonstrating differences between HCAL and conventional HE with respect to the underlying mortality information. All the discussed conceptual and health data-related issues would apply likewise to other self-reported health indicators from surveys such as EU-SILC.
HCAL is a summary measure of health and mortality based on the Sullivan method. Using proportions of cohort survivors instead of period mortality rates requires a long time series of mortality data. We have demonstrated several advantages of HCAL which suggest it is an attractive measure for population health. First, HCAL yields a coherent quantity, i.e., combining the health of the real cohort survivors with mortality of the real cohorts. Second, HCAL offers an alternative perspective on health and mortality. Previous approaches have focused either on one single period or on one single cohort. HCAL, on the other hand, is the sum over all cohorts, of the probability surviving and being in good health at the time of observation. In this sense, HCAL is also a measure of population dynamics and, thus, provides new insights into the evolution of the healthy/unhealthy shares in a population.
The empirical analysis suggests that the quality of health data is much more important than the decision between CAL and LE as basis for the total number of life years. We have shown that the prevalence of being in the (un)healthy state varies notably between populations and across time. These differences have by far the strongest impact on healthy life years derived with the Sullivan method. The overall trend in healthy life years is similar in conventional HE and HCAL and taking into account the uncertainty stemming from the health data does not result in statistically significant differences between both indicators.
Nevertheless, conventional HE and HCAL should not be treated as interchangeable as they correspond to two different concepts. The analysis of the gender gap in healthy life years demonstrates that the choice of the survivorship function can indeed affect the result. By taking into account the past mortality experiences of males and females, HCAL gives a more favourable picture for women compared to conventional HE. Also regarding the compression-expression debate we found that the two indicators can suggest different trends as in the case among Italian females. Accordingly, researchers should consider using HCAL especially in applications where period mortality differs strongly from the actual cohort experience. It is important to note, however, that we do not argue that conventional HE should be replaced by HCAL. Yet, given that HCAL is the conceptually more coherent approach, it is worth to be used in addition to conventional HE whenever it is possible to estimate both indicators because it widens the spectrum of empirical analyses and serves for verification of results based on the highly sensitive HE indicator.
All data used in this paper is publicly available. Mortality data can be found at www.mortality.org and on www.destatis.de for Germany. Health data is available on https://ec.europa.eu/eurostat/web/microdata/european-union-statistics-on-income-and-living-conditions.
CAL:
Cross-sectional average length of life
EU-SILC:
European Union Statistics on Income and Living Conditions
GALI:
Global Activity Limitation Indicator
HCAL:
Cross-sectional average length of healthy life
HMD:
Human Mortality Database
LE:
WWII:
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The authors thank Vladimir Canudas-Romo for providing helpful comments to an earlier version of this manuscript and Werner Richter for language editing.
This project has received funding from the European Research Council under the EU's Horizon 2020 Research and Innovation Programme, grant agreement no. 725187 (LETHE).
Vienna Institute of Demography, Austrian Academy of Sciences, Vienna, Austria
Markus Sauerberg & Marc Luy
Wittgenstein Centre for Demography and Global Human Capital (IIASA, OeAW, University of Vienna), Vienna, Austria
Population Studies Center, University of Pennsylvania, Philadelphia, PA, USA
Michel Guillot
French Institute for Demographic Studies (INED), Paris, France
Markus Sauerberg
Marc Luy
ML developed the research idea and designed the study together with MS. MS carried out the analyses and wrote the paper. ML supervised the analyses and gave inputs to the manuscript. MG contributed to the interpretation of the data and commented on all parts of the paper with corresponding inputs to the text. The authors read and approved the final manuscript.
Correspondence to Markus Sauerberg.
The authors have no competing interests.
Additional file 1.
R code example for estimating HE and HCAL.
Table 4 HE and HCAL with 95% confidence intervals (approximated) for five European countries from 2008 to 2014
Table 5 HE and HCAL with 95% confidence intervals (approximated) for four European countries in 2008 and 2014
Cross-sectional prevalence rates from a cohort perspective
The proportions of cohort survivors, pc(x, t − x), can be derived on the basis of cohort- and age-specific survival probabilities. Since death is irreversible, this proportion is monotonically decreasing with age. In other words, the stock of cohort survivors at a given point in time is the product of past survival probabilities. The proportions of the unhealthy population, π(x, t), on the other hand, are more complex in the sense that they are a function of all the past transitions between the states "healthy", "unhealthy", and "death" [36]. Unfortunately, detailed data on such transitions are not available for a long time series. Still, cross-sectional health data provides information about the proportion of healthy individuals for a given cohort at a particular point in time as a result of these transitions. For example, a representative health survey conducted in the year 2015 allows us to estimate the fraction of healthy individuals over all individuals for each birth cohort alive in 2015, i.e. cohorts born between 1915 and 2015 (assuming the survey includes the population at age zero to 100). In this way, the exact health trajectories for cohorts remain unobserved but have been implicitly included in the π(x, t) function. This is illustrated in Fig. 5 for the 1935 and 1965 birth cohorts. Guillot and Yu [45] provide an equation for calculating the population proportion of unhealthy individuals at a particular point in time on the basis of transition probabilities between health states and death ([45], p. 508). This equation has been applied to simulated data for the sake of demonstrating the relationship between in- and outflows, i.e. transitions to the (un)healthy state, and the proportion of the unhealthy population as a stock. Please note that assuming an exponential trend for health transition probabilities is in line with previous research [46,47,48]. The transitions (left panel) produce the prevalence of being unhealthy (right panel). This is 96 percent for the 1935 birth cohort at age 80 (in 2015). The younger 1965 birth cohort turns 50 in 2015 and reaches a proportion of 77% unhealthy individuals. The younger cohort has been exposed to favourable health and mortality conditions (indicated by favourable transition probabilities). This is why a higher share has been found unhealthy at age 50 for the 1935 birth cohort compared to the 1965 birth cohort (over 80% vs. 77%). The observed cross-sectional prevalence at time t, i.e. the π(x, t) function, does not provide any information about the exact morbidity and mortality trajectories of cohorts but gives an estimate of the unhealthy population stock for the cohorts reaching age x in time t (born in time t − x) in accordance with their underlying multistate process.
Relationship between transition probabilities and prevalence data. The figure shows simulated age-specific transition probabilities between the states of being healthy, unhealthy, and dead for the cohorts born in 1935 and 1965 (depicted on the left side of the figure). The proportions of the unhealthy population for the birth cohorts 1935 and 1965 are calculated from these age-specific transition probabilities (depicted on the right side of the figure)
HCAL and HE in relative terms
Differences in the pattern of the pc(x, t − x)function compared to the p(x, t) affect the outcome of the measure. First, the less rectangular pattern of the pc(x, t − x) function causes lower absolute values for HCAL. Obviously, the lower age-specific survival probabilities result in a lower total number of person-years lived and therefore, HE will exceed HCAL in absolute terms. Interestingly, the choice of the survivorship function also affects the measure in relative terms, i.e. the ratio HCAL/CAL vs. the ratio HE/LE. The reason for that can be revealed by looking at person-years lived in relative terms (dividing the age-specific count of person-years lived by the total number of person-years lived). As already mentioned, the total number of person-years lived is lower in CAL. However, at young ages, both functions show similar number of person-years lived. This is why dividing the healthy life years at young ages by the total number of life years yields greater ratios for HCAL/CAL compared to HE/LE (Fig. 6). This relationship inverses at older ages so that the ratios for HE now exceed the ones for HCAL. In other words, the pc(x, t − x) function (in relative terms) gives more weights on young ages and less weights on old ages compared to the p(x, t) function. Since individuals are mostly healthy at young ages and become increasingly more unhealthy at older ages, the share of healthy life years on total life years is higher in HCAL.
Person-years lived in relative terms on the basis of period LE and CAL. The figure shows the relative number of age-specific person-years lived between age zero and 100, i.e. the share of age-specific person-years lived on the total number of person-years lived, using the concept of period LE (solid line) and CAL (dashed line)
Interpreting HE and HCAL results
The main purpose of HE is to combine the health and mortality conditions prevailing in one particular period to one single measure. The mortality information in the p(x, t) function refers solely to time t so that LE is a pure period mortality measure [10]. What about the π(x, t) function? This function refers to the age-specific prevalence of the population in healthy and unhealthy states at time t and, therefore, can be seen as a period estimate as well. In this sense, HE(t) is a period measure that reflects the health composition of the real population at time t adjusted for period mortality [49]. However, it is not a pure period indicator in a synthetic cohort fashion [5]. We have illustrated that the health state-specific prevalence depends on past mortality rates and health transitions rates. Therefore, HE cannot be interpreted as an average number of healthy life years, lived by currently newborns under the assumption that they are solely exposed to the health and mortality conditions observed in one single period. In order to interpret HE in the same manner as LE, one would need to replace the observed prevalence of being in the (un)healthy state with a constructed "period" or "equilibrium" prevalence of being in the (un)healthy state [50]. This is a synthetic prevalence calculated from transitions between health states and death observed in a given period. Lièvre et al. [48] have shown that this modelled prevalence is currently lower than the observed prevalence in the USA. Only if health and mortality conditions are constant over time, i.e. constant transition rates, the period prevalence equals the observed prevalence [50]. In this scenario, the Sullivan method yields a period HE estimate, which is equal to the results based on the multistate life table method [51].
CAL differs from period and cohort LE because it does not refer to one single cohort. While period and cohort life tables give an estimate of the average number of person-years lived for one cohort (real or synthetic), CAL is the average number of person-years, which has been lived by all cohorts alive in a given period (assuming a closed population with a constant inflow of annual births). Thus, HCAL refers to the average number of healthy person-years lived by these cohorts. In other words, HCAL reflects the health and mortality conditions that the cohorts present in a population at a given point in time have been exposed to during their past life course. Alternatively, CAL(t) can also be seen as the relative population size at time t in a constant-birth population [15]. Imagine a population where all cohorts have the same initial size, i.e. a population with a constant number of births each year. Formally, this model population is expressed as follows:
$$ N(t)={\int}_0^{\infty }B\cdotp {p}_c\left(x,t-x\right) dx $$
with N(t) being the total population at time t and B the number of the annual inflow of births. Rewriting the equation and substituting CAL(t) for integrated pc(x, t − x) function yields the population size at time t as the product of CAL(t) and B.
$$ N(t)=B\cdotp {\int}_0^{\infty }{p}_c\left(x,t-x\right) dx $$
$$ N(t)=B\cdotp CAL(t) $$
Let us assume that the initial size of each cohort is 100,000 (B = 100,000). By applying the cohort survivor probabilities (given by the pc(x, t − x) function) to this modelled population, each cohort is exposed to the mortality regime, which actually took place during its past life course (from birth up to time t). Since CAL(t) summarizes the cohort survivor proportions in time t, multiplying the CAL value by 100,000 yields the population size in time t. Applying the same population model to HCAL allows to interpret HCAL(t) as the relative size of the healthy population in time t.
$$ {N}^{\mathrm{Healthy}}(t)={\int}_0^{\infty }B\cdotp \mathrm{HCAL} $$
In this perspective, looking at HCAL estimates can be seen as looking at population data, while controlling for fluctuations in births. Comparing CAL and HCAL estimates for a given population over time indicates clearly how the healthy and total share of individuals in a population has evolved. In the case of an increasing HCAL, changes in transition rates promote a situation in which the healthy population is increasing in size over time. This is the clearest sign that people are indeed living longer and healthier lives.
Sauerberg, M., Guillot, M. & Luy, M. The cross-sectional average length of healthy life (HCAL): a measure that summarizes the history of cohort health and mortality. Popul Health Metrics 18, 21 (2020). https://doi.org/10.1186/s12963-020-00220-5
Sullivan method
Healthy life years | CommonCrawl |
March 2002 , Volume 44, Issue 1, pp 111–126 | Cite as
Sublattices of regular elements
D. D. Anderson
E. W. Johnson
Richard L. Spellerberg II
Let L be an r-lattice, i.e., a modular multiplicative lattice that is compactly generated, principally generated, and has greatest element 1 compact. We consider certain subsets of L consisting of "regular elements": \(L_f = \left\{ 0 \right\} \cup \left\{ A \right. \in \left. L \right|\left. {(0:A) = 0} \right\},L_{sr} = \left\{ 0 \right\} \cup \left\{ A \right. \in \left. L \right|\) there is a compact element \(X \leqslant A\) with \(\left. {(0:X) = 0} \right\},L_r = \left\{ 0 \right\} \cup \left\{ A \right. \in \left. L \right|\) there is a principal element \(X \leqslant A{\text{ with }}\left. {(0:X) = 0} \right\},{\text{ and }}L_{rg} = \left\{ 0 \right\} \cup \left\{ A \right. \in \left. L \right|A = \bigvee _\alpha X_\alpha {\text{ where each }}X_\alpha \) is a principal element with (0:X_{\alpha })=0\} . The first three subsets L_{f}, L_{sr}, and L_{r} are augmented filters \(\mathcal{L}^0 \) on L, i.e., \(\mathcal{L}^0 = \mathcal{L} \cup \left\{ 0 \right\}{\text{ where }}\mathcal{L}\) is a multiplicatively closed subset of $L$ with $A\in \(\mathcal{L}\) and $B\geq A$ with $B\in L$ implies $B\in \(\mathcal{L}\) and hence are sublattices of $L$ closed under multiplication. We first consider the more general situation of augmented filters on $L.$ These results are then applied to study the four previously defined subsets for $L$ an $r$-lattice or Noether lattice (i.e., an $r$-lattice with ACC). Finally, we give a brief discussion of how the results for augmented lattices can be applied to subsets of $L$ which are "regular" with respect to an $L$-module.
General Situation Regular Element Great Element Principal Element Compact Element
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© Kluwer Academic Publishers 2002
1.Department of MathematicsThe University of IowaIowa CityU.S.A.
2.Department of MathematicsSimpson CollegeIndianolaU.S.A.
Anderson, D.D., Johnson, E.W. & Spellerberg II, R.L. Periodica Mathematica Hungarica (2002) 44: 111. https://doi.org/10.1023/A:1014932204184
DOI https://doi.org/10.1023/A:1014932204184 | CommonCrawl |
\begin{document}
\pagestyle{empty} \title{Operational axioms for a C${}^*$-algebraic formulation of Quantum Mechanics} \author{Giacomo Mauro D'Ariano\refnote{1}} \affiliation{\affnote{1}Dipartimento di Fisica ``A. Volta'', via Bassi 6, 27100 Pavia, Italy} \begin{abstract}
A C${}^*$-algebra formulation of Quantum Mechanics is derived from purely operational axioms in
which the primary role is played by the {\em transformations} that the system undergoes in the
course of an {\em experiment}. The notion of the {\em adjoint} of a transformation is based on the
postulated existence of {\em faithful states} that allows one to calibrate the experimental
apparatus. \end{abstract} \section{Introduction} In a set of recent papers \cite{darianoVax2006,dariano-beyond} I recently showed how it is possible to derive the mathematical formulation of Quantum Mechanics in terms of complex Hilbert spaces or in terms of C${}^*$-algebras, starting from five purely operational Postulates concerning {\em
experimental accessibility and simplicity}. The starting point for the axiomatization is a seminal definition of {\em physical experiment} which entails the thorough series of notions that are at the basis of the axiomatization. In the present short account I will briefly review the derivation of a C${}^*$-algebra formulation from only two Postulates on the physical experiment, based on a operational notion of the {\em adjoint} of a transformation, which follows from the postulated existence of {\em faithful states}. Such states are crucial for calibrating the experimental apparatus, and their basic idea comes from modern Quantum Tomography \cite{tomo_lecture,calib}. The quantum C${}^*$-algebra representation of the transformations (for generally infinite dimensions) is then derived from the Postulates via a Gelfand-Naimark-Segal (GNS) construction\cite{GNS}. \section{The postulates} The general premise of the present axiomatization is the fact that one performs experiments to get information on the {\em state} of an {\em object} physical system, and the knowledge of such a state will then enable to predict the results of forthcoming experiments. Moreover, since we necessarily work with only partial {\em a priori} knowledge of both system and experimental apparatus, the rules for the experiment must be given in a probabilistic setting. {\em What we mean by experiment?} An experiment on an object system consists in making it interact with an apparatus: the interaction between object and apparatus produces one of a set of possible transformations of the object, each one occurring with some probability. Information on the state of the object at the beginning of the experiment is gained from the knowledge of which transformation occurred, which is the "outcome" of the experiment signaled by the apparatus. \par We can now introduce the two postulates. \begin{postulate}[Independent systems]\label{p:independent} There exist independent physical systems. \end{postulate} \begin{postulate}[Symmetric faithful state]\label{p:faith} For every composite system made of two identical
physical systems there exist a symmetric joint state that is both dynamically and preparationally
faithful. \end{postulate}
\section{The statistical and dynamical structure} The starting point of the axiomatization is the identification {\bf experiment }$\equiv${\em set of
transformations} $\mathbb A}\def\AB{\mathbb B}\def\AC{\mathbb C}\def\AL{\mathbb L\equiv\{\tA_j\}$ that can occur on the object. The apparatus signals which transformation actually occurs. Now, since the knowledge of the state of a physical system allows us to predict the results of forthcoming experiments on the object, then it would allow us to evaluate the probability of any possible transformation in any conceivable experiment. Therefore, by definition, a {\bf state} $\omega$ of a system is a rule providing probabilities of transformation, and $\omega(\tA)$ is the probability that the transformation $\tA$ occurs. We clearly have the completeness $\sum_{\tA_j\in\mathbb A}\def\AB{\mathbb B}\def\AC{\mathbb C}\def\AL{\mathbb L}\omega(\tA_j)=1$, and assume $\omega(\trnsfrm I}\def\tT{\trnsfrm T}\def\tU{\trnsfrm U}\def\tM{\trnsfrm M}\def\tX{\trnsfrm X)=1$ for the identical transformation $\trnsfrm I}\def\tT{\trnsfrm T}\def\tU{\trnsfrm U}\def\tM{\trnsfrm M}\def\tX{\trnsfrm X$, corresponding to adopting $\trnsfrm I}\def\tT{\trnsfrm T}\def\tU{\trnsfrm U}\def\tM{\trnsfrm M}\def\tX{\trnsfrm X$ as the free evolution (this is the {\em Dirac
picture}, \ie a suitable choice of the lab reference frame). In the following for a given physical system we will denote by ${\mathfrak S}}\def\Wset{{\mathfrak W}$ the set of all possible states and by ${\mathfrak T}}\def\Actset{{\mathfrak A}}\def\trnset{{\mathfrak t}$ the set of all possible transformations.
\par When composing two transformations $\tA$ and $\tB$, the probability $p(\tB|\tA)$ that $\tB$
occurs conditional on the previous occurrence of $\tA$ is given by the Bayes rule for conditional probabilities $p(\tB|\tA)=\omega(\tB\circ\tA)/\omega(\tA)$. This sets a new probability rule corresponding to the notion of {\bf conditional state} $\omega_\tA$ which gives the probability that a transformation $\tB$ occurs knowing that the transformation $\tA$ has occurred on the object in the state $\omega$, namely $\omega_\tA\doteq\omega(\cdot\circ\tA)/\omega(\tA)$\footnote{Throughout
the paper we will make extensive use of the functional notation with the central dot corresponding
to a variable transformation}. We can see that the notion of ``state'' itself logically implies the identification {\em evolution}$\equiv${\em state-conditioning}, entailing a {\em linear
action of transformations on states} (apart from normalization) $\tA\omega:=\omega(\cdot\circ\tA)$: this is the same concept of {\bf operation} that we have in Quantum Mechanics, which gives the conditioning $\omega_\tA=\tA\omega/\tA\omega(\trnsfrm I}\def\tT{\trnsfrm T}\def\tU{\trnsfrm U}\def\tM{\trnsfrm M}\def\tX{\trnsfrm X)$. In other words, this is the analogous of the Schr\"{o}dinger picture evolution of states in Quantum Mechanics (clearly such identification of evolution as state-conditioning also includes the deterministic case $\tU\omega=\omega(\cdot\circ\tU)$ of transformations $\tU$ with $\omega(\tU)=1\,\forall\omega\in{\mathfrak S}}\def\Wset{{\mathfrak W}$---the analogous of quantum unitary evolutions and channels.
From the Bayes conditioning it follows that we can define two complementary types of equivalences for transformations: {\em dynamical} and {\em informational}. The transformations $\tA_1$ and $\tA_2$ are {\bf dynamically equivalent} when $\omega_{\tA_1}=\omega_{\tA_2}$ $\forall\omega\in{\mathfrak S}}\def\Wset{{\mathfrak W}$, whereas they are {\bf informationally equivalent} when $\omega(\tA_1)=\omega(\tA_2)$ $\forall\omega\in{\mathfrak S}}\def\Wset{{\mathfrak W}$. The two transformations are then completely equivalent (write $\tA_1=\tA_2$) when they are both dynamically and informationally equivalent, corresponding to the identity $\omega(\tB\circ\tA_1)=\omega(\tB\circ\tA_2)$, $\forall\omega\in{\mathfrak S}}\def\Wset{{\mathfrak W},\;\forall\tB\in{\mathfrak T}}\def\Actset{{\mathfrak A}}\def\trnset{{\mathfrak t}$. We call {\bf effect} the informational equivalence class of transformations (this is the same notion introduced by Ludwig\cite{Ludwig-axI}). In the following we will denote effects with the underlined symbols ${\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT}$, $\cB$, etc., or as $[\tA]_\eff$, and we will write $\tA_0\in{\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT}$ meaning that "the transformation $\tA$ belongs to the equivalence class ${\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT}$", or "$\tA_0$ has effect ${\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT}$'', or "$\tA_0$ is informationally equivalent to $\tA$". Since, by definition one has $\omega(\tA)\equiv\omega({\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT})$, we will legitimately write $\omega({\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT})$ instead of $\omega(\tA)$. Similarly, one has $\omega_\tA(\tB)\equiv \omega_\tA(\cB)$, which implies that $\omega(\tB\circ\tA)=\omega(\cB\circ\tA)$, leading to the chaining rule $\cB\circ\tA\in\underline{\tB\circ\tA}$ corresponding to the "Heisenberg picture" evolution of transformations acting on effects (notice how transformations act on effects from the right). Now, by definitions effects are linear functionals over states with range $[0,1]$, and, by duality, we have a convex structure over effects, and we will denote their convex set as ${\mathfrak P}}\def\Ball{{\mathfrak B}_1$. An {\bf
observable} is just a complete set of effects $\AL=\{l_i\}$ of an experiment $\mathbb A}\def\AB{\mathbb B}\def\AC{\mathbb C}\def\AL{\mathbb L=\{\tA_j\}$, namely one has $l_i=\underline{\tA_j}$ $\forall j$ (clearly, one has the completeness relation $\sum_il_i=1$). We will call the observable $\AL=\{l_i\}$ is {\bf informationally complete} when each effect $l$ can be written as a linear combination $l=\sum_ic_i(l)l_i.$ of elements of $\AL$, and when these are linearly independent we will call the informationally complete observable {\em
minimal}.
\par The fact that we necessarily work in the presence of partial knowledge about both object and apparatus corresponds to the possibility of incomplete specification of both states and transformations, entailing the convex structure on states and the addition rule for {\em coexistent
transformations}, namely for transformations $\tA_1$ and $\tA_2$ for which $\omega(\tA_1)+\omega(\tA_2)\leq 1,\;\forall\omega\in{\mathfrak S}}\def\Wset{{\mathfrak W}$ (\ie transformations that can in principle occur in the same experiment). The addition of the two coexistent transformations is the transformation $\trnsfrm S}\def\tG{\trnsfrm G=\tA_1+\tA_2$ corresponding to the event $e=\{1,2\}$ in which the apparatus signals that either $\tA_1$ or $\tA_2$ occurred, but does not specify which one. Such transformation is uniquely determined by the informational and dynamical classes as $\forall\omega\in{\mathfrak S}}\def\Wset{{\mathfrak W}$: $\omega(\tA_1+\tA_2)=\omega(\tA_1)+\omega(\tA_2),\; (\tA_1+\tA_2)\omega=\tA_1\omega+ \tA_2\omega$. The composition "$\circ$" of transformations is distributive with respect to the addition "$+$". We can also define the multiplication $\lambda\tA$ of a transformation $\tA$ by a scalar $0\leq\lambda\leq 1$ as the transformation dynamically equivalent to $\tA$, but occurring with rescaled probability $\omega(\lambda\tA)=\lambda\omega(\tA)$. Now, since for every couple of transformation $\tA$ and $\tB$ the transformations $\lambda\tA$ and $(1-\lambda)\tB$ are coexistent for $0\leq\lambda\leq 1$, the set of transformations also becomes a convex set. Moreover, since the composition $\tA\circ\tB$ of two transformations $\tA$ and $\tB$ is itself a transformation and there exists the identical transformation $\trnsfrm I}\def\tT{\trnsfrm T}\def\tU{\trnsfrm U}\def\tM{\trnsfrm M}\def\tX{\trnsfrm X$ satisfying $\trnsfrm I}\def\tT{\trnsfrm T}\def\tU{\trnsfrm U}\def\tM{\trnsfrm M}\def\tX{\trnsfrm X\circ\tA=\tA\circ\trnsfrm I}\def\tT{\trnsfrm T}\def\tU{\trnsfrm U}\def\tM{\trnsfrm M}\def\tX{\trnsfrm X=\tA$ for every transformation $\tA$, the transformations make a semigroup with identity, \ie a {\em monoid}. Therefore, the set of physical transformations ${\mathfrak T}}\def\Actset{{\mathfrak A}}\def\trnset{{\mathfrak t}$ is a convex monoid.
It is obvious that we can extend the notions of coexistence, sum and multiplication by a scalar from transformations to effects via equivalence classes.
\par A purely dynamical notion of {\bf independent systems} coincides with the possibility of performing local experiments. More precisely, we say that two physical systems are {\em independent} if on the two systems 1 and 2 we can perform {\em local experiments} $\mathbb A}\def\AB{\mathbb B}\def\AC{\mathbb C}\def\AL{\mathbb L^{(1)}$ and $\mathbb A}\def\AB{\mathbb B}\def\AC{\mathbb C}\def\AL{\mathbb L^{(2)}$ whose transformations commute each other (\ie $\tA^{(1)}\circ\tB^{(2)}=\tB^{(2)}\circ\tA^{(1)},\;\forall \tA^{(1)}\in\mathbb A}\def\AB{\mathbb B}\def\AC{\mathbb C}\def\AL{\mathbb L^{(1)},\,\forall \tB^{(2)}\in\AB^{(2)}$). Notice that the above definition of independent systems is purely dynamical, in the sense that it does not contain any statistical requirement, such as the existence of factorized states. Indeed, the present notion of dynamical independence is so minimal that it can be satisfied not only by the quantum tensor product, but also by the quantum direct sum. As shown in Ref. \cite{darianoVax2006}, it is an additional Postulate---the {\em local observability
principle}---which selects the tensor product. In the following, when dealing with more than one independent system, we will denote local transformations as ordered strings of transformations as follows $\tA,\tB,\tC,\ldots:=\tA^{(1)}\circ\tB^{(2)}\circ\tC^{(3)}\circ\ldots$. For effects one has the locality rule $([\tA]_\eff,[\tB_\eff)\in[(\tA,\tB)]_\eff$. The notion of independent systems now entails the notion of {\em local state}---the equivalent of partial trace in Quantum Mechanics. For two independent systems in a joint state $\Omega$, we define the {\bf local state} $\Omega|_1$
(and similarly $\Omega|_2$) as the probability rule $\Omega|_1(\tA)\doteq\Omega(\tA,\trnsfrm I}\def\tT{\trnsfrm T}\def\tU{\trnsfrm U}\def\tM{\trnsfrm M}\def\tX{\trnsfrm X)$ of the joint state $\Omega$ with a local transformation $\tA$ acting only on system $1$ and with all other systems untouched.
\section{The C${}^*$-algebra of transformations} We have seen that the physical transformations make a convex monoid. It is easy to extend it to a real algebra ${\mathfrak T}}\def\Actset{{\mathfrak A}}\def\trnset{{\mathfrak t}_\mathbb R}\def\Cmplx{\mathbb C}\def\Naturals{\mathbb N}\def\Intgrs{\mathbb Z$ by taking differences of physical transformations, and multiply them by scalars $\lambda>1$. We will call the elements of ${\mathfrak T}}\def\Actset{{\mathfrak A}}\def\trnset{{\mathfrak t}_\mathbb R}\def\Cmplx{\mathbb C}\def\Naturals{\mathbb N}\def\Intgrs{\mathbb Z$ that are not in ${\mathfrak T}}\def\Actset{{\mathfrak A}}\def\trnset{{\mathfrak t}$ {\em
generalized transformations}. Likewise, we can introduce generalized effects, and denote their linear space as ${\mathfrak P}}\def\Ball{{\mathfrak B}_1_\mathbb R}\def\Cmplx{\mathbb C}\def\Naturals{\mathbb N}\def\Intgrs{\mathbb Z$. Now that we have a real algebra of generalized transformations and a linear space of generalized effects we want to introduce a positive bilinear form over them, by which we will be able to introduce a scalar product via the GNS construction\cite{GNS}. The role of such bilinear form will be played by a {\em faithful state}.
\par We say that a state $\Phi$ of a bipartite system is {\bf dynamically faithful} for system 1 when for every transformation $\tA$ the map $\tA\leftrightarrow(\tA,\trnsfrm I}\def\tT{\trnsfrm T}\def\tU{\trnsfrm U}\def\tM{\trnsfrm M}\def\tX{\trnsfrm X)\Phi$ is one-to-one. This means that for every bipartite effect $\cB$ one has $\Phi(\cB\circ(\tA,\trnsfrm I}\def\tT{\trnsfrm T}\def\tU{\trnsfrm U}\def\tM{\trnsfrm M}\def\tX{\trnsfrm X))=0\quad\Longleftrightarrow\quad\tA=0$. Clearly the correspondence remains one-to-one when extended to ${\mathfrak T}}\def\Actset{{\mathfrak A}}\def\trnset{{\mathfrak t}_\mathbb R}\def\Cmplx{\mathbb C}\def\Naturals{\mathbb N}\def\Intgrs{\mathbb Z$. On the other hand, we will call a state $\Phi$ of a bipartite system {\bf preparationally faithful} for system 1 if every joint bipartite state $\Omega$ can be achieved by a suitable local transformation $\tT_\Omega$ on system 1 occurring with nonzero probability. Clearly a bipartite state $\Phi$ that is preparationally faithful is also {\em locally} preparationally faithful, namely every local state $\omega$ of system 2 can be achieved by a suitable local transformation $\tT_\omega$ on system 1. \par In Postulate \ref{p:faith} we also use the notion of {\bf symmetric joint state}. This is simply defined as a joint state of two identical systems such that for any couple of transformations $\tA$ and $\tB$ one has $\Phi(\tA,\tB)=\Phi(\tB,\tA)$.
Clearly both notions of faithfulness hold for both systems for a symmetrical state. For a {\em
faithful} bipartite state $\Phi$, the {\em transposed transformation} $\tA'$ of the transformation $\tA$ is the generalized transformation which when applied to the second component system gives the same conditioned state and with the same probability as the transformation $\tA$ operating on the first system, namely $(\tA,\trnsfrm I}\def\tT{\trnsfrm T}\def\tU{\trnsfrm U}\def\tM{\trnsfrm M}\def\tX{\trnsfrm X)\Phi=(\trnsfrm I}\def\tT{\trnsfrm T}\def\tU{\trnsfrm U}\def\tM{\trnsfrm M}\def\tX{\trnsfrm X,\tA')\Phi$ or, equivalently $\Phi(\cB\circ\tA,\cC)=\Phi(\cB,\cC\circ\tA')$ $\forall \cB,\cC\in{\mathfrak P}}\def\Ball{{\mathfrak B}_1$. Clearly the transposed is unique, due to injectivity of the map $\tA\leftrightarrow(\tA,\trnsfrm I}\def\tT{\trnsfrm T}\def\tU{\trnsfrm U}\def\tM{\trnsfrm M}\def\tX{\trnsfrm X)\Phi$, and it is easy to check the axioms of transposition ($(\tA+\tB)'= \tA'+\tB'$, $(\tA')'=\tA$, $(\tA\circ\tB)'= \tB'\circ\tA'$) and that $\trnsfrm I}\def\tT{\trnsfrm T}\def\tU{\trnsfrm U}\def\tM{\trnsfrm M}\def\tX{\trnsfrm X'=\trnsfrm I}\def\tT{\trnsfrm T}\def\tU{\trnsfrm U}\def\tM{\trnsfrm M}\def\tX{\trnsfrm X$.
The main ingredient of a GNS construction for representing transformations would be a positive form
$\varphi$ by which one can construct a scalar product as $\<\tA|\tB\>:=\varphi(\tA^\dagger}\def\eff{{\rm eff}}\def\dyn{{\rm dyne}\circ\tB)$, in terms of which we then have
$\<\tA|\tC\circ\tB\>=\<\tC^\dagger}\def\eff{{\rm eff}}\def\dyn{{\rm dyne}\circ\tA|\tB\>\equiv\varphi(\tA^\dagger}\def\eff{{\rm eff}}\def\dyn{{\rm dyne}\circ\tC\circ\tB) =\varphi((\tC^\dagger}\def\eff{{\rm eff}}\def\dyn{{\rm dyne}\circ\tA)^\dagger}\def\eff{{\rm eff}}\def\dyn{{\rm dyne}\circ\tB)$. However, we don't have a definition for the adjoint, and it is not easy to devise a positive form over generalized transformations ${\mathfrak T}}\def\Actset{{\mathfrak A}}\def\trnset{{\mathfrak t}_\mathbb R}\def\Cmplx{\mathbb C}\def\Naturals{\mathbb N}\def\Intgrs{\mathbb Z$ such that the transposition plays the role of the adjoint on a real Hilbert space. Indeed, if we take
$\varphi$ as the local state of a symmetric faithful state $\varphi=\Phi|_2\equiv\Phi|_1$ we have $\varphi(\tA'\circ\tB)=\Phi(\tA',\tB')\equiv\Phi({\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT}',\cB')$ (notice that the bilinear form $\Phi$ is actually defined on effects), but the fact that $\Phi$ is positive over the convex set ${\mathfrak T}}\def\Actset{{\mathfrak A}}\def\trnset{{\mathfrak t}$ of physical transformations doesn't guarantee that its extension to generalized transformations
${\mathfrak T}}\def\Actset{{\mathfrak A}}\def\trnset{{\mathfrak t}_\mathbb R}\def\Cmplx{\mathbb C}\def\Naturals{\mathbb N}\def\Intgrs{\mathbb Z$ is still positive. One can, however, extract from $\Phi$ a positive bilinear form over ${\mathfrak P}}\def\Ball{{\mathfrak B}_1_\mathbb R}\def\Cmplx{\mathbb C}\def\Naturals{\mathbb N}\def\Intgrs{\mathbb Z$ in terms of its absolute value $|\Phi|:=\Phi_+-\Phi_-$. Indeed, the absolute value can be defined thanks to the fact that $\Phi$ is real symmetric, whence it can be diagonalized over ${\mathfrak P}}\def\Ball{{\mathfrak B}_1_\mathbb R}\def\Cmplx{\mathbb C}\def\Naturals{\mathbb N}\def\Intgrs{\mathbb Z$. Upon denoting by $\map{P}_\pm$ the orthogonal projectors over the linear space corresponding to positive and negative eigenvalues, respectively, \footnote{The existence of
the orthogonal space decomposition corresponding to positive and negative eigenvalues is
guaranteed for finite dimensions. For infinite dimensions $\Phi$ is just a symmetric form over a
Banach space, and the existence of such decomposition remains to be seen.} one has
$|\Phi|({\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT},\cB)=\Phi(\varsigma({\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT}),\cB)$, where $\varsigma({\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT}):=(\map{P}_+-\map{P}_-)({\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT})$. The map $\varsigma$ is an involution, namely $\varsigma^2=\map{I}$. The fact that the state is also preparationally faithful implies that the bilinear form is {\em strictly}
positive\cite{darianoVax2006} (namely $|\Phi|(\cC,\cC)=0$ implies that $\cC=0$). We can extend the involution $\varsigma$ to generalized transformations by considering the absolute value of $\Phi$ regarded as bilinear form over generalized transformations ${\mathfrak T}}\def\Actset{{\mathfrak A}}\def\trnset{{\mathfrak t}_\mathbb R}\def\Cmplx{\mathbb C}\def\Naturals{\mathbb N}\def\Intgrs{\mathbb Z$. In this way we have $\varphi(\varsigma(\tA')\circ\tB)$ as a positive form, and we can identify $\varsigma(\tA')\equiv\tA^\dagger}\def\eff{{\rm eff}}\def\dyn{{\rm dyne}$ as the adjoint, namely as the composition of the transposition and the {\em complex conjugation} $\varsigma$. We need to choose the extension of $\varsigma$ to transposition to be composition-preserving, \ie $\varsigma(\tB\circ\tA)=\tB^\varsigma\circ\tA^\varsigma$,\footnote{The involution $\varsigma$ is
composition-preserving if $\varsigma({\mathfrak T}}\def\Actset{{\mathfrak A}}\def\trnset{{\mathfrak t})={\mathfrak T}}\def\Actset{{\mathfrak A}}\def\trnset{{\mathfrak t}$ namely if the involution preserves physical
transformations. Indeed, for such an involution one can consider its action on transformations
induced by the involutive isomorphism $\omega\to\omega^\varsigma$ of the convex set of states
${\mathfrak S}}\def\Wset{{\mathfrak W}$ defined as $\omega^\varsigma(\tA):=\omega(\varsigma(\tA))$,
$\forall\omega\in{\mathfrak S}}\def\Wset{{\mathfrak W},\;\forall\tA\in{\mathfrak T}}\def\Actset{{\mathfrak A}}\def\trnset{{\mathfrak t}$. Consistency with state-reduction
$\omega_\tA^\varsigma(\tB)\equiv\omega_{\tA^\varsigma}(\tB^\varsigma)$
$\forall\omega\in{\mathfrak S}}\def\Wset{{\mathfrak W},\;\forall\tA,\tB\in{\mathfrak T}}\def\Actset{{\mathfrak A}}\def\trnset{{\mathfrak t}$ is then equivalent to
$\omega(\varsigma(\tB\circ\tA))=\omega(\tB^\varsigma\circ\tA^\varsigma)$
$\forall\omega\in{\mathfrak S}}\def\Wset{{\mathfrak W},\;\forall\tA,\tB\in{\mathfrak T}}\def\Actset{{\mathfrak A}}\def\trnset{{\mathfrak t}$. The involution $\varsigma$ of ${\mathfrak S}}\def\Wset{{\mathfrak W}$ is
just the inversion of the principal axes corresponding to negative eigenvalues of the symmetric
bilinear form $\Phi$.\cite{darianoVax2006})} in such a way that $(\tB\circ\tA)^\dagger}\def\eff{{\rm eff}}\def\dyn{{\rm dyne}=\tA^\dagger}\def\eff{{\rm eff}}\def\dyn{{\rm dyne}\circ\tB^\dagger}\def\eff{{\rm eff}}\def\dyn{{\rm dyne}$ (\ie it is the transposition that takes care of ordering). Now, following the GNS construct, we introduce the scalar product as
${}_\Phi\!\<{\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT}|\cB\>_\Phi:=\varphi(\tA^\dagger}\def\eff{{\rm eff}}\def\dyn{{\rm dyne}\circ\tB)= \Phi(\varsigma({\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT}'),\cB')$, and we can verify that $\tA^\dagger}\def\eff{{\rm eff}}\def\dyn{{\rm dyne}:=\varsigma(\tA')$ works as an adjoint for such scalar product, namely
${}_\Phi\!\<\tC^\dagger}\def\eff{{\rm eff}}\def\dyn{{\rm dyne}\circ{\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT}|\cB\>_\Phi={}_\Phi\!\<{\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT}|\tC\circ\cB\>_\Phi$.\footnote{Clearly in this
way one recovers the customary operator-like action of transformations from the left
$|\underline{\tC\circ\tA}\>_\Phi= |\tC\circ{\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT}\>_\Phi$ which follows from the fact that the scalar
product is defined in terms of the positive bilinear form $|\Phi|$ over transposed transformations
${}_\Phi\!\<\tC\circ{\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT}|\tB\>_\Phi=|\Phi|({\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT}'\circ\tC',\cB')=\Phi(\varsigma({\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT}'\circ\tC'),\cB')$.} In the following we will equivalently write the entries of the scalar product as generalized transformations or as generalized effects, with
${}_\Phi\!\<\tA|\tB\>_\Phi:={}_\Phi\!\<{\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT}|\cB\>_\Phi$, the generalized effects being the actual vectors of the linear factor space of generalized transformations modulo informational equivalence.
Now, by taking complex linear combinations of generalized transformations and defining $\varsigma(c\tA)=c^*\varsigma(\tA)$ for $c\in\Cmplx$, we can extend the adjoint to complex linear combinations of generalized transformations, whose linear space will be denoted by ${\mathfrak T}}\def\Actset{{\mathfrak A}}\def\trnset{{\mathfrak t}_\Cmplx$, which is a complex algebra that we will also denote as $\aA$. On the other hand, we can trivially extend the real pre-Hilbert space of generalized effects ${\mathfrak P}}\def\Ball{{\mathfrak B}_1_\mathbb R}\def\Cmplx{\mathbb C}\def\Naturals{\mathbb N}\def\Intgrs{\mathbb Z$ to a complex pre-Hilbert space ${\mathfrak P}}\def\Ball{{\mathfrak B}_1_\Cmplx$ by just considering complex linear combinations of generalized effects.
The remaining setting up of the C${}^*$-algebra representation of $\aA$ is just standard GNS construction. We now have a scalar product ${}_\Phi\!\<\tA|\tB\>_\Phi=\Phi_2(\tA^\dagger}\def\eff{{\rm eff}}\def\dyn{{\rm dyne}\circ\tB)$ between transformations. Symmetry and positivity imply the bounding\cite{darianoVax2006}
${}_\Phi\!\<\tA|\tB\>_\Phi\leq\n{\tA}_\Phi\n{\tB}_\Phi$, where we introduced the norm induced by the scalar product $\n{\tA}_\Phi^2\doteq{}_\Phi\!\<\tA|\tA\>_\Phi$. By taking the equivalence classes
$\aA/\aI$ with respect to the zero-norm elements $\aI\subseteq\aA$ we thus obtain a complex pre-Hilbert space equipped with a symmetric scalar product, and, since the scalar product is strictly positive over generalized effects, the elements of $\aA/\aI$ are indeed the generalized effects, \ie $\aA/\aI\simeq{\mathfrak P}}\def\Ball{{\mathfrak B}_1_\Cmplx$ as linear spaces. Moreover, from the bounding for the scalar product it follows that the set $\aI\subseteq\aA$ of zero norm elements $\tX\in\aA$ is a left ideal (\ie $\tX\in\aI$, $\tA\in\aA$ implies $\tA\circ\tX\in\aI$), whence using our scalar product defined as ${}_\Phi\!\<\tA|\tB\>_\Phi=\Phi_2(\tA^\dagger}\def\eff{{\rm eff}}\def\dyn{{\rm dyne}\circ\tB)$ we can represent elements of $\aA$ ($\aA\equiv{\mathfrak T}}\def\Actset{{\mathfrak A}}\def\trnset{{\mathfrak t}_\Cmplx$ are the generalized complex transformations) as operators over the pre-Hilbert space of effects and make $\aA$ a C${}^*$-algebra. We just need to introduce the norm on transformations as $\n{\tA}_\Phi:=\sup_{\cB\in{\mathfrak P}}\def\Ball{{\mathfrak B}_1_\Cmplx,\n{\cB}_\Phi\leq
1}\n{\tA\circ\cB}_\Phi$. Completion of $\aA/\aI\simeq{\mathfrak P}}\def\Ball{{\mathfrak B}_1_\Cmplx$ in the norm topology will make it a Hilbert space that we will denote by $\set{H}}\def\sK{\set{K}}\def\sR{\set{R}}\def\sW{\set{W}_\Phi$. Such completion also implies that ${\mathfrak T}}\def\Actset{{\mathfrak A}}\def\trnset{{\mathfrak t}_\Cmplx\simeq\aA$ can be completed to a complex C$^*$-algebra (\ie a Banach algebra satisfying the identity $\n{\tA^\dagger}\def\eff{{\rm eff}}\def\dyn{{\rm dyne}\circ\tA}=\n{\tA}^2$), as it can be easily proved by standard techniques\cite{darianoVax2006}.
The product in $\aA$ defines the action of $\aA$ on the vectors in $\aA/\aI$, by associating to each element $\tA\in\aA$ the linear operator $\pi_\Phi(\tA)$ defined on the dense domain
$\aA/\aI\subseteq\set{H}}\def\sK{\set{K}}\def\sR{\set{R}}\def\sW{\set{W}_\Phi$ as $\pi_\Phi(\tA)|\cB\>_\Phi\doteq|\underline{\tA\circ\tB}\>_\Phi$. The fact that $\aA$ is a Banach algebra also implies that the domain of definition of $\pi_\Phi(\tA)$ can be easily extended to the whole $\set{H}}\def\sK{\set{K}}\def\sR{\set{R}}\def\sW{\set{W}_\Phi$ by continuity. From the definition of the scalar product, and using the fact that the state $\Phi$ is also preparationally faithful according to Postulate \ref{p:faith}, the Born rule can be written in the GNS representation as
$\omega({\underline{\tA}}}\def\cB{\underline{\tB}}\def\cC{\underline{\tC}}\def\cT{\underline{\tT})={}_\Phi\<\underline{\tA^\dagger}\def\eff{{\rm eff}}\def\dyn{{\rm dyne}}|\varrho\>_\Phi$, with representation of state $\varrho=\cT_\omega'/\Phi(\cT_\omega,\trnsfrm I}\def\tT{\trnsfrm T}\def\tU{\trnsfrm U}\def\tM{\trnsfrm M}\def\tX{\trnsfrm X)$ \cite{darianoVax2006}, $\cT_\omega$ denoting the transformation on system 2 corresponding to the local state $\omega$ on system 1. Then, the representation of transformations is
$\omega(\cB\circ\tA)={}_\Phi\<\underline{\tB^\dagger}\def\eff{{\rm eff}}\def\dyn{{\rm dyne}}|\tA|\rho\>_\Phi:=
{}_\Phi\<\underline{\tB^\dagger}\def\eff{{\rm eff}}\def\dyn{{\rm dyne}}|\tA\circ\rho\>_\Phi\equiv
{}_\Phi\<\tA^\dagger}\def\eff{{\rm eff}}\def\dyn{{\rm dyne}\circ\underline{\tB^\dagger}\def\eff{{\rm eff}}\def\dyn{{\rm dyne}}|\rho\>_\Phi$.
\ack I acknowledge illuminating discussions with M. Ozawa. This work has been supported by Ministero Italiano dell'Universit\`a e della Ricerca (MIUR) through PRIN 2005.
\end{document} | arXiv |
Quiz 2: Polynomial, Power, and Rational Functions
The Manager of 100 Apartments Knows That At $600 \$ 600 $600
The manager of 100 apartments knows that at $600 \$ 600 $600 rent per month, all apartments will be rented. For each \$25 increase, one apartment will not be occupied. Let x x x represent the number of $25 \$ 25 $25 increases to the rent.
(a) Write the revenue as a function of x x x .
(b) What rent per unit will yield maximum revenue?
(c) What is the maximum revenue? ch02_p13_20 3/31/10 8:49 AM Page 15
Solve the rational equation x(2x+1)x−2=10x−2−52\frac{x(2 x+1)}{x-2}=\frac{10}{x-2}-\frac{5}{2}x−2x(2x+1)=x−210−25
The formula h=−16t2+v0t+s0h=-16 t^{2}+v_{0} t+s_{0}h=−16t2+v0t+s0 gives the height of an object tossed upward where v0v_{0}v0 represents the initial velocity, S0\boldsymbol{S}_{0}S0 represents the initial height, and t represents time. A golf ball is hit straight up from the ground level with an initial velocity of 72 ft/sec. Find the maximum height that the ball reaches and the number of seconds it takes to reach that height.
Which of the following gives the zeros of the graph and their multiplicity? A) 1 (multiplicity 1), 3 (multiplicity 2) B) 1 (multiplicity 3), 2 (multiplicity 1) C) 1 (multiplicity 3 ), 3 (multiplicity 1 ) D) 1 (multiplicity 2), 3 (multiplicity 1) E) 1 (multiplicity 1), 2 (multiplicity 3)
Solve the inequality 3x+2(x+1)(2x)≤0\frac{3 x+2}{(x+1)(2 x)} \leq 0(x+1)(2x)3x+2≤0
A contractor purchases a new bulldozer for $45,000 \$ 45,000 $45,000 . After 15 years the bulldozer will be outdated and have no value. Write a linear equation giving the value V V V of the equipment during the 15 years it will be used, where t t t is the number of years after purchase. ch02_p13_20 3/31/10 8:49 AM Page 14
Use long division to find the remainder when x4−3x2+5x−1x^{4}-3 x^{2}+5 x-1x4−3x2+5x−1 is divided by x2−3x^{2}-3x2−3
Use the Remainder Theorem to find the remainder when x3−6x2+5x−2x^{3}-6 x^{2}+5 x-2x3−6x2+5x−2 is divided by x-6 .
Draw the graph of f(x)=0.05x3+6x2−2x−3f(x)=0.05 x^{3}+6 x^{2}-2 x-3f(x)=0.05x3+6x2−2x−3 in the [-15,10] by [-100,175] viewing rectangle. How many real zeros are evident from this graph? A) 1 B) 2 C) 3 D) 0 E) Infinitely many
Find all rational zeros of f(x)=2x3−x2−23x−20f(x)=2 x^{3}-x^{2}-23 x-20f(x)=2x3−x2−23x−20
Find all the zeros of f(x)=x4−x3−x2−x−2f(x)=x^{4}-x^{3}-x^{2}-x-2f(x)=x4−x3−x2−x−2 | CommonCrawl |
You are playing a game which consists of $n$ rooms. Each room has a teleporter to some other room (or the room itself).
You have to process $q$ queries of the form: You are now in room $a$ and want to reach room $b$. What is the minimum number of teleportations?
The first input line contains two integers $n$ and $q$: the number of rooms and queries. The rooms are numbered $1,2,\ldots,n$.
The second line contains $n$ integers $t_1,t_2,\ldots,t_n$: for each room, the destination of the teleporter in that room.
Finally, there are $q$ lines that describe the queries. Each line has two integers $a$ and $b$: you are now in room $a$ and want to reach room $b$.
For each query, print the minimum number of teleportations. If it is not possible to reach the destination, print $-1$. | CommonCrawl |
Fundamental theorem of calculus questions and answers
Use the Fundamental Theorem of Calculus to find G'(x) if: \\ G(x)=\int_1^{x^2} \cos t dt
Use the Fundamental Theorem of Calculus to find {eq}G'(x){/eq} if:
{eq}G(x)=\int_1^{x^2} \cos t dt{/eq}
Fundamental Theorem of Calculus:
The fundamental theorem of calculus is very helpful in finding the derivative of a definite integral without actually computing the definite integral. This theorem states:
$$\frac{d}{d x} \int_{a}^{x} f(t) d t=f(x) $$
The given integral is:
$$G(x)=\int_1^{x^2} \cos t dt $$
To apply the fundamental theorem of calculus for finding its derivative, the upper limit of the integral should be a variable of degree 1.
We make this by substitution.
$$\begin{align} &\text{Let } x^2 =t \\ &\text{Then } 2x\, dx = dt \\ \\ &\text{Limits:} \\ &\text{Upper limit: }: t=x^2 \\ &\text{Lower limit: } t=1 \Rightarrow x^2=1 \Rightarrow x=1 \end{align} $$
Substitute these values in the given integral:
$$G(x) = \int_1^t \cos (x^2) \, (2x \, dx) = \int_1^t 2x \cos(x^2) dx $$
Now we interchange the variables x and t in the above integral:
$$G(x) = \int_1^x 2t \cos(t^2) dt $$
Now we use the fundamental theorem of calculus to find its derivative which states:
$$\dfrac{d}{d x} \int_{a}^{x} f(t) d t=f(x) $$
Then we get:
$$G'(x) =\dfrac{d}{d x} \int_1^x 2t \cos(t^2) dt = 2x \cos(x^2) dx $$
Therefore: {eq}\boxed{\mathbf{G'(x)= 2x \cos(x^2) dx}} {/eq}
The Fundamental Theorem of Calculus
from Math 104: Calculus
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The best way to calculate variations between 2 datasets?
I am trying to write an application which identifies SQL query patterns. The program will trace all the queries executed in a database. When it identifies a considerable pattern change in the database, it will ask the DBA to tune the database. My tracing data looks like this.
The images show the query execution count from 2 different months. According to the images there is a considerable pattern chnage in query execution. Query2 is executed most of the times in the first dataset and Query3 is executed most of the time in second dataset. I want to mathamatically calcuate this pattern difference. One solution I can think of is calculating the percentage differene of execution count of each query and sum it all. If the value is greater than the predefined thresold I can send a notification to the DBA. Is there a better way to do that? Any data analysis technique that can be applied? Any libraries to do that?
data-mining dataset data
What you are attempting to do is calculate the distance between two discrete probability density functions. The standard way to calculate this distance is via the Kullback-Leibler Divergence. It is defined as such:
$$D(P||Q) = \sum_iP(i)*log(\frac{P(i)}{Q(i)})$$
To be technically correct, this is not a metric in a mathematical sense because it is not symmetric and does not abide by the triangle inequality. But never the less it is a "distance" that is used a lot to compare distributions. From an information theoretical perspective, the KL divergence (given that you're using base 2 for the log) represents the number of bits it would take to encode distribution P as Q.
In your case, you would have to normalize each distribution to ensure that the sum is 1, and then calculate distance via the KL divergence. If you want a real metric you can use the Jensen-Shannon Divergence which is defined as: $${{\rm {JSD}}}(P\parallel Q)={\frac {1}{2}}D(P\parallel M)+{\frac {1}{2}}D(Q\parallel M)$$$$ {\displaystyle M={\frac {1}{2}}(P+Q)} M={\frac {1}{2}}(P+Q)$$
Armen AghajanyanArmen Aghajanyan
$\begingroup$ Just one question, is it feasible to simply calculate the vector different using the standard euclidean distance formula? $\endgroup$
– Syed Ali Hamza
$\begingroup$ In theory you could use any distance. KL is interesting because it has a interpretation in information theory. It is also used a lot to compare distributions in various fields. $\endgroup$
– Armen Aghajanyan
Not the answer you're looking for? Browse other questions tagged data-mining dataset data or ask your own question.
Datasets understanding best practices
How do I determine the best statistical way for data transformation for standardization (like log, sq root) to remove bias between different datasets?
Best way to gather musical data
What is the best way to organize the datasets for my task?
What's the best way to do classification basing on two given datasets (annual data and daily data)?
What is the best way to visualize range of data?
Best way to preprocess data | CommonCrawl |
Is there a neat formula for the volume of a tetrahedron on $S^3$?
There is a nice formula for the area of a triangle on the 2-dimensional sphere; If the triangle is the intersection of three half spheres, and has angles $\alpha$, $\beta$ and $\gamma$, and we normalize the area of the whole sphere to be $4\pi$ then the area of the triangle is $$ \alpha + \beta + \gamma - \pi. $$ The proof is a cute application of inclusion-exclusion of three sets, and involves the fact that the area we want to calculate appears on both sides of the equation, but with opposite signs.
However, when trying to copy the proof to the three dimensional sphere the parity goes the wrong way and you get 0=0.
Is there a simple formula for the volume of the intersection of four half-spheres of $S^3$ in terms of the 6 angles between the four bounding hyperplanes?
Note that the 3-dimensional formula has to be much more complicated. The 2-dimensional formula comes from Euler characteristic and Gauss-Bonnet, but the Euler characteristic of the 3-sphere, or any odd-dimensional manifold, vanishes. In fact every characteristic class of a 3-sphere vanishes, because the tangent bundle is trivial. There can't be a purely linear treatment of volumes in isotropic spaces in odd dimensions. In even dimensions, there is always a purely linear extension from lower dimensions using generalized Gauss-Bonnet.
Nice answer, Greg. I looked at the linked paper and was sufficiently intimidated. I just want to point out, again, that for those (like me) who have a phobia of differential geometry, and hence don't want to use (generalized)Gauss-Bonnet, it is easy to see, using inclusion-exclusion, that the formula in even dimensions is a neat linear combination of the formulas in lower dimensions.
Not the answer you're looking for? Browse other questions tagged mg.metric-geometry or ask your own question.
Is there a general formula for calculating the volume of elliptical simplex on the surface of $S^n$?
Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry? | CommonCrawl |
CLs method (particle physics)
In particle physics, CLs[1] represents a statistical method for setting upper limits (also called exclusion limits[2]) on model parameters, a particular form of interval estimation used for parameters that can take only non-negative values. Although CLs are said to refer to Confidence Levels, "The method's name is ... misleading, as the CLs exclusion region is not a confidence interval."[3] It was first introduced by physicists working at the LEP experiment at CERN and has since been used by many high energy physics experiments. It is a frequentist method in the sense that the properties of the limit are defined by means of error probabilities, however it differs from standard confidence intervals in that the stated confidence level of the interval is not equal to its coverage probability. The reason for this deviation is that standard upper limits based on a most powerful test necessarily produce empty intervals with some fixed probability when the parameter value is zero, and this property is considered undesirable by most physicists and statisticians.[4]
Upper limits derived with the CLs method always contain the zero value of the parameter and hence the coverage probability at this point is always 100%. The definition of CLs does not follow from any precise theoretical framework of statistical inference and is therefore described sometimes as ad hoc. It has however close resemblance to concepts of statistical evidence[5] proposed by the statistician Allan Birnbaum.
Definition
Let X be a random sample from a probability distribution with a real non-negative parameter $\theta \in [0,\infty )$. A CLs upper limit for the parameter θ, with confidence level $1-\alpha '$, is a statistic (i.e., observable random variable) $\theta _{up}(X)$ which has the property:
${\frac {\mathbb {P} (\theta _{up}(X)<\theta |\theta )}{\mathbb {P} (\theta _{up}(X)<\theta |0)}}\leq \alpha '{\text{ for all }}\theta .$
(1)
The inequality is used in the definition to account for cases where the distribution of X is discrete and an equality can not be achieved precisely. If the distribution of X is continuous then this should be replaced by an equality. Note that the definition implies that the coverage probability $\mathbb {P} (\theta _{up}(X)\geq \theta |\theta )$ is always larger than $1-\alpha '$.
An equivalent definition can be made by considering a hypothesis test of the null hypothesis $H_{0}:\theta =\theta _{0}$ against the alternative $H_{1}:\theta =0$. Then the numerator in (1), when evaluated at $\theta _{0}$, correspond to the type-I error probability ($\alpha $) of the test (i.e., $\theta _{0}$ is rejected when $\theta _{up}(X)<\theta _{0}$) and the denominator to the power ($1-\beta $). The criterion for rejecting $H_{0}$ thus requires that the ratio $\alpha /(1-\beta )$ will be smaller than $\alpha '$. This can be interpreted intuitively as saying that $\theta _{0}$ is excluded because it is $\alpha '$ less likely to observe such an extreme outcome as X when $\theta _{0}$ is true than it is when the alternative $\theta =0$ is true.
The calculation of the upper limit is usually done by constructing a test statistic $q_{\theta }(X)$ and finding the value of $\theta $ for which
${\frac {\mathbb {P} (q_{\theta }(X)\geq q_{\theta }^{*}|\theta )}{\mathbb {P} (q_{\theta }(X)\geq q_{\theta }^{*}|0)}}=\alpha '.$
where $q_{\theta }^{*}$ is the observed outcome of the experiment.
Usage in high energy physics
Upper limits based on the CLs method were used in numerous publications of experimental results obtained at particle accelerator experiments such as LEP, the Tevatron and the LHC, most notable in the searches for new particles.
Origin
The original motivation for CLs was based on a conditional probability calculation suggested by physicist G. Zech[6] for an event counting experiment. Suppose an experiment consists of measuring $n$ events coming from signal and background processes, both described by Poisson distributions with respective rates $s$ and $b$, namely $n\sim {\text{Poiss}}(s+b)$. $b$ is assumed to be known and $s$ is the parameter to be estimated by the experiment. The standard procedure for setting an upper limit on $s$ given an experimental outcome $n^{*}$ consists of excluding values of $s$ for which $\mathbb {P} (n\leq n^{*}|s+b)\leq \alpha $, which guarantees at least $1-\alpha $ coverage. Consider, for example, a case where $b=3$ and $n^{*}=0$ events are observed, then one finds that $s+b\geq 3$ is excluded at 95% confidence level. But this implies that $s\geq 0$ is excluded, namely all possible values of $s$. Such a result is difficult to interpret because the experiment cannot essentially distinguish very small values of $s$ from the background-only hypothesis, and thus declaring that such small values are excluded (in favor of the background-only hypothesis) seems inappropriate. To overcome this difficulty Zech suggested conditioning the probability that $n\leq n^{*}$ on the observation that $n_{b}\leq n^{*}$, where $n_{b}$ is the (unmeasurable) number of background events. The reasoning behind this is that when $n_{b}$ is small the procedure is more likely to produce an error (i.e., an interval that does not cover the true value) than when $n_{b}$ is large, and the distribution of $n_{b}$ itself is independent of $s$. That is, not the over-all error probability should be reported but the conditional probability given the knowledge one has on the number of background events in the sample. This conditional probability is
$\mathbb {P} (n\leq n^{*}|n_{b}\leq n^{*},s+b)={\frac {\mathbb {P} (n\leq n^{*},n_{b}\leq n^{*}|s+b)}{\mathbb {P} (n_{b}\leq n^{*}|s+b)}}={\frac {\mathbb {P} (n\leq n^{*}|s+b)}{\mathbb {P} (n\leq n^{*}|b)}}.$
which correspond to the above definition of CLs. The first equality just uses the definition of Conditional probability, and the second equality comes from the fact that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle n \leq n^* \Rightarrow n_b \leq n^*} and the number of background events is by definition independent of the signal strength.
Generalization of the conditional argument
Zech's conditional argument can be formally extended to the general case. Suppose that $q(X)$ is a test statistic from which the confidence interval is derived, and let
$p_{\theta }=\mathbb {P} (q(X)>q^{*}|\theta )$
where $q*$ is the outcome observed by the experiment. Then $p_{\theta }$ can be regarded as an unmeasurable (since $\theta $ is unknown) random variable, whose distribution is uniform between 0 and 1 independent of $\theta $. If the test is unbiased then the outcome $q*$ implies
$p_{\theta }\leq \mathbb {P} (q(X)>q^{*}|0)\equiv p_{0}^{*}$
from which, similarly to conditioning on $n_{b}$ in the previous case, one obtains
$\mathbb {P} (q(X)\geq q^{*}|p_{\theta }\leq p_{0}^{*},\theta )={\frac {\mathbb {P} (q(X)\geq q^{*}|\theta )}{\mathbb {P} (p_{\theta }\leq p_{0}^{*}|\theta )}}={\frac {\mathbb {P} (q(X)\geq q^{*}|\theta )}{p_{0}^{*}}}={\frac {\mathbb {P} (q(X)\geq q^{*}|\theta )}{\mathbb {P} (q(X)>q^{*}|0)}}.$
Relation to foundational principles
The arguments given above can be viewed as following the spirit of the conditionality principle of statistical inference, although they express a more generalized notion of conditionality which do not require the existence of an ancillary statistic. The conditionality principle however, already in its original more restricted version, formally implies the likelihood principle, a result famously shown by Birnbaum.[7] CLs does not obey the likelihood principle, and thus such considerations may only be used to suggest plausibility, but not theoretical completeness from the foundational point of view. (The same however can be said on any frequentist method if the conditionality principle is regarded as necessary).
Birnbaum himself suggested in his 1962 paper that the CLs ratio $\alpha /(1-\beta )$ should be used as a measure of the strength of statistical evidence provided by significance tests, rather than $\alpha $ alone. This followed from a simple application of the likelihood principle: if the outcome of an experiment is to be only reported in a form of a "accept"/"reject" decision, then the overall procedure is equivalent to an experiment that has only two possible outcomes, with probabilities $\alpha $,$(1-\beta )$ and $1-\alpha $,$(\beta )$ under $H_{1},(H_{2})$. The likelihood ratio associated with the outcome "reject $H_{1}$" is therefore $\alpha /(1-\beta )$ and hence should determine the evidential interpretation of this result. (Since, for a test of two simple hypotheses, the likelihood ratio is a compact representation of the likelihood function). On the other hand, if the likelihood principle is to be followed consistently, then the likelihood ratio of the original outcome should be used and not $\alpha /(1-\beta )$, making the basis of such an interpretation questionable. Birnbaum later described this as having "at most heuristic, but not substantial, value for evidential interpretation".
A more direct approach leading to a similar conclusion can be found in Birnbaum's formulation of the Confidence principle, which, unlike the more common version, refers to error probabilities of both kinds. This is stated as follows:[8]
"A concept of statistical evidence is not plausible unless it finds 'strong evidence for $H_{2}$ as against $H_{1}$' with small probability $(\alpha )$ when $H_{1}$ is true, and with much larger probability $\ (1-\beta )\ $ when $H_{2}$ is true."
Such definition of confidence can naturally seem to be satisfied by the definition of CLs. It remains true that both this and the more common (as associated with the Neyman-Pearson theory) versions of the confidence principle are incompatible with the likelihood principle, and therefore no frequentist method can be regarded as a truly complete solution to the problems raised by considering conditional properties of confidence intervals.
Calculation in the large sample limit
If certain regularity conditions are met, then a general likelihood function will become a Gaussian function in the large sample limit. In such case the CLs upper limit at confidence level $1-\alpha '$ (derived from the uniformly most powerful test) is given by[9]
$\theta _{up}={\hat {\theta }}+\sigma \Phi ^{-1}(1-\alpha '\Phi ({\hat {\theta }}/\sigma )),$
where $\Phi $ is the standard normal cumulative distribution, ${\hat {\theta }}$ is the maximum likelihood estimator of $\theta $ and $\sigma $ is its standard deviation; the latter might be estimated from the inverse of the Fisher information matrix or by using the "Asimov"[9] data set. This result happens to be equivalent to a Bayesian credible interval if a uniform prior for $\theta $ is used.
References
1. Read, A. L. (2002). "Presentation of search results: The CL(s) technique". Journal of Physics G: Nuclear and Particle Physics. 28 (10): 2693–2704. Bibcode:2002JPhG...28.2693R. doi:10.1088/0954-3899/28/10/313.
2. Particle Physics at the Tercentenary of Mikhail Lomonosov, p. 13, at Google Books
3. Amnon Harel. "Statistical methods in CMS searches" (PDF). indico.cern.ch. Retrieved 2015-04-10.
4. Mark Mandelkern (2002). "Setting Confidence Intervals for Bounded Parameters". Statistical Science. 17 (2): 149–159. doi:10.1214/ss/1030550859. JSTOR 3182816.
5. Ronald N. Giere (1977). "Allan Birnbaum's Conception of Statistical Evidence". Synthese. 36 (1): 5–13. doi:10.1007/bf00485688. S2CID 46973213.
6. G. Zech (1989). "Upper limits in experiments with background or measurement errors" (PDF). Nucl. Instrum. Methods Phys. Res. A. 277 (2–3): 608–610. Bibcode:1989NIMPA.277..608Z. doi:10.1016/0168-9002(89)90795-X.
7. Birnbaum, Allan (1962). "On the foundations of statistical inference". Journal of the American Statistical Association. 57 (298): 269–326. doi:10.2307/2281640. JSTOR 2281640. MR 0138176. (With discussion.)
8. Birnbaum, Allan (1977). "The Neyman-Pearson Theory as Decision Theory, and as Inference Theory; with a Criticism of the Lindley-Savage Argument for Bayesian Theory". Synthese. 36 (1): 19–49. doi:10.1007/bf00485690. S2CID 35027844.
9. G. Cowan; K. Cranmer; E. Gross; O. Vitells (2011). "Asymptotic formulae for likelihood-based tests of new physics". Eur. Phys. J. C. 71 (2): 1554. arXiv:1007.1727. Bibcode:2011EPJC...71.1554C. doi:10.1140/epjc/s10052-011-1554-0.
Further reading
• Leon Jay Gleser (2002). "[Setting Confidence Intervals for Bounded Parameters]: Comment". Statistical Science. 17 (2): 161–163. doi:10.1214/ss/1030550859. JSTOR 3182818.
• Fraser, D. A. S.; Reid N.; Wong, A. C. M. (2004). "Inference for bounded parameters". Phys. Rev. D. 69 (3): 033002. arXiv:physics/0303111. doi:10.1103/PhysRevD.69.033002. S2CID 18947032.
• Robert D. Cousins (2011). "Negatively Biased Relevant Subsets Induced by the Most-Powerful One-Sided Upper Confidence Limits for a Bounded Physical Parameter". arXiv:1109.2023 [physics.data-an].
External links
• The Particle Data Group (PDG) review of statistical methods
| Wikipedia |
A ring on which all finitely generated projectives modules are free but not all projectives are free?
Sorry if the question is naive: any nice example of such a ring or, better, of a class of such rings?
Cher Michel, these rings are uncommon.
1) Over a local ring ALL projective modules are free : this is a celebrated theorem due to Kaplansky.
And now for the good news: the rings you are after are uncommon but they exist. Bass in the article just quoted shows that the ring $R=\mathcal C([0,1])$ of continuous functions on the unit interval has all its finitely generated projective modules free. Nevertheless the ideal consisting of functions vanishing in a neighbourhood of zero (depending on the function) is projective, not finitely generated and not free. Bass attributes the result to Kaplansky.
Not the answer you're looking for? Browse other questions tagged projective-modules or ask your own question.
A finitely generated, locally free module over a domain which is not projective?
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Nonfree projective module over a regular UFD?
Must a finitely generated projective module over a group ring with vanishing coinvariants be trivial?
Is projectivity preserved by invariants?
When is countable direct-product of projective modules again projective ?
Finitely generated submodules of projectives lie inside f. g. projectives? | CommonCrawl |
\begin{definition}[Definition:Tangent Circles]
{{EuclidSaid}}
:''{{:Definition:Euclid's Definitions - Book III/3 - Tangent Circles}}''
{{EuclidDefRefNocat|III|3|Tangent Circles}}
:420px
In the above diagram, the two circles are '''tangent''' to each other at the point $C$.
Category:Definitions/Circles
Category:Definitions/Tangents
\end{definition} | ProofWiki |
\begin{document}
\title {On the distribution of imaginary parts of zeros of the Riemann zeta function, II}
\titlerunning{Imaginary parts of zeros of $\zeta(s)$, II}
\author{Kevin Ford \thanks{The first author is supported by National Science Foundation Grant DMS-0555367} \and K. Soundararajan \thanks{The second author is partially supported by the National Science Foundation and the American Institute of Mathematics (AIM)} \and Alexandru Zaharescu \thanks{The third author is supported by
National Science Foundation Grant DMS-0456615}}
\institute{\textsc{Kevin Ford and Alexandru Zaharescu} \at Department of Mathematics, 1409 West Green Street, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA \and \textsc{K. Soundararajan} \at
Department of Mathematics, 450 Serra Mall, Bldg. 380, Stanford University, Stanford, CA 94305, USA}
\maketitle
\begin{abstract} \subclass{Primary 11M26; Secondary 11K38} We continue our investigation of the distribution of the fractional parts of $\ensuremath{\alpha} \ensuremath{\gamma}$, where $\ensuremath{\alpha}$ is a fixed non-zero real number and $\ensuremath{\gamma}$ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. We establish
some connections to Montgomery's pair correlation function and the distribution of primes in short intervals. We also discuss analogous results for a more general $L$-function. \keywords{Riemann zeta function -- zeros, fractional parts -- primes in
short intervals -- pair correlation functions} \end{abstract}
\section{Introduction and Statement of Results}
In this paper we continue the study of the distribution of the fractional parts $\{\ensuremath{\alpha} \ensuremath{\gamma}\}$ initiated by the first and third authors in \cite{FZ}, where $\ensuremath{\alpha}$ is a fixed positive real number and $\ensuremath{\gamma}$ runs over the positive ordinates of zeros of the Riemann zeta function $\zeta(s)$. We extend and generalize the results from \cite{FZ} in several directions, establishing connections between these fractional parts, the pair correlation of zeros of $\zeta(s)$ and the distribution of primes in short intervals. It is known \cite{Hl} that for any fixed $\ensuremath{\alpha}$, the fractional parts $\{\ensuremath{\alpha} \ensuremath{\gamma}\}$ are uniformly distributed $\pmod 1$. That is, for all continuous functions $f:{\mathbb T} \to {\mathbb C}$, as $T\to \infty$ we have \begin{equation}\label{uniform} \sum_{0< \ensuremath{\gamma} \le T} f(\ensuremath{\alpha}\ensuremath{\gamma}) = N(T) \int_{\mathbb T} f(x) dx + o(N(T)). \end{equation} Here ${\mathbb T}$ is the torus ${\mathbb R}/{\mathbb Z}$ and $N(T)$ denotes the number of ordinates $0 < \ensuremath{\gamma} \le T$; it is well-known that \begin{equation}\label{NT} N(T) = \frac{T}{2\pi} \log \frac{T}{2\pi e} +O(\log T). \end{equation} We are interested in the lower order terms in the asymptotic \eqref{uniform}. For a general continuous function $f$ the asymptotic \eqref{uniform} can be attained arbitrarily slowly so that no improvement of the error term there is possible. But if we assume that $f$ has nice smoothness properties then we can isolate a second main term of size about $T$. More precisely, we define the function $g_{\ensuremath{\alpha}}:{\mathbb T} \to {\mathbb C}$ as follows. If
$\ensuremath{\alpha}$ is not a rational multiple of $\frac{\log p}{2\pi}$ for some prime $p$, then $g_{\ensuremath{\alpha}}$ is identically zero. If $\ensuremath{\alpha}= \frac{a}{q} \frac{\log p}{2\pi} $ for some rational number $a/q$ with $(a,q)=1$ then we set
\begin{equation}\label{galpha} g_{\ensuremath{\alpha}}(x) = -\frac{\log p}{\pi} \Re \sum_{k=1}^{\infty} \frac{e^{-2\pi i qkx}}{p^{ak/2}} = - \frac{(p^{a/2} \cos 2\pi q x -
1)\log p}{\pi(p^a - 2p^{a/2}\cos 2\pi q x + 1)}. \end{equation} Then, we expect (for suitable $f$) that as $T\to \infty$ \begin{equation}\label{FZ} \sum_{0< \ensuremath{\gamma} \le T} f(\ensuremath{\alpha} \ensuremath{\gamma}) = N(T) \int_{\mathbb T} f(x) dx + T \int_{\mathbb T} f(x) g_{\ensuremath{\alpha}}(x) dx + o(T). \end{equation} As remarked above, certainly \eqref{FZ} does not hold for all continuous functions $f$. In Corollary 2 of \cite{FZ}, it is shown that \eqref{FZ} holds for all $f\in C^2 ({\mathbb T})$, and if the Riemann Hypothesis (RH) is true then \eqref{FZ} holds for all absolutely continuous functions $f$ (see Corollary 5 there). Moreover it is conjectured there (see Conjecture A there) that \eqref{FZ} does hold when $f$ is the characteristic function of an interval in ${\mathbb T}$.
\begin{conjecture}\label{conj1} Let ${\mathbb I}$ be an interval of ${\mathbb T}$. Then $$ \sum_{\substack{ 0< \ensuremath{\gamma} \le T \\ \{\ensuremath{\alpha} \ensuremath{\gamma}\} \in {\mathbb I}}}
1 = |{\mathbb I}| N(T) + T \int_{{\mathbb I}} g_{\ensuremath{\alpha}}(x) dx + o(T), $$ uniformly in ${\mathbb I}$. \end{conjecture}
We define the discrepancy of the sequence $\{\ensuremath{\alpha} \ensuremath{\gamma}\}$ (for $0<\ensuremath{\gamma} \le T$) as $$
D_{\ensuremath{\alpha}}(T) = \sup_{{\mathbb I}} \Big| \frac{1}{N(T)} \sum_{\substack{
0 < \ensuremath{\gamma} \le T \\ \{\ensuremath{\alpha} \ensuremath{\gamma}\} \in {\mathbb I}}} 1 - |{\mathbb I}|\Big|, $$ where the supremum is over all intervals ${\mathbb I}$ of ${\mathbb T}$. Unconditionally, Fujii \cite{F76} proved that $D_\ensuremath{\alpha}(T) \ll \frac{\log\log T}{\log T}$ for every $\ensuremath{\alpha}$. On RH, Hlawka \cite{Hl}
showed that $D_\ensuremath{\alpha}(T) \ll \frac{1}{\log T}$, which is best possible
for $\ensuremath{\alpha}$ of the form $\frac{a}{q} \frac{\log p}{2\pi}$ (\cite{FZ},
Corollary 3). Conjecture 1 clearly implies the following conjecture for the discrepancy (see Conjecture A and Corollary 6 of \cite{FZ}).
\begin{conjecture}\label{conj2} We have $$
D_{\ensuremath{\alpha}}(T) = \frac{T}{N(T)} \sup_{{\mathbb I}} \Big| \int_{{\mathbb I}} g_{\ensuremath{\alpha}}(x) dx \Big| + o\Big(\frac{1}{\log T} \Big). $$ \end{conjecture}
Even assuming RH, we are unable to establish Conjectures \ref{conj1} and \ref{conj2}. We show here some weaker results towards these conjectures, and how these conjectures would follow from certain natural assumptions on the zeros of $\zeta(s)$, or the distribution of prime numbers.
\begin{theorem}\label{theorem1} (i) We have unconditionally $$
D_{\ensuremath{\alpha}}(T) \ge \frac{T}{N(T)} \sup_{{\mathbb I}} \Big| \int_{{\mathbb I}} g_{\ensuremath{\alpha}}(x) dx \Big|
+ o\Big(\frac{1}{\log T} \Big).
$$
(ii) Assuming RH, for any interval ${\mathbb I}$ of ${\mathbb T}$ we have $$
\Big| \sum_{\substack{ 0<\ensuremath{\gamma} \le T \\ \{\ensuremath{\alpha} \ensuremath{\gamma}\} \in {\mathbb I}}} 1
- |{\mathbb I}| N(T) - T\int_{{\mathbb I}} g_{\ensuremath{\alpha}}(x) dx \Big| \le \frac{\ensuremath{\alpha}}{2} T + o(T). $$
\end{theorem}
The left side of \eqref{uniform} depends strongly on the behavior of the sums $\sum_{0<\ensuremath{\gamma} \le T} x^{i\ensuremath{\gamma}}$.
\begin{conjecture}\label{conj3} Let $A >1$ be a fixed real number. Uniformly for all
$\frac{T^2}{(\log T)^{5}} \le x\le T^{A}$ we have \begin{equation}\label{sumxg} \sum_{0<\ensuremath{\gamma} \le T} x^{i\ensuremath{\gamma}} = o(T). \end{equation} \end{conjecture}
\begin{theorem}\label{123} Assume RH. Then Conjecture \ref{conj3} implies Conjectures \ref{conj1} and \ref{conj2}. \end{theorem}
{\bf Remarks}.
Assuming RH,
\eqref{sumxg} holds for $x\to \infty$ and $x = o (T^2/\log^{4} T)$ as $T\to\infty$ by uniform versions of Landau's formula for $\sum_{0<\ensuremath{\gamma} \le T} x^{\rho}$ \cite{La}. For example, Lemma 1 of \cite{FZ} implies, for $x>1$ and $T\ge 2$, that (unconditionally) \begin{equation}\label{Landau} \sum_{0<\ensuremath{\gamma} \le T} x^\rho =
-\frac{\Lambda(n_x)}{2\pi} \frac{e^{iT\log(x/n_x)}-1} {i\log(x/n_x)} + O\( x \log^2 (Tx) + \frac{\log T}{\log x}\), \end{equation} where $n_x$ is the nearest prime power to $x$, and the main term is to be interpreted as $-T \frac{\Lambda(x)}{2\pi}$ if $x=n_x$. This main term is always $\ll T \log x$. On RH, divide both sides of \eqref{Landau} by $x^{1/2}$ to obtain \eqref{sumxg}. Unconditionally, one can use Selberg's zero-density estimate to deduce $$
\Big| \sum_{0<\ensuremath{\gamma}\le T} (x^{i\ensuremath{\gamma}} - x^{\rho-1/2}) \Big| \ll \frac{T\log^2 (2x)}{\log T}; $$ see e.g. (3.8) of \cite{FZ}. This gives \eqref{sumxg} when $\log x = o(\sqrt{\log T})$.
We next relate Conjecture \ref{conj3} to the distribution of primes in short intervals.
\begin{conjecture}\label{primeshort} For any $\ensuremath{\varepsilon}>0$, if $x$ is large and $y\le x^{1-\epsilon}$, then $$ \psi(x+y) - \psi(x) = y + o(x^{\frac 12}/\log \log x). $$ \end{conjecture}
\begin{theorem}\label{theoremshort} Assume RH. Conjecture \ref{primeshort} implies Conjecture \ref{conj3}, and hence Conjectures \ref{conj1} and \ref{conj2}. Conversely, if RH and Conjecture \ref{conj3} holds, then for all fixed $\ensuremath{\varepsilon}>0$, large $x$ and $y\le x^{1-\ensuremath{\varepsilon}}$, $$ \psi(x+y) - \psi(x) = y + o(x^{\frac12} \log x). $$ \end{theorem}
{\bf Remarks.} Whereas the behavior of the left side of \eqref{Landau} is governed by a single prime when $x$ is small, for larger $x$ the sum is governed by the primes in an interval. It has been conjectured (\cite{MS}, Conjecture 2) that for $x^\ensuremath{\varepsilon} \le h\le x^{1-\ensuremath{\varepsilon}}$, $\psi(x+h)-\psi(x)-h$ is normally distributed with mean 0 and variance
$h\log (x/h)$. Thus, it is reasonable to conjecture that for every $\ensuremath{\varepsilon}>0$, \begin{equation}\label{psixy} \psi(x+y)-\psi(x)-y \ll_\ensuremath{\varepsilon} y^{1/2} x^{\ensuremath{\varepsilon}} \qquad (1\le y\le x), \end{equation} a far stronger assertion than Conjecture \ref{primeshort}. It is known that RH implies $\psi(x)=x+O(x^{1/2}\log^2 x)$ (von Koch, 1900).
A statement similar to the second part of Theorem \ref{theoremshort} has been given by Gonek (\cite{Go}, Theorem 4). Assuming RH, Gonek showed that if $$ \sum_{0<\ensuremath{\gamma}\le T} x^{i\ensuremath{\gamma}} \ll_\ensuremath{\varepsilon} T x^{-1/2+\ensuremath{\varepsilon}} + T^{1/2} x^{\ensuremath{\varepsilon}} $$ holds uniformly for all $x,T\ge 2$ and for each fixed $\ensuremath{\varepsilon}>0$, then \eqref{psixy} follows.
We also want to describe how to bound the sum $\sum_{0<\ensuremath{\gamma} \le T} x^{i\ensuremath{\gamma}}$ in terms of the pair correlation function \begin{equation}\label{PCF} \mathcal{F}(x,T) = \sum_{0<\ensuremath{\gamma},\ensuremath{\gamma}'\le T} \frac{4 x^{i(\ensuremath{\gamma}-\ensuremath{\gamma}')}}{4+(\ensuremath{\gamma}-\ensuremath{\gamma}')^2}. \end{equation} Such bounds have been given by Gallagher and Mueller \cite{GM}, Mueller \cite{Mue}, Heath-Brown \cite{HB}, and Goldston and Heath-Brown \cite{GH}. First we state a strong version of the Pair Correlation Conjecture for $\zeta(s)$.
\begin{conjecture}\label{conjPC} Fix a real number $A>1$. Uniformly for all
$\frac{T^2}{(\log T)^{6}} \le x\le T^{A}$ we have $$ \mathcal{F}(x,T)=N(T)+o\left(\frac{T}{\log T}\right) \qquad (T\to\infty). $$ \end{conjecture}
\begin{theorem}\label{theoremPC} Assume RH. Then Conjecture \ref{conjPC} implies Conjecture \ref{conj3}, and therefore also Conjectures \ref{conj1} and \ref{conj2}. \end{theorem}
{\bf Remarks.}
The original pair correlation conjecture of Montgomery \cite{M1} states that $$ \mathcal{F}(x,T) \sim N(T) \qquad (T\to\infty) $$ uniformly for $T\le x \le T^A$, where $A$ is any fixed real number. Tsz Ho Chan \cite{Ch} has made an even stronger conjecture than Conjecture \ref{conjPC}, namely he conjectured that for any $\epsilon>0$ and any large $A>1$, $$ \mathcal{F}(x,T) = N(T) + O\left(T^{1-\epsilon_1}\right) $$ if $T^{1+\epsilon}\le x\le T^A$, where $\epsilon_1>0$ may depend on $\epsilon$, and the implicit constant may depend on $\epsilon$ and $A$.
In the next section, we prove Theorems \ref{theorem1}--\ref{theoremPC}. In section \ref{sec:genF} we discuss analogous results for general $L$-functions.
\section{Proof of Theorems \ref{theorem1}--\ref{theoremPC}}
\emph{Proof of Theorem \ref{theorem1} (i)}.
Let ${\mathbb I}$ denote an interval of ${\mathbb T}$ for which
$|\int_{\mathbb I} g_{\ensuremath{\alpha}}(x) dx|$ attains its maximum. Let $\epsilon$ be a small positive number, and let $h_{\epsilon}:{\mathbb T} \to {\mathbb R}$ be a smooth function satisfying $h_{\epsilon}(x) \ge 0$ for all $x$, $h_{\epsilon}(x)=0$ for $\epsilon <x \le 1$, and $\int_{\mathbb T} h_{\epsilon}(x)dx=1$. Set $f(x) =\int_{{\mathbb T}} h_{\epsilon}(y)\chi_{{\mathbb I}}(x-y) dy$, where ${\chi}_{\mathbb I}$ denotes the characteristic function of the interval ${\mathbb I}$. Then $f$ is smooth, and so \eqref{FZ} holds for $f$.
Therefore
\begin{equation}\label{heps} \int_{\mathbb T} h_{\epsilon}(y) \Big( \sum_{\substack{ 0<\ensuremath{\gamma} \le T \\ \{\ensuremath{\alpha} \ensuremath{\gamma}\}
\in {\mathbb I}+y}} 1 - N(T) |{\mathbb I}| \Big)\, dy = T \int_0^\ensuremath{\varepsilon} h_{\epsilon}(y) \int_{{\mathbb I}+y} g_{\ensuremath{\alpha}}(x) dx\, dy + o(T). \end{equation} By \eqref{galpha}, $g_{\ensuremath{\alpha}}$ is bounded and it follows that $$
\Big| \int_{{\mathbb I}+y}g_{\ensuremath{\alpha}}(x) dx -\int_{{\mathbb I}} g_{\ensuremath{\alpha}}(x)dx \Big| \ll \epsilon $$ for $0\le y\le \ensuremath{\varepsilon}$. Therefore the right side of \eqref{heps} equals $$ T \int_{\mathbb I} g_{\ensuremath{\alpha}}(x) dx + o(T) +O(\epsilon T). $$ It follows that for some choice of $y\in (0,\epsilon)$ one must have $$
\Big| \sum_{\substack{ 0< \ensuremath{\gamma} \le T\\ \{\ensuremath{\alpha} \ensuremath{\gamma} \} \in {\mathbb I}+y}} 1 -N(T) |{\mathbb I}|
\Big| \ge T\Big|\int_{{\mathbb I}} g_{\ensuremath{\alpha}}(x) dx \Big| + o(T) + O(\epsilon T). $$ Letting $\epsilon \to 0$, we obtain our lower bound for the discrepancy.
\emph{Proof of Theorem \ref{theorem1} (ii) and Theorem \ref{123}}.
Let $$ h(u) = \begin{cases} 1 & \{ u\} \in {\mathbb I} \\ 0 & \text{else} \end{cases} $$ and let $J$ be a positive integer. There are trigonometric polynomials $h^+$ and $h^-$, depending on $J$ and ${\mathbb I}$, satisfying \begin{equation*} \begin{split} h^-(u) &\le h(u) \le h^+(u) \qquad (u\in {\mathbb R}), \\
h^{\pm}(u) &= \sum_{|j| \le J} c_j^{\pm} e^{2\pi i j u}, \\
c_0^\pm &= |{\mathbb I}| \pm \frac{1}{J+1}, \qquad |c_j^{\pm}| \le
\frac{1}{|j|} \quad (j\ge 1). \end{split} \end{equation*} For proofs, see Chapter 1 of \cite{M2}, for example. These trigonometric polynomials are optimal in the sense that with $J$
fixed, $|c_0^\pm-|{\mathbb I}||$ cannot be made smaller. We have
$$ \sum_{0<\ensuremath{\gamma}\le T} h^-(\ensuremath{\alpha}\ensuremath{\gamma}) \le \sum_{\substack{ 0< \ensuremath{\gamma} \le T \\ \{\ensuremath{\alpha} \ensuremath{\gamma}\} \in {\mathbb I}}} 1 \le \sum_{0<\ensuremath{\gamma}\le T} h^+(\ensuremath{\alpha}\ensuremath{\gamma}). $$ For integers $j$, let $x_j=e^{2\pi j \ensuremath{\alpha}}$ and for positive $j$ put
$$ V_j = \frac{-\Lambda (n_{x_j})}{2\pi x_j^{1/2}} \frac{e^{iT\log(x_j/n_{x_j})}-1}{i\log(x_j/n_{x_j})}. $$ Also define $V_{-j} = \overline{V_j}$.
By \eqref{Landau}, for nonzero $j$ we have $$
\sum_{0<\ensuremath{\gamma}\le T} x_j^{i \ensuremath{\gamma}} = V_j + O\( x_{|j|}^{1/2} \log^2 (x_{|j|} T) \). $$ This will be used for
$$
1 \le |j| \le J_0 := \left\lfloor \frac{2\log T - 5\log\log T}{2\pi \ensuremath{\alpha}} \right\rfloor. $$ Suppose that $J\ge J_0$. We obtain (implied constants depend on $\ensuremath{\alpha}$)
\begin{align*}
\sum_{0<\ensuremath{\gamma}\le T} & h^{\pm}(\ensuremath{\alpha}\ensuremath{\gamma}) = c_0^{\pm} N(T) + \sum_{1\le |j| \le
J} c_j^{\pm} \sum_{0<\ensuremath{\gamma}\le T} x_j^{i \ensuremath{\gamma}} \\ &= c_0^{\pm} N(T) + 2 \Re \sum_{1\le j \le J_0}
c_j^{\pm} \biggl[V_j + O(x_j^{1/2} \log^2 T) \biggr] \\ &\qquad +
\sum_{J_0 < |j| \le J} O\pfrac{1}{|j|} \Big| \sum_{0<\ensuremath{\gamma}\le T}
x_j^{i\ensuremath{\gamma}} \Big| \\
&\!\!\!\!\!= |{\mathbb I}| N(T)+ \sum_{j\ne 0} c_j^{\pm} V_j \pm \frac{N(T)}{J+1} + o(T) + \!\!
\sum_{J_0<|j| \le J} O(|j|^{-1}) \Big| \sum_{0<\ensuremath{\gamma}\le T}
x_j^{i\ensuremath{\gamma}} \Big|, \end{align*} where the term $o(T)$ is uniform in ${\mathbb I}$. If $\ensuremath{\alpha} = \frac{a}{q} \frac{\log p}{2\pi}$ for a prime $p$ and coprime positive $a,q$, then $x_j = p^{aj/q}$ and consequently
$$ V_{kq}=-\frac{T\log p}{2\pi p^{ak/2}} $$ for nonzero integers $k$. Thus,
$$
\sum_{\substack{j\ne 0 \\ q|j }} c_j^{\pm} V_j = T \int_{\mathbb T} h^{\pm} g_\ensuremath{\alpha}. $$ If $q\nmid j$, then $x_j$ is not an integer. Hence $$ \sum_{\substack{j\ne 0 \\ q\nmid j}} c_j^{\pm} V_j \ll T
\sum_{\substack{1\le |j| \le J \\ q \nmid j}}
\frac{1}{e^{\pi j \ensuremath{\alpha}}} \, \left| \frac{e^{iT\log(x_j/n_{x_j})}-1}
{iT\log(x_j/n_{x_j})} \right|. $$ The sum on the right converges uniformly in $T$, and each summand is $o(1)$ as $T\to\infty$, hence the left side is $o(T)$. We conclude \begin{equation}\label{sumcjVj} \sum_{j\ne 0} c_j^{\pm} V_j = T \int_{\mathbb T} h^{\pm} g_\ensuremath{\alpha} + o(T). \end{equation} When $\ensuremath{\alpha}$ is not of the form $\frac{a}{q} \frac{\log p}{2\pi}$, $x_j$ is never an integer (for nonzero $j$), and a similar argument yields \eqref{sumcjVj}.
Since $h-h^\pm$ has constant sign, $$
\Big| \int_{\mathbb T} (h-h^{\pm})g_\ensuremath{\alpha} \Big| \le \max_{x\in {\mathbb T}} |g_\ensuremath{\alpha}(x)|
\int_{\mathbb T} |h-h^\pm| = \frac{\max_{x\in{\mathbb T}} |g_\ensuremath{\alpha}(x)|}{J+1} \ll \frac{1}{\log T}. $$ Therefore, \begin{align*}
\sum_{0<\ensuremath{\gamma}\le T} h^{\pm}(\ensuremath{\alpha}\ensuremath{\gamma}) &= |{\mathbb I}| N(T) + T \int_{{\mathbb T}} h g_\ensuremath{\alpha} + o(T) \pm \frac{N(T)}{J+1} \\
&\qquad + \sum_{J_0<|j| \le J} O\pfrac{1}{|j|}
\Big| \sum_{0<\ensuremath{\gamma}\le T} x_j^{i\ensuremath{\gamma}} \Big|. \end{align*}
For Theorem \ref{theorem1} (ii), we take $J=J_0$. For Theorem \ref{123}, take $J=\lfloor \ensuremath{\lambda} \log T \rfloor$ with $\ensuremath{\lambda}$ fixed, and then let $\ensuremath{\lambda}\to\infty$.
\emph{Proof of Theorem \ref{theoremshort}}.
We first construct a function $F$ which is a good approximation of the characteristic function of the interval $[0,1]$ and whose Fourier transform is supported on $[-K,K]$, where $K$ is a parameter to be specified later. Consider the entire function $$ H(z)=\pfrac{\sin \pi z}{\pi}^2 \biggl( \sum_{n=1}^\infty \frac{1}{(z-n)^2} - \sum_{n=1}^\infty \frac{1}{(z+n)^2} + \frac{2}{z} \biggr) $$ for complex $z$, and set $$ F(z) = \frac{H(Kz) + H(K-Kz)}{2}. $$ The function $H(z)$ is related to the so-called Beurling-Selberg functions, and basic facts about $H$ can be found in \cite{V}. In particular, for real $x$, (i) $H(x)$ is an odd function; (ii) the Fourier transform $\widehat{H}$ is supported on $[-1,1]$; (iii) $H(x) =
\operatorname{sgn}(x) + O(\frac{1}{1+|x|^3})$, where $\operatorname{sgn}(x)=1$ if $x>0$, $\operatorname{sgn}(x)=-1$ if $x<0$ and $\operatorname{sgn}(0)=0$; (iv) $H'(x)
=O(\frac{1}{1+|x|^3})$. Item (iii) follows from (2.26) of \cite{V} and the Euler-Maclaurin summation formula, and (iv) follows from Theorem 6 of \cite{V}. Let $I$ be the indicator function of the interval $[0,1]$. It follows that the Fourier transform $\widehat{F}$ of $F$ is supported on $[-K,K]$ and \begin{equation}\label{Fest}
| F(x) - I(x) | \ll \frac{1}{1+K^3 |x|^3} + \frac{1}{1+K^3|1-x|^3}. \end{equation} Since $$ \widehat{I}(t) = \frac{1-e^{-2\pi i t}}{2\pi i t}, $$ it follows readily that $\widehat{F}(t) \ll 1$, uniformly in $K$, and \begin{align*}\label{Fhatprime} \widehat{F}'(t) &= \frac{1-(1+2\pi i t) e^{-2\pi i t}}{-2\pi i t^2} +
O\biggl( \int_{-\infty}^{\infty} \frac{|x|}{1+K^3 |x|^3} +
\frac{|x|}{1+K^3 |1-x|^3}\, dx \!\biggr) \\
&= O\( \frac{1}{1+|t|} + \frac{1}{K} \). \end{align*}
Next, let $T\ge 2$ and $T \le x \le T^A$. Write $$
\sum_{0<\ensuremath{\gamma}\le T} x^{i\ensuremath{\gamma}} = \sum_{|\ensuremath{\gamma}| \le x} x^{i\ensuremath{\gamma}} F(\ensuremath{\gamma}/T) +
\sum_{|\ensuremath{\gamma}| \le x} x^{i\ensuremath{\gamma}} \bigl[ I(\ensuremath{\gamma}/T)-F(\ensuremath{\gamma}/T) \bigr]. $$ By \eqref{NT} and \eqref{Fest}, the second sum on the right is \begin{align*} &\ll N\pfrac{T}{K} + \( N\(T + \frac{T}{K}\) - N\(T - \frac{T}{K}\)\) \\
&\qquad + \frac{T^3}{K^3} \biggl( \; \sum_{|\ensuremath{\gamma}| > T/K}
\frac{1}{|\ensuremath{\gamma}|^3} + \sum_{|\ensuremath{\gamma}-T| \ge T/K}
\frac{1}{|\ensuremath{\gamma}-T|^3} \; \biggr) \\ &\ll \frac{T\log T}{K}. \end{align*} Also, \begin{align*}
\sum_{|\ensuremath{\gamma}| \le x} x^{i\ensuremath{\gamma}} &F(\ensuremath{\gamma}/T) \sum_{|\ensuremath{\gamma}| \le x} x^{i\ensuremath{\gamma}}
\int_{-K}^K e^{2\pi i v \ensuremath{\gamma}/T} \widehat{F}(v) \, dv \\
&= x^{-1/2} \int_{-K}^K e^{-\pi v/T} \widehat{F}(v) \sum_{|\ensuremath{\gamma}|\le x} \( x
e^{2\pi v/T} \)^\rho\, dv \\
&= -\frac{T}{2\pi x^{1/2}} \int_{-K}^K e^{-\pi v/T} \( \widehat{F}'(v) - \frac{\pi}{T}
\widehat{F}(v) \) \sum_{|\ensuremath{\gamma}| \le x} \frac{\( x e^{2\pi v/T}
\)^\rho}{\rho}\, dv, \end{align*} where the last line follows from the previous line using integration by parts. The final sum on $\ensuremath{\gamma}$ is evaluated using the explicit formula (see e.g. \cite{Da}, \S 17) \begin{equation}\label{explicit}
G(x) := \psi(x) - x = -\sum_{|\ensuremath{\gamma}| \le M} \frac{x^\rho}{\rho} + O\( \log x + \frac{x\log^2 (Mx)}{M} \), \end{equation} valid for $x\ge 2$, $M\ge 2$. Since $$ \int_{-K}^K e^{-\pi v/T} \(\widehat{F}'(v)-\frac{\pi}{T} \widehat{F}(v) \)\, dv =0, $$ we obtain \begin{align*}
\sum_{|\ensuremath{\gamma}| \le x} x^{i\ensuremath{\gamma}} F(\ensuremath{\gamma}/T) &= \frac{-T}{2\pi \sqrt{x}} \int_{-K}^K \widehat{F}'(v) \( G(xe^{2\pi v/T})-G(x) \)\, dv \\ &\qquad + O\( K \( 1 + T x^{-1/2} \) \log^2 x\). \end{align*} Altogether, this gives \begin{align*}
\sum_{|\ensuremath{\gamma}| \le T} x^{i\ensuremath{\gamma}} &\ll \frac{T\log K}{\sqrt{x}} \max_{xe^{-2\pi
K/T} \le y \le x e^{2\pi K/T}} | G(y)-G(x) | \\ &\qquad + \frac{T\log T}{K} +
K \( 1 + T x^{-1/2} \) \log^2 x. \end{align*} Take $K=\log^2 T$ and assume Conjecture \ref{primeshort}. The first part of Theorem \ref{theoremshort} follows.
The second part is straightforward, starting with the explicit formula \eqref{explicit} in the form $$
\psi(x+y)-\psi(x)-y = - \sum_{|\ensuremath{\gamma}| \le x} \frac{(x+y)^\rho -
x^\rho}{\rho} +O(\log^2 x). $$ Fix $\ensuremath{\varepsilon}>0$ and apply Conjecture \ref{conj3} with $A=2/\ensuremath{\varepsilon}$. By partial summation, \begin{align*}
\Big|\sum_{x^{\ensuremath{\varepsilon}/2} < |\ensuremath{\gamma}| \le x} \frac{x^{\rho}}{\rho} \Big|
&=2 \Big| \Re \sum_{x^{\ensuremath{\varepsilon}/2} < \ensuremath{\gamma} \le x} \frac{x^{\rho}}{\rho} \Big| \\ &\le 2 x^{1/2}
\biggl| \frac{1}{\frac12 + ix} \sum_{0<\ensuremath{\gamma}\le x} x^{i\ensuremath{\gamma}} +
i \int_{x^{\ensuremath{\varepsilon}/2}}^x
\frac{1}{(\frac12+it)^2} \sum_{0<\ensuremath{\gamma}\le t} x^{i\ensuremath{\gamma}} \, dt \biggr|\\ &= o(x^{1/2} \log x). \end{align*} The smaller zeros are handled in a trivial way. We have, for $y\le x$, $$
(x+y)^\rho - x^\rho = x^\rho \( \rho \frac{y}{x} + O\pfrac{|\rho|^2
y^2}{x^2} \), $$ whence $$
\sum_{|\ensuremath{\gamma}| \le x^{\ensuremath{\varepsilon}/2}} \frac{(x+y)^\rho - x^\rho}{\rho} \ll N(x^{\ensuremath{\varepsilon}/2}) x^{1/2} \( \frac{y}{x} + x^{\ensuremath{\varepsilon}/2} \frac{y^2}{x^2} \) \ll x^{\frac{1}{2} - \frac{\ensuremath{\varepsilon}}{2}} \log x. $$ Therefore, $\psi(x+y)-\psi(x) - y = o(x^{1/2}\log x)$, as claimed.
\emph{Proof of Theorem \ref{theoremPC}}.
It will be convenient to work with the normalized sum $$ \mathcal{D}(x,T) = \frac{\mathcal{F}(x,T)}{N(T)}. $$
\begin{lemma}\label{PCY} Suppose $T\ge 10$ and $1\le\ensuremath{\beta}\le \frac{T}{2\log T}$. Then \begin{align*}
\sum_{0<\ensuremath{\gamma} \le T} &x^{i\ensuremath{\gamma}} \ll T \pfrac{\log T}{\ensuremath{\beta}}^{\frac12} \! \biggl( 1 + \max_{\frac{T}{\ensuremath{\beta}\log T}\le t\le T}
|\mathcal{D}(x,t)| \\
&\qquad + \ensuremath{\beta}^3 \biggl|\int_{-\infty}^\infty (\mathcal{D}(xe^{u},t)-\mathcal{D}(x,t))
e^{-2\ensuremath{\beta}|u|} \, du \biggr| \biggr)^{\frac12} \\ &\ll \frac{T(\log T)^{\frac12}}{\ensuremath{\beta}^{1/2}} \( 1 + \max_{\frac{T}{\ensuremath{\beta}\log T}\le t\le T}
|\mathcal{D}(x,t)| \)^{1/2}
+T (\ensuremath{\beta} \log T)^{1/2} \\ &\quad \times \biggl( \max_{\frac{T}{\ensuremath{\beta}\log T}\le t\le T} \; \max_{0\le u \le \frac{1}{\ensuremath{\beta}}\log
(\ensuremath{\beta}\log T)} |\mathcal{D}(xe^u,t)+\mathcal{D}(xe^{-u},t)-2\mathcal{D}(x,t)| \biggr)^{\frac12}. \end{align*} \end{lemma}
\begin{proof} We follow \cite{GH} by estimating $\sum_{0<\ensuremath{\gamma} \le T} x^{i\ensuremath{\gamma}}$ in terms of $$ G_\beta(x,T) = \sum_{0<\ensuremath{\gamma},\ensuremath{\gamma}'\le T} \frac{4\beta^2 x^{i(\ensuremath{\gamma}-\ensuremath{\gamma}')}}{4\beta^2+(\ensuremath{\gamma}-\ensuremath{\gamma}')^2}. $$
In particular, $G_1(x,T) = \mathcal{F}(x,T)$, and
by \eqref{NT}, we have $G_\ensuremath{\beta}(x,T) \ll (1+\ensuremath{\beta}) T \log^2 T$. By Lemma 1 of \cite{GH}, uniformly for $1\le \ensuremath{\beta} \le T$ and $1\le V\le T$, we have \begin{equation}\label{YG} \begin{split} \sum_{0<\ensuremath{\gamma} \le T} x^{i\ensuremath{\gamma}} &\ll
\Big(T\ensuremath{\beta}^{-1} \max_{t\le T}G_\ensuremath{\beta}(x,t)\Big)^{1/2} \\ &\ll
\frac{T\log T}{V^{1/2}} + \Big(T\ensuremath{\beta}^{-1} \max_{T/V \le t
\le T} G_\ensuremath{\beta}(x,t) \Big)^{1/2}. \end{split} \end{equation}
Using Lemma 2 of \cite{GH}, we have \begin{align*} G_\ensuremath{\beta}(x,t) &= \ensuremath{\beta}^2 \mathcal{F}(x,t) + \ensuremath{\beta}(1-\ensuremath{\beta}^2) \int_{-\infty}^\infty \mathcal{F}(xe^{u},t)
e^{-2\ensuremath{\beta}|u|}\, du \\ &= \mathcal{F}(x,t) + \ensuremath{\beta}(1-\ensuremath{\beta}^2) \int_{-\infty}^\infty (\mathcal{F}(xe^u,t)-\mathcal{F}(x,t))
e^{-2\ensuremath{\beta}|u|}\, du, \end{align*}
from which the first inequality in the lemma follows upon taking $V=\ensuremath{\beta} \log T$. For the second inequality, combine the terms in the integral with $u=v$ and $u=-v$ for
$0\le v \le \frac{\log (\ensuremath{\beta} \log T)}{\ensuremath{\beta}}$, and use the trivial bound $\mathcal{D}(z,t) \ll \log t$ when $|u| \ge \frac{\log (\beta \log T)}{\beta}$ ($z=x$ and $z=xe^u$). \end{proof}
In order to finish the proof of Theorem \ref{theoremPC}, suppose that $\log T \le \ensuremath{\beta} \le \log^2 T$. From Conjecture \ref{conjPC} it follows that the terms $\mathcal{D}(xe^u,t)$, $\mathcal{D}(xe^{-u},t)$, and $\mathcal{D}(x,t)$, in the ranges from the statement of the above lemma, are all of the form $1+o\left( (\log T)^{-2}\right)$. Therefore, $$ \sum_{0<\ensuremath{\gamma} \le T} x^{i\ensuremath{\gamma}}= O\left(T\frac{(\log T)^{1/2}}{\beta^{1/2}}\right) +o\left(T\frac{\beta^{1/2}}{(\log T)^{1/2}}\right). $$ Thus, taking $\beta$ slightly larger than $\log T$ produces the desired result.
\section{General $L$-functions}\label{sec:genF}
Consider a Dirichlet series $F(s)= \sum_{n=1}^\infty a_F(n) n^{-s} $ satisfying the following axioms: \par \noindent (i) there exists an integer $m \geq 0$ such that $(s-1)^mF(s)$ is an entire function of finite order; \par \noindent (ii) $F$ satisfies a functional equation of the type: $$ \Phi(s) = w\overline{\Phi}(1-s), $$ where $$ \Phi(s) = Q^s \prod_{j=1}^r \Gamma(\lambda_j s + \mu_j) F(s) $$
with $Q>0$, $\lambda_j > 0$, $\Re(\mu_j) \geq 0$ and $|w|=1$. (Here, $\overline{f}(s) = \overline{f(\overline{s})}$); \par \noindent (iii) $F(s)$ has an Euler product, which we write as $$ -\frac{F'}{F}(s) =\sum_{n=1}^\infty \Lambda_F(n) n^{-s}, $$ where $\Lambda_F(n)$ is supported on powers of primes.
We also need some growth conditions on the coefficients $a_F(n)$ and $\Lambda_F(n)$. Although stronger than we require, for convenience we impose the conditions (iv) $\Lambda_F(n) \ll n^{\theta_F}$ for some $\theta_F < \frac12$ and (v) for every $\ensuremath{\varepsilon}>0$, $a_F(n)\ll_\ensuremath{\varepsilon} n^{\ensuremath{\varepsilon}}$. Together, conditions (i)--(v) define the \emph{Selberg class} $\mathcal S$ of Dirichlet series. For a survey of results and conjectures concerning the Selberg class, the reader may consult Kaczorowski and Perelli's paper \cite{KP}. In particular, $\mathcal S$ includes the Riemann zeta function, Dirichlet $L$-functions, and $L$-functions attached to number fields and elliptic curves. The Selberg class is conjectured to equal the class of all automorphic $L$-functions, suitably normalized so that their nontrivial zeros have real parts between 0 and 1.
The functional equation is not uniquely determined in light of the duplication formula for $\Gamma$-function, however the real sum $$ d_F = 2 \sum_{j=1}^r \ensuremath{\lambda}_j $$ is well-defined and is known as the degree of $F$. Analogous to \eqref{NT}, we have (cf. \cite{S3}, (1.6)) \begin{equation}\label{NFT} \begin{split}
N_F(T) &= \left| \{ \rho=\beta+i\gamma : F(\rho)=0, 0<\beta<1, 0<\gamma\le T\} \right| \\ &= \frac{d_F}{2\pi} T \log T + c_1 T + O(\log T) \end{split} \end{equation} for some constant $c_1=c_1(F)$. A function $F\in \mathcal S$ is said to be \emph{primitive} if it cannot be written as a product of two or more elements of $\mathcal S$. We henceforth assume that $F$ is primitive. The extension of our results to non-primitive $F$ is straightforward. It is expected that all zeros of $F$ with real part between 0 and 1 have real part $\frac12$, a hypothesis we abbreviate as RH$_F$. Although we shall assume RH$_F$\, for many of the results in this section, sometimes a weaker hypothesis suffices, that most zeros of $F$ are close to the critical line.
\noindent {\bf Hypothesis} $Z_F$. There exist constants $A>0, B>0$
(depending on $F$) such that \begin{align*}
N_F(\sigma,T) &= \left| \left\{ \beta+i\gamma: \frac12 \le \beta \le \sigma,
0<\gamma \le T \right\} \right| \\ &\ll T^{1-A(\sigma-1/2)}\log^B T, \end{align*} uniformly for $\sigma\ge 1/2$ and $T\ge 2$.
Hypothesis $Z_F$ is known, with $B=1$, for the Riemann zeta function and Dirichlet $L$-functions (Selberg \cite{S1}, \cite{S2}), and certain degree 2 $L$-functions attached to cusp forms (Luo \cite{Luo}).
The next tool we require is an analog of \eqref{Landau}. It is very similar to Proposition 1 of \cite{MZ}, and with small modifications to that proof we obtain the following result, which is nontrivial provided $x^{1/2+\ensuremath{\theta}_F}+x^{1/2+\ensuremath{\varepsilon}} \ll T$.
\begin{lemma}\label{lem1} Let $F\in\mathcal S$, $x>1$, $T\geq2,$ and let $n_x$ be a nearest integer to $x$. Then, for any $\ensuremath{\varepsilon} > 0$, \begin{align*} \sum_{0<\gamma\leq T}x^{\rho} &= -\frac{\Lambda_F(n_x)}{2\pi} \frac{e^{iT\log(x/n_x)}-1} {i\log(x/n_x)} \\ &\qquad +O_{\ensuremath{\varepsilon}}\(x^{1+\theta_F}\log (2x) + x^{1+\ensuremath{\varepsilon}} \log T + \frac{\log T}{\log
x} \). \end{align*} \end{lemma}
Using Lemma \ref{lem1} in place of Lemma 1 of \cite{FZ}, Hypothesis $Z_F$ in place of Lemma 2 of \cite{FZ}, and following the proof of Theorem 1 of \cite{FZ}, we obtain a generalization of \eqref{FZ}.
\begin{theorem}\label{theorem0F} Let $F\in \mathcal S$. If $\alpha= \frac{a \log p}{2\pi q}$ for some prime number $p$ and positive integers $a,q$ with $(a,q)=1$, define $$ g_{F,\alpha}(t)= -\frac1{\pi} \Re \sum_{k=1}^\infty \frac{\Lambda_F(p^{ak})}{p^{ak/2}}e^{-2\pi i qkt}. $$ For other $\ensuremath{\alpha}$, define $g_{F,\ensuremath{\alpha}}(t)=0$ for all $t$. If Hypothesis $Z_F$ holds, then \begin{equation}\label{uniformF} \sum_{0< \ensuremath{\gamma} \le T} f(\ensuremath{\alpha}\ensuremath{\gamma}) = N_F(T) \int_{\mathbb T} f(x) \, dx + T \int_{\mathbb T} f(x) g_{F,\ensuremath{\alpha}}(x)\, dx + o(T) \end{equation} for all $f \in C^2({\mathbb T})$. Assuming RH$_F$, \eqref{uniformF} holds for all absolutely continuous $f$. \end{theorem}
Since Hypothesis $Z_F$ holds for Dirichlet $L$-functions $L(s,\chi)$, we obtain the following.
\begin{corollary}\label{Dirichlet} Unconditionally, for Dirichlet $L$-functions $F$, \eqref{uniformF} holds for all $f \in C^2({\mathbb T})$. \end{corollary}
When $F(s)=L(s,\chi)$ and $\ensuremath{\alpha} = \frac{a\log p}{2\pi q}$ with $p$ prime, $(a,q)=1$, we have $$ g_{F,\alpha}(t)= -\frac{\log p}{\pi} \Re \left( \frac{e^{2\pi i(qt+a\xi)}}{p^{a/2}-e^{2\pi i(qt+a\xi)}}\right), $$ where $\chi(p)=e^{2\pi i \xi}$. It follows that there is a shortage of zeros of $L(s,\chi)$ with $\{\alpha\ensuremath{\gamma}\}$ near $\frac{k-a\xi}{q}$, $k=0,\cdots,q-1$. We illustrate this phenomenon with three histograms of $M_F(y;T)$, where $$ M_F(y) = \frac{T}{N_F(T)}
\Bigg| \sum_{\substack{ 0< \ensuremath{\gamma} \le T \\ \{\ensuremath{\alpha} \ensuremath{\gamma}\} < y}}
1 - y N_F(T) \Bigg|, $$ $F$ a Dirichlet $L$-function associated with a character of conductor 5 and $T=500,000$. For both characters, $N_F(T)=946488$. The list of zeros was taken from Michael Rubinstein's data files on his Web page. In Figure 1 we plot for each subinterval $I=[y,y+\frac{1}{500})$ the value of $500 (M_F(y+\frac{1}{500})-M_F(y))$ and also the graph of $g_{F,\ensuremath{\alpha}}(y)$. The characters are identified by their value at 2.
\afterpage{
} \begin{figure}\end{figure}
We conjecture that \eqref{uniformF} holds when $f$ is the indicator function of an interval, and are thus led to the following generalizations of Conjectures \ref{conj1} and \ref{conj2}. Here $D_{F,\ensuremath{\alpha}}$ is the natural generalization of the discrepancy function $D_\ensuremath{\alpha}$.
\begin{conjecture}\label{conj1F} Let ${\mathbb I}$ be an interval of ${\mathbb T}$. Then $$ \sum_{\substack{ 0< \ensuremath{\gamma} \le T \\ \{\ensuremath{\alpha} \ensuremath{\gamma}\} \in {\mathbb I}}}
1 = |{\mathbb I}| N_F(T) + T \int_{{\mathbb I}} g_{F,\ensuremath{\alpha}}(x) dx + o(T). $$ \end{conjecture}
\begin{conjecture}\label{conj2F} We have $$
D_{F,\ensuremath{\alpha}}(T) = \frac{T}{N_F(T)} \sup_{{\mathbb I}} \Big| \int_{{\mathbb I}} g_{F,\ensuremath{\alpha}}(x)
\, dx \Big| + o\Big(\frac{1}{\log T} \Big). $$ \end{conjecture}
Combining Theorem \ref{theorem0F} and the proof of Theorem \ref{theorem1}, we obtain the following. The only difference in the proof is that here we take $$
J_0 = \left\lfloor \frac{\tfrac{\log T}{1/2+\theta_F}-5\log\log T}{2\pi \ensuremath{\alpha}} \right\rfloor. $$
\begin{theorem}\label{theorem1F} (i) Assuming Hypothesis $Z_F$, we have $$
D_{F,\ensuremath{\alpha}}(T) \ge \frac{T}{N_F(T)} \sup_{{\mathbb I}} \Big| \int_{{\mathbb I}}
g_{F,\ensuremath{\alpha}}(x) \, dx \Big| + o\Big(\frac{1}{\log T} \Big). $$
(ii) Assuming RH$_F$, for any interval ${\mathbb I}$ of ${\mathbb T}$ we have $$
\Big| \sum_{\substack{ 0<\ensuremath{\gamma} \le T \\ \{\ensuremath{\alpha} \ensuremath{\gamma}\} \in {\mathbb I}}} 1
- |{\mathbb I}| N_F(T) - T\int_{{\mathbb I}} g_{F,\ensuremath{\alpha}}(x) dx \Big| \le \ensuremath{\alpha}(1/2+\theta_F) T + o(T). $$ \end{theorem}
We can prove a direct analog of Theorem \ref{123}, by requiring a slightly larger range of $T$ in the analog of Conjecture \ref{conj3}, since $\theta_F$ may be large.
\begin{conjecture}\label{conj3F} Let $A >1$ be a fixed real number. Uniformly for
$$ \frac{T^{1/(1/2+\theta_F)}}{\log^{5} T} \le x\le T^{A}, $$ we conjecture that \begin{equation}\label{sumxgF} \sum_{0<\ensuremath{\gamma} \le T} x^{i\ensuremath{\gamma}} = o(T). \end{equation} \end{conjecture}
\begin{theorem}\label{123F} Assume RH$_F$. Then Conjecture \ref{conj3F} implies Conjectures \ref{conj1F} and \ref{conj2F}. \end{theorem}
The analog of Theorem \ref{theoremshort} holds for $F\in \mathcal S$, by following the proof given in the preceding section. Here we need an explicit formula similar to \eqref{explicit}. By standard contour integration methods, one obtains $$
G_F(x) := \sum_{n\le x} \Lambda_F(n) - d_F x = - \sum_{|\rho| \le Q} \frac{x^\rho}{\rho} + O(x^{\theta_F}\log x) $$ provided $Q \ge x\log x$. Since $\theta_F<\frac12$, the error term is acceptable.
\begin{conjecture}\label{primeshortF} For every $\ensuremath{\varepsilon}>0$, if $x$ is large and $y\le x^{1-\epsilon}$, then $$ G_F(x+y) - G_F(x) = o(x^{\frac 12}/\log \log x). $$ \end{conjecture}
\begin{theorem}\label{theoremshortF} Assume RH$_F$. Conjecture \ref{primeshortF} implies Conjecture \ref{conj3F}, and hence Conjectures \ref{conj1F} and \ref{conj2F}. Conversely, if RH$_F$\, and Conjecture \ref{conj3F} holds, then for all fixed $\ensuremath{\varepsilon}>0$, large $x$ and $y\le x^{1-\ensuremath{\varepsilon}}$, $$ G_F(x+y) - G_F(x) = o(x^{\frac12} \log x). $$ \end{theorem}
In order to address an analog of Theorem \ref{theoremPC}, we first quote a Pair Correlation Conjecture for $F$, due to Murty and Perelli \cite{MP}.
\begin{conjecture}\label{conjPCF} Define $$ \mathcal{F}_F(x,T) = \sum_{0<\ensuremath{\gamma},\ensuremath{\gamma}'\le T} \frac{4 x^{i(\ensuremath{\gamma}-\ensuremath{\gamma}')}}{4+(\ensuremath{\gamma}-\ensuremath{\gamma}')^2} $$ and $\mathcal{D}_F(x,T)=\mathcal{F}_F(x,T)/N_F(T)$. We have $\mathcal{D}_F(T^{\ensuremath{\theta} d_F},T) \sim \ensuremath{\theta}$ for $0<\ensuremath{\theta}\le 1$ and $\mathcal{D}(T^{\ensuremath{\theta} d_F},T) \sim 1$ for $\ensuremath{\theta} \ge 1$. \end{conjecture}
Notice that, as a function of $x$, $\mathcal{F}_F(x,T)$ is conjectured to undergo a change of behavior in the vicinity of $x=T^{d_F}$. In order to deduce Conjecture \ref{conj3F}, we can postulate a stronger version of Conjecture \ref{conjPCF}, with error terms of relative order $o(1/\log^2 T)$. We succeed, as in the proof of Theorem \ref{theoremPC}, when $d_F=1$. When $d_F \ge 2$, however, this transition zone lies outside the range in which Lemma \ref{lem1} is useful (Kaczorowski and Perelli recently proved that $1<d_F<2$ is impossible \cite{KP2}; it is conjectured that $d_F$ is always an integer). We can use an analog of Lemma \ref{lem1}, which follows by the same method (replace $\mathcal{D}(x,T)$ with $\mathcal{D}_F(x,T)$). However, in order to prove the right side is small, we require that $\mathcal{D}_F(x,T)$ has small \emph{variation}, even through the transition zone $x\approx T^{d_F}$. Tsz Ho Chan \cite{Ch} studied the behavior of $\mathcal{D}(x,T)$ (for $\zeta(s)$) in the vicinity of $x=T$ assuming RH plus a quantitative version of the twin prime conjecture with strong error term. His analysis leads to a pair correlation conjecture with $\mathcal{D}(x,T)$ smoothly varying through the transition zone. We conjecture that the same holds for other $F\in \mathcal S$.
\begin{conjecture}\label{conjPCF2} For $F\in \mathcal S$, $\mathcal{D}_F(x,T)\ll 1$ uniformly in $x$ and $T$, and for any $A>0$ there is a $c>0$ so that $$
| \mathcal{D}_F(x+\delta x,T) + \mathcal{D}_F(x-\delta x,T) - 2 \mathcal{D}_F(x,T)| = o (T/\log T) $$ uniformly for $T \le x\le T^A$ and $0 \le \delta \le (\log T)^{c-1}$. \end{conjecture}
Following the proof of Theorem \ref{theoremPC} (take $\beta=\log T \log\log T$, for example), we arrive at the following.
\begin{theorem}\label{theoremPCF} Assume RH$_F$. Then Conjecture \ref{conjPCF2} implies Conjecture \ref{conj3F}, and therefore also Conjectures \ref{conj1F} and \ref{conj2F}. \end{theorem}
{\bf Acknowledgement.} The authors thank the referee for carefully
reading the paper and for pointing out several misprints and minor errors.
\end{document} | arXiv |
12.5: Derivatives
[ "article:topic", "instantaneous velocity", "average rate of change", "Differentiability (two variables)", "Tangent Line", "secant", "Derivatives", "license:ccby", "showtoc:no", "authorname:openstaxjabramson" ]
Book: Precalculus (OpenStax)
12: Introduction to Calculus
Contributed by Jay Abramson
Principal Lecturer (School of Mathematical and Statistical Sciences) at Arizona State University
Publisher: OpenStax CNX
Finding the Average Rate of Change of a Function
Understanding the Instantaneous Rate of Change
Derivatives: Interpretations and Notation
Finding Derivatives of Rational Functions
Finding Derivatives of Functions with Roots
Finding Instantaneous Rates of Change
Using Graphs to Find Instantaneous Rates of Change
Using Instantaneous Rates of Change to Solve Real-World Problems
Finding Points Where a Function's Derivative Does Not Exist
Finding an Equation of a Line Tangent to the Graph of a Function
Finding the Instantaneous Speed of a Particle
Section Exercises
Chapter Review Exercises
Finding Limits: A Numerical and Graphical Approach
Finding Limits: Properties of Limits
The average teen in the United States opens a refrigerator door an estimated 25 times per day. Supposedly, this average is up from 10 years ago when the average teenager opened a refrigerator door 20 times per day1. It is estimated that a television is on in a home 6.75 hours per day, whereas parents spend an estimated 5.5 minutes per day having a meaningful conversation with their children. These averages, too, are not the same as they were 10 years ago, when the television was on an estimated 6 hours per day in the typical household, and parents spent 12 minutes per day in meaningful conversation with their kids. What do these scenarios have in common? The functions representing them have changed over time. In this section, we will consider methods of computing such changes over time.
The functions describing the examples above involve a change over time. Change divided by time is one example of a rate. The rates of change in the previous examples are each different. In other words, some changed faster than others. If we were to graph the functions, we could compare the rates by determining the slopes of the graphs.
A tangent line to a curve is a line that intersects the curve at only a single point but does not cross it there. (The tangent line may intersect the curve at another point away from the point of interest.) If we zoom in on a curve at that point, the curve appears linear, and the slope of the curve at that point is close to the slope of the tangent line at that point.
Figure \(\PageIndex{1}\) represents the function \(f(x)=x^3−4x\). We can see the slope at various points along the curve.
slope at \(x=−2\) is 8
slope at \(x=2\) is 8
Figure \(\PageIndex{1}\): Graph showing tangents to curve at –2, –1, and 2.
Let's imagine a point on the curve of function \(f\) at \(x=a\) as shown in Figure \(\PageIndex{1}\). The coordinates of the point are \((a,f(a))\). Connect this point with a second point on the curve a little to the right of \(x=a\), with an x-value increased by some small real number \(h\). The coordinates of this second point are \((a+h,f(a+h))\) for some positive-value \(h\).
Figure \(\PageIndex{2}\): Connecting point \(a\) with a point just beyond allows us to measure a slope close to that of a tangent line at \(x=a\).
We can calculate the slope of the line connecting the two points \((a,f(a))\) and \((a+h,f(a+h))\), called a secant line, by applying the slope formula,
\[ \mathrm{slope = \dfrac{change \; in \; y}{change \; in \; x}} \]
We use the notation \(m_{sec}\) to represent the slope of the secant line connecting two points.
\[\begin{align} m_{sec} &= \dfrac{f(a+h)−f(a)}{(a+h)−(a) } \\ &= \dfrac{f(a+h)−f(a)}{\cancel{a}+h−\cancel{a}} \end{align}\]
The slope \(m_{sec}\) equals the average rate of change between two points \((a,f(a))\) and \((a+h,f(a+h)).\)
\[m_{sec}=\dfrac{f(a+h)−f(a)}{h}\]
the AVERAGE RATE OF CHANGE BETWEEN TWO POINTS ON A CURVE
The average rate of change (AROC) between two points \((a,f(a))\) and \((a+h,f(a+h))\) on the curve of \(f\) is the slope of the line connecting the two points and is given by
\[\text{AROC}=\dfrac{f(a+h)−f(a)}{h}\]
Example \(\PageIndex{1}\): Finding the Average Rate of Change
Find the average rate of change connecting the points \((2,−6)\) and \((−1,5)\).
We know the average rate of change connecting two points may be given by
If one point is \((2,−6)\), or \((2,f(2))\), then \(f(2)=−6.\)
The value \(h\) is the displacement from \(2\) to \(−1\), which equals \(−1−2=−3.\)
For the other point, \(f(a+h)\) is the y-coordinate at \(a+h\), which is \(2+(−3)\) or \(−1,\) so \(f(a+h)=f(−1)=5\).
\[\begin{align} \text{AROC} &= \dfrac{f(a+h)−f(a)}{h} \\ &=\dfrac{5−(−6)}{−3} \\&=\dfrac{11}{−3} \\ &=−\dfrac{11}{3} \end{align}\]
Find the average rate of change connecting the points \((−5,1.5)\) and \((−2.5,9)\)
Now that we can find the average rate of change, suppose we make \(h\) in Figure \(\PageIndex{2}\) smaller and smaller. Then \(a+h\) will approach \(a\) as \(h\) gets smaller, getting closer and closer to 0. Likewise, the second point \((a+h,f(a+h))\) will approach the first point, \((a,f(a))\). As a consequence, the connecting line between the two points, called the secant line, will get closer and closer to being a tangent to the function at \(x=a\), and the slope of the secant line will get closer and closer to the slope of the tangent at \(x=a\) (Figure \(\PageIndex{3}\)).
Figure \(\PageIndex{3}\): The connecting line between two points moves closer to being a tangent line at \(x=a\).
Because we are looking for the slope of the tangent at \(x=a\), we can think of the measure of the slope of the curve of a function \(f\) at a given point as the rate of change at a particular instant. We call this slope the instantaneous rate of change, or the derivative of the function at \(x=a.\) Both can be found by finding the limit of the slope of a line connecting the point at \(x=a\) with a second point infinitesimally close along the curve. For a function \(f\) both the instantaneous rate of change of the function and the derivative of the function at \(x=a\) are written as \(f'(a),\) and we can define them as a two-sided limit that has the same value whether approached from the left or the right.
\[f′(a)= \lim \limits_{h \to 0} \dfrac{f(a+h)−f(a)}{h}\]
The expression by which the limit is found is known as the difference quotient.
DEFINITION OF INSTANTANEOUS RATE OF CHANGE AND DERIVATIVE
The derivative, or instantaneous rate of change, of a function \(f\) at \(x=a\), is given by
\[ f'(a)= \lim \limits_{h \to 0} \dfrac{f(a+h)−f(a)}{h}\]
The expression \(\frac{f(a+h)−f(a)}{h}\) is called the difference quotient.
We use the difference quotient to evaluate the limit of the rate of change of the function as \(h\) approaches 0.
The derivative of a function can be interpreted in different ways. It can be observed as the behavior of a graph of the function or calculated as a numerical rate of change of the function.
The derivative of a function \(f(x)\) at a point \(x=a\) is the slope of the tangent line to the curve \(f(x)\) at \(x=a\). The derivative of \(f(x)\) at \(x=a\) is written \(f′(a)\).
The derivative \(f′(a)\) measures how the curve changes at the point \((a,f(a))\).
The derivative \(f′(a)\) may be thought of as the instantaneous rate of change of the function \(f(x)\) at \(x=a\).
If a function measures distance as a function of time, then the derivative measures the instantaneous velocity at time \(t=a\).
NOTATIONS FOR THE DERIVATIVE
The equation of the derivative of a function \(f(x)\) is written as \(y′=f′(x)\), where \(y=f(x)\). The notation \(f′(x)\) is read as "\(f\) prime of \(x\)." Alternate notations for the derivative include the following:
\[f′(x)=y′=\dfrac{dy}{dx}=\dfrac{df}{dx}=\dfrac{d}{dx} f(x)=Df(x)\]
The expression \(f′(x)\) is now a function of \(x\); this function gives the slope of the curve \(y=f(x)\) at any value of \(x\). The derivative of a function \(f(x)\) at a point \(x=a\) is denoted \(f′(a)\).
how to: Given a function \(f\), find the derivative by applying the definition of the derivative.
Calculate \(f(a+h)\).
Calculate \(f(a)\).
Substitute and simplify \(\frac{f(a+h)−f(a)}{h}\).
Evaluate the limit if it exists: \(f′(a)=\lim \limits_{h \to 0} \frac{f(a+h)−f(a)}{h}\).
Example \(\PageIndex{1}\): Finding the Derivative of a Polynomial Function
Find the derivative of the function \(f(x)=x^2−3x+5\) at \(x=a.\)
\[ f′(a)= \lim \limits_{h \to 0} \dfrac{f(a+h)−f(a)}{h} \;\;\;\;\;\;\;\; \text{Definition of a derivative}\]
Substitute \(f(a+h)=(a+h)^2−3(a+h)+5\) and \(f(a)=a^2−3a+5.\)
\[ \begin{align} f′(a) &= \lim \limits_{h \to 0} \dfrac{(a+h)(a+h)−3(a+h)+5−(a^2−3a+5)}{h} \\ &= \lim \limits_{h \to 0} \dfrac{a^2+2ah+h^2−3a−3h+5−a^2+3a−5}{h} && \text{Evaluate to remove parentheses.} \\ & = \lim \limits_{h \to 0} \dfrac{\cancel{a^2}+2ah+h^2−\cancel{3a}−3h+\cancel{5}−\cancel{a^2}+\cancel{3a}−\cancel{5}}{h} && \text{Simplify.} \\ & = \lim \limits_{h \to 0} \dfrac{2ah+h^2−3h}{h} && \text{Factor out an }h. \\ & =2a+0−3 && \text{Evaluate the limit.} \\ &=2a−3 \end{align} \]
Find the derivative of the function \(f(x)=3x^2+7x\) at \(x=a\)
\(f′(a)=6a+7\)
To find the derivative of a rational function, we will sometimes simplify the expression using algebraic techniques we have already learned.
Example \(\PageIndex{1}\): Finding the Derivative of a Rational Function
Find the derivative of the function\(f(x)=\dfrac{3+x}{2−x}\) at \(x=a.\)
\[\begin{align} f′(a) &= \lim \limits_{h \to 0} \dfrac{f(a+h)−f(a)}{h} \\ &= \lim \limits_{h \to 0} \dfrac{\frac{3+(a+h)}{2−(a+h)}−(\frac{3+a}{2−a})}{h} && \text{Substitute }f(a+h) \text{ and }f(a) \\ &= \lim \limits_{h \to 0} \dfrac{(2−(a+h))(2−a)[ \frac{3+(a+h)}{2−(a+h)}−(\frac{3+a}{2−a}) ]}{(2−(a+h))(2−a)(h)} && \text{Multiply numerator and denominator by } (2−(a+h))(2−a) \\ & =\lim \limits_{h \to 0}\dfrac{(\cancel{2−(a+h)})(2−a)(\frac{3+(a+h)}{\cancel{(2−(a+h))}})−(2−(a+h))\cancel{(2−a)}(\frac{3+a}{\cancel{2−a}})}{(2−(a+h))(2−a)(h)} && \text{Distribute} \\ & =\lim \limits_{h \to 0} \dfrac{6−3a+2a−a^2+2h−ah−6+3a+3h−2a+a^2+ah}{(2−(a+h))(2−a)(h)} && \text{Multiply} \\ &=\lim \limits_{h \to 0} \dfrac{5 \cancel{h}}{(2−(a+h))(2−a)(\cancel{h})} && \text{Combine like terms} \\ & = \lim \limits_{h \to 0} \dfrac{5}{(2−(a+h))(2−a)} && \text{Cancel like factors} \\ & =\dfrac{5}{(2−(a+0))(2−a)}=\dfrac{5}{(2−a)(2−a)}=\dfrac{5}{(2−a)^2} && \text{Evaluate the limit} \end{align}\]
Exercise \(\PageIndex{1}\):
Find the derivative of the function \(f(x)=\frac{10x+11}{5x+4}\) at \(x=a\).
\[f′(a)=\dfrac{−15}{(5a+4)^2}\]
To find derivatives of functions with roots, we use the methods we have learned to find limits of functions with roots, including multiplying by a conjugate.
Example \(\PageIndex{1}\): Finding the Derivative of a Function with a Root
Find the derivative of the function \(f(x)=4\sqrt{x}\) at \(x=36.\)
\[\begin{align} f′(a) &=\lim \limits_{h \to 0} \dfrac{f(a+h)−f(a)}{h} \\ &= \lim \limits_{h \to 0} \dfrac{4\sqrt{a+h}−4\sqrt{a}}{h} && \text{Substitute }f(a+h) \text{ and }f(a) \end{align}\]
Multiply the numerator and denominator by the conjugate: \(\frac{4\sqrt{a+h}+4\sqrt{a}}{4\sqrt{a+h}+4\sqrt{a}}\).
\[\begin{align} f′(a) &= \lim \limits_{h \to 0}\bigg( \dfrac{4\sqrt{a+h}−4\sqrt{a}}{h} \bigg)⋅ \bigg(\dfrac{4\sqrt{a+h}+4\sqrt{a}}{4\sqrt{a+h}+4\sqrt{a}} \bigg) \\ &=\lim \limits_{h \to 0} \bigg( \dfrac{16(a+h)−16a}{h4(\sqrt{a+h}+4\sqrt{a})} \bigg) && \text{Multiply.} \\ &=\lim \limits_{ h \to 0} \bigg( \dfrac{\cancel{16a}+16h\cancel{−16a}}{h4(\sqrt{a+h}+4\sqrt{a})} \bigg) && \text{Distribute and combine like terms.} \\ &= \lim \limits_{h \to 0}\bigg(\dfrac{16\cancel{h}}{\cancel{h}(4\sqrt{a+h}+4\sqrt{a})} \bigg) && \text{Simplify.} \\ & = \lim \limits_{h \to 0} \bigg( \dfrac{16}{4\sqrt{a+h}+4\sqrt{a}} \bigg) && \text{Evaluate the limit by letting } h=0. \\ & =\dfrac{16}{8\sqrt{a}}=\dfrac{2}{\sqrt{a}} \\ f′(36) &= \dfrac{2}{\sqrt{36}} && \text{Evaluate the derivative at } x=36. \\ &=\dfrac{2}{6} \\ & =\dfrac{1}{3} \end{align}\]
Find the derivative of the function \(f(x)=9\sqrt{x}\) at \(x=9.\)
\(\frac{3}{2}\)
Many applications of the derivative involve determining the rate of change at a given instant of a function with the independent variable time—which is why the term instantaneous is used. Consider the height of a ball tossed upward with an initial velocity of 64 feet per second, given by \(s(t)=−16t^2+64t+6\), where \(t\) is measured in seconds and \(s(t)\) is measured in feet. We know the path is that of a parabola. The derivative will tell us how the height is changing at any given point in time. The height of the ball is shown in Figure as a function of time. In physics, we call this the "s-t graph."
Example \(\PageIndex{1}\): Finding the Instantaneous Rate of Change
Using the function above, \(s(t)=−16t^2+64t+6\),what is the instantaneous velocity of the ball at 1 second and 3 seconds into its flight?
The velocity at \(t=1\) and \(t=3\) is the instantaneous rate of change of distance per time, or velocity. Notice that the initial height is 6 feet. To find the instantaneous velocity, we find the derivative and evaluate it at \(t=1\) and \(t=3\):
\[\begin{align} f′(a) &= \lim \limits_{h \to 0} \dfrac{f(a+h)−f(a)}{h} \\ &= \lim \limits_{h \to 0} \dfrac{−16(t+h)^2+64(t+h)+6−(−16t^2+64t+6)}{h} && \text{Substitute } s(t+h) \text{ and } s(t). \\ &= \lim \limits_{h \to 0} \dfrac{−16t^2−32ht−h^2+64t+64h+6+16t^2−64t−6}{h} && \text{Distribute} \\ & =\lim \limits_{h \to 0} \dfrac{−32ht−h^2+64h}{h} && \text{Simplify.} \\ &= \lim \limits_{h \to 0} \dfrac{\cancel{h}(−32t−h+64)}{\cancel{h}} && \text{Factor the numerator.} \\ & =\lim \limits_{h \to 0}−32t−h+64 && \text{ Cancel out the common factor} h. \\ s′(t) &=−32t+64 && \text{Evaluate the limit by letting} h=0. \end{align}\]
For any value of \(t\), \(s′(t)\) tells us the velocity at that value of \(t\).
Evaluate \(t=1\) and \(t=3\).
\[\begin{align}s′(1) &=−32(1)+64=32 \\ s′(3) &=−32(3)+64=−32 \end{align}\]
The velocity of the ball after 1 second is 32 feet per second, as it is on the way up.
The velocity of the ball after 3 seconds is −32 feet per second, as it is on the way down.
The position of the ball is given by\(s(t)=−16t^2+64t+6.\) What is its velocity 2 seconds into flight?
We can estimate an instantaneous rate of change at \(x=a\) by observing the slope of the curve of the function \(f(x)\) at \(x=a\). We do this by drawing a line tangent to the function at \(x=a\) and finding its slope.
how to: Given a graph of a function \(f( x )\), find the instantaneous rate of change of the function at \(x=a\).
Locate \(x=a\) on the graph of the function \(f(x)\).
Draw a tangent line, a line that goes through \(x=a\) at \(a\) and at no other point in that section of the curve. Extend the line far enough to calculate its slope as
\[\dfrac{\text{change in }y}{\text{change in }x.}\]
Example \(\PageIndex{1}\): Estimating the Derivative at a Point on the Graph of a Function
From the graph of the function \(y=f(x)\) presented in Figure, estimate each of the following:
\(f(0) ; f(2) ; f'(0) ; f'(2)\)
To find the functional value, \(f(a)\), find the y-coordinate at \(x=a\).
To find the derivative at \(x=a, f′(a),\) draw a tangent line at \(x=a,\) and estimate the slope of that tangent line. See Figure.
\(f(0)\) is the y-coordinate at \(x=0\). The point has coordinates \((0,1)\), thus \(f(0)=1\).
\(f′(0)\) is found by estimating the slope of the tangent line to the curve at \(x=0\). The tangent line to the curve at \(x=0\) appears horizontal. Horizontal lines have a slope of 0, thus \(f′(0)=0\).
\(f′(2)\) is found by estimating the slope of the tangent line to the curve at \(x=2\). Observe the path of the tangent line to the curve at \(x=2\). As the \(x\) value moves one unit to the right, the \(y\) value moves up four units to another point on the line. Thus, the slope is 4, so \(f′(2)=4\).
Using the graph of the function\(f(x)=x^3−3x\) shown in Figure, estimate: \(f(1), f′(1), f(0)\),and \(f′(0)\).
−2,−2,0, 0, −3
Another way to interpret an instantaneous rate of change at \(x=a\) is to observe the function in a real-world context. The unit for the derivative of a function \(f(x)\) is
\[\dfrac{\text{output units}}{\text{input unit}}\]
Such a unit shows by how many units the output changes for each one-unit change of input. The instantaneous rate of change at a given instant shows the same thing: the units of change of output per one-unit change of input.
One example of an instantaneous rate of change is a marginal cost. For example, suppose the production cost for a company to produce \(x\) items is given by \(C(x)\), in thousands of dollars. The derivative function tells us how the cost is changing for any value of \(x\) in the domain of the function. In other words, \(C′(x)\) is interpreted as a marginal cost, the additional cost in thousands of dollars of producing one more item when \(x\) items have been produced. For example, \(C′(11)\) is the approximate additional cost in thousands of dollars of producing the 12th item after 11 items have been produced. \(C′(11)=2.50\) means that when 11 items have been produced, producing the 12th item would increase the total cost by approximately $2,500.00.
Example \(\PageIndex{1}\): Finding a Marginal Cost
The cost in dollars of producing \(x\) laptop computers in dollars is \(f(x)=x^2−100x.\) At the point where 200 computers have been produced, what is the approximate cost of producing the 201stunit?
If \(f(x)=x^2−100x\) describes the cost of producing \(x\) computers, \(f′(x)\) will describe the marginal cost. We need to find the derivative. For purposes of calculating the derivative, we can use the following functions:
\[\begin{align} f(a+b) &=(x+h)^2−100(x+h) \\ f(a) &=a ^2−100a \end{align}\]
\[\begin{align} f′(x) &=\dfrac{f(a+h)−f(a)}{h} && \text{Formula for a derivative} \\ &=\dfrac{(x+h)^2−100(x+h)−(x^2−100x)}{h} \\ \text{Substitute }f(a+h) \text{ and }f(a). \\ & =\dfrac{x^2+2xh+h^2−100x−100h−x^2+100x}{h} && \text{Multiply polynomials, distribute.} \\ &= \text{2xh+h^2−100h}{h} && \text{Collect like terms.} \\ &=\dfrac{\cancel{h}(2x+h−100)}{\cancel{h}} && \text{Factor and cancel like terms.} \\ &=2x+h−100 && \text{Simplify.} \\ &=2x−100 && \text{Evaluate when }h=0. \\ f′(x) &=2x−100 && \text{Formula for marginal cost} \\ f′(200) &=2(200)−100=300 && \text{Evaluate for 200 units.} \end{align}\]
The marginal cost of producing the 201st unit will be approximately $300.
Example \(\PageIndex{1}\):Interpreting a Derivative in Context
A car leaves an intersection. The distance it travels in miles is given by the function \( f(t)\), where \(t\) represents hours. Explain the following notations:
\(f(0)=0 f′(1)=60 f(1)=70 f(2.5)=150\)
First we need to evaluate the function \(f(t)\) and the derivative of the function \(f′(t)\), and distinguish between the two. When we evaluate the function \(f(t)\), we are finding the distance the car has traveled in \(t\) hours. When we evaluate the derivative f′(t), f′(t), we are finding the speed of the car after \(t\) hours.
\(f(0)=0\) means that in zero hours, the car has traveled zero miles.
\(f′(1)=60\) means that one hour into the trip, the car is traveling 60 miles per hour.
\(f(1)=70\) means that one hour into the trip, the car has traveled 70 miles. At some point during the first hour, then, the car must have been traveling faster than it was at the 1-hour mark.
\(f(2.5)=150\) means that two hours and thirty minutes into the trip, the car has traveled 150 miles.
A runner runs along a straight east-west road. The function \(f(t)\) gives how many feet eastward of her starting point she is after \(t\) seconds. Interpret each of the following as it relates to the runner.
\(f(0)=0 ; f(10)=150 ; f′(10)=15 ; f′(20)=−10 ; f(40)=−100\)
After zero seconds, she has traveled 0 feet.
After 10 seconds, she has traveled 150 feet east.
After 10 seconds, she is moving eastward at a rate of 15 ft/sec.
After 20 seconds, she is moving westward at a rate of 10 ft/sec.
After 40 seconds, she is 100 feet westward of her starting point.
To understand where a function's derivative does not exist, we need to recall what normally happens when a function \(f(x)\) has a derivative at \(x=a\). Suppose we use a graphing utility to zoom in on \(x=a\). If the function \(f(x)\) is differentiable, that is, if it is a function that can be differentiated, then the closer one zooms in, the more closely the graph approaches a straight line. This characteristic is called linearity.
Look at the graph in Figure. The closer we zoom in on the point, the more linear the curve appears.
Figure_12_04_009">Figure, the graph does not approach a straight line. No matter how close we zoom in, the graph maintains its sharp corner.
Graph of the function \(f(x)=| x |\),with x-axis from –0.1 to 0.1 and y-axis from –0.1 to 0.1.
What are the characteristics of a graph that is not differentiable at a point? Here are some examples in which function \(f(x)\) is not differentiable at \(x=a\).
In Figure, we see the graph of
\[f(x)=\begin{cases} x^2, &&x≤2 \\ 8−x, &&x>2.\end{cases} .\]
Notice that, as \(x\) approaches 2 from the left, the left-hand limit may be observed to be 4, while as \(x\) approaches 2 from the right, the right-hand limit may be observed to be 6. We see that it has a discontinuity at \(x=2\).
The graph of \(f(x)\) has a discontinuity at \(x=2\).
In Figure, we see the graph of \(f(x)=|x|\). We see that the graph has a corner point at \(x=0\).
The graph of \(f(x)=| x |\) has a corner point at \(x=0\).
In Figure, we see that the graph of \(f(x)=x^{\frac{2}{3}}\) has a cusp at \(x=0\). A cusp has a unique feature. Moving away from the cusp, both the left-hand and right-hand limits approach either infinity or negative infinity. Notice the tangent lines as \(x\) approaches 0 from both the left and the right appear to get increasingly steeper, but one has a negative slope, the other has a positive slope.
The graph of \(f(x)=x^\frac{2}{3}\) has a cusp at \(x=0\).
In Figure, we see that the graph of \(f(x)=x^{frac{1}{3}}\) has a vertical tangent at \(x=0\). Recall that vertical tangents are vertical lines, so where a vertical tangent exists, the slope of the line is undefined. This is why the derivative, which measures the slope, does not exist there.
The graph of \(f(x)=x^\frac{1}{3}\) has a vertical tangent at \(x=0\).
differentiability
A function \(f(x)\) is differentiable at \(x=a\) if the derivative exists at \(x=a\),which means that \(f′(a)\) exists.
There are four cases for which a function \(f(x)\) is not differentiable at a point \(x=a\).
When there is a discontinuity at \(x=a\).
When there is a corner point at \(x=a\).
When there is a cusp at \(x=a\).
Any other time when there is a vertical tangent at \(x=a\).
Example \(\PageIndex{1}\): Determining Where a Function Is Continuous and Differentiable from a Graph
Using Figure, determine where the function is
differentiable
not differentiable
At the points where the graph is discontinuous or not differentiable, state why.
Figure_12_04_016">Figure.
Three intervals where the function is continuous
The graph of is differentiable on \((−∞,−2)∪(−2,−1)∪(−1,1)∪(1,2)∪(2,∞)\). The graph of \(f(x)\) is not differentiable at \(x=−2\) because it is a point of discontinuity, at \(x=−1\) because of a sharp corner, at \(x=1\) because it is a point of discontinuity, and at \(x=2\) because of a sharp corner. See Figure.
Five intervals where the function is differentiable
Determine where the function \(y=f(x)\) shown in Figure is continuous and differentiable from the graph.
The graph of \(f\) is continuous on \((−∞,1)∪(1,3)∪(3,∞).\) The graph of f f is discontinuous at \(x=1\) and \(x=3\). The graph of \(f\) is differentiable on \((−∞,1)∪(1,3)∪(3,∞)\). The graph of \(f\) is not differentiable at \(x=1\) and \(x=3\).
The equation of a tangent line to a curve of the function \(f(x)\) at \(x=a\) is derived from the point-slope form of a line, \(y=m(x−x_1)+y_1\). The slope of the line is the slope of the curve at \(x=a\) and is therefore equal to \(f′(a),\) the derivative of \(f(x)\) at \(x=a.\) The coordinate pair of the point on the line at \(x=a\) is \((a,f(a))\).
If we substitute into the point-slope form, we have
The equation of the tangent line is
\[y=f'(a)(x−a)+f(a)\]
THE EQUATION OF A LINE TANGENT TO A CURVE OF THE FUNCTION F
The equation of a line tangent to the curve of a function \(f\) at a point \(x=a\) is
how to: Given a function \(f\), find the equation of a line tangent to the function at \(x=a\).
Find the derivative of \(f(x)\) at \(x=a\) using \(f′(a)=\lim \limits_{h \to 0} \frac{f(a+h)−f(a)}{h}.\)
Evaluate the function at \(x=a\). This is \(f(a)\).
Substitute \((a,f(a))\) and \(f′(a)\) into \(y=f'(a)(x−a)+f(a)\).
Write the equation of the tangent line in the form \(y=mx+b\).
Example \(\PageIndex{1}\): Finding the Equation of a Line Tangent to a Function at a Point
Find the equation of a line tangent to the curve \(f(x)=x^2−4x\) at \(x=3.\)
\[f'(a)= \lim \limits_{h \to 0} \dfrac{f(a+h)−f(a)}{h}\]
Substitute \(f(a+h)=(a+h)^2−4(a+h)\) and \(f(a)=a^2−4a.\)
\[\begin{align} f′(a) &= \lim \limits_{h \to 0}\dfrac{(a+h)(a+h)−4(a+h)−(a2−4a)}{h} \\ &= \lim \limits_{h \to 0} \dfrac{a^2+2ah+h^2−4a−4h−a^2+4a}{h} && \text{Remove parentheses.} \\ &= \lim \limits_{h \to 0} \dfrac{\cancel{a^2}+2ah+h^2−\cancel{4a}−4h−\cancel{a^2}+\cancel{4a}}{h} && \text{Combine like terms.} \\ &= \lim \limits_{h \to 0} \dfrac{2ah+h^2−4h}{h} \\ &= \lim \limits_{h \to 0} \dfrac{\cancel{h}(2a+h−4)}{h} && \text{Factor out }h. \\ &=2a+0−4 \\ f′(a)&=2a−4 && \text{Evaluate the limit.} \\ f′(3)&=2(3)−4=2 \end{align}\]
Equation of tangent line at \(x=3\):
\[\begin{align} y &= f'(a)(x−a)+f(a) \\ y &=f'(3)(x−3)+f(3) \\ y &=2(x−3)+(−3) \\ y &=2x−9 \end{align}\]
We can use a graphing utility to graph the function and the tangent line. In so doing, we can observe the point of tangency at \(x=3\) as shown in Figure.
Graph confirms the point of tangency at \(x=3\).
Find the equation of a tangent line to the curve of the function \(f(x)=5x^2−x+4\) at \(x=2\).
\(y=19x−16\)
If a function measures position versus time, the derivative measures displacement versus time, or the speed of the object. A change in speed or direction relative to a change in time is known as velocity. The velocity at a given instant is known as instantaneous velocity.
In trying to find the speed or velocity of an object at a given instant, we seem to encounter a contradiction. We normally define speed as the distance traveled divided by the elapsed time. But in an instant, no distance is traveled, and no time elapses. How will we divide zero by zero? The use of a derivative solves this problem. A derivative allows us to say that even while the object's velocity is constantly changing, it has a certain velocity at a given instant. That means that if the object traveled at that exact velocity for a unit of time, it would travel the specified distance.
INSTANTANEOUS VELOCITY
Let the function \(s(t)\) represent the position of an object at time \(t.\) The instantaneous velocity or velocity of the object at time \(t=a\) is given by
\[s′(a)= \lim \limits_{h \to 0} \dfrac{s(a+h)−s(a)}{h}\]
Example \(\PageIndex{1}\): Finding the Instantaneous Velocity
A ball is tossed upward from a height of 200 feet with an initial velocity of 36 ft/sec. If the height of the ball in feet after \(t\) seconds is given by \(s(t)=−16t^2+36t+200,\) find the instantaneous velocity of the ball at \( t=2\).
First, we must find the derivative \(s′(t)\). Then we evaluate the derivative at \(t=2\), using \(s(a+h)=−16(a+h)^2+36(a+h)+200\) and \(s(a)=−16a^2+36a+200.\)
\[\begin{align} s′(a)= \lim \limits_{h \to 0} \dfrac{s(a+h)−s(a)}{h} \\ &= \lim \limits_{h \to 0}\dfrac{−16(a+h)^2+36(a+h)+200−(−16a^2+36a+200)}{h} \\ &= \lim \limits_{h \to 0} \dfrac{−16(a^2+2ah+h^2)+36(a+h)+200−(−16a^2+36a+200)}{h} \\ &= \lim \limits_{h \to 0}\dfrac{−16a^2−32ah−16h^2+36a+36h+200+16a^2−36a−200}{h} \\ &= \lim \limits_{h \to 0}\dfrac{\cancel{−16a^2}−32ah−16h^2+\cancel{36a}+36h+\cancel{200}+\cancel{16a^2}−\cancel{36a}−\cancel{200}}{h} \\ &= \lim \limits_{h \to 0}\dfrac{−32ah−16h^2+36h}{h} \\ &= \lim \limits_{h \to 0}\dfrac{\cancel{h}(−32a−16h+36)}{\cancel{h}} \\ &= \lim \limits_{h \to 0}(−32a−16h+36) \\&=−32a−16⋅0+36 \\ s′(a) &=−32a+36 \\ s′(2) &=−32(2)+36 \\ & =−28 \end{align}\]
This result means that at time \(t=2\) seconds, the ball is dropping at a rate of 28 ft/sec.
A fireworks rocket is shot upward out of a pit 12 ft below the ground at a velocity of 60 ft/sec. Its height in feet after \(t\) seconds is given by \(s=−16t^2+60t−12.\) What is its instantaneous velocity after 4 seconds?
–68 ft/sec, it is dropping back to Earth at a rate of 68 ft/s.
Access these online resources for additional instruction and practice with derivatives.
Estimate the Derivative
Estimate the Derivative Ex. 4
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average rate of change \(\text{AROC}=\frac{f(a+h)−f(a)}{h}\)
derivative of a function \(f′(a)=\lim \limits_{h \to 0} \frac{f(a+h)−f(a)}{h}\)
The slope of the secant line connecting two points is the average rate of change of the function between those points. See Example.
The derivative, or instantaneous rate of change, is a measure of the slope of the curve of a function at a given point, or the slope of the line tangent to the curve at that point. See Example, Example, and Example.
The difference quotient is the quotient in the formula for the instantaneous rate of change:
\(\frac{f(a+h)−f(a)}{h}\)
Instantaneous rates of change can be used to find solutions to many real-world problems. See Example.
The instantaneous rate of change can be found by observing the slope of a function at a point on a graph by drawing a line tangent to the function at that point. See Example.
Instantaneous rates of change can be interpreted to describe real-world situations. See Example and Example.
Some functions are not differentiable at a point or points. See Example.
The point-slope form of a line can be used to find the equation of a line tangent to the curve of a function. See Example.
Velocity is a change in position relative to time. Instantaneous velocity describes the velocity of an object at a given instant. Average velocity describes the velocity maintained over an interval of time.
Using the derivative makes it possible to calculate instantaneous velocity even though there is no elapsed time. See Example.
How is the slope of a linear function similar to the derivative?
The slope of a linear function stays the same. The derivative of a general function varies according to \(x\). Both the slope of a line and the derivative at a point measure the rate of change of the function.
What is the difference between the average rate of change of a function on the interval \([ x,x+h ]\) and the derivative of the function at \(x\)?
A car traveled 110 miles during the time period from 2:00 P.M. to 4:00 P.M. What was the car's average velocity? At exactly 2:30 P.M., the speed of the car registered exactly 62 miles per hour. What is another name for the speed of the car at 2:30 P.M.? Why does this speed differ from the average velocity?
Average velocity is 55 miles per hour. The instantaneous velocity at 2:30 p.m. is 62 miles per hour. The instantaneous velocity measures the velocity of the car at an instant of time whereas the average velocity gives the velocity of the car over an interval.
Explain the concept of the slope of a curve at point \(x\).
Suppose water is flowing into a tank at an average rate of 45 gallons per minute. Translate this statement into the language of mathematics.
The average rate of change of the amount of water in the tank is 45 gallons per minute. If \(f(x)\) is the function giving the amount of water in the tank at any time \(t\),then the average rate of change of \(f(x)\) between \(t=a\) and \(t=b\) is \(f(a)+45(b−a).\)
For the following exercises, use the definition of derivative \(\lim \limits_{h to 0} \frac{f(x+h)−f(x)}{h}\) to calculate the derivative of each function.
\(f(x)=3x−4\)
\(f(x)=−2x+1\)
\(f′(x)=−2\)
\(f(x)=x^2−2x+1\)
\(f(x)=2x^2+x−3
\(f′(x)=4x+1\)
\(f(x)=2x^2+5\)
\(f(x)=\frac{−1}{x−2}\)
\(f′(x)=\frac{1}{(x−2)^2}\)
\(f(x)=\frac{2+x}{1−x}\)
\(f(x)=\frac{5−2x}{3+2x}\)
\(\frac{−16(}{3+2x)^2}\)
\(f(x)=\sqrt{1+3x}\)
\(f(x)=3x^3−x^2+2x+5\)
\(f′(x)=9x^2−2x+2\)
\(f(x)=5\)
\(f(x)=5π\)
\(f′(x)=0\)
For the following exercises, find the average rate of change between the two points.
\((−2,0)\) and \((−4,5)\)
\((4,−3)\) and \((−2,−1)\)
\(−\frac{1}{3}\)
\((0,5)\) and \((6,5)\)
\((7,−2)\) and \((7,10)\)
For the following polynomial functions, find the derivatives.
\(f(x)=x^3+1\)
\(f(x)=−3x^2−7x=6\)
\(f′(x)=−6x−7\)
\(f(x)=7x^2\)
\(f(x)=3x^3+2x^2+x−26\)
\(f′(x)=9x^2+4x+1\)
For the following functions, find the equation of the tangent line to the curve at the given point \(x\) on the curve.
\(f(x)=2x^2−3x \;\;\; x=3\)
\(f(x)=x^3+1 \;\;\;\; x=2\)
\(f(x)=\sqrt{x} \;\;\;\; x=9\)
For the following exercise, find \(k\) such that the given line is tangent to the graph of the function.
\(f(x)=x^2−kx, \;\;\; y=4x−9\)
\(k=−10\) or \(k=2\)
For the following exercises, consider the graph of the function \(f\) and determine where the function is continuous/discontinuous and differentiable/not differentiable.
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Discontinuous at \(x=−2\) and \(x=0\). Not differentiable at –2, 0, 2.
Discontinuous at \(x=5\). Not differentiable at -4, –2, 0, 1, 3, 4, 5.
For the following exercises, use Figure to estimate either the function at a given value of \(x\) or the derivative at a given value of \(x\), as indicated.
\(f(−1)\)
\(f(0)\)
\(f(0)=−2\)
\(f(2)=−6f(2)=−6
\(f′(−1)\)
\(f′(−1)=9\)
\(f′(0)\)
\(f′(1)=−3\)
\(f′(3)=9\)
Sketch the function based on the information below:
\(f′(x)=2x, f(2)=4\)
Numerically evaluate the derivative. Explore the behavior of the graph of \(f(x)=x^2\) around \(x=1\) by graphing the function on the following domains: \([ 0.9,1.1 ], [ 0.99,1.01 ], [ 0.999,1.001 ],\) and \([0.9999, 1.0001]\). We can use the feature on our calculator that automatically sets Ymin and Ymax to the Xmin and Xmax values we preset. (On some of the commonly used graphing calculators, this feature may be called ZOOM FIT or ZOOM AUTO). By examining the corresponding range values for this viewing window, approximate how the curve changes at \(x=1,\) that is, approximate the derivative at \(x=1.\)
Answers vary. The slope of the tangent line near \(x=1\) is 2.
For the following exercises, explain the notation in words. The volume \(f(t)\) of a tank of gasoline, in gallons, \(t\) minutes after noon.
\(f(0)=600\)
\(f'(30)=−20\)
At 12:30 p.m., the rate of change of the number of gallons in the tank is –20 gallons per minute. That is, the tank is losing 20 gallons per minute.
\(f(30)=0\)
\(f'(200)=30\)
At 200 minutes after noon, the volume of gallons in the tank is changing at the rate of 30 gallons per minute.
\(f(240)=500\)
For the following exercises, explain the functions in words. The height, \(s\), of a projectile after \(t\) seconds is given by \(s(t)=−16t^2+80t.\)
\(s(2)=96\)
The height of the projectile after 2 seconds is 96 feet.
\(s'(2)=16\)
The height of the projectile at \(t=3\) seconds is 96 feet.
\(s'(3)=−16\)
\(s(0)=0,s(5)=0.\)
The height of the projectile is zero at \(t=0\) and again at \(t=5\). In other words, the projectile starts on the ground and falls to earth again after 5 seconds.
For the following exercises, the volume \(V\) of a sphere with respect to its radius \(r\) is given by \(V=\frac{4}{3}πr^3.\)
Find the average rate of change of \(V\) as \(r\) changes from 1 cm to 2 cm.
Find the instantaneous rate of change of \(V\) when r=3 cm. r=3 cm.
\(36π\)
For the following exercises, the revenue generated by selling \(x\) items is given by \(R(x)=2x^2+10x\).
Find the average change of the revenue function as \(x\) changes from \(x=10\) to \(x=20\).
Find \(R'(10)\) and interpret.
$50.00 per unit, which is the instantaneous rate of change of revenue when exactly 10 units are sold.
Find \(R'(15)\) and interpret. Compare \(R'(15)\) to \(R'(10),\) and explain the difference.
For the following exercises, the cost of producing \(x\) cellphones is described by the function \(C(x)=x^2−4x+1000.\)
Find the average rate of change in the total cost as \(x\) changes from \(x=10\) to \(x=15.\)
Find the approximate marginal cost, when 15 cellphones have been produced, of producing the 16th cellphone.
Find the approximate marginal cost, when 20 cellphones have been produced, of producing the 21st cellphone.
For the following exercises, use the definition for the derivative at a point \(x=a\), \(\lim \limits_{x \to a}\frac{f(x)−f(a)}{x−a},\) to find the derivative of the functions.
\(f(x)=\frac{1}{x^2}\)
\(f(x)=5x^2−x+4\)
\(f'(x)=10a−1\)
\(f(x)=−x^2+4x+7\)
\(f(x)=\frac{−4}{3−x^2}\)
\(\frac{4}{(3−x)^2}\)
For the following exercises, use Figure.
\(\lim \limits_{x \to −1^+}f(x)\)
\(\lim \limits_{x \to −1^−}f(x)\)
\(\lim \limits_{x \to −1} f(x)\)
\(\lim \limits_{x \to 3}f(x)\)
At what values of \(x\) is the function discontinuous? What condition of continuity is violated?
Discontinuous at \(x=−1\) (\(\lim \limits_{x \to a} f(x)\) does not exist), \(x=3\) (jump discontinuity),and \(x=7\) ((\lim \limits_{x \to a} f(x)\) does not exist).
Using Table, estimate \(\lim \limits_{x \to 0}f(x).\)
\(x\) \(F(x)\)
−0.1 2.875
−0.01 2.92
−0.001 2.998
0 Undefined
0.001 2.9987
0.01 2.865
0.1 2.78145
For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as \(x\) approaches \(a\). If the function has limit as \(x\) approaches \(a\), state it. If not, discuss why there is no limit.
\(f(x)=\begin{cases} | x |−1, && \text{if }x≠1 \\ x^3, \text{if }x=1 \end{cases} a=1\)
\(f(x)=\begin{cases} \frac{1}{x+1}, && \text{if }x=−2 \\ (x+1)^2, && \text{if }x≠−2 \end{cases} a=−2\)
\(\lim \limits_{x \to −2} f(x)=1\)
\(f(x)= \begin{cases} \sqrt{x+3} && \text{if } x < 1 \\ −\sqrt[3]{x} && \text{if }x>1 \end{cases} a=1\)
For the following exercises, find the limits if \(\lim \limits_{x \to c} f(x)=−3\) and \(\lim \limits_{x \to c} g(x)=5\).
\(\lim \limits_{x \to c} (f(x)+g(x))\)
\(\lim \limits_{x \to c} \frac{f(x)}{g(x)}\)
\(\lim \limits_{x to c}(f(x)⋅g(x))\)
\(−15\)
\(\lim \limits_{x \to 0^+}f(x),f(x)= \begin{cases} 3x^2+2x+1 && x>0 \\ 5x+3 && x<0 \end{cases}\)
\(\lim \limits_{x \to 0^-}f(x),f(x)= \begin{cases} 3x^2+2x+1 && x>0 \\ 5x+3 && x<0 \end{cases}\)
\(\lim \limits_{x \to 3^+}(3x−〚x〛)\)
For the following exercises, evaluate the limits using algebraic techniques.
\(\lim \limits_{h \to 0}(\frac{(h+6)^2−36}{h})\)
\(\lim \limits_{x \to 25}(\frac{x^2−625}{\sqrt{x}−5)}\)
\(\lim \limits_{x \to 1}(\frac{−x^2−9x}{x})\)
\(\lim \limits_{x \to 4}\frac{7−\sqrt{12x+1}}{x−4}\)
\(\lim \limits_{x \to −3}(\frac{\frac{1}{3}+\frac{1}{x}}{3+x})\)
For the following exercises, use numerical evidence to determine whether the limit exists at \(x=a\). If not, describe the behavior of the graph of the function at \(x=a\).
\(f(x)=\frac{−2}{x−4}; a=4\)
\(f(x)=\frac{−2}{(x−4)^2}; a=4\)
At \(x=4\),the function has a vertical asymptote.
\(f(x)=\frac{−x}{x^2−x−6}; a=3\)
\(f(x)=\frac{6x^2+23x+20}{4x^2−25}; a=−\frac{5}{2}\)
removable discontinuity at \(a=−\frac{5}{2}\)
\(f(x)=\frac{\sqrt{x}−3}{9−x}; a=9\)
For the following exercises, determine where the given function \(f(x)\) is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities.
\(f(x)=x^2−2x−15\)
continuous on \((−∞,∞)\)
\(f(x)=\frac{x^2−2x−15}{x−5}\)
\(f(x)=\frac{x^2−2x}{x^2−4x+4}\)
removable discontinuity at \(x=2. f(2)\) is not defined, but limits exist.
\(f(x)=\frac{x^3−125}{2x^2−12x+10}\)
\(f(x)=\frac{x^2−\frac{1}{x}}{2−x}\)
discontinuity at \(x=0\) and \(x=2\). Both \(f(0)\) and \(f(2)\) are not defined.
\(f(x)=\frac{x+2}{x^2−3x−10}\)
\(f(x)=\frac{x+2}{x^3+8}\)
removable discontinuity at \(x=–2. f(–2)\) is not defined.
For the following exercises, find the average rate of change \(\frac{f(x+h)−f(x)}{h}\).
\(f(x)=3x+2\)
\(f(x)=\frac{1}{x+1}\)
\(f(x)= \ln (x)\)
\(\frac{\ln (x+h)− \ln (x)}{h}\)
\(f(x)=e^{2x}\)
For the following exercises, find the derivative of the function.
\(=4\)
\(f(x)=5x^2−3x\)
Find the equation of the tangent line to the graph of \(f(x)\) at the indicated \(x\) value.
\(f(x)=−x^3+4x; x=2.\)
\(y=−8x+16\)
For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.
\(f(x)=\frac{x}{| x |}\)
Given that the volume of a right circular cone is \(V=\frac{1}{3}πr^2h\) and that a given cone has a fixed height of 9 cm and variable radius length, find the instantaneous rate of change of volume with respect to radius length when the radius is 2 cm. Give an exact answer in terms of \(π\)
For the following exercises, use the graph of \(f\) in Figure.
\(\lim \limits_{x \to −1^-}f(x)\)
\(\lim \limits_{x \to −1}f(x)\)
\(−1\)
At what values of \(x\) is \(f\) discontinuous? What property of continuity is violated?
For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as \(x\) approaches \(a\). If the function has a limit as \(x\) approaches \(a\),state it. If not, discuss why there is no limit
\(f(x)=\begin{cases} \frac{1}{x}−3, && \text{if }x≤2 \\ x^3+1, && \text{if } x>2 \end{cases} a=2\)
\(\lim \limits_{x \to 2^−} f(x)=−\frac{5}{2}a\) and \(\lim \limits_{x \to 2^+} f(x)=9\) Thus, the limit of the function as \(x\) approaches 2 does not exist.
\(f(x)=\begin{cases} x^3+1, && \text{if }x<1 \\ 3x^2−1, && \text{if } x=1 \\ −\sqrt{x+3}+4, && \text{if } x>1 \end{cases} a=1\)
For the following exercises, evaluate each limit using algebraic techniques.
\(\lim \limits_{x \to −5}(\frac{\frac{1}{5}+\frac{1}{x}}{10+2x})\)
\(−\frac{1}{50}\)
\(\lim \limits_{h \to 0} (\frac{\sqrt{h^2+25}−5}{h^2})\)
\(\lim \limits_{h \to 0} (\frac{1}{h}−\frac{1}{h^2+h})\)
For the following exercises, determine whether or not the given function \(f\) is continuous. If it is continuous, show why. If it is not continuous, state which conditions fail.
\(f(x)=\sqrt{x^2−4}\)
\(f(x)=\frac{x^3−4x^2−9x+36}{x^3−3x^2+2x−6}\)
removable discontinuity at \(x=3\)
For the following exercises, use the definition of a derivative to find the derivative of the given function at \(x=a\).
\(f(x)=\frac{3}{5+2x}\)
\(f(x)=\frac{3}{\sqrt{x}}\)
\(f'(x)=−\frac{3}{2a^{\frac{3}{2}}}\)
\(f(x)=2x^2+9x\)
discontinuous at –2,0, not differentiable at –2,0, 2.
\(f(x)=| x−2 |−| x+2 |\)
\(f(x)=\frac{2}{1+e^{\frac{2}{x}}}\)
not differentiable at \(x=0\) (no limit)
For the following exercises, explain the notation in words when the height of a projectile in feet, \(s\), is a function of time t t in seconds after launch and is given by the function \(s(t)\).
\(s(0)\)
the height of the projectile at \(t=2\) seconds
\(s'(2)\)
\(\frac{s(2)−s(1)}{2−1}\)
the average velocity from \(t=1\) to \(t=2\)
\(s(t)=0\)
For the following exercises, use technology to evaluate the limit.
\(\lim \limits_{x \to 0} \frac{\sin (x)}{3x}\)
\(\lim \limits_{x \to 0} \frac{\tan ^2 (x)}{2x}\)
\(\lim \limits_{x \to 0}\frac{\sin (x)(1−\cos (x))}{2x^2}\)
Evaluate the limit by hand.
\(\lim \limits_{x \to 1}f(x), \text{where } f(x)= \begin{cases} 4x−7 && x≠1 \\ x^2−4 &&x=1 \end{cases}\)
At what value(s) of \(x\) is the function below discontinuous?
\(f(x)= \begin{cases} 4x−7 && x≠1 \\ x^2−4 &&x=1 \end{cases}\)
For the following exercises, consider the function whose graph appears in Figure.
Find the average rate of change of the function from \(x=1\) to \(x=3\).
Find all values of \(x\) at which \(f'(x)=0\).
\(x=1\)
Find all values of \(x\) at which \(f'(x)\) does not exist.
Find an equation of the tangent line to the graph of \(f\) the indicated point: \(f(x)=3x^2−2x−6, x=−2\)
\(y=−14x−18\)
For the following exercises, use the function \(f(x)=x(1−x)^{\frac{2}{5}}\).
Graph the function \(f(x)=x(1−x)^{\frac{2}{5}}\) by entering \(f(x)=x((1−x)^2)^{\frac{1}{5}}\) and then by entering \(f(x)=x((1−x)^{\frac{1}{5}})^2\).
Explore the behavior of the graph of \(f(x)\) around \(x=1\) by graphing the function on the following domains, [0.9, 1.1], [0.99, 1.01], [0.999, 1.001], and [0.9999, 1.0001]. Use this information to determine whether the function appears to be differentiable at \(x=1\).
The graph is not differentiable at \(x=1\) (cusp).
For the following exercises, find the derivative of each of the functions using the definition: \(\lim \limits_{h \to 0} \frac{f(x+h)−f(x)}{h}\)
\(f(x)=4x^2−7\)
\(f′(x)=8x\)
\(f(x)=x−\frac{1}{2}x^2\)
\(f'(x)=−\frac{1}{(2+x)^2}\)
\(f(x)=\frac{3}{x−1}\)
\(f(x)=−x^3+1\)
\(f′(x)=−3^x2\)
\(f(x)=x^2+x^3\)
\(f(x)=\sqrt{x−1}\)
\(f'(x)=\frac{1}{2\sqrt{x−1}}\)
1 http://www.csun.edu/science/health/d...tv&health.html Source provided.
the slope of the line connecting the two points \((a,f(a))\) and \((a+h,f(a+h))\) on the curve of \(f(x)\); it is given by
\[\text{AROC}=\dfrac{f(a+h)−f(a)}{h}.\]
the slope of a function at a given point; denoted \(f′(a)\),at a point \(x=a\) it is \(f′(a)=\lim \limits_{h \to 0}\frac{f(a+h)−f(a)}{h}\),providing the limit exists.
a function \(f(x)\) for which the derivative exists at \(x=a.\) In other words, if f′(a) f′(a) exists.
the slope of a function at a given point; at \(x=a\) it is given by \(f′(a)=\lim \limits_{h \to 0} \frac{f(a+h)−f(a)}{h}\).
the change in speed or direction at a given instant; a function \(s(t)\) represents the position of an object at time \(t\),and the instantaneous velocity or velocity of the object at time \(t=a\) is given by \(s′(a)=\lim \limits_{h \to 0}\frac{s(a+h)−s(a)}{h}\).
secant line
a line that intersects two points on a curve
tangent line
a line that intersects a curve at a single point
12.4: Continuity
12.E: Introduction to Calculus (Exercises)
Jay Abramson
Differentiability (two variables) | CommonCrawl |
For how many integers $n$ between 1 and 11 (inclusive) is $\frac{n}{12}$ a repeating decimal?
Recall that a simplified fraction has a terminating decimal representation if and only if the denominator is divisible by no primes other than 2 or 5.
The prime factorization of $12$ is $2^2 \cdot 3$. Therefore, $n/12$ terminates if and only if the numerator has a factor of $3$ in it to cancel out the $3$ in the denominator. Since $3$ integers from $1$ to $11$ are divisible by $3$, there are $11-3=\boxed{8}$ integers $n$ for which the fraction is a repeating decimal. | Math Dataset |
Labs septic
In mathematics, the Labs septic surface is a degree-7 (septic) nodal surface with 99 nodes found by Labs (2006). As of 2015, it has the largest known number of nodes of a degree-7 surface, though this number is still less than the best known upper bound of 104 nodes given by Varchenko (1983).[1]
See also
• Barth surface
• Endrass surface
• Sarti surface
• Togliatti surface
References
1. The upper bound in degree 7 given by Giventalʹ (1983) is 106.
• Giventalʹ, A. B. (1983), "The maximum number of singular points on a projective hypersurface", Funktsionalʹnyĭ Analiz i ego Prilozheniya, 17 (3): 73–74, MR 0714227
• Labs, Oliver (2006), "A septic with 99 real nodes", Rend. Semin. Mat. Univ. Padova, 116: 299–313, arXiv:math/0409348, Bibcode:2004math......9348L, MR 2287352
• Varchenko, A. N. (1983), "Semicontinuity of the spectrum and an upper bound for the number of singular points of the projective hypersurface", Doklady Akademii Nauk SSSR, 270 (6): 1294–1297, MR 0712934
External links
• Bothmer, video of Labs septic
| Wikipedia |
\begin{definition}[Definition:Continuous Mapping (Normed Vector Space)]
Let $M_1 = \struct{X_1, \norm {\,\cdot\,}_{X_1} }$ and $M_2 = \struct{X_2, \norm {\,\cdot\,}_{X_2} }$ be normed vector spaces.
Let $f: X_1 \to X_2$ be a mapping from $X_1$ to $X_2$.
Let $a \in X_1$ be a point in $X_1$.
\end{definition} | ProofWiki |
\begin{definition}[Definition:Additive Semiring/Axioms]
An additive semiring is an algebraic structure $\struct {R, *, \circ}$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions:
{{begin-axiom}}
{{axiom | n = \text A 0
| q = \forall a, b \in S
| m = a * b \in S
| rc= Closure under $*$
}}
{{axiom | n = \text A 1
| q = \forall a, b, c \in S
| m = \paren {a * b} * c = a * \paren {b * c}
| rc= Associativity of $*$
}}
{{axiom | n = \text A 2
| q = \forall a, b \in S
| m = a * b = b * a
| rc= Commutativity of $*$
}}
{{axiom | n = \text M 0
| q = \forall a, b \in S
| m = a \circ b \in S
| rc= Closure under $\circ$
}}
{{axiom | n = \text M 1
| q = \forall a, b, c \in S
| m = \paren {a \circ b} \circ c = a \circ \paren {b \circ c}
| rc= Associativity of $\circ$
}}
{{axiom | n = \text D
| q = \forall a, b, c \in S
| m = a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}
| rc= $\circ$ is distributive over $*$
}}
{{axiom | m = \paren {a * b} \circ c = \paren {a \circ c} * \paren {a \circ c}
| rc=
}}
{{end-eqn}}
These criteria are called the '''additive semiring axioms'''.
\end{definition} | ProofWiki |
Method of undetermined coefficients
In mathematics, the method of undetermined coefficients is an approach to finding a particular solution to certain nonhomogeneous ordinary differential equations and recurrence relations. It is closely related to the annihilator method, but instead of using a particular kind of differential operator (the annihilator) in order to find the best possible form of the particular solution, an ansatz or 'guess' is made as to the appropriate form, which is then tested by differentiating the resulting equation. For complex equations, the annihilator method or variation of parameters is less time-consuming to perform.
Differential equations
Scope
Fields
• Natural sciences
• Engineering
• Astronomy
• Physics
• Chemistry
• Biology
• Geology
Applied mathematics
• Continuum mechanics
• Chaos theory
• Dynamical systems
Social sciences
• Economics
• Population dynamics
List of named differential equations
Classification
Types
• Ordinary
• Partial
• Differential-algebraic
• Integro-differential
• Fractional
• Linear
• Non-linear
By variable type
• Dependent and independent variables
• Autonomous
• Coupled / Decoupled
• Exact
• Homogeneous / Nonhomogeneous
Features
• Order
• Operator
• Notation
Relation to processes
• Difference (discrete analogue)
• Stochastic
• Stochastic partial
• Delay
Solution
Existence and uniqueness
• Picard–Lindelöf theorem
• Peano existence theorem
• Carathéodory's existence theorem
• Cauchy–Kowalevski theorem
General topics
• Initial conditions
• Boundary values
• Dirichlet
• Neumann
• Robin
• Cauchy problem
• Wronskian
• Phase portrait
• Lyapunov / Asymptotic / Exponential stability
• Rate of convergence
• Series / Integral solutions
• Numerical integration
• Dirac delta function
Solution methods
• Inspection
• Method of characteristics
• Euler
• Exponential response formula
• Finite difference (Crank–Nicolson)
• Finite element
• Infinite element
• Finite volume
• Galerkin
• Petrov–Galerkin
• Green's function
• Integrating factor
• Integral transforms
• Perturbation theory
• Runge–Kutta
• Separation of variables
• Undetermined coefficients
• Variation of parameters
People
List
• Isaac Newton
• Gottfried Leibniz
• Jacob Bernoulli
• Leonhard Euler
• Józef Maria Hoene-Wroński
• Joseph Fourier
• Augustin-Louis Cauchy
• George Green
• Carl David Tolmé Runge
• Martin Kutta
• Rudolf Lipschitz
• Ernst Lindelöf
• Émile Picard
• Phyllis Nicolson
• John Crank
Undetermined coefficients is not as general a method as variation of parameters, since it only works for differential equations that follow certain forms.[1]
Description of the method
Consider a linear non-homogeneous ordinary differential equation of the form
$\sum _{i=0}^{n}c_{i}y^{(i)}+y^{(n+1)}=g(x)$
where $y^{(i)}$ denotes the i-th derivative of $y$, and $c_{i}$ denotes a function of $x$.
The method of undetermined coefficients provides a straightforward method of obtaining the solution to this ODE when two criteria are met:[2]
1. $c_{i}$ are constants.
2. g(x) is a constant, a polynomial function, exponential function $e^{\alpha x}$, sine or cosine functions $\sin {\beta x}$ or $\cos {\beta x}$, or finite sums and products of these functions (${\alpha }$, ${\beta }$ constants).
The method consists of finding the general homogeneous solution $y_{c}$ for the complementary linear homogeneous differential equation
$\sum _{i=0}^{n}c_{i}y^{(i)}+y^{(n+1)}=0,$
and a particular integral $y_{p}$ of the linear non-homogeneous ordinary differential equation based on $g(x)$. Then the general solution $y$ to the linear non-homogeneous ordinary differential equation would be
$y=y_{c}+y_{p}.$[3]
If $g(x)$ consists of the sum of two functions $h(x)+w(x)$ and we say that $y_{p_{1}}$ is the solution based on $h(x)$ and $y_{p_{2}}$ the solution based on $w(x)$. Then, using a superposition principle, we can say that the particular integral $y_{p}$ is[3]
$y_{p}=y_{p_{1}}+y_{p_{2}}.$
Typical forms of the particular integral
In order to find the particular integral, we need to 'guess' its form, with some coefficients left as variables to be solved for. This takes the form of the first derivative of the complementary function. Below is a table of some typical functions and the solution to guess for them.
Function of xForm for y
$ke^{ax}\!$$Ce^{ax}\!$
$kx^{n},\;n=0,1,2,\ldots \!$
$\sum _{i=0}^{n}K_{i}x^{i}\!$
$k\cos(ax){\text{ or }}k\sin(ax)\!$
$K\cos(ax)+M\sin(ax)\!$
$ke^{ax}\cos(bx){\text{ or }}ke^{ax}\sin(bx)\!$
$e^{ax}(K\cos(bx)+M\sin(bx))\!$
$\left(\sum _{i=0}^{n}k_{i}x^{i}\right)\cos(bx){\text{ or }}\ \left(\sum _{i=0}^{n}k_{i}x^{i}\right)\sin(bx)\!$
$\left(\sum _{i=0}^{n}Q_{i}x^{i}\right)\cos(bx)+\left(\sum _{i=0}^{n}R_{i}x^{i}\right)\sin(bx)$
$\left(\sum _{i=0}^{n}k_{i}x^{i}\right)e^{ax}\cos(bx){\text{ or }}\left(\sum _{i=0}^{n}k_{i}x^{i}\right)e^{ax}\sin(bx)\!$
$e^{ax}\left(\left(\sum _{i=0}^{n}Q_{i}x^{i}\right)\cos(bx)+\left(\sum _{i=0}^{n}R_{i}x^{i}\right)\sin(bx)\right)$
If a term in the above particular integral for y appears in the homogeneous solution, it is necessary to multiply by a sufficiently large power of x in order to make the solution independent. If the function of x is a sum of terms in the above table, the particular integral can be guessed using a sum of the corresponding terms for y.[1]
Examples
Example 1
Find a particular integral of the equation
$y''+y=t\cos t.$
The right side t cos t has the form
$P_{n}e^{\alpha t}\cos {\beta t}$
with n = 2, α = 0, and β = 1.
Since α + iβ = i is a simple root of the characteristic equation
$\lambda ^{2}+1=0$
we should try a particular integral of the form
${\begin{aligned}y_{p}&=t\left[F_{1}(t)e^{\alpha t}\cos {\beta t}+G_{1}(t)e^{\alpha t}\sin {\beta t}\right]\\&=t\left[F_{1}(t)\cos t+G_{1}(t)\sin t\right]\\&=t\left[\left(A_{0}t+A_{1}\right)\cos t+\left(B_{0}t+B_{1}\right)\sin t\right]\\&=\left(A_{0}t^{2}+A_{1}t\right)\cos t+\left(B_{0}t^{2}+B_{1}t\right)\sin t.\end{aligned}}$
Substituting yp into the differential equation, we have the identity
${\begin{aligned}t\cos t&=y_{p}''+y_{p}\\&=\left[\left(A_{0}t^{2}+A_{1}t\right)\cos t+\left(B_{0}t^{2}+B_{1}t\right)\sin t\right]''+\left[\left(A_{0}t^{2}+A_{1}t\right)\cos t+\left(B_{0}t^{2}+B_{1}t\right)\sin t\right]\\&=\left[2A_{0}\cos t+2\left(2A_{0}t+A_{1}\right)(-\sin t)+\left(A_{0}t^{2}+A_{1}t\right)(-\cos t)+2B_{0}\sin t+2\left(2B_{0}t+B_{1}\right)\cos t+\left(B_{0}t^{2}+B_{1}t\right)(-\sin t)\right]\\&\qquad +\left[\left(A_{0}t^{2}+A_{1}t\right)\cos t+\left(B_{0}t^{2}+B_{1}t\right)\sin t\right]\\&=[4B_{0}t+(2A_{0}+2B_{1})]\cos t+[-4A_{0}t+(-2A_{1}+2B_{0})]\sin t.\end{aligned}}$
Comparing both sides, we have
${\begin{cases}1=4B_{0}\\0=2A_{0}+2B_{1}\\0=-4A_{0}\\0=-2A_{1}+2B_{0}\end{cases}}$
which has the solution
$A_{0}=0,\quad A_{1}=B_{0}={\frac {1}{4}},\quad B_{1}=0.$
We then have a particular integral
$y_{p}={\frac {1}{4}}t\cos t+{\frac {1}{4}}t^{2}\sin t.$
Example 2
Consider the following linear nonhomogeneous differential equation:
${\frac {dy}{dx}}=y+e^{x}.$
This is like the first example above, except that the nonhomogeneous part ($e^{x}$) is not linearly independent to the general solution of the homogeneous part ($c_{1}e^{x}$); as a result, we have to multiply our guess by a sufficiently large power of x to make it linearly independent.
Here our guess becomes:
$y_{p}=Axe^{x}.$
By substituting this function and its derivative into the differential equation, one can solve for A:
${\frac {d}{dx}}\left(Axe^{x}\right)=Axe^{x}+e^{x}$
$Axe^{x}+Ae^{x}=Axe^{x}+e^{x}$
$A=1.$
So, the general solution to this differential equation is:
$y=c_{1}e^{x}+xe^{x}.$
Example 3
Find the general solution of the equation:
${\frac {dy}{dt}}=t^{2}-y$
$t^{2}$ is a polynomial of degree 2, so we look for a solution using the same form,
$y_{p}=At^{2}+Bt+C,$
Plugging this particular function into the original equation yields,
$2At+B=t^{2}-(At^{2}+Bt+C),$
$2At+B=(1-A)t^{2}-Bt-C,$
$(A-1)t^{2}+(2A+B)t+(B+C)=0.$
which gives:
$A-1=0,\quad 2A+B=0,\quad B+C=0.$
Solving for constants we get:
$y_{p}=t^{2}-2t+2$
To solve for the general solution,
$y=y_{p}+y_{c}$
where $y_{c}$ is the homogeneous solution $y_{c}=c_{1}e^{-t}$, therefore, the general solution is:
$y=t^{2}-2t+2+c_{1}e^{-t}$
References
1. Ralph P. Grimaldi (2000). "Nonhomogeneous Recurrence Relations". Section 3.3.3 of Handbook of Discrete and Combinatorial Mathematics. Kenneth H. Rosen, ed. CRC Press. ISBN 0-8493-0149-1.
2. Zill, Dennis G., Warren S. Wright (2014). Advanced Engineering Mathematics. Jones and Bartlett. p. 125. ISBN 978-1-4496-7977-4.{{cite book}}: CS1 maint: multiple names: authors list (link)
3. Dennis G. Zill (14 May 2008). A First Course in Differential Equations. Cengage Learning. ISBN 978-0-495-10824-5.
Differential equations
Classification
Operations
• Differential operator
• Notation for differentiation
• Ordinary
• Partial
• Differential-algebraic
• Integro-differential
• Fractional
• Linear
• Non-linear
• Holonomic
Attributes of variables
• Dependent and independent variables
• Homogeneous
• Nonhomogeneous
• Coupled
• Decoupled
• Order
• Degree
• Autonomous
• Exact differential equation
• On jet bundles
Relation to processes
• Difference (discrete analogue)
• Stochastic
• Stochastic partial
• Delay
Solutions
Existence/uniqueness
• Picard–Lindelöf theorem
• Peano existence theorem
• Carathéodory's existence theorem
• Cauchy–Kowalevski theorem
Solution topics
• Wronskian
• Phase portrait
• Phase space
• Lyapunov stability
• Asymptotic stability
• Exponential stability
• Rate of convergence
• Series solutions
• Integral solutions
• Numerical integration
• Dirac delta function
Solution methods
• Inspection
• Substitution
• Separation of variables
• Method of undetermined coefficients
• Variation of parameters
• Integrating factor
• Integral transforms
• Euler method
• Finite difference method
• Crank–Nicolson method
• Runge–Kutta methods
• Finite element method
• Finite volume method
• Galerkin method
• Perturbation theory
Applications
• List of named differential equations
Mathematicians
• Isaac Newton
• Gottfried Wilhelm Leibniz
• Leonhard Euler
• Jacob Bernoulli
• Émile Picard
• Józef Maria Hoene-Wroński
• Ernst Lindelöf
• Rudolf Lipschitz
• Joseph-Louis Lagrange
• Augustin-Louis Cauchy
• John Crank
• Phyllis Nicolson
• Carl David Tolmé Runge
• Martin Kutta
• Sofya Kovalevskaya
• Boyce, W. E.; DiPrima, R. C. (1986). Elementary Differential Equations and Boundary Value Problems (4th ed.). John Wiley & Sons. ISBN 0-471-83824-1.
• Riley, K. F.; Bence, S. J. (2010). Mathematical Methods for Physics and Engineering. Cambridge University Press. ISBN 978-0-521-86153-3.
• Tenenbaum, Morris; Pollard, Harry (1985). Ordinary Differential Equations. Dover. ISBN 978-0-486-64940-5.
• de Oliveira, O. R. B. (2013). "A formula substituting the undetermined coefficients and the annihilator methods". Int. J. Math. Educ. Sci. Technol. 44 (3): 462–468. arXiv:1110.4425. Bibcode:2013IJMES..44..462R. doi:10.1080/0020739X.2012.714496. S2CID 55834468.
| Wikipedia |
\begin{document}
\title[Optimal Hardy--Littlewood type inequalities]{Optimal Hardy--Littlewood type inequalities for polynomials and multilinear operators} \author[Albuquerque]{N. Albuquerque} \address{Departamento de Matem\'{a}tica, \newline \indent Universidade Federal da Para\'{\i}ba, \newline \indent 58.051-900 - Jo\~{a}o Pessoa, Brazil.} \email{[email protected]}
\author[Bayart]{F. Bayart} \address{Laboratoire de Math\'ematiques, \newline\indent Universit\'e Blaise Pascal Campus des C\'ezeaux, \newline\indent
F-63177 Aubiere Cedex, France.} \email{[email protected]}
\author[Pellegrino]{D. Pellegrino} \address{Departamento de Matem\'{a}tica, \newline\indent Universidade Federal da Para\'{\i}ba, \newline\indent 58.051-900 - Jo\~{a}o Pessoa, Brazil.} \email{[email protected] and [email protected]}
\author[Seoane]{J. B. Seoane-Sep\'{u}lveda} \address{Departamento de An\'{a}lisis Matem\'{a}tico,\newline\indent Facultad de Ciencias Matem\'{a}ticas, \newline\indent Plaza de Ciencias 3, \newline\indent Universidad Complutense de Madrid,\newline\indent Madrid, 28040, Spain.\newline \indent \textsc{ and }\newline \indent \noindent Instituto de Ciencias Matem\'aticas -- ICMAT \newline \indent calle Nicol\'as Cabrera 13--15, \newline \indent Madrid, 28049, Spain.} \email{[email protected]}
\subjclass[2010]{46G25, 47H60} \keywords{Absolutely summing operators, multilinear operators, Bohnenblust-Hille inequality} \thanks{D. Pellegrino and J.B. Seoane-Sep\'{u}lveda was supported by CNPq Grant 401735/2013-3 (PVE - Linha 2). N. Albuquerque was supported by CAPES}
\begin{abstract} In this paper we obtain quite general and definitive forms for Hardy--Littlewood type inequalities. Moreover, when restricted to the original particular cases, our approach provides much simpler and straightforward proofs and we are able to show that in most cases the exponents involved are optimal. The technique we used is a combination of probabilistic tools and of an interpolative approach; this former technique is also employed in this paper to improve the constants for vector-valued Bohnenblust--Hille type inequalities. \end{abstract} \maketitle
\section{Introduction}
In 1930 Littlewood \cite{LW} has shown the following result on bilinear forms on $c_{0}\times c_{0}$, now called Littlewood's $4/3$ inequality: for any bounded bilinear form $A:c_{0}\times c_{0} \rightarrow \mathbb{C}$, \[
\left( \sum_{i,j=1}^{+\infty}|A(e_{i},e_{j})|^{\frac{4}{3}}\right) ^{\frac {3}{4}}\leq\sqrt{2}\Vert A\Vert \] and, moreover, the exponent $4/3$ is optimal. From now on, $m\geq1$ is a positive integer, $\mathbf{p}:=\left( p_{1},\dots,p_{m}\right) \in\left[ 1,+\infty\right] ^{m}$ and \[ \left\vert \frac{1}{\mathbf{p}}\right\vert :=\frac{1}{p_{1}}+\dots+\frac {1}{p_{m}}. \] For $1\leq p<+\infty$, let us set $X_{p}:=\ell_{p}$ and let us define $X_{\infty}=c_{0}$. As soon as Littlewood's $4/3$ inequality appeared, it was rapidly extended to more general frameworks. For instance:
\begin{itemize} \item (Bohnenblust and Hille, \cite[Theorem I]{BH}, 1931 (see also \cite{dddsss})) There exists a constant $C=C(m)\geq1$ such that \begin{equation} \left( \sum\limits_{i_{1},...,i_{m}=1}^{+\infty}\left\vert A(e_{i_{^{1}} },...,e_{i_{m}})\right\vert ^{\frac{2m}{m+1}}\right) ^{\frac{m+1}{2m}}\leq C\left\Vert A\right\Vert \label{juui} \end{equation} for all continuous $m$-linear forms $A:c_{0}\times\cdots\times c_{0} \rightarrow\mathbb{C}$ and the exponent $\frac{2m}{m+1}$ is optimal.
\item (Hardy and Littlewood, \cite{hardy.lw}, 1934 (see also \cite[page 224]{hardy})/Praciano-Pereira, \cite[Theorems A and B]{praciano}, 1981) Let $\mathbf{p}\in\left[ 1,+\infty\right] ^{m}$ with \[ \left\vert {\frac{1}{\mathbf{p}}}\right\vert \leq\frac{1}{2}, \] then there exists a constant $C>0$ such that, for every continuous $m$-linear form $A:X_{p_{1}}\times\dots\times X_{p_{m}}\rightarrow\mathbb{C}$, \begin{equation}
\left( \sum_{i_{1},\dots,i_{m}=1}^{+\infty}|A(e_{i_{1}},\dots,e_{i_{m}
})|^{\frac{2m}{m+1-2\left\vert {\frac{1}{\mathbf{p}}}\right\vert }}\right) ^{\frac{m+1-2\left\vert {\frac{1}{\mathbf{p}}}\right\vert }{2m}}\leq C\Vert A\Vert.\label{juui2} \end{equation}
\item (Defant and Sevilla-Peris, \cite[Theorem 1]{df}, 2009) If $1\leq s\leq q\leq2,$ there exists a constant $C>0$ such that, for every continuous $m$-linear mapping $A:c_{0}\times \dots\times c_{0}\rightarrow\ell_{s}$, then \[ \left( \sum_{i_{1},\dots,i_{m}=1}^{+\infty}\Vert A(e_{i_{1}},\dots,e_{i_{m} })\Vert_{\ell_{q}}^{\frac{2m}{m+2\left( \frac{1}{s}-\frac{1}{q}\right) }}\right) ^{\frac{m+2\left( \frac{1}{s}-\frac{1}{q}\right) }{2m}}\leq C\Vert A\Vert. \]
\end{itemize}
Very recently the previous results were generalized by the authors and by Dimant and Sevilla-Peris:
\begin{itemize} \item (\cite[Corollary 1.3]{abps}, 2013) Let $1\leq s\leq q\leq2$ and $\mathbf{p}\in\left[ 1,+\infty\right] ^{m}$ such that \begin{equation} \frac{1}{s}-\frac{1}{q}-\left\vert{\frac{1}{\mathbf{p}}}\right\vert \geq0.\label{assump} \end{equation} Then there exists a constant $C>0$ such that, for every continuous $m$-linear mapping $A:X_{p_1} \times\dots\times X_{p_m} \rightarrow {X_{s}}$, we have \[ \left( \sum_{i_{1},\dots,i_{m}=1}^{+\infty}\Vert A(e_{i_{1}},\dots,e_{i_{m} })\Vert_{\ell_{q}}^{\frac{2m}{m+2\left( \frac{1}{s}-\frac{1}{q}-\left\vert {\frac {1}{\mathbf{p}}}\right\vert \right) }}\right) ^{\frac{m+2\left( \frac{1} {s}-\frac{1}{q}-\left\vert {\frac{1}{\mathbf{p}}}\right\vert \right) }{2m} }\leq C\Vert A\Vert \] and the exponent is optimal. \end{itemize}
\begin{itemize} \item (Dimant and Sevilla-Peris, \cite[Proposition 4.4]{dsp}, 2013) Let $\mathbf{p}\in\left[ 1,+\infty\right] ^{m}$ and $s, q \in [1,+\infty]$ be such that $s\leq q$. Then there exists a constant $C>0$ such that, for every continuous $m$-linear mapping $A : X_{p_1} \times\cdots\times X_{p_m} \rightarrow X_s$, we have \[ \left( \sum_{i_1,\dots,i_m=1}^{+\infty}
\left\Vert A \left( e_{i_1}, \ldots, e_{i_m}\right) \right\Vert_{\ell_q}^\rho \right)^\frac1\rho \leq C\Vert A\Vert, \] where $\rho$ is given by
\begin{itemize} \item[(i)] If $s \leq q \leq2$, and
\begin{itemize} \item[(a)] if $0 \leq \left|\frac1{\mathbf p}\right| <\frac1s -\frac1q$, then $\frac{1}{\rho} = \frac12 + \frac1m \left( \frac1s - \frac1q - \left|\frac1{\mathbf p}\right| \right)$.
\item[(b)] if $\frac1s - \frac1q \leq \left|\frac1{\mathbf p}\right| < \frac12 + \frac1s - \frac1q$, then $\frac1\rho = \frac12 + \frac1s - \frac1q - \left|\frac1{\mathbf p}\right|$. \end{itemize}
\item[(ii)] If $s \leq2 \leq q$, and
\begin{itemize} \item[(a)] if $0 \leq \left|\frac1{\mathbf p}\right| < \frac1s - \frac12$, then $\frac1\rho = \frac12 + \frac1m \left( \frac1s - \frac12 - \left|\frac1{\mathbf p}\right|\right)$.
\item[(b)] if $\frac1s - \frac12 \leq \left|\frac1{\mathbf p}\right| < \frac1s$, then $\frac1\rho = \frac1s - \left|\frac1{\mathbf p}\right|$. \end{itemize}
\item[(iii)] If $2 \leq s \leq q$ and $0 \leq \left|\frac1{\mathbf p}\right| < \frac1s$, then $\frac1\rho = \frac1s - \left|\frac1{\mathbf p}\right|$. \end{itemize}
Moreover, the exponents in the cases (ia),(iib) and (iii) are optimal. Also, the exponent in (ib) is optimal for $\frac1s-\frac1q\leq\ \left|\frac1{\mathbf p}\right| <\frac12$. \end{itemize}
Our main intention, in this paper, is to improve the previous theorems in three directions. \begin{enumerate} \item We study in depth the remaining cases of the Dimant and Sevilla-Peris result. Surprisingly, we show that in case (iia), the exponent given above is optimal whereas it is not optimal in case (ib) when $\left|\frac1{\mathbf p}\right|>\frac 12$. We give a better exponent in that case and show a necessary condition on it. These two bounds coincide when $s=1$. We can summarize this into the two following statements.
\begin{theorem} \label{nov1} Let $\mathbf{p}\in\left[ 1,+\infty\right] ^{m}$ and let $\rho>0$. Assume moreover that either $q\geq 2$ or $q<2$ and $\left|\frac1{\mathbf p}\right|< \frac 12$. Let
\[ \frac{1}{\lambda} := \frac{1}{2}+\frac{1}{s}-\frac{1}{\min\{q,2\}} - \left\vert{\frac{1}{\mathbf{p}}}\right\vert > 0. \]
Then there exists $C>0$ such that, for every continuous $m$-linear operator $A:X_{p_{1}}\times\dots\times X_{p_{m}}\rightarrow X_{s}$, we have \[ \left( \sum_{i_{1},\dots,i_{m}=1}^{+\infty}\Vert A(e_{i_{1}},\dots,e_{i_{m} })\Vert_{\ell_{q}}^{\rho}\right) ^{\frac{1}{\rho}}\leq C\Vert A\Vert \] if and only if \[ \frac{m}{\rho}\leq\frac{1}{\lambda}+\frac{m-1}{\max\{\lambda,s,2\}}. \]
\end{theorem}
The following table summarizes the optimal value of $\frac{1}{\rho}$ following the respective values of $s,q,p_{1},...,p_{m}$: \[ \begin{array}
[c]{c|c} 1\leq s\leq q\leq2,\ \lambda<2 & \displaystyle\frac{1}{2}+\frac{1}{ms} -\frac{1}{mq}-\frac{1}{m}\times\left\vert {\frac{1}{\mathbf{p}}}\right\vert \\[3mm]\hline & \\ 1\leq s\leq q\leq2,\ \lambda\geq2,\ \left|\frac1{\mathbf p}\right| < \frac12 & \displaystyle\frac{1}{2}+\frac{1}{s} -\frac{1}{q}-\left\vert {\frac{1}{\mathbf{p}}}\right\vert \\[3mm]\hline & \\ 1\leq s\leq2\leq q,\ \lambda<2 & \displaystyle\frac{m-1}{2m}+\frac{1} {ms}-\frac{1}{m}\times\left\vert {\frac{1}{\mathbf{p}}}\right\vert \\[3mm]\hline & \\ 1\leq s\leq2\leq q,\ \lambda\geq2 & \displaystyle\frac{1}{s}-\left\vert {\frac{1}{\mathbf{p}}}\right\vert \\[3mm]\hline & \\ 2\leq s\leq q & \displaystyle\frac{1}{s}-\left\vert {\frac{1}{\mathbf{p}} }\right\vert \\ & \end{array} \]
We note that (\ref{juui}) and (\ref{juui2}) are recovered by Theorem \ref{nov1} just by choosing $s=1$ and $q=2$.
When $q<2$ and $\left|\frac1{\mathbf p}\right|>\frac 12$ (observe that this automatically implies $\lambda\geq 2$), the situation is more difficult and we get the following statement. \begin{theorem}\label{nov2} Let $\mathbf{p}\in\left[ 1,+\infty\right] ^{m}$, $\left|\frac1{\mathbf p}\right|>\frac 12$, $1\leq s\leq q\leq 2$ and let $\rho>0$. Let us consider the following property. \begin{quote} There exists $C>0$ such that, for every continuous $m$-linear operator $A:X_{p_{1}}\times\dots\times X_{p_{m}}\rightarrow X_{s}$, we have \[ \left( \sum_{i_{1},\dots,i_{m}=1}^{+\infty}\Vert A(e_{i_{1}},\dots,e_{i_{m} })\Vert_{\ell_{q}}^{\rho}\right) ^{\frac{1}{\rho}}\leq C\Vert A\Vert. \] \end{quote} \begin{itemize} \item[(A)] The property is satisfied as soon as $$\frac 1\rho\leq \frac{\left(\frac 1s-\frac 1q\right)\left(\frac 1s-\left|\frac1{\mathbf p}\right|\right)}{\frac 12-\frac 1s}.$$ \item[(B)] If the property is satisfied, then $$\frac 1\rho\leq 2\left(1-\left|\frac1{\mathbf p}\right|\right)\left(\frac 1s-\frac 1q\right).$$ \end{itemize} \end{theorem} In particular, if $s=1$, then the property is satisfied if and only if $$\frac 1\rho\leq 2\left(1-\left|\frac1{\mathbf p}\right|\right)\left(1-\frac1q\right).$$
\item We give a simpler proof of the sufficient part of the Dimant and Sevilla-Peris theorem. It turns out that it is easier to prove a more general result.
\begin{theorem} \label{main2}Let $\mathbf{p}\in\left[ 1,+\infty\right] ^{m}$ and $1\leq s\leq q\leq\infty$ be such that \[ \left\vert \frac{1}{\mathbf{p}}\right\vert <\frac{1}{2}+\frac{1}{s}-\frac {1}{\min\{q,2\}}. \]
Let
\[ \frac{1}{\lambda}:=\frac{1}{2}+\frac{1}{s}-\frac{1}{\min\{q,2\}}-\left\vert \frac{1}{\mathbf{p}}\right\vert. \]
If $\lambda>0$ and $t_{1},\ldots,t_{m}\in\left[ \lambda,\max\left\{ \lambda,s,2\right\} \right] $ are such that \begin{equation} \frac{1}{t_{1}}+\cdots+\frac{1}{t_{m}}\leq\frac{1}{\lambda}+\frac{m-1} {\max\{\lambda,s,2\}},\label{dez11} \end{equation}
then there exists $C>0$ satisfying, for every continuous $m$-linear map $A:X_{p_{1}}\times\cdots\times X_{p_{m}}\rightarrow X_{s}$, \begin{equation} \left( \sum_{i_{1}=1}^{+\infty}\left( \ldots\left( \sum_{i_{m}=1}^{+\infty }\left\Vert A\left( e_{i_{1}},\ldots,e_{i_{m}}\right) \right\Vert _{\ell_{q}}^{t_{m}}\right) ^{\frac{t_{m-1}}{t_{m}}}\ldots\right) ^{\frac{t_{1} }{t_{2}}}\right) ^{\frac{1}{t_{1}}}\leq C\Vert A\Vert. \label{q3} \end{equation} Moreover, the exponents are optimal except eventually if $q\leq 2$ and $\left|\frac1{\mathbf p}\right|>\frac 12$. \end{theorem}
\begin{remark}
The optimality in the above theorem shall be understood in a strong sense: when $\lambda<2,$ we prove that if $t_{1},\ldots,t_{m}\in\left[ 1,+\infty\right) $ are so that (\ref{q3}) holds then (\ref{dez11}) is valid. When $\lambda\geq2$, note that $\lambda=\max\left\{ \lambda ,s,2\right\} $ and we prove that if $t=t_{1}=\cdots=t_{m}$ are in $\left[ 1,+\infty\right)$ and (\ref{q3}) is valid, then we have (\ref{dez11}) and, as a direct consequence, $t\geq\lambda$. \end{remark}
\item We prove similar results for $m$-linear mappings with arbitrary codomains which assume their cotype. For a Banach space $X$, let $q_X=\inf\{q\geq 2;\ X$ has cotype $q\}.$
The proof that $(B)$ implies $(A)$ in the theorem below appears in \cite[Proposition 4.3]{dsp}. \begin{theorem}\label{000aaa} Let $\mathbf{p\in}\left[ 2,{+\infty}\right] ^{m}$, let $X$ be an infinite dimensional Banach space with cotype $q_{X}$, $\left\vert \frac{1}{\mathbf{p}}\right\vert <\frac{1}{q_{X}}$, and let $\rho>0.$ The following assertions are equivalent:
(A) Every bounded $m$-linear operator $A:X_{p_{1}}\times\dots\times X_{p_{m} }\rightarrow X$ is such that \[
\sum\limits_{i_{1},...,i_{m}=1}^{{+\infty}}\left\| A(e_{i_{^{1}}}
,...,e_{i_{m}})\right\| ^{\rho}<{+\infty}. \]
(B) $\frac{1}{\rho}\leq\frac{1}{q_{X}}-\left\vert \frac{1}{\mathbf{p} }\right\vert .$ \end{theorem}
\end{enumerate}
Finally, in the last section of the paper we obtain better estimates for the constants of vector-valued Bohnenblust--Hille inequalities.
We conclude this introduction by noting that our theorems can be naturally stated in the context of homogeneous polynomials. Given an $m$-homogeneous polynomial $P:X\to Y$, we denote its coefficients $(c_\alpha(P))$. In \cite[Lemma 5]{df}, it is shown that an inequality
$$\left(\sum_{\alpha}\|c_\alpha(P)\|^\rho\right)^{\frac 1\rho}\leq C\|P\|$$ holds for every $m$-homogeneous polynomial $P:X\to Y$ if and only if a similar inequality
$$\left(\sum_{i_1,\dots,i_m}\|T(e_{i_1},\dots,e_{i_m})\|^\rho\right)^{\frac 1\rho}\leq C'\|T\|$$ is satisfied for every $m$-linear mapping $A:X\times\dots\times X\to Y$, where $X$ is a Banach sequence space.
\noindent\textsc{Notations.} For two positive integers $n,k$, we set
\begin{eqnarray*} \mkn{k}&:=&\big\{\mathbf i=(i_1,\dots,i_k); \ i_1,\dots,i_k\in\{1,\dots,n\}\big\}.\\ \end{eqnarray*} For $q\in [1,+\infty]$, $q^*$ will denote its conjugate exponent.
\section{Proof of Theorem \ref{main2} (sufficiency)\label{88}}
Let $1\leq q\leq{+\infty}$. We recall that a Banach space $X$ has \emph{cotype} $q$ if there is a constant $\kappa>0$ such that, no matter how we select finitely many vectors $x_{1},...,x_{n}\in X$, \[ \left( \sum_{k=1}^{n}\Vert x_{k}\Vert^{q}\right) ^{\frac{1}{q}}\leq \kappa\left( \int_{I}\left\Vert \sum_{k=1}^{n}r_{k}(t)x_{k}\right\Vert ^{2}dt\right) ^{\frac{1}{2}} \] where $I=[0,1]$ and $r_{k}$ denotes the $k$-th Rademacher function. To cover the case $q={+\infty}$, the left hand side should be replaced by $\max_{1\leq k\leq n}\Vert x_{k}\Vert$. The smallest of all these constants is denoted by $C_{q}(X)$ and named the cotype $q$ constant of $X$.
An operator between Banach spaces $v:X\to Y$ is $(r,s)$-summing (with $s\leq r\leq{+\infty}$) if there exists $C>0$ such that, for all $n\geq1$ and for all vectors $x_{1},\dots,x_{n}\in X$, \[
\left( \sum_{k=1}^{n} \|vx_{k}\|^{r}\right) ^{\frac1r}\leq C\sup_{x^{*}\in B_{X^{*}}}\left( \sum_{k=1}^{n}|x^{*}(x_{k})|^{s}\right) ^{\frac1s}. \] The smallest constant in this inequality is denoted by $\pi_{r,s}(v)$.
We need a cotype $q$ version of \cite[Proposition 4.1]{abps}, whose proof can be found in \cite[Proposition 3.1]{dsp}:
\begin{lemma} \label{cot.q} Let $X$ be a Banach space, let $Y$ be a cotype $q$ space, let $r\in[1,q]$ and let $\mathbf p\in [1,+\infty]^m$ with \[ \left\vert {\frac{1}{\mathbf{p}}}\right\vert <\frac{1}{r}-\frac{1}{q}. \] Define \[ \frac{1}{\lambda}:=\frac{1}{r}-\left\vert {\frac{1}{\mathbf{p}}}\right\vert . \] Then, for every continuous $m$-linear map $A:X_{p_{1}}\times\cdots\times X_{p_{m}}\rightarrow X$ and every $\left( r,1\right) $-summing operator $v:X\rightarrow Y$, we have \begin{equation} \left( \sum_{i_{k}}\left( \sum_{\widehat{i_{k}}}\Vert vA(e_{i_{1}} ,\cdots,e_{i_{m}})\Vert_{Y}^{q}\right) ^{\lambda/q}\right) ^{1/\lambda} \leq\left( \sqrt{2}C_{q}(Y)\right) ^{m-1}\pi_{r,1}(v)\Vert A\Vert\label{q2} \end{equation} for all $k=1,...,m.$ \end{lemma}
The symbol $\sum_{\widehat{i_{k}}}$ means that we are fixing the $k$-th index and that we are summing over all the remaining indices.
We shall deduce from this lemma the following theorem, which extends results of \cite{abps} and \cite{dsp}:
\begin{theorem} \label{thmain} Let $\mathbf{p}\in\left[ 1,+\infty\right] ^{m}$, $X$ be a Banach space, $Y$ be a cotype $q$ space and $1\leq r\leq q$, with $\left\vert \frac{1}{\mathbf{p}}\right\vert <\frac{1}{r}$. Define \[ \frac{1}{\lambda}:=\frac{1}{r}-\left\vert \frac{1}{\mathbf{p}}\right\vert . \] If $t_{1},\ldots,t_{m}\in\left[ \lambda,\max\left\{ \lambda,q\right\} \right] $ are such that \[ \frac{1}{t_{1}}+\cdots+\frac{1}{t_{m}}\leq\frac{1}{\lambda}+\frac{m-1} {\max\{\lambda,q\}}, \] then, for every continuous $m$-linear map $A:X_{p_{1}}\times\cdots\times X_{p_{m}}\rightarrow X$ and every $\left( r,1\right) $-summing operator $v:X\rightarrow Y$, we have {\small \begin{equation} \left( \sum_{i_{1}=1}^{{+\infty}}\left( \ldots\left( \sum_{i_{m}=1}^{{+\infty} }\left\Vert vA\left( e_{i_{1}},\ldots,e_{i_{m}}\right) \right\Vert _{Y}^{t_{m}}\right) ^{\frac{t_{m-1}}{t_{m}}}\ldots\right) ^{\frac{t_{1} }{t_{2}}}\right) ^{\frac{1}{t_{1}}}\leq\left( \sqrt{2}C_{\max\left\{ \lambda,q\right\} }(Y)\right) ^{m-1}\pi_{r,1}(v)\left\Vert A\right\Vert . \label{q1} \end{equation} } \end{theorem}
\begin{proof} If $\lambda<q$, from Lemma \ref{cot.q}, we have (\ref{q1}) for \[ \left( t_{1},...,t_{m}\right) =\left( \lambda,q,...,q\right) . \] Since $\lambda<q$, the mixed $\left( \ell_{\lambda},\ell_{q}\right) -$ norm inequality (see \cite[Proposition 3.1]{abps}), we also have (\ref{q1}) for the exponents \[ \left( t_{1},...,t_{m}\right) =\left( q,...,q,\lambda,q,...,q\right) \] with $\lambda$ in the $k$-th position, for all $k=1,...,m$. Now, using a general version of H\"older's inequality (see \cite[Theorem 2.1]{fou}), or interpolating \[ \left( \lambda,q,...,q\right) ,\left( q,\lambda,q,...,q\right) ,\ldots,\left( q,...,q,\lambda\right) \] in the sense of \cite{abps}, we get (\ref{q1}) for all $\left( t_{1} ,...,t_{m}\right) \in[\lambda,q]^{m} $ such that \[ \frac{1}{t_{1}}+\cdots+\frac{1}{t_{m}}=\frac{1}{\lambda}+\frac{m-1}{q} =\frac{1}{\lambda}+\frac{m-1}{\max\{\lambda,q\}}. \] If $\lambda\geq q$, for any $\varepsilon>0$, let $q_{\varepsilon} =\lambda+\varepsilon$. So $\lambda<q_{\varepsilon}$ and this automatically implies that $$\left|\frac1{\mathbf p}\right|<\frac 1r-\frac1{q_{\varepsilon}}.$$ Since $Y$ has cotype $q_{\varepsilon}>q$, we may apply Lemma \ref{cot.q} to get \[ \left( \sum_{i_{1}=1}^{N}\left( \sum_{i_{2},...,i_{m}=1}^{N}\left\Vert vA\left( e_{i_{1}},\ldots,e_{i_{m}}\right) \right\Vert ^{\lambda +\varepsilon}\right) ^{\frac{\lambda}{^{\lambda+\varepsilon}}}\right) ^{\frac{1}{\lambda}}\leq\left( \sqrt{2}C_{\lambda+\varepsilon}(Y)\right) ^{m-1}\pi_{r,1}(v)\Vert A\Vert \] for all positive integer $N$. Making $\varepsilon\rightarrow0$, we get \[ \left( \sum_{i_{1},...,i_{m}=1}^{N}\left\Vert vA\left( e_{i_{1}} ,\ldots,e_{i_{m}}\right) \right\Vert ^{\lambda}\right) ^{\frac{1}{\lambda} }\leq\left( \sqrt{2}C_{\lambda}(Y)\right) ^{m-1}\pi_{r,1}(v)\Vert A\Vert, \] for all $N$ and the proof is done. \end{proof}
\begin{remark} If we take $t_{1}=\dots=t_{m}$, then, upon polarization, we recover exactly \cite[Theorem 1.2]{dsp} with a much simpler proof due to the fact that the inequality is simpler to prove for the extremal values of $(t_{1},\dots ,t_{m})$. \end{remark}
We are now ready for the proof of the sufficient part of Theorem \ref{main2}. We split the proof into three cases, and we combine Theorem \ref{thmain} with the Bennett-Carl inequalities (\cite{ben, carl}): for $1\leq s\leq q\leq {+\infty}$, the inclusion map $\ell_{s}\hookrightarrow\ell_{q}$ is $\left( r,1\right) $-summing, where the optimal $r$ is given by \[ \frac{1}{r}:=\frac{1}{2}+\frac{1}{s}-\frac{1}{\min\{2,q\}}. \]
\noindent\emph{(i) $s\leq q\leq2$:} The Bennet-Carl-inequalities ensure that the inclusion map $\ell_{s} \hookrightarrow\ell_{q}$ is $\left( r,1\right) $-summing with $\frac{1}{r} = \frac{1}{2} + \frac{1}{s} - \frac{1}{q}$, so the results follow from Theorem \ref{thmain}, with $t_{1},\ldots,t_{m}$ satisfying \[ \frac{1}{t_{1}}+\cdots+\frac{1}{t_{m}}=\frac{1}{2}+\frac{1}{s}-\frac{1} {q}-\left\vert \frac{1}{\mathbf{p}}\right\vert +\frac{m-1}{\max\{\lambda ,2\}}. \]
\noindent\emph{(ii) $s\leq2\leq q$:} Also by using Bennet-Carl inequalities, $\ell_{s}\hookrightarrow\ell_{2}$ is $\left( s,1\right) $-summing, thus we get (\ref{q3}) applying Theorem \ref{thmain}, with $t_{1},\ldots,t_{m}$ satisfying \[ \frac{1}{t_{1}}+\cdots+\frac{1}{t_{m}}=\frac{1}{s}-\left\vert \frac {1}{\mathbf{p}}\right\vert +\frac{m-1}{\max\{\lambda,2\}}. \]
\noindent\emph{(iii) $2\leq s\leq q$:} Since $\ell_{s}\hookrightarrow\ell_{s}$ is $\left( s,1\right) $-summing, the result follows from Theorem \ref{thmain}, with $t_{1}=\dots=t_{m}=\lambda$ and $\lambda\geq s$, since $r=s$ and \[ \frac{1}{\lambda}:=\frac{1}{s}-\left\vert \frac{1}{\mathbf{p}}\right\vert \leq\frac{1}{s}. \]
\begin{remark}
Let us set \[ c_{qs}:=\left\{ \begin{array} [c]{ll} q\,, & \mbox{ if }s\leq q\leq2\,,\\ 2\,, & \mbox{ if }s\leq2\leq q\,,\\ s\,, & \mbox{ if }2\leq s\leq q. \end{array} \right. \]
With the above notations, a careful look at the proof shows that the constant $C$ which appears in Theorem \ref{main2} is dominated by \[ \left( \sqrt{2}C_{\max\left\{ \lambda,s,2\right\} }(\ell_{c_{qs}})\right) ^{m-1}\pi_{r,1}(\ell_{s}\hookrightarrow\ell_{c_{qs}}) \]
\end{remark}
\section{Proof of Theorem \ref{main2} (optimality) \label{99}} In this section we show that the exponents in Theorem \ref{main2} are optimal except when $q\leq 2$ and $\left|\frac1{\mathbf p}\right|>\frac 12$. More precisely, if $(t_1,\dots,t_m)\in [1,+\infty)^m$ are such that there exists $C>1$ satisfying, for any continuous multilinear map $X_{p_1}\times\dots\times X_{p_m}\to X_{s}$, \begin{equation} \left( \sum_{i_{1}=1}^{{+\infty}}\left( \ldots\left( \sum_{i_{m}=1}^{{+\infty} }\left\Vert A\left( e_{i_{1}},\ldots,e_{i_{m}}\right) \right\Vert _{\ell_{q}}^{t_{m}}\right) ^{\frac{t_{m-1}}{t_{m}}}\ldots\right) ^{\frac{t_{1} }{t_{2}}}\right) ^{\frac{1}{t_{1}}}\leq C \Vert A\Vert, \label{q4} \end{equation} then we prove that (\ref{dez11}) holds. When $\lambda\geq 2$, we will always assume that $t_1=\dots=t_m=t$, since $\lambda=\max\left\{ \lambda ,s,2\right\} $ and our inequality holds true when all the exponents are equal. We split the proof into several cases. Most of the cases are a consequence of a random construction. The main tool is the following lemma, from \cite[Lemma 6.2]{abps}. \begin{lemma}\label{LEMPROBA} Let $d,n\geq 1$, $q_1,\dots,q_{d+1}\in [1,+\infty]^{d+1}$ and let, for $q\geq 1$, $$\alpha(q)=\left\{ \begin{array}{ll} \frac12-\frac 1q&\textrm{if }q\geq 2\\ 0&\textrm{otherwise.} \end{array}\right.$$ Then there exists a $d$-linear mapping $A:\ell_{p_1}^n\times\dots\times\ell_{p_d}^n\to \ell_{p_{d+1}}^n$ which may be written $$A\big(x^{(1)},\dots,x^{(d)}\big)=\sum_{i_1,\dots,i_{d+1}=1}^n \pm x_{i_1}^{(1)}\cdots x_{i_d}^{(d)}e_{i_{d+1}}$$ such that
$$\|A\|\leq C_d n^{\frac 12+\alpha(p_1)+\dots+\alpha(p_d)+\alpha(p_{d+1}^*)}.$$ \end{lemma}
\subsection{Case 1: $1\leq s\leq q\leq 2$ and $\lambda<2$} This case has already been solved in \cite[Section 6.2]{abps}, using Lemma \ref{LEMPROBA} with $d=m$ and $(q_1,\dots,q_{m+1})=(p_1,\dots,p_m,s)$.
\subsection{Case 2: $1\leq s\leq q\leq 2$, $\lambda\geq 2$ and $\left|\frac1{\mathbf p}\right|\leq \frac 12$} This case has already been solved in \cite[Proposition 4.4(ib)]{dsp} using a Fourier matrix. We shall give an alternative probabilistic proof. Let $p\in [2,+\infty]$ be such that $\frac 1p=\left|\frac1{\mathbf p}\right|$. By Lemma \ref{LEMPROBA}, there exists a linear map $T:\ell_p^n\to\ell_s^n$ which may be written $T(x)=\sum_{i,j}\varepsilon_{i,j}x_ie_j$ with $\varepsilon_{i,j}=\pm 1$ and such that
$$\|T\|\leq Cn^{\frac 12+\frac 12-\frac 1p+\frac 12-\frac 1{s^*}}=Cn^{\frac 12+\frac 1s-\left|\frac1{\mathbf p}\right|}.$$ Let $A:\ell_{p_1}^n\times\cdots\times \ell_{p_m}^n\to\ell_{s}^n$ defined by $$A\big(x^{(1)},\dots,x^{(m)}\big):=\sum_{i,j}\varepsilon_{i,j}x_i^{(1)}\cdots x_i^{(m)}e_j.$$
By H\"older's inequality, it is plain that $\|A\|\leq \|T\| \leq Cn^{\frac 12+\frac 1s-\left|\frac1{\mathbf p}\right|}$. On the other hand, since $A(e_{i_{1}},\dots,e_{i_{m}})\neq 0$ if and only if $i_{1}=\ldots=i_{m}$, and
$$\|A(e_i,\dots,e_i)\|_{\ell_{q}}=n^{1/q},$$ we have $$\left( \sum_{\mathbf i\in\mkn{m}}\left\Vert A\left( e_{i_{1}},\ldots,e_{i_{m}}\right) \right\Vert _{\ell_{q}}^{t}\right) ^{\frac{1}{t}}=n^{\frac 1q+\frac1{t}}.$$ This clearly implies
$$\frac 1t\leq \frac 12+\frac 1s-\frac 1q-\left|\frac1{\mathbf p}\right|.$$
\subsection{Case 3: $1\leq s\leq 2\leq q$ and $\lambda<2$} Let $p\in [0,+\infty]$ be defined by $$\frac 1p=\frac 1{p_m}+\frac 1{s^*}.$$ Since $\lambda<2$, it is easy to check that $p\geq 2$ and that $p_i\geq 2$ for any $i=1,\dots,m$. We then apply Lemma \ref{LEMPROBA} with $d=m-1$ and $(q_1,\dots,q_m)=(p_1,\dots,p_{m-1},p^*)$. We get an $(m-1)$-linear form $T:\ell_{p_1}^n\times\cdots\times \ell_{p_{m-1}}^n\to\ell_{p^*}^n$ which can be written $$T\big(x^{(1)},\dots,x^{(m-1)}\big)=\sum_{i_1,\dots,i_m}\varepsilon_{i_1,\dots,i_m}x_{i_1}^{(1)}\cdots x_{i_{m-1}}^{(m-1)}e_{i_m}$$ and such that
$$\|T\|\leq C n^{\frac 12+\frac m2-\left|\frac1{\mathbf p}\right|-\frac 1{s^*}}=Cn^{\frac{m-1}2-\left|\frac1{\mathbf p}\right|+\frac 1s}.$$ We then define $A:\ell_{p_1}^n\times\cdots\times \ell_{p_m}^n\to\ell_{s}^n$ by $$A\big(x^{(1)},\dots,x^{(m)}\big)=\sum_{i_1,\dots,i_m}\varepsilon_{i_1,\dots,i_m}x^{(1)}_{i_1}\cdots x_{i_m}^{(m)} e_{i_m}.$$ Then, for any $x^{(1)},\dots,x^{(m)}\in B_{\ell_{p_1}^n}\times\dots\times B_{\ell_{p_m}^n}$, \begin{eqnarray*}
\|A\big(x^{(1)},\dots,x^{(m)}\big)\|&=&\sup_{y\in B_{\ell^n_{s^*}}}\left |\sum_{i_1,\dots,i_m}\varepsilon_{i_1,\dots,i_m}x_{i_1}^{(1)}\cdots x_{i_m}^{(m)}y_{i_m}\right|\\
&\leq&\sup_{z\in B_{\ell^n_{p}}}\left |\sum_{i_1,\dots,i_m}\varepsilon_{i_1,\dots,i_m}x_{i_1}^{(1)}\cdots x_{i_{m-1}}^{(m-1)}z_{i_m}\right|\\
&\leq&\|T\|. \end{eqnarray*}
Moreover, given any $\mathbf i\in\mkn{m}$, $\|A(e_{i_1},\dots,e_{i_m})\|_q=\|e_{i_m}\|_q=1$, so that $$\left( \sum_{i_{1}=1}^{{+\infty}}\left( \ldots\left( \sum_{i_{m}=1}^{{+\infty} }\left\Vert A\left( e_{i_{1}},\ldots,e_{i_{m}}\right) \right\Vert _{\ell_{q}}^{t_{m}}\right) ^{\frac{t_{m-1}}{t_{m}}}\ldots\right) ^{\frac{t_{1} }{t_{2}}}\right) ^{\frac{1}{t_{1}}}=n^{\frac {1}{t_1}+\dots+\frac{1}{t_m}}.$$ Hence, provided (\ref{q4}) is satisfied, $(t_1,\dots,t_m)$ has to satisfy
$$\frac 1{t_1}+\dots+\frac 1{t_m}\leq \frac{m-1}2+\frac 1s-\left|\frac1{\mathbf p}\right|.$$
\subsection{Case 4 and Case 5: $1\leq s\leq2\leq q$ and $\lambda\geq2$, $2\leq s\leq q$} These cases have a deterministic proof, as noted in \cite[Proposition 4.4 (iib), (iii)]{dsp}, considering $A:\ell_{p_{1}}^n\times\cdots\times \ell_{p_{m}}^n \rightarrow \ell^n_{s}$ given by \[ A \big(x^{(1)},\dots,x^{(m)}\big):=\sum_{i=1}^{n}x_{i}^{(1)}\cdots x_{i}^{(m)}e_{i}. \]
\subsection{ The proof of Theorem \ref{nov1}}
From Theorem 1.3, by choosing $t_{1}=\ldots=t_{m}$ we conclude that provided \[ \left\vert \frac{1}{\mathbf{p}}\right\vert <\frac{1}{2}+\frac{1}{s}-\frac {1}{\min\{q,2\}}, \] the best exponent $\rho$ in Theorem \ref{nov1} satisfies \[ \frac{m}{\rho}=\frac{1}{\lambda}+\frac{m-1}{\max\{\lambda,s,2{\}}}. \] To conclude the proof, it remains to prove that, whenever \[ \left\vert \frac{1}{\mathbf{p}}\right\vert \geq\frac{1}{2}+\frac{1}{s} -\frac{1}{\min\{q,2\}}, \] we cannot find an exponent $\rho>0$ such that (\ref{nov1}) is satisfied for all $m$-linear operators $A:X_{p_{1}}\times\cdots\times X_{p_{m}}\rightarrow X_{s}$. In fact, everything has already been done before: if $q\leq 2$, then we have just to follow the lines of Case 2 and if $q\geq 2$, then we may consider the $m$-linear mapping of Cases 4 and 5.
\section{The case $1\leq s\leq q\leq 2$, $\lambda\geq 2$ and $\left|\frac1{\mathbf p}\right|>\frac12$}
\subsection{A reformulation of the Hardy-Littlewood type inequalities}
We shall improve in this section the bound given by Theorem \ref{nov1}. We shall proceed by interpolation. To do this, we need a reformulation of the result of this theorem, as Villanueva and Perez-Garcia reformulated the Bohnenblust-Hille inequality in \cite{PV2}. The proof is a combination of \cite[Corollary 3.20]{PV2} and Proposition 2.2 and will be omitted.
\begin{theorem}\label{THMREFORMULATION} Let $1\leq p_{1},...,p_{m}\leq+\infty$, $1\leq s\leq q\leq\infty$ and let $\rho>0$. The following assertions are equivalent. \begin{itemize} \item[(A)] There exists $C>0$ such that, for every continuous $m$-linear mapping $A:X_{p_1}\times\cdots\times X_{p_m}\to X_s$, we have
$$\left(\sum_{i_1,\dots,i_m} \left\|A(e_{i_1},\dots,e_{i_m})\right\|_{\ell_q}^{\rho}\right)^{1/\rho}\leq C\|A\|.$$ \item[(B)] There exists $C>0$ such that, for any $n\geq 1$, for any Banach spaces $Y_1,\dots,Y_m$, for any continuous $m$-linear mapping $S:Y_1\times\cdots\times Y_m\to X_s$, the induced operator \begin{eqnarray*} T:\ell_{p_1^*,w}^n(Y_1)\times\cdots\times \ell_{p_m^*,w}^n(Y_m)&\to&\ell_{\rho}^{n^m}(X_q)\\ \big(x^{(1)},\dots,x^{(m)}\big)&\mapsto&\big(S(x_{i_1}^{(1)},\dots,x_{i_m}^{(m)})\big)_{\mathbf i\in\mkn{m}} \end{eqnarray*}
satisfies $\|T\|\leq C\|S\|$. \end{itemize} \end{theorem} We recall that, for any $p\in[1,+\infty]$ and any Banach space $Y$,
$$\ell_{p,w}^n(Y)=\left\{(x_j)_{j=1}^n\subset Y;\ \|(x_j)\|_{w,p}:=\sup_{\varphi\in B_{Y^*}}\left(\sum_{j=1}^n |\varphi(x_j)|^p\right)^{1/p}<+\infty\right\}$$ with the appropriate modifications for $p=\infty$.
\subsection{Proof of the sufficient condition} We now prove our better upper bound in the case $1\leq s\leq q\leq 2$, $\left|\frac1{\mathbf p}\right|>\frac 12$ (namely we prove the first part of Theorem \ref{nov2}). Let $n\geq 1$, let $Y_1,\dots,Y_m$ be Banach spaces and let $S:Y_1\times\dots\times Y_m\to X_s$ be bounded. Let $n\geq 1$ and let $T$ be the operator induced by $S$ on $\mathcal Y=\ell_{p_1^*,w}^n(Y_1)\times\cdots\times \ell_{p_m^*,w}(Y_m)$, defined by $$T\big(x^{(1)},\dots,x^{(m)}\big)=\big(S(x_{i_1}^{(1)},\dots,x_{i_m}^{(m)}\big)).$$ Then $T$ is bounded as an operator from $\mathcal Y$ into $\ell_{\infty}^{n^m}(X_s)$ (this is trivial). $T$ is also bounded as an operator from $\mathcal Y$ into $\ell_\rho^{n^m}(X_s)$ with $\frac 1\rho=\frac 1s-\left|\frac1{\mathbf p}\right|$ (this is Theorem \ref{nov1} for $1\leq s\leq 2$ and $q\geq 2$). We can interpolate between these two extreme situations. Hence, let $q\in [s,2]$ and let $\theta\in [0,1]$ be such that $$\frac 1q=\frac{1-\theta}s+\frac \theta2\iff \theta=\frac{\frac 1s-\frac 1q}{\frac 1s-\frac 12}.$$ By \cite[Theorem 4.4.1]{berg.lofst}, $T$ is bounded as an operator from $\mathcal Y$ into $\ell_t^{n^m}(X_q)$ where $$\frac 1t=\frac{1-\theta}{\infty}+\frac{\theta}{\rho}=\frac{\left(\frac 1s-\frac 1q\right)\left(\frac 1s-\left|\frac1{\mathbf p}\right|\right)}{\frac 1s-\frac 12}.$$ \begin{remark} It is easy to check that, for $1\leq s\leq q\leq 2$ and $\left|\frac1{\mathbf p}\right|\geq\frac 12$, then the bound $\frac{\left(\frac 1s-\frac 1q\right)\left(\frac 1s-\left|\frac1{\mathbf p}\right|\right)}{\frac 1s-\frac 12}$ is always better (namely larger) than the bound $\frac 12+\frac 1s-\frac 1q-\left|\frac1{\mathbf p}\right|$ obtained in Theorem \ref{nov1}. \end{remark}
\subsection{The necessary condition} We now prove the second part of Theorem \ref{nov2}. It also uses a probabilistic device for linear maps when the two spaces do not need to have the same dimension. The forthcoming lemma can be found in \cite[Proposition 3.2]{ben}. \begin{lemma}\label{LEMBENNETT} Let $n,d\geq 1$, $1\leq p,s\leq 2$. There exists $T:\ell_p^d\to\ell_s^n$, $T(x)=\sum_{i,j}\pm x_je_i$ such that
$$\|T\|\leq C_{p,s}\max \left(d^{1/s},n^{1-\frac 1p}d^{\frac 1s-\frac 12}\right).$$ \end{lemma} Coming back to the proof of Theorem \ref{nov2}, we first observe that we may always assume that $\left|\frac1{\mathbf p}\right|<1$. Otherwise, we can consider the $m$-linear map $A:X_{p_1}\times\cdots\times X_{p_m}\to X_s$ defined by \[ A\big(x^{(1)},\dots,x^{(m)}\big)=\sum_{i\geq 1}x_i^{(1)}\dots x_i^{(m)}e_0 \] and observe that it is bounded whereas it has infinitely many coefficients equal to 1. We then define $p\in [1,2]$ by $\frac 1p=\left|\frac1{\mathbf p}\right|$ and we consider $T:\ell_p^d\to\ell_s^n$, $T(x)=\sum_{i,j}\varepsilon_{i,j}x_je_i$ the map given by Lemma \ref{LEMBENNETT}. We then define \begin{eqnarray*} A:\ell_{p_1}^d\times\cdots\times\ell_{p_m}^d&\to&\ell_s^n\\ \big(x^{(1)},\dots,x^{(m)}\big)&\mapsto&\sum_{i,j}\varepsilon_{i,j}x_j^{(1)}\cdots x_j^{(m)}e_i \end{eqnarray*}
and we observe that, by H\"older's inequality, $\|A\|\leq \|T\|$. Furthermore,
$$\left(\sum_{i_1,\dots,i_m}\|A(e_{i_1},\dots,e_{i_m})\|_{\ell_q}^t\right)^{1/t}=n^{1/t}d^{1/q}.$$ Taking $d^{1/2}=n^{1-\frac 1p}$ (this is the optimal relation between $d$ and $n$), we get that if
$$\left(\sum_{i_1,\dots,i_m}\|A(e_{i_1},\dots,e_{i_m})\|_{\ell_q}^t\right)^{1/t}\leq C\|A\|,$$ then it is necessary that $$\frac 1t\leq 2\left(1-\frac 1p\right)\left(\frac 1s-\frac 1q\right).$$ \begin{remark} This last condition is optimal when $s=1$ or when $\left|\frac1{\mathbf p}\right|=\frac 12$ (with, in fact, the same proof as in Case 2 above). When $1<s<2$, another necessary condition is $$\frac 1t\leq\frac 1s-\left|\frac1{\mathbf p}\right|$$ (see Case 4 or Case 5 above). \end{remark}
\section{Optimal estimates under cotype assumptions}\label{s8}
For a Banach space $X$, let $q_X:=\inf\{q\geq 2;\ X \textrm{ has cotype } q\}$. For scalar-valued multilinear operators it is easy to observe that summability in multiple indexes behaves in a quite different way than summability in just one index. For instance, for any bounded bilinear form $A:c_{0}\times c_{0}\rightarrow\mathbb{C}$, \[
\left( \sum_{i,j=1}^{{+\infty}}|A(e_{i},e_{j})|^{\frac{4}{3}}\right) ^{\frac {3}{4}}\leq\sqrt{2}\Vert A\Vert \] and the exponent $4/3$ is optimal. But, if we sum diagonally $\left( i=j\right) $ the exponent $4/3$ can be reduced to $1$ since \[
\sum_{i=1}^{{+\infty}}|A(e_{i},e_{i})|\leq\Vert A\Vert \] for any bounded bilinear form $A:c_{0}\times c_{0}\rightarrow\mathbb{C}$. Now we prove Theorem \ref{000aaa} which shows that when replacing the scalar field by infinite-dimensional spaces the situation is quite different.
\begin{proof}
$(A)\Rightarrow(B).$ From a deep result of Maurey and Pisier (\cite{pisier} and \cite[Section 14]{djt}), $\ell_{q_{X}}$ is finitely representable in $X$, which means that, for any $n\geq1$, one may find unit vectors $z_{1},\dots,z_{n}\in X$ such that, for any $a_{1},\dots,a_{n} \in\mathbb{C}$, \[ \sum_{i=1}^{n}\Vert a_{i}z_{i}\Vert_{X}\leq2\left( \sum_{i=1}^{n}
|a_{i}|^{q_{X}}\right) ^{1/q_{X}}. \] We then consider the $m$-linear map $A:\ell_{p_{1}}^{n}\times\cdots \times\ell_{p_{m}}^{n}\rightarrow X$ defined by \[ A\left( x^{(1)},\cdots,x^{(m)}\right) :=\sum_{i=1}^{n}x_{i}^{(1)}\cdots x_{i}^{(m)}z_{i}. \] Then, for any $(x^{(1)},\dots,x^{(m)})$ belonging to $B_{\ell_{p_{1}}^{n} }\times\dots\times B_{\ell_{p_{m}}^{n}}$, \begin{align*} \left\Vert A\left( x^{(1)},\cdots,x^{(m)}\right) \right\Vert & \leq2\left(
\sum_{i=1}^{n}|x_{i}^{(1)}|^{q_{X}}\cdots|x_{i}^{(m)}|^{q_{X}}\right) ^{1/{q_{X}}}\\ & \leq2n^{\frac{1}{q_{X}}-\left\vert \frac{1}{\mathbf{p}}\right\vert } \end{align*} where the last inequality follows from H\"{o}lder's inequality applied to the exponents $$\frac{p_{1}}{q_{X}},...,\frac{p_{m}}{q_{X}},\left( 1-q_{X}\left\vert \frac{1}{\mathbf{p}}\right\vert \right) ^{-1}.$$
On the other hand, \[ \left( \sum_{i=1}^{n}\Vert A\left( e_{i},\dots,e_{i}\right) \Vert^{\rho }\right) ^{1/\rho}=n^{\frac{1}{\rho}} \] and we obtain $\left( 3\right) $.
$(B)\Rightarrow(A).$ This implication is proved in \cite[Proposition 4.3]{dsp}. \end{proof}
If $X$ does not have cotype $q_X$, the condition remains necessary. But now we just have the following sufficient condition: \[
\frac{m}{\rho}<\frac{1}{q_X}-\left|\frac{1}{\mathbf p}\right|. \]
Of course, it would be nice to determine what happens in this case. A look at \cite[page 304]{djt} shows that the situation does not look simple.
As a consequence of the previous result we conclude that under certain circumstances the concepts of absolutely summing multilinear operator and multiple summing multilinear operator (see \cite{ir, matos, PV}) are precisely the same.
\begin{corollary}
Let $p\in[ 2,{+\infty}] $, let $X$ be an infinite dimensional Banach space with cotype $q_{X}<\frac{p}{m}$ and let $\rho>0$. The following assertions are equivalent:
(A) Every bounded $m$-linear operator $A:X_{p}\times\dots\times X_{p} \rightarrow X$ is absolutely $\left( \rho;p^{\ast}\right) $-summing.
(B) Every bounded $m$-linear operator $A:X_{p}\times\dots\times X_{p}\rightarrow X$ is multiple $\left( \rho;p^{\ast}\right) $-summing.
(C) $\frac{1}{\rho}\leq\frac{1}{q_{X}}- \frac{m}{p}.$ \end{corollary}
We stress the equivalence between $(A)$ and $(B)$ is not true, in general. For instance, every bounded bilinear operator $A:\ell_{2}\times\ell_{2}\rightarrow\ell_{2}$ is absolutely $\left( 1;1\right) $-summing but this is no longer true for multiple summability.
\section{Constants of vector-valued Bohnenblust--Hille inequalities}
A particular case of our main result is the following vector-valued Bohnenblust--Hille inequality (see \cite[Lemma 3]{df} and also \cite[Section 2.2]{ursula}):
\begin{theorem} \label{61} Let $X$ be a Banach space, $Y$ a cotype $q$ space and $v:X\rightarrow Y$ an $(r,1)$-summing operator with $1\leq r\leq q$. Then, for all $m$-linear operators $T:c_{0}\times\cdots\times c_{0}\rightarrow X$, \[ \left( \sum_{i_{1},...,i_{m}=1}^{+\infty}\Vert vT\left( e_{i_{1}} ,\ldots,e_{i_{m}}\right) \Vert_{Y}^{\frac{qrm}{q+(m-1)r}}\right) ^{\frac{q+\left( m-1\right) r}{qrm}}\leq C_{Y,m}\pi_{r,1}(v)\Vert T\Vert \] with $C_{Y,m}=\left( \sqrt{2}C_{q}\left( Y\right) \right) ^{m-1}.$ \end{theorem}
In this section, in Theorem \ref{ttrr}, we improve the above estimate for $C_{Y,m}$. The proof of Theorem \ref{ttrr} follows almost word by word the proof of \cite[Proposition 3.1]{bps} using \cite[Lemma 2.2]{defant33} and Kahane's inequality instead of the Khinchine inequality. We present the proof for the sake of completeness. We need the following inequality due to Kahane:
\begin{kahane} Let $0<p,q<+\infty$. Then there is a constant $K_{p,q}>0$ for which \[ \left( \int_{I}\Vert\sum_{k=1}^{n}r_{k}(t)x_{k}\,dt\Vert^{q}\right) ^{\frac{1}{q}}\leq K_{p,q}\left( \int_{I}\Vert\sum_{k=1}^{n}r_{k} (t)x_{k}\,dt\Vert^{p}\right) ^{\frac{1}{p}}, \] regardless of the choice of a Banach space $X$ and of finitely many vectors $x_{1},...,x_{n}\in X$. \end{kahane}
\begin{theorem}\label{ttrr} \label{62}For all $m$ and all $1\leq k<m,$ \[ C_{Y,m}\leq\left( C_{q}(Y)K_{\frac{qrk}{q+(k-1)r},2}\right) ^{m-k}C_{Y,k}. \]
\end{theorem}
\begin{proof} Let $\rho:=\frac{qrm}{q+(m-1)r}$ and to simplify notation let us write \[ vTe_{\mathbf{i}}=vT\left( e_{i_{1}},\ldots,e_{i_{m}}\right) . \] Let us make use of \cite[Remark 2.2]{bps} with $m\geq2,\ 1\leq k\leq m-1$ and $s=\frac{qrk}{q+(k-1)r}$. So we have \begin{equation} \left( \sum_{\mathbf{i}}\Vert vTe_{\mathbf{i}}\Vert_{Y}^{\rho}\right) ^{\frac{1}{\rho}}\leq\prod_{S\in P_{k}(m)}\left( \sum_{\mathbf{i}_{S}}\left( \sum_{\mathbf{i}_{\hat{S}}}\Vert vT\left( e_{\mathbf{i}_{S}},e_{\mathbf{i} _{\hat{S}}}\right) \Vert_{Y}^{q}\right) ^{\frac{s}{q}}\right) ^{\frac {1}{s\binom{m}{k}}},\label{55f} \end{equation} where $P_{k}(m)$ denotes the set of all subsets of $\{1,...,m\}$ with cardinality $k.$ For sake of clarity, we shall assume that $S=\{1,\dots,k\}$. By the multilinear cotype inequality (see \cite[Lemma 2.2]{defant33}) and the Kahane inequality, we have {\small \begin{align*} & \left( \sum_{\mathbf{i}_{\hat{S}}}\Vert vT\left( e_{\mathbf{i}_{S} },e_{\mathbf{i}_{\hat{S}}}\right) \Vert_{Y}^{q}\right) ^{\frac{s}{q}}\\ & \leq\left( C_{q}(Y)K_{s,2}\right) ^{s(m-k)}\int_{I^{m-k}}\left\Vert \sum_{\mathbf{i}_{\hat{S}}}r_{\mathbf{i}_{\hat{S}}}(t_{\hat{S}})vT\left( e_{\mathbf{i}_{S}},e_{\mathbf{i}_{\hat{S}}}\right) \right\Vert _{Y} ^{s}\,dt_{\hat{S}}\\ & =\left( C_{q}(Y)K_{s,2}\right) ^{s(m-k)}\int_{I^{m-k}}\left\Vert vT\left( e_{\mathbf{i}_{S}},\sum_{\mathbf{i}_{\hat{S}}}r_{\mathbf{i}_{\hat {S}}}(t_{\hat{S}})e_{\mathbf{i}_{\hat{S}}}\right) \right\Vert _{Y} ^{s}\,dt_{\hat{S}}\\ & =\left( C_{q}(Y)K_{s,2}\right) ^{s(m-k)}\int_{I^{m-k}}\left\Vert v\left( T\left( e_{i_{1}},\dots,e_{i_{k}},\sum_{i_{k+1}}r_{k+1}(t_{k+1})e_{k+1} ,\dots,\sum_{i_{m}}r_{m}(t_{m})e_{m}\right) \right) \right\Vert _{Y} ^{s}\,dt_{k+1}\dots dt_{m}\\ & \end{align*} } But for a fixed choice of $\left( t_{k+1},\dots,t_{m}\right) \in I^{m-k}=[0,1]^{m-k}$, we know, by Theorem \ref{61}, that \[ \sum_{i_{1},\dots,i_{k}}\left\Vert v\left( T\left( e_{i_{1}},\dots,e_{i_{k} },\sum_{i_{k+1}}r_{k+1}(t_{k+1})e_{k+1},\dots,\sum_{i_{m}}r_{m}(t_{m} )e_{m}\right) \right) \right\Vert _{Y}^{s}\newline\leq\left( C_{Y,k} \pi_{r,1}(v)\Vert T\Vert\right) ^{s}. \] Thus, \begin{align} & \sum_{\mathbf{i}_{S}}\left( \sum_{\mathbf{i}_{\hat{S}}}\Vert vT\left( e_{\mathbf{i}_{S}},e_{\mathbf{i}_{\hat{S}}}\right) \Vert_{Y}^{q}\right) ^{\frac{s}{q}}\nonumber\\ & \leq\left( C_{q}(Y)K_{s,2}\right) ^{s(m-k)}\cdot\sum_{i_{1},\dots,i_{k} }\left\Vert v\left( T\left( e_{i_{1}},\dots,e_{i_{k}},\sum_{i_{k+1}} r_{k+1}(t_{k+1})e_{k+1},\dots,\sum_{i_{m}}r_{m}(t_{m})e_{m}\right) \right) \right\Vert _{Y}^{s}\nonumber\\ & \leq\left( \left( C_{q}(Y)K_{s,2}\right) ^{m-k}\pi_{r,1}(v)C_{Y,k}\Vert T\Vert\right) ^{s},\label{estrela} \end{align} namely \[ \left( \sum_{\mathbf{i}_{S}}\left( \sum_{\mathbf{i}_{\hat{S}}}\Vert vT\left( e_{\mathbf{i}_{S}},e_{\mathbf{i}_{\hat{S}}}\right) \Vert_{Y} ^{q}\right) ^{\frac{s}{q}}\right) ^{\frac{1}{s}}\leq\left( C_{q} (Y)K_{s,2}\right) ^{m-k}\pi_{r,1}(v)C_{Y,k}\Vert T\Vert. \] From (\ref{55f}) we conclude that \[ \left( \sum_{\mathbf{i}}\Vert vTe_{\mathbf{i}}\Vert_{Y}^{\rho}\right) ^{\frac{1}{\rho}}\leq\left( C_{q}(Y)K_{s,2}\right) ^{m-k}C_{Y,k}\pi _{r,1}(v)\Vert T\Vert. \]
\end{proof}
When $m$ is even, the case $k=\frac{m}{2}$ recovers the constants from \cite{ursula}.
\begin{corollary} For all $m,$ \[ C_{Y,m}\leq C_{q}(Y)^{m-1}
{\textstyle\prod\limits_{k=1}^{m-1}}
K_{\frac{qrk}{q+(k-1)r},2}. \]
\end{corollary}
\section{Other exponents}
From now on $1\leq r\leq q$ and $\left( q_{1},\ldots,q_{m}\right) \in\lbrack r,q]^{m}$ so that \[ \frac{1}{q_{1}}+\cdots+\frac{1}{q_{m}}=\frac{q+(m-1)r}{qr}=\frac{1}{r} +\frac{m-1}{q} \] are called vector-valued Bohnenblust--Hille exponents. From Theorem \ref{thmain} we have:
\begin{theorem} [Multiple exponent vector-valued Bohnenblust--Hille inequality]Let $X$ be a Banach space and $Y$ a cotype $q$ space with $1\leq r\leq q$. If $\left( q_{1},\ldots,q_{m}\right) \in\lbrack r,q]^{m}$ are vector-valued Bohnenblust--Hille exponents, then there exists $C_{Y,q_{1},\dots,q_{m}}\geq1$ such that, for all $m$-linear operators $T:c_{0}\times\cdots\times c_{0}\rightarrow X$ and every $(r,1)$-summing operator $v:X\rightarrow Y$, we have \begin{equation} \left( \sum_{i_{1}=1}^{+\infty}\dots\left( \sum_{i_{m}=1}^{+\infty}\Vert vTe_{\mathbf{i}}\Vert_{Y}^{q_{m}}\right) ^{\frac{q_{m-1}}{q_{m}}} \dots\right) ^{\frac{1}{q_{1}}}\leq C_{Y,q_{1},\ldots,q_{m}}\pi_{r,1}(v)\Vert T\Vert,\label{eq} \end{equation} with $C_{Y,q_{1},\ldots,q_{m}}=\left( \sqrt{2}C_{q}\left( Y\right) \right) ^{m-1}.$ \end{theorem}
Our final result gives better estimates for the constants $C_{Y,q_{1} ,\ldots,q_{m}}:$
\begin{theorem} If $\left( q_{1},\dots,q_{m}\right) $ is a vector-valued Bohnenblust--Hille exponent, then \[ C_{Y,q_{1}\dots,q_{m}}\leq\prod_{k=1}^{m}\left( \left( C_{q}(Y)K_{\frac {kqr}{q+(k-1)r},2}\right) ^{m-k}C_{Y,k}\right) ^{\theta_{k}} \] with \begin{equation} \theta_{m}=m\left( \frac{1}{r}-\frac{1}{q}\right) ^{-1}\left( \frac {1}{q_{m}}-\frac{1}{q}\right) \label{222} \end{equation} and \begin{equation} \theta_{k}=k\left( \frac{1}{r}-\frac{1}{q}\right) ^{-1}\left( \frac {1}{q_{k}}-\frac{1}{q_{k+1}}\right) ,\ \text{ for }k=1,\dots,m-1. \label{333} \end{equation}
\end{theorem}
\begin{proof} It suffices to consider $q_{i}\leq q_{j}$ whenever $i<j$. For each $k=1,\dots,m$, define \[ s_{k}=\frac{kqr}{q+(k-1)r}. \] From the proof of Theorem \ref{62} we have (\ref{eq}) for each exponent $\left( s_{k},\overset{k\text{ times}}{\dots},s_{k},q\dots,q\right) .$ More precisely, from (\ref{estrela}) we have \[ \left( \sum_{i_{1},\dots,i_{k}}\left( \sum_{i_{k+1},\dots,i_{m}}\Vert vTe_{\mathbf{i}}\Vert_{Y}^{q}\right) ^{\frac{s_{k}}{q}}\right) ^{\frac {1}{s_{k}}}\leq\left( C_{q}(Y)K_{s_{k},2}\right) ^{m-k}C_{Y,k}\pi _{r,1}(v)\Vert T\Vert. \] Consequently, for each $k=1,\dots,m$ we have \[ C_{Y,s_{k},\overset{k\text{ times}}{\dots},s_{k},q\dots,q}\leq\left( C_{q}(Y)K_{s_{k},2}\right) ^{m-k}C_{Y,k}. \] Since every vector-valued Bohnenblust--Hille exponent $\left( q_{1} ,\ldots,q_{m}\right) $ with $q_{1}\leq\cdots\leq q_{m}$ is obtained by interpolation of $\alpha_{1},...,\alpha_{m}$ with $\alpha_{k}=\left( s_{k},\overset{k\text{ times}}{\dots},s_{k},q\dots,q\right) $, and $\theta_{1},...,\theta_{m}$ as in (\ref{222}) and (\ref{333}), we conclude that \[ C_{Y,q_{1},\dots,q_{m}}\leq\prod_{k=1}^{m}\left( C_{Y,s_{k},\overset{k\text{ times}}{\dots},s_{k},q,\dots,q}\right) ^{\theta_{k}}\leq\prod_{k=1} ^{m}\left( \left( C_{q}(Y)K_{s_{k},2}\right) ^{m-k}C_{Y,k}\right) ^{\theta_{k}} \]
\end{proof}
A particular case of Kahane's inequality is Khintchine's inequality: if $(\varepsilon_{i})$ is a sequence of independent Rademacher variables, then, for any $p\in\lbrack1,2]$, there exists a constant $\mathrm{A}_{\mathbb{R},p}$ such that, for any $n\geq1$ and any $a_{1}\dots,a_{n}\in\mathbb{R}$, \[
\left( \sum_{i=1}^{n}|a_{i}|^{2}\right) ^{\frac{1}{2}}\leq\mathrm{A} _{\mathbb{R},p}\left( \int_{\Omega}\left\vert \sum_{i=1}^{n}a_{i} \varepsilon(\omega)\right\vert ^{p}d\omega\right) ^{\frac{1}{p}}. \] It has a complex counterpart: for any $p\in\lbrack1,2]$, there exists a constant $\mathrm{A}_{\mathbb{C},p}$ such that, for any $n\geq1$ and any $a_{1}\dots,a_{n}\in\mathbb{C}$, \[
\left( \sum_{i=1}^{n}|a_{i}|^{2}\right) ^{\frac{1}{2}}\leq\mathrm{A} _{\mathbb{C},p}\left( \int_{\mathbb{T}^{n}}\left\vert \sum_{i=1}^{n} a_{i}z_{i}\right\vert ^{p}dz\right) ^{\frac{1}{p}}. \] The best constants $\mathrm{A}_{\mathbb{R},p}$ and $\mathrm{A}_{\mathbb{C},p}$ are known (see \cite{haagerup} and \cite{kk}):
\begin{itemize} \item $\mathrm{A}_{\mathbb{R},p} = \begin{cases} 2^{\frac{1}{p}-\frac{1}{2}}, \text{ if } 0<p \leq p_{0} \approx1.847\\ \frac{1}{\sqrt{2}} \left( \frac{\Gamma\left( \frac{1+p}{2}\right) } {\sqrt{\pi}} \right) ^{-\frac{1}{p}}, \text{ if } p > p_{0} ; \end{cases} $
\item $\mathrm{A}_{\mathbb{C},p} = \Gamma\left( \frac{1+p}{2}\right) ^{-\frac{1}{p}}$, if $1 <p \leq2$. \end{itemize}
Taking $X=Y=\mathbb{K}$ and $r=1$ we obtain estimates for the constants of the scalar-valued Bohnenblust--Hille inequality with multiple exponents:
\begin{corollary} If $\left( q_{1},\ldots,q_{m}\right) \in\lbrack1,2]^{m}$ are so that \[ \frac{1}{q_{1}}+\cdots+\frac{1}{q_{m}}=\frac{m+1}{2}, \] then \[ \left( \sum_{i_{1}=1}^{+\infty}\dots\left( \sum_{i_{m}=1}^{+\infty}\left\vert T\left( e_{i_{1}},...,e_{i_{m}}\right) \right\vert ^{q_{m}}\right) ^{\frac{q_{m-1}}{q_{m}}}\dots\right) ^{\frac{1}{q_{1}}}\leq C_{\mathbb{K} ,m}^{2m\left( \frac{1}{q_{m}}-\frac{1}{2}\right) }\left( \prod_{k=1} ^{m-1}\left( \mathrm{A}_{\mathbb{K},\frac{2k}{k+1}}^{m-k}C_{\mathbb{K} ,k}\right) ^{2k\left( \frac{1}{q_{k}}-\frac{1}{q_{k+1}}\right) }\right) \Vert T\Vert \] for all $m$-linear operators $T:c_{0}\times\cdots\times c_{0}\rightarrow \mathbb{K}.$ In particular, for complex scalars, the left hand side of the above inequality can be replaced by \[ \left(
{\displaystyle\prod\limits_{j=1}^{m}}
\Gamma\left( 2-\frac{1}{j}\right) ^{\frac{j}{2-2j}}\right) ^{2m\left( \frac{1}{q_{m}}-\frac{1}{2}\right) }\left( \prod_{k=1}^{m-1}\left( \Gamma\left( \frac{3k+1}{2k+2}\right) ^{\left( \frac{-k-1}{2k}\right) \left( m-k\right) }
{\displaystyle\prod\limits_{j=1}^{k}}
\Gamma\left( 2-\frac{1}{j}\right) ^{\frac{j}{2-2j}}\right) ^{2k\left( \frac{1}{q_{k}}-\frac{1}{q_{k+1}}\right) }\right) \Vert T\Vert. \]
\end{corollary}
Acknowledgement. The authors thank the referee for important comments and suggestions.
\end{document} | arXiv |
Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds
Polynomial and linearized normal forms for almost periodic differential systems
January 2016, 36(1): 323-344. doi: 10.3934/dcds.2016.36.323
Intermediate $\beta$-shifts of finite type
Bing Li 1, , Tuomas Sahlsten 2, and Tony Samuel 3,
Department of Mathematics, South China University of Technology, Guangzhou, 510641, China
Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel
Fachbereich 3 Mathematik, Universität Bremen, 28359 Bremen, Germany
Received March 2014 Revised March 2015 Published June 2015
An aim of this article is to highlight dynamical differences between the greedy, and hence the lazy, $\beta$-shift (transformation) and an intermediate $\beta$-shift (transformation), for a fixed $\beta \in (1, 2)$. Specifically, a classification in terms of the kneading invariants of the linear maps $T_{\beta,\alpha} \colon x \mapsto \beta x + \alpha \bmod 1$ for which the corresponding intermediate $\beta$-shift is of finite type is given. This characterisation is then employed to construct a class of pairs $(\beta,\alpha)$ such that the intermediate $\beta$-shift associated with $T_{\beta, \alpha}$ is a subshift of finite type. It is also proved that these maps $T_{\beta,\alpha}$ are not transitive. This is in contrast to the situation for the corresponding greedy and lazy $\beta$-shifts and $\beta$-transformations, for which both of the two properties do not hold.
Keywords: subshifts of finite type, $\beta$-transformations, transitivity..
Mathematics Subject Classification: Primary: 37B10; Secondary: 11A67, 11R0.
Citation: Bing Li, Tuomas Sahlsten, Tony Samuel. Intermediate $\beta$-shifts of finite type. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 323-344. doi: 10.3934/dcds.2016.36.323
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\begin{document}
\baselineskip=16pt
\title[]{Tautological ring of the moduli space of generalised parabolic line bundles on a curve}
\author[J. N. Iyer]{Jaya NN Iyer}
\address{The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India}
\email{[email protected]}
\footnotetext{Mathematics Classification Number: 14C25, 14D05, 14D20, 14D21 } \footnotetext{Keywords: Parabolic line bundles, Nodal curve, Chow groups.} \begin{abstract} In this paper, we consider the \textit{tautological ring} containing the extended Brill-Noether algebraic classes on the normalization of the compactified Jacobian of a complex nodal projective curve (with one node). This smallest $\Q$-subalgebra of algebraic classes under algebraic equivalence, stable under extensions of the maps induced by multiplication maps, Pontrayagin product and Fourier transform, is shown to be generated by pullback of the Brill-Noether classes of the Jacobian of the normalized curve and some natural classes.
\end{abstract} \maketitle
\section{Introduction}
Suppose $X$ is a connected smooth projective curve of genus $g$ and defined over the complex numbers. Let $D=x_1+x_2$ be an effective divisor on the curve $X$, such that the point $x_1$ is different from the point $x_2$. We recall the notion of generalised parabolic line bundles, due to U. Bhosle \cite{Bhosle}. A generalised parabolic bundle of rank one on $(X,D)$ consists of the data (\cite[p.187]{Bhosle}):
1) $E$ is a line bundle on $X$
2) $F^1(E)_D\subset E_{x_1}\oplus E_{x_2}$ is a rank one subspace of the direct sum of the fibres of $E$ at $x_1$ and $x_2$.
Th moduli space $P$ of generalised parabolic line bundles on $X$ is a smooth projective variety of dimension $g+1$ and is in fact a $\p^1$-bundle over the Jacobian variety $Jac(X)$, see \cite[Proposition 2.2]{Bhosle}.
U. Bhosle and A.J. Parameswaran \cite{BhosleParam} have defined natural subvarieties $\widetilde{W}_{i}\subset P$, for $i=1,2,...,g$. These are defined as the Brill-Noether loci, similar to the naturally defined subvarieties $W_i\subset Jac(X)$, and they have proved a Poincar\'e formula in terms of the classes $\widetilde{W}_i$ in the group of algebraic cycles modulo numerical equivalence.
In this paper, we would like to understand the Poincar\'e relations in the rational ring of algebraic classes of $P$ modulo algebraic equivalence. More precisely, consider the group $A^k(P)$ of algebraic cycles of codimension $k$ on $P$, modulo algebraic equivalence and denote $A^k(P)_\Q:=A^k(P)\otimes \Q$. Then the direct sum $A^*(P)_\Q:=\oplus_{k\geq 0}A^k(P)_\Q$ is a commutative ring with the intersection product. The first question that arises is whether the classical Poincar\'e formula for the classes $\widetilde{W}_i$, holds in the ring $A^*(P)_\Q$.
In this context, for the Jacobian $Jac(X)$ of a smooth projective curve $X$ of genus $g$, a result of Ceresa \cite{Ceresa} says that the Poincar\'e formula does not hold in the ring $A^*(Jac(X))_\Q$. To study other relations between the classes $W_i$, A. Beauville \cite{Beauville2} considered the tautological ring $\cR\subset A^*(Jac(X))_\Q$. The ring $\cR$ is the smallest $\Q$-subalgebra containing the classes $W_i$, $1\leq i\leq g$, and stable under the pullback maps $\textbf{n}^*$, pushforward maps $\textbf{n}_*$ and closed under the Pontryagin product $*$. Here $\textbf{n}:Jac(X)\rar Jac(X)$ is multiplication by the integer $n$, for any $n$. The Pontryagin product $*$ is defined in \cite{Beauville}, and we recall it in \S \ref{prelim}. He proved that the ring $\cR$ is in fact generated by the classes $W_i$, for $1\leq i\leq g$.
To obtain similar results on $P$, we note that the multiplication maps, Pontryagin product and Fourier transform can be extended on the ring $A^*(P)_\Q$. We consider the tautological ring $\widetilde{\cR}\subset A^*(P)_\Q$ containing the classes $\widetilde{W}_i, S_y, c_1(\cO_P(1))$ and the smallest $\Q$-subalgebra stable under natural analogues of $\textbf{n}^*,\textbf{n}_*$ and the Pontryagin product $*$, see \S \ref{extPP}, \ref{extFT}. Here $S_y$ is a section of the $\p^1$-bundle $\pi:P\rar Jac(X)$, defined in \S \ref{specialsection}. We show
\begin{theorem} The $\Q$-algebra $\widetilde{\cR}$ is generated by $\pi^{-1}{W_i}$ and by $c_1(\cO_P(1))$. \end{theorem}
The proof is given in \S \ref{finaltheorem}, and depends on the structure of the classes $\widetilde{W_i}$ and action of the extended Pontryagin product and Fourier transform on the ring of algebraic classes.
{\Small Acknowledgements: We thank A.J.Parameswaran for useful discussions on \cite{BhosleParam} in Dec 2009. Thanks are also due to the referee for the comments. }
\section{Preliminaries}\label{prelim}
In this section, we recall some of the basic properties of the group of algebraic classes on the Jacobian of a smooth projective curve of genus $g$.
Suppose $X$ is a smooth projective curve of genus $g$. It can be embedded into the Jacobian variety $Jac(X)$, fixing a base point on $X$. Using the group law on $Jac(X)$, we define certain natural subvarieties $W_i := X + X + \ldots + X (i $ times). These subvarieties are well-defined up to translation. The classical Poincare's formula gives $$ W_i = \frac{1}{(g-i)!} \, \theta^{g-i} $$ and we see that $W_{g-1}$ is the theta divisor $\theta$ on $Jac(X)$.
A. Beauville \cite{Beauville2} studied the tautological subring $\cR \subset A^*(Jac(X))_\Q$ which is the smallest $\Q$-subalgebra containing the classes $W_i$, $1 \leq i \leq g$ and stable under the pullback maps $\textbf{n}^*$, pushforward maps $\textbf{n}_*$, closed under the intersection product and the Pontryagin product $*$. The group $A^*(Jac(X))_\Q$ is a $\Q$-vector space, and graded by the codimension of the cycle classes. The addition (or multiplication) map $m: Jac(X)\times Jac(X)\rar Jac(X)$, $(x,y)\rar x+y$, induces the pushforward map $m_*$ on the group of cycles on $Jac(X)\times Jac(X)$. This helps us to define the Pontryagin product as below.
The group $A^*(Jac(X)_\Q$ has two natural multiplication laws which are associative and commutative: namely the intersection product and the Pontryagin product defined by: $$ x*y := m_*(p^*x \cdot q^*y). $$
Here $p^*,q^*: A^*(Jac(X))_\Q \rar A^*(Jac(X))_\Q \times A^*(Jac(X))_\Q$ are the maps induced by the first and the second projections $Jac(X)\times Jac(X)\rar Jac(X)$, respectively. Note that $*$ is homogeneous of degree $-g$.
We will identify $Jac(X)$ with it's dual using the principal polarization on it. We define $W^{g-i} \in A^{g-i}(Jac(X)) $ as the class of $W_i$ in $A^*(Jac(X))$.
\begin{lemma} There is a second graduation on $A^p(Jac(X)) = \oplus_{s}A^p(Jac(X))_{(s)}$ such that $$ \textbf{n}^*x = n^{2p-s}x \,\, , \,\, \textbf{n}_*x = n^{2g-2p+s}x $$ for $x \in A^p(Jac(X))_{(s)} $ and for all $ n \in \Z.$ Also, $A^p(Jac(X))_{(s)} \neq 0 $ only if $ s < p \leq g+s $. Both products are homogeneous with respect to the second graduation. \end{lemma} \begin{proof}
See \cite[Proposition 1 and Proposition 4]{Beauville}. \end{proof}
We recall some of the properties of the Fourier transform for algebraic cycles on the Jacobian $Jac(X)$.
Let \begin{equation}\label{Poincareclass} \ell := p^*\theta + q^*\theta - m^*\theta \in A^1(Jac(X) \times Jac(X)) \end{equation}
which is the class of \emph{Poincare line bundle $\cL$} on $Jac(X) \times Jac(X)$. Fourier transform $\cF: A^*(Jac(X)) \rightarrow A^*(Jac(X)) $ defined by $\cF x = q_*(p^*x \cdotp e^{\ell}) $ satisfies following properties:
\subsection{}\label{Property2.1} $\cF \circ \cF = (-1)^g(-1)^*$ \subsection{}\label{Property2.2} $\cF(x * y) = \cF x \cdotp \cF y$ and $\cF(x \cdotp y) = (-1)^g \cF x * \cF y$ \subsection{}\label{Beauvillefourier} $\cF A^p(X)_{(s)} = A^{g-p+s}(X)_{(s)}$ (see \cite[Proposition 1]{Beauville}).
The main theorem in \cite{Beauville2} is:
\begin{theorem}\label{Beauville} $\cR$ is the $\Q$-subalgebra of $A^*(Jac(X))_\Q$ generated by the algebraic classes $W^1, W^2, \ldots ,W^{g-1}$. \end{theorem}
\section{Algebraic cycles on the moduli space of generalised parabolic line bundles on a curve}
Suppose $X$ is a connected nodal curve, with a single node at $x\in X$. The Jacobian $Jac(X)$ of $X$ is a non-compact variety, namely an extension of the Jacobian of the normalized curve by $\comx^*$. Oda and Seshadri \cite{OdaSeshadri} defined a compactification $\overline{Jac(X)}$ of $Jac(X)$ by adding rank one torsion free sheaves on $X$. However, this is a singular variety and hence it is difficult to understand algebraic classes of naturally defined subvarieties. So it is convenient to look at good compactifications of $Jac(X)$ and try to define extensions suitably. For this purpose, we recall U.Bhosle's work \cite{Bhosle} on rank one Parabolic sheaves and subsequent relations.
\subsection{Generalised Parabolic line bundles on a curve}
Suppose $X$ is a connected smooth projective curve of genus $g$. U. Bhosle \cite{Bhosle} defined the notion of a generalised parabolic bundle on a smooth curve. This is relevant to the study of torsion free rank one sheaves on nodal curves, as we will see below.
Fix an effective divisor $D$ on $X$, such that the points are distinct. For the sake of simplicity, in this paper, we will assume that $D=x_1+x_2$, $x_1\neq x_2$.
A generalised parabolic line bundle on $(X,D)$ is the data:
1) $E$ is a line bundle on $X$.
2) $F^1(E)_D\subset E_{x_1}\oplus E_{x_2}$ is a rank one vector subspace. Here $E_{x_i}$ denotes the fibre of $E$ at the point $x_i$.
A generalised parabolic line bundle as above will be denoted by the pair $(E,F^1(E)_D)$.
We have the following result: \begin{proposition}
The moduli space of generalised parabolic line bundles on $(X,D)$, where $D=x_1+x_2$, $x_1\neq x_2$, is a smooth projective variety. It is in fact a $\p^1$-bundle over the Jacobian of the normalized curve. \end{proposition} \begin{proof}
See \cite[Proposition 2.2]{Bhosle}. \end{proof}
\subsection{Relationship with torsion free sheaves on a nodal curve}\label{specialsection}
Suppose $X$ is an irreducible nodal curve of arithmetic genus $g+1$. Denote the nodal point by $x\in X$. Consider the desingularisation $p:X'\rar X$ of the curve $X$. Let $p^{-1}(x)=\{y,z\}$, where $y,z\in X'$ are smooth points. Consider the effective divisor $D:=y+z$ on the smooth projective curve $X'$ of genus $g$.
Firstly, we note that the Jacobian variety $Jac(X)$ of $X$ is a smooth group variety. In fact, it can be expressed as a central extension: $$ 1\rar \comx^* \rar Jac(X) \sta{p^*}{\rar} Jac(X')\rar 0. $$ Here $p^*$ is the morphism defined by the pullback of line bundles via $p:X'\rar X$. In particular, the group variety $Jac(X)$ is not a compact variety. Oda and Seshadri \cite{OdaSeshadri} defined a natural compactification $\ov{Jac(X)}$ of $Jac(X)$. The variety $\ov{Jac(X)}$ is constructed as the moduli space of torsion free rank one sheaves on the curve $X$. However, the variety $\ov{Jac(X)}$ is not a smooth variety. It was noticed in \cite{Bhosle}, that there is a morphism $$ h: P\rar \ov{Jac(X)}. $$ It is defined as follows. Given a generalised parabolic data $(E,F^1(E)_D))$ on $X'$, consider the exact sequence of sheaves on $X$: $$ 0\rar F \rar p_*(E) \sta{q}{\rar} p_*(\frac{E_{y}\oplus E_{z}}{F^1(E)_D}) \rar 0. $$ The term at the right end of the exact sequence is a skyscraper sheaf supported at the nodal point $x$. Then the sheaf $F$ is defined as the kernel of the natural restriction map, and it is a torsion free rank one sheaf on $X$. Furthermore, $F$ is locally free as long as the map $q$ is not the projection to either $E_{y}$ or $E_{z}$. In other words, $F$ is locally free if and only if $F^1(E)_D\neq E_{z} \m{ or } E_{y}$, see \cite[Proposition 1.8]{Bhosle}, \cite[Lemma 2.2]{BhosleParam}.
\begin{proposition} There is a natural inclusion of the Jacobian $J(X)\subset P$ and the morphism $h$ restricts to an isomorphism on $J(X)$. \end{proposition} \begin{proof} See \cite[889]{BhosleParam}. The inclusion $Jac(X)\sta{i}{\hookrightarrow} P$ is given as: given $L_0\in Jac(X)$, consider the exact sequence, $$ 0\rar L_0\rar p_*p^*L_0\rar Q \rar 0. $$ Here $Q$ is a rank one skyscraper sheaf supported at the node $x$. Consider the kernel $F^1:=\,ker \{(p_*p^*L_0)_{x}=(p^*L_0)_y\oplus (p^*L_0)_z\rar Q\}$. Then $i(L_0):=(p^*L_0,F^1)$. By construction $h$ is an isomorphism over $Jac(X)$.
\end{proof}
Let $S_y\subset P$ (resp. $S_z\subset P$) denote the section which corresponds to the points parametrizing $(p^*L,p^*(L)_y)\in P$ (resp. $(p^*L,p^*(L)_z$)).
We now look for subvarieties of $P$, which are an analogue of the subvarieties $W_i$ of the Jacobian of a smooth projective curve. These are defined in \cite{BhosleParam} as follows.
Denote $X_0$ the smooth locus of the nodal curve $X$, with one node at $x\in X$. Fix a basepoint $p\in X_0$.
Define the morphism $f_d$, for $1\leq d\leq g$: $$ \begin{array}{ccc} \mbox{Sym}^d(X_0) & \rar & P \\ (x_1,x_2,...,x_d)& \mapsto & \cO_{X}(x_1+x_2+...+x_d - d.p). \\ \end{array} $$
Denote $\widetilde{W_d}$ the closure of the image of the morphism $f_d$, with the reduced scheme structure.
We will use the following decomposition from \cite{BhosleParam}, for the projection $\pi:P\rar Jac(X')$. Recall that $X'\rar X$ is the normalization of $X$ such that $y,z\in X'$ lie over $x\in X$.
\begin{lemma}\label{genceresa} There is a decomposition of cycles in the Chow group $CH_{g-d}(P)$: $$ \widetilde{W}_{g-d}\,=\, \pi^{-1}(W_{g-d}).S_{y} + \pi^{-1}W_{g-d-1}. $$ \end{lemma} \begin{proof} See \cite[Lemma 3.5,p.892]{BhosleParam}. \end{proof}
We now have good analogues of the naturally defined subvarieties $\{W_i\}$. Our next aim is to extend the basic operations $\textbf{n}_*,\textbf{n}^*$, the Pontryagin product and Fourier transform on the group of cycles on $P$. This is done in the next few subsections.
\subsection{Extension of the natural operators $\textbf{n}^*,\,\textbf{n}_*$ on $A^*(P)$}
Given an integer $n\in \Z$, consider the multiplication map $$ \textbf{n}:Jac(X)\rar Jac(X),\, x\mapsto n.x. $$ This map induces a group homomorphism $\textbf{n}^*:A^k(Jac(X))_\Q\rar A^k(Jac(X))_\Q$. Since the variety $Jac(X)$ is not a compact variety, the pushforward map $\textbf{n}_*$ is not defined. In this subsection, we would like to define natural extensions of the maps $\textbf{n}^*$ and the pushforward map $\textbf{n}_*$ on the group $A^*(P)_\Q$.
\begin{proposition}\label{extendedmult} Given an integer $n\in \Z$, the multiplication map $\textbf{n}$ extends on $P$ and defines the pushforward $\textbf{n}_*$ and pullback map $\textbf{n}^*$ on $A^k(P)_\Q$. \end{proposition} \begin{proof} We need to check that the map $\textbf{n}:Jac(X)\to Jac(X)$ extends to $\textbf{n}:P\to P$. This can be checked fibre wise. The natural multiplication map on the fibres $\comx^*$ of $Jac(X)\to Jac(X')$ is the usual map $a\mapsto a^n$ on $\comx^*$. This map obviously extends to $\p^1$ fixing the complementary points $0$ and $\infty$. \end{proof}
Consider the eigenspace decomposition of the groups of algebraic cycles: $$ A^k(P)_\Q=\bigoplus_{s}A^k(P)_{(s)}, $$ where $$ A^k(P)_{(s)}:= \{\alpha\in A^k(P)_\Q: \textbf{n}_*\alpha = n^{2g-2k+s}.\alpha, \,\textbf{n}^* \alpha = n^{2k-s}.\alpha, \m{ for all } n\in \Z \}. $$
Consider the projection $\pi:P\rar Jac(X')$.
\begin{lemma} The group of algebraic cycles $A^k(P)_\Q$ can be expressed as: \begin{equation}\label{PBformula} A^k(P)_\Q\,=\, A^{k}(Jac(X'))_\Q\oplus H.A^{k-1}(Jac(X'))_\Q. \end{equation} Here $H:=c_1(\cO_P(1))$. \end{lemma} \begin{proof} This is a consequence of the projective bundle formula \cite{Fulton} applied to the $\p^1$-bundle $\pi: P\rar Jac(X')$. \end{proof}
\begin{lemma}\label{eigendecomp} The pushforward map $\textbf{n}_*$ and the pullback map $\textbf{n}^*$ are compatible with the decomposition in \eqref{PBformula}. In particular, the eigenspace $A^k(P)_{(s)}$ can be written as: $$ A^k(P)_{(s)}=A^{k}(Jac(X'))_{(s)}\oplus H.A^{k-1}(Jac(X'))_{(s)}. $$ \end{lemma} \begin{proof} Clear. \end{proof}
\subsection{Pontryagin product on the ring $A^*(P)_\Q$}\label{extPP}
We first consider the multiplication (or the addition) map $$ m:Jac(X)\times Jac(X) \rar Jac(X),\, (a,b)\mapsto a+b. $$ We first note that the map $m$ does not extend on the compactification $P$ (otherwise $P$ will be a group variety, which is not the case). Hence, we consider the rational map: $$ m:P\times P\rar P. $$
After suitable blow-ups, we can resolve the rational map $m$ to get a commutative diagram: \begin{eqnarray}\label{blowmultiplication} \tilde{P} & & \\ \downarrow \!f & \sta{\tilde{m}}{\searrow} & \\ P \times P & \sta{m}{\rar} & P \end{eqnarray}
\begin{lemma} There is a Pontryagin product $*$ on the ring of algebraic cycles $A^*(P)_\Q$: $$ *:A^k(P)_\Q\times A^l(P)_\Q\rar A^{k+l-g}(P)_\Q. $$ In particular the ring $A^*(P)_\Q$ has two products, the intersection product and the Pontryagin product. \end{lemma} \begin{proof} Consider the above resolution of the rational map $m$. Note that the map $f$ is a sequence of blow-ups. In fact it needs only one blow up: one needs to blow up only $S_y\times S_z$ and $S_z\times S_y$ in $P\times P$. Here $S_y\subset P$ (resp. $S_z\subset P$) denotes the section which corresponds to the points parametrizing $(p^*L,p^*(L)_y)\in P$ (resp. $(p^*L,p^*(L)_z$)).
Hence, by the blow-up formula \cite{Fulton}, we have $$ A^k(\tilde{P})_\Q\,=\,A^k(P\times P)_\Q \oplus A^k(S). $$ Here $S\subset \tilde{P}$ is a proper closed subvariety, and determined by the centre of blow-ups. Denote the two projections on $P\times P$ by $p_1$ and $p_2$. We now define the Pontryagin product $*$ as follows: $$ *\,:\, A^k(P)_\Q\times A^l(P)_\Q\rar A^{k+l-g}(P)_\Q $$ $$ *(\alpha,\beta)= \tilde{m}_*(f^*(p_1^*\alpha.p_2^*\beta))\,\in A^{k+l-g}(P)_\Q $$ This gives the Pontryagin product on the ring $A^*(P)_\Q$. \end{proof}
\begin{lemma}\label{compatiblePP} The Pontryagin product $*$ on $A^*(P)_\Q$ is compatible with the decomposition \eqref{PBformula}. \end{lemma} \begin{proof} We just need to note that the section $c_1(\cO_P(1))$ and any fibre of the $\p^1$-bundle $P$ is preserved under multiplication $\tilde{m}$. Hence $\tilde{m}_*$ reduces to $m'_*$ on the decomposition \eqref{PBformula}. Here $m'_*$ is induced by the multiplication $m'$ on $Jac(X')$.
\end{proof}
\subsection{Fourier transform on the ring $A^*(P)_\Q$}\label{extFT}
We would like to now define Fourier transform on $P$. Recall that on the usual Jacobian $Jac(C)$ of a smooth projective curve $C$, the Fourier transform is defined via the first Chern class $c_1(\cP)$ of the Poincar\'e line bundle $\cP$, on the product $Jac(C)\times Jac(C)$. The class $c_1(\cP)$ is $p^*\theta + q^*\theta - m^*\theta$, where $\theta\subset Jac(C)$ is the theta divisor, see \eqref{Poincareclass}.
On the variety $P$ with projection $\pi:P\rar Jac(X')$, we consider the extended theta divisor class (see \cite[Lemma 3.5, p.892]{BhosleParam}): \begin{equation}\label{extendedthetaclass} \widetilde{W}_{g}:= S_{y} + \pi^{-1}W_{g-1}. \end{equation} Here $g+1$ is the arithmetic genus of $X$ and $g$ is the genus of the normalization $X'$.
We define the extended Poincar\'e class as follows: \begin{equation}\label{extendedPoincareclass} \widetilde{\ell}:= p^*\widetilde{W}_g +q^*\widetilde{W}_g-f_*\widetilde{m}^*\widetilde{W}_g. \end{equation} Here $\widetilde{m}$ and $f$ are defined in \eqref{blowmultiplication}.
\begin{lemma}\label{extendedPoincare} The extended Poincar\'e class is $$ \widetilde{\ell}\,=\,(\pi\times \pi)^*\ell. $$ \end{lemma} \begin{proof} We first note, using Lemma \ref{genceresa}, $$ \begin{array}{ccc} \widetilde{\ell} & = & p^*(S_y) +p^*\pi^{-1}W_{g-1} + q^*(S_y) + q^*\pi^{-1}W_{g-1} - f_*\widetilde{m}^*S_y - f_*\widetilde{m}^*(\pi^{-1}W_{g-1})\\
&=& p^*S_y +q^* S_y - f_*\widetilde{m}^*S_y + (\pi\times \pi)^* \ell\\
&=& p^*S_y +q^*S_y -p^*S_y -q^*S_y + (\pi\times \pi)^*\ell \\
&=& (\pi\times \pi)^*\ell. \end{array} $$
\end{proof}
\begin{definition} The Fourier transform $\widetilde{F}$ on $A^*(P)_\Q$ is defined as: $$ \widetilde{\cF}: A^*(P)_\Q \rightarrow A^*(P)_\Q, $$ for $x\in A^*(P)_\Q$, let $\widetilde{\cF} x = q_*(p^*x \cdotp e^{\widetilde\ell})$. Here $p,q: P\times P\rar P$ are the first and second projections respectively. \end{definition}
\begin{lemma}
The Fourier transform $\widetilde{\cF}$
satisfies following properties:
1) $\widetilde{\cF} \circ \widetilde{\cF} = (-1)^g(-1)^*$
2) $\widetilde{\cF}(x * y) = \widetilde{\cF} x \cdotp \widetilde{\cF} y$ and $\widetilde{\cF}(x \cdotp y) = (-1)^g \widetilde{\cF} x * \widetilde{\cF} y$. \end{lemma}
\begin{proof} From Lemma \ref{extendedPoincare}, we note that $e^{\widetilde{\ell}}=(\pi\times \pi)^*e^\ell$ and $\widetilde{\cF}$ is defined by this correspondence cycle. Hence, using the decomposition \eqref{PBformula}, compatibility of Pontryagin product Lemma \ref{compatiblePP}, and Properties \eqref{Property2.1}, \eqref{Property2.2}, the assertion follows.
\end{proof}
\begin{lemma} We have $$ \widetilde{\cF}(A^p(P)_\Q)_{(s)}\, =\, A^{g-p+s}(P)_\Q)_{(s)} $$ \end{lemma} \begin{proof} Use the decomposition in Lemma \ref{eigendecomp} and apply Proposition \ref{Beauvillefourier}.
\end{proof}
\section{The tautological ring $\widetilde{R}$ of $P$}\label{finaltheorem}
As in \cite{Beauville2}, consider the \textit{tautological} subring $\widetilde{\cR} \subset A(P)_\Q$ which is the smallest $\Q$-subalgebra containing the classes $\widetilde{W}_i$, $1 \leq i \leq g$, $S_y$ and $c_1(\cO_P(1))$, and stable under the pullback maps $\textbf{n}^*$, pushforward maps $\textbf{n}_*$, closed under the intersection product and the Pontryagin product $*$.
\begin{theorem} The $\Q$-algebra $\widetilde{\cR}$ is generated by the classes $\pi^{-1}W_i,\,1\leq i\leq g-1$, $S_y$ and $c_1(\cO_P(1))$. \end{theorem} \begin{proof} We just need to note that $\widetilde{\cR}$ is generated by the $\Q$-subalgebra $\cR$ and the classes $S_y$, $c_1(\cO_P(1))$. Indeed, the maps $\textbf{n}_*$, $\textbf{n}^*$ induced by multiplication by $\textbf{n}$, preserve $\cR$ and the classes $S_y$, $c_1(\cO_P(1))$, by Proposition \ref{extendedmult}. Similarly, it is now straightforward to check that the Pontryagin product and Fourier transform preserve the $\Q$-subalgebra $<\cR, S_y,c_1(\cO_P(1))>$. By Theorem \ref{Beauville}, the assertion follows. \end{proof}
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\bibitem[Ce]{Ceresa} Ceresa, G. {\em $C$ is not algebraically equivalent to $C^{-}$ in its Jacobian.} Ann. of Math. (2) 117 (1983), no. \textbf{2}, 285--291.
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\end{document} | arXiv |
K-tree
In graph theory, a k-tree is an undirected graph formed by starting with a (k + 1)-vertex complete graph and then repeatedly adding vertices in such a way that each added vertex v has exactly k neighbors U such that, together, the k + 1 vertices formed by v and U form a clique.[1][2]
Characterizations
The k-trees are exactly the maximal graphs with a treewidth of k ("maximal" means that no more edges can be added without increasing their treewidth).[2] They are also exactly the chordal graphs all of whose maximal cliques are the same size k + 1 and all of whose minimal clique separators are also all the same size k.[1]
Related graph classes
1-trees are the same as unrooted trees. 2-trees are maximal series–parallel graphs,[3] and include also the maximal outerplanar graphs. Planar 3-trees are also known as Apollonian networks.[4]
The graphs that have treewidth at most k are exactly the subgraphs of k-trees, and for this reason they are called partial k-trees.[2]
The graphs formed by the edges and vertices of k-dimensional stacked polytopes, polytopes formed by starting from a simplex and then repeatedly gluing simplices onto the faces of the polytope, are k-trees when k ≥ 3.[5] This gluing process mimics the construction of k-trees by adding vertices to a clique.[6] A k-tree is the graph of a stacked polytope if and only if no three (k + 1)-vertex cliques have k vertices in common.[7]
References
1. Patil, H. P. (1986), "On the structure of k-trees", Journal of Combinatorics, Information and System Sciences, 11 (2–4): 57–64, MR 0966069.
2. Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2008), "Structural Properties of Sparse Graphs" (PDF), in Grötschel, Martin; Katona, Gyula O. H. (eds.), Building Bridges: between Mathematics and Computer Science, Bolyai Society Mathematical Studies, vol. 19, Springer-Verlag, p. 390, ISBN 978-3-540-85218-6.
3. Hwang, Frank; Richards, Dana; Winter, Pawel (1992), The Steiner Tree Problem, Annals of Discrete Mathematics (North-Holland Mathematics Studies), vol. 53, Elsevier, p. 177, ISBN 978-0-444-89098-6.
4. Distances in random Apollonian network structures Archived 2011-07-21 at the Wayback Machine, talk slides by Olivier Bodini, Alexis Darrasse, and Michèle Soria from a talk at FPSAC 2008, accessed 2011-03-06.
5. Koch, Etan; Perles, Micha A. (1976), "Covering efficiency of trees and k-trees", Proceedings of the Seventh Southeastern Conference on Combinatorics, Graph Theory, and Computing (Louisiana State Univ., Baton Rouge, La., 1976), Utilitas Math., Winnipeg, Man., pp. 391–420. Congressus Numerantium, No. XVII, MR 0457265. See in particular p. 420.
6. Below, Alexander; De Loera, Jesús A.; Richter-Gebert, Jürgen. "The Complexity of Finding Small Triangulations of Convex 3-Polytopes". arXiv:math/0012177..
7. Kleinschmidt, Peter (1 December 1976). "Eine graphentheoretische Kennzeichnung der Stapelpolytope". Archiv der Mathematik. 27 (1): 663–667. doi:10.1007/BF01224736.
| Wikipedia |
\begin{document}
\title{Efficient Folding Algorithms for Regular Polyhedra \thanks{A part of this research was presented at CCCG 2020. A part of this research is supported by JSPS KAKENHI Grant Number JP17H06287 and 18H04091.}}
\author{ Tonan Kamata$^1$ \and Akira Kadoguchi$^2$ \and Takashi Horiyama$^3$ \and Ryuhei Uehara$^1$ } \date{1 School of Information Science, Japan Advanced Institute
of Science and Technology (JAIST), Ishikawa, Japan \\
\texttt{\{kamata,uehara\}@jaist.ac.jp} \\ 2 Intelligent Vision \&{} Image Systems (IVIS), Tokyo, Japan \\
\texttt{[email protected]} \\ 3 Faculty of Information Science and Technology, Hokkaido
University, Hokkaido, Japan \\
\texttt{[email protected]}} \maketitle
\begin{abstract} We investigate the folding problem that asks if a polygon $P$ can be folded to a polyhedron $Q$ for given $P$ and $Q$. Recently, an efficient algorithm for this problem has been developed when $Q$ is a box. We extend this idea to regular polyhedra, also known as Platonic solids. The basic idea of our algorithms is common, which is called stamping. However, the computational complexities of them are different depending on their geometric properties. We developed four algorithms for the problem as follows. (1) An algorithm for a regular tetrahedron, which can be extended to a tetramonohedron. (2) An algorithm for a regular hexahedron (or a cube), which is much efficient than the previously known one. (3) An algorithm for a general deltahedron, which contains the cases that $Q$ is a regular octahedron or a regular icosahedron. (4) An algorithm for a regular dodecahedron. Combining these algorithms, we can conclude that the folding problem can be solved pseudo-polynomial time when $Q$ is a regular polyhedron and other related solid.
\noindent {\it Keywords:} Computational origami \and folding problem \and pseudo-polynomial time algorithm \and regular polyhedron (Platonic solids) \and stamping \end{abstract}
\section{Introduction} \label{sec:intro}
In 1525, the German painter Albrecht D\"{u}rer published his masterwork on geometry~\cite{durer1525underweysung}, whose title translates as ``On Teaching Measurement with a Compass and Straightedge for lines, planes, and whole bodies.'' In the book, he presented each polyhedron by drawing a {\em net}, which is an unfolding of the surface of the polyhedron to a planar layout without overlapping by cutting along its edges.
To this day, it remains an important open problem whether every convex polyhedron has a net by cutting along its edges. On the other hand, when we allow to cut anywhere, any convex polyhedron can be unfolded to a planar layout without overlapping. There are two known algorithms; one is called \emph{source unfolding}, and the other is called \emph{star unfolding} (see \cite{DemaineORourke2007}).
\begin{figure}
\caption{A Latin cross made by six unit squares.
For any real number $x$ with $0<x<1$, folding along dotted lines, we can obtain a doubly-covered fat cross.}
\label{fig:cross}
\end{figure}
In order to understand unfolding, it is interesting to look at the inverse: what kind of polyhedra can be folded from a given polygonal sheet of paper? For example, the Latin cross, which is one of the eleven nets of a cube, can be folded to 23 different convex polyhedra by 85 distinct ways of folding \cite{DemaineORourke2007} and an infinite number of doubly covered concave polygons (\figurename~\ref{fig:cross}). Comprehensive surveys of folding and unfolding can be found in \cite{DemaineORourke2007}. In this simple example, we can find that the convexity of a polyhedron plays an important role in this context.
We investigate the folding problem when both a polygon $P$ and a polyhedron $Q$ are explicitly given. That is, for a given polygon $P$ and a polyhedron $Q$, the folding problem asks if $P$ can fold to $Q$ or not.
This is a natural problem, however, there are a few results so far. When $Q$ is a regular tetrahedron, we have a mathematical characterization of its net \cite{Akiyama2007}; according to this result, $P$ can fold to $Q$ if and only if $P$ is a kind of tiling.
Recently, the folding problem was investigated for the case that $Q$ is a box. Some special cases were investigated in \cite{HoriyamaMizunashi2016,XHSU2017}, and the problem for a box $Q$ is solved in \cite{MHU2020} in general. The running time of the algorithm in \cite{MHU2020} is $O(D^{11}n^2(D^5+\log n))$, where $D$ is the diameter of $P$. In the algorithm, $Q$ is given as just a ``box'' without size, and the algorithm tried all feasible sizes. If $Q$ is explicitly given as a ``cube'', the running time of the algorithm in \cite{MHU2020} is reduced to $O(D^7 n^2(D^5+\log n))$ time.
In this paper, we will show that the folding problem can be solved efficiently for a regular polyhedron, which is also known as a Platonic solid. While there are five Platonic solids, our main result consists of four algorithms. That is, we investigate four problems depending on the target solid and show pseudo-polynomial time algorithms for them.
The first algorithm solves the folding problem for a regular tetrahedron. We give a bit stronger algorithm that solves the folding problem for a tetramonohedron, which is a tetrahedron that consists of four congruent acute triangles. As we have already mentioned, a mathematical characterization of a net of a regular tetrahedron is given in \cite{Akiyama2007}, and its extension to a tetramonohedron is given by Akiyama and Matsunaga \cite{AkiyamaMatsunaga2020}\footnote{In the literature, a tetramonohedron is called an \emph{isotetrahedron}.}. However, they only gave mathematical characterizations for the solids, and as far as the authors know, any algorithmic result for checking these characterizations has never been given explicitly\footnote{The authors thank an anonymous referee of \cite{KKHU2020}, who mentioned this point.}.
We give a pseudo-polynomial time algorithm for the folding problem for a tetramonohedron.
The next algorithm solves the folding problem for a regular hexahedron, or a cube. We give a bit stronger algorithm that solves the folding problem for a box of size $a\times b\times c$, where $a$, $b$, and $c$ are integers. As mentioned above, a known algorithm for this problem in \cite{MHU2020} runs in $O(D^7 n^2(D^5+\log n))$ time when $a,b,c$ are explicitly given. We improve this running time to $O(D^2 n^3)$ time. (In our context, the running time can be represented as $O(L (L+n) n^2)$ time, where $L$ is the perimeter of $P$.)
The third algorithm solves the folding problem for a regular dodecahedron.
The last algorithm solves the folding problem for a set of special convex deltahedra. Usually, a deltahedron means a polyhedron whose faces are congruent equilateral triangles. (More precisely, it is a regular tetrahedron, a regular octahedron, a regular icosahedron, a triangular bipyramid, a pentagonal bipyramid, a snub disphenoid, a triangulated triangular prism, or a gyroelongated square bipyramid.)
This algorithm cannot deal with a special case that all vertices have curvature $180^\circ$.
Therefore, it cannot deal with a regular tetrahedron among the set of deltahedron. On the other hand, the algorithm can deal with a vertex of curvature $360^\circ$, where six equilateral triangles make a flat hexagonal face. That is, we allow each face to consist of coplanar regular triangles, or each face can be any convex polyiamond, which consists of convex polygon obtained by gluing a collection of equal-sized equilateral triangles arranged with coincident sides (see \figurename~\ref{fig:HiDelta} for example). Thus there are an infinite of non-strictly convex deltahedra that our algorithm can deal with. In summary, if $Q$ has a non-strictly convex deltahedra with at least two vertices of curvature not equal to $180^\circ$, our algorithm solves the folding problem in pseudo-polynomial time. This set includes a regular octahedron and a regular icosahedron. Therefore, using these algorithms, we can efficiently solve the folding problem for Platonic solids (and more).
\begin{figure}
\caption{Examples of non-strictly convex deltahedra}
\label{fig:HiDelta}
\end{figure}
\begin{table}\centering
\begin{tabular}{|rc|}\hline
Platonic solids & Running times\\ \hline
Regular tetrahedron & $O(L(L+n)n^2)$ \\
Regular hexahedron (cube) & $O(L(L+n)n^2)$ \\
Regular octahedron & $O(L(L+n)n^2)$ \\
Regular dodecahedron & $O((L+n)^4n^2 )$ \\
Regular icosahedron & $O(L(L+n)n^2)$ \\\hline \end{tabular} \caption{Running times of our algorithms for Platonic solids, where $n$ is the number of vertices of $P$ and $L$ is the perimeter of $P$.} \label{tab:alg} \end{table}
The running times of our algorithms can be summarized in Table \ref{tab:alg}.
\section{Preliminaries} \label{sec:pre}
We first state our problem. For a given polygon $P$ and a polyhedron $Q$, the \emph{folding problem} asks if $P$ can fold to $Q$ or not. Since we will design an algorithm for each specific $Q$,
the input is a polygon $P=(p_0,p_1,\ldots,p_{n-1},p_n=p_0)$. Let $x(p)$ and $y(p)$ be the $x$-coordinate and $y$-coordinate of a point $p$, respectively. For a line segment $\ell$, $\msize{\ell}$ denotes its length (precisely, the length of a geodesic line). We assume the real RAM model for computation; each coordinate is an exact real number, and the running time is measured by the number of mathematical operations.
Since the length of the edges of $Q$ can be computed from the area of $P$, we assume that the length of an edge of $Q$ is 1 when $Q$ is a regular polyhedron and $P$ has an area that is consistent with $Q$ without loss of generality.
Each algorithm has the information of $Q$ which is represented in the standard form in computational geometry (see \cite{BCKO}). Precisely, for each $i,j,k$, the polyhedron $Q$ consists of vertices $q_i=(x(q_i),y(q_i),z(q_i))$, edges $\{q_i,q_j\}$, and faces $f_1,\ldots,f_k$, where each $f_i$ is represented by a cycle of vertices in counterclockwise-order in relation to the normal vector of the face.
We will construct a one-to-one correspondence between each point on $Q$ with a point on $P$ on the $xy$-plane where $P$ is placed. To deal with that, we define a \emph{local coordinate} of a point $q_j$ on a face $f_i$ of $Q$ by $q_j=(f_i;x(q_j),y(q_j))$. To simplify, when $Q$ is a regular polyhedron, we suppose that each face $f_i$ of $Q$ has a vertex of $f_i$ with local coordinate $(f_i;0,0)$ and another vertex with local coordinate $(f_i;1,0)$. That is, each face $f_i$ has the edge $((f_i;0,0),(f_i;1,0))$ of unit length. We note that each point inside of $f_i$ has a unique local coordinate, a point on an edge of $f_i$ except its endpoints has two local coordinates $(f_i;x,y)$ and $(f_{i'};x',y')$, where $f_{i'}$ is the face sharing the edge, and each vertex of the polygon $f_i$ has $d$ local coordinates, where $d$ is the number of the faces sharing the vertex on $Q$. (Namely, the value of $d$ is 3, 4, or 5 in this paper.)
\begin{figure}
\caption{An unfolding of a unit cube having long edges}
\label{fig:longl}
\end{figure}
In order to estimate time complexity, for a given polygon $P=(p_0,p_1,\ldots,p_{n-1},p_n=p_0)$, we define its \emph{diameter} $D_P$ and \emph{perimeter} $L_P$ as follows: \begin{eqnarray*} D_P &=& \max_{p,p' \mbox{\scriptsize on }\partial P } \msize{p p'}\\ L_P &=& \sum_{0\le i<n}\msize{p_i p_{i+1}}, \end{eqnarray*} where $\msize{p q}$ is the distance between two points $p$ and $q$, and $\partial P$ is the boundary of the polygon $P$. When the polygon $P$ is clear, the subscript $P$ is omitted. We also denote by $\ell_{\max}$ the length of the longest edge of $P$ defined by $\max_{0\le i<n}\msize{p_i p_{i+1}}$. We observe that $\ell_{\max}\le D_P$ and $2D_P<L_P$ for any simple polygon $P$ of positive area. In some $P$, $\ell_{\max}$ can be quite long compared to the face of $Q$ (see \figurename~\ref{fig:longl}). We give the upper bound of the number of faces that an edge of length $\ell_{\max}$ can go through on $Q$. \begin{lemma} \label{lem:traverse} Let $Q$ a polyhedron that consists of convex polyiamonds, integral rectangles, or regular pentagons as its faces. Then an edge of length $\ell_{\max}$ can go through $M$ faces on $Q$, where $M=O(D_P)$. \end{lemma} \begin{proof} We first focus on $Q$ that consists of faces of convex polyiamonds. Then the minimum angle of a face is $60^\circ$. Therefore, an edge of length $\sqrt{3}$ can penetrate at most 4 faces on $P$ since
the minimum distance between $p$ and $q$ in \figurename~\ref{fig:traverse}(a) is $\sqrt{3}$. In other words, the number of faces that an edge of length $\ell_{\max}$ has intersections is at most $\ell_{\max}\times \frac{4}{\sqrt{3}}$.
Next we turn to the integral rectangular faces. Let $a$ be the length of the shortest edge of the rectangles. Then $a$ is an integer by assumption. Thus, by a similar argument with \figurename~\ref{fig:traverse}(b), we can show that the number of faces that an edge of length $\ell_{\max}$ has intersections is at most $\ell_{\max}\times \frac{2}{a}$. When a face is a regular pentagon, we can use similar arguments.
By $\ell_{\max}\le D_P$ and $2D_P<L_P$, we have $M=O(D_P)$ and $M=O(L_P)$ in both cases. \qed\end{proof}
We note that when the minimum angle in the faces and the minimum distance between two non-adjacent edges of $Q$ are bounded by some constants, we have the same claim with some geometric parameters given by the bounds.
\begin{figure}
\caption{(a) An edge of length $\sqrt{3}$ can penetrate at most 4 regular triangles, and (b) an edge of length $a$ can penetrate at most 2 rectangles.}
\label{fig:traverse}
\end{figure}
Let $Q$ be a polyhedron and $q$ be a point on $Q$. The \emph{curvature} $\cur{q}$ at $q$ is the angle defined by the value $360^\circ-A$ and \emph{co-curvature} at $q$ is the angle $A$, where $A$ is the total angle of the angles on faces of $Q$ adjacent to $q$. That is, the curvature of $q$ on a convex polyhedron $Q$ is less than $360^\circ$ if and only if $q$ is a vertex of $Q$.
We will use the following Gauss-Bonnet Theorem: \begin{theorem}[{Gauss-Bonnet Theorem}] \label{th:gauss} The total sum of the curvature of all vertices of a convex polyhedron is $720^\circ$. \end{theorem} See \cite[Sec.~21.3]{DemaineORourke2007} for details.
\begin{figure}
\caption{A simple development of a cube overlaps.}
\label{fig:cube-overlap}
\end{figure}
Let $Q$ be a convex polyhedron. A \emph{development} of a convex polyhedron $Q$ results when we cut $Q$ along a set of polygonal lines and unfold on a plane. If the development produces a connected simple non-overlapping polygon $P$, we call $P$ a \emph{net} of $Q$. We note that the property of non-overlapping is counterintuitive; we may have it even for a simple development of a cube (\figurename~\ref{fig:cube-overlap}).
We assume that any cut ends at a point with curvature less than $360^\circ$. Otherwise, since $Q$ is convex, it makes a redundant cut on $P$, which can be eliminated (the proof can be found in \cite[Theorem 3]{MU2008}).
Let $T$ be the set of cut lines on $Q$ to obtain a net $P$. Then the following is well known as a basic property of unfolding (see \cite[Sec.~22.1.3]{DemaineORourke2007} for details): \begin{theorem} \label{th:spanning} $T$ forms a spanning tree of the vertices of $Q$. \end{theorem}
\subsection{Properties of Unfolding}
A \emph{tetramonohedron} is a tetrahedron that consists of four congruent acute triangles. We here note that a vertex of curvature $180^\circ$ on a polyhedron can be on an edge on its unfolding. Therefore, we need some ideas to find vertices of curvature $180^\circ$ of $Q$ on a polygon $P$ which may be a net of $Q$. In this context, a tetramonohedron has a special property:
\begin{lemma} \label{lem:tetra} Let $Q$ be a convex polyhedron. Then $Q$ is a tetramonohedron if and only if the curvature of every vertex is $180^\circ$. \end{lemma}
\begin{proof} If $Q$ is a tetramonohedron, by its symmetric property, each vertex $q$ consists of three distinct angles of a congruent triangle. (Otherwise, we cannot glue corresponding edges to form $Q$ from four acute congruent triangles.)
Thus the curvature at $q$ is $180^\circ$.
In order to show the opposite, we assume that every vertex of a convex polyhedron $Q$ has curvature $180^\circ$. Then, by the Gauss-Bonnet Theorem, $Q$ has four vertices. Let $q_0,q_1,q_2,q_3$ be these four vertices. We cut along three shortest straight lines $q_0q_1$, $q_0q_2$, and $q_0q_3$ on $Q$, respectively. (We note that it is called star unfolding in literature \cite[Sec.~24.3]{DemaineORourke2007}.) Since the curvature at any point on $Q$ except $q_0,q_1,q_2,q_3$ is $360^\circ$, we can take three non-crossing straight lines from $q_0$ to $q_1$, $q_2$, and $q_3$ on $Q$.
By developing $Q$ from $q_0$ along these three cut lines, we obtain a polygon $P=(q_0,q_1,q_0',q_2,q_0'',q_3,q_0)$. Then, by assumption, curvatures at $q_1$, $q_2$, $q_3$ are all $180^\circ$. That is, $P$ is a triangle with three vertices $q_0$, $q_0'$, and $q_0''$. Moreover, each edge of the triangle consists of two cut lines which form an edge on $Q$. Therefore, $q_1$, $q_2$, and $q_3$ are all the middle points of three edges $q_0q_0'$, $q_0'q_0''$, and $q_0''q_0$ of the triangle $P=(q_0,q_0',q_0'',q_0)$, respectively. Thus all four triangles $q_0q_1q_3$, $q_1q_0'q_2$, $q_3q_2q_0''$, and $q_2q_3q_1$ are congruent. If they are not acute, $P$ cannot fold to any solid by these crease lines (see, e.g., \cite{Uehara2020}). Therefore, we can conclude that $Q$ is a tetramonohedron. \qed\end{proof}
Combining Lemma \ref{lem:tetra} and Theorem \ref{th:gauss}, we have the following corollary. \begin{cor} \label{cor:nontetra} Let $Q$ be a convex polyhedron. When $Q$ is not a tetramonohedron, $Q$ has at least two vertices of curvature not equal to $180^\circ$. \end{cor} \begin{proof} Let $q_0,\ldots,q_k$ be the vertices of $Q$. Since $Q$ is a convex polyhedron, $k\ge 3$. When $k=3$, the only possible solid is a doubly covered triangle. Then $Q$ satisfies the claim. If $k>4$, by Theorems \ref{th:gauss}, at least two vertices have curvature not equal to $180^\circ$. Thus we focus on the case $k=4$. By Lemma \ref{lem:tetra}, since $Q$ is not a tetramonohedron, four vertices cannot have curvatures equal to $180^\circ$. By the Gauss-Bonnet Theorem, it is impossible that three vertices have curvature $180^\circ$ except one.
Thus $Q$ has at least two vertices $q$ and $q'$ of curvature not equal to $180^\circ$. \qed\end{proof}
The following simple lemma is basic but important: \begin{lemma}[{\cite{DemaineORourke2007}}] \label{lem:boundary} Let $Q$ be a convex polyhedron, and $P$ be a net of $Q$. Then all vertices of $Q$ appear on $\partial P$. \end{lemma} \begin{proof} Since $Q$ is convex, each vertex of $Q$ has positive curvature. Hence it cannot correspond to an interior point of $P$. Thus we have the lemma. \qed\end{proof}
The following lemma is useful for dealing with polyhedra except tetramonohedra. \begin{lemma} \label{lem:non180} Let $Q$ be a convex polyhedron that is not a tetramonohedron, and $P$ be a net of $Q$. Then $P$ has at least two vertices of angle not equal to $180^\circ$ that correspond to distinct vertices of $Q$. Moreover, if $Q$ has no vertex of curvature $180^\circ$, $P$ has at least two vertices such that each of the vertices on $P$ is glued to the corresponding vertex of $Q$ without extra angle from $P$. \end{lemma}
\begin{proof} By Corollary \ref{cor:nontetra}, $Q$ has at least two vertices $q$ and $q'$ of curvature not equal to $180^\circ$. Then, by Lemma \ref{lem:boundary}, $q$ and $q'$ correspond to distinct vertices of $P$. Now consider the set $S_q$ of vertices of $P$ that are glued together to form $q$. Then, since the curvature of $q$ is not equal to $180^\circ$ and less than $360^\circ$, at least one of the elements in $S_q$ has an angle not equal to $180^\circ$. Thus $q$ produces at least one vertex on $\partial P$ of angle not equal to $180^\circ$. We have another vertex on $\partial P$ produced by $q'$ by the same argument.
Now we assume that $Q$ has no vertex of curvature $180^\circ$. By Theorem \ref{th:spanning}, the set of cut lines forms a spanning tree of the vertices of $Q$. Then each leaf of the tree corresponds to a vertex of $Q$, and this vertex forms a vertex of $P$ since the curvature is not $180^\circ$. Since any tree has at least two leaves, we have the lemma. \qed \end{proof}
\subsection{Outline of Algorithms}
Lemma \ref{lem:non180} plays an important role in this paper except Section \ref{sec:tetra}. For each (regular) polyhedron except tetramonohedron, we can use at least two vertices $p_i$ and $p_{i'}$ of $P$ such that these vertices can fold to two vertices $q_j$ and $q_{j'}$ without any extra vertices of $P$, respectively. That is, intuitively, we can start gluing at the vertices $p_i$ and $p_{i'}$ of $P$ to fold $q_j$ and $q_{j'}$ of $Q$ and each gluing can be done by zipping from the vertex locally. In the context of stamping, the outline of the algorithms can be described as follows.
\begin{algorithm}[h]
\caption{Common outline of our folding algorithm (except tetramonohedron)}
\label{alg:common}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input{A polygon $P=(p_0,p_1,\ldots,p_{n-1},p_0)$ and a convex polyhedron $Q$}
\Output{All ways of folding $P$ to $Q$ (if one exists)}
Let $\{q_0,\ldots,q_{m-1}\}$ be the set of vertices of $Q$\;
\ForEach{pair of two vertices $\{p_{i},p_{i'}\}$ of $P$}{
\ForEach{vertex $q_j$ of $Q$}{
Check if $Q$ is reachable from $p_i$ to $p_{i'}$ on $P$ by stamping $Q$
so that $p_i$ and $p_{i'}$ correspond to $q_j$ and $q_{j'}$, respectively,
for some $q_{j'}$ with $j\neq j'$ on $Q$\;
Check if $P$ is a net of $Q$ by folding and gluing $P$ based on the partition of $P$ by stamping of $Q$\;
}
} \end{algorithm} The stamping operation and the check of the gluing are key aspects in the algorithm. We describe the details of each of them.
\subsection{Stamping} \label{sec:stamping}
When we fold $P$ to $Q$, we have two viewpoints depending on which we focus on. One viewpoint is that we fix $Q$ in 3D space and ``wrap'' $Q$ by folding $P$ on it. The other is that we put $P$ on a plane and ``roll'' $Q$ on it. The second idea is called \emph{stamping} and used to give a characterization of a net of a regular tetrahedron in \cite{Akiyama2007}. In \cite{Akiyama2007}, Akiyama rolls a regular tetrahedron on a plane as a \emph{stamper} and obtains a tiling by the stamping. The key property of the stamping in \cite{Akiyama2007} is that a regular tetrahedron has the same direction and position when it returns to the original position, no matter what the route is. Therefore, the cut lines of any net on the surface of a regular tetrahedron tile plane, or the net tiles plane based on $180^\circ$ rotation symmetry.
In \cite{MHU2020}, the authors extend the idea of stamping to the box folding problem. When the stamper is a box, the positions and directions are changed by the rolling, however, all faces on the plane are always orthogonal,
or each edge of a face is always parallel to one of the $xy$-axes
defined by the first face on the plane.
In this paper, we extend the idea to the general polyhedra. We first give the details of the stamping. It consists of two steps.
In the first step, we put $Q$ on $P$. Precisely, we put the vertex $q_0=(0,0,0)$ of $Q$ on the vertex $p_0=(0,0)$ of $P$ on the plane. We adjust their relative angle suitably, which will be discussed later for each case. (In fact, this ``first position'' is the first crucial point.) We assume that the face $f_0$ of $Q$ is put on the plane. (The local coordinate of $q_0$ on $f_0$ is $(f_0;0,0)$.) We consider the intersection of $f_0$ and $P$. In general, the intersection is not connected; it is a set of simple polygons. Among them, we define a simple polygon $F_0$ by a simple polygon that contains $p_0=q_0$. When two or more simple polygons share the point $p_0$ in the intersection, we choose any one of them. This $F_0$ is said to be the \emph{initial region} of $P$. From $F_0$, we obtain a sequence of regions $F_i$ with $i>0$ by stamping.
Now we consider the intersection of two boundaries $\partial F_0 \cap \partial P$. We classify each edge $e={q_i,q_{i'}}$ of $f_0$ to two groups. We first note that by Lemma \ref{lem:boundary}, if $q_i$ and $q_{i'}$ should be outside of $P$ or on $\partial P$. Otherwise, we cannot fold $Q$ by $P$ from this first position. If $e \cap (\partial F_0 \cap \partial P)$ contains no point in $P\setminus \partial P$, we say that $e$ is \emph{closed}. Intuitively, the face $f_0$ contains no more points in $P$ beyond the closed edge $e$, and hence we do not need to roll $Q$ on the edge $e$ to extend the net. Otherwise, we say that $e$ is \emph{open}. For each open edge $e$ on $F_0$, the net should be extended beyond $e$, and hence we roll $Q$ along $e$ from $F_0$ to obtain the next face. We note that $e \cap (\partial F_0 \cap \partial P)$ can consist of two or more line segments. Let $e_1$ and $e_2$ be two distinct line segments in $e \cap (\partial F_0 \cap \partial P)$. Then we obtain two different faces $F_1$ and $F_2$ sharing $e_1$ and $e_2$ with $F_0$, respectively. After rolling $Q$ along $e_0$, we have the following claim: $F_1\cap F_2=\emptyset$. Otherwise, $P$ should contain a hole surrounded by $F_0,F_1,F_2$, which contradicts the assumption that $P$ is a simple polygon. Therefore, for each line segment in $e \cap (\partial F_0 \cap \partial P)$ for each edge $e$ of $f_0$, we obtain a new set of faces $F_1,F_2,\ldots$ surrounding $F_0$.
For the notational simplicity, we number the faces in $f_i\cap P$ in this manner. That is, from $F_0$, we give the index of each intersection $f_i\cap P$ as $F_1,F_2,\ldots,$ in the breadth-first search (BFS) manner. We here define the distance $\dist(F_0,F_i)$ for each $i>0$ by the number of rollings made to reach to $F_i$ in the shortest way. That is, $\dist(F_0,F_0)=0$ and $\dist(F_0,F_i)=\min_{i'}\dist(F_0,F_{i'})+1$, where $F_{i'}$ is a region that sharing an edge $e$ with $F_i$. Intuitively, $F_i$ is obtained by rolling $Q$ from $F_{i'}$ along the edge $e$. By the property of the BFS, we have $i'<i$ for them.
Aside the numbering, the implementation of the stamping is simpler when we apply the depth-first search (DFS) manner rather than the BFS manner. That is, starting from $F_0$, as the second step of the stamping, we repeat rolling $Q$ as long as we can along open edges, and back to the last unvisited open edge when we get stuck. Therefore, hereafter, we will describe the algorithm in DFS style, while the indices of regions $F$ are numbered in BFS manner, and these indices are supposed to be precomputed in the same time bound of the algorithm.
In the stamping, the following observation is important: \begin{obs} \label{obs:out} Let $P$ be a net of $Q$. When we perform the stamping of $Q$ on $P$ starting from the initial region $F_0$, and it is the right place and the right direction to fold $Q$, no vertex of $Q$ is placed inside of $P$. \end{obs} \begin{proof} If a vertex $q$ is put inside of $P$, $q$ cannot be folded from $P$ since it has curvature less than $360^\circ$. It contradicts to the assumption that $P$ is a net of a convex polyhedron $Q$. \qed \end{proof}
Let $\calF=\{F_0,F_1,\ldots\}$ be the set of regions starting from the initial region $F_0$. Now we define a graph $T(P,Q,F_0)=(\calF,E)$ be a contact graph of $\calF$. That is, $\{F_i,F_j\}\in E$ if and only if they share an open edge $e$ such that $F_j$ is stamped by rolling $Q$ from $F_i$ along the edge $e$ or vice versa. Then the following lemma states that the graph $T(P,Q,F_0)$ is a well-defined tree if $P$ is a net of $Q$ and $F_0$ is the right initial region. \begin{lemma} \label{lem:tree} Let $P$ be a net of $Q$. We assume that we perform the stamping of $Q$ on $P$ starting from an initial region $F_0$, and it is the right place and the right direction to fold $Q$. Then the contact graph $T(P,Q,F_0)$ is a tree. \end{lemma} \begin{proof} By Observation \ref{obs:out}, each region $F_i$ with $i>0$ is defined by some rolling. Thus the contact graph is trivially connected. To show that $T(P,Q,F_0)$ is acyclic by contradictions, we consider two cases that may produce a cycle in the contact graph.
The first case is that a region $F_i$ from $F_{i'}$ by rolling and the other region $F_j$ from $F_{j'}$ have a collision; that is, $F_i\cap F_j\neq\emptyset$ in $P$. In this case, we have $i'<i$ and $j'<j$. We assume that this pair $(i',j')$ is the minimum in the lexicographical ordering. (Note that $i'=j'$ is possible.) Without loss of generality, we assume that $i<j$. The contact graph induced by $\{F_0,F_1,\ldots,F_{j-1}\}$ is a tree that contains $F_{i'}$, $F_{j'}$, and $F_{i}$, and the one induced by $\{F_0,F_1,\ldots,F_{j}\}$ has a cycle by adding $F_j$, which overlaps with $F_i$. In this case, if $F_i$, $F_j$, and other regions surround a space, it contradicts that $P$ is a simple polygon with no hole. On the other hand, when $F_i$ and $F_j$, and other regions share a point (for example, $F_{i'}=F_{j'}$, $F_i$ and $F_j$ share a vertex of $Q$), it means that a vertex of $Q$ is put inside of $P$, which contradicts Lemma \ref{lem:boundary}.
The second case is that a region $F_k$ is obtained from $F_{k'}$ and $F_{k''}$ with $k>k'>k''$. That is, $F_k$ and $F_{k'}$ share an edge $e'$, $F_k$ and $F_{k''}$ share an edge $e''$, and $e'\neq e''$ with $k'\neq k''$. In this case, we can use a similar argument in the first case by considering as $F_{k'}=F_{i'}$, $F_{k''}=F_{j'}$, and $F_{k}=F_{i}=F_{j}$. That is, we have a hole in $P$ or a vertex of $Q$ is inside of $P$.
By arguments above, we can conclude that $T(P,Q,F_0)$ forms a tree if $P$ is a net of $Q$ and $F_0$ is the right initial region. \qed \end{proof}
Hereafter, we assume that $T(P,Q,F_0)$ is a tree (otherwise, the algorithm rejects the input) rooted at $F_0$. Therefore, we can uniquely define the parent $F_i$ for each region $F_j$ in $\calF$ (except $F_0$), and it is easy to observe that $F_j$ is obtained by rolling $Q$ from $F_i$ along an open edge $e$ shared by $F_i$ and $F_j$. Thus we obtain the following theorem: \begin{theorem} \label{th:stamping} Let $P$ be a net of $Q$. We assume that we perform the stamping of $Q$ on $P$ starting from an initial region $F_0$, and it is the right place and the right direction to fold $Q$. Let $\calF$ be the set of regions obtained by the stamping of $Q$. Then the contact graph $T(P,Q,F_0)=(\calF,E)$ is a tree, and it can be computed in $O(\msize{\calF}n)$ time in general. When $Q$ is a regular polyhedron, the running time is reduced to $O(\msize{\calF}+n)$ time. \end{theorem} \begin{proof} From $F_0$, we traverse and construct the tree $T(P,Q,F_0)$ and obtain the sequence of regions $\calF=(F_0,F_1,F_2,\ldots)$ incrementally (in the DFS or BFS manner). At a region $F_i$, we roll $Q$ along an open edge and put a face $f$ of $Q$ on $P$, then we obtain the next region $F_{i'}$. In order to compute $F_{i'}$ that is the intersection of a face $f$ of $Q$ and $P$, it takes $O(n)$ time to traverse the vertices in $P$. The construction part of the tree $T(P,Q,F_0)$ requires time proportional to $\msize{\calF}$. Therefore, the running time of $T(P,Q,F_0)$ is $O(\msize{\calF}n)$ time in general.
When $Q$ is a regular polyhedron and each face $f$ is a fixed regular polygon, by performing DFS incrementally on $T(P,Q,F_0)$, we can construct it so that each vertex of $P$ is touched constant time. Then the running time can be reduced to $O(\msize{\calF}+n)$ time. \qed \end{proof}
The size $\msize{\calF}$ will be discussed in each case, which depends on the polyhedron $Q$. During the stamping, each region $F_i$ is stamped by a face of $Q$. At that time, each vertex $v$ of the polygonal region $F_i$ can also obtain the corresponding local coordinate $(f_i;x(v),y(v))$ on $Q$. Therefore, in the next phase, we can use the local coordinate of each vertex of a region $F_i$ in a constant time.
By the stamping of $Q$, the boundary $\partial P$ of the polygon $P=(p_0,p_1,\ldots,p_{n-1},p_n=p_0)$ is also partitioned into line segments which are intersections of faces of $Q$. Precisely, each point $p$ on $\partial P$ is (1) an original vertex $p_i$ for some $i$ or (2) an internal point of an edge $p_{i}p_{i+1}$ from the viewpoint of $\partial P$, and (a) a vertex $q_j$ for some face of $Q$, (b) an internal point of an edge of $Q$, or (c) an internal point of a face of $Q$. We consider $\partial P$ as a new polygon $P'=(p'_0,p'_1,\ldots,p'_{n'-1},p'_{n'}=p'_0)$, where $p'_{i}$ is a vertex in cases (1), (b), or (c). Intuitively, when $P'$ is a net of $Q$, each vertex of $P'$ is either a crease point of $P$ which should be folded to form $Q$ or a corner of $P$ of angle not equal to $180^\circ$.
As shown in Lemma \ref{lem:non180}, if $Q$ is not a tetramonohedron, we have at least two vertices on $P'$ that directly (i.e., without extra angle) correspond to the vertices of $Q$. During the stamping, we can obtain $P'$ and the set of such vertices without extra computation time within a constant factor. Let $S$ be the set of the vertices of $P'$, and call them \emph{gluing points} of $P$.
\subsection{Check of gluing} \label{sec:glue-check}
In this phase, we check if we can fold $Q$ from $P$ by folding along the crease lines given in the first stamping phase. It may seem to be the first phase is enough. However, we have not yet checked if some regions in $\calF$ cause overlap on a face of $Q$. In other words, we have to check each face of $Q$ is made by a certain set of regions of $P$ by gluing without overlap or hole.
This can be done by checking the new polygon $P'=(p'_0,p'_1,\ldots,p'_{n'-1},p'_{n'}=p'_0)$, which is constructed by the stamping. By Theorem \ref{th:spanning}, when $P$ is an unfolding of $Q$, the set $T$ of cut lines on $Q$ forms a spanning tree. Therefore, each line segment $\ell$ in $T$ appears twice as $\ell'$ and $\ell''$ on $P'$ (except its endpoints) and the pairs of $\ell'$ and $\ell''$ form a nest structure on $P'$. This nest structure is discussed in \cite[Sec.~25.2.1]{DemaineORourke2007} for a dynamic programming formulation of checking of all ways of edge-to-edge gluings. Their algorithm checks all ways of edge-to-edge gluing based on dynamic programming. On the other hand, in our glue checking, we can find each pair of line segments in the polygon $P'=(p'_0,p'_1,\ldots,p'_{n'-1},p'_{n'}=p'_0)$ by using their local coordinates. Precisely, for the given polygon $P'$, each point $p'_i$ has its local coordinate $(f;x,y)$ (or it has a list of local coordinates if it corresponds to a vertex of $Q$ or a point on an edge of $Q$).
\begin{theorem} \label{th:glue} Let $P'=(p'_0,p'_1,\ldots,p'_{n'-1},p'_{n'}=p'_0)$ be the polygon given by the stamping. Then the gluing check of $P'$ that asks if $P'$ can fold to $Q$ can be done in $O(n')$ time. \end{theorem} \begin{proof} We maintain $P'$ by a doubly linked list; each item corresponds to $p'_i$ which stores $p'_{i-1}$, $p'_{i+1}$, $\msize{p'_i p'_{i-1}}$, $\msize{p'_i p'_{i+1}}$, and $\angle p'_{i-1} p'_{i} p'_{i+1}$. We also inherit the set $S$ of gluing points in $P'$ from the stamping step
such that each gluing point $p$ has the angle $A$ at the point on $\partial P'$
that corresponds to a vertex $q$ of $Q$ with $\cur{q}=360^\circ-A$. Intuitively, each gluing point should be zipped up from the point to fold $Q$ from $P'$. In other words, each gluing point corresponds to a leaf of the spanning tree $T$ of the vertices of $Q$ to cut and unfold to $P$.
Therefore, our gluing starts from any gluing point. We pick up arbitrary gluing point $p'_i$ in $S$. Then glue two line segments $p'_{i-1}p'_i$ and $p'_ip'_{i+1}$ from $p'_i$. Now we have two cases. The first case is $\msize{p'_i p'_{i-1}} \neq \msize{p'_i p'_{i+1}}$. Without loss of generality, we assume that $\msize{p'_i p'_{i-1}} < \msize{p'_i p'_{i+1}}$. In this case, we glue up to $p'_{i-1}$ from $p'_i$. That is, we remove $p'_i$ from $S$ and $P'$, replace $p'_i$ in $p'_{i-1}$ by $p'_{i+1}$ with length $\msize{p'_{i} p'_{i+1}}-\msize{p'_{i-1} p'_{i}}$, and replace $p'_i$ in $p'_{i+1}$ by $p'_{i-1}$ with length $\msize{p'_{i} p'_{i+1}}-\msize{p'_{i-1} p'_{i}}$. The angle $\angle p'_{i-2} p'_{i-1} p'_{i}$ in $p'_{i-1}$ is replaced by $\angle p'_{i-2} p'_{i-1} p'_{i+1}=\angle p'_{i-2} p'_{i-1} p'_{i}+180^\circ$.
The second case is $\msize{p'_i p'_{i-1}} = \msize{p'_i p'_{i+1}}$. In this case, we glue $p'_{i-1} p'_i$ and $p'_{i} p'_{i+1}$ completely, remove $p'_i$ from $S$ and $P'$, and we merge $p'_{i-1}$ and $p'_{i+1}$ to a new vertex $p'$ on $P'$ so that $\msize{p'_{i-2} p'}=\msize{p'_{i-2} p'_{i-1}}$, $\msize{p',p'_{i+2}}=\msize{p'_{i+1} p'_{i+2}}$, and the angle at $p'$ is given by $\angle p'_{i-2} p' p'_{i+2}=\angle p'_{i} p'_{i-1} p'_{i-2}+ \angle p'_{i} p'_{i+1} p'_{i+2}$. If the angle at $p'$ is equal to the co-curvature at the corresponding point on $Q$ (which can be done by checking the corresponding local coordinate in a constant time), $p'$ is put into $S$ as a new gluing point. Otherwise, $p'$ is just a new vertex on $P'$.
From the viewpoint of the spanning tree $T$, each gluing step from a gluing point in $S$ corresponds to removal of one leaf from $T$. Therefore, it is not difficult to see that the algorithm correctly works. Each gluing process decreases at least one edge from $P'$ in a constant time. Therefore, the running time of the gluing check is $O(n')$ time. \qed \end{proof}
\section{Regular Tetrahedron and Tetramonohedron} \label{sec:tetra}
In this section, we give a pseudo-polynomial time algorithm for solving the folding problem for a tetramonohedron $Q$. We can solve the folding problem for a regular tetrahedron as a special case of it. Precisely, the input of the folding problem in this section is $P$ and three lengths $a,b,c$ of an acute triangle of four congruent faces of $Q$. To simplify, we denote by $Q(a,b,c)$ the tetramonohedron defined by the edge lengths. By the Heron's formula, the surface area of $Q$ is given by $4\sqrt{s(s-a)(s-b)(s-c)}$, where $s=(a+b+c)/2$. Thus, without loss of generality, we assume that the area of $P$ is equal to the surface area of $Q(a,b,c)$.
The reason why this special case is difficult is that each vertex of $Q(a,b,c)$ has curvature $180^\circ$ (Lemma \ref{lem:tetra}). When the cut line ends at a vertex and $Q(a,b,c)$ is unfolded, the vertex makes $180^\circ$ on $\partial P$. Thus we cannot find the vertex of $Q(a,b,c)$ on a straight line of $\partial P$ straightforwardly. The mathematical characterization of a net of a tetramonohedron is done by a tiling, which is called \emph{p2 tiling} in which rotation symmetry plays an important role. However, finding rotation centers is not easy since the point has curvature $180^\circ$. In order to construct the pseudo-polynomial time algorithm, we use the notion of Conway condition \cite{Schattschneider1980}.
\begin{figure}
\caption{A Conway tiling.}
\label{fig:Conway}
\end{figure}
Let $P$ be a polygon and $p,q$ be two points in $\partial P$. Then $[p,q]$ denotes the part of boundary $\partial P$ starting from $p$ to $q$ in counterclockwise. Then, a polygon $P$ is called \emph{Conway tile} if it has six points $A$, $B$, $C$, $D$, $E$, and $F$ in $\partial P$ in counterclockwise such that (1) $[A,B]$ can be moved to $[D,E]$ by translation $\tau$ with $\tau(A)=E$ and $\tau(B)=D$, (2) each of $[B,C]$, $[C,D]$, $[E,F]$, and $[F,A]$ is rotation symmetry with respect to the midpoint of it, and (3) at least three of these six points are distinct (\figurename~\ref{fig:Conway}). In \cite{AkiyamaMatsunaga2020}, they prove that $P$ is a net of a tetramonohedron if and only if $P$ is a Conway tile. Using this characterization, we show the following theorem:
\begin{theorem} \label{th:tetra} Let $P$ be a polygon of $n$ vertices, and $a,b,c$ be positive real numbers. We can decide whether $P$ can fold to a tetramonohedron $Q(a,b,c)$ or not in $O(L(L+n)n^2)$ time. \end{theorem}
Without loss of generality, we assume that the area of $P$ is equal to the surface area of $Q(a,b,c)$.
We prove the theorem by following the proof of the characterization in \cite{AkiyamaMatsunaga2020}.
Let four vertices of a given tetramonohedron $Q$ be $v_1,v_2,v_3,v_4$. As shown in Theorem \ref{th:spanning}, the set of cut lines of $Q$ to unfold $P$ forms a tree $T(Q)$ spanning all four vertices $v_1,v_2,v_3,v_4$. Thus $T(Q)$ contains at least two leaves of degree 1. We first note that no point $p$ on the surface of $Q$ cannot make a leaf in $T(Q)$ except on four vertices $v_1,v_2,v_3,v_4$ since the curvature at $p$ is $360^\circ$ (such a point is made by a ``redundant'' cut and reduced on $P$ when it is unfolded). That is, every leaf of $T(Q)$ corresponds to one of the four vertices of $Q$. The vertices of degree greater than 2 are also defined in the same manner as the graph theory. We now define the vertex of degree 2 more carefully. Let $p$ be a point in $T(Q)$ such that two line segments $\ell$ and $\ell'$ are incident to $p$. Then, we have two angles around $p$ between $\ell$ and $\ell'$. The point $p$ is a \emph{vertex} $p$ on $T(Q)$ of degree 2 if and only if one of these two angles is not equal to $180^\circ$. In other words, each point on $T(Q)$ is not considered as a vertex of degree 2 when the point is surrounded by two $180^\circ$ angles. Intuitively, two consecutive line segments in a kinked line on the surface of $Q$ share a ``vertex'' of $T(Q)$ of degree 2 if they make non-$180^\circ$ angle at the point. \begin{obs} \label{obs:deg2} Each vertex $v_i$ of $Q$ is always a vertex of $T(Q)$. \end{obs} \begin{proof} If $v_i$ has degree 1 or greater than 2, it is a vertex of $T(Q)$. If it has degree 2, it should be a vertex of $T(Q)$ since the curvature of $v_i$ is $180^\circ$. \qed\end{proof}
\begin{figure}
\caption{The two angles $a_R(\ell)$ and $a_L(\ell)$ for a leaf $\ell$.}
\label{fig:angles}
\end{figure}
Now we focus on the spanning tree $T(Q)$ of $Q$; every vertex $v_i$ of $Q$ is also a vertex of $T(Q)$, and every leaf of $T(Q)$ is a vertex of $Q$. For this tree $T(Q)$, we introduce some notations. For each leaf $\ell$ of $T(Q)$, the \emph{associated edge} $e(\ell)$ of $\ell$ is the unique edge incident to $\ell$. When $e(\ell)=\{u,\ell\}$, $u$ is the \emph{parent} of $\ell$. On $T(Q)$, $a_R(\ell)$ and $a_L(\ell)$ are the angles made by $e(\ell)$ with its neighbor edges sharing the parent of $\ell$ in clockwise and counterclockwise,
respectively (see \figurename~\ref{fig:angles}).
A leaf $\ell$ of $T(Q)$ corresponds to a vertex of a tetramonohedron $Q$. Therefore, the leaf is mapped to a unique point in $\partial P$. However, since $Q$ is a tetramonohedron, the curvature at $\ell$ is $180^\circ$ by Lemma \ref{lem:tetra}, hence it is an inner point on an edge (or a line segment) of $P$. Thus we need some tricks to find it for a given $P$. By the definition of vertices of $T(Q)$, if the leaf $\ell$ satisfies the following property $(*)$, $\ell$ is the midpoint of an edge:
\noindent{\sf Property $(*)$}: Both of $a_R(\ell)\neq 180^\circ$ and $a_L(\ell)\neq 180^\circ$.
If a leaf $\ell$ of $T(Q)$ has Property $(*)$, the neighbor points on $\partial P$ are vertices of $P$. That is, for the parent $u$ of $\ell$, we have an edge (or straight line) $u'u''$ on $\partial P$ such that $u'$ and $u''$ are vertices of $P$ and they are glued together to make the vertex $u$ on $Q$ and the midpoint of the line segment $u'u''$ in $\partial P$ corresponds to $\ell$ on $T(Q)$. In other words, when the curvatures at $u'$ and $u''$ are both not equal to $180^\circ$, it is easy to find the corresponding vertex $\ell$ of $T(Q)$, or a vertex $v_i$ on $Q$ of curvature $180^\circ$.
Since $T(Q)$ is a tree, it has at least two leaves. We first consider the case that two leaves have Property $(*)$ as Type 1. We have two exceptional cases, which will be handled as Type 2 and Type 3 later.
\paragraph{Type 1:} In Type 1, the $T(Q)$ has at least two leaves that satisfy Property $(*)$. Let $\ell$ be a leaf satisfying Property $(*)$ and $v$ be the parent of $\ell$. Then it is easy to see that there is a line segment made from two copies of the edge $v \ell$ of $T(Q)$ on $\partial P$. In other words, $\partial P$ has two consecutive vertices $p_i$ and $p_{i+1}$ such that the midpoint of the line segment $p_i p_{i+1}$ will be folded to $\ell$, which is one of $v_1,v_2,v_3,v_4$, and $p_i$ and $p_{i+1}$ are glued together to make the vertex $v$ on $Q$. Since $T(Q)$ has two leaves, we can obtain two of $v_1,v_2,v_3,v_4$. Thus we obtain the following algorithm:
\begin{algorithm}[h]
\caption{Folding algorithm for Type 1}
\label{alg:type1}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input{A polygon $P=(p_0,p_1,\ldots,p_{n-1},p_0)$ and three positive numbers $a,b,c$}
\Output{All ways of folding $P$ to a tetramonohedron $Q(a,b,c)$ in Type 1 (if one exists)}
\ForEach{pair of two edges $\{e,e'\}$ of $P$}{
take midpoints $m$ of $e$ and $m'$ of $e'$\;
\ForEach{triangular lattice defined by $(a,b,c)$ having two grid points $m$ and $m'$}{
Perform the stamping of $Q$ on $P$ along the lattice\;
Check if $P$ is a net of $Q$ by folding and gluing $P$ based on the partition of $Q$ by stamping\;
}
} \end{algorithm}
\begin{figure}
\caption{A polygon $P$ on a lattice by $(a,b,c)$.}
\label{fig:lattice}
\end{figure}
Since the algorithm checks all combinations, the correctness of Algorithm \ref{alg:type1} is trivial: It decides whether $P$ can fold to a tetramonohedron $Q$ when $T(Q)$ is in Type 1. In order to evaluate running time of Algorithm \ref{alg:type1}, we show a technical lemma: \begin{lemma} \label{lem:lattice} Let $P$ be a simple polygon $(p_0,p_1,\ldots,p_{n-1},p_0)$, and $m, m'$ be two midpoints of two edges $e,e'$ of coordinates $(x,y)$ and $(x',y')$. Let $Q(a,b,c)$ be a tetramonohedron having the surface area equal to the area of $P$. Then the number of triangular lattice defined by $(a,b,c)$ having two grid points $m$ and $m'$ is $O(L)$. \end{lemma} \begin{proof} An illustration of the desired lattice is depicted in \figurename~\ref{fig:lattice}. That is, a feasible lattice for $m$ and $m'$ has grid points on $m$ and $m'$. Then, we introduce two vectors $\vec{v_a}$ and $\vec{v_b}$ that span two edges of the unit lattice triangle of length $a$ and $b$, respectively. Then, it is easy to see that $m$ and $m'$ are on grid points if and only if there are two integers $k_a$ and $k_b$ such that $\vec{mm'}=k_a \vec{v_a}+k_b \vec{v_b}$. Then we have $-L/a \le k_a \le L/a$. Thus the number of possible $k_a$ is $O(L+n)$. Once $k_a$ is fixed, we can compute if $k_b$ is a reasonable integer or not. Therefore, we can conclude that the number of triangular lattices defined by $(a,b,c)$ having two grids $m$ and $m'$ is $O(L)$. \qed\end{proof}
Now we show the running time: \begin{lemma} \label{lem:type1} Algorithm \ref{alg:type1} runs in $O(L(L+n)n^2)$ time. \end{lemma} \begin{proof} The number of pairs of two edges is $O(n^2)$. The stamping of $Q$ on $P$ and gluing check of $P$ takes $O(L+n)$ time as shown in Theorem \ref{th:stamping} and Theorem \ref{th:glue}. By Lemma \ref{lem:lattice}, the number of possible triangular lattices for a given pair of grid points is $O(L)$. Thus Algorithm \ref{alg:type1} runs in $O(L(L+n)n^2)$. \qed \end{proof}
We here give two other exceptional cases, and we show that any $T(Q)$ is in one of Types 1, 2, and 3 later.
\begin{figure}
\caption{Each leaf $v_i$ satisfies $a_R(v_i)=180^\circ$ or $a_L(v_i)=180^\circ$ in Type 2.}
\label{fig:type2}
\end{figure}
\paragraph{Type 2:} In Type 2, the set of cut lines of $Q$ contains two independent line segments, say, $v_1 v_2$ and $v_3 v_4$. When we first cut along these lines, we obtain a cylinder, which is called the ``rolling belt'' in \cite{DemaineORourke2007}. After that, the cylinder is cut, unfolded, and $P$ is obtained from it. If the last cut line(s) does not touch any of four vertices of $Q$, we can observe that for each leaf $\ell$, one of $a_R(\ell)$ and $a_L(\ell)$ is $180^\circ$ (\figurename~\ref{fig:type2}).
Therefore, there are two edges $e$ and $e'$ in $\partial P$ corresponding to $v_1 v_2$ and $v_3 v_4$. Thus $e$ is parallel to $e'$ on $P$ and the length of $\msize{e}=\msize{e'}=2\msize{v_1v_2}=2\msize{v_3v_4}$. In this case, since a cylinder is obtained when $P$ is glued except $e$ and $e'$ to a tetramonohedron $Q$, we can obtain (another) tetramonohedron $Q'$ when we fold along the crease line joining two midpoints of $e$ and $e'$. In other words, when $P$ is given, we can fold infinitely many distinct tetramonohedra. Hence, without loss of generality, we assume that one endpoint of $e$ and another endpoint of $e'$ are vertices of curvature $180^\circ$ of a tetramonohedron $Q'$. Thus we can determine if $P$ can fold to a tetramonohedron by checking if $P$ can fold to a cylinder by gluing except $e$ and $e'$.
\begin{algorithm}[h]
\caption{Folding algorithm for Type 2}
\label{alg:type2}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input{A polygon $P=(p_0,p_1,\ldots,p_{n-1},p_0)$}
\Output{A way of folding $P$ to a tetramonohedron in Type 2 (if one exists)}
\ForEach{pair of two edges $\{e,e'\}$ of $P$}{
\If{if $e=\{v_i,v_{i+1}\}$ and $e'=\{v_{j},v_{j+1}\}$ are parallel and $\msize{e}=\msize{e'}$}{
Check if the path $(v_{i+1},\ldots,v_{j})$ can be glued to $(v_{j+1},\ldots,v_{i})$\;
\lIf{Two paths are glued}{output ``Yes''}
}
} \end{algorithm}
The matching of the paths $(v_{i+1},\ldots,v_{j})$ and $(v_{j+1},\ldots,v_{i})$ can be done by checking the following conditions: $\angle v_{i-1}v_{i}v_{i+1}+\angle v_{i}v_{i+1}v_{i+2}=180^\circ$, $\angle v_{j-1}v_{j}v_{j+1}+\angle v_{j}v_{j+1}v_{j+2}=180^\circ$, $\angle v_{i+k}v_{i+k+1}v_{i+k+2}+\angle v_{j-k}v_{j-k+1}v_{j-k+2}=360^\circ$ for each $k=0,1,2,\ldots$, $\msize{v_{i+k+1}v_{i+k+2}}=\msize{v_{j-k+1} v_{j-k}}$ for each $k=0,1,2,\ldots$, and the number of vertices in $v_{i+1},\ldots,v_j$ and the number of vertices in $v_{j+1},\ldots,v_{i}$ are equal.
The correctness of Algorithm \ref{alg:type2} in this case is trivial. \begin{lemma} \label{lem:type2} Algorithm \ref{alg:type2} runs in $O(n^3)$ time. \end{lemma} \begin{proof} The number of pairs of two edges is $O(n^2)$. The matching of two paths can be checked in linear time. Thus Algorithm \ref{alg:type2} runs in $O(n^3)$ time. \qed \end{proof}
\begin{figure}
\caption{Type 3: Only $v_3$ satisfies Condition $(*)$.}
\label{fig:type3}
\end{figure}
\paragraph{Type 3:} In Type 3, the set of cut lines of $Q$ contains two independent line segments, say, $v_1 v_2$ and $x v_4$ with $x\neq v_3$. In this case, only $v_3$ satisfies Property $(*)$. On $T(Q)$, we have the situation shown in \figurename~\ref{fig:type3}; $v_1$ and $v_2$ are joined by a straight line and there are three vertices $x,y,w$ such that $v_4$ and $x$ are joined by a straight line, $y$ is on the line segment $v_4 y$, and $v_3$ is joined to $y$ with some cut lines. (Note: $v_3$ can be $w$.) Therefore, on $\partial P$, $v_3$ is the midpoint of an edge of $P$, we have the same sequence of length from $v_3$ to both sides, and when we find the first pair $e$ and $e'$ of $\msize{e}>\msize{e'}$, then $e'=(w,y')$ and $e=(y'',v_4,y''',x)$, where $y',y'',y'''$ are the three vertices forming $y$ on $Q$. Hence we can find the point $v_4$ on $\partial P$ uniquely. Thus we can determine if $P$ can fold to a tetramonohedron $Q$ in Type 3.
\begin{algorithm}[h]
\caption{Folding algorithm for Type 3}
\label{alg:type3}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input{A polygon $P=(p_0,p_1,\ldots,p_{n-1},p_0)$}
\Output{A way of folding $P$ to a tetramonohedron in Type 3 (if one exists)}
\ForEach{an edge $e$ of $P$}{
Take the midpoint of $e$ as $v_3$\;
Find $v_4$ from $v_3$\;
Glue from $v_3$ and $v_4$ and obtain the points $x$, $y$, and $w$\;
Continue gluing from $y$ to the last edge\;
Check if $P$ is a net of a tetramonohedron by comparing $\msize{v_3v_4}$ and the last unglued edge\;
} \end{algorithm}
In the last step, when the algorithm obtains an open end of a cylinder by cutting along the edge $v_1v_2$, it concludes that if $P$ can fold to a tetramonohedron if $2\msize{v_3 v_4}$ is equal to the length of the open end (or cycle). The correctness of Algorithm \ref{alg:type3}, or if $P$ is a polygon that can fold to a tetramonohedron $Q$ when $T(Q)$ is in Type 3, follows the arguments in \cite{AkiyamaMatsunaga2020}. Thus we give the running time: \begin{lemma} \label{lem:type3} Algorithm \ref{alg:type3} runs in $O(n^2)$ time. \end{lemma} \begin{proof} The number of edges of $P$ is $O(n)$. The vertex $v_4$ can be found in linear time by just following both sides from $v_3$. The gluing can be done in $O(n)$ time from $v_3$ and $v_4$. Therefore, Algorithm \ref{alg:type3} runs in $O(n^2)$ time. \qed \end{proof}
Before the proof of the main theorem, we give a technical lemma for Property $(*)$: \begin{lemma} \label{lem:property} (1) For any leaf $\ell$, it satisfies Property $(*)$ if its parent $v$ has degree 2. (2) If two leaves $\ell$ and $\ell'$ share their parent $v$ of degree 3
and the angle between $e(\ell)$ and $e(\ell')$ is not $180^\circ$,
then at least one of $\ell$ and $\ell'$ satisfies Property $(*)$. (3) When four leaves share their parent $r$, at least two leaves satisfy Property $(*)$. \end{lemma} \begin{proof} (1) If the parent $v$ is a vertex of $Q$, we have $a_R(\ell)+a_L(\ell)=180^\circ$ and $0<a_R(\ell),a_L(\ell)<180^\circ$. Thus Property $(*)$ holds.
\noindent (2) Since $\ell$ and $\ell'$ share the angle, $a_R(\ell)=a_L(\ell')$ or $a_R(\ell')=a_L(\ell)$. Without loss of generality, we assume that $a_R(\ell)=a_L(\ell')$. Then, $a_R(\ell)\neq 180^\circ$ by assumption. Since $a_R(\ell)+a_R(\ell')+a_L(\ell)=360^\circ$, $a_R(\ell')+a_L(\ell)\neq 180^\circ$. Therefore, at least one of $a_R(\ell')$ and $a_L(\ell)$ is not equal to $180^\circ$. Thus at least one of $\ell$ and $\ell'$ satisfies Property $(*)$.
\noindent (3) Since $r$ has 4 children leaves, at most one angle can be equal to or greater than $180^\circ$, and the other three angles are consecutively less than $180^\circ$. Let $\ell$ and $\ell'$ be the leaves between these three consecutive angles less than $180^\circ$. Then these two leaves satisfy Property $(*)$.
\qed \end{proof}
We now turn to the proof of the main theorem in this section: \begin{proof} (of Theorem \ref{th:tetra}) For a given polygon $P$, we perform three Algorithms \ref{alg:type1}, \ref{alg:type2}, and \ref{alg:type3} one by one. By Lemmas \ref{lem:type1}, \ref{lem:type2}, and \ref{lem:type3}, the running time is $O(L(L+n)n^2)$ time in total. Thus it is sufficient to show that any cut lines of $Q$ to obtain $P$ should be one of these types.
\begin{figure}
\caption{X-shape, Y-shape, U-shape, F-shape, and H-shape.}
\label{fig:Shapes}
\end{figure}
From now on, we consider the topological structure of $T(Q)$, and show that most cases are in Type 1 except two special cases, which imply Type 2 and Type 3. According to the case analysis in \cite{AkiyamaMatsunaga2020}, $T(Q)$ has one of the following topological structures of X-shape, Y-shape, U-shape, F-shape, and H-shape (\figurename~\ref{fig:Shapes}):
\paragraph{X-shape:} All of $v_1,v_2,v_3,v_4$ are leaves, and there is a vertex $r$ in $T(Q)$ with $\deg(r)=4$. By Lemma \ref{lem:property}(3), this case is in Type 1.
\paragraph{Y-shape:} Three of $v_1,v_2,v_3,v_4$ are leaves, and the last one is a vertex of degree 3. Without loss of generality, we assume that $\deg(v_1)=3$ and $\deg(v_2)=\deg(v_3)=\deg(v_4)=1$. Then none of three angles $\angle v_2 v_1 v_3$, $\angle v_3 v_1 v_4$ and $\angle v_4 v_1 v_2$ is equal to $180^\circ$ since $v_1$ is a vertex of $Q$ of curvature is $\angle v_2 v_1 v_3+\angle v_3 v_1 v_4+\angle v_4 v_1 v_2=180^\circ$, and $0<\angle v_2 v_1 v_3,\angle v_3 v_1 v_4,\angle v_4 v_1 v_2$. Hence this case is in Type 1.
\paragraph{U-shape:} Two of $v_1,v_2,v_3,v_4$ are leaves, and the other two are vertices of degree 2. Without loss of generality, we assume that $\deg(v_1)=\deg(v_2)=2$ and $\deg(v_3)=\deg(v_4)=1$, and $v_1$ is closer to $v_4$ than $v_2$. If $T(Q)$ has a vertex $u$ of degree 2 between $v_1$ and $v_4$, $v_4$ has Property $(*)$ by Lemma \ref{lem:property}(1). Thus we consider the case that $v_1$ is the parent of $v_4$. However, then $v_4$ has Property $(*)$ by Lemma \ref{lem:property}(1) for the parent $v_1$. The leaf $v_3$ also satisfies Property $(*)$ by the same argument with $v_2$. Thus this case is in Type 1.
\paragraph{F-shape:} Three of $v_1,v_2,v_3,v_4$ are leaves, and the last one is a vertex of degree 2, and $T(Q)$ has another vertex $r$ of $\deg(r)=3$. Without loss of generality, $v_1$ is the vertex of degree 2, and $v_2$ is the leaf reachable to $v_1$ without through $r$. Then, by the same argument of U-shape, $v_2$ satisfies Property $(*)$. In the same way, if $v_3$ or $v_4$ has other vertices of degree 2 on the way to $r$, it satisfies Property $(*)$. Thus, we consider the other case that both $v_3$ and $v_4$ are children of $r$. If the angle $\angle v_3 r v_4\neq 180^\circ$, by Lemma \ref{lem:property}(2), one of $v_3$ and $v_4$ satisfies Property $(*)$. On the other hand, when $\angle v_3 r v_4 = 180^\circ$, we have two independent line segments $v_1 v_2$ and $v_3 v_4$. This case is in Type 2. Intuitively, the cut lines $v_1 v_2$ and $v_3 v_4$ open $Q$ to a cylinder, and the cylinder is open by cutting line segment(s) joining $v_1$ and $r$.
\paragraph{H-shape:} All of $v_1,v_2,v_3,v_4$ are leaves, and there are two vertices $r_1$ and $r_2$ in $T(Q)$ with $\deg(r_1)=\deg(r_2)=3$. We assume that $r_1$ has children $v_1$ and $v_2$, and $r_2$ has children $v_3$ and $v_4$. (When the other vertices are between them, we can reduce to the other cases above.) If $\angle v_1 r_1 v_2\neq 180^\circ$ and $\angle v_3 r_2 v_4\neq 180^\circ$, by Lemma \ref{lem:property}(2), two vertices $r_1$ and $r_2$ have at least one leaf satisfying Property $(*)$. On the other hand, when $\angle v_1 r_1 v_2 =180^\circ$ and $\angle v_3 r_2 v_4 =180^\circ$, we have the case in Type 2. The last case is that, without loss of generality,
$\angle v_1 r_1 v_2 = 180^\circ$ and $\angle v_3 r_2 v_4\neq 180^\circ$. Let $r'$ be the third neighbor of $r_2$ other than $v_3$ and $v_4$ (it can be $r'=r_2$). If $\angle r' r_2 v_3\neq 180^\circ$ and $\angle r' r_2 v_4\neq 180^\circ$, both of $v_3$ and $v_4$ satisfy Property $(*)$, which is in Type 1. Type 3 is the last remaining case that $\angle v_1 r_1 v_2 =180^\circ$ and $\angle v_4 r_2 r'=180^\circ$.
Since all cases are covered by the results in \cite{AkiyamaMatsunaga2020}, we can conclude that three Algorithms \ref{alg:type1}, \ref{alg:type2}, and \ref{alg:type3} can determine whether $P$ can fold to $Q$ or not. \qed \end{proof}
\begin{cor} \label{cor:tetra} Let $P$ be a polygon of $n$ vertices. We can decide whether $P$ can fold to a regular tetrahedron or not in $O(L(L+n)n^2)$ time. \end{cor} \begin{proof} When $Q$ is a regular tetrahedron, by letting $a=b=c$, we obtain the same results as for the tetramonohedron. Thus the running time of Algorithm \ref{alg:type1} is $O(L(L+n)n^2)$. Algorithms \ref{alg:type2} and \ref{alg:type3} run in this bound, which completes the proof. \qed \end{proof}
\section{Cube} \label{sec:cube}
In this section, the main goal in our context is the case of a cube $Q$, which consists of six squares and the co-curvature of each vertex is $270^\circ$. We however show a stronger result for general orthogonal boxes.
\begin{theorem} \label{th:cube} Let $a,b,c$ be any positive natural numbers. Then the folding problem of a box $Q$ of size $a\times b\times c$ from a given simple polygon $P$ can be solved in $O(D^2 n^3)$ time, where $n$ is the number of vertices in $P$, and $D$ is the diameter of $P$.
In our context, the running time can be represented as $O(L (L+n) n^2)$ time, where $L$ is the perimeter of $P$. \end{theorem}
In \cite{MHU2020}, the authors investigated the same folding problem for a box $Q$, and they gave a pseudo-polynomial time algorithm that runs in $O(D^{11}n^2(D^5+\log n))$ time. It checks all combinations of $a,b$, and $c$ in $O(D^4)$ time. Therefore, the algorithm in \cite{MHU2020} gives us $O(D^{7}n^2(D^5+\log n))$ time algorithm when $a$, $b$, and $c$ are explicitly given. Our algorithm drastically improves its running time to $O(D^2 n^3)$.
\begin{cor} \label{cor:cube} The folding problem of a cube $Q$ from a given simple polygon $P$ can be solved in $O(D^2 n^3)$ time, where $n$ is the number of vertices in $P$, and $D$ is the diameter of $P$. In our context, the running time can be represented as $O(L(L+n) n^2)$ time, where $L$ is the perimeter of $P$. \end{cor}
In our algorithm, the fact $a=b=c$ is not useful to improve the running time. Therefore, hereafter, we assume that $Q$ is a box of size $a\times b\times c$ (the size is explicitly given as a part of the input), $P$ is a simple polygon with diameter $D$ and perimeter $L$, and let $\ell_{\max}$ be the length of the longest edge of $P$.
\subsection{Algorithm}
\begin{algorithm}[th] \caption{Outline of the algorithm in \cite{MHU2020} (when $a,b,c$ are given).} \label{alg:1}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input{A polygon $P=(p_0,p_1,\ldots,p_{n-1},p_0)$}
\Output{A set $S=\{Q_0,Q_1,\ldots,Q_k\}$ of boxes of size $a\times b\times c$ that can be folded from $P$}
\For{$i\leftarrow 0$ \KwTo $n-1$}{
\If{curvature at $p_i$ is $270^\circ$}{
Find a position of $Q$ on $P$ such that $p_i$ is a vertex of $Q$,
and all vertices of $Q$ are not inside of $P$ by stamping\;
Check if $P$ can fold $Q$ by gluing, and output if it can\;
Find the next position by rotating $Q$ on $P$ at $p_i$ and repeat the check\;
}
} \end{algorithm}
The algorithm in \cite{MHU2020} can be outlined as Algorithm \ref{alg:1}. That is, the algorithm checks all possible points $p_i$ if it makes $270^\circ$. The key issue is to find the next position of $Q$ on $P$ by rotating at one point $p_i$ of curvature $270^\circ$. In \cite{MHU2020}, the upper bound of the number of possible rotations is the main factor of the running time.
By Lemma \ref{lem:non180}, if $Q$ can be folded from $P$, there are at least \emph{two} vertices on $P$ that fold to the vertices of $Q$. Our main idea is to use the second vertex as shown in Algorithm \ref{alg:2}.
\begin{algorithm}[th] \caption{Outline of our algorithm.} \label{alg:2}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input{A polygon $P=(p_0,p_1,\ldots,p_{n-1},p_0)$}
\Output{A set $S=\{Q_0,Q_1,\ldots,Q_k\}$ of boxes of size $a\times b\times c$ that can be folded from $P$}
\For{$i\leftarrow 0$ \KwTo $n-1$}{
\For{$j\leftarrow i+1$ \KwTo $n-1$}{
\If{curvature at $p_i$ and $p_j$ are $270^\circ$}{
Let $\ell:=(x(p_i)-x(p_j))^2+(y(p_i)-y(p_j))^2$\;
\For{$X\leftarrow 0$ \KwTo $\floor{\sqrt{\ell}}$}{
Let $Y:=\sqrt{\ell -X^2}$\;
\If{$Y$ is an integer}{
Set $x$-axis and $y$-axis by $X$ and $Y$\;
Put $Q$ on $P$ such that $p_i$ is a vertex of $Q$ along the axes\;
Check if all vertices of $Q$ are not inside of $P$ by stamping\;
Check if $P$ can fold to $Q$ by gluing, and output if it can\;
}
}
}
} } \end{algorithm}
We can use the same arguments in \cite{MHU2020} to show the correctness of our algorithm. Essentially, we check all the possible cases, which is guaranteed by Lemma \ref{lem:non180}. That is, if $P$ can fold to $Q$, there are two vertices $p_i$ and $p_j$ of curvature $270^\circ$ on the boundary of $P$. Our algorithm checks all combinations of them in $O(n^2)$ time. When we fix the right pair of $p_i$ and $p_j$, the edge $p_i p_j$ should be the oblique side of some right triangle $\angle p_i q p_j$ such that $x(q)=x(p_j)$ and $y(q)=y(p_i)$. Since we can assume an integral grid for the box $Q$, we have $\msize{p_i q}=X$ and $\msize{p_j q}=Y$ for some positive integers $X$ and $Y$, which give the $y$-axis and $x$-axis for box stamping. Then we have a lemma: \begin{lemma} \label{lem:XY} Assume that we fix the right pair of $p_i$ and $p_j$. Then the number of the corresponding pairs of $x$-axis and $y$-axis is $O(D)$. Since $2D\le L$, the number is also $O(L)$. \end{lemma} \begin{proof} As discussed above, $X$ is a positive integer with $0\le X\le \msize{p_i p_j}$ for the given $p_i$ and $p_j$. Since $\msize{p_i p_j}\le D$, we have the lemma. \qed\end{proof}
Once we fix the $x$-axis and $y$-axis with letting $p_i=(0,0)$ and $p_j=(X,Y)$, we now put $Q$ at $p_i$. Then, since the curvature at $p_i$ is $270^\circ$, at least one of four orthants is included in $P$. More precisely, for sufficiently small $\epsilon>0$, all points $(x,y)$ with $x^2+y^2\le \epsilon$ are in $P$ for at least one of the following four conditions: (1) $x\ge 0$ and $y\ge 0$, (2) $x\ge 0$ and $y\le 0$, (3) $x\le 0$ and $y\ge 0$, and (4) $x\le 0$ and $y\le 0$. Without loss of generality, we assume that we have case (1). Then we can put the box $Q$ at $p_i$ on this orthant. There are six possible cases. That is, the first rectangle occupies $(x,y)$ such that (1) $0\le x\le a$ and $0\le y\le b$, (2) $0\le x\le a$ and $0\le y\le c$, (3) $0\le x\le b$ and $0\le y\le c$, (4) $0\le x\le b$ and $0\le y\le a$, (5) $0\le x\le c$ and $0\le y\le a$, or (6) $0\le x\le c$ and $0\le y\le b$. We check these 6 cases one by one. From each of these six initial positions, we start stamping by rolling the box $Q$ on $P$. As discussed in Sections \ref{sec:stamping} and \ref{sec:glue-check}, we can check all possible ways of folding of $Q$ from $P$ in this way. Now we give a crucial lemma for the estimation of time complexity: \begin{lemma} \label{lem:rolling} The number of rollings of $Q$ on $P$ in each loop is $O(Dn)$. The number is also $O(L+n)$. \end{lemma}
\begin{proof} The stamping is done in the DFS manner. Then, by Observation \ref{obs:out}, any vertex of $Q$ cannot be inside of $P$ during the rolling (otherwise, $P$ is not a net of $Q$ at that point, and we refuse). Therefore, two rollings contribute at least length $a$ to consume the length of an edge of $P$ in the proof of Lemma \ref{lem:traverse}. That is, each edge of $P$ is consumed by $O(D)$ rollings of $Q$. Since $P$ has $n$ edges, the total number of rolling of $Q$ to consume all the edges of $P$ is $O(Dn)$. When all edges of $P$ are consumed, or covered by $Q$, the stamping is finished. Thus we have the lemma. In the same argument, each edge $e$ requires $O(\msize{e}/a)$ rolling. Thus the number is also $O(L+n)$ in total. \qed\end{proof} We now turn to the proof of the main theorem in this section:
\begin{proof}(of Theorem \ref{th:cube}) We owe the previous work in \cite{MHU2020} the proof of the correctness of our algorithm. Essentially, both algorithms check all feasible combinations that contain correct answers if they exist. Therefore, our remaining task is to show the time complexity of our algorithm.
As shown already, the first loop for $p_i$ and $p_j$ takes $O(n^2)$ time. For each pair of $p_i$ and $p_j$, the number of possible combinations of $X$ and $Y$ is $O(D)$ by Lemma \ref{lem:XY}. The number of ways of putting the box $Q$ is six, and for each of them, the number of rolling takes $O(Dn)$ time by Lemma \ref{lem:rolling}.
By Theorem \ref{th:stamping}, each stamping can be done in $O(Dn+n)$ time. By Theorem \ref{th:glue}, the gluing check takes $O(n')$ time, where $n'$ is the number of the vertices of $P'$. In the box case, we can see that $n'=O(D+n)$.
Therefore, total running time of our algorithm is $O(n^2\times D\times(Dn+n+D+n)\log n))=O(D^2 n^3)$.
When we use $L$ instead of $D$, the number of rolling is $O(L+n)$ by Lemma \ref{lem:rolling} and hence this step requires $O(L+n)$ time to traverse $\partial P$. Therefore, the running time becomes $O(L(L+n) n^2)$. \qed\end{proof}
\section{Deltahedra and Regular Dodecahedron} \label{sec:delta}
The common outline of our algorithms for a regular dodecahedron and a non-concave deltahedron is given in Algorithm \ref{alg:common}.
That is, the algorithm checks all possible combinations of pairs $\{p_{i},p_{i'}\}$ and $q_j$. By Lemma \ref{lem:non180}, if $Q$ can be folded from $P$, there are at least two vertices $p_{i},p_{i'}$ of $P$ that correspond to $q_{j},q_{j'}$ of $Q$ for some $q_{j'}$, respectively, with $i\neq i'$ and $j\neq j'$. Hereafter, we assume that the vertex $p_0$ of $P$ corresponds to the vertex $q_0$ of $Q$, and $p_{i}$ of $P$ corresponds to the vertex $q_j$ of $Q$, respectively, without loss of generality.
The key point is how to decide the relative orientation of $Q$ and $P$, which has an influence of time complexity of the algorithm. Intuitively, for this issue, we also try all possible cases. The time complexity (or the number of trials) is different depending on the shape of $Q$. For the remainder of Section \ref{sec:delta}, we assume that the orientation of $Q$ relative to $P$ is fixed.
\subsection{Regular Dodecahedron} \label{sec:dodeca}
In this section, we assume that $Q$ is a regular dodecahedron and the length of each edge is 1. Since the area of a pentagon of unit edge is $\frac{\sqrt{25+10\sqrt{5}}}{4}$, we assume the area of $P$ is $12\times \frac{\sqrt{25+10\sqrt{5}}}{4}=3\sqrt{25+10\sqrt{5}}$
without loss of generality. Since we know $Q$, the input of this problem is just a polygon $P=(p_0,p_1,\ldots,p_{n-1},p_0)$ of area $3\sqrt{25+10\sqrt{5}}$, and we will decide if $P$ can be folded to a unit-size regular dodecahedron $Q$. By Lemma \ref{lem:non180}, we also know that two vertices $p_0$ and $p_i$ of $P$ correspond to two distinct vertices, say $q_0$ and $q_j$, of $Q$. Then the main theorem in this section is as below.
\begin{theorem} \label{th:penta} Let $P$ be a simple polygon with $n$ vertices. We denote by $L$ the perimeter of $P$. Then the folding problem of a regular dodecahedron from $P$ can be solved in $O((L+n)^4 n^2)$ time. \end{theorem}
\begin{figure}
\caption{An example of overlapping stamping. Some pentagons are overlapping by stamping of $Q$ along a feasible net $P$.}
\label{fig:overlap}
\end{figure}
\subsubsection{Stamping}
By assumption, $Q$ can reach from $p_0$ to $p_i$ on $P$ by stamping $Q$ such that $p_0$ and $p_i$ are corresponding to two different vertices of $Q$. By rotation of $P$, we have a sequence of regular pentagonal faces $(\hat{f}_0,\hat{f}_1,\ldots,\hat{f}_k)$ such that (1) $\hat{f}_0$ contains the edge joining points $p_0=(0,0)$ and $(1,0)$ as its base edge, (2) $p_i=(x_i,y_i)$ is a vertex of $\hat{f}_k$, and (3) two consecutive pentagons $\hat{f}_{j'}$ and $\hat{f}_{j'+1}$ share an edge for each $j'$ with $0\le j'<k$. Intuitively, the sequence gives us the shortest way of stamping of $Q$ joining $p_0$ and $p_i$ on $P$. In other words, if we put $Q$ on $P$ with a proper relative angle, $Q$ can be unfolded to $P$, and we can reach from $p_0$ to $p_i$ by traversing the edges of these regular pentagons. We note that two consecutive pentagons do not overlap (without their shared edge), but nonconsecutive pentagons can overlap by stamping (see \figurename~\ref{fig:overlap}).
\begin{figure}
\caption{Four unit vectors for a unit pentagon.}
\label{fig:vectors}
\end{figure}
\begin{figure}
\caption{Two points $p_i$ and $p_j$ are close, however, they can be spanned by many vectors.}
\label{fig:penta}
\end{figure}
When we consider each edge of the pentagons as a unit vector, this traverse can be represented by a linear combination of the following four vectors (\figurename~\ref{fig:vectors}): $\vec{b_0}=(0,1)$, $\vec{b_1}=(\cos\frac{\pi}{5},\sin\frac{\pi}{5})$, $\vec{b_2}=(\cos\frac{2\pi}{5},\sin\frac{2\pi}{5})$, and $\vec{b_3}=(\cos\frac{3\pi}{5},\sin\frac{3\pi}{5})$. Note that $(\cos\frac{4\pi}{5},\sin\frac{4\pi}{5})= -\vec{b_0}+\vec{b_1}-\vec{b_2}+\vec{b_3}$. Thus, $Q$ can be folded from $P$ only if we have four integers $B_0,B_1,B_2,B_3$ such that \[ \vec{p_i}-\vec{p_0}=B_0\vec{b_0}+B_1\vec{b_1}+B_2\vec{b_2}+B_3\vec{b_3}, \] and hence \[ \msize{\vec{p_i}-\vec{p_0}}=\msize{B_0\vec{b_0}+B_1\vec{b_1}+B_2\vec{b_2}+B_3\vec{b_3}}. \]
We here note that $\msize{B_k}$ does not necessarily small even if $\msize{\vec{p_i}-\vec{p_0}}$ is small (\figurename~\ref{fig:penta}). When we consider a grid (as a triangular grid in Lemma \ref{lem:lattice} and a square grid in Lemma \ref{lem:XY}) which is spanned by two unit vectors, we can say that $\msize{B_k}\le \msize{\vec{p_i}-\vec{p_0}}$. In the case of a regular pentagon, we have no such a grid, and hence we bound the number $\msize{B_k}$ by the number of the stamping. That is, by Theorem \ref{th:stamping}, we have $\msize{B_k}=O(L+n)$. Thus, it is enough to check $O((L+n)^4)$ combinations of four integers $B_0,B_1,B_2,B_3$.
For each possible integers $B_0,B_1,B_2,B_3$, we can compute $p_i=(x_i,y_i)$ by rotation of $P$. After putting $P$ on the proper place so that $p_0=(0,0)$ and $p_i=(x_i,y_i)$, we perform the stamping of $Q$ on $P$ and obtain the partition of $P$. We here note that we use the commutative law of vectors. Thus the first relative position of $Q$ is one of four positions along $\vec{b_0}, \vec{b_1}, \vec{b_2}, \vec{b_3}$.
For each position, we perform the second phase for checking gluing.
\subsubsection{Gluing check}
By stamping of $Q$ on $P$, $P$ is partitioned into regions $\calF=\{F_0,F_1,\ldots,F_{h-1}\}$. More precisely, $F_0$ is the intersection of $P$ and $Q$ on an initial position such that $p_0=q_0=(0,0)$. Since it is a valid stamping, there are no vertices of $Q$ inside of $F_0$. As discussed in Lemma \ref{lem:tree}, the contact graph $T=(P,Q,F_0)$ is a tree. For notational convenience, we consider $F_0$ is the root of $T$, and the elements in $\calF$ are numbered from $F_0$ in the way of the BFS manner.
First, we glue $F_0$ on $Q$ so that the corresponding vertex $p_0$ on $P$ (or $F_0$) comes to a vertex $q_0$ of $Q$. Then the gluing process is done on $Q$ from $F_0$ in the BFS manner. As shown in Theorem \ref{th:stamping}, the stamping can be done in $O(\msize{\calF}n)$ time. We have the following upper bound of $\msize{\calF}$: \begin{theorem} \label{th:upper-penta} $\msize{\calF}=O(L+n)$. \end{theorem}
\begin{proof} The number of stampings of $Q$ on $P$ is given by the total number of visits of each region $F_i$. On the other hand, $\msize{\calF}$ is the number of $F_i$s. Thus, precisely, $(\msize{\calF}-1)$ is the number of the first visiting each $F_i$ by $Q$ except $F_0$. The stamping of $Q$ is done along the BFS tree. Therefore, since each edge of the BFS tree is traversed twice, the number of stampings made by $Q$ is $2(\msize{\calF}-1)$. Thus $\msize{\calF}$ is proportional to the number of stampings.
\begin{figure}
\caption{An edge $e$ can be covered by $O(\msize{e})$ pentagons since each angle of a pentagon is $108^\circ$.}
\label{fig:pentagons}
\end{figure}
Let $e$ be an edge of $P$. By stamping of $Q$ along the edge $e$, since each pentagonal face of $Q$ has the unit size, the number of pentagons $\hat{f}_i$ to cover $e$ is $O(\msize{e})$ by Lemma \ref{lem:traverse} with \figurename~\ref{fig:pentagons}. Thus, the number of pentagonal faces of $Q$ as stamps to cover all the edges $e$ of $P$ is $O(L+n)$ in total. Therefore, we obtain $\msize{\calF}=O(L+n)$. \qed \end{proof}
\subsubsection{Time complexity}
Now we consider the time complexity of our algorithm for a regular dodecahedron. For a given polygon $P=(p_0,p_1,\ldots,p_{n-1},p_0)$, the algorithm first generates all possible combinations of $(p_i,p_{i'})$, which produce $O(n^2)$ cases. We here note that we essentially have one way of choosing $q_0$ by the symmetry of $Q$. For this $q_0$, we have a constant number (precisely, it is 7) of cases of $q_j$. Thus we do not need to consider this constant factor for a regular dodecahedron. For each pair $(p_i,p_{i'})$, we construct a vector $\vec{p_{i'}}-\vec{p_i}=B_0\vec{b_0}+B_1\vec{b_1}+B_2\vec{b_2}+B_3\vec{b_3}$ by checking all possible values of $B_0,B_1,B_2,B_3$ with $B_k=O(L+n)$ for $k=0,1,2,3$. This step generates $O((L+n)^4)$ combinations if we check all combinations in a straightforward way. However, when $B_0,B_1,B_2$ are fixed, since $\msize{\vec{p_i}-\vec{p_0}}=\msize{B_0\vec{b_0}+B_1\vec{b_1}+B_2\vec{b_2}+B_3\vec{b_3}}$ is given, we have two possible values depending on $B_3\ge 0$ or $B_3<0$, and they can be computed in a constant time. Thus it is enough to check $O((L+n)^3)$ combinations by computing two candidates of $B_3$ from $B_0,B_1,B_2$. For each case, the algorithm performs stamping of $Q$. During the stamping, we check if each vertex of a face of $Q$ is inside $P$ or not. It is done along the traverse of the tree in BFS manner, and hence it can be done in $O(n)$ time in total. Thus the running time of stamping is $O(\msize{\calF}+n)$, where $(\msize{\calF}-1)$ is the number of stampings. By Theorem \ref{th:upper-penta}, we have $\msize{\calF}=O(L+n)$.
After the (valid) stamping, we obtain a partition $\calF=\{F_0,F_1,\ldots,F_{\msize{\calF}}\}$ of $P$. During the stamping, as discussed in Section \ref{sec:glue-check}, we have already constructed a refined polygon $P'=(p'_0,p'_1,\ldots,p'_{n'-1},p'_{n'}=p'_0)$ with the set $S$ of gluing points in $P'$. By Theorem \ref{th:glue}, checking the gluing of elements in $\calF$ onto $Q$ takes $O(n')$ time. Since the tree $T=(P,Q,F_0)$ has $\msize{\calF}$ vertices and $(\msize{\calF}-1)$ edges, we have $\sum_{i=0}^{\msize{\calF}-1}\msize{F_i}=O(\msize{\calF}+n)$, which is $O(L+n)$ by Theorem \ref{th:upper-penta}. Therefore, in total, the algorithm runs in $O((L+n)^4 n^2)$ time. It completes the proof of Theorem \ref{th:penta}.
\subsection{Non-concave Deltahedron}
In this section, we assume that $Q$ is a non-concave deltahedron such that it has at least two vertices of curvature not equal to $180^\circ$. We assume that each face of $Q$ consists of some unit regular triangles; each unit triangle has three edges of unit length 1 and area $\frac{\sqrt{3}}{4}$. Let $t$ be the total number of unit triangles on the surface of $Q$. That is, the surface area of $Q$ is $\frac{\sqrt{3}}{4}t$. Let $\{q_0,q_1,\ldots,q_{m-1}\}$ be the set of vertices of $Q$. We assume that (1) the set of faces $\{f_0,f_1,\ldots,f_{l-1}\}$ of $Q$ is given, where $l$ is the number of faces of $Q$, (2) each vertex $q_j$ has its coordinate $(x_j,y_j,z_j)$, and (3) each face has its vertices in clockwise order. The basic idea of the algorithm is the same as in Section \ref{sec:dodeca}; we here consider the differences.
\begin{theorem} \label{th:tri} Let $P$ be a simple polygon with $n$ vertices of perimeter $L$. Let $Q$ be a non-concave deltahedron\footnote{For simplicity, we call ``non-concave deltahedron'' a polyhedron that is either a convex deltahedron or a non-strictly-convex deltahedron, and we assume that it is not a regular tetrahedron.} with $m$ vertices. Then the folding problem of $Q$ from $P$ can be solved in $O(L(L+n) m n^2)$ time. \end{theorem}
We still have the property that we can reach from $p_0$ to $p_i$ on $P$ by stamping $Q$ on it. However, now we have $O(m n^2)$ combinations for pairs of pair $(p_i,p_{i'})$ and $q_j$. Hereafter, we assume that the vertex $p_{i}$ of $P$ forms a vertex $q_{j}$ on $Q$ and the vertex $p_{i'}$ forms some vertex $q_{j'}$ on $Q$. In the same argument in the tetrahedron case, for two vectors $\vec{b_0}=(1,0)$ and $\vec{b'_1}=(\cos\frac{\pi}{3},\sin\frac{\pi}{3})=(\frac{1}{2},\frac{\sqrt{3}}{2})$, $Q$ can be folded from $P$ only if we have two integers $B'_0$ and $B'_1$ such that \[ \msize{\vec{p_{i'}}-\vec{p_i}}=\msize{B'_0\vec{b_0}+B'_1\vec{b'_1}}. \] We have that $\msize{B'_0}$ and $\msize{B'_1}$ are at most $L$ by Lemma \ref{lem:lattice}, and hence we have $O(L^2)$ combinations to be checked. However, once we fix $B'_0$, then $B'_1$ has two possible values. Thus this step requires $O(L)$ combinations.
Each partition of $P$ by stamping of $Q$ takes $O(L+n)$ time by the same argument in the case of a dodecahedron.
For gluing, almost all arguments are the same as the pentagonal case since they do not use the fact that the shape of a face is a pentagon. The only difference is that we stamp all (possibly different) faces of $Q$; this fact gives us an additional lower bound $l$ of the number of stampings. Therefore, the time complexity of this algorithm for a non-concave deltahedron is $O(L(L+l+n)m n^2)$. Here, by the Euler characteristic, we have $l=2+e-m$, where $e$ is the number of edges of $Q$. When we consider $Q$ as a graph, it is a planar graph, which implies that $e=O(m)$, or $l=O(m)$. By Theorem \ref{th:gauss}, $Q$ has at most four vertices of curvature $180^\circ$. Thus we have $m=O(n)$. Therefore, the time complexity of this algorithm is $O(L(L+n)m n^2)$, which completes the proof of Theorem \ref{th:tri}.
Since $m$ is a constant for each of a regular octahedron and a regular icosahedron, we have the time complexities in Table \ref{tab:alg}.
\section{Concluding Remarks}
In this paper, we give a series of design scheme of pseudo-polynomial algorithms for solving the folding problem for given simple polygon $P$ and convex polyhedron $Q$. When $Q$ is a regular polyhedra (also known as a Platonic solid) or some variants, our algorithm runs efficiently. We have some open problems for extension.
The extension to convex polyhedra that consist of finite regular polygons is not so difficult. Most results in this paper work except the estimation of the number of possible vectors. If we allow to use non-regular polygons, it is not easy to estimate the number of possible vectors joining two common vertices of $P$ and $Q$. Thus we may need a different approach.
The extension to concave polyhedra is more challenging. In our algorithms, we use the convexity of a polyhedron in several places. For example, the contact graph of the faces of a polyhedron is not necessarily acyclic for a concave polyhedron (a simple example is given in \cite[Fig.~22.6]{DemaineORourke2007}). In such a case, the set of cut lines is not connected, and the contact graph is not a tree. Moreover, a vertex of $Q$ may have curvature $360^\circ$. Finding a nontrivial set of concave polyhedra that allows us to solve the folding problem efficiently is another interesting open problem.
\end{document}
\begin{figure}
\caption{Please write your figure caption here}
\label{fig:1}
\end{figure}
\begin{figure*}
\caption{Please write your figure caption here}
\label{fig:2}
\end{figure*}
\begin{table}
\caption{Please write your table caption here} \label{tab:1}
\begin{tabular}{lll} \hline\noalign{
} first & second & third \\ \noalign{
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} number & number & number \\ number & number & number \\ \noalign{
}\hline \end{tabular} \end{table}
\end{document} | arXiv |
Generalized Autoregressive Score (GAS) Models: EViews Plays with Python
Starting with EViews 11, users can take advantage of communication between EViews and Python. This means that workflow can begin in EViews, switch over to Python, and be brought back into EViews seamlessly. To demonstrate this feature, we will use U.S. macroeconomic data on the unemployment rate to fit a GARCH model in EViews, transfer the data over and estimate a GAS model equivalent of the GARCH model in Python, transfer the data back to EViews, and compare the results.
GAS Models
Example Description
Preparatory Work
Data Analysis in EViews
Data Analysis in Python
Back to EViews
Historically, time varying parameters have received an enormous amount of attention and the literature is saturated with numerous specifications and estimation techniques. Nevertheless, many of these specifications are often difficult to estimate, such as the family of stochastic volatility models, among which GARCH is a canonical example. In this regard, Creal, Koopman, and Lucas (2013) and Harvey (2013) proposed a novel family of time-varying parametric models estimated using the familiar maximum likelihood framework with the score of the conditional density function driving the updating mechanism. The family has now come to be known as the generalized autoregressive score (GAS) family or model.
GAS models are agnostic as to the type of data under consideration as long as the score function and the Hessian are well defined. In particular, the model assumes an input vector of random variables at time $ t $, say $ \pmb{y}_{t} \in \mathbf{R}^{q} $, where $ q=1 $ if the setting is univariate. Furthermore, the model assumes a conditional distribution at time $ t $ specified as: $$ \pmb{y}_{t} | \pmb{y}_{1}, \ldots, \pmb{y}_{t-1} \sim p(\pmb{y}_{t}; \pmb{\theta}_{t}) $$ where $ \pmb{\theta}_{t} \equiv \pmb{\theta}_{t} (\pmb{y}_{1}, \ldots, \pmb{y}_{t-1}, \pmb{\xi}) \in \Theta \subset \mathbf{R}^{r}$ is a vector of time varying parameters which fully characterize $ p(\cdot) $ and are functions of past data and possibly time invariant parameters $ \pmb{\xi} $.
What distinguishes GAS models from the rest of the literature is that dynamics in $ \pmb{\theta}_{t} $ are driven by an autoregressive mechanism augmented with the score of the conditional distribution of $ p(\cdot) $. In particular, $$ \pmb{\theta}_{t+1} = \pmb{\omega} + \pmb{A}\pmb{s}_{t} + \pmb{B}\pmb{\theta}_{t} $$ where $ \pmb{\omega}, \pmb{A}, $ and $ \pmb{B} $ are matrix coefficients collected in $ \pmb{\xi} $, and $ \pmb{s}_{t} $ is a vector proportional to the score of $ p(\cdot) $: $$ \pmb{s}_{t} = \pmb{S}_{t}(\pmb{\theta}_{t}) \pmb{\nabla}_{t}(\pmb{y}_{t}, \pmb{\theta}_{t}) $$ Above, $ \pmb{S}_{t} $ is an $ r\times r $ positive definite scaling matrix known at time $ t $, and $$ \pmb{\nabla}_{t}(\pmb{y}_{t}, \pmb{\theta}_{t}) \equiv \frac{\partial \log p(\pmb{y}_{t}; \pmb{\theta}_{t})}{\partial \pmb{\theta}_{t}}$$ It turns out that different choices of $ \pmb{S}_{t} $ produce different GAS models. For instance, setting $ \pmb{S}_{t} $ to some power $ \gamma > 0 $ of the information matrix of $ \pmb{\theta}_{t} $ will change how the variance of $ \pmb{\nabla}_{t} $ impacts the model. In particular, consider: $$ \pmb{S}_{t} = \pmb{\mathcal{I}}_{t}(\pmb{\theta}_{t})^{-\gamma} $$ where $$ \pmb{\mathcal{I}}_{t}(\pmb{\theta}_{t}) = E_{t-1}\left\{ \pmb{\nabla}_{t}(\pmb{y}_{t}, \pmb{\theta}_{t}) \pmb{\nabla}_{t}(\pmb{y}_{t}, \pmb{\theta}_{t})^{\top} \right\} $$ Typical choices for $ \gamma $ are 0, 1/2, and 1. For instance, if $ \gamma=0 $, $ \pmb{S}_{t} = \pmb{I} $ and no scaling occurs. Alternatively, when $ \gamma = 1/2 $, the scaling results in $ Var_{t-1}(\pmb{s}_{t}) = \pmb{I} $; in other words, standardization occurs.
Regardless of the choice of $ \gamma $, $ \pmb{s}_{t} $ is a martingale difference with respect to the distribution $ p(\cdot) $, and $ E_{t-1}\left\{ \pmb{s}_{t} \right\} = 0 $ for all $ t $. This latter property further implies that $ \pmb{\theta}_{t} $ is in fact a stationary process with long-term mean value $ (\pmb{I}_{t} - \pmb{B})^{-1}\pmb{\omega} $, whenever the spectral radius of $ \pmb{B} $ is less than one. Thus, $ \pmb{\omega} $ and $ \pmb{B} $ are respectively responsible for controlling the level and the persistence of $ \pmb{\theta}_{t} $, whereas $ \pmb{A} $ controls for the impact of $ \pmb{s}_{t} $. In other words, $ \pmb{s}_{t} $ denotes the direction of updating $ \pmb{\theta}_{t} $ to $ \pmb{\theta}_{t+1} $, acting as the the steepest ascent algorithm for improving the model's local fit.
With the above frameowrk established, Creal, Koopman, and Lucas (2013) show that various choices for $ p(\cdot) $ and $ \pmb{S}_{t} $ lead to various GAS specifications, some of which reduce to very familiar and well established existing models. For instance, let $ y_{t} = \sigma_{t}\epsilon_{t} $, and suppose $ \epsilon_{t} $ is a Gaussian random variable with mean zero and unit variance. It is readily shown that setting $ S_{t} = \mathcal{I}_{t}^{-1} $ and $ \theta_{t} = \sigma_{t}^{2} $, the GAS updating equation reduces to: $$ \theta_{t+1} = \omega + A(y_{t}^{2} - \theta_{t}) + B\theta_{t} $$ which is equivalent to the standard GARCH(1,1) model $$ \sigma_{t+1}^{2} = \alpha + \beta y_{t}^{2} + \eta \sigma_{t}^{2} $$ where $ \alpha = \omega $, $ \beta = A $, and $ \eta = B - A $. There is of course a number of other examples and configurations, and we refer the reader to the original texts for more details.
Our objective here is to communicate between EViews and Python to estimate a GAS model in Python and compare the results back in EViews. In particular, we will work with U.S. monthly civil unemployment rate, defined as the number of unemployed as a percentage of the labor force -- Labor force data are restricted to people 16 years of age and older, who currently reside in 1 of the 50 states or the District of Columbia, who do not reside in institutions (e.g., penal and mental facilities, homes for the aged), and who are not on active duty in the Armed Forces. See the FRED database at https://fred.stlouisfed.org/series/UNRATE) -- to which we will fit a GARCH(1,1) model using the traditional method as well as the GAS approach.
It is well known that unemployment rates are typically very volatile and persistent, particularly in contractionary economic cycles. This is because major firm decisions, such as workforce expansions and contractions, are often accompanied by large sunk costs (e.g. job advertisements, screening, training), and are usually irreversible in the immediate short term (e.g. wage frictions such as labour contracts and dismissal costs). Thus, in contractionary periods, firms typically prefer to defer hiring decisions until more favourable conditions return, resulting in strong unemployment persistence known as spells. On the other hands, these periods are often characterized by frequent labour force transitions and increased search activities, both of which contribute to unemployment volatility.
In light of the above, measuring the volatility of unemployment requires the use of econometric models which are designed to capture both volatility and persistence. While several such models exist in the literature, here we focus on perhaps the most well known such model proposed by Engle (1982) and Bollerslev (1986), the generalized autoregressive conditional heteroskedasticity (GARCH) model described earlier. In particular, if we let $ y_{t} $ denote the monthly unemployment rate, we are interested in obtaining an estimate $ \widehat{\sigma}_{t} $ of $ \sigma_{t} $, at each point in time, effectively tracing the evolution of unemployment volatility for the period under consideration. Since the GAS model above reduces to the GARCH model when the conditional distribution $ p(\cdot) $ is Gaussian and the time varying parameter is the volatility of the process, we would like to compare the estimates from the GAS model to those generated by EViews' internal GARCH estimation. Note here that while EViews can estimate numerous (G)ARCH models, it cannot yet natively estimate GAS models. Accordingly, we will fit a GARCH model in EViews, transfer our data over to Python, and estimate a GAS model using the Python package PyFlux. We will then compare our findings.
Before getting started, please make sure that you have Python 3 installed from https://www.python.org/downloads/release/python-368/ on your system, and that you also have the following Python packages installed:
PyFlux
One (certainly not the only) way to install said packages, is to open up a command prompt on your system and navigate to the directory where Python was installed; this is usually C:\Users\USER_NAME\AppData\Local\Programs\Python\Python36_64 if you have a 64-bit version. From there, issue the following commands:
python -m pip install PACKAGE_NAME
Next, make sure that the path to Python is specified in your EViews options. Specifically, in EViews, go to Options/General Options... and on the left tree select External program interface and ensure that Home Path is correctly pointing to the directory where Python is installed. Usually, you will not have to touch this setting since EViews populates this field by searching your system for the install directory.
Finally, please note that as of writing, the analysis that follows was tested with Python version 3.6.8 and PyFlux version 0.4.15.
Turning to data analysis, in EViews, create a new monthly workfile. To do so, click on File/New/Workfile. Under Frequency select Monthly, and set the Start date to 2006M12 and the End date to 2013M12, and hit OK. Next, fetch the unemployment rate data from the FRED database by clicking on File/Open/Database.... From here, select FRED Database from the Database/File Type dropdown, and hit OK. This opens the FRED database window. To get the series of interest from here, click on the Browse button. This opens a new window with a folder-like overview. Here, click on All Series Search and then type UNRATE in the Search For textbox. This will list a series called Civilian Unemployment Rate (M,SA,%). Drag the series over to the workfile to make it available for analysis. This will fetch the series UNRATE from the FRED database and place it in the workfile. In particular, we are grabbing data from the period of December 2006 to December 2013 -- effectively the recessionary period characterized by the recent housing loan crisis in the United States.
Figure 1A: Workfile Dialog
Figure 1B: Database Dialog
Figure 1C: FRED Browse
Figure 1C: FRED Search
Also, restrict the sample to the period from January 2007 to December 2013. Why we do this will become apparent later. To do so, issue the following command in EViews:
smpl 2007M01 @last
To see what the data looks like, double click on a UNRATE in the workfile to open the series object. Next, click on View/Graph.... This will open a graph options window. We will stick with the defaults so click on OK. The output is reproduced below.
Figure 2: Time Series Plot of UNRATE
We will now estimate a basic GARCH model on UNRATE. To do this, click on Quick/Estimate Equation..., and under Method choose ARCH - Autoregressive Conditional Heteroskedasticity. In the Mean Equation text box type UNRATE and leave everything else as their default values. Click on OK.
Figure 3A: GARCH Estimation Dialog
Figure 3B: GARCH Estimation Output
From the estimation output we can see that model parameters have the following estimates:
$ \alpha = 1.068302 $
$ \beta = 1.236277 $
$ \eta = -0.247753 $
We can also see the path of the volatility process by clicking on View/Garch Graph/Conditional Variance. This produces a plot of $ \widehat{\sigma}^{2}_{t} $. In fact, we will also create a series object from the data points used to produce the GARCH conditional variance. To do this, from the GARCH conditional variance window, click on Proc/Make GARCH Variance Series... and in the Conditional Variance textbox enter EVGARCH and hit OK. This produces a series object called EVGARCH and places it in the workfile. We will use it a bit later.
Figure 4A: GARCH Conditional Variance of UNRATE
Figure 4B: GARCH Conditional Variance Proc
To estimate the GAS equivalent of this model we must first transfer our data over to Python. To do so, issue the following command in EViews:
xopen(p)
This tells EViews to open an instance of Python within EViews and open up bi-directional communication. In fact you should see a new command window appear, titled Log: Python Output. Here you can issue commands into Python directly as if you had opened a Python instance at any command prompt. You can also send commands to Python using EViews command prompt. In fact, we will use the latter approach to import packages into our Python instance as follows:
xrun "import numpy as np"
xrun "import pandas as pd"
xrun "import pyflux as pf"
xrun "import matplotlib.pyplot as plt"
For instance, the first command above tells eviews to issue the command import numpy as np in the open Python instance, thereby importing the NumPy package. In fact, all results will be echoed in the Python instance.
Figure 5: Python Output Log
Next, transfer the UNRATE series over to Python by issuing the following command in EViews:
xput(ptype=dataframe) unrate
The command above sends the series UNRATE to Python and transforms that data into a Pandas DataFrame object.
We now follow the PyFlux documentation and estimate the GAS model by issuing the following commands from EViews:
xrun "model = pf.GAS(ar=1, sc=1, data=unrate, family=pf.Normal())"
xrun "fit = model.fit('MLE')"
xrun "fit.summary()"
The first command above tells PyFlux to create a GAS model object that has one autoregressive and one scaling parameter, sets $ p(\cdot) $ to the Gaussian distribution, and uses the series UNRATE as $ y_{t} $. In other words, the autoregressive and scaling parameters respectively corresponds to the coefficients $ A $ and $ B $ in the first section of this document. The second command tells Python to create a variable FIT which will hold the output from an estimated GAS model which uses maximum likelihood as the estimation technique. We display the output of this estimation by invoking the third command. In particular, we have the following estimates:
$ \omega = 0.0027 $
$ A = 1.2973 $
$ B = 0.9994 $
In fact, we can also obtain a distributional plot of the autoregressive coefficient $ B $ across the period of estimation. To do this, invoke the following command within EViews:
xrun "model.plot_z([1], figsize=(15,5))"
The latter command tells Python to plot the distribution of the 2nd estimated coefficient (the AR coefficient) and to display a figure which is of size $ 15\times 5 $ inches. This is the distribution of the evolution of $ B $ and is not the time path of the estimated coefficient.
Figure 6: Python GAS Distribution of AR Parameter
While we can obtain a distribution of the estimated parameters, unfortunately, PyFlux does not offer a way to extract the time path as a Python data object. Thankfully, we can recreate it manually and easily as a series in EViews.
To create the time path of the estimated GAS coefficient, we first need to transfer the coefficients from the estimated GAS model back into EViews. To do this, we invoke the following command in EViews:
xget(name=gascoefs, type=vector) fit.results.x[0:3]
This tells Python to send the first three estimated coefficients back to EViews, and saves the result as a vector called GASCOEFS.
Next, create a new series in the workfile called GASGARCH by issuing the following command in the EViews:
series gasgarch
Also, since this is an autoregressive process, we need to set an initial value for GASGARCH. We do this by setting the December 2006 observation to 0.7 -- the default value EViews uses to initialize its internal GARCH estimation. We do this by typing the following commands in EViews:
smpl 2006M12 2006M12
gasgarch = 0.7
Next, we set the sample back to the period of interest and fill the values of GASGARCH using the GARCH formula with the coefficients from the GAS model. To do this, issue the following commands in EViews again:
gasgarch = gascoefs(1) + gascoefs(3)*(unrate(-1)^2 - gasgarch(-1)) + gascoefs(2)*gasgarch(-1)
At last, we plot the GARCH conditional variance path from the internal estimation, EVGARCH along with the newly created series GASGARCH. We can do this programatically by issuing the following commands in EViews:
plot evgarch gasgarch
Figure 7: GARCH Conditional Variance Comparison with GAS
It is clear that the two estimation techniques produce the same path despite having different estimates for the coefficients. At last, note that while GARCH models are estimated using maximum likelihood procedures, parameter estimates are typically numerically unstable and often fail to converge. This often requires a re-specification of the convergence criterion and / or a change in starting values. These drawbacks are also an issue with GAS models.
The workfile and program files can be downloaded here.
seasuroot.WF1
seasuroot.prg
1 Tim Bollerslev. Generalized autoregressive conditional heteroskedasticity. Journal of econometrics, 31(3):307--327, 1986. [ bib ]
2 Drew Creal, Siem Jan Koopman, and André Lucas. Generalized autoregressive score models with applications. Journal of Applied Econometrics, 28(5):777--795, 2013. [ bib ]
3 Robert F Engle. Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation. Econometrica: Journal of the Econometric Society, pages 987--1007, 1982. [ bib ]
4 Andrew C Harvey. Dynamic models for volatility and heavy tails: with applications to financial and economic time series, volume 52. Cambridge University Press, 2013. [ bib ]
Generalized Autoregressive Score (GAS) Models: EVi...
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