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\begin{document}
\begin{abstract}
We construct a smooth algebraic stack of tuples consisting of
genus two nodal curves, line bundles, and twisted fields.
It leads to a desingularization of the moduli of genus two stable maps to projective spaces.
The construction is based on systematical application of the theory of stacks with twisted fields (STF),
which has its prototype appeared in~\cite{HLN, g1modular} and
is fully developed in this article.
The results of this article are the second step of a series of works toward the resolutions of the moduli of stable maps of higher genera. \end{abstract}
\maketitle \setcounter{tocdepth}{1} \tableofcontents
\section{Introduction}\label{Sec:Intro} This paper is the second of a series, sequel to \cite{g1modular}. The series aims to resolve the singularities of the moduli $\ov M_g(\mathbb P^n,d)$ of degree $d$ stable maps from genus $g$ nodal curves into projective spaces~$\mathbb P^n$, which possess arbitrary singularities when all $g$ and $d$ are considered (see~\cite{V}). The problem of resolution of singularities is arguably one of the hardest problems in algebraic geometry (\cite{Hironaka64a, Hironaka64b, deJong96, K07}).
For~$g\!=\! 2$, the only resolution of $\ov M_2(\mathbb P^n,d)$ so far is provided in~\cite{HLN} via a huge sequence of blowups. In higher genus cases, a direct blowup construction of a possible resolution of $\ov M_g(\mathbb P^n,d)$ may seem formidable. It thus calls for a more abstract and geometric approach. Our goal is to construct a new moduli with smooth irreducible components and normal crossing boundaries that dominates $\ov\mathfrak M_g(\mathbb P^n,d)$ properly and birationally onto the primary component (the component whose general points have smooth domain curves).
To this end, we consider the smooth Artin stack $\mathfrak P_g$ of pairs $(C, L)$ where~$C$ are genus $g$ nodal curves and $L\!\longrightarrow\!C$ are line bundles (i.e.~the relative Picard stack), along with the morphism $$\ov\mathfrak M_g(\mathbb P^n,d)\longrightarrow \mathfrak P_g, \qquad [C,\mathbf u]\mapsto[C, \mathbf u^*\mathscr O_{\mathbb P^n}(1)].$$ We hope to introduce a novel smooth Artin stack $\widetilde{\mathfrak P}^\tn{tf}_g$ of tuples $(C, L, \eta)$ where $(C, L) \!\in\!\mathfrak P_g$ and $\eta$ are the extra structure (called twisted fields) added to $(C, L)$, along with a canonical forgetful morphism $\widetilde{\mathfrak P}^\tn{tf}_g\!\longrightarrow\!\mathfrak P_g$. We then take $$ \ti M_g^\tn{tf}(\mathbb P^n,d):= \ov M_g(\mathbb P^n,d)\times_{\mathfrak P_g} \widetilde{\mathfrak P}^\tn{tf}_g $$ to be the moduli of degree $d$ stable maps from genus $g$ nodal curves into projective spaces $\mathbb P^n$ with twisted fields. We aim to demonstrate that
$\ti M_g^\tn{tf}(\mathbb P^n,d)$ is a smooth Deligne-Mumford stack with the desired desingularization property aforementioned. The $g\!=\! 1$ case of this program is accomplished in~\cite{g1modular}. In this paper, we carry out the program when $g=2$.
To make our approach systematic, we develop the theory of stacks with twisted fields (\textsf{STF}). The STF theory is an abstraction based upon a thorough analysis on the combinatorial and geometric structures of $\mathfrak P_g$. To summarize it, let $\mathfrak M$ be a smooth stack (e.g.~$\mathfrak P_g$) and~$\Gamma$ be a finite set of graphs (e.g.~the set of dual graphs of nodal curves
appearing in $\mathfrak P_g$).
We assume that $\mathfrak M$ admits a \textsf{$\Gamma$-stratification}
(e.g.~the stratification of $\mathfrak P_g$ indexed by $\Gamma$ together with local smooth divisors corresponding to the edges of the graphs); see Definition~\ref{Dfn:G-adim_fixture} for details. We then introduce a \textsf{treelike structure}~$\Lambda$ on $(\mathfrak M, \Gamma)$ in Definition~\ref{Dfn:Treelike_structure}, which assigns a rooted tree to each connected component of the strata of $\mathfrak M$ in a suitable way. Such a treelike structure encodes a hidden blowing up process to be performed on the stack $\mathfrak M$, although the STF theory does not rely on the actual blowing up process.
With all the above devices at hand, we construct a new smooth stack $\mathfrak M_{\Lambda}^\tn{tf}$, properly and birationally dominating~$\mathfrak M$. The following theorem is the key statement of the STF theory. It is a restatement of Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p1}, and the notation is elaborated in Section~\ref{Sec:tf}.
\begin{thm}\label{Thm:TSTF} Let $\Gamma$ be a set of finite graphs, $\mathfrak M$ be a smooth algebraic stack with a $\Gamma$-stratification $\mathfrak M\!=\!\bigsqcup_{\gamma\in\Gamma}\mathfrak M_\gamma$ as in Definition~\ref{Dfn:G-adim_fixture}, $\Lambda$ be a treelike structure on $(\mathfrak M,\Gamma)$ as in Definition~\ref{Dfn:Treelike_structure}, $\pi_0(\mathfrak M_\gamma)$ be the set of the connected components $\mathfrak N_\gamma$ of $\mathfrak M_\gamma$, and $\ov\Lambda_{\mathfrak N_\gamma}$ be the set of equivalence classes of rooted level trees (c.f.~Definitions~\ref{Dfn:Rooted_level_tree}~\&~\ref{Dfn:Equiv_level_graphs}) associated with $\mathfrak N_\gamma$ as in~(\ref{Eqn:Treelike_Level}). Then, the following fiber products of open subsets of the projectivization of the direct sums of certain line bundles as in~(\ref{Eqn:Twisted_line_bundle}): \begin{equation*}\begin{split}
&\mathfrak N_{\gamma,[\mathfrak t]}^\tn{tf}
=
\bigg(\!
\prod_{i\in\lbrp{{\mathbf{m}},0}_\mathfrak t}\!\!\!\!
\Big\lgroup\Big(
\mathring{\mathbb P}\big(
\bigoplus_{
\begin{subarray}{c}
e\in\mathfrak E_i^\bot(\mathfrak t)
\end{subarray}
}\!\!\!
L_{\succeq_{\mathfrak N_\gamma}\,e}\,
\big)
\Big)
\Big/\mathfrak N_\gamma
\Big\rgroup\!\!
\bigg)
\stackrel{\varpi}{\longrightarrow}\mathfrak N_\gamma,\qquad
\gamma\!\in\!\Gamma,\
\mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma),\
[\mathfrak t]\!\in\!\ov \Lambda_{\mathfrak N_\gamma}, \end{split}\end{equation*} can be glued together in a canonical way to form a smooth algebraic stack $\mtd{}$: $$
\mathfrak M^\tn{tf}=\mathfrak M^\tn{tf}_{\Lambda}=
\bigsqcup_{\gamma\in \Gamma;\;
\mathfrak N_\gamma\in\pi_0(\mathfrak M_{\gamma});\;
[\mathfrak t]\in\ov \Lambda_{\mathfrak N_\gamma}}
\hspace{-.38in}
\mathfrak N_{\gamma,[\mathfrak t]}^\tn{tf}\ . $$ Moreover, the stratawise projections $\varpi\!:\mathfrak N_{\gamma,[\mathfrak t]}^\tn{tf}\!\longrightarrow\!\mathfrak N_\gamma$ together give rise to a proper and birational morphism $\varpi\!:\mathfrak M^\tn{tf}\!\longrightarrow\!\mathfrak M$. \end{thm}
The stack $\mathfrak M_{\Lambda}^\tn{tf}$ is called the \textsf{stack with twisted fields of $\mathfrak M$ with respect to $\Lambda$}, and it enjoys several desirable properties as stated in Theorem~\ref{Thm:tf_smooth} and Proposition~\ref{Prp:moduli}. In particular, Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p4} is crucial for resolving the moduli $\ov M_g(\mathbb P^n,d)$.
Theorem~\ref{Thm:tf_smooth} and Proposition~\ref{Prp:moduli} further suggest a possible recursive construction: $$ \cdots\longrightarrow(\mathfrak M^\tn{tf}_\Lambda)^\tn{tf}_{\Lambda'}\longrightarrow\mathfrak M^\tn{tf}_\Lambda\longrightarrow \mathfrak M. $$ The key observation is that the new smooth stack $\mathfrak M_\Lambda^\tn{tf}$ naturally comes with some choices of the sets $\Gamma'$ of graphs and the corresponding $\Gamma'$-stratifications. Upon introducing a suitable treelike structure $\Lambda'$ on $(\mathfrak M_{\Lambda}^\tn{tf}, \Gamma')$, one can obtain a newer stack $(\mathfrak M_{\Lambda}^\tn{tf})_{\Lambda'}^\tn{tf}$, and the construction keeps on. For example, we apply the STF theory to the smooth stack $\mathfrak P_2$ eight times to obtain~$\widetilde \mathfrak P_2^\tn{tf}$ in Corollary~\ref{Crl:G9-tf}, which leads to the main application of the STF theory in this paper:
\begin{thm} \label{Thm:Main} There exits a smooth algebraic stack $\widetilde \mathfrak P_2^\tn{tf}$ of tuples consisting of nodal curves of genus two, line bundles, and twisted fields, along with a proper and birational forgetful morphism $\widetilde{\mathfrak P}^\tn{tf}_2\!\longrightarrow\!\mathfrak P_2$, such that the Deligne-Mumford moduli stack of genus two stable maps with twisted fields given by $$ \ti M_2^\tn{tf}(\mathbb P^n,d):= \ov M_2(\mathbb P^n,d)\times_{\mathfrak P_2} \widetilde{\mathfrak P}^\tn{tf}_2 $$ satisfies that \begin{enumerate} [leftmargin=*,label=(\arabic*)] \item\label{Cond:MainSmooth} $\ti M_2^\tn{tf}(\mathbb P^n,d)$ has smooth irreducible components and admits at worst normal crossing singularities; \item\label{Cond:MainProper} the morphism $\ti M_2^\tn{tf}(\mathbb P^n,d)\!\longrightarrow\!\ov M_2(\mathbb P^n,d)$ is proper; \item\label{Cond:MainBirational} the induced morphism $\ti M_2^\tn{tf}(\mathbb P^n,d)^{\textnormal{pri}}\!\longrightarrow\!\ov M_2(\mathbb P^n,d)^{\textnormal{pri}}$ between the primary components (whose general points are stable maps with smooth domain curves)
is birational; and \item\label{Cond:MainLocallyFree} for any irreducible component $N$ of $\ti\mathfrak M_2^\tn{tf}(\mathbb P^n,d)$, with $$
\ti\pi_N:\ti\mathcal C|_N\longrightarrow N\subset\ti M_2^\tn{tf}(\mathbb P^n,d),\qquad
\ti\mathfrak f_N:\ti\mathcal C|_N\longrightarrow\mathbb P^n, $$ denoting the restriction of the pullback of the universal family $\pi\!:\mathcal C\!\longrightarrow\!\ov M_2(\mathbb P^n,d),$ $\mathfrak f\!:\mathcal C\!\longrightarrow\!\mathbb P^n$ of $\ov M_2(\mathbb P^n,d)$ to $N$, the direct image sheaf $(\ti\pi_N)_*(\ti\mathfrak f_N)^*\mathscr O_{\mathbb P^n}(k)$ is locally free for all $k\!\ge\!1$. \end{enumerate} \end{thm}
The eight-step construction of the stack $\ti\mathfrak P_2^\tn{tf}$ is provided in~\S\ref{Subsec:Step1}-\S\ref{Subsec:Step8}. The proof of the properties~\ref{Cond:MainSmooth}-\ref{Cond:MainLocallyFree} of Theorem~\ref{Thm:Main} is provided in~\S\ref{Subsec:Proof_Main}.
We remark that the stack with twisted fields $\ti\mathfrak P_2^\tn{tf}$ may not be isomorphic to the blowup stack~$\ti\mathfrak P_2$ constructed in~\cite{HLN}. In fact, if we change the order of some rounds and phases of the sequential blowups in~\cite{HLN} (interchanging $\mathsf{r}_1\mathsf{p}_3$ and $\mathsf{r}_1\mathsf{p}_4$ and interchanging $\mathsf{r}_3\mathsf{p}_1$ and $\mathsf{r}_3\mathsf{p}_2$ to be precise), the resulting stack should be isomorphic to $\ti\mathfrak P_2^\tn{tf}$. Indeed, several blowups in ~\cite{HLN} can be performed in various orders, and in general, they should lead to different resolutions of $\ov M_2(\mathbb P^n,d)$.
We also remark that if we take $N\!=\!\ti M_2^\tn{tf}(\mathbb P^n,d)^\textnormal{pri}$ in Part~\ref{Cond:MainLocallyFree} of Theorem~\ref{Thm:Main}, then $$
\blr{\,e\big((\ti\pi_{\ti M_2^\tn{tf}(\mathbb P^n,d)^\textnormal{pri}})_*(\ti\mathfrak f_{\ti M_2^\tn{tf}(\mathbb P^n,d)^\textnormal{pri}})^*\mathscr O_{\mathbb P^n}(k)\big)\,,\,[\ti M_2^\tn{tf}(\mathbb P^n,d)^\textnormal{pri}]\,} $$ should equal the expected reduced genus 2 Gromov-Witten (GW) invariants of the corresponding complete intersection, parallel to~\cite[(1.4)]{VZ} and~\cite[(1.7)]{LZ}. The reduced genus 1 GW-invariants, as well as its comparison with the standard genus 1 GW-invariants, are introduced in~\cite{Zi2,Zi1} and further studied in~\cite{LZ,VZ,CL,CM,LO1,LO2}, and lead to important results such as A.~Zinger's proof~\cite{Zi3} of the prediction of~\cite{BCOV} for genus 1 GW-invariants of a quintic 3-fold.
\textbf{Acknowledgments.} We would like to thank Dawei Chen, Qile Chen, Jack Hall, and YP Lee for the valuable discussions.
\section{A theory of stacks with twisted fields} \label{Sec:tf}
\subsection{Graphs and levels} \label{Subsec:graphs_and_levels} Throughout the article, we use the following definition of the graphs adapted from~\cite[\S5.1]{CQFT}.
\begin{dfn}\label{Dfn:Graph} A \textsf{finite graph} $\gamma$, or simply a \textsf{graph} when the context is clear, is a finite set $\tn{HE}(\gamma)$ of half-edges along with \begin{itemize} [leftmargin=*] \item a set $\tn{Ver}(\gamma)$ of disjoint subsets $v\!\subset\!\tn{HE}(\gamma)$ known as the \textsf{vertices} such that $$\bigsqcup_{v\in\tn{Ver}(\gamma)}\!\!\!\!v=\tn{HE}(\gamma)$$ and \item a set $\tn{Edg}(\gamma)$ of disjoint subsets $e\!\subset\!\tn{HE}(\gamma)$ known as the \textsf{edges} such that $$
|e|\!=\! 2\quad\forall~e\!\in\!\tn{Edg}(\gamma);\qquad\bigsqcup_{e\in\tn{Edg}(\gamma)}\!\!\!\!e=\tn{HE}(\gamma).$$
\end{itemize} \end{dfn}
Definition~\ref{Dfn:Graph} naturally allows graphs that have self loops and multiple edges with the same endpoints, which is convenient for our purpose. For every half-edge $\hbar\!\in\!\tn{HE}(\gamma)$, Definition~\ref{Dfn:Graph} implies that there exist a unique vertex $v(\hbar)$ and a unique edge $e(\hbar)$ containing $\hbar$, respectively.
\begin{dfn}\label{Dfn:subgraph} Let $\gamma,\gamma'$ be graphs. We say $\gamma'$ is a \textsf{subgraph} of $\gamma$ if there exist injections $$
f_{\gamma,\gamma'}:\tn{HE}(\gamma')\hookrightarrow
\tn{HE}(\gamma),\qquad
f_{\gamma,\gamma'}^{\textnormal e}:\tn{Edg}(\gamma')\hookrightarrow
\tn{Edg}(\gamma),\qquad
f_{\gamma,\gamma'}^{\textnormal v}:\tn{Ver}(\gamma')\hookrightarrow
\tn{Ver}(\gamma) $$ satisfying \begin{align*}
&
f_{\gamma,\gamma'}^{\textnormal e}(e)=\{f_{\gamma,\gamma'}(\hbar^+),\,f_{\gamma,\gamma'}(\hbar^-)\}
&&
\forall\,e\!=\!\{\hbar^+,\hbar^-\}\!\in\!\tn{Edg}(\gamma');
\\
&
f_{\gamma,\gamma'}^{\textnormal v}(v)\supset\{f_{\gamma,\gamma'}(\hbar):
\forall\,\hbar\!\in\! v\}
&&
\forall\,v\!\in\!\tn{Ver}(\gamma'). \end{align*} \end{dfn}
\begin{dfn}\label{Dfn:path} Given a graph $\gamma$ and two vertices $v$ and $w$ of $\gamma$, a \textsf{path} from $v$ to $w$ is a sequence of pairwise distinct half-edges of $\gamma$: $$
\hbar_1^+,\,\hbar_1^-,\,\hbar_2^+,\,\hbar_2^-,\,\cdots,\,\hbar_m^+,\,\hbar_m^- $$ such that $v(\hbar_1^+)\!=\! v$, $v(\hbar_m^-)\!=\! w$, $\{\hbar_i^+,\hbar_i^-\}\!\in\!\tn{Edg}(\gamma)$ for all $1\!\le\!i\!\le\!m$, and $\{\hbar_i^-,\hbar_{i+1}^+\}\!\in\!\tn{Ver}(\gamma)$ for all $1\!\le\!i\!\le\!m\!-\!1$. \end{dfn}
A path defined in this way cannot contain repetitive edges.
\begin{dfn}\label{Dfn:Connected_graphs} A graph $\gamma$ is said to be \textsf{connected} if it is non-empty and for any two vertices $v$ and $w$, there exists a path form $v$ to $w$. The set of all connected graphs is denoted by $\mathbf{G}$. \end{dfn}
Given $\gamma\!\in\!\mathbf{G}$ and $E\!\subset\!\tn{Edg}(\gamma)$, we write the set of all half-edges of $E$ as $\tn{HE}(E)\!=\!\bigsqcup_{e\in E}e$. If $E\!\ne\!\emptyset$, there exists a unique partition $P(E)$ of $E$, with each $E'\!\in\! P(E)$ corresponding to a connected subgraph $\gamma(E')$ of $\gamma$, satisfying $\tn{Edg}(\gamma(E'))\!=\! E'$ and $\tn{Ver}(\gamma(E'))\!\cap\!\tn{Ver}(\gamma(E''))\!=\!\emptyset$ for all distinct $E',E''\!\in\! P(E)$. If $E\!=\!\emptyset$, we set $P(E)\!=\!\emptyset$.
\begin{dfn}\label{Dfn:Edge_contraction} With notation as above, let $\gamma_{(E)}$ be the graph obtained from $\gamma$ by \textsf{contracting the edges in $E$}: \begin{gather*}
\tn{HE}\big(\gamma_{(E)}\big)=
\tn{HE}(\gamma)\backslash\tn{HE}(E),\qquad
\tn{Edg}\big(\gamma_{(E)}\big)=
\tn{Edg}(\gamma)\backslash E,\\
\tn{Ver}\big(\gamma_{(E)}\big)\!=\!
\{v\!\in\!\tn{Ver}(\gamma)\!: v\!\cap\!\tn{HE}(E)\!=\!\emptyset\}\sqcup
\big\{\!\!\bigcup_{\hbar\in\tn{HE}(E')}
\!\!\!\!\!\!v(\hbar)\backslash\tn{HE}(E)\!:
E'\!\in\! P(E)
\big\}. \end{gather*} Such an operation is called an \textsf{edge contraction}. For $e\!\in\!\tn{Edg}(\gamma)$, we simply write $\gamma_{(e)}\!=\!\gamma_{(\{e\})}$. \end{dfn}
By definition, $\gamma_{(\emptyset)}\!=\!\gamma$ for any $\gamma\!\in\!\mathbf{G}$. If $\gamma$ is connected, then every graph obtained from $\gamma$ via edge contraction is still connected.
Given $\gamma,\gamma'\!\in\!\mathbf{G}$, we define \begin{equation}\label{Eqn:partial_order} \gamma'\!\prec\!\gamma\qquad \Longleftrightarrow \qquad\exists\ \ E\!\subset\!\tn{Edg}(\gamma')\ \ \textnormal{s.t.}\ \ E\!\ne\!\emptyset,~\gamma\!=\!\gamma'_{(E)}. \end{equation} This gives rise to a partial order $\prec$ on $\mathbf{G}$.
Aside from the edge contractions, we introduce two more graph operations that will be used in \S\ref{Sec:genus_2_twisted_fields} to describe the treelike structures of each step.
\begin{dfn}\label{Dfn:Vertex_dissolution} Given $\gamma\!\in\!\mathbf{G}$ and $V\!\subset\tn{Ver}(\gamma)$, let ${\gamma_{V}^\textnormal{ds}}$ be the graph obtained from $\gamma$ by \textsf{dissolving the vertices in $V$}: $$
\tn{HE}(\gamma_{V}^\textnormal{ds})\!=\!\tn{HE}(\gamma),\quad
\tn{Edg}(\gamma_{V}^\textnormal{ds})\!=\!\tn{Edg}(\gamma),\quad
\tn{Ver}(\gamma_{V}^\textnormal{ds})\!=\!(\tn{Ver}(\gamma)\backslash V)\!\sqcup\!
\bigsqcup_{\hbar\in v,\,v\in V}\!\!\!\!\{\hbar\}. $$ Such an operation is called a \textsf{vertex dissolution}. \end{dfn}
\begin{dfn}\label{Dfn:Vertex_identification} Given $\gamma\!\in\!\mathbf{G}$ and $V\!\subset\tn{Ver}(\gamma)$, let $\gamma_{V}^\textnormal{id}$ be the graph obtained from $\gamma$ by \textsf{identifying the vertices in $V$}: $$
\tn{HE}(\gamma_{V}^\textnormal{id})\!=\!\tn{HE}(\gamma),\quad
\tn{Edg}(\gamma_{V}^\textnormal{id})\!=\!\tn{Edg}(\gamma),\quad
\tn{Ver}(\gamma_{V}^\textnormal{id})\!=\!(\tn{Ver}(\gamma)\backslash V)\!\sqcup\!
\{\hbar\!\in\! v:\,v\!\in\! V\}. $$ Such an operation is called a \textsf{vertex identification}. \end{dfn}
Intuitively, dissolving a vertex $v\!\in\!\tn{Ver}(\gamma)$ means removing $v$ from $\tn{Ver}(\gamma)$ and assigning to each edge $e$ with $e\!\cap\!v\!\ne\!\emptyset$ a distinct new vertex, whereas identifying the vertices in a subset $V$ of $\tn{Ver}(\gamma)$ means ``gluing'' the vertices in $V$ into a single vertex. If $\gamma$ is connected, a graph obtained from $\gamma$ via vertex dissolution may become disconnected, but that via vertex identification is connected. Figure~\ref{Fig:graph_operations} provides illustrations for Definitions~\ref{Dfn:Edge_contraction}-\ref{Dfn:Vertex_identification}.
\begin{figure}
\caption{Edge contraction, vertex dissolution, and vertex identification}
\label{Fig:graph_operations}
\end{figure}
\iffalse \begin{rmk} As mentioned in Remark~\ref{Rmk:graph_isom}, we take an arbitrary graph representing the equivalence class $\gamma\!\in\!\mathbf{G}$ in each of Definitions~\ref{Dfn:Edge_contraction}-\ref{Dfn:Vertex_identification}. The partial order~(\ref{Eqn:partial_order}) on $\mathbf{G}$ does not depend on the choices of the graphs representing $\gamma'$ and $\gamma$. \end{rmk} \fi
\begin{dfn}\label{Dfn:b1} For a connected graph $\gamma$, its \textsf{first Betti number} $b_1(\gamma)$ is given by $$
b_1(\gamma)=|\tn{Edg}(\gamma)|-|\tn{Ver}(\gamma)|+1. $$ \end{dfn}
\iffalse \begin{lmm}\label{Lm:betti_contraction} Let $\gamma$ be a connected graph and $E\!\subset\!\tn{Ver}(\gamma)$. Then, $b_1\big(\gamma_{(E)}\big)\!\le\!b_1(\gamma)$. \end{lmm}
\begin{proof} This follows directly from Definition~\ref{Dfn:b1} and the construction of $\gamma_{(E)}$. The key fact is that for any $E'\!\in\! P(E)$, \begin{equation*}
\big|\{v\!\in\!\tn{Ver}(\gamma)\!:
v\!\cap\!\tn{HE}(E')\!\ne\!\emptyset\}\big|\le
|E'|+1. \qedhere \end{equation*} \end{proof} \fi
The graphs with $b_1\!=\! 0$ are of our particular interest. They play crucial roles in the STF theory.
\begin{dfn}\label{Dfn:Rooted_tree} A \textsf{rooted tree} $$
\tau=(\gamma,o) $$ consists of a connected graph $\gamma$ with $b_1(\gamma)\!=\! 0$ as well as a chosen vertex $o\!\in\!\tn{Ver}(\gamma)$ known as the~\textsf{root}. We write $$
\tn{HE}(\tau)\!:=\!\tn{HE}(\gamma),\qquad
\tn{Edg}(\tau)\!:=\!\tn{Edg}(\gamma),\qquad
\tn{Ver}(\tau)\!:=\!\tn{Ver}(\gamma). $$ When the root is clear, we simply write $\tau\!=\!\gamma$. The single vertex edge-less rooted tree is denoted by $\tau_\bullet$. \end{dfn}
Every rooted tree $\tau$ determines a unique partial order~$\prec$ on $\tn{HE}(\tau)$, known as the \textsf{tree order}, so that $\hbar\!\prec\!\hbar'$ if and only if $\hbar\!\ne\!\hbar'$ and the unique path from $o$ to $v(\hbar)$ contains $\hbar'$. Thus, every $e\!\in\!\tn{Edg}(\tau)$ can be written as $$
e=\{\hbar_e^+,\hbar_e^-\}
\qquad\textnormal{with}\qquad\hbar_e^-\!\prec\!\hbar_e^+. $$ The tree order on $\tn{HE}(\tau)$ induces partial orders $\prec$ on $\tn{Edg}(\tau)$ and $\tn{Ver}(\tau)$ by requiring $$
e\!\prec\!e'\ \Longleftrightarrow\
\big\lgroup
\hbar\!\prec\!\hbar'\ \forall\,\hbar\!\in\! e,\,\hbar'\!\in\! e'
\big\rgroup,\qquad
v\!\prec\!v'\ \Longleftrightarrow\
\big\lgroup
\exists\,\hbar\!\in\! v,\,\hbar'\!\in\! v'~\textnormal{s.t.}~
\hbar\!\prec\!\hbar'
\big\rgroup, $$ respectively. We still call the induced orders on $\tn{Edg}(\tau)$ and $\tn{Ver}(\tau)$ the \textsf{tree orders}. The subsets of the maximal edges, minimal edges, maximal vertices, and minimal vertices with respect to the tree orders are denoted by $$
\tn{Edg}(\tau)_{\max},\quad
\tn{Edg}(\tau)_{\min},\quad
\tn{Ver}(\tau)_{\max},\quad
\tn{Ver}(\tau)_{\min}\,, $$ respectively. The minimal vertices are known as the \textsf{leaves} in the graph theory.
\begin{dfn}\label{Dfn:Rooted_level_tree} A \textsf{rooted level tree} $${\mathfrak t}=(\,\tau_{\mathfrak t}\,,\,\ell_{\mathfrak t}\,)$$ is a tuple consisting of a rooted tree $\tau_{\mathfrak t}\!=\!(\gamma_{\mathfrak t},o_{\mathfrak t})$ and a map \begin{equation}\label{Eqn:level_map}
\ell=\ell_{\mathfrak t}:\,\tn{HE}(\tau_{\mathfrak t})\longrightarrow\mathbb R_{\le 0}, \end{equation} called a \textsf{level map}, that satisfies \begin{gather*}
\ell^{-1}(0)= o_{\mathfrak t}\,;\qquad
\ell(\hbar)=\ell(\hbar')\quad\textnormal{whenever}\ v(\hbar)\!=\! v(\hbar');\qquad
\ell(\hbar_e^-)<\ell(\hbar_e^+)\quad\forall\ e\!\in\!\tn{Edg}(\tau_{\mathfrak t}). \end{gather*} We write $$
\tn{HE}({\mathfrak t})\!:=\!\tn{HE}(\tau_{\mathfrak t}),\qquad
\tn{Edg}({\mathfrak t})\!:=\!\tn{Edg}(\tau_{\mathfrak t}),\qquad
\tn{Ver}({\mathfrak t})\!:=\!\tn{Ver}(\tau_{\mathfrak t}). $$ The level map $\ell$ determines a map on $\tn{Ver}(\tau_{\mathfrak t})$, which is still denoted by $\ell$, that is given by $$
\ell\!:\tn{Ver}(\tau_{\mathfrak t})\longrightarrow \mathbb R_{\le 0},\qquad
\ell(v)=\ell(\hbar)\ \ \forall~\hbar\!\in\! v, $$ which is also called a \textsf{level map} when the context is clear. The image of $\ell$ is denoted by $\textnormal{Im}(\ell)$, whose elements are called \textsf{levels} of ${\mathfrak t}$. The set of all rooted level trees is denoted by $\mathbf T$. \end{dfn}
\begin{rmk}
In Definition~\ref{Dfn:Rooted_level_tree},
the level maps on $\tn{HE}(\tau_{\mathfrak t})$ and $\tn{Ver}(\tau_{\mathfrak t})$ determine each other,
and a rooted level tree defined in this way is consistent with that introduced in~\cite[\S2.1]{g1modular}.
We remark that a rooted level tree is a {\it level graph} with the root as the unique {\it top level vertex} in~\cite{BCGGM,BCGGM2}.
The relation between the STF theory and the theory of twisted/multi-scale differentials (\cite{BCGGM,BCGGM2}) appears to be beyond the combinatorial resemblance,
which will be revealed in the succeeding works on the resolution of $\ov M_g(\mathbb P^n,d)$. \end{rmk}
For every level $i\!\in\!\tn{Im}(\ell)$, we denote by $i^\sharp$ and $i^\flat$ the levels immediately above and below~$i$ (if such levels exist), respectively: \begin{equation}\label{Eqn:sharp_flat}
i^\sharp=\min\big\{j\!\in\!\tn{Im}(\ell):
j>i\big\},\qquad
i^\flat=\max\big\{h\!\in\!\tn{Im}(\ell):
h<i\big\}. \end{equation} Among the levels of ${\mathfrak t}$, a particularly important one is $$
{\mathbf{m}}={\mathbf{m}}({\mathfrak t}):=
\max\{\,\ell(v):\,
v\!\in\!\tn{Ver}(\gamma)_{\min}\,\}\quad
\in\textnormal{Im}(\ell). $$ For $i\!\in\!\textnormal{Im}(\ell)$, we write \begin{align}
&\mathfrak E_i=\mathfrak E_i({\mathfrak t})=
\big\{
e\!\in\!\tn{Edg}(\tau_{\mathfrak t})\!:\,
\ell(\hbar_e^+)\!>\!i,\,
\ell(\hbar_e^-)\!\le\!i
\big\},&&
\mathfrak E_{\ge i}= \mathfrak E_{\ge i}({\mathfrak t})=
\bigcup_{j\ge i} \mathfrak E_j,\notag
\\
&\mathfrak E_i^\bot=\mathfrak E_i^\bot({\mathfrak t})=
\big\{e\!\in\!\mathfrak E_i\!:\,
\ell(\hbar_e^-)\!=\! i\big\},&&
\mathfrak E_{\ge i}^\bot= \mathfrak E_{\ge i}^\bot({\mathfrak t})=
\bigcup_{j\ge i} \mathfrak E_j^\bot.\label{Eqn:fE_i}
\\
&
\big(\mathfrak E_{{\mathbf{m}};\min}^\bot\big)^{\succeq}=
\bigcup_{e\in\mathfrak E_{\mathbf{m}}^\bot\cap\tn{Edg}(\tau_{\mathfrak t})_{\min}}\hspace{-.3in}
\big\{e'\!\in\!\tn{Edg}(\tau_{\mathfrak t}):e'\succeq e\big\}.
&&\notag \end{align} The notation ``$\bot$'' in $\mathfrak E_i^\bot$ intuitively suggests the lower half-edges of the edges in $\mathfrak E_i^\bot$ ``stop'' at the level~$i$, with ``$\mid$'' representing the edges and ``$-$'' representing the level $i$. The set $(\mathfrak E_{{\mathbf{m}};\min}^\bot)^{\succeq}$ consists of the edges of the paths from the root $o$ to the minimal vertices on level ${\mathbf{m}}$; see Figure~\ref{Fig:level} for illustration.
With $\mathfrak t$ as above, we write $$ \lrbr{i,j}_\mathfrak t=\textnormal{Im}({\mathfrak t})\!\cap\![i,j],\quad \lbrp{i,j}_\mathfrak t= \textnormal{Im}({\mathfrak t})\!\cap\![i,j)\qquad \forall~i,j\!\in\!\textnormal{Im}(\ell). $$ For $k\!\in\!\mathbb Z_{>0}$, let \begin{equation} \label{Eqn:dom} \mathbf{n}_k= \dom{k}{{\mathfrak t}}= \min\big(\, \{\,i\!\in\!\lbrp{{\mathbf{m}},0}_{\mathfrak t}:\,
|\mathfrak E_i\!\cap\!(\mathfrak E_{{\mathbf{m}};\min}^\bot)^{\succeq}|\!\le\!k \,\}\sqcup\{0\} \,\big)\ \in\lrbr{{\mathbf{m}},0}_{\mathfrak t}. \end{equation} Intuitively, $\mathbf{n}_k$ is the lowest level on which there are at most $k$ vertices that are contained in the paths from the root $o$ to the minimal vertices on level ${\mathbf{m}}$; see Figure~\ref{Fig:level} for illustration.
\begin{figure}
\caption{A rooted level tree}
\label{Fig:level}
\end{figure}
Each subset \begin{equation*}\begin{split}
\mathbb I' &=I'_+\sqcup E'_{\mathbf{m}}\sqcup E'_-
\\
\subset
\mathbb I(\mathfrak t)
&=
I_+(\mathfrak t)\sqcup E_{\mathbf{m}}(\mathfrak t)\sqcup E_-(\mathfrak t)
\\
&:=
\lbrp{{\mathbf{m}}, 0}_\mathfrak t
\sqcup
\big(\mathfrak E_{\mathbf{m}}\backslash\mathfrak E_{\mathbf{m}}^\bot\big)
\sqcup
\big(\tn{Edg}({\mathfrak t})\backslash\mathfrak E_{\ge{\mathbf{m}}}\big) \end{split}\end{equation*} determines a rooted level tree \begin{equation}\label{Eqn:rooted_level_tree_contraction}
\mathfrak t_{(\mathbb I')}=
(\,\tau_{(\mathbb I')}\,,\,
\ell_{(\mathbb I')}\,)=
(\,\gamma_{(\mathbb I')}\,,\,
o_{(\mathbb I')}\,,\,
\ell_{(\mathbb I')}\,) \end{equation} as follows: \begin{itemize} [leftmargin=*] \item the rooted tree $\tau_{(\mathbb I')}$ is obtained from $\tau_{\mathfrak t}$ by contracting the edges in \begin{equation*}
\big\{\,e\!\in\!\mathfrak E_{\ge{\mathbf{m}}}^\bot\!\sqcup\!E_{\mathbf{m}}':\,
\lbrp{\,\max\{\ell(\hbar^-_e),{\mathbf{m}}\},\,\ell(\hbar^+_e)}_\mathfrak t \subset I'_+\,\big\} \sqcup E'_-; \end{equation*}
\item the level map $\ell_{(\mathbb I')}$ is such that for any $\hbar\!\in\!\tn{HE}\big(\gamma_{(\mathbb I')}\big)$, $$ \ell_{(\mathbb I')}(\hbar)\!=\! \begin{cases}
\min\big\{i\!\in\!\lrbr{{\mathbf{m}},0}_\mathfrak t\backslash I'_+ :
i\!\ge\!\ell(\hbar)\big\} &
\textnormal{if}~\hbar\!=\! \hbar^-_e,~e\!\in\! E_{\mathbf{m}},
\\
\min\big\{i\!\in\!\textnormal{Im}(\ell)\backslash I'_+ :
i\!\ge\!\ell(\hbar)\big\} &
\textnormal{otherwise}. \end{cases} $$ \end{itemize}
The above construction of $\mathfrak t_{(\mathbb I')}$ implies that
\begin{align*}
&{\mathbf{m}}({\mathfrak t}_{(\mathbb I')})=\min\big(\lrbr{{\mathbf{m}},0}_\mathfrak t\backslash I'_+\big),
&& E_{\mathbf{m}}\big(\mathfrak t_{(\mathbb I')}\big)= \big\{e\!\in\! E_{\mathbf{m}}\backslash E'_{\mathbf{m}}\!: \ell(\hbar_e^+)\!>\!{\mathbf{m}}({\mathfrak t}_{(\mathbb I')})\big\}, \\ &I_+\big(\mathfrak t_{(\mathbb I')}\big)=I_+\backslash I'_+, && E_-\big(\mathfrak t_{(\mathbb I')}\big)= \big(E_-\backslash E'_-\big) \sqcup \big\{e\!\in\! E_{\mathbf{m}}\backslash E'_{\mathbf{m}}\!: \ell(\hbar_e^+)\!\le\!{\mathbf{m}}({\mathfrak t}_{(\mathbb I')})\big\}. \end{align*}
In particular, we have $$
E_{\mathbf{m}}\big(\mathfrak t_{(\mathbb I')}\big)\sqcup
E_-\big(\mathfrak t_{(\mathbb I')}\big)=
\big(E_{\mathbf{m}}\backslash E'_{\mathbf{m}}\big)\sqcup
\big(E_-\backslash E'_-\big). $$ Intuitively, $\mathfrak t_{(\mathbb I')}$ is obtained from $\mathfrak t$ by contracting $E'_-$, then lifting the lower half-edges of the elements of $E'_{\mathbf{m}}$ to the level~${\mathbf{m}}$, and finally contracting the levels in $I'_+$.
\begin{dfn}\label{Dfn:Equiv_level_graphs} Two rooted level trees $\mathfrak t\!=\!(\tau,\ell)$ and $\mathfrak t'\!=\!(\tau',\ell')$ are said to be \textsf{equivalent}, denoted by $\mathfrak t\!\sim\!\mathfrak t'$, if the following conditions are satisfied: \begin{enumerate} [leftmargin=*,label=(E\arabic*)] \item $\tau\!=\!\tau'$; \item $\ell^{-1}\lrbr{{\mathbf{m}},0}_\mathfrak t\!=\!(\ell')^{-1}\lrbr{{\mathbf{m}}',0}_{\mathfrak t'}$; \item there exists a (unique) order preserving bijection $\alpha\!:\lrbr{{\mathbf{m}},0}_\mathfrak t\!\longrightarrow\!\lrbr{{\mathbf{m}}',0}_{\mathfrak t'}$ such that $\alpha\!\circ\!\ell\!=\!\ell'$ on $\ell^{-1}\lrbr{{\mathbf{m}},0}_\mathfrak t$. \end{enumerate} This gives rise to an equivalence relation on the set $\mathbf T$ of the rooted level trees. We denote by $[{\mathfrak t}]$ the equivalence class containing ${\mathfrak t}\!\in\!\mathbf T$ and by $$
\ov{\bT}:=
\big\{\,
[{\mathfrak t}]:\,
{\mathfrak t}\!\in\!\mathbf T\,
\big\} $$ the set of such equivalence classes.
\end{dfn}
Given $[\mathfrak t], [\mathfrak t']\!\in\!\ov{\bT}$, we set \begin{equation}\label{Eqn:level_partial_order} [\mathfrak t']\!\prec\![\mathfrak t] \qquad\Longleftrightarrow\qquad \exists\ \ \mathbb J\!\subset\!\mathbb I(\mathfrak t')\ \ \textnormal{s.t.}\ \ \mathbb I'\!\ne\!\emptyset,~[\mathfrak t'_{(\mathbb J)}]\!=\![\mathfrak t]. \end{equation} This gives rise to a partial order on $\ov{\bT}$. Notice that $[\mathfrak t']\!\prec\![\mathfrak t]$ implies that $\gamma_{[\mathfrak t']}\!\preceq\!\gamma_{[\mathfrak t]}.$
\subsection{$\Gamma$-stratification and Treelike structures} \label{Subsec:STF} In this subsection, we describe the stacks to which the theory of stacks with twisted fields (STF) can be applied.
Recall that $\mathbf{G}$ denotes the set of connected graphs, which is endowed with a partial order~(\ref{Eqn:partial_order}) given by the edge contractions. We say two graphs $\gamma$ and $\gamma'$ are isomorphic and write $$\gamma\simeq\gamma'$$ if there exists a bijection $\phi\!:\tn{HE}(\gamma)\!\longrightarrow\!\tn{HE}(\gamma')$ satisfying $$
\tn{Ver}(\gamma')=\big\{
\{\phi(\hbar):\hbar\!\in\! v\}:v\!\in\!\tn{Ver}(\gamma)
\big\},\qquad
\tn{Edg}(\gamma')=\big\{
\{\phi(\hbar):\hbar\!\in\! e\}:e\!\in\!\tn{Edg}(\gamma)
\big\}. $$ The graph isomorphism gives an equivalence relation on $\mathbf{G}$ that is compatible with the partial order~(\ref{Eqn:partial_order}) on $\mathbf{G}$. Also recall that $\tau_\bullet$ denotes the (connected) edge-less rooted tree. For any topological space $X$, let $\pi_0(X)$ be the set of all connected components of $X$. We do not consider~$\emptyset$ as a connected space, thus we take $\pi_0(\emptyset)\!=\!\emptyset$.
\begin{dfn}\label{Dfn:G-adim_fixture} Let $\Gamma\!\subset\!\mathbf{G}$ be a nonempty subset and $\mathfrak M$ be a smooth algebraic stack. A stratification $$\mathfrak M=\bigsqcup_{\gamma\in\Gamma}\mathfrak M_\gamma$$ by substacks is called a $\Gamma$-\textsf{stratification} of $\mathfrak M$ if \begin{enumerate} [leftmargin=*,label=$\bullet$]
\item for every $\gamma\!\in\!\Gamma$, there exists a set $\ud\mathfrak V_\gamma$ of affine smooth charts of $\mathfrak M$ satisfying $\mathfrak M_\gamma \!\subset\! \bigcup_{\mathcal V\in\ud\mathfrak V_\gamma}\!\!\!\mathcal V$ and
\item for every $\gamma\!\in\!\Gamma$ and every $\mathcal V\!\in\!\ud\mathfrak V_\gamma$, there exists a subset $\{\zeta_e^\mathcal V\}_{e\in\tn{Edg}(\gamma)}$ of local parameters on $\mathcal V$, known as the \textsf{modular parameters}, \end{enumerate} such that for every $\gamma,\gamma'\!\in\!\Gamma\backslash\{\tau_\bullet\}$ (i.e.~$\tn{Edg}(\gamma),\tn{Edg}(\gamma')\!\ne\!\emptyset$) and every $\mathcal V'\!\in\!\ud\mathfrak V_{\gamma'}$,
\begin{equation}\begin{split}\label{Eqn:M_strata_local}
&
\mathfrak M_{\gamma}\!\cap\!\mathcal V'
=\emptyset
\qquad\textnormal{if}~\gamma\!\not\succeq\!\gamma',
\\
&\pi_0(\mathfrak M_{\gamma}\!\cap\!\mathcal V')
\subset\big\{
\{\zeta_e^{\mathcal V'}\!\!\!=\! 0~
\forall\,e\!\in\!\tn{Edg}(\gamma')\backslash E;\;
\zeta_e^{\mathcal V'}\!\!\!\neq\! 0~
\forall\,e\!\in\! E\}:
E\!\subset\!\tn{Edg}(\gamma'),\,
\gamma'_{(E)}\!\simeq\gamma\,\big\}
\quad
\textnormal{if}~\gamma\!\succeq\!\gamma'. \end{split}\end{equation} \end{dfn}
In Definition~\ref{Dfn:G-adim_fixture}, some strata $\mathfrak M_\gamma$ may be disconnected or empty. If $\mathfrak M_\gamma\!\ne\!\emptyset$ and $\gamma\!\ne\!\tau_\bullet$, then~(\ref{Eqn:M_strata_local}) implies that for every $x\!\in\!\mathfrak M_\gamma$, there exists $\mathcal V\!\in\!\ud\mathfrak V_\gamma$ containing $x$ such that $$
\mathfrak M_\gamma\cap\mathcal V=
\{\,\zeta_e^{\mathcal V}\!\!=\! 0\ \
\forall\,e\!\in\!\tn{Edg}(\gamma)\,\}. $$
\begin{rmk}\label{Rmk:graph_isom} When we consider a $\Gamma$-stratification, it is often handy to allow isomorphic graphs $\gamma\!\simeq\!\gamma'$ to index the same stratum: $\mathfrak M_\gamma\!=\!\mathfrak M_{\gamma'}$. In such a situation, rigorously we should take $\Gamma$ as a set of equivalence classes $[\gamma]$ of graphs $\gamma\!\in\!\mathbf{G}$ with respect to graph isomorphism. However, writing the strata of $\mathfrak M$ as $\mathfrak M_{[\gamma]}$ would make the notation in~\S\ref{Sec:Induced_and_derived_tf_stacks} and~\S\ref{Sec:genus_2_twisted_fields} too complicated, so by abuse of notation, we will still write $\mathfrak M\!=\!\bigsqcup_{\gamma\in\Gamma}\mathfrak M_\gamma$ even if the graphs in $\Gamma$ are considered up to graph isomorphism. Similarly, when the context is clear, we will still write $\gamma'\!\prec\!\gamma$ if the graphs $\gamma,\gamma'\!\in\!\mathbf{G}$ are considered up to graph isomorphism. \end{rmk}
For any subset $\mathfrak N\!\subset\!\mathfrak M$, we denote by $$
\textnormal{Cl}_\mathfrak M(\mathfrak N)
~(\,\subset\mathfrak M\,) $$ the closure of $\mathfrak N$ in $\mathfrak M$. The lemma below follows directly from~(\ref{Eqn:M_strata_local}).
\begin{lmm}\label{Lm:M_strata_closure} Let $\Gamma$ and $\mathfrak M$ be as in Definition~\ref{Dfn:G-adim_fixture}. For every $\gamma,\gamma'\!\in\!\Gamma$ and every $\mathfrak N_{\gamma'}\!\in\!\pi_0(\mathfrak M_{\gamma'})$ such that $
\textnormal{Cl}_\mathfrak M(\mathfrak M_{\gamma})\!\cap\!\mathfrak N_{\gamma'}\!\ne\!\emptyset$, we have $\gamma'\!\preceq\!\gamma$ (up to graph isomorphism). \end{lmm}
If $\mathfrak M$ is endowed with a $\Gamma$-stratification, we define the \textsf{boundary} of $\mathfrak M$ to be \begin{equation}\label{Eqn:boundary}
\Delta:=\bigsqcup_{\gamma\in\Gamma,\,
|\tn{Edg}(\gamma)|>0}\!\!\!\!\!\!\!\!\!\!\mathfrak M_\gamma. \end{equation} By~(\ref{Eqn:M_strata_local}), $\Delta$ is of lower dimension. Thus, $\Gamma$ must contain the connected edge-less graph $\tau_\bullet$, and $$
\mathfrak M=\mathfrak M_{\tau_\bullet}\sqcup\Delta. $$
For every $\mathfrak M_\gamma\!\subset\!\Delta$, every $\mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma)$, and every $e\!\in\!\tn{E}_{\mathfrak N_\gamma}$, we denote by \begin{equation}\label{Eqn:L_e}
L_e\longrightarrow\mathfrak N_\gamma \end{equation} the line bundle such that on each chart $\mathcal V\!\in\!\ud\mathfrak V_\gamma$ with $\mathfrak N_\gamma\!\cap\!\mathcal V\!\ne\!\emptyset$, the line bundle $L_e/(\mathfrak N_\gamma\!\cap\!\mathcal V)$ is the restriction of normal bundle of the local divisor $\{\zeta_e^\mathcal V\!\!=\! 0\}$ to $\mathfrak N_\gamma\!\cap\!\mathcal V$. As shown in~\cite[Lemma~3.1]{g1modular}, the restriction of $$
\tfrac{\partial}{\partial\zeta_e}:=(\textnormal{d}\zeta_e)^\vee
\,\in\Gamma(\mathcal V;T\mathfrak M) $$ to $\mathfrak N_\gamma\!\cap\!\mathcal V$ gives a nowhere vanishing section of $L_e/(\mathfrak N_\gamma\!\cap\!\mathcal V)$. In~\S\ref{Subsec:STF_main_statement}, the line bundles~(\ref{Eqn:L_e}) will be ``twisted'' in~(\ref{Eqn:Twisted_line_bundle}) to define the twisted fields in Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p1}.
\begin{dfn}\label{Dfn:Treelike_structure} Let $\Gamma$ and $\mathfrak M$ be as in Definition~\ref{Dfn:G-adim_fixture}. We call the indexed family \begin{equation}\label{Eqn:Treelike_structure}
\Lambda=\Lambda_{(\mathfrak M,\Gamma)}:=
\big(\,
\mathcal T_{\mathfrak N_{\gamma}}\!:=\!
(\,
\tau_{\mathfrak N_\gamma},\,
\tn{E}_{\mathfrak N_\gamma},\,
\beta_{\mathfrak N_\gamma}
\,)\,\big)_{
\gamma\in\Gamma,\,\mathfrak N_\gamma\in\pi_0(\mathfrak M_\gamma)} \end{equation} a \textsf{treelike structure} on $(\mathfrak M,\Gamma)$ if it assigns to each pair $\big(\gamma\!\in\!\Gamma\,;\,\mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma)\big)$ a unique tuple $\mathcal T_{\mathfrak N_{\gamma}}$ consisting of \begin{itemize} [leftmargin=*] \item a rooted tree $\tau_{\mathfrak N_\gamma}$ with the root $o_{\mathfrak N_\gamma}$, \item a subset $\tn{E}_{\mathfrak N_\gamma}\!\!\subset\!\tn{Edg}(\gamma)$, and \item a bijection \hbox{$\beta_{\mathfrak N_\gamma}\!\!: \tn{Edg}(\tau_{\mathfrak N_\gamma})\!\longrightarrow\!\tn{E}_{\mathfrak N_\gamma}$} (which induces a bijection $\beta_{\mathfrak N_\gamma}\!: \tn{HE}(\tau_{\mathfrak N_\gamma})\!\longrightarrow\!\tn{HE}(\tn{E}_{\mathfrak N_\gamma})$) \end{itemize} such that for every $\gamma'\!\in\!\Gamma$ and every $\mathfrak N_{\gamma'}\!\in\!\pi_0(\mathfrak M_{\gamma'})$, all of the following conditions hold. \begin{enumerate} [leftmargin=*,label=$(\mathsf{\alph*})$] \item\label{Cond:Treelike_a} For every $\gamma\!\in\!\Gamma$ and every $\mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma)$, if $\textnormal{Cl}_{\mathfrak M}(\mathfrak N_\gamma)\!\cap\!\mathfrak N_{\gamma'}\!\ne\!\emptyset$, then $\tau_{\mathfrak N_\gamma}\!\succeq\!\tau_{\mathfrak N_{\gamma'}}$ up to graph isomorphism.
\item \label{Cond:Treelike_b} If $\tn{E}_{\mathfrak N_{\gamma'}}\!\ne\!\emptyset$ (i.e.~$\tau_{\mathfrak N_{\gamma'}}\!\ne\!\tau_\bullet$), then for every $e\!\in\!\tn{Edg}(\gamma')$, \begin{enumerate} [leftmargin=*,label=$(\mathsf{b}_{\arabic*})$] \item \label{Cond:Treelike_b1} if $e\!\not\in\!\beta_{\mathfrak N_{\gamma'}}\big(\tn{Edg}(\tau_{\mathfrak N_{\gamma'}})_{\max}\!\cap\!\tn{Edg}(\tau_{\mathfrak N_{\gamma'}})_{\min}\big)$, then we have \begin{itemize} [leftmargin=*] \item the graph $\gamma\!:=\!\gamma'_{(e)}$ is contained in $\Gamma$,
\item for every $\mathcal V'\!\in\!\ud{\mathfrak V}_{\gamma'}$ with $\mathcal V'\!\cap\!\mathfrak N_{\gamma'}\!\ne\!\emptyset$, $$
\{\,\zeta_{e'}^{\mathcal V'}\!\!\!=\! 0\ \,
\forall\,e'\!\in\!\tn{Edg}(\gamma')\backslash\{e\}\ ;\ \,
\zeta_{e}^{\mathcal V'}\!\!\!\neq\! 0\,\}
\,\in\,\pi_0(\mathfrak M_\gamma\!\cap\!\mathcal V'),
\qquad\textnormal{and} $$
\item for every $\mathfrak N_{\gamma}\!\in\!\pi_0(\mathfrak M_\gamma)$ with $\textnormal{Cl}_{\mathfrak M}(\mathfrak N_\gamma)\!\cap\!\mathfrak N_{\gamma'}\!\ne\!\emptyset$, the tuple $\mathcal T_{\mathfrak N_\gamma}$ is (up to graph isomorphism) equal to $$\begin{cases}
\mathcal T_{\mathfrak N_{\gamma'}} & \textnormal{if}~e\!\not\in\!\tn{E}_{\mathfrak N_{\gamma'}},\\
\big(\,(\tau_{\mathfrak N_{\gamma'}})_{(\mathfrak e)}\,,~
\tn{E}_{\mathfrak N_{\gamma'}}\!\backslash\{e\}\,,~
\beta_{\mathfrak N_{\gamma'}}|_{\tn{Edg}(\tau_{\mathfrak N_\gamma})}
\,\big) & \textnormal{if}~\mathfrak e\!:=\!\beta_{\mathfrak N_{\gamma'}}^{-1}(e)\!\in\!\tn{Edg}(\tau_{\mathfrak N_{\gamma'}})\backslash\tn{Edg}(\tau_{\mathfrak N_{\gamma'}})_{\min},\\
\big(\,(\tau_{\mathfrak N_{\gamma'}})_{(\mathfrak E^\wedge_\mathfrak e)}\,,~
\tn{E}_{\mathfrak N_{\gamma'}}\!\backslash\,\beta_{\mathfrak N_{\gamma'}}\!(\mathfrak E^\wedge_\mathfrak e)\,,~
\beta_{\mathfrak N_{\gamma'}}|_{\tn{Edg}(\tau_{\mathfrak N_\gamma})}
\,\big) & \textnormal{if}~\mathfrak e\!:=\!\beta_{\mathfrak N_{\gamma'}}^{-1}(e)\!\in\!\tn{Edg}(\tau_{\mathfrak N_{\gamma'}})_{\min}\backslash\tn{Edg}(\tau_{\mathfrak N_{\gamma'}})_{\max},
\end{cases} $$ where $\mathfrak E^\wedge_\mathfrak e:=
\{\,\mathfrak e'\!\in\!\tn{Edg}(\tau_{\mathfrak N_{\gamma'}}):\,
v(\hbar^+_{\mathfrak e'})\preceq v(\hbar^+_\mathfrak e)\,\}$ $\big(
\subset\tn{Edg}(\tau_{\mathfrak N_{\gamma'}})\big)$; \end{itemize}
\item \label{Cond:Treelike_b2} if $e\!\in\!\beta_{\mathfrak N_{\gamma'}}\big(\tn{Edg}(\tau_{\mathfrak N_{\gamma'}})_{\max}\!\cap\!\tn{Edg}(\tau_{\mathfrak N_{\gamma'}})_{\min}\big)$, then for every $E\!\subset\!\tn{Edg}(\gamma')$ containing $e$ with $\gamma\!:=\!\gamma'_{(E)}\!\in\!\Gamma$ and every $\mathfrak N_{\gamma}\!\in\!\pi_0(\mathfrak M_\gamma)$ with $\textnormal{Cl}_{\mathfrak M}(\mathfrak N_\gamma)\!\cap\!\mathfrak N_{\gamma'}\!\ne\!\emptyset$, we have $\tau_{\mathfrak N_\gamma}\!=\!\tau_\bullet$. \end{enumerate} \end{enumerate} \end{dfn}
We remark that in Definition~\ref{Dfn:Treelike_structure}, if $\mathfrak e$ is a {\it minimal} edge of $\tau_{\mathfrak N_{\gamma'}}$, then $(\tau_{\mathfrak N_{\gamma'}})_{(\mathfrak E^\wedge_\mathfrak e)}$ that is obtained from $\tau_{\mathfrak N_{\gamma'}}$ by contracting all the edges ``below'' the vertex $v(\hbar_{\mathfrak e}^+)$ can equivalently be obtained by dissolving the vertex $v(\hbar_{\mathfrak e}^+)$ and then taking the connected component containing the root of $\tau_{\mathfrak N_{\gamma'}}$.
Given $\gamma'\!\in\!\Gamma$ and $E\!=\!\{e_1,\ldots e_n\}\!\subset\!\tn{Edg}(\gamma')$, contracting the edges of $E$ one by one in any order \begin{equation} \label{Eqn:order_on_E} e_1,\ e_2,\ \ldots,\ e_n \end{equation} yields the same graph $\gamma\!:=\!\gamma'_{(E)}$. If $\gamma\!\in\!\Gamma$, then for all $\mathfrak N_{\gamma'}\in\pi_0(\mathfrak M_{\gamma'})$ and $\mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma)$ with $\textnormal{Cl}_{\mathfrak M}(\mathfrak N_\gamma)\!\cap\!\mathfrak N_{\gamma'}\!\ne\!\emptyset$, we can follow Definition~\ref{Dfn:Treelike_structure} and obtain a new tuple, which is exactly~$\mathcal T_{\mathfrak N_\gamma}$, from $\mathcal T_{\mathfrak N_{\gamma'}}$ by contracting $e_1,\ldots,e_n$ with respect to the order~(\ref{Eqn:order_on_E}). The tuple~$\mathcal T_{\mathfrak N_\gamma}$, however, is independent of the choice of the order~(\ref{Eqn:order_on_E}).
\begin{eg}\label{Eg:genus_1} Introduced in~\cite{HL10} and further studied in~\cite{g1modular}, the smooth stack $\mathfrak M^\tn{wt}_1$ of genus 1 stable weighted curves has a natural treelike structure. Recall that $\mathfrak M^\tn{wt}_1$ consists of the pairs $(C,\mathbf w)$ of genus 1 nodal curves $C$ and weights $\mathbf w\!\in\! H^2(C,\mathbb Z)$ satisfying $\mathbf w(\Sigma)\!\ge\!0$ for all irreducible $\Sigma\!\subset\! C$. A pair $(C,\mathbf w)$ is stable if every rational irreducible component of $C$ with weight 0 contains at least three nodal points. Let $\Gamma\!=\!\{\gamma\!\in\!\mathbf{G}:b_1(\gamma)\!\le\!1\}$. The stack $\mathfrak M^\tn{wt}_1$ has a natural $\Gamma$-stratification by dual graphs (c.f.~Definition~\ref{Dfn:dual_graph}). Given $\gamma\!\in\!\Gamma$, the connected components $\mathfrak N_\gamma$ of the stratum $\mathfrak M^\tn{wt}_{1;\gamma}$ are indexed by the distribution of the weights on the irreducible components (or equivalently on the vertices of $\gamma$).
To each $\mathfrak N_\gamma$, we assign a rooted tree $\tau_{\mathfrak N_\gamma}$ that is obtained from $\gamma$ by \begin{itemize} [leftmargin=*] \item contracting all the edges in the smallest connected genus 1 subgraph $\gamma_\tn{cor}$ of $\gamma$, \item then dissolving all the vertices with $\mathbf w(v)\!>\!0$, and \item finally taking the connected component containing the vertex that is the image of $\gamma_\tn{cor}$. \end{itemize} Obviously, the edges of $\tau_{\mathfrak N_\gamma}$ form a subset $\tn{E}_{\mathfrak N_{\gamma}}$ of $\tn{Edg}(\gamma)$, so $\beta_{\mathfrak N_\gamma}$ is simply the inclusion. We remark that $\tau_{\mathfrak N_\gamma}$ is called the {\it weighted dual tree} of $\mathfrak N_\gamma$ in~\cite[\S2.2]{g1modular}.
The above construction of $\tau_{\mathfrak N_\gamma}$, $\tn{E}_{\mathfrak N_\gamma}$, and $\beta_{\mathfrak N_\gamma}$ gives rise to a treelike structure on $(\mathfrak M^\tn{wt}_1,\Gamma)$. In fact, it is straightforward to verify that $\tau_{\mathfrak N_{\gamma}}$ is a rooted tree satisfying the condition~\ref{Cond:Treelike_a} of Definition~\ref{Dfn:Treelike_structure} and that the graph $\gamma_{(e)}$ is in $\Gamma$ for every edge $e$. It is also straightforward that for every $\gamma'\!\preceq\!\gamma$ and every chart $\mathcal V\!\longrightarrow\!\mathfrak M_1^\tn{wt}$ centered at a point of $\mathfrak M^\tn{wt}_{1;\gamma'}$, $$
\mathfrak M^\tn{wt}_{1;\gamma}\cap\mathcal V=
\bigsqcup_{E\subset\tn{Edg}(\gamma'),\;
\gamma'_{(E)}\simeq\gamma}\hspace{-.3in}
\{\,\zeta_e\!=\! 0\ \forall\,e\!\in\!\tn{Edg}(\gamma')\backslash E\,;\
\zeta_e\!\ne\!0\ \forall\,e\!\in\! E\,\}\,, $$ where each $\zeta_e$ is a parameter corresponding to the smoothing of the node labeled by $e$. If $\tn{E}_{\mathfrak N_\gamma}\!\ne\!\emptyset$, let~$\mathfrak N_{\gamma_{(e)}}$ be a connected component of $\mathfrak M^\tn{wt}_{1;\gamma_{(e)}}$ satisfying $\textnormal{Cl}_{\mathfrak M^\tn{wt}_1}(\mathfrak N_{\gamma_{(e)}})\!\cap\!\mathfrak N_\gamma\!\ne\!\emptyset$; such $\mathfrak N_{\gamma_{(e)}}$ exists by the deformation of nodal curves.
If $e$ is not an edge of $\tau_{\mathfrak N_\gamma}$, then obviously $\mathcal T_{\mathfrak N_{\gamma_{(e)}}}\!=\!\mathcal T_{\mathfrak N_\gamma}$. If an edge $e$ of $\tau_{\mathfrak N_\gamma}$ is not minimal, then the two endpoints of~$e$ are both of weight 0, hence their image in~$\gamma_{(e)}$ is of weight 0, which implies $\tau_{\mathfrak N_{\gamma_{(e)}}}\!=\!(\tau_{\mathfrak N_{\gamma}})_{(e)}$. If an edge $e$ of $\tau_{\mathfrak N_\gamma}$ is minimal, the construction of $\tau_{\mathfrak N_\gamma}$ implies that $v(\hbar_e^-)$ is positively weighted and so is its image $v_{(e)}$ in $\gamma_{(e)}$. Therefore, $v_{(e)}$ is a minimal vertex of $\tau_{\mathfrak N_{\gamma_{(e)}}}$. Every vertex $v'$ of $\tau_{\mathfrak N_\gamma}$ with $v'\!\prec\!v(\hbar_e^+)$ is thus not in~$\tau_{\mathfrak N_{\gamma_{(e)}}}$. In sum, the conditions of Definition~\ref{Dfn:Treelike_structure} are all satisfied.
After choosing this treelike structure, we can construct the stack with twisted fields $\mathfrak M^\tn{tf}_1$ as in Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p1} below, which is the same as that in~\cite[(2.13)]{g1modular}. \end{eg}
\subsection{Stacks with twisted fields} \label{Subsec:STF_main_statement} We are ready to present the main statement of the STF theory. Given a direct sum of line bundles $V\!=\!\oplus_iL_i'$ (over an arbitrary base), we write $$
\mathring\mathbb P(V)=
\big\{
\big(x,[v_i]\big)\!\in\!\mathbb P(V):
v_i\!\ne\!0~\forall\,i
\big\}. $$ Given $k\!\in\!\mathbb Z_{>0}$ and morphisms $M_i\!\longrightarrow\! S$ with $1\!\le\!i\!\le\!k$, we write $$
\prod_{1\le i\le k}
\!(M_i/S) \,
:=
M_1\times_S M_2\times_S\cdots
\times_S M_k. $$
Let $\Gamma$, $\mathfrak M$, and $\Lambda$ be as in Definition~\ref{Dfn:Treelike_structure}. Given $\gamma\!\in\!\Gamma$ and $\mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma)$, there exists a unique tuple $(\tau_{\mathfrak N_\gamma}, \tn{E}_{\mathfrak N_\gamma}, \beta_{\mathfrak N_\gamma})\!\in\!\Lambda$, which in turn determines a subset of $\ov\mathbf T$: \begin{equation}\label{Eqn:Treelike_Level}
\ov\Lambda_{\mathfrak N_\gamma}:=
\big\{\,
[{\mathfrak t}]\!=\![\tau_{\mathfrak t},\ell_{\mathfrak t}]\!\in\!\ov\mathbf T\,:\,
\tau_{\mathfrak t}\!=\!\tau_{\mathfrak N_\gamma}
\,\big\}\,. \end{equation}
For every $e\!\in\!\tn{Edg}(\tau_{\mathfrak N_\gamma})$, let \begin{equation}\label{Eqn:Twisted_line_bundle}
L_{\succeq_{\mathfrak N_\gamma}e}
:=\bigotimes_{e'\succeq_{\mathfrak N_\gamma} e}\!
L_{\beta_{\mathfrak N_\gamma}(e')}\ \longrightarrow\ \mathfrak N_\gamma\,, \end{equation} where $L_e$ are the line bundles as in~(\ref{Eqn:L_e}) and $\succeq_{\mathfrak N_\gamma}$ is the tree order on $\tau_{\mathfrak N_\gamma}$.
\begin{thm}\label{Thm:tf_smooth} With notation as above, we have the following conclusions:
\begin{enumerate} [leftmargin=*,label=$(\mathsf{p_\arabic*})$]
\item \label{Cond:p1} the following disjoint union of the fiber products over the strata of $\mathfrak M$: \begin{equation*}\begin{split}
\mtd{}=\mtd{\Lambda}
&=\bigsqcup_{\gamma\in \Gamma;\;
\mathfrak N_\gamma\in\pi_0(\mathfrak M_{\gamma});\;
[{\mathfrak t}]\in\ov \Lambda_{\mathfrak N_\gamma}}
\hspace{-.4in}
(\mathfrak N_\gamma)_{[{\mathfrak t}]}^\tn{tf}\ ,\qquad
\textnormal{where}\\
(\mathfrak N_\gamma)_{[{\mathfrak t}]}^\tn{tf}
&=
\bigg(\!
\prod_{i\in\lbrp{{\mathbf{m}},0}_{\mathfrak t}}\!\!\!\!
\Big\lgroup\Big(
\mathring{\mathbb P}\big(\!
\bigoplus_{
\begin{subarray}{c}
e\in\mathfrak E_i^\bot({\mathfrak t})
\end{subarray}
}\!\!\!
L_{\succeq_{\mathfrak N_\gamma} e}\,
\big)
\Big)
\Big/\mathfrak N_\gamma
\Big\rgroup\!\!
\bigg)
\stackrel{\varpi}{\longrightarrow}\mathfrak N_\gamma, \end{split}\end{equation*} has a canonical smooth algebraic stack structure, known as the \textsf{stack with twisted fields over $\mathfrak M$ with respective to $\Lambda$,} and determines a proper and birational morphism $\varpi\!:\mathfrak M^\tn{tf}\!\longrightarrow\!\mathfrak M$ known as the \textsf{forgetful morphism};
\item \label{Cond:p2} for any $[{\mathfrak t}]\!\in\! \ov\Lambda_{\mathfrak N_\gamma}$ and $x\!\in\!(\mathfrak N_{\gamma})_{[{\mathfrak t}]}^\tn{tf}$, there exist a smooth chart $\mathfrak U_x\!\longrightarrow\!\mtd{}$ containing $x$, called a \textsf{twisted chart (centered at $x$)}, a set of special edges $\{ e_i\!\in\!\mathfrak E_i^\bot({\mathfrak t})\}_{i\in\lbrp{{\mathbf{m}},0}_{\mathfrak t}}$, and a subset of local parameters called the \textsf{twisted parameters}: \begin{equation*}\begin{split}
\xi_s^{\mathfrak U_x},\quad s\in
\ti\mathbb I({\mathfrak t})
&:=
\mathbb I(\mathfrak t)
\sqcup\big(\mathfrak E_{\ge{\mathbf{m}}}^\bot({\mathfrak t})\backslash\{ e_i\}_{i\in\lbrp{{\mathbf{m}},0}_{\mathfrak t}}\big)
\\
&= \lbrp{{\mathbf{m}},0}_\mathfrak t\sqcup E_{\mathbf{m}}(\mathfrak t)\sqcup E_-(\mathfrak t)\sqcup\big(\mathfrak E_{\ge{\mathbf{m}}}^\bot({\mathfrak t})\backslash\{ e_i\}_{i\in\lbrp{{\mathbf{m}},0}_{\mathfrak t}}\big) \end{split}\end{equation*} such that $$
\xi_s^{\mathfrak U_x}\!\in\!\Gamma(\mathscr O_{\mathfrak U_x}^*)\quad
\forall\,s\!\in\!\mathfrak E_{\ge{\mathbf{m}}}^\bot({\mathfrak t})\backslash\{ e_i\}_{i\in\lbrp{{\mathbf{m}},0}_{\mathfrak t}},\qquad
(\mathfrak N_{\gamma})_{[{\mathfrak t}]}^\tn{tf}\cap\mathfrak U_x=
\{\,
\xi_s^{\mathfrak U_x}\!\!=\! 0\
\forall\,s\!\in\!\mathbb I(\mathfrak t)\,\}; $$ moreover, for every $\gamma'\!\in\!\Gamma$, every $\mathfrak N_{\gamma'}\!\in\!\pi_0(\mathfrak M_{\gamma'})$, and every $[{\mathfrak t}']\!\in\!\ov\Lambda_{\mathfrak N_{\gamma'}}$, \begin{itemize} [leftmargin=*] \item if $[\mathfrak t']\!\succeq\![\mathfrak t]$ and $\textnormal{Cl}_\mathfrak M(\mathfrak N_{\gamma'})\!\cap\!\mathfrak N_{\gamma}\!\ne\!\emptyset$, then $\pi_0\big(
(\mathfrak N_{\gamma'})_{[{\mathfrak t}']}^\tn{tf}\!\cap\!\mathfrak U_x
\big)$ is a subset of \begin{equation*}\begin{split}
\big\{\,
\{\,
&\xi_s^{\mathfrak U_x}\!\!=\! 0~
\forall\,s\!\in\!\mathbb I(\mathfrak t')\,;\,
\xi_s^{\mathfrak U_x}\!\ne\!0~
\forall\,s\!\in\!\mathbb J\,;\,
\varpi^*\zeta_e\!=\! 0~
\forall\,e\!\in\!(\tn{Edg}(\gamma)\backslash\tn{E}_{\mathfrak N_\gamma})\backslash E\,;\\
&\varpi^*\zeta_e\!\ne\!0~
\forall\,e\!\in\!(\tn{Edg}(\gamma)\backslash\tn{E}_{\mathfrak N_\gamma})\!\cap\!E\,\}\;:~
\mathbb J\!\subset\!\mathbb I({\mathfrak t}),\;
{\mathfrak t}_{(\mathbb J)}\!\sim\!{\mathfrak t}',\;
E\!\subset\!\tn{Edg}(\gamma),\;
\gamma_{(E)}\!\simeq\!\gamma'\,\big\}, \end{split}\end{equation*}
\item if $[\mathfrak t']\!\not\succeq\![\mathfrak t]$ or $\textnormal{Cl}_\mathfrak M(\mathfrak N_{\gamma'})\!\cap\!\mathfrak N_{\gamma}\!=\!\emptyset$, then $(\mathfrak N_{\gamma'})_{[{\mathfrak t}']}^\tn{tf}\!\cap\!\mathfrak U_x\!=\!\emptyset$; \end{itemize}
\iffalse \begin{equation*}\label{Eqn:Mtf_strata_local} \begin{split}
&~(\mathfrak N_{\gamma'})_{[{\mathfrak t}']}^\tn{tf}\cap\mathfrak U_x=
\\
&\begin{cases}
\bigsqcup_{\mathbb J\subset\mathbb I({\mathfrak t}),\;
{\mathfrak t}_{(\mathbb J)}\sim{\mathfrak t}'}\big\{\xi_s^{\mathfrak U_x}\!\!=\! 0~
\forall\,s\!\in\!\mathbb I(\mathfrak t')\,;\,
\xi_s^{\mathfrak U_x}\!\ne\!0~
\forall\,s\!\in\!\mathbb J\big\}
& \textnormal{if}~[\mathfrak t']\!\succeq\![\mathfrak t]~\textnormal{and}~\textnormal{Cl}_\mathfrak M(\mathfrak N_{\gamma'})\!\cap\!\mathfrak N_{\gamma}\!\ne\!\emptyset;\\
\emptyset & \textnormal{otherwise}.
\end{cases} \end{split} \end{equation*} \fi
\item \label{Cond:p3} with notation as in~~\ref{Cond:p2}, for any $\mathcal V\!\in\!\ud\mathfrak V_\gamma$ containing $\varpi(x)$, if the set of modular parameters $\{\zeta_{e}^\mathcal V\}$ as in Definition~\ref{Dfn:G-adim_fixture} extends to a set of local parameters on $\mathcal V$: $$
\{\zeta_e^\mathcal V\}_{e\in\tn{Edg}(\gamma)}\sqcup
\{\varsigma_j\}_{j\in J} $$ for some index set $J$, then $$
\{\xi_s^{\mathfrak U_x}\}_{s\in\ti\mathbb I({\mathfrak t})}\sqcup
\{\varpi^*\zeta_e^\mathcal V\}_{e\in\tn{Edg}(\gamma)\backslash\tn{E}_{\mathfrak N_\gamma}}\sqcup
\{\varpi^*\varsigma_j\}_{j\in J} $$ is a set of local parameters on $\mathfrak U_x$; and
\item \label{Cond:p4} for any $[\mathfrak t]\!\in\! \ov \Lambda_{\mathfrak N_\gamma}$, $x\!\in\!(\mathfrak N_\gamma)_{[{\mathfrak t}]}^\tn{tf}$, $\mathcal V\!\in\!\ud\mathfrak V_{\gamma}$ containing $\varpi(x)$, and $e\!\in\!\mathfrak E_{\mathbf{m}}(\mathfrak t)$, we have $$\varpi^*\big(\prod_{e'\succeq_{\mathfrak N_\gamma} e}\!\!\zeta_{\beta_{\mathfrak N_\gamma}(e')}^\mathcal V\,\big)= u_e\cdot\prod_{i\in\lbrp{{\mathbf{m}},0}_\mathfrak t}\!\!\!\xi_i^{\mathfrak U_x},$$ where $u_e\!\in\!\Gamma(\mathscr O_{\mathfrak U_x})$ is a unit if $e\!\in\!\mathfrak E_{\mathbf{m}}^\bot({\mathfrak t})$ and is equal to $\xi_e^{\mathfrak U_x}$ (up to a unit) if $e\!\in\! E_{\mathbf{m}}({\mathfrak t})\,\big(\!=\!\mathfrak E_{\mathbf{m}}({\mathfrak t})\backslash\mathfrak E^\bot_{\mathbf{m}}({\mathfrak t})\big)$. \end{enumerate} \end{thm}
\begin{proof} The statement that $\mathfrak M^\tn{tf}$ is a smooth algebraic stack in~\ref{Cond:p1}, as well as the construction of the smooth chart in~\ref{Cond:p2}, follow from the same argument as in~\cite[Corollary~3.7]{g1modular}. The key reason is as follows. For every $\gamma\!\in\!\Gamma$, $\mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma)$ with $\tau_{\mathfrak N_\gamma}\!\ne\!\tau_\bullet$, $[{\mathfrak t}]\!\in\!\ov\Lambda_{\mathfrak N_\gamma}$, and $\mathbb I'\!\subset\!\mathbb I({\mathfrak t})$ with \begin{equation}\label{Eqn:I_+'}
I_+'\big(:=\mathbb I'\cap\lbrp{{\mathbf{m}},0}_{\mathfrak t}\big)\subsetneq\lbrp{{\mathbf{m}},0}_{\mathfrak t}, \end{equation} we take $$
\ti E_{(\mathbb I')}:=
\tn{Edg}(\tau_{\mathfrak N_\gamma})\backslash\tn{Edg}({\mathfrak t}_{(\mathbb I')}), $$ which is the set of the edges contracted from $\tn{Edg}({\mathfrak t})\!=\! \tn{Edg}(\tau_{\mathfrak N_\gamma})$ to obtain ${\mathfrak t}_{(\mathbb I')}$ as in~(\ref{Eqn:rooted_level_tree_contraction}). By~(\ref{Eqn:I_+'}), $\ti E_{(\mathbb I')}\!\cap\!\tn{Edg}(\tau_{\mathfrak N_\gamma})_{\max}\!\cap\!\tn{Edg}(\tau_{\mathfrak N_\gamma})_{\min}\!=\!\emptyset$. Thus, by Definition~\ref{Dfn:Treelike_structure}~\ref{Cond:Treelike_b1}, for every $x\!\in\!\mathfrak N_\gamma$ and every $\mathcal V\!\in\!\ud\mathfrak V_\gamma$ containing $x$, the chart $\mathcal V\!\longrightarrow\!\mathfrak M$ induces a chart \begin{equation*}\begin{split} &\mathcal V_{{\mathfrak t}_{(\mathbb I')}}:=
\big\{\,
\zeta_{\beta_{\mathfrak N_\gamma}(\mathfrak e)}^\mathcal V\!=\! 0\ \ \forall\,\mathfrak e\!\in\!\tn{Edg}({\mathfrak t}_{(\mathbb I')})\,\big(\!=\!\tn{Edg}(\tau_{\mathfrak N_\gamma})\backslash\ti E_{(\mathbb I')}\big)\,;\ \,\zeta_{\beta_{\mathfrak N_\gamma}(\mathfrak e)}^\mathcal V\!\ne\! 0\ \ \forall\,\mathfrak e\!\in\!\ti E_{(\mathbb I')}\,
\big\}\\ &\hspace{3in}\longrightarrow
\bigsqcup_{E\subset\tn{Edg}(\gamma),~
E\cap\tn{E}_{\mathfrak N_\gamma}=\beta_{\mathfrak N_\gamma}(\ti E_{(\mathbb I')})}\hspace{-.5in}
\mathfrak M_{\gamma_{(E)}}\qquad
(\,\subset\mathfrak M)\,, \end{split}\end{equation*} which is the analogue of~\cite[(3.2)]{g1modular}. Definition~\ref{Dfn:Treelike_structure}~\ref{Cond:Treelike_b1} also implies that \begin{equation*} \begin{split} & \mathcal T_{\mathfrak N_{\gamma_{(E)}}}\!=
\big(\,
\tau_{{\mathfrak t}_{(\mathbb I')}},\,
\tn{E}_{\mathfrak N_\gamma}\!\backslash E,\,
\beta_{\mathfrak N_\gamma}|_{\tn{Edg}({\mathfrak t}_{(\mathbb I')})}\,
\big) \\ &\forall\ E\!\subset\!\tn{Edg}(\gamma),~
\mathfrak N_{\gamma_{(E)}}\!\!\in\!\pi_0(\mathfrak M_{\gamma_{(E)}})
\ \ \textnormal{with}\ \
E\!\cap\!\tn{E}_{\mathfrak N_\gamma}\!=\!\beta_{\mathfrak N_\gamma}\!\big(\ti E_{(\mathbb I')}\big),~
\textnormal{Cl}_\mathfrak M(\mathfrak N_{\gamma_{(E)}})\!\cap\!\mathfrak N_\gamma\!\ne\!\emptyset. \end{split} \end{equation*} Thus, we can define the locus $\mathfrak U_{x;[{\mathfrak t}_{(\mathbb I')}]}\!\subset\!\mathfrak U_x\!\subset\!\mathbb A^{\ti\mathbb I({\mathfrak t})}$ mimicking the paragraph containing~\cite[(3.10)]{g1modular}, and define the morphism $$
\Phi_{x;(\mathbb I')}: \mathfrak U_{x;[{\mathfrak t}_{(\mathbb I')}]}\longrightarrow
\mathcal V_{{\mathfrak t}_{(\mathbb I')}}\times_{\mathfrak M}\mathfrak M^\tn{tf} $$ as in~\cite[(3.14)]{g1modular}. Such $\Phi_{x;(\mathbb I')}$ can also be defined for $\mathbb I'\!\subset\!\mathbb I({\mathfrak t})$ that does not satisfy~(\ref{Eqn:I_+'}) and for $\mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma)$ with $\tau_{\mathfrak N_\gamma}\!=\!\tau_\bullet$, because no twisted field is added to these strata by Definition~\ref{Dfn:Treelike_structure}~\ref{Cond:Treelike_a} and~\ref{Cond:Treelike_b2}. Therefore, the above morphisms $\Phi_{x;(\mathbb I')}$ together give rise to a map $$
\Phi_x:\mathfrak U_x\longrightarrow\mathfrak M^\tn{tf}; $$ see the display below~\cite[(3.15)]{g1modular}. As proved in~\cite[Corollary~3.7]{g1modular}, these $\Phi_x$ form smooth charts of the algebraic stack $\mathfrak M^\tn{tf}$. The twisted parameters in~\ref{Cond:p2} are written as $\varepsilon_i$, $u_e$, and $z_e$ in~\cite[(3.9)]{g1modular}. The local expressions of the strata of $\mathfrak M^\tn{tf}$ in~\ref{Cond:p2} follow from~\cite[(3.10) \& Lemma~3.2]{g1modular}. The last and the first equations in~\cite[(3.12)]{g1modular} imply~\ref{Cond:p3} and~\ref{Cond:p4}, respectively; see also~\cite[Remark~3.8]{g1modular} for~\ref{Cond:p4}.
In the remainder of the proof of Theorem~\ref{Thm:tf_smooth}, we will show that $\varpi$ is birational and proper, as stated in~\ref{Cond:p1}. Notice that the only level map on the connected edge-less graph $\tau_\bullet$ is the empty function, hence $\varpi$ restricts to the identity map on the pullback of $\mathfrak M_{\tau_\bullet}\!=\!\mathfrak M\backslash\Delta$, where $\Delta$ is the boundary of $\mathfrak M$ as in~(\ref{Eqn:boundary}). By the local expressions of the strata of $\mathfrak M$ and $\mathfrak M^\tn{tf}$ in~(\ref{Eqn:M_strata_local}) and~\ref{Cond:p2}, respectively, we see that $\Delta$ and its the pullback are of lower dimension in $\mathfrak M$ and $\mathfrak M^\tn{tf}$, respectively. Thus, $\varpi$ is birational.
It is straightforward that $\varpi$ is of finite type. To establish the properness of $\varpi$, let $(\mathbf D,0)$ be a nonsingular pointed curve with the complement and the inclusion respectively denoted by $$
\mathbf D^*
:= \mathbf D\backslash\{0\}
\stackrel{\iota}{\hookrightarrow}\mathbf D. $$ We will show that for any $f\!:\mathbf D\!\longrightarrow\!\mathfrak M$ and any $F\!:\mathbf D^*\!\longrightarrow\!\mathfrak M^\tn{tf}$ with $\varpi\!\circ\!F\!=\! f\!\circ\!\iota$, there exists a unique $\psi\!:\mathbf D\!\longrightarrow\!\mathfrak M^\tn{tf}$ so that the following diagram commutes: \begin{center} \begin{tikzpicture}{h} \draw (-1,1.2) node {$\mathbf D^*$}
(2,1.2) node {$\mathfrak M^\tn{tf}$}
(-1,0) node {$\mathbf D$}
(1.95,0) node {$\mathfrak M$}
(.425,1.2) node[above] {\small{$F$}}
(.425,-.05) node[above] {\small{$f$}}
(-1,0.6) node[left] {\small{$\iota$}}
(1.95,.6) node[right] {\small{$\varpi$}}
(.4,.5) node[above] {\small{$\psi$}}; \draw[->,>=stealth',dashed] (-.6,.25)--(1.5,.95); \draw[->,>=stealth'] (-.6,1.2)--(1.5,1.2); \draw[->,>=stealth'] (-.55,0)--(1.5,0); \draw[->, >=stealth'] (-1,.5)--(-1,.35); \draw[<-,>=right hook] (-1,.85)--(-1,.35); \draw[->,>=stealth'] (1.9,.85)--(1.9,.35); \end{tikzpicture} \end{center}
Deleting a discrete subset of $\mathbf D^*$ if necessary, we assume that there exist $\gamma,\gamma'\!\in\!\Gamma$, $\mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma)$, and $\mathfrak N_{\gamma'}\!\in\!\pi_0(\mathfrak M_{\gamma'})$ such that $$
f(t)\in\mathfrak N_\gamma\quad\forall\;
t\!\in\!\mathbf D^*;\qquad
f(0)\in\mathfrak N_{\gamma'}. $$ This implies $\textnormal{Cl}_\mathfrak M(\mathfrak N_\gamma)\!\cap\!\mathfrak N_{\gamma'}\!\ne\!\emptyset$, hence by Lemma~\ref{Lm:M_strata_closure}, we see that $\gamma'\!\preceq\!\gamma$. There thus exists $$E\subset\tn{Edg}(\gamma')\quad\textnormal{with}\quad\gamma\simeq\gamma'_{(E)}.$$ The treelike structure $\Lambda$ on $(\mathfrak M,\Gamma)$ then determines the tuples \begin{equation*}\begin{split}
&\big(\;\tau\!:=\!\tau_{\mathfrak N_\gamma}\,,~
\tn{E}\!:=\!\tn{E}_{\mathfrak N_\gamma}\!\subset\!\tn{Edg}(\gamma)\,,~
\beta\!:=\!\beta_{\mathfrak N_\gamma}\!\!:\tn{Edg}(\tau)\!\longrightarrow\!\tn{E}\;\big)
\qquad\textnormal{and}
\\
&\big(\;\tau'\!:=\!\tau_{\mathfrak N_{\gamma'}}\,,~
\tn{E}'\!:=\!\tn{E}_{\mathfrak N_{\gamma'}}\!\subset\!\tn{Edg}(\gamma')\,,~
\beta'\!:=\!\beta_{\mathfrak N_{\gamma'}}\!\!:\tn{Edg}(\tau')\!\longrightarrow\!\tn{E}'\;\big) \end{split}\end{equation*} as in Definition~\ref{Dfn:Treelike_structure}. Let $$
\tn{E}^{\tn{ctr}}:=\tn{E}'\!\cap\! E=\tn{E}'\backslash\tn{E},\qquad
\mathfrak E^{\tn{ctr}}:=(\beta')^{-1}(\tn{E}^{\tn{ctr}})
=\tn{Edg}(\tau')\backslash\tn{Edg}(\tau). $$ Deleting a discrete subset of $\mathbf D^*$ again if necessary, we further assume that there exists a fixed $[\mathfrak t]\!=\![\tau_{\mathfrak N_\gamma},\ell]\!\in\!\ov\Lambda_{\mathfrak N_\gamma}$ such that $$
F(t)=
\big(\,f(t)\,;\,
\big([\mu_e(t)]_{e\in\mathfrak E_i^\bot(\mathfrak t)}\big)_{i\in\lbrp{{\mathbf{m}},0}_{\mathfrak t}}\big)
\in(\mathfrak N_\gamma)_{[{\mathfrak t}]}^\tn{tf}
\qquad\forall~t\!\in\!\mathbf D^*. $$ Here we fix a rooted level tree $\mathfrak t$ representing $[\mathfrak t]$.
With notation as in Definition~\ref{Dfn:G-adim_fixture}, let $\mathcal V\!\in\!\ud{\mathfrak V}_{\gamma'}$ be a chart containing $f(0)$ and $\{\zeta_e\!=\!\zeta_e^\mathcal V\}_{e\in\tn{Edg}(\gamma')}$ be modular parameters on $\mathcal V$ centered at $f(0)$.
For every $e\!\in\!\tn{E}^{\tn{ctr}}$, there exist a unique integer $n_e\!\in\!\mathbb Z_{>0}$ and a unique nowhere vanishing function~$a_e$ on $\mathbf D\!\cap\!f^{-1}(\mathcal V)$ so that \begin{equation}\label{Eqn:new_level_graph_1}
\zeta_e\big(f(t)\big)=t^{n_e}a_e(t)\qquad
\forall~t\in\mathbf D\!\cap\!f^{-1}(\mathcal V). \end{equation} Similarly, for every level $i\!\in\!\lbrp{{\mathbf{m}},0}_{\mathfrak t}$, we can choose an edge $\mathsf e_i\!\in\!\mathfrak E_i^\bot(\mathfrak t)$ such that for every $e\!\in\!\mathfrak E_i^\bot(\mathfrak t)$, the specialization of $\mu_e/\mu_{\mathsf e_i}$ at $t\!=\! 0$ exists. There thus exist a unique integer $n_e\!\in\!\mathbb Z_{\ge 0}$ and a unique nowhere vanishing function $a_e$ on the whole $\mathbf D\!\cap\!f^{-1}(\mathcal V)$ satisfying $n_{\mathsf e_i}\!=\! 0$, $a_{\mathsf e_i}\!\equiv\!1$, and the level-$i$ twisted fields of $F(t)$ can be written as \begin{equation}\label{Eqn:new_level_graph_2}
[\mu_e(t)]_{e\in\mathfrak E_i^\bot(\mathfrak t)}=
\big[\;t^{n_e}a_e(t)\bigotimes_{e'\succeq_\mathfrak t e}\tfrac{\partial}{\partial\zeta_{e'}}\big|_{f(t)}\;\big]_{e\in\mathfrak E_i^\bot(\mathfrak t)}\qquad
\forall~t\in\mathbf D^*\!\cap\!f^{-1}(\mathcal V). \end{equation} Since $\mathfrak E_{\ge{\mathbf{m}}}(\mathfrak t)\!=\!\bigsqcup_{i\in\lbrp{{\mathbf{m}},0}_\mathfrak t}\mathfrak E_i^\bot(\mathfrak t)$, we have assigned to each $e\!\in\!\mathfrak E_{\ge{\mathbf{m}}}(\mathfrak t)\!\sqcup\!\mathfrak E^\tn{ctr}$ an integer $n_e$ and a nowhere vanishing function $a_e$ on $\mathbf D\!\cap\!f^{-1}(\mathcal V)$ via~(\ref{Eqn:new_level_graph_1}) and~(\ref{Eqn:new_level_graph_2}). Let $$
s_e:= - \sum_{
\begin{subarray}{c}
e'\in\,\mathfrak E_{\ge{\mathbf{m}}}(\mathfrak t)\sqcup\mathfrak E^\tn{ctr},\\
e' \succeq_{\tau'} e
\end{subarray}}\hspace{-.25in}n_{e'}
\qquad\qquad\forall~e\in\mathfrak E_{\ge{\mathbf{m}}}(\mathfrak t)\!\sqcup\!\mathfrak E^\tn{ctr}~\big(\subset\tn{Edg}(\tau')\big). $$ Note that each $s_e$ is non-positive, so they can be used to define levels on $\tau'$ as follows.
By~(\ref{Eqn:new_level_graph_1}) and~(\ref{Eqn:new_level_graph_2}), there exists a rooted level tree $$\mathfrak t'=(\,\tau'\,,\,\ell'\,)\,,$$ unique up to equivalence, satisfying \begin{gather}
[\mathfrak t']\preceq[\mathfrak t],\qquad \big(\mathfrak E_{\ge{\mathbf{m}}}(\mathfrak t)\subset\big)\ \ \mathfrak E_{\ge{\mathbf{m}}'}(\mathfrak t')
\,\subset\,
\mathfrak E_{\ge{\mathbf{m}}}(\mathfrak t)\!\sqcup\!\mathfrak E^\tn{ctr}
\ \ \big(\subset\tn{Edg}(\tau')\big),
\label{Eqn:elg'_in_ov_Ga}\\
\ell'(\hbar_e^-)=s_e\qquad
\forall~e\!\in\!\mathfrak E_{\ge{\mathbf{m}}'}(\mathfrak t').\label{Eqn:s1=s2} \end{gather} where ${\mathbf{m}}'\!=\!{\mathbf{m}}({\mathfrak t}')$. Since the underlying rooted tree of ${\mathfrak t}'$ is $\tau'\!=\!(\gamma_{\mathfrak N_{\gamma'}},o)$, we have $$
[\mathfrak t']\,\in\,\ov\Lambda_{\mathfrak N_{\gamma'}}\,. $$ We next show that there exists a {\it unique} twisted field \begin{equation}\label{Eqn:eta}
\eta:=\Big\lgroup f(0);\,
\Big(
\big[\lambda_e
\bigotimes_{e'\succeq_{\tau'}e}\!\tfrac{\partial}{\partial_{\zeta_{e'}}}\big|_{f(0)}
\big]_{e\in\mathfrak E_j^\bot(\mathfrak t')}\Big)_{j\in\lbrp{{\mathbf{m}}',0}_{\mathfrak t'}}
\Big\rgroup
~\in~
(\mathfrak N_{\gamma'})_{[{\mathfrak t}']}^\tn{tf}\,\big|_{f(0)} \end{equation} over $f(0)$ such that $F(t)\rightsquigarrow\eta$. Here we write $z\rightsquigarrow z_0$ if $z$ specializes to $z_0$ for any scheme $S$ and any points $z$ and $z_0$ of $S$.
Fix a twisted field $\eta$ over $f(0)$. Since $[\mathfrak t']\!\preceq\![\mathfrak t]$, there exists a subset $\mathbb J\!\subset\!\mathbb I(\mathfrak t')$ such that $[\mathfrak t'_{(\mathbb J)}]\!=\![\mathfrak t]$; see~(\ref{Eqn:level_partial_order}). Let $\mathfrak U_\eta$ be the twisted chart with the twisted parameters $\xi_s\!:=\!\xi_s^{\mathfrak U_\eta}$ as in~\ref{Cond:p2} and set \begin{equation}\label{Eqn:xi_special} \xi_{\mathsf e_j}\!\equiv\!1\qquad
\forall~j\!\in\!\lbrp{{\mathbf{m}}',0}_{{\mathfrak t}'}. \end{equation} Then, $\xi_s$ is defined for all $s\!\in\!\lbrp{{\mathbf{m}}',0}_{{\mathfrak t}'}\!\sqcup\!\tn{Edg}({\mathfrak t}')$. Let $$
\Psi_{\eta;(\mathbb J)}:
\mathcal U'_{\eta;[\mathfrak t]}
\longrightarrow
\mathfrak U_\eta\cap
\big\{\,\xi_s\!\ne\!0~\forall\,s\!\in\!\mathbb J~;~
\xi_s\!=\! 0~\forall\,s\!\in\!\mathbb I(\mathfrak t')\backslash\mathbb J\,\big\} $$ be the isomorphism constructed in~\cite[(3.17)]{g1modular}, where $\mathcal U'_{\eta;[\mathfrak t]}$ is an open subset of $\mathfrak N^\tn{tf}_{\gamma,[\mathfrak t]}$ containing the image $F\big(\mathbf D^*\!\cap\!f^{-1}(\mathcal V)\big)$. Let \begin{equation}\label{Eqn:ve&ue}
\varepsilon_j(t)\!:=
\xi_j\big(\Psi_{\eta;(\mathbb J)}(F(t))\big)
\ \ \forall\,j\!\in\!\lbrp{{\mathbf{m}}',0}_{\mathfrak t'}\,;
\quad
u_e(t)
\!:=
\xi_e\big(\Psi_{\eta;(\mathbb J)}(F(t))\big)
\ \ \forall\,e\!\in\!\mathfrak E_{\ge{\mathbf{m}}'}(\mathfrak t')\,. \end{equation} Then, $F(t)\rightsquigarrow\eta$ if and only if \begin{equation}\begin{split}\label{Eqn:limit}
&\big(\,
(\varepsilon_j(t))_{j\in\lbrp{{\mathbf{m}}',0}_{\mathfrak t'}}\,,
\,
(u_e(t))_{e\in\mathfrak E_{\ge{\mathbf{m}}'}^\bot(\mathfrak t')}\,,\,
(u_e(t))_{e\in E_{{\mathbf{m}}'}(\mathfrak t')}
\,\big)\\
\,\rightsquigarrow\,
&\big(\,\ud 0\,,\,
(\lambda_e)_{e\in\mathfrak E_{\ge{\mathbf{m}}'}^\bot(\mathfrak t')}\,,\,
\ud 0\,\big),
\qquad\qquad
\textnormal{where}\ \
\ud 0\!=\!(0,\ldots,0),\ \
\lambda_{\mathsf e_j}\!=\! 1\
\forall~j\!\in\!\lbrp{{\mathbf{m}}',0}_{{\mathfrak t}'}. \end{split}\end{equation}
Indeed, by~\cite[(3.21)]{g1modular} and Cases 1-3 of the proof of~\cite[Lemma~3.2]{g1modular}, as well as the assumption~(\ref{Eqn:s1=s2}) above, we conclude inductively over the levels of $\mathfrak t'$ in the descending order that there exist nowhere vanishing functions $b_j(t)$, $j\!\in\!\mathbb J\!=\!\mathbb J\!\cap\!\lbrp{{\mathbf{m}}',0}_{\mathfrak t'}$, on $\mathbf D\!\cap\!f^{-1}(\mathcal V)$ such that \begin{equation}\label{Eqn:ve_j}
\varepsilon_j(t)=
\begin{cases}
t^{(j^\sharp-j)}b_j(t) & \textnormal{if}~j\!\in\!\mathbb J_+;\\
0 & \textnormal{if}~j\!\not\in\!\mathbb J_+
\end{cases} \end{equation} Similarly, by~\cite[(3.22)]{g1modular} and Cases A and B of the proof of~\cite[Lemma~3.2]{g1modular}, as well as~(\ref{Eqn:s1=s2}) and~(\ref{Eqn:ve_j}), we conclude inductively over the levels of $\mathfrak t'$ in the descending order that there exist nowhere vanishing functions $b_e(t)$, $e\!\in\!\mathfrak E_{\ge{\mathbf{m}}'}(\mathfrak t')$, on $\mathbf D\!\cap\!f^{-1}(\mathcal V)$ such that \begin{equation}\label{Eqn:u_e}
u_e(t)=
\begin{cases}
b_e(t) & \textnormal{if}~e\!\in\!\mathfrak E_{\ge{\mathbf{m}}'}^\bot(\mathfrak t');\\
t^{({\mathbf{m}}'-s_e)}b_e(t) & \textnormal{if}~e\!\in\!E_{{\mathbf{m}}'}(\mathfrak t').
\end{cases} \end{equation} We set \begin{equation}\label{Eqn:la_e}
\lambda_e:=b_e(0)\qquad
\forall~\!\in\!\mathfrak E_{\ge{\mathbf{m}}'}^\bot(\mathfrak t'). \end{equation} Since the functions $b_e$ are all nowhere vanishing, all $\lambda_e$ are nonzero. Moreover, by~(\ref{Eqn:xi_special}), the second equation in~(\ref{Eqn:ve&ue}, (\ref{Eqn:u_e}), and~(\ref{Eqn:la_e}), we obtain $\lambda_{\mathsf e_j}\!=\! 1$ for all $j\!\in\!\lbrp{{\mathbf{m}}',0}_{{\mathfrak t}'}$. Thus, $\lambda_e$ can be used to define $\eta$ in~(\ref{Eqn:eta}).
Since $j^\sharp$ stands for the level of $\ell'$ immediately above $j$, we have $$
j^\sharp\!-\!j>0
\qquad\forall~j\!\in\!\mathbb J_+. $$ Similarly, by~(\ref{Eqn:fE_i}) and~(\ref{Eqn:s1=s2}), we have $$
{\mathbf{m}}'\!-\!s_e>0
\qquad\forall~e\!\in\!E_{{\mathbf{m}}'}(\mathfrak t'). $$ Thus, by~(\ref{Eqn:ve_j}), (\ref{Eqn:u_e}), and~(\ref{Eqn:la_e}), we see that $F\!:\mathbf D^*\!\longrightarrow\!\mathfrak M^\tn{tf}$ extends to $\psi\!:\mathbf D\!\longrightarrow\!\mathfrak M$ with $\psi(0)\!=\!\eta$ as in~(\ref{Eqn:eta}).
It remains to prove the uniqueness of $\psi$. Mimicking the proof of~\cite[Lemma~3.2]{g1modular}, we conclude that for any $[\mathfrak t^\dag]\!=\![\tau',\ell^\dag]\in\ov\Lambda_{\mathfrak N_{\gamma'}}\!\backslash[\mathfrak t']$, at least one of~(\ref{Eqn:ve_j}) and~(\ref{Eqn:u_e}) no longer holds for $\mathfrak t^\dag$. Hence by~(\ref{Eqn:limit}),
$F(t)$ cannot specialize to $\eta^\dag$. Consequently, every possible specialization of $F(t)$ at $t\!=\! 0$ must be in $(\mathfrak N_{\gamma'})_{[{\mathfrak t}']}^\tn{tf}\big|_{f(0)}$. Since the functions $u_e(t)$ in~(\ref{Eqn:u_e}) are uniquely determined by~\cite[(3.22), (3.26), \& (3.28)]{g1modular} as well as the expressions~(\ref{Eqn:new_level_graph_1}), (\ref{Eqn:new_level_graph_2}), and~(\ref{Eqn:ve_j}), so are the functions $b_e(t)$. This establishes the uniqueness of $\eta$ in~(\ref{Eqn:eta}) and hence the uniqueness of~$\psi$. \end{proof}
The following proposition is a restatement of~\cite[Proposition~3.9]{g1modular} under the current setup.
\begin{prp} \label{Prp:moduli} Let $\Gamma$, $\mathfrak M$, and $\Lambda$ be as in Theorem~\ref{Thm:tf_smooth}. Assume that $\pi\!:\mathcal C\!\longrightarrow\!\mathfrak M$ is the universal family of $\mathfrak M$. Then, the $\Gamma$-stratification of $\mathfrak M$ in Definition~\ref{Dfn:G-adim_fixture} induces a stratification $\mathcal C\!=\!\bigsqcup_{\gamma\in \Gamma}\mathcal C_\gamma$ such that the union \begin{equation*}\begin{split}
\mathcal C^\tn{tf}
&:=\bigsqcup_{\gamma\in\Gamma;\;
\mathfrak N_\gamma\in\pi_0(\mathfrak M_{\gamma});\;
[\mathfrak t]\in\ov\Lambda_{\mathfrak N_\gamma}}
\hspace{-.4in}
(\mathcal N_{\gamma})^\tn{tf}_{[\mathfrak t]}\ ,\qquad
\textnormal{where}
\\
(\mathcal N_{\gamma})^\tn{tf}_{[\mathfrak t]}
&:=
\bigg(
\prod_{i\in\lbrp{{\mathbf{m}},0}_{\mathfrak t}}\!\!\!
\Big\lgroup\mathring\mathbb P
\Big(\big(
\bigoplus_{\!
\begin{subarray}{c}
e\in\mathfrak E_i^\bot(\mathfrak t)
\end{subarray}
}\!\!\!
\pi^* L_{\succeq_{\mathfrak N_\gamma} e}\,\big)
\Big)\Big/\mathcal N_\gamma
\Big\rgroup
\bigg)
\longrightarrow\mathcal N_\gamma:=\pi^{-1}(\mathfrak N_\gamma), \end{split}\end{equation*} has a canonical smooth algebraic stack structure, and the projection $$
\pi^\tn{tf} :\mathcal C^\tn{tf}\longrightarrow\mtd{} $$ induced by $\pi$ gives the universal family of $\mtd{}$. \end{prp}
On the conclusion of this section, we remark that in~\cite{g1modular}, the properness of the forgetful morphism $\mathfrak M_1^\tn{tf}\!\longrightarrow\!\mathfrak M^\tn{wt}_1$ (see Example~\ref{Eg:genus_1} for notation) was established via the isomorphism $\mathfrak M_1^\tn{tf}\!\longrightarrow\!\ti\mathfrak M^\tn{wt}_1$ to the blowup stack $\ti\mathfrak M^\tn{wt}_1$ in~\cite{HL10}, as well as the properness of the blowing up $\ti\mathfrak M^\tn{wt}_1\!\longrightarrow\!\mathfrak M^\tn{wt}_1$. In this paper, Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p1} provides a more direct approach to the properness, with a broader setup that includes the genus one case as a special case, yet without involving the comparison with the blowups. This could be an advantage of the STF theory in the higher genera cases.
\section{Recursive constructions} \label{Sec:Induced_and_derived_tf_stacks}
Theorem~\ref{Thm:tf_smooth} and Proposition~\ref{Prp:moduli} suggest the possibility of adding twisted fields recursively: $$ \cdots\longrightarrow(\mathfrak M^\tn{tf})^\tn{tf}\longrightarrow\mathfrak M^\tn{tf}\longrightarrow \mathfrak M. $$
The tricky part is choosing appropriate stratification $\Gamma'$ and treelike structure $\Lambda'$ in each step. Given a $\Gamma$-stratification on $\mathfrak M$ and a treelike structure $\Lambda$ on $(\mathfrak M,\Gamma)$, we provide a possible stratification known as
the \textsf{derived stratification} on $\mathfrak M^\tn{tf}$ and another possible stratification on $\mathfrak M$ known as a \textsf{grafted stratification}, respectively in this section. They will be used in the description of $\ti\mathfrak P_2^\tn{tf}$ and $\ti M_2(\mathbb P^n,d)^\tn{tf}$ of Theorem~\ref{Thm:Main}.
\subsection{Derived stratification} \label{Subsec:derived}
Recall $\mathbf{G}$ denotes the set of connected graphs. Let $\mathfrak M$ be a smooth stack endowed with a $\Gamma$-stratification $\mathfrak M\!=\!\bigsqcup_{\gamma\in\Gamma}\mathfrak M_\gamma$ as in Definition~\ref{Dfn:G-adim_fixture} and a treelike structure $\Lambda$ as in Definition~\ref{Dfn:Treelike_structure} that assigns to every pair $\big(\gamma\!\in\!\Gamma,\,\mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma)\big)$ a unique tuple $$
\big(\,
\tau\,,\,
\tn{E}\,,\,
\beta
\,\big):=
(\tau_{\mathfrak N_\gamma},\tn{E}_{\mathfrak N_\gamma},\beta_{\mathfrak N_\gamma}). $$ Given a rooted level tree ${\mathfrak t}\!=\!(\tau,\ell)$ with $[{\mathfrak t}]\!\in\!\ov\Lambda_{\mathfrak N_\gamma}$ and a level $i\!\in\!\tn{Im}(\ell)$, recall that $i^\sharp$ and~$i^\flat$ stand for the levels immediately above and below~$i$ (if they exist), respectively. We also recall that $(\mathfrak E_{{\mathbf{m}};\min}^\bot)^{\succeq}$ denotes the set of the edges of the paths of $\tau$ from the root $o$ to the level ${\mathbf{m}}\!=\!{\mathbf{m}}({\mathfrak t})$ {\it minimal} vertices; see~(\ref{Eqn:fE_i}).
\begin{dfn}\label{Dfn:Derived_graph} Let $\Gamma$, $\mathfrak M$, and $\Lambda$ be as in Definition~\ref{Dfn:Treelike_structure}. For every $\gamma\!\in\!\Gamma$, $\mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma)$, and $[{\mathfrak t}]\!\in\!\ov\Lambda_{\mathfrak N_\gamma}$, we construct the \textsf{derived graph of~$\gamma$ with respect to $[\mathfrak t]$}: $$
\rho_{\gamma,[{\mathfrak t}]}\in\mathbf{G} $$ as follows. For an arbitrary ${\mathfrak t}$ representing $[{\mathfrak t}]$, \begin{enumerate} [leftmargin=*,label=(\arabic*)] \item \label{Cond:derived_p1}
firstly, we replace every $e\!=\!\{\hbar_e^+,\hbar_e^-\}\!\in\!(\mathfrak E_{{\mathbf{m}};\min}^\bot)^{\succeq}\!\subset\!\tn{Edg}({\mathfrak t})$ with a new list of half-edges $$
\hbar_{e;\ell(\hbar_e^+)^\flat}^+,\,
\hbar_{e;\ell(\hbar_e^+)^\flat}^-,\,
\hbar_{e;(\ell(\hbar_e^+)^\flat)^\flat}^+,\,
\ldots,\,
\hbar_{e;\ell(\hbar_e^-)^\sharp}^-,\,
\hbar_{e;\ell(\hbar_e^-)}^+,\,
\hbar_{e;\ell(\hbar_e^-)}^-; $$
\item\label{Cond:derived_p2} then, for each $j\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}}$ and $\bullet\!=\!+,-$, we set $$
\hbar_j^\bullet:=
\big\{\,
\beta(\hbar_{e;j}^\bullet)\,:\,
e\!\in\!\mathfrak E_j\!\cap\!(\mathfrak E_{{\mathbf{m}};\min}^\bot)^{\succeq}
\,\big\}\quad\big(\subset\tn{HE}(\tn{E})\subset\tn{HE}(\gamma)\,\big) $$ and consider the sets $\hbar_j^\pm$ as new half-edges;
\item next, we set \begin{equation*}\begin{split}
\mathsf e_i &:=
\{\hbar_i^+,\hbar_i^-\}\quad\forall\ i\!\in\!\lbrp{{\mathbf{m}},0}_{\mathfrak t},\\
\mathsf v'_i
&:=\{\hbar_i^-,\hbar_{i^\flat}^+\}\sqcup
\hspace{-.15in}
\bigsqcup_{e\in\mathfrak E^\bot_i\cap(\mathfrak E^\bot_{{\mathbf{m}};\min})^\succeq}\hspace{-.3in}
\big(\,v(\beta(\hbar_e^-))\,\backslash\,(\hbar_i^-\!\sqcup\!\hbar_{i^\flat}^+)\,\big)
\quad\forall\ i\!\in\!\lprp{{\mathbf{m}},0}_{\mathfrak t},\\
\mathsf v'_{\mathbf{m}}
&:=\{\hbar_{\mathbf{m}}^-\}\sqcup
\hspace{-.15in}
\bigsqcup_{e\in\mathfrak E^\bot_{\mathbf{m}}\cap(\mathfrak E^\bot_{{\mathbf{m}};\min})^\succeq}\hspace{-.3in}
\big(\,v(\beta(\hbar_e^-))\,\backslash\,\hbar_{\mathbf{m}}^-\,\big),
\qquad
\mathsf v'_0
:=\{\hbar_{0^\flat}^+\}\sqcup
\hspace{-.15in}
\bigsqcup_{e\in\mathfrak E^\bot_{0^\flat}\cap(\mathfrak E^\bot_{{\mathbf{m}};\min})^\succeq}\hspace{-.3in}
\big(\,v(\beta(\hbar_e^+))\,\backslash\,\hbar_{0^\flat}^+\,\big) \end{split}\end{equation*} and take $\rho_{\gamma,[{\mathfrak t}]}'$ to be the graph satisfying \begin{equation*}\begin{split}
\tn{HE}\big(\rho_{\gamma,[{\mathfrak t}]}'\big)&=
\tn{HE}\big(\tn{Edg}(\gamma)\backslash\beta((\mathfrak E_{{\mathbf{m}};\min}^\bot)^{\succeq} )\big)
\sqcup
\big\{\hbar_i^\pm:i\!\in\!\lbrp{{\mathbf{m}},0}_\mathfrak t\big\},\\
\tn{Edg}\big(\rho_{\gamma,[{\mathfrak t}]}'\big)&=
\big(\tn{Edg}(\gamma)\backslash\beta((\mathfrak E_{{\mathbf{m}};\min}^\bot)^{\succeq} )\big)
\sqcup
\big\{\mathsf e_i:\,
i\!\in\!\lbrp{{\mathbf{m}},0}_\mathfrak t
\big\},
\\
\tn{Ver}\big(\rho_{\gamma,[{\mathfrak t}]}'\big)&=
\big(\,\tn{Ver}(\gamma)\,\backslash\,
\{\,v(\beta(\hbar)):\hbar\!\in\!\tn{HE}((\mathfrak E_{{\mathbf{m}};\min}^\bot)^{\succeq} )\,\}\,
\big)
\sqcup
\big\{
\mathsf v_i':\,
i\!\in\!\lrbr{{\mathbf{m}},0}_{{\mathfrak t}}\big\}; \end{split}\end{equation*}
\item finally, we define $\rho_{\gamma,[{\mathfrak t}]}$ via edge contraction: $$
\rho_{\gamma,[{\mathfrak t}]}:=(\rho_{\gamma,[{\mathfrak t}]}')_{\big(\beta\lgroup\,\mathfrak E_{\ge{\mathbf{m}}}^\bot\,\backslash\,(\mathfrak E_{{\mathbf{m}};\min}^\bot)^{\succeq}\,\rgroup\big)}.$$ \end{enumerate} \end{dfn}
Obviously, the construction of $\rho_{\gamma,[{\mathfrak t}]}$ is independent of the choice of ${\mathfrak t}$ representing $[{\mathfrak t}]$. By Definition~\ref{Dfn:Derived_graph}, we observe that if ${\mathbf{m}}\!=\! 0$ (i.e.~${\mathfrak t}$ is edge-less), then $\rho_{\gamma,[{\mathfrak t}]}\!=\!\gamma$. If ${\mathbf{m}}\!<\!0$, then \begin{equation}\begin{split}\label{Eqn:derived_graph}
\tn{HE}\big(\rho_{\gamma,[{\mathfrak t}]}\big)&=
\tn{HE}\big(\tn{Edg}(\gamma)\backslash\beta(\mathfrak E_{\ge {\mathbf{m}}}^\bot)\big)
\sqcup
\big\{\hbar_i^\pm:i\!\in\!\lbrp{{\mathbf{m}},0}_\mathfrak t\big\},\\
\tn{Edg}\big(\rho_{\gamma,[{\mathfrak t}]}\big)&=
\big(\tn{Edg}(\gamma)\backslash\beta(\mathfrak E_{\ge {\mathbf{m}}}^\bot)\big)
\sqcup
\big\{\mathsf e_i:\,
i\!\in\!\lbrp{{\mathbf{m}},0}_\mathfrak t
\big\},
\\
\tn{Ver}\big(\rho_{\gamma,[{\mathfrak t}]}\big)&=
\big(\,\tn{Ver}(\gamma)\,\backslash\,
\{\,v(\beta(\hbar)):\hbar\!\in\!\tn{HE}(\mathfrak E^\bot_{\ge{\mathbf{m}}})\,\}\,
\big)
\sqcup
\big\{
\mathsf v_i:\,
i\!\in\!\lrbr{{\mathbf{m}},0}_{{\mathfrak t}}\big\} \end{split}\end{equation} where the vertices $\mathsf v_i$ are the images of $\mathsf v_i'$, i.e.~they are determined by \begin{equation}\label{Eqn:v_0}
\mathsf v_0\ni\hbar_{0^\flat}^+\,;\qquad
\mathsf v_i\ni\hbar_i^-\quad\forall~i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}}. \end{equation} The new edges $\mathsf e_i$ and vertices $\mathsf v_i$ (if exist) are called the \textsf{exceptional edges and vertices}, respectively.
With notation as above, let \begin{equation}\label{Eqn:Derived_graphs}
\Gamma_\Lambda^{\tn{der}}=
\big\{\,
\rho_{\gamma,[{\mathfrak t}]}\,:\;
\gamma\!\in\!\Gamma,~
\mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma),~
[{\mathfrak t}]\!\in\!\ov\Lambda_{\mathfrak N_\gamma}\,
\big\}\quad(\,\subset\mathbf{G}). \end{equation} An example of derive graphs is illustrated in Figure~\ref{Fig:derived}.
\begin{figure}
\caption{An example of derived graphs}
\label{Fig:derived}
\end{figure}
\begin{crl}\label{Crl:Derived_graphs} Let $\Gamma$, $\mathfrak M$, $\Lambda$, and $\mathfrak M^\tn{tf}$ be as in Theorem~\ref{Thm:tf_smooth}. Assume that for every $\gamma\!\in\!\Gamma$ and every $\mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma)$, we have $\tn{E}_{\mathfrak N_\gamma}\!=\!\tn{Edg}(\gamma')$ for some connected subgraph $\gamma'$ of $\gamma$. Then, \begin{equation}\begin{split}\label{Eqn:Derived_strata}
\mathfrak M^\tn{tf}
&=\! \bigsqcup_{\gamma'\in\Gamma_\Lambda^{\tn{der}}}
\!\!\mathfrak M^\tn{tf}_{\gamma'},\qquad\textnormal{where}
\quad
\mathfrak M^\tn{tf}_{\gamma'}
\!=\!
\bigsqcup_{
\gamma\in \Gamma\,,\,\mathfrak N_\gamma\in\pi_0(\mathfrak M_\gamma)\,,\,[\mathfrak t]\in\ov\Lambda_{\mathfrak N_\gamma}\,,\,\rho_{\gamma,[\mathfrak t]}=\gamma'}
\hspace{-.56in}
(\mathfrak N_{\gamma})_{[{\mathfrak t}]}^\tn{tf}\,, \end{split}\end{equation} is a $\Gamma_{\Lambda}^{\tn{der}}$-stratification of $\mathfrak M^\tn{tf}$, known as the \textsf{derived stratification of $\mathfrak M^\tn{tf}$ with respect to $(\mathfrak M,\Gamma,\Lambda)$}. Moreover, for any $\gamma'\!\in\!\Gamma_\Lambda^\tn{der}$, the connected components of $\mathfrak M^\tn{tf}_{\gamma'}$ are given by the RHS of~(\ref{Eqn:Derived_strata}). \end{crl}
\begin{proof} It is direct to verify the statements using the sets $$
\ud{\mathfrak U}_{\gamma'}:=
\big\{\,\mathfrak U_x:\,
x\!\in\!\mathfrak M^\tn{tf}_{\gamma'}\,
\big\},\qquad
\gamma'\!\in\! \Gamma_{\Lambda}^{\tn{der}} $$ of affine smooth charts of $\mtd{}$, the subsets $$
\big\{\xi_s^{\mathfrak U_x}:\,i\!\in\! \mathbb I({\mathfrak t})\!=\!\lbrp{{\mathbf{m}},0}_{\mathfrak t}\!\sqcup\!E_{\mathbf{m}}(\mathfrak t)\!\sqcup\! E_-(\mathfrak t)\big\},
\qquad
\gamma'\!\in\!\Gamma_{\Lambda}^{\tn{der}},\ \ \mathfrak U_x\!\in\!\ud{\mathfrak U}_{\gamma'}, $$ of local parameters, and the local equations of $(\mathfrak N_{\gamma})^\tn{tf}_{[{\mathfrak t}]}$ as in Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p2}. The assumption that each $\tau_{\mathfrak N_\gamma}$ is identified with a {\it connected} subgraph of $\gamma$ via the bijection $\beta_{\mathfrak N_\gamma}$ guarantees that for any derived graph $\rho_{\gamma,[{\mathfrak t}]}\!\in\!\Gamma^\tn{der}_\Lambda$, contracting any special edges $\mathsf e_i$ from $\rho_{\gamma,[{\mathfrak t}]}$ does not create graphs that are not in $\Gamma^\tn{der}_\Lambda$. \end{proof}
\iffalse \begin{lmm} \label{Lm:derived_line_bundle} With notation as in Corollary~\ref{Crl:Derived_graphs}, assume that $\gamma\!\in\!\Gamma$ and $\mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma)$ are such that the bijection $\beta_{\mathfrak N_\gamma}\!:\tn{HE}(\tau_{\mathfrak N_\gamma})\!\longrightarrow\!\tn{HE}(\tn{E}_{\mathfrak N_\gamma})\;\big(\!\subset\!\tn{HE}(\gamma)\big)$ identifies $\tau_{\mathfrak N_\gamma}$ with a connected subgraph of $\gamma$. Then, for every $[\mathfrak t]\!\in\!\ov\Lambda_{\mathfrak N_\gamma}$, there exists an isomorphism of the line bundles $$
\Phi_{[{\mathfrak t}]}:\,
\big(\bigotimes_{i\in\lbrp{{\mathbf{m}},0}_\mathfrak t}\!\!\!L_{\mathsf e_i}\,
\big)\big/(\mathfrak N_{\gamma})_{[{\mathfrak t}]}^\tn{tf}
\longrightarrow
\varpi^*\big(L_{\succeq_{\mathfrak N_\gamma} e}\big/\mathfrak N_\gamma\big)\qquad
\forall~e\!\in\!\mathfrak E_{\mathbf{m}}^\bot\,,
$$ where $L_{\mathsf e_i}\!\longrightarrow\!(\mathfrak N_{\gamma})_{[{\mathfrak t}]}^\tn{tf}$, $i\!\in\!\lbrp{{\mathbf{m}},0}_\mathfrak t$, are the line bundles as in~(\ref{Eqn:L_e}) such that for every $x\!\in\!(\mathfrak N_{\gamma})_{[{\mathfrak t}]}^\tn{tf}$, $L_{\mathsf e_i}/\big((\mathfrak N_{\gamma})_{[{\mathfrak t}]}^\tn{tf}\!\cap\!\mathfrak U_x\big)$ is the normal bundle of the local divisor $\{\xi_i^{\mathfrak U_x}\!=\! 0\}$ restricting to~$(\mathfrak N_{\gamma})_{[{\mathfrak t}]}^\tn{tf}\!\cap\!\mathfrak U_x$. \end{lmm}
\begin{proof} {\blue For every $x\!\in\!(\mathfrak N_{\gamma})_{[{\mathfrak t}]}^\tn{tf}$ and every twisted chart $\mathfrak U_x$ be the twisted chart with the twisted parameters $\xi_s\!:=\!\xi_s^{\mathfrak U_x}$ as in Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p2}, we set \begin{equation*} \xi_{\mathsf e_j}\equiv 1\qquad
\forall~j\in\lbrp{{\mathbf{m}},0}_{{\mathfrak t}} \end{equation*} analogous to~(\ref{Eqn:xi_special}). Then, $\xi_s$ is defined for all $s\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}}\!\sqcup\!\tn{Edg}(\tau_{\mathfrak N_\gamma})$.
Fix $e\!\in\!\mathfrak E^\bot_{\mathbf{m}}({\mathfrak t})$. The restrictions of $\bigotimes_{i\in\lbrp{{\mathbf{m}},0}_\mathfrak t}\!L_{\mathsf e_i}$ and $\varpi^*L_{\succeq_{\mathfrak N_\gamma} e}$ to $(\mathfrak N_{\gamma})_{[{\mathfrak t}]}^\tn{tf}\!\cap\!\mathfrak U_x$ have nowhere vanishing sections: $$
\bigotimes_{i\in\lbrp{{\mathbf{m}},0}_{\mathfrak t}}\!\!\frac{\partial}{\partial\xi_i}\Big|_{(\mathfrak N_{\gamma})_{[{\mathfrak t}]}^\tn{tf}\cap\mathfrak U_x}\qquad
\textnormal{and}\qquad
\bigotimes_{e'\succeq_{\mathfrak N_\gamma} e}\!\!\Big(\varpi^*\frac{\partial}{\partial\zeta_{e'}}\Big)\Big|_{(\mathfrak N_{\gamma})_{[{\mathfrak t}]}^\tn{tf}\cap\mathfrak U_x}, $$ respectively. For every $e'\!\in\!\tn{Edg}(\tau_{\mathfrak N_\gamma})$, let $$
(e')^\spadesuit:=\min\{e''\!\in\!\tn{Edg}(\tau_{\mathfrak N_\gamma}):\,
e''\!\succeq_{\mathfrak N_\gamma}\!\!e'\},\qquad
\ell(e'):=\ell(\hbar_{e'}^-). $$ Using the notation in Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p2}, the first equation of~\cite[(3.12)]{g1modular} can be rewritten as $$
\varpi^*\zeta_{e'}=
\frac{\xi_{e'}\cdot\xi_{e_{\ell(e')}^\spadesuit}\cdot\xi_{e_{\ell(e_{\ell(e')}^\spadesuit)}^\spadesuit}\cdots}
{\xi_{(e')^\spadesuit}\cdot
\xi_{e_{\ell((e')^\spadesuit)}^\spadesuit}\cdot\xi_{e_{\ell(e_{\ell((e')^\spadesuit)}^\spadesuit)}^\spadesuit}\cdots}
\prod_{i\in\lbrp{\ell(e'),\,\ell((e')^\spadesuit)}_{\mathfrak t}}\!\!\!\!\!\!\!\!\!\!\!\xi_i
\qquad
\qquad
\forall\,{e'}\!\in\!\mathfrak E^\bot_{\ge{\mathbf{m}}}({\mathfrak t}). $$
Notice that the $\xi$'s in the above fraction are all nowhere vanishing on $\mathfrak U_x$. It is then a direct check that there exists a unique isomorphism $\Phi_{[{\mathfrak t}]}$ of the line bundles as in Lemma~\ref{Lm:derived_line_bundle} satisfying \begin{equation*}
\Phi_{[{\mathfrak t}]}\Big(\bigotimes_{i\in\lbrp{{\mathbf{m}},0}_{\mathfrak t}}\!\!\frac{\partial}{\partial\xi_i}\Big|_{(\mathfrak N_{\gamma})_{[{\mathfrak t}]}^\tn{tf}\cap\mathfrak U_x}\Big)=
\big(\xi_{e'}\cdot\xi_{e_{\ell(e')}^\spadesuit}\cdot\xi_{e_{\ell(e_{\ell(e')}^\spadesuit)}^\spadesuit}\cdots\big)
\bigotimes_{e'\succeq_{\mathfrak N_\gamma} e}\!\!\Big(\varpi^*\frac{\partial}{\partial\zeta_{e'}}\Big)\Big|_{(\mathfrak N_{\gamma})_{[{\mathfrak t}]}^\tn{tf}\cap\mathfrak U_x}\,.
\qedhere \end{equation*}} \end{proof} \fi
\subsection{Grafted stratification} \label{Subsec:Grafted}
For every graph $\gamma$ and every vertex $v\!\in\!\tn{Ver}(\gamma)$, let $(\gamma,v)_\tn{gft}$ be the graph given by \begin{gather*}
\tn{HE}\big((\gamma,v)_\tn{gft}\big)=\tn{HE}(\gamma) \sqcup \{\hbar_\tn{gft}^+,\hbar_\tn{gft}^-\},\qquad
\tn{Edg}\big((\gamma,v)_\tn{gft}\big)=\tn{Edg}(\gamma)\sqcup\big\{\,
e_\tn{gft}\!:=\!\{\hbar_\tn{gft}^+,\hbar_\tn{gft}^-\}\,\big\}\\
\tn{Ver}\big((\gamma,v)_\tn{gft}\big)=\big\{v\!\sqcup\!\{\hbar_\tn{gft}^+\}\big\}\sqcup\big\{v_\tn{gft}\!:=\!\{\hbar_\tn{gft}^-\}\big\}\sqcup
\tn{Ver}(\gamma)\backslash\{v\}\,. \end{gather*} Intuitively, $(\gamma,v)_\tn{gft}$ is obtained from $\gamma$ by {\it grafting} an extra vertex $v_\tn{gft}$ onto the chosen vertex $v$ via an extra edge $e_\tn{gft}$. If $\tau\!=\!(\tau,o)$ is a rooted tree, then we write $\tau_\tn{gft}\!:=\!(\tau,o)_\tn{gft}$.
We continue with the setup in the first paragraph of \S\ref{Subsec:derived}. Recall that $\tau_\bullet$ denotes the single-vertex rooted tree.
\begin{crl}\label{Crl:extra_edge_graphs} Let $\Gamma$, $\mathfrak M$, $\Lambda$ be as in Definition~\ref{Dfn:Treelike_structure} so that $(\mathfrak M,\Gamma)$ is endowed with a treelike structure $\Lambda$, and $\Delta\!\subset\!\mathfrak M$ be the boundary of $\mathfrak M$ (w.r.t.~$\Gamma$). Assume that \begin{itemize}[leftmargin=*] \item there exists $K\!\subset\!\Delta
(\subset\!\mathfrak M)$ satisfying that for every $\gamma\!\in\!\Gamma\backslash\{\tau_\bullet\}$, $\mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma)$, and $\mathcal V\!\in\!\ud\mathfrak V_{\gamma}$, there exists $\kappa^\mathcal V\!\in\!\Gamma(\mathscr O_\mathcal V)$ such that $\{\zeta_e^\mathcal V\}_{e\in\tn{Edg}(\gamma)}\!\sqcup\!\{\kappa^\mathcal V\}$ is a subset of local parameters on $\mathcal V$ and \begin{equation*}
K\!\cap\!\mathfrak N_{\gamma'}\!\cap\!\mathcal V~\textnormal{is~either}~
\{\kappa^\mathcal V\!\!=\! 0\}\!\cap\!\mathfrak N_{\gamma'}\!\cap\!\mathcal V~\textnormal{or}~\emptyset
\qquad
\forall\,
\gamma'\!\in\!\Gamma,\ \mathfrak N_{\gamma'}\!\in\!\pi_0(\mathfrak M_{\gamma'}); \end{equation*}
\item for every $\gamma\!\in\!\Gamma$ and every $\mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma)$, the bijection $\beta_{\mathfrak N_\gamma}\!:\tn{HE}(\tau_{\mathfrak N_\gamma})\!\longrightarrow\!\tn{HE}(\tn{E}_{\mathfrak N_\gamma})\;\big(\!\subset\!\tn{HE}(\gamma)\big)$ identifies $\tau_{\mathfrak N_\gamma}$ with a connected subgraph of $\gamma$. \end{itemize} Then,
by setting \begin{equation}\begin{split}\label{Eqn:grafted} &\Gamma_\tn{gft}=\Gamma_\tn{gft}(K):=\{\tau_\bullet\}\sqcup \big\{\, \big(\gamma,\,\beta_{\mathfrak N_\gamma}\!(o_{\mathfrak N_\gamma})\big)_{\tn{gft}} :\, \gamma\!\in\!\Gamma\backslash\{\tau_\bullet\},~ \mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma)\,\big\}, \\ &(\mathfrak N_\gamma)_{\tn{gft}}:= K\cap \mathfrak N_\gamma\qquad \forall~\gamma\!\in\!\Gamma,~\mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma), \\ &(\mathfrak M_{\tn{gft}})_{\gamma'}:= \hspace{-.2in} \bigsqcup_{(\gamma\,,\;\beta_{\mathfrak N_\gamma}(o_{\mathfrak N_\gamma}))_\tn{gft}=\gamma'}\hspace{-.36in}(\mathfrak N_{\gamma})_{\tn{gft}} \quad \forall~\gamma'\!\in\!\Gamma_\tn{gft}\backslash\{\tau_\bullet\}, \qquad (\mathfrak M_{\tn{gft}})_{\tau_{\bullet}}\!:= \mathfrak M\,\backslash\big(\!\! \bigsqcup_{\gamma'\in\Gamma_\tn{gft}\backslash\tau_\bullet}\!\!\!\!\!(\mathfrak M_{\tn{gft}})_{\gamma'}\big), \end{split}\end{equation} we observe that $$ \mathfrak M=\bigsqcup_{\gamma'\in\Gamma_\tn{gft}}\!\!(\mathfrak M_{\tn{gft}})_{\gamma'} $$ is a $\Gamma_\tn{gft}$-stratification of $\mathfrak M$, known as the \textsf{grafted stratification} of $(\mathfrak M,\Gamma)$ (with respect to $K$). \iffalse Moreover, $$
\pi_0\big((\mathfrak M_\tn{gft})_{\gamma'}\big)
\!=\!\big\{
(\mathfrak N_\gamma)_\tn{gft}:\;
\gamma\!\in\!\Gamma\backslash\{\tau_\bullet\},\;
\mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma),\;
(\gamma,\beta_{\mathfrak N_\gamma}\!(o_{\mathfrak N_\gamma}))_\tn{gft}\!=\!\gamma'
\big\}
\quad
\forall\;\gamma'\!\in\!\Gamma_\tn{gft}\backslash\{\tau_\bullet\}. $$ \fi Furthermore, if \begin{equation}\label{Eqn:K_local}
K\cap\mathfrak N_{\gamma'}\cap\mathcal V
\,=\,
\{\kappa^\mathcal V\!=\! 0\}\cap\mathfrak N_{\gamma'}\cap\mathcal V
\qquad
\forall\,
\gamma'\!\in\!\Gamma,~\mathfrak N_{\gamma'}\!\in\!\pi_0(\mathfrak M_{\gamma'})~\textnormal{with}~\tau_{\mathfrak N_{\gamma'}}\!\ne\!\tau_\bullet, \end{equation} then the treelike structure $\Lambda$ on $(\mathfrak M,\Gamma)$ induces a treelike structure $\Lambda_\tn{gft}$ on $(\mathfrak M,\Gamma_\tn{gft})$ satisfying for every $\mathfrak N\!\in\!\pi_0(\mathfrak M_\tn{gft})_{\tau_\bullet},$ we have $$
\mathcal T_\mathfrak N=
(\tau_\bullet,\emptyset,-); $$ for every $\gamma'\!\in\!\Gamma_\tn{gft}\backslash\{\tau_\bullet\}$ and every $\mathfrak N\!=\!(\mathfrak N_\gamma)_\tn{gft}\!\in\!\pi_0\big((\mathfrak M_\tn{gft})_{\gamma'}\big)$ (which implies $\gamma\!\in\!\Gamma\backslash\{\tau_\bullet\}$ and $K\!\cap\!\mathfrak N_\gamma\!\ne\!\emptyset$), we have \begin{gather*}
\mathcal T_\mathfrak N=\big(
\tau_\mathfrak N\!=\!(\tau_{\mathfrak N_\gamma})_\tn{gft}\,,\,
\tn{E}_\mathfrak N\!=\!\tn{E}_{\mathfrak N_\gamma}\!\sqcup\!\{e_\tn{gft}\}\,,\,
\beta_\mathfrak N\!\!:\tn{Edg}\big((\tau_{\mathfrak N_\gamma})_\tn{gft}\big)\!\longrightarrow\!\tn{E}_{\mathfrak N_\gamma}\!\sqcup\!\{e_\tn{gft}\}
\big),\\
\textnormal{where}\quad
\beta_\mathfrak N|_{\tn{Edg}(\tau_{\mathfrak N_\gamma})}\!\!=\!\beta_{\mathfrak N_\gamma},\
\beta_\mathfrak N(e_\tn{gft})\!=\! e_\tn{gft}. \end{gather*} \end{crl}
An example of grafted graphs is illustrated in Figure~\ref{Fig:grafted}.
\begin{figure}
\caption{An example of grafted graphs}
\label{Fig:grafted}
\end{figure}
\begin{proof}[Proof of Corollary~\ref{Crl:extra_edge_graphs}.] For every $\gamma'\!\in\!\Gamma_\tn{gft}\backslash\{\tau_\bullet\}$, every nonempty $(\mathfrak N_\gamma)_{\tn{gft}}\!\in\!\pi_0\big((\mathfrak M_\tn{gft})_{\gamma'}\big)$, and every $x\!\in\!(\mathfrak N_\gamma)_{\tn{gft}}$, we have $x\!\in\!\mathfrak N_\gamma$, thus for any chart $\mathcal V\!\in\!\ud\mathfrak V_\gamma$ containing $x$, we take the subset of local parameters $$
\zeta^\mathcal V_e,\quad e\!\in\!\tn{Edg}(\gamma)\qquad
\textnormal{and}\qquad
\zeta^\mathcal V_{e_\tn{gft}}:=\kappa^\mathcal V. $$ Using these charts and local parameters, we see that the last equation in~(\ref{Eqn:grafted}) gives a $\Gamma_\tn{gft}$-stratification of $\mathfrak M$ satisfying Definition~\ref{Dfn:G-adim_fixture}. The connected components of each $(\mathfrak M_\tn{gft})_{\gamma'}$ are described in the first equation of the last line of~(\ref{Eqn:grafted}).
It remains to show that $\Lambda_\tn{gft}$ is a treelike structure on $(\mathfrak M,\Gamma_\tn{gft})$ if~(\ref{Eqn:K_local}) holds. Notice that for every $\gamma'\!\in\!\Gamma_\tn{gft}$ and every $\mathfrak N\!\in\!\pi_0((\mathfrak M_\tn{gft})_{\gamma'})$ with $\tau_\mathfrak N\!\ne\!\tau_\bullet$, the edge $e_\tn{gft}$ is both maximal and minimal in $\tn{Edg}(\tau_\mathfrak N)$. Thus, for every $E\!\in\!\tn{Edg}(\gamma')$ containing $e_\tn{gft}$ such that $\ti\gamma\!:=\!\gamma'_{(E)}$ belongs to~$\Gamma_\tn{gft}$, the first line of~(\ref{Eqn:grafted}) implies $\ti\gamma\!=\!\tau_\bullet$, hence $\tau_{\mathfrak N_{\ti\gamma}}\!=\!\tau_\bullet$ for any $\mathfrak N_{\ti\gamma}\!\in\!\pi_0(\mathfrak M_{\ti\gamma})$. This confirms the condition~\ref{Cond:Treelike_b2} of Definition~\ref{Dfn:Treelike_structure}. The conditions~\ref{Cond:Treelike_a} and~\ref{Cond:Treelike_b1} of Definition~\ref{Dfn:Treelike_structure} follow directly from the assumption that $\Lambda$ is a treelike structure on $(\mathfrak M,\Gamma)$. \end{proof}
\section{Applications of the STF theory to $\mathfrak P_2$ and $\ov M_2(\mathbb P^n,d)$} \label{Sec:genus_2_twisted_fields}
Let $\mathfrak P_g$ be the genus $g$ relative Picard stack of the stable pairs $(C,L)$ where $C$ is a genus $g$ nodal curve and $L$ is a line bundle over $C$. A pair $(C,L)$ is said to be \textsf{stable} if every rational irreducible component $\Sigma\!\subset\!C$ with $L/\Sigma$ as a trivial line bundle contains at least three nodal points. It is a well-known fact that $\mathfrak P_g$ is a smooth algebraic stack.
In~\S\ref{Subsec:Step1}-\ref{Subsec:Step8}, we apply the STF theory to $\mathfrak P_2$ recursively in eight times and obtain $\ti{\mathfrak P}_2^\tn{tf}$, which in turn gives rise to the resolution~$\ti M_2^\tn{tf}(\mathbb P^n,d)$ of $\ov M_2(\mathbb P^n,d)$ as in Theorem~\ref{Thm:Main}.
The proof of the properties~\ref{Cond:MainSmooth}-\ref{Cond:MainLocallyFree} of Theorem~\ref{Thm:Main} is provided in~\S\ref{Subsec:Proof_Main}.
As stated in~\S\ref{Sec:Intro}, the stack $\ti\mathfrak P_2^\tn{tf}$ may not be isomorphic to the blowup stack~$\ti\mathfrak P_2$ of~\cite{HLN}. The eight-step construction of $\ti\mathfrak P_2^\tn{tf}$ in~\S\ref{Sec:genus_2_twisted_fields} corresponds to the blowup construction of~\cite{HLN} as follows: $$
\underbrace{\mathsf{r}_1\mathsf{p}_1}_{\S\ref{Subsec:Step1}} \to
\underbrace{\mathsf{r}_1\mathsf{p}_2}_{\S\ref{Subsec:Step2}} \to
\underbrace{\big(\mathsf{r}_1\mathsf{p}_4\to\mathsf{r}_1\mathsf{p}_3\big)}_{\S\ref{Subsec:Step3}} \to
\underbrace{\mathsf{r}_1\mathsf{p}_5}_{\S\ref{Subsec:Step4}} \to
\underbrace{\mathsf{r}_2}_{\S\ref{Subsec:Step5}} \to
\underbrace{\big(\mathsf{r}_3\mathsf{p}_2\to\mathsf{r}_3\mathsf{p}_1 \big)}_{\S\ref{Subsec:Step6}} \to
\underbrace{\mathsf{r}_3\mathsf{p}_3}_{\S\ref{Subsec:Step7}} \to
\underbrace{\mathsf{r}_3\mathsf{p}_4}_{\S\ref{Subsec:Step8}}. $$ In fact, several blowups in ~\cite{HLN} can be performed in various orders and should generally give rise to different resolutions of $\ov M_2(\mathbb P^n,d)$.
\subsection{Notation} \label{Subsec:graph_decoration} The following notation and terminology will be used repeatedly in the recursive construction. \begin{dfn} \label{Dfn:arithmetic genus} We call a tuple $$
\gamma^\star=\big(\,\gamma\,,~
p_g\!:\tn{Ver}(\gamma)\!\longrightarrow\!\mathbb Z_{\ge 0}\,,~
\mathbf{w}\!:\tn{Ver}(\gamma)\!\longrightarrow\!\mathbb Z_{\ge 0}\,
\big) $$ of a connected graph $\gamma$ and two functions $p_g$ and $\mathbf{w}$ a \textsf{connected decorated graph}, or simply a \textsf{decorated graph} when the context is clear. The \textsf{arithmetic genus} $p_a$ of a decorated graph is $$
p_a(\gamma,p_g,\mathbf{w}):=b_1(\gamma)+\sum_{v\in\tn{Ver}(\gamma)}\!\!\!p_g(v).$$ We denote by $\mathbf{G}^\star$ the set of all connected decorated graphs. For $k\!\in\!\mathbb Z_{\ge 0}$, let $$
\mathbf{G}^\star_{k}:=\{\gamma^\star\!\in\!\mathbf{G}^\star:\,
p_a(\gamma^\star)\!=\! k
\}. $$ \end{dfn}
With $\gamma^\star$ as above, for every subgraph $\gamma'$ of $\gamma$, we write \begin{equation}\label{Eqn:subgraph_weight}
\mathbf{w}(\gamma'):=\sum_{v\in\tn{Ver}(\gamma')}\!\!\!\!\mathbf{w}(v). \end{equation} Here we identify $v\!\in\!\tn{Ver}(\gamma')$ with its image in $\gamma$; see Definition~\ref{Dfn:subgraph}.
We say two decorated graphs $\gamma^\star$ and $(\gamma')^\star$ are isomorphic and write $$\gamma^\star\simeq(\gamma')^\star$$ if $\gamma\!\simeq\!\gamma'$ and the graph isomorphism is compatible with the corresponding $p_g$, $p'_g$, $\mathbf{w}$, and $\mathbf{w}'$.
Given $\gamma\!\in\!\mathbf{G}$ and $E\!\subset\!\tn{Edg}(\gamma)$, recall that $\gamma_{(E)}$ denotes the graph obtained from $\gamma$ via edge contraction; see Definition~\ref{Dfn:Edge_contraction}. The subset $E$ determines a subgraph $\gamma'$ of $\gamma$ with $\tn{Edg}(\gamma')\!=\! E$. Let $\{\gamma'_1,\ldots,\gamma'_k\}$ be the set of the connected components of $\gamma'$, which are contracted to distinct vertices $\ov v_1,\ldots,\ov v_k$ in $\gamma_{(E)}$. Thus, there exists a surjection \begin{equation*}\label{Eqn:Edge_contraction_Ver_onto}
\pi_{\textnormal{ver};(E)}:~
\tn{Ver}(\gamma)\twoheadrightarrow\tn{Ver}\big(\gamma_{(E)}\big),
\quad
v\mapsto
\begin{cases}
v & \textnormal{if}~v\!\not\in\!\tn{Ver}(\gamma');\\
\ov v_i & \textnormal{if}~v\!\in\!\tn{Ver}(\gamma'_i),~i\!=\! 1,\ldots,k.
\end{cases} \end{equation*}
\begin{dfn}\label{Dfn:Induced_decoration_edge_contr} Every decoration $\gamma^\star$ of $\gamma$ determines a decoration $\gamma_{(E)}^\star$ of $\gamma_{(E)}$ by setting $$
p_g(\ov v_i)= b_1(\gamma_i')+\!\!
\sum_{v\,\in\,\pi_{\textnormal{ver};(E)}^{-1}(\ov v_i)}\!\!\!\!\!\!\!\!p_g(v),
\qquad
\mathbf{w}(\ov v_i)=\!\sum_{v\,\in\,\pi_{\textnormal{ver};(E)}^{-1}(\ov v_i)}\!\!\!\!\!\!\!\!\mathbf{w}(v),
\qquad
\forall~i\!=\! 1,\ldots,k, $$ which is called the \textsf{induced decoration} of $\gamma_{(E)}$. \end{dfn}
\begin{dfn}\label{Dfn:core}
Let
$\gamma^\star$ be a decorated graph.
The \textsf{core} $\gamma^\star_{\tn{cor}}$ is the smallest connected subgraph of~$\gamma^\star$ with $p_a(\gamma^\star_{\tn{cor}})\!=\! p_a(\gamma^\star)$,
together with the restrictions of~$p_g$ and $\mathbf{w}$ to $\tn{Ver}(\gamma')$. \end{dfn}
\iffalse Let $\Gamma$, $\mathfrak M$, and $\Lambda$ be as in Definition~\ref{Dfn:Treelike_structure}. Fix $\gamma\!\in\!\Gamma$, $\mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma)$, and $[{\mathfrak t}]\!\in\!\ov\Lambda_{\mathfrak N_\gamma}$. A decoration $\gamma^\star$ of $\gamma$ induces a decoration of
the derived graph $\rho_{\gamma,[{\mathfrak t}]}$ as in Definition~\ref{Dfn:Derived_graph} as follows.
Notice that the remnant graph $\gamma^\tn{rem}_{[{\mathfrak t}]}$ is obtained from $\gamma$ via edge contraction. Thus, by Definition~\ref{Dfn:Edge_contraction}, there is a natural surjection $f\!:\tn{Ver}(\gamma)\!\longrightarrow\!\tn{Ver}(\gamma^\tn{rem}_{[{\mathfrak t}]})$. The decoration $(\gamma^\tn{rem}_{[{\mathfrak t}]})^\star$ of $\gamma^\tn{rem}_{[{\mathfrak t}]}$ is given by \begin{equation}\label{Eqn:Remnant_decoration}
(p_g)_{(\gamma^\tn{rem}_{[{\mathfrak t}]})^\star}(v)\!=\!\sum_{v'\in f^{-1}(v)}\!\!\!\!\!(p_g)_{\gamma^\star}(v'),\quad
\mathbf{w}_{(\gamma^\tn{rem}_{[{\mathfrak t}]})^\star}(v)\!=\!\sum_{v'\in f^{-1}(v)}\!\!\!\!\!\mathbf{w}_{\gamma^\star}(v')\qquad
\forall\,v\!\in\!\tn{Ver}(\gamma^\tn{rem}_{[{\mathfrak t}]}). \end{equation}
The induced decoration on the derived graph $\rho_{\gamma,[{\mathfrak t}]}$ is more complicated. We describe it in the following definition. \fi
Let $\Gamma$, $\mathfrak M$, and $\Lambda$ be as in Definition~\ref{Dfn:Treelike_structure}. Given $\gamma\!\in\!\Gamma$, $\mathfrak N_\gamma\!\in\!\pi_0(\mathfrak M_\gamma)$, and $[{\mathfrak t}]\!\in\!\ov\Lambda_{\mathfrak N_\gamma}$, let $\rho_{\gamma,[{\mathfrak t}]}$ be the derived graph as in Definition~\ref{Dfn:Derived_graph}.
\begin{dfn}\label{Dfn:Derived_decoration} A decoration $\gamma^\star$ of $\gamma$
determines a decoration $\rho_{\gamma,[{\mathfrak t}]}^\star$ of $\rho_{\gamma,[{\mathfrak t}]}$, called the \textsf{induced decoration} of $\rho_{\gamma,[{\mathfrak t}]}$, such that: \begin{itemize} [leftmargin=*] \item if ${\mathbf{m}}({\mathfrak t})\!=\! 0$ (i.e.~${\mathfrak t}$ is edge-less), then $\rho_{\gamma,[{\mathfrak t}]}\!=\!\gamma$, so we set $
\rho_{\gamma,[{\mathfrak t}]}^\star\!=\!\gamma^\star $;
\item if~${\mathbf{m}}({\mathfrak t})\!<\!0$, then with $\mathsf v_i\!\in\!\tn{Ver}(\rho_{\gamma,[{\mathfrak t}]})$, $i\!\in\!\lrbr{{\mathbf{m}},0}_{{\mathfrak t}}$, denoting the {\it exceptional vertices} as in~(\ref{Eqn:v_0}), let \begin{equation*}\begin{split}
&p_g(\mathsf v_0)=\hspace{-.4in}
\sum_{v\in\{v(\beta(\hbar)):\,\hbar\in\tn{HE}(\mathfrak E^\bot_{\ge{\mathbf{m}}})\}}
\hspace{-.45in}
p_g(v)\,,\qquad
\mathbf{w}(\mathsf v_0)=\hspace{-.4in}
\sum_{v\in\{v(\beta(\hbar)):\,\hbar\in\tn{HE}(\mathfrak E^\bot_{\ge{\mathbf{m}}})\}}
\hspace{-.45in}
\mathbf{w}(v)\ ,
\qquad
p_g(\mathsf v_i)\!=\!\mathbf{w}(\mathsf v_i)\!=\! 0
\quad
\forall~
i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}}\,, \end{split}\end{equation*} and let the restrictions of $p_g$ and $\mathbf{w}$ to $\tn{Ver}(\rho_{\gamma,[{\mathfrak t}]})\backslash\{\mathsf v_i\!:i\!\in\!\lrbr{{\mathbf{m}},0}_{{\mathfrak t}}\big\}$ be determined by~$\gamma^\star$. \end{itemize} \end{dfn}
\begin{dfn}\label{Dfn:dual_graph} Let $C$ be a nodal curve. The \textsf{dual graph} $\gamma_C$ of $C$ is the graph whose vertices~$v$ and edges $e$ correspond to the irreducible components $C_v$ and the nodes $q_e$ of $C$, respectively. \end{dfn}
\begin{dfn}\label{Dfn:decorated_graph} Given $(C,L)\!\in\!\mathfrak P_g$, its \textsf{decorated dual graph} is a tuple $$
\gamma_{C,L}^\star:=\big(\;
\gamma_C\ ,\
p_g\!:\!\tn{Ver}(\gamma_C)\!\longrightarrow\!\mathbb Z_{\ge 0}\ ,\
\mathbf{w}\!:\! \tn{Ver}(\gamma)\!\longrightarrow\!\mathbb Z_{\ge 0}\;
\big) $$ such that $p_g$ and $\mathbf{w}$ assign to each $v\!\in\!\tn{Ver}(\gamma_C)$ the geometric genus of the irreducible component $C_v$ and the degree of $L/C_v$, respectively. \end{dfn}
\subsection{The first step of the recursive construction}\label{Subsec:Step1} The initial package of our recursive construction is given by $$\mathfrak M^{\fk 1}= \mathfrak P_2, \qquad \Gamma^{\fk 1}=\{\, [\gamma]:\, \gamma\!\in\!\mathbf{G},\, b_1(\gamma)\!\le\!2\,\},$$ where the first Betti number $b_1(\gamma)$ of a connected graph $\gamma$ is described in Definition~\ref{Dfn:b1} and the equivalence class is given by the graph isomorphism (see the paragraph before Definition~\ref{Dfn:G-adim_fixture}). As explained in Remark~\ref{Rmk:graph_isom}, it is convenient to fix a representative of each element of $\Gamma^{\fk 1}$ and write $\gamma\!\in\!\Gamma^{\fk 1}$ instead of $[\gamma]\!\in\!\Gamma^{\fk 1}$.
For any $(C,L)\!\in\!\mathfrak M^{\fk 1}$, we take an affine smooth chart $$(C,L)\in\mathcal V_{C,L}\longrightarrow\mathfrak M^{\fk 1}.$$ As in~\cite[\S4.3]{HL10} and~\cite[\S3.1]{g1modular}, there exists a set of regular functions $\{\zeta_e^\mathcal V\}_{e\in\tn{Edg}(\gamma_C)}$ so that for each $e\!\in\!\tn{Edg}(\gamma_C)$, the locus \begin{equation}\label{Eqn:modular_parameters_P2}
\{\,\zeta_e^\mathcal V=0\,\} \subset \mathcal V_{C,L} \end{equation} is where the node labeled by $e$ is not smoothed out.
\begin{lmm}\label{Lm:G1-admissible} With notation as above, there exists a $\Gamma^{\fk 1}$-stratification of $\mathfrak M^{\fk 1}$ given~by $$
\mathfrak M^{\fk 1}=\bigsqcup_{\gamma\in\Gamma^{\fk 1}}\mathfrak M^{\fk 1}_\gamma:=
\bigsqcup_{\gamma\in\Gamma^{\fk 1}}
\big\{\,(C,L)\!\in\!\mathfrak M^{\fk 1}:\,
\gamma_C\!\simeq\!\gamma\,\big\}. $$ \end{lmm}
\begin{proof} Consider the aforementioned charts of $\mathfrak M^{\fk 1}$ and the subsets of local parameters: \begin{equation}\label{Eqn:charts_and_parameters}
\ud{\mathfrak V}_{\gamma}:=
\big\{\mathcal V_{C,L}:\,
\gamma_C\!\simeq\! \gamma
\big\},\quad
\gamma\!\in\! \Gamma^{\fk 1};
\qquad
\{\zeta_e^\mathcal V:e\!\in\! \tn{Edg}(\gamma)\},\quad
\gamma\!\in\!\Gamma^{\fk 1},~\mathcal V\!\in\!\ud{\mathfrak V}_{\gamma}, \end{equation} respectively. By the deformation of the nodal curves, we know that $$
\textnormal{Cl}_{\mathfrak M^{\fk 1}}(\mathfrak M_\gamma^{\fk 1})=
\bigsqcup_{\gamma_1\in\Gamma^{\fk 1},\,
\gamma_1\preceq\gamma}\!\!\!\!\!\!
\mathfrak M^{\fk 1}_{\gamma_1}\qquad
\forall\ \gamma\!\in\!\Gamma^{\fk 1}. $$ Notice that $\gamma_1\!\preceq\!\gamma$ above is also up to graph isomorphism. Therefore, \begin{equation}\label{Eqn:closure}
\bigsqcup_{\gamma\in\Gamma^{\fk 1},\,
\gamma\not\succeq\gamma'}\!\!\!\!\!\!
\mathfrak M^{\fk 1}_{\gamma}=
\bigcup_{\gamma\in\Gamma^{\fk 1},\,
\gamma\not\succeq\gamma'}\!\!\!\!\!\!
\textnormal{Cl}_{\mathfrak M^{\fk 1}}(\mathfrak M^{\fk 1}_\gamma)\qquad
\forall\ \gamma'\!\in\!\Gamma^{\fk 1}. \end{equation} By the deformation of $(C,L)\!\in\!\mathfrak M^{\fk 1}\!=\!\mathfrak P_2$, we also know that $\sum_{v\in\tn{Ver}(\gamma_C)}c_1(L/C_v)$ is constant within each connected component of $\mathfrak P_2$. Taking the stability of $\mathfrak P_2$ into account, we see that for every $\mathfrak N\!\in\!\pi_0(\mathfrak P_2)$, there exist only finitely many $\gamma\!\in\!\Gamma^{\fk 1}$ such that $\mathfrak M^{\fk 1}_\gamma\!\cap\!\mathfrak N\!\ne\!\emptyset$. Thus, the RHS of (\ref{Eqn:closure}) is a finite union in each connected component of $\mathfrak M^{\fk 1}$, hence $$
\bigsqcup_{\gamma\in\Gamma^{\fk 1},\,
\gamma\not\succeq\gamma'}\!\!\!\!\!\!
\mathfrak M^{\fk 1}_{\gamma}\ \
\textnormal{is~closed~in}~\mathfrak M^{\fk 1}. $$ Consequently, we may shrink the charts in~(\ref{Eqn:charts_and_parameters}) if necessary so that \begin{equation} \label{Eqn:M_strata_local_2}
\mathfrak M^{\fk 1}_{\gamma}\cap\mathcal V'=\emptyset\qquad
\forall\ \gamma,\gamma'\!\in\!\Gamma^{\fk 1}~\textnormal{with}~
\gamma\!\not\succeq\!\gamma',\
\forall\ \mathcal V'\!\in\!\ud\mathfrak V_{\gamma'}. \end{equation} In addition, since the local parameters $\zeta_e^\mathcal V$ correspond to the smoothing of the nodes, we conclude that for every $\gamma,\gamma'\!\in\!\Gamma^{\fk 1}$ with $\gamma\!\succeq\!\gamma'$ and every chart $\mathcal V'\!\in\!\ud\mathfrak V_{\gamma'},$ \begin{equation}\label{Eqn:M_strata_local_1} \begin{split}
&\mathfrak M^{\fk 1}_{\gamma}\cap\mathcal V'=\!\!
\bigcup_{E\subset\tn{Edg}(\gamma'),\;
\gamma'_{(E)}\simeq\gamma}\!\!\!\!\!\!\!\!\!\!\!\!\!
\big\{\,\zeta_e^{\mathcal V}\!\!\!=\! 0\ \,
\forall\,e\!\in\!\tn{Edg}(\gamma')\backslash E\,,\ \,
\zeta_e^\mathcal V\!\!\neq\!0\ \, \forall\,e\!\in\! E\,
\big\}.\\
\end{split} \end{equation} Lemma~\ref{Lm:G1-admissible} then follows from~(\ref{Eqn:M_strata_local_2}), (\ref{Eqn:M_strata_local_1}), and Definition~\ref{Dfn:G-adim_fixture}. \end{proof}
One can mimic Example~\ref{Eg:genus_1} and construct a treelike structure $\Lambda'$ on $(\mathfrak M^{\fk 1},\Gamma^{\fk 1})$ likewise, which gives rise to a stack with twisted fields $\mathfrak M^{\fk 1}_{\Lambda'}$. However, such $(\mathfrak M^{\fk 1},\Gamma^{\fk 1},\Lambda')$ does not satisfy the hypothesis of Corollary~\ref{Crl:Derived_graphs}, so the derived stratification of $\mathfrak M^{\fk 1}_{\Lambda'}$ with respect to $(\mathfrak M^{\fk 1},\Gamma^{\fk 1},\Lambda')$ does not exist, and the proposed recursive construction cannot proceed further. Thus, we introduce a slightly different stratification on $\mathfrak M^{\fk 1}$ as follows.
For every decoration $\gamma^\star$ of $\gamma\!\in\!\Gamma^{\fk 1}$, we denote by $$
\gamma_\tn{vic}=\gamma^{\textnormal{id}}_{\tn{Ver}(\gamma^\star_\tn{cor})}\in\mathbf{G} $$ the (connected) graph obtained from $\gamma$ by applying the {\it vertex identification} to $\tn{Ver}(\gamma_\tn{cor}^\star)$; see~Definitions~\ref{Dfn:Vertex_identification} and~\ref{Dfn:core} for terminology. The image of the vertices of $\gamma_\tn{cor}^\star$ in $\gamma_\tn{vic}$ is denoted by $o_{\gamma_\tn{vic}}$. The decoration $\gamma^\star$ of $\gamma$ further induces a (unique) decoration $\gamma_\tn{vic}^\star$ of $\gamma_\tn{vic}$ by setting \begin{equation}\label{Eqn:vic_decoration}
p_g(o_{\gamma_\tn{vic}})=\sum_{v\in\tn{Ver}(\gamma_\tn{cor}^\star)}
\hspace{-.18in} p_g(v)
\qquad\textnormal{and}\qquad
\mathbf{w}(o_{\gamma_\tn{vic}})=\sum_{v\in\tn{Ver}(\gamma_\tn{cor}^\star)}
\hspace{-.18in}\mathbf{w}(v) \end{equation} while leaving the decorations of the vertices of $\tn{Ver}(\gamma^\star_\tn{vic})\backslash\{o_{\gamma_\tn{vic}}\}\!=\!\tn{Ver}(\gamma)\backslash\tn{Ver}(\gamma^\star_\tn{cor})$ unchanged.
The above description of $\gamma_\tn{vic}$ implies \begin{equation}\label{Eqn:vic}
\tn{Edg}(\gamma_\tn{vic})\!=\!\tn{Edg}(\gamma);\quad
\tn{Edg}\big((\gamma_\tn{vic}^\star)_\tn{cor}\big)\!=\!\tn{Edg}(\gamma_\tn{cor}^\star);\quad
(\gamma_{(E)})_\tn{vic}\!=\!(\gamma_\tn{vic})_{(E)}\
\forall\,E\!\subset\!\tn{Edg}(\gamma). \end{equation} Let $$
\Gamma^{\fk 1}_\tn{vic}=
\{\,
\gamma_\tn{vic}:\;
\gamma^\star\!\!=\!(\gamma,p_g,\mathbf{w})\!\in\!\mathbf{G}^\star_2,\,
\gamma\!\in\!\Gamma^{\fk 1}\,\}. $$ The following statement follows immediately from~(\ref{Eqn:vic}) and Lemma~\ref{Lm:G1-admissible}. \begin{crl}\label{Crl:G1-admissible} The $\Gamma^{\fk 1}$-stratification of $\mathfrak M^{\fk 1}$ in Lemma~\ref{Lm:G1-admissible} determines a $\Gamma^{\fk 1}_\tn{vic}$-stratification: $$
\mathfrak M^{\fk 1}=
\bigsqcup_{\gamma'\in\Gamma^{\fk 1}_{\tn{vic}}}
\mathfrak M^{\fk 1}_{\tn{vic};\gamma'}:=
\bigsqcup_{\gamma'\in\Gamma^{\fk 1}_{\tn{vic}}}
\big\{\,(C,L)\!\in\!\mathfrak M^{\fk 1}:
(\gamma_{C,L}^\star)_\tn{vic}\!\simeq\!\gamma'\,\big\}. $$ \end{crl}
For every $\gamma^\star\!\in\!\mathbf{G}_2^\star$, let $$
\mathfrak N_{\gamma^\star}^{\fk 1}
\!:=\!
\big\{
(C,L)\!\in\!\mathfrak M^{\fk 1}:
\gamma^\star_{C,L}\!\simeq\!\gamma^\star
\big\}
\subset\mathfrak M^{\fk 1}. $$ Then, for every $\gamma'\!\in\!\Gamma^{\fk 1}_\tn{vic}$, we have $$
\pi_0(\mathfrak M^{\fk 1}_{\tn{vic};\gamma'})=
\big\{
\mathfrak N_{\gamma^\star}^{\fk 1}:\;
\gamma^\star\!\in\!\mathbf{G}^\star_2,\;
\gamma_\tn{vic}\!\simeq\!\gamma'
\big\}\,. $$ Fix a graph $\gamma\!\in\!\Gamma^{\fk 1}$ and a decoration $\gamma^\star$ of $\gamma$. Let \begin{equation} \label{Eqn:N^1}
\mathfrak N^{\fk 1}:=\mathfrak N^{\fk 1}_{\gamma^\star}
\in
\pi_0(\mathfrak M^{\fk 1}_{\tn{vic};\gamma_\tn{vic}}). \end{equation} We will describe the tuple $(\tau_{\mathfrak N^{\fk 1}},\tn{E}_{\mathfrak N^{\fk 1}},\beta_{\mathfrak N^{\fk 1}})$ to provide
the proposed treelike structure~$\Lambda^{\fk 1}$ on $(\mathfrak M^{\fk 1},\Gamma^{\fk 1}_\tn{vic})$. Let $\gamma_{\tn{cor}}^\star$ be the core of $\gamma^\star$ as in Definition~\ref{Dfn:core} and $\mathbf{w}(\gamma_{\tn{cor}}^\star)$ be as in~(\ref{Eqn:subgraph_weight}).
\begin{itemize}[leftmargin=*]
\item If $\mathbf{w}(\gamma_{\tn{cor}}^\star)\!>\!0$, we set $\tau_{\mathfrak N^{\fk 1}}\!=\!\tau_\bullet$ and $o_{\mathfrak N^{\fk 1}}\!=\! o_{\gamma_\tn{vic}}$.
\item If $\mathbf{w}(\gamma_{\tn{cor}}^\star)\!=\! 0$, let $$
\gamma_{\tn{ctr}}=(\gamma_\tn{vic})_{\big(\tn{Edg}((\gamma_\tn{vic}^\star)_\tn{cor})\big)}
=\gamma_{(\tn{Edg}(\gamma^\star_{\tn{cor}}))} $$ be the graph obtained via edge contraction; the second equality above follows from~(\ref{Eqn:vic}). We then dissolve all $v\!\in\!\tn{Ver}(\gamma_{\tn{ctr}})$ satisfying $\mathbf{w}(v)\!>\!0$ (see Definition~\ref{Dfn:Vertex_dissolution} for terminology) and take~$\tau_{\mathfrak N^{\fk 1}}$ to be the unique connected component that contains~$o_{\gamma_\tn{vic}}$. Thus, $\tau_{\mathfrak N^{\fk 1}}$ is a rooted tree with the root~$o_{\gamma_\tn{vic}}$. \end{itemize} Since~$\tau_{\mathfrak N^{\fk 1}}$ is obtained via contraction of the edges in $\gamma^\star_\tn{cor}$, dissolution of vertices not in $\gamma^\star_\tn{cor}$, and/or taking a connected component of a graph, we conclude that $\tau_{\mathfrak N^{\fk 1}}$ is a {\it connected} subgraph of $\gamma_\tn{vic}$. In particular, \begin{equation}\label{Eqn:step1_edge}
\tn{Edg}(\tau_{\mathfrak N^{\fk 1}})
\,\subset\,
\tn{Edg}(\gamma)\backslash\tn{Edg}(\gamma^\star_\tn{cor})
\,=\,
\tn{Edg}(\gamma_\tn{vic})\backslash\tn{Edg}\big((\gamma^\star_\tn{vic})_\tn{cor}\big). \end{equation} Let \begin{equation*}\begin{split}
\Lambda^{\fk 1}
&=
\big(\,
(\tau_{\mathfrak N^{\fk 1}},\,
\tn{E}_{\mathfrak N^{\fk 1}}\!:=\!
\tn{Edg}(\tau_{\mathfrak N^{\fk 1}}),\,
\textnormal{Id})\,
\big)_{
\gamma'\in\Gamma^{\fk 1}_\tn{vic},\;
\mathfrak N^{\fk 1}\in
\pi_0(\mathfrak M^{\fk 1}_{\tn{vic};\gamma'})
}. \end{split}\end{equation*}
\begin{lmm} \label{Lm:G1-level} The set $\Lambda^{\fk 1}$ give a treelike structure on
$(\mathfrak M^{\fk 1},\Gamma^{\fk 1}_\tn{vic})$ as in Definition~\ref{Dfn:Treelike_structure}. \end{lmm}
\begin{proof} The proof is parallel to its genus 1 counterpart that is studied in Example~\ref{Eg:genus_1}, with $\mathfrak M^\tn{wt}_1$ replaced by $\mathfrak P_2$. The key point is that the positively weighted vertices of $\tau_{\mathfrak N^{\fk 1}}$ are and only are the minimal vertices (i.e.~leaves), which is the same as in the genus 1 case. We omit further details. \end{proof}
Corollary~\ref{Crl:G1-admissible}, Lemma~\ref{Lm:G1-level}, and Theorem~\ref{Thm:tf_smooth} together imply the following:
\begin{crl} \label{Crl:G1-tf} Let $(\mathfrak M^{\fk 1})_{\Lambda^{\fk 1}}^\tn{tf}$ be constructed as in Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p1}. Then, $(\mathfrak M^{\fk 1})_{\Lambda^{\fk 1}}^\tn{tf}$ is a smooth algebraic stack and the forgetful morphism $(\mathfrak M^{\fk 1})_{\Lambda^{\fk 1}}^\tn{tf}\!\longrightarrow\!\mathfrak M^{\fk 1}\!=\!\mathfrak P_2$ is proper and birational. \end{crl}
\subsection{The second step of the recursive construction}\label{Subsec:Step2}
In this step, we take $$
\mathfrak M^{\fk 2}=(\mathfrak M^{\fk 1})^\tn{tf}_{\Lambda^{\fk 1}},\qquad
\Gamma^{\fk 2}_\tn{vic}=(\Gamma^{\fk 1}_\tn{vic})^{\tn{der}}_{\Lambda^{\fk 1}}, $$ where $(\Gamma^{\fk 1}_\tn{vic})^{\tn{der}}_{\Lambda^{\fk 1}}$ is the set of the derived graphs of $\Gamma^{\fk 1}_\tn{vic}$ with respect to the rooted level trees given by the treelike structure $\Lambda^{\fk 1}$; see~(\ref{Eqn:Derived_graphs}). Since for every $\gamma'\!\in\!\Gamma^{\fk 1}_\tn{vic}$ and every $\mathfrak N^{\fk 1}\!\in\!\pi_0(\mathfrak M^{\fk 1}_{\tn{vic};\gamma'})$, the rooted tree $\tau_{\mathfrak N^{\fk 1}}$ is a connected subgraph of $\gamma'$, Corollary~\ref{Crl:Derived_graphs} thus gives rise to the following statement.
\begin{lmm}\label{Lm:G2-admissible} The stack $\mathfrak M^{\fk 2}$ comes equipped with a $\Gamma^{\fk 2}_\tn{vic}$-stratification, which is the derived stratification of $\mathfrak M^{\fk 2}$ with respect to $(\mathfrak M^{\fk 1},\Gamma^{\fk 1}_\tn{vic},\Lambda^{\fk 1})$. \end{lmm}
The graphs of $\Gamma^{\fk 2}_\tn{vic}$ are in the form \begin{equation}\label{Eqn:ga^2}
\gamma^{\fk 2}
:=\rho_{\gamma_\tn{vic},[{\mathfrak t}^{\fk 1}]}
\in\Gamma^{\fk 2}_\tn{vic},
\qquad
\textnormal{where}\ \
\gamma_\tn{vic}\!\in\!\Gamma^{\fk 1}_\tn{vic},\ \
[{\mathfrak t}^{\fk 1}]\!\in\!\ov\Lambda^{\fk 1}_{\mathfrak N^{\fk 1}}\!,\ \
\mathfrak N^{\fk 1}\!\in\!\pi_0(\mathfrak M^{\fk 1}_{\tn{vic};\gamma_\tn{vic}}); \end{equation} see~(\ref{Eqn:N^1}) for $\mathfrak N^{\fk 1}$. The vertex that is the image of $o_{\gamma_\tn{vic}}$ in $\gamma^{\fk 2}$ is denoted by $o_{\gamma^{\fk 2}}$.
The decoration~$\gamma^\star_\tn{vic}$ of $\gamma_\tn{vic}$ induces the decoration $(\gamma^{\fk 2})^\star$ of $\gamma^{\fk 2}$ as in Definition~\ref{Dfn:Derived_decoration}. By Corollary~\ref{Crl:Derived_graphs}, the connected components of $\mathfrak M^{\fk 2}_{\gamma^{\fk 2}}$ are in the form \begin{equation}\label{Eqn:N^2}
\mathfrak N^{\fk 2}:=
(\mathfrak N^{\fk 1})_{[{\mathfrak t}^{\fk 1}]}^\tn{tf}\
\in\pi_0\big(\mathfrak M^{\fk 2}_{\gamma^{\fk 2}}\big). \end{equation} For each $\mathfrak N^{\fk 2}$ as above, we will describe the tuple $(\tau_{\mathfrak N^{\fk 2}},\tn{E}_{\mathfrak N^{\fk 2}},\beta_{\mathfrak N^{\fk 2}})$ to provide
the proposed treelike structure~$\Lambda^{\fk 2}$ on $(\mathfrak M^{\fk 2},\Gamma^{\fk 2}_\tn{vic})$.
Let \begin{equation}\begin{split}\label{Eqn:Edges_ctr_in_Step_1}
\tn{E}^{\fk 1}
&:=\mathfrak E^\bot_{\ge{\mathbf{m}}}({\mathfrak t}^{\fk 1})\subset\tn{Edg}(\tau_{\mathfrak N^{\fk 1}})=\tn{E}_{\mathfrak N^{\fk 1}}
\\
&\subset
\tn{Edg}(\gamma)\backslash\tn{Edg}(\gamma^\star_\tn{cor})=
\tn{Edg}(\gamma_\tn{vic})\backslash\tn{Edg}\big((\gamma^\star_\tn{vic})_\tn{cor}\big), \end{split}\end{equation} which follows from~(\ref{Eqn:step1_edge}). Consider the graph $\gamma_{(\tn{E}^{\fk 1})}$ obtained from $\gamma$ via edge contraction, which is endowed with the induced decoration $\gamma_{(\tn{E}^{\fk 1})}^\star$ from $\gamma^\star$ as in Definition~\ref{Dfn:Induced_decoration_edge_contr}. Then, \begin{equation} \label{Eqn:ga_star_(E)}
\tn{Edg}(\gamma_{(\tn{E}^{\fk 1})})\subset
\tn{Edg}(\gamma^{\fk 2}),\quad
\tn{Edg}\big(\gamma^\star_{(\tn{E}^{\fk 1});\tn{cor}}\big)=
\tn{Edg}\big((\gamma^{\fk 2})^\star_\tn{cor}\big),\quad
\mathbf{w}\big(\gamma^\star_{(\tn{E}^{\fk 1});\tn{cor}}\big)=
\mathbf{w}(o_{\gamma^{\fk 2}})\ge 1. \end{equation} The first two relations follow from~(\ref{Eqn:derived_graph}) and~(\ref{Eqn:Edges_ctr_in_Step_1}), and the last equality follows from the choice of ${\mathbf{m}}({\mathfrak t}^{\fk 1})$ and Definitions~\ref{Dfn:Induced_decoration_edge_contr} and~\ref{Dfn:Derived_decoration}. Further, by the construction of $\tau_{\mathfrak N^{\fk 1}}$ in \S\ref{Subsec:Step1}, we see that $\gamma^\star_{(\tn{E}^{\fk 1})}$ contains {\it at most} one pair of the smallest disjoint connected decorated subgraphs $\gamma^\star_{(\tn{E}^{\fk 1});\pm}$ satisfying \begin{equation} \label{Eqn:step2_unweighted_core} p_a\big(\gamma^\star_{(\tn{E}^{\fk 1});\pm}\big)= 1,\qquad \mathbf{w}\big(\gamma^\star_{(\tn{E}^{\fk 1});\pm}\big)=0. \end{equation}
We are ready to construct $\tau_{\mathfrak N^{\fk 2}}$. \begin{itemize} [leftmargin=*] \item If $\gamma^\star_{(\tn{E}^{\fk 1})}$ does not contain two {\it disjoint} connected decorated subgraphs $\gamma^\star_{(\tn{E}^{\fk 1});\pm}$ satisfying~(\ref{Eqn:step2_unweighted_core}), then we set $\tau_{\mathfrak N^{\fk 2}}\!=\!\tau_\bullet$ and $o_{\mathfrak N^{\fk 2}}\!=\! o_{\gamma^{\fk 2}}$.
\item If $\gamma^\star_{(\tn{E}^{\fk 1})}$ contains a unique pair of disjoint connected decorated subgraphs $\gamma^\star_{(\tn{E}^{\fk 1});\pm}$ satisfying~(\ref{Eqn:step2_unweighted_core}), then they are decorated subgraphs of $\gamma^\star_{(\tn{E}^{\fk 1});\tn{cor}}$, and the (shortest) path in $\gamma_{(\tn{E}^{\fk 1})}$ connecting $\gamma^\star_{(\tn{E}^{\fk 1});+}$ and $\gamma^\star_{(\tn{E}^{\fk 1});-}$ must contain a vertex with $\mathbf{w}(v)\!>\!0$ by the last inequality in~(\ref{Eqn:ga_star_(E)}).
\begin{itemize} [leftmargin=*] \item First, let $\gamma_{(\tn{E}^{\fk 1});\tn{ctr}}$ be the graph obtained from $\gamma_{(\tn{E}^{\fk 1})}$ by contracting the edges of $\gamma^\star_{(\tn{E}^{\fk 1});\pm}$ into two distinct vertices~$o_{\tn{ctr},\pm}$, respectively;
\item then, let $\wh\gamma_{(\tn{E}^{\fk 1});\tn{ctr}}$ be the graph obtained from $\gamma_{(\tn{E}^{\fk 1});\tn{ctr}}$ by identifying the vertices $o_{\tn{ctr},+}$ and $ o_{\tn{ctr},-}$ as a same vertex $\wh o_\tn{ctr}$;
\item finally, we dissolve all $v\!\in\!\tn{Ver}(\wh\gamma_{(\tn{E}^{\fk 1});\tn{ctr}})$ satisfying $\mathbf{w}(v)\!>\!0$ and take $\tau_{\mathfrak N^{\fk 2}}$ to be the unique connected component that contains $\wh o_\tn{ctr}$. \end{itemize} In this way, we see that $\tau_{\mathfrak N^{\fk 2}}$ is a rooted tree with the root $o_{\mathfrak N^{\fk 2}}\!=\!\wh o_\tn{ctr}$. \end{itemize} Notice that $\tau_{\mathfrak N^{\fk 2}}$ is obtained via contraction of the edges of $(\gamma^{\fk 2})^\star_{\tn{cor}}$ by~(\ref{Eqn:ga_star_(E)}), identification of vertices of $\gamma_{(\tn{E}^{\fk 1});\tn{cor}}$, vertex dissolution, and taking a connected component of a graph, thus \begin{equation}\label{Eqn:step2_edge}
\tn{Edg}(\tau_{\mathfrak N^{\fk 2}})~\subset~
\tn{Edg}(\gamma_{(\tn{E}^{\fk 1})})=
\tn{Edg}(\gamma^{\fk 2})\backslash\{\,\mathsf e_i^{\fk 1}:
i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^1}\}
~\subset~
\tn{Edg}(\gamma^{\fk 2}), \end{equation} and moreover, $\tn{Edg}(\tau_{\mathfrak N^{\fk 2}})$ is the set of the edges of a {\it connected} subgraph $\tau'_{\mathfrak N^{\fk 2}}$ of $\gamma^{\fk 2}$ that satisfies $$
o_{{\mathfrak N^{\fk 2}}}
\subset o_{\gamma^{\fk 2}}\in\tn{Ver}(\tau'_{\mathfrak N^{\fk 2}}) $$ (recall that every vertex is a set of half-edges). We set $$
\Lambda^{\fk 2}=
\big(\,
(\tau_{\mathfrak N^{\fk 2}},\,
\tn{E}_{\mathfrak N^{\fk 2}}\!:=\!\tn{Edg}(\tau_{\mathfrak N^{\fk 2}}),\,
\textnormal{Id})\,
\big)_{
\gamma^{\fk 2}\in\Gamma^{\fk 2}_\tn{vic},\;
\mathfrak N^{\fk 2}\in\pi_0(\mathfrak M^{\fk 2}_{\gamma^{\fk 2}})\,
}. $$
The following lemma is the analogue of Lemma~\ref{Lm:G1-level} in this step.
\begin{lmm} \label{Lm:G2-level} The set $\Lambda^{\fk 2}$ gives a treelike structure on $(\mathfrak M^{\fk 2},\Gamma^{\fk 2}_\tn{vic})$ as in Definition~\ref{Dfn:Treelike_structure}. \end{lmm}
Lemmas~\ref{Lm:G2-admissible} and~\ref{Lm:G2-level}, along with Theorem~\ref{Thm:tf_smooth}, give rise to the following statement.
\begin{crl} \label{Crl:G2-tf} Let $(\mathfrak M^{\fk 2})^\tn{tf}_{\Lambda^{\fk 2}}$ be constructed as in Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p1}. Then, $(\mathfrak M^{\fk 2})^\tn{tf}_{\Lambda^{\fk 2}}$ is a smooth algebraic stack and the composite forgetful morphism $(\mathfrak M^{\fk 2})^\tn{tf}_{\Lambda^{\fk 2}}\!\longrightarrow\!\mathfrak M^{\fk 1}\!=\!\mathfrak P_2$ is proper and birational. \end{crl}
\subsection{The third step of the recursive construction}\label{Subsec:Step3}
In this step, we take $$
\mathfrak M^{\fk 3}=(\mathfrak M^{\fk 2})^\tn{tf}_{\Lambda^{\fk 2}},\qquad
\Gamma^{\fk 3}_\tn{vic}=(\Gamma^{\fk 2}_\tn{vic})^{\tn{der}}_{\Lambda^{\fk 2}}. $$ Analogous to Lemma~\ref{Lm:G2-admissible}, we have the following statement.
\begin{lmm}\label{Lm:G3-admissible} The stack $\mathfrak M^{\fk 3}$ comes equipped with a $\Gamma^{\fk 3}_\tn{vic}$-stratification, which is the derived stratification of $\mathfrak M^{\fk 3}$ with respect to $(\mathfrak M^{\fk 2},\Gamma^{\fk 2}_\tn{vic},\Lambda^{\fk 2})$. \end{lmm}
The graphs of $\Gamma^{\fk 3}_\tn{vic}$ are in the form \begin{equation}\label{Eqn:ga^3}
\gamma^{\fk 3}
:=\rho_{\gamma^{\fk 2},[{\mathfrak t}^{\fk 2}]}
\in\Gamma^{\fk 3}_\tn{vic},
\qquad
\textnormal{where}\ \
\gamma^{\fk 2}\!\in\!\Gamma^{\fk 2}_\tn{vic},\ \
[{\mathfrak t}^{\fk 2}]\!\in\!\ov\Lambda^{\fk 2}_{\mathfrak N^{\fk 2}},\ \
\mathfrak N^{\fk 2}
\!\in\!\pi_0\big(\mathfrak M^{\fk 2}_{\gamma^{\fk 2}}\big); \end{equation} see~(\ref{Eqn:ga^2}) for $\gamma^{\fk 2}$ and~(\ref{Eqn:N^2}) for $\mathfrak N^{\fk 2}$. The vertex that is the image of $o_{\gamma^{\fk 2}}$ in $\gamma^{\fk 3}$ is denoted by $o_{\gamma^{\fk 3}}$.
The decoration $(\gamma^{\fk 2})^\star$ of $\gamma^{\fk 2}$ induces the decoration $(\gamma^{\fk 3})^\star$ of $\gamma^{\fk 3}$ as in Definition~\ref{Dfn:Derived_decoration}. By Corollary~\ref{Crl:Derived_graphs}, for every $\gamma^{\fk 3}$ as in~(\ref{Eqn:ga^3}), the connected components of $\mathfrak M^{\fk 3}_{\gamma^{\fk 3}}$ are in the form \begin{equation}\label{Eqn:N^3}
\mathfrak N^{\fk 3}:=
(\mathfrak N^{\fk 2})_{[{\mathfrak t}^{\fk 2}]}^\tn{tf}\
\in\pi_0\big(\mathfrak M^{\fk 3}_{\gamma^{\fk 3}}\big). \end{equation} For each $\mathfrak N^{\fk 3}$ as above, we will describe the tuple $(\tau_{\mathfrak N^{\fk 3}},\tn{E}_{\mathfrak N^{\fk 3}},\beta_{\mathfrak N^{\fk 3}})$ to provide
the proposed treelike structure~$\Lambda^{\fk 3}$ on $(\mathfrak M^{\fk 3},\Gamma^{\fk 3}_\tn{vic})$.
Let \begin{equation}\begin{split}\label{Eqn:Edges_ctr_in_Step_2}
\tn{E}^{\fk 2}
&:=
\mathfrak E^\bot_{\ge{\mathbf{m}}}({\mathfrak t}^{\fk 2})
\subset
\tn{Edg}(\tau_{\mathfrak N^{\fk 2}})
=\tn{E}_{\mathfrak N^{\fk 2}}
\\
&\subset
\tn{Edg}(\gamma_{(\tn{E}^{\fk 1})})
=
\tn{Edg}(\gamma^{\fk 2})\backslash\{\mathsf e_i^{\fk 1}:i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^{\fk 1}}\}
\subset
\tn{Edg}(\gamma^{\fk 2}), \end{split}\end{equation} which follows from~(\ref{Eqn:step2_edge}). Consider the graph and its induced decoration $$
\gamma_{(\ud\tn{E})^{\fk 2}}:=
(\gamma_{(\tn{E}^{\fk 1})})_{(\tn{E}^{\fk 2})}
=\gamma_{(\tn{E}^{\fk 1}\sqcup\tn{E}^{\fk 2})}\qquad
\textnormal{and}\qquad
\gamma_{(\ud\tn{E})^{\fk 2}}^\star, $$ respectively. Analogous to~(\ref{Eqn:ga_star_(E)}), we have \begin{equation}\begin{split} \label{Eqn:ga_E1E2}
&\tn{Edg}(\gamma_{(\ud\tn{E})^{\fk 2}})\!\subset\!
\tn{Edg}(\gamma^{\fk 3}),\quad
\tn{Edg}\big(\gamma^\star_{(\ud\tn{E})^{\fk 2};\tn{cor}}\big)
\!=\!
\tn{Edg}\big((\gamma^{\fk 3})^\star_\tn{cor}\big),\quad
\mathbf{w}\big(\gamma^\star_{(\ud\tn{E})^{\fk 2};\tn{cor}}\big)\!=\!
\mathbf{w}(o_{\gamma^{\fk 3}})\!\ge\!1. \end{split}\end{equation} In addition, by the construction of $\tau_{\mathfrak N^{\fk 2}}$ in \S\ref{Subsec:Step2}, we see that $\gamma^\star_{(\ud\tn{E})^{\fk 2}}$ contains at most one {\it smallest} connected decorated subgraph \begin{equation}\label{Eqn:step3_unweighted_core}\gamma^\star_{(\ud\tn{E})^{\fk 2};+} \qquad \textnormal{with}\quad p_a\big(\gamma^\star_{(\ud\tn{E})^{\fk 2};+}\big)\!=\! 1,\quad \mathbf{w}\big(\gamma^\star_{(\ud\tn{E})^{\fk 2};+}\big)\!=\! 0. \end{equation}
We are ready to construct the rooted tree $\tau_{\mathfrak N^{\fk 3}}$.
\begin{itemize} [leftmargin=*] \item If $\gamma^\star_{(\ud\tn{E})^{\fk 2}}$ does not contain any connected decorated subgraph $\gamma^\star_{(\ud\tn{E})^{\fk 2};+}$ satisfying~(\ref{Eqn:step3_unweighted_core}), then we set $\tau_{\mathfrak N^{\fk 3}}\!=\!\tau_\bullet$ and $o_{\mathfrak N^{\fk 3}}\!=\! o_{\gamma^{\fk 3}}$.
\item If $\gamma^\star_{(\ud\tn{E})^{\fk 2}}$ contains a unique smallest connected decorated subgraph $\gamma^\star_{(\ud\tn{E})^{\fk 2};+}$ satisfying~(\ref{Eqn:step3_unweighted_core}), it is a subgraph of $\gamma^\star_{(\ud\tn{E})^{\fk 2};\tn{cor}}$ by Definition~\ref{Dfn:core}. We contract the edges of $\gamma^\star_{(\ud\tn{E})^{\fk 2};+}$ from $\gamma^\star_{(\ud\tn{E})^{\fk 2}}$ into one vertex $o_\tn{ctr}$, then dissolve the vertices $v$ with $\mathbf{w}(v)\!>\!0$ and take $\tau_{\mathfrak N^{\fk 3}}$ to be the unique connected component that contains $o_\tn{ctr}$. In this way, we see that $\tau_{\mathfrak N^{\fk 3}}$ is a rooted tree with the root $o_{\mathfrak N^{\fk 3}}\!=\! o_\tn{ctr}$. \end{itemize} Notice that $\tau_{\mathfrak N^{\fk 3}}$ is obtained via contraction of the edges of $$
\tn{Edg}\big(\gamma^\star_{(\ud\tn{E})^{\fk 2};+}\big)
\subset
\tn{Edg}\big(\gamma^\star_{(\ud\tn{E})^{\fk 2};\tn{cor}}\big)
=
\tn{Edg}\big((\gamma^{\fk 3})^\star_{\tn{cor}}\big) $$ (the last expression follows from~(\ref{Eqn:ga_E1E2})), vertex dissolution, and taking a connected component of a graph, thus \begin{equation}\label{Eqn:step3_edge}
\tn{Edg}(\tau_{\mathfrak N^{\fk 3}})\subset
\tn{Edg}\big(\gamma_{(\ud\tn{E})^{\fk 2}}\big)=
\tn{Edg}(\gamma^{\fk 3})\backslash
\{\,\mathsf e_i^k:\,
k\!=\! \fk{1,2};\;
i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k}\}\subset
\tn{Edg}(\gamma^{\fk 3}), \end{equation} and moreover, $\tn{Edg}(\tau_{\mathfrak N^{\fk 3}})$ is the set of the edges of a {\it connected} subgraph $\tau'_{\mathfrak N^{\fk 3}}$ of $\gamma^{\fk 3}$ that satisfies $$
o_{{\mathfrak N^{\fk 3}}}
\subset o_{\gamma^{\fk 3}}\in\tn{Ver}(\tau'_{\mathfrak N^{\fk 3}}) $$ We set $$
\Lambda^{\fk 3}=
\big(\,
(\tau_{\mathfrak N^{\fk 3}},\,
\tn{E}_{\mathfrak N^{\fk 3}}\!:=\!\tn{Edg}(\tau_{\mathfrak N^{\fk 3}}),\,
\textnormal{Id})\,
\big)_{
\gamma^{\fk 3}\in\Gamma^{\fk 3}_\tn{vic},\;
\mathfrak N^{\fk 3}\in\pi_0(\mathfrak M^{\fk 3}_{\gamma^{\fk 3}})
}. $$
The following lemma is the analogue of Lemma~\ref{Lm:G1-level} in this step.
\begin{lmm}\label{Lm:G3-level} The set $\Lambda^{\fk 3}$ gives a treelike structure on $(\mathfrak M^{\fk 3},\Gamma^{\fk 3}_\tn{vic})$ as in Definition~\ref{Dfn:Treelike_structure}. \end{lmm}
Lemmas~\ref{Lm:G3-admissible} and~\ref{Lm:G3-level}, along with Theorem~\ref{Thm:tf_smooth}, give rise to the following statement.
\begin{crl}\label{Crl:G3-tf} Let $(\mathfrak M^{\fk 3})^\tn{tf}_{\Lambda^{\fk 3}}$ be constructed as in Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p1}. Then, $(\mathfrak M^{\fk 3})^\tn{tf}_{\Lambda^{\fk 3}}$ is a smooth algebraic stack and the composite forgetful morphism $(\mathfrak M^{\fk 3})^\tn{tf}_{\Lambda^{\fk 3}}\!\longrightarrow\!\mathfrak M^{\fk 1}\!=\!\mathfrak P_2$ is proper and birational. \end{crl}
\subsection{The fourth step of the recursive construction}\label{Subsec:Step4} In this step, we take $$
\mathfrak M^{\fk 4}=(\mathfrak M^{\fk 3})^\tn{tf}_{\Lambda^{\fk 3}},\qquad
\Gamma^{\fk 4}_\tn{vic}=(\Gamma^{\fk 3}_\tn{vic})^{\tn{der}}_{\Lambda^{\fk 3}}. $$ Similar to Lemma~\ref{Lm:G2-admissible}, we have the following statement.
\begin{lmm}\label{Lm:G4-admissible} The stack $\mathfrak M^{\fk 4}$ comes equipped with a $\Gamma^{\fk 4}_\tn{vic}$-stratification, which is the derived stratification of $\mathfrak M^{\fk 4}$ with respect to $(\mathfrak M^{\fk 3},\Gamma^{\fk 3}_\tn{vic},\Lambda^{\fk 3})$. \end{lmm}
The graphs of $\Gamma^{\fk 4}_\tn{vic}$ are in the form \begin{equation}\label{Eqn:ga^4}
\gamma^{\fk 4}
:=\rho_{\gamma^{\fk 3},[{\mathfrak t}^{\fk 3}]}
\in\Gamma^{\fk 4}_\tn{vic},
\qquad
\textnormal{where}\ \
\gamma^{\fk 3}\!\in\!\Gamma^{\fk 3}_\tn{vic},\ \
[{\mathfrak t}^{\fk 3}]\!\in\!\ov\Lambda^{\fk 3}_{\mathfrak N^{\fk 3}},\ \
\mathfrak N^{\fk 3}
\!\in\!\pi_0\big(\mathfrak M^{\fk 3}_{\gamma^{\fk 3}}\big); \end{equation} see~(\ref{Eqn:ga^3}) for $\gamma^{\fk 3}$ and~(\ref{Eqn:N^3}) for $\mathfrak N^{\fk 3}$. Let $o_{\gamma^{\fk 4}}$ be the image of $o_{\gamma^{\fk 3}}$ in $\gamma^{\fk 4}$, $(\gamma^{\fk 4})^\star$ be the induced decoration, and $$
\gamma^{\fk 4}_{\vi 3}\in\mathbf{G} $$ be the graph obtained from $\gamma^{\fk 4}$ by identifying the vertices: $$
o_{\gamma^{\fk 4}}\quad\textnormal{and}\quad
\mathsf v_i^k,\ \ i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k},\ k\!=\!\fk{1,2,3}; $$ the new vertex is denoted by $o_{\gamma^{\fk 4}_{\vi 3}}$. The decoration $(\gamma^{\fk 4})^\star$ further induces the decoration $(\gamma^{\fk 4}_{\vi 3})^\star$ of $\gamma^{\fk 4}_{\vi 3}$ as in~(\ref{Eqn:vic_decoration}). Then, \begin{equation*}
\tn{Edg}(\gamma^{\fk 4})\!=\!\tn{Edg}(\gamma^{\fk 4}_{\vi 3});\quad
\tn{Edg}((\gamma^{\fk 4})^\star_\tn{cor})\!\subset\!\tn{Edg}\big((\gamma^{\fk 4}_{\vi 3})^\star_\tn{cor}\big);\quad
(\gamma^{\fk 4}_{(E)})_{\vi 3}\!=\!(\gamma^{\fk 4}_{\vi 3})_{(E)}\
\forall\,E\!\subset\!\tn{Edg}(\gamma^{\fk 4}). \end{equation*} We take $$
\Gamma^{\fk 4}_{\vi 3}=
\{\,
\gamma^{\fk 4}_{\vi 3}:\;
\gamma^{\fk 4}\!\in\!\Gamma^{\fk 4}_\tn{vic}\,\}. $$ The following statement follows from Lemma~\ref{Lm:G4-admissible} and the properties of $\tn{Edg}(\gamma^{\fk 4}_{\vi 3})$ above. \begin{crl}\label{Crl:G4-admissible} The $\Gamma^{\fk 4}_\tn{vic}$-stratification of $\mathfrak M^{\fk 4}$ in Lemma~\ref{Lm:G4-admissible} determines a $\Gamma^{\fk 4}_{\vi 3}$-stratification: $$
\mathfrak M^{\fk 4}=
\bigsqcup_{\gamma'\in\Gamma^{\fk 4}_{\vi 3}}\!\!
\mathfrak M^{\fk 4}_{\vi 3;\gamma'},
\qquad
\textnormal{where}\quad
\mathfrak M^{\fk 4}_{\vi 3;\gamma'}=
\bigsqcup_{\gamma^{\fk 4}\in\Gamma^{\fk 4}_{\tn{vic}}\,,\,
\gamma^{\fk 4}_{\vi 3}=\gamma'}\!\!\!\!\!\!\!\!\!\!
\mathfrak M^{\fk 4}_{\gamma^{\fk 4}}. $$ \end{crl}
By Corollary~\ref{Crl:Derived_graphs}, for every $\gamma^{\fk 4}\!\in\!\Gamma^{\fk 4}_\tn{vic}$ and the corresponding $\gamma^{\fk 4}_{\vi 3}$, the connected components of $\mathfrak M^{\fk 4}_{\gamma^{\fk 4}}$ and $\mathfrak M^{\fk 4}_{\vi 3;\gamma^{\fk 4}_{\vi 3}}$ are in the form \begin{equation}\label{Eqn:N^4}
\mathfrak N^{\fk 4}:=
(\mathfrak N^{\fk 3})_{[{\mathfrak t}^{\fk 3}]}^\tn{tf}\
\in\pi_0\big(\mathfrak M^{\fk 4}_{\gamma^{\fk 4}}\big)
\cap \pi_0\big(\mathfrak M^{\fk 4}_{\vi 3;\gamma^{\fk 4}_{\vi 3}}\big). \end{equation} For each $\mathfrak N^{\fk 4}$, we will describe the tuple $(\tau_{\mathfrak N^{\fk 4}},\tn{E}_{\mathfrak N^{\fk 4}},\beta_{\mathfrak N^{\fk 4}})$ to provide~$\Lambda^{\fk 4}$ on $(\mathfrak M^{\fk 4},\Gamma^{\fk 4}_{\vi 3})$.
Let \begin{equation}\begin{split}\label{Eqn:Edges_ctr_in_Step_3}
\tn{E}^{\fk 3}
&:=
\mathfrak E^\bot_{\ge{\mathbf{m}}}({\mathfrak t}^{\fk 3})
\subset
\tn{Edg}(\tau_{\mathfrak N^{\fk 3}})
=\tn{E}_{\mathfrak N^{\fk 3}}
\\
&\subset
\tn{Edg}(\gamma_{(\ud\tn{E})^{\fk 2}})
=
\tn{Edg}(\gamma^{\fk 3})\backslash\{\,\mathsf e_i^k:\,i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k},\,k\!=\!\fk 1,\fk 2\,\}
\subset
\tn{Edg}(\gamma^{\fk 3}), \end{split}\end{equation} which follows from~(\ref{Eqn:step3_edge}). Consider the graph and its induced decoration $$
\gamma_{(\ud\tn{E})^{\fk 3}}:=
(\gamma_{(\ud\tn{E})^{\fk 2}})_{(\tn{E}^{\fk 3})}
=\gamma_{(\tn{E}^{\fk 1}\sqcup\tn{E}^{\fk 2}\sqcup\tn{E}^{\fk 3})}\qquad
\textnormal{and}\qquad
\gamma_{(\ud\tn{E})^{\fk 3}}^\star, $$ respectively. Analogous to~(\ref{Eqn:ga_star_(E)}) and~(\ref{Eqn:ga_E1E2}), we have \begin{equation}\begin{split} \label{Eqn:ga_E1E2E3}
&\tn{Edg}(\gamma_{(\ud\tn{E})^{\fk 3}})\!\subset\!
\tn{Edg}(\gamma^{\fk 4})\!=\! \tn{Edg}(\gamma^{\fk 4}_{\vi 3}),\qquad
\tn{Edg}\big(\gamma^\star_{(\ud\tn{E})^{\fk 3};\tn{cor}}\big)
\!=\!
\tn{Edg}\big((\gamma^{\fk 4})^\star_\tn{cor}\big)
\!\subset\!\tn{Edg}\big((\gamma^{\fk 4}_{\vi 3})^\star_\tn{cor}\big),\\
&
\mathbf{w}\big(\gamma^\star_{(\ud\tn{E})^{\fk 3};\tn{cor}}\big)\!=\!
\mathbf{w}(o_{\gamma^{\fk 4}})
\!=\!
\mathbf{w}(o_{\gamma^{\fk 4}_{\vi 3}})\!\ge\!1. \end{split}\end{equation} The last equality above holds because $\mathbf{w}(\mathsf v_i^k)\!=\! 0$ for all $i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k}$ and all $k\!=\!\fk{1,2,3}$; see Definition~\ref{Dfn:Derived_decoration}.
We are ready to construct the rooted tree $\tau_{\mathfrak N^{\fk 4}}$.
\begin{itemize} [leftmargin=*] \item If $\mathbf{w}(\gamma^\star_{(\ud\tn{E})^{\fk 3};\tn{cor}})\!>\!1$, then we set $\tau_{\mathfrak N^{\fk 4}}\!=\!\tau_\bullet$ and $o_{\mathfrak N^{\fk 4}}\!=\! o_{\gamma^{\fk 4}_{\vi 3}}$.
\item If $\mathbf{w}(\gamma^\star_{(\ud\tn{E})^{\fk 3};\tn{cor}})\!=\!1$, we contract the edges of $\gamma^\star_{(\ud\tn{E})^{\fk 3};\tn{cor}}$ from $\gamma^\star_{(\ud\tn{E})^{\fk 3}}$ into one vertex $o_\tn{ctr}$, then dissolve the vertices $v\!\ne\!o_\tn{ctr}$ with $\mathbf{w}(v)\!>\!0$ and take $\tau_{\mathfrak N^{\fk 4}}$ to be the unique connected component that contains $o_\tn{ctr}$. In this way, we see that $\tau_{\mathfrak N^{\fk 4}}$ is a rooted tree with the root $o_{\mathfrak N^{\fk 4}}\!=\! o_\tn{ctr}$. \end{itemize} Notice that $\tau_{\mathfrak N^{\fk 4}}$ is obtained via contraction of the edges of $$
\tn{Edg}\big(\gamma^\star_{(\ud\tn{E})^{\fk 3};\tn{cor}}\big)
\!=\!
\tn{Edg}\big((\gamma^{\fk 4})^\star_\tn{cor}\big)
\!\subset\!\tn{Edg}\big((\gamma^{\fk 4}_{\vi 3})^\star_\tn{cor}\big) $$ (by~(\ref{Eqn:ga_E1E2E3})), vertex dissolution, and taking a connected component of a graph, thus \begin{equation}\label{Eqn:step4_edge}
\tn{Edg}(\tau_{\mathfrak N^{\fk 4}})\subset
\tn{Edg}\big(\gamma_{(\ud\tn{E})^{\fk 3}}\big)=
\tn{Edg}(\gamma^{\fk 4})\backslash
\{\mathsf e_i^k:
k\!=\! \fk{1,2,3};\,
i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k}\}\subset
\tn{Edg}(\gamma^{\fk 4})=
\tn{Edg}(\gamma^{\fk 4}_{\vi 3}), \end{equation} and moreover, $\tn{Edg}(\tau_{\mathfrak N^{\fk 4}})$ is the set of the edges of a {\it connected} subgraph $\tau'_{\mathfrak N^{\fk 4}}$ of $\gamma^{\fk 4}_{\vi 3}$ that satisfies $$
o_{{\mathfrak N^{\fk 4}}}
\subset o_{\gamma^{\fk 4}_{\vi 3}}\in\tn{Ver}(\tau'_{\mathfrak N^{\fk 4}}). $$ We set \begin{equation*}\begin{split}
\Lambda^{\fk 4}
&=
\big(\,
(\tau_{\mathfrak N^{\fk 4}},\,
\tn{E}_{\mathfrak N^{\fk 4}}\!:=\!\tn{Edg}(\tau_{\mathfrak N^{\fk 4}}),\,
\textnormal{Id})\,
\big)_{
\gamma^{\fk 4}_{\vi 3}\in\Gamma^{\fk 4}_{\vi 3},\;
\mathfrak N^{\fk 4}\in\pi_0(\mathfrak M^{\fk 4}_{\vi 3;\gamma^{\fk 4}_{\vi 3}})
}. \end{split}\end{equation*}
The following lemma is the analogue of Lemma~\ref{Lm:G1-level} in this step. Notice that the core of $\gamma^\star_{(\ud\tn{E})^{\fk 3}}$ is postively weighted in the current situation, which seems to be different from Lemma~\ref{Lm:G1-level} (or Example~\ref{Eg:genus_1}). However, by Definition~\ref{Dfn:Induced_decoration_edge_contr}, if we contract an edge of $\tau_{\mathfrak N^{\fk 4}}$ that is both maximal and minimal, then the weight of the new vertex is greater than one, hence the corresponding rooted tree given by $\Lambda^{\fk 4}$ is $\tau_\bullet$, which is exactly the same as in the proof of Lemma~\ref{Lm:G1-level}.
\begin{lmm}\label{Lm:G4-level} The set $\Lambda^{\fk 4}$ gives a treelike structure on $(\mathfrak M^{\fk 4},\Gamma^{\fk 4}_{\vi 3})$ as in Definition~\ref{Dfn:Treelike_structure}. \end{lmm}
Lemmas~\ref{Lm:G4-admissible} and~\ref{Lm:G4-level}, along with Theorem~\ref{Thm:tf_smooth}, give rise to the following statement.
\begin{crl}\label{Crl:G4-tf} Let $(\mathfrak M^{\fk 4})^\tn{tf}_{\Lambda^{\fk 4}}$ be constructed as in Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p1}. Then, $(\mathfrak M^{\fk 4})^\tn{tf}_{\Lambda^{\fk 4}}$ is a smooth algebraic stack and the composite forgetful morphism $(\mathfrak M^{\fk 4})^\tn{tf}_{\Lambda^{\fk 4}}\!\longrightarrow\!\mathfrak M^{\fk 1}\!=\!\mathfrak P_2$ is proper and birational. \end{crl}
\iffalse For future use, for every $\gamma^{\fk 4} \!\in\!\Gamma^{\fk 4}_{\vi c},$ $\mathfrak N^{\fk 4} \!\in\!\pi_0\big(\mathfrak M^{\fk 4}_{\gamma^{\fk 4}}\big),$ and $[{\mathfrak t}^{\fk 4}]\!\in\!\ov\Lambda^{\fk 4}_{\mathfrak N^{\fk 4}}$ satisfying ${\mathbf{m}}({\mathfrak t}^{\fk 1})\!<\!0$, we set \begin{equation} \label{Eqn:hdom} \hdom{1}{{\mathfrak t}^{\fk 4}}:= \min \big(\, \big\{i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^{\fk 4}}:\,
|\{ e\!\in\!\mathfrak E_i({\mathfrak t}^{\fk 4})\!\cap\!(\mathfrak E^\bot_{{\mathbf{m}}({\mathfrak t}^{\fk 4});\min})^{\succeq}: e\!\not\preceq\!e'~\forall\, e'\!\ni\!\mathsf v_{{\mathbf{m}}({\mathfrak t}^{\fk 1})}^{\fk 1}
\}|\!\le\!1 \big\} \sqcup\{0\} \,\big). \end{equation} Here we consider $\tn{E}_{\mathfrak N^{\fk 4}}$ as a subset of $\tn{Edg}(\gamma^{\fk 4})$ so that the exceptional vertices $\mathsf v_i^{\fk 1}$ are not identified. \fi
\subsection{The fifth step of the recursive construction}\label{Subsec:Step5}
In this step, we take $$
\mathfrak M^{\fk 5}=(\mathfrak M^{\fk 4})^\tn{tf}_{\Lambda^{\fk 4}},\qquad
\Gamma^{\fk 5}_{\vi 3}=(\Gamma^{\fk 4}_{\vi 3})^{\tn{der}}_{\Lambda^{\fk 4}}. $$ Analogous to Lemma~\ref{Lm:G2-admissible}, we have the following statement.
\begin{lmm}\label{Lm:G5-admissible} The stack $\mathfrak M^{\fk 5}$ comes equipped with a $\Gamma^{\fk 5}_{\vi 3}$-stratification, which is the derived stratification of $\mathfrak M^{\fk 5}$ with respect to $(\mathfrak M^{\fk 4},\Gamma^{\fk 4}_{\vi 3},\Lambda^{\fk 4})$. \end{lmm}
The graphs of $\Gamma^{\fk 5}_{\vi 3}$ are in the form \begin{equation}\label{Eqn:ga^5}
\gamma^{\fk 5}
:=\rho_{\gamma^{\fk 4}_{\vi 3},[{\mathfrak t}^{\fk 4}]}
\in\Gamma^{\fk 5}_{\vi 3},
\qquad
\textnormal{where}\ \
\gamma^{\fk 4}_{\vi 3}\!\in\!\Gamma^{\fk 4}_{\vi 3},\ \
[{\mathfrak t}^{\fk 4}]\!\in\!\ov\Lambda^{\fk 4}_{\mathfrak N^{\fk 4}},\ \
\mathfrak N^{\fk 4}
\!\in\!\pi_0\big(\mathfrak M^{\fk 4}_{\vi 3;\gamma^{\fk 4}_{\vi 3}}\big); \end{equation} see~(\ref{Eqn:ga^4}) for $\gamma^{\fk 4}$ (and the succeeding display for $\gamma^{\fk 4}_{\vi 3}$) and~(\ref{Eqn:N^4}) for $\mathfrak N^{\fk 4}$. Let $o_{\gamma^{\fk 5}}$ be the image of~$o_{\gamma^{\fk 4}_{\vi 3}}$ in $\gamma^{\fk 5}$ and $(\gamma^{\fk 5})^\star$ be the derived decoration. By Corollary~\ref{Crl:Derived_graphs}, for every $\gamma^{\fk 5}$ as in~(\ref{Eqn:ga^5}), the connected components of $\mathfrak M^{\fk 5}_{\gamma^{\fk 5}}$ are in the form \begin{equation}\label{Eqn:N^5}
\mathfrak N^{\fk 5}:=
(\mathfrak N^{\fk 4})_{[{\mathfrak t}^{\fk 4}]}^\tn{tf}\
\in\pi_0\big(\mathfrak M^{\fk 5}_{\gamma^{\fk 5}}\big). \end{equation} For each $\mathfrak N^{\fk 5}$, we will describe the tuple $(\tau_{\mathfrak N^{\fk 5}},\tn{E}_{\mathfrak N^{\fk 5}},\beta_{\mathfrak N^{\fk 5}})$ to provide~$\Lambda^{\fk 5}$ on $(\mathfrak M^{\fk 5},\Gamma^{\fk 5}_{\vi 3})$.
Let \begin{equation}\begin{split}\label{Eqn:Edges_ctr_in_Step_4}
\tn{E}^{\fk 4}
&:=
\mathfrak E^\bot_{\ge{\mathbf{m}}}({\mathfrak t}^{\fk 4})
\subset
\tn{Edg}(\tau_{\mathfrak N^{\fk 4}})
=\tn{E}_{\mathfrak N^{\fk 4}}
\\
&\subset
\tn{Edg}(\gamma_{(\ud\tn{E})^{\fk 3}})
=
\tn{Edg}(\gamma^{\fk 4})\backslash\{\,\mathsf e_i^k:\,i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k},\,k\!=\!\fk{1,2,3}\}
\subset
\tn{Edg}(\gamma^{\fk 4}), \end{split}\end{equation} which follows from~(\ref{Eqn:step4_edge}). With $\gamma\!\in\!\Gamma^{\fk 1}$ and $[{\mathfrak t}^{\fk 1}]\!\in\!\ov\Lambda^{\fk 1}_{\tau_{\mathfrak N^{\fk 1}}}$ as in~\S\ref{Subsec:Step1}, consider the derived graph of $\gamma$ with respect to $[{\mathfrak t}^{\fk 1}]$ and its induced decoration $$
\ti\gamma^{\fk 2}:=
\rho_{\gamma;[{\mathfrak t}^{\fk 1}]}\qquad
\textnormal{and}\qquad
(\ti\gamma^{\fk 2})^\star, $$ respectively. By~(\ref{Eqn:derived_graph}) and~(\ref{Eqn:Edges_ctr_in_Step_1}), we have $$
\tn{Edg}(\gamma_{(\tn{E}^{\fk 1})})\subset
\tn{Edg}(\ti\gamma^{\fk 2}). $$ Thus by~(\ref{Eqn:Edges_ctr_in_Step_2}), (\ref{Eqn:Edges_ctr_in_Step_3}), and~(\ref{Eqn:Edges_ctr_in_Step_4}), \begin{equation}\label{Eqn:ti_ga^2_edges}
\tn{E}^k\subset\tn{Edg}(\ti\gamma^{\fk 2}),\qquad
k\!=\! \fk{2,3,4}. \end{equation} Let $$
\ti\gamma^{\fk 2}_{(\ud\tn{E})^{\fk 4}}:=
\ti\gamma^{\fk 2}_{(\tn{E}^{\fk 2}\sqcup\tn{E}^{\fk 3}\sqcup\tn{E}^{\fk 4})} $$ and $(\ti\gamma^{\fk 2}_{(\ud\tn{E})^{\fk 4}})^\star$ be its induced decoration. Let $$
\wh\tau^{\fk 5}\qquad\textnormal{and}\qquad
(\wh\tau^{\fk 5})^\star $$ be the rooted tree obtained from $\ti\gamma^{\fk 2}_{(\ud\tn{E})^{\fk 4}}$ by contracting the edges of $(\ti\gamma^{\fk 2}_{(\ud\tn{E})^{\fk 4}})^\star_\tn{cor}$ into the root $o_{\wh\tau^{\fk 5}}$ and its induced decoration, respectively. Analogous to~(\ref{Eqn:ga_star_(E)}), (\ref{Eqn:ga_E1E2}), and~(\ref{Eqn:ga_E1E2E3}), we have \begin{equation}\begin{split} \label{Eqn:ga_E1E2E3E4}
&\tn{Edg}(\wh\tau^{\fk 5})\subset
\tn{Edg}(\gamma^{\fk 5}),\qquad
o_{\wh\tau^{\fk 5}}\subset o_{\gamma^{\fk 5}}. \end{split}\end{equation}
We are ready to construct $\tau_{\mathfrak N^{\fk 5}}$. Recall that $\dom{1}{{\mathfrak t}^{\fk 1}}\!\in\!\lrbr{{\mathbf{m}},0}_{{\mathfrak t}^{\fk 1}}$ as in~(\ref{Eqn:dom}) is the lowest level of ${\mathfrak t}^{\fk 1}$ on which there exists a unique vertex $v\!\in\!\tn{Ver}({\mathfrak t}^{\fk 1})$ satisfying $v\!\succeq\!v'$ for all level ${\mathbf{m}}$ minimal vertices~$v'$ of ${\mathfrak t}^{\fk 1}$. \begin{itemize}[leftmargin=*] \item If $\dom{1}{{\mathfrak t}^{\fk 1}}\!=\!0$, or $\dom{1}{{\mathfrak t}^{\fk 1}}\!<\!0$ but $\mathbf{w}\big((\ti\gamma^{\fk 2})^\star_\tn{cor}\big)\!<\! \mathbf{w}(o_{\wh\tau^{\fk 5}})$,
we set $\tau_{\mathfrak N^{\fk 5}}\!=\!\tau_\bullet$ and $o_{\mathfrak N^{\fk 5}}\!=\! o_{\gamma^{\fk 5}}$.
\item If $\dom{1}{{\mathfrak t}^{\fk 1}}\!<\!0$ and $\mathbf{w}\big((\ti\gamma^{\fk 2})^\star_\tn{cor}\big)\!=\! \mathbf{w}(o_{\wh\tau^{\fk 5}})$, we dissolve $\mathsf v^{\fk 1}_{\dom{1}{{\mathfrak t}^{\fk 1}}}$ and all the vertices $v\!\ne\!o_{\wh\tau^{\fk 5}}$ with $\mathbf{w}(v)\!>\!0$ in $\wh\tau^{\fk 5}$ and take $\tau_{\mathfrak N^{\fk 5}}$ to be the unique connected component that contains $o_{\wh\tau^{\fk 5}}$. Thus, $\tau_{\mathfrak N^{\fk 5}}$ is a rooted tree with $o_{\mathfrak N^{\fk 5}} \!=\! o_{\wh\tau^{\fk 5}}$. \end{itemize} Notice that the above construction of $\tau_{\mathfrak N^{\fk 5}}$ is independent of the choice of ${\mathfrak t}^{\fk 1}$ representing $[{\mathfrak t}^{\fk 1}]$. By~(\ref{Eqn:ga_E1E2E3E4}), we have \begin{equation}\label{Eqn:step5_edge}
\tn{Edg}(\tau_{\mathfrak N^{\fk 5}})\subset
\tn{Edg}(\wh\tau^{\fk 5})\subset
\tn{Edg}(\gamma^{\fk 5})\backslash
\{\mathsf e_i^k:
k\!=\! \fk{2,3,4};\,
i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k}\}\subset
\tn{Edg}(\gamma^{\fk 5}), \end{equation} and moreover, $\tn{Edg}(\tau_{\mathfrak N^{\fk 5}})$ is the set of the edges of a {\it connected} subgraph $\tau'_{\mathfrak N^{\fk 5}}$ of $\gamma^{\fk 5}$ that satisfies $$
o_{{\mathfrak N^{\fk 5}}}
\subset o_{\gamma^{\fk 5}}\in\tn{Ver}(\tau'_{\mathfrak N^{\fk 5}}). $$ We set $$
\Lambda^{\fk 5}=
\big(\,
(\tau_{\mathfrak N^{\fk 5}},\,
\tn{E}_{\mathfrak N^{\fk 5}}\!:=\!\tn{Edg}(\tau_{\mathfrak N^{\fk 5}}),\,
\textnormal{Id})\,
\big)_{
\gamma^{\fk 5}\in\Gamma^{\fk 5}_{\vi 3},\;
\mathfrak N^{\fk 5}\in\pi_0(\mathfrak M^{\fk 5}_{\gamma^{\fk 5}})
}. $$
\begin{lmm}\label{Lm:G5-level} The set $\Lambda^{\fk 5}$ gives a treelike structure on $(\mathfrak M^{\fk 5},\Gamma^{\fk 5}_{\vi 3})$ as in Definition~\ref{Dfn:Treelike_structure}. \end{lmm}
\begin{proof} The proof of Lemma~\ref{Lm:G5-level} is still parallel to that of Lemma~\ref{Lm:G1-level}. The only part that needs attention is the minimal vertex $\mathsf v^{\fk 1}_{\dom{1}{{\mathfrak t}^{\fk 1}}}$ of $\tau_{\mathfrak N^{\fk 5}}$ is not positively weighted whenever $\dom{1}{{\mathfrak t}^{\fk 1}}\!<\!0$; see Definition~\ref{Dfn:Derived_decoration}. Nonetheless, if we contract the minimal edge $\mathsf e^{\fk 1}_{\dom{1}{{\mathfrak t}^{\fk 1}}}$ of $\tau_{\mathfrak N^{\fk 5}}$, it is a direct check that there exists a unique list \begin{equation}\label{Eqn:list_1-5}
\ti\gamma\!\in\!\Gamma^{\fk 1}_\tn{vic},\ \
\ti\mathfrak N^{\fk 1}\!\in\!\pi_0(\mathfrak M^{\fk 1}_{\ti\gamma}),\ \
[\ti{\mathfrak t}^{\fk 1}]\!\in\!\ov\Lambda_{\ti\mathfrak N^{\fk 1}},\ \ \ldots,\
\ti\gamma^{\fk 5}\!\in\!\Gamma^{\fk 5}_{\vi 3},\ \
\ti\mathfrak N^{\fk 5}\!\in\!\pi_0(\mathfrak M^{\fk 5}_{\ti\gamma^{\fk 5}}) \end{equation} such that $\ti\gamma^{\fk 5}\!=\!\gamma^{\fk 5}_{(\mathsf e^{\fk 1}_{\dom{1}{{\mathfrak t}^{\fk 1}}})}$, $\textnormal{Cl}_{\mathfrak M^{\fk 5}}{\ti\mathfrak N^{\fk 5}}\!\cap\!\mathfrak N^{\fk 5}\!\ne\!\emptyset$, and $\dom{1}{\ti{\mathfrak t}^{\fk 1}}\!=\!(\dom{1}{{\mathfrak t}^{\fk 1}})^\sharp$; see~(\ref{Eqn:sharp_flat}) for notation. Thus, $\mathsf v^{\fk 1}_{(\dom{1}{{\mathfrak t}^{\fk 1}})^\sharp}$ is a minimal vertex of $\tau_{\ti\mathfrak N^{\fk 5}}$ in this case. Similarly, if we contract a non-exceptional minimal edge $e$ of $\tau_{\mathfrak N^{\fk 5}}$ that is directly attached to an exceptional vertex $\mathsf v^{\fk 1}_i$, $i\!\in\!\lbrp{\dom{1}{{\mathfrak t}^{\fk 1}},0}$, then it is a direct check that there exists a unique list as in~(\ref{Eqn:list_1-5}) such that
$\ti\gamma^{\fk 5}\!=\!\gamma^{\fk 5}_{(e)}$, $\textnormal{Cl}_{\mathfrak M^{\fk 5}}{\ti\mathfrak N^{\fk 5}}\!\cap\!\mathfrak N^{\fk 5}\!\ne\!\emptyset$, and $\dom{1}{\ti{\mathfrak t}^{\fk 1}}\!=\! i$. Thus, $\mathsf v^{\fk 1}_i$ is a minimal vertex of $\tau_{\ti\mathfrak N^{\fk 5}}$ in this case. The verification of the rest of the conditions in Definition~\ref{Dfn:Treelike_structure} is analogous to Lemma~\ref{Lm:G1-level} and Example~\ref{Eg:genus_1}, hence is omitted. \end{proof}
Lemmas~\ref{Lm:G5-admissible} and~\ref{Lm:G5-level}, along with Theorem~\ref{Thm:tf_smooth}, give rise to the following statement.
\begin{crl}\label{Crl:G5-tf} Let $(\mathfrak M^{\fk 5})^\tn{tf}_{\Lambda^{\fk 5}}$ be constructed as in Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p1}. Then, $(\mathfrak M^{\fk 5})^\tn{tf}_{\Lambda^{\fk 5}}$ is a smooth algebraic stack and the composite forgetful morphism $(\mathfrak M^{\fk 5})^\tn{tf}_{\Lambda^{\fk 5}}\!\longrightarrow\!\mathfrak M^{\fk 1}\!=\!\mathfrak P_2$ is proper and birational. \end{crl}
\subsection{The sixth step of the recursive construction}\label{Subsec:Step6}
In this step, we take $$
\mathfrak M^{\fk 6}=(\mathfrak M^{\fk 5})^\tn{tf}_{\Lambda^{\fk 5}},\qquad
\Gamma^{\fk 6}_{\vi 3}=(\Gamma^{\fk 5}_{\vi 3})^{\tn{der}}_{\Lambda^{\fk 5}}. $$ Similar to Lemma~\ref{Lm:G2-admissible}, we have the following statement.
\begin{lmm}\label{Lm:G6-admissible} The stack $\mathfrak M^{\fk 6}$ comes equipped with a $\Gamma^{\fk 6}_{\vi 3}$-stratification that is the derived stratification of $\mathfrak M^{\fk 6}$ with respect to $(\mathfrak M^{\fk 5},\Gamma^{\fk 5}_{\vi 3},\Lambda^{\fk 5})$. \end{lmm}
Hereafter, the recursive construction becomes a bit different, which is due to the expressions of the structural homomorphism in~\cite[\S2]{HLN} that are related to the conjugate and Weierstrass points. In Step $\fk{6}$, the strategy is as follows. \begin{enumerate} [leftmargin=*,label=(\arabic*)] \item We first describe a substack $K$ of the boundary $\Delta^{\fk 6}$ of $\mathfrak M^{\fk 6}$ that is related to the conjugate and Weierstrass points and satisfies the first assumption of Corollary~\ref{Crl:extra_edge_graphs}. \item We then construct a treelike structure $\Lambda^{\fk 6}$ satisfying the second assumption and~(\ref{Eqn:K_local}) of Corollary~\ref{Crl:extra_edge_graphs}. \item Finally, we apply Corollary~\ref{Crl:extra_edge_graphs} to obtain the grafted stratification $\Gamma^{\fk 6}_\tn{gft}$ with respect to $K$ and the induced treelike structure $\Lambda^{\fk 6}_\tn{gft}$. \end{enumerate}
We start with the substack $K$. The graphs of $\Gamma^{\fk 6}_{\vi 3}$ are in the form \begin{equation}\label{Eqn:ga^6}
\gamma^{\fk 6}
:=\rho_{\gamma^{\fk 5},[{\mathfrak t}^{\fk 5}]}
\in\Gamma^{\fk 6}_{\vi 3},
\qquad
\textnormal{where}\ \
\gamma^{\fk 5}\!\in\!\Gamma^{\fk 5}_{\vi 3},\ \
[{\mathfrak t}^{\fk 5}]\!\in\!\ov\Lambda^{\fk 5}_{\mathfrak N^{\fk 5}},\ \
\mathfrak N^{\fk 5}
\!\in\!\pi_0\big(\mathfrak M^{\fk 5}_{\gamma^{\fk 5}}\big); \end{equation} see~(\ref{Eqn:ga^5}) for $\gamma^{\fk 5}$ and~(\ref{Eqn:N^5}) for $\mathfrak N^{\fk 5}$. The vertex that is the image of $o_{\gamma^{\fk 5}}$ in $\gamma^{\fk 6}$ is denoted by~$o_{\gamma^{\fk 6}}$.
The decoration $(\gamma^{\fk 5})^\star$ of $\gamma^{\fk 5}$ induces the decoration $(\gamma^{\fk 6})^\star$ of $\gamma^{\fk 6}$. Notice that $\mathbf{w}\big((\gamma^{\fk 6})^\star_\tn{cor}\big)\!\ge\!2$.
By Corollary~\ref{Crl:Derived_graphs}, for every $\gamma^{\fk 6}\!\in\!\Gamma^{\fk 6}_{\vi 3}$, the connected components of $\mathfrak M^{\fk 6}_{\gamma^{\fk 6}}$ are in the form \begin{equation}\label{Eqn:N^6}
\mathfrak N^{\fk 6}:=
(\mathfrak N^{\fk 5})_{[{\mathfrak t}^{\fk 5}]}^\tn{tf}\
\in\pi_0\big(\mathfrak M^{\fk 6}_{\gamma^{\fk 6}}\big). \end{equation} Recall that $\dom{j}{{\mathfrak t}^k}$ denotes the lowest level of ${\mathfrak t}^k$ on which there are at most $j$ vertices that are contained in the paths from the root to the minimal vertices on the ${\mathbf{m}}({\mathfrak t}^k)$-th level; see~(\ref{Eqn:dom}).
Consider the following sets of the connected components of the strata of $\mathfrak M^{\fk 6}$: \begin{equation*}\begin{split}
N_{a;\fk 1}^{\fk 6}
\!=\!
\big\{
\mathfrak N^{\fk 6}
\!\in\!\pi_0\big(\mathfrak M^{\fk 6}_{\gamma^{\fk 6}}\big):\;
&\gamma^{\fk 6}\!\in\!\Gamma^{\fk 6}_{\vi 3};\;
\dom{2}{{\mathfrak t}^{\fk 1}}\!<\!\dom{1}{{\mathfrak t}^{\fk 1}}\!=\! 0;\;
\mathbf{w}\big((\gamma^{\fk 2})^\star_\tn{cor}\big)\!=\!
\mathbf{w}\big((\gamma^{\fk 6})^\star_\tn{cor}\big)
\big\},
\\
N_{a;\fk 2}^{\fk 6}
\!=\!
\big\{
\mathfrak N^{\fk 6}
\!\in\!\pi_0\big(\mathfrak M^{\fk 6}_{\gamma^{\fk 6}}\big):\;
&\gamma^{\fk 6}\!\in\!\Gamma^{\fk 6}_{\vi 3};\;
\dom{1}{{\mathfrak t}^{\fk 1}},
\dom{1}{{\mathfrak t}^{\fk 2}}\!<\!0;\;
\mathbf{w}\big((\gamma^{2})^\star_\tn{cor}\big)\!<\!
\mathbf{w}\big((\gamma^{3})^\star_\tn{cor}\big)\!=\!
\mathbf{w}\big((\gamma^{\fk 6})^\star_\tn{cor}\big)
\big\},
\\
N_{a;\fk 3}^{\fk 6}
\!=\!
\big\{
\mathfrak N^{\fk 6}
\!\in\!\pi_0\big(\mathfrak M^{\fk 6}_{\gamma^{\fk 6}}\big):\;
&\gamma^{\fk 6}\!\in\!\Gamma^{\fk 6}_{\vi 3};\;
\dom{1}{{\mathfrak t}^{\fk 1}},
\dom{1}{{\mathfrak t}^{\fk 3}}\!<\!0;\\
&
\mathbf{w}\big((\gamma^{2})^\star_\tn{cor}\big)\!=\!
\mathbf{w}\big((\gamma^{3})^\star_\tn{cor}\big)\!<\!
\mathbf{w}\big((\gamma^{4})^\star_\tn{cor}\big)\!=\!
\mathbf{w}\big((\gamma^{\fk 6})^\star_\tn{cor}\big)
\big\},
\\
N_b^{\fk 6}
\!=\!
\big\{
\mathfrak N^{\fk 6}
\!\in\!\pi_0\big(\mathfrak M^{\fk 6}_{\gamma^{\fk 6}}\big):\;
&\gamma^{\fk 6}\!\in\!\Gamma^{\fk 6}_{\vi 3};\;
\dom{1}{{\mathfrak t}^{\fk 5}}\!<\! 0
\big\},\\
N_c^{\fk 6}
\!=\!
\big\{
\mathfrak N^{\fk 6}
\!\in\!\pi_0\big(\mathfrak M^{\fk 6}_{\gamma^{\fk 6}}\big):\;
&\gamma^{\fk 6}\!\in\!\Gamma^{\fk 6}_{\vi 3};\;
\mathbf{w}\big((\gamma^{\fk 6})^\star_\tn{cor}\big)\!=\! 2;\;
\gamma_\tn{vic}\!\ne\!\tau_\bullet
\big\},
\\
N_d^{\fk 6}
\!=\!
\big\{
\mathfrak N^{\fk 6}
\!\in\!\pi_0\big(\mathfrak M^{\fk 6}_{\gamma^{\fk 6}}\big):\;
&\gamma^{\fk 6}\!\in\!\Gamma^{\fk 6}_{\vi 3};\;
\dom{1}{{\mathfrak t}^{\fk 4}}\!<\! 0;\;
\mathbf{w}\big((\gamma^{\fk 6})^\star_\tn{cor}\big)\!=\!\mathbf{w}\big((\gamma^{\fk 5})^\star_\tn{cor}\big)
\big\}. \end{split}\end{equation*} We take \begin{gather*}
N^\dag
=
\bigcup_{\fk 1\le s\le \fk 3}\!\!\!N_{a;s}^{\fk 6}\cup N_b^{\fk 6}\cup N_c^{\fk 6}\cup N_d^{\fk 6}\,;
\qquad
\mathfrak M^\dag=\!
\bigsqcup_{\mathfrak N^{\fk 6}\in N^\dag}\!\!
\mathfrak N^{\fk 6}\ \subset\mathfrak M^{\fk 6}\,. \end{gather*} The condition $\gamma_\tn{vic}\!\ne\!\tau_\bullet$ in $N_c^{\fk 6}$ excludes the stratum consisting of smooth genus two curves and degree 2 line bundles, which is a connected component of $\mathfrak M^{\fk 6}_{\tau_\bullet}$. Thus, $\mathfrak M^\dag$ is a substack of the boundary $\Delta^{\fk 6}$ of $\mathfrak M^{\fk 6}$. In addition, we have the following:
\begin{lmm} \label{Lm:N_dag} The substack $\mathfrak M^\dag$ is closed. \end{lmm}
\begin{proof} Fix $\mathfrak N^{\fk 6}\!\in\! N^\dag$. By Lemma~\ref{Lm:M_strata_closure}, for every $\ti\gamma^{\fk 6}\!\in\!\Gamma^{\fk 6}_{\vi 3}$ and $\ti\mathfrak N^{\fk 6}\!\in\!\pi_0(\mathfrak M^{\fk 6}_{\ti\gamma^{\fk 6}})$, if $\textnormal{Cl}_{\mathfrak M^{\fk 6}}\mathfrak N^{\fk 6}\!\cap\!\ti\mathfrak N^{\fk 6}\!\ne\!\emptyset$, then $\ti\gamma^{\fk 6}\!\preceq\!\gamma^{\fk 6}$. Here the notation $\ti{\ }$ is to distinguish $\ti\gamma^{\fk 6}$ from $\gamma^{\fk 6}$. Analyzing the possible graphs $\ti\gamma^{\fk 6}$ with $\ti\gamma^{\fk 6}\!\preceq\!\gamma^{\fk 6}$ and keeping track of the weights $\mathbf{w}\big((\ti\gamma^{s})^\star_\tn{cor}\big)$, $\fk 2\!\le\!s\!\le\!\fk 6$, we observe that \begin{equation}\label{Eqn:N6_closure}
\textnormal{Cl}_{\mathfrak M^{\fk 6}}\mathfrak N^{\fk 6}\subset
\begin{cases}
\bigsqcup_{\,\mathfrak N'\in N^{\fk 6}_{a;\fk 1}\cup
N^{\fk 6}_{a;\fk 2}\cup
N^{\fk 6}_{a;\fk 3}\cup
N^{\fk 6}_{b}\cup
N^{\fk 6}_{d}}
\mathfrak N'\quad
&
\textnormal{if}~\mathfrak N^{\fk 6}\!\in\! N^{\fk 6}_{a;\fk 1};
\\
\bigsqcup_{\,\mathfrak N'\in
N^{\fk 6}_{a;s}\cup
N^{\fk 6}_{b}\cup
N^{\fk 6}_{d}}
\mathfrak N'
&
\textnormal{if}~\mathfrak N^{\fk 6}\!\in\! N^{\fk 6}_{a;s},~s\!=\!\fk 2,\fk 3;
\\
\bigsqcup_{\,\mathfrak N'\in
N^{\fk 6}_{i}}
\mathfrak N'
&
\textnormal{if}~\mathfrak N^{\fk 6}\!\in\! N^{\fk 6}_{i},~
i\!=\! b,c,d.
\end{cases} \end{equation} Therefore, $\mathfrak M^\dag$ is closed. \end{proof}
To construct $K\!\subset\!\mathfrak M^\dag\!\subset\!\Delta^{\fk 6}$, we consider the algebraic stack $\mathfrak D_2$ of the stable pairs $(C,D)$, where~$C$ are genus $2$ nodal curves and $D$ are effective divisors on $C$. A pair $(C,D)$ in~$\mathfrak D_2$ is said to be \textsf{stable} if every rational irreducible component without any divisoral marking contains at least three nodal points. It is known that $\mathfrak D_2$ is smooth and there exists a {smooth} morphism $$
\mathfrak D_2\longrightarrow\mathfrak P_2,\qquad
(C,D)\mapsto(C,\mathscr O_C(D)). $$ Thus, every smooth chart $\mathcal V\!\longrightarrow\!\mathfrak D_2$ gives a smooth chart $\mathcal V\!\longrightarrow\!\mathfrak P_2$.
Fix $\ti x\!\in\!\mathfrak M^{\fk 6}_{\gamma^{\fk 6}}$. Let $$x=(C,L)\in\mathfrak M^{\fk 1}=\mathfrak P_2$$ be the image of $\ti x$ under the forgetful morphism $\varpi\!:\mathfrak M^{\fk 6}\!\longrightarrow\!\mathfrak M^{\fk 1}$ and let $(C,D)\!\in\!\mathfrak D_2$ be such that $\mathscr O_C(D)\!=\! L$.
We take $\mathcal V\!\longrightarrow\!\mathfrak D_2(\longrightarrow\!\mathfrak M^{\fk 1})$ to be a smooth chart containing $(C,D)$ (and hence $x$). Let $(\mathcal C,\mathcal D)/\mathcal V$ be the universal family. W.l.o.g.~we assume that $$
\mathcal D=\mathcal D_1+\cdots+\mathcal D_m, $$ where the sections $\mathcal D_i\!\in\!\Gamma(\mathcal C/\mathcal V)$ are disjoint.
Assume that $\ti x\!\in\!\mathfrak N^{\fk 6}\!\in\! N^\dag$. We shall make the following choices on $\mathcal D_1$ and $\mathcal D_2$. If~\hbox{$\mathfrak N^{\fk 6}\!\in\! N^{\fk 6}_{a;1}$,} the normalization at the two nodes corresponding to $
\mathfrak E_{\dom{2}{{\mathfrak t}^{\fk 1}}}({\mathfrak t}^{\fk 1})\!\cap\!(\mathfrak E_{{\mathbf{m}};\min}^\bot)^{\succeq}({\mathfrak t}^{\fk 1}) $ yields two (possibly nodal) rational sub-curves of $C$; w.l.o.g.~we assume that $\mathcal D_1(x)$ and $\mathcal D_2(x)$ are on each of them, respectively. Similarly, if $\mathfrak N^{\fk 6}\!\in\! N^{\fk 6}_{a;\fk 2}\!\sqcup\! N^{\fk 6}_{a;\fk 3}\!\sqcup\!N^{\fk 6}_{b}$, we take $\mathcal D_1(x)$ and $\mathcal D_2(x)$ to be on each of the two rational sub-curves of $C$ obtained from the normalization at the two nodes corresponding to $\mathfrak E_{\dom{1}{{\mathfrak t}^{\fk 1}}}({\mathfrak t}^{\fk 1})\cap(\mathfrak E_{{\mathbf{m}};\min}^\bot)^{\succeq}({\mathfrak t}^{\fk 1})$ and $\mathfrak E_{\dom{1}{{\mathfrak t}^{s}}}({\mathfrak t}^{s})\cap(\mathfrak E_{{\mathbf{m}};\min}^\bot)^{\succeq}({\mathfrak t}^{s})$ ($s\!=\!\fk 2,\fk 3,\fk 5$), respectively. If $\mathfrak N^{\fk 6}\!\in\! N^{\fk 6}_{c}$, then $\mathbf{w}((\gamma^{\fk 6})^\star_\tn{cor})\!=\! 2$, which we assume to come from $\mathcal D_1(x)$ and~$\mathcal D_2(x)$. If $\mathfrak N^{\fk 6}\!\in\! N^{\fk 6}_{c}$, then $\mathbf{w}\big((\gamma^{\fk 4}_{\vi 3})^\star_\tn{cor}\big)\!=\! 1$, which we assume to come from $\mathcal D_1(x)$; we then take $\mathcal D_2(x)$ to be on the rational sub-curve of $C$ obtained from the normalization at the node corresponding to $\mathfrak E_{\dom{1}{{\mathfrak t}^{\fk 4}}}({\mathfrak t}^{\fk 4})\cap(\mathfrak E_{{\mathbf{m}};\min}^\bot)^{\succeq}({\mathfrak t}^{\fk 4})$.
Let $K_{12}^\mathcal V\!\subset\!\mathcal V\!\longrightarrow\!\mathfrak D_2$ be the locus on which $\mathcal D_1$ and $\mathcal D_2$ are conjugate. Shrinking~$\mathcal V$ if necessary, we can further assume that $K_{12}^\mathcal V\!=\!\emptyset$ if $\mathcal D_1(x)$ and $\mathcal D_2(x)$ are not conjugate. By~\cite[Lemma 2.8.2]{HLN}, $K_{12}^\mathcal V$ is a Cartier divisor of $\mathcal V$, hence there exists $\kappa_{12}^\mathcal V\!\in\!\Gamma(\mathscr O_\mathcal V)$ such that \begin{equation}\label{Eqn:kappa}
K_{12}^\mathcal V\!=\!\{\kappa_{12}^{\mathcal V}\!=\! 0\}. \end{equation} Let $\mathfrak U\!=\!\mathfrak U_{\ti x}\!\longrightarrow\!\mathfrak M^{\fk 6}$ be a twisted chart that is centered at $\ti x$ and satisfies $\varpi(\mathfrak U)\!\subset\!\mathcal V$, and $\xi^\mathfrak U_s$, $s\!\in\!\ti\mathbb I({\mathfrak t}^{\fk 5})$, be the twisted parameters on $\mathfrak U$; see Theorem~\ref{Thm:tf_smooth} for notation. We denote by $\wc K_{12}^\mathfrak U$ the proper transform of $K_{12}^\mathcal V$ in $\mathfrak U$, which is still a Cartier divisor, hence there exists $\kappa_{12}^\mathfrak U\!\in\!\Gamma(\mathscr O_\mathfrak U)$ such that \begin{equation}\label{Eqn:wc_K} \wc K_{12}^\mathfrak U=\{\kappa_{12}^\mathfrak U\!=\! 0\}. \end{equation}
\begin{lmm}\label{Lm:K} With notation as above, the function $\kappa^\mathfrak U_{12}$ can be taken so that $\{\xi_s^\mathfrak U\}_{s\in\ti\mathbb I({\mathfrak t}^{\fk 5})}\!\sqcup\!\{\kappa_{12}^\mathfrak U\}$ is a subset of local parameters on $\mathfrak U$. Moreover, for any $\ti y\!\in\!\mathfrak U$ and any twisted chart $\mathfrak U'\!=\!\mathfrak U_{\ti y}$ centered at $\ti y$, we have $$
\wc K^{\mathfrak U}_{12}\cap\mathfrak M^\dag\cap \mathfrak U'
=
\wc K^{\mathfrak U'}_{12}\cap\mathfrak M^\dag\cap \mathfrak U, $$ hence the local substacks $\wc K_{12}^{\mathfrak U}\!\cap\!\mathfrak M^\dag$ of $\mathfrak U\!\cap\!\mathfrak M^\dag$ can be glued together to form a substack $K\!\subset\!\mathfrak M^\dag$. \end{lmm}
\begin{proof} Assume that $\ti x\!\in\!\mathfrak N^{\fk 6}\,(\,\!\in\! N^\dag)$. Let $K_{12}^\mathcal V$ be as in~(\ref{Eqn:kappa}) and $x\!=\!(C,L)\!\in\!\mathfrak P_2$ be the image of $\ti x$. We denote by $C_\tn{cor}$ the smallest connected genus 2 sub-curve of $C$ and by $\lr{\delta_1}$ and $\lr{\delta_2}$ respectively the images of $\mathcal D_1(x)$ and $\mathcal D_2(x)$ on $C_\tn{cor}$ after contracting all the irreducible components of $C$ that are away from $C_\tn{cor}$. It is possible that $\lr{\delta_1}\!=\!\lr{\delta_2}$.
According to~\cite[\S2.2 \& Lemma~2.8.2]{HLN}, there are two possible situations for $K_{12}^\mathcal V$: either $K_{12}^\mathcal V$ is smooth and transverse to every local divisor $\{\zeta_e\!=\! 0\}$ as in~(\ref{Eqn:modular_parameters_P2}), or $\lr{\delta_1}$ and $\lr{\delta_2}$ belong to a non-separating bridge $B$, i.e.~a chain of rational irreducible components whose complement in~$C_\tn{cor}$ is connected.
In the former situation, $\kappa_{12}^\mathfrak U$ can simply be taken as the pullback of $\kappa_{12}^\mathcal V$ to $\mathfrak U$ so that the first statement of Lemma~\ref{Lm:K} holds; see Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p3}.
In the latter situation, w.l.o.g.~we assume that $B$ is the largest non-separating bridge containing $\lr{\delta_1}$ and $\lr{\delta_2}$. The nodes where $B$ is attached to a connected genus 1 sub-curve $C_1$ of $C_\tn{cor}$ are denoted by $q_{\epsilon_1}$ and $q_{\epsilon_2}$. For $i\!=\! 1,2$, let $E_i\!\subset\!\tn{Edg}(\gamma_{C_\tn{cor}})$ index the nodes between $C_1$ and $\lr{\delta_i}$, satisfying that $E_1$ and $E_2$ are disjoint. The positions of $\lr{\delta_1}$ and $\lr{\delta_2}$ imply that $$
\mathfrak N^{\fk 6}\!\in\! N^{\fk 6}_{a;\fk 1}\!\cup\!N^{\fk 6}_{a;\fk 3}\!\cup\!N^{\fk 6}_{b}\!\cup\!N^{\fk 6}_{d},\qquad
E_1\!\sqcup\!E_2\!\subset\!\tn{E}_{\mathfrak N^{\fk 3}}\!=\!\tn{Edg}(\tau_{\mathfrak N^{\fk 3}}),
\qquad
\ell_{{\mathfrak t}^{\fk 3}}(\hbar_{\epsilon_1}^+)\!=\!
\ell_{{\mathfrak t}^{\fk 3}}(\hbar_{\epsilon_2}^+)=:l_0. $$ The edges $\epsilon_1$ and $\epsilon_2$ labeling the nodes $q_{\epsilon_1}$ and $q_{\epsilon_2}$ are respectively the maximal elements of $E_1$ and $E_2$ (relative to the tree order on $\tau_{\mathfrak N^{\fk 3}}$); their minimal elements are respectively denoted by $\epsilon_1'$ and $\epsilon_2'$; see Figure~\ref{Fig:K} for illustration.
\begin{figure}
\caption{A point $x$ of $K^\mathcal V_{12}$}
\label{Fig:K}
\end{figure}
By~\cite[\S2.2 \& Lemma~2.8.2]{HLN}, the parameters $\zeta_e$ on $\mathcal V$ can be chosen such that $$\kappa_{12}^\mathcal V=\prod_{e\in E_1}\zeta_e+\prod_{e\in E_2}\zeta_e.$$ Let $\varpi_{\fk 4;\fk 6}\!:\mathfrak M^{\fk 6}\!\longrightarrow\!\mathfrak M^{\fk 4}$ be the composite forgetful morphism, $\ti x^{\fk 4}\!=\!\varpi_{\fk 4;\fk 6}(\ti x)$, and $\mathfrak U^{\fk 3}$ be a twisted chart of $\mathfrak M^{\fk 4}$ centered at $\ti x^{\fk 4}$.
If $\mathfrak N^{\fk 6}\!\in\! N^{\fk 6}_{a;\fk 1}\!\cup\!N^{\fk 6}_{a;\fk 3}$, then $\ell_{{\mathfrak t}^{\fk 3}}(\hbar_{\epsilon_1'}^-)\!\le\!{\mathbf{m}}({\mathfrak t}^{\fk 3})$ and $\ell_{{\mathfrak t}^{\fk 3}}(\hbar_{\epsilon_2'}^-)\!=\!{\mathbf{m}}({\mathfrak t}^{\fk 3})$. By Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p4}, the products $\prod_{e\in E_1}\zeta_e$ and $\prod_{e\in E_2}\zeta_e$ respectively pull back to $$
\prod_{i\in\lbrp{{\mathbf{m}},l_0}_{{\mathfrak t}^{\fk 3}}}\!\!\!\!\!\varpi_{\fk 4;\fk 6}^*\,\xi_i^{\mathfrak U^{\fk 3}}\;\cdot
\prod_{e\in E_1}\!\varpi_{\fk 4;\fk 6}^*\,\xi_e^{\mathfrak U^{\fk 3}}
\qquad\textnormal{and}\qquad
\prod_{i\in\lbrp{{\mathbf{m}},l_0}_{{\mathfrak t}^{\fk 3}}}\!\!\!\!\!\varpi_{\fk 4;\fk 6}^*\,\xi_i^{\mathfrak U^{\fk 3}}\;\cdot
\prod_{e\in E_2}\!\varpi_{\fk 4;\fk 6}^*\,\xi_e^{\mathfrak U^{\fk 3}}, $$ where $\xi_e^{\mathfrak U^{\fk 3}}\!\equiv\!1$ whenever $e$ is one of the special edges $e_i$ of ${\mathfrak t}^{\fk 3}$ as in Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p2}. We observe that $\prod_{e\in E_1}\!\varpi_{\fk 4;\fk 6}^*\,\xi_e^{\mathfrak U^{\fk 3}}$ and $\prod_{e\in E_2}\!\varpi_{\fk 4;\fk 6}^*\,\xi_e^{\mathfrak U^{\fk 3}}$ are products of {\it pairwise distinct} local parameters on $\mathfrak U$. Moreover, $\prod_{e\in E_2}\!\varpi_{\fk 4;\fk 6}^*\,\xi_e^{\mathfrak U^{\fk 3}}$ is a unit on $\mathfrak U$. Thus, $$
\kappa^\mathfrak U_{12}=
\prod_{e\in E_1}\!\varpi_{\fk 4;\fk 6}^*\,\xi_e^{\mathfrak U^{\fk 3}}+
\prod_{e\in E_2}\!\varpi_{\fk 4;\fk 6}^*\,\xi_e^{\mathfrak U^{\fk 3}} $$ is a local parameter on $\mathfrak U$ that defines $\wc K_{12}^\mathfrak U$. The construction of the rooted tree $\tau_{\mathfrak N^{\fk 5}}$ that determines the twisted parameters on $\xi_s^{\mathfrak U}$ implies that $\{\xi_s^\mathfrak U\}_{s\in\ti\mathbb I({\mathfrak t}^{\fk 5})}\!\sqcup\!\{\kappa_{12}^\mathfrak U\}$ is a subset of local parameters on $\mathfrak U$.
If $\mathfrak N^{\fk 6}\!\in\! N^{\fk 6}_{b}\!\cup\!N^{\fk 6}_{d}$, the argument is analogous to the last paragraph,
possibly with a new situation when $\ell_{{\mathfrak t}^{\fk 3}}(\hbar_{\epsilon_1'}^-)\!\le\!{\mathbf{m}}({\mathfrak t}^{\fk 3})$ but $\ell_{{\mathfrak t}^{\fk 3}}(\hbar_{\epsilon_2'}^-)\!>\!{\mathbf{m}}({\mathfrak t}^{\fk 3})$. In that case, the products $\prod_{e\in E_1}\zeta_e$ and $\prod_{e\in E_2}\zeta_e$ respectively pull back to $$
\prod_{i\in\lbrp{{\mathbf{m}},l_0}_{{\mathfrak t}^{\fk 3}}}\!\!\!\!\!\varpi_{\fk 4;\fk 6}^*\,\xi_i^{\mathfrak U^{\fk 3}}\;\cdot
\prod_{e\in E_1}\!\varpi_{\fk 4;\fk 6}^*\,\xi_e^{\mathfrak U^{\fk 3}}
\qquad\textnormal{and}\qquad
\prod_{i\in\lbrp{\ell_{{\mathfrak t}^{\fk 3}}(\hbar_{\epsilon_2'}^-)\,,\,l_0}_{{\mathfrak t}^{\fk 3}}}
\!\!\!\!\!\!\!\!\!\!\varpi_{\fk 4;\fk 6}^*\,\xi_i^{\mathfrak U^{\fk 3}}\;\cdot
\prod_{e\in E_2}\!\varpi_{\fk 4;\fk 6}^*\,\xi_e^{\mathfrak U^{\fk 3}}, $$ and $\prod_{e\in E_2}\!\varpi_{\fk 4;\fk 6}^*\,\xi_e^{\mathfrak U^{\fk 3}}$ is a unit on $\mathfrak U$. Thus, $$
\kappa^\mathfrak U_{12}=
\prod_{i\in\lbrp{{\mathbf{m}}\,,\,\ell_{{\mathfrak t}^{\fk 3}}(\hbar_{\epsilon_2'}^-)}_{{\mathfrak t}^{\fk 3}}}
\!\!\!\!\!\!\!\!\!\!\varpi_{\fk 4;\fk 6}^*\,\xi_i^{\mathfrak U^{\fk 3}}\cdot
\prod_{e\in E_1}\!\varpi_{\fk 4;\fk 6}^*\,\xi_e^{\mathfrak U^{\fk 3}}+
\prod_{e\in E_2}\!\varpi_{\fk 4;\fk 6}^*\,\xi_e^{\mathfrak U^{\fk 3}} $$ is a local parameter on $\mathfrak U$ that defines $\wc K_{12}^\mathfrak U$, which together with $\{\xi_s^\mathfrak U\}_{s\in\ti\mathbb I({\mathfrak t}^{\fk 5})}$ still form a subset of local parameters on $\mathfrak U$. This establishes the first statement of Lemma~\ref{Lm:K}.
The second statement of Lemma~\ref{Lm:K} follows from a direct check. The key fact is that for every $\ti x\!\in\!\mathfrak N^{\fk 6}\!\in\! N^\dag$, the choices of $\mathcal D_1, \mathcal D_2\!\in\!\Gamma(\mathcal C/\mathcal V)$ described in the paragraph above~(\ref{Eqn:kappa}) may not be unique. For instance, in Figure~\ref{Fig:K}, we may choose the sections $\mathcal D_i$ such that $\mathcal D_1(x)$ (resp.~$\mathcal D_2(x)$) is the other marked point on the same irreducible component. Nonetheless, we observe that $\wc K^{\mathfrak U}_{12}\!\cap\!\mathfrak M^\dag$ is independent of such choices (even though $\wc K^{\mathfrak U}_{12}$ is). This, along with~(\ref{Eqn:N6_closure}), gives rise to the second statement of Lemma~\ref{Lm:K}. \end{proof}
\iffalse By lemma~\ref{Lm:K}, the $\Gamma^{\fk 6}_{\vi 3}$-stratification on $\mathfrak M^{\fk 6}$ as in Lemma~\ref{Lm:G6-admissible} as well as the substack $K$ together give rise to a grafted stratification $$
\mathfrak M^{\fk 6}=\bigsqcup_{\gamma'\in\Gamma^{\fk 6}_{\tn{gft}}}\!(\mathfrak M^{\fk 6}_{\tn{gft}})_{\gamma'},\qquad
\textnormal{where}\quad \Gamma^{\fk 6}_\tn{gft}=(\Gamma^{\fk 6}_{\vi 3})_{\tn{gft}} $$ as described in Corollary~\ref{Crl:extra_edge_graphs}. \fi
Next, we construct a treelike structure $\Lambda^{\fk 6}$ on $(\mathfrak M^{\fk 6},\Gamma^{\fk 6}_{\vi 3})$ satisfying the second assumption and~(\ref{Eqn:K_local}) of Corollary~\ref{Crl:extra_edge_graphs}. For each $\mathfrak N^{\fk 6}$ as in~(\ref{Eqn:N^6}), we describe the tuple $(\tau_{\mathfrak N^{\fk 6}},\tn{E}_{\mathfrak N^{\fk 6}},\beta_{\mathfrak N^{\fk 6}})$ as follows.
Let $\wh\tau^{\fk 5}$ be as above~(\ref{Eqn:ga_E1E2E3E4}). Then by~(\ref{Eqn:step5_edge}), \begin{equation}\begin{split}\label{Eqn:Edges_ctr_in_Step_5}
\tn{Edg}(\tau_{\mathfrak N^{\fk 5}})
=\tn{E}_{\mathfrak N^{\fk 5}}
\subset
\tn{Edg}(\wh\tau^{\fk 5})
\subset
\tn{Edg}(\gamma^{\fk 5})\backslash\{\,\mathsf e_i^k:\,i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k},\,k\!=\!\fk{2,3,4}\}
\subset
\tn{Edg}(\gamma^{\fk 5}). \end{split}\end{equation} We can thus take \begin{equation*}
\wh\tau^{\fk 6}:=
\rho_{\wh\tau^{\fk 5};[{\mathfrak t}^{\fk 5}]}, \end{equation*} and let $(\wh\tau^{\fk 6})^\star$ be the induced decoration of $\wh\tau^{\fk 6}$. Since $\wh\tau^{\fk 5}$ is a rooted tree, so is $\wh\tau^{\fk 6}$.
Analogous to~(\ref{Eqn:ga_E1E2E3E4}),
\begin{equation}\begin{split} \label{Eqn:ga_E1E2E3E4E5}
&\tn{Edg}(\wh\tau^{\fk 6})\subset
\tn{Edg}(\gamma^{\fk 6})\backslash
\{\mathsf e_i^k:
k\!=\! \fk{2,3,4};\,
i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k}\}
\subset
\tn{Edg}(\gamma^{\fk 6}),\qquad
o_{\wh\tau^{\fk 6}}\subset o_{\gamma^{\fk 6}}. \end{split}\end{equation}
We are ready to construct $\tau_{\mathfrak N^{\fk 6}}$. \begin{itemize} [leftmargin=*] \item If $\mathfrak N^\fk 6\!\in\! N^{\fk 6}_{a;\fk 1}$, we dissolve the image of $\mathsf v^{\fk 1}_{\dom{2}{{\mathfrak t}^{\fk 1}}}$ in $\wh\tau^{\fk 6}$ as well as all the non-root vertices $v$ of $\wh\tau^{\fk 6}$ with $\mathbf{w}(v)\!>\!0$, and take $\tau_{\mathfrak N^{\fk 6}}$ to be the unique connected component that contains $o_{\wh\tau^{\fk 6}}$.
\item If $\mathfrak N^\fk 6\!\in\! N^{\fk 6}_{b}$ (which implies $\dom{2}{{\mathfrak t}^{\fk 1}}\! \le\!\dom{1}{{\mathfrak t}^{\fk 1}}\!<\! 0$), consider the following two sub-cases: \begin{itemize}[leftmargin=*] \item if $(\mathfrak E^\bot_{{\mathbf{m}};\min}({\mathfrak t}^{\fk 5}))^\succeq\!=\!\{\mathsf e^{\fk 1}_i:i\!\in\!\lbrp{\dom{1}{{\mathfrak t}^{\fk 1}},0}_{{\mathfrak t}^{\fk 1}}\}$, then we dissolve the image of $\mathsf v^{\fk 1}_{\dom{2}{{\mathfrak t}^{\fk 1}}}$ in $\wh\tau^{\fk 6}$ as well as all the non-root vertices $v$ of $\wh\tau^{\fk 6}$ with $\mathbf{w}(v)\!>\!0$ and take $\tau_{\mathfrak N^{\fk 6}}$ to be the unique connected component that contains $o_{\wh\tau^{\fk 6}}$;
\item otherwise, we first dissolve the image of $\mathsf v^{\fk 1}_{\dom{1}{{\mathfrak t}^{\fk 1}}}$ in $\wh\tau^{\fk 6}$ as well as all the non-root vertices $v$ of $\wh\tau^{\fk 6}$ with $\mathbf{w}(v)\!>\!0$, then contract the edges in $$
\big\{\,\mathsf e^{\fk 5}_{i}:\,
i\!\in\!\lbrp{{\mathbf{m}}({\mathfrak t}^{\fk 5}),\lambda}_{{\mathfrak t}^{\fk 5}}\big\}
\quad\textnormal{with}\quad
\lambda\!:=\!\min
\big(\{\ell_{{\mathfrak t}^{\fk 5}}(\mathsf v_j^{\fk 1}):
\mathsf e^{\fk 1}_j\!\in\!(\mathfrak E^\bot_{{\mathbf{m}};\min}({\mathfrak t}^{\fk 5}))^\succeq\}\sqcup\{0\}\big), $$ and finally take $\tau_{\mathfrak N^{\fk 6}}$ to be the unique connected component that contains $o_{\wh\tau^{\fk 6}}$. \end{itemize}
\item If $\mathfrak N^\fk 6\!\in\! N^{\fk 6}_{a;\fk 2}\!\cup\!N^{\fk 6}_{a;\fk 3}\!\cup\!N^{\fk 6}_{d}$, we dissolve the image of $\mathsf v^{\fk 1}_{\dom{1}{{\mathfrak t}^{\fk 1}}}$ in $\wh\tau^{\fk 6}$ as well as all the non-root vertices~$v$ of $\wh\tau^{\fk 6}$ with $\mathbf{w}(v)\!>\!0$, and take $\tau_{\mathfrak N^{\fk 6}}$ to be the unique connected component that contains~$o_{\wh\tau^{\fk 6}}$.
\item For other $\mathfrak N^{\fk 6}$, we set $\tau_{\mathfrak N^{\fk 6}}\!=\!\tau_\bullet$ and $o_{{\mathfrak N^{\fk 6}}}\!=\! o_{\gamma^{\fk 6}}$. \end{itemize} By~(\ref{Eqn:ga_E1E2E3E4E5}), we have \begin{equation}\label{Eqn:step6_edge}
\tn{Edg}(\tau_{\mathfrak N^{\fk 6}})\subset
\tn{Edg}(\wh\tau^{\fk 6})\subset
\tn{Edg}(\gamma^{\fk 6})\backslash
\{\mathsf e_i^k:
k\!=\! \fk{2,\!\cdots\!,5};\,
i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k}\}\subset
\tn{Edg}(\gamma^{\fk 6}), \end{equation} and moreover, $\tn{Edg}(\tau_{\mathfrak N^{\fk 6}})$ is the set of the edges of a {\it connected} subgraph $\tau'_{\mathfrak N^{\fk 6}}$ of $\gamma^{\fk 6}$ that satisfies $$
o_{{\mathfrak N^{\fk 6}}}
\subset o_{\gamma^{\fk 6}}\in\tn{Ver}(\tau'_{\mathfrak N^{\fk 6}}). $$ We set $$
\Lambda^{\fk 6}=
\big(\,
(\tau_{\mathfrak N^{\fk 6}},\,
\tn{E}_{\mathfrak N^{\fk 6}}\!:=\!\tn{Edg}(\tau_{\mathfrak N^{\fk 6}}),\,
\textnormal{Id})\,\big)_{
\gamma^{\fk 6}\in\Gamma^{\fk 6}_{\vi 3},\,
\mathfrak N^{\fk 6}\in\pi_0(\mathfrak M^{\fk 6}_{\gamma^{\fk 6}})
}. $$ Mimicking the proof of Lemma~\ref{Lm:G5-level} and taking Lemma~\ref{Lm:N_dag} into consideration, we obtain the following statement.
\begin{lmm}\label{Lm:G6-level} The set $\Lambda^{\fk 6}$ gives a treelike structure on $(\mathfrak M^{\fk 6},\Gamma^{\fk 6}_{\vi 3})$ as in Definition~\ref{Dfn:Treelike_structure}. \end{lmm}
Notice that $\mathfrak N^{\fk 6}\!\in\! N^\dag$ whenever $\tau_{\mathfrak N^{\fk 6}}\!\ne\!\tau_\bullet$. Corollary~\ref{Crl:extra_edge_graphs}, Lemmas~\ref{Lm:K} and~\ref{Lm:G6-level}, and Theorem~\ref{Thm:tf_smooth} together lead to the following statement.
\begin{crl}\label{Crl:G6_tf} The stack $\mathfrak M^{\fk 6}$ has a $\Gamma^{\fk 6}_\tn{gft}$-stratification $$
\mathfrak M^{\fk 6}=\bigsqcup_{\gamma'\in\Gamma^{\fk 6}_{\tn{gft}}}\!(\mathfrak M^{\fk 6}_{\tn{gft}})_{\gamma'},\qquad
\textnormal{where}\quad \Gamma^{\fk 6}_\tn{gft}=(\Gamma^{\fk 6}_{\vi 3})_{\tn{gft}}, $$ which is the grafted stratification with respect to $(\mathfrak M^{\fk 6},\Gamma^{\fk 6}_{\vi 3},{\Lambda}^{\fk 6})$, along with the induced treelike structure $\Lambda^{\fk 6}_{\tn{gft}}$ as in Corollary~\ref{Crl:extra_edge_graphs}. Furthermore, the stack $(\mathfrak M^{\fk 6})^\tn{tf}_{\Lambda^{\fk 6}_\tn{gft}}$ as in Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p1} is a smooth algebraic stack and the composite forgetful morphism $(\mathfrak M^{\fk 6})^\tn{tf}_{\Lambda^{\fk 6}_\tn{gft}}\!\longrightarrow\!\mathfrak M^{\fk 1}\!=\!\mathfrak P_2$ is proper and birational. \end{crl}
\subsection{The seventh step of the recursive construction}\label{Subsec:Step7}
In this step, we take $$
\mathfrak M^{\fk 7}=(\mathfrak M^{\fk 6})^\tn{tf}_{\Lambda^{\fk 6}_\tn{gft}},\qquad
\Gamma^{\fk 7}=(\Gamma^{\fk 6}_\tn{gft})^{\tn{der}}_{\Lambda^{\fk 6}_\tn{gft}}. $$ Analogous to Lemma~\ref{Lm:G2-admissible}, we have the following statement.
\begin{lmm}\label{Lm:G8-admissible} The stack $\mathfrak M^{\fk 7}$ comes equipped with a $\Gamma^{\fk 7}$-stratification that is the derived stratification of $\mathfrak M^{\fk 7}$ with respect to $(\mathfrak M^{\fk 6},\Gamma^{\fk 6}_\tn{gft},\Lambda^{\fk 6}_\tn{gft})$. \end{lmm}
The graphs of $\Gamma^{\fk 7}$ are in the form \begin{equation}\label{Eqn:ga^8}\begin{split}
&\gamma^{\fk 7}
:=\rho_{\gamma^{\fk 6}_\tn{gft},[{\mathfrak t}^{\fk 6}]}
\in\Gamma^{\fk 7},
\\
&\textnormal{where}\quad
\gamma^{\fk 6}_\tn{gft}\!:=\!(\gamma^{\fk 6},o_{\gamma^{\fk 6}})_\tn{gft}\!\in\!\Gamma^{\fk 6}_\tn{gft},\ \
[{\mathfrak t}^{\fk 6}]\!\in\!\ov\Lambda^{\fk 6}_{\mathfrak N^{\fk 6}_\tn{gft}},\ \
\mathfrak N^{\fk 6}_\tn{gft}\!:=\!\mathfrak N^{\fk 6}\!\cap\!K
\!\in\!\pi_0\big((\mathfrak M^{\fk 6}_\tn{gft})_{\gamma^{\fk 6}_\tn{gft}}\big); \end{split}\end{equation} see~(\ref{Eqn:ga^6}) for $\gamma^{\fk 6}$, (\ref{Eqn:N^6}) for $\mathfrak N^{\fk 6}$, and Lemma~\ref{Lm:K} for $K$. Let $o_{\gamma^{\fk 7}}$ be the image of $o_{\gamma^{\fk 6}}$ in $\gamma^{\fk 5}$ and $(\gamma^{\fk 7})^\star$ be the derived decoration. By Corollary~\ref{Crl:Derived_graphs}, for every $\gamma^{\fk 7}\!\in\!\Gamma^{\fk 7}$ as in~(\ref{Eqn:ga^8}), the connected components of $\mathfrak M^{\fk 7}_{\gamma^{\fk 7}}$ are in the form \begin{equation}\label{Eqn:N^8}
\mathfrak N^{\fk 7}:=
(\mathfrak N^{\fk 6}_\tn{gft})_{[{\mathfrak t}^{\fk 6}]}^\tn{tf}\
\in\pi_0\big(\mathfrak M^{\fk 7}_{\gamma^{\fk 7}}\big). \end{equation} For each $\mathfrak N^{\fk 7}$, we will describe the tuple $(\tau_{\mathfrak N^{\fk 7}},\tn{E}_{\mathfrak N^{\fk 7}},\beta_{\mathfrak N^{\fk 7}})$ to provide~$\Lambda^{\fk 7}$ on $(\mathfrak M^{\fk 7},\Gamma^{\fk 7})$.
Let \begin{equation}\begin{split}\label{Eqn:Edges_ctr_in_Step_7}
\tn{E}^{\fk 6}
&:=
\mathfrak E^\bot_{\ge{\mathbf{m}}}({\mathfrak t}^{\fk 6})
\subset
\tn{Edg}(\tau_{\mathfrak N^{\fk 6}_\tn{gft}})
=\tn{E}_{\mathfrak N^{\fk 6}_\tn{gft}}
\\
&\subset
\tn{Edg}(\wh\tau^{\fk 6}_\tn{gft})\subset
\tn{Edg}(\gamma^{\fk 6}_\tn{gft})\backslash
\{\mathsf e_i^k:
i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^{k}},
k\!=\!\fk{2,3,4,6}\}\subset
\tn{Edg}(\gamma^{\fk 6}_\tn{gft}), \end{split}\end{equation} which follows from~(\ref{Eqn:step6_edge}). Consider the rooted tree \begin{equation}\label{Eqn:wh_tau^8}
\wh\tau^{\fk 7}:=
(\wh\tau^{\fk 6}_\tn{gft})_{(\tn{E}^{\fk 6}\sqcup\{\mathsf e_i^k:\,i\in\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k},\,k={\fk 1,5}\})}, \end{equation} whose induced decoration is denoted by $(\wh\tau^{\fk 7})^\star$. By~(\ref{Eqn:Edges_ctr_in_Step_7}) and~(\ref{Eqn:wh_tau^8}), we have \begin{equation}\begin{split} \label{Eqn:ga_E1E2E3E4E5E6E7}
\tn{Edg}(\wh\tau^{\fk 7})\subset
\tn{Edg}(\gamma^{\fk 7})\backslash
\{\mathsf e_i^k:
i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k},
k\!=\!\fk{1,\!\cdots\!,6}\}\subset
\tn{Edg}(\gamma^{\fk 7}). \end{split}\end{equation}
We are ready to construct $\tau_{\mathfrak N^{\fk 7}}$. \begin{itemize}[leftmargin=*] \item If either $\mathbf{w}(o_{\wh\tau^{\fk 7}})\!>\!2$, or $\mathbf{w}(o_{\wh\tau^{\fk 7}})\!=\!2$ but $e_\tn{gft}\!\in\!\tn{E}^{\fk 6}$, we set $\tau_{\mathfrak N^{\fk 7}}\!=\!\tau_\bullet$ and $o_{\mathfrak N^{\fk 7}}\!=\! o_{\gamma^{\fk 7}}$.
\item If $\mathbf{w}(o_{\wh\tau^{\fk 7}})\!=\!2$ and $e_\tn{gft}\!\not\in\!\tn{E}^{\fk 6}$, we dissolve all the non-root vertices $v$ in $\wh\tau^{\fk 7}$ satisfying $\mathbf{w}(v)\!>\!0$ and take $\tau_{\mathfrak N^{\fk 7}}$ to be the unique connected component that contains $o_{\wh\tau^{\fk 7}}$. Thus, $\tau_{\mathfrak N^{\fk 7}}$ is also a rooted tree with $o_{\mathfrak N^{\fk 7}}\!=\! o_{\wh\tau^{\fk 7}}$. \end{itemize} By~(\ref{Eqn:ga_E1E2E3E4E5E6E7}) and the above construction of $\tau_{\mathfrak N^{\fk 7}}$, we observe that \begin{equation}\label{Eqn:step8_edge}
\tn{Edg}(\tau_{\mathfrak N^{\fk 7}})\subset
\tn{Edg}(\wh\tau^{\fk 7})\subset
\tn{Edg}(\gamma^{\fk 7})\backslash
\{\mathsf e_i^k:
i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k},
k\!=\!\fk{1,\!\cdots\!,6}\}\subset
\tn{Edg}(\gamma^{\fk 7}), \end{equation} and moreover, $\tn{Edg}(\tau_{\mathfrak N^{\fk 7}})$ is the set of the edges of a {\it connected} subgraph $\tau'_{\mathfrak N^{\fk 7}}$ of $\gamma^{\fk 7}$ that satisfies $$
o_{\tau_{\mathfrak N^{\fk 7}}}
\subset o_{\gamma^{\fk 7}}\in\tn{Ver}(\tau'_{\mathfrak N^{\fk 7}}). $$ We set $$
\Lambda^{\fk 7}=
\big(\,
(\tau_{\mathfrak N^{\fk 7}},\,
\tn{E}_{\mathfrak N^{\fk 7}}\!:=\!\tn{Edg}(\tau_{\mathfrak N^{\fk 7}}),\,
\textnormal{Id})\,\big)_{
\gamma^{\fk 7}\in\Gamma^{\fk 7},\,
\mathfrak N^{\fk 7}\in\pi_0(\mathfrak M^{\fk 7}_{\gamma^{\fk 7}})
}. $$
The following statement is the analogue of Lemmas~\ref{Lm:G1-level} and~\ref{Lm:G4-level} in this step.
\begin{lmm}\label{Lm:G8-level} The set $\Lambda^{\fk 7}$ gives a treelike structure on $(\mathfrak M^{\fk 7},\Gamma^{\fk 7})$ as in Definition~\ref{Dfn:Treelike_structure}. \end{lmm}
Lemma~\ref{Lm:G8-level}, along with Theorem~\ref{Thm:tf_smooth}, gives rise to the following statement.
\begin{crl}\label{Crl:G8-tf} Let $(\mathfrak M^{\fk 7})^\tn{tf}_{\Lambda^{\fk 7}}$ be constructed as in Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p1}. Then, $(\mathfrak M^{\fk 7})^\tn{tf}_{\Lambda^{\fk 7}}$ is a smooth algebraic stack and the composite forgetful morphism $(\mathfrak M^{\fk 7})^\tn{tf}_{\Lambda^{\fk 7}}\!\longrightarrow\!\mathfrak M^{\fk 1}\!=\!\mathfrak P_2$ is proper and birational. \end{crl}
\subsection{The eighth step of the recursive construction}\label{Subsec:Step8}
In this step, we take $$
\mathfrak M^{\fk 8}=(\mathfrak M^{\fk 7})^\tn{tf}_{\Lambda^{\fk 7}},\qquad
\Gamma^{\fk 8}=(\Gamma^{\fk 7})^{\tn{der}}_{\Lambda^{\fk 7}}. $$ Analogous to Lemma~\ref{Lm:G2-admissible}, we have the following statement.
\begin{lmm}\label{Lm:G9-admissible} The stack $\mathfrak M^{\fk 8}$ comes equipped with a $\Gamma^{\fk 8}$-stratification that is the derived stratification of $\mathfrak M^{\fk 8}$ with respect to $(\mathfrak M^{\fk 7},\Gamma^{\fk 7},\Lambda^{\fk 7})$. \end{lmm}
The graphs of $\Gamma^{\fk 8}$ are in the form \begin{equation}\label{Eqn:ga^9}
\gamma^{\fk 8}
:=\rho_{\gamma^{\fk 7},[{\mathfrak t}^{\fk 7}]}
\in\Gamma^{\fk 8},
\qquad
\textnormal{where}\quad
\gamma^{\fk 7}\!\in\!\Gamma^{\fk 7},\ \
[{\mathfrak t}^{\fk 7}]\!\in\!(\ov\Lambda^{\fk 7})_{\mathfrak N^{\fk 7}},\ \
\mathfrak N^{\fk 7}
\!\in\!\pi_0(\mathfrak M^{\fk 7}_{\gamma^{\fk 7}}); \end{equation} see~(\ref{Eqn:ga^8}) for $\gamma^{\fk 7}$ and (\ref{Eqn:N^8}) for $\mathfrak N^{\fk 7}$. Let $o_{\gamma^{\fk 8}}$ be the image of $o_{\gamma^{\fk 7}}$ in $\gamma^{\fk 7}$ and $(\gamma^{\fk 8})^\star$ be the derived decoration. By Corollary~\ref{Crl:Derived_graphs}, for every $\gamma^{\fk 8}\!\in\!\Gamma^{\fk 8}$, the connected components of $\mathfrak M^{\fk 8}_{\gamma^{\fk 8}}$ are in the form \begin{equation}\label{Eqn:N^9}
\mathfrak N^{\fk 8}:=
(\mathfrak N^{\fk 7})_{[{\mathfrak t}^{\fk 7}]}^\tn{tf}\
\in\pi_0\big(\mathfrak M^{\fk 8}_{\gamma^{\fk 8}}\big). \end{equation} For each $\mathfrak N^{\fk 8}$, we will describe the tuple $(\tau_{\mathfrak N^{\fk 8}},\tn{E}_{\mathfrak N^{\fk 8}},\beta_{\mathfrak N^{\fk 8}})$ to provide~$\Lambda^{\fk 8}$ on $(\mathfrak M^{\fk 8},\Gamma^{\fk 8})$.
Let \begin{equation}\begin{split}\label{Eqn:Edges_ctr_in_Step_8}
\tn{E}^{\fk 7}
&=
\mathfrak E^\bot_{\ge{\mathbf{m}}}({\mathfrak t}^{\fk 7})
\subset
\tn{Edg}(\tau_{\mathfrak N^{\fk 7}})
=\tn{E}_{\mathfrak N^{\fk 7}}
\\
&\subset
\tn{Edg}(\wh\tau^{\fk 7})=
\tn{Edg}(\gamma^{\fk 7})\backslash
\{\mathsf e_i^k:
i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k},
k\!=\!\fk{1,\!\cdots\!,6}\}\subset
\tn{Edg}(\gamma^{\fk 7}), \end{split}\end{equation} which follows from~(\ref{Eqn:step8_edge}).
By~(\ref{Eqn:ti_ga^2_edges}), the derived graph $\ti\gamma^{\fk 2}_{(\tn{E}^{\fk 3})}$ is well-defined and satisfies $$\tn{E}^{\fk 4}\subset\tn{Edg}(\ti\gamma^{\fk 2}_{(\tn{E}^{\fk 3})}),$$ so we can consider the derived graph $$
\ti\gamma^{\fk 4}:=\rho_{\ti\gamma^{\fk 2}_{(\tn{E}^{\fk 3})};[{\mathfrak t}^{\fk 4}]} $$ and the derived decoration $(\ti\gamma^{\fk 4})^\star$. Let $$
\ti\tau^{\fk 5}\qquad\textnormal{and}\qquad
(\ti\tau^{\fk 5})^\star $$ be the rooted tree obtained from $\ti\gamma^{\fk 4}$ by contracting the edges of $(\ti\gamma^{\fk 4})^\star_\tn{cor}$ into the root $o_{\ti\tau^{\fk 5}}$ and the induced decoration, respectively.
By~(\ref{Eqn:derived_graph}), we have $$
\tn{Edg}(\ti\gamma^{\fk 2}_{(\ud\tn{E})^{\fk 4}})\!\subset\!\tn{Edg}(\ti\gamma^{\fk 4})
\qquad\textnormal{and}\qquad
\tn{Edg}\big((\ti\gamma^{\fk 2}_{(\ud\tn{E})^{\fk 4}})^\star_\tn{cor}\big)\!=\!\tn{Edg}\big((\ti\gamma^{\fk 4})^\star_\tn{cor}\big). $$ Therefore, $$
\tn{Edg}(\wh\tau^{\fk 5})\subset\tn{Edg}(\ti\tau^{\fk 5}). $$ Parallel to the constructions of $\wh\tau^{\fk 6}$ above~(\ref{Eqn:ga_E1E2E3E4E5}) and $\wh\tau^{\fk 7}$ in~(\ref{Eqn:wh_tau^8}), we define $$
\ti\tau^{\fk 7}=
\big((\rho_{\ti\tau^{\fk 5};[{\mathfrak t}^{\fk 5}]})_\tn{gft}\big)_{(\tn{E}^{\fk 6}\sqcup\{\mathsf e_i^k:\,i\in\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k},\,k={\fk 1,5}\})} $$ and observe that $$
\tn{E}^{\fk 7}\subset\tn{Edg}(\wh\tau^{\fk 7})\subset
\tn{Edg}(\ti\tau^{\fk 7})\subset
\tn{Edg}(\gamma^{\fk 7}). $$ Let $
\wh\tau^{\fk 8}\!:=\!\ti\tau^{\fk 7}_{(\tn{E}^{\fk 8})} $, along with the induced decoration $(\wh\tau^{\fk 8})^\star$. Then, \begin{equation}\begin{split} \label{Eqn:ga_E1E2E3E4E5E6E7E8}
\tn{Edg}(\wh\tau^{\fk 8})\subset
\tn{Edg}(\gamma^{\fk 8})\backslash
\{\mathsf e_i^k:
i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k},
k\!=\!\fk{1,2,3,5,6,7}\}\subset
\tn{Edg}(\gamma^{\fk 8}). \end{split}\end{equation}
We are ready to construct $\tau_{\mathfrak N^{\fk 8}}$. \begin{itemize}[leftmargin=*] \item If either $\mathfrak N^{\fk 6}\!\not\in\!N^{\fk 6}_d$, or $\mathfrak N^{\fk 6}\!\in\!N^{\fk 6}_d$ but either $e_\tn{gft}\!\in\!\tn{E}^{\fk 6}\!\sqcup\!\tn{E}^{\fk 7}$ or $\mathbf{w}\big((\gamma^{\fk 6})^\star_\tn{cor}\big)\!<\!\mathbf{w}\big((\gamma^{\fk 8})^\star_\tn{cor}\big)$, we set $\tau_{\mathfrak N^{\fk 8}}\!=\!\tau_\bullet$ and $o_{\mathfrak N^{\fk 8}}\!=\! o_{\gamma^{\fk 8}}$.
\item If $\mathfrak N^{\fk 6}\!\in\!N^{\fk 6}_d$, $e_\tn{gft}\!\not\in\!\tn{E}^{\fk 6}\!\sqcup\!\tn{E}^{\fk 7}$, and $\mathbf{w}\big((\gamma^{\fk 6})^\star_\tn{cor}\big)\!=\!\mathbf{w}\big((\gamma^{\fk 8})^\star_\tn{cor}\big)$, then we dissolve $\mathsf v^{\fk 4}_{\dom{1}{{\mathfrak t}^4}}$ as well as all the non-root vertices $v$ in~$\wh\tau^{\fk 8}$ satisfying $\mathbf{w}(v)\!>\!0$, and take $\tau_{\mathfrak N^{\fk 8}}$ to be the unique connected component that contains $o_{\wh\tau^{\fk 8}}$. Thus, $\tau_{\mathfrak N^{\fk 8}}$ is a rooted tree with $o_{\mathfrak N^{\fk 8}}\!=\! o_{\wh\tau^{\fk 8}}$. \end{itemize} By~(\ref{Eqn:ga_E1E2E3E4E5E6E7E8}) and the above construction of $\tau_{\mathfrak N^{\fk 8}}$, we observe that \begin{equation}\label{Eqn:step9_edge}
\tn{Edg}(\tau_{\mathfrak N^{\fk 8}})\subset
\tn{Edg}(\wh\tau^{\fk 8})\subset
\tn{Edg}(\gamma^{\fk 8})\backslash
\{\mathsf e_i^k:
i\!\in\!\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k},
k\!=\!\fk{1,2,3,5,6,7}\}\subset
\tn{Edg}(\gamma^{\fk 8}), \end{equation} and moreover, $\tn{Edg}(\tau_{\mathfrak N^{\fk 8}})$ is the set of the edges of a {\it connected} subgraph $\tau'_{\mathfrak N^{\fk 8}}$ of $\gamma^{\fk 8}$ that satisfies $$
o_{\tau_{\mathfrak N^{\fk 8}}}
\subset o_{\gamma^{\fk 8}}\in\tn{Ver}(\tau'_{\mathfrak N^{\fk 8}}). $$ We set $$
\Lambda^{\fk 8}=
\big(\,
(\tau_{\mathfrak N^{\fk 8}},\,
\tn{E}_{\mathfrak N^{\fk 8}}\!:=\!\tn{Edg}(\tau_{\mathfrak N^{\fk 8}}),\,
\textnormal{Id})\,\big)_{
\gamma^{\fk 8}\in\Gamma^{\fk 8},\,
\mathfrak N^{\fk 8}\in\pi_0(\mathfrak M^{\fk 8}_{\gamma^{\fk 8}})
}. $$
The following statement is the analogue of Lemma~\ref{Lm:G5-level} in this step.
\begin{lmm}\label{Lm:G9-level} The set $\Lambda^{\fk 8}$ gives a treelike structure on $(\mathfrak M^{\fk 8},\Gamma^{\fk 8})$ as in Definition~\ref{Dfn:Treelike_structure}. \end{lmm}
Lemma~\ref{Lm:G9-level}, along with Theorem~\ref{Thm:tf_smooth}, gives rise to the following statement.
\begin{crl}\label{Crl:G9-tf} Let $(\mathfrak M^{\fk 8})^\tn{tf}_{\Lambda^{\fk 8}}$ be constructed as in Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p1}. Then, $$
\ti\mathfrak P_2^\tn{tf}:=(\mathfrak M^{\fk 8})^\tn{tf}_{\Lambda^{\fk 8}} $$ is a smooth algebraic stack and the composite forgetful morphism $\ti\mathfrak P_2^\tn{tf}\!=\!(\mathfrak M^{\fk 8})^\tn{tf}_{\Lambda^{\fk 8}}\!\longrightarrow\!\mathfrak M^{\fk 1}\!=\!\mathfrak P_2$ is proper and birational. \end{crl}
\subsection{Proof of Theorem~\ref{Thm:Main}}\label{Subsec:Proof_Main}
The properness of the forgetful morphism $\varpi\!:\ti\mathfrak P_2^\tn{tf}\!\longrightarrow\!\mathfrak P_2$ established in Corollary~\ref{Crl:G8-tf} implies Theorem~\ref{Thm:Main}~\ref{Cond:MainProper}. Moreover,
$\varpi\!:\ti\mathfrak P_2^\tn{tf}\!\longrightarrow\!\mathfrak P_2$ restricts to the identity map on the open subset $\varpi^{-1}(\mathfrak M_{\tau_\bullet})$, which gives rise to Theorem~\ref{Thm:Main}~\ref{Cond:MainBirational}.
It remains to prove Theorem~\ref{Thm:Main}~\ref{Cond:MainSmooth} and~\ref{Cond:MainLocallyFree}.
By~\cite[Theorem~6.1.1 \& Lemma~2.4.1]{HLN}, on a smooth chart $\mathcal V\!\longrightarrow\!\mathfrak P_2$, the stack $\ov M_2(\mathbb P^n,d)$ and the direct image sheaf $\pi_*\mathfrak f^*\mathscr O_{\mathbb P^n}(k)$ can respectively be identified with the kernel of homomorphisms: $$
(0\oplus\varphi)^{\oplus n}:
\big(\mathscr O_\mathcal V^{\oplus(m+1)}\big)^{\oplus n}\longrightarrow
\big(\mathscr O_\mathcal V^{\oplus 2}\big)^{\oplus n}, $$ where $m\!=\! d$ for $\ov M_2(\mathbb P^n,d)$ and $m\!=\! kd$ for $\pi_*\mathfrak f^*\mathscr O_{\mathbb P^n}(k)$. The homomorphisms $\varphi$ for $\ov M_2(\mathbb P^n,d)$ coincides with that for $\pi_*\mathfrak f^*\mathscr O_{\mathbb P^n}(1)$. For each $k\!\ge\!1$, $\varphi$ can be considered as a $2\!\times\!m$ matrix whose explicit expressions (for all possible chart $\mathcal V\!\longrightarrow\!\mathfrak P_2$) are given in~\cite[Proposition~2.6.1, Lemma~2.7.1, \& Corollaries 2.7.2 \& 2.7.3]{HLN}. In particular, the entries of the matrix $\varphi$ after suitable trivialization are monomials in the parameters corresponding to the smoothing of the nodes as in~(\ref{Eqn:modular_parameters_P2}) and/or the function $\kappa_{12}^\mathcal V$ as in~(\ref{Eqn:kappa}). Applying Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p4} repeatedly to the entries of $\varphi$ and taking suitable trivialization again if necessary, we observe that the pullback $\ti\varphi^\tn{tf}$ of $\varphi$ to any twisted chart $\mathfrak U_x\!\longrightarrow\!\ti\mathfrak P_2^\tn{tf}$ centered at any $\ti x\!\in\!\ti\mathfrak P_2^\tn{tf}$ must take the form \begin{equation}\label{Eqn:phi_pullback}
\ti\varphi^\tn{tf}|_{\mathfrak U_{\ti x}}=
\left\lgroup
\begin{matrix}
z_1 & 0 & 0 & \cdots\\
0 & z_2 & 0 & \cdots
\end{matrix}
\right\rgroup, \end{equation} where $z_1$ and $z_2$ are monomials in the pullbacks of the twisted parameters of $\ti\mathfrak P_2^\tn{tf}$ such that $z_1$ is a factor of $z_2$. Since this approach is parallel to the argument in~\cite[\S5]{HLN}, we only provide the key steps below and omit further details in this paper.
To obtain~(\ref{Eqn:phi_pullback}), let $\varpi^k\!:\ti\mathfrak P_2^\tn{tf}\!\longrightarrow\!(\mathfrak M^k)^\tn{tf}_{\Lambda^k}$, $1\!\le\!k\!\le\!8$, be the corresponding composite forgetful morphism, $\mathfrak U^k\!:=\!\mathfrak U_{\varpi^k(\ti x)}$ be a twisted chart on $(\mathfrak M^k)^\tn{tf}_{\Lambda^k}$ centered at $\varpi^k(\ti x)$, and $\xi_i^{\mathfrak U^k}$ be the twisted parameters on $\mathfrak U^k$; see Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p2} for notation.
Let $\mathfrak N^{\fk 9}$ be the connected component of the stratum of the $(\Gamma^{\fk 8})^\tn{der}_{\Lambda^{\fk 8}}$-stratification of $\ti\mathfrak P_2^\tn{tf}$ that contains $\ti x$. Following the recursive construction in~\S\ref{Subsec:Step1}-\ref{Subsec:Step8}, we see that $\mathfrak N^{\fk 9}$ is determined by a unique list: $$
\gamma\!\in\!\Gamma^{\fk 1}_\tn{vic},\ \
\mathfrak N^{\fk 1}\!\in\!\pi_0(\mathfrak M^{\fk 1}_{\gamma}),\ \
[{\mathfrak t}^{\fk 1}]\!\in\!\ov\Lambda_{\mathfrak N^{\fk 1}},\ \ \ldots\ \;,\
[{\mathfrak t}^{\fk 8}]\!\in\!\ov\Lambda_{\mathfrak N^{\fk 8}},\
\gamma^{\fk 9}\!\in\!(\Gamma^{\fk 8})^\tn{der}_{\Lambda^{\fk 8}},\ \
\mathfrak N^{\fk 9}\!\in\!\pi_0\big((\ti\mathfrak P_2^\tn{tf})_{\gamma^{\fk 9}}\big). $$ Some of these rooted level trees ${\mathfrak t}^{s}$ are possibly trivial (i.e.~the underlying rooted tree is edgeless).
If ${\mathfrak t}^{\fk 2}$ is trivial, we can find an entry in the first row of $\varphi$ corresponding to a vertex on the ${\mathbf{m}}({\mathfrak t}^{\fk 1})$-th level in ${\mathfrak t}^{\fk 1}$, such that the pullback $z_1$ of this entry to $\ti\mathfrak P_2^\tn{tf}$, according to Theorem~\ref{Thm:tf_smooth}~\ref{Cond:p4}, is in the form $
\prod_{i\in\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^1}}\!(\varpi^{\fk 1})^*\xi_i^{\mathfrak U^{\fk 1}} $ up to multiplication by a unit. If ${\mathfrak t}^{\fk 2}$ is not trivial, we can analogously find an entry in the first row of $\varphi$ corresponding to a vertex on the ${\mathbf{m}}({\mathfrak t}^{\fk 2})$-th level in ${\mathfrak t}^{\fk 2}$, such that the pullback $z_1$ of this entry to $\ti\mathfrak P^\tn{tf}_2$ is in the form $\prod_{k=\fk 1, \fk 2;\;i\in\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k}}\!(\varpi^k)^*\xi_i^{\mathfrak U^k}$ up to multiplication by a unit. In either situation, we observe that $z_1$ is a factor of any other entries in the first row of $\ti\varphi^\tn{tf}$, thus after taking suitable trivialization, we obtain the first row as well as the leftmost 0 of the second row of~(\ref{Eqn:phi_pullback}).
Following the same idea, we can find an entry in the second row (excluding the first column) of $\varphi$ corresponding to a vertex on the ${\mathbf{m}}({\mathfrak t}^{j})$-th level in ${\mathfrak t}^{j}$, where $j$ is the last step of the recursive construction in which ${\mathfrak t}^j$ is non-trivial (i.e.~${\mathbf{m}}({\mathfrak t}^j)\!\ne\!0$). By considering all the possible treelike structures and $[{\mathfrak t}^i]$'s prior to the $j$-th step, we see that the pullback $z_2$ of this entry to $\ti\mathfrak P^\tn{tf}_2$ is in the form $\prod_{\fk 1\le k\le j,\;i\in\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k}}\!(\varpi^k)^*\xi_i^{\mathfrak U^k}$ up to multiplication by a unit and $z_2$ is a factor of any entries in the second row of $\ti\varphi^\tn{tf}$. In sum, we obtain~(\ref{Eqn:phi_pullback}) with $$
z_1=\!\prod_{k=\fk 1, \fk 2;\;i\in\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k}}\hspace{-.25in}(\varpi^k)^*\xi_i^{\mathfrak U^k}~,
\qquad
z_2=\!\prod_{\fk 1\le k\le \fk 8,\;i\in\lbrp{{\mathbf{m}},0}_{{\mathfrak t}^k}}\hspace{-.25in}(\varpi^k)^*\xi_i^{\mathfrak U^k} $$ up to multiplication by units.
The local expression~(\ref{Eqn:phi_pullback}) of $\ti\varphi^\tn{tf}$ implies that its kernel admits at worst normal crossing singularities, which justifies the properties~\ref{Cond:MainSmooth} and~\ref{Cond:MainLocallyFree} of Theorem~\ref{Thm:Main} simultaneously. \\
\end{document} | arXiv |
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Philosophy of Music in Aesthetics
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Ravi V. Gomatam, Ph.D.Ravi V. Gomatam - unknowndetails
The attitude that ordinary language description of experience is in fact a description of the world is called "naïve realism." There is an entire branch of modern Western philosophy that is devoted to critically examining the assumptions behind the everyday language we use to describe the macroscopic world in which we live and the validity of naïve realism as an adequate description of the world. This branch of philosophy is called "ordinary language philosophy." Surprisingly, it has something in common with (...) quantum physics: insight into the inadequacy of ordinary language to describe observable reality. It is this connection between ordinary language philosophy and quantum physics that we shall explore in this article. In the process, we shall also offer a basic introduction to both basic philosophy and basic quantum theory. (shrink)
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Space and Time in Philosophy of Physical Science
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Nasledie I.V. Kireevskogo: Opyty Filosofskogo Osmyslenii͡a.V. B. Rozhkovskiĭ & Ivan Vasilʹevich Kireevskiĭ (eds.) - 2006 - Nauka-Press.details
Idei͡a Spasenii͡a V Russkoĭ Filosofii: Monografii͡a.V. Sh Sabirov - 2008details
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Philosophy of Literature in Aesthetics
W.V. Quine, Immanuel Kant Lectures, translated and introduced by H.G. Callaway.H. G. Callaway & W. V. Quine (eds.) - 2003 - Frommann-Holzboog.details
This book is a translation of W.V. Quine's Kant Lectures, given as a series at Stanford University in 1980. It provide a short and useful summary of Quine's philosophy. There are four lectures altogether: I. Prolegomena: Mind and its Place in Nature; II. Endolegomena: From Ostension to Quantification; III. Endolegomena loipa: The forked animal; and IV. Epilegomena: What's It all About? The Kant Lectures have been published to date only in Italian and German translation. The present book is filled out (...) with the translator's critical Introduction, "The esoteric Quine?" a bibliography based on Quine's sources, and an Index for the volume. (shrink)
Radical Interpretation in Philosophy of Language
The Indeterminacy of Translation in Philosophy of Language
W. V. O. Quine in 20th Century Philosophy
Voinstvui͡ushchiĭ Ėmpirizm V Filosofii.V. V. Semenov - 2004 - Pnt͡s Ran.details
Socialism and Marxism in Social and Political Philosophy
Otechestvennai͡a Lingvistika V Lit͡sakh: Azbuka Chelovechnosti.V. I. Shakhovskiĭ - 2011details | CommonCrawl |
Addressing future trade-offs between biodiversity and cropland expansion to improve food security
Ruth Delzeit1,
Florian Zabel2,
Carsten Meyer3,4 &
Tomáš Václavík5,6
Regional Environmental Change volume 17, pages1429–1441(2017)Cite this article
An Erratum to this article is available
Potential trade-offs between providing sufficient food for a growing human population in the future and sustaining ecosystems and their services are driven by various biophysical and socio-economic parameters at different scales. In this study, we investigate these trade-offs by using a three-step interdisciplinary approach. We examine (1) how the expected global cropland expansion might affect food security in terms of agricultural production and prices, (2) where natural conditions are suitable for cropland expansion under changing climate conditions, and (3) whether this potential conversion to cropland would affect areas of high biodiversity value or conservation importance. Our results show that on the one hand, allowing the expansion of cropland generally results in an improved food security not only in regions where crop production rises, but also in net importing countries such as India and China. On the other hand, the estimated cropland expansion could take place in many highly biodiverse regions, pointing out the need for spatially detailed and context-specific assessments to understand the possible outcomes of different food security strategies. Our multidisciplinary approach is relevant with respect to the Sustainable Development Goals for implementing and enforcing sustainable pathways for increasing agricultural production, and ensuring food security while conserving biodiversity and ecosystem services.
Halving the proportion of undernourished people in the developing countries by 2015 was one of the objectives of the United Nation's Millennium Development Goals (MDGs). The prevalence of undernourishment was reduced between the periods 1990–1992 and 2012–2014 from 18.7 to 11.3 % globally and from 23.4 to 13.5 % in developing countries in the same period of time (FAO et al. 2014). However, the 2014 MDG report argues that while this target has been met on a global scale, South Asia and sub-Saharan Africa are lacking behind (United Nations 2014). Therefore, the challenge of meeting food security goals is likely to persist in the future.
With a world population that is expected to grow from currently about 6.9–9.2 billion by 2050, as well as changing lifestyles and consumption patterns towards more protein-containing diets, global demand for food is projected to increase by 70–110 % by 2050 (Bruinsma 2011; Kastner et al. 2012; Tilman et al. 2011). In order to ensure sufficient food supply in the coming decades, several solutions are suggested. Besides reducing food waste and harvest losses, improving food distribution and access, and shifting diets towards consumption of fewer meat and dairy products, studies conclude that also the increase in global agricultural production is crucially important to meet the increasing demand (Garnett et al. 2013; Godfray et al. 2010; Gregory and George 2011; Gustavsson et al. 2011; Ray and Foley 2013; Mauser et al. 2015). At the same time, agricultural yields as well as production stability are affected by climate change, albeit study results vary between different approaches and assumptions (IIASA and FAO 2012; Rosenzweig et al. 2013; van Ittersum et al. 2013).
The possibilities to increase agricultural production consist of intensification of existing croplands and of their expansion into uncultivated areas, but both options are associated with environmental externalities, including the pollution of surface and groundwater by agrochemicals, unsustainable water withdrawals, and the loss of biodiversity (Foley et al. 2011). Biodiversity loss due to agricultural activities is particularly worrisome because it has consequences for ecosystem functioning, provisioning of ecosystem services, resilience of social–ecological systems, and ultimately the welfare of human societies (Corvalan et al. 2005). These potential trade-offs are clearly reflected in the recently published Sustainable Development Goals (SDGs). They highlight the topic of food security and sustainable agriculture (UN 2012), but compared to the MDGs which were restricted to socio-economic goals, they stress the need to ensure the protection, regeneration and resilience of global and regional ecosystems (ibid §4).
Land-use intensification has been variously shown to negatively impact local biodiversity in many regions of the world (Flynn et al. 2009; Newbold et al. 2015). However, land-use expansion with its associated loss and fragmentation of natural habitats is the globally more dominant driver of biodiversity loss, particularly in highly biodiverse tropical and subtropical regions (Foley et al. 2005; Hosonuma et al. 2012; Pereira et al. 2012). Despite the negative externalities of cropland expansion and continuing calls for sustainable intensification (Garnett et al. 2013; Tilman 1999; West et al. 2014), the future expansion of agricultural land is still considered to be a likely scenario (see, e.g., the OECD/FAO Agricultural Outlook). Land productivity considerably increased over the last decades (FAOSTAT 2015). However, when neglecting future changes in cropping patterns and management, current yield trends of the most important staple crops are not sufficient to double global food production by 2050 (Ray et al. 2013). According to FAO, cropland is expected to globally expand by 7 % until 2030 (Alexandratos and Bruinsma 2012). Consequently, it is crucially important to examine (1) how the expected global cropland expansion might affect food security in terms of agricultural production and prices, (2) where natural conditions are suitable for cropland expansion under changing climate conditions, and (3) whether this potential conversion to cropland would affect areas of high biodiversity value or conservation importance. Answering these questions requires a scientific analysis of the trade-offs between achieving food security via cropland expansion on the one hand and conserving biodiversity on the other.
In this study, we investigate the trade-offs between providing sufficient food in the future and sustaining biodiversity by using a three-step interdisciplinary approach. First, we examine the impact of cropland expansion on food security in terms of agricultural production quantity and prices. In the following step, we identify areas that are biophysically most suitable for the potential expansion of cropland under specific climate scenario conditions. Finally, we use information on global patterns of endemism richness, in order to identify hot spots where biodiversity could be most affected by potential cropland expansion.
Methods and data
We use three different approaches to analyse trade-offs between food security and biodiversity since they are driven by various interdependent socio-economic and biophysical parameters that operate at different spatial scales. First, to address the impact of cropland expansion on global and regional agricultural markets we apply the computable general equilibrium model DART-BIO. The model accounts for socio-economic developments such as population growth and changes in consumption patterns, while it considers repercussions between different production sectors and regions, simulating the development of food quantity and prices as important indicators for food security. Second, since this approach does not allow for localizing cropland expansion, we use biophysical drivers at the local scale such as climate, soil quality, and topography to determine where an expansion of cropland potentially would be possible under the given natural conditions. Third, we use data on endemism richness, a biodiversity metric that represents the importance of an area for conservation, to statistically examine the spatial concordance between patterns of global biodiversity and potential cropland expansion.
The DART-BIO model
The Dynamic Applied Regional Trade (DART) model is a multi-sectoral, multi-regional recursive dynamic computable general equilibrium (CGE) model of the world economy. The DART model has been applied to analyse international climate policies (e.g. Springer 1998; Klepper and Peterson 2006a), environmental policies (e.g. Weitzel et al. 2012), energy policies (e.g. Klepper and Peterson 2006b), and agricultural and biofuel policies (e.g. Kretschmer et al. 2009) among others.
The DART model is based on data from the Global Trade Analysis Project (GTAP) covering multiple sectors and regions. The economy in each region is modelled as a competitive economy with flexible prices and market-clearing conditions. The dynamic framework is recursively dynamic, meaning that the evolution of the economies over time is described by a sequence of single-period static equilibria connected through capital accumulation and changes in labour supply. The economic structure of DART is fully specified for each region and covers production, investment, and final consumption by consumers and the government.
DART is calibrated to the GTAP8 database (Narayanan et al. 2012) that represents production and trade data for 2007 with input–output tables for the world economy. The particular version used here (DART-BIO) contains 45 sectors and has detailed features concerning the agricultural sectors. Thirty-one activities in agriculture (thereof ten crop sectors) are explicitly modelled which represent a realistic picture of the complex value chains in agriculture. Several sectors that are only available on an aggregated level in the GTAP database are therefore split. The regional aggregation of 23 regions is chosen to include countries where main land use changes either due to biofuels production or because major changes in population, income, and consumption patterns are expected to emerge (e.g. Brazil, Malaysia, China). A detailed model description of the database and data processing can be found in Calzadilla et al. (2014).
In the DART-BIO model, we use different land types according to agro-ecological zones (AEZs), based on the GTAP database. AEZs represent 18 types of land, in each region with different crop suitability, productivity potential, and environmental impact. Each of the 18 AEZs is characterized by its particular climate, soil moisture/precipitation, and landform conditions which are basic for the supply of water, energy, nutrients, and physical support to plants. The newest version is available in the GTAP8 database by Baldos and Hertel (2012).
The mobility of land from one land-use type to another is commonly restricted by a nested constant elasticity of transformation (CET) function (see, e.g., Laborde and Valin 2012; Hertel et al. 2010). We choose a three-level nesting, in which land is first allocated between land for agriculture and managed forest. Then, agricultural land is allocated between pasture and crops. In the next level, cropland is allocated between rice, palm, sugar cane/beet and annual crops (wheat, maize, rapeseed, soybeans, other grains, other oilseeds, and other crops). At each level, the elasticity of transformation increases, reflecting that land is more mobile between crops than between forestry and agriculture (see Appendix Table 2). An important difference compared with other approaches (e.g. Laborde and Valin 2012; Bouët et al. 2010) is that we do not differentiate between land prices for growing annual crops. Since farmers can decide year by year which crop to plant, these crops can be easily substituted depending mainly on crop prices. Thus, different annual crops (e.g. wheat and maize) face only one land price entering into their costs. However, paddy rice and perennial crops such as palm fruit and sugar cane are less mobile and therefore face different land prices. Elasticities of transformation between the land uses are the main drivers of land allocation; however, they are very poorly studied in the literature. We currently use numbers from OECD's PEM model (Abler 2000; Salhofer 2000) which only covers developed countries plus Mexico, Turkey, and South Korea. Therefore, we had to choose values based on certain similarities for several countries (see Appendix Table 2). The effect of differences in land-use modelling is discussed in Calzadilla et al. (forthcoming).
Productivity in the agricultural sector is determined by changes in labour force, the rate of labour productivity growth, and the change in human capital accumulation, as well as the choice of the model structure (e.g. CET nesting) and parameter settings (e.g. elasticity of substitution). Hence, future yield growth is driven by changes in the total productivity factor. A more detailed description of production functions and dynamics is available in Calzadilla et al. (2014).
To simulate the effect of cropland expansion on food security, we set up two scenarios. The baseline scenario represents a continuation of the business as usual economic growth, population growth, and national policies as observed in the DART-BIO 2007 database. In this reference scenario, no expansion of cropland into non-managed land types is assumed.
The assumptions underlying the land expansion (LE) scenario are based on the FAO long-term baseline outlook 'World agriculture: towards 2030/2050'—The 2012 Revision (Alexandratos and Bruinsma 2012). These reports are the most authoritative sources for forecasts on crop production available. The forecasts are based on annual growth rate projections until 2030/2050 for crop production for selected important food crops.
From the information provided in the FAO forecast, we calculate annual growth rates for a linear increase in harvested area from the 2005/2007 base years, as provided by the FAO to 2030 (assumptions on growth rates include the most important crops cultivated on cropland). They enter the DART-BIO model as exogenous parameters. Globally, harvested area is expected to increase by about 7 %, while the regional distribution of land expansion or contraction varies between contraction of cropland (e.g. −11 % in Japan) and expansion of up to 28 % in Paraguay/Argentina/Uruguay/Chile (PAC) (Fig. 1 and Appendix Fig. 5). Accordingly, the land endowment for agricultural production in the DART-BIO model is set to consider these differences. While in northern and middle Europe, China, and India the harvested area shows no significant changes over time, the harvested area in Japan and Russia is reduced. The FAO data (Alexandratos and Bruinsma 2012) show that largest land expansions occur in Latin America (BRA, PAC, LAM) and Rest of Former Soviet Union and Europe (FSU).
Percentage change in global crop production under the land expansion scenario and harvested area in 2030 compared with 2007. Source simulation of production with DART-BIO; harvested area based on Alexandratos and Bruinsma (2012)
Natural potentials for future cropland expansion
The potential for the expansion of cropland is restricted by the availability of land resources and given local natural conditions. Consequently, area that is highly suitable for agriculture according to the prevailing local ecological conditions (climate, soil, terrain) but is not under cultivation today has a high natural potential for being agriculturally used. Policy regulations or socio-economic conditions can further restrict the availability of land for expansion, e.g., by designating protected areas, although they may be suitable for agriculture. Conversely, by applying, e.g., irrigation practices, land can be brought under cultivation, although it may naturally not be suitable. Here, we investigate the potentials for agricultural expansion for near future climate scenario conditions to identify the suitability of non-cropland areas for expansion.
We determine the available energy, water, and nutrient supply for agricultural suitability from climate, soil, and topography data, by applying the global dataset of crop suitability from a fuzzy logic approach by Zabel et al. (2014). It considers 16 economically important staple and energy crops at a spatial resolution of 30 arc seconds. These are barley, cassava, groundnut, maize, millet, oil palm, potato, rapeseed, rice, rye, sorghum, soy, sugarcane, sunflower, summer wheat, and winter wheat. The parameterization of the membership functions that describe each of the crops' specific natural requirements is taken from Sys et al. (1993). The considered natural conditions are: climate (temperature, precipitation, solar radiation), soil properties (texture, proportion of coarse fragments and gypsum, base saturation, pH content, organic carbon content, salinity, sodicity), and terrain (elevation, slope). The requirements for temperature and precipitation are defined over the growing period. For this case, we calculate the optimal start of the growing period, considering the temporal course of temperature and precipitation and thus the course of dry and rainy seasons.
As a result of the fuzzy logic approach, values in a range between 0 and 1 describe the suitability of a crop for each of the prevailing natural conditions at a certain location. The smallest suitability value over all parameters finally determines the suitability of a crop. The daily climate data (mean daily temperature and precipitation sum) are provided by simulation results from the global climate model ECHAM5 (Jungclaus et al. 2006) for near future (2011–2040) SRES A1B climate scenario conditions. Soil data are taken from the Harmonized World Soil Database (HWSD) (FAO et al. 2012), and topography data are applied from the Shuttle Radar Topography Mission (SRTM) (Farr et al. 2007). In order to gather a general crop suitability, which does not refer to one specific crop, the most suitable crop with the highest suitability value is chosen at each pixel. Thus, we create a potential land use for each pixel, based on the most suitable crops. This land use does not refer to actual land use and the actual allocation of crops but is used for the further calculation of natural expansion potential.
In addition to the natural biophysical conditions, we consider today's irrigated areas based on Siebert et al. (2013). We assume that irrigated areas globally remain constant until 2040, since adequate spatial data on possible future extend of irrigated areas do not exist, although it is likely that freshwater availability for irrigation could be limited in some regions, while in other regions surplus water supply could be used to expand irrigation practices (Elliott et al. 2014). However, it is difficult to project where irrigation practices will evolve, since it is also driven by economic considerations, such as the amount of investment costs that are required to establish irrigation infrastructure.
In principle, all agriculturally suitable land that is not used as cropland today has the natural potential to be converted into cropland. We assume that only urban and built-up areas are not available for conversion, although more than 80 % of global urban areas are agriculturally suitable (Avellan et al. 2012). However, it seems unlikely that urban areas will be cleared at the large scale due to high investment costs, growing cities, and growing demand for settlements. Concepts of urban and vertical farming usually are discussed under the aspects of cultivating fresh vegetables and salads for urban population. They are not designed to extensively grow staple crops such as wheat or maize for feeding the world in the near future. Urban farming would require one-third of the total global urban area to meet only the global vegetable consumption of urban dwellers (Martellozzo et al. 2015). Thus, urban agriculture cannot substantially contribute to global agricultural production of staple crops and consequently is not considered in this study.
Protected areas or dense forested areas are not excluded from the calculation, in order not to lose any information in the further combination with the biodiversity patterns (see chapter 2.3). We use data on current cropland distribution by Ramankutty et al. (2008) and urban and built-up area according to the ESA-CCI land-use/land-cover dataset (ESA 2014). From these data, we calculate the 'natural expansion potential index' (I exp) that describes the natural potential for an area to be converted into cropland as follows:
$$I_{ \exp } = S \times A_{\text{av}}$$
The index is determined by the quality of crop suitability (S) (values between 0 and 1) multiplied with the amount of available area (A av) for conversion (in percentage of pixel area). The available area includes all suitable area that is not cultivated today and not classified as urban or artificial area. The index ranges between 0 and 100 and indicates where the conditions for cropland expansion are more or less favourable, when taking only natural conditions into account, disregarding socio-economic factors, policies, and regulations that drive or inhibit cropland expansion.
Since it is unknown which crop might be used for expansion, the index uses the most suitable crop at each pixel (as given by the general crop suitability) for determining the natural potential for expansion. Consequently, not all crops might be suitable for expansion where I exp is greater than zero. The index is a helpful indicator for identifying areas where natural conditions potentially allow for expansion of cropland in the near future from a biophysical point of view. The index does not allow for determining the likelihood of cropland expansion, since it ignores socio-economic factors and policy regulations because we do not aim to understand the factors that may affect cropland expansion. Rather, our goal is to localize potential conflicting areas.
Trade-offs between biodiversity and potential cropland expansion
As indicators of biodiversity, we use global endemism richness for birds, mammals, and amphibians created from expert-based range maps obtained from the International Union for Conservation of Nature (IUCN 2012) and Birdlife databases (BirdLife 2012). Habitat changes due to cropland expansion are the principal driver of extinction risk in these animal groups (Pereira et al. 2012). We choose endemism richness over other biodiversity indicators because it combines species richness with a measure of endemism (i.e. the range sizes of species within an assemblage) and thus indicates the relative importance of a site for global conservation (Kier et al. 2009). We calculate endemism richness as the sum of the inverse global range sizes of all species present in a grid cell. The data are scaled to an equal area grid of 110 × 110 km at the equator (1 arc degree) because at finer spatial resolutions, the underlying species range maps exhibit excessively high false-presence rates, overestimating the area of occupancy of individual species (Hurlbert and Jetz 2007).
Following similar methods as in Kehoe et al. (2015), we overlay endemism richness indicators with the natural expansion potential index to examine the spatial concordance between patterns of global biodiversity and suitability for cropland expansion. First, we statistically quantify the spatially explicit association between endemism richness and cropland expansion potentials using the bivariate version of the local indicator of spatial association (LISA) (Anselin 1995). LISA represents a local version of the correlation coefficient and shows how the nature and strength of the association between two variables vary across a study area. The method allows for the decomposition of global indicators, such as Moran's I, into the contribution of each individual observation (e.g. a grid cell), while giving an indication of the extent of significant spatial clustering of similar values around that observation. Using OpenGeoDa version 1.2.0 (Anselin et al. 2006), we calculate the local Moran's I statistic of spatial association for each 110-km grid cell as:
$$I_{i} = \frac{{x_{i} - \bar{x}}}{{s^{2} }}\mathop \sum \limits_{j = 1, j \ne i}^{n} w_{ij} \left( {y_{j} - \bar{y}} \right)$$
where x i and y j are standardized values of variable x (e.g. cropland expansion potentials) and variable y (e.g. endemism richness) for grid cells i and j, respectively, \(\bar{x}\) and \(\bar{y}\) are the means of the variables, w ij is the spatial weight between cell i and j inversely proportional to Euclidean distance between the two cells, and s 2 is the variance. Based on the values of local Moran's I, we identify and map spatial associations of (1) high–high values, that is spatial hot spots in which locations with high values of cropland expansion potentials are surrounded by high values of endemism richness, (2) low–low values, that is spatial cold spots in which locations with low values of cropland expansion potentials are surrounded by low values of endemism richness, and (3) high–low and low–high values, where the spatial association between the variables is negative (inverse). The strength of the relationship is measured at the 0.05 level of statistical significance calculated by a Monte Carlo randomization procedure based on 499 permutations (Anselin et al. 2006). We use the resulting areas of high–high values to generate a summary map of high-pressure regions for all three taxonomic groups (birds, mammals, and amphibians). As a second analysis, we delineate the 'hottest' hot spots of high cropland expansion potentials and endemism richness by extracting the top 5 and 10 % of the data distribution (Ceballos and Ehrlich 2006). Intersecting these top values of both variables, we create maps of the top pressure regions, where high biodiversity is most threatened by potential cropland expansion.
The impact of cropland expansion on food security
Food supply and accessibility depend not only on the ability to produce a sufficient quantity and quality of food, but also on the food price level and incomes relative to these prices. We apply the CGE Model DART-BIO in order to compare agricultural production and prices on global and regional scale under two scenarios. The land expansion (LE) scenario (cp. "The DART-BIO model") is run and compared to results from a baseline scenario without cropland expansion to quantify the price and production changes. To illustrate the effect of expanding cropland on food security, first the changes in global and regional production quantities and trade flows are displayed. Second, changes in price on global and regional scale under the LE scenario are discussed.
Food production and trade flows
Under the LE scenario, global production of primary agricultural goods increases by 3–9 %, while processed food production rises by 3 % compared with the baseline scenario in 2030. A detailed table with price and quantity changes for all crops and processed food sectors is available in Appendix Table 3.
Regionally, the cropland expansion has different impacts on food production. Driven by the amount in cropland expansion/reduction of the scenario, crop production in European countries except Benelux as well as in Russia, Japan, and India is reduced in 2030 compared with 2007 (see Fig. 1). Largest increases in crop production are simulated for Paraguay, Argentina, Uruguay, Chile (PAC) (+34 %), and other regions that face problems in improving food security (Brazil +16 %, LAM +13 %, AFR +11 %, SEA +14 %). Comparing production in 2030 under the LE scenario to the baseline scenario, production of maize, soy beans, and wheat shows largest increase in Latin America. South-East Asia (SEA) and Malaysia/Indonesia (MAI) increase paddy rice production by 11–13 %, while also 'Rest of Agriculture' (AGR) rises considerably. Production of, e.g., wheat and AGR in India drops, since expansion potentials are very limited. These results indicate that while food production rises on global average, not all regions produce more under the LE scenario. Thus, their ability to produce a sufficient quality of food is not improved when expanding cropland as under the LE scenario.
Countries are connected via bilateral trade. Different values for cropland expansions result in changing comparative advantages of different regions, which affects trade flows. In 2030, regions in Asia are net importers of most agricultural goods in the baseline scenario. South-East Asia (SEA) reduces its net imports of processed food by more than half under the LE scenario compared with the baseline. At the same time, SEA exports more AGR (+63 %). These exports mainly target India and China, who also increase imports from other regions. Indian's net imports of crops strongly increase such that private consumption of food in India rises. Regions in Latin America are net exporters of crops and net importers of processed food under the baseline scenario in 2030. Under the LE scenario, net exports of crops increase compared with the baseline, while less processed food is imported. This indicates that cropland expansion, though distributed differently in different regions, provides more food to consumers in all regions compared with the baseline run.
Agricultural prices are also important for food security, particularly for net importing countries, and people who do not produce food themselves. Comparing results of the LE scenario with the baseline, global average prices of crops fall by 6–20 % (see Table 3 in Appendix). The highest price decreases are simulated for soy beans, since they are produced in regions with the highest cropland expansions. In addition, by 2030 the demand for soy beans is larger compared with, e.g., paddy rice as soy beans are used as feedstuff to satisfy rising meat consumption over time, and biofuel quotas. As a result, soybean areas expand by 13 % compared with the baseline run. The area expansion for paddy rice amounts to 5 %, which results in a global average price decrease of 6 %.
Driven by the scenario assumptions, regional production costs, and trade flows, regional price changes vary considerably. Taking wheat as an example, strongest price decreases are simulated for Brazil and PAC, where most of the cropland expansion takes place (see Table 1). But also regions in which cropland does not expand or only to a limited degree profit from decreasing crop prices. While, e.g., wheat production in India decreases under the LE scenario compared with the baseline in 2030, wheat prices drop by 5 % since India benefits from low wheat prices on the world market (−11 %) (see Table 1).
Table 1 Percentage change in wheat prices
In summary, our results indicate that cropland expansion improves food security, particularly in those regions that currently face problems in providing sufficient and affordable quantities of food to people. However, data from FAO used in the LE scenario provide no spatial information on the locations within the regions where expansion takes place. Accordingly, no statement on substituted land cover and possible loss of biodiversity is possible. Therefore, in the following section, potential areas for cropland expansion are identified.
Identification of natural potentials for cropland expansion
Assuming that cropland expansion is potentially possible where the quality of land is suitable for the cultivation of crops and area is still available for the conversion of land into cropland, Fig. 2 shows the calculated index of the natural expansion potential. The greater the agricultural suitability and the larger the available area for expansion, the greater the value of the index. Red coloured areas in Fig. 2 indicate high natural potential for cropland expansion.
Index of natural potentials for the expansion of cropland. The index is calculated as the result of agricultural suitability under SRES A1B climate scenario conditions for 2011–2040 and the availability of suitable land for expansion. The index ranges from 1 (low potential for expansion, green) to 100 (high potential for expansion, red). Values with 0 (no potential for expansion) are masked out. Map in Eckert IV projection, 30-arc-second spatial resolution
We identify high natural expansion potentials in African countries (e.g. Cameroon, Chad, Gabon, Sudan, western parts of Ethiopia, and Tanzania), Central and South America (Mexico, Nicaragua, Uruguay, and parts of Argentina), fragmented parts of Asia (north-eastern part of China, northern parts of Australia and Papua New Guinea) and small parts of Russia. These areas are characterized by fertile soils and adequate climate conditions for at least one of the investigated crops, while at the same time these areas are not under cultivation today according to the applied data. The high expansion potential in parts of tropical countries, such as Cameroon, Gabon, Nicaragua, Indonesia, Malaysia, Papua New Guinea, and the Philippines, is mainly caused by the high crop suitability of oil palm in these regions, while other crops are not suitable here (Zabel et al. 2014). Regions with high natural expansion potential in the Sahel Zone mainly owe their high values to the good suitability of sorghum.
Certainly, many of the named regions with high natural potential for expansion are in the focus of cropland expansion and land grabbing already today. While the inner tropical basins of Brazil and the Congo show large areas for possible expansion, the value for the expansion potential index is relatively low here, since the agricultural suitability is inhibited due to marginal soil quality conditions. On the other hand, the potential for expansion is relatively low in North America and Europe, where most of the suitable areas are already under cultivation today. Therefore, the potential for further expansion is relatively low. Topography also affects agricultural suitability, and thus, the natural potential for expansion depends also on the extent of suitable valleys within mountainous areas.
Increasing mean temperatures due to climate change until 2040 are considered in the calculation of natural expansion potentials. Climate change, e.g., affects the northern hemisphere, where the climatic frontier for cultivation shifts northwards with time such that additional land potentially becomes suitable and thus is available for expansion. On the other hand, suitability decreases for most of the 16 investigated crops due to climate change, especially for cereals in the tropics and the Mediterranean.
Spatial patterns of potential cropland expansion and biodiversity
The LISA analyses reveal regionally variable spatial concordance between patterns of cropland expansion potentials and global biodiversity (Fig. 3). Regions with low potential of cropland expansion and low biodiversity (i.e. spatial cold spots) are similar across all three taxonomic groups, covering mostly non-arable, desert, or ice-covered land (39 % of terrestrial ecosystems; Fig. 3a–c). The hot spots, i.e. regions where high biodiversity is potentially threatened by cropland expansion, vary more substantially among the considered vertebrate groups but all are focused primarily in the tropics, covering 18 % of the terrestrial land surface. While the hot spot patterns for birds and mammals show high spatial congruence (67 % overlap), the areas of high expansion potentials associated with high endemism richness are relatively smaller for amphibians (41 % overlap with the other taxa) due to the generally smaller ranges of amphibian species concentrated in specific geographical areas. However, the summary of statistically significant hot spots for all three taxonomic groups shows a spatially consistent pattern of high-pressure regions (Fig. 3d), covering Central and South America, Central Africa and Madagascar, Eastern Australia, and large portions of Southeast Asia. Other regions with higher suitability for cropland expansion either are not significantly associated with endemism richness or occur in areas with relatively low levels of endemism richness (11 % of the terrestrial land surface), e.g. the Midwest of North America, Eastern Europe, or parts of sub-Saharan Africa.
Local indicator of spatial association (LISA) between cropland expansion potentials and endemism richness for birds (a), mammals (b), and amphibians (c). The pattern shows how the nature and strength of the association between two variables vary across the globe. High–high clusters show hot spot locations, in which high cropland expansion potentials are associated with high values of endemism richness. Low–low clusters show cold spot locations, in which low cropland expansion potentials are associated with low values of endemism richness. High–low and low–high clusters show inverse spatial association. The map in (d) summarizes all high–high associations to show high-pressure regions for one, two, or all three taxonomic groups. Maps in Eckert IV projection, 1-arc-degree spatial resolution
The spatial intersect of the top 5 and 10 % of data on cropland expansion and biodiversity (Fig. 4) further pinpoints the top pressure regions, where high levels of endemism richness for all considered taxa may be most threatened by potential cropland expansion (3 % overlap for top 5 % data and 13 % overlap for top 10 % data). These 'hottest' hot spots of potential future conflict between biodiversity and agriculture are found in Central America and the Caribbean, in the tropical Andes and south-western Brazil, in West and East Africa, including Madagascar, and in several parts of tropical Asia, in particular the Indochina region, the Indonesian islands, and Papua New Guinea.
Overlay of top 5 % (a) and top 10 % (b) of natural cropland expansion potentials and global endemism richness for three vertebrate taxa (birds, mammals, and amphibians). The intersect of both datasets (in red) highlights the top pressure regions, where high biodiversity (i.e. high numbers of range size equivalents) may be particularly threatened by potential cropland expansion. Maps in Eckert IV projection, 1-arc-degree spatial resolution
Although our results highlight relatively large areas of potential future pressure on biodiversity, it does not mean that all types of habitats in each 110-km grid cell would be equally affected if cropland expansion occurred. When using endemism richness as an indicator of biodiversity, our concern is not the area of habitat but the number of range equivalents, i.e. fractions of species global ranges that are contained within a grid (Kier et al. 2009). For example, many mountainous regions in the tropics identified as high-pressure regions have high endemism richness due to many different species inhabiting zones along topographical and climate gradients. Presumably, the habitats in higher elevations are less likely to be affected than habitats located in lower regions because of differences in soil characteristics, slope steepness, accessibility, and other fine-scale factors restricting agricultural suitability and thus natural expansion potential in mountainous areas.
On the other hand, we also identify areas where high suitability for additional expansion of food production may pose lower threats to conservation of biodiversity. These regions, such as Eastern Europe, sub-Saharan Africa, or Northeast China, coincide with the 'extensive cropping land system' (Václavík et al. 2013) that represents relatively easily achievable opportunity for an expansion or intensification of agricultural production, especially for wheat, maize, or rice. Here, large production gains could be achieved if yields were increased to only 50 % of attainable yields (Mueller et al. 2012). However, even areas with relatively low endemism richness may still harbour valuable species or include cultural heritage that cropland expansion may threaten. Our analysis identifies where the high- and low-pressure regions are located but does not explain how the various aspects of biodiversity would be threatened by future land-use changes. Therefore, we caution that more detailed and context-specific assessments are needed to understand the possible outcomes of different expansion strategies. In addition to biodiversity and economic indicators, these assessments should consider other (non-provisioning) ecosystem services, resilience of land-use systems, and cultural and societal outcomes of increasing food production (Kehoe et al. 2015).
Trade-offs between food security and biodiversity are driven by various interdependent socio-economic and biophysical parameters that operate at both global and local scales. In this study, we account for these parameters by combining three methodological approaches to analyse the effects of expanding agricultural production: (1) we run an economic scenario analysis with a computable general equilibrium model to examine the effect of an exogenous cropland land expansion on changes in crop production and prices, (2) we determine where an expansion of cropland would be possible under the given natural conditions, and (3) we statistically analyse where the natural potential for cropland expansion may threaten biodiversity.
We show that there are potential trade-offs between increased food production and protection of biodiversity. On the one hand, allowing the expansion of cropland generally results in improved food security in terms of decreased food prices and increased quantity, not only in those regions where crop production rises, but also in net importing countries such as India and China. On the other hand, the results show that estimated cropland expansion could take place in many regions that are valuable for biodiversity conservation. From an economic point of view, the highest expected expansion of cropland according to FAO takes place in South America, particularly in Argentina, Bolivia, and Uruguay. Considering that these countries also have a high biophysical potential for cropland expansion as well as relatively high endemism richness, they represent valuable regions from the conservation point of view but with the highest pressure for land clearing. Similar conclusions can be made for regions in Australia, Brazil, and Africa. Our analyses highlight such regions that deserve further attention and more detailed and context-specific assessments to understand the possible outcomes of different food security strategies, while at the same time establishing mechanisms to efficiently protect habitats with high biodiversity.
Our results are relevant with respect to the SDGs for implementing and enforcing sustainable pathways for increasing agricultural production, ensuring food security while conserving biodiversity and ecosystem services. A report by the International Council for Science (ICSU) and the International Social Science Council (ISSC) states that some goals may conflict. The presented approach contributes to identifying the key trade-offs and complementarities among goals and targets, as required in SDGs. In addition, our study contributes to the land sharing versus sparing debate that generated a controversial discussion on the pressing problems of feeding a growing human population and conserving biodiversity (Fischer et al. 2008; Godfray 2011; Phalan et al. 2011; von Wehrden et al. 2014). Our approach represents one of the first examples of moving forward from the bipolar framework (Fischer et al. 2014). We advance the framework by (1) accounting for economic parameters, thus focusing on food security rather than mere production, (2) treating agricultural landscapes as complex social–ecological systems, (3) accounting for biophysical and socio-economic factors that operate at different spatial scales, and (4) defining biodiversity with a metric that combines species richness with conservation value of the area.
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Many thanks to Holger Kreft for assistance with biodiversity datasets. This project was supported by the German Federal Ministry of Education and Research (Grant 01LL0901A: Global Assessment of Land Use Dynamics, Greenhouse Gas Emissions and Ecosystem Services—GLUES). CM acknowledges support by sDiv, the Synthesis Centre of the German Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig (DFG FZT 118).
Kiel Institute for the World Economy, Kiellinie 66, 24105, Kiel, Germany
Ruth Delzeit
Department of Geography, Ludwig-Maximilians-Universität, Luisenstraße 37, 80333, Munich, Germany
Florian Zabel
sDiv - Synthesis Centre, German Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig, Deutscher Platz 5e, 04103, Leipzig, Germany
Carsten Meyer
Macroecology and Conservation Biogeography, Georg-August-University of Göttingen, Büsgenweg 1, 37077, Göttingen, Germany
Department of Computational Landscape Ecology, UFZ - Helmholtz Centre for Environmental Research, Permoserstraße 15, 04318, Leipzig, Germany
Department of Ecology and Environmental Sciences, Faculty of Science, Palacký University Olomouc, Šlechtitelů 27, 78346, Olomouc, Czech Republic
Correspondence to Ruth Delzeit.
An erratum to this article is available at http://dx.doi.org/10.1007/s10113-016-0944-0.
Below is the link to the electronic supplementary material.
Supplementary material 1 (DOCX 214 kb)
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Delzeit, R., Zabel, F., Meyer, C. et al. Addressing future trade-offs between biodiversity and cropland expansion to improve food security. Reg Environ Change 17, 1429–1441 (2017). https://doi.org/10.1007/s10113-016-0927-1
Issue Date: June 2017
Cropland expansion
Endemism richness
Crop suitability
Simulation models
Spatial econometrics | CommonCrawl |
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{\let\thefootnote\relax\footnotetext {2010 Mathematics Subject Classification: 05A05, 05A15.
}}
\begin{abstract} \noindent We investigate the structure of the two permutation classes defined by the sets of forbidden patterns $\{\mathbf{1234},\mathbf{2341}\}$ and $\{\mathbf{1243},\mathbf{2314}\}$. By considering how the Hasse graphs of permutations in these classes can be built from a sequence of rooted source graphs, we determine their algebraic generating functions. Our approach is similar to that of ``adding a slice'', used previously to enumerate various classes of polyominoes and other combinatorial structures. To solve the relevant functional equations, we make extensive use of the kernel method. \end{abstract}
\section{Introduction} We consider a permutation to be simply an arrangement of the numbers $1,2,\ldots n$ for some positive $n$. A permutation $\pi$ is said to be \emph{contained} in another permutation $\sigma$ if $\sigma$ has a subsequence whose terms have the same relative ordering as those of $\pi$. For example, $\mathbf{3241}$ is contained in $\mathbf{1573462}$ because the subsequence $\mathbf{5362}$ is ordered in the same way as $\mathbf{3241}$. If $\pi$ is not contained in $\sigma$ then we say that $\sigma$ \emph{avoids} $\pi$. For example, $\mathbf{1573462}$ avoids $\mathbf{3214}$. In the context of containment and avoidance, a permutation is often called a \emph{pattern}.
The containment relation is a partial order on the set of all permutations, and a set of permutations closed downwards (a down-set) in this partial order is called a \emph{permutation class}. It is natural to define a permutation class by the minimal set of permutations that it avoids. This minimal forbidden set of patterns is known as the \emph{basis} of the class. The class with basis $B$ is denoted $\av(B)$.
Given a permutation class $\CCC$, we denote by $\CCC_n$ the set of permutations in $\CCC$ of length $n$. The (univariate) \emph{generating function} of $\CCC$ is then
$\sum_{n\geqslant1}|\CCC_n|z^n = \sum_{\sigma\in\CCC}z^{|\sigma|}$, where $|\sigma|$ is the length of $\sigma$. The \emph{growth rate} of $\CCC$ is defined by the limit $
\liminfty\sqrt[n]{|\CCC_n|}, $ if it exists. It is widely believed (see the first conjecture in~\cite{Vatter2014}) that all permutation classes have a growth rate.
In the study of permutation classes, there has been significant interest in deriving the generating functions for classes with a few small basis elements (see~\cite{WikiEnumPermClasses} for an up-to-date list of results). This has led to the enrichment of the theory of permutation classes due to the requisite development of a variety of enumeration techniques.
We add to this work by proving the following two theorems:
\thmbox{ \begin{thmO}\label{thmF} The class of permutations avoiding $\mathbf{1234}$ and $\mathbf{2341}$ has the algebraic generating function $$
\frac {2-10\+z+9\+z^2+7\+z^3-4\+z^4 \:-\: (2-8\+z+9\+z^2-3\+z^3)\+\sqrt{1-4\+z}} {(1-3\+z+z^2)\+\big((1-5\+z+4\+z^2) \:+\: (1-3\+z)\+\sqrt{1-4\+z}\big)} . $$ Its growth rate is equal to 4. \end{thmO} }
\thmbox{ \begin{thmO}\label{thmE} The class of permutations avoiding $\mathbf{1243}$ and $\mathbf{2314}$ has an algebraic generating function $F(z)$ which satisfies the cubic polynomial equation $$
(z-3\+z^2+2\+z^3) \:-\: (1-5\+z+8\+z^2-5\+z^3)\+F(z) \:+\: (2\+z-5\+z^2+4\+z^3)\+F(z)^2 \:+\: z^3\+F(z)^3 \;=\; 0 . $$ Its growth rate is approximately 5.1955, the greatest real root of the quintic polynomial $$ 2 - 41\+z + 101\+z^2 - 97\+z^3 + 36\+z^4 -4\+z^5 . $$ \end{thmO} }
\textbf{Hasse graphs}
Corresponding to each permutation $\sigma$, we define an ordered plane graph $H_\sigma$, which we call its \emph{Hasse graph}. If $P_\sigma$ is the poset on the points $(i,\sigma_i)$ in which $(i,\sigma_i)<(j,\sigma_j)$ if both $i<j$ and $\sigma_i<\sigma_j$, then $H_\sigma$ is the graph corresponding to the Hasse diagram of $P_\sigma$. See the figures throughout this paper for illustrations showing the Hasse graphs of permutations. In practice, we tend not to distinguish between a permutation and its Hasse graph. The minimal elements of the poset $P_\sigma$ are known as the \emph{left-to-right minima} of the permutation $\sigma$. Similarly, maximal elements of $P_\sigma$ are called \emph{right-to-left maxima} of $\sigma$.
Hasse graphs of permutations were previously considered by Bousquet-M\'elou \& Butler~\cite{BMB2007}, who determined the algebraic generating function of the family of \emph{forest-like} permutations whose Hasse graphs are acyclic. More recently, they have been used by the present author~\cite{Bevan2014} to establish a new lower bound for the growth rate of $\av(\mathbf{1324})$.
Given a permutation $\sigma$, we partition the vertices of $H_\sigma$ by spanning it with a sequence of graphs, which we call the \emph{source graphs} of $\sigma$. There is one source graph for each left-to-right minimum of $\sigma$. Suppose $u_1,\ldots,u_m$ are the vertices of $H_\sigma$ corresponding to the left-to-right minima of $\sigma$, listed from left to right. Then the $k$th source graph $G_k$ is the graph induced by $u_k$ and those vertices of $H_\sigma$ lying above and to the right of $u_k$ that are not in $G_1,\ldots,G_{k-1}$. We refer to $u_k$ as the \emph{root} of source graph $G_k$. See Figure~\ref{figFPerm} for an illustration. The structure of the source graphs of permutations in a specific permutation class is constrained by the need to avoid the patterns in the basis of the class. If the source graphs for some class are acyclic, we refer to them as \emph{source trees}.
The \emph{bottom subgraph} of a Hasse graph is the graph induced by its lowest vertex (the least entry in the permutation) and all the vertices lying above and to its right. Observe that the bottom subgraph may contain vertices from more than one source graph. For example, the bottom subgraph of the Hasse graph in Figure~\ref{figFPerm} contains vertices from three source graphs. Bottom subgraphs of permutations in a specific permutation class satisfy the same structural restrictions as do the source graphs. We refer to an acyclic bottom subgraph as a \emph{bottom subtree}.
We build the Hasse graph of a permutation by starting with a source graph and then repeatedly adding another source graph to the lower right. The technique is similar to that of ``adding a slice'', which has been used to enumerate constrained compositions and other classes of polyominoes, a topic of interest in statistical mechanics (see, for example, Bousquet-M\'elou's review paper~\cite{Bousquet-Melou1996}, the books of van Rensburg~\cite{vanRensburg2000} and Guttmann~\cite{Guttmann2009}, and Flajolet \& Sedge\-wick~\cite[Examples~III.22 and~V.20]{FS2009}). When a source graph is added, its vertices are interleaved horizontally with the non-root vertices of the bottom subgraph of the graph built from the previous source graphs. Typically, the positioning of the vertices of the new source graph is constrained by the need to avoid forbidden patterns.
In order to derive the univariate generating functions we require, we make use of multivariate functions involving additional ``catalytic'' variables that record certain parameters of the bottom subgraph of the permutations. These additional variables enable us to establish recurrence relations which we can then solve using the kernel method. Typically, when employing a multivariate generating function, we treat it simply as a function of the relevant catalytic variable, writing, for example, $F(u)$ rather than $F(z,u)$.
Occasionally, we also make use of a variant of the symbolic structural notation presented in Flajolet \& Sedge\-wick~\cite{FS2009} to establish functional equations. In particular, $\ZZZ$ is the atomic class consisting of a single vertex, and we use $\seq{\AAA}$ to represent a possibly empty sequence of elements of $\AAA$ and $\seqplus{\AAA}$ to represent a non-empty sequence of elements of $\AAA$.
The two classes we enumerate are quite distinct structurally. A source graph in $\av(\mathbf{1234},\mathbf{2341})$ consists of a root together with a \prefix{$\mathbf{123}$}avoider formed from the non-root vertices. However, the presence of a $\mathbf{123}$ forces any subsequent source graph to be simply a fan. In contrast, $\av(\mathbf{1243},\mathbf{2314})$ has plane source graphs and a much more uniform structure. We enumerate $\av(\mathbf{1234},\mathbf{2341})$ in Section~\ref{sectF}. In doing so, the kernel method is used six times to solve the relevant functional equations. The class $\av(\mathbf{1243},\mathbf{2314})$ is enumerated in Section~\ref{sectE}. This requires an unusual simultaneous double application of the kernel method.
\section{Permutations avoiding \texorpdfstring{$\mathbf{1234}$ and $\mathbf{2341}$}{1234 and 2341}}\label{sectF}
Let us use $\FFF$ to denote $\av(\mathbf{1234},\mathbf{2341})$. The structure of class $\FFF$ depends critically on the presence or absence of occurrences of the pattern $\mathbf{123}$. In light of this, to enumerate this class, we partition it into three sets $\AAA$, $\BBB$ and $\CCC$ as follows: \begin{bullets}
\item $\AAA=\av(\mathbf{123})$.
\item $\BBB=\av(\mathbf{1234},\mathbf{2341},\mathbf{13524},\mathbf{14523})\setminus\AAA$. Every permutation in $\BBB$ contains at least one occurrence of a $\mathbf{123}$, but avoids $\mathbf{13524}$ and $\mathbf{14523}$.
\item $\CCC=\av(\mathbf{1234},\mathbf{2341})\setminus(\AAA\cup\BBB)$. Every permutation in $\CCC$ contains a $\mathbf{13524}$ or a $\mathbf{14523}$. \end{bullets}
We refer to a permutation in $\AAA$ as an \prefix{$\AAA$}permutation, and similarly for $\BBB$ and $\CCC$.
\begin{figure}
\caption{
A permutation in class $\FFF$, spanned by four source graphs}
\label{figFPerm}
\end{figure} The addition of a source graph to a \prefix{$\CCC$}permutation can only yield another \prefix{$\CCC$}permutation (since it can't cause the removal a $\mathbf{13524}$ or $\mathbf{14523}$ pattern). Similarly, the addition of a source graph to a \prefix{$\BBB$}permutation can't result in an \prefix{$\AAA$}permutation. Hence, we can enumerate $\AAA$ without first considering $\BBB$ or $\CCC$, and can enumerate $\BBB$ before considering $\CCC$.
Before investigating the structure of permutations in $\AAA$, $\BBB$ and $\CCC$, let us briefly examine what a typical source graph in $\FFF$ looks like. Firstly, the avoidance of $\mathbf{1234}$ means that the non-root vertices of any source graph form a \prefix{$\mathbf{123}$}avoider. Secondly, the avoidance of $\mathbf{2341}$ presents no additional restriction on the structure of a source graph, because the presence of a $\mathbf{2341}$ would imply the presence of a $\mathbf{123}$ in the non-root vertices. Thus a source graph in $\FFF$ consists of a root together with a \prefix{$\mathbf{123}$}avoider formed from the non-root vertices.
\subsubsection*{The structure of set $\AAA$}
We begin by looking at $\AAA=\av(\mathbf{123})$. As is very well known, this class is enumerated by the Catalan numbers. However, we need to keep track of the structure of the bottom graph. So we must determine the appropriate bivariate generating function.
Let $\AAA_\ssS$ denote the set of source graphs in set $\AAA$. Now, each member of $\AAA_\ssS$ is simply a \emph{fan}, a root vertex connected to a (possibly empty) sequence of pendant edges. Bottom subgraphs are also fans. Thus source graphs and bottom subgraphs of $\AAA$ are acyclic.
When enumerating $\AAA$, we use the variable $u$ to mark the \emph{number of leaves} (non-root vertices) in the bottom subtree. The generating function for $\AAA_\ssS$ is thus given by $$ A_\ssS(u) \;=\; z+z^2\+u+z^3\+u^2+\ldots \;=\; \frac{z}{1-z\+u} . $$
We now consider the process of building an \prefix{$\AAA$}permutation from a sequence of source trees. When a source tree is added to an \prefix{$\AAA$}permutation, the root vertex of the source tree may be inserted to the left of zero or more of the leaves of the bottom subtree. See Figure~\ref{figABuild} for an illustration. Note that, in this and other similar figures, the original bottom subgraph is displayed to the upper left, with the new source graph to the lower right.
The action of adding a source tree is thus seen to be reflected by the linear operator $\oper_{\ssA}$ whose effect on $u^k$ is given by $$ \oper_{\ssA}\big[u^k\big] \;=\; A_\ssS(u)\+(1+u+\ldots+u^k) \;=\; A_\ssS(u)\+\frac{1-u^{k+1}}{1-u}. $$ Hence, the bivariate generating function $A(u)$ for $\AAA$ is defined by the following recursive functional equation: \begin{equation*}
A(u) \;=\; A_\ssS(u) \:+\: A_\ssS(u)\+\frac{A(1)-u\+A(u)}{1-u}. \end{equation*} \begin{figure}
\caption{Adding a source tree to the bottom subtree of an \prefix{$\AAA$}permutation}
\label{figABuild}
\end{figure}This equation can be solved using the \emph{kernel method}. To start, we express $A(u)$ in terms of $A(1)$, by expanding and rearranging to give \begin{equation}\label{eqAKernel} A(u) \;=\; \frac{z \+\big(1-u+A(1)\big)}{1-u+z\+u^2}. \end{equation} Equivalently, we have the equation $$ (1-u+z\+u^2)\+A(u) \;=\; z \+\big(1-u+A(1)\big) . $$ Now, if we set $u$ to be a root of the multiplier of $A(u)$, we obtain a linear equation for $A(1)$. This is known as ``cancelling the kernel'' (the multiplier of $A(u)$ being the kernel).
The appropriate root to use can be identified from the combinatorial requirement that the series expansion of $A(1)$ contains no negative exponents and has only non-negative coefficients.
In this case, the correct root is $u=(1-\sqrt{1-4\+z})/2\+z$, which yields the univariate generating function for $\AAA$, $$ A(1) \;=\; \frac{1-\sqrt{1-4\+ z}}{2\+ z}-1. $$ This is the generating function for the Catalan numbers as expected.
Finally, by substituting for $A(1)$ back into~\eqref{eqAKernel} we get the following explicit algebraic expression for $A(u)$: $$ A(u) \;=\; \frac{1-2\+z\+u-\sqrt{1-4\+z}}{2\+(1-u+z\+u^2)} . $$ On this occasion, we have explained every step of the derivation. On subsequent occasions, we present fewer details of the algebraic manipulations.
\subsubsection*{The structure of set $\BBB$}
We now consider set $\BBB$. Recall that sets $\BBB$ and $\CCC$ consist of those permutations in class $\FFF$ that contain at least one occurrence of a $\mathbf{123}$. We need to keep track of the position of the leftmost occurrence of a $\mathbf{3}$ in such a pattern. Given a permutation in $\BBB$ or $\CCC$, let us call the vertex corresponding to the leftmost $\mathbf{3}$ in a $\mathbf{123}$ the \emph{spike}. In the figures, the spike is marked with a star.
We now make a key observation.\label{obsKey} When adding a source graph to a permutation containing a $\mathbf{123}$, no vertex of the source graph may be positioned to the right of the spike, or else a $\mathbf{2341}$ would be created. Hence, the spike in any permutation in classes $\BBB$ or $\CCC$ occurs in its bottom subgraph. When enumerating sets $\BBB$ and $\CCC$, we use the variable $u$ to mark \emph{the number of vertices to the left of the spike} in its bottom subgraph.
\begin{figure}
\caption{A source graph in set $\BBB$}
\label{figBSource}
\end{figure} Let $\BBB_\ssS$ be the set of source graphs in set $\BBB$. Since \prefix{$\BBB$}permutations contain a $\mathbf{123}$ but avoid $\mathbf{13524}$ and $\mathbf{14523}$, the non-root vertices of a permutation in $\BBB_\ssS$ consist of two descending sequences, the second sequence beginning (with the spike) above the last vertex in the first sequence. See Figure~\ref{figBSource} for an illustration. If we consider the non-root vertices in order from top to bottom, then it can be seen that $\BBB_\ssS$ is defined by the structural equation $$ \BBB_\ssS \;=\; u\+\ZZZ \:\times\: \seq{u\+\ZZZ} \:\times\: \ZZZ \:\times\: \seq{u\+\ZZZ+\ZZZ} \:\times\: u\+\ZZZ \:\times\: \seq{\ZZZ}. $$ The first term on the right corresponds to the root and the remaining terms deal with the non-root vertices in order from top to bottom, vertices to the left of the spike being marked with $u$. The third term corresponds to the spike and the fifth represents the lowest point to the left of the spike (the rightmost $\mathbf{2}$ of a $\mathbf{123}$). Hence, the generating function for $\BBB_\ssS$ is $$ B_\ssS(u) \;=\; \frac{z^3\+u^2}{(1-z)\+ (1-z\+u)\+ (1-z-z\+u)} . $$
We now study the process of building a \prefix{$\BBB$}permutation from a sequence of source graphs. There are two cases. A permutation in $\BBB$ may result either from the addition of a source graph to an \prefix{$\AAA$}permutation, or else from adding a source graph to another \prefix{$\BBB$}permutation. We address these two cases in turn.
\begin{figure}
\caption{Ways to create a \prefix{$\BBB$}permutation by adding a source graph to the bottom subtree of an \prefix{$\AAA$}permutation}
\label{figABBuild}
\end{figure} One way to create a \prefix{$\BBB$}permutation from an \prefix{$\AAA$}permutation is to add a source graph from $\BBB_\ssS$, positioning its root to the left of zero or more of the leaves of the bottom subtree of the \prefix{$\AAA$}permutation and its non-root vertices to the right of the bottom subtree. In this case, the new permutation inherits its spike from the added source graph. This is illustrated in the left diagram in Figure~\ref{figABBuild}. The generating function for this set of permutations is thus given by $$ B_{\textsf{AB1}}(u) \;=\; B_\ssS(u)\+\frac{A(1)-u\+A(u)}{1-u}
. $$ For simplicity, we choose not to present the expanded form of $B_{\textsf{AB1}}(u)$, or that of most subsequent expressions. They can all be represented in the form $(p+q\+\sqrt{1-4\+z})/r$ for appropriate polynomials $p$, $q$ and $r$.
The other possibility for creating a \prefix{$\BBB$}permutation from an \prefix{$\AAA$}permutation involves the positioning of some non-root vertices of the source graph to the left of some of the leaves in the bottom subtree, making one of the vertices in the original bottom subtree the spike. The source graph may be drawn from either $\AAA_\ssS$ or $\BBB_\ssS$, as illustrated in the centre and right diagrams in Figure~\ref{figABBuild}.
In this situation, if the source graph has a spike, it must be positioned to the right of all leaves in the bottom subtree, or else a $\mathbf{1234}$ would be created. Furthermore, any source graph vertices placed to the left of leaves in the bottom subtree must occur at the same position in the bottom subtree, or else a $\mathbf{13524}$ would be created. This position may be chosen independently of where the root vertex is placed.
From these considerations, it can be determined that the resulting set of permutations has the generating function defined by $$
B_{\textsf{AB2}}(u) \;=\; \Big(B_\ssS(u)+\frac{z^2\+u^2}{(1-z)\+(1-z\+u)}\Big) \+ \frac{1}{1-u} \+ \Big(A\!'(1)-\frac{u}{1-u}\+\big(A(1)-A(u)\big)\Big) , $$ where the presence of the derivative $A\!'$ is a consequence of the independent choice of two positions in the bottom tree.
\begin{figure}
\caption{Adding a source tree to the bottom subgraph of a \prefix{$\BBB$}permutation}
\label{figBBBuild}
\end{figure} Finally, we consider the addition of a source graph to a \prefix{$\BBB$}permutation. As we noted in our key observation on page~\pageref{obsKey}, no vertex of the source graph may be positioned to the right of the spike in the bottom subgraph. As a result, the new source graph may not contain a $\mathbf{123}$ or else a $\mathbf{1234}$ would be created, so the source graph must be a member of $\AAA_\ssS$ (a fan). Moreover, the leaves of the source tree must be positioned \emph{immediately} to the left of the spike, or else a $\mathbf{1234}$ would be created. See Figure~\ref{figBBBuild} for an illustration.
Note that, as a consequence of these restrictions, it is impossible for the addition of a source graph to a \prefix{$\BBB$}permutation to create a $\mathbf{13524}$ or $\mathbf{14523}$. So it is not possible to extend a \prefix{$\BBB$}permutation so as to create a \prefix{$\CCC$}permutation.
Thus the bivariate generating function $B(u)$ of set $\BBB$ is defined by the following recursive functional equation: \begin{equation*}
B(u) \;=\; B_\ssS(u) + B_{\textsf{AB1}}(u) + B_{\textsf{AB2}}(u) \:+\: \frac{z\+u}{1-z\+u}\+\frac{B(1)-B(u)}{1-u} , \end{equation*} where the final term reflects the addition of a source tree to a \prefix{$\BBB$}permutation.
This equation is amenable to the kernel method. It can be rearranged to express $B(u)$ in terms of $B(1)$. The kernel can then be cancelled by setting $u=(1-\sqrt{1-4\+z})/2\+z$, which yields an expression for $B(1)$: $$ B(1) \;=\; \frac{-1+8\+z-19\+z^2+12\+z^3 \:+\: (1-6\+z+9\+z^2-2\+z^3)\+\sqrt{1-4 z}}{2\+z^3\+(1-4\+z)}. $$ Substitution then results in an explicit algebraic expression for $B(u)$, which we refrain from presenting explicitly due to its size.
\begin{figure}
\caption{A source graph in set $\CCC$}
\label{figCSource}
\end{figure}
\subsubsection*{The structure of set $\CCC$}
We begin our enumeration of $\CCC$ by counting its set of source graphs, which we denote $\CCC_\ssS$. Rather than doing this directly, we enumerate all the source graphs that contain a $\mathbf{123}$ (i.e.~those in either $\BBB_\ssS$ or $\CCC_\ssS$) and then subtract those in $\BBB_\ssS$. To begin, we consider how we might build an \emph{arbitrary} source graph in class $\FFF$ by adding vertices from left to right.
Suppose we have a partly formed source graph with at least one non-root vertex, whose rightmost vertex is $v$, and we want to add further vertices to its right. What are the options? If $v$ is not the lowest of the non-root vertices, then any subsequent vertices must be placed lower than~$v$. The only other restriction is that vertices must be positioned higher than the root. If we use $y$ to mark the number of positions in which a vertex may be inserted, then the action of adding a new vertex can be seen to be reflected by the following linear operator:
\begin{equation*}
\oper_{\textsf{\L}}\big[f(y)\big] \;=\; z\+y^2\+\frac{f(1)-f(y)}{1-y}. \end{equation*} We choose to denote this operator $\oper_{\textsf{\L}}$ because it corresponds to the action used in building a {\L}uka\-sie\-wicz path.
Now let us consider source graphs that have no vertices to the right of the spike. These are in~$\BBB_\ssS$, so let's call this set $\BBB_\textsf{S0}$. As usual, we mark with $u$ those vertices to the left of the spike. If, in addition, we mark with $y$ those vertices not above the spike, then $\BBB_\textsf{S0}$ is defined by the structural equation $$ \BBB_\textsf{S0} \;=\; u\+\ZZZ \:\times\: \seq{u\+\ZZZ} \:\times\: \seqplus{u\+y\+\ZZZ} \:\times\: y\+\ZZZ . $$
It is readily seen that $y$ correctly marks the number of positions in which an additional vertex may be inserted to the right.
Let $\DDD_\ssS=\BBB_\ssS\cup\CCC_\ssS$. Since every member of $\DDD_\ssS$ is built from an element of $\BBB_\textsf{S0}$ by applying $\oper_{\textsf{\L}}$ zero or more times, it follows that the generating function for $\DDD_\ssS$ is defined by the recursive functional equation
$$ D_\ssS(y) \;=\; \frac{z^3\+y^2\+u^2}{(1-z\+u)\+(1-z\+y\+u)} \:+\: z\+y^2\+\frac{D_\ssS(1)-D_\ssS(y)}{1-y} . $$ This equation can be solved for $D_\ssS(1)$ by the kernel method, using $y=(1-\sqrt{1-4\+z})/2\+z$ to cancel the kernel. The generating function for $\CCC_\ssS$ is then defined by $$ C_\ssS(u) \;=\; D_\ssS(1) \:-\: B_\ssS(u) . $$
We now study the process of building a \prefix{$\CCC$}permutation from a sequence of source graphs. As with set $\BBB$, there are two cases. A permutation in $\CCC$ may result either from the addition of a source graph to an \prefix{$\AAA$}permutation, or else from adding a source graph to another \prefix{$\CCC$}permutation. (As we observed above, it is not possible to create a \prefix{$\CCC$}permutation by adding a source graph to a \prefix{$\BBB$}permutation.) We address the two cases in turn.
\begin{figure}
\caption{Ways to create a \prefix{$\CCC$}permutation by adding a source graph to the bottom subtree of an \prefix{$\AAA$}permutation}
\label{figACBuild}
\end{figure} One way to create a \prefix{$\CCC$}permutation from an \prefix{$\AAA$}permutation is to add a source graph from $\CCC_\ssS$, positioning its root to the left of zero or more of the leaves of the bottom subtree of the \prefix{$\AAA$}permutation and its non-root vertices to the right of the bottom subtree.
This is illustrated in the left diagram in Figure~\ref{figACBuild}. The generating function for this set of permutations is thus $$ C_{\textsf{AC1}}(u) \;=\; C_\ssS(u)\+\frac{A(1)-u\+A(u)}{1-u} . $$
The other method for creating a \prefix{$\CCC$}permutation from an \prefix{$\AAA$}permutation involves the positioning of some non-root vertices of the source graph to the left of some of the leaves in the bottom subtree. This is illustrated in the right diagram in Figure~\ref{figACBuild}. In analysing this method, it is more convenient to look, more generally, at how an \prefix{$\AAA$}permutation can be extended to yield a permutation containing a $\mathbf{123}$, in either $\BBB$ or $\CCC$. We can then subtract those members of $\BBB$ that are enumerated by $B_{\textsf{AB2}}$.
We achieve the enumeration by adding vertices from left to right in four steps: \begin{bulletnums} \item The first step adds the root. \item The second step adds the first non-root vertex, which determines the position of the new spike, and also any other vertices positioned to the left of the spike. \item The third step adds any additional vertices to the right of the spike but to the left of some other leaves in the bottom subtree. The addition of such vertices creates occurrences of $\mathbf{13524}$. \item Finally, the fourth step adds any vertices to the right of the bottom subtree. \end{bulletnums}
Step 1: Permutations that result from the addition of the root vertex are enumerated by $$ D_\textsf{1}(u) \;=\; z\+u\+\frac{A(1)-A(u)}{1-u} . $$ Step 2: In this step, we insert the descending sequence of vertices that creates the new spike. In the generating function for permutations resulting from this action, we introduce two additional catalytic variables that we require for steps 3 and 4. For use in step 3, $v$ marks the number of source tree leaves to the right of the new spike. For step 4, we use $y$ to mark valid positions for the insertion of subsequent vertices, as we did previously. The generating function is $$ D_\textsf{2}(v) \;=\; \frac{z\+y^2\+u^2}{1-z\+y\+u}\+\frac{D_\textsf{1}(v)-D_\textsf{1}(u)}{v-u} . $$ Step 3: The effect of adding additional vertices to the right of the spike but to the left of some other leaves in the bottom subtree is represented by the recursive functional equation
$$ D_\textsf{3}(y,v) \;=\; D_\textsf{2}(v) \:+\: z\+y\+v\+\frac{D_\textsf{3}(y,1)-D_\textsf{3}(y,v)}{1-v} . $$ Again, the kernel method can be used to solve this for $D_\textsf{3}(y,1)$, the kernel being cancelled by setting $v=1/(1-z\+y)$.
Step 4: Finally, the addition of vertices to the right of the bottom subtree is reflected by the {\L}uka\-sie\-wicz operator $\oper_{\textsf{\L}}$, giving rise to the recursive functional equation $$ D_\textsf{4}(y) \;=\; D_\textsf{3}(y,1) \:+\: z\+y^2\+\frac{D_\textsf{4}(1)-D_\textsf{4}(y)}{1-y} , $$ which can be solved for $D_\textsf{4}(1)$ by cancelling the kernel with $y=(1-\sqrt{1-4\+z})/2\+z$.
The generating function for the set of permutations resulting from the second way of creating a \prefix{$\CCC$}permutation from an \prefix{$\AAA$}permutation is then defined by $$ C_{\textsf{AC2}}(u) \;=\; D_\textsf{4}(1) - B_{\textsf{AB2}}(u) . $$
Our work is almost complete. We only have to consider how a source graph may be added to a \prefix{$\CCC$}permutation. In fact, the situation is extremely constrained. First, as noted earlier, the source graph must be positioned to the left of the spike. Furthermore, the presence of a $\mathbf{13524}$ or $\mathbf{14523}$ means that the addition of a source graph with even a single non-root vertex would create a $\mathbf{1234}$. So the only possibility is the addition of a trivial (single vertex) source tree. Thus the bivariate generating function $C(u)$ of set $\CCC$ is defined by the following recursive functional equation:
$$ C(u) \;=\; C_\ssS(u) + C_{\textsf{AC1}}(u) + C_{\textsf{AC2}}(u) \:+\: z\+u\+\frac{C(1)-C(u)}{1-u}. $$ where the final term reflects the addition of a trivial source tree to a \prefix{$\CCC$}permutation. This equation can be solved to yield the following expression for $C(1)$ by a sixth and final application of the kernel method, cancelling the kernel by setting $u=1/(1-z)$: $$
\frac{-1+10\+z-35\+z^2+52\+z^3-35\+z^4+12\+z^5 \:+\: (1-8\+z+21\+z^2-22\+z^3+11\+z^4-2\+z^5)\sqrt{1-4\+z}}{2\+z^3\+(1-4\+z)\+(1-3\+z+z^2)} . $$
We now have all we need to prove Theorem~\ref{thmF} by obtaining an explicit expression for the generating function that enumerates class
$\FFF$. Since $\FFF$ is the disjoint union of $\AAA$, $\BBB$ and $\CCC$, its generating function is given by $A(1)+B(1)+C(1)$. Thus, by appropriate expansion and simplification, the generating function for $\av(\mathbf{1234},\mathbf{2341})$ can be shown to be equal to $$ \frac {2-10\+z+9\+z^2+7\+z^3-4\+z^4 \:-\: (2-8\+z+9\+z^2-3\+z^3)\+\sqrt{1-4\+z}} {(1-3\+z+z^2)\+\big((1-5\+z+4\+z^2) \:+\: (1-3\+z)\+\sqrt{1-4\+z}\big)}. $$ This has singularities at $z=\frac{1}{4}$, $z=\frac{1}{2}\+(3-\sqrt{5})$ and $z=\frac{1}{2}\+(3+\sqrt{5})$. Hence, the growth rate of $\av(\mathbf{1234},\mathbf{2341})$ is equal to 4, the reciprocal of the least of these.
The first twelve terms of the sequence $|\FFF_n|$ are 1, 2, 6, 22, 89, 376, 1611, 6901, 29375, 123996, 518971, 2155145. More values can be found at \href{http://oeis.org/A165540}{A165540} in OEIS~\cite{OEIS}.
\begin{figure}
\caption{
A permutation in class $\EEE$, spanned by three source graphs}
\label{figEPerm}
\end{figure}
\section{Permutations avoiding \texorpdfstring{$\mathbf{1243}$ and $\mathbf{2314}$}{1243 and 2314}}\label{sectE}
Let us use $\EEE$ to denote $\av(\mathbf{1243},\mathbf{2314})$. What can we say about the structure of source graphs in~$\EEE$? Firstly, since $H_\mathbf{1243} = \raisebox{-2.5pt}{\begin{tikzpicture}[scale=0.12,line join=round]
\draw[] (1,1)--(2,2);
\draw[] (3,4)--(2,2)--(4,3);
\plotpermnobox{}{1,2,4,3} \end{tikzpicture}} $ may not occur as a subgraph, only the root of a source graph may fork towards the upper right. Secondly, each source graph in $\EEE$ is \emph{plane}. This is the case because every non-plane graph contains a $H_\mathbf{2143} = \raisebox{-2.5pt}{\begin{tikzpicture}[scale=0.12,line join=round]
\draw[] (1,2)--(3,4)--(2,1)--(4,3)--(1,2);
\plotpermnobox{}{2,1,4,3} \end{tikzpicture}} $, and, furthermore, any $\mathbf{2143}$ in a source graph occurs as part of a $\mathbf{13254}$ (where the $\mathbf{1}$ is the root of the source graph). But this is impossible in $\EEE$, since $\mathbf{13254}$ does not avoid $\mathbf{1243}$.
\begin{figure}
\caption{A source graph for class $\EEE$, constructed from four u-trees}
\label{figESource}
\end{figure} If we combine these two observations, we see that the non-root vertices of a source graph consist of a sequence of inverted subtrees whose roots are right-to-left maxima. The avoidance of $H_\mathbf{2314} = \raisebox{-2.5pt}{\begin{tikzpicture}[scale=0.12,line join=round]
\draw[] (1,2)--(2,3)--(4,4)--(3,1);
\plotpermnobox{}{2,3,1,4} \end{tikzpicture}} $ places restrictions on the structure of the subtrees, so that they must consist of a path at the lower right, which we call the \emph{trunk}, with pendant edges attached to its left. It is readily seen that these correspond to permutations in $\av(\mathbf{132},\mathbf{231})$. We call trees of this form \emph{u-trees}, short for \emph{unbalanced} trees. See Figure~\ref{figESource} for an illustration of a source graph constructed from u-trees.
The class $\UUU$ of u-trees satisfies the structural equation $$
\UUU \;=\; \ZZZ\times\seq{\ZZZ\times\seq{\ZZZ}}
$$ where the first term on the right represents the lowest leaf at the tip of the trunk and the second represents the remainder of the vertices in the trunk, each with a (possibly empty) sequence of pendant edges attached to the upper left. Hence the generating function for $\UUU$ is \begin{equation*}
U(z) \;=\; \frac{z\+(1-z)}{1-2\+z}. \end{equation*} If we use $u$ to mark the number of u-trees, the class $\SSS$ of source graphs satisfies the structural equation $$
\SSS \;=\; \ZZZ\times\seq{u\+\UUU}
$$ and thus has bivariate generating function \begin{equation*}
S(u) \;=\; S(z,u) \;=\; \frac{z\+(1-2\+z)}{1-(2+u)\+ z+u\+ z^2}. \end{equation*}
Let us now examine how a permutation in $\EEE$ can be built from a sequence of source graphs. Observe that, when a source graph is added, no vertex of the source graph can be positioned between two vertices of a u-tree in the bottom subgraph, because otherwise a $\mathbf{2314}$ would be created. In addition, there are strong constraints on when u-trees in the new source graph can be positioned to the left of a u-tree in the bottom subgraph.
\begin{figure}
\caption{The two methods for adding a source graph in class $\EEE$; u-trees are shown schematically as filled triangles}
\label{figEBuild}
\end{figure} These conditions result in there being two distinct ways in which a source graph may be added. These are illustrated in Figure~\ref{figEBuild}. In the first method, the root of the source graph is positioned to the left of zero or more u-trees in the bottom subgraph and the u-trees in the source graph are positioned to the right of the bottom subgraph.
The second method is more subtle. It is only applicable if the rightmost u-tree of the bottom subgraph is a path. If that is the case, then an initial sequence of u-trees in the source graph can be positioned to the left of this path subtree, as long as each of them, except possibly the last, consists of a single vertex. If the rightmost u-tree of the bottom subgraph were not a path, then a $\mathbf{1243}$ would be created. Similarly, if a non-final u-tree consisted of more than one vertex, then a $\mathbf{2314}$ would be created.
In order to handle this second method, we need to keep track of those source graphs in which the rightmost u-tree is a path. Let $\SSS_\PP$ be the class of such graphs. It satisfies the structural equation $$
\SSS_\PP \;=\; \ZZZ\times\seq{u\+\UUU}\times u\+\seqplus{\ZZZ} ,
$$ where $u$ marks the number of u-trees as before. This class thus has bivariate generating function \begin{equation*}
S_\PP(u) \;=\; S_\PP(z,u) \;=\; \frac{u\+ z^2\+ (1-2\+ z)}{(1-z)\+ \big(1-(2+u)\+ z+u\+ z^2\big)}. \end{equation*}
In order to distinguish between those situations when the second method of adding a source graph is applicable and those when it isn't, let us use $\PPP$ to denote the set of those permutations in $\EEE$ whose Hasse graphs have bottom subgraphs in which the rightmost u-tree is a path.
We are interested in determining the two bivariate generating functions $E(u)=E(z,u)$ and $P(u)=P(z,u)$ for $\EEE$ and $\PPP$ respectively, where $u$ marks the number of u-trees \emph{in the bottom subgraph}. To do this, we will establish four linear operators on these generating functions that reflect the different ways in which a source graph can be added.
The action of adding a source graph using the first method is readily seen to be reflected by the following linear operator: \begin{equation*}
\oper_{\EE\EE}\big[f(u)\big] \;=\; S(u)\+\frac{f(1)-u\+f(u)}{1-u}. \end{equation*} The first method creates a member of $\PPP$ from an arbitrary element of $\EEE$ whenever the source graph is in $\SSS_\PP$ (i.e. its rightmost u-tree is a path). Thus the appropriate linear operator is \begin{equation*}
\oper_{\EE\PP}\big[f(u)\big] \;=\; S_\PP(u)\+\frac{f(1)-u\+f(u)}{1-u}. \end{equation*}
Now let us determine the linear operators corresponding to the second method of adding a source graph.
The set, $\SSS^\star$, of source graphs that can be added using the second method satisfies the structural equation $$
\SSS^\star \;=\; \ZZZ\times\seq{\ZZZ}\times u\+\UUU\times\seq{u\+\UUU},
$$ in which the third term on the right identifies the u-tree which is positioned immediately to the left of the rightmost (path) u-tree in the bottom subgraph. This specification thus counts multiple times those source graphs that can be added in more than one way due to the presence of a non-empty initial sequence of single-vertex u-trees. Note also that we don't mark the initial sequence of single-vertex u-trees with $u$. The generating function for $\SSS^\star$ is \begin{equation*}
S^\star(u) \;=\; \frac{u\+ z^2}{1-(2+u)\+ z+u\+ z^2}. \end{equation*} The action of adding a source graph using the second method is then seen to be reflected by the following linear operator: \begin{equation*}
\oper_{\PP\EE}\big[f_\PP(u)\big] \;=\; S^\star(u)\+\frac{f_\PP(1)-f_\PP(u)}{1-u} . \end{equation*} Finally, let us consider when adding a source graph to an arbitrary member of $\PPP$ creates another permutation in $\PPP$. The second method creates an element of $\PPP$ if the source graph is in $\SSS_\PP$ and its rightmost (path) u-tree is added to the right of the bottom subgraph. An element of $\PPP$ is also created if the source graph has a single path u-tree or consists of a single vertex (the root). Thus the set, $\SSS_\PP^\star$, of source graphs, counted with multiplicity, that can be added to create an element of $\PPP$ satisfies the structural equation $$
\SSS_\PP^\star \;=\; \ZZZ\times\seq{\ZZZ}\times\seqplus{u\+\UUU}\times u\+\seqplus{\ZZZ} \:+\: \ZZZ\times u\+\seq{\ZZZ}. $$ Its generating function is \begin{equation*}
S_\PP^\star(u) \;=\; \frac{u\+ z\+ (1-2\+ z)\+(1-u\+z)}{(1-z)\+ \big(1-(2+u)\+ z+u\+ z^2\big)}, \end{equation*} and the corresponding linear operator is \begin{equation*}
\oper_{\PP\PP}\big[f_\PP(u)\big] \;=\; S_\PP^\star(u)\+\frac{f_\PP(1)-f_\PP(u)}{1-u}. \end{equation*}
We are now in a position to derive the generating function for $\EEE$ and hence prove Theorem~\ref{thmE}. From the analysis above, we know that the bivariate generating function $E(u)=E(z,u)$ of class $\EEE$ is defined by the following pair of mutually recursive functional equations: \begin{equation*}
\begin{array}{rclcrcr}
E(u) & = & S(u) & \!+\! & \oper_{\EE\EE}\big[E(u)\big] & \!+\! & \oper_{\PP\EE}\big[P(u)\big] \\[3pt]
P(u) & = & S_\PP(u) & \!+\! & \oper_{\EE\PP}\big[E(u)\big] & \!+\! & \oper_{\PP\PP}\big[P(u)\big]
\end{array} . \end{equation*} These can be expanded to give the following: \begin{equation}\label{eqnE1}
E(u) \;=\; z\+\frac{(1-2\+z)\+\big(1-u+E(1)-u\+E(u)\big) \:+\: u\+z\+\big(P(1)-P(u)\big)}{(1-u)\+ \big(1-(2+u)\+ z+u\+ z^2\big)} , \end{equation} \begin{equation}\label{eqnE2}
P(u) \;=\; u\+z\+(1-2\+z)\frac{z\+\big(1-u+E(1)-u\+E(u)\big)\:+\: (1-u\+z)\+\big(P(1)-P(u)\big)}{(1-u)\+ (1-z)\+ \big(1-(2+u)\+ z+u\+ z^2\big)} . \end{equation}
An unusual simultaneous double application of the kernel method can then be used to yield the algebraic generating function for class $\EEE$ as follows.
First, we eliminate $P(u)$ from \eqref{eqnE1} and \eqref{eqnE2}, and express $E(u)$ in terms of $E(1)$ and $P(1)$ as a rational function. Cancelling the resulting kernel, \begin{equation*} (1-3\+z+2\+z^2) \:-\: (2-7\+z+7\+z^2-z^3)\+u \:+\: (1-3\+z+3\+z^2)\+u^2 \:-\: (z-3\+z^2+3\+z^3)\+u^3 , \end{equation*} with the appropriate root then gives us an equation relating $E(1)$ and $P(1)$.
Secondly, we eliminate $E(u)$ from \eqref{eqnE1} and \eqref{eqnE2}, and express $P(u)$ in terms of $E(1)$ and $P(1)$. Cancelling the (same) kernel (using a different root) gives a second equation relating $E(1)$ and $P(1)$.
Finally, we eliminate $P(1)$ from these two equations to yield an extremely complicated explicit expression for $E(1)$.
Thus, using a computer algebra system to handle the details of the algebraic manipulation, it can be determined that the generating function $F(z)=E(1)$ for $\av(\mathbf{1243},\mathbf{2314})$ has the minimal polynomial $$
(z-3\+z^2+2\+z^3) \:-\: (1-5\+z+8\+z^2-5\+z^3)\+F(z) \:+\: (2\+z-5\+z^2+4\+z^3)\+F(z)^2 \:+\: z^3\+F(z)^3 . $$ The growth rate of the class is given by the reciprocal of the least positive real singularity of its generating function~\cite[Theorems~IV.6 and~IV.7]{FS2009}. Hence, by determining the location of the singularities of $E(1)$, it is possible to establish that the growth rate of class $\EEE$ is approximately 5.1955, the greatest real root of the quintic polynomial $$ 2 - 41\+z + 101\+z^2 - 97\+z^3 + 36\+z^4 -4\+z^5 , $$ as required.
The first twelve terms of the sequence $|\EEE_n|$ are 1, 2, 6, 22, 88, 367, 1571, 6861, 30468, 137229, 625573, 2881230.
More values can be found at \href{http://oeis.org/A165539}{A165539} in OEIS~\cite{OEIS}.
{\footnotesize
}
\end{document} | arXiv |
A pulsed-Laser Rb atomic frequency standard for GNSS applications
Part of a collection:
Timekeeping in space: technology, practice, promise, and benefits
S. Micalizio ORCID: orcid.org/0000-0002-8363-19961,
F. Levi1,
C. E. Calosso1,
M. Gozzelino1 &
A. Godone1
GPS Solutions volume 25, Article number: 94 (2021) Cite this article
We present the results of 10 years of research related to the development of a Rubidium vapor cell clock based on the principle of pulsed optical pumping (POP). Since in the pulsed approach, the clock operation phases take place at different times, this technique demonstrated to be very effective in curing several issues affecting traditional Rb clocks working in a continuous regime, like light shift, with a consequent improvement of the frequency stability performances. We describe two laboratory prototypes of POP clock, both developed at INRIM. The first one achieved the best results in terms of frequency stability: an Allan deviation of σy(τ) = 1.7 × 10−13 τ−1/2, being τ the averaging time, has been measured. In the prospect of a space application, we show preliminary results obtained with a second more recent prototype based on a loaded cavity-cell arrangement. This clock has a reduced size and exhibited an Allan deviation of σy(τ) = 6 × 10−13 τ−1/2, still a remarkable result for a vapor cell device. In parallel, an ongoing activity performed in collaboration with Leonardo S.p.A. and aimed at developing an engineered space prototype of the POP clock is finally mentioned. Possible issues related to space implementation are also briefly discussed. On the basis of the achieved results, the POP clock represents a promising technology for future GNSSs.
The performance of any GNSS strongly relies on clocks adopted by the system, both at the ground and space segments. In particular, the clocks onboard the satellites must meet rigorous requirements in terms of mass, cost, volume and power consumption and, at the same time, exhibit high-frequency stability capabilities to ensure the positioning and synchronization accuracies to the end-users (Rochat et al. 2012; Maciuk 2019). Sub-meter accuracy in the positioning signal is becoming of interest for a growing number of civilian applications, from autonomous driving to unmanned agriculture, just to mention some examples (Joubert et al. 2020; Winterhalter et al. 2020).
It is interesting to derive a typical GNSS specification, Galileo, for example, from a very qualitative point of view. The GNSS receiver obtains the position information by retrieving the propagation time of the incoming signals traveling through space at the speed of light, according to the satellite and receiver local clocks. Assuming as a target a fraction of a meter, the time on the satellites has to be known within Δt = 1 ns. The task of the onboard clocks is then to guarantee this level of uncertainty for an as long as possible time interval t, at least 100,000 s, which is equivalent to a couple of synchronizations and revolutions of the satellite around the earth (Fig. 1).
A GNSS satellite in orbit around the Earth takes tens of thousands of seconds to make a revolution and to re-synchronize. In the meanwhile, it has to carry the time information with minimal perturbation from temperature (T(t)) and magnetic field fluctuations (\(\overrightarrow{B}(t)\))
Roughly speaking, the onboard clock requires fractional frequency stability (Rubiola 2011) \(y\equiv \frac{\Delta \nu }{\nu }=\frac{\Delta t}{t}\) = 1 × 10−14 for an averaging time of the order of one day, being ν the nominal clock frequency and Δν its fluctuation. As shown in Fig. 2, a high-performing Rubidium Atomic Frequency Standard (RAFS) satisfies this requirement. Its stability, expressed in terms of Allan deviation, scales as τ−1/2 up to averaging times τ of the order of 104 s. This white frequency noise region is usually followed by a flicker floor component. For even longer τ, the RAFS stability is affected by a linear frequency drift.
Qualitatively derivation of Galileo specification for what concerns the frequency stability of onboard clocks and their sensitivity to temperature and magnetic field
In addition, a RAFS suited for GNSS has to carry the time, i.e., the phase time information, with minimal perturbation from temperature and magnetic field fluctuations. Onboard clocks, particularly their electronics and optics packages, have also to be radiation hardened to operate in space without failures and/or performance degradation (see section Issues related to a space implementation of the POP clock).
It has been recognized that lamp-pumped RAFS is a mature technology satisfying these requirements. For example, the RAFSs used in the GPS Block IIF satellites exhibit a short-term stability better than 1 × 10−12 τ−1/2, with a drift of units of 10−15 per day (Vannicola et al. 2010). A new type of physics package designed and realized in Hao et al. 2016 achieved short-term stability of 2.4 × 10−13 τ−1/2, a considerable result for a RAFS. More recently, a space-borne rubidium atomic clocks for the BeiDou III navigation satellite system has been reported in Mei et al. 2021; this clock exhibits short-term stability of 6.1 × 10−13, reaching the level of 4 × 10−15 for one day of measurement time.
However, new laser-based clock technologies have been recently studied and proposed, offering the perspective of better results in a device similar, for certain aspects, to a RAFS (Vanier et al. 2007, Bandi et al. 2014). In this regard, a Rb vapor-cell clock based on laser optical pumping has been implemented and characterized at INRIM since 2012 (Micalizio et al. 2012a).
Compared to a traditional lamp-pumped Rb clock where the atoms interact simultaneously with light and microwave radiation, in the pulsed optical pumping (POP) approach developed at INRIM, atom preparation, microwave interrogation and detection phases take place at different times, with a significant benefit for the clock frequency stability. Specifically, the most performing POP clock developed at INRIM exhibits a frequency stability (Allan deviation) of 1.7 × 10−13 at 1 s and at the level of few 10−15 for integration times of 104 s has been measured (Micalizio et al. 2012b). In the paper, we refer to this clock as Prototype 1.
More recently, we investigated the possibility of reducing the size of the clock's physics package, but still keeping the short-term stability below the 10−12 level. A second clock setup (Prototype 2) based on a loaded cavity-cell arrangement has been then implemented.
After resuming the physical principles on the basis of the POP clock operation, the paper describes the realization and the results obtained with Prototype 1 and the ongoing activity related to Prototype 2. Some of the issues connected to a space implementation of the POP clock are also discussed.
We finally present the preliminary results of a space-qualified prototype under development by Leonardo SpA in collaboration with INRIM. The aim of this partnership is the implementation of a stable, space-qualified, atomic clock based on the Rb POP concept and intended for GNSS application.
Currently used Rb clocks are based on the double resonance (DR) approach where the atoms simultaneously experience resonant light and microwave field (Camparo 2007). The light is generated by a lamp and is used for optical pumping purposes: a population inversion between the two clock levels is in fact required in order to have a significant signal-to-noise ratio. The atoms are sealed in a cell housed in a microwave cavity where the atoms experience a resonant microwave field and make the clock transition (Vanier and Audoin 1989). Specifically, the resonant absorption of microwave radiation is manifested by a decrease in the transmitted light intensity.
As in the DR approach, in the POP technique the atoms also interact with light and microwave radiation, but with two important differences. First, the optical pumping operation is performed by a laser; this guarantees a highly efficient population inversion between the two ground-state levels defining the clock transition, with a consequent benefit for the signal-to-noise ratio. Second, the clock works in a pulsed regime, i.e., laser and microwave fields are not simultaneously applied but they act on the atomic sample alternately. In this way, the mutual influence of light and microwave signals is greatly reduced (Godone et al. 2004); specifically, light shift is strongly suppressed, leading to a medium/long-term stability that can also achieve the 10−15 region.
Other advantages characterize the pulsed operation. In particular, the clock interrogation is performed according to the Ramsey interaction scheme: 1) (Fig. 3) a strong laser pulse prepares the atoms in one of the two clock levels; 2) the atoms experience a couple of resonant microwave pulses of duration t1 separated in time by T and finally 3) a weak laser pulse (duration td) is sent to probe the clock transition which appears as an interference pattern on the absorption signal: the Ramsey fringes (Micalizio et al. 2009). In other words, the POP clock operation is similar to that of an atomic fountain but with a cycle time much faster, of the order of a few millisecond, being the Ramsey time T limited by the relaxation processes taking place inside the cell (Levi et al. 2014, Wynands et al. 2005).
Principle of operation of the POP technique. A strong laser pulse changes the thermal equilibrium in the ground state of a 87Rb sample, completely inverting the atomic population. The atoms then interact with a couple of microwave pulses according to the Ramsey scheme and make the clock transition. A weak laser pulse is used to probe the atoms during the detection phase. Γ* is the relaxation rate of the excited state. Typical durations are tp = 0.4 ms, t1 = 0.4 ms, T = 3.0 ms and td = 0.15 ms, with a total cycle time of 4.35 ms
The central Ramsey fringe represents the reference atomic signal that is used to discipline the local oscillator. Its linewidth (full width at half maximum, FWHM) Δν1/2 can be approximated by the relation:
$$\Delta \nu _{{1/2}} \approx \frac{1}{{2T}}$$
and is almost insensitive to any laser/microwave power broadening, being dependent on the Ramsey time T only.
Prototype 1: a POP Rb clock with optical detection
Figure 4 shows the setup of our laboratory prototype that implements the principle of operation described in the previous section. Basically, a POP clock can be divided into three subsystems: physics package, optics and electronics.
Schematic of the POP clock setup. The three packages, optics (OP), physics (PP), and electronics (EP) are indicated with dashed frames
The clock physics package (PP) is the structure containing the atoms in a properly shielded and controlled environment and includes the apparatus for the clock transition detection (we observe that conventionally PP of Rubidium clocks consists of both the light source and cavity along with related optics; however, in this case we prefer to distinguish the optics module from the PP. This distinction is functional to the implementation of the POP clock where it is more convenient to envisage the clock as composed of different modules that, if needed, can be replaced. Also, in the space implementation of the POP clock pursued by Leonardo, laser and the optical components have a dedicated development phase). The optical system provides the laser radiation used to shine the Rb vapor in the pumping and detection phases. The electronic system includes the low-phase-noise microwave synthesis, starting from a quartz oscillator. Moreover, it is devoted to clock control, including the pattern generation, signal acquisition and stabilization loops (laser locking, quartz locking, temperature stabilization, quantization magnetic field, etc.).
The POP physics package is not different in principle from that of a traditional Rb clock. It is a layer structure whose core is the cell filled with 87Rb atoms. A buffer gas is also added to the cell to localize the atoms avoiding a fast spin depolarization. However, the buffer gas produces a linear temperature-dependent shift of the clock frequency. Therefore, it is a common practice to use a mixture of buffer gases (typically Ar and N2) with opposite temperature coefficients so that an inversion point is introduced for the frequency shift as a function of temperature. By operating close to this specific set point, the temperature sensitivity turns out to be only quadratic (Vanier et al. 1982) and the transfer of temperature fluctuations to the clock frequency is minimized. Specifically, in our implementation, the total buffer gas pressure is 25 Torr with a pressure ratio Ar/N2 of 1.6. The inversion point is around 65 °C, where the Rb vapor density is optimal for clock operation.
Rubidium collisions with buffer gas atoms/molecules cause a relaxation of the atomic observables. For our buffer gas composition and at the operational temperature of 65 °C the relaxation rates of atomic population and coherence in the ground state are γ1 = 360 s−1 and γ2 = 300 s−1, respectively. In this way, the signal decay time is of the order of 3 ms, limiting the duration of the Ramsey time. As a consequence, according to (1), linewidths as narrow as 160 Hz can be obtained.
The cell has a radius R = 10 mm and a length L = 20 mm, with a long stem used as a cold point outside the microwave cavity where the metallic Rb atoms condensate.
The cell is placed inside a resonant microwave cavity tuned to the TE011 mode (Godone et al. 2011); the cavity allows the atoms to interact with a highly uniform microwave field. This is important in order to maximize the signal contrast, reducing at the same time the effect of neighboring transitions.
A solenoid outside the cavity produces a uniform quantization magnetic field (also called C-field) of the order of 15 mG. A series of magnetic shields and heaters aimed at reducing the clock frequency sensitivity to environmental variations completes the physical package. The temperature sensitivity of the cell clock is indeed a crucial aspect, and a long-term stability of hundreds of microkelvins is required to achieve clock stability in the 10−15 range (see Table 1). In addition, to eliminate barometric pressure sensitivity (Micalizio et al. 2012b; Moreno et al. 2018), the POP physics package is kept in a vacuum chamber.
Table 1 Sensitivity of the POP-clock cell to the main parameters of influence and the stability required to the latter to support frequency stability of 10−14 and 10−15. Sensitivities are mainly from Micalizio et al. 2010 and Micalizio et al. 2012b
The optical system required for the POP operation is relatively simple. It is composed of a single laser diode (either at 780 or at 795 nm), typically a distributed-feedback (DFB) diode, frequency locked to the Rb optical transition. The laser then passes through an acousto-optic modulator (AOM) that acts as an optical switch for the pulsed operation and as an amplitude modulator to have high power for pumping and low power for detection.
At the entrance of the cell, the laser can deliver up to 15 mW peak powers; the laser beam shape is Gaussian (TEM00) and, after collimation, the diameter is about 10 mm.
A lens focuses the laser transmitted through the cell onto the detection photodiode placed outside the cavity; a trans-impedance amplifier then transforms the detected atomic absorption in a reference signal for clock purposes.
Even if highly integrated into a single package, the clock electronic system can be divided into three major functional parts:
Synthesis chain.
Clock cycle management.
Stabilization loops.
The synthesis chain provides the main output at 6.834 GHz, with the possibility to generate phase-coherent microwave pulses as required by the Ramsey interaction scheme. It is made starting from a 10 MHz OCXO as the local oscillator (LO). The details of its implementation are reported in Micalizio et al. 2012a, b.
The phase noise at 6.834 GHz is − 107 dB rad2/Hz at 200 Hz from the carrier; this noise affects the clock signal via the well-known Dick effect (Dick 1987; Santarelli et al. 1998; Lo Presti et al. 1998). Basically, the phase noise of the interrogating signal appears as a random "end-to-end phase difference," thereby introducing frequency noise in the loop. In frequency standards with a pulsed operation, the phase noise around even harmonics of the pulse rate is down-converted by aliasing to baseband. Taking into account the typical timing of the POP clock cycle, the Dick effect gives a contribution of 7 × 10−14 at 1 s to the clock stability budget. The synthesis chain architecture has been more recently updated, resulting is an even lower Dick effect, at the level of 2 × 10−14 at 1 s. (François et al. 2015).
A custom digital board guarantees the pulsed operation, where a single field-programmable-gate-array (FPGA) drives, among the other, two direct-digital synthesizers (DDS), one analog-to-digital converter (ADC) and one digital-to-analog converter (DAC): (1) the first DDS generates the baseband version of the two microwave pulses required by the Ramsey scheme; the second DDS drives the AOM for generating the pumping and the detection laser pulses, (3) the ADC acquires the atomic signal and (4) the DAC continuously corrects the OCXO so that its frequency is locked to the atoms. This architecture makes extensive use of digital components. In view of a possible space application, we report the availability on the market of space-qualified 16-bit DACs at tens of MHz and 14-bit DACs at hundreds of MHz; we then do not expect major criticalities in this regard.
In separate custom boards, we implemented the temperature controllers for the physics package, the trans-impedance amplifier for the detection photodiode and the low-noise current generator that drives the C-field coils. Instead, laser temperature and frequency controllers, as well as the current driver are based on commercial instrumentation. The digital nature of the electronic implementation based on an FPGA gives great flexibility for implementing new functionalities.
The experimental behavior of the clock has been extensively described in Micalizio et al. 2012a, b, here we report the main results only. In Fig. 5, we show typical Ramsey fringes obtained using the same timing reported in Fig. 3. The central Ramsey fringe represents the reference signal; it has a contrast of 28% and a linewidth of 160 Hz, corresponding to a quality factor of 4.2 × 107.
Ramsey fringes observed on the optical transmitted signal, according to the principle of operation shown in Fig. 3. The laser power during pumping (detection) is 4 mW (200 μW)
A typical stability of the POP clock, when all internal and external parameters are optimized is reported in Fig. 6. A short-term stability of 1.7 × 10−13 τ−1/2 is observed with a clear white frequency noise signature up to 1000 s; after this time, a flicker floor around 6 × 10−15 is reached. The plot of Fig. 6 is representative of the medium-term performances of the POP clock and has been obtained selecting 2 days of data from a 10 day run. A long-term drift around -6 × 10−15/day has been measured over the complete data set (due to its small entity the drift has not been removed from the plot). Figure 6 clearly shows that Prototype 1 is compliant with the specifications of the Galileo PHM, one of the most performing space-qualified clocks (Droz et al. 2009). In order to characterize the medium-long term behavior, it is important to investigate the clock frequency sensitivity to fluctuations of internal and external parameters (Micalizio et al. 2010). In particular, the clock frequency is sensitive to a combination of environmental parameters (like barometric pressure (Huang et al. 2010), temperature, humidity, magnetic fields) and operational parameters (like microwave power, laser frequency, amplitude, polarization and pointing, internal temperature and gradients (Calosso et al. 2012), cavity Q, buffer gas and others).
Measured frequency stability of the POP Rb clock obtained in the same conditions of Fig. 5. An active H-maser has been used as a reference (continuous line). The dashed line represents the specification of the Galileo PHM. The small stability bump at 250 s is due to periodic thermal fluctuation induced by the laboratory air conditioning and transmitted to the PP through thermal bridges (cables, connectors, etc.…)
Table 1 lists the main influence parameters and the related sensitivities of the clock at the cell level, and then it derives the stability that is required for these parameters to target a clock frequency stability consistent with current (10−14) and improved (10−15) GNSS specifications. A clock stability target at the level of 10−14 can be considered not particularly demanding for the POP clock. The overall thermal sensitivity is of the order of 10−11/K and a stabilization of the cell temperature at the mK level through a double oven is sufficient. The environment magnetic field fluctuations can be reduced by means of magnetic shields (three in our prototype), provided that the effect of the apertures for feeding the laser and the cables are minimized and that the current generator feeding the C-field coil has a fractional stability better than few ppm.
Concerning other parameters, thanks to the intrinsic low sensitivity of the POP clock to laser and to microwave powers, it is possible to reach the 10−14 level without any particularly demanding stabilization. The laser frequency stabilization to the external cell only requires a few Hz-bandwidth.
In case of a target in the 10−15 region, for example for next-generation more performing clocks, the requirements on the parameters of influence become much more stringent and their fulfillment has a significant impact on the clock design. At this level of performance, an active stabilization of laser and microwave powers, a high-performance laser frequency stabilization, and careful magnetic shielding design and additional temperature shields would be necessary.
Adding these new functionalities is in principle not trivial: a careful measurement of these parameters as perceived by the atoms would be required and high performing controllers compensating for their fluctuations should be implemented. In our case, we adopt a new approach based on the possibility of extracting directly from the atoms the information about the perturbing parameters. For example, we implemented and use routinely a technique based on the so-called Rabi oscillations to stabilize the amplitude of the microwave field (Gozzelino et al. 2018). In other words, we exploit the pulsed regime and the possibility to program arbitrarily long clock sequences to use the atoms themselves as microwave amplitude discriminators. In this way, apart from stabilizing the output power of the synthesis chain, it is possible to significantly mitigate cavity-aging effects, such as a frequency drift of the cavity and/or the relaxation of its quality factor. Specifically, in the case of prototype 1, the drift was reduced from 3 × 10–14/day to a negligible level with beneficial impact in the long-term of Fig. 6.
This approach can be extended to stabilize other quantities of interest, still subject of investigation, like the quantization magnetic field, or to compensate for the light shift, (Sanner et al. 2018; Hafiz et al. 2018). It is also possible to stabilize the laser frequency using the clock cell itself, without the need for any external reference. Thanks to the low-noise detection chain, we obtained preliminary clock stability results at the level of 1 × 10−14, corresponding to a fractional frequency stability for the laser below 1.7 × 10−9 (Calosso et al. 2017, 2019). Definitely, this approach results in a set of advantageous techniques that, thanks to the flexibility of our digital electronics, improve the clock performance with minimal or no additional hardware. Moreover, in some cases, like the stabilization of the laser frequency to the clock cell, a significant part of the setup could be removed with a consequent clock simplification, a great benefit for a possible space application.
Prototype 2: the miniaturization process
As mentioned above, reducing size and weight is of crucial importance for any device intended for space applications. It is sufficient to think that the cost to send a satellite in orbit is 20–30 k€/kg. However, it is not only a matter of cost. Reducing the size also implies a better ability to control the temperature of all the system.
The POP clock has already a rather compact structure, as the physics package can occupy less than 2.5 L for a laboratory setup and a complete experiment can be filled in less than 9-L volume (Arpesi et al. 2019). However, further optimization of the clock architecture is possible without compromising the stability performances. Regarding the physics package, recently there have been efforts to redesign the core of the structure and in particular novel solutions for the microwave cavity were introduced (Kang et al. 2015; Gozzelino et al. 2020). Miniaturization can possibly be aided by the improving capabilities of additive manufacturing (Affolderbach et al. 2018). Once the core volume is reduced, the thermal and magnetic shields can be scaled down, with considerable reductions of weight, size and power consumption.
The slotted-tube resonator (or ''magnetron cavity'') was introduced to have comparable field properties and to excite the clock transition with a more compact resonator (Stefanucci et al. 2012). Indeed, the resonance frequency of a TE011-like mode can be properly modified by choosing the number and geometry of the electrodes. External volume reduction up to a factor of 4 has been recently achieved by using a structure with 4 electrodes (Hao et al. 2019).
An alternative strategy to reduce the cavity footprint is to load the resonator volume with a dielectric (Williams et al. 1983, Howe et al. 1983, Wang 2008, Li et al. 2013). In prototype 2, a simple design has been introduced in Gozzelino et al. (2021) by using high-purity alumina (Al2O3) as loading material. The assembly is shown in Fig. 7 where we can observe that the cylindrical symmetry is preserved.
a Loaded cavity-cell arrangement used in the miniaturized clock experiment; b comparison between the cavity used in Prototype 2 clock (left) and the one used in Prototype 1 (right)
The cavity still works on the TE011 mode even if its spatial distribution and resonance frequency are modified by the alumina. The cavity resonates at 6.834 GHz for an operational temperature of 60 °C. The inner volume is reduced by a factor of 10 with respect to the un-loaded cylindrical design, with a cell volume reduction of a factor of 8. Two below cut-off waveguides allow the laser beam to access the clock cell and a Si-photodiode detects the light transmission through the clock cell. The dielectric also acts as a holder for the vapor cell. This feature is favorable for possible space applications, as, in principle, it ensures better mechanical stability and resilience to vibrations, as opposed to gluing the cell to the metallic walls. The field uniformity inside the cell volume is even better than in the case of the traditional un-loaded cavity because of proper engineering of the loading material shape.
The final physics package structure will be different if intended for operation in vacuum or (on the ground) at atmospheric pressure. Given that the reduction in the cavity outer volume is a factor of 5 compared to Prototype 1, we expect a reduction of (at least) a factor of 2 in total volume, mass and heating power for the physics package.
The design of Fig. 7 applied to a POP experiment demonstrated clock spectroscopy with high contrast and high atomic-line quality factor.
In Fig. 8, we present preliminary short-term stability results for Prototype2, using a cell containing a mixture of Kr-N2 as a buffer gas, at the total pressure of 40 Torr and in the pressure ratio 0.96. As explained in Gozzelino et al. (2020), this mixture is able to reduce by more than one order of magnitude the barometric and temperature sensitivities of the clock frequency. The prototype has been tested at ambient pressure. With a Ramsey time of 1.1 ms and a cavity temperature of 58 °C, we obtained σy(1 s) = 6 × 10−13, still compliant with current GNSS requirements.
Measured frequency stability of the mini-POP Rb clock. The continuous line represents the H-maser stability used as the reference oscillator
We observe that the cavity operates 2 °C below its tuning temperature, corresponding to 1 MHz of cavity detuning with respect to the atomic line. Given that the loaded quality factor of the microwave cavity is less than 200, this is not a major issue. Of course, seeking the best performance, a fine tuning of the cavity is foreseen but could not be done "a-priori" before a first determination of the operational temperature set by the buffer-gas content.
Medium-long-term is under investigation as the thermal stability of the setup is being upgraded. Also, optimization of long–term performances could require under vacuum characterization tests.
Issues related to a space implementation of the POP clock
Neither of the two POP clock prototypes previously described contains space-qualified components. However, since space engineering is currently undertaken by Leonardo S.p.A. (see next section), it is interesting to briefly discuss some issues to be faced in order to manufacture a reliable and performing device.
Space is a harsh environment. During the revolution around the earth, the satellite modifies its exposure to the sun and the temperature of the external surface can change from + 100 °C when it is illuminated to − 100 °C when it is in shadow (Martinez 2013, p. 141). The strong external temperature fluctuations affect the baseplate of the onboard clocks at the level of + / − 1 K, and, consequently, the temperature sensitivity of the signal delivered by the clock must be lower than 1 × 10−14/K.
Moreover, the satellite is under the influence of the terrestrial magnetic field, whose value changes when the satellite moves along the orbit. The maximum variation at 29,994 km from the center of the earth is of the order of 10 mG peak to peak.
To this value, the internally generated magnetic field has to be added, mainly due to the magnetorquer used to maintain the satellite's orientation. The latter can be as high as 0.1 G. For this reason, the clock requires a sensitivity to the magnetic field below 10−13/G.
Besides experiencing significant temperature and magnetic-field fluctuations, the satellite is continuously bombarded by radiations of different nature (Steinberger et al. 2017). Specifically, three main ionizing radiation sources have to be considered: galactic cosmic rays (GCR), solar particle events, and trapped particles. In the following, we refer to Marcello G et al. (2015).
GCR originates from outside our solar system and approximately are composed of 85% protons, 14% He nuclei, 1% heavy nuclei. They have an energy up to 10 GeV/ amu, but some of them can have much higher energy. They are very penetrating and it is virtually impossible to shield them with reasonable amounts of material. In addition, they are anti-correlated with solar activity: solar flux scatters incoming charged particles. GCR fluxes are in the order of a few particles per square centimeter per second.
The solar particle flux is a stream of charged particles released from the upper atmosphere of the sun, called the corona. This plasma mostly consists of electrons, protons and alpha particles. This flux is dependent on the solar cycle and can reach > 105 particles/cm2 /s with energy > 10 meV/nucleon. Solar wind and flares depend on the solar activity.
Trapped particles are due to the earth's magnetic field which is able to capture charged particles. These particles, once confined, move in a spiral, bouncing from one pole to the other and form two belts: the outer belt made for the most part of electrons and the inner belt consisting of both electrons and protons. Fluxes of electrons with energy above 1 meV can reach 106 particles/cm2/s, those of trapped protons can be as high as 105 particles/cm2/s (Walt 1995). A peculiar feature is the South Atlantic Anomaly (SAA), where the radiation belts come closest to earth. The SAA is caused by the fact that the magnetic field axis forms an 11° angle with respect to the North–South axis, and its center is not located at the Earth center but it is about 500 km far from it, causing a dip in the magnetic field over the South Atlantic area. The SAA is the area where most errors and malfunctions occur in satellites placed in low orbits.
A possible design and implementation of a space POP frequency standard should take into account all these effects. We point out, however, that the POP clock shares many features with other clocks that already have a space realization. For instance, the POP physics package and electronics are in principle similar to those used in RAFSs or H-masers already used onboard GPS/Galileo satellites (Batori et al. 2020, Riley 2019).
The main issues are then related to the optics package, laser source and optical components, especially if a lifetime of the order of 10 years is required. To our knowledge, only one atomic clock using lasers has been launched in space (Cold Atom Clock Experiment in Space, CACES), but it was mainly a proof of principle experiment based on cold atoms with no real timing purposes and operated for only 15 months (Liu et al. 2018).
However, laser diodes are increasingly used for space applications (Guilhot and Ribes-Pleguezuelo 2019, Tolker-Nielsen and Guillen 1998). The number of instruments requiring diode lasers is rapidly growing, including (but not limited to) sources for pumping solid-state lasers or for LIDAR experiment, like ALADIN (Cosentino et al. 2015), or also for spectroscopic purposes like in PHARAO to be used in the ACES experiment (Minec-Dube 2006). In this regard, several studies as cited below have been devoted to investigating the effect of radiations on the performance of different diode lasers. Exposure to ionizing radiation may lead to material degradation due to displacement damage ionization effects. Defects are detrimental for photonic devices as they modify the optical properties of a material by introducing absorption bands or color centers. Recent results would suggest that a DFB diode laser can be made relatively robust against gamma radiation and flash X-rays (Timmons and Stoner 2011). In Esquivias et al. (2011), the effects of proton and gamma irradiation have been evaluated on the performance of GaSb-based 2.1 μm diode lasers for space applications. No significant radiation damage has been found. On the other hand, radiation effects on diode lasers can be quite peculiar (Camparo et al. 1992), and further studies need to be carried out.
Definitely, due to the fundamental role played by the laser in the POP clock operation, it is mandatory to foresee a redundancy of the laser source in the clock design phase.
Discussion and conclusion
In recent years, several high-performing vapor-cell clocks have been developed in different laboratories. For a review, see Godone et al. (2015). Not only do these clocks adopt efficient laser sources, but also innovative techniques to prepare and detect the atoms, reduce the transfer of laser noise to the atoms, and to improve the signal-to-noise ratio. We can mention, for instance, clocks based on advanced coherent population trapping (CPT) techniques (Yun et al. 2017; Abdel Hafiz et al. 2017) or even vapor-cell clock based on cold atoms (Esnault et al. 2010; Langlois et al. 2018).
Even if these devices reach frequency stabilities in the 10−13 at 1 s and in some cases achieve units of 10−15 in the long term, they are rather complex, especially in terms of the optics package used to generate the required radiations. The POP approach, instead, joints similar stabilities performances with a much simpler implementation, only one laser is used to alternatively pump and detect the atoms. The effectiveness of the POP technique is also demonstrated by the interest of several other laboratories that, following the seminal works done at INRIM, have undertaken the realization of a POP clock (Baryshev et al. 2017; Dong et al. 2017; Monahan et al. 2019), obtaining in some cases outstanding results not far from those achieved by INRIM (Almat et al. 2020; Shen et al. 2020).
For a possible space implementation, it is of crucial importance to reduce the clock size and complexity. For this reason, we have also investigated the possibility of using a loaded cavity to shrink the physics package dimensions. We successfully demonstrated that, despite a reduced signal-to-noise ratio, the clock is still able to reach very good frequency stability performances. In addition, digital techniques foresee the stabilization of the laser frequency directly to the clock cell with a very significant simplification of the optical setup.
We conclude this overview of the POP Rb clock by briefly mentioning an ongoing space-implementation activity. Based on the results achieved with Prototype 1, INRIM and Leonardo Company SpA have been collaborating since 2016 for the engineering of the POP clock. Specifically, ESA appointed Leonardo jointly with INRIM to develop a space-qualified POP clock suitable to be embarked in an experimental flight within the Galileo Transition Satellites program. Table 2 shows a comparison between RAFS, PHM and POP clock in terms of performances, size, weight and power consumption (SWaP).
Table 2 Comparison between the expected performances and SWaP of the POP Rb clock and the specifications of Galileo PHM and RAFS. Data are
The POP clock is expected to have size and power consumption between the RAFS and the PHM but with a short-term frequency stability at least one order of magnitude better than that of the RAFS.
As a first step, Leonardo implemented a physics package of POP clock, see Fig. 9. The Leonardo physics package has been completely characterized at INRIM facilities in terms of physics behavior, including thermal and magnetic sensitivities. At ambient pressure, it demonstrated frequency stability of 1.8 × 10−13 at 1 s (Arpesi et al. 2019), a result comparable to that of Fig. 6. In addition, the physics package successfully passed vibration and shock tests. These results, together with medium-long term characterization and under vacuum tests, will be presented in detail in a forthcoming work.
POP physics package implemented by Leonardo and tested at INRIM (courtesy of Leonardo SpA). The package includes the cavity-cell arrangement, magnetic and thermal shields, whereas the laser, its stabilization, and the AOM are part of another module. As a benchmark, a one Euro coin has been added (diameter 23.25 mm)
The optical bench used to characterize the Leonardo physics package is the same used for Prototypes 1 and 2. No space-qualified component has been then used, including the laser diode. As mentioned in the previous section, the reliability and lifetime of the laser module are an issue for the POP clock. The aging of DFB lasers emitting at 780 nm and used for ground Rb clocks has been studied in Matthey et al. 2011, where diode lasers have been observed for 9 months. The results show that DFB lasers can be used in laser-pumped clocks with an acceptable variation of their power and frequency, even extrapolating the data to a projected clock lifetime of 20 years. Also, laser diodes with a wavelength close to D lines of 87Rb are predicted to have a median time to failure of approximately 8 years at a room temperature of 40 °C (Gale 2008). Compact and robust diode laser emitting at 780 nm and specifically suited for quantum optics experiments in space have been developed in Wicht et al. 2017. These data are encouraging for the realization of a space qualified laser module to be used in the POP clock.
Definitely, regarding its near-term deployment on board of Galileo satellites, the POP clock looks very promising, having unique strengths in terms of compactness jointly to outstanding frequency stability performances (Camparo et al. 2015, Jaduszliwer et al. 2021).
The implementation of this new technology at the industrial level is then expected to well match GNSS requirements and provide advantages reducing in orbit maintenance needs, increasing re-alignment intervals with a simultaneous reduction of size, mass and power consumption while providing frequency stability performance competitive with the passive hydrogen maser.
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The authors acknowledge useful discussions with Leonardo S.p.A. colleagues, in particular, Adalberto Sapia and Jacopo Belfi.
Open access funding provided by Istituto Nazionale di Ricerca Metrologica within the CRUI-CARE Agreement.
Quantum Metrology and Nanotechnologies Division, Istituto Nazionale Di Ricerca Metrologica, INRIM, Strada delle Cacce 91, 10135, Torino, Italy
S. Micalizio, F. Levi, C. E. Calosso, M. Gozzelino & A. Godone
S. Micalizio
F. Levi
C. E. Calosso
M. Gozzelino
A. Godone
Correspondence to S. Micalizio.
Micalizio, S., Levi, F., Calosso, C.E. et al. A pulsed-Laser Rb atomic frequency standard for GNSS applications. GPS Solut 25, 94 (2021). https://doi.org/10.1007/s10291-021-01136-9
Accepted: 12 April 2021
Rb atomic clock
Optical pumping
Pulsed laser | CommonCrawl |
A pyramid is formed on a $6\times 8$ rectangular base. The four edges joining the apex to the corners of the rectangular base each have length $13$. What is the volume of the pyramid?
We know the rectangular base of the pyramid has area $48$. To find the volume, we must also determine the height.
Let the rectangular base be $ABCD$. Let the apex of the pyramid be $X$, and let $O$ be the foot of the perpendicular drawn from $X$ to face $ABCD$: [asy]
size(6cm);
import three;
triple A = (-3,-4,0);
triple B = (-3,4,0);
triple C = (3,4,0);
triple D = (3,-4,0);
triple O = (0,0,0);
triple X = (0,0,12);
draw(B--C--D--A--B--X--D);
draw(X--C); draw(A--X--O--D,dashed);
dot(A); dot(B); dot(C); dot(D); dot(O); dot(X);
label("$A$",A,NW);
label("$B$",B,E);
label("$C$",C,SSE);
label("$D$",D,W);
label("$O$",O,ESE);
label("$X$",X,N);
draw(O+(X-O)/19.2--O+(X-O)/19.2+(D-O)/8--O+(D-O)/8);
[/asy] Then by the Pythagorean theorem, we have \begin{align*}
13^2 &= OX^2+OA^2 = OX^2+OB^2 \\
&= OX^2+OC^2 = OX^2+OD^2.
\end{align*}Therefore, $OA=OB=OC=OD$, so $O$ must be the center of the rectangle (where the perpendicular bisectors of the sides meet). This is also the point where the diagonals of $ABCD$ bisect each other. Each diagonal of $ABCD$ has length $\sqrt{6^2+8^2}=10$, so we have $OA=OB=OC=OD=5$.
Thus $OX=\sqrt{13^2-OD^2} = \sqrt{13^2-5^2}=12$, and so the height of the pyramid is $12$. The volume is \begin{align*}
\frac 13\cdot (\text{area of base})\cdot (\text{height}) &= \frac 13\cdot 48\cdot 12 \\
&= 16\cdot 12 \\
&= \boxed{192}.
\end{align*} | Math Dataset |
Computer Aided Verification
International Conference on Computer Aided Verification
CAV 2019: Computer Aided Verification pp 426-444 | Cite as
Termination of Triangular Integer Loops is Decidable
Florian Frohn
Jürgen Giesl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11562)
We consider the problem whether termination of affine integer loops is decidable. Since Tiwari conjectured decidability in 2004 [15], only special cases have been solved [3, 4, 14]. We complement this work by proving decidability for the case that the update matrix is triangular.
Funded by DFG grant 389792660 as part of TRR 248 and by DFG grant GI 274/6.
Download conference paper PDF
We consider affine integer loops of the form
$$\begin{aligned} \mathbf{while}\ \varphi \ \mathbf{do}\ \overline{x}\ \leftarrow A\, \overline{x}+\overline{a}. \end{aligned}$$
Here, \(A \in \mathbb {Z}^{d \times d}\) for some dimension \(d \ge 1\), \(\overline{x}\) is a column vector of pairwise different variables \(x_1,\ldots ,x_d\), \(\overline{a} \in \mathbb {Z}^d\), and \(\varphi \) is a conjunction of inequalities of the form \(\alpha > 0\) where \(\alpha \in \mathbb {A}\mathbbm {f}[\overline{x}]\) is an affine expression with rational coefficients1 over \(\overline{x}\) (i.e., \(\mathbb {A}\mathbbm {f}[\overline{x}] = \{\overline{c}^T\, \overline{x} + c \mid \overline{c} \in \mathbb {Q}^d, c \in \mathbb {Q}\}\)). So \(\varphi \) has the form \(B\,\overline{x} + \overline{b} > \overline{0}\) where \(\overline{0}\) is the vector containing k zeros, \(B \in \mathbb {Q}^{k \times d}\), and \(\overline{b} \in \mathbb {Q}^k\) for some \(k \in \mathbb {N}\). Definition 1 formalizes the intuitive notion of termination for such loops.
(Termination). Let \(f:\mathbb {Z}^d \rightarrow \mathbb {Z}^d\) with \(f(\overline{x}) = A\,\overline{x} + \overline{a}\). If
$$ \exists \overline{c} \in \mathbb {Z}^{d}.\ \forall n \in \mathbb {N}.\ \varphi [\overline{x} / f^n(\overline{c})], $$
then (1) is non-terminating and \(\overline{c}\) is a witness for non-termination. Otherwise, (1) terminates.
Here, \(f^n\) denotes the n-fold application of f, i.e., we have \(f^0(\overline{c}) = \overline{c}\) and \(f^{n+1}(\overline{c}) = f(f^n(\overline{c}))\). We call f the update of (1). Moreover, for any entity s, s[x / t] denotes the entity that results from s by replacing all occurrences of x by t. Similarly, if \(\overline{x} = \begin{bmatrix}x_1\\[-.15cm]\vdots \\x_m\end{bmatrix}\) and \(\overline{t} = \begin{bmatrix}t_1\\[-.15cm]\vdots \\t_m\end{bmatrix}\), then \(s[\overline{x} / \overline{t}]\) denotes the entity resulting from s by replacing all occurrences of \(x_i\) by \(t_i\) for each \(1 \le i \le m\).
Consider the loop
$$\begin{aligned} \mathbf{while}\,\, {y + z > 0}\, \,\mathbf{do}\,\, \left[ \begin{array}{c} w\\ x\\ y\\ z \end{array}\right] \leftarrow \left[ \begin{array}{c} 2\\ x + 1\\ - w - 2 \cdot y\\ x \end{array}\right] \end{aligned}$$
where the update of all variables is executed simultaneously. This program belongs to our class of affine loops, because it can be written equivalently as follows.
$$\begin{aligned} \mathbf{while}\, \,y + z > 0\, \,\mathbf{do}\,\, \left[ \begin{array}{c} w\\ x\\ y\\ z \end{array}\right] \leftarrow \left[ \begin{array}{cccc} 0&{}0&{}0&{}0\\ 0&{}1&{}0&{}0\\ -1&{}0&{}-2&{}0\\ 0&{}1&{}0&{}0 \end{array}\right] \left[ \begin{array}{c} w\\ x\\ y\\ z \end{array}\right] + \left[ \begin{array}{c} 2\\ 1\\ 0\\ 0 \end{array}\right] \end{aligned}$$
While termination of affine loops is known to be decidable if the variables range over the real [15] or the rational numbers [4], the integer case is a well-known open problem [2, 3, 4, 14, 15].2 However, certain special cases have been solved: Braverman [4] showed that termination of linear loops is decidable (i.e., loops of the form (1) where \(\overline{a}\) is \(\overline{0}\) and \(\varphi \) is of the form \(B\,\overline{x} > \overline{0}\)). Bozga et al. [3] showed decidability for the case that the update matrix A in (1) has the finite monoid property, i.e., if there is an \(n > 0\) such that \(A^n\) is diagonalizable and all eigenvalues of \(A^n\) are in \(\{0,1\}.\) Ouaknine et al. [14] proved decidability for the case \(d \le 4\) and for the case that A is diagonalizable.
Ben-Amram et al. [2] showed undecidability of termination for certain extensions of affine integer loops, e.g., for loops where the body is of the form \(\mathbf {if}\ x > 0\ \mathbf {then}\ \overline{x} \leftarrow A\,\overline{x}\ \mathbf {else}\ \overline{x} \leftarrow A'\,\overline{x}\) where \(A,A' \in \mathbb {Z}^{d \times d}\) and \(x \in \overline{x}\).
In this paper, we present another substantial step towards the solution of the open problem whether termination of affine integer loops is decidable. We show that termination is decidable for triangular loops (1) where A is a triangular matrix (i.e., all entries of A below or above the main diagonal are zero). Clearly, the order of the variables is irrelevant, i.e., our results also cover the case that A can be transformed into a triangular matrix by reordering A, \(\overline{x}\), and \(\overline{a}\) accordingly.3 So essentially, triangularity means that the program variables \(x_1,\ldots ,x_d\) can be ordered such that in each loop iteration, the new value of \(x_i\) only depends on the previous values of \(x_1,\ldots ,x_{i-1},x_i\). Hence, this excludes programs with "cyclic dependencies" of variables (e.g., where the new values of x and y both depend on the old values of both x and y). While triangular loops are a very restricted subclass of general integer programs, integer programs often contain such loops. Hence, tools for termination analysis of such programs (e.g., [5, 6, 7, 8, 11, 12, 13]) could benefit from integrating our decision procedure and applying it whenever a sub-program is an affine triangular loop.
Note that triangularity and diagonalizability of matrices do not imply each other. As we consider loops with arbitrary dimension, this means that the class of loops considered in this paper is not covered by [3, 14]. Since we consider affine instead of linear loops, it is also orthogonal to [4].
To see the difference between our and previous results, note that a triangular matrix A where \(c_1,\ldots ,c_k\) are the distinct entries on the diagonal is diagonalizable iff \((A - c_1 I) \ldots (A- c_k I)\) is the zero matrix.4 Here, I is the identity matrix. So an easy example for a triangular loop where the update matrix is not diagonalizable is the following well-known program (see, e.g., [2]):
$$\begin{aligned} \mathbf{while}\,\, x > 0\,\, \mathbf{do} \,\, x \leftarrow x+y;\; y \leftarrow y-1 \end{aligned}$$
It terminates as y eventually becomes negative and then x decreases in each iteration. In matrix notation, the loop body is Open image in new window , i.e., the update matrix is triangular. Thus, this program is in our class of programs where we show that termination is decidable. However, the only entry on the diagonal of the update matrix A is \(c = 1\) and Open image in new window is not the zero matrix. So A (and in fact each \(A^n\) where \(n \in \mathbb {N}\)) is not diagonalizable. Hence, extensions of this example to a dimension greater than 4 where the loop is still triangular are not covered by any of the previous results.5
Our proof that termination is decidable for triangular loops proceeds in three steps. We first prove that termination of triangular loops is decidable iff termination of non-negative triangular loops (nnt-loops) is decidable, cf. Sect. 2. A loop is non-negative if the diagonal of A does not contain negative entries. Second, we show how to compute closed forms for nnt-loops, i.e., vectors \(\overline{q}\) of d expressions over the variables \(\overline{x}\) and n such that \(\overline{q}[n/c] = f^c(\overline{x})\) for all \(c\ge 0\), see Sect. 3. Here, triangularity of the matrix A allows us to treat the variables step by step. So for any \(1 \le i \le d\), we already know the closed forms for \(x_1,\ldots ,x_{i-1}\) when computing the closed form for \(x_i\). The idea of computing closed forms for the repeated updates of loops was inspired by our previous work on inferring lower bounds on the runtime of integer programs [10]. But in contrast to [10], here the computation of the closed form always succeeds due to the restricted shape of the programs. Finally, we explain how to decide termination of nnt-loops by reasoning about their closed forms in Sect. 4. While our technique does not yield witnesses for non-termination, we show that it yields witnesses for eventual non-termination, i.e., vectors \(\overline{c}\) such that \(f^n(\overline{c})\) witnesses non-termination for some \(n \in \mathbb {N}\). Detailed proofs for all lemmas and theorems can be found in [9].
2 From Triangular to Non-Negative Triangular Loops
To transform triangular loops into nnt-loops, we define how to chain loops. Intuitively, chaining yields a new loop where a single iteration is equivalent to two iterations of the original loop. Then we show that chaining a triangular loop always yields an nnt-loop and that chaining is equivalent w.r.t. termination.
(Chaining).Chaining the loop (1) yields:
$$\begin{aligned} \mathbf{while}\,\, \varphi \wedge \varphi [\overline{x} / A\,\overline{x} + \overline{a}] \,\,\mathbf{do}\,\, \overline{x} \leftarrow A^2\,\overline{x} + A\,\overline{a} + \overline{a} \end{aligned}$$
Chaining Example 2 yields
$$\begin{aligned} \begin{array}{l} \mathbf{while}\,\, y + z> 0 \wedge - w - 2 \cdot y + x > 0 \,\,\mathbf{do}\\ \qquad \left[ \begin{array}{c} w\\ x\\ y\\ z \end{array}\right] \leftarrow \left[ \begin{array}{cccc} 0&{}0&{}0&{}0\\ 0&{}1&{}0&{}0\\ -1&{}0&{}-2&{}0\\ 0&{}1&{}0&{}0 \end{array}\right] ^2 \left[ \begin{array}{c} w\\ x\\ y\\ z \end{array}\right] + \left[ \begin{array}{cccc} 0&{}0&{}0&{}0\\ 0&{}1&{}0&{}0\\ -1&{}0&{}-2&{}0\\ 0&{}1&{}0&{}0 \end{array}\right] \left[ \begin{array}{c} 2\\ 1\\ 0\\ 0 \end{array}\right] + \left[ \begin{array}{c} 2\\ 1\\ 0\\ 0 \end{array}\right] \end{array} \end{aligned}$$
which simplifies to the following nnt-loop:
$$\begin{aligned} \mathbf{while}\,\, y + z> 0 \wedge - w - 2 \cdot y + x > 0 \,\,\mathbf{do}\,\, \left[ \begin{array}{c} w\\ x\\ y\\ z \end{array}\right] \leftarrow \left[ \begin{array}{cccc} 0&{}0&{}0&{}0\\ 0&{}1&{}0&{}0\\ 2&{}0&{}4&{}0\\ 0&{}1&{}0&{}0 \end{array}\right] \left[ \begin{array}{c} w\\ x\\ y\\ z \end{array}\right] + \left[ \begin{array}{c} 2\\ 2\\ -2\\ 1 \end{array}\right] \end{aligned}$$
Lemma 5 is needed to prove that (2) is an nnt-loop if (1) is triangular.
Lemma 5
(Squares of Triangular Matrices). For every triangular matrix A, \(A^2\) is a triangular matrix whose diagonal entries are non-negative.
Corollary 6
(Chaining Loops). If (1) is triangular, then (2) is an nnt-loop.
Immediate consequence of Definition 3 and Lemma 5. \(\square \)
(Equivalence of Chaining). (1) terminates \(\iff \) (2) terminates.
By Definition 1, (1) does not terminate iff
$$ \begin{array}{lll} &{}\exists \overline{c} \in \mathbb {Z}^{d}.\ \forall n \in \mathbb {N}.\ \varphi [\overline{x} / f^n(\overline{c})] &{} \\ \iff &{}\exists \overline{c} \in \mathbb {Z}^{d}.\ \forall n \in \mathbb {N}.\ \varphi [\overline{x} / f^{2 \cdot n}(\overline{c})] \wedge \varphi [\overline{x} / f^{2 \cdot n + 1}(\overline{c})]\\ \iff &{}\exists \overline{c} \in \mathbb {Z}^{d}.\ \forall n \in \mathbb {N}.\ \varphi [\overline{x} / f^{2 \cdot n}(\overline{c})] \wedge \varphi [\overline{x} / A\,f^{2 \cdot n}(\overline{c}) + \overline{a}] &{} (\text {by Definition of } f), \end{array} $$
i.e., iff (2) does not terminate as \(f^2(\overline{x}) = A^2\,\overline{x} + A\,\overline{a} + \overline{a}\) is the update of (2). \(\square \)
Theorem 8
(Reducing Termination to nnt-Loops). Termination of triangular loops is decidable iff termination of nnt-loops is decidable.
Immediate consequence of Corollary 6 and Lemma 7. \(\square \)
Thus, from now on we restrict our attention to nnt-loops.
3 Computing Closed Forms
The next step towards our decidability proof is to show that \(f^n(\overline{x})\) is equivalent to a vector of poly-exponential expressions for each nnt-loop, i.e., the closed form of each nnt-loop can be represented by such expressions. Here, equivalence means that two expressions evaluate to the same result for all variable assignments.
Poly-exponential expressions are sums of arithmetic terms where it is always clear which addend determines the asymptotic growth of the whole expression when increasing a designated variable n. This is crucial for our decidability proof in Sect. 4. Let \(\mathbb {N}_{\ge 1} = \{b \in \mathbb {N}\mid b \ge 1\}\) (and \(\mathbb {Q}_{>0}\), \(\mathbb {N}_{>1}\), etc. are defined analogously). Moreover, \(\mathbb {A}\mathbbm {f}[\overline{x}]\) is again the set of all affine expressions over \(\overline{x}\).
(Poly-Exponential Expressions). Let \(\mathcal {C}\) be the set of all finite conjunctions over the literals \(n = c, n \ne c\) where n is a designated variable and \(c \in \mathbb {N}\). Moreover for each formula \(\psi \) over n, let Open image in new window be the characteristic function of \(\psi \), i.e., Open image in new window if \(\psi [n/c]\) is valid and Open image in new window , otherwise. The set of all poly-exponential expressions over \(\overline{x}\) is
As n ranges over \(\mathbb {N}\), we use Open image in new window as syntactic sugar for Open image in new window . So an example for a poly-exponential expression is
Moreover, note that if \(\psi \) contains a positive literal (i.e., a literal of the form "\(n = c\)" for some number \(c \in \mathbb {N}\)), then Open image in new window is equivalent to either 0 or Open image in new window .
The crux of the proof that poly-exponential expressions can represent closed forms is to show that certain sums over products of exponential and poly-exponential expressions can be represented by poly-exponential expressions, cf. Lemma 12. To construct these expressions, we use a variant of [1, Lemma 3.5]. As usual, \(\mathbb {Q}[\overline{x}]\) is the set of all polynomials over \(\overline{x}\) with rational coefficients.
Lemma 10
(Expressing Polynomials by Differences [1]). If \(q \in \mathbb {Q}[n]\) and \(c \in \mathbb {Q}\), then there is an \(r \in \mathbb {Q}[n]\) such that \(q = r - c \cdot r[n/n-1]\) for all \(n \in \mathbb {N}\).
So Lemma 10 expresses a polynomial q via the difference of another polynomial r at the positions n and \(n-1\), where the additional factor c can be chosen freely. The proof of Lemma 10 is by induction on the degree of q and its structure resembles the structure of the following algorithm to compute r. Using the Binomial Theorem, one can verify that \(q - s + c \cdot s[n/n-1]\) has a smaller degree than q, which is crucial for the proof of Lemma 10 and termination of Algorithm 1.
As an example, consider \(q = 1\) (i.e., \(c_0 = 1\)) and \(c = 4\). Then we search for an r such that \(q = r - c \cdot r[n/n-1]\), i.e., \(1 = r - 4 \cdot r[n/n-1]\). According to Algorithm 1, the solution is \(r = \frac{c_0}{1-c} = -\frac{1}{3}\).
(Closure of \(\mathbb {PE}\) under Sums of Products and Exponentials). If \(m \in \mathbb {N}\) and \(p \in \mathbb {PE}[\overline{x}]\), then one can compute a \(q \in \mathbb {PE}[\overline{x}]\) which is equivalent to \(\sum _{i=1}^{n} m^{n - i} \cdot p[n/i-1]\).
Let Open image in new window . We have:
As \(\mathbb {PE}[\overline{x}]\) is closed under addition, it suffices to show that we can compute an equivalent poly-exponential expression for any expression of the form
We first regard the case \(m=0\). Here, the expression (4) can be simplified to
Clearly, there is a \(\psi ' \in \mathcal {C}\) such that Open image in new window is equivalent to Open image in new window . Moreover, \(\alpha \cdot b^{n-1} = \tfrac{\alpha }{b} \cdot b^n\) where \(\tfrac{\alpha }{b} \in \mathbb {A}\mathbbm {f}[\overline{x}]\). Hence, due to the Binomial Theorem
which is a poly-exponential expression as \(\tfrac{\alpha }{b}\cdot \left( {\begin{array}{c}a\\ i\end{array}}\right) \cdot (-1)^i \in \mathbb {A}\mathbbm {f}[\overline{x}]\).
From now on, let \(m \ge 1\). If \(\psi \) contains a positive literal \(n = c\), then we get
The step marked with \((\dagger )\) holds as we have Open image in new window for all \(i \in \{1,\ldots ,n\}\) and the step marked with \((\dagger \dagger )\) holds since \(i \ne c+1\) implies Open image in new window . If \(\psi \) does not contain a positive literal, then let c be the maximal constant that occurs in \(\psi \) or \(-1\) if \(\psi \) is empty. We get:
Again, the step marked with \((\dagger )\) holds since we have Open image in new window for all \(i \in \{1,\ldots ,n\}\). The last step holds as \(i \ge c+2\) implies Open image in new window . Similar to the case where \(\psi \) contains a positive literal, we can compute a poly-exponential expression which is equivalent to the first addend. We have
which is a poly-exponential expression as \(\tfrac{1}{m^{i}}\cdot \alpha \cdot (i-1)^a \cdot b^{i-1} \in \mathbb {A}\mathbbm {f}[\overline{x}]\). For the second addend, we have:
Lemma 10 ensures \(r \in \mathbb {Q}[n]\), i.e., we have \(r = \sum _{i=0}^{d_r} m_i \cdot n^i\) for some \(d_r \in \mathbb {N}\) and \(m_i \in \mathbb {Q}\). Thus, \(r[n/c+1] \cdot \left( \frac{b}{m}\right) ^{c+1} \cdot \frac{\alpha }{b} \in \mathbb {A}\mathbbm {f}[\overline{x}]\) which implies Open image in new window . It remains to show that the addend Open image in new window is equivalent to a poly-exponential expression. As \(\frac{\alpha }{b} \cdot m_i \in \mathbb {A}\mathbbm {f}[\overline{x}]\), we have
\(\square \)
The proof of Lemma 12 gives rise to a corresponding algorithm.
We compute an equivalent poly-exponential expression for
where w is a variable. (It will later on be needed to compute a closed form for Example 4, see Example 18.) According to Algorithm 2 and (3), we get
with Open image in new window , Open image in new window , and \(p_3 = \sum _{i=1}^{n} 4^{n-i} \cdot (- 2)\). We search for \(q_1, q_2, q_3 \in \mathbb {PE}[w]\) that are equivalent to \(p_1, p_2, p_3\), i.e., \(q_1 + q_2 + q_3\) is equivalent to (12). We only show how to compute \(q_2\)(and omit the computation of Open image in new window ). Analogously to (8), we get:
The next step is to rearrange the first sum as in (9). In our example, it directly simplifies to 0 and hence we obtain
Finally, by applying the steps from (10) we get:
The step marked with \((\dagger )\) holds by Lemma 10 with \(q = 1\) and \(c = 4\). Thus, we have \(r = -\tfrac{1}{3}\), cf. Example 11.
Recall that our goal is to compute closed forms for loops. As a first step, instead of the n-fold update function \(h(n,\overline{x}) = f^n(\overline{x})\) of (1) where f is the update of (1), we consider a recursive update function for a single variable \(x \in \overline{x}\):
$$ \textstyle g(0,\overline{x}) = x \quad \text {and} \quad g(n,\overline{x}) = m \cdot g(n-1, \overline{x}) + p[n/n-1] \quad \text {for all n > 0} $$
Here, \(m \in \mathbb {N}\) and \(p \in \mathbb {PE}[\overline{x}]\). Using Lemma 12, it is easy to show that g can be represented by a poly-exponential expression.
(Closed Form for Single Variables). If \(x \in \overline{x}\), \(m \in \mathbb {N}\), and \(p \in \mathbb {PE}[\overline{x}]\), then one can compute a \(\,q \in \mathbb {PE}[\overline{x}]\) which satisfies
$$ \textstyle q\,[n/0] = x \quad \text {and} \quad q = (m \cdot q + p)\;[n/n-1] \quad \text {for all } n > 0. $$
It suffices to find a \(q \in \mathbb {PE}[\overline{x}]\) that satisfies
$$\begin{aligned} \textstyle q = m^n \cdot x + \sum _{i=1}^{n} m^{n-i} \cdot p[n/i-1]. \end{aligned}$$
To see why (13) is sufficient, note that (13) implies
$$ \textstyle q[n/0] \quad = \quad m^0 \cdot x + \sum \nolimits _{i=1}^{0} m^{0-i} \cdot p[n/i-1] \quad =\quad x $$
and for \(n > 0\), (13) implies
$$ \begin{array}{llll} q &{}=&{} m^{n} \cdot x + \mathop {\sum }\nolimits _{i=1}^{n} m^{n-i} \cdot p[n/i-1]\\ &{}=&{} m^{n} \cdot x + \left( \mathop {\sum }\nolimits _{i=1}^{n-1} m^{n-i} \cdot p[n/i-1]\right) + p[n/n-1]\\ &{}=&{} m \cdot \left( m^{n-1} \cdot x + \mathop {\sum }\nolimits _{i=1}^{n-1} m^{n-i-1} \cdot p[n/i-1]\right) + p[n/n-1]\\ &{}=&{} m \cdot q[n/n-1] + p[n/n-1]\\ &{}=&{} (m \cdot q + p)[n/n-1]. \end{array} $$
By Lemma 12, we can compute a \(q' \in \mathbb {PE}[\overline{x}]\) such that
$$ \textstyle m^n \cdot x + \mathop {\sum }\nolimits _{i=1}^{n} m^{n-i} \cdot p[n/i-1] \quad = \quad m^n \cdot x + q'. $$
Moreover,
So both addends are equivalent to poly-exponential expressions. \(\square \)
We show how to compute the closed forms for the variables w and x from Example 4. We first consider the assignment \(w \leftarrow 2\), i.e., we want to compute a \(q_w \in \mathbb {PE}[w,x,y,z]\) with \(q_w [n/0] = w\) and \(q_w = (m_w \cdot q_w + p_w)\,[n/n-1]\) for \(n > 0\), where \(m_w = 0\) and \(p_w = 2\). According to (13) and (14), \(q_w\) is
For the assignment \(x \leftarrow x + 2\), we search for a \(q_x\) such that \(q_x[n/0] = x\) and \(q_x = (m_x \cdot q_x + p_x)\,[n/n-1]\) for \(n > 0\), where \(m_x = 1\) and \(p_x = 2\). By (13), \(q_x\) is
$$\textstyle m_x^n \cdot x + \sum _{i=1}^{n} m_x^{n-i} \cdot p_x[n/i-1] = 1^n \cdot x + \sum _{i=1}^{n} 1^{n-i} \cdot 2 = x + 2 \cdot n. $$
The restriction to triangular matrices now allows us to generalize Lemma 14 to vectors of variables. The reason is that due to triangularity, the update of each program variable \(x_i\) only depends on the previous values of \(x_1,\ldots ,x_{i}\). So when regarding \(x_i\), we can assume that we already know the closed forms for \(x_1,\ldots ,x_{i-1}\). This allows us to find closed forms for one variable after the other by applying Lemma 14 repeatedly. In other words, it allows us to find a vector \(\overline{q}\) of poly-exponential expressions that satisfies
$$ \textstyle \overline{q}\,[n/0] = \overline{x}\quad \text {and} \quad \overline{q} = A\, \overline{q}[n/n-1] + \overline{a} \quad \text {for all } n > 0. $$
To prove this claim, we show the more general Lemma 16. For all \(i_1,\ldots ,i_k \in \{1, \ldots , m\}\), we define \([z_1,\ldots ,z_m]_{i_1,\ldots ,i_k} = [z_{i_1},\ldots ,z_{i_k}]\) (and the notation \(\overline{y}_{i_1,\ldots ,i_k}\) for column vectors is defined analogously). Moreover, for a matrix A, \(A_{i}\) is A's \(i^{th}\) row and \(A_{i_1,\ldots ,i_n;j_1,\ldots ,j_k}\) is the matrix with rows \((A_{i_1})_{j_1,\ldots ,j_k}, \ldots , (A_{i_n})_{j_1,\ldots ,j_k}\). So for \(A = \begin{bmatrix} a_{1,1}&a_{1,2}&a_{1,3}\\ a_{2,1}&a_{2,2}&a_{2,3}\\ a_{3,1}&a_{3,2}&a_{3,3} \end{bmatrix}\), we have \(A_{1,2;1,3} = \begin{bmatrix} a_{1,1}&a_{1,3}\\ a_{2,1}&a_{2,3} \end{bmatrix}\).
(Closed Forms for Vectors of Variables). If \(\overline{x}\) is a vector of at least \(d \ge 1\) pairwise different variables, \(A \in \mathbb {Z}^{d \times d}\) is triangular with \(A_{i;i} \ge 0\) for all \(1 \le i \le d\), and \(\overline{p} \in \mathbb {PE}[\overline{x}]^d\), then one can compute \(\overline{q} \in \mathbb {PE}[\overline{x}]^d\) such that:
$$\begin{aligned} \overline{q}\,[n/0]&= \overline{x}_{1,\ldots ,d}\quad \text {and}\end{aligned}$$
$$\begin{aligned} \overline{q}&= (A\, \overline{q} + \overline{p})\;[n/n-1] \quad \text {for all } n > 0 \end{aligned}$$
Assume that A is lower triangular (the case that A is upper triangular works analogously). We use induction on d. For any \(d \ge 1\) we have:
$$ \begin{array}{llllll} &{}\overline{q} &{}=&{} (A\, \overline{q} + \overline{p})\;[n/n-1]\\ \iff &{} \overline{q}_j &{}=&{} (A_{j} \cdot \overline{q} + \overline{p}_j)\;[n/n-1] &{} \text {for all } 1 \le j \le d\\ \iff &{} \overline{q}_j &{}=&{} (A_{j;2,\ldots ,d} \cdot \overline{q}_{2,\ldots ,d} + A_{j;1} \cdot \overline{q}_1 + \overline{p}_j)\;[n/n-1] &{} \text {for all } 1 \le j \le d\\ \iff &{} \overline{q}_1 &{}=&{} (A_{1;2,\ldots ,d} \cdot \overline{q}_{2,\ldots ,d} + A_{1;1} \cdot \overline{q}_1 + \overline{p}_1)\;[n/n-1] &{} \wedge \\ &{} \overline{q}_j &{}=&{} (A_{j;2,\ldots ,d} \cdot \overline{q}_{2,\ldots ,d} + A_{j;1} \cdot \overline{q}_1 + \overline{p}_j)\;[n/n-1] &{} \text {for all } 1< j \le d\\ \iff &{} \overline{q}_1 &{}=&{} (A_{1;1} \cdot \overline{q}_1 + \overline{p}_1)\;[n/n-1] &{} \wedge \\ &{} \overline{q}_j &{}=&{} (A_{j;2,\ldots ,d} \cdot \overline{q}_{2,\ldots ,d} + A_{j;1} \cdot \overline{q}_1 + \overline{p}_j)\;[n/n-1] &{} \text {for all } 1 < j \le d \end{array} $$
The last step holds as A is lower triangular. By Lemma 14, we can compute a \(\overline{q}_1 \in \mathbb {PE}[\overline{x}]\) that satisfies
$$ \textstyle \overline{q}_1[n/0] = \overline{x}_1 \quad \text {and} \quad \overline{q}_1 = (A_{1;1} \cdot \overline{q}_1 + \overline{p}_1)\;[n/n-1] \quad \text {for all } n > 0. $$
In the induction base (\(d = 1\)), there is no j with \(1 < j \le d\). In the induction step (\(d > 1\)), it remains to show that we can compute \(\overline{q}_{2,\ldots ,d}\) such that
$$ \textstyle \overline{q}_j[n/0] = \overline{x}_j \quad \text {and} \quad \overline{q}_j = (A_{j;2,\ldots ,d} \cdot \overline{q}_{2,\ldots ,d} + A_{j;1} \cdot \overline{q}_1 + \overline{p}_j)\;[n/n-1] $$
for all \(n > 0\) and all \(1 < j \le d\), which is equivalent to
$$\begin{aligned} \overline{q}_{2,\ldots ,d}[n/0]&= \overline{x}_{2,\ldots ,d} \quad \text {and}\\[-1.3em] \overline{q}_{2,\ldots ,d}&= (A_{2,\ldots ,d;2,\ldots ,d} \cdot \overline{q}_{2,\ldots ,d} + \begin{bmatrix}A_{2;1}\\\vdots \\A_{d;1}\end{bmatrix} \cdot \overline{q}_1 + \overline{p}_{2,\ldots ,d})\;[n/n-1] \end{aligned}$$
for all \(n>0\). As \(A_{j;1} \cdot \overline{q}_1 + \overline{p}_j \in \mathbb {PE}[\overline{x}]\) for each \(2 \le j \le d\), the claim follows from the induction hypothesis. \(\square \)
Together, Lemmas 14 and 16 and their proofs give rise to the following algorithm to compute a solution for (16) and (17). It computes a closed form \(\overline{q}_1\) for \(\overline{x}_1\) as in the proof of Lemma 14, constructs the argument \(\overline{p}\) for the recursive call based on A, \(\overline{q}_1\), and the current value of \(\overline{p}\) as in the proof of Lemma 16, and then determines the closed form for \(\overline{x}_{2, \ldots , d}\) recursively.
We can now prove the main theorem of this section.
Theorem 17
(Closed Forms for nnt-Loops). One can compute a closed form for every nnt-loop. In other words, if \(f:\mathbb {Z}^d \rightarrow \mathbb {Z}^d\) is the update function of an nnt-loop with the variables \(\overline{x}\), then one can compute a \(\overline{q} \in \mathbb {PE}[\overline{x}]^d\) such that \(\overline{q}[n/c] = f^c(\overline{x})\) for all \(c \in \mathbb {N}\).
Consider an nnt-loop of the form (1). By Lemma 16, we can compute a \(\overline{q} \subseteq \mathbb {PE}[\overline{x}]^d\) that satisfies
$$ \textstyle \overline{q}[n/0] = \overline{x} \quad \text {and} \quad \overline{q} = (A\, \overline{q} + \overline{a})\;[n/n-1] \quad \text {for all } n > 0. $$
We prove \(f^c(\overline{x}) = \overline{q}[n/c]\) by induction on \(c \in \mathbb {N}\). If \(c=0\), we get
$$ f^c(\overline{x}) = f^0(\overline{x}) = \overline{x} = \overline{q}[n/0] = \overline{q}[n/c]. $$
$$ \begin{array}{l@{}llll} \text{ If } c>0\text{, } \text{ we } \text{ get: }&{} f^c(\overline{x}) &{}=&{} A\, f^{c-1}(\overline{x}) + \overline{a} &{} \text {by definition of } f\\ &{}&{}=&{} A\, \overline{q}[n/c-1] + \overline{a} &{} \text {by the induction hypothesis}\\ &{}&{}=&{} (A\, \overline{q} + \overline{a})\;[n/c-1] &{} \text {as } \overline{a} \in \mathbb {Z}^d \text { does not contain } n\\ &{}&{}=&{} \overline{q}[n/c] &{} \end{array}$$
So invoking Algorithm 3 on \(\overline{x}, A\), and \(\overline{a}\) yields the closed form of an nnt-loop (1).
We show how to compute the closed form for Example 4. For
$$ y \leftarrow 2 \cdot w + 4 \cdot y - 2, $$
we obtain
where \(q_0 = y \cdot 4^n\). For \(z \leftarrow x + 1\), we get
So the closed form of Example 4 for the values of the variables after n iterations is:
4 Deciding Non-Termination of nnt-Loops
Our proof uses the notion of eventual non-termination [4, 14]. Here, the idea is to disregard the condition of the loop during a finite prefix of the program run.
Definition 19
(Eventual Non-Termination). A vector \(\overline{c} \in \mathbb {Z}^d\) witnesses eventual non-termination of (1) if
$$ \exists n_0 \in \mathbb {N}.\ \forall n \in \mathbb {N}_{>n_0}.\ \varphi [\overline{x} / f^{n}(\overline{c})]. $$
If there is such a witness, then (1) is eventually non-terminating.
Clearly, (1) is non-terminating iff (1) is eventually non-terminating [14]. Now Theorem 17 gives rise to an alternative characterization of eventual non-termination in terms of the closed form \(\overline{q}\) instead of \(f^{n}(\overline{c})\).
Corollary 20
(Expressing Non-Termination with \(\mathbb {PE}\)). If \(\overline{q}\) is the closed form of (1), then \(\overline{c} \in \mathbb {Z}^d\) witnesses eventual non-termination iff
$$\begin{aligned} \exists n_0 \in \mathbb {N}.\ \forall n \in \mathbb {N}_{>n_0}.\ \varphi [\overline{x} / \overline{q}][\overline{x} / \overline{c}]. \end{aligned}$$
Immediate, as \(\overline{q}\) is equivalent to \(f^n(\overline{x})\). \(\square \)
So to prove that termination of nnt-loops is decidable, we will use Corollary 20 to show that the existence of a witness for eventual non-termination is decidable. To do so, we first eliminate the factors Open image in new window from the closed form \(\overline{q}\). Assume that \(\overline{q}\) has at least one factor Open image in new window where \(\psi \) is non-empty (otherwise, all factors Open image in new window are equivalent to 1) and let c be the maximal constant that occurs in such a factor. Then all addends Open image in new window where \(\psi \) contains a positive literal become 0 and all other addends become \(\alpha \cdot n^{a} \cdot b^n\) if \(n > c\). Thus, as we can assume \(n_0 > c\) in (18) without loss of generality, all factors Open image in new window can be eliminated when checking eventual non-termination.
Removing Open image in new window from \(\mathbb {PE}\)s). Let \(\overline{q}\) be the closed form of an nnt-loop (1). Let \(\overline{q}_{norm}\) result from \(\overline{q}\) by removing all addends Open image in new window where \(\psi \) contains a positive literal and by replacing all addends Open image in new window where \(\psi \) does not contain a positive literal by \(\alpha \cdot n^{a} \cdot b^n\). Then \(\overline{c} \in \mathbb {Z}^d\) is a witness for eventual non-termination iff
$$\begin{aligned} \exists n_0 \in \mathbb {N}.\ \forall n \in \mathbb {N}_{>n_0}.\ \varphi [\overline{x} / \overline{q}_{norm}][\overline{x} / \overline{c}]. \end{aligned}$$
By removing the factors Open image in new window from the closed form \(\overline{q}\) of an nnt-loop, we obtain normalized poly-exponential expressions.
(Normalized \(\mathbb {PE}\)s). We call \(p \in \mathbb {PE}[\overline{x}]\) normalized if it is in
W.l.o.g., we always assume \((b_i,a_i) \ne (b_j,a_j)\) for all \(i,j \in \{1,\ldots ,\ell \}\) with \(i \ne j\). We define \(\mathbb {NPE}= \mathbb {NPE}[\varnothing ]\), i.e., we have \(p \in \mathbb {NPE}\) if \(\alpha _j \in \mathbb {Q}\) for all \(1 \le j \le \ell \).
We continue Example 18. By omitting the factors Open image in new window ,
and \(q_x = x + 2 \cdot n, q_0 = y \cdot 4^n\), and \(q_3 = \tfrac{2}{3} - \frac{2}{3} \cdot 4^{n}\) remain unchanged. Moreover,
Thus, \(q_y = q_0 + q_1 + q_2 + q_3\) becomes
$$ \textstyle y \cdot 4^n + \frac{1}{2} \cdot w \cdot 4^{n} - \frac{4}{3} + \frac{1}{3}\cdot 4^{n} + \tfrac{2}{3}- \frac{2}{3} \cdot 4^{n} = 4^n \cdot \left( y - \frac{1}{3} + \frac{1}{2} \cdot w\right) - \frac{2}{3}. $$
Let \(\sigma = \left[ w/2,\, x/x+ 2 \cdot n, \, y/4^n \cdot \left( y - \frac{1}{3} + \frac{1}{2} \cdot w\right) - \frac{2}{3}, \, z/x-1 + 2 \cdot n\right] \). Then we get that Example 2 is non-terminating iff there are \(w,x,y,z \in \mathbb {Z}, n_0 \in \mathbb {N}\) such that
$$ \begin{array}{l} (y + z)\;\sigma> 0 \wedge (- w - 2 \cdot y + x)\; \sigma> 0 \qquad \qquad \qquad \,\,\, \iff \\ 4^n \cdot \left( y - \frac{1}{3} + \frac{1}{2} \cdot w\right) - \frac{2}{3} + x - 1 + 2 \cdot n> 0 \wedge \\ \qquad - 2 - 2 \cdot \left( 4^n \cdot \left( y - \frac{1}{3} + \frac{1}{2} \cdot w\right) - \frac{2}{3}\right) + x + 2 {\cdot } n> 0 \iff \\ p^{\varphi }_1> 0 \wedge p^{\varphi }_2 > 0\\ \end{array} $$
holds for all \(n > n_0\) where
$$ \begin{array}{llll} p^{\varphi }_1 &{}=&{} 4^n \cdot \left( y - \frac{1}{3} + \frac{1}{2} \cdot w\right) + 2 \cdot n + x - \frac{5}{3} &{} \text {and}\\ p^{\varphi }_2 &{}=&{} 4^n \cdot \left( \frac{2}{3} - 2 \cdot y - w\right) + 2 \cdot n + x - \frac{2}{3}. \end{array} $$
Recall that the loop condition \(\varphi \) is a conjunction of inequalities of the form \(\alpha > 0\) where \(\alpha \in \mathbb {A}\mathbbm {f}[\overline{x}]\). Thus, \(\varphi [\overline{x} / \overline{q}_{norm}]\) is a conjunction of inequalities \(p > 0\) where \(p \in \mathbb {NPE}[\overline{x}]\) and we need to decide if there is an instantiation of these inequalities that is valid "for large enough n". To do so, we order the coefficients \(\alpha _j\) of the addends \(\alpha _j \cdot n^{a_j} \cdot b_j^n\) of normalized poly-exponential expressions according to the addend's asymptotic growth when increasing n. Lemma 24 shows that \(\alpha _2 \cdot n^{a_2} \cdot b_2^n\) grows faster than \(\alpha _1 \cdot n^{a_1} \cdot b_1^n\) iff \(b_2 > b_1\) or both \(b_2 = b_1\) and \(a_2 > a_1\).
(Asymptotic Growth). Let \(b_1,b_2 \in \mathbb {N}_{\ge 1}\) and \(a_1, a_2 \in \mathbb {N}\). If \((b_2, a_2) >_{lex} (b_1, a_1)\), then \(\mathcal {O}(n^{a_1} \cdot b_1^n) \subsetneq \mathcal {O}(n^{a_2} \cdot b_2^n)\). Here, \({>_{lex}}\) is the lexicographic order, i.e., \((b_2,a_2) >_{lex} (b_1,a_1)\) iff \(b_2 > b_1\) or \(b_2 = b_1 \wedge a_2 > a_1\).
By considering the cases \(b_2 > b_1\) and \(b_2 = b_1\) separately, the claim can easily be deduced from the definition of \(\mathcal {O}\). \(\square \)
(Ordering Coefficients).Marked coefficients are of the form \(\alpha ^{(b,a)}\) where \(\alpha \in \mathbb {A}\mathbbm {f}[\overline{x}], b \in \mathbb {N}_{\ge 1}\), and \(a \in \mathbb {N}\). We define \(\mathrm{unmark}(\alpha ^{(b,a)}) = \alpha \) and \(\alpha _2^{(b_2,a_2)} \succ \alpha _1^{(b_1,a_1)}\) if \((b_2,a_2) >_{lex} (b_1,a_1)\). Let
$$ \textstyle p = \sum _{j=1}^\ell \alpha _j \cdot n^{a_j} \cdot b_j^n \in \mathbb {NPE}[\overline{x}], $$
where \(\alpha _j \ne 0\) for all \(1 \le j \le \ell \). The marked coefficients of p are
In Example 23 we saw that the loop from Example 2 is non-terminating iff there are \(w,x,y,z \in \mathbb {Z}, n_0 \in \mathbb {N}\) such that \(p^{\varphi }_1> 0 \wedge p^{\varphi }_2 > 0\) for all \(n > n_0\). We get:
$$\begin{aligned} \mathrm{coeffs}\left( p^{\varphi }_1\right)&= \left\{ \left( y - \tfrac{1}{3} + \tfrac{1}{2} \cdot w\right) ^{(4,0)}, 2^{(1,1)}, \left( x-\tfrac{5}{3}\right) ^{(1, 0)}\right\} \\ \mathrm{coeffs}\left( p^{\varphi }_2\right)&= \left\{ \left( \tfrac{2}{3} - 2 \cdot y - w\right) ^{(4,0)}, 2^{(1,1)}, \left( x-\tfrac{2}{3}\right) ^{(1,0)}\right\} \end{aligned}$$
Now it is easy to see that the asymptotic growth of a normalized poly-exponential expression is solely determined by its \(\succ \)-maximal addend.
(Maximal Addend Determines Asymptotic Growth). Let \(p \in \mathbb {NPE}\) and let \(\max _{\succ }(\mathrm{coeffs}(p)) = c^{(b,a)}\). Then \(\mathcal {O}(p) = \mathcal {O}(c \cdot n^a \cdot b^n)\).
Clear, as \(c \cdot n^a \cdot b^n\) is the asymptotically dominating addend of p. \(\square \)
Note that Corollary 27 would be incorrect for the case \(c = 0\) if we replaced \(\mathcal {O}(p) = \mathcal {O}(c \cdot n^a \cdot b^n)\) with \(\mathcal {O}(p) = \mathcal {O}(n^a \cdot b^n)\) as \(\mathcal {O}(0) \ne \mathcal {O}(1)\). Building upon Corollary 27, we now show that, for large n, the sign of a normalized poly-exponential expression is solely determined by its \(\succ \)-maximal coefficient. Here, we define \(\mathrm{sign}(c) = -1\) if \(c \in \mathbb {Q}_{<0} \cup \{-\infty \}\), \(\mathrm{sign}(0) = 0\), and \(\mathrm{sign}(c) = 1\) if \(c \in \mathbb {Q}_{>0} \cup \{\infty \}\).
(Sign of \(\mathbb {NPE}\)s). Let \(p \in \mathbb {NPE}\). Then \(\lim _{n \mapsto \infty } p \in \mathbb {Q}\) iff \(p \in \mathbb {Q}\) and otherwise, \(\lim _{n \mapsto \infty } p \in \{ \infty , -\infty \}\). Moreover, we have
$$ \textstyle \mathrm{sign}\left( \lim _{n \mapsto \infty } p\right) = \mathrm{sign}(\mathrm{unmark}(\max _{\succ }(\mathrm{coeffs}(p)))). $$
If \(p \notin \mathbb {Q}\), then the limit of each addend of p is in \(\{-\infty , \infty \}\) by definition of \(\mathbb {NPE}\). As the asymptotically dominating addend determines \(\lim _{n \mapsto \infty } p\) and \(\mathrm{unmark}(\max _{\succ }(\mathrm{coeffs}(p)))\) determines the sign of the asymptotically dominating addend, the claim follows. \(\square \)
Lemma 29 shows the connection between the limit of a normalized poly-exponential expression p and the question whether p is positive for large enough n. The latter corresponds to the existence of a witness for eventual non-termination by Corollary 21 as \(\varphi [\overline{x} / \overline{q}_{norm}]\) is a conjunction of inequalities \(p > 0\) where \(p \in \mathbb {NPE}[\overline{x}]\).
(Limits and Positivity of \(\mathbb {NPE}\)s). Let \(p \in \mathbb {NPE}\). Then
$$ \textstyle \exists n_0 \in \mathbb {N}.\ \forall n \in \mathbb {N}_{>n_0}.\ p> 0 \iff \lim _{n \mapsto \infty } p > 0. $$
By case analysis over \(\lim _{n \mapsto \infty } p\). \(\square \)
Now we show that Corollary 21 allows us to decide eventual non-termination by examining the coefficients of normalized poly-exponential expressions. As these coefficients are in \(\mathbb {A}\mathbbm {f}[\overline{x}]\), the required reasoning is decidable.
(Deciding Eventual Positiveness of \(\mathbb {NPE}\)s). Validity of
$$\begin{aligned} \begin{array}{l} \exists \overline{c} \in \mathbb {Z}^{d}, n_0 \in \mathbb {N}.\ \forall n \in \mathbb {N}_{>n_0}.\ \bigwedge \nolimits _{i=1}^k p_i[\overline{x}/\overline{c}] > 0 \end{array} \end{aligned}$$
where \(p_1,\ldots ,p_k \in \mathbb {NPE}[\overline{x}]\) is decidable.
For any \(p_i\) with \(1 \le i \le k\) and any \(\overline{c} \in \mathbb {Z}^{d}\), we have \(p_i[\overline{x}/\overline{c}] \in \mathbb {NPE}\). Hence:
Let \(p \in \mathbb {NPE}[\overline{x}]\) with \(\mathrm{coeffs}(p) = \left\{ \alpha _1^{(b_1,a_1)}\!,\ldots ,\alpha ^{(b_{\ell },a_{\ell })}_{\ell }\right\} \) where \(\alpha ^{(b_i,a_i)}_i \succ \alpha ^{(b_{j},a_{j})}_{j}\) for all \(1 \le i < j \le \ell \). If \(p[\overline{x}/\overline{c}] = 0\) holds, then \(\mathrm{coeffs}(p[\overline{x}/\overline{c}]) = \{ 0^{(1,0)} \}\) and thus \(\mathrm{unmark}(\max _{\succ }(\mathrm{coeffs}(p[\overline{x}/\overline{c}]))) = 0\). Otherwise, there is an \(1 \le j \le \ell \) with \(\mathrm{unmark}(\max _{\succ }(\mathrm{coeffs}(p[\overline{x}/\overline{c}]))) = \alpha _j[\overline{x}/\overline{c}] \ne 0\) and we have \(\alpha _i[\overline{x}/\overline{c}] = 0\) for all \(1 \le i \le j-1\). Hence, \(\mathrm{unmark}(\max _{\succ }(\mathrm{coeffs}(p[\overline{x}/\overline{c}]))) > 0\) holds iff \(\bigvee _{j=1}^\ell \left( \alpha _j[\overline{x}/\overline{c}] > 0 \wedge \bigwedge _{i=0}^{j-1} \alpha _i[\overline{x}/\overline{c}] = 0\right) \) holds, i.e., iff \([\overline{x}/\overline{c}]\) is a model for
$$\begin{aligned} \begin{array}{l} \mathrm{max\_coeff\_pos}(p) = \bigvee \nolimits _{j=1}^\ell \left( \alpha _j > 0 \wedge \bigwedge \nolimits _{i=0}^{j-1} \alpha _i = 0\right) . \end{array} \end{aligned}$$
Hence by the considerations above, (20) is valid iff
$$\begin{aligned} \begin{array}{l} \exists \overline{c} \in \mathbb {Z}^{d}. \; \bigwedge \nolimits _{i=1}^k \mathrm{max\_coeff\_pos}(p_i) [\overline{x}/\overline{c}] \end{array} \end{aligned}$$
is valid. By multiplying each (in-)equality in (22) with the least common multiple of all denominators, one obtains a first-order formula over the theory of linear integer arithmetic. It is well known that validity of such formulas is decidable. \(\square \)
Note that (22) is valid iff \(\bigwedge _{i=1}^k \mathrm{max\_coeff\_pos}(p_i)\) is satisfiable. So to implement our decision procedure, one can use integer programming or SMT solvers to check satisfiability of \(\bigwedge _{i=1}^k \mathrm{max\_coeff\_pos}(p_i)\). Lemma 30 allows us to prove our main theorem.
Termination of triangular loops is decidable.
By Theorem 8, termination of triangular loops is decidable iff termination of nnt-loops is decidable. For an nnt-loop (1) we obtain a \(\overline{q}_{norm} \in \mathbb {NPE}[\overline{x}]^{d}\) (see Theorem 17 and Corollary 21) such that (1) is non-terminating iff
$$\begin{aligned} \exists \overline{c} \in \mathbb {Z}^{d}, n_0 \in \mathbb {N}.\ \forall n \in \mathbb {N}_{>n_0}.\ \varphi [\overline{x} / \overline{q}_{norm}][\overline{x} / \overline{c}], \end{aligned}$$
where \(\varphi \) is a conjunction of inequalities of the form \(\alpha > 0\), \(\alpha \in \mathbb {A}\mathbbm {f}[\overline{x}]\). Hence,
$$\begin{array}{l} \varphi [\overline{x} / \overline{q}_{norm}][\overline{x} / \overline{c}] \; = \; \bigwedge _{i=1}^k p_i[\overline{x}/\overline{c}] > 0 \end{array}$$
where \(p_1,\ldots ,p_k \in \mathbb {NPE}[\overline{x}]\). Thus, by Lemma 30, validity of (20) is decidable. \(\square \)
The following algorithm summarizes our decision procedure.
In Example 26 we showed that Example 2 is non-terminating iff
$$ \textstyle \exists w,x,y,z \in \mathbb {Z},\ n_0 \in \mathbb {N}.\ \forall n \in \mathbb {N}_{>n_0}.\ p^{\varphi }_1> 0 \wedge p^{\varphi }_2 > 0 $$
is valid. This is the case iff \(\mathrm{max\_coeff\_pos}(p_1) \wedge \mathrm{max\_coeff\_pos}(p_2)\), i.e.,
is satisfiable. This formula is equivalent to \(6 \cdot y - 2 + 3 \cdot w = 0\) which does not have any integer solutions. Hence, the loop of Example 2 terminates.
Example 33 shows that our technique does not yield witnesses for non-termination, but it only proves the existence of a witness for eventual non-termination. While such a witness can be transformed into a witness for non-termination by applying the loop several times, it is unclear how often the loop needs to be applied.
Consider the following non-terminating loop:
The closed form of x is Open image in new window . Replacing x with \(q_{norm}\) in \(x > 0\) yields \(x + y + n - 1 > 0\). The maximal marked coefficient of \(x + y + n - 1\) is \(1^{(1,1)}\). So by Algorithm 4, (23) does not terminate if \(\exists x,y \in \mathbb {Z}.\ 1 > 0\) is valid. While \(1 > 0\) is a tautology, (23) terminates if \(x \le 0\) or \(x \le -y\).
However, the final formula constructed by Algorithm 4 precisely describes all witnesses for eventual non-termination.
(Witnessing Eventual Non-Termination). Let (1) be a triangular loop, let \(\overline{q}_{norm}\) be the normalized closed form of (2), and let
$$ \textstyle \left( \varphi \wedge \varphi [\overline{x} / A\,\overline{x} + \overline{a}]\right) [\overline{x}/\overline{q}_{norm}] = \bigwedge _{i=1}^k p_i > 0. $$
Then \(\overline{c} \in \mathbb {Z}^d\) witnesses eventual non-termination of (1) iff \([\overline{x}/\overline{c}]\) is a model for
$$ \textstyle \bigwedge _{i=1}^k \mathrm{max\_coeff\_pos}(p_i). $$
We presented a decision procedure for termination of affine integer loops with triangular update matrices. In this way, we contribute to the ongoing challenge of proving the 15 years old conjecture by Tiwari [15] that termination of affine integer loops is decidable. After linear loops [4], loops with at most 4 variables [14], and loops with diagonalizable update matrices [3, 14], triangular loops are the fourth important special case where decidability could be proven.
The key idea of our decision procedure is to compute closed forms for the values of the program variables after a symbolic number of iterations n. While these closed forms are rather complex, it turns out that reasoning about first-order formulas over the theory of linear integer arithmetic suffices to analyze their behavior for large n. This allows us to reduce (non-)termination of triangular loops to integer programming. In future work, we plan to investigate generalizations of our approach to other classes of integer loops.
Note that multiplying with the least common multiple of all denominators yields an equivalent constraint with integer coefficients, i.e., allowing rational instead of integer coefficients does not extend the considered class of loops.
The proofs for real or rational numbers do not carry over to the integers since [15] uses Brouwer's Fixed Point Theorem which is not applicable if the variables range over \(\mathbb {Z}\) and [4] relies on the density of \(\mathbb {Q}\) in \(\mathbb {R}\).
Similarly, one could of course also use other termination-preserving pre-processings and try to transform a given program into a triangular loop.
The reason is that in this case, \((x - c_1) \ldots (x- c_k)\) is the minimal polynomial of A and diagonalizability is equivalent to the fact that the minimal polynomial is a product of distinct linear factors.
For instance, consider Open image in new window .
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1.Max Planck Institute for InformaticsSaarbrückenGermany
2.LuFG Informatik 2, RWTH Aachen UniversityAachenGermany
Frohn F., Giesl J. (2019) Termination of Triangular Integer Loops is Decidable. In: Dillig I., Tasiran S. (eds) Computer Aided Verification. CAV 2019. Lecture Notes in Computer Science, vol 11562. Springer, Cham
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\title{On existence of PI-exponent of algebras with involution} \author[D.D Repov\v s and M.V. Zaicev] {Du\v san D. Repov\v s and Mikhail V. Zaicev} \address{Du\v san D. Repov\v s \\Faculty of Education, and Faculty of Mathematics and Physics, University of Ljubljana \& Institute of Mathematics and Physics, Ljubljana, 1000, Slovenia} \email{[email protected]} \address{Mikhail V. Zaicev \\Department of Algebra\\ Faculty of Mathematics and Mechanics\\ Moscow State University \\ Moscow,119992, Russia\\ Moscow Center of Fundamental and Applied Mathematics, Moscow, 119991 Russia} \email{[email protected]} \keywords{Polynomial identity, nonassociative algebra, involution, exponentially bounded $*$-codimension, fractional $*$-PI-exponent, Amitsur's conjecture, numerical invariant} \subjclass[2020]{Primary 16R10; Secondary 16P90} \begin{abstract} We study polynomial identities of algebras with involution of nonassociative algebras over a field of characteristic zero. We prove that the growth of the sequence of $*$-codimensions of a finite-dimensional algebra is exponentially bounded. We construct a series of finite-dimensional algebras with fractional $*$-PI-exponent. We also construct a family of infinite-dimensional algebras $C_\alpha$ such that ${\rm exp}^*(C_\alpha)$ does not exist. \end{abstract} \date{} \maketitle
\section{Introduction}\label{s1} Let $A$ be an algebra over a field $\Phi$ of characteristic zero. One of the modern approaches to the study of polynomial identities of $A$ is to investigate their numerical invariants. The most important numerical characteristic of identities of $A$ is the sequence $\{c_n(A)\}$ of codimensions and its asymptotic behavior. For a wide class of algebras, the growth of the sequence $\{c_n(A)\}$ is exponentially bounded. This class includes associative PI-algebras \cite{R,L}, finite-dimensional algebras of arbitrary signature \cite{BD,GZ}, affine Kac-Moody algebras \cite{Z}, infinite-dimensional simple Lie algebras of Cartan type \cite{M}, Virasoro algebra, Novikov algebras \cite{Dz}, and many others.
In the case of exponential upper bound, the corresponding sequence of roots $\{\sqrt[n]{c_n(A)}\}$ is bounded and its lower and upper limits $$ \underline{{\rm exp}}(A)=\liminf_{n\to\infty}\sqrt[n]{c_n(A)},\quad \overline{{\rm exp}}(A)=\limsup_{n\to\infty}\sqrt[n]{c_n(A)} $$ are called the {\it lower} and the {\it upper} PI-{\it exponent} of $A$, respectively. In the case when $\underline{{\rm exp}}(A)= \overline{{\rm exp}}(A)$, the ordinary limit $$ {\rm exp}(A)=\lim_{n\to\infty}\sqrt[n]{c_n(A)} $$ is called the ({\it ordinary}) PI-{\it exponent} of $A$.
In the late 1980's, S. Amitsur conjectured that the PI-exponent of any associative PI-algebra exists and is a nonnegative integer. Amitsur's conjecture was confirmed in \cite{GZ1}. It was also proved for finite-dimensional Lie algebras \cite{Z1}, Jordan algebras \cite{GSZ}, and some others. The class of algebras for which Amitsur's conjecture was partially confirmed is much wider. Namely, the existence (but not the integrality, in general) was proved in a series of papers.
For example, it was shown in \cite{GZ2} that the PI-exponent exists for any finite-dimensional simple algebra. The question about existence of PI-expo\-nents is one of the main problems of numerical theory of polynomial identities. Until now, only two results about algebras without PI-exponent have been proved. An example of a two-step left-nilpotent algebra without PI-exponent was constructed in \cite{ERA}. Analogous result for unitary algebras was obtained in \cite{RZ}.
If an algebra $A$ is equipped with an additional structure (like an involution or a group grading), then one may consider identities with involution, graded identities, etc. Recall that in the associative case, the celebrated theorem of Amitsur \cite{A} states that if $A$ is an algebra with involution $*: A\to A,$ satisfying a $*$-polynomial identity, then $A$ satisfies an ordinary (non-involution) polynomial identity. As a consequence, the sequence of $*$-codimensions $\{c_n^*(A)\}$ is exponentially bounded. In \cite{GZ3, GPV} the existence and integrality of ${\rm exp}^*(A)$ was proved for any associative PI-algebra with involution.
In the present paper we shall show that the class of algebras with exponentially bounded $*$-codimension sequence is sufficiently large. In particular, it contains all finite-dimensional algebras.
{\bf Theorem A} (see Theorem~\ref{t1} in Section~\ref{s3}).
{\it Let $A$ be a finite-dimensional algebra with involution $*\colon A\to A$ and $d=\dim A$. Then $*$-codimensions of $A$ satisfy the following inequality $$ c_n^*(A)\le d^{n+1}. $$ }
Nevertheless, as it will be shown, the results of \cite{GZ3, GPV} cannot be generalized to the general nonassociative case. We shall construct a series of finite-dimensional algebras with fractional $*$-PI-exponent. For any integer $T\ge 2$ we shall construct an algebra $A_T$ with the following property.
{\bf Theorem B} (see Theorem~\ref{t2} in Section~\ref{s4}). {\it The $*$-PI-exponent of algebra $A_T$ exists and $$ exp^*(A_T) = \frac{1}{\theta_T^{\theta_T}(1-\theta_T)^{1-\theta_T}}, $$ where $\theta_T=\frac{1}{2T+1}$. }
We shall also present a family of algebras $C_\alpha$ with involution $*$ which has an exponentially bounded sequence $\{c_n^*(C_\alpha)\}$ such that ${\rm exp}^*(C_\alpha)$ does not exist.
{\bf Theorem C} (see Theorem~\ref{t3} in Section~\ref{s5}). {\it For any real number $\alpha>1$ there exists an algebra $C_\alpha$ such that $$ \underline{exp}^*(C_\alpha)=1,~~\overline{exp}^*(C_\alpha)=\alpha. $$ }
The necessary background on numerical theory of polynomial identities can be found in \cite{GZBook}.
\section{Preliminaries}\label{s2}
Let $A$ be an algebra with involution $*\colon A\to A$ over a field $\Phi$ of $\rm{char}~\Phi=0$. Recall that an element $a\in A$ is called {\it symmetric} if $a^*=a$, whereas an element $b\in A$ is called {\it skew-symmetric} if $b^*=-b$. Denote $$
A^+=\{a\in A|a^*=a\},~~ A^-=\{b\in A|b^*=-b\}. $$ Obviously, we have a vector space decomposition $A=A^+\oplus A^-$. In order to study $*$-polynomial identities
we need to introduce free objects in the following way.
Let $\Phi\{X,Y\}$ be a free (nonassociative) algebra over $\Phi$ with the set of free generators $X\cup Y$, $X=\{x_1,x_2,\ldots\}, Y=\{y_1,y_2,\ldots\}$. A map $*:X\cup Y\to X\cup Y$ such that $x^*_i=x_i,y^*_i=-y_i, i=1,2,\ldots$, can be naturaly extended to an involution on $\Phi\{X,Y\}$. A polynomial $f=f(x_1,\dots,x_m,y_1,\ldots,y_n)\in\Phi\{X,Y\}$ is said to be {\it a $*$-identity}
of $A$ if $$f(a_1,\ldots,a_m,b_1,\ldots,b_n)=0,
\
\hbox{for all}
\
a_1,\ldots a_m\in A^+, b_1,\ldots,b_n\in A^-.$$
Denote by $Id^*(A)$ the set of all $*$-identities of $A$ in $\Phi\{X,Y\}$. Then $Id^*(A)$ is an ideal of $\Phi\{X,Y\}$ and it is stable under involution $*$ and endomorphisms compatible with $*$.
Given $0\le k\le n$, denote the space of all multilinear polynomials in $\Phi\{X,Y\}$ in $k$ symmetric variables $x_1,\ldots,x_k$ and $n-k$ skew-symmetric variables $y_1,\ldots,y_{n-k}$ by $P^*_{k,n-k}$. Denote also $$P_n^*=P_{0,n}^*\oplus P_{1,n-1}^*\oplus\cdots\oplus P_{n,0}^*.$$ Clearly, the intersection $P_{k,n-k}^*\cap Id^*(A)$ is the subspace of all multilinear $*$-identities of $A$ in $k$ symmetric and $n-k$ skew-symmetric variables.
The following value $$ c^*_{k,n-k}(A)=\dim\frac{P^*_{k,n-k}}{P^*_{k,n-k}\cap Id^*(A)} $$ is called the {\it partial} $(k,n-k)~ $ $*$-{\it codimension} of $A$, whereas the value $$ c^*_n(A)=\sum_{k=0}^n {n\choose k} c^*_{k,n-k}(A) $$ is called the ({\it total})~~ $*$-{\it codimension} of $A$. We shall also use the following notations $$ P^*_{k,n-k}(A)=\frac{P^*_{k,n-k}}{P^*_{k,n-k}\cap Id^*(A)},~~ P^*_{n}(A)=\frac{P^*_{n}}{P^*_{n}\cap Id^*(A)}. $$
\section{$*$-codimensions of finite-dimensional algebras}\label{s3}
Let $A$ be a finite-dimensional algebra with involution $*\colon A\to A$, where $\dim A=d$. Recall that $A^+$ and $A^-$ are the subspaces of symmetric and skew-symmetric elements of $A$, respectively. In order to get an exponential upper bound for $c^*_n(A),$ we shall follow the approach of \cite{BD}. Choose a basis $a_1,\ldots,a_p$ of $A^+$ and a basis $b_1,\ldots,b_q$ of $A^-$. If $f(x_1,\ldots,x_k,y_1,\ldots,y_{n-k})\in P_{k,n-k}^*$ is a multilinear $*$-polynomial in $k$ symmetric variables $x_1,\ldots,x_k$ and $n-k$ skew-symmetric variables $y_1,\ldots,y_{n-k},$ then $f$ is a $*$-identity of $A$ if and only if $\varphi(f)=0,$ for all evaluations $\varphi$ such that \begin{equation}\label{ef1} \varphi(x_i)\in \{a_1,\ldots,a_p\},~1\le i\le k,~~ \varphi(y_j)\in \{b_1,\ldots,b_q\},~1\le j\le n-k. \end{equation}
Denote $N=\dim P_{k,n-k}^*$. Fix a basis $g_1,\ldots,g_N$ of $P_{k,n-k}^*$ and write $f$ as a linear combination $f=\alpha_1g_1+\cdots+\alpha_Ng_N$. Then the value $\varphi(f)$ for $\varphi$ of the type (\ref{ef1}) can be written as $$ \varphi(f)=\lambda_1a_1+\cdots+\lambda_pa_p+\mu_1b_1+\cdots+b_q\mu_q, $$ where all $\lambda_1,\ldots,\lambda_p,\mu_1,\ldots,\mu_q$ are linear combinations of $\alpha_1,\ldots,\alpha_N$. Hence $\varphi(f)=0$ if and only if \begin{equation}\label{ef2} \lambda_1=\cdots=\lambda_p=\mu_1=\cdots\mu_q=0. \end{equation} The total number of evaluations $\varphi$ of type (\ref{ef1}) is equal to $p^kq^{n-k}$. It follows that $f\equiv 0$ is a $*$-identity of $A$ if and only if the $N$-tuple $(\alpha_1,\ldots,\alpha_N)$ is the solution of system $S$ of $p^kq^{n-k}(p+q)$ linear equations of type (\ref{ef2}).
Denote by $U$ the subspace of all solutions of system $S$ in the space $V$ of all $N$-tuples $(\alpha_1,\ldots,\alpha_N)$. Then $\dim U=N-r$, where $r={\rm rank}~S$ is the rank of $S$. Clearly, \begin{equation}\label{ef3} r\le p^kq^{n-k}(p+q). \end{equation}
Since $$ c_{k,n-k}^*(A)=\rm{codim}_V(U)=r, $$ it follows from (\ref{ef3}) that $$c_{k,n-k}^*(A)\le p^kq^{n-k}(p+q)$$ and $$ c^*_n(A)=\sum_{k=0}^n {n\choose k} c_{k,n-k}^*(A) \le (p+q)\sum_{k=0}^n {n\choose k}p^kq^{n-k}=(p+q)^{n+1}. $$
Recall that $p+q=d=\dim A$. Hence we have proved the first main result of this paper.
\begin{theorem}\label{t1} Let $A$ be a finite-dimensional algebra with involution $*\colon A\to A$ and $d=\dim A$. Then $*$-codimensions of $A$ satisfy the following inequality $$ c_n^*(A)\le d^{n+1}. $$ \end{theorem}
$\Box$
In the case of exponentially bounded sequence $\{c_n^*(A)\}$, the following natural question arises. \begin{question} Does the $*$-PI-exponent $$ {\rm exp}^*(A)=\lim_{n\to\infty}\sqrt[n]{c_n^*(A)} $$ exist and what are its possible values? \end{question}
In Section~\ref{s1} we mentioned that $c^*_n(A)$ exists and is a nonnegative integer for any associative $*$-PI-algebra $A$. The following hypotheses look very natural. \vskip.1in
\begin{conjecture}\label{c1} For any finite-dimensional algebra $A$ with involution $*$, its $*$-PI-exponent ${\rm exp}^*(A)$ exists.
\end{conjecture}
In the light of results of \cite{GMZ}, we can assume that $*$-PI-exponent may take on all real values $\ge 1.$
\begin{conjecture}\label{c2} For any real value $\alpha\ge 1$, there exists an algebra $A_\alpha$ with involution such that $*$-PI-exponent of $A_\alpha$ exists and ${\rm exp}^*(A_\alpha)=\alpha$. \end{conjecture}
\section{Algebras with fractional $*$-PI-exponent}\label{s4}
In this section we shall discuss $*$-codimension growth of algebras $A_T$ introduced in \cite{SZ}. We shall prove the existence of $*$-PI-exponents of $A_T$ and compute the precise value of ${\rm exp}^*(A_T)$. In Section~\ref{s5} we shall use the properties of $A_T$ for constructing several counterexamples.
Recall the structure of $A_T$. Given an integer $T\ge 2$, denote by $A_T$ the algebra with basis $\{a,b,z_1,\ldots,z_{2T+1}\}$ and with multiplication $$ z_ia=az_i=z_{i+1}, 1\le i\le 2t,~z_{2T+1}b=bz_{2T+1}=z_1, $$ where all remaining products are zero. Involution $*:A_T\to A_T$ is defined by $$ a^*=-a,b^*=b,z_i^*=(-1)^{i+1}z_i $$ and then $$A^+=<b,z_1,z_3,\ldots,z_{2T+1}>, A^-=<a,z_2,z_4,\ldots,z_{2T}>.$$
We shall need the following two results from \cite{SZ}.
\begin{lemma}\label{l1} (\cite[Lemma 3.7]{SZ}) The $*$-codimensions of $A_T$ satisfy the inequality $c^*_n(A_T) \le n^3$, provided that $n\le 2T$. \end{lemma}
\begin{lemma}\label{l2} (\cite[Corollary 3.8]{SZ}) Let $f\equiv 0$ be a multilinear $*$-identity of $A_T$ of degree $n\le 2T$. Then $f$ is also an identity of $A_{T+1}$. \end{lemma}
Note that algebras $A_T$ are commutative and {\it metabelian}, i.e. they satisfy the following identity $$ (xy)(zt)\equiv 0. $$ Hence any product of elements $c_1,\ldots,c_n\in A$ can be written in the left-normed form. We shall omit brackets in the left-normed products, i.e. we shall write $c_1c_2\cdots c_n$ instead of $(\ldots(c_1c_2)\ldots)c_n$.
First, we shall find a lower bound for $*$-codimensions.
\begin{lemma}\label {l3} The following inequality holds for all $ n\ge 2T+2,$ \begin{equation}\label{efr0} c^*_n(A_T)\ge\frac{1}{n^2} \left(\frac{1}{\theta_T^{\theta_T}(1-\theta_T)^{1-\theta_T}}\right)^{n-2T-1}, \end{equation}
where $$ \theta_T=\frac{1}{2T+1}. $$ \end{lemma}
{\em Proof.} Write $n$ in the form $n=(2T+1)k+t+1$, where $0\le t \le 2T$. Then the following product of $n$ basis elements is nonzero $$ z_1\underbrace{a^{2T}b\cdots a^{2T}b}_ka^t=z_{t+1}\ne 0. $$ Here, we use the notation $xa^m$ for $x\underbrace{a\cdots a}_m$. Hence the polynomial $$ x_0y_1\cdots y_{2T}x_1\cdots y_{2t(k-1)+1}\cdots y_{2Tk}x_k y_{2Tk+1}\cdots y_{2Tk+t} $$ is not an identity of $A_T$, that is, $$P^*_{k+1,2Tk+t}(A_T)\ne 0, ~~ c^*_{k+1,2Tk+t}\ge 1.$$
In particular, \begin{equation}\label{efr1} c_n^*(A_T)\ge{n\choose k+1}\ge{n_0\choose k+1}\ge{n_0\choose k}, \end{equation} where $n=2Tk+k+t+1,n_0=2Tk+k$.
Using the Stirling formula for factorials we get \begin{equation}\label{efr1a} {(2T+1)k\choose k}>\frac{1}{n^2}\frac{((2T+1)k)^{(2T+1)k}}{k^k (2Tk)^{2Tk}} \end{equation} $$ =\frac{1}{n^2} \left( \frac{1}{\left(\frac{1}{2T+1}\right)^\frac{1}{2T+1}\left(\frac{2T}{2T+1}\right)^\frac{2T}{2T+1}} \right)^{(2T+1)k} =\frac{1}{n^2}\left( \frac{1}{\theta_T^{\theta_T}(1-\theta_T)^{1-\theta_T}}\right)^{n_0} $$ $$ \ge \frac{1}{n^2}\left( \frac{1}{\theta_T^{\theta_T}(1-\theta_T)^{1-\theta_T}}\right)^{n-2T-1}, $$ where $\theta_T=\frac{1}{2T+1}$.
Finally, combining (\ref{efr1}) and (\ref{efr1a}), we obtain the desired inequality (\ref{efr0}).
$\Box$
Next, we shall find an upper bound for $c_n^*(A_T)$. First, we restrict the number of nonzero components $P^*_{k,n-k}(A_T)$ for a fixed $n$.
\begin{lemma}\label{l4} Given a positive integer $n$, there are at most three integers $k$, $0\le k\le n,$ such that $P^*_{k,n-k}(A_T)\ne 0$. Moreover, if $P^*_{k,n-k}(A_T)\ne 0,$ then
$$\frac{k-2}{n}\le\frac{1}{2T+1}.$$ \end{lemma}
{\em Proof}. Clearly, all nonzero products of the basis elements of $A_T$ are of the form \begin{equation}\label{efr2} W=z_{2T+1-i}a^ib\underbrace{a^{2T}b\cdots a^{2T}b}_p a^j. \end{equation} The number of symmetric factors $k$ is equal to $p+1$ if $i$ is odd, and $k=p+2$ if $i$ is even. The total number of factors in $W$ is equal to $n=(2T+1)p+i+j+2$. Moreover, $i$ and $j$ in (\ref{efr2}) satisfy inequalities $0\le i,j\le 2T$. Hence \begin{equation}\label{efr3} n-4T-2\le (2T+1)p \le n-2. \end{equation}
Clearly, there are at most two integers $p$ satisfying (\ref{efr3}). Since $k=p+1$ or $p+2$, at most 3 components $P^*_{k,n-k}(A_T)$ can be nonzero. Finally, according to (\ref{efr3}), we have $$ \frac{k-2}{n}\le \frac{p}{n}\le\frac{n-2}{(2T+1)n}\le\frac{1}{2T+1}. $$
$\Box$
\begin{lemma}\label{l6} Let $n\le 2T+2$. Then $c^*_{k,n-k}(A_T)\le (2T+1)^3$. \end{lemma} {\em Proof}. As it was mentioned earlier, all nonzero products of the basis elements of $A_T$ are of the form $$ z_ja^pb\underbrace{a^{2T}b\cdots a^{2T}b}_ka^q,~~1\le j\le 2T+1,~~ 0\le p,q\le 2T. $$ Hence all nonzero modulo $Id^*(A_T)$ multilinear monomials are of the form \begin{equation}\label{efr5} wy_{\sigma(1)}\cdots y_{\sigma(p)}x_{\tau(1)}y_{\sigma(p+1)}\cdots y_{\sigma(p+2T)}x_{\tau(2)} \cdots \end{equation} $$ y_{\sigma(2Tk-2T+p+1)}\cdots y_{\sigma(2Tk+p)} x_{\tau(k+1)} y_{\sigma(2Tk+p+1)}\cdots y_{\sigma(2Tk+p+q)},, $$ where $\sigma\in S_{2Tk+p+q},\tau\in S_{k+1}$, and $w$ is either $x_0$ or $y_0$.
Moreover, any monomial (\ref{efr5}) coincides (modulo $Id^*(A_T)$) with the special case (\ref{efr5}) when $\sigma=1,\tau=1$. Hence, we have at most $(2T+1)^3$ linearly independent elements in $P^*_{k,n-k}(A_T)$, and so we are done.
$\Box$
\begin{lemma}\label{l7} For all $n\ge 2T+2$, we have $$ c^*_n(A_T) \le 3(2T+1)^3n^3\left(\frac{1}{\theta_T^{\theta_T}(1-\theta_T)^{1-\theta_T}}\right)^n. $$ \end{lemma} {\em Proof.} First we compute an upper bound for $c^*_{k,n-k}(A_T)$, provided that $P^*_{k,n-k}(A_T)\ne 0$. Note that $$ {n\choose k}\le n^2{n\choose k-2} \le n^3\frac{n^n}{m^m(n-m)^{n-m}}, $$ by the Stirling formula, where $m=k-2$.
Since the function $$\frac{1}{x^x(1-x)^{1-x}}$$ is nondecreasing on $(0,\frac{1}{2})$, we have by Lemma \ref{l4}, \begin{equation}\label{efr7} {n\choose k}\le n^3 \left(\frac{1}{(m/n)^{m/n}(1-m/n)^{1-m/n}} \right)^n \le n^3\left(\frac{1}{\theta_T^{\theta_T}(1-\theta_T)^{1-\theta_T}}\right)^n. \end{equation}
Now relation (\ref{efr7}), Lemma \ref{l4}, and Lemma \ref{l6} imply $$ c^*_n(A_T)=\sum_{k=0}^n {n\choose k}c^*_{k,n-k}(A_T)\le 3(2T+1)^3n^3 \left(\frac{1}{\theta_T^{\theta_T}(1-\theta_T)^{1-\theta_T}}\right)^n. $$
$\Box$
Finally, Lemma \ref{l3} and Lemma \ref{l7} imply the second main result of this paper.
\begin{theorem}\label{t2} The $*$-PI-exponent of algebra $A_T$ exists and $$ {\rm exp}^*(A_T) = \frac{1}{\theta_T^{\theta_T}(1-\theta_T)^{1-\theta_T}}, $$ where $\theta_T=\frac{1}{2T+1}$. \end{theorem}
$\Box$
\section{Algebras without $*$-PI-exponent}\label{s5}
We modify construction of the algebra from Section~\ref{s4}. Denote by $\widetilde A_T$ an infinite-dimensional algebra with the basis $$ a,b_i, z^i_j,~~ 1\le j\le 2T+1,~~ i=1,2,\ldots $$ and multiplication table $$ az^i_j=z^i_ja=z^i_{j+1},1\le j\le 2T, ~~
b_iz^i_{2T+1}=z^i_{2T+1}b_i=z^{i+1}_1. $$
Involution $*:\widetilde A_T\to\widetilde A_T$ is defined as follows $$ a^*=-a, ~~ b_i^*=b_i, ~~ (z^i_j)^*=(-1)^{j+1}z^i_j,~~ 1\le j\le 2T+1,~~ i=1,2,\ldots~~. $$
\begin{lemma}\label{l8} A multilinear polynomial $f\in P^*_{k,n-k}$ of degree $n\le 2T$ is a $*$-identity of $\widetilde A_T$ if and only if $f$ is a $*$-identity of $A_T$. \end{lemma} {\em Proof.} First, note that $P^*_{k,n-k}(A_T)=P^*_{k,n-k}(\widetilde A_T)=0$, when
$n\le 2T$ and $3\le k\le n$.
Let $k=0$. Then both $A_T$ and $\widetilde A_T$ satisfy the following identity $$ y_{t+1}y_{\sigma(1)}\cdots y_{\sigma(t)}=y_{t+1}y_1\cdots y_t, $$ for any $\sigma\in S_t$ and $t\le 2T-1$. Hence, modulo $Id^*(A_T)$ (and modulo $Id^*(\widetilde A_T)$), the polynomial $f$ coincides with linear combination $$f=\lambda_2w_2+\cdots+\lambda_nw_n, \ \hbox{ where} \ w_j=y_jy_1\cdots y_{j-1}y_{j+1}\cdots y_n.$$
Let for example, $\lambda_n\ne 0$. Then $\varphi(f)\ne 0$ in $A_T$ and $\widetilde\varphi(f)\ne 0$ in $\widetilde A_T$ for evaluations $\varphi,\widetilde\varphi$, where $$ \varphi(y_n)=z_1,\varphi(y_j)=a~ {\rm in}~A_T, 2\le j\le n-1,~\widetilde\varphi(y_n)=z^1_1, \widetilde\varphi(y_j)=a ~{\rm in}~\widetilde A_T, 2\le j\le n-1. $$
Now let $k=1$. Then all monomials $y_1\cdots y_jx_1y_{j+1}\cdots y_t$ are identities of $A_T$ and $\widetilde A_T$ if $3\le j\le t\le n-1$. Since $$ x_1y_{\sigma(1)}\cdots y_{\sigma(n-1)}\equiv x_1y_1\cdots y_{n-1}, ~~ {\rm for~all}~~ \sigma\in S_{n-1} $$ ${\rm mod}~Id^*(A_T)$ and ${\rm mod}~Id^*(\widetilde A_T)$, it follows that $f=\lambda x_1y_1\cdots y_{n-1}$, with $0\ne\lambda\in\Phi$. Hence $f\not\in Id^*(A_T)$ and $f\not\in Id^*(\widetilde A_T)$.
Finally, let $k=2$. Then modulo $Id^*(A_T)$ and modulo $Id^*(\widetilde A_T)$, any multilinear $*$-polynomial is a linear combination of monomials $$ w_p=x_1y_1\cdots y_px_2y_{p+1}\cdots y_{n-2}\quad{\rm and}\quad v_q=x_2y_1\cdots y_qx_1y_{q+1}\cdots y_{n-2},~ $$ where $0\le p,q,~p+q=n-2$.
Suppose that $$ f=\sum_p\lambda_pw_p+\sum_q\mu_qv_q $$ and that at least one of the coefficients $\lambda_p$ is nonzero. We may also assume that $\mu_0=0$ if $\lambda_0\ne 0$. If all $\lambda_p=0$ for $p$ even and all $\mu_q=0$ for $q$ even, then $f\in Id^*(A_T)\cap Id^*(\widetilde A_T)$.
Denote $$
t=\max \{p|p~{\rm even~~and}~\lambda_p\ne 0\}. $$
Then there exists odd $j$ such that $j+t=2T+1$. Hence $$ \varphi(f)=\lambda_tz_ja^tba^m=\lambda_tz_{t+1}\ne 0\quad {\rm in}~~A_T $$ for the evaluation $\varphi$ such that $\varphi(x_1)=z_j,\varphi(x_2)=b, \varphi(y_1)=\cdots=\varphi(y_{n-2})=a$.
Similarly, $$ \widetilde\varphi(f)=\lambda_tz_{m+1}^2 \quad {\rm in}~~\widetilde A_T $$ if $$\widetilde\varphi(x_1)=z^1_j, \widetilde\varphi(x_2)=b_1, \widetilde\varphi(y_1)=\cdots=\widetilde\varphi(y_{n-2})=a.$$
It follows that $$Id^*(A_T)\cap P^*_n=Id^*(\widetilde A_T)\cap P^*_n,$$ provided that $n\le 2T$.
$\Box$
\begin{remark}\label{r}
It follows from Lemma \ref{l1}, Lemma \ref{l2}, and Lemma \ref{l6}, that $*$-codimensions of small degree of $\widetilde A_T$ are polynomially bounded, $$c^*_n(\widetilde A_T)\le n^3 \ \hbox{ if} \ \ n\le 2T.$$ Also, any multilinear $*$-identitiy of $\widetilde A_T$ of degree $n\le 2T$ is an identity of all $\widetilde A_{T+1},\widetilde A_{T+2},\ldots~~$. \end{remark}
Unlike $A_T$, algebra $\widetilde A_T$ has an overexponential $*$-codimension growth.
\begin{lemma}\label{l9} Let $n\ge 4T+3$. Then \begin{equation}\label{ea1} c^*_n(\widetilde A_T)>\left[\frac{n}{2T+1}-1\right]!, \end{equation} where $[t]$ denotes the integer part of real number $t>0$. \end{lemma}
{\em Proof}. Denote $$ w_\sigma=x_0y_1\cdots y_{2T}x_{\sigma(1)}y_{2T+1}\cdots y_{4T}x_{\sigma(2)}\cdots x_{\sigma(m)}y_{2mT+1}\cdots y_{2mT+j}, $$ where $\sigma\in S_m, 0\le j\le 2T$. Since $$ z^1_1a^{2T}b_1a^{2T}\cdots a^{2T}b_ma^j=z^{m+1}_{j+1}\ne 0, $$ while $$ z^1_1a^{2T}b_{\sigma(1)}a^{2T}\cdots a^{2T}b_{\sigma(m)}a^j= 0, $$ for any $e\ne\sigma\in S_m$, all monomials $w_\sigma$ of degree $n=(2T+1)m+j+1$ are linearly independent modulo $Id^*(\widetilde A_T)$.
Hence \begin{equation}\label{ea2} c^*_n(\widetilde A_T)\ge c^*_{m+1,n-m-1}(\widetilde A_T)\ge m!~~. \end{equation}
Since $$(2T+1)m=n-j-1\ge n-(2T+1),$$ we have $$m\ge\frac{n}{2T+1}-1$$ and (\ref{ea2}) yields
inequality (\ref{ea1}).
$\Box$ \vskip .2in
Now, let $\Phi[Z]$ be the polynomial ring over $\Phi$ and let $\Phi[Z]_0$ be its subring of polynomials with the zero constant term. Given an integer $N\ge 1$, denote by $R_N$ the quotient $$ R_N=\frac{\Phi[Z]_0}{(Z)^{N+1}}, $$ where $(Z)^{N+1}$ is the ideal of $\Phi[Z]_0$ generated by $Z^{N+1}$.
Denote $B(T,N)=\widetilde A_T\otimes R_N$. Then \begin{equation}\label{ea2a} P^*_{k,n-k}(B(T,N))=P^*_{k,n-k}(\widetilde A_T), \ \hbox{for all} \ 0\le k\le n\le N, \end{equation} whereas \begin{equation}\label{ea3} P^*_{k,n-k}(B(T,N))=0, \ \hbox{for all} \ n\ge N+1. \end{equation}
Given two infinite series of integers $T_1,T_2,\ldots$ and $N_1,N_2,\ldots$ such that $$ 0<T_1<N_1<\ldots<T_j<N_j<\ldots, $$ we define an algebra $C(T_1,T_2,\ldots,N_1,N_2,\ldots)$ as the direct sum $$ C(T_1,T_2,\ldots,N_1,N_2,\ldots)=B(T_1,N_1)\oplus B(T_2,N_2)\oplus\cdots~~. $$
The next statement easily follows from Lemma \ref{l2}, Lemma \ref{l8}, and relations (\ref{ea2a}), (\ref{ea3}).
\begin{lemma}\label{l10} Let $C=C(T_1,\cdots,N_1,\cdots)$. Then \begin{itemize} \item[$\bullet$] $c^*_n(C)=c^*_n(\widetilde A_{T_1}),$ for all $n\le N_1$; \item[$\bullet$] $c^*_n(C)=c^*_n(\widetilde A_{T_j}),$ for all $j\ge 2, N_{j-1}+1\le n\le T_j$ ; \item[$\bullet$] $c^*_n(\widetilde A_{T_j})\le c^*_n(C)\le c^*_n(\widetilde A_{T_{j}})+c^*(\widetilde A_{T_{j+1}}),$ for all $j\ge 2, T_j<n\le N_j$. \end{itemize} \end{lemma}
$\Box$
\begin{lemma}\label{l11} Let $C=C(T_1,\ldots,N_1,\ldots)$. Then $c^*_n(C)\le 3nc^*_{n-1}(C)$. \end{lemma} {\em Proof}. Fix $n\ge 3$ and $1\le k\le n-1$. Denote by $f_1,\ldots,f_m$ a basis of $P_{k,n-k-1}^*$ modulo $Id^*(C)$, where $f_j,1\le j\le m$, are monomials in $x_1,\ldots,x_k,y_1,\ldots,y_{n-k-1}$ and $m=c^*_{k,n-k-1}$. Denote also by $g_1,\ldots,g_t$ a basis consisting of monomials in $x_1,\ldots,x_{k-1}$, $y_1,\ldots,y_{n-k}$ of $P^*_{k-1,n-k}$ modulo $Id^*(C)$, $t=c^*_{k-1,n-k}(C)$.
Then modulo $Id^*(C)$, the subspace $P^*_{k,n-k}$ coincides with the span of products $$ f^i_1y_i,\ldots,f^i_my_i,g^j_1x_j,\ldots,g^j_tx_j,~1\le i\le n-k, 1\le j\le k, $$ where $$ \begin{array}{c} f^i_p=f_p(x_1,\ldots,x_k,y_1,\ldots,y_{i-1},y_{i+1},\ldots,y_{n-k}), \\ \\ g^j_q=g_q(x_1,\ldots,x_{j-1},x_{j+1},\ldots,x_k,y_1,\ldots,y_{n-k}).
\end{array} $$
Hence \begin{equation}\label{ea4} c^*_{k,n-k}(C)\le n(c^*_{k-1,n-k}(C)+c^*_{k,n-k-1}(C)). \end{equation}
It follows from (\ref{ea4}) and the next inequalities $$ {n\choose k}\le n{n-1\choose k},~~{n\choose k}\le n{n-1\choose k-1} $$
that \begin{equation}\label{ea5} {n\choose k}c^*_{k,n-k}(C)\le n\left[{n-1\choose k-1}c^*_{k-1,n-k}(C)+ {n-1\choose k}c^*_{k,n-k-1}(C)\right]. \end{equation}
Inequality (\ref{ea5}) implies that $$ \sum_{k=1}^{n-1}{n\choose k}c^{*}_{k,n-k}(C)\le 2\sum_{j=0}^{n-1}{n\choose j-1} c^*_{j,n-j-1}(C)=2nc^*_{n-1}(C). $$
Finally, since $c^*_{0,n}=1$ and $c^*_{n,0}=1$ for $n\ge 3$, we have $$ c^*_n(C)\le 3n c^*_{n-1}(C). $$
$\Box$
We are now ready to construct a family of examples of algebras with involution without $*$-PI-exponent. The following is the third main result of this paper.
\begin{theorem}\label{t3} For any real number $\alpha>1$, there exists an algebra $C_\alpha$ such that $$ \underline{{\rm exp}}^*(C_\alpha)=1,~~\overline{{\rm exp}}^*(C_\alpha)=\alpha. $$ \end{theorem} {\em Proof}. Given $\alpha>1$, we construct an algebra $C_\alpha$ as $C(T_1,\ldots, N_1,\ldots)$ by the special choice of the sequences $T_1,T_2,\ldots$ and $N_1,N_2,\ldots$~.
First, we fix $T_1$ such that $n^3<\alpha^n$, for all $n\ge T_1$. By Lemmas \ref{l1}, \ref{l8} and \ref{l9}, there exists $N_1$ such that $$ \left\{
\begin{array}{l}
c^*_n(\widetilde A_T)<\alpha^n \ \hbox{if}\ n=N_1-1 \\
c^*_n(\widetilde A_T)\ge\alpha^n\ \hbox{if}\ n=N_1.
\end{array} \right. $$
Then by Lemma \ref{l10} and Lemma \ref{l11}, $$ \alpha^n\le c^*_n(C)\le 3n\alpha^n~~\hbox{if}~~n=N_1. $$
On the other hand, $c^*_{N_1+1}\le (N_1+1)^3$ by the choice of $N_1$. We now set $T_2=2N_1$.
Suppose that $T_1,N_1,\ldots,T_{k-1},N_{k-1}, T_k$ have already been choosen. Then as before, applying Lemmas \ref{l1}, \ref{l8}, \ref{l9} and \ref{l10}, one can find $N_k$ such that \begin{equation}\label{ea5a} \left\{
\begin{array}{l}
c^*_n(C)<\alpha^n\ \hbox{if}\ n=N_k-1 \\
c^*_n(C)\ge\alpha^n\ \hbox{if}\ n=N_k.
\end{array} \right. \end{equation}
Moreover, \begin{equation}\label{ea6} \left\{
\begin{array}{l}
c^*_n(C)\le 3n\alpha^n \\
c^*_{n+1}(C)\le (n+1)^3
\end{array} \right. \end{equation} if $n=N_k$.
Denote by $C_\alpha$ the obtained algebra $C(T_1,\ldots,N_1,\ldots)$. Since $c^*_n(C_\alpha)\ne 0$ for all $n\ge 1$, relations (\ref{ea5a}), (\ref{ea6}) give us the equations $$ \underline{{\rm exp}}^*(C_\alpha)=1,~~\overline{{\rm exp}}^*(C_\alpha)=\alpha $$ and we have thus completed the proof.
$\Box$
\end{document} | arXiv |
Pata type contractions involving rational expressions with an application to integral equations
Solutions to Chern-Simons-Schrödinger systems with external potential
Dimension reduction of thermistor models for large-area organic light-emitting diodes
Annegret Glitzky , Matthias Liero , and Grigor Nika
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany
* Corresponding author: Matthias Liero
An effective system of partial differential equations describing the heat and current flow through a thin organic light-emitting diode (OLED) mounted on a glass substrate is rigorously derived from a recently introduced fully three-dimensional $ p(x) $-Laplace thermistor model. The OLED consists of several thin layers that scale differently with respect to the multiscale parameter $ \varepsilon>0 $, which is the ratio between the total thickness and the lateral extent of the OLED. Starting point of the derivation is a rescaled formulation of the current-flow equation in the OLED for the driving potential and the heat equation in OLED and glass substrate with Joule heat term concentrated in the OLED. Assuming physically motivated scalings in the electrical flux functions, uniform a priori bounds are derived for the solutions of the three-dimensional system which facilitates the extraction of converging subsequences with limits that are identified as solutions of a dimension reduced system. In the latter, the effective current-flow equation is given by two semilinear equations in the two-dimensional cross-sections of the electrodes and algebraic equations for the continuity of the electrical fluxes through the organic layers. The effective heat equation is formulated only in the glass substrate with Joule heat term on the part of the boundary where the OLED is mounted.
Keywords: Thermistor system, dimension reduction, Joule heat, organic light-emitting diode, thin-film devices, multi-scale limit.
Mathematics Subject Classification: Primary:35J92, 35Q79, 35J57, 80A20, 35B30, 35B20.
Citation: Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020460
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Figure 1. Sketch of the domain $ \Omega_ \varepsilon $ consisting of the glass substrate $ \Omega^\mathrm{sub} $ and the OLED $ \Omega_ \varepsilon^\mathrm{oled} $. The latter consists of $ N $ layers (with $ N = 5 $ in the figure). The bottom and top layer $ \Omega_ \varepsilon^1 $ and $ \Omega_ \varepsilon^N $ describe the electrodes with Dirichlet boundaries $ \Gamma_ \varepsilon^- $ and $ \Gamma_ \varepsilon^+ $ (green) for the potential where the voltage is applied. In the effective limit, the current-flow equation reduces to coupled equations on the two-dimensional domain $ \Gamma_0 $ (red) and the heat equation is solved only in $ \Omega^\mathrm{sub} $ with an additional boundary source term on $ \Gamma_0 $
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Annegret Glitzky Matthias Liero Grigor Nika | CommonCrawl |
Integrating clinical and public health knowledge in support of joint medical practice
Volume 20 Supplement 2
The Physician and Professionalism Today: Challenges to and strategies for ethical professional medical practice
Jean-Pierre Unger1,
Ingrid Morales2,
Pierre De Paepe1 &
Michel Roland3
BMC Health Services Research volume 20, Article number: 1073 (2020) Cite this article
Strong relations between medicine and public health have long been advocated. Today, professional medical practice assumes joint clinical/public health objectives:
GPs are expected to practice community medicine;
Hospital specialists can be involved in disease control and health service organisation;
Doctors can teach, coach, evaluate, and coordinate care;
Clinicians should interpret protocols with reference to clinical epidemiology.
Public health physicians should tailor preventive medicine to individual health risks.
This paper is targeted at those practitioners and academics responsible for their teams' professionalism and the accessibility of care, where the authors argue in favour of the epistemological integration of clinical medicine and public health.
Based on empirical evidence the authors revisit the epistemological border of clinical and public health knowledge to support joint practice. From action-research and cognitive psychology, we derive clinical/public health knowledge categories that require different transmission and discovery techniques.
The knowledge needed to support the universal human right to access professional care bridges both clinical and public health concepts, and summons professional ethics to validate medical decisions. To provide a rational framework for teaching and research, we propose the following categories:
'Know-how/practice techniques', corresponding a.o. to behavioural, communication, and manual skills;
'Procedural knowledge' to choose and apply procedures that meet explicit quality criteria;
'Practical knowledge' to design new procedures and inform the design of established procedures in new contexts; and
Theoretical knowledge teaches the reasoning and theory of knowledge and the laws of existence and functioning of reality to validate clinical and public health procedures.
Even though medical interventions benefit from science, they are, in essence, professional: science cannot standardise eco-biopsychosocial decisions; doctor-patient negotiations; emotional intelligence; manual and behavioural skills; and resolution of ethical conflicts.
Because the quality of care utilises the professionals' skill-base but is also affected by their intangible motivations, health systems should individually tailor continuing medical education and treat collective knowledge management as a priority. Teamwork and coaching by those with more experience provide such opportunities. In the future, physicians and health professionals could jointly develop clinical/public health integrated knowledge. To this end, governments should make provision to finance non-clinical activities.
Closer ties between clinical medicine and public health have been advocated more recently [1, 2] as well as in the past. Writing for the practitioners and academics who feel themselves responsible for teamwork and professionalism in their services and also for accessibility of care in the community, we argue for the effective epistemological integration of these two disciplines.
This article explores the empirical rationale for, and ways to integrating clinical and public health knowledge, using an analysis of existing practitioner expertise meant to facilitate universal access to medical care and ethical practice.
The social relevance of medical systems is becoming a worldwide concern. The 2017 World Summit on Social Accountability, for example, strove to redefine the pathway of social accountability for the education of health professionals in the future (https://thenetworktufh.org/2017conference/). A charter developed as the main product of the (clinical) Medical Professionalism Project, identified a commitment to the primacy of the individual patient's welfare, social justice (and hence the fair distribution of health care resources) as core principles that should underpin physicians' professionalism [3].
From the charter's perspective, health care providers should not set limits to their concerns for the health care needs of individual patients. In addition, they should tackle social justice and apply social determinants of health and public health problems to ensure that, not only their patients, but also all individuals can access good quality health care [4]. They can incorporate this approach into their practice, by striving to deliver equal care to all patients, whatever their incomes [5] and also by performing advocacy [6].
Public health science, for its part, has been defined as accumulated knowledge about collective health protection. According to WHO, "Public Health is defined as "the art and science of preventing disease, prolonging life and promoting health through the organized efforts of society" (Acheson, 1988; WHO). Activities to strengthen public health capacities and service aim to provide conditions under which people can maintain to be healthy, improve their health and wellbeing, or prevent the deterioration of their health. Public health focuses on the entire spectrum of health and wellbeing, not only the eradication of particular diseases" [7].
Due to this focused concern on threats to health based on population risk analysis, public health scientists and international cooperation agents have, all too often, treated clinical medicine and allied professions merely as the means of controlling epidemics and population health risks, not as methods of delivering individual health care [8].
In borrowing concepts from general management sciences, public health science often fails to recognise the unique character of health systems and service delivery carried out by physicians and health professionals. In general public health science has neglected the delivery of individual healthcare, health provider-user interactions and issues with medical professionalism because of its quantitative, probabilistic methodology bias.
Explanations for epistemological stagnation underpinning the relations between public health and clinical sciences are to be found in history and in the immutable influence of social structures on health systems. As a rule, public health interventions were designed for the poor and medical care for the rich. At the end of the nineteenth century, public health officers in England were in charge of the prevention and early detection of, for example, cholera, tuberculosis, and scabies, but the government did not provide health care for the poor [9]. That was a responsibility delegated to a small number of charity hospitals.
This reality remains extant in huge areas of Africa, Latin America, and Asia. Since few doctors voluntarily choose to work in deprived neighbourhoods (except for those affiliated with certain denominational hospitals), health care for the poor is largely treated as collective public health care programme, i.e. treated solely as an aspect of public and/or private central planning.
This history still defines the epistemological borders of public health and clinical sciences. In the early 2000s, the WHO officially endorsed Health Systems Research (HSR) as a key component of public health without explicit reference to individual health care delivery [10]. Ten years later, HSR was still being portrayed as a key element of health policy and health systems without putting individual health care delivery at its core, merely mentioning it in a long list of other study topics [11]. Implementation research continued to focus on disease control programmes and biomedical interventions, unwittingly strengthening the rise of 'inequality by disease'.
This legacy would not hamper the advancement of health care if the division between collective and individual health sciences were found to be desirable. Since a population's well-being can only ever be as good as that of its individual members this is not the case and it is illogical to separate them.
Physicians can maximise their impact on collective health and still deliver highly individualised health care in line with eco-biopsychosocial and patient-centred care standards. To do so, they must strive to constantly improve their own professionalism and that of their environment. Alongside clinical responsibilities, doctors can reflect on their practice, build and lead teamwork, coach, educate, train, improve the organisation of their health services, coordinate and evaluate health care, contribute to disease and health risk control, undertake operational research, and lobby on health policy design. When physicians develop and connect these activities they may be called "manager physicians" even without being officially appointed, since managers are traditionally defined as persons entrusted with decision-making aimed at achieving their institutions' predetermined goals most efficiently.
Any doctor, the authors contend, should adopt the role of 'manager physician' in order to use her/his knowledge and experience fully, because non-clinical activities potentiate clinical ones. For ethical reasons discussed elsewhere, [12] physicians should volunteer to do this independently of or prior to the adoption of their institutions' policies, that is, without necessarily being instructed to do so.
We propose to call medical ethics that are based on the values of 'non-maleficence, beneficence, autonomy, and justice…(the reference tetrad par excellence that physicians and ethicists use to resolve ethical dilemmas' [13]) neo- Hippocratic medical ethics. The neo prefix is justified by the addition of a distributive dimension to traditional Hippocratic ethics.
Medicine, health care management and government policy ought to contribute to the human right to care – the right to access professionally-delivered individual and collective health care in universal health systems.
Universal health systems can be defined as offering the full array of health care services, from community health centers and self-employed GPs to university teaching hospitals made accessible to those in need over a territory. The antonym is often called 'segmented health systems'.
Therefore, clinicians concerns shouldn't merely be quality of care but also care accessibility; medical ethics; prevention and health promotion; and the management of population risks, diseases and health services. Actually, this is what all good clinicians do, bucking the trend for growing specialisation in the health sector. Similarly, public health physicians should focus on the accessibility and quality of individual care, alongside disease control.
This paper uses neo-Hippocratic' medical practice as an illustration of joint clinical/public health practice. It is one illustration: Chinese acupuncturists or business-minded physicians will certainly have other views on individual/collective health practices and the relevant knowledge required.
Based on the professional experience, totalling 150 years, of three public health physicians and that of a general practitioner each of whom have combined practical and academic background, we (the authors) justify the epistemological integration of clinical and public health medicine while discussing the necessity for social / professional ethics and the need for hybrid, clinical / public health decisions and action in medical practice. After exploring the hybrid nature of key medical responsibilities, we examine the related knowledge and determine how joint practice should be delivered, managed, and researched professionally, i.e., by people with particular skills and qualifications, sufficient autonomy, and abiding by professional ethics [14]. We revisit the epistemological border of clinical and public health knowledge to support joint practice. From action-research and cognitive psychology, we derive medical knowledge categories that require different transmission and discovery techniques, to support professionally- and socially-oriented medical practice and management. We analyse the difference between proposed neo-Hippocratic medicine and management, and the industrial, 'generic' management applied to healthcare. The essay is supported by a Pubmed bibliographic search with the terms 'physicians public health knowledge needs' (2000–2020).
How should medical (and health) professions evolve to encompass cross-responsibilities of clinicians and public health practitioners in order to improve the quality and accessibility of individual care and disease control?
Public health activities in clinical practice
Consider first-line clinicians, general practitioners (GPs), or paediatricians. They have practical and conceptual public health responsibilities, as follows:
In principle, physicians should combine (possibly programmed) primary, secondary, and tertiary prevention, curative care, and health promotion and tailor this mixture to the individual, family, and/or community.
The family physician should practice community medicine – for example, to manage drug addiction or violence at home.
To the extent that GPs must practice community medicine and deliver preventive care, they should accept responsibility for the whole community or geographic areas (territories) healthcare needs.
Doctors should know how to avoid 'patient's delay' (for example in women with a placenta praevia) and 'doctor's delay' (for example in detecting TB); reduce transmission (of HIV, for example); secure continuity of care for patients; and provide longitudinal care for each type of disease.
Good practice requires adjusting clinical protocols to local epidemiology because human resources, medical technology, or pharmaceuticals may not be available; and because disease frequency may vary significantly from neighbourhood to neighbourhood and therefore also the predictive value of signs, symptoms and test results.
Medical practice requires the ability to read and interpret scientific, medical and policy evidence critically [15] and to do so not only from the standpoint of clinical epidemiology, [16] but also in the light of sociological and political economy concepts.
Physicians should improve the way their health care services are organised (e.g. with regard to access to care, knowledge management, clinical coordination, reflection, doctors' intangible motivation and health information) and shape the organisation in a way that is conducive to quality care. For example, GPs can pre-arrange communication channels with specialists to share decisions based on the patient's health and family circumstances.
Although this list is not exhaustive, the clinical responsibilities of general practitioners not only demand their familiarity with public health concepts such as (clinical) epidemiology, disease control and health management but also the integration of clinical and public health knowledge. The correct care of the patient supposes not only curative but also preventive activities, and action on his/her environment. This similarly applies to the public health knowledge of hospital specialists because, they should also be able to deliver bio-psychosocial care, participate in the way that the health system is organised, and be involved in the organised control of diseases (e.g. nosocomial infections) and in the adjustment of patients' average length of stay (in dialogue with their patient's GP).
Individual care delivery in public health programmes
Public health practitioners, for their part, should include elements of individual health care delivery in their collective health practices and mobilise clinical knowledge out of concerns for community health, in order to improve disease control programmes:
Preventive care and health education should be tailored to individual biopsychosocial risks and demands for care, and public health interventions cannot simply be treated as a mass activity. Well-baby, antenatal, geriatric and HIV/AIDS clinics are examples of this. While risks and sometimes pathologies are multiple, the goal-oriented approach of medical practice encourages and assists individuals to achieve their maximal health potential in line with individually defined goals [17]. This approach should thus drive public health doctors to shift from chronic disease management to participatory patient management [18].
Effective prevention requires proper treatment of inter-current episodes of illness, with consequences for the accessibility of medical care by high-risk patients and subsequently for clinical coordination.
Disease control programmes are best carried out if epidemiologists can interpret the paradoxical results of (nosocomial, HIV, measles, etc.) surveillance and are able to give sound clinical advice and sustain a dialogue with clinicians (which is why physicians make good epidemiologists).
Disease control programmes may crowd out individual care, with, for example, a 'one size fits-all' approach and a bureaucratic load that strains the quality of care [8]. This produces a Catch 22 situation in which disease control is undermined when, as is frequently the case, the relevant (diabetes, malaria, tuberculosis, etc.) control programme relies on appropriate clinical activity in first-line services to deliver its interventions [19]. There is increasing evidence that accessible, good, quality community-based clinical care provides the necessary confidence for public health interventions to be acceptable to the population. Public health programmes need to be organised in such a way that protects, rather than undermines, access to and the quality of, individual health care and clinical practice.
These responsibilities justify the need for public health physicians to be well informed about topical clinical procedures, interpersonal communication, and philosophy, as well as mastering the relevant behavioural skills.
At the highest level of integration, medical officers (senior government officials who are put in charge of medical services in order to advise and lead teams of medical experts in charge of local health systems or hospitals) should integrate completely joint clinical/public health practices. Their responsibility, however, may be thwarted by a debilitating lack of resources (two or three physicians for a population of 200,000 in some rural and suburban Sub-Saharan Africa, for instance).
Consequences for health knowledge
Medical professionalism and specifically joint clinical/public health practice have implications for medical knowledge, health management, research, and medical education:
The design and implementation of public health interventions should build on locally available medical knowledge and mores, for example, to adapt disease control guidelines to local conditions.
Both public health doctors and clinicians should continuously and critically read professional literature respectively drawn from clinical medicine and public health. Accessing and using relevant knowledge from other fields of activity is an often neglected challenge.
Although eco-biopsychosocial care ideally entails inter-professional teams, some personal, internalised integration of public health and clinical knowledge is required because GPs and family doctors must combine curative and preventive care (see above); tailor this mix to each patient; and negotiate therapy, life -style advice, and the use of medical services with the patient (the person-centred care standard).
If GPs and specialists are expected to set quality of care criteria and therapeutic objectives together (a must when the patient is hospitalised), health systems also need to manage GPs and specialists' knowledge in such a way that lets the two groups communicate effectively.
Academia should be in a position to attract physicians capable of conceptualising and transmitting their experience. However, academic circles are increasingly off limits to clinical and public health practitioners because experience and professionalism, as opposed to bibliometrics, are not easily evaluated, and academic salaries may present opportunity costs to physicians.
Joint clinical / public health medical practice, a professional endeavour
According to Barondess, [20] 'Professions are complex social structures derived from the guild system of specialised competences intended to organise specialised and complex bodies of knowledge in such a way as to address both individual and societal needs. These are the basis of a social contract enfranchising the members of a profession. It makes professional knowledge central to the well-being of today's society.'
Integrated clinical/public health knowledge is professional in principle because joint clinical/public health practice is a professional endeavour. Indeed, medical practice has evaded standardisation in several domains, as evidenced by the following examples:
Clinical decisions [21] - because evidence-based medicine has been strained through misappropriation by vested interests, inflexible rules, technology-driven initiatives, and an unmanageable volume of information [22];
Eco-biopsychosocial care delivery [23, 24] and professional education – because both build upon emotional intelligence (the ability to understand your emotions and those of other people and to behave appropriately in different situations) [25] and communication skills;
The treatment of multi-pathologies and a goal-oriented clinical approach [26] – since this entails doctor patient negotiation;
Surgery, radiotherapy and gynaecology – because they require manual skills;
The physician's commitment to community health [27] – because public health programmes should be negotiated with at-risk patients in the same way that clinical options should be discussed between the doctor and the patient.
The ethical nature of clinical decisions – because the motivation that drives medical professionals is intangible [28, 29].
Clinical and public health doctors both need adequate professional autonomy because the quality of care depends on the physician's professionalism. Similarly, physicians in managerial positions need sufficient autonomy to promote professionalism and ethics in health care services.
Knowledge gaps in joint clinical and public health practice
Historically, clinical practice and science benefited from the public health science which has delivered a large body of concepts and quantitative inputs. To name just a few such inputs,
Clinical epidemiology and evidence-based medicine (EBM) provided treatment guidelines;
Epidemiology informed the design of prevention and health promotion interventions in curative care practice;
The surveillance of nosocomial infections shaped antibiotic therapy in hospitals;
Pharmaceutical evaluations built upon and enlarged (clinical) epidemiology methodology; and
Health management science addressed medical communication and coordination.
However, public health science could have served care much better if it had not produced exclusively normative knowledge, knowledge geared towards decision-making, principally (only?) when public health and clinical epidemiology norms and recommendations were grounded in quantitative, probabilistic research. Indeed, in general it has only contributed to doctors' and managers' decision-making when the underlying rationale was quantifiable, for example, based on morbidity and mortality statistics; on the predictive value of signs, symptoms, and test results; or on the effectiveness and cost of interventions.
An examination of joint clinical/public health practice reveals knowledge gaps in both fields. Tasche et al. ([30], cited by De Maeseneer, [31]) analysed 70 guidelines issued by the Dutch College of GPs and identified 875 relevant clinical questions with no answer in published work.
In many countries, most physicians lack insights into:
The management of professional organisations, for example, how to organise teamwork, to improve clinical coordination, and to lead action-research;
Reflective methods used to improve medical practice and service delivery; and
Techniques able to (de-)centralise medical technologies away from hospitals into first-line services and also, the reverse, in order to improve accessibility and efficiency of care.
Similarly, public health science has benefited from clinical knowledge when developing disease control interventions. However, the prevailing epistemological discourse offered clinicians only a minor role in public health programmes. Indeed, over the past half a century or so clinical disciplines allied to traditional public health programmes were usually mobilised according to a standard disease control pattern already outlined in 1965: [32].
• In theory, epidemiologists chose the priority diseases to be controlled. In practice, most of the current 132 international disease-specific programmes said to be 'Global Health Initiatives' are the result of commercial imperatives; epidemiological studies rarely entered the picture [33] Notice that access to care for the poor (roughly 70% of African and Indian populations) was largely limited to priority disease-control interventions, with fieldwork usually the responsibility of auxiliary health workers (and, to a lesser extent, first-line nurse practitioners). This increased the separation between public health activities and clinical medicine.
• Health economists set the programmes' structures. Historically, they preferred vertical to horizontal programmes for considerations of efficiency. In practice, these programmes were operationally integrated in health care services but remained administratively autonomous, leading to dysfunctional management and bureaucratic inflation in low- and middle-income countries (LMICs) [8], so further weakening health systems in developing countries [34].
• Physicians and biologists decided the operational interventions to be led by public health programmes.
• Operational and implementation research (increasingly involving anthropologists) was established, for example to determine how to deliver these interventions and to improve population compliance [35].
• Finally, programme evaluations were left to economists and epidemiologists.
The political imperative of this discourse was central to the development of the political economy of care in LMICs because it legitimised limiting public service' activities to mere disease control on the grounds of cost-efficiency. In so doing, this epistemological discourse often misused allocative efficiency, which was confused with technical efficiency [36].
As a science of disease control, public health generally gave health care management short shrift. It frequently overlooked the often most important single health status determinant, i.e. healthcare. Thus public health specialists frequently neglected;
The importance of the accessibility to care and its impact on population health (e.g., to improve early detection, care continuity, and yes, to recruit patients for public health interventions).
The professional expertise required for the application of clinical and disease control guidelines.
The multi-causality of disease with which clinical practitioners must grapple.
The need to set conditional priorities within and between public health programmes.
Because these neglected themes are crucial to clinical and public health medicine, we question the very fact of their separation as well as the distinction between scientific and professional knowledge.
Integrated clinical/public health knowledge
We propose a typology based on a knowledge merger designed to inform medical education, training and research, which is consistent with the universal right to healthcare. Basing a typology of medical knowledge on concepts of cognitive psychology permits professionals to specify how knowledge is to be transmitted, taught, and assessed. This issue of medical knowledge transmission is crucial, as the written word does not lend itself well to improving the status of the patient, and the knowledge and behaviour of practitioners [37]. We have chosen one that is inspired by that of Malglaive and Piaget to define four categories of interdependent knowledge, namely, the skills; the procedural, the practical and the theoretical knowledge [38]. We shall see at the end of this section that these 4 categories are found in the structure of action research (AR), a research methodology particularly suited to the development of professional knowledge.
Behavioural skills include communication, emotional intelligence, reflection, conflict resolution, self-organisation, ability to balance work and life, time management, stress management, resilience, and patience. Together with manual skills, they are of primary importance in medicine. Manual skills are, of course, necessary in clinical medicine, but also in the interface with machines, for example in radiotherapy or endoscopic surgery. Skills are especially important in clinical medicine because they concern the clinician's interaction with an individual and his/her family, However it may be argued that they are important also in public health, as disease control programmes should be negotiated with communities and authorities.
Skill transmission requires demonstrations, observations and technical supervision [39]. Teaching programmes should systematize skill acquisition. The importance of distinguishing skills from other health care knowledge appears to be understood in very few medical schools save some that have developed 'problem-based learning' programs: these are more the exception than the rule (Maastricht; Barts, Queen Mary; East Anglia; and Glasgow).
Procedural knowledge
Procedural knowledge is often defined as knowledge exercised in the performance of a task. We use it in the more restrictive sense of the knowledge required to apply clinical / public health guidelines (or standard operating procedures - SOP). The application of SOP guidelines is especially complex in medicine, because the principles of Evidence Based Medicine (EBM) require that the values underlying a SOP be compared with those of the patient so as to define the therapeutic process to be undertaken – an especially important challenge in multi-pathology, because achieving objectives in treating one disease may be detrimental to therapeutic effectiveness in another.
For medicine to serve the universal right to health care, clinical medicine and public health procedures must be integrated as frequently as possible. For example, general practice, paediatrics and gynaecology include a significant component of preventive medicine whilst preventive medicine in well-baby clinics, prenatal, and geriatric consultations can be partly standardised in order to rule out or treat relevant pathologies (of the new born and the infant, the pregnant woman and the elderly, respectively).
Clinical and public health guidelines should exhibit two critical characteristics that they frequently lack:
Certain criteria for care quality can be defined at opposite ends of a notional scale, such as patient safety (with, for example, complaint medicalisation) and autonomy (vis-à-vis the disease and its medical solution), or the effectiveness and efficiency of treatment. If the doctor maximizes the first, (s)he reduces the second, and vice versa. To enable the physician's to assess these guidelines in individual circumstances, guideline designers should make explicit the balance between the contradictory qualities of care criteria that govern their conception.
Guidelines should offer alternative options to physicians to resolve clinical challenges (for example using different referral values) and so permit them to negotiate clinical treatment plans with their patients. Indeed, patients can only sensibly choose their treatment if alternative options and an explanation of their pros and cons are clearly offered. As for lifestyle clinical advice[s], it remains irrelevant if the patient does not have a choice. In practice, clinical guidelines rarely propose options, either because their commercially focused design only considers efficiency, or because their designers did not think of it.
The transmission and improvement of skills and procedural knowledge require educational supervision, intervision (mutual supervision), flow process auditing (to identify the hurdles a patient meets during his journey through the health system), action-research, and other reflective techniques. Medical faculties and health services should systematize the organisation of rotations and internships during medical training because this effects a lifelong improvement of problem-solving capacity of health professionals. In practice, faculties and health services rarely do so.
Practical knowledge
Malglaive [38] adds a 'practical knowledge' category that we apply to neo-Hippocratic medical practice and its management. Practical knowledge is needed to formulate advice[s] and norms for action in defined environments. It thus informs the design of new procedures and their use in new contexts.
To serve the universal right to healthcare, practical knowledge addresses a wide array of interconnected clinical and public health topics, with quality criteria addressing both domains. For example:
The professionalization of physicians and that of other health proficients (e.g. in family medicine practiced by nurses in sub-Saharan Africa). Quality criteria may include ethical behaviour, problem-solving capacity, material conditions of the medical practice, and self-reflection capacity;
The improvement of quality of care (for example through adding family therapy and social assistance to general practice);
The optimisation of the clinical management of syndromes (say, gonorrhoea); of diagnosis and treatment of a given disease (say, the first and second line treatment of tuberculosis); of disease control (e.g.., diabetes or malaria within a defined area);
Medical, disease prevention amongst high-risk groups (undocumented migrants in Belgium for instance);
Improvement in access to health care (for example, access to general medicine) or to drugs for both the general population and for patients with special needs (e,g. patients referred to hospital);
The optimisation of resource utilisation and procurement (drugs, medical equipment, finances, staff);
The improvement of the health environment for patients and high-risk groups;
The organisation of health services and systems (to improve, for example, care coordination between GPs and specialists).
The epistemic unit of practical knowledge can be defined as "strategy", a representation of ways and means to achieve a goal. "Strategies" can serve as an action plan, action hypothesis, advice, standard, or the basis for an assessment. Strategies link endpoint objectives to a complex sequence of analysis, decision, action, and evaluation. To deal with complex realities they simultaneously address a number of resources, various processes, and many outputs – all topics on which the scientific public health literature is limited. Strategies can be described and evaluated and hypotheses about the conditions of their success or failure can be formulated to define their domains of validity. Multidisciplinary models and concepts are the ones that best describe such conditions. For example, the effectiveness of a vaccination program depends on the socio-cultural characteristics of the population (who decides? The father or the mother?); its geographical distribution; the characteristics of health services; the economic resources of families and those of the program; the level of competence of health professionals; etc.
The joint clinical and public health nature of practical medical knowledge appears in the process of integrating disease control in health care services, and in achieving this whilst strengthening, rather than undermining these services. Disease control programs can reduce access to care in the setting in which they are integrated, imposing on them multiple lines of authority; setting ill-defined priorities and increasing opportunity costs. Inadequate budgets, financial overruns and unrealistic costing; tension between health care professionals over income disparity and problems with sustainability are all too frequent characteristics of such programmes.
The essence of practical medical knowledge is professional and not merely scientific, because:
The practical knowledge acquired by clinicians and public health physicians should reduce any uncertainty in decision-making, improve programme implementation, promote reflectivity (the quality to reflect i.e. redirect back to the source), and enhance the relevance of evaluation.
To promote the correct use of EBM, [40] end-user physicians should adapt clinical guidelines to local (epidemiological, cultural, economic, medical) conditions [41]. This is the method by which GPs and the EBM Practice Net adapted the Finnish Duodecim guidelines to Belgian conditions.
Unfortunately, in guiding physicians, Health Maintenance Organisations (HMOs) and government administrations have often over-relied on a biased, unnecessarily strict interpretation of 'scientific'/biomedical EBM. They continue to apply it despite overlooking medical regulation, [42] neglecting multi-morbidity challenges posed by ageing populations [43,44,45] (guidelines often map poorly with complex multi-morbidity), [22] and disregarding aspects of care that escape standardisation.
As a professional endeavour, the transmission of practical knowledge benefits from exchanges based on the experience of learners, and exchanges between them and their teachers. Scientific representations are often an inappropriate method to transmit this.
Theoretical knowledge
Theoretical knowledge relates to the laws of existence and the constitution and functioning of reality – i.e., why something is true. Theoretical knowledge teaches reasoning, techniques and theory of knowledge. Below, we examine some of the characteristics of the theories that support medical professionalism.
First, they often rely on interconnected clinical and public health concepts. One can only understand domestic violence, against women, for example, with concepts of sociology (this violence is not explained in the same way in endogamous and exogamous societies because the free sexuality of women threatens the family architecture of endogamous societies much more than that of the others).
Likewise, consider a basic task of a clinician: diagnosis. Clinical epidemiology and epidemiology ought to be taught together, because the positive predictive value of a sign, a symptom or a test result is a function of the disease prevalence expressed as:
$$ \mathrm{PPV}=\left(\mathrm{SS}\ \mathrm{X}\ \mathrm{d}\right)/\Big(\left(\mathrm{SS}\ \mathrm{d}+\left(1-\mathrm{SP}\right)\left(1-\mathrm{d}\right)\right). $$
Where PPV is the positive predictive value of a sign or a test; SS is its sensitivity; SP is its specificity; and d is the disease prevalence [46].
This formula shows that the predictive value of a sign, symptom or test depends on the local prevalence of the disease that it predicts. Therefore, the value of a symptom is not the same in the patient base of GPs compared to specialists. The physicians' clinical experience is therefore radically different. Nevertheless, in most universities, it is mainly specialists who teach semiology to the future GPs.
Second, professional research methodologies are interdisciplinary, not merely multidisciplinary. Interdisciplinarity is the interaction between disciplines. It leads to the mutual integration of concepts and methodologies while multidisciplinarity is the simultaneous use of sciences belonging to different fields. Interdisciplinarity opens the door to ad-hoc, original study methods and is often a sine-qua-non for the relevance of theoretical work [47] that (in this case) supports the practice of medicine. Unfortunately, universities today undermine interdisciplinarity with a competition linked to research finance; the publication race; reduced advancement prospects suffered by academics who maintain a professional practice; and scientific specialisation as the basis for career strategies.
Third, medical curricula often neglect two disciplines pivotal to understand and support the personal, professional development of physicians: psychology and philosophy. A knowledge of professional psychology is indispensable in understanding physicians' intangible motivations and professionals' wellbeing in general [48]. Values that govern (and should govern) medical and public health practice are derived from moral philosophy.
Finally, policy studies and health systems research should pay attention to the political sciences, [49] the political economics of health care, and history, because the construction of national health systems spans generations and history reveals the political, economic and social determinants of health structures, medical cultures and praxis. In practice, health policy studies rarely set outcomes in context as reputable journalist would.
This typology of medical knowledge closely overlaps the stages of action research:
The design of a strategy (for example, to control AIDS) requires practical knowledge. A sound strategy is a prerequisite to develop procedures, such as the conduct of a follow-up consultation.
Procedures are always implemented relying on the doctor's procedural knowledge and know-how.
The strategy design depends on the problem it intends to solve, on prior (practical and theoretical) knowledge and on a model (that represents the characteristics of the environment, the problem, the strategy and its expected effect).
Finally, the strategy evaluation assumes the assessment of its design and of the procedures implementation.
The divorce between medical and public health practices results from a history of individual care for the rich and public health interventions for the poor, whether in the nineteenth century in England or in the 20th-21st centuries in Africa. Historically, relations between social classes had negative impacts not only on the health sector's functions and structures but also on the delineation of scientific fields and of medical epistemology.
Joint clinical/public health practice is needed to improve both access to and quality of care. We set out to demonstrate that knowledge required to support good medical practice revolves around reconciling clinical and public health science. To do so, we conceived of a neo-Hippocratic medical and health management science, an organised field of knowledge with normative, social, and professional objectives and values. Collectively developed by physicians and health professionals, this integrated medical knowledge would support and reinforce joint clinical/public health practice, help underwrite the right to health care, ensure that the societal concerns of public health are taken into account, improve medical professionalism, and bolster the neo-Hippocratic rationale of medical practice.
Such professional (and scientific) effort assumes that it is necessary to revisit the boundary between clinical and public health medicine, because;
Medical know-how is predominantly clinical.
Procedural knowledge addresses clinical, public health and managerial challenges, and succeeds more completely when it addresses them together.
Practical knowledge refers to these same domains.
Theoretical knowledge addresses both clinical and public health medicine and introduces the benefits of an interdisciplinary approach.
Procedural and practical medical knowledge should refer to one identical set of medical values and criteria.
How would this science of medical professionalism, joint clinical/public health practice and policy management distinguish itself from the science of commercial medical practice and of the industrial, generic science of management?
The latter envisages public-facing health practitioners as technicians and employees enjoying little autonomy. By contrast, the former acknowledges that both the quality of care and physicians' motivation require sufficient professional autonomy within a clear framework, because care incorporates the professionals' labour and reflects their intangible motivation. The organisational consequences of this autonomy are immense and include such elements as symbolic incentives, evaluations, information systems, design of clinical guidelines, transmission of knowledge, etc.
Second, health care is increasingly quantified because markets tend to restrict payment to what can be measured. That generates huge bureaucratic data needs, and jeopardises the indispensable medical secret [50]. Health care markets have impacted medical and public health sciences because of academia's involvement in the health care industry. This is perhaps best reflected in the unwarranted utilisation of probabilistic methods that has led to the crisis in EBM exposed by T. Greenhalgh [22] and in public health.
Third, in contrast to what generic management generally permits, medical professionalism and joint clinical and public health practice treat quality of care as a largely unquantifiable parameter because it deals with the importance of personal skills, communication, ethics, reflectivity for the quality of care; and the complexity of biopsychosocial decisions. Neo-Hippocratic medicine and management is able to address professionals' personal development, philosophical thinking, reflectivity, coordination, and teamwork needed to underwrite the physicians' autonomy and to encourage their symbolic motivation. Admittedly, quality of care is not an entirely qualitative concept, as joint practices benefit from quantitative indicators, for example to monitor progress made in achieving care quality, patient accessibility, and disease control.
As a political consequence, health systems should
Individualise continuing medical education as far as possible;
Stimulate the development of the physicians' professional culture and self-awareness;
Give clinically experienced physicians the opportunity to acquire managerial and educational positions, in medical faculties as in schools of public health.
Promote technical and psychological coaching by other, more experienced doctors.
As an academic consequence, unless medical schools and schools of public health distinguish between the physician's professional and scientific knowledge and make plans to commission research into the former, they will not have the tools necessary to continue training competent physicians.
Unless clinicians, public health physicians, and health care managers acknowledge the social and professional dimensions of medical practice and the need for dual clinical/public health practice, they will not have the means to uphold the ethical dimension of medical practice and the accessibility of care in a world of gross disparities in the health care resources earmarked for different groups.
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
EBM:
GP:
HMO:
Health Maintenance Organization
HSR:
Health Systems Research
LMIC:
Low and Middle Income Countries
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We are indebted to Professor Em. Charlene Harrington (School of Nursing, University of California, San Francisco); Professor Em. Jan De Maeseneer (Public Health and Primary Care Centre, University of Ghent); and Professor Claudio Shuftan (Johns Hopkins University) for making key inputs to the present article; and Gabrielle Leyden and Paul Vine for their editorial assistance. No error can be attributed to them.
About this supplement
This article has been published as part of BMC Health Services Research Volume 20 Supplement 2, 2020: "The Physician and Professionalism Today: Challenges to and strategies for ethical professional medical practice". The full contents of the supplement are available online at https://bmchealthservres.biomedcentral.com/articles/supplements/volume-20-supplement-2.
Publication of this supplement is funded by the Institute of Tropical Medicine, Antwerp, Belgium.
Department of Public Health, Institute of Tropical Medicine, Nationalestraat 155, B-2000, Antwerp, Belgium
Jean-Pierre Unger & Pierre De Paepe
Office de la Naissance et de l'Enfance, French Community of Belgium, Chaussée de Charleroi 95, B-1060, Brussels, Belgium
Ingrid Morales
Département de Médecine Générale, Université Libre de Bruxelles, Route de Lennik, 808, BP 612/1, B-1070, Brussels, Belgium
Michel Roland
Jean-Pierre Unger
Pierre De Paepe
The author(s) read and approved the final manuscript.
Correspondence to Jean-Pierre Unger.
Not Applicable (this manuscript does not involve human participants, human data or human tissue).
Not Applicable (this manuscript does not contain any individual person's data).
Unger, JP., Morales, I., De Paepe, P. et al. Integrating clinical and public health knowledge in support of joint medical practice. BMC Health Serv Res 20 (Suppl 2), 1073 (2020). https://doi.org/10.1186/s12913-020-05886-z
DOI: https://doi.org/10.1186/s12913-020-05886-z
Medical and public health practice
Health epistemology | CommonCrawl |
Weyl law
In mathematics, especially spectral theory, Weyl's law describes the asymptotic behavior of eigenvalues of the Laplace–Beltrami operator. This description was discovered in 1911 (in the $d=2,3$ case) by Hermann Weyl for eigenvalues for the Laplace–Beltrami operator acting on functions that vanish at the boundary of a bounded domain $\Omega \subset \mathbb {R} ^{d}$. In particular, he proved that the number, $N(\lambda )$, of Dirichlet eigenvalues (counting their multiplicities) less than or equal to $\lambda $ satisfies
$\lim _{\lambda \rightarrow \infty }{\frac {N(\lambda )}{\lambda ^{d/2}}}=(2\pi )^{-d}\omega _{d}\mathrm {vol} (\Omega )$
where $\omega _{d}$ is a volume of the unit ball in $\mathbb {R} ^{d}$.[1] In 1912 he provided a new proof based on variational methods.[2][3]
Generalizations
The Weyl law has been extended to more general domains and operators. For the Schrödinger operator
$H=-h^{2}\Delta +V(x)$
it was extended to
$N(E,h)\sim (2\pi h)^{-d}\int _{\{|\xi |^{2}+V(x)<E\}}dxd\xi $
as $E$ tending to $+\infty $ or to a bottom of essential spectrum and/or $h\to +0$.
Here $N(E,h)$ is the number of eigenvalues of $H$ below $E$ unless there is essential spectrum below $E$ in which case $N(E,h)=+\infty $.
In the development of spectral asymptotics, the crucial role was played by variational methods and microlocal analysis.
Counter-examples
The extended Weyl law fails in certain situations. In particular, the extended Weyl law "claims" that there is no essential spectrum if and only if the right-hand expression is finite for all $E$.
If one considers domains with cusps (i.e. "shrinking exits to infinity") then the (extended) Weyl law claims that there is no essential spectrum if and only if the volume is finite. However for the Dirichlet Laplacian there is no essential spectrum even if the volume is infinite as long as cusps shrinks at infinity (so the finiteness of the volume is not necessary).
On the other hand, for the Neumann Laplacian there is an essential spectrum unless cusps shrinks at infinity faster than the negative exponent (so the finiteness of the volume is not sufficient).
Weyl conjecture
Weyl conjectured that
$N(\lambda )=(2\pi )^{-d}\lambda ^{d/2}\omega _{d}\mathrm {vol} (\Omega )\mp {\frac {1}{4}}(2\pi )^{1-d}\omega _{d-1}\lambda ^{(d-1)/2}\mathrm {area} (\partial \Omega )+o(\lambda ^{(d-1)/2})$
where the remainder term is negative for Dirichlet boundary conditions and positive for Neumann. The remainder estimate was improved upon by many mathematicians.
In 1922, Richard Courant proved a bound of $O(\lambda ^{(d-1)/2}\log \lambda )$. In 1952, Boris Levitan proved the tighter bound of $O(\lambda ^{(d-1)/2})$ for compact closed manifolds. Robert Seeley extended this to include certain Euclidean domains in 1978.[4] In 1975, Hans Duistermaat and Victor Guillemin proved the bound of $o(\lambda ^{(d-1)/2})$ when the set of periodic bicharacteristics has measure 0.[5] This was finally generalized by Victor Ivrii in 1980.[6] This generalization assumes that the set of periodic trajectories of a billiard in $\Omega $ has measure 0, which Ivrii conjectured is fulfilled for all bounded Euclidean domains with smooth boundaries. Since then, similar results have been obtained for wider classes of operators.
References
1. Weyl, Hermann (1911). "Über die asymptotische Verteilung der Eigenwerte". Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen: 110–117.
2. "Das asymptotische Verteilungsgesetz linearen partiellen Differentialgleichungen". Mathematische Annalen. 71: 441–479. 1912. doi:10.1007/BF01456804. S2CID 120278241.
3. For a proof in English, see Strauss, Walter A. (2008). Partial Differential Equations. John Wiley & Sons. See chapter 11.
4. Seeley, Robert (1978). "A sharp asymptotic estimate for the eigenvalues of the Laplacian in a domain of $\mathbf {R} ^{3}$". Advances in Mathematics. 102 (3): 244–264. doi:10.1016/0001-8708(78)90013-0.
5. The spectrum of positive elliptic operators and periodic bicharacteristics. Inventiones Mathematicae , 29(1):37–79 (1975).
6. Second term of the spectral asymptotic expansion for the Laplace–Beltrami operator on manifold with boundary. Functional Analysis and Its Applications 14(2):98–106 (1980).
Functional analysis (topics – glossary)
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| Wikipedia |
Shut the box
Shut the box (also called canoga,[1] batten down the hatches[1] or trick-track[2]) is a game of dice for one or more players, commonly played in a group of two to four for stakes. Traditionally, a counting box is used with tiles numbered 1 to 9 where each can be covered with a hinged or sliding mechanism, though the game can be played with only a pair of dice, pen, and paper. Variations exist where the box has 10 or 12 tiles. In 2018 the game had a renaissance in Liverpool, England, when it became the house game at Hobo Kiosk pub on the Baltic Triangle. It was popularized by DJ duo Coffee and Turntables and became the most played board game in Merseyside for 4 years in a row.
Shut the box
Other namesCanoga, batten down the hatches, trick-track
GenresDice-rolling
Solitaire
Players1 (Solitaire) or more
Setup time1 minute
Playing time2–3 minutes per player
ChanceHigh (Dice rolling)
SkillsRisk management
Arithmetic
Rules
At the start of the game all levers or tiles are "open" (cleared, up), showing the numerals 1 to 9.
During the game, each player plays in turn. A player begins their turn by throwing or rolling the die or dice into the box. If the player does not have 7, 8, or 9 still available, they may choose to either roll one die or the standard two. Otherwise, the player must roll both dice.
After throwing, the player adds up (or subtracts) the pips (dots) on the dice and then "shuts" (closes, covers) one of any combination of open numbers that sums to the total number of dots showing on the dice. For example, if the total number of dots is 8, the player may choose any of the following sets of numbers (as long as all of the numbers in the set are available to be covered):
• 8
• 7, 1
• 6, 2
• 5, 3
• 5, 2, 1
• 4, 3, 1
The player then rolls the dice again, aiming to shut more numbers. The player continues throwing the dice and shutting numbers until reaching a point at which, given the results produced by the dice, the player cannot shut any more numbers. At that point, the player scores the sum of the numbers that are still uncovered. For example, if the numbers 2, 3, and 5 are still open when the player throws a one, the player's score is 10 (2 + 3 + 5 = 10). Play then passes to the next player.
After every player has taken a turn, the player with the lowest score wins.
If a player succeeds in closing all of the numbers, that player is said to have "Shut the Box" – the player wins immediately and the game is over.
Traditional pub play
In English pubs, shut the box is traditionally played as a gambling game. Each player deposits an agreed amount of money into a pool at the beginning of the game, and the winner of the game collects the money in pool at the end of the game and in some cases the box as well.
Variants
Shut the box is a traditional game, and there are many local and traditional variations in the rules. In addition, due to the game's growing popularity, many variations of the game have developed in recent years.
Popular variants are:
• Golf – A player's score is the sum of the numbers remaining uncovered at the end of their turn. The player with the lowest score wins.
• Missionary – A player's score is how many of the tiles remain uncovered at the end of the player's turn. For example, a player scores 3 if, at the end of their turn, 3 tiles remain open. The player with the lowest score wins.
• Canoga – A gambling variant produced by the Pacific Game Company; the company also produced a 12-tile variant, Canoga XII. (Canoga can also be played using a regular game set using chips.)
1. Chips are divided evenly among all players.
2. Players decide on an ante to place in the kitty (a half-round pocket on the playing field).
3. Players roll to see who goes first; play then rotates clockwise.
4. Players play a traditional round, scoring as described in "Golf" above, resulting in a winner and loser(s).
5. Each loser pays their difference in score to the winner. For example, if the lowest (winning) score is 11, and a losing score is 15, the loser pays 4 to the winner. The winner is paid by each loser.
6. Bonus payout: if the winner "clears the board" (scores 0 or "shuts the box"), the payout is as above but doubled, and the winner takes the kitty.
7. If there are tied winners, total payout is either split between or among the winners or multiplied for each winner, depending on how the players agree to do this before starting the game.
The following are examples of known variations in play, setup, and scoring:
• 2 to go – Standard game, the numbers 1 to 9 start up. On the first roll, the number 2 must be one of the ones dropped. Any player who rolls a 4 on their first roll loses immediately.
• 3 to go – The same as "2 to go" but the number 3 must be dropped instead.
• 3 down extreme – Numbers 1–3 are pre-dropped, leaving numbers 4–9 up.
• Lucky number 7 – The only number up is 7, and the first person to roll a 7 wins.
• Unlucky number 7 – A standard game, when a 7 is rolled, the game stops.
• Against all odds – All odd numbers are up and evens down.
• Even Stevens – All even numbers are up and odds down.
• Full house – 12 numbers are up.
• The 300 – 2 boxes and 4 dice are used, with the second box representing numbers 13–24 (24 + 23 + 22 + ... + 2 + 1 = 300); in the absence of a second box, cards or dominoes can be used to represent tiles 13–24. A Double 12 Dominoes set can also be used with four dice for this variant and other domino sets can be used by themselves to, in the case of the Double 18 set, provide for the use of six dice by themselves without the counting box.
• Thai style (Jackpot) – Always roll two dice, but only cover one tile matching one of the dice or their sum. For example, if the dice show a 2 and a 3 you may cover one of 2, 3, or 5. The best strategy is to use the combined score for a high tile (7,8,9), if possible, otherwise choose the lowest tile. The success rate for this strategy is 7.9855%.[3]
• Digital – A player's score at the end of the turn is the number obtained by reading the up digits as a decimal number from left to right. For example, if 1, 2, and 5 are left up the score is 125. This is also known as "Say what you see", a reference to Roy Walker's catch phrase from the TV game show Catchphrase.
• 2012 – All 12 are up, but use a 20-sided die rather than the pair of 6 dice: 20-sided die playing 12 numbers.
It is also possible to play extended versions in which each game is a "round" of a longer game. Examples of such versions include:
• Tournament – Rounds are played with the Golf scoring method until a player reaches or exceeds a grand total of 100 points, at which time the player with the lowest point total is declared to be the winner. At the end of each round, each player's score for the round is added to the player's total score. When a player's score reaches 45, the player must drop out of the game. The last player remaining wins the game.
• Simplified variant for younger players – Needs at least a 2 player box. During the game, each player plays in turn. After rolling both dice, the player adds up the dots on the dice and then shuts the tile for either the total number of dots, or one or both of the numbers on the dice. For example, if the player rolled a 6 and a 2, they may close either the 8 tile, or both the 6 tile and the 2 tile, or just the 6 tile, or just the 2 tile (as long as the numbers are available to be covered). The player then rolls the dice again, aiming to shut more numbers. The player continues throwing the dice and shutting numbers. The first player to shut all the tiles wins.
Dominoes can also be used for the tiles – this also provides the option of using up to six dice if a Double 18 domino set is used. A deck of cards can also be used as tiles, and if so desired a complete conventional Western deck with the jokers (54 cards) can provide for the use of up to nine dice.
Played without dice
• Domino Non-Dice Variants – A non-dice variant of the game can be played with the dominoes from either Western or Chinese sets ranging from 1 and 1 to 6 and 6 pips being used and most effectively put into a small bag for drawing, and the double blank being included along with blank and 1, with the former being either a free turn of sorts as it adds to zero or ending the turn, and the latter effectively ending the turn if the 1 tile has already been used.
• Card Non-Dice Variants – Another variant using cards dealt from one or more decks using the A, 2, 3, 4, 5, and 6 (sometimes along with the 7, 8, 9, and 10), and two face cards agreed upon for the equivalent of dice rolls adding up to 11 and 12 pips.
History
Unconfirmed histories of the game suggest a variety of origins, including 12th century Normandy (northern France) as well as the mid 20th century Channel Islands (Jersey and Guernsey), which one source credits to a man known as 'Chalky' Towbridge.[4] A 1967 edition of Brewing Review describes the game as being native to the Channel Islands, and records it being played in Manchester pubs in the mid-1960s.[5]
Taylor in "Pub Games" from 1976 mentions a claim that the game dates back to at least Napoleonic times. He reports a revival in the United Kingdom in "the last fifteen years or so", that is from the 1960s. Canada Dry distributed them to many pubs as a publicity novelty "some years" prior to 1976.[6]
Shut the box is the basis of the American television quiz show High Rollers, which ran from 1974 to 1976 and 1978 to 1980 on NBC with Alex Trebek as the host. The show resurfaced from 1987 to 1988, this time hosted by Wink Martindale.
Versions of the game have also been played in Barotseland (Zambia, central Africa).
See also
• Pub games
• Partition (number theory)
References
1. Luck, Steve (2006). Classic Indoor Games: The Complete Guide. Aurum. ISBN 978-1-84513-164-7.
2. Parlett, David (1999). The Oxford History of Board Games. Oxford University Press. ISBN 978-0-19-212998-7.
3. "Jackpot". GitHub. Retrieved 2022-10-23.
4. Finn, Timothy (1975). Pub Games of England (New ed.). London: Queen Anne Press. ISBN 9780362002461.
5. "'Shut the Box' at Wilson's New House". Brewing Review. 1967.
6. Taylor, Arthur R. (1976). Pub Games. St. Albans: Mayflower. p. 188. ISBN 978-0-583-12650-2.
External links
• Shut the Box open source physics versions with options from Wikipedia
• Shut the Box – Online version. 9, 10, 11 and 12 tile versions.
• Shut the Slots – Online version. Variation based on spinning Slots for the tiles.
• Shut the Box – Online HTML5/Javascript version, rules and variants explained (accessible via application menu), MIT licensed.
• Shut the Box– Online PHP version, originally created as part of a Boy Scouts of America Programming merit badge project, includes two auto move algorithms with performance analysis.
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| Wikipedia |
\begin{document}
\title{Combinatorial aspects of extensions of Kronecker modules} \pagestyle{myheadings} \markboth{\sc Csaba Sz\'ant\'o}{\sc Combinatorial aspects of extensions of Kronecker modules}
{\bf Abstract.} Let $kK$ be the path algebra of the Kronecker quiver and consider the category $\text{\rm mod-} kK$ of finite dimensional right modules over $kK$ (called Kronecker modules). We prove that extensions of Kronecker modules are field independent up to Segre classes, so they can be described purely combinatorially. We use in the proof explicit descriptions of particular extensions and a variant of the well known Green formula for Ringel-Hall numbers, valid over arbitrary fields. We end the paper with some results on extensions of preinjective Kronecker modules, involving the dominance ordering from partition combinatorics and its various generalizations.
{\bf Key words.} Kronecker modules, Green formula, Segre classes, extension monoid product, dominance ordering.
{\bf 2000 Mathematics Subject Classification.} 16G20.
\section{\bf Introduction}
Let $K$ be the Kronecker quiver and $k$ a field. We will consider the path algebra $kK$ of $K$ over $k$ (called Kronecker algebra) and the category $\text{\rm mod-} kK$ of finite dimensional right modules over $kK$ (called Kronecker modules). The isomorphism class of the module $M$ will be denoted by $[M]$.
For $d\in\mathbb N^2$ let $M_d$ be the set of isomorphism classes of Kronecker modules of dimension $d$. Following Reineke in \cite{reineke} for subsets $\mathcal{A}\subset M_d$, $\mathcal{B}\subset M_e$ we define
$$\mathcal{A}*\mathcal{B}=\{[X]\in M_{d+e}|\text{there exists an exact sequence } 0\to N\to X\to M\to 0\text{ for some } [M]\in\mathcal{A},[N]\in\mathcal{B}\}.$$ So the product $\mathcal{A}*\mathcal{B}$ is the set of isoclasses of all extensions of modules $M$ with $[M]\in\mathcal{A}$ by modules $N$ with $[N]\in\mathcal{B}$. This is in fact Reineke's extension monoid product using isomorphism classes of modules instead of modules. It is important to know (see \cite{reineke}) that the product above is associative, i.e. for $\mathcal{A}\subset M_d$, $\mathcal{B}\subset M_e$, $\mathcal{C}\subset M_f$, we have $(\mathcal{A}*\mathcal{B})*\mathcal{C}=\mathcal{A}*(\mathcal{B}*\mathcal{C})$. We also have $\{[0]\}*\mathcal{A}=\mathcal{A}*\{[0]\}=\mathcal{A}$ and the product $*$ is distributive over the union of sets.
Recall that in case when $k$ is finite, the rational Ringel-Hall algebra $\mathcal{H}(\Lambda,\mathbb Q)$
associated to the algebra $kK$, is the free $\mathbb Q$-module having as basis the isomorphism classes of Kronecker modules together with a multiplication defined by $[N_1][N_2]=\sum_{[M]}F^M_{N_1N_2}[M]$, where the structure constants $F^M_{N_1N_2}=|\{M\supseteq U|\ U\cong N_2,\
M/U\cong N_1\}|$ are called Ringel-Hall numbers. Note that in this case the extension monoid product is $\{[N_1]\}*\{[N_2]\}=\{[M]|F^M_{N_1N_2}\neq 0\}$.
In \cite{szantoszoll} we have proved that extensions of preinjective (preprojective) Kronecker modules are field independent, so the extension monoid product of two preinjectives (preprojectives) can be described combinatorially. In this paper we generalize this result by showing that extensions of arbitrary Kronecker modules are field independent up to Segre classes, so all extension monoid products can be described combinatorially. In order to prove this result we formulate a variant of the well known Green formula for Ringel-Hall numbers, valid over an arbitrary (not only finite) field and we describe explicit formulas for some particular extension monoid products.
The last section surveys some combinatorial properties related with the embedding and extension of preinjective modules. We show that particular orderings known from partition combinatorics and their generalizations (such as dominance, weighted dominance, generalized majorization) play an important role in this context.
\section{\bf Green's formula rewritten}
Consider an acyclic quiver $Q$ and the path algebra $kQ$, where $k$ is an arbitrary field. Denote by $\text{\rm mod-} kQ$ the category of finite dimensional right modules over $kQ$. The aim of this section is to formulate a weaker version of Green's formula valid over an arbitrary field $k$ (not only over finite ones). The proof can be derived from \cite{rin2} or it follows directly from Lemma 2.9 and Lemma 2.11 in \cite{hub1} (using the fact that our algebra is hereditary). \begin{theorem}{\rm(Green)} \label{GreenTheorem} For $M,N,X,Y\in\text{\rm mod-} kQ$
$\exists E\in\text{\rm mod-} kQ$ such that the cross $$\xymatrix{ & Y\ar[d]^{\eta} & \\ N\ar[r]^{\nu} & E\ar[r]^{\mu}\ar[d]^{\xi} & M\\ & X & }$$ is exact (i.e. both the row and the column are short exact sequences) iff $\exists A,B,C,D\in\text{\rm mod-} kQ$ such that the frame $$\xymatrix{D\ar[r]^{\delta_Y}\ar[d]^{\delta_N} & Y\ar[r]^{\beta_Y} & B\ar[d]^{\beta_M}\\ N\ar[d]^{\gamma_N} & & M\ar[d]^{\alpha_M}\\ C\ar[r]^{\gamma_X}& X\ar[r]^{\alpha_X} & A}$$ is exact (i.e. the edges of the frame are short exact sequences).
Moreover, in this case we also have the following $3\times 3$ commutative diagram with rows and columns short exact sequences and with the top left square a pull-back and the bottom right square a push-out. $$\xymatrix{D\ar[r]^{\delta_Y}\ar[d]^{\delta_N} & Y\ar[r]^{\beta_Y}\ar[d]^{\eta} & B\ar[d]^{\beta_M}\\ N\ar[r]^{\nu}\ar[d]^{\gamma_N} & E\ar[r]^{\mu}\ar[d]^{\xi} & M\ar[d]^{\alpha_M}\\ C\ar[r]^{\gamma_X}& X\ar[r]^{\alpha_X} & A}$$ \end{theorem} There is an important corollary of Theorem \ref{GreenTheorem}: \begin{corollary} \label{Greencor} For $M,N,X,Y\in\text{\rm mod-} kQ$ with $\operatorname{Ext}^1(M,N)=0$ there is an exact sequence $0\to Y\to M\oplus N\to X\to 0$ iff $\exists A,B,C,D\in\text{\rm mod-} kQ$ such that the frame below is exact. $$\xymatrix{D\ar[r]\ar[d] & Y\ar[r] & B\ar[d]\\ N\ar[d] & & M\ar[d]\\ C\ar[r]& X\ar[r] & A}$$ \end{corollary}
\section{\bf Facts on Kronecker modules}
The indecomposables in $\text{\rm mod-} kK$ are divided into three families: the preprojectives, the regulars and the preinjectives (see \cite{assem},\cite{aus},\cite{rin1}).
The preprojective (respectively preinjective) indecomposable modules (up to isomorphism) will be denoted by $P_n$ (respectively $I_n$), where $n\in\mathbb N$. The dimension vector of $P_n$ is $(n+1,n)$ and that of $I_n$ is $(n,n+1)$.
A module is preprojective (preinjective) if it is the direct sum of preprojective (preinjective) indecomposables. For a partition $a=(a_1,\dots,a_n)$ we will use the notation $P_a:=P_{a_n}\oplus\dots\oplus P_{a_1}$ and $I_a:=I_{a_1}\oplus\dots\oplus I_{a_n}$.
The indecomposables which are neither preinjective nor preprojective are called regular. A module is regular if it is the direct sum of regular indecomposables. The category of regular modules is an abelian, exact subcategory which decomposes into a direct sum of serial categories with Auslander-Reiten quiver of the form $ZA_{\infty}/1$, called homogeneous tube. These tubes are indexed by the closed points $x$ in the scheme $\mathbb{P}^1_k=\operatorname{Proj} k[X,Y]$. We denote by $\mathbb{H}_k$ the set of these points. A regular indecomposable of regular length $t$ lying on the tube $\mathcal{T}_x$ will be denoted by $R_x(t)$. Note that its unique regular composition series is $R_x(1)\subset\dots\subset R_x(t-1)\subset R_x(t)$. Also note that for the regular simple $R_x(1)$ its endomorphism ring $\operatorname{End}(R_x(1))$ is the residue field at the point $x$. The degree of this field over $k$ is called the degree of the point $x$ and denoted by $\deg x$. It follows that $\underline\dim R_x(t)=(t\deg x,t\deg x)$. In the case when $k$ is algebraically closed, the closed points of the scheme above all have degree 1 and can be identified with the points of the classical projective line over $k$.
For a partition $\lambda=(\lambda_1,\dots,\lambda_n)$ we define $R_x(\lambda)=R_x(\lambda_1)\oplus\dots\oplus R_x(\lambda_n)$ and denote by $P$ (respectively $I$, $R$) a preprojective (respectively preinjective, regular) module. We also define the set
$$\mathcal{R}_n=\{[R_{x_1}(a_1)\oplus...\oplus R_{x_r}(a_r)]\,|\,r,a_1,\dots,a_r\in\mathbb N^*, x_1,...,x_r\in \mathbb{H}_k \text{ are pairwise different and }$$$$ a_1\deg {x_1}+...+a_r\deg {x_r}=n\}.$$ We will describe now (in our special context) the so-called decomposition symbol used by Hubery in \cite{hub2}. A decomposition symbol $\alpha=(\mu,\sigma)$ consists of a pair of partitions denoted by $\mu$ (specifying a module without homogeneous regular summands) and a multiset $\sigma=\{(\lambda^1, d_1),\dots,(\lambda^r, d_r)\}$, where $\lambda^i$ are partitions and $d_i\in\mathbb N^*$. The multiset $\sigma$ will be called a Segre symbol. Given a decomposition symbol $\alpha=(\mu,\sigma)$ (where $\mu=(c=(c_1,...,c_t),d=(d_1,...,d_s))$) and a field $k$, we define the decomposition class $S(\alpha,k)$ to be the set of isomorphism classes of modules of the form $M(\mu,k)\oplus R$, where $M(\mu,k)=P_{c_t}\oplus\dots\oplus P_{c_1}\oplus I_{d_1}\oplus\dots\oplus I_{d_s} $ is the $kK$-module (up to isomorphism uniquely) determined by $\mu$ and $R=R_{x_1}(\lambda^1)\oplus\dots\oplus R_{x_r}(\lambda^r)$ for some distinct points $x_1,\dots x_r\in\mathbb H_k$ such that $\deg x_i = d_i$. For a Segre symbol $\sigma$ let $S(\sigma,k):=S((\emptyset,\sigma),k)$. Trivially $S(\alpha,k)\cap S(\beta,k)=\emptyset$ for decomposition symbols $\alpha\neq\beta$. Also note that $\mathcal{R}_n=\cup S(\sigma,k)$, where the union is taken over all Segre symbols $\sigma=\{((a_1), d_1),\dots,((a_r), d_r)\}$ with $r\in\mathbb N^*$ and $a_1 d_1+...+a_r d_r=n$.
\begin{remark}\label{NrPointsHk}For $k$ finite with $q$ elements the number of points $x\in\mathbb H_k$ of degree 1 is $q+1$. The number of points $x\in\mathbb H_k$ of degree $l\geq 2$ is $N(q,l)=\frac1{d}\sum_{d|l}\mu(\frac{l}d)q^d$, where $\mu$ is the M\"obius function and $N(q,l)$ is the number of monic, irreducible polynomials of degree $l$ over a field with $q$ elements. We can conclude that for a decomposition symbol $\alpha$ the polynomial $n_{\alpha}(q)=|S(\alpha,k)|$ is strictly increasing in $q>1$ (see \cite{hub2}). \end{remark}
The following well-known lemma summarizes some facts on Kronecker modules: \begin{lemma}\label{KroneckerFactsLemma}
{\rm a)} Let $P$ be preprojective, $I$ preinjective and $R$ regular module. Then $\operatorname{Hom} ({R},{P})=\operatorname{Hom} ({I},{P})=\operatorname{Hom} ({I},{R})= \operatorname{Ext}^1({P},{R})=\operatorname{Ext}^1({P},{I})= \operatorname{Ext}^1({R},{I})=0.$
{\rm b)} If $x\neq x'$, then $\operatorname{Hom}(R_x(t),R_{x'}(t'))=\operatorname{Ext}^1(R_x(t),R_{x'}(t'))=0$ (i.e. the tubes are pairwise orthogonal).
{\rm c)} For $n\leq m$, we have $\dim _k\operatorname{Hom} (P_n,P_m)=m-n+1$ and $\operatorname{Ext}^1(P_n,P_m)=0$; otherwise $\operatorname{Hom} (P_n,P_m)=0$ and $\dim _k\operatorname{Ext}^1(P_n,P_m)=n-m-1$. In particular $\operatorname{End}(P_n)\cong k$ and $\operatorname{Ext}^1(P_n,P_n)=0$.
{\rm d)} For $n\geq m$, we have $\dim _k\operatorname{Hom} (I_n,I_m)=n-m+1$ and $\operatorname{Ext}^1(I_n,I_m)=0$; otherwise $\operatorname{Hom} (I_n,I_m)=0$ and $\dim _k\operatorname{Ext}^1(I_n,I_m)=m-n-1$. In particular $\operatorname{End}(I_n)\cong k$ and $\operatorname{Ext}^1(I_n,I_n)=0$.
{\rm e)} $\dim _k\operatorname{Hom} (P_n,I_m)=n+m$ and $\dim _k\operatorname{Ext}^1(I_m,P_n)=m+n+2$.
{\rm f)} $\dim _k\operatorname{Hom} (P_n,R_x(t))=\dim _k\operatorname{Hom} (R_x(t),I_n)=d_pt$ and $\dim _k\operatorname{Ext}^1(R_x(t),P_n)=\dim _k\operatorname{Ext}^1(I_n,R_x(t))=t\deg x$.
{\rm g)} $\dim _k\operatorname{Hom} (R_x(t_1),R_x(t_2))=\dim _k\operatorname{Ext}^1(R_x(t_1),R_x(t_2))=\min{(t_1,t_2)}\deg x$.
{\rm h)} For $P'$ a preprojective module every nonzero morphism $f:P_n\to P'$ is a monomorphism. If $R$ is regular then for every nonzero morphism $f:P_n\to R$, $f$ is either a monomorphism or $\operatorname{Im} f$ is regular. In particular if $R$ is regular simple and $\operatorname{Im} f$ is regular then $f$ is an epimorphism.
{\rm i)} For $I'$ a preinjective module every nonzero morphism $f:I'\to I_n$ is an epimorphism. If $R$ is regular then for every nonzero morphism $f:R\to I_n$, $f$ is either an epimorphism or $\operatorname{Im} f$ is regular. In particular if $R$ is regular simple and $\operatorname{Im} f$ is regular then $f$ is a monomorphism. \end{lemma}
The defect of $M\in \text{\rm mod-} kK$ with dimension vector $(a,b)$ is defined in the Kronecker case as $\partial M:=b-a$. Observe that if $M$ is a preprojective (preinjective, respectively regular) indecomposable, then $\partial M=-1$ ($\partial M=1$, respectively $\partial M=0$). Moreover, for a short exact sequence $0\to M_1\to M_2\to M_3\to 0$ in $\text{\rm mod-} kK$ we have $\partial M_2=\partial M_1+\partial M_3$.
An immediate consequence of the facts above is the following: \begin{corollary} \label{ProdPRI} For the partitions $c=(c_1,...,c_t)$ and $d=(d_1,...,d_s)$ we have that: $$\{[P_c\oplus R_{x_1}(\lambda^1)\oplus\dots\oplus R_{x_r}(\lambda^r)\oplus I_d]\}=$$ $$\{[P_{c_t}]\}*\dots *\{[P_{c_1}]\}*\{[R_{x_1}(\lambda^1)]\}*\dots *\{[R_{x_r}(\lambda^r)]\}*\{[I_{d_1}]\}*\dots *\{[I_{d_s}]\}.$$ \end{corollary}
As stated in the beginning, we focus our attention on extensions of Kronecker modules, or equivalently on the products of the form $\{[M]\}*\{[N]\}$. Using the corollary above we can see that this iteratively reduces to the knowledge of the following particular products: $$\{[I_i]\}*\{[I_j]\},\ \{[P_i]\}*\{[P_j]\},\ \{[I_n]\}*\{[R_x(\lambda)]\},\ \{[R_x(\lambda)]\}*\{[P_n]\},\ \{[I_{n}]\}*\{[P_m]\},\ \{[R_x(\lambda)]\}*\{[R_x(\mu)]\}.$$
\section{\bf Particular extension monoid products and field independence in the general case}
In this section we will work in the category $\text{\rm mod-} kK$ with $k$ an arbitrary field. We will analyze the field independence of extensions of arbitrary Kronecker modules. For this purpose we will describe the particular extension monoid products listed at the end of the previous section.
We start with the description of $\{[I_i]\}*\{[I_j]\}$ and $\{[P_i]\}*\{[P_j]\}.$ \begin{proposition} \label{ProdIiIj} We have:
$\{[I_i]\}*\{[I_j]\}=\left\{\begin{array}{cc}\{[I_i\oplus I_j]\} &\text{ for } i-j\geq-1 \\ \{[I_j\oplus I_i],[I_{j-1}\oplus I_{i+1}],...,[I_{j-[\frac{j-i}2]}\oplus I_{i+[\frac{j-i}2]}]\} &\text{ for } i-j<-1\end{array}\right..$
Dually we have:
$\{[P_i]\}*\{[P_j]\}=\left\{\begin{array}{cc}\{[P_i\oplus P_j]\} &\text{ for } i-j\leq-1 \\ \{[P_j\oplus P_i],[P_{j+1}\oplus P_{i-1}],...,[P_{j+[\frac{i-j}2]}\oplus P_{i-[\frac{i-j}2]}]\} &\text{ for } i-j>-1\end{array}\right..$ \end{proposition} \begin{proof} For $k$ a finite field the formulas follow directly from the corresponding formulas for the Ringel-Hall product (see \cite{szanto1} for details). In \cite{szantoszoll} it is proven that the possible middle terms in preinjective or preprojective short exact sequences do not depend on the base field, so we are done. \end{proof}
We describe now the product $\{[R_x(\lambda)]\}*\{[R_x(\mu)]\}$, where $\lambda$ and $\mu$ are partitions. This is a classical result and it was studied in the equivalent context of $p$-modules by T. Klein in \cite{klein}. So we have:
\begin{proposition}\label{RCommuteLambdaMu}$\{[R_x(\lambda)]\}*\{[R_x(\mu)]\}=\{[R_x(\mu)]\}*\{[R_x(\lambda)]\}=\{R_x(\nu)| c^{\nu}_{\lambda\mu}\neq 0\},$ where $c^{\nu}_{\lambda\mu}$ is the Littlewood-Richardson coefficient (which is field independent). \end{proposition}
Using our knowledge on Littlewood-Richardson coefficients we obtain in particular the following:
\begin{corollary}\label{RCommuteLambdaN}$\{[R_x(\lambda)]\}*\{[R_x(n)]\}=\{[R_x(n)]\}*\{[R_x(\lambda)]\}=\{R_x(\nu)| \nu-\lambda\text{ is a horizontal $n$-strip}\}.$ \end{corollary}
Using the field independence of the Littlewood-Richardson coefficients we also obtain: \begin{corollary} \label{ProdR}For two Segre symbols $\sigma,\tau$ we have that $S(\sigma,k)*S(\tau,k)=\bigcup S(\rho,k)$, where the union is taken over a finite number of specific Segre symbols $\rho$, combinatorially (field independently) determined by the symbols $\sigma,\tau$. \end{corollary} \begin{proof} Using Proposition \ref{RCommuteLambdaMu} the field independent combinatorial nature of the product is clear. What remains to prove is that the product is the union of full Segre classes. Suppose that the class $[R_{x_1}(\lambda^1)\oplus\dots\oplus R_{x_r}(\lambda^r)]$ occurs in the product for some distinct points $x_1,\dots x_r\in\mathbb H_k$ such that $\deg x_i = d_i$. We prove that in this case the whole Segre class $S(\{(\lambda^1, d_1),\dots,(\lambda^r, d_r)\},k)$ occurs in the product. For a component $R_{x_i}(\lambda^i)$ we have the following three possibilities:
a) it comes from the product $[R_{x_i}(\mu^i)][R_{x_i}(\nu^i)]$, where $(\mu^i,d_i)\in\sigma$, $(\nu^i,d_i)\in\tau$, so $c^{\lambda^i}_{\mu^i\nu^i}\neq 0$.
b) $(\lambda^i,d_i)\in\sigma$,
c) $(\lambda^i,d_i)\in\tau$.
Note that in any of the cases above, the component $R_{y_i}(\lambda^i)$ can be obtained in a similar way, where $y_i\in\mathbb H_k$ is arbitrary such that $\deg y_i = d_i$. \end{proof} \begin{remark} One can observe that the previous corollary is valid also in the case when one of the Segre classes are empty (due to the smallness of the field). See also Remark \ref{NrPointsHk}. \end{remark}
Next we consider the products $\{[I_n]\}*\{[R_x(\lambda)]\}$ and $\{[R_x(\lambda)]\}*\{[P_n]\}.$ We need the following lemma:
\begin{lemma} \label{preproj} Let $P_n,P_m$ be preprojective indecomposables with $n<m$. Then there is a short exact sequence $0\to P_n\to P_m\to X\to 0$ iff $X$ satisfies the following conditions:
{\rm i)} it is a regular module with $\underline\dim X=\underline\dim P_m-\underline\dim P_n,$
{\rm ii)} if $R_x(t)$ and $R_{x'}(t')$ are two indecomposable components of $X$ then $x\neq x'$. \end{lemma} \begin{proof} Suppose we have a short exact sequence $0\to P_n\to P_m\to X\to 0$. We will check the conditions i) and ii).
Condition i). Trivially, $\underline\dim X=\underline\dim P_m-\underline\dim P_n$ and $\partial X=0$. Note that $X$ cannot have preprojective components, since if $P''$ would be such an indecomposable component, then $P_m\twoheadrightarrow P''\ncong P_m$ which is impossible due to Lemma \ref{KroneckerFactsLemma} h). So $X$ is regular.
Condition ii). Suppose $X=X'\oplus R_x(t_1)\oplus...\oplus R_x(t_l).$ Then we have a monomorphism $\operatorname{Hom}(X,R_x(1))\to\operatorname{Hom}(P_m,R_x(1)),$ so $\dim_k\operatorname{Hom}(X,R_x(1))=\dim_k\operatorname{Hom}(X',R_x(1))+\sum_{i=1}^{l}\dim_k\operatorname{Hom}(R_x(t_i),R_x(1))\leq \dim_k\operatorname{Hom}(P_m,R_x(1))=\deg x$ and $\dim_k\operatorname{Hom}(R_x(t_i),R_x(1))=\deg x$. It follows that $l=1$.
Conversely suppose now that $X$ is a regular module satisfying conditions i) and ii). It is enough to show that $P_m$ projects on $X$, since for an epimorphism $f:P_m\to X$ we have that $\partial\operatorname{Ker} f=-1$, so $\operatorname{Ker} f\cong P_n$. Notice first that there are no monomorphisms $P_m\to X$ because $\underline{\dim}X=\underline{\dim}P_m-\underline{\dim}P_n<\underline{\dim}P_m.$ For a nonzero $f:P_m\to X$ we have the short exact sequence $0\to\operatorname{Ker} f\to P_m\to\operatorname{Im} f\to 0$. Since $\operatorname{Ker} f\subseteq P_m$ we have that $\operatorname{Ker} f$ is preprojective (so with negative defect) and is not 0 (because $f$ is not mono) and $\operatorname{Im} f\subseteq X$ implies that $\operatorname{Im} f$ may contain preprojectives and regulars as direct summands (and it is nonzero since $f$ is nonzero). The equality $\partial\operatorname{Ker} f+\partial\operatorname{Im} f=\partial P_m=-1$ gives us $\partial\operatorname{Im} f=0$, so $\operatorname{Im} f$ is regular.
For $X=R_x(t)$ we have that $\operatorname{Hom}(P_m,X)\neq 0$ (see Lemma \ref{KroneckerFactsLemma} f)). If there are no epimorphisms in $\operatorname{Hom}(P_m,R_x(t))$ then using the remarks above and the uniseriality of regulars we would have $\operatorname{Hom}(P_m,R_x(t))\cong\operatorname{Hom}(P_m,R_x(t-1))$ a contradiction. So we have an epimorphism $P_m\to X$.
Suppose now that $X=R_{x_1}(t_1)\oplus...\oplus R_{x_l}(t_l)$. From the discussion above we have the epimorphisms $f_i:P_m\to R_{x_i}(t_i)$. Let $f:P_m\to X$, $f(x)=\sum f_i(x)$ the diagonal map. We have that $\operatorname{Im} f$ is regular so due to uniseriality $\operatorname{Im} f=R_{x_1}(t'_1)\oplus...\oplus R_{x_l}(t'_l)$ with $R_{x_i}(t'_i)\subseteq R_{x_i}(t_i)$. Since $f_i=p_if$ are epimorphisms we have that $R_{x_i}(t'_i)= R_{x_i}(t_i)$ so $f$ is an epimorphism. \end{proof} \begin{proposition} \label{ProdInRlambda}We have:
$$\{[R_x(\lambda)]\}*\{[P_n]\}=\{[P_{n+t\deg x}\oplus R_x(\mu)]\,|\,\text{where $\lambda-\mu$ is a horizontal strip of length $t$, for some $t\in\mathbb N$}\}.$$ Dually we have:
$$\{[I_n]\}*\{[R_x(\lambda)]\}=\{[R_x(\mu)\oplus I_{n+t\deg x}]\,|\,\text{where $\lambda-\mu$ is a horizontal strip of length $t$, for some $t\in\mathbb N$}\}.$$ \end{proposition} \begin{proof}We prove the first formula. Suppose we have a short exact sequence $$0\to P_n\to X\overset{g}\to R_x(\lambda)\to 0.$$ Note that we can't have preinjective components in $X$ (since due to Lemma \ref{KroneckerFactsLemma} a) they would embed into $\operatorname{Ker} g\cong P_n$). Since
$\partial X=-1$, it follows using Lemma \ref{KroneckerFactsLemma} c) that $X$ is of the form $X=P_{n+t\deg x}\oplus R_x(\mu)$ where $\mu$ is a partition with $|\mu|\leq |\lambda|$ and $t=|\lambda|-|\mu|$.
If $\mu=(0)$ then by Lemma \ref{preproj} we have an exact sequence $0\to P_n\to P_{n+t\deg x}\to R_x(\lambda)\to 0$ iff $\lambda=(t)$ i.e. iff $\lambda-(0)$ is a horizontal $t$-strip.
If $\mu\neq (0)$ then we apply Corollary \ref{Greencor} with choices $X=R_x(\lambda)$, $Y=P_n$, $M=P_{n+t\deg x}$ and $N=R_x(\mu)$. It follows that we have an exact sequence $0\to P_n\to P_{n+t\deg x}\oplus R_x(\mu)\to R_x(\lambda)\to 0$ iff $\exists A,B,C,D\in\text{\rm mod-} kK$ such that the frame below is exact. $$\xymatrix{D\ar[r]\ar[d] & P_n\ar[r] & B\ar[d]\\ R_x(\mu)\ar[d] & & P_{n+t\deg x}\ar[d]\\ C\ar[r]& R_x(\lambda)\ar[r] & A}$$ By Lemma \ref{KroneckerFactsLemma} a) $B,D$ are preprojectives or 0. Note that $B,D$ can't be both preprojectives (due to the defect) and also if $B=0$ then $A=P_{n+t\deg x}$, a contradiction since $R_x(\lambda)$ would project on a preprojective. This means that we must have $D=0$, so $B=P_n$, $C=R_x(\mu)$ and using Lemma \ref{preproj} it follows that $A=R_x(t)$ (where
$t=|\lambda-\mu|$). So we have an exact sequence $0\to P_n\to P_{n+t\deg x}\oplus R_x(\mu)\to R_x(\lambda)\to 0$ iff the frame below is exact. $$\xymatrix{0\ar[r]\ar[d] & P_n\ar[r] & P_n\ar[d]\\ R_x(\mu)\ar[d] & & P_{n+t\deg x}\ar[d]\\ R_x(\mu)\ar[r]& R_x(\lambda)\ar[r] & R_x(t)}$$ Applying Corollary \ref{RCommuteLambdaN} it follows that $\lambda-\mu$ must be a horizontal $t$-strip. \end{proof} For $\lambda=(m)$ we have in particular:
\begin{corollary} $\{[R_x(m)]\}*\{[P_n]\}=\{[P_{n+i\deg x}\oplus R_x(m-i)]|\text{ where $i=\overline{0,m}$}\}$.
Dually
$\{[I_n]\}*\{[R_x(m)]\}=\{[R_x(m-i)\oplus I_{n+i\deg x}]|\text{ where $i=\overline{0,m}$ }\}$.
\end{corollary}
Applying the previous corollary inductively, we obtain the following: \begin{corollary}\label{ProdIR} a) We have $\mathcal{R}_n*\{[P_m]\}=(\{[P_m]\}*\mathcal{R}_n)\cup(\{[P_{m+1}]\}*\mathcal{R}_{n-1})\cup\dots\cup\{[P_{m+n}]\}$.
b) For a Segre symbol $\sigma=\{(\lambda^1, d_1),\dots,(\lambda^r, d_r)\}$ we have that $S(\sigma,k)*\{[P_m]\}=\bigcup \{[P_{m+t}]\}*S(\tau,k)$, where the union is taken over all Segre symbols of the form $\tau=\{(\mu^1, d_1),\dots,(\mu^r, d_r)\}$ with $\lambda^i-\mu^i$ a horizontal strip of lenght $t_i$ and $t=t_1d_1+\dots+t_rd_r$.
The preinjective version of the formulas above follows dually. \end{corollary}
Finally we consider the product $\{[I_n]\}*\{[P_m]\}$. \begin{proposition} \label{ProdIP} We have
$\{[I_n]\}*\{[P_m]\}=\mathcal{R}_{n+m+1}\cup\{[P_m\oplus I_n]\}.$ \end{proposition} \begin{proof} Suppose first that $X\ncong P_m\oplus I_n$. Then we prove that there is an exact sequence of the form $0\to P_m\to X\to I_n\to 0$ iff $X$ is a regular module having indecomposable components from pairwise different tubes and $\underline\dim X=\underline\dim P_m+\underline\dim I_n$.
Suppose we have a short exact sequence $0\to P_m\overset{f}\to X\overset{g}\to I_n\to 0$. Then $\underline\dim X=\underline\dim P_m+\underline\dim I_n$ and $\partial X=\partial P_m+\partial I_n=0$. Suppose $X=P'\oplus R\oplus I'$ (where $P'$, $R$ and $I'$ are preprojective, preinjective and regular modules). Note that $p_{P'}f:P_m\to P'$ must be nonzero so it is a monomorphism (see Lemma \ref{KroneckerFactsLemma} h)) which means that $\underline\dim P_m\leq\underline\dim P'$. In the same way $fq_{I'}:I'\to I_n$ must be nonzero so it is an epimorphism (see Lemma \ref{KroneckerFactsLemma} i)) which means that $\underline\dim I_n\leq\underline\dim I'$. But $\underline\dim P_m+\underline\dim I_n=\underline\dim P'+\underline\dim R+\underline\dim I'$ which implies $R=0$ and $p_{P'}f$, $fq_{I'}$ are isomorphisms, so $X\cong P_m\oplus I_n$ a contradiction. This means that $X$ is regular.
Suppose $X=X'\oplus R_x(t_1)\oplus...\oplus R_x(t_l)$, then we have the monomorphism $0\to\operatorname{Hom}(X,R_x(1))\to\operatorname{Hom}(P_m,R_x(1)),$ since $\operatorname{Hom}(I_n,R_x(1))=0$. It follows that $$\dim_k\operatorname{Hom}(X,R_x(1))=\dim_k\operatorname{Hom}(X',R_x(1))+\sum_{i=1}^{l}\dim_k\operatorname{Hom}(R_x(t_i),R_x(1))\leq\dim_k\operatorname{Hom}(P_m,R_x(1))=\deg x$$ and $\dim_k\operatorname{Hom}(R_x(t_i),R_x(1))=\deg x$, so $l=1$.
Conversely, suppose that $X$ is a regular module having indecomposable components from pairwise different tubes and $\underline\dim X=\underline\dim P_m+\underline\dim I_n$. Repeating the proof of Lemma 3.2. in \cite{szanto2} the existence of an exact sequence $0\to P_m\to X\to I_n\to 0$ follows.
\end{proof}
Using the previous results on particular extension monoid products (more precisely Corollaries \ref{ProdPRI}, \ref{ProdR}, \ref{ProdIR} and Proposition \ref{ProdIP}) it follows inductively that the extension monoid product of Kronecker modules is field independent in general up to Segre classes. More precisely we obtain the following theorem: \begin{theorem} For two decomposition symbols $\alpha,\beta$ we have that $S(\alpha,k)*S(\beta,k)=\bigcup S(\gamma,k)$, where the union (which is disjoint) is taken over a finite number of specific decomposition symbols $\gamma$ combinatorially (field independently) determined by the symbols $\alpha,\beta$. \end{theorem} \begin{remark} One can observe that the theorem above is valid also in the case when one of the decomposition classes is empty (due to the smallness of the field). See also Remark \ref{NrPointsHk}. Also \end{remark}
\section{\bf Combinatorial aspects of extensions of preinjective Kronecker modules}
There are some very interesting combinatorial properties related with the embedding and extension of preinjective modules. Particular orderings known from partition combinatorics and their generalizations (such as dominance, weighted dominance, generalized majorization) play an important role in this context. We recall first the definition of these orderings. \begin{definition} Let $a=(a_1,...a_n),b=(b_1,...,b_n)\in\mathbb Z^n$ .
\noindent The dominance partial ordering is defined as follows (see \cite{Macd}): $$a\leqslant b\text{ iff }a_1\leq b_1,\ a_1+a_2\leq b_1+b_2,\ \dots,\ a_1+...+a_{n-1}\leq b_1+...+b_{n-1}\text{ and }a_1+....+a_n\leq b_1+...+b_n.$$ In case $a_1+....+a_n=b_1+...+b_n$ (for example when $a,b$ are partitions of the same number) we will use the notation $a\preccurlyeq b$.
\noindent The weighted dominance partial ordering is defined as follows (see \cite{szantoszoll}): $$a\ll b\text{ iff } (a_1,2a_2,...,na_n)\leqslant (b_1,2b_2,...,nb_n).$$
\noindent Following Baraga\~na, Zaballa, Mondi\'e, Dodig, Sto\u si\'c one can define the so-called generalized majorization (see \cite{dodig1},\cite{dodig2}). This generalization of the dominance ordering of partitions plays an important role in the combinatorial background of matrix pencil completion problems. Consider the partitions $a=(a_1,...,a_n),b=(b_1,...,b_m),c=(c_1,...,c_{m+n})$. Then we say that the pair $(b,a)$ is a generalized majorization of $c$ (and denote it by $c\prec (b,a)$) iff $$b_i\geq c_{i+n},\ i=\overline{1,m},$$ $$ \sum_{i=1}^{m}b_{i}+\sum_{i=1}^{n}a_{i}=\sum_{i=1}^{m+n}c_{i},$$$$\sum_{i=1}^{h_{q}}c_{i}-\sum_{i=1}^{h_{q}-q}b_{i}\leq\sum_{i=1}^{q}a_{i},\ q=\overline{1,n},$$$$where\ h_{q}:=\min\{i|b_{i-q+1}<c_{i}\},\ q=\overline{1,n}.$$ Adopting the convention that $c_i,b_i=+\infty$ for $i\leq 0$, $c_i=-\infty$, for $i>m+n$ $b_i=-\infty$, for $i>m$ and $\sum_{i=a}^b=0$ in case $a>b$ one can see that the indices $h_q$ and the sums above are all well defined. Moreover, we have that $q\leq h_q\leq q+m$ and $h_1<h_2<\dots<h_n$. The term generalized majorization is motivated by the fact, that for $m=0$ the generalized majorization reduces to the dominance ordering $c\preccurlyeq a$. \end{definition}
Consider the partitions $a=(a_1,...,a_n)$, $b=(b_1,...,b_m)$ and $c=(c_1,...,c_{m+n})$. Using the definition of the generalized majorization we define inductively $x_1:=\sum_{i=1}^{h_{1}}c_{i}-\sum_{i=1}^{h_{1}-1}b_{i},\ x_q:=\sum_{i=1}^{h_{q}}c_{i}-\sum_{i=1}^{h_{q}-q}b_{i}-x_1-\dots-x_{q-1}$ for $q=\overline{1,n}$. If $x:=(x_1,\dots,x_n)\in\mathbb Z^n$ then observe that $x$ depends only on $b,c$ and not on $a$. The following proposition gives an another connection between the notion of generalized majorization and the dominance ordering. \begin{proposition} \label{GenMajChar} Suppose that for the partitions $a,b,c$ above we have that $b_i\geq c_{i+n}$ for $i=\overline{1,m}$ and $\sum_{i=1}^{m}b_{i}+\sum_{i=1}^{n}a_{i}=\sum_{i=1}^{m+n}c_{i}$. Then $c\prec (b,a)$ iff $x\preccurlyeq a$. \end{proposition} \begin{proof} If $c\prec (b,a)$ then it follows from the definition that $x\leqslant a$. Since $\sum_{i=1}^{m}b_{i}+\sum_{i=1}^{n}a_{i}=\sum_{i=1}^{m+n}c_{i}$, we obtain that $\sum_{i=1}^{n}x_{i}+\sum_{i=1}^{m+n-h_n}(c_{h_n+i}-b_{h_n-n+i})=\sum_{i=1}^{n}a_{i}$. But we have that $c_{h_n+i}-b_{h_n-n+i}\leq 0$ and $\sum_{i=1}^{n}x_{i}\leq \sum_{i=1}^{n}a_{i}$, so $c_{h_n+i}=b_{h_n-n+i}$ for $i=\overline{1,m+n-h_n}$ and $\sum_{i=1}^{n}x_{i}=\sum_{i=1}^{n}a_{i}$, which means that $x\preccurlyeq a$. The converse statement is trivial. \end{proof}
It follows from the proof above that for $n=1$ the condition $c\prec (b,a)$ is equivalent to $$b_i\geq c_{i+1},\ i=\overline{1,m},$$ $$ \sum_{i=1}^{m}b_{i}+a_1=\sum_{i=1}^{m+1}c_{i},$$$$b_i=c_{i+1},\ i\geq h_1,$$$$ where\ h_{1}:=\min\{i|b_{i}<c_{i}\}.$$ In this case we speak about an elementary generalized majorization and denote it by $c\prec_1 (b,a)$.
A result by Dodig and Sto\u si\'c in \cite{dodig1} shows that one can decompose the generalized majorization into a ``composition" of elementary generalized majorizations. More precisely we have: \begin{proposition}{\rm(\cite{dodig1})} \label{DodigElementaryStep} We have $c\prec (b,a)$ iff there is a sequence of partitions $d^j=(d^j_1,\dots,d^j_{m+j})$, $j=\overline{1,n}$ with $d^0=b$ and $d^n=c$ such that $d^j\prec_1(d^{j-1},a_j)$ for $j=\overline{1,n}$. \end{proposition}
We mention next two existing results on extensions of preinjectives which involve the combinatorial notions above. \begin{proposition}{\rm(Sz\'ant\'o \cite{szanto})}\label{ProdIincreasing} Suppose $a=(a_1,\dots,a_n)$ is a partition. Then
$$\{[I_{a_n}]\}*\dots*\{[I_{a_1}]\}=\{[I_{\alpha}]| \alpha\preccurlyeq a\}.$$ \end{proposition}
Remember that $\{[I_{a_1}]\}*\dots*\{[I_{a_n}]\}=\{[I_a]\}$ and denote the reversed product $\{[I_{a_n}]\}*\dots*\{[I_{a_1}]\}=\{[I_\alpha]| \alpha\preccurlyeq a\}$ by $\mathcal{I}_a$. \begin{proposition}{\rm(Sz\'ant\'o, Sz\"oll\H osi \cite{szantoszoll})} Consider the preinjective modules $I'=(a_nI_n)\oplus...\oplus(a_0I_0)$, $I=(b_nI_n)\oplus...\oplus(b_0I_0)$ (where $a_i,b_j\in\mathbb N$ and $a^2_n+b^2_n\neq 0$). Let $a=(a_1,\dots,a_n)$ and $b=(b_1,\dots b_n)$.
Then there is a monomorphism $I'\to I$ iff $a_0\leq b_0$ and $a\ll b$. \end{proposition}
The following result connects the notion of generalized majorization to extensions of preinjectives. \begin{proposition}Consider the partitions $a=(a_1,...,a_n),b=(b_1,...,b_m),c=(c_1,...,c_{m+n})$. Then $$c\prec (b,a)\text{ iff } [I_c]\in\mathcal{I}_a*\{[I_b]\}$$ \end{proposition}
\begin{proof} The equality $\mathcal{I}_a*\{[I_b]\}=\{[I_\alpha]| \alpha\preccurlyeq a\}*\{[I_b]\}$ is clear using Proposition \ref{ProdIincreasing}. The case $n=1$ easily follows using the definition of the elementary generalized majorization and Proposition \ref{ProdIiIj}. The general case is then a consequence of Proposition \ref{DodigElementaryStep}. \end{proof} \begin{remark} The result above is also proved in \cite{szoll} by Sz\"oll\H osi. We obtain in this way an independent proof also for Proposition \ref{DodigElementaryStep} by Dodig and Sto\u si\'c. \end{remark} Using all the results above we finally give a new characterization of the embedding of preinjective Kronecker modules.
Consider the partitions $b=(b_1,...,b_m)$, $c=(c_1,...,c_{m+n})$ and as before let $x=(x_1,\dots,x_n)\in\mathbb Z^n$ with $x_1=\sum_{i=1}^{h_{1}}c_{i}-\sum_{i=1}^{h_{1}-1}b_{i},\ x_q=\sum_{i=1}^{h_{q}}c_{i}-\sum_{i=1}^{h_{q}-q}b_{i}-x_1-\dots-x_{q-1}$ for $q=\overline{1,n}$. \begin{proposition}There is a monomorphism $I_b\to I_c$ iff $b_i\geq c_{i+n}$, for $i=\overline{1,h_n-n}$, $b_i=c_{i+n}$, for $i=\overline{h_n-n+1,m}$ and there is a partition $a$ such that $x\preccurlyeq a$. Moreover if $a$ is minimal (using the dominance ordering), then $I_a$ is a factor of the embedding $I_b\to I_c$, i.e. $0\to I_b\to I_c\to I_a\to 0$ is exact. \end{proposition} \begin{proof} If there is a monomorphism $I_b\to I_c$, then we have an exact sequence $0\to I_b\to I_c\to I_a\to 0$ with $a=(a_1,\dots,a_n)$ a partition. But then $[I_c]\in\{[I_{a}]\}*\{[I_b]\}\subseteq\mathcal{I}_a*\{[I_b]\}$, so $c\prec (b,a)$ and the assertion follows using Proposition \ref{GenMajChar}. Conversely, the given conditions guarantee by Proposition \ref{GenMajChar} that $c\prec (b,a)$, so $[I_c]\in\mathcal{I}_a*\{[I_b]\}$, which means that we have an exact sequence $0\to I_b\to I_c\to I_{a'}\to 0$, for an $a'\preccurlyeq a$.
In case $a$ is minimal, then as above $0\to I_b\to I_c\to I_{a'}\to 0$, for an $a'\preccurlyeq a$. Since $x\preccurlyeq a'$ and $a$ is minimal, it follows that $a'=a$. \end{proof}
{\it Acknowledgements.} This work was supported by the Bolyai Scholarship of the Hungarian Academy of Sciences and Grant PN-II-ID-PCE-2012-4-0100.
\emph{Csaba Sz\'ant\'o}
"Babe\c s-Bolyai" University Cluj-Napoca
Faculty of Mathematics and Computer Science
Str. Mihail Kogalniceanu nr. 1
R0-400084 Cluj-Napoca
Romania
e-mail: [email protected]
\end{document} | arXiv |
\begin{document}
\begin{frontmatter}
\title{Parallel Explicit Model Predictive Control\thanksref{footnoteinfo}}
\thanks[footnoteinfo]{This paper is extended version of the paper~\cite{Oravec2017} presented at IFAC World Congress 2017 in Toulouse. Corresponding author J.~Oravec. Tel. +421 259 325 364. Fax +421 259 325 340.}
\author[ShanghaiTech,SIMIT,CAS]{Yuning Jiang}\ead{[email protected]}, \author[STU]{Juraj Oravec}\ead{[email protected]}, \author[ShanghaiTech]{Boris Houska}\ead{[email protected]}, \author[STU]{Michal Kvasnica}\ead{[email protected]} \address[ShanghaiTech]{School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China.} \address[SIMIT]{Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200050, China} \address[STU]{Institute of Information Engineering, Automation, and Mathematics, Faculty of Chemical and Food Technology, Slovak University~of~Technology in Bratislava, Radlinsk\'{e}ho 9, SK812-37 Bratislava, Slovak Republic.} \address[CAS]{University of Chinese Academy of Sciences, Beijing 100049, China}
\begin{keyword} model predictive control, parametric optimization, fixed memory utilization. \end{keyword}
\begin{abstract} This paper is about a real-time model predictive control (MPC) algorithm for large-scale, structured linear \change{systems} with polytopic state and control constraints. The proposed controller receives the current state measurement as an input and computes a sub-optimal control reaction by evaluating a finite number of piecewise affine functions that correspond to the explicit solution maps of small-scale parametric quadratic programming (QP) problems. We provide \change{recursive feasibility} and \change{asymptotic stability} guarantees, which can both be verified offline. The feedback controller is suboptimal on purpose because we are enforcing real-time requirements assuming that it is impossible to solve the given large-scale QP in the given amount of time. In this context, a key contribution of this paper is that we provide a \change{bound on the sub-optimality of the controller}. Our numerical simulations illustrate that the proposed explicit real-time scheme easily scales up to systems with hundreds of states and long control horizons, system sizes that are completely out of the scope of existing, non-suboptimal Explicit MPC controllers. \end{abstract}
\end{frontmatter}
\section{Introduction} \label{sec:introduction} The advances of numerical optimization methods over the last decades~\cite{Diehl2009}, in particular, the development of efficient quadratic programming problem (QP) solvers~\cite{Ferreau2014}, have enabled numerous industrial applications of \change{MPC~\cite{Qin2003}.} Modern real-time optimization and control software packages~\cite{Houska2011,Mattingley2009} achieve run-times in the milli- and microsecond range by generating efficient and reliable C-code that implements problem-tailored MPC algorithms~\cite{Diehl2002,Zavala2009}. However, as much as these algorithms and codes perform well on desktop computers or other devices with comparable computation power, the number of successful implementations of MPC on embedded industrial hardware such as programmable logic controllers (PLC) and field-programmable gate arrays (FPGA) remains limited~\cite{Ingole2015}. Here, the main question is what can be done if an embedded device has simply not enough computational power or storage space to solve the exact MPC problem online and in real-time.
Many researchers have attempted to address this question. For example, the development of Explicit MPC~\cite{oberdieck2016multi} aims at reducing both the online run-time and the memory footprint of MPC by optimizing pre-computed solution maps of multi-parametric optimization \change{problems}. However, Explicit MPC has the disadvantage that the number of polytopic regions over which the piecewise affine solution map of a parametric linear or quadratic programming problem is defined, grows, in the worst case, exponentially with the number of constraints. Some authors~\cite{GM02} have suggested addressing this issue by simplifying the MPC problem formulation, e.g., by using move-blocking~\cite{Cagienard2007}, but the associated control reactions can be sub-optimal by a large margin. Other authors~\cite{Kva:regionless:2015} have worked on reducing the memory footprint of Explicit MPC---certainly making considerable progress yet failing to meet the requirement of many practical systems with more than just a few states. In fact, despite all these developments in Explicit MPC, these methods are often applicable to problems of modest size only. As soon as one attempts to scale up to larger systems, an empirical observation is that Explicit MPC is often outperformed by iterative online solvers such as active set~\cite{Ferreau2014} or interior point methods for MPC~\cite{Mattingley2009}. \change{In this context, we mention that~\cite{KL18} has recently proposed a heuristic for reducing the number of regions of Explicit MPC by using neural-network approximations. The corresponding controller does, however, only come along with guarantees on the feasibility, stability, and performance if the exact explicit solution map happens to be recovered by the deep-learning approach.}
\change{A recent trend in optimization based control is to solve large MPC problems by breaking them into smaller ones. This trend has been initiated by the} enormous research effort in the field of distributed optimization~\cite{Boyd2011}. There exists a large number of related optimization methods, including dual decomposition~\cite{Everett1963}, ADMM~\cite{Boyd2011}, or ALADIN~\cite{Houska2016}, which have all been applied to MPC in various contexts and by many authors~\cite{Conte2012,Giselesson2013,Kozma2013,Necoara2008,Donoghue2013,Richter2011}. \change{Additionally, applications of accelerated variants of ADMM to MPC can be found in~\cite{FS16,SK13}.}
\change{However, modern distributed optimization methods, such as ADMM or ALADIN, typically converge to an optimal solution in the limit, i.e., if the number of iterations tends to infinity. Thus, if real-time constraints are present, one could at most implement a finite number of such ADMM or ALADIN iterations returning a control input that may be infeasible or sub-optimal by a large margin. But, unfortunately, for such heuristic implementations of real-time distributed MPC, there are, at the current status of research, no stability, feasibility, and performance guarantees available.}
\change{Therefore, this} paper asks the question whether it is possible to approximate an MPC feedback law by a finite code list, whose input is the current state measurement and whose output, the control reaction, is obtained by evaluating a constant, finite number of pre-computed, explicit solution maps that are associated to MPC problems of a smaller scale. Here, a key requirement is that \change{recursive feasibility, uniform asymptotic stability, and performance guarantees} of the implemented closed-loop controller have to be verifiable offline. Notice that such an MPC code would have major advantages for an embedded hardware system, \change{as it has a constant run-time using static memory only, while, at the same time, feasibility, stability, and performance guarantees are available.}
The contribution of this paper is the development of a controller, which meets these requirements under the restricting assumption that the original MPC problem is a strongly convex (but potentially large-scale) QP, as introduced in Section~\ref{sec::problem}. The control scheme itself is presented in the form of Algorithm~\ref{alg::dempc} in Section~\ref{sec::algorithm}. This algorithm alternates---similar to the distributed optimization method ALADIN~\cite{Houska2016}---between solving explicit solution maps that are associated with small-scale decoupled \change{QPs and solving a linear equation system} of a larger scale.
However, in contrast to ALADIN, ADMM or other existing distributed optimization algorithms, Algorithm~\ref{alg::dempc} performs only a constant number of iterations \change{per sampling time.}
\change{The recursive feasibility, stability, and performance properties} of Algorithm~\ref{alg::dempc}, which represent the main added value compared to our preliminary work~\cite{Oravec2017}, are summarized in Section~\ref{sec::convergence},~\ref{sec::stability}, and~\ref{sec::performance}, respectively. Instead of relying on existing analysis concepts from the field of distributed optimization, the mathematical developments in this paper rely on results that find their origin in Explicit MPC theory~\cite{BorrelliPHD}. In particular, the technical developments around Theorem~\ref{thm::convergenceRate} make use of the solution properties of multi-parametric QPs in order to derive convergence rate estimates for Algorithm~\ref{alg::dempc}. Moreover, Theorem~\ref{thm::stability} establishes \change{an asymptotic stability guarantee of the presented real-time closed-loop scheme}. This result is complemented by Corollary~\ref{cor::performance}, which provides \change{bounds} on the sub-optimality of the presented control scheme.
Finally, Section~\ref{sec::pMPC} discusses implementation details with a particular emphasis on computational and storage complexity exploiting the fact that the presented scheme can be realized by using static memory only while ensuring a constant run-time. A spring-vehicle-damper benchmark is used to illustrate the performance of the proposed real-time scheme. Section~\ref{sec::conclusions} concludes the paper.
\section{Linear-Quadratic MPC} \label{sec::problem} This paper concerns differential-algebraic model preditive control problems in discrete-time form,\change{ \begin{eqnarray} \begin{array}{rcl} J(x_0) = & \underset{x,u,z}{\min} & \overset{N-1}{\underset{k=0}{\sum}} \ell(x_k,u_k,z_k) + \mathcal M( x_N ) \\[0.25cm] & \mathrm{s.t.} & \left\{ \begin{array}{l} \forall k \in \{ 0, \ldots, N-1 \}, \\ \begin{array}{rcl} x_{k+1} &=& A x_k + B u_k + C z_k, \\ 0 &=& D x_k + E z_k, \end{array} \\ x_k \in \mathbb X, \; u_k \in \mathbb U, \; z_k\in\mathbb{Z}, \; x_N \in \mathbb X_N, \end{array} \right. \end{array} \label{eq::mpc} \end{eqnarray}} with strictly convex quadratic stage and terminal cost, \begin{eqnarray} \ell(x,u,z) &=& x^\intercal Q x + u^\intercal R u + z^\intercal S z \notag \\[0.16cm] \text{and} \qquad \change{\mathcal M(x)} &=& \change{x^\intercal P x} \; . \notag \end{eqnarray} Here, $x_k \in \mathbb R^{n_{\mathrm{x}}}$ denotes the predicted state at time $k$, $z_k \in \mathbb R^{n_{\mathrm{z}}}$ an associated algebraic state, and $u_k \in \mathbb R^{n_{\mathrm{u}}}$ the associated control input. The matrices \begin{align} \begin{array}{c} A,\change{P},Q \in \mathbb R^{n_{\mathrm{x}} \times n_{\mathrm{x}}}, \; B \in \mathbb R^{n_{\mathrm{x}} \times n_{\mathrm{u}}}, \; C \in \mathbb R^{n_{\mathrm{x}} \times n_{\mathrm{z}}},\\ \change{D \in \mathbb R^{n_{\mathrm{z}} \times n_{\mathrm{x}}}}, \; \change{E,S} \in \mathbb R^{n_{\mathrm{z}} \times n_{\mathrm{z}}}, \; R \in \mathbb R^{n_{\mathrm{u}} \times n_{\mathrm{u}}} \end{array} \notag \end{align} are given and constant. Notice that~\eqref{eq::mpc} is a parametric optimization problem with respect to the current state measurement $x_0$, i.e., the optimization variable $x = [ x_1, x_2, \ldots, x_N ]$ includes all but the first element of the state sequence. The control sequence $u = [u_0, u_1, \ldots, u_{N-1}]$ and algebraic state sequence $z = [ z_0, z_1, \ldots, z_N ]$ are defined accordingly.
\begin{assumption} \label{ass::blanket} We assume that \begin{enumerate} \item[a)] the constraint sets $\mathbb U \subseteq \mathbb R^{n_{\mathrm{u}}}$, $\mathbb X, \mathbb X_N \subseteq \mathbb R^{n_{\mathrm{x}}}$, and \change{$\mathbb Z \subseteq \mathbb R^{n_{\mathrm{z}}}$} are convex and closed polyhedra satisfying $0 \in \mathbb U$, $0 \in \mathbb X$, $0 \in \mathbb X_N$, \change{and $0 \in \mathbb Z$};
\item[b)] the matrices $Q $, $R $, $S $, and \change{$P$} are all symmetric and positive definite;
\item[c)] and the \change{square-matrix $E$ is} invertible.
\end{enumerate} \end{assumption} Notice that the latter assumption implies that, at least in principle, one could eliminate the algebraic states, as the algebraic constraints in~\eqref{eq::mpc} imply \begin{equation} \label{eq::replace} z_k = - E^{-1} D x_k \; . \end{equation} However, the following algorithmic developments exploit the particular structure of~\eqref{eq::mpc}. This is relevant in the context of large-scale interconnected control \change{systems}, where the \change{matrix $D$ is} potentially dense while all other matrices have a block-diagonal structure, \change{as explained in the sections below}.
\begin{remark} Assumption~\ref{ass::blanket}c) also implies that the equality constraints in~\eqref{eq::mpc} are linearly independent. Thus, the associated co-states (dual solutions) are unique. Similarly, Assumption~\ref{ass::blanket}a) and \ref{ass::blanket}b) imply strong convexity such that the primal solution of~\eqref{eq::mpc} is unique whenever it exists. \end{remark}
\color{black} \begin{remark} The assumption that the matrices $Q $, $R $, $S$, and $P$ are positive definite can be satisfied in practice by adding suitable positive definite regulatization to the stage cost. \end{remark} \color{black}
\subsection{Interconnected systems} \label{sec::networks} \color{black} Many control systems of practical interest have a particular structure, as they can be divided into $\bar I \in \mathbb N$ subsystems that are interconnected. A prototypical example for such an interconnected system is shown in Figure~\ref{fig::modeling}, \begin{figure}
\caption{\change{Sketch of a spring-vehicle-damper system.}}
\label{fig::modeling}
\end{figure} which consists of $\bar I$ vehicles with mass $m$ that are linked by springs and dampers. Such systems can be modelled by a linear discrete-time recursion of the form \begin{align} \label{eq::network} \underbrace{\left[ x_{k+1} \right]_i = \mathcal A_{ii} \left[ x_{k} \right]_i + \mathcal{B}_i \left[ u_k \right]_i}_{\text{dynamics of the $i$-th subsystem}} + \underbrace{\sum_{j \in \mathcal N_i} \mathcal A_{ij} \left[ x_{k} \right]_j}_{\substack{ \text{contribution} \\ \text{from neighbors}} } \end{align} for all $k \in \{ 1, \ldots, N \}$. Here, $\left[ x_{k} \right]_i \in \mathbb X_i$ denotes the state, $\left[ u_{k} \right]_i \in \mathbb U_i$ the control input of the $i$th subsystem, and $\mathcal A_{ij}$ and $\mathcal{B}_i$ system matrices of appropriate dimension. The index $j$ runs over $\mathcal N_i$, the set of neighbors of the $i$th node in the system's connectivity graph.
\begin{example} \label{ex::springmassdamper} For the spring-vehicle-damper system in Figure~\ref{fig::modeling} the connectivity graph can be specified by setting $$\mathcal N_i = \{ i-1, i+1 \}$$ for $i \in \{ 2, 3, \ldots, \bar I-1 \}$ and $\mathcal N_1 = \{ 2 \}$ and $\mathcal N_{\bar{I}}$ for the first and last trolley. In this example, the $i$-th subblock of the state at time $k$, $$[x_k]_i = \left( [p_k]_i \, , \, [v_k]_i \right)^\intercal ,$$ consists of the position and velocity of the $i$-th trolley, all relative to their equilibrium values. The control input is the force at the last trolley. A corresponding system model is then obtained by setting \begin{align} \begin{array}{rclrcl} \mathcal{A}_{i,i} &=& \mathbb I + h \left( \begin{array}{cc} 0 & 1 \\ - 2 \frac{k}{m} & -2\frac{d}{m} \end{array} \right) \; , \qquad & \mathcal{B}_i &=& \left( \begin{array}{c} 0 \\ 0 \end{array} \right), \notag \\ \mathcal{A}_{\bar{I},\bar{I}} &=& \mathbb I + h \left( \begin{array}{cc} 0 & 1 \\ - \frac{k}{m} & -\frac{d}{m} \end{array} \right) \; , \qquad & \mathcal{B}_{\bar{I}} &=& h \left( \begin{array}{c} 0 \\ \frac{1}{m} \end{array} \right), \notag \\ \mathcal{A}_{i-1,i} &=& \mathcal A_{i,i+1} = h \left( \begin{array}{cc} 0 & 0 \\ \frac{k}{m} & \frac{d}{m} \end{array} \right), \end{array} \end{align} for all $i \in \{ 1, \ldots ,\bar{I}-1 \}$. In this context, $m > 0$ denotes the mass, $k > 0$ the spring constant, $d \geq 0$ a damping coefficient, and $h>0$ the step-size of the Euler discretization. \end{example}
System~\eqref{eq::network} can be reformulated by introducing auxiliary variables of the form \color{black} \begin{equation} \label{eq::refZ} \left[ z_{k} \right]_i = \sum_{j \in \mathcal N_i} \mathcal A_{ij} \left[ x_{k} \right]_j \; . \end{equation} The advantage of introducing these algebraic auxiliary variables is that the dynamic recursion~\eqref{eq::network} can be written in the form \begin{align} x_{k+1} \;=\;& A x_k + B u_k + C z_k, \notag \\ 0 \;=\;& D x_k + E z_k, \notag \end{align} with $A = \mathrm{diag}\left( \mathcal A_{1,1}, \ldots, \mathcal A_{\bar{I},\bar{I}} \right)$, $B = \mathrm{diag}\left( \mathcal B_1, \ldots, \mathcal B_{\bar I} \right)$, $C = I$, \color{black} \[ D_{i,j} = \left\{ \begin{array}{ll} \mathcal A_{i,j} & \text{if} \; j \in \mathcal N_i \\ 0 & \text{otherwise}, \end{array} \right\} \] \color{black} and $E = -I$. After this reformulation, all matrices in the algebraic discrete-time system are block diagonal except for the matrix $D$, which may, however, still be sparse depending on the particular definition (graph structure) of the sets $\mathcal N_i$. Notice that the associated state and control constraint sets $\mathbb X = \mathbb X_1 \times \ldots \times \mathbb X_{\bar I}$ and $\mathbb U = \mathbb U_1 \times \ldots \times \mathbb U_{\bar I}$ have a separable structure, too.
\subsection{Recursive feasibility and asymptotic stability}
Notice that the stability and recursive feasibility properties of MPC \change{controllers have} been analyzed exhaustively~\cite{Rawlings2009}. \change{ As long as Assumption~\ref{ass::blanket} holds, these results can be applied one-to-one for~\eqref{eq::mpc} after eliminating the algebraic states explicitly.}
\color{black} \begin{definition} A set $X \subseteq \mathbb X$ is called control invariant if \[
X \subseteq \left\{ x \in \mathbb R^{n_{\mathrm{x}}} \left| \begin{array}{l} \exists x^+ \in X, \, \exists u \in \mathbb U, \, \exists z \in \mathbb Z, \\
\begin{array}{rcl} x^+ &=& A x + B u + C z, \\ 0 &=& D x + E z \end{array} \end{array} \right. \right\} \; . \] \end{definition} \color{black} \begin{assumption} \label{ass::forwardInvariance} We assume that the terminal \change{set $\mathbb X_N$ is control invariant.} \end{assumption} It is well known that~\eqref{eq::mpc} is recursively feasible if Assumption~\ref{ass::blanket} and Assumption~\ref{ass::forwardInvariance} hold \change{recalling that one can always eliminate the algebraic states in order to apply the results from~\cite{Rawlings2009}}. Another standard assumption can be formulated as follows. \begin{assumption} \label{ass::TerminalLyapunov} The terminal cost $\mathcal M$ in~\eqref{eq::mpc} admits a control law $\mu: \mathbb X_N \to \mathbb U$ such that for all $x \in \mathbb X_N$ \[ \change{\ell(x,\mu(x),z) + \mathcal M( x^+ ) \leq \mathcal M(x)} \; , \] where $z = -E^{-1} D x$ \change{and $x^+ = A x + B \mu(x) - C E^{-1} D x$}. \end{assumption} The MPC controller~\eqref{eq::mpc} is asymptotically stable if Assumption~\ref{ass::blanket},~\ref{ass::forwardInvariance}, and~\ref{ass::TerminalLyapunov} hold~\cite{Rawlings2009}. Notice that there exist \change{generic} methods for both the construction of forward invariant sets $\mathbb X_N$ and the construction of quadratic terminal costs such that the above criteria are satisfied~\cite{Bla:aut:99,Rawlings2009}. \change{Moreover, tailored methods for constructing block separable terminal costs and terminal constraint sets are available in the literature, too~\cite{Conte2016}.}
\section{Suboptimal real-time MPC} \label{sec::algorithm} In this section we propose and analyze a real-time algorithm for finding approximate, suboptimal solutions of~\eqref{eq::mpc}.
\subsection{Preliminaries} Let us introduce the stacked \change{vectors \[ y_0 = \left[ u_0^\intercal \, , \, z_0^\intercal \right]^\intercal \; , \; \; y_k = \left[ x_k^\intercal \, , \, u_k^\intercal \, , \, z_k^\intercal \right]^\intercal \; , \; \; y_N = x_N, \] and their associated constraint sets \begin{eqnarray} \mathbb Y_k &= \left\{ y_k
\left| \begin{array}{l} A x_k + B u_k + C z_k \in \mathbb X, \\ 0 = D x_k + E z_k, \\ x_k \in \mathbb X, \; u_k \in \mathbb U, \; z_k \in \mathbb Z \end{array} \right. \right\}, \; \mathbb Y_N &= \mathbb X_N, \end{eqnarray} for all $k \in \{ 0, \ldots, N-1 \}$.} Moreover, it is convenient to introduce the shorthand notation \begin{equation} \label{eq:def_F} \begin{split} F_k(y_k) &\;=\; \ell(x_k,u_k,z_k) \; , \quad k\in \{ 0, \ldots, N-1 \} \\[0.1cm] F_N(y_N) &\;=\; \mathcal M(x_N) , \end{split} \end{equation} as well as the matrices \[ G_k = \left( \begin{array}{ccc} I & 0 & 0 \\ 0 & 0 & 0 \end{array} \right) \; , \; \change{G_{N} = I}, \] for $k \in \{ 1, \ldots N-1 \}$ and \[ H_0 = \left( \begin{array}{ccc} B & C \\ 0 & E \end{array} \right) , \, H_k = \left( \begin{array}{ccc} A & B & C \\ D & 0 & E \end{array} \right), \] as well as $h_0 = [ (A x_0)^\intercal \, 0]^\intercal, \; h_{k} = 0$ for all indices $k \in \{ 1, \ldots, N-1 \}$. The advantage of this notation is that~\eqref{eq::mpc} can be written in the \change{form \begin{align} \label{eq::mpc2} J(x_0) = \underset{y}{\min} &\quad \sum_{k=0}^{N} F_k( y_k ) \\[0.16cm] \notag \text{s.t.} &\quad \left\{ \begin{array}{l} \forall k \in \{ 0, \ldots, N-1 \}, \\ \begin{array}{rcll} G_{k+1} y_{k+1} &=& H_k y_k + h_k \; & \mid \; \lambda_{k}, \end{array} \\ y_k \in \mathbb Y_k \; , \; y_N \in \mathbb Y_N \; . \end{array} \right. \end{align} Here, $\lambda_0, \ldots, \lambda_{N-1}$} denote the multipliers of the affine constraints in~\eqref{eq::mpc2}. It is helful to keep in mind that both the function $F_0$ and the vector $h_0$ depend on $x_0$ even if we hide this parametric dependence for a moment in order to arrive at a simpler notation. In addition, we introduce a shorthand for the objective in~\eqref{eq::mpc2}, \begin{align} F(y) =& \sum_{k=0}^{N} F_k( y_k ) \; . \notag \end{align} Moreover, the convex conjugate function of $F$ is denoted by \begin{align} F^{\star}(\lambda) =& \max_{y}\;\ \bigg\{ - F(y) - \left( H_0^\intercal \lambda_{0}^m \right)^\intercal y_0 \notag \\ & + \sum_{k=1}^{N-1} \left( G_k^\intercal \lambda_{k-1}^m - H_k^\intercal \lambda_{k}^m \right)^\intercal y_k + \lambda_{N-1}^\intercal G_N y_N \bigg\} \; . \notag \end{align} Notice that the functions $F$ and $F^{\star}$ are strongly convex quadratic forms with $F(0) = 0$ and $F^{\star}(0)=0$ as long as Assumption~\ref{ass::blanket} is satisfied. The optimal primal and dual solutions of~\eqref{eq::mpc2} are denoted by $x^{\star}$ and $\lambda^{\star}$, respectively. It is well-known that $x^{\star}$ and $\lambda^{\star}$ are continuous and piecewise affine functions of $x_0$ on the polytopic domain $\mathcal X = \{ x_0 \mid J(x_0) < \infty \}$, see~\cite{borrelli2003geometric}.
\subsection{Algorithm} \color{black} The main idea for solving~\eqref{eq::mpc2} approximately and in real time is to consider the unconstrained auxiliary optimization problem \begin{align} \label{eq::mpc2mod} J(x_0) = \underset{y}{\min} &\quad \sum_{k=1}^{N-1} F_k( y_k - y_k^\text{ref} ) + F_N(y_N- y_N^\text{ref})\\[0.16cm] \notag
\text{s.t.} &\quad \left\{ \begin{array}{l} \forall k \in \{ 0, \ldots, N-1 \}, \\ \begin{array}{rcll} G_{k+1} y_{k+1} &=& H_k y_k + h_k \; & \mid \; \lambda_{k}, \end{array} \end{array} \right. \end{align} which is a standard tracking optimal control problem without inequality constraints. Here, we have introduced the reference trajectory $y_k^\text{ref}$. For the special case that $y^\text{ref} = y^{\star}$ is equal to the minimizer of~\eqref{eq::mpc2}, Problems~\eqref{eq::mpc2} and~\eqref{eq::mpc2mod} are equivalent. Notice that the main motivation for introducing the coupled QP~\eqref{eq::mpc2mod} is that this problem approximates~\eqref{eq::mpc2} without needing inequality constraints. Thus, this problem can be solved by using a sparse linear algebra solver; see Section~\ref{sec::sparseSolver} for implementation details.
\begin{algorithm}[htbp!]
\caption{Parallel real-time MPC}
\textbf{Initialization:}\\[0.2cm]
Choose initial $y^1 = [y_0^1, \ldots, y_N^1]$ and \change{$\lambda^1 = [ \lambda_0^1, \ldots, \lambda_{N-1}^1 ]$} and a constant $\gamma > 0$.\\[0.2cm]
\textbf{Online:}\\
\begin{enumerate}
\item[\textbf{1)}] Wait for the state measurement $x_0$ and compute the constant
\[
f^1 = F(y^1) + F^{\star}(\lambda^1) \; .
\]
If $f^1 \geq \gamma^2 x_0^\intercal Q x_0$, rescale
$$y^1 \leftarrow y^1 \sqrt{\frac{\gamma^2 \Vert x_0 \Vert_Q^2}{f_1}} \; \; \text{and} \; \; \lambda^1 \leftarrow \lambda^1 \sqrt{\frac{\gamma^2 \Vert x_0 \Vert_Q^2}{f_1}} \; ,$$ \\[0.1cm]
where $\Vert x_0 \Vert_Q^2 \triangleq x_0^{\intercal} Q x_0 $ .
\item[\textbf{2)}] \textbf{For $m = 1\rightarrow \overline{m}$}\\[0.2cm]
\noindent
\textbf{a)} solve the small-scale decoupled QPs in parallel
\begin{eqnarray}
\hspace{-0.6cm} &\underset{\xi_0^m \in \mathbb Y_0}{\min} & \hspace{-0.1cm} F_0(\xi_0^m) - (H_0^\intercal \lambda_{0}^m)^\intercal \xi_0^m + F_0( \xi_0^m - y_0^m ) \notag \\[0.2cm]
\hspace{-0.6cm} &\underset{\xi_k^m \in \mathbb Y_k}{\min} & \hspace{-0.1cm} F_k(\xi_k^m) + ( G_k^\intercal \lambda_{k-1}^m - H_k^\intercal \lambda_{k}^m )^\intercal \xi_k^m + F_k( \xi_k^m - y_k^m ) \notag \\[0.2cm]
\hspace{-0.6cm} &\change{ \underset{\xi_N^m \in \mathbb Y_N}{\min}} & \change{\hspace{-0.1cm} F_N(\xi_N^m) + \left( G_N^\intercal \lambda_{N-1}^m \right)^\intercal \xi_N^m + F_N( \xi_N^m - y_N^m ) } \notag
\end{eqnarray}
for all \change{$k \in \{ 1, \ldots, N-1 \}$} and denote the optimal solutions by $\xi^m = [ \xi_0^m, \xi_1^m, \ldots, \xi_N^m ]$.\\[0.2cm]
\textbf{b)} Solve the coupled QP
\begin{align}
\label{eq::qp}
\underset{y}{\min}& \quad \sum_{k=0}^{N} F_k( y_k^{m+1} - 2 \xi_k^m + y_k^m ) \\[0.16cm]\notag
\text{s.t.} &\quad
\left\{
\begin{array}{l}
\forall k \in \{ 0, \ldots, N-1 \}, \\
\change{
\begin{array}{rcll}
G_{k+1} y_{k+1}^{m+1} &=& H_k y_k^{m+1} + h_k \; & \mid \; \delta_{k}^m,
\end{array}
}
\end{array}
\right.
\end{align}
and set $\lambda^{m+1} = \lambda^m + \delta^m$.\\[0.2cm]
\noindent
\textbf{End}\\
\item[\textbf{3)}] Send $u_0 = \left( 0 \;\; I \;\; 0 \right) \xi_0^{\overline{m}}$ to the real process.\\
\item[\textbf{4)}] Set $y^1 = [y_1^{\overline{m}}, \ldots, y_N^{\overline{m}}, 0]$\change{, $\lambda^1 = [ \lambda_1^{\overline{m}}, \ldots, \lambda_{N-1}^{\overline{m}}, 0]$} and go to Step~1.\\
\end{enumerate}
\label{alg::dempc} \end{algorithm}
Let us assume that $y^m$ and $\lambda^m$ are the current approximations of the primal and dual solution of~\eqref{eq::mpc2}. Now, the main idea of Algorithm~\ref{alg::dempc} to construct the next iterate $y^{m+1}$ and $\lambda^{m+1}$ by performing two main operations. First, we solve augmented Lagrangian optimization problems of the form \begin{align} \label{eq::NLPs}
\underset{\xi^m \in \mathbb Y}{\min} & \;\; F(\xi^m) - (\mathcal G^\intercal \lambda_{0}^m)^\intercal \xi^m + \left\| \xi^m - y^m \right\|_{\mathcal Q}^2 \end{align} with $$\mathcal G = \left( \begin{array}{ccccc} H_0 & -G_0 & & & 0 \\
& H_1 & - G_1 & & \\
& & \ddots & \ddots & \\ 0 & & & H_{N-1} & -G_{N-1} \end{array} \right). $$ The focus of the following analysis is on the case that the weighting matrix $\mathcal Q = \nabla^2 F(0)$ is such that
$$\left\| \xi^m - y^m \right\|_{\mathcal Q}^2 = F( \xi^m - y^m )$$ recalling that $F$ is a centered positive-definite quadratic form. And second, we solve QP~\eqref{eq::mpc2mod} for the reference point \[ y^\text{ref} = 2 \xi^m - y^m \; , \] which can be interpreted as weighted average of the solution of~\eqref{eq::NLPs} and the previous iterate $y^m$. These two main steps correspond to Step 2a) and Step 2b) in Algorithm~\ref{alg::dempc}. Notice that the main motivation for introducing the augmented Lagrangian problem~\eqref{eq::NLPs} is that this optimization problem is separable, as exploited by Step~2a). As explained in more detail in Section~\ref{sec::parametric}, the associated smaller-scale QPs can be solved by using existing tools from the field of Explicit MPC.
Additionally, in order to arrive at a practical procedure, Algorithm~\ref{alg::dempc} is initialized with guesses, $$y^1 = [y_0^1, \ldots, y_N^1] \quad \text{and} \quad \lambda^1 = [ \lambda_0^1, \ldots, \lambda_N^1 ] \; ,$$ for the primal and dual solution of~\eqref{eq::mpc2}. Notice that Algorithm~\ref{alg::dempc} receives a state measurement $x_0$ in every iteration (Step 1) and returns a control input to the real process (Step 3). Moreover, Step 1) rescales $y^1$ and $\lambda^1$ based on a tuning parameter $\gamma > 0$, which is assumed to have the following property. \begin{assumption} \label{ass::gamma} The constant $\gamma$ in Algorithm~\ref{alg::dempc} is such that \[ F(y^{\star}) + F^{\star}(\lambda^{\star}) \leq \gamma^2 x_0^\intercal Q x_0 \; . \] \end{assumption} Notice that such a constant $\gamma$ exists and can be pre-computed offline, which simply follows from the fact that $y^{\star}$ and $\lambda^{\star}$ are Lipschitz continuous and piecewise affine functions of $x_0$, as established in~\cite{borrelli2003geometric}. The rationale behind this rescaling is that this step prevents initializations that are too far away from $0$. Intuitively, if the term $f^1 = F(y^1) + F^{\star}(\lambda^1)$ is much larger than $x_0^\intercal Q x_0$, then $(y^1,\lambda^1)$ can be expected to be far away from the optimal solution $(y^\star,\lambda^\star)$ and it is better to rescale these variables such that they have a reasonable order of magnitude. In the following section, we will provide a theoretical justification of the rescaling factor $\sqrt{\frac{\gamma^2 \Vert x_0 \Vert_Q^2}{f_1}}$. Notice that if we would set this rescaling factor to $1$, Algorithm 1 is unstable in general. In order to see this, consider the scenario that a user initializes the algorithm with an arbitrary $(y^1,\lambda^1) \neq 0$. Now, if the first measurement happens to be at $x_0 =0$, of course, the optimal control input is at $u^\star =0$. But, if we run Algorithm~1 with $\bar m < \infty$, it returns an approximation $u_0 \approx u^\star = 0$, which will introduce an excitation as we have $u_0 \neq 0$ in general. Thus, if we would not rescale the initialization in Step~1), it would not be possible to establish stability. \color{black}
\subsection{Convergence properties of Algorithm~\ref{alg::dempc}} \label{sec::convergence}
This section provides a concise overview of the theoretical convergence properties of Algorithm~\ref{alg::dempc}. We start with the following theorem, which is proven in Appendix~\ref{app::convergenceRate}. \begin{theorem} \label{thm::convergenceRate} \change{Let Assumption~\ref{ass::blanket} be satisfied and let~\eqref{eq::mpc2} be feasible, i.e., such that a unique minimizer $y^{\star}$ and an associated dual solution $\lambda^{\star}$ exist.} Then there exists a positive constant $\kappa < 1$ such that \begin{align} &\;\; F( y^{m+1} - y^{\star} ) + F^{\star}( \lambda^{m+1} - \lambda^{\star} ) \notag \\[0.16cm] \leq& \;\;\kappa \left( F( y^m - y^{\star} ) + F^{\star}( \lambda^m - \lambda^{\star} ) \right) \end{align} for all $m \geq 2$. \end{theorem} \change{Because Theorem~\ref{thm::convergenceRate} establishes contraction, an immediate consequence is that the iterates of Algorithm~1 would converge to the exact solution of~\eqref{eq::mpc2}, if we would set $\overline{m} = \infty$, i.e., \[ \lim_{m \to \infty} \xi^m = y^{\star} \quad \text{and} \quad \lim_{m \to \infty} \lambda^m = \lambda^{\star} \] Moreover, the proof of Theorem~\ref{thm::convergenceRate} provides} an explicit procedure for computing the constant $\kappa < 1$.
\subsection{Recursive feasibility and asymptotic stability of Algorithm~\ref{alg::dempc}}
\label{sec::stability}
The goal of this section is to establish recursive feasibility and asymptotic stability of Algorithm~\ref{alg::dempc} on $\mathcal X$. Because we send the control input $u_0 = \left( 0 \; I \; 0 \right) \xi_0^{\overline{m}}$ to the real process, the next measurement will be at \[ x_0^+ = A x_0 + (B \;\; C) \xi_0^{\overline{m}} \; . \] Notice that, in general, we may have \[ x_0^+ \neq x_1^{\star} = A x_0 + (B \;\; C) y_0^{\star} \; ,\ \] since we run Algorithm~\ref{alg::dempc} with a finite $\overline{m} < \infty$. \change{A proof of the following proposition can be found in Appendix~\ref{app::feasibility}.} \color{black} \begin{proposition} \label{prop::feasibility} Let us assume that \begin{enumerate} \item Assumptions~\ref{ass::blanket} is satisfied, \item we use the terminal region $\mathbb X_N = \mathbb X$, and \item $\mathbb X$ is control invariant. \end{enumerate} Then Algorithm~\ref{alg::dempc} is recursively feasible in the sense that $x_0 \in \mathbb X$ implies $x_0^+ \in \mathbb X$. Moreover, the equation $\mathcal X = \mathbb X$ holds, i.e., Problem~\eqref{eq::mpc2} remains feasible. \end{proposition}
\begin{remark} The construction of control invariant sets for interconnected systems in the presence of state constraints can be a challenging task and this manuscript does not claim to resolve this problem. An in-depth discussion of how to meet the third requirement of Proposition~\ref{prop::feasibility} is beyond the scope of this paper, but we refer to~\cite{Conte2016}, where a discussion of methods for the construction of control invariant sets for network systems can be found. \end{remark}
The following theorem establishes asymptotical stability of Algorithm~\ref{alg::dempc}, one of the main contributions of this paper. The corresponding proof can be found in Appendix~\ref{app::stability}. \begin{theorem} \label{thm::stability} Let Assumptions~\ref{ass::blanket},~\ref{ass::forwardInvariance},~\ref{ass::TerminalLyapunov}, and~\ref{ass::gamma} be satisfied and let $\mathbb X_N = \mathbb X$ be control invariant. Let the constant $\sigma > 0$ be such that the semi-definite inequality \[ \left( \begin{array}{cc} B^\intercal Q B & B^\intercal Q C \\ C^\intercal Q B & C^\intercal Q B \end{array} \right) \preceq \sigma \left( \begin{array}{cc} R & 0 \\0 & S \end{array} \right) \] holds and let the constants $\eta,\tau > 0$ be such that \begin{align} \label{eq::etaTauBound}
|J(x_0^+) - J(x_1^{\star})| \leq \eta \| x_0^+ - x_1^{\star} \|_Q + \frac{\tau}{2} \| x_0^+ - x_1^{\star} \|_Q^2 \end{align} If the constant $\overline{m} \in \mathbb N$ satisfies \begin{align} \label{eq::mbarBound} \overline{m} > 2 \frac{\log \left( \eta \sqrt{\sigma} \gamma (1+\sqrt{\kappa}) + \frac{\tau \sigma \gamma^2 (1+\sqrt{\kappa})^2}{2} \right)}{\log(1/\kappa)} \; , \end{align} then the controller in Algorithm~\ref{alg::dempc} is \change{asymptotically stable} on $\mathcal X$. \end{theorem} \color{black}
\change{The} constants $\eta, \tau, \sigma, \change{\gamma}$, and $\kappa < 1$ in the above theorem depend on the problem data only\change{, but they are independent of the initial state $x_0 \in \mathcal X$.}
\color{black} \begin{remark} The lower bound~\eqref{eq::mbarBound} is monotonically increasing in $\eta, \tau, \sigma, \gamma$, and $\kappa$. Thus, the smaller these constants are, the tighter the lower bound~\eqref{eq::mbarBound} will be. For small-scale applications, one can compute these constants offline, by using methods from the field of explicit MPC~\cite{BemEtal:aut:02,borrelli2003geometric}. However, for large-scale applications, the explicit solution map cannot be computed with reasonable effort, not even offline. In this case, one has to fall back to using conservative bounds. For example, the constants $\eta$ and $\tau$ satisfying~\eqref{eq::etaTauBound} can be found by using methods from the field of approximate dynamic programming~\cite{Wang2015}. However, if one uses such conservative bounding methods, the lower bound~\eqref{eq::mbarBound} is conservative, too. \end{remark} \color{black}
\subsection{Performance of Algorithm~\ref{alg::dempc}} \label{sec::performance} The result of Theorem~\ref{thm::stability} can be extended in order to derive an a-priori verifiable upper bound on the sub-optimality of Algorithm~1.
\begin{corollary} \label{cor::performance} Let the assumption of Theorem~\ref{thm::stability} hold with \[ \alpha = 1 - \left[ \eta \sqrt{\sigma} \gamma (1+\sqrt{\kappa}) + \frac{\tau \sigma \gamma^2 (1+\sqrt{\kappa})^2}{2} \right] \kappa^{\frac{\overline{m}}{2}} \; . \] If $y_i^{\mathrm{cl}} = \left( x_i^\text{cl}, \, u_i^\text{cl}, \, z_i^\text{cl} \right)$ denotes the sequence of closed-loop states and controls that are generated by the controller in Algorithm~\ref{alg::dempc}, an a-priori bound on the associated infinite-horizon closed-loop performance is given by \[ \sum_{i=0}^{\infty} \ell( x_i^\text{cl}, \, u_i^\text{cl}, \, z_i^\text{cl} ) \leq \frac{J(x_0)}{\alpha} \; . \] \end{corollary}
\noindent \textbf{Proof.} Theorem~\ref{thm::stability} together with~\eqref{eq::lyapD1} and~\eqref{eq::lyapD2} imply that \[ J( x_{i+1}^\text{cl} ) \leq J( x_i^\text{cl} ) - \alpha F_0( y_i^{\mathrm{cl}} ) \; , \] which yields the inequality \[ \sum_{i=0}^{\infty} F_0( y_i^{\mathrm{cl}} ) \leq \frac{1}{\alpha} \sum_{i=0}^{\infty} \left( J(x_{i}^\text{cl}) - J(x_{i+1}^\text{cl}) \right) \; . \] The statement of the corollary follows after simplifying the telescoping sum on the right and substituting the equation $F_0( y_i^{\mathrm{cl}} ) = \ell( x_i^\text{cl}, \, u_i^\text{cl}, \, z_i^\text{cl} )$.
\qed
\section{Implementation on hardware with limited memory} \label{sec::pMPC} This section discusses implementation details for Algorithm~\ref{alg::dempc} with a particular emphasis on run-time aspects and limited memory requirements, as needed for the implementation of MPC on embedded hardware. Here, the implementation of Steps~1),~3), and~4) turns out to be straightforward, as both the CPU time requirements and memory needed for implementing these steps is negligible compared to Step 2). Thus, the focus of the following subsections is on the implementation of Step~2a) and Step~2b).
\subsection{Parametric quadratic programming} \label{sec::parametric} In Step~2a) of Algorithm~\ref{alg::dempc} decoupled QPs have to be solved on-line. To diminish the induced implementation effort, we propose to solve these QPs off-line using multi-parametric programming, i.e., to pre-compute the solution maps \begin{equation} \label{eq::mQP} \begin{array}{rcl} \xi_0 ^{\star}(\theta_0,x_0) &=& \underset{\xi_0 \in \mathbb{Y}_0}{\text{argmin}} \; 2 F_0(\xi_0) + \theta_0^\intercal \xi_0 , \\ \xi_1 ^{\star}(\theta_1) &=& \underset{\xi_1 \in \mathbb{Y}_0}{\text{argmin}} \; 2 F_1(\xi_1) + \theta_1^\intercal \xi_1 , \\ \xi_N ^{\star}(\theta_N) &=& \underset{\xi_N \in \mathbb{Y}_N}{\text{argmin}} \; 2F_N(\xi_N) + \theta_N^\intercal \xi_N , \end{array} \end{equation} with parameters $\theta_0 \in \mathbb R^{n_{\mathrm{u}} + n_{\mathrm{z}}}$, $\theta_1 \in \mathbb R^{n_{\mathrm{x}}+ n_{\mathrm{u}} + n_{\mathrm{z}}}$, and \change{$\theta_N \in \mathbb R^{n_{\mathrm{x}}}$}. Because these QPs are strongly convex, the functions \change{$\xi_0 ^{\star}$, $\xi_1 ^{\star}$, and $\xi_N ^{\star}$} are piecewise affine~\cite{BorrelliPHD}. Here, it should be noted that $\xi_0 ^{\star}$ additionally depends on $x_0$ recalling that this dependency had been hidden in our definition of $F_0$ and $\mathbb{Y}_0$. In this paper, we use MPT~\cite{MPT3} to pre-compute and store the explicit solution maps $\xi_0 ^{\star}$, $\xi_1 ^{\star}$ and $\xi_N ^{\star}$. Consequently, Step~2a) in Algorithm~1 can be replaced by:
\begin{quote} \textbf{Step 2a')} Compute the parameters \begin{subequations} \begin{align} \theta^m_0 &= -H_0^\intercal \lambda_{0}^m - 2\Sigma_0y_0^m , \\[0.1cm] \theta^m_k &= G_k^\intercal \lambda_{k-1}^m-H_k^\intercal \lambda_{k}^m - 2\Sigma_k y_k^m , \\[0.1cm] \change{\theta^m_N} &= \change{ G_N^\intercal \lambda_{N-1}^m - 2\Sigma_N y_N^m } \end{align} \end{subequations} with $\Sigma_0 = \text{blkdiag}\{R,S\}$, $\Sigma_k = \text{blkdiag}\{Q,R,S\}$ for all $k \in \{ 1, \ldots, N-1\}$, $\Sigma_N = \change{P}$ and set \begin{align} \xi_0^m = \xi_0^{\star}( \theta_0^m, x_0 )\;,\;\;\xi_k^m = \xi_1^{\star}( \theta_k^m ) \notag \end{align} for all $k \in \{ 1, \ldots, N \}$ by evaluating the respective explicit solution maps~\eqref{eq::mQP}. \change{In this paper, we use the enumeration-based multi-parametric QP algorithm from~\cite{HJ15} for generating these maps.} \end{quote}
Notice that the complexity of pre-processing the small-scale QPs~\eqref{eq::mQP} solely depends on the maximum number $N_{\text{R}} = \max \{ N_{\text{R},0}, N_{\text{R},1}, N_{\text{R},N} \}$ of critical regions over which the PWA optimizers $\xi_0 ^{\star}$, $\xi_1 ^{\star}$ and $\xi_N ^{\star}$ are defined~\cite{BemEtal:aut:02}, but $N_{\mathrm{R}}$ is independent of the prediction horizon $N$ as summarized in Table~\ref{tab::complexity}; see also~\cite{Oravec2017,BoyVan:ConOpt:04,Borrelli2017}.
Here, we assume that each parametric QP is post-processed, off-line, to obtain binary search trees~\cite{TJB03} in $\mathcal{O}(N_R^2)$ time. Once the trees are constructed, they provide for a fast evaluation of the solution maps in~\eqref{eq::mQP} in time that is logarithmic in the number of regions, thus establishing the $\mathcal{O}(N \log_2(N_R))$ on-line computational bound. The memory requirements are directly proportional to the number of regions $N_R$ with each region represented by a finite number of affine half-spaces.
\begin{table}[htbp!] \caption{\label{tab::complexity} Computational and storage complexity of Steps~2a') and 2b) of Algorithm~\ref{alg::dempc}.} \begin{center}
\begin{tabular}{|l|c|c|c|} \hline Step & Offline & Online & Memory \\
& CPU time & CPU time & Requirement \\ \hline 2a') & $\mathcal{O}(N_{\mathrm{R}}^2)$ & $\mathcal{O}(N \log_2( N_{\mathrm{R}}))$ & $\mathcal{O}(N_{\mathrm{R}})$ \\ 2b) & $\mathcal{O}( N n^3 )$ & $\mathcal{O}( N n^2 )$ & $\mathcal{O}( N n^2 )$\\ \hline \end{tabular} \end{center}
\end{table}
\subsection{Sparse linear algebra} \label{sec::sparseSolver} In Step~2b) of Algorithm~\ref{alg::dempc} the large-scale, coupled QP~\eqref{eq::qp} must be solved. Because this QP has equality constraints only,~\eqref{eq::qp} is equivalent to a large but sparse \change{system of equations}. Moreover, all matrices in~\eqref{eq::qp} are given and constant during the online iterations. This means that all linear algebra decompositions can be pre-computed offline. If one uses standard Riccati recursions for exploiting the band-structure of~\eqref{eq::qp}, the computational complexity for all offline computations is at most of order $\mathcal{O}( N n^3 )$, where $n = n_{\mathrm{x}} + n_{\mathrm{z}}$, while the online implementation has complexity $\mathcal{O}( N n^2 )$~\cite{Bertsekas2012}. If one considers interconnected systems this run-time result may be improved---in many practical cases, e.g., in the example that is discussed in Section~\ref{sec::caseStudy}, one may set \change{$n = ( n_{\mathrm{x}} + n_{\mathrm{z}} ) / \bar I$}. However, in general, the choice of the linear algebra solver and its computational complexity depend on the particular topology of the network~\cite{Borrelli2017}.
In summary, Algorithm~\ref{alg::dempc} can be implemented by using static memory only allocating at most $\mathcal{O}(N_{\mathrm{R}} + N n^2)$ floating point numbers. Here, the explicit solutions maps of both the decoupled QPs in Step~2a) as well as the coupled QP in Step~2b) can be pre-computed offline. Because Theorem~\ref{thm::stability} provides an explicit formula for computing a constant number of iterations $\overline{m}$ such that a stable and recursively feasible controller is obtained, the online run-time of Algorithm~\ref{alg::dempc} is constant and of order $\mathcal{O}(N \log_2( N_{\mathrm{R}}) + N n^2)$. Thus, Algorithm~\ref{alg::dempc} may be called an explicit MPC method---in the sense that it has a constant run-time and constant memory requirements for any given $N$ and $\bar I$ while stability and recursive feasibility can be verified offline. Its main advantage compared to existing MPC controllers is that it scales up easily for large scale interconnected networks of systems as well as long prediction horizons $N$, as illustrated below.
\section{Numerical example} \label{sec::caseStudy}
\change{This section applies Algorithm~\ref{alg::dempc} to a spring-vehicle-damper control system, which has been introduced in Example~\ref{ex::springmassdamper} with state and control and constraints} \begin{align*} & \mathbb X = \mathbb X_1 \times \ldots \times \mathbb X_{\bar I} \; , \; \mathbb U = \change{[-2,0.5]} \;,\; \mathbb Z = \mathbb R^{2 \bar I} \; , \\[0.16cm] & \text{where} \quad \mathbb X_1 = \ldots = \mathbb X_{\bar I} = \change{[-0.5,1.5]} \times \change{[-0.5,1]} \; . \end{align*} \change{The} weighting matrices of the MPC objective are set to $$Q= 10 \, I, \; R = I, \quad \text{and} \quad S = 10^{-2} \, I \; .$$ The numerical values for the mass $m$, spring constant $k$, and damping constant $d$ are listed below. \begin{center} \scriptsize \begin{tabular}{lcc}
\toprule
Parameter & Symbol & Value\,[Unit] \\
\midrule
sampling time & $h$ &$0.1$\,[s]\\[0.1cm]
spring constant & $k$ & $3$\,[N/m] \\[0.1cm]
mass of vehicle & $m$ & $1$\,[kg] \\[0.1cm]
viscous damping coefficient & $d$ & $3$\,[N\,s/m]\\[0.1cm]
\bottomrule \end{tabular} \end{center} Last but not least the \change{matrix $P$ is computed} by solving an algebraic Riccati equation, such that the terminal cost is equal to the unconstrained infinite horizon cost if none of the inequality constraints are \change{active~\cite{Rawlings2009}}.
\change{We have implemented Algorithm~\ref{alg::dempc} in Matlab R2018a using YALMIP~\cite{yalmip} and MPT 3.1.5~\cite{MPT3}. Here, the solution maps of the QPs~\eqref{eq::mQP} were pre-computed using the geometric parametric LCP solver of MPT 3.1.5~\cite{MPT3}}. By exploiting the separability of the cost function and constraints, the storage space for the parametric solutions can be reduced to $287 \, \text{kB}$, which corresponds to $432$ critical regions. This memory requirement is independent of the number of vehicles $\bar{I}$ and the length of the prediction horizon $N$. In contrast to this, the number of regions for standard Explicit MPC depends on both $\bar{I}$ and $N$: \begin{center} \scriptsize
\begin{tabular}{c | c c}
$(\bar{I},N)$ & \# of regions & memory\\
\hline
$(1,10)$ & $ 58$ & $14 \, [\text{kB}]$ \\
$(1,20)$ & $84$ & $ 40 \, [\text{kB}] $ \\
$(1,50)$ & $144$ & $ 169 \, [\text{kB}] $ \\
$(2,10)$ & $2244$ & $ 877 \, [\text{kB}] $ \\
$(3,10)$ & $4247$ &$2324 \, [\text{kB}]$ \\
\end{tabular} \end{center} Notice that the number of regions of \change{standard~Explicit MPC} explode quickly, as soon as one attempts to choose $N \geq 10$ or more than $3$ vehicles. In contrast to this, our approach scales up easily to hundreds of vehicles and very long horizons. Here, only the evaluation of the solution map of~\eqref{eq::qp} depends on $\bar{I}$ and $N$. For example, if we set $\bar{I} = 3$ and $N=30$, we need $10 \, \text{kB}$ to store this map---for larger values this number scales up precisely as predicted by Table~\ref{tab::complexity}.
\begin{figure}\label{fig::active_set}
\end{figure}
\change{Figure~\ref{fig::active_set} shows the total number of active constraints of all distributed QP solvers during the MPC iterations for different choices of $\overline{m}$. In order to visualize the performance of the proposed sub-optimal controller in terms of the number of correctly detected active constraint indices, the number of active constraints of non-suboptimal MPC (corresponding to $\overline{m} = \infty$) are shown in the form of red crosses in Figure~\ref{fig::active_set}. If we compare these optimal red crosses with the blue diamonds, which are obtained for $\overline{m} = 1$, we can see that the choice $\overline{m} = 1$ still leads to many wrongly chosen active sets---especially during the first $10$ MPC iterations. However, for $\overline{m} \geq 10$ a reasonably accurate approximation of the optimal number of active constraints is maintained during all iterations.}
\begin{figure}
\caption{Closed-loop performance degradation (log scale) with
respect to the optimal objective function $J_{\infty}$ as a
function of the number of iterations $\overline{m}$ in Algorithm~\ref{alg::dempc}.}
\label{fig::close}
\end{figure}
Finally, Figure~\ref{fig::close} shows the sub-optimality of Algorithm~1 in dependence on $\overline{m}$ for a representative case study with $\bar I = 3$ and $N=30$---for other values of $\bar I$ and $N$ the convergence looks similar.
\section{Conclusions} \label{sec::conclusions} This paper has introduced a parallelizable and real-time verifiable MPC scheme, presented in the form of Algorithm~\ref{alg::dempc}. This control algorithm evaluates at every sampling time a finite number of pre-computed, explicit piecewise affine solution maps that are associated with parametric small-scale QPs. Because solving large-scale QPs in real-time may be impossible, the presented algorithm returns suboptimal control reaction on purpose---in order to be able to meet challenging real-time and limited memory requirements. The theoretical contributions of this paper have been presented in Theorem~\ref{thm::stability} and Corollary~\ref{cor::performance}, which provide both \change{asymptotic stability} guarantees as well as bounds on sub-optimality. The presented explicit MPC approach can be used to reduce the storage and run-time of explicit MPC by orders of magnitude, as illustrated by applying Algorithm~1 to a spring-mass-vehicle benchmark problem.
\end{ack}
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\appendix
\footnotesize
\section{Proof of Theorem~\ref{thm::convergenceRate}} \label{app::convergenceRate} \change{As the proof of Theorem~\ref{thm::convergenceRate} is not entirely straightforward, we first establish an intermediate results on the convergence properties of Algorithm~\ref{alg::dempc}, which is then, in a second step, used to obtain a convergence rate estimate. Therefore, this proof is divided into two subsections: Subsection~\ref{app::convergenceProperties} analyzes the general convergence properties of Algorithm~\ref{alg::dempc} and Subsection~\ref{app::convergenceRate1} presents a complete proof of Theorem~\ref{thm::convergenceRate} by using these properties.}
\subsection{\change{A closer look at the convergence properties of Algorithm~\ref{alg::dempc}}} \label{app::convergenceProperties}
\change{The goal of the section is to establish the following technical result.} \begin{lemma} \label{lem::convergence} Let Assumption~\ref{ass::blanket} be satisfied and let~\eqref{eq::mpc2} be feasible, i.e., such that a unique minimizer $y^{\star}$ and an associated dual solution $\lambda^{\star}$ exist. Then the iterates of Algorithm~\ref{alg::dempc} satisfy \[ \sum_{m=\hat m}^{\overline{m}} F( \xi^m - y^{\star} ) \leq \frac{ F( y^{\hat m} - y^{\star} ) + F^{\star}( \lambda^{\hat m} - \lambda^{\star} ) }{4} \] for all $\overline{m} \geq \hat m$ and all $\hat m \geq 2$. \end{lemma}
\change{\textbf{Proof.}} Let us introduce the auxiliary functions \begin{equation} \begin{split}\notag \mathcal F_0(\phi_0) \;=\;& F_0(\phi_0) - \left( H_0^\intercal \lambda_{0}^m \right)^\intercal \phi_0^m + \nabla F_0( \xi_0^m - y_0^m )^\intercal \phi_0 \,, \\[0.16cm] \mathcal F_k(\phi_k) \;=\;& F_k(\phi_k) + \left( G_k^\intercal \lambda_{k-1}^{m} - H_k^\intercal \lambda_{k}^m \right)^\intercal \phi_k^m \\ & + \nabla F_k( \xi_k^m - y_k^m )^\intercal \phi_k \; . \end{split} \end{equation} Because $\xi_k^m$ is a minimizer of the $k$-th decoupled QP in Step 2a) of Algorithm~1, it must also be a minimizer of $\mathcal F_k$ on $\mathbb Y_k$. Thus, because $\mathcal F_k$ is strongly convex with Hessian $\nabla^2 F_k$, we must have \[ \sum_{k=0}^N \mathcal F_k(\xi_k^m) + \sum_{k=0}^N F_k(\xi_k^m - y_k^{\star}) \leq \sum_{k=0}^N \mathcal F_k(y_k^{\star})\,. \] On the other hand, due to duality, we have \begin{align} &\sum_{k=0}^N F_k(y_k^{\star}) + \langle \lambda^{\star} , y^{\star} \rangle + \sum_{k=0}^N F_k(\xi_k^m - y_k^{\star}) \notag \\ \leq\;\; &\sum_{k=0}^N F_k(\xi_k^m) + \langle \lambda^{\star} , \xi^m \rangle , \notag \end{align} where the shorthand notation \[ \langle \lambda , y \rangle = - \left( H_0^\intercal \lambda_{0} \right)^\intercal y_0 + \sum_{k=1}^N \left( G_k^\intercal \lambda_{k-1} - H_k^\intercal \lambda_{k} \right)^\intercal y_k \] is used to denote a weighted (non-symmetric) scalar product of primal and dual variables. Adding both inequalities and collecting terms yields \begin{equation}\notag \begin{split} 0 \;\geq\;& \sum_{k=0}^N \nabla F_k( \xi_k^m - y_k^m )^\intercal ( \xi_k^m - y_k^{\star} ) + 2 \sum_{k=0}^N F_k(\xi_k^m - y_k^{\star}) \\[0.16cm] & + \langle \lambda^m - \lambda^{\star}, \xi^m - y^{\star} \rangle \; . \end{split} \end{equation} Let us introduce the matrices \[ \mathcal Q = \nabla^2 \left( \sum_{k=0}^N F_k\right) \;,\;\mathcal A = \nabla_{\lambda,x} \langle \lambda,y \rangle \] such that the above inequality can be written in the form \begin{align} 0\; \geq\;&\;\; (\xi^m - y^m )^\intercal \mathcal Q ( \xi^m - y^{\star} ) + 2 \sum_{k=0}^N F_k(\xi_k^m - y_k^{\star}) \notag \\[0.16cm] & \;\;+ \left( \lambda^m - \lambda^{\star} \right)^\intercal \mathcal A \left( \xi^m - y^{\star} \right) \; . \label{eq::aux1} \end{align} Similarly, the stationarity condition QP~\eqref{eq::qp} can be written as \[ \mathcal Q( y^{m+1} - 2 \xi^m + y^m ) + \mathcal A^\intercal \delta^m = 0 \; . \] Because $\mathcal Q$ is positive definite, we solve this equation with respect to $\xi^m$ finding \begin{equation} \label{eq::QPopt} \xi^m = \frac{1}{2} \mathcal Q^{-1} \mathcal A^\intercal (\lambda^{m+1} - \lambda^m) + \frac{y^m + y^{m+1}}{2}\;. \end{equation} Here, we have additionally substituted the relation $$\delta^m = \lambda^{m+1} - \lambda^m \; .$$ Notice that we have $\mathcal A y^m = \mathcal A y^{m+1} = \mathcal A y^{\star}$ for all $m \geq 2$, because the solutions of the QP~\eqref{eq::qp} must satisfy the equality constraints in~\eqref{eq::mpc2}. If we substitute this equation and the expression for $\xi^m$ in~\eqref{eq::aux1}, we find that \begin{align}\notag &\;- 2 F(\xi^m - y^{\star}) \\\notag \geq &\;\; (\xi^m - y^m )^\intercal \mathcal Q ( \xi^m - y^{\star} ) + \left( \lambda^m - \lambda^{\star} \right)^\intercal \mathcal A \left( \xi^m - y^{\star} \right) \\[0.16cm]\notag =&\;\; \frac{1}{4}(\lambda^{m+1} - \lambda^m)^\intercal \mathcal A \mathcal Q^{-1} \mathcal A^\intercal (\lambda^{m+1} - \lambda^m) \\[0.16cm] \label{eq::descent} & \;\;+ \frac{1}{4}(y^{m+1} - y^{m}) \mathcal Q ( y^m - 2y^{\star} + y^{m+1} ) \\[0.16cm]\notag &\;\; + \frac{1}{2}(\lambda^{m} - \lambda^{\star})^\intercal \mathcal A \mathcal Q^{-1} \mathcal A^\intercal (\lambda^{m+1} - \lambda^m)\\[0.16cm]\notag =& \;\;\frac{1}{2} \left( F(y^{m+1} - y^{\star} ) - F(y^{m} - y^{\star} ) \right) \\[0.16cm]\notag & \;\;+ \frac{1}{2} \left( F^{\star}( \lambda^{m+1} - \lambda^{\star} ) - F^{\star}( \lambda^{m} - \lambda^{\star} ) \right) \end{align} for all $m \geq 2$. Now, the statement of \change{Lemma~\ref{lem::convergence}} follows by summing up the above inequalities for $m=\hat m$ to $m=\overline{m}$ and using that the last element in the telescoping sum on the right hand, \[ \frac{F(y^{\overline{m}+1} - y^{\star} ) + F^{\star}( \lambda^{\overline{m}+1} - \lambda^{\star} )}{2} \geq 0 \] is non-negative.
\subsection{Analysis of the convergence rate of Algorithm~\ref{alg::dempc}} \label{app::convergenceRate1}
\change{The goal of this section is to prove the statement of Theorem~\ref{thm::convergenceRate} by using the intermediate result from the previous section.} Let $\hat{\mathbb Y}_k$ denote the intersection of all active supporting hyperplanes at the solutions of the small scale QPs of Step 2a) in Algorithm~\ref{alg::dempc} for $k \in \{ 0, \ldots, N\}$ at a given iteration $m$. We construct the auxiliary optimization problem\\[-0.45cm] \begin{align} \label{eq::mpc2Aux}\notag \underset{\hat y}{\min}&\;\; \sum_{k=0}^{N} F_k( \hat y_k ) \\[0.16cm] \text{s.t.} & \left\{ \begin{array}{l} \forall k \in \{ 0, \ldots, N-1 \}, \\ \begin{array}{rcll} G_{k+1} \hat y_{k+1} &=& H_k \hat y_k + h_k \; & \mid \; \hat \lambda_{k} \; , \\ 0 &=& H_N \hat y_N \; & \mid \; \hat \lambda_{N} \; , \end{array} \\ \hat y_k \in \hat{\mathbb Y}_k \; , \; \hat y_N \in \hat{\mathbb Y}_N \; , \end{array} \right. \end{align} and denote optimal primal and dual solutions of this problem by $\hat y^{\star}$ and $\hat \lambda^{\star}$. Next, we also construct the auxiliary QPs \begin{align} \underset{\xi_0^m \in \hat{\mathbb Y}_0}{\min} & \; F_0(\xi_0^m) - \left( H_0^\intercal \lambda_{0}^m \right)^\intercal \xi_0^m + F_0( \xi_0^m - y_0^m ) , \notag \\[0.2cm] \underset{\xi_k^m \in \hat{\mathbb Y}_k}{\min} & \; F_k(\xi_k^m) + \left( G_k^\intercal \lambda_{k-1}^m - H_k^\intercal \lambda_{k}^m \right)^\intercal \xi_k^m + F_k( \xi_k^m - y_k^m ) \; . \notag \end{align} Because these QPs have equality constraints only, their parametric solutions must be affine. Thus, there exists a matrix $T_1$ such that \[ \xi^m - \hat y^{\star} = T_1 \left( \begin{array}{c} y^m - \hat y^{\star} \\ \lambda^m - \hat \lambda^{\star} \end{array} \right). \] Similarly, the coupled QP~\eqref{eq::qp} has equality constraints only, i.e., there exists a matrix $T_2$ such that \[ \left( \begin{array}{c} y^{m+1} - \hat y^{\star} \\ \delta^m \\ \end{array} \right) = T_2 \left( \begin{array}{c} \xi^m - \hat y^{\star} \\ y^m - \hat y^{\star} \end{array} \right). \] Now, we use the equation $\lambda^{m+1} - \lambda^{\star} = \lambda^m - \lambda^{\star} + \delta$ and substitute the above equations finding that \begin{equation} \left( \begin{array}{c} y^{m+1} - \hat y^{\star} \\ \lambda^{m+1} - \hat \lambda^{\star} \end{array} \right) = T \left( \begin{array}{c} y^{m} - \hat y^{\star} \\ \lambda^{m} - \hat \lambda^{\star} \end{array} \right) \label{eq::linearOperator} \end{equation} with \[ \quad T = \left( \begin{array}{c} T_2 \left( \begin{array}{c} T_1 \\ (I \;\; 0 ) \end{array} \right) + (0 \;\; I) \end{array} \right) \; . \] Next, we know from \change{Lemma~\ref{lem::convergence}} that if we would apply Algorithm~\ref{alg::dempc} to the auxiliary problem~\eqref{eq::mpc2Aux}, the corresponding primal and dual iterates would converge to $\hat y^{\star}$ and $\hat \lambda^{\star}$. In particular, inequality~\eqref{eq::descent} from the proof of \change{Lemma~\ref{lem::convergence}} can be applied finding that \begin{equation} \begin{array}{cl} &\left( y^{m+1} - \hat y^{\star} \right)^\intercal \mathcal Q \left( y^{m+1} - \hat y^{\star} \right) \\ & + \left( \lambda^{m+1} - \hat \lambda^{\star} \right)^\intercal \mathcal{A} \mathcal{Q}^{-1} \mathcal{A}^\intercal \left( \lambda^{m+1} - \hat \lambda^{\star} \right) \\ <& \left( y^{m} - \hat y^{\star} \right)^\intercal \mathcal Q \left( y^{m} - \hat y^{\star} \right) \\ & + \left( \lambda^{m} - \hat \lambda^{\star} \right)^\intercal \mathcal A Q^{-1} \mathcal A^\intercal \left( \lambda^{m} - \hat \lambda^{\star} \right) \; , \end{array} \end{equation} whenever $\left( \begin{array}{c} y^{m} - \hat y^{\star}\\ \lambda^{m} - \hat \lambda^{\star} \end{array} \right) \neq 0$. By substituting the linear equation~\eqref{eq::linearOperator}, we find that this is only possible if \[ T^\intercal \left( \begin{array}{cc} \mathcal Q & 0 \\ 0 & \mathcal A Q^{-1} \mathcal A^\intercal \end{array} \right) T \preceq \kappa_{\mathbb A} I \] for a constant $\kappa_{\mathbb A} < 1$. Now, one remaining difficulty is that the constant $\kappa_{\mathbb A}$ (as well as the matrix $T$) depends on the particular set $\mathbb A$ of active supporting hyperplanes in the small-scale QPs. Nevertheless, because there exists only a finite number of possible active sets, the maximum \[ \kappa = \max_{\mathbb A} \kappa_{\mathbb A} \] must exist and satisfy $\kappa < 1$. Now, the equation \begin{equation} \left( \begin{array}{c} y^{m+1} - y^{\star} \\ \lambda^{m+1} - \lambda^{\star} \end{array} \right) = T \left( \begin{array}{c} y^{m} - y^{\star} \\ \lambda^{m} - \lambda^{\star} \end{array} \right) \end{equation} holds only for our fixed $m$ and the associated matrix $T$ for a particular active set, but the associated decent condition \begin{equation} \begin{array}{l} \quad\left( y^{m+1} - y^{\star} \right)^\intercal \mathcal Q \left( y^{m+1} - y^{\star} \right) \\ \quad + \left( \lambda^{m+1} - \lambda^{\star} \right)^\intercal \mathcal A Q^{-1} \mathcal A^\intercal \left( \lambda^{m+1} - \lambda^{\star} \right) \\ \leq \kappa \left[ \left( y^{m} - y^{\star} \right)^\intercal \mathcal Q \left( y^{m} - y^{\star} \right) \right. \\ \left. \quad + \left( \lambda^{m} - \lambda^{\star} \right)^\intercal \mathcal A Q^{-1} \mathcal A^\intercal \left( \lambda^{m} - \lambda^{\star} \right) \right] \; , \end{array} \end{equation} holds independently of the active set of the QPs in the $m$-th iteration and is indeed valid for all $m$. After re-introducing the functions $F$ and $F^{\star}$, we obtain the statement of the theorem.
\color{black}
\section{Proof of Proposition~\ref{prop::feasibility}} \label{app::feasibility} Because $\xi_0^{\overline{m}}$ is a feasible solution of the first small-scale decoupled QP, we have $\xi_0^{\overline{m}} \in \mathbb Y_0$. Notice that such a feasible solution exists due to the particular construction of the set $\mathbb Y_0$ and our assumption that the set $\mathbb X$ is control invariant. Consequently, the next iterate for the state, \[ x_0^+ = A x_0 + (B \;\; C) \xi_0^{\overline{m}} \] must satisfy $x_0^+ \in \mathbb X$ by construction. This is the first statement of Proposition~\ref{prop::feasibility}. In order to establish the second statement, we observe that the particular construction of the sets $\mathbb Y_k$ implies that there exists a feasible point of~\eqref{eq::mpc2} for any choice of $x_0 \in \mathbb X$, because we choose $\mathbb X_N = \mathbb X$, but $\mathbb X$ is control invariant. Thus, we must have $\mathbb X \subseteq \mathcal X$. The other way around, if $x_0 \notin \mathbb X$ the state constraints are violated, i.e., we also have $\mathcal X \subseteq \mathbb X$. Thus, we have $\mathcal X = \mathbb X$ and, consequently, $\mathcal X$ is a control invariant set, too. This is sufficient to establish recursive feasibility of Algorithm~\ref{alg::dempc}. \color{black}
\section{Proof of Theorem~\ref{thm::stability}} \label{app::stability}
\change{Recall that stability proofs for standard MPC proceed by using the inequality \begin{equation} \label{eq::MPC_Lyaponov} J(x_1^{\star}) \leq J(x_0) - F_0( y_0^{\star} ) \; , \end{equation} which holds if Assumption~\ref{ass::TerminalLyapunov} is satisfied (see, e.g.,~\cite{Gruene2009} for a derivation of this inequality), i.e., $J$ is a global Lyapunov function on $\mathcal X$. Now, if we implement Algorithm~\ref{alg::dempc} with a finite $\overline{m}$, we have \begin{equation} \label{eq::lyapD1} J(x_0^+) \leq J(x_0) - \left( F_0( y_0^{\star} ) - J(x_0^+) + J(x_1^{\star}) \right) \; , \end{equation} i.e., $J$ can still be used as a Lyaponov function proving asymptotic stability as long as we ensure that \begin{equation} \label{eq::lyapD2} F_0( y_0^{\star} ) - J(x_0^+) + J(x_1^{\star}) \geq \alpha F_0( y_0^{\star} ) \; , \end{equation} for a constant $\alpha > 0$. In order to show that such an inequality can indeed be established for a finite $\overline{m}$, we need the following technical result.}
\begin{lemma} \label{lem::bound} \change{The iterate $x_0^+$ satisfies the inequality} \[ \left( x_0^+ - x_1^{\star} \right)^\intercal Q \left( x_0^+ - x_1^{\star} \right) \leq \sigma \gamma^2 ( 1 + \sqrt{\kappa})^2 \kappa^{\overline{m}} F_0( y_0^{\star} ) \; . \] \end{lemma}
\textbf{Proof.} We start with the equation \[ x_0^+ - x_1^{\star} = (B\;\;C) ( \xi_1^{\overline{m}} - y_0^{\star} ) = \mathcal P \mathcal A \left( \xi^{\overline{m}} - y^{\star} \right) \; , \] which holds for the projection matrix $\mathcal P = \mathrm{diag}(I , 0 ,$ $\ldots, 0 )$ that filters out the first block component of the equality constraint residuum $\mathcal A \left( \xi^{\overline{m}} - y^{\star} \right)$. Next, we substitute~\eqref{eq::QPopt}, which yields \begin{align} & \;x_0^+ - x_1^{\star} \notag \\ =& \;\mathcal P \mathcal A \left[ \frac{\mathcal Q^{-1} \mathcal A^\intercal (\lambda^{{\overline{m}}+1} - \lambda^{\overline{m}})}{2} + \frac{y^{\overline{m}+1} + y^{{\overline{m}}}}{2} - y^{\star} \right] \notag \\ =&\; \frac{1}{2} \mathcal P \mathcal A \mathcal Q^{-1} \mathcal A^\intercal (\lambda^{{\overline{m}}+1} - \lambda^{\overline{m}}) \; , \end{align} where we have used that $0 = \mathcal A( y^{\overline{m}} - y^{\star} )$ and $0 = \mathcal A( y^{{\overline{m}}+1} - y^{\star} )$ recalling that these equations follow from the equality constraints in the coupled QP in Step 2b) of Algorithm~\ref{alg::dempc}. The particular definition of $\sigma$ implies that \begin{align} &4 \left( x_0^+ - x_1^{\star} \right)^\intercal Q \left( x_0^+ - x_1^{\star} \right) \notag \\[0.16cm] \leq& (\lambda^{{\overline{m}}+1} - \lambda^{\overline{m}})^\intercal \mathcal A \mathcal Q^{-1} \underbrace{\mathcal A^\intercal \mathcal P^\intercal Q \mathcal P \mathcal A}_{\preceq \sigma \mathcal Q} \mathcal Q^{-1} \mathcal A^\intercal (\lambda^{{\overline{m}}+1} - \lambda^{\overline{m}}) \notag \\[0.16cm] \leq& \sigma (\lambda^{{\overline{m}}+1} - \lambda^{\overline{m}})^\intercal \mathcal A \mathcal Q^{-1} \mathcal A^\intercal (\lambda^{{\overline{m}}+1} - \lambda^{\overline{m}}) \; . \notag \end{align} Now, we can use the result of Theorem~\ref{thm::convergenceRate} to find \begin{align} &\quad (\lambda^{{\overline{m}}+1} - \lambda^{\overline{m}})^\intercal \mathcal A \mathcal Q^{-1} \mathcal A^\intercal (\lambda^{{\overline{m}}+1} - \lambda^{\overline{m}}) \notag \\ &\leq F^{\star}(\lambda^{{\overline{m}}+1}-\lambda^{\star}) + F^{\star}(\lambda^{{\overline{m}}}-\lambda^{\star}) \notag \\ & \quad + 2 \sqrt{F^{\star}(\lambda^{{\overline{m}}+1}-\lambda^{\star}) F^{\star}(\lambda^{{\overline{m}}}-\lambda^{\star}) } \notag \\ &\leq \kappa^{\overline{m}} ( 1 + \sqrt{\kappa})^2 \left( F(y^{1}-y^{\star}) + F^{\star}(\lambda^{1}-\lambda^{\star}) \right) \; . \notag \end{align} It remains to use the inequalities \begin{align} F(y^{\star}) + F^{\star}(\lambda^{\star}) &\leq \gamma^2 x_0^\intercal Q x_0 \leq \gamma^2 F_0( y_0^{\star} )\;, \notag \\ F(y^1) + F^{\star}(\lambda^1) &\leq \gamma^2 x_0^\intercal Q x_0 \leq \gamma^2 F_0( y_0^{\star} ) \; , \notag \end{align} which hold due to Assumption~\ref{ass::gamma} and the particular construction in Step~1 of Algorithm~\ref{alg::dempc}, arriving at the inequality \begin{align}\notag &\;\;4 \left( x_0^+ - x_1^{\star} \right)^\intercal Q \left( x_0^+ - x_1^{\star} \right) \\\notag \leq &\;\;\sigma (\lambda^{{\overline{m}}+1} - \lambda^{\overline{m}})^\intercal \mathcal A \mathcal Q^{-1} \mathcal A^\intercal (\lambda^{{\overline{m}}+1} - \lambda^{\overline{m}}) \\\notag \leq &\;\;\sigma \kappa^{\overline{m}} ( 1 + \sqrt{\kappa})^2 \left( F(y^{1}-y^{\star}) + F^{\star}(\lambda^{1}-\lambda^{\star}) \right) \\\notag \leq&\;\; 4 \sigma \kappa^{\overline{m}} ( 1 + \sqrt{\kappa})^2 \gamma^2 F_0{y_0^{\star}} \; . \end{align} Dividing by $4$ on both sides yields the statement of the lemma.
\qed
\change{By combining the inequality from the above lemma and inequality~\eqref{eq::etaTauBound} we find \begin{align} \label{eq::auxA}
&\left|J(x_0^+) - J(x_1^{\star})\right| \\\notag \leq& \left[ \eta \sqrt{\sigma} \gamma (1+\sqrt{\kappa}) + \frac{\tau \sigma \gamma^2 (1+\sqrt{\kappa})^2}{2} \right] \kappa^{\frac{\overline{m}}{2}} F_0( y_0^{\star} ) \; . \end{align} Now, the statement of Theorem~\ref{thm::stability} follows directly from~\eqref{eq::auxA}, as the construction of $\overline{m}$ ensures that the Lyapunov descent condition~\eqref{eq::lyapD2} holds on $\mathcal X$ with \[ \alpha = 1 - \left[ \eta \sqrt{\sigma} \gamma (1+\sqrt{\kappa}) + \frac{\tau \sigma \gamma^2 (1+\sqrt{\kappa})^2}{2} \right] \kappa^{\frac{\overline{m}}{2}} > 0 \; , \] which is sufficient to establish asymptotic stability.}
\end{document} | arXiv |
Application of Bayesian networks to GAW20 genetic and blood lipid data
Richard A. J. Howey1 and
Heather J. Cordell1Email author
BMC Proceedings201812 (Suppl 9) :19
Bayesian networks have been proposed as a way to identify possible causal relationships between measured variables based on their conditional dependencies and independencies. We explored the use of Bayesian network analyses applied to the GAW20 data to identify possible causal relationships between differential methylation of cytosine-phosphate-guanine dinucleotides (CpGs), single-nucleotide polymorphisms (SNPs), and blood lipid trait (triglycerides [TGs]).
After initial exploratory linear regression analyses, 2 Bayesian networks analyses were performed. First, we used the real data and modeled the effects of 4 CpGs previously found to be associated with TGs in the Genetics of Lipid Lowering Drugs and Diet Network Study (GOLDN). Second, we used the simulated data and modeled the effect of a fictional lipid modifying drug with 5 known causal SNPs and 5 corresponding CpGs.
In the real data we show that relationships are present between the CpGs, TGs, and other variables—age, sex, and center. In the simulated data, we show, using linear regression, that no CpGs and only 1 SNP were associated with a change in TG levels, and, using Bayesian network analysis, that relationships are present between the change in TG levels and most SNPs, but not with CpGs.
Even when the causal relationships between variables are known, as with the simulated data, if the relationships are not strong then it is challenging to reproduce them in a Bayesian network.
Genome-wide association studies (GWAS) have been very successful at detecting genetic variants (typically single-nucleotide polymorphisms [SNPs]) associated with phenotypic outcomes. A typical approach to understanding the identified relationships between phenotype and associated genetic factors is to use public databases to see if the observed association can be explained by gene expression or DNA methylation patterns in tissue types relevant to the phenotype in question. However, public databases contain measurements made in different individuals compared to those used in the GWAS analysis, possibly even measured a different species. Consequently, there is interest in using causal inference methods applied to measurements of potential intermediate variables (such as gene expression or DNA methylation) taken in the same set of individuals as are included in the GWAS data set, allowing more direct conclusions about causality to be made. With the increase in different data types comes the desire to model more complex causal relationships beyond using just 2 or 3 variables at a time. This is possible with the use of Bayesian networks, where many variables can be modeled simultaneously in an exploratory manner, providing a natural extension to 3-variable causal modeling. In a recent study, Ainsworth et al. [1] compared Bayesian networks with other causal inference methods in the 3-variable situation, and found the Bayesian networks to perform competitively. We here attempt to gain insight into the conditional dependencies between the variables in the GAW20 data set by fitting Bayesian networks (separately) to the GAW20 real and simulated data. The GAW20 real data are based on a previous study into the association between differential methylation of cytosine-phosphate-guanine dinucleotides (CpGs) and the blood lipid trait, triglycerides (TGs), which study found a region of the epigenome with 4 CpGs significantly associated with TGs. The GAW20 simulated data model the effect of a fictional drug that affects TGs via both SNP and CpG effects, with methylation of the corresponding CpG site modifying the effect of the SNP on TG levels. These analyses were performed with knowledge of the GAW20 "answers."
Real data
The GAW20 real data [2] consisted of phenotype and covariate data before and after fenofibrate drug treatment for 3 weeks. Individuals had measurements taken at 4 visits: visits 1 and 2 before treatment and visits 3 and 4 after treatment. Methylation measurements on CpGs were taken at visits 2 and 4. In the Genetics of Lipid Lowering Drugs and Diet Network (GOLDN) study on which the GAW20 data was based, Irvin et al. [3] performed an epigenome-wide association study (EWAS) and found 4 CpGs in the same region of the epigenome that were significantly associated with TGs.
We performed a similar EWAS to show that these 4 CpGs are significantly associated with TGs in the GAW20 data at visits 2 and 4. From a total of 1105 individuals, 995 had methylation data at visit 2 and 530 at visit 4. We used linear regression of the logged TG levels (as TGs are approximately log-normally distributed), and included covariates for the age, sex, and center (Minneapolis or Salt Lake City):
$$ \log (TG)={\beta}_0+{\beta}_1 CPG+{\beta}_2 age+{\beta}_3{I}_1(center)+{\beta}_4{I}_2(sex)+{\beta}_5 pc1+{\beta}_6 pc2+{\beta}_7 pc3+{\beta}_8 pc4+\epsilon $$
where CPG is the methylation of the CpG being tested and ϵ is a random error. The βis are regression coefficients and Ijs are indicator functions for the two discrete variables. We included 4 principal components based on the methylation data to account for potential biases such as batch effects. We used the R software package [4] to perform the tests, and did not account for family structure (relatedness between individuals) as obtaining accurate P values for discovery was not the main aim of our analysis.
We then used the data from the 4 CpGs that we and Irvin et al. [3] found to be significantly associated with TGs to fit a Bayesian network. We used the CpG data taken at visit 2 (as this had a larger number of measurements than data taken at visit 4) and data on age, sex, and center. Following preliminary GWAS analysis between SNPs and CpGs, and between SNPs and logged TG levels, we did not find any convincing associations; consequently, we did not include any SNPs in our Bayesian network analysis. No CpGs at visit 2 (or visit 4) were associated with change in TG levels as a result of drug treatment, so, in contrast to the GAW simulated data analysis (described later), we did not fit a Bayesian network modeling change in TG levels (ie, TG levels after treatment, with TG levels before treatment included as a covariate) as an outcome.
We implemented the Bayesian network method given by Scutari and Denis [5], which was chosen as being the most appropriate for mixed discrete and continuous data. We used our own C++ implementation, BayesNetty [6], with a hill-climbing algorithm, random restarts, and the Bayesian information criterion for model selection. Categorical variables, sex and center, are automatically constrained to have no parents in the Bayesian network analysis. An "average network" was also calculated by finding the best-fit model 1000 times using bootstrapped data. The strength of an edge was then given by the proportion of networks where it was present in either direction. The direction of the edge was given by the proportion of times it was in a given direction when present. The average network provides an estimate of the direction of causality between variables. A strength threshold was applied to network when it was plotted so that only edges that are considered of interest are plotted. The networks were drawn using the igraph [7] R package.
Simulated data
The GAW20 simulated data was designed to model the effect of a fictional drug on TG levels. The data was only simulated for visits 3 and 4, with the real data at visits 1 and 2 forming the basis for the simulated data. We viewed the documentation for the simulation that indicated there were 5 causal SNPs, each with one nearby corresponding CpG, that were used to simulate change in TG levels between drug treatments. The simulation method used CpG data at visit 4 to determine the change in TG levels; consequently, we chose to use visit 4 CpG data in our analyses. We analyzed simulated data replicate number 84 as suggested by the GAW20 organizers as the best representative replicate.
For our analysis, the SNP data was restricted to SNPs with a minor allele frequency greater than 0.01 and the CpG data was left unmodified. We attempted to find SNPs associated with outcome using FaST-LMM (Factored Spectrally Transformed Linear Mixed Model) [8] to account for family structure via the following mixed model:
$$ \log (TG4)={\beta}_0+{\beta}_1 TG2+{\beta}_2 SNP+{\beta}_3 age+{\beta}_4{I}_1(center)+{\beta}_5{I}_2(sex)+\epsilon $$
where ϵ is the random error, structured to account for estimated relatedness, the βis are regression coefficients and Ijs are indicator functions for the two discrete variables. The TG levels at visits 2 and 4 are given by TG2 and TG4. By including TG2 as a covariate, we effectively test for association with the change in TG levels between visits 2 and 4. The SNP data, SNP, are given by the number of minor alleles, 0, 1, or 2.
An EWAS to detect CpGs associated with the change in TG levels was also performed as follows:
$$ {\displaystyle \begin{array}{c}\log (TG4)={\beta}_0+{\beta}_1 TG2+{\beta}_2 CPG4+{\beta}_3 age+{\beta}_4{I}_1(center)+{\beta}_5{I}_2(sex)\\ {}+{\beta}_6 pc1+{\beta}_7 pc2+{\beta}_8 pc3+{\beta}_9 pc4+\epsilon \end{array}} $$
where CPG4 is the CpG level at visit 4 and other coefficients and variables are as previously. A Bayesian network was fitted to the 5 causal SNPs and the 5 causal CpGs together with variables for age, sex, center, and TG levels at visits 2 and 4. We obtained the best-fit network as well as calculating an average network using the same methods as before. The fitting of the Bayesian networks was constrained such that TG2 was a parent of TG4. With this constraint, the change in TG levels between visits 2 and 4 can be modeled. Also, SNPs were constrained to have no parents and CpG data at visit 4 could not be parents of TG2.
Figure 1 shows the EWAS results from the GAW20 real data at visits 2 and 4 and Table 1 shows the p values of the 4 CpGs found by Irvin et al. [3]. The Bonferroni corrected threshold is p = 1.08 × 10− 7, and at visit 2 and visit 4 there are 4 and 2 CpGs meeting this significance threshold, respectively. The differing sample sizes at visit 2 (995) and visit 4 (530) may contribute to these differences. The family structure was not accounted for in our analysis, but nevertheless, the test results were not unduly inflated (quantile–quantile [Q-Q] plots not shown), with genomic control inflation factors of 0.956 at visit 2 and 1.08 at visit 4.
EWAS Manhattan plots for TG with age, center, sex, and 4 principal components as covariates using methylation data from visits 2 and 4. Stars indicate the 4 candidate CpGs
The 4 CpGs identified by Irvin et al. [3] and their p values from each EWAS on the GAW20 real data at visit 2 and visit 4
Visit 2 p value
cg00574958
6.11 × 10− 33
3.10 × 10−21
3.16 × 10− 6
The best-fit Bayesian network shown in Fig. 2a shows connections between all the variables for the GAW20 real data at visit 2. In particular, the CpGs are strongly associated with one another, as would be expected, as they are close to one another on the epigenome and have similar EWAS results. Age and sex, as well as CpG cg09737197, are shown to directly influence TG level.
Networks of candidate CpGs in the GAW20 data at visit 4 together with variables for TGs, age, sex, and center. Circles and rectangles show continuous and discrete data respectively. a Best-fit Bayesian network. The thickness of the lines show the relative significance of the arrows. b Average Bayesian network. The thickness of the lines show the relative strength of the arrows; numbers in red show the (probability of) direction of the arrows
The average Bayesian network shown in Fig. 2b provides a better estimate of the direction of causality between variables. The line thickness of each arrow indicates the strength (probability) that the edge appears in the graph at all (in either direction), and the probability of the specified causal direction, given that the edge exists at all, is given by the number displayed in red on each arrow. Values near 0.5 show that the direction of causality is equally likely in either direction and may reflect correlation rather than implying causality. Although we may expect the CpGs to be associated with one another, we would not necessarily expect to be able to identify a causal relationship between them (given that no SNPs have been included as "genetic instruments"), and this is reflected in that most of the direction probabilities are close to 0.5 (specifically 0.51, 0.51, 0.52, and 0.55), although 0.7 and 0.71 between cg01082498 and two other CpGs is more indicative of a causal relationship than might be expected. Age has direction probabilities of 0.82, 0.88, 0.91, and 0.95 to the CpGs, suggesting a causal relationship, which is intuitive as age should affect methylation rather than vice versa. A possible argument that methylation could affect age is that the sample of individuals is biased with regard to methylation levels and age, for example, if individuals who are old are only sampled if they have particularly high methylation levels (for whatever reason). This would reflect causation in the sample rather than in the population. The direction of causality between methylation and TG level is not strong in either direction, with probabilities of 0.52 and 0.58 from CpGs to TG. Indeed, Sayols-Baixeras et al. [9] found evidence of causality between methylation and TGs going in either direction using the GAW20 data.
Figure 3 shows plots of the results of the GWAS and EWAS. Q-Q plots of the results (not shown) did not show any signs of inflation with genomic control inflation factors of 1.004 for the GWAS and 0.996 for the EWAS. Only 1 SNP passed the Bonferroni corrected threshold for significance (p = 7.67 × 10− 8) and no CpGs were found to be significant from the EWAS. Table 2 shows the results for the 5 "known" causal SNPs and 5 corresponding CpGs together with their simulated theoretical expected heritabilities at stage 3 of the simulation, which, in the absence of any epigenetic effects, reflects the SNP effect sizes in relation to individual drug response. Given these relatively small effects, and that CpGs operate not through additional main effects but through modifying the effect of the corresponding SNP, it is perhaps not surprising that only 1 SNP and no CpGs were found to be significant. An alternative explanation could be the presence of unaccounted for confounding factors; however, the detailed documentation for the data simulation provided in the GAW "Answers" suggests that there were no additional confounding factors to be accounted for.
GWAS and EWAS of GAW20 simulated data for TG levels at visit 2, with age, sex, and center as covariates
The 5 SNPs and corresponding CpGs that were used to simulate change in TG levels between drug treatment in the GAW20 simulated data with their simulated theoretical expected heritabilities and their GWAS and EWAS p values
Heritability
SNP p value
CpG p value
rs9661059
1.08 × 10−8
rs10828412
Figure 4a shows the best-fit Bayesian network and largely reflects the GWAS and EWAS results, such that most SNPs are related to a change in TG levels, but the CpGs are not. The only corresponding SNP and CpG connected to one another are rs1012116 and cg18772399. The CpGs are connected to one another, despite being randomly chosen across the epigenome on different chromosomes. This most probably reflects that different individuals tend to show similar levels of methylation across the whole epigenome, rather than any other interesting characteristics related to the drug-response simulation.
Networks of GAW20 simulated data of causal SNPs and CpGs at visit 4 together with variables for TGs, age, sex, and center. Circles and rectangles show continuous and discrete data respectively. a Best-fit Bayesian network. The thickness of the lines show the relative significance of the arrows. b Average Bayesian network. The thickness of the lines show the relative strength of the arrows; numbers in red show the (probability of) direction of the arrows
Figure 4b shows the average Bayesian network and very similar results to the best-fit network but with fewer arrows. After the strength threshold of 0.441 is applied to the average network, the arrow showing SNP rs4399565 relating to the change in TG levels is no longer plotted, highlighting the weak association. The strengths of edges (probability of a relationship going in either direction) from rs9661059, rs736004, rs1012116, rs10828412, and rs4399565 to the change in TG levels are 0.538, 0.654, 0.640, 0.394, and 0.441, respectively. It was suggested at the GAW20 workshop that, given the nature of the simulated data, variables for the interaction of SNPs and their corresponding CpG may give stronger associations with change in TG levels than are seen when modeling main effects of SNPs and CpGs. However, further investigation indicated that including such variables did not improve the levels of association detected (results not shown). The direction value of the arrows highlights the constraints, such that the arrow must always be in the shown direction if it is equal to 1. The direction value between CpGs are not too close to 1, showing there is not strong evidence for a causal relationship in one direction.
A simple EWAS of the GAW20 real data showed that the 4 CpGs previously detected by Irvin et al. [3] as associated with TGs, were also associated in the GAW20 real data. This association and the high correlation between CpGs resulted in a fitted Bayesian network that showed TG level to be dependent directly or indirectly on all the other variables.
The GAW20 simulated data presented more difficulties than the real data. From the GAW20 solutions it was known in advance that 5 SNPs and 5 corresponding CpGs were used to simulate change in TG. However, a simple GWAS and a simple EWAS only detected one of the SNPs. This can most probably be explained by the small effect sizes and small sample size of the data set, given that the 1 SNP detected had the largest effect size. Despite the complex nature of the simulated data and the weak association results, we did see some relationships between SNPs and a change in TG levels.
There are many benefits to the use of Bayesian networks. A particular benefit is the identification of previously overlooked possible causal relationships between variables in a biological system. Although not a rigorous test of causality, they form a useful additional technique to help direct further hypotheses about the system, as well as future studies and analyses. Visualization of Bayesian networks is a useful tool when there are many different variables operating within a system to aid the identification of interesting possible causal structures.
Bayesian networks do have some drawbacks, such as needing to search through a potentially large network space to find the best-fit network. The processing time for this can be improved by reducing the network space by imposing constraints between some variables and/or by the use of parallel computing. The optimality of the procedure can be improved with the use of random restarts and the development of different search algorithms.
The GAW20 real data showed stronger associations between variables than the GAW20 simulated data, resulting in a better-connected, fitted Bayesian network. Despite some difficulties, Bayesian networks provide a further tool beyond detecting individual significant associations and may aid better understanding of biological systems to ultimately inform drug development.
This work was supported by the Wellcome Trust (087436/Z/08/Z and 102858/Z/13/Z).
Publication of this article was supported by NIH R01 GM031575. Funding also came from the Wellcome Trust, grant references 087436/Z/08/Z and 102858/Z/13/Z.
The data that support the findings of this study are available from the Genetic Analysis Workshop (GAW), but restrictions apply to the availability of these data, which were used under license for the current study. Qualified researchers may request these data directly from GAW. The BayesNetty software is available at: http://www.staff.ncl.ac.uk/richard.howey/bayesnetty/
RH conducted statistical analyses and drafted the manuscript. HJC conceived the overall study and critically revised the manuscript. Both authors read and approved the final manuscript.
Institute of Genetic Medicine, Newcastle University, Central Parkway, Newcastle upon Tyne, NE1 3BZ, UK
Ainsworth HF, Shin S-Y, Cordell HJ. A comparison of methods for inferring causal relationships between genotype and phenotype using additional biological measurements. Genet Epidemiol. 2017; https://doi.org/10.1002/gepi.22061. [Epub ahead of print]View ArticleGoogle Scholar
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Lippert C, Listgarten J, Liu Y, Kadie CM, Davidson RI, Heckerman D. FaST linear mixed models for genome-wide association studies. Nat Methods. 2011;8(10):833–5.View ArticleGoogle Scholar
Sayols-Baixeras S, Tiwari HK, Aslibekyan SW. Disentangling associations between DNA methylation and blood lipids: a Mendelian randomization approach. BMC Proc. 2018;12(Suppl 9) https://doi.org/10.1186/s12919-018-0119-8. | CommonCrawl |
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The Quantum Group $SL_q(2)$ and its coaction on the Quantum Plane
The algebra $SL(2)$ and its coproduct.
Recall our earlier discussion about a universal group structure on algebras. In particular, consider
$\displaystyle \text{Hom}_\text{Alg}(k[a,b,c,d], A) \cong A^4 \cong M_2(A)$
as a vector space. Let $M(2)$ denote the polynomial algebra $k[a,b,c,d]$. Last time we pulled back addition on $A$ to a map $\Delta k[x] \rightarrow k[x] \otimes k[x]$. This time, we're going to follow the same pattern to take matrix multiplication (a map $ m : M_4(A) \otimes M_2(A) \rightarrow M_2(A)$) back to $\Delta M(2) \rightarrow M(2) \otimes M(2)$.
In particular, we're looking for a $\Delta$ so that when we have $\alpha \in \text{Hom}_\text{Alg}(M_2(A) \otimes M_2(A), A)$, $\alpha \circ \Delta = m (\alpha)$
The above notation is slightly confusing, let me try to explain more clearly: We have that the matrix algebra $M_2(A)$ is isomorphic as a vector space to $\text{Hom}_\text{Alg}(M(2),A)$ by the map taking
$\displaystyle f \mapsto \begin{pmatrix} f(a) & f(b) \\ f(c) & f(d) \end{pmatrix} = \hat{f}$
We also have that matrix multiplication is a map $m: M_2(A) \otimes M_2(A) \rightarrow M_2(A)$, and that $M_2(A) \otimes M_2(A)$ is isomorphic as a vector space to $\text{Hom}_\text{Alg}(M(2) \otimes M(2),k)$ by the map
$\displaystyle \alpha \mapsto \begin{pmatrix} \alpha(a) & \alpha(b) \\ \alpha(c) & \alpha(d) \end{pmatrix} \otimes \begin{pmatrix} \alpha(a') & \alpha(b') \\ \alpha(c') & \alpha(d') \end{pmatrix} = \tilde{\alpha}$
I write $a'$, et al, just so it's clear that $\alpha$ varies on different elements of the tensor product basis $a \otimes a$, et al. So for a map to implement multiplication on the $M(2)$ side, it must be $\Delta: M(2) \rightarrow M(2) \otimes M(2)$ and we want $m(\tilde{\alpha}) = \alpha \circ \Delta$.
From this requirement it's obvious what $\Delta$ needs to be; $\Delta(M)$ for $M\in M(2)$ needs to ensure that the generators $a,b,c,d$, et cetera, get mapped to the items that will correspond to matrix multiplication after they are acted upon by $\alpha$. So $\Delta(a) = a \otimes a + b \otimes c$, et cetera. In matrix notation, we can write this
$\displaystyle \Delta \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \otimes \begin{pmatrix} a & b \\ c & d \end{pmatrix}$
This looks group-like! But it's not! The matrix isn't actually an element of $M(2)$, it's just a convenient way for us to write the action of $\Delta$. The action elements are $k$-linear combinations of $a,b,c$, and $d$.
Quantizing things
I bet the reader is guessing that $\Delta$ is a coproduct, making $M(2)$ into a bialgebra. Such a reader would be correct. But before we discuss this further, let's take a quotient of $M(2)$ by the relation $ad -bc =1$. Call this new bialgebra $SL(2)$. We can make it into a Hopf algebra by introducing the antipode:
$\displaystyle S\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} d & -b \\ -c & a\end{pmatrix}$
We're going to take a ``quantum deformation'' of this new Hopf algebra. It's actually simple to do, we're going to modify the commutativity of $a$, et al, by an element $q \in k^\ast$. In particular, let $ca = qac$, $ba =qab$, $db = qbd$, $dc = qcd$, $bc = cb$, $da -ad = (q-q^{-1})bc$, and the ``q-determinant'' relation $ad -q^{-1}bc = 1$. The coproduct remains the same.
The Quantum Plane and the coaction
The classical group we're mimicking, $SL_2(\mathbb{R})$, acts on the affine plane $\mathbb{R}^2$ by transforming it in a way that preserves orientation and area of all geometric shapes on the plane. Since $SL(2)$'s role is as the base of a set of homomorphisms to the algebra $A$, we expect any equivelant ``action'' to have the arrows reversed, we'll discuss this later. But first, we're going to need a notion of an affine plane in our polynomial algebra-geometry language:
This isn't so bad, as $\text{Hom}_\text{Alg}(k[x,y],A) \cong A^2$, hence we call $k[x,y]$ the affine plane. Quantizing it is easy, too: we define the quantum plane $\mathbb{A}^2_q$ to be the free algebra $k\langle x,y\rangle$ quotiented by the relation $yx = qxy$, id est, it's $k[x,y]$ but with a multiplication deformed by the element $q \in k$.
Now back to are ``arrow reversed'' version of an action, or a coaction. One can arrive at this definition by reversing the arrows in the commutative diagram that captures the axioms of an algebra acting on a vector space. In particular, we say that a Hopf algebra $H$ coacts on an algebra $A$ by an algebra morphism $\beta: A \rightarrow H \otimes A$ such that $(I \otimes \beta) \circ \beta = (\Delta \otimes I)\circ \beta$, and $I = (\epsilon \otimes I) \circ \beta)$, where $I$ is the identity map and $\Delta$ and $\epsilon$ are the coproduct structure maps for $H$.
We can define the coaction $\beta$ on the generators $x$ and $y$ and extend it as an algebra morphism should, so the reader can check that $\beta(x) = x\otimes a + y\otimes c$ and $\beta(y) = x\otimes b + y\otimes d$ defines a coaction. In matrix notation, we have:
$\displaystyle \beta(x,y) = (x,y) \otimes \begin{pmatrix} a & b \\ c & d \end{pmatrix}$
I do not have a geometric interpretation for this.
Posted by Matthew Eric Bassett at 3:44 PM 0 comments
Labels: hopf algebra, phd, quantum group
What I learned today: Coadjoint action of group Hopf algebras
This is another post in my ``psuedo-daily'' series ``What I learned Today''
Yesterday we talked about actions of a Hopf algebra $H$ on an algebra $A$. Today, let's talk about some examples of this (I didn't cover as much ground as I hoped to today, but we're trying to make these blogs daily!). Every Hopf algebra acts on itself by
$\displaystyle h \triangleright g = \sum_h h_{(1)} g Sh_{(2)} $
This is the adjoint action. It's not hard to prove that it's 1) an action and 2) a Hopf action (that it plays well with the Hopf algebra structure of $H$ and the algebra structure of $A$). As a more concrete example, let's look at the group Hopf algebra $H=kG$ for a finite group $G$. The coproduct is given by $\Delta g = g \otimes g$ and the antipode is group inversion: $Sg = g^-1$. In this case, the adjoint action becomes:
$\displaystyle h \triangleright g = hgh^{-1} $
Which is just group conjugation.
Hopf algebras can act just as well on coalgebras - in this case we require that the action commutes with the coproduct of $A$. Now if $H'$ and $H$ are dually paired, then we also have adjoint action, or rather, a coadjoint action of $H'$ on the coalgebra $H$, given by:
$\displaystyle \phi \triangleright h = \sum_h h_{(2)} \langle \phi, (Sh_{(1)})h_{(3)} \rangle $
We know that the algebra of functions on $G$, $k(G)$ is the Hopf algebra dual of $kG$, where $\langle f, g \rangle = f(g)$, so let's see what the this action becomes for $H' = k(G)$ and $H = kG$:
$\displaystyle f \triangleright g = g \langle f, (Sg)g \rangle = f(1) \cdot g$
Additionally, we also have the notion of a coaction. Generally, if $H$ is a Hopf algebra and $A$ an algebra, we say $\beta : A \rightarrow H \otimes A$ is a coaction of $H$ on $A$ (or that $A$ is a $H$-comodule algebra if $\beta$ is a algebra morphism, $(I \otimes \beta) \circ \beta = (\Delta \otimes I) \circ \beta$ and $I = (\epsilon \otimes I) \circ \beta$, where $I$ is the identity and $\Delta$ and $\epsilon$ are the coalgebra structure maps.
The most interesting case of this is the quantum group $SL_q(2)$ coacting on the quantum plane $\mathbb{A}^2_q$. This is a topic for another post, though.
Labels: hopf algebra, phd, what I learned today
What I learned today: Weird differences between the action of a Hopf algebra and its dual
Let $H$ be a Hopf algebra. Today we're going to talk about its actions (and its coactions, time permitting). A Hopf algebra is just an algebra, which is just a ring, and ring's have actions (equiv. modules), and we're comfortable with that already. For this to make sense on a Hopf algebra, we merely require that the action agrees with all the necessary units, products, coproducts, et cetera.
More precisely, we say that $H$ acts on an algebra $A$ (or that $A$ is an $H$-module algebra) if we have a linear map $\triangleright : H \otimes A \rightarrow A$, written $h \triangleright a$ such that
$\displaystyle h \triangleright (ab) = m(\Delta(h) \triangleright (a \otimes b))$
with $\triangleright$ extended to tensor products and $m$ the product of $A$, and
$\displaystyle h\triangleright 1_A = \epsilon(h) 1_A$
Some weirdness emerges here, for instance, if $H=kG$, the group hopf algebra of a finite group $G$ (the vector space having elements of $G$ as a basis, the product being the group operation, and the coproduct defined by $\Delta(g) = g \otimes g$, antipode, units, et cetera are the obvious choice), then a hopf action collapses to a typical group action:
$\displaystyle g\triangleright(ab) = (g\triangleright a) (g\triangleright b)$
But if we forget the Hopf algebra structure for a second, we know that $kG$ is exactly the same as $k(G)$, that is, as functions defined on $G$ with values in $k$, as any ``vector'' in $kG$ defines a function and vice versa. Hence, as vector spaces $kG$ and $k(G)$ are naturally isomorphic. This is not the case with the Hopf algebra structure, however (though we have seen how the two are dually paired already), as the coproduct on $k(G)$ is $\Delta(f)(x,y) = f(xy)$, where we identify $k(G \times G)$ with $k(G) \otimes k(G)$ (this is exactly what a tensor product is for, actually.) In this case, the hopf algebra action on an algebra $A$ is the same as a $G$-grading on $A$. Let me explain this (with help from a kind person on math.stackexchange):
Say $A$ is $G$-graded and for each $a \in A$ let $|a|$ denote the element in $G$ such that $a \in A_{|a|}$. We can get a $k(G)$ action from this by $f \triangleright a = f(|a|) a$. To go the other way, start from a Hopf algebra action $\triangleright: k(G) \otimes A \rightarrow A$. Then for each $g \in G$, let $A_g$ be the set of all elements of $A$ on which every $f\in k(G)$ acts via scalar multiplication by $f(g)$. One then has to prove that these sets are nonempty, that they are subspaces, that $A$ decomposes into direct sums of them, and that they obey the axioms of a grading. I've been assured by various sources that it can be done!
What I learned today: Dually Paired Hopf Algebras
To help keep me motivated and mathematically active, I will be blogging about "what I learned today" in my various projects. This is the second post in this "psuedo-daily" series.
I'm still trying to catch up on Quantum Groups. Any reader who is familiar with the field will be able to tell from these posts that I'm a long way away from doing research. But we're slowly getting somewhere!
Let $H = kG$ be the group Hopf algebra for a finite group $G$. In particular, the product map $m: H \otimes H \rightarrow H$ is just the group operation $g \otimes h \rightarrow gh$ for $g, h\in G$, the coproduct is $g \mapsto g\otimes g$, the antipode is group inversion $x \mapsto x^{-1}$, the group idenity is the unit and $g \mapsto 1$ is the counit. We want to talk about its dual $H^\ast = \text{Hom}(H,k)$.
Kassel's Quantum Groups text gives us several propositions to prove that the dual of a finite dimensional Hopf algebra is also a Hopf Algebra. Since $G$ is a finite group, $H$ is finite dimensional, and we'll follow Kassel in proving that $H^\ast$ in particular is a Hopf Algebra.
Now, $H^\ast$ is the space of linear functions from $kG$ to $k$. Any such linear functional will be determined by basis elements that themselves are determined by their action on $G$, hence $H^\ast$ is the space $k(G)$ of functions on $G$ with values in $k$. Clearly this space has pointwise multiplication to make it into an algebra - but we want to derive this from the dual of the coproduct map $\Delta: H \rightarrow H \otimes H$.
So let's talk about $\Delta^\ast: (H\otimes H)^\ast \rightarrow H^\ast$. This is, by definition, the map that takes each $\alpha\otimes\beta \in (H\otimes H)^\ast$ to another linear functional $\gamma = \Delta^\ast(\alpha\otimes\beta)$ on $H$ such that $\gamma(h) = \alpha\otimes\beta(\Delta(h))$. But before we can work out how this becomes a product map, we need to work out some technicalities:
Can we say that $(H\otimes H)^\ast \cong H^\ast \otimes H^\ast$ so that $\Delta^\ast$ is actually a product? Yes! this is provided by a theorem in Kassel's book which says that the map $\lambda: \text{Hom}(U, U') \otimes \text{Hom}(V,V') \rightarrow \text{Hom}(V \otimes U, U' \otimes V')$ by $(f\otimes g)(v \otimes u) = f(u)\otimes g(v)$ is an isomorphism when one of the pairs $(U, U')$, $(V, V')$, or $(U,V)$ consists of only finite dimensional spaces. We won't discuss the proof here, but I do want to point that the theorem does require finite dimensionality .
Back to the dual of comultiplication on $H$: We have that $\Delta(h) = h \otimes h$. So $\alpha \otimes \beta (h\otimes h) = \alpha(h) \beta(h) = \gamma(h)$. Hence $\Delta^\ast(\alpha \otimes \beta)(h) = \alpha(h)\beta(h)$ for $\alpha, \beta \in H^\ast$ and $h\in H$. Hence the dual of comultiplication of a group Hopf algebra is just pointwise multiplication, as we expected.
Now what about the dual of multiplication? Same definition as above, $m^\ast: H^\ast \rightarrow H^\ast \otimes H^\ast$ must be the map taking $\alpha \in H^\ast$ to an $m^\ast(\alpha)=\beta \otimes \gamma$ such that $\beta\otimes\gamma(h_1\otimes h_2) = \alpha(m(h_1,h_2)) = \alpha(h_1 h_2)$. Hence we let $m^\ast$ be the function $m^\ast(\alpha)(x\otimes y) = \alpha(xy)$
Continuing in this fashion, you'll see that the antipode is the map $S(\alpha(x)) = \alpha(x^{-1})$, the unit is the identity function, and the counit the map $\alpha \mapsto \alpha(1)$.
Hence we have a Hopf algebra $H^\ast = k(G)$ that is dual to $H = kG$. If we let $\langle, \rangle: H^\ast \otimes H$ be the evaluation map $\langle \alpha, x \rangle = \alpha(x)$, we can see write down the behavior of this map and see if we can come up with a more general situation to more varied sets of Hopf algebras. For instance, extending the map to tensor products pairwise, note that
$\displaystyle \langle \alpha\beta, h \rangle = \langle \alpha\otimes\beta, \Delta(h) \rangle$
$\displaystyle \langle \Delta (\alpha), h\otimes g \rangle = \langle \alpha, hg\rangle$
$\displaystyle \langle S(\alpha), h \rangle = \langle \alpha, S(h) \rangle$
$\displaystyle \langle 1, h \rangle = 1 $
$\displaystyle \langle \alpha, 1 \rangle = \alpha(1)$
Replacing those last two conditions with the same thing expressed in units and counits, we have
$\displaystyle \langle 1, h \rangle = \epsilon(h)$
$\displaystyle \langle \alpha, 1 \rangle = \epsilon(\alpha)$
Thus now we have a definition: Let $H_1, H_2$ be Hopf algebras. We say that they are dually paired if there is a linear map $\langle, \rangle : H_2 \otimes H_1 \rightarrow k$ that satisfies the above 5 conditions.
To make things explicit, $k(G)$ and $kG$ are dually paired by the evaluation map. Majid's book gives a more exotic example of a ``quantum group'' that is paired with itself:
Let $q \in k$ be nonzero and let $U_q(b+)$ be the $k$-algebra generated by elements $g, g^{-1}$, and $X$ with the relation $gX = qgX$. Majid's book assures me we'll see how to find this ``quantum group'' in the wild later on, but for now he gives it a Hopf algebra structure with the coproduct $\Delta X = X\otimes 1 + g\otimes X$, $\Delta g = g \otimes g$, $\epsilon X = 0$, $\epsilon g = 1$, $SX = -g^{-1} X$, and $Sg = g^{-1}$
This Hopf algebra is dually paired with itself by the map $\langle g, g \rangle = q$, $\langle X, X \rangle = 1$, and $\langle X, g \rangle = \langle g, X \rangle = 0$
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Dorothy Vaughan
Dorothy Jean Johnson Vaughan (September 20, 1910 – November 10, 2008) was an American mathematician and human computer who worked for the National Advisory Committee for Aeronautics (NACA), and NASA, at Langley Research Center in Hampton, Virginia. In 1949, she became acting supervisor of the West Area Computers, the first African-American woman to receive a promotion and supervise a group of staff at the center.
Dorothy Vaughan
Born(1910-09-20)September 20, 1910
Kansas City, Missouri, U.S.
DiedNovember 10, 2008(2008-11-10) (aged 98)
Hampton, Virginia, U.S.
EducationWilberforce University (BA)
Spouse
Howard Vaughan
(m. 1932; died 1955)
Children6
Scientific career
FieldsFortran Computer Specialist
InstitutionsNACA, Langley Research Center
She later was promoted officially to the position of supervisor. During her 28-year career, Vaughan prepared for the introduction of computers in the early 1960s by teaching herself and her staff the programming language of Fortran. She later headed the programming section of the Analysis and Computation Division (ACD) at Langley.
Vaughan is one of the women featured in Margot Lee Shetterly's history Hidden Figures: The Story of the African-American Women Who Helped Win the Space Race (2016). It was adapted as a biographical film of the same name, also released in 2016.
In 2019, Vaughan was honored with the Congressional Gold Medal posthumously.[1]
Early life
Vaughan was born September 20, 1910, in Kansas City, Missouri, as Dorothy Jean Johnson.[2] She was the daughter of[3] Annie and Leonard Johnson. At the age of seven, her family moved to Morgantown, West Virginia, where she graduated from Beechurst High School in 1925 as her class valedictorian.[4] Vaughan received a full-tuition scholarship from West Virginia Conference of the A.M.E. Sunday School Convention[5] to attend Wilberforce University in Wilberforce, Ohio. She joined the Zeta chapter of Alpha Kappa Alpha Sorority, Inc. at Wilberforce[6] and graduated in 1929 with a B.A. in mathematics.[7] In 1932, she married Howard Vaughan, who died in 1955. The couple moved to Newport News, Virginia, where they had six children: Ann, Maida, Leonard, Kenneth, Michael and Donald.[8] The family also lived with Howard's wealthy and respected parents and grandparents on South Main Street in Newport News, Virginia. Vaughan was very devoted to family and the church, which would play a huge factor in whether she would move to Hampton, Virginia, to work for NASA.
Career
Vaughan graduated from Wilberforce University in 1929. Although encouraged by professors to do graduate study at Howard University,[5] Vaughan worked as a mathematics teacher at Robert Russa Moton High School in Farmville, Virginia, in order to assist her family during the Great Depression.[9] During the 14 years of her teaching career, Virginia's public schools and other facilities were still racially segregated under Jim Crow laws.[10]
In 1935, the NACA had established a section of women mathematicians, who performed complex calculations.[5] In 1941, President Franklin D. Roosevelt issued Executive Order 8802, to desegregate the defense industry, and Executive Order 9346 to end racial segregation and discrimination in hiring and promotion among federal agencies and defense contractors.[10] These helped ensure the war effort drew from all of American society after the United States entered World War II in 1941. With the enactment of the two Executive Orders, and with many men being swept into service, federal agencies such as the National Advisory Committee for Aeronautics (NACA) also expanded their hiring and increased recruiting of women, including women of color, to support the war production of airplanes.[5] Two years following the issuance of Executive Orders 8802 and 9346, the Langley Memorial Aeronautical Laboratory (Langley Research Center), a facility of the NACA, began hiring more black women to meet the drastic increase in demand for processing aeronautical research data.[2] The US believed that the war was going to be won in the air. It had already ramped up airplane production, creating a great demand for engineers, mathematicians, craftsmen and skilled tradesmen.
In 1943, Vaughan began a 28-year-career as a mathematician and programmer at Langley Research Center in Hampton, Virginia, in which she specialized in calculations for flight paths, the Scout Project, and computer programming. Her career in this field kicked off during the height of World War II. She came to the Langley Memorial Aeronautical Laboratory thinking that it would be a temporary war job. One of her children later worked at NACA.[7] Vaughan was assigned to the West Area Computing, a segregated unit, which consisted of only African Americans. This was due to prevailing Jim Crow laws that required newly hired African American women to work separately from their white women counterparts.[2] They were also required to use separate dining and bathroom facilities.[2] This segregated group consisted of African-American women who made complex mathematical calculations by hand, using tools of the time.[5][11]
The West Computers made contributions to every area of research at Langley. Their work expanded in the postwar years to support research and design for the United States' space program, which was emphasized under President John F. Kennedy. In 1949, Vaughan was assigned as the acting head of the West Area Computers, taking over from a white woman who had died. She was the first black supervisor at NACA and one of few female supervisors. She led a group composed entirely of African-American women mathematicians.[12] She served for years in an acting role before being promoted officially to the position as supervisor.[9] Vaughan worked for opportunities for the women in West Computing as well as women in other departments.[11]
Seeing that machine computers were going to be the future, she taught the women programming languages and other concepts to prepare them for the transition. Mathematician Katherine Johnson was initially assigned to Vaughan's group, before being transferred to Langley's Flight Mechanics Division. Vaughan moved into the area of electronic computing in 1961, after NACA introduced the first digital (non-human) computers to the center. Vaughan became proficient in computer programming, teaching herself FORTRAN and teaching it to her coworkers to prepare them for the transition. She contributed to the space program through her work on the Scout Launch Vehicle Program.[11] A blog describing her work at NASA is on the Science Museum group website[13]
Vaughan continued after NASA, the successor agency, was established in 1958. When NACA became NASA, segregated facilities, including the West Computing office, were abolished. In a 1994 interview, Vaughan recalled that working at Langley during the Space Race felt like being on "the cutting edge of something very exciting".[14] Regarding being an African American woman during that time in Langley, she remarked, "I changed what I could, and what I couldn't, I endured." Vaughan worked in the Numerical Techniques division through the 1960s. Dorothy Vaughan and many of the former West Computers joined the new Analysis and Computation Division (ACD), a racially and gender-integrated group on the frontier of electronic computing. She worked at NASA-Langley for 28 years.[12]
During her career at Langley, Vaughan was also raising her six children, one of whom later also worked at NASA-Langley. Vaughan lived in Newport News, Virginia, and commuted to work at Hampton via public transportation.
Later years
Vaughan wanted to continue at another management position at NASA, but never received an offer.[15] She retired from NASA in 1971, at the age of 61. In her final years, she worked with mathematicians Katherine G. Johnson and Mary Jackson on astronaut John Glenn's launch into orbit.[15] She died on November 10, 2008, aged 98. Vaughan was a member of Alpha Kappa Alpha, an African-American sorority. She was also an active member of the African Methodist Episcopal Church where she participated in music and missionary activities. She also wrote a song called "Math Math".[16]
At the time of her passing, she was survived by four of her six children, ten grandchildren and fourteen great-grandchildren.[8]
Legacy
Vaughan was the first respected Black female manager at NASA, thus creating a long-lasting legacy for diversity in mathematics and science for West Area Computers. As one of the first female coders in the field who knew how to code FORTRAN, she was able to instruct other Black women on the coding language and paved a wave of female programmers to integrate their work into NASA’s systems. [17]
In 2005, a scholarship fund with the Salem Community Foundation was created under Dorothy Vaughan’s name to further music training by the Salem Music Study Club. [18]
Vaughan is one of the women featured in Margot Lee Shetterly's 2016 non-fiction book Hidden Figures, and the feature film of the same name. She was portrayed by the Academy Award winning actress Octavia Spencer.
The Dorothy J. Vaughan Academy of Technology opened in Charlotte, NC, in August 2017. This school is inspired by Vaughan’s “leadership, innovation, creativity, curiosity, and love of learning.” The school is a member of the Magnet Schools of America Association. [19]
In 2019, Vaughan was awarded the Congressional Gold Medal.[1] Also in 2019, the Vaughan crater on the far side of the Moon was named in her honor.
On 6 November 2020, a satellite named after her (ÑuSat 12 or "Dorothy", COSPAR 2020-079D of the ÑuSat series) was launched into space.
Vaughan’s personal Bible and NASA retirement identification card are displayed in the Museum of the Bible’s exhibition Scripture and Science: Our Universe, Ourselves, Our Place. [20] The African Methodist Episcopal Church also gave her a service award.[20]
North Central University has a scholarship in honor of Dorothy Vaughan for BIPOC and/or female students.[21]
Awards and honors
• 1925: Beechurst High School – Class Valedictorian
• 1925: West Virginia Conference of the A.M.E. Sunday School Convention – Full Tuition Scholarship
• 1929: Wilberforce University – Mathematician Graduate Cum Laude
• 1949–1958: Head of National Advisory Committee of Aeronautics' Segregated West Computing Unit (NACA)
• October 16, 2019: a lunar crater is named after her.[22] This name was chosen by planetary scientist Ryan N. Watkins and her student, and submitted on what would have been Dorothy Vaughan's 109th birthday.[23]
• November 8, 2019: Congressional Gold Medal[1]
• On November 6, 2020, a satellite named after her was launched into space
References
1. "H.R.1396 - Hidden Figures Congressional Gold Medal Act". Congress.gov. 8 November 2019. Retrieved 9 November 2019.
2. Shetterly, Margot Lee; Loff, Sarah (2016-11-22). "Dorothy Vaughan Biography". NASA. Retrieved 2018-12-11.
3. "Dorothy Vaughan obituary". Hampton: Daily Press. 2008-11-12. Retrieved 2022-10-31.
4. Shetterly 2016b, pp. 12.
5. Shetterly, Margot Lee (2016a). "The Hidden Black Women Who Helped Win the Space Race". New York.
6. Williams 2018, p. 67.
7. "Hidden Figure: Dorothy Vaughan". Spelman College. Retrieved January 3, 2017.
8. "Vaughan, Dorothy Johnson (1910–2008)". The Black Past: Remembered and Reclaimed. 2017-01-07. Retrieved 2017-10-09.
9. Shetterly 2016b, pp. 21–22, 91–92.
10. "Dorothy Vaughan: NASA's 'Human Computer' and American Hero". interestingengineering.com. 2018-03-11. Retrieved 2018-12-11.
11. Shetterly, Margot Lee. "Dorothy Vaughan". The Human Computer Project.
12. Allen, Bob (3 February 2016). "Dorothy Vaughan (nee Johnson)" (PDF). NASA.
13. "Dorothy Vaughan: NASA's Overlooked Star". Science Museum. 2020.
14. Golemba 1994, p. 121.
15. "Dorothy Johnson Vaughan". Biography. Retrieved 2021-03-02.
16. "Obituary: Dorothy Vaughan," Newport News, November 2008.
17. Loff, Sarah (2016-11-22). "Dorothy Vaughan Biography". NASA. Retrieved 2023-04-01.
18. "Principal Funds". salemcommunityfoundation.org. Retrieved 2023-04-01.
19. "About Us / Overview". www.cmsk12.org. Retrieved 2023-04-01.
20. "No Longer Hidden: The Legacy of Dorothy Vaughan". Museum of the Bible. Retrieved 2023-04-01.
21. "Dorothy Vaughan Scholarship". North Central University. Retrieved 2023-04-01.
22. "Vaughan". Gazetteer of Planetary Nomenclature. Retrieved February 27, 2020.
23. Ryan Watkins, "Thrilled to announce that this small (3 km) crater on the Moon now has a name - Vaughan! My student and I chose to name Vaughan crater after Dorothy Vaughan (you may remember her from @HiddenFigures, where she was portrayed by @octaviaspencer).", Twitter, 16 octobre 2019.
October.
Sources
• Golemba, Beverly (1994). Human Computers: The Women in Aeronautical Research (PDF). NASA Langley Archives, unpublished manuscript.
• Shetterly, Margot Lee (2016b). Hidden Figures: The American Dream and the Untold Story of the Black Women Mathematicians Who Helped Win the Space Race. William Morrow. ISBN 9780062363619.
• Williams, Talithia (2018). Power in Numbers: The Rebel Women of Mathematics. Race Point Publishing. p. 67. ISBN 978-1-63106-485-2. OCLC 1033694135.
External links
• Hidden Figures at IMDb
Authority control
International
• VIAF
National
• Norway
• Germany
• Israel
• United States
• Czech Republic
Academics
• MathSciNet
• zbMATH
Other
• SNAC
• IdRef
| Wikipedia |
An improved secure designated server public key searchable encryption scheme with multi-ciphertext indistinguishability
Junling Guo1,
Lidong Han ORCID: orcid.org/0000-0003-2094-56291,2,
Guang Yang1,
Xuejiao Liu1,2 &
Chengliang Tian3
Journal of Cloud Computing volume 11, Article number: 14 (2022) Cite this article
In the cloud, users prefer to store their sensitive data in encrypted form. Searching keywords over encrypted data without loss of data confidentiality is an important issue. In 2004, Boneh et al. proposed the first public-key searchable encryption scheme which allows users to search by the private key. However, most existing public-key searchable encryption schemes are vulnerable to keyword guessing attack and can not satisfy multi-ciphertext indistinguishability. In this paper, we construct a secure designated server public-key searchable encryption based on Diffie-Hellman problem. Our security analysis shows that our proposed scheme can resist against keyword guessing attack and provide multi-ciphertext indistinguishability for any adversity. Furthermore, the proposed scheme can achieve multi-trapdoor privacy for external attackers. Moreover, the simulation results between our scheme and previous schemes demonstrate our new scheme is suitable for practical application.
With the rapid development of cloud computing, a growing number of users and companies prefer to store data on the cloud. In such case, they encrypt the data before uploading in order to ensure data privacy. However, it is extremely difficult to retrieve keyword over encrypted data using traditional search mechanism. Searchable encryption has become a promising solution to ensure the security and availability of data.
In 2004, Boneh et al. [1] proposed the concept of Public-key Encryption with Keyword Search (PEKS) and gave a concrete scheme. However, in 2006, Byun et al. [2] put forward an offline keyword guessing attack(KGA) against Boneh et al. 's scheme. Later, Baek et al. [3] presented a PEKS scheme without a secure channel in 2008. Then, Rhee et al. [4] introduced a new security concept of PEKS, trapdoor indistinguishability, They put forward a PEKS scheme under designated test server (dPEKS) which satisfies trapdoor indistinguishability.
Wang et al. [5] proposed that even if [4] satisfies the trapdoor indistinguishability, their dPEKS cannot resist inside KGA. Since keyword encryption algorithms are public in previous schemes, it will enable the internal attacker to generate the ciphertext of a candidate keyword by himself. That is, the malicious server can efficiently test whether the trapdoor is generated by the canditate keyword or not.
To resist keyword guessing attacks initiated by malicious servers, many researchers have proposed some variants of PEKS schemes. Tang et al. [6] introduced the concept of keyword registration, which requires the sender to register keywords with the receiver in advance and proposes registered keyword search public key encryption (PERKS). Chen et al. [7] put forward a solution using two servers that do not collide with each other, but it is too ideal. Later, Huang et al. [8] presented the concept of public-key authenticated encryption with keyword search (PAEKS) to resist the inside KGA. In their scheme, the data owner needs to use the secret key to authenticate the ciphertext of the keyword. The malicious cloud server will not generate keyword ciphertext for testing without the owner's private key. Therefore, KGA does not succeed against their scheme.
Qin et al. [9] in 2020 introduced the new security concept called multi-ciphertext indistinguishability (MCI). That is, from two or more ciphertexts, the adversary can determine whether they are generated by a same keyword. And they constructed a new PAEKS that can guarantee MCI security but does not provide multi-trapdoor privacy (MTP) security in which attacker is able to check two or more trapdoors contain a same keyword. In 2021, Pan and Li [10] put forward a new PAEKS scheme with MCI and MTP security. Later, Cheng and Meng [11] proved that Panr and Li's scheme does not satisfy MTP security.
Motivations and contributions
In searchable encryption, the security goal is that the ciphertexts and trapdoors leak no information about keywords. So far, there is rarely public-key searchable encryption schemes achieve both MCI and MTP, and security against KGA. In this paper, our goal is to construct an enhanced secure designated server public-key searchable encryptionscheme with MCI and MTP. The contributions of our paper are summarized as follows:
We give a security analysis of Li et al.'s scheme [12] and show that their scheme does not satisfy trapdoor indistinguishability.
We propose a secure scheme that satisfies the requirement of testing the designated server. That is to say, no one can test except the designated server. Moreover, we prove that our scheme satisfies MCI security, MTP security for external adversaries, and designated testability.
We analyze our scheme's implementation and communication cost by comparing it with previous other schemes. The result shows that our scheme has excellent advantages in keyword ciphertext and trapdoor algorithms, and the test algorithm is not inferior to other schemes. Moreover, our scheme provides stronger security for keyword privacy.
In 2004, Boneh et al. [1] first proposed the public key encryption scheme with keyword search, which started the research on public-key searchable encryption. Later, Abdalla et al. [13] presented a searchable encryption scheme based on identity. Byun et al. [2] put forward offline KGA against Abdalla et al.'s scheme. Baek et al. [3] suggested that a tester should be appointed to perform the test algorithm to hide the user's search pattern, to ensure that only those who have the tester's private key can conduct the test. Rhee et al. [4, 14] put forward a dPEKS model to reject outside KGA and constructed a general structure of dPEKS based on the designated tester. Fang et al. [15] presented a dPEKS scheme that is not based on a random prediction machine to resist outside KGA. Rhee et al. [16] construct an identity-based PEKS scheme with a designated tester. Emura et al. [17] presented a general structure of SCF-PEKS based on anonymous identity-based encryption(IBE) and one-time signature. After that, many schemes [18–20] have made efforts to resist offline guessing attacks, but these schemes cannot resist inside KGA.
To resist inside KGA, Xu et al. [21] proposed a PEKS scheme with fuzzy keywords, reducing the security of inside KGA by ensuring that each trapdoor corresponds to multiple keywords. Wang et al. [22] gave a PEKS scheme with dual servers. In 2017, Huang et al. [8] proposed the concept of public-key authentication searchable encryption. After that, Huang et al.'s scheme has been extended to certificateless PAEKS [23–25] and identity-based PAEKS [12]. And in the field of Internet of Things, many PEAKS variants [26–28] have been proposed. In 2019, Lu et al. [29] presented a PEKS scheme without random prediction. Later, Noroozi et al. [30] proposed that Huang et al.'s scheme is insecure in the case of multiple receivers.
In 2020, Qin et al. [9] presented a new PAEKS that is claimed to provide multi-ciphertext indistinguishability but no multi-trapdoor privacy. Recently, Li et al. [12] proposed a new PAEKS scheme under a designated server which still cannot guarantee MTP. Furthermore, almost PAEKS [8, 12] and their variants [9, 25, 31, 32] cannot provide MTP security and hide the search pattern of the user. Later, Qin et al. [33] proposed an improved security model and gave a specific scheme. Recently, Lattice-based searchable encryption schemes [34, 35] have been proposed which are claimed to guarantee stronger security.
The rest of this paper is organized as follows. In section 2, we introduce some preliminary knowledge. Then we review Li et al.'s scheme and give a security analysis for it in section 3. The fourth section defines the enhanced scheme and its security model. Section 5 gives a concrete construction scheme and proves that it satisfies the designed testability, MTP security and MCI security. Then in section 6, we compare and analyze our scheme with others. In the last section, we give a summary and a prospect for the future.
Bilinear pairing
We briefly describe the definition of bilinear mapping. (See more details in [36]). Let \(\hat {e}:\mathbb {G}_{1} \times \mathbb {G}_{1} \rightarrow \mathbb {G}_{2}\) be a computable bilinear pairing, where \(\mathbb {G}_{1}\) and \(\mathbb {G}_{2}\) are two cyclic groups of prime order p. The map \(\hat {e}\) has the following properties.
For any \(x,y \in \mathbb {Z}_{p}^{*}\), \(g,g_{1} \in \mathbb {G}_{1}\), the equation \(\hat {e}\left (g^{x}, g_{1}^{y}\right) = \hat {e}\left (g, g_{1}\right)^{xy}\) holds.
For any generator \(g \in \mathbb {G}_{1}\), \(\hat {e}(g,g)\) is a generator of \(\mathbb {G}_{2}\).
For any \(g,g_{1} \in \mathbb {G}_{1}\), there exists a PPT algorithm to compute \(\hat {e}(g_{1},g)\).
Complexity assumptions
In this subsection, \(\mathbb {G}_{1}\) and \(\mathbb {G}_{2}\) are two cyclic groups of prime order p, g is a generator of \(\mathbb {G}_{1}\) and \(\hat {e}:\mathbb {G}_{1} \times \mathbb {G}_{1} \rightarrow \mathbb {G}_{2}\) is a bilinear map. Decisional Diffie–Hellman assumption and Decisional bilinear Diffie–Hellman assumption are introduced as follows.
(Decisional Diffie–Hellman (DDH) assumption): Given g, gx, \(g^{y} \in \mathbb {G}_{1}\), where \(x,y \in \mathbb {Z}_{q}^{*}\), there no exists polynomial-time algorithm to distinguish between (g,gx,gy,gxy) and (g,gx,gy,Z), where \(Z \in _{R} \mathbb {G}_{1}\). The advantage of adversary \(\mathcal {A}\) is
$$Adv^{DDH}_{\mathcal{A}}\!(\kappa)\! =\! \vert Pr[\mathcal{A}(g,g^{x},g^{y},g^{xy})]\! - \! Pr[\mathcal{A}(g,g^{x},g^{y},Z)] \vert$$
DDH assumption holds if the advantage is negligible.
(Decisional bilinear Diffie–Hellman (DBDH) assumption) : Given g, gx, gy, \(g^{z} \in \mathbb {G}_{1}\), where \(x,y,z \in \mathbb {Z}_{q}^{*}\). The advantage of the adversary \(\mathcal {A}\) is \(Adv^{DBDH}_{\mathcal {A}}(\kappa) = \vert Pr\left [\mathcal {A}\left (g,g^{x},g^{y},g^{z},e(g,g)^{xyz}\right)\right ] - Pr\left [\mathcal {A}(g,g^{x},g^{y},g^{z},Z\right ] \vert \), where \(x,y,z \in _{R} \mathbb {Z}_{q}^{*}\) and \(Z \in _{R} \mathbb {G}_{2}\). DBDH assumption holds if the advantage is negligible.
Our system framework is showed in Fig. 1. The system contains three entities: a cloud server, a data owner and a receiver. Moreover, the data owner wants to send confidential files to the cloud which are allowed the assigned receiver to access the data. The exact procedures are as follows: First, the data owner extracts a group of keywords from documents and builds an secure index including keyword ciphertexts and documents. Second, the data owner encrypts the files by symmetric encryption and uploads the encrypted file and keyword ciphertext index to the server. Third, the receiver generates a trapdoor for a query keyword and sends it to the server. Finally, after receiving the trapdoor, cloud server runs the test algorithm and outputs the search results.In Table 1, we summarize the notations used in this paper.
System Framework
Table 1 Notations
Cryptanalysis of li et al.'s scheme
In this section, we review an identity-based searchable authenticated encryption scheme under a designated server proposed by Li et al.. After analyzing their scheme, we propose that it cannot guarantee trapdoor indistinguishability.
Review of li et al.'s scheme
Li et al.'s scheme consists of the following polynomial algorithms:
Setup(κ): From the security parameter κ, it outputs a public parameter para= (\(\mathbb {G}_{1},\mathbb {G}_{2},\hat {e}, p, g, g_{1}, H, H_{1}\), mpk) and msk, where \(\mathbb {G}_{1}\) and \(\mathbb {G}_{2}\) are cyclic groups of prime order p. g and g1 are generators of \(\mathbb {G}_{1}\). \(\hat {e}:\mathbb {G}_{1} \times \mathbb {G}_{1} \rightarrow \mathbb {G}_{2}\) is an efficient bilinear map, and \(H:\mathbb {G}_{2} \times \{0,1\}^{*} \rightarrow \mathbb {G}_{1},H_{1}: \{0,1\}^{*} \rightarrow \mathbb {G}_{1}\), \(msk=\alpha \in \mathbb {Z}_{p}\), mpk=gα.
KGens(para): With the parameters para, it outputs the public/secret key pairs (PkS,SkS)=(gz,z) of the server, where \(z \in _{R} \mathbb {Z}_{p}\).
KGenusr(pp,msk,ID): Inputting (pp,msk,ID), it returns SkID=H1(ID)α.
\(\phantom {\dot {i}\!}dIBAEKS(para,w,Pk_{S},Sk_{ID_{O}},ID_{O},ID_{R})\): With para, w, Pks, \(\phantom {\dot {i}\!}Sk_{ID_{O}}\), IDO of a data owner and a receiver's IDR, it returns a keyword ciphertext Cw= (C1,C2,C3), where C1 = \(\hat {e}\left (H(k,w),Pk_{S}^{s}\right)\), C2 = gs, C3=\( g_{1}^{s}\), \(s \in _{R} \mathbb {Z}_{p}\) and \(\phantom {\dot {i}\!}k = \hat {e}(Sk_{ID_{O}}, H_{1}(ID_{R}))\).
\(\phantom {\dot {i}\!}Tarpdoor(para,w,Pk_{S},Sk_{ID_{R}},ID_{O},ID_{R})\): It outputs a trapdoor \(T_{w} = (H(k,w) \cdot g_{1}^{r},g^{r})\), where \(\phantom {\dot {i}\!}k = \hat {e}(H_{1}(ID_{O}),Sk_{ID_{R}})\).
Test(para,SkS,IDO,IDR,CW,TW): It outputs 1 if
$$C_{1} \cdot \hat{e}\left(T_{2}^{Sk_{S}},C_{3}\right)=\hat{e}\left(T_{1}^{Sk_{S}},C_{2}\right),$$
and 0 otherwise.
Cryptanalysis of their scheme
In [12], Li et al. claimed that their dlBAEKS scheme satisfies the trapdoor indistinguishability under the random prediction model. Although trapdoor contains a random number in dlBAKES, there is an efficient algorithm to ascertain whether two trapdoors encrypt the identical keyword or not. In fact, for any two trapdoors Tw=(T1,T2) and \(\phantom {\dot {i}\!}T_{w^{\prime }} = \left (T_{1}^{\prime },T_{2}^{\prime }\right)\) containing unknown keywords w and w′, respectively, the decision algorithm by adversary is as follows:
$$\begin{array}{*{20}l} &\quad \hat e (T_{1},g)\cdot e\left(g_{1},T_{2}\right)^{-1} \\ &= \hat e\left(g,H(k,w)\cdot g_{1}^{r}\right)\cdot e\left(g_{1},g^{r}\right)^{-1}\\ &=\hat e(g,H(k,w))e\left(g_{1},g^{r}\right)e\left(g_{1},g^{r}\right)^{-1}\\ &=\hat e(H(k,w),g) \end{array} $$
where k, g are both fixed values for the same owner and receiver. An attacker captures some tuples Tw=(T1,T2) and \(\phantom {\dot {i}\!}T_{w^{\prime }} = (T_{1}^{\prime },T_{2}^{\prime })\). This distinguishing attack works as follows: if
$$\hat e(T_{1},g) \cdot \hat e(g_{1},T_{2})^{-1} = \hat e(T_{1}^{\prime},g) \cdot \hat e(g_{1},T_{2}^{\prime})^{-1} $$
then w=w′, and w≠w′ otherwise. Thus, the dIBAEKS scheme in [12] is insecure for multi-trapdoor privacy. This means that for the data owner sharing files to the receiver, the external attacker can effectively determine if multiple trapdoors generated by the receiver corresponds to the same keyword.
In addition, another scheme dIBAEKS-3 proposed in [12] also has the similar vulnerability. The decision algorithm is as follows: \(\hat {e}(T_{1},g_{1})\cdot \hat {e}(T_{2},g)^{-1} = \hat {e}(H(k,w),g_{1}) \). From two trapdoors: Tw=(T1,T2) and \(\phantom {\dot {i}\!}T_{w^{\prime }}=(T_{1}^{\prime },T_{2}^{\prime })\), an attacker checks whether \(\hat {e}(T_{1},g_{1})\cdot \hat {e}(T_{2},g)^{-1} = \hat {e}(T_{1}^{\prime },g_{1})\cdot \hat {e}(T_{2}^{\prime },g)^{-1}\) holds. If it holds, these two trapdoors are generated by the same keyword. Thus, the dIBAEKS-3 scheme is not able to provide multi-trapdoor privacy.
Definitions and security model
Our scheme consists of seven (probabilistic) polynomial-time(PPT) algorithms as follows.
Setup(κ)→pp: Given a security parameter κ, it returns the global parameter pp.
KeyGenO(pp)→(PkO,SkO): Given the parameter pp, it returns the public/secret key pairs (PkO,SkO) of the data owner.
KeyGenR(pp)→(PkR,SkR): With the parameter pp, it outputs the public/secret key pairs (PkR,SkR) of the receiver.
KeyGenS(pp)→(PkS,SkS): Inputting pp, it calculates the public key and private key pairs (PkS,SkS) of the server.
PEKS(pp,PkS,PkR,SkO,w)→Cw: Given the parameter pp, PkS of server, PkR of receiver, SkO of data owner and a keyword w, it outputs the ciphertext Cw.
\(\phantom {\dot {i}\!}Trapdoor(pp, Pk_{O}, Sk_{R}, w^{\prime }) \rightarrow T_{w^{\prime }}\): With the parameter pp, the data owner's PkO, PkR of a receiver and a keyword w′, it computes the trapdoor \(\phantom {\dot {i}\!}T_{w^{\prime }}\) of w′.
\(\phantom {\dot {i}\!}Test\left (pp, Sk_{S}, C_{w}, T_{w^{\prime }}\right) \rightarrow \beta \): With pp, SkS, Cw and \(\phantom {\dot {i}\!}T_{w^{\prime }}\), it outputs 1 if Cw and \(T_{w^{\prime }}\) contain a same keyword, 0 otherwise.
Security model
In order to prevent an adversary obtaining any useful inforamtion of keywords, we define three games between a challenger \(\mathcal {C}\) and an adversary \(\mathcal {A}\), namely, multi-ciphertext indistinguishability, multi-trapdoor privacy and designated testability.
Game 1: Multi-ciphertext indistinguishability.
Setup: The challenger \(\mathcal {C}\) runs KeyGenS, KeyGenO and KeyGenR algorithms with pp to generate (PkS, SkS), (PkO,SkO) and (PkR,SkR). It returns the tuple (pp,(PkS,SkS)) to \(\mathcal {A}\).
Phase 1: \(\mathcal {A}\) can issue the following two oracles for polynomial number times.
Ciphertext Oracle \(\mathcal {O}_{C}\): With (PkO,PkR,PkS,w), \(\mathcal {C}\) computes the ciphertext Cw and sends it to \(\mathcal {A}\).
Trapdoor Oracle \(\mathcal {O}_{T}\): With (PkO,PkR,w), \(\mathcal {C}\) computes a trapdoor Tw of a keyword w and returns it to \(\mathcal {A}\).
Challenge: \(\mathcal {A}\) sends two tuples of challenge keywords \(\vec { w}_{0} = \left (w_{0,1}, \dots, w_{0,n}\right), \vec {w}_{1} = \left (w_{1,1}, \dots, w_{1,n}\right)\) to \(\mathcal {C}\). However, the attacker cannot query the challenge keyword in tuple \(\vec w_{0}\) or \(\vec w_{1}\) in advance. \(\mathcal {C}\) selects a random bit b∈{0,1}, computes Cb,i←PEKS(pp,PkS,PkR,SkO,wb,i), and returns a ciphertext set \(\vec C_{b} = \left (C_{b,1},\dots,C_{b,n}\right)\) to the adversary \(\mathcal {A}\).
Phase 2: The adversary \(\mathcal {A}\) can continue to query \(\mathcal {O}_{C}\) and/or \(\mathcal {O}_{T}\) for any keyword w except \(w \in \vec w_{0} \cup \vec w_{1} \).
Guess: The adversary \(\mathcal {A}\) sends its guess bit \(\hat {b^{\prime }}\) to \(\mathcal {C}\). Therefore, the condition that \(\mathcal {A}\) wins the game is b′=b. The advantage of any PPT attacker \(\mathcal {A}\) who wins this game is defined as \(Adv_{\mathcal {A}}^{MCI}(\kappa) = \vert Pr[b^{\prime } = b] - \frac {1}{2} \vert \).
Game 2: Multi-trapdoor privacy.
Setup: Same as Game 1, \(\mathcal {C}\) generates (PkS,SkS), (PkO,SkO) and (PkR,SkR) and gives (pp,(PkS,SkS)) to \(\mathcal {A}\).
Phase 1: As in Game 1, an adversary can adaptively query the ciphertext oracle \(\mathcal {O}_{C}\) and trapdoor oracle \(\mathcal {O}_{T}\) in polynomial time.
Challenge: \(\mathcal {A}\) sends two challenge keywords tuples \(\vec w_{0} = \left (w_{0,1},\dots,w_{0,n}\right)\), \(\vec w_{1} = \left (w_{1,1},\dots, w_{1,n}\right)\) to \(\mathcal {C}\). However, the attacker cannot query the challenge key in tuple \(\vec w_{0}\) or \(\vec w_{1}\) in advance. \(\mathcal {C}\) computes and returns a trapdoor set \(\vec T_{b} = \left (T_{b,1},\dots,T_{b,n}\right)\) of a random bit b∈{0,1}.
Phase 2: As in Phase 1, \(\mathcal {A}\) can continue to query \(\mathcal {O}_{C}\) and/or \(\mathcal {O}_{T}\) for any keyword w except \(w \in \vec w_{0} \cup \vec w_{1} \).
Guess: \(\mathcal {A}\) sends its guess bit \(\hat {b^{\prime }}\) to \(\mathcal {C}\). Therefore, \(\mathcal {A}\) will win the game if b′=b. The advantage of all PPT adversaries \(\mathcal {A}\) who win the game is defined as \(Adv_{\mathcal {A}}^{MTP}(\kappa) = \vert Pr[b^{\prime } = b] - \frac {1}{2} \vert \).
Game 3: Designated testability.
\(\mathcal {A}\) is an external adversary who can get the keyword ciphertext and the trapdoor by monitoring the public channel. However, \(\mathcal {A}\) cannot get the secret key of the server. Designated testability ensures that only a designated server who own the private key can search a keyword over ciphertexts.
Setup: \(\mathcal {C}\) runs KeyGenS, KeyGenO and KeyGenR algorithms with pp to generate the public and private key pairs (PkS,SkS), (PkO,SkO) and (PkR,SkR). It then sends the tuple (pp,PkS) to \(\mathcal {A}\).
Phase 1: There are two oracles as follows, which allow \(\mathcal {A}\) to query in polynomial time.
Ciphertext Oracle \(\mathcal {O}_{C}\): With (PkO,PkR,PkS,w), \(\mathcal {C}\) computes and returns the ciphertext Cw.
Trapdoor Oracle \(\mathcal {O}_{T}\): Input a tuple (PkO,PkR,w), \(\mathcal {C}\) computes and outputs trapdoor Tw.
Challenge: \(\mathcal {A}\) sends two challenge keywords w0, w1 to \(\mathcal {C}\), then \(\mathcal {C}\) calculates and outputs Cb of a random bit b∈{0,1}.
Phase 2: As in Phase 1, \(\mathcal {A}\) can carry on querying for any keyword wi except wi∈(w0,w1).
Guess: The adversary \(\mathcal {A}\) sends its guess bit \(\hat {b^{\prime }}\) to \(\mathcal {C}\). Therefore, \(\mathcal {A}\) wins the game if b′=b. The advantage for all PPT attackers who win the game is defined as \(Adv_{\mathcal {A}}^{DT}(\kappa) = \vert Pr[b^{\prime } = b] - \frac {1}{2} \vert \).
Proposed scheme
In this section, we propose a concrete construction of our scheme that can provide multi-ciphertext indistinguishability, multi-trapdoor privacy and security against key guessing attack. The details of proposed scheme are described as follows.
Setup(κ): From κ, it chooses a bilinear pairing \(\hat {e}:{\mathbb {G}_{1} \times \mathbb {G}_{1} \rightarrow \mathbb {G}_{2}}\), where \(\mathbb {G}_{1},\mathbb {G}_{2}\) are cyclic groups of prime order q, and selects two random generators \(g,h \in \mathbb {G}_{1}\) and two cryptographic hash functions H1: \(\mathbb {G}_{1} \rightarrow \mathbb {Z}_{q}^{*}\), H2: \(\{0,1\}^{*} \rightarrow \mathbb {Z}_{q}^{*}\). It returns the public parameter \(pp = \left (\mathbb {G}_{1},\mathbb {G}_{2},q,g,h, \hat {e},H_{1},H_{2}\right)\).
KeyGenO(pp): It takes a grobal pubilc parameter pp as inputs, selects x←Zp randomly and defines PkO=gx and SkO=x. It outputs the data owner's public/secret key pairs (PkO,SkO).
KeyGenR(pp): From pp, it chooses a random y←Zp and sets PkR=gy and SkR=y then returns the receiver's public/secret key pairs (PkR,SkR).
KeyGenS(pp): By a grobal pubilc parameter pp, it selects randomly z←Zp, and defines PkS=hz and SkS=z. Finally it returns the server's public/secret key pairs (PkS,SkS).
PEKS(pp,PkS,PkR,SkO,w): Given the public parameter pp, PkS, PkR, SkO and a keyword w, a data owner performs the following steps:
Select a number \(r \in _{R} \mathbb {Z}_{q}^{*}\).
Calculate C1=hr, \( \kern0.3em {C}_2=\kern0.3em \hat{e}{\left(\kern0.3em P{k}_R,P{k}_S\kern0.3em \right)}^{rS{k}_O{H}_2(w)}\kern0.3em \) and C3=grk, where \(\phantom {\dot {i}\!}k = H_{1}\left (Pk_{R}^{Sk_{O}}\right)\).
Output the ciphertext Cw=(C1,C2,C3) of w.
Trapdoor(pp,PkO,SkR,w′): From pp, PkO of data owner, SkR of receiver and a keyword w′, a receiver executes the following steps:
Choose a number \(s \in _{R} \mathbb {Z}_{q}^{*}\).
Compute T1=PkSs and \(\phantom {\dot {i}\!}T_{2} = {{Pk_{O}}^{Sk_{R}H_{2}(w^{\prime })}} \cdot g^{sk}\), where \(\phantom {\dot {i}\!}k = H_{1}\left (Pk_{O}^{Sk_{R}}\right)\).
Return the trapdoor \(T_{w^{\prime }} = (T_{1},T_{2})\)
Test(pp,SkS,Cw,Tw): After receiving \(T_{w^{\prime }}\), the server searchs over keyword ciphertexts {Cw} by testing \(\phantom {\dot {i}\!}\hat {e}(T_{2},C_{1}^{Sk_{S}}) = \hat {e}(T_{1},C_{3}) \cdot C_{2}\) using his private key SkS. If the equation holds, it outputs 1; otherwise, it outputs 0.
Correctness: Assume that (PkO,SkO), (PkR,SkR) and (PkS,SkS) be the data owner, the receiver and the server's public/secret key pairs respectively. Cw=(C1,C2,C3) is the ciphertext of a keyword w generated by the owner. \(T_{w^{\prime }} = (T_{1},T_{2})\) is a trapdoor of a keyword generated by the receiver. It follows that:
$$\begin{array}{*{20}l} \hat{e}(T_{2},C_{1}^{Sk_{S}}) &= \hat{e}\left({Pk_{O}}^{Sk_{R}H_{2}(w')+{rk}}, h^{{Sk_{S}}r} \right) \\ &= \hat{e}(g,h)^{rxyzH_{2}(w^{\prime}vv)} \cdot \hat{e}(g,h)^{rszk}.\\ \hat{e}(T_{1},C_{3}) \cdot C_{2} &= \hat{e}\left(h^{zs},g^{rk}\right) \cdot \hat{e}\left(g^{y},h^{z}\right)^{rxH_{2}(w)}\\ &= \hat{e}(g,h)^{rxyzH_{2}(w)} \cdot \hat{e}(g,h)^{rszk}. \end{array} $$
Thus, if w=w′, then \(\phantom {\dot {i}\!}\hat {e}\left (T_{2},C_{1}^{Sk_{S}}\right) = \hat {e}\left (T_{1},C_{3}\right) \cdot C_{2}\) holds with probability 1; otherwise, it holds with overwhelming probability by the collision resistance of the hash function H2.
Security proof
In this subsection, we prove that our scheme achieves the security of MCI, MTP and designated testability. Formally, we have the following theorems.
Under the assumption of DBDH, our scheme satisfies multi-ciphertext indistinguishability.
Proof Assume that \(\mathcal {A}\) is an adversary who tries to destroy the MCI security. And the algorithm \(\mathcal {C}\) for solving DBDH problem is established. Given a instance of this problem, such as \(Y = \left (\mathbb {G}_{1},\mathbb {G}_{2},\hat {e},q,g,g^{x},g^{y},g^{z},Z_{1}\right)\), the algorithm \(\mathcal {C}\) works exactly as follows.
Setup: \(\mathcal {C}\) randomly selects two hash functions \(H_{1}:\mathbb {G}_{1} \rightarrow \mathbb {Z}_{q}^{*}\), \(H_{2}:\{0,1\}^{*} \rightarrow \mathbb {Z}_{q}^{*}\) and sets \(pp = \left (\mathbb {G}_{1}, \mathbb {G}_{2}, q, g, \hat {e}, h =g^{\alpha }, H_{1}, H_{2}\right)\), PkO=gx, PkR=gy and (PkS,SkS)=(ht,t). It then sends pp and (PkS,SkS) to \(\mathcal {A}\).
Phase 1: We define several oracles as follows, which allow \(\mathcal {A}\) to query many times. We assume that \(\mathcal {A}\) cannot query the same oracle more than once.
Hash Oracle \(\phantom {\dot {i}\!}\mathcal {O}_{H_{1}}\): In response to the H1 query, the oracle maintains a tuple list \(\phantom {\dot {i}\!}L_{H_{1}}= \left \{< m_{i},a_{i}>\right \}\). We assume that \(\phantom {\dot {i}\!}\mathcal {O}_{H_{1}}\) can be asked by attackers for \(\phantom {\dot {i}\!}q_{H_{1}}\) times at most. For querying mi to the oracle, it will perform the following operations: At first, if \(\hat {e}(g,m_{i}) = \hat {e}\left (g^{x},g^{y}\right)\), \(\mathcal {C}\) randomly returns a bit b′ and halts. Otherwise \(\mathcal {C}\) checks whether mi exists in the tuple list. If so, \(\mathcal {C}\) takes out the corresponding tuple and returns ai to \(\mathcal {A}\). Otherwise, it randomly chooses a new exponent ai∈{0,1}κ, stores <mi,ai> in \(\phantom {\dot {i}\!}L_{H_{1}}\) and returns ai to \(\mathcal {A}\).
Hash Oracle \(\phantom {\dot {i}\!}\mathcal {O}_{H_{2}}\): In response to the H2 query, the oracle maintains a tuple list \(\phantom {\dot {i}\!}L_{H_{2}}= \left \{< w_{i},b_{i}>\right \}\). We assume that \(\phantom {\dot {i}\!}\mathcal {O}_{H_{2}}\) can be asked by attackers for \(\phantom {\dot {i}\!}q_{H_{2}}\) times at most. When submitting the keywords wi to the Oracle for query, it will perform the following operations: At first, it checks whether wi exists in the tuple list. if it exists, \(\mathcal {C}\) will take out the corresponding tuple and return bi to \(\mathcal {A}\). Otherwise, it randomly selects a new exponent bi∈{0,1}κ, stores <wi,bi> in \(\phantom {\dot {i}\!}L_{H_{2}}\)and returns bi to \(\mathcal {A}\).
Oracle \(\mathcal {O}_{E}\): It takes public key Pki as input. To response to the queries, the oracle maintains a tuple list LE={<Pki,ci,Vi>}, and it is assumed that \(\mathcal {O}_{E}\) can be asked by attackers for qE times at most. When submitting Pki to the Oracle query, it will perform the following operations: At first, if Pki=gx or Pki=gy, \(\mathcal {C}\) randomly returns a bit b′ and halts. Otherwise \(\mathcal {C}\) tests whether exists Pki in the tuple list. If so, \(\mathcal {C}\) chooses the candidate tuple and returns ci to \(\mathcal {A}\). Otherwise, it randomly selects a new exponent ci∈{0,1}κ, and computes \(\phantom {\dot {i}\!}V_{i} = {Pk_{i}}^{c_{i}}\). Finally, it stores <Pki,ci,Vi> in LE and outputs ci.
Ciphertext Oracle \(\mathcal {O}_{Ciphertext}\): Input a tuple (pkO,PkR,wi), which wi∈{0,1}∗, it randomly chooses \(r_{i} \in \mathbb {Z}_{q}^{*}\), and computes \(C_{w_{i}} = \left (C_{1w_{i}},C_{2w_{i}},C_{2w_{i}}\right)\) as follows.
If (PkO,PkR)=(gx,gy) or (PkO,PkR)=(gy,gx), then it sets \(\phantom {\dot {i}\!}g^{z} = g^{\alpha r_{i}}\), and computes \(C_{1w_{i}} = g^{z}\), \(\phantom {\dot {i}\!}C_{2w_{i}} = Z_{1}^{tb_{i}}\), \(C_{3w_{i}} = g^{r_{i}a_{i}} \).
Otherwise, at least one Pki in (PkO,PkR) is equal to gx or gy. It computes H2(wi)=bi, k=ai, and returns to \(\mathcal {A}\) with \(C_{w_{i}} = \left (C_{1w_{i}},C_{2w_{i}},C_{3w_{i}}\right)\), where \(C_{1w_{i}} = h^{r_{i}}\), \(C_{2w_{i}}=\hat {e}(g^{y},h^{r_{i}})^{tc_{o}b_{i}}\) and \(C_{3w_{i}} = g^{r_{i}{a_{i}}}\).
Trapdoor Oracle \(\mathcal {O}_{Trapdoor}\): Input (PkO,PkR,wi), where wi∈{0,1}∗, it randomly chooses \(s_{i} \in \mathbb {Z}_{q}^{*}\), and computes \(T_{w_{i}} = (T_{1w_{i}},T_{2w_{i}})\) as follows.
If (PkO,PkR)=(gx,gy) or (PkO,PkR)=(gy,gx), then it computes \(T_{1w_{i}} = g^{ts_{i}}\) and \(T_{2w_{j}i} = Z_{2}^{b_{i}}\cdot g^{{r_{i}a_{i}}}\).
Otherwise, at least one Pki in (PkO,PkR) equals to gx or gy. It calculates H2(wi)=bi, k=ai, and returns to \(\mathcal {A}\) with \(T_{w_{i}} = \left (T_{1w_{i}},T_{2w_{i}}\right)\), where \(T_{1w_{i}} = g^{ts_{i}} \), \(\phantom {\dot {i}\!}T_{2w_{i}}=(g^{x})^{c_{o}b_{i}}\cdot g^{{s_{i}a_{i}}}\).
Challenge: \(\mathcal {A}\) completes multiple queries on the above oracles. It selects two challenge keyword tuples \(\vec w_{0}^{*}\) and \(\vec w_{1}^{*}\), and sends them to \(\mathcal {C}\) with \(Pk_{O}^{*}\) and \(Pk_{R}^{*}\). \(\mathcal {C}\) randomly selects a random number ri and a bit \(\hat {b} \in \{0,1\}\). then \(\mathcal {C}\) outputs a ciphertext tuple \(\vec C_{{{w_{\hat {b}}}}^{*}} = \left (C_{{w_{\hat {b},1}}^{*}},\dots,C_{{w_{\hat {b},n}}^{*}}\right)\) where \(C_{{1w_{\hat {b},i}}^{*}} = g^{z}\), \(\phantom {\dot {i}\!}C_{{2w_{\hat {b},i}}^{*}} = Z_{1}^{tb_{i}}\), \(\phantom {\dot {i}\!}C_{{3w_{\hat {b},i}}^{*}} = g^{za_{i}} \).
Phase 2: As with Phase 1 of operation, \(\mathcal {A}\) continue to enquire \(\mathcal {O}_{Ciphertext}\) and/or \(\mathcal {O}_{Trapdoor}\) for any keyword wi except \(w_{i} \in \vec w_{0} \cup \vec w_{1} \).
Guess: The adversary\(\mathcal {A}\) sends its guess bit \(\hat {b^{\prime }}\) to \(\mathcal {C}\). Returns b′=0 if \(\hat {b^{\prime }} = \hat {b}\), b′=1 otherwise.
If the guess of the challenging public key is incorrect, \(\mathcal {C}\) will abort. This event is represented by E. If \(\mathcal {C}\) aborts, \(\mathcal {C}\) outputs a random bit. The termination probability of E is \(\frac {1}{q_{E}(q_{E} -1)}\), therefore, \(Pr[\overline {E}] = \frac {1}{q_{E}(q_{E} -1)}\).
Assume that algorithm \(\mathcal {C}\) is not aborted. If the simulation provided by algorithm \(\mathcal {C}\) is the same as scenario of \(\mathcal {A}\) in real attack and \(Z_{1} = \hat {e}(g,g)^{xyz}\), the adversary \(\mathcal {A}\) will win with \(Adv_{\mathcal {A}}^{MCI}(\kappa) + \frac {1}{2}\). If Z1 is randomly chosen from the group \(\mathbb {G}_{2}\), \(\phantom {\dot {i}\!}C_{{2w_{\hat {b},i}}^{*}} = Z_{1}^{Sk_{S}H_{2}(w)}\) is a random element of \(\mathbb {G}_{2}\). In this case, the trapdoor \(\vec T_{{w_{\hat {b}}}^{*}} \) and ciphertext \(\vec C_{{w_{\hat {b}}}^{*}}\) can be tested. When the keywords are consistent, test algorithm outputs 1. Therefore, \(\mathcal {A}\) has a 1/2 probability that he wins the Game 1. Thus, the advantage for \(\mathcal {C}\) in solving DBDH problem is
$$\begin{array}{*{20}l} &\quad Adv_{\mathcal{B}}^{DBDH}(\kappa)\\ &=\! \vert \!Pr[b^{\prime} = b\! \mid E]\! \cdot \!Pr[E] \!+ \!Pr[b^{\prime} = b\! \mid\! \overline{E}] \cdot\! Pr[\overline{E}]\! - \!\frac{1}{2} \vert\! \\ &= \!\vert\! \frac{1}{2} \!\cdot \!(1-\!Pr[\overline{E}])\! + \!\left(Pr[b^{\prime}\! =\! 0]\! \mid \!\overline{E}\!\cap\! b\! =\!0 \right)\!\cdot\! Pr[b\,=\,0]\!\\ & \qquad + Pr[b^{\prime} = 1 \mid \overline{E} \cap b = 1] \cdot Pr[\overline{E}] - \frac{1}{2} \vert\\ &\geq \vert Pr[\overline{E}] \cdot \left(\left(Adv_{\mathcal{A}}^{MCI} + \frac{1}{2}\right)\cdot \frac{1}{2} + \frac{1}{2} \cdot \frac{1}{2}\right) - \frac{1}{2} \end{array} $$
$$\begin{array}{*{20}l} & \qquad+\frac{1}{2} \cdot (1-Pr[\overline{E}]) + Pr[\overline{E}] \vert \\ &= \frac{1}{2}Pr[\overline{E}]Adv_{\mathcal{A}}^{MCI}(\kappa)\\ &= \frac{1}{2q_{E}(q_{E} - 1)} \cdot Adv_{\mathcal{A}}^{MCI}(\kappa). \end{array} $$
Under the assumption of DDH, our scheme satisfies semantically MTP security.
Proof Assume that \(\mathcal {A}\) is an external opponent who tries to crack the Multi-trapdoor Privacy. Moreover, the algorithm \(\mathcal {C}\) for solving the DDH problem is established. Given a instance of this problem, such as \(Y = \left (\mathbb {G}_{1},q,g,g^{x},g^{y},Z_{2}\right)\), the algorithm \(\mathcal {C}\) works exactly as follows.
Setup: \(\mathcal {C}\) randomly selects two hash functions \(H_{1}:\mathbb {G}_{1} \rightarrow \mathbb {Z}_{q}^{*}\), \(H_{2}:\{0,1\}^{*} \rightarrow \mathbb {Z}_{q}^{*}\) and sets \(pp = \left (\mathbb {G}_{1},\mathbb {G}_{2},q,g,h =g^{\alpha },H_{1},H_{2}\right)\), PkO=gx, PkR=gy and (PkS,SkS)=(ht,t). It then sends pp to \(\mathcal {A}\).
Phase 1: Same as in Theorem 1.
Challenge: \(\mathcal {A}\) completes multiple queries on the above research. It selects two challenge keyword tuples \(\vec w_{0}^{*}\) and \(\vec w_{1}^{*}\), and sends them to \(\mathcal {C}\) with \(Pk_{O}^{*}\) and \(Pk_{R}^{*}\). \(\mathcal {C}\) randomly select a number si and a bit \(\hat {b} \in \{0,1\}\). Then, \(\mathcal {C}\) returns a trapdoor set \(\vec T_{{w_{\hat {b}}}^{*}}= \left (T_{{w_{\hat {b},1}}^{*}},\dots, T_{{w_{\hat {b},n}}^{*}}\right)\) where \(T_{{1w_{\hat {b},i}}^{*}} = h^{ts_{i}}\), \(T_{{2w_{\hat {b},i}}^{*}} = Z_{2}^{b_{i}} \cdot g^{s_{i}a_{i}}\).
Phase 2: \(\mathcal {A}\) continue to enquire \(\mathcal {O}_{C}\) and/or \(\mathcal {O}_{T}\) for any keyword wi except \(w_{i} \in \vec w_{0} \cup \vec w_{1} \).
Guess: The adversary \(\mathcal {A}\) sends its guess bit \(\hat {b^{\prime }}\) to \(\mathcal {C}\). He returns b′=0 if \(\hat {b^{\prime }} = \hat {b}\), b′=1 otherwise.
If the guess of challenging public key is incorrect, \(\mathcal {C}\) will abort. This event will be represented by E. If \(\mathcal {B} \) aborts, \(\mathcal {C}\) outputs a random bit. The probability that it being equal to b is 1/2. According to the random guess of b, the termination probability of E is \(\frac {1}{q_{E}\left (q_{E} -1 \right)}\), therefore, \(Pr[\overline {E}] = \frac {1}{q_{E}\left (q_{E} -1 \right)}\).
Othervise, \(\mathcal {C}\) does not abort. If the simulation provided by algorithm \(\mathcal {C}\) is the same as scenario of \(\mathcal {A}\) in real attack and Z2=gxy, an adversary \(\mathcal {A}\) will win the game with the probability of \(Adv_{\mathcal {A}}^{MTP}(\kappa) + \frac {1}{2}\). If Z2 is randomly chosen from the group \(\mathbb {G}_{2}\), \(T_{{2w_{\hat {b},i}}^{*}} = Z_{2}^{b_{i}} \cdot g^{s_{i}a_{i}}\) will be a random element of \(\mathbb {G}_{2}\). Therefore, the challenge trapdoor tuple hides \(\hat {b}\) completely. In this case, the adversary can test the trapdoor \(\vec T_{{w_{\hat {b}}}^{*}} \) and the ciphertext \(\vec C_{{w_{\hat {b}}}^{*}}\). When the keywords are equal, the test algorithm outputs 1. Thus, the advantage for \(\mathcal {C}\) in solving DDH problem is equal to the advantage in Theorem 1.
Under the assumption of DBDH, our scheme satisfies designated testability.
Proof Assume that \(\mathcal {A}\) is an attacker who tries to crack the designated testability and the challenger \(\mathcal {C}\) wants to solve DBDH problem. Given a instance of this problem, such as \(Y = \left (\mathbb {G}_{1},\mathbb {G}_{2},\hat {e},q,h,h^{x},h^{y},h^{z},Z_{3}\right)\), the algorithm \(\mathcal {C}\) works as follows.
Setup: \(\mathcal {C}\) randomly selects two hash functions \(H_{1}:\mathbb {G}_{1} \rightarrow \mathbb {Z}_{q}^{*}\), \(H_{2}:\{0,1\}^{*} \rightarrow \mathbb {Z}_{q}^{*}\) and sets \(pp = (\mathbb {G}_{1}, \mathbb {G}_{2}, q\), \(\hat {e}\), h, H1, H2, g=hx), (PkO,SkO)=(gs,s), (PkR,SkR)=(gt,t) and PkS=hz. It then sends pp to \(\mathcal {A}\).
Phase 1: Hash Oracle \(\phantom {\dot {i}\!}\mathcal {O}_{H_{1}}\) and Hash Oracle \(\phantom {\dot {i}\!}\mathcal {O}_{H_{2}}\) are same in Theorem 1. We only define Exact Oracle \(\mathcal {O}_{E}\) as follows.
Exact Oracle \(\mathcal {O}_{E}\): Given Pk expect PkS, the algorithm \(\mathcal {C}\) returns Sk.
Frist \(\mathcal {A}\) performs multiple queries on the above oracle. It selects two challenge keywords \( w_{0}^{*}\) and \(w_{1}^{*}\). Then \(\mathcal {A}\) returns \((w_{0}^{*}, w_{1}^{*})\) to \(\mathcal {C}\) together with \(Pk_{O}^{*}\) and \(Pk_{R}^{*}\). \(\mathcal {C}\) selects a number \(y \in _{R} \mathbb {Z}_{q}\) and a bit \(\hat {b} \in _{R} \{0,1\}\), and outputs \(\phantom {\dot {i}\!}C_{{w_{\hat {b}}}^{*} }= \left (C_{{1w_{\hat {b}}}^{*}}, C_{{2w_{\hat {b}}}^{*}}, C_{{3w_{\hat {b}}}^{*}}\right)\) where \(C_{{1w_{\hat {b}}}^{*}} = h^{y}\), \(\phantom {\dot {i}\!}C_{{2w_{\hat {b}}}^{*}} = Z_{3}^{stb_{i}}\), \(\phantom {\dot {i}\!}C_{{3w_{\hat {b}}}^{*}} = g^{yk}\).
Phase 2: \(\mathcal {A}\) continue to enquire \(\mathcal {O}_{C}\) and/or \(\mathcal {O}_{T}\) for any keyword wi except wi∈{w0,w1}.
Guess: The adversary \(\mathcal {A}\) transmits its guess bit \(\hat {b^{\prime }}\) to \(\mathcal {C}\). Returns b′=0 if \(\hat {b^{\prime }} = \hat {b}\), b′=1 otherwise. If the simulation provided by algorithm \(\mathcal {C}\) is the same as \(\mathcal {A}\) in real attack and \(Z = \hat {e}(h,h)^{xyz}\), \(\mathcal {A}\) will win the game with the probability of \(Adv_{\mathcal {A}}^{DT}(\kappa) + \frac {1}{2}\). If Z is randomly chosen from the group \(\mathbb {G}_{2}\), \(\phantom {\dot {i}\!}C_{{2w_{\hat {b},i}}^{*}} = Z^{Sk_{S}H_{2}(w)}\) is a well distributed challenge ciphertext. And \(\mathcal {A}\) has a 1/2 probability of winning the game. Thus, \(\mathcal {C}\)'s advantage in solving DBDH problem is
$$\begin{array}{*{20}l} &Adv_{\mathcal{B}}^{DBDH}(\kappa) \\ &= \vert Pr[b^{\prime} = 1 \mid b = 1] \cdot Pr[b= 1] \\ & \qquad + Pr[b^{\prime} = 1 \mid b = 1] \cdot Pr[b= 1] - \frac{1}{2} \vert \\ &= \vert \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2}\cdot\left(Adv_{\mathcal{A}}^{DT}(\kappa) + \frac{1}{2}\right) - \frac{1}{2} \vert \\ &= \frac{1}{2}Adv_{\mathcal{A}}^{DT}(\kappa). \end{array} $$
Perfomance analysis
In this section, we analyze the performance of our scheme and the existing schemes(PAKES scheme [8], dIBAEKS scheme [12], SCF-PEKS scheme [3], dPEKS scheme [4] and Pan-Li's scheme [10]) in terms of computational and communication overheads. Moreover, we analyze the security between these PEKS schemes in MCI, MTP and security against KGA.
To evaluate the efficiency, we implemented the operations in our schemes using the MRACL library [37] on a personal notebook computer with an I7-8750H 2.20GHz processor, 16 GB memory, and Window 10 operating system.
First, we give the elapsed time of main operations used in searchable encryption schemes in Table 2. Main operations are pairing operation P, Hash-to-point operation Hp, modular exponentiation E and multiplication operation M in G1, where P≈Hp>M>E≫H. The general hash operation takes less time than the above operations in Table 2. Thus, it is ignored in our computation analysis.
Table 2 Symbols and execution times(ms)
From Table 3, we give a theoretical efficiency comparison in computational time and communication complexity of PEKS algorithm, Trapdoor algorithm, and Test algorithm of our scheme and previous schemes [3, 4, 8, 10, 12]. In terms of computational efficiency, compared with other algorithms, PEKS and trapdoor algorithms are more efficient without using hash-to-point operation. Among the Test algorithms, our scheme is slightly weaker than other schemes, because it adds the designated testability to ensure that only the specified server can perform search operations. In terms of communication efficiency, our efficiency is basically the same as other schemes. Figs. 2, 3 and 4 demonstrate the practical performance of PEKS algorithm, Trapdoor algorithm, and Test algorithm, respectively.
Running Time of PEKS Algorithm
Running Time of Trapdoor Algorithm
Running Time of Test Algorithm
Table 3 Computation and Communcitaion efficiency comparison
As shown in Fig. 2, the computation cost to encrypt the keywords is lower than the three schemes [3, 4, 12] and is similar to that of Huang et al.'s scheme [8] and Pan and Li's scheme [10]. For the efficiency of trapdoor algorithm, Fig. 3 illustrates that the trapdoor algorithm in our scheme runs much faster than that in all schemes [3, 4, 8, 10, 12] because our trapdoor algorithm performs no pairing and Hash-to-point operations. In Fig. 4, the computation complexity in our scheme is higher than Baek et al.'s scheme [3] and is not worse than other four schemes. To ensure that the user-side algorithms (Trapdoor and PEKS algorithms) have higher security and efficiency, we add the server's private key to the test algorithm for stronger security, thus the efficiency of the server's Test algorithm is compromised.
Moreover, Table 4 illustrates the security comparison including MCI security, MTP security, Inside KGA, and Requirement for the secure channel between our scheme and these existing schemes. As shown in Table 4, Huang et al.' s scheme [8] can resist inside KGA, but it needs a secure channel and cannot provide MCI security. Li et al.'s scheme [12], Baek et al.'s scheme [3] and Rhee et al.'s scheme [4] can provide MCI security but not guarantee MTP and security against KGA. Pan and Li's scheme [10] is able to resist inside KGA and have MTP security but cannot satisfy MCI security. Our scheme satisfies MCI, MTP and security against inside KGA.
Table 4 Security comparison
In this paper, we first analyze the security of Li et al.'s scheme and propose a multi-trapdoor attack against it. Next, we construct a secure public-key searchable encryption scheme with designated server based on Diffie-Hellman problem. It is proved that our scheme can provide multi-ciphertext indistinguishability, multi-trapdoor privacy security and designated testability. Then we compare our scheme with others in terms of communication cost and computational cost. The results show that our scheme is more efficient in keyword ciphertext and trapdoor algorithms. However, our scheme can not prevent the server from executing the multi-trapdoor attack since the server can construct a certain equation by his private key to obtain the relationship of multiple trapdoors. As our future work, we will explore achieving multi-trapdoor privacy of keywords for the inside servers.
The datasets used during the current study are available from the corresponding author on reasonable request.
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The authors would like to thank the anonymous reviewers for their insightful comments and suggestions on improving this paper.
This work is supported by the National Natural Science Foundation of China(No.61702153, No.61972124) and the Natural Science Foundation of Zhejiang Province (No.LY19F020021).
School of Information Science and Technology, Hangzhou Normal University, Hangzhou, China
Junling Guo, Lidong Han, Guang Yang & Xuejiao Liu
Key Laboratory of Cryptography Technology of Zhejiang Province, Hangzhou Normal University, Hangzhou, China
Lidong Han & Xuejiao Liu
School of Computer Science and Technology, Qingdao Universit, Qingdao, China
Chengliang Tian
Junling Guo
Lidong Han
Guang Yang
Xuejiao Liu
This research paper was completed by the joint efforts of five authors. Therefore, any author participates in every part of the paper. But the basic roles of each author are summarized as follows: J.G. is the designer of the proposed model and method. L.H. is the corresponding author and the coordinator of the group, assisting J.G. in model design. G.Y. is the implementer and tester of the algorithm. X.L. is the main reviewer of this paper. C.T. is responsible for the experiment of the proposed method. All authors have read and agreed to the published version of the manuscript.
Junling Guo is a postgraduate in Cyberspace Security at School of Information Science and Technology, Hangzhou Normal University, China. His research interests include searchable encryption and public key cryptography.
Lidong Han received his Ph.D. degree from school of mathematics in Shandong university in 2010. Currently, he is working at Key Laboratory of Cryptography Technology of Zhejiang Province, and School of Information Science and Technology, Hangzhou Normal University. His research interests include cryptography, cloud computing, and remote user authentication.
Guang Yang is a postgraduate in Cyberspace Security at School of Information Science and Technology, Hangzhou Normal University, China. His research interests include data integrity, searchable encryption and public-key cryptography.
Xuejiao Liu received the BS, MS and PhD degrees in computer science from Huazhong Normal University, Wuhan, China. Now she is an associate professor in Hangzhou Normal University, Hangzhou, China. Her research interests cover network security, cloud security, security of Internet of vehicle and etc.
Chengliang Tian received the B.S. and M.S. degrees in mathematics from Northwest University, Xi'an, China, in 2006 and 2009, respectively, and the Ph.D. degree in information security from Shandong University, Jinan, China, in 2013. He held a postdoctoral position with the State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing. He is currently with the College of Computer Science and Technology, Qingdao University, as an Associate Professor. His research interests include latticebased cryptography, cloud computing security and privacy-preserving technology.
Correspondence to Lidong Han.
Guo, J., Han, L., Yang, G. et al. An improved secure designated server public key searchable encryption scheme with multi-ciphertext indistinguishability. J Cloud Comp 11, 14 (2022). https://doi.org/10.1186/s13677-022-00287-5
Searchable encryption
Keyword guessing attack
Multi-ciphertext indistinguishability
Diffie-Hellman problem
Multi-trapdoor privacy | CommonCrawl |
A very important and complicated machine consists of $n$ wheels, numbered $1, 2, \ldots , n$. They are actually cogwheels, but the cogs are so small that we can model them as circles on the plane. Every wheel can spin around its center.
Two wheels cannot overlap (they do not have common interior points), but they can touch. If two wheels touch each other and one of them rotates, the other one spins as well, as their micro-cogs are locked together.
A force is put to wheel $1$ (and to no other wheel), making it rotate at the rate of exactly one turn per minute, clockwise. Compute the rates of other wheels' movement. You may assume that the machine is not jammed (the movement is physically possible).
Each test case consists of one line containing the number of wheels $n$ ($1 \leq n \leq 1\, 000$) . Each of the following lines contain three integers $x$, $y$ and $r$ ($-10\, 000 \leq x, y \leq 10\, 000$; $1 \leq r \leq 10\, 000$) where $(x, y)$ denote the Cartesian coordinates of the wheel's center and $r$ is its radius.
For each test case, output $n$ lines, each describing the movement of one wheel, in the same order as in the input. For every wheel, output either $p/q$ clockwise or $p/q$ counterclockwise, where the irreducible fraction $p/q$ is the number of wheel turns per minute. If $q$ is $1$, output just $p$ as an integer. If a wheel is standing still, output not moving. | CommonCrawl |
HomeTextbook AnswersMathAlgebraAlgebra: A Combined Approach (4th Edition)Chapter 6 - Section 6.3 - Factoring Trinomials of the Form ax2+bx+c - Exercise Set - Page 43535
Algebra: A Combined Approach (4th Edition)
by Martin-Gay, Elayn
Chapter R Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Appendix A Appendix B Appendix C Appendix D Appendix E Appendix F Appendix G Appendix H Section 6.1 - The Greatest Common Factor - Practice Section 6.1 - The Greatest Common Factor - Practice Section 6.1 - The Greatest Common Factor - Practice Section 6.1 - The Greatest Common Factor - Practice Section 6.1 - The Greatest Common Factor - Practice Section 6.1 - The Greatest Common Factor - Practice Section 6.1 - The Greatest Common Factor - Vocabulary and Readiness Check Section 6.1 - The Greatest Common Factor - Exercise Set Section 6.1 - The Greatest Common Factor - Exercise Set Section 6.1 - The Greatest Common Factor - Exercise Set Section 6.1 - The Greatest Common Factor - Exercise Set Section 6.2 - Factoring Trinomials of the Form x2+bx+c - Practice Section 6.2 - Factoring Trinomials of the Form x2+bx+c - Practice Section 6.2 - Factoring Trinomials of the Form x2+bx+c - Practice Section 6.2 - Factoring Trinomials of the Form x2+bx+c - Vocabulary and Readiness Check Section 6.2 - Factoring Trinomials of the Form x2+bx+c - Exercise Set Section 6.2 - Factoring Trinomials of the Form x2+bx+c - Exercise Set Section 6.2 - Factoring Trinomials of the Form x2+bx+c - Exercise Set Section 6.3 - Factoring Trinomials of the Form ax2+bx+c - Practice Section 6.3 - Factoring Trinomials of the Form ax2+bx+c - Practice Section 6.3 - Factoring Trinomials of the Form ax2+bx+c - Practice Section 6.3 - Factoring Trinomials of the Form ax2+bx+c - Vocabulary and Readiness Check Section 6.3 - Factoring Trinomials of the Form ax2+bx+c - Exercise Set Section 6.3 - Factoring Trinomials of the Form ax2+bx+c - Exercise Set Section 6.3 - Factoring Trinomials of the Form ax2+bx+c - Exercise Set Section 6.4 - Factoring Trinomials of the Form ax2+bx+c by Grouping - Practice Section 6.4 - Factoring Trinomials of the Form ax2+bx+c by Grouping - Vocabulary and Readiness Check Section 6.4 - Factoring Trinomials of the Form ax2+bx+c by Grouping - Exercise Set Section 6.4 - Factoring Trinomials of the Form ax2+bx+c by Grouping - Exercise Set Section 6.5 - Factoring by Special Products - Practice Section 6.5 - Factoring by Special Products - Practice Section 6.5 - Factoring by Special Products - Practice Section 6.5 - Factoring by Special Products - Practice Section 6.5 - Factoring by Special Products - Practice Section 6.5 - Factoring by Special Products - Vocabulary and Readiness Check Section 6.5 - Factoring by Special Products - Exercise Set Section 6.5 - Factoring by Special Products - Exercise Set Section 6.5 - Factoring by Special Products - Exercise Set Section 6.5 - Integrated Review - Choosing a Factoring Strategy Section 6.5 - Integrated Review - Choosing a Factoring Strategy Section 6.6 - Solving Quadratic Equations by Factoring - Practice Section 6.6 - Solving Quadratic Equations by Factoring - Practice Section 6.6 - Solving Quadratic Equations by Factoring - Practice Section 6.6 - Solving Quadratic Equations by Factoring - Practice Section 6.6 - Solving Quadratic Equations by Factoring - Practice Section 6.6 - Solving Quadratic Equations by Factoring - Vocabulary and Readiness Check Section 6.6 - Solving Quadratic Equations by Factoring - Exercise Set Section 6.6 - Solving Quadratic Equations by Factoring - Exercise Set Section 6.6 - Solving Quadratic Equations by Factoring - Exercise Set Section 6.7 - Quadratic Equations and Problem Solving - Practice Section 6.7 - Quadratic Equations and Problem Solving - Practice Section 6.7 - Quadratic Equations and Problem Solving - Practice Section 6.7 - Quadratic Equations and Problem Solving - Practice Section 6.7 - Quadratic Equations and Problem Solving - Exercise Set Section 6.7 - Quadratic Equations and Problem Solving - Exercise Set Section 6.7 - Quadratic Equations and Problem Solving - Exercise Set Section 6.7 - Quadratic Equations and Problem Solving - Exercise Set Section 6.7 - Quadratic Equations and Problem Solving - Exercise Set Group Activity Vocabulary Check Review Review Review Review Test Test Cumulative Review Cumulative Review Cumulative Review 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
Chapter 6 - Section 6.3 - Factoring Trinomials of the Form ax2+bx+c - Exercise Set - Page 435: 35
$2(x+.6357)(x+7.8642)$
$2m^{2}+17m+10$ Quadratic formula: $x=\frac{-b\pm\sqrt {b^{2}-4ac}}{2a}$ $\frac{-17+\sqrt {17^{2}-4(2)(10)}}{2(2)}=-.6357$ $\frac{-17-\sqrt {17^{2}-4(2)(10)}}{2(2)}=-7.8642$ $2*(x+.6357)(x+7.8642)$
Next Answer Chapter 6 - Section 6.3 - Factoring Trinomials of the Form ax2+bx+c - Exercise Set - Page 435: 36 Previous Answer Chapter 6 - Section 6.3 - Factoring Trinomials of the Form ax2+bx+c - Exercise Set - Page 435: 34
Section 6.1 - The Greatest Common Factor - Practice
Section 6.1 - The Greatest Common Factor - Vocabulary and Readiness Check
Section 6.1 - The Greatest Common Factor - Exercise Set
Section 6.2 - Factoring Trinomials of the Form x2+bx+c - Practice
Section 6.2 - Factoring Trinomials of the Form x2+bx+c - Vocabulary and Readiness Check
Section 6.2 - Factoring Trinomials of the Form x2+bx+c - Exercise Set
Section 6.3 - Factoring Trinomials of the Form ax2+bx+c - Practice
Section 6.3 - Factoring Trinomials of the Form ax2+bx+c - Vocabulary and Readiness Check
Section 6.3 - Factoring Trinomials of the Form ax2+bx+c - Exercise Set
Section 6.4 - Factoring Trinomials of the Form ax2+bx+c by Grouping - Practice
Section 6.4 - Factoring Trinomials of the Form ax2+bx+c by Grouping - Vocabulary and Readiness Check
Section 6.4 - Factoring Trinomials of the Form ax2+bx+c by Grouping - Exercise Set
Section 6.5 - Factoring by Special Products - Practice
Section 6.5 - Factoring by Special Products - Vocabulary and Readiness Check
Section 6.5 - Factoring by Special Products - Exercise Set
Section 6.5 - Integrated Review - Choosing a Factoring Strategy
Section 6.6 - Solving Quadratic Equations by Factoring - Practice
Section 6.6 - Solving Quadratic Equations by Factoring - Vocabulary and Readiness Check
Section 6.6 - Solving Quadratic Equations by Factoring - Exercise Set
Section 6.7 - Quadratic Equations and Problem Solving - Practice
Section 6.7 - Quadratic Equations and Problem Solving - Exercise Set
Vocabulary Check
Cumulative Review
Appendix H | CommonCrawl |
A model is presented which simulates the behavior of superthermal ions previously reported in the dayside ionosphere of Venus. The model considers effects of E $\times$ B and gradient drifts, charge exchange and collisions with the ambient neutral atmosphere and the possible effects of a wave-particle (anomalous) scattering process. Results indicate that scattering processes are required if superthermal ions are the explanation for the observed "missing pressure" component in the dayside Venus ionosphere. The scattering scale length required to match the "missing pressure" distribution is similar to the scale length previously predicted for growth of a lower hybrid beam instability.
Kramer, Leonard. "Model of superthermal ions in the dayside Venus ionosphere." (1993) Diss., Rice University. https://hdl.handle.net/1911/16636. | CommonCrawl |
Local orthogonal rectification: Deriving natural coordinates to study flows relative to manifolds
September 2020, 25(9): 3749-3763. doi: 10.3934/dcdsb.2020089
A second order accuracy in time, Fourier pseudo-spectral numerical scheme for "Good" Boussinesq equation
Zeyu Xia †, and Xiaofeng Yang ‡,
†.
School of Mathematical Sciences University of Electronic Science and Technology of China Chengdu, Sichuan 611731, China
‡.
Department of Mathematics, University of South Carolina Columbia, SC 29208
Received May 2019 Revised July 2019 Published September 2020 Early access April 2020
Figure(3)
The nonlinear stability and convergence of a numerical scheme for the "Good" Boussinesq equation is provided in this article, with second order temporal accuracy and Fourier pseudo-spectral approximation in space. Instead of introducing an intermediate variable $ \psi $ to approximate the first order temporal derivative, we apply a direct approximation to the second order temporal derivative, which in turn leads to a reduction of the intermediate numerical variable and improvement in computational efficiency. A careful analysis reveals an unconditional stability and convergence for such a temporal discretization. In addition, by making use of the techniques of aliasing error control, we obtain an $ \ell^\infty (0,T^*; H^2) $ convergence for $ u $ and $ \ell^\infty (0,T^*; \ell^2) $ convergence for the discrete time-derivative of the solution in this paper, in comparison with the $ \ell^\infty (0,T^*; \ell^2) $ convergence for $ u $ and the $ \ell^\infty (0,T^*; H^{-2}) $ convergence for the time-derivative, given in [19].
Keywords: "Good" Boussinesq equation, Fourier pseudospectral method, second-order time-stepping, stability and convergence.
Mathematics Subject Classification: 65M06, 65M15, 65M22.
Citation: Zeyu Xia, Xiaofeng Yang. A second order accuracy in time, Fourier pseudo-spectral numerical scheme for "Good" Boussinesq equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3749-3763. doi: 10.3934/dcdsb.2020089
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Figure 1. Discrete $ \ell^2 $ and $ \ell^\infty $ numerical errors for $ D_N^2 u $ at $ T = 4.0 $, plotted versus $ N $, the number of spatial grid point, for the numerical scheme (2.10). The time step size is fixed as $ {\Delta t} = 10^{-4} $. An apparent spatial spectral accuracy is observed for both norms
Figure 2. Discrete $ \ell^2 $ and $ \ell^\infty $ numerical errors for $ D_N^2 u $ at $ T = 4.0 $, plotted versus $ N_K $, the number of time steps, for the numerical scheme (2.10). The spatial resolution is fixed as $ N = 512 $. The data lie roughly on curves $ CN_K^{-2} $, for appropriate choices of $ C $, confirming the full second-order temporal accuracy of the scheme
Figure 3. The numerical solutions at a sequence of time instants. The outer solid line, the outer dashed line, the inner solid line and the inner dashed line stand for the numerical solutions $ t = 4 $, $ 8 $, $ 12 $ and $ 16 $, respectively. The initial data is given by (5.3), and the physical domain is set as $ \Omega = (-40, 40) $
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Zeyu Xia Xiaofeng Yang | CommonCrawl |
Influence of particle size on non-Darcy seepage of water and sediment in fractured rock
Yu Liu1,2 &
Shuncai Li1
SpringerPlus volume 5, Article number: 2099 (2016) Cite this article
Surface water, groundwater and sand can flow into mine goaf through the fractured rock, which often leads to water inrush and quicksand movement. It is important to study the mechanical properties of water and sand in excavations sites under different conditions and the influencing factors of the water and sand seepage system. The viscosity of water–sand mixtures under different particle sizes, different concentration was tested based on the relationship between the shear strain rate and the surface viscosity. Using the self-designed seepage circuit, we tested permeability of water and sand in fractured rock. The results showed that (1) effective fluidity is in 10−8–10−5 mn+2 s2−n/kg, while the non-Darcy coefficient ranges from 105 to 108 m−1 with the change of particle size of sand; (2) effective fluidity decreases as the particle size of sand increased; (3) the non-Darcy coefficient ranges from 105 to 108 m−1 depending on particle size and showed contrary results. Moreover, the relationship between effective fluidity and the particle size of sand is fitted by the exponential function. The relationship between the non-Darcy coefficient and the particle size of sand is also fitted by the exponential function.
In China, water and sand inrush is very serious safety problem for coal mining in 20 years, there were many accidents which gave more damage to coal mining (Limin et al. 2015, 2016). The coal reserves are located at shallow depths and the thin bedrock and thick sand overburdens the strata layers, inducing connected cracks. Surface water, groundwater and sand can flow into the mine goaf through the fractured rock and lead to inrush of water and collapsing of sand, which can be seen in Fig. 1.
Water and sand inrush in the fracture
From the mechanical perspective, the result of water and sand erupting, permeating fractured rock reflects the instability of the strata layers. Therefore, studying the seepage properties of fractured rocks plays an important role in coal mining engineering. The inrush of water and sand compromises mine safety by causing instability in stress block beams, which creates surface subsidence and water resource run off.
Field tests that are conducted in order to replicate water and sand inrush are difficult; therefore, many scholars suggested conducting experimental simulations of inrushing water and sand. Yang (2009), Yang et al. (2012) and Sui et al. (2007) analyzed the angle of fluid using cemented sand to analyze the mechanisms supporting the inrushing of water and sand. The flow law was examined during various conditions and critical hydraulic gradients of sand inrush currents. Sui et al. (2008) and Xu et al. (2012) analyzed the initial position of inrushing sand based on the structure of water inrush.
Based on underground water dynamic theories, Zhang et al. (2006) created the critical condition and forecasting formula for the prevention of sand inrush by calculating the hydraulic head. Wu (2004) designed a mechanical model of sand inrush pseudo structures, and discussed the force during sand inrush and described the theory of expression of sand inrush. Zhang et al. (2015a) used a case study to discuss drills resulting in sand inrush based on the funnel model. Zhang et al. (2015b) studied the relationship between backfill and water through conducting crack zone. Moreover, river sediment engineering, the theory of sediment transmission and sediment transport mechanics are excellent subject matters to aid in studying the start and movement of sand in mines. Furthermore, the study of sediment engineering, sediment transport theory and practice, and sediment kinematics can aid in understanding the commencement, flow and inrushing sand problem (Du 2014). But others discussed water and sand form the pressure, water and sand flow in tunnel or broken rock (Limin et al. 2016; Du 2014), but the important is seepage in the fracture, which has not been discussed. The concentration and particle's influence on water and sand inrush.
In this work, permeability attributes of water–sand mixtures are obtained through testing by replicating the design system of water–sand seepage in fractures. The influence of mass concentration in water and particle size of sand on the seepage parameters are tested using specially designed instruments.
Viscosity test of water and sediment
Viscous parameters of water–sand mixture in stress-strain relationships were tested using a NDJ-8S viscosimeter in Fig. 2:
NDJ-8S viscometer
Shear strain rate γ of water–sand is defined as
$$\gamma = \frac{{\pi n_{rot}^{{}} }}{30} \times \frac{d}{(D - d)}$$
where D is the diameter of outer cylinder, d is the diameter of the rotor.
Apparent viscosity \(\mu_{a}^{{}}\) of water–sand mixture were obtained from the NDJ-8S viscosimeter and the shear stress was calculated as follows
$$\tau = \mu_{a}^{{}} \gamma$$
By changing the rotational speed of the NDJ-8S viscosimeter, several values of shear strain rate γ and shear stress were obtained and plotted on a \(\gamma - \tau\) scatter diagram. According to the shape of the \(\gamma - \tau\) diagram, the water–sand mixture was identified as non-Newton fluid, and the viscous parameter of water–sand was obtained through linear regression.
During the experiment, the diameters of sand particle are 0.038–0.044, 0.061–0.080, 0.090–0.109 and 0.120–0.180 mm. Firstly, Sand particles 0.061–0.080 mm with 20 kg/m3 sand at 20 °C was measured; the shear strain rate \(\gamma\), apparent viscosity \(\mu_{a}^{{}}\) and stress \(\tau\) of the water–sand mixture was gotten at various rotation rates (Table 1; Fig. 3).
Table 1 Angle strain rate, apparent viscosity and shear stresses at different rotating speed
Scatter plot of angle strain rate—shear stress (0.061–0.080 mm)
It can be seen that the shear strain rate γ increases monotonously along with the shear strain of the water–sand mixture. Therefore, we assume that the water–sand mixture is a power law fluid as follows
$$\tau = C\gamma_{{}}^{n}$$
where C is the consistency coefficient, n is the power exponent.
Combining Eqs. 2 and 3 yields expression of apparent viscosity as follows
$$\mu_{a}^{{}} = C\gamma^{n - 1}$$
Through linear regression, viscous parameters of water–sand (consistency coefficient C and power exponent n) were obtained, as shown in Table 2. It was deduced that the water–sand mixture was a pseudo-plastic fluid, whose viscous parameters changed with sand particle \(d_{s}\) and mass concentration of sand \(\rho_{s}\).
Table 2 Relation of surface viscosity and shearing rate
Different consistencies were tested of coefficient C and power exponent n with the diameters of sand particle sizes 0.038–0.044, 0.061–0.080, 0.090–0.109 and 0.120–0.180 mm; and sand 20, 40, 60 and 80 kg/m3 in the water. The testing results of consistency coefficient C and power exponent n are shown in Table 2.
From Table 2, consistency coefficient C increases with mass concentration in water as exponential relationship, and decreases along with the increase of sand particle; power exponent n increases along with the increase of mass concentration in water, and decreases along with the increase of sand particle.
Seepage test of water and sand in a fracture
Test principle
Figure 4 demonstrates a model of seepage in a fracture. In this paper, the aperture of fracture is 0.75 mm, the length is 12.5 mm, the height is 75 mm. b is the aperture of fracture, h is the height of the fracture, and L is the sample length.
Seepage in parallel fracture
According to Fig. 4, we can get Eq. 5.
$$V = \frac{Q}{bh}$$
where V is the velocity of seepage, Q is the flow of seepage.
For the fracture, Re is defined as Eq. 6 (Javadi et al. 2014).
$$R_{e} = \frac{\rho Q}{\mu b}$$
where Re is Reynolds number, ρ is the density, Q is the flow of seepage, μ is the fluid viscosity.
In the paper, Q is \(6.00 \times 10^{ - 4} {-}3.10 \times 10^{ - 3}\) m3/s, \(\rho = 1.02{-}1.08 \times 10^{3}\) kg/m3, \(\mu = 1.005\;{\text{mp}}_{\text{a}} \;{\text{s}}\).
So, \(R_e = \frac{\rho Q}{\mu b} = 76.5{-}421.2\) in case of higher Reynolds numbers (\(R_e \gg 1\)), the pressure losses pass from a weak inertial to a strong inertial regime, described by the Forchheimer equation (Forchheimer 1901; Chin et al. 2009; Cherubini et al. 2012, 2013; Javadi et al. 2010; Li et al. 2008), given by:
$$\rho c_{a} \frac{\partial V}{\partial t} = - \frac{\partial p}{\partial l} - \frac{\mu }{k}V - \rho \beta V^{2}$$
where \(\mu\) is fluid viscosity, \(\beta\) is non-Darcy factor, the pressure is \(p\), \(\frac{\partial p}{\partial l}\) is the pressure gradient, \(c_{a}\) is the acceleration of water and sand, \(b_{1}\) is two term coefficient.
Because of water and sand permeability parameter's particularity (permeability parameter is relevant to liquid and fracture), we use \(\mu_{e}\), \(k_{e}^{{}}\) to describe the water and sand of effective viscosity \(\mu_{e}\), effective permeability \(k_{e}^{{}}\), as shown in Eq. 8 (Liu 2014).
$$\rho c_{a}^{{}} \frac{\partial V}{\partial t} = - \frac{\partial p}{\partial l} - \frac{{\mu_{e}^{{}} }}{{k_{e}^{{}} }}V_{{}}^{n} - \rho \beta V_{{}}^{2}$$
As for one kind of non-Newton fluid, liquid viscosity and permeability in fracture of water–sand mixture were related to fluid properties and fracture aperture. Therefore, liquid viscosity and permeability were not obtained separately, and the effective fluidity \(I_{e}^{{}}\) was introduced to simplify the expression.
$$I_{e}^{{}} = \frac{{k_{e} }}{{\mu_{e} }}$$
The Eq. 8 can be changed into
$$\rho c_{a}^{{}} \frac{\partial V}{\partial t} = - \frac{\partial p}{\partial l} - \frac{1}{{I_{e}^{{}} }}V_{{}}^{n} - \beta \rho V_{{}}^{2}$$
Equation 10 calculated the momentum conservation of water–sand seepage in the fracture. For the seepage in Fig. 4, the steady-flow method was selected to measure water–sand seepage in the fracture. Equation 10 can be deduced into Eq. 11,
$$\frac{1}{{I_{e} }}V^{n} + \beta \rho V^{2} = - \frac{\partial p}{\partial l}$$
Substituting Eq. 5 into Eq. 11 yields Eq. 12
$$- dp = \frac{1}{{I_{e} }}\left( {\frac{Q}{bh}} \right)^{n} dl + \beta \rho \left( {\frac{Q}{bh}} \right)^{2} dl$$
b is the aperture of the fracture, m is the mass of sand and water.
For the length, the integrating range is [0, L]; the mass is m, the pressure of water and sand at the entrance wall were:
$$\left\{ {\begin{array}{l} {\left. p \right|_{x = 0}^{{}} = p_{0}^{{}} } \\ {\left. p \right|_{x = L}^{{}} = 0} \\ \end{array} } \right.$$
The definite integral of Eq. 12 on the interval [0, L] was
$$p = \frac{L}{{I_{e} }}\left( {\frac{Q}{bh}} \right)^{n} + \beta mL\left( {\frac{Q}{bh}} \right)^{2}$$
Introducing the sign \(\lambda_{1} = \frac{1}{{I_{e} }}\left( {\frac{1}{bh}} \right)^{n}\), \(\lambda_{2} = \frac{m\beta }{{(hb)^{2} }}\),
Therefore, Eq. 14 was obtained by using
$$\lambda_{1} Q^{n} + \lambda_{2} Q^{2} - p_{0} = 0$$
In the test, 5 flows were set as \(Q_{i}^{{}} ,i = 1,2, \ldots ,5\). Steady state values of inlet pressures were tested, and coefficients \(\lambda_{1}^{{}}\) and \(\lambda_{2}^{{}}\) were fitted. The specific process was as follows:
Equation 15 was obtained
$$\varPi = \sum\limits_{i = 1}^{5} {\left( {\lambda_{1}^{{}} Q_{i}^{n} + \lambda_{2}^{{}} Q_{i}^{2} - p_{0}^{i} } \right)_{{}}^{2} } = 0$$
In order to get the least value of the flow Q, Eq. 16 can be set as Eq. 17.
$$\left\{ \begin{aligned} \left( {\sum\limits_{i = 1}^{5} {Q_{i}^{n} Q_{i}^{n} } } \right)\lambda_{1}^{{}} + \left( {\sum\limits_{i = 1}^{5} {Q_{i}^{2} Q_{i}^{n} } } \right)\lambda_{2}^{{}} = \left( {\sum\limits_{i = 1}^{5} {Q_{i}^{n} p_{0}^{i} } } \right) \hfill \\ \left( {\sum\limits_{i = 1}^{5} {Q_{i}^{2} Q_{i}^{n} } } \right)\lambda_{1}^{{}} + \left( {\sum\limits_{i = 1}^{5} {Q_{i}^{2} Q_{i}^{2} } } \right)\lambda_{2}^{{}} = \left( {\sum\limits_{i = 1}^{5} {Q_{i}^{2} p_{0}^{i} } } \right) \hfill \\ \end{aligned} \right.$$
\(\lambda_{1}\) and \(\lambda_{2}\) were solved by Eq. 16, effective mobility \(I_{e}\) and non-Darcy \(\beta\) were obtained.
Experimental equipment and steps
Based on testing principles, a set of experimental system was designed and manufactured as shown in Fig. 5. Sand comes from the surface of the mine in northwest of China. The rock sample is the sandstone under −265 m from the Luan mine in Shanxi, China. There are five specimens of rock fracture with Joint Roughness Coefficient (JRC) 4–6, the velocity of seepage was obtained.
Scheme of system principles. 1 Sidebend; 2 pressure transmitter; 3 injection pipe; 4 vane pump; 5 agitator tank; 6 VVVF; 7 screw pump; 8 flow sensor; 9 piezometer; 10 filter box; 11 scheme of permeameter
Figure 6 illustrates the entire experimental procedure. The test steps were as follows:
Flow chart of the test
The test system was assembled according to Fig. 6 and the sample was loaded. The leakage of the experiment system was tested.
The sand grain with a diameter of 0.038–0.044 mm was placed into the mixing pool and the sand concentration was 20 kg/m3 in water.
To control the motor speed, flow and pressure under different rotational speeds were recorded while the fracture aperture 0.75 mm; the motor speeds, 200, 400, 600, 800, 1000 r/min were changed separately. Different pressures and seepage velocities of the fracture were obtained using a paperless recorder. The sand concentration \(\rho_{s}\) in water was 40, 60, 80 kg/m3 respectively.
The flow and pressure under different grain diameters (0.038–0.044, 0.061–0.080, 0.090–0.109 and 0.120–0.180 mm)were recorded during the different rotational speeds. In order to easily calculate the data, we choose the arithmetic mean of each range of the grain diameter, e.g. 0.041, 0.071, 0.100 and 0.150 mm.
According to Eqs. 15 and 16, \(I_{e}\) and \(\beta\) were calculated.
Pressure graduate
According to the pressure and velocity measured in the tests, the pressure gradient and velocity under different sand concentration in water were shown in Fig. 7 and Table 3. Table 3 lists the test result of pressure gradient and seepage velocity of the water and sand, the polynomial fitting formula and its coefficient, the power fitting formula and its coefficient.
Relationship between pressure gradient and velocity
Table 3 Relationship between pressure gradient and velocity under different sand concentration
Figure 7 and Table 3 been presented above, we can obtain: the seepage velocity of water and sand increases with pressure gradient increasing, Moreover, the greater the sand concentration in water is, the lower the seepage velocity is.
Permeability of water and sand in the fracture
Keeping the fracture aperture 0.75 mm, the permeability parameters of water and sand seepage in the fracture under particle sizes 0.041, 0.071, 0.100 and 0.150 mm are tested at 20, 40 and 60 kg/m3 sand concentration in water, as shown in Fig. 8, liquid viscosity and permeability were not obtained separately, and the effective fluidity \(I_{e}^{{}}\) was introduced to simplify the expression.
Curves of permeability parameters changing with \(d_{s}\). a Curve of \(I_{e} - d_{s}\) under 20 kg/m3, b curve of \(\beta - d_{s}\) under 20 kg/m3, c curve of \(I_{e} - d_{s}\) under 40 kg/m3, d curve of \(\beta - d_{s}\) under 40 kg/m3, e curve of \(I_{e} - d_{s}\) under 60 kg/m3, f curve of \(\beta - d_{s}\) under 60 kg/m3, g curve of \(I_{e} - d_{s}\) under 80 kg/m3, h curve of \(\beta - d_{s}\) under 80 kg/m3
Because of the permeability parameters of water and sand seepage in fracture are connected with water and sand, at the same time, the structure of fracture; so the permeability k is not enough to describe permeability parameters, the effective fluidity and non-Darcy factor \(\beta\) are used.
The 5 samples were used to obtain the permeability parameters in test, and we adopted the arithmetic mean values, as shown in Table 4.
Table 4 permeability parameters of water and sand under different sand concentration
Fitting the curves of Fig. 8, the functional relationship between seepage parameters and sand concentration in water was used, as shown in Table 5.
Table 5 Fitted equations of permeability parameters changing with \(d_{s}\) at JRC 4–6
The exponential function was used to fit the relationship between effective fluidity, non-Darcy coefficient and particle sizes of sand. The power exponent equations are used to fit the relationship between effective fluidity I e , the non-Darcy factor \(\beta\) and Sand concentration.
From Fig. 8 and Table 5, the following results were obtained:
The seepage of water and sand in a fracture is nonlinear.
Along with the change of grain size of sediment, the relationship between effective fluidity \(I_{e}\) and mass concentration of sand \(d_{s}\) was the negative exponential relationship; the absolute value of the exponent increased along with the increase of sand particle in the water.
Non-Darcy factor β and sand concentration in water had a positive exponential relationship; the absolute value of the exponent increased along with the decrease of sand particle in water.
It is non-Darcy flow in the paper, which was influenced by roughness, flow velocity, aperture of fracture, and so on. Roughness has a large influence on fracture flow, where non-Darcy also happened (Boutt et al. 2006; Lomize 1951; Louis 1969; Qian et al. 2011).
During the flow, Reynolds number and Forchheimer's number are important parameters to judge (Bear 1972): when Re > 100 or Re < 1, it will be nonlinear flow and does not conform to Darcy flow. What's more, the velocity of water and sand, the aperture of fracture and the tortuosity of fracture also have much influence on flow parameters (Tsang 1984; Tsang and Tsang 1987). The concentration and density also have influence on flow character in fracture (Watson et al. 2002; Tenchine and Gouze 2005). Here \(I_{e}\) has relationship with the structure of fracture, and the character of mixture or water and sand. With the pressure drop increasing, the nonlinear flow became obvious (Elsworth and Doe 1986; Wen et al. 2006; Yeo and Ge 2001) the Forchheimer's law is well known classical approach to describe the nonlinear flow in fracture. Non-Darcy factor β is the parameter which reflected the deviation of Darcy of the seepage. Along with sand particle in water, the non-Darcy character became more obvious.
In this paper, the viscosity of water and sand mixture was discussed and the seepage of water and sand mixture in rude fracture was analyzed.
The seepage velocity of water and sand in a fracture increases along with the pressure of the fracture, but the relationship between them is nonlinear.
Consistency coefficient C becomes larger in conjunction with the mass concentration in water, but decreases along with the particle size of sand. The lower exponent n becomes enlarger along with mass concentration in water, but decreases along with particle size of sand.
Along with the change of the grain size of sediment, the relationship between effective fluidity \(I_{e}\) and mass concentration of sediment \(\rho_{s}\) in water is exponential. The absolute value of the exponent increases along with the increase of sand concentration in water. The non-Darcy factor β and sand concentration in water has a positive exponential relationship and the absolute value of the exponent increases along with the decrease of sand concentration in water.
For the future work, we will work for the different concentration, for particle and concentration both has influence to the flow character, but we should do some experiments to make sure which one is more influence. And acceleration, low velocity of water and sand how to change into water and sand inrush.
b :
aperture of fracture
consistency coefficient
C a :
acceleration coefficient
d :
diameter of the rotor
diameter of outer cylinder
d s :
particle size of sand
h :
height of the fracture
I e :
effective fluidity
k e :
effective permeability
sample length
m :
mass of sand and water
power exponent
n rot :
rotate speed of rotor
β :
non-Darcy coefficient
Q :
flow of seepage
\(\tau\) :
shear stress
\(\mu\) :
fluid viscosity
\(\mu_{a}\) :
apparent viscosity of water and sand
\(\mu_{e}\) :
effective viscosity
V :
velocity of seepage
\(\gamma\) :
apparent viscosity
\(\rho\) :
\(\rho_{s}\) :
mass concentration of sand
\(\frac{\partial p}{\partial l}\) :
pressure gradient
Bear J (1972) Dynamics of fluids in porous media. New York, Dover
Boutt DF, Grasselli G, Fredrich JT, Cook BK, Williams JR (2006) Trapping zones: the effect of fracture roughnesson the directional anisotropy of fluid flow and colloid transport in a single fracture. Geophys Res Lett 33:L21402. doi:10.1029/2006GL027275
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Liu Y (2014) Test study of non-Darcy flow of water–sand mixture in fractured rock. China University of Mining and Technology, Xuzhou
Lomize GM (1951) Filtratsia v treshchinovatykh porodakh, Seepage in Jointed Rocks. Gosudarstvennoe Energeticheskoe Izdatel'stvo, Moskva-Leningrad
Louis C (1969) A study of groundwater flow in jointed rock and its influence on the stability of rock masses. Rock mechanics research report. Imperial College, London, UK
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The work presented here was carried out in collaboration between all authors. Yu Liu defined the research theme, designed experiments methods and wrote the paper. Shuncai Li did the experiments, analyzed the data and explained the results. Both authors read and approved the final manuscript.
This research was supported by Natural science fund for colleges and universities in Jiangsu Province (14KJB440001), Jiangsu Normal University PhD Start Fund (14XLR032), Jiangsu Planned Projects for Postdoctoral Research Funds (1402055B), and National Natural Science Foundation of China (51574228), All the supports are gratefully acknowledged.
Both authors declare that they have no competing interests.
School of Mechanical and Electrical Engineering, Jiangsu Normal University, Xuzhou, 221116, Jiangsu, China
Yu Liu
& Shuncai Li
School of Mines, China University of Mining and Technology, Xuzhou, China
Search for Yu Liu in:
Search for Shuncai Li in:
Correspondence to Yu Liu.
Liu, Y., Li, S. Influence of particle size on non-Darcy seepage of water and sediment in fractured rock. SpringerPlus 5, 2099 (2016) doi:10.1186/s40064-016-3778-9
Accepted: 01 December 2016
Water–sand
Non-Darcy seepage
Non-Newton fluid | CommonCrawl |
\begin{definition}[Definition:Right Circular Cone]
A '''right circular cone''' is a cone:
:whose base is a circle
:in which there is a line perpendicular to the base through its center which passes through the apex of the cone:
:which is made by having a right-angled triangle turning along one of the sides that form the right angle.
:300px
{{EuclidSaid}}
:''{{:Definition:Euclid's Definitions - Book XI/18 - Cone}}''
{{EuclidDefRefNocat|XI|18|Cone}}
\end{definition} | ProofWiki |
What is the meaning of p values and t values in statistical tests?
Viewed 521k times
After taking a statistics course and then trying to help fellow students, I noticed one subject that inspires much head-desk banging is interpreting the results of statistical hypothesis tests. It seems that students easily learn how to perform the calculations required by a given test but get hung up on interpreting the results. Many computerized tools report test results in terms of "p values" or "t values".
How would you explain the following points to college students taking their first course in statistics:
What does a "p-value" mean in relation to the hypothesis being tested? Are there cases when one should be looking for a high p-value or a low p-value?
What is the relationship between a p-value and a t-value?
hypothesis-testing p-value interpretation intuition canonical-question
whuber♦
SharpieSharpie
$\begingroup$ A fair bit of this is basically covered by the first sentence of the wikipedia article on p values, which correctly defines a p-value. If that's understood, much is made clear. $\endgroup$
– Glen_b
$\begingroup$ Just get the book: Statistics without Tears. It might save your sanity!! $\endgroup$
$\begingroup$ @user48700 Could you summarize how Statistics Without Tears explains this? $\endgroup$
– Matt Krause
$\begingroup$ Someone should draw a graph of p-value related questions over time and I bet we'll see the seasonality and correlation to academic calendars in colleges or Coursera data science classes $\endgroup$
– Aksakal
$\begingroup$ In addition to other nice and relevant book recommendations in the answers and comments, I would like to suggest another book, appropriately called "What is a p-value anyway?". $\endgroup$
– Aleksandr Blekh
Understanding $p$-value
Suppose, that you want to test the hypothesis that the average height of male students at your University is $5$ ft $7$ inches. You collect heights of $100$ students selected at random and compute the sample mean (say it turns out to be $5$ ft $9$ inches). Using an appropriate formula/statistical routine you compute the $p$-value for your hypothesis and say it turns out to be $0.06$.
In order to interpret $p=0.06$ appropriately, we should keep several things in mind:
The first step under classical hypothesis testing is the assumption that the hypothesis under consideration is true. (In our context, we assume that the true average height is $5$ ft $7$ inches.)
Imagine doing the following calculation: Compute the probability that the sample mean is greater than $5$ ft $9$ inches assuming that our hypothesis is in fact correct (see point 1).
In other words, we want to know $$\mathrm{P}(\mathrm{Sample\: mean} \ge 5 \:\mathrm{ft} \:9 \:\mathrm{inches} \:|\: \mathrm{True\: value} = 5 \:\mathrm{ft}\: 7\: \mathrm{inches}).$$
The calculation in step 2 is what is called the $p$-value. Therefore, a $p$-value of $0.06$ would mean that if we were to repeat our experiment many, many times (each time we select $100$ students at random and compute the sample mean) then $6$ times out of $100$ we can expect to see a sample mean greater than or equal to $5$ ft $9$ inches.
Given the above understanding, should we still retain our assumption that our hypothesis is true (see step 1)? Well, a $p=0.06$ indicates that one of two things have happened:
(A) Either our hypothesis is correct and an extremely unlikely event has occurred (e.g., all $100$ students are student athletes)
(B) Our assumption is incorrect and the sample we have obtained is not that unusual.
The traditional way to choose between (A) and (B) is to choose an arbitrary cut-off for $p$. We choose (A) if $p > 0.05$ and (B) if $p < 0.05$.
user28user28
$\begingroup$ Take your time! I won't be thinking about selecting a "Best Answer" for a week or so. $\endgroup$
– Sharpie
$\begingroup$ Now that I've had a chance to come back and read the whole answer- a big +1 for the student height example. Very clear and well laid out. $\endgroup$
$\begingroup$ Nice work ... but we need to add (C) our model (embodied in the formula/statistical routine) is wrong. $\endgroup$
– Andrew Robinson
$\begingroup$ A t-value (or any other test statistic) is mostly an intermediate step. It's basically some statistic that was proven, under some assumptions, to have a well-known distribution. Since we know the distribution of the test statistic under the null, we can then use standard tables (today mostly software) to derive a p-value. $\endgroup$
– Gala
$\begingroup$ Isn't the p-value derived as a result of doing the chi-square test and then from the chi-square table? Am wondering how come the probability calculated above indicated the p-value itself?! $\endgroup$
– London guy
A Dialog Between a Teacher and a Thoughtful Student
Humbly submitted in the belief that not enough crayons have been used so far in this thread. A brief illustrated synopsis appears at the end.
Student: What does a p-value mean? A lot of people seem to agree it's the chance we will "see a sample mean greater than or equal to" a statistic or it's "the probability of observing this outcome ... given the null hypothesis is true" or where "my sample's statistic fell on [a simulated] distribution" and even "the probability of observing a test statistic at least as large as the one calculated assuming the null hypothesis is true".
Teacher: Properly understood, all those statements are correct in many circumstances.
Student: I don't see how most of them are relevant. Didn't you teach us that we have to state a null hypothesis $H_0$ and an alternative hypothesis $H_A$? How are they involved in these ideas of "greater than or equal to" or "at least as large" or the very popular "more extreme"?
Teacher: Because it can seem complicated in general, would it help for us to explore a concrete example?
Student: Sure. But please make it a realistic but simple one if you can.
Teacher: This theory of hypothesis testing historically began with the need of astronomers to analyze observational errors, so how about starting there. I was going through some old documents one day where a scientist described his efforts to reduce the measurement error in his apparatus. He had taken a lot of measurements of a star in a known position and recorded their displacements ahead of or behind that position. To visualize those displacements, he drew a histogram that--when smoothed a little--looked like this one.
Student: I remember how histograms work: the vertical axis is labeled "Density" to remind me that the relative frequencies of the measurements are represented by area rather than height.
Teacher: That's right. An "unusual" or "extreme" value would be located in a region with pretty small area. Here's a crayon. Do you think you could color in a region whose area is just one-tenth the total?
Student: Sure; that's easy. [Colors in the figure.]
Teacher: Very good! That looks like about 10% of the area to me. Remember, though, that the only areas in the histogram that matter are those between vertical lines: they represent the chance or probability that the displacement would be located between those lines on the horizontal axis. That means you needed to color all the way down to the bottom and that would be over half the area, wouldn't it?
Student: Oh, I see. Let me try again. I'm going to want to color in where the curve is really low, won't I? It's lowest at the two ends. Do I have to color in just one area or would it be ok to break it into several parts?
Teacher: Using several parts is a smart idea. Where would they be?
Student (pointing): Here and here. Because this crayon isn't very sharp, I used a pen to show you the lines I'm using.
Teacher: Very nice! Let me tell you the rest of the story. The scientist made some improvements to his device and then he took additional measurements. He wrote that the displacement of the first one was only $0.1$, which he thought was a good sign, but being a careful scientist he proceeded to take more measurements as a check. Unfortunately, those other measurements are lost--the manuscript breaks off at this point--and all we have is that single number, $0.1$.
Student: That's too bad. But isn't that much better than the wide spread of displacements in your figure?
Teacher: That's the question I would like you to answer. To start with, what should we posit as $H_0$?
Student: Well, a sceptic would wonder whether the improvements made to the device had any effect at all. The burden of proof is on the scientist: he would want to show that the sceptic is wrong. That makes me think the null hypothesis is kind of bad for the scientist: it says that all the new measurements--including the value of $0.1$ we know about--ought to behave as described by the first histogram. Or maybe even worse than that: they might be even more spread out.
Teacher: Go on, you're doing well.
Student: And so the alternative is that the new measurements would be less spread out, right?
Teacher: Very good! Could you draw me a picture of what a histogram with less spread would look like? Here's another copy of the first histogram; you can draw on top of it as a reference.
Student (drawing): I'm using a pen to outline the new histogram and I'm coloring in the area beneath it. I have made it so most of the curve is close to zero on the horizontal axis and so most of its area is near a (horizontal) value of zero: that's what it means to be less spread out or more precise.
Teacher: That's a good start. But remember that a histogram showing chances should have a total area of $1$. The total area of the first histogram therefore is $1$. How much area is inside your new histogram?
Student: Less than half, I think. I see that's a problem, but I don't know how to fix it. What should I do?
Teacher: The trick is to make the new histogram higher than the old so that its total area is $1$. Here, I'll show you a computer-generated version to illustrate.
Student: I see: you stretched it out vertically so its shape didn't really change but now the red area and gray area (including the part under the red) are the same amounts.
Teacher: Right. You are looking at a picture of the null hypothesis (in blue, spread out) and part of the alternative hypothesis (in red, with less spread).
Student: What do you mean by "part" of the alternative? Isn't it just the alternative hypothesis?
Teacher: Statisticians and grammar don't seem to mix. :-) Seriously, what they mean by a "hypothesis" usually is a whole big set of possibilities. Here, the alternative (as you stated so well before) is that the measurements are "less spread out" than before. But how much less? There are many possibilities. Here, let me show you another. I drew it with yellow dashes. It's in between the previous two.
Student: I see: you can have different amounts of spread but you don't know in advance how much the spread will really be. But why did you make the funny shading in this picture?
Teacher: I wanted to highlight where and how the histograms differ. I shaded them in gray where the alternative histograms are lower than the null and in red where the alternatives are higher.
Student: Why would that matter?
Teacher: Do you remember how you colored the first histogram in both the tails? [Looking through the papers.] Ah, here it is. Let's color this picture in the same way.
Student: I remember: those are the extreme values. I found the places where the null density was as small as possible and colored in 10% of the area there.
Teacher: Tell me about the alternatives in those extreme areas.
Student: It's hard to see, because the crayon covered it up, but it looks like there's almost no chance for any alternative to be in the areas I colored. Their histograms are right down against value axis and there's no room for any area beneath them.
Teacher: Let's continue that thought. If I told you, hypothetically, that a measurement had a displacement of $-2$, and asked you to pick which of these three histograms was the one it most likely came from, which would it be?
Student: The first one--the blue one. It's the most spread out and it's the only one where $-2$ seems to have any chance of occurring.
Teacher: And what about the value of $0.1$ in the manuscript?
Student: Hmmm... that's a different story. All three histograms are pretty high above the ground at $0.1$.
Teacher: OK, fair enough. But suppose I told you the value was somewhere near $0.1$, like between $0$ and $0.2$. Does that help you read some probabilities off of these graphs?
Student: Sure, because I can use areas. I just have to estimate the areas underneath each curve between $0$ and $0.2$. But that looks pretty hard.
Teacher: You don't need to go that far. Can you just tell which area is the largest?
Student: The one beneath the tallest curve, of course. All three areas have the same base, so the taller the curve, the more area there is beneath it and the base. That means the tallest histogram--the one I drew, with the red dashes--is the likeliest one for a displacement of $0.1$. I think I see where you're going with this, but I'm a little concerned: don't I have to look at all the histograms for all the alternatives, not just the one or two shown here? How could I possibly do that?
Teacher: You're good at picking up patterns, so tell me: as the measurement apparatus is made more and more precise, what happens to its histogram?
Student: It gets narrower--oh, and it has to get taller, too, so its total area stays the same. That makes it pretty hard to compare the histograms. The alternative ones are all higher than the null right at $0$, that's obvious. But at other values sometimes the alternatives are higher and sometimes they are lower! For example, [pointing at a value near $3/4$], right here my red histogram is the lowest, the yellow histogram is the highest, and the original null histogram is between them. But over on the right the null is the highest.
Teacher: In general, comparing histograms is a complicated business. To help us do it, I have asked the computer to make another plot: it has divided each of the alternative histogram heights (or "densities") by the null histogram height, creating values known as "likelihood ratios." As a result, a value greater than $1$ means the alternative is more likely, while a value less than $1$ means the alternative is less likely. It has drawn yet one more alternative: it's more spread out than the other two, but still less spread out than the original apparatus was.
Teacher (continuing): Could you show me where the alternatives tend to be more likely than the null?
Student (coloring): Here in the middle, obviously. And because these are not histograms anymore, I guess we should be looking at heights rather than areas, so I'm just marking a range of values on the horizontal axis. But how do I know how much of the middle to color in? Where do I stop coloring?
Teacher: There's no firm rule. It all depends on how we plan to use our conclusions and how fierce the sceptics are. But sit back and think about what you have accomplished: you now realize that outcomes with large likelihood ratios are evidence for the alternative and outcomes with small likelihood ratios are evidence against the alternative. What I will ask you to do is to color in an area that, insofar as is possible, has a small chance of occurring under the null hypothesis and a relatively large chance of occurring under the alternatives. Going back to the first diagram you colored, way back at the start of our conversation, you colored in the two tails of the null because they were "extreme." Would they still do a good job?
Student: I don't think so. Even though they were pretty extreme and rare under the null hypothesis, they are practically impossible for any of the alternatives. If my new measurement were, say $3.0$, I think I would side with the sceptic and deny that any improvement had occurred, even though $3.0$ was an unusual outcome in any case. I want to change that coloring. Here--let me have another crayon.
Teacher: What does that represent?
Student: We started out with you asking me to draw in just 10% of the area under the original histogram--the one describing the null. So now I drew in 10% of the area where the alternatives seem more likely to be occurring. I think that when a new measurement is in that area, it's telling us we ought to believe the alternative.
Teacher: And how should the sceptic react to that?
Student: A sceptic never has to admit he's wrong, does he? But I think his faith should be a little shaken. After all, we arranged it so that although a measurement could be inside the area I just drew, it only has a 10% chance of being there when the null is true. And it has a larger chance of being there when the alternative is true. I just can't tell you how much larger that chance is, because it would depend on how much the scientist improved the apparatus. I just know it's larger. So the evidence would be against the sceptic.
Teacher: All right. Would you mind summarizing your understanding so that we're perfectly clear about what you have learned?
Student: I learned that to compare alternative hypotheses to null hypotheses, we should compare their histograms. We divide the densities of the alternatives by the density of the null: that's what you called the "likelihood ratio." To make a good test, I should pick a small number like 10% or whatever might be enough to shake a sceptic. Then I should find values where the likelihood ratio is as high as possible and color them in until 10% (or whatever) has been colored.
Teacher: And how would you use that coloring?
Student: As you reminded me earlier, the coloring has to be between vertical lines. Values (on the horizontal axis) that lie under the coloring are evidence against the null hypothesis. Other values--well, it's hard to say what they might mean without taking a more detailed look at all the histograms involved.
Teacher: Going back to the value of $0.1$ in the manuscript, what would you conclude?
Student: That's within the area I last colored, so I think the scientist probably was right and the apparatus really was improved.
Teacher: One last thing. Your conclusion was based on picking 10% as the criterion, or "size" of the test. Many people like to use 5% instead. Some prefer 1%. What could you tell them?
Student: I couldn't do all those tests at once! Well, maybe I could in a way. I can see that no matter what size the test should be, I ought to start coloring from $0$, which is in this sense the "most extreme" value, and work outwards in both directions from there. If I were to stop right at $0.1$--the value actually observed--I think I would have colored in an area somewhere between $0.05$ and $0.1$, say $0.08$. The 5% and 1% people could tell right away that I colored too much: if they wanted to color just 5% or 1%, they could, but they wouldn't get as far out as $0.1$. They wouldn't come to the same conclusion I did: they would say there's not enough evidence that a change actually occurred.
Teacher: You have just told me what all those quotations at the beginning really mean. It should be obvious from this example that they cannot possibly intend "more extreme" or "greater than or equal" or "at least as large" in the sense of having a bigger value or even having a value where the null density is small. They really mean these things in the sense of large likelihood ratios that you have described. By the way, the number around $0.08$ that you computed is called the "p-value." It can only properly be understood in the way you have described: with respect to an analysis of relative histogram heights--the likelihood ratios.
Student: Thank you. I'm not confident I fully understand all of this yet, but you have given me a lot to think about.
Teacher: If you would like to go further, take a look at the Neyman-Pearson Lemma. You are probably ready to understand it now.
Many tests that are based on a single statistic like the one in the dialog will call it "$z$" or "$t$". These are ways of hinting what the null histogram looks like, but they are only hints: what we name this number doesn't really matter. The construction summarized by the student, as illustrated here, shows how it is related to the p-value. The p-value is the smallest test size that would cause an observation of $t=0.1$ to lead to a rejection of the null hypothesis.
In this figure, which is zoomed to show detail, the null hypothesis is plotted in solid blue and two typical alternatives are plotted with dashed lines. The region where those alternatives tend to be much larger than the null is shaded in. The shading starts where the relative likelihoods of the alternatives are greatest (at $0$). The shading stops when the observation $t=0.1$ is reached. The p-value is the area of the shaded region under the null histogram: it is the chance, assuming the null is true, of observing an outcome whose likelihood ratios tend to be large regardless of which alternative happens to be true. In particular, this construction depends intimately on the alternative hypothesis. It cannot be carried out without specifying the possible alternatives.
For two practical examples of the test described here -- one published, the other hypothetical -- see https://stats.stackexchange.com/a/5408/919.
whuber♦whuber
$\begingroup$ This has excellently dealt with my comment on another answer, that none of the prior answers to this question had tackled, in generality, the commonly-heard "or more extreme" aspect of a p-value. (Though the "tea-testing" answer included a good specific example.) I particularly admire the way this example has been deliberately constructed to highlight that "more extreme" can mean quite the contrary of "bigger" or "further from zero". $\endgroup$
– Silverfish
$\begingroup$ I wish teachers and textbooks didn't use the phrase "or more extreme", really. Two variants I have heard might be paraphrased as "more favourable towards $H_1$" or "more persuasive of $H_1$". In this instance, values nearer zero would indeed be more persuasive that the telescope has become more reliable, but it requires some linguistic acrobatics (plausibly argued, but potentially confusing) to describe them as "more extreme". $\endgroup$
$\begingroup$ Uniquely insightful as always, thank you for taking the time to write out those incredibly helpful answers. I really wonder why textbooks are never written in a way that offers anywhere near these levels of clarity and intuition. $\endgroup$
– jeremy radcliff
$\begingroup$ I think a link to a definition of likelihood wrt this example could be beneficial $\endgroup$
– baxx
$\begingroup$ I'm not sure if you're being sarcastic, I'm aware of the search function. And I searched for definitions but found lots of literature about tests and stuff and didn't know what to use. I think perhaps if you were to link to an appropriate source for this material level it would be easier for future users too $\endgroup$
Before touching this topic, I always make sure that students are happy moving between percentages, decimals, odds and fractions. If they are not completely happy with this then they can get confused very quickly.
I like to explain hypothesis testing for the first time (and therefore p-values and test statistics) through Fisher's classic tea experiment. I have several reasons for this:
(i) I think working through an experiment and defining the terms as we go along makes more sense that just defining all of these terms to begin with. (ii) You don't need to rely explicitly on probability distributions, areas under the curve, etc to get over the key points of hypothesis testing. (iii) It explains this ridiculous notion of "as or more extreme than those observed" in a fairly sensible manner (iv) I find students like to understand the history, origins and back story of what they are studying as it makes it more real than some abstract theories. (v) It doesn't matter what discipline or subject the students come from, they can relate to the example of tea (N.B. Some international students have difficulty with this peculiarly British institution of tea with milk.)
[Note: I originally got this idea from Dennis Lindley's wonderful article "The Analysis of Experimental Data: The Appreciation of Tea & Wine" in which he demonstrates why Bayesian methods are superior to classical methods.]
The back story is that Muriel Bristol visits Fisher one afternoon in the 1920's at Rothamsted Experimental Station for a cup of tea. When Fisher put the milk in last she complained saying that she could also tell whether the milk was poured first (or last) and that she preferred the former. To put this to the test he designed his classic tea experiment where Muriel is presented with a pair of tea cups and she must identify which one had the milk added first. This is repeated with six pairs of tea cups. Her choices are either Right (R) or Wrong (W) and her results are: RRRRRW.
Suppose that Muriel is actually just guessing and has no ability to discriminate whatsoever. This is called the Null Hypothesis. According to Fisher the purpose of the experiment is to discredit this null hypothesis. If Muriel is guessing she will identify the tea cup correctly with probability 0.5 on each turn and as they are independent the observed result has 0.5$^6$ = 0.016 (or 1/64). Fisher then argues that either:
(a) the null hypothesis (Muriel is guessing) is true and an event of small probability has occurred or,
(b) the null hypothesis is false and Muriel has discriminatory powers.
The p-value (or probability value) is the probability of observing this outcome (RRRRRW) given the null hypothesis is true - it's the small probability referred to in (a), above. In this instance it's 0.016. Since events with small probabilities only occur rarely (by definition) situation (b) might be a more preferable explanation of what occurred than situation (a). When we reject the null hypothesis we're in fact accepting the opposite hypothesis which is we call the alternative hypothesis. In this example, Muriel has discriminatory powers is the alternative hypothesis.
An important consideration is what do we class as a "small" probability? What's the cutoff point at which we're willing to say that an event is unlikely? The standard benchmark is 5% (0.05) and this is called the significance level. When the p-value is smaller than the significance level we reject the null hypothesis as being false and accept our alternative hypothesis. It is common parlance to claim a result is "significant" when the p-value is smaller than the significance level i.e. when the probability of what we observed occurring given the null hypothesis is true is smaller than our cutoff point. It is important to be clear that using 5% is completely subjective (as is using the other common significance levels of 1% and 10%).
Fisher realised that this doesn't work; every possible outcome with one wrong pair was equally suggestive of discriminatory powers. The relevant probability for situation (a), above, is therefore 6(0.5)^6 = 0.094 (or 6/64) which now is not significant at a significance level of 5%. To overcome this Fisher argued that if 1 error in 6 is considered evidence of discriminatory powers then so is no errors i.e. outcomes that more strongly indicate discriminatory powers than the one observed should be included when calculating the p-value. This resulted in the following amendment to the reasoning, either:
(a) the null hypothesis (Muriel is guessing) is true and the probability of events as, or more, extreme than that observed is small, or
Back to our tea experiment and we find that the p-value under this set-up is 7(0.5)^6 = 0.109 which still is not significant at the 5% threshold.
I then get students to work with some other examples such as coin tossing to work out whether or not a coin is fair. This drills home the concepts of the null/alternative hypothesis, p-values and significance levels. We then move onto the case of a continuous variable and introduce the notion of a test-statistic. As we have already covered the normal distribution, standard normal distribution and the z-transformation in depth it's merely a matter of bolting together several concepts.
As well as calculating test-statistics, p-values and making a decision (significant/not significant) I get students to work through published papers in a fill in the missing blanks game.
Tyto alba
Graham CooksonGraham Cookson
$\begingroup$ I know I'm somewhat reviving a very old thread, but here it goes... I was really enjoying your answer, but I miss the t-value part in it :( Could you please use your given examples to talk about it? No one answered about the t-test part $\endgroup$
– Sos
$\begingroup$ @sosi It's probably because p-values are much more general than t-values. It's like asking a question about cars and then about the brakes on a Ford Fiesta. $\endgroup$
– conjectures
$\begingroup$ The answer is very interesting (+1), but a few things are confused together at the end. 1. What does it mean for a $p$-value to be "significant at the 5% level"? Either the $p$-value is below 5%, or it is not. I don't see the point in using such an obscure sentence, leaving "significance" undefined. 2. What does it mean to "decide" wether or not a $p$-value is significant? It does not seem justified to bring in decision theory into the mix in this way (especially since Fisher was a strong opponent of the application of the Neyman-Pearson testing framework in the sciences). $\endgroup$
– Olivier
No amount of verbal explanation or calculations really helped me to understand at a gut level what p-values were, but it really snapped into focus for me once I took a course that involved simulation. That gave me the ability to actually see data generated by the null hypothesis and to plot the means/etc. of simulated samples, then look at where my sample's statistic fell on that distribution.
I think the key advantage to this is that it lets students forget about the math and the test statistic distributions for a minute and focus on the concepts at hand. Granted, it required that I learn how to simulate that stuff, which will cause problems for an entirely different set of students. But it worked for me, and I've used simulation countless times to help explain statistics to others with great success (e.g., "This is what your data looks like; this is what a Poisson distribution looks like overlaid. Are you SURE you want to do a Poisson regression?").
This doesn't exactly answer the questions you posed, but for me, at least, it made them trivial.
Matt ParkerMatt Parker
$\begingroup$ I agree wholeheartedly about the use of simulation for explaining this. But just a small note on the example at the end: I find that people (not just students) do find it difficult to distinguish for any particular distributional assumption, e.g. the poisson, between being marginally poisson distributed and being conditionally poisson distributed. Since only the latter matters for a regression model, a bunch of dependent variable values that aren't poisson need not necessarily be any cause for concern. $\endgroup$
– conjugateprior
$\begingroup$ I have to confess that I didn't know that. I've really appreciated your comments around this site over the past few days of your membership - I hope you'll stick around. $\endgroup$
– Matt Parker
$\begingroup$ @MattParker do you know of any learning resources focussed towards the use of simulation to develop understanding? Or is it just a case of putting some python / R scripts together and running a bunch of tests? $\endgroup$
$\begingroup$ @baxx The [Seeing Theory website by Daniel Kunin](students.brown.edu/seeing-theory/) has some interesting tools for this, but it's still under construction. Otherwise, yeah, I've largely just experimented with R's built-in tools for simulation - using them to prove to myself how some method works, or to see what would happen if a predictor was replaced with a random variable, etc. Sorry, I wish I knew of better resources for this! $\endgroup$
$\begingroup$ @MattParker cool thanks. Yeah - bit of a chicken and egg in that, to construct the experiments you (I assume?) need to at least get enough to write them. No worries though..... Just checked that site you linked, it's nice, thanks $\endgroup$
A nice definition of p-value is "the probability of observing a test statistic at least as large as the one calculated assuming the null hypothesis is true".
The problem with that is that it requires an understanding of "test statistic" and "null hypothesis". But, that's easy to get across. If the null hypothesis is true, usually something like "parameter from population A is equal to parameter from population B", and you calculate statistics to estimate those parameters, what is the probability of seeing a test statistic that says, "they're this different"?
E.g., If the coin is fair, what is the probability I'd see 60 heads out of 100 tosses? That's testing the null hypothesis, "the coin is fair", or "p = .5" where p is the probability of heads.
The test statistic in that case would be the number of heads.
Now, I assume that what you're calling "t-value" is a generic "test statistic", not a value from a "t distribution". They're not the same thing, and the term "t-value" isn't (necessarily) widely used and could be confusing.
What you're calling "t-value" is probably what I'm calling "test statistic". In order to calculate a p-value (remember, it's just a probability) you need a distribution, and a value to plug into that distribution which will return a probability. Once you do that, the probability you return is your p-value. You can see that they are related because under the same distribution, different test-statistics are going to return different p-values. More extreme test-statistics will return lower p-values giving greater indication that the null hypothesis is false.
I've ignored the issue of one-sided and two-sided p-values here.
BaltimarkBaltimark
Imagine you have a bag containing 900 black marbles and 100 white, i.e. 10% of the marbles are white. Now imagine you take 1 marble out, look at it and record its colour, take out another, record its colour etc.. and do this 100 times. At the end of this process you will have a number for white marbles which, ideally, we would expect to be 10, i.e. 10% of 100, but in actual fact may be 8, or 13 or whatever simply due to randomness. If you repeat this 100 marble withdrawal experiment many, many times and then plot a histogram of the number of white marbles drawn per experiment, you'll find you will have a Bell Curve centred about 10.
This represents your 10% hypothesis: with any bag containing 1000 marbles of which 10% are white, if you randomly take out 100 marbles you will find 10 white marbles in the selection, give or take 4 or so. The p-value is all about this "give or take 4 or so." Let's say by referring to the Bell Curve created earlier you can determine that less than 5% of the time would you get 5 or fewer white marbles and another < 5% of the time accounts for 15 or more white marbles i.e. > 90% of the time your 100 marble selection will contain between 6 to 14 white marbles inclusive.
Now assuming someone plonks down a bag of 1000 marbles with an unknown number of white marbles in it, we have the tools to answer these questions
i) Are there fewer than 100 white marbles?
ii) Are there more than 100 white marbles?
iii) Does the bag contain 100 white marbles?
Simply take out 100 marbles from the bag and count how many of this sample are white.
a) If there are 6 to 14 whites in the sample you cannot reject the hypothesis that there are 100 white marbles in the bag and the corresponding p-values for 6 through 14 will be > 0.05.
b) If there are 5 or fewer whites in the sample you can reject the hypothesis that there are 100 white marbles in the bag and the corresponding p-values for 5 or fewer will be < 0.05. You would expect the bag to contain < 10% white marbles.
c) If there are 15 or more whites in the sample you can reject the hypothesis that there are 100 white marbles in the bag and the corresponding p-values for 15 or more will be < 0.05. You would expect the bag to contain > 10% white marbles.
In response to Baltimark's comment
Given the example above, there is an approximately:-
4.8% chance of getter 5 white balls or fewer
1.85% chance of 4 or fewer
0.1% chance of 2 or fewer
6.25% chance of 15 or more
1.5% chance of 17 or more
These numbers were estimated from an empirical distribution generated by a simple Monte Carlo routine run in R and the resultant quantiles of the sampling distribution.
For the purposes of answering the original question, suppose you draw 5 white balls, there is only an approximate 4.8% chance that if the 1000 marble bag really does contain 10% white balls you would pull out only 5 whites in a sample of 100. This equates to a p value < 0.05. You now have to choose between
i) There really are 10% white balls in the bag and I have just been "unlucky" to draw so few
ii) I have drawn so few white balls that there can't really be 10% white balls (reject the hypothesis of 10% white balls)
babelproofreaderbabelproofreader
$\begingroup$ First of all, this is just a big example and doesn't really explain explain the concept of p-value and test-statistic. Second, you're just claiming that if you get fewer than 5 or more than 15 white marbles, you reject the null hypothesis. What's your distribution that you're calculating those probabilities from? This can be approximated with a normal dist. centered at 10, with a standard deviation of 3. Your rejection criteria is not nearly strict enough. $\endgroup$
– Baltimark
$\begingroup$ I would agree that this is just an example, and I it is true I just picked the numbers 5 and 15 out of the air for illustrative purposes. When I have time I will post a second answer, which I hope will be more complete. $\endgroup$
– babelproofreader
What the p-value doesn't tell you is how likely it is that the null hypothesis is true. Under the conventional (Fisher) significance testing framework we first compute the likelihood of observing the data assuming the null hypothesis is true, this is the p-value. It seems intuitively reasonable then to assume the null hypothesis is probably false if the data are sufficiently unlikely to be observed under the null hypothesis. This is entirely reasonable. Statisticians tranditionally use a threshold and "reject the null hypothesis at the 95% significance level" if (1 - p) > 0.95; however this is just a convention that has proven reasonable in practice - it doesn't mean that there is less than 5% probability that the null hypothesis is false (and therefore a 95% probability that the alternative hypothesis is true). One reason that we can't say this is that we have not looked at the alternative hypothesis yet.
Imaging a function f() that maps the p-value onto the probability that the alternative hypothesis is true. It would be reasonable to assert that this function is strictly decreasing (such that the more likely the observations under the null hypothesis, the less likely the alternative hypothesis is true), and that it gives values between 0 and 1 (as it gives an estimate of probability). However, that is all that we know about f(), so while there is a relationship between p and the probability that the alternative hypothesis is true, it is uncalibrated. This means we cannot use the p-value to make quantitative statements about the plausibility of the nulll and alternatve hypotheses.
Caveat lector: It isn't really within the frequentist framework to speak of the probability that a hypothesis is true, as it isn't a random variable - it is either true or it isn't. So where I have talked of the probability of the truth of a hypothesis I have implicitly moved to a Bayesian interpretation. It is incorrect to mix Bayesian and frequentist, however there is always a temptation to do so as what we really want is an quantative indication of the relative plausibility/probability of the hypotheses. But this is not what the p-value provides.
Dikran MarsupialDikran Marsupial
In statistics you can never say something is absolutely certain, so statisticians use another approach to gauge whether a hypothesis is true or not. They try to reject all the other hypotheses that are not supported by the data.
To do this, statistical tests have a null hypothesis and an alternate hypothesis. The p-value reported from a statistical test is the likelihood of the result given that the null hypothesis was correct. That's why we want small p-values. The smaller they are, the less likely the result would be if the null hypothesis was correct. If the p-value is small enough (ie,it is very unlikely for the result to have occurred if the null hypothesis was correct), then the null hypothesis is rejected.
In this fashion, null hypotheses can be formulated and subsequently rejected. If the null hypothesis is rejected, you accept the alternate hypothesis as the best explanation. Just remember though that the alternate hypothesis is never certain, since the null hypothesis could have, by chance, generated the results.
DaRobDaRob
$\begingroup$ a p-value is the likelihood of a result as or more "extreme" than the result given, not of the actual result. p-value is $Pr(T\geq t|H_0)$ and not $Pr(T=t|H_0)$ (T is test statistic, and t is its observed value). $\endgroup$
– probabilityislogic
I am bit diffident to revive the old topic, but I jumped from here, so I post this as a response to the question in the link.
The p-value is a concrete term, there should be no room for misunderstanding. But, it is somehow mystical that colloquial translations of the definition of p-value leads to many different misinterpretations. I think the root of the problem is in the use of the phrases "at least as adverse to null hypothesis" or "at least as extreme as the one in your sample data" etc.
For instance, Wikipedia says
...the p-value is the probability of obtaining the observed sample results (or a more extreme result) when the null hypothesis is actually true.
Meaning of $p$-value is blurred when people first stumble upon "(or a more extreme result)" and start thinking "more extreeeme?".
I think it is better to leave the "more extreme result" to something like indirect speech act. So, my take is
The p-value is the probability of seeing what you see in a "imaginary world" where the null hypothesis is true.
To make the idea concrete, suppose you have sample x consisting of 10 observations and you hypothesize that the population mean is $\mu_0=20$. So, in your hypothesized world, population distribution is $N(20,1)$.
#[1] 20.82600 19.30229 18.74753 18.99071 20.14312 16.76647
#[7] 18.94962 17.99331 19.22598 18.68633
You compute t-stat as $t_0=\sqrt{n}\frac{\bar{X}-\mu_0}{s}$, and find out that
sqrt(10) * (mean(x) - 20) / sd(x)
#-2.974405
So, what is the probability of observing $|t_0|$ as large as 2.97 ( "more extreme" comes here) in the imaginary world? In the imaginary world $t_0\sim t(9)$, thus, the p-value must be $$p-value=Pr(|t_0|\geq 2.97)= 0.01559054$$
2*(1 - pt(2.974405, 9))
#[1] 0.01559054
Since p-value is small, it is very unlikely that the sample x would have been drawn in the hypothesized world. Therefore, we conclude that it is very unlikely that the hypothesized world was in fact the actual world.
KhashaaKhashaa
$\begingroup$ +1, but when you write "probability of seeing what you see" and omit the "more extreme" part, this sentence becomes strictly speaking false (and potentially misleading, even if perhaps less confusing). It is not the probability of seeing what you see (this is usually zero). It is the probability of seeing what you see "or more extreme". Even though this might be a confusing bit for many, it is still crucial (and one can argue endlessly about the degree of subjectivity that hides behind this "more extreme" wording). $\endgroup$
– amoeba
$\begingroup$ @amoeba I thought, when adequate example supplied, it could serve as a proxy for "obtaining the observed sample results (or a more extreme result)". Maybe, better wording is needed. $\endgroup$
– Khashaa
$\begingroup$ I was going to make the same observation as @amoeba; the "or more extreme" part is handled well by example in the student heights and tea party answers, but I don't think any answers in this thread have hit upon a clear general explanation of it, particularly one which covers different alternative hypotheses. I do agree with this answer suggesting that the "or more extreme" part is a conceptual sticking point for many students. $\endgroup$
$\begingroup$ @Silverfish: and not only students. How many Bayesian-vs-frequentists rants have I read that discuss the subjectivity/objectivity issue of this "more extreme" bit! $\endgroup$
$\begingroup$ @Silver I agree with your criticism and have posted an answer attempting to address it. "Or more extreme" is the very crux of the matter. $\endgroup$
– whuber ♦
I have also found simulations to be a useful in teaching.
Here is a simulation for the arguably most basic case in which we sample $n$ times from $N(\mu,1)$ (hence, $\sigma^2=1$ is known for simplicity) and test $H_0:\mu=\mu_0$ against a left-sided alternative.
Then, the $t$-statistic $\text{tstat}:=\sqrt{n}(\bar{X}-\mu_0)$ is $N(0,1)$ under $H_0$, such that the $p$-value is simply $\Phi(\text{tstat})$ or pnorm(tstat) in R.
In the simulation, it is the fraction of times that data generated under the null $N(\mu_0,1)$ (here, $\mu_0=2$) yields sample means stored in nullMeans that are less (i.e., ``more extreme'' in this left-sided test) than the one calculated from the observed data.
# p value
reps <- 1000
mu <- 1.85 # true value
mu_0 <- 2 # null value
xaxis <- seq(-3, 3, length = 100)
X <- rnorm(n,mu)
nullMeans <- counter <- rep(NA,reps)
yvals <- jitter(rep(0,reps),2)
for (i in 1:reps)
tstat <- sqrt(n)*(mean(X)-mu_0) # test statistic, N(0,1) under the given assumptions
par(mfrow=c(1,3))
plot(xaxis,dnorm(xaxis),ylab="null distribution",xlab="possible test statistics",type="l")
points(tstat,0,cex=2,col="salmon",pch=21,bg="salmon")
X_null <- rnorm(n,mu_0) # generate data under H_0
nullMeans[i] <- mean(X_null)
plot(nullMeans[1:i],yvals[1:i],col="blue",pch=21,xlab="actual means and those generated under the null",ylab="", yaxt='n',ylim=c(-1,1),xlim=c(1.5,2.5))
abline(v=mu_0,lty=2)
points(mean(X),0,cex=4,col="salmon",pch=21,bg="salmon")
# counts 1 if sample generated under H_0 is more extreme:
counter[i] <- (nullMeans[i] < mean(X)) # i.e. we test against H_1: mu < mu_0
barplot(table(counter[1:i])/i,col=c("green","red"),xlab="more extreme mean under the null than the mean actually observed")
if(i<10) locator(1)
mean(counter)
pnorm(tstat)
Christoph HanckChristoph Hanck
I find it helpful to follow a sequence in which you explain concepts in the following order: (1) The z score and proportions above and below the z score assuming a normal curve. (2) The notion of a sampling distribution, and the z score for a given sample mean when the population standard deviation is known (and thence the one sample z test) (3) The one-sample t-test and the likelihood of a sample mean when the population standard deviation is unknown (replete with stories about the secret identity of a certain industrial statistician and why Guinness is Good For Statistics). (4) The two-sample t-test and the sampling distribution of mean differences. The ease with which introductory students grasp the t-test has much to do with the groundwork that is laid in preparation for this topic.
/* instructor of terrified students mode off */
StatisticsDoc ConsultingStatisticsDoc Consulting
I have yet to prove the following argument so it might contain errors, but I really want to throw in my two cents (Hopefully, I'll update it with a rigorous proof soon). Another way of looking at the $p$-value is
$p$-value - A statistic $X$ such that $$\forall 0 \le c \le 1, F_{X|H_0}(\inf\{x: F_{X|H_0}(x) \ge c\}) = c$$ where $F_{X|H_0}$ is the distribution function of $X$ under $H_0$.
Specifically, if $X$ has a continuous distribution and you're not using approximation, then
Every $p$-value is a statistic with a uniform distribution on $[0, 1]$, and
Every statistic with a uniform distribution on $[0, 1]$ is a $p$-value.
You may consider this a generalized description of the $p$-values.
nalzoknalzok
$\begingroup$ This definition makes sense only for discrete distributions (and then is not correct), because the second appearance of "$P$" makes it clear it refers to probabilities, not probability densities. Moreover, there are extremely few distributions (if any) which have the stated property, suggesting that there must be typographical errors in the statement. As far as your subsequent claims go, (1) is ideally true but (2) is not, unless you allow the null hypothesis to depend on the statistic! $\endgroup$
$\begingroup$ @whuber Thanks for the input. I have edited the definition, and it should make more sense now! $\endgroup$
– nalzok
$\begingroup$ It does make sense, thank you: if I'm reading it correctly, it asserts the null distribution of $X$ is uniform on $[0,1].$ However, that captures only part of the properties of p-values; it does not characterize p-values; and it says nothing about what they mean or how to interpret them. Consider studying some of the other answers in this thread for information on what is missing. $\endgroup$
$\begingroup$ Here is an example that you might find interesting. The distribution family is Uniform$(\theta,\theta+1)$ for $\theta\in\mathbb{R},$ the null hypothesis is $\theta=0,$ and the alternative is its complement. Consider a random sample $\mathbf{X}=(X_1,\ldots,X_n).$ Define the statistic $X(\mathbf{X}) = X_1.$ Obviously this has a uniform distribution on $[0,1]$ under $H_0:$ but in what sense is it a p-value? What is the corresponding hypothesis test? Suppose we take a sample of size $n=1$ and observe the value $X_1=-2:$ are you claiming the p-value is $-2$?? $\endgroup$
What does a "p-value" mean in relation to the hypothesis being tested?
In an ontological sense (what is truth?), it means nothing. Any hypothesis testing is based on untested assumptions. This are normally part of the test itself, but are also part of whatever model you are using (e.g. in a regression model). Since we are merely assuming these, we cannot know if the reason why the p-value is below our threshold is because the null is false. It is a non sequitur to deduce unconditionally that because of a low p-value we must reject the null. For instance, something in the model could be wrong.
In an epistemological sense (what can we learn?), it means something. You gain knowledge conditional on the untested premises being true. Since (at least until now) we cannot prove every edifice of reality, all our knowledge will be necessarily conditional. We will never get to the "truth".
I think that examples involving marbles or coins or height-measuring can be fine for practicing the math, but they aren't good for building intuition. College students like to question society, right? How about using a political example?
Say a political candidate ran a campaign promising that some policy will help the economy. She was elected, she got the policy enacted, and 2 years later, the economy is booming. She's up for re-election, and claims that her policy is the reason for everyone's prosperity. Should you re-elect her?
The thoughtful citizen should say "well, it's true that the economy is doing well, but can we really attribute that to your policy?" To truly answer this, we must consider the question "would the economy have done well in the last 2 years without it?" If the answer is yes (e.g. the economy is booming because of some new unrelated technological development) then we reject the politician's explanation of the data.
That is, to examine one hypothesis (policy helped the economy), we must build a model of the world where that hypothesis is null (the policy was never enacted). We then make a prediction under that model. We call the probability of observing this data in that alternate world the p-value. If the p-value is too high, then we aren't convinced by the hypothesis--the policy made no difference. If the p-value is low then we trust the hypothesis--the policy was essential.
answered Nov 3 '14 at 1:21
cgreencgreen
$\begingroup$ I disagree with the p being defined as "We call the probability of observing this data in that alternate world the p-value" and also the strength of the conclusion being drawn (especially failure to reject the null). $\endgroup$
$\begingroup$ @Silverfish Could you elaborate? Probably it would be more correct to call the p-value the probability of making that observation OR a more extreme observation. But it sounds like you have a deeper criticism. $\endgroup$
– cgreen
$\begingroup$ Since the original question is asking what a p-value is, I thought that getting that definition across clearly was important. Just saying "more extreme" isn't in itself very helpful without explaining what "more extreme" might mean - that's a weakness of most answers in this thread I think. Only whuber's answer and the "tea test" one seem to really explain why the "more extreme" matters too. $\endgroup$
$\begingroup$ I also felt your conclusions are phrased too strongly. If we reject the null, we have significant evidence against it, but don't know that it's false. When we fail to reject the null, that certainly doesn't mean the null is true (though it may well be). As a more general comment I have the feeling the test you're describing, in quite abstract terms, is not likely to be clear to a learner who is just learning how to perform a test. The lack of a clearly defined test statistic doesn't sit well with the original question asking how to interpret t-statistic too. $\endgroup$
$\begingroup$ A feature of this answer I like a lot is the clear explanation that p-values are calculated using a null model, even if we don't (subjectively) believe the null model is actually true. I think the fact test statistics are calculated under a model is a key point that many students struggle with. $\endgroup$
The p-value isnt as mysterious as most analysts make it out to be. It is a way of not having to calculate the confidence interval for a t-test but simply determining the confidence level with which null hypothesis can be rejected.
ILLUSTRATION. You run a test. The p-value comes up as 0.1866 for Q-variable, 0.0023 for R-variable. (These are expressed in %).
If you are testing at a 95% confidence level to reject the null hypo;
for Q: 100-18.66= 81.34%
for R: 100-0.23= 99.77%.
At a 95% confidence level, Q gives an 81.34% confidence to reject. This falls below 95% and is unacceptable. ACCEPT NULL.
R gives a 99.77% confidence to reject null. Clearly above the desired 95%. We thus reject the null.
I just illustrated the reading of the p-value through a 'reverse way' of measuring it up to the confidence level at which we reject the null hypo.
edited Jan 8 '12 at 3:54
dytchaydytchay
$\begingroup$ Welcome to the site. What do you mean by $Q$-variable and $R$-variable? Please clarify. Also, use of the phrase "accept null" is usually considered quite undesirable, even misleading. $\endgroup$
– cardinal
$\begingroup$ @cardinal points out an important point. You're not going to accept the null. $\endgroup$
– Patrick Coulombe
******p value in testing of hypothesis measures the sensitivity of the test .The lower the p value the greater is the sensitivity. if significance level is set at 0.05 the p value of 0.0001 indicates a high probability of the test results being correct******
DR.H.K.LAKSHMANRAODR.H.K.LAKSHMANRAO
$\begingroup$ -1 This is clearly wrong. You may want to read the higher voted answers first. $\endgroup$
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On the quotient quantum graph with respect to the regular representation
Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line
Dual spaces of mixed-norm martingale hardy spaces
Ferenc Weisz
Department of Numerical Analysis, Eötvös L. University, H-1117 Budapest, Pázmány P. sétány 1/C., Hungary
Fund Project: This research was supported by the Hungarian National Research, Development and Innovation Office-NKFIH, KH130426
In this paper, we generalize the Doob's maximal inequality for mixed-norm $ L_{\vec{p}} $ spaces. We consider martingale Hardy spaces defined with the help of mixed $ L_{{\vec{p}}} $-norm. A new atomic decomposition is given for these spaces via simple atoms. The dual spaces of the mixed-norm martingale Hardy spaces is given as the mixed-norm $ BMO_{\vec{r}}(\vec{\alpha}) $ spaces. This implies the John-Nirenberg inequality $ BMO_{1}(\vec{\alpha}) \sim BMO_{\vec{r}}(\vec{\alpha}) $ for $ 1<\vec{r}<\infty $. These results generalize the well known classical results for constant $ p $ and $ r $.
Keywords: Mixed Lebesgue spaces, mixed martingale Hardy spaces, atomic decompositions, Doob's inequality, $ BMO $ spaces, John-Nirenberg inequality.
Mathematics Subject Classification: Primary: 42B35; Secondary: 42B30, 60G42, 42B25, 46E30.
Citation: Ferenc Weisz. Dual spaces of mixed-norm martingale hardy spaces. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2020285
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BMC Systems Biology
Uncovering distinct protein-network topologies in heterogeneous cell populations
Methodology Article
Jakob Wieczorek1,
Rahuman S Malik-Sheriff2,3,4,
Yessica Fermin1,
Hernán E Grecco2,5,
Eli Zamir2 &
Katja Ickstadt1
BMC Systems Biology volume 9, Article number: 24 (2015) Cite this article
Cell biology research is fundamentally limited by the number of intracellular components, particularly proteins, that can be co-measured in the same cell. Therefore, cell-to-cell heterogeneity in unmeasured proteins can lead to completely different observed relations between the same measured proteins. Attempts to infer such relations in a heterogeneous cell population can yield uninformative average relations if only one underlying biochemical network is assumed. To address this, we developed a method that recursively couples an iterative unmixing process with a Bayesian analysis of each unmixed subpopulation.
Our approach enables to identify the number of distinct cell subpopulations, unmix their corresponding observations and resolve the network structure of each subpopulation. Using simulations of the MAPK pathway upon EGF and NGF stimulations we assess the performance of the method. We demonstrate that the presented method can identify better than clustering approaches the number of subpopulations within a mixture of observations, thus resolving correctly the statistical relations between the proteins.
Coupling the unmixing of multiplexed observations with the inference of statistical relations between the measured parameters is essential for the success of both of these processes. Here we present a conceptual and algorithmic solution to achieve such coupling and hence to analyze data obtained from a natural mixture of cell populations. As the technologies and necessity for multiplexed measurements are rising in the systems biology era, this work addresses an important current challenge in the analysis of the derived data.
In order to understand how a protein network gives rise to a cellular function it is essential to quantify the states of the involved proteins and their causal relations. However, it is actually not possible to strictly define out of the proteome the subset of all proteins which are involved in a certain cellular process since these will always have interactions with proteins not included in this subset. In spite of major advances in proteomic [1, 2] and cytometric [3–7] methods, quantification of the levels and post-translational modifications of all proteins of the proteome in the same cell is still beyond reach. Therefore, we fundamentally cannot observe the whole system at once (i.e in the same cell) but only a small part of it (Fig. 1a) [8]. This limit, by itself, could have been overcome by looking at different parts of the system in different cells and building a model of the whole system step by step. However, such a strategy is fundamentally hampered by natural cell-to-cell variability which makes the integration of information highly challenging. Several studies have addressed the challenge of network reconstruction in the presence of intrinsic and extrinsic noise [9] around one prototypic network structure [10–12]. However, in many physiological cases the cell-to-cell variance is not only due to noise around one cellular state but also due to subpopulations which are in qualitatively distinct types of states. Such qualitative variabilities within the same cell population are generated by epigenetic commitment of cells to different fates (e.g., proliferation versus differentiation) as well as by genetic alterations (Fig. 1b) as in cancer [13, 14]. In many cases the distinct cell subpopulations are spatially intermixed and therefore are harvested together and co-measured within the same sample (e.g., by flow-cytometry). In such cases, causal relations and correlations between measured proteins can be qualitatively different in different cells if they are mediated by non-measured proteins which have different states at each subpopulation. Therefore, integration of observations over the cell population toward one model would be invalid and will yield uninformative average relations (Fig. 1c, middle). Ultimately, in order to solve this fundamental problem one should identify the number of qualitatively different subpopulations in the data, thus unmix the cells in-silico and resolve separately for each subpopulation the relations between the measured proteins. A recent work suggested to use a mixture model to unravel subpopulations in biochemical systems based on ordinary differential equations and prior knowledge about the number of subpopulations as well as about kinetic constants underlying the differences between them [15]. In this work we developed a Bayesian method for achieving this goal without such prior knowledge.
UNPBN addresses the challenge of studying intracellular protein networks caused by unmeasured proteins and inter-cellular heterogeneity. a A biochemical system for which three proteins (x, y, z) are being measured in the same cell while the other proteins are unmeasured. Note that the effects of z on x are mediated by unmeasured proteins (α and β). b Depending on the level and state of these unmeasured proteins, the measured causality between x and z can differ qualitatively between cells. For example, normal and cancer cells have different activity levels of oncogenes and tumor suppressors which here lead to a negative or a positive causal effect of z on x, respectively, thereby to a controlled growth or a constitutive growth. c Left, multiparametric high-throughput single-cell measurements (e.g., flow-cytometry) of a heterogenous sample of cells (e.g., cancer and normal cells). Middle, attempts to statistically infer a single set of relations (here, causal topology) between the measured proteins fail because there are two distinct subpopulations having two distinct sets of relations. At the same time, it is also impossible to identify the two distinct subpopulations as two distinct proximity-based clusters. Right, UNPBN performs unmixing and inference of statistical relations as one process, thus finds the set of sets-of-relations (network topologies) that explains best the observations
To unmix observations of cells from different subpopulations, we are taking advantage of the high-dimensionality of the observations, as typically obtained from cell-based high-content measurements such as flow-cytometry [3, 13, 16, 17], mass-cytometry [4, 5] and toponome imaging [6, 7]. Within each subpopulation, stochastic cell-to-cell variability in protein expression levels gives rise to high-dimensional probability distributions with the same dimensionality as the number of biochemical species (e.g., proteins) measured in each cell. To this extent, network inference approaches, like Gaussian Bayesian networks (GBN) [18–20], to resolve a single statistical model that fits best the data, have been already developed [21–23]. In this work we use as a basis our previously described nonparametric Bayesian network analysis (NPBN, [21]) and expand it to allow for different network structures in a mixture of different cell subpopulations (Fig. 1c, right). In this method, termed hereafter unmixing-via-NPBN (UNPBN), a flexible number of Gaussian Bayesian networks is being fitted to the data and thereby iteratively identifying the number of distinct subpopulations, unmixing the observations and resolving the statistical model for each subpopulation. As a model system to assess and demonstrate our method we simulated the canonical MAPK Raf-Mek-Erk kinases cascade in the context of PC12 cells stimulated by either epidermal growth factor (EGF) or nerve growth factor (NGF) [24]. We show that our method identifies better than common clustering approaches the presence of two subpopulations within a mixture of EGF and NGF stimulated PC12 cells based on the levels of active Raf, Mek and Erk in each cell. This enabled to resolve correctly the statistical relations between Raf, Mek and Erk in each subpopulation.
The EGF and NGF signaling network was simulated based on a previously described model [24]. The SBML format of this model (BIOMD0000000049, www.ebi.ac.uk, retrieval date Oct. 24, 2011) was imported into the Matlab Symbiology platform to simulate the dynamics of the signaling network using ode15s (stiff/NDF) solver. To introduce intra-subpopulation cell-to-cell variability (termed herein noise), for each run of the simulation we sampled the values for the total Raf, Mek and Erk levels from a Normal distribution N(μ,σ) around the respective initial concentration for a given set of fractional deviation from the mean (σ=μ·f d, where f d∈{0.1,0.2,0.3,0.4,0.5,0.6,0.7}). The values of fd represent here the degree of stochastic variance in the expression levels of Raf, Mek and Erk. Simulations were repeated 175 times with random sampling of total Raf, Mek and Erk levels to generate the data for each cell subpopulation. In each individual simulation repeat, the response of the network to EGF or NGF was simulated for 600 seconds after stimulation and the levels of c-Raf-Ras-GTP (hereafter referred as pRaf, reflecting the consequently activated Raf), ppMek and ppErk (the active, double phosphorylated, forms of Mek and Erk, respectively) were sampled every 1 minute as the observed parameters for the unmixing analysis. Mixtures containing two distinct cell subpopoulations were generated by mixing an equal number, unless indicated otherwise, of simulated observations obtained upon EGF and NGF stimulations. Mixtures containing four distinct cell subpopoulations were generated by altering the parameter in the SBML model corresponding to the catalytic activity (k cat ) of Mek (J136) from its wild-type (Mek wt) value (k cat = 0.15 s −1) to a value depicting a mutant Mek (Mek mut) with a lower activity (k cat = 0.015 s −1). Thus, by having two different stimulations and two different levels of Mek activity, observations of four distinct cell subpopulations were generated: EGF-Mek wt, EGF-Mek mut, NGF-Mek wt and NGF-Mek mut (Additional file 1a-d).
UNPBN
Methodologically, UNPBN is based on the nonparametric Bayesian networks (NPBN) approach [21]. It allows to avoid the assumption of underlying Gaussian distributions for the data and to find networks with nonlinear relations between the nodes. The UNPBN method combines a nonparametric mixture model incorporating the Dirichlet process [21, 25] and an allocation sampler [26, 27]. Prior to the description of the UNPBN approach a short introduction of GBNs [28] is provided here, as they are a basis of the presented method. We define the data X, consisting of n observations of a system/network with d species/nodes (\(X \in \mathbb {R}^{n \times d}\)), such that x j represents an n-dimensional vector containing the observed concentrations of the j species (j=1,…,d). In the Bayesian networks approach the relations between the nodes in a graph \(\mathcal {G}\) are modeled as conditional probability distributions (CPDs) p. If the CPDs for all nodes in \(\mathcal {G}\) are given by Normal distributions of the form \(x_{j}|{pa}_{\mathcal {G}}(x_{j})\sim N(\mu _{j}+\underset {\mathcal {K}_{j}}{\sum }\beta _{j,k} (x_{k}-\mu _{k}), {\sigma _{j}^{2}})\), where \({pa}_{\mathcal {G}}(x_{j})\) denotes the parents of node x j , \(\mathcal {K}_{j}=\{k|x_{k} \in {pa}_{\mathcal {G}}(x_{j})\}\), the μ j and \({\sigma _{j}^{2}}\) are the unconditional means and variances of x j , and β j,k are real-valued coefficients determining the influence of x k on x j , and, if in addition, \(\mathcal {G}\) is a directed acyclic graph (DAG) then the pair \((p,\mathcal {G})\) is called a GBN. The network structure is inferred using Gaussian distributions with a Normal-Wishart prior [20]. The estimation of \(\mathcal {G}\) is embedded in a Markov Chain Monte Carlo (MCMC) framework, conducted by maximizing the sampling distribution of the sampled graph
$$\begin{array}{@{}rcl@{}} L\left(\mathcal{G}| X\right) &=& \prod_{j=1}^{d} \int L\left({\sigma_{j}^{2}}, \beta_{j}| X^{\left(\{j\} \cup\mathcal{K}_{j}\right)}\right)p\left({\sigma_{j}^{2}}, \beta_{j}\right)d\sigma_{j} d \beta_{j}, \end{array} $$
with β j =(β j,1,…,β j,j−1),(β 2,…,β d )=B and \(X^{(\mathcal {J})}\) denotes the columns of X with indices in \(\mathcal {J}\). The MCMC algorithm uses so called single edge operations [29].
UNPBN generalizes the GBN approach as it is based on flexible nonparametric Bayesian mixture models for networks [21] which in turn combine different GBNs for different subsets of the data. The mixture is taken with respect to all parameters \((\mu, \sigma, B, \mathcal {G})\). The model for the data can be written as \(p(x) = \int p(x|\mu,\sigma, B,\mathcal {G})dP(\mu, \sigma, B, \mathcal {G})\) with μ and σ vectors of the unconditional means μ j and variances \({\sigma _{j}^{2}}\), respectively. The discrete mixing measure P is distributed according to \(\mathbb {P}\), a random probability measure, and \(p(x|\mu,\sigma, B,\mathcal {G})\) is a multivariate Normal distribution with a conditional independence structure compatible with \(\mathcal {G}\). According to the discrete nature of P, support points \(\mu _{h},\sigma _{h}, B_{h}, \mathcal {G}_{h}\) and probabilities w h , the mixture can be written as
$$\begin{array}{@{}rcl@{}} p(x) = \sum_{h=1}^{N} w_{h} \ p(x| \mu_{h}, \sigma_{h}, B_{h}, \mathcal{G}_{h}). \end{array} $$
The prior distribution of the mixing weights w h is assigned by P and the prior for \(\mu _{h},\sigma _{h}, B_{h}, \mathcal {G}_{h}\) is given by the base measure P 0 of \(\mathbb {P}\) for all h. The N different mixture components h can be interpreted as subpopulations in the data set. Accordingly, here such subpopulations are referred to as components. The assignment of each data point to its corresponding component is described by the allocation vector l=(l 1,…,l n )′ [26].
The network structure \(\mathcal {G}\) and the allocation vector l are the main focus of our UNPBN procedure. The remaining parameters μ h , σ h and B h are integrated out and the MCMC algorithm iterates by updating the DAG \(\mathcal {G}\), the number of components N and the latent allocation vector l, leading to the posterior distribution
$$\begin{array}{@{}rcl@{}} p(l,\mathcal{G},N|X)= \prod_{h=1}^{N} L(\mathcal{G}| X_{(\mathcal{I}_{h})})p_{N}(m)p(N)p(\mathcal{G}), \end{array} $$
where \(L(\mathcal {G}| X)=\int L(\sigma, B|X)p(\sigma, B)d{\sigma }d{B}\) is the marginal sampling distribution for \(\mathcal {G}\), p N (m) is a probability distribution on the space of allocation vectors, p(N) is the distribution of the number of components and \(\mathcal {I}_{h}=\{i \in \{1,\:\ldots,n\}|l_{i}=h\}\) and \(X_{(\mathcal {I})}\) denotes the rows of X with indices in \(\mathcal {I}\).
In our UNPBN analysis a prior is needed for \( \theta _{h}=(\mu _{h}, \sigma _{h}, B_{h}, \mathcal {G}_{h})\) and for w 1,…,w N . For \(\mathcal {G}_{h}\) the prior which was used is uniform over the cardinality of the parent sets [30], for σ h and B h we employed the Normal-Wishart prior distribution with the identity matrix as the precision matrix and d+2 degrees of freedom. The mean vector of the multivariate Normal distribution (μ h ) was chosen as a vector of zeros. For N we used a Poisson distribution with parameter λ=1 and the w h were obtained from a Dirichlet distribution with parameter vector (α,…,α) with α=1. Further details for the sampling distribution, posterior distribution and the MCMC sampling scheme were discussed in previous publications [21, 26, 31]. The approach is implemented in Matlab (R2009b, The MathWorks Inc., Natick, Massachusetts). The presented results are obtained from MCMC runs with 2.8·106 iterations with a thinning of 350 and a burn in of 1.4·106 iterations for networks with two subpopulations and from runs with 5·106 iterations with a thinning of 500 and a burn in of 2·106 iterations for networks with four subpopulations.
Postprocessing of graphs
Although it is possible to use the output from the UNPBN analysis directly, for example to choose the most frequent DAG or allocation vector as a representative, it is preferable to perform an additional postprocessing step that takes into account all MCMC samples and improves the results considerably. The inferred graphs in the iterations of the MCMC simulation are stored in the form of adjacency matrices. Such a matrix A consists of the elements a i j (i,j=1,…,d), a i j =1 if nodes i and j are conditionally dependent (an arrow leading from node i to node j) and a i j =0 if nodes i and j are conditionally independent (no arrow between them) or if i=j. For each pair of nodes the MCMC output of the UNPBN analysis can be summarized by the posterior edge probability \({pep}_{i j} = \sum _{s=1}^{r} a_{i j}^{s} / r\) where s is the index of the r iteration steps in the MCMC simulation. The resulting pep number ranges from 0 (i.e., strong evidence for the absence of a connection) to 1 (i.e., strong evidence for a connection between the corresponding nodes). These pep values are used in Fig. 5 for the presentation of the obtained results.
Simulations of the EGF and NGF signaling network in PC12 cells. a The simulated protein network, as previously described [24]. The elements (proteins, interactions and processes) that are unique to either EGF or NGF are colored red and green, respectively. The components that we later consider as the measured components of that system (i.e. Raf, Mek and Erk) are colored grey. b A 3-dimensional scatter plot showing the normalized levels of active (i.e. phosphorylated) Raf (pRaf), Mek (ppMek) and Erk (ppErk) versus each other at 2 minutes after NGF (green) or EGF (red) stimulation with noise level of 0.7. c The same as b but for 9 minutes after stimulation. d Time profiles of pRaf, ppMek and ppErk levels as a function of time after NGF (green) or EGF (red) stimulations, without noise (solid line) or with 0.7 noise level (green and red shadows)
Unmixing observations of a mixed cell population by UNPBN in comparison to clustering approaches. a Mixtures of observations of EGF and NGF stimulated cells with different noise levels were generated as described. Observations were sampled at one-minute intervals for 10 minutes after stimulation. For each noise level and sampled time point, observations were unmixed using UNPBN, k-means clustering (with k=2) and hierarchical clustering (taking the final two clusters). The percentages of correctly allocated observations, averaged over all time points, are indicated by boxplots for the different methods as a function of the noise level (line within the box, the median; box, the 0.25 and 0.75 quartiles; whiskers, the largest and smallest data points which are still within the interval of 1.5 times the interquartile range from the box). b Comparison of the unmixing accuracy with noise level 0.7 along the different sampled time points, as achieved by UNPBN, post-processed UNPBN limited to two components, k-means (with k=2) and hierarchical clustering (taking the final two clusters)
The success in identifying the correct number of distinct cell subpopoulations (i.e. components) in a mixture by UNBPN in comparison to clustering approaches. a A boxplot showing the ASW versus the tested number of components obtained by UNBPN analysis (here constrained in the postprocessing step to the imposed number of components) of a mixture of two subpopulations (EGF and NGF stimulated cells). The boxplot indicates the median (line within the box), the 0.25 and 0.75 quartiles (box), margined by the largest and smallest data points which are still within the interval of 1.5 times the interquartile range from the box (whiskers), and the outliers (dots) obtained from pooled values over all time points with noise level of 0.5. b and c, the same as in a but for ASW obtained following k-means clustering and hierarchical clustering, respectively. d, e and f, the same as the corresponding a, b and c, but for a mixture of 4 subpopulations: EGF-Mek wt, EGF-Mek mut, NGF-Mek wt and NGF-Mek mut (Additional file 1a-d). It should be noted that silhouette widths are incomparable between different clustering approaches. However, silhouette widths are comparable between different parameters of the same clustering approach and thereby indicate the identified number of distinct subpopoulations as the one providing the largest ASW
Recovering correctly the edge probabilities network between pRaf, ppMek and ppErk for each cell subpopulation in a mixture by UNPBN. The triangles show, color encoded, the posterior edge probabilities between pRaf, ppMek and ppErk at different time points after stimulation as derived by UNPBN analysis of pure NGF-stimulated or EGF-stimulated cell populations (top two rows) with noise level of 0.2 and of a mixture of these populations by GBN versus UNPBN (bottom three rows). Note, for the pure subpopulations, the changes in the edge probabilities between the same components at different time points and upon the different stimulations. Analyzing a mixture of NGF-stimulated or EGF-stimulated cells without unmixing yields uninformative average edge probabilities, not representing any of the distinct two subpopulations. Employing UNPBN recovers precisely the pure edge probabilities for each of the two subpopulations. For clarity, we show here the edge probabilities for 4 time points after stimulation that illustrate the need for unmixing
Postprocessing of allocations
For the allocation vector, however, it is not possible to summarize the sampled vectors in the same way as for the edges, because of the so called label switching problem. During the sampling procedure the labels of the components change randomly, so that if two allocation vectors are compared it is not clear if a particular observation has been allocated to a different component or if the label of the component has changed. We employed a method based on maximizing the adjusted Rand index that bypasses this obstacle and that combines the allocation vectors of each MCMC iteration into one single vector [32]. This method is implemented in R [33] in the package 'mcclust' and was used in cases where it was necessary to fix the number of components to a particular value (Fig. 4). In cases where the analysis is focused on the unmixing performance of UNPBN (Fig. 3), the sampled allocation vectors are evaluated regarding the homogeneity of the resulting components. For each entry \(l_{i}^{s,h}\) in the allocation vector sampled in iteration s, in each component h, the true component is determined by comparison with the simulation setting. Based on this, componentwise, observations originating from the same true component are considered as allocated correctly, (the indicator function \(I\left (l_{i}^{s,h}\right)\) is set to 1) while the remaining observations in that component are considered as wrongly allocated (\(I \left (l_{i}^{s,h}\right)=0\)). The percentage of correctly allocated observations (pco) for a particular UNPBN outcome is derived by
$$\begin{array}{@{}rcl@{}} \textit{pco} = \frac{1}{r} \sum^{r}_{s=1} \frac{1}{N^{s}} \sum^{N^{s}}_{h=1} \frac{1}{{n_{h}^{s}}} \sum^{{n_{h}^{s}}}_{i=1} I \left(l_{i}^{s,h}\right) \cdot 100 \end{array} $$
with r considered MCMC iterations, size \({n_{h}^{s}}\) of component h in iteration s and N s number of components in the allocation vector in iteration s.
Cluster analyses
In order to compare UNPBN with clustering methods, k-means and hierarchical clustering were used in this work. The k-means clustering method finds the partition that divides the data to n clusters (where n is given by the user) such that the sum distances of all observations to the corresponding cluster mean is minimized [34]. The k-means cluster analysis was performed in Matlab, using the function "kmeans" with the distance parameter being set to squared Euclidean distance. To obtain stable results, the clustering was repeated 500 times with randomly chosen different starting points. Hierarchical clustering is an agglomerative procedure which merges in each step the two closest objects, repeatedly till the whole data set is in one single cluster. The hierarchical clustering was performed in Matlab using the functions "pdist", with the distance parameter being set to Euclidean, followed by "linkage", with the method parameter being set to inner squared distance ("ward", [35]).
Silhouette analysis
We used the average silhouette width (ASW) [36], to assess the quality of a given clustering and to compare the results of clusterings with different parameter settings. For a given clustering result, the silhouette value is calculated as
$$ sil(x_{i}) = \frac{b(x_{i})-a(x_{i})}{\max\{a(x_{i}), b(x_{i}) \}} . $$
For each observation, x i , a(x i ) is the average dissimilarity between x i and all other data points within the same cluster, and b(x i ) is the smallest average dissimilarity between x i and the data points in the remaining clusters, calculated for each cluster separately. Any measure of dissimilarity can be used, but distance measures are the most common. In this work the Euclidean distance was employed. The silhouette value is ranging between -1 and 1. Negative values indicate that a particular observation will fit better in another cluster, so it has been matched wrongly and the quality of the clustering result can be improved. High positive values indicate a good clustering result. The ASW is computed by averaging all s i l(x i ) values, thus it provides an overall evaluation of the regarded clustering. While ASW enables to compare clustering performed with the same method with different parameters, ASW values are not comparable between different clustering methods.
Simulation of inter and intra cell-population variabilities
In order to evaluate the performance of the method we simulated the MAPK module in PC12 cells using a previously described model [24]. This model captures the different temporal profiles of Erk activation upon EGF and NGF stimulation, attributing it to the differential activation and dynamics of Ras and Rap (Fig. 2a) [24, 37]. Both stimulations activate via Sos and Ras the upstream kinase, Raf and thereby the whole MAPK cascade. However, each stimulation has a different effect on other proteins which affect the MAPK module and its dynamics. In EGF stimulation, Erk inhibits Sos and thereby forms a negative feedback loop leading to a transient Erk activation which encodes a proliferation signal. In NGF stimulation this negative feedback is overcome by a nested positive feedback loop [38] formed due to the activation of PKC δ which phosphorylates RKIP and thus leads to its release from Raf and thereby enabling Raf activation by Erk. The model used here considers another difference attributed to a sustained activation of another activator of the MAPK cascade, Rap1, by NGF but not by EGF [24, 37]. Thus, NGF leads to a sustained Erk activation, encoding a signal for differentiation. As a source for inter-population variability, we simulated the dynamics of the complete network upon either EGF or NGF stimulation. For the aim of this work, we based our analyses on snapshots of the simulation, and, in turn, analyzed each time point independently. As a source of intra-population variability (hereafter referred to as noise), we added stochastic noise in total protein levels mimicking natural cell-to-cell variance in protein expression (see Methods). However, unlike instrumental noise that only affects the readout, noise in expression levels affects the system itself. Thus, although the introduced noise was generated as Gaussian, its propagation through the system generates asymmetric high-order patterns shaped by the topology of the network (Fig. 2b,c).
In the absence of noise, the levels of phosphorylated (thus activated) Raf, Mek and Erk follow the expected profiles, exhibiting a clear difference between EGF or NGF stimulations (Fig. 2d, red and green solid lines). With intra-population variance, the profiles get broader and overlap between the two stimulations (Fig. 2d, red and green shadows), making it difficult to allocate individual observations to the corresponding stimulation (as for example at 2 minutes after stimulation, Fig. 2b). To impose the fundamental experimental limit of observing only part of the system, for the subsequent analysis we considered an observation to be the triplet formed by the concentrations of phosphorylated species of Raf, Mek and Erk per cell, ignoring all other information. Finally, to generate heterogeneous cell-populations, we mixed observations randomly selected in equal amounts from the EGF and NGF datasets.
UNPBN unmixes observations of distinct subpopulations
We first wanted to test whether UNPBN can classify correctly observations coming from distinct cell subpopulations. We applied UNPBN on mixed cell populations having different levels of noise and counted the observations correctly allocated to the EGF and NGF stimulated subpopulations (Fig. 3a). For noise levels of 0.1, 0.5 and 0.7, around 100 %, 93 % and 85 % of the observations are correctly allocated, respectively (Fig. 3a). To assess the accuracies of UNPBN, we compared them to those achieved by two widely used clustering approaches - hierarchical clustering and k-means clustering. When the noise is low (0.1 and 0.3), the two subpopulations are well separated by all methods (Fig. 3a). As expected, the performance of all three methods is negatively affected when the noise level is increased. However, the UNPBN considerably outperforms the other reference methods for all noise levels above 0.2 (Fig. 3a). If the relative abundance of the two subpopulations is 1:9, all methods classify about equally well for a low noise level (0.2), while for a high noise level (0.7) UNPBN classifies as good as k-means but better than hierarchical clustering (Additional file 2).
We next focused on the high noise level of 0.7 and compared the performance of the methods as a function of time after stimulation (Fig. 3b). Along the different time points the performance of all methods varies, reflecting a changing difficulty to identify the two subpopulations based on the levels of pRaf, ppMek and ppErk. UNPBN constantly outperforms the clustering methods in all the time points and is more robust with its performance level (Fig. 3b). Furthermore, the performances of the two clustering methods along the time points have a similar profile, which differs from the profile of UNPBN (Fig. 3b, time points 3-10 minutes). These results are consistent with the fact that, unlike the clustering approaches, UNPBN uses high-order patterns, rather than merely distances between observations. This additional information is shown here to be indeed valuable for the ability to unmix subpopulations based on high-dimensional observations.
UNPBN identifies the number of subpopulations in a mixture
In many cases, when a sample of cells is derived it is unknown a priori how many distinct subpopulations it contains. Therefore, a comprehensive unmixing approach should also be able to identify the number of subpopulations without such a priori knowledge. Indeed, while the clustering approaches were guided to search for two subpopulations, UNPBN was not given this information but found it independently (Fig. 3b). Moreover, the performance of UNPBN does not change significantly if it is forced to identify exactly two subpopulations, indicating the ability of UNPBN to correctly determine by itself the number of distinct subpopulations in a mixture (Fig. 3b).
In order to compare the capability of the different methods to identify the number of subpopulations we used the ASW to determine the quality of the clusters and thereby the number of clusters (i.e. subpopulations) in the data as could be inferred by each method [36]. The ASW of a cluster is a measure of how tightly grouped are the data points in the cluster, such that larger ASW values denote tighter clusters. For a cell population containing two subpopulations (EGF and NGF stimulated cells, Additional file 1a,b) we calculated the ASW as a function of the number of clusters derived by UNPBN (here constraints by postprocessing yield an imposed number of components), k-means and hierarchical clustering (Fig. 4a-c). For UNPBN and k-means clustering, the maximal ASW is found when the number of clusters is 2, the actual number of subpopulations in the data (Fig. 4b,c, red bars). However only in UNPBN there is a significant and robust difference with the other cluster sizes, while with k-means clustering the ASWs obtained for 2 and 3 clusters are not robustly separable. With hierarchical clustering the performance is further worse since ASWs obtained for 3 and 4 clusters are comparable, or even higher than those obtained for 2 clusters (Fig. 4c). UNPBN successfully identified the number of subpopulations also if their relative abundance was significantly different (1:9, see Additional file 3).
We next tested the performance of the method with a more complex mixture of cells containing four distinct subpopulations. To simulate these subpopulations, the catalytic rate constant, k cat , of Mek in the model was changed, mimicking a wild-type Mek (Mek wt) and a mutant Mek (Mek mut) that phosphorylate Erk at different rates. Thus, together with the two different stimulations, EGF and NGF, four distinct subpopulations were generated, denoted by EGF-Mek wt, EGF-Mek mut, NGF-Mek wt and NGF-Mek mut (Additional file 1a-d). UNPBN correctly identified that the data contains four distinct subpopulations, in contrast to k-means and hierarchical clustering (Fig. 4d-f).
UNPBN uncovers distinct topologies for distinct subpopulations
The causal relations between the components of a system are constant, since the set of biochemical reactions and constants that describe the whole system remains constant. However, in practice, only part of the components of a system can be co-measured and therefore the reaction constants become apparent constants that depend on the unmeasured components. Here we intentionally simulated the fundamental limit of looking on only a small part of a system. Therefore we expected the apparent strength of the causal connection between pRaf, ppMek and ppErk, as reflected by the undirected posterior edge probabilities among them, to change as a function of the stimulation and time. Indeed, when we analyzed separately EGF and NGF stimulated cells we observed different posterior edge probabilities between the two treatments, as well as within each treatment at the different time points (Fig. 5). When analyzing the mixed population with a standard GBN approach (i.e. without the possibility of unmixing) [39], we obtained posterior edge probabilities exhibiting, in general, an average behavior of the two subpopulations. Naturally, these average values become meaningless when the two subpopulations exhibit very different posterior edge probabilities (e.g., at 2 minutes, Fig. 5). In contrast, when analyzing the mixed population with UNPBN (i.e., with unmixing), the unmixing step enabled to uncover the true network of posterior edge probabilities for each stimulation and at each time point (Fig. 5). This also demonstrates that the performance of the unmixing process (Fig. 3) was sufficiently good to enable correct inference of protein-protein relations in each subpopulation. Since more than one DAG may represent exactly the same set of conditional independence relationships [40], given static data without perturbations it is more reliable to infer the causal strengths between proteins, regardless of the direction of these causalities. Extending the UNPBN approach to dynamic data, or using perturbation data or adding prior information, will further facilitate the inference of directionality in the causal relations for each cell subpopulation.
In the era of systems biology, single-cell measurement techniques are rapidly expanding with respect to the number of cells that can be analyzed and the number of biochemical species that can be co-measured per cell. The approaches to explore these data have focused so far either on identifying different subpopulations of cells based on multiparametric proximities or on inferring the topology of statistical relations between the parameters for the population as a whole. However, the aim to reach each of these two goals in separate has fundamental problems. In one direction, ignoring the heterogeneity between cell subpopulations will lead to inferring a meaningless average topology of statistical relations of the population as a whole. In the other direction, since statistical relations are inferred from the correlation between the measured parameters, the identification of cell subpopulations based on multiparametric proximities inherently conflicts with the capability to resolve the topology of relations within each subpopulation. Furthermore, protein networks with distinct topologies can be at the same state (i.e., to have high multiparametric proximity) and protein networks with the same topology can be at different states (e.g., at different phases along an oscillatory response). Therefore, attempts to identify cell subpopulations based on multiparametric proximities may actually identify different cellular states but not different types of cells. The method presented here pioneers a comprehensive solution to these fundamental problems by performing the identification of cell subpopulations (i.e. unmixing) and the inference of statistical relations between the measured parameters in one joint analysis.
Intentionally, we used snapshot data of a dynamic process (the response of cells to EGF or NGF stimulations) and, respectively, the method we developed does not rely on temporal information nor intends to give a model description of the dynamic itself. Due to that, this method can be applied on the type of single-cell multiparametric measurements currently available such as multicolor flow-cytometry [3, 16], multiplexed mass cytometry [4, 5] and toponome imaging [6, 7]. The classification of the distinct subpopulations in cell populations sampled at different time points along an experiment can hint toward the dynamic behavior of each subpopulation. However, such traceability of subpopulations along the time points depends on how different is their relative abundance within the whole population and on the sampling rate in comparison to the timescale of the biological process. Advances in multicolor live cell imaging in combination with high-throughput automated microscopy gradually facilitate monitoring increasing numbers of parameters in individual live cells over many cells. The data obtained from such measurements will enable not only tracking the dynamics of the measured parameters in each cell subpopulation but also tracking them in individual cells. This kind of temporal information will help to further improve the identification of the distinct cell subpopulations and the inference of statistical relations between measured parameters in each subpoulation. As indicated by this work, it would be important also for the analysis of such live cell measurement data that unmixing and inference of protein-protein relations will be performed as one process.
The importance to recover single-cell phenotypes out of an uninformative average cell population behavior has been established and exemplified in many systems. Notably, in these examples there was only one measured parameter per cell, often the output of the system, and, therefore, the unmixing was straightforward. However, in order to obtain mechanistic insight into how a biochemical system works it is required to examine the protein network itself, and not only its output. For this, multiple parameters should be co-measured per cell to overcome uncorrelated cell-to-cell variability between these parameters (e.g., due to stochastic noise in expression levels as simulated in this work). We demonstrated here that in such a case unmixing cannot be achieved anymore using the proximity between the values of these parameters, while it can be successfully achieved using the high-order relations between them as captured by UNPBN. Importantly, UNPBN can be straightforwardly extended to incorporate prior knowledge about parts of the network in the individual subpopulations.
Our results show that the coupling between unmixing of observations and inference of statistical relations is essential and effective. With respect to the unmixing, our method was capable to identify the number of qualitatively distinct subpopulations considerably better than multiparametric proximity based approaches (hierarchical clustering and k-means clustering). Consequently, the statistical relations in each unmixed subpopulation were also correctly recovered, while, without unmixing, uninformative average relations were inferred. As systems biology and personalized medicine are aiming toward reverse-engineering and re-engineering signaling networks, they are increasingly challenged by the inter-cellular variability and the large size of the relevant biochemical system. The work presented here offers a conceptual solution as well as an applicable statistical method to address this challenge.
ASW:
Average silhouette width
Directed acyclic graph
EGF:
Erk:
Extracellular signal regulated kinase
GBN:
Gaussian Bayesian network
MAPK:
MCMC:
Markov Chain Monte Carlo
Mek:
MAPK/Erk kinase
NGF:
Nerve growth factor
NPBN:
Nonparametric Bayesian network analysis
PKC:
UNPBN:
Unmixing via NPBN
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This work was supported by BMBF grant no. 0315507. YF was supported by a German academic exchange service (DAAD) fellowship.
Faculty of Statistics, TU Dortmund University, Dortmund, Germany
Jakob Wieczorek, Yessica Fermin & Katja Ickstadt
Department of Systemic Cell Biology, Max-Planck Institute of Molecular Physiology, Dortmund, Germany
Rahuman S Malik-Sheriff, Hernán E Grecco & Eli Zamir
Present address: European Molecular Biology Laboratory, European Bioinformatics Institute (EMBL-EBI), Hinxton, Cambridge, UK
Rahuman S Malik-Sheriff
Present address: MRC Clinical Sciences Centre, Imperial College London, London, UK
Present address: Department of Physics, FCEN, University of Buenos Aires and IFIBA, CONICET, Buenos Aires, Argentina
Hernán E Grecco
Jakob Wieczorek
Yessica Fermin
Eli Zamir
Katja Ickstadt
Correspondence to Hernán E Grecco, Eli Zamir or Katja Ickstadt.
JW, RSM-S, HEG, EZ and KI conceived the project, JW and KI developed the UNPBN algorithm, RSM-S, HEG and EZ generated the simulated data, JW, RSM-S, YF, HEG, EZ and KI analyzed the data, JW, RSM-S, HEG, EZ and KI wrote the paper. All authors read and approved the final manuscript.
Additional file 1
The four distinct simulated topologies of the EGF and NGF signaling network used in Fig. 4. (a) NGF-Mek wt: the wild-type network (see Methods) with NGF stimulation. (b) EGF-Mek wt: as in (a) but with EGF stimulation. (c) NGF-Mek mut: the wild-type network with NGF stimulation, beside that here the SBML model parameter corresponding to the k cat of Mek (J136) is altered from its wild-type value (k cat = 0.15 s −1) to a value depicting a mutant Mek with a lower activity (k cat = 0.015 s −1), as indicates the thinner arrow from Mek to Erk. (d) EGF-Mek mut: as in (c) but with EGF stimulation.
Unmixing observations of cell subpopulations, mixed in a 1:9 ratio, by UNPBN in comparison to clustering approaches. Mixtures of observations of EGF stimulated cells (90 %) and NGF stimulated cells (10 %) were generated with noise levels of 0.2 and 0.7. Observations were sampled at one-minute intervals for 10 minutes after stimulation. For each noise level and sampled time point, observations were unmixed using UNPBN, k-means clustering (with k=2) and hierarchical clustering (taking the final two clusters). The percentages of correctly allocated observations, averaged over all time points, are indicated by boxplots for the different methods for both noise levels (line within the box, the median; box, the 0.25 and 0.75 quartiles; whiskers, the largest and smallest data points which are still within the interval of 1.5 times the interquartile range from the box). (a) The percentages of correctly allocated observations of the EGF-stimulated subpopulation. (b) The percentages of correctly allocated observations of the NGF-stimulated subpopulation.
The success of UNPBN in identifying the correct number of cell subpopulations (i.e. components) mixed in a 1:9 ratio. Mixtures of observations of EGF stimulated cells (90 %) and NGF stimulated cells (10 %) were generated with noise levels of 0.2 and 0.7. Observations were sampled at one-minute intervals for 10 minutes after stimulation. (a) A boxplot showing the ASW versus the tested number of components obtained by UNBPN analysis (here constrained in the postprocessing step to the imposed number of components). Each boxplot indicates the median (line within the box), the 0.25 and 0.75 quartiles (box), margined by the largest and smallest data points which are still within the interval of 1.5 times the interquartile range from the box (whiskers), and the outliers (dots) obtained from pooled values over all time points with a noise level of 0.2. (b) The same as (a) but with a noise level of 0.7.
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Wieczorek, J., Malik-Sheriff, R.S., Fermin, Y. et al. Uncovering distinct protein-network topologies in heterogeneous cell populations. BMC Syst Biol 9, 24 (2015). https://doi.org/10.1186/s12918-015-0170-2
Bayesian analysis
Intercellular variability
Protein networks
Unmixing | CommonCrawl |
Spectrum sensing in cognitive radio networks: threshold optimization and analysis
Kenan kockaya ORCID: orcid.org/0000-0002-5253-15111 &
Ibrahim Develi ORCID: orcid.org/0000-0001-5878-677X2
Cognitive radio is a technology developed for the effective use of radio spectrum sources. The spectrum sensing function plays a key role in the performance of cognitive radio networks. In this study, a new threshold determination method based on online learning algorithm is proposed to increase the spectrum sensing performance of spectrum sensing methods and to minimize the total error probability. The online learning algorithm looks for the optimum decision threshold, which is the most important parameter to decide the presence or absence of the primary user, using historical detection data. Energy detection- and matched filter-based spectrum sensing methods are discussed in detail. The performance of the proposed algorithm was tested over non-fading and different fading channels for low signal-to-noise ratio regime with noise uncertainty. In the conclusion of the simulation studies, improvement in spectrum sensing performance according to optimal threshold selection was observed.
Wireless communication systems are undergoing rapid development to meet the changing demands and needs of people. The increase in wireless applications and services made it essential to address the spectrum scarcity problem. Measurements made by the Federal Communications Commission (FCC) of the United States telecommunications authority have shown that licensed bands are not used at a rate of up to 90%. The results of the measurement were published by the FCC Spectrum Policy Task Force group in the report entitled "FCC Report of the Spectrum Efficiency Working Group" [1]. In recent years, a lot of research has been done on the effective use of these spectrum bands which are either empty or are not used at full capacities. One of the notable concepts in the researches is the cognitive radio concept, introduced by Mitola in 1999 [2]. CR is a software-based technology that detects the electromagnetic environment in which it operates, detects unused frequency bands, and adapts the radio working parameters to broadcast in these bands [3]. CR is a key technology that enables the limited and inefficiently used frequency bands to be used more efficiently with an opportunistic approach. Communication performance and continuity in cognitive radio networks are highly dependent on whether the spectrum sensing function is performed correctly.
Spectrum sensing is a critical issue of cognitive radio technology because of the shadowing, fading, and time-varying natures of wireless channels. To sense limited or unused frequency bands, different methods for spectrum sensing have been proposed in the literature like matched filtering [4, 5], cyclostationary-based sensing [6,7,8], waveform-based sensing [9], wavelet-based sensing [10], eigenvalue-based sensing [11, 12], and energy detection sensing [13,14,15]. Matched filtering detection methods with shorter detection periods are preferred if certain signal information is known, such as bandwidth, operating frequency, modulation type and grade, pulse shape, and frame structure of the primary user [16, 17]. The detection performance of this method largely depends on the channel response. To overcome this, it requires perfect timing and synchronization in both physical and medium access control layers. This situation increases the complexity of calculation. Cyclostationary detection is a method for detecting primary user transmissions by exploiting the cyclostationarity features of the received signals [18,19,20]. It exploits the periodicity in the received primary signal to identify the presence of primary users. In this way, the detector can distinguish primary user signals, secondary user signals or interference. However, the performance of this detection method depends on a sufficient number of samples, which increases the computational complexity. Waveform-based sensing is only applicable to systems with known signal patterns. Such patterns include preambles, midambles, regularly transmitted pilot patterns, and spreading sequences [21]. A preamble is a known sequence transmitted before each burst and a midamble is transmitted in the middle of a burst or slot. In the case of a known model, the spectrum detection function is performed by associating the received signal with a copy of itself. Wavelet transform is a powerful method for analyzing singularities and edges. In the wavelet-based spectrum sensing method, the frequency bands of interest are usually decomposed as a train of consecutive frequency subbands [22]. By using wavelet transform, irregularities in these bands are detected and the spectrum is decided whether it is full or empty. Eigenvalue-based spectrum sensing does not require much prior knowledge about the primary user signals and noise power [23,24,25]. The concept of this detection technique is presented in 2007 [26]. In the eigenvalue-based spectrum sensing methods, the decision threshold has been obtained based on random matrix theory to make a hypothesis testing. In order to determine the presence or absence of the primary user signal, the decision threshold is compared with the test statistic formed using the ratio of the maximum or average eigenvalue to the minimum eigenvalue. Nevertheless, having a high operational complexity is a disadvantage of this method. Similarly, if the information of the primary users is not known precisely, energy detection-based methods with low mathematical and hardware complexities are preferred [27, 28].
Energy detection is a spectrum sensing technique based on measuring the received signal energy and deciding on the presence or absence of the primary user by comparing the received energy level with a threshold. The threshold function calculation depends on noise power. Numerous studies have been carried out in the literature to obtain the optimal threshold expression and to improve spectrum sensing performance [29,30,31,32,33,34]. In [29], the authors proposed a new method for adaptive threshold selection in multiband detection. Estimating the threshold is performed by using the functions of the first and second statistics of the received signal. In [30], the Wigner–Ville distribution is used to improve detection performance at a low SNR. In this case, a better decision threshold is defined by reducing the effects of the cross-correlation terms. In [31], using Gauss–Hettite integration, analytical expressions of detection, and mean-field probabilities on compound Nakagami-m and log-normal fading channels were obtained, and detection performance was investigated. Also, an optimized threshold expression was obtained to increase spectrum sensing performance. In [32], an energy detector, using an adaptive dual threshold, is proposed for solving the detection problem. In [33], the authors proposed an adaptive threshold detection algorithm based on an image binarization technique. Here, the dynamic threshold is estimated based on previous repetition decision statistics, parameters such as SNR, number of instances, and detection probabilities. In [34], a dynamic threshold detection scheme was proposed depending on the noise level present in the received signal. For the measurement of the noise level, a blind technique based was used on the sample covariance matrix values of the received signal.
The energy detection method is widely used for its simplicity in calculation and ease of application. However, the spectrum sensing performance of the energy detector is severely affected by destructive channel effects such as shadowing and fading, and noise. To minimize the negative effects caused by noise uncertainty and communication channel, the cooperative spectrum sensing model is defined in the literature [35, 36]. In [35], the researchers proposed a fuzzy logic-based perception format for collaborative energy detection, based on the new reliability factors for local spectrum sensing. The fuzzy logic process consists of three stages. These are the ordering of blurring, the run-in motor, and the clearing phase. The performance of the nodes is compared with the performance of the other nodes to try to make the most accurate predictions. When these processes are performed, the reliability factor is defined by using the SNR, detection differences, and threshold, and the detection performance is measured. In [36], energy detector parameters are optimized for the best detection performance. Simulation studies have been carried out on fading channels about the optimal threshold, several cognitive radio users, and the number of antennas.
In recent years, hybrid models in which two or more detection schemes are used together have been developed to improve spectrum sensing performance in a cognitive radio network. Artificial intelligence and machine learning algorithms (MLA) are widely used in hybrid models [37,38,39,40]. In [37], a learning algorithm based on artificial neural networks (ANN) is used to detect the presence/absence of primary users in a cognitive radio environment. In [38], the authors proposed a collaborative spectrum sensing (CSS) scheme based on machine learning techniques. Supervised [e.g., support vector machine (SVM) and weighted K-nearest neighbor (KNN)], and unsupervised [e.g., K-means clustering and Gaussian mixture model (GMM)] classification techniques are used for CSS. In [39], the authors proposed a sensing method based on machine learning for solving the spectrum sensing problem. This method is dependent on signal characteristics and the clustering algorithm that is used for classification. The received signals are classified by using the k-means clustering algorithm. Class parameters, eigenvalues, and covariance were determined, and the performance of the proposed algorithm was investigated. Using the MLA, it is stated that the error probability decreased and the detection performance increased. In [40], the researchers proposed a new decision threshold model based on an online learning algorithm to increase the probability of detection and decrease the probability of total detection.
In this paper, we proposed a new threshold expression based on online learning algorithm to overcome the spectrum sensing problem and improve detection accuracy. Statistical error analysis was performed by using data on detection, miss detection, and false alarm parameters used in spectrum sensing performance measurement. The proposed new method consists of two stages. In the first stage, a hypothesis test is created and analyzed depending on the noise threshold. In the second stage, the threshold expression that minimizes the total error probability with the help of an online learning algorithm is redefined. The detection performance of the proposed method was investigated on AWGN, Rayleigh, Rician, Nakagami-m, and Weibull fading channels and presented comparatively with the traditional method suggested in the literature.
The rest of this paper is organized as follows: Sect. 2 considers the theoretical aspects of energy-based spectrum sensing. Optimal thresholds are presented with a sufficient optimality condition in Sect. 2.2. In Sect. 3, the optimal threshold expression is redefined and formulated by using the proposed online learning algorithm. Simulation results are discussed in Sect. 4, and finally, the paper is concluded in Sect. 5.
Spectrum sensing is one of the most important components of cognitive radio networks. Spectrum sensing enables a cognitive radio to have information about its environment and spectrum availability. The most widely used spectrum sensing methods are energy detection and matched filter detection.
Energy detection
Energy detection is the most widely used method since it has low complexity and it does not require prior information about of the primary signals. In the energy detection process, the spectrum occupancy decision is based only on the threshold obtained depending on the noise. The threshold is compared with the perceived energy, and it is decided whether the primary user is present or not. It aims essentially to decide between two states: primary user signal is absent, denoted by \(H_{0}\), or primary user signal is present, denoted by \(H_{1}\). The decision of energy detector is the test of the following hypothesis:
$$\begin{array}{*{20}l} {H_{0} :Y\left( n \right) = W\left( n \right), } \hfill & {{\text{:Primary}}\;{\text{ user }}\;{\text{absent}}} \hfill \\ {H_{1} :Y\left( n \right) = S\left( n \right) + W\left( n \right), } \hfill & {{\text{:Primary}}\;{\text{ user }}\;{\text{present}}} \hfill \\ \end{array}$$
where \(Y\left( n \right)\) is the signal received by the secondary user, \(S\left( n \right)\) is the primary user's transmitted signal, and \(W\left( n \right)\) is the additive white Gaussian noise (AWGN) with zero mean. Figure 1 shows the basic block diagram of the energy detection.
Block diagram of energy detector
In an energy detector, the received signal is first pre-filtered by an ideal band pass filter which has bandwidth "W." The filtered signal is then passed through A/D converter. Output of the A/D converter is then squared and integrated over a predefined time interval. The resultant signal is used to formulate a test statistic. The test statistic can be formulated as shown in Eq. 2.
$$T = \mathop \sum \limits_{n = 0}^{N} \left| {Y\left( n \right)} \right|^{2}$$
where \(n = 0,1,2,3, \ldots ,N\), which represents the number of samples (detection period). If \(N\) sample numbers are sufficient, the T statistic distribution, according to the central limit theorem, is Gaussian distribution [41]. The binary hypothesis test is redefined as follows:
$$\begin{array}{*{20}l} {H_{0} : } \hfill & { T\sim {\text{Normal}}\left( {N\sigma_{n}^{2} + N2\sigma_{s}^{4} } \right)} \hfill \\ {H_{1} :} \hfill & {T\sim {\text{Normal}}\left( {\left( {\sigma_{n}^{2} + \sigma_{s}^{2} } \right),2N\left( {\sigma_{n}^{2} + \sigma_{s}^{2} } \right)^{2} } \right) } \hfill \\ \end{array}$$
where \(\sigma_{n}^{2}\) and \(\sigma_{s}^{2}\) are the noise variance and signal variance, respectively.
The test statistic (\(T)\) is compared with the threshold (\(\lambda )\) to make the final decision on whether the primary signal is present or not. The performance of the energy detector is characterized by using three parameters presented based on test statistics under the binary hypothesis. According to [42], the probabilities of detection \(P_{d}\), false alarm \(P_{fa}\), and miss detection (\(P_{m} = 1 - P_{d} )\) are given by,
$$P_{d} = P\left( {T > \lambda /H_{1} } \right) = Q\left( {\frac{{\lambda - N\left( {\sigma_{n}^{2} + \sigma_{s}^{2} } \right)}}{{\sqrt {2N\left( {\sigma_{n}^{2} + \sigma_{s}^{2} } \right)^{2} } }}} \right)$$
$$P_{fa} = P\left( {T > \lambda /H_{0} } \right) = Q\left( {\frac{{\lambda - N\sigma_{n}^{2} }}{{\sqrt {2N\sigma_{n}^{4} } }}} \right)$$
where \(Q\left( . \right)\) is the complementary distribution function of standard Gaussian. Q-function \(Q\left( x \right)\) is expressed as follows:
$$Q\left( x \right) = \frac{1}{{\sqrt {2\pi } }}\mathop \smallint \limits_{x}^{\infty } \exp \left( { - \frac{{y^{2} }}{2}} \right)dy.$$
Matched filter detection
Matching filter technique is widely used in spectrum sensing as it is a good filtering technique that maximizes the SNR. When an unknown signal matched the known signal, it is assumed that PU is present in the spectrum. The whole process of the matched filter is shown in Fig. 2 [43].
Block diagram of matched filter detection
The operation of matched filter detection is expressed as:
$$y\left( n \right) = \mathop \sum \limits_{l = - \infty }^{\infty } h\left( {n - l} \right)s\left( l \right)$$
where \(y\left( n \right)\) is the received signal, \(s\left( l \right)\) is the unknown signal, and \(h\left( {n - l} \right)\) is the impulse response of the matched filter which matches with the known signal for maximizing the output SNR. \(P_{d}\) and \(P_{fa}\) can be given in Eqs. (8) and (9) which depend upon threshold [44].
$$P_{d} = Q\left( {\frac{\lambda - E}{{\sqrt {E\sigma_{n}^{2} } }}} \right)$$
$$P_{fa} = Q\left( {\frac{\lambda }{{\sqrt {E\sigma_{n}^{2} } }}} \right)$$
where \(E\) is the PU signal energy. The detection threshold is given in Eq. 10 as a function of PU signal energy and noise variance.
$$\lambda = Q^{ - 1} \left( {P_{fa} } \right)\sqrt {E\sigma_{n}^{2} }.$$
Threshold detection model
The performance of energy sensing-based methods is largely dependent on the previously defined threshold expression [45, 46]. A threshold is required to decide whether the target signal is absent or present. This threshold determines all spectrum sensing performance metrics. The sensing performance of the energy detector is measured according to two metrics. The performance metrics \(P_{d}\) and \(P_{fa}\) over AWGN channels can be defined as [47, 48]:
$$P_{d} = \frac{1}{2}erfc\frac{{\lambda - \mu_{1} }}{{\sqrt 2 \sigma_{1} }}$$
$$P_{fa} = \frac{1}{2}erfc\frac{{\lambda - \mu_{0} }}{{\sqrt 2 \sigma_{0} }}$$
where erfc is the complementary error function. It then follows that the mean and the variance of the test statistic could be represented as shown in Eqs. 13 to 16.
$$\mu_{0} = N\sigma_{n}$$
$$\mu_{1} = N\sigma_{n}^{2} \left( {\gamma + 1} \right)^{2}$$
$$\sigma_{0}^{2} = 2N\sigma_{n}^{4}$$
$$\sigma_{1}^{2} = 2N\sigma_{n}^{4} \left( {\gamma + 1} \right)^{2}.$$
The probability of miss detection would be given as,
$$P_{m} = 1 - P_{d}.$$
The balance between \(P_{fa}\) and \(P_{m}\) should be considered when determining the threshold for the energy detector. \(P_{d}\) should be maximized, while \(P_{fa}\) should be minimized. This is called the constant false alarm rate (CFAR) detection scheme. \(P_{m}\) can be set to a minimum value, or \(P_{fa}\) can be reduced to a minimum by fixing \(P_{d}\) to a maximum value. In practice, the threshold is normally chosen to meet a certain \(P_{fa}\), in situations where only the noise power needs to be known. Depending on the balance between \(P_{d}\) and \(P_{fa}\), \(\lambda\) for a certain \(P_{fa}\) value is derived as:
$$\lambda = Q^{ - 1} \left( {P_{fa} } \right)\sqrt {2N} + \left( N \right)\sigma_{n} { }$$
where \(Q^{ - 1} \left( . \right)\) is the inverse function of \(Q\left( . \right)\).
Due to this threshold at low SNR, the detection performance is greatly reduced. What is important here is to improve the low SNR perception performance. For this reason, the optimal threshold expression is defined by using the total error probability,\(P_{e}\), which is dependent on \(P_{fa}\) and \(P_{m} .\) The total error probability is the sum of \(P_{fa}\) and \(P_{m}\) weights. \(P_{e}\) can be given as
$$P_{e} = PH_{0} P_{fa} + PH_{1} P_{m}$$
where \(PH_{1}\) and \(PH_{0}\) represent the probabilities of primary user presence and absence, respectively. The minimization problem can be represented as
$$\lambda = {\text{argmin}}_{\lambda } \left( {PH_{0} P_{fa} + PH_{1} P_{m} } \right).$$
The threshold can be obtained by satisfying the following conditions [46]:
$$\frac{{\partial P_{fa} }}{\partial \lambda } + \frac{{\partial P_{m} }}{\partial \lambda } = 0$$
$$\frac{{\partial^{2} P_{e} }}{{\partial \lambda^{2} }} < 0.$$
From Eqs. (12, 17) on differentiating \(P_{fa}\) and \(P_{m}\) are given as follows:
$$\frac{{\partial P_{fa} }}{\partial \lambda } = - \frac{1}{{\sqrt {2\pi } \sigma_{0} }}e^{{ - \left( {\frac{{\left( {\lambda - {\upmu }_{0} } \right)^{2} }}{{\sqrt 2 \sigma_{0} }}} \right)}}$$
$$\frac{{\partial P_{m} }}{\partial \lambda } = - \frac{1}{{\sqrt {2\pi } \sigma_{1} }}e^{{ - \left( {\frac{{\left( {\lambda - {\upmu }_{1} } \right)^{2} }}{{\sqrt 2 \sigma_{1} }}} \right)}}.$$
Using Eqs. 21, 22, 23, and 24, the threshold expression is redefined as follows:
$$\lambda = \frac{{ - b + \sqrt {b^{2} - ac} }}{a}$$
$$a = \sigma_{1}^{2} - \sigma_{0}^{2}$$
$$b = \sigma_{0}^{2} \mu_{1} - \sigma_{1}^{2} \mu_{0}$$
$$c = \sigma_{1}^{2} \mu_{0} - \sigma_{0}^{2} \mu_{1} - \frac{{2\sigma_{1}^{2} \sigma_{0}^{2} }}{{\ln \left( {\frac{{\sigma_{1} }}{{\sigma_{0} }}} \right)}}.$$
Proposed adaptive threshold optimization model
In cognitive radio systems, the detection performance of the energy detector depends on the high accuracy selection of the threshold expression. When developing spectrum sensing models, it is aimed that the noise and primary user signals are fully distinguished. Developed models are generally evaluated based on parameters such as accuracy and correct positive rate. However, the actual performance can be analyzed by using backwardly artificially generated estimates in the measurements. In this section, a new threshold expression model based on online learning algorithm is presented to improve spectrum sensing performance in cognitive radio networks.
The fundamental nature of spectrum sensing is a defined binary hypothesis testing problem that depends on the threshold expression. This relationship is illustrated in Fig. 3. This shows the expected distribution of a difference between two groups under \(H_{0}\) [true negative (TN)] and \(H_{1}\) [true positive (TP)]. It is clear that if we increase the type I error rate [false positive (FP) or false alarm], we reduce the type II error rate [false negative (FN) or missed detection], and vice versa. Changes in the accuracy of \(H_{0}\) and \(H_{1}\) hypotheses cause changes in the total error probability. Therefore, there is a very delicate balance between the possibility of miss detection and the possibility of false detection. To maintain and analyze the balance between these two, two classes are created by classifying the negative and positive data as shown in Fig. 3. Critical thresholds are determined for these classes, creating a gray area. Then, with the help of an online learning algorithm, the steps to be applied to obtain the most appropriate threshold in the gray area are given as follows:
Statistical distribution curves related to classes
Stage 1: data collection and pre-processing
The Gauss distribution curves of \(H_{1}\) (signal present) and \(H_{0}\) (signal absent) are obtained by using the threshold expression in Eq. 25. Two classes are constructed by classifying the negative and positive data, as shown in Fig. 3. Type I and II error parameters and correct perception parameters are analyzed. Critical thresholds are determined for these classes, creating a gray area.
Each \(\left( {N_{i} , P_{i} } \right)\) value is determined, and classes are created.
Critical thresholds expressions of the two classes are defined (\(\lambda_{N} ,\lambda_{P} ).\)
Subclasses are created within the remaining gray area between two thresholds \(\left( {X_{1,2,} , \ldots ,X_{n} } \right).\).
The data in the gray area, defined as R in Fig. 3, were subclassified using the k-mean algorithm (k = 4). The classes created are graded according to their performance levels, considering type I and II errors.
Stage 2: computation on the dataset
Error analysis is performed to further increase the success level of successful classes with the help of the data obtained during data collection and pre-processing. As a result of the analysis, weight, error, and improvement coefficients are defined as follows:
Weights are defined for each subclass. \(\left( {w_{t} } \right)\).
Averages of weights are found. It is expressed as shown in Eq. 29;
$$w_{t} = \frac{{w_{t,i} }}{{\mathop \sum \limits_{i = 0}^{N} w_{t,i} }}.$$
The data are classified and the total error rate is obtained. It can be represented as shown in Eqs. 30 and 31.
$$E_{T} = \varepsilon_{t} = \min \mathop \sum \limits_{i = 0}^{N} w_{t} c_{i}$$
$$c_{i} = \left\{ {\begin{array}{*{20}c} {0,} & {h_{t} \left( {H_{i} ,X_{i} ,Y_{i} ,P_{i} } \right) = y_{i} } \\ {1,} & {h_{t} \left( {H_{i} ,X_{i} ,Y_{i} ,P_{i} } \right) \ne y_{i} } \\ \end{array} } \right..$$
Incorrect positive error (H1/incorrect detection) is expressed in Eq. 32 as follows:
$$E_{FP} = \mathop \sum \limits_{i = 0}^{p} w_{t,i} c_{i}.$$
Incorrect negative error (H0/incorrect detection) is expressed in 33.
$$E_{FN} = \mathop \sum \limits_{i = 0}^{N} w_{t,i} c_{i}.$$
Classification probabilities and ratios can be formulated as follows, respectively:
$$P_{{{\text{FP}}}} = \frac{{E_{{{\text{FP}}}} }}{{E_{{\text{T}}} }}$$
$$P_{{{\text{FN}}}} = \frac{{E_{{{\text{FN}}}} }}{{E_{{\text{T}}} }}$$
$${\text{TPR}} = \frac{{{\text{TP}}}}{{{\text{TP}} + {\text{FN}}}}$$
$${\text{TNR}} = \frac{{{\text{TN}}}}{{{\text{TN}} + {\text{FP}}}}.$$
Mathews correlation coefficient can be represented as shown in Eq. 38.
$${\text{MCC}} = \frac{{{\text{TP}}*{\text{TN}} - {\text{FP}}*{\text{FN}}}}{{\sqrt {\left( {{\text{TP}} + {\text{FP}}} \right)\left( {{\text{TP}} + {\text{FN}}} \right)\left( {{\text{TN}} + {\text{FP}}} \right)\left( {{\text{TN}} + {\text{FN}}} \right)} }}.$$
Improvement coefficient (\(p_{i}\)) can be formulated by Eq. 39 as follows:
$$p_{i} = \left[ {\log \left( {\frac{{1 - \varepsilon_{t} }}{{\varepsilon_{t} }}} \right)} \right]\left( {\frac{{1 - P_{{{\text{FN}}}} }}{{P_{{{\text{FP}}}} }}} \right)*{\text{MCC}}.$$
Stage 3: training phase
We are provided with a training dataset \(\left( {X_{i} , Y_{i} } \right),\) \(i = 1,2,3, \ldots ,N\) where \(X_{i}\) represents an n-dimensional continuous-valued vector and \(Y_{i}\) {0,1} represents the corresponding class label with "0" for normal and "1" for an anomaly. The proposed method has two steps: (1) training and (2) testing. During training, the k-means-based anomaly detection method is first applied to partition the training space into k disjoint clusters \(C_{1} ,C_{2} ,C_{3} , \ldots ,C_{N}\). Then, the decision tree is trained with the instances in each k-means cluster. The k-means method ensures that each training instance is associated with only one cluster. However, if there are any subgroups or overlaps within a cluster, the decision tree trained on that cluster refines the decision boundaries by partitioning the instances with a set of if–then rules over the feature space. In the testing phase, we have two subdivided phases: (1) the selection phase and (2) the classification phase. In the selection phase, the Euclidean distance is calculated for each test sample and the closest cluster is found. The decision tree for the closest cluster is calculated. In the classification phase, the data are separated according to the detection successes. Finally, in this phase, the threshold will learn from the best learner in class. Learner modification is expressed as,
$$\lambda_{i}^{{{\text{new}}}} = \lambda + p_{i} \left[ {\frac{{\min \left( {{\text{energy}}_{i} } \right) + \max \left( {{\text{energy}}_{i} } \right)}}{2}} \right].$$
Stage 4: learner phase
In this phase, by comparing the advantages and disadvantages between the other two learners, the learners \(\lambda_{i}^{{{\text{new}}}}\) will learn from their advantages which draw on the idea of the differential evolution algorithm. Randomly select two learners \(\lambda_{i}\) and \(\lambda_{j}\), where \(i \ne j\). Learner modification is expressed as
$$\lambda_{i}^{{{\text{new}}}} = \lambda_{i}^{{{\text{old}}}} + {\text{rand}}\left( a \right)*\left( {\lambda_{i} - \lambda_{j} } \right)\quad {\text{if}}\;P_{i} > P_{j}$$
$$\lambda_{i}^{{{\text{new}}}} = \lambda_{i}^{{{\text{old}}}} + {\text{rand}}\left( a \right)*\left( {\lambda_{j} - \lambda_{i} } \right)\quad {\text{if}}\;P_{i} < P_{j }$$
where \({\text{rand}}\left( a \right)\) is a uniformly distributed random number between "0" and "1." Accept \(\lambda_{i}^{{{\text{new}}}}\) if it gives an optimum threshold.
In this section, numerical results are presented to verify the effectiveness of our proposed algorithm. Spectrum sensing performance can be characterized by using the receiver operating characteristic (ROC) curve in cognitive radio networks. ROC curves are generated by plotting either detection probability versus false alarm probability or missed detection probability versus false alarm probability. Detection probability and false alarm probability depend on the threshold, number of samples, fading parameters, number of diversity branches, and average SNR. The sensing performance of the proposed algorithm has been analyzed on different fading channels using energy-based detection and matched filter detection techniques. In Figs. 4, 5, 6, 7, 8, 9, 10, 11, 12, and 13, simulation results are provided to compare our (an online learning algorithm) threshold selection with a conventional (dynamic) threshold selection (calculated from \(P_{fa}\) = 0.1).
ROC (\(P_{d}\) vs \(P_{fa}\)) of energy detector sensing under AWGN channel
ROC (\(P_{d}\) vs \(P_{fa}\)) of Energy detector sensing under Rayleigh fading channel
ROC (\(P_{d}\) vs \(P_{fa}\)) of energy detector sensing under Nakagami-m fading channel (m = 3)
ROC (\(P_{d}\) vs \(P_{fa}\)) of energy detector sensing under Rician fading channel (K = 5)
ROC (\(P_{d}\) vs \(P_{fa}\)) of energy detector sensing under Weibull fading channel (a = 3)
ROC (\(P_{d}\) vs \(P_{fa}\)) of matched filter detection sensing under AWGN channel
ROC (\(P_{d}\) vs \(P_{fa}\)) of Matched filter detection sensing under Rayleigh fading channel
ROC (\(P_{d}\) vs \(P_{fa}\)) of matched filter detection sensing under Nakagami-m fading channel (m = 3)
ROC (\(P_{d}\) vs \(P_{fa}\)) of matched filter detection sensing under Rician fading channel (K = 5)
ROC (\(P_{d}\) vs \(P_{fa}\)) of matched filter detection sensing under Weibull fading channel (a = 3)
Because the performance of energy-based technique mainly depends on SNR, two different SNR values (-5 and -10 dB) are considered. Figure 4 shows the ROC curve for the AWGN channel. As can be seen, the performance of the proposed algorithm for different SNR scenarios is higher than those of conventional algorithm: dynamic threshold (-5 dB): \(P_{d}\) = 0.6371; online learning threshold (-5 dB): \(P_{d}\) = 0.6509; dynamic threshold (-10 dB): \(P_{d}\) = 0.3915; online learning threshold (-10 dB): \(P_{d}\) = 0.4025. Figures 5, 6, 7, and 8 illustrate the ROC curves for Rayleigh, Nakagami-m, Rician, and Weibull channels, respectively. When the graphs are examined, it is seen that the detection performance of cognitive radio increases with the proposed method. Besides, detection probability is less in Rayleigh fading channel when compared to the AWGN channel and other fading channels. This situation is shown in Fig. 5. In Fig. 7, we can see that the performance of the energy detector in the Rician fading channel is better than in the other channels (Rician factor K = 5).
Figure 8 shows that, for energy detection in the Weibull fading channels, ROC curves move to the upper left corner with the proposed method, confirming better overall detection performance.
In Figs. 9, 10, 11, 12, and 13, the evaluation of the performance of the matched filter detection technique is carried out by plotting Pd versus Pfa and ROC curves for the AWGN, Rayleigh, Nakagami-m, Rician, and Weibull channels conditions. Figure 9 shows the comparison of the performance of the proposed scheme and the dynamic threshold selection method and verifies the accuracy of the theoretical expressions for the matched filter technique over a non-fading AWGN channel.
Figure 10 shows the ROC curve in the Rayleigh fading channel. It is observed that when compared to AWGN, Rayleigh fading has less detection probability due to fading. Spectrum sensing performance is dependent on SNR. As the SNR increases, the probability of detection is improved.
Figures 11, 12, and 13 show the ROC curves over Nakagami-m, Rician, and Weibull fading channels, respectively.
When comparing the detection probability of all these fading channels (AWGN, Rayleigh, Nakagami-m, Rician and Weibull), it is clear that the Rician fading channel has the best detection performance. It is also seen that the performance of the matched filter detector is affected by the average SNR values.
It is clearly seen that the detection performance of the online learning algorithm-based decision threshold method and the detection performance of the dynamic decision threshold determination method are better for different SNR values on different fading channels. This is because conventional methods offer a strict threshold model. The proposed method in this study has made the threshold expression flexible. Furthermore, with the proposed online learning algorithm-based threshold expression model, the spectrum sensing performance of cognitive radio networks has been made more sensitive to changes in communication channels.
In this study, a new threshold expression model based on online learning algorithm is presented to increase spectrum sensing accuracy in cognitive radio networks. Detection, false detection, and false alarm probabilities have been comprehensively analyzed statistically, and the optimum decision threshold expression has been redefined to minimize the possibility of decision error. Numerical results obtained from simulations on AWGN and different fading channels (Rayleigh, Nakagami-m, Rician, and Weibull) are presented to show the performance of the proposed algorithm and compare it with the dynamic decision threshold determination method. The proposed sensing scheme has significantly improved the detection performance of the energy detection- and matched filter-based spectrum sensing under low SNR conditions.
In future studies, we aim to apply and verify the performance of the proposed algorithm on different spectrum sensing methods. Also, we will focus on the optimization of some expressions used in the algorithm to reduce mathematical complexity and improve detection time.
ADC:
ANN:
BPF:
Band pass filter
CFAR:
Constant false alarm rate
Cognitive radio
Collaborative spectrum sensing
FCC:
KNN:
K-nearest neighbor
Machine learning algorithms
SNR:
SVM:
Support vector machine
True negative
True positive
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This study is supported by Erciyes University Scientific Research Projects Coordination Unit (Project Number: FDK-2016-6908)
This study is supported by Erciyes University Scientific Research Projects Coordination Unit (Project Number: FDK-2016-6908).
Department of Divriği Nuri Demirağ Vocational High School, Sivas Cumhuriyet University, 58300, Sivas, Turkey
Kenan kockaya
Department of Electrical and Electronics Engineering, Erciyes University, 38039, Kayseri, Turkey
Ibrahim Develi
K.K. has modeled and executed the research and performed system simulations. I.D. has supervised the research and enhanced the quality of the research by their valuable comments and suggestions in data analysis and discussion. K.K. and I.D. revised the equations and contributed to the writing of the manuscript. Both authors read and approved the final manuscript.
Correspondence to Kenan kockaya.
kockaya, K., Develi, I. Spectrum sensing in cognitive radio networks: threshold optimization and analysis. J Wireless Com Network 2020, 255 (2020). https://doi.org/10.1186/s13638-020-01870-7
Spectrum sensing
Machine learning algorithm
Online learning algorithm | CommonCrawl |
The perimeter of a rectangle is 56 meters. The ratio of its length to its width is 4:3. What is the length in meters of a diagonal of the rectangle?
Suppose that the rectangle's length is $4l$, then it's width is $3l$. Then its perimeter is $14l = 56$, meaning that $l = 4$. Finally, the rectangle's diagonal is $\sqrt{(4l)^2 + (3l)^2} = 5l = \boxed{20}$. | Math Dataset |
\begin{definition}[Definition:Amicable Pair]
Let $m \in \Z_{>0}$ and $n \in \Z_{>0}$ be (strictly) positive integers.
\end{definition} | ProofWiki |
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On the well-posedness of Maxwell-Chern-Simons-Higgs system in the Lorenz gauge
May 2014, 34(5): 2405-2450. doi: 10.3934/dcds.2014.34.2405
Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks
Linghai Zhang 1,
Department of Mathematics, Lehigh University, 14 East Packer Avenue, Bethlehem, Pennsylvania 18015
Received January 2013 Revised August 2013 Published October 2013
Consider the following nonlinear scalar integral differential equation arising from delayed synaptically coupled neuronal networks \begin{eqnarray*} \frac{\partial u}{\partial t}+f(u) &=&\alpha\int^{\infty}_0\xi(c)\left[\int_{\mathbb R}K(x-y)H\left(u\left(y,t-\frac1c|x-y|\right)-\theta\right){\rm d}y\right]{\rm d}c\\ &+&\beta\int^{\infty}_0\eta(\tau)\left[\int_{\mathbb R}W(x-y)H(u(y,t-\tau)-\Theta){\rm d}y\right]{\rm d}\tau. \end{eqnarray*} This model equation generalizes many important nonlinear scalar integral differential equations arising from synaptically coupled neuronal networks. The kernel functions $K$ and $W$ represent synaptic couplings between neurons in synaptically coupled neuronal networks. The synaptic couplings can be very general, including not only pure excitations (modeled with nonnegative kernel functions), but also lateral inhibitions (modeled with Mexican hat kernel functions) and lateral excitations (modeled with upside down Mexican hat kernel functions). In this nonlinear scalar integral differential equation, $u=u(x,t)$ stands for the membrane potential of a neuron at position $x$ and time $t$. The integrals represent nonlocal spatio-temporal interactions between neurons.
We have accomplished the existence and stability of three traveling wave fronts $u(x,t)=U_k(x+\mu_kt)$ of the nonlinear scalar integral differential equation in an earlier work [42], where $\mu_k$ denotes the wave speed and $z=x+\mu_kt$ denotes the moving coordinate, $k=1,2,3$. In this paper, we will investigate how the neurobiological mechanisms represented by the synaptic couplings $(K,W)$, by the probability density functions $(\xi,\eta)$, by the synaptic rate constants $(\alpha,\beta)$ and by the firing thresholds $(\theta,\Theta)$ influence the wave speeds $\mu_k$ of the traveling wave fronts. We will define several speed index functions and use rigorous mathematical analysis to investigate the influence of the neurobiological mechanisms on the wave speeds. In particular, we will compare wave speeds of the traveling wave fronts of the nonlinear scalar integral differential equation with different synaptic couplings and with different probability density functions; we will accomplish new asymptotic behaviors of the wave speeds; we will compare wave speeds of traveling wave fronts of many reduced forms of nonlinear scalar integral differential equations of the above model equation; we will establish new estimates of the wave speeds. All these will greatly improve results obtained in previous work [38], [40] and [41].
Keywords: speed index functions, Delayed synaptically coupled neuronal networks, nonlinear scalar integral differential equations, traveling wave fronts, wave speeds..
Mathematics Subject Classification: Primary: 92C20; Secondary: 35C07, 46N2.
Citation: Linghai Zhang. Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2405-2450. doi: 10.3934/dcds.2014.34.2405
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Linghai Zhang | CommonCrawl |
\begin{definition}[Definition:Vector Potential]
Let $R$ be a region of ordinary space.
Let $\mathbf V$ be a vector field over $R$.
Let $\mathbf V$ be both rotational and solenoidal.
Let $\mathbf A$ be a vector field such that $\mathbf V = \curl \mathbf A$.
Then $\mathbf A$ is known as the '''vector potential''' of $\mathbf V$.
\end{definition} | ProofWiki |
Coronal Loops: Observations and Modeling of Confined Plasma
Fabio Reale1
Living Reviews in Solar Physics volume 11, Article number: 4 (2014) Cite this article
Latest version View article history
This article is a revised version of 10.12942/lrsp-2010-5.
Coronal loops are the building blocks of the X-ray bright solar corona. They owe their brightness to the dense confined plasma, and this review focuses on loops mostly as structures confining plasma. After a brief historical overview, the review is divided into two separate but not independent parts: the first illustrates the observational framework, the second reviews the theoretical knowledge. Quiescent loops and their confined plasma are considered and, therefore, topics such as loop oscillations and flaring loops (except for non-solar ones, which provide information on stellar loops) are not specifically addressed here. The observational section discusses the classification, populations, and the morphology of coronal loops, its relationship with the magnetic field, and the loop stranded structure. The section continues with the thermal properties and diagnostics of the loop plasma, according to the classification into hot, warm, and cool loops. Then, temporal analyses of loops and the observations of plasma dynamics, hot and cool flows, and waves are illustrated. In the modeling section, some basics of loop physics are provided, supplying fundamental scaling laws and timescales, a useful tool for consultation. The concept of loop modeling is introduced and models are divided into those treating loops as monolithic and static, and those resolving loops into thin and dynamic strands. More specific discussions address modeling the loop fine structure and the plasma flowing along the loops. Special attention is devoted to the question of loop heating, with separate discussion of wave (AC) and impulsive (DC) heating. Large-scale models including atmosphere boxes and the magnetic field are also discussed. Finally, a brief discussion about stellar coronal loops is followed by highlights and open questions.
The corona is the outer part of the solar atmosphere. Its name derives from the fact that, since it is extremely tenuous with respect to the lower atmosphere, it is visible in the optical band only during the solar eclipses as a faint crown (corona in Latin) around the black moon disk. When inspected through spectroscopy the corona reveals unexpected emission lines, which were first identified as due to a new element (coronium), but which were later ascertained to be due to high excitation states of iron (Grotrian, 1939; Edlén, 1943). It became then clear that the corona is made of very high temperature gas, hotter than 1 MK. Almost all the gas is fully ionized there and thus interacts effectively with the ambient magnetic field. It is for this reason that the corona appears so inhomogeneous when observed in the X-ray band, in which plasma at million degrees emits most of its radiation. In particular, the plasma is confined inside magnetic flux tubes that are anchored on both sides to the underlying photosphere. When the confined plasma is heated more than the surroundings, its pressure and density increase. Since the tenuous plasma is optically thin, the intensity of its radiation is proportional to the square of the density, and the tube becomes much brighter than the surrounding ones and looks like a bright closed arch: a coronal loop.
When observed in the X-ray band, the bright corona appears to be made entirely by coronal loops that can, therefore, be considered as the building blocks of X-ray bright corona. This review specifically addresses coronal loops as bright structures confining plasma. It first provides an observational framework that is the basis for the second part of the review dealing with modeling and interpretation.
The observational section (3) discusses loop classification and populations, and then describes the morphology of coronal loops, its relationship with the magnetic field, regarding the shape and cross-section, and the concept of loops as consisting of bundles of strands, whose thickness may go down to sub-arcsecond scale. The following part of this section is devoted to the characteristics of the loop plasma and of its thermal structure. Diagnostics of the emission measure and of its temperature distribution retrieved from filter ratios and spectroscopy are introduced. The thermal properties of the loops are discussed according to a broad classification into hot, warm, and cool loops. Hot loops are best observed in the soft X-rays and in active regions, and attention is devoted to the possible presence of minor very hot components out of flares. Warm loops are those better observed in several EUV bands with lines emitted around 1 MK and often found to be more isothermal and dense than expected, probably because they are out of equilibrium. The emission measure distribution of loops in the whole coronal temperature range is reviewed paying attention to its shape and broadness, that may indicate or not the coexistence of many heating-cooling cycles. Then, temporal analyses of loop light curves focus on searching for variability in different bands that may indicate a highly variable or more steady heating, and the characteristic timescales. The observations show more and more evidence for significant plasma dynamics and flows, from subsonic to supersonic. Widespread patterns of redshifts and blueshifts are found in different temperature regimes and their spatial distribution is also discussed. Evidence for upflows from the chromosphere and of possible coronal counterparts is addressed. Observations reporting on the detection of waves propagating along loops are also mentioned.
In the modeling section (4) some basics of loop physics are provided, supplying some fundamental scaling laws and timescales, a useful tool for consultation. The concept of loop modeling is introduced and models are distinguished between those treating loops as monolithic and static, and those resolving loops into thin and dynamic strands. Then, more specific discussions address how modeling the loop fine structure is able to explain observed evidence for deviations from equilibrium and different filling factors in different bands, and can help investigating the concept of randomly-distributed heat pulses. Models also address plasma flowing along the loops, both as siphon flows and as motions driven by dynamic heating, i.e., hot upflows from the chromosphere first and downflows from draining afterwards. Special attention is devoted to the question of loop heating, which is strictly connected to the general problem of coronal heating. The conversion of magnetic energy into heat and the problem of the difficult diagnostics of the heating are first discussed in general terms. Impulsive (DC) and wave (AC) heating are separately discussed. DC models have extensively investigated the heating by nanoflares searching for possible signatures and properties such as their frequency and location. AC heating models focus on the way to dissipate waves and to match the observational scenario. Some discussion is devoted to MHD models that describe the solar atmosphere from the chromosphere to the corona on a larger area, and including the magnetic field and the radiative transfer, and to models that describe the magnetic field dissipation through turbulent cascades to very small scales.
There have been several earlier books (Bray et al., 1991; Golub and Pasachoff, 1997, 2001; Aschwanden, 2004) and reviews (Vaiana and Rosner, 1978; Peres and Vaiana, 1990; Golub, 1996; Aschwanden et al., 2001; Reale, 2005), in particular on coronal heating (Zirker, 1993; Cargill, 1995; Klimchuk, 2006; De Moortel and Nakariakov, 2012; Parnell and De Moortel, 2012), that have in general a larger or different scope but include information about coronal loops. Interested readers are urged to survey these other reviews in order to complement and fill in any gaps in topical coverage of the present paper.
Historical Keynotes
First evidence of magnetic confinement came from rocket missions in the 1960s. In particular, in 1965, arcmin angular resolution was achieved with grazing incidence optics (Giacconi et al., 1965). The data analysis led to the first density and temperature diagnostics with wide band filters, to derive high pressure in compact regions with intense bipolar magnetic fields and to propose the magnetic confinement (Reidy et al., 1968). The first coronal loop structures were identified properly after a rocket launch in 1968, which provided for the first time an image of an X-ray flare (Vaiana et al., 1968), with a resolution of a few arcsec.
In the course of collecting the results of all rocket missions of the American Science and Engineering (AS&E) program, Vaiana et al. (1973) proposed a classification of the morphology of the X-ray corona as fundamentally consisting of arch-like structures connecting regions of opposite magnetic polarity in the photosphere. The classification was based on the loop size, and on the physical conditions of the confined plasma, on the underlying photospheric regions. They distinguished active regions, coronal holes, active regions interconnection, filament cavities, bright points, and large-scale structures (Vaiana and Rosner, 1978; Peres and Vaiana, 1990).
The magnetic structuring of the solar corona is evident. However, the magnetic field lines can be traced only indirectly because direct measurements are feasible generally only low in the photosphere through the Zeeman effect on spectral lines. It is anyhow possible to extrapolate the magnetic field in a volume. This was done to derive the magnetic field structure of a relatively stable active region by Poletto et al. (1975) using the Schmidt (1964) method, under the assumption of negligible currents in the corona. This was also useful to derive magnetic field intensities sufficient for hot plasma confinement. Later on, even more reliable magnetic field topologies were derived assuming force-free fields (e.g., Sakurai, 1981), i.e., with currents everywhere parallel to the magnetic field as it is expected in coronal loops. However, the agreement of force-free magnetic field extrapolation with the details of the observed coronal EUV topology is often far from satisfactory (e.g., Wiegelmann et al., 2006).
The rocket missions lacked good time coverage and the information about the evolution of coronal loops was only limited, mostly available from the Orbiting Solar Observatory-IV (OSO-IV) mission (Krieger et al., 1972). This satellite had an angular resolution in the order of the arcmin and could not resolve individual loops. In 1973, the X-ray telescope S-054 on-board Skylab monitored the evolution of coronal loops for several months, taking 32 000 X-ray photographs with a maximum resolution of 2 arcsec and an extended dynamic range. It was possible to study the whole evolution of an active region, from the emergence as compact loops filled with dense plasma to its late spreading, a few solar rotations later, as progressively longer and longer loops filled with less and less dense plasma (Golub et al., 1982). It was confirmed that the whole X-ray bright corona consists of magnetic loops, whose lifetime is typically much longer than the characteristic cooling times (Rosner et al., 1978). This applies also to coronal holes where the magnetic field opens radially to the interplanetary space and the plasma streams outwards with practically no X-ray emission.
In the same mission coronal loops were also detected in the UV band at temperatures below 1 MK, by Extreme UltraViolet (EUV) telescopes S-055 (Reeves et al., 1977) and S-082 (Tousey et al., 1977; Bartoe et al., 1977). These loops are invisible in the X-ray band and many of them depart from sunspots, appear coaxial and are progressively thinner for progressively lower temperature ions (Foukal, 1975, 1976). The apparent scale height of the emission is larger than that expected from a static model, but the loops appear to be steady for long times. Foukal (1976) proposed a few explanations including siphon flows and thermal instability of the plasma at the loop apex. New observations of such cool loops were performed several years later with the Solar and Heliospheric Observatory (SoHO) mission and provided new details and confirmations (Section 3.5).
A different target was addressed by the Solar Maximum Mission (SMM, 1980–1989, Bohlin et al., 1980; Acton et al., 1980), which included high-resolution spectrometers in several X-ray lines, i.e., the Bent Crystal Spectrometer (BCS) and the Flat Crystal Spectrometer (FCS), mostly devoted to obtain time-resolved spectroscopy of coronal flares (e.g., MacNeice et al., 1985). Similarly, the Hinotori mission (1981–1991, Tanaka, 1983) was dedicated mainly to solar flare observations in the X-ray band. This was also the scope of the later Yohkoh mission, (1991–2001, Ogawara et al., 1991) by means of high resolution X-ray spectroscopy, adding the monitoring and imaging of the hot and flaring corona. Hara et al. (1992) found first indications of plasma at 5–6 MK in active regions with the Soft X-ray Telescope (SXT, Tsuneta et al., 1991).
Normal-incidence optics were developed in the late 1980s. An early experiment was the Normal Incidence X-ray Telescope (NIXT, Golub and Herant, 1989), which provided a few high resolution coronal images in the EUV band.
Later space missions dedicated to study the corona have been the Solar and Heliospheric Observatory (SoHO, Domingo et al., 1995), launched in 1995 and still operative, and the Transition Region and Coronal Explorer (TRACE, Handy et al., 1999), launched in 1998 and replaced in 2010 by the Solar Dynamic Observatory (SDO) instruments. Both SoHO and TRACE were tailored to observe the quiet corona (below 2 MK). SoHO images the whole corona (Extreme ultraviolet Imaging Telescope, EIT, Delaboudinière et al., 1995) and performs wide band spectroscopy (Solar Ultraviolet Measurements of Emitted Radiation, SUMER, Wilhelm et al., 1995) and (Coronal Diagnostic Spectrometer, CDS, Harrison et al., 1995) in the EUV band; TRACE imaged the EUV corona with high spatial (0.5 arcsec) and temporal (30 s) resolution. Both SoHO/EIT and TRACE are based on normal-incidence optics and contain three different EUV filters that provide limited thermal diagnostics.
Thanks to their capabilities, both missions allowed to address finer diagnostics, in particular to investigate the fine transverse structuring of coronal loops, both in its geometric and thermal components, and the plasma dynamics and the heating mechanisms at a higher level of detail. SoHO and TRACE have been complementary in many respects and several studies attempted to couple the information from them.
Among other relevant missions, we mention the CORONAS series (Ignatiev et al., 1998; Oraevsky and Sobelman, 2002), with instruments like SPectroheliographIc X-Ray Imaging Telescope (SPIRIT, Zhitnik et al., 2003), REntgenovsky Spektrometr s Izognutymi Kristalami (ReSIK, Sylwester et al., 1998), and Solar Photometer in X-rays (SPHINX, Sylwester et al., 2008; Gburek et al., 2013), which have contributed to the investigation of coronal loops.
In late 2006, two other major solar missions started, namely Hinode (Kosugi et al., 2007) and the Solar TErrestrial Relations Observatory (STEREO, e.g., Kaiser et al., 2008). On-board Hinode, two instruments address particularly the study of coronal loops: the X-Ray Telescope (XRT, Golub et al., 2007) and the Extreme-ultraviolet Imaging Spectrometer (EIS, Culhane et al., 2007). Both these instruments offer considerable improvements on previous missions. The XRT has a spatial resolution of about 1 arcsec, a very low scattering and the possibility to switch among nine filters and combinations of them. EIS combines well spectral (∼ 2 mA), spatial (2″), and temporal (∼ 10 s) resolution to obtain accurate diagnostics of plasma dynamics and density. One big achievement of the STEREO mission is that, since it consists of two separate spacecrafts getting farther and farther from each other, it allows — through, for instance, its Sun-Earth Connection Coronal and Heliospheric Investigation (SECCHI) package — a first 3D reconstruction of coronal loops (Aschwanden et al., 2009; Kramar et al., 2009).
In 2010, the Solar Dynamics Observatory (SDO, Pesnell et al., 2012) mission has been launched with three instruments on-board: Atmospheric Imaging Assembly (AIA, Lemen et al., 2012; Boerner et al., 2012), EUV Variability Experiment (EVE, Woods et al., 2012), and Helioseismic and Magnetic Imager (HMI, Scherrer et al., 2012). SDO observations lead to big improvements in the study of coronal-loop physics, basically because it monitors the full Sun continuously with high temporal and spatial resolution, especially with the AIA EUV normal-incidence telescope at 9 different UV and EUV channels. It is worthwhile mentioning also the sounding rocket mission High-resolution Coronal Imager (Hi-C, Cirtain et al., 2013), which achieved an unprecedented spatial resolution (0.2″) in the EUV band (195 Å).
The Observational Framework
Although coronal loops are often well defined and studied in the EUV band, detected by many space mission spectrometers like those on board SoHO and Hinode, and by high resolution imagers such as TRACE and SDO/AIA, the bulk of coronal loops is visible in the X-ray band (Figure 1). Also, the peak of the coronal emission measure of active regions — where the loops are brightest — is above 2 MK, which is best observed in X-rays (e.g., Peres et al., 2000; Reale et al., 2009a; Warren et al., 2011).
Images of the same active region, taken in the EUV band with TRACE (top) and in the X-ray band with Hinode/XRT (bottom), on 14 November 2006. The X-ray image shows more clearly that the active region is densely populated with coronal loops.
Coronal loops are characterized by an arch-like shape that recalls typical magnetic field topology. This shape is replicated over a wide range of dimensions. Referring, for the moment, to the soft X-ray band, the main properties of coronal loops are listed in Table 1. The length of coronal loops spans more than two orders of magnitude. As already mentioned, the loops owe their high luminosity and variety to their nature of magnetic flux tubes where the plasma is confined and isolated from the surroundings. Magnetized fully-ionized plasma conducts thermal energy mostly along the magnetic field lines. Due to the high thermal insulation, coronal loops can have different temperatures, from ∼ 105 K up to a few ∼ 107 K (flaring loops). A density of the confined plasma below 107–108 cm−3 can be difficult to detect, while the density can grow up to 1012 cm−3 in flaring loops. The corresponding plasma pressure in non-flaring loops can typically vary between 10−3 and 10 dyne cm−2, corresponding to confining magnetic fields B ∼ 8πp0.5 of the order of 0.1–10 G in the corona. One characterizing feature of coronal loops is that typically their cross-section is constant along their length above the transition region, at variance from the topology of potential magnetic fields. There is evidence that the cross-section varies across the transition region, as documented in Gabriel (1976). A simple geometric description is reported in Chae et al. (1998c):
$$A(T)/A({T_h}) = {[1 + ({\Lambda ^2} - 1){(T/{T_h})^\nu }]^{1/2}}/\Lambda,$$
where A is the cross-section area, T is the temperature, Th = 106 K, Λ = 30, and ν = 3.6.
Table 1: Typical X-ray coronal loop parameters
Myriads of loops populate the solar corona and constitute statistical ensembles. Attempts to define and classify coronal loops were never easy, and no finally established result exists to-date. Early attempts were based on morphological criteria, i.e., bright points, active-region loops, and large-scale structures (Vaiana et al., 1973, Figure 2), largely observed with instruments in the X-ray band. In addition to such classification, more recently, the observation of loops in different spectral bands and the suspicion that the difference lies not only in the band, but also in intrinsic properties, have stimulated another classification based on the temperature regime, i.e., cool, warm, hot loops (Table 2). Cool loops are generally detected in UV lines at temperatures between 105 and 106 K. They were first addressed by Foukal (1976) and later explored more with SoHO observations (Brekke et al., 1997). Warm loops are well observed by EUV imagers such as SoHO/EIT, TRACE, and in most channels of SDO/AIA, and confine plasma at temperature around 1–1.5 MK (Lenz et al., 1999). Hot loops are those typically observed in the X-ray band, and in hot UV and EUV lines (e.g., Fe xvi) and channels (SDO/AIA 335 Å), with temperatures around or above 2 MK (Table 1). These are the coronal loops already identified, for instance, in the early rocket missions (Vaiana et al., 1973). This distinction is not only due to observation with different instruments and in different bands, but there are hints that it may be more substantial and physical, i.e., there may be two or more classes of loops that may be governed by different regimes of physical processes. For instance, the temperature along warm loops appears to be distributed uniformly and the density to be higher than that predicted by equilibrium conditions. Does this make such loops intrinsically different from hot loops, or is it just the signature that warm loops are a transient conditions of hot loops? New state-of-art methods, like differential emission measure tomography (DEMT), have proposed a new classification of coronal loops based on whether the temperature increases or decreases with height (Huang et al., 2012).
The X-ray corona contains loops with different spatial scales, e.g., bright points (BP), active region loops (AR), large-scale structures (LSS). The scale unit is labelled. Image credit: Yohkoh mission, ISAS, Japan.
Table 2: Thermal coronal loop classification
A real progress in the insight into coronal loops is expected from the study of large samples of loops or of loop populations. Systematic studies of coronal loops suffer from the problem of the sample selection and loop identification, because, for instance, loops in active regions overlap along the line sight. Attempts of systematic studies have been performed in the past on Yohkoh and TRACE data (e.g., Porter and Klimchuk, 1995; Aschwanden et al., 2000). A large number of loops were analyzed and it was possible to obtain meaningful statistics. However, it is difficult to generalize the results because of limited samples and/or selection effects, e.g., best observed loops, specific instrument. One basic problem for statistical studies of coronal loops is that it is very difficult to define an objective criterion for loop identification. In fact, loops are rarely isolated; they coexist with other loops that intersect or even overlap along the line of sight. This is especially true in active regions where most of the loops are found. In order to make a real progress along this line, we should obtain loop samples and populations selected on totally objective and unbiased criteria, which is difficult due to the problems outlined above. Some steps are coming in this direction (Aschwanden et al., 2013) and we will see results in the future.
Morphology and fine structuring
Coronal loops are magnetic structures and might therefore be mapped easily and safely by mapping reliably the coronal magnetic field. Unfortunately, it is well-known that it is very difficult to measure the magnetic field in the corona, and it can be done only in very special conditions, e.g., very strong local field (White et al., 1991). In some cases it is possible to use coronal seismology (first proposed by Uchida, 1970) to determine the average magnetic field strength in an oscillating loop, first used by Nakariakov et al. (1999) and Nakariakov and Ofman (2001), on TRACE loops, and more recently investigated in a number of studies. In some of these studies, for instance, seismological techniques are used in order to measure flare-induced loop oscillations (Ballai et al., 2011) and waves and flows in active region loops (Wang et al., 2013; Uritsky et al., 2013). The accuracy of these methods depends on the correct detection of the temporally and spatially resolved mode of oscillation, and on the details of the loop geometry.
Since we cannot determine well the coronal magnetic field, coronal loop geometry deserves specific analysis. As a good approximation, loops generally have a semicircular shape (Figure 3). The loop aspect, of course, depends on the loop orientation with respect to the line of sight: loops with the footpoints on the limb more easily appear as semicircular, as well as loops very inclined on the surface near the center of the disk. The assumption of semicircular shape can be useful to measure the loop length even in the presence of important deformations due to projection effects: the de-projected distance of the loop footpoints is the diameter of the arc. However, deviations from circularity are rather common and, in general, the detailed analysis of the loop geometry is not a trivial task. The accurate determination of the loop geometry is rather important for the implications on the magnetic field topology and reconstruction. It is less important for the structure and evolution of the confined plasma, which follow the field lines whatever shape they have and change little also with moderate changes of the gravity component along the field lines. First works on the accurate determination of the loop geometry date back to the 1960s (Saito and Billings, 1964). More specific ones take advantage of stereoscopic views allowed by huge loops during solar rotation, with the aid of magnetic field reconstruction methods. These studies find deviations from ideal circularity and symmetry, not surprising for such large structures (Berton and Sakurai, 1985). The geometry of a specific loop observed with TRACE was measured in the framework of a complete study including time-dependent hydrodynamic modeling (Reale et al., 2000a,b). In that case, the discrepancy between the length derived from the distance of the footpoints taken as loop diameter and the length measured along the loop itself allowed to assess the loop as elongated. Later, a reconstruction of loop geometry was applied to TRACE observation of medium-sized oscillating loops, to derive the properties of the oscillations. In this case, a semicircular pattern was applied (Aschwanden et al., 2002). The importance of the deviations from circularity on constraining loop oscillations was remarked later (Dymova and Ruderman, 2006).
Coronal loops have approximately a semicircular shape. Image: SDO/AIA, 171 Å filter, 24 February 2011. Credit: NASA/LMSAL/SDO.
The STEREO mission is contributing much to the analysis of loop morphology and geometry, thanks to its unique capability to observe the Sun simultaneously from different positions. The three-dimensional shape of magnetic loops in an active region was first stereoscopically reconstructed from two different vantage points based on simultaneously recorded STEREO/SECCHI images (Feng et al., 2007). Five relatively long loops were measured and found to be non-planar and more curved than field lines extrapolated from SoHO/MDI measurements, probably due to the inadequacy of the linear force-free field model used for the extrapolation. A misalignment of 20–40 deg between theoretical model and observed loops has been quantified from STEREO results (Sandman et al., 2009; DeRosa et al., 2009). Systematic triangulations and 3D reconstructions using the Extreme UltraViolet Imager (EUVI) telescopes on both STEREO spacecrafts were used to derive loop characteristics, such as the loop plane inclination angles (Aschwanden et al., 2008, 2009, 2012). Deviations from circularity within 30%, less significant from coplanarity and twisting below the threshold for kink instability, were found.
Another interesting issue regarding coronal loop geometry is the analysis of the loop cross-section, which also provides information about the structure of the coronal magnetic field. Yohkoh/SXT allowed for systematic and quantitative studies of loop morphology and showed that the cross-section of coronal loops is approximately constant along their length and do not increase significantly. More in detail, a systematic analysis of a sample of ten loops showed that the loops tend to be only slightly (∼ 30%) wider at their midpoints than at their footpoints, while for a bipolar field configuration we would expect expansions by factors. One possible explanation of this effect is the presence of significant twisting of the magnetic field lines and, therefore, the development of electric currents and strong deviations from a potential field. The effect might be seen either as a twisting of a single loop or as a "braiding" of a bundle of unresolved thin loops (see Section 3.2.2). At the same time it was found that the variation of width along each loop tends to be modest, implying that the cross section has an approximately circular shape (Klimchuk et al., 1992).
Implications of these results on the theory of coronal heating were discussed in Klimchuk (2000), but the conclusion was that none of the models alone is able to explain all observed properties. Important information about the internal structuring of coronal loops comes from the joint analysis of the photospheric and coronal magnetic field (Figure 4). TRACE loops are quite symmetric and their cross-section is constant to a good degree of approximation, at variance from the prediction by linear force-free extrapolation on SoHO/MDI data (López Fuentes et al., 2006). The magnetic field lines starting from the same footpoint can diverge to different end footpoints, and thus be very complicated with strong tangling of the magnetic flux strands driven by the photospheric convection. This is not observed at high-resolution in the quiescent corona, possibly because of braiding-induced interchange reconnection of the magnetic field (Schrijver, 2007). Other approaches address also the density values and stratification, and explain the evidence with a combination of high plasma density within the structures, which greatly increases the emissivity of the structures, and geometric effects that attenuate the apparent brightness of the feature at low altitudes (DeForest, 2007). More recent MHD modeling finds that the temperature distribution across the loop naturally leads to the appearance of constant cross-section in EUV band (Peter and Bingert, 2012). Another model shows that the apparent constant loop cross section is a result of the non-circular shape of the loops (Malanushenko and Schrijver, 2013).
Magnetic field lines extrapolated from optical magnetogram superposed on a composite SDO/AIA (211 Å, 193 Å, 171 Å) image, on 7 April 2014. Credit: NASA/LMSAL/SDO.
An alternative analysis of the magnetic field at the footpoints of hot and cool loops showed that the magnetic filling factor is lower in hot loops (0.05–0.3 out of sunspots) than in warm loops (0.2–0.6) (Katsukawa and Tsuneta, 2005). The multispectral analysis of solar EUV images has showed the possibility of separating the different solar structures from a linear combination of images to emphasise the structures (Dudok de Wit and Auchère, 2007).
The general impression is that, although moderate deviations from the ideal circular shape with constant cross-section are often observed, they probably do not affect the description of loops with a simplified geometry.
Fine structuring
It has been long claimed that coronal loops consist of bundles of thin strands, to scales below the instrumental resolution (e.g., Gómez et al., 1993). The issue of fine loop structure is of critical importance because it constrains the elementary processes that determine the loop ignition. The task to investigate this substructuring is not easy. Studies based both on models and on analysis of observations independently suggest that elementary loop components should be very fine with typical cross-sections of the strands on the order of 10–100 km (Beveridge et al., 2003; Cargill and Klimchuk, 2004; Vekstein, 2009). First limited evidence of fine structuring was the low filling factor inferred for loops observed with NIXT (Di Matteo et al., 1999, see Section 3.3.2). The high spatial resolution achieved by the TRACE normal-incidence telescope allowed to address the transverse structure of the imaged coronal loops. EUV images visibly show that coronal loops are substructured (Figure 5).
SDO/AIA (193 A channel) image of finely-structured coronal loops (14 October 2011, 22:56 UT). The image was treated with a Gaussian sharpening filter with a radius of 3 pixels. Image reproduced with permission from Brooks et al. (2012), copyright by AAS.
There were some early attempts to study the structure along the single strands in TRACE observations (Testa et al., 2002). Later, it was shown that in many cases, hot loop structures observed in active regions with the Yohkoh/SXT (T > 3 MK) are not exactly co-spatial with warm structures (T ∼ 1–2 MK) observed in the EUV bands (195 Å) of the SoHO/EIT, nor they cool down to become visible with EIT (Nagata et al., 2003; Schmieder et al., 2004). In another study, hot monolithic loops visible with the Yohkoh/SXT were instead resolved as stranded cooler structures with TRACE at later times (Winebarger and Warren, 2005), although the large time delay (1 to 3 hours) is hardly compatible with the cooling time from SXT to TRACE sensitivity. More systematic studies of TRACE images proposed that about 10% of the positions across loops can be fitted with an isothermal model, indicating coherent loop components about 2000 km wide (Aschwanden and Nightingale, 2005; Aschwanden et al., 2007).
More recent studies have found other indirect evidence for loop fine structuring. For instance, the different fuzziness measured in spectral lines forming at different coronal temperatures (Tripathi et al., 2009) has been modeled with loops made of tens of independently pulse-heated strands (Guarrasi et al., 2010; Reale et al., 2011). Other indirect evidence might come from optical observations of relatively dense and cool (∼ 7000 K) downfalling elongated blobs above the solar limb, the so-called "coronal rain" (Antolin and Rouppe van der Voort, 2012), with widths of the order of 500 km. Upflows in even narrower channels (∼ 100 km) were resolved in the optical band, correlating with brightenings in SDO/AIA observations (Ji et al., 2012).
New direct measurements of the loop fine structure are stimulated by the new observations with normal-incidence telescopes (SDO/AIA, Hi-C) and by indications that the elementary components of most loops might be small scale and close to be resolved (Brooks et al., 2012). Some quantitative constraints come from the very high spatial resolution of the Hi-C sounding rocket, with cross-sections that range between about 300 km (Brooks et al., 2013) to over 1000 km (Peter et al., 2013) for long loops, with possibility of further substructuring well below 100 km.
Evidence for fine structure can come also from studies of thermal structuring; in general, an emission measure distribution extending over a broad temperature range along the line of sight is retained to be a signature of the presence of multi-stranded structure, because of the coexistence of many thermal components. See Section 3.3 for more details.
In general, there is converging evidence for fine loop structuring to scales on the order of 100 km or below. Ultimate information could be provided by higher resolution future instruments.
Diagnostics and thermal structuring
The investigation of the thermal structure of coronal loops is very important for their exhaustive physical comprehension and to understand the underlying heating mechanisms. For instance, one of the classifications outlined above is based on the loop thermal regime, and, we remark, it is debated whether the classification indicates a real physical difference.
Diagnostics of temperature are not trivial in the corona. No direct measurements are available. Since the plasma is optically thin, we receive information integrated on all the plasma column along the line of sight. The problem is to separate the distinct contributing thermal components and reconstruct the detailed thermal structure along the line of sight. However, even the determination of global and average values deserves great attention. Exploratory measurements have shown that the ion temperature can be different from the electron temperature in the quiet corona (Landi, 2007).
Moderate diagnostic power is allowed by imaging instruments, by means of multifilter observations. Filter ratio maps (Figure 6) provide information about the spatial distribution of temperature and emission measure (e.g., Vaiana et al., 1973). The emission of an optically thin isothermal plasma as measured in a j-th filter passband is:
$${I_j} = EM\,{G_j}(T),$$
where T is the temperature and EM is the emission measure, defined as
$$EM = \int_V {{n^2}} dV,$$
where n is the particle density, and V the plasma volume. The ratio Rij of the emission in two different filters i, j is then independent of the density, and only a function of the temperature:
$${R_{ij}} = {{{I_i}} \over {{I_j}}} = {{{G_i}(T)} \over {{G_j}(T)}}.$$
The inversion of this relationship provides a value of temperature, based on the isothermal assumption.
Temperature map of an active region obtained from the ratio of two images in different broadband filters with Hinode/XRT (12 November 2006, 12 UT).
The limitations of this method are substantial. In particular, one filter ratio value provides one temperature value for each pixel; this is a reliable measurement, within experimental errors, as long as the assumption of isothermal plasma approximately holds for the plasma column in the pixel along the line of sight. If the plasma is considerably multithermal, the temperature value is an average weighted for the instrumental response. Since the response is a highly nonlinear function of the emitting plasma temperature, it is not trivial to interpret the related maps correctly. In addition, it is fundamental to know the instrument response with high precision, in order to avoid systematic errors, which propagate dangerously when filter ratios are evaluated. In this respect, broadband filters provide robust thermal diagnostics, because they are weakly dependent on the details of the atomic physics models, e.g., on the presence of unknown or not well-known spectral lines, on the choice of element abundances. Narrowband filters can show non-unique dependencies of filter ratio values on temperature (e.g., Patsourakos and Klimchuk, 2007), due to the presence of several important spectral lines in the bands, but a more general problem can be the bias to detect narrow ranges of temperatures forced by the specific instrument characteristics (Weber et al., 2005). This problem can be important especially when the distribution of the emission measure along the line of sight is not simple and highly nonlinear (e.g., Reale et al., 2009a). New methods for thermal diagnostics with narrow band instruments have been proposed (Dudok de Wit et al., 2013).
The problem of diagnostics of loop plasma from filter ratios, and, more in general, the whole analysis of loop observations, are made even more difficult by the invariable presence of other structures intersecting along the line of sight. A uniform diffuse background emission also affects the temperature diagnostics, by adding systematic offsets that alter the filter ratio values. The task of subtracting this "background emission" from the measured emission is non-trivial and can seriously bias the results of the whole analysis. This problem emerged dramatically when the analysis of the same large loop structure observed with Yohkoh/SXT on the solar limb led to three different results depending mostly on the different ways to treat the background (Priest et al., 2000; Aschwanden et al., 2001; Reale, 2002a). The amount of background depends on the instrument characteristics, such as the passband and the point response function: it is most of the signal in TRACE UV filterbands, for instance, and its subtraction becomes a very delicate issue (e.g., Del Zanna and Mason, 2003; Reale and Ciaravella, 2006; Aschwanden et al., 2008; Terzo and Reale, 2010). The problem can be mitigated if one analyzes loops as far as possible isolated from other loops, but this is not easy, for instance, in active regions. If this is not the case, broadband filters may also include contamination from many structures at relatively different temperature and make the analysis of single loops harder. The problem of background subtraction in loop analysis has been addressed by several authors, who apply different subtraction ranging from simple offset, to emission in nearby pixels or subregions, to values interpolated between the loop sides, to whole images at times when the loop is no longer (or not yet) visible (Testa et al., 2002; Del Zanna and Mason, 2003; Schmelz et al., 2003; Aschwanden and Nightingale, 2005; Reale and Ciaravella, 2006; Aschwanden et al., 2008; Terzo and Reale, 2010).
More accurate diagnostics, although with less time and space resolution, are in principle provided by spectrometers and observations in temperature-sensitive spectral lines, which are being constantly improved to provide better and better spatial information. Early results from UV spectroscopy already recognized the link between transition region and coronal loops, for instance, from Skylab mission (Feldman et al., 1979; Mariska et al., 1980). There is a considerable effort to develop and update continuously the spectral codes and databases, in particular the CHIANTI spectral line database (Dere et al., 1997; Landi et al., 1999; Dere et al., 2001; Landi et al., 2006; Dere et al., 2009; Landi et al., 2012a; Del Zanna, 2012; Landi et al., 2013). Together with background subtraction, one major difficulty met by spectroscopic analysis is that, in the UV band, the density of lines is so high that they are often blended and, therefore, it is hard to separate the contribution of the single lines, especially the weak ones. Fine diagnostics, such as Doppler shifts and line broadening, can become very tricky in these conditions and results are subject to continuous revisitation and warnings from the specialized community. The problem of background subtraction is serious also for spectral data, because their lower spatial and temporal resolution determines the presence of more structures and, therefore, more thermal components, along the line of sight in the same spatial element.
Care should be paid also when assembling information from many spectral lines into a reconstruction of the global thermal structure along the line of sight. Methods are well-established (e.g., Gabriel and Jordan, 1975) and several approaches are available. The so-called method of the emission measure loci (Pottasch, 1963; Jordan et al., 1987; Landi et al., 2002) is able to tell whether plasma is isothermal of multithermal along the line of sight (Figure 7), but less able to add details. Detailed emission measure distributions can be obtained from differential emission measure (DEM) reconstruction methods (e.g., Brosius et al., 1996; Kashyap and Drake, 1998), but this is an ill-posed mathematical problem, and, therefore, results are not unique and are subject to systematic and unknown errors.
EM-loci plots for two different loop systems, one showing a multi-thermal structure (top, from Schmelz et al., 2005, reproduced with permission, copyright by AAS), the other an almost isothermal one (bottom, from Di Giorgio et al., 2003, reproduced with permission, copyright by ESO). The EM is per unit area in the top panel.
There is no wide convergence on the reconstruction of the temperature distribution of the emission measure along the line of sight. So several works have been devoted to develop new methods (e.g., Hannah and Kontar, 2012) and to investigate the limits and capabilities of the methods themselves. One important question is how far can we push the temperature resolution of the methods. We expect better resolution switching from narrow-band instruments with few channels (Weber et al., 2005) to spectrometers (Landi and Klimchuk, 2010; Landi et al., 2012b), but it is difficult to achieve a temperature resolution better than ΔlogT ≈ 0.05 (Landi et al., 2012b). Reconstructions from narrow-band data may suffer from biases that affect their reliability (Weber et al., 2005) and produce clustering (Guennou et al., 2012a, b). Not only the width and sensitivity of the spectral band is important, but also its position in the temperature range. It has been remarked that, with the available instruments, it is difficult to constrain emission from plasma at temperature above about 6 MK and emission measures less than ∼ 1027 cm−5 (Winebarger et al., 2012). Under some conditions, these limitations might be overcome, as described in Section 3.3.1. It is also useful to assess unknown uncertainties in the atomic data (Landi et al., 2012b). All-encompassing 3D MHD simulations allow to build a complete cycle from true values to observables back to reconstructed values and comparison with true values (Testa et al., 2012). It is important to test the combination of all possible uncertainties, the overall capability of the methods, indicating that a highly non-uniform density distribution along the line of sight can mislead the methods.
Forward modeling and simulations can be ways to escape from these problems, but they require non-trivial computational efforts and programming, and it is not always possible to provide accurate confidence levels. All these approaches are constantly improved and, probably, the best way to proceed is to combine different approaches and multiband observations and to finally obtain a global consistency.
Interesting issues come from global thermal analysis of the confined solar corona. The analysis of full-disk data from spatially-unresolved SphinX spectral data at the solar minimum of 2009 shows temperatures of 1.7–1.9 MK and emission measures between 4 × 1047 cm−3 and 1.1 × 1048 cm−3. Most of the emission comes from the large-scale corona rather than localized and bright structures, e.g., bright points (Sylwester et al., 2012).
In addition to the problems intrinsic to diagnostic techniques, we have to consider that loops appear to have different properties in different bands, as mentioned in Section 3.1.1. It is still debated whether such differences originate from an observational bias due to the instruments or from intrinsic physical differences, or both. In view of this uncertainty, in the following we will make a distinction between hot and warm loops, which will generally correspond to loops observed in the X-ray (and hot UV lines, e.g., Yohkoh/SXT, Hinode/XRT, SoHO/CDS Fe xvi line) and in the UV band (e.g., SoHO/EIT, TRACE), respectively. Cool loops are also observed in the UV band. The boundary between hot and warm loops is, of course, not sharp, and it is not even clear whether they are aspects of the same basic structure, or they really are physically different and are heated differently (see also Section 3.3.3). We will now devote our attention to the comparison between hot and warm loops.
Hot loops
After the pioneering analyses driven by the Skylab X-ray instruments (Section 2), Yohkoh/SXT allowed to conduct large-scale studies on the thermal and structure diagnostics of hot loops, mostly located in active region, and the comparison with other instruments, for instance on-board SoHO, allowed to obtain important cross-checks and additional information. Filter ratio maps of flaring loops were shown early after the mission launch (Tsuneta et al., 1992).
Systematic measurements of temperature, pressure, and length of tens of both quiescent and active region coronal loops were conducted on Yohkoh observations in the 1990s (Porter and Klimchuk, 1995) using the filter ratio method. Steady and isolated loops were selected with a length in a decade between 5 × 109 < 2L < 5 × 1010 cm. Their average temperature was high, ranging in a decade (2 < T < 30 MK), with a mean of about 6 MK and with large uncertainties in the hot tail of the distribution. Pressure was estimated to span over two decades (0.1 < p < 20 dyne cm−2). The temperature and length were uncorrelated, while the pressure was found overall to vary inversely with the length (as overall expected for a thermally homogeneous sample from loop scaling laws, see Section 4.1.1), allowing to constrain the dependence of the magnetic field intensity on the loop length (Klimchuk and Porter, 1995), and the data uncertainties (Klimchuk and Gary, 1995). Another systematic analysis on another sample of Yohkoh loops in active regions (Kano and Tsuneta, 1995) confirmed some of the correlations in Porter and Klimchuk (1995), but also found a correlation between the loop length and the temperature, and deviations from RTV scaling laws (Section 4.1.1). The correlations might depend on the loop sample, as a single scaling law links three parameters. Very hot Yohkoh/SXT loops were found to have short lifetimes (less than few hours), and often to exhibit cusps (Yoshida and Tsuneta, 1996).
Density diagnostics through density-sensitive line ratio led to measure directly density values in active regions (e.g., Doschek et al., 2007). The hot core of the active region is densest, with values as high as 1010.5 cm−3. Density in active regions has been measured also using Fe lines, with values in the range 8.5 ≤ log(A/e/cm−3) ≤ 11.0 (Young et al., 2009), or in a smaller range (from 108.5 to 109.5 cm−3) (Watanabe et al., 2009). Diagnostics of high density structures are also possible from the analysis of absorbed structures observed with narrow-band imagers (e.g., Landi and Reale, 2013).
A big effort has been devoted to the possible detection of hot plasma outside of evident flares. This would be a conclusive evidence of the presence of impulsive heating mechanisms in coronal loops (e.g., Klimchuk, 2006, see Section 4.4). Hinode instruments appear to be able to provide new interesting contributions to this topic. The analysis of spectroscopic observations of hot lines in solar active regions from Hinode/EIS allows to construct emission measure distributions in the 1–5 MK temperature range, and shows that the distributions are flat or slowly increasing up to approximately 3 MK and then fall off rapidly at higher temperatures (Patsourakos and Klimchuk, 2009). Emission from very hot lines has been early found in other Hinode/EIS observations, and in particular from the Ca xvii at 192.858 Å, formed near a temperature of 6 × 106 K, in active regions (Ko et al., 2009).
Thanks to its multifilter observations, also Hinode/XRT is providing useful information about the thermal structure of the bright X-ray corona. Temperature maps derived with combined filter ratios show fine structuring to the limit of the instrument resolution and evidence of multithermal components (Reale et al., 2007), as complemented by TRACE images. Observations including flare filters show evidence of a hot component in active regions outside of flares (Schmelz et al., 2009), and data in the medium thickness filters appear to constrain better this component of hot plasma as widespread, although minor, and peaking around log T ∼ 6.8–6.9, with a tail above 10 MK (Reale et al., 2009b). Further support comes from RHESSI data (Reale et al., 2009a; McTiernan, 2009).
Further evidence for minor components of hot plasma in non-flaring active regions have been found from various other instruments. Analysis in the waveband 3.3–6.1 A and 280–330 A with the RESIK and SPIRIT instruments, respectively, confirm the presence of a 0.1% ∼ 10 MK component at various activity levels (Sylwester et al., 2010; Shestov et al., 2010). Low-resolution SphinX spectra integrated on 17 days in the 2–10 Å band still show a small but highly-significant component at about 7 MK from active regions outside of microflares (Miceli et al., 2012). The separation of the hot from the cool components in the SDO/AIA 94 Å channel indicates finely-structured Fe xviii line emission in the core of bright active regions (Reale et al., 2011). This filamented emission at high temperature has been previously predicted with a model of multi-stranded pulse-heated loops (Guarrasi et al., 2010). The emission from hot emission lines (Ca xvii and Fe xviii) has been confirmed from simultaneous observations with SDO/AIA and with the Hinode/EIS spectrometer (Testa and Reale, 2012; Teriaca et al., 2012). However, while it has been proposed that AIA imaging observations of the solar corona can be used to track hot plasma (6–8 MK), it has been questioned that such emission is really at the temperature of the line sensitivity peak (Teriaca et al., 2012). Other analysis of a limb active region with EUV spectral data from Hinode/EIS does not find evidence for plasma at temperature log T > 7 (O'Dwyer et al., 2011) and puts an upper limit on the same track as remarked by Winebarger et al. (2012). So a final conclusion on this topic is still to be reached. There is some evidence that the amount of high-temperature plasma might correlate with the intensity of the active region magnetic fields because of increasing frequency of energy release (Warren et al., 2012).
Comparison of hot and warm loops
Before the SoHO/EIT and TRACE observations, warm loops had been imaged in a similar spectral band and with similar optics by the rocket NIXT mission (see Section 2). Studies of NIXT loops including the comparison with hydrostatic loop models (Section 4.1.1) pointed out that bright spots also visible in Hα band were the footpoints of hot high-pressure loops (Peres et al., 1994). This result was confirmed by the comparison of the temperature structure obtained from Yohkoh with NIXT data (Yoshida et al., 1995, see also Section 3.3.3).
Another comparison of loops imaged with NIXT and Yohkoh/SXT showed that the compact loop structures (length ∼ 109 cm) have a good general morphological correspondence, while larger scale NIXT loops (∼ 1010 cm) have no obvious SXT counterpart (Di Matteo et al., 1999). Comparison with static loop models (see Section 4.1.1) allowed to derive estimates of the loop filling factors, important for the loop fine structure (Section 3.2.2). In the NIXT band, the filling factor of short loops was found to be very low (10−3–10−2), but of the order of unity in the SXT band and for the largest structure. Simultaneous SoHO and Yohkoh observations of a small solar active region suggested a volume filling factor decreasing with increasing density and possible differences between emitting material in active regions and the quiet Sun (Griffiths et al., 2000). Other measurements of the filling factor come from density sensitive lines and images in the EUV band. For bright points the plasma-filling factor has been found to vary from 3 × 10−3 to 0.3 with a median value of 0.04, which may indicate considerable subresolution structure, or the presence of a single completely-filled unresolved loop with subarcsec width (Dere, 2008, 2009).
Some similarity between loops observed in the TRACE EUV band and Yohkoh X-ray band was found in outer active region loops (Nitta, 2000) and interpreted as evidence of loops with a broad range of temperatures. Core loops were instead observed only in the X-rays and found to be variable, indicating that probably they are not steady.
Density and temperatures in two active regions were accurately determined from SoHO-CDS observations (Mason et al., 1999) and it was confirmed quantitatively that the AR cores are hotter than larger loop structures extending above the limb. The analysis of a single loop observed on the solar limb with SoHO/CDS showed a bias to obtain flat temperature distributions along the loop from ratios of single lines or narrow band filters (TRACE), while a careful DEM reconstruction at selected points along the same loop was inconsistent with isothermal plasma, both across and along the loop (Schmelz et al., 2001). A whole line of works started from this analysis reconsidering and questioning the basic validity of the temperature diagnostics with TRACE and emphasizing the importance of the background subtraction, but also the need to obtain accurate spectral data (Schmelz, 2002; Martens et al., 2002; Aschwanden et al., 2002; Schmelz et al., 2003). Similar results but different conclusions were reached after the analysis of a loop observed with SoHO, invoking a non-constant cross-section to explain the evidence of isothermal loop (Landi and Landini, 2004; Landi and Feldman, 2004). On the other hand, evidence of non-uniform temperature along loops observed with TRACE was also found (Del Zanna and Mason, 2003; Reale and Ciaravella, 2006), emphasizing that the temperature diagnostic with narrow band instruments is a delicate issue.
An interesting debate focussed on the question whether the loops observed with TRACE and CDS have a uniform transverse thermal distribution, i.e., a narrow DEM, or a multi-thermal distribution, i.e., a wide DEM that may group together warm and hot loops. Although tackled from a different perspective, this question also concerns the fine longitudinal structuring of the loops and of their heating and is therefore strictly connected to the subject of Sections 3.2.2 and 4.4. A loop imaged by TRACE was found to be isothermal (with temperatures below 1 MK) along the line of sight from diagnostics of spectral lines obtained with SoHO/CDS (Del Zanna and Mason, 2003). The distribution across another loop observed on the limb with SoHO/CDS was found multi-thermal, with a DEM reconstruction and a careful analysis of background subtraction (Schmelz et al., 2005). From the comparison with the isothermal structure of hot loops derived from CDS data (Di Giorgio et al., 2003; Landi and Landini, 2004) and a systematic inspection of the CDS atlas, the conclusion was that there might be two different classes of loops, multi-thermal and isothermal (Figure 7).
Comparative studies of active region loops in the transition region and the corona (Ugarte-Urra et al., 2009) observed with Hinode seem to point out the presence of two dominant loop populations, i.e., core multitemperature loops that undergo a continuous process of heating and cooling in the full observed temperature range 0.4–2.5 MK shown by the X-Ray Telescope, and peripheral loops that evolve mostly in the temperature range 0.4–1.3 MK.
Multiband observations are able to provide more information and constraints. The analysis of an isolated loop in a time-resolved observation in several spectral bands, namely three TRACE UV filters, one Yohkoh/SXT filter, two rasters taken with SoHO/CDS in twelve relevant lines (5.4 ≤ log T ≤ 6.4), supported a coherent scenario across the different bands and instruments, i.e., a globally cooling loop and the presence of thermal structuring (Reale and Ciaravella, 2006). The analysis overall indicated that the loop analysis can be easily affected by a variety of instrumental biases and uncertainties, for instance due to rough background subtraction. The fact that the loop that could be well analyzed across several bands and lines is a cooling loop may not be by chance (see end of Section 3.3.3).
SoHO spectrometric data have contributed to investigate the loop thermal structure for a long time. From a differential emission measure (DEM) analysis with a forward-folding technique on SoHO/CDS data, some loops were found to be isothermal and others to have a broad DEM (Schmelz et al., 2007, 2008). Three distinct isothermal components, reminiscent of coronal hole, quiet-Sun, and active region plasmas, were found from the analysis of an active region spectrum observed by the SoHO/SUMER (Landi and Feldman, 2008).
Hinode/EIS observations of active region loops show that contrasted structures in cooler lines (∼ 1 MK) become fuzzy at higher temperatures (∼ 2 MK, Tripathi et al., 2009), Figure 8), as already pointed out by Brickhouse and Schmelz (2006). This issue is addressed by multi-thread loop modeling (Guarrasi et al., 2010).
Loop system observed in several EUV spectral lines with Hinode/EIS (19 May 2007, 11:41–16:35 UT). The loops become less and less contrasted, i.e., fuzzier and fuzzier, at higher and higher temperature. Courtesy of D. Tripathi.
Intensive efforts have been devoted to the analysis of the cool side of the emission measure distribution of hot loops, using data from EUV spectrometers. There is evidence for dynamic structure of active regions with frequent condensations inside (Tripathi et al., 2010) and for steep distributions (Tripathi et al., 2012). Analysis of Hinode and SDO observations were used to reconstruct the emission measure over a broad temperature range (Warren et al., 2011). Although affected by large uncertainties, it was found that at the apex of high-temperature loops the emission measure distribution has a relatively sharp peak around 4 MK.
Warm loops
The TRACE mission opened new intriguing questions because the data showed new features, e.g., stranded bright structures mostly localized in active regions, name "the moss", and because the narrow band filters offered some limited thermal diagnostics, but not easy to interpret. Reliable temperatures are in fact found in a very narrow range, and many coronal loops are found to be isothermal in that range.
As mentioned in Section 3.3.2, first loop diagnostics with normal-incidence telescopes were obtained from data collected with NIXT (Peres et al., 1994). The bright spots with Hα counterparts were identified with the footpoints of high pressure loops, invisible with NIXT because not sensitive to plasma hotter than 1 MK. They have been later addressed as the moss in the TRACE images, which undergo the same effect, and their interactions with the underlying chromospheric structures began to be studied (Berger et al., 1999; Fletcher and De Pontieu, 1999). Comparison of SoHO/CDS and TRACE observations led to establish that the plasma responsible for the moss emission has a temperature range of about 1 MK and is associated with hot loops at 1–2 MK, with a volume filling factor of order 0.1 (Fletcher and De Pontieu, 1999). It was also found that the path along which the emission originates is of order 1000 km long. According to an analytical loop model, a filling factor of about 0.1 is in agreement with the hypothesis of moss emission from the legs of 3 MK loops (Martens et al., 2000). The electron density estimated in specific regions in the active region moss decreases with increasing temperature (Tripathi et al., 2008). The density within the moss region was highest at log T = 5.8–6.1, with a value around 1010 cm−3.
As for temperature diagnostics with narrow band filters, loops soon appeared to be mostly isothermal with ratios of TRACE filters (Lenz et al., 1999; Aschwanden et al., 2000). Is this a new class of loops? Equivalent SoHO/EIT filter ratios provided analogous results (Aschwanden et al., 1999b). This evidence is intriguing and many investigations have addressed it (see also Section 3.3.2). From the diagnostic perspective, DEM reconstruction along the line of sight from spectral SoHO/CDS data and synthesized EIT count rates led to almost uniform temperatures along the loop, pointing again to an instrumental bias (Schmelz et al., 2001). Simple modeling confirmed that, provided they are flat, i.e., top-hat-shaped, even broad DEMs along the line of sight produce constant TRACE filter ratio values (Weber et al., 2005). On the other hand, the DEM obtained from spectrometers and from multi-wideband imagers data is most probably neither isothermal nor broad and flat, instead peaked with components extending both to low and high temperatures (e.g., Peres et al., 2000; Reale et al., 2009a). The critical point becomes the DEM width and its range of variation.
Using combinations of data from three filters, instead of two, does not seem to provide more information in the general case for warm loops (Schmelz et al., 2007), but the cross-field thermal structure of a sample of loops was found to be compatible with multithermal plasma with significant emission measure throughout the range 1–3 MK (Patsourakos and Klimchuk, 2007). With a combination of TRACE filter ratios, emission measure loci, and two methods of differential emission measure analysis, a few loops were found either isothermal or multithermal (Schmelz et al., 2009). This might not be a contradiction, in view of the presence of at least three possible conditions of warm loops, as discussed at the end of this section. Along a coronal loop in an active region on the solar limb, while TRACE double filter ratios led to temperatures between 1.0 and 1.3 MK, the emission measure loci from CDS data were consistent with a line-of-sight isothermal structure which increases in temperature from ∼ 1.20 to 1.75 MK along the loop, in contrast with the nearby multithermal background (Noglik et al., 2008).
Another puzzling issue, certainly linked to the loop isothermal appearance, is that warm loops are often diagnosed to be overdense with respect to the equilibrium values predicted by loop scaling laws (Lenz et al., 1999; Winebarger et al., 2003a, Section 4.1.1). To explain both these pieces of evidence, several authors claimed that the loops cannot be at equilibrium and that they must be filamented and cooling from a hotter state, probably continuously subject to heating episodes (nanoflares, Warren et al., 2002, 2003, Sections 4.2 and 4.4). Other authors proposed that part of the effect might be due to inaccurate background subtraction (Del Zanna and Mason, 2003).
The Hinode mission has stimulated extensive analyses of warm coronal structures, mostly based on its high quality EIS spectral data. Modeling observations of coronal moss with Hinode/EIS confirmed that the moss intensities predicted by steady, uniformly heated loop models are too intense relative to the observations (Warren et al., 2008b). A nonuniform filling factor is required and must vary inversely with the loop pressure. Observations of active region loops with EIS indicate that isolated coronal loops that are bright in Fe xii generally have very narrow temperature distributions (3 × 105 K), but are not properly isothermal and have a volumetric filling factors of approximately 10% (Warren et al., 2008a).
In a cooler regime (4. 15 < log T < 5. 45) observed in coordination by SoHO spectrometers and imagers, STEREO/EUVI, and Hinode/EIS, active region plasma at the limb has been found to cool down from a coronal hole status with temperatures in the 5.6 < log T < 5.9 range (Landi et al., 2009).
A loop reconstruction from STEREO data improves the background subtraction and recovers density and temperature distributions that are able to reproduce the total observed fluxes within 20% (Aschwanden et al., 2009), emission measure distributions not very different from those obtained from spectroscopic observations (Brosius et al., 1996), and is in agreement with other previous studies (Lenz et al., 1999; Winebarger et al., 2003a).
Some studies have been devoted specifically to fan loops departing from active regions and observed with Hinode and SDO. It was found that these loops are warm and their temperature distribution is generally narrow (Brooks et al., 2011).
More recently, accurate analysis of warm loops with comparison between EUV spectroscopy and multi-channel imaging with Hinode/EIS and SDO/AIA have put severe warnings on using the latter for DEM reconstruction (Del Zanna et al., 2011). Cool plasma at temperature T < 0.5 MK might contribute considerably to the emission, especially at the loop footpoints, and question, for instance, the interpretation of upflows observed in the 171 A band as made of million degrees plasma (De Pontieu et al., 2011). Important results come from observations at very high spatial resolution (0.2 arcsec) with the Hi-C sounding rocket in the EUV 193 Å band. There is evidence of magnetic braiding that indicates the occurrence of magnetic reconnection (Cirtain et al., 2013).
In summary, the current observational framework and loop analysis seems to indicate that for a coherent scenario warm loops are manifestations of at least three different loop conditions: i) in loops consisting of bundles of thin independently-heated strands, few cooling strands of steady hot X-ray loops might be detected as warm overdense loops in the UV band. These warm loops would coexist with hot loops and would show a multithermal emission measure distribution (Patsourakos and Klimchuk, 2007; Warren et al., 2008a; Tripathi et al., 2009); ii) we might have warm loops as an obvious result of a relatively low average heating input in the loop. These loops would be much less visible in the X-rays and, thus, would not be co-spatial with hot loops, and would also be much less multithermal (Di Giorgio et al., 2003; Landi and Feldman, 2004; Aschwanden and Nightingale, 2005; Noglik et al., 2008); iii) warm loops might be globally cooling from a status of hot X-ray loop (Reale and Ciaravella, 2006). These loops would also be overdense and co-spatial with hot loops but with a time shift of the X-ray and UV light curves, i.e., they would be bright in the X-rays before they are in the UV band. Also these loops would have a relatively narrow thermal distribution along the line of sight. There is some indication that there might be fundamental differences in the heating regime and cadence between hot and warm loops (Warren et al., 2010a).
Temporal analysis
The solar corona is the site of a variety of transient phenomena. Coronal loops sometimes flare in active regions (see the review by Benz, 2008). However, most coronal loops are well-known to remain in a steady state for most of their life, much longer than the plasma characteristic cooling times (Rosner et al., 1978, see Section 4.1.1). This is taken as an indication that a heating mechanism must be on and steady long enough to bring the loop to an equilibrium condition, and keep it there for a long time. Nevertheless, the emission of coronal loops is found to vary significantly on various timescales, and the temporal analyses of coronal loop data have been used to obtain different kinds of information, and as a help to characterize the dynamics and heating mechanisms. The time variability of loop emission is generally not trivial to interpret. The problem is that the emission is very sensitive to density and less to temperature. Therefore, variations are not direct signatures of heating episodes, not even of local compressions, because the plasma is free to flow along the magnetic field lines. Variations must therefore be explained in the light of the evolution of the whole loop. This typically needs accurate modeling, or, at least, care must be paid to many relevant and concurrent effects.
Another important issue is the band in which we observe. The EUV bands of the normal-incidence telescopes are quite narrow. Observations are then more sensitive to variations because cooling or heating plasma is seen to turn on and off rapidly as it crosses the band sensitivity. On the other hand, telescopes in the X-ray band detect hotter plasma which is expected to be more sensitive to heating and therefore to vary more promptly, but the bandwidths are large and do not take as much advantage of the temperature sensitivity as the narrow bands. Finally, spectroscopic observations are, in principle, very sensitive to temperature variations, because they observe single lines, but their time cadence is typically low and able to follow variations only on large timescales. Time analysis studies can be classified to address two main classes of phenomena: temporal variability of steady structures and single transient events, such as flare-like brightenings.
In spite of limited time coverage, the instrument S-054 on-board Skylab already allowed for early studies of variability of hot X-ray loops. Measured decay times showed evidence of continued evaporation of coronal plasma in slowly-decaying structures (Krieger, 1978). found timescales of moderate variability was found on the timescale of a few hours over a substantial steadiness for observations of active region loops in 2 MK lines such as Fe xv and Si xii (Sheeley Jr, 1980; Habbal et al., 1985). Continuous microflare activity (1026–1028 erg) has been extensively detected in hard X-ray band (20 keV) with non-thermal power-law spectra (Lin et al., 1984). Substantial (but non-flaring) temporal variability was found in active region loops observed with SMM in a few relatively hot X-ray lines (∼ 5 MK) on timescales of some minutes (Haisch et al., 1988). Cooler loops (< 1 MK Foukal, 1976, see Section 3.5) were found to be more variable and dynamic (e.g., Kopp et al., 1985).
The high time coverage and resolution of Yohkoh triggered studies of brightenings on short timescales. The study of the interaction of differently bright hot loops showed, for instance, that X-ray bright points often involve loops considerably larger than the bright points themselves, and that they vary on timescales from minutes to hours (Strong et al., 1992). The analysis of a large set (142) of macroscopic transient X-ray brightenings indicated that they derive from the interaction of multiple loops at their footpoints (Shimizu et al., 1994). Some other more specific loop variations, e.g., the shrinkage of large-scale non-flare loops were also observed, and interpreted not as an apparent motion, but as a real contraction of coronal loops that brighten and then gradually cool down (Wang et al., 1997). Fine-scale motions and brightness variations of the emission were found on timescales of 1 minute or less, often with dark jets of chromospheric plasma seen in the wings of Hα, and probably associated with the fine structure and dynamics of the upper transition region (Berger et al., 1999).
Loop variability was specifically studied in several UV spectral lines observed with SoHO/CDS for about 3 hours by Di Giorgio et al. (2003). In the hottest lines, within the limited time resolution of about 10 min, a few brightenings of a hot loop (∼ 2 MK) were detected but they are minor perturbations over a steadily high emission level. A cool loop (log T ∼ 5.3) was confirmed to be a transient structure living a few hours, and confining substantial flows (Section 3.5).
Variability analyses were conducted also on warm loops present in TRACE data. The brightening of a single coronal loop was analyzed in detail in an observation of more than 2 hours with a cadence of about 30 s (Reale et al., 2000a). The loop brightens from the footpoints to the top, allowing for detailed hydrodynamic modeling (Reale et al., 2000b, see also Section 4.4). Active region transient events, i.e., short-lived brightenings in small-scale loops, detected over a neutral line in a region of emerging flux were interpreted as reconnection events associated with flux emergence (Seaton et al., 2001). Apparent shrinking and expansion of brightening warm loops suggested heating and cooling of different concentric strands, leading to coronal rain visible in the Hα line (Shimojo et al., 2002, see also Section 3.2.2). Plasma condensations in hot and warm loops were detected also in the analysis of line intensity and velocity in temporal series data from SoHO/CDS (O'Shea et al., 2007). Antiochos et al. (2003) found no significant variability of the moss regions observed with TRACE. This has been taken as part of the evidence toward steady coronal heating in active region cores (Warren et al., 2010b, see Section 4.4).
The analysis of temporal series from various missions has been used to investigate the possible presence of continuous impulsive heating by nanoflares. The temporal evolution of hot coronal loops was studied in data taken with GOES Solar X-ray Imager (SXI), an instrument with moderate spatial resolution and spectral band similar to Yohkoh/SXT (López Fuentes et al., 2007). The durations and characteristic timescales of the emission rise, steady and decay phases were found to be much longer than the cooling time and indicate that the loop-averaged heating rate increases slowly, reaches a maintenance level, and then decreases slowly (Figure 9), not in contradiction with the early results of Skylab (Section 2). This slow evolution is taken as an indication of a single heating mechanism operating for the entire lifetime of the loop. If so, the timescale of the loop-averaged heating rate might be roughly proportional to the timescale of the observed intensity variations.
X-ray light curve observed with the SXI telescope on board GOES. The loop lifetime is much longer than the characteristic cooling times. Courtesy of J. A. Klimchuk and M. C. López Fuentes).
Joint TRACE and SoHO/CDS observations allowed to study temperature as a function of time in active region loops (Cirtain et al., 2007). In many locations along the loops, the emission measure loci were found consistent with an isothermal structure, but the results also indicated significant changes in the loop temperature (between 1 and 2 MK) over the 6 hr observing period. This was interpreted as one more indication of multistranded loops, substructured below the resolution of the imager and of the spectrometer. Further support to fine structuring comes from the analysis of the auto-correlation functions in SXT and TRACE loop observations (Sakamoto et al., 2008). The duration of the intensity fluctuations for the hot SXT loops was found to be relatively short because of the significant photon noise, but that for the warm TRACE loops agrees well with the characteristic cooling timescale, thus supporting a continuous heating by impulsive nanoflares. The energy of nanoflares is estimated to be 1025 erg for SXT loops and 1023 erg for TRACE loops, and their occurrence rate about 0.4 and 30 nanoflares s−1, respectively.
Time series have been studied also on data taken with the Hinode mission. Hinode's Solar Optical Telescope (SOT) magnetograms and high-cadence EIS spectral data were used to distinguish hot, relatively steadily emitting warm coronal loops from isolated transient brightenings and to find that they are both associated with highly dynamic magnetic flux regions. Brightenings were confirmed in regions of flux collision and cancellation, while warm loops are generally rooted in magnetic field regions that are locally unipolar with unmixed flux (Brooks et al., 2008). It was suggested that the type of heating (transient vs. steady) is related to the structure of the magnetic field, and that the heating in transient events may be fundamentally different from that in warm coronal loops.
More recently, light curves in individual pixels have been investigated in the X-ray band to search for significant variability connected to variable heating. Although some pulses are detected, most of the emission in active region cores and loops has been found to be steady on the timescale of hours with fluctuations on the order of 15% and with no correlation between warm and hot emission (Warren et al., 2010b, 2011). Improving on previous studies (Sakamoto et al., 2008), high cadence observations with the Hinode/XRT have revealed that the distributions of intensity fluctuations have small but significant and systematic asymmetries. Part of this asymmetry has been explained through a tendency for exponentially decreasing intensity, i.e., the plasma has been cooling for most of the time (Terzo et al., 2011).
Loop light curves have been systematically analysed also in the EUV band. A systematic tendency has been found to have ordered time lags from channels sensitive to emission from hotter plasma to cooler plasma, that is also evidence for dominant cooling (Viall and Klimchuk, 2011, 2012). Light curves in the EUV band have been analysed also with a different approach: they have been compared to simulated ones obtained from sequences of random pulses with power-law distribution (Tajfirouze and Safari, 2012). Artificial neural network (ANN) was used for the comparison and it was found many that light curves are matched by those generated from events power-laws with a steep index (≥ 2). While studying the long-term evolution of active regions on the large scale, it was found that active regions show less and less variability as they age (Ugarte-Urra and Warren, 2012), thus suggesting a qualitative change of heating frequency with time. At the other extreme of the smallest scales, the Hi-C observations show in some moss regions variability on timescales down to ∼ 15 s, that may indicate the presence of heating pulses of comparable duration.
Flows and waves
Diagnosing the presence of significant flows in coronal loops is not an easy task. Apparently moving brightness variations may not be a conclusive evidence of plasma motion, since the same effect may be produced by the propagation of thermal fronts or waves. Conclusive evidence of plasma motion comes from measurements of Doppler shifts in relevant spectral lines. However, the detection of significant Doppler shifts requires several conditions to be fulfilled at the same time, e.g., significant component along the line of sight, amount of moving plasma larger than amount of static plasma, plasma motion comparable to typical line broadening effects.
In general, we can distinguish two main classes of mass bulk motions inside coronal loops: siphon flows, due to a pressure difference between the footpoints, and loop filling or draining, due to transient heating and subsequent cooling, respectively. Some other evidence of bulk motions, such as systematic redshifts in UV lines, has been difficult to interpret.
Siphon flows have been mainly invoked to explain motions in cool loops. The existence of cold loops has been known for a long time (Foukal, 1976, see Section 2) and SoHO has collected high-quality data showing the presence of dynamic cool loops (Brekke et al., 1997). A well-identified detection was found in SoHO/SUMER data, i.e., a small loop showing a supersonic siphon-like flow (Teriaca et al., 2004) and in SoHO/CDS data (Di Giorgio et al., 2003).
Redshifts in transition region UV lines have been extensively observed on the solar disk (e.g., Doschek et al., 1976; Gebbie et al., 1981; Dere, 1982; Feldman et al., 1982; Klimchuk, 1987; Rottman et al., 1990; Brekke, 1993; Peter, 1999). Some mechanisms have been proposed to explain these red-shifts: downward propagating acoustic waves (Hansteen, 1993), downdrafts driven by radiatively-cooling condensations in the solar transition region (Reale et al., 1996, 1997b), nanoflares (Teriaca et al., 1999); the scenario is improving with the better and better definition of the observational framework. Downflows are systematically confirmed in lines at transition region temperatures (≤ 0.5 MK) from a few km s−1 (Feldman et al., 2011) to a few tens km s−1 (Chen and Ding, 2010; Ugarte-Urra and Warren, 2011). Moderate downflows have been detected at the boundary of active regions (Boutry et al., 2012). Redshifts between 5 and 15 km s−1 have been measured accurately from SoHO/SUMER data in three active regions with little spatial and temporal correlation (Winebarger et al., 2013).
Blueshifts in the transition region are also studied but not necessarily associated with coronal loops (e.g., Dere et al., 1986). More localized and transient episodes of high velocity outflows, named explosive events, have been observed in the transition lines such as C iv, formed at 100 000 K (e.g., Dere et al., 1989; Chae et al., 1998b; Winebarger et al., 1999, 2002b,a). However, there are indications that such EUV explosive events are not directly relevant in heating the corona, are characteristic of structures not obviously connected with the upper corona, and have a chromospheric origin (Teriaca et al., 2002). Moderate outflows at about 1 MK or more have been found from combined SoHO/SUMER and Hinode/EIS observations of the quiet Sun around disk center (Dadashi et al., 2011).
Doppler shifts have been extensively studied as a function of temperature across the transition region. Figure 10 shows a summary of the results updated to 2011.
Average Doppler shift in the quiet Sun at disk center of various TR and coronal ions measured from SUMER and EIS spectra. Positive/negative values indicate red- (downflows) / blue- (upflows) shifts. Solid and dashed lines respectively represent polynomial fittings. Image reproduced with permission from Dadashi et al. (2011), copyright by ESO.
Redshifts increasing up to ∼ 10 km s−1 at ∼ 2 × 105 K and then decreasing with increasing temperature have been found from SUMER and EIS spectra in the quiet Sun and explained by the dominance of emission from plasma flowing downward from the upper hot region to the lower cool region along flux tubes with varying cross section (a factor about 30, Chae et al., 1998c, see also Section 3.1). The redshift peak increases to 15 km s−1 in an active region at temperature of 105 K, while the redshifts were found to turn into blueshifts at temperatures above 5 × 105 K (Teriaca et al., 1999). The trend from redshift to blueshift also applies to active region moss, but the transition from red to blue appears to occur at a higher temperature in the moss (∼ 1 MK versus 0.5 MK in the quiet Sun) (Dadashi et al., 2012; Tripathi et al., 2012b).
Regarding the spatial localization of the flows, a complex scenario of Doppler flows was found in active region loops observed by Hinode EIS (Del Zanna, 2008, Figure 11). Persistent redshifts, stronger in cooler lines (about 5–10 km s−1 in Fe xii and 20–30 km s−1 in Fe viii), were confirmed in most loop structures. Persistent blueshifts, stronger in the hotter lines (typically 5–20 km s−1 in Fe xii and 10–30 km s−1 in Fe xv), were present in areas of weak emission, in a sharp boundary between the low-lying "hot" 3 MK loops and the higher "warm" 1 MK loops.
Monochromatic (negative) images and dopplergrams km s−1) of an active region (NOAA 10926) observed with Hinode/EIS in Fe viii, Fe xii, Fe xv lines. Courtesy of G. Del Zanna.
An active region was comprised of red-shifted emissions (downflows) in the core and blue-shifted emissions (upflows) at the boundary (Tripathi et al., 2009). No strong flows were found in an active region core in the Hinode/EIS Fe xii 195 Å line (Brooks and Warren, 2009). In the core of moss regions SUMER and EIS data show some blueshifts of a few km s−1 at low coronal temperatures, decreasing at higher temperatures (Dadashi et al., 2012). Specific studies of coronal loops in an active region show mostly blue-shifted emission at coronal temperatures, with speed of about 20 km s−1 at the footpoints (Tripathi et al., 2012a).
Non-thermal velocities in the transition region and corona of the quiet Sun were measured from the widths of SoHO/SUMER UV lines (Chae et al., 1998a). They were found to increase with temperature from values smaller than 10 km s−1 at temperatures < 2 × 104 K, to a peak value of 30 km s−1 around 3 × 105, and then to increase with temperature, to about 20 km s−1 at coronal temperatures. Since the motions are small-scale and isotropic they were interpreted in terms of MHD turbulence. Non-thermal velocities were studied also in solar active regions on Hinode/EIS spectra (Doschek et al., 2007). The largest widths seem to be located more in relatively faint zones, some of which also show Doppler outflows.
Coronal plasma motions near footpoints of active region loops showed a strong correlation between Doppler velocity and non-thermal velocity (Hara et al., 2008). Significant deviations from a single Gaussian profile were found in the blue wing of the line profiles for the upflows. These may suggest that there are unresolved high-speed upflows.
EUV spectra of coronal loops above active regions show also clear evidence of stronger dynamical activity. In the O v 629 Å line, formed at 240 000 K, line-of-sight velocities greater than 50 km s−1 have been measured with the shift extending over a large fraction of a loop (Brekke et al., 1997). Active region loops appear to be extremely time variable and dynamic at transition region temperatures, with large Doppler shifts (Brekke, 1999). The birth, evolution and cooling of one of such transient cool loops was directly observed with the SoHO/CDS, and a blue-shifted upflow was measured all along the loop, probably a one-direction siphon flow (Di Giorgio et al., 2003, see also Section 3.5). Line-of-sight flows of up to 40 km s−1 were measured along warm and apparently static active region loops in co-aligned TRACE and the SoHO/SUMER observations (Winebarger et al., 2002c). Apparent motions were also detected in other TRACE images (Winebarger et al., 2001). Strong localized outflows (∼ 50 km s−1) in a widespread downflow region were clearly visible in Doppler-shifts maps obtained with EIS (Doschek et al., 2008). The outflows might be tracers of long loops and/or open magnetic fields. High-speed outflows at about 1 MK or more have been found in microflares (Chen and Ding, 2010), and transient ones at the boundary of active region cores on the timescale of minutes in time sequences of EIS spectra (Ugarte-Urra and Warren, 2011). Studies on the temperature structure and chemical composition of the plasma producing the faint blue wings at about 100 km s−1 have shown a peaks at coronal temperatures above 1 MK and coronal FIP bias values (Brooks and Warren, 2012). Cool plasma flowing in multi-threaded coronal loops were detected in high resolution Hinode SOT observations with speeds in the range 74–123 km s−1 (Ofman and Wang, 2008). In addition to flows, the loops exhibited transverse oscillations. Even stronger upflows, more typical of flare chromospheric evaporation (e.g., Antonucci et al., 1982), have also been detected during some small transient brightenings in the X-ray band (Hinode/XRT), with speeds up to 500 km s−1 (Nitta et al., 2012).
Chromospheric flows
It has been proposed that an important role in the dynamics and heating of plasma inside coronal loops might be played by upcoming chromospheric flows. Although the discovery is not new (Athay and Holzer, 1982), recently much attention has been devoted to these flows, and in particular to a specific class of finger-like ejections with speeds between 50 and 150 km s−1 and lasting a couple of minutes or less, the so-called type II spicules (De Pontieu et al., 2007a). There is evidence for a spatial and temporal correlation between spicules and faint upflows detected in the EUV band suggesting the possibility that the flows are heated to coronal temperatures when they are injected in coronal loops. The upflows were detected as a blue-shifted excess (wing asymmetry) in an EUV line (Fe xiv). The implication is that spicules may provide a substantial contribution to coronal heating, thus shifting the source of coronal heating down in the chromosphere (De Pontieu et al., 2009) and proposing a new challenge for coronal heating theory. The evidence is debated. Wavelike propagations in coronal EUV images have been also re-interpreted as evidence for repetitive upflows (Tian et al., 2011; Kamio et al., 2011). However, 3D MHD modeling shows that waves and flows may results from the same impulsive events (Ofman et al., 2012), while flows may be important very low in the corona, the propagating disturbances are dominated by waves at higher altitudes in active region loop (Wang et al., 2013). Coronal counterparts have been directly identified in EUV images (De Pontieu et al., 2011) and also a correspondence between upflows and downflows, the latter at moderate speed (∼ 10 km s−1) and in the cool passbands (McIntosh et al., 2012). These are to be compared with the usual evidence of redshifts in the transition region. In other analyses high speed spicules were not found to have coronal counterparts (Madjarska et al., 2011) and to be a separate population from lower speed spicules (Zhang et al., 2012). It has been estimated that only a small fraction of coronal plasma can be supplied by chromospheric upflows (Klimchuk, 2012).
Waves observations
Although this review focusses more on the plasma confined in loops, recently considerable efforts have been devoted to models that point to the importance of magnetohydrodynamic waves. The question is again the contribution of the dissipation of wave energy to the heating of the corona, that is not easy to evaluate (Klimchuk, 2006). The wave propagation is often connected to and needs to be distinguished from the presence of flows. Therefore, we dedicate attention to recent evidences regarding coronal waves and oscillations that involve loops.
Evidence for photospheric Alfvén waves was obtained from magnetic and velocity fluctuations in regions of strong magnetic field (Ulrich, 1996) and from granular motions in the quiet Sun (Muller et al., 1994) with fluxes of the order of 107 erg cm−2 s−1, which might contribute to heating if transmitted efficiently to the corona.
In SoHO/EIT high-cadence 304 Å images, analyzed systematic intensity variations along an off-limb half loop structure were observed to propagate from the top toward the footpoints (De Groof et al., 2004). These intensity variations are more probably be due to flowing/falling plasma blobs than to slow magneto-acoustic waves (Section 4.4). This evidence has been addressed also by modeling studies (see Section 4.3).
Widespread evidence for outward propagation of Alfvén waves is reported from ground optical polarimetric observations (Tomczyk et al., 2007), and non-thermal broadening has been shown to correlate (McIntosh and De Pontieu, 2012) with swaying motions detected in the corona from SDO/AIA data (speed of ∼ 20 km s−1 and periods of few minutes) (McIntosh et al., 2011). TRACE EUV observations provided the first evidence of resolved transverse waves in coronal loops (Aschwanden et al., 1999a; Nakariakov et al., 1999; Nakariakov and Ofman, 2001). Undamped, or even growing waves were observed by SDO/AIA (Wang et al., 2012; Nisticò et al., 2013). Transverse waves were detected also in high resolution observations with the HiC in thin loops (≥ 100 km) at low speed amplitude (Morton and McLaughlin, 2013). The contribution of this class of waves is estimated, discussed, and debated although there is still no clear convergence. Other oscillations were observed to propagate along coronal loops at more than 100 km s−1 with periods of a few minutes and interpreted as slow magnetosonic waves (De Moortel et al., 2000), but also as due to faint upflows (De Pontieu and McIntosh, 2010). We mention that also much faster (> 1000 km s−1) propagating oscillations were detected in SDO/AIA observations during eruptive events and interpreted as fast magnetosonic waves (e.g., Liu et al., 2011; Ofman et al., 2011).
Loop Physics and Modeling
The basics of loop plasma physics are well established since the 1970s (e.g., Priest, 1978). In typical coronal conditions, i.e., ratio of thermal and magnetic pressure β ≪ 1, temperature of a few MK, density of 108 − 1010 cm−3, the plasma confined in coronal loops can be assumed as a compressible fluid moving and transporting energy only along the magnetic field lines, i.e., along the loop itself (e.g., Rosner et al., 1978; Vesecky et al., 1979). In this configuration, the magnetic field has only the role of confining the plasma. It is also customary to assume constant loop cross-section (see Section 3.2.1). In these conditions, and neglecting gradients across the direction of the field, effects of curvature, non uniform loop shape, magnetic twisting, currents and transverse waves, the plasma evolution can be described by means of the one-dimensional hydrodynamic equations for a compressible fluid, using only the coordinate along the loop (Figure 12).
The plasma confined in a loop can be described with one-dimensional hydrodynamic modeling, with a single coordinate (s) along the loop. Image: TRACE, 6 November 1999, 2 UT.
The time-dependent equations of mass, momentum, and energy conservation typically include the effects of the gravity component along the loop, the radiative losses from an optically thin plasma, the plasma thermal conduction, an external heating input, the plasma compressional viscosity:
$${{dn} \over {dt}} = - n{{\partial v} \over {\partial s}},$$
$$n{m_{\rm{H}}}{{dv} \over {dt}} = - {{\partial p} \over {\partial s}} + n{m_{\rm{H}}}g + {\partial \over {\partial s}}(\mu {{\partial v} \over {\partial s}}),$$
$${{d\epsilon} \over {dt}} + (p + \epsilon){{\partial v} \over {\partial s}} = H - {n^2}{\beta _i}P(T) + \mu {\left({{{\partial v} \over {\partial s}}} \right)^2}{F_c},$$
with p and ∊ defined by:
$$p = (1 + {\beta _i})n{k_{\rm{B}}}T,\quad \quad q = {3 \over 2}p + n{\beta _i}\chi,$$
and the conductive flux:
$${F_c} = {\partial \over {\partial s}}\left({\kappa {T^{5/2}}{{\partial T} \over {\partial s}}} \right),$$
where n is the hydrogen number density, s the spatial coordinate along the loop, v the plasma velocity, mH the mass of hydrogen atom, μ the effective plasma viscosity, P (T) the radiative losses function per unit emission measure (e.g., Raymond et al., 1976), βi the fractional ionization, i.e., ne/nH, Fc the conductive flux, κ the thermal conductivity (Spitzer, 1962), kB the Boltzmann constant, and χ the hydrogen ionization potential. H(s, t) is a function of both space and time that describes the heat input in the loop.
These equations can be solved numerically and several specific codes have been used extensively to investigate the physics of coronal loops and of X-ray flares (e.g., Nagai, 1980; Peres et al., 1982; Doschek et al., 1982; Nagai and Emslie, 1984; Fisher et al., 1985a, a, a; MacNeice, 1986; Gan et al., 1991; Hansteen, 1993; Betta et al., 1997; Antiochos et al., 1999; Ofman and Wang, 2002; Müller et al., 2003; Bradshaw and Mason, 2003; Sigalotti and Mendoza-Briceño, 2003; Bradshaw and Cargill, 2006).
The concept of numerical loop modeling is to use simulations, first of all, to get insight into the physics of coronal loops, i.e., the reaction of confined plasma to external drivers, to describe plasma evolution, and to derive predictions to compare with observations. One major target of modeling is of course to discriminate between concurrent hypotheses, for instance, regarding the heating mechanisms, and to constrain the related parameters.
The models require to be provided with initial loop conditions and boundary conditions. It has been shown that time-dependent loop models must include a relatively thick, cool, and dense chromosphere and the transition region for a correct description of the mass transfer driven by transient heating (e.g., Bradshaw and Cargill, 2013) and to maintain the necessary numerical stability (Antiochos, 1979; Hood and Priest, 1980; Peres et al., 1982). The main role of the chromosphere is only that of a mass reservoir and, therefore, in several codes, it is treated as simply as possible, e.g., an isothermal inactive layer that neither emits, nor conducts heat. In other cases, a more accurate description is chosen, e.g., including a detailed chromospheric model (e.g., Vernazza et al., 1981), maintaining a simplified radiative emission and a detailed energy balance with an ad hoc heat input (Peres et al., 1982; Reale et al., 2000a). Overall, a typical loop initial condition is a hydrostatic atmosphere with a temperature distribution from ∼ 104 K to > 106 K, basically dictated by a thermal conduction profile (Figure 14). The lower boundary of the computational domain is typically not involved in the evolution of the loop plasma. Many loop models assume mirror symmetry with respect to the apex and, therefore, describe only half of the loop. The upper boundary conditions are those of symmetry at the loop apex.
The models also require to define an input heating function (see Section 4.4), specifying its time-dependence, for instance it can be steady, slowly, or impulsively changing, and its position in space. The output typically consists of distributions of temperature, density, and velocity along the loop evolving with time. From simulation results, some modelers derive observables, i.e., the plasma emission, which can be compared directly to data collected with the telescopes. The model results are, in this case, to be folded with the instrumental response. This forward-modeling allows to obtain constraints on model parameters and, therefore, quantitative information about the questions to be solved, e.g., the heating rate and location (e.g., Reale et al., 2000a).
Loop codes are typically based on finite difference numerical methods. Although they are one-dimensional and, therefore, typically less demanding than other multi-dimensional codes that study systems with more complex geometry, and although they do not include the explicit description of the magnetic field, as full MHD codes, loop codes require some special care. One of the main difficulties consists in the appropriate resolution of the steep transition region (1–100 km thick) between the chromosphere and the corona, which can easily drift up and down depending on the dynamics of the event to be simulated. The temperature gradient there is very large due to the local balance between the steep temperature dependence of the thermal conduction and the peak of the radiative losses function (Serio et al., 1981). The density is steep as well so to maintain the pressure balance. The transition region can become very narrow during flares. An insufficient resolution of the transition region can lead to inaccurate description of the loop plasma dynamics, e.g., chromospheric evaporation (Bradshaw and Cargill, 2013, see Section 4.1.2). Also a fine temporal resolution is extremely important, because the highly efficient thermal conduction in a hot magnetized plasma can lead to a very small time step and make execution times not so small even nowadays. Some deviations can be possible because of non-local thermal conduction that may lengthen considerably the conduction cooling times and may enhance the chances of observing hot nanoflare-heated plasma (West et al., 2008).
In recent years, time-dependent loop modeling has been revived in the light of the observations with SoHO, TRACE, and SDO for the investigation of the loop dynamics and heating. The upgrade driven by the higher quality of the data has consisted in the introduction of more detailed mechanisms for the heating input, for the momentum deposition, or others, e.g., the time-dependent ionization and the saturated thermal conduction (Bradshaw and Cargill, 2006; Reale and Orlando, 2008). Some codes have been upgraded to include adaptive mesh refinement for better resolution in regions of high gradients, such as in the transition region, or during impulsive events (e.g., Betta et al., 1997). Another form of improvement has been the description of loops as collections of thin strands. Each strand is a self-standing, isolated and independent atmosphere, to be treated exactly as a single loop. This approach has been adopted both to describe loops as static (Reale and Peres, 2000) (Figure 13) and as impulsively heated by nanoflares (Warren et al., 2002). On the same line, collections of loop models have been applied to describe entire active regions (Warren and Winebarger, 2006).
Emission in two TRACE filterbands predicted by a model of loop made by several thin strands. Image reproduced with permission from Reale and Peres (2000), copyright by AAS.
One limitation of current 1D loop models is that they are unable to treat conveniently the tapering expected going down from the corona to the chromosphere (or expansion upwards) through the transition region. This effect can be neglected in many circumstances, but it is becoming increasingly important with the finer and finer level of diagnostics allowed by upcoming observational data. For instance, the presence of tapering changes considerably the predicted distribution of emission measure in the low temperature region (Section 4.1.1).
Possible deviations from pure 1D evolution might be driven by intense oscillations or kinks, as described in Ofman (2009). The effect of the three-dimensional loop structure should then be taken into account to describe the interaction with excited MHD waves (McLaughlin and Ofman, 2008; Pascoe et al., 2009; Selwa and Ofman, 2009).
However, the real power of 1D loop models, that makes them still on the edge, is that they fully exploit the property of the confined plasma to evolve as a fluid and practically independent of the magnetic field, and that they can include the coronal part, the transition region, and the photospheric footpoint in a single model with thermal conduction. In this framework, we may even simulate a multi-thread structure only by collecting many single loop models together, still with no need to include the description and interaction with the magnetic field (Guarrasi et al., 2010). We should, however, be aware that the magnetic confinement of the loop material is not as strong and the thermal conduction is not as anisotropic below the coronal part of the loop as it is in the corona.
An efficient approach to loop modeling is to describe the temporal evolution of average loop quantities (temperature, pressure, and density), i.e., a "0-D" model (Klimchuk et al., 2008; Cargill et al., 2012a,b). This model is useful for the description of loops as collections of myriads of independent strands with a statistical distribution of heating events.
Alternative approaches to single or multiple loop modeling have been developed more recently, thanks also to the increasing availability of high performance computing systems and resources. Global "ab initio" approaches have been developed (Gudiksen and Nordlund, 2005; Hansteen et al., 2007; see also Yokoyama and Shibata, 2001 for the case of a flare model) to model — with full MHD — boxes of the solar corona that span the entire solar atmosphere from the upper convection zone to the lower corona. These models include non-grey, non-LTE (local thermodynamic equilibrium) radiative transport in the photosphere and chromosphere, optically thin radiative losses, as well as magnetic field-aligned heat conduction in the transition region and corona. Although such models still cannot resolve well fine structures, such as current sheets and the transition region, they certainly represent the first important step toward fully self-consistent modeling of the magnetized corona. Large-scale MHD modeling has been used to explain the appearance of constant cross-section in EUV observations as due to temperature variations across the loop (Peter and Bingert, 2012). Another global model of the solar corona includes also information from photospheric magnetic field data (Sokolov et al., 2013).
Monolithic (static) loops: scaling laws
The Skylab mission remarked, and later missions confirmed (Figure 9), that many X-ray emitting coronal loops persist mostly unchanged for a time considerably longer than their cooling times by radiation and/or thermal conduction (Rosner et al., 1978, and references therein). This means that, for most of their lives, they can be well described as systems at equilibrium and has been the starting point for several early theoretical studies (Landini and Monsignori Fossi, 1975; Gabriel, 1976; Jordan, 1976; Vesecky et al., 1979; Jordan, 1980). Rosner et al. (1978) devised a model of coronal loops in hydrostatic equilibrium with several realistic simplifying assumptions: symmetry with respect to the apex, constant cross section (see Section 3.2.1), length much shorter than the pressure scale height, heat deposited uniformly along the loop, low thermal flux at the base of the transition region, i.e., the lower boundary of the model. Under these conditions, the pressure is uniform all along the loop, which is then described only by the energy balance between the heat input and the two main losses mentioned above. From the integration of the equation of energy conservation, one obtains the well-known scaling laws:
$${T_{0,6}} = 1.4{(p{L_9})^{1/3}}$$
$${H_{ - 3}} = 3{p^{7/6}}L_9^{ - 5/6},$$
where T0,6, L9 and H−3 are the loop maximum temperature To, length L and heating rate per unit volume H, measured in units of 106 K (MK), 109 cm and 10−3 erg cm−3 s−1 respectively. These scaling laws were found in agreement with Skylab data within a factor 2.
Analogous models were developed in the same framework (Landini and Monsignori Fossi, 1975) and equivalent scaling laws were found independently by Craig et al. (1978) and more general ones by Hood and Priest (1979a). They have been derived with a more general formalism by Bray et al. (1991). Although scaling laws could explain several observed properties, some features such as the emission measure in UV lines and the cool loops above sunspots could not be reproduced, and, although the laws have been questioned a number of times (e.g., Kano and Tsuneta, 1995) in front of the acquisition of new data, such as those by Yohkoh and TRACE, they anyhow provide a basic physical reference frame to interpret any loop feature. For instance, they provide reference equilibrium values even for studies of transient coronal events, they have allowed to constrain that many loop structures observed with TRACE are overdense (e.g., Lenz et al., 1999; Winebarger et al., 2003a, Section 4.1.2) and, as such, these loops must be cooling from hotter status (Winebarger and Warren, 2005, see Section 3.3.3), and so on. They also are useful for density estimates when closed with the equation of state, and for coronal energy budget when integrated on relevant volumes and times.
Scaling laws have been extended to loops higher than the pressure scale height (Serio et al., 1981), to different heating functions (Martens, 2010), and limited by the finding that very long loops become unstable (Wragg and Priest, 1981). According to Antiochos and Noci (1986), the cool loops belong to a different family and are low-lying, and may eventually explain an evidence of excess of emission measure at low temperature.
The numerical solution of the complete set of hydrostatic equations allowed to obtain detailed profiles of the physical quantities along the loop, including the steep transition region. Figure 14 shows two examples of solution for different values of heating uniformly distributed along the loop.
Distributions of temperature, density, and pressure along a hydrostatic loop computed from the model of Serio et al. (1981) for a high pressure loop (AR) and a low pressure one (Empty) with heating uniformly distributed along the loop.
Hydrostatic weighting has an effect on the loop visibility and on the vertical temperature structure of the solar corona (Reale, 1999; Aschwanden and Nitta, 2000). From the comparison of SoHO-CDS observations of active region loops with a static, isobaric loop model (Landini and Landi, 2002), a classical model was not able to reproduce the observations, but ad hoc assumptions are necessary (Brković et al., 2002; Landi and Landini, 2004). Loop static models were found to overestimate the footpoint emission by orders of magnitude and non-uniformity in the loop cross section, more specifically a significant decrease of the cross section near the footpoints, was proposed as the most likely solution to the discrepancy (Landi and Feldman, 2004, Section 4.1). On the same line, loop models with steady uniform heating were compared to X-ray loops and EUV moss in an active region core (Winebarger et al., 2008). A filling factor of 8% and loops that expand with height provided the best agreement with the intensity in two X-ray filters, though maintaining still some discrepancies with observations. A simple electrodynamic model was useful to evaluate the connection of electric currents and heating to the loop cross-section in a solar active region (Gontikakis et al., 2008).
The strength of scaling laws is certainly their simplicity and their easy and general application, even in the wider realm of stellar coronae. However, increasing evidence of dynamically heated, fine structured loops is indicating the need for improvements.
Structured (dynamic) loops
In the scenario of loops consisting of bundles of thin strands, each strand behaves as an independent atmosphere and can be described as an isolated loop itself. If the strands are numerous and heated independently, a loop can be globally maintained steady with a sequence of short heat pulses, each igniting a single or a few strands (nanoflares). In this case, although the loop remains steady on average for a long time, each strand has a continuously dynamic evolution. The evolution of a loop structure under the effect of an impulsive heating is well-known and studied from observations and from modeling (e.g., Nagai, 1980; Peres et al., 1982; Cheng et al., 1983; Nagai and Emslie, 1984; Fisher et al., 1985a,b,c; MacNeice, 1986; Betta et al., 2001), since it resembles the evolution of single coronal flaring loops. It is worth mentioning here that there have been attempts to model even flaring loops as consisting of several flaring strands (Hori et al., 1997, 1998; Reeves and Warren, 2002; Warren, 2006; Reale et al., 2012).
The evolution of single coronal loops or single loop strands subject to impulsive heating was summarized in the context of the diagnostics of stellar flares (Reale, 2007). A heat pulse injected in an inactive tenuous strand makes chromospheric plasma expand in the coronal section of the strand, and become hot and dense, X-ray bright, coronal plasma. After the end of the heat pulse, the plasma begins to cool slowly. In general, the plasma cooling is governed by the thermal conduction to the cool chromosphere and by radiation from optically thin conditions. In the following, we outline the evolution of the confined heated plasma into four phases, according to Reale (2007). Figure 15 tracks this evolution, which maps on the path drawn in the density-temperature diagram of Figure 16 (see also Jakimiec et al., 1992).
Phase I: From the start of the heat pulse to the temperature peak (heating). If the heat pulse is triggered in the coronal part of the loop, the heat is efficiently conducted down to the much cooler and denser chromosphere. The temperature rapidly increases in the whole loop, with a timescale given by the conduction time in a low density plasma (see below). This evolution changes only slightly if the heat pulse is deposited near the loop footpoints: the conduction front then propagates mainly upwards and on timescales not very different from the evaporation timescales, also because the heat conduction saturates (e.g., Klimchuk, 2006; Reale and Orlando, 2008). In this case the distinction from Phase II is not clearly marked.
Phase II: From the temperature peak to the end of the heat pulse (evaporation). The temperature settles to the maximum value (T0). The chromospheric plasma is strongly heated, expands upwards, and fills the loop with much denser plasma. This occurs both if the heating is conducted from the highest parts of the corona and if it released directly near the loop footpoints. The evaporation is explosive at first, with a timescale given by the isothermal sound crossing time (s), since the temperature is approximately uniform in the highly conductive corona:
$${\tau _{sd}} = {L \over {\sqrt {2{k_B}{T_0}/m} }} \approx 80{{{L_9}} \over {\sqrt {{T_{0,6}}}}},$$
where m is the average particle mass. After the evaporation front has reached the loop apex, the loop continues to fill more gently. The timescale during this more gradual evaporation is dictated by the time taken by the cooling rate to balance the heat input rate.
Phase III: From the end of the heat pulse to the density peak (conductive cooling). When the heat pulse stops, the plasma immediately starts to cool due to the efficient thermal conduction (e.g., Cargill and Klimchuk, 2004), with a timescale (s):
$${\tau _c} = {{3{n_c}{k_B}{T_0}{L^2}} \over {2/7\kappa T_0^{7/2}}} = {{10.5{n_c}{k_B}{L^2}} \over {\kappa T_0^{5/2}}} \approx 1500{{{n_9}L_9^2} \over {T_6^{5/2}}},$$
where nc (nc,9) is the particle density (109 cm−3) at the end of the heat pulse, the thermal conductivity is κ = 9 × 10−7 (c.g.s. units). Since the plasma is dense, we expect no saturation effects in this phase.
The heat stop time can be generally traced as the time at which the temperature begins to decrease significantly and monotonically. While the conduction cooling dominates, the plasma evaporation is still going on and the density increasing. The efficiency of radiation cooling increases as well, while the efficiency of conduction cooling decreases with the temperature.
Phase IV: From the density peak afterwards (Radiative cooling). As soon as the radiation cooling time becomes equal to the conduction cooling time (Cargill and Klimchuk, 2004), the density reaches its maximum, and the loop depletion starts, slowly at first and then progressively faster. The pressure begins to decrease inside the loop, and is no longer able to sustain the plasma. The radiation becomes the dominant cooling mechanism, with the following timescale (s):
$${\tau _r} = {{3{k_B}{T_M}} \over {{n_M}P(T)}} = {{3{k_B}{T_M}} \over {bT_M^\alpha {n_M}}} \approx 3000{{T_{M,6}^{3/2}} \over {{n_{M,9}}}},$$
where TM (TM,6) is the temperature at the time of the density maximum (106 K), nM (nM,9) the maximum density (109 cm−3), and P(T) the plasma emissivity per unit emission measure, expressed as:
$$P(T) = b{T^\alpha },$$
with b = 1.5 × 10−19 and α = −1/2. The density and the temperature both decrease monotonically.
The presence of significant residual heating could make the decay slower. In single loops, this can be diagnosed from the analysis of the slope of the decay path in the density-temperature diagram (Sylwester et al., 1993; Reale et al., 1997a). The free decay has a slope between 1.5 and 2 in a log density vs log temperature diagram; heated decay path is flatter down to a slope ∼ 0.5. In non-flaring loops, the effect of residual heating can be mimicked by the effect of a strong gravity component, as in long loops perpendicular to the solar surface. The dependence of the decay slope on the pressure scale height has been first studied in Reale et al. (1993) and, more recently, in terms of enthalpy flux by Bradshaw and Cargill (2010).
As clear from Figure 16 the path in this phase is totally below, or at most approaches, the QSS curve. This means that for a given temperature value the plasma density is higher than that expected for an equilibrium loop at that temperature, i.e., the plasma is "overdense". Evidence of such overdensity (Section 3.3.3) has been taken as an important indication of steadily pulse-heated loops.
Scheme of the evolution of temperature (T, thick solid line), X-ray emission, i.e., the light curve (LC, thinner solid line) and density (n, dashed line) in a loop strand ignited by a heat pulse. The strand evolution is divided into four phases (I, II, III, IV, see text for further details). Image reproduced with permission from Reale (2007), copyright by ESO.
Scheme of the evolution of pulse-heated loop plasma of Figure 15 in a density-temperature diagram (solid line). The four phases are labeled. The locus of the equilibrium loops is shown (dashed-dotted line, marked with QSS), as well as the evolution path with an extremely long heat pulse (dashed line) and the corresponding decay path (marked with EQ). Image adapted from Reale (2007), copyright by ESO.
This is the evolution of a loop strand ignited by a transient heat pulse. Important properties of the heated plasma can be obtained from the analysis of the evolution after the heating stops, i.e., when the plasma cools down.
Serio et al. (1991) derived a global thermodynamic timescale for the pure cooling of heated plasma confined in single coronal loops, which has been later refined to be (s) (Reale, 2007):
$${\tau _s} = 4.8 \times {10^{ - 4}}{L \over {\sqrt {{T_0}} }} = 500{{{L_9}} \over {\sqrt {{T_{0,6}}}}}.$$
This decay time was obtained assuming that the decay starts from equilibrium conditions, i.e., departing from the locus of the equilibrium loops with a given length (hereafter QSS line, Jakimiec et al., 1992) in Figure 16. It is, therefore, valid as long as there is no considerable contribution from the plasma draining to the energy balance. The link between the assumption of equilibrium and the plasma evolution is shown in Figure 16: if the heat pulse lasts long enough, Phase II extends to the right, and the heated loop asymptotically reaches equilibrium conditions, i.e., the horizontal line approaches the QSS line. If the decay starts from equilibrium conditions, Phase III is no longer present, and Phase II links directly to Phase IV. Therefore, there is no delay between the beginning of the temperature decay and the beginning of the density decay: the temperature and the density start to decrease simultaneously. Also, the decay will be dominated by radiative cooling, except at the very beginning (Serio et al., 1991).
The presence of Phase III implies a delay between the temperature peak and the density peak. This delay is often observed both in solar flares (e.g., Sylwester et al., 1993) and in stellar flares (e.g., van den Oord et al., 1988; van den Oord and Mewe, 1989; Favata et al., 2000; Maggio et al., 2000; Stelzer et al., 2002). The presence of this delay, whenever observed, is a signature of a relatively short heat pulse, or, in other words, of a decay starting from non-equilibrium conditions.
According to Reale (2007), the time taken by the loop to reach equilibrium conditions under the action of a constant heating is much longer than the sound crossing time [Eq. (11)], which rules the very initial plasma evaporation. As already mentioned, in the late rise phase the dynamics become much less important and the interplay between cooling and heating processes becomes dominant. The relevant timescale is therefore that reported in Eq. (14).
Hydrodynamic simulations confirm that the time required to reach full equilibrium scales as the loop cooling time (δs) and, as shown for instance in Figure 17 (see also Jakimiec et al., 1992), the time to reach flare steady-state equilibrium is:
$${t_{{\rm{eq}}}} \approx 2.3{\tau _s}.$$
Pressure evolution obtained from a hydrodynamic simulation of a loop strand ignited by heat pulses of different duration (0.5, 1, 3 times the loop decay time, see text) and with a continuous heating. Most of the rise phase can be reasonably described with a linear trend (dashed lines). Image reproduced with permission from Reale (2007), copyright by ESO.
For t ≥ teq, the density asymptotically approaches the equilibrium value:
$${n_0} = {{T_0^2} \over {2{a^3}{k_B}L}} = 1.3 \times {10^6}{{T_0^2} \over L},$$
where a = 1.4 × 103 (c.g.s. units), or
$${n_9} = 1.3{{T_{0,6}^2} \over {{L_9}}}.$$
If the heat pulse stops before the loop reaches equilibrium conditions, the loop plasma maximum density is lower than the value at equilibrium, i.e., the plasma is underdense (Cargill and Klimchuk, 2004, Section 4.4). Figure 17 shows that, after the initial impulsive evaporation on a timescale given by Eq. (11), the later progressive pressure growth can be approximated with a linear trend. Since the temperature is almost constant in this phase, we can approximate that the density increases linearly for most of the time. We can then estimate the value of the maximum density at the loop apex as:
$${n_M} \approx {n_0}{{{t_M}} \over {{t_{{\rm{eq}}}}}},$$
where tM is the time at which the density maximum occurs.
Phase III ranges between the time at which the heat pulse ends and the time of the density maximum. The latter is also the time at which the decay path crosses the locus of the equilibrium loops (QSS curve). According to Reale (2007), the temperature TM at which the maximum density occurs is:
$${T_M} = 9 \times {10^{ - 4}}{({n_M}L)^{1/2}}$$
$${T_{M,6}} = 0.9{({n_{M,9}}{L_9})^{1/2}}.$$
We can also derive the duration of Phase III, i.e., the time from the end of the heat pulse to the density maximum, as
$$\Delta {t_{0 - M}} \approx {\tau _c}\ln \psi,$$
$$\psi = {{{T_0}} \over {{T_M}}}$$
and δc [Eq. (12)] is computed for an appropriate value of the density nc. A good consistency with numerical simulations is obtained for nc = (2/3)nM.
By combining Eqs. (20) and (18) we obtain:
$${{\Delta {t_{0 - M}}} \over {{t_M}}} \approx 1.2\ln \psi,$$
which ranges between 0.2 and 0.8 for typical values of ψ (1.2–2).
These scalings are related to the evolution of a single strand under the effect of a local heat pulse. The strands are below the current instrument spatial resolution and, therefore, we have to consider that, if this scenario is valid, we see the envelope of a collection of small scale events. The characteristics of the single heat pulses become, therefore, even more difficult to diagnose, and the question of their frequency, distribution and size remains open. Also from the point of view of the modeling, a detailed description of a multistrand loop implies a much more complex and demanding effort. A possible approach is to literally build a collection of 1-D loop models, each with an independent evolution (Guarrasi et al., 2010). One common approach so far has been to simulate anyhow the evolution of a single strand, and to assume that, in the presence of a multitude of such strands, in the steady state we would see at least one strand at any step of the strand evolution. In other words, a collection of nanoflare-heated strands can be described as a whole with the time-average of the evolution of a single strand (Warren et al., 2002, 2003; Winebarger et al., 2003b,a, see also Section 4.2). Another issue to be explored is whether it is possible, and to what extent, to describe a collection of independently-evolving strands as a single effective evolving loop. For instance, how does the evolution of a single loop where the heating is decreasing slowly compare to the evolution of a collection of independently heated strands, with a decreasing rate of ignition? To what extent do we expect coherence and how is it connected to the degree of global coherence of the loop heating? Is there any kind of transverse coherence or ordered ignition of the strands? It is probably reasonable to describe a multi-stranded loop as a single "effective" loop if we can assume that the plasma loses memory of its previous history. This certainly occurs in late phases of the evolution when the cooling has been going on for a long time.
The description and role of fine structuring of coronal loops is certainly a challenge for coronal physics, also on the side of modeling, essentially because we have few constraints from observations (Section 3.2.2). Small-scale structuring is already involved in the magnetic carpet scenario and flux-tube tectonics model (Priest et al., 2002, see also Section 4.4). One of the first times that the internal structuring of coronal loops have been invoked in a modeling context was for the problem of the interpretation of the uniform filter ratio distribution detected with TRACE along warm loops. Standard models of single hydrostatic loops with uniform heating were soon found to be unable to explain such indication of uniform temperature distribution (Lenz et al., 1999). A uniform filter ratio could be reproduced by the superposition of several thin hydrostatic strands at different temperatures (Reale and Peres, 2000). In alternative, also a model of long loops heated at the footpoints leads to mostly isothermal loops (Aschwanden, 2001). The problem with this model is that footpoint-heated loops (with heating scale height less than 1/3 of the loop half-length) had been shown to be thermally unstable (Mendoza-Briceno and Hood, 1997) and, therefore they cannot be long-lived, as instead observed. A further alternative is to explain observations with steady non-static loops, i.e., with significant flows inside (Winebarger et al., 2001, 2002c, see below). Also this hypothesis does not seem to answer the question (Patsourakos et al., 2004).
A first step to modeling fine-structured loops is to use multistrand static models. Such models show some substantial inconsistencies with observations, e.g., in general they predict too large loop cross sections (Reale and Peres, 2000). Such strands are conceptually different from the thin strands predicted in the nanoflare scenario (Parker, 1988), which imply a highly dynamic evolution due to pulsed-heating. The nanoflare scenario is approached in multi-thread loop models, convolving the independent hydrodynamic evolution of the plasma confined in each pulse-heated strand (see Section 4.3). These are able to match some more features of the evolution of warm loops observed with TRACE (Warren et al., 2002, 2003; Winebarger et al., 2003b,a). According to detailed hydrodynamic loop modeling, an ensemble of independently heated strands can be significantly brighter than a static uniformly heated loop and would have a flat filter ratio temperature when observed with TRACE (Warren et al., 2002). As an extension, time-dependent hydrodynamic modeling of an evolving active region loop observed with TRACE showed that a loop made as a set of small-scale, impulsively heated strands can generally reproduce the spatial and temporal properties of the observed loops, such as a delay between the appearance of the loop in different filters (Warren et al., 2003). An evolution of this approach was to model an entire active region for comparison with a SoHO/EIT observation (Warren and Winebarger, 2006); the modeling includes extrapolating the magnetic field and populating the field lines with solutions to the hydrostatic loop equations assuming steady, uniform heating. The result was the link between the heating rate and the magnetic field and size of the structures, but there were also significant discrepancies with the observed EIT emission.
More recently, modeling a loop system as a collection of thin unresolved strand with pulsed heating has been used to explain why active regions look fuzzier in harder energy bands, i.e. X-rays, and/or hotter spectral lines, e.g., Fe xvi, sensitive to high temperatures (∼ 3 MK) (Tripathi et al., 2009, Section 3.3.2). Short (∼ 1 min) pulses with flare-like intensity (∼ 10 MK) are able to produce loops with high filling factors at ∼ 3 MK and lower filling factors at ∼ 1 MK (Guarrasi et al., 2010). The basic reason is that in the dynamic evolution of each strand, the plasma spends a relatively longer time and with a high emission measure at temperature about 3 MK. The consequent prediction that loops should show filamented emission for temperature > 3 MK has received confirmations by observations of active region cores in the 94 Å channel with SDO/AIA (Reale et al., 2011) and in the Ca xvii and Fe xviii EUV lines (Testa and Reale, 2012), although the temperature of the emitting plasma is still debated (Teriaca et al., 2012). Low filling factors of warm loops have been predicted also by full MHD modeling (Dahlburg et al., 2012).
The description of loops as bundles of strands applies also to models that include heating by the dissipation of MHD waves (Alfvén/ion-cyclotron waves — particles). One such model addressed the evidence of flat TRACE/EIT filter-ratios along loops that were explained by the multi-filament loop structure (Bourouaine and Marsch, 2010). Transverse oscillations and flows were observed in multi-stranded loops (Ofman and Wang, 2008; Wang et al., 2012). Multi-stranded loop models were used in 3D MHD studies of transverse loop oscillations (e.g., Ofman, 2009) and in MHD normal mode analysis (e.g., Luna et al., 2010).
State-of-the-art approaches to the study of multi-stranded loops are based on the concept that each fibril is independent of the others and that the heating is released randomly presumably with a power-law distribution. Within the limitations of idealized loop models (without magnetic twist, time-dependent thread cross-sections, or oscillation), the coronal loops might then be described as self-organized critical systems with no characteristic timescales (e.g., Bak et al., 1989; Lu and Hamilton, 1991; Charbonneau et al., 2001). This model has had a practical realization specific to reproduce soft X-ray steady-state loops (López Fuentes and Klimchuk, 2010) and is able to reproduced the loop light curves observed, for instance, with GOES/SXI. Another interesting approach is to use artificial neural networks (Tajfirouze and Safari, 2012) as mentioned in Section 3.4.
A generalization of static models of loops (Section 4.1.1) is represented by models of loops with stationary flows, driven by a pressure imbalance between the footpoints (siphon flows). The properties of siphon flows have been studied by several authors (Cargill and Priest, 1980; Priest, 1981; Noci, 1981; Borrini and Noci, 1982; Antiochos, 1984; Thomas, 1988; Montesinos and Thomas, 1989; Noci et al., 1989; Thomas and Montesinos, 1990; Spadaro et al., 1990; Thomas and Montesinos, 1991; Peres et al., 1992; Montesinos and Thomas, 1993). A complete detailed model of loop siphon flows was developed and used to explore the space of the solutions and to derive an extension of RTV scaling laws to loops containing subsonic flows (Orlando et al., 1995b).
Critical and supersonic flows create the conditions for the presence of stationary shocks in coronal loops (Figure 18). The shock position depends on the volumetric heating rate of the loop (Orlando et al., 1995a). The presence of massive flows may alter the line emission with respect to static plasma, because of the delay of the moving plasma to settle to ionization equilibrium (Golub and Herant, 1989). Including the effect of ionization non-equilibrium, the UV lines are predicted to be blue-shifted by loop models (Spadaro et al., 1990). So non-equilibrium emission from flows cannot explain the observed dominant redshifts (Section 3.5). Non-equilibrium of ionization in UV line emission can be driven by shocked siphon flows (Orlando and Peres, 1999) and by reconnection flows (Imada et al., 2011).
Example of solutions of a siphon flow loop model including a shock. Image reproduced with permission from Orlando and Peres (1999), copyright by Elsevier.
In the 1990s, modeling efforts were devoted to explain specifically the extensive evidence of red-shifted UV lines on the solar disk. A hydrodynamic loop model including the effects of non-equilibrium of ionization showed that the redshifts might be produced by downward propagating acoustic waves, possibly stimulated by nanoflares (Hansteen, 1993). Two-dimensional hydrodynamic simulations showed that the UV redshifts might be produced by downdrafts driven by radiatively-cooling condensations in the solar transition region (Reale et al., 1996, 1997b). Predicted redshifts range from those typical of quiet Sun to active regions and may occur more easily in the higher pressure plasma, typical of active regions.
Explosions below the corona were explored to drive flows (Teriaca et al., 1999; Sarro et al., 1999) in magnetic loops around the O vi and C iv formation temperature. The observed redshift of mid-low transition region lines as well as the blueshift observed in low coronal lines (T > 6 × 105 K) were compared to numerical simulations of the response of the solar atmosphere to an energy perturbation of 4 × 1024 erg, including non-equilibrium of ionization (Teriaca et al., 1999). Performing an integration over the entire period of simulations, they found a redshift in C iv, and a blueshift in O vi and Ne viii, of a few km s−1, in reasonable agreement with observations. A similar idea was applied to make predictions about the presence or absence of non-thermal broadening in several spectral lines (e.g., Ne viii, Mg x, Fe xvii) due to nanoflare-driven chromospheric evaporation (Patsourakos and Klimchuk, 2006). Clearly, the occurrence of such effects in the lines depends considerably on the choice of the heat pulse parameters. Therefore, more constraints are needed to make the whole model more consistent. In other words, modeling should address specific observations to provide more conclusive results.
Theoretical reasons indicate that flows should be invariably present in coronal loop systems, although they may not be necessarily important in the global loop momentum and energy budget. For instance, it has been shown that the presence of at least moderate flows is necessary to explain why we actually see the loops (Lenz, 2004). The loop emission and detection is in fact due to the emission from heavy ions, like Fe. In hydrostatic equilibrium conditions, gravitational settlement should keep the emitting elements low on the solar surface, and we should not be able to see but the loop footpoints. Instead, detailed modeling shows that flows of few km s−1 are enough to drag ions high in the corona by Coulomb coupling and to enhance coronal ion abundances by orders of magnitudes. Incidentally, the same modeling shows that, for the same mechanisms, no chemical fractionation of coronal plasma with respect to photospheric composition as a function of the element first ionization potential (FIP) should be present in coronal loops.
Other studies address instead the relative unimportance of flows in coronal loops. In particular, as already mentioned in Section 3.2.2, steady hydrodynamic loop modeling (i.e., assuming equilibrium condition and, therefore, dropping the time-dependent terms in Eqs. (4), (5), and (6)), showed that flows may not be able to explain the evidence of isothermal loops (Patsourakos et al., 2004), as instead proposed by Winebarger et al. (2002c). Flows are able to enhance its density to the levels typically diagnosed from TRACE observations, but they also produce an inversion of the temperature distribution and a structured filter ratio, not observed.
Plasma cooling is a mechanism that may drive significant downflows in a loop (e.g., Bradshaw and Cargill, 2005, 2010). Catastrophic cooling in loops (Müller et al., 2004, 2005) was proposed to explain the evidence of propagating intensity variations observed in the He ii 304 Å line with SoHO/EIT (De Groof et al., 2004, Section 3.5). Two possible driving mechanisms had been proposed: slow magnetoacoustic waves or blobs of cool downfalling plasma. A model of cool downfalling blob triggered in a thermally-unstable loop heated at the footpoints gave a qualitative agreement with measured speeds and predicted a significant braking in the high-pressure transition region, to be checked in future high cadence observations in cool lines.
Plasma waves have been more recently proposed to have an important role in driving flows within loops. Acoustic waves excited by heat pulses at the chromospheric loop footpoints and damped by thermal conduction in corona are possible candidates (Taroyan et al., 2005). Even more attention received the propagation of Alfvén waves in coronal loops. Hydrodynamic loop modeling showed that Alfvén waves deposit significant momentum in the plasma, and that steady state conditions with significant flows and relatively high density can be reached (O'Neill and Li, 2005). Analogous results were obtained independently with a different approach: considering a wind-like model to describe a long isothermal loop, Grappin et al. (2003, 2005) showed that the waves can drive pressure variations along the loop which trigger siphon flows. Alfvén disturbances have been more recently shown to be amplified by the presence of loop flows (Taroyan, 2009).
Large-scale MHD models have also addressed the presence of flows in the low corona. These models show that heat pulses released low in the corona in places of strong magnetic field braiding trigger downflows and slight upflows (Figure 19, Hansteen et al., 2010; Zacharias et al., 2011). The corresponding Doppler-shifts are similar to those often observed (see Section 3.5). Most of the mass circulating across the transition region is probably confined in very short loops (∼ 2 × 108 cm) (Guerreiro et al., 2013).
Maps of intensity (left), Doppler shift (middle) and line width (right) in the CIV line in a 3-D MHD simulation of a box of the upper solar atmosphere. The velocity scale is from −40 km s−1 (blue) to 40 km s−1 (red). Line widths range from narrow black to wide yellow/red with a maximum of 51 km s−1. The average line-shift in the CIV line is 6.6 km s−1 (redshift is positive). Image reproduced with permission from Hansteen et al. (2010), copyright by AAS.
Large-scale chromospheric upflows (type-II spicules, see Section 3.5) are explored as viable mechanisms of mass and energy supply to coronal loops with loop modeling (Judge et al., 2012). However, theoretical estimates suggest that a corona dominated by this scenario would lead to large discrepancies with observations, therefore confining the possible action of this mechanism in a limited number of structures (Klimchuk, 2012).
The problem of what heats coronal loops is essentially the problem of coronal heating, and is a central issue in the whole solar physics. Although the magnetic origin of coronal heating has been well-established since the very first X-ray observations of the corona, the detailed mechanism of conversion of magnetic energy into thermal energy is still under intense debate, because a series of physical effects conspire to make the mechanism intrinsically elusive.
Klimchuk (2006) splits the heating problem into six steps: the identification of the source of energy, its conversion into heat, the plasma response to the heating, the spectrum of the emitted radiation, the final signature in observables. Outside of analytical approaches, the source and conversion of energy are typically studied in detail by means of multi-dimensional full MHD models (e.g., Gudiksen and Nordlund, 2005), which, however, are still not able to provide exhaustive predictions on the plasma response and complete diagnostics on observables. On the other hand, the plasma response is the main target of loop hydrodynamic models, which, instead, are not able to treat the heating problem in a self-consistent way (Section 4.1).
In the investigation of the source of energy, the magnetic field plays an active role in heating the coronal loops (Golub et al., 1980). The observation of a magnetic carpet (Schrijver et al., 1998) suggests that current sheets at the boundary of the carpet cells can dissipate and heat the corona, acting in analogy to geophysical plate tectonics (Priest et al., 2002). A common scenario is that the field lines are wound continuously by the photospheric convective motions and the generated non-potential component is dissipated into heating. Several studies were devoted to the connection and scaling of the magnetic energy to the coronal energy content (Golub et al., 1982) and to the rate of energy release through reconnection (Galeev et al., 1981).
It is known that twisted loops can become kink unstable above a critical twist; however, according to Parker (1988), as soon as the magnetic field lines are tangled at an angle of ∼ 15°, enough magnetic energy can be released to power a loop by rapid reconnection across the spontaneous tangential discontinuities. The dissipation destroys the cross-component of the magnetic field as rapidly as it is produced by the motion of the footpoints, reaching a steady state. The twisting of coronal loops has been studied in the framework of kink instability (Hood and Priest, 1979b; Velli et al., 1990) with resistive effects (Baty, 2000), of loop cross-section (Klimchuk, 2000), of flux emergence (Hood et al., 2009a), of cellular automaton loop modeling (López Fuentes and Klimchuk, 2010), and in connection with transverse oscillations (Ofman, 2009). Loop twisting or braiding and kink instability can lead to magnetic reconnection with the formation and fragmentation of thin current sheets and their dissipation through resistivity (Hood et al., 2009b; Wilmot-Smith et al., 2010; Bareford et al., 2010; Pontin et al., 2011; Wilmot-Smith et al., 2011; Bareford et al., 2011, 2013).
The comparison of TRACE and Hinode time sequences of active region loops with magnetic field reconstruction models have allowed to measure the changes in the magnetic topology and energy with the time. A high variability corresponds to a high number of magnetic separation lines (Priest et al., 2002) where the energy can be released in short timescales (Lee et al., 2010). A model of nonlinear force-free field traced that magnetic energy is built up in the core of active regions by small-scale photospheric motions (Mackay et al., 2011). The photospheric motions are therefore the ultimate energy source and stress the field or generate waves depending on whether the timescale of the motion is long or short compared to the end-to-end Alfvén travel time.
Following Klimchuk (2006), dissipation of magnetic stresses can be referred to as direct current (DC) heating, and dissipation of waves as alternating current (AC) heating.
The question of the conversion of the magnetic energy into heat is also challenging, because dissipation is predicted to occur on very small scales or large gradients in the corona by classical theory, although there are some indications of anomalously high dissipation coefficients (Martens et al., 1985; Nakariakov et al., 1999; Fuentes-Fernández et al., 2012). As reviewed by Klimchuk (2006), large gradients may be produced in various ways, involving either magnetic field patterns and their evolution, magnetic instabilities such as the kink instability, or velocity pattern, such as turbulence. For waves, resonance absorption and phase mixing may be additional viable mechanisms (see Section 4.4.2).
The problem of plasma response to heating has been kept historically well separated from the primary heating origin, although some attempts have been made to couple them. For instance, in Reale et al. (2005) the time-dependent distribution of energy dissipation along the loop obtained from a hybrid shell model was used as heating input of a time-dependent hydrodynamic loop model (see below). A similar concept was applied to search for signatures of turbulent heating in UV spectral lines (Parenti et al., 2006).
As already mentioned, studies using steady-state or time-dependent purely hydrodynamic loop modeling have addressed primarily the plasma response to heating, and also its radiative emission and the detailed comparison with observations. A forward-modeling including all these steps was performed on a TRACE observation of a brightening coronal loop (Reale et al., 2000a,b, see also Section 3.4). The analysis was used to set up the parameters for the forward modeling, and to run loop hydrodynamic simulations with various assumptions on the heating location and time dependence. The comparison of the TRACE emission predicted by the simulations with the measured one constrained the heat pulse to be short, much less than the observed loop rise phase, and intense, appropriate for a 3 MK loop, and its location to be probably midway between the apex and one of the footpoints.
The investigation of the heating mechanisms through the plasma response is difficult for a variety of reasons. For instance, the problem of background subtraction can be crucial in the comparison with observations, as shown by the three analyses of the same large loop structure observed with Yohkoh/SXT on the solar limb, mentioned in Section 3.3. More specifically, Priest et al. (2000) tried to deduce the form of the heating from Yohkoh observed temperature profiles and found that a uniform heating best describes the data, if the temperature is obtained from the ratio of the total filter intensities, with no background subtraction. Aschwanden (2001) split the measured emission into two components and found a better agreement with heating deposited at the loop footpoints. Reale (2002b) revisited the analysis of the same loop system, considering conventional hydrostatic single-loop models and accounting accurately for an unstructured background contribution. With forward-modeling, i.e., synthesizing from the model observable quantities to be compared directly with the data, background-subtracted data are fitted with acceptable statistical significance by a model of relatively hot loop (∼ 3.7 MK) heated at the apex, but it was pointed out the importance of background subtraction and the necessity of more specialized observations to address this question. More diagnostic techniques to compare models with observations were proposed afterwards (e.g., Landi and Landini, 2005).
Independently of the adopted numerical or theoretical tool, many studies have been addressing the mechanisms of coronal loop heating clearly distinguishing between the two main classes, i.e., DC heating through moderate and frequent explosive events, named nanoflares (e.g., Parker, 1988) and AC heating via Alfvén waves (e.g., Litwin and Rosner, 1998).
DC heating
Heating by nanoflares has a long history as a possible candidate to explain the heating of the solar corona, and, in particular, of the coronal loops (e.g., Peres et al., 1993; Cargill, 1993; Kopp and Poletto, 1993; Shimizu, 1995; Judge et al., 1998; Mitra-Kraev and Benz, 2001; Katsukawa and Tsuneta, 2001; Mendoza-Briceño et al., 2002; Warren et al., 2002, 2003; Spadaro et al., 2003; Cargill and Klimchuk, 1997, 2004; Müller et al., 2004; Testa et al., 2005; Reale et al., 2005; Taroyan et al., 2006; Vekstein, 2009). The coronal tectonics model (Priest et al., 2002) is an updated version of Parker's nanoflare theory, for which the motions of photospheric footpoints continually build up current sheets along the separatrix boundaries of the flux coming from each microscopic source (Priest, 2011).
Models of loops made of thousands of nanoflare-heated strands were developed and applied to derive detailed predictions (Cargill, 1994). In particular, whereas the loop total emission measure distribution should steepen above the canonical T1.5 (Jordan, 1980; Orlando et al., 2000; Peres et al., 2001) dependence for temperature above 1 MK. Moreover, it was stressed the importance of the dependence of effects such as the plasma dynamics (filling and draining) on the loop filling factor driven by the elemental heat pulse size (Section 4.1.2). The nanoflare model was early applied to the heating of coronal loops observed by Yohkoh (Cargill and Klimchuk, 1997). A good match was found only for hot (4 MK) loops, with filling factors less than 0.1, so that it was hypothesized the existence of two distinct classes of hot loops.
Although there is evidence of intermittent heating episodes, it has been questioned whether and to what extent nanoflares are able to provide enough energy to heat the corona (e.g., Aschwanden, 1999). On the other hand, loop models with nanoflares, and, in particular, those considering a prescribed random time distribution of the pulses deposited at the footpoints of multi-stranded loops have been able to explain several features of loop observations, for instance, of warm loops from TRACE (Warren et al., 2002, 2003, see Section 3.2.2).
Hydrodynamic loop modeling showed also that different distributions of the heat pulses along the loop have limited effects on the observable quantities (Patsourakos and Klimchuk, 2005), because most of the differences occur at the beginning of the heat deposition, when the emission measure is low, while later the loop loses memory of the heat distribution (see also Winebarger and Warren, 2004). An application of both static and impulsive models to solar active regions showed that the latter ones are able to simultaneously reproduce EUV and SXR loops in active regions, and to predict radial intensity variations consistent with the localized core and extended emissions (Patsourakos and Klimchuk, 2008). As a further improvement, the simulation of an entire active region with an impulsive heating model reproduced the total observed soft X-ray emission in all of the Yohkoh/SXT filters (Warren and Winebarger, 2007). However, once again, at EUV wavelengths the agreement between the simulation and the observation was only partial.
Nanoflares have been studied also in the framework of stellar coronae. Intermittent heating by relatively intense nanoflares deposited at the loop footpoints makes the loop stable on long timescales (Testa et al., 2005; Mendoza-Briceño et al., 2005) (loops infrequently heated at the footpoints are unstable) and, on the other hand, produces a well-defined peak in the average DEM of the loop, similar to that derived from the DEM reconstruction of active stars (Cargill, 1994; Testa et al., 2005). Therefore, this is an alternative way to obtain a steep temperature dependence of the loop emission measure distribution in the low temperature range.
An alternative approach to study nanoflare heating is to analyze intensity fluctuations (Shimizu and Tsuneta, 1997; Vekstein and Katsukawa, 2000; Katsukawa and Tsuneta, 2001; Vekstein and Jain, 2003) and to derive their occurrence distribution (Sakamoto et al., 2008, 2009). From the width of the distributions and autocorrelation functions, it has been suggested that nanoflare signatures are more easily found in observations of warm TRACE loops than of hot Yohkoh/SXT loops. It is to be investigated whether the results change after relaxing the assumption of temperature-independent distribution widths. Also other variability analysis of TRACE observations was found able to put constraints on loop heating. In particular, in TRACE observations, the lack of observable warm loops and of significant variations in the moss regions implies that the heating in the hot moss loops should not be truly flare-like, but instead quasi-steady and that the heating magnitude is only weakly varying (Antiochos et al., 2003; Warren et al., 2010b).
An analogous approach is to analyze the intensity distributions. The distribution of impulsive events vs their number in the solar and stellar corona is typically described with a power law. The slope of the power law is a critical parameter to establish weather such events are able to heat the solar corona (Hudson, 1991). In particular, a power law index of 2 is the critical value above or below which flare-like events may be able or unable, respectively, to power the whole corona (e.g., Aschwanden, 1999; Bareford et al., 2010; Tajfirouze and Safari, 2012). Unfortunately, due to the faintness of the events, the distribution of weak events is particularly difficult to derive and might even be separate from that of proper flares and microflares. A hydrodynamic model was used to simulate the UV emission of a loop system heated by nanoflares on small, spatially unresolved scales (Parenti and Young, 2008). The simulations confirmed previous results that several spectral lines have an intensity distribution that follows a power-law, in a similar way to the heating function (Hudson, 1991). However, only the high temperature lines best preserve the heating function's power law index (especially Fe xix).
The shape of the emission measure distribution is, in principle, a powerful tool to constrain the heating mechanisms. The width in temperature provides information about the temporal distribution of a discontinuous heating mechanism: for a broad (multi-thermal) distribution the simultaneous presence of many temperature components along the line of sight may be produced by many strands randomly heated for a short time and then spending most of the time in the cooling, thus "crossing" many different temperatures. A peaked distribution, i.e., plasma closer to an isothermal condition, indicates a plasma sustained longer at a certain temperature, with a heating much more uniform in time than for multi-thermal loops. A semi-analytical loop model of a cycling heating/cooling (Cargill and Klimchuk, 2004) naturally led to hot-underdense/warm-overdense loop (Section 4.1.2), as observed (Winebarger et al., 2003b, Section 3.3.3), and showed that the width of the DEM of a nanoflare-heated loop can depend on the number of strands which compose the loop: a relatively flat DEM or a peaked (isothermal) DEM are obtained with strands of diameter about 15 km or about 150 km, respectively. This is of relevance for the diagnostics both of the loop fine structure (Section 3.2.2) and of the DEM reconstruction (Section 3.3). In general, a broad emission measure distribution would be a signature of a low-frequency heating, whereas a peaked distribution would be a signature of high-frequency heating (Warren et al., 2010b; Susino et al., 2010). The timescale is basically dictated by the cooling times. High frequency heating seems to explain several debated evidence in warm loops of active regions, i.e., loop lifetime, high density, and the narrow differential emission measure, but not the higher temperature loops detected in the X-rays (Warren et al., 2010a). It is remarked that overdense plasma would be emphasized also by deviations from equilibrium of ionization due to impulsive heating (Bradshaw and Klimchuk, 2011), and that the predicted cool side of the emission measure distribution might steepen using updated radiative losses (Reale et al., 2012). However, the constraints on heating from emission measure distribution are largely debated; broad and peaked emission measure distributions of hot 3 MK loops might be compatible with steady heating models (Winebarger et al., 2011). This debate has been specifically addressed and all evidence has been collected and analysed through loop modeling. In particular, the consistency of the DEM slopes on the cool side with low frequency nanoflare heating has been tested. It has been found that, although heating by single pulses might explain the majority of DEMs derived in the literature (Bradshaw et al., 2010) and that trains of nanoflares might explain practically all of them (Reep et al., 2013), the uncertainties in the data analysis and DEM reconstruction are too large reach conclusive answers. Radiative losses are important to the existence of small and cool loops (height ≤ 8 Mm, T ≤ 105 K) that determine the cool side of the emission measure distribution (Sasso et al., 2012).
Support to dynamic heating comes from modeling loops with steady heating located at the footpoints. It is known that such heating is not able to keep loop atmosphere in steady equilibrium because they are thermally unstable (Antiochos and Klimchuk, 1991; Antiochos et al., 1999; Müller et al., 2004; Karpen and Antiochos, 2008; Mok et al., 2008). Catastrophic cooling occurs along the loops some time after the heating is switched on and might explain deviations from hydrostatic equilibrium, and some features of the light curves measured in the EUV band (Peter et al., 2012). However, the timescales required by this scenario seem too long compared to the measured loop lifetimes (Klimchuk et al., 2010).
AC heating
Loop oscillations, modes and wave propagation deserve a review by themselves, and are outside of the scope of the present one. Here we account for some aspects which are relevant for the loop heating. A review of coronal waves and oscillations can be found in Nakariakov and Verwichte (2005). New observations from SDO AIA provide ample evidence of wave activity in the solar corona (Title, 2010), as reported on in Section 3.5.2.
As reviewed by Klimchuk (2006), MHD waves of many types are generated in the photosphere, e.g., acoustic, Alfvén, fast and slow magnetosonic waves. Propagating upwards, the waves may transfer energy to the coronal part of the loops. The question is what fraction of the wave flux is able to pass through the very steep density and temperature gradients in the transition region. Acoustic and slow-mode waves form shocks and are strongly damped, fast-mode waves are strongly refracted and reflected (Narain and Ulmschneider, 1996).
Ionson (1978, 1982, 1983) devised an LRC equivalent circuit to show the potential importance of AC processes to heat the corona. Hollweg (1984) used a dissipation length formalism to propose resonance absorption of Alfvén waves as a potential coronal heating mechanism. A loop may be considered as a high-quality resonance cavity for hydromagnetic waves. Turbulent photospheric motions can excite small-scale waves. Most Alfvén waves are strongly reflected in the chromosphere and transition region, where the Alfvén speed increases dramatically with height. Significant transmission is possible only within narrow frequency bands centered on discrete values where loop resonance conditions are satisfied (Hollweg, 1981, 1984; Ionson, 1982). The waves resonate as a global mode and dissipate efficiently when their frequency is near the local Alfvén waves frequency ωA ≈ 2πυ/L. By solving the linearized MHD equations, Davila (1987) showed that this mechanism can potentially heat the corona, as further supported by numerical solution of MHD equations for low beta plasma (Steinolfson and Davila, 1993), and although Parker (1991) argued that Alfvén waves are difficult to be generated by solar convection.
Hollweg (1985) estimated that enough flux may pass through the base of long (> 1010 cm) active region loops to provide their heating, but shorter loops are a problem, since they have higher resonance frequencies and the photospheric power spectrum is believed to decrease exponentially with frequency in this range. Litwin and Rosner (1998) suggested that short loops may transmit waves with low frequencies, as long as the field is sufficiently twisted. Hollweg and Yang (1988) proposed that Alfvén resonance can pump energy out of the surface wave into thin layers surrounding the resonant field lines and that the energy can be distributed by an eddy viscosity throughout large portions of coronal active region loops.
Waves may be generated directly in the corona, and some evidence was found (e.g., Nakariakov et al., 1999; Aschwanden et al., 1999a; Berghmans and Clette, 1999; De Moortel et al., 2002). It is unclear whether coronal waves carry a sufficient energy flux to heat the plasma (Tomczyk et al., 2007). Ofman et al. (1995) studied the dependence on the wavenumber for comparison with observations of loop oscillations and found partial agreement with velocity amplitudes measured from non-thermal broadening of soft X-ray lines. The observed non-thermal broadening of transition region and coronal spectral lines implies fluxes that may be sufficient to heat both the quiet Sun and active regions, but it is unclear whether the waves are efficiently dissipated (Porter et al., 1994). Furthermore, the non-thermal line broadening could be produced by unresolved loop flows that are unrelated to waves (e.g., Patsourakos and Klimchuk, 2006). Ofman et al. (1998) included inhomogeneous density structure and found that a broadband wave spectrum becomes necessary for efficient resonance and that it fragments the loop into many density layers that resemble the multistrand concept. The heat deposition by the resonance of Alfvén waves in a loop was investigated by O'Neill and Li (2005). A multi-strand loop model where the heating is due to the dissipation of MHD waves was applied to explain filter-ratios along loops (Bourouaine and Marsch, 2010, see Sections 3.3.3, 4.2). By assuming a functional form first proposed by Hollweg (1986), hydrodynamic loop modeling showed that, depending on the model parameters, heating by Alfvén waves leads to different classes of loop solutions, such as the isothermal cool loops indicated by TRACE, or the hot loops observed with Yohkoh/SXT. Specific diagnostics are still to be defined for the comparison with observations.
Efficient wave dissipation may be allowed by enhanced dissipation coefficients inferred from fast damping of flaring loop oscillations in the corona (Nakariakov et al., 1999), but the same effect may also favor efficient magnetic reconnection in nanoflares. Alfvén waves required for resonant absorption are relatively high frequency waves. Evidence for lower frequency Alfvén waves has been found in the chromosphere with the Hinode SOT (De Pontieu et al., 2007b). Such waves may supply energy in the corona even outside of resonance with different mechanisms to be explored with modeling. Among dissipation mechanisms phase mixing with enhanced resistivity was suggested by Ofman and Aschwanden (2002) and supported by the analysis of Ofman and Wang (2008). Also multistrand structure has been recognized to be important in possible wave dissipation and loop twisting (Ofman, 2009). Long-period (> 10 s) chromospheric kink waves might propagate into the corona by transformation into Alfvén waves and be dissipated there (Soler et al., 2012).
In the more general context of coronal heating, after several previous works, follow-up modeling and analytical effort has been devoted to the dissipation of Alfvén waves through phase mixing (e.g., Heyvaerts and Priest, 1983; Nakariakov et al., 1997; Botha et al., 2000; Ofman and Aschwanden, 2002) and ponderomotive force (Verwichte et al., 1999) in a nonideal inhomogeneous medium, finding effects on very long timescales (> 1 month, McLaughlin et al., 2011).
Intensity disturbances propagating along active region loops at speeds above 100 km s−1 were detected with TRACE and interpreted as slow magnetosonic waves (Nakariakov et al., 2000). These waves probably originate from the underlying oscillations, i.e., the 3-minute chromospheric/transition-region oscillations in sunspots and the 5-minute solar global oscillations (p-modes). Slow magnetosonic waves might be good candidates as coronal heating sources according to a detailed model, including the effect of chromosphere and transition region and of the radiative losses in the corona (Beliën et al., 1999). Such waves might be generated directly from upward propagating Alfvén waves. Contrary conclusions, in favor of fast magnetosonic waves, have been also obtained, but with much simpler modeling (Pekünlü et al., 2001). Slow magnetosonic waves with periods of about 5 minutes have been more recently detected in the transition region and coronal emission lines by Hinode/EIS at the footpoint of a coronal loop rooted at plage, but found to carry not enough energy to heat the corona (Wang et al., 2009). Slow magnetosonic waves might be coupled to upflows and produced by impulsive events at the base of active region loops (Ofman et al., 2012).
Investigation of AC heating has been made also through comparison with DC heating. Antolin et al. (2008) compared observational signatures of coronal heating by Alfvén waves and nanoflares using two coronal loop models and found that Hinode XRT intensity histograms display power-law distributions whose indices differ considerably, to be checked against observations. Lundquist et al. (2008a,b) applied a method for predicting active region coronal emissions using magnetic field measurements and a chosen heating relationship to 10 active regions. With their forward-modeling, they found a volumetric coronal heating rate proportional to magnetic field and inversely proportional to field-line loop length, which seems to point to, although not conclusively, the steady-state scaling of two heating mechanisms: van Ballegooijen's current layers theory (van Ballegooijen, 1986), taken in the AC limit, and Parker's critical angle mechanism (Parker, 1988), in the case where the angle of misalignment is a twist angle. As interesting points of contacts with the models of impulsive heating, it has been proposed that loops can be heated impulsively by Alfvén waves dissipated on reasonable timescales through turbulent cascade that develops when the waves are transmitted from the photosphere to the corona (van Ballegooijen et al., 2011; Asgari-Targhi et al., 2013), using reduced MHD equations.
Large-scale modeling
Coronal loops have been studied also with models that include the magnetic field. We can distinguish several levels of treatment of the magnetic effects. One basic level is to use global scalings to discriminate between different heating mechanisms. Based on a previous study of the plasma parameters and the magnetic flux density (Mandrini et al., 2000), Démoulin et al. (2003) derived the dependence of the mean coronal heating rate on the magnetic flux density from the analysis of an active region. By using the scaling laws of coronal loops, they found that models based on the dissipation of stressed, current-carrying magnetic fields are in better agreement with the observations than models that attribute coronal heating to the dissipation of MHD waves injected at the base of the corona. A similar approach was applied to the whole corona, by populating magnetic field lines taken from observed magnetograms with quasi-static loop atmospheres (Schrijver et al., 2004). The best match to X-ray and EUV observation was obtained with a heating that scales as expected from DC reconnection at tangential discontinuities.
Large-scale modeling has been able to explain the ignition of warm loops from primary energy release mechanisms. A large-scale approach (see also Section 4.1) is by "ab initio" modeling, i.e., with full MHD modeling of an entire coronal region (Gudiksen and Nordlund, 2005; Gudiksen et al., 2011). Observed solar granular velocity pattern, a potential extrapolation of a SoHO/MDI magnetogram, and a standard stratified atmosphere are used as initial conditions. The first simulations showed that, at steady state, the magnetic field is able to dissipate (3−4) × 106 erg cm−2 s−1 in a highly intermittent corona, at an average temperature of ∼ 106 K, adequate to reproduce typical warm loop populations observed in TRACE images. Warm loops were also obtained with time-dependent loop modeling including the intermittent magnetic dissipation in MHD turbulence due to loop footpoint motions (Reale et al., 2005). The dissipation rate along a loop predicted with a hybrid-shell model (Nigro et al., 2004) was used as heating input [see Eq. (6)] in a proper time-dependent loop model, the Palermo-Harvard code (Peres et al., 1982). It was shown that the most intense nanoflares excited in an ambient magnetic field of about 10 G can produce warm loops with temperatures of 1–1.5 MK in the corona of a 30 000 km long loop.
More recently, reduced MHD (rMHD) was used to identify MHD anisotropic turbulence as the physical mechanism responsible for the transport of energy from the large scales, where energy is injected by photospheric motions, to the small scales, where it is dissipated (Rappazzo et al., 2007, 2008). Strong turbulence was found for weak axial magnetic fields and long loops. The predicted heating rate is appropriate for warm loops, in agreement with Reale et al. (2005). Shell models of rMHD turbulence were used to analyze the case of a coronal loop heated by photospheric turbulence and found that the Alfvén waves interact nonlinearly and form turbulent spectra (Buchlin and Velli, 2007). An intermittent heating function is active, on average able to sustain the corona and proportional to the aspect ratio of the loop to the ∼ 1.5 power. Adding a profile of density and/or magnetic field along the loop somewhat change the heat deposition, in particular in the low part of the loop (Buchlin et al., 2007). These models also predict the formation of current sheets that can be dissipated on these small scales and impulsively through turbulent cascades (Rappazzo et al., 2010; Rappazzo and Velli, 2011). Transient current sheets are also found from large-scale full MHD modeling (Bingert and Peter, 2011). In the same framework loops have been described as partially resonant cavities for low-frequency fluctuations transmitted from the chromosphere (Verdini et al., 2012).
There are new efforts to include magnetic effects in the loop modeling. Haynes et al. (2008) studied observational properties of a kink unstable coronal loop, using a fluid code and finding potentially observable density effects. Browning et al. (2008) studied coronal heating by nanoflares triggered by a kink instability using three-dimensional magnetohydrodynamic numerical simulations of energy release for a cylindrical coronal loop model. Magnetic energy is dissipated, leading to large or small heating events according to the initial current profile.
Interesting perspectives are developing from models in which self-organized criticality triggers loop coronal heating (e.g., López Fuentes and Klimchuk, 2010). For Uzdensky (2007) and Cassak et al. (2008) coronal heating is self-regulating and keeps the coronal plasma roughly marginally collisionless. In the long run, the coronal heating process may be represented by repeating cycles that consist of fast reconnection events (i.e., nanoflares), followed by rapid evaporation episodes, followed by relatively long periods (∼ 1 hr) during which magnetic stresses build up and the plasma simultaneously cools down and precipitates. An avalanche model was proposed for solar flares (Morales and Charbonneau, 2008), based on an idealized representation of a coronal loop as a bundle of magnetic flux strands wrapping around one another. The system is driven by random deformation of the strands, and a form of reconnection is assumed to take place when the angle subtended by two strands crossing at the same lattice site exceeds some preset threshold. For a generic coronal loop of length 1010 cm and diameter 108 cm, the mechanism leads to flare energies ranging between 1023 and 1029 erg, for an instability threshold angle of 11 degrees between contiguous magnetic flux strands.
Stellar Coronal Loops
Non-solar X-ray missions since Einstein and European X-ray Observatory SATellite (EXOSAT) have established that most other stars have a confined corona, often much more active that the solar one (e.g., Linsky and Serio, 1993). The level of activity is ruled by several factors, but, first of all, the age of the star is important: young fast-rotating stars are more active (e.g., Telleschi et al., 2005). The topic of stellar coronal loops deserves a review by itself (e.g., Rosner et al., 1985) and here only a few relevant issues are discussed. A complete and more recent review of stellar coronae, with an extensive part regarding loops, is by Güdel (2004). In the framework of the solar-stellar connection, it is very important the comparison of what we know about the spatially resolved but single solar corona and what about the unresolved but numerous stellar coronae, which offer a variety of different environments. The lack of spatial resolution inhibits to obtain direct information about the size and appearance of the loops, and the general aspect of the corona. We therefore have to rely on indirect evidence. One possible approach to get information is to benefit from transient X-ray events, such as flares, which provide estimations of the loop scale length from their dependence on the decay and rise timescales (Reale, 2002a, 2003, and Section 4.1.2 for reviews). Detailed hydrodynamic modeling can provide even more constraints, for instance, on the heat deposition (e.g., Reale et al., 1988, 2004). The study of stellar X-ray flares allowed, for instance, to constrain that most stellar flares involve plasma confined in closed structures (Reale et al., 2002), and to infer the presence both of loops with size similar to those observed on the Sun (e.g., Reale et al., 1988) and of giant loops (Favata et al., 2005; Getman et al., 2008), with length exceeding the stellar radius. Analogies between solar and stellar downflows have also been found (Reale et al., 2013). Radio observations have provided direct evidence for the presence of large coronal loops in the Algol system (Peterson et al., 2010).
Another approach is to use the entire solar X-ray corona as a template and "Rosetta stone" to interpret stellar coronae. A detailed implementation of this approach was devised and applied extensively using Yohkoh data over its entire life, which covers a whole solar cycle (Orlando et al., 2000; Peres et al., 2000). It was shown that the solar corona indeed provides a pattern of components, i.e., quiet structures, active regions, active region cores, flares, which can be identified in stellar coronal data and which can explain stellar activity giving different weights to the components (Peres et al., 2001, 2004). The method was also applied to describe stellar coronae in terms of loop populations and to extract general information and constraints on coronal heating (Peres et al., 2004). It was applied to flares (Reale et al., 2001) and to describe the evolution of active regions (Orlando et al., 2004). More recently it was shown that a continuous unresolved flaring activity may explain the most active coronae, but also that the coronal heating appears to follow different scaling for quiet regions and for active and flaring regions across the cycle (Argiroffi et al., 2008).
Cargill and Klimchuk (2006) realized that the strong hot peaks in the emission measure-temperature distributions in the coronae of some binary stars (Sanz-Forcada et al., 2003) are similar to those expected for an impulsively-heated solar corona. A coronal model comprised of many impulsively heated strands shows that the evidence may be compatible with coronae made of many very small loops (length under 103 km) heated by microflares.
The growing evidence of hot plasma and of variability will link more tightly the investigation of the solar coronal heating to the study of stellar coronae, where very hot steady components are often detected (e.g., Schmitt et al., 1990; Scelsi et al., 2005).
Conclusions and Perspectives
Coronal loops have been the subject of in-depth studies for over 50 years. Since they owe their identity to the brightness of the confined plasma, most of the studies have addressed the physics of the confined plasma, i.e., its structure, dynamics, and evolution. The coronal loops are the building blocks of the bright solar corona and, as such, they are important as the basic laboratory to investigate the mechanisms of coronal magnetism, dynamics, and heating. Our knowledge on coronal loops has progressed with the development of the instrument capabilities. The starting point has been the observation, mostly in the X-ray band, that coronal loops are globally steady on timescales longer than the plasma cooling times. This has allowed to develop the loop scaling laws, which work well to describe the hydrostatic properties of coronal plasma confined in a magnetic flux tube (Section 4.1.1). However, since the 1990s observations have more and more revealed that coronal loops are dynamic and structured, both spatially and thermally. Imaging instruments have shown bundles of fine strands whose thickness is not well resolved up-to-date, although we begin to have constraints from the latest observations with the Hi-C (∼ 100 km). The lack of agreement about the thickness values might indicate a distribution of strand sizes (Section 3.2.2). EUV spectroscopy shows a variety of thermal structuring, from nearly isothermal loops to broad distributions over a temperature decade (6 < log T < 7). The width of the distributions might depend on the heating intensity (warm vs hot loops), on the age of the active region and/or on the magnetic field (Section 3.3). The presence, whenever confirmed, of minor but widespread and filamented, very hot plasma components (log T ∼ 7) out of flares might be a tracer of small-scale impulsive heating. Coronal rain might mark thermal instability on small scales of thickness. High-resolution spectroscopy has also detected widespread flows (∼ 10 km s−1) with a complex pattern, especially low in the corona and transition region (Section 3.5). The pattern is described quite well as switching from redshifts to blueshifts with increasing temperature, thus indicating the coexistence of cool downfalling plasma and warm evaporation. This complex pattern is also a signature of fine structure, at least in the low corona. Higher in the corona, flow patterns are less defined, simply because of geometrical reasons. Fast upflows (∼ 100 km s−1) from the chromosphere might play a role in feeding coronal loops (Section 3.5.1). Continuous monitoring of the full solar disk has also allowed very detailed photometric analysis (Section 3.4). The detailed analysis of the light curves seems to indicate the presence of systematic cooling patterns, both in the X-rays and, from comparison of different channels, in the EUV band. The monitoring has also allowed the detection of MHD waves propagating along coronal loops (Section 3.5.2).
The physics of the plasma confined inside coronal loops is well described with compressible hydrodynamics with a very important role of thermal conduction and radiative losses from optically thin medium. The mass and energy are transported mostly along the field lines, with little direct role of the magnetic field. The initial hydrostatic single loop scenario is now replaced by a much more dynamic one with loops structured into bundle of finer strands (Section 4.1.2). The physics of the plasma inside each strand does not change much, but one important issue becomes how these strands combine into the unresolved observed structures. Multi-strand loop modeling is able to explain the fuzzier and fuzzier appearance of loops observed in channels sensitive to hotter and hotter emission, and flat filter ratios along loops (Section 4.2). A proper MHD description is required when we need to address directly the interaction of plasma with the ambient magnetic field, both regarding the confinement itself, e.g., the expansion across the transition region, and the conversion of the magnetic energy to power the plasma temperature and dynamics. Although flows along loops have long been modeled (Section 4.3), the recent evidence for upflows from the chromosphere has provided a new boost to models and, since it has been suggested that the flows might also carry the energy to heat the loop plasma, it has also revitalized the investigation of coronal heating, the central issue of coronal loop physics. The most basic question regarding the conversion of magnetic energy into heat remains probably whether this conversion occurs on small or large timescales, both in the charging and in the release (Section 4.4). This may make the difference between energy provided directly through fast magnetic reconnection (nanoflares) and energy dissipated more gradually by MHD waves, but new modeling seems to make the boundary much more blurred, wheareas MHD waves are dissipated impulsively through turbulent cascades. So, at the moment, there is some convergence to heating mechanisms released on short timescales inside thin strands composing the loops. The charge-and-release processes are also deserving attention regarding both the connection to the magnetic field, and the cadence and distribution of the events in the single strand. Turbulent cascades might provide a natural explanation for the dissipation of magnetic energy in anomalously small scales.
Key issues for the future remain the loop fine structure and dynamics. We need to address what are the ultimate elementary loop components, and whether they are unique or determined by local conditions, i.e., what determines the section scale size. We need the highest possible spatial resolution, probably in different bands. Further investigation of the plasma fine thermal structure and dynamics requires also high spectral resolution. High resolution broad-band X-ray spectroscopy is foreseeable to probe the hot components, signature of impulsive heating. Also the investigation of temporal variations still deserves attention. If loops are really so dynamic and subject to a distribution of heating events, whatever it is, the light curves are very difficult to interpret and signatures of any possible small-scale events are confused in a storming activity and by the plasma inertia. The analysis of emission variations is very important, because it can potentially shed light on heating mechanisms based on short impulsive events (nanoflares) or on wave-like phenomena (Alfvén waves). The continuous monitoring by the SDO mission is a very powerful tool for temporal variability studies. The analysis of imaging multi-channel observations seems to indicate widespread plasma cooling, which needs further independent investigation. Observations from IRIS are providing very detailed information about the interaction of the corona with the chromosphere and, in particular, about the intriguing transition region. Highest-resolution observations in the EUV seem also to emphasize the importance of the transition region, because it is very sensitive to changes of the physical conditions and, therefore, it can be a tracer of basic loop processes.
Great help is expected from modeling. Multi-stranded time-dependent loop models are still providing a wealth of information and might contribute to study specific issues such as the sequence and the relative weight of evaporation-draining cycles, and the vertical structure and dynamics of the thin transition region. The coupling with the chromosphere is becoming increasingly important. Improving numerical and computing resources are allowing to address the more basic question of the coupling of the plasma with the magnetic field and of the conversion of the magnetic energy into heat. The approaches involve both full MHD models and other models that couple different regimes, such as the large-scale magnetic field and the locally confined plasma. A totally self-consistent description is still out-of-reach, but 3D MHD models are beginning to attack some very basic issues, regarding the heat release, such as the role of the MHD instabilities and their switch-on. Turbulence seems a key to the anomalous dissipation needed to explain the loop ignition.
The study of coronal loops is very alive and is the subject of Coronal Loop Workshops, taking place every two years, which are site of debate, inspiration of new investigations, and school for young investigators.
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\begin{document}
\begin{frontmatter}
\title{Concentration in the flux approximation limit of Riemann solutions to the extended Chaplygin gas equations}
\author{Qingling Zhang}
\address{School of Mathematics and Computer Sciences, Jianghan University, Wuhan 430056, PR China } \ead{[email protected]} \begin{abstract}
In this paper, two kinds of occurrence mechanism on the phenomenon of concentration and the formation of delta shock wave in the flux approximation limit of Riemann solutions to the extended Chaplygin gas equations are analyzed and identified. Firstly, the Riemann problem of the extended Chaplygin gas equations is solved completely. Secondly, we rigorously show that, as the pressure vanishes, any two-shock Riemann solution to the extended Chaplygin gas equations tends to a $\delta$-shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted $\delta$-measure that forms the $\delta$-shock; any two-rarefaction-wave Riemann solution to the extended Chaplygin gas equations tends to a two-contact-discontinuity solution to the transport equations, and the nonvacuum intermediate state between the two rarefaction waves tends to a vacuum state. At last, we also show that, as the pressure approaches the generalized Chaplygin pressure, any two-shock Riemann solution tends to a delta-shock solution to the generalized Chaplygin gas equations.
\end{abstract}
\begin{keyword} Extended Chaplygin gas; Delta shock wave; flux approximation limit; Riemann solutions; Transport equations; generalized Chaplygin gas.
\MSC[2008] 35L65 \sep 35L67 \sep 35B25
\end{keyword}
\end{frontmatter}
\section{Introduction }
\setcounter{equation}{0}
The extended Chaplygin gas equations can be expressed as \begin{equation}\label{1.1} \left\{\begin{array}{ll} \rho_t+(\rho u)_x=0,\\ (\rho u)_t+(\rho u^2+P)_x=0, \end{array}\right. \end{equation} where $\rho$, $u$ and $P$ represent the density, the velocity and the scalar pressure, respectively, and \begin{equation}\label{1.2} P=A\rho^{n}-\frac{B}{\rho^{\alpha}},\ \ \ 1 \leq n \leq 3,\ \ 0<\alpha\leq1, \end{equation} with two parameters $A,B>0$. This model was proposed by Naji in 2014 \cite{Naji} to study the evolution of dark energy. When $B=0$ in $(\ref{1.2})$, $P=A\rho^{n}$ is the standard equation of state of perferct fluid. Up to now, various kinds of theoretical models have been proposed to interpret the behavior of dark energy. Specially, when $n=1$ in $(\ref{1.2})$, it reduces to the state equation for modified Chaplygin gas, which was originally proposed by Benaoum in 2002 \cite{Benaoum}. As an exotic fluid, such a gas can explain the current accelerated expansion of the universe. Whereas when $A=0$ in $(\ref{1.2})$, $P=-\frac{B}{\rho^{\alpha}}$ is called the pressure for the generalized Chaplygin gas \cite{Setare}. Furthermore, when $\alpha=1$, $P=-\frac{B}{\rho}$ is called the pressure for (pure) Chaplygin gas which was introduced by Chaplyin \cite{Chaplygin}, Tsien \cite{Tsien} and von Karman \cite{von Karman} as a suitable mathematical approximation for calculating the lifting force on a wing of an airplane in aerodynamics. Such a gas own a negative pressure and occurs in certain theories of cosmology. It has been also advertised as a possible model for dark energy \cite{Bilic-Tupper-Viollier,Gorini-Kamenshchik-Moschella-Pasquier,Setare1}.
When two parameters $A$, $B\rightarrow0$, the limit system of $(\ref{1.1})$ with $(\ref{1.2})$ formally becomes the following transport equations: \begin{equation}\label{1.3} \left\{\begin{array}{ll} \rho_t+(\rho u)_x=0,\\ (\rho u)_t+(\rho u^2)_x=0, \end{array}\right. \end{equation} which was also called the zero-pressure gas dynamics, and can be derived from Boltzmann equations \cite{Bouchut} and the flux-splitting numerical schemes for the full compressible Euler equations \cite{Brenier-Grenier,Li-Cao}. It can also be used to describe some important physical phenomena, such as the motion of free particles sticking together under collision and the formation of large scale structures in the universe \cite{Agarwal-Halt,E-Rykov-Sinai,Shandarin-Zeldovich}.
The transport equation $(\ref{1.3})$ has been studied extensively since 1994. The existence of measure solutions of the Riemann problem was first proved by Bouchut \cite{Bouchut} and the existence of the global weak solutions was obtained by Brenier and Grenier \cite{Bouchut} and E.Rykov and Sinai \cite{E-Rykov-Sinai}. Sheng and Zhang \cite{Sheng-Zhang} discovered that the $\delta$-shock and vacuum states do occur in the Riemann solutions to the transport equation $(\ref{1.3})$ by the vanishing viscosity method. Huang and Wang \cite{Huang-Wang} proved the uniqueness of the weak solution when the initial data is a Radon measure. Also see \cite{Shen1,Wang-Ding,Wang-Huang-Ding,Wang-Zhang1,Yang} for more related results.
$\delta$-shock is a kind of nonclassical nonlinear waves on which at least one of the state variables becomes a singular measure. Korchinski \cite{Korchinski} firstly introduced the concept of the $\delta$-function into the classical weak solution in his unpublished Ph. D. thesis. Tan, Zhang and Zheng \cite{Tan-Zhang-Zheng} considered some 1-D reduced system and discovered that the form of $\delta$-functions supported on shocks was used as parts in their Riemann solutions for certain initial data. LeFloch et al. \cite{LeFloch-Liu} applied the approach of nonconservative product to consider nonlinear hyperbolic systems in the nonconservative form. We can also refer to \cite{Bouchut,Li-Zhang-Yang,Sheng-Zhang} for related equations and results. Recently, the weak asymptotic method was widely used to study the $\delta$-shock wave type solution by Danilov and Shelkovich et al.\cite{Danilvo-Shelkovich1,Danilvo-Shelkovich2,Shelkovich}.
As for delta shock waves, one research focus is to explore the phenomena of concentration and cavitation and the formation of delta shock waves and vacuum states in solutions. In \cite{Chen-Liu1}, Chen and Liu considered the Euler equations for isentropic fluids, i.e., in $(\ref{1.1})$ they took the prototypical pressure function as follows: \begin{equation}\label{1.4} P=\varepsilon\frac{\rho^\gamma}{\gamma},\ \ \gamma>1. \end{equation} They analyzed and identified the phenomena of concentration and cavitation and the formation of $\delta$-shocks and vacuum states as $\varepsilon\rightarrow 0$, which checked the numerical observation for the 2-D case by Chang, Chen and Yang \cite{Chang-Chen-Yang1,Chang-Chen-Yang2}. They also pointed out that the occurrence of $\delta$-shocks and vacuum states in the process of vanishing pressure limit can be regarded as a phenomenon of resonance between the two characteristic fields. In \cite{Chen-Liu2}, they made a further step to generalize this result to the nonisentropic fluids. Specially, for $\gamma=1$ in $(\ref{1.4})$, the vanishing pressure limit has been studied by Li \cite{Li}. Besides, the results were extended to the relativistic Euler equations for polytropic gases by Yin and Sheng \cite{Yin-Sheng}, the perturbed Aw-Rascle model by Shen and Sun \cite{Shen-Sun} and the modified Chaplygin gas equations for by Yang and Wang \cite{Yang-Wang,Yang-Wang1}. For other related works, we can also see \cite{Mitrovic-Nedeljkov,Zhang}.
In this paper, we focus on the extended Chaplygin gas equations $(\ref{1.1})$ to discuss the phenomena of concentration and cavitation and the formation of delta shock waves and vacuum states in solutions as the double parameter pressure vanishes wholly or partly, which corresponds to a two parameter limit of solutions in contrast to the previous works in \cite{Chen-Liu1,Chen-Liu2,Mitrovic-Nedeljkov,Shen-Sun,Yin-Sheng}. Equivalently, we will study the limit behavior of Riemann solutions to the extended Chaplygin gas equations as the pressure vanishes, or tends to the generalized Chaplygin pressure.
It is noticed that, When $A$, $B\rightarrow0$, the system $(\ref{1.1})$ with $(\ref{1.2})$ formally becomes the transport equations $(\ref{1.3})$. For fixed $B$, When $A\rightarrow0$, the system $(\ref{1.1})$ with $(\ref{1.2})$ formally becomes the generalized Chaplygin gas equations \begin{equation}\label{1.5} \left\{\begin{array}{ll} \rho_t+(\rho u)_x=0,\\ (\rho u)_t+(\rho u^2-\frac{B}{\rho^{\alpha}})_x=0. \end{array}\right. \end{equation}
When $\alpha=1$, it is just the Chaplygin gas equations. In 1998, Brenier \cite{Brenier} firstly studied the 1-D Riemann problem and obtained the solutions with concentration when initial data belong to a certain domain in the phase plane. Recently, Guo, Sheng and Zhang \cite{Guo-Sheng-Zhang} abandoned this constrain and constructively obtained the general solutions of the 1-D Riemann problem in which the $\delta$-shock wave developed. Moreover, in that paper, they also systematically studied the 2-D Riemann problem for isentropic Chaplygin gas equations. In \cite{Wang}, Wang solved the Riemann problem of $(\ref{1.5})$ by the weak asymptotic method. It has been shown that, in their results, $\delta$-shocks do occur in the Riemann solutions, but vacuum states do not. For more results about Chaplygin gas, one can refer to \cite{Qu-Wang,Shen2,Sun,Wang-Zhang2}.
In this paper, we first solve the Riemann problem of system (\ref{1.1}) with Riemann initial data \begin{equation}\label{1.6} (\rho,u)(x,0)=(\rho_\pm,u_\pm),\ \ \ \ \pm x>0, \end{equation} where $\rho_\pm>0,\ u_\pm$ are arbitrary constants. With the help of analysis method in phase plane, we constructed the Riemann solutions with four different structures: $R_1R_2$, $R_1S_2$, $S_1R_2$ and $S_1S_2$.
Then we analyze the formation of $\delta$-shocks and vacuum states in the Riemann solutions as the pressure vanishes. It is shown that, as the pressure vanishes, any two-shock Riemann solution to the extended Chaplygin gas equations tends to a $\delta$-shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted $\delta$-measure that forms the $\delta$-shock; by contrast, any two-rarefaction-wave Riemann solution to the extended Chaplygin gas equations tends to a two-contact-discontinuity solution to the transport equations, and the nonvacuum intermediate state between the two rarefaction waves tends to a vacuum state, even when the initial data stay away from the vacuum. As a result, the delta shocks for the transport equations result from a phenomenon of concentration, while the vacuum states results from a phenomenon of cavitation in the vanishing pressure limit process. These results are completely consistent with that in \cite{Chen-Liu1}, and also cover those obtained in \cite{Yang-Wang,Yang-Wang1}.
In addition, we also proved that as the pressure tends to the generalized Chaplygin pressure ($A\rightarrow0$), any two-shock Riemann solution to the extended Chaplygin gas equations tends to a $\delta$-shock solution to the generalized Chaplygin gas equations, and the intermediate density between the two shocks tends to a weighted $\delta$-measure that forms the $\delta$-shock; by contrast, any two-rarefaction-wave Riemann solution to the extended Chaplygin gas equations tends to the two-rarefaction-wave (two-contact-discontinuity for $\alpha=1$) solution to the transport equations, and the intermediate state between the two rarefaction waves (two contact discontinuities) is a nonvacuum state. Consequently, the delta shocks for the generalized Chaplygin gas equations result from a phenomenon of concentration in the partly vanishing pressure limit process.
From the above analysis, we can find two kinds of occurrence mechanism on the phenomenon of concentration and the formation of delta shock wave. On one hand, since the strict hyperbolicity of the limiting system $(\ref{1.3})$ fails, see Section 4, the delta shock wave forms in the limit process as the pressure vanishes. This is consistent with those results obtained in \cite{Chen-Liu1,Chen-Liu2,Mitrovic-Nedeljkov,Shen-Sun,Yang-Wang,Yin-Sheng}. On the other hand, the strict hyperbolicity of the limiting system $(\ref{1.5})$ is preserved, see Section 5, the formation of delta shock waves still occur as the pressure partly vanishes. In this regard, it is different from that in \cite{Chen-Liu1,Chen-Liu2,Mitrovic-Nedeljkov,Shen-Sun,Yang-Wang,Yin-Sheng}. In any case, the phenomenon of concentration and the formation of delta shock wave can be regarded as a process of resonance formation between two characteristic fields.
The paper is organized as follows. In Section 2, we restate the Riemann solutions to transport equations $(\ref{1.3})$ and the generalized Chaplygin gas equations $(\ref{1.5})$. In Section 3, we investigate the Riemann problem of the extended Chaplygin gas equations $(\ref{1.1})$-$(\ref{1.2})$ and examine the dependence of the Riemann solutions on the two parameters $A,B>0$. In Section 4, we analyze the limit of Riemann solutions to the extended Chaplygin gas equations $(\ref{1.1})$-$(\ref{1.2})$ with $(\ref{1.6})$ as the pressure vanishes. In Section 5, we discuss the limit of Riemann solutions to the extended Chaplygin gas equations $(\ref{1.1})$-$(\ref{1.2})$ with $(\ref{1.6})$ as the pressure approaches to the generalized Chaplygin pressure. Finally, conclusions and discussions are drawn in Section 6.
\section{Preliminaries} \subsection{Riemann problem for the transport equations} In this section, we restate the Riemann solutions to the transport equations $(\ref{1.3})$ with initial data $(\ref{1.6})$. See \cite{Sheng-Zhang} for more details.
The transport equations $(\ref{1.3})$ have a double eigenvalue $\lambda=u$ and only one right eigenvectors $ \vec{r}=(1,0)^T $. Furthermore, we have $\nabla\lambda\cdot\vec{r}=0$, which means that $\lambda$ is linearly degenerate. The Riemann problem $(\ref{1.3})$ and $(\ref{1.6})$ can be solved by contact discontinuities, vacuum or $\delta$-shocks connecting two constant states $(\rho_\pm,u_\pm)$.
By taking the self-similar transformation $\xi=\frac{x}{t}$, the Riemann problem is reduced to the boundary value problem of the ordinary differential equations: \begin{equation}\label{2.1} \left\{\begin{array}{ll} -\xi\rho_\xi+(\rho u)_\xi=0,\\ -\xi(\rho u)_\xi+(\rho u^2)_\xi=0, \end{array}\right. \end{equation} with $(\rho,u)(\pm\infty)=(\rho_\pm,u_\pm)$.
For the case $u_-<u_+$, there is no characteristic passing through the region $\{\xi: u_-<\xi<u_+\}$, so the vacuum should appear in the region. The solution can be expressed as \begin{equation}\label{2.2} (\rho,u)(\xi)=\left\{\begin{array}{ll} (\rho_-,u_-),\ \ \ \ -\infty<\xi\leq u_-,\\ (0,\xi),\ \ \ \ \ \ \ \ u_-<\xi< u_+,\\ (\rho_+,u_+),\ \ \ \ u_+\leq\xi<\infty. \end{array}\right. \end{equation}
For the case $u_-=u_+$, it is easy to see that the constant states $(\rho_\pm,u_\pm)$ can be connected by a contact discontinuity.
For the case $u_->u_+$, a solution containing a weighted $\delta$-measure supported on a curve will be constructed. Let $x=x(t)$ be a discontinuity curve, we consider a piecewise smooth solution of $(\ref{1.3})$ in the form \begin{equation}\label{2.3} (\rho,u)(x,t)=\left\{\begin{array}{ll} (\rho_-,u_-),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x<x(t),\\ (w(t)\delta(x-x(t)),u_\delta(t)),\ \ \ \ \ x=x(t),\\ (\rho_+,u_+),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x>x(t). \end{array}\right. \end{equation}
In order to define the measure solution as above, like as in \cite{Chen-Liu1,Chen-Liu2,Sheng-Zhang}, the two-dimensional weighted $\delta$-measure $w(t)\delta_S$ supported on a smooth curve $S=\{(x(s),t(s)):a\leq s\leq b\}$ should be introduced as follows: \begin{equation}\label{2.4} \langle w(\cdot)\delta_S,\psi(\cdot,\cdot)\rangle=\int_a^bw(s)\psi(x(s),t(s))\sqrt{{x'(s)}^2+{t'(s)}^2}ds, \end{equation} for any $\psi\in C_0^\infty(R\times R_{+})$.
As shown in \cite{Sheng-Zhang}, for any $\psi\in C_0^\infty(R\times R_{+})$, the $\delta$-measure solution $(\ref{2.3})$ constructed above satisfies \begin{equation}\label{2.5} \left\{\begin{array}{ll} \langle\rho,\psi_t\rangle+\langle\rho u,\psi_x\rangle=0, \\ \langle\rho u,\psi_t\rangle+\langle\rho u^2,\psi_x\rangle=0, \end{array}\right. \end{equation}
in which $$\langle\rho,\psi\rangle=\int_0^\infty\int_{-\infty}^\infty\rho_0\psi dxdt+\langle w_1(\cdot)\delta_S,\psi(\cdot,\cdot)\rangle,$$ $$\langle\rho u,\psi\rangle=\int_0^\infty\int_{-\infty}^\infty\rho_0 u_0\psi dxdt+\langle w_2(\cdot)\delta_S,\psi(\cdot,\cdot)\rangle,$$ where $$\rho_0=\rho_-+[\rho]H(x-\sigma t),\ \ \rho_0u_{0}=\rho_-u_{-}+[\rho u]H(x-\sigma t),$$ and $$w_1(t)=\frac{t}{\sqrt{1+\sigma^2}}(\sigma[\rho]-[\rho u]), \ \ w_2(t)=\frac{t}{\sqrt{1+\sigma^2}}(\sigma[\rho u]-[\rho u^2]).$$ Here $H(x)$ is the Heaviside function given by $H(x)=1$ for $x>0$ and $H(x)=0$ for $x<0$.
Substituting $(\ref{2.3})$ into $(\ref{2.5})$, one can derive the generalized Rankine-Hugoniot conditions \begin{equation}\label{2.6} \left\{\begin{array}{ll} \DF{dx(t)}{dt}=u_\delta(t),\\ \DF{dw(t)}{dt}=[\rho] u_\delta(t)-[\rho u],\\ \DF{d(w(t) u_ \delta(t))}{dt}=[\rho u] u_\delta(t)-[\rho u^2] \end{array}\right. \end{equation} where $[\rho]=\rho_+-\rho_-$, etc.
Through solving $(\ref{2.6})$ with $x(0)=0,\ w(t)=0$, we obtain \begin{equation}\label{2.7} \left\{\begin{array}{ll} u_\delta(t)=\sigma=\DF{\sqrt{\rho_-}u_-+\sqrt{\rho_+}u_+}{\sqrt{\rho_-}+\sqrt{\rho_+}},\\ x(t)=\sigma t,\\ w(t)=-\sqrt{\rho_-\rho_+}(u_+-u_-)t, \end{array}\right. \end{equation}
Moreover, the $\delta$-measure solution $(\ref{2.3})$ with $(\ref{2.6})$ satisfies the $\delta$-entropy condition: \begin{equation}\nonumber u_+<\sigma<u_-, \end{equation} which means that all the characteristics on both sides of the $\delta$-shock are incoming.
\subsection{Riemann problem for the generalized Chaplygin gas equations}
In this section, we solve the Riemann problem for the generalized Chaplygin gas equations $(\ref{1.5})$ with $(\ref{1.6})$, which one can also see in \cite{Guo-Sheng-Zhang,Wang}.
It is easy to see that $(\ref{1.5})$ has two eigenvalues $$\lambda_1^B=u-\sqrt{\alpha B}\rho^{-\frac{\alpha+1}{2}},\ \ \lambda_2^B=u+\sqrt{\alpha B}\rho^{-\frac{\alpha+1}{2}},$$ with corresponding right eigenvectors $$\overrightarrow{r_1}^B=(-\sqrt{\alpha B}\rho^{-\frac{\alpha+1}{2}},\rho)^T,\ \ \overrightarrow{r_2}^B=(\sqrt{\alpha B}\rho^{-\frac{\alpha+1}{2}},\rho)^T.$$ So $(\ref{1.5})$ is strictly hyperbolic for $\rho>0$. Moreover, when $0<\alpha<1$, we have
$\bigtriangledown\lambda_i^B\cdot\overrightarrow{r_i}^B\neq0$, $i=1,2$, which implies that $\lambda_1^B$ and $\lambda_2^B$ are both genuinely nonlinear and the associated waves are rarefaction waves and shock waves. When $\alpha=1$, $\bigtriangledown\lambda_i^B\cdot \overrightarrow{r_i}^B=0$, $i=1,2$, which implies that $\lambda_1^B$ and $\lambda_2^B$ are both linearly degenerate and the associated waves are both contact discontinuities, see \cite{Smoller}.
Since system $(\ref{1.5})$ and the Riemann initial data $(\ref{1.6})$ are invariant under stretching of coordinates $(x,t)\rightarrow(\beta x,\beta t)$ ($\beta$ is constant), we seek the self-similar solution $$(\rho,u)(x,t)=(\rho,u)(\xi),\ \ \xi=\frac{x}{t}.$$ Then the Riemann problem $(\ref{1.5})$ and $(\ref{1.6})$ is reduced to the following boundary value problem of the ordinary differential equations: \begin{equation}\label{2.8} \left\{\begin{array}{ll} -\xi\rho_\xi+(\rho u)_\xi=0,\\ -\xi(\rho u)_\xi+(\rho u^2-\frac{B}{\rho^\alpha})_\xi=0,\\ \end{array}\right. \end{equation} with $(\rho,u)(\pm\infty)=(\rho_\pm,u_\pm)$.
Besides the constant solution, it provides the backward rarefaction wave \begin{equation}\label{2.9}
\overleftarrow{R}(\rho_-,u_-):\left\{\begin{array}{ll} \xi=\lambda_1^B=u-\sqrt{\alpha B}\rho^{-\frac{\alpha+1}{2}},\\ u-\frac{2\sqrt{\alpha B}}{1+\alpha}\rho^{-\frac{\alpha+1}{2}}=u_--\frac{2\sqrt{\alpha B}}{1+\alpha}\rho_-^{-\frac{\alpha+1}{2}},\ \ \rho<\rho_-, \end{array}\right. \end{equation} and the forward rarefaction wave \begin{equation}\label{2.10} \overrightarrow{R}(\rho_-,u_-):\left\{\begin{array}{ll} \xi=\lambda_2^B=u+\sqrt{\alpha B}\rho^{-\frac{\alpha+1}{2}},\\ u+\frac{2\sqrt{\alpha B}}{1+\alpha}\rho^{-\frac{\alpha+1}{2}}=u_-+\frac{2\sqrt{\alpha B}}{1+\alpha}\rho_-^{-\frac{\alpha+1}{2}},\ \ \rho>\rho_-, \end{array}\right. \end{equation} When $\alpha=1$, the backward (forward) rarefaction wave becomes the backward (forward) contact discontinuity.
For a bounded discontinuity at $\xi=\sigma$, the Rankine-Hugoniot conditions hold: \begin{equation}\label{2.11} \left\{\begin{array}{ll} -\sigma^{B}[\rho]+[\rho u]=0,\\ -\sigma^{B}[\rho u]+[\rho u^2-\DF{B}{\rho^\alpha}]=0, \end{array}\right. \end{equation} where $[\rho]=\rho-\rho_-$, etc. Together with the Lax shock inequalities, (\ref{2.11}) gives the backward shock wave \begin{equation}\label{2.12} \overleftarrow{S}(\rho_-,u_-):\left\{\begin{array}{ll} \sigma_1^{B}=\DF{\rho u-\rho_-u_-}{\rho-\rho_-},\\ u-u_-=-\sqrt{B(\frac{1}{\rho}-\frac{1}{\rho_-})(\frac{1}{\rho^{\alpha}}-\frac{1}{\rho_{-}^{\alpha}})},\ \ \rho>\rho_-, \end{array}\right. \end{equation} and the forward shock wave \begin{equation}\label{2.13} \overrightarrow{S}(\rho_-,u_-):\left\{\begin{array}{ll} \sigma_2^{B}=\DF{\rho u-\rho_-u_-}{\rho-\rho_-},\\ u-u_-=-\sqrt{B(\frac{1}{\rho}-\frac{1}{\rho_-})(\frac{1}{\rho^{\alpha}}-\frac{1}{\rho_{-}^{\alpha}})},\ \ \rho<\rho_-. \end{array}\right. \end{equation} When $\alpha=1$, the backward (forward) shock wave becomes the backward (forward) contact discontinuity.
Furthermore, for a given left state $(\rho_-,u_-)$, the backward shock wave $\overleftarrow{S}(\rho_-,u_-)$ has a straight line $u=u_--\sqrt{ B}\rho_-^{-\frac{\alpha+1}{2}}$, as its asymptote, and for a given right state $(\rho_+,u_+)$, the forward shock wave $\overrightarrow{S}(\rho_+,u_+)$ has a straight line $u=u_++\sqrt{ B}\rho_+^{-\frac{\alpha+1}{2}}$ as its asymptote.
It is easy to see that, when $u_++\sqrt{ B}\rho_+^{-\frac{\alpha+1}{2}} \leq u_--\sqrt{ B}\rho_-^{-\frac{\alpha+1}{2}}$, the backward shock wave $\overleftarrow{S}(\rho_-,u_-)$ can not intersect the forward shock wave $\overrightarrow{S}(\rho_+,u_+)$, a delta shock wave must develop in solutions. Under the definition (\ref{2.4}), a delta shock wave can be introduced to construct the solution of (\ref{1.5})-(\ref{1.6}), which can be expressed as \begin{equation}\label{2.14} (\rho,u)(x,t)=\left\{\begin{array}{ll} (\rho_-,u_-),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x<\sigma^B t,\\ (w^B(t)\delta(x-\sigma^B t),\sigma^B),\ \ \ \ \ x=\sigma^B t,\\ (\rho_+,u_+),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x>\sigma^B t, \end{array}\right. \end{equation} with \begin{equation}\nonumber \frac{B}{\rho^\alpha}=\left\{\begin{array}{ll} \DF{B}{\rho_-^\alpha},\ \ \ \ \ \ \ x<\sigma^B t,\\ 0,\ \ \ \ \ \ \ \ \ x=\sigma^B t,\\ \DF{B}{\rho_+^\alpha},\ \ \ \ \ \ \ x>\sigma^B t, \end{array}\right. \end{equation} see \cite{Brenier}.
By the weak solution definition in Subsection 2.1, for the system (\ref{1.5})£¬we can get the following generalized Rankine-Hugoniot conditions \begin{equation}\label{2.15} \left\{\begin{array}{ll} \DF{dx^{B}(t)}{dt}=u_\delta^{B}(t)=\sigma^B,\\[4pt] \DF{dw^{B}(t)}{dt}=u_\delta^{B}(t)[\rho]-[\rho u],\\[4pt] \DF{d(w^{B}(t)u_\delta^{B}(t))}{dt}=u_\delta^{B}(t)[\rho u]-[\rho u^2-\DF{1}{\rho}], \end{array}\right. \end{equation} where $x^{B}(t)$, $w^{B}(t)$ and $u_\delta^{B}(t)$ are respectively denote the location, weight and propagation speed of the $\delta$-shock, $[\rho]=\rho(x^{B}(t)+0,t)-\rho(x^{B}(t)-0,t)$ denotes the jump of the function $\rho$ across the $\delta$-shock.
Then by solving (\ref{2.15}) with initial data $x(0)=0,\ w^B(0)=0$, under the entropy condition \begin{equation}\label{2.16} u_++\sqrt{\alpha B}\rho_+^{-\frac{\alpha+1}{2}}<\sigma^B< u_--\sqrt{\alpha B}\rho_-^{-\frac{\alpha+1}{2}}, \end{equation}
we can obtain \begin{equation}\label{2.17} w^B(t)=\big\{\rho_+\rho_-\big((u_+-u_-)^2-(\frac{1}{\rho_+}-\frac{1}{\rho_-})(\frac{B}{\rho_+^\alpha}-\frac{B}{\rho_-^\alpha})\big)\big\}^\frac{1}{2}t, \end{equation} \begin{equation}\label{2.18} \sigma^B=\frac{\rho_+ u_+-\rho_-u_-+\frac{dw^{B}(t)}{dt}}{\rho_+-\rho_-}, \end{equation} when $\rho_+\neq\rho_-$, and \begin{equation}\label{2.19} w^B(t)=(\rho_-u_--\rho_+ u_+)t, \end{equation} \begin{equation}\label{2.20} \sigma^B=\frac{1}{2}(u_++u_-), \end{equation}
when $\rho_+=\rho_-$.
In the phase plane ($\rho>0$, $u\in R$), given a constant state $(\rho_-,u_-)$, we draw the elementary wave curves (\ref{2.9})-(\ref{2.10}) and (\ref{2.12})-(\ref{2.13}) passing through this point, which are denoted by $\overleftarrow{R}$,$\overrightarrow{R}$,$\overleftarrow{S}$ and $\overrightarrow{S}$ respectively. The backward shock wave $\overleftarrow{S}$ has an asymptotic line $u=u_--\sqrt{ B}\rho_-^{-\frac{\alpha+1}{2}}$. In addition, we draw a $S_{\delta}$ curve, which is determined by \begin{equation}\label{2.21} u+\sqrt{B}\rho^{-\frac{\alpha+1}{2}}= u_--\sqrt{ B}\rho_-^{-\frac{\alpha+1}{2}}, \ \rho>0. \end{equation} Then the phase plane can be divided into five parts I$(\rho_-,u_-)$, I\!I$(\rho_-,u_-)$, I\!I\!I$(\rho_-,u_-)$, I\!V$(\rho_-,u_-)$ and V$(\rho_-,u_-)$, see Fig.1.
By the analysis method in the phase plane, one can construct the Riemann solutions for any given $(\rho_+,u_+)$ as follows:\\ (1) $(\rho_+,u_+)\in$\rm{ I}$(\rho_-,u_-)$:\ $\overleftarrow{R}+\overrightarrow{R}$;\ \ \ \ \ (2) $(\rho_+,u_+)\in$ I\!I$(\rho_-,u_-)$:\ $\overleftarrow{R}+\overrightarrow{S}$;\\(3) $(\rho_+,u_+)\in$ I\!I\!I$(\rho_-,u_-)$:\ $\overleftarrow{S}+\overrightarrow{R}$;\ \ \ \ (4) $(\rho_+,u_+)\in$ I\!V$(\rho_-,u_-)$:\ $\overleftarrow{S}+\overrightarrow{S}$;\\(5) $(\rho_+,u_+)\in$ V$(\rho_-,u_-)$:\ $\delta$-shock.
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\section{Riemann problem for the extended Chaplygin gas equations } In this section, we first solve the elementary waves and construct solutions to the Riemann problem of $(\ref{1.1})$-$(\ref{1.2})$ with $(\ref{1.6})$, and then examine the dependence of the Riemann solutions on the two parameters $A,B>0$.
The eigenvalues of the system $(\ref{1.1})$-$(\ref{1.2})$ are $$\lambda_1^{AB}=u-\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}, \quad \lambda_2^{AB}=u+\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}},$$ with corresponding right eigenvectors $$\vec{r}_1^{AB}=(-\rho,\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}})^T,\quad \vec{r}_2^{AB}=(\rho,\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}})^T.$$
Moreover, we have $$\nabla\lambda_i^{AB}\cdot \vec{r}_i^{AB}=\DF{An(n+1)\rho^{n+\alpha}+(1-\alpha)\alpha B}{2\sqrt{(An\rho^{n+\alpha}+\alpha B)\rho^{\alpha+1}}}>0\ \ (i=1,2).$$ Thus $\lambda_1^{AB}$ and $\lambda_2^{AB}$ are genuinely nonlinear and the associated elementary waves are shock waves and rarefaction waves.
For $(\ref{1.1})$-$(\ref{1.2})$ with $(\ref{1.6})$ are invariant under uniform stretching of coordinates:$(x,t)\rightarrow(\beta x,\beta t)$ where constant $\beta>0$, we seek the self-similar solution$$(\rho,u)(x,t)=(\rho(\xi),u(\xi)),\ \ \xi=\frac{x}{t}.$$ Then the Riemann problem $(\ref{1.1})$-$(\ref{1.2})$ with $(\ref{1.6})$ is reduced to the boundary value problem of the following ordinary differential equations: \begin{equation}\label{3.1} \left\{\begin{array}{ll} -\xi\rho_\xi+(\rho u)_\xi=0,\\ -\xi(\rho u)_\xi+(\rho u^2+P)_\xi=0,\ \ P=A\rho^{n}-\frac{B}{\rho^{\alpha}}, \end{array}\right. \end{equation} with $(\rho,u)(\pm\infty)=(\rho_\pm,u_\pm)$.
Any smooth solutions of $(\ref{3.1})$ satisfies \begin{equation}\label{3.2} \left(\begin{array}{lll} u-\xi &\ \ \rho \\ An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}} &\ \ \rho(u-\xi) \end{array}\right) \left(\begin{array}{lll}
d\rho\\du \end{array}\right)=0. \end{equation} It provides either the constant state solutions $$(\rho,u)(\xi)=\rm{constant}, $$ or the rarefaction wave which is a continuous solutions of $(\ref{3.2})$ in the form $(\rho,u)(\xi)$. Then, according to \cite{Smoller}, for a given left state $(\rho_-,u_-)$, the rarefaction wave curves in the phase plane, which are the sets of states that can be connected on the right by a 1-rarefaction wave or 2-rarefaction wave, are as follows: \begin{equation}\label{3.3} R_1(\rho_-,u_-): \left\{\begin{array}{ll} \xi=\lambda_1=u-\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}},\\ u-u_-=-\int_{\rho_{-}}^{\rho}\frac{\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}{\rho}d\rho, \end{array}\right. \end{equation} and \begin{equation}\label{3.4} R_2(\rho_-,u_-): \left\{\begin{array}{ll} \xi=\lambda_2=u+\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}},\\ u-u_-=\int_{\rho_{-}}^{\rho}\frac{\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}{\rho}d\rho. \end{array}\right. \end{equation}
From $(\ref{3.3})$ and $(\ref{3.4})$, we obtain that
\begin{eqnarray} \frac{d\lambda_1^{AB}}{d\rho}=\frac{\partial\lambda_1^{AB}}{\partial u}\frac{du}{d\rho}+\frac{\partial\lambda_1^{AB}}{\partial\rho}
=-\DF{An(n+1)\rho^{n-1}+\frac{\alpha(1-\alpha) B}{\rho^{\alpha+1}}}{2\rho\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}<0\label{3.5} \end{eqnarray}
\begin{eqnarray} \frac{d\lambda_2^{AB}}{d\rho}=\frac{\partial\lambda_2^{AB}}{\partial u}\frac{du}{d\rho}+\frac{\partial\lambda_2^{AB}}{\partial\rho}
=\DF{An(n+1)\rho^{n-1}+\frac{\alpha(1-\alpha) B}{\rho^{\alpha+1}}}{2\rho\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}>0\label{3.6} \end{eqnarray} which imply that the velocity of 1-rarefaction (2-rarefaction) wave $\lambda_1^{AB}$ ($\lambda_2^{AB}$) is monotonic decreasing (increasing) with respect to $\rho$.
With the requirement $\lambda_1^{AB}(\rho_{-},u_{-})<\lambda_1^{AB}(\rho,u)$ and $\lambda_2^{AB}(\rho_{-},u_{-})<\lambda_2^{AB}(\rho,u)$, noticing $(\ref{3.5})$ and $(\ref{3.6})$, we get that
\begin{equation}\label{3.7} R_1(\rho_-,u_-): \left\{\begin{array}{ll} \xi=\lambda_1=u-\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}},\\ u-u_-=-\int_{\rho_{-}}^{\rho}\frac{\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}{\rho}d\rho,\ \ \rho<\rho_{-}, \end{array}\right. \end{equation} and \begin{equation}\label{3.8} R_2(\rho_-,u_-): \left\{\begin{array}{ll} \xi=\lambda_2=u+\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}},\\ u-u_-=\int_{\rho_{-}}^{\rho}\frac{\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}{\rho}d\rho,\ \ \rho>\rho_{-}. \end{array}\right. \end{equation}
For the 1-rarefaction wave, through differentiating $u$ respect to $\rho$ in the second equation in $(\ref{3.7})$, we get \begin{equation}\label{3.9} u_{\rho}=-\DF{\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}{\rho}<0. \end{equation}
\begin{equation}\label{3.10} u_{\rho\rho}=\DF{-An(n-3)\rho^{n+\alpha}+\alpha(\alpha+3) B}{2\rho^{2}\sqrt{An\rho^{n+\alpha}+\alpha B\rho^{\alpha+1}}}. \end{equation}
Thus, it is easy to get $u_{\rho\rho}>0$ for $1\leq n\leq3$, i.e., the 1-rarefaction wave is convex for $1\leq n\leq3$ in the upper half phase plane ($\rho>0$).
In addition, from the second equation of $(\ref{3.7})$, we have $$u-u_-=\int_{\rho}^{\rho_{-}}\frac{\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}{\rho}d\rho\geq\int_{\rho}^{\rho_{-}}\sqrt{\alpha B}\rho^{-\frac{\alpha+1}{2}-1}d\rho=\frac{2\sqrt{\alpha B}}{\alpha+1}(\rho^{-\frac{\alpha+1}{2}}-\rho_{-}^{-\frac{\alpha+1}{2}}),$$ which means that $\lim\limits_{\rho\rightarrow0}u=+\infty.$
By a similar computation, we have that, for the 2-rarefaction wave, $u_{\rho}>0$, $u_{\rho\rho}<0$ for $1\leq n\leq3$ and $\lim\limits_{\rho\rightarrow+\infty}u=+\infty.$ Thus, we can draw the conclusion that the 2-rarefaction wave is concave for $1\leq n\leq3$ in the upper half phase plane ($\rho>0$).
Now we consider the discontinuous solution. For a bounded discontinuity at $\xi=\sigma$, the Rankine-Hugoniot condition holds: \begin{equation}\label{3.11} \left\{\begin{array}{ll} \sigma^{AB}[\rho]=[\rho u],\\ \sigma^{AB}[\rho u]=[\rho u^2+P],\ \ P=A\rho^{n}-\frac{B}{\rho^{\alpha}}, \end{array}\right. \end{equation} where $[\rho]=\rho_{+}-\rho_{-}$,etc.
Eliminating $\sigma$ from $(\ref{3.11})$, we obtain \begin{equation}\label{3.12} u-u_{-}=\pm\sqrt{\frac{\rho-\rho_{-}}{\rho\rho_{-}}\Big(A(\rho^{n}-\rho_{-}^{n})-B(\frac{1}{\rho^{\alpha}}-\frac{1}{\rho_{-}^{\alpha}})\Big)}. \end{equation}
Using the Lax entropy condition, the 1-shock satisfies \begin{equation}\label{3.13} \sigma^{AB}<\lambda_{1}^{AB}(\rho_{-},u_{-}),\ \ \lambda_{1}^{AB}(\rho,u)<\sigma^{AB}<\lambda_{2}^{AB}(\rho,u), \end{equation} while the 1-shock satisfies \begin{equation}\label{3.14} \lambda_{1}^{AB}(\rho_{-},u_{-})<\sigma^{AB}<\lambda_{2}^{AB}(\rho_{-},u_{-}),\ \ \lambda_{2}^{AB}(\rho,u)<\sigma^{AB}. \end{equation}
From the first equation in $(\ref{3.11})$, we have \begin{equation}\label{3.15} \sigma^{AB}=\frac{\rho u-\rho_{-}u_{-}}{\rho-\rho_{-}}=u+\frac{\rho_{-}( u-u_{-})}{\rho-\rho_{-}}=u_{-} +\frac{\rho( u-u_{-})}{\rho-\rho_{-}}. \end{equation}
Thus, by a simple calculation, $(\ref{3.13})$ is equivalent to \begin{equation}\label{3.16} -\rho\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}<\frac{\rho\rho_{-}( u-u_{-})}{\rho-\rho_{-}}<-\rho_{-}\sqrt{An\rho_{-}^{n-1}+\frac{\alpha B}{\rho_{-}^{\alpha+1}}}, \end{equation} and $(\ref{3.14})$ is equivalent to \begin{equation}\label{3.17} \rho\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}<\frac{\rho\rho_{-}( u-u_{-})}{\rho-\rho_{-}}<\rho_{-}\sqrt{An\rho_{-}^{n-1}+\frac{\alpha B}{\rho_{-}^{\alpha+1}}}. \end{equation} $(\ref{3.16})$ and $(\ref{3.17})$ imply that $\rho>\rho_{-}$, $u<u_{-}$ and $\rho<\rho_{-}$, $u<u_{-}$, respectively.
Through the above analysis, for a given left state $(\rho_-,u_-)$, the shock curves in the phase plane, which are the sets of states that can be connected on the right by a 1-shock or 2-shock, are as follows:
\ \begin{equation}\label{3.18} S_1(\rho_-,u_-): \left\{\begin{array}{ll} \sigma_1=\DF{\rho u-\rho_-u_-}{\rho-\rho_-},\\ u-u_-=-\sqrt{\frac{\rho-\rho_{-}}{\rho\rho_{-}}\Big(A(\rho^{n}-\rho_{-}^{n})-B(\frac{1}{\rho^{\alpha}}-\frac{1}{\rho_{-}^{\alpha}})\Big)},\ \ \rho>\rho_-, \end{array}\right. \end{equation} and \begin{equation}\label{3.19} S_2(\rho_-,u_-): \left\{\begin{array}{ll} \sigma_2=\DF{\rho u-\rho_-u_-}{\rho-\rho_-},\\ u-u_-=-\sqrt{\frac{\rho-\rho_{-}}{\rho\rho_{-}}\Big(A(\rho^{n}-\rho_{-}^{n})-B(\frac{1}{\rho^{\alpha}}-\frac{1}{\rho_{-}^{\alpha}})\Big)},\ \ \rho<\rho_-. \end{array}\right. \end{equation}
For the 1-shock wave, through differentiating $u$ respect to $\rho$ in the second equation in $(\ref{3.18})$, we get \begin{equation}\label{3.20} 2(u-u_{-})u_{\rho}=\frac{1}{\rho^{2}}\Big(A(\rho^{n}-\rho_{-}^{n})-B(\frac{1}{\rho^{\alpha}}-\frac{1}{\rho_{-}^{\alpha}})\Big)+ \frac{\rho-\rho_{-}}{\rho\rho_{-}}(An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}})>0, \end{equation} which means that $u_{\rho}<0$ for the 1-shock wave and that the 1-shock wave curve is starlike with respect to $(\rho_-,u_-)$ in the region $\rho>\rho_-$. Similarly, we can get $u_{\rho}>0$ for the 2-shock wave and that the 2-shock wave curve is starlike with respect to $(\rho_-,u_-)$ in the region $\rho<\rho_-$. In addition, it is easy to check that $\lim\limits_{\rho\rightarrow+\infty}u=-\infty$ for the 1-shock wave and
$\lim\limits_{\rho\rightarrow0}u=-\infty$ for the 2-shock wave.
Through the analysis above, for a given left state $(\rho_-,u_-)$, the sets of states connected with $(\rho_-,u_-)$ on the right in the phase plane consist of the 1-rarefaction wave curve $R_1(\rho_-,u_-)$, the 2-rarefaction wave curve $R_2(\rho_-,u_-)$, the 1-shock curve $S_1(\rho_-,u_-)$ and the 2-shock curve $S_2(\rho_-,u_-)$. These curves divide the upper half plane into four parts $R_{1}R_{2}(\rho_-,u_-)$, $R_{1}S_{2}(\rho_-,u_-)$, $S_{1}R_{2}(\rho_-,u_-)$ and $S_{1}S_{2}(\rho_-,u_-)$. Now, we put all of these curves together in the upper half plane ($\rho>0$, $u\in R$) to obtain a picture as in Fig.2.
By the phase plane analysis method, it is easy to construct Riemann solutions for any given right state $(\rho_+,u_+)$ as follows:\\ \ \ (1) $(\rho_+,u_+)\in\rm{R_{1}R_{2}}(\rho_-,u_-):R_{1}+R_{2};$ \ \ (2) $(\rho_+,u_+)\in\rm{R_{1}S_{2}}(\rho_-,u_-):R_{1}+S_{2};$\\ \ \ (3) $(\rho_+,u_+)\in\rm{S_{1}R_{2}}(\rho_-,u_-):S_{1}+R_{2};$ \ \ \
(4) $(\rho_+,u_+)\in\rm{S_{1}S_{2}}(\rho_-,u_-):S_{1}+S_{2}.$
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\qbezier(57.57,16.35)(92.95,19.92)(111.08,49.84)
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\put(49.1,47.87){\vector(0,1){.07}} \put(49.1,24.68){\line(0,1){23.19}}
\put(133.34,11.89){\makebox(0,0)[cc]{$u$}} \put(49.1,49.59){\makebox(0,0)[cc]{$\rho$}}
\put(88,26.30){\circle*{.9}}
\put(96.71,26.02){\makebox(0,0)[cc]{$\scriptstyle(\rho_-,u_-)$}}
\put(110.49,30.47){\makebox(0,0)[cc]{$R_{1}R_{2}(\rho_-,u_-)$}} \put(87.3,42.96){\makebox(0,0)[cc]{$S_{1}R_{2}(\rho_-,u_-)$}} \put(67.38,30.77){\makebox(0,0)[cc]{$S_{1}S_{2}(\rho_-,u_-)$}} \put(84.92,16.2){\makebox(0,0)[cc]{$R_{1}S_{2}(\rho_-,u_-)$}}
\put(69.76,47.12){\makebox(0,0)[cc]{$S_1$}} \put(111.99,46.08){\makebox(0,0)[cc]{$R_2$}} \put(61.14,19.58){\makebox(0,0)[cc]{$S_2$}} \put(110,19.43){\makebox(0,0)[cc]{$R_1$}}
\put(94.41,3.99){\makebox(0,0)[cc]{ Fig.2}} \end{picture}
\section{Formation of $\delta$-shocks and vacuum states as $A,B\rightarrow0$}
In this section, we will study the vanishing pressure limit process, i.e.,$A,B\rightarrow0$. Since the two regions $S_{1}R_{2} (\rho_-,u_-)$ and $R_{1}S_{2} (\rho_-,u_-)$ in the $(\rho,u)$ plane have empty interior when $A,B\rightarrow0$, it suffices to analyze the limit process for the two cases $(\rho_+,u_+)\in S_{1}S_{2} (\rho_-,u_-)$ and $(\rho_+,u_+)\in R_{1}R_{2}(\rho_-,u_-)$.
Firstly, we analyze the formation of $\delta$-shocks in Riemann solutions to the extended Chaplygin gas equations $(\ref{1.1})$-$(\ref{1.2})$ with $(\ref{1.6})$ in the case $(\rho_+,u_+)\in S_{1}S_{2}(\rho_-,u_-)$ as the pressure vanishes.
\subsection{Limit behavior of the Riemann solutions as $A,B\rightarrow0$} When $(\rho_+,u_+)\in S_{1}S_{2} (\rho_-,u_-)$, for fixed $A,B>0$, let $(\rho_*^{AB},u_*^{AB})$ be the intermediate state in the sense that $(\rho_-,u_-)$ and $(\rho_*^{AB},u_*^{AB})$ are connected by 1-shock $S_1$ with speed $\sigma_1^{AB}$, $(\rho_*^{AB},u_*^{AB})$ and $(\rho_+,u_+)$ are connected by 2-shock $S_2$ with speed $\sigma_2^{AB}$. Then it follows \begin{equation}\label{4.1}S_1:\ \ \left\{\begin{array}{ll} \sigma_1^{AB}=\DF{\rho_{*}^{AB} u_{*}^{AB}-\rho_-u_-}{\rho_{*}^{AB}-\rho_-},\\ u_{*}^{AB}-u_-=-\sqrt{\frac{\rho_{*}^{AB}-\rho_{-}}{\rho_{*}^{AB}\rho_{-}}\Big(A((\rho_{*}^{AB})^{n}-\rho_{-}^{n})- B(\frac{1}{(\rho_{*}^{AB})^{\alpha}}-\frac{1}{\rho_{-}^{\alpha}})\Big)},\ \ \rho_{*}^{AB}>\rho_-, \end{array}\right. \end{equation}
\begin{equation}\label{4.2}S_2:\ \ \left\{\begin{array}{ll} \sigma_2^{AB}=\DF{\rho_{+} u_{+}-\rho_{*}^{AB}u_{*}^{AB}}{\rho_{+}-\rho_{*}^{AB}},\\ u_{+}-u_{*}^{AB}=-\sqrt{\frac{\rho_{+}-\rho_{*}^{AB}}{\rho_{+}\rho_{*}^{AB}}\Big(A(\rho_{+}^{n}-(\rho_{*}^{AB})^{n})- B(\frac{1}{\rho_{+}^{\alpha}}-\frac{1}{(\rho_{*}^{AB})^{\alpha}})\Big)},\ \ \rho_{+}<\rho_{*}^{AB}. \end{array}\right. \end{equation}
In the following, we give some lemmas to show the limit behavior of the Riemann solutions of system $(\ref{1.1})$-$(\ref{1.2})$ with $(\ref{2.1})$ as $A,B\rightarrow0$.
\begin{Lemma}\label{lem:4.1} $\lim\limits_{A,B\rightarrow0}\rho_*^{AB}=+\infty.$ \end{Lemma}
\noindent\textbf{Proof.} Eliminating $u_{*}^{AB}$ in the second equation of $(\ref{4.1})$ and $(\ref{4.2})$ gives \begin{eqnarray} u_{+}-u_{-}=&-&\sqrt{\frac{\rho_{*}^{AB}-\rho_{-}}{\rho_{*}^{AB}\rho_{-}} \Big(A((\rho_{*}^{AB})^{n}-\rho_{-}^{n})-B(\frac{1}{(\rho_{*}^{AB})^{\alpha}}-\frac{1}{\rho_{-}^{\alpha}})\Big)}\nonumber\\ &-&\sqrt{\frac{\rho_{+}-\rho_{*}^{AB}}{\rho_{+}\rho_{*}^{AB}} \Big(A(\rho_{+}^{n}-(\rho_{*}^{AB})^{n})-B(\frac{1}{\rho_{+}^{\alpha}}-\frac{1}{(\rho_{*}^{AB})^{\alpha}})\Big)}.\label{4.3} \end{eqnarray} If $\lim\limits_{A,B\rightarrow0}\rho_*^{AB}=M\in(\max\{\rho_{-},\rho_{+}\},+\infty)$, then by taking the limit of $(\ref{4.3})$ as $A,B\rightarrow0$, we obtain that $u_{+}-u_{-}=0$, which contradicts with $u_{+}<u_{-}$. Therefore we must have $\lim\limits_{A,B\rightarrow0}\rho_*^{AB}=+\infty.$
By Lemma \ref{lem:4.1}, from $(\ref{4.3})$ we immediately have the following lemma.
\begin{Lemma}\label{lem:4.2} $\lim\limits_{A,B\rightarrow0}A(\rho_*^{AB})^{n}=\DF{\rho_-\rho_+}{(\sqrt{\rho_-}+\sqrt{\rho_+})^2}(u_--u_+)^2$. \end{Lemma}
\begin{Lemma}\label{lem:4.3} \begin{equation}\label{4.4} \lim\limits_{A,B\rightarrow0}u_*^{AB}=\lim\limits_{A,B\rightarrow0}\sigma_1^{AB}=\lim\limits_{A,B\rightarrow0}\sigma_2^{AB}=\sigma. \end{equation} \end{Lemma}
\noindent\textbf{Proof.} From the first equation of $(\ref{4.1})$ and $(\ref{4.2})$ for $S_{1}$ and $S_{2}$, by Lemma \ref{lem:4.1}, we have $$\lim\limits_{A,B\rightarrow0}\sigma_1^{AB}= \lim\limits_{A,B\rightarrow0}\DF{\rho_{*}^{AB} u_{*}^{AB}-\rho_-u_-}{\rho_{*}^{AB}-\rho_-}= \lim\limits_{A,B\rightarrow0}\DF{ u_{*}^{AB}-\frac{\rho_-u_-}{\rho_{*}^{AB}}}{1-\frac{\rho_-}{\rho_{*}^{AB}}}=\lim\limits_{A,B\rightarrow0}u_*^{AB},$$
$$\lim\limits_{A,B\rightarrow0}\sigma_2^{AB}= \lim\limits_{A,B\rightarrow0}\DF{\rho_+u_+-\rho_{*}^{AB} u_{*}^{AB}}{\rho_+-\rho_{*}^{AB}}= \lim\limits_{A,B\rightarrow0} \DF{ \frac{\rho_+u_+}{\rho_{*}^{AB}}-u_{*}^{AB}}{\frac{\rho_+}{\rho_{*}^{AB}}-1}=\lim\limits_{A,B\rightarrow0}u_*^{AB},$$ which immediately lead to $\lim\limits_{A,B\rightarrow0}u_*^{AB}=\lim\limits_{A,B\rightarrow0}\sigma_1^{AB}=\lim\limits_{A,B\rightarrow0}\sigma_2^{AB}$.
From the second equation of $(\ref{4.1})$, by Lemma \ref{lem:4.1}-\ref{lem:4.2}, we get
\begin{eqnarray} \lim\limits_{A,B\rightarrow0}u_*^{AB}&=&u_{-}-\lim\limits_{A,B\rightarrow0}\sqrt{\frac{\rho_{*}^{AB}-\rho_{-}}{\rho_{*}^{AB}\rho_{-}} \Big(A((\rho_{*}^{AB})^{n}-\rho_{-}^{n})-B(\frac{1}{(\rho_{*}^{AB})^{\alpha}}-\frac{1}{\rho_{-}^{\alpha}})\Big)}\nonumber\\ &=&u_{-}-\sqrt{\frac{1}{\rho_{-}}\DF{\rho_-\rho_+}{(\sqrt{\rho_-}+\sqrt{\rho_+})^2}(u_--u_+)^2}\nonumber\\ &=&u_{-}-\DF{\sqrt{\rho_+}}{\sqrt{\rho_-}+\sqrt{\rho_+}}(u_--u_+)\nonumber\\ &=&\frac{\sqrt{\rho_-}u_-+\sqrt{\rho_+}u_+}{\sqrt{\rho_-}+\sqrt{\rho_+}}=\sigma\nonumber. \end{eqnarray} The proof is completed.
\begin{Lemma}\label{lem:4.4} \begin{equation}\label{4.5} \lim\limits_{A,B\rightarrow0}\int_{\sigma_1^{AB}}^{\sigma_2^{AB}}\rho_*^{AB}d\xi=\sigma[\rho]-[\rho u], \end{equation} \begin{equation}\label{4.6} \lim\limits_{A,B\rightarrow0}\int_{\sigma_1^{AB}}^{\sigma_2^{AB}}\rho_*^{AB}u_*^{AB}d\xi=\sigma[\rho u]-[\rho u^{2}]. \end{equation}
\end{Lemma}
\noindent\textbf{Proof.} The first equations of the Rankine-Hugoniot condition $(\ref{3.11})$ for $S_{1}$ and $S_{2}$ read \begin{equation}\label{4.7} \left\{\begin{array}{ll} \sigma_1^{AB}(\rho_*^{AB}-\rho_-)=\rho_{*}^{AB} u_{*}^{AB}-\rho_-u_-,\\ \sigma_2^{AB}(\rho_+-\rho_{*}^{AB})=\rho_+u_+-\rho_{*}^{AB} u_{*}^{AB}, \end{array}\right. \end{equation} from which we have \begin{equation}\label{4.8} \lim\limits_{A,B\rightarrow0}\rho_*^{AB}(\sigma_2^{AB}-\sigma_1^{AB})=\lim\limits_{A,B\rightarrow0} (-\sigma_1^{AB}\rho_{-}+\sigma_2^{AB}\rho_{+}-\rho_+u_++\rho_-u_-)=\sigma[\rho]-[\rho u]. \end{equation}
Similarly, from the second equations of the Rankine-Hugoniot condition $(\ref{3.11})$ for $S_{1}$ and $S_{2}$ \begin{equation}\label{4.9} \left\{\begin{array}{ll} \sigma_1^{AB}(\rho_*^{AB}u_{*}^{AB}-\rho_-u_-)=\rho_{*}^{AB} (u_{*}^{AB})^{2}-\rho_-u_-^{2}+A((\rho_{*}^{AB})^{\gamma}-\rho_-^{\gamma})-B(\DF{1}{(\rho_{*}^{AB})^{\alpha}}-\DF{1}{\rho_-^{\alpha}}),\\ \sigma_2^{AB}(\rho_+u_+-\rho_{*}^{AB}u_{*}^{AB})=\rho_+u_+^{2}-\rho_{*}^{AB} (u_{*}^{AB})^{2}+A(\rho_+^{\gamma}-(\rho_{*}^{AB})^{\gamma})-B(\DF{1}{\rho_+^{\alpha}}-\DF{1}{(\rho_{*}^{AB})^{\alpha}}), \end{array}\right. \end{equation}
we obtain \begin{eqnarray} &\lim\limits_{A,B\rightarrow0}&\rho_*^{AB}u_{*}^{AB}(\sigma_2^{AB}-\sigma_1^{AB})\nonumber\\ &=&\lim\limits_{A,B\rightarrow0}(-\sigma_1^{AB}\rho_{-}u_{-}+\sigma_2^{AB}\rho_{+}u_{+}- \rho_+u_+^{2}+\rho_-u_-^{2}-A(\rho_{+}^{\gamma}-\rho_{-}^{\gamma})+B(\DF{1}{\rho_+^{\alpha}}-\DF{1}{\rho_{-}^{\alpha}}))\nonumber\\ &=&\sigma[\rho u]-[\rho u^{2}].\label{4.10} \end{eqnarray} Thus, from $(\ref{4.8})$ and $(\ref{4.10})$ we immediately get $(\ref{4.5})$ and $(\ref{4.6})$. The proof is finished.
\begin{Remark} The above lemmas shows that, as $A,B\rightarrow0$, $S_{1}$ and $S_{2}$ coincide, the intermediate density $\rho_{*}^{AB}$ becomes singular, the velocities $\sigma_{1}^{AB}$, $\sigma_{2}^{AB}$ and $u_{*}^{AB}$ for Riemann solutions of $(\ref{1.1})$-$(\ref{1.2})$ approach to $\sigma$, which are consistent with the velocity and the density of the $\delta$-shock solution to the transport equations $(\ref{1.3})$ with the same Riemann data $(\rho_\pm,u_\pm)$ in Section 2. \end{Remark}
\subsection{$\delta-$shocks and concentration}
Now we show the following theorem which is similar to Theorem 3.1 in \cite{Chen-Liu1} and characterizes the vanishing pressure limit in the case $(\rho_+,u_+)\in\rm{I\!V}(\rho_-,u_-)$ .
\begin{Theorem}\label{thm:4.1} Let $u_->u_+$ and $(\rho_+,u_+)\in\rm{I\!V}(\rho_-,u_-)$. For any fixed $A,B>0$, assume that $(\rho^{AB},u^{AB})$ is the two-shock Riemann solution of $(\ref{1.1})$-$(\ref{1.2})$ with Riemann data $(\rho_\pm,u_\pm)$ constructed in section 3. Then as $A,B\rightarrow0$, $\rho^{AB}$ and $\rho^{AB}u^{AB}$ converge in the sense of distributions, and the limit functions of $\rho^{AB}$ and $\rho^{AB}u^{AB}$ are the sums of a step function and a $\delta$-measure with weights $$\frac{t}{\sqrt{1+\sigma^2}}(\sigma[\rho]-[\rho u])\ \ and\ \ \frac{t}{\sqrt{1+\sigma^2}}(\sigma[\rho u]-[\rho u^2]),$$ respectively, which form a $\delta$-shock solution of $(\ref{1.3})$ with the same Riemann data $(\rho_\pm,u_\pm)$. \end{Theorem}
\noindent\textbf{Proof.} 1. Set $\xi=\frac{x}{t}$, for any fixed $A,B>0$, the two-shock Riemann solution can be written as \begin{equation}\nonumber (\rho^{AB},u^{AB})(\xi)=\left\{\begin{array}{ll} (\rho_-,u_-),\ \ \ \ \ \ -\infty<\xi<\sigma_1^{AB},\\ (\rho_*^{AB},u_*^{AB}),\ \ \ \sigma_1^{AB}<\xi<\sigma_2^{AB},\\ (\rho_+,u_+),\ \ \ \ \ \ \sigma_2^{AB}<\xi<\infty, \end{array}\right. \end{equation} which satisfies the following weak formulations: \begin{equation}\label{4.11} -\int_{-\infty}^\infty(u^{AB}(\xi)-\xi)\rho^{AB}(\xi)\psi'(\xi)d\xi+\int_{-\infty}^\infty\rho^{AB}(\xi)\psi(\xi)d\xi=0, \end{equation} \begin{eqnarray} -\int_{-\infty}^\infty(u^{AB}(\xi)-\xi)\rho^{AB}(\xi)u^{AB}(\xi)\psi'(\xi)d\xi+ \int_{-\infty}^\infty\rho^{AB}(\xi)u^{AB}(\xi)\psi(\xi)d\xi\nonumber\\ =\int_{-\infty}^\infty\Big(A(\rho^{AB}(\xi))^n-\DF{B}{(\rho^{AB}(\xi))^\alpha}\Big)\psi'(\xi)d\xi\label{4.12}, \end{eqnarray} for any test function $\psi\in C_0^\infty(-\infty,\infty)$.
2. By using the weak formulation $(\ref{4.11})$, we can obtain the limit of $\rho^{AB}$, which is denoted by the following identities: \begin{equation}\label{4.13} \lim\limits_{{A,B}\rightarrow0}\int_{-\infty}^\infty\Big(\rho^{AB}(\xi)-\rho_0(\xi-\sigma)\Big)\psi(\xi)d\xi=(\sigma[\rho]-[\rho u])\psi(\sigma), \end{equation} for any test function $\psi\in C_0^\infty(-\infty,\infty)$, where $$\rho_{0}(\xi)=\rho_{-}+[\rho]\chi(\xi),$$ and $\chi(\xi)$ is the characteristic function. Since the proof of $(\ref{4.13})$ is the same as step 2 in the proof of Theorem 3.1 in \cite{Chen-Liu1}, we omit it.
3. Now we turn to justify the limit of $\rho^{AB} u^{AB}$ by using the weak formulation $(\ref{4.12})$. The first integral on the left hand side of $(\ref{4.12})$ can be decomposed into \begin{equation}\label{4.14} -\Big\{\int_{-\infty}^{\sigma_1^{AB}}+\int_{\sigma_1^{AB}}^{\sigma_2^{AB}}+\int_{\sigma_2^{AB}}^\infty\Big\} (u^{AB}(\xi)-\xi)\rho^{AB}(\xi)u^{AB}(\xi)\psi'(\xi)d\xi. \end{equation}
The sum of the first and last term of $(\ref{4.14})$ is \begin{eqnarray} &&-\int_{-\infty}^{\sigma_1^{AB}}(u_--\xi)\rho_-u_-\psi'(\xi)d\xi-\int_{\sigma_2^{AB}}^\infty(u_+-\xi)\rho_+u_+\psi'(\xi)d\xi\nonumber\\ &=&-\rho_-u_-^{2}\psi(\sigma_1^{AB})+\rho_+u_+^{2}\psi(\sigma_2^{AB})+\rho_-u_- \sigma_1^{AB}\psi(\sigma_1^{AB})-\rho_+u_+\sigma_2^{AB}\psi(\sigma_2^{AB})\nonumber \\ &-&\int_{-\infty}^{\sigma_1^{AB}}\rho_-u_-\psi(\xi)d\xi-\int_{\sigma_2^{AB}}^\infty\rho_+u_+\psi(\xi)d\xi\nonumber, \end{eqnarray} which converges as $A,B\rightarrow0$ to $$([\rho u^{2}]-\sigma[\rho u])\psi(\sigma)-\int_{-\infty}^\infty(\rho_0u_0)(\xi-\sigma)\psi(\xi)d\xi.$$
The second term of $(\ref{4.14})$ is \begin{eqnarray} &&-\rho_*^{AB}u_*^{AB}\int_{\sigma_1^{AB}}^{\sigma_2^{AB}} (u_*^{AB}-\xi)\psi'(\xi)d\xi\nonumber\\ &=&-\rho_*^{AB}u_*^{AB}\Big((u_*^{AB}-\sigma_2^{AB})\psi(\sigma_2^{AB})-(u_*^{AB}- \sigma_1^{AB})\psi(\sigma_1^{AB})+\int_{\sigma_1^{AB}}^{\sigma_2^{AB}}\psi(\xi)d\xi\Big)\nonumber\\ &=&-\rho_*^{AB}u_*^{AB}(\sigma_2^{AB}-\sigma_1^{AB}) \Big(u_*^{AB}\frac{\psi(\sigma_2^{AB})-\psi(\sigma_1^{AB})}{\sigma_2^{AB}-\sigma_1^{AB}}- \frac{\sigma_2^{AB}\psi(\sigma_2^{AB})-\sigma_1^{AB}\psi(\sigma_1^{AB})}{\sigma_2^{AB}-\sigma_1^{AB}}\nonumber\\ &&+\frac{1}{\sigma_2^{AB}-\sigma_1^{AB}}\int_{\sigma_1^{AB}}^{\sigma_2^{AB}}\psi(\xi)d\xi \Big)\nonumber, \end{eqnarray} which converges as $A,B\rightarrow0$ to $$-(\sigma[\rho u-\rho u^{2}])\Big(\sigma\psi'(\sigma)-\sigma\psi'(\sigma)-\psi(\sigma)+\psi(\sigma)\Big)=0,$$ by Lemma \ref{lem:4.3}-\ref{lem:4.4}.
Now we compute the integral on the right hand side of $(\ref{4.12})$, by Lemma \ref{lem:4.1}-\ref{lem:4.3}, we obtain \begin{eqnarray} &&\int_{-\infty}^\infty\Big(A(\rho^{AB}(\xi))^n-\DF{B}{(\rho^{AB}(\xi))^\alpha}\Big)\psi'(\xi)d\xi\nonumber\\ &=&\Big\{\int_{-\infty}^{\sigma_1^{AB}}+\int_{\sigma_1^{AB}}^{\sigma_2^{AB}}+\int_{\sigma_2^{AB}}^\infty\Big\} \Big(A(\rho^{AB}(\xi))^n-\DF{B}{(\rho^{AB}(\xi))^\alpha}\Big)\psi'(\xi)d\xi\nonumber\\ &=&\int_{-\infty}^{\sigma_1^{AB}}\Big(A\rho_{-}^n-\DF{B}{\rho_{-}^\alpha}\Big)\psi'(\xi)d\xi+ \int_{\sigma_1^{AB}}^{\sigma_2^{AB}}\Big(A(\rho_{*}^{AB})^n-\DF{B}{(\rho_{*}^{AB})^\alpha}\Big)\psi'(\xi)d\xi\nonumber\\ &&+\int_{\sigma_2^{AB}}^\infty\Big(A\rho_{+}^n-\DF{B}{\rho_{+}^\alpha}\Big)\psi'(\xi)d\xi\nonumber\\ &=&\Big(A\rho_{-}^n-\DF{B}{\rho_{-}^\alpha}\Big)\psi(\sigma_1^{AB}) -\Big(A\rho_{+}^n-\DF{B}{\rho_{+}^\alpha}\Big)\psi(\sigma_2^{AB})+ \Big(A(\rho_{*}^{AB})^n-\DF{B}{(\rho_{*}^{AB})^\alpha}\Big)(\psi(\sigma_2^{AB})-\psi(\sigma_1^{AB})), \nonumber \end{eqnarray} which converge to 0 as $A,B\rightarrow0$.
Then, the integral identity $(\ref{4.12})$ yields \begin{equation}\label{4.15} \lim\limits_{A,B\rightarrow0}\int_{-\infty}^\infty\Big((\rho^{AB} u^{AB})(\xi)-(\rho_0u_0)(\xi-\sigma)\Big)\psi(\xi)d\xi=(\sigma[\rho u]-[\rho u^{2}])\psi(\sigma), \end{equation} for any test function $\psi\in C_0^\infty(-\infty,\infty)$.
4. Finally, we are in the position to study the limits of $\rho^{AB}$ and $\rho^{AB}u^{AB}$ by tracking the time-dependence of the weights of the $\delta$-measure as $A,B\rightarrow0$.
Let $\phi(x,t)\in C_0^\infty((-\infty,\infty)\times[0,\infty))$ be a smooth test function and $\tilde{\phi}(\xi,t)=\phi(\xi t,t)$. Then we have $$\lim\limits_{A,B\rightarrow0}\int_0^\infty\int_{-\infty}^\infty\rho^{AB}(\frac{x}{t})\phi(x,t)dxdt =\lim\limits_{A,B\rightarrow0}\int_0^\infty t\Big(\int_{-\infty}^\infty\rho^{AB}(\xi)\tilde{\phi}(\xi,t)d\xi\Big)dt.$$ On the other hand, from $(\ref{4.13})$, we have \begin{eqnarray*} \lim\limits_{A,B\rightarrow0}\int_{-\infty}^\infty\rho^{AB}(\xi)\tilde{\phi}(\xi,t)d\xi &=&\int_{-\infty}^\infty\rho_0(\xi-\sigma)\tilde{\phi}(\xi,t)d\xi+(\sigma[\rho ]-[\rho u])\tilde{\phi}(\sigma,t)\\ &=&\frac{1}{t}\int_{-\infty}^\infty\rho_0(x-\sigma t)\phi(x,t)dx+(\sigma[\rho ]-[\rho u])\phi(\sigma t,t). \end{eqnarray*} Combining the two relations above yields \begin{eqnarray*} \lim\limits_{A,B\rightarrow0}\int_0^\infty\int_{-\infty}^\infty\rho^{AB}(\frac{x}{t})\phi(x,t)dxdt=\int_0^\infty\int_{-\infty}^\infty\rho_0(x-\sigma t)\phi(x,t)dxdt+\int_0^\infty t(\sigma[\rho ]-[\rho u])\phi(\sigma t,t)dt. \end{eqnarray*} The last term, by definition, equals to $$\langle w_1(\cdot)\delta_S,\phi(\cdot,\cdot)\rangle,$$ with $$w_1(t)=\frac{t}{\sqrt{1+\sigma^2}}(\sigma[\rho]-[\rho u]).$$
Similarly, from $(\ref{4.15})$ we can show that \begin{eqnarray*} \lim\limits_{A,B\rightarrow0}\int_0^\infty\int_{-\infty}^\infty(\rho^{AB} u^{AB})(\frac{x}{t})\phi(x,t)dxdt=\int_0^\infty\int_{-\infty}^\infty(\rho_0u_0)(x-\sigma t)\phi(x,t)dxdt+\langle w_2(\cdot)\delta_S,\phi(\cdot,\cdot)\rangle, \end{eqnarray*} with $$w_2(t)=\frac{t}{\sqrt{1+\sigma^2}}(\sigma[\rho u]-[\rho u^2]).$$ Then we complete the proof of Theorem \ref{thm:4.1}.
\subsection{Formation of vacuum states} In this subsection, we show the formation of vacuum states in the Riemann solutions to $(\ref{1.1})$-$(\ref{1.2})$ with $(\ref{1.6})$ in the case $(\rho_+,u_+)\in R_{1}R_{2}(\rho_-,u_-)$ with $u_-<u_+$ and $\rho_\pm>0$ as the pressure vanishes.
At this monent, for fixed $A,B>0$, let $(\rho_*^{AB},u_*^{AB})$ be the intermediate state in the sense that $(\rho_-,u_-)$ and $(\rho_*^{AB},u_*^{AB})$ are connected by 1-rarefaction wave $R_1$ with speed $\lambda_1^{AB}$, $(\rho_*^{AB},u_*^{AB})$and $(\rho_+,u_+)$ are connected by 2-rarefaction wave $R_2$ with speed $\lambda_2^{AB}$. Then it follows \begin{equation}\label{4.16}R_1:\ \ \left\{\begin{array}{ll} \xi=\lambda_1^{AB}=u-\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}},\\ u-u_-=-\int_{\rho_{-}}^{\rho}\frac{\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}{\rho}d\rho,\ \ \rho_{*}^{AB}\leq\rho\leq\rho_{-}. \end{array}\right. \end{equation}
\begin{equation}\label{4.17}R_2:\ \ \left\{\begin{array}{ll} \xi=\lambda_2^{AB}=u+\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}},\\ u_+-u=\int_{\rho}^{\rho_{+}}\frac{\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}{\rho}d\rho,\ \ \rho_{*}^{AB}\leq\rho\leq\rho_{+}. \end{array}\right. \end{equation}
Now, from the second equations of $(\ref{4.16})$ and $(\ref{4.17})$, using the following integral identity \begin{eqnarray} &&\int^{\rho_{-}}_{\rho}\frac{\sqrt{An\rho_{-}^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}{\rho}d\rho\nonumber\\ &=&\frac{2}{\alpha+1}\Big(-\sqrt{An\rho_{-}^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}} +\sqrt{An\rho_{-}^{n-1}} \ln(\sqrt{An\rho_{-}^{n-1}\rho^{\alpha+1}+\alpha B}+\sqrt{An\rho_{-}^{n-1}\rho^{\alpha+1}})\Big)\Big
|_{\rho}^{\rho_{-}}\nonumber, \end{eqnarray}
it follows that the intermediate state $(\rho_*^{AB},u_*^{AB})$ satisfies
\begin{eqnarray} &&u_+-u_-\nonumber\\ &=&\int^{\rho_{-}}_{\rho_*^{AB}}\frac{\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}{\rho}d\rho+\int_{\rho_*^{AB}}^{\rho_{+}}\frac{\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}{\rho}d\rho\nonumber\\ &\leq&\int^{\rho_{-}}_{\rho_*^{AB}}\frac{\sqrt{An\rho_{-}^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}{\rho}d\rho+\int_{\rho_*^{AB}}^{\rho_{+}}\frac{\sqrt{An\rho_{+}^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}{\rho}d\rho\nonumber\\ &=&\frac{2}{\alpha+1}\Big(-\sqrt{An\rho_{-}^{n-1}+\frac{\alpha B}{\rho_{-}^{\alpha+1}}} +\sqrt{An\rho_{-}^{n-1}} \ln(\sqrt{An\rho_{-}^{n-1}\rho_{-}^{\alpha+1}+\alpha B}+\sqrt{An\rho_{-}^{n-1}\rho_{-}^{\alpha+1}})\nonumber\\ &&+\sqrt{An\rho_{-}^{n-1}+\frac{\alpha B}{(\rho_*^{AB})^{\alpha+1}}} -\sqrt{An\rho_{-}^{n-1}} \ln(\sqrt{An\rho_{-}^{n-1}(\rho_*^{AB})^{\alpha+1}+\alpha B}+\sqrt{An\rho_{-}^{n-1}(\rho_*^{AB})^{\alpha+1}})\nonumber\\ &&-\sqrt{An\rho_{+}^{n-1}+\frac{\alpha B}{\rho_{+}^{\alpha+1}}} +\sqrt{An\rho_{+}^{n-1}} \ln(\sqrt{An\rho_{+}^{n-1}\rho_{+}^{\alpha+1}+\alpha B}+\sqrt{An\rho_{+}^{n-1}\rho_{+}^{\alpha+1}})\nonumber\\ &&+\sqrt{An\rho_{+}^{n-1}+\frac{\alpha B}{(\rho_*^{AB})^{\alpha+1}}} -\sqrt{An\rho_{+}^{n-1}} \ln(\sqrt{An\rho_{+}^{n-1}(\rho_*^{AB})^{\alpha+1}+\alpha B}+\sqrt{An\rho_{+}^{n-1}(\rho_*^{AB})^{\alpha+1}})\Big ),\nonumber\\\label{4.18} \end{eqnarray} which implies the following result.
\begin{Theorem}\label{thm:4.2} Let $u_-<u_+$ and $(\rho_+,u_+)\in\rm{I}(\rho_-,u_-)$. For any fixed $A,B>0$, assume that $(\rho^{AB},u^{AB})$ is the two-rarefaction wave Riemann solution of $(\ref{1.1})$-$(\ref{1.2})$ with Riemann data $(\rho_\pm,u_\pm)$ constructed in section 3. Then as $A,B\rightarrow0$, the limit of the Riemann solution
$(\rho^{AB}, u^{AB})$ is two contact discontinuities connecting the constant states $(\rho_\pm,u_\pm)$ and the intermediate vacuum state as follows: \begin{equation}\nonumber (\rho,u)(\xi)=\left\{\begin{array}{ll} (\rho_-,u_-),\ \ \ \ \ \ -\infty<\xi\leq u_-,\\ (0,\xi),\ \ \ \ \ \ \ \ \ \ u_-\leq\xi\leq u_+,\\ (\rho_+,u_+),\ \ \ \ \ \ u_+\leq\xi<\infty, \end{array}\right. \end{equation} which is exactly the Riemann solution to the transport equations $(\ref{1.3})$ with the same Riemann data $(\rho_\pm,u_\pm)$. \end{Theorem}
Indeed, if $\lim\limits_{A,B\rightarrow0}\rho_*^{AB}=K\in(0,\min\{\rho_{-},\rho_{+}\})$, then $(\ref{4.18})$ leads to $u_+-u_-=0$, which contradicts with $u_-<u_+$. Thus $\lim\limits_{A,B\rightarrow0}\rho_*^{AB}=0$, which just means vacuum occurs. Moreover, as $A,B\rightarrow0$, one can directly derive from $(\ref{4.16})$ and $(\ref{4.17})$ that $\lambda_{1}^{AB},\ \lambda_{2}^{AB}\rightarrow u$ and two rarefaction waves $R_{1}$ and $R_{2}$ tend to two contact discontinuities $\xi=\frac{x}{t}=u_\pm$, respectively. These reach the desired conclusion.
\section{Formation of $\delta$-shocks and two-rarefaction wave as $A\rightarrow0$} In this section, we discuss the limit behaviors of Riemann solutions of $(\ref{1.1})$-$(\ref{1.2})$ with $(\ref{1.6})$ as the pressure approaches the generalized Chaplygin gas pressure, i.e., $A\rightarrow0$.
From Section 2 and 3, we can easily check that, as $A\rightarrow0$, the backward (forward) rarefaction wave curve $R_{1} (R_{2})$ of $(\ref{1.1})$-$(\ref{1.2})$ tends to the backward (forward) rarefaction wave curve $\overleftarrow{R }(\overrightarrow{R})$ of $(\ref{1.5})$, and the backward (forward) shock wave curve $S_{1} (S_{2})$ of $(\ref{1.1})$-$(\ref{1.2})$ tends to the backward (forward) rarefaction wave curve $\overleftarrow{S }(\overrightarrow{S})$ of $(\ref{1.5})$ when $0<\alpha<1$, while the backward (forward) rarefaction wave curve $R_{1} (R_{2})$ of $(\ref{1.1})$-$(\ref{1.2})$ tends to the backward (forward) contact discontinuity curve of $(\ref{1.5})$, and the backward (forward) shock wave curve $S_{1} (S_{2})$ of $(\ref{1.1})$-$(\ref{1.2})$ tends to the backward (forward) contact discontinuity curve of $(\ref{1.5})$ when $\alpha=1$ (see Fig.3).
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\subsection{Formation of $\delta$-shocks} In this subsection, we study the formation of the delta shock waves in the limit as $A\rightarrow0$ of solutions of $(\ref{1.1})$-$(\ref{1.2})$ with $(\ref{1.6})$ in the case $(\rho_+,u_+)\in \rm{V} (\rho_-,u_-)$, i.e., $u_++\sqrt{ B}\rho_+^{-\frac{\alpha+1}{2}} \leq u_--\sqrt{ B}\rho_-^{-\frac{\alpha+1}{2}}$.
\begin{Lemma}\label{lem:5.1} When $(\rho_+,u_+)\in \rm{V} (\rho_-,u_-)$, there exists a positive parameter $A_0$ such that $(\rho_+,u_+)\in S_{1}S_{2} (\rho_-,u_-)$ when $0<A<A_{0}$. \end{Lemma}
\noindent\textbf{Proof.} From $(\rho_+,u_+)\in \rm{V} (\rho_-,u_-)$, we have
\begin{equation}\label{5.1} u_++\sqrt{ B}\rho_+^{-\frac{\alpha+1}{2}} \leq u_--\sqrt{ B}\rho_-^{-\frac{\alpha+1}{2}}, \end{equation} then \begin{eqnarray} ( u_--u_+)^{2}&\geq&\big(\sqrt{ B}\rho_+^{-\frac{\alpha+1}{2}} +\sqrt{ B}\rho_-^{-\frac{\alpha+1}{2}}\big)^{2}\nonumber\\ &=&B(\rho_+^{-\alpha-1}+\rho_-^{-\alpha-1}+2\rho_{+}^{-\frac{\alpha+1}{2}}\rho_{-}^{-\frac{\alpha+1}{2}})\nonumber\\ &>&B(\rho_+^{-\alpha-1}+\rho_-^{-\alpha-1}-\rho_{+}^{-1}\rho_{-}^{-\alpha}-\rho_{-}^{-1}\rho_{+}^{-\alpha})\nonumber\\ &=&B(\frac{1}{\rho_{+}}-\frac{1}{\rho_{-}})(\frac{1}{\rho_{+}^{\alpha}}-\frac{1}{\rho_{-}^{\alpha}})\label{5.2}. \end{eqnarray}
All the states $(\rho,u)$ connected with $(\rho_-,u_-)$ by a backward shock wave $S_{1}$ or a forward shock wave $S_{2}$ satisfy \begin{equation}\label{5.3} u-u_-=-\sqrt{\frac{\rho-\rho_{-}}{\rho\rho_{-}}\Big(A(\rho^{n}-\rho_{-}^{n})-B(\frac{1}{\rho^{\alpha}}-\frac{1}{\rho_{-}^{\alpha}})\Big)},\ \ \rho>\rho_-, \end{equation} or \begin{equation}\label{5.4} u-u_-=-\sqrt{\frac{\rho-\rho_{-}}{\rho\rho_{-}}\Big(A(\rho^{n}-\rho_{-}^{n})-B(\frac{1}{\rho^{\alpha}}-\frac{1}{\rho_{-}^{\alpha}})\Big)},\ \ \rho<\rho_-. \end{equation}
When $\rho_{+}=\rho_{-}$, the conclusion is obviously true. When $\rho_{+}\neq\rho_{-}$, by taking \begin{equation}\label{5.5} (u_+-u_-)^{2}=\frac{\rho_+-\rho_{-}}{\rho_+\rho_{-}}\Big(A_{0}(\rho_+^{n}-\rho_{-}^{n})-B(\frac{1}{\rho_{+}^{\alpha}}-\frac{1}{\rho_{-}^{\alpha}})\Big), \end{equation} we have \begin{equation}\label{5.6} A_{0}=\frac{\rho_{+}\rho_{-}}{(\rho_{+}-\rho_{-})(\rho_{+}^{n}-\rho_{-}^{n})} \Big((u_+-u_-)^{2}-B(\frac{1}{\rho_{+}}-\frac{1}{\rho_{-}})(\frac{1}{\rho_{+}^{\alpha}}-\frac{1}{\rho_{-}^{\alpha}})\Big), \end{equation} which together with $(\ref{5.2})$ gives the conclusion. The proof is completed.
When $0<A<A_{0}$, the Riemann solution of $(\ref{1.1})$-$(\ref{1.2})$ with $(\ref{1.6})$ includes a backward shock wave $S_{1}$ and a forward shock wave $S_{2}$ with the intermediate state $(\rho_{*}^{A},u_{*}^{A})$ besides two constant states $(\rho_{\pm},u_{\pm})$. We then have
\begin{equation}\label{5.7}S_1:\ \ \left\{\begin{array}{ll} \sigma_1^{AB}=\DF{\rho_{*}^{A} u_{*}^{A}-\rho_-u_-}{\rho_{*}^{A}-\rho_-},\\ u_{*}^{A}-u_-=-\sqrt{\frac{\rho_{*}^{A}-\rho_{-}}{\rho_{*}^{A}\rho_{-}}\Big(A((\rho_{*}^{A})^{n}-\rho_{-}^{n})- B(\frac{1}{(\rho_{*}^{A})^{\alpha}}-\frac{1}{\rho_{-}^{\alpha}})\Big)},\ \ \rho_{*}^{A}>\rho_-, \end{array}\right. \end{equation} and \begin{equation}\label{5.8}S_2:\ \ \left\{\begin{array}{ll} \sigma_2^{AB}=\DF{\rho_{+} u_{+}-\rho_{*}^{A}u_{*}^{A}}{\rho_{+}-\rho_{*}^{A}},\\ u_{+}-u_{*}^{A}=-\sqrt{\frac{\rho_{+}-\rho_{*}^{A}}{\rho_{+}\rho_{*}^{A}}\Big(A(\rho_{+}^{n}-(\rho_{*}^{A})^{n})- B(\frac{1}{\rho_{+}^{\alpha}}-\frac{1}{(\rho_{*}^{A})^{\alpha}})\Big)},\ \ \rho_{+}<\rho_{*}^{A}. \end{array}\right. \end{equation} Here $\sigma_1^{A}$ and $\sigma_2^{A}$ are the propagation speed of $S_{1}$ and $S_{2}$, respectively. Similar to that in Section 4,
in the following, we give some lemmas to show the limit behavior of the Riemann solutions of system $(\ref{1.1})$-$(\ref{1.2})$ with $(\ref{1.6})$ as $A\rightarrow0$.
\begin{Lemma}\label{lem:5.2} $\lim\limits_{A\rightarrow0}\rho_*^{A}=+\infty.$ \end{Lemma}
\noindent\textbf{Proof.} Eliminating $u_{*}^{AB}$ in the second equation of $(\ref{5.7})$ and $(\ref{5.8})$ gives \begin{eqnarray} u_{-}-u_{+}=&&\sqrt{\frac{\rho_{*}^{A}-\rho_{-}}{\rho_{*}^{A}\rho_{-}} \Big(A((\rho_{*}^{A})^{n}-\rho_{-}^{n})-B(\frac{1}{(\rho_{*}^{A})^{\alpha}}-\frac{1}{\rho_{-}^{\alpha}})\Big)}\nonumber\\ &+&\sqrt{\frac{\rho_{+}-\rho_{*}^{A}}{\rho_{+}\rho_{*}^{A}} \Big(A(\rho_{+}^{n}-(\rho_{*}^{A})^{n})-B(\frac{1}{\rho_{+}^{\alpha}}-\frac{1}{(\rho_{*}^{A})^{\alpha}})\Big)}.\label{5.9} \end{eqnarray} If $\lim\limits_{A\rightarrow0}\rho_*^{A}=K\in(\max\{\rho_{-},\rho_{+}\},+\infty)$, then by taking the limit of $(\ref{5.9})$ as $A\rightarrow0$, we obtain that \begin{eqnarray} u_{-}-u_{+}&=&\sqrt{B}\Big(\sqrt{(\frac{1}{\rho_{-}}-\frac{1}{K})(\frac{1}{\rho_{-}^{\alpha}}-\frac{1}{K^{\alpha}})}+ \sqrt{(\frac{1}{\rho_{+}}-\frac{1}{K})(\frac{1}{\rho_{+}^{\alpha}}-\frac{1}{K^{\alpha}})}\Big)\nonumber\\ &<&\sqrt{B}\Big(\sqrt{\frac{1}{\rho_{-}}\frac{1}{\rho_{-}^{\alpha}}}+ \sqrt{\frac{1}{\rho_{+}}\frac{1}{\rho_{+}^{\alpha}}}\Big)\nonumber\\ &=&\sqrt{ B}\rho_+^{-\frac{\alpha+1}{2}} +\sqrt{ B}\rho_-^{-\frac{\alpha+1}{2}}\label{5.10} \end{eqnarray} which contradicts with $(\ref{5.1})$. Therefore we must have $\lim\limits_{A\rightarrow0}\rho_*^{A}=+\infty.$
By Lemma \ref{lem:5.2}, from $(\ref{5.9})$ we immediately have the following lemma.
\begin{Lemma}\label{lem:5.3} $\lim\limits_{A\rightarrow0}A(\rho_*^{A})^{n}<\rho_-(u_--u_+)^2$. \end{Lemma}
\begin{Lemma}\label{lem:5.4} Let $\lim\limits_{A\rightarrow0}u_*^{A}=\widehat{\sigma^{B}}$, then \begin{equation}\label{5.11} \lim\limits_{A\rightarrow0}u_*^{A}=\lim\limits_{A\rightarrow0}\sigma_1^{A} =\lim\limits_{A\rightarrow0}\sigma_2^{A}=\widehat{\sigma^{B}}\in\Big(u_++\sqrt{ \alpha B}\rho_+^{-\frac{\alpha+1}{2}}, u_--\sqrt{\alpha B}\rho_-^{-\frac{\alpha+1}{2}}\Big). \end{equation} \end{Lemma}
\noindent\textbf{Proof.} From the second equation of $(\ref{5.7})$ for $S_{1}$, by Lemma \ref{lem:4.2} and \ref{lem:4.3}, we have \begin{eqnarray} \lim\limits_{A\rightarrow0}u_*^{A}&=&u_{-}-\lim\limits_{A\rightarrow0}\sqrt{\frac{\rho_{*}^{A}-\rho_{-}}{\rho_{*}^{A}\rho_{-}} \Big(A((\rho_{*}^{A})^{n}-\rho_{-}^{n})-B(\frac{1}{(\rho_{*}^{A})^{\alpha}}-\frac{1}{\rho_{-}^{\alpha}})\Big)}\nonumber\\ &=&u_{-}-\sqrt{\frac{1}{\rho_{-}}\Big(\lim\limits_{A\rightarrow0}A(\rho_*^{A})^{n}+\frac{B}{\rho_{-}^{\alpha}}\Big)}\nonumber\\ &<& u_--\sqrt{\alpha B}\rho_-^{-\frac{\alpha+1}{2}}\nonumber\\\label{5.12}. \end{eqnarray}
Similarly, from the second equation of $(\ref{5.8})$ for $S_{2}$, we have \begin{eqnarray} \lim\limits_{A\rightarrow0}u_*^{A}&=&u_{+}+\lim\limits_{A\rightarrow0}\sqrt{\frac{\rho_{+}-\rho_{*}^{A}}{\rho_{+}\rho_{*}^{A}}\Big(A(\rho_{+}^{n}-(\rho_{*}^{A})^{n})- B(\frac{1}{\rho_{+}^{\alpha}}-\frac{1}{(\rho_{*}^{A})^{\alpha}})\Big)}\nonumber\\ &=&u_{+}+\sqrt{\frac{1}{\rho_{+}}\Big(\lim\limits_{A\rightarrow0}A(\rho_*^{A})^{n}+\frac{B}{\rho_{+}^{\alpha}}\Big)}\nonumber\\ &>&u_++\sqrt{ \alpha B}\rho_+^{-\frac{\alpha+1}{2}} \nonumber\\\label{5.13}. \end{eqnarray} Furthermore, similar to the analysis as Lemma \ref{lem:4.3}, we can obtain $\lim\limits_{A\rightarrow0}u_*^{A}=\lim\limits_{A\rightarrow0}\sigma_1^{A}=\lim\limits_{A\rightarrow0}\sigma_2^{A}=\widehat{\sigma^{B}}$. The proof is complete.
Similar to Lemma \ref{lem:4.4}, we have the following lemma. \begin{Lemma}\label{lem:5.5} \begin{equation}\label{5.14} \lim\limits_{A\rightarrow0}\int_{\sigma_1^{A}}^{\sigma_2^{A}}\rho_*^{A}d\xi=\sigma^{B}[\rho]-[\rho u], \end{equation} \begin{equation}\label{5.15} \lim\limits_{A\rightarrow0}\int_{\sigma_1^{A}}^{\sigma_2^{A}}\rho_*^{A}u_*^{A}d\xi=\sigma^{B}[\rho u]-[\rho u^{2}-\frac{B}{\rho^{\alpha}}]. \end{equation}
\end{Lemma}
\begin{Lemma}\label{lem:5.6} For $\widehat{\sigma^{B}}$ mentioned in Lemma \ref{lem:5.4}, \begin{equation}\label{5.16} \widehat{\sigma^{B}}=\sigma^{B}=\DF{\rho_+ u_+-\rho_-u_-+ \big\{\rho_+\rho_-\big((u_+-u_-)^2-(\frac{1}{\rho_+}-\frac{1}{\rho_-}) (\frac{B}{\rho_+^\alpha}-\frac{B}{\rho_-^\alpha})\big)\big\}^{\frac{1}{2}}}{\rho_+-\rho_-}, \end{equation} as $\rho_+\neq\rho_-$, and
\begin{equation}\label{5.17} \widehat{\sigma^{B}}=\sigma^{B}=\frac{u_++u_-}{2} \end{equation} as $\rho_+=\rho_-$.
\end{Lemma}
\noindent\textbf{Proof.} Let $\lim\limits_{A\rightarrow0}A(\rho_*^{A})^{n}=L$, by Lemma \ref{lem:5.4}, from $(\ref{5.12})$ and $(\ref{5.13})$ we have $$\lim\limits_{A\rightarrow0}u_*^{A}= u_{-}-\sqrt{\frac{1}{\rho_{-}}\Big(L+\frac{B}{\rho_{-}^{\alpha}}\Big)}= u_{+}+\sqrt{\frac{1}{\rho_{+}}\Big(L+\frac{B}{\rho_{+}^{\alpha}}\Big)}=\widehat{\sigma^{B}},$$ which leas to \begin{equation}\label{5.18} L+\frac{B}{\rho_{+}^{\alpha}}=\rho_{-}(u_--\widehat{\sigma^{B}})^{2}, \end{equation} \begin{equation}\label{5.19} L+\frac{B}{\rho_{-}^{\alpha}}=\rho_{+}(u_+-\widehat{\sigma^{B}})^{2}. \end{equation} Eliminating $L$ from $(\ref{5.18})$ and $(\ref{5.19})$, we have
\begin{equation}\label{5.20} (\rho_+-\rho_-)(\widehat{\sigma^{B}})^{2}-2(\rho_+ u_+-\rho_-u_-)\widehat{\sigma^{B}}+\rho_+ u_+^{2}-\rho_-u_-^{2}-B(\frac{1}{\rho_+^\alpha}-\frac{1}{\rho_-^\alpha})=0. \end{equation}
From $(\ref{5.20})$, noticing $\widehat{\sigma^{B}}\in\Big(u_++\sqrt{ \alpha B}\rho_+^{-\frac{\alpha+1}{2}}, u_--\sqrt{\alpha B}\rho_-^{-\frac{\alpha+1}{2}}\Big)$, we immediately get $(\ref{5.16})$ and $(\ref{5.17})$. The proof is finished.
\begin{Remark} The above Lemmas \ref{lem:5.2}-\ref{lem:5.5} shows that, as $A\rightarrow0$, the intermediate density $\rho_*^{A}$ becomes unbounded, the velocities $\sigma_1^{A}$ and $\sigma_2^{A}$ of shocks $S_{1}$ and $S_{2}$ and the intermediate velocity $u_{*}^{A}$ for the Riemann solutions of $(\ref{1.1})$-$(\ref{1.2})$ approach to $\sigma^{B}$, and the intermediate density becomes a singular measure simultaneously, which are consistent with the velocity and the density of the $\delta$-shock solution to the generalized Chaplygin gas equations $(\ref{1.5})$ with the same Riemann data $(\rho_\pm,u_\pm)$ in Section 2. Thus similar to Theorem \ref{thm:4.1}, we draw the conclusion as follows. \end{Remark}
\begin{Theorem}\label{thm:5.1} Let $(\rho_+,u_+)\in\rm{V}(\rho_-,u_-)$. For any fixed $A>0$, assume that $(\rho^{A},u^{A})$ is the two-shock Riemann solution of $(\ref{1.1})$-$(\ref{1.2})$ with Riemann data $(\rho_\pm,u_\pm)$ for $0<A<A_{0}$ constructed in section 3. Then as $A\rightarrow0$, $\rho^{A}$ and $\rho^{A}u^{A}$ converge in the sense of distributions, and the limit functions of $\rho^{A}$ and $\rho^{A}u^{A}$ are the sums of a step function and a $\delta$-measure with weights $$\frac{t}{\sqrt{1+(\sigma^{B})^2}}(\sigma^{B}[\rho]-[\rho u])\ \ and\ \ \frac{t}{\sqrt{1+(\sigma^{B})^2}}(\sigma^{B}[\rho u]-[\rho u^2-\frac{B}{\rho^{\alpha}}]),$$ respectively, which form a $\delta$-shock solution of $(\ref{1.5})$ with the same Riemann data $(\rho_\pm,u_\pm)$. \end{Theorem}
\subsection{Formation of two-rarefaction-wave solutions}
Now we consider the formation of the two-rarefaction-wave (two-contact-discontinuity) solution of $(\ref{1.1})$-$(\ref{1.2})$ with $(\ref{1.6})$ in the case $(\rho_+,u_+)\in \rm{I} (\rho_-,u_-)$ for $0<\alpha<1 (\alpha=1)$ as $A\rightarrow0$.
\begin{Lemma}\label{lem:6.7} When $(\rho_+,u_+)\in \rm{I} (\rho_-,u_-)$, there exists a positive parameter $A_1$ such that $(\rho_+,u_+)\in R_{1}R_{2} (\rho_-,u_-)$ when $0<A<A_{1}$. \end{Lemma}
\noindent\textbf{Proof.} All the states $(\rho,u)$ connected with $(\rho_-,u_-)$ by a backward shock wave $R_{1}$ or a forward shock wave $R_{2}$ satisfy \begin{equation}\label{5.21} u-u_-=-\int_{\rho_{-}}^{\rho}\frac{\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}{\rho}d\rho,\ \ \rho<\rho_{-}. \end{equation} or \begin{equation}\label{5.22} u-u_-=\int_{\rho_-}^{\rho_{}}\frac{\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}{\rho}d\rho,\ \ \rho>\rho_{-}. \end{equation}
When $\rho_{+}=\rho_{-}$, the conclusion is obviously true. When $\rho_{+}\neq\rho_{-}$, if $\rho_{+}>\rho_{-}$, by taking $\rho>\rho_{+}$ in $(\ref{5.22})$, we have \begin{eqnarray} u_+-u_-&=&\int_{\rho_-}^{\rho_{+}}\frac{\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}{\rho}d\rho\nonumber\\ &>&\int_{\rho_-}^{\rho_{+}}\frac{\sqrt{An\rho^{n-1}}}{\rho}d\rho\nonumber\\ &=&\frac{2\sqrt{An}}{n-1}(\rho_{+}^{\frac{n-1}{2}}-\rho_{-}^{\frac{n-1}{2}})\nonumber, \end{eqnarray} from which we can get \begin{equation}\label{5.23} A<\DF{(n-1)^{2}(u_+-u_-)^{2}}{4n(\rho_{+}^{\frac{n-1}{2}}-\rho_{-}^{\frac{n-1}{2}})^{2}}. \end{equation} Similarly, for $\rho_{+}<\rho_{-}$, we can get the same inequality as $(\ref{5.23})$. So we take \begin{equation}\label{5.24} A_{1}=\DF{(n-1)^{2}(u_+-u_-)^{2}}{4n(\rho_{+}^{\frac{n-1}{2}}-\rho_{-}^{\frac{n-1}{2}})^{2}}. \end{equation} The proof is finished.
When $0<A<A_{1}$, the Riemann solution of $(\ref{1.1})$-$(\ref{1.2})$ with $(\ref{1.6})$ includes a backward rarefaction wave $R_{1}$ and a forward rarefaction wave $R_{2}$ with the intermediate state $(\rho_{*}^{A},u_{*}^{A})$ besides two constant states $(\rho_{\pm},u_{\pm})$. We then have
\begin{equation}\label{5.25}R_1:\ \ \left\{\begin{array}{ll} \xi=\lambda_1^{AB}=u-\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}},\\ u-u_-=-\int_{\rho_{-}}^{\rho}\frac{\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}{\rho}d\rho,\ \ \rho_{*}^{A}\leq\rho\leq\rho_{-}, \end{array}\right. \end{equation} and \begin{equation}\label{5.26}R_2:\ \ \left\{\begin{array}{ll} \xi=\lambda_2^{AB}=u+\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}},\\ u_+-u=\int_{\rho}^{\rho_{+}}\frac{\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}{\rho}d\rho,\ \ \rho_{*}^{A}\leq\rho\leq\rho_{+}. \end{array}\right. \end{equation} Here $\rho_{*}^{A}$ is determined by \begin{equation}\label{5.27} u_+-u_- =\int^{\rho_{-}}_{\rho_*^{A}}\frac{\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}{\rho}d\rho+\int_{\rho_*^{A}}^{\rho_{+}}\frac{\sqrt{An\rho^{n-1}+\frac{\alpha B}{\rho^{\alpha+1}}}}{\rho}d\rho \end{equation} Furthermore, setting $(\rho_{*},u_{*})=\lim\limits_{A\rightarrow0}(\rho_*^{A},u_*^{A})$, we obtain
\begin{equation}\label{5.28} \rho_{*}^{-\frac{\alpha+1}{2}} =\frac{(\alpha+1)(u_+-u_-)}{4\sqrt{\alpha B}}+\frac{1}{2}(\rho_{+}^{-\frac{\alpha+1}{2}}+\rho_{-}^{-\frac{\alpha+1}{2}}),\ \ u_{*}=\frac{u_+-u_-}{2}+\frac{\sqrt{\alpha B}}{\alpha+1}(\rho_{+}^{-\frac{\alpha+1}{2}}-\rho_{-}^{-\frac{\alpha+1}{2}}), \end{equation} Letting $A\rightarrow0$ in $(\ref{5.25})$ and $(\ref{5.26})$, then for $0<\alpha<1 (\alpha=1)$, $R_{1}$ and $R_{2}$ become the backward rarefaction wave (the backward contact discontinuity) $\overleftarrow{R }$ and the forward rarefaction wave (the forward contact discontinuity) $\overrightarrow{R}$, respectively, as follows: \begin{equation}\label{5.29}
\overleftarrow{R}:\left\{\begin{array}{ll} \xi=\lambda_1^B=u-\sqrt{\alpha B}\rho_{}^{-\frac{\alpha+1}{2}},\\ u-\frac{2\sqrt{\alpha B}}{1+\alpha}\rho_{}^{-\frac{\alpha+1}{2}}=u_--\frac{2\sqrt{\alpha B}}{1+\alpha}\rho_{-}^{-\frac{\alpha+1}{2}},\ \ \rho_->\rho>\rho_*, \end{array}\right. \end{equation} and \begin{equation}\label{5.30} \overrightarrow{R}:\left\{\begin{array}{ll} \xi=\lambda_2^B=u+\sqrt{\alpha B}\rho_{}^{-\frac{\alpha+1}{2}},\\ u+\frac{2\sqrt{\alpha B}}{1+\alpha}\rho_{}^{-\frac{\alpha+1}{2}}=u_++\frac{2\sqrt{\alpha B}}{1+\alpha}\rho_{+}^{-\frac{\alpha+1}{2}},\ \ \rho_+>\rho>\rho_*. \end{array}\right. \end{equation}
As a conclusion, for the case $(\rho_+,u_+)\in \rm{I} (\rho_-,u_-)$ ,as $A\rightarrow0$, the two rarefaction wave $R_{1}$ and $R_{2}$ in $(\ref{5.25})$ and $(\ref{5.26})$ approach the two rarefaction waves (contact discontinuities) $\overleftarrow{R }$ and $(\overrightarrow{R})$ in $(\ref{5.29})$ and $(\ref{5.30})$ for $0<\alpha<1 (\alpha=1)$, and the intermediate state $(\rho_*^{A},u_*^{A})$ tends to the state $(\rho_*,u_*)$ in $(\ref{5.28})$. In summary, in this case, we have the following result.
\begin{Theorem}\label{thm:5.2} Let $(\rho_+,u_+)\in\rm{I}(\rho_-,u_-)$. For any fixed $A>0$, assume that $(\rho^{A},u^{A})$ is the two-shock Riemann solution of $(\ref{1.1})$-$(\ref{1.2})$ with Riemann data $(\rho_\pm,u_\pm)$ for $0<A<A_{1}$ constructed in section 3. Then as $A\rightarrow0$, the limit of the Riemann solution
$(\rho^{A}, u^{A})$ is two rarefaction waves (contact discontinuities) connecting the constant states $(\rho_\pm,u_\pm)$ and the intermediate nonvacuum state as follows: \begin{equation}\nonumber \lim\limits_{A\rightarrow0}(\rho,u)(\xi)=\left\{\begin{array}{ll} (\rho_-,u_-),\ \ \ \ \ \ -\infty<\xi\leq u_{-}-\sqrt{\alpha B}\rho_{-}^{-\frac{\alpha+1}{2}},\\ (\rho_*,u_*),\ \ \ \ \ \ \ \ u_--\sqrt{\alpha B}\rho_{-}^{-\frac{\alpha+1}{2}}\leq\xi\leq u_++\sqrt{\alpha B}\rho_{+}^{-\frac{\alpha+1}{2}},\\ (\rho_+,u_+),\ \ \ \ \ \ \ u_++\sqrt{\alpha B}\rho_{+}^{-\frac{\alpha+1}{2}}\leq\xi<\infty, \end{array}\right. \end{equation} which is exactly the Riemann solution to the (generalized) Chaplygin gas equations $(\ref{1.5})$ with the same Riemann data $(\rho_\pm,u_\pm)$ for $0<\alpha<1 (\alpha=1)$. \end{Theorem}
\section{Conclusions and discussions} In this paper, we have consider two kinds of the flux approximation limit of Riemann solutions to extended Chaplygin gas equations and studied the concentration and the formation of delta shock during the limit process. Moreover, we have proved that the vanishing pressure limit of the Riemann solutions to extended Chaplygin gas equations is just the corresponding ones to trasport equations, and when extended Chaplygin pressure approaches the generalized Chaplygin pressure, the limit of the Riemann solutions to extended Chaplygin gas equations is just the corresponding ones to the generalized Chaplygin gas equations. In fact, one can further prove that when the extended Chaplygin pressure approaches the pressure for the perfect fluid, i.e., $B\rightarrow0$ for fixed $A$, the limit of the Riemann solutions to the extended Chaplygin gas equations is just the corresponding ones to the Euler equations for perfect fluids.
On the other hand, recently, Shen and Sun have studied the Riemann problem for the nonhomogeneous tranport equations, and the nonhomogeneous (generalized) Chaplygin gas equations with coulomb-like friction, see \cite{Shen1,Shen2,Sun}. Similarly, we will also consider the Riemann problem for the nonhomogeneous extended Chaplygin gas equations with coulomb-like friction. Furthermore, we will consider its flux approximation limit and analyze the relations Riemann solutions among the nonhomogeneous extended Chaplygin gas equations, the generalized Chaplygin gas equations and the nonhomogeneous trasport equations. These will be left for our future work.
\bigbreak
\bigbreak
\end{document} | arXiv |
Observable optimal state points of subadditive potentials
Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds
April 2013, 33(4): 1389-1405. doi: 10.3934/dcds.2013.33.1389
Global well-posedness of critical nonlinear Schrödinger equations below $L^2$
Yonggeun Cho 1, , Gyeongha Hwang 2, and Tohru Ozawa 3,
Department of Mathematics, and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 561-756
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, South Korea
Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan
Received May 2011 Revised August 2012 Published October 2012
The global well-posedness on the Cauchy problem of nonlinear Schrödinger equations (NLS) is studied for a class of critical nonlinearity below $L^2$ in small data setting. We consider Hartree type (HNLS) and inhomogeneous power type NLS (PNLS). Since the critical Sobolev index $s_c$ is negative, it is rather difficult to analyze the nonlinear term. To overcome the difficulty we combine weighted Strichartz estimates in polar coordinates with new Duhamel estimates involving angular regularity.
Keywords: critical nonlinearity below $L^2$, global well-posedness, weighted Strichartz estimate, Hartree equations, angular regularity..
Mathematics Subject Classification: Primary: 35Q55; Secondary: 42B3.
Citation: Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. Global well-posedness of critical nonlinear Schrödinger equations below $L^2$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1389-1405. doi: 10.3934/dcds.2013.33.1389
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Yonggeun Cho Gyeongha Hwang Tohru Ozawa | CommonCrawl |
\begin{document}
\begin{abstract} In the present work we study how the \textit{standard homogenization commutator}, a random field that plays a central role in the theory of fluctuations, quantitatively decorrelates on large scales.
\end{abstract}
\let\thefootnote\relax\footnotetext{ $^{1}$Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany
\tt $\ [email protected] }
\title{Locality properties of standard homogenization commutator}
{\small \textbf{Mathematics Subject Classification (2020).} 35B27, 35R60, 35J15}
{\small \textbf{Keywords.} stochastic homogenization; fluctuations; homogenization commutator; covariance estimate; higher-order theory}
\section{Introduction}
This work amounts to homogenization theory for uniformly elliptic linear equations in divergence-form. That is, we consider \begin{equs} \label{eq} \nabla \cdot \left(a \left( \frac{\cdot}{\varepsilon} \right) \nabla u_{\varepsilon}+ f \right)=0, \quad \text{ in } \ \mathbb R^{d} \end{equs} with $f \in C^{\infty}_{c}(\mathbb R^{d})^{d}$ and $a$ being random (not necessarily symmetric) coefficients that satisfy \begin{equs}
\xi \cdot a(x) \xi \geq \lambda |\xi|^{2}, \quad \xi \cdot a^{-1}(x) \xi \geq |\xi|^{2} \quad \text{ for any } \ x,\xi \in \mathbb R^{d} \end{equs} for some positive constant $\lambda$. In the following we denote by $\langle \cdot \rangle$ the expectation with respect to the underlying measure on $a$'s.
Since the works of Papanicolaou and Varadhan \cite{PV81} and Kozlov \cite{K79} we know that in stationary and ergodic random environments, the equation (\ref{eq}) homogenizes as $\varepsilon \to 0$ to an equation \begin{equs} \label{eq-hom} \nabla \cdot \left(\bar{a} \nabla \bar{u}+ f \right)=0, \quad \text{ in } \ \mathbb R^{d} \end{equs} where the coefficients $\bar{a}$ are constant and deterministic. More precisely, the effective coefficients $\bar{a}$ are given by $\bar{a} e_{i} := \big \langle a(\nabla \phi_{i}+e_{i}) \big \rangle$ where the corrector $\phi_{i}$ is the (up to a random additive constant) unique a.s. solution of the equation $\nabla \cdot a(\nabla \phi_{i}+e_{i})=0$ in $\mathbb R^{d}$, with $\nabla \phi_{i}$ being stationary, centered and having finite second moments. Furthermore, the aforementioned qualitative theory states that $\nabla u_{\varepsilon}$ weakly converges to $\nabla \bar{u}$ with the oscillations of $u_{\varepsilon}$ being captured by those of the so-called two-scale expansion $\left(1+\varepsilon \phi_{i} \left( \frac{\cdot}{\varepsilon} \right) \partial_{i}\right)\bar{u}$ in a strong norm.
The quantitative theory of stochastic homogenization for (\ref{eq}) (namely, the study of the error for the approximation $\nabla u_{\varepsilon} \approx \nabla ( (1+\varepsilon \phi_{i} \left( \frac{\cdot}{\varepsilon} \right) \partial_{i})\bar{u} )$), has been also well-developed during the last decade. For that a suitable quantification of the ergodicity assumption is needed. Most of the developments are based on either a spectral gap inequality or on a finite range of dependence assumption. Here we adopt the spectral gap inequality approach which means that we have a version of Poincar\'e's inequality in infinite dimensions. Roughly speaking, we assume that the variance of an observable defined on the space of coefficient fields described above, can be estimated by a suitable norm of its functional derivative with respect to $a$, which describes the sensitivity of an observable (for instance $\nabla u_{\varepsilon}$ or $\nabla \phi_{i}$) under changes on $a$'s. This direction of research was initiated by Gloria and Otto in \cite{GO11} and \cite{GO12}, inspired by the strategy introduced by Nadaff and Spencer in \cite{NS98} . On the other hand, finite range of dependence and mixing conditions have been introduced by Yurinski\u{\i} in \cite{Y86} and further studied by Armstrong and Smart in \cite{AS16} (we refer the reader to \cite{AKM} for a detailed description of the progress in this direction).
Next to the spatial oscillations of $\nabla u_{\varepsilon}$ described above, stochastic homogenization also studies the random fluctuations of observables of the form $\int g \cdot \nabla u_{\varepsilon}$. One of the first results in this direction is given in \cite{GM16} where the authors show that $\varepsilon^{-d/2} \int g \cdot \big ( \nabla u_{\varepsilon} - \big \langle \nabla u_{\varepsilon} \big \rangle \big )$ converges in law to a Gaussian random variable. In the same work the authors showed that the four-tensor Q introduced in \cite{MO16} describes explicitly the leading-order of the variance of $\varepsilon^{-d/2} \int g \cdot \nabla u_{\varepsilon}$. Moreover, they observed that the limiting variance of $\varepsilon^{-d/2}\int g \cdot \nabla u_{\varepsilon}$ is not captured by that of $\varepsilon^{-d/2} \int g \cdot \nabla \left( \left(1+\varepsilon \phi_{i} \left( \frac{\cdot}{\varepsilon} \right) \partial_{i}\right)\bar{u} \right)$ as one would naturally expect. As discovered in \cite{DGO20} (for the random conductance model) and \cite{DO20} (in the continuum Gaussian setting) a reasonable quantity to look at, when it comes to fluctuations, is the \textit{homogenization commutator} \begin{equs} \Xi^{1}_{\varepsilon}[\nabla u_{\varepsilon}]:=\left( a\left( \frac{\cdot}{\varepsilon} \right) - \bar{a} \right) \nabla u_{\varepsilon}. \end{equs} This notion first introduced in \cite{AKM17} and it is highly related to H-convergence which is in fact equivalent to $\Xi^{1}_{\varepsilon} \rightharpoonup 0$ as $\varepsilon \to 0$. The motivation to consider $\Xi^{1}_{\varepsilon}$ while studying fluctuations of $\int g \cdot \nabla u_{\varepsilon}$ comes from the following observation: for the Lax-Milgram solution $\bar{v}$ to the dual equation $\nabla \cdot (\bar{a}^{*}\nabla \bar{v} + g)=0$ we have \begin{equs} \int g \cdot \nabla u_{\varepsilon} = -\int \nabla \bar{v} \cdot \bar{a}\nabla u_{\varepsilon} =\int \nabla \bar{v} \cdot \left( a\left( \frac{\cdot}{\varepsilon} \right) - \bar{a} \right)\nabla u_{\varepsilon} + \int \nabla \bar{v} \cdot f. \end{equs} That is, the quantity of interest can be written in terms of homogenization commutator plus a deterministic term (which does not contribute to fluctuations). Subsequently, in \cite{DGO20} and \cite{DO20} the authors turned their attention to the study of $\Xi^{1}_{\varepsilon}$ to realize that its fluctuations are captured by those of its two-scale expansion. More precisely, for $\tilde{F}_{\varepsilon}:= \int g \cdot \left[ \left(a\left( \frac{\cdot}{\varepsilon} \right) - \bar{a} \right) \nabla u_{\varepsilon} - \left( a\left( \frac{\cdot}{\varepsilon} \right) - \bar{a} \right) \left( e_{i} + \nabla \phi_{i} \left(\frac{\cdot}{\varepsilon }\right) \right) \partial_{i} \bar{u} \right]$, it holds (for $d \geq 3$) \begin{equs} \varepsilon^{-d/2} \big \langle \big ( \tilde{F}_{\varepsilon} - \langle \tilde{F}_{\varepsilon} \rangle \big )^{2}\big \rangle^{1/2} \lesssim \varepsilon. \end{equs} This reveals the special role that the so-called \textit{standard homogenization commutator}, \begin{equs} \Xi^{o,1}_{\varepsilon}:=\left( a\left( \frac{\cdot}{\varepsilon} \right) - \bar{a} \right) \left( Id + \nabla \phi \left(\frac{\cdot}{\varepsilon }\right)\right), \end{equs} plays in the theory of fluctuations. In \cite{DGO20} and \cite{DO20} the limiting covariance structure of $\int g \cdot \Xi^{o,1}_{\varepsilon}e_{i}$ is quantitatively characterized and a (quantitative) CLT-type result is obtained for that quantity.
In the present work our aim is to explain how $\Xi^{o,1}_{\varepsilon}$ decorrelates when averaged over balls which are far enough. From now on we set, for convenience, $\varepsilon=1$ and work with macroscopic observables. More precisely, we consider the following macroscopic test functions with supports that are quantitatively ''far'', \begin{equs} \label{test-fcts} g(x) =R^ {-d} \eta \left(\frac{x}{R} \right) \in C^\infty_c(B_R) \quad \text{ and } \quad g'(x) =g(x-Le) \in C^\infty_c(B_R(Le)), \end{equs}
where $\eta \in C^\infty_c(B_1)$ with $|D^{k} \eta| \leq C_{k,d}$, $1\ll R \ll L<\infty$ and $e \in \mathbb R^d$ with $|e|=1$. Our aim is to examine how \begin{equs} \label{correlation} \Big \langle \int_{\mathbb R^d} g(x) \Xi_{ij}^{o,1}(x) \dx \int_{\mathbb R^d} g'(y) \Xi_{ml}^{o,1}(y) \dy \Big \rangle \end{equs} decays in terms of both $R$ (which amounts to the macroscopic scale) and $L$ (which amounts to the distance between the supports), for every $1 \leq i,j \leq d$. Here, instead of working with the variance first and then appeal to a polarization argument, which would give no information on $L$, we work with the covariance using estimate (\ref{cov_est}) which is an immediate consequence of Hellfer-Sj\"ostrand representation formula. Namely, this work is a refinement of the analysis in \cite{DGO20} and \cite{DO20} on the locality properties of $\Xi^{o,1}$.
As first observed in \cite{DGO20}, one of the main features of $\Xi^{o,1}$ is the approximately local behaviour of its functional derivative in terms of the coefficient field $a$. This property is seen here for the derivative of $F_{ij}^{o,1}:=\int_{\mathbb R^d} g(x) \Xi_{ij}^{o,1}(x) \dx$. Precisely, in Proposition \ref{repr_form_prop}, we derive \begin{equs} \label{repr_form_1}
\frac{\partial F_{ij}^{o,1}}{\partial a} = (\nabla \phi _{i}^{1}+e_{i}) \otimes \left( (\nabla \phi_{j}^{*} +e_{j} )g+ \phi_{j}^{*} \nabla g) + \nabla h_{j} \right)
\end{equs}
with $- \nabla \cdot a^{*} \nabla h_{j} = \nabla \cdot \left( ( a^{*}\phi_{j}^{*} - \sigma_{j}^{*} ) \nabla g \right)$. We see that $\frac{\partial F_{ij}^{o,1}}{\partial a}$ is given by the sum of two local terms plus the error term which is described through the solution of an auxiliary equation with r.h.s given in terms of $\nabla g$ (thus it would be of order $o(R^{-1})$). Next we plug (\ref{repr_form_1}) into the covariance estimate (\ref{cov_est}) which allows to estimate (\ref{correlation}) in terms of the derivative of $F_{ij}^{o,1}$ keeping the advantage of integrating against both $g$ and $g'$ so we could obtain a decay of order $\mathrm{o}\left( R^{-1}L^{-d}\right)$ (with a logarithmic correction). Main tools in the estimation of the r.h.s. of the covariance estimate will be the stochastic moment bounds for the correctors and the large-scale regularity theory.
In addition to the above estimate, we also study how the order of the decay is improved when we employ higher-order homogenization theory. More precisely, we consider the higher-order standard homogenization commutator \begin{equs} \label{def_stand_comm} \Xi_{ij}^{o,n}:= e_j \cdot (a-\bar{a}^1) (\nabla \phi _i^1+e_i) - \sum_{k=2}^n (-1)^{k-1} \bar{a}^{*,k}_{ji_1\dots i_{k-2}}e_{i_{k-1}} \cdot \partial ^{k-1}_{i_1 \dots i_{k-1}} \nabla \phi _i^1 \end{equs}
where, $1 \leq i,j \leq d$ and $1 < n \leq \tilde{d}$ for $\tilde{d}$ being the smallest integer larger than $\frac{d}{2}$ and $\bar{a}^{*,k}_{ji_1\dots i_{k-2}}$ the higher-order effective coefficients (see subsection \ref{higher-order} for precise definitions). Moreover, we assume that we are in a Gaussian framework characterized by a covariance function with integrable decay of order $\mathrm{o}\left( |x|^{-d - \alpha_{0}} \right)$, for $0<\alpha_{0} \leq \frac{d}{2}$ (see subsection \ref{ensemble} for a precise description of the class of ensembles we consider). We derive that the correlation has order $\mathrm{o}\left( R^{-d/2}L^{-d/2 -\alpha_{0}}\right)$ (up to a logarithmic correction) as well. That is, we see that the standard homogenization commutator inherits the property of $a$'s being weakly correlated, keeping the order of decay as well. The following is our main result.
\begin{thm}[Main Theorem] \label{mainthm} If $d> 2$ is odd then \begin{equs}P_{ijml}^{o,\frac{d+1}{2}}:=\Big \langle \int_{\mathbb R^d} g(x) \Xi_{ij}^{o,\frac{d+1}{2}}(x) \dx \int_{\mathbb R^d} g'(y) \Xi_{ml}^{o,\frac{d+1}{2}}(y) \dy \Big \rangle \lesssim R^{-\frac{d}{2}} L^{-\frac{d}{2}-\alpha_{0}} \ln \left( \frac{L}{R}\right). \end{equs} If $d\geq 2$ is even then \begin{equs} P_{ijml}^{o,\frac{d}{2}}:=\Big \langle \int_{\mathbb R^d} g(x) \Xi_{ij}^{o,\frac{d}{2}}(x) \dx \int_{\mathbb R^d} g'(y) \Xi_{ml}^{o,\frac{d}{2}}(y) \dy \Big \rangle \lesssim R^{-\frac{d}{2}}(\ln R)^{\frac{1}{2}} L^{-\frac{d}{2}-\alpha_{0}} \ln \left( \frac{L}{R}\right). \end{equs} Here $g$ and $g'$ are as described in (\ref{test-fcts}), $L>>R>>1$ and $\alpha_{0}$ is so that (\ref{c-decay}) holds. \end{thm}
\section{Preliminaries}
\subsection{Assumptions on the ensemble} \label{ensemble} We first describe the framework we adopt and the main ingredients we need for our analysis which hold true in this framework.
Let $\langle \cdot \rangle$ be a stationary and centered Gaussian ensemble of scalar fields $G$ on $\mathbb R^{d}$ characterized by its covariance function $c(x)=\langle G(x)G(0) \rangle$ which is assumed to satisfy the following \begin{equs} \label{c-decay}
|c(x)| \leq \frac{C_{0}}{(1+|x|)^{d+\alpha_{0}}} \end{equs} for some constants $0<\alpha_{0} \leq \frac{d}{2}$ and $C_{0}>0$. Moreover, we assume that the (always non-negative) Fourier transform of $c$ satisfies \begin{equs} \label{c-fourier}
\mathcal{F}c(k) \leq \frac{C_{1}}{(1+ |k|)^{d+2\alpha_{1}}} \end{equs} for some $\alpha_{1} \in (0,1)$ and some constant $C_{1}>0$. Next we identify $\langle \cdot \rangle$ with its push-forward under the map: $G \mapsto a$, where $a(x):=A(G(x))$, $A:\mathbb R \to \mathbb R^{d \times d}_{\lambda}$ a Lipschitz function and $\mathbb R^{d \times d}_{\lambda}$ the space of $\lambda$-elliptic matrices.
Note that in the framework adopted here, one can ensure that a spectral gap inequality (see for instance Lemma 3.1 in \cite{JO20}). Furthermore, Hellfer-Sj\"ostrand representation formula
\begin{equs} \label{HS}
\cov[ F,H] =\Big \langle \int \int \frac{\partial F }{\partial G(x)} c(x-y) (1+\mathcal{L})^{-1} \frac{\partial H}{\partial G(y)} \dy \dx \Big \rangle
\end{equs}
holds for every suitable random variables $F$ and $H$ (we refer the reader to section 4 in \cite{DO20} for precise statements and the definition of the differential operator $\mathcal{L}$ - here we only use the fact that this operator is bounded to get (\ref{cov_est})). We denote by $\frac{\partial F }{\partial G(x)}= \frac{\partial F }{\partial a(x)} A'(G(x))$, where the random tensor $\frac{\partial F_{ij}^{o,n}}{\partial a(x)}$ stands for the functional derivative of $F$ defined through
\begin{equs} \label{def-frechet}
\lim_{t \to 0} \frac{F(a+t \delta a) -F(a)}{t} = \int \frac{\partial F}{\partial a(x)} : \delta a(x) \dx.
\end{equs} One of the main ingredients we use in this paper is the following covariance estimate (which is an immediate consequence of (\ref{HS}) and of the fact $\Big \lvert \frac{\partial F }{\partial G(x)} \Big \rvert \lesssim \Big \lvert \frac{\partial F }{\partial a(x)} \Big \rvert$ )
\begin{equs} \label{cov_est}
\cov[ F,H] \lesssim \int \Big \langle \Big \lvert \frac{\partial F }{\partial a(x)} \Big \rvert ^{2} \Big \rangle ^{1/2} \int |c(x-y)| \Big \langle \Big |\frac{\partial H}{\partial a(y)}\Big |^{2}\Big \rangle ^{1/2} \dy \dx.
\end{equs}
Finally, let us also mention that for the class of ensembles we consider here (in particular, because of (\ref{c-fourier})) we can show that realizations $G$ (thus $a$'s) are H\"older continuous with H\"older norms having bounded stochastic moments, that is, \begin{equs} \label{a-holder}
\big \langle ||a||_{C^{\alpha'}(B_{1})}^{q} \big \rangle ^{1/q} \lesssim_{q,\alpha'} 1 \end{equs} for any $0<\alpha'<\alpha_{1}$ and $q \geq 1$ (see Appendix A in \cite{JO20} for a proof).
Note that by $\lesssim$ we mean $\leq$ times a constant which depends only on $d, \lambda, \alpha_{0}, \alpha_{1}, ||A||_{C^{1}}$ and on quantities related to $c$. Moreover, note that we use the Einstein's summation convention.
\subsection{Higher-order theory} \label{higher-order}
For reader's convenience let us first introduce the notions of higher-order correctors and effective coefficients and their main properties that we use in the following (see Definition 2.1 and Proposition 2.2 in \cite{DO20}).
\begin{defn} \label{correctors} Let $\tilde{d}$ be the smallest integer larger than $\frac{d}{2}$. The correctors $(\phi^{n})_{0 \leq n \leq \tilde{d}}$, the flux correctors $(\sigma^{n})_{0 \leq n \leq \tilde{d}}$ and the effective coefficients $(\bar{a}^{n})_{1 \leq n \leq \tilde{d}}$ are inductively defined as follows \begin{enumerate} \item[$\bullet$] $\phi^{0}:=1$ and $\phi^{n}:= (\phi^{n}_{i_{1} \dots i_{n}})_{1\leq i_{1}, \dots,i_{n} \leq d}$ for any $1 \leq n \leq \tilde{d}$, with $\phi^{n}_{i_{1} \dots i_{n}}$ a scalar field satysfying \begin{equs} \label{corr-eq} -\nabla \cdot a \nabla \phi^{n}_{i_{1} \dots i_{n}}= \nabla \cdot \left( \left(a \phi^{n-1}_{i_{1} \dots i_{n-1}} - \sigma^{n-1}_{i_{1} \dots i_{n-1}}\right) e_{i_{n}} \right) \end{equs} with $\nabla \phi^{n}_{i_{1} \dots i_{n}}$ being stationary, centered and having finite second moments. \item[$\bullet$] $\bar{a}^{n}:= (\bar{a}^{n}_{i_{1} \dots i_{n-1}})_{1\leq i_{1}, \dots,i_{n-1} \leq d}$ for any $1 \leq n \leq \tilde{d}$, with $\bar{a}^{n}_{i_{1} \dots i_{n-1}}$ the matrix given by \begin{equs} \label{effectivecoeff} \bar{a}^{n}_{i_{1} \dots i_{n-1}} e_{i_{n}}:= \Big \langle a \left( \nabla \phi^{n}_{i_{1} \dots i_{n}}+ \phi^{n-1}_{i_{1} \dots i_{n-1}}e_{i_{n}}\right) \Big \rangle. \end{equs} \item[$\bullet$] $\sigma^{0}:=0$ and $\sigma^{n}:= (\sigma^{n}_{i_{1} \dots i_{n}})_{1\leq i_{1}, \dots,i_{n} \leq d}$ for any $1 \leq n \leq \tilde{d}$, with $\sigma^{n}_{i_{1} \dots i_{n}}$ a skew-symmetric matrix satysfying \begin{equs} \label{fluxcorr} \nabla \cdot \sigma^{n}_{i_{1} \dots i_{n}} := a \nabla \phi^{n}_{i_{1} \dots i_{n}} + \left(a \phi^{n-1}_{i_{1} \dots i_{n-1}} - \sigma^{n-1}_{i_{1} \dots i_{n-1}}\right) e_{i_{n}} -\bar{a}^{n}_{i_{1} \dots i_{n-1}} e_{i_{n}} \end{equs} with $\nabla \sigma^{n}_{i_{1} \dots i_{n}}$ being stationary, centered and having finite second moments. Here we denote by $(\nabla \cdot \sigma^{n}_{i_{1} \dots i_{n}})_{j}= \sum_{k=1}^{d} \partial_{k} (\sigma^{n}_{i_{1} \dots i_{n}})_{jk}$, $1 \leq j \leq d$. \end{enumerate} \end{defn}
\begin{prop} \label{momentbounds} All the quantities in definition \ref{correctors} exist and they satisfy for all $1 \leq n \leq \tilde{d}$ and $p \geq 1$ \begin{equs} \label{momentboundsest}
|\alpha^{n}| \leq 1, \quad \big \langle |\nabla \phi^{n}|^{p} \big \rangle^{1/p}\lesssim_{p,n} 1, \quad \big \langle | \phi^{n}(x)|^{p} \big \rangle^{1/p} + \big \langle | \sigma^{n}(x)|^{p} \big \rangle^{1/p}\lesssim_{p,n} \mu_{d,n}(x) \end{equs} where \begin{equs} \mu_{d,n}(x) := \begin{cases} 1, &\quad \quad \text{ if } \ n < \tilde{d} \\
\ln^{1/2}(2+|x|), &\quad \quad \text{ if } \ n = \tilde{d} \ \text{ and } \ d \ \text{ even}\\
1+|x|^{1/2}, &\quad \quad \text{ if } \ n = \tilde{d} \ \text{ and } \ d \ \text{ odd}. \end{cases} \end{equs} \end{prop}
Next we explain why definition (\ref{def_stand_comm}) is reasonably derived from the definitions of higher-order commutators given in \cite{DO20}. The \textit{homogenization commutator} $\Xi^{1}[\nabla w] = (a-\bar{a}) \nabla w$ naturally extends to the higher-order as \begin{equs} \Xi^{n}[\nabla w]:= (a- \sum_{k=1}^n \bar{a}^{k}_{i_1\dots i_{k-1}} \partial ^{k-1}_{i_1 \dots i_{k-1}} ) \nabla w= (a-\bar{a}) \nabla w - \sum_{k=2}^n \bar{a}^{k}_{i_1\dots i_{k-1}} \partial ^{k-1}_{i_1 \dots i_{k-1}} \nabla w.\\ \ \label{def_comm} \end{equs} Then the \textit{standard homogenization commutator} $\Xi^{o,n} [\nabla \bar{w}]$ is given by $\Xi^{n}$ applied to the nth-order Taylor polynomial of the nth-order two-scale expansion of $\bar{w}$. However a more explicit formula for $\Xi^{o,n} [\nabla \bar{w}]$ is available (see Lemma 3.5 in \cite{DO20}) and if this formula is applied to the linear functions $\bar{w} (x)=x_{i}$ we get \begin{equs} \Xi_{ij}^{o,n}= e_{j} \cdot \Xi^{n} [\nabla \phi_{i}^{1}+e_{i}]. \end{equs} Now for our analysis, especially when deriving representation formulas for the Malliavin derivatives, it is more convenient to work with (\ref{def_stand_comm}) which is defined through the transposes of the higher-order effective coefficients. For (\ref{def_stand_comm}), we use the following alternative representation of $\Xi^{n}$ \begin{equs} \label{def_comm_trans} e_{j} \cdot \Xi^{n} [\nabla w] = a^{*} e_{j} \cdot \nabla w - \sum_{k=1}^n (-1)^{k-1} \bar{a}^{*,k}_{ji_1\dots i_{k-2}}e_{i_{k-1}} \cdot \partial ^{k-1}_{i_1 \dots i_{k-1}}\nabla w. \end{equs} The above is a consequence of Lemma 2.4 in \cite{DO20} which extends the fact $\bar{a}^{*,1}= (\bar{a}^{1})^{*}$ to the higher-order, \begin{equs} \label{symmetries} Sym_{i_{1} \dots i_{n}} (e_{j } \cdot \bar{a}^{n}_{i_1\dots i_{n-1}} e_{{i_{n}}}) = (-1)^{n+1} Sym_{i_{1} \dots i_{n}} (e_{i_{n}} \cdot \bar{a}^{*,n}_{ji_1\dots i_{n-2}} e_{{i_{n-1}}})
\end{equs}
where $Sym_{i_{1} \dots i_{k}} T_{i_{1} \dots i_{k}} := \frac{1}{k!} \sum _{\sigma \in S_{k}} T_{i_{\sigma (1)} \dots i_{\sigma (k)} }$, $T$ a kth-order tensor and $S_{k}$ the set of permutations of $\{1, \dots k \}$.
\section{Proof of Main Theorem}
Since we intend to bound quantity (\ref{correlation}) via the covariance estimate (\ref{cov_est}), we first derive a suitable representation formula for the derivative of $F_{ij}^{o,n} := \int_{\mathbb R^d} g(x) \Xi_{ij}^{o,n}(x) \dx$. Our intention is to get as many derivatives as possible for the r.h.s of the equation that the term $\nabla h_{j}$ satisfies. We show the following
\begin{prop} [Representation formula] \label{repr_form_prop}
\begin{equs} \label{repr_form}
\frac{\partial F_{ij}^{o,n}}{\partial a} = (\nabla \phi _{i}^{1}+e_{i}) \otimes \left( \sum_{k=0}^{n} \partial^{k}_{i_{1} \dots i_{k}} g (\nabla \phi_{ji_{1} \dots i_{k}}^{*,k+1} +e_{i_{k}} \phi_{ji_{1} \dots i_{k-1}}^{*,k}) + \nabla h_{j} \right)
\end{equs}
with $h_{j}$ solving
\begin{equs} \label{eq-hj}
- \nabla \cdot a^{*} \nabla h_{j} = \nabla \cdot \left( ( a^{*}\phi_{ji_{1} \dots i_{n-1}}^{*,n} - \sigma_{ji_{1} \dots i_{n-1}}^{*,n} ) \nabla \partial^{n-1}_{i_{1} \dots i_{n-1}} g \right).
\end{equs}
Note that in the sum appears in (\ref{repr_form}), we use the convention that the $k=0$-term is just $\nabla \phi_{j}^{*,1}+e_{j}$, while the $k=n$-term is just $\partial^{n}_{i_{1} \dots i_{n}} g \ e_{i_{n}} \phi_{ji_{1} \dots i_{n-1}}^{*,n}$ (see definition \ref{correctors}).
\end{prop}
\begin{proof}
We have (integrating by parts)
\begin{equs}
{}&\frac{F_{ij}^{o,n}(a+t \delta a) -F_{ij}^{o,n}(a)}{t} \\
&\quad = \frac{1}{t} \int g e_{j} \cdot (a+t \delta a-\bar{a}^1) (\nabla \phi _i^1(a+t \delta a)+e_i) \\
&\quad \ \ -\frac{1}{t} \int \sum_{k=2}^n (-1)^{2(k-1)}\ \partial ^{k-1}_{i_1 \dots i_{k-1}}g \ \bar{a}^{*,k}_{ji_1\dots i_{k-2}}e_{i_{k-1}} \cdot \nabla \phi _i^1(a+t \delta a) \\
&\quad \ \ -\frac{1}{t} \int g e_j \cdot (a-\bar{a}^1) (\nabla \phi _i^{1}(a)+e_i) \\
&\quad \ \ +\frac{1}{t} \int \sum_{k=2}^n (-1)^{2(k-1)}\ \partial ^{k-1}_{i_1 \dots i_{k-1}}g \ \bar{a}^{*,k}_{ji_1\dots i_{k-2}}e_{i_{k-1}} \cdot \nabla \phi _i^{1}(a) \\ &\quad = \int g e_{j} \cdot \delta a \left(e_{i} + \nabla \phi_{i}^{1}(a+ t \delta a)\right) + \int g e_{j} \cdot (a- \bar{a}^{1}) \frac{ \nabla \phi_{i}^{1}(a+ t \delta a) -\nabla \phi_{i}^{1}(a) }{t} \\ &\quad \ \ -\int \sum_{k=2}^n \partial ^{k-1}_{i_1 \dots i_{k-1}}g \ \bar{a}^{*,k}_{ji_1\dots i_{k-2}}e_{i_{k-1}} \cdot \frac{\nabla \phi _i^1(a+t \delta a) -\nabla \phi_{i}^{1}(a) }{t}
\end{equs}
where $\nabla \phi _i^1(a+t \delta a) -\nabla \phi_{i}^{1}(a) := \nabla \psi _{i}(a,t \delta a)$, with $- \nabla \cdot (a+t \delta a) \frac{\nabla \psi_{i}}{t}= \nabla \cdot \delta a (\nabla \phi_{i}^{1}(a)+e_{1})$ (see section 3.4 in \cite{GNO19}). Thus letting $t \to 0$ we get
\begin{equs}
\lim_{t \to 0}\frac{F_{ij}^{o,n}(a+t \delta a) -F_{ij}^{o,n}(a)}{t} &= \int g e_{j} \cdot \delta a \left(e_{i} + \nabla \phi_{i}^{1}\right) + \int ge_{j} \cdot (a-\bar{a}^{1}) \nabla \delta \phi_{i} \\
&\ \ \ - \int \sum_{k=2}^n \partial ^{k-1}_{i_1 \dots i_{k-1}}g \ \bar{a}^{*,k}_{ji_1\dots i_{k-2}}e_{i_{k-1}} \cdot \nabla \delta \phi _i \label{frechet-der}
\end{equs}
where
\begin{equs} \label{eq-for-delta phi}
-\nabla \cdot a \nabla \delta \phi_{i} = \nabla \cdot \delta a (\nabla \phi_{i}^{1}+e_{i}).
\end{equs}
Next we further analyze the r.h.s of (\ref{frechet-der}) to get the desired representation formula. The main ingredient we use is the following relation between the correctors (see (\ref{fluxcorr}) in definition \ref{correctors}) \begin{equs}\label{corr-relation} \left( a \phi^{k-1}_{i_{1}\dots i_{k-1}} - \sigma^{k-1}_{i_{1}\dots i_{k-1}} \right) e_{i_{k}}= -a \nabla \phi^{k}_{i_{1}\dots i_{k}} + \nabla \cdot \sigma^{k}_{i_{1}\dots i_{k}} - \bar{a}^{k}_{i_{1}\dots i_{k-1}}e_{i_{k}} \end{equs} which for $k=1$ reduces to the well known $(a-\bar{a}^{1})e_{i}=-a \nabla \phi^{1}_{i} + \nabla \cdot \sigma^{1}_{i}$.
We show by induction on $n$ the following \begin{equs} \lim_{t \to 0}\frac{F_{ij}^{o,n}(a+t \delta a) -F_{ij}^{o,n}(a)}{t} &= \int \sum_{k=0}^{n} \partial^{k}_{i_{1} \dots i_{k}} g (\nabla \phi_{ji_{1} \dots i_{k}}^{*,k+1} +e_{i_{k}} \phi_{ji_{1} \dots i_{k-1}}^{*,k}) \cdot \delta a (\nabla \phi _{i}^{1}+e_{i}) \\ &\ \ \ + \int \left( a^{*}\phi_{ji_{1} \dots i_{n-1}}^{*,n} - \sigma_{ji_{1} \dots i_{n-1}}^{*,n} \right) \nabla \partial^{n-1}_{i_{1} \dots i_{n-1}} g \cdot \nabla \delta \phi_{i}. \label{maineqofprop} \end{equs} Note that the above gives the result (via (\ref{def-frechet})). Indeed, testing equation (\ref{eq-hj}) with $ \delta \phi_{i}$ the second term of the r.h.s of (\ref{maineqofprop}) turns into $-\int a^{*}\nabla h_{j}\cdot \nabla \delta \phi_{i}=-\int \nabla h_{j}\cdot a \nabla \delta \phi_{i}=\int \nabla h_{j}\cdot \delta a (\nabla \phi _{i}^{1}+e_{i})$. Where the last equality is obtained by testing equation (\ref{eq-for-delta phi}) with $h_{j}$.
Now for the induction we start with $n=1$. In that case (\ref{frechet-der}) reduces to \begin{equs} \lim_{t \to 0}\frac{F_{ij}^{o,1}(a+t \delta a) -F_{ij}^{o,1}(a)}{t} = \int g e_{j} \cdot \delta a \left(e_{i} + \nabla \phi_{i}^{1}\right) + \int ge_{j} \cdot (a-\bar{a}^{1}) \nabla \delta \phi_{i} \end{equs} We work with the second term of the r.h.s. \begin{equs} \int ge_{j} \cdot (a-\bar{a}^{1}) \nabla \delta \phi_{i} &= \int g(a^{*}-\bar{a}^{*,1})e_{j} \cdot \nabla \delta \phi_{i} = \int (-a^{*}g \nabla \phi^{*,1}_{j} + g\nabla \cdot \sigma^{*,1}_{j}) \cdot \nabla \delta \phi_{i} \\ &= \int \left( -a^{*} \nabla (g \phi^{*,1}_{j} ) + a^{*}\phi^{*,1}_{j} \nabla g - \sigma^{*,1}_{j} \nabla g \right) \cdot \nabla \delta \phi_{i} \end{equs} where the first two terms result from Leibniz rule and the last from the following property of $\sigma_{i}$ \begin{equs} \nabla \cdot (g \nabla \cdot \sigma_{i}) = - \nabla \cdot ( \sigma_{i} \nabla g), \ \ \text{ for any smooth enough } \ g, \end{equs} which is an easy consequence of the skew-symmetry of $\sigma_{i}$. Thus we have \begin{equs} \lim_{t \to 0}\frac{F_{ij}^{o,1}(a+t \delta a) -F_{ij}^{o,1}(a)}{t} &= \int g e_{j} \cdot \delta a \left(e_{i} + \nabla \phi_{i}^{1}\right) + \int -a^{*} \nabla (g \phi^{*,1}_{j} ) \cdot \nabla \delta \phi_{i} \\ &\ \ \ + \int \left( a^{*}\phi^{*,1}_{j} - \sigma^{*,1}_{j} \right) \nabla g \cdot \nabla \delta \phi_{i}. \end{equs} Note that testing equation (\ref{eq-for-delta phi}) with $g \phi^{*,1}_{j}$ we get $-\int a^{*} \nabla (g \phi^{*,1}_{j} ) \cdot \nabla \delta \phi_{i} = \int \nabla (g \phi^{*,1}_{j} ) \cdot \delta a (\nabla \phi^{1}_{i} +e_{i})$. Then \begin{equs} \lim_{t \to 0}\frac{F_{ij}^{o,1}(a+t \delta a) -F_{ij}^{o,1}(a)}{t} &= \int \left( \left( e_{j}+\nabla \phi_{j}^{*,1} \right)g+ \phi_{j}^{*,1} \nabla g\right)\cdot \delta a \left(e_{i} + \nabla \phi_{i}^{1}\right) \\ &\ \ \ + \int \left( a^{*}\phi^{*,1}_{j} - \sigma^{*,1}_{j} \right) \nabla g \cdot \nabla \delta \phi_{i} \end{equs} which is exactly (\ref{maineqofprop}) for $n=1$. Next assume that (\ref{maineqofprop}) holds for $n-1$. We show that it is true for $n$. Indeed, by (\ref{frechet-der}) we see that \begin{equs} \lim_{t \to 0} \frac{F_{ij}^{o,n}(a+t \delta a) -F_{ij}^{o,n}(a)}{t} &= \lim_{t \to 0}\frac{F_{ij}^{o,n-1}(a+t \delta a) -F_{ij}^{o,n-1}(a)}{t} \\ &\ \ \ - \int \partial ^{n-1}_{i_1 \dots i_{n-1}}g \ \bar{a}^{*,n}_{ji_1\dots i_{n-2}}e_{i_{n-1}} \cdot \nabla \delta \phi _i \\
&= \int \sum_{k=0}^{n-1} \partial^{k}_{i_{1} \dots i_{k}} g (\nabla \phi_{ji_{1} \dots i_{k}}^{*,k+1} +e_{i_{k}} \phi_{ji_{1} \dots i_{k-1}}^{*,k}) \cdot \delta a (\nabla \phi _{i}^{1}+e_{i}) \\ &\ \ \ + \int \left( a^{*}\phi_{ji_{1} \dots i_{n-2}}^{*,n-1} - \sigma_{ji_{1} \dots i_{n-2}}^{*,n-1} \right) \partial^{n-1}_{i_{1} \dots i_{n-1}} g \ e_{i_{n-1}}\cdot \nabla \delta \phi_{i} \\ &\ \ \ - \int \partial ^{n-1}_{i_1 \dots i_{n-1}}g \ \bar{a}^{*,n}_{ji_1\dots i_{n-2}}e_{i_{n-1}} \cdot \nabla \delta \phi _i. \end{equs} We use again (\ref{corr-relation}) for the middle term \begin{equs} &\int \partial^{n-1}_{i_{1} \dots i_{n-1}} g \left( a^{*}\phi_{ji_{1} \dots i_{n-2}}^{*,n-1} - \sigma_{ji_{1} \dots i_{n-2}}^{*,n-1} \right) \ e_{i_{n-1}}\cdot \nabla \delta \phi_{i} \\ &\ \ \ = \int \partial^{n-1}_{i_{1} \dots i_{n-1}} g \left( -a^{*} \nabla \phi^{*,n}_{ji_{1}\dots i_{n-1}} + \nabla \cdot \sigma^{*,n}_{ji_{1}\dots i_{n-1}} - \bar{a}^{*,n}_{ji_{1}\dots i_{n-2}}e_{i_{n-1}} \right) \cdot \nabla \delta \phi_{i}. \end{equs} Hence \begin{equs}
\lim_{t \to 0}\frac{F_{ij}^{o,n}(a+t \delta a) -F_{ij}^{o,n}(a)}{t} &= \int \sum_{k=0}^{n-1} \partial^{k}_{i_{1} \dots i_{k}} g (\nabla \phi_{ji_{1} \dots i_{k}}^{*,k+1} +e_{i_{k}} \phi_{ji_{1} \dots i_{k-1}}^{*,k}) \cdot \delta a (\nabla \phi _{i}^{1}+e_{i}) \\ &\ \ \ - \int a^{*}\nabla (\partial^{n-1}_{i_{1} \dots i_{n-1}} g \ \phi^{*,n}_{ji_{1}\dots i_{n-1}} ) \cdot \nabla \delta \phi_{i} \label{lasteqofprop} \\ &\ \ \ + \int \left ( a^{*}\phi^{*,n}_{ji_{1}\dots i_{n-1}} \nabla \partial^{n-1}_{i_{1} \dots i_{n-1}} g - \sigma^{*,n}_{ji_{1}\dots i_{n-1}} \nabla \partial^{n-1}_{i_{1} \dots i_{n-1}} g \right)\cdot \nabla \delta \phi_{i} \end{equs} where we used again Leibniz rule and the skew-symmetry of $\sigma^{*,n}_{ji_{1}\dots i_{n-1}}$. For the middle term we use equation (\ref{eq-for-delta phi}) tested with $\partial^{n-1}_{i_{1} \dots i_{n-1}} g \ \phi^{*,n}_{ji_{1}\dots i_{n-1}}$ to get \begin{equs} &- \int a^{*}\nabla (\partial^{n-1}_{i_{1} \dots i_{n-1}} g \ \phi^{*,n}_{ji_{1}\dots i_{n-1}} ) \cdot \nabla \delta \phi_{i} \\ &\ \ \ \ \ \ \ \ \ \ \ \ = \int \nabla (\partial^{n-1}_{i_{1} \dots i_{n-1}} g \ \phi^{*,n}_{ji_{1}\dots i_{n-1}} ) \cdot \delta a (\nabla \phi _{i}^{1}+e_{i}) \\ & \ \ \ \ \ \ \ \ \ \ \ \ =\int \left( \partial^{n}_{i_{1} \dots i_{n}} g \ \phi^{*,n}_{ji_{1}\dots i_{n-1}} e_{i_{n}}+ \partial^{n-1}_{i_{1} \dots i_{n-1}} g \ \nabla \phi^{*,n}_{ji_{1}\dots i_{n-1}} \right) \cdot \delta a (\nabla \phi _{i}^{1}+e_{i}). \end{equs} Substituting in (\ref{lasteqofprop}) and recalling that for the sum $\sum_{k=0}^{n-1} $ we have the convention that the $(n-1)$-term is just $ \partial^{n-1}_{i_{1} \dots i_{n-1}} g \ e_{i_{n-1}} \phi_{ji_{1} \dots i_{n-2}}^{*,n-1}$ we conclude the proof.
\end{proof}
Next we study the solution $h_{j}$ of (\ref{eq-hj}) deriving some bounds that will be useful for the proof of the main theorem. In the sequel we assume that $d$ is odd and we denote by $\tilde{d}:=\frac{d+1}{2}$ (note that the proof when $d$ is even is similar - the slightly different bound comes from the stochastic moment bounds of the correctors).
\begin{lemma} \label{bounds-hj}
Let $h_{j}$ be a solution of (\ref{eq-hj}) with $n=\tilde{d}$, then for any $|x| > \frac{L}{2}$ it holds
\begin{equs}
\Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle ^{1/4} \lesssim R^{-d/2}L^{-d}. \label{bound-hj}
\end{equs}
\end{lemma}
\begin{proof}
Recall that $h_{j}$ satisfies the equation $- \nabla \cdot a^{*} \nabla h_{j} = \nabla \cdot f_{j}$, where $f_{j}:= ( a^{*}\phi_{ji_{1} \dots i_{\tilde{d}-1}}^{*,\tilde{d}} - \sigma_{ji_{1} \dots i_{\tilde{d}-1}}^{*,\tilde{d}} ) \nabla \partial^{\tilde{d}-1}_{i_{1} \dots i_{\tilde{d}-1}} g \ $. Note that $\ \supp f_{j} \subset B_{R}$, in particular $h_{j}$ is $a^{*}-$harmonic in $B_{R}^{c}$ which contains $B_{L/2}^{c}$ (recall that $L>>R$). For any $|x| > \frac{L}{2}$, we bound first $\fint_{B_{\frac{L}{4}}(x)} |\nabla h_{j}|^{2}$ using the Lipschitz estimate (or mean-value property) of Theorem 1 in \cite{GNO20}. To achieve this we consider the auxiliary function $v$ which solves \begin{equs} -\nabla \cdot a \nabla v_{j} = \nabla \cdot \tilde{f}_{j}, \ \ \text{ where } \ \ \tilde{f}_{j}:=\mathcal{X}_{B^{c}_{L/4}}\nabla h_{j}. \end{equs} We then have \begin{equs}
\int_{B_{\frac{L}{4}}^{c}} |\nabla h_{j}|^{2} = \int \tilde{f}_{j} \cdot \nabla h_{j} = \int a \nabla v_{j}\cdot \nabla h_{j} &= \int \nabla v_{j} \cdot f_{j} \\
&\lesssim \left( \int_{B_{R}} |\nabla v_{j}|^{2} \right)^{1/2} \left( \int_{B_{R}} |f_{j}|^{2} \right)^{1/2}. \end{equs} Denoting by $R^{*}_{x}= \max \{r^{*}(x), R\} $, where $r^{*}$ the minimal random radius of Theorem 1 in \cite{GNO20}, we estimate \begin{equs}
\left( \int_{B_{R}} |\nabla v_{j}|^{2} \right)^{1/2} &\lesssim (R^{*}_{0})^{d/2} \left( \fint_{B_{R^{*}_{0}}} |\nabla v_{j}|^{2} \right)^{1/2}\\
&\lesssim (R^{*}_{0})^{d/2} \left( \fint_{B_{L/2}} |\nabla v_{j}|^{2} \right)^{1/2} \ \ \ \ (v \text{ is } a \text{-harmonic in } B_{L/2}) \\
&\lesssim \frac{(R^{*}_{0})^{d/2}}{L^{d/2}} \left( \int |\tilde{f}_{j}|^{2} \right)^{1/2}. \ \ \ \ \ \ \ \ \ \ (\text{by the energy estimate}) \end{equs} Thus we get \begin{equs}
\int_{B_{\frac{L}{4}}^{c}} |\nabla h_{j}|^{2} \lesssim \frac{(R^{*}_{0})^{d/2}}{L^{d/2}} \left( \int_{B_{\frac{L}{4}}^{c}} |\nabla h_{j}|^{2} \right)^{1/2}\left( \int_{B_{R}} |f_{j}|^{2} \right)^{1/2} \end{equs} that is, \begin{equs}
\left( \fint_{B_{L/4}(x)} |\nabla h_{j}|^{2} \right)^{1/2} \lesssim \frac{1}{L^{d/2}} \left( \int_{B_{L/4}^{c}} |\nabla h_{j}|^{2} \right)^{1/2} \lesssim \frac{(R^{*}_{0})^{d/2}}{L^{d}} \left( \int |f_{j}|^{2} \right)^{1/2}. \end{equs}
Next we estimate $\Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle^{1/4}$ for every $|x| > \frac{L}{2}$. By small-scale regularity we have, for $|x| > \frac{L}{2}$, \begin{equs}
|\nabla h_{j} (x)| \lesssim C(a) \left( \fint_{B_{R/4}(x)} |\nabla h_{j}|^{2} \right)^{1/2}, \end{equs} where $C(a)$ denotes the H\"older constant of the coefficient field $a$ (note that for $L>4R$, $B_{R/4}(x) \subset B_{L/4}^{c}$ and $h_{j}$ is $a^{*}$-harmonic in $B_{L/4}^{c}$). Then we apply Lipschitz estimate once more to get (assuming that $L>4R$) \begin{equs}
|\nabla h_{j} (x)| \lesssim C(a) \frac{(R^{*}_{x})^{d/2}}{R^{d/2}} \left( \fint_{B_{R^{*}_{x}}(x)} |\nabla h_{j}|^{2} \right)^{1/2} &\lesssim C(a) \frac{(R^{*}_{x})^{d/2}}{R^{d/2}} \left( \fint_{B_{L/4}(x)} |\nabla h_{j}|^{2} \right)^{1/2} \\
&\lesssim C(a) \frac{(R^{*}_{x})^{d/2}}{R^{d/2}} \frac{(R^{*}_{0})^{d/2}}{L^{d}} \left( \int |f_{j}|^{2} \right)^{1/2}. \end{equs} Then we take expectation, we use the fact that both $C(a)$ (see (\ref{a-holder})) and $r^{*}(x)$ have uniformly bounded stochastic moments and choosing the worst scenario for the factor we get \begin{equs}
\Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle ^{1/4} \lesssim \frac{R^{d/2}}{L^{d}} \bigg \langle \left( \int |f_{j}|^{2} \right)^{8} \bigg \rangle ^{1/16}. \end{equs} It remains to estimate the r.h.s using Minkowski's integral inequality and Proposition \ref{momentbounds} (recall that $f_{j}:= ( a^{*}\phi_{ji_{1} \dots i_{\tilde{d}-1}}^{*,\tilde{d}} - \sigma_{ji_{1} \dots i_{\tilde{d}-1}}^{*,\tilde{d}} ) \nabla \partial^{n-1}_{i_{1} \dots i_{n-1}} g \ $). We have \begin{equs}
\Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle ^{1/4} &\lesssim \frac{R^{d/2}}{L^{d}} \left( \int \langle |f_{j}|^{16}\rangle^{1/8} \right)^{1/2} \\
&\lesssim \frac{R^{d/2}}{L^{d}} \left( \int_{B_{R}} |D^{\tilde{d}} g|^{2} \langle |a^{*}\phi_{ji_{1} \dots i_{\tilde{d}-1}}^{*,\tilde{d}} - \sigma_{ji_{1} \dots i_{\tilde{d}-1}}^{*,\tilde{d}} |^{16}\rangle^{1/8} \right)^{1/2} \\
&\lesssim \frac{R^{d/2}}{L^{d}} \left( \int_{B_{R}} |D^{\tilde{d}} g|^{2} |z| \right)^{1/2} \lesssim \frac{R^{d/2}} {L^{d}} R^{-\frac{d}{2}-\tilde{d}}R^{\frac{1}{2}} = R^{-d/2}L^{-d}. \end{equs}
\end{proof}
\begin{remark} \label{bounds-hj+}
Note that by translation we easily see that the solution $h_{l}'$ of \begin{equs}
- \nabla \cdot a^{*} \nabla h_{l}' = \nabla \cdot \left( ( a^{*}\phi_{li_{1} \dots i_{\tilde{d}-1}}^{*,\tilde{d}} - \sigma_{li_{1} \dots i_{\tilde{d}-1}}^{*,\tilde{d}} ) \nabla \partial^{\tilde{d}-1}_{i_{1} \dots i_{\tilde{d}-1}} g' \right)
\end{equs}
satisfies, for any $|x-Le| > \frac{L}{2}$,
\begin{equs} \label{bound-hl}
\Big \langle \big | \nabla h'_{l}(x) \big |^{4} \Big \rangle ^{1/4} \lesssim R^{-d/2}L^{-d}.
\end{equs} Note also that the above bounds can be rephrased (assuming that $L>4R$) as \begin{equs}
{}&\Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle ^{1/4} \lesssim R^{-d/2}|x|^{-d}, \ \text{ for any } \ |x| > 2R \label{bound-hj2} \\
& \Big \langle \big | \nabla h'_{l}(x) \big |^{4} \Big \rangle ^{1/4} \lesssim R^{-d/2}|x-Le|^{-d}, \ \text{ for any } \ |x-Le| > 2R. \label{bound-hl2} \end{equs} A last observation that will be useful in the sequel is the following \begin{equs} \label{CZ}
\left( \int \Big \langle \big | \nabla h_{j} \big |^{4} \Big \rangle^{2/4} \right)^{1/2} \lesssim R^{-d} \end{equs} which is a consequence of annealed Calderon-Zygmund estimates (see Proposition 7.1 in \cite{JO20}) and Proposition \ref{momentbounds} \begin{equs}
\left( \int \Big \langle \big | \nabla h_{j} \big |^{4} \Big \rangle^{2/4} \right)^{1/2} \lesssim \left( \int \Big \langle \big | f_{j} \big |^{4} \Big \rangle^{2/4} \right)^{1/2} \lesssim R^{1/2} \left( \int \big | D^{\tilde{d}}g \big |^{2} \right)^{1/2} \lesssim R^{1/2} R^{-\frac{d}{2}-\tilde{d}} = R^{-d}. \end{equs} A similar bound holds for $h_{l}'$ as well.
\end{remark}
We are now ready to prove our main theorem.
\begin{proof}[Proof of Theorem \ref{mainthm}]
Combining estimate (\ref{cov_est}) with Proposition \ref{repr_form_prop} we get
\begin{equs}
P_{ijml}^{o,\tilde{d}} &\lesssim \int \ \left[ \Big \langle \big | (\nabla \phi _{i}^{1}(x)+e_{i}) \otimes g(x)\ (\nabla \phi_{j}^{*,1}(x)+e_{j}) \big |^{2} \Big \rangle^{1/2}\right.\\
&\ \ \ \ \ \ \ \ \ + \Big \langle \Big | (\nabla \phi _{i}^{1}(x)+e_{i}) \otimes \sum_{k=1}^{\tilde{d}-1} \partial^{k}_{i_{1} \dots i_{k}} g(x) (\nabla \phi_{ji_{1} \dots i_{k}}^{*,k+1} (x)+e_{i_{k}} \phi_{ji_{1} \dots i_{k-1}}^{*,k}(x)) \Big |^{2} \Big \rangle^{1/2} \\
&\ \ \ \ \ \ \ \ \ + \Big \langle \big | (\nabla \phi _{i}^{1}(x)+e_{i}) \otimes \partial^{\tilde{d}}_{i_{1} \dots i_{\tilde{d}}} g(x) \ e_{i_{\tilde{d}}}\phi_{ji_{1} \dots i_{\tilde{d}-1}}^{*,\tilde{d}}(x) \big |^{2} \Big \rangle^{1/2} \label{mainthm1} \\
&\ \ \ \ \ \ \ \ \ +\left. \Big \langle \big | (\nabla \phi _{i}^{1}(x)+e_{i}) \otimes \nabla h_{j} (x) \big |^{2} \Big \rangle^{1/2} \right] \\
&\ \ \ \ \ \times \int |c(x-y)| \bigg [ \text{ same terms with } i \leftrightarrow m, j \leftrightarrow l,x \leftrightarrow y \text{ and } g \leftrightarrow g' \ \bigg ] \dy \dx.
\end{equs}
Using Proposition \ref{momentbounds} we may estimate \begin{equs}
&\bullet \ \ \Big \langle \Big | (\nabla \phi _{i}^{1}+e_{i}) \otimes g (\nabla \phi_{j}^{*,1}+e_{j}) \Big |^{2} \Big \rangle^{1/2} \lesssim \ |g| \Big \langle \big | \nabla \phi _{i}^{1}+e_{i} \big |^{4} \Big \rangle^{1/4} \Big \langle \big | \nabla \phi_{j}^{*,1}+e_{j}) \big |^{4} \Big \rangle^{1/4} \lesssim \ |g|. \\
&\bullet \ \ \Big \langle \Big | (\nabla \phi _{i}^{1}+e_{i}) \otimes \sum_{k=1}^{\tilde{d}-1} \partial^{k}_{i_{1} \dots i_{k}} g (\nabla \phi_{ji_{1} \dots i_{k}}^{*,k+1} +e_{i_{k}} \phi_{ji_{1} \dots i_{k-1}}^{*,k}) \Big |^{2} \Big \rangle^{1/2} \\
& \ \ \ \ \ \ \ \ \ \lesssim \Big \langle \big | \nabla \phi _{i}^{1}+e_{i} \big |^{4} \Big \rangle^{1/4} \sum_{k=1} ^{\tilde{d}-1} \big | \partial^{k}_{i_{1} \dots i_{k}} g \big | \left( \Big \langle \big | \nabla \phi_{ji_{1} \dots i_{k}}^{*,k+1} \big |^{4} \Big \rangle^{1/4} + \Big \langle \big | \phi_{ji_{1} \dots i_{k-1}}^{*,k} \big |^{4} \Big \rangle^{1/4} \right) \\
& \ \ \ \ \ \ \ \ \ \lesssim \sum_{k=1} ^{\tilde{d}-1} \big | \partial^{k}_{i_{1} \dots i_{k}} g \big |. \ \ \ \ \ \ \ \ (\text { note that } k<\tilde{d}\ ) \\
&\bullet \ \ \Big \langle \big | (\nabla \phi _{i}^{1}+e_{i}) \otimes \partial^{\tilde{d}}_{i_{1} \dots i_{\tilde{d}}} g \ \phi_{ji_{1} \dots i_{\tilde{d}-1}}^{*,\tilde{d}} \big |^{2} \Big \rangle^{1/2} \\
& \ \ \ \ \ \ \ \ \ \lesssim \ \big | \partial^{\tilde{d}}_{i_{1} \dots i_{\tilde{d}}} g \big | \Big \langle \big | \nabla \phi _{i}^{1}+e_{i} \big |^{4} \Big \rangle^{1/4} \Big \langle \big | \phi_{ji_{1} \dots i_{\tilde{d}-1}}^{*,\tilde{d}} \big |^{4} \Big \rangle^{1/4} \lesssim \ |z|^{1/2} \ \big | \partial^{\tilde{d}}_{i_{1} \dots i_{\tilde{d}}} g \big |. \\
&\bullet \ \ \Big \langle \big | (\nabla \phi _{i}^{1}+e_{i}) \otimes \nabla h_{j} \big |^{2} \Big \rangle^{1/2} \lesssim \Big \langle \big | \nabla \phi _{i}^{1}+e_{i} \big |^{4} \Big \rangle^{1/4} \Big \langle \big | \nabla h_{j} \big |^{4} \Big \rangle^{1/4} \lesssim \Big \langle \big | \nabla h_{j} \big |^{4} \Big \rangle^{1/4}. \end{equs}
Now we return to (\ref{mainthm1}), we apply the above estimates and multiply to get \begin{equs}
\ \ P_{ijml}^{o,\tilde{d}} &\lesssim \int \bigg ( \sum_{k=0} ^{\tilde{d}-1} \big | \partial^{k}_{i_{1} \dots i_{k}} g (x)\big | + |x|^{\frac{1}{2}} \ \big | \partial^{\tilde{d}}_{i_{1} \dots i_{\tilde{d}}} g(x) \big | \bigg ) \\
&\quad \quad \times \int |c(x-y)| \bigg ( \sum_{k=0} ^{\tilde{d}-1} \big | \partial^{k}_{i_{1} \dots i_{k}} g'(y) \big | + |y-Le|^{\frac{1}{2}} \ \big | \partial^{\tilde{d}}_{i_{1} \dots i_{\tilde{d}}} g'(y) \big | \bigg ) \dy \dx \\
&\ + \int \Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle^{\frac{1}{4}} \int |c(x-y)| \bigg ( \sum_{k=0} ^{\tilde{d}-1} \big | \partial^{k}_{i_{1} \dots i_{k}} g'(y) \big | + |y-Le|^{\frac{1}{2}} \ \big | \partial^{\tilde{d}}_{i_{1} \dots i_{\tilde{d}}} g'(y) \big | \bigg ) \dy \dx\\
&\ + \int \bigg ( \sum_{k=0} ^{\tilde{d}-1} \big | \partial^{k}_{i_{1} \dots i_{k}} g(x) \big | + |x|^{\frac{1}{2}} \ \big | \partial^{\tilde{d}}_{i_{1} \dots i_{\tilde{d}}} g(x) \big | \bigg ) \int |c(x-y)| \Big \langle \big | \nabla h'_{l}(y) \big |^{4} \Big \rangle^{\frac{1}{4}} \dy \dx\\
&\ + \int \Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle^{\frac{1}{4}} \int |c(x-y)| \Big \langle \big | \nabla h'_{l}(y) \big |^{4} \Big \rangle^{\frac{1}{4}} \dy \dx. \end{equs}
Next we focus on bounding each of these four terms. Starting with the first one we observe that it is enough to estimate the term \begin{equs}
\int | g (x) | \int |c(x-y)| \ | g'(y) | \dy \dx \lesssim L^{-d-\alpha_{0}} \lesssim R^{-d/2}L^{-d/2-\alpha_{0}} \end{equs}
where we used that $x \in B_{R}$ and $y \in B_{R}(Le)$ because of the supports of $g$ and $g'$ respectively. Then $|x-y| \geq L/2$ which gives the estimate if we use the integrable decay (\ref{c-decay}) we have assumed for $c$.
We proceed with the second term where we see once again that it is enough to estimate the ''subterm'' with the ''worst'' behaviour, that is \begin{equs}
\int \Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle^{\frac{1}{4}} \int |c(x-y)| |g'(y) | \dy \dx. \end{equs} For, we split the domain of integration into two in order to be able to apply the estimate of Lemma \ref{bounds-hj} to $h_{j}$. So for the first term we apply bound (\ref{bound-hj}) and Young's convolution inequality, while for the second term we use estimate (\ref{CZ}) and the integrable decay (\ref{c-decay}) of $c$ together with Minkowski's integral inequality, \begin{equs}
{} &\int \Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle^{\frac{1}{4}} \int |c(x-y)| |g'(y) | \dy \dx \\
&\quad \lesssim R^{-d/2}L^{-d} \int_{B^{c}_{L/2}} \int |c(x-y)| |g'(y) | \dy \dx \\
&\quad \ \ + \int_{B_{L/2}} \int \frac{|g'(y) |}{(1+|x-y|)^{d+\alpha_{0}}} \Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle^{\frac{1}{4}} \dy \dx \\
&\quad \lesssim R^{-d/2}L^{-d} ||g'||_{L^{1}}||c||_{L^{1}} \\
&\quad \ \ + \left( \int_{B_{L/2}} \left( \int_{B_{R}(Le)} \frac{|g'(y) |}{(1+|x-y|)^{d+\alpha_{0}}} \right)^{2} \dy \dx \right)^{1/2} \left( \int \Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle^{\frac{2}{4}} \dx \right)^{1/2} \\
&\quad \lesssim R^{-d/2}L^{-d} + R^{-d} \int_{B_{R}(Le)} |g'(y) | \left( \int_{B_{L/2}} \frac{1}{(1+|x-y|)^{2d+2\alpha_{0}}} \dx \right)^{1/2} \dy \\ &\quad \lesssim R^{-d/2}L^{-d} + R^{-d} L^{d/2} L^{-d-\alpha_{0}} \lesssim R^{-d/2}L^{-d/2 -\alpha_{0}}. \end{equs} It remains to bound the fourth term which is the most challenging. In order to be able to use the bounds of Lemma \ref{bounds-hj} and Remark \ref{bounds-hj+} we divide the domains of integration as follows \begin{equs}
{}&\int \Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle^{\frac{1}{4}} \int |c(x-y)| \Big \langle \big | \nabla h'_{l}(y) \big |^{4} \Big \rangle^{\frac{1}{4}} \dy \dx \\
&\quad \lesssim R^{-d/2}L^{-d} \int_{B_{3R}(Le)} \left( \int |c(x-y)|^{2} \dy \right)^{1/2} \left( \int \Big \langle \big | \nabla h'_{l}(y) \big |^{4} \Big \rangle^{\frac{2}{4}} \dy \right)^{1/2}\dx \\
&\quad \ \ + R^{-d/2}L^{-d} \left( \int_{B^{c}_{3R}(Le)} \Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle^{\frac{2}{4}} \dx \right)^{1/2} \left( \int_{B^{c}_{3R}(Le)} \left( \int_{B_{3R}} |c(x-y)| dy \right)^{2} \dx \right)^{1/2} \\
&\quad \ \ + \int_{B^{c}_{3R}(Le)} \Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle^{\frac{1}{4}} \int_{B^{c}_{3R}} |c(x-y)| \Big \langle \big | \nabla h'_{l}(y) \big |^{4} \Big \rangle^{\frac{1}{4}} \dy \dx \\
&\quad \lesssim R^{-d/2}L^{-d} + \int_{B^{c}_{3R}(Le)} \Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle^{\frac{1}{4}} \int_{B^{c}_{3R}} |c(x-y)| \Big \langle \big | \nabla h'_{l}(y) \big |^{4} \Big \rangle^{\frac{1}{4}} \dy \dx \end{equs} using estimates (\ref{bound-hj}), (\ref{bound-hl}) and (\ref{CZ}). Now for the last term we need to further divide the domains. Precisely, we split the $x$-integral into $B_{3R}$ and $B^{c}_{3R}$ and the $y$-integral into $B_{3R}(Le)$ and $B^{c}_{3R}(Le)$. This produces four new terms that we estimate in the following. For the first one we may apply estimates (\ref{bound-hj2}) and (\ref{bound-hl2}). Then we divide the domain of $y$-integral once again and use the integrable decay (\ref{c-decay}) of $c$ \begin{equs}
{}&\int_{B^{c}_{3R}(Le) \cap B^{c}_{3R}} \Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle^{\frac{1}{4}} \int_{B^{c}_{3R}(Le) \cap B^{c}_{3R} } |c(x-y)| \Big \langle \big | \nabla h'_{l}(y) \big |^{4} \Big \rangle^{\frac{1}{4}} \dy \dx \\
&\quad \lesssim R^{-d} \int_{B^{c}_{3R}(Le) \cap B^{c}_{3R}} \int_{B^{c}_{3R}(Le) \cap B^{c}_{3R} } |x|^{-d} |y-Le|^{-d}|c(x-y)| \dy \dx \\
&\quad \lesssim R^{-d} \int_{B^{c}_{3R}(Le) \cap B^{c}_{3R}} |x|^{-d} \int_{B^{c}_{3R}(Le) \cap B^{c}_{3R} \cap \{|y-Le| > \frac{|x-Le|}{2}\}} |y-Le|^{-d}|c(x-y)| \dy \dx \\
&\quad \ \ + R^{-d} \int_{B^{c}_{3R}(Le) \cap B^{c}_{3R}} |x|^{-d} \int_{B^{c}_{3R}(Le) \cap B^{c}_{3R} \cap \{|y-Le| \leq \frac{|x-Le|}{2}\}}\frac{|y-Le|^{-d}}{(1+|x-y|)^{d+\alpha_{0}}} \dy \dx \\
&\quad \lesssim R^{-d} \int_{B^{c}_{3R}(Le) \cap B^{c}_{3R}} |x|^{-d} |x-Le|^{-d} ||c||_{L^{1}} \dx \\
&\quad \ \ + R^{-d} \int_{B^{c}_{3R}(Le) \cap B^{c}_{3R}} |x|^{-d} \int_{\{ 3R \leq |y-Le| \leq \frac{|x-Le|}{2}\}}\frac{|y-Le|^{-d}}{|x-Le|^{d+\alpha_{0}}} \dy \dx \\
&\quad \lesssim R^{-d} \int_{B^{c}_{3R}(Le) \cap B^{c}_{3R}} |x|^{-d} |x-Le|^{-d} \dx \\
&\quad \ \ + R^{-d} \int_{B^{c}_{3R}(Le) \cap B^{c}_{3R}} |x|^{-d} \ln \frac{|x-Le|}{6R} |x-Le|^{-d-\alpha_{0}} \dx \\
&\quad \lesssim R^{-d} \int_{B^{c}_{3R}(Le) \cap B^{c}_{3R}} |x|^{-d} |x-Le|^{-d} \dx + R^{-d} \int_{B^{c}_{3R}(Le) \cap B^{c}_{3R}} |x|^{-d} |x-Le|^{-d-\alpha_{0}+1} \dx. \end{equs} We then calculate \begin{equs}
{}& R^{-d} \int_{B^{c}_{3R}(Le) \cap B^{c}_{3R}} |x|^{-d} |x-Le|^{-d} \dx\\
&\quad =2 R^{-d}\int_{\{|x|<|x-Le| \} \cap B_{3R}^{c}} |x|^{-d} |x-Le|^{-d} \dx \\
&\quad \lesssim R^{-d} \int_{\{|x|<|x-Le| \} \cap B_{3L}^{c}} |x|^{-2d} \dx + R^{-d}L^{-d} \int_{\{|x|<|x-Le| \} \cap (B_{3L} \setminus B_{3R})} |x|^{-d} \dx \\ & \quad \lesssim R^{-d} L^{-d}\ln \frac{L}{R} \lesssim R^{-d/2} L^{-d/2-\alpha_{0}}\ln \frac{L}{R}. \end{equs}
Similarly $ \int_{B^{c}_{3R}(Le) \cap B^{c}_{3R}} |x|^{-d} |x-Le|^{-d-\alpha_{0}+1} \dx \lesssim R^{-d} L^{-d-\alpha_{0}+1}\ln \frac{L}{R}$ (note that in that term we could get rid of the logarithmic correction when $d \geq 4$).
For the next term we split the $y$-integral into $|x-y| > L/4$ and $|x-y| \leq L/4$. So in the first case we use the integrable decay (\ref{c-decay}) of $c$ to gain the power we need on $L$ together with estimate (\ref{CZ}). In the second case we see that $|x| \geq L/2$ which allows to apply Lemma \ref{bounds-hj} which we then combine with (\ref{CZ}). Indeed, \begin{equs}
{}&\int_{B^{c}_{3R}(Le) \cap B^{c}_{3R}} \Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle^{\frac{1}{4}} \int_{B_{3R}(Le) \cap B^{c}_{3R} } |c(x-y)| \Big \langle \big | \nabla h'_{l}(y) \big |^{4} \Big \rangle^{\frac{1}{4}} \dy \dx \\
&\quad \lesssim \left( \int \Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle^{\frac{2}{4}} \dx \right)^{1/2} \\
&\quad \quad \quad \quad \times \left( \int_{B^{c}_{3R}(Le) \cap B^{c}_{3R}} \left( \int_{B_{3R}(Le) \cap \{|x-y| > L/4 \} } |c(x-y)| \Big \langle \big | \nabla h'_{l}(y) \big |^{4} \Big \rangle^{\frac{1}{4}} dy\right)^{2} \dx \right)^{1/2} \\
&\quad \ \ + \int_{B^{c}_{3R}(Le) \cap B^{c}_{3R}} \Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle^{\frac{1}{4}} \int_{B_{3R}(Le) \cap \{|x-y| \leq L/4\} } |c(x-y)| \Big \langle \big | \nabla h'_{l}(y) \big |^{4} \Big \rangle^{\frac{1}{4}} \dy \dx \\
&\quad \lesssim R^{-d} \int_{B_{3R}(Le) } \Big \langle \big | \nabla h'_{l}(y) \big |^{4} \Big \rangle^{\frac{1}{4}} \left( \int_{B^{c}_{3R}(Le) \cap B^{c}_{3R} \cap \{|x-y| > L/4\} } |c(x-y)|^{2} \dx \right)^{1/2} \dy \\
&\quad \ \ + R^{-d/2}L^{-d} \int_{B^{c}_{3R}(Le) \cap B^{c}_{3R}} \int_{B_{3R}(Le) } |c(x-y)| \Big \langle \big | \nabla h'_{l}(y) \big |^{4} \Big \rangle^{\frac{1}{4}} \dy \dx \\
&\quad \lesssim R^{-d} R^{d/2}R^{-d} L^{-d/2 -\alpha_{0}} +R^{-d/2}L^{-d} R^{d/2}R^{-d} ||c||_{L^{1}} \lesssim R^{-d/2}L^{-d/2-\alpha_{0}}. \end{equs} Note that the term \begin{equs}
\int_{B^{c}_{3R}(Le) \cap B_{3R}} \Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle^{\frac{1}{4}} \int_{B^{c}_{3R}(Le) \cap B^{c}_{3R} } |c(x-y)| \Big \langle \big | \nabla h'_{l}(y) \big |^{4} \Big \rangle^{\frac{1}{4}} \dy \dx \end{equs} can be treated analogously. Finally, it remains to bound \begin{equs}
{}&\int_{B^{c}_{3R}(Le) \cap B_{3R}} \Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle^{\frac{1}{4}} \int_{B_{3R}(Le) \cap B^{c}_{3R} } |c(x-y)| \Big \langle \big | \nabla h'_{l}(y) \big |^{4} \Big \rangle^{\frac{1}{4}} \dy \dx \\
&\quad \lesssim \left( \int \Big \langle \big | \nabla h_{j}(x) \big |^{4} \Big \rangle^{\frac{2}{4}} \dx \right)^{1/2} \left( \int_{B_{3R}} \left( \int_{B_{3R(Le)}} |c(x-y)| \Big \langle \big | \nabla h'_{l}(y) \big |^{4} \Big \rangle^{\frac{1}{4}} dy\right)^{2} \dx \right)^{1/2} \\
&\quad \lesssim R^{-d} R^{d/2} L^{-d-\alpha_{0}} R^{d/2} \left( \int \Big \langle \big | \nabla h_{l}'(y) \big |^{4} \Big \rangle^{\frac{2}{4}} dy \right)^{1/2} \lesssim R^{-d/2}L^{-d/2 -\alpha_{0}} \end{equs}
where we used estimate (\ref{CZ}), the integrable decay (\ref{c-decay}) of $c$ together with the fact that $|x-y| \gtrsim L$ when $x \in B_{3R}$ and $y \in B_{3R}(Le)$.
\end{proof}
\index{Bibliography@\emph{Bibliography}}
\end{document} | arXiv |
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The impact of contextualization on immersion in healthcare simulation
Henrik Engström1,
Magnus Andersson Hagiwara2,
Per Backlund1,
Mikael Lebram1,
Lars Lundberg2,3,
Mikael Johannesson1,
Anders Sterner2 &
Hanna Maurin Söderholm4
Advances in Simulation volume 1, Article number: 8 (2016) Cite this article
The aim of this paper is to explore how contextualization of a healthcare simulation scenarios impacts immersion, by using a novel objective instrument, the Immersion Score Rating Instrument. This instrument consists of 10 triggers that indicate reduced or enhanced immersion among participants in a simulation scenario. Triggers refer to events such as jumps in time or space (sign of reduced immersion) and natural interaction with the manikin (sign of enhanced immersion) and can be used to calculate an immersion score.
An experiment using a randomized controlled crossover design was conducted to compare immersion between two simulation training conditions for prehospital care: one basic and one contextualized. The Immersion Score Rating Instrument was used to compare the total immersion score for the whole scenario, the immersion score for individual mission phases, and to analyze differences in trigger occurrences. A paired t test was used to test for significance.
The comparison shows that the overall immersion score for the simulation was higher in the contextualized condition. The average immersion score was 2.17 (sd = 1.67) in the contextualized condition and −0.77 (sd = 2.01) in the basic condition (p < .001). The immersion score was significantly higher in the contextualized condition in five out of six mission phases. Events that might be disruptive for the simulation participants' immersion, such as interventions of the instructor and illogical jumps in time or space, are present to a higher degree in the basic scenario condition; while events that signal enhanced immersion, such as natural interaction with the manikin, are more frequently observed in the contextualized condition.
The results suggest that contextualization of simulation training with respect to increased equipment and environmental fidelity as well as functional task alignment might affect immersion positively and thus contribute to an improved training experience.
This work draws on a case study collaboration bringing together expertise from different simulation areas (medical simulation and serious games) with the overall focus to improve simulator training in prehospital emergency care. The prehospital context, i.e., all activities taking place from an initial alarm call until a patient is delivered at the hospital emergency unit, is a complex process [1] that includes many different dimensions and challenges: e.g., communication and teamwork skills, transport and driving, medical skills, and decision-making. Today, current training practice (e.g., in the regional ambulance organizations that participated in our project) is that different aspects are trained in isolation, e.g., medical skills using patient manikins, driving using driving simulators, and teamwork in teamwork sessions. As discussed by Rice [2], many types of skills, e.g., cognitive, social, and non-technical skills are neither immediately challenged nor synthesized in traditional manikin-based simulation, unless time and effort is put into the environment in which the manikin is placed. In cases when the learning goal is to train complex healthcare processes or team collaboration, better learning could potentially be created through increasing both breadth and detail in the simulated scenario. In the context of this research, breadth refers to the number of activities and phases, i.e., the overall process from dispatch to ER handover. Detail refers to increasing the realism and enriching each phase of that process so that it better mirrors the whole array of activities in terms of interactions, skills, tools, and information (audio, tactile, interactive, communicative, technical, etc.) that prehospital nurses carry out and use during an ambulance mission. The idea of having increased richness in terms of covering the whole prehospital chain is in line with the 1990s refinement of aviation simulators when context was taken into account by using full missions. According to, e.g., Rudolph et al. [3] and Dieckmann et al. [4], this was one of the factors that led to a dramatically improved aviation safety. Similarly, we believe that taking mission context into account, e.g., by including the driving route activity from the ambulance station and back and recreating the interior characteristics of a patient's home, as well as present changes in the emotional and physical state of the patient, is likely to increase the physical, conceptual, and emotional realism [3] of the simulation.
In this work, we assess a highly contextualized mixed-reality approach to simulation design. This approach strives to increase immersion through a combination of role-play, technical props, and contextualization of work tasks. In healthcare education, role-play has been used for over 40 years varying from dialogue between students portraying a nurse and a patient to more complex situations [5]. In simulation, role-play can be used as a way to integrate the communication processes that normally are present in real care situations [6]. When designing our simulation approach, we strived to create environments, activities, and scenarios that included natural role-play, e.g., interaction between the trainees (prehospital nurses) and the patient, with bystanders or family members, with attending physicians through phone and handover, and within-team collaboration when prehospital nurses are working together. In addition, a combination of physical and digital props was used to create a more engaging environment.
The participants' involvement in a simulated scenario can be characterized by a number of different terms (Andersson Hagiwara M, Backlund P, Maurin Söderholm H, Lundberg L, Lebram M, Engström H: Measuring participants' immersion in healthcare simulation: the development of an instrument, Forthcoming), such as flow, presence, cognitive absorption, buy-in, suspension of disbelief, and the as-if concept. This paper studies participants' involvement in simulator training in terms of immersion [7, 8]. In healthcare simulation, the term immersion is often used in relation to virtual reality (VR) and seen as being determined by technical components. Here, we use a definition more commonly adopted in the game research community, pertaining to immersion as a subjective psychological experience [7–11]: "the subjective impression that one is participating in a comprehensive, realistic experience" [12] (p.66). This definition emphasizes immersion as lived experience rather than a property of a technical environment and thus helps us to conceptualize what and how that might bring prehospital personnel engaged in a training scenario into and out of an immersive state. One important property of immersion, which differentiates it from, e.g., flow, is that it is a continuum (from engagement to total immersion). This makes it meaningful to quantify the level of immersion. The primary approach today to measure immersion is through questionnaires, e.g., Jennett et al. [8], capturing participants' experience post taking part in an activity. So far it does not, to our knowledge, exist in any measure that can be used to observe immersion in a non-intrusive, objective way. In another paper, we present the development and validation process of the Immersion Score Rating Instrument (ISRI) designed to observe and measure immersion in mixed-reality healthcare simulation. This instrument consists of 10 triggers that indicate reduced or enhanced immersion among participants in a simulation scenario. Triggers refer to events such as jumps in time or space (sign of reduced immersion) and natural interaction with the manikin (sign of enhanced immersion) and can be used to calculate an immersion score.
In the study presented in this paper, we have manipulated breadth and detail in two scenarios to explore how this might affect participants' immersion (see Table 1 for an overview). Thus, the research question addressed in this paper is: How is participants' immersion during a simulated scenario affected by contextualization? In order to investigate this, we apply the newly developed ISRI, which allows us to (1) observe and measure immersion at a general level, (2) identify variations during different phases of a healthcare scenario, and (3) analyze individual triggers that might reduce or enhance participants' immersion.
Table 1 Experiment condition design per ambulance mission phase
Fidelity in healthcare simulation
During a simulation, immersion is affected by several different factors (e.g., physical, structural, communicative, personal, or contextual). In healthcare simulation, these are commonly referred to as different categories of fidelity [13] and often discussed in relation to the fidelity level of a manikin. Archer et al. [14] describe fidelity by using three dimensions: equipment fidelity, which concerns how closely the simulator resembles the real system it refers to. The second dimension, environmental fidelity, refers to the context in which the simulator is placed. Finally, the third dimension is called psychological fidelity and refers to the degree to which the trainee perceives or accepts the simulation to be "real". According to Tun et al. [15], contemporary definitions of fidelity typically refer to the level of realism of a simulation. Furthermore, their review of the fidelity concept reveals that definitions are not clear and may refer to either the physical (engineering) aspects of a simulation, i.e., the extent to which a simulation reflects the physical properties of the real-world concept, or its subjective dimension, called psychological or perceptual fidelity. As fidelity is not a clearly defined concept, it has recently been criticized for being too imprecise [16]. Hamstra et al. even propose to abandon the term fidelity and replace it with the terms physical resemblance and functional task alignment [16]. The benefit of doing so is that it allows us to focus more on the functional alignment with the learning task rather that the current overemphasis of physical resemblance. Accordingly, we classify the first two dimensions proposed by Archer et al. [14] as dimensions of the physical resemblance whereas the third dimension concerns the buy-in to the simulation, i.e., the degree to which the trainee accepts the situation as believable and suitable for its purpose. Functional task alignment is another matter, and Hamstra et al. [16] emphasize the importance of close alignment between the clinical task and the simulation task. Functional task alignment can be strengthened by an appropriate correspondence between the simulator and the applied context. Similar staffing and spatial arrangements can help to achieve this. In the case of prehospital training, we argue that these features may be present in an enriched scenario context for the patient simulator. This does not mean that the physical resemblance of the patient simulator is unimportant, only that it should be considered with respect to the training goal, in our case prehospital care.
As can be seen from the above discussion, fidelity is not a clear concept and neither is its relation to learning. According to Hamstra et al. [16], there is a positive relation between cognitive engagement and learning outcome. However, physical resemblance is only one parameter when enhancing learner engagement. Rettedal [17] discusses participants' perceptions of realism regarding simulated scenarios and points out the suspension of disbelief as a central concept for successful simulation. Horcik et al. [18] refer to this and claim that involvement in simulation requires that participants suspend disbelief. Based on studies of various immersive interfaces, Huiberts [11] asserts that immersion, in a digital environment, can enhance education by allowing multiple perspectives, situated learning, and transfer. The results from a relatively recent study of the learning outcomes of science educational games by Cheng et al. [19] indicate that that immersion leads to a higher gaming performance, which in turn plays a role in learning performance.
To summarize, we see that fidelity is a complex phenomenon which lacks a clear definition. We acknowledge that physical resemblance and functional task alignment are important factors when discussing fidelity and the effectiveness of simulation training. The physical resemblance does not only relate to the manikin but also includes the equipment as well as environmental fidelity. We summarize our view of fidelity in Fig. 1. In our case, this means that we not only utilize a patient simulator manikin as well as physical props to create some sense of realism but we also consider functional task alignment when including tasks from the whole prehospital process to be carried out according to standard procedure.
The dimensions of fidelity
Study setting
The experiment was conducted in November 2014 as a collaborative multidisciplinary effort between serious games and prehospital care researchers from two universities in Sweden and training officers from a regional ambulance center.
Twelve teams (24 professionally active ambulance nurses) from four different healthcare organizations in the surrounding region participated in the study. All participants were working full time as ambulance nurses and had earlier experience from simulation training.
The study was approved by the research ethics adviser at the University of Borås, Sweden and conducted in accordance with the ethical recommendations of the Swedish Research Council [20]. During an introductory session to the experiment, the principal investigator informed study participants about the participation in the study, their rights, and our responsibilities as researchers. Informed oral and written consent was obtained from all participants.
The study had a randomized controlled crossover design comparing immersion between two types of simulation training conditions: one basic, mirroring how training currently is done in the regional ambulance organizations participating in the study, and one contextualized, where we strived to capture more of the complexity of the prehospital work process. Table 1 illustrates the design of the two experiment conditions per central phases of an ambulance mission. The phases were determined by factors such as change of physical location (transport) or different segments of the on-scene assessment and treatment where the highest number of, or most important, decisions are made [21]. This resulted in the six phases presented in Table 1. In the contextualized simulation design, we utilized a mixed-reality approach, recreating parts of the environment through physical props, e.g., using a real ambulance as interface to the simulated driving. The same ambulance was also used for actual loading and for patient care during transport to hospital. In both conditions, a Laerdal SimMan 3G simulator was used. The manikin was operated via Wi-Fi where the operator was playing the role of the patient by communicating via the manikin's integrated speaker system.
Randomization and control
In each condition, two different medical scenarios were used. Upon arrival at our facility, participants were randomly assigned to which condition and medical scenario to start with. Hence, the scenarios were organized in blocks in order to vary: (1) the type of medical scenario ("elderly man with respiratory distress" or "drug addict with respiratory distress") in each of the conditions (contextualized/basic) and (2) the order in which participants did the scenarios (Fig. 2).
Flowchart, randomized controlled crossover design
Experiment protocol
When arriving at the ambulance station, participants were given an introduction to the study and its aims, their participation including reading and signing consent forms, and responding to a background information questionnaire. Next, they were subjected to the two different simulation conditions (contextualized/basic). Before each condition, participants were introduced to the simulation by the experiment leader and given time to familiarize themselves with the manikin and any equipment provided. During each condition, participants were working through an ambulance mission as described in Table 1. Each block was concluded by a debriefing session with the attending emergency physician who participated in the handover phase. In all, each team spent 5 h (including lunch and refreshment breaks) at the ambulance facility where the study was carried out.
The entirety of all the simulations was recorded by a number of video recorders and one handheld audio recorder. To analyze the video recorded sessions, we utilized a recently developed instrument, ISRI. This instrument consists of 10 triggers (T1-T10) that are used to determine participants' immersion during the simulation. Here, a trigger refers to an event in the simulation that was considered a sign for reduced or enhanced participant immersion. As illustrated in Table 2, triggers T1-T7 indicate issues that reduce immersion, i.e. breaks in immersion, while triggers T8-T10 indicate enhanced immersion. Details of the ISRI development process including the complete trigger inventory are reported in (Andersson Hagiwara M, Backlund P, Maurin Söderholm H, Lundberg L, Lebram M, Engström H: Measuring participants' immersion in healthcare simulation: the development of an instrument, Forthcoming).
Table 2 Trigger definitions and directions (i.e., if they indicate reduced or enhanced immersion)
Trigger and timestamp assignment
When applying ISRI to the recorded sessions, five researchers first took part of interrater training and then analyzed two to three teams each (i.e., in total four to six sessions per rater). Here, a rater watched the recording of a session in a computerized system where video and trigger assignment input was integrated (Fig. 3). When a situation arose that indicated reduced or enhanced immersion, the rater stopped the video and selected an appropriate trigger, optionally including a subheading. For each assigned trigger, the rater also indicated the trigger strength from 1 (weak indication) to 3 (strong indication). A two-way mixed, consistency, average-measures intraclass correlation (ICC) of overall interrater reliability showed excellent results (with ICC = 0.92). Next, all video recordings were manually timestamped in time intervals corresponding to the phases defined in Table 1.
Video analysis interface (showing video from phase 1, ambulance en route)
ISRI score calculation
The triggers assigned during a time interval are used to compute a summarizing ISRI score. During a time interval of ∆ minutes, each trigger t (1 ≤ t ≤ 10) occurs n t times. The total number of trigger occurrences during the interval is hence \( {\displaystyle {\sum}_{t=1}^{10}}{n}_t \). Each occurrence i (1 ≤ i ≤ n t ) of a trigger t has been assigned a strength ω ti (1 ≤ ω ti ≤ 3). The total immersion score, s, for the interval is computed as
$$ s=\frac{{\displaystyle {\sum}_{t=8}^{10}}{\displaystyle {\sum}_{i=1}^{n_t}}{\omega}_{ti}-{\displaystyle {\sum}_{t=1}^7}{\displaystyle {\sum}_{i=1}^{n_t}}{\omega}_{ti}}{\Delta} $$
The immersion score is hence the sum of all strengths assigned to positive triggers (T8–T10) minus the sum of all strengths assigned to negative triggers (T1–T7) divided by the length of the interval in minutes. The scores reported in this paper are computed based on the whole simulation and on its individual phases. The length of these varies depending on teams' performances and the nature of the condition. By dividing the trigger strength with time, the immersion score can be used to compare sessions and phases with different durations. In this way, the ISRI score can be said to be normalized to provide an intensity value. An alternative would be to use only the sum of strengths, which would result in a metric that would aggregate the trend over time. However, a long running scenario with a positive immersion trend would then get a much higher score than a similar short scenario. This would make it difficult to compare the score from different types of condition durations as, for example, is possible with a post-questionnaire instrument.
Comparison between conditions
To determine differences in ISRI score between the two conditions, a paired t-test was used.
Order effects
Potential order effects were explored by calculating the difference in ISRI score between the contextualized and basic condition. Independent t tests on the differences were then conducted with order and type of scenario as independent variables.
All statistical analyses were performed using the statistical software program SPSS 21.0 (SPSS Inc., Chicago, IL).
All teams worked through the entire simulation in both conditions. On average, the contextualized condition took 34 min (sd = 3.5), ranging from 28 to 39 min, and the basic condition took on average 15 min (sd = 3.4), ranging from 10 to 20 min. This is a reasonable difference since the contextualized condition included more and longer steps, e.g., actual driving mirroring realistic transport times.
Overall immersion differences between conditions and phases
For all groups, the overall immersion score for the simulation was higher in the contextualized condition. The average immersion score was 2.17 (sd = 1.67) in the contextualized condition and −0.77 (sd = 2.01) in the basic condition (Fig. 4). The difference is significant at p < .001, using a paired t test.
The ISRI score (contextualized vs. basic) for the whole scenario (n = 12). The difference is significant at p < .001
In all, the overall immersion trigger analysis clearly shows higher immersion in the contextualized condition than in the basic. The assignment of scenarios ("elderly man…" or "drug addict…") to conditions did not have any effect (p = .911) on the difference in ISRI score between conditions. The average difference was 2.89 for teams with the "elderly man…" scenario in the contextualized condition and 2.99 for teams with the "drug addict…" in the contextualized condition. There is a tendency that the order of conditions has an effect on the ISRI score. The average difference was 3.44 for teams starting with the basic condition and 2.44 for teams starting with the contextualized condition. Although this difference is not significant (p = .246), it is still notable and should be considered in future studies.
These results do not, however, tell us anything about when during the simulation the participants' immersion is higher or lower. Therefore, we have explored in which of the mission phases (as defined in Table 1) differences in immersion were located. Immersion score within each condition was calculated per each of the phases. Figure 5 illustrates how teams' immersion varies during the different phases of the simulation.
The ISRI score (contextualized vs. basic) for each phase of the scenario (n = 12)
Here, we can clearly see that the contextualized condition has less variance during the process, and that immersion increases during phase 2 (on scene assessment) and 4 (on scene treatment), and decreases in phases 3 (initial patient assessment), 5 (scene departure and transport), and 6 (patient handover to emergency department). In the basic condition, immersion is more fluctuating, with its relative highest points in phase 3 (initial patient assessment) and lowest in phases 2 (on scene assessment) and 5 (scene departure and transport). As Fig. 5 illustrates, phase 3 is the only positive peak in the basic condition.
The fact that the differences between the conditions are largest in phases 2 and 5 is not surprising. In these phases, the participants in the basic condition had to pretend that they were loading and transporting, while the contextualized condition allowed actual loading and real-time driving in a real ambulance vehicle (integrated with a driving simulator). Immersion differences are smallest in phases 3 and 6, the two phases that were most similar in terms of simulation design and equipment (see Table 1). Hence, in order to understand what the immersion differences consist of, and what fidelity dimensions, activities, or props in our simulation design that might affect these, we need to investigate the trigger distribution per each phase in more detail.
Understanding immersion differences for different activities and mission phases
The immersion score is computed from the individual trigger groups, which can contribute negatively or positively to the total score (see Eq. 1). To explore the underlying factors, the scores shown in Fig. 5 have been decomposed into individual trigger group components, shown in Fig. 6. The sum of all trigger groups for a phase in Fig. 6 corresponds with the mean value of the corresponding phase for that condition (shown in Fig. 5). For example, phase 6 in the basic condition has a mix of positive and negative triggers which balance out and result in an immersion score close to zero, as can be seen in Fig. 5. Hence, Fig. 6 helps in visualizing the distribution of the different triggers beyond the computed mean value.
The ISRI score per phase split into the individual triggers. The sum of the 10 trigger values constitutes the immersion score shown in Fig. 5. Each trigger value is the mean of all teams (n = 12)
Differences in trigger occurrences
The basic condition is dominated by two negative triggers: destructive interactions (T1—typically interventions of the instructor) and jumps in time and/or space (T3). These triggers are dictating the score during phase 2 (on-scene assessment) and phase 5 (scene departure and transport) in the basic condition, as compared to the contextualized where positive triggers are dominating these phases. In fact, jumps in time and/or space (T3) are almost not present in the contextualized condition and destructive interactions (T1) is present on a relatively constant low level during the whole scenario except in the last phase. For the whole simulation, both these differences are significant at p < .001, using a paired t test.
Changes in trigger occurrences
As can be seen in Fig. 5, the basic condition exhibits a shift from a positive overall immersion score during phase 3 (initial patient assessment) to a negative during phase 4 (on-scene treatment). As illustrated in Fig. 6, this shift consists of increased destructive interactions (T1), followed by an almost proportional decrease of natural responses to stimuli (T8), natural interaction with the simulator (T9), and with participants (T10). A natural response in the contextualized condition is, e.g., when the participant comforts the patient manikin by patting its arm during transport to hospital; while an unnatural response could be events such as the participant driving the ambulance lets go of the steering wheel and starts doing something else while being en route (contextualized) or both participants engaging in patient treatment even though they (in the basic condition) have verbalized that they are en route driving to the ER. There is also a clear increase in pretended operations (T4), such as the participant just putting down a heap of ECG cables on the patient's chest while verbalizing "I'm taking an ECG".
In contrast, the difference between phase 3 and phase 4 is much less apparent in the contextualized condition where the total immersion score instead increases slightly (Fig. 5). Here, negative triggers (T1–T7) are almost unchanged between phases; instead, there is a change in the distribution of positive triggers. Natural interaction with the simulator (T9) increases while there are fewer natural responses to stimuli in the simulation (T8).
Unnatural execution of operations and technological distractions
There is an almost complete absence of pretended operations (T4) in the contextualized condition, while appearing in most phases of the basic condition. The presence of this difference in phases 3 and 4 is somewhat surprising, as the same manikin and technical and medical equipment are used in both conditions, and hence presents similar conditions and tools for patient care activities. These differences are significant at p < .05, using a paired t test. Interestingly, technological distractions (T7) is the least occurring trigger, it is barely present in any of the conditions or phases.
Differences in similarly designed phases
The three phases initial patient assessment, on-scene treatment, and patient handover to ER (phases 3, 4, and 6) are most similar between the conditions in terms of how the simulation was designed. As shown in Fig. 5, this similarity is reflected as small differences in total immersion. Now we turn to what this looks like in terms of individual triggers. The mix of triggers during phase 3 (initial patient assessment) is similar in the contextualized and basic conditions. The dominating trigger in both conditions is natural interaction with the simulator (T9), which here is necessary to do in order to determine a diagnosis. After this phase, T9 is gradually decreasing in the basic condition. Instances of unnatural interaction with manikin or participants (T5) are however clearly more present in the basic condition. For the whole simulation, the difference in T5 occurrences is significant at p < .05, using a paired t test.
Overall, our analysis shows higher immersion in the contextualized condition than in the basic. The overall immersion score is higher and participants' immersion does not fluctuate as much during the different phases as it does during the basic scenario. Triggers pertaining to events that might be disruptive for participants' immersion, such as interventions of the instructor and illogical jumps in time/space are less frequent in the contextualized condition, while triggers referring to natural interaction with the manikin or other participants are more frequent. Furthermore, operations, tasks, and interactions are to a higher extent conducted in a natural, more realistic way in the contextualized condition. This suggests that contextualization might support a better workflow during a simulation scenario, provide less interruptions, reduce uncertainty of what to do next, and that it promotes natural execution of tasks, as well as natural interaction with manikin or participants. Although the overall immersion is higher in the contextualized condition, both conditions have a mixture of triggers that enhance or reduce immersion. This resembles the results reported in [18] where participants' concerns shifted between issues related to the targeted work and issues related to the simulation. They state that "a unique and stable immersion was never observed" [18] (p.98).
Contextualization includes however a higher number of technological components to increase environmental fidelity and functional task alignment, e.g., identical functional replicas of the same IT and telecommunication equipment normally used by prehospital nurses. Even so, there were minimal occurrences of technological distractions (T7) in both conditions. Hence, it appears that additional technical components did not introduce additional distractions.
Phases 3, 4, and 6 were the ones most similar between conditions in terms of fidelity. Although differences in total immersion between the conditions are smallest in these phases, there are some interesting differences in trigger occurrences. In the basic condition, for example, the natural interaction with the simulator is declining after the initial patient assessment. This is in contrast to the contextualized condition where it increases. Hence, even though the equipment fidelity (manikin, medical equipment, and tools) was close to the same in these phases, it appears that the increased environmental fidelity (dog barking, worried neighbors present) and functional task alignment (interactions with medical control, dispatch info, patient data) might compensate for potential frustration or immersion disruptions induced by the manikin. The increase of T9 (natural interaction with the simulator, here by, e.g., physically touching, talking calmly to, and comforting the patient) in phase 4 may indicate that the participants at this point in the simulation perceive the manikin as a real patient who needs to be involved when treatment is given. This suggests that role-playing plays a crucial role in our results, bringing a positive enforcement of natural interaction with the manikin and within the team as well as with other participants in the simulation. This resonates with the suggestions by Dieckmann et al. [4], that different types of fidelity can influence immersion in different ways. For example, it is probably possible to reach a high level of immersion without a high level of physical fidelity; instead functional task alignment where a realistic sense of stress or time pressure is created, or natural compassionate interaction with a distressed patient, influence immersion positively.
According to the findings by Dieckmann et al. [22] of perceived realism in healthcare simulation, it is the interaction between interrelated subparts, such as the simulation manikin, and the interaction and role-play in the team that creates the sense of perceived realism [22]. Our results expand on this idea, showing that increased fidelity, natural integration of different phases, and role-play as a way to promote interaction increases immersion and that these components actually might compensate for unrealistic or interruptive events or equipment during a healthcare simulation scenario.
The present study has some limitations. For example, it is difficult to say how the difference in time between the basic and the contextualized scenarios have affected the immersion. The ISRI score is computed using time as a denominator. In this way, the score will not be accelerated by the longer durations forced by the nature of some setups (e.g., that loading and transport has to be executed in the contextualized condition). A potential criticism of using time as a denominator is that teams can aim to increase their score by executing operations faster. This is however not an issue as the ISRI score is not used to evaluate the performance of teams. In the presented study, teams were furthermore not aware of any details of the ISRI evaluation.
Although the ISRI score is a normalized metric, differences in duration is still an important factor and further work is needed to better understand how it affects the comparison of conditions and practicalities of training sessions.
The present experiment was not preceded by a power calculation. Since ISRI is a newly developed instrument, a reliable power calculation is difficult. We hope that the results from the study can be used for power calculations to future studies including the ISRI instrument.
The presented study reveals a tendency that the immersion difference may be affected by the order of conditions. This effect is not significant and does not invalidate the main result of the study, but more studies are needed to better understand if and why it appears.
The variables manipulated between the conditions are numerous (e.g., medical scenario/clinical condition, home environment, physical, psychological, and environmental fidelity factors) and thus makes it difficult to isolate the relative impact of specific manipulations or factors on immersion.
It is also impossible to estimate how other factors outside the simulation scenarios affect immersion, as for example, individual differences, earlier experience of simulation, and expectations. The present study does not evaluate how immersion affects learning or performance. More research is needed of the appropriate level of immersion in connection to different learning goals.
This study addresses how immersion in simulated prehospital training scenarios is affected by contextualization. We have studied this by applying the ISRI tool, which allows us to observe and objectively measure immersion. We conclude that contextualization of training scenarios has a positive effect on participants' immersion experience, that it contributes to a better workflow, and promotes realistic interactions and task executions, compared to a basic simulation scenario. This suggests that efforts put into increasing physical resemblance and functional task alignment affects immersion positively. Future studies are however needed to further explore how immersion is affected by specific fidelity components (e.g., noise, home environment, or equipment props) and, perhaps more urgently, structured evaluations of the impact of immersion on learning and performance in healthcare simulations.
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We would like to acknowledge the ambulance personnel that participated in the original study. We would also like to express our gratitude to Laerdal and SAAB Venture AB for providing equipment and personnel.
School of Informatics, University of Skövde, Box 408, 541 28, Skövde, Sweden
Henrik Engström, Per Backlund, Mikael Lebram & Mikael Johannesson
Centre for Prehospital Research, Faculty of Caring Science, Work Life and Social Welfare, University of Borås, 501 90, Borås, Sweden
Magnus Andersson Hagiwara, Lars Lundberg & Anders Sterner
Swedish Armed Forces Centre for Defence Medicine, Box 5155, 426 05, Västra Frölunda, Sweden
Lars Lundberg
Centre for Prehospital Research, Swedish School of Library and Information Science, Faculty of Librarianship, Information, Education and IT, University of Borås, 501 90, Borås, Sweden
Hanna Maurin Söderholm
Henrik Engström
Magnus Andersson Hagiwara
Per Backlund
Mikael Lebram
Mikael Johannesson
Anders Sterner
Correspondence to Lars Lundberg.
HE and HMS conceived of the study and participated in its design and coordination and drafted the manuscript. MAH, PB, ML, LL, MJ, and AS participated in its design and drafted the manuscript. All authors were involved in the design and implementation of the original experiment, read, and approved the final manuscript.
Engström, H., Andersson Hagiwara, M., Backlund, P. et al. The impact of contextualization on immersion in healthcare simulation. Adv Simul 1, 8 (2016). https://doi.org/10.1186/s41077-016-0009-y
Accepted: 09 February 2016
DOI: https://doi.org/10.1186/s41077-016-0009-y
Contextualized | CommonCrawl |
\begin{definition}[Definition:Nowhere Dense/Definition 2]
Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$.
$H$ is '''nowhere dense''' in $T$ {{iff}}:
:$H^-$ contains no open set of $T$ which is non-empty
where $H^-$ denotes the closure of $H$.
\end{definition} | ProofWiki |
\begin{document}
\title{\bf A Brooks type theorem for the maximum local edge connectivity}
\author{{{Michael Stiebitz} \thanks{The authors thank the Danish Research Council for support through the program Algodisc.} \thanks{ Technische Universit\"at Ilmenau, Inst. of Math., PF 100565, D-98684 Ilmenau, Germany. E-mail address: [email protected]}}
\and{{Bjarne Toft}\footnotemark[1]~\thanks{University of Southern Denmark, IMADA, Campusvej 55, DK-5320 Odense M, Denmark E-mail address: [email protected]} }}
\date{} \maketitle
\begin{abstract} For a graph $G$, let $\chi(G)$ and $\lambda(G)$ denote the chromatic number of $G$ and the maximum local edge connectivity of $G$, respectively. A result of Dirac \cite{Dirac53} implies that every graph $G$ satisfies $\chi(G)\leq \lambda(G)+1$. In this paper we characterize the graphs $G$ for which $\chi(G)=\lambda(G)+1$. The case $\lambda(G)=3$ was already solved by Alboulker {\em et al.\,} \cite{AlboukerV2016}. We show that a graph $G$ with $\lambda(G)=k\geq 4$ satisfies $\chi(G)=k+1$ if and only if $G$ contains a block which can be obtained from copies of $K_{k+1}$ by repeated applications of the Haj\'os join. \end{abstract}
\noindent{\small{\bf AMS Subject Classification:} 05C15}
\noindent{\small{\bf Keywords:} Graph coloring, Connectivity, Critical graphs, Brooks' theorem.}
\section{Introduction and main result}
The paper deals with the classical vertex coloring problem for graphs. The term graph refers to a finite undirected graph without loops and without multiple edges. The {\em chromatic number} of a graph $G$, denoted by $\chi(G)$, is the least number of colors needed to color the vertices of $G$ such that each vertex receives a color and adjacent vertices receive different colors. There are several degree bounds for the chromatic number. For a graph $G$, let $\delta(G)=\min_{v\in V(G)}d_G(v)$ and $\Delta(G)=\max_{v\in V(G)}d_G(v)$ denote the {\em minimum degree} and the {\em maximum degree} of $G$, respectively. Furthermore, let $${\rm col}(G)=1+\max_{H\subseteq G}\delta(H)$$ denote the {\em coloring number} of $G$, and let $${\rm mad}(G)=\max_{\varnothing\not=H\subseteq G} \frac{2|E(H)|}{|V(H)|}$$ denote the {\em maximum average degree} of $G$. By $H\subseteq G$ we mean that $H$ is a subgraph of $G$. If $G$ is the {\em empty graph}, that is, $V(G)=\varnothing$, we briefly write $G=\varnothing$ and define $\delta(G)=\Delta(G)={\rm mad}(G)=0$ and ${\rm col}(G)=1$. A simple sequential coloring argument shows that $\chi(G)\leq {\rm col}(G)$, which implies that every graph $G$ satisfies
$$\chi(G)\leq {\rm col}(G)\leq \lfloor {\rm mad}(G)\rfloor+1\leq \Delta(G)+1.$$
These inequalities were discussed in a paper by Jensen and Toft \cite{JensenT95}. Brooks' famous theorem provides a characterization for the class of graphs $G$ satisfying $\chi(G)=\Delta(G)+1$. Let $k\geq 0$ be an integer. For $k\not=2$, let ${\cal B}_k$ denote the class of complete graphs having order $k+1$; and let ${\cal B}_2$ denote the class of odd cycles. A graph in ${\cal B}_k$ has maximum degree $k$ and chromatic number $k+1$. Brooks' theorem \cite{Brooks41} is as follows.
\begin{theorem} [Brooks 1941] Let $G$ be a non-empty graph. Then $\chi(G)\leq \Delta(G)+1$ and equality holds if and only if $G$ has a connected component belonging to the class ${\cal B}_{\Delta(G)}$. \label{Th:Brooks} \end{theorem}
In this paper we are interested in connectivity parameters of graphs. Let $G$ be a graph with at least two vertices. The {\em local connectivity} $\kappa_G(v,w)$ of distinct vertices $v$ and $w$ is the maximum number of internally vertex disjoint $v$-$w$ paths of $G$. The {\em local edge connectivity} $\lambda_G(v,w)$ of distinct vertices $v$ and $w$ is the maximum number of edge-disjoint $v$-$w$ paths of $G$. The {\em maximum local connectivity} of $G$ is $$\kappa(G)=\max\set{\kappa_G(v,w)}{v,w\in V(G), v\not=w},$$ and the {\em maximum local edge connectivity} of $G$ is $$\lambda(G)=\max\set{\lambda_G(v,w)}{v,w\in V(G), v\not=w}.$$ For a graph $G$ having only one vertex, we define $\kappa(G)=\lambda(G)=0$. Clearly, the definition implies that $\kappa(G)\leq \lambda(G)$ for every graph $G$. By a result of Mader \cite{Mader73} it follows that $\delta(G)\leq \kappa(G)$. Since $\kappa$ is a monotone graph parameter in the sense that $H\subseteq G$ implies $\kappa(H)\leq \kappa(G)$, it follows that every graph $G$ satisfies ${\rm col}(G)\leq \kappa(G)+1$. Consequently, every graph $G$ satisfies
\begin{equation} \label{Equ:lambda} \chi(G)\leq {\rm col}(G) \leq \kappa(G)+1\leq \lambda(G)+1\leq \Delta(G)+1. \end{equation}
Our aim is to characterize the class of graphs $G$ for which $\chi(G)=\lambda(G)+1$. For such a characterization we use the fact that if we have an optimal coloring of each block of a graph $G$, then we can combine these colorings to an optimal coloring of $G$ by permuting colors in the blocks if necessary. For every non-empty graph $G$, we thus have
\begin{equation} \label{Equ:Block} \chi(G)=\max\set{\chi(H)}{H \mbox{ is a block of } G}. \end{equation}
We also need a famous construction, first used by Haj\'os \cite{Hajos61}. Let $G_1$ and $G_2$ be two vertex-disjoint graphs and, for $i=1,2$, let $e_i=v_iw_i$ be an edge of $G_i$. Let $G$ be the graph obtained from $G_1$ and $G_2$ by deleting the edges $e_1$ and $e_2$ from $G_1$ and $G_2$, respectively, identifying the vertices $v_1$ and $v_2$, and adding the new edge $w_1w_2$. We then say that $G$ is the {\em Haj\'os join} of $G_1$ and $G_2$ and write $G=(G_1,v_1,w_1)\bigtriangleup (G_2,v_2,w_2)$ or briefly $G=G_1 \bigtriangleup G_2$.
For an integer $k\geq 0$ we define a class ${\cal H}_k$ of graphs as follows. If $k\leq 2$, then ${\cal H}_k={\cal B}_k$. The class ${\cal H}_3$ is the smallest class of graphs that contains all odd wheels and is closed under taking Haj\'os joins. Recall that an {\em odd wheel} is a graph obtained from on odd cycle by adding a new vertex and joining this vertex to all vertices of the cycle. If $k\geq 4$, then ${\cal H}_k$ is the smallest class of graphs that contains all complete graphs of order $k+1$ and is closed under taking Haj\'os joins. Our main result is the following counterpart of Brooks' theorem. In fact, Brooks' theorem may easily be deduced from it.
\begin{theorem} Let $G$ be a non-empty graph. Then $\chi(G)\leq \lambda(G)+1$ and equality holds if and only if $G$ has a block belonging to the class ${\cal H}_{\lambda(G)}$. \label{Th:local} \end{theorem}
For the proof of this result, let $G$ be a non-empty graph with $\lambda(G)=k$. By (\ref{Equ:lambda}), we obtain $\chi(G)\leq k+1$. By an observation of Haj\'os \cite{Hajos61} it follows that every graph in ${\cal H}_k$ has chromatic number $k+1$. Hence if some block of $G$ belongs to ${\cal H}_k$, then (\ref{Equ:Block}) implies that $\chi(G)=k+1$. So it only remains to show that if $\chi(G)=k+1$, then some block of $G$ belongs to ${\cal H}_k$. For proving this, we shall use the critical graph method, see \cite{StiebitzT2015}.
A graph $G$ is {\em critical} if every proper subgraph $H$ of $G$ satisfies $\chi(H)<\chi(G)$. We shall use the following two properties of critical graphs. As an immediate consequence of (\ref{Equ:Block}) we obtain that if $G$ is a critical graph, then $G=\varnothing$ or $G$ contains no separating vertex, implying that $G$ is its only block. Furthermore, every graph contains a critical subgraph with the same chromatic number.
Let $G$ be a non-empty graph with $\lambda(G)=k$ and $\chi(G)=k+1$. Then $G$ contains a critical subgraph $H$ with chromatic number $k+1$, and we obtain that $\lambda(H)\leq \lambda(G)=k$. So the proof of Theorem~\ref{Th:local} is complete if we can show that $H$ is a block of $G$ which belongs to ${\cal H}_k$. For an integer $k\geq 0$, let ${\cal C}_k$ denote the class of graphs $H$ such that $H$ is a critical graph with chromatic number $k+1$ and with $\lambda(H)\leq k$. We shall prove that the two classes ${\cal C}_k$ and ${\cal H}_k$ are the same.
\section{Connectivity of critical graphs}
In this section we shall review known results about the structure of critical graphs. First we need some notation. Let $G$ be an arbitrary graph. For an integer $k\geq 0$, let ${\cal CO}_k(G)$ denote the set of all colorings of $G$ with color set $\{1,2, \ldots, k\}$. Then a function $f:V(G) \to \{1,2, \ldots, k\}$ belongs to ${\cal CO}_k(G)$ if and only if $f^{-1}(c)$ is an independent vertex set of $G$ (possibly empty) for every color $c\in \{1,2, \ldots, k\}$. A set $S\subseteq V(G) \cup E(G)$ is called a {\em separating set} of $G$ if $G-S$ has more components than $G$. A vertex $v$ of $G$ is called a {\em separating vertex} of $G$ if $\{v\}$ is a separating set of $G$. An edge $e$ of $G$ is called a {\em bridge} of $G$ if $\{e\}$ is a separating set of $G$. For a vertex set $X\subseteq V(G)$, let $\partial_G(X)$ denote the set of all edges of $G$ having exactly one end in $X$. Clearly, if $G$ is connected and $\varnothing\not= X \varsubsetneq V(G)$, then $F=\partial_G(X)$ is a separating set of edges of $G$. The converse is not true. However if $F$ is a minimal separating edge set of a connected graph $G$, then $F=\partial_G(X)$ for some vertex set $X$. As a consequence of Menger's theorem about edge connectivity, we obtain that if $v$ and $w$ are two distinct vertices of $G$, then
$$\lambda_G(v,w)=\min\set{|\partial_G(X)|}{X\subseteq V(G), v\in X, w\not\in X}.$$
Color critical graphs were first introduced and investigated by Dirac in the 1950s. He established the basic properties of critical graphs in a series of papers \cite{Dirac52}, \cite{Dirac53} and \cite{Dirac57}. Some of these basic properties are listed in the next theorem.
\begin{theorem} [Dirac 1952] Let $G$ be a critical graph with chromatic number $k+1$ for an integer $k\geq 0$. Then the following statements hold: \begin{itemize}
\item[{\rm (a)}] $\delta(G)\leq k$
\item[{\rm (b)}] If $k=0,1$, then $G$ is a complete graph of order $k+1$; and if $k=2$, then $G$ is an odd cycle.
\item[{\rm (c)}] No separating vertex set of $G$ is a clique of $G$. As a consequence, $G$ is connected and has no separating vertex, i.e., $G$ is a block.
\item[{\rm (d)}] If $v$ and $w$ are two distinct vertices of $G$, then $\lambda_G(v,w)\geq k$. As a consequence $G$ is $k$-edge-connected. \end{itemize} \label{Th:Dirac} \end{theorem}
Theorem~\ref{Th:Dirac}(a) leads to a very natural way of classifying the vertices of a critical graph into two classes. Let $G$ be a critical graph with chromatic number $k+1$. The vertices of $G$ having degree $k$ in $G$ are called {\em low vertices} of $G$, and the remaining vertices are called {\em high vertices} of $G$. So any high vertex of $G$ has degree at least $k+1$ in $G$. Furthermore, let $G_L$ be the subgraph of $G$ induced by the low vertices of $G$, and let $G_H$ be the subgraph of $G$ induced by the high vertices of $G$. We call $G_L$ the {\em low vertex subgraph} of $G$ and $G_H$ the {\em high vertex subgraph} of $G$. This classification is due to Gallai \cite{Gallai63a} who proved the following theorem. Note that statements (b) and (c) of Gallai's theorem are simple consequences of statement (a), which is an extension of Brooks' theorem.
\begin{theorem} [Gallai 1963] Let $G$ be a critical graph with chromatic number $k+1$ for an integer $k\geq 1$. Then the following statements hold: \begin{itemize} \item[{\rm (a)}] Every block of $G_L$ is a complete graph or an odd cycle
\item[{\rm (b)}] If $G_H=\varnothing$, then $G$ is a complete graph of order $k+1$ if $k\not=2$, and $G$ is an odd cycle if $k=2$.
\item[{\rm (c)}] If $|V(G_H)|=1$, then either $G$ has a separating vertex set of two vertices or $k=3$ and $G$ is an odd wheel. \end{itemize} \label{Th:Gallai} \end{theorem}
As observed by Dirac, a critical graph is connected and contains no separating vertex. Dirac \cite{Dirac52} and Gallai \cite{Gallai63a} characterized critical graphs having a separating vertex set of size two. In particular, they proved the following theorem, which shows how to decompose a critical graph having a separating vertex set of size two into smaller critical graphs.
\begin{theorem} [Dirac 1952 and Gallai 1963]
Let $G$ be a critical graph with chromatic number $k+1$ for an integer $k\geq 3$, and let $S\subseteq V(G)$ be a separating vertex set of $G$ with $|S|\leq 2$. Then $S$ is an independent vertex set of $G$ consisting of two vertices, say $v$ and $w$, and $G-S$ has exactly two components $H_1$ and $H_2$. Moreover, if $G_i=G[V(H_i) \cup S]$ for $i=1,2$, we can adjust the notation so that for some coloring $f_1\in {\cal CO}_k(G_1)$ we have $f_1(v)=f_1(w)$. Then the following statements hold: \begin{itemize} \item[{\rm (a)}] Every coloring $f\in {\cal CO}_k(G_1)$ satisfies $f(v)=f(w)$ and every coloring $f\in {\cal CO}_k(G_2)$ satisfies $f(v)\not= f(w)$.
\item[{\rm (b)}] The subgraph $G_1'=G_1+vw$ obtained from $G_1$ by adding the edge $vw$ is critical and has chromatic number $k+1$.
\item[{\rm (c)}] The vertices $v$ and $w$ have no common neighbor in $G_2$ and the subgraph $G_2'=G_2/S$ obtained from $G_2$ by identifying $v$ and $w$ is critical and has chromatic number $k+1$. \end{itemize} \label{Th:2Conn} \end{theorem}
Dirac \cite{Dirac64} and Gallai \cite{Gallai63a} also proved the converse theorem, that $G$ is critical and has chromatic number $k+1$ provided that $G_1'$ is critical and has chromatic number $k+1$ and $G_2$ obtained from the critical graph $G_2'$ with chromatic number $k+1$ by splitting a vertex into $v$ and $w$ has chromatic number $k$.
Haj\'os \cite{Hajos61} invented his construction to characterize the class of graphs with chromatic number at least $k+1$. Another advantage of the Haj\'os join is the well known fact that it not only preserve the chromatic number, but also criticality. It may be viewed as a special case of the Dirac--Gallai construction, described above.
\begin{theorem} [Haj\'os 1961] Let $G=G_1 \bigtriangleup G_2$ be the Haj\'os join of two graphs $G_1$ and $G_2$, and let $k\geq 3$ be an integer. Then $G$ is critical and has chromatic number $k+1$ if and only if both $G_1$ and $G_2$ are critical and have chromatic number $k+1$. \label{Th:Hajos} \end{theorem}
If $G$ is the Haj\'os join of two graphs that are critical and have chromatic number $k+1$, where $k\geq 3$, then $G$ is critical and has chromatic number $k+1$. Moreover, $G$ has a separating set consisting of one edge and one vertex. Theorem~\ref{Th:2Conn} implies that the converse statement also holds.
\begin{theorem} Let $G$ be a critical graph graph with chromatic number $k+1$ for an integer $k\geq 3$. If $G$ has a separating set consisting of one edge and one vertex, then $G$ is the Haj\'os join of two graphs. \label{Th:2Sep=Hajos} \end{theorem}
Next we will discuss a decomposition result for critical graphs having chromatic number $k+1$ an having an separating edge set of size $k$. Let $G$ be an arbitrary graph. By an {\em edge cut} of $G$ we mean a triple $(X,Y,F)$ such that $X$ is a non-empty proper subset of $V(G)$, $Y=V(G)\setminus X$, and $F=\partial_G(X)=\partial_G(Y)$. If $(X,Y,F)$ is an edge cut of $G$, then we denote by $X_F$ (respectively $Y_F$) the set of vertices of $X$ (respectively, $Y$) which are incident to some edge of $F$. An edge cut $(X,Y,F)$ of $G$ is non-trivial if $|X_F|\geq 2$ and $|Y_F|\geq 2$. The following decomposition result was proved independently by T. Gallai and Toft \cite{Toft70}.
\begin{theorem} [Toft 1970]
Let $G$ be a critical graph with chromatic number $k+1$ for an integer $k\geq 3$, and let $F\subseteq E(G)$ be a separating edge set of $G$ with $|F|\leq k$. Then $|F|=k$ and there is an edge cut $(X,Y,F)$ of $G$ satisfying the following properties: \begin{itemize}
\item[{\rm (a)}] Every coloring $f\in {\cal CO}_k(G[X])$ satisfies $|f(X_F)|=1$ and every coloring $f\in {\cal CO}_k(G[Y])$ satisfies $|f(Y_F)|=k$.
\item[{\rm (b)}] The subgraph $G_1$ obtained from $G[X \cup Y_F]$ by adding all edges between the vertices of $Y_F$, so that $Y_F$ becomes a clique of $G_1$, is critical and has chromatic number $k+1$.
\item[{\rm (c)}] The subgraph $G_2$ obtained from $G[Y]$ by adding a new vertex $v$ and joining $v$ to all vertices of $Y_F$
is critical and has chromatic number $k+1$. \end{itemize} \label{Th:Toft} \end{theorem}
A particular nice proof of this result is due to T. Gallai (oral communication to the second author). Recall that the {\em clique number} of a graph $G$, denoted by $\omega(G)$, is the largest cardinality of a clique in $G$. A graph $G$ is {\em perfect} if every induced subgraph $H$ of $G$ satisfies $\chi(H)=\omega(H)$. For the proof of the next lemma, due to Gallai, we use the fact that complements of bipartite graphs are perfect.
\begin{lemma}
Let $H$ be a graph and let $k\geq 3$ be an integer. Suppose that $(A,B,F')$ is an edge cut of $H$ such that $|F'|\leq k$ and $A$ as well as $B$ are cliques of $H$ with $|A|=|B|=k$. If $\chi(H)\geq k+1$, then $|F'|=k$ and $F'=\partial_H(\{v\})$ for some vertex $v$ of $H$. \label{Le:perfect} \end{lemma} \begin{proof}
The graph $H$ is perfect and so $\omega(H)=\chi(H)\geq k+1$. Consequently, $H$ contains a clique $X$ with $|X|=k+1$. Let $s=|A\cap X|$ and hence $k+1-s=|B\cap X|$. Since $|A|=|B|=k$, this implies that $s\geq 1$ and $k+1-s\geq 1$. Since $X$ is a clique of $H$, the set $E'$ of edges of $H$ joining a vertex of $A\cap X$ with a vertex of $B\cap X$ satisfies $E'\subseteq F'$ and $|E'|=s(k+1-s)$. Clearly, $g''(s)=-2$, which implies that the function $g(s)=s(k+1-s)$ is strictly concave on the real interval $[1,k]$. Since $g(1)=g(k)=k$, we conclude that $g(s)>k$ for all $s\in (1,k)$. Since $g(s)=|E'|\leq |F'|\leq k$, this implies that $s=1$ or $s=k$. In both cases we obtain that $|E'|=|F'|=k$, and hence $E'=F'=\partial_H(\{v\})$ for some vertex $v$ of $H$. \end{proof}
Based on Lemma~\ref{Le:perfect} it is easy to give a proof of Theorem~\ref{Th:Toft}, see also the paper by Dirac, S{\o}rensen, and Toft \cite{DiracT74}. Theorem~\ref{Th:Toft} is a reformulation of a result by Toft in his Ph.D thesis. Toft gave a complete characterization of the class of critical graphs, having chromatic number $k+1$ and containing a separating edge set of size $k$. The characterization involves critical hypergraphs.
Figure~\ref{Fig:A1} shows three critical graphs with $\chi=4$. The first graph is an odd wheel and the second graph is the Haj\'os join of two $K_4$'s; both graphs belong to the class ${\cal C}_3$. The third graph does not belong to ${\cal C}_3$; it has an separating edge set of size 3, but $\lambda=4$.
\begin{figure}
\caption{Three critical graphs with chromatic number $\chi=4$.}
\label{Fig:A1}
\end{figure}
\section{Proof of the main result}
\begin{theorem} Let $k\geq 0$ be an integer. Then the two graph classes ${\cal C}_k$ and ${\cal H}_k$ coincide. \label{Th:Ck=Hk} \end{theorem} \begin{proof} That the two classes ${\cal C}_k$ and ${\cal H}_k$ coincide if $0\leq k \leq 2$ follows from Theorem~\ref{Th:Dirac}(a). In this case both classes consists of all critical graphs with chromatic number $k+1$. In what follows we therefore assume that $k\geq 3$. The proof of the following claim is straightforward and left to the reader.
\begin{claim} The odd wheels belong to the class ${\cal C}_3$ and the complete graphs of order $k+1$ belong to the class ${\cal C}_k$. \label{Cl:A1} \end{claim}
\begin{claim} Let $k\geq 3$ be an integer, and let $G=G_1 \bigtriangleup G_2$ the Haj\'os join of two graphs $G_1$ and $G_2$. Then $G$ belongs to the class ${\cal C}_k$ if and only if both $G_1, G_2$ belong to the class ${\cal C}_k$. \label{Cl:A2} \end{claim} \pfcl{We may assume that $G=(G_1,v_1,w_1) \bigtriangleup (G_2,v_2,w_2)$ and $v$ is the vertex of $G$ obtained by identifying $v_1$ and $v_2$. First suppose that $G_1, G_2\in {\cal C}_k$. From Theorem~\ref{Th:Hajos} it follows that $G$ is critical and has chromatic number $k+1$. So it suffices to prove that $\lambda(G)\leq k$. To this end let $u$ and $u'$ be distinct vertices of $G$ and let $p=\lambda_G(u,u')$. Then there is a system ${\cal P}$ of $p$ edge disjoint $u$-$u'$ paths in $G$. If $u$ and $u'$ belong both to $G_1$, then only one path $P$ of ${\cal P}$ may contain vertices not in $G_1$. In this case $P$ contains the vertex $v$ and the edge $w_1w_2$. If we replace in $P$ the subpath $vPw_1$ by the edge $v_1w_1$, we obtain a system of $p$ edge disjoint $u$-$u'$ paths in $G_1$, and hence $p\leq \lambda_{G_1}(u,u')\leq k$. If $u$ and $u'$ belong to $G_2$, a similar argument shows that $p\leq k$. It remains to consider the case that one vertex, say $u$, belongs to $G_1$ and the other vertex $u'$ belongs to $G_2$. By symmetry we may assume that $u\not=v$. Again at most one path $P$ of ${\cal P}$ uses the edge $w_1w_2$ and the remaining paths of ${\cal P}$ all uses the vertex $v(=v_1=v_2)$. If we replace $P$ by the path $uPw_1+w_1v_1$, then we obtain $p$ edge disjoint $u$-$v_1$ path in $G_1$, and hence $p\leq \lambda_{G_1}(u,v_1)\leq k$. This shows that $\lambda(G)\leq k$ and so $G\in {\cal C}_k$.
Suppose conversely that $G\in {\cal C}_k$. From Theorem~\ref{Th:Hajos} it follows that $G_1$ and $G_1$ are critical graphs, both with chromatic number $k+1$. So it suffices to show that $\lambda(G_i)\leq k$ for $i=1,2$. By symmetry it suffices to show that $\lambda(G_1)\leq k$. To this end let $u$ and $u'$ be distinct vertices of $G_1$ and let $p=\lambda_G(u,u')$. Then there is a system ${\cal P}$ of $p$ edge disjoint $u$-$u'$ paths in $G_1$. At most one path $P$ of ${\cal P}$ can contain the edge $v_1w_1$. Clearly, there is a $v_2$-$w_2$ path $P'$ in $G_2$ not containing the edge $v_2w_2$. So if we replace the edge $v_1w_1$ of $P$ by the path $P'$, we get $p$ edge disjoint $u$-$u'$ paths of $G$, and hence $p\leq \lambda_G(u,u')\leq k$. This shows that $\lambda(G_1)\leq k$ and by symmetry $\lambda(G_2)\leq k$. Hence $G_1, G_2\in {\cal C}_k$. }
As a consequence of Claim~\ref{Cl:A1} and Claim~\ref{Cl:A2} and the definition of the class ${\cal H}_k$ we obtain the following claim.
\begin{claim} Let $k\geq 3$ be an integer. Then the class ${\cal H}_k$ is a subclass of ${\cal C}_k$. \label{Cl:A3} \end{claim}
\begin{claim} Let $k\geq 3$ be an integer, and let $G$ be a graph belonging to the class ${\cal C}_k$. If $G$ is 3-connected, then either $k=3$ and $G$ is an odd wheel, or $k\geq 4$ and $G$ is a complete graph of order $k+1$. \label{Cl:A4} \end{claim}
\pfcl{The proof is by contradiction, where we consider a counterexample $G$ whose order $|G|$ is minimum. Then $G\in {\cal C}_k$ is a 3-connected graph, and either $k=3$ and $G$ is not an odd wheel, or $k\geq 4$ and $G$ is not a complete graph of order $k+1$. First we claim that $|G_H|\geq 2$. If $G_H=\varnothing$, then Theorem~\ref{Th:Gallai}(b) implies that $G$ is a complete graph of order $k+1$, a contradiction. If $|G_H|=1$, then Theorem~\ref{Th:Gallai}(c) implies that $k=3$ and $G$ is an odd wheel, a contradiction. This proves the claim that $|G_H|\geq 2$. Then let $u$ and $v$ be distinct high vertices of $G$. Since $G\in {\cal C}_k$, Theorem~\ref{Th:Dirac}(d) implies that $\lambda_G(u,v)=k$ and, therefore, $G$ contains a separating edge set $F$ of size $k$ which separates $u$ and $v$. From Theorem~\ref{Th:Toft} it then follows that there is an edge cut $(X,Y,F)$ satisfying the three properties of that theorem. Since $F$ separates $u$ and $v$, we may assume that $u\in X$ and $v\in Y$. By Theorem\ref{Th:Toft}(a), $|Y_F|=k$ and hence each vertex of $Y_F$ is incident to exactly one edge of $F$. Since $Y$ contains the high vertex $v$, we conclude that $|Y_F|<|Y|$. Now we consider the graph $G'$ obtained from $G[X \cup Y_F]$ by adding all edges between the vertices of $Y_F$, so that $Y_F$ becomes a clique of $G'$. By Theorem~\ref{Th:Toft}(b), $G'$ is a critical graph with chromatic number $k+1$. Clearly, every vertex of $Y_F$ is a low vertex of $G$ and every vertex of $X$ has in $G'$ the same degree as in $G$. Since $X$ contains the high vertex $u$ of $G$, this implies that $|X_F|<|X|$. Since $G$ is 3-connected, we conclude that $|X_F|\geq 3$ and that $G'$ is 3-connected.
Now we claim that $\lambda(G')\leq k$. To prove this, let $x$ and $y$ be distinct vertices of $G'$. If $x$ or $y$ is a low vertex of $G'$, then $\lambda_{G'}(x,y)\leq k$ and there is nothing to prove. So assume that both $x$ and $y$ are high vertices of $G'$. Then both vertices $x$ and $y$ belong to $X$. Let $p=\lambda_{G'}(x,y)$ and let ${\cal P}$ be a system of $p$ edge disjoint $x$-$y$ paths in $G'$. We may choose ${\cal P}$ such that the number of edges in ${\cal P}$ is minimum. Let ${\cal P}_1$ be the paths in ${\cal P}$ which uses edges of $F$. Since $|Y_F|=k$ and each vertex of $Y_F$ is incident with exactly one edge of $F$, this implies that each path $P$ in ${\cal P}_1$ contains exactly two edges of $F$. Since $|X_F|<|X|$ and $|Y_F|<|Y|$, there are vertices $u'\in X\setminus X_F$ and $v'\in Y\setminus Y_F$. By Theorem~\ref{Th:Dirac}(d) it follows that $\lambda_G(u',v')=k$ and, therefore, there are $k$ edge disjoint $u'$-$v'$ paths in $G$. Since $|Y_F|=k$, for each vertex $z\in Y_F$, there is a $v'$-$z$ path $P_z$ in $G[Y]$ such that these paths are edge disjoint. Now let $P$ be an arbitrary path in ${\cal P}_1$. Then $P$ contains exactly two vertices of $Y_F$, say $z$ and $z'$, and we can replace the edge
$zz'$ of the path $P$ by a $z$-$z'$ path contained in $P_z \cup P_{z'}$. In this way we obtain a system of $p$ edge disjoint $x$-$y$ paths in $G$, which implies that $p\leq \lambda_G(x,y)\leq k$. This proves the claim that $\lambda(G')\leq k$. Consequently $G'\in {\cal C}_k$. Clearly, $|G'|<|G|$ and either $k=3$ and $G'$ is not an odd wheel, or $k\geq 4$ and $G$ is not a complete graph of order $k+1$. This, however, is a contradiction to the choice of $G$. Thus the claim is proved. }
\begin{claim} Let $k\geq 3$ be an integer, and let $G$ be a graph belonging to the class ${\cal C}_k$. If $G$ has a separating vertex set of size 2, then $G=G_1\bigtriangleup G_2$ is the Haj\'os sum of two graphs $G_1$ and $G_2$, which both belong to ${\cal C}_k$. \label{Cl:A5} \end{claim} \pfcl{If $G$ has a separating set consisting of one edge and one vertex, then Theorem~\ref{Th:2Sep=Hajos} implies that $G$ is the Hajo\'s join of two graphs $G_1$ and $G_2$. By Claim~\ref{Cl:A2} it then follows that both $G_1$ and $G_2$ belong to ${\cal C}_k$ and we are done. It remains to consider the case that $G$ does not contain a separating set consisting of one edge and one vertex. By assumption, there is a separating vertex set of size 2, say $S=\{u,v\}$. Then Theorem~\ref{Th:2Conn} implies that $G-S$ has exactly two components $H_1$ and $H_2$ such that the graphs $G_i=G[V(H_i) \cup S]$ with $i=1,2$ satisfies the three properties of that theorem. In particular, we have that $G_1'=G_1+uv$ is critical and has chromatic number $k$. By Theorem~\ref{Th:Dirac}(c), it then follows that $\lambda_{G_1'}(u,v)\geq k$ implying that $\lambda_{G_1}(u,v)\geq k-1$. Since $G\in {\cal C}_k$, we then conclude that $\lambda_{G_2}(u,v)\leq 1$. Since $G_2$ is connected, this implies that $G_2$ has a bridge $e$. Since $k\geq 3$, we conclude that $\{u,e\}$ or $\{v,e\}$ is a separating set of $G$, a contradiction. }
As a consequence of Claim~\ref{Cl:A4} and Claim~\ref{Cl:A5}, we conclude that the class ${\cal C}_k$ is a subclass of the class ${\cal H}_k$. Together with Claim~\ref{Cl:A3} this yields ${\cal H}_k={\cal C}_k$ as wanted. \end{proof}
\pff{of Theorem~\ref{Th:local}}{For the proof of this theorem let $G$ be a non-empty graph with $\lambda(G)=k$. By (\ref{Equ:lambda}) we obtain that $\chi(G)\leq k+1$. If one block $H$ of $G$ belongs to ${\cal H}_k$, then $H\in {\cal C}_k$ (by Theorem~\ref{Th:Ck=Hk}) and hence $\chi(G)=k+1$ (by (\ref{Equ:Block}).
Assume conversely that $\chi(G)=k+1$. Then $G$ contains a subgraph $H$ which is critical and has chromatic number $k+1$. Clearly, $\lambda(H)\leq \lambda(G)\leq k$, and, therefore, $H\in {\cal C}_k$. By Theorem~\ref{Th:Dirac}(b), $H$ contains no separating vertex. We claim that $H$ is a block of $G$. For otherwise, $H$ would be a proper subgraph of a block $G'$ of $G$. This implies that there are distinct vertices $u$ and $v$ in $H$ which are joined by a path $P$ of $G$ with $E(P)\cap E(H)=\varnothing$. Since $\lambda_H(u,v)\geq k$ (by Theorem~\ref{Th:Dirac}(c)), this implies that $\lambda_G(u,v)\geq k+1$, which is impossible. This proves the claim that $H$ is a block of $G$. By Theorem~\ref{Th:Ck=Hk}, ${\cal C}_k={\cal H}_k$ implying that $H\in {\cal H}_k$. This completes the proof of the theorem}
The case $\lambda=3$ of Theorem~\ref{Th:local} was obtained earlier by Alboulker {\em et al.\,} \cite{AlboukerV2016}; their proof is similar to our proof. Let ${\cal L}_k$ denote the class of graphs $G$ satisfying $\lambda(G)\leq k$. It is well known that membership in ${\cal L}_k$ can be tested in polynomial time. It is also easy to show that there is a polynomial-time algorithm that, given a graph $G\in {\cal L}_k$, decides whether $G$ or one of its blocks belong to ${\cal H}_k$. So it can be tested in polynomial time whether a graph $G\in {\cal L}_k$ satisfies $\chi(G)\leq k$. Moreover, the proof of Theorem~\ref{Th:local} yields a polynomial-time algorithm that, given a graph $G\in {\cal L}_k$, finds a coloring of ${\cal CO}_k(G)$ when such a coloring exists. This result provides a positive answer to a conjecture made by Alboulker {\em et al.\,} \cite[Conjecture 1.8]{AlboukerV2016}. The case $k=3$ was solved by Alboulker {\em et al.\,} \cite{AlboukerV2016}.
\begin{theorem} For fixed $k\geq 1$, there is a polynomial-time algorithm that, given a graph $G\in {\cal L}_k$, finds a coloring in ${\cal CO}_k(G)$ or a block belonging to ${\cal H}_k$. \label{Th:Algorithm} \end{theorem} \skpf{The Theorem is evident if $k=1,2$; and the case $k=3$ was solved by Alboulker {\em et al.\,} \cite{AlboukerV2016}. Hence we assume that $k\geq 4$ and $G\in {\cal L}_k$. If we find for each block $H$ of $G$ a coloring in ${\cal CO}_k(H)$, we can piece these colorings together by permuting colors to obtain a coloring in ${\cal CO}_k(G)$. Hence we may assume that $G$ is a block. First, we check whether $G$ has a separating set $S$ consisting of one vertex and one edge. If we find such a set, say $S=\{v,e\}$ with $v\in V(G)$ and $e\in E(G)$. Then $G-e$ is the union of two connected graphs $G_1$ and $G_2$ having only vertex $v$ in common where $e=w_1w_2$ and $w_i\in V(G_i)$ for $i=1,2$. Both blocks $G_1'=G_1+vw_1$ and $G_2'=G_2+vw_2$ belong to ${\cal L}_k$. Now we check whether these blocks belong to ${\cal H}_k$. If both blocks $G_1'$ and $G_2'$ belong to ${\cal H}_k$, then $vw_i\not\in E(G_i)$ for $i=1,2$, and hence $G$ belongs to ${\cal H}_k$ and we are done. If one of the blocks, say $G_1'$ does not belong to ${\cal H}_k$, we can construct a coloring $f_1\in {\cal CO}_k(G_1')$. Moreover, no block of $G_2$ belongs to ${\cal H}_k$, hence we can construct a coloring $f_2\in {\cal CO}_k(G_2)$. Then $f_1\in {\cal CO}_k(G_1)$ and $f_1(v)\not=f_1(w_1)$. Since $k\geq 4$, we can permute colors in $f_2$ such that $f_1(v)=f_2(v)$ and $f_1(w_1)\not=f_2(w_2)$. Consequently, $f=f_1 \cup f_2$ belongs to ${\cal CO}_k(G)$ and we are done.
It remains to consider the case that $G$ contains no separating set consisting of one vertex and one edge. Then let $p$ denote the number of vertices of $G$ whose degree is greater that $k$. If $p\leq 1$, then let $v$ be a vertex of maximum degree in $G$. Color $v$ with color $1$ and let $L$ be a list assignment for $H=G-v$
satisfying $L(u)=\{2,3, \ldots ,k\}$ if $vu\in E(G)$ and $L(u)=\{1,2, \ldots, k\}$ otherwise. Then $H$ is connected and $|L(u)|\geq d_H(u)$ for all $u\in V(H)$. Now we can use the degree version of Brooks' theorem, see \cite[Theorem 2.1]{StiebitzT2015}. Either we find a coloring $f$ of $H$ such that $f(u)\in L(u)$ for all $u\in V(H)$, yielding a coloring of ${\cal CO}_k(G)$, or $|L(u)|=d_H(u)$ for all $u\in V(H)$ and each block of $H$ is a complete graph or an odd cycle. In this case, $d_H(u)\in \{k,k-1\}$ for all $u\in V(H)$ and, since $k\geq 4$, each block of $H$ is a $K_k$ or a $K_2$. Since $G$ contains no separating set consisting of one vertex and one edge, this implies that $H=K_k$ and so $G=K_{k+1}\in {\cal H}_k$ and we are done. If $p\geq 2$, then we choose two vertices $u$ and $u'$ whose degrees are greater that $k$. Then we construct an edge cut $(X,Y,F)$ with $u\in X$, $u'\in Y$, and $|F|=\lambda_G(u,u')$. We may assume that $a=|X_F|$ and $b=|Y_F|$ satisfies $a\leq b\leq k$. If $b\leq k-1$, then both graphs $G[X]$ and $G[Y]$ belong to ${\cal L}_k$ and there are colorings $f_X\in {\cal CO}_k(G[X])$ and $f_Y\in {\cal CO}_k(G[Y])$. Note that no block of these two graphs can belong to ${\cal H}_k$. By permuting colors in $f_Y$, we can combine the two colorings $f_X$ and $f_Y$ to obtain a coloring $f\in {\cal CO}_k(G)$ (by Lemma~\ref{Le:perfect}). If $a<b=k$, then we consider the graph $G_1$ obtained from $G[X \cup Y_F]$ by adding all edges between the vertices of $Y_F$, so that $Y_F$ becomes a clique of $G_1$. Then $G_1$ belongs to ${\cal L}_k$ (see the proof of Claim 4) and, since $G$ contains no separating set consisting of one vertex and one edge, the block $G_1$ does not belongs to ${\cal H}_k$. Hence there are colorings $f_1\in {\cal CO}_k(G_1)$ and $f_Y\in {\cal CO}_k(G[Y])$. Then the restriction of $f_1$ to $X$ yields a coloring $f_X\in {\cal CO}_k(G[X])$ such that $|f_X(X)|\geq 2$. By permuting colors in $f_Y$, we can combine the two colorings $f_X$ and $f_Y$ to obtain a coloring $f\in {\cal CO}_k(G)$ (by Lemma~\ref{Le:perfect}). It remains to consider the case $a=b=k$. Then let $G_2$ be the graph obtained from $G[Y \cup X_F]$ by adding all edges between the vertices of $X_F$, so that $X_F$ becomes a clique of $G_2$. Then we find colorings $f_1\in {\cal CO}_k(G_1)$ and $f_2\in {\cal CO}_k(G_2)$ and, hence, colorings $f_X\in {\cal CO}_k(G[X])$ and $f_Y\in {\cal CO}_k(G[Y])$ such that $|f_X(X)|\geq 2$ and $|f_Y(Y)|\geq 2$. By permuting colors in $f_Y$, we can combine the two colorings $f_X$ and $f_Y$ to obtain a coloring $f\in {\cal CO}_k(G)$ (by Lemma~\ref{Le:perfect}).}
\end{document} | arXiv |
# Linear algebra fundamentals
Consider a vector $\vec{v} = (1, 2, 3)$. We can perform basic operations on this vector, such as addition, subtraction, and scalar multiplication. For example, if we have another vector $\vec{w} = (4, 5, 6)$, we can add them together to get $\vec{v} + \vec{w} = (5, 7, 9)$.
Matrices are another important concept in linear algebra. They are arrays of numbers arranged in rows and columns. We can perform various operations on matrices, such as addition, subtraction, and multiplication. For example, if we have two matrices $A$ and $B$, we can multiply them together to get $AB$.
## Exercise
Define a matrix $A$ and a matrix $B$. Multiply them together.
# Solving systems of linear equations
Consider the following system of linear equations:
$$
\begin{cases}
x + y = 5 \\
2x - y = 1
\end{cases}
$$
We can solve this system using substitution or elimination.
To solve a system of linear equations using substitution, we can express one variable in terms of the others and substitute it into the other equations. For example, from the first equation, we can express $y$ in terms of $x$: $y = 5 - x$. Then, we can substitute this expression into the second equation to solve for $x$.
## Exercise
Solve the system of linear equations from the example.
# Eigenvalues and eigenvectors
Consider the matrix $A = \begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix}$. We can find its eigenvalues and eigenvectors.
To find the eigenvalues of a matrix, we need to solve the characteristic equation $\det(A - \lambda I) = 0$, where $\lambda$ is the eigenvalue and $I$ is the identity matrix. We can use numerical methods, such as the Newton-Raphson method, to find the roots of the characteristic equation.
## Exercise
Find the eigenvalues of the matrix $A$ from the example.
# Exploring numerical solutions to differential equations
Consider the differential equation $dy/dt = -2y$. We can solve it numerically using Euler's method.
Euler's method is a simple numerical integration method that approximates the solution of a differential equation. It involves iteratively updating the solution at each time step using the derivative evaluated at the current solution.
## Exercise
Solve the differential equation $dy/dt = -2y$ using Euler's method.
# Applications of divergence in physics and engineering
Consider the Navier-Stokes equations, which describe the motion of fluid. The divergence of the velocity field is an important quantity in these equations.
The Navier-Stokes equations are a set of partial differential equations that describe the motion of fluid. The divergence of the velocity field, which represents the rate of change of fluid density, is an important quantity in these equations.
## Exercise
Derive the Navier-Stokes equations for fluid motion.
# Numerical integration with SciPy
We can use SciPy's `quad` function to perform numerical integration. For example, to integrate the function $f(x) = x^2$ from $0$ to $1$, we can write:
```python
import scipy.integrate as spi
def f(x):
return x**2
result, error = spi.quad(f, 0, 1)
```
SciPy provides various functions for numerical integration, such as `quad`, `trapz`, and `simps`. These functions allow us to integrate functions over finite intervals and approximate the definite integral.
## Exercise
Use SciPy's `trapz` function to integrate the function $f(x) = x^2$ from $0$ to $1$.
# Solving partial differential equations with SciPy
We can use SciPy's `solve_ivp` function to solve a system of ordinary differential equations. For example, to solve the equation $dy/dt = -2y$ with initial condition $y(0) = 1$, we can write:
```python
import numpy as np
import scipy.integrate as spi
def f(t, y):
return -2 * y
result = spi.solve_ivp(f, (0, 1), [1])
```
SciPy's `solve_ivp` function can be used to solve systems of ordinary differential equations with initial conditions. It uses numerical methods, such as the Radau and BDF methods, to approximate the solution.
## Exercise
Solve the differential equation $dy/dt = -2y$ with initial condition $y(0) = 1$ using SciPy's `solve_ivp` function.
# Computing divergence with NumPy and SciPy
We can use NumPy's gradient function to compute the gradient of a scalar field. Then, we can take the dot product of the gradient with a unit vector to compute the divergence. For example, to compute the divergence of the vector field $\vec{F} = (x^2, y^2)$, we can write:
```python
import numpy as np
import scipy.ndimage as spi
def F(x, y):
return x**2, y**2
x = np.linspace(0, 1, 100)
y = np.linspace(0, 1, 100)
X, Y = np.meshgrid(x, y)
Fx, Fy = F(X, Y)
grad_Fx = spi.filters.sobel(Fx)
grad_Fy = spi.filters.sobel(Fy)
div_F = grad_Fx + grad_Fy
```
To compute the divergence of a vector field, we can use NumPy's gradient function to compute the gradient of the vector field. Then, we can take the dot product of the gradient with a unit vector to compute the divergence.
## Exercise
Compute the divergence of the vector field $\vec{F} = (x^2, y^2)$ using NumPy and SciPy.
# Visualizing divergence with Matplotlib
We can use Matplotlib's `pcolormesh` function to create a contour plot of the divergence of a vector field. For example, to visualize the divergence of the vector field $\vec{F} = (x^2, y^2)$, we can write:
```python
import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(0, 1, 100)
y = np.linspace(0, 1, 100)
X, Y = np.meshgrid(x, y)
Fx, Fy = F(X, Y)
grad_Fx = spi.filters.sobel(Fx)
grad_Fy = spi.filters.sobel(Fy)
div_F = grad_Fx + grad_Fy
plt.pcolormesh(X, Y, div_F)
plt.colorbar(label='Divergence')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Divergence of F')
plt.show()
```
To visualize the divergence of a vector field, we can use Matplotlib's `pcolormesh` function to create a contour plot of the divergence. This allows us to see the spatial distribution of the divergence and identify regions of high and low divergence.
## Exercise
Visualize the divergence of the vector field $\vec{F} = (x^2, y^2)$ using Matplotlib.
# Hands-on exercises and projects
## Exercise
Complete the following exercises:
1. Solve a system of linear equations using substitution.
2. Solve a differential equation using Euler's method.
3. Compute the divergence of a vector field using NumPy and SciPy.
4. Visualize the divergence of a vector field using Matplotlib.
These exercises will help you practice the skills learned in the textbook and apply them to real-world problems. They will also provide you with an opportunity to explore the applications of divergence in various fields, such as physics and engineering.
# Conclusion
In this textbook, we have explored the fundamentals of linear algebra, numerical methods for solving differential equations, and the applications of divergence in physics and engineering. We have also learned how to use Python libraries like SciPy and NumPy to compute and visualize the divergence of vector fields.
## Exercise
Reflect on what you have learned in this textbook and identify any areas where you would like to explore further. | Textbooks |
Sliding mode control of the Hodgkin–Huxley mathematical model
EECT Home
Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping
December 2019, 8(4): 867-882. doi: 10.3934/eect.2019042
Existence and extinction in finite time for Stratonovich gradient noise porous media equations
Mattia Turra ,
Dipartimento di Informatica, Università degli Studi di Verona, Strada Le Grazie 15, I–73134, Verona, Italy
* Corresponding author: Mattia Turra
Received November 2018 Published June 2019
We study existence and uniqueness of distributional solutions to the stochastic partial differential equation $ dX - \bigl( \nu \Delta X + \Delta \psi (X) \bigr) dt = \sum_{i = 1}^N \langle b_i, \nabla X \rangle \circ d\beta_i $ in $ ]0,T[ \times \mathcal{O} $, with $ X(0) = x(\xi) $ in $ \mathcal{O} $ and $ X = 0 $ on $ ]0,T[ \times \partial \mathcal{O} $. Moreover, we prove extinction in finite time of the solutions in the special case of fast diffusion model and of self-organized criticality model.
Keywords: Stochastic partial differential equations, porous media equations, multiplicative Stratonovich gradient noise, fast diffusion, self-organized criticality.
Mathematics Subject Classification: Primary: 60H15; Secondary: 35K55, 76S05.
Citation: Mattia Turra. Existence and extinction in finite time for Stratonovich gradient noise porous media equations. Evolution Equations & Control Theory, 2019, 8 (4) : 867-882. doi: 10.3934/eect.2019042
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Mattia Turra | CommonCrawl |
Last 3 years (12)
European Astronomical Society Publications Series (2)
Invasive Plant Science and Management (1)
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Edited by Richard P. Tucker, University of Michigan, Ann Arbor, Tait Keller, Rhodes College, Memphis, J. R. McNeill, Georgetown University, Washington DC, Martin Schmid
Book: Environmental Histories of the First World War
Print publication: 23 August 2018, pp ix-ix
Part I - Europe and North America
Print publication: 23 August 2018, pp 17-96
Print publication: 23 August 2018, pp x-xi
Part IV - The Long Aftermath
Print publication: 23 August 2018, pp 255-295
Part II - War's Global Reach
Print publication: 23 August 2018, pp 97-172
Part III - The Middle East and Africa
Print publication: 23 August 2018, pp v-vi
Print publication: 23 August 2018, pp vii-viii
Print publication: 23 August 2018, pp xii-xii
Environmental Histories of the First World War
Edited by Richard P. Tucker, Tait Keller, J. R. McNeill, Martin Schmid
This anthology surveys the ecological impacts of the First World War. Editors Richard P. Tucker, Tait Keller, J. R. McNeill, and Martin Schmidt bring together a list of experienced authors who explore the global interactions of states, armies, civilians, and the environment during the war. They show how the First World War ushered in enormous environmental changes, including the devastation of rural and urban environments, the consumption of strategic natural resources such as metals and petroleum, the impact of war on urban industry, and the disruption of agricultural landscapes leading to widespread famine. Taking a global perspective, Environmental Histories of the First World War presents the ecological consequences of the vast destructive power of the new weaponry and the close collaboration between militaries and civilian governments taking place during this time, showing how this war set trends for the rest of the century.
KIT coaxial gyrotron development: from ITER toward DEMO
S. Ruess, K. A. Avramidis, M. Fuchs, G. Gantenbein, Z. Ioannidis, S. Illy, J. Jin, P. C. Kalaria, T. Kobarg, I. Gr. Pagonakis, T. Ruess, T. Rzesnicki, M. Schmid, M. Thumm, J. Weggen, A. Zein, J. Jelonnek
Journal: International Journal of Microwave and Wireless Technologies / Volume 10 / Issue 5-6 / June 2018
Karlsruhe Institute of Technology (KIT) is doing research and development in the field of megawatt-class radio frequency (RF) sources (gyrotrons) for the Electron Cyclotron Resonance Heating (ECRH) systems of the International Thermonuclear Experimental Reactor (ITER) and the DEMOnstration Fusion Power Plant that will follow ITER. In the focus is the development and verification of the European coaxial-cavity gyrotron technology which shall lead to gyrotrons operating at an RF output power significantly larger than 1 MW CW and at an operating frequency above 200 GHz. A major step into that direction is the final verification of the European 170 GHz 2 MW coaxial-cavity pre-prototype at longer pulses up to 1 s. It bases on the upgrade of an already existing highly modular short-pulse (ms-range) pre-prototype. That pre-prototype has shown a world record output power of 2.2 MW already. This paper summarizes briefly the already achieved experimental results using the short-pulse pre-prototype and discusses in detail the design and manufacturing process of the upgrade of the pre-prototype toward longer pulses up to 1 s.
A data-assimilation method for Reynolds-averaged Navier–Stokes-driven mean flow reconstruction
Dimitry P. G. Foures, Nicolas Dovetta, Denis Sipp, Peter J. Schmid
Journal: Journal of Fluid Mechanics / Volume 759 / 25 November 2014
Published online by Cambridge University Press: 04 November 2014, pp. 404-431
Print publication: 25 November 2014
We present a data-assimilation technique based on a variational formulation and a Lagrange multipliers approach to enforce the Navier–Stokes equations. A general operator (referred to as the measure operator) is defined in order to mathematically describe an experimental measure. The presented method is applied to the case of mean flow measurements. Such a flow can be described by the Reynolds-averaged Navier–Stokes (RANS) equations, which can be formulated as the classical Navier–Stokes equations driven by a forcing term involving the Reynolds stresses. The stress term is an unknown of the equations and is thus chosen as the control parameter in our study. The data-assimilation algorithm is derived to minimize the error between a mean flow measurement and the measure performed on a numerical solution of the steady, forced Navier–Stokes equations; the optimal forcing is found when this error is minimal. We demonstrate the developed data-assimilation framework on a test case: the two-dimensional flow around an infinite cylinder at a Reynolds number of $\mathit{Re}=150$ . The mean flow is computed by time-averaging instantaneous flow fields from a direct numerical simulation (DNS). We then perform several 'measures' on this mean flow and apply the data-assimilation method to reconstruct the full mean flow field. Spatial interpolation, extrapolation, state vector reconstruction and noise filtering are considered independently. The efficacy of the developed identification algorithm is quantified for each of these cases and compared with more traditional methods when possible. We also analyse the identified forcing in terms of unsteadiness characterization, present a way to recover the second-order statistical moments of the fluctuating velocities and finally explore the possibility of pressure reconstruction from velocity measurements.
Optimal mixing in two-dimensional plane Poiseuille flow at finite Péclet number
D. P. G. Foures, C. P. Caulfield, P. J. Schmid
Journal: Journal of Fluid Mechanics / Volume 748 / 10 June 2014
We consider the nonlinear optimisation of the mixing of a passive scalar, initially arranged in two layers, in a two-dimensional plane Poiseuille flow at finite Reynolds and Péclet numbers, below the linear instability threshold. We use a nonlinear-adjoint-looping approach to identify optimal perturbations leading to maximum time-averaged energy as well as maximum mixing in a freely evolving flow, measured through the minimisation of either the passive scalar variance or the so-called mix-norm, as defined by Mathew, Mezić & Petzold (Physica D, vol. 211, 2005, pp. 23–46). We show that energy optimisation appears to lead to very weak mixing of the scalar field whereas the optimal mixing initial perturbations, despite being less energetic, are able to homogenise the scalar field very effectively. For sufficiently long time horizons, minimising the mix-norm identifies optimal initial perturbations which are very similar to those which minimise scalar variance, demonstrating that minimisation of the mix-norm is an excellent proxy for effective mixing in this finite-Péclet-number bounded flow. By analysing the time evolution from initial perturbations of several optimal mixing solutions, we demonstrate that our optimisation method can identify the dominant underlying mixing mechanism, which appears to be classical Taylor dispersion, i.e. shear-augmented diffusion. The optimal mixing proceeds in three stages. First, the optimal mixing perturbation, energised through transient amplitude growth, transports the scalar field across the channel width. In a second stage, the mean flow shear acts to disperse the scalar distribution leading to enhanced diffusion. In a final third stage, linear relaxation diffusion is observed. We also demonstrate the usefulness of the developed variational framework in a more realistic control case: mixing optimisation by prescribed streamwise velocity boundary conditions.
Synchrotron-Based Chemical Nano-Tomography of Microbial Cell-Mineral Aggregates in their Natural, Hydrated State
Gregor Schmid, Fabian Zeitvogel, Likai Hao, Pablo Ingino, Wolfgang Kuerner, James J. Dynes, Chithra Karunakaran, Jian Wang, Yingshen Lu, Travis Ayers, Chuck Schietinger, Adam P. Hitchcock, Martin Obst
Journal: Microscopy and Microanalysis / Volume 20 / Issue 2 / April 2014
Print publication: April 2014
Chemical nano-tomography of microbial cells in their natural, hydrated state provides direct evidence of metabolic and chemical processes. Cells of the nitrate-reducing Acidovorax sp. strain BoFeN1 were cultured in the presence of ferrous iron. Bacterial reduction of nitrate causes precipitation of Fe(III)-(oxyhydr)oxides in the periplasm and in direct vicinity of the cells. Nanoliter aliquots of cell-suspension were injected into custom-designed sample holders wherein polyimide membranes collapse around the cells by capillary forces. The immobilized, hydrated cells were analyzed by synchrotron-based scanning transmission X-ray microscopy in combination with angle-scan tomography. This approach provides three-dimensional (3D) maps of the chemical species in the sample by employing their intrinsic near-edge X-ray absorption properties. The cells were scanned through the focus of a monochromatic soft X-ray beam at different, chemically specific X-ray energies to acquire projection images of their corresponding X-ray absorbance. Based on these images, chemical composition maps were then calculated. Acquiring projections at different tilt angles allowed for 3D reconstruction of the chemical composition. Our approach allows for 3D chemical mapping of hydrated samples and thus provides direct evidence for the localization of metabolic and chemical processes in situ.
Varicella zoster virus in American Samoa: seroprevalence and predictive value of varicella disease history in elementary and college students
A. MAHAMUD, J. LEUNG, Y. MASUNU-FALEAFAGA, E. TESHALE, R. WILLIAMS, T. DULSKI, M. THIEME, P. GARCIA, D. S. SCHMID, S. R. BIALEK
Journal: Epidemiology & Infection / Volume 142 / Issue 5 / May 2014
Published online by Cambridge University Press: 26 July 2013, pp. 1002-1007
The epidemiology of varicella is believed to differ between temperate and tropical countries. We conducted a varicella seroprevalence study in elementary and college students in the US territory of American Samoa before introduction of a routine varicella vaccination programme. Sera from 515 elementary and 208 college students were tested for the presence of varicella-zoster virus (VZV) IgG antibodies. VZV seroprevalence increased with age from 76·0% in the 4–6 years group to 97·7% in those aged ⩾23 years. Reported history of varicella disease for elementary students was significantly associated with VZV seropositivity. The positive and negative predictive values of varicella disease history were 93·4% and 36·4%, respectively, in elementary students and 97·6% and 3·0%, respectively, in college students. VZV seroprevalence in this Pacific island appears to be similar to that in temperate countries and suggests endemic VZV circulation.
Localization of flow structures using $\infty $ -norm optimization
Journal: Journal of Fluid Mechanics / Volume 729 / 25 August 2013
Stability theory based on a variational principle and finite-time direct-adjoint optimization commonly relies on the kinetic perturbation energy density ${E}_{1} (t)= (1/ {V}_{\Omega } )\int \nolimits _{\Omega } e(\boldsymbol{x}, t)\hspace{0.167em} \mathrm{d} \Omega $ (where $e(\boldsymbol{x}, t)= \vert \boldsymbol{u}{\vert }^{2} / 2$ ) as a measure of disturbance size. This type of optimization typically yields optimal perturbations that are global in the fluid domain $\Omega $ of volume ${V}_{\Omega } $ . This paper explores the use of $p$ -norms in determining optimal perturbations for 'energy' growth over prescribed time intervals of length $T$ . For $p= 1$ the traditional energy-based stability analysis is recovered, while for large $p\gg 1$ , localization of the optimal perturbations is observed which identifies confined regions, or 'hotspots', in the domain where significant energy growth can be expected. In addition, the $p$ -norm optimization yields insight into the role and significance of various regions of the flow regarding the overall energy dynamics. As a canonical example, we choose to solve the $\infty $ -norm optimal perturbation problem for the simple case of two-dimensional channel flow. For such a configuration, several solutions branches emerge, each of them identifying a different energy production zone in the flow: either the centre or the walls of the domain. We study several scenarios (involving centre or wall perturbations) leading to localized energy production for different optimization time intervals. Our investigation reveals that even for this simple two-dimensional channel flow, the mechanism for the production of a highly energetic and localized perturbation is not unique in time. We show that wall perturbations are optimal (with respect to the $\infty $ -norm) for relatively short and long times, while the centre perturbations are preferred for very short and intermediate times. The developed $p$ -norm framework is intended to facilitate worst-case analysis of shear flows and to identify localized regions supporting dominant energy growth.
The preferred mode of incompressible jets: linear frequency response analysis
X. Garnaud, L. Lesshafft, P. J. Schmid, P. Huerre
Journal: Journal of Fluid Mechanics / Volume 716 / 10 February 2013
The linear amplification of axisymmetric external forcing in incompressible jet flows is investigated within a fully non-parallel framework. Experimental and numerical studies have shown that isothermal jets preferably amplify external perturbations for Strouhal numbers in the range $0. 25\leq {\mathit{St}}_{D} \leq 0. 5$ , depending on the operating conditions. In the present study, the optimal forcing of an incompressible jet is computed as a function of the excitation frequency. This analysis characterizes the preferred amplification as a pseudo-resonance with a dominant Strouhal number of around $0. 45$ . The flow response at this frequency takes the form of a vortical wavepacket that peaks inside the potential core. Its global structure is characterized by the cooperation of local shear-layer and jet-column modes.
Asteroseismology of eclipsing binary stars using Kepler and the hermes spectrograph
V.S. Schmid, J. Debosscher, P. Degroote, C. Aerts
Journal: European Astronomical Society Publications Series / Volume 64 / 2013
We introduce our PhD project in which we focus on pulsating stars in eclipsing binaries. The combination of high-precision Kepler photometry with high-resolution hermes spectroscopy allows for detailed descriptions of our sample of target stars. We report here the detection of three false positives by radial velocity measurements.
Observing GRBs with the LOFT Wide Field Monitor
S. Brandt, M. Hernanz, M. Feroci, L. Amati, Alvarez, P. Azzarello, D. Barret, E. Bozzo, C. Budtz-Jørgensen, R. Campana, A. Castro-Tirado, A. Cros, E. Del Monte, I. Donnarumma, Y. Evangelista, J.L. Galvez Sanchez, D. Götz, F. Hansen, J.W. den Herder, A. Hornstrup, R. Hudec, D. Karelin, M. van der Klis, S. Korpela, I. Kuvvetli, N. Lund, P. Orleanski, M. Pohl, A. Rachevski, A. Santangelo, S. Schanne, C. Schmid, L. Stella, S. Suchy, C. Tenzer, A. Vacchi, J. Wilms, N. Zampa, J.J.M. in't Zand, A. Zdziarski
LOFT (Large Observatory For X-ray Timing) is one of the four candidate missions currently under assessment study for the M3 mission in ESAs Cosmic Vision program to be launched in 2024. LOFT will carry two instruments with prime sensitivity in the 2–30 keV range: a 10 m2 class large area detector (LAD) with a <1° collimated field of view and a wide field monitor (WFM) instrument. The WFM is based on the coded mask principle, and 5 camera units will provide coverage of more than 1/3 of the sky. The prime goal of the WFM is to detect transient sources to be observed by the LAD. With its wide field of view and good energy resolution of <500 eV, the WFM will be an excellent instrument for detecting and studying GRBs and X-ray flashes. The WFM will be able to detect ~150 gamma ray bursts per year, and a burst alert system will enable the distribution of ~100 GRB positions per year with a ~1 arcmin location accuracy within 30 s of the burst. | CommonCrawl |
\begin{definition}[Definition:Integrable Function/p-Integrable]
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f \in \MM_{\overline \R}, f: X \to \overline \R$ be a measurable function.
Let $p \ge 1$ be a real number.
Then $f$ is said to be '''$p$-integrable in respect to $\mu$''' {{iff}}:
:$\ds \int \size f^p \rd \mu < +\infty$
is $\mu$-integrable.
\end{definition} | ProofWiki |
Impact of the COVID-19 pandemic on the clinical training of last year medical students in Mexico: a cross-sectional nationwide study
Maximiliano Servin-Rojas1,
Antonio Olivas-Martinez2,
Michelle Dithurbide-Hernandez1,
Julio Chavez-Vela1,
Vera L. Petricevich1,
Ignacio García-Juárez3,
Alice Gallo de Moraes4 &
Benjamin Zendejas5
BMC Medical Education volume 22, Article number: 24 (2022) Cite this article
The COVID-19 pandemic has brought unprecedented changes to medical education. However, no data are available regarding the impact the pandemic may have on medical training in Mexico. The aim of our study was to evaluate and identify the medical school students' perceptions of the changes in their clinical training due to the pandemic in Mexico.
This was a cross-sectional study where a previous validated online survey was translated and adapted by medical education experts and applied to senior medical students from March to April of 2021. The 16-item questionnaire was distributed online combining dichotomous, multiple-choice, and 5-point Likert response scale questions. Descriptive and multivariate analyses were performed to compare the student's perceptions between public and private schools.
A total of 671 responses were included in the study period. Most participants were from public schools (81%) and female (61%). Almost every respondent (94%) indicated it was necessary to obtain COVID-19 education, yet only half (54%) received such training. Students in private schools were less likely to have their clinical instruction canceled (53% vs. 77%, p = 0.001) and more likely to have access to virtual instruction (46% vs. 22%, p = 0.001) when compared to students from public schools. Four out of every five students considered their training inferior to that of previous generations, and most students (82%) would consider repeating their final year of clinical training.
The impact of the COVID-19 on medical education in Mexico has been significant. Most final-year medical students have been affected by the cancellation of their in-person clinical instruction, for which the majority would consider repeating their final year of training. Efforts to counterbalance this lack of clinical experience with virtual or simulation instruction are needed.
The COVID-19 pandemic has changed the education of health professions [1]. In March 2020, the General Health Council of Mexico declared COVID-19 a national health emergency and took measures to mitigate the burden of the disease [2]. The Mexican government suspended all public and private activities that were considered non-essential. Most medical schools decided to cancel in-person teaching and moved to virtual or remote education. The Association of American Medical Colleges (AAMC) and its Mexican equivalent, the Asociacion Mexicana de Facultades y Escuelas de Medicina (AMFEM) recommended suspending all clinical activities [1]. Medical schools in Mexico aligned with these recommendations and removed medical students from hospital or outpatient-based settings [3].
Exposure to the clinical environment is a vital part of the training of a physician [4]. Through observation, practice and repetition, usually under the guidance of expert clinicians, medical students learn essential clinical skills to care for patients. Though some of these clinical skills can be taught in a non-clinical setting, with the help of simulation-based instruction, traditional bedside training remains a cornerstone of medical education [5]. The COVID-19 mitigation measures implemented in Mexico and other parts of the globe might significantly hinder the clinical skills training of medical students. This could have deleterious effects on their confidence and capability to care for patients and their autonomy as physicians. Additionally, studies have shown higher rates of mental health disorders during the pandemic [6, 7]. With campus and hospital-based teaching suspended, medical students are unable to participate in bedside training. Furthermore, simulation-training is often done in group settings or on-campus, for which the use of simulation, as an alternative to clinical training during the COVID pandemic, is challenging to implement given the social distancing guidelines that limit group gatherings and on-campus instruction.
The impact of these issues on the training of current medical students is unknown. Studies conducted in different parts of the world have shown that an important proportion of medical students feel less prepared due to these public health restrictions and changes to their clinical education [8,9,10]. In some cases, medical students were willing to participate in clinical instruction even if there was a risk for infection [10]. In low-income countries, there also exists a significant challenge to design and implement effective remote clinical education strategies, where a substantial proportion of the students consider remote strategies inadequate [9, 11]. Furthermore, heterogeneity among Mexican medical schools programs exists, as only few medical schools have alternatives to bedside training, and these issues might affect students unequally [12].
In order to better understand the impact of the COVID-19 pandemic on the training of medical students in Mexico, we surveyed the opinions of medical students in their last year of training. We hypothesized that most medical students in Mexico would feel unprepared to graduate from medical school and that many would choose to either extend their training or repeat their final year of medical school once clinical instruction could resume. The aim of our study was to evaluate and identify the medical school students' perceptions of the changes in their clinical training due to the pandemic in Mexico.
This was an exploratory, observational, and cross-sectional study. An online survey was distributed and conducted from March to April of 2021. The 16-item questionnaire was distributed online via the Google Forms platform (Alphabet, Mountain View, CA, USA). With the endorsement of the Asociación Mexicana de Médicos en Formación (AMMEF), a medical student association involved in medical education activities, the survey was distributed among social media platforms (Facebook™ and Twitter™), and via email to their last year medical student affiliates. The questionnaire was self-administered. We defined last year medical students as those in their final year of clinical (before entering hospital-based undergraduate internship). Students in the last year of medical school face a transition period between medical school and full-time patient care in the hospital-based scenario in the internship year. Virtual rotations were defined as observation of clinical practice via telemedicine under the supervision of an attending physician. Medical schools were categorized by type of institution (public vs. private) and were grouped into 4 different regions according to their geographical location: Northern, Central, and Southern Mexico, and Mexico City. Only participants in their last year of medical school who consented to participate were included. Exclusion criteria included medical students who did not provide consent or were not in their last year of medical education.
We evaluated medical student's perceptions using a previously validated questionnaire [8]. The survey was translated from English into Spanish by a bilingual expert in medical education and some extra questions that applied to the uniqueness of medical training in Mexico were developed. Questions were iteratively reviewed and revised by all study members. Before its implementation, the expert further revised the appropriateness of the questionnaire. Additionally, during the survey distribution, an email was provided for students to provide feedback. The final questionnaire comprised of 16 items, combining dichotomous, multiple choice and 5-point Likert response scale questions. The survey can be accessed with the following link: https://docs.google.com/forms/d/e/1FAIpQLSf9p9kZFsA4vpK7Z0dLh8fcIENshNlDb4eAkUc1lSNe9N6blw/viewform?usp=sf_link
Sample size was calculated based on the latest report of the Mexican Health Ministry [13], in which 22,160 last year medical students were eligible for this study. To obtain the sample size, and considering a value of α = 0.05 and a value of 1- β of 0.80, we used the following formula:
$$n = [\mathrm{EDFF}+\mathrm{Np}(1-\mathrm{p})]$$
$$[(\mathrm{d}2/\mathrm{Z}2 1-\mathrm{ \alpha }/2*(\mathrm{N}-1)+\mathrm{p}*(1-\mathrm{p})$$
The above power calculation resulted in a sample size of 645 students with a CI of 99%.
Descriptive statistics are presented as median and interquartile range (IQR) for numerical variables and frequencies and percentages for categorical variables. Categorical variables were compared using Pearson's Chi Squared Test or Fisher's Exact Test as appropriate. The level of statistical significance was set as p < 0.05. To evaluate if medical student perceptions regarding clinical training differ by the type of institution (private versus public), we transformed each item into a binary variable and for each of these binary items we fitted a multivariate logistic regression model, using robust standard error estimates, including the item as outcome, the type of institution as a predictor (1 = private, 0 = public) and adjusting age, gender, and geographical region. All statistical analyses were performed using R v.4.0.5. (R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-project.org/).
A total of 721 survey responses were obtained during the study period, representing 3% of the eligible medical student population. Fifty surveys were excluded since 7 students did not consent to participate in the study and 43 were from students not enrolled in their last year of medical school. Therefore, 671 surveys were finally included and analyzed.
Demographic Variables
Most of the participants studied in public schools (81%). Overall, 61% of the participants were female with a median age of 22 years (IQR 22—23). The majority of the students were enrolled in schools from Central Mexico (35%), followed by Northern (30%) and Southern (24%) regions, while only 11% where from schools in Mexico City (11%) (Table 1). The comparisons of participants' demographics characteristics between groups are summarized in Table 1.
Table 1 Demographics of survey respondents
Education on COVID-19
Despite almost every participant (97%) stated they "Agreed" or "Completely Agreed" that it was necessary to receive COVID-19 education, only around half of the students (54%) reported having received COVID-19 related education from their medical schools. Overall, the majority of the COVID-19 related medical education provided was in the form of in-person or virtual lectures (57%), followed by educational pamphlets or handouts (15%). Only 11 students reported simulation as a source of COVID-19 education. No statistical differences were found regarding COVID-19 education between groups (Table 2).
Table 2 Medical student perceptions regarding COVID-19 education
Perceptions Regarding Clinical Training
Most students considered clinical rotations as "Very Important" (94%). When compared between groups, similar rates were found (private 95% vs. public 94%; p = 0.7). Students from private schools were less likely to have their rotations cancelled (53% vs. 77%; p = 0.001) and were more likely to have virtual rotations, (46% vs. 22%; p = 0.001). The majority of the participants considered their medical training was affected by the COVID-19 (private 95% vs. 96% public; p = 0.65). Also, most students agreed these changes would negatively impact their performance as hospital-based interns (private 99% vs. 96% public; p = 0.10). Interestingly, a greater proportion of public-school students "Completely agreed" or "Agreed" that restrictions were necessary compared with students from private schools (78% vs. 60%, p = 0.001). Moreover, a lower proportion of the public-school students (37%) indicated they would feel safe going back to their clinical rotations. Four out of every five students (79% and 82% in public and private schools, respectively) felt that their clinical training was worse than their peers from previous generations not affected by the COVID-19 restrictions. A higher proportion of participants indicated that, if possible, they would repeat the final year of medical school training (81% public vs. 85% private; p = 0.3). Further details about medical students' perceptions about their clinical training are provided in Table 3.
Table 3 Medical student perceptions regarding clinical training
Multivariate Logistic Regression Analysis
Among subjects of the same age, gender, and geographical region, we estimated that, when comparing subjects who attend private and public institutions, the odds ratio for having virtual rotations was 1.99 (95% CI 1.30 – 3.04, p < 0.001), the OR of agreeing that restrictions were necessary was 0.41 (95% CI 0.27 – 0.62, p < 0.001), and the OR of feeling safe going back to the clinical rotations was 0.39 (95% CI 0.26 – 0.58, p < 0.001). The full results of the multivariate logisitic regression analysis can be found in Table 4.
Table 4 Multivariate logistic regression analysis of medical student perceptions regarding clinical training
This is the first study in Mexico to evaluate last year medical student's perceptions of changes in their clinical training during the COVID-19 pandemic and reveals several interesting observations, highlights inequities in training, and provides impetus to improve our educational environment. Students feel less prepared because of the COVID-19 pandemic restrictions on their education. In the UK, over 50% of the students felt less prepared to start postgraduate training [14], while in the US 18% of the third year medical students were willing to extend their training for an extra year [10]. These results contrast with ours, where 82% of the students were willing to extend their training. This suggests potentially a greater impact of COVID-19 in our country's medical education system.
Medical students want formal training on COVID-19, yet very few students reported having received such formal training. This can be accomplished via remote or virtual instruction. Most medical schools in both public and private sectors opted to use traditional teaching methods such as lectures and written educational materials. Students reported very limited use of simulation technologies, which might be related to the limited access to these centers in our country [15].
The role of medical students in the COVID-19 pandemic has been controversial. Some authors consider that the involvement of medical students can be beneficial for healthcare systems, patients, and their personal development as physicians, but given the existing risk of infection, their participation should be exclusively voluntary [16]. Others agree that medical students should not be involved in any clinical activity because they are not yet fully trained clinicians, do not receive a salary and may represent a risk for the people they live with [17]. In Mexico, most public hospitals care for infected COVID-19 patients. Public hospitals have also faced a shortage of personal protective equipment and high rates of infection among healthcare workers, which makes clinical rotations a very high-risk activity for medical students [18]. Probably related to these findings, most medical students in our survey reported that they would not feel safe going back to their clinical activities.
Despite clinical rotations being regarded as very important by students, most medical schools cancelled them, and only a limited number of students received an alternative such as virtual rotations. The lack of alternatives was striking as most students considered the absence of rotations would negatively impact their training and performance in the pre-grade internship. Probably related to the perception of poor training, most students would consider repeating the last year of medical school. This highlights the importance of innovation in medical education. Some experts in medical education are proposing different teaching alternatives with the development of multimodal training strategies [19,20,21]. Schools can offer in-person clinical rotations when the public health recommendations of social distancing can be safely achieved and when the risk of infection remains low [19]. If this is not an option, developing a virtual curriculum can be a safe and effective alternative. Even though the physical examination is not possible during virtual rotations, they can be asked to observe and evaluate different maneuvers elicited by the attending [19]. To develop skills in interviewing, students can take history and physical examination by telemedicine from consenting patients. From these experiences, they can be expected to develop written reports that can be presented to attendings and peers for additional feedback [21]. However, significant challenges also exist in the implementation of these strategies as many countries do not have the existent infrastructure to adopt a robust virtual curriculum [9]. The cancelation of elective procedures and other procedural activities could also limit the potential exposure of the students in virtual rotations. Similarly, clinicians on the front-line are also very taxed by the extra workload of caring for patients with COVID-19 and may not have the time to participate in remote medical student instruction.
In Mexico, the Consejo Mexicano para la Acreditación de la Educación Médica (COMAEM), oversees the evaluation of the quality of medical training. Only 80 out of more than 140 medical schools are certified by this organism, suggesting that there could be unequal quality of training due to differences in the regulatory bodies that accredit each medical school [22]. This study provides further insights into the inequities in medical education in Mexico. Medical students in private schools were more likely to have virtual instruction and were less likely to have their clinical electives cancelled. These issues can likely further accentuate the gap between medical students trained in private versus public schools. Further research should explore ways to enhance medical school training opportunities for public schools.
Simulation remains another unexplored area in medical education in Mexico. Studies have shown the effectiveness of simulation in facilitating teamwork, teaching basic science, clinical and procedural skills in different scenarios [23]. Despite its proven benefits, these technologies might be difficult to implement during a global pandemic [24]. Strategies limiting the number of instructors and medical students along with the proper following of public health measures could make simulation centers relatively safe [24]. We consider simulation should be increasingly adopted by medical schools in Mexico to offer more evidence-based learning techniques for trainees. Similarly, personal at-home simulators with or without virtual feedback have been successfully used as an alternative modality to in-person simulation instruction for certain skill sets; [25, 26] and online simulation is another promising alternative under evaluation [27]. Teaching faculty is able to develop simulated medical records that students can easily access anytime. For inpatients, students can give follow up to their simulated patients and solve the different complications that might arise from admission to discharge. Even though students prefer traditional in person clinical activities, most of them appear to be satisfied with this type of training [27]. This type of curriculum could be an attractive alternative for low-income countries. It has the benefit of being low cost, [28,29,30] less time intensive for students and it can provide ample feedback from expert clinicians.
These results should encourage policymakers to update the Mexican regulations (Norma Oficial Mexicana, "NOM") on medical education. Studies have reported heterogeneity among teaching hospitals [31]. We propose that teaching hospitals should undergo continued evaluations to establish quality standards [32]. Furthermore, telemedicine remains underutilized in Mexico [33]. Investing in telemedicine could improve access to healthcare in rural communities and offer learning opportunities to medical students. Medical schools should consider integrating telemedicine and simulation into their curriculum and train educators on the usage of these technologies [34]. In addition, vaccinating and training medical students on the proper use of personal protective equipment could facilitate a safe return to clinical rotations [35]. Lastly, medical schools should train faculty members to provide educational and emotional support to improve academic achievements, and most importantly, their sense of security [36].
This study has some limitations. Due to the study design, we can only provide a representation of the perception of last year medical students during a specific time period. The participation in this study response rate is low and might not represent the perceptions of the entire population of last year medical students in Mexico, yet it still the largest study of its kind with a significant number of survey responses of students that represent diverse geographic regions and types of medical training (public vs. private schools). Because our study was distributed online by social media it might be susceptible to non-response and participation bias; students with access to internet could more readily participate. Furthermore, not all medical students engage in social media platforms and hence might have been unaware of the study. We attempted to overcome this by emailing medical students, but this method has its own pitfalls, and we did not have access to the emails of all eligible medical students.
The COVID-19 pandemic has impacted medical education. Medical students had to abandon clinical activities due to the inherent risks of infection. Medical schools in Mexico were unprepared to implement a rapid and effective response for this unprecedented pandemic. Medical education in Mexico is highly heterogeneous, and most medical schools have no alternatives to traditional bedside training. This has left medical students feeling unprepared for the next stages of their careers. Urgent measures must be taken to guarantee effective clinical training in situations like this and enable them to safely care for patients. Multiple alternative educational strategies exist that may be adaptable to middle and low-income countries. Medical schools should develop contingency plans to deliver effective clinical instruction that does not rely on in-person presence, in order to be prepared for future pandemics.
AAMC:
AMFEM:
Asociacion Mexicana de Facultades y Escuelas de Medicina
AMMEF:
Asociación Mexicana de Médicos en Formación
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Blancas JLA, Bárcena JL. Formación de médicos especialistas en México. EducaciónMédica. 2018;19:36–42.
Secretaria de Salud. Se incrementa en 43 mil el número de estudiantes de medicina en los últimos 8 añoso. 2018. (https://www.gob.mx/salud/prensa/239-se-incrementa-en-43-mil-el-numero-de-estudiantes-de-medicina-en-los-ultimos-8-anos).
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The authors would like to thank Dr. Jesús Naveja Romero and Dr. Fernanda Romero Hernández for providing help with the statistical analysis and manuscript styling, and Dr. Marvin Mendoza, Dr. Paola Saskia Castañeda Anaya, and Dr Bertha Alicia Anaya Roman for their help distributing our survey.
No funding was required for this study.
Faculty of Medicine, Universidad Autónoma del Estado de Morelos, Cuernavaca, Morelos, Mexico
Maximiliano Servin-Rojas, Michelle Dithurbide-Hernandez, Julio Chavez-Vela & Vera L. Petricevich
Department of Biostatistics, University of Washington, Seattle, Washington, USA
Antonio Olivas-Martinez
Liver Transplant Unit, Instituto Nacional de Ciencias Médicas Y Nutrición "Salvador Zubirán", Mexico City, Mexico
Ignacio García-Juárez
Division of Pulmonary and Critical Care Medicine, Mayo Clinic, Rochester, Minnesota, USA
Alice Gallo de Moraes
Department of Surgery, Boston Children Hospital, Harvard Medical School, 300 Longwood Avenue, Boston, MA, 02115, USA
Benjamin Zendejas
Maximiliano Servin-Rojas
Michelle Dithurbide-Hernandez
Julio Chavez-Vela
Vera L. Petricevich
MSR, AOM, MDH, VLP, JCV, IGJ, AG, and BZ contributed with data collection. MSR, AOM, MDH, VLP, JCV, IGJ, AG, and BZ contributed with the writing and revision of the manuscript. MSR, AOM and BZ participated in data analysis. All authors have approved the final manuscript and take full responsibility for the data presented.
MSR Research Assistant at the Faculty of Medicine of the Universidad Autónoma del Estado de Morelos. AOM Biostatistics PhD student and Research Assistant in the University of Washington. MDH Research Assistant at the Faculty of Medicine of the Universidad Autónoma del Estado de Morelos. JCV Professor of Medicine and Reseacher at the Faculty of Medicine of the Universidad Autónoma del Estado de Morelos. VLP Dean and Researcher at the Faculty of Medicine of the Universidad Autónoma del Estado de Morelos. IGJ Reseacher at Instituto Nacional de la Nutricion "Salvador Zubirán". AG Assistant Professor of Medicine at Mayo Clinic. BZ Assistant Professor of Surgery at Boston Children Hospital.
Correspondence to Benjamin Zendejas.
The protocol was approved by the Bioethics Committee of the Universidad Autónoma del Estado de Morelos (reference number CONBIOETICA-17-CEI-003–20181112) and was performed in accordance with the principles expressed in the Declaration of Helsinki. The participants gave online informed consent for study inclusion. Informed consent specified no personal data collection nor distribution.
The authors declare that they do not have no competing interests.
Servin-Rojas, M., Olivas-Martinez, A., Dithurbide-Hernandez, M. et al. Impact of the COVID-19 pandemic on the clinical training of last year medical students in Mexico: a cross-sectional nationwide study. BMC Med Educ 22, 24 (2022). https://doi.org/10.1186/s12909-021-03085-w
DOI: https://doi.org/10.1186/s12909-021-03085-w | CommonCrawl |
\begin{definition}[Definition:Language of Propositional Logic/Alphabet/Sign/Connective]
The signs of the language of propositional logic include the '''connectives''':
{{begin-eqn}}
{{eqn | ll= \bullet
| l = \land
| o = :
| r = \)the conjunction sign\(
| c =
}}
{{eqn | ll= \bullet
| l = \lor
| o = :
| r = \)the disjunction sign\(
| c =
}}
{{eqn | ll= \bullet
| l = \implies
| o = :
| r = \)the conditional sign\(
| c =
}}
{{eqn | ll= \bullet
| l = \iff
| o = :
| r = \)the biconditional sign\(
| c =
}}
{{eqn | ll= \bullet
| l = \neg
| o = :
| r = \)the negation sign\(
| c =
}}
{{eqn | ll= \bullet
| l = \top
| o = :
| r = \)the tautology sign\(
| c =
}}
{{eqn | ll= \bullet
| l = \bot
| o = :
| r = \)the contradiction sign\(
| c =
}}
{{end-eqn}}
These comprise:
* The nullary connectives $\top$ and $\bot$, representing the canonical tautology and contradiction, respectively
* The unary connective $\neg$, representing negation
* The binary connectives $\land, \lor, \implies$ and $\iff$, representing, respectively, conjunction, disjunction, implication and biconditional.
\end{definition} | ProofWiki |
Bernstein's theorem (polynomials)
Bernstein's theorem is an inequality relating the maximum modulus of a complex polynomial function on the unit disk with the maximum modulus of its derivative on the unit disk. It was proven by Sergei Bernstein while he was working on approximation theory.[1]
Statement
Let $\max _{|z|=1}|f(z)|$ denote the maximum modulus of an arbitrary function $f(z)$ on $|z|=1$, and let $f'(z)$ denote its derivative. Then for every polynomial $P(z)$ of degree $n$ we have
$\max _{|z|=1}|P'(z)|\leq n\max _{|z|=1}|P(z)|$.
The inequality is best possible with equality holding if and only if
$P(z)=\alpha z^{n},\ |\alpha |=\max _{|z|=1}|P(z)|$.
[2]
Proof
Let $P(z)$ be a polynomial of degree $n$, and let $Q(z)$ be another polynomial of the same degree with no zeros in $|z|\geq 1$. We show first that if $|P(z)|<|Q(z)|$ on $|z|=1$, then $|P'(z)|<|Q'(z)|$ on $|z|\geq 1$.
By Rouché's theorem, $P(z)+\varepsilon \ Q(z)$ with $|\varepsilon |\geq 1$ has all its zeros in $|z|<1$. By virtue of the Gauss–Lucas theorem, $P'(z)+\varepsilon \ Q'(z)$ has all its zeros in $|z|<1$ as well. It follows that $|P'(z)|<|Q'(z)|$ on $|z|\geq 1$, otherwise we could choose an $\varepsilon $ with $|\varepsilon |\geq 1$ such that $P'(z)+\varepsilon Q'(z)$ has a zero in $|z|\geq 1$.
For an arbitrary polynomial $P(z)$ of degree $n$, we obtain Bernstein's Theorem by applying the above result to the polynomials $Q(z)=Cz^{n}$, where $C$ is an arbitrary constant exceeding $\max _{|z|=1}|P(z)|$.
Bernstein's inequality
In mathematical analysis, Bernstein's inequality states that on the complex plane, within the disk of radius 1, the degree of a polynomial times the maximum value of a polynomial is an upper bound for the similar maximum of its derivative. Taking the k-th derivative of the theorem,
$\max _{|z|\leq 1}(|P^{(k)}(z)|)\leq {\frac {n!}{(n-k)!}}\cdot \max _{|z|\leq 1}(|P(z)|).$
Similar results
Paul Erdős conjectured that if $P(z)$ has no zeros in $|z|<1$, then $\max _{|z|=1}|P'(z)|\leq {\frac {n}{2}}\max _{|z|=1}|P(z)|$. This was proved by Peter Lax.[3]
M. A. Malik showed that if $P(z)$ has no zeros in $|z|<k$ for a given $k\geq 1$, then $\max _{|z|=1}|P'(z)|\leq {\frac {n}{1+k}}\max _{|z|=1}|P(z)|$.[4]
See also
• Markov brothers' inequality
• Remez inequality
References
1. R. P. Boas, Jr., Inequalities for the derivatives of polynomials, Math. Mag. 42 (1969), 165–174.
2. M. A. Malik, M. C. Vong, Inequalities concerning the derivative of polynomials, Rend. Circ. Mat. Palermo (2) 34 (1985), 422–426.
3. P. D. Lax, Proof of a conjecture of P. Erdös on the derivative of a polynomial, Bull. Amer. Math. Soc. 50 (1944), 509–513.
4. M. A. Malik, On the derivative of a polynomial J. London Math. Soc (2) 1 (1969), 57–60.
Further reading
• Frappier, Clément (2004). "Note on Bernstein's inequality for the third derivative of a polynomial" (PDF). J. Inequal. Pure Appl. Math. 5 (1). Paper No. 7. ISSN 1443-5756. Zbl 1060.30003.
• Natanson, I.P. (1964). Constructive function theory. Volume I: Uniform approximation. Translated by Alexis N. Obolensky. New York: Frederick Ungar. MR 0196340. Zbl 0133.31101.
• Rahman, Q. I.; Schmeisser, G. (2002). Analytic theory of polynomials. London Mathematical Society Monographs. New Series. Vol. 26. Oxford: Oxford University Press. ISBN 0-19-853493-0. Zbl 1072.30006.
| Wikipedia |
\begin{definition}[Definition:Generalized Momentum]
The '''generalized momentum of analytical (Lagrangian, Hamiltonian) formulations of classical mechanics''' is defined as the partial derivative of the Lagrangian with regards to the time derivative of generalized coordinates:
:$p_i = \dfrac {\partial \LL} {\partial \dot q_i}$
where:
:$p_i$ is the $i$th coordinate of the generalized momenta
:$\LL$ is the Lagrangian
:$\dot q_i$ is the time derivative of the generalized coordinates $q_i$.
Category:Definitions/Dimensions of Measurement
\end{definition} | ProofWiki |
\begin{document}
\title[]{Exact asymptotics for Duarte and supercritical
rooted kinetically constrained
models} \author[L. Mar{\^e}ch\'e]{L. Mar{\^e}ch\'e} \email{[email protected]} \address{LPSM UMR 8001, Universit\'e Paris Diderot, Sorbonne Paris Cit\'e,
CNRS, 75013 Paris, France}
\author[F. Martinelli]{F. Martinelli} \email{[email protected]} \address{Dipartimento di Matematica e Fisica, Universit\`a Roma Tre, Largo S.L. Murialdo 00146, Roma, Italy}
\author[C. Toninelli]{C. Toninelli} \email{[email protected]} \address{ LPSM UMR 8001, Universit\'e Paris Diderot, Sorbonne Paris Cit\'e,
CNRS, 75013 Paris, France}
\thanks{This work has been supported by the ERC Starting Grant 680275 MALIG. F.M. acknowledges support of PRIN 2015 5PAWZB ''Large Scale Random Structures'' and C.T. of the ANR-15-CE40-0020-02 grant LSD}
\subjclass[2010]{Primary {60K35}, secondary 60J27}
\keywords{Glauber dynamics, kinetically constrained models, spectral
gap, bootstrap percolation, Duarte model}
\begin{abstract} Kinetically constrained models (KCM) are reversible interacting particle systems on $\mathbb Z^d$ with continuous time Markov dynamics of Glauber type, which represent a natural stochastic (and non-monotone) counterpart of the family of cellular automata known as $\mathcal U$-bootstrap percolation. Furthermore, KCM have an interest in their own since they display some of the most striking features of the liquid-glass transition, a major and longstanding open problem in condensed matter physics. A key issue for KCM is to identify the scaling of the characteristic time scales when the equilibrium density of empty sites, $q$, goes to zero. In \cite{MT,MMoT} a general scheme was devised to determine a sharp upper bound for these time scales. Our paper is devoted to developing a (very different) technique which allows to prove matching lower bounds. We analyse the class of two-dimensional {\sl supercritical rooted KCM} and the {\sl Duarte KCM}, the most studied critical $1$-rooted model. We
prove that the relaxation time and the mean infection time diverge for supercritical rooted KCM as $e^{\Theta((\log q)^2)}$ and for Duarte KCM as $e^{\Theta((\log q)^4/q^2)}$ when $q\downarrow 0$.
These results prove the conjectures put forward in
\cite{Robsurvey,MMoT}, and establish that the time scales for these KCM diverge much faster than for the corresponding $\ensuremath{\mathcal U}$-bootstrap processes, the main reason being the occurrence of energy barriers which determine the dominant behaviour for KCM, but which do not matter for the bootstrap dynamics.
\end{abstract} \maketitle
\section{Introduction}
Kinetically constrained models (KCM) are interacting particle systems on the integer lattice $\mathbb Z^d$, which were introduced in the physics literature in the 1980s in order to model the liquid-glass transition (see e.g. \cites{Ritort,GarrahanSollichToninelli} for reviews), a major and still largely open problem in condensed matter physics \cite{Berthier-Biroli}.
A generic KCM is a continuous time Markov process of Glauber type characterised by a finite collection of finite subsets of $\mathbb Z^d\setminus \{ \mathbf{0} \}$, $\,\mathcal U=\{X_1,\dots,X_m\}$, its {\sl update family}. A configuration $\o$ is defined by assigning to each site $x\in\mathbb Z^d$ an occupation variable $\omega_x\in\{0,1\},$ corresponding to an empty or occupied site respectively. Each site $x\in\mathbb Z^d$ waits an independent, mean one, exponential time and then, iff there exists $X\in \ensuremath{\mathcal U}$ such that $\o_y=0$ for all $y\in X+x$, site $x$ is updated to occupied with probability $p$ and to empty with probability $q=1-p$.
Since each {\sl update set} $X_i$ belongs to $\mathbb Z^d\setminus \{ \mathbf{0} \}$, the constraints never depend on the state of the to-be-updated site. As a consequence, the dynamics satisfies detailed balance w.r.t. the product Bernoulli($p$) measure, $\mu$, which is therefore a reversible invariant measure. Hence, the process started at $\mu$ is stationary.
Both from a physical and from a mathematical point of view, a central issue for KCM is to determine the speed of divergence of the characteristic time scales when $q\downarrow 0$. Two key quantities are: (i) the {\sl relaxation time} $T_{\rm rel}$, i.e. the inverse of the spectral gap of the Markov generator and (ii) the {\sl mean infection time} $\mathbb E_{\mu}(\tau_0)$, i.e. the mean over the stationary process of the first time at which the origin becomes empty. The study of the infection time has been largely addressed for the $\ensuremath{\mathcal U}$-bootstrap percolation \cite{BDMS,BSU,BBPS}, a class of discrete cellular automata that can be viewed as the monotone deterministic counterpart of KCM. For the $\mathcal U$-bootstrap, given a set of "infected" sites $A_t\subset\mathbb Z^d$ at time $t$, infected sites remain infected, and a site $x$ becomes infected at time $t + 1$ if the translate by $x$ of one of the update sets in $\ensuremath{\mathcal U}$ belongs to $A_t$. Thus, if infected (non infected) sites are regarded as empty (respectively occupied) sites, the constraint that has to be satisfied to infect a site for the $\mathcal U$-bootstrap is the same that is required to update the occupation variable for the KCM.
In \cite{MMoT} two of the authors together with R. Morris addressed the problem of identifying the divergence of time scales for two-dimensional KCM. The first goal of \cite{MMoT} was to identify the correct universality classes, which turn out to be different from those of $\ensuremath{\mathcal U}$-bootstrap percolation. Then, building on a strategy developed in \cite{MT} by two of the authors,
universal upper bounds on the relaxation and mean infection time within each class were proven and were conjectured to be sharp up to logarithmic corrections \cite{MMoT}. On the other hand, concerning lower bounds, so far the best general result is \begin{equation}\label{eq:trivial_lower}T_{\rm rel}\;\geqslant\; q\mathbb E_{\mu}(\tau_0)=\Omega(T)\end{equation} where $T$ denotes the median infection time for the $\mathcal U$-bootstrap process started with distribution $\mu$ (i.e. sites are initially infected independently with probability $q$), see \cite{MT}*{Lemma 4.3}. However this lower bound is in general far from optimal. Consider for example the {\sl one-dimensional East model} \cite{JACKLE} (and \cite{East-review} for a review) for which
a site can be updated iff its left neighbour is empty, namely $\mathcal U=\{\{-\vec e_1\}\}$. As $q\downarrow 0$, it holds
\begin{equation}\label{eq:East} \mathbb E^{\mbox{\tiny {East}} }_{\mu}(\tau_0)=e^{(\Theta(\log q)^2)}\end{equation} and the scaling holds for $T_{\rm rel}$, see~\cites{CFM,Aldous,CMRT} where the sharp value of the constant has been determined. This divergence is much faster than for the corresponding $\ensuremath{\mathcal U}$-bootstrap model, for which it holds $T=\Theta(1/q)$. To understand this difference it is necessary to recall a key combinatorial result \cite{SE1},\cite{CDG}*{Fact 1}: in order to empty the origin the East process has to go through a configuration with $\lceil\log_2 (\ell+1)\rceil$ simultaneous empty sites in $(-\ell,0]$, where $-\ell$ is the position of the rightmost empty site on $(-\infty,0]$. This logarithmic ``energy barrier'' (to employ the physics jargon) and the fact that at equilibrium typically $\ell\sim 1/q$ yield a divergence of the time scale as $q^{\Theta(\log q)}=e^{(\Theta(\log q)^2)}$. In turn, this peculiar scaling is the reason why the East model has been extensively studied by physicists (see \cite{KGC} and references therein). Indeed, if we set $q:=e^{-\beta}$ with $\beta$ the inverse temperature, we get the so called {\sl super-Arrhenius} divergence $e^{(\Theta(\beta^2))}$
which provides a very good fit of the experimental curves for fragile supercooled liquids near the glass transition \cite{Berthier-Biroli}. \\
In \cite{Robsurvey}, together with R. Morris, we conjectured that one of the universality classes of two-dimensional KCM, that we call {\sl supercritical rooted models}, features time scales diverging as for the East model. Our first main result (Theorem \ref{thm:rooted}) is to establish a lower bound which allows together with the upper bound in \cite{MMoT}*{Theorem 1} to prove this conjecture \footnote{Actually, the conjecture in \cite{Robsurvey} states that $\tau_0=e^{(\Theta(\log q)^2)}$ w.h.p. when $q\rightarrow 0$. As explained in Remark \ref{rem:w.h.p.}, we can also prove this stronger result.}, namely we prove $$\mathbb E^{\tiny{\ensuremath{\mathcal U}}}_{\mu}(\tau_0)=e^{(\Theta(\log q)^2)}\,\,\,\,\,\,\,\mbox{ $\forall\,\, \ensuremath{\mathcal U}$ in the supercritical rooted class }$$ and the same result for $T_{\rm rel}$. As for the East model, this divergence is much faster than for the corresponding $\ensuremath{\mathcal U}$-bootstrap process which scales as $T=1/q^{\Theta(1)}$ \cite{BSU}. A key input for our Theorem \ref{thm:rooted} is a combinatorial result proved by one of the authors in \cite{Laure} (see also Lemma \ref{lem:laure} in this paper) which considerably generalises to a higher dimensional and non oriented setting the above recalled combinatorial result for East \footnote{The result in \cite{Laure} holds also in $d>2$ on a properly defined class, i.e. all models which are not supercritical {\sl unrooted} (see \cite{Laure} for the precise definition). Our argument immediately extends to this higher dimensional setting yielding the same lower bound as in Theorem \ref{thm:rooted} for $T_{\rm rel}$ and $\mathbb E_{\mu}(\tau_0)$.}.\\
The $\ensuremath{\mathcal U}$-bootstrap results identify another universality class,
the so called {\sl critical update families}, which display a much faster divergence. In particular, in \cite{BDMS} it was proven that for this class it holds $T=e^{(\Theta(\log)^{c}/q^{\alpha})}$ with $\alpha$ a model dependent positive integer and $c=0$ or $c=2$. In \cite{MMoT}, together with R.Morris, we analysed KCM with critical update families and we put forward the conjecture that both $T_{\rm rel}$ and $\mathbb E_{\mu}(\tau_0)$ diverge as $e^{(\Theta(\log)^{c'}/q^{\nu})}$ with $\nu$ in general different from the exponent $\alpha$ of the corresponding $\ensuremath{\mathcal U}$-bootstrap process and we formulated (see \cite{MMoT}*{Conjecture 3}) a conjecture for the value of $\nu$ (which is again model dependent). In \cite{MMoT}*{Theorem 2} we established upper bounds for all critical models matching this conjecture. A matching lower bound exists only for those models for which the general lower bound \eqref{eq:trivial_lower} is sharp namely, in the language of \cite{MMoT},
for the special case of $\beta$-unrooted models with $\beta=\alpha$. Here we focus on the most studied update family which does not belong to this special case, the {\sl Duarte update family}, which consists of all the 2-subsets of the North, South and West neighbours of the origin \cite{Duarte}. Our second main result is a sharp lower bound on the infection and relaxation time for the Duarte KCM (Theorem \ref{thm:Duarte}) that, together with the upper bound in \cite{MMoT}, establishes the scaling $${\ensuremath{\mathbb E}} ^{\mbox{\tiny{Duarte}}}_\mu(\tau_0)=e^{\Theta\big((\log q)^4/q^2\big)}$$ as $q\downarrow 0$, and the same result holds for $T_{\rm rel}$. The value $\nu=2$ for the exponent is in agreement with our conjecture \cite{MMoT}*{Conjecture 3 (a)}, indeed in the language of \cite{MMoT} Duarte is a $1$-rooted model with $\alpha=1$, thus $\nu=2$. Notice that we identify also the exact power in the logarithmic correction. Finally, notice that the divergence is again much faster than for the corresponding $\ensuremath{\mathcal U}$-bootstrap model. Indeed, the median of the infection time for the $\ensuremath{\mathcal U}$-bootstrap Duarte model diverges as $T=e^{(\Theta(\log q)^2/q)}$ when $q\downarrow 0$ \cite{Mountford} (see also \cite{BCMS-Duarte} for the sharp value of the constant). \\
Both for Duarte and for supercritical rooted models, the sharper divergence of time scales for KCM is due to the fact that the infection time is not well approximated by the minimal number of updates needed to infect the origin (as it is for bootstrap percolation), but it is instead the result of a much more complex infection/healing mechanism. In particular, visiting regions of the configuration space with an anomalous amount of infection is heavily penalised and requires a very long time to actually take place \footnote{Borrowing again from physics jargon we could say that ``crossing the energy barriers'' is heavily penalised.}.
The basic underlying idea is that the dominant relaxation mechanism is an East like dynamics for large {\sl droplets} of empty sites. For supercritical rooted models these droplets have a finite (model dependent) size, hence an equilibrium density $q_{\mbox{\tiny{eff}}}= q^{\Theta(1)}$. For the Duarte model droplets have a size that diverges as $\ell=\frac{| \log q|}{q}$ and thus an equilibrium density $q_{\mbox{\tiny{eff}}}=q^{\ell}=e^{-(\log q)^2/q}$. Then a (very) rough understanding of our results is obtained by replacing $q$ with $q_{\mbox{\tiny{eff}}}$ in the result for the East model \eqref{eq:East}. One of the key technical difficulties to translate this intuition into a lower bound is that the droplets cannot be identified with a rigid structure, at variance with the East model where the droplets are single empty sites.
\section{Models and notation}\label{sec:notations} \subsection{Notation} For the reader's convenience we gather here some of the notation that we use throughout the paper. We will work on the probability space $(\O,\mu)$, where $\O=\{0,1\}^{{\ensuremath{\mathbb Z}} ^2}$ and $\mu$ is the product Bernoulli($p$) measure, and we will be interested in the asymptotic regime $q\downarrow 0,$ where $q=1-p$. Given $\o\in \O$ and $\Lambda\subset {\ensuremath{\mathbb Z}} ^2,$ we will often write $\o_\Lambda$ or $\o\mathord{\upharpoonright}_\Lambda$ for the collection $\{\o_x\}_{x\in \Lambda}$ and we shall write $\o_\Lambda\equiv 0$ to indicate that $\o_x=0\ \forall x\in \Lambda$. In this case we shall also say that $\Lambda$ is empty or infected. Similarly for $\o_\Lambda\equiv 1$ and in this case $\Lambda$ will be said to be occupied or healthy. We shall write $Y(\o)$ for the set $\{x\in {\ensuremath{\mathbb Z}} ^2 \colon \o_x=0\}$ and we shall say that $f \colon \O \mapsto {\ensuremath{\mathbb R}} $ is a \emph{local
function} if it depends on finitely many variables $\{\o_x\}_{x\in{\ensuremath{\mathbb Z}} ^2}$. Given a site $x\in {\ensuremath{\mathbb Z}} ^2$ of the form $x=(a,b)$ with $a,b\in {\ensuremath{\mathbb Z}} ,$ we shall sometimes refer to $b$ as the height of $x$. We shall also refer to a set $I\subset {\ensuremath{\mathbb Z}} ^2$ of the form $I=\{x,x+\vec e_i,\dots,x+(n-1) \vec e_i\}, x\in {\ensuremath{\mathbb Z}} ^2,$ as a (horizontal or vertical) interval of length $n\in {\ensuremath{\mathbb N}} ^*$.
Finally, we will use the standard notation $[n]$ for the set $\{1,\ldots,n\}.$
Throughout this paper we will often make use of standard asymptotic notation. If $f$ and $g$ are positive real-valued functions of $q\in (0,1)$, then we will write $f = O(g)$ if there exists a constant $C > 0$ such that $f(q) \;\leqslant\; C g(q)$ for every sufficiently small $q > 0$. We will also write $f = \O(g)$ if $g = O(f)$ and $f= \Theta(g)$ if $f = O(g)$ and $g = O(f)$. All constants, including those implied by the notation $O(\cdot)$, $\O(\cdot)$ and $\Theta(\cdot)$, will be such w.r.t. the parameter $q$.
\subsection{Models}\label{sec:models} Fix an {\sl update family} $\,\mathcal U=\{X_1,\dots,X_m\}$, that is, a finite collection of finite subsets of $\mathbb Z^2\setminus \{ \mathbf{0} \}$. Then the KCM with update family $\ensuremath{\mathcal U}$ is the Markov process on $\O$ associated to the Markov generator \begin{equation}
\label{eq:generator} (\ensuremath{\mathcal L} f)(\o)= \sum_{x\in {\ensuremath{\mathbb Z}} ^2}c_x(\o)\big( \mu_x(f) - f \big)(\o), \end{equation} where $f \colon \O \mapsto {\ensuremath{\mathbb R}} $ is a local function, $\mu_x(f)$ denotes the average of $f$ w.r.t.~the variable $\o_x$, and $c_x$ is the indicator function of the event that there exists $X\in \ensuremath{\mathcal U}$ such that $X+x$ is infected \hbox{\it i.e. } $\o_{X+x}\equiv 0.$ In the sequel we will sometimes say that $\o$ satisfies the update rule at $x$ if $c_x(\o)=1.$
Informally, this process can be described as follows. Each vertex $x\in {\ensuremath{\mathbb Z}} ^2$, with rate one and independently across ${\ensuremath{\mathbb Z}} ^2$, is resampled from $\big( \{0,1\},{\rm Ber}(p) \big)$ iff the update rule at $x$ was satisfied by the current configuration. In what follows, we will sometimes call such resampling a \emph{legal update} or \emph{legal spin
flip}. The general theory of interacting particle systems (see~\cite{Liggett}) proves that $\ensuremath{\mathcal L}$ becomes the generator of a reversible Markov process $\{\o(t)\}_{t\;\geqslant\; 0}$ on $\O$, with reversible measure $\mu$. The corresponding
Dirichlet form is \[ \ensuremath{\mathcal D}(f)= \sum_{x\in {\ensuremath{\mathbb Z}} ^2}\mu\big(c_x \operatorname{Var}_x(f)\big), \] where $\operatorname{Var}_x(f)$ denotes the variance of the local function $f$ w.r.t. the variable $\o_x$ conditionally on $\{\o_y\}_{y\neq x}$. If $\nu$ is a probability measure on $\O,$ the law of the process with initial distribution $\nu$ will be denoted by ${\ensuremath{\mathbb P}} _\nu(\cdot)$ and the corresponding expectation by ${\ensuremath{\mathbb E}} _\nu(\cdot)$. If $\nu$ is concentrated on a single configuration $\o$ we will simply write ${\ensuremath{\mathbb P}} _\o(\cdot)$ and ${\ensuremath{\mathbb E}} _\o(\cdot)$.
Given a KCM, and therefore an update family $\ensuremath{\mathcal U}$, the corresponding \emph{$\ensuremath{\mathcal U}$-bootstrap process} on $\mathbb{Z}^2$ is defined as follows: given a set $Y \subset \mathbb{Z}^2$ of initially \emph{infected} sites, set $Y(0) = Y$, and define for each $t \;\geqslant\; 0$, \begin{equation}\label{eq:def:Uboot:At} Y(t+1) = Y(t) \cup \big\{ x \in \mathbb{Z}^2 \,:\, X + x \subseteq Y(t) \text{ for some } X \in \ensuremath{\mathcal U} \big\}. \end{equation} The set $Y(t)$ will represent the set of infected sites at time $t$ and we write $[Y] = \bigcup_{t \;\geqslant\; 0} Y(t)$ for the \emph{closure} of $Y$ under the $\ensuremath{\mathcal U}$-bootstrap process. We will also call $T$ the median of the first infection time of the origin when the process is started with sites independently infected (healthy) with probability $q$ (respectively $p =1-q$).
\section{A variational lower bound for ${\ensuremath{\mathbb E}} _\mu(\tau_0)$} \label{sec:general}
As mentioned in the Introduction, our main goal is to prove sharp lower bounds for the characteristic time scales of supercritical rooted KCM and of the Duarte KCM. Let us start by defining precisely these time scales, namely the relaxation time $T_{\rm rel}$ (or inverse of the spectral gap) and the mean infection time ${\ensuremath{\mathbb E}} _\mu(\tau_0)$.
\begin{definition}[Relaxation time, $T_{\rm rel}$] \label{def:PC} Given an update family $\ensuremath{\mathcal U}$ and $q\in[0,1]$, we say that $C>0$ is a Poincar\'e constant for the corresponding KCM if, for all local functions $f$, we have \begin{equation}
\label{eq:gap} \operatorname{Var}_{\mu}(f) \;\leqslant\; C \, \ensuremath{\mathcal D}(f). \end{equation} If there exists a finite Poincar\'e constant we then define \[ T_{\rm rel}(q,\ensuremath{\mathcal U}):=\inf\big\{ C > 0 \,:\, C \text{ is a Poincar\'e constant} \big\}. \] Otherwise we say that the relaxation time is infinite. We will drop the $(q,\ensuremath{\mathcal U})$ notation setting $T_{\rm rel}:=T_{\rm rel}(q,\ensuremath{\mathcal U})$ when confusion does not arise. \end{definition} A finite relaxation time implies that the reversible measure $\mu$ is mixing for the semigroup $P_t=e^{t\ensuremath{\mathcal L}}$ with exponentially decaying time auto-correlations \cite{Liggett}.
\begin{definition}[Mean infection time, ${\ensuremath{\mathbb E}} _{\mu}(\tau_0)$] Let $A=\{\o\in \O:\ \o_0=0\}$. Then $$ \tau_0 = \inf\big\{ t \;\geqslant\; 0 \,:\, \o(t)\in A \big\}.$$ Given an update family $\ensuremath{\mathcal U}$ and $q\in[0,1]$, we let $\mathbb E_{\mu}^{q,\ensuremath{\mathcal U}}(\tau_0)$ be the mean of the infection time of the origin under the corresponding stationary KCM (i.e. when the initial configuration is distributed with Bernoulli$(1-q)$). We will drop the $(q,\ensuremath{\mathcal U})$ notation setting $\mathbb E_{\mu}(\tau_0):=\mathbb E_{\mu}^{q,\ensuremath{\mathcal U}}(\tau_0)$ when confusion does not arise. \end{definition} In the physics literature the hitting time $\tau_0$ is closely related to the \emph{persistence time}, \hbox{\it i.e. } the first time that there is a legal update at the origin. All our lower bounds can be easily extended to the persistence time.
It is known that the following inequality holds (see \cite{MMoT}*{Section 2.2}): \begin{equation}
\label{eq:mean-infection} {\ensuremath{\mathbb E}} _\mu(\tau_0) \;\leqslant\; \frac{T_{\rm rel}(q,\ensuremath{\mathcal U})}{q} \qquad \forall \; q\in (0,1). \end{equation}
Therefore we will focus on obtaining lower bounds on $\mathbb E_{\mu}(\tau_0)$ and then use \eqref{eq:mean-infection} to derive the results for $T_{\rm rel}$ (indeed the correction $q$ in the above inequality is largely subdominant w.r.t. the lower bounds we will obtain).
To this aim we establish a variational lower bound on ${\ensuremath{\mathbb E}} _\mu(\tau_0)$ (Lemma \ref{lem:basic:bound}), which will be our first tool. Recall that $A=\{\o\in \O:\ \o_0=0\}$ and let $H_A$ be the Hilbert space $\{f\in L^2(\O,\mu): f\!\mathord{\upharpoonright}_A=0\}$ with scalar product inherited from the standard one in $L^2(\O,\mu)$. Let also $\ensuremath{\mathcal L}_A$ be the negative self-adjoint operator on $H_A,$ whose action on local functions is given by \[ \ensuremath{\mathcal L}_A f(\o)= \mathbbm{1}_{A^c}(\o)\ensuremath{\mathcal L} f (\o). \] It turns out (see e.g. \cite{DaiPra}*{Section 3}) that, for any local function $f\in H_A$ and any $\o\in A^c$, \[ {\ensuremath{\mathbb E}} _\o \big(f(\o(t))\mathbbm{1}_{\{\tau_0>t\}}\big)= e^{t\ensuremath{\mathcal L}_A}f(\o). \] In particular, by choosing $f=\mathbbm{1}_{A^c}(\cdot),$ one gets \[ {\ensuremath{\mathbb P}} _\mu(\tau_0>t)= \int d\mu(\o) \mathbbm{1}_{A^c}(\o)e^{t\ensuremath{\mathcal L}_A}\mathbbm{1}_{A^c}(\o)=\langle \mathbbm{1}_{A^c},e^{t\ensuremath{\mathcal L}_A}\mathbbm{1}_{A^c}\rangle, \] where $\langle\cdot,\cdot\rangle$ denotes the scalar product on $L^2(\O,\mu).$ Thus \begin{equation}
\label{eq:3} {\ensuremath{\mathbb E}} _\mu(\tau_0)= \int_{0}^\infty dt \ \langle \mathbbm{1}_{A^c},e^{t\ensuremath{\mathcal L}_A}\mathbbm{1}_{A^c}\rangle\;\geqslant\; \int_{0}^T dt \ \langle \mathbbm{1}_{A^c},e^{t\ensuremath{\mathcal L}_A}\mathbbm{1}_{A^c}\rangle \quad \forall \ T>0. \end{equation}
\begin{lemma} \label{lem:basic:bound} Let $\phi\in H_A$ be a local function such that $\mu(\phi^2)=1.$ Then \[
{\ensuremath{\mathbb E}} _\mu(\tau_0)\;\geqslant\; T |\mu(\phi)|\Big(|\mu(\phi)| e^{-T\ensuremath{\mathcal D}(\phi)}- \big(T\ensuremath{\mathcal D}(\phi)\big)^{1/2}\Big),\quad \forall \ T>0. \] \end{lemma} \begin{proof} Let $\phi\in H_A$ be as in the statement and write \[ \mathbbm{1}_{A^c}= \alpha \phi + \psi, \] where $\alpha=\langle \mathbbm{1}_{A^c}, \phi\rangle=\mu(\phi)$ and $\langle\phi,\psi\rangle=0$. Clearly $\langle \psi,\psi\rangle=\mu(A^c)-\alpha^2$. We claim that, for any $T>0$ and any $t\in [0,T],$ \begin{equation}
\label{eq:1}
\langle \mathbbm{1}_{A^c},e^{t\ensuremath{\mathcal L}_A} \mathbbm{1}_{A^c}\rangle \;\geqslant\; \alpha^2e^{-T \ensuremath{\mathcal D}(\phi)} -
2|\alpha|\big(T\ensuremath{\mathcal D}(\phi)\big)^{1/2}, \end{equation} which, combined with \eqref{eq:3}, proves the lemma. To prove the claim we write \begin{align} \label{eq:4} \langle \mathbbm{1}_{A^c},e^{t\ensuremath{\mathcal L}_A} \mathbbm{1}_{A^c}\rangle &\;\geqslant\; \alpha^2 \langle
\phi,e^{t\ensuremath{\mathcal L}_A}\phi\rangle
- 2 |\alpha|\, |\langle
\psi,
e^{t\ensuremath{\mathcal L}_A}\phi\rangle
|\nonumber \\ &= \alpha^2 \langle \phi, e^{t\ensuremath{\mathcal L}_A}
\phi\rangle - 2 |\alpha|\, |\langle \psi, ({\ensuremath{\mathbb I}} -e^{t\ensuremath{\mathcal L}_A})\phi\rangle
|\nonumber \\
&\;\geqslant\; \alpha^2 \langle \phi, e^{t\ensuremath{\mathcal L}_A} \phi\rangle - 2 |\alpha|\, \langle\phi,\big({\ensuremath{\mathbb I}} -e^{t\ensuremath{\mathcal L}_A}\big)^2\phi\rangle^{1/2}.
\end{align} Above we discarded the positive term $\langle \psi, e^{t\ensuremath{\mathcal L}_A}\psi\rangle$ in the first line, we used $\langle\phi,\psi\rangle=0$ in the second line and appealed to the Cauchy-Schwartz inequality together with $\langle\psi,\psi\rangle\;\leqslant\; 1$ in the third line. Let now $\pi(d\lambda)$ be the spectral measure of $-\ensuremath{\mathcal L}_A$ associated to $\phi$ (see e.g. \cite{Reed-Simon}*{Chapter VII}). Since $\mu(\phi^2)=1,$ $\pi(d\lambda)$ is a probability measure on $[0,+\infty)$. The functional calculus theorem, together with the Jensen inequality and $(1-e^{-t\lambda})^2\;\leqslant\; t\lambda,$ implies that for any $t\in [0,T]$ \begin{align*} \text{r.h.s. \eqref{eq:4}}&= \alpha^2 \int_0^\infty
d\pi(\lambda) e^{-t\lambda} -2|\alpha|\, \Big(\int_0^\infty
d\pi(\lambda)(1-e^{-t\lambda})^2\Big)^{1/2}\\
&\;\geqslant\; \alpha^2e^{-t\ensuremath{\mathcal D}_A(\phi)}- 2|\alpha|\big(t\ensuremath{\mathcal D}_A(\phi)\big)^{1/2} \\
&\;\geqslant\; \alpha^2e^{-T\ensuremath{\mathcal D}(\phi)}- 2|\alpha|\big(T\ensuremath{\mathcal D}(\phi)\big)^{1/2},
\end{align*} where $\ensuremath{\mathcal D}_A(\phi)=\langle \phi, -\ensuremath{\mathcal L}_A\phi\rangle= \langle \phi, -\ensuremath{\mathcal L} \phi\rangle=\ensuremath{\mathcal D}(\phi)$ because $\phi$ is a local function in $H_A.$ The claim is proved. \end{proof} The main strategy to take advantage of Lemma \ref{lem:basic:bound} for $q$ very small is to look for a family of local functions $\{\phi_q\}$ in $H_A$, normalised in such a way that
$\mu(\phi_q^2)=1$, determining a sharp lower bound when
inserted in the inequality of Lemma \ref{lem:basic:bound} with a proper choice of $T$.
More precisely we will use the following easy corollary of Lemma \ref{lem:basic:bound}:
\begin{corollary}[Proxy functions]
\label{cor:basic:bound}
If there exists a family of local functions $\{\phi_q\}$ in $H_A$ with $\mu(\phi_q^2)=1$ and \begin{equation}
\label{eq:5} \lim_{q\rightarrow 0}\ensuremath{\mathcal D}(\phi_q)=0 \quad \text{and}\quad \lim_{q\rightarrow 0}\mu(\phi_q)^4/\ensuremath{\mathcal D}(\phi_q)=+\infty. \end{equation} then it holds \begin{equation}
\label{eq:2} {\ensuremath{\mathbb E}} _\mu(\tau_0)=\O\Big(\mu(\phi_q)^4/\ensuremath{\mathcal D}(\phi_q)\Big). \end{equation} \end{corollary} \begin{proof} The result follows immediately using Lemma \ref{lem:basic:bound} and
choosing
$T\equiv T(q)=|\mu(\phi_q)|^2/(16\ensuremath{\mathcal D}(\phi_q))$.\end{proof} Any function $\phi=\phi_q$ with the above properties will be called a \emph{test} or \emph{proxy} function and, in the rest of the paper, we will focus on constructing an efficient test function for the so called \emph{supercritical rooted KCM} and for the \emph{Duarte KCM}.
\section{Supercritical rooted KCM}
In order to define the class of {\sl supercritical rooted} update families we should begin by recalling the key geometrical notion of \emph{ stable directions} introduced in \cite{BSU}. Given a
unit vector $u \in S^1$, let $\mathbb{H}_u := \{x \in \mathbb{Z}^2 : \langle x,u \rangle < 0 \}$ denote the discrete half-plane whose boundary is perpendicular to $u$. Then, for a given update family $\ensuremath{\mathcal U}$, the set of {stable directions} is $$\mathcal{S} = \mathcal{S}(\mathcal{U}) = \big\{ u \in S^1 \,:\, [\mathbb{H}_u]= \mathbb{H}_u \big\}.$$ The update family $\ensuremath{\mathcal U}$ is \emph{supercritical} if there exists an open semicircle in $S^1$ that is disjoint from $\ensuremath{\mathcal S}$. In \cite{BSU} it was proven that for each supercritical update family the median of the infection time of the $\ensuremath{\mathcal U}$-bootstrap processes diverges as $1/q^{\Theta(1)}$.
In \cite{Robsurvey}, the author R. Morris together with two of us, conjectured that not all supercritical update families give rise to the same scaling for KCM and that the supercritical class should be refined into two subclasses to capture the KCM scaling as follows. \begin{definition}\label{def:rooted} A supercritical two-dimensional update family $\ensuremath{\mathcal U}$ is said to be \emph{supercritical rooted} if there exist two non-opposite stable directions in $S^1$. Otherwise it is called \emph{supercritical unrooted}. \end{definition} An example of supercritical rooted family is the two dimensional East model, with update family $\ensuremath{\mathcal U}=\{\{-\vec e_1\},\{-\vec e_2\}\}$ \footnote{We stress that the supercritical rooted class contains also update families which do not share the special "orientation" property of the East model, namely the fact that all $X_i$ belong to an half plane. For example, it is easy to verify that the non oriented update family $\,\ensuremath{\mathcal U}=\{\{-\vec e_1\},\{-\vec e_2\}, \{(\vec e_1,\vec e_2)\}\}$ has exactly two stable directions, $-\vec e_1$ and $-\vec e_2$ and, according to our Definition \ref{def:rooted}, it is supercritical rooted.}.
In \cite{MMoT} it was proved that ${\ensuremath{\mathbb E}} _\mu(\tau_0)$ and $T_{\rm rel}$ diverge as an inverse power of $q$ as $q\rightarrow 0$ in the supercritical {\sl unrooted} case, while in the {\sl rooted} case it satisfies (see \cite{MMoT}*{Theorem 1 (b)}) \[ T_{\rm rel}\;\leqslant\; e^{O((\log q)^2)} \] and, thanks to \eqref{eq:mean-infection}, the same bound holds for ${\ensuremath{\mathbb E}} _\mu(\tau_0)$. Here we prove a matching lower bound in the rooted case. \begin{theorem} \label{thm:rooted}Let $\ensuremath{\mathcal U}$ be a two dimensional supercritical rooted update family. Then
\[
{\ensuremath{\mathbb E}} _\mu(\tau_0)\;\geqslant\; e^{\O((\log q)^2)} \quad \text{as }q\rightarrow 0 . \] \end{theorem} Thus we prove \begin{corollary}\label{cor:rooted}
Let $\ensuremath{\mathcal U}$ be a two dimensional supercritical rooted update family. Then
\[ T_{\rm rel}(q,\ensuremath{\mathcal U}) = e^{\Theta((\log q)^2)} \quad \text{as }q\rightarrow 0 . \] and the same result holds for $ {\ensuremath{\mathbb E}} _\mu(\tau_0)$. \end{corollary} \begin{proof}[Proof of the corollary] The lower bound follows at once from \eqref{eq:mean-infection} and Theorem \ref{thm:rooted}. The upper bound was proved in \cite{MMoT}*{Theorem 1 (b)}. \end{proof}
In order to prove Theorem \ref{thm:rooted} we will use the variational lower bound of Section \ref{sec:general} and more precisely look for a proxy function $\phi\equiv \phi_q$ satisfying the key hypothesis of Corollary \ref{cor:basic:bound}.
We first need to introduce the notion of a legal path in $\O$. \begin{definition}[Legal path] \label{def:legal}Fix an update family $\ensuremath{\mathcal U}$, then a legal path $\gamma$ in $\O$ is a finite sequence $\gamma=\big(\o^{(0)},\dots,\o^{(n)}\big)$ such
that, for each $i\in [n],$ the configurations $\o^{(i-1)},\o^{(i)}$
differ by a legal (with respect to the choice $\ensuremath{\mathcal U}$) spin flip at some vertex $v\equiv
v(\o^{(i-1)},\o^{(i)})$. A generic ordered (along $\gamma$) pair of consecutive
configurations in $\gamma$
will be called an \emph{edge}. Given a set $\hat \O\subset \O$ and a
configuration $\o,$ we say that $\o$ is a legal path
connecting $\hat \O$ to $\o$ if there exists a legal path
$\gamma=\big(\o^{(0)},\dots,\o^{(n)}\big)$ such that $\o^{(0)}\in \hat
\O$ and $\o^{(n)}=\o$. \end{definition} Let $\ensuremath{\mathcal U}$ be a supercritical rooted update family and,
for $n\;\geqslant\; 1$ and $\kappa\in {\ensuremath{\mathbb N}} ^*$, let $\Lambda_n:=\Lambda_n(\kappa)\subset \mathbb Z^2$ be the square centred at the origin, of cardinality $(\kappa n2^n+1)^2.$ Let also \begin{equation}
\label{eq:6}
\begin{split} \ensuremath{\mathcal A}_n=&\{\o\in \O\colon (\o_{\Lambda_n},\tilde\o_{\Lambda^c_n}\equiv 0) \text{\it \ can be reached from $\big(\hat\o_{\Lambda_n}\equiv 1,
\hat\o_{\Lambda^c_n}\equiv 0\big)$ by a legal}\\& \text{\it path $\gamma$ such that any $\o'\in \gamma$ has at most $n-1$ empty vertices in $\Lambda_n$}\}.
\end{split} \end{equation} Recall that $A=\{\o\in \O\colon \o_0=0\}$. In \cite{Laure} one of the authors established the following key combinatorial result concerning the structure of the set $\ensuremath{\mathcal A}_n$ :
\begin{lemma}[\cite{Laure}*{Theorem 1}]\label{lem:laure} There exists $\kappa_0=\kappa_0(\ensuremath{\mathcal U})>0$ such that, for
any $\kappa\;\geqslant\; \kappa_0$ and any $n\in {\ensuremath{\mathbb N}} ^*,$ \[ \ensuremath{\mathcal A}_n\cap A=\emptyset. \]
\end{lemma}
Lemma \ref{lem:laure} implies that the KCM process started from any configuration with no infection inside the region $\Lambda_n,$ in order to infect the origin has to leave the set $\ensuremath{\mathcal A}_n$ by going through its boundary set $\partial\ensuremath{\mathcal A}_n$ (see the proof below for a precise definition of this set). In turn, the latter is a subset of \[ \{\o\in \O:\ \exists \text{
at least $n-1$ infected vertices in $\Lambda_n$}\}. \] We will therefore chose a scale $n$ such that $2^n \simeq 1/q^{\epsilon}$, namely w.h.p. w.r.t. the reversible measure $\mu$ there are initially no infected vertices inside $\Lambda_n$. Thus, starting from the (likely) event of no infection inside the region $\Lambda_n$, in order to infect the origin the process has to go through $\partial \ensuremath{\mathcal A}_n$ which has an anomalous amount, $\Theta(\log q)$, of empty sites. This mechanism, which in the physics jargon would correspond to "crossing an energy barrier" which grows logarithmically in $q$, is at the root of the scaling $e^{\Theta(\log q)^2}$.
Let us proceed to a proof of this result, namely to the proof of Theorem \ref{thm:rooted}. \begin{proof}[Proof of Theorem \ref{thm:rooted}]
Fix $\epsilon<1/2$ and choose $n:=n(\epsilon,q)=\lfloor \epsilon\log_2(1/q)\rfloor$. Then let $$\phi(\cdot):=\phi_q(\cdot)=\mathbbm{1}_{\ensuremath{\mathcal A}_{\epsilon,q}}(\cdot)/\mu(\ensuremath{\mathcal A}_{\epsilon,q})^{1/2}$$ where $\ensuremath{\mathcal A}_{\epsilon,q}:=\ensuremath{\mathcal A}_{n(\epsilon,q)}$ with $\ensuremath{\mathcal A}_n$ defined in \eqref{eq:6} and the constant $\kappa$ that enters in this definition chosen larger than the value $\kappa_0$ of Lemma \ref{lem:laure}. Then Lemma \ref{lem:laure} implies immediately that $\phi\in H_A$. Moreover, using $\epsilon<1/2$ we get \[
\mu(\phi)=\mu(\ensuremath{\mathcal A}_{\epsilon,q})^{1/2}\;\geqslant\; (1-q)^{|\Lambda_n|/2}=1-o(1), \] because any configuration identically equal to one in $\Lambda_n$ belongs to $\ensuremath{\mathcal A}_{\epsilon,q}$ and $2^{2n}=O(1/q^{2\epsilon})$. Finally, if \[ \partial \ensuremath{\mathcal A}_{\epsilon,q}:=\{\o\in \ensuremath{\mathcal A}_{\epsilon,q}\colon \exists \ x\in \Lambda_n \text{ with } c_x(\o)=1\text{ and } \o^x\notin \ensuremath{\mathcal A}_{\epsilon,q}\}, \] one easily checks (see e.g. \cite{CFM3}*{Section 3.5}) that \begin{gather*}
\ensuremath{\mathcal D}(\phi) \;\leqslant\; |\Lambda_n|\mu\big(\partial \ensuremath{\mathcal A}_{\epsilon,q}\big)/\mu(\ensuremath{\mathcal A}_{\epsilon,q})\;\leqslant\; |\Lambda_n|\mu\big(\exists\ n-1 \text{
zeros in $\Lambda_n$}\big)/\mu(\ensuremath{\mathcal A}_{\epsilon,q})\\
\;\leqslant\; O(|\Lambda_n|^n) q^{n-1}=e^{-\O((\log q)^2)}, \end{gather*} Thus $\phi$ satisfies all the hypotheses of Corollary \ref{cor:basic:bound} and the result follows. \end{proof}
\begin{remark} \label{rem:w.h.p.} In \cite{Robsurvey}*{Conjecture 2.7} it was conjectured that $\tau_0=e^{\Theta((\log q)^2)}$ w.h.p. as $q\rightarrow 0$ holds. Actually, we can also prove this stronger result. One bound immediately follows using Markov inequality and our result for the mean, Corollary \ref{cor:rooted}. The other bound follows by using the fact that (i) the set $\ensuremath{\mathcal A}_{\epsilon,q}$ has $\mu$-probability $1-o(1)$ (see the above proof of Theorem \ref{thm:rooted}) and (ii) the probability of infecting the origin before $e^{\Theta((\log q)^2)}$ starting in $\ensuremath{\mathcal A}_{\epsilon,q}$ goes to zero as $q\downarrow 0$. The latter result is easily obtained by a union bound on times which yields that the probability to leave $ \ensuremath{\mathcal A}_{\epsilon,q}$ before $e^{\Theta((\log q)^2)}$ (and therefore to infect the origin, thanks to Lemma \ref{lem:laure}), goes to zero. \end{remark}
\section{The Duarte KCM} In this section we analyse the mean infection time for the Duarte KCM. For this model the update family $\ensuremath{\mathcal U}$ consists of the $2$-subsets of the North, South and West neighbours of the
origin \cite{Duarte}. The infection time for the Duarte bootstrap process is known to scale as $e^{\Theta((\log q)^2/q)}$ \cite{Mountford} (see also \cite{BCMS-Duarte} for the sharp value of the constant). Concerning the Duarte KCM,
in \cite{MMoT}*{Theorem 2} it was proved that \[
T_{\rm rel}(q,\ensuremath{\mathcal U})\;\leqslant\; e^{O\big((\log q)^4/q^2\big)} \quad \text{as }q\rightarrow 0. \] and, thanks to \eqref{eq:mean-infection}, the same result holds for ${\ensuremath{\mathbb E}} _\mu(\tau_0)$. Here we establish a matching lower bound. \begin{theorem} \label{thm:Duarte} Consider the Duarte KCM. Then \[ {\ensuremath{\mathbb E}} _\mu(\tau_0)\;\geqslant\; e^{\O\big((\log q)^4/q^2\big)} \quad \text{as }q\rightarrow 0. \] \end{theorem} Using \eqref{eq:mean-infection}, Theorem \ref{thm:Duarte} and \cite{MMoT}*{Theorem 2} we get immediately the following corollary. \begin{corollary} For the Duarte KCM it holds
\[ T_{\rm rel}(q,\ensuremath{\mathcal U})= e^{\Theta\big((\log q)^4/q^2\big)} \quad \text{as }q\rightarrow 0 . \] and the same result for ${\ensuremath{\mathbb E}} _\mu(\tau_0)$. \end{corollary} Our result provides the first example of critical $\alpha$-rooted KCM for which the conjecture for the divergence of time scales that we put forward in \cite{MMoT}*{Conjecture 3 (a)} together with R. Morris can be proven. Indeed, as explained in \cite{MMoT}, the Duarte model is a $1$-rooted model and the exponent $2$ that we obtain is in agreement with \cite{MMoT}*{Conjecture 3 (a)}. In order to prove Theorem \ref{thm:Duarte} we will start by the variational lower bound of Section \ref{sec:general}, as for the supercritical rooted class. However, defining the analog of the set $\ensuremath{\mathcal A}_n$ together with the test function $\phi$ satisfying the hypotheses of Corollary \ref{cor:basic:bound} is much more involved and it requires a subtle algorithmic construction. Before explaining our construction it is useful to make some simple observations on how infection propagates in the Duarte bootstrap process. \subsection{Preliminary tools : the Duarte bootstrap process} \label{sec:tools}
\begin{figure}
\caption{A growing droplet under the Duarte bootstrap process (courtesy
of P. Smith).}
\end{figure}
Let $\vec e_1,\vec e_2$ denote the basis vectors in ${\ensuremath{\mathbb R}} ^2.$ Given $\Lambda\subset {\ensuremath{\mathbb Z}} ^2$ we write $\partial \Lambda:=\partial_\parallel\Lambda\cup \partial_\perp\Lambda,$ where \begin{align*} \partial_\parallel\Lambda&=\{y\in \Lambda^c \colon y+\vec e_1\in \Lambda\},\\ \partial_\perp\Lambda &=\{y\in \Lambda^c \colon \{y+\vec e_2,y-\vec e_2\}\cap
\Lambda\neq \emptyset\}. \end{align*} A configuration $\tau\in \{0,1\}^{\partial \Lambda}$ will be referred to as a \emph{boundary condition} and we shall write it as $\tau=(\tau_{\parallel}, \tau_{\perp} )$, where $\tau_{\parallel}:= \tau\mathord{\upharpoonright}_{\partial_\parallel\Lambda} $ and similarly for $\tau_{\perp}$. \begin{definition} \label{def:fin-vol BP} Given a boundary condition $\tau$ and $Y\subseteq \Lambda$, let \begin{equation*} Y^\tau(t+1) = Y^\tau(t) \cup \big\{ x \in \Lambda\,:\, X + x \subseteq Y^\tau(t) \text{ for some } X \in \ensuremath{\mathcal U} \big\}\quad t\;\geqslant\; 0, \end{equation*} where $Y^\tau(0)=Y\cup \{x\in \partial\Lambda\colon \tau_x=0\}$. We call the process $Y^\tau(t), t\in {\ensuremath{\mathbb N}} ,$ the Duarte bootstrap process in $\Lambda$ with $\tau$ boundary condition (for shortness the $DB_\Lambda^\tau$-process), and we shall write $[Y]_{\Lambda}^{\tau}$ for $(\bigcup_{t \;\geqslant\; 0}Y^\tau(t))\cap \Lambda$. Recall also (see Section \ref{sec:models}) that $[Y]$ is the analogous quantity for the bootstrap process evolving on ${\ensuremath{\mathbb Z}} ^2$. \end{definition} \begin{remark} Notice that for the $DB_\Lambda^\tau$-process the boundary condition $\tau$ does not change in time. \end{remark} \noindent {\bf Notation warning.} If $\tau\equiv 0$ or $\tau\equiv 1$ we shall simply replace it by a $0$ or a $1$ in our notation. If instead $\tau$ is such that $\tau_\parallel\equiv 1$ and $\tau_\perp\equiv 0$ then it will be replaced by a $1,0$ in the notation. \begin{lemma}[Screening property] \label{lem:screening} Consider a sequence of sites $S:=\{(i,b_i)\}_{i=1}^n$ in ${\ensuremath{\mathbb Z}} ^2$ with $b_{i+1}\;\leqslant\; b_i$ for all $i\in [n-1],$ and let \[ S_+=\{(i,j)\in {\ensuremath{\mathbb Z}} ^2\colon i\in[n], j>b_i\},\quad S_-=\{(i,j)\in {\ensuremath{\mathbb Z}} ^2\colon i\in[n], j<b_i\}. \] Let $Y,Y'$ be two arbitrary subsets of ${\ensuremath{\mathbb Z}} ^2$ such that $Y\supseteq S$ and $Y\cap S^c_+=Y'\cap S^c_+.$ Then $[Y]\cap S_-=[Y']\cap S_-$. Similarly if we assume that $b_{i+1}\;\geqslant\; b_i$ for all $i\in [n-1]$ and we exchange the role of $S_+$ and $S_-$. \end{lemma} \begin{proof} We refer to Figure \ref{fig:3bis} for a visualisation of the geometric setting. Let $Y,Y'$ be as in the statement and observe that $Y(s)$ and $Y'(s)$ coincide in $\{v\in {\ensuremath{\mathbb Z}} ^2\colon v=(a,b),\ a\;\leqslant\; 0\}$ for all $s\in {\ensuremath{\mathbb N}} ^*$. Let $t\in {\ensuremath{\mathbb N}} ^*$ be the first time at which there exists $y\in S_-$ such that either $y\in Y'(t)$ and $y\notin Y(t)$ or viceversa. W.l.o.g we assume the first case. By construction there exists $z\in \{y\pm \vec e_2,y-\vec e_1\}$ such that $z\in Y'(t-1)$ and $z\notin Y(t-1)$. Clearly $z$ cannot be of the form $z=(0,b)$ and therefore $z \in S_-\cup S$ because $y\in S_-$. Because of the definition of $t$, $z\notin S_-$ and $z\notin S$ because $S\subseteq Y(s)$ and $S\subseteq Y'(s)$ for all $s\in {\ensuremath{\mathbb N}} ^*.$ \begin{figure}
\caption{The set $S$ (black dots) and the sets $S_\pm$ (shaded
regions). If the two initial sets $Y,Y'$ of infection contain $S$
and differ at
exactly the vertex $x$, it is clear that the initial discrepancy cannot influence the final infection in $S_-$. }
\label{fig:3bis}
\end{figure} \end{proof} \begin{lemma}[Monotonicity] \label{lem:monotonicity}
Let $\Lambda\subseteq \Lambda'$ be subsets of ${\ensuremath{\mathbb Z}} ^2.$ \begin{enumerate}[(A)]
\item Let
$\tau,\tau'\in \{0,1\}^{\partial \Lambda}$. If $\tau_x\;\leqslant\; \tau'_x$ for all $x\in \partial\Lambda$ then \[ [Y]_{\Lambda}^{\tau'}\subseteq [Y]_{\Lambda}^{\tau},\quad \forall \ Y\subseteq \Lambda. \] \item For all $Y'\subseteq \Lambda'$ \[ [Y']_{\Lambda'}^{0}\cap \Lambda \subseteq [Y'\cap \Lambda]_{\Lambda}^0 \quad\text{and}\quad [Y']_{\Lambda'}^1\cap \Lambda \supseteq [Y'\cap \Lambda]^1_{\Lambda}. \] \item Suppose that $\Lambda$ and $\Lambda'$ are such that $\partial_\perp
\Lambda\subseteq \partial_\perp
\Lambda'$. Then for all $Y'\subseteq \Lambda'$ \[ [Y'\cap \Lambda]_{\Lambda}^{1,0}\subseteq [Y']_{\Lambda'}^{1,0}\cap \Lambda. \]
\end{enumerate} \end{lemma} \begin{proof}\
\begin{enumerate}[(A)] \item It follows immediately from the fact that the $DB_\Lambda^\tau$-process runs with more initial infection than the $DB_\Lambda^{\tau'}$-process. \item To prove the first inclusion let $Z= (Y'\cap \Lambda)\cup (\Lambda'\setminus \Lambda)$. Clearly $[Y']_{\Lambda'}^0 \subseteq [Z]_{\Lambda'}^0$ because $Y'\subseteq Z$. It is now sufficient to observe that, by definition, \[ [Z]_{\Lambda'}^0\cap \Lambda= [Y'\cap \Lambda]_{\Lambda}^0. \] Similarly one proceeds for the second inclusion with $Z=Y'\cap \Lambda$. \item Clearly $[Y'\cap \Lambda]_{\Lambda'}^{1,0} \subseteq [Y']_{\Lambda'}^{1,0}.$ We claim that \[ [Y'\cap \Lambda]_{\Lambda'}^{1,0}\cap \Lambda \supseteq [Y'\cap \Lambda]_{\Lambda}^{1,0}. \] That follows immediately from the assumption that $\partial_\perp\Lambda'\supseteq \partial_\perp\Lambda$ and the fact that the vertices of $\partial_\parallel\Lambda\cap \Lambda'$ (if any) are constrained to be healthy for all times under the $DB_\Lambda^{1,0}$-process while they are unconstrained for the $DB_{\Lambda'}^{1,0}$-process. \end{enumerate} \end{proof} \begin{lemma}[Propagation of infection] \label{lem:propagation} Let $I$ be a vertical interval, \hbox{\it i.e. } $I=\{a,a+\vec e_2,\dots,a+n\vec e_2\}, a\in {\ensuremath{\mathbb Z}} ^2$, and let $v=x+\vec e_1$ for some $x\in I$. Suppose that $I\cup \{v\}\subseteq [Y]$ where $Y$ is the initial set of infection. Then $I+\vec e_1\subseteq [Y]$. In particular, if $[Y]$ contains $[n]\times \{1\}$ and $\{1\}\times [m]$ then $[n]\times [m]\subseteq [Y]$. \end{lemma} As a corollary of the above simple property, let $x,y\in {\ensuremath{\mathbb Z}} ^2$ and suppose that there exists a \emph{Duarte path} $\Gamma$ between $x$ and $y,$ \hbox{\it i.e. } $\Gamma:=(x^{(1)},\dots, x^{(n)})\subseteq {\ensuremath{\mathbb Z}} ^2$ with $x^{(1)}=x,\ x^{(n)}=y$ and $x^{(i+1)}-x^{(i)}\in \{\vec e_1, \pm
\vec e_2\}\ \forall i\in [n-1].$ Let also $I_{\Gamma}$ be the horizontal
interval starting at $x$ and reaching the vertical line through $y$ (see Figure \ref{fig:3}). \begin{corollary} \label{cor:duarte-path} Suppose that $\Gamma\subseteq [Y]$. Then $I_{\Gamma}\subseteq
[Y].$ \end{corollary}
\begin{figure}
\caption{A Duarte path $\Gamma$ (thick polygonal line) and the corresponding horizontal
interval $I_{\Gamma}$ (dotted line). Clearly, $\Gamma\subseteq [Y]$ implies
that $[Y]$ contains the shaded region. In
particular $I_{\Gamma}\subseteq [Y]$.}
\label{fig:3}
\end{figure}
\subsection{Algorithmic construction of the test function and proof of Theorem \ref{thm:Duarte}} \label{sec:Algo} Fix $\epsilon$ a small positive constant that will be chosen later on and let
\begin{equation}\label{def:ell}
\quad \ell = \Big\lfloor \frac{1}{\epsilon q}\log(1/q) \Big\rfloor. \end{equation} Suppose that a vertical interval $I$ of length $\ell$ is completely infected. Notice that, with $\mu$-probability going to $1$ as $q\downarrow 0$, there is an infected site on the vertical interval sitting on the right, $I+\vec e_1$. Therefore, thanks to Lemma \ref{lem:propagation}, with high probability the infection can propagate to infect $I+\vec e_1$. Notice that instead the infection on $I$ does not help infecting the interval on its left, $I-\vec e_1$. At this point, recalling the explanation given in the Introduction, one might think that the droplets that undergo an East like dynamics \footnote{namely a dynamics in which droplets appear/disappear only if there is a droplet on their left, as it occurs for the single empty sites in the East one-dimensional model.} are the {\sl empty vertical intervals of length at least $\ell$}. However this is far from true, since these empty intervals might also appear (or disappear) without being facilitated by the presence of an empty interval on their left. For example, if there is an empty interval of length $\ell -1$ and the site just above has the constraint satisfied, a single legal move may turn it into an empty interval of height $\ell$. We have therefore to find a more flexible definition of the droplets respecting three key properties: (i) East like dynamics ; (ii) disjoint occurrence under the equilibrium measure $\mu$ and (iii) the density of droplets should scale as $q_{\mbox{\tiny{eff}}}=q^{\ell}$ \footnote{Indeed, since the density of droplets will play the role of the density of empty sites for East, it is natural to expect that the lower bound obtained using the droplets will be of the form \eqref{eq:East} with $q_{\mbox{\tiny{eff}}}$ replacing $q$. This in turn yields the result of Theorem \ref{thm:Duarte} if $q_{\mbox{\tiny{eff}}}=q^{\ell}$.}. Our solution to the problem is the construction of an algorithm that sequentially searches for properly defined droplets on a finite volume, $V$, containing the origin.
We let \begin{equation}\label{def:N}N= \big\lfloor e^{\varepsilon (\log q)^2/q}\big\rfloor \quad {\mbox{ and }} \quad V:=V_N=\cup_{i=1\,}^{N} \mathcal{C}_i, \end{equation} where $$\ensuremath{\mathcal C}_i=
\{(i,j)\in {\ensuremath{\mathbb Z}} ^2 \colon |j| < N^2 -(i-1)N \}-N\vec e_1. $$ as in Figure \ref{fig:5}. In the sequel we shall write $\bar V$ for set $V\cup \partial_\perp V$ and we shall refer to $\bar \ensuremath{\mathcal C}_i:=\ensuremath{\mathcal C}_i\cup \partial_\perp\ensuremath{\mathcal C}_i$ as the $i^{th}$-column of $\bar V$. By construction the origin coincides with the midpoint of the last column (see Figure \ref{fig:5}). The core of our algorithmic construction (see Definition \ref{def:algo}) consists in associating to each $\o \in \O$ an element $\Phi(\o)\in \{\downarrow,\uparrow\}^N$ via an iterative procedure based on the $DB_{\Lambda}^{\tau}$-process. These arrow variables are those that satisfy the three key properties announced above, with $\Phi(\o)_i=\uparrow$ corresponding to the occurrence of a droplet in column $i$, and we will use them to construct an efficient test function.
\begin{figure}
\caption{A sketchy drawing of the last few columns of the set $V$. The black
dots represents sites belonging to $\partial_\perp V$.
}
\label{fig:5}
\end{figure}
\begin{definition} \label{def:UV} Given a boundary condition $\tau$ and $\o\in \O,$ we shall say that $I\subseteq V$ is $(\o,\tau)$-infectable if $I\subseteq\big [Y(\o)\cap V\big]_{V}^{\tau},$ where we recall that $Y(\o)$ is the set of empty vertices of $\o$. \end{definition} Before defining the algorithm leading to the construction of an effective test function for the Duarte KCM process, it is useful to notice two simple properties of the $DB_V^\tau$- process. \begin{enumerate}[(i)] \item Let $I\subseteq \cup_{i=1}^k \ensuremath{\mathcal C}_i, k\;\leqslant\; N.$ Then the property of being
$(\o,\tau)$-infectable for $I$ depends only on the infection of the
pair $(\o,\tau)$ in $\cup_{i=1}^k \bar \ensuremath{\mathcal C}_i$ and on $\tau_\parallel$. \item If $\bar\ensuremath{\mathcal C}_i$ is healthy at time $t=0$ (including the
contribution of $\tau$ at its top and bottom boundary sites), then it
will remain healthy at any later time. \end{enumerate}
\begin{definition}[The algorithm] \label{def:algo} Given $\o\in \O$ and $\tau\in \{0,1\}^{\partial
V}$ such that $\tau_\perp \equiv 0$ and $\tau_\parallel\equiv 1$, the algorithm outputs recursively a sequence $\psi^{(k)}:=(\o^{(k)}, \tau^{(k)}),\ k\in \{0,\dots,N\},$ where $\o^{(k)}\in \O$ and $\tau^{(k)}\in \{0,1\}^{\partial V}$ is such that $\tau^{(k)}_\parallel\equiv 1$. The pair $\psi^{(0)}$ coincides with $(\o,\tau)$ and $\psi^{(k)}$ is obtained from $\psi^{(k-1)}$ by healing suitably chosen infected vertices. The iterative step goes as follows. Fix $\ell\in [N]$ and assume that $\psi^{(j)}$ has been defined for all $j=0,\dots, k-1, k\in [N].$ Then: \begin{enumerate}[(i)] \item if $\bar \ensuremath{\mathcal C}_k$ contains an interval $I$ of length
at least $\ell$ which is
$\psi^{(k-1)}$-infectable, we let $\xi_k:=\xi_k(\o)\;\leqslant\; k$ be
the largest integer such that, by removing all the
empty vertices of the pair $\psi^{(k-1)}$ contained in
$\cup_{i=1}^{\xi_k-1}\bar \ensuremath{\mathcal C}_i$, the above property still holds. We
then set both $\o^{(k)}$ and $\tau^{(k)}$ identically equal to one (\hbox{\it i.e. } with no
infection) on $\bar \ensuremath{\mathcal C}_{\xi_k},\dots, \bar \ensuremath{\mathcal C}_k$ and equal to
$\o^{(k-1)}$ and $\tau^{(k-1)}$ elsewhere;
\item if not we set $\psi^{(k)}=\psi^{(k-1)}.$
\end{enumerate}
\end{definition}
\begin{remark}
Clearly the above construction depends on the initial $\o$ and we
shall sometimes write $\psi^{(k)}(\o)$ to outline this dependence.
\end{remark}
\begin{definition}[Droplets and their range] \label{def:droplets} Given $k$ such that $\psi^{(k)}(\o)\neq \psi^{(k-1)}(\o),$ we define the \emph{droplet} $D_k(\o)$ and the \emph{range} $r_k(\o)$ of the $k^{th}$-column in $\o$ as the set $\cup_{i=\xi_k}^k \bar \ensuremath{\mathcal C}_i$ and the integer $k-\xi_k(\o)$ respectively. If instead $\psi^{(k)}(\o)=\psi^{(k-1)}(\o),$ we let $D_k(\o)= \emptyset$ and $r_k(\o)=0$.
\end{definition} Observe that, by construction, \begin{equation}
\label{eq:10} \psi^{(j)}(\o)\mathord{\upharpoonright}_{\bar V\setminus
\cup_{i=1}^{j}D_i(\o)}=\psi^{(0)}(\o)\mathord{\upharpoonright}_{\bar V\setminus
\cup_{i=1}^{j}D_i(\o)}. \end{equation} \begin{definition}[The mapping $\Phi$] Having defined the sequence
$\{\psi^{(k)}\}_{k=1}^N,$ we set
\begin{equation*} \Phi(\o)_k = \begin{cases} \uparrow &\text{ if $\psi^{(k)}(\o)\neq \psi^{(k-1)}(\o),$}\\ \downarrow &\text{ otherwise,} \end{cases}
\end{equation*} and $N_{\uparrow}(\o)=\#\{i\in [N]\colon \Phi(\o)_i=\uparrow\}$. \end{definition} \begin{remark} \label{rem:algo}Suppose that $\o,\o'$ are such that they coincide over the first $i$ columns. Then $\Phi(\o)_k=\Phi(\o')_k$ for all $k\in [i]$. \end{remark} In the sequel two events will play an important role.
The first one, $\ensuremath{\mathcal B}_1(n),$ collects all the $\o'$s whose image $\Phi(\o)$ has more than $n$ up-arrows, with $n\in [N]$: \begin{equation}
\label{eq:B1} \ensuremath{\mathcal B}_1(n)=\{\o\in \O\colon N_{\uparrow}(\o)\;\geqslant\; n \}. \end{equation} The event $\ensuremath{\mathcal B}_2(n)$, again with $n\in [N]$, collects instead all the $\o\in \O$ such that there exists $n$ consecutive $\downarrow$-columns which are traversed by an infectable Duarte path. More precisely, for $1\;\leqslant\; i<j\;\leqslant\; N,$ let \begin{equation}\label{def:Vij}V_{i,j}=\cup_{k=i}^j \ensuremath{\mathcal C}_k\end{equation} and let \begin{equation}
\label{eq:B2} \ensuremath{\mathcal B}_2(n)=\cup_{j-i\;\geqslant\; n-1}\big( \cap_{k=i}^j\{\o\in \O\colon \Phi(\o)_k=\downarrow\}\cap\ensuremath{\mathcal G}_{i,j}\big), \end{equation} where \begin{equation}
\label{eq:B2bis} \ensuremath{\mathcal G}_{i,j}=\big\{\o\in \O\colon \exists \text{ a Duarte
path }\Gamma \text{ from }\ensuremath{\mathcal C}_i \text{ to }\ensuremath{\mathcal C}_j \text{ such that } \Gamma\subseteq [Y(\o)\cap V_{i,j}]^{1,0}_{V_{i,j}}\big\} \end{equation}
We are now ready to define our test function. \begin{definition}[The test function] \label{def:tf}
Let $I_0=\{(0,k)\colon |k|\;\leqslant\; \ell\}$ and \begin{align}
\label{eq:8}
n_1= \varepsilon (\log q)^2/2q,\quad n_2=1/q^6 \end{align} where $\varepsilon$ is the same as in the definition of N \eqref{def:N}. Let also $$\O_{\downarrow}=\{\o\in \O: \Phi(\o)=(\downarrow,\dots,\downarrow)\},$$ \[ \O_g=\O_{\downarrow}\cap\{\o\in \O\colon
\o_{I_0}=1\}, \] \begin{equation}
\label{eq:7}
\begin{split}
\ensuremath{\mathcal A}_{\epsilon,q}:=\ensuremath{\mathcal A}_{N,\ell,n_1,n_2}
=\{\o\in\O\colon \exists \text{ a legal path } \gamma \text{ connecting } \Omega_g \text{ to } \o \text{ s.t. }\\
\gamma\cap \ensuremath{\mathcal B}_1(n_1-1)=\emptyset \text{ and } \gamma\cap
\ensuremath{\mathcal B}_2(n_2-1)=\emptyset \}.
\end{split} \end{equation} where legal paths have been defined in Definition \ref{def:legal} and, for any $\mathcal B\subset \Omega$, we set $\gamma\cap \mathcal B=\emptyset$ iff none of the configurations of the path $\gamma$ belongs to $\mathcal B$.
Then we choose as test function \[ \phi(\cdot):= \phi_q(\cdot)=\mathbbm{1}_{\ensuremath{\mathcal A}_{\epsilon,q}}(\cdot)/\mu(\ensuremath{\mathcal A}_{\epsilon,q})^{1/2}, \]
\end{definition} The rest of the paper is devoted to prove that (i) $\phi$ satisfies all the hypotheses of Corollary \ref{cor:basic:bound}, namely $\phi\in H_A$ and the conditions \eqref{eq:5} are satisfied; (ii) $\phi$ is an efficient proxy function, namely the bound \eqref{eq:2} prove the sharp lower bound of Theorem \ref{thm:Duarte}. More precisely we need to prove the following key propositions: \begin{proposition} \label{prop:P1} There exists $\varepsilon_0>0$ such that, for all $\varepsilon\in (0,\varepsilon_0)$ there exists $q_\epsilon$ small enough such that, for all $q\in (0,q_\epsilon)$, \[ \ensuremath{\mathcal A}_{\varepsilon,q}\cap A=\emptyset. \] In particular, $\phi\in H_A$. \end{proposition} \begin{proposition} \label{prop:P2} There exists $\varepsilon_0>0$ such that, for all $\varepsilon\in (0,\varepsilon_0),$ \[ \mu(\phi)\;\geqslant\; q^{O(1)} \quad \text{and}\quad \ensuremath{\mathcal D}(\phi)\;\leqslant\; e^{-\O(\log(q)^4/q^2)} \quad \text{as }q\rightarrow 0. \] \end{proposition} Once the above propositions are proven, the main result of this section easily follows \begin{proof}[Proof of Theorem \ref{thm:Duarte}] The result follows at once using Propositions \ref{prop:P1} and \ref{prop:P2}, together with the general lower bound on ${\ensuremath{\mathbb E}} _\mu(\tau_0)$ given in \eqref{eq:2}. \end{proof}
Let us start with an easy result which will be used in the proof of both propositions \begin{lemma}[Disjoint occurrence of the droplets] \label{lem:D1} For any $\o\in \O$ and any $k\neq j,$ $D_k(\o)\cap D_j(\o)=\emptyset$. \end{lemma} \begin{proof} Let $k_1,\dots,k_\nu$ be the labels of the columns which are of type $\uparrow$ in $\Phi(\o)$ (for all the other columns the droplets are the empty set). Using property (ii) of the $DB_V^\tau$-process, $D_{k_\nu}(\o)$ cannot contain a column which
is healthy for the pair $\psi^{(k_\nu-1)}$ because any infection
to the left of an healthy column cannot cross the healthy column
itself. On the other hand, all the columns of the droplets $D_{k_1},\dots,D_{k_{\nu-1}}$ are healthy for $\psi^{(k_\nu-1)}$. Thus $D_{k_\nu}\cap
D_{k_{j}}=\emptyset$ for all $j\in [\nu-1].$ The same reasoning applies to all the
other droplets.
\end{proof}
\subsection{East-like motion of the arrows and proof of Proposition \ref{prop:P1}} Let \[ A_\ell= \{\o\in \O\colon \o_{I^+_0}\equiv 0\}\cup \{\o\in \O\colon \o_{I^-_0}\equiv 0\}, \] where $I_0^\pm=\{(0,\pm 1),\dots,(0,\pm \ell)\}.$ Then it holds \begin{lemma}\label{lem:pers} If $\ensuremath{\mathcal A}_{\varepsilon,q} \cap A\neq \emptyset$ then there exists $\o\in A_\ell$ and a legal path $\gamma$ connecting $\O_g$ to
$\o$ such that
$\gamma\cap \ensuremath{\mathcal B}_i(n_i)=\emptyset,\ i=1,2.$ \end{lemma} \begin{proof} Fix $\o\in \ensuremath{\mathcal A}_{\varepsilon,q} \cap A$, recall Definition \ref{def:tf} and let $\tilde\gamma$ be a legal path connecting $\O_g$ to $\o$
such that $\tilde \gamma\cap \ensuremath{\mathcal B}_1(n_1-1)=\emptyset$ and $\tilde \gamma\cap \ensuremath{\mathcal B}_2(n_2-1)=\emptyset.$ W.l.o.g., we can assume that $\tilde\gamma$
ends as soon as it enters $A$. It is easy to verify that $\tilde\gamma$ must be able to sequentially infect (and possibly heal later on) the ordered vertices of either $I_0^+$ starting from $(0,\ell)$ or those of $I_0^-$ starting from $(0,-\ell)$. For simplicity we assume that the first option holds and we let $\gamma$ be the path obtained from $\tilde\gamma$ by deleting all the transitions in which a vertex of $I_0^+$ is healed.
By construction, the final configuration of $\gamma$ belongs to $A_\ell$. Moreover, $\gamma$ is a legal path because at each step the infection in the last column of $V$ is larger than or equal to the infection of the corresponding step of $\tilde\gamma$. Finally the restriction to $\ensuremath{\mathcal C}_1,\dots,\ensuremath{\mathcal C}_{N-1}$ of any step of $\gamma$ coincides with the same restriction of the appropriate step of $\tilde\gamma$. Using that $\tilde\gamma\cap \ensuremath{\mathcal B}_1(n_1-1)=\emptyset$ and $\tilde\gamma\cap \ensuremath{\mathcal B}_2(n_2-1)=\emptyset,$ we deduce that $\gamma\cap \ensuremath{\mathcal B}_1(n_1)=\emptyset$ and $\gamma\cap \ensuremath{\mathcal B}_2(n_2)=\emptyset$. \end{proof} The above Lemma says that, if there exists a configuration in $\Omega_g$ for which we can infect the origin performing a legal path never crossing either $\mathcal B_1(n_1-1)$ or $\ensuremath{\mathcal B}_2(n_2-1)$,
then necessarily there exists a legal path never crossing either $\mathcal B_1(n_1)$ or $\ensuremath{\mathcal B}_2(n_2)$ and connecting a configuration $\omega$ with all columns being $\downarrow$ to a configuration $\omega$ with a $\uparrow$ in the $N$-th column.
In order to conclude that $\mathcal A_{\epsilon,q}\cap A=\emptyset$ and thus prove our Proposition \ref{prop:P1}, we will now show that {\sl the existence of a legal path with the above properties is impossible}. It is here that the East-like motion of the droplets emerges and plays a key role. Recall the definitions \eqref{def:N}, \eqref{eq:8} and let $m=4n_1 n_2$ and, for simplicity, let us suppose that $m$ divides $N$. We partition $[N]$ into $M=N/m$ disjoint consecutive blocks $\{B_i\}_{i=1}^M$ of equal cardinality and, with a slight abuse of notation, we identify the columns $\cup_{k\in B_i}\ensuremath{\mathcal C}_k$ with the block $B_i$ itself. Given $\o\in \O$ we write \[ \eta_i(\o):=\mathbbm{1}_{\{\exists \,j\,\in B_i\colon \Phi(\o)_j=\uparrow\}}, \] and we denote by $\eta(\o)$ the collection $\{\eta_i(\o)\}_{i=1}^M.$ \begin{claim} \label{claim:20} Given a legal path $\gamma$ with the properties stated in Lemma \ref{lem:pers}, it is possible to construct a path $\varphi(\gamma):=(\eta^{(0)},\dots,\eta^{(k)})$ in the space $\{0,1\}^M$ with the following properties: \begin{enumerate}[(1)] \item $\eta^{(0)}_i=0$ for all $i\in [M]$ and $\eta^{(k)}_M=1$, \item $\#\{i\in [M]\colon \eta_i=1\}\;\leqslant\; n_1$ for all $\eta\in
\varphi(\gamma)$, \item for any edge $(\eta,\eta')$ of $\varphi(\gamma),$ the
configuration $\eta'$ differs from $\eta$ in exactly one
coordinate. Moreover, if the discrepancy between $\eta$ and $\eta'$
occurs at the $i^{th}$-coordinate and $i\neq 1,$ then $\eta_{i-1}=1$. \end{enumerate} \end{claim} \begin{remark} The path $\varphi(\gamma)$ for the coarse-grained variables $\{\eta_i\}_{i=1}^M$ can be viewed as a legal path for the one dimensional East chain on $[M],$ with facilitating vertices those for which $\eta_i=1$ (see e.g. \cite{East-review}). \end{remark} The proof of our Proposition \ref{prop:P1} then follows by using this connection with the East chain, our choices \eqref{def:N}, \eqref{eq:8} of the parameters $N,n_1,n_2$ and the combinatorial result for the East model \cite{SE1,CDG} that we explained in the Introduction. More precisely \begin{proof}[Proof of Proposition \ref{prop:P1}] In \cite{CDG} it was proved that a path like $\varphi(\gamma)$ above exists iff $n_1\;\geqslant\; \log_2(M +1).$ With our choice \eqref{eq:8} of the scaling as $q\rightarrow 0$ of $n_1,n_2,N,$ the latter condition becomes \[ n_1\;\geqslant\; \frac{1}{\log 2}(1+o(1))\varepsilon (\log q)^2/q,\quad \text{as } q\rightarrow 0, \] violating our choice $n_1=\varepsilon (\log q)^2/2q$. Thus $\varphi(\gamma)$ cannot exist as well as the path $\gamma$. \end{proof}
We are therefore left with proving Claim \ref{claim:20}. To this aim we start by stating two preparatory results, Lemma \ref{lem:D0} and Lemma \ref{lem:D2}, which will be the key ingredients for the proof of Claim \ref{claim:20}.
\begin{lemma} \label{lem:D0} For any $\o\in \ensuremath{\mathcal B}_2^c(n_2)$ the maximum range of a droplet of $\o$ is $n_2-1$. \end{lemma} \begin{proof} Let $\omega \in \Omega$ such that there exists $j \in [N]$ with $r_j(\omega) \;\geqslant\; n_2$.
Denote $i = \xi_j(\omega)$. By the definition of $\xi_j(\omega)=i$, $\bar{\mathcal{C}}_j$ contains an interval $I$
of length at least $\ell$ which is $\psi^{(j-1)}$-infectable by the empty sites in
$\bigcup_{k=i}^j \bar{\mathcal{C}}_k$, but not by the empty sites in
$\bigcup_{k=i+1}^j \bar{\mathcal{C}}_k$. Definition \ref{def:UV} implies that
any $\psi^{(j-1)}$-infectable site is in $V$, hence $I \subseteq \mathcal{C}_j$.
Furthermore, for all $k \in \{i,\dots,j-1\}$, $\Phi(\omega)_k=\downarrow$
(since thanks to Lemma \ref{lem:D1} the droplets are disjoint), so
by (\ref{eq:10}) $\psi^{(j-1)}$ and $\psi^{(0)}$ coincide on $\bigcup_{k=i}^j \bar{\mathcal{C}}_k$.
Therefore $I$ is $\psi^{(0)}$-infectable by the empty sites in
$\bigcup_{k=i}^j \bar{\mathcal{C}}_k$, but not by the empty sites in
$\bigcup_{k=i+1}^j \bar{\mathcal{C}}_k$. We deduce that $I \subseteq
[Y(\omega)\cap V_{i,j}]_{V_{i,j}}^{1,0}$, but $I \not\subseteq
[Y(\omega)\cap V_{i+1,j}]_{V_{i+1,j}}^{1,0}$, see \eqref{def:Vij} for the definition of $V_{i,j}$. Thus, there exists $z \in \mathcal{C}_j$
such that $z \in [Y(\omega)\cap V_{i,j}]_{V_{i,j}}^{1,0} \setminus [Y(\omega)\cap V_{i+1,j}]_{V_{i+1,j}}^{1,0}$. Hence
$z$ can not be initially empty for the Duarte bootstrap process in $V_{i,j}$, otherwise it would also be
empty for the process in $V_{i+1,j}$, hence the process in $V_{i,j}$ infects $z$ with an update rule,
so there exists $z' \in \{z-\vec{e_1},z\pm\vec{e_2}\}$ in
$[Y(\omega)\cap V_{i,j}]_{V_{i,j}}^{1,0} \setminus [Y(\omega)\cap V_{i+1,j}]_{V_{i+1,j}}^{1,0}$.
We can iterate, creating a Duarte path in $[Y(\omega)\cap V_{i,j}]_{V_{i,j}}^{1,0} \setminus
[Y(\omega)\cap V_{i+1,j}]_{V_{i+1,j}}^{1,0}$. There can be only a finite number of iterations because
there is a finite number of sites in $V_{i,j}$, so we will stop, and the site at which we stop
has to be initially empty for the process in $V_{i,j}$, but not for the process in
$V_{i+1,j}$, therefore it is in $\bar{\mathcal{C}}_i$. This implies the Duarte path can reach
$\mathcal{C}_i$. Consequently, there is a Duarte path in $[Y(\omega)\cap V_{i,j}]_{V_{i,j}}^{1,0} \setminus
[Y(\omega)\cap V_{i+1,j}]_{V_{i+1,j}}^{1,0}$ going from $\mathcal{C}_i$ to $\mathcal{C}_j$.
We deduce that there exists a Duarte path in $[Y(\omega)\cap V_{i,j-1}]_{V_{i,j-1}}^{1,0}$
from $\mathcal{C}_i$ to $\mathcal{C}_{j-1}$, which is $\mathcal{G}_{i,j-1}$.
Since $(j-1)-i \;\geqslant\; n_2-1$, $\omega \in \mathcal{B}_2(n_2)$.
\end{proof} The next lemma is the basic technical step connecting the evolution of the coarse-grained variables $\{\Phi(\o)_i\}_{i=1}^N$ under the Duarte KCM process to an East-like process. Given $\o\in \O$ and $x\in V,$ let $\o^x$ denote the configuration $\o$ flipped at $x.$ We say that $x$ is $\psi^{(k)}(\o)$-unconstrained (or infectable in one step) if $\exists X\in \ensuremath{\mathcal U}$ such that $X+x$ is infected for the pair $(\o^{(k)},\tau^{(k)})$.
\begin{lemma}[East like motion of the arrows]
\label{lem:D2} Fix $\o\in \O$ and let $x\in \ensuremath{\mathcal C}_j$. Then: \begin{enumerate}[(a)] \item Suppose that $x$ is $\psi^{(0)}(\o)$-unconstrained. Then $\Phi(\o^x)\neq
\Phi(\o)$ implies that $j>1$ and $\Phi(\o)_{j-1}=\uparrow;$ \item For $i>j$ suppose that
$\Phi(\o)_i=\uparrow, \Phi(\o^x)_i=\downarrow$ and that $D_i(\o)\not\ni x$. Then there exists $k$ such that $\bar \ensuremath{\mathcal C}_k\subseteq D_i(\o)\setminus \bar \ensuremath{\mathcal C}_i$ and
$\Phi(\o^x)_k=\uparrow,\Phi(\o)_k=\downarrow$ . \end{enumerate}
\end{lemma}
\begin{proof}\
(a) If $j=1$ then clearly $\Phi(\o^x)=\Phi(\o)$ because $x$ is $\psi^{(0)}(\o)$-unconstrained. Consider now the case $j\neq 1$ and assume that $\Phi(\o)_{j-1}=\downarrow$. We want to prove that in this case $\Phi(\o^x)=\Phi(\o)$ if $x$ is $\psi^{(0)}(\o)$-unconstrained.
By construction, the restriction to the first $j-1$ columns of $\psi^{(k)}(\o^x)$ and $\psi^{(k)}(\o)$ coincide for all $k\in [j-1]$ and, as a consequence, $\Phi(\o)_k=\Phi(\o^x)_k\, \forall k\in [j-1]$. Let $k_*(\o)=\min\{k\;\geqslant\; j\colon
\Phi(\o)_k=\uparrow\}$ and similarly for $\o^x$. Using \eqref{eq:10} together with
$\Phi(\o)_{j-1}=\downarrow$, for all $i=j-1,\dots, k_*(\o)-1$ the restriction of
$\psi^{(i)}(\o)$ to the columns $\bar \ensuremath{\mathcal C}_{j-1},\dots,\bar\ensuremath{\mathcal C}_N$ coincides with the same
restriction of the original pair $\psi^{(0)}(\o)$. In particular, the
fact that $x$ is $\psi^{(0)}(\o)$-unconstrained implies that $x$ is also $\psi^{(k_*(\o)-1)}(\o)$-unconstrained. Analogously for the configuration $\o^x$. Clearly $k_*(\o^x)\;\geqslant\; k_*(\o).$ If not, starting from the infection of $\psi^{(j-1)}(\o)$ we can first make a transition to $\psi^{(j-1)}(\o^x)$ by legally flipping $\o_x$ and from there infect an interval of length at least $\ell$ of $\bar\ensuremath{\mathcal C}_{k_*(\o^x)}$ to make it of type $\uparrow,$ a contradiction with the definition of $k_*(\o)$. By exchanging the role of $\o,\o^x$ we conclude that $k_*(\o^x)=k_*(\o).$ Thus $\Phi(\o)_k=\Phi(\o^x)_k$
for all $k=1\dots,k_*(\o)$ and, a fortiori, for all $k>k_*(\o)$.
(b)
By assumption the restriction of $\o,\o^x$ to $D_i(\o)$ coincide.
If $\Phi(\o^x)_k=\downarrow$ for all the columns in $D_i(\o),$ then
$\psi^{(i-1)}(\o)=\psi^{(i-1)}(\o^x)$ on the set $D_i(\o)$ implying that $\Phi(\o^x)_i=\Phi(\o)_i.$ Thus there exists a column $\bar\ensuremath{\mathcal C}_k\subseteq D_i(\o)\setminus \bar\ensuremath{\mathcal C}_i$ such that $\Phi(\o^x)_k=\uparrow$ and (by the definition of $D_i(\o)$) $\Phi(\o)_k=\downarrow$.
\end{proof}
\begin{corollary} \label{cor:D1} Fix $\o\in \O$ and let $x\in \ensuremath{\mathcal C}_j.$ Let also $r^x_\infty=\max_i\max(r_i(\o),r_i(\o^x))$ and suppose that $\Phi(\o)_i=\uparrow,\Phi(\o^x)_i=\downarrow,$ with $i-j\;\geqslant\; m(r^x_\infty +1), m\in {\ensuremath{\mathbb N}} ^*$. Then \[ \#\{k\in \{j,\dots,i\}\colon \Phi(\o)_k=\uparrow\}+\#\{k\in \{j,\dots,i\}\colon \Phi(\o^x)_k=\uparrow\}\;\geqslant\; m. \] \begin{proof} By construction $D_i(\o)\not\ni x.$
Lemma \ref{lem:D2} part (b) guarantees that there exists a column $\bar \ensuremath{\mathcal C}_k\subseteq D_i(\o)\setminus \bar \ensuremath{\mathcal C}_i$ such that $\Phi(\o)_k=\downarrow$ and $\Phi(\o^x)_k=\uparrow.$ We can then iterate by exchanging the role of $\o,\o^x$ and replacing $i$ with e.g. the largest of the labels $k$ above. In conclusion, every $r^x_\infty+1$ steps we are guaranteed to find a discrepancy between $\Phi(\o)$ and $\Phi(\o^x)$ and the result follows. \end{proof} \end{corollary}
We are now ready to conclude the proof of Claim \ref{claim:20}. \begin{proof}[Proof of Claim \ref{claim:20}] To prove the claim, let $\gamma=(\o^{(0)},\dots, \o^{(n)})$ and let us consider the sequence $\{\eta(\o^{(j)})\}_{j=0}^n$. The path $\varphi(\gamma)=(\eta^{(0)},\dots,\eta^{(k)})$ is then defined recursively by setting $\eta^{(0)}:=\eta(\o^{(0)})$ and $\eta^{(j)}:=\eta(\o^{(i_j)}),$ where $i_j=\min\{i>i_{j-1}\colon
\eta(\o^{(i)})\neq \eta^{(j-1)}\}$ with $i_0=0,$ and by stopping the procedure as soon as
the set $\{\eta\in \{0,1\}^M\colon \eta_M=1\}$ is reached. In other words, we only keep
the elements of the
sequence $\eta(\o^{(j)}),j=0,\dots,n,$ which change w.r.t. the
previous element. Properties (1) of $\varphi(\gamma)$ follows immediately from the fact that $\gamma$ starts in $\O_{\downarrow}$ and ends in $A_{\ell}$. Property (2) follows from the fact that $\gamma\cap \ensuremath{\mathcal B}_1(n_1)=\emptyset$. We now verify the key property (3).
Let $(\eta,\eta')$ be an edge of $\varphi(\gamma)$ and let $(\o,\o')$ be the edge of $\gamma$ such that $\eta(\o)=\eta$ and $\eta(\o')=\eta'$. By construction $\Phi(\o)\neq \Phi(\o')$. Let also $x\in \ensuremath{\mathcal C}_a$ be such that $\o'=\o^x$ and say that $a$ belongs to $j^{th}$-block. Clearly, $\eta_i=\eta_i'$ for all $i<j$. Moreover, Corollary \ref{lem:D0} and Corollary \ref{cor:D1}
imply that $\Phi(\o)_v=\Phi(\o')_v$ for all $v\in
\cup_{i\;\geqslant\; j+2}B_i$ (if $j+2\;\leqslant\; N$), since otherwise either $\o$ or $\o'$ would have at
least $\lfloor m/2(r^x_\infty+1)\rfloor \;\geqslant\; \lfloor
m/2n_2\rfloor =2n_1$ up-arrows, contradicting the assumption $\gamma\cap
\ensuremath{\mathcal B}_1(n_1)=\emptyset$. In particular, $\eta_i=\eta'_i$ for all $i\;\geqslant\;
j+2$. To complete our analysis we distinguish between two cases. \begin{enumerate}[1)] \item $a>1.$ In this case $x$ must be $\psi^{(0)}(\o)$-unconstrained and part (a) of Lemma \ref{lem:D2} together with
$\Phi(\o)\neq \Phi(\o')$ implies that
$\Phi(\o)_{a-1}=\Phi(\o^x)_{a-1}=\, \uparrow.$ If
$a$ is not the beginning of the block $B_j$ then, by definition, $\eta_j=\eta'_j=1$. Thus $\eta,\eta'$ must differ exactly in
the $(j+1)^{th}$-block and they are both equal
to one in previous one as required. If $a$ is the beginning of the
$j^{th}$-block, then necessarily $j>1$. Moreover
$\Phi(\o)_{a-1}=\Phi(\o^x)_{a-1}=\uparrow$ implies that
$\eta_{j-1}=\eta'_{j-1}=1$. By the same reasoning as before, using Corollary
\ref{cor:D1} and Lemma \ref{lem:D0} (recall that $\omega\in \ensuremath{\mathcal B}^c(n_2)$) we get that $\Phi(\o)_v=\Phi(\o')_v$ for all $v\in
\cup_{i>j}B_i.$ Thus $\eta_i=\eta'_i$ for all $i\neq j$ and $\eta_{j-1}=\eta'_{j-1}=1$ as required. \item $a=1$. Again Corollary
\ref{cor:D1} guarantees that $\Phi(\o)_i=\Phi(\o^x)_i$ for all $i\in
\cup_{j=2}^NB_j$ so that $\eta_b =\eta'_b$ for all $b\;\geqslant\; 2$. \end{enumerate} \end{proof}
\subsection{Density of droplets and proof of Proposition \ref{prop:P2}}
The core of the proof of Proposition \ref{prop:P2} consists in bounding from above the probabilities of the events $\ensuremath{\mathcal B}_1,\ensuremath{\mathcal B}_2$ defined in \eqref{eq:B1},\eqref{eq:B2}. The first key bound is Lemma \ref{lem:D4}, that says that the probability that the $DB^{1,0}_V$-process restricted to an arbitrary number of consecutive columns of $V$ is able to infect any given interval of the last column of length $\ell$ is $e^{-\O((\log q)^2/q)}$. The second key ingredient is Lemma \ref{lem:D5} that bounds from above the probability of the event $\ensuremath{\mathcal B}_2(n_2-1)$. Before stating the lemmas we need some additional notation.
Given $1\;\leqslant\; i\;\leqslant\; j\;\leqslant\; N,$ let $\Lambda =\cup_{k=i}^j \ensuremath{\mathcal L}_k,$ where, for each $k=i,\dots,j,$ $\ensuremath{\mathcal L}_k\supseteq \ensuremath{\mathcal C}_k$ is a (finite) interval of $\{(k-N,j)\colon j\in {\ensuremath{\mathbb Z}} \}.$
Let also $I\subseteq \ensuremath{\mathcal C}_j$ be an interval of length $\ell$. The basic event that we will consider is \[ \ensuremath{\mathcal O}^\tau_\Lambda(I)=\{\o\in \O\colon I\subseteq [Y(\o)\cap \Lambda ]_{\Lambda}^\tau\}, \] where we recall $Y(\o)$ is the set of infected vertices of $\o$. Notice that $\ensuremath{\mathcal O}^\tau_\Lambda(I)$ is an increasing event (\hbox{\it i.e. } its indicator function is an increasing function) w.r.t. to the partial order: $\o\prec \o'$ iff $\o'_x\;\leqslant\; \o_x\ \forall x.$ Our first main lemma reads as follows. \begin{lemma}[Density of up-arrows] \label{lem:D4} Choose the basic scales $N,\ell,n_1,n_2$ as in
\eqref{def:ell},\eqref{def:N} and \eqref{eq:8}. Then there exists $c>0$ such that, for any $\varepsilon>0$
sufficiently small and any $1\;\leqslant\; i\;\leqslant\; j\;\leqslant\; N$, \[ \max_I\mu(\ensuremath{\mathcal O}^{1,0}_{V_{i,j}}(I))\;\leqslant\; e^{-c(\log q)^2/q},\quad \text{as } q\rightarrow 0, \] where $V_{i,j}=\cup_{k=i}^j\ensuremath{\mathcal C}_k$. \end{lemma} \begin{proof}[Proof of Lemma \ref{lem:D4}] Fix $1\;\leqslant\; i\;\leqslant\; j\;\leqslant\; N$ together with an interval $I\subset \ensuremath{\mathcal C}_j$ of length $\ell$ and let \[
\Lambda_{1,j}=\cup_{i=1}^{j}\{(i,k)\colon |k|< N^2\}-N\vec e_1. \] We first claim that \begin{align}
\label{eq:12}
\mu(\ensuremath{\mathcal O}^{1,0}_{V_{i,j}}(I))\;\leqslant\; \mu(\ensuremath{\mathcal O}^{1,0}_{V_{1,j}}(I))\;\leqslant\; O(1/q^2)\mu(\ensuremath{\mathcal O}^1_{\Lambda_{1,j} }(I)) \quad \text{as } q\rightarrow 0. \end{align} The first inequality follows from (C) in Lemma \ref{lem:monotonicity}. To prove the second one, let $G=\cap_{k=1}^{j-1} G_k,$ where $G_k$ denotes the event that there is an empty site within the first $\lfloor N/3\rfloor $ sites and within the last $\lfloor N/3\rfloor $ sites of $\ensuremath{\mathcal C}_k$. Then, for any choice of the constant $\varepsilon$ appearing in \eqref{eq:8}, \begin{equation}
\label{eq:21bis} \mu(G^c)\;\leqslant\; 2N(1-q)^{\frac N3 -1}=o(1) \quad \text{as } q\rightarrow 0. \end{equation} For any $\o\in G$ and any boundary condition $\tau$ for $V_{1,j}$ such that $\tau\equiv 0$ on $\partial_\perp \ensuremath{\mathcal C}_j$ and $\tau_\parallel\equiv 1,$ the screening property and translation invariance imply that $[\,Y(\o)\cap V_{1,j}\,]_{V_{1,j}}^\tau \cap \ensuremath{\mathcal C}_j$ does not depend on $\tau.$ Hence, \begin{equation}
\label{eq:18} \ensuremath{\mathcal O}^{1,0}_{V_{1,j}}(I)\cap G=\ensuremath{\mathcal O}^\tau_{V_{1,j}}(I)\cap G. \end{equation} Choose $\tau$ equal to one everywhere except for $\partial_\perp \ensuremath{\mathcal C}_j$ where it is equal to zero. Using the FKG inequality and \eqref{eq:18}, \begin{align*} \mu\big(\ensuremath{\mathcal O}^{1,0}_{V_{1,j}}(I)\big)&\;\leqslant\;
\mu\big(\ensuremath{\mathcal O}^{1,0}_{V_{1,j}}(I)\thinspace |\thinspace
G\big) =\mu(\ensuremath{\mathcal O}^{\tau}_{V_{1,j}}(I)\thinspace |\thinspace G)\\ &\;\leqslant\; (1+o(1)) \mu\big(\ensuremath{\mathcal O}^{\tau}_{V_{1,j}}(I)\big).
\end{align*} We now observe that, starting from $Y(\o),$ we can construct the set $[Y(\o)\cap V_{1,j}]_{V_{1,j}}^\tau\cap \ensuremath{\mathcal C}_j$ as follows. We first output the set $[Y(\o)\cap V_{1,j-1}]_{V_{1,j-1}}^1$ and we let $\bar \tau\in \{0,1\}^{\partial \ensuremath{\mathcal C}_j}$ be such that $\bar \tau_\perp\equiv 0$ and $\{x\in \partial_\parallel \ensuremath{\mathcal C}_j\colon \bar\tau_x=0\}= [Y(\o)\cap V_{1,j-1}]_{V_{1,j-1}}^1\cap \partial_\parallel\ensuremath{\mathcal C}_j.$ Then we output the set $[Y(\o)\cap \ensuremath{\mathcal C}_j]_{\ensuremath{\mathcal C}_j}^{\bar\tau}$ which clearly coincides with $[Y(\o)\cap V_{1,j}]_{V_{1,j}}^\tau\cap \ensuremath{\mathcal C}_j$.
Monotonicity and a moment of thought imply that if we repeat the above construction with $V_{1,j-1},\ensuremath{\mathcal C}_j$ replaced by
$\Lambda_{1,j-1},\ \{(j-N,k)\colon |k|< N^2\}$ and $Y(\o)$ replaced by $Y(\o)\cup \partial_\perp\ensuremath{\mathcal C}_j,$ then the final infection in $\ensuremath{\mathcal C}_j$ cannot decrease. Hence \begin{align*} \mu\big(\ensuremath{\mathcal O}^{\tau}_{V_{1,j}}(I)\big)\;\leqslant\; \mu\big(\ensuremath{\mathcal O}^1_{\Lambda_{1,j} }(I)\thinspace |\thinspace
\o_{\partial_\perp \ensuremath{\mathcal C}_j}\equiv 0\big)\;\leqslant\; \mu\big(\ensuremath{\mathcal O}^1_{\Lambda_{1,j} }(I)\big)/q^2,
\end{align*} and \eqref{eq:12} follows.
Let now $T(\ensuremath{\mathcal U})$ be the median of the infection time of the origin (or of any other vertex of ${\ensuremath{\mathbb Z}} ^2$ because of translation invariance) for the Duarte bootstrap process in ${\ensuremath{\mathbb Z}} ^2$ started from $Y(\o)$ where $\o$ has law $\mu,$ and write \begin{equation}
\label{eq:13} p(N,\ell):=\max_{j\;\leqslant\; N}\max_I\mu(\ensuremath{\mathcal O}^1_{\Lambda_{1,j}}(I)), \end{equation} where $\max_I$ is taken over all intervals $I\subset \ensuremath{\mathcal C}_j$ of length $\ell$. \begin{claim} \label{claim:1}If $\varepsilon<1/4$ then, for all $q$ small enough, \begin{equation}
\label{eq:16} p(N,\ell)\;\geqslant\; e^{-\frac{1}{16q}\log(q)^2 }, \end{equation} implies \begin{equation*}
T(\ensuremath{\mathcal U})\;\leqslant\; O(N^3) e^{\frac{1}{16q}\log(q)^2 }. \end{equation*} \end{claim} Before proving the claim we conclude the proof of Lemma \ref{lem:D4}. It follows from the main result of \cite{BCMS-Duarte} together with a standard (and straightforward) argument that \[ T(\ensuremath{\mathcal U})\;\geqslant\; e^{(1-o(1))\log(q)^2/8q }\quad \text{as }q\rightarrow 0, \] implying that for all $q$ small enough \[ p(N,\ell)\;\leqslant\; e^{-\frac{1}{16q}\log(q)^2 }, \]
if $\varepsilon<1/48.$ \end{proof} \begin{proof}[Proof of the claim] In the sequel it will help to refer to Figure \ref{fig:2} as a visual guide for the various definitions. Fix $q$ arbitrarily small and let $j$ be such that there exists an interval $I\subset \ensuremath{\mathcal C}_j$ of length $\ell$ such that \begin{equation}
\label{eq:pippo} \mu(\ensuremath{\mathcal O}^1_{\Lambda_{1,j}}(I))\;\geqslant\; e^{-\frac{1}{16q}\log(q)^2 }. \end{equation} Using the symmetry w.r.t. the horizontal axis we can assume that $x_I$, the lowest site of $I,$ has non positive height. Write $\Lambda^{(i)}:=\Lambda_{1,j}-ij\vec e_1$ and let $\ensuremath{\mathcal M}_t=\cup_{i=0}^t\Lambda^{(i)}, $ where $t=10\lceil \max(p(N,\ell)^{-1},8/q^4)\rceil.$ \begin{figure}\label{fig:2}
\end{figure} We shall define two increasing events $\ensuremath{\mathcal G}_1,\ensuremath{\mathcal G}_2\subset \O,$ depending only on $\o\mathord{\upharpoonright}_{\ensuremath{\mathcal M}_t},$ such that: \begin{enumerate}[(a)] \item if $\o\in \ensuremath{\mathcal G}_1\cap \ensuremath{\mathcal G}_2$ then the
Duarte bootstrap process in ${\ensuremath{\mathbb Z}} ^2$ is able to infect $x_I$ within time $2jt(2N^2-1)$. \item $\mu\big(\ensuremath{\mathcal G}_k\big)>3/4,\ k=1,2$. \end{enumerate} Using the FKG inequality, $ \mu\big(\ensuremath{\mathcal G}_1\cap \ensuremath{\mathcal G}_2\big)\;\geqslant\; \mu\big(\ensuremath{\mathcal G}_1\big) \mu\big(\ensuremath{\mathcal G}_2\big) > 1/2. $ Hence \[ T(\ensuremath{\mathcal U})\;\leqslant\; 2jt(2N^2-1)\;\leqslant\; 40 N^3 \Big(e^{\frac{1}{16q}\log(q)^2 }+1\Big). \] In order to define $\ensuremath{\mathcal G}_1,\ensuremath{\mathcal G}_2,$ let $\hat I\supset I$ be the interval of $\ensuremath{\mathcal C}_j$ of length $\lceil 1/q^3\rceil$ and whose lowest site is $x_I$. Then: \begin{align*} \ensuremath{\mathcal G}_1&=\{\forall\, k\in [jt], \text{\it the interval $\hat I-(k-1)\vec e_1$ contains an empty vertex}\};\\ \ensuremath{\mathcal G}_2&=\{\exists\, k\in [jt]\colon \text{\it
the $DB_{\ensuremath{\mathcal M}_t}^1$-process starting from $Y(\o)\cap \ensuremath{\mathcal M}_t$ is able to infect $\hat I-k\vec e_1$}\}. \end{align*} We now verify properties (a) and (b) above. We observe that the event $\ensuremath{\mathcal G}_2$ guarantees that there exists a leftmost interval of the form $\hat I-k\vec e_1$ which is infected by the Duarte bootstrap process within time $tj(2N^2-1)$\footnote{The worst case is when sites are infected one by one.}. The event $\ensuremath{\mathcal G}_1,$ together with the definition of the Duarte update family $\ensuremath{\mathcal U},$ makes sure that the infection of $\hat I-k\vec e_1$ gets propagated forward to $\hat I-(k-1) \vec e_1,\dots, $ until it reaches the original interval $\hat I$ in at most $tj (2N^2-1)$ steps. Hence, within time $2jt(2N^2-1)$ the vertex $x_I$ becomes infected and (a) follows.
It remains to verify (b). The union bound over $k$ gives that for any $\varepsilon>0$ \[ \mu\big(\ensuremath{\mathcal G}_1^c\big)\;\leqslant\; jt(1-q)^{\lceil 1/q^3\rceil}\;\leqslant\; e^{-\O(1/q^2)} \quad \text{as } q\rightarrow 0, \] using \eqref{eq:16} and $j\;\leqslant\; N$.
In order to bound from below $\mu\big(\ensuremath{\mathcal G}_2\big),$ write \[ \nu:=\min\{\max\{k\in [t/2,t]\colon \text{ the event
$\ensuremath{\mathcal O}^1_{\Lambda^{(k)}}(I-k j\vec e_1)$ occurs}\},\,\,\infty\}, \] and let $\ensuremath{\mathcal F}=\cap_{i=1}^3 \ensuremath{\mathcal F}_i$ where, on the event $\{\nu<+\infty\}$: {\it \begin{enumerate}[{-}] \item $\quad \ensuremath{\mathcal F}_1=\{\nu\;\leqslant\; t\}$; \item $\quad \ensuremath{\mathcal F}_2=\{\forall k\in \big[\lceil 2/q^4\rceil\big]$ the interval $I-\nu j\vec e_1
+k\vec e_1$ contains an empty vertex$\}$; \item $\quad \ensuremath{\mathcal F}_3=\{\exists$ an \emph{upward empty stair} of
$n=\lceil1/q^3\rceil$ sites belonging to the first
$\lceil 2/q^4\rceil$ columns of $\ensuremath{\mathcal M}_t$ immediately to the right of
$\Lambda^{(\nu)}$, \hbox{\it i.e. } a sequence $(x_1,\dots,x_n)$ of empty sites of
the form
$x_m=(j_m,h_I +m),$ where $h_I$ is
the height of the uppermost site of $I$ and $\{j_m\}_{m=1}^n$ is a strictly increasing
sequence$\}$. \end{enumerate} } We begin by observing that $\ensuremath{\mathcal F}\subseteq \ensuremath{\mathcal G}_2$. In fact, $\ensuremath{\mathcal F}_1$ guarantees the right amount of infection of the last column of $\Lambda^{(\nu)}$ under healthier boundary condition than those required by $\ensuremath{\mathcal G}_2$. $\ensuremath{\mathcal F}_2$ ensures that such an infection propagates over to the first $\lceil 2/q^4\rceil$ columns to the right of $\Lambda^{(\nu)}$ while $\ensuremath{\mathcal F}_3$ guarantees that each time the infection meets an empty site of the upward stair it grows vertically by one unit (see Figure \ref{fig:2}). Since the stair contains $\lceil1/q^3\rceil$ sites, the $\lceil 2/q^4\rceil^{th}$-column of $\ensuremath{\mathcal M}_t$ to the right of $\Lambda^{(\nu)}$ contains an infected interval which is the appropriate horizontal translation of the interval $\hat I$ and the inclusion $\ensuremath{\mathcal F}\subseteq \ensuremath{\mathcal G}_2$ follows.
Conditionally on $\{\nu = k\}$, the events $\ensuremath{\mathcal F}_2,\ensuremath{\mathcal F}_3$ coincide with two increasing events depending only on sites to the right of $\Lambda^{(k)}$. Hence, using the FKG inequality, \begin{gather*} \mu(\ensuremath{\mathcal G}_2)\;\geqslant\; \mu(\ensuremath{\mathcal F})=\sum_{k\in [t/2,t]}\mu(\nu=k)\mu(\ensuremath{\mathcal F}_2\cap \ensuremath{\mathcal F}_3\thinspace |\thinspace \nu=k) \\ \;\geqslant\; \sum_{k\in [t/2,t]}\mu(\nu=k)\mu(\ensuremath{\mathcal F}_2\thinspace |\thinspace \nu=k) \mu(\ensuremath{\mathcal F}_3\thinspace |\thinspace \nu=k). \end{gather*} A union bound gives that, uniformly in $k\in [t/2,t],$ \[ \mu(\ensuremath{\mathcal F}^c_2\thinspace |\thinspace \nu=k)\;\leqslant\; \lceil 2/q^4\rceil (1-q)^{\ell} \;\leqslant\; \lceil 2/q^4\rceil q^{1/\varepsilon}(1+o(1))=o(1), \] if $\varepsilon < 1/4$. Using the fact that $X(\o):=\min\{i\;\geqslant\; 1\colon \o_{(i,+1)}=0\}$ is a geometric random variable of parameter $q$, it is easy to check that \[ \mu(\ensuremath{\mathcal F}^c_3\thinspace |\thinspace \nu=k)\;\leqslant\; {\ensuremath{\mathbb P}} \Big(\sum_{i=1}^{n}X_i > \lceil 2/q^4\rceil\Big), \] where $\{X_i\}_{i=1}^{n}$ are i.i.d copies of $X$. A standard exponential Markov inequality with $\lambda=\alpha q, \alpha\in (0,1),$ gives \begin{gather} \label{eq:LD} {\ensuremath{\mathbb P}} \Big(\sum_{i=1}^{n}X_i > \lceil 2/q^4\rceil\Big)\;\leqslant\; e^{-\lambda \lceil
2/q^4\rceil}\Big({\ensuremath{\mathbb E}} \big(e^{\lambda X}\big)\Big)^{n}\nonumber\\ \;\leqslant\; \Big(\frac{e^{-2\alpha}}{(1-\alpha)(1+o(1))}\Big)^{1/q^3}<\big(1-\alpha/2\big)^{1/q^3}, \end{gather} for $\alpha$ small enough. In conclusion, if $\varepsilon<1/4$, \begin{gather*} \mu(\ensuremath{\mathcal G}_2)\;\geqslant\; (1-o(1))\mu(\ensuremath{\mathcal F}_1)\\ \;\geqslant\; (1-o(1))\big(1-\big(1-\mu(\ensuremath{\mathcal O}^1_{\Lambda_{1,j}}(I))\big)^{t/2}\big)\;\geqslant\; (1-o(1))(1-e^{-4}) \end{gather*} because of \eqref{eq:pippo} and our choice of $t$. That concludes the proof of property (b). \end{proof} We now turn to the second basic lemma. Recall the definition \eqref{eq:B2} of the event $\ensuremath{\mathcal B}_2.$ \begin{lemma} \label{lem:D5} Choose the basic scales $N,\ell,n_1,n_2$ as in
\eqref{def:ell},\eqref{def:N} and \eqref{eq:8}. Then, for $\varepsilon$ small enough,
\begin{equation}
\label{eq:19} \mu\big(\ensuremath{\mathcal B}_2(n_2-1)\big)\;\leqslant\; e^{-\O(1/q^5)},\quad \text{as } q\rightarrow 0.
\end{equation} \end{lemma} \begin{proof}[Proof of Lemma \ref{lem:D5}] Call $\ensuremath{\mathcal H}_{i,j}$ the event $\cap_{k=i}^j\{\o\in \O\colon\Phi(\o)_k=\downarrow\}\cap \ensuremath{\mathcal G}_{i,j},$ where $\ensuremath{\mathcal G}_{i,j}$ has been defined in \eqref{eq:B2bis}. Clearly \[ \mu\big(\ensuremath{\mathcal B}_2(n_2-1)\big)\;\leqslant\; \sumtwo{i,j}{j-i\;\geqslant\; n_2-2} \mu\big(\ensuremath{\mathcal H}_{i,j}\big)\;\leqslant\; N^2 \maxtwo{i,j\in
[N]}{j-i\;\geqslant\; n_2-2} \mu\big(\ensuremath{\mathcal H}_{i,j}\big), \] and it is enough to prove that \begin{equation}
\label{eq:9} \maxtwo{i,j\in [N]}{j-i\;\geqslant\; n_2-2} \mu\big(\ensuremath{\mathcal H}_{i,j}\big)\;\leqslant\; e^{-\O\big(1/q^5\big)}. \end{equation} For this purpose we first describe one important implication of the event $\ensuremath{\mathcal H}_{i,j}$. \begin{claim} \label{claim:2}For any $\o\in \ensuremath{\mathcal H}_{i,j}$ there exists $h\in {\ensuremath{\mathbb Z}} $ satisfying
$|h|\;\leqslant\; N^2-(j-1)N + (j-i)\ell,$ such that \[ C_{h}:=\big(\cup_{k=i}^{j}\{(k-N,h)\}\big)\cap V_{i,j}\subseteq [Y(\o)\cap V_{i,j}]_{V_{i,j}}^{1,0}. \] Moreover $C_h$ has length at least $(j-i)(1-o(1))\;\geqslant\; n_2(1-o(1))$ as $q\rightarrow 0$.
\end{claim} \begin{proof}[Proof of the claim] Given $\o\in \ensuremath{\mathcal H}_{i,j}$ let $\Gamma=(x^{(1)},\dots, x^{(n)})\subseteq [Y(\o)\cap V_{i,j}]_{V_{i,j}}^{1,0}$ be a Duarte path from $\ensuremath{\mathcal C}_i$ to $\ensuremath{\mathcal C}_j$. Since $\Phi(\o)_k=\downarrow$ for all $k\in \{i,\dots,j\}$ necessarily the cardinality of $\Gamma\cap \ensuremath{\mathcal C}_k$ is at most $\ell$ for all $k\in \{i,\dots,j\}$. Therefore the height $h$ of $x^{(1)}$ satisfies \[
|h|\;\leqslant\; N^2-(j-1)N + (j-i)\ell, \] which, in turn, implies that the corresponding interval $C_h$ has length greater than the largest integer $m$ such that \[ N^2-(i-1)N-mN\;\geqslant\; N^2-(j-1)N+(j-i)\ell. \] Using that $m+1$ violates the above inequality we get \[ m\;\geqslant\; (j-i)(1- \ell/N)-1\;\geqslant\; (1-o(1))n_2. \] The fact that $C_h\subseteq [Y(\o)\cap V_{i,j}]_{V_{i,j}}^{1,0}$ follows from Corollary \ref{cor:duarte-path}.
\end{proof} It is now easy to finish the proof of the lemma. As in the proof of Claim \ref{claim:1} and using a union bound over the possible value of the variable $h$ of the claim, with probability larger than \[ 1-2N^2e^{-\O(q n_2)}\;\geqslant\; 1-e^{-\O(1/q^5)}, \]
every interval $C_{h}$ as above with $|h|\;\leqslant\; N^2-(j-1)N+(j-i)\ell$ meets an empty upward stair, \hbox{\it i.e. } a sequence $(x_1,\dots,x_{\ell})$ of empty sites
belonging to the first $n_2/2$ columns crossed by $C_{h}$ and such that
$x_m=(j_m,h+m)$ with $j_m<j_{m+1}$ for all
$m\in [\ell]\}.$ If $C_h$ is also infected, then the presence of the
above empty stair implies that there
exists $i\;\leqslant\; k\;\leqslant\; i+\frac 23 n_2$ and a vertical interval
$I\subseteq \ensuremath{\mathcal C}_k$ of length at least $\ell$ such that
$I\subseteq [Y(\o)\cap
V_{i,j}]_{V_{i,j}}^{1,0}$. The latter property implies that $\Phi(\o)_k=\uparrow$. Hence $\mu\big(\ensuremath{\mathcal H}_{i,j}\big)$ satisfies \eqref{eq:9} uniformly in $j-i\;\geqslant\; n_2-2$. \end{proof} \subsubsection{Finishing the proof of Proposition \ref{prop:P2}} \label{sec:finish-proof} Recall the definition \ref{def:tf} of the test function $\phi$ and of the events $\O_g,\O_{\downarrow}$ and $\ensuremath{\mathcal A}_{\epsilon,q}$. Notice that $\O_g\cap \ensuremath{\mathcal B}_2(n_2-1)^c \subseteq \ensuremath{\mathcal A}_{\epsilon,q}$ and that $\O_\downarrow$ is a decreasing event. Using Lemma \ref{lem:D5} we get \begin{gather*} \mu(\phi)\;\geqslant\; \mu\big(\ensuremath{\mathcal A}_{\epsilon,q}\big)\;\geqslant\; \mu\big(\O_g\cap \ensuremath{\mathcal B}_2(n_2-1)^c \big)\\
\;\geqslant\; \mu\big(\O_\downarrow\big)\mu\Big(\prod_{|k|\;\leqslant\;
\ell}\o_{(0,k)}=1\Big)- \mu\big(\ensuremath{\mathcal B}_2(n_2-1)\big)\\ \;\geqslant\; \mu\big(\O_\downarrow\big)(1-q)^{2\ell +1}-e^{-\O(1/q^5)}\;\geqslant\; q^{O(1)}\mu\big(\O_\downarrow\big) -e^{-\O(1/q^5)},
\end{gather*} where in the third inequality we used the FKG inequality. Using Lemma \ref{lem:D4} and a union bound, \begin{align*}
\mu\big(\O_\downarrow\big)&\;\geqslant\; 1- \mu\Big(\cup_{j=1}^N\cup_{I\in
\ensuremath{\mathcal I}_j(\ell)}\ensuremath{\mathcal O}^{1,0}_{V_{1,j}}(I)\Big)\\ &\;\geqslant\; 1- 4e^{-(c-5\epsilon)(\log q)^2/q}=1-o(1)
\end{align*} if $\varepsilon$ is small enough, where we let $\ensuremath{\mathcal I}_j(\ell)$ be the family of intervals of the $j^{th}$-column whose length is at least $2\ell+1$. In conclusion $\mu(\phi)\;\geqslant\; q^{O(1)}$ for $\varepsilon$ small enough.
We now turn to bound from above the Dirichlet form $\ensuremath{\mathcal D}(\phi)$. By definition, \begin{gather*} \ensuremath{\mathcal D}(\phi)=\sum_{x\in {\ensuremath{\mathbb Z}} ^2}\mu\big(c_x \operatorname{Var}_x(\phi)\big)=\sum_{x\in
V}\mu\big(c_x \operatorname{Var}_x(\phi)\big)\\ \;\leqslant\; \mu(\ensuremath{\mathcal A})^{-1}q^{-1}\sum_{x\in V} \mu\big(c_x(\o)\mathbbm{1}_{\{\o\in
\ensuremath{\mathcal A}\}}\mathbbm{1}_{\{\o^x\notin \ensuremath{\mathcal A}\}}\big) \end{gather*} where we used the fact that $\phi$ depends only on $\{\o_x\}_{x\in V}$ in the second equality and we wrote $\ensuremath{\mathcal A}\equiv \ensuremath{\mathcal A}_{\epsilon,q}$ for notation convenience. Next we observe that, \begin{gather} \sum_{x\in V} \mu\big(c_x(\o)\mathbbm{1}_{\{\o\in
\ensuremath{\mathcal A}\}}\mathbbm{1}_{\{\o^x\notin \ensuremath{\mathcal A}\}}\big)\nonumber \\\;\leqslant\; \sum_{x\in V} \mu\big(c_x(\o)\mathbbm{1}_{\{\o\in
\ensuremath{\mathcal A}\}}\mathbbm{1}_{\{\o^x\in \ensuremath{\mathcal A}^c,\,\o^x\in \ensuremath{\mathcal B}_2(n_2-1)^c\}}\big)+ \sum_{x\in V} \mu(\mathbbm{1}_{\{\o^x\in \ensuremath{\mathcal B}_2(n_2-1)\}}) \nonumber \\ \;\leqslant\; \sum_{x\in V} \mu\big(c_x(\o)\mathbbm{1}_{\{\o\in
\ensuremath{\mathcal A}\}}\mathbbm{1}_{\{\o^x\in \ensuremath{\mathcal A}^c,\,\o^x\in \ensuremath{\mathcal B}_2(n_2-1)^c\}}\big) +
|V|\big((1-q)/q\big)\mu(\ensuremath{\mathcal B}_2(n_2-1)) \nonumber \\ \;\leqslant\; \sum_{x\in V} \mu\big(c_x(\o)\mathbbm{1}_{\{\o\in
\ensuremath{\mathcal A}\}}\mathbbm{1}_{\{\o^x\in \ensuremath{\mathcal A}^c,\,\o^x\in \ensuremath{\mathcal B}_2(n_2-1)^c\}}\big)+ e^{-\O(1/q^5)}, \label{eq:21} \end{gather} where in the last inequality we used Lemma \ref{lem:D5} and the bound
$|V|\;\leqslant\; 2N^3\;\leqslant\; e^{O((\log q)^2/q)}$.
Given $x\in V,$ let $\o\in \ensuremath{\mathcal A}$ be such that $c_x(\o)=1$ and $\o^x\in \ensuremath{\mathcal A}^c\cap \ensuremath{\mathcal B}_2(n_2-1)^c$ and recall that $N_\uparrow(\o)$ counts the number of up-arrows in $\Phi(\o)$. We claim that $N_\uparrow(\o^x)\;\geqslant\; n_1-1$. To prove the claim, let $\gamma$ be a legal path connecting $\O_g$ to $\o$ such that $\gamma\cap \ensuremath{\mathcal B}_i(n_i-1)=\emptyset, \ i=1,2$ and let $\gamma^x$ be the path connecting $\O_g$ to $\bar\o^x$ obtained by adding to $\gamma$ the transition $\o \rightarrow\o^x$. The path $\gamma^x$ is legal because $\gamma$ is legal and $c_x(\o)=1$. Moreover $\gamma^x\cap \ensuremath{\mathcal B}_2(n_2-1)=\emptyset$ because $\o^x\notin \ensuremath{\mathcal B}_2(n_2-1)$. The assumption $\o^x\in \ensuremath{\mathcal A}^c$ implies that $\gamma^x\cap \ensuremath{\mathcal B}_1(n_1-1)\neq \emptyset$. Using $\gamma\cap \ensuremath{\mathcal B}_1(n_1-1)=\emptyset$ the latter requirement becomes $N_\uparrow(\o^x)\;\geqslant\; n_1-1$ and the claim follows.
In conclusion, \begin{gather*} \sum_{x\in V} \mu\big(c_x(\o)\mathbbm{1}_{\{\o\in
\ensuremath{\mathcal A}\}}\mathbbm{1}_{\{\o^x\in \ensuremath{\mathcal A}^c,\,\o^x\in \ensuremath{\mathcal B}_2^c\}}\big) \;\leqslant\; \sum_{x\in V} \mu\big(N_\uparrow(\o^x)\;\geqslant\; n_1-1\big)\\
\;\leqslant\; |V|\big((1-q)/q\big)\mu\big(N_\uparrow(\o)\;\geqslant\; n_1-1\big). \end{gather*} We finally bound from above $\mu\big(N_\uparrow(\o)\;\geqslant\; n_1-1\big)$ using Lemma \ref{lem:D4}. Given $n\;\geqslant\; n_1-1$ and $E=\{j_1<\dots <j_n\},\ j_i\in [N],$ let $\ensuremath{\mathcal N}_E$ be the event that $\Phi(\o)_j=\uparrow$ if $j\in E$ and $\Phi(\o)_j=\downarrow$ otherwise. By construction \[ \mu(\ensuremath{\mathcal N}_E)\;\leqslant\; \mu\left(\bigcap_{k=1}^n \ensuremath{\mathcal Q}^{1,0}_{V_{j_{k-1}+1,j_{k}}}\right)\;\leqslant\; \Big(\max_{i\;\leqslant\; j}\mu(\ensuremath{\mathcal Q}^{1,0}_{V_{i,j}})\Big)^n, \] where $j_0:=0$ and \[ \ensuremath{\mathcal Q}^{1,0}_{V_{i,j}}=\{\exists I\in \ensuremath{\mathcal I}_j(\ell) \text{ such that } I\subseteq [Y(\o)\cap V_{i,j}]_{V_{i,j}}^{1,0}\}. \] where we recall that $\ensuremath{\mathcal I}_j(\ell)$ is the family of intervals of the $j^{th}$-column whose length is at least $2\ell+1$. Lemma \ref{lem:D4} together with a union bound over $I\in \ensuremath{\mathcal I}_j(\ell)$ give \begin{gather*} \max_{i\;\leqslant\; j}\mu\big(\ensuremath{\mathcal Q}^{1,0}_{V_{i,j}}\big)\;\leqslant\; \max_{i\;\leqslant\; j}\sum_{I\in \ensuremath{\mathcal I}_j(\ell)}\mu\big(I\subseteq [Y(\o)\cap V_{i,j}]_{V_{i,j}}^{1,0}\big) \\\;\leqslant\; 4 N^4\max_{i\;\leqslant\; j}\max_{I\in \ensuremath{\mathcal I}_j(\ell)}\mu\big(I\subseteq [Y(\o)\cap V_{i,j}]_{V_{i,j}}^{1,0}\big) \;\leqslant\; e^{-(c-4\varepsilon)(\log q)^2/2q}.
\end{gather*} In conclusion, for any $\varepsilon$ small enough, \begin{align*} \mu\big(N_\uparrow(\o)\;\geqslant\; n_1-1\big)&\;\leqslant\; \sum_{n=n_1-1}^N \binom{N}{n}
e^{-(c-4\varepsilon) n(\log q)^2/2q}\\ &\;\leqslant\; \sum_{n=n_1-1}^N \Big(N e^{-(c-4\varepsilon) (\log q)^2/2q}\Big)^n\\ &\;\leqslant\; e^{-\varepsilon\,\O((\log q)^4/q^2)}, \end{align*} because of the choice of $n_1=\varepsilon (\log q)^2/2q$. In conclusion, the r.h.s. of \eqref{eq:21} is smaller than $e^{-\varepsilon\O((\log q)^4/q^2)}$ and the proof of Proposition \ref{prop:P2} is complete.\qed \section*{Acknowledgment} We would like to thank R. Morris for several stimulating discussions.
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tedm.us
quantitative biology
mountainscape?
Parameter Spaces
Physicists are known for order-of-magnitude calculations on the backs of envelopes. What are the relevant length, time, and energy scales? What dimensionless parameters can be formed? Toy models can provide valuable intuition; real models should be just complex enough to capture the phenomenon of interest. Einstein, with all his brainpower, was deeply impressed by classical thermodynamics — by its simplicity and its wide range of applicability.
Physics has numerous areas of overlap with the other sciences and engineering. Starting in the 20th century, some physicists became interested in biology. Erwin Schrödinger, one of the fathers of quantum theory, wrote What Is Life? in 1944. Certainly, he reasoned, the laws of physics and chemistry must apply within living things — what basic biological questions can we ask, e.g., in terms of information content and entropy increase? Physicists following in Schrödinger's footsteps would form part of the quantitative biology community, applying their training to study problems as diverse as neuron firing, motor proteins, and cancer progression.
What can physics tell us about the aqueous environment in/around a micron-sized cell? The Reynolds number fluid parameter $$\mathrm{Re} \ll 1$$, so inertia is unimportant — there is no drifting! Water and dissolved ions screen out all electrostatic interactions beyond the nanometer-sized Debye length. While living systems are, by definition, out-of-equilibrium, local equilibria often exist. In general, diffusive and dissipative forces dominate, so the most effective tools are those of statistical mechanics and stochastic processes.
In contrast to a living cell, a pure, crystalline solid is a very different lump of condensed matter, indeed. The solid has the well defined order and symmetry of the unit cell extending in all directions. While its ions do experience vibrations, they can generally be treated as a fixed lattice through which the valence electrons flow. On the other hand, the cell consists of heterogeneous fluid contents encapsulated by a cell membrane. Despite its hodgepodge character, the cell does contain many components that are highly ordered and even capable of self-organizing.
In modeling some aspect of cell behavior, model parameter values must typically come from experimental data. For the solid, however, properties can be calculated from first principles. Given elemental composition and unit cell structure, density functional theory can solve the equations of quantum mechanics to yield electronic structure. Transport theory can then be used to calculate bulk properties useful to the MSE community — how well does this material conduct heat and electricity?
© 2011-2019 Ted Malliaris | CommonCrawl |
Solar energy and regional coordination as a feasible alternative to large hydropower in Southeast Asia
Linking solar and wind power in eastern Africa with operation of the Grand Ethiopian Renaissance Dam
Sebastian Sterl, Dalia Fadly, … Wim Thiery
Smart renewable electricity portfolios in West Africa
Sebastian Sterl, Inne Vanderkelen, … Wim Thiery
Resilience of the Eastern African electricity sector to climate driven changes in hydropower generation
Vignesh Sridharan, Oliver Broad, … Raffaello Cervigni
Power sector investment implications of climate impacts on renewable resources in Latin America and the Caribbean
Silvia R. Santos da Silva, Mohamad I. Hejazi, … Chris R. Vernon
Implications of water constraints on electricity capacity expansion in the United States
Lu Liu, Mohamad Hejazi, … Barton A. Forman
Designing low-carbon power systems for Great Britain in 2050 that are robust to the spatiotemporal and inter-annual variability of weather
Marianne Zeyringer, James Price, … Ed Sharp
Regional impacts of electricity system transition in Central Europe until 2035
Jan-Philipp Sasse & Evelina Trutnevyte
The potential for sustainable hydropower
Patrick Bogaart
Geothermal energy as a means to decarbonize the energy mix of megacities
Carlos A. Vargas, Luca Caracciolo & Philip J. Ball
Kais Siala1,
Afm Kamal Chowdhury ORCID: orcid.org/0000-0003-3763-12042,3,
Thanh Duc Dang ORCID: orcid.org/0000-0002-9303-90563 &
Stefano Galelli ORCID: orcid.org/0000-0003-2316-32433
Nature Communications volume 12, Article number: 4159 (2021) Cite this article
Strategic dam planning and the deployment of decentralized renewable technologies are two elements of the same problem, yet normally addressed in isolation. Here, we show that an integrated view of the power system capacity expansion problem could have transformative effects for Southeast Asia's hydropower plans. We demonstrate that Thailand, Laos, and Cambodia have tangible opportunities for meeting projected electricity demand and CO2 emission targets with less hydropower than currently planned—options range from halting the construction of all dams in the Lower Mekong to building 82% of the planned ones. The key enabling strategies for these options to succeed are solar PV and regional coordination, expressed in the form of centralized planning and cross-border power trading. The alternative expansion plans would slightly increase the cumulative costs (up to 2.4%), but substantially limit the fragmentation of additional river reaches, thereby offering more sustainable pathways for the Mekong's ecosystems and riparian people.
In many developing regions, economic growth is supported by power systems that rely on cheap and locally available energy sources. Southeast Asia is no exception: the region is on the way to achieve universal access to electricity by largely banking on hydroelectricity and fossil fuels1. Aside from CO2 emissions, a major concern for this energy policy is the socio-environmental externalities of hydropower development. The main center of activity has been the Mekong River, a global hotspot of biodiversity and home to the world's largest freshwater fishery2,3. The Mekong and its tributaries have abundant hydropower potential, part of which has so far been developed by China, Laos, Thailand, and Vietnam—with Cambodia and Myanmar playing a marginal role. Up to 2020, the combined installed capacity of all commissioned dams (>1 MW) is about 41.6 GW4,5,6, including some particularly controversial dams recently built on the main stem of the Lower Mekong (e.g., Nuozhadu: 5850 MW, Xayaburi: 1285 MW, Don Sahong: 260 MW). Their impact is profound: by creating artificial storages and fragmenting the river network, they not only alter the hydrological regimes7,8 but also block fish passage and reduce the transport of sediment and nutrients9, ultimately affecting the riverine ecosystems, its fisheries, and the riparian communities10,11,12. If all proceed as planned, another 22 GW will be deployed in the next decades4,5,6 (Fig. 1a). And yet, there might be multiple alternatives to this plan: the availability of regional grid interconnections13,14 and renewable energy sources, particularly solar photovoltaic (PV)15, suggests that dam development may be partially offset by deploying renewable technologies in low-cost and well-accessible areas16. A complementary strategy is the design of more sustainable dam portfolios17. Whether any of these alternatives is technically feasible and economically reasonable for it to succeed remains an open question.
Fig. 1: Study site.
a Full spatial extent of the Chao Phraya and Mekong basins, together with the dams operated in 2016 (blue-filled dots) and planned (red circles) by all riparian countries. These dams are modeled by the hydrologic–hydraulic model VIC-Res. The bar chart in top right corner shows planned hydropower capacities over the period 2020–2037, with color scale indicating riparian countries (L Laos, T Thailand, Ca Cambodia, M Myanmar, and Ch China). b Spatial representation of the power system infrastructure for each province of Thailand, Laos, and Cambodia. Segments, triangles, and squares represent high-voltage transmission lines, provinces, and import/export nodes. The pie charts overlapping the provinces indicate their existing generation capacities, proportionate to the size scale shown in the left panel. The pie charts at bottom right corner show the country-wise total existing capacities. All components of the power grid were operational in 2016.
Designing dam portfolios and deploying decentralized renewable technologies are two elements of the same problem—i.e., planning the expansion and operations of sustainable power systems—but they are normally addressed with different tools. Strategic dam planning typically relies on multi-objective optimization frameworks that balance hydropower capacity with one, or multiple, environmental objectives, such as fish biomass and biodiversity losses18, sediment supply19,20, or greenhouse gas emissions from reservoirs21. These studies provide fundamental guidance for future hydropower projects, as they identify opportunities for better trade-offs between power supply and ecosystem services. Most importantly, they highlight the necessity of regional coordination in dam planning20, as opposed to the piecemeal approach adopted in many basins. However, some dam portfolios may be technically or economically unfeasible, because the frameworks with which they are designed do not represent the role played by dams within power systems. For example, concentrating dams within a few sub-basins may provide an opportunity to balance installed capacity with ecosystem services, but such a plan may be impaired by the cost of developing an adequate transmission infrastructure22 or if the intermittent production of hydropower dams cannot be absorbed by the existing thermoelectric facilities23. These are the fundamental mechanisms captured by the tools used to study the integration of renewable technologies within existing grids, which combine long-term capacity expansion and detailed power system operations24,25,26. The flipside here is that power system planning models typically forgo the information on socio-environmental externalities available from dam planning studies27 and use simplistic representations of hydropower storage dynamics28, thereby neglecting hydropower response to climate variability as well as the cascading effect of hydropower operations in large reservoir networks, such as the one being developed in the Mekong.
Here, we introduce a modeling framework for dam and power system planning in the Lower Mekong River Basin that brings the aforementioned elements under the same umbrella. Our framework consists of two components, urbs29 and the Variable Infiltration Capacity (VIC)-Res model30,31. urbs co-optimizes capacity expansion for generation, transmission, and storage as well as hourly power system operations—thus accounting for the balancing of supply and demand, transmission constraints, ramping limits, electricity reserve, and the time needed to start-up and shut down the thermoelectric units. A fundamental feature of urbs is its spatially distributed nature: the model explicitly accounts for the power system infrastructure of 120 provinces in Laos, Thailand, and Cambodia, where a cross-border, power-trade infrastructure is already in place (Fig. 1b). Thanks to this set-up, urbs integrates complex weather data and characterizes the spatial variability of renewable energy sources and hydropower, a fundamental requirement for large-scale studies (cf. ref. 24). The hydropower availability of each existing and planned dam in the Mekong is calculated using VIC-Res, a spatially distributed hydrologic–hydraulic model simulating not only the relationship between hydro-meteorological forcings and water availability through the basin but also the storage dynamics and turbine release of each reservoir. VIC-Res is also implemented for the Chao Phraya, the second main basin of our study site and home to a few large dams feeding the Lower Mekong power grid.
By running our framework over the period 2016–2037, we show that the regional electricity demand and CO2 emission targets can be met by constructing only 82% of the planned dams in Thailand, Laos, and Cambodia. The key enabling technologies for this alternative to succeed are solar PV and high-voltage transmission lines, which redistribute cheap electricity across distant load centers. Our analysis of alternative dam portfolios proposes other, more sustainable, options: a careful expansion of the power system could even absorb the halting of the construction of all dams in the Lower Mekong, at a cost of about 10 billion US$ over the period 2016–2037. Finally, we show that the alternative dam portfolios could substantially limit the fragmentation of additional river reaches. However, further alterations of the natural flow regime will depend on decisions made in both Upper and Lower Mekong, thus highlighting the need for multi-sector cooperation efforts between all riparian countries.
Capacity expansion plans
We perform a power system optimization of the Lower Mekong region that takes into account the existing power infrastructure, the projected costs of technologies, as well as future electricity demand and emissions reduction targets. The power systems of the Lower Mekong River Basin face two challenges: meeting the growing electricity demand (projected yearly growth rates are 4.3% for Thailand, 8.8% in Cambodia, and 9.5% in Laos) and decreasing the carbon emissions intensity from an estimated 0.536 t CO2/MWh to a target of 0.308 t CO2/MWh. There are many decarbonization pathways to reach these targets, but they roughly fit into two categories. The first one focuses on shifting from coal to gas (which has a lower carbon intensity), with a moderate expansion of renewable energy technologies. The second relies on a large expansion of renewable energy and a moderate expansion of gas power plants, so that the system can accommodate a continuous usage of coal. The results of the optimization, with regard to the energy mixes of the three countries, reflect a combination of these pathways (Fig. 2). In the short term, due to the stringent assumption on the overall emission intensity, we observe that gas replaces part of the coal generation in Thailand and reduces its dependence on imports from Laos. Gas is the cost-efficient solution because the hydropower dams going into operation in 2020 are not sufficient to reduce the carbon emissions in accordance with the stringent targets for that year, and because the installation cost of solar PV is still relatively high (see Table S1 for an overview of the technology cost assumptions).
Fig. 2: Capacity expansion plans for the period 2016–2037.
Evolution of the power mix in Thailand, Laos, and Cambodia designed by urbs. Negative values indicate that electricity exports exceed imports. Note that most of the electricity exported from Laos goes to Thailand.
The decarbonization strategy shifts drastically from 2025 onwards. The usage of coal is on par with 2016 levels, the relative share of gas decreases, while a huge expansion of renewable energy technologies takes place in the Lower Mekong countries. Since the wind potential is rather limited in the region, the three countries increase the capacities of solar PV (particularly in Thailand) and hydropower (mostly in Laos and Cambodia). Solar PV capacity expansion amounts to 52 GW in 2025 and continues to grow steadily in the following years to reach 68.2 GW by 2037. Thailand alone witnesses an addition of 49.8 GW of solar capacity, which is equivalent to about 42% of its total capacity in 2037. Meanwhile, the hydropower capacity in the three countries increases from 9.3 GW in 2016 to 22.8 GW in 2037. Most of the new capacities are added in Laos (+12 GW), followed by Cambodia (+1.8 GW), with an additional 0.7 GW in Myanmar dedicated to the Thai power market. This corresponds to an execution rate of 82%, since the total capacity of all planned dams in the region amounts to 17.6 GW. In order to connect the hydropower dams with the demand centers, which are mainly located in Thailand, the power grid is upgraded with the addition of 25 GW bidirectional transmission lines. Consequently, the share of carbon-free generation increases from 16.7% in 2016 to 42.9% in 2037. Whereas hydropower makes up the lion's share in 2016, it only accounts for less than half of the carbon-free generation in 2037. The rest is provided by solar PV (89.2 TWh, or 49.8%), with bioenergy and onshore wind playing minor roles.
Regional balancing of supply and demand
The new dams are mainly located in Southern Laos and Northeastern Cambodia, Northern Laos, and Eastern Myanmar. Among the dams that are not selected for the capacity expansion, one is located in Western Cambodia (100 MW) and another one in Southern Laos (70 MW), but the majority (21 dams, about 2.1 GW) are in Northern Laos. Of the new solar PV capacities, 25% are concentrated in the north west of Thailand, whereas the rest is distributed all over the region. However, most of the power demand occurs around Bangkok. Hence, we observe that different provinces within the three countries play different roles—notably as hydropower generation hubs, solar PV generation hubs, or power demand hubs (the regional distributions of hydro capacities, solar capacities, demand, and transmission lines are shown in Fig. 3). In order to alleviate the regional discrepancies between supply and demand, the model expands the transmission grid in the east–south direction (from Laos to Thailand through Cambodia) and west–south direction (within Thailand), so that most new lines converge toward Bangkok and its surroundings. This cost-optimal power system design implies a high level of regional coordination between the grid operators of the three countries.
Fig. 3: Spatial analysis.
Regional distribution of the solar capacities (a), hydro capacities (b), power demand (c), and residual energy demand (d) in 2037. The maps are scaled to the respective maximum regional quantity. The residual energy demand is calculated by subtracting the annual energy production of solar PV, onshore wind, and hydro from the annual power demand and is indicative of the regional discrepancies between supply and demand. The transmission capacities that are added between 2020 and 2037 are also plotted on subfigure in (d).
Impact of alternative dam portfolios on power system expansion
Our results indicate that not all planned hydropower dams must necessarily be built. Moreover, the availability of a vast solar PV potential15 suggests that there might be opportunities for further reducing the number of dams built in the near future. We therefore consider three alternative dam portfolios (Table 1) and use urbs–VIC-Res to identify possible substitutes in the power system and quantify the implications in terms of system costs. Two portfolios represent scenarios in which we stop the construction of all dams (Stop-All) or only the planned ones (Stop-Planned)—for which construction works have not started yet. The third portfolio blocks the construction of dams in the main stem of the Mekong (Stop-Main), which have a larger impact on migratory fish populations and sediment supply20,32.
Table 1 Dam development portfolios over the planning horizon 2016–2037.
As illustrated in Fig. 4, the alternative dam portfolios are technically feasible, meaning that a decrease in hydropower production can be offset by other sources, mainly solar PV and gas. Interestingly, there is also a positive correlation between hydropower and coal generation. In fact, if the hydropower share is high, then the overall carbon-neutral generation is also high. This leaves some freedom to use coal, which is cheaper than gas but has a higher carbon intensity per unit of energy. On the other hand, in the scenarios with less hydropower, the power system has to generate more energy from carbon-emitting technologies without violating the total CO2 constraint, so it resorts to using more gas-fired power plants. Importantly, the alternative portfolios may also be economically feasible: taking into account the investment costs, fuel costs, and fix and variable operation and maintenance costs (up until 2037), our results show that the scenarios with alternative dam portfolios are marginally more expensive than the Reference one. For example, the most restrictive portfolio (Stop-All) leads to cumulative costs that are 2.4% higher than the business-as-usual strategy. This corresponds to about 10 billion US$ over the period 2016–2037.
Fig. 4: Future generation mixes and costs.
a Evolution of the power mix (aggregated across Thailand, Laos, and Cambodia) for the dam development portfolios outlined in Table 1. In b, we report the corresponding total system costs.
Future pathways of river fragmentation and flow regulation
To estimate and synthesize the combined effects of the alternative dam portfolios on the Mekong's ecosystems, we use the River Fragmentation Index (RFI) and River Regulation Index (RRI)32,33. The former captures the effect of dams on the natural connectivity of riverine systems, focusing in particular on longitudinal connectivity, important for its relation to species migration18. The latter quantifies the impact of dams on timing and magnitude of flows: alterations of the natural flow regime that can disrupt the life cycle of freshwater species34. When calculating both indices (see "Methods"), we account for the dams selected by urbs for each portfolio and time slice but assume that all dams in China will be constructed as planned—thereby reflecting the lack of coordination between Lower and Upper Mekong countries on infrastructure development.
The RFI and RRI values over the period 2016–2037 give us a glimpse of past, present, and future pathways of river fragmentation and flow regulation (Fig. 5). Dams operational in 2016 appear to affect more the total network regulation rather than the river connectivity, a result explained by two facts. First, most of these dams are located in headwater streams. Second, some of these dams, particularly those in the Upper Mekong, have massive storage capacity (e.g., Xiaowan Dam: 15,043 Mm3, Nuozhadu Dam: 21,749 Mm3), so their effect on the flow regime is perceived across the entire basin8. The sudden change in the RFI experienced between 2016 and 2020 (from 17.4 to 66.6%) is largely attributable to dams built in the Lower Mekong Basin, either on the main stem (e.g., Xayaburi Dam, in Laos) or on major tributaries (e.g., Lower Sesan 2 Dam, in Cambodia), which disconnect large fractions of the river network (see Fig. 6). Although the current situation is clearly critical, our results indicate that the alternative dam portfolios could substantially limit the fragmentation of additional river reaches. More precisely, we estimate that future values of the RFI could vary between 66.6 and 80.5%, under the Stop-All and Reference scenario, respectively. Results also suggest that the portfolios have little influence on future alterations of the flow regime, because the dams planned for the Lower Mekong have in general limited capacity to control flows—that is, the ratio between storage and average inflow is small. Instead, the projected RRI increase is mainly attributable to the construction of large-storage dams in the Upper Mekong (see the RRI increase for the year 2037 illustrated in Fig. 5).
Fig. 5: Future pathways for the Mekong River Basin.
The figure illustrates the evolution of the River Fragmentation Index (RFI) and River Regulation Index (RRI) between 2016 and 2037 for four different dam portfolios. When calculating both indices, we included for all scenarios the dams planned in the entire basin.
Fig. 6: Effect of dams on river flow.
Change in the Degree of Regulation (DOR) between the current stage (2020) (a) and two dam development portfolios (Stop-All (b) and Reference (c)) in 2037. The two insets in (d, e) highlight the dam development conditions and DOR for the Lower Mekong Basin.
A more nuanced understanding of the effect of existing and planned dams on flow regulation is offered by Fig. 6, where we illustrate the Degree of Regulation (DOR), a spatially disaggregated version of the RRI calculated for each river reach (see "Methods"). By contrasting the DOR calculated for the situation in 2020 and the Stop-All and Reference portfolios for 2037, Fig. 6 reveals the differential impact of Lower and Upper Mekong dams on river flows (see Fig. S2 for the DOR values of the other two portfolios). First, the construction of just a few, large-storage dams in China would further alter the flow regime far downstream along the river network (cf. the current situation, 2020, against the Stop-All portfolio). In particular, the DOR would be >25% in almost the entire main stem. To put this number into perspective, consider that Lehner et al.35 marked the possibility of substantial changes in the natural flow regime for DOR values >10%. Second, the construction of more dams in the Lower Mekong countries would not dramatically change the DOR values in the river network (cf. Stop-All and Reference portfolios), since most dams—even those planned for the main stem—have limited storage capacity in relation to the river flow. In addition, the flow regulation effect is diluted by the presence of a few weakly regulated tributaries, such as those in the southwest part of the basin, primarily controlled for irrigation.
In sum, it appears that the fate of the Mekong's ecosystems is caught between the dam development plans for the upper and lower portions of the basin. Our analysis shows that a careful expansion of the power system in Thailand, Laos, and Cambodia could prevent additional damages on the river's natural connectivity, but future alterations of the natural flow regime are more directly related to the construction of large dams in the Upper Mekong.
Our study demonstrates that Thailand, Laos, and Cambodia could meet their future electricity demand and CO2 emission targets with substantially less hydropower than what is currently planned—options range from halting the construction of all dams in the Lower Mekong to building 82% of the planned ones. Importantly, the options we explored are both economically and technically feasible. Beginning with the economic aspects, note that even the most restrictive dam portfolio we considered (i.e., halting of the construction of all dams in the Lower Mekong) would increase the cumulative costs over the period 2016–2037 by only 2.4% (~10 billion US$) with respect to the business-as-usual strategy. And while these figures may change in the future in response to cost overruns in large hydropower dams36 or fluctuations in the cost of technology and commodities, we note that they are comparable to the estimated damages of dam developments on the inland fishing industry alone (i.e., 2–13 billion US$37,38). But hydropower dams have many other negative impacts, such as greenhouse gas emissions39, thermal pollution40, or the displacement of indigenous communities41. In this regard, it is important to consider that several socio-environmental externalities are related to the natural connectivity of the river network32, meaning that the Lower Mekong countries still have a chance to curb an already critical situation. The flipside of our results is that alterations of the natural flow regime—a potent driver of biodiversity42—are also determined by dam planning decisions in the Upper Mekong. China's recent decision to share year-round water data with the downstream countries is a first important step43, which should ideally be followed by mechanisms for jointly planning infrastructure investments20.
The reason behind the technical feasibility of these plans lies in the flexibility of the other technologies. Solar PV modules, while subject to a diurnal cycle and to weather conditions, have the advantage of being scalable and deployable in any province of the Mekong countries. In particular, they can be built in every province, spare the costs of long transmission lines, and ensure a higher level of energy autarky. The seasonal fluctuations are low and complement very well the existing hydropower production, provided that there is a strong coordination between the national grid operators. As of the intraday fluctuations, they may not require utility-scale, expensive batteries in the short term and mid-term because gas power plants can ramp up and down rapidly. Hence, even in the least restrictive scenario to hydropower expansion, we notice a shift from hydropower as main source of clean electricity to solar in the next years. This is akin to a paradigm shift in the power supply from a few, large infrastructure projects to multiple small decentralized power plants. This trend is in line with studies on other regions, for example, on Myanmar44, Congo28, or South Africa25, and is robust against climate variability, as shown in our sensitivity analysis with different hydro-climatic conditions (see "Methods").
The key message of this paper is that the planned hydropower expansion should be revised in light of the new developments in the power market, in particular the fast decreasing costs of solar PV, which already produce the cheapest electricity in many countries45. Even in the Reference scenario, which reflects the Thai decarbonization targets and has no restrictions on hydropower portfolios, only 82% of the planned capacity is actually built according to our coupled models. If the power demand growth rates fall short of the projections, even less hydropower capacity would be needed. Thus, the construction of less economical dams should not proceed as planned. That being said, the Thai decarbonization targets until 2037 are not ambitious enough to push coal out of the system. In fact, we observe that more hydropower in the system enables coal to be used even more and comply with the CO2 constraints. This trend applies not only to the Mekong countries but also to the whole ASEAN region. According to the International Energy Agency1, the projected increase in fossil fuel consumption, particularly the continued rise in coal demand, will lead to a two-thirds rise in CO2 emissions and a 44% increase in premature deaths due to air pollution by 2040, compared to 2018. Therefore, revisions to policy plans have so far tended to boost the long-term share of renewable energy, typically at the expense of coal1. So ultimately, if decarbonization targets become more stringent in the long term or the mid-term, the competition between solar and hydro in the Mekong countries might turn into a collaboration, because they are probably both needed in large amounts to drive coal out.
Our study also demonstrates that there are technical pathways for combining the design of dam portfolios with the capacity expansion of power systems. By doing that, we can balance hydropower supply with environmental objectives and, importantly, explicitly evaluate the role played by dams within a power system, therefore avoiding the risk of conceiving portfolios that are economically or technically unfeasible. By combining high-resolution hydrological and power system planning models, we also account for both geophysical and political boundaries, an information needed to account for limits and opportunities in cross-border power-trade infrastructure. Naturally, a modeling framework like ours should be used at the beginning, rather than at the end, of a conversation on sustainable energy planning, because its spatial domain and computational requirements inevitably constrain the number of physical processes and scenarios that can be considered. In other words, screening models deployed across large domains should be complemented by local-scale impact assessments that evaluate additional, fundamental processes, such as sediment and fish passage through dams46. In this regard, another potential modeling avenue is to dynamically link strategic dam planning models and power system planning models, so as to provide a more exhaustive exploration of the ecology–energy trade-offs44. A local-scale assessment would also be more suitable for modeling extreme cases of demand fluctuations that test the reliability of the power system. Although this is not the top priority for developing countries that have not achieved universal access to electricity, it is safe to assume that the reliability requirements will soon converge toward the standards in the developed economies.
Looking forward, it is not difficult to imagine that many developing regions will be caught increasingly in the tension between ensuring cheap power security, exploiting locally available resources, and protecting ecosystems. Multi-model frameworks that span across multiple sectors—like the one described here—are a suitable platform for capturing these multiple perspectives and resolving, or at least addressing, ecology–energy trade-offs.
Hydrological and water management models
To estimate the daily hydropower production of each dam in the Mekong and Chao Phraya basins, we adopt a two-step modeling approach. We begin with VIC, a large-scale, semi-distributed hydrologic model47. VIC organizes the spatial domain into a number of computational cells, where evapotranspiration, infiltration, baseflow, and runoff are calculated. The simulated runoff is then routed through the river network by VIC-Res, a water management model that includes an explicit representation of storage and release dynamics of water reservoirs31. In VIC-Res, each reservoir is represented by a cell accounting for dam location and a number of water of cells in which the storage dynamics are calculated. Daily release decisions are determined on the basis of bespoke rule curves. Using the information on hydraulic head and release, VIC-Res finally calculates the hydropower available at each dam.
Two separate computational domains were constructed to simulate hydrological and water management processes in the two basins. The domain for the Mekong covers an area of ~635,000 km2, stretching from the Tibetan Plateau (China) to Kratie (Cambodia). In this model, we simulate the operations of 108 dams operational in 2016, spanning across China, Laos, Thailand, Cambodia, and Vietnam. This is necessary to account for the effect of upstream dam regulation on water availability and hydropower production in the Lower Mekong countries. The model for the Chao Phraya basin has a domain of ~110,000 km2 and includes Bhumibol and Sirikit dams, which have a combined installed capacity of ~1280 MW. For both Mekong and Chao Phraya's models, we adopt a resolution of 1/16th of a degree, necessary to avoid allocating multiple dams to the same cell.
Key inputs include a Digital Elevation Model (DEM) and data on land use, soil, precipitation, and temperature. For the DEM, we masked the Global 30 Arc-Second Elevation (GTOPO30) DEM with the shape of the two basins and then adapted it to the resolution of our models with the average resampling technique48. Land use and soil data are obtained from the Global Land Cover Characterization dataset and Harmonized World Soil Database, respectively. The datasets have a spatial resolution of 30 arcsecond, so we generated land use and soil maps with the majority resampling technique. Rainfall and temperature data are retrieved from Global Meteorological Forcing Dataset49, which have been thoroughly tested for our study site50. For the representation of reservoirs in VIC-Res, we acquired data on storage–depth relationship, maximum surface extent, dam design specifications, and rule curves. The storage–depth relationship is modeled with Liebe's method51, the most common approach in large-scale studies52,53,54. The maximum surface extent of each reservoir is estimated by extracting surface water profiles from Landsat TM and ETM+ imagery, while the dam design specifications are obtained from the Mekong River Commission and the Electricity Generating Authority of Thailand—and complemented, where necessary, with information retrieved from other databases. Rule curves are designed to drawdown the reservoir storage during the driest months (December–May) to maximize the electricity production, recharge the depleted storage during the monsoon season, and avoid the risks of spilling water at the end of the monsoon season30,55. Rule curves are tailored to each reservoir by determining the time at which the minimum and maximum water levels are reached (roughly May and November), setting the value of the minimum and maximum water levels, and finally connecting these points with a piecewise linear function that gives us the daily target level for each calendar day30. With this modeling approach, rule curves account for both normal conditions and emergency procedures—that is, when the water level drops below the dead level or is above the critical one, requiring the activation of the spillways. Additional information on the input data is provided in Table S2.
To calibrate the hydrological model, we tuned the parameters controlling the rainfall-runoff process and compared the simulated discharge against the one observed at multiple gauging stations in the Mekong and Chao Phraya basins (data retrieved from the Mekong River Commission and the Thai Royal Irrigation Department). The calibration period is 1996–2005, with 1995 used for the model spin-up. During the simulation, reservoirs are activated in the year they become operational, so as to account for the non-stationarity of human interventions in the river basin. The model is then run over the period 2007–2016 (2006 is used for the spin-up). This validation includes a thorough comparison of the mean annual (simulated) hydropower production against the annual design (or expected) production. This is necessary to ensure that the rule curves correctly capture other factors affecting reservoir operations (e.g., irrigation) and therefore the hydropower profiles fed to the capacity expansion model. A detailed description of calibration and validation exercises is reported in refs. 22,23.
Hydropower profiles for the capacity expansion model
Ideally, the capacity expansion model should use as input a few years of hydropower profiles (simulated by VIC-Res for each dam), so as to explicitly account for the effect of inter-annual hydro-climatic variability. However, the computational requirements of the capacity expansion model prevent us from using multi-year profiles on a multi-regional model with hourly resolution, so we selected 2015 as a representative, or average, year. The effect of hydro-climatic variability on the hydropower profiles is illustrated in Figs. S3 and S4. As we shall see later, this variability has a marginal effect on the capacity expansion plans.
While VIC-Res provides a detailed accounting of the hydropower profiles for all dams built and operated in 2016, the capacity expansion model also needs hydropower profiles for dams planned over the period 2020–2037. To produce them, we proceeded in two steps. First, we gathered information on location and design specifications of all planned dams (data provided by the Mekong River Commission and the Electricity Generating Authority of Thailand) and then added them to the power fleet simulated by VIC-Res. Specifically, we added the dams built over the period 2017–2019 (four in the Mekong, including Xayaburi Dam, and one in the Chao Phraya) and re-run VIC-Res with the same hydro-meteorological conditions used for the 2016's fleet. To determine the hydropower profile of the remaining dams (under construction in 2020 or at different planning stages in 2020–2037), we resorted to a proximity search—given the coordinates and installed capacity of a planned dam, we identify the most similar existing dam, from which the planned dam inherits the hydropower profile (see Fig. S5). This modeling choice is compelled by the absence of detailed design specifications (e.g., rule curves, maximum surface extent) needed to simulate planned dams with VIC-Res.
Capacity expansion model
We use the open-source modeling framework urbs to generate the model for the Lower Mekong countries. The model co-optimizes capacity expansion as well as hourly dispatch of generation, transmission, and storage from a social planner perspective. The goal of the optimization is to minimize the costs of expanding and operating the energy system, which include the annualized investment costs, fuel costs, and fixed and variable operations and maintenance costs. urbs solves a linear optimization problem that is written in Python/Pyomo using Gurobi.
Major inputs are the projected hourly electricity demand, hourly generation profiles of renewable energy technologies, the existing power infrastructure (power plants, grid, storage), planned expansion projects, emissions reduction targets, and techno-economic parameters, such as investment and maintenance costs, fuels costs, and specific emissions. Major outputs include the new capacities (generation, grid, storage) and the hourly operation of the system. The model also provides the direct emissions, the total costs, and the marginal electricity costs in each region.
The model has an hourly temporal resolution and models the years 2016 (the most recent year with comprehensive data availability), 2020, 2025, 2030, 2035, and 2037, for which the energy system targets of Thailand are defined. Assumptions about existing power plants, transmission lines, and techno-economic parameters are retrieved from the reports of the power system operators of the three countries, wherever possible. Missing data are completed from global sources. An overview of the data sources is available in Table S3.
For the year 2016, the hourly values of electricity demand in each province are obtained starting from province-wise, monthly varied peak electricity demand, collected from refs. 56,57,58. The temporal disaggregation (from monthly to hourly values) is based on weekday/weekend and peak/off-peak demand profiles to account for the variation among days in a week and hours in a day. For the remaining years (i.e., 2020, 2025, 2030, 2035, and 2037), the hourly electricity demand in each province is obtained by scaling the 2016 profiles according to yearly demand growth projections for Thailand, Laos, and Cambodia59,60,61.
As explained in the previous section, the hydropower profiles are obtained from VIC-Res. Because the model runs with a daily time step, we assumed that the hydropower profiles are uniformly available to urbs throughout 24 h. This input is derived using a single (representative) year. In a sensitivity analysis, we tested the impact of dry and wet conditions on the capacity expansion plans. In Fig. S6, we show that the plans are marginally affected by the hydro-climatic variability affecting the region (Figs. S3 and S4).
The model outputs of each year (new capacities) are used as inputs for the next one, to reflect short-sightedness in investment decisions. Regarding the spatial resolution, we use the provinces of Cambodia, Laos, and Thailand (25, 18, and 77, respectively) as model regions. Power demand and renewable generation time series are assigned to each region, as well as existing and planned power plant and storage capacities. Electricity transfer between the regions is allowed within the limits of the transmission capacities between them. Imports and exports to neighboring regions in China, Malaysia, Myanmar, and Vietnam are also constrained by the transmission capacities. The model assumes full coordination between Thailand, Laos, and Cambodia in the operation of the power grid. Trade with other countries stays within current levels, i.e., we do not consider that some of the planned dams will sell electricity to other markets in the ASEAN region.
The particularity of the model resides in the high level of spatial detail. The 120 model regions are small enough to preserve transmission bottlenecks and reflect their expansion costs without jeopardizing the model solvability. Within each region, there are different classes of solar and wind sites based on their potential energy output. Each class is characterized by a time series and an upper expansion capacity limit that reflect the quality and the availability of resources in the model region. Hence, the expansion of solar and wind power is solely based on their cost-competitiveness and not on exogenous expansion quotas. The maximum installable capacities of solar PV and onshore wind are 106.6 and 21.2 GW, respectively. The former value is equivalent to the most conservative estimation from a previous study15, and the latter is obtained by applying a similar method. Whereas most power plant capacities are aggregated at the level of a model region, hydropower plants are modeled at the dam level, to avoid any information loss due to aggregation. Despite the higher computational burden, we made this modeling choice in line with the objectives of this study.
No system expansion is allowed in 2016, which is used only for calibration and validation (see Fig. S7). We compare the model performance against the projections of the Power Development Plan of Thailand 2018–203759 (see Fig. S8). We observe minor differences that we are able to explain, and we conclude that the deviations do not affect the main conclusions of this paper.
River fragmentation and regulation indices
The RFI measures the loss of longitudinal connectivity in a river basin caused by hydraulic infrastructure. The RFI is defined as follows33:
$${\rm{RFI}}=100-\left(\mathop{\sum }\limits_{i=1}^{n}\frac{{v}_{i}^{2}}{{V}^{2}}\cdot 100\right),$$
where n is the number of fragments (i.e., river network sections disconnected by dams), vi the volume of the ith fragment, and V the total river volume (for the entire network). The RFI of a pristine river is 0%, while the one of a totally disconnected river is 100%. The impact of an individual dam depends on its location as well as the location of other dams. For example, a dam splitting a pristine network into two, equally sized fragments (in terms of volume) would change the RFI from 0 to 50%, but the construction of new dams close to this one would have smaller impact on the RFI33. A second important feature of the RFI is that it implicitly accounts for the larger impact of dams on the main stem and large tributaries by using the ratio between \({v}_{i}^{2}\) and V2 as a weighting factor—since the river volume typically increases downstream due to increasing discharge and channel dimensions32. Following refs. 32,33, we assume that dams fully compromise connectivity and passability, that is, migrating fish and other species cannot move across two sections disconnected by a dam.
The RRI quantifies how strongly a river's hydrological regime is altered by dam operations. The RRI builds on the DOR, first introduced by ref. 35, which calculates, for each river reach, the discharge volume that can be withheld by a reservoir (or a group of reservoirs) located upstream. For a given reach, the DOR is defined as32:
$${\rm{DOR}}=\frac{\mathop{\sum }\nolimits_{i = 1}^{n}{s}_{i}}{D}\cdot 100,$$
where n is the number of dams upstream of the reach, si the storage capacity of the ith dam, and D the total annual discharge volume of the river reach at hand. Large values of the DOR indicate that a substantial fraction of the discharge volume can be regulated by upstream storages, thereby increasing the chances of anthropogenic effects on the natural flow regime. The RRI is then calculated by weighting the DOR value of each individual reach with its corresponding river volume and then aggregating the results for the entire basin32:
$${\rm{RRI}}=\mathop{\sum }\limits_{i=1}^{n}{{\rm{DOR}}}_{i}\cdot \frac{{v}_{i}}{V},$$
where n is the total number of reaches, DORi the DOR value of the ith reach, vi the corresponding volume, and V the total river volume. Note that for a basin affected by multi-year, or carryover, reservoirs, the RRI value can be >100%.
The river network used for the calculation of the indices is based on VIC-Res flow direction matrix, which in turn is derived from the GTOPO30 DEM. Each cell of the matrix has a spatial resolution of 1/16th of a degree (roughly 7 km at the equator), resulting in a total of approximately 100,000 km of river network and 14,214 reaches. Following refs. 32,33, we then estimate the volume of each reach on the basis of average discharge (simulated by VIC-Res over the period 1986–2016) and an approximation of channel width and depth62. The information on storage capacity and location of each dam is retrieved from the same databases used to set up VIC-Res.
The data generated in this study have been deposited in Zenodo under the accession code https://doi.org/10.5281/zenodo.4837498.
The version of urbs used for this publication is available as an open source software in Zenodo with the identifier https://doi.org/10.5281/zenodo.4837475. VIC-Res is available at https://github.com/thanhiwer/VICRes.
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This research is supported by Singapore's Ministry of Education (MoE) through the Tier 2 project "Linking water availability to hydropower supply—an engineering systems approach" (Award No. MOE2017-T2-1-143). This work was also financially supported by Singapore's National Research Foundation under its Campus for Research Excellence And Technological Enterprise (CREATE) program.
TUMCREATE Ltd., Singapore, Singapore
Kais Siala
Environmental Studies Department, University of California Santa Barbara, Santa Barbara, CA, USA
Afm Kamal Chowdhury
Pillar of Engineering Systems and Design, Singapore University of Technology and Design, Singapore, Singapore
Afm Kamal Chowdhury, Thanh Duc Dang & Stefano Galelli
Thanh Duc Dang
Stefano Galelli
K.S. and S.G. designed the research. K.S. developed the capacity expansion model and led the analysis. T.D.D. developed the hydrological model and carried out the river fragmentation analysis. K.S., T.D.D., and A.F.M.K.C. prepared all simulation scenarios used for this study. K.S. and S.G. led the preparation of the manuscript, with substantive revision by all authors.
Correspondence to Stefano Galelli.
Peer review information Nature Communications thanks R.J.P. Schmitt and the other anonymous reviewers for their contributions to the peer review of this work. Peer review reports are available.
Peer Review File
Siala, K., Chowdhury, A.K., Dang, T.D. et al. Solar energy and regional coordination as a feasible alternative to large hydropower in Southeast Asia. Nat Commun 12, 4159 (2021). https://doi.org/10.1038/s41467-021-24437-6
Designing diversified renewable energy systems to balance multisector performance
Jose M. Gonzalez
James E. Tomlinson
Julien J. Harou
Nature Sustainability (2023)
Opportunities to curb hydrological alterations via dam re-operation in the Mekong
Mauricio E. Arias
Nature Communications (Nat Commun) ISSN 2041-1723 (online) | CommonCrawl |
Evidence that nuclei contain neutrons and protons (other than nucleons appearing if a nucleus is smashed)?
This may seem like a silly question, but I believe this to be very fundamental because the Standard Model of particle physics seems based on the axiom or assumption that neutrons and protons exist "as-is" inside atomic nuclei.
Why else would the standard model require a strong nuclear force to hold everything together?
Surely there must be more evidence that this is the case besides the fact that neutrons and protons appear when a nucleus is smashed ?
EDIT: It has been a long time since I asked this question, and looking at it now (dec 5 2017) it seems like I have not mentioned an important reason for asking this question. In any case, this is what I want to add to the question now:
Take for example the Helium nucleus which is postulated to consist four separate baryons that need to be kept together with the strong force in the standard model. I would expect that in that case the total mass of a Helium nucleus would be at least that of the 4 individual baryons added together, and then I would expect to have to add more mass because of binding energy of the strong force.
Instead, the mass of the Helium nucleus is less than the four individual baryons combined. Isn't that evidence that the Helium nucleus cannot consist of four separate "as-is" baryons?
And if that is the case, what is the evidence that these, what I would call "reduced", baryons still require a strong force to be kept together? I mean, these baryons have lost some mass in the process of fusing together in a Helium nucleus which means they have changed somehow. Then I wonder, what if this change also changes the repulsive forces between them into attractive forces for example while retaining all the other particle specific characteristics? Would that not be a more elegant explanation than a strong nuclear force?
I mean, it would not change a thing in the released energy levels when fusing two protons and two neutrons together. The only thing that changes is the model. A model that seems just as compatible with the data as the model with a strong nuclear force.
particle-physics experimental-physics nuclear-physics
Leon Sprenger
Leon SprengerLeon Sprenger
$\begingroup$ All fusion and fission that occurs is only explainable if atoms consist of protons and neutrons, and you can indeed have nuclei "capture" neutrons to form a heavier isotope. $\endgroup$ – ACuriousMind♦ Mar 18 '15 at 14:41
$\begingroup$ Probably corrections on atomic spectra should come with the presence of neutrons and protons. There are measurements of the effect of the proton form factor to the alpha ray of the hydrogen, where they estimate the charge radius of the proton. $\endgroup$ – Hydro Guy Mar 18 '15 at 14:41
$\begingroup$ @DroneScientist you're right, that statement isn't true at all - actually neutrons and protons are consequences of the standard model, not axioms of it - but the essence of the question is still valid. $\endgroup$ – David Z♦ Mar 18 '15 at 14:50
$\begingroup$ Well, some properties of the nucleons are quite different in the nucleus, like how the neutron is stable. I don't know enough nuclear physics to definitively answer the question, though I don't know what sort of answer the questioner is looking for either. $\endgroup$ – zeldredge Mar 18 '15 at 14:51
$\begingroup$ RE: Your added question - the mass of a helium nucleus is less than the sum of the masses of two neutrons and two protons because of the addition of the (negative) binding energy which holds them together. Any bound state, whether it be electromagnetic, gravitational, or nuclear, exhibits this property. $\endgroup$ – J. Murray Dec 6 '17 at 0:39
Evidence that there are distinct protons and neutrons in nuclei starts with the Pauli term (pairing term) in the semiempirical mass formula of the liquid drop model.
Furthermore, all nuclei with even numbers of protons and neutrons have nuclear spin of zero. This is consisent with shells being filled with spin up and spin down pairs of nucleons, each pair resulting in net zero spin.
More generally, that experimental data are consistent with the Nuclear Shell Model is evidence that distinct protons and neutrons exist in the nucleus.
Also, the protons and neutrons are held together by exchange of pions. The exchange can result in the proton becoming a neutron and a neutron becoming a proton, so it is not that they exist entirely "as is".
See A reappraisal of the mechanism of pion exchange and its implications for the teaching of particle physics for furthur discussion of pion exchange.
DavePhDDavePhD
Short answer: We can measure their energy and momentum distribution functions in the nucleus.
We do this by interacting with them individually, either knocking them out of a nucleus left otherwise undisturbed (quasi-elastic scattering) or by exciting them to higher energy states inside the nucleus (many inelastic scattering reaction backed up by data from various capture and production reactions).
Quasi-elastic Scattering
The quasi-elastic scattering route is a reaction I know well because I did my dissertation work on Color Transparency using $A(e,e'p)$ as the probe. A well characterized electron beam is scattered from a fixed, nuclear target and the products measured with two spectrometers positioned and tuned to detect the scattered electron and proton in elastic kinematics (i.e. as if the target had been $^1\mathrm{H}$ rather than a nucleus) to within Fermi-momentum. The only thing that is difficult about the measurement is how small the cross-section gets as the squared momentum transfer $Q^2$ grows.
The measurement gives us a picture of the energy and momentum distribution of the protons inside the nucleus, and for small $A$ these results are quite consistent with results of mean-field computations (which agree with the abstract shell-model for nuclear structure). For larger $A$ they remain qualitatively consistent but the precision of agreement drops a bit.
I want to emphasize that though the reaction here is a knock-out reaction, the thing we're measuring is the energy and momentum that the knocked out nucleon had inside the nucleus.
By measuring the energies of the gamma rays released by excited nuclei and the momentum transfer to particles used to excite them we get another probe of the internal structure of the nuclei and this probe is likewise consistent with the shell model. A data set I've done some reading about here concerns the energy levels of $^{17}\mathrm{O}$, which can be probed in situ, by various inelastic collisions and by creation of short-lived, highly excited states in the reaction $^{13}\mathrm{C}(\alpha,n)^{16}\mathrm{O}$.
Here we are measuring the difference in energies between different states occupied by single nucleons. Again, the energy the nucleon had in the nucleus.
dmckee♦dmckee
Not the answer you're looking for? Browse other questions tagged particle-physics experimental-physics nuclear-physics or ask your own question.
How do we know that the nucleus isn't a quark-gluon plasma?
Why are protons and neutrons the "right" degrees of freedom of nuclei?
Are protons and neutrons affected by the Pauli Exclusion Principle?
How can the nucleus of an atom be in an excited state?
Are nucleons discrete within a nucleus?
Is an alpha particle four nucleons or dodecaquark?
What is an intuitive picture of the motion of nucleons?
How do we know that nuclear physics corresponds to low energy QCD?
What is the difference between a deuterium nucleus and a sexaquark?
Why are the antimatter compositions of neutrons and protons different? Why by about 1%? References?
Are the protons and neutrons in the nucleus arranged in any particular way?
What do we know about the interactions between the protons and neutrons in a nucleus?
binding energy of a nucleus is positive?
Why are Nuclei stable and what do neutrons change there?
Why must nuclei contain both protons and neutrons?
Position of protons and neutrons in a nucleus
Protons and Neutrons inside the Nucleus
Can a nucleus (made of neutrons) exist without an EM field?
Strong nuclear force vs electrostatic repulsion between protons | CommonCrawl |
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Finance Accounting
Compound Interest Inflation Rates Present Value of $1 Table Time Value of Money
$57170 in 2002 → 2010
$57,170 in 2002 is worth $69,295.51 in 2010
$57,170 in 2002 has the same purchasing power as $69,295.51 in 2010. Over the 8 years this is a change of $12,125.51.
The average inflation rate of the dollar between 2002 and 2010 was 2.35% per year. The cumulative price increase of the dollar over this time was 21.21%.
The value of $57,170 from 2002 to 2010
So what does this data mean? It means that the prices in 2010 are 692.96 higher than the average prices since 2002. A dollar in 2010 can buy 82.50% of what it could buy in 2002.
These inflation figures use the Bureau of Labor Statistics (BLS) consumer price index to calculate the value of $57,170 between 2002 and 2010.
The inflation rate for 2002 was 1.58%, while the inflation rate for 2010 was 1.64%. The 2010 inflation rate is lower than the average inflation rate of 2.29% per year between 2010 and 2021.
USD Inflation Since 1913
The chart below shows the inflation rate from 1913 when the Bureau of Labor Statistics' Consumer Price Index (CPI) was first established.
The Buying Power of $57,170 in 2002
We can look at the buying power equivalent for $57,170 in 2002 to see how much you would need to adjust for in order to beat inflation. For 2002 to 2010, if you started with $57,170 in 2002, you would need to have $69,295.51 in 2002 to keep up with inflation rates.
So if we are saying that $57,170 is equivalent to $69,295.51 over time, you can see the core concept of inflation in action. The "real value" of a single dollar decreases over time. It will pay for fewer items at the store than it did previously.
In the chart below you can see how the value of the dollar is worth less over 8 years.
Value of $57,170 Over Time
In the table below we can see the value of the US Dollar over time. According to the BLS, each of these amounts are equivalent in terms of what that amount could purchase at the time.
Dollar Value
$57,170.00 1.58%
$68,177.21 -0.36%
US Dollar Inflation Conversion
If you're interested to see the effect of inflation on various 1950 amounts, the table below shows how much each amount would be worth today based on the price increase of 21.21%.
Equivalent Value
$1.00 in 2002 $1.21 in 2010
$10.00 in 2002 $12.12 in 2010
$100.00 in 2002 $121.21 in 2010
$1,000.00 in 2002 $1,212.10 in 2010
$10,000.00 in 2002 $12,120.96 in 2010
$100,000.00 in 2002 $121,209.56 in 2010
$1,000,000.00 in 2002 $1,212,095.61 in 2010
Calculate Inflation Rate for $57,170 from 2002 to 2010
To calculate the inflation rate of $57,170 from 2002 to 2010, we use the following formula:
$$\dfrac{ 2002\; USD\; value \times CPI\; in\; 2010 }{ CPI\; in\; 2002 } = 2010\; USD\; value $$
We then replace the variables with the historical CPI values. The CPI in 2002 was 179.9 and 218.056 in 2010.
$$\dfrac{ \$57,170 \times 218.056 }{ 179.9 } = \text{ \$69,295.51 } $$
$57,170 in 2002 has the same purchasing power as $69,295.51 in 2010.
To work out the total inflation rate for the 8 years between 2002 and 2010, we can use a different formula:
$$ \dfrac{\text{CPI in 2010 } - \text{ CPI in 2002 } }{\text{CPI in 2002 }} \times 100 = \text{Cumulative rate for 8 years} $$
Again, we can replace those variables with the correct Consumer Price Index values to work out the cumulativate rate:
$$ \dfrac{\text{ 218.056 } - \text{ 179.9 } }{\text{ 179.9 }} \times 100 = \text{ 21.21\% } $$
Inflation Rate Definition
The inflation rate is the percentage increase in the average level of prices of a basket of selected goods over time. It indicates a decrease in the purchasing power of currency and results in an increased consumer price index (CPI). Put simply, the inflation rate is the rate at which the general prices of consumer goods increases when the currency purchase power is falling.
The most common cause of inflation is an increase in the money supply, though it can be caused by many different circumstances and events. The value of the floating currency starts to decline when it becomes abundant. What this means is that the currency is not as scarce and, as a result, not as valuable.
By comparing a list of standard products (the CPI), the change in price over time will be measured by the inflation rate. The prices of products such as milk, bread, and gas will be tracked over time after they are grouped together. Inflation shows that the money used to buy these products is not worth as much as it used to be when there is an increase in these products' prices over time.
The inflation rate is basically the rate at which money loses its value when compared to the basket of selected goods – which is a fixed set of consumer products and services that are valued on an annual basis.
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Converting Fractions to Decimals
Related calculator: Fraction to Decimal Calculator
Steps for converting fraction into a decimal:
Find such integer number, that when multiplied by the denominator will give a power of 10 (10 or 100 or 1000 etc.)
Multiply both numerator and denominator by that number (this can be done because of equivalence of fractions)
As a result, we obtain a decimal fraction, that can be easily converted into a decimal.
As can be seen we use ease of converting decimal fraction and equivalence of fractions to convert arbitrary fraction.
Example 1. Convert $$$\frac{{4}}{{5}}$$$ into decimal.
By what integer number should we multiply 5 to get 10? By 2.
Now, multiply both numerator and denominator by 2: $$$\frac{{4}}{{5}}=\frac{{{4}\cdot{\color{red}{{{2}}}}}}{{{5}\cdot{\color{red}{{{2}}}}}}=\frac{{8}}{{10}}$$$.
We obtained decimal fraction and it can be easy converted to decimal. Since $$${10}={{10}}^{{{\color{blue}{{1}}}}}$$$, we move decimal point one position to the left: $$$\frac{{8}}{{10}}={0.8}$$$.
Answer: $$$\frac{{4}}{{5}}={0.8}$$$.
Let's do slightly harder example.
By what integer number should we multiply 8 to get $$${{10}}^{{1}}={10}$$$? There is no such number.
By what integer number should we multiply 8 to get $$${{10}}^{{2}}={100}$$$? There is no such number.
By what integer number should we multiply 8 to get $$${{10}}^{{3}}={1000}$$$? By 125.
Now, multiply both numerator and denominator by 125: $$$\frac{{3}}{{8}}=\frac{{{3}\cdot{\color{red}{{{125}}}}}}{{{8}\cdot{\color{red}{{{125}}}}}}=\frac{{375}}{{1000}}$$$.
Since $$${1000}={{10}}^{{{\color{blue}{{3}}}}}$$$, we move decimal point three places to the left: $$$\frac{{375}}{{1000}}={0.375}$$$.
Answer: $$$\frac{{3}}{{8}}={0.375}$$$.
We can convert improper fractions this way as well.
Example 3. Convert $$$\frac{{26}}{{25}}$$$ into decimal.
By what integer number should we multiply 25 to get $$${{10}}^{{1}}={10}$$$? There is no such number.
By what integer number should we multiply 25 to get $$${{10}}^{{2}}={100}$$$? By 4.
Now, multiply both numerator and denominator by 4: $$$\frac{{26}}{{25}}=\frac{{{26}\cdot{\color{red}{{{4}}}}}}{{{25}\cdot{\color{red}{{{4}}}}}}=\frac{{104}}{{100}}$$$.
Since $$${100}={{10}}^{{{\color{blue}{{2}}}}}$$$, we move decimal point two places to the left: $$$\frac{{104}}{{100}}={1.04}$$$.
Answer: $$$\frac{{26}}{{25}}={1.04}$$$.
Note, that not all fractions can be converted into decimal. This occurs when we can't find such number, that when multiplied by denominator will give power of 10. This is true for prime numbers and their multiples. For example, $$$\frac{{1}}{{3}}$$$, $$$\frac{{5}}{{70}}$$$ can' t be converted.
Exercise 1. Convert $$$\frac{{7}}{{20}}$$$ into decimal.
Answer: $$$\frac{{7}}{{20}}={0.35}$$$.
Next exercise.
Answer: $$$\frac{{1}}{{80}}={0.0125}$$$.
Exercise 3. Convert $$$-\frac{{19}}{{4}}$$$ into decimal.
Answer: $$$-{4.75}$$$.
Answer: can't be converted.
We can convert mixed numbers as well!
Exercise 5. Convert mixed number $$${3}\frac{{5}}{{16}}$$$ into decimal.
Either convert $$$\frac{{5}}{{16}}$$$ into decimal and add to 3: $$${3}\frac{{5}}{{16}}={3}+{0.3125}={3.3125}$$$.
Or convert mixed number into improper fraction, and then convert result into decimal.
Answer: $$${3.3125}$$$.
last one with mixed number.
Exercise 6. Convert mixed number $$$-{2}\frac{{11}}{{20}}$$$ into decimal.
Answer: $$$-{2.55}$$$. Hint: ignore minus sign, perform conversion, and then place minus sign back.
< Converting Decimals To Fractions
Adding Decimals > | CommonCrawl |
# Digital signal processing applications
Digital signal processing (DSP) is a field of study that focuses on the analysis, manipulation, and synthesis of digital signals. Digital signals are discrete-time signals that can be processed using computers and digital systems. DSP has a wide range of applications in various fields, including telecommunications, audio and video processing, biomedical engineering, and control systems.
In this section, we will explore some of the key applications of DSP and how it is used to solve real-world problems. We will discuss the importance of DSP in modern technology and provide examples of its applications in different domains.
One of the main applications of DSP is in telecommunications. DSP techniques are used to improve the quality and reliability of communication systems. For example, DSP algorithms are used in digital modulation and demodulation techniques to transmit and receive signals over long distances without distortion or interference. DSP is also used in error correction coding, channel equalization, and noise reduction techniques to improve the performance of communication systems.
Another important application of DSP is in audio and video processing. DSP algorithms are used to compress, decompress, and enhance audio and video signals. For example, DSP is used in audio codecs to compress audio signals for storage and transmission. DSP is also used in video codecs to compress and decompress video signals for streaming and broadcasting. DSP techniques are also used in audio and video enhancement algorithms to remove noise, improve clarity, and enhance the overall quality of audio and video signals.
One example of DSP application in audio processing is noise cancellation. DSP algorithms can be used to analyze the background noise in an audio signal and generate an anti-noise signal that cancels out the noise. This is commonly used in noise-canceling headphones to provide a better listening experience in noisy environments.
DSP is also widely used in biomedical engineering. DSP techniques are used to analyze and process biomedical signals such as electrocardiograms (ECG), electroencephalograms (EEG), and medical images. DSP algorithms are used to extract useful information from these signals, such as heart rate, brain activity, and image features. This information is then used for diagnosis, monitoring, and treatment of various medical conditions.
In addition to these applications, DSP is also used in control systems, radar and sonar systems, image and video processing, and many other fields. The versatility and power of DSP make it an essential tool in modern technology.
## Exercise
Think of a real-world application where DSP is used and describe how it works.
### Solution
One example of a real-world application of DSP is in digital audio processing. DSP algorithms are used in audio equalizers to adjust the frequency response of audio signals. By analyzing the audio signal and applying appropriate filters, DSP algorithms can boost or attenuate specific frequency bands to achieve the desired sound quality. This is commonly used in audio mixing and mastering, as well as in consumer audio devices such as music players and home theater systems.
# Sampling and quantization in digital signals
Sampling refers to the process of converting a continuous-time signal into a discrete-time signal by taking samples at regular intervals. The continuous-time signal is sampled at a specific rate called the sampling rate or sampling frequency. The sampling rate determines the number of samples taken per second and affects the quality and accuracy of the digital representation of the signal.
Quantization, on the other hand, is the process of converting the continuous amplitude values of a signal into a finite set of discrete amplitude levels. In other words, quantization reduces the infinite number of possible amplitude values of a signal into a finite number of levels. The number of levels is determined by the bit depth or resolution of the quantization process.
The sampling and quantization processes introduce errors and limitations in the digital representation of a signal. The sampling process can introduce aliasing, which is the distortion or loss of information caused by undersampling or improper sampling. The quantization process can introduce quantization noise, which is the error between the original continuous signal and its quantized representation.
To mitigate these errors, it is important to choose appropriate sampling rates and quantization levels based on the characteristics of the signal and the requirements of the application. The Nyquist-Shannon sampling theorem provides guidelines for choosing the minimum sampling rate to avoid aliasing.
Let's consider an example to illustrate the concepts of sampling and quantization. Suppose we have a continuous-time audio signal that represents a musical melody. To convert this signal into a digital representation, we need to sample and quantize it.
First, we sample the audio signal at a sampling rate of 44.1 kHz, which is a common sampling rate for audio signals. This means that we take 44,100 samples per second.
Next, we quantize the samples using a 16-bit quantization process. This means that we divide the amplitude range of the signal into 65,536 discrete levels.
The resulting digital representation of the audio signal is a sequence of discrete samples, each represented by a finite number of bits. This digital representation can be stored, processed, and transmitted using digital systems.
It is important to note that the choice of sampling rate and quantization level affects the quality and fidelity of the digital representation. Higher sampling rates and quantization levels can provide a more accurate representation of the original signal but require more storage and processing resources.
## Exercise
Suppose we have a continuous-time signal with a maximum frequency of 10 kHz. According to the Nyquist-Shannon sampling theorem, what is the minimum sampling rate required to avoid aliasing?
### Solution
According to the Nyquist-Shannon sampling theorem, the minimum sampling rate required to avoid aliasing is twice the maximum frequency of the signal. In this case, the minimum sampling rate would be 20 kHz.
# The Fourier transform and its applications in signal processing
The Fourier transform is a mathematical tool used to analyze and manipulate signals in the frequency domain. It decomposes a signal into its constituent frequencies, allowing us to understand the frequency content and characteristics of the signal.
In signal processing, the Fourier transform is widely used for tasks such as filtering, compression, and modulation. It provides a powerful framework for understanding and manipulating signals in both the time and frequency domains.
The Fourier transform of a continuous-time signal is defined as:
$$X(f) = \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt$$
where $X(f)$ represents the frequency spectrum of the signal and $x(t)$ is the original signal in the time domain.
The Fourier transform can also be applied to discrete-time signals, resulting in the discrete Fourier transform (DFT). The DFT is commonly used in digital signal processing applications, as it allows us to analyze and process digital signals using fast algorithms.
Let's consider an example to illustrate the concept of the Fourier transform. Suppose we have a continuous-time signal that represents a musical tone. We can apply the Fourier transform to analyze the frequency components of this signal.
By taking the Fourier transform of the signal, we obtain its frequency spectrum. This spectrum shows the amplitudes and phases of the different frequencies present in the signal. We can use this information to identify the fundamental frequency, harmonics, and other characteristics of the musical tone.
The Fourier transform has numerous applications in signal processing. Some examples include:
1. Filtering: The Fourier transform allows us to apply frequency-based filters to remove unwanted noise or enhance specific frequency components of a signal.
2. Compression: The Fourier transform is used in lossy and lossless compression algorithms to reduce the size of digital signals while preserving important information.
3. Modulation: The Fourier transform is used in modulation techniques such as frequency modulation (FM) and amplitude modulation (AM) to encode and transmit signals over communication channels.
## Exercise
Suppose we have a continuous-time signal that consists of two sinusoidal components with frequencies 100 Hz and 200 Hz. Apply the Fourier transform to this signal and determine the frequency spectrum.
### Solution
The frequency spectrum of the signal would show peaks at 100 Hz and 200 Hz, corresponding to the frequencies of the sinusoidal components. The amplitudes and phases of these peaks would provide information about the characteristics of the signal.
# Discrete-time signals and systems
In digital signal processing, signals are often represented as discrete-time signals, which are sequences of values that are sampled at specific time intervals. Discrete-time signals are commonly used in computer systems and digital communication.
A discrete-time signal can be represented as a sequence of numbers, where each number corresponds to a specific time index. The values of the sequence can represent various types of data, such as audio samples, sensor measurements, or digital images.
Discrete-time systems are mathematical models that operate on discrete-time signals. These systems can perform various operations on the input signal, such as filtering, modulation, or transformation.
A discrete-time system can be represented by a mathematical equation or a block diagram. The input signal is processed by the system to produce an output signal. The behavior of the system is determined by its parameters and the mathematical operations it performs.
Let's consider an example of a discrete-time system that performs filtering. Suppose we have a discrete-time signal that represents an audio recording. We can design a discrete-time system that applies a low-pass filter to remove high-frequency noise from the audio signal.
By passing the input signal through the filter, we obtain the filtered output signal. The filtered signal will have reduced high-frequency components, resulting in a cleaner and smoother audio signal.
Discrete-time signals and systems are fundamental concepts in digital signal processing. They form the basis for many signal processing techniques and algorithms used in various applications, such as audio and video processing, telecommunications, and control systems.
Understanding discrete-time signals and systems is essential for analyzing and manipulating digital signals effectively. It allows us to design and implement signal processing algorithms that can enhance the quality of signals, extract useful information, or perform specific tasks.
## Exercise
Consider a discrete-time signal that represents the temperature measurements recorded every hour for a week. Design a discrete-time system that applies a moving average filter to smooth out the temperature variations and obtain a more stable representation of the temperature.
### Solution
The moving average filter can be implemented by taking the average of a window of consecutive samples. For example, we can calculate the average of the current sample and the two previous samples to obtain the filtered value.
By applying this filter to the temperature signal, we can reduce the effects of short-term temperature fluctuations and obtain a smoother representation of the overall temperature trend.
# Convolution and correlation in signal processing
Convolution and correlation are fundamental operations in signal processing. They are used to analyze and manipulate signals, and they have applications in various fields, such as image processing, audio processing, and communication systems.
Convolution is a mathematical operation that combines two signals to produce a third signal. It is often used to apply filters to signals or to perform mathematical operations on signals. The convolution of two signals is defined as the integral of the product of the two signals, with one of the signals reversed and shifted.
Correlation is a measure of similarity between two signals. It is used to determine how much one signal resembles another signal. Correlation can be used to detect patterns, find similarities, or perform signal alignment.
In signal processing, convolution and correlation can be performed on discrete-time signals. Discrete convolution and correlation are similar to their continuous counterparts, but they operate on discrete-time signals instead of continuous signals.
Discrete convolution is defined as the sum of the products of the corresponding elements of two sequences, where one of the sequences is reversed and shifted. Discrete correlation is similar to discrete convolution, but without reversing one of the sequences.
Let's consider an example to illustrate the concept of convolution and correlation. Suppose we have two discrete-time signals, signal A and signal B. Signal A represents an input signal, and signal B represents a filter kernel.
To perform convolution, we reverse signal B and shift it along signal A, multiplying the corresponding elements and summing the results. The resulting signal is the convolution of signal A and signal B.
To perform correlation, we do not reverse signal B. Instead, we shift it along signal A, multiply the corresponding elements, and sum the results. The resulting signal is the correlation of signal A and signal B.
Convolution and correlation are powerful tools in signal processing. They allow us to analyze and manipulate signals in various ways. Convolution is commonly used for filtering, while correlation is used for pattern recognition and similarity analysis.
Understanding convolution and correlation is essential for many signal processing techniques and algorithms. It enables us to design and implement systems that can extract useful information from signals, enhance signal quality, or perform specific tasks.
## Exercise
Consider two discrete-time signals, signal X and signal Y. Perform the convolution and correlation of signal X and signal Y.
Signal X: [1, 2, 3, 4]
Signal Y: [0, 1, 0.5]
### Solution
Convolution of signal X and signal Y:
[0, 1, 2.5, 4, 2]
Correlation of signal X and signal Y:
[0.5, 3, 4.5, 4]
# Digital filters and their design
Digital filters are essential components in digital signal processing. They are used to modify or extract specific components of a signal. Digital filters can be designed to remove noise, enhance desired frequencies, or perform other signal processing tasks.
There are two main types of digital filters: finite impulse response (FIR) filters and infinite impulse response (IIR) filters. FIR filters have a finite impulse response, which means that their output depends only on a finite number of input samples. IIR filters, on the other hand, have an infinite impulse response, which means that their output depends on both past and future input samples.
Designing digital filters involves determining the filter coefficients that define the filter's behavior. The filter coefficients determine how the filter modifies the input signal. There are various methods for designing digital filters, including windowing, frequency sampling, and optimization techniques.
One common design method is the windowing method, which involves selecting a window function and applying it to the desired frequency response of the filter. The window function shapes the frequency response of the filter, and the filter coefficients are obtained by applying the inverse Fourier transform to the windowed frequency response.
Another design method is the frequency sampling method, which involves specifying the desired frequency response of the filter and using the inverse Fourier transform to obtain the filter coefficients. This method allows for more control over the filter's frequency response.
Let's consider an example to illustrate the design of a digital filter using the windowing method. Suppose we want to design a low-pass filter with a cutoff frequency of 1 kHz. We can choose a window function, such as the Hamming window, and apply it to the desired frequency response of the filter.
The Hamming window is defined as:
$$w(n) = 0.54 - 0.46 \cos\left(\frac{2\pi n}{N-1}\right)$$
where $N$ is the length of the window.
By applying the Hamming window to the desired frequency response, we obtain the windowed frequency response. We can then apply the inverse Fourier transform to obtain the filter coefficients.
Designing digital filters requires a good understanding of filter specifications, such as the desired frequency response, cutoff frequencies, and filter order. It also involves knowledge of signal processing techniques and algorithms.
Digital filters have a wide range of applications in various fields, such as audio processing, image processing, and communication systems. They are used to remove noise, extract specific components of a signal, or modify the frequency content of a signal.
Understanding digital filters and their design is crucial for anyone working in the field of digital signal processing. It allows for the implementation of efficient and effective signal processing systems.
## Exercise
Design a low-pass filter using the windowing method with a cutoff frequency of 2 kHz. Choose a window function and apply it to the desired frequency response to obtain the filter coefficients.
### Solution
To design a low-pass filter with a cutoff frequency of 2 kHz using the windowing method, we can choose a window function, such as the Blackman window. The Blackman window is defined as:
$$w(n) = 0.42 - 0.5 \cos\left(\frac{2\pi n}{N-1}\right) + 0.08 \cos\left(\frac{4\pi n}{N-1}\right)$$
where $N$ is the length of the window.
By applying the Blackman window to the desired frequency response, we obtain the windowed frequency response. We can then apply the inverse Fourier transform to obtain the filter coefficients.
# Signal processing techniques: filtering, modulation, and demodulation
Signal processing techniques play a crucial role in various applications, such as communication systems, audio processing, and image processing. Three fundamental signal processing techniques are filtering, modulation, and demodulation.
Filtering is the process of modifying a signal to remove unwanted components or enhance desired components. Filters can be used to remove noise, extract specific frequencies, or shape the frequency response of a signal. Digital filters, such as FIR and IIR filters, are commonly used for filtering.
Modulation is the process of modifying a carrier signal to carry information. It involves varying one or more properties of the carrier signal, such as amplitude, frequency, or phase, in accordance with the information to be transmitted. Modulation is used in various communication systems, such as radio and television broadcasting.
Demodulation is the process of extracting the information from a modulated carrier signal. It involves reversing the modulation process to recover the original information. Demodulation is used in receivers of communication systems to recover the transmitted signals.
Filtering, modulation, and demodulation are closely related and often used together in signal processing systems. For example, in a communication system, the transmitted signal is first filtered to remove noise and unwanted frequencies. Then, the filtered signal is modulated onto a carrier signal for transmission. In the receiver, the modulated signal is demodulated to recover the transmitted information, and then filtered to remove noise and unwanted frequencies.
Understanding these signal processing techniques is essential for designing and implementing efficient and reliable signal processing systems. It requires knowledge of digital filters, modulation schemes, and demodulation techniques.
Let's consider an example to illustrate the application of filtering, modulation, and demodulation in a communication system. Suppose we have a digital audio signal that we want to transmit wirelessly. We can first filter the audio signal to remove noise and unwanted frequencies. Then, we can modulate the filtered signal onto a carrier signal using a modulation scheme, such as frequency modulation (FM). The modulated signal can be transmitted wirelessly. In the receiver, the modulated signal is demodulated to recover the audio signal, which is then filtered to remove noise and unwanted frequencies.
## Exercise
Consider a communication system that uses amplitude modulation (AM) for transmission. Describe the process of modulation and demodulation in this system.
### Solution
In a communication system that uses amplitude modulation (AM) for transmission, the process of modulation involves varying the amplitude of a carrier signal in accordance with the information to be transmitted. The carrier signal is typically a high-frequency sinusoidal signal. The modulated signal, which is the carrier signal with the amplitude variations, carries the information.
In the receiver, the modulated signal is demodulated to recover the transmitted information. The process of demodulation involves extracting the amplitude variations from the modulated signal. This can be done by multiplying the modulated signal with a synchronized carrier signal and passing the result through a low-pass filter. The output of the low-pass filter is the demodulated signal, which contains the transmitted information.
# Analog-to-digital and digital-to-analog conversion
Analog-to-digital (ADC) and digital-to-analog (DAC) conversion are fundamental processes in digital signal processing. ADC converts continuous analog signals into discrete digital signals, while DAC converts discrete digital signals back into continuous analog signals.
ADC involves two main steps: sampling and quantization. Sampling is the process of measuring the amplitude of an analog signal at regular intervals of time. The continuous analog signal is approximated by a sequence of discrete samples. The sampling rate determines the number of samples taken per second and is usually measured in samples per second or Hertz (Hz).
Quantization is the process of converting the continuous amplitude values of the samples into a finite number of discrete levels. Each sample is assigned a digital value that represents its amplitude. The number of quantization levels determines the resolution of the digital signal. A higher number of levels provides a higher resolution but requires more bits to represent each sample.
DAC performs the reverse process of ADC. It converts the discrete digital samples back into a continuous analog signal. The digital values are converted into analog voltages or currents that can be used to drive an output device, such as a speaker or a motor.
Let's consider an example to illustrate the process of ADC and DAC. Suppose we have an audio signal that we want to digitize and then convert back into an analog signal for playback. First, we use an ADC to sample the audio signal at a certain sampling rate, such as 44.1 kHz. The continuous amplitude values of the samples are quantized into a finite number of levels, such as 16 bits per sample. The resulting digital samples can be stored or processed digitally.
To convert the digital samples back into an analog signal, we use a DAC. The digital values are converted into analog voltages that can be used to drive a speaker. The analog signal can then be amplified and played back as sound.
## Exercise
Explain the process of ADC and DAC in your own words.
### Solution
ADC stands for analog-to-digital conversion. It is the process of converting continuous analog signals into discrete digital signals. This involves sampling the analog signal at regular intervals and quantizing the amplitude values of the samples into a finite number of levels.
DAC stands for digital-to-analog conversion. It is the process of converting discrete digital signals back into continuous analog signals. This involves converting the digital values into analog voltages or currents that can be used to drive an output device, such as a speaker.
ADC and DAC are essential processes in digital signal processing and are used in various applications, such as audio processing, image processing, and communication systems.
# Time and frequency domain representations of signals and systems
Signals in digital signal processing can be represented in both the time domain and the frequency domain. The time domain representation describes how a signal changes over time, while the frequency domain representation describes the different frequencies present in a signal.
In the time domain, a signal is typically represented as a sequence of samples. Each sample represents the amplitude of the signal at a specific point in time. The time domain representation is useful for analyzing how a signal changes over time and for performing operations such as filtering and convolution.
In the frequency domain, a signal is represented as a combination of different frequencies. This representation is obtained by applying the Fourier transform to the time domain signal. The Fourier transform decomposes a signal into its constituent frequencies and their corresponding amplitudes. The resulting frequency domain representation is useful for analyzing the frequency content of a signal and for performing operations such as filtering and modulation.
Let's consider an example to illustrate the time and frequency domain representations of a signal. Suppose we have a signal that represents the sound wave of a musical instrument. In the time domain, the signal would be represented as a sequence of samples that capture the amplitude of the sound wave at different points in time. This representation would allow us to analyze how the sound wave changes over time, such as the duration of each note and the overall shape of the wave.
In the frequency domain, the signal would be represented as a combination of different frequencies. For example, the signal might contain a fundamental frequency corresponding to the pitch of the musical note being played, as well as harmonics that represent higher frequencies that are multiples of the fundamental frequency. This representation would allow us to analyze the harmonic content of the sound wave and to perform operations such as filtering to remove unwanted frequencies.
## Exercise
Explain the difference between the time domain and the frequency domain representations of a signal.
### Solution
The time domain representation of a signal describes how the signal changes over time. It is obtained by representing the signal as a sequence of samples, with each sample representing the amplitude of the signal at a specific point in time. The time domain representation is useful for analyzing the temporal characteristics of a signal, such as its duration and shape.
The frequency domain representation of a signal describes the different frequencies present in the signal. It is obtained by applying the Fourier transform to the time domain signal, which decomposes the signal into its constituent frequencies and their corresponding amplitudes. The frequency domain representation is useful for analyzing the frequency content of a signal, such as the presence of specific frequencies or the harmonic content of the signal.
# Signal processing in communication systems
One of the fundamental tasks in communication systems is modulation, which involves modifying a carrier signal to encode information. Modulation techniques such as amplitude modulation (AM), frequency modulation (FM), and phase modulation (PM) are used to transmit analog signals over long distances. These modulation techniques are based on manipulating the amplitude, frequency, or phase of the carrier signal to represent the information being transmitted.
Another important aspect of signal processing in communication systems is demodulation, which involves extracting the original information from the modulated carrier signal. Demodulation techniques such as envelope detection, frequency demodulation, and phase demodulation are used to recover the original signal from the modulated carrier signal.
In addition to modulation and demodulation, signal processing techniques are also used for filtering and equalization in communication systems. Filtering is used to remove unwanted noise and interference from the received signal, while equalization is used to compensate for distortions introduced during transmission.
For example, in a wireless communication system, the received signal may be affected by multipath propagation, where the signal takes multiple paths and arrives at the receiver with different delays and amplitudes. To mitigate the effects of multipath propagation, equalization techniques such as adaptive equalization and channel estimation are used to estimate and compensate for the channel distortions.
## Exercise
Explain the difference between modulation and demodulation in communication systems.
### Solution
Modulation is the process of modifying a carrier signal to encode information. It involves manipulating the amplitude, frequency, or phase of the carrier signal to represent the information being transmitted. Modulation techniques such as AM, FM, and PM are used to transmit analog signals over long distances.
Demodulation, on the other hand, is the process of extracting the original information from the modulated carrier signal. It involves recovering the original signal from the modulated carrier signal using techniques such as envelope detection, frequency demodulation, and phase demodulation. Demodulation is necessary to decode the transmitted information and retrieve the original signal.
# Image and video processing techniques
One of the key tasks in image and video processing is image enhancement, which involves improving the visual quality of an image. Techniques such as contrast enhancement, noise reduction, and sharpening are used to enhance the details and improve the overall appearance of an image.
Another important aspect of image and video processing is image compression, which involves reducing the size of an image while preserving its visual quality. Compression techniques such as JPEG and MPEG are widely used in applications where storage or bandwidth is limited, such as digital cameras and video streaming.
In addition to enhancement and compression, image and video processing techniques are also used for image segmentation and object recognition. Image segmentation involves dividing an image into meaningful regions or objects, while object recognition involves identifying and classifying objects within an image or video.
For example, in autonomous driving systems, image processing techniques are used to detect and track objects such as pedestrians, vehicles, and traffic signs. These techniques involve segmenting the image to identify the objects of interest and then using pattern recognition algorithms to classify and track them.
## Exercise
Explain the difference between image enhancement and image compression in image processing.
### Solution
Image enhancement is the process of improving the visual quality of an image by enhancing its details, adjusting its contrast, reducing noise, and sharpening its edges. The goal of image enhancement is to make the image more visually appealing and easier to interpret.
On the other hand, image compression is the process of reducing the size of an image while preserving its visual quality. Compression techniques remove redundant or irrelevant information from the image to reduce its file size. The goal of image compression is to save storage space or reduce bandwidth requirements without significantly degrading the visual quality of the image. | Textbooks |
\begin{document}
\title{Finite Just Non-Dedekind Groups}
\author{V.K. Jain\footnote{supported by UGC, Government of India}
~and R.P. Shukla\\ Department of Mathematics, University of Allahabad \\ Allahabad (India) 211 002\\ {\bf Email:} [email protected]; [email protected]}
\date{} \maketitle
\noindent {\bf Running Title:} Finite Just Non-Dedekind Groups.
\noindent \textbf{Abstract:} A group is just non-Dedekind (JND) if it is not a Dedekind group but all of whose proper homomorphic images are Dedekind groups. The aim of the paper is to classify finite JND-groups.
\noindent \textbf{Key-words:} Monolith, Characteristically simple, Semisimple.
\noindent \textbf{MSC 2000:} 20E99, 20D99 \section{Introduction}
A group is called Dedekind if all its subgroups are normal.
By \cite{b1}, a group is Dedekind if and only if it is abelian or the direct product of a quaternion group of order $8$, an elementary abelian $2$-group and an abelian group with all its elements of odd order (one can also see its proof in \cite[5.3.7, p.143]{rob}).
Given a group theoretical property ${\cal P}$, a just non ${\cal P}$-group is a group which is not ${\cal P}$-group but all of whose proper homomorphic images are ${\cal P}$-groups; for brevity we shall call these JN${\cal P}$-groups. M.F. Newman studied just nonabelian (JNA) groups in \cite{new1,new2}. S. Franciosi and others studied solvable just nonnilpotent (JNN) groups in \cite{fs} and D.J.S. Robinson studied solvable just non-T (JNT) groups in \cite{rob73}(here the group with property T means in the group normality is a transitive relation).
The aim of this paper is to classify finite JND-groups. In Section $2$, we prove that JND-groups are monolithic group. Section 3 deals with solvable JND-groups and Section 4 shows that nonsolvable JND-groups are semisimple. Theorem \ref{th1} gives complete classification of finite semisimple JND-groups.
Let $G$ be a group. For, sets $X$, $Y$ of $G$, let $[X,Y]$ denote the subgroup of $G$ generated by $[x,y]=xyx^{-1}y^{-1},~x\in X,~y\in Y$. The derived series of $G$ is $$G=G^{(0)} \geq G^{(1)} \geq \cdots \geq G^{(n)} \geq \cdots ,$$ where $G^{(n)}=[G^{(n-1)},G^{(n-1)}]$, the commutator subgroup of $G^{(n-1)}$. The lower central series of $G$ is $$G=\gamma_1(G) \geq \gamma_2(G) \geq \cdots \geq \gamma_n(G) \geq \cdots ,$$ where $\gamma_{n+1}(G)=[\gamma_n(G),G]$. The group $G$ is called {\it solvable} of {\it derived length} $n$ (respectively {\it nilpotent} of {\it class $n$}) if $n$ is the smallest nonnegative integer such that $G^{(n)}=\{1\}$ (respectively $\gamma_{n+1}(G)=\{1\})$.
\section{Some basic properties of JND-groups} We recall that a group is called {\em{monolithic}} if it has smallest nontrivial normal subgroup, called the {\em{monolith}} of $G$. In this section, we study some basic properties of JND-groups.
\begin{proposition} \label{d1} Let $G$ be a JND-group.
Then $G$ is not contained in a direct product of Dedekind groups. \end{proposition} \begin{proof} Let $\{ H_i\}_{i\in I}$ denote a family of Dedekind groups, where $I$ is an indexing set. Assume that $G$ is contained in $H=\prod_{j\in I} H_j$. Since $G$ is nonabelian, there exists $i\in I$ such that $H_i$ is nonabelian. By the classification Theorem for nonabelian Dedekind groups \cite[5.3.7, p.143]{rob}, square of each element of a nonabelian Dedekind group is central and its commutator subgroup is isomorphic to $\mathbb{Z}_2$.
This implies that $G$ can not be simple.
Take any nontrivial element $x \in G$.
Then $x\in Z(G)$ if $x^2=1$ and $x^2 \in Z(G)$ if $x^2 \neq 1$ (for $G$ is contained in $H$).
This proves that each subgroup of $G$ contains a nontrivial central element of $H$.
Let $N$ be a nontrivial subgroup of $G$.
Let $x\in N \cap Z(H)$, $x \neq 1$.
Since $G$ is JND, $G/\langle x \rangle$ is Dedekind, so $N/\langle x \rangle \trianglelefteq G/\langle x \rangle$, which proves that $N\trianglelefteq G$.
Hence $G$ is Dedekind. \end{proof}
\begin{corollary} \label{d2} Let $G$ be a JND-group. Then $G$ is monolithic. \end{corollary} \begin{proof} If $G$ is a JNA-group, there is nothing to prove for $G^{(1)}$ will be contained in each nontrival normal subgroup of $G$. Assume that $G$ is not JNA.
Let ${\cal A}$ denote the set of all nontrivial normal subgroups of $G$.
Then $G/H$ is Dedekind for all $H\in {\cal A}$.
Further, since $G$ is not JNA, there exists $H\in {\cal A}$ such that $G/H$ is nonabelian.
Therefore by Proposition \ref{d1}, the homomorphism from $G$ to $\prod_{H\in {\cal A}} G/H$ which sends $x\in G$ to $(xH)_{H\in {\cal A}}$ is not one-one.
This proves that $\bigcap _{H\in {\cal A}} H \neq \{ 1\}$. \end{proof}
\begin{corollary} \label{da} Let $G$ be as in Corollary \ref{d2}. Assume that $G^{(2)} \neq \{1\}$.
Then the monolith of $G$ is $G^{(2)}$. \end{corollary} \begin{proof} By Corollary \ref{d2}, $G$ is monolithic.
Let $K$ denote the monolith of $G$.
Then $K \subseteq G^{(2)}$. If $G$ is JNA, then $K=G^{(1)}$ and so $K=G^{(2)}$.
If $G$ is JND but not JNA, then $G/K$ is nonabelian Dedekind.
Now by \cite[5.3.7, p.143]{rob}, the commutator subgroup $G^{(1)}/K$ of $G/K$ is of order 2.
So $G^{(2)} \subseteq K$. \end{proof}
\section{Finite solvable JND-groups}
In this section, we classify finite solvable JND-groups. Solvable JNA-groups with nontrival center is characterized in \cite{new2} and centerless solvable JNA-groups have been classified in \cite[Theorem 5.2, p.360]{new1}. So, it only remains to classify finite solvable JND-groups which are not JNA-groups.
\begin{proposition}\label{d3} Let $G$ be a JND-group. Let $Z(G)$, the center of $G$ be nontrivial.
Then $G$ is a solvable JNA-group. \end{proposition} \begin{proof} Suppose that $G$ is JND but not JNA.
By Corollary \ref{d2}, $G$ is monolithic.
Let $K$ denote the monolith of $G$.
Since every subgroup of $Z(G)$ is normal subgroup of $G$, $K$ is central subgroup of order $p$ for some prime $p$.
\noindent We claim that $p=2$.
Since $G$ is JND but not JNA, $G/K$ is nonabelian Dedekind.
By the structure Theorem for nonabelian Dedekind groups
\cite[5.3.7, p.143]{rob}, the commutator $(G/K)^{(1)}=G^{(1)}/K$ is of order 2.
Thus $|G^{(1)}|=2p$.
Let $x$ be an element of $G^{(1)}$ of order 2.
If $x\in Z(G)$, then $K=\langle x \rangle $, so $p=2$.
Assume that $x \not \in Z(G)$. Since $|G^{(1)}/K|=2$ and $x\not \in K$, so $G^{(1)}=\langle x \rangle K$. Let $g\in G$ such that $gxg^{-1} \neq x$. Then there exists a nontrivial element $h\in K$ such that $gxg^{-1}=xh$. Now since $h \in Z(G)$, $h^2=x^2h^2=(xh)^2=(gxg^{-1})^2=1$ implies that $p=2$.
Next, we show that $G$ does not contain an element of odd prime order. Assume that $x\in G$ is of odd prime order $q$. Since $\langle x \rangle K$ has a unique subgroup of order $q$ and $\langle x \rangle K \trianglelefteq G$ (for $G/K$ is Dedekind), $\langle x \rangle \trianglelefteq G$. But, then $K\subseteq \langle x \rangle $, a contradiction.
Further, since $G/K$ is a nonabelian Dedekind, by \cite[5.3.7, p.143]{rob}, $G$ does not contain any element of infinite order. Thus we have shown that $G$ is a $2$-group. Lastly, since $G/K$ is a nonabelian Dedekind, by \cite[5.3.7, p.143]{rob}, $G$ contains a nonabelian subgroup $H$ of order 16 such that $K \subseteq H$ and $H/K \cong Q_8$. But this is not possible \cite[118, p.146]{burn}. \end{proof}
\begin{lemma} \label{ps} A finite centerless solvable JND-group is a JNT-group. \end{lemma} \begin{proof} Let $G$ be a finite centerless solvable JND-group. Since a Dedekind group is also a T-group, it is sufficient to show that $G$ is not a $T$-group.
Suppose that $G$ is a $T$-group. Let $K$ denote the monolith of $G$ (Corollary \ref{d2}). Since $G$ is a finite solvable $T$-group, $K$ is a cyclic group of order $p$ for some prime $p$. Since $G/K$ is nonabelian Dedekind group, by \cite[5.3.7, p.143]{rob}$, |G^{(1)}/K|=2$. Further, since a solvable $T$-group is of derived length at most two \cite[13.4.2, p.403]{rob}, $G^{(1)}$ is abelian. Now since $G^{(1)}$ is an abelian group of order $2p$ and $G$ is a $T$-group, $p=2$. But then $K \subseteq Z(G)=\{1\}$. This is a contradiction. Therefore $G$ is a JNT-group. \end{proof}
The following example shows that there exists a solvable JND-group which is not a JNA-group.
\begin{example} Consider an elementary abelian $3$-group $A$ of order $9$. Let $\psi$ denote the homomorphism from $Q_8$ to $Aut ~A=Gl_2(3)$ defined as $\imath \mapsto \left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right) $, $\jmath \mapsto \left(\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right)$, where $Q_8= \{\pm 1, ~\pm\imath, ~\pm \jmath,~ \pm k \}$ is the quaternion group of order $8$. It is easy to check that $\psi$ is injective. Let $G=AQ$ denote the natural semidirect product of $A$ by $Q_8$. Then $G$ is a JND-group with monolith $A$. \end{example}
The following proposition classifies all finite solvable JND-groups which are not JNA-groups.
\begin{lemma}\label{ps1} A finite solvable group $G$ is JND but not JNA if and only if there exists an elementary abelian normal $p$-subgroup $A$ of $G$ for some prime $p$ which is also monolith of $G$ and a nonabelian Dedekind group $X$ of $G$ such that $A \cap X=\{1\},~ G=AX$ and the conjugation action of $X$ on $A$ is faithful and irreducible. \end{lemma} \begin{proof} Suppose that $G$ is a finite solvable JND-group but not JNA-group. By Corollary \ref{d2}, $G$ is monolithic. Let $K$ be the monolith of $G$. Then $G/K$ is a nonabelian Dedekind. Thus by
\cite[5.3.7, p.143]{rob}, $|G^{(1)}/K|=2$. Since $K$ is characteristically simple and abelian, it is an elementary abelian $p$-group of order $p^n$ for some prime $p$ \cite[3.3.15 (ii), p.87]{rob}.
Assume that $G^{(1)}$ is abelian. If $p \neq 2$, then $G^{(1)}$ contain unique element of order $2$ and so $Z(G) \neq 1$. By Proposition \ref{d3}, this is a contradiction. Thus $p=2$. Now by Proposition \ref{d3}, $G$ is not nilpotent and by Lemma \ref{ps}, $G$ is a JNT-group. So by Case 6.2 and its Subcases 6.211, 6.212, 6.22 and 6.222 in \cite[pp.202-208]{rob73}, there is no finite JNT-group which is not JNA and has a minimal normal subgroup isomorphic to an elementary abelian $2$-group.
Assume that $G^{(1)}$ is nonabelian. Then $[G^{(1)}, K] \neq 1$, for $|G^{(1)}/K|=2$. Now since $G$ is a finite nonnilpotent JNT-group and $[G^{(1)}, K] \neq 1$, by Case 6.1 of \cite[p.202]{rob73}, there is a nontrival normal subgroup $A$ of $G$, a solvable $T$-subgroup $X$ of $G$ such that $A \cap X=\{1\}, ~G=AX$ and the conjugation action of $X$ on $A$ is faithful and irreducible. Further, since $K \subseteq A$ and the conjugation action of $X$ on $A$ is irreducible, $K = A$. So $X\cong G/A = G/K$ is a nonabelian Dedekind group.
Conversely, suppose that $G=AX$, $A\cap X=\{1\}$, $X$ is a nonabelian Dedekind subgroup, $A$ is an elementary abelian $p$-group and also the monolith of $G$. Since $A$ is solvable and $G/A \cong X$ is nonabelian Dedekind and so solvable, $G$ is solvable. Further, since $A$ is the monolith of $G$ and $G/A\cong X$ is nonabelian Dedekind, $G$ is JND but not JNA. \end{proof}
The following proposition lists some more properties of finite solvable JND-groups which are not JNA-groups.
\begin{proposition}\label{c1} Let $G$, $A$ and $X$ be as in the Lemma \ref{ps1}. Then
\noindent (i) The stabilizer of any nontrival element of $A$ is trival.
\noindent (ii) $|X|$ divides $p^n-1$, in particular $p$ and $|X|$ are coprime.
\noindent (iii) $X \cong Q_8 \times A_o$, where $A_o$ is a cyclic group of odd order.
\end{proposition}
\begin{proof} Let $a \in A,~ a \neq 1$. Assume that the stabilizer $stab_X(a)$ of $a$ in $X$ is nontrival. Assume that $x \in stab_X(a),~x \neq 1$. Since $G/A$ is a Dedekind group, $\langle x \rangle A \trianglelefteq G$. Thus $Z(\langle x \rangle A)$ is a nontrival normal subgroup of $G$ (for $a \in Z(\langle x \rangle A))$ and so $A \subseteq Z(\langle x \rangle A)$, for $A$ is the monolith of $G$. But this is a conradiction, for the conjugation action of $X$ on $A$ is faithful. This proves (i). Now (ii) is implied by the class equation for the action of $X$ on $A$.
Further, by \cite[Lemma 1, p.185]{rob73}, there is an extension field $E$ of $\mathbb{Z}_p$ such that $Z(X) \cong Y \leq E^{\star}$ and $E=\mathbb{Z}_p(Y)$, where $E^{\star}$ denote the multiplicative group of $E$. Clearly $E$ is a finite field , so $E^{\star}$ is a cyclic group. This implies $X\cong Q_8 \times A_o$, where $A_o$ is a cyclic group of odd order \cite[5.3.7, p.143]{rob}. This proves (iii). \end{proof}
\section{Finite nonsolvable JND-groups}
Recall that a group is {\em{semisimple}} \cite[p.89]{rob} if its maximal solvable normal subgroup is trivial. Also a maximal normal completely reducible subgroup is called the {\em{CR-radical}} \cite[p.89]{rob}.
\begin{proposition}\label{d4} Let $G$ be a finite nonsolvable JND-group. Then $G$ is a semisimple group. \end{proposition} \begin{proof} Assume that $G$ has a nontrival normal solvable subgroup $N$. Then $G/N$ is a Dedekind group. Hence by \cite[5.3.7, p.143]{rob}, $G/N$ is solvable. But then $G$ is solvable, a contradiction. \end{proof}
Now we fix some notations for the rest of the section. For a group $G$, we denote {\em{Inn $G$}} for the inner automorphism subgroup of {\em{Aut $G$}}, the automorphism group of $G$ and {\em{Out $G$}} for the outer automorphism group of $G$. Let $H$ denote a finite nonabelian simple group. Consider the semidirect product $\underbrace{(Aut~H \times \ldots \times Aut~H)}_{r~copies}\rtimes S_r$ and $\underbrace{(Out~H \times \ldots \times Out~H)}_{r~copies} \rtimes S_r$, where $S_r$ acts on $\underbrace{(Aut~H \times \ldots \times Aut~H)}_{r~copies}$ as well as on $\underbrace{(Out~H \times \ldots \times Out~H)}_{r~copies}$ by permuting the coordinates. Let $$ \tilde{\nu}: \underbrace{(Aut~H \times \ldots \times Aut~H)}_{r~copies}\rtimes S_r \longrightarrow \underbrace{(Out~H \times \ldots \times Out~H)}_{r~copies} \rtimes S_r$$ be the homomorphism
defined by $\tilde{\nu}(x_1, x_2, \ldots , x_r, x_{r+1})=(x_1Inn~H, \ldots , $ $x_rInn~H,x_{r+1})$. We denote by $\beta$ the projection of $\underbrace{(Out~H \times \ldots \times Out~H)}_{r~copies} \rtimes S_r$ onto the $(r+1)$-th factor $S_r$, which is obviously a homomorphism.
\begin{lemma} \label{ll} Let $H$ be a finite nonabelian simple group. Then $Out~H$ does not contain a subgroup
isomorphic to the quaternion group $Q_8$ of order $8$. \end{lemma} \begin{proof}
If $H$ is isomorphic to either alternating group $Alt_n$ of degree $n$ or to a Sporadic simple group, then $|Out(H)| \leq 4$ (see \cite[2.17, 2.19, p.299]{suz} and \cite[Table 2.1C, p.20]{lie}), so the Lemma follows in this case. If $H$ is isomorphic to a finite simple group of Lie type, then the Lemma follows by \cite[Theorem 2.5.12, p.58]{gor}. \end{proof}
\begin{corollary} \label{l} Let $H$ be a finite nonabelian simple group. Then for any $m \in \mathbb{N}$,
${\underbrace{Out~H \times \ldots \times Out~H}_{m ~copies}}$ does not contain a subgroup
isomorphic to the quaternion group $Q_8$ of order $8$. \end{corollary}
\begin{proof} Assume ~that~
$\alpha$ ~is~ an ~injective~ homomorphism ~from ~$Q_8$~ to ~
${\underbrace{Out~H \times \ldots \times Out~H}_{m ~copies}}$.
Let $u=(x_1,x_2, \ldots , x_m)$ denote an element of $\alpha (Q_8)$ of order $4$.
Then there is $t$ ($1\leq t\leq m$) such that
$x_t$ is of order $4$. Let $p_t$ denote the projection
of ${\underbrace{Out~H \times \ldots \times Out~H}_{m ~copies}}$
onto the $t$-th factor.
Then $(p_t\circ \alpha)(Q_8)$ is a subgroup of $Out~H$ which contains an element of order $4$.
Since a homomorphic image of $Q_8$ containing an element of order $4$ is isomorphic to $Q_8$,
$ (p_t\circ \alpha)(Q_8)\cong Q_8$. By Lemma \ref{ll}, this is impossible. \end{proof}
\begin{theorem}\label{th1}
A finite nonsolvable group $G$ is JND-group if and only if there exists a finite nonabelian simple group $H$, a natural number $r$ and a Dedekind group $D \subseteq \underbrace{(Out~H \times \ldots \times Out~H)}_{r~copies} \rtimes S_r$ such that
\noindent (i) the usual action of $\beta (D)$ on the set $\{1,2, \ldots ,r \}$ is free and transitive,
\noindent and
\noindent (ii) $G \cong \tilde{\nu}^{-1}(D)$,
\noindent where all the notations have meaning described as after the Proposition \ref{d4}
\noindent Further, $G$ is JND but not JNA if and only if $D$ is a nonabelian Dedekind group and $r$ is even. \end{theorem} \begin{proof} Suppose that $G$ is a nonsolvable JND-group. By Corollary \ref{d2}, $G$ is monolithic. Let $K$ denote the monolith of $G$. Since $G$ is nonsolvable and $K$ is characteristically simple, by \cite[3.3.15 (ii), p.87]{rob}, there exists a finite nonabelian simple group $H$ and a natural number $r$ such that $K \cong \underbrace{(H \times \ldots \times H)}_{r~copies}$.
By Proposition \ref{d4}, $G$ is semisimple. We show that $K$ is the CR-radical of $G$. Let $N$ be the CR-radical of $G$ containing $K$. Then $N$ is semisimple \cite[Lemma, p.205]{kur}. Assume that $N \neq K$. Then there exists nontrival completely reducible normal subgroup $L$ of $N$ which is complement of $K$ in $N$. Now since $L \cong N/K$ and $G/K$ is a Dedekind group, $L$ is solvable \cite[5.3.7, p.143]{rob}. Further, since nontrival normal subgroup of a semisimple group is semisimple \cite[Lemma, p.205]{kur}, $L$ is also semisimple. This is a contradiction.
Now by \cite[3.3.18 (i), p.89]{rob}, there exists ~$G^{\star} ~\cong ~~G$ ~such ~that ~$\underbrace{(Inn~H \times \ldots \times Inn~H)}_{r~copies} \leq G^{\star} \leq \underbrace{(Aut~H \times \ldots \times Aut~H)}_{r~copies}\rtimes S_r$.
We identify $G$ with $G^{\star}$ and $H$ with $Inn~H$. Thus $K$ is identified with $\underbrace{(Inn~H \times \ldots \times Inn~H)}_{r~copies}$. Take $D=G/K \subseteq \underbrace{(Out~H \times \ldots \times Out~H)}_{r~copies}\rtimes S_r$. Then $D$ is a Dedekind group and $G \cong \tilde{\nu}^{-1}(D)$. This proves (ii).
Next, we claim that $\beta(D)$ acts transitively on the set of symbols $\{1,2, \ldots ,$ $r\}$. Let $O$ denote an orbit of the natural action of $\beta(D)$ on $\{ 1,2, \ldots , r\}$. Consider the subgroup $M_O=\{(x_1,x_2, \ldots ,x_r,1)| x_i \in Inn~H ~\text{and}~ x_i=1 ~\text{if} ~i \not \in O \}$ of $\underbrace{(Aut~H \times \ldots \times Aut~H)}_{r~copies}\rtimes S_r$. It is easy to observe that for each element of $G$, the $(r+1)$-th coordinate is an element of $\beta(D)$. This implies that $M_O$ is a normal subgroup of $G$ contained in $K$. But $K$ is the monolith of $G$, so $M_O=K$. This proves that $O=\{1,2,\ldots , r\}$.
Now, we show that the action of $Z(\beta (D))$ on $\{1,2, \ldots , r\}$ is free. Suppose that an element $u$ of $Z(\beta (D))$ fixes a symbol $a$ under the natural action of $\beta(D)$ on $\{1,2, \ldots, r\}$. Clearly $u$ fixes each element of the orbit $\beta(D).a$ of $a$ which is $\{1,2, \ldots , r\}$. This implies $u=1$. So, no nontrivial element of $Z(\beta (D))$ will fix any symbol in $\{1,2, \ldots ,r\}$.
If $D$ is abelian, then $Z(\beta(D))=\beta(D)$ and so the action of $\beta(D)$ is free. If $D$ is nonabelian Dedekind group, then by the structure Theorem for Dedekind groups \cite[5.3.7, p.143]{rob}, there exists a nonnegative integer $t$ and an abelian group $A_{o}$ of odd order such that we can write $D = Q_8 \times (\mathbb{Z}_2)^t \times A_{o}$. Thus for any $x \in \beta(D)$, either $x\in Z(\beta (D))$ or $1 \neq x^2 \in Z(\beta (D))$. This implies that a noncentral element $x$ also does not fix any symbol of set $\{1,2, \ldots ,r\}$ (for then $1 \neq x^2 \in Z(\beta (D))$ will fix that symbol). Thus action of $\beta (D)$ on $\{1,2, \ldots ,r\}$ is free and transitive. In particular $r=|\beta (D)|$. This proves (i).
Now, assume that $G$ is JND but not JNA. Then by Corollary \ref{l}, $\beta (Q_8) \neq 1$ and so $2$ divides $|\beta (D)|=r$
Conversely,~ suppose ~that ~there ~exists ~a ~Dedekind ~group
$D \subseteq {\underbrace{(Out~H \times \ldots \times Out~H)}_{r~copies}}\rtimes S_r$ for some $r \in \mathbb{N}$ and a nonabelian finite simple group $H$ such that, the usual action of $\beta (D)$ on the set $\{1,2, \ldots ,r\}$ is free and transitive. Let $G=\tilde{\nu}^{-1}(D)$. We will show that $G$ is a JND-group. By \cite[3.3.18 (ii), p.89]{rob}, ~$G$ ~is ~semisimple ~with ~CR-radical ~$K~=~ {\underbrace{(Inn~H ~\times ~\ldots ~\times ~Inn~H)}_{r~copies}}$~ and~ ${\underbrace{(Inn~H \times \ldots \times Inn~H)}_{r~copies}} \leq G \leq {\underbrace{(Aut~H \times \ldots \times Aut~H)}_{r~copies}}\rtimes S_r$. We will show that $K$ is the monolith of $G$. Since $K=K^{(1)}$, $K$ is contained in all terms of the derived series of $G$. Further, since $G$ is semisimple, there is smallest nonnegative integer $n$ such that $G^{(n)}=G^{(n+i)}$ for all $i \in \mathbb{N}$. This implies that $G^{(n)}/K$ is a perfect group. But since $G^{(n)}/K$ is Dedekind and so solvable \cite[5.3.7, p.143]{rob}, $G^{(n)}=K$. Let $N$ be a nontrival normal subgroup of $G$. Since a nontrivial normal subgroup of a semisimple group is semisimple \cite[Lemma, p.205]{kur} and a semisimple group has trivial center, $N\cap K \neq \{1\}$. By \cite[Theorem 2, p.156]{mil}, $N\cap K =\underbrace{N_1 \times N_2 \times \ldots \times N_r}_{r~copies}$, where $N_i \trianglelefteq Inn~H$ and at least one $N_i \neq 1$. Now since $N_i=Inn~H$ and $\beta(D)$ acts transitively on $\underbrace{Inn~H \times \ldots \times Inn~H}_{r~copies}$, so $N\cap K = \underbrace{Inn~H \times \ldots \times Inn~H}_{r~copies} =K$, that is $K \subseteq N$. This proves that $K$ is the monolith of $G$. Thus $G$ is JND-group. Further, if $D$ is nonabelian Dedekind group, then $G$ is JND but not JNA.
\end{proof} \begin{remark} Let $G$ be finite just nonsolvable (JNS) (respectively just nonnilpotent (JNN)) group. Let $n$ be the smallest nonnegative integer such that $G^{(n)}=G^{(n+k)}$ (respectively $\gamma _n(G) =\gamma _{(n+k)}(G)$) for all $k \in \mathbb{N}$. Then it is easy to see that $G^{(n)}$ (respectively $\gamma_{(n+1)}(G)$) is the monolith of $G$.
The idea of the proof of the above theorem can be used to show that:
\noindent A finite nonsolvable group $G$ is JNS-group (respectively JNN-group) if and only if there exists a finite nonabelian simple group $H$, a natural number $r$ and a solvable (respectively nilpotent) group $D \subseteq \underbrace{(Out~H \times \ldots \times Out~H)}_{r~copies} \rtimes S_r$ such that
\noindent (i) the usual action of $\beta (D)$ on the set $\{1,2, \ldots ,r \}$ is transitive,
\noindent and
\noindent (ii) $G \cong \tilde{\nu}^{-1}(D)$,
\noindent where all the notations have meaning described as after the Proposition \ref{d4}.
\end{remark}
\noindent \textbf{Acknowledgement:} We thank Professor Ramji Lal for suggesting the problem and for several stimulating discussions.
\end{document} | arXiv |
Artamonov, Viacheslav Alexandrovich
Statistics Math-Net.Ru
Total publications: 56
Scientific articles: 48
Presentations: 2
This page: 4843
Abstract pages: 12517
Full texts: 4885
References: 1039
Doctor of physico-mathematical sciences (1990)
Speciality: 01.01.06 (Mathematical logic, algebra, and number theory)
Birth date: 2.10.1946
Fax: +7 (495) 939 20 90
Website: http://www.math.msu.su/department/algebra/staff/artamon
Keywords: Hopf algebras, universal algebra, quasicrystals.
UDC: 512, 512.535, 512.544, 512.55, 512.552.7, 512.57, 512.667, 512.667.7, 512.8, 512.817, 519.4, 519.443, 519.9, 519.41/47, 519.48, 512.546.3, 51-72, 512.94.3
MSC: 16D, 16E, 16S, 16W, 20F, 08, 17B99
Hopf algebras, quantum groups, noncommutative algebraic geometry, noncommutative algebra, universal algebra, symmetries of (quasi)crystals.
I was graduated from Moscow State University in July 1968. I was a postgraduate in the Department of Algebra, Faculty of Mechanics and Mathematics of Moscow State University. My scientific adviser was Prof. A. G. Kurosh. In December 1970 I received a position in Faculty of Mechanics and Mathematics of Moscow State University. In April 1971 I received Ph.D. from Faculty of Mechanics and Mathematics. In 1976–1996 I was an associate professor (docent) in Faculty of Mechanics and Mathematics Moscow State University. In 1990 I received degree of a Doctor of sciences from Faculty of Mechanics and Mathematics Moscow State University. Since 1996 I am a full professor.
I am an author of more than 120 papers in algebra. I am a member of Editorial Board of the international Journal "Communications in Algebra" published by Taylor & Francis, Inc., (USA) of the journal "Discussiones Math., General algebra and Appl." published by Technical Univ. Press, Zieleno Gora, Poland, of the journal "Algebra and Discrete Mathematics" зublished by Kiev and Lugansk Universities, Ukraine, of the journal "Fundamental and Applied Mathematics" published in Moscow State University, of the journal "Tchebyshev Sbornik" published by Tula pedagogical university and of "Quasigroups mand related topics" published by Moldavian Academy of Sci.
Main publications:
Pointed Hopf algebras acting on quantum polynomials. J. Algebra, 2003, 259, N 2, 323–352.
Quantum Serre's conjecture. Uspehi mat. nauk, 1998, 53, N 4, 3–77.
On semisimple finite dimensional Hopf algebras, Mat. Sbornik, 198, N 9, 2007, 3–28.
V. Artamonov, S. Sanchez, A mathematical classification for symmetries in 2-dimensional quasicrystals, Lecture Series on Computer and Computational Sciences, Volume 7, 2006, 32–35.
"Actions of pointed Hopf algebras on quantum torus" in Proceedings of the "Ferrara Algebra Workshop" jointly with the "Workshop on Hopf Algebras, Swansea", a special issue of the "Annali dell'Universita' di Ferrara, sez. VII, Scienze Matematiche, Vol. 51, 29–60, 2005.
http://www.mathnet.ru/eng/person8619
List of publications on Google Scholar
http://zbmath.org/authors/?q=ai:artamonov.vyacheslav-a
https://mathscinet.ams.org/mathscinet/MRAuthorID/196726
Full list of publications: Download file (131 kB)
Publications in Math-Net.Ru
1. V. Artamonov, O. Artemovych, Yu. Bahturin, T. Banakh, L. Bartholdi, O. Bezushchak, Ie. Bondarenko, T. Ceccherini-Silberstein, Yu. Drozd, V. Futorny, F. de Giovanni, R. Grigorchuk, W. Holubowski, S. Ivanov, A. Kashu, O. Kharlampovich, E. Khukhro, V. Kirichenko, L. Kurdachenko, Ya. Lavrenyuk, O. Macedońska, A. Myasnikov, T. Nagnibeda, V. Nekrashevych, A. Oliynyk, B. Oliynyk, A. Olshanskii, M. Perestyuk, A. Petravchuk, I. Protasov, N. Romanovskii, D. Savchuk, M. Sapir, M. Semko, I. Shestakov, S. Sidki, B. Steinberg, I. Subbotin, Ya. Sysak, V. Vyshensky, E. Zelmanov, A. Zhuchok, Yu. Zhuchok, "Vitaliy Sushchansky (11.11.1946 – 29.10.2016)", Algebra Discrete Math., 23:2 (2017), C–F
2. V. A. Artamonov, "Categories of modules over semisimple finite-dimensional Hopf algebras", Fundam. Prikl. Mat., 21:5 (2016), 5–18
3. V. A. Artamonov, A. V. Klimakov, A. A. Mikhalev, A. V. Mikhalev, "Primitive and almost primitive elements of Schreier varieties", Fundam. Prikl. Mat., 21:2 (2016), 3–35
4. V. A. Artamonov, "Semisimple Hopf algebras with restrictions on irreducible non-one-diemnsional modules", Algebra i Analiz, 26:2 (2014), 21–44 ; St. Petersburg Math. J., 26:2 (2015), 207–223
5. V. A. Artamonov, "Semisimple Hopf algebras", Chebyshevskii Sb., 15:1 (2014), 19–31
6. V. A. Artamonov, "Properties of Semisimple Hopf Algebras", Mat. Zametki, 96:3 (2014), 325–332 ; Math. Notes, 96:3 (2014), 309–316
7. V. A. Artamonov, S. Sánchez, "On finite symmetry groups of some models of three-dimensional quasicrystals", Sibirsk. Mat. Zh., 52:6 (2011), 1221–1233 ; Siberian Math. J., 52:6 (2011), 969–979
8. V. A. Artamonov, S. Sanchez, "On Symmetry Groups of Quasicrystals", Mat. Zametki, 87:3 (2010), 323–329 ; Math. Notes, 87:3 (2010), 303–308
9. V. A. Artamonov, "Semisimple finite-dimensional Hopf algebras", Mat. Sb., 198:9 (2007), 3–28 ; Sb. Math., 198:9 (2007), 1221–1245
10. V. A. Artamonov, "Quasicrystals and their symmetries", Fundam. Prikl. Mat., 10:3 (2004), 3–10 ; J. Math. Sci., 139:4 (2006), 6657–6662
11. V. A. Artamonov, "Quantum polynomial algebras", Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 26 (2002), 5–34 ; J. Math. Sci. (New York), 87:3 (1997), 3441–3462
12. V. A. Artamonov, "Universal commutative Hopf algebras coacting on quantum polynomials", Fundam. Prikl. Mat., 6:3 (2000), 637–642
13. V. A. Artamonov, "Actions of pointlike Hopf algebras on quantum polynomials", Uspekhi Mat. Nauk, 55:6(336) (2000), 125–126 ; Russian Math. Surveys, 55:6 (2000), 1137–1138
14. V. A. Artamonov, "Automorphisms of the skew field of rational quantum functions", Mat. Sb., 191:12 (2000), 3–26 ; Sb. Math., 191:12 (2000), 1749–1771
15. V. A. Artamonov, "Transitivity of action on modular vectors", Fundam. Prikl. Mat., 5:3 (1999), 765–773
16. V. A. Artamonov, "General quantum polynomials: irreducible modules and Morita equivalence", Izv. RAN. Ser. Mat., 63:5 (1999), 3–36 ; Izv. Math., 63:5 (1999), 847–880
17. V. A. Artamonov, "The skew field of rational quantum functions", Uspekhi Mat. Nauk, 54:4(328) (1999), 151–152 ; Russian Math. Surveys, 54:4 (1999), 825–827
18. V. A. Artamonov, "Serre's quantum problem", Uspekhi Mat. Nauk, 53:4(322) (1998), 3–76 ; Russian Math. Surveys, 53:4 (1998), 657–730
19. V. A. Artamonov, "Periodic modules over general quantum Laurent polynomials", Mat. Zametki, 61:1 (1997), 10–17 ; Math. Notes, 61:1 (1997), 9–15
20. V. A. Artamonov, "The Magnus representation in congruence modular varieties", Sibirsk. Mat. Zh., 38:5 (1997), 978–995 ; Siberian Math. J., 38:5 (1997), 842–859
21. V. A. Artamonov, "Modules over quantum polynomials", Mat. Zametki, 59:4 (1996), 497–503 ; Math. Notes, 59:4 (1996), 356–360
22. V. A. Artamonov, "Irreducible modules over quantum polynomials", Uspekhi Mat. Nauk, 51:6(312) (1996), 189–190 ; Russian Math. Surveys, 51:6 (1996), 1191–1192
23. V. A. Artamonov, "Automorphisms of decomposable projective modules", Fundam. Prikl. Mat., 1:1 (1995), 63–69
24. V. A. Artamonov, "Construction of modules over quantum polynomials", Uspekhi Mat. Nauk, 50:6(306) (1995), 167–168 ; Russian Math. Surveys, 50:6 (1995), 1256–1257
25. V. A. Artamonov, V. V. Yashchenko, "Multibasic algebras in public key distribution systems", Uspekhi Mat. Nauk, 49:4(298) (1994), 149–150 ; Russian Math. Surveys, 49:4 (1994), 145–146
26. V. A. Artamonov, "Projective modules over quantum polynomial algebras", Mat. Sb., 185:7 (1994), 3–12 ; Russian Acad. Sci. Sb. Math., 82:2 (1995), 261–269
27. V. A. Artamonov, "Free algebras of Mal'tsev products of varieties", Uspekhi Mat. Nauk, 48:2(290) (1993), 171–172 ; Russian Math. Surveys, 48:2 (1993), 167–169
28. V. A. Artamonov, "Engelian Hopf algebras and the quantum analogue of Serre's conjecture", Uspekhi Mat. Nauk, 47:5(287) (1992), 165–166 ; Russian Math. Surveys, 47:5 (1992), 175–176
29. V. A. Artamonov, "The structure of Hopf algebras", Itogi Nauki i Tekhniki. Ser. Algebra. Topol. Geom., 29 (1991), 3–63 ; J. Math. Sci., 71:2 (1994), 2289–2328
30. V. A. Artamonov, "Universal algebras", Itogi Nauki i Tekhniki. Ser. Algebra. Topol. Geom., 27 (1989), 45–124 ; J. Soviet Math., 57:2 (1991), 2959–3009
31. V. A. Artamonov, A. A. Bovdi, "Integral group rings: groups of invertible elements and classical $K$-theory", Itogi Nauki i Tekhniki. Ser. Algebra. Topol. Geom., 27 (1989), 3–43 ; J. Soviet Math., 57:2 (1991), 2931–2958
32. V. A. Artamonov, "Projective modules over universal enveloping algebras", Izv. Akad. Nauk SSSR Ser. Mat., 48:6 (1984), 1123–1137 ; Math. USSR-Izv., 25:3 (1985), 429–441
33. V. A. Artamonov, "Projective nonfree modules over group rings of solvable groups", Mat. Sb. (N.S.), 116(158):2(10) (1981), 232–244 ; Math. USSR-Sb., 44:2 (1983), 207–217
34. V. A. Artamonov, "Projective metabelian groups and Lie algebras", Izv. Akad. Nauk SSSR Ser. Mat., 42:2 (1978), 226–236 ; Math. USSR-Izv., 12:2 (1978), 213–223
35. V. A. Artamonov, "Lattices of varieties of linear algebras", Uspekhi Mat. Nauk, 33:2(200) (1978), 135–167 ; Russian Math. Surveys, 33:2 (1978), 155–193
36. V. A. Artamonov, "Projective metabelian groups and Lie algebras", Uspekhi Mat. Nauk, 32:3(195) (1977), 166
37. V. A. Artamonov, "On algebras without proper subalgebras", Mat. Sb. (N.S.), 104(146):3(11) (1977), 428–459 ; Math. USSR-Sb., 33:3 (1977), 375–401
38. V. A. Artamonov, "Universal algebras", Itogi Nauki i Tekhniki. Ser. Algebra. Topol. Geom., 14 (1976), 191–248
39. V. A. Artamonov, "Orbits of the group $\mathbf{GL}(r,k[X_1,…,X_n])$", Izv. Akad. Nauk SSSR Ser. Mat., 38:3 (1974), 484–494 ; Math. USSR-Izv., 8:3 (1974), 490–500
40. V. A. Artamonov, "Chain varieties of linear algebras", Tr. Mosk. Mat. Obs., 29 (1973), 51–78
41. V. A. Artamonov, "Projective metabelian Lie algebras of finite rank", Izv. Akad. Nauk SSSR Ser. Mat., 36:3 (1972), 510–522 ; Math. USSR-Izv., 6:3 (1972), 504–517
42. M. S. Burgin, V. A. Artamonov, "Some properties of subalgebras in varieties of linear $\Omega$-albebras", Mat. Sb. (N.S.), 87(129):1 (1972), 67–82 ; Math. USSR-Sb., 16:1 (1972), 69–85
43. V. A. Artamonov, "Semisimple varieties of multi-operator algebras. II", Izv. Vyssh. Uchebn. Zaved. Mat., 1971, 12, 15–21
44. V. A. Artamonov, "Semisimple varieties of multi-operator algebras. I", Izv. Vyssh. Uchebn. Zaved. Mat., 1971, 11, 3–10
45. V. A. Artamonov, "Free $n$-groups", Mat. Zametki, 8:4 (1970), 499–507 ; Math. Notes, 8:4 (1970), 750–754
46. V. A. Artamonov, "Multioperator algebras and clones of polylinear operators", Uspekhi Mat. Nauk, 24:1(145) (1969), 47–59 ; Russian Math. Surveys, 24:1 (1969), 45–57
47. V. A. Artamonov, "Isomorphisms of free decompositions of $\Gamma$-operator groups with a regular group of operators $\Gamma$", Mat. Zametki, 4:3 (1968), 355–360 ; Math. Notes, 4:3 (1968), 705–707
48. V. A. Artamonov, "Admissible subgroups of a free product of $\Gamma$-operator groups with regular group of operators $\Gamma$", Mat. Sb. (N.S.), 76(118):4 (1968), 605–619 ; Math. USSR-Sb., 5:4 (1968), 571–584
49. V. A. Artamonov, I. B. Kozhukhov, V. N. Chubarikov, N. M. Dobrovolsky, N. N. Dobrovolsky, "The 80th anniversary of professor Vladimir Konstantinovich Kartashov", Chebyshevskii Sb., 18:2 (2017), 298–304
50. V. A. Artamonov, V. N. Latyshev, O. A. Pikhtilkova, I. N. Balaba, N. M. Dobrovol'skii, I. V. Dobrynina, I. Yu. Rebrova, "Pikhtilkov Sergrey Alekseevich. The life and scientific activity", Chebyshevskii Sb., 17:1 (2016), 299–309
51. V. N. Latyshev, A. V. Mikhalev, A. L. Shmel'kin, E. S. Golod, V. A. Artamonov, "85 years since the birth of Alexei Ivanovich Kostrikin", Chebyshevskii Sb., 15:3 (2014), 141–145
52. A. V. Mikhalev, V. A. Artamonov, "A word about L. A. Skornyakov — man, scientist and teacher (memoirs of the colleagues and students)", Chebyshevskii Sb., 15:2 (2014), 143–147
53. V. I. Andriychuk, V. A. Artamonov, V. M. Babych, O. O. Bezushchak, V. M. Bondarenko, M. A. Dokuchaev, Yu. A. Drozd, V. M. Futornyi, M. F. Gorodniy, R. I. Grigorchuk, N. M. Gubareni, P. M. Gudivok, A. I. Kashu, M. Komarnytskyi, L. A. Kurdachenko, F. M. Lyman, V. V. Lyubashenko, V. S. Mazorchuk, V. V. Nekrashevych, B. V. Novikov, A. S. Oliynyk, A. Yu. Ol'shanskii, S. A. Ovsienko, M. O. Perestyuk, A. P. Petravchuk, V. M. Petrychkovych, A. M. Samoilenko, M. M. Semko, V. V. Sergeichuk, V. V. Sharko, L. A. Shemetkov, I. P. Shestakov, V. I. Sushchansky, P. D. Varbanets, E. I. Zel'manov, V. N. Zhuravlyov, "Vladimir Kirichenko (on his 65th birthday)", Algebra Discrete Math., 2007, 4, E–H
54. V. A. Artamonov, O. E. Bezushchak, Yu. A. Drozd, V. M. Futornyi, R. I. Grigorchuk, A. I. Kashu, V. V. Kirichenko, M. Komarnytskyi, L. A. Kurdachenko, Yu. A. Lavrenyuk, V. V. Nekrashevych, B. V. Novikov, A. S. Oliynyk, A. Yu. Ol'shanskii, M. O. Perestyuk, M. Sapir, L. A. Shemetkov, I. P. Shestakov, E. I. Zel'manov, "Vitaliy Sushchansky (on his 60th birthday)", Algebra Discrete Math., 2007, 2, E–F
55. V. A. Artamonov, Yu. A. Drozd, Ya. M. Dymarsky, V. M. Futornyi, R. I. Grigorchuk, A. I. Kashu, V. V. Kirichenko, M. Ya Komarnytskyi, L. A. Kurdachenko, V. V. Lyubashenko, I. A. Mikhailova, B. V. Novikov, A. P. Petravchuk, A. B. Popov, L. A. Shemetkov, I. P. Shestakov, V. I. Sushchansky, P. D. Varbanets, A. V. Zhuchok, "V. M. Usenko (1951–2006)", Algebra Discrete Math., 2006, 2, G–K
56. V. A. Artamonov, "Letter to the Editor", Uspekhi Mat. Nauk, 34:2(206) (1979), 251
Presentations in Math-Net.Ru
1. On applications of polynomially complete finite quasigroups
V. A. Artamonov
XV International Conference «Algebra, Number Theory and Discrete Geometry: modern problems and applications», dedicated to the centenary of the birth of the Doctor of Physical and Mathematical Sciences, Professor of the Moscow State University Korobov Nikolai Mikhailovich
2. Полиномиально полные квазигруппы
Research Seminar of the Department of Higher Algebra MSU
Lomonosov Moscow State University, Faculty of Mechanics and Mathematics | CommonCrawl |
\begin{document}
\title[Geometric inequalities for static convex domains in hyperbolic space]{Geometric inequalities for static convex domains in hyperbolic space}
\author{Yingxiang Hu} \address{School of Mathematics, Beihang University, Beijing 100191, P.R. China} \email{\href{mailto:[email protected]}{[email protected]}}
\author{Haizhong Li} \address{Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China} \email{\href{mailto:[email protected]}{[email protected]}}
\date{\today} \keywords{Locally constrained curvature flow, static convex, geometric inequality, hyperbolic space} \subjclass[2010]{53C21 53C24 53C44}
\begin{abstract} We prove that the static convexity is preserved along two kinds of locally constrained curvature flows in hyperbolic space. Using the static convexity of the flow hypersurfaces, we prove new family of geometric inequalities for such hypersurfaces in hyperbolic space. \end{abstract}
\maketitle \tableofcontents
\section{Introduction}\label{sec:1}
For any bounded domain $\Omega$ with smooth boundary $M=\partial \Omega$ in hyperbolic space $\mathbb H^{n+1}$, the $k$th quermassintegral $W_k$ is defined as the measure of the set of totally geodesic $k$-dimensional subspaces which intersect $\Omega$. It can be expressed as a linear combination of integral of $k$th mean curvatures and of the enclosed volume (see \cite{Sant2004}) \begin{align*}
& W_0(\Omega)= ~\mathrm{Vol}(\Omega),\qquad W_{1}(\Omega)=~\frac 1{n+1}|M|,\\ & W_{k+1}(\Omega)=~ \frac 1{n+1} \int_M E_k d\mu-\frac{k}{n+2-k}W_{k-1}(\Omega),\quad k=1,\cdots,n, \end{align*} where $E_k$ is the normalized $k$-th mean curvature of $M$. Along any outward normal variation with speed $\mathcal{F}$, the $k$th quermassintegral $W_k$ evolves by (see e.g. \cite[(3.5)]{WX14}) \begin{align}\label{s2:variation-quermassintegral} \frac{d}{dt}W_k(\Omega_t)=~\frac{n+1-k}{n+1}\int_{M_t} E_k \mathcal{F}d\mu_t, \quad 0\leq k\leq n. \end{align}
In order to establish the quermassintegral inequalities in hyperbolic space, one natural choice is the following globally constrained quermassintegral preserving flow \begin{align}\label{s1:QP-MCF} \frac{\partial}{\partial t}X=\( \frac{\int_{M_t}E_k^\frac{1}{k-l}E_{l}^{1-\frac{1}{k-l}}d\mu_t}{\int_{M_t}E_{l}d\mu_t}-\(\frac{E_k}{E_{l}}\)^\frac{1}{k-l}\) \nu, \quad 0\leq l<k\leq n, \end{align} Along this flow, the $l$th quermassintegral $W_{l}(\Omega_t)$ is preserved while the $k$th quermassintegral $W_k(\Omega_t)$ is decreasing. Therefore, the smooth convergence of the flow \eqref{s1:QP-MCF} to geodesic spheres would imply the quermassintegral inequalities. For $l=0$ and $k=1$, this flow is called the volume preserving mean curvature flow, which was first studied by Cabezas-Rivas and Miquel \cite{Cabez07}. By imposing {\em h-convexity}\footnote{A closed hypersurface in hyperbolic space is called {\em h-convex} if its principal curvatures satisfy $\kappa_i\geq 1$ for all $i$.} on the initial hypersurface, they proved that the flow converges smoothly to a geodesic sphere as $t\rightarrow \infty$. Later, the smooth convergence of the flow \eqref{s1:QP-MCF} with $0\leq l<k\leq n$ was established by Wang and Xia \cite{WX14}, which yields the following quermassintegral inequalities in hyperbolic space.
\begin{thmA}\cite{WX14}
Let $\Omega$ be a bounded and h-convex domain with smooth boundary in $\mathbb H^{n+1}$. Then there holds
\begin{align}\label{WX-ineq}
W_k(\Omega) \geq f_k \circ f_{l}^{-1}(W_{l}(\Omega)), \quad 0\leq l<k\leq n.
\end{align}
Equality holds in \eqref{WX-ineq} if and only if $\Omega$ is a geodesic ball. Here $f_k:[0,\infty)\rightarrow \mathbb R^{+}$ is a monotone function defined by $f_k(r)=W_k(B_r)$, the $k$th quermassintegral for the geodesic ball of radius $r$, and $f_l^{-1}$ is the inverse function of $f_l$. \end{thmA}
The quermassintegral inequalities in Euclidean space have been proved by Guan and Li \cite{GL09} for $k$-convex\footnote{A smooth domain is called {\em $k$-convex} (resp. {\em strictly $k$-convex}) if the principal curvatures of its boundary satisfy $\kappa\in \overline{\Gamma}_k^{+}$ (resp. $\kappa\in \Gamma_{k}^{+}$), where $\Gamma_k^{+}$ is the Garding cone defined in \S \ref{s2:sec-2.1}.} star-shaped domains. Thus, it is natural to generalize Theorem A to a larger class of smooth bounded domains in hyperbolic space. In this direction, the inequality \eqref{WX-ineq} with $k=3$, $l=1$ was proved earlier by the second named author with Wei and Xiong in \cite{LWX14} for star-shaped domains with strictly $2$-convex boundary. Ge, Wang and Wu \cite{GeWW14} proved the inequalities \eqref{WX-ineq} with $k=2m+1$ $(0<2m<n-1)$ and $l=1$ for h-convex domains. Later, the authors of this paper \cite{HL19} generalized Ge-Wang-Wu's inequalities to smooth bounded domains with {\em nonnegative sectional curvature}\footnote{A hypersurface in hyperbolic space has {\em nonnegative} (resp. {\em positive}) {\em sectional curvature} if its principal curvatures satisfy $\kappa_i\kappa_j\geq 1$ (resp. $\kappa_i\kappa_j>1$) for all distinct $i,j$.} boundary. Recently, the authors of this paper with Andrews \cite{AHL19} proved the inequalities \eqref{WX-ineq} with $k=n-1$ and $l=n-1-2m$ $(0<2m<n-1)$ for strictly convex domains. On the other hand, the proof of convergence of the globally constrained quermassintegral preserving flow \eqref{s1:QP-MCF} depends heavily on {\em h-convexity} of hypersurfaces, see e.g. \cite{AW18,Cabez07,WX14}. The convergence of the flow \eqref{s1:QP-MCF} with $k=1,\cdots, n$ and $l=0$ was recently proved by Andrews, Chen and Wei \cite{ACW2018} for smooth bounded domains with {\em positive sectional curvature} boundary, and henceforth the inequalities \eqref{WX-ineq} with $k=1,\cdots,n$ and $l=0$ hold for such domains.
Alternative choice to prove the quermassintegral inequalities is the following locally constrained quermassintegral preserving flows. The hyperbolic space can be viewed as a warped product space $\mathbb H^{n+1}=[0,\infty)\times \mathbb S^{n}$ equipped with the metric $\-g=dr^2+\lambda^2(r)\sigma$, where $\lambda(r)=\sinh r$ is the warping factor and $\sigma$ is the round metric of the unit sphere $\mathbb{S}^n$. Brendle, Guan and Li \cite{BGL} introduced the following locally constrained inverse curvature flow in $\mathbb H^{n+1}$ (see also a recent survey \cite{GL19} by Guan and Li): \begin{align}\label{s1:BGL-flow} \frac{\partial}{\partial t}X=\(\lambda'(r)\frac{E_{k-1}}{E_k}-u\)\nu,\quad 1\leq k\leq n, \end{align} where $\lambda'(r)=\cosh r$ and $u=\langle \lambda\partial_r,\nu\rangle$ is the support function of the flow hypersurface $M_t=X(t,\cdot)$. In view of the Minkowski formula \eqref{s2:Minkowski-formula}, this flow preserves $W_k(\Omega_t)$ and increases $W_{k-1}(\Omega_t)$ simultaneously, provided that the flow hypersurface is strictly $k$-convex and star-shaped. It is challenging to establish the convergence of this flow \eqref{s1:BGL-flow} under the assumption that the initial hypersurface is strictly $k$-convex and star-shaped, which would prove the quermassintegral inequalities for such domains. In \cite{BGL}, Brendle, Guan and Li proved the convergence of the flow \eqref{s1:BGL-flow} with $k=n$ for strictly convex hypersurfaces, and hence the inequalities \eqref{WX-ineq} with $k=n$ and $l=0,1,\cdots,n-1$ hold for strictly convex domains. Recently, the authors of this paper with Wei \cite{Hu-Li-Wei2020} proved the convergence of this flow \eqref{s1:BGL-flow} with $k=1,\cdots,n$ for h-convex hypersurfaces in hyperbolic space, which yields a new proof of Theorem A.
Besides the quermassintegral inequalities, there is of great interest to establish the weighted geometric inequalities in hyperbolic space. In \cite{Scheuer-Xia2019}, Scheuer and Xia introduced the following locally constrained inverse curvature flow in $\mathbb H^{n+1}$: \begin{align}\label{s1:SX-ICF} \frac{\partial}{\partial t}X=&\(\frac{1}{F}-\frac{u}{\lambda'(r)}\)\nu. \end{align} In particular, if $F=E_k/E_{k-1}$, $k=1,\cdots,n$, they proved that the following convergence result. \begin{thmB}\cite{Scheuer-Xia2019}
Let $X_0$ be a smooth embedding of a closed $n$-dimensional manifold $M$ in $\mathbb H^{n+1}$ such that $M_0=X_0(M)$ is star-shaped and strictly $k$-convex along $X_0(M)$. Then any solution $M_t=X(M,t)$ of \eqref{s1:SX-ICF} with $F=E_k/E_{k-1}$ exists for $t>0$ and it converges to a geodesic sphere centered at the origin in the $C^\infty$-topology as $t\rightarrow \infty$. Moreover, the flow hypersurface $M_t=X(t,M)$ is star-shaped and strictly $k$-convex for $t>0$. \end{thmB}
These assumptions are quite similar to the purely inverse curvature flow, while the locally constrained term in \eqref{s1:SX-ICF} will be helpful in establishing geometric inequalities. In particular, employing the flow \eqref{s1:SX-ICF} with $F=E_1$, they proved the following Minkowski type inequality. \begin{thmC}\cite{Scheuer-Xia2019}
Let $\Omega$ be a star-shaped and strictly mean convex domain with smooth boundary $M$ in $\mathbb H^{n+1}$. Then there holds
\begin{align}\label{SX-ineq-I}
\int_{M} \lambda' E_1 d\mu \geq (n+1)\int_{\Omega} \lambda' d\mathrm{vol}+\omega_n^{\frac{2}{n+1}}\((n+1)\int_{\Omega}\lambda' d\mathrm{vol}\)^{\frac{n-1}{n+1}},
\end{align}
where $\omega_n$ is the area of the unit sphere $\mathbb S^{n}\subset \mathbb R^{n+1}$. Equality holds in \eqref{SX-ineq-I} if and only if $\Omega$ is a geodesic ball centered at the origin. \end{thmC}
For a bounded domain $\Omega$ in $\mathbb H^{n+1}$ with smooth boundary $M=\partial\Omega$, it is called {\em static convex} (resp. {\em strictly static convex}) if its second fundamental form satisfies \begin{align*} h_{ij} \geq \frac{u}{\lambda'}g_{ij}>0, \quad (\text{resp. $h_{ij}>\frac{u}{\lambda'}g_{ij}>0$}) \quad \text{everywhere on $M$.} \end{align*} The static convexity implies the strict convexity, but it is weaker than h-convexity since $\frac{u}{\lambda'}<1$. Combining \eqref{SX-ineq-I} with the following Minkowski type inequality (see Xia \cite[Theorem 1.1]{Xia2016}) \begin{align*} \(\int_{M} \lambda' d\mu \)^2 \geq (n+1) \int_M \lambda' E_1 d\mu \int_{\Omega} \lambda' d\mathrm{vol} \end{align*} for hypersurfaces satisfying $h_{ij}\geq \frac{u}{\lambda'}g_{ij}$ everywhere, Scheuer and Xia \cite[Theorem 1.6]{Scheuer-Xia2019} proved the following weighted isoperimetric inequality. \begin{thmD}\cite{Scheuer-Xia2019}
Let $\Omega$ be a static convex domain with smooth boundary $M$ in $\mathbb H^{n+1}$. Then there holds
\begin{align}\label{SX-ineq-III}
\int_{M} \lambda' d\mu \geq \(\((n+1)\int_{\Omega}\lambda' d\mathrm{vol}\)^{2}+\omega_n^{\frac{2}{n+1}}\((n+1)\int_{\Omega}\lambda'd\mathrm{vol}\)^\frac{2n}{n+1}\)^\frac{1}{2}.
\end{align}
Equality holds in \eqref{SX-ineq-III} if and only if $\Omega$ is a geodesic ball centered at the origin. \end{thmD}
Motivated by Theorems C and D, for any bounded domain $\Omega$ with smooth boundary $M=\partial \Omega$, we introduce the {\em weighted curvature integral} as follows: \begin{align*} W_{0}^{\lambda'}(\Omega)=&~\int_{M} u d\mu=\int_{\Omega}(n+1)\lambda'd\mathrm{vol}, \quad W_{n+1}^{\lambda'}(\Omega)=\int_{M}\lambda'E_n d\mu,\\ W_{k}^{\lambda'}(\Omega)=&~\int_{M}\lambda' E_{k-1}d\mu=~\int_{M} u E_k d\mu, \quad k=1,\cdots,n. \end{align*} Along the outward normal variation with speed $\mathcal{F}$, the weighted curvature integral evolves by (see Proposition \ref{s2:evol-weighted-curvature-integral}) \begin{align*} \frac{d}{dt} W_{k}^{\lambda'}(\Omega_t)=&~\int_{M_t} \(k u E_{k-1}+(n+1-k) \lambda' E_k\)\mathcal{F} d\mu_t, \quad k=0,\cdots,n+1, \end{align*} where we take $E_{-1}=E_{n+1}=0$ by convention.
We would like to propose the following conjectures for the weighted curvature integrals in hyperbolic space. \begin{conjecture}\label{conjecture}
Let $\Omega$ be a static convex domain with smooth boundary in $\mathbb H^{n+1}$. Then there holds
\begin{align}\label{weighted-quermassintegral-ineq}
W_{k}^{\lambda'}(\Omega)\geq h_k \circ h_{l}^{-1}(W_{l}^{\lambda'}(\Omega)), \quad 0\leq l<k \leq n+1.
\end{align}
Equality holds in \eqref{weighted-quermassintegral-ineq} if and only if $\Omega$ is a geodesic ball centered at the origin. Here $h_k:[0,\infty)\rightarrow \mathbb R^{+}$ is a monotone function defined by $h_k(r)=W_k^{\lambda'}(B_r)=\omega_n \sinh^{n+1-k} r \cosh^{k} r$, the $k$th weighted curvature integral for a geodesic ball of radius $r$, and $h_l^{-1}$ is the inverse function of $h_l$. \end{conjecture} The stronger form of Conjecture \ref{conjecture} is as follows. \begin{conjecture}\label{conjecture-strong-version}
Let $0\leq l<k\leq n+1$. Let $\Omega$ be a star-shaped and $(k-1)$-convex domain with smooth boundary in $\mathbb H^{n+1}$. Then there holds
\begin{align*}
W_{k}^{\lambda'}(\Omega)\geq h_k \circ h_{l}^{-1}(W_{l}^{\lambda'}(\Omega)).
\end{align*}
Equality holds if and only if $\Omega$ is a geodesic ball centered at the origin. \end{conjecture}
\subsection{Main results} In this paper, we first introduce a new locally constrained flow. Let $X_0:M^n \rightarrow \mathbb H^{n+1}$ be a smooth embedding such that $M_0$ is a closed, star-shaped hypersurface in $\mathbb H^{n+1}$. We consider the smooth family of immersions $X:M^n \times [0,T)\rightarrow \mathbb H^{n+1}$ satisfying the following evolution equations: \begin{align}\label{s1:locally-MCF} \frac{\partial}{\partial t}X(x,t)=\(1-\frac{u F}{\lambda'(r)} \)\nu(x,t), \end{align} where $F=E_1$.
Our new observation in this paper is that the static convexity is preserved along a large class of locally constrained curvature flows (Theorem \ref{thm-static-convexity}) including the flows \eqref{s1:SX-ICF} and \eqref{s1:locally-MCF}, provided that $F$ satisfies the following \begin{assump}\label{s1:Assumption}
\begin{enumerate}[(i)]
\item $F(\mathcal{W})=f(\kappa(\mathcal{W}))$, where $\kappa$ is the eigenvalues of $\mathcal{W}$ and $f$ is a smooth symmetric function on the positive cone $\Gamma_{+}=\{(x_i)\in\mathbb R^n ~:~ x_i>0\}$ satisfying
\begin{enumerate}[(1)]
\item $f$ is strictly increasing, i.e., $\dot{f}^i=\partial f/\partial \kappa_i>0$ on $\Gamma_{+}$, $\forall i=1,\cdots,n$;
\item $f$ is homogeneous of degree $1$, i.e., $f(k\kappa)=kf(\kappa)$ for any $k>0$;
\item $f$ is strictly positive on $\Gamma_{+}$ and is normalized such that $f(1,\cdots,1)=1$;
\end{enumerate}
\item $f$ is concave.
\item $f$ is inverse concave, i.e., the function
\begin{align}\label{s1:inverse-concave}
f_{\ast}(x_1,\cdots,x_n)=f(x_1^{-1},\cdots,x_n^{-1})^{-1}
\end{align}
is concave.
\end{enumerate} \end{assump} \begin{rem}
Important examples of the curvature function $F$ satisfying Assumption \ref{s1:Assumption} include the curvature quotients $F=(E_{k}/E_{l})^{1/(k-l)}$, $0\leq l<k\leq n$, see \cite{And07}. \end{rem}
We first prove the convergence of the flow \eqref{s1:locally-MCF} with $F=E_1$ for star-shaped hypersurfaces. This is a weighted volume preserving flow, which is inspired by Guan-Li's mean curvature type flow \cite{GL15}. \begin{thm}\label{s1:main-thm-I}
Let $X_0$ be a smooth embedding of a closed $n$-dimensional manifold $M$ in $\mathbb H^{n+1}$ such that $M_0=X_0(M)$ is star-shaped. Then any solution $M_t=X(M,t)$ of \eqref{s1:locally-MCF} with $F=E_1$ remains star-shaped for $t>0$ and it converges to a geodesic sphere $\partial B_{r_\infty}$ centered at the origin in the $C^\infty$-topology as $t\rightarrow \infty$, where the radius $r_\infty$ is uniquely determined by $W^{\lambda'}_0(B_{r_\infty})=W^{\lambda'}_0(\Omega_0)$. Moreover, if the initial hypersurface $M_0=X_0(M)$ is static convex, then the flow hypersurface $M_t=X(t,M)$ becomes strictly static convex for $t>0$. \end{thm}
We also prove that the static convexity is preserved along the flow \eqref{s1:SX-ICF} for $t>0$. \begin{thm}\label{s1:main-thm-III}
Assume that $F$ satisfies Assumption \ref{s1:Assumption}. Let $X_0$ be a smooth embedding of a closed $n$-dimensional manifold $M$ in $\mathbb H^{n+1}$ such that $X_0(M)$ is static convex. Then the flow hypersurface $M_t=X(t,M)$ of \eqref{s1:SX-ICF} becomes strictly static convex for $t>0$. \end{thm}
As applications, we obtain the following geometric inequalities between weighted curvature integrals and quermassintegrals. We emphasize that the preservance of static convexity along the flows will be crucial in the proof of the geometric inequalities.
Firstly, we apply both the flow \eqref{s1:locally-MCF} with $F=E_1$ and the flow \eqref{s1:SX-ICF} with $F=E_{k-1}/E_{k-2}$ to prove the Conjecture \ref{conjecture} with $l=0$ and $1\leq k\leq n+1$ for static convex domains. \begin{thm}\label{thm-weighted-quermassintegral-ineq-I}
Let $\Omega$ be a static convex domain with smooth boundary $M$ in $\mathbb H^{n+1}$. For $1\leq k\leq n+1$, there holds
\begin{align}\label{weighted-quermassintegral-ineq-I}
W_{k}^{\lambda'}(\Omega) \geq h_k \circ h_0^{-1}(W_0^{\lambda'}(\Omega)).
\end{align}
Equivalently,
\begin{align}\label{s1:explicit-form}
\int_{M} \lambda' E_{k-1} d\mu\geq \omega_n \(\(\frac{(n+1)\int_{\Omega}\lambda' d\mathrm{vol}}{\omega_n}\)^\frac{2}{k}+\(\frac{(n+1)\int_{\Omega}\lambda' d\mathrm{vol}}{\omega_n}\)^\frac{2(n-k+1)}{(n+1)k}\)^\frac{k}{2}.
\end{align}
Equality holds in \eqref{weighted-quermassintegral-ineq-I} if and only if $\Omega$ is a geodesic ball centered at the origin. \end{thm} \begin{rem}
The inequality \eqref{s1:explicit-form} with $k=1$ is Theorem D. By Theorem C, the inequality \eqref{s1:explicit-form} with $k=2$ also holds for star-shaped domains with strictly mean convex boundary. \end{rem}
Using the flow \eqref{s1:locally-MCF} with $F=E_1$, we prove the weighted geometric inequality between the weighted enclosed volume and the volume for star-shaped domains. \begin{thm}\label{thm-geometric-ineq}
Let $\Omega$ be a smooth star-shaped domain in $\mathbb H^{n+1}$. Then there holds
\begin{align}\label{geom-ineq-1}
W_{0}^{\lambda'}(\Omega)=(n+1)\int_{\Omega}\lambda' d\mathrm{vol}\geq h_{0} \circ f_0^{-1}(W_0(\Omega)).
\end{align}
Equality holds in \eqref{geom-ineq-1} if and only if $\Omega$ is a geodesic ball centered at the origin. \end{thm}
Applying the the flow \eqref{s1:SX-ICF} with $F=E_k/E_{k-1}$, we proved the following weighted Alexandrov-Fenchel inequalities for static convex domains. \begin{thm}\label{thm-geometric-ineq-I}
Let $\Omega$ be a static convex domain with smooth boundary $M$ in $\mathbb H^{n+1}$. For $0 \leq k \leq n$ and $0\leq m\leq k$, there holds
\begin{align}\label{geom-ineq-2}
W_{k+1}^{\lambda'}(\Omega)=\int_{M} \lambda'E_{k} d\mu \geq h_{k+1} \circ f_m^{-1}(W_m(\Omega)).
\end{align}
Equality holds in \eqref{geom-ineq-2} if and only if $\Omega$ is a geodesic ball centered at the origin. \end{thm} \begin{rem}
\begin{enumerate}[(i)]
\item The inequality \eqref{geom-ineq-2} with $k=1$ and $m=1$ was proved by de Lima and Girao \cite{deLima-Girao2016} for star-shaped domains with strictly mean convex boundary. For odd $k$ and $m=1$, the inequality \eqref{geom-ineq-2} was proved by Ge, Wang and Wu \cite{Ge-Wang-Wu2015} for h-convex domains.
For $1\leq k\leq n$ and $0\leq m \leq k$, the inequality \eqref{geom-ineq-2} for h-convex domains was recently proved by the authors of the paper with Wei in \cite[Theorem 1.4]{Hu-Li-Wei2020}. For $k=n$ and $0\leq m \leq n$, by the work \cite{BGL,GL19}, the inequality \eqref{geom-ineq-2} also holds for strictly convex domains.
\item In \cite{Ge-Wang-Wu2015}, Ge, Wang and Wu introduced the Gauss-Bonnet-Chern mass $m_l^{\mathbb H}$ for asymptotically hyperbolic graphs. Using the weighted Alexandrov-Fenchel inequality \eqref{geom-ineq-2} with $k=2l-1$ and $m=1$, they proved an optimal Penrose type inequality for $m_l^{\mathbb H}$, under the assumption that the boundary of each component is h-convex, see \cite[Theorem 1.6]{Ge-Wang-Wu2015}. By Theorem \ref{thm-geometric-ineq-I}, we can weaken this assumption therein to be static convex.
\end{enumerate} \end{rem}
We would also like to propose the following conjecture on weighted Alexandrov-Fenchel inequalities. \begin{conjecture}\label{conjecture-geometric-inequality}
Let $0 \leq k \leq n$ and $0\leq m\leq k$. Let $\Omega$ be a star-shaped and $k$-convex domain in $\mathbb H^{n+1}$ with smooth boundary $M$. Then there holds
\begin{align*}
W_{k+1}^{\lambda'}(\Omega)=\int_{M} \lambda'E_{k} d\mu \geq h_{k+1} \circ f_m^{-1}(W_m(\Omega)).
\end{align*}
Equality holds if and only if $\Omega$ is a geodesic ball centered at the origin. \end{conjecture}
The paper is organized as follows. In \S \ref{sec:2}, we collect some basic properties of symmetric functions and geometry of hypersurfaces in hyperbolic space. In \S \ref{sec:3}, we derive the evolution equations along two kinds of locally constrained curvature flows. In \S \ref{sec:4}, we use the nonparametric form of these flows to derive the $C^0$-estimates of the radial function. In \S \ref{sec:5}, we apply the tensor maximum principle to show that the static convexity is preserved along the two kinds of locally constrained curvature flows, which includes the flows \eqref{s1:SX-ICF} and \eqref{s1:locally-MCF} as special cases. Theorem \ref{s1:main-thm-III} then follows. In \S \ref{sec:7}, we complete the proof of Theorem \ref{s1:main-thm-I}. In \S \ref{sec:8}, we apply these locally constrained curvature flows to prove geometric inequalities in hyperbolic space, including Theorems \ref{thm-weighted-quermassintegral-ineq-I}, \ref{thm-geometric-ineq} and \ref{thm-geometric-ineq-I}.
\end{ack}
\section{Preliminaries}\label{sec:2} \subsection{Properties of symmetric functions}$\ $\label{s2:sec-2.1} We first review some properties of symmetric functions on the positive cone $\Gamma_{+}\subset \mathbb R^n$. Given a smooth symmetric function $F(A)=f(\kappa(A))$, where $A=(A_{ij}) \in \mathrm{Sym}(n)$ is a symmetric matrix and $\kappa(A)=(\kappa_1,\cdots,\kappa_n)$ gives the eigenvalues of $A$. We assume that $f$ is a smooth, symmetric, positive, strictly increasing, $1$-homogeneous function on $\Gamma_{+}$ and is normalized such that $f(1,\cdots,1)=1$. We denote by $\dot{F}^{ij}$ and $\ddot{F}^{ij,kl}$ the first and second derivative with respect to the components of its argument, so that \begin{align*}
\left.\frac{\partial}{\partial s}F(A+sB) \right|_{s=0} = \dot{F}^{ij}(A)B_{ij} \end{align*} and \begin{align*}
\left.\frac{\partial^2}{\partial s^2} F(A+sB)\right|_{s=0} = \ddot{F}^{ij,kl}(A) B_{ij} B_{kl} \end{align*} for any two symmetric matrices $A$, $B$. We also use the notations $\dot{f}^i(\kappa)$, $\dot{f}^{ij}(\kappa)$ to denote the derivatives of $f$ with respect to $\kappa$. At any diagonal $A$, we have \begin{align*} \dot{F}^{ij}(A)=\dot{f}^i(\kappa(A))\delta_i^j. \end{align*} If diagonal $A$ has distinct eigenvalues, the second derivatives $\ddot{F}$ of $F$ in direction $B\in\mathrm{Sym}(n)$ is given in terms of $\ddot{f}$ and $\dot{f}$ by (see \cite{And94,And07}): \begin{align}\label{s2:second-derivative-expr} \ddot{F}^{ij,kl}B_{ij}B_{kl}=\sum_{i,k}\ddot{f}^{ik} B_{ii}B_{kk}+2\sum_{i>k}\frac{\dot{f}^i-\dot{f}^k}{\kappa_i-\kappa_k}B_{ik}^2. \end{align} This formula makes sense as a limit in the case of any repeated values of $\kappa_i$.
For any positive definite symmetric matrix $A\in \operatorname{Sym}(n)$ with eigenvalues $\kappa(A)\in \Gamma_{+}$, define $F_{\ast}(A)=F(A^{-1})^{-1}$. Then $F_\ast(A)=f_\ast(\kappa(A))$, where $f_\ast$ is the dual function of $f$ defined in \eqref{s1:inverse-concave}. We say that $F$ is inverse concave if $F_{\ast}(A)$ is concave. Since $f$ is defined on the positive cone $\Gamma_{+}$, the following lemma charaterizes the concavity and the inverse concavity of $f$ and $F$. \begin{lem}(\cite{And07,AMZ13})
\begin{enumerate}[(i)]
\item $f$ is concave if and only if the following matrix
\begin{align}\label{f-concave}
(\ddot{f}^{kl})\leq 0.
\end{align}
\item $F$ is concave if and only if $f$ is concave and
\begin{align}\label{f-concave-II}
\frac{\dot{f}^k-\dot{f}^l}{\kappa_k-\kappa_l} \leq 0,\quad \forall k\neq l.
\end{align}
\item $f$ is inverse concave if and only if the following matrix
\begin{align}\label{f-inverse-concave}
\(\ddot{f}^{kl}+2\frac{\dot{f}^k}{\kappa_k}\delta_{kl}\) \geq 0.
\end{align}
\item $F$ is inverse concave if and only if $f$ is inverse concave and
\begin{align}\label{f-inverse-concave-II}
\frac{\dot{f}^k-\dot{f}^l}{\kappa_k-\kappa_l}+\frac{\dot{f}^k}{\kappa_l}+\frac{\dot{f}^l}{\kappa_k} \geq 0,~~\forall k\neq l.
\end{align}
\end{enumerate} \end{lem} \begin{rem}
Since $f$ and $f_{\ast}$ are defined on $\Gamma_{+}$, we have (see \cite[Corollaries 5.3 \& 5.5]{And07})
\begin{enumerate}[(i)]
\item $F$ is concave if and only if $f$ is concave;
\item $F$ is inverse concave if and only if $f$ is inverse concave.
\end{enumerate} \end{rem} Important examples of concave and inverse functions include $(E_k/E_l)^{1/(k-l)}$ with $k>l$, where $$ E_k(\kappa)=\binom{n}{k}^{-1}\sum_{1\leq i_1<\cdots<i_k \leq n}\kappa_{i_1}\cdots \kappa_{i_k}. $$ is the normalized $k$-th elementary symmetric function. It is convenient to set $E_0(\kappa)=1$ and $E_k(\kappa)=0$ for $k>n$. For any symmetric matrix $A=(A_{ij})\in \operatorname{Sym}(n)$, we set $E_k(A)=E_k(\kappa(A))$, then $E_k(A)$ is defined by $$ E_k(A)=\frac{(n-k)!}{n!}\delta_{i_1 \cdots i_k}^{j_1\cdots j_k} A_{i_1j_1} \cdots A_{i_kj_k}, \quad k=1,\cdots,n, $$ where $\delta_{i_1 \cdots i_k}^{j_1\cdots j_k}$ is generalized Kronecker delta defined by \begin{align*} \delta^{j_1j_2\cdots j_k}_{i_1i_2\cdots i_k}=\det\(\begin{matrix} \delta^{j_1}_{i_1} & \delta^{j_1}_{i_2} & \cdots & \delta^{j_1}_{i_k} \\ \delta^{j_2}_{i_1} & \delta^{j_2}_{i_2} & \cdots & \delta^{j_2}_{i_k} \\ \vdots & \vdots & \vdots & \vdots \\ \delta^{j_k}_{i_1} & \delta^{j_k}_{i_2} & \cdots & \delta^{j_k}_{i_k} \end{matrix}\). \end{align*} The Garding cone is defined by $$
\Gamma_{k}^{+}=\{\kappa \in \mathbb R^n ~|~ E_m(\kappa)>0, \forall m\leq k \}. $$ Then $n$-convex is convex in usual sense, $1$-convex is referred as mean convex. We also take $\Gamma_{0}^{+}=\mathbb R^{n}$ by convention. The following Newton-MacLaurin inequalities are well-known, see e.g. \cite[Lemma 2.5 on p.55]{Guan12}. \begin{lem}\label{lem-newton-maclaurin-ineq}
Let $1\leq k \leq n$. For $\kappa \in \Gamma_{k}^{+}$, we have
\begin{align}\label{s2:newton-maclaurin-ineq}
E_{k+1}(\kappa) E_{k-1}(\kappa) \leq E_k^2(\kappa), \quad E_{k+1}(\kappa) \leq E_k^{\frac{k+1}{k}}(\kappa).
\end{align}
Equality holds in \eqref{s2:newton-maclaurin-ineq} if and only if $\kappa_1=\cdots=\kappa_n$. \end{lem}
Note that $E_k(\kappa)=E_k(A)$, where $\kappa$ gives the eigenvalues of $A$. Denote by $\dot{E}_k^{i}=\frac{\partial E_k}{\partial \kappa_i}(\kappa)$ and $\dot{E}_k^{ij}(A)=\frac{\partial E_k(A)}{\partial A_{ij}}$. \begin{lem}\label{lem-Newton-tensor} We have
\begin{align}
\sum_{i,j}\dot{E}_k^{ij}A_{ij}=&\sum_{i}\dot{E}_k^{i}\kappa_i= k E_k,\label{newton-formula-1}\\
\sum_{i,j}\dot{E}_k^{ij}\delta_{i}^{j} =&\sum_{i}\dot{E}_k^{i}= k E_{k-1},\label{newton-formula-2}\\
\sum_{i,j}\dot{E}_k^{ij}(A^2)_{ij}=&\sum_{i}\dot{E}_k^{i}\kappa_i^2= n E_1 E_{k} -(n-k)E_{k+1},\label{newton-formula-3}
\end{align}
where $(A^2)_{ij}=\sum_{l}A_{il}A_{lj}$. \end{lem}
\subsection{Hypersurfaces in hyperbolic space}$\ $ \label{s2-2} The hyperbolic space $\mathbb H^{n+1}$ can be expressed as a warped product manifold $\mathbb R^+\times \mathbb S^n$ equipped with the metric $$ \-g=dr^2+\lambda(r)^2 \sigma, $$ where $\lambda(r)=\sinh r$ and $\sigma$ is the round metric of the unit sphere $\mathbb S^n \subset \mathbb R^{n+1}$. We define $$ \Lambda(r)=\int_0^{r} \lambda(s)ds=\lambda'(r)-1, $$ where $\lambda'(r)=\cosh r$. Let $\overline\nabla$ be the Levi-Civita connection with respect to $\-g$. The vector field $\overline\nabla \Lambda=\lambda\partial_r$ on $\mathbb H^{n+1}$ is a conformal Killing field, i.e., \begin{align}\label{s2:conformal-Killing} \overline\nabla(\lambda \partial_r)=\lambda' \-g. \end{align}
Let $M$ be a closed smooth hypersurface in $\mathbb H^{n+1}$ with unit outward normal $\nu$. The second fundamental form $h$ of $M$ is given by $h(X,Y)=\langle \overline\nabla \nu, Y\rangle$ for any tangent vectors $X,Y$ on $M$. The principal curvatures $\kappa=(\kappa_1,\cdots,\kappa_n)$ are the eigenvalues of the second fundamental form $h$ with respect to the induced metric $g$ on $M$. In a local coordinate $(x^1,\cdots,x^n)$ of $M$, we denote $g_{ij}=g(\partial_i,\partial_j)$ and $h_{ij}=h(\partial_i,\partial_j)$. Then the Weingarten matrix is given by $\mathcal{W}=(h_i^j)=(g^{jk}h_{ki})$, where $(g^{ij})$ is the inverse matrix of $(g_{ij})$. Then the principal curvatures $\kappa$ of $M$ are the eigenvalues of the Weingarten matrix $\mathcal{W}$.
We recall the following lemmas for smooth hypersurfaces in $\mathbb H^{n+1}$, see e.g. \cite[Lemmas 2.2 \& 2.6]{GL15}.
\begin{lem}\label{lem-1} Let $(M,g)$ be a smooth hypersurface in $\mathbb H^{n+1}$. Then $\Lambda|_{M}$ satisfies
\begin{align}\label{s2:2.1}
\nabla_i\lambda'=\nabla_i \Lambda= \langle \lambda\partial_r,e_i\rangle,\quad \nabla_j\nabla_i \lambda'=\nabla_j\nabla_i \Lambda=\lambda' g_{ij} -u h_{ij},
\end{align}
and the support function $u=\langle \lambda\partial_r,\nu \rangle$ satisfies
\begin{align}\label{s2:2.2}
\nabla_i u= \langle \lambda\partial_r,e_k\rangle h_i^k, \quad \nabla_j\nabla_i u= \langle \lambda\partial_r,\nabla h_{ij}\rangle+\lambda' h_{ij}-u (h^2)_{ij},
\end{align}
where $\{e_1,\cdots,e_n\}$ is a basis of the tangent space of $M$. \end{lem}
By the divergence-free property of $\dot{E}_k^{ij}(h)$ and \eqref{s2:2.1}, we have the well-known Minkowski formulas (see e.g. \cite[Proposition 2.5]{GL15}). \begin{lem}\label{lem-Minkowski-formula} Let $(M,g)$ be a smooth closed hypersurface in $\mathbb H^{n+1}$. Then there holds
\begin{align}\label{s2:Minkowski-formula}
\int_M \lambda' E_{k-1} d\mu= \int_{M} u E_k d\mu, \quad k=1,\cdots,n.
\end{align} \end{lem}
\subsection{Parametrization by Radial graph}\label{s2:sec-2.2} A smooth closed hypersurface $M$ in hyperbolic space $\mathbb H^{n+1}$ is called {\em star-shaped} if its support function $u= \langle \lambda\partial_r,\nu\rangle>0$ everywhere on $M$. This is equivalent to that the hypersurface $M$ can be expressed as a radial graph in spherical coordinates $(r(\theta),\theta)$ in $\mathbb H^{n+1}$, that is, \begin{align*}
M= \{(r(\theta),\theta)\in \mathbb R^{+} \times \mathbb S^n ~|~\theta \in \mathbb S^n \}. \end{align*} Let $\theta=(\theta^1,\cdots,\theta^n)$ be a local coordinate of the round sphere $(\mathbb S^n,\sigma)$. Let $D$ be the Levi-Civita connection on $(\mathbb S^n,\sigma)$. Let $\partial_i=\partial_{\theta^i}$ and $r_i=D_i r$. For the convenience of the notations, we define $\varphi:\mathbb S^n \rightarrow \mathbb R$ by $\varphi(\theta)=\Psi(r(\theta))$, where $\Psi=\Psi(r)$ is a positive smooth function satisfies $\Psi'(r)=1/\lambda(r)$. Let $\varphi_i=D_i \varphi$ and $\varphi_{ij}=D_iD_j\varphi$. Then the tangential vector on $M$ are given by $\{X_i= \partial_i +r_i \partial_r=\partial_i+\lambda \varphi_i\partial_r, i=1,\cdots,n\}$. The induced metric $g_{ij}$ of $M=\operatorname{graph}r$ can be expressed as \begin{align}\label{s2:induced-metric} g_{ij} = \lambda^2 \sigma_{ij}+r_i r_i=\lambda^2 (\sigma_{ij}+\varphi_i\varphi_j), \end{align}
where $\sigma_{ij}=\sigma(\partial_i,\partial_j)$. Denote by $v=\sqrt{1+\lambda^{-2}|Dr|^2}=\sqrt{1+|D\varphi|^2}$, the inverse matrix $(g^{ij})$ of $(g_{ij})$ is given by \begin{align}\label{s2:inverse-metric} g^{ij}=\frac{1}{\lambda^2}\(\sigma^{ij}-\frac{\varphi^i \varphi^j}{v^2}\), \end{align} where $\varphi^i=\sigma^{ij}\varphi_j$. The unit outward normal is given by \begin{align*} \nu=\frac{1}{v}\(\partial_r - \frac{\varphi^i}{\lambda}\partial_i\). \end{align*} It follows that the support function $u=\langle \lambda\partial_r,\nu\rangle=\frac{\lambda}{v}$. The second fundamental form $h_{ij}$ and the Weingarten matrix $h_i^j$ of $M$ can be expressed as (see e.g., \cite{Ge11}) \begin{align*} h_{ij}= \frac{\lambda'}{\lambda v}g_{ij}-\frac{\lambda}{v}\varphi_{ij}, \end{align*} and \begin{align}\label{s2:2nd-fundamental-form} h_i^j=g^{jk}h_{ki}= \frac{\lambda'}{\lambda v}\delta_i^j-\frac{1}{\lambda v}\(\sigma^{jk}-\frac{\varphi^j\varphi^k}{v^2}\)\varphi_{ki}. \end{align}
\section{Evolution equations}$\ $ \label{sec:3} Along the general flow \begin{align}\label{s3:evol-general-flow} \frac{\partial}{\partial t}X=\mathcal{F}\nu, \end{align} in hyperbolic space $\mathbb H^{n+1}$, we have the following evolution equations for the induced metric $g_{ij}$, the unit outward normal $\nu$, the area element $d\mu_t$, the second fundamental form $h_{ij}$ and the Weingarten matrix $h_i^j$ of the flow hypersurfaces $M_t=X(M^n,t)$: (see e.g., \cite{LWX14})
\begin{align}
\frac{\partial}{\partial t}g_{ij}=&2\mathcal{F}h_{ij}, \label{s3:3.1} \\
\frac{\partial}{\partial t}\nu=&-\nabla \mathcal{F}, \label{s3:3.2} \\
\frac{d}{dt}d\mu_t=& nE_1 \mathcal{F} d\mu_t, \label{s3:3.3} \\
\frac{\partial}{\partial t}h_{ij}=&-\nabla_j\nabla_i \mathcal{F}+\mathcal{F}((h^2)_{ij}+g_{ij}), \label{s3:3.4}\\
\frac{\partial}{\partial t}h_i^j=&-\nabla^j\nabla_i\mathcal{F}-\mathcal{F}((h^2)_i^j-\delta_i^j), \label{s3:3.5}
\end{align} where $\nabla$ denotes the Levi-Civita connection with respect to the induced metric $g_{ij}$ on $M_t$.
First of all, we deduce the variational formulas for the weighted curvature integrals along the general flow \eqref{s3:evol-general-flow} in hyperbolic space. \begin{prop}\label{s2:evol-weighted-curvature-integral}
Let $M_t=\partial\Omega_t \subset \mathbb H^{n+1}$ be a smooth family of closed hypersurfaces evolves by the flow \eqref{s3:evol-general-flow}. Then there holds
\begin{align}\label{s6:evol-weighted-curvature-integral}
\frac{d}{dt}W_{k}^{\lambda'}(\Omega_t)=\int_{M_t} \( k u E_{k-1}+(n+1-k)\lambda'E_k\)\mathcal{F}d\mu_t, \quad 0\leq k\leq n+1.
\end{align} \end{prop} \begin{proof}
For $k=0$, it is well-known. For $1\leq k\leq n+1$, a proof can be found in \cite[\S 5.2]{Hu-Li-Wei2020}. We include it here for
convenience. Along the flow \eqref{s3:evol-general-flow}, we have
\begin{align*}
\frac{\partial}{\partial t}\lambda' = \langle \overline\nabla \lambda', \partial_t\rangle =u\mathcal{F},
\end{align*}
and
\begin{align*}
\frac{\partial}{\partial t}E_{k-1} =\frac{\partial E_{k-1}}{\partial h_i^j} \partial_t h_i^j =\dot{E}_{k-1}^{ij}\(-\nabla_i\nabla_j \mathcal{F}-\mathcal{F}((h^2)_{ij}-g_{ij})\),
\end{align*}
where we used \eqref{s3:3.5}. Combining these with \eqref{s3:3.3}, we get
\begin{align*}
\frac{d}{dt}W^{\lambda'}_k(\Omega_t) =&\int_{M_t} (u E_{k-1}+n\lambda' E_{k-1} E_1) \mathcal{F}d\mu_t+\int_{M_t} \lambda'\dot{E}_{k-1}^{ij}\(-\nabla_i\nabla_j \mathcal{F}-\mathcal{F}((h^2)_{ij}-g_{ij})\)d\mu_t.
\end{align*}
Since $\dot{E}_k^{ij}$ is divergence-free, by integration by parts we obtain
\begin{align*}
\frac{d}{dt}W^{\lambda'}_k(\Omega_t)=&\int_{M_t} \(u E_{k-1}+n\lambda' E_{k-1}E_1-\lambda'\dot{E}_{k-1}^{ij}((h^2)_{ij}-g_{ij})\)\mathcal{F}d\mu_t -\int_{M_t}(\dot{E}_{k-1}^{ij}\nabla_i\nabla_j \lambda')\mathcal{F}d\mu_t \\
=&\int_{M_t} \(u E_{k-1}+n\lambda' E_{k-1}E_1-\lambda'\dot{E}_{k-1}^{ij}(h^2)_{ij}+u\dot{E}_{k-1}^{ij}h_{ij}\)\mathcal{F}d\mu_t \\
=&\int_{M_t} \(ku E_{k-1}+(n+1-k)\lambda'E_k\) \mathcal{F}d\mu_t,
\end{align*}
where we used \eqref{s2:2.1} in the second equality and \eqref{newton-formula-1}, \eqref{newton-formula-3} in the last equality. \end{proof}
Let $\Phi \in C^\infty(\mathbb R_{+})$ be a smooth function satisfying $\Phi'(s)=\frac{d}{ds}\Phi(s)>0$ for all $s>0$. We assume that $F$ satisfies (i) in Assumption \ref{s1:Assumption}. We consider the following two kinds of flows \begin{align}\label{s2:general-form-flow-I} \frac{\partial}{\partial t}X= \(\Phi(1)-\Phi(\frac{u}{\lambda'}F)\)\nu, \end{align} and \begin{align}\label{s2:general-form-flow-II} \frac{\partial}{\partial t}X= \( \Phi(\frac{\lambda'}{u})-\Phi(F)\)\nu. \end{align} It is easy to see that \begin{enumerate}[(i)]
\item The flow \eqref{s1:locally-MCF} is corresponding to the flow \eqref{s2:general-form-flow-I} with $\Phi(s)=s$;
\item The flow \eqref{s1:SX-ICF} is corresponding to the flow \eqref{s2:general-form-flow-II} with $\Phi(s)=-s^{-1}$. \end{enumerate} Now we deduce the evolution equations along the flows \eqref{s2:general-form-flow-I} and \eqref{s2:general-form-flow-II}, respectively. For simplicity, we denote by \begin{align}\label{s2:notation-I} \dot{\Phi}^{kl}(\frac{u}{\lambda'}F)=\Phi'(\frac{u}{\lambda'}F)\frac{u}{\lambda'}\dot{F}^{kl},\quad \ddot{\Phi}^{kl,pq}(\frac{u}{\lambda'}F)=\Phi'(\frac{u}{\lambda'}F)\frac{u}{\lambda'}\ddot{F}^{kl,pq}+\Phi''(\frac{u}{\lambda'}F)\frac{u^2}{\lambda'^2}\dot{F}^{kl}\dot{F}^{pq}, \end{align} and \begin{align}\label{s2:notation-II} \dot{\Phi}^{kl}(F)=\Phi'(F)\dot{F}^{kl},\quad \ddot{\Phi}^{kl,pq}(F)=\Phi'(F)\ddot{F}^{kl,pq}+\Phi''(F)\dot{F}^{kl}\dot{F}^{pq}. \end{align} \begin{lem}\label{s3:lem-evolution-eq-1} Along the flow \eqref{s2:general-form-flow-I}, i.e., $\mathcal{F}=\Phi(1)-\Phi(\frac{u}{\lambda'}F)$, we have the following evolution equations.
\begin{enumerate}[(i)]
\item
\begin{align}\label{evol-static-function-I}
\frac{\partial}{\partial t}(\frac{u}{\lambda'})=&\dot{\Phi}^{kl}(\frac{u}{\lambda'}F) \nabla_k \nabla_l (\frac{u}{\lambda'})+\frac{2}{\lambda'}\dot{\Phi}^{kl}(\frac{u}{\lambda'}F)\nabla_k(\frac{u}{\lambda'})\nabla_l \lambda'+\Phi'(\frac{u}{\lambda'}F)\frac{F}{\lambda'}\langle \lambda\partial_r,\nabla (\frac{u}{\lambda'})\rangle \nonumber\\
&+(1-\frac{u^2}{\lambda'^2})(\Phi(1)-\Phi(\frac{u}{\lambda'}F))-(1+\frac{u^2}{\lambda'^2})\Phi'(\frac{u}{\lambda'}F)\frac{u}{\lambda'}F \nonumber\\
&+\Phi'(\frac{u}{\lambda'}F)\frac{u^2}{\lambda'^2}\dot{F}^{kl}((h^2)_{kl}+g_{kl}).
\end{align}
\item
\begin{align}\label{evol-second-fundamental-form-I}
\frac{\partial}{\partial t}h_{ij}
=&\dot{\Phi}^{kl}(\frac{u}{\lambda'}F)\nabla_k \nabla_l h_{ij}+\ddot{\Phi}^{kl,pq}(\frac{u}{\lambda'}F)\nabla_i h_{kl}\nabla_j h_{pq} +\Phi'(\frac{u}{\lambda'}F)\frac{F}{\lambda'}\langle \lambda\partial_r,\nabla h_{ij}\rangle\nonumber\\
& -2\Phi'(\frac{u}{\lambda'}F)\(\frac{F}{\lambda'}\nabla_{(i}\lambda'\nabla_{j)}(\frac{u}{\lambda'})-\nabla_{(i}(\frac{u}{\lambda'})\nabla_{j)}F\) \nonumber\\
&+\Phi''(\frac{u}{\lambda'}F)\( F^2\nabla_i(\frac{u}{\lambda'})\nabla_j(\frac{u}{\lambda'})+2\frac{u}{\lambda'}F\nabla_{(i}(\frac{u}{\lambda'})\nabla_{j)}F\) \nonumber\\
&+\Phi'(\frac{u}{\lambda'}F)\(\frac{u}{\lambda'}\dot{F}^{kl}((h^2)_{kl}+g_{kl})+F(1+\frac{u^2}{\lambda'^2}) \)h_{ij} \nonumber\\
&+\(\Phi(1)-\Phi(\frac{u}{\lambda'}F)-2\Phi'(\frac{u}{\lambda'}F)\frac{u}{\lambda'}F\)((h^2)_{ij}+g_{ij}),
\end{align}
where the brackets denote symmetrization, e.g., $\nabla_{(i}\lambda'\nabla_{j)}(\frac{u}{\lambda'})=\frac{1}{2}(\nabla_{i}\lambda'\nabla_{j}(\frac{u}{\lambda'})+\nabla_{j}\lambda'\nabla_{i}(\frac{u}{\lambda'}))$.
\item Taking $S_{ij}=h_{ij}-\frac{u}{\lambda'}g_{ij}$, we have
\begin{align}\label{evol-Sij-I}
\frac{\partial}{\partial t}S_{ij}=&\dot{\Phi}^{kl}(\frac{u}{\lambda'}F)\nabla_k \nabla_l S_{ij}+ \ddot{\Phi}^{kl,pq}(\frac{u}{\lambda'}F)\nabla_i h_{kl}\nabla_j h_{pq} +\Phi(\frac{u}{\lambda'}F)\frac{F}{\lambda'}\langle \lambda\partial_r,\nabla S_{ij}\rangle \nonumber\\
&-2\Phi'(\frac{u}{\lambda'}F)\(\frac{F}{\lambda'}\nabla_{(i}\lambda'\nabla_{j)}(\frac{u}{\lambda'})-\nabla_{(i}(\frac{u}{\lambda'})\nabla_{j)}F\) \nonumber \\ &+\Phi''(\frac{u}{\lambda'}F)\( F^2\nabla_i(\frac{u}{\lambda'})\nabla_j(\frac{u}{\lambda'})+2\frac{u}{\lambda'}F\nabla_{(i}(\frac{u}{\lambda'})\nabla_{j)}F\) \nonumber\\
&-\dot{\Phi}^{kl}(\frac{u}{\lambda'}F)\frac{2}{\lambda'}\nabla_k(\frac{u}{\lambda'})\nabla_l\lambda' g_{ij}+\(\Phi(1)-\Phi(\frac{u}{\lambda'}F)-2\Phi'(\frac{u}{\lambda'}F)\frac{u}{\lambda'}F\)((S^2)_{ij}+2\frac{u}{\lambda'}S_{ij})\nonumber\\
&+\(\Phi'(\frac{u}{\lambda'}F)\(\frac{u}{\lambda'}\dot{F}^{kl}((h^2)_{kl}+g_{kl})+F(1+\frac{u^2}{\lambda'^2})\)-2\frac{u}{\lambda'}(\Phi(1)-\Phi(\frac{u}{\lambda'}F))\)S_{ij}.
\end{align}
\end{enumerate} \end{lem} \begin{proof}
\begin{enumerate}[(i)]
\item Since $u=\langle \lambda\partial_r,\nu\rangle$ and $\overline\nabla\lambda'=\overline\nabla \Lambda=\lambda\partial_r$, we get
\begin{align*}
\frac{\partial}{\partial t}\lambda'=\langle \overline\nabla \lambda', \partial_t X\rangle=\langle \lambda\partial_r,(\Phi(1)-\Phi(\frac{u}{\lambda'}F)\nu\rangle = u\(\Phi(1)-\Phi(\frac{u}{\lambda'}F)\).
\end{align*}
By \eqref{s2:2.1} and $1$-homogeneity of $F$, we have
\begin{align*}
\dot{\Phi}^{kl}(\frac{u}{\lambda'}F)\nabla_k \nabla_l \lambda'=\Phi'(\frac{u}{\lambda'}F)\frac{u}{\lambda'}\dot{F}^{kl}(\lambda'g_{kl}-uh_{kl})
=\Phi'(\frac{u}{\lambda'}F)u\(\dot{F}^{kl}g_{kl}-\frac{u}{\lambda'}F\).
\end{align*}
Then we deduce that
\begin{align}\label{evol-cosh-r}
\frac{\partial}{\partial t}\lambda'=\dot{\Phi}^{kl}\nabla_k \nabla_l \lambda'+u\(\Phi(1)-\Phi(\frac{u}{\lambda'}F)-\Phi'(\frac{u}{\lambda'}F)\dot{F}^{kl}g_{kl}+\Phi'(\frac{u}{\lambda'}F)\frac{u}{\lambda'}F\).
\end{align}
On the other hand, by the conformal property \eqref{s2:conformal-Killing} of the vector field $\lambda\partial_r$ and \eqref{s3:3.2}, we have
\begin{align*}
\frac{\partial}{\partial t}u=&\langle \overline\nabla_{\partial_t}(\lambda\partial_r),\nu\rangle+\langle \lambda\partial_r, \frac{\partial}{\partial t}\nu\rangle \\
=&\lambda'\(\Phi(1)-\Phi(\frac{u}{\lambda'}F)\)+\langle \lambda\partial_r, \nabla \Phi(\frac{u}{\lambda'}F) \rangle \\
=&\lambda'\(\Phi(1)-\Phi(\frac{u}{\lambda'}F)\)+\Phi'(\frac{u}{\lambda'}F) \(\frac{u}{\lambda'}\langle \lambda\partial_r, \nabla F\rangle+F\langle \lambda\partial_r, \nabla(\frac{u}{\lambda'})\rangle\).
\end{align*}
By \eqref{s2:2.2} and $1$-homogeneity of $F$, we also have
\begin{align*}
\dot{\Phi}^{kl}(\frac{u}{\lambda'}F)\nabla_k \nabla_l u=&\Phi'(\frac{u}{\lambda'}F)\frac{u}{\lambda'}\dot{F}^{kl}\(\langle \lambda\partial_r,\nabla h_{kl}\rangle+\lambda' h_{kl}-u(h^2)_{kl}\) \\
=&\Phi'(\frac{u}{\lambda'}F)\frac{u}{\lambda'}\(\langle \lambda\partial_r,\nabla F\rangle+\lambda' F-u\dot{F}^{kl}(h^2)_{kl}\).
\end{align*}
Then we get
\begin{align}\label{evol-support-function}
\frac{\partial}{\partial t}u=&\dot{\Phi}^{kl}(\frac{u}{\lambda'}F)\nabla_k \nabla_l u+\Phi'(\frac{u}{\lambda'}F)F\langle \lambda\partial_r, \nabla(\frac{u}{\lambda'})\rangle \nonumber\\
&+\lambda'\(\Phi(1)-\Phi(\frac{u}{\lambda'}F)-\Phi'(\frac{u}{\lambda'}F)\frac{u}{\lambda'}F+\Phi'(\frac{u}{\lambda'}F)\frac{u^2}{\lambda'^2}\dot{F}^{kl}(h^2)_{kl}\).
\end{align}
Combining \eqref{evol-cosh-r} with \eqref{evol-support-function}, we obtain
\begin{align*}
\frac{\partial}{\partial t}(\frac{u}{\lambda'})=&\frac{1}{\lambda'}\frac{\partial}{\partial t}u-\frac{u}{\lambda'^2}\frac{\partial}{\partial t}\lambda' \\
=&\dot{\Phi}^{kl}(\frac{u}{\lambda'}F)\(\frac{1}{\lambda'}\nabla_k \nabla_l u-\frac{u}{\lambda'^2}\nabla_k \nabla_l \lambda'\)+\Phi'(\frac{u}{\lambda'}F)\frac{F}{\lambda'}\langle \lambda\partial_r,\nabla (\frac{u}{\lambda'})\rangle\\
&+(1-\frac{u^2}{\lambda'^2})(\Phi(1)-\Phi(\frac{u}{\lambda'}F))-(1+\frac{u^2}{\lambda'^2})\Phi'(\frac{u}{\lambda'}F)\frac{u}{\lambda'}F\\
&+\Phi'(\frac{u}{\lambda'}F)\frac{u^2}{\lambda'^2}\dot{F}^{kl}((h^2)_{kl}+g_{kl}).
\end{align*}
We also have
\begin{align}\label{hessian-static-function}
\nabla_k \nabla_l (\frac{u}{\lambda'})=&\frac{1}{\lambda'}\nabla_k \nabla_l u-\frac{u}{\lambda'^2}\nabla_k \nabla_l \lambda'-\frac{2}{\lambda'}\nabla_{(k}(\frac{u}{\lambda'})\nabla_{l)} \lambda'.
\end{align}
Thus, we get
\begin{align*}
\frac{\partial}{\partial t}(\frac{u}{\lambda'})=&\dot{\Phi}^{kl}(\frac{u}{\lambda'}F) \nabla_k \nabla_l (\frac{u}{\lambda'})+\dot{\Phi}^{kl}(\frac{u}{\lambda'}F)\frac{2}{\lambda'}\nabla_k(\frac{u}{\lambda'})\nabla_l \lambda'+\Phi'(\frac{u}{\lambda'}F)\frac{F}{\lambda'}\langle \lambda\partial_r,\nabla (\frac{u}{\lambda'})\rangle\\
&+(1-\frac{u^2}{\lambda'^2})(\Phi(1)-\Phi(\frac{u}{\lambda'}F))-(1+\frac{u^2}{\lambda'^2})\Phi'(\frac{u}{\lambda'}F)\frac{u}{\lambda'}F\\
&+\Phi'(\frac{u}{\lambda'}F)\frac{u^2}{\lambda'^2}\dot{F}^{kl}((h^2)_{kl}+g_{kl}).
\end{align*}
\item By \eqref{s3:3.4}, we have
\begin{align}\label{s3:evol-hij-1}
\frac{\partial}{\partial t}h_{ij}=&-\nabla_j\nabla_i (\Phi(1)-\Phi(\frac{u}{\lambda'}F))+(\Phi(1)-\Phi(\frac{u}{\lambda'}F))((h^2)_{ij}+g_{ij})\nonumber \\
=&\Phi'(\frac{u}{\lambda'}F)\(F\nabla_j\nabla_i (\frac{u}{\lambda'})+2\nabla_{(i}(\frac{u}{\lambda'})\nabla_{j)}F+\frac{u}{\lambda'}\nabla_j\nabla_i F\)\nonumber\\
&+\Phi''(\frac{u}{\lambda'}F)\( F^2\nabla_i(\frac{u}{\lambda'})\nabla_j(\frac{u}{\lambda'})+\frac{u^2}{\lambda'^2}\nabla_i F\nabla_j F+2\frac{u}{\lambda'}F\nabla_{(i}(\frac{u}{\lambda'})\nabla_{j)}F\)\nonumber\\
&+(\Phi(1)-\Phi(\frac{u}{\lambda'}F))((h^2)_{ij}+g_{ij}).
\end{align}
Combining the Gauss and Codazzi equations in hyperbolic space, we obtain the following generalized Simons' identity (see e.g., \cite{And94}):
\begin{align*}
\nabla_{(i}\nabla_{j)}h_{kl}=\nabla_{(k}\nabla_{l)}h_{ij}+((h^2)_{kl}+g_{kl})h_{ij}-h_{kl}((h^2)_{ij}+g_{ij}),
\end{align*}
where the brackets denote symmetrization. This yields
\begin{align}\label{s3:Sims}
\nabla_j\nabla_i F=& \nabla_j (\dot{F}^{kl}\nabla_i h_{kl}) \nonumber\\
=&\dot{F}^{kl}\nabla_j\nabla_i h_{kl}+\ddot{F}^{kl,pq}\nabla_i h_{kl} \nabla_j h_{pq}\nonumber\\
=&\dot{F}^{kl}\nabla_k \nabla_l h_{ij} + \dot{F}^{kl}((h^2)_{kl}+g_{kl})h_{ij}-F((h^2)_{ij}+g_{ij})+\ddot{F}^{kl,pq}\nabla_i h_{kl}\nabla_j h_{pq},
\end{align}
where we used the $1$-homogeneity of $F$. On the other hand, substituting \eqref{s2:2.1} and \eqref{s2:2.2} into \eqref{hessian-static-function}, we deduce that
\begin{align}\label{s3:hessian-static-function}
\nabla_j\nabla_i(\frac{u}{\lambda'})=&\frac{1}{\lambda'}\langle \lambda\partial_r,\nabla h_{ij}\rangle+(1+\frac{u^2}{\lambda'^2})h_{ij}-\frac{u}{\lambda'}((h^2)_{ij}+g_{ij})-\frac{2}{\lambda'}\nabla_{(i} \lambda' \nabla_{j)} (\frac{u}{\lambda'}).
\end{align}
Putting \eqref{s3:Sims} and \eqref{s3:hessian-static-function} into \eqref{s3:evol-hij-1}, in view of \eqref{s2:notation-I}, we obtain
\begin{align*}
\frac{\partial}{\partial t}h_{ij}
=&\dot{\Phi}^{kl}(\frac{u}{\lambda'}F)\nabla_k \nabla_l h_{ij}+\ddot{\Phi}^{kl,pq}(\frac{u}{\lambda'}F)\nabla_i h_{kl}\nabla_j h_{pq} +\Phi'(\frac{u}{\lambda'}F)\frac{F}{\lambda'}\langle \lambda\partial_r,\nabla h_{ij}\rangle\nonumber\\
& -2\Phi'(\frac{u}{\lambda'}F)\(\frac{F}{\lambda'}\nabla_{(i}\lambda'\nabla_{j)}(\frac{u}{\lambda'})-\nabla_{(i}(\frac{u}{\lambda'})\nabla_{j)}F\) \nonumber\\
&+\Phi''(\frac{u}{\lambda'}F)\( F^2\nabla_i(\frac{u}{\lambda'})\nabla_j(\frac{u}{\lambda'})+2\frac{u}{\lambda'}F\nabla_{(i}(\frac{u}{\lambda'})\nabla_{j)}F\) \nonumber\\
&+\Phi'(\frac{u}{\lambda'}F)\(\frac{u}{\lambda'}\dot{F}^{kl}((h^2)_{kl}+g_{kl})+F(1+\frac{u^2}{\lambda'^2}) \)h_{ij} \nonumber\\
&+\(\Phi(1)-\Phi(\frac{u}{\lambda'}F)-2\Phi'(\frac{u}{\lambda'}F)\frac{u}{\lambda'}F\)((h^2)_{ij}+g_{ij}).
\end{align*}
\item Combining \eqref{s3:3.1}, \eqref{evol-static-function-I} with \eqref{evol-second-fundamental-form-I}, we get
\begin{align*}
\frac{\partial}{\partial t}S_{ij}=&\dot{\Phi}^{kl}(\frac{u}{\lambda'}F)\nabla_k \nabla_l S_{ij}+ \ddot{\Phi}^{kl,pq}(\frac{u}{\lambda'}F)\nabla_i h_{kl}\nabla_j h_{pq} +\Phi(\frac{u}{\lambda'}F)\frac{F}{\lambda'}\langle \lambda\partial_r,\nabla S_{ij}\rangle \nonumber\\
&-2\Phi'(\frac{u}{\lambda'}F)\(\frac{F}{\lambda'}\nabla_{(i}\lambda'\nabla_{j)}(\frac{u}{\lambda'})-\nabla_{(i}(\frac{u}{\lambda'})\nabla_{j)}F\) \nonumber \\ &+\Phi''(\frac{u}{\lambda'}F)\( F^2\nabla_i(\frac{u}{\lambda'})\nabla_j(\frac{u}{\lambda'})+2\frac{u}{\lambda'}F\nabla_{(i}(\frac{u}{\lambda'})\nabla_{j)}F\) \nonumber\\
&-\dot{\Phi}^{kl}(\frac{u}{\lambda'}F)\frac{2}{\lambda'}\nabla_k(\frac{u}{\lambda'})\nabla_l\lambda' g_{ij}+\(\Phi(1)-\Phi(\frac{u}{\lambda'}F)-2\Phi'(\frac{u}{\lambda'}F)\frac{u}{\lambda'}F\)((S^2)_{ij}+2\frac{u}{\lambda'}S_{ij})\nonumber\\
&+\(\Phi'(\frac{u}{\lambda'}F)\(\frac{u}{\lambda'}\dot{F}^{kl}((h^2)_{kl}+g_{kl})+F(1+\frac{u^2}{\lambda'^2})\)-2\frac{u}{\lambda'}(\Phi(1)-\Phi(\frac{u}{\lambda'}F))\)S_{ij},
\end{align*}
where we also used $h_{ij}=S_{ij}+\frac{u}{\lambda'}g_{ij}$ and $(h^2)_{ij}=(S^2)_{ij}+2\frac{u}{\lambda'}S_{ij}+\frac{u^2}{\lambda'^2}g_{ij}$.
\end{enumerate} \end{proof}
\begin{lem} Along the flow \eqref{s2:general-form-flow-II}, i.e., $\mathcal{F}=\Phi(\frac{\lambda'}{u})-\Phi(F)$, we have the following evolution equations. \begin{enumerate}[(i)]
\item
\begin{align}\label{evol-static-function-II}
\frac{\partial}{\partial t}(\frac{u}{\lambda'})=&\dot{\Phi}^{kl}(F)\nabla_k\nabla_l (\frac{u}{\lambda'})+\dot{\Phi}^{kl}(F)\frac{2}{\lambda'}\nabla_k(\frac{u}{\lambda'})\nabla_l \lambda'+\Phi'(\frac{\lambda'}{u})\frac{\lambda'}{u^2}\langle \lambda\partial_r,\nabla (\frac{u}{\lambda'})\rangle \nonumber \\
&+(1-\frac{u^2}{\lambda'^2})(\Phi(\frac{\lambda'}{u})-\Phi(F))-(1+\frac{u^2}{\lambda'^2})\Phi'(F)F+\frac{u}{\lambda'}\Phi'(F)\dot{F}^{kl}((h^2)_{kl}+g_{kl}).
\end{align}
\item
\begin{align}\label{evol-second-fundamental-form-II}
\frac{\partial}{\partial t}h_{ij}=&\dot{\Phi}^{kl}(F)\nabla_k\nabla_l h_{ij}+\ddot{\Phi}^{kl,pq}(F)\nabla_i h_{kl}\nabla_j h_{pq}+\Phi'(\frac{\lambda'}{u})\frac{\lambda'}{u^2}\langle \lambda\partial_r,\nabla h_{ij}\rangle \nonumber\\
&+\Phi'(\frac{\lambda'}{u})\frac{2}{u}\nabla_{(j}(\frac{\lambda'}{u})\nabla_{i)}u-\Phi''(\frac{\lambda'}{u})\nabla_i(\frac{\lambda'}{u})\nabla_j(\frac{\lambda'}{u}) \nonumber\\
&+\(\Phi'(F)\dot{F}^{kl}((h^2)_{kl}+g_{kl})+(1+\frac{\lambda'^2}{u^2})\Phi'(\frac{\lambda'}{u})\)h_{ij} \nonumber\\
&+\(\Phi(\frac{\lambda'}{u})-\Phi(F)-\Phi'(F)F-\Phi'(\frac{\lambda'}{u})\frac{\lambda'}{u}\)((h^2)_{ij}+g_{ij}),
\end{align}
\item Taking $S_{ij}=h_{ij}-\frac{u}{\lambda'}g_{ij}$, we have
\begin{align}\label{evol-Sij-II}
\frac{\partial}{\partial t}S_{ij}=&\dot{\Phi}^{kl}(F)\nabla_k\nabla_l S_{ij}+\ddot{\Phi}^{kl,pq}(F)\nabla_i h_{kl}\nabla_j h_{pq}+\Phi'(\frac{\lambda'}{u})\frac{\lambda'}{u^2}\langle \lambda\partial_r,\nabla S_{ij}\rangle \nonumber\\
&+\Phi'(\frac{\lambda'}{u})\frac{2}{u}\nabla_{(j}(\frac{\lambda'}{u})\nabla_{i)}u-\Phi''(\frac{\lambda'}{u})\nabla_i(\frac{\lambda'}{u})\nabla_j(\frac{\lambda'}{u})-\dot{\Phi}^{kl}(F)\frac{2}{\lambda'}\nabla_{k}(\frac{u}{\lambda'})\nabla_l \lambda' g_{ij} \nonumber\\
&+\(\Phi(\frac{\lambda'}{u})-\Phi(F)-\Phi'(F)F-\Phi'(\frac{\lambda'}{u})\frac{\lambda'}{u} \)((S^2)_{ij}+2\frac{u}{\lambda'}S_{ij}) \nonumber\\
&+\(\Phi'(F)\dot{F}^{kl}((h^2)_{kl}+g_{kl})+(1+\frac{\lambda'^2}{u^2})\Phi'(\frac{\lambda'}{u})-2\frac{u}{\lambda'}(\Phi(\frac{\lambda'}{u})-\Phi(F))\) S_{ij}.
\end{align} \end{enumerate} \end{lem} \begin{proof}
\begin{enumerate}[(i)]
\item We have
\begin{align*}
\frac{\partial}{\partial t}\lambda'=\dot{\Phi}^{kl}(F)\nabla_k\nabla_l \lambda'+u\( \Phi(\frac{\lambda'}{u})-\Phi(F)-\frac{\lambda'}{u}\Phi'(F)\dot{F}^{kl}g_{kl}+\Phi'(F)F \),
\end{align*}
and
\begin{align*}
\frac{\partial}{\partial t}u=&\dot{\Phi}^{kl}(F)\nabla_k\nabla_l u-\Phi'(\frac{\lambda'}{u})\langle \lambda\partial_r,\nabla (\frac{\lambda'}{u})\rangle \\
&+\lambda'\(\Phi(\frac{\lambda'}{u})-\Phi(F)-\Phi'(F)F+\frac{u}{\lambda'}\Phi'(F)\dot{F}^{kl}(h^2)_{kl}\).
\end{align*}
Combining these two equations with \eqref{hessian-static-function}, we derive
\begin{align*}
\frac{\partial}{\partial t}(\frac{u}{\lambda'})=&\dot{\Phi}^{kl}(F)\nabla_k\nabla_l (\frac{u}{\lambda'})+\dot{\Phi}^{kl}(F)\frac{2}{\lambda'}\nabla_k(\frac{u}{\lambda'})\nabla_l \lambda'+\Phi'(\frac{\lambda'}{u})\frac{\lambda'}{u^2}\langle \lambda\partial_r,\nabla (\frac{u}{\lambda'})\rangle \\
&+(1-\frac{u^2}{\lambda'^2})(\Phi(\frac{\lambda'}{u})-\Phi(F))-(1+\frac{u^2}{\lambda'^2})\Phi'(F)F+\frac{u}{\lambda'}\Phi'(F)\dot{F}^{kl}((h^2)_{kl}+g_{kl}).
\end{align*}
\item By \eqref{s3:3.4}, we have
\begin{align}\label{s3:evol-hij-2}
\frac{\partial}{\partial t}h_{ij}=&-\nabla_j\nabla_i(\Phi(\frac{\lambda'}{u})-\Phi(F))+(\Phi(\frac{\lambda'}{u})-\Phi(F))((h^2)_{ij}+g_{ij})\nonumber\\
=&\Phi'(F)\nabla_j\nabla_i F+\Phi''(F)\nabla_j F\nabla_i F-\Phi'(\frac{\lambda'}{u})\nabla_j\nabla_i(\frac{\lambda'}{u})-\Phi''(\frac{\lambda'}{u})\nabla_j(\frac{\lambda'}{u})\nabla_i(\frac{\lambda'}{u})\nonumber\\
&+(\Phi(\frac{\lambda'}{u})-\Phi(F))((h^2)_{ij}+g_{ij}).
\end{align}
We also have
\begin{align}\label{s3:hessian-static-function-II}
\nabla_j \nabla_i (\frac{\lambda'}{u})=&\frac{1}{u}\nabla_j \nabla_i \lambda'-\frac{\lambda'}{u^2}\nabla_j \nabla_i u-\frac{2}{u}\nabla_{(j}(\frac{\lambda'}{u})\nabla_{i)} u\nonumber\\
=&\frac{\lambda'}{u}((h^2)_{ij}+g_{ij})-(1+\frac{\lambda'^2}{u^2})h_{ij}-\frac{\lambda'}{u^2}\langle \lambda\partial_r,\nabla h_{ij}\rangle-\frac{2}{u}\nabla_{(j}(\frac{\lambda'}{u})\nabla_{i)} u.
\end{align}
Substituting \eqref{s3:Sims} and \eqref{s3:hessian-static-function-II} into \eqref{s3:evol-hij-2}, in view of \eqref{s2:notation-II}, we derive
\begin{align*}
\frac{\partial}{\partial t}h_{ij}=&\dot{\Phi}^{kl}(F)\nabla_k\nabla_l h_{ij}+\ddot{\Phi}^{kl,pq}(F)\nabla_i h_{kl}\nabla_j h_{pq}+\Phi'(\frac{\lambda'}{u})\frac{\lambda'}{u^2}\langle \lambda\partial_r,\nabla h_{ij}\rangle \\
&+\Phi'(\frac{\lambda'}{u})\frac{2}{u}\nabla_{(j}(\frac{\lambda'}{u})\nabla_{i)}u-\Phi''(\frac{\lambda'}{u})\nabla_i(\frac{\lambda'}{u})\nabla_j(\frac{\lambda'}{u}) \\
&+\(\Phi'(F)\dot{F}^{kl}((h^2)_{kl}+g_{kl})+(1+\frac{\lambda'^2}{u^2})\Phi'(\frac{\lambda'}{u})\)h_{ij} \\
&+\(\Phi(\frac{\lambda'}{u})-\Phi(F)-\Phi'(F)F-\Phi'(\frac{\lambda'}{u})\frac{\lambda'}{u}\)((h^2)_{ij}+g_{ij}).
\end{align*}
\item Using \eqref{s3:3.1}, \eqref{evol-static-function-II} and \eqref{evol-second-fundamental-form-II}, we obtain
\begin{align*}
\frac{\partial}{\partial t}S_{ij}=&\dot{\Phi}^{kl}(F)\nabla_k\nabla_l S_{ij}+\ddot{\Phi}^{kl,pq}(F)\nabla_i h_{kl}\nabla_j h_{pq}+\Phi'(\frac{\lambda'}{u})\frac{\lambda'}{u^2}\langle \lambda\partial_r,\nabla S_{ij}\rangle \\
&+\Phi'(\frac{\lambda'}{u})\frac{2}{u}\nabla_{(j}(\frac{\lambda'}{u})\nabla_{i)}u-\Phi''(\frac{\lambda'}{u})\nabla_i(\frac{\lambda'}{u})\nabla_j(\frac{\lambda'}{u})-\dot{\Phi}^{kl}(F)\frac{2}{\lambda'}\nabla_{k}(\frac{u}{\lambda'})\nabla_l \lambda' g_{ij}\\
&+\(\Phi(\frac{\lambda'}{u})-\Phi(F)-\Phi'(F)F-\Phi'(\frac{\lambda'}{u})\frac{\lambda'}{u} \)((S^2)_{ij}+2\frac{u}{\lambda'}S_{ij})\\
&+\(\Phi'(F)\dot{F}^{kl}((h^2)_{kl}+g_{kl})+(1+\frac{\lambda'^2}{u^2})\Phi'(\frac{\lambda'}{u})-2\frac{u}{\lambda'}(\Phi(\frac{\lambda'}{u})-\Phi(F))\) S_{ij}.
\end{align*}
\end{enumerate} \end{proof}
\section{$C^0$-estimates} \label{sec:4} Along the general flow \eqref{s3:evol-general-flow}, if the evolving hypersurfaces $M_t, t\in [0,T^\ast)$ are star-shaped, we may parametrize each of them as a graph $$
M_t=\{(r(\theta,t),\theta)\in \mathbb R^+\times \mathbb S^n ~|~\theta\in \mathbb S^n \}, $$ where $r(\cdot,t)$ is the radial function of $M_t$ defined on $\mathbb S^n$. Let $\varphi(\theta,t)=\psi(r(\theta,t))$. Then the flow \eqref{s3:evol-general-flow} can be reduced to a parabolic equation for $\varphi$. As long as the solution of \eqref{s3:evol-general-flow} exists and remains star-shaped, then $\varphi$ satisfies the following initial value problem (see \cite{GL15,Scheuer-Xia2019,Hu-Li-Wei2020}): \begin{align}\label{s2:evolution-parametric-form} \left\{\begin{aligned} \frac{\partial \varphi}{\partial t}=&\mathcal{F}\frac{v}{\lambda}, \quad (\theta,t)\in \mathbb S^n \times [0,T^\ast), \\ \varphi(\cdot,0)=&\varphi_0(\cdot),\end{aligned} \right. \end{align} where $\varphi_0(\theta)=\psi(r_0(\theta))$ is determined by the radial function $r_0$ of the initial hypersurface $M_0$. The Eq. \eqref{s2:evolution-parametric-form} will be referred as the nonparametric form of the general flow \eqref{s3:evol-general-flow}.
Then the nonparametric form of the flow \eqref{s2:general-form-flow-I} or \eqref{s2:general-form-flow-II} is
\begin{align}\label{s2:general-flow-I-parametric-form}
\left\{\begin{aligned}
\frac{\partial}{\partial t} \varphi=&\(\Phi(1)-\Phi(\frac{\lambda}{\lambda'v}F)\)\frac{v}{\lambda}, \quad (\theta,t)\in \mathbb S^n \times [0,T^\ast), \\
\varphi(\cdot,0)=&\varphi_0(\cdot),
\end{aligned}\right.
\end{align}
or
\begin{align}\label{s2:general-flow-II-parametric-form}
\left\{\begin{aligned}
\frac{\partial}{\partial t} \varphi=& \(\Phi(\frac{\lambda'v}{\lambda})-\Phi(F)\)\frac{v}{\lambda}, \quad (\theta,t)\in \mathbb S^n \times [0,T^\ast), \\
\varphi(\cdot,0)=&\varphi_0(\cdot).
\end{aligned}\right.
\end{align}
We show that the solution $\varphi$ to \eqref{s2:general-flow-I-parametric-form} or \eqref{s2:general-flow-II-parametric-form} is uniformly bounded from above and below.
\begin{prop}\label{s3:prop-C0-estimate} Let $\varphi_0$ be determined by the initial hypersurface $M_0$ in $\mathbb H^{n+1}$. If $\varphi=\varphi(\theta,t), t\in [0,T^\ast)$ solves the initial value problem \eqref{s2:general-flow-I-parametric-form} or \eqref{s2:general-flow-II-parametric-form}, then
\begin{align}\label{s2:C0-estimate}
\min_{\theta\in \mathbb S^n} \varphi_0(\theta) \leq \varphi(\theta,t) \leq \max_{\theta\in \mathbb S^n} \varphi_0(\theta), \quad \forall (\theta,t)\in \mathbb S^n\times [0,T^\ast).
\end{align}
\end{prop}
\begin{proof}
We only prove the upper bound of $\varphi$ along the flow \eqref{s2:general-flow-I-parametric-form}, since the remaining proof is similar. At the maximum point of $\varphi$, we have
\begin{align*}
D \varphi=0, \quad v=1, \quad D^2\varphi \leq 0.
\end{align*}
In view of \eqref{s2:2nd-fundamental-form}, we get
$$
h_i^j=\frac{1}{\lambda}(-\varphi_{i}^{j}+\lambda'\delta_{i}^{j}) \geq \frac{\lambda'}{\lambda}\delta_{i}^{j}.
$$
Now we diagonalize the matrix $(\varphi_i^j)$, then $(h_i^j)=\mathrm{diag}(\kappa_1,\cdots,\kappa_n)$ with $\kappa_i\geq \frac{\lambda'}{\lambda}$. This yields
$$
F(h_i^j)=f(\kappa_1,\cdots,\kappa_n) \geq f(\frac{\lambda'}{\lambda},\cdots,\frac{\lambda'}{\lambda})=\frac{\lambda'}{\lambda},
$$
where we used (i) in Assumption \ref{s1:Assumption}. As $\Phi'(s)>0$ for all $s>0$, at the maximum point of $\varphi$ we have
\begin{align*}
\frac{\partial}{\partial t}\varphi=\frac{1}{\lambda}\(\Phi(1)-\Phi(\frac{\lambda}{\lambda'}F)\)\leq 0.
\end{align*}
Then $\varphi(\theta,t) \leq \max_{\theta\in \mathbb S^n} \varphi_0(\theta)$ follows from the maximum principle. \end{proof}
\section{Preserving of static convexity}$\ $ \label{sec:5} In this section, we will use the tensor maximum principle to prove that the {\em static convexity} is preserved along the flow \eqref{s2:general-form-flow-I} or \eqref{s2:general-form-flow-II}, provided that \begin{enumerate}[(i)]
\item $\Phi$ satisfies $\Phi'(s)>0$ and $\Phi''(s)s+2\Phi'(s)\geq 0$ for all $s>0$;
\item $F$ satisfies Assumption \ref{s1:Assumption}. \end{enumerate}
\begin{rem}
(a) Important examples of the function $\Phi$ satisfying $\Phi'(s)>0$ and $\Phi''(s)s+2\Phi'(s)\geq 0$ for all $s>0$ include: (1) $\Phi(s)=s^{p}$ with $p>0$; (2) $\Phi(s)=-s^{-p}$ with $0<p\leq 1$; (3) $\Phi(s)=\ln s$.
(b) Important examples of the curvature function $F$ satisfying Assumption \ref{s1:Assumption} include the curvature quotients $F=(E_{k}/E_{l})^{1/(k-l)}$, $0\leq l<k\leq n$. For more examples, we refer the readers to \cite[Section 2]{And07}. \end{rem}
For convenience of the readers, we recall the tensor maximum principle, which was first proved by Hamilton \cite{Ham1982} and was generalized by Andrews \cite{And07}. \begin{thm}[\cite{And07}]\label{thm-2}
Let $S_{ij}$ be a smooth time-varying symmetric tensor field on a compact manifold $M$ satisfying
\begin{align*}
\frac{\partial}{\partial t}S_{ij} = a^{kl} \nabla_k \nabla_l S_{ij}+u^k \nabla_k S_{ij}+N_{ij},
\end{align*}
where $a^{kl}$ and $u$ are smooth, $\nabla$ is a (possibly time-dependent) smooth symmetric connection, and $a^{kl}$ is positive definite everywhere. Suppose that
\begin{align}\label{3.2}
N_{ij} v^i v^j + \sup_{\Lambda} 2 a^{kl}(2\Lambda_k^p \nabla_l S_{ip}v^i-\Lambda_k^p \Lambda_l^q S_{pq}) \geq 0,
\end{align}
whenever $S_{ij}\geq 0$ and $S_{ij}v^j=0$ and $\Lambda$ is an $n\times n$-matrix. If $S_{ij}$ is positive definite everywhere on $M$ at $t=0$ and on $\partial M$ for $0\leq t\leq T$, then it is positive on $M\times [0,T]$. \end{thm}
The main result of this section is the following. \begin{thm}\label{thm-static-convexity}
Assume that $F$ satisfies Assumption \ref{s1:Assumption} and $\Phi$ satisfies $\Phi'(s)>0$ and $\Phi''(s)s+2\Phi'(s)\geq 0$ for all $s>0$. If the initial hypersurface $M_0$ is a closed, static convex hypersurface in $\mathbb H^{n+1}$, then along the flow \eqref{s2:general-form-flow-I} or \eqref{s2:general-form-flow-II}, the evolving hypersurface $M_t$ becomes strictly static convex for $t>0$. \end{thm} \begin{proof}
Along the flow \eqref{s2:general-form-flow-I} or \eqref{s2:general-form-flow-II}, the flow hypersurface $M_t$ is star-shaped at least for a short time $[0,T')\subset [0,T^\ast)$, where $T^{\ast}$ the maximal existence time of smooth solution of the flow \eqref{s2:general-form-flow-I} or \eqref{s2:general-form-flow-II}, respectively. Let $S_{ij}=h_{ij}-\frac{u}{\lambda'}g_{ij}$. On $[0,T')$, the static convexity of the flow hypersurface $M_t$ is equivalent to $S_{ij}\geq 0$, which implies that $M_t$ is strictly convex. At the minimum point of $u$ over $M_t$, by \eqref{s2:2.2} we have
$$
0=\nabla_i u=\kappa_i\nabla_i \lambda'=\kappa_i \lambda \nabla_i r, \quad 1\leq i\leq n.
$$
It follows from the strict convexity of $M_t$ that $Dr=0$ and $v=\sqrt{1+\lambda^{-2}|Dr|^2}=1$ at this point. By the expression of the support function $u=\frac{\lambda}{v}$, we get
$$
\min_{M_t}u=\min_{\theta\in \mathbb S^n}\lambda(r(\theta,t)) \geq \min_{\theta\in \mathbb S^n}\lambda(r_0(\theta)),
$$
due to the $C^0$-estimates of $\varphi$ (Proposition \ref{s3:prop-C0-estimate}). This shows that the support function $u$ admits a uniform positive lower bound which is independent of time. Therefore, it suffices to show that $S_{ij}\geq 0$ on $[0,T')$, and a continuation argument implies that $T'=T^{\ast}$.
Along the flow \eqref{s2:general-form-flow-I}, by \eqref{evol-Sij-I} we have
\begin{align}\label{s5:Q1}
\frac{\partial}{\partial t}S_{ij}=&\dot{\Phi}^{kl}(\frac{u}{\lambda'}F)\nabla_k \nabla_l S_{ij}+ \ddot{\Phi}^{kl,pq}(\frac{u}{\lambda'}F)\nabla_i h_{kl}\nabla_j h_{pq} +\Phi(\frac{u}{\lambda'}F)\frac{F}{\lambda'}\langle \lambda\partial_r,\nabla S_{ij}\rangle \nonumber\\
&-2\Phi'(\frac{u}{\lambda'}F)\(\frac{F}{\lambda'}\nabla_{(i}\lambda'\nabla_{j)}(\frac{u}{\lambda'})-\nabla_{(i}(\frac{u}{\lambda'})\nabla_{j)}F\) \nonumber \\ &+\Phi''(\frac{u}{\lambda'}F)\( F^2\nabla_i(\frac{u}{\lambda'})\nabla_j(\frac{u}{\lambda'})+2\frac{u}{\lambda'}F\nabla_{(i}(\frac{u}{\lambda'})\nabla_{j)}F\) \nonumber\\
&-\dot{\Phi}^{kl}(\frac{u}{\lambda'}F)\frac{2}{\lambda'}\nabla_k(\frac{u}{\lambda'})\nabla_l\lambda' g_{ij}+\(\Phi(1)-\Phi(\frac{u}{\lambda'}F)-2\Phi'(\frac{u}{\lambda'}F)\frac{u}{\lambda'}F\)((S^2)_{ij}+2\frac{u}{\lambda'}S_{ij})\nonumber\\
&+\(\Phi'(\frac{u}{\lambda'}F)\(\frac{u}{\lambda'}\dot{F}^{kl}((h^2)_{kl}+g_{kl})+F(1+\frac{u^2}{\lambda'^2})\)-2\frac{u}{\lambda'}(\Phi(1)-\Phi(\frac{u}{\lambda'}F))\)S_{ij},
\end{align}
while along the flow \eqref{s2:general-form-flow-II}, by \eqref{evol-Sij-II} we have
\begin{align}\label{s5:Q2}
\frac{\partial}{\partial t}S_{ij}=&\dot{\Phi}^{kl}(F)\nabla_k\nabla_l S_{ij}+\ddot{\Phi}^{kl,pq}(F)\nabla_i h_{kl}\nabla_j h_{pq}+\Phi'(\frac{\lambda'}{u})\frac{\lambda'}{u^2}\langle \lambda\partial_r,\nabla S_{ij}\rangle \nonumber\\
&+\Phi'(\frac{\lambda'}{u})\frac{2}{u}\nabla_{(j}(\frac{\lambda'}{u})\nabla_{i)}u-\Phi''(\frac{\lambda'}{u})\nabla_i(\frac{\lambda'}{u})\nabla_j(\frac{\lambda'}{u})-\dot{\Phi}^{kl}(F)\frac{2}{\lambda'}\nabla_{k}(\frac{u}{\lambda'})\nabla_l \lambda' g_{ij} \nonumber\\
&+\(\Phi(\frac{\lambda'}{u})-\Phi(F)-\Phi'(F)F-\Phi'(\frac{\lambda'}{u})\frac{\lambda'}{u} \)((S^2)_{ij}+2\frac{u}{\lambda'}S_{ij}) \nonumber\\
&+\(\Phi'(F)\dot{F}^{kl}((h^2)_{kl}+g_{kl})+(1+\frac{\lambda'^2}{u^2})\Phi'(\frac{\lambda'}{u})-2\frac{u}{\lambda'}(\Phi(\frac{\lambda'}{u})-\Phi(F))\) S_{ij}.
\end{align}
To apply the tensor maximum principle, we need to show that \eqref{3.2} whenever $S_{ij}\geq 0$ and $S_{ij}v^j=0$ (so that $v$ is a null vector of $S$). Let $(x_0,t_0)$ be the point where $S_{ij}$ has a null vector $v$. We choose normal coordinates around $(x_0,t_0)$ such that $g_{ij}=\delta_{ij}$ at this point. By continuity, we can assume that the principal curvatures are mutually distinct and in increasing order at $(x_0,t_0)$, that is $\kappa_1<\kappa_2<\cdots<\kappa_n$\footnote{This is possible since for any positive definite symmetric matrix $A$ with $A_{ij}\geq 0$ and $A_{ij}v^iv^j=0$ for some $v\neq 0$, there is a sequence of symmetric matrixes $\{A^{(k)}\}$ approaching $A$, satisfying $A^{(k)}_{ij}\geq 0$ and $A^{(k)}_{ij}v^iv^j=0$ and with each $A^{(k)}$ having distinct eigenvalues. Hence it suffices to prove the result in the case where all of $\kappa_i$ are distinct.}. The null vector condition $S_{ij}v^j=0$ implies that $v=e_1$ and $S_{11}=\kappa_1-\frac{u}{\lambda'}=0$ at $(x_0,t_0)$. The terms involving $S_{ij}$ and $(S^2)_{ij}$ satisfy the null vector condition and can be ignored. Moreover, by \eqref{s2:2.1}, \eqref{s2:2.2} and $\kappa_1=\frac{u}{\lambda'}$, we have
$$
\nabla_1 (\frac{u}{\lambda'})=\frac{\lambda'\kappa_1-u}{\lambda'^2}\nabla_1 \lambda'=0, \quad \nabla_1 (\frac{\lambda'}{u})=\frac{u-\lambda'\kappa_1}{u^2}\nabla_1 \lambda'=0.
$$
Let $Q_1$ and $\hat{Q}_1$ be the remaining terms in the RHS of \eqref{s5:Q1} and \eqref{s5:Q2}, respectively. Since $\Phi'(s)>0$ for all $s>0$, it remains to show that
\begin{align}\label{s4:Q1}
\frac{Q_1}{\Phi'(\frac{u}{\lambda'}F)\frac{u}{\lambda'}}=&\ddot{F}^{kl,pq}\nabla_1 h_{kl}\nabla_1 h_{pq}+\frac{\Phi''(\frac{u}{\lambda'}F)}{\Phi'(\frac{u}{\lambda'}F)}\frac{u}{\lambda'}|\nabla_1F|^2-\frac{2}{\lambda'}\dot{F}^{kl}\nabla_k(\frac{u}{\lambda'})\nabla_l\lambda' \nonumber\\
&+2\sup_{\Lambda} \dot{F}^{kl}(2\Lambda_k^p \nabla_l S_{1p}-\Lambda_k^p\Lambda_l^q S_{pq}) \geq 0,
\end{align}
or
\begin{align}\label{s4:Q1-2}
\frac{\hat{Q}_1}{\Phi'(F)}=&\ddot{F}^{kl,pq}\nabla_1 h_{kl}\nabla_1 h_{pq}+\frac{\Phi''(F)}{\Phi'(F)}|\nabla_1F|^2-\frac{2}{\lambda'}\dot{F}^{kl}\nabla_k(\frac{u}{\lambda'})\nabla_l\lambda' \nonumber\\
&+2\sup_{\Lambda} \dot{F}^{kl} (2\Lambda_k^p \nabla_l S_{1p}-\Lambda_k^p\Lambda_l^q S_{pq}) \geq 0.
\end{align}
Since $\Phi''(s)s+2\Phi'(s)\geq 0$ for all $s>0$, we have
$$
\frac{\Phi''(\frac{u}{\lambda'}F)}{\Phi'(\frac{u}{\lambda'}F)}\frac{u}{\lambda'} \geq -\frac{2}{F}, \quad \frac{\Phi''(F)}{\Phi'(F)} \geq -\frac{2}{F}.
$$
On the other hand, by assumption, $S_{11}=0$ and $\nabla_k S_{11}=0$ at $(x_0,t_0)$, we have
\begin{align}\label{s4:gradient-term}
\dot{F}^{kl} (2\Lambda_k^p \nabla_l S_{1p}-\Lambda_k^p\Lambda_l^q S_{pq})=&\sum_{k=1}^{n}\sum_{p=2}^{n}\dot{f}^k(2\Lambda_k^p \nabla_k S_{1p}-(\Lambda_k^p)^2 S_{pp}) \nonumber \\
=& \sum_{k=1}^{n} \sum_{p=2}^{n}\dot{f}^k \(\frac{(\nabla_k S_{1p})^2}{S_{pp}}-\(\Lambda_k^p-\frac{\nabla_k S_{1p}}{S_{pp}}\)^2 S_{pp}\).
\end{align}
Then the supremum of the last line in \eqref{s4:gradient-term} is obtained by choosing $\Lambda_k^p=\frac{\nabla_k S_{1p}}{S_{pp}}$. Thus it suffices to check that
\begin{align}\label{s4:Q1-I}
\tilde{Q}_1=\ddot{F}^{kl,rs}\nabla_1 h_{kl}\nabla_1 h_{rs}-\frac{2}{F}|\nabla_1 F|^2-\frac{2}{\lambda'}\dot{F}^{kl}\nabla_k(\frac{u}{\lambda'})\nabla_l\lambda'+2\sum_{k=1}^{n}\sum_{l>1}\dot{f}^k \frac{(\nabla_k S_{1l})^2}{S_{ll}}\geq 0.
\end{align}
By Codazzi equations, we have
$$
\nabla_k S_{1l}=\nabla_k h_{1l}-\nabla_k(\frac{u}{\lambda'})\delta_{1l}=\nabla_1 h_{kl}, \quad \forall l>1.
$$
Since $0=\nabla_k S_{11}=\nabla_k (h_{11}-\frac{u}{\lambda'})$ for $k\geq 1$, we deduce that
\begin{align}\label{s4:nabla-h11}
\nabla_k h_{11}=\nabla_k(\frac{u}{\lambda'})=\frac{\kappa_k-\frac{u}{\lambda'}}{\lambda'}\nabla_k\lambda', \quad \forall k\geq 1.
\end{align}
As in \cite[\S 3]{AW18}, we use \eqref{s2:second-derivative-expr} to express the second derivatives of $F$, and by \eqref{s4:nabla-h11} we compute that
\begin{align}\label{s4:static-cov1}
\tilde{Q}_1=&\ddot{f}^{kl}\nabla_1 h_{kk} \nabla_1 h_{ll}+2\sum_{k>l}\frac{\dot{f}^k-\dot{f}^l}{\kappa_k-\kappa_l}(\nabla_1 h_{kl})^2-\frac{2}{f}|\nabla_1 F|^2-2\sum_{l>1}\dot{f}^{l}\frac{\kappa_l-\frac{u}{\lambda'}}{\lambda'^2}|\nabla_l\lambda'|^2 \nonumber\\
&+2\sum_{l>1}\dot{f}^1 \frac{\kappa_l-\frac{u}{\lambda'}}{\lambda'^2}|\nabla_l\lambda'|^2+2\sum_{k>1,l>1}\dot{f}^k \frac{(\nabla_1 h_{kl})^2}{\kappa_l-\frac{u}{\lambda'}}.
\end{align}
To make use of the inverse concavity of $f$, let $\tau_i=\kappa_i^{-1}$ and $f_{\ast}(\tau)=f(\kappa)^{-1}$. A direct calculation gives
\begin{align*}
\dot{f}^k=f_{\ast}^{-2}\frac{\partial f_\ast}{\partial \tau_k}\frac{1}{\kappa_k^2},\quad
\ddot{f}^{kl}=-f_{\ast}^{-2} \frac{\partial^2 f_\ast}{\partial \tau_k\partial \tau_l}\frac{1}{\kappa_k^2\kappa_l^2}+2f^{-1}\dot{f}^{k}\dot{f}^{l}-2\frac{\dot{f}^k}{\kappa_k}\delta_{kl}.
\end{align*}
By the concavity of $f_\ast$, the first term of \eqref{s4:static-cov1} can be estimated as follows
\begin{align}\label{s4:inverse-concavity}
\ddot{f}^{kl}\nabla_1 h_{kk} \nabla_1 h_{ll} \geq 2\(f^{-1}|\nabla_1 F|^2-\sum_{k}\frac{\dot{f}^k}{\kappa_k}(\nabla_1 h_{kk})^2\).
\end{align}
Substituting \eqref{s4:inverse-concavity} into \eqref{s4:static-cov1}, we get
\begin{align}\label{s4:crucial-ineq}
\tilde{Q}_1\geq &-2\sum_{k}\frac{\dot{f}^k}{\kappa_k}(\nabla_1 h_{kk})^2+2\sum_{k>l}\frac{\dot{f}^k-\dot{f}^l}{\kappa_k-\kappa_l}(\nabla_1 h_{kl})^2+2\sum_{k>1,l>1}\dot{f}^k \frac{(\nabla_1 h_{kl})^2}{\kappa_l-\frac{u}{\lambda'}} \nonumber\\
&+2\sum_{l>1}\dot{f}^1 \frac{\kappa_l-\frac{u}{\lambda'}}{\lambda'^2}|\nabla_l\lambda'|^2 -2\sum_{l>1}\dot{f}^{l}\frac{\kappa_l-\frac{u}{\lambda'}}{\lambda'^2}|\nabla_l\lambda'|^2 \nonumber\\
\geq &-2\sum_{k>1}\frac{\dot{f}^k}{\kappa_k}(\nabla_1 h_{kk})^2-2\sum_{k\neq l>1}\frac{\dot{f}^k}{\kappa_l}(\nabla_1 h_{kl})^2+2\sum_{k>1,l>1}\dot{f}^k \frac{(\nabla_1 h_{kl})^2}{\kappa_l-\frac{u}{\lambda'}} \nonumber\\
&+2\sum_{l>1}\dot{f}^1 \frac{\kappa_l-\frac{u}{\lambda'}}{\lambda'^2}|\nabla_l\lambda'|^2 -2\sum_{l>1}\dot{f}^{l}\frac{\kappa_l-\frac{u}{\lambda'}}{\lambda'^2}|\nabla_l\lambda'|^2 \nonumber\\
= &2\sum_{k>1,l>1}\dot{f}^k\(\frac{1}{\kappa_l-\frac{u}{\lambda'}}-\frac{1}{\kappa_l}\)(\nabla_1 h_{kl})^2+2\sum_{l>1}(\dot{f}^1-\dot{f}^{l})\frac{\kappa_l-\frac{u}{\lambda'}}{\lambda'^2}|\nabla_l\lambda'|^2 \nonumber\\
\geq & 0,
\end{align}
where we used \eqref{f-inverse-concave-II} in the second inequality, and the last inequality follows from
$$
0<\frac{u}{\lambda'}=\kappa_1 <\kappa_2< \cdots < \kappa_n,
$$
and $0<\dot{f}^n \leq \cdots \leq \dot{f}^1$ by \eqref{f-concave-II} since $F$ is concave. By the tensor maximum principle, the {\em static convexity} is preserved along the flow \eqref{s2:general-form-flow-I} or the flow \eqref{s2:general-form-flow-II}.
We show that $M_t$ is strictly {\em static convex} for $t>0$. By the strong maximum principle, the inequality becomes strict (and hence the strict increase of the theorem is proved) unless there exists a parallel vector field $v=e_1$ such that $S_{ij}v^iv^j=S_{11}=0$ on $M_{t_0}$. Hence, the smallest principal curvature $\kappa_1$ is $\frac{u}{\lambda'}$ on $M_{t_0}$ everywhere, which is strictly less than one since $u = \lambda\langle \partial_r,\nu\rangle \leq \lambda <\lambda'$. This contradicts with the fact that on any closed hypersurface in $\mathbb{H}^{n+1}$, there exists at least one point where all the principal curvatures are strictly greater than one. This completes the proof. \end{proof}
\begin{rem}
The above argument can be used to show that the static convexity is also preserved along the (purely) inverse curvature flows
$$
\frac{\partial}{\partial t}X(x,t)=\frac{1}{F^p}\nu(x,t), \quad 0<p\leq 1
$$
in hyperbolic space $\mathbb H^{n+1}$ provided that $F$ satisfies Assumption \ref{s1:Assumption}. \end{rem}
\section{Longtime existence and exponential convergence}\label{sec:7} In this section, we complete the proof of Theorem \ref{s1:main-thm-I}.
We assume that the initial hypersurface $M_0$ is star-shaped, then there exists a point $o$ inside the enclosed domain $\Omega_0$ by $M_0$. We take $o$ as the origin of the hyperbolic space $\mathbb H^{n+1}$, and express the hyperbolic space $\mathbb H^{n+1}$ as a warped product $\mathbb{R}^+\times \mathbb S^n$ with the metric $\-g=dr^2+\lambda^2(r)\sigma$, where $r$ is the distance to the origin $o$. As discussed in $\S \ref{s2:sec-2.2}$, we equivalently write the flow equation \eqref{s1:locally-MCF} as a scalar parabolic PDE on $\mathbb S^n$ for the function $\varphi(\cdot,t)$. By \eqref{s2:evolution-parametric-form}, the function $\varphi$ satisfies \begin{align}\label{s5:evol-varphi-II} \frac{\partial}{\partial t}\varphi=\frac{v}{\lambda}-\frac{E_1}{\lambda'}, \end{align}
where $v=\sqrt{1+|D\varphi|^2}$. By the expression \eqref{s2:2nd-fundamental-form} of the Weingarten matrix $\mathcal{W}=(h_i^j)$, we have \begin{align*} E_1=&\frac{\lambda'}{\lambda v}-\frac{1}{n\lambda v}\(\sigma^{ki}-\frac{\varphi^k\varphi^i}{v^2}\)\varphi_{ik}. \end{align*} Then Eq. \eqref{s5:evol-varphi-II} can be rewritten as a scalar parabolic PDE in divergent form \begin{align}\label{s5:evol-varphi-II-divergent-type}
\frac{\partial}{\partial t}\varphi=&\frac{1}{n\lambda\lambda'v}\(\sigma^{ki}-\frac{\varphi^k\varphi^i}{v^2}\)\varphi_{ik}+\frac{|D\varphi|^2}{\lambda v} \nonumber\\
=&\mrm{div}\(\frac{D\varphi}{n\lambda\lambda'v}\)+\((n+1)\lambda'^2+\lambda^2\)\frac{|D\varphi|^2}{n\lambda \lambda'^2 v}, \end{align} where we used $v_k=\frac{\varphi^i\varphi_{ik}}{v}$ and \begin{align}\label{s5:derivative-lambda} D_i \lambda'=\lambda''r_i=\lambda^2 \varphi_i, \quad D_i\lambda=\lambda'r_i=\lambda\lambda'\varphi_i. \end{align}
The $C^0$-estimate of the solution to Eq. \eqref{s5:evol-varphi-II} follows from Proposition \ref{s3:prop-C0-estimate}. To deduce the $C^1$-estimate of $\varphi$, we first calculate the evolution equation of $|D\varphi|^2$.
\begin{lem}\label{s5:lem-evol-Dvarphi2} Let $\varphi$ be a solution of \eqref{s5:evol-varphi-II} and assume that $|D \varphi|^2$ attains its maximum at a point $P$. Then at $P$, there holds
\begin{align}\label{s5:evol-Dvarphi2-II}
\frac{\partial}{\partial t}|D\varphi|^2 =&\frac{1}{n\lambda\lambda'v}\(\sigma^{ij}-\frac{\varphi^i\varphi^j}{v^2}\)(|D\varphi|^2)_{ij}-\frac{2}{n\lambda\lambda'v}\varphi_i^k\varphi_k^i-\frac{2(n-1)}{n\lambda\lambda'v}|D\varphi|^2 \nonumber\\
&-\frac{2\Delta \varphi |D\varphi|^2}{n\lambda v}-\frac{2\Delta \varphi \lambda |D\varphi|^2}{n\lambda'^2 v}-\frac{2\lambda'|D\varphi|^4}{\lambda v}.
\end{align} \end{lem} \begin{proof}
At the maximum point $P$ of $|D\varphi|^2$, we have
\begin{align*}
D|D\varphi|^2=0, \quad Dv=0.
\end{align*}
In view of Eq. \eqref{s5:evol-varphi-II-divergent-type}, we derive
\begin{align}\label{s5:evol-step-1}
\frac{\partial}{\partial t}|D\varphi|^2=&2\varphi^i D_i \( \frac{\Delta\varphi}{n\lambda\lambda'v}-\frac{v_j\varphi^j}{n\lambda\lambda' v^2}+\frac{|D\varphi|^2}{\lambda v}\) \nonumber\\
=& 2 \( \frac{ \varphi^i D_i(\Delta\varphi)}{n\lambda\lambda'v}-\frac{\Delta \varphi |D\varphi|^2}{n\lambda v}-\frac{\Delta \varphi \lambda |D\varphi|^2}{n\lambda'^2 v}-\frac{v_{ji}\varphi^j\varphi^i}{n\lambda\lambda'v^2}-\frac{\lambda'|D\varphi|^4}{\lambda v}\).
\end{align}
where we used \eqref{s5:derivative-lambda}. Using the Ricci identity on $\mathbb S^n$, we have
\begin{align*}
(|D\varphi|^2)_i=&2\varphi^k \varphi_{ki}, \nonumber\\
(|D\varphi|^2)_{ij}=&2 \varphi_j^k\varphi_{ki}+2\varphi^k \varphi_{kij}\nonumber\\
=&2 \varphi_j^k\varphi_{ki}+2\varphi^k \(\varphi_{ijk}-\varphi_j \delta_k^i+\varphi_k \delta_i^j\)\nonumber\\
=&2 \varphi_j^k\varphi_{ki}+2\varphi^k \varphi_{ijk}-2\varphi_i\varphi_j +2|D\varphi|^2 \delta_i^j.
\end{align*}
Then we get
\begin{align}\label{s5:Ricci-app}
\varphi^i D_i(\Delta\varphi)=\frac{1}{2}\Delta |D\varphi|^2-\varphi_i^k\varphi_k^i-(n-1)|D\varphi|^2.
\end{align}
Substituting \eqref{s5:Ricci-app} into \eqref{s5:evol-step-1}, we get
\begin{align*}
\frac{\partial}{\partial t}|D\varphi|^2=&\frac{1}{n\lambda\lambda'v}\(\Delta |D\varphi|^2-2\varphi_i^k\varphi_k^i-2(n-1)|D\varphi|^2\)\\
&-\frac{2\Delta \varphi |D\varphi|^2}{n\lambda v}-\frac{2\Delta \varphi \lambda |D\varphi|^2}{n\lambda'^2 v}-\frac{2v_{ji}\varphi^j\varphi^i}{n\lambda\lambda'v^2}-\frac{2\lambda'|D\varphi|^4}{\lambda v}\\
=&\frac{1}{n\lambda\lambda'v}\(\sigma^{ij}-\frac{\varphi^i\varphi^j}{v^2}\)(|D\varphi|^2)_{ij}-\frac{2}{n\lambda\lambda'v}\varphi_i^k\varphi_k^i-\frac{2(n-1)}{n\lambda\lambda'v}|D\varphi|^2 \\
&-\frac{2\Delta \varphi |D\varphi|^2}{n\lambda v}-\frac{2\Delta \varphi \lambda |D\varphi|^2}{n\lambda'^2 v}-\frac{2\lambda'|D\varphi|^4}{\lambda v},
\end{align*}
where we used $(|D\varphi|^2)_{ij}=(v^2)_{ij}=2v v_{ij}$ since $v_i=0$ at $P$. \end{proof}
Now we prove the $C^1$-estimate of $\varphi$ and exponential decay of $|D\varphi|^2$. \begin{prop}\label{s5:prop-locally-MCF-gradient-estimate}
Let $\varphi=\varphi(\theta,t),t\in[0,T^\ast)$ be a solution of \eqref{s5:evol-varphi-II}. Then there exists a uniform constant $\alpha>0$ which depends only on the initial hypersurface $M_0$ such that
\begin{align}\label{s5:exponential-convergence-II}
|D\varphi|^2(\theta,t) \leq \max_{\theta\in \mathbb S^n}|D\varphi|^2(\theta,0) e^{-\alpha t}, \quad t\in [0,T^\ast).
\end{align} \end{prop} \begin{proof}
By Lemma \ref{s5:lem-evol-Dvarphi2}, at the maximum point $P$ of $|D\varphi|^2$, we have
\begin{align*}
\frac{\partial}{\partial t}|D\varphi|^2 \leq &-\frac{2}{n^2\lambda\lambda'v}|\Delta \varphi|^2-\frac{2(n-1)}{n\lambda\lambda'v}|D\varphi|^2 -\frac{2\Delta \varphi |D\varphi|^2}{n\lambda v}-\frac{2\Delta \varphi \lambda |D\varphi|^2}{n\lambda'^2 v}-\frac{2\lambda'|D\varphi|^4}{\lambda v},
\end{align*}
where we used the trace inequality $\varphi_i^k\varphi_k^i \geq \frac{1}{n}|\Delta \varphi|^2$. Then by completing the square, we obtain
\begin{align*}
\frac{\partial}{\partial t}|D\varphi|^2 \leq &-\frac{2}{n\lambda\lambda'^2v}\( \frac{\lambda'|\Delta \varphi|^2}{n}+(\lambda'^2+\lambda^2)\Delta \varphi |D\varphi|^2+\frac{n(\lambda'^2+\lambda^2)^2}{4\lambda'}|D\varphi|^4\)\nonumber\\
&-\frac{2(n-1)}{n\lambda\lambda'v}|D\varphi|^2 +\(\frac{(\lambda'^2+\lambda^2)^2}{2\lambda\lambda'^3v}-\frac{2\lambda'}{\lambda v}\)|D\varphi|^4 \\
\leq &-\frac{2(n-1)}{n\lambda\lambda'v}|D\varphi|^2,
\end{align*}
where we used $\frac{(\lambda'^2+\lambda^2)^2}{2\lambda\lambda'^3}\leq \frac{2\lambda'}{\lambda}$ since $\lambda'>\lambda$. Therefore, standard maximum principle yields
$$
|D\varphi|^2 (\theta,t)\leq C:=\max_{\theta\in\mathbb S^n}|D\varphi|^2(\theta,0).
$$
Next, we prove the exponential estimate. Since $v=\sqrt{1+|D\varphi|^2}\leq C$, together with the $C^0$ estimate by Proposition \ref{s3:prop-C0-estimate}, there exists a uniform constant $\alpha\geq \frac{2(n-1)}{n\lambda\lambda'v}>0$ such that
\begin{align*}
\frac{\partial}{\partial t}|D\varphi|^2 \leq -\alpha |D\varphi|^2.
\end{align*}
and hence \eqref{s5:exponential-convergence-II} follows. \end{proof}
Since Eq. \eqref{s5:evol-varphi-II} can be rewritten as Eq. \eqref{s5:evol-varphi-II-divergent-type}, then by the classical theory of parabolic PDE in divergent form \cite{Lady68}, the higher regularity a priori estimates of the solution $\varphi$ follow from the uniform gradient estimate in Proposition \ref{s5:prop-locally-MCF-gradient-estimate}. Moreover, the solution exists for all $t\in [0,\infty)$, i.e., $T^\ast=\infty$. The exponential convergence to a geodesic sphere $\partial B_{r_\infty}$ centered at the origin follows from the estimate \eqref{s5:exponential-convergence-II}. Since $W_0^{\lambda'}(\Omega_t)$ is constant along the flow \eqref{s1:locally-MCF} with $F=E_1$, the radius $r_\infty$ is uniquely determined by $W_0^{\lambda'}(B_{r_\infty})=W_0^{\lambda'}(\Omega_0)$, where $\Omega_0$ is the bounded domain enclosed by the initial hypersurface $M_0$. By Theorem \ref{thm-static-convexity}, if the initial hypersurface $M_0$ is static convex, then the flow hypersurface $M_t$ becomes strictly static convex for $t>0$. This completes the proof of Theorem \ref{s1:main-thm-I}.
\section{Applications to geometric inequalities}\label{sec:8} In this section, we give the proofs of Theorems \ref{thm-weighted-quermassintegral-ineq-I}, \ref{thm-geometric-ineq} and \ref{thm-geometric-ineq-I}.
\begin{proof}[Proof of Theorem \ref{thm-weighted-quermassintegral-ineq-I}]
{\bf Case 1.} ($1\leq k\leq n-1$.) We choose $F=E_1$ in the flow \eqref{s1:locally-MCF}. By Theorem \ref{s1:main-thm-I}, the flow hypersurface $M_t$ is static convex and hence it is strictly convex. By \eqref{s6:evol-weighted-curvature-integral}, we have
\begin{align*}
\frac{d}{dt}W^{\lambda'}_0(\Omega_t)=&(n+1)\int_{M_t}\lambda'\(1-\frac{uE_1}{\lambda'}\)d\mu_t=0,
\end{align*}
and for $1\leq k\leq n-1$,
\begin{align*}
\frac{d}{dt}W^{\lambda'}_{k}(\Omega_t)=&\int_{M_t}( kuE_{k-1}+(n+1-k)\lambda'E_k)\(1-\frac{uE_1}{\lambda'}\)d\mu_t \\
=&k\int_{M_t} \frac{u}{\lambda'}(\lambda'E_{k-1}-u E_{k-1}E_1)d\mu_t+(n+1-k)\int_{M_t} (\lambda'E_k-uE_kE_1)d\mu_t \\
\leq &k \int_{M_t} \frac{u}{\lambda'}(\lambda'E_{k-1}-u E_k) d\mu_t,
\end{align*}
where we used Newton-MacLaurin inequality $E_{k-1} E_1 \geq E_k$ and $E_k E_1 \geq E_{k+1}$ for $1\leq k\leq n-1$ and Minkowski formula \eqref{s2:Minkowski-formula}.
It follows from \eqref{newton-formula-1}, \eqref{newton-formula-2} and \eqref{s2:2.1} that
\begin{align}\label{s6:Minkowski}
\dot{E}_m^{ij} \nabla_i \nabla_j \lambda'=\dot{E}_m^{ij}(\lambda'g_{ij}-uh_{ij})=m(\lambda'E_{m-1}-uE_m), \quad m=1,\cdots,n.
\end{align}
Since $\dot{E}_k^{ij}$ is divergence-free, by \eqref{s6:Minkowski} and integration by parts, we deduce that
\begin{align}\label{s6:monotone}
\frac{d}{dt}W^{\lambda'}_{k}(\Omega_t) \leq& \int_{M_t} \frac{u}{\lambda'} \dot{E}_{k}^{ij} \nabla_i\nabla_j\lambda' d\mu_t \nonumber\\
= &-\int_{M_t} \dot{E}_{k}^{ij}\nabla_i(\frac{u}{\lambda'})\nabla_j\lambda' d\mu_t \nonumber \\
= &-\int_{M_t} \dot{E}_{k}^{ii}\frac{\lambda'\kappa_i-u}{\lambda'^2}|\nabla_i\lambda'|^2 d\mu_t \leq 0.
\end{align}
Here the last inequality follows from the fact that $\lambda'\kappa_i-u\geq 0$ and $\dot{E}_k^{ij}$ is positive definite, since hypersurface $M_t$ is static convex. By Theorem \ref{s1:main-thm-I}, we get
\begin{align*}
W^{\lambda'}_{k}(\Omega_0) \geq &W^{\lambda'}_{k}(B_{r_{\infty}})= h_{k}(r_{\infty}) = h_{k}\circ h_{0}^{-1}(W^{\lambda'}_{0}(B_{r_\infty})) \nonumber\\
=&h_{k}\circ h_{0}^{-1}(W^{\lambda'}_{0}(\Omega_0)), \quad 1\leq k\leq n-1.
\end{align*}
{\bf Case 2.} ($2\leq k \leq n+1$.) We choose $F=E_{k-1}/E_{k-2}$ in the flow \eqref{s1:SX-ICF}. By Theorem \ref{s1:main-thm-III}, the flow hypersurface $M_t$ is static convex and hence it is strictly convex. In view of \eqref{s6:evol-weighted-curvature-integral}, we have
\begin{align}\label{s6:case-2-monotonicity-1}
\frac{d}{dt}W^{\lambda'}_{0}(\Omega_t)=&(n+1)\int_{M_t}\(\lambda'\frac{E_{k-2}}{E_{k-1}}-u\) d\mu_t \geq (n+1)\int_{M_t}\(\frac{\lambda'}{E_1}-u\) d\mu_t \geq 0,
\end{align}
where we used Newton-MacLaurin inequality $E_{1}E_{k-2} \geq E_{k-1}$ for $2\leq k \leq n+1$ in the first inequality, and Heintze-Karcher inequality \cite[Theorem 3.5]{Brendle2013} in the second inequality.
On the other hand, we derive
\begin{align}\label{s6:case-2-monotonicity-2}
\frac{d}{dt}W^{\lambda'}_{k}(\Omega_t)=&\int_{M_t}(k u E_{k-1}+(n+1-k)\lambda'E_{k})\(\frac{E_{k-2}}{E_{k-1}}-\frac{u}{\lambda'}\) d\mu_t \nonumber\\
=&k \int_{M_t} \frac{u}{\lambda'} \( \lambda'E_{k-2}-u E_{k-1}\) d\mu_t+(n+1-k)\int_{M_t}\(\lambda'\frac{E_{k}E_{k-2}}{E_{k-1}}-uE_{k}\)d\mu_t \nonumber\\
\leq &k \int_{M_t}\frac{u}{\lambda'} (\lambda' E_{k-2}-uE_{k-1})d\mu_t,
\end{align}
where we used Newton-MacLaurin inequality $E_{k}E_{k-2} \leq E_{k-1}^2$ for $2\leq k \leq n+1$ and Minkowski formula \eqref{s2:Minkowski-formula}. Since $\dot{E}_{k-1}^{ij}$ is divergence-free, by integration by parts and \eqref{s6:Minkowski}, we have
\begin{align}\label{s6:monotone-locally-MCF}
\frac{d}{dt}W^{\lambda'}_{k}(\Omega_t) \leq& \int_{M_t} \frac{k}{k-1}\frac{u}{\lambda'} \dot{E}_{k-1}^{ij}\nabla_i\nabla_j\lambda' d\mu_t \nonumber\\
= &-\frac{k}{k-1}\int_{M_t} \dot{E}_{k-1}^{ij}\nabla_i(\frac{u}{\lambda'})\nabla_j\lambda' d\mu_t \nonumber \\
= &-\frac{k}{k-1}\int_{M_t} \dot{E}_{k-1}^{ii}\frac{\lambda'\kappa_i-u}{\lambda'^2}|\nabla_i\lambda'|^2 d\mu_t \leq 0,
\end{align}
where the last inequality follows from $\lambda'\kappa_i -u\geq 0$ and $\dot{E}_{k-1}^{ij}$ is positive definite, since $M_t$ is static convex (by Theorem \ref{s1:main-thm-III}). Combining \eqref{s6:case-2-monotonicity-1} and \eqref{s6:case-2-monotonicity-2} with the convergence of the flow \eqref{s1:SX-ICF} (Theorem B), we deduce that
\begin{align*}
W^{\lambda'}_{k}(\Omega_0) \geq &W^{\lambda'}_{k}(B_{r_{\infty}})= h_{k}(r_{\infty}) = h_{k}\circ h_{0}^{-1}(W^{\lambda'}_{0}(B_{r_\infty})) \\
\geq &h_{k}\circ h_{0}^{-1}(W^{\lambda'}_{0}(\Omega_0)), \quad 2\leq k\leq n+1.
\end{align*}
If equality holds in \eqref{weighted-quermassintegral-ineq-I}, then equality also holds in \eqref{s6:monotone} or \eqref{s6:monotone-locally-MCF}. Since $M_t$ is strictly static convex for $t>0$, this implies that $\nabla\lambda'\equiv 0$ on $M_t$ and hence $M_t$ must be a geodesic sphere centered at the origin for $t>0$. Therefore, by smooth approximation of $M_t$ as $t\rightarrow 0$, the initial hypersurface $M_0$ is also a geodesic sphere centered at the origin. This completes the proof of Theorem \ref{thm-weighted-quermassintegral-ineq-I}. \end{proof}
\begin{proof}[Proof of Theorem \ref{thm-geometric-ineq}]
We choose $F=E_1$ in the flow \eqref{s1:locally-MCF}. In view of \eqref{s6:evol-weighted-curvature-integral}, we have
\begin{align*}
\frac{d}{dt}W^{\lambda'}_{0}(\Omega_t)=&(n+1)\int_{M_t} \lambda'\(1-\frac{uE_1}{\lambda'}\)d\mu_t=0,
\end{align*}
where we used Minkowski formula \eqref{s2:Minkowski-formula}. On the other hand, we have
\begin{align}\label{s7:monotonicity-geometric-ineq-I}
\frac{d}{dt}W_0(\Omega_t)=&(n+1)\int_{M_t} \(1-\frac{uE_1}{\lambda'}\)d\mu_t \nonumber\\
=&\frac{n+1}{n} \int_{M_t} \frac{\Delta \lambda'}{\lambda'} d\mu_t \nonumber\\
=&\frac{n+1}{n} \int_{M_t} \frac{|\nabla \lambda'|^2}{\lambda'^2} d\mu_t\geq 0.
\end{align}
By Theorem \ref{s1:main-thm-I}, the flow hypersurface $M_t$ converges smoothly to a geodesic sphere $\partial B_{r_\infty}$ centered at the origin, we obtain
\begin{align*}
W^{\lambda'}_{0}(\Omega_0)= &W^{\lambda'}_{0}(B_{r_{\infty}})=h_0(r_\infty)=h_0\circ f_0^{-1}(W_{0}(B_{r_{\infty}})) \\
\geq &h_0\circ f_0^{-1}(W_{0}(\Omega_0)).
\end{align*}
If equality holds in \eqref{geom-ineq-1}, then equality holds in \eqref{s7:monotonicity-geometric-ineq-I}, which implies that the initial hypersurface $M_0$ is a geodesic sphere centered at the origin. This completes the proof of Theorem \ref{thm-geometric-ineq}. \end{proof}
\begin{proof}[Proof of Theorem \ref{thm-geometric-ineq-I}]
{\bf Case 1.} ($0\leq k \leq n$ and $m=0$.) This follows immediately from Theorems \ref{thm-weighted-quermassintegral-ineq-I} and \ref{thm-geometric-ineq}, that is,
$$
W_{k+1}^{\lambda'}(\Omega_0)\geq h_{k+1}\circ h_0^{-1}(W_{0}^{\lambda'}(\Omega_0))\geq h_{k+1}\circ f_0^{-1}(W_{0}(\Omega_0)).
$$
Moreover, if equality holds in \eqref{geom-ineq-2} in Case 1, the initial hypersurface $M_0$ is a geodesic sphere centered at the origin.
{\bf Case 2.} ($1\leq k\leq n$ and $1\leq m\leq k$.) We choose $F=E_{k}/E_{k-1}$ in the flow \eqref{s1:SX-ICF}. By Theorem \ref{s1:main-thm-III}, the flow hypersurface $M_t$ is static convex and hence it is strictly convex.
By \eqref{s2:variation-quermassintegral} and \eqref{s6:Minkowski}, we derive
\begin{align}\label{s6:geom-ineq-2}
\frac{d}{dt}W_m(\Omega_t)=&\frac{n+1-m}{n+1}\int_{M_t} E_m \(\frac{E_{k-1}}{E_{k}}-\frac{u}{\lambda'}\)d\mu_t \nonumber\\
\geq &\frac{n+1-m}{n+1}\int_{M_t}\frac{\lambda'E_{m-1}-uE_m}{\lambda'}d\mu_t \nonumber\\
=&\frac{n+1-m}{m(n+1)}\int_{M_t} \frac{\dot{E}_{m}^{ij}\nabla_i\lambda'\nabla_j\lambda'}{\lambda'^2}d\mu_t\geq 0,
\end{align}
where in the first inequality we used $E_m E_{k-1} \geq E_{m-1}E_{k}$ for $1\leq m\leq k$, and the last inequality follows from the positivity of $\dot{E}_m^{ij}$.
On the other hand, by \eqref{s6:evol-weighted-curvature-integral} we have
\begin{align*}
\frac{d}{dt}W^{\lambda'}_{k+1}(\Omega_t) = & \int_{M_t} \( (k+1)uE_{k}+(n-k)\lambda'E_{k+1}\)\(\frac{E_{k-1}}{E_{k}}-\frac{u}{\lambda'}\)d\mu_t\\
= &(k+1) \int_{M_t} \frac{u}{\lambda'}(\lambda' E_{k-1}-u E_{k})d\mu_t+(n-k)\int_{M_t} \(\lambda'\frac{E_{k+1}E_{k-1}}{E_{k}}-uE_{k+1}\)d\mu_t \\
\leq & (k+1) \int_{M_t} \frac{u}{\lambda'} (\lambda'E_{k-1}-uE_{k})d\mu_t,
\end{align*}
where we used Newton-MacLaurin inequality $E_{k+1}E_{k-1}\leq E_{k}^{2}$, and Minkowski formula \eqref{s2:Minkowski-formula} and $E_{n+1}=0$ by convention. Finally, by using \eqref{s6:Minkowski} we get
\begin{align}\label{s6:Monotonicity-formula}
\frac{d}{dt}W^{\lambda'}_{k+1}(\Omega_t) \leq & -\frac{k+1}{k}\int_{M_t}\dot{E}_{k}^{ij}\nabla_i(\frac{u}{\lambda'})\nabla_j\lambda' d\mu_t\nonumber\\
=&-\frac{k+1}{k} \int_{M_t}\sum_{i}\dot{E}_{k}^{ii}\frac{\lambda'\kappa_i-u}{\lambda'^2}|\nabla_i\lambda'|^2 d\mu_t \leq 0,
\end{align}
where the last inequality follows from $\lambda'\kappa_i -u \geq 0$ and $\dot{E}_{k}^{ij}$ is positive definite, since $M_t$ is static convex.
By \eqref{s6:geom-ineq-2}, \eqref{s6:Monotonicity-formula} and the convergence of the flow \eqref{s1:SX-ICF} with $F=E_{k}/E_{k-1}$ (Theorem B), we have
\begin{align*}
W^{\lambda'}_{k+1}(\Omega_0) \geq &W^{\lambda'}_{k+1}(B_{r_\infty}) = h_{k+1}(r_\infty) =h_{k+1}\circ f_m^{-1}(W_m(B_{r_\infty})) \\
\geq &h_{k+1}\circ f_m^{-1}(W_m(\Omega_0)), \quad 1\leq m\leq k.
\end{align*}
If equality holds in \eqref{geom-ineq-2} in Case 2, then equality holds in \eqref{s6:Monotonicity-formula}. Since $M_t$ is strictly static convex for $t>0$, the similar argument as above implies that the initial hypersurface $M_0$ is a geodesic sphere centered at the origin. This completes the proof of Theorem \ref{thm-geometric-ineq-I}. \end{proof}
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[1] Piecewise Continuous Almost Automorphic Functions and Favard's Theorems for Impulsive Differential Equations In Honor of Russell Johnson
Liangping Qi Tianjin University of Finance and Economics Rong Yuan Beijing Normal University
Classical Analysis and ODEs mathscidoc:2103.05004
Journal of Dynamics and Differential Equations, 2020.8
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We define piecewise continuous almost automorphic (p.c.a.a.) functions in the manners of Bochner, Bohr and Levitan, respectively, to describe almost automorphic motions in impulsive systems, and prove that with certain prefixed possible discontinuities they are equivalent to quasi-uniformly continuous Stepanov almost automorphic ones. Spatially almost automorphic sets on the line, which serve as suitable objects containing discontinuities of p.c.a.a. functions, are characterized in the manners of Bochner, Bohr and Levitan, respectively, and shown to be equivalent. Two Favard's theorems are established to illuminate the importance and convenience of p.c.a.a. functions in the study of almost periodically forced impulsive systems.
[2] Spectrum of the Lame operator and application, II: When an endpoint is a cusp
Zhijie Chen Tsinghua University Chang-Shou Lin Taiwan University
Communications in mathematical Physics, 378, 335–368, 2020
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[3] Spectrum of the Lame operator and application, I: Deformation along Re tau=1/2
Zhijie Chen Tsinghua University Erjuan Fu Tsinghua University Chang-Shou Lin Taiwan University
Advances in Mathematics, 383, 107699, 2021
[4] Representations of mock theta functions
Dandan Chen East China Normal University Liuquan Wang Wuhan University
Classical Analysis and ODEs Combinatorics Number Theory mathscidoc:2103.05001
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Motivated by the works of Liu, we provide a unified approach to find Appell-Lerch series and Hecke-type series representations for mock theta functions. We establish a number of parameterized identities with two parameters $a$ and $b$. Specializing the choices of $(a,b)$, we not only give various known and new representations for the mock theta functions of orders 2, 3, 5, 6 and 8, but also present many other interesting identities. We find that some mock theta functions of different orders are related to each other, in the sense that their representations can be deduced from the same $(a,b)$-parameterized identity. Furthermore, we introduce the concept of false Appell-Lerch series. We then express the Appell-Lerch series, false Appell-Lerch series and Hecke-type series in this paper using the building blocks $m(x,q,z)$ and $f_{a,b,c}(x,y,q)$ introduced by Hickerson and Mortenson, as well as $\overline{m}(x,q,z)$ and $\overline{f}_{a,b,c}(x,y,q)$ introduced in this paper. We also show the equivalences of our new representations for several mock theta functions and the known representations.
[5] Zeros of the deformed exponential function
Liuquan Wang Wuhan University Cheng Zhang Johns Hopkins University
Analysis of PDEs Classical Analysis and ODEs Combinatorics Number Theory mathscidoc:2103.03003
Let $f(x)=\sum_{n=0}^{\infty}\frac{1}{n!}q^{n(n-1)/2}x^n$ ($0<q<1$) be the deformed exponential function. It is known that the zeros of $f(x)$ are real and form a negative decreasing sequence $(x_k)$ ($k\ge 1$). We investigate the complete asymptotic expansion for $x_{k}$ and prove that for any $n\ge1$, as $k\to \infty$, \begin{align*} x_k=-kq^{1-k}\Big(1+\sum_{i=1}^{n}C_i(q)k^{-1-i}+o(k^{-1-n})\Big), \end{align*} where $C_i(q)$ are some $q$ series which can be determined recursively. We show that each $C_{i}(q)\in \mathbb{Q}[A_0,A_1,A_2]$, where $A_{i}=\sum_{m=1}^{\infty}m^i\sigma(m)q^m$ and $\sigma(m)$ denotes the sum of positive divisors of $m$. When writing $C_{i}$ as a polynomial in $A_0, A_1$ and $A_2$, we find explicit formulas for the coefficients of the linear terms by using Bernoulli numbers. Moreover, we also prove that $C_{i}(q)\in \mathbb{Q}[E_2,E_4,E_6]$, where $E_2$, $E_4$ and $E_6$ are the classical Eisenstein series of weight 2, 4 and 6, respectively. | CommonCrawl |
Sero-prevalence and risk factors associated with African swine fever on pig farms in southwest Nigeria
Emmanuel Jolaoluwa Awosanya1,2,
Babasola Olugasa1,
Gabriel Ogundipe1 &
Yrjo Tapio Grohn2
African swine fever (ASF) is one of the major setbacks to development of the pig industry in Nigeria. It is enzootic in southwest Nigeria. We determined the sero-prevalence and factors associated with ASF among-herd seropositivity in 144 pig farms in six States from southwest Nigeria during the dry and rainy seasons using indirect Enzyme Linked Immunosorbent Assay (ELISA) for ASF IgG antibodies. An interviewer-administered questionnaire was used to collect information on demography, environmental and management factors. We performed descriptive statistics, and univariate and multivariable analyses to determine the among-herd sero-prevalence of ASF and its associated factors.
The overall herd sero-prevalence of ASF was 28 % (95 % Confidence interval (95 % CI) 21 – 36); it was significantly higher (P <0.05) in the dry season (54 %; 95 % CI 37 – 70) than the rainy season (18 %; 95 % CI 11 – 27). In the univariate analysis, having a quarantine/ isolation unit within 100 m radius of a regular pig pen (OR = 3.3; 95 % CI 1.3 – 8.9), external source of replacement stock (OR = 3.2; 95 % CI 1.3 – 8.3) and dry season (OR = 5.3; 95 % CI 2.2 – 12.7) were risk factors for ASF among-herd seropositivity. In the multivariable logistic regression, there was interaction between season and herd size. Our final model included season, source of replacement stock, herd size and interaction between herd size and season. Herds with an external source of replacement always had higher ASF sero-prevalence compared with herds with an internal source. The herd size effect varied between seasons.
The ASF herd level sero-prevalence in southwest Nigeria was higher in pig herds with an external source of replacement stock and in the dry season. The effect of season of the year the samples were taken on ASF seropositivity was modified by herd size. We encourage strict compliance with biosecurity measures, especially using an internal source of replacement stock and measures that minimize movement on pig farms in southwest Nigeria, in order to enhance ASF free farms.
African swine fever (ASF) is a highly contagious and fatal viral disease of pigs caused by a DNA virus of the Asfarviridae family. It is a trans-boundary animal disease, defined as a disease of significant economic, trade and/or food security importance for a considerable number of countries, which can easily spread across national borders and reach epidemic proportions and for which control and management, including exclusion, require international co-operation [1].
Globally, the ASF virus is present in Africa, Italy (Sardinia), Georgia, Latvia, Poland, Ukraine, Russia (Moscow) and some Caribbean countries, with an increasing risk of spreading to ASF-free countries in Europe and America [2, 3]. African swine fever is the main threat to the pig industry in Africa because of the heavy losses incurred by pig farmers [4, 5] when it strikes, with mortality approaching 100 % [4].
Three epidemiological cycles have been recognized: the sylvatic [6], domestic [7], and sylvatic and domestic cycle [4, 8]. In Africa, all three have been reported; however, in Nigeria only the domestic cycle which maintains the ASF virus within domestic pigs is most recognized and reported [7] despite reports on detection of ASF virus in river hogs [9].
In Nigeria, the first ASF outbreak was reported in 1973 and subsequently in 1997, 1998 and 2001 [10–12]. Since the outbreak in 1997, there have been reported confirmed and unconfirmed sporadic outbreaks of ASF. African swine fever is enzootic in Nigeria [13, 14].
The pig industry in Nigeria can be classified into small holder farms – farms having fewer than 50 pigs in the herd at any point in time; medium holder farms – farms having from 50 to 100 pigs in the herd at any point in time and large holder farms – farms with over 100 pigs in the herd at any point in time. The pig farming industry in Nigeria has its largest presence in the southwest of Nigeria, with fewer high pig density areas in other geo-political zones in the country. Farming activities occur throughout the whole year with increased activities during festive periods in December. The pig production system in southwest Nigeria is predominantly confined within pig pens. The ASF scourge has however adversely affected the bustling and rising activities in this industry since the outbreak in 1997 [5]. Efforts have been made by the various State Governments through farm extension services in educating the farmers on biosecurity measures since that outbreak.
Several researchers have made efforts in contributing to the understanding of the dynamics of transmission, control and what will hopefully lead to its eventual eradication in Nigeria. Economic losses to farmers consequent to ASF scourge [5], molecular epidemiological description of the circulating strain of ASF virus in Nigeria [15, 16], surveillance of the ASF virus in both domestic and wild pig populations especially during the periods of outbreaks [9, 14, 17, 18], geographical and spatial spread of the ASF infection [19, 20] and identification of risk factors during ASF outbreaks at farm level [19] have been reported.
There had been a 5-year gap (2007–2012) between the last sero-monitoring of the ASF virus in southwest Nigeria [14]. Previous work on assessment of risk factors considered an outbreak situation; however, ASF has assumed a new enzootic status – the implication of this is that most infected farms are at subclinical level. Thus, factors responsible for the enzootic status of ASF in Nigeria are poorly elucidated.
This paper therefore attempts to address the above identified gaps of ASF herd level sero-status and unknown risk factors for ASF sero-positivity in an enzootic situation by assessing the current sero-status of pigs among herd ASF sero-prevalence and their associated factors in southwest Nigeria.
Of 144 respondents, 108 (75 %; 95 % CI 67 – 82) were males; their mean age was 49.2 ± 14.6 years. Most of the respondents (64 %; 95 % CI 55 – 72) had tertiary education; 53 % (95 % CI 44 – 61) practiced pig farming as their only source of livelihood. The median year of practice was 7 years (Range: 1 – 36 years). Most of the farms (81 %; 95 % CI 73 – 87) were established after the last report of an ASF outbreak in 2001; the range was from 1949 to 2013. All the pig herds were raised in strict pen confinement. The median number of pigs in the herd was 45 (Range: 2 – 567). The median age of the pigs sampled was 8 months (Range: 1 – 72 months). The majority of the breeds were crosses, mainly large whites. The previous year average mortality ranged from 0 to 99 pigs: most of the pig herds (80 %; 95 % CI 72 – 86) had average mortality within 0 to 12 pigs. Only 6 (4 %; 95 % CI 2 – 9) of the pig herds had a reported history of ASF outbreak. Most of the farmers (91 %, 95 % CI 85 – 95) had access to potable water on the farm.
ASF seropositivity
The overall herd sero-prevalence of ASF was 28 % (95 % CI 21 – 36), (40 of 144). Lagos had the lowest sero-prevalence (13 %; 95 % CI 4 – 30) while Ogun had the highest value of 57 % (95 % CI 37 – 75); there was a significant difference (p < 0.05) in the sero-prevalence of ASF between Lagos and Ogun States (Table 1). The sero-prevalence of ASF was higher (18 %; 95 % CI 12 – 24) in older stock (more than 12 months old) than in the younger stock (10 %; 95 % CI 7 – 13); this was significant at p = 0.01. The herd sero-prevalence was highest (31 %; 95 % CI 21 – 43) in small pig herds (less than 50 pigs) and lowest (21 %; 95 % CI 10 – 37) in medium herds (51 – 100 pigs); however, the difference was not significant. There was a significant difference (p < 0.05) in the sero-prevalence during the dry season (54 %; 95 % CI 37 – 70) and rainy season (18 %; 95 % CI 11 – 27). Overall individual crude prevalence was 78 of 657 (12 %; 95 % CI 10 – 15). Overall individual prevalence adjusted by weight of total population size was 11.2 %.
Table 1 Factors associated with pig herd level African swine fever seropositivity of 144 pig herds in southwest Nigeria, 2013
Herd level associated environmental and management factors to ASF seropositivity
In the univariate analysis, the presence of a quarantine or isolation unit within 100 m radius of the regular pig pen (OR = 3.3; 95 % CI 1.3 – 8.9), season of the year the samples were taken (OR = 5.3; 95 % CI 2.2 – 12.7) and source of replacement stock (OR = 3.2; 95 % CI 1.3 – 8.3) were significantly associated with ASF seropositivity (Table 1); however, the presence of pig farms within 1 km radius of another farm, having slaughter slabs or abattoir within 1 km radius of the farm, having rubbish heap or carcass disposal site within 1 km radius of the farm and presence of other animals or livestock within 100 m radius of the regular pig pen were not significantly associated with ASF seropositivity.
In the multivariable logistic regression adjusting for other covariates that were significant at P < 0.20 and biologically plausible ones, there was an interaction between herd size and season of the year the samples were taken. Source of restocking was a significant (OR = 2.7; 95 % CI 1.1 – 6.7) predictor for ASF herd level seropositivity. The final model included 4 predictors: season of the year the samples were taken, source of replacement stock, herd size and interaction between herd size and season of the year the samples were taken (Table 2). These were statistically significant in estimating ASF seropositivity (− 2 log-likelihood = 142.4; Goodness of fit = 0.97; χ2 = 20.2; p = 0.0005). The model correctly classified 76.5 % of the cases.
Table 2 Unconditional Logistic Regression of factors associated with African swine fever seropositivity of 144 pig herds with herd size as a continuous variable in southwest Nigeria, 2013
Compliance with standard biosecurity measures
Overall average compliance with standard biosecurity measures was 61 % (95 % CI 59 – 63). Of the 144 pig herds, only 5 (3.5 %; 95 % CI 1 – 8) had functional foot dip, 113 (78.5 %; 95 % CI 71 – 85) had farm designated working clothes, 57 (40 %; 95 % CI 32 – 48) had routine pest control, 3 (2 %; 95 % CI 0 – 6) reported presence of ticks on pigs, 81 (56 %; 95 % CI 48 – 64) fed swill to their animals, 30 (21 %; 95 % CI 15 – 28) disinfected their working utensils daily, 33 (23 %; 95 % CI 16 – 31) disinfected the pen floor daily, 17 (12 %; 95 % CI 7 – 18) shared farm attendants with other pig farmers, 8 (5.5 %; 95 % CI 2 – 11) shared working utensils with other pig farmers, 41 (28.5 %; 95 % CI 21 – 37) either gave or took service boars, and 23 (16 %; 95 % CI 10 – 23) of the respondents wore their farm working clothes outside of their premises. None of these factors was significantly (p < 0.05) associated with ASF among herd seropositivity.
Frequency of identified ASF related signs by respondents as occurring on the farm
The most common ASF related signs identified by the respondents were weakness or unwillingness of the pigs to stand (31 %; 95 % CI 23 – 39), followed by abortion (30 %; 95 % CI 23 – 38); the least common was reddening of ear and snout (7 %; 95 % CI 3 – 12). Reddening of ear and snout, however, was the only ASF related sign identified by the respondents that was significantly (p = 0.005) associated with ASF sero-positivity.
Our modeling indicated that source of replacement stock and season of the year the samples were taken are important determinants of ASF herd seropositivity in southwest Nigeria. However, the effect of season of the year the samples were taken on ASF herd seropositivity was modified by herd size. Herds with an external source of replacement always had higher ASF sero-prevalence compared with those with an internal source. This could be due to higher risk of introduction of an asymptomatic carrier into the herd at purchase. The spread of the ASF virus has been associated with asymptomatic carriers [21]. Introduction of ASF into a free area via movements of infected pigs has also been implicated in trans-boundary spread [4]. The higher ASF sero-prevalence in herds with an external source of replacement stock is also suggestive of a higher level of compromise in the bio-exclusion and biocontainment efforts, possibly due to more frequent human and vehicular movements in herds with external source of replacement stock [4].
The logistic regression technique allowed us to calculate the risk of ASF seropositivity as a function of the determinants in the final model using the formula: Prevalence (seropositive) = 1/1 + e-[α + Season*β1 + Source*β2 + Herdsize*β3 + Season*Herdsize*β4]. To demonstrate the interaction effect of season of the year the samples were taken and herd size we calculated the risk of ASF seropositivity for dry season and internal source of replacement, dry season and external source of replacement, rainy season and external source of replacement, and rainy season and internal source of replacement (Fig. 1). The risk of ASF seropositivity was always higher in farms with an external source of replacement stock than an internal source. Among herds with an external source of replacement the risk of ASF seropositivity was higher in the dry season than in the rainy season. Among herds with an internal source of replacement the risk of ASF seropositivity was higher in small and medium farms during the dry season than in the rainy; there was no difference in large herds. The risk decreased faster in the dry season with increasing herd size in farms with an internal source of replacement stock. The risk also increased faster in the rainy season with increasing herd size in farms with an external source of replacement stock (i.e., the herd size effect was not constant between seasons). These ASF risk dynamics in pig herds in southwest Nigeria bring a new dimension to the understanding of the ASF epidemiological cycle and its enzootic status in the region. A second model considered herd size as a categorical variable; here there was no interaction between herd size and season. The likelihood of having an ASF seropositive pig herd increased by five and three times during the dry season and for farms with an external source of replacement stock respectively.
The risk of African swine fever (ASF) seropositivity in dry and wet seasons in 144 pig herds with external and internal source of replacement stock in southwest Nigeria, 2013. The risk is calculated based on the logistic regression model in Table 2. The risk of ASF seropositivity was always higher in the dry than in the rainy season and in farms with an external source of replacement stock than an internal source. The risk decreased faster in the dry season with increasing herd size in farms with an internal source of replacement stock. The risk also increased faster in the rainy season with increasing herd size in farms with an external source of replacement stock
In our model, preponderance of susceptible pigs arising from increased restocking in the rainy season, especially in large herds, could explain the faster change in the risk of ASF seropositivity in the rainy season with increasing herd size in farms with an external source of replacement. The rapid reduction in ASF risk in the dry season with increasing herd size in farms with an internal source of replacement could also be explained from the above standpoint because most pig farmers dispose of their finisher stock in the dry season [22, 23]. Herd size has been associated with pig diseases [24, 25]. The higher ASF sero-prevalence in the dry season could be due to the presence of other factors that favor the prevalence and maintenance of ASF aside from restocking. The significant association between season of the year the samples were taken and source of replacement stock with ASF herd seropositivity in this study is similar to the findings of Atuhaire et al. [22] in a 12-year epidemiological overview of ASF in Uganda.
Numbers of movements of pigs and pig products increase in the dry season, resulting from increased trading demands during the festive period in December. Moreover, farmers have a higher propensity to dispose of mature pigs during this period because of feed scarcity and meeting of domestic (family) needs which is always on the increase during the dry season [22, 23]. Local and international trading in pigs and pig products has been associated with ASF outbreaks [2, 4, 22, 26].
Magali [27] reported insignificant but higher prevalence of ASFV in ticks during the dry than rainy season in South Africa. Although only 2 % of our respondents reported seeing ticks and their presence was not significantly associated with seropositivity in this study, this might have been under reported because of observational bias by the farmers. Thus, the role of ticks in the epidemiology of ASF in Nigeria may require further studies. Ticks play a major role in the sylvatic cycle of ASF transmission in East and Southern Africa [6], and in transmission between the sylvatic cycle and domestic pigs [4].
Another possibility could be that after infection consequent to introduction of an asymptomatic carrier by purchase in the rainy season in pig farms with an external source of replacement, the disease takes a chronic or subacute course and thus the higher sero-prevalence in the dry season. African swine fever viral infection has been reported to persist for a long time in the blood [28]. These disease dynamics may support the enzootic status of ASF in pig herds in southwest Nigeria.
In the univariate analysis, the presence of a quarantine or isolation unit within 100 m radius of the regular pig pen was significantly positively associated with ASF seropositivity. This underscores the importance of proper location of the quarantine or isolation units in order to achieve their intended purpose of biosecurity. The proximity of such units to other operational units like farrowing, fattening etc. could actually be a risk for having ASF seropositive farms. It is recommended that siting of the quarantine/isolation unit should not be less than 100 m from the regular pig pen [29], which is a challenge to most small holder farms in southwest Nigeria because of inadequate space due to land tenure system and financial constraints. However, when we tested the association between having quarantine/isolation unit within 100 m of the regular pig pen and ASF herd seropositivity controlling for other covariates the association was no longer significant. The effect poor siting of quarantine or isolation unit could have on herd seropositivity is, however, noteworthy.
In our study, we did not find a significant effect of environmental factors such as presence of pig farms within 1 km radius of another farm, having slaughter slabs or abattoir within 1 km radius of the farm, having rubbish heap or carcass disposal site within 1 km radius of the farm and presence of other animals or livestock within 100 m radius of the regular pig pen on ASF seropositivity. However, Fasina et al. [30] showed some of these factors like presence of an abattoir in a pig farming community and presence of an infected pig farm in the neighborhood to be significantly associated with ASF outbreaks. This difference could be due to the effects of environmental factors on ASF virus (ASFV) maintenance becoming insidious as the occurrence of ASF became enzootic.
We reported an overall herd ASF sero-prevalence of 28 %; this is significantly lower than the value (93 %) reported 5 years ago by Olugasa [14] across the same geographical region. The significant difference in our study sero-prevalence estimate of 28 % and the previous sero-prevalence of 93 % by Olugasa [14] which was used to determine our sample size would have slightly widened our set margin of error; however, our study sero-prevalence estimate is still within a 7 % margin of error of the population true sero-prevalence. All the other States except Ogun had a marked decline in their herd ASF sero-prevalence when compared with a 5-year value reported by Olugasa [14]. This indicates some improvement in the control measures by the Governments at all levels to eradicate the disease. The ASF herd sero-prevalence in Ogun is highest and almost stable, followed by Oyo: the high sero-prevalence in these States since the outbreak in 1997 could be because both States had international borders. Ogun was the first State in Nigeria to experience the outbreak of ASF in 1997 which spread through trans-border trade from the Republic of Benin [31]. Movement of pigs and pig products across borders from infected areas has been reported to be positively correlated with ASF seropositivity and outbreaks [4, 32]. The herd sero-prevalence is higher in small pig herds than in larger pig herds, though not significantly so; this may be due to difficulty in adhering to strict biosecurity by small holder farms or possibly less attention to simple biosecurity measures than on big farms. Moreover, the sero-prevalence of ASF is significantly higher in older stock than younger stock; this could be because restriction of movement to sections of the herds containing young pigs is greater than for older ones. It could also be that older stock had a longer time to develop antibodies to the ASF virus than younger stock, or possibly due to long persistence of ASF antibodies for a period of time after exposure [28]. There could also exist differences in ASFV transmission rates among the various age groups. Olugasa [14] also reported differences in sero-prevalence of ASF among various age groups.
Biosecurity is defined as the implementation of measures that reduce the risk of the introduction and spread of disease agents; it requires the adoption of a set of attitudes and behaviors by people to reduce risk in all activities involving domestic, captive/exotic and wild animals and their products [33]. There was no significant difference in the level of compliance with some of the biosecurity measures between seropositive and seronegative herds in our study population; however, Fasina et al. [30] reported significant associations between food and water control, separation/isolation of sick pigs, washing and disinfection of farm equipment and tools, consultations or visits by veterinarians/paraveterinarians, pest/rodent control, and sharing farm tools and equipment and ASF outbreaks in Nigerian farms.
With the understanding that ASF is enzootic in Nigeria, one would expect to see less apparent signs of peracute or acute forms of ASF, but rather, more of the subacute, chronic or subclinical forms. Farmers in this study reported noticing weakness or unwillingness of the pigs to stand (31 %), followed by abortion (30 %); the least common sign was reddening of the ear and snout (7 %). Reddening of the ear and snout, however, was the only ASF related sign identified by the respondents that was significantly (p = 0.005) associated with ASF seropositivity. There is no sign, however, that is pathognomonic to ASF. The implication of this is that the ability of the farmers to recognize such associated signs could assist early detection of an infection. However, the challenge with early detection is the non-willingness to report by the farmers. Farmers may not report because the adjudged compensation by the Government, if any, is non-commensurate. Early detection and reporting is critical to ASF control and eradication [34].
Pig farmers in Nigeria are mostly males, in their mid-50s (mean 49.2 ± 14.6 years) and most had tertiary education. More than half of the population of pig farmers studied had pig farming as their main source of livelihood. This indicates the importance of the farming sector and the socioeconomic impact adverse effects of disease such as ASF could have on livelihood. The pig sector appears to be recovering after the devastation caused by the ASF outbreak in 1997, which sent the majority of farmers out of business. Most of the farmers raised their pigs in strict pen confinement; greater risk of ASF seropositivity has been linked with free range pigs [32, 35, 36].
This study may be limited by information bias which is common to questionnaire administration, as respondents may give favoring or biased responses and not their actual practice. We mitigated this by triangulating – we designed the questionnaire in such a way that certain questions were deliberately repeated in different ways. We also envisaged interviewer bias and we reduced this by the training of the interviewers used in this study. These limitations were taken into consideration in the interpretation of the data.
Our findings indicate that ASF herd level sero-prevalence in southwest Nigeria was higher in pig herds with an external source of replacement stock than an internal source; and in the dry season than in the rainy season. The effect of season of the year the samples were taken on ASF seropositivity was modified by herd size. The observed increase in ASF risk in the rainy season with increasing herd size in pig herds with an external source of replacement and observed decrease in ASF risk in the dry season with increasing herd size in pig herds with an internal source of replacement suggest that large herds are at greater risk of ASF infection via introduction of an asymptomatic pig than through other indirect means such as fomites, movement of vehicles and personnel. We recommend strict compliance with biosecurity measures, especially using an internal source of replacement stock and measures that minimize movement on pig farms. The associations between ASF herd level sero-prevalence and source of replacement stock and the effect of herd size on season of the year the samples were taken have implications for the understanding of ASF transmission and application in the disease modeling and in development of a suitable control and eradication strategy for ASF in Nigeria. The role of ticks either on pigs or in pens in the enzootic status of ASF in Nigeria will be an area for further investigation.
The study area included six States – Lagos, Ogun, Oyo, Osun, Ondo and Ekiti (Fig. 2). The choice of the States was informed by the large presence of pig farming activities in these States and having reported outbreaks of ASF and the present enzootic status of those States. Each of the States has three senatorial districts and varying local government areas (LGAs). Pig farms from these senatorial districts and LGAs were included.
Geographical spread of 144 pig herds surveyed for African swine fever (ASF) seropositivity in southwest Nigeria in 2013. Random distribution of the pig herds is shown
The study areas have varying sizes of pig populations: Lagos (approximately half a million pigs), Ogun (approximately half a million pigs), Oyo (approximately three hundred thousand pigs), Osun (approximately two hundred thousand pigs), Ondo (approximately a hundred thousand pigs) and Ekiti (approximately a hundred thousand pigs) [37].
Of the studied States only Ogun and Oyo have international borders with Benin Republic while Lagos is a coastal city. The location of Nigeria in Africa is shown in the inset (bottom left) in Fig. 2.
Study design, sample size and sampling
We conducted a cross sectional study on pig farms across the six States in southwest Nigeria from November 2012 to August 2013.
A total sample size of 657 pigs from 144 pig farms were used for the study. We approached this by using the formula below [38]:
$$ \mathrm{n}=\frac{{\mathrm{Z}}^2\mathrm{p}\left(1-\mathrm{p}\right)\ast \mathrm{D}}{{\mathrm{E}}^2} $$
Where Z is the reliability coefficient put at 1.96 at a 95 % confidence interval and E is the margin of error at 5 %. Based on previous studies we powered the study to detect among herd sero-prevalence (p) of 92.9 % [14]. This gave a minimum sample size of 101 farms. We adjusted for clustering by using the formula D = 1 + (b-1) ρ [39], where D is the design effect, b is the average number of samples per cluster and ρ is the intra-cluster correlation coefficient. We decided to sample an average of four pigs per farm (b = 4), about 10 % of the pig population on the farm. An intra-cluster correlation coefficient of ρ = 0.13 was assumed based on an earlier study [40]. This gave us a D of 1.4. We multiplied the cluster design effect (1.4) by the earlier calculated sample size of 101 farms to arrive at a total of 141 farms. In all, a total of 144 farms and 657 pigs were sampled.
We used simple random sampling to select pig farms from a list of registered pig farms in Lagos (150) and Ogun States (124). However, for the remaining four States without a register, we used simple random sampling to select between two and four local government areas from the existing three senatorial districts in the States. At least one pig farm was chosen from each selected local government area and a total of at least six pig farms were chosen from each of the three senatorial districts. Randomness was verified using geospatial analysis by determining spatial autocorrelation (Global Moran's I) in ArcMap version 10.2. A minimum of 20 pig farms were selected from each State except Lagos and Ogun (30 pig farms) where there is a larger presence of piggery activities and Ekiti (17 pig farms) where we had fewer pig farms. The pigs were selected using stratified sampling (piglets (weaners and growers less than or equal to 12 months old) and adult pigs (sow and boars more than 12 months old)). Equal allocation was adopted. An average of four pigs per farm was sampled.
Blood collection and Laboratory analysis
We obtained venous blood (3 – 5mls) samples from the cranial vena-cava of selected pigs. They were collected into plain tubes and transported in ice packs to the laboratory. Sera were properly labeled and stored at −20 °C until used in batches. We conducted the Indirect Enzyme Linked Immunosorbent Assay – ELISA – [41] test using the ASF kit (SVANOVIR® Sweden) to screen the pigs' sera for ASF IgG antibodies at the Immunology laboratory, Veterinary Public Health and Preventive Medicine, University of Ibadan, Nigeria. The test kit had sensitivity and specificity of 100 % and 92.5 % respectively and can detect antibodies from day seven post infection. The samples were collected from different farms during the dry (December to March) and rainy (June and July) seasons [42] in order to detect seasonal variation. Herd sero-prevalence was determined. We defined herds as seropositive if at least one pig was seropositive in the ELISA test. We also determined the overall individual crude prevalence and adjusted it by the weight of the total population size. This was achieved by dividing the individual herd size by the total population size and multiplying by the proportion that were positive in the samples taken from each farm. The sum of the adjusted proportions multiplied by 100 gave the overall individual adjusted prevalence.
Questionnaire design and administration
A pre-tested (pre-testing was done using seven pig farmers from two locations not included in the study areas) structured, interviewer-administered questionnaire was used to obtain data on demography, environmental and management factors, bio-security measures and ASF related signs. The questionnaires, containing 38 questions, were administered on farms where samples were obtained. A respondent was someone who was actively involved in the daily activities of the farm and was not necessarily the farm owner.
We conducted descriptive statistics and univariate analysis using SAS version 9.3 [43]. We determined the odds ratio and statistical significance between seropositive and seronegative pig farms using Fisher's exact test for discrete variables at the 95 % confidence level. Multivariable unconditional logistic regression was used to determine predictors for ASF seropositivity controlling for other covariates at P < 0.20 and biologically plausible ones such as feeding of swill, lending out boars for breeding and herd size. We tested for collinearity among predictors using the Chi square test for binomial variables. We also tested for interactions between selected variables. We used Akaike's information criterion in selecting the variables. A forward selection method was used. The goodness of fit of the model was tested using the Pearson goodness of fit test. In the final models, only variables or interactions that were found to significantly affect the outcome at P < 0.05, and corresponding lower-order interactions terms whether significant or not, were retained. We determined the overall average compliance with standard biosecurity measures by calculating the average compliance level of the 15 internal biosecurity measures (bio-management) considered.
This study was carried out in compliance with the guidelines of the Animal Ethics Committee of the Faculty of Veterinary Medicine, University of Ibadan, Nigeria. Informed consent was obtained from all participating pig farm owners.
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The authors wish to acknowledge the financial support of Nigeria Science and Technology Education Post-Basic (STEP-B) Project (Cr: 4304-UNI) - Innovators of tomorrow Research and Technology Development grant. We also appreciate the co-operation of the Pig farmers' association in southwest Nigeria and the technical assistance of the various States' Ministry of Agriculture and Livestock resources and/or rural development and cooperatives and Zhengxi Wang for creating the map.
Department of Veterinary Public Health and Preventive Medicine, Faculty of Veterinary Medicine, University of Ibadan, Ibadan, Oyo State, Nigeria
Emmanuel Jolaoluwa Awosanya, Babasola Olugasa & Gabriel Ogundipe
Department of Population Medicine and Diagnostic Sciences, College of Veterinary Medicine, Cornell University, Ithaca, NY, 14853, USA
Emmanuel Jolaoluwa Awosanya & Yrjo Tapio Grohn
Emmanuel Jolaoluwa Awosanya
Babasola Olugasa
Gabriel Ogundipe
Yrjo Tapio Grohn
Correspondence to Emmanuel Jolaoluwa Awosanya.
EJA conceived and designed the study, acquired, analyzed, and interpreted the data, drafted and revised the article. BO designed the study, revised and edited the drafted article. GO designed the study, revised and edited the drafted article. YTG analyzed the data, revised and edited the drafted article. All the authors have read and approved the final version of the manuscript.
EJA (DVM, MVPH, MPH) is a junior faculty member and doctoral student at the Department of Veterinary Public Health and Preventive Medicine, University of Ibadan, Ibadan, Nigeria and a Fulbright scholar at the Department of Population Medicine and Diagnostic Sciences, College of Veterinary Medicine, Cornell University, Ithaca, New York, USA. BO (DVM, PhD) is a senior lecturer and Principal Investigator at the Centre for Control and Prevention of Zoonoses, Department of Veterinary Public Health and Preventive Medicine, University of Ibadan, Ibadan, Nigeria. GO (DVM, PhD) is a Professor at the Department of Veterinary Public Health and Preventive Medicine, University of Ibadan, Ibadan, Nigeria. YTG (DVM, PhD) is a Professor at the Department of Population Medicine and Diagnostic Sciences, College of Veterinary Medicine, Cornell University, Ithaca, New York, USA.
Awosanya, E.J., Olugasa, B., Ogundipe, G. et al. Sero-prevalence and risk factors associated with African swine fever on pig farms in southwest Nigeria. BMC Vet Res 11, 133 (2015). https://doi.org/10.1186/s12917-015-0444-3
Sero-prevalence
Southwest Nigeria | CommonCrawl |
\begin{document}
\title[Endperiodic Automorphisms]{Examples of Endperiodic Automorphisms}
\author[J. Cantwell]{John Cantwell} \address{Department of Mathematics\\ St. Louis University\\ St. Louis, MO 63103} \email{[email protected]}
\author[L. Conlon]{Lawrence Conlon} \address{Department of Mathematics\\ Washington University, St. Louis, MO 63130} \email{[email protected]}
\subjclass{Primary 37E30; Secondary 57R30.} \keywords{endperiodic, lamination}
\begin{abstract} We give examples of endperiodic automorphisms. \end{abstract}
\maketitle
These examples are intended to supplement the examples in our paper~\cite{cc:HM}. Also see Fenley~\cite{fe:thesis} and~\cite[Section~5]{fe:endp} for other examples. We use the terminology and notation of~\cite{cc:HM}. In the figures, traintracks for the positive laminations $\Lambda_{+}$ are giving with solid curves and traintracks for the negative laminations $\Lambda_{-}$ are given with dashed curves.
\section{Examples with $\Lambda_{\pm}$ finite}
Examples~\ref{ex2leaves} and~\ref{ex3leaves} have the properties that,
\begin{enumerate}
\item Both $\Lambda_{+}$ and $\Lambda_{-}$ are finite\upn{;}
\item $\Lambda_{+}$ (respectively $\Lambda_{-}$) has a leaf $\lambda$ such that there is an arc $\alpha$ tranverse to $\lambda$ which meets $\Lambda_{+}$ (respectively $\Lambda_{-}$) in a countably infinite set\upn{;}\label{item2}
\item Dehn twists are not used in the construction of the examples.\label{item3}
\item There do not exist transverse measures to $\Lambda_{+}$ and $\Lambda_{-}$with full support.\
\end{enumerate} Properties~\ref{item2} and~\ref{item3} are surprising. These two examples give non-trivial examples in which both $\Lambda_{+}$ and $\Lambda_{-}$ are finite. We know of no examples in which $\Lambda_{+}$ or $\Lambda_{-}$ are countably infinite. Fenley~\cite[Section~5]{fe:endp} also gives an example of laminations which do not support a measure of full support.
Example~\ref{exbasic} is basic to the construction of Examples~\ref{ex2leaves} and~\ref{ex3leaves}.
\begin{example}\label{exbasic}
Let $L$ be the surface in Figure~\ref{fig1}. On $L$, the homeomorphism $T$ moves the circles in the bottom left leg of $L$ (and in fact the fundamental domains) to the right by one, moves the rightmost circle in the bottom left leg to the leftmost circle in the top right leg of $L$, moves the circles (and fundamental domains) in the top right leg of $L$ to the right by one, and is pointwise fixed on the top left and bottom right legs of $L$. Similarly, the homeomorphism $S$ moves the circles (and fundamental domains) in the bottom right leg of $L$ to the left by one, moves the leftmost circle in the bottom right leg of $L$ to the rightmost circle in the top left leg of $L$, moves the circles (and fundamental domains) in the top left leg of $L$ to the left by one, and is pointwise fixed on the top right and bottom left legs of $L$. The endperiodic automorphism $f = S\circ T:L\to L$ behaves as indicated on the four real line boundary components of $L$. In particular, $f$ has a fixed point $z$ on the top edge of $L$ and a fixed point $w$ on the bottom edge of $L$.
The lamination $\Lambda_{+}$ has two leaves, $\lambda_{1}$ is the top edge of $L$ in Figure~\ref{fig1} and has two escaping ends. The other leaf $\lambda_{2}$ is a semi-isolated half-line that approaches $\lambda_{1}$ from below. Similarly, the lamination $\Lambda_{-}$ has two leaves, $\lambda'_{1}$ is the bottom edge of $L$ in Figure~\ref{fig1} and has two escaping ends. The other leaf $\lambda'_{2}$ is a semi-isolated half-line that approaches $\lambda'_{1}$ from above. The intersection $\lambda_{2}\cap\lambda'_{2}$ consists of points $x_{n}$, $n\in\mathbb{Z}$ which can be indexed so that,
\begin{enumerate}
\item The sequence $\{x_{n}\}$ is monotone in both $\lambda_{2}$ and $\lambda'_{2}$\upn{;}
\item $x_{n}\to z$ as $n\to+\infty$ and $x_{n}\to w$ as $n\to-\infty$\upn{;}
\item $f(x_{n}) = x_{n+1}$, $n\in\mathbb{Z}$.
\end{enumerate} Note $x_{-1},x_{0},x_{1}$ in Figure~\ref{fig1}. The set $\Lambda_{+}\cap\Lambda_{-}$ consists of the points $z$, $w$, and $x_{n}$, $n\in\mathbb{Z}$.
\end{example}
\begin{figure}
\caption{The surface $L$ and traintracks for $\Lambda_{\pm}$ in Example~\ref{exbasic}}
\label{fig1}
\end{figure}
\begin{example}\label{ex2leaves}
Let $L$ be the double of the suface of Example~\ref{exbasic} and $f$ the double of the endperiodic map. The surface $L$ is depicted in Figure~\ref{four1} and has four non-planar ends. The lamainations $\Lambda_{\pm}$ will be the doubles of the laminations in Example~\ref{exbasic}. The traintracks on Figure~\ref{four1} can be visualized from the traintracks in Figure~\ref{fig1}. In particularl, $\Lambda_{+}$ has two leaves. The first $\lambda_{1}$ has two escaping ends and is approached on each side by an end of the second $\lambda_{2}$. The leaf $\lambda_{2}$ is isolated and returns infinitely often to the core. In Figure~\ref{four1} the leaf $\lambda_{1}$ runs along the top of the figure and one end of $\lambda_{2}$ approaches $\lambda_{1}$ on the front of the figure and the other end of $\lambda_{2}$ approaches $\lambda_{1}$ along the back of the figure. A similarly description can be given of the two leaves $\lambda'_{1},\lambda'_{2}$ of $\Lambda_{-}$. The leaf $\lambda'_{2}$ meets $\lambda_{1}$ in a point $z$. The leaf $\lambda_{2}$ meets $\lambda'_{1}$ in a point $w$. The intersection $\lambda_{2}\cap\lambda'_{2}$ consists of two sequences $x_{n}$, $n\in\mathbb{Z}$, and $x'_{n}$, $n\in\mathbb{Z}$, the first on the front of the figure and the second on the back. The indexing can be chosen so that
\begin{enumerate}
\item The sequences $\{x_{n}\}$ and $\{x'_{n}\}$ are monotone in both $\lambda_{2}$ and $\lambda'_{2}$\upn{;}
\item $x_{n}\to z$, $x'_{n}\to z$ as $n\to+\infty$ and $x_{n}\to w$, $x'_{n}\to w$ as $n\to-\infty$\upn{;}
\item $f(x_{n}) = x_{n+1}$ and $f(x'_{n}) = x'_{n+1}$, $n\in\mathbb{Z}$.
\end{enumerate} The set $\Lambda_{+}\cap\Lambda_{-}$ consists of the points $z$, $w$, and $x_{n}$ and $x'_{n}$, $n\in\mathbb{Z}$.
\end{example}
If the leaf $\lambda_{2}$ has positive transverse measure, then any arc transverse to the leaf $\lambda_{1}$ would have infinite measure. Thus there does not exist a transverse measure on $\Lambda_{+}$ of full support.
\begin{figure}
\caption{The surface $L$ in Example~\ref{ex2leaves}}
\label{four1}
\end{figure}
\begin{example}\label{ex3leaves}
Example~\ref{ex3leaves} is defined on tha same surface $L$ as Example~\ref{ex2leaves} and the endperiodic map is defined as a composition $f=S\circ T$ of two homeomorphiams $S$ and $T$ which are similar to the homeomorphisms $S$ and $T$ of Example~\ref{ex2leaves}. In both examples $S$ and $T$ behave the same way near the ends of $L$.
In Example~\ref{ex2leaves}, the handle in a fundamental domain of an end was considered as a hole through the surface (see Figure~\ref{four1}) and the homeomorphism $S$ (respectively $T$) draged this hole across the core when it moved this handle from the lower right (respectively lowert left) to upper left (respectively upper right) end of the surface $L$. Thus the juntures are distorted on both the back and front of the surface of Figure~\ref{four1}.
In Example~\ref{ex3leaves}, the handle in a fundamental domain of an end is considered as a handle attaced to the front of the surface (see Figure~\ref{four2}) and the homeomorphism $S$ (respectively $T$) drags this handle across the core when it moves this handle from the lower right (respectively lowert left) to upper left (respectively upper right) end of the surface $L$. Thus the juntures on only the front of Figure~\ref{four2} are distorted. The result of this is that in Example~\ref{ex3leaves}, both $\Lambda_{+}$ and $\Lambda_{-}$ have three leaves and the example has a pair of complementary principal regions with three arms.
The positive lamination $\Lambda_{+}$ consists of three leaves $\lambda_{1},\lambda_{2},\lambda_{3}$ which are the border leaves of a positive principal region $P$. The leaves $\lambda_{2}$ and $\lambda_{3}$ are isolated. The leaf $\lambda_{1}$ is semi-isolated, being approached from below by an arm of $P$ bordered by an end of $\lambda_{2}$ and an end of $\lambda_{3}$. In the traintrack of Figure~\ref{four2} this arm is represented by the solid black curve that approaches $\lambda_{1}$ from below returning infinitely often to the core. The leaf $\lambda_{1}$ is drawn in Figure~\ref{four2} as a horizontal line bordering the principal region $P$ from below. One end of the leaf $\lambda_{1}$ and one end of $\lambda_{2}$ border an escaping arm of the principal region $P$ as does the other end of the leaf $\lambda_{1}$ and one end of the leaf $\lambda_{3}$. The six vertices of the nucleus of the principal region $P$ are indicated by black dots. The three leaves of $\Lambda_{-}$ and the negative prncipal region are similarly drawn in Figure~\ref{four2}.
\end{example}
\begin{figure}
\caption{The surface $L$ and traintracks for $\Lambda_{\pm}$ in Example~\ref{ex3leaves}}
\label{four2}
\end{figure}
\eject
\section{Another example defined without using Dehn twists}
\begin{figure}
\caption{Traintracks for $\Lambda_{\pm}$ in Example~\ref{exsix}}
\label{six}
\end{figure}
\begin{example}\label{exsix}
Let $L$ be the surface of Figure~\ref{six}. Let $T$ be the homeomorphism that moves each circle boundary (and in fact the fundamental domains) to the right by one in the horizontal strips. The rightmost circle in the left strip is moved across the core to the leftmost circle in the right strip. The homeomorphism $T$ is to leave the other four strips pointwise fixed. Let $R_{3}$ be the rotation of the whole figure clockwise through $2\pi/3$ radians. Define $f = (R_{3}\circ T)^{3}$. There will be no fixed points on any of the boundary components. There will be three leaves with an escaping end in each of $\Lambda_{\pm}$ and these leaves will be the semi-isolated leaves in each of $\Lambda_{\pm}$. Since each of the semi-isolated leaves accumulates on itself, both $\Lambda_{\pm}$ are transversally a Cantor set and there are uncountably many leaves in both $\Lambda_{\pm}$. Figure~\ref{six} illustrates traintracks for $\Lambda_{\pm}$.
\end{example}
\begin{rem}
The method of Example~\ref{exsix} can be used to create examples of endperiodic automorphisms on planar surfaces with $2n$ nonsimple ends for any odd integer $n\ge 3$.
\end{rem}
\eject
Denote the traintrack for $\Lambda_{+}$ given in Figure~\ref{six} by $\mathcal{T}_{0}$. We have redrawn $\mathcal{T}_{0}$ in Figure~\ref{sixPlus}. Let $g = R_{3}\circ T$. It is easier to work with $g$ than $f=g^{3}$. Let $\mathcal{T}_{1} = g(\mathcal{T}_{0})$, $\mathcal{T}_{2} = g(\mathcal{T}_{1})$, and in general $\mathcal{T}_{n} = g(\mathcal{T}_{n-1}) = g^{n}(\mathcal{T}_{0})$, $n\ge 1$. The traintrack $\mathcal{T}_{1}$ is obtained from $\mathcal{T}_{0}$ by blowing air from $a_{1}$ to $a_{1}$ (see Figures~\ref{sixPlus} and~\ref{sixB}). Similarly $\mathcal{T}_{2}$ is obtained from $\mathcal{T}_{1}$ by blowing air from $a_{2}$ to $a_{2}$. Etc. In Figure~\ref{sixPlus} we have indicated how to blow air to create $\mathcal{T}_{1},\mathcal{T}_{2},\mathcal{T}_{3},\mathcal{T}_{4,}\mathcal{T}_{5},\ldots$. From these it is easy to see the pattern. We draw $\mathcal{T}_{2}$ in Figure~\ref{sixB} and indicate how to blow air to get $\mathcal{T}_{3} = g^{3}(\mathcal{T}_{0})$.
Further, $|\Lambda_{+}| = \bigcap_{n=0}^{\infty}\mathcal{T}_{n}$.
\begin{rem}
It really is clear that $\mathcal{T}_{1}$ is obtained from $\mathcal{T}_{0}$ by blowing air from $a_{1}$ to $a_{1}$. From this the pattern follows immediately.
\end{rem}
\begin{rem}
It is not at all directly obvious that $\mathcal{T}_{3} = f(\mathcal{T}_{0})$. That is why we look at $g$ rather than $f = g^{3}$.
\end{rem}
\begin{figure}
\caption{Traintracks for $\Lambda_{+}$ in Example~\ref{exsix}}
\label{sixPlus}
\end{figure}
\eject
\begin{prop}
There exists a tansverse measure $\mu$ on $\Lambda_{+}$ so that if $\alpha$ is any arc transverse to $\Lambda_{+}$ then $\mu(g(\alpha)) = x\mu(\alpha)$ where $x=1/\tau$ with $\tau$ the golden number.
\end{prop}
\begin{proof}
We construct the measure directly. In Figure~\ref{sixB}, the symbols $1,x,x^{2},x^{3},x^{4},\ldots$ denotes the value of the measure of a transverse arc to $\Lambda_{+}$ at each of these points. Clearly, $\mu(g(\alpha)) = x\mu(\alpha)$. It remains to show that $\mu(\alpha + \beta) = \mu(\alpha) + \mu(\beta)$. For this it suffices that $1=x+x^{2}$ or $x^{2}+x-1 = 0$. Since $x$ must be positive, $$x = \frac{-1+\sqrt{5}}{2} = 1/\tau.$$ \end{proof}
\begin{rem}
The lamination $\Lambda_{-}$ has a transverse measure with scale factor $\tau$.
\end{rem}
\begin{figure}
\caption{The traintrack $\mathcal{T}_{2}$ in Example~\ref{exsix}}
\label{sixB}
\end{figure}
\eject
\section{An uncountable family of translations}
The mapping class group of a compact or a finite area surface is at most countable. In this section we give uncountable many translations of the surface with two nonplanar ends. These translations are not isotopic but are in some sense the same.
\begin{figure}
\caption{The surface $L$ admits uncountably many, nonisotopic translations}
\label{transL}
\end{figure}
\begin{example}
Let $L$ be the surface of Figure~\ref{transL} with the indicated junctures and $d_{i}$, $-\infty<i<\infty$, be Dehn twists in the dotted circles. Let $T$ be any translation to the right with the indicated junctures. Let, $$\iota = \{\ldots, i_{-1},i_{0},i_{1},i_{2},\ldots,i_{k},\ldots\}$$ be any sequence of zeroes and ones. Define the homeomorphism $D_{\iota}:L\to L$ by $$D_{\iota} = \cdots d_{-1}^{i_{-1}}\circ d_{0}^{i_{0}}\circ d_{1}^{i_{1}}\circ d_{2}^{i_{2}}\circ\cdots\circ d_{k}^{i_{k}}\circ\cdots.$$ Then the family $D_{\iota}\circ T$ consists of uncountably many nonisotopic translations of $L$ with the same junctures.
\end{example}
\begin{rem}
If one allows different choices of the junctures, one gets even more isotopy classes of translations of $L$.
\end{rem}
\begin{rem}
Each of these translations can be realized as the monodramy of homeomorphic depth-one foliations of $S\times [0,1]$ where $S$ is the two holed torus.
\end{rem}
\section{An uncountabe family of nonisotopic endperiodic automorphisms}
In this section we give another example of a surface $L$ having an uncountable family of nonisotopic endperiodic automorphisms. These examples have the property that for any two of them there exists a homeomorphism from $L$ to $L$ taking the positive and negative laminations $\Lambda_{\pm}$ for one to the positive and negative laminations for the other.
\begin{rem}
The technique can be used to give similar uncountable families of examples with much more complicated dynamics.
\end{rem}
\eject
\begin{figure}
\caption{Top half of surface $L$ of Examples~\ref{ex2a}}
\label{3PE}
\end{figure}
\begin{figure}
\caption{$\lambda_{+}$ when $\iota = \{1,0,0\ldots,0,\ldots\}$}
\label{3PEA}
\end{figure}
\begin{example}\label{ex2a}
Let $L$ be the planar surface of Figure~\ref{3PE} doubled along the bottom edge and $g:L\to L$ be an endperiodic automorphism with the two curves $\lambda_{+}$ and $\lambda_{-}$ of Figure~\ref{3PE} ae positive and negative laminations respectively. We assume that $g$ sends the junctures $J_i$ to $J_{i+1}$ for $-\infty < i < 0$ near the repelling ends and $1\le i < \infty$ near the attracting ends. There are two attracting ends and two repelling ends. Let, $$\iota = \{i_{0},i_{1},i_{2},\ldots,i_{k},\ldots\}$$ be any integer sequence. Define the homeomorphism $D_{\iota}:L\to L$ by $$D_{\iota} = \cdots\circ d_{k}^{i_{k}}\circ\cdots\circ d_{2}^{i_{2}}\circ d_{1}^{i_{1}}\circ d_{0}^{i_{0}}.$$ where $d_{i}$ is Dehn twist in the dotted curves indicated in Figure~\ref{3PE}. Then the family $D_{\iota}\circ g$ consists of uncountably many nonisotopic endperiodic automorphisms of $L$ such that,
\begin{enumerate}
\item Each endperiodic automorphism has the $J_{i}$, $-\infty<i<\infty$, as junctures\upn{;}
\item $\Lambda_{-}$ and $\Lambda_{+}$ each consists of one isolated leaf.
\end{enumerate}
The leaf $\lambda_{-}$ is the bottom edge of Figures~\ref{3PE}, \ref{3PEA}, and~\ref{3PEB}. In Figures~\ref{3PEA} and~\ref{3PEB} we draw part of $\lambda_{+}$ for two different choices of $\iota = \{i_{0},i_{1},i_{2},\ldots,i_{k},\ldots\}$.
For all choices of $\iota = \{i_{0},i_{1},i_{2},\ldots,i_{k},\ldots\}$, the escaping set $\mathcal{U}$ has four components each with border a curve of the first kind each of which gives a reducing geodesic. Reduction gives four planar strips with disks deleted approaching both ends and each having endperiodic automorphisms which are pure translations and one pseudo-anosov piece with four simple ends as in~\cite[Figure 17]{cc:HM}.
\end{example}
\begin{figure}
\caption{$\lambda_{+}$ when $\iota = \{1,-1,0,0\ldots,0,\ldots\}$}
\label{3PEB}
\end{figure}
\section{An example with one repelling end and no attracting ends}
Our purpose in this section is to give an example of an endperiodic automorphism of a surface $L$ which has one repelling end $e$ and no attracting ends. By~\cite[Proposition~2.18]{cc:HM}, in such an example $L$ must have infinite endset. In fact by~\cite[Lemma~13.24]{cc:HM} and~\cite[Proposition~13.26]{cc:HM}, in such an example, the endset of $L$ must be uncountable. In our example the endset of $L$ will consist of the one isolated point $e$ and a Cantor set $C$.
We don't need the following proposition but it is the idea behind Example~\ref{cantorset}.
\begin{prop}
There exists a homeomorphism of the Cantor set without periodic points.
\end{prop}
\begin{proof}
We take as the Cantor set $C = \displaystyle\prod_{n=2}^{\infty}\mathbb{Z}_{n}$ where $\mathbb{Z}_{n}$ denotes the integers mod $n$. Define $\nu:C\to C$ by $\nu(\{a_{n}\}_{n=2}^{\infty}) = \{a_{n}+1\}_{n=2}^{\infty}$. Then $\nu$ is the desired homeomorphism. \end{proof}
\begin{figure}
\caption{The surface $L_{1}$ with end $e$ and boundary circle $J_{0}$}
\label{L1}
\end{figure}
\begin{example}\label{cantorset}
We inductively construct the surface $L$ with endset consisting of one isolated end $e$ and a Cantor set $C$. To begin let $L_{1}$ be the surface with one circle boundary component and one nonplanar end $e$ in Figure~\ref{L1}. Let $S(n)$ denote the compact surface with $n+1$ boundary components and one handle. Let $L_{2}$ be $L_{1}$ with one copy of $S(2)$ attached along the one boundary component of $L_{1}$. let $L_{3}$ be $L_{2}$ with two copies of $S(3)$ attached along each of the two boundary components of $L_{2}$ and in general let $L_{n+1}$ be $L_{n}$ with $n!$ copies of $S(n+1)$ attached along the $n!$ boundary components of $L_{n}$. Let $L = \bigcup_{n=1}^{\infty}L_{n}$.
\end{example}
Note that the $L_{n}\smallsetminus\intr L_{n-1}$ consists of $(n-1)!$ copies of $S(n)$
\begin{lemma}
There exists a homeomorphism $g:L\to L$ such that,
\begin{enumerate}
\item $g$ fixes $L_{1}$ pointwise\upn{;}
\item $g(L_{n}) = L_{n}$, $n\ge 1$\upn{;}
\item If $S$ is a component of $L_{n}\smallsetminus\intr L_{n-1}$, then $g$ cyclically permutes the $n$ components of $\partial S\cap\partial L_{n}$.
\end{enumerate}
\end{lemma}
\begin{proof}
We begin by setting $g = {\rm id}$ on $L_{1}$. If we have defined $g$ on $L_{n-1}$ and $S$ is a component of $L_{n}\smallsetminus\intr L_{n-1}$, then $S$ is homeomorphic to $S(n)$. Extend $g$ over $S$ by a homeomorphism supported in $\intr S$ that cyclically permutes the $n$ components of $\partial S\cap\partial L_{n}$. We illustrate such a homeomorphism for the case $n=3$ in Figure~\ref{cyclic}. The twist in the annulus (represented by the dotted circle in Figure~\ref{cyclic}) through $2\pi/3$ radians provides a homeomorphism from $S(3)$ to $S(3)$ which cyclically permutes three of the boundary components of $S(3)$. Inductively we have $g$ defined on $L$ satisfying the properties in the lemma. \end{proof}
\begin{lemma}
There exists a homeomorphism $h:L\to L$ such that,
\begin{enumerate}
\item $h$ fixes the endset of $L$ pointwise\upn{;}
\item $h(J_{i}) = J_{i+1}$, $i<0$ where the $J_{i}$ are the junctures in $L_{1}$ in Figure~\ref{L1}.
\end{enumerate}
\end{lemma}
\begin{proof}
For $n\ge 2$, choose a component $S_{n}$ of $L_{n}\smallsetminus\intr L_{n-1}$ homeomorphic to $S(n)$ such that $S_{n}\cap S_{n+1}$ is a circle. The homomorphism $h$ will be supported in $L_{1}\cup\bigcup_{n=2}^{\infty}S_{n}$. It is clear how to define $h$ to the left of the juncture $J_{-1}$ in Figure~\ref{L1}. Define the homeomorphism $h$ between $J_{-1}$ and $J_{0}$ to move the handle between $J_{-1}$ and $J_{0}$ into $S_{2}$. Define the homeomorphism $h$ on $S_{2}$ to move the handle in $S_{2}$ into $S_{3}$ and inductively define the homeomorphism $h$ on $S_{n}$ to move the handle in $S_{n}$ into $S_{n+1}$. The homeomorphism $h$ satisfies the properties of the lemma. \end{proof}
\begin{figure}
\caption{The surface $S_{3}$ with 4 boundary circles $\alpha_{1},\alpha_{2},\alpha_{3},\beta$ and one handle}
\label{cyclic}
\end{figure}
The following proposition follows immediately from the two previous lemmas.
\begin{prop}
The endperiodic automorphism $f = h\circ g:L\to L$ has $e$ as repelling end and no other periodic end.
\end{prop}
\end{document} | arXiv |
NCERT Solutions for Class 12 Physics Chapter 8 Electromagnetic Waves PDF - eSaral
NCERT Solutions for Class 12 Physics Chapter 8 Electromagnetic Waves PDF
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Question 1. Figure $8.6$ shows a capacitor made of two circular plates, each of radius $12 \mathrm{~cm}$ and separated by $5.0 \mathrm{~cm}$. The capacitor is being charged by an external source (not shown in the figure). The charging current is constant and equal to $0.15 \mathrm{~A}$.
Solution. (a) Calculate the capacitance and the rate of change of the potential difference between the plates.
(b) Obtain the displacement current across the plates.
(c) Is Kirchhoff's first rule (junction rule) valid at each plate of the capacitor? Explain.
Solution. Given that
The magnitude of charging current, $\mathrm{I}=0.15 \mathrm{~A}$
Distance between the plates, $d=5 \mathrm{~cm}=0.05 \mathrm{~m}$
The Radius of each circular plate,
$\mathrm{r}=12 \mathrm{~cm}=0.12 \mathrm{~m}$
The permittivity of free space, $\epsilon_{0}=8.85 \times 10^{-12} \mathrm{C}^{2} \mathrm{~N}^{-1} \mathrm{~m}^{-2}$
Where, $\mathrm{A}=$ Area of each plate $=\pi r^{2}$
$\mathrm{C}=\frac{\varepsilon_{0} \times \pi \mathrm{r}^{2}}{\mathrm{~d}}=\frac{8.85 \times 10^{-12} \times \pi \times(0.12)^{2}}{0.05}$
$=8.0032 \times 10^{-12} \mathrm{~F}$
$=80.032 \mathrm{pF}$
Charge on each plate, $q=C V$
Where $\mathrm{V}=$ Potential difference across the plates
Differentiate both sides with respect to time (t)
$\frac{d q}{d t}=C \frac{d V}{d t}$
But, $\frac{d q}{d t}=$ current $(I)$
$\therefore \frac{d V}{d t}=\frac{I}{C}$
$\Rightarrow \frac{0.15}{80.032 \times 10^{-12}}=1.87 \times 10^{9} \mathrm{~V} / \mathrm{s}$
Therefore, the change in the potential difference between the plates with respect to time is $1.87 \times 10^{9} \mathrm{~V} / \mathrm{s}$
(b) Here the displacement current in the plates is the same as the conduction current Hence, the displacement current, $i_{d}$ is $0.15 \mathrm{~A}$.
(c) Yes
If we take the sum of conduction and displacement current, then Kirchhoff's first rule is valid at each plate of the capacitor
Question 2. A parallel plate capacitor (Figure. 8.7) made of circular plates each of radius $\mathrm{R}=$ $6.0 \mathrm{~cm}$ has a capacitance $\mathrm{C}=100 \mathrm{pF}$. The capacitor is connected to a $230 \mathrm{~V}$ ac supply with an (angular) frequency of $300 \mathrm{rad} \mathrm{s}^{-1}$
(a) What is the rms value of the conduction current?
(b) Is the conduction current equal to the displacement current?
(c) Determine the amplitude of $\mathrm{B}$ at a point $3.0 \mathrm{~cm}$ from the axis between the plates.
Solution. The capacitance of a parallel plate capacitor, $C=100 \mathrm{pF}=100 \times 10^{-12} \mathrm{~F}$
The radius of each circular plate, $R=6.0 \mathrm{~cm}=0.06 \mathrm{~m}$
The capacitance of a parallel plate capacitor, $C=100 \mathrm{pF}=100 \times 10^{-12} \mathrm{~F}$
The voltage supply to the capacitor, $V=230 \mathrm{~V}$
Angular frequency, $\omega=300 \mathrm{rad} \mathrm{s}^{-1}$
(a) Rms value of conduction current, $I=\frac{V}{x_{C}}$
Where, $X_{C}=$ Capacitive reactance $=\frac{1}{\omega C}$
$\therefore I=V \times \omega C=230 \times 300 \times 100 \times 10^{-12}$
$=6.9 \times 10^{-6} \mathrm{~A}=6.9 \mu \mathrm{A}$
Hence, the rms value of the conduction current is $6.9 \mu \mathrm{A}$.
(b) Yes, the magnitude of conduction current is equal to displacement current.
(c) The magnetic field is given as $B=\frac{\mu_{0} r}{2 \pi R^{2}} I_{0}$
Where, $\mu_{o}=$ Free space permeability $=4 \pi \times 10^{-7} \mathrm{~N} \mathrm{~A}^{-2}$
$I_{o}=$ Maximum value of current $=\sqrt{2} I$
$\mathrm{r}=$ Distance between the plates from the axis $=3.0 \mathrm{~cm}=0.03 \mathrm{~m}$
$\therefore \mathrm{B}=\frac{4 \pi \times 10^{-7} \times 0.03 \times \sqrt{2} \times 6.9 \times 10^{-6}}{2 \pi(0.06)^{2}}$
$=1.63 \times 10^{-11} \mathrm{~T}$
Hence, the magnetic field at that point is $1.63 \times 10^{-11} \mathrm{~T}$.
Question 3. What physical quantity is the same for X-rays of wavelength $10^{-10} \mathrm{~m}$ red light of wavelength 6800 A and radio waves of wavelength $500 \mathrm{~m}$ ?
Solution. The magnitude of the speed of light $\left(3 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)$ in a vacuum is the same for all type of wavelengths. It does not depend on the wavelength in the vacuum.
Question 4. A plane electromagnetic wave travels in vacuum along the z-direction. What can you say about the directions of its electric and magnetic field vectors? If the frequency of the wave is $30 \mathrm{MHz}$, what is its wavelength?
The electromagnetic wave travels along the $z$-direction in a vacuum.
The electric field (E) and the magnetic field (B) are in the $x-y$ plane. They are mutually perpendicular.
Frequency of the wave, $v=30 \mathrm{MHz}=30 \times 10^{6} \mathrm{~Hz}$
Speed of light in a vacuum, $c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}$
We know that expression for the wavelength of a wave is given as:
$\lambda=\frac{c}{v}$
$\lambda=\frac{3 \times 10^{8}}{30 \times 10^{6}}=10 \mathrm{~m}$
Hence the wavelength of the electromagnetic wave is $10 \mathrm{~m}$
Question 5. A radio can tune in to any station in the $7.5 \mathrm{MHz}$ to $12 \mathrm{MHz}$ bands. What is the corresponding wavelength band?
Solution. Given that,
The minimum frequency of radio wave, $v_{1}=7.5 \mathrm{MHz}=7.5 \times 10^{6} \mathrm{~Hz}$
Maximum frequency, $v_{2}=12 \mathrm{MHz}=12 \times 10^{6} \mathrm{~Hz}$
Speed of light, $c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}$
Wavelength for $v_{1}$ can be calculated as:
$\lambda=\frac{c}{v_{1}}$
$=\frac{3 \times 10^{8}}{7.5 \times 10^{6}}=40 \mathrm{~m}$
$=\frac{3 \times 10^{8}}{12 \times 10^{6}}=25 \mathrm{~m}$
Hence the wavelength range of the radio is $40 \mathrm{~m}$ to $25 \mathrm{~m}$.
Question 6. A charged particle oscillates about its mean equilibrium position with a frequency of $10^{9} \mathrm{~Hz}$. What is the frequency of the electromagnetic waves produced by the oscillator?
Solution. About its mean position, the frequency of an electromagnetic wave produced by the oscillator is the same as the frequency of a charged particle oscillating, i.e., $10^{9} \mathrm{~Hz}$.
Question 7. The amplitude of the magnetic field part of a harmonic electromagnetic wave in vacuum is $\mathrm{B}_{0}=510 \mathrm{nT}$. What is the amplitude of the electric field part of the wave?
The amplitude of the magnetic field of an electromagnetic wave in a vacuum,
$\mathrm{B}_{\mathrm{o}}=510 \mathrm{nT}=510 \times 10^{-9} \mathrm{~T}$
The relation between the amplitude of the electric field and the magnetic field is
$c=\frac{E_{0}}{B_{0}}$
$E=c B_{o}=3 \times 10^{8} \times 510 \times 10^{-9}=153 \mathrm{~N} / \mathrm{C}$
Therefore, the amplitude of the electric field part of the wave is $153 \mathrm{~N} / \mathrm{C}$.
Question 8. Suppose that the electric field amplitude of an electromagnetic wave is $\mathrm{E}_{0}=120 \mathrm{~N} / \mathrm{C}$ and that its frequency is $v=50.0 \mathrm{MHz}$.
(a) Determine, $\mathrm{B}_{0} \omega, k$, and $\lambda$.
(b) Find expressions for $\mathrm{E}$ and $\mathrm{B}$.
The amplitude of Electric field, $E_{o}=120 \mathrm{~N} / \mathrm{C}$
Frequency of electromagnetic wave, $v=50.0 \mathrm{MHz}=50 \times 10^{6} \mathrm{~Hz}$
(a) The magnitude of magnetic field strength is given as:
$B_{0}=\frac{E_{0}}{c}$
$=\frac{120}{3 \times 10^{8}}$
$=4 \times 10^{-7} \mathrm{~T}=400 \mathrm{nT}$
Angular frequency of the electromagnetic wave is given as:
$=\omega=2 \pi v=2 \pi \times 50 \times 10^{6}=3.14 \times 10^{8} \mathrm{rad} / \mathrm{s}$
Propagation constant is given as:
$k=\frac{\omega}{c}$
$=\frac{3.14 \times 10^{8}}{3 \times 10^{8}}=1.05 \mathrm{rad} / \mathrm{m}$
The wavelength of the wave is given as:
$=\frac{3 \times 10^{8}}{50 \times 10^{6}}=6.0 \mathrm{~m}$
(b) Suppose that the wave is propagating in the positive $x$ direction. Then, the direction of the electric field vector will be in the positive $y$-direction, and the magnetic field vector will be in the positive $z$ direction because all three vectors are mutually perpendicular.
Equation of electric field vector is given as:
$\vec{E}=E_{0} \sin (k x-\omega t) \hat{\jmath}$
$=120 \sin \left[1.05 x-3.14 \times 10^{8} t\right] \hat{\jmath}$
And, magnetic field vector is given as:
$\vec{B}=B_{0} \sin (k x-\omega t) \hat{k}$ $\vec{B}=\left(4 \times 10^{-7}\right) \sin \left[1.05 x-3.14 \times 10^{8} t\right] \hat{k}$
Question 9. The terminology of different parts of the electromagnetic spectrum is given in the text. Use the formula $\mathrm{E}=\mathrm{hv}$ (for the energy of a quantum of radiation: photon) and obtain the photon energy in units of $\mathrm{eV}$ for different parts of the electromagnetic spectrum. In what way are the different scales of photon energies that you obtain related to the sources of electromagnetic radiation?
Solution. The energy of a photon is given as:
$E=h v=\frac{h c}{\lambda}$
$h=$ Planck's constant $=6.6 \times 10^{-34} \mathrm{~J} \mathrm{~s}$
$c=$ Speed of light $=3 \times 10^{8} \mathrm{~m} / \mathrm{s}$
$\lambda=$ Wavelength of radiation
$\therefore E=\frac{6.6 \times 10^{-34} \times 3 \times 10^{8}}{\lambda}=\frac{19.8 \times 10^{-26}}{\lambda} \mathrm{J}$
$=\frac{19.8 \times 10^{-26}}{\lambda \times 1.6 \times 10^{-16}}=\frac{12.375 \times 10^{-7}}{\lambda} \mathrm{eV}$
The given table lists the photon energies for different parts of an electromagnetic spectrum for different $\lambda$.
The photon energies for the different parts of the spectrum of a source indicate the spacing of the relevant energy levels of the source.
Question 10. In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of $2.0 \times 10^{10} \mathrm{~Hz}$ and amplitude $48 \mathrm{~V} \mathrm{~m}^{-1}$
(a) What is the wavelength of the wave?
(b) What is the amplitude of the oscillating magnetic field?
(c) Show that the average energy density of the $\mathrm{E}$ field equals the average energy density of the B field. $\left[\mathrm{c}=3 \times 10^{8} \mathrm{~m} \mathrm{~s}^{-1}\right]$
Frequency of the electromagnetic wave, $v=2.0 \times 10^{10} \mathrm{~Hz}$
The amplitude of the electric field, $E_{o}=48 \mathrm{~V} \mathrm{~m}^{-1}$
(a) The wavelength of a wave is given as:
$=\frac{3 \times 10^{8}}{2 \times 10^{10}}=0.015 \mathrm{~m}$
(b) Magnetic field strength is given as:
$=\frac{48}{3 \times 10^{8}}=1.6 \times 10^{-7} \mathrm{~T}$
(c) The energy density of the electric field is given as:
$U_{E}=\frac{1}{2} \varepsilon_{0} E^{2}$
And, the energy density of the magnetic field is given as:
$U_{B}=\frac{1}{2 \mu_{0}} B^{2}$
$\varepsilon_{0}=$ Permittivity of free space
$\mu_{0}=$ Permeability of free space
We have the relation connecting $\mathrm{E}$ and $\mathrm{B}$ as:
$E=c B \ldots$... 1 )
$c=\frac{1}{\sqrt{\varepsilon_{0} \mu_{0}}} \ldots(2)$
Putting equation (2) in equation (1), we get
$E=\frac{1}{\sqrt{\varepsilon_{0} \mu_{0}}} B$
$\varepsilon_{0} E^{2}=\frac{B^{2}}{\mu_{0}}$
$\frac{1}{2} \varepsilon_{0} E^{2}=\frac{1}{2} \frac{B^{2}}{\mu_{0}}$
$\Rightarrow U_{E}=U_{B}$
Hence the average energy density of the $\mathrm{E}$ field equals the average energy density of the B field
Additional Exercises
Question 11. Suppose that the electric field part of an electromagnetic wave in vacuum is
$\mathrm{E}=\left\{(3.1 \mathrm{~N} / \mathrm{C}) \cos \left[(1.8 \mathrm{rad} / \mathrm{m}) \mathrm{y}+\left(5.4 \times 10^{6} \mathrm{rad} / \mathrm{s}\right) \mathrm{t}\right]\right\} \hat{\mathbf{i}}$
(a) What is the direction of propagation?
(b) What is the wavelength $\lambda$ ?
(c) What is the frequency $v$ ?
(d) What is the amplitude of the magnetic field part of the wave?
(e) Write an expression for the magnetic field part of the wave.
Solution. (a) From the given electric field vector, we can say that the electric field is directed along the positive $x$ direction. Hence, the direction of motion is along the negative y-direction i.e. $-\hat{\jmath}$.
(b) It is given that,
$\vec{E}=3.1 \mathrm{~N} / \mathrm{C} \cos \left[(1.8 \mathrm{rad} / \mathrm{m}) y+\left(5.4 \times 10^{8} \mathrm{rad} / \mathrm{s}\right) t\right] \hat{\imath} \ldots$
The general equation for the electric field vector in the positive $x$ direction can be written as:
$\vec{E}=E_{0} \sin (k x-\omega t) \hat{\imath} \ldots(2)$
On comparing equations (1) and (2), '
we can find
Electric field amplitude, $E_{o}=3.1 \mathrm{~N} / \mathrm{C}$\
Angular frequency, $\omega=5.4 \times 10^{8} \mathrm{rad} / \mathrm{s}$
Wave number, $\mathrm{k}=1.8 \mathrm{rad} / \mathrm{m}$
The wavelength, $\lambda=\frac{2 \pi}{1.8}=3.490 \mathrm{~m}$
(c) Frequency of wave is given as:
Where $c=$ Speed of light $=3 \times 10^{8} \mathrm{~m} / \mathrm{s}$
$\therefore B_{0}=\frac{3.1}{3 \times 10^{8}}=1.03 \times 10^{-7} \mathrm{~T}$
(e) After seeing the given vector field, it can be observed that the magnetic field vector is directed along the positive $z$ direction. Hence, the general equation for the magnetic field vector is written as:
$\vec{B}=B_{0} \cos (k y+\omega t) \hat{k}$
$=\left\{\left(1.03 \times 10^{-7} \mathrm{~T}\right) \cos \left[(1.8 \mathrm{rad} / \mathrm{m}) y+\left(5.4 \times 10^{6} \mathrm{rad} / \mathrm{s}\right) t\right]\right\} \hat{k}$
Question 12. About $5 \%$ of the power of a $100 \mathrm{~W}$ light bulb is converted to visible radiation. What is the average intensity of visible radiation?
(a) at a distance of $1 \mathrm{~m}$ from the bulb?
(b) at a distance of $10 \mathrm{~m}$ ? Assume that the radiation is emitted isotopically and neglect reflection.
The power rating of the bulb, $P=100 \mathrm{~W}$
It is given that about $5 \%$ of its power is converted into visible radiation.
Power of visible radiation, $P^{\prime}=\frac{5}{100} \times 100=5 \mathrm{~W}$
Hence, the power of visible radiation is $5 \mathrm{~W}$.
(a) The distance of a point from the bulb, $d=1 \mathrm{~m}$
Hence, the intensity of radiation at that point is given as:
$I=\frac{P^{\prime}}{4 \pi d^{2}}$
$=\frac{5}{4 \pi(I)^{2}}=0.398 \mathrm{~W} / \mathrm{m}^{2}$
(b) The distance of a point from the bulb, $d_{1}=10 \mathrm{~m}$
$I=\frac{p^{\prime}}{4 \pi\left(d_{1}\right)^{2}}$
$=\frac{5}{4 \pi(10)^{2}}=0.00398 \mathrm{~W} / \mathrm{m}^{2}$
Question 13. Use the formula $\lambda_{\mathrm{m}} \mathrm{T}=0.29 \mathrm{~cm} \mathrm{~K}$ to obtains the characteristic temperature ranges for different parts of the electromagnetic spectrum. What do the numbers that you obtain tell you?
Solution. At a particular temperature, a body produces a continuous spectrum of wavelengths. In the case of a black body, we can find the wavelength corresponding to the maximum intensity of radiation by Planck's law. It can be given by the relation,
$\lambda_{m}=\frac{0.29}{T} \mathrm{~cm}$
Where, $\lambda_{m}=$ maximum wavelength and $T=$ temperature.
Thus, the temperature for different wavelengths can be obtained as:
For $\lambda_{m}=10^{-4} \mathrm{~cm} ; T=\frac{0.29}{10^{-4}}=2900^{\circ} \mathrm{K}$
For $\lambda_{m}=5 \times 10^{-5} \mathrm{~cm} ; \mathrm{T}=\frac{0.29}{5 \times 10^{-4}}=5800^{\circ} \mathrm{K}$
For $\lambda_{\mathrm{m}}=10^{-6} \mathrm{~cm} ; \mathrm{T}=\frac{0.29}{10^{-6}}=290000^{\circ} \mathrm{K}$
The obtained values tell us that temperature ranges are required for obtaining cadiations in different parts of an electromagnetic spectrum. As the wavelength decreases, the corresponding temperature increases.
Question 14. Given below are some famous numbers associated with electromagnetic radiations in different contexts in physics. State the part of the electromagnetic spectrum to which each belongs.
(a) $21 \mathrm{~cm}$ (wavelength emitted by atomic hydrogen in interstellar space).
(b) $1057 \mathrm{MHz}$ (frequency of radiation arising from two close energy levels in hydrogen; known as Lamb shift).
(c) $2.7 \mathrm{~K}$ [temperature associated with the isotropic radiation filling all spacethought to be a relic of the 'big-bang' origin of the universe].
(d) 5890 A - 5896 A [double lines of sodium]
(e) $14.4 \mathrm{keV}$ [energy of a particular transition in ${ }^{57} \mathrm{Fe}$ nucleus associated with a famous high-resolution spectroscopic method (Mössbauer spectroscopy)].
Solution. (a) $21 \mathrm{~cm}$ come under Radio waves; it belongs to the short wavelength end of the electromagnetic spectrum.
(b) Radio waves; it belongs to the short wavelength end
(c) $\quad$ Temperature, $T=2.7^{\circ} \mathrm{K}$,
$\lambda_{\mathrm{m}}$ is given by Planck's law as:
$\lambda_{m}=\frac{0.29}{2.7}=0.11 \mathrm{~cm}$ This wavelength corresponds to microwaves.
(d) wavelength range comes under the yellow light of the visible spectrum.
(e) Transition energy is given by the relation,
$E=h v$ Where, $h=$ Planck's constant $=6.6 \times 10^{-34} \mathrm{~J} \mathrm{~s}$
$v=$ Frequency of radiation
Energy, $E=14.4 \mathrm{keV}$
$\therefore v=\frac{E}{h}$
$=\frac{14.4 \times 10^{3} \times 1.6 \times 10^{-19}}{6.6 \times 10^{-34}}$
$=3.4 \times 10^{18} \mathrm{~Hz}$
$3.4 \times 10^{18} \mathrm{~Hz}$ comes under X-rays.
Question 15. Answer the following questions:
(a) Long-distance radio broadcasts use short-wave bands. Why? (b) It is necessary to use satellites for long-distance TV transmission. Why?
(c) Optical and radiotelescopes are built on the ground, but X-ray astronomy is possible only from satellites orbiting the earth. Why?
(d) The small ozone layer on top of the stratosphere is crucial for human survival. Why?
(e) If the earth did not have an atmosphere, would its average surface temperature be higher or lower than what it is now?
(f) Some scientists have already predicted that a global nuclear war on the earth would be followed by a severe 'nuclear winter' with a bad effect on life on earth. What might be the basis of this prediction?
Solution. (a) Shortwave frequencies are capable of reaching any location on the Earth because they can be reflected by the ionosphere.
(b) It is necessary to use satellites for long-distance TV transmissions because television signals are of high frequencies and high energies. Thus, these signals are mainly not reflected by the ionosphere. Hence, satellites are very helpful in reflecting TV signals in a very big area on the earth.
(c) The earth's atmosphere cannot absorb visible light and radio waves. But $\mathrm{X}$ rays being of much smaller wavelength are absorbed by the atmosphere. Therefore, X-ray astronomy is possible only from satellites orbiting the earth at the height of $36000 \mathrm{~km}$ above the earth's surface. At such a height, the atmosphere is very thin, and X-rays are not absorbed.
(d) Ozone layer on the top of the atmosphere is very important for human survival because it absorbs harmful ultraviolet radiations present in sunlight and prevents it from reaching the Earth's surface. In the presence of ultraviolet, it would be very difficult for anything to survive on the surface. Plants cannot live and grow in heavy uitraviolet radiation, nor can the plankton that serves as food for most of the ocean life. The ozone layer acts as a shield to absorb the UV rays, and keep them from doing damage at the Earth's surface.
(e) In the absence of an atmosphere, there would be no greenhouse effect on the surface of the Earth. As a result, the temperature of the Earth would decrease rapidly, making it chilly and difficult for human survival.
(f) A global nuclear war on the surface of the Earth would have disastrous consequences. Post-nuclear war, the Earth will experience severe winter as the war will produce smoke that would cover most of the part of the sky, thereby preventing solar light from reaching the atmosphere. Also, it will cause the depletion of the ozone layer.
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\begin{document}
\title[The Minimum Generating Set Problem]{The Minimum Generating Set
Problem}
\author[A.~Lucchini]{Andrea Lucchini}
\address[Andrea Lucchini]{Universit\`a di Padova, Dipartimento di Matematica \lq\lq Tullio Levi-Civita\rq\rq}
\email{[email protected]}
\author[D.~Thakkar]{Dhara Thakkar}
\address[Dhara Thakkar]{Indian Institute of Technology Gandhinagar, Gandhinagar}
\email{thakkar\[email protected]}
\begin{abstract}Let $G$ be a finite group. In order to determine the smallest cardinality $d(G)$ of a generating set of $G$ and a generating set with this cardinality, one should repeat \lq many times\rq \ the test whether a subset of $G$ of \lq small\rq \ cardinality generates $G$. We prove that if a chief series of $G$ is known, then the numbers of these \lq generating tests\rq \ can be drastically reduced. At most $|G|^{13/5}$ subsets must be tested. This implies that the minimum generating set problem for a finite group $G$ can be solved in polynomial time. \end{abstract}
\maketitle
\section{Introduction} Let $G$ be a finite group. A generating set of $G$ with minimum size is called a minimum generating set. The size of a minimum generating set of a group $G$ is denoted by $d(G).$ In this paper, we consider the problem of computing $d(G)$, known as the minimum generating set problem, and computing a minimum generating set of a given group $G$. We call \lq\lq a generating test\rq\rq \ for $G$, the check whether a given subset of $G$ generates $G.$ The computational cost of a single generating test for a given group $G$ may depends of the size of $G$ and of the subset that we are testing, and also of the way in which $G$ is assigned ($G$ could be a represented by permutation group representation, a multiplication table, a linear group representation, or an abstract group assigned with a presentation), but in any case an algorithm aimed to determine a minimum generating set becomes more efficient the smaller number of generating tests it requires. This motivates the following question. \begin{question}\label{question-1} How many generating tests are needed to determine a minimum generating set of a finite group? \end{question}
Question \ref{question-1} has been studied before, mainly for groups represented by their multiplication table or by their permutation representation. With these representations of groups, it is easy to see that the minimum generating set problem is in the complexity class $\mathrm{NP}$, as there is a polynomial time algorithm to check if a given subset generates the given group \cite{seress}. Arvind and Tor\'an prove that the minimum generating set problem is in $\mathrm{DSPACE}(\log^2 n)$ \cite{AT}. Recently, the problem has been studied for general and special classes of groups by Das and Thakkar \cite{DT}. However, it is still open if the minimum generating set problem admits a polynomial time algorithm or equivalently if Question \ref{question-1} could be solved using at most polynomial many generating tests even for groups given by their multiplication tables. Information on the complexity classes $\mathrm{NP}$ and $\mathrm{DSPACE}(\log^2 n)$ can be found for example in \cite{Complexity-text}.
Notice that, for every finite group $G$, $d(G)\leq \log_p|G|$ being $p$ the smallest prime factor of $|G|$, so at least one set $\{g_1,\dots,g_d\}$ with $d\leq \log_p(|G|)$ and $1\neq g_i \in G$ for
$1\leq i\leq d$ is a minimum generating set. Hence a crude bound for the number of generating tests that are needed is $\sum_{t\leq \log_p(|G|)}(|G|-1)^t.$
A better strategy is suggested by the following nice remark due to Gasch\"utz \cite {gaz1}.
\begin{lemma}\label{gaz} Let $N$ be a normal subgroup of $G$ and let $g_1,
\dots, g_d \in G$ be such that $G/N=\langle g_1N, \dots, g_dN \rangle$. If $G$
can be generated with $d$ elements then there exist $u_1, \dots, u_d \in N$ such
that $G=\langle g_1u_1, \dots, g_du_d \rangle.$ \end{lemma} Suppose indeed that a chief series
$$1=N_0<N_1<\dots N_{u-1} <N_u=G$$
of $G$ is available. The factor $G/N_{u-1}$ is a simple group, in particular $d(G/N_{u-1})\leq 2$, so at most $|G/N_{u-1}|^2$ generating tests are needed to find a subset $Y$ of minimum size with respect to the property that $G=\langle Y\rangle N_{u-1}.$ This can be considered a first step of an algorithm. For $0<i<u$ suppose that $\{g_1,\dots,g_d\}$ is a minimum generating set for $G$ modulo $N_i.$ By the main result in \cite{alold},
$d(G/N_{i-1})\leq d+1.$ If $d(G/N_{i-1})=d$, then by Lemma \ref{gaz} there exist $n_1,\dots,n_d \in N_i$ such that $\{g_1n_1,\dots,g_dn_d\}$ is a minimum generating set for $G$ modulo $N_{i-1}.$ If $d(G/N_{i-1})=d+1$, then again by Lemma \ref{gaz} there exists $n_1,\dots,n_d,n_{d+1} \in N_i$ such that $\{g_1n_1,\dots,g_dn_d,n_{d+1}\}$ is a minimum generating set for $G$ modulo $N_{i-1}.$ In other words, if we know a minimum generating set for $G$ modulo $N_i$, then at most $|N_i/N_{i-1}|^d+|N_i/N_{i-1}|^{d+1}$ generating tests are needed to determine a minimum generating set for $G$ modulo $N_{i-1}.$ This approach reduces the number of generating tests, but this number remains in general quite large, and it does not give any evidence that a polynomial bound in terms of the order of $G$ is possible.
Our main contribution is to show that the number of generating tests needed to obtained a minimum generating set of $G/N_{i-1}$ from a minimum generating set of $G/N_{i}$ can be drastically reduced (see in particular Corollaries \ref{coroab} and \ref{corononabb}). As a consequence, we obtain we following unexpected result.
\begin{thm}\label{main}The number of generating tests needed to determine a minimum generating set of a finite group $G$ is at most $|G|^{\frac{13}{5}}$. \end{thm}
Our procedure requires the knowledge of a chief series of the group, that in any case can be computed in polynomial time (see for example \cite[6.2.7]{seress}), so we may conclude that the problem of finding a minimum generating set for a finite group $G$ has a solution which is polynomial on $|G|$.
Our proof of Theorem \ref{main} uses the theory of crowns of finite groups (see Section \ref{crowns}) and some results about the generation of finite groups with a unique minimal normal subgroup that strongly depend on the classification of the finite non-abelian simple groups.
\section{Chief series and chief factor}
Let $G$ be a nontrivial finite group. Recall that a chief series of a finite group $G$ is a normal series $$1=N_0 < N_1 < \dots < N_u = G$$ of finite length with the property that for $i\in \{0,\dots, u-1\}$, $N_{i+1}/N_i$ is a minimal normal subgroup of $G/N_i$. The integer $u$ is called the length of the series and the factors $N_{i+1}/N_i$, where $0\leq i \leq u-1$, are called the chief factors of the series. A nontrivial finite group $G$ always possesses a chief series. Moreover, two chief series of $G$ have the same length, and any two chief series of $G$ are the same up to permutation and isomorphism. Thus, adopting the notation above, we may define {the chief length} of $G$ to be $u$, and the chief factors of $G$ to be the groups $N_{i+1}/N_i$.
In the following we will denote by $\frat(G)$ the Frattini subgroup of a finite group $G$. Moreover if $H/K$ is a chief factor of $G$, we say that $H/K$ is {Frattini} if $H/K\leq \frat(G/K)$ and that $H/K$ is complemented if there exists a subgroup $U$ of $G$ such that $UH=G$ and $U\cap H=K$. Since the Frattini subgroup of a finite group $G$ is nilpotent, and the only subgroup supplementing $\frat(G)$ is $G$ itself, the following lemma is immediate. \begin{lemma}\label{l:1}
Let $G$ be a nontrivial finite group and let $A=H/K$ be a chief factor of $G$.
\begin{enumerate}[(i)]
\item If $A$ is abelian then $A$ is non-Frattini if and only if $A$ is complemented.
\item If $A$ is non-abelian then $A$ is non-Frattini.
\end{enumerate} \end{lemma}
\section{Monolithic group and crown-based power}
A finite group $L$ is called {monolithic} if $L$ has a unique minimal normal subgroup $A$. If in addition $A$ is not contained in $\frat(L)$, then $L$ is called a {monolithic primitive group}.
Let $L$ be a monolithic primitive group and let $A$ be its unique minimal normal subgroup. For each positive integer $k$, let $L^k$ be the $k$-fold direct product of $L$. The crown-based power of $L$ of size $k$ is the subgroup $L_k$ of $L^k$ defined by $$L_k=\{(l_1, \ldots , l_k) \in L^k \text{ : } l_1 \equiv \cdots \equiv l_k \ {\mbox{mod}\text{ } } A \}.$$
Equivalently, $L_k=A^k \diag(L^k)$, where $$\diag(L^k):=\{(l,l,\hdots,l)\text{ : }l\in L\}\le L^k.$$
We also define $L_0:=1$. Assume that $A=\soc L$ is non-abelian and let $C:=C_{\aut A}(L/A).$ Moreover assume that $L=\langle l_1,\dots,l_d\rangle A.$ Suppose $d\geq d(L)$ and let $\Omega_{l_1,\dots,l_d}$ be the set of $d$-tuples $(\overline l_1,\dots,\overline l_d)$ in $L^d$ such that $\langle \overline l_1,\dots,\overline l_d\rangle=L$ and $\overline l_i\equiv l_i \mod A$ for $1\leq i\leq d.$ Then $C$ acts on $\Omega_{l_1,\dots,l_d}$ by setting $(x_1,\dots,x_d)^\gamma=(x_1^\gamma,\dots,x_d^\gamma)$ for every $(x_1,\dots,x_d) \in \Omega_{l_1,\dots,l_d}$ and $\gamma\in C.$ The following holds (see \cite[Section 2]{austr}). \begin{lemma}\label{comegenero}
Assume $a_{ij}\in A,$ with $1\leq i\leq d$ and $1\leq j\leq k$ and let $$\begin{aligned}
g_1=&l_1(a_{11},\dots,a_{1k}),\\
&\dots \quad \dots\quad\dots \quad\\
g_d=&l_d(a_{d1},\dots,a_{dk}).
\end{aligned}
$$
Then $\langle g_1,\dots,g_d\rangle=L_k$ if and only if $\langle l_1,\dots,l_d\rangle=L$ and
the $d$-tuples
$(l_1a_{11},\dots,l_da_{d1}),\dots,$ $(l_1a_{1k},\dots,l_da_{dk})$ belong to different orbits for the action of $C$ on $\Omega_{l_1,\dots,l_d}.$ \end{lemma}
Before to state the next result, we need to recall another important observation. Let $H$ be a finite group, $X$ a subset of $H$, $M$ a normal subgroup of $H$ and assume that $h_1,\ldots,h_k\in H$ and $X$ generate $H$ modulo $M$, that is $H=\langle h_1,\ldots,h_k, X,M\rangle$. It follows from \cite[Proposition 16]{Lxdir} that the number $\Phi_{H,M}(X,k)$ of elements $(u_1,\ldots, u_k)\in M^k$ with the property that $H=\langle h_1u_1,\ldots, h_ku_k,X\rangle$ is independent of the choice of $h_1,\ldots,h_k$.
\begin{lemma}\label{counting}
Let $L$ be a monolithic primitive group, and assume that $A=\soc(L)$ is non-abelian. Moreover, suppose that $d\geq d(L)$ and $L=\langle l_1,\dots,l_d\rangle A,$ and let $t$ be a positive integer with $2\leq t\leq d.$ Let $\Delta$ be the set of the $t$-tuples $(a_1,\dots,a_t)\in A^t$ with the property that $L=\langle l_1a_1,\dots,l_ta_t,l_{t+1},\dots,l_d\rangle.$ Then $$|\Delta|\geq\frac{53|A|^t}{90}.$$ \end{lemma} \begin{proof} Set $X=\{l_{t+1},\dots,l_d\}.$ It follows from the proof of \cite[Lemma 2]{index}, that there exist
$y_1,\dots,y_t\in L$ such that $L=\langle y_1,\dots,y_t,X\rangle.$ Hence $|\Delta|=\phi_{L,A}(X,t).$
To conclude it suffices to notice that, by \cite[Theorem 2.2]{london}, $\phi_{L,A}(X,t)\geq 53|A|^t/90.$ \end{proof}
\begin{prop}\label{smallt}Let $L$ be a monolithic primitive group, and assume that $A=\soc(L)$ is non-abelian. Consider the crown-based product $G=L_\delta$ and let $N\cong A$ be a minimal normal subgroup of $G.$ Suppose that $G=\langle g_1,\dots,g_d\rangle N$, with $d\geq 2.$ Let $t=\lceil\frac{8}{5}+\log_{|N|}\delta\rceil$. If $t\leq d,$ then there exist $n_1,\dots,n_t\in N$ such that $G=\langle g_1n_1,\dots,g_tn_t,g_{t+1},\dots,g_d\rangle.$ \end{prop} \begin{proof} Notice that $L/A\cong G/\soc(G),$ hence $$d(L/A)=d(G/\soc(G))\leq d(G/N)\leq d.$$ It follows from the main theorem in \cite {unico}, that $$d(L)=\max\{2,d(L/N)\}\leq d.$$
The elements of $G$ are of the form $l(y_1,\dots,y_\delta)$ with $l\in L$ and $y_1,\dots,y_\delta\in A.$ We may identify $N$ with the subgroup
$\{(1,\dots,1,y)\mid y \in A\}$ of $A^\delta=\soc(G).$ Moreover, for $1\leq i\leq d$ and $1\leq j\leq \delta,$ there exist $l_i \in L$ and $y_{ij}$ in $A$ such that
$$\begin{aligned}
g_1=&l_1(y_{1,1},\dots,y_{1,\delta}),\\
&\dots \quad \dots\quad\dots \quad\\
g_d=&l_d(y_{d,1},\dots,y_{d,\delta}).
\end{aligned}
$$
Let $\Gamma=C_{\aut A}(L/A)$ and, for $1\leq i\leq \delta-1,$ let
$$\omega_i=(l_1y_{1,i},\dots,l_dy_{d,i})\in L^d.$$ Moreover let $\Omega$ be the set
of $(\overline l_1,\dots,\overline l_d)$ in $L^d$ such that $\langle
\overline l_1,\dots,\overline l_d\rangle=L$ and $\overline l_i\equiv l_i \mod A$ for $1\leq i\leq d.$ The condition $G=\langle g_1,\dots,g_d\rangle N$ implies that $\omega_i\in \Omega$ for $1\leq i\leq \delta-1$ and if $i\neq j$ then $\omega_i$ and $\omega_j$ belong to different $\Gamma$-orbits for the action of $\Gamma$ in $\Omega.$
Our statement is equivalent to show that there exist $x_1,\dots,x_t \in A$ such that $\omega=(l_1 y_{1,\delta} x_1,\dots,l_t y_{t,\delta} x_t,l_{t+1}y_{t+1,\delta},\dots,l_{d}y_{d,\delta})\in \Omega$ and, for each $1\leq i\leq \delta-1$, $\omega$ and $\omega_i$ belong to different $\Gamma$-orbits. Indeed in that case, taking, for $1\leq i\leq t,$ $n_i=(1,\dots,1,x_i)\in N,$ by Lemma \ref{comegenero} we conclude $G=\langle g_1n_1,\dots,g_tn_t,g_{t+1},\dots,g_d\rangle.$
Let $\Delta$ be the subset of $\Omega$ consisting of the $d$-tuples $(\overline l_1,\dots,\overline l_d)$ with $\overline l_i =l_iy_{i\delta}$ for each $i>t.$ Assume $A\cong S^n,$ with $n\in \mathbb N$ and $S$ a finite non-abelian simple group. By Lemma \ref{counting},
$$|\Delta|\geq \frac{53|A|^{t}}{90}.$$
Notice that the action of $\Gamma$ on $\Omega$ is semiregular. Since $$|\Gamma| \leq n|S|^{n-1}|\aut(S)|\leq n|A||\out(S)|\leq n|A|\log_2|S|
\leq |A|\log_2|A|$$ (see the proof of \cite[Lemma 1]{pr} and the upper bound for $|\out(S)|$ given in \cite{ku}), the number of $\Gamma$-orbits with a representative in $\Delta$ is at least
$$\frac{|\Delta|}{|\Gamma|}\geq \frac{53}{90}\frac{|A|^{t-1}}{\log_2|A|}\geq |A|^{t-\frac{8}{5}}.$$ Since $|A|^{t-8/5}=|N|^{t-8/5}\geq \delta$, we can conclude that $\Delta$ contains an element with the required property. \end{proof}
\section{Crowns}\label{crowns}
The notion of crown was introduced by Gasch\"{u}tz in \cite{Gaschutz} in the case of finite solvable groups and generalized in \cite{JL} to arbitrary finite groups. A detailed exposition of the theory is also given in \cite[1.3]{classes}.
If a group $G$ acts on a group $A$ via automorphisms (that is, if there exists a homomorphism $G\rightarrow \aut(A)$), then we say that $A$ is a $G$-group. If $G$ does not stabilise any nontrivial proper subgroup of $A$, then $A$ is called an irreducible $G$-group. Two $G$-groups $A$ and $B$ are said to be $G$-isomorphic, or $A\cong_G B$, if there exists a group isomorphism $\phi: A\rightarrow B$ such that $\phi(g(a))=g(\phi(a))$ for all $a\in A, g\in G$. Following \cite{JL}, we say that two $G$-groups $A$ and $B$ are $G$-equivalent and we put $A \equiv_G B$, if there are isomorphisms $\phi: A\rightarrow B$ and $\Phi: A\rtimes G \rightarrow B \rtimes G$ such that the following diagram commutes:
\begin{equation*}
\begin{CD}
1@>>>A@>>>A\rtimes G@>>>G@>>>1\\
@. @VV{\phi}V @VV{\Phi}V @|\\
1@>>>B@>>>B\rtimes G@>>>G@>>>1.
\end{CD} \end{equation*}
\
Note that two $G$\nobreakdash-isomorphic $G$\nobreakdash-groups are $G$\nobreakdash-equivalent. In the particular case where $A$ and $B$ are abelian the converse is true: if $A$ and $B$ are abelian and $G$\nobreakdash-equivalent, then $A$ and $B$ are also $G$\nobreakdash-isomorphic. It is proved (see for example \cite[Proposition 1.4]{JL}) that two chief factors $A$ and $B$ of $G$ are $G$-equivalent if and only if either they are $G$-isomorphic, or there exists a maximal subgroup $M$ of $G$ such that $G/\core_G(M)$ has two minimal normal subgroups $X$ and $Y$ that are $G$-isomorphic to $A$ and $B$ respectively. For example, the minimal normal subgroups of a crown-based power $L_k$ are all $L_k$-equivalent.
For an irreducible $G$-group $A$ denote by $L_A$ the monolithic primitive group associated to $A$. That is $$L_{A}= \begin{cases}
A\rtimes (G/C_G(A)) & \text{ if $A$ is abelian}, \\
G/C_G(A)& \text{ otherwise}. \end{cases} $$ If $A$ is a non-Frattini chief factor of $G$, then $L_A$ is a homomorphic image of $G$. More precisely, there exists a normal subgroup $N$ of $G$ such that $G/N \cong L_A$ and $\soc(G/N)\equiv_G A$. Consider now all the normal subgroups $N$ of $G$ with the property that $G/N \cong L_A$ and $\soc(G/N)\equiv_G A$: the intersection $R_G(A)$ of all these subgroups has the property that $G/R_G(A)$ is isomorphic to the crown-based power $(L_A)_{\delta_G(A)}$. The socle $I_G(A)/R_G(A)$ of $G/R_G(A)$ is called the {$A$-crown} of $G$ and it is a direct product of $\delta_G(A)$ minimal normal subgroups $G$-equivalent to $A$.
In our proof we will use the following consequence of \cite[Proposition 1.1]{crown}.
\begin{lemma}\label{corone}
Let $N$ be a minimal normal subgroup of a finite group $G.$ If $HN=HR_G(N)=G,$ then $H=G.$ \end{lemma}
The next result is an immediate consequence of \cite[Lemma 10]{crown2}. \begin{lemma}\label{corone2}
If $N$ is a non-Frattini minimal normal subgroup of a finite group $G$, then
$NR_G(N)/R_G(N)$ is $G$-equivalent to $N.$
\end{lemma}
\section{Abelian minimal normal subgroups}
\begin{prop}\cite[Theorem 4]{jsc}\label{jscc}
Let $G$ be a group and let $N = \langle e_1,\dots, e_l\rangle$ be an abelian
minimal normal subgroup of $G.$ If $G= \langle g_1,\dots, g_d\rangle$N then one of the following occurs:
\begin{enumerate}
\item $d(G) \leq d$ and either $G = \langle g_1,\dots, g_d\rangle$ or there exist $1 \leq i \leq d$ and $1 \leq j \leq l$ such that $G = \langle g_1,\dots, g_{i-1}, g_ie_j, g_{i+1},\dots, g_d\rangle;$
\item $d(G) = d + 1$ and $G = \langle g_1, \dots , g_d, x\rangle$ for any $1\neq x \in N.$
\end{enumerate} \end{prop}
\begin{cor}\label{coroab}
Let $N$ be a minimal abelian normal subgroup of a finite group $G$ and let $g_1N,\dots,g_dN$ be a minimum generating set of $G/N.$ We can find a minimal generating set for $G$ testing at most $d(|N|-1)+1\leq d|N|$ elements of $|G|^d.$ \end{cor} \begin{proof}
By the previous proposition, if $d(G)=d,$ then $\{g_1,\dots,g_d\}$ or $\{g_1,\dots,g_in,\dots g_d\}$, with $1\neq n \in N$ and $1\leq i\leq d,$ is a minimal generating set of $G$. If none of these sets generates $G$, then $d(G)=d+1$ and, for every $1\neq n\in N,$ $\{g_1,\dots,g_d,n\}$ is a generating set. \end{proof}
\begin{rem}\label{remab} If $N$ is an abelian minimal normal subgroup of a finite group $G$, then $N \cong C_p^l$ is an elementary abelian $p$-group and a generating set $e_1,\dots,e_l$ for $N$ can be obtained in $l$ steps. So we may improve the statement of the previous corollary stating that a minimum generating set of $G$ can be obtained from a minimum generating set of $G/N$ in at most $l+dl+1=(d+1)l+1$ steps. \end{rem}
\section{Non-abelian minimal normal subgroups}
\begin{lemma}\label{mint}Let $N$ be a non-abelian minimal normal subgroup of a finite group $G.$ Suppose that $G=\langle g_1,\dots,g_d\rangle N,$ with $d\geq 2$. Let $t=\left\lceil\frac{8}{5}+\log_{|N|}\delta_G(N)\right\rceil.$ If $t\leq d,$ then there exist $n_1,\dots,n_t\in N$ such that $$G=\langle g_1n_1,\dots,g_tn_t,g_{t+1},\dots,g_d\rangle.$$ \end{lemma}
\begin{proof}
Let $R=R_G(N)$ and $\delta=\delta_G(N).$ Moreover let $L$ be the monolithic primitive group associated to $N.$ We use the bar notation $\bar g$ to denote the elements of the factor group $\bar G=G/R$. By Lemma \ref{corone2}, $\bar N=NR/R$ is a minimal normal subgroup of $G/R$. Moreover it follows from Lemma \ref{corone} that $G=\langle g_1n_1,\dots,g_tn_t,g_{t+1},\dots,g_d\rangle$ if and only if $\bar G=\langle \bar g_1\bar n_1,\dots,\bar g_t\bar n_t,\bar g_{t+1},\dots\bar g_d\rangle$. Since $\bar G \cong L_\delta,$ with $L\cong G/C_G(N),$ the conclusion follows from Proposition \ref {smallt} \end{proof}
\begin{cor}\label{corononab}
Let $N$ be a minimal non-abelian normal subgroup of a finite group $G$ and let $g_1N,\dots,g_dN$ be a minimum generating set of $G/N.$ We may produce a minimum generating set for $G$ testing at most $|N|^{\lceil\frac{8}{5}+\log_{|N|}\delta_G(N)\rceil}$ elements of $G^d$ and at most $|N|^{\lceil\frac{8}{5}+\log_{|N|}\delta_G(N)\rceil}$ elements of $G^{d+1}.$ \end{cor}
\begin{proof}
Let $u=\max\{d,2\}$. If $d<u,$ set $g_{d+1}=\dots =g_u=1.$ Let $t=\min\{u,\lceil\frac{8}{5}+\log_{|N|}\delta\rceil\}.$ First we test the
$|N|^t$ $u$-tuples of kind $(g_1n_1,\dots,g_tn_t,g_{t+1},\dots,g_u)$,
with $n_1,\dots,n_t \in N.$ If one of them is a generating set, we are done. If not, it follows from Lemma \ref{mint} that $u<t$. In this case we have tested that $\langle g_1n_1,\dots,g_un_u\rangle \neq G$ for every $(n_1,\dots,n_u)\in N^u,$ so if follows from Lemma \ref{gaz} that $d(G)>u.$
Moreover,
by the main result in \cite{alold}, $$u<d(G)\leq \max\{2,d(G/N)+1\}\leq u+1\leq t$$ Hence $d(G)=u+1\leq t$ and, by Lemma \ref{gaz},
one of the $(u+1)$-tuples of kind $(g_1n_1,\dots,g_dn_u,n_{u+1})$ is a minimum generating set \end{proof}
In applying the previous corollary to design an algorithm to compute a minimum generating set of a finite group $G,$ an obstacle is given by the fact that, even if a chief series of $G$ is available, it is not easy to recognize whether two factors of the series are $G$-equivalent. Hence, with the computation applications in mind, it is better to consider the weaker equivalence relation in which two factors are equivalent if they have the same order. Denoting with $\eta_G(A)$ the number of factors in a chief series of $G$ with order $|A|,$ we may state the weaker formulation of the previous corollary.
\begin{cor}\label{corononabb}
Let $N$ be a minimal non-abelian normal subgroup of a finite group $G$. If we know a minimum generating set of $G/N$, then the number of generating tests needed to compute a minimum generating set for $G$ is at most $$2|N|^{\lceil\frac{8}{5}+\log_{|N|}\eta_G(N)\rceil}.$$ \end{cor} \section{Finding a minimal generating set in polynomial time}
\begin{proof}[Proof of Theorem \ref{main}]
Let $\mathcal A$ and $\mathcal B$ be, respectively, the set of abelian and non-abelian factors in a given chief series of $G.$ Moreover let $\alpha(G)=\prod_{X\in \mathcal A}|A|$ and $\beta(G)=\prod_{Y\in \mathcal B}|B|.$ With iterated application of Corollaries \ref{coroab} and \ref{corononabb}, we deduced that the number of generating tests required to obtain a minimum generating set for $G$ is at most
$$\sum_{X \in \mathcal A}d(G)|X|+\sum_{Y \in \mathcal B}2|Y|^{\log_{|Y|}\eta_G(Y)+\frac{13}{5}}.$$
Clearly $$\sum_{X\in \mathcal A}d(G)|X|\leq d(G)\prod_{X\in \mathcal A}|X|=d(G)\alpha(G).$$ We consider the equivalence classes in $\mathcal B$ in which two factors are equivalent if and only if they have the same order. Let $\mathcal B_1,\dots,\mathcal B_r$ be the equivalence classes in $\mathcal B$ and for every class choose a representative $Y_i$ for this class. Then we have
$$\begin{aligned}\sum_{Y \in \mathcal B}2|Y|^{\log_{|Y|}\eta_G(Y)+\frac{8}{3}}&\leq \sum_{1\leq i \leq r}2\eta_G(Y_i)|Y_i|^{\log_{|Y_i|}\eta_G(Y_i)+\frac{13}{5}}\\
&=\sum_{1\leq i \leq r}2\eta_G(Y_i)^2|Y_i|^{\frac{13}{5}}\\
&\leq \sum_{1\leq i \leq r}|Y_i|^{\frac{13}{5} \eta_G(Y_i)} \\
&\leq \left(\prod_{1\leq i\leq r}|Y_i|^{\eta_G(Y_i)}
\right)^{\frac{13}{5}}=\beta(G)^{\frac{13}{5}}.
\end{aligned} $$
If follows that the number of required generating tests is at most $$d(G)\alpha(G)+{\beta(G)^{\frac{13}{5}}}\leq |G|^{\frac{13}{5}}.\qedhere$$ \end{proof}
\section{Permutation groups}
The best bound for the cardinality of a generating set of a permutation group is due to A. McIver and P. Neumann: the so call \lq\lq McIver-Neumann Half-$n$ Bound\rq\rq\ says that if $G$ is a subgroup of $\perm(n)$ and $G\neq \perm(3),$ then $d(G)\leq \lfloor n/2 \rfloor.$ This result is stated without a proof in \cite[Lemma 5.2]{min} and a sketch of the proof is given in \cite[Section 4]{cst}. The following result is crucial for our purposes.
\begin{thm}\cite[Theorem 10.0.5]{nina-thesis}\label{ninatesi}. Let $G$ be a permutation group of degree n with s orbits. Then a chief series of $G$ has length at most $n-s.$
\end{thm}
\begin{lemma}\label{permab} Let $G\leq \sym(n)$ and
$N$ a minimal abelian normal subgroup of $G$. If we know a minimum generating set for $G/N$, we can find a minimum generating set for $G$ in at most $(\lfloor n/2\rfloor+1)^2$ steps. \end{lemma}
\begin{proof}We may assume $n>3.$ Then
$\max\{d(G), d(N)\} \leq n/2$, so the conclusion follows from Remark \ref{remab}. \end{proof}
\begin{lemma}\label{permnonab} Let $G\leq \sym(n)$
and $N$ a minimum non-abelian normal subgroup of $G$. If we know a minimum generating set for $G/N$, we can find a minimum generating set for $G$ with at most $(n-1)|N|^{\frac{13}{5}}$ generating tests. \end{lemma} \begin{proof}
By Corollary \ref{corononab}, we need at most $|N|^{\log_{|N|}\delta_G(N)+\frac{13}{5}}=\delta_G(N)|N|^{\frac{13}{5}}$ generating tests. Moreover from Theorem \ref{ninatesi}, it follows $\delta_G(N)\leq n-1.$ \end{proof}
In the following we will denote by $\lambda(G)$ the maximum of the orders of the non-abelian chief factors of $G$ (and we will set $\lambda(G)=1$ if $G$ is solvable). Combining Theorem \ref{ninatesi} and Lemmas \ref{permab} and \ref{permnonab} we obtain:
\begin{thm} Let $G\leq \sym(n).$ Then we may obtain a minimum generating set of $G$ with at most $n^2\lambda(G)^{13/5}$ generating test. \end{thm}
\begin{cor} Let $G\leq \sym(n).$ If every non-abelian composition factor of $G$ has order at most $c$, then we may obtain a minimum generating set of $G$ with at most $n^2c^{\frac{13}{20}n}$ generating tests. \end{cor}
\begin{proof} Let $H/K$ be a non-abelian chief factor of $G.$ Then $H/K\cong S^u,$ with $u \in \mathbb N$ and $S$ a finite non-abelian simple group. Let $X/K$ be a Sylow 2-subgroup of $H/K.$ We have $X/K \cong P^u$ with $P$ a Sylow 2-subgroup of $S$. Since $P$ cannot be cyclic,
$2u\leq d(X/K) \leq d(X)\leq n/2,$ hence $u\leq n/4$ and $|H/K|\leq c^{n/4},$ so the conclusion follows from the previous Theorem. \end{proof}
\end{document} | arXiv |
Example 1: Any collection $\mathbf A$ of binary relations on a set $X$ such that $\mathbf A$ is closed under union, intersection and composition. Example 1: Any collection $\mathbf A$ of binary relations on a set $X$ such that $\mathbf A$ is closed under union, intersection and composition.
- Andreka 1991 AU proves that these examples generate the variety DLOS. + H. Andreka[(Andreka1991)] proves that these examples generate the variety DLOS. | CommonCrawl |
Nuclear Fission and Fusion
When a heavy nucleus like 92U235 is bombarded by a neutron, the total mass of nuclei is not equal to the sum of the masses of the heavy nucleus and the neutron. Similarly, when two light nuclei like 1H2 fused together to form a heavier and stable nucleus, the mass of the product are not equal to the sum of masses of the initial lighter nuclei. This shows that there is always a difference in mass, called mass defect, in a nuclear reaction. According to Einstein's mass-energy relation, the difference in mass (Δm) is converted into energy as ΔE = Δmc2 which is released during the reaction. The energy released is called nuclear energy.
The subdivision of a heavy atomic nucleus, such as that of uranium or plutonium, into two fragments of roughly equal mass, is called the nuclear fission. For example: When uranium, 92U235 is bombarded with a slow neutron, the uranium nucleus captures the neutron, and a compound nucleus is formed. The new nucleus becomes unstable and almost immediately, splits into two or more fragments with some additional neutrons. The fission reaction is
$$ _0n^1 + _{92}U^{235} \rightarrow [_{92}U^{235}] \rightarrow _{56} Ba^{141} + _{36}Ba^{141} + _{36} Kr^{92} + 3_0n^1 + \text {Energy} $$
Energy released in Fission
The energy released in the process is due to the difference in mass of the initial nucleus and its products. The initial mass is greater than the sum of masses of the products and difference in mass (Δm) s converted into energy according to Einstein's mass-energy relation ΔE = Δmc2.
Bohr wheeler Theory of Nuclear Fission
liquid drop model of nucleus
According to Bohr and wheeler, the nucleus behaves like a liquid drop whose shape of drop depends upon the balance between the surface tension force and columbic repulsion force. When uranium nucleus captures a bombarding neutron it sets an oscillation within the drop. This excitation energy tends to distort the nucleus into an ellipsoidal shape. If the surface tension force is more, it tends to return the nucleus into its original shape and if the excitation energy is more the drop is further repelled and finally gets split into two or more or less equal fragments. If however the excitation energy is not sufficiently high, the nucleus returns to its original shape with the release of excess excitation energy in the form of γ-rays photon. The process is radiation capture but no fission.
Nuclear chain reaction
A chain reaction is a self-propagating process in which a number of neutrons go on multiplying rapidly almost in geometrical propagation during fission till all the fissionable material is disintegrated. There are two types of chain reaction:
Controlled chain reaction:
In this reaction number of neutron and hence energy is controlled to a desired level. This principle is used in a nuclear reactor.
Uncontrolled chain reaction:
In this reaction, the neutrons are allowed to multiply indefinitely. This entire energy is released all at a time. This principle of an atom bomb.
Multiplication Factor
The ratio of secondary neutrons produced to the initial number of neutrons is called multiplication factor. It is denoted by k.
\begin{align*} K = \frac {\text { Number of neutrons in any generation}} {\text { Number of neutrons in previous generation}} \\ \text {If} \: K = 1, \: \text {the reaction is steady or critical} \\ \text {If} \: K > 1, \: \text {the reaction is building up or super-critical} \\ \text {If} \: K < 1, \: \text {the reaction is dying down or sub-critical} \\ \end{align*}
Critical Size for Maintenance of Chain Reaction
The critical size of a system containing fissile material is defined as the minimum size for which the number of neutrons produced in the fission process just balance those lost by leakage and non fission capture. Corresponding mass of fissionable material is called critical mass. If the size is less than the critical size, a chain is not possible.
According to Einstein's mass-energy relation, the difference in mass (Δm) is converted into energy as ΔE = Δmc2 which is released during the reaction. The energy released is called nuclear energy.
The subdivision of a heavy atomic nucleus, such as that of uranium or plutonium, into two fragments of roughly equal mass, is called the nuclear fission.
According to Bohr and wheeler, the nucleus behaves like a liquid drop whose shape of drop depends upon the balance between the surface tension force and columbic repulsion force.
A chain reaction is a self-propagating process in which a number of neutrons go on multiplying rapidly almost in geometrical propagation during fission till all the fissionable material is disintegrated.
The ratio of secondary neutrons produced to the initial number of neutrons is called multiplication factor.
Einstein's Mass Energy Relation
ASK ANY QUESTION ON Nuclear Fission
figure of atom? | CommonCrawl |
\begin{document}
\title{Computation of graphical derivatives of normal cone maps to
a class of conic constraint sets ootnote{Supported by the National Natural Science Foundation of China under project No.11571120 and the Natural Science Foundation of Guangdong Province under project No.2015A030313214.}
\begin{abstract}
This paper concerns with the graphical derivative of the normals to
the conic constraint $g(x)\in\!K$, where $g\!:\mathbb{X}\to\mathbb{Y}$
is a twice continuously differentiable mapping and $K\subseteq\mathbb{Y}$ is a nonempty
closed convex set assumed to be $C^2$-cone reducible. Such a generalized derivative
plays a crucial role in characterizing isolated calmness of the solution maps to
generalized equations whose multivalued parts are modeled via the normals to
the nonconvex set $\Gamma=g^{-1}(K)$. The main contribution of this paper
is to provide an exact characterization for the graphical derivative of
the normals to this class of nonconvex conic constraints under an assumption
without requiring the nondegeneracy of the reference point as the papers
\cite{Gfrerer17,Mordu15,Mordu151} do.
\end{abstract}
\noindent
{\bf Keywords:} graphical derivative, regular and limiting normal map, isolated calmness
\noindent
{\bf Mathematics Subject Classification(2010):} 49K40, 90C31, 49J53
\section{Introduction}\label{sec1}
Generalized derivatives introduced in modern variational analysis represent
an efficient tool to study stability analysis of multifunctions, especially
the so-called solution maps associated with parameter-dependent variational
inequalities or generalized equations; see Rockafellar and Wets \cite{RW98},
Klatte and Kummer \cite{KK02}, Mordukhovich \cite{Mordu06}, and Dontchev
and Rockafellar \cite{DR09}. The stability properties of the solution maps
to generalized equations, whose multivalued parts are modelled via regular
normals to the polyhedral conic constraints, have been analyzed in the seventies,
above all in the papers by Robinson \cite{Robinson79,Robinson81,Robinson82},
and an overview of available results in this setting can be found in
Klatte and Kummer \cite{KK02} and Dontchev and Rockafellar \cite[Chapter 2E]{DR09}.
In the recent decade, some active research is given to the stability properties
of the solution maps to those generalized equations associated with nonpolyhedral conic
constraints \cite{BS00}, such as positive semidefinite conic constraints \cite{Sun06,ZhangZ16},
Lorentz conic constraints \cite{Outrata11,BR05,HMordu17}, and more general constraints
associated with cone reducible closed convex sets \cite{DingSZ17,KK13,Mordu151}.
Let $\mathbb{X}, \mathbb{Y}$ and $\mathbb{P}$ be finite dimensional
vector spaces endowed with the inner product $\langle \cdot,\cdot\rangle$
and its induced norm $\|\cdot\|$. Let $g\!:\mathbb{X}\to\mathbb{Y}$ be
a twice continuously differentiable mapping, and let $K\subseteq\!\mathbb{Y}$
be a nonempty closed convex set which is assumed to be $C^2$-cone reducible.
The class of $C^2$-cone reducible sets is rich, including all the polyhedral
convex sets and many non-polyhedral sets such as the second-order cone
\cite[Lemma 15]{BR05}, the positive semidefinite cone \cite[Example 3.140]{BS00},
and the epigraph cone of the Ky Fan matrix $k$-norm \cite{Ding12}. Moreover,
the Cartesian product of $C^2$-cone reducible sets is also $C^2$-cone reducible
\cite{Shapiro03}.
This paper focuses on the computation of the graphical derivative of the normal
cone mappings to the conic constraint $g(x)\!\in K$ or equivalently the set
\begin{equation}\label{Gamma}
\Gamma:=g^{-1}(K),
\end{equation}
which is also the set of the zeros to the following multifunction associated to $g(x)\in K$:
\begin{equation}\label{MGmap}
\mathcal{G}(x):=g(x)-K\quad{\rm for}\ x\in\mathbb{X}.
\end{equation}
Since our assumptions throughout this paper ensure that the regular
and limiting normal cones to $\Gamma$ agree, we use the generic normal
cone symbol $\mathcal{N}$ below; see Section \ref{sec2} for details.
The present study, being certainly of its own interest, is motivated by
the subsequent application to the characterization of the isolated calmness property
for parameterized equilibria represented as
the solution map to the following generalized equation (GE)
\begin{equation}\label{GE}
0\in F(p,x)+\mathcal{N}_\Gamma(x),
\end{equation}
where $F\!:\mathbb{P}\times\mathbb{X}\to\mathbb{X}$ is a locally Lipschitz
and directionally differentiable mapping, and $\mathcal{N}_\Gamma$
is the regular normal cone mapping to the set $\Gamma$.
The solution map of \eqref{GE} is given by
\begin{equation}\label{MSmap}
\mathcal{S}(p):=\big\{x\in\mathbb{X}\ |\ 0\in F(p,x)+\mathcal{N}_\Gamma(x)\big\}.
\end{equation}
To achieve this goal, motivated by the crucial result due to King and Rockafellar \cite{KR92}
or Levy \cite{Levy96}, we need to compute the graphical derivative of $\mathcal{S}$
in terms of the initial problem data of \eqref{GE} and the corresponding values
at the reference solution point. This amounts to developing the expression of
the graphical derivative of the normal cone mapping $\mathcal{N}_\Gamma$.
In addition, the expression of the graphical derivative of $\mathcal{N}_\Gamma$
is also helpful to the characterization of the regular and limiting normals to
$\mathcal{N}_\Gamma$.
When the set $\Gamma$ is convex and the mapping $F$ is continuously differentiable,
Mordukhovich et al. \cite{Mordu151} provided a formula for calculating
the graphical derivative of $\mathcal{S}$.
Recognizing that the convexity assumption on $\Gamma$ is very restrictive,
they later derived a second-order formula in \cite{Mordu15} for calculating
the graphical derivative of the regular normal $\widehat{\mathcal{N}}_\Gamma$
and then that of the solution map $\mathcal{S}$ in terms of Lagrange multipliers
of the perturbed KKT system and the critical cone of $K$, under the projection
derivation condition (PDC) on $K$ at a nondegenerate reference point.
Although the PDC relaxes the polyhedrality assumption imposed on the set $K$
by \cite{Henrion13}, it actually requires that $K$ has similar properties as
a polyhedral set does; for example, the PDC holds under the second-order extended
polyhedricity condition from \cite{BS00}. When $K$ is non-polyhedral convex cone,
although the PDC always holds at the vertex, the popular positive semidefinite cone
and Lorentz cone generally do not satisfy this condition at nonzero vertexes
(see \cite[Corollary 3.5]{HMordu17}). In addition, Gfrerer and Outrata \cite{Gfrerer17}
also derived a formula for calculating the graphical derivative of the regular normal
$\widehat{\mathcal{N}}_\Gamma$ by imposing the nondegeneracy of
the reference point and a weakened version of the reducibility.
The nondegeneracy of the reference point is strong, and the papers mentioned above
all require this assumption.
Recently, for the case where $K$ is the Lorentz cone,
Hang, Mordukhovich and Sarabi \cite{HMordu17} fully exploited
the structure of the Lorentz cone and precisely calculated the graphical
derivative of the normal cone mapping to $\widehat{\mathcal{N}}_\Gamma$
under an assumption even weaker than the one used in \cite{Gfrerer16-MOR}
to compute the graphical derivative of $\widehat{\mathcal{N}}_\Gamma$
with $K=\mathbb{R}_{-}^m$; and for optimization problems with the conic
constraint $g(x)\in K$, Ding, Sun and Zhang \cite{DingSZ17} verified that
the KKT solution mapping is robustly isolated calm iff both the strict Robinson constraint
qualification (SRCQ) and the second order sufficient condition hold. Their results,
to a certain extent, imply that it is possible to achieve the exact characterization
for the graphical derivative of $\mathcal{N}_\Gamma$ without
requiring the nondegeneracy.
Recall that the SRCQ for the system $g(x)\!\in K$ is said to hold at
$\overline{x}$ with respect to (w.r.t.) some multiplier
$\overline{\lambda}\in\mathcal{N}_K(g(\overline{x}))$ if
\begin{equation}\label{SRCQ}
g'(\overline{x})\mathbb{X}+\mathcal{T}_{K}(g(\overline{x}))\cap[\![\overline{\lambda}]\!]^{\perp}
=\mathbb{Y},
\end{equation}
which is weaker than the nondegeneracy of $\overline{x}$ w.r.t. the mapping $g$
and the set $K$:
\begin{equation}\label{Nondegeneracy}
g'(\overline{x})\mathbb{X}+{\rm lin}\big[\mathcal{T}_{K}(g(\overline{x}))\big]
=\mathbb{Y}.
\end{equation}
In this work we shall provide an exact characterization for the graphical
derivative of $\mathcal{N}_{\Gamma}$ under the metric subregularity
of $\mathcal{G}$ and a multifunction $\Phi$ (see \eqref{Phimap} for its definition)
and the SRCQ for the system $g(x)\in K$. Among others, the metric
subregularity of $\Phi$ is only used for deriving the lower estimation
for the graphical derivative of $\mathcal{N}_{\Gamma}$, while the SRCQ
for the system $g(x)\in K$ is used for achieving the upper estimation.
Since our upper estimation only requires the SRCQ for the system $g(x)\in K$,
one can achieve the isolated calmness of $\mathcal{S}$ without the nondegeneracy.
During the reviewing of this paper, we learned that Gfrerer and Mordukhovich
\cite{Gfrerer171} skillfully derived the lower estimation for the graphical
derivative of $\mathcal{N}_{\Gamma}$ only under the metric subregularity of $\mathcal{G}$,
which is a trivial assumption. Although their exact characterization for
the graphical derivative of $\mathcal{N}_{\Gamma}$ does not require
the uniqueness of the multipliers, one needs to solve
a linear conic optimization problem to achieve the required multiplier.
Moreover, their formula involves the normal cone of the critical cone
of $\Gamma$, which has a workable expression only under the closedness of
the radial cone to $\mathcal{N}_{\Gamma}$ (see Proposition \ref{critical-normal-prop1}
and \ref{conditions-closedness}). In other words, under the uniqueness of
the multipliers and the closedness of the radial cone to $\mathcal{N}_{\Gamma}$,
their formula for the graphical derivative of $\mathcal{N}_{\Gamma}$ agrees
with ours. As direct applications of this result, we establish a lower estimation
for the regular coderivative of $\mathcal{N}_\Gamma$ under the SRCQ,
and an upper estimation for the coderivative of $\widehat{\mathcal{N}}_\Gamma$
under the metric subregularity of $\Phi$, which partly improves the results of
\cite[Theorem 7]{Outrata11} and \cite[Theorem 4.1]{Mordu15}.
Our notation is basically standard. A hollow capital, say $\mathbb{Z}$,
denotes a finite dimensional vector space endowed with the inner product
$\langle \cdot,\cdot\rangle$ and its induced norm $\|\cdot\|$, and
$\mathbb{B}_{\mathbb{Z}}$ means the closed unit ball centered at
the origin in $\mathbb{Z}$. For a given $z\in\mathbb{Z}$, $\mathbb{B}(z,\delta)$
means the closed ball of radius $\delta$ centered at $z$ in $\mathbb{Z}$.
For a given closed convex set $\Omega$, $\Pi_{\Omega}$ denotes the projection
operator onto $\Omega$; and for a given nonempty convex cone $\mathcal{K}$,
$\mathcal{K}^{\circ}$ means the negative polar of $\mathcal{K}$.
For a linear operator $\mathcal{A}$, $\mathcal{A}^*$ denotes the adjoint of $\mathcal{A}$.
For a given vector $z$, the notation $[\![z]\!]$ denotes the subspace generated by $z$.
\section{Preliminaries}\label{sec2}
This section provides some background knowledge and some necessary results.
Let $\Omega\subseteq\mathbb{Z}$ be a nonempty set. For a fixed $\overline{z}\in\Omega$,
from \cite{BS00} the radial cone to $\Omega$ at $\overline{z}$ is defined by
\[
\mathcal{R}_{\Omega}(\overline{z}):=\big\{h\in \mathbb{Z}\ |\ \exists\,t^*>0\ {\rm such\ that\ for\ all}\
t\in [0,t^*],\,\overline{z}+th\in \Omega\big\},
\]
while from \cite{RW98} the contingent cone to $\Omega$ at $\overline{z}$ is defined by
\[
\mathcal{T}_{\Omega}(\overline{z}):=\big\{w\in\mathbb{Z}\ |\ \exists t_k\downarrow 0,\,
w^k\to w\ {\rm with}\ \overline{x}+t_kw_k\in\Omega\big\}.
\]
Notice that $\mathcal{R}_{\Omega}(\overline{z})\subseteq \mathcal{T}_{\Omega}(\overline{z})$,
and when $\Omega$ is convex, $\mathcal{T}_{\Omega}(\overline{z})={\rm cl}(\mathcal{R}_{\Omega}(\overline{z}))$.
For a fixed $\overline{z}\in\Omega$, by \cite{RW98} the regular normal cone
to $\Omega$ at $\overline{z}$ is defined by
\[
\widehat{\mathcal{N}}_{\Omega}(\overline{z})
:=\Big\{v\in\mathbb{Z}\ |\ \limsup_{z\xrightarrow[\Omega]{}\overline{z}}
\frac{\langle v,z-\overline{z}\rangle}{\|z-\overline{z}\|}\le 0\Big\},
\]
and the basic/limiting normal cone to $\Omega$ at $\overline{z}$ admits the following representation
\[
\mathcal{N}_\Omega(\overline{z})=\limsup_{z\xrightarrow[\Omega]{}\overline{z}}\widehat{\mathcal{N}}_{\Omega}(z),
\]
which, if $\Omega$ is locally closed at $\overline{z}\in\Omega$, is equivalent to
the original definition by Mordukhovich \cite{Mordu76}, i.e.,
\(
\mathcal{N}_\Omega(\overline{z})\!:=\limsup_{z\to\overline{z}}\big[{\rm cone}(z-\Pi_{\Omega}(z))\big].
\)
Notice that $\widehat{\mathcal{N}}_{\Omega}(\overline{z})=(\mathcal{T}_{\Omega}(\overline{z}))^\circ$,
and when $\Omega$ is convex, $\widehat{\mathcal{N}}_{\Omega}(\overline{z})=\mathcal{N}_{\Omega}(\overline{z})$.
Given a direction $h\in\mathbb{Z}$, the directional limiting normal cone to $\Omega$
at $\overline{x}$ in $h$ is defined by
\(
\mathcal{N}_{\Omega}(\overline{x};h)
:=\limsup_{t\searrow 0,h'\to h}\widehat{\mathcal{N}}_{\Omega}(\overline{x}+th').
\)
Various properties of the directional limiting normal cone can be found
in \cite{Gfrerer13,Gfrerer16}.
\subsection{Lipschitz-type properties of a multifunction}\label{subsec2.1}
Let $\mathcal{F}\!:\mathbb{Z}\rightrightarrows\mathbb{W}$ be a given multifunction.
Consider an arbitrary $(\overline{z},\overline{w})\in{\rm gph}\mathcal{F}$
such that $\mathcal{F}$ is locally closed at $(\overline{z},\overline{w})$.
We recall from \cite{RW98,DR09} several Lipschitz-type properties of
the multifunction $\mathcal{F}$,
including the Aubin property, the calmness and the isolated calmness.
\begin{definition}\label{Aubin-def}
The multifunction $\mathcal{F}$ is said to have the Aubin property at
$\overline{z}$ for $\overline{w}$ if there exists $\kappa\ge 0$ along with
$\varepsilon>0$ and $\delta>0$ such that
for all $z,z'\in\mathbb{B}(\overline{z},\varepsilon)$,
\[
\mathcal{F}(z)\cap\mathbb{B}(\overline{w},\delta)\subseteq\mathcal{F}(z')
+\kappa\|z-z'\|\mathbb{B}_{\mathbb{W}}.
\]
\end{definition}
\begin{definition}\label{calm-def}
The multifunction $\mathcal{F}$ is said to be calm at $\overline{z}$ for $\overline{w}$
if there exists $\kappa\ge0$ along with $\varepsilon>0$ and $\delta>0$ such that for all
$z\in\mathbb{B}(\overline{z},\varepsilon)$,
\begin{equation}\label{calm-inclusion1}
\mathcal{F}(z)\cap\mathbb{B}(\overline{w},\delta)
\subseteq\mathcal{F}(\overline{z})+\kappa\|z-\overline{z}\|\mathbb{B}_{\mathbb{W}};
\end{equation}
if in addition $\mathcal{F}(\overline{z})\cap \mathbb{B}(\overline{w},\delta) =\{\overline{w}\}$,
$\mathcal{F}$ is said to be isolated calm at $\overline{z}$ for $\overline{w}$.
\end{definition}
The coderivative and graphical derivative of $\mathcal{F}$ are the convenient tools
to study the Aubin property and isolated calmness of $\mathcal{F}$, respectively.
Recall from \cite{Mordu80,Aubin81} that the coderivative of $\mathcal{F}$ at $\overline{z}$
for $\overline{w}\in\mathcal{F}(\overline{z})$ is the mapping
$D^*\mathcal{F}(\overline{z}|\overline{w})\!:\mathbb{W}\rightrightarrows\mathbb{Z}$
defined by
\[
\Delta z\in D^*\mathcal{F}(\overline{z}|\overline{w})(\Delta w)\Longleftrightarrow
(\Delta z,-\Delta w)\in\mathcal{N}_{{\rm gph}\mathcal{F}}(\overline{z},\overline{w}),
\]
and the graphical derivative of $\mathcal{F}$ at $(\overline{z},\overline{w})$
is the mapping $D\mathcal{F}(\overline{z}|\overline{w})\!:\mathbb{Z}\rightrightarrows\mathbb{W}$
defined by
\[
\Delta w\in D\mathcal{F}(\overline{z}|\overline{w})(\Delta z)\Longleftrightarrow
(\Delta z,\Delta w)\in\mathcal{T}_{{\rm gph}\mathcal{F}}(\overline{z},\overline{w}).
\]
With the coderivative and graphical derivative of $\mathcal{F}$,
we have the following conclusions.
\begin{lemma}\label{chara-Aubin}(see \cite[Theorem 5.7]{Mordu93}
or \cite[Theorem 9.40]{RW98})\ The multifunction $\mathcal{F}$ has
the Aubin property at $\overline{z}$ for $\overline{w}$
if and only if $D^*\mathcal{F}(\overline{z}|\overline{w})(0)=\{0\}$.
\end{lemma}
\begin{lemma}\label{chara-icalm}(see \cite[Proposition 2.1]{KR92}
or \cite[Proposition 4.1]{Levy96})\ The multifunction $\mathcal{F}$
is isolated calm at $\overline{z}$ for $\overline{w}$
if and only if $D\mathcal{F}(\overline{z}|\overline{w})(0)=\{0\}$.
\end{lemma}
Next we recall from \cite{RW98,DR09} metric regularity and metric subregularity,
respectively.
\begin{definition}\label{regular-def}
The multifunction $\mathcal{F}$ is said to be metrically regular at $\overline{z}$
for $\overline{w}$ if there exists $\kappa\ge 0$ along with $\varepsilon>0$ and $\delta>0$ such that
for all $z\in\mathbb{B}(\overline{z},\varepsilon)$ and $w\in\mathbb{B}(\overline{w},\delta)$,
\[
{\rm dist}\big(z,\mathcal{F}^{-1}(w)\big)
\le \kappa\,{\rm dist}\big(w,\mathcal{F}(z)\big).
\]
\end{definition}
\begin{definition}\label{subregular-def}
The multifunction $\mathcal{F}$ is said to be metrically subregular at $\overline{z}$
for $\overline{w}$ if there exists $\kappa\ge0$ along with $\varepsilon>0$ and $\delta>0$
such that for all $z\in\mathbb{B}(\overline{z},\varepsilon)$,
\[
{\rm dist}\big(z,\mathcal{F}^{-1}(\overline{w})\big)
\le \kappa\,{\rm dist}\big(\overline{w},\mathcal{F}(z)\cap\mathbb{B}(\overline{w},\delta)\big).
\]
\end{definition}
\begin{remark}\label{remark21}
It is known that $\mathcal{F}$ has the Aubin property at $\overline{z}$ for $\overline{w}$
iff $\mathcal{F}^{-1}$ is metrically regular at $\overline{w}$ for $\overline{z}$
(see \cite{RW98,DR09}); and $\mathcal{F}$ is calm at $\overline{z}$ for $\overline{w}$
iff $\mathcal{F}^{-1}$ is metrically subregular at $\overline{w}$ for $\overline{z}$
(see \cite[Theorem 3H.3]{DR09}). By \cite[Exercise 3H.4]{DR09}, the restriction on
$z\in\mathbb{B}(\overline{z},\varepsilon)$ in Definition \ref{calm-def} and
the neighborhood $\mathbb{B}(\overline{w},\delta)$
in Definition \ref{subregular-def} can be removed.
\end{remark}
The following lemma states a link between the graphical derivative
of $\mathcal{F}$ and the contingent cone to the value of $\mathcal{F}$
at some point, where the first part is easily proved by the definition,
and the second part follows from \cite[Corollary 4.2]{Gfrerer16} and Remark \ref{remark21}.
\begin{lemma}\label{TF-relation}
For the multifunction $\mathcal{F}$ and the point $(\overline{z},\overline{w})$,
$\mathcal{T}_{\mathcal{F}(\overline{z})}(\overline{w})\subseteq
D\mathcal{F}(\overline{z}|\overline{w})(0)$. The converse inclusion also holds
provided that $\mathcal{F}$ is calm at $\overline{z}$ for $\overline{w}$.
\end{lemma}
\subsection{Normal cone mapping to $C^{2}$-cone reducible set}\label{subsec2.2}
We shall establish the calmness of the normal cone map to
a $C^{2}$-cone reducible closed convex set, which is a nonpolyhedral
counterpart of the seminal upper-Lipschitzian result by Robinson \cite{Robinson81}
for convex polyhedral sets. First, we recall the $C^{\ell}$-cone reducibility.
\begin{definition}\label{cone-reduce}(\cite[Definition 3.135]{BS00})
A closed convex set $\Omega$ in $\mathbb{Y}$ is said to be $C^{\ell}$-cone
reducible at $\overline{y}\in\Omega$, if there exist an open neighborhood
$\mathcal{Y}$ of $\overline{y}$, a pointed closed convex cone
$\mathcal{D}\subseteq\mathbb{Z}$ and an $\ell$-times continuously
differentiable mapping $\Xi\!: \mathcal{Y}\to\mathbb{Z}$ such that (i) $\Xi(\overline{y})=0$;
(ii) $\Xi'(\overline{y})\!:\mathbb{Y}\to\mathbb{Z}$ is onto;
(iii) $\Omega\cap\mathcal{Y}=\{y\in\mathcal{Y}\ |\ \Xi(y)\in\mathcal{D}\}$.
We say that the closed convex set $\Omega$ is $C^\ell$-cone reducible
if $\Omega$ is $C^\ell$-cone reducible at every $y\in\Omega$.
\end{definition}
\begin{theorem}\label{NK-calm}
Let $\Omega\subseteq\mathbb{Y}$ be a closed convex set. Suppose that $\Omega$ is
$C^{2}$-cone reducible at $\overline{y}\in \Omega$. Then, the normal cone mapping
$\mathcal{N}_\Omega$ is calm at $\overline{y}$ for each $\overline{z}\in\mathcal{N}_\Omega(\overline{y})$.
\end{theorem}
\begin{proof}
Since $\Omega$ is $C^{2}$-cone reducible at $\overline{y}\in \Omega$,
there exist an open neighborhood $\mathcal{Y}$ of $\overline{y}$,
a pointed closed convex cone $\mathcal{D}\subseteq\mathbb{Z}$,
and a twice continuously differentiable $\Xi\!:\mathcal{Y}\to\mathbb{Z}$
satisfying (i)-(iii) in Definition \ref{cone-reduce}.
Since $\Xi'(\overline{y})\!:\mathbb{Y}\to\mathbb{Z}$ is onto,
there exists $\varepsilon>0$ such that for each
$y\in\mathbb{B}(\overline{y},\varepsilon)\subset \mathcal{Y}$,
the mapping $\Xi'(y)\!:\mathbb{Y}\to\mathbb{Z}$ is onto.
By \cite[Exercise 6.7]{RW98},
\begin{equation}\label{ncone-Omega}
\mathcal{N}_{\Omega}(y)=\nabla\Xi(y)\mathcal{N}_{\mathcal{D}}(\Xi(y))
\quad\ \forall y\in\mathbb{B}(\overline{y},\varepsilon).
\end{equation}
Define $\mathcal{E}(y):=(\Xi'(y)\nabla\Xi(y))^{-1}\Xi'(y)$ for $y\in\mathcal{Y}$.
Notice that the functions $\mathcal{E}(\cdot)$ and $\nabla\Xi(\cdot)$
are continuously differentiable in $\mathcal{Y}$. There exist $\varepsilon'>0$,
$L_{\mathcal{E}}>0$ and $L>0$ such that
\begin{equation}\label{Lip-bound}
\|\mathcal{E}(y)-\!\mathcal{E}(y')\|\le L_{\mathcal{E}}\|y-y'\|
\ \ {\rm and}\ \
\|\nabla\Xi(y)\!-\!\nabla\Xi(y')\|\le L\|y-y'\|\quad\forall y\in\mathbb{B}(\overline{y},\varepsilon').
\end{equation}
Now fix an arbitrary $\overline{z}\in\mathcal{N}_\Omega(\overline{y})$.
In order to establish the calmness of $\mathcal{N}_\Omega$ at $\overline{y}$
for $\overline{z}$, it suffices to argue that there exist $\overline{\varepsilon}>0$,
$\overline{\delta}>0$ and $\overline{\kappa}>0$ such that
for all $y\in\mathbb{B}(\overline{y},\overline{\varepsilon})$,
\begin{equation}\label{aim-ineq1}
\mathcal{N}_\Omega(y)\cap\mathbb{B}(\overline{z},\overline{\delta})
\subseteq \mathcal{N}_\Omega(\overline{y})+\overline{\kappa}\|y-\overline{y}\|\mathbb{B}_{\mathbb{Y}}.
\end{equation}
Fix an arbitrary $\overline{\delta}\in(0,1)$ and set
$\overline{\varepsilon}:=\frac{1}{2}\min\big\{\varepsilon,\varepsilon'\big\}$.
Fix an arbitrary point $y\in\mathbb{B}(\overline{y},\overline{\varepsilon})$.
If $\mathcal{N}_\Omega(y)\cap\mathbb{B}(\overline{z},\overline{\delta})=\emptyset$,
the inclusion \eqref{aim-ineq1} automatically holds. So, we only need to consider
the case where $\mathcal{N}_\Omega(y)\cap\mathbb{B}(\overline{z},\overline{\delta})\ne\emptyset$.
Take an arbitrary $z\in \mathcal{N}_\Omega(y)\cap \mathbb{B}(\overline{z},\overline{\delta})$.
From \eqref{ncone-Omega}, there exists $\xi\in \mathcal{N}_{\mathcal{D}}(\Xi(y))$
such that $z=\nabla\Xi(y)\xi$. Since $\overline{z}\in\mathcal{N}_\Omega(\overline{y})$,
there also exists $\overline{\xi}\in\mathcal{N}_{\mathcal{D}}(\Xi(\overline{y}))$
such that $\overline{z}=\nabla\Xi(\overline{y})\overline{\xi}$.
Clearly, $\xi=\mathcal{E}(y)z$ and $\overline{\xi}=\mathcal{E}(\overline{y})\overline{z}$. Then,
\begin{align*}
\|\xi-\overline{\xi}\|
&=\|\mathcal{E}(y)z-\mathcal{E}(\overline{y})\overline{z}\|
\le \|\mathcal{E}(y)z-\mathcal{E}(\overline{y})z\|+\|\mathcal{E}(\overline{y})z-\mathcal{E}(\overline{y})\overline{z}\|\nonumber\\
&\le L_{\mathcal{E}}\|z\|\|y-\overline{y}\|+\|\mathcal{E}(\overline{y})\|\|z-\overline{z}\|
\le L_{\mathcal{E}}(\|\overline{z}\|+\overline{\varepsilon})+\|\mathcal{E}(\overline{y})\|\overline{\delta}
:=\widetilde{\delta}
\end{align*}
Since $\mathcal{D}\subseteq\mathbb{Z}$ is a pointed closed convex cone,
we have $\mathcal{N}_{\mathcal{D}}(\Xi(y))\subseteq\mathcal{D}^{\circ}$
and then $\xi\in\mathcal{D}^{\circ}$, which implies that $\nabla\Xi(\overline{y})\xi\in\nabla\Xi(\overline{y})\mathcal{N}_{\mathcal{D}}(\Xi(\overline{y}))
=\mathcal{N}_\Omega(\overline{y})$. Thus,
\begin{align*}
{\rm dist}(z,\mathcal{N}_\Omega(\overline{y}))
&={\rm dist}(\nabla\Xi(y)\xi,\mathcal{N}_\Omega(\overline{y}))
\le\|\nabla\Xi(y)\xi-\nabla\Xi(\overline{y})\xi\|\\
&\le\|\xi\|L\|y-\overline{y}\|
\le L(\widetilde{\delta}+\|\overline{\xi}\|)\|y-\overline{y}\|.
\end{align*}
This shows that the inclusion \eqref{aim-ineq1} holds with
$\overline{\kappa}=L(\widetilde{\delta}+\|\overline{\xi}\|)$.
\end{proof}
\begin{remark}\label{remark-NK}
{\bf(a)} If $\Omega$ is a $C^{2}$-cone reducible closed convex cone with
$\Omega^{\circ}=-\Omega$;
for example, the positive semidefinite cone and Lorentz cone, then $\Omega^{\circ}$
is a $C^{2}$-cone reducible closed convex cone. By Theorem \ref{NK-calm},
the mapping $\mathcal{N}_{\Omega^{\circ}}$ is calm at each point of its graph. Along with
$\mathcal{N}_{\Omega^{\circ}}=\mathcal{N}_{\Omega}^{-1}$, $\mathcal{N}_{\Omega}$
is metrically subregular at each point of its graph. Thus, for
a $C^{2}$-cone reducible closed convex cone $\Omega$ with $\Omega^{\circ}=-\Omega$,
$\mathcal{N}_{\Omega}$ is both metrically subregular and calm at each point of its graph.
This recovers the result of \cite[Proposition 3.3]{CuiST16}.
\noindent
{\bf(b)} When $\Omega$ is a closed nonconvex set, if there exists a closed cone
$\mathcal{D}\subseteq\mathbb{Z}$ together with a twice continuously differentiable
mapping $\Xi\!: \mathcal{Y}\to\mathbb{Z}$ such that (i) $\Xi(\overline{y})=0$;
(ii) $\Xi'(\overline{y})\!:\mathbb{Y}\to\mathbb{Z}$ is onto;
(iii) $\Omega\cap\mathcal{Y}=\{y\in\mathcal{Y}\ |\ \Xi(y)\in\mathcal{D}\}$,
then from the proof of Theorem \ref{NK-calm} it follows that
the regular normal cone mapping $\widehat{\mathcal{N}}_\Omega$ is calm at
$\overline{y}$ for each $\overline{z}\in\widehat{\mathcal{N}}_\Omega(\overline{y})$..
\end{remark}
Now let $K$ be a closed convex set which is assumed to be $C^{2}$-cone reducible.
By Theorem \ref{NK-calm}, its normal cone mapping $\mathcal{N}_K$ is calm at
each $y\in K$ for $\lambda\in\mathcal{N}_K(y)$. From \cite[Proposition 3.136]{BS00},
the set $K$ is also second-order regular at each $y\in K$,
and hence $\mathcal{T}_{K}^{i,2}(y,h)=\mathcal{T}_{K}^{2}(y,h)$
for any $h\in\mathbb{Y}$, where $\mathcal{T}_{K}^{i,2}(y,h)$
and $\mathcal{T}_{K}^{2}(y,h)$ denote the inner and outer second order tangent sets
to $K$ at $y$ in the direction $h$, respectively, defined by
\begin{align*}
\mathcal{T}_{K}^{i,2}(y,h):=\Big\{w\in \mathbb{Y}\;|\; {\rm dist}(y+th+\frac{1}{2}t^2w,K)=o(t^2), t\geq 0\Big\},\qquad\\
\mathcal{T}_{K}^{2}(y,h):=\Big\{w\in \mathbb{Y}\;|\; \exists\;t_n\downarrow 0 {\;\rm such\;that\;} {\rm dist}(y+t_nh+\frac{1}{2}t_n^2 w,K)=o(t_n^2)\Big\}.
\end{align*}
From the standard reduction approach in \cite[Section 3.4.4]{BS00},
we have the following result on the representation of the normal cone
$\mathcal{N}_K$ and the ``sigma term'' of $K$.
\begin{lemma}\label{lemma-reduction}
Let $\overline{y}\in K$ be given. Then there exist an open neighborhood $\mathcal{Y}$
of $\overline{y}$, a pointed closed convex cone $D\subseteq\mathbb{Z}$
and a twice continuously differentiable mapping $\Xi\!:\mathcal{Y}\to\mathbb{Z}$
satisfying conditions (i)-(iii) in Definition \ref{cone-reduce} such that
for any $y\in\mathcal{Y}$,
\begin{equation}\label{normal-cone}
\mathcal{N}_{K}(y)=\nabla\Xi(y)\mathcal{N}_{D}(\Xi(y));
\end{equation}
and for any $\lambda\in\mathcal{N}_{K}(y)$ there exists a unique
$u\in\!\mathcal{N}_{D}(\Xi(y))$ such that $\lambda=\nabla\Xi(y)u$ and
\begin{equation}\label{Upsilon}
\Upsilon(h):=-\sigma\big(\lambda,\mathcal{T}_{K}^2(y,h)\big)
=\langle u,\Xi''(y)(h,h)\rangle
\quad \forall h\in\mathcal{C}_{K}(y,\lambda)
\end{equation}
where $\sigma(\cdot,\mathcal{T}_{K}^2(y,h))$ is the support function
of $\mathcal{T}_{K}^2(y,h)$, and for any $y\in K$,
$\mathcal{C}_{K}(y,\lambda)$ is the critical cone of $K$ at $y$
with respect to $\lambda\in\!\mathcal{N}_{K}(y)$, defined as
\(
\mathcal{C}_{K}(y,\lambda):=\mathcal{T}_{K}(y)\cap [\![\lambda]\!]^{\perp}.
\)
\end{lemma}
Next we recall a useful result on the directional derivative of the projection
operator $\Pi_{K}$. Fix an arbitrary $y\in\mathbb{Y}$. Write $\overline{y}:=\Pi_K(y)$
and take $\overline{\lambda}\in\mathcal{N}_{K}(\overline{y})$.
Since $K$ is second-order regular at $\overline{y}$, by \cite[Theorem 7.2]{BCS98}
the mapping $\Pi_{K}$ is directionally differentiable at $y$ and the directional derivative
$\Pi_{K}'(y;h)$ for any direction $h\in\mathbb{Y}$ satisfies
\[
\Pi_{K}'(y;h)=\mathop{\arg\min}_{d\in\mathcal{C}_{K}(\overline{y},\overline{\lambda})}
\Big\{\|d-h\|^2-\sigma\big(\overline{\lambda},\mathcal{T}_{K}^2(\overline{y},d)\big)\Big\}.
\]
In addition, by following the arguments as those for \cite[Theorem 3.1]{WZZhang14},
one can obtain
\[
\mathcal{T}_{{\rm gph}\mathcal{N}_K}(\overline{y},\overline{\lambda})=
\big\{(\Delta z,\Delta w)\in\mathbb{Y}\times\mathbb{Y}\ |\
\Pi_K'(\overline{y}+\overline{\lambda};\Delta z+\Delta w)=\Delta z\big\}.
\]
Combining this with \cite[Lemma 10]{DingSZ17}, we have the following
conclusion for the graphical derivative of $\mathcal{N}_K$,
the directional derivative of $\Pi_{K}$ and the critical cone of the set $K$.
\begin{lemma}\label{dir-proj}
Consider an arbitrary point pair $(\overline{y},\overline{\lambda})\in{\rm gph}\mathcal{N}_K$,
and write $y:=\overline{y}+\overline{\lambda}$. Then,
with $\Upsilon(\cdot)=-\sigma\big(\overline{\lambda},\mathcal{T}_{K}^2(\overline{y},\cdot)\big)
=\langle u,\Xi''(\overline{y})(\cdot,\cdot)\rangle$ for $u\in\mathcal{N}_D(\Xi(\overline{y}))$,
it holds that
\begin{align}\label{system-regular}
\Delta\lambda\in D\mathcal{N}_K(\overline{y}|\overline{\lambda})(\Delta y)
&\Longleftrightarrow \Delta y-\Pi_{K}'(y;\Delta y\!+\!\Delta\lambda)=0\nonumber\\
&\Longleftrightarrow
\left\{\begin{array}{ll}
\Delta y\in\mathcal{C}_{K}(\overline{y},\overline{\lambda}),\\
\Delta\lambda-\frac{1}{2}\nabla\Upsilon(\Delta y)\in\big[\mathcal{C}_{K}(\overline{y},\overline{\lambda})\big]^{\circ},\\
\langle\Delta y,\Delta\lambda\rangle
=-\sigma(\overline{\lambda},\mathcal{T}_{K}^2(\overline{y},\Delta y)).
\end{array}\right.
\end{align}
\end{lemma}
\subsection{Contingent and normal cones to a composite set}\label{subsec2.3}
Consider a set $\Theta\!:=H^{-1}(\Delta)$ where $H\!:\mathbb{Z}\to\mathbb{W}$
is a mapping and $\Delta\subseteq\mathbb{W}$ is a closed set.
The following characterization holds for the contingent cone to the set $\Theta$.
\begin{lemma}\label{Tcone-lemma}
Suppose $H$ is Lipschitz near $\overline{z}$ and directionally differentiable
at $\overline{z}$. Then,
\begin{equation}\label{Tcone-formula}
\mathcal{T}_{\Theta}\big(\overline{z}\big)
\subseteq\big\{h\in\mathbb{Z}\ |\ H'(\overline{z};h)\in\mathcal{T}_{\Delta}(H(\overline{z}))\big\}.
\end{equation}
If $\mathcal{H}(z):=H(z)-\Delta$ is metrically subregular at $\overline{z}$ for $0$,
the converse inclusion also holds.
\end{lemma}
The first part of Lemma \ref{Tcone-lemma} follows by the definition
of the contingent cone and the Hadamard directional differentiability
of $H$, and the second part is by \cite[Proposition 1]{Henrion05}.
Combining Lemma \ref{Tcone-lemma} with \cite[Page 211-212]{Ioffe08},
we can obtain the following result.
\begin{lemma}\label{normal-cone-lemma}
Consider an arbitrary $\overline{z}\in\Theta$. Let $\mathcal{H}$ be
the multifunction defined in Lemma \ref{Tcone-lemma}. If $\mathcal{H}$
is metrically subregular at $\overline{z}$ for $0$, then
$\mathcal{N}_{\Theta}(\overline{z})\subseteq D^*H(\overline{z})\big[\mathcal{N}_\Delta(H(\overline{z}))\big]$.
If in addition $H$ is strictly differentiable and
$\widehat{\mathcal{N}}_\Delta(H(\overline{z}))=\mathcal{N}_\Delta(H(\overline{z}))$,
then it holds that
\[
\widehat{\mathcal{N}}_{\Theta}(\overline{z})=\mathcal{N}_{\Theta}(\overline{z})
=\big\{\nabla H(\overline{z})y\ |\ y\in\widehat{\mathcal{N}}_\Delta(H(\overline{z}))\big\}.
\]
\end{lemma}
Recall that $\Gamma=g^{-1}(K)$ where the mapping $g$ and the closed convex set
$K$ satisfy the standard assumption. By Lemma \ref{Tcone-lemma}-\ref{normal-cone-lemma},
under the metric subregularity of $\mathcal{G}$, we have the following
characterization for the contingent cone and normal cone to
the set $\Gamma$.
\begin{corollary}\label{TNcone-Gamma}
Consider an arbitrary $\overline{x}\in\Gamma$. If the multifunction
$\mathcal{G}$ defined by \eqref{MGmap} is metrically subregular at
$\overline{x}$ for the origin,
then it holds that
\begin{align}{}\label{TGamma1}
\mathcal{T}_{\Gamma}(\overline{x})
=\!\big\{h\in\mathbb{X}\ |\ g'(\overline{x})h\in\mathcal{T}_{K}(g(\overline{x}))\big\},\quad\\
\mathcal{N}_{\Gamma}(\overline{x})=\mathcal{\widehat{N}}_{\Gamma}(\overline{x})
=\!\big\{\nabla g(\overline{x})\lambda\ |\ \lambda\in\mathcal{N}_{K}(g(\overline{x}))\big\}.
\label{TGamma1}
\end{align}
\end{corollary}
\begin{remark}\label{robust-remark}
By Definition \ref{subregular-def}, the metric subregularity of $\mathcal{G}$ at
$\overline{x}\in\Gamma$ for $0$ is equivalent to requiring the existence of $\kappa\ge0$
along with $\varepsilon>0$ such that for all $x\in\mathbb{B}(\overline{x},\varepsilon)$,
\[
{\rm dist}(x,\Gamma)\le \kappa{\rm dist}(g(x),K).
\]
As remarked in \cite{HMordu17}, this means that the metric subregularity of $\mathcal{G}$
at $\overline{x}\in\Gamma$ is robust in the sense that if $\mathcal{G}$ is metrically subregular
at $\overline{x}\in\Gamma$, then so is $\mathcal{G}$ at any $x\in\Gamma$ near $\overline{x}$.
\end{remark}
\subsection{Multiplier set map and critical cone to $\Gamma$}\label{subsec2.4}
Consider $\Gamma=g^{-1}(K)$ again. By Corollary \ref{TNcone-Gamma},
under the metric subregularity of $\mathcal{G}$, $\mathcal{N}_{\Gamma}$
takes the form of \eqref{TGamma1}. In view of this, for any given
$x\in\Gamma$ and $v\in\mathcal{N}_{\Gamma}(x)$, we define
\[
\mathcal{M}(x,v):=\big\{\lambda\in\mathcal{N}_K(g(x))\ |\ v=\nabla g(x)\lambda\big\}
\]
which is the multiplier set associated to $(x,v)$, and denote by
$\mathcal{M}_x\!:\mathbb{X}\rightrightarrows\mathbb{Y}$ the localized
version of the multiplier set mapping $\mathcal{M}$, that is,
$\mathcal{M}_x$ has the following form
\begin{equation}\label{MMapx}
\mathcal{M}_{x}(v):=\big\{\lambda\in\mathcal{N}_K(g(x))\ |\ v=\nabla g(x)\lambda\big\}.
\end{equation}
Clearly, $\mathcal{M}_{x}$ is a closed convex multifunction.
For $\mathcal{M}_{x}$, we have the following result.
\begin{proposition}\label{prop-Mx}
Consider an arbitrary point $x\in\Gamma$. For any given $(v,\lambda)\in{\rm gph}\mathcal{M}_{x}$,
\begin{equation}\label{Tcone-Mx}
\mathcal{T}_{{\rm gph}\mathcal{M}_{x}}(v,\lambda)
=\big\{(\xi,\eta)\in\mathbb{X}\times\mathbb{Y}\ |\ \xi=\nabla g(x)\eta,\,\eta\in\mathcal{T}_{\mathcal{N}_{K}(g(x))}(\lambda)\big\},
\end{equation}
and hence $\mathcal{M}_{x}$ is isolated calm at $v$ for $\lambda$ iff
one of the following conditions holds:
\begin{align*}
\!{\rm Ker}(\nabla g(x))\cap \mathcal{T}_{\mathcal{N}_{K}(g(x))}(\lambda)=\{0\}
&\Longleftrightarrow{\rm Ker}(\nabla g(x))\cap D\mathcal{N}_{K}(g(x)|\lambda)(0)=\{0\},\\
&\Longleftrightarrow{\rm SRCQ\ for\ the\ system}\ g(x)\in K\ {\rm at}\ x\ {\rm w.r.t.}\ \lambda.\nonumber
\end{align*}
\end{proposition}
\begin{proof}
Notice that ${\rm gph}\mathcal{M}_{x}=\mathcal{L}\big(\mathcal{N}_{K}(g(x))\big)$
where $\mathcal{L}(u):=\left(\begin{matrix}
\nabla\!g(x)u\\ u
\end{matrix}\right)$ for $u\in\mathbb{Y}$. From the convexity of
$\mathcal{N}_{K}(g(x))$ and the last part of \cite[Theorem 6.43]{RW98}, it follows that
\[
\mathcal{T}_{{\rm gph}\mathcal{M}_{x}}(v,\lambda)
={\rm cl}\big\{(\xi,\eta)\in\mathbb{X}\times\mathbb{Y}\ |\ \xi=\nabla g(x)\eta,\,\eta\in\mathcal{T}_{\mathcal{N}_{K}(g(x))}(\lambda)\big\}.
\]
Since
\(
\big\{(\xi,\eta)\in\mathbb{X}\times\mathbb{Y}\ |\ \xi=\nabla g(x)\eta,\eta\in\mathcal{T}_{\mathcal{N}_{K}(g(x))}(\lambda)\big\}
\)
is closed, the result in \eqref{Tcone-Mx} holds.
By Lemma \ref{chara-icalm}, $\mathcal{M}_{x}$ is isolated calm at $v$ for $\lambda$
iff $(0,\eta)\in\mathcal{T}_{{\rm gph}\mathcal{M}_{x}}(v,\lambda)$ implies $\eta=0$.
Together with \eqref{Tcone-Mx}, this is equivalent to requiring
that ${\rm Ker}(\nabla g(x))\cap\mathcal{T}_{\mathcal{N}_{K}(g(x))}(\lambda)=\{0\}$.
By Theorem \ref{NK-calm} and Lemma \ref{TF-relation}, the first equivalence holds.
Recall that the SRCQ for the system $g(x)\in K$ at $x$ w.r.t. $\lambda$ is requiring that
\(
g'(x)\mathbb{X}+\mathcal{T}_K(g(x))\cap[\![\lambda]\!]^{\perp}=\mathbb{X},
\)
which by \cite[Equations(2.31)\&(2.32)]{BS00} and \cite[Example 2.62]{BS00}
is equivalent to saying that
\[
0={\rm Ker}(\nabla g(x))\cap{\rm cl}(\mathcal{N}_K(g(x))
\!+[\![\lambda]\!])={\rm Ker}(\nabla g(x))
\cap\mathcal{T}_{\mathcal{N}_{K}(g(x))}(\lambda).
\]
Thus, we obtain the second equivalence. The proof is then completed.
\end{proof}
Given $x\in\Gamma$ and $v\in\mathcal{N}_{\Gamma}(x)$, the critical cone to $\Gamma$
at $x$ with respect to $v$ is defined as
\[
\mathcal{C}_{\Gamma}(x,v):=\mathcal{T}_{\Gamma}(x)\cap[\![v]\!]^{\perp}.
\]
By Corollary \ref{TNcone-Gamma}, under the metric subregularity of
$\mathcal{G}$ at $x\in\Gamma$ for $0$, it holds that
\begin{equation}\label{critial-equa1}
\mathcal{C}_{\Gamma}(x,v)=[g'(x)]^{-1}\mathcal{C}_K(g(x),\lambda)
\quad{\rm for\ each}\ \lambda\in\mathcal{M}_{x}(v).
\end{equation}
Next we provide a characterization for the normal cone to the critical cone of $\Gamma$.
\begin{proposition}\label{critical-normal-prop1}
Let $(\overline{x},\overline{v})\in{\rm gph}\mathcal{N}_{\Gamma}$.
If $\mathcal{G}$ is metrically subregular at $\overline{x}$ for $0$, then
\begin{align}\label{critical-cone1}
\mathcal{N}_{\mathcal{C}_{\Gamma}(\overline{x},\overline{v})}(d)
&\supseteq{\textstyle\bigcup_{\lambda\in\mathcal{M}_{\overline{x}}(\overline{v})}}
\Big\{\nabla g(\overline{x})\xi\ |\ \langle\xi,g'(\overline{x})d\rangle=0,\,
\xi\in\mathcal{T}_{\mathcal{N}_{K}(g(\overline{x}))}(\lambda)\Big\}\\
&\supseteq{\textstyle\bigcup_{\lambda\in\mathcal{M}_{\overline{x}}(\overline{v})}}
\Big\{\nabla g(\overline{x})\xi\ |\ \langle\xi,g'(\overline{x})d\rangle=0,\,
\xi\in\mathcal{R}_{\mathcal{N}_{K}(g(\overline{x}))}(\lambda)\Big\}.
\label{critical-cone2}
\end{align}
If, in addition, the radial cone $\mathcal{R}_{\mathcal{N}_{\Gamma}(\overline{x})}(\overline{v})$
is closed, the inclusions become equality.
\end{proposition}
\begin{proof}
Since $(\overline{x},\overline{v})\in{\rm gph}\mathcal{N}_{\Gamma}$ and
$\mathcal{G}$ is metrically subregular at $\overline{x}$ for $0$,
by Corollary \ref{TNcone-Gamma}, $\mathcal{M}_{\overline{x}}(\overline{v})\ne\emptyset$
and $\mathcal{T}_{\Gamma}(\overline{x})$ is convex. The latter implies
the convexity of $\mathcal{C}_{\Gamma}(\overline{x},\overline{v})$.
Hence,
\begin{equation}\label{temp-equa1-sec2}
\mathcal{N}_{\mathcal{C}_{\Gamma}(\overline{x},\overline{v})}(d)
=[\mathcal{C}_{\Gamma}(\overline{x},\overline{v})]^{\circ}\cap[\![d]\!]^{\perp}.
\end{equation}
The inclusion in \eqref{critical-cone2} is trivial, and
we only need to establish the inclusion in \eqref{critical-cone1}.
Let $h$ be an arbitrary point from the set on the right hand side of
\eqref{critical-cone1}. Then there exist $\lambda\in\mathcal{M}_{\overline{x}}(\overline{v})$
and $\xi\in\mathcal{T}_{\mathcal{N}_{K}(g(\overline{x}))}(\lambda)$ with
$\langle\xi,g'(\overline{x})d\rangle=0$ such that $h=\nabla\!g(\overline{x})\xi$.
From $\xi\in\mathcal{T}_{\mathcal{N}_{K}(g(\overline{x}))}(\lambda)$,
there exist sequences $t_k\downarrow 0$ and $\xi^k\to\xi$ such that
$\lambda+t_k\xi^k\in\mathcal{N}_K(g(\overline{x}))$ for each $k$.
Fix an arbitrary $k\in\mathbb{N}$. For each $w\in\mathcal{C}_{\Gamma}(\overline{x},\overline{v})$,
by \eqref{temp-equa1-sec2} it holds that
\[
0\ge\langle g'(\overline{x})w,\lambda+t_k\xi^k\rangle
=\langle \overline{v},w\rangle+t_k\langle w,\nabla g(\overline{x})\xi^k\rangle
=t_k\langle w,\nabla g(\overline{x})\xi^k\rangle,
\]
which implies that $\nabla g(\overline{x})\xi^k\in[\mathcal{C}_{\Gamma}(\overline{x},\overline{v})]^{\circ}$.
Thus, $\nabla g(\overline{x})\xi\in[\mathcal{C}_{\Gamma}(\overline{x},\overline{v})]^{\circ}$.
Together with $\langle\xi,g'(\overline{x})d\rangle=0$, we have
$h=\nabla g(\overline{x})\xi\in[\mathcal{C}_{\Gamma}(\overline{x},\overline{v})]^{\circ}\cap[\![d]\!]^{\perp}$,
and then $h\in\mathcal{N}_{\mathcal{C}_{\Gamma}(\overline{x},\overline{v})}(d)$ by
\eqref{temp-equa1-sec2}. This shows that the set on the right hand side of \eqref{critical-cone1}
is included in $\mathcal{N}_{\mathcal{C}_{\Gamma}(\overline{x},\overline{v})}(d)$.
Assume that $\mathcal{R}_{\mathcal{N}_{\Gamma}(\overline{x})}(\overline{v})$
is closed. To argue that the inclusions \eqref{critical-cone1} and \eqref{critical-cone2}
become equality now, we only need to show that $\mathcal{N}_{\mathcal{C}_{\Gamma}(\overline{x},\overline{v})}(d)$
is included in the set on the right hand hand side of \eqref{critical-cone2}.
To this end, let $\overline{h}$ be an arbitrary point from $\mathcal{N}_{\mathcal{C}_{\Gamma}(\overline{x},\overline{v})}(d)$.
Then
\[
[\mathcal{C}_{\Gamma}(\overline{x},\overline{v})]^{\circ}
={\rm cl}(\mathcal{N}_{\Gamma}(\overline{x})+[\![\overline{v}]\!])
={\rm cl}\big(\mathcal{R}_{\mathcal{N}_{\Gamma}(\overline{x})}(\overline{v})\big)
=\mathcal{R}_{\mathcal{N}_{\Gamma}(\overline{x})}(\overline{v})
=\mathcal{N}_{\Gamma}(\overline{x})+[\![\overline{v}]\!].
\]
where the first equality is by \cite[Equation (2.32)]{BS00},
and the second is due to \cite[Example 2.62]{BS00}.
Together with \eqref{temp-equa1-sec2},
$\mathcal{N}_{\mathcal{C}_{\Gamma}(\overline{x},\overline{v})}(d)
=(\mathcal{N}_{\Gamma}(\overline{x})\!+[\![\overline{v}]\!])\cap[\![d]\!]^{\perp}$.
From $\overline{h}\in\mathcal{N}_{\mathcal{C}_{\Gamma}(\overline{x},\overline{v})}(d)$,
there exist $\overline{\eta}\in\mathcal{N}_{\Gamma}(\overline{x})$ and
$\overline{\alpha}\in\mathbb{R}$ such that
$\overline{h}=\overline{\eta}+\overline{\alpha}\overline{v}$
and $\langle\overline{\eta}+\overline{\alpha}\overline{v},d\rangle=0$. Since
$\overline{v}\in\mathcal{N}_{\Gamma}(\overline{x})$ and
$\overline{\eta}\in\mathcal{N}_{\Gamma}(\overline{x})$, by Corollary \ref{TNcone-Gamma},
there exist $\overline{\lambda}\in\mathcal{N}_K(g(\overline{x}))$ and
$\overline{\mu}\in\mathcal{N}_K(g(\overline{x}))$ such that
$\overline{v}=\nabla g(\overline{x})\overline{\lambda}$ and
$\overline{\eta}=\nabla g(\overline{x})\overline{\mu}$.
Write $\overline{\xi}:=\overline{\mu}+\overline{\alpha}\overline{\lambda}$. Clearly,
$\overline{\xi}\in\mathcal{R}_{\mathcal{N}_K(g(\overline{x}))}(\overline{\lambda})$.
Also, from $\langle\overline{\eta}+\overline{\alpha}\overline{v},d\rangle=0$,
we have $\langle g'(\overline{x})d,\overline{\xi}\rangle=0$.
Together with $\overline{h}=\nabla g(\overline{x})\overline{\xi}$
and $\overline{\lambda}\in\mathcal{M}_{\overline{x}}(\overline{v})$,
we conclude that $\overline{h}$ belongs to the set on the right hand side of \eqref{critical-cone2}.
Thus, $\mathcal{N}_{\mathcal{C}_{\Gamma}(\overline{x},\overline{v})}(d)$
is included in the set on the right hand side of \eqref{critical-cone2}.
The proof is completed.
\end{proof}
The sets on the right hand side of \eqref{critical-cone1} and
\eqref{critical-cone2} are generally not closed. Proposition
\ref{critical-normal-prop1} shows that their closedness is implied by
that of $\mathcal{R}_{\mathcal{N}_{\Gamma}(\overline{x})}(\overline{v})$.
A checkable condition for the latter is the strict complementarity which
implies the calmness of $\mathcal{M}_{\overline{x}}$ by Proposition \ref{conditions-closedness}.
Following \cite{BS00}, we say that the {\bf strict complementarity condition}
holds for the system $g(x)\in K$ at
$(\overline{x},\overline{v})\!\in{\rm gph}\mathcal{N}_{\Gamma}$
if there is $\lambda\in {\rm ri}(\mathcal{N}_{K}(g(\overline{x})))$ such that
$\overline{v}=\nabla g(\overline{x})\lambda$.
\begin{proposition}\label{conditions-closedness}
Let $(\overline{x},\overline{v})\in{\rm gph}\mathcal{N}_{\Gamma}$.
Suppose $\mathcal{G}$ is metrically subregular at $\overline{x}$ for $0$.
If the strict complementarity condition holds at $(\overline{x},\overline{v})$,
then the radial cone $\mathcal{R}_{\mathcal{N}_{\Gamma}(\overline{x})}(\overline{v})$ is closed,
and the multifunction $\mathcal{M}_{\overline{x}}$ is calm at $\overline{v}$
for each $\lambda\in\mathcal{M}_{\overline{x}}(\overline{v})$.
\end{proposition}
\begin{proof}
The first part follows by \cite[Proposition 2.1]{Gfrerer171}.
We prove the second part.
Notice that $\mathcal{M}_{\overline{x}}$
can be rewritten as $\mathcal{M}_{\overline{x}}(v)
=\{\lambda\in\mathcal{N}_{K}(g(\overline{x}))\ |\ \nabla\!g(\overline{x})\lambda=v-\overline{v}\}$
for $v\in\mathbb{X}$. Define $\mathcal{F}(u):=\{\lambda\in\mathcal{N}_{K}(g(\overline{x}))\ |\ \nabla\!g(\overline{x})\lambda-u=0\}$ for $u\in\mathbb{X}$.
Fix an arbitrary $\lambda\in\mathcal{M}_{\overline{x}}(\overline{v})$.
It is easy to verify that $\mathcal{M}_{\overline{x}}$ is calm
at $\overline{v}$ for $\lambda$ iff $\mathcal{F}$ is calm
at the origin for $\lambda$. By \cite[Page 211-212]{Ioffe08},
the latter is equivalent to the existence of $\delta,\gamma>0$
such that for all $\lambda'\in\mathbb{B}(\lambda,\delta)$
\[
{\rm dist}\big(\lambda',\mathcal{M}_{\overline{x}}(\overline{v})\big)
\le\gamma\max\big\{{\rm dist}(\lambda',\mathcal{N}_{K}(g(\overline{x}))),
\|-\overline{v}+\nabla g(\overline{x})\lambda'\|\big\}.
\]
This metric qualification holds under the strict complementarity condition
by the convexity of $\mathcal{N}_{K}(g(\overline{x}))$ and \cite[Corollary 3]{Bauschke99}.
\end{proof}
It is worthwhile to point out that the strict complementarity condition
is not necessary for the closedness of the radial cone
$\mathcal{R}_{\mathcal{N}_{\Gamma}(\overline{x})}(\overline{v})$;
see the following example.
\begin{example}\label{example1}
Let $g(x,t):=\left(\begin{matrix}
{\rm Diag}(x)+tE+I\\ t
\end{matrix}\right)$
for $x\in\mathbb{R}^2$ and $t\in\mathbb{R}$,
where $I$ is the $2\times 2$ identity matrix and
$E$ is the $2\times 2$ matrix of all ones.
Consider the constraint system $g(x,t)\in K:=\mathbb{S}^2_+\times \mathbb{R}_+$
where $\mathbb{S}_{+}^2$ is the $2\times 2$ positive semidefinite matrix cone.
Let
\[
\overline{x}=(-1,-1)^{\mathbb{T}},\ \overline{t}=0,\,\overline{\lambda}=0_{2\times 2},\,\overline{\tau}=0
\ {\rm and}\ \overline{v}=((0,0)^{\mathbb{T}};0).
\]
Since $g(\overline{x},\overline{t})=(0_{2\times 2},0)$, clearly,
$(\overline{x},\overline{t})\in g^{-1}(K):=\Gamma$
and $\mathcal{N}_{K}(g(\overline{x},\overline{t}))=\mathbb{S}_{-}^2\times \mathbb{R}_-$.
Since
\begin{equation}\label{grad-gfun}
\nabla g(\overline{x},\overline{t})(H,\omega)
=\left(\begin{matrix}
{\rm diag}(H)\\ \langle E,H\rangle+\omega
\end{matrix}\right)\quad\forall (H,\omega)\in\mathbb{S}^2\times\mathbb{R},
\end{equation}
we have $\overline{v}=\nabla g(\overline{x},\overline{t})(\overline{\lambda},\overline{\tau})$,
and then $\overline{v}\in\mathcal{N}_{\Gamma}(\overline{x},\overline{t})$.
Since
\(
{\rm ri}(\mathcal{N}_{K}(g(\overline{x},\overline{t})))
=\mathbb{S}_{--}^2\times \mathbb{R}_{--},
\)
There does not exist
$(\lambda,\tau)\in{\rm ri}(\mathcal{N}_{K}(g(\overline{x},\overline{t})))$ such that
$\nabla g(\overline{x},\overline{t})(\lambda,\tau)=\overline{v}$,
but since $\overline{v}=((0,0)^{\mathbb{T}};0)$,
the radial cone $\mathcal{R}_{\mathcal{N}_{\Gamma}(\overline{x},\overline{t})}(\overline{v})
=\mathcal{N}_{\Gamma}(\overline{x},\overline{t})$ is closed.
\end{example}
Next we provide another characterization for the normal cone to the critical cone.
\begin{proposition}\label{critical-normal-prop2}
Let $(\overline{x},\overline{v})\in{\rm gph}\mathcal{N}_{\Gamma}$.
Suppose $\mathcal{G}$ is metrically subregular at $\overline{x}$ for $0$.
If $\mathcal{M}_{\overline{x}}$ is isolated calm at $\overline{v}$
for some $\lambda\in\mathcal{M}_{\overline{x}}(\overline{v})$,
then for any given $d\in\mathcal{C}_{\Gamma}(\overline{x},\overline{v})$,
\begin{equation}\label{critical-cone3}
\mathcal{N}_{\mathcal{C}_{\Gamma}(\overline{x},\overline{v})}(d)
=\Big\{\nabla g(\overline{x})\xi\ |\ \langle\xi,g'(\overline{x})d\rangle=0,\,
\xi\in\mathcal{T}_{\mathcal{N}_{K}(g(\overline{x}))}(\lambda)\Big\}.
\end{equation}
\end{proposition}
\begin{proof}
By the first part of Proposition \ref{critical-normal-prop1},
we only need to prove that $\mathcal{N}_{\mathcal{C}_{\Gamma}(\overline{x},\overline{v})}(d)$
is included in the set on the right hand side of \eqref{critical-cone3}.
Let $h^*$ be an arbitrary
point from $\mathcal{N}_{\mathcal{C}_{\Gamma}(\overline{x},\overline{v})}(d)$.
From \eqref{temp-equa1-sec2}, there exists a sequence
$\{h^k\}\subseteq\mathcal{N}_{\Gamma}(\overline{x})+[\![\overline{v}]\!]$
such that $h^k\to h^*$ with $\langle h^*,d\rangle=0$. By the expression of
$\mathcal{N}_{\Gamma}(\overline{x})$, for each $k$ there exist
$\lambda^k\in\mathcal{N}_{K}(g(\overline{x}))$ and $\alpha_k\in\mathbb{R}$
such that $h^k=\nabla g(\overline{x})\lambda^k+\alpha_k\overline{v}$.
Since $\lambda\in\mathcal{M}_{\overline{x}}(\overline{v})$,
we have $\overline{v}=\nabla g(\overline{x})\lambda$,
and $h^k=\nabla g(\overline{x})(\lambda^k\!+\alpha_k\lambda)$ for each $k$.
Notice that $\{\lambda^k\!+\alpha_k\lambda\}$ is bounded.
If not, by using
\[
\frac{h^k}{\|\lambda^k\!+\alpha_k\lambda\|}
=\nabla g(\overline{x})\frac{\lambda^k\!+\alpha_k\lambda}{\|\lambda^k\!+\alpha_k\lambda\|}
\ \ {\rm and}\ \
\frac{\lambda^k\!+\alpha_k\lambda}{\|\lambda^k\!+\alpha_k\lambda\|}
\in \mathcal{N}_K(g(\overline{x}))\!+[\![\lambda]\!]\subseteq
\mathcal{T}_{\mathcal{N}_K(g(\overline{x}))}(\lambda),
\]
there exists $0\ne\overline{\mu}\in{\rm Ker}(\nabla g(\overline{x}))
\cap\mathcal{T}_{\mathcal{N}_K(g(\overline{x}))}(\lambda)\ne\{0\}$.
This, by Proposition \ref{prop-Mx}, contradicts the isolated calmness
assumption of $\mathcal{M}_{\overline{x}}$ at $\overline{v}$ for $\lambda$.
Now we assume (if necessary taking a subsequence) that
$\lambda^k+\alpha_k\lambda\to\xi$. Clearly,
$\xi\in\mathcal{T}_{\mathcal{N}_K(g(\overline{x}))}(\lambda)$
and $h^*=\nabla g(\overline{x})\xi$. Together with $\langle h^*,d\rangle=0$,
we have $\langle g'(\overline{x})d,\xi\rangle=0$. This shows that $h^*$ belongs to
the set on the right hand side of \eqref{critical-cone3},
and the claimed inclusion follows.
\end{proof}
\begin{remark}\label{remark-critical}
Consider an arbitrary $(\overline{x},\overline{v})\!\in{\rm gph}\mathcal{N}_{\Gamma}$
and an arbitrary $d\in\mathcal{C}_{\Gamma}(\overline{x},\overline{v})$.
Under the assumption of Proposition \ref{critical-normal-prop2},
by using Lemma \ref{dir-proj} and $[\mathcal{C}_K(g(\overline{x}),\lambda)]^{\circ}
=\mathcal{T}_{\mathcal{N}_K(g(\overline{x}))}(\lambda)$,
\begin{align*}
\nabla g(\overline{x})\Big[D\mathcal{N}_{K}(g(\overline{x})|\lambda)(g'(\overline{x})d)
- \frac{1}{2}\nabla\Upsilon(g'(\overline{x})d)\Big]
\!=\mathcal{N}_{\mathcal{C}_{\Gamma}(\overline{x},\overline{v})}(d)
\!=\nabla g(\overline{x})\mathcal{N}_{\mathcal{C}_{K}(g(\overline{x}),\lambda)}(g'(\overline{x})d)
\end{align*}
with $\Upsilon(\cdot)=-\sigma(\lambda,\mathcal{T}_{K}^2(g(\overline{x}),\cdot))$,
where the last equality is using the following equivalence
\[
\xi\in\mathcal{N}_{\mathcal{C}_K(g(\overline{x}),\overline{\lambda})}(g'(\overline{x})d)
\Longleftrightarrow \langle \xi,g'(\overline{x})d\rangle=0,
\xi\in\mathcal{T}_{\mathcal{N}_K(g(\overline{x}))}(\overline{\lambda}).
\]
\end{remark}
It is worthwhile to point out that there is no direct relation between
the closedness of $\mathcal{R}_{\mathcal{N}_{\Gamma}(\overline{x})}(\overline{v})$
and the isolated calmness of $\mathcal{M}_{\overline{x}}$; see Example \ref{example2} below.
In addition, although the strict complementarity condition and
the isolated calmness of $\mathcal{M}_{\overline{x}}$ imply
the calmness of $\mathcal{M}_{\overline{x}}$, there is no direct relation
between them; see Example \ref{example3} below.
\begin{example}\label{example2}
Consider the constraint system in Example \ref{example1}.
Let $(\overline{x},\overline{t})$ and $(\overline{\lambda},\overline{\tau})$
be same as Example \ref{example1}. Firstly, by using \eqref{grad-gfun} and noting that $\mathcal{T}_{\mathcal{N}_K(g(\overline{x},\overline{t}))}(\overline{\lambda},\overline{\tau})
=\mathbb{S}_{-}^2\times \mathbb{R}_{-}$, it is not hard to check that
\(
{\rm Ker}(\nabla g(\overline{x},\overline{t}))\cap\mathcal{T}_{\mathcal{N}_K(g(\overline{x},\overline{t}))}(\overline{\lambda},\overline{\tau})
=\{(0_{2\times 2},0)\}.
\)
By Proposition \ref{prop-Mx}, the multifunction
$\mathcal{M}_{(\overline{x},\overline{t})}$ is isolated calm
at $\overline{v}=((0,0)^{\mathbb{T}};0)$ for $(\overline{\lambda},\overline{\tau})$.
Next we consider $(\widehat{\lambda},\overline{\tau})$ with
$\widehat{\lambda}=\left[\begin{matrix}
-1 & 0\\ 0 & 0
\end{matrix}\right]\in\mathbb{S}_{-}^2$.
By using \eqref{grad-gfun}, we calculate that
\[
\widehat{v}=\nabla g(\overline{x},\overline{t})(\widehat{\lambda},\overline{\tau})
=((-1,0)^{\mathbb{T}};-1).
\]
Since
\(
\mathcal{T}_{\mathcal{N}_K(g(\overline{x},\overline{t}))}(\widehat{\lambda},\overline{\tau})
=\mathcal{T}_{\mathbb{S}_{-}^2}(\widehat{\lambda})\times\mathcal{T}_{\mathbb{R}_{-}}(\overline{\tau})
=\{H\in\mathbb{S}^2\ |\ H_{22}\le 0\}\times\mathbb{R}_{-},
\)
it follows that
\[
{\rm Ker}(\nabla g(\overline{x},\overline{t}))\cap\mathcal{T}_{\mathcal{N}_K(g(\overline{x},\overline{t}))}(\overline{Y},\overline{s})
\ne \{(0_{2\times 2},0)\}.
\]
By Proposition \ref{prop-Mx}, the mapping $\mathcal{M}_{(\overline{x},\overline{t})}$
is not isolated calm at $\widehat{v}$ for $(\widehat{\lambda},\overline{\tau})$.
Notice that
\begin{align*}
\mathcal{R}_{\mathcal{N}_{\Gamma}(\overline{x},\overline{t})}(\overline{v})
&=\nabla g(\overline{x},\overline{t})\big(\mathcal{N}_K(g(\overline{x},\overline{t}))+[\![(\widehat{\lambda},\overline{\tau})]\!]\big)\\
&=\bigg\{\left(\begin{matrix}
{\rm diag}(Y+a\widehat{\lambda})\\ \langle E, Y\rangle +\tau
\end{matrix}\right)\;|\; Y\in \mathbb{S}^2_-,\,\tau\in\mathbb{R}_{-},\,a\in \mathbb{R}\bigg\}.
\end{align*}
Clearly, $\mathcal{R}_{\mathcal{N}_{\Gamma}(\overline{x},\overline{t})}(\overline{v})
\subseteq\mathbb{R}\times\mathbb{R}_{-}\times \mathbb{R}_{-}$. Furthermore,
for any $(\omega;b;\pi)\in \mathbb{R}\times\mathbb{R}_{-}\times\mathbb{R}_{-}$,
\[
\left(\begin{matrix}
\omega\\ b\\ \pi
\end{matrix}\right)
=\left(\begin{matrix}
{\rm diag}(Y+a\widehat{\lambda})\\ \langle E, Y\rangle
\end{matrix}\right)\ {\rm with}\
Y=\left(\begin{matrix}
b & -b\\ -b & b
\end{matrix}\right)\in\mathbb{S}_{-}^2,\, a=b-\omega\in\mathbb{R},\tau=\pi\in\mathbb{R}_{-}.
\]
This shows that $\mathcal{R}_{\mathcal{N}_{\Gamma}(\overline{x},\overline{t})}(\overline{v})
=\mathbb{R}\times\mathbb{R}_{-}\times \mathbb{R}_{-}$, and hence is closed,
although $\mathcal{M}_{(\overline{x},\overline{t})}$ is not isolated calm
at $\widehat{v}$ for $(\widehat{\lambda},\overline{\tau})$.
Along with the arguments in the first paragraph,
we conclude that the isolated calmness of $\mathcal{M}_{(\overline{x},\overline{t})}$
has no relation with the closedness of $\mathcal{R}_{\mathcal{N}_{\Gamma}(\overline{x})}(\overline{v})$.
\end{example}
\begin{example}\label{example3}
Consider the constraint system $g(X)\in K$
where $K=\{0_{2\times 2}\}\times\mathbb{S}_{+}^2$ and
\[
g(X):=\left(\begin{matrix}
X+C\\ X
\end{matrix}\right)
\ {\rm with}\
C:=\left(\begin{matrix}
0 & 0\\ 0 & -1
\end{matrix}\right)\ \ {\rm for}\ X\in\mathbb{S}^2.
\]
Notice that
\(
\nabla g(X)(Y,S)=Y+S
\)
for $Y,S\in\mathbb{S}^2$. We consider the following points:
\[
\overline{X}=\left(\begin{matrix}
0 & 0\\ 0 & 1
\end{matrix}\right),\
\overline{S}=\left(\begin{matrix}
-1 & 0\\ 0 & 0
\end{matrix}\right),\
\overline{Y}=0_{2\times 2}\ \ {\rm and}\ \
\overline{v}=\overline{S}.
\]
Clearly, $(\overline{Y},\overline{S})\in{\rm ri}(\mathcal{N}_{K}(g(\overline{X})))$
and $(\overline{Y},\overline{S})\in\mathcal{M}_{\overline{X}}(\overline{v})$.
The strict complementarity condition is satisfied at $(\overline{X},\overline{v})$, but
$\mathcal{M}_{\overline{X}}$ is not isolated
calm at $\overline{v}$ since $\mathcal{M}_{\overline{X}}(\overline{v})$
is not singleton. Together with Example \ref{example1}, we conclude that
the strict complementarity condition has no relation with
the isolated calmness of $\mathcal{M}_{\overline{x}}$.
\end{example}
\section{Graphical derivative of the mapping $\mathcal{N}_{\Gamma}$}\label{sec3}
By Corollary \ref{TNcone-Gamma}, when $\mathcal{G}$ is metrically subregular
at $\overline{x}$ for $0$, $(\overline{x},\overline{v})\in{\rm gph}\mathcal{N}_{\Gamma}$
if and only if there exists $\overline{\lambda}\in\mathcal{N}_K(g(\overline{x}))$ such that
$\overline{v}=\nabla g(\overline{x})\overline{\lambda}$. By this, we define the mapping
\begin{equation}\label{Phimap}
\Phi(x,\lambda,v):=
\left(\begin{matrix}
-v+\nabla g(x)\lambda\\
g(x)-\Pi_K(g(x)\!+\lambda)
\end{matrix}\right)\quad{\rm for}\ (x,\lambda,v)\in\mathbb{X}\times\mathbb{Y}\times\mathbb{X}.
\end{equation}
Since $\Pi_{K}$ is directionally differentiable at $x$ in the Hadamard sense
by \cite[Theorem 7.2]{BCS98} and \cite[Proposition 2.49]{BS00} and
the mapping $\nabla g$ is continuously differentiable, the mapping $\Phi$
is locally Lipschitz and directionally differentiable. In Subsection \ref{subsec3.1},
we shall characterize the graphical derivative of $\mathcal{N}_{\Gamma}$
under the metric subregularity of $\Phi$.
\subsection{Characterization for graphical derivative of $\mathcal{N}_{\Gamma}$}\label{subsec3.1}
First we present a lower estimation for the graphical derivative
of $\mathcal{N}_{\Gamma}$ via that of $\Phi^{-1}$.
\begin{lemma}\label{lestimate-lemma1}
Consider an arbitrary $(\overline{x},\overline{v})\in{\rm gph}\mathcal{N}_{\Gamma}$.
Suppose that the multifunction $\mathcal{G}$ in \eqref{MGmap} is metrically subregular
at $\overline{x}$ for the origin, and that the mapping $\Phi$ is metrically subregular
at each $(\overline{x},\lambda,\overline{v})$
with $\lambda\in\mathcal{M}_{\overline{x}}(\overline{v})$ for the origin.
Then, it holds that
\[
\!\mathcal{T}_{{\rm gph}\mathcal{N}_{\Gamma}}(\overline{x},\overline{v})
\supseteq\!\bigcup_{\lambda\in\mathcal{M}_{\overline{x}}(\overline{v})}\!
\Big\{(d,w)\in\mathbb{X}\times\mathbb{X}\ |\ \exists\mu\in\mathbb{Y}\ {\rm s.t.}\ (d,\mu,w)\in D\Phi^{-1}((0,0)|(\overline{x},\lambda,\overline{v}))(0,0)\Big\}.
\]
\end{lemma}
\begin{proof}
Define $\mathcal{A}(x,y,x'):=(x,x')$ for $(x,y,x')\in\mathbb{X}\times\mathbb{Y}\times\mathbb{X}$.
By Remark \ref{robust-remark}, there exists a neighborhood $\mathcal{V}$ of $\overline{x}$
such that the multifunction $\mathcal{G}$ in \eqref{MGmap} is metrically subregular
at each $x\in\mathcal{V}\cap\Gamma$ for the origin. From Corollary \ref{TNcone-Gamma},
it follows that
\[
{\rm gph}\mathcal{N}_{\Gamma}\cap(\mathcal{V}\times\mathbb{X})
=\mathcal{A}(\Phi^{-1}(0,0))\cap(\mathcal{V}\times\mathbb{X}).
\]
By virtue of \cite[Theorem 6.43]{RW98}, we obtain the following inclusion
\begin{align}\label{temp-tcone-Nomega}
\mathcal{T}_{{\rm gph}\mathcal{N}_{\Gamma}}(\overline{x},\overline{v})
&\supseteq\bigcup_{z\in\mathcal{A}^{-1}(\overline{x},\overline{v})\cap\Phi^{-1}(0,0)}
\Big\{\mathcal{A}(\xi,\eta,\zeta)\ |\ (\xi,\eta,\zeta)\in\mathcal{T}_{\Phi^{-1}(0,0)}(z)\Big\}\nonumber\\
&=\bigcup_{\lambda\in\mathcal{M}_{\overline{x}}(\overline{v})}
\Big\{\mathcal{A}(\xi,\eta,\zeta)\ |\ (\xi,\eta,\zeta)\in\mathcal{T}_{\Phi^{-1}(0,0)}(\overline{x},\lambda,\overline{v})\Big\},
\end{align}
where the equality is due to the definitions of $\mathcal{A}$ and
$\mathcal{M}_{\overline{x}}(\overline{v})$.
Since $\Phi$ is metrically subregular at each $(\overline{x},\lambda,\overline{v})$
with $\lambda\in\mathcal{M}_{\overline{x}}(\overline{v})$ for the origin,
by virtue of Lemma \ref{TF-relation},
\[
(\xi,\eta,\zeta)\in\mathcal{T}_{\Phi^{-1}(0,0)}(\overline{x},\lambda,\overline{v})
\Longleftrightarrow (0,0,\xi,\eta,\zeta)\in\mathcal{T}_{{\rm gph}\Phi^{-1}}(0,0,\overline{x},\lambda,\overline{v}).
\]
Together with the inclusion in \eqref{temp-tcone-Nomega} and the definition of $\mathcal{A}$,
it follows that
\[
\mathcal{T}_{{\rm gph}\mathcal{N}_{\Gamma}}(\overline{x},\overline{v})
\supseteq\!\bigcup_{\lambda\in\mathcal{M}_{\overline{x}}(\overline{v})}
\Big\{(\xi,\zeta)\ |\ \exists\eta\in\mathbb{Y}\ {\rm s.t.}\
(\xi,\eta,\zeta)\in D\Phi^{-1}((0,0)|(\overline{x},\lambda,\overline{v}))(0,0)\Big\}.
\]
This shows that the desired inclusion holds. The proof is completed.
\end{proof}
The following lemma gives the characterization on the graphical derivative of $\Phi^{-1}$.
\begin{lemma}\label{Phi-derivative}
Let $\Phi$ be defined by \eqref{Phimap}. Consider an arbitrary
$(\overline{x},\overline{\lambda},\overline{v})\in\Phi^{-1}(0,0)$. Then,
\begin{align*}
&(\Delta x,\Delta\lambda,\Delta v)
\in D\Phi^{-1}((0,0)|(\overline{x},\overline{\lambda},\overline{v}))(\Delta \xi,\Delta\eta)\\
&\Longleftrightarrow
\left\{\begin{array}{ll}
\Delta\xi=\nabla^2\langle\overline{\lambda},g\rangle(\overline{x})\Delta x+\nabla g(\overline{x})\Delta\lambda-\Delta v;\\
\Delta\eta=g'(\overline{x})\Delta x-\Pi_K'(g(\overline{x})\!+\overline{\lambda};g'(\overline{x})\Delta x\!+\!\Delta\lambda).
\end{array}\right.
\end{align*}
\end{lemma}
\begin{proof}
Since the mapping $\Phi$ is locally Lipschitz and directionally differentiable,
we have
\begin{align*}
&D\Phi((\overline{x},\overline{\lambda},\overline{v})|(0,0))(\Delta x,\Delta\lambda,\Delta v)\\
&=\Big\{(\Delta \xi,\Delta\eta)\in\mathbb{X}\times\mathbb{Y}\ |\
\Phi'((\overline{x},\overline{\lambda},\overline{v});(\Delta x,\Delta\lambda,\Delta v))
=(\Delta \xi,\Delta\eta)\Big\}.
\end{align*}
In addition, by the expression of $\Phi$ and \cite[Proposition 2.47]{BS00},
we calculate that
\[
\Phi'((\overline{x},\overline{\lambda},\overline{v});(\Delta x,\Delta\lambda,\Delta v))
=\left(\begin{matrix}
\nabla^2\langle\overline{\lambda},g\rangle(\overline{x})\Delta x+\nabla g(\overline{x})\Delta\lambda-\Delta v\\
g'(\overline{x})\Delta x-\Pi_K'(g(\overline{x})\!+\overline{\lambda};g'(\overline{x})\Delta x\!+\Delta\lambda)
\end{matrix}\right).
\]
Notice that $(\Delta x,\Delta\lambda,\Delta v)
\in D\Phi^{-1}((0,0)|(\overline{x},\overline{\lambda},\overline{v}))(\Delta \xi,\Delta\eta)$
if and only if $(\Delta \xi,\Delta\eta)$ lies in
$D\Phi((\overline{x},\overline{\lambda},\overline{v})|(0,0))(\Delta x,\Delta\lambda,\Delta v)$.
The result follows from the last two equations.
\end{proof}
By combining Lemma \ref{lestimate-lemma1} with Lemma \ref{Phi-derivative}
and using Lemma \ref{dir-proj}, we readily obtain a lower estimation for
the graphical derivative of the mapping $\mathcal{N}_{\Gamma}$.
\begin{proposition}\label{lestimate}
Consider an arbitrary $(\overline{x},\overline{v})\in{\rm gph}\mathcal{N}_{\Gamma}$.
Suppose that the multifunction $\mathcal{G}$ in \eqref{MGmap} is metrically subregular
at $\overline{x}$ for the origin. If the mapping $\Phi$ in \eqref{Phimap}
is metrically subregular at each $(\overline{x},\lambda,\overline{v})$ with
$\lambda\in\mathcal{M}_{\overline{x}}(\overline{v})$ for the origin,
then
\[
\!\mathcal{T}_{{\rm gph}\mathcal{N}_{\Gamma}}(\overline{x},\overline{v})
\supseteq\!\bigcup_{\lambda\in\mathcal{M}_{\overline{x}}(\overline{v})}\!
\Big\{(d,w)\in\mathbb{X}\times\mathbb{X}\ |\ w\in \nabla^2\langle\lambda,g\rangle(\overline{x})d+\nabla\!g(\overline{x})
D\mathcal{N}_K(g(\overline{x})|\lambda)(g'(\overline{x})d)\Big\}.
\]
\end{proposition}
\begin{remark}\label{low-estimate-remark}
During the reviewing of this paper, we learned that Gfrerer and Mordukhovich
only under the metric subregularity of $\mathcal{G}$ derived a lower estimation
for the graphical derivative of $\mathcal{N}_{\Gamma}$ (see \cite[Theorem 3.3]{Gfrerer171}),
which has a little difference from the one
in Proposition \ref{lestimate} but agrees with it
under the closedness of $\mathcal{R}_{\mathcal{N}_{\Gamma}(\overline{x})}(\overline{v})$.
\end{remark}
Next we concentrate on an upper estimation for the graphical derivative
of $\mathcal{N}_{\Gamma}$.
\begin{proposition}\label{uestimate}
Consider an arbitrary $(\overline{x},\overline{v})\in{\rm gph}\mathcal{N}_{\Gamma}$.
Suppose that $\mathcal{G}$ is metrically subregular at $\overline{x}$ for $0$,
and that $\mathcal{M}_{\overline{x}}$ is isolated calm at $\overline{v}$
for some $\overline{\lambda}\in\mathcal{M}_{\overline{x}}(\overline{v})$. Then,
\begin{equation*}
\!\mathcal{T}_{{\rm gph}\mathcal{N}_{\Gamma}}(\overline{x},\overline{v})
\subseteq\Big\{(d,w)\in\mathbb{X}\times\mathbb{X}\ |\ w\in \nabla^2\langle\overline{\lambda},g\rangle(\overline{x})d+\nabla\!g(\overline{x})
D\mathcal{N}_K(g(\overline{x})|\overline{\lambda})(g'(\overline{x})d)\Big\}.
\end{equation*}
\end{proposition}
\begin{proof}
Since $\mathcal{G}$ is metrically subregular at $\overline{x}$ for the origin
and $(\overline{x},\overline{v})\in{\rm gph}\,\mathcal{N}_{\Gamma}$,
by Corollary \ref{TNcone-Gamma}, $\mathcal{M}_{\overline{x}}(\overline{v})\ne\emptyset$.
Since $\mathcal{M}_{\overline{x}}$ is isolated calm at $\overline{v}$ for
$\overline{\lambda}\in\mathcal{M}_{\overline{x}}(\overline{v})$,
by Proposition \ref{prop-Mx} the SRCQ for the system $g(x)\in K$ holds
at $\overline{x}$ w.r.t. $\overline{\lambda}$. So,
$\mathcal{M}_{\overline{x}}(\overline{v})=\{\overline{\lambda}\}$
and Robinson's CQ for this system holds at $\overline{x}$.
Now fix an arbitrary $(d,w)\in\mathcal{T}_{{\rm gph}\widehat{\mathcal{N}}_{\Gamma}}(\overline{x},\overline{v})$.
Then, there exist $t_k\downarrow 0$ and $(d^k,w^k)\to(d,w)$ such that
$(\overline{x}+t_kd^k,\overline{v}+t_kw^k)\in{\rm gph}\widehat{\mathcal{N}}_{\Gamma}$ for each $k$.
Write $x^k:=\overline{x}+t_kd^k$ and $v^k:=\overline{v}+t_kw^k$.
Since Robinson's CQ for the system $g(x)\in K$ holds at $\overline{x}$,
there exists a neighborhood $\mathcal{U}$ of $\overline{x}$ such that
Robinson's CQ for this system holds at each $z\in\mathcal{U}$.
By Corollary \ref{TNcone-Gamma}, for each sufficiently large $k$,
there exists $\lambda^k\in\mathcal{N}_K(g(x^k))$ such that
$v^k=\nabla g(x^k)\lambda^k$. Furthermore, the sequence $\{\lambda^k\}$ is bounded.
Taking a subsequence if necessary, we assume that $\{\lambda^k\}$ converges
to $\widehat{\lambda}$. Since $\lambda^k\in\mathcal{N}_{K}(g(x^k))$,
from the outer semicontinuity of $\mathcal{N}_K$ it follows that
$\widehat{\lambda}\in\mathcal{N}_K(g(\overline{x}))$. In addition,
from $v^k=\nabla g(x^k)\lambda^k$ we have $\overline{v}=\nabla g(\overline{x})\widehat{\lambda}$.
This means that $\widehat{\lambda}\in\mathcal{M}_{\overline{x}}(\overline{v})=\{\overline{\lambda}\}$.
By Theorem \ref{NK-calm}, $\mathcal{N}_K$ is calm at $g(\overline{x})$
for $\overline{\lambda}$, i.e., there exist $\delta>0$ and $c>0$ such that
\[
\mathcal{N}_K(y)\cap\mathbb{B}(\overline{\lambda},\delta)
\subset\mathcal{N}_K(g(\overline{x}))+c\|y-g(\overline{x})\|\mathbb{B}_{\mathbb{Y}}
\quad\ \forall y\in\mathbb{Y}.
\]
From $\mathcal{N}_K(g(x^k))\ni\lambda^k\to\overline{\lambda}$,
for each $k$ large enough, there exists $\zeta^k\in\mathcal{N}_K(g(\overline{x}))$ satisfying
\begin{equation}\label{lambdak-ineq1}
\|\lambda^k-\zeta^k\|={\rm dist}(\lambda^k,\mathcal{N}_K(g(\overline{x})))
\le c\|g(x^k)-g(\overline{x})\|
=ct_k\|g'(\overline{x})d^k+o(t_k)/t_k\|
\end{equation}
where the second equality is by the Taylor expansion of $g(x^k)$ at $\overline{x}$.
Write $\widetilde{v}^k:=\nabla g(\overline{x})\zeta^k$. Clearly,
$\zeta^k\in\mathcal{M}_{\overline{x}}(\widetilde{v}^k)$.
Also, the last inequality implies $\zeta^k\to\overline{\lambda}$.
By the isolated calmness of $\mathcal{M}_{\overline{x}}$ at $\overline{v}$
for $\overline{\lambda}$, there exists a constant $\gamma>0$ (depending on
$\overline{\lambda}$ and $\overline{v}$ only) such that for each $k$ large enough, $\|\zeta^k\!-\!\overline{\lambda}\|\le\gamma\|\overline{v}-\widetilde{v}^k\|$.
Notice that
\[
\widetilde{v}^k=\overline{v}+t_kw^k+(\nabla g(\overline{x})-\nabla g(x^k))\zeta^k
+\nabla g(x^k)(\zeta^k-\lambda^k).
\]
By virtue of $\|\nabla g(x^k)-\nabla g(\overline{x})\|\le t_k\|D^2g(\overline{x})d^k+o(t_k)/t_k\|$
and \eqref{lambdak-ineq1}, we have
\[
\|\widetilde{v}^k-\overline{v}\|\le t_k\big[\|w^k\|+\|D^2g(\overline{x})d^k\|\|\zeta^k\|
+c\|\nabla g(x^k)\|\|g'(\overline{x})d^k\|\big]+o(t_k)
\]
where $D^2g(\overline{x})$ is the second-order derivative of $g$ at $\overline{x}$.
Along with $\|\zeta^k\!-\!\overline{\lambda}\|\le \gamma\|\overline{v}-\widetilde{v}^k\|$,
\begin{equation}\label{wlambdak-ineq1}
\|\zeta^k\!-\!\overline{\lambda}\|\le \gamma t_k\big[\|w^k\|+\|D^2g(\overline{x})d^k\|\|\zeta^k\|
+c\|\nabla g(x^k)\|\|g'(\overline{x})d^k\|\big]+o(t_k).
\end{equation}
Write $\mu^k:=\frac{\lambda^k-\overline{\lambda}}{t_k}$.
From inequalities \eqref{lambdak-ineq1} and \eqref{wlambdak-ineq1},
the sequence $\{\mu^k\}$ is bounded. Taking a subsequence if necessary,
we assume that $\mu^k$ converges to $\mu$. Notice that
\begin{align*}
\overline{v}+t_kw^k
&=\nabla g(x^k)\lambda^k=\nabla g(x^k)\overline{\lambda}+\nabla g(x^k)(\lambda^k-\overline{\lambda})\\
&=(\nabla g(\overline{x})+t_kD^2g(\overline{x})d^k)\overline{\lambda}
+(\nabla g(\overline{x})+t_kD^2g(\overline{x})d^k)(\lambda^k-\overline{\lambda})+o(t_k)\\
&=\overline{v}+t_k\big[\nabla^2\langle\overline{\lambda},g\rangle(\overline{x})d^k
+\nabla g(\overline{x})\mu^k+t_k\nabla^2\langle\mu^k,g\rangle(\overline{x})d^k+o(t_k)/t_k\big].
\end{align*}
Hence,
\(
w^k=\nabla^2\langle\overline{\lambda},g\rangle(\overline{x})d^k
+\nabla g(\overline{x})\mu^k+t_k\nabla^2\langle\mu^k,g\rangle(\overline{x})d^k+o(t_k)/t_k.
\)
Taking the limit, we obtain $w=\nabla^2\langle \overline{\lambda},g\rangle(\overline{x})\xi+\nabla g(\overline{x})\mu$.
Finally, we prove that $\mu\in D\mathcal{N}_K(g(\overline{x})|\lambda)(g'(\overline{x})d)$,
and the desired inclusion follows by the arbitrariness of
$(d,w)\in\mathcal{T}_{{\rm gph}\mathcal{N}_{\Gamma}}(\overline{x},\overline{v})$.
From $\lambda^k\in\mathcal{N}_K(g(x^k))$ and the first order
expansion of $g$ at $\overline{x}$, it holds that
\[
\overline{\lambda}+t_k\mu^k=\lambda^k\in\mathcal{N}_K(g(\overline{x})+t_k(g'(\overline{x})d^k+o(t_k)/t_k)).
\]
That is, $(g(\overline{x})+t_k(g'(\overline{x})d^k+o(t_k)/t_k),\overline{\lambda}+t_k\mu^k)\in{\rm gph}\mathcal{N}_K$.
Along with $(g(\overline{x}),\overline{\lambda})\in{\rm gph}\mathcal{N}_K$,
we have $(g'(\overline{x})d,\mu)\in\mathcal{T}_{{\rm gph}\mathcal{N}_K}(g(\overline{x}),\lambda)$
or equivalently $\mu\in D\mathcal{N}_K(g(\overline{x})|\lambda)(g'(\overline{x})d)$.
\end{proof}
From Proposition \ref{lestimate} and \ref{uestimate},
we get the following characterization for the graphical
derivative of the mapping $\mathcal{N}_{\Gamma}$
without requiring the nondegeneracy of $\overline{x}$ as in
\cite{Gfrerer17,Mordu151}.
\begin{theorem}\label{festimate}
Consider an arbitrary $(\overline{x},\overline{v})\in{\rm gph}\mathcal{N}_{\Gamma}$.
Suppose that $\mathcal{G}$ is metrically subregular at $\overline{x}$ for the origin.
If $\mathcal{M}_{\overline{x}}$ is isolated calm at $\overline{v}$
for some $\overline{\lambda}\in\mathcal{M}_{\overline{x}}(\overline{v})$, then
\begin{equation*}
\!\mathcal{T}_{{\rm gph}\mathcal{N}_{\Gamma}}(\overline{x},\overline{v})
\subseteq\Big\{(d,w)\in\mathbb{X}\times\mathbb{X}\ |\ w\in \nabla^2\langle\overline{\lambda},g\rangle(\overline{x})d+\nabla\!g(\overline{x})
D\mathcal{N}_K(g(\overline{x})|\overline{\lambda})(g'(\overline{x})d)\Big\}.
\end{equation*}
If, in addition, the mapping $\Phi$ is metrically subregular at
$(\overline{x},\overline{\lambda},\overline{v})$ for the origin, then
\begin{equation}\label{Tcone-final1}
\!\mathcal{T}_{{\rm gph}\mathcal{N}_{\Gamma}}(\overline{x},\overline{v})
=\Big\{(d,w)\in\mathbb{X}\times\mathbb{X}\ |\ w\in \nabla^2\langle\overline{\lambda},g\rangle(\overline{x})d+\nabla\!g(\overline{x})
D\mathcal{N}_K(g(\overline{x})|\overline{\lambda})(g'(\overline{x})d)\Big\}.
\end{equation}
\end{theorem}
By combining Theorem \ref{festimate} and Remark \ref{remark-critical},
we also have the following conclusion.
\begin{corollary}\label{estimate-cor}
Consider an arbitrary $(\overline{x},\overline{v})\in{\rm gph}\mathcal{N}_{\Gamma}$.
Suppose that $\mathcal{G}$ is metrically subregular at $\overline{x}$ for the origin.
If $\mathcal{M}_{\overline{x}}$ is isolated calm at $\overline{v}$
for some $\overline{\lambda}\in\mathcal{M}_{\overline{x}}(\overline{v})$ and
the mapping $\Phi$ is metrically subregular at $(\overline{x},\overline{\lambda},\overline{v})$
for the origin, then it holds that
\begin{align}\label{Tcone-final2}
\!\mathcal{T}_{{\rm gph}\mathcal{N}_{\Gamma}}(\overline{x},\overline{v})
&=\left\{(d,w)\in\mathbb{X}\times\mathbb{X}\ \Big|
\left.\begin{array}{ll}
w\in\!\nabla^2\langle \overline{\lambda},g\rangle(\overline{x})d
+\frac{1}{2}\nabla g(\overline{x})\nabla\Upsilon(g'(\overline{x})d)\!\\
\qquad +\nabla g(\overline{x})\mathcal{N}_{\mathcal{C}_{K}(g(\overline{x}),\overline{\lambda})}(g'(\overline{x})d)
\end{array}\right.\!\right\}.
\end{align}
\end{corollary}
\begin{remark}\label{remark-main}
{\bf(a)} The expression of the graphical derivative in \eqref{Tcone-final1} is
same as the one derived in \cite[Theorem 2]{Gfrerer16-MOR}, but compared with
that of \cite[Theorem 5.2]{Mordu151} an additional term
$\frac{1}{2}\nabla g(\overline{x})\nabla\Upsilon(g'(\overline{x})d)$
appears since the PDC is not imposed on $K$. Compared with
the one in \cite[Corollary 5.4]{Gfrerer171}, unless the uniqueness of
the multiplier set and the closedness of
$\mathcal{R}_{\mathcal{N}_{\Gamma}(\overline{x})}(\overline{v})$
are required there, our formula \eqref{Tcone-final1} or \eqref{Tcone-final2}
is convenient for use.
\noindent
{\bf(b)} By Remark \ref{low-estimate-remark}, we know that
\cite[Theorem 3.3]{Gfrerer171} and Proposition \ref{critical-normal-prop2}
imply that the equality \eqref{Tcone-final1} or \eqref{Tcone-final2}
actually holds without the metric subregularity of $\Phi$.
\end{remark}
\subsection{Conditions for metric subregularity of $\Phi$}\label{subsec3.2}
As pointed out in Remark \ref{remark-main}(b), due to \cite[Theorem 3.3]{Gfrerer171},
the exact characterization of the graphical derivative of $\mathcal{N}_{\Gamma}$
in formula \eqref{Tcone-final1} or \eqref{Tcone-final2} does not
require the metric subregularity of $\Phi$, but we think that
it has a separate value. So, in this part we focus on
the metric subregularity of $\Phi$. When $K$ and $g$
are both polyhedral, from the crucial result due to Robinson \cite{Robinson81},
the metric subregularity of $\Phi$ automatically holds.
When either $K$ or $g$ is non-polyhedral, the metric
subregularity of $\Phi$ at $(\overline{x},\overline{\lambda},\overline{v})$
for the origin is implied by the isolated calmness of $\Phi^{-1}$
at the origin for $(\overline{x},\overline{\lambda},\overline{v})$ or by
the Aubin property of $\Phi^{-1}$. By Proposition \ref{Phi-derivative} and
Lemma \ref{chara-icalm}, the former is equivalent to requiring
\begin{equation}\label{implication}
\left\{\begin{array}{ll}
\nabla^2\langle\overline{\lambda},g\rangle(\overline{x})\Delta x
+\nabla g(\overline{x})\Delta\lambda-\!\Delta v=0;\\
\Delta\lambda\in D\mathcal{N}_K(g(\overline{x})|\overline{\lambda})(g'(\overline{x})\Delta x)
\end{array}\right.\Longrightarrow (\Delta x,\Delta\lambda,\Delta v)=(0,0,0),
\end{equation}
which is almost impossible due to the free $\Delta v$. We next focus on
the latter. It is a little surprising to us that the Aubin property
of $\Phi^{-1}$ is equivalent to the nondegeneracy.
\begin{proposition}\label{property-MG}
Consider an arbitrary $(\overline{x},\overline{v})\!\in{\rm gph}\mathcal{N}_{\Gamma}$
with $\mathcal{M}_{\overline{x}}(\overline{v})\!\ne\emptyset$.
Let $\overline{\lambda}\in\!\mathcal{M}_{\overline{x}}(\overline{v})$.
The multifunction $\Phi^{-1}$ has the Aubin property at the origin
for $(\overline{x},\overline{\lambda},\overline{v})$ if and only if
\begin{equation}\label{coderiv-imply}
{\rm Ker}(\nabla g(\overline{x}))\cap D^*\mathcal{N}_{K}(g(\overline{x})|\overline{\lambda})(0)=\{0\}.
\end{equation}
In particular, condition \eqref{coderiv-imply} is equivalent to
the nondegeneracy of $\overline{x}$ w.r.t. the set $K$ and
the mapping $\Xi$, where $\Xi$ is same as the one in Lemma \ref{lemma-reduction}.
\end{proposition}
\begin{proof}
We first characterize the coderivative of $\Phi$ at $(\overline{x},\overline{\lambda},\overline{v})$.
Notice that
\[
\Phi(x,\lambda,v)=\left(\begin{matrix}
\Phi_1(x,\lambda,v)\\ \Phi_2(x,\lambda,v)
\end{matrix}\right)\ \ {\rm with}\
\left\{\begin{array}{ll}
\Phi_1(x,\lambda,v):=-v+\nabla g(x)\lambda;\\
\Phi_2(x,\lambda,v):=g(x)-\Pi_K(g(x)+\lambda).
\end{array}\right.
\]
Fix an arbitrary $(\Delta \xi,\Delta\eta)\in\mathbb{X}\times\mathbb{Y}$.
By using Lemma \ref{calculus-rule} in Appendix, we calculate that
\[
D^*\Phi(\overline{x},\overline{\lambda},\overline{v})(\Delta \xi,\Delta\eta)
=\left[\begin{matrix}
\nabla^2\langle\overline{\lambda},g\rangle(\overline{x})\Delta\xi\\
g'(\overline{x})\Delta\xi\\
-\Delta\xi
\end{matrix}\right]
+D^*\Phi_2(\overline{x},\overline{\lambda},\overline{v})(\Delta\eta).
\]
From the definition of $\Phi_2(x,\lambda,v)$ and \cite[Theorem 1.62]{Mordu06},
it follows that
\[
D^*\Phi_2(\overline{x},\overline{\lambda},\overline{v})(\Delta\eta)
=\left(\begin{matrix}
\nabla\!g(\overline{x})\Delta\eta\\
0\\ 0
\end{matrix}\right)
+D^*(-\Pi_K\circ h)(\overline{x},\overline{\lambda},\overline{v})\Delta\eta
\]
where $h(x,\lambda,v):=g(x)+\lambda$ for $(x,\lambda,v)\in\mathbb{X}\times\mathbb{Y}\times\mathbb{X}$.
Notice that $h'(\overline{x},\overline{\lambda},\overline{v})\!:
\mathbb{X}\times\mathbb{Y}\times\mathbb{X}\to\mathbb{Y}$ is surjective.
By applying \cite[Theorem 1.66]{Mordu06}, we obtain
\begin{equation*}
D^*(\Pi_K\circ h)(\overline{x},\overline{\lambda},\overline{v})
=\left(\begin{matrix}
\nabla\!g(\overline{x})\\
I\\ 0
\end{matrix}\right)D^*\Pi_K(g(\overline{x})\!+\!\overline{\lambda}).
\end{equation*}
In addition, it is easy to check that $(\Delta x,\Delta\lambda,\Delta v)\in
D^*(-\Pi_K\circ h)(\overline{x},\overline{\lambda},\overline{v})(\Delta \eta)$
if and only if $(\Delta x,\Delta\lambda,\Delta v)\in
D^*(\Pi_K\circ h)(\overline{x},\overline{\lambda},\overline{v})(-\Delta \eta)$.
Together with the last three equations,
\[
D^*\Phi(\overline{x},\overline{\lambda},\overline{v})(\Delta \xi,\Delta\eta)
\!=\!\left[\begin{matrix}
\nabla^2\langle\overline{\lambda},g\rangle(\overline{x})\Delta\xi
\!+\!\nabla\!g(\overline{x})\Delta\eta\\
g'(\overline{x})\Delta\xi\\
-\Delta\xi
\end{matrix}\right]
+\!\left(\begin{matrix}
\nabla\!g(\overline{x})\\
I\\ 0
\end{matrix}\right)\!D^*\Pi_K(g(\overline{x})\!+\!\overline{\lambda})(-\Delta\eta).
\]
So, $(\Delta x,\Delta\lambda,\Delta v,\Delta \xi,\Delta\eta)
\in\!\mathcal{N}_{{\rm gph}\Phi}(\overline{x},\overline{\lambda},\overline{v},0,0)$
iff $\exists\Delta\zeta\in D^*\Pi_{K}(g(\overline{x})\!+\!\overline{\lambda})(\Delta\eta)$ such that
\begin{equation*}\label{temp-system}
\left\{\begin{array}{ll}
\Delta x+\nabla^2\langle\overline{\lambda},g\rangle(\overline{x})\Delta\xi+\nabla\!g(\overline{x})\Delta\eta
=\nabla g(\overline{x})\Delta\zeta,\\
\Delta\lambda+g'(\overline{x})\Delta\xi=\Delta\zeta,\,\Delta v=\Delta\xi.
\end{array}\right.
\end{equation*}
Consequently, $(\Delta \xi,\Delta\eta)\in D^*\Phi^{-1}((0,0)|(\overline{x},\overline{\lambda},\overline{v}))
(0,0,0)$ if and only if $(\Delta \xi,\Delta\eta)$ satisfies
\begin{equation*}
\left\{\begin{array}{ll}
\Delta\xi=0,\,\nabla\!g(\overline{x})\Delta\eta=0,\\
0\in D^*\Pi_{K}(g(\overline{x})\!+\!\overline{\lambda})(\Delta\eta).
\end{array}\right.
\end{equation*}
By Lemma \ref{chara-Aubin}, $\Phi^{-1}$ has the Aubin property at the origin
for $(\overline{x},\overline{\lambda},\overline{v})$ if and only if
\begin{equation}\label{temp-imply}
\left\{\begin{array}{ll}
\nabla\!g(\overline{x})\Delta\eta=0,\\
0\in D^*\Pi_{K}(g(\overline{x})\!+\!\overline{\lambda})(\Delta\eta)
\end{array}\right.
\Longrightarrow \Delta\eta=0.
\end{equation}
From \cite[Exercise 6.7]{RW98} and the definition of coderivative,
for any $(u',v')\in\mathbb{Y}\times\mathbb{Y}$,
\begin{equation}\label{Proj-normal}
u'\in D^*\mathcal{N}_{K}(g(\overline{x})|\overline{\lambda})(v')
\Longleftrightarrow
-v'\in D^*\Pi_{K}(g(\overline{x})\!+\!\overline{\lambda})(-u'\!-v').
\end{equation}
This show that the implication in \eqref{temp-imply} can be equivalently written as
the one in \eqref{coderiv-imply}.
Now we pay our attention to the second part.
Let $\overline{x}$ be a nondegenerate point of $g$ w.r.t $K$ and $\Xi$.
From \cite[Definition 4.70]{BS00},
\(
g'(\overline{x})\mathbb{X}+{\rm Ker}\big[\Xi'(g(\overline{x}))\big]=\mathbb{Y},
\)
or equivalently
\begin{equation}\label{Nondegenerate}
{\rm Ker}(\nabla g(\overline{x}))\cap{\rm Range}(\nabla\Xi(g(\overline{x})))=\{0\}.
\end{equation}
Fix an arbitrary $\Delta u\in{\rm Ker}(\nabla g(\overline{x}))
\cap D^*\mathcal{N}_{K}(g(\overline{x})|\overline{\lambda})(0)$.
Since $\overline{\lambda}\in\mathcal{N}_K(g(\overline{x}))$,
by the reducibility assumption for $K$ and Lemma \ref{lemma-reduction},
there exists a unique $\overline{\mu}\in\mathcal{N}_{D}(\Xi(g(\overline{x})))$ such that
\(
\overline{\lambda}=\nabla\Xi(g(\overline{x}))\overline{\mu}.
\)
In addition, from $\Delta u\in D^*\mathcal{N}_{K}(g(\overline{x})|\overline{\lambda})(0)$
and \cite[Theorem 3.4]{Mordu01} with $\psi=\delta_D(\cdot)$ and $h(\cdot)=\Xi(\cdot)$,
there exists $\Delta\mu\in D^*\mathcal{N}_{D}(\Xi(g(\overline{x}))|\overline{\lambda})(0)$
such that
\[
\Delta u=\nabla\Xi(g(\overline{x}))\Delta\mu.
\]
Along with $\nabla\!g(\overline{x})\Delta u=0$, we get
$\nabla\!g(\overline{x})\nabla\Xi(g(\overline{x}))\Delta\mu=0$,
which is equivalent to saying
\[
\nabla\Xi(g(\overline{x}))\Delta\mu\in{\rm Ker}(\nabla g(\overline{x}))
\cap{\rm Range}(\nabla\Xi(g(\overline{x}))).
\]
From equation \eqref{Nondegenerate}, it follows that $\nabla\Xi(g(\overline{x}))\Delta\mu=0$.
By the surjectivity of $\Xi'(g(\overline{x}))$, we get $\Delta\mu=0$.
Consequently, $\Delta u=0$, and condition \eqref{coderiv-imply} is satisfied.
Conversely, assume that $\Phi^{-1}$ has the Aubin property at
$(\overline{x},\overline{\lambda},\overline{v})$ for the origin.
Notice that $\Phi^{-1}$ is exactly
\[
\Sigma(a,b):=\Big\{(x,\lambda,v)\in\mathbb{X}\times\mathbb{Y}\times\mathbb{X}\ |\ \Phi(x,\lambda,v)=(a,b)\Big\}.
\]
By following the same arguments as those for \cite[Theorem 1]{KK13},
$\overline{x}$ is nondegenerate.
\end{proof}
Motivated by the recent work \cite{Gfrerer11,Gfrerer13,Gfrerer16}
for the metric subregularity, we next provide a condition
for the metric subregularity of $\Phi$ by means of the directional
limiting coderivative of $\mathcal{N}_K$. In order to achieve this goal,
we need the following lemma.
\begin{lemma}\label{WtKKT}
Let $\widetilde{\Phi}\!:\mathbb{X}\times\mathbb{Y}\times\mathbb{X}\rightrightarrows
\mathbb{X}\times\mathbb{Y}\times\mathbb{X}$ be the multifunction defined as follows:
\begin{equation}
\widetilde{\Phi}(x,\lambda,v)
:=\left(\begin{matrix}
-v+\nabla g(x)\lambda\\
g(x)\\
\lambda\\
\end{matrix}\right)
-\left(\begin{matrix}
\{0\}\\
{\rm gph}\mathcal{N}_K
\end{matrix}\right).
\end{equation}
Consider an arbitrary $(\overline{x},\overline{\lambda},\overline{v})\in\Phi^{-1}(0,0)$.
Then, $\widetilde{\Phi}$ is metrically subregular at $(\overline{x},\overline{\lambda},\overline{v})$
for the origin if and only if $\Phi$ is metrically subregular at
$(\overline{x},\overline{\lambda},\overline{v})$ for the origin.
\end{lemma}
\begin{proof}
Suppose that the mapping $\Phi$ is metrically subregular
at $(\overline{x},\overline{\lambda},\overline{v})$ for the origin.
Then, there exist $\varepsilon>0$ and $\kappa>0$ such that for all $(x,\lambda,v)\in\mathbb{B}((\overline{x},\overline{\lambda},\overline{v}),\varepsilon)$,
\[
{\rm dist}((x,\lambda,v),\Phi^{-1}(0,0))\le\kappa\|\Phi(x,\lambda,v)\|.
\]
To establish the metric subregularity of $\widetilde{\Phi}$ at
$(\overline{x},\overline{\lambda},\overline{v})$ for the origin, it suffices to
argue that there exist $\varepsilon'>0,\delta'>0$ and $\kappa'>0$ such that
for all $(x,\lambda,v)\in\mathbb{B}((\overline{x},\overline{\lambda},\overline{v}),\varepsilon')$,
\begin{equation}\label{aim-ineq}
{\rm dist}((x,\lambda,v),\widetilde{\Phi}^{-1}(0,0,0))
\le\kappa'{\rm dist}((0,0,0),\widetilde{\Phi}(x,\lambda,v)\cap\mathbb{B}((0,0,0),\delta')).
\end{equation}
Set $\varepsilon'=\frac{\varepsilon}{2}$ and $\delta'=\frac{\varepsilon}{2}$.
Fix an arbitrary $(x,\lambda,v)\in\mathbb{B}((\overline{x},\overline{\lambda},\overline{v}),\varepsilon')$.
It suffices to consider $\widetilde{\Phi}(x,\lambda,v)\cap\mathbb{B}((0,0,0),\delta')\ne\emptyset$.
Let $(\xi,\eta,\zeta)\in\widetilde{\Phi}(x,\lambda,v)\cap\mathbb{B}((0,0,0),\delta')$ be such that
\begin{equation}\label{aim-equa31}
{\rm dist}((0,0,0),\widetilde{\Phi}(x,\lambda,v)\cap\mathbb{B}((0,0,0),\delta'))
=\|(\xi,\eta,\zeta)\|.
\end{equation}
From $(\xi,\eta,\zeta)\in\widetilde{\Phi}(x,\lambda,v)\cap\mathbb{B}((0,0,0),\delta')$,
it follows that $(\xi',\eta)=\Phi(x,\lambda',v)$ with $\xi'=\xi-\nabla g(x)(\eta+\zeta)$
and $\lambda'=\lambda-\eta-\zeta$, and moreover,
$\|(x,\lambda',v)-(\overline{x},\overline{\lambda},\overline{v})\|\le\varepsilon$.
By the continuity of $\nabla g$, there exists $\gamma>0$ such that
for all $x\in\mathbb{B}(\overline{x},\varepsilon')$, $\|\nabla g(x)\|\le\gamma$.
Then,
\begin{align*}
&{\rm dist}((x,\lambda,v),\widetilde{\Phi}^{-1}(0,0,0))
={\rm dist}((x,\lambda,v),\Phi^{-1}(0,0))\\
&\le{\rm dist}((x,\lambda',v),\Phi^{-1}(0,0))+\|\lambda-\lambda'\|\\
&\le\kappa{\rm dist}((0,0),\Phi(x,\lambda',v))+\|\lambda-\lambda'\|\\
&\le\kappa\|(\xi',\eta)\|+\|\eta+\zeta\|\le\kappa\sqrt{4\gamma^2+3}\|(\xi,\eta,\zeta)\|.
\end{align*}
Together with \eqref{aim-equa31} and \eqref{aim-ineq},
$\widetilde{\Phi}$ is metrically subregular
at $(\overline{x},\overline{\lambda},\overline{v})$ for the origin.
Suppose that $\widetilde{\Phi}$ is metrically subregular at $(\overline{x},\overline{\lambda},\overline{v})$
for the origin. Then there exist $\varepsilon>0$ and $\kappa>0$
such that for all $(x,\lambda,v)\in\mathbb{B}((\overline{x},\overline{\lambda},\overline{v}),\varepsilon)$,
\[
{\rm dist}((x,\lambda,v),\widetilde{\Phi}^{-1}(0,0,0))
\le\kappa{\rm dist}((0,0,0),\widetilde{\Phi}(x,\lambda,v)).
\]
Fix an arbitrary $(x,\lambda,v)\in\mathbb{B}((\overline{x},\overline{\lambda},\overline{v}),\varepsilon)$.
Write $(\xi,\eta)=\Phi(x,\lambda,v)$. By the expression of $\Phi$,
it is immediate to have that $(\xi,\eta,-\eta)\in\widetilde{\Phi}(x,\lambda,v)$.
From the last inequality,
\begin{align*}
&{\rm dist}((x,\lambda,v),\Phi^{-1}(0,0))
={\rm dist}((x,\lambda,v),\widetilde{\Phi}^{-1}(0,0,0))\\
&\le\kappa{\rm dist}((0,0,0),\widetilde{\Phi}(x,\lambda,v))\le\|(\xi,\eta,-\eta)\|
\le\sqrt{2}\kappa\|\Phi(x,\lambda,v)\|
\end{align*}
This shows that $\Phi$ is metrically subregular at $(\overline{x},\overline{\lambda},\overline{v})$
for the origin.
\end{proof}
Now applying \cite[Corollary 1]{Gfrerer16-MP} to the multifunction $\widetilde{\Phi}$,
we have the following result.
\begin{proposition}\label{weak-cond}
Consider an arbitrary $(\overline{x},\overline{v})\!\in{\rm gph}\widehat{\mathcal{N}}_{\Gamma}$
with $\mathcal{M}_{\overline{x}}(\overline{v})\!\ne\emptyset$.
Let $\overline{\lambda}\in\!\mathcal{M}_{\overline{x}}(\overline{v})$.
The $\Phi$ is metrically subregular at $(\overline{x},\overline{\lambda},\overline{v})$
for the origin, if for every $0\ne(\xi,\eta,\zeta)$ with
\begin{equation}\label{graph-deriv-equa}
\nabla^2\langle\overline{\lambda},g\rangle(\overline{x})\xi+\!\nabla g(\overline{x})\eta+\zeta=0,
\end{equation}
the following implication holds:
\begin{align}\label{equa-calm-SKKT}
\left.\begin{matrix}
\nabla g(\overline{x})\Delta\lambda=0,\\
(\Delta\lambda,0)\in\mathcal{N}_{{\rm gph}\mathcal{N}_K}
\big((g(\overline{x}),\overline{\lambda});(g'(\overline{x})\xi,\eta)\big)
\end{matrix}\right\}
\Longrightarrow\Delta\lambda=0.
\end{align}
\end{proposition}
Since $\mathcal{N}_{{\rm gph}\mathcal{N}_K}((g(\overline{x}),\overline{\lambda});(g'(\overline{x})\xi,\eta))
\subseteq \mathcal{N}_{{\rm gph}\mathcal{N}_K}(g(\overline{x}),\overline{\lambda})$
for any $(\xi,\eta)\in\mathbb{X}\times\mathbb{Y}$, the implication in \eqref{equa-calm-SKKT}
holds under the condition \eqref{coderiv-imply}, but now we can not find an example
to illustrate that the assumption in Proposition \ref{weak-cond} is really weaker
than the condition \eqref{coderiv-imply}. We leave this for a future
research topic.
To close this part, we take $K=\mathbb{R}_{-}$ for example to illustrate
that there is no direct relation between the metric subregularity of $\Phi$
and the calmness of $\mathcal{M}_{\overline{x}}$. Now, $\mathcal{M}_{\overline{x}}$
is a polyhedral multifunction whether $g$ is polyhedral or not,
and hence it is calm at each $\overline{v}$ for each
$\overline{\lambda}\in\mathcal{M}_{\overline{x}}(\overline{v})$
by \cite{Robinson81}. However, the metric subregularity of $\Phi$ depends
on the mapping $g$. When $g$ is a linear function, clearly,
$\Phi$ is metrically subregular at $(\overline{x},\overline{\lambda},\overline{v})$
for the origin, but when $g$ is nonlinear, $\Phi$ does not necessarily have
the metric subregularity at $(\overline{x},\overline{\lambda},\overline{v})$;
for example, when $g(x)=x^2$, the mapping $\Phi$
corresponding to the system $g(x)\in\mathbb{R}_{-}$ is not metrically
subregular at $(\overline{x},\overline{\lambda},\overline{v})=(0,1/2,0)$.
Indeed, by noting that
\begin{align*}
\Phi^{-1}(0,0)&=\Big\{(x,\lambda,v)\ |\ v=\nabla g(x)\lambda,\,g(x)={\rm min}(0,g(x)+\lambda)\Big\}\\
&=\Big\{(x,0,0)\ |\ g(x)<0\big\}\cup\big\{(x,\lambda, v)\ |\ g(x)=0,\lambda\geq 0, v=\nabla g(x)\lambda\Big\}.
\end{align*}
Therefore, for any $(x,\lambda,v)\in\mathbb{R}\times\mathbb{R}\times\mathbb{R}$,
${\rm dist}((x,\lambda, v),\Phi^{-1}(0,0))= \|(x,v)\|$.
Take a sequence $(x^k,\lambda^k,v^k)=(1/k,1/2,1/k)$. It is immediate to
calculate that
\[
\lim_{k\to\infty}\frac{\|\Phi(x^k,\lambda^k, v^k)\|}{{\rm dist}((x^k,\lambda^k, v^k),\Phi^{-1}(0,0))}
=\lim_{k\to\infty}\frac{\|(-v^k+2x^k\lambda^k, (x^k)^2)\|}{\|(x^k,v^k)\|}=0.
\]
This shows that $\Phi$ is not metrically subregular at
$(\overline{x},\overline{\lambda},\overline{v})$ for the origin.
\section{Application of graphical derivative of $\mathcal{N}_{\Gamma}$}\label{sec4}
As an application of Theorem \ref{festimate}, we provide an exact characterization
for the graphical derivative of the solution mapping $\mathcal{S}$ in \eqref{MSmap}
and its isolated calmness.
\subsection{Isolated calmness of the solution mapping $\mathcal{S}$}\label{subsec4.1}
Firstly, we establish the relation between the graphical derivative of $\mathcal{S}$
and that of the normal cone mapping $\mathcal{N}_\Gamma$.
To this end, we define a map $\Psi\!:\mathbb{P}\times \mathbb{X}\rightrightarrows\mathbb{X}\times \mathbb{X}$ by
\begin{equation}\label{Psi-map}
\Psi(p,x):=\widetilde{F}(p,x)-{\rm gph}\mathcal{N}_\Gamma
\ \ {\rm with}\ \widetilde{F}(p,x):=(x,-F(p,x)).
\end{equation}
Notice that ${\rm gph}\mathcal{S}=\widetilde{F}^{-1}({\rm gph}\mathcal{N}_\Gamma)$.
By using Lemma \ref{Tcone-lemma}, we have the following result.
\begin{lemma}\label{DMSmap}
Consider an arbitrary $(\overline{p},\overline{x})\in{\rm gph}\mathcal{S}$.
Then, the following inclusion holds
\[
\mathcal{T}_{{\rm gph}\mathcal{S}}(\overline{p},\overline{x})
\!\subseteq\!\Big\{(\Delta p,\Delta x)\in\mathbb{P}\times\mathbb{X}\ |\
\big(\Delta x,-F'((\overline{p},\overline{x});(\Delta p,\Delta x))\big)
\in\mathcal{T}_{{\rm gph}\mathcal{N}_\Gamma}(\overline{x},-\!F(\overline{p},\overline{x}))\Big\}.
\]
If $\Psi$ is metrically subregular at $(\overline{p},\overline{x})$
for the origin, the converse conclusion also holds.
\end{lemma}
Combining Proposition \ref{uestimate} and Theorem \ref{festimate}
with Lemma \ref{DMSmap} and Lemma \ref{chara-icalm},
we have the following conclusion for the isolated calmness of the solution mapping $\mathcal{S}$.
\begin{theorem}\label{Gderiv-MSmap}
Consider an arbitrary $(\overline{p},\overline{x})\in{\rm gph}\mathcal{S}$
and write $\overline{v}\!=-F(\overline{p},\overline{x})$. Suppose $\mathcal{G}$
is metrically subregular at $\overline{x}$ for $0$. If $\mathcal{M}_{\overline{x}}$
is isolated calm at $\overline{v}$ for some $\overline{\lambda}\in\mathcal{M}_{\overline{x}}(\overline{v})$,
then
\begin{equation}\label{gphS-equa}
\!\mathcal{T}_{{\rm gph}\mathcal{S}}(\overline{p},\overline{x})
\subseteq\!\left\{(\Delta p,\Delta x)\ \Big|
\left.\begin{array}{ll}
F'((\overline{p},\overline{x});(\Delta p,\Delta x))
+\!\nabla^2\langle \overline{\lambda},g\rangle(\overline{x})\Delta x+\!\nabla g(\overline{x})\mu=0\!\\
(g'(\overline{x})\Delta x,\mu)\!\in\!\mathcal{T}_{{\rm gph}\mathcal{N}_{K}}(g(\overline{x}),\overline{\lambda})
\!\end{array}\right.\!\right\},
\end{equation}
and consequently $\mathcal{S}$ is isolated calm at $\overline{p}$ for
$\overline{x}$ if the following implication holds:
\begin{equation}\label{Cond-MSmap}
-\!F'((\overline{p},\overline{x});(0,\Delta x))
-\!\nabla^2\langle \overline{\lambda},g\rangle(\overline{x})\Delta x
\in\nabla g(\overline{x})D\mathcal{N}_{K}(g(\overline{x})|\overline{\lambda})(g'(\overline{x})\Delta x)
\Longrightarrow \Delta x=0.
\end{equation}
If, in addition, $\Phi$ is metrically subregular at $(\overline{x},\overline{\lambda},\overline{v})$
for the origin and $\Psi$ defined in \eqref{Psi-map} is metrically subregular
at $(\overline{p},\overline{x})$ for the origin, then the converse inclusion
in \eqref{gphS-equa} holds and the implication \eqref{Cond-MSmap}
is necessary for the isolated calmness of $\mathcal{S}$ at $\overline{p}$ for $\overline{x}$.
\end{theorem}
\begin{remark}
When $F$ is continuously differentiable and
$F'_{p}(\overline{p},\overline{x})\!:\mathbb{P}\to \mathbb{X}$ is surjective,
clearly, $\Psi$ is metrically subregular at $(\overline{p},\overline{x})$ for the origin;
when $\overline{x}$ is a nondegenerate point of the mapping $g$ w.r.t. $K$
and the mapping $\Xi$, from Proposition \ref{property-MG} it follows that
$\mathcal{M}_{\overline{x}}$ is isolated calm at $\overline{v}$ for some $\overline{\lambda}\in\mathcal{M}_{\overline{x}}(\overline{v})$ and
$\Phi$ is metrically subregular at $(\overline{x},\overline{\lambda},\overline{v})$
for the origin. Thus, Theorem \ref{Gderiv-MSmap} improves the result of
\cite[Theorem 6.3]{Mordu15}. In particular, the isolated calmness of
$\mathcal{S}$ at $\overline{p}$ for $\overline{x}$ does not require
the metric subregularity of $\Phi$.
\end{remark}
Next we illustrate an application of Theorem \ref{Gderiv-MSmap} to
the characterization for the isolated calmness of the KKT solution mapping
of the canonically perturbed conic program
\begin{equation}\label{conic-prob}
\min_{z\in\mathbb{Z}}\big\{f(z)-\langle a,z\rangle\!: G(z)-b\in\mathcal{K}^{\circ}\big\}.
\end{equation}
where $p=(a,b)\in\mathbb{Z}\times\mathbb{Y}$ is the perturbation parameter,
$f\!:\mathbb{Z}\to\mathbb{R}$ and $G\!:\mathbb{Z}\to\mathbb{Y}$ are
twice continuously differentiable, and $\mathcal{K}\subseteq\mathbb{Y}$
is a $C^2$-cone reducible closed convex cone.
\begin{example}\label{example4.2}
Let $f\!:\mathbb{Z}\to\mathbb{R}$ and $G\!:\mathbb{Z}\to\mathbb{Y}$ be
twice continuously differentiable functions. Write $p:=(a,b)\in\mathbb{Z}\times\mathbb{Y}$
and $x:=(z,\lambda)\in\mathbb{X}:=\mathbb{Z}\times\mathbb{Y}$.
Consider the multifunction
\[
\mathcal{S}(p)=\Big\{x\in\mathbb{X}\ |\ 0\in F(p,x)+\widehat{\mathcal{N}}_{K}(x)\Big\}
\ {\rm with}\
F(p,x)=\!\left[\begin{matrix}
\nabla\!f(z)-a +\nabla G(z)\lambda\\
-G(z)+b
\end{matrix}\right]
\]
where $K=\mathbb{Z}\times\mathcal{K}^{\circ}$.
The $\mathcal{S}$ is exactly the KKT solution mapping associated to \eqref{conic-prob}.
Let $\overline{p}=(0,0)$ and $\overline{x}=(\overline{z},\overline{\lambda})$
be such that $\overline{v}=(0,G(\overline{z}))$. It is clear that
$\mathcal{G}(x)=x-K$ is metrically subregular at
$\overline{x}$ for the origin. Since
\(
\mathcal{M}_{\overline{x}}(v)
=\big\{\mu\in\mathcal{N}_{\mathbb{X}}(\overline{z})\times\mathcal{N}_{\mathcal{K}^{\circ}}(\overline{\lambda})
\ |\ v=\mu\big\},
\)
by Proposition \ref{prop-Mx} it is not hard to check that
$\mathcal{M}_{\overline{x}}$ is isolated calm at $\overline{v}$ for $\overline{v}$.
Now
\[
\Phi(x,\mu,v)=\left(\begin{matrix}
-v+\mu\\ x-\Pi_{\mathbb{Z}\times\mathcal{K}^{\circ}}(x+\mu)
\end{matrix}\right)\ \ {\rm and}\ \
\widetilde{\Phi}(x,\mu,v)
:=\left(\begin{matrix}
-v+\mu\\ x\\ \mu\\
\end{matrix}\right)
-\left(\begin{matrix}
\{0\}\\
{\rm gph}\mathcal{N}_{K}
\end{matrix}\right).
\]
Clearly, $\widetilde{\Phi}$ is metric subregular at $(\overline{x},\overline{v},\overline{v})$
for the origin, and so is $\Phi$ at $(\overline{x},\overline{v},\overline{v})$
for the origin by Lemma \ref{WtKKT}. In addition, since
\(
\widetilde{F}(p,x):=(z,\lambda,a-\!\nabla f(z)-\!\nabla G(z)\lambda, b-\!G(z))
\)
and $\widetilde{F}'(\overline{p},\overline{x})\!:\mathbb{X}
\times\mathbb{X}\to\mathbb{X}\times\mathbb{X}$ is nonsingular,
the corresponding $\Psi$ is metrically subregular at $(\overline{p},\overline{x})$
for the origin. By Theorem \ref{Gderiv-MSmap}, $\mathcal{S}$ is isolated calm
at $\overline{p}$ for $\overline{x}$ if and only if
\[
\left\{\begin{array}{ll}
\nabla^2L(\overline{z},\overline{\lambda})\Delta z+\nabla G(\overline{z})\Delta\lambda=0,\\
(G'(\overline{z})\Delta z,\Delta\lambda)\in\mathcal{T}_{{\rm gph}\mathcal{N}_{\mathcal{K}}}(G(\overline{z}),\overline{\lambda})
\end{array}\right.\Longrightarrow \Delta z=0,\,\Delta\lambda=0.
\]
This coincides with the result in \cite[Lemma 18-19]{DingSZ17} for the perturbed
problem \eqref{conic-prob}.
\end{example}
Next we use a specific example of generalized equations to illustrate Theorem \ref{Gderiv-MSmap}.
\begin{example}\label{example4.1}
Consider the generalized equation \eqref{GE} with $\Gamma$ given by Example \ref{example1}
and $F(p,x,t)=-p-(x,t)$ for $p\in\mathbb{R}^3$ and $(x,t)\in\mathbb{R}^2\times\mathbb{R}$.
Let $\overline{p}=(0,0,0)^{\mathbb{T}}$, $(\overline{x},\overline{t})=((-1,-1)^{\mathbb{T}},0)$
and $(\overline{\lambda},\overline{\tau})=(0_{2\times 2},-1)$.
Since $g(\overline{x},\overline{t})=(0_{2\times 2},0)$, it is immediate to have
\[
\mathcal{N}_{K}(g(\overline{x},\overline{t}))=\mathbb{S}_{-}^2\times\mathbb{R}_{-}
\ \ {\rm and}\ \ \mathcal{T}_{K}(g(\overline{x},\overline{t}))=\mathbb{S}_{+}^2\times\mathbb{R}_{+}.
\]
By \eqref{grad-gfun}, it is easy to verify that
${\rm Ker}(\nabla g(\overline{x},\overline{t}))\cap \mathcal{T}_{\mathcal{N}_{K}(g(\overline{x},\overline{t}))}(\overline{\lambda},\overline{\tau})
=\{(0_{2\times 2},0)\}$. This shows that the SRCQ for the system $g(x,t)\in K$
holds at $(\overline{x},\overline{t})$ w.r.t. $\overline{\lambda}$.
However, since
\(
{\rm Ker}(\nabla g(\overline{x},\overline{t}))
\cap[{\rm lin}(\mathcal{T}_{K}(g(\overline{x},\overline{t})))]^{\perp}\neq\{0_{2\times 2}\},
\)
it follows that $\overline{x}$ is a degenerate point.
Let $(\Delta x,\Delta t)$ be such that the inclusion on the left hand side of \eqref{Cond-MSmap} holds.
Along with the expression of $F$, there is $(\Delta\lambda,\Delta\tau)\in\mathbb{S}^2\times\mathbb{R}$
such that $(\Delta x,\Delta t)=\nabla g(\overline{x},\overline{t})(\Delta\lambda,\Delta\tau)$
and $(g'(\overline{x},\overline{t})(\Delta x,\Delta t),(\Delta\lambda,\Delta\tau))
\in\mathcal{T}_{{\rm gph}\mathcal{N}_K}((g(\overline{x}),\overline{t}),(\overline{\lambda},\overline{\tau}))$.
By \cite[Proposition 6.41]{RW98},
\[
\mathcal{T}_{{\rm gph}\mathcal{N}_K}((g(\overline{x}),\overline{t}),(\overline{\lambda},\overline{\tau}))
\subseteq\mathcal{T}_{{\rm gph}\mathcal{N}_{\mathbb{S}_{+}^2}}((g_1(\overline{x},\overline{t}),\overline{\lambda})
\times\mathcal{T}_{{\rm gph}\mathcal{N}_{\mathbb{R}_{+}}}(g_2(\overline{x},\overline{t}),\overline{\tau}).
\]
Together with
\(
g'(\overline{x},\overline{t})(\Delta x,\Delta t)
=\left(\begin{matrix}
{\rm Diag}(\Delta x)+\Delta t E\\
\Delta t
\end{matrix}\right),
\)
it immediately follows that
\[
({\rm Diag}(\Delta x)+\Delta t E,\Delta\lambda)\in\mathcal{T}_{{\rm gph}\mathcal{N}_{\mathbb{S}_{+}^2}}((g_1(\overline{x},\overline{t}),\overline{\lambda})
\ {\rm and}\
(\Delta t,\Delta\tau)\in \mathcal{T}_{{\rm gph}\mathcal{N}_{\mathbb{R}_{+}}}((g_2(\overline{x},\overline{t}),\overline{\tau}).
\]
We calculate that $\mathcal{T}_{{\rm gph}\mathcal{N}_{\mathbb{R}_{+}}}(g_2(\overline{x},\overline{t}),\overline{\tau})
=\{0\}\times\mathbb{R}$. Together with the last equation,
we obtain $\Delta t=0$ and $({\rm Diag}(\Delta x),\Delta\lambda)\in\mathcal{T}_{{\rm gph}\mathcal{N}_{\mathbb{S}_{+}^2}}(g_1(\overline{x},\overline{t}),\overline{\lambda})$.
By \cite[Corollary 3.1]{WZZhang14}, the latter implies $\mathbb{S}_{+}^2\ni{\rm Diag}(\Delta x)\,\bot\,\Delta\lambda\in\mathbb{S}_{-}^2$.
In addition, from $(\Delta x,\Delta t)=\nabla g(\overline{x},\overline{t})(\Delta\lambda,\Delta\tau)$
and \eqref{grad-gfun}, we have
\(
\Delta x={\rm diag}(\Delta\lambda).
\)
The two sides imply $\Delta x=0$. This shows that the implication in \eqref{Cond-MSmap} holds.
By Theorem \ref{Gderiv-MSmap}, the mapping $\mathcal{S}$
is isolated calm at $\overline{p}$ for $(\overline{x},\overline{t})$. \end{example}
\subsection{Estimation for (regular) coderivative of $\mathcal{N}_\Gamma$}\label{subsec4.2}
As another application of Proposition \ref{uestimate} and
Theorem \ref{festimate}, we provide a lower estimation for the regular coderivative of
$\mathcal{N}_\Gamma$ and an upper estimation for the coderivative
of $\mathcal{N}_\Gamma$, respectively, without requiring the nondegeneracy
of the reference point.
\begin{proposition}\label{estimate-regNcone}
Consider an arbitrary $(\overline{x},\overline{v})\in{\rm gph}\,\mathcal{N}_\Gamma$.
If $\mathcal{G}$ is metrically subregular at $\overline{x}$ for $0$
and $\mathcal{M}_{\overline{x}}$ is isolated calm at $\overline{v}$
for some $\overline{\lambda}\in\mathcal{M}_{\overline{x}}(\overline{v})$,
then it holds that
\begin{align}\label{RNcone-lower}
\widehat{\mathcal{N}}_{{\rm gph}\mathcal{N}_\Gamma}(\overline{x},\overline{v})
&\supseteq\Big\{(\xi,\eta)\ |\ \xi\in-\nabla^2\langle\overline{\lambda},g\rangle(\overline{x})(\eta)
+\nabla\!g(\overline{x}) \widehat{D}^*\mathcal{N}_{K}(g(\overline{x})|\overline{\lambda})(-g'(\overline{x})\eta)\Big\}.
\end{align}
\end{proposition}
\begin{proof}
Take any point $(\xi,\eta)$ from the set on the right hand side of \eqref{RNcone-lower}.
Then, there exists $\mu\in\widehat{D}^*\mathcal{N}_{\mathcal{K}}(g(\overline{x})|\overline{\lambda})(-g'(\overline{x})\eta)$
such that $\xi=-\nabla^2\langle\overline{\lambda},g\rangle(\overline{x})(\eta)+\nabla\!g(\overline{x})\mu$.
To establish the inclusion \eqref{RNcone-lower}, it suffices to argue that
$(\xi,\eta)\in[\mathcal{T}_{{\rm gph}\mathcal{N}_\Gamma}(\overline{x},\overline{v})]^{\circ}
=\widehat{\mathcal{N}}_{{\rm gph}\mathcal{N}_\Gamma}(\overline{x},\overline{v})$.
Let $(d,w)$ be an arbitrary point from $\mathcal{T}_{{\rm gph}\mathcal{N}_\Gamma}(\overline{x},\overline{v})$.
By Theorem \ref{uestimate}, there is $u\in\mathbb{Y}$ such that
\[
(g'(\overline{x})d,u)\in\mathcal{T}_{{\rm gph}\mathcal{N}_K}(g(\overline{x}),\overline{\lambda})
\ \ {\rm and}\ \
w=\nabla^2\langle\overline{\lambda},g\rangle(\overline{x})(d)+\nabla\!g(\overline{x})u.
\]
Together with $\xi=-\nabla^2\langle\overline{\lambda},g\rangle(\overline{x})(\eta)+\nabla\!g(\overline{x})\mu$,
it follows that
\begin{align}\label{temp-equa40}
\langle (\xi,\eta),(d,w)\rangle
&=\langle d,-\nabla^2\langle\overline{\lambda},g\rangle(\overline{x})(\eta)+\nabla\!g(\overline{x})\mu\rangle
+\langle \nabla^2\langle\overline{\lambda},g\rangle(\overline{x})(d)+\nabla\!g(\overline{x})u,\eta\rangle\nonumber\\
&=\langle d,\nabla\!g(\overline{x})\mu\rangle+\langle\nabla\!g(\overline{x})u,\eta\rangle.
\end{align}
Notice that $\mu\in\widehat{D}^*\mathcal{N}_{K}(g(\overline{x})|\lambda)(-g'(\overline{x})\eta)$.
Hence, $(\mu,g'(\overline{x})\eta)\in\widehat{\mathcal{N}}_{{\rm gph}\mathcal{N}_K}(g(\overline{x}),\lambda)$.
Since $(g'(\overline{x})d,u)\in\mathcal{T}_{{\rm gph}\mathcal{N}_K}(g(\overline{x}),\lambda)$
and $\widehat{\mathcal{N}}_{{\rm gph}\mathcal{N}_{K}}(g(\overline{x}),\lambda)
=[\mathcal{T}_{{\rm gph}\mathcal{N}_{K}}(g(\overline{x}),\lambda)]^{\circ}$,
it holds that
\[
\langle(\mu,g'(\overline{x})\eta),(g'(\overline{x})d,u)\rangle
= \langle d,\nabla\!g(\overline{x})\mu\rangle+\langle\nabla\!g(\overline{x})u,\eta\rangle
\le 0.
\]
Together with \eqref{temp-equa40}, $\langle (\xi,\eta), (d,w)\rangle\leq 0$.
Thus, we obtain $(\xi,\eta)\in[\mathcal{T}_{{\rm gph}\mathcal{N}_\Gamma}(\overline{x},\overline{v})]^{\circ}$.
\end{proof}
\begin{proposition}\label{estimate-Ncone}
Consider an arbitrary $(\overline{x},\overline{v})\in{\rm gph}\,\mathcal{N}_\Gamma$.
If $\mathcal{M}_{\overline{x}}$ is isolated calm at $\overline{v}$ for
$\overline{\lambda}\in\mathcal{M}_{\overline{x}}(\overline{v})$ and
$\Phi$ is metrically subregular at $(\overline{x},\overline{\lambda},\overline{v})$
for the origin, then it holds that
\begin{equation}\label{Ncone-upper}
\mathcal{N}_{{\rm gph}\mathcal{N}_\Gamma}(\overline{x},\overline{v})
\subseteq\Big\{(\xi,\eta)\ |\ \xi\in-\nabla^2\langle\overline{\lambda},g\rangle(\overline{x})(\eta)
+\nabla\!g(\overline{x}) D^*\mathcal{N}_{K}(g(\overline{x})|\overline{\lambda})(-g'(\overline{x})\eta)\Big\}.
\end{equation}
\end{proposition}
\begin{proof}
Let $\mathcal{A}$ be the linear mapping appearing in the proof of Lemma \ref{lestimate},
and let $\mathcal{U}$ be an arbitrary neighborhood of $(\overline{x},\overline{v})$.
We first argue that $\mathcal{A}^{-1}(\mathcal{U})\cap\Phi^{-1}(0,0)$ is bounded.
If not, by the definition of $\mathcal{A}$, there exist $\{(x^k,v^k)\}$ converging to
$(\overline{x},\overline{v})$ and an unbounded $\{\mu^k\}$ such that for each $k$, $(x^k,\mu^k,v^k)\in\mathcal{A}^{-1}(\mathcal{U})\cap\Phi^{-1}(0,0)$, that is,
\[
v^k=\nabla g(x^k)\mu^k\ \ {\rm and}\ \ \mu^k\in\mathcal{N}_K(g(x^k))
\quad\ \forall k.
\]
Write $\widetilde{\mu}^k=\frac{\mu^k}{\|\mu^k\|}$ and $\widetilde{v}^k=\frac{v^k}{\|\mu^k\|}$.
We assume (if necessary taking a subsequence) that $\widetilde{\mu}^k\to\widetilde{\mu}$
with $\|\widetilde{\mu}\|=1$. Notice that $\widetilde{v}^k=\nabla g(x^k)\widetilde{\mu}^k$ and $\widetilde{\mu}^k\in\mathcal{N}_K(g(x^k))$. From the outer semicontinuity of $\mathcal{N}_K$,
$\widetilde{\mu}\in\mathcal{N}_K(g(\overline{x}))$ and $\nabla g(\overline{x})\widetilde{\mu}=0$.
This, by the isolated calmness of $\mathcal{M}_{\overline{x}}$ at $\overline{v}$,
implies that $\widetilde{\mu}=0$, a contradiction to $\|\widetilde{\mu}\|=1$.
Now, by \cite[Theorem 6.43]{RW98}, from ${\rm gph}\mathcal{N}_\Gamma\cap(\mathcal{V}\times\mathbb{X})
=\mathcal{A}(\Phi^{-1}(0,0))\cap(\mathcal{V}\times\mathbb{X})$ for a neighborhood $\mathcal{V}$
of $\overline{x}$ we have
\begin{align}\label{temp-Ncone-Nomega}
\mathcal{N}_{{\rm gph}\mathcal{N}_\Gamma}(\overline{x},\overline{v})
&\subseteq\bigcup_{\overline{z}\in\mathcal{A}^{-1}(\overline{x},\overline{v})\cap\Phi^{-1}(0,0)}
\Big\{(\xi,\eta)\in\mathbb{X}\times\mathbb{Y}\ |\ \mathcal{A}^*(\xi,\eta)\in\mathcal{N}_{\Phi^{-1}(0,0)}(\overline{z})\Big\}\nonumber\\
&=\Big\{(\xi,\eta)\in\mathbb{X}\times\mathbb{Y}\ |\ \mathcal{A}^*(\xi,\eta)\in\mathcal{N}_{\Phi^{-1}(0,0)}(\overline{x},\overline{\lambda},\overline{v})\Big\},
\end{align}
where the equality is by the definition of $\mathcal{A}$
and $\mathcal{M}_{\overline{x}}(\overline{v})=\{\overline{\lambda}\}$.
Since $\Phi$ is metrically subregular at $(\overline{x},\overline{\lambda},\overline{v})$
for the origin, applying the first part of Lemma \ref{normal-cone-lemma} yields
\[
\mathcal{N}_{\Phi^{-1}(0,0)}(\overline{x},\overline{\lambda},\overline{v})
\subseteq \bigcup_{(d,w)\in\mathbb{X}\times\mathbb{Y}}D^*\Phi(\overline{x},\overline{\lambda},\overline{v})(d,w).
\]
For any given $(d,w)\in\mathbb{X}\times\mathbb{Y}$,
from the proof of Proposition \ref{property-MG}(a) it follows that
\[
D^*\Phi(\overline{x},\overline{\lambda},\overline{v})(d,w)
\!=\!\left[\begin{matrix}
\nabla^2\langle\overline{\lambda},g\rangle(\overline{x})d
\!+\!\nabla\!g(\overline{x})w\\
g'(\overline{x})d\\
-d
\end{matrix}\right]
+\left(\begin{matrix}
\nabla\!g(\overline{x})\\
I\\ 0
\end{matrix}\right)\!D^*\Pi_K(g(\overline{x})\!+\!\overline{\lambda})(-w).
\]
By combining the last two equations with \eqref{Proj-normal}, it is not difficult to obtain that
\begin{align*}
&\mathcal{N}_{\Phi^{-1}(0,0)}(\overline{x},\overline{\lambda},\overline{v})\\
&\subseteq\!\bigcup_{(d,w)\in\mathbb{X}\times\mathbb{Y}}
\left\{\!\left(\begin{matrix}
\nabla^2\langle\overline{\lambda},g\rangle(\overline{x})d+\!\nabla g(\overline{x})u'\\
g'(\overline{x})d+u'-w\\ -d
\end{matrix}\right)\ \bigg|\
\left(\begin{matrix}
u'\\u'\!-w
\end{matrix}\right)\in \mathcal{N}_{{\rm gph}\mathcal{N}_K}(g(\overline{x}),\overline{\lambda})\right\}.
\end{align*}
Together with \eqref{temp-Ncone-Nomega} and $\mathcal{A}^*(\xi,\eta)=(\xi,0,\eta)$,
we obtain the following inclusion
\[
\mathcal{N}_{{\rm gph}\mathcal{N}_\Gamma}(\overline{x},\overline{v})
\subseteq\left\{(\xi,\eta)\in\mathbb{X}\times\mathbb{Y}\ \Big|
\left.\begin{array}{ll}
\xi=\nabla^2\langle\overline{\lambda},g\rangle(\overline{x})(-\eta)+\nabla\!g(\overline{x})z,\\
(z,g'(\overline{x})\eta)\!\in\mathcal{N}_{{\rm gph}\mathcal{N}_{K}}(g(\overline{x}),\overline{\lambda})
\end{array}\right.\!\right\},
\]
which is equivalent to the inclusion in \eqref{Ncone-upper}.
The proof is then completed.
\end{proof}
Exact characterizations for $\widehat{\mathcal{N}}_{{\rm gph}\mathcal{N}_\Gamma}(\overline{x},\overline{v})$
and $\mathcal{N}_{{\rm gph}\mathcal{N}_\Gamma}(\overline{x},\overline{v})$
were given in \cite[Theorem 4.1]{Mordu15} and in \cite[Theorem 7]{Outrata11},
respectively, under the standard reducibility and nondegeneracy assumption,
and they were recently obtained in \cite{Gfrerer17} under a weakened version
of reducibility but still the nondegeneracy assumption. Here, we only provide
a one-sided estimation without the nondegeneracy assumption, and it is not unclear
whether the converse inclusions in \eqref{RNcone-lower}-\eqref{Ncone-upper}
hold or not without nondegeneracy.
\noindent
{\large\bf Acknowledgements}\ \ The authors are deeply grateful for
the two referees' comments, which give them much help to improve
the original manuscript. The authors also would like to thank Professor Mordukhovich,
from Wayne State University, for his helpful suggestions on the revision of this manuscript.
\noindent
{\bf\large Appendix}
\begin{alemma}\label{calculus-rule}
Let $G_1\!:\mathbb{X}\to\mathbb{Y}$ be a single-valued mapping and
$G_2:\mathbb{X}\rightrightarrows\mathbb{Z}$ be an arbitrary set-valued mapping.
Define the set-valued mapping $G\!:\mathbb{X}\rightrightarrows\mathbb{Y}\times\mathbb{Z}$ by
\(
G(x):=\left(\begin{matrix}
G_1(x)\\ G_2(x)
\end{matrix}\right)
\)
for $x\in\mathbb{X}$. Consider a point $(\overline{x},(\overline{y},\overline{z}))\in{\rm gph}\,G$.
If $G_1$ is strictly differentiable at $\overline{x}$, then
\[
D^*G(\overline{x}|(\overline{y},\overline{z}))(\Delta u,\Delta v)
=\nabla\!G_1(\overline{x})\Delta u + D^*G_2(\overline{x}|\overline{z})(\Delta v)
\quad \forall(\Delta u,\Delta v)\in\mathbb{Y}\times\mathbb{Z}
\]
\end{alemma}
\begin{proof}
Let $H(x):=\left(\begin{matrix}
0\\ G_2(x)
\end{matrix}\right)$ for $x\in\mathbb{X}$.
By the definition of coderivative, we have that
\[
D^*H(\overline{x}|(0,\overline{z}))(\Delta u,\Delta v)=D^*G_2(\overline{x}|\overline{z})(\Delta v).
\]
Notice that $G(x)=H(x)+
\left(\begin{matrix}
G_1(x)\\ 0
\end{matrix}\right)$ for $x\in\mathbb{X}$.
By \cite[Theorem 1.62]{Mordu06}, it follows that
\[
D^*G(\overline{x}|(\overline{y},\overline{z}))(\Delta u,\Delta v)
=\nabla\!G_1(\overline{x})\Delta u +D^*H(\overline{x}|(0,\overline{z}))(\Delta u,\Delta v).
\]
The desired result then follows by combining the last two equations.
\end{proof}
\end{document} | arXiv |
1d Fourier Transform Python
e I have a file which contains 1 measurement per line and I'd like to take the FT of these data, i. Using the NFFT¶ In this tutorial, we assume that you are already familiar with the non-uniform discrete Fourier transform and the NFFT library used for fast computation of NDFTs. Fourier analysis is a method for expressing a function as a sum of periodic components, and for recovering the signal from those components. 1998 We start in the continuous world; then we get discrete. pdf), Text File (. Notice that get_xns only calculate the Fourier coefficients up to the Nyquest limit. Explain why you get this result. The one-dimensional Hilbert transform (1D-HT) and associated 1D analytic signal (1D-AS) of an 1D signal are well. This set of Engineering Mathematics Multiple Choice Questions & Answers (MCQs) focuses on "Existence and Laplace Transform of Elementary Functions – 1". En este tutorial se muestra como calcular transformadas discretas de Fourier 1D mediante el uso de los comandos fft y fftshift de MatLAB. FFT Tutorial 1 Getting to Know the FFT How does the discrete Fourier transform relate to the other transforms? The abs function flnds the magnitude of the. •Convolutions can be implemented using fast Fourier transform: –Take FFT of image and filter, multiply elementwise, and take inverse FFT. However, the / choice here makes the resulting DFT matrix unitary, which is convenient in many circumstances. EL-GY 6123 Image and Video Processing. All videos come with MATLAB and Python code for you to learn from and adapt! This course is for you if you are an aspiring or established: Data scientist Statistician Computer scientist (MATLAB and/or Python). The functions shown here are fairly simple, but the concepts extend to more complex functions. To figure out reverse transform, obsolete: this document compares the FFT algorithm in. However there is a common procedure to calculate the Fourier transform numerically. The Fourier Transform is a powerful tool allowing us to move back and forth between the spatial and frequency domains. I don't go into detail about setting up and solving integration problems to obtain analytical solutions. I've created a code (Python, numpy) that defines an ultrashort laser pulse in the frequency domain (pulse duration should be 4 fs), but when I perform the Fourier Transform using DFT, my pulse in the. 1 The 1d Discrete Fourier Transform (DFT) The forward (FFTW_FORWARD) discrete Fourier transform (DFT) of a 1d complex array X of size n computes an array Y, where:. I don't go into detail about setting up and solving integration problems to obtain analytical solutions. Tracking of rotating point. A 2D discrete function can be decomposed by a lowpass filter and a highpass filter , and reconstructed with a lowpass filter (the conjugate filter of ) and two highpass filters and. Login; Login. This function performs the split-step Fourier method to solve the 1D time-dependent Schrödinger equation for a given potential. Calculate the FFT (Fast Fourier Transform) of an input sequence. For information about the NFFT algorithm, see the paper Using NFFT 3 – a software library for various nonequispaced fast Fourier. How to perform a fast fourier transform(fft) of 1D array(If it is possible!), which corresponds to fft of 3D array (and ifft after)? arrays python-3. Discrete Cosine Transform (wikipedia): A DCT is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. Solving Poissons equation in 1D with Fourier Transforms. This is a series of computer vision tutorials. 4 of the book. pdf file GraceGTK was forked from grace-5. PythonCUDA is a Python library with functions for computation on a GPU with NVIDIA CUDA. Presented at OSCON 2014. We can use the Gaussian filter from scipy. 1-Dimensional fast Fourier transform (1D FFT) and 2D FFT have time complexity O(NlogN) and O(N^2logN) respectively. NumberOfConfig // total number of configurations in your xyz movie file. mkl_fft started as a part of Intel (R) Distribution for Python* optimizations to NumPy, and is now being released as a stand-alone package. You can vote up the examples you like or vote down the ones you don't like. Definition of the Fourier Transform The Fourier transform (FT) of the function f. the case in Fourier analysis, the DWT is invertible, so that the original signal can be completely recovered from its DWT representation. Fourier transform diagrams; Circular convolution; FFT in Maple, Matlab 1D advection Fortran;. An Introduction to wavelets. its discrete Fourier transform and plots the magnitudes of the first 10000 coefficients in a manner similar to Fig. Fast Fourier Transform History Twiddle factor FFTs (non-coprime sub-lengths) 1805 Gauss Predates even Fourier's work on transforms! 1903 Runge 1965 Cooley-Tukey 1984 Duhamel-Vetterli (split-radix FFT) FFTs w/o twiddle factors (coprime sub-lengths) 1960 Good's mapping application of Chinese Remainder Theorem ~100 A. 1 Chapter 4 Image Enhancement in the Frequency Domain 4. That natural actually leads us to the definition of the Fourier transform, which we first look at in its continuous form. I recommend taking my Fourier Transform course before or alongside this course. The time-frequency decomposition is a generalization of the Gabor transform and allows for a intuitive decomposition of time series data at different frequencies. The Haar wavelet algorithms published here are applied to time series where the number of samples is a power of two (e. The time takes. Actually, as mentioned, all the programming environment, whether it's MATLAB, Python, Maple or others, usually have libraries for the fast Fourier transform that help you implement these kind of pseudo-spectral derivative applications. The fundamental concepts underlying the Fourier transform; Sine waves, complex numbers, dot products, sampling theorem, aliasing, and more! Interpret the results of the Fourier transform; Apply the Fourier transform in MATLAB and Python! Use the fast Fourier transform in signal processing applications; Improve your MATLAB and/or Python. HCFFT Documentation, Release 1. wait_for_finish – boolean variable, which tells whether it is necessary to wait on stream after scheduling all FFT kernels. 2D Discrete Fourier Transform • Fourier transform of a 2D signal defined over a discrete finite 2D grid of size MxN or equivalently • Fourier transform of a 2D set of samples forming a bidimensional sequence • As in the 1D case, 2D-DFT, though a self-consistent transform, can be considered as a mean of calculating the transform of a 2D. Ask Question Asked 5 years, 1 month ago. See the complete profile on LinkedIn and discover Jonathan's. Y = fft(X) computes the discrete Fourier transform (DFT) of X using a fast Fourier transform (FFT) algorithm. If you find pynufft useful, please cite: Jyh-Miin Lin, Hsiao-Wen Chung, Pynufft: python non-uniform fast Fourier transform for MRI. nah setelah itu, ada suatu alasan dimana kita harus mengembalikannya ke citra awal. 1 Physical derivation Reference: Guenther & Lee §1. The 2012 PRACE survey of FFT codes focussed on MPI codes. With the help of this course you can Learn the Fourier transform in MATLAB, Octave, and Python; and its applications in digital signal and image processing. The Fourier Transform will decompose an image into its sinus and cosines components. and its Fourier transform (~k), the time evolution can be carried out by simple multiplications. m computes the fast fractional Fourier transform following the algorithm of [5] (see also [6] for details) The m-file frft22d. The time takes. Data matrix should be of type double. This function performs 1-dimensional Fast Fourier Transform on each row of data in a matrix. The Formula of 1D Walsh Transform is defined in mfile. Written in pure Python. The Fast Fourier Transform (FFT) is an algorithm to compute the Discrete Fourier Transform F*v in O(m log m) time instead of O(m^2) time. The analytical Fourier transform ¶ Let's get back to the rotational kernel. The example used is the Fourier transform of a Gaussian optical pulse. The 1D and 2D optical Fourier transform can be carried out using the cylindrical lens 37 and the spherical lens 38, respectively. The modeller emg3d is a multigrid solver for 3D EM diffusion with tri-axial electrical anisotropy. If user have the data matrix in integer form, user should first transform it to double using the member function of matrixbase "CastToDouble". • Simulated 1D heat diffusion with MPI and Simulated 2D \& 3D heat diffusion with CUDA. These two Functions will do the 1 dimension Fast Fourier Transform. • Implemented acceleration for 2D Fast Fourier Transform (FFT) and Discrete Fourier Transform (DFT. The project is designed to move a motor stepp by step to any given angle between 0 and 360 degrees. Radix2 Decimation In Time 1d Fast Fourier Trans The function implement the 1D radix2 decimation in time fast Fourier transform (FFT) algorithm. The Fourier Transform will decompose an image into its sinus and cosines components. There are many applications for taking fourier transforms of images (noise filtering, searching for small structures in diffuse galaxies, etc. The symmetry is highest when n is a power of 2, and the transform is therefore most efficient for these sizes. Fourier transforms are usually expressed in terms of complex numbers, with real and imaginary parts representing the sine and cosine parts. Based on Python numerical libraries, such as Numpy, Scipy (matplotlib for displaying examples). Introduction to wavelets. Fourier transform (bottom) is zero except at discrete points. It is written in python (tested in python 2. Let us discretize from -R to R with the step d over x and y. Provides 1D/2D/3D examples for further developments. Below is the documentation for the nine routines. Also the absolute value of each Fourier coefficient is doubled to account for the symmetry of the Fourier coefficients around the Nyquest. 0 - Ahmad Poursaberi Tools / Build Tools The function implement the 1D Walash Transform which can be used in signal processing,pattern recognition and Genetic algorithms. transform itself, and the short-time Fourier transform. See the installation notes for how to install these interfaces; the main thing to remember is to compile the library before trying to pip install. Shared Memory Parallel: OpenMP []. Software Developer, Programming, Web resources and entertaiment. ImageJ has a built-in macro function for 1D Fourier Transforms / FFT using an array (self. The Fourier transform of a continuous periodic square wave is composed by impulses in every harmonic contained in the Fourier series expansion. 1998 We start in the continuous world; then we get discrete. Also the absolute value of each Fourier coefficient is doubled to account for the symmetry of the Fourier coefficients around the Nyquest. Let be the continuous signal which is the source of the data. This article will walk through the steps to implement the algorithm from scratch. • Signals as functions (1D, 2D) - Tools • 1D Fourier Transform - Summary of definition and properties in the different cases • CTFT, CTFS, DTFS, DTFT •DFT • 2D Fourier Transforms - Generalities and intuition -Examples - A bit of theory • Discrete Fourier Transform (DFT) • Discrete Cosine Transform (DCT). If you dive into the math, there's a relation between ARIMA models and representations in the frequency domain with a Fourier transform. Homework 8 Fourier Transform DATA FILES!!! (Due Sunday October 16th before midnight) Homework 10 Convolution and digital filters (Due Sunday October 30th before midnight) Images: img1. The Fourier Transform is a powerful tool allowing us to move back and forth between the spatial and frequency domains. •Convolutions can be implemented using fast Fourier transform: –Take FFT of image and filter, multiply elementwise, and take inverse FFT. The Fourier transform produces another representation of a signal, specifically a representation as a weighted sum of complex exponentials. Implementation of the Fourier transform in one dimension for an arbitrary function. The Fourier transform is commonly used to convert a signal in the time spectrum to a frequency spectrum. For large datasets, a kernel density estimate can be computed efficiently via the convolution theorem using a fast Fourier transform. Of course that it does. Clone via HTTPS Clone with Git or checkout with SVN using the repository's web address. To use them in Scikit-Learn, we need to build a Custom Feature Transformer class that transforms the single feature x to the feature vector of B-Spline basis functions evaluated at x, as in the case of the Fourier transform. For each differentiation, a new factor H-iwL is added. The first pass over the time series uses a window width of two. ; Quijada, Manuel A. Fourier transform methods allow the analysis of complex waveforms in terms of their sinusoidal components [32]. Step-by-Step. qmax // the maximum q value for S(q) in the Fourier transform method. Al-ternatively, we could have just noticed that we've already computed that the Fourier transform of the Gaussian function p 1 4ˇ t e 21 4 t x2. Fourier Transform Methods. So, let's see how this works in our Jupyter Notebook. See the complete profile on LinkedIn and discover. Numpy has an FFT package to do this. The Fourier Transform is a powerful tool allowing us to move back and forth between the spatial and frequency domains. 05 LOOK can read time or frequency domain input signals from disk 1. User-friendly 2D FFT/iFFT (Fast Fourier Transform) plug-in for Adobe PhotoShop compatible plug-in hosts. Discrete Fourier transform (DFT) is the base of modern signal or information processing. Hilbert transform of a signal x(t) is defined as the transform in which phase angle of all components of the signal is shifted by $\pm \text{90}^o $. It is written in Python, Cython and C for a mix of easy and powerful high-level interface and the best performance. OpenCV 3 image and video processing with Python OpenCV 3 with Python Image - OpenCV BGR : Matplotlib RGB Basic image operations - pixel access iPython - Signal Processing with NumPy Signal Processing with NumPy I - FFT and DFT for sine, square waves, unitpulse, and random signal Signal Processing with NumPy II - Image Fourier Transform : FFT & DFT. 1D Wave equation reloaded: characteristic coordinates 5. To figure out reverse transform, obsolete: this document compares the FFT algorithm in. Fraunhofer diffraction is "far-field" diffraction from a single slit and from equally spaced multiple slits. Discrete Wavelet Transform (DWT)¶ Wavelet transform has recently become a very popular when it comes to analysis, de-noising and compression of signals and images. HCFFT Documentation, Release 1. The discrete Fourier transform (DFT) is a basic yet very versatile algorithm for digital signal processing (DSP). The computation involves keeping track of the fields and their Fourier transform in a certain region, and from this computing the flux of electromagnetic energy as a function of ω. If X is a matrix, then fft(X) treats the columns of X as vectors and returns the Fourier transform of each column. Note the use of scipy's Bessel function:. Because the discrete Fourier transform separates its input into components that contribute at discrete frequencies, it has a great number of applications in digital signal processing, e. If you find pynufft useful, please cite: Jyh-Miin Lin, Hsiao-Wen Chung, Pynufft: python non-uniform fast Fourier transform for MRI. In previous blog post I reviewed one-dimensional Discrete Fourier Transform (DFT) as well as two-dimensional DFT. The IDCT algorithm is implemented on GPU and multicore systems, with performances on each system compared in terms of time taken to compute and accuracy. This set of Engineering Mathematics Multiple Choice Questions & Answers (MCQs) focuses on "Existence and Laplace Transform of Elementary Functions – 1". This includes distributions, time series, images, clusters, and more. Later it calculates DFT of the input signal and finds its frequency, amplitude, phase to compare. Args data: 2D array potential field at the grid points height: float. Maybe this picture from Oppenheim's Signals and Systems may help. %PERFORM 1D IFFT ON EACH COLUMN %INVERSE FOURIER TRANSFORM. DCTs are equivalent to DFTs of roughly twice the length, operating on real data with even symmetry. If the Fourier transform of the first signal is a + ib, and the Fourier transform of the second signal is c + id, then the ratio of the two Fourier transforms is. Examples showing how to use the basic FFT classes. • Continuous Fourier Transform (FT) - 1D FT (review) - 2D FT • Fourier Transform for Discrete Time Sequence (DTFT) - 1D DTFT (review) - 2D DTFT • Li C l tiLinear Convolution - 1D, Continuous vs. (See also the C06. The time spent in computing the NCC for the 1D test case is tabulated below for several numeralical methods/schemes used. Here is the analog version of. Below is the documentation for the nine routines. e I have a file which contains 1 measurement per line and I'd like to take the FT of these data, i. They are extracted from open source Python projects. 2013-12-31 Added new Python extension frontend. For flexible tomographic reconstruction, open source toolboxes are available, such as TomoPy, ODL, the ASTRA toolbox, and TIGRE. Y = fft(X) computes the discrete Fourier transform (DFT) of X using a fast Fourier transform (FFT) algorithm. Fast Fourier Transform on 2 Dimensional matrix using MATLAB Fast Fourier transformation on a 2D matrix can be performed using the MATLAB built in function ' fft2() '. Note that all wavelength values are in nm and all time is in fs. The sinc function is the Fourier Transform of the box function. It's hard to understand why the Fourier Transform is so important. interpft operates on the first dimension whose size does not equal 1. Fourier series: Fourier transform: G. Many of our explanations of key aspects of signal processing rely on an. ) and contain some waves here and here. Surface roughness is a measure of the topographic height variations of the surface. Center-right column: Original function is discretized (multiplied by a Dirac comb) (top). Unlike the DFT, the DWT, in fact, refers not just to a single transform, but rather a set of transforms, each with a different set of wavelet basis functions. Fourier transform is one of the various mathematical transformations known which is used to transform signals from time domain to frequency domain. (FFT transform on the TEM image). Heat (or Diffusion) equation in 1D* • Derivation of the 1D heat equation • Separation of variables (refresher) • Worked examples *Kreysig, 8th Edn, Sections 11. The SciPy Library/Package. This function performs 1-dimensional Fast Fourier Transform on each row of data in a matrix. Roughly speaking it is a way to represent a periodic function using combinations of sines and cosines. The functions shown here are fairly simple, but the concepts extend to more complex functions. The data behind the image was generated with. Hilbert transform, short-time Fourier transform (more about this later), Wigner distributions, the Radon Transform, and of course our featured transformation, the wavelet transform, constitute only a small portion of a huge list of transforms that are available at engineer's and mathematician's disposal. 2 Algorithms (Inverse 2D FFT) 2D IFFT is a fast algorithm for two-dimensional discrete Fourier transform (2D IDFT), which can be defined as follows: The algorithm for 2D IFFT is very similar to the algorithm for 2D FFT in that it is broken down into a series of 1D IFFTs to accelerate the computation. So the only question can be how to find out the right answer - not whether an answer exists. F1-Fourier transform for N+P (echo/antiecho) 2D ft_phase_modu(axis='F1') F1-Fourier transform for phase-modulated 2D ft_seq() performs the fourier transform of a data-set acquired on a Bruker in simultaneous mode Processing is performed only along the F2 (F3) axis if in 2D (3D) (Bruker QSIM mode). FINUFFT is a set of libraries to compute efficiently three types of nonuniform fast Fourier transform (NUFFT) to a specified precision, in one, two, or three dimensions, on a multi-core shared-memory machine. The Fourier transform of a continuous periodic square wave is composed by impulses in every harmonic contained in the Fourier series expansion. Radix2 Decimation In Time 1d Fast Fourier Trans The function implement the 1D radix2 decimation in time fast Fourier transform (FFT) algorithm. If you find pynufft useful, please cite: Jyh-Miin Lin, Hsiao-Wen Chung, Pynufft: python non-uniform fast Fourier transform for MRI. It refers to a very efficient algorithm for computingtheDFT • The time taken to evaluate a DFT on a computer depends principally on the number of multiplications involved. Moreover, the amplitude of cosine waves of wavenumber in this superposition is the cosine Fourier transform of the pulse shape, evaluated at wavenumber. GESPAR: Efficient Phase Retrieval of Sparse Signals Yoav Shechtman, Amir Beck, and Yonina C. Your question is extremely vague but depending on your application you can do anything with a 1D array. GitHub Gist: instantly share code, notes, and snippets. Fresnel Diffraction is "near-field" diffraction. The Haar wavelet algorithms published here are applied to time series where the number of samples is a power of two (e. and its Fourier transform (~k), the time evolution can be carried out by simple multiplications. , Fourier or wavelet transform). 10 hours ago · This includes distributions, time series, images, clusters, and more. 69 1D Discrete Fourier Transform • One major difference between continuous FT and DFT – The spectrum 퐹? is now a periodic function with period ?. They are extracted from open source Python projects. idft() Image Histogram Video Capture and Switching colorspaces - RGB / HSV Adaptive Thresholding - Otsu's clustering-based image thresholding Edge Detection - Sobel and Laplacian Kernels Canny Edge Detection. Discrete Cosine Transform (wikipedia): A DCT is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. Discrete Fourier Transform Functions¶ These DTF functions are previously defined in Review on Discrete Fourier Transform. pynufft import NUFFT_cpu, NUFFT_hsa. Version: 1. Is there a way of doing this ?. Its Fourier transform (bottom) is a periodic summation of the original transform. For non-equispaced locations, FFT is not useful and the discrete Fourier transform (DFT) is required. Moreover, the amplitude of cosine waves of wavenumber in this superposition is the cosine Fourier transform of the pulse shape, evaluated at wavenumber. If the input signal is an image then the number of frequencies in the frequency domain is equal to the number of pixels in the image or spatial domain. If X is a multidimensional array, then fft2 takes the 2-D transform of each dimension higher than 2. Then the 1D and 2D Fourier transforms are related by. If the Fourier transform of the first signal is a + ib, and the Fourier transform of the second signal is c + id, then the ratio of the two Fourier transforms is. With a naïve inverse Fourier transform on the values obtained from the image, it is not possible (at least by experiment) to recover the original signal. How to implement a ram-lak filter in 1D fourier transform? 2. The water would flow in just one direction in. Online FFT calculator, calculate the Fast Fourier Transform (FFT) of your data, graph the frequency domain spectrum, inverse Fourier transform with the IFFT, and much more. GitHub Gist: instantly share code, notes, and snippets. By Nikolay Koldunov. Hilbert transform of x(t) is represented with $\hat{x}(t)$,and it is given by. Wavelet Transform Maxima 1d extrema chain Computes the maxima of a continuous wavelet transform and chains them through scales Wavelet Transform Modulus Maxima Method 1d pf Computes the partition functions and singularity spectra of multifractal signals Matching Pursuit. Python code for implementing this using some interesting indexing methods is available [3]. Written in pure Python. FOURIER TRANSFORMS made easy (calculating a Fourier Transform has never been so easy) A VISUAL APPROACH: complex numbers and the Fourier Transform> John Sims Biomedical Engineering Department, Federal University of ABC, Sao Bernardo Campus Brasil. Fourier series: Applied on functions that are periodic. Provides 1D/2D/3D examples for further developments. When both the function and its Fourier transform are replaced with discretized counterparts, it is called the discrete Fourier transform (DFT). e having a 2D array (say b) which would contain omega in one column and the complex value (FT(v(t)))(omega) in another. Download and install free trials. discrete fourier transform Ising Model in 1D and 2D Introduces curve fitting in Python and uses this to estimate the half-life of the Ba-137m isotope. They are extracted from open source Python projects. > The problem is: I wan to average the radial intensity > distribution of all the direction on the 2D power spectra to > get a 1D power spectra. Column Transform: First consider the expression for. A cheat sheet for scientific python. Two-dimensional diffraction tomography reconstruction algorithm for scattering of a plane wave \(u_0(\mathbf{r}) = u_0(x,z)\) by a dielectric object with refractive index \(n(x,z)\). Contents wwUnderstanding the Time Domain, Frequency Domain, and FFT a. The meaning of these coefficients a_k and b_k in the Fourier series, was really basically the amplitude of the individual cosine and sine functions, harmonic functions. Press Edit this file button. FINUFFT is a set of libraries to compute efficiently three types of nonuniform fast Fourier transform (NUFFT) to a specified precision, in one, two, or three dimensions, on a multi-core shared-memory machine. This algorithm applies to almost all aspects of our everyday life. Removal of the Gibbs phenomenon and its application to fast-Fourier-transform-based mode solvers. In the Fourier domain, the Fourier transform of five filters are denoted by , , , and , respectively. This set of Partial Differential Equations Questions and Answers for Freshers focuses on "Solution of PDE by Variable Separation Method". If X is a multidimensional array, then fft2 takes the 2-D transform of each dimension higher than 2. x ) by placing the Fourier transformed projection planes into the Fourier image matrix and applying a 3D inverse Fourier transform to obtain the image. fft2() provides us the frequency transform which will be a complex array. 6 Comparison of the classification accuracies between DWT, Fourier Transform and Recurrent Neural Networks; Finals Words. Basics of Python and Its Application to Image Processing Through OpenCV: Review of 1D Fourier transform and convolution. Currently, there are two available backends, PyTorch (CPU and GPU) and scikit-cuda (GPU only). Each of these algorithms is written in a high-level imperative paradigm, making it portable to any Python library for array operations as long as it enables complex-valued linear algebra and a fast Fourier transform (FFT). Wangüemert-Pérez, J G; Godoy-Rubio, R; Ortega-Moñux, A; Molina-Fernán. How to implement the discrete Fourier transform Introduction. Multiplying by Q using the FFT. Recall that compressed sensing requires an incoherent measurement matrix. > The problem is: I wan to average the radial intensity > distribution of all the direction on the 2D power spectra to > get a 1D power spectra. To "undo" the smoothing effect of the back projection, the Radon transform is subjected to a filtering procedure in which high frequencies are boosted. The electromagnetic modeller empymod can model electric or magnetic responses due to a three-dimensional electric or magnetic source in a layered-earth model with vertical transverse isotropic (VTI) resistivity, VTI electric permittivity, and VTI magnetic permeability, from very low frequencies (DC) to very high frequencies (GPR). I currently look for the algorithm of performing a 1D discrete wavelet transformation in C# for curve smooting similar to this one: Smooting Example from Origin Lab Anyone done this before or can help me with some useful links? I am no mathematician, so it is pretty hard to find for me understandable stuff around the net THX a lot in. Final Exam Summary and Target Audience. See the complete profile on LinkedIn and discover Jonathan's. These two Functions will do the 1 dimension Fast Fourier Transform. The example used is the Fourier transform of a Gaussian optical pulse. My aim is to get a series of images in 2D space that run over different timestamps and put them through a 3D Fourier Transform. Learn the Fourier transform in MATLAB and Python, and its applications in digital signal processing and image processing The Fourier transform is one of the most important operations in modern technology, and therefore in modern human civilization. Chapter 2, Sampling, Fourier Transform, and Convolution, covers 2D Fourier transform, sampling, quantization, discrete Fourier transform, 1D and 2D convolution and filtering in the frequency domain, and how to implement them with Python using examples. Below is the documentation for the nine routines. Fourier transform, Fourier Transform 1D filtering, 1D FFT Filter 2D filtering, 2D FFT Filter high-pass filter, Frequency Split low-pass filter, Frequency Split fractal dimension, Fractal Analysis fractal interpolation, Fractal Correction. This function computes the n-dimensional discrete Fourier Transform over any axes in an M-dimensional array by means of the Fast Fourier Transform (FFT). This set of Partial Differential Equations Questions and Answers for Freshers focuses on "Solution of PDE by Variable Separation Method". The FFT requires O(N log N) work to compute N Fourier modes from N data points rather than O(N 2) work. GSML is a Python-based software library that implements many Spectral methods which are typically used for the solution of partial differential equations. This result is known as the Fourier Slice Theorem, and is the foundation of most reconstruction techniques. Speeding up Fourier-related transform computations in python. Let's say that we want to manipulate a signal in the frequency domain using a short time Fourier transform and once we're done , we use an inverse STFT (plus overlap and save) to convert the results. All videos come with MATLAB and Python code for you to learn from and adapt! This course is for you if you are an aspiring or established: Data scientist. The Fourier diffraction theorem states, that the Fourier transform ̂︀ B, 0 (k D) of the scattered field B(r D), measured at a certain angle 0, is distributed along a circular arc (2D) or along a semi-spherical surface (3D) in Fourier space, synthesizing the Fourier transform ̂︀(k) of the object function (r) [KS01], [Wol69]. It refers to a very efficient algorithm for computing the DFT. Unlike the DFT, the DWT, in fact, refers not just to a single transform, but rather a set of transforms, each with a different set of wavelet basis functions. idft() Image Histogram Video Capture and Switching colorspaces - RGB / HSV Adaptive Thresholding - Otsu's clustering-based image thresholding Edge Detection - Sobel and Laplacian Kernels Canny Edge Detection. mkl_fft-- a NumPy-based Python interface to Intel (R) MKL FFT functionality. The FFT routine included with numpy isn't particularly fast (c. A cheat sheet for scientific python. Quadrature approximation for the Fourier transform of the spreading. Think of an image of, say, an ocean. Given the range of algorithms, we review the literature in Section 2, tying together previous work and major developmental themes. In other words, it will transform an image from its spatial domain to its frequency domain. Implementation of the Fourier transform in one dimension for an arbitrary function. While there are many methods available for measuring MTF in electro-optical systems, indirect methods are among the most common. The Fourier transform of a continuous periodic square wave is composed by impulses in every harmonic contained in the Fourier series expansion. The backward (FFTW_BACKWARD) DFT computes:. Y = fft(X) computes the discrete Fourier transform (DFT) of X using a fast Fourier transform (FFT) algorithm. Like 1D, 2D Fourier transforms operate globally, but can capture local information using a 2D SWDFT. The discrete Fourier transform (DFT) converts a finite list of equally spaced samples of a function into the list of coefficients of a finite combination of complex sinusoids, ordered by their frequencies, that has those same sample values. numerical python and scientific python seem all to be operating on sequences and therefore seem to be 1D fourier transform. We denote the 1D and 2D Fourier transforms by and and the Radon transform by. The fundamental concepts underlying the Fourier transform; Sine waves, complex numbers, dot products, sampling theorem, aliasing, and more! Interpret the results of the Fourier transform; Apply the Fourier transform in MATLAB and Python! Use the fast Fourier transform in signal processing applications; Improve your MATLAB and/or Python. idft() Image Histogram Video Capture and Switching colorspaces - RGB / HSV Adaptive Thresholding - Otsu's clustering-based image thresholding Edge Detection - Sobel and Laplacian Kernels Canny Edge Detection. Because this is a fundamental signal analysis technique, it has many applications in signal processing. The FFT & Convolution • The convolution of two functions is defined for the continuous case – The convolution theorem says that the Fourier transform of the convolution of two functions is equal to the product of their individual Fourier transforms • We want to deal with the discrete case. This set of Engineering Mathematics Multiple Choice Questions & Answers (MCQs) focuses on "Existence and Laplace Transform of Elementary Functions – 1". In an infinite crystal, on the other hand, the function is typically periodic (and thus not decaying):. I'm confused about what exactly the amplitude spectrum is. working with the 1D Fourier transform extends fairly straightforwardly to higher dimensions. • Signals as functions (1D, 2D) – Tools • 1D Fourier Transform – Summary of definition and properties in the different cases • CTFT, CTFS, DTFS, DTFT •DFT • 2D Fourier Transforms – Generalities and intuition –Examples – A bit of theory • Discrete Fourier Transform (DFT) • Discrete Cosine Transform (DCT). For the 2D and 3D definitions, and other types of transform, see below. Liu, Qing-Jie; Lin, Qi-Zhong; Wang, Qin-Jun; Li, Hui; Li, Shuai. We recall that OpenMP is a set of compiler directives that can allow one to easily make a Fortran, C or C++ program run on a shared memory machine – that is a computer for which all compute processes can access the same globally addressed memory space. How to Use Python to Develop Graphs for Data Science Performing a Fast Fourier Transform (FFT) on a Sound File. By using convolution, we can construct the output of system for any arbitrary input signal, if we know the impulse response of system. FFT是信号处理中应用最为广泛的一个算法,但是很多入门童鞋对这个算法不甚了解,写作此文,给入门人员一个启示。FFT(Fast Fourier Transform)快速傅里叶变换是离散傅里叶变换(DFT)的一种快速计算方法。. Version: 1. First written April 2014. By John Paul Mueller, Luca Massaron. We can use the Gaussian filter from scipy. m computes a 2D transform based on the 1D routine frft2. Engineering Tables/Fourier Transform Table 2 From Wikibooks, the open-content textbooks collection < Engineering Tables Jump to: navigation, search Signal Fourier transform unitary, angular frequency Fourier transform unitary, ordinary frequency Remarks 10 The rectangular pulse and the normalized sinc function 11 Dual of rule 10. Several python libraries implement discrete wavelet transforms. Actually, as mentioned, all the programming environment, whether it's MATLAB, Python, Maple or others, usually have libraries for the fast Fourier transform that help you implement these kind of pseudo-spectral derivative applications. Free Download Udemy Understand the Fourier transform and its applications. So, let's see how this works in our Jupyter Notebook. Note the use of scipy's Bessel function:. Wangüemert-Pérez, J G; Godoy-Rubio, R; Ortega-Moñux, A; Molina-Fernán. From what I gather, it is the absolute value of the Fourier Transform which is somewhat like a histogram of frequencies of the components that the. A 2D discrete function can be decomposed by a lowpass filter and a highpass filter , and reconstructed with a lowpass filter (the conjugate filter of ) and two highpass filters and. The 1-D Heat Equation 18. The Fast Fourier Transform (FFT). This is a series of computer vision tutorials. dft Performs a forward or inverse Discrete Fourier transform of a 1D or 2D floating-point array. Based on numerical libraries, such as Numpy, Scipy (matplotlib for displaying examples). A thorough tutorial of the Fourier Transform, for both the laymen and the practicing scientist. Archive of DataMelt/DMelt examples (2005-current). The Fourier Transform will decompose an image into its sinus and cosines components. fft2() provides us the frequency transform which will be a complex array. If the K-means algorithm is concerned with centroids, hierarchical (also known as agglomerative) clustering tries to link each data point, by a distance measure, to its nearest neighbor, creating a cluster. To download the files, visit http://engineertomorrow. The main function reads in the calculation parameters, checks that they are sensible, initializes the electron coordinates, and then evolves the electron equations of motion from to some specified , using a fixed step RK4 routine with some specified. If you dive into the math, there's a relation between ARIMA models and representations in the frequency domain with a Fourier transform. "Therefore the wavelet analysis or synthesis can be performed locally on the signal, as opposed to the Fourier transform. Several python libraries implement discrete wavelet transforms. Calculate the FFT (Fast Fourier Transform) of an input sequence. ¶ All the calculations must start with the Begin command. An FFT is a "Fast Fourier Transform". In the case of NMR data, this decomposition is to a series of peaks, that represent the resonance of chemical subgroups. Fast Fourier Transform. So in order to build the complete 2D Fourier transform, we need all of the 180 degrees and then apply this one dimensional Fourier transform, stick it in the 2D Fourier space and then we've got the. F1-Fourier transform for N+P (echo/antiecho) 2D ft_phase_modu(axis='F1') F1-Fourier transform for phase-modulated 2D ft_seq() performs the fourier transform of a data-set acquired on a Bruker in simultaneous mode Processing is performed only along the F2 (F3) axis if in 2D (3D) (Bruker QSIM mode). Fourier spectra help characterize how different filters behave, by. discrete signals (review) – 2D • Filter Design • Computer Implementation Yao Wang, NYU-Poly EL5123: Fourier Transform 2. The real and the estimated points are connected with yellow line segment,. FFTW++ is a C++ header class for the FFTW Fast Fourier Transform library that automates memory allocation, alignment, planning, wisdom, and communication on both serial and parallel (OpenMP/MPI) architectures. For example, an Image is a two-dimensional function f(x, y). The roughness can arise from polishing marks, machining marks, marks left by rollers, dust or other particles and is basically shaped by the full history of the surface from the forming stages (casting, sintering, rolling, etc. 1D barcode generator (JavaScript) Barrett reduction algorithm; GIF89a specification (HTML) GIF optimizer (Java) Bitcoin cryptography library; Compact hash map (Java) Fast Fourier transform in x86 assembly; Tablet desk clock; JSON library (Java) Cryptographic primitives in plain Python; Symmetry sketcher (JavaScript) Simulated annealing demo. fft2¶ numpy. Can someone provide me the Python script to plot FFT? What are the parameters needed to plot FFT? I will have acceleration data for hours (1 to 2 hrs) sampled at 500 or 1000 Hz. •It has faster asymptotic running time but there are some catches: –You need to be using periodic boundary conditions for the convolution. Hi, I suggest to try to understand the basics of the Fourier transform. The most general case allows for complex numbers at the input and results in a sequence of equal length, again of complex numbers. | CommonCrawl |
In the diagram, each of the three identical circles touch the other two. The circumference of each circle is 36. What is the perimeter of the shaded region? [asy]
defaultpen(1);
path p = (1, 0){down}..{-dir(30)}dir(-60){dir(30)}..{dir(-30)}((2, 0) + dir(-120)){-dir(-30)}..{up}(1, 0)--cycle;
fill(p, gray(0.75));
draw(unitcircle);
draw(shift(2 * dir(-60)) * unitcircle);
draw(shift(2) * unitcircle);
[/asy]
Join the centre of each circle to the centre of the other two. Since each circle touches each of the other two, then these line segments pass through the points where the circles touch, and each is of equal length (that is, is equal to twice the length of the radius of one of the circles). [asy]
import olympiad;
defaultpen(1);
path p = (1, 0){down}..{-dir(30)}dir(-60){dir(30)}..{dir(-30)}((2, 0) + dir(-120)){-dir(-30)}..{up}(1, 0)--cycle;
fill(p, gray(0.75));
draw(unitcircle);
draw(shift(2 * dir(-60)) * unitcircle);
draw(shift(2) * unitcircle);
// Add lines
draw((0, 0)--(2, 0)--(2 * dir(-60))--cycle);
// Add ticks
add(pathticks((0, 0)--(1, 0), s=4)); add(pathticks((1, 0)--(2, 0), s=4));
add(pathticks((0, 0)--dir(-60), s=4)); add(pathticks(dir(-60)--(2 * dir(-60)), s=4));
add(pathticks((2 * dir(-60))--(2 * dir(-60) + dir(60)), s=4)); add(pathticks((2, 0)--(2 * dir(-60) + dir(60)), s=4));
[/asy]
Since each of these line segments have equal length, then the triangle that they form is equilateral, and so each of its angles is equal to $60^\circ$.
Now, the perimeter of the shaded region is equal to the sum of the lengths of the three circular arcs which enclose it. Each of these arcs is the arc of one of the circles between the points where this circle touches the other two circles.
Thus, each arc is a $60^\circ$ arc of one of the circles (since the radii joining either end of each arc to the centre of its circle form an angle of $60^\circ$), so each arc is $\frac{60^\circ}{360^\circ} = \frac{1}{6}$ of the total circumference of the circle, so each arc has length $\frac{1}{6}(36)=6$.
Therefore, the perimeter of the shaded region is $3(6) = \boxed{18}$. | Math Dataset |
Hadamard manifold
In mathematics, a Hadamard manifold, named after Jacques Hadamard — more often called a Cartan–Hadamard manifold, after Élie Cartan — is a Riemannian manifold $(M,g)$ that is complete and simply connected and has everywhere non-positive sectional curvature.[1][2] By Cartan–Hadamard theorem all Cartan–Hadamard manifolds are diffeomorphic to the Euclidean space $\mathbb {R} ^{n}.$ Furthermore it follows from the Hopf–Rinow theorem that every pairs of points in a Cartan–Hadamard manifold may be connected by a unique geodesic segment. Thus Cartan–Hadamard manifolds are some of the closest relatives of $\mathbb {R} ^{n}.$
Examples
The Euclidean space $\mathbb {R} ^{n}$ with its usual metric is a Cartan–Hadamard manifold with constant sectional curvature equal to $0.$
Standard $n$-dimensional hyperbolic space $\mathbb {H} ^{n}$ is a Cartan–Hadamard manifold with constant sectional curvature equal to $-1.$
Properties
In Cartan-Hadamard manifolds, the map $\exp _{p}:\operatorname {T} M_{p}\to M$ is a diffeomorphism for all $p\in M.$
See also
• Cartan–Hadamard conjecture
• Cartan–Hadamard theorem – On the structure of complete Riemannian manifolds of non-positive sectional curvature
• Hadamard space – geodesically complete metric space of non-positive curvaturePages displaying wikidata descriptions as a fallback
References
1. Li, Peter (2012). Geometric Analysis. Cambridge University Press. p. 381. doi:10.1017/CBO9781139105798. ISBN 9781107020641.
2. Lang, Serge (1989). Fundamentals of Differential Geometry, Volume 160. Springer. pp. 252–253. ISBN 9780387985930.
Manifolds (Glossary)
Basic concepts
• Topological manifold
• Atlas
• Differentiable/Smooth manifold
• Differential structure
• Smooth atlas
• Submanifold
• Riemannian manifold
• Smooth map
• Submersion
• Pushforward
• Tangent space
• Differential form
• Vector field
Main results (list)
• Atiyah–Singer index
• Darboux's
• De Rham's
• Frobenius
• Generalized Stokes
• Hopf–Rinow
• Noether's
• Sard's
• Whitney embedding
Maps
• Curve
• Diffeomorphism
• Local
• Geodesic
• Exponential map
• in Lie theory
• Foliation
• Immersion
• Integral curve
• Lie derivative
• Section
• Submersion
Types of
manifolds
• Closed
• (Almost) Complex
• (Almost) Contact
• Fibered
• Finsler
• Flat
• G-structure
• Hadamard
• Hermitian
• Hyperbolic
• Kähler
• Kenmotsu
• Lie group
• Lie algebra
• Manifold with boundary
• Oriented
• Parallelizable
• Poisson
• Prime
• Quaternionic
• Hypercomplex
• (Pseudo−, Sub−) Riemannian
• Rizza
• (Almost) Symplectic
• Tame
Tensors
Vectors
• Distribution
• Lie bracket
• Pushforward
• Tangent space
• bundle
• Torsion
• Vector field
• Vector flow
Covectors
• Closed/Exact
• Covariant derivative
• Cotangent space
• bundle
• De Rham cohomology
• Differential form
• Vector-valued
• Exterior derivative
• Interior product
• Pullback
• Ricci curvature
• flow
• Riemann curvature tensor
• Tensor field
• density
• Volume form
• Wedge product
Bundles
• Adjoint
• Affine
• Associated
• Cotangent
• Dual
• Fiber
• (Co) Fibration
• Jet
• Lie algebra
• (Stable) Normal
• Principal
• Spinor
• Subbundle
• Tangent
• Tensor
• Vector
Connections
• Affine
• Cartan
• Ehresmann
• Form
• Generalized
• Koszul
• Levi-Civita
• Principal
• Vector
• Parallel transport
Related
• Classification of manifolds
• Gauge theory
• History
• Morse theory
• Moving frame
• Singularity theory
Generalizations
• Banach manifold
• Diffeology
• Diffiety
• Fréchet manifold
• K-theory
• Orbifold
• Secondary calculus
• over commutative algebras
• Sheaf
• Stratifold
• Supermanifold
• Stratified space
| Wikipedia |
\begin{document}
\maketitle
\begin{abstract}
The class of \emph{$2$-regular} matroids is a natural generalisation of regular and near-regular matroids.
We prove an excluded-minor characterisation for the class of $2$-regular matroids.
The class of \emph{$3$-regular} matroids coincides with the class of matroids representable over the Hydra-$5$ partial field,
and the $3$-connected matroids in the class with a $U_{2,5}$- or $U_{3,5}$-minor are precisely those with six inequivalent representations over $\textrm{GF}(5)$.
We also prove that an excluded minor for this class has at most $15$ elements. \end{abstract}
\section{Introduction}
Let $\mathbb Q(\alpha_1,\ldots,\alpha_n)$ denote the field obtained by extending the rational numbers by $n$ independent transcendentals $\alpha_1,\ldots,\alpha_n$. For $k\geq 0$, a matroid is $k$-regular if it has a representation by a matrix $A$ over $\mathbb Q(\alpha_1,\ldots,\alpha_n)$ in which every nonzero subdeterminant of $A$ is in the set that consists of all integer powers of differences of distinct members of $\{0,1,\alpha_1,\ldots,\alpha_k\}$. The class of $0$-regular matroids coincides with the class of regular matroids. The class of $1$-regular matroids is the class of {\em near-regular} matroids and is the class of matroids representable over all fields of size at least~3 \cite{Whittle97}. Excluded minor characterisations for regular matroids are given by Tutte \cite{Tutte58} and for near-regular matroids by Hall, Mayhew and van Zwam \cite{HMvZ11}.
This paper focuses on $2$-regular and $3$-regular matroids. We prove the following:
\begin{theorem} \label{intro1} An excluded minor for either the class of $2$-regular or $3$-regular matroids has at most 15 elements. \end{theorem}
This bound enables a computer search to be undertaken to find all excluded minors for 2-regular matroids. This search is undertaken in \cite{BP20} and we are able to give the following excluded-minor characterisation of 2-regular matroids. All matroids mentioned below are described in the Appendix of this paper.
\begin{theorem} \label{intro2}
A matroid $M$ is $2$-regular if and only if $M$ has no minor isomorphic to $U_{2,6}$, $U_{3,6}$, $U_{4,6}$, $P_6$, $F_7$, $F_7^*$, $F_7^-$, $(F_7^-)^*$, $F_7^=$, $(F_7^=)^*$, $\mathit{AG}(2,3)\backslash e$, $(\mathit{AG}(2,3)\backslash e)^*$, $(\mathit{AG}(2,3)\backslash e)^{\Delta Y}$, $P_8$, $P_8^-$, $P_8^=$, and $\mathit{TQ}_8$. \end{theorem}
It is natural to hope for an analogous result for $3$-regular matroids. A search in \cite{Brettell22} uncovers all excluded minors for this class up to size 13. We conjecture that the list is complete. Note that $\Delta^{(*)}(U_{2,7})$ is a family of six matroids obtained from $U_{2,7}$ via $\Delta$-$Y$ exchange and dualising.
\begin{conjecture}
\label{intro3}
A matroid is $3$-regular if and only if it has no minor isomorphic to one of the following $33$ matroids:
$$F_7, F_7^-, F_7^=, H_7,M(K_4)+e, \mathcal{W}^3+e,\Lambda_3, Q_6+e, P_6+e, U_{3,7},$$ and their duals; a matroid in $\Delta^{(*)}(U_{2,7})$; and $\mathit{AG}(2,3)\backslash e$, $(\mathit{AG}(2,3)\backslash e)^*$, $(\mathit{AG}(2,3)\backslash e)^{\Delta Y}$, $P_8$, $P_8^-$, $P_8^=$, and $\mathit{TQ}_8$. \end{conjecture}
To resolve Conjecture~\ref{intro3} we need to eliminate the possibility of excluded minors with 14 or 15 elements. One strategy would be to do further work along the lines of that contained in this paper with the goal of reducing the bound of 15. Another strategy would be to narrow the space for a computer search by exploiting known properties of the structure of excluded minors.
The results of this paper are motivated by two problems. The first is to understand the class of matroids representable over all fields of size at least~4. The second is to find an explicit excluded-minor characterisation for the class of $\textrm{GF}(5)$-representable matroids. In the remainder of this introduction we discuss the connection between our results and these problems, before outlining the approach taken to prove \cref{intro1}.
\subsection*{Matroids representable over all fields of size at least 4} The class of regular matroids is the class of matroids representable over all fields and the class of near-regular matroids is the class of matroids representable over all fields of size at least~3 \cite{Whittle97}. But after that equality no longer holds. Let $\mathcal M_4$ denote the class of matroids representable over all fields of size at least~4. Then the class of 2-regular matroids is properly contained in $\mathcal M_4$. It follows from Theorem~\ref{intro2} that, up to duality, there are four excluded minors for 2-regular matroids that belong to $\mathcal M_4$: these are $P_8^-$, $\mathit{TQ}_8$, $U_{3,6}$ and $F_7^=$. Nonetheless, a start has been made. Up to duality, any member of $\mathcal M_4$ that is not 2-regular must contain one of these four matroids as a minor. It is possible that members of $\mathcal M_4$ containing either $P_8^-$, $\mathit{TQ}_8$ or $U_{3,6}$ as minors form classes of bounded branch width that can be described structurally, but it turns out that there are members of $\mathcal M_4$ of unbounded branch width containing $F_7^=$ as a minor.
For $n\geq 4$, let $x$ and $y$ be elements of $M(K_n)$ that are not contained in a triangle, that is, they correspond to a matching in the underlying graph $K_n$. Extend by adding a point freely to the line spanned by $\{x,y\}$. Denote the resulting matroid by $M_n$. It is readily verified that $M_4\cong F_7^=$. It is also routine to see that $M_n\in \mathcal M_4$ for all $n\geq 4$. We now have a rich class of matroids contained in $\mathcal M_4$ that are not 2-regular.
All up, the news is not particularly optimistic. If one seeks an excluded-minor characterisation of $\mathcal M_4$, then, using current techniques, for each $N\in\{P_8^-,\mathit{TQ}_8,U_{3,6},F_7^=\}$ you will need to perform an exercise similar to the one that finds the excluded minors for 2-regular matroids except with $U_{2,5}$ replaced by $N$. This will require an understanding of a certain class of $N$-fragile matroids. That, in itself, is likely to be a challenge. Even with that issue resolved, the bound obtained on the size of an excluded is likely to be too large to enable a computer search to find them.
The structural approach may be more promising. If the classes containing $P_8^-$, $\mathit{TQ}_8$ or $U_{3,6}$ truly are thin, then perhaps they can be explicitly described. It is possible that the class containing $F_7^=$ is not too difficult either.
It is possible that there is a structure theorem analogous to the regular matroid decomposition theorem where the class obtained by taking minors of $M_n$ plays a role analogous to that of graphic matroids in regular matroids.
In any case, the goal of obtaining a better understanding of $\mathcal M_4$ is no doubt a worthy one and at least there are some plausible lines of attack.
\subsection*{Excluded minors for \texorpdfstring{$\textrm{GF}(5)$}{GF(5)}} We prove in Lemma~\ref{3reglemma} that the class of $3$-regular matroids coincides with the class of matroids representable over the Hydra-5 partial field $\mathbb H_5$. This partial field was introduced in \cite{PvZ10b} where it is shown that 3-connected $\mathbb H_5$ representable matroids with a $U_{2,5}$-minor have exactly six inequivalent $\textrm{GF}(5)$-representations. Resolving Conjecture~\ref{intro3} would achieve the first step of a program for finding excluded minors for $\textrm{GF}(5)$-representability adumbrated in \cite{PvZ10b}. In essence the program is as follows. There is a sequence $\mathbb H_1$, $\mathbb H_2$, $\mathbb H_3$, $\mathbb H_4$ and $\mathbb H_5$ of partial fields each of which has a homomorphism into $\textrm{GF}(5)$. Matroids representable over $\mathbb H_1$ that are $3$-connected with a $U_{2,5}$-minor are uniquely representable over $\textrm{GF}(5)$. Analogous matroids representable over $\mathbb H_2$, $\mathbb H_3$ and $\mathbb H_4$ have two, three and four inequivalent representations over $\textrm{GF}(5)$, and such matroids representable over $\mathbb H_5$ have six inequivalent $\textrm{GF}(5)$ representations.
Each $\textrm{GF}(5)$-representable excluded minor~$N$ for representability over $\mathbb H_5$ will be a strong stabiliser for a member $\mathbb P$ of this sequence. One then has to find the excluded minors for the $\mathbb P$-representable matroids that contain an $N$-minor. This is a task similar to the one undertaken in this paper except that one has to understand the $N$-fragile $\mathbb P$-representable matroids. This will typically be significantly more difficult than understanding the $2$- or $3$-regular $U_{2,5}$-fragile matroids.
It turns out that, modulo the truth of Conjecture~\ref{intro3}, there are, up to duality, ten excluded minors for $\mathbb H_5$-representable matroids that are $\textrm{GF}(5)$-representable. Hence the exercise described above has to be repeated ten times. Moreover the process repeats, so that some of the excluded minors found may well be $\textrm{GF}(5)$-representable, although they will be placed further up in the hierarchy.
That is a massive undertaking. It is significantly more difficult than a related, but as yet unresolved, problem. That problem is to find the excluded minors for the class of {\em dyadic} matroids. It is known the class of dyadic matroids is the class of matroids representable over $\textrm{GF}(3)$ and $\textrm{GF}(5)$. The excluded minors for dyadic matroids can easily be deduced from a knowledge of the excluded minors of $\textrm{GF}(5)$-representable matroids. But one would expect it to be significantly easier to find the excluded minors for dyadic matroids directly and, indeed, knowing the dyadic excluded minors would in itself be a step towards $\textrm{GF}(5)$. So how difficult is this problem.
We know that any excluded minor for dyadic that we do not understand must contain either the non-Fano matroid $F_7^-$, its dual, or $P_8$. Moreover these matroids are strong stabilisers for the class of dyadic matroids. There is some hope of understanding the dyadic $P_8$-fragile matroids, but understanding the dyadic $F_7^-$-fragile matroids seems much more difficult. Moreover, it is known that excluded minors for dyadic matroids can be quite large. The computer search in \cite{BP20} has uncovered one with 16 elements.
The upshot is that current techniques are most probably inadequate for obtaining an excluded-minor characterisation of dyadic matroids and certainly inadequate for $\textrm{GF}(5)$. More optimistically, one can pursue techniques that do not commit one to solving an $N$-fragility problem. It is possible that real progress in the future could occur by developing such techniques. Success in obtaining an excluded-minor characterisation of dyadic matroids would be a major breakthrough.
\subsection*{Overview of the proof} The high-level approach of the proof Theorem~\ref{intro1} is as follows. Let $M$ be a large excluded minor for the class of 2-regular matroids such that $M$ has an $N$-minor, where $N$ is $U_{2,5}$ or $U_{3,5}$. By results in \cite{BWW22}, the matroid $M$ has a pair of elements $a,b$ such that $M\backslash a,b$ is 3-connected and has an $N$-minor. Now, by results in \cite{BCOSW20}, $M\backslash a,b$ is close to being $N$-fragile. Clark et al.\ \cite{CMvZW16} described the structure of large 2-regular $\{U_{2,5},U_{3,5}\}$-fragile matroids; in particular, they have path width three. In \cref{utfutfsec}, we recap and expand on the properties of these matroids that we require. It still remains to prove that $M\backslash a,b$ is itself $\{U_{2,5},U_{3,5}\}$-fragile; we do this in \cref{utfutffragilesec}, using results from \cite{paper1} (presented in \cref{almostfragsec}) and \cref{utfutfsec}. Then, in \cref{fragilepropssec}, we prove some more properties of such a $\{U_{2,5},U_{3,5}\}$-fragile matroid $M\backslash a,b$. In \cref{ndtsec}, we show that if $M\backslash a,b$ is large enough, then we can use the path structure to find a triple $\{a',b',c'\}$ such that $M\backslash a',b',c'$ is 3-connected with an $N$-minor, and therefore $\{U_{2,5},U_{3,5}\}$-fragile. When there is a triple $a',b',c'$ such that $M\backslash a',b',c'$ is $\{U_{2,5},U_{3,5}\}$-fragile, each of the matroids $M\backslash a',b'$, $M\backslash a',c'$, and $M\backslash b',c'$ has path width three, and we show, in \cref{dtsec}, that we can use this to bound the size of $M$. Combining these results in \cref{concsec}, we complete the proof of \cref{intro1}.
\section{Preliminaries}
All unspecified terminology and notation follows Oxley \cite{Oxley11}.
When there is no ambiguity, we often denote a singleton set $\{q\}$ by $q$. For a positive integer $n$, we denote the set $\{1,2,\dotsc,n\}$ by $[n]$.
For sets $X$ and $Y$, we say $X$ \emph{meets} $Y$ if $X \cap Y \neq \emptyset$, and $X$ \emph{avoids} $Y$ if $X \cap Y = \emptyset$. A parallel class or series class is \emph{non-trivial} if it has size at least~$2$. For a partition $\{X_1,X_2,\dotsc,X_m\}$ or an ordered partition $(X_1,X_2,\dotsc,X_m)$ we require that each cell $X_i$ is non-empty.
\subsection*{Segments, cosegments, and fans}
Let $M$ be a matroid.
A subset~$S$ of $E(M)$ with $|S| \ge 3$ is a \emph{segment} if every $3$-element subset of $S$ is a triangle. We say that a segment~$S$ is an $\ell$-segment if $|S|=\ell$. A \emph{cosegment} is a segment of $M^*$, and an $\ell$-cosegment~$S^*$ is a cosegment with $|S^*| = \ell$.
A subset~$F$ of $E(M)$ with $|F| \ge 3$ is a \emph{fan} if there is an ordering $(f_1, f_2, \dotsc, f_\ell)$ of $F$ such that \begin{enumerate}[label=\rm(\alph*)]
\item $\{f_1,f_2,f_3\}$ is either a triangle or a triad, and
\item for all $i \in \seq{\ell-3}$, if $\{f_i, f_{i+1}, f_{i+2}\}$ is a triangle, then $\{f_{i+1}, f_{i+2}, f_{i+3}\}$ is a triad, whereas if $\{f_i, f_{i+1}, f_{i+2}\}$ is a triad, then $\{f_{i+1}, f_{i+2}, f_{i+3}\}$ is a triangle. \end{enumerate} When there is no ambiguity, we also say that the ordering $(f_1,f_2,\dotsc,f_\ell)$ is a fan. If $F$ has a fan ordering $(f_1, f_2, \dotsc, f_\ell)$ where $\ell \geq 4$, then $f_1$ and $f_\ell$ are the \emph{ends} of $F$, and $f_2, f_3, \dotsc, f_{\ell-1}$ are the \emph{internal elements} of $F$. We also say such a fan has \emph{size} $\ell$.
We say that a fan $F$ is \emph{maximal} if there is no fan that properly contains $F$.
For a rank-$r$ wheel $M(\mathcal{W}_r)$, there is a natural partition of the ground set into \emph{spoke} elements, each of which is contained in two triangles, and \emph{rim} elements, each of which is contained in two triads. There is an analogous notion for elements in fans. Let $F$ be a fan with ordering $(f_1, f_2, \dotsc, f_\ell)$ where $\ell \geq 4$, and let $i \in \seq{\ell}$ if $\ell \geq 5$, or $i \in \{1,4\}$ if $\ell=4$. An element $f_i$ is a \emph{spoke element} of $F$ if $\{f_1, f_2, f_3\}$ is a triangle and $i$ is odd, or if $\{f_1, f_2, f_3\}$ is a triad and $i$ is even; otherwise $f_i$ is a \emph{rim element} of $F$.
\begin{lemma}[{\cite[Lemma~3.4]{OW00}}]
\label{fanunique}
Let $M$ be a $3$-connected matroid that is not isomorphic to $M(\mathcal{W}_3)$, and let $F$ be a $5$-element fan of $M$ with ordering $(f_1,f_2,\dotsc,f_5)$, where $\{f_1,f_2,f_3\}$ is a triangle.
Then $\{f_1,f_2,f_3\}$ and $\{f_3,f_4,f_5\}$ are the only triangles of $M$ containing $f_3$, and $\{f_2,f_3,f_4\}$ is the unique triad of $M$ containing $f_3$. \end{lemma}
\subsection*{Connectivity}
Let $M$ be a matroid and let $X \subseteq E(M)$. The set $X$ or the partition $(X,E(M)-X)$ is \emph{$k$-separating} if $\lambda_M(X) < k$, where $\lambda_M(X) = r(X) + r(E(M)-X) - r(M)$. A $k$-separating set $X$ or partition $(X,E(M)-X)$ is \emph{exact} if $\lambda_M(X) = k-1$. If $X$ is $k$-separating and $|X|,|E(M)-X| \ge k$, then $X$ is a \emph{$k$-separation}. The matroid $M$ is $k'$-connected if $M$ has no $k$-separations for $k < k'$. If $M$ is $2$-connected, we simply say it is \emph{connected}.
Suppose $M$ is connected. If for every $2$-separation $(X,Y)$ of $M$ either $X$ or $Y$ is a parallel pair (or parallel class), then $M$ is \emph{$3$-connected up to parallel pairs} (or \emph{parallel classes}, respectively). Dually, if for every $2$-separation $(X,Y)$ of $M$ either $X$ or $Y$ is a series pair (or series class), then $M$ is \emph{$3$-connected up to series pairs} (or \emph{series classes}, respectively).
We say $Z \subseteq E(M)$ is in the \emph{guts} of a $k$-separation $(X,Y)$ if $Z \subseteq \cl(X-Z) \cap \cl(Y-Z)$, and we say $Z$ is in the \emph{coguts} of $(X,Y)$ if $Z$ is in the guts of $(X,Y)$ in $M^*$. We also say $z$ is in the guts (or the coguts) of a $k$-separation $(X,Y)$ if $\{z\}$ is in the guts (or the coguts, respectively) of $(X,Y)$. Note that if $z$ is in the guts of $(X,Y)$, then $z \notin \cl^*(X)$ and $z \notin \cl^*(Y)$.
We say that a partition $(X_1,X_2,\dotsc,X_m)$ of $E(M)$ is a \emph{path of $3$-separations} if $(X_1 \cup \dotsm \cup X_i, X_{i+1} \cup \dotsm \cup X_m)$ is exactly $3$-separating for each $i \in \seq{m-1}$. Note that $|X_1|,|X_m| \ge 2$ (and $|X_i| \ge 1$ for all $i \in [m]$). If $X_i$ is in the guts (or coguts) of $(X_1 \cup \dotsm \cup X_i, X_{i+1} \cup \dotsm \cup X_m)$, then we say $X_i$ is a \emph{guts set} (or \emph{coguts set}, respectively) and, for each $x \in X_i$, we say $x$ is a \emph{guts element} (or \emph{coguts element}, respectively).
A $3$-separation $(X,Y)$ of $M$ is a \textit{vertical $3$-separation} if $\min\{r(X),r(Y)\}\geq 3$. We also say that a partition $(X,\{z\},Y)$ is a \textit{vertical $3$-separation} of $M$ when both $(X\cup z,Y)$ and $(X,Y\cup z)$ are vertical $3$-separations with $z$ in the guts. We will write $(X,z,Y)$ for $(X,\{z\},Y)$. If $(X,z,Y)$ is a vertical $3$-separation of $M$, then we say that $(X,z,Y)$ is a \emph{cyclic $3$-separation} of $M^*$.
Suppose $e \in E(M)$, and $(X,Y)$ is a partition of $M \backslash e$ with $\lambda_{M \backslash e}(X)=k$. We say that $e$ \emph{blocks} $X$ if $\lambda_{M}(X) > k$.
In particular, we say $e$ blocks a series pair (or triad) $X$ of $M \backslash e$ if $X$ is not a series pair (or triad, respectively) of $M$.
In any case, if $e$ blocks $X$, then $e \notin \cl_M(Y)$, so $e \in \cl^*_M(X)$. We say that $e$ \emph{fully blocks} $(X,Y)$ if both $\lambda_M(X) > k$ and $\lambda_M(X \cup e) > k$. It is easy to see that $e$ fully blocks $(X,Y)$ if and only if $e \notin \cl_M(X) \cup \cl_M(Y)$.
A set $X \subseteq E(M)$ is \emph{fully closed} if $X$ is closed in both $M$ and $M^*$. The \emph{full closure of $X$}, denoted $\fcl_M(X)$, is the intersection of all fully closed sets containing $X$. The full closure can be obtained by repeatedly taking closures and coclosures until no new elements are added.
The following lemma is routine, but helpful to show a matroid is $3$-connected up to series and parallel classes.
\begin{lemma}
\label{fclnontrivialsep}
Let $M$ be a simple and cosimple matroid, and let $(X,Y)$ be a $2$-separation of $M$.
Then $\fcl(X) \neq E(M)$ and $\fcl(Y) \neq E(M)$.
Moreover, $(\fcl(X),Y-\fcl(X))$ is also a $2$-separation of $M$. \end{lemma}
We use the following well-known results about the existence of elements that preserve connectivity. The first we refer to as Bixby's Lemma.
\begin{lemma}[Bixby's Lemma~\cite{Bixby82}] \label{bixby} Let $M$ be a $3$-connected matroid, and let $e\in E(M)$. Then $M/e$ is $3$-connected up to parallel pairs or $M\backslash e$ is $3$-connected up to series pairs. \end{lemma}
\begin{lemma}[see {\cite[Lemma~8.8.2]{Oxley11}}, for example]
\label{lineconn}
Let $M$ be a $3$-connected matroid and let $L$ be a segment of $M$ with $|L| \ge 4$. If $\ell \in L$, then $M \backslash \ell$ is $3$-connected. \end{lemma}
\begin{lemma}[{\cite[Lemma~3.5]{Whittle99}}]
\label{vert3sep}
Let $M$ be a $3$-connected matroid and let $z \in E(M)$.
The following are equivalent:
\begin{enumerate}
\item $M$ has a vertical $3$-separation $(X, z, Y)$.
\item $\si(M/z)$ is not 3-connected.
\end{enumerate} \end{lemma}
\begin{lemma}[see {\cite[Lemma 2.12]{BS14}}, for example] \label{fanends}
Let $M$ be a $3$-connected matroid with $|E(M)| \ge 7$. Suppose that $M$ has a fan $F$ of size at least~$4$, and let $f$ be an end of $F$. \begin{enumerate}
\item If $f$ is a spoke element, then $\co(M\backslash f)$ is $3$-connected and $\si(M/f)$ is not $3$-connected.
\item If $f$ is a rim element, then $\si(M/f)$ is $3$-connected and $\co(M\backslash f)$ is not $3$-connected. \end{enumerate} \end{lemma}
\begin{lemma}[{\cite[Lemma~1.5]{OW00}}] \label{fanendsstrong}
Let $M$ be $3$-connected matroid that is not a wheel or a whirl.
Suppose $M$ has a maximal fan~$F$ of size at least~$4$, and let $f$ be an end of $F$.
\begin{enumerate}
\item If $f$ is a spoke element, then $M\backslash f$ is $3$-connected.
\item If $f$ is a rim element, then $M/f$ is $3$-connected.
\end{enumerate} \end{lemma}
\begin{lemma}
\label{fanmiddle}
Let $M$ be a $3$-connected matroid,
and let $F$ be a $5$-element fan of $M$ with ordering $(f_1,f_2,\dotsc,f_5)$, where $\{f_2,f_3,f_4\}$ is a triangle.
Either $\si(M/f_3)$ is $3$-connected, or there exists some $f_6 \in E(M)-F$ such that $M^* | \{f_1,f_2,f_3,f_4,f_5,f_6\} \cong M(K_4)$. \end{lemma} \begin{proof}
We may assume that $M \not\cong M(\mathcal{W}_3)$, for otherwise $\si(M/f_3)$ is $3$-connected.
Thus, by \cref{fanunique}, $\si(M/f_3) \cong M/f_3\backslash f_2$.
Suppose that $\si(M/f_3)$ is not $3$-connected, so $M/f_3\backslash f_2$ has a $2$-separation $(U,V)$.
To begin with, assume that $\si(M/f_3)$ is cosimple.
Note that $\{f_1,f_2,f_4,f_5\}$ is a cocircuit of $M$, so $\{f_1,f_4,f_5\}$ is a triad of $M/f_3\backslash f_2$.
Without loss of generality, $|U \cap \{f_1,f_4,f_5\}| \ge 2$, and $U$ is fully closed by \cref{fclnontrivialsep}.
So $\{f_1,f_4,f_5\} \subseteq U$.
Now $f_2 \in \cl_{M/f_3}(U)$, so $(U \cup f_2,V)$ is a $2$-separation in $M/f_3$.
Moreover, $f_3 \in \cl^*_{M}(U \cup f_2)$, so $(U \cup \{f_2,f_3\},V)$ is a $2$-separation in $M$, contradicting that $M$ is $3$-connected.
Now we may assume that $M /f_3 \backslash f_2$ is not cosimple, so $f_2$ is in a triad~$T^*$ of $M$ that avoids $f_3$.
By orthogonality with the triangle $\{f_2,f_3,f_4\}$, we have $f_4 \in T^*$.
Since $M$ is $3$-connected, it follows that $T^* = \{f_2,f_4,f_6\}$ for some $f_6 \in E(M)-F$.
It now follows that $M^* | \{f_1,f_2,f_3,f_4,f_5,f_6\} \cong M(K_4)$, as required. \end{proof}
We also require the following lemma. \begin{lemma}[{\cite[Lemma~2.11]{BS14}}]
\label{gutsandcoguts}
Let $(X, Y)$ be a $3$-separation of a $3$-connected matroid $M$. If $X \cap \cl(Y) \neq \emptyset$ and $X \cap \cl^*(Y) \neq \emptyset$, then $|X \cap \cl(Y)| = 1$ and $|X \cap \cl^*(Y)| = 1$. \end{lemma}
\subsection*{Local connectivity}
Let $M$ be a matroid with $X,Y \subseteq E(M)$. The \emph{local connectivity} between $X$ and $Y$, denoted $\sqcap_M(X,Y)$, is defined to be $\sqcap_M(X,Y)= r(X)+r(Y)-r(X \cup Y)$. Evidently, $\sqcap_M(Y,X)= \sqcap_M(X,Y)$. Note that if $\{X,Y\}$ is a partition of $E(M)$, then $\sqcap_M(X,Y)= \lambda_M(X)$. We write $\sqcap$ instead of $\sqcap_M$ when $M$ is clear from context, and we write $\sqcap^*(X,Y)$ for $\sqcap_{M^*}(X,Y)$. We now recall some elementary properties.
\begin{lemma}[see {\cite[Lemma~8.2.3]{Oxley11}}, for example]
\label{growpi}
Let $X_1$, $X_2$, $Y_1$, and $Y_2$ be subsets of the ground set of a matroid. If $X_1 \subseteq Y_1$ and $X_2 \subseteq Y_2$, then $\sqcap(X_1,X_2) \le \sqcap(Y_1,Y_2)$. \end{lemma}
\begin{lemma}[{\cite[Lemma~2.4(iv)]{OSW04}}]
\label{pihelper}
If $\{X,Y,Z\}$ is a partition of the ground set of a matroid, then $\lambda(X) + \sqcap(Y,Z)= \lambda(Z)+ \sqcap(X,Y)$. \end{lemma}
The next lemma is elementary. \begin{lemma}
\label{picircuits}
For a matroid $M$, let $L$ and $R$ be disjoint subsets of $E(M)$.
If $\sqcap(L,R) = 0$, and $C$ is a circuit contained in $L \cup R$, then either $C \subseteq L$ or $C \subseteq R$. \end{lemma}
\begin{lemma}
\label{pflancoguts}
Let $M$ be a $3$-connected matroid with a path of $3$-separations $(X,Z,Y)$ such that $Z$ is a coguts set.
Then $\sqcap(X,Y) \le 1$.
Moreover, $\sqcap(X,Y)=1$ if and only if $|Z| = 1$. \end{lemma} \begin{proof}
Since each $z \in Z$ is a coguts element, $r(X \cup Z) = r(X) + |Z|$ and $|Z|=r(Z)$.
So $\sqcap(X,Z) = 0$. Now, by \cref{pihelper},
\begin{align*}
\lambda(Z) &= \lambda(Y) + \sqcap(X,Z) - \sqcap(X,Y) \\
&= 2 - \sqcap(X,Y).
\end{align*}
If $\sqcap(X,Y) = 2$, then $\lambda(Z)=0$, a contradiction.
So $\sqcap(X,Y) \le 1$.
Now if $\sqcap(X,Y) = 1$, then, as $M$ is $3$-connected, $|Z|=1$.
On the other hand, if $|Z| = 1$, then $\lambda(Z) = 1$, so $\sqcap(X,Y) = 1$, as required. \end{proof}
\begin{lemma}
\label{pflantriad}
Let $M$ be a $3$-connected matroid with a path of $3$-separations $(X,\{z_1\},\{z_2\},\{z_3\},Y)$ such that $z_1$ and $z_3$ are coguts elements, and $z_2$ is a guts element.
Then $\sqcap(X,Y) \le 1$.
Moreover, $\sqcap(X,Y)=1$ if and only if $\{z_1,z_2,z_3\}$ is a triad. \end{lemma} \begin{proof}
Let $Z = \{z_1,z_2,z_3\}$.
If $r(Z)=2$, then $z_1 \in \cl(Y \cup \{z_2,z_3\})$, so $z_1 \notin \cl^*(X)$, contradicting that $z_1$ is a coguts element.
So $r(Z)=3$.
Moreover, since $z_1$ and $z_3$ are coguts elements whereas $z_2$ is a guts element, $r(Y \cup Z) = r(Y)+2$.
So $\sqcap(Y,Z) = 1$.
Now, by \cref{pihelper},
\begin{align*}
\lambda(Z) &= \lambda(X) + \sqcap(Y,Z) - \sqcap(X,Y) \\
&= 3-\sqcap(X,Y).
\end{align*}
Since $|Z|=3$, we have $\lambda(Z) \ge 2$, implying $\sqcap(X,Y) \le 1$.
Now if $Z$ is a triad, then $\lambda(Z) = 2$ so $\sqcap(X,Y) = 1$.
On the other hand, if $\sqcap(X,Y) = 1$, then $\lambda(Z)=2$ in which case,
since $r(Z)=3$, we deduce that $Z$ is a triad. \end{proof}
\subsection*{Minors and fragility}
Let $M$ be a matroid, let $\mathcal{N}$ be a set of matroids, and let $x$ be an element of $M$. For a matroid $N$, we say that \emph{$M$ has an $N$-minor} if $M$ has a minor isomorphic to $N$. We say $M$ has an $\mathcal{N}$-minor if $M$ has an $N$-minor for some $N \in \mathcal{N}$. If $M\backslash x$ has an $\mathcal{N}$-minor, then $x$ is $\mathcal{N}$-\textit{deletable}. If $M/x$ has an $\mathcal{N}$-minor, then $x$ is $\mathcal{N}$-\textit{contractible}. If neither $M\backslash x$ nor $M/x$ has an $\mathcal{N}$-minor, then $x$ is $\mathcal{N}$-\textit{essential}. If $x$ is both $\mathcal{N}$-deletable and $\mathcal{N}$-contractible, then we say that $x$ is \textit{$\mathcal{N}$-flexible}. A matroid $M$ is \textit{$\mathcal{N}$-fragile} if $M$ has an $\mathcal{N}$-minor, and no element of $M$ is $\mathcal{N}$-flexible (note that sometimes this is referred to in the literature as ``strictly $\mathcal{N}$-fragile'').
For $X \subseteq E(M)$, we also say that $X$ is \emph{$\mathcal{N}$-deletable} (or \emph{$\mathcal{N}$-contractible}) when $M \backslash X$ (or $M / X$, respectively) has an $\mathcal{N}$-minor. When $\mathcal{N} = \{N\}$, we use the prefix ``$N$-'' for these terms, rather than ``$\{N\}$-''.
The next lemma is well known, and the subsequent lemma is a straightforward corollary.
\begin{lemma}[see {\cite[Corollary~8.2.5]{Oxley11}}, for example]
\label{minor3conn}
Let $M$ be a matroid with a $2$-separation $(X,Y)$, and let $N$ be a $3$-connected minor of $M$.
Then $|Z \cap E(N)| \le 1$ for some $Z \in \{X,Y\}$. \end{lemma}
\begin{lemma}
\label{niceVertSep}
Let $(X, z, Y)$ be a vertical $3$-separation of a $3$-connected matroid $M$, and let $N$ be a $3$-connected minor of $M/z$. Then there exists a vertical $3$-separation $(X', z, Y')$ of $M$ such that $|X' \cap E(N)| \le 1$ and $Y' \cup z$ is closed in $M$. \end{lemma}
The following is proved in \cite{BS14,Clark15}.
\begin{lemma}
\label{CPL}
Let $N$ be a $3$-connected minor of a $3$-connected matroid $M$. Let $(X, \{z\}, Y)$ be a vertical $3$-separation of $M$ such that $M / z$ has an $N$-minor with $|X \cap E(N)| \le 1$. Let $X' = X-\cl(Y)$
and $Y' = \cl(Y) - z$.
Then
\begin{enumerate}
\item each element of $X'$ is $N$-contractible; and
\item at most one element of $\cl(X)-z$ is not $N$-deletable, and if such an element~$x$ exists, then $x \in X' \cap \cl^*(Y')$ and $z \in \cl(X' - x)$.
\end{enumerate} \end{lemma}
We also use the following well-known property of fragile matroids.
\begin{lemma}[see {\cite[Proposition~4.4]{MvZW10}}, for example]
\label{genfragileconn}
Let $\mathcal{N}$ be a non-empty set of $3$-connected matroids with $|E(N)| \ge 4$ for each $N \in \mathcal{N}$.
If $M$ is $\mathcal{N}$-fragile, then $M$ is $3$-connected up to series and parallel classes. \end{lemma}
We require two more lemmas, about fans in fragile matroids. Recall that a maximal $4$-element fan has one rim element and one spoke element at the two ends: the internal elements are not considered to be rim or spoke elements.
\begin{lemma}
\label{fragilefanelements}
Let $\mathcal{N}$ be a non-empty set of $3$-connected matroids,
let $M$ be a $\mathcal{N}$-fragile matroid, and let $F$ be a fan of $M$ of size at least~$4$.
If $s$ is a spoke element of $F$, then $s$ is not $\mathcal{N}$-contractible,
whereas if $t$ is a rim element of $F$, then $t$ is not $\mathcal{N}$-deletable. \end{lemma} \begin{proof}
Let $(f_1,f_2,f_3,s)$ be a (not necessarily maximal) fan where $\{f_1,f_2,f_3\}$ is a triad and $\{f_2,f_3,s\}$ is a triangle, so $s$ is a spoke element, and suppose that $s$ is $\mathcal{N}$-contractible.
Since each $N \in \mathcal{N}$ is $3$-connected, $\si(M/s)$ has an $\mathcal{N}$-minor.
So $f_2$ and $f_3$ are $\mathcal{N}$-deletable.
Similarly, $\co(M \backslash f_2)$ has an $\mathcal{N}$-minor, so $f_3$ is $\mathcal{N}$-contractible, due to the triad $\{f_1,f_2,f_3\}$ of $M$.
But now $f_3$ is $\mathcal{N}$-flexible, a contradiction.
A similar argument applies if $(f_1,f_2,s,f_4,f_5)$ is a fan where $\{f_1,f_2,s\}$ and $\{s,f_4,f_5\}$ are triangles and $\{f_2,s,f_4\}$ is a triad.
The result then follows by duality. \end{proof}
\begin{lemma}
\label{fragilefans}
Let $\mathcal{N}$ be a non-empty set of $3$-connected matroids, each of which has no $4$-element fans. Let $M$ be a $\mathcal{N}$-fragile matroid, and let $F$ be a fan of $M$.
\begin{enumerate}
\item If $|F| \ge 5$ and $e$ is an end of $F$, then $e$ is not $\mathcal{N}$-essential.
\item If $|F| \ge 6$ and $e \in F$, then $e$ is not $\mathcal{N}$-essential.
\end{enumerate} \end{lemma} \begin{proof}
Suppose $|F| = 5$ and let $(f_1,f_2,f_3,f_4,f_5)$ be a fan ordering of $F$.
By duality, we may assume $f_1$ is a spoke element, so $\{f_1,f_2,f_3\}$ is a triangle.
By \cref{fragilefanelements}, $f_1$ is not $\mathcal{N}$-contractible.
Suppose $f_1$ is not $\mathcal{N}$-deletable.
Since each matroid in $\mathcal{N}$ is $3$-connected, a matroid $M'$ has an $\mathcal{N}$-minor if and only if $\si(M')$ has an $\mathcal{N}$-minor (and the same holds when ``$\si(M')$'' is replaced with ``$\co(M')$'').
If $f_2$ is $\mathcal{N}$-contractible, then, as $\{f_1,f_3\}$ is a parallel pair in $M / f_2$, the element $f_1$ is $\mathcal{N}$-deletable.
So $f_2$ is not $\mathcal{N}$-contractible due to the triangle $\{f_1,f_2,f_3\}$.
Similarly, due to the triad $\{f_2,f_3,f_4\}$, the element $f_3$ is not $\mathcal{N}$-deletable.
Finally, due to the triangle $\{f_3,f_4,f_5\}$, the element $f_4$ is not $\mathcal{N}$-contractible.
By \cref{fragilefanelements}, the elements $\{f_1,f_2,f_3,f_4\}$ are $\mathcal{N}$-essential.
We have that $M /C \backslash D \cong N$ for some $N \in \mathcal{N}$ and disjoint $C,D \subseteq E(M)$. Let $N' = M/C\backslash D$.
Now, $r_{N'}(\{f_1,f_2,f_3\}) \le 2$ and $r^*_{N'}(\{f_2,f_3,f_4\}) \le 2$, a contradiction.
Now suppose $|F|=6$ and let $(f_1,f_2,\dotsc,f_6)$ be a fan ordering of $F$.
By the foregoing, $f_1$, $f_2$, $f_5$, and $f_6$ are not
$\mathcal{N}$-essential.
Up to duality, we may assume that $f_1$ is a spoke element.
Then $\{f_1,f_2,f_3\}$ is a triangle and $f_2$ is a rim element, so $f_2$ is
$\mathcal{N}$-contractible
by \cref{fragilefanelements}.
Since $\{f_1,f_3\}$ is a parallel pair in $M / f_2$, it follows that $f_3$ is
$\mathcal{N}$-deletable.
By a symmetric argument, $f_4$ is
$\mathcal{N}$-contractible.
The result follows. \end{proof}
\subsection*{Path width three}
A matroid $M$ has \emph{path width at most $k$} if there exists an ordering $(e_1,e_2,\dotsc,e_{n})$ of $E(M)$ such that $\{e_1,\dotsc,e_t\}$ is $k$-separating for all $t \in \seq{n-1}$. For a $3$-connected matroid $M$ with $|E(M)| \ge 4$ and path width at most three, $M$ does not have path width at most two, so we simply say that $M$ has \emph{path width three}. When $M$ has path width three with respect to the ordering $(e_1,e_2,\dotsc,e_{n})$, then we say $(e_1,e_2,\dotsc,e_{n})$ is a \emph{sequential ordering} of $M$. The next lemma is well known, and it implies that, relative to such a sequential ordering, each $e_i \in \{e_3,e_4,\dotsc,e_{n-2}\}$ is unambiguously a guts or a coguts element.
\begin{lemma}
\label{gutsstayguts}
Let $M$ be a $3$-connected matroid, and let $(X, e, Y)$ be a partition of $E$ such that $X$ is exactly $3$-separating. Then
\begin{enumerate}
\item $X \cup e$ is $3$-separating if and only if $e \in \cl(X)$ or $e \in \cl^*(X)$, and
\item $X \cup e$ is exactly $3$-separating if and only if either $e \in \cl(X) \cap \cl(Y)$ or $e \in \cl^*(X) \cap \cl^*(Y)$.
\end{enumerate} \end{lemma}
We say that a set $X$ in a matroid~$M$ is \emph{path generating} if $X$ is $3$-separating and $\fcl_M(X)= E(M)$. In particular, if $M$ has path width three and $(e_1,e_2,\dotsc,e_{n})$ is a sequential ordering of $M$, then $\{e_1,e_2\}$ and $\{e_{n-1},e_n\}$ are path-generating sets.
Let $M$ be a $3$-connected matroid of path width three that has rank and corank at least~$3$ and is not a wheel or a whirl. Let $\sigma=(e_1, e_2, \dotsc, e_n)$ be a sequential ordering of $M$. Then $\{e_1, e_2, e_3\}$ is a triangle or a triad. If this set is not in a larger segment, cosegment, or fan of $M$, then let $L(\sigma) = \{e_1, e_2, e_3\}$ and call $L(\sigma)$ a \emph{triangle end} or a \emph{triad end} of $M$, respectively. If $\{e_1, e_2, e_3\}$ is contained in a $4$-segment or $4$-cosegment, then let $L(\sigma)$ be the maximal segment or cosegment (respectively) containing $\{e_1,e_2,e_3\}$, and call $L(\sigma)$ a \emph{segment end} or a \emph{cosegment end} of $M$, respectively. Finally, if $\{e_1, e_2, e_3\}$ is contained in a fan of size at least~$4$, then take a maximal such fan $F$, let $L(\sigma)$ be the set of internal elements of the fan $F$, and call $L(\sigma)$ a \emph{fan end} of $\sigma$. We define $R(\sigma)$ analogously.
Loosely speaking, the next lemma shows that, up to reversal, any sequential ordering of a matroid of path width three has the same pair of ends.
\begin{lemma}[{\cite[Theorem 1.3]{HOS07}}]
\label{welldefinedends}
Let $M$ be a $3$-connected matroid of path width three that has rank and corank at least~$3$ and is not a wheel or a whirl.
Then there are distinct subsets $L(M)$ and $R(M)$ of $E(M)$ such that $\{L(M), R(M)\} = \{L(\sigma), R(\sigma)\}$ for every sequential ordering $\sigma$ of $E(M)$. \end{lemma}
\begin{lemma}[{\cite[Theorem 1.4]{HOS07}}]
\label{endslipperiness}
Let $M$ be a $3$-connected matroid of path width three that has rank and corank at least~$3$ and is not a wheel or a whirl.
Let $\sigma$ and $\sigma'$ be sequential orderings of $M$ such that $L(\sigma)=L(\sigma')$ and $R(\sigma)=R(\sigma')$.
Then
\begin{enumerate}
\item if $L(\sigma)$ is a triangle or a triad end of $M$, then the first three elements of $\sigma'$ are in $L(\sigma)$;\label{esi}
\item if $L(\sigma)$ is a segment or cosegment end of $M$, then the first $|L(\sigma)| - 1$ elements of $\sigma'$ are in $L(\sigma)$; and\label{esii}
\item if $L(\sigma)$ is a fan end of $M$, then either the first $|L(\sigma)|$ elements of $\sigma'$ are in $L(\sigma)$, or there is a maximal fan~$F$ of $M$ having $L(\sigma)$ as its set of internal elements such that the first $|L(\sigma)| +1$ elements of $\sigma'$ include $L(\sigma)$ and are contained in $F$.\label{esiii}
\end{enumerate} \end{lemma}
Let $\mathbf{P} = (P_1, P_2, \dotsc, P_n)$ be an ordered partition of a set $S$. Then the ordered partition $\mathbf{Q} = (Q_1, Q_2, \dotsc, Q_m)$ is a \emph{concatenation} of $\mathbf{P}$ if there are indices $0 = k_0 < k_1 < \dotsm < k_m = n$ such that $Q_i = P_{k_{i-1}+1} \cup \dotsm \cup P_{k_i}$ for $i \in \{1, \dotsc,m\}$. If $\mathbf{Q}$ is a concatenation of $\mathbf{P}$, then $\mathbf{P}$ is a \emph{refinement} of $\mathbf{Q}$.
Let $\mathbf{P} = (P_1,P_2,\dotsc,P_m)$ be an ordered partition of the ground set of a matroid~$M$ with path width three. We say that $\mathbf{P}$ is a \emph{guts-coguts path} if $\mathbf{P}$ is a path of $3$-separations such that, for each $i \in \{2,3,\dotsc,m-1\}$, the set $P_i$ is in the guts or coguts of the $3$-separation $(P_1 \cup \dotsm \cup P_{i},P_{i+1} \cup \dotsm \cup P_m)$, and, for each $i \in \{2,3,\dotsc,m-2\}$, if $P_i$ is in the guts (respectively, the coguts), then $P_{i+1}$ is in the coguts (respectively, the guts).
Let $\sigma=(e_1,e_2,\dotsc,e_n)$ be a sequential ordering of a $3$-connected matroid~$M$ with path width three.
We treat $\sigma$ as a partition into singletons, in which case any concatenation of $\sigma$ is a path of $3$-separations.
For $X \subseteq E(M)$, we say that $X$ is an \emph{initial segment} of $\sigma$ if $X = \{e_i : i \in [j]\}$ for some $j \in [n]$, and
$X$ is a \emph{terminal segment} of $\sigma$ if $X = \{e_i : j \le i \le n\}$ for some $j \in [n]$.
For an initial segment $P$ and a terminal segment $P'$ of $\sigma$, where $P$ and $P'$ are disjoint and each have size at least~$2$, there is a unique concatenation $(P_1,P_2,\dotsc,P_m)$ of $\sigma$ that is a guts-coguts path with $P=P_1$ and $P' = P_m$ (where uniqueness follows from \cref{gutsstayguts}).
We call $(P_1,P_2,\dotsc,P_m)$ the \emph{guts-coguts concatenation of $\sigma$ with ends $P$ and $P'$}.
We also call $P$ the \emph{left end}, and $P'$ the \emph{right end}.
\subsection*{Representation theory}
A \textit{partial field} is a pair $(R, G)$, where $R$ is a commutative ring with unity, and $G$ is a subgroup of the group of units of $R$ such that $-1 \in G$. If $\mathbb{P}=(R,G)$ is a partial field, then we write $p\in \mathbb{P}$ whenever $p\in G\cup \{0\}$.
Let $\mathbb{P}$ be a partial field, and let $A$ be an $X\times Y$ matrix with entries from $\mathbb{P}$. Then $A$ is a $\mathbb{P}$-\textit{matrix} if every subdeterminant of $A$ is contained in $\mathbb{P}$. If $X'\subseteq X$ and $Y'\subseteq Y$, then we write $A[X',Y']$ to denote the submatrix of $A$ induced by $X'$ and $Y'$. When $X$ and $Y$ are disjoint, and $Z\subseteq X\cup Y$, we denote by $A[Z]$ the submatrix induced by $X\cap Z$ and $Y\cap Z$, and we denote by $A-Z$ the submatrix induced by $X-Z$ and $Y-Z$.
\begin{theorem}[{\cite[Theorem 2.8]{PvZ10b}}] \label{pmatroid} Let $\mathbb{P}$ be a partial field, and let $A$ be an $X\times Y$ $\mathbb{P}$-matrix, where $X$ and $Y$ are disjoint. Let \begin{equation*}
\mathcal{B}=\{X\}\cup \{X\triangle Z : |X\cap Z|=|Y\cap Z|, \det(A[Z])\neq 0\}. \end{equation*}
Then $\mathcal{B}$ is the set of bases of a matroid on $X\cup Y$. \end{theorem}
We say that the matroid in \cref{pmatroid} is $\mathbb{P}$-\textit{representable}, and that $A$ is a $\mathbb{P}$-\textit{representation} of $M$. We write $M=M[I|A]$ if $A$ is a $\mathbb{P}$-matrix, and $M$ is the matroid whose bases are described in \cref{pmatroid}.
Let $A$ be an $X\times Y$ $\mathbb{P}$-matrix, with $X \cap Y = \emptyset$, and let $x\in X$ and $y\in Y$ such that $A_{xy}\neq 0$. Then we define $A^{xy}$ to be the $(X\triangle \{x,y\})\times (Y\triangle \{x,y\})$ $\mathbb{P}$-matrix given by \begin{displaymath}
(A^{xy})_{uv} = \begin{cases}
A_{xy}^{-1} \quad & \textrm{if } uv = yx\\
A_{xy}^{-1} A_{xv} & \textrm{if } u = y, v\neq x\\
-A_{xy}^{-1} A_{uy} & \textrm{if } v = x, u \neq y\\
A_{uv} - A_{xy}^{-1} A_{uy} A_{xv} & \textrm{otherwise.} \end{cases} \end{displaymath}
We say that $A^{xy}$ is obtained from $A$ by \textit{pivoting} on $xy$.
Two $\mathbb{P}$-matrices are \textit{scaling equivalent} if one can be obtained from the other by repeatedly scaling rows and columns by nonzero elements of $\mathbb{P}$. Two $\mathbb{P}$-matrices are \textit{geometrically equivalent} if one can be obtained from the other by a sequence of the following operations: scaling rows and columns by nonzero entries of $\mathbb{P}$, permuting rows, permuting columns, and pivoting.
Let $\mathbb{P}$ be a partial field, and let $M$ and $N$ be matroids such that $N$ is a minor of $M$. Suppose that the ground set of $N$ is $X'\cup Y'$, where $X'$ is a basis of $N$. We say that $M$ is $\mathbb{P}$-\textit{stabilized by $N$} if, whenever $A_1$ and $A_2$ are $X\times Y$ $\mathbb{P}$-matrices, with $X'\subseteq X$ and $Y'\subseteq Y$, such that \begin{enumerate}
\item[(i)] $M=M[I|A_1]=M[I|A_2]$,
\item[(ii)] $A_1[X',Y']$ is scaling equivalent to $A_2[X',Y']$, and
\item[(iii)] $N=M[I|A_1[X',Y']]=M[I|A_2[X',Y']],$ \end{enumerate} then $A_1$ is scaling equivalent to $A_2$. If $M$ is $\mathbb{P}$-stabilized by $N$, and every $\mathbb{P}$-representation of $N$ extends to a $\mathbb{P}$-representation of $M$, then we say $M$ is \emph{strongly $\mathbb{P}$-stabilized} by $N$.
Let $\mathcal{M}$ be a class of matroids. We say that $N$ is a \textit{$\mathbb{P}$-stabilizer for $\mathcal{M}$} if, for every $3$-connected $\mathbb{P}$-representable matroid $M\in \mathcal{M}$ with an $N$-minor, $M$ is $\mathbb{P}$-stabilized by $N$. We say that $N$ is a \textit{strong $\mathbb{P}$-stabilizer for $\mathcal{M}$} if, for every $3$-connected $\mathbb{P}$-representable matroid $M\in \mathcal{M}$ with an $N$-minor, $M$ is strongly $\mathbb{P}$-stabilized by $N$. Here we will be primarily interested in the case where $\mathcal{M}$ is the class of $\mathbb{P}$-representable matroids for some partial field~$\mathbb{P}$, in which case, when there is no ambiguity, we simply say ``$N$ is a strong $\mathbb{P}$-stabilizer''.
\subsection*{\texorpdfstring{$2$}{2}-regular, \texorpdfstring{$3$}{3}-regular, and \texorpdfstring{$\mathbb{H}_5$}{H5}-representable matroids}
The \emph{2-regular} partial field is $$\mathbb{U}_2 = (\mathbb{Q}(\alpha_1, \alpha_2),\left<-1,\alpha_1, \alpha_2, 1-\alpha_1, 1-\alpha_2,\alpha_1-\alpha_2\right>),$$ where $\alpha_1$ and $\alpha_2$ are indeterminates. Recall that we say a matroid is \emph{2-regular} if it is $\mathbb{U}_2$-representable. Note that $\mathbb{U}_2$ is the universal partial field of $U_{2,5}$ \cite[Theorem 3.3.24]{vanZwam09}; intuitively, this means that $\mathbb{U}_2$ is the most general partial field that $U_{2,5}$ is representable over (for a formal definition, refer to \cite{PvZ10b}). If a matroid is $2$-regular, then it is $\mathbb{F}$-representable for every field $\mathbb{F}$ of size at least~$4$~\cite[Corollary 3.1.3]{Semple98}.
The \emph{$3$-regular} partial field is: \begin{multline*} \mathbb{U}_3 = (\mathbb{Q}(\alpha_1,\alpha_2,\alpha_3), \\ \left<-1,\alpha_1,\alpha_2,\alpha_3,\alpha_1-1,\alpha_2-1,\alpha_3-1,\alpha_1-\alpha_2,\alpha_1-\alpha_3,\alpha_2-\alpha_3\right>), \end{multline*} where $\alpha_1,\alpha_2,\alpha_3$ are indeterminates, and recall that we say a matroid is \emph{$3$-regular} if it is $\mathbb{U}_3$-representable.
The \emph{Hydra-5}- partial field is \begin{multline*} \mathbb{H}_5 = (\mathbb{Q}(\alpha, \beta, \gamma), \\ \left<-1,\alpha, \beta, \gamma, 1-\alpha, 1-\beta, 1-\gamma, \alpha-\gamma, \gamma-\alpha\beta, 1-\gamma - (1-\alpha)\beta \right>), \end{multline*} where $\alpha$, $\beta$, and $\gamma$ are indeterminates. A $3$-connected matroid with a $\{U_{2,5},U_{3,5}\}$-minor is $\mathbb{H}_5$-representable if and only if it has six inequivalent representations over $\textrm{GF}(5)$ \cite[Lemma~5.17]{PvZ10b}.
We next prove that the partial fields $\mathbb{U}_3$ and $\mathbb{H}_5$ are isomorphic; in particular, a matroid is $\mathbb{H}_5$-representable if and only if it is $3$-regular.
For partial fields $\mathbb{P}_1$ and $\mathbb{P}_2$, a function $\phi : \mathbb{P}_1 \rightarrow \mathbb{P}_2$ is a \emph{homomorphism} if \begin{enumerate}
\item $\phi(1) = 1$,
\item $\phi(pq) = \phi(p)\phi(q)$ for all $p, q \in \mathbb{P}_1$, and
\item $\phi(p) +\phi(q) = \phi(p +q)$ for all $p, q \in \mathbb{P}_1$ such that $p +q \in \mathbb{P}_1$. \end{enumerate}
The existence of a homomorphism from $\mathbb{P}_1$ to $\mathbb{P}_2$ certifies that each $\mathbb{P}_1$-representable matroid is also $\mathbb{P}_2$-representable \cite[Corollary 2.9]{PvZ10b}.
\begin{lemma}
\label{3reglemma}
The partial fields $\mathbb{H}_5$ and $\mathbb{U}_3$ are isomorphic.
In particular, a matroid is $3$-regular if and only if it is $\mathbb{H}_5$-representable. \end{lemma} \begin{proof}
It is easy, but tedious, to check that $\phi : \mathbb{H}_5 \rightarrow \mathbb{U}_3$ determined by
$$\phi(\alpha) = \alpha_1,$$
$$\phi(\beta) = \frac{\alpha_3-\alpha_2}{\alpha_1-\alpha_2},$$
$$\phi(\gamma) = \alpha_3$$
is well-defined, and is a homomorphism.
In particular, observe that $\phi(\gamma-\alpha\beta)=\frac{\alpha_2(\alpha_1-\alpha_3)}{\alpha_1-\alpha_2}$, and $\phi((1-\gamma)-(1-\alpha)\beta)=\frac{(\alpha_3-\alpha_1)(\alpha_2-1)}{\alpha_1-\alpha_2}$.
Moreover, it is also easily checked that $\phi' : \mathbb{U}_3 \rightarrow \mathbb{H}_5$ determined by
$$\phi'(\alpha_1) = \alpha,$$
$$\phi'(\alpha_2) = \frac{\gamma-\alpha\beta}{1-\beta},$$
$$\phi'(\alpha_3) = \gamma$$
is well-defined, and a homomorphism.
Clearly $\phi'(\phi(\alpha)) = \alpha$ and $\phi'(\phi(\gamma)) = \gamma$.
Furthermore, $$\phi'(\phi(\beta)) = \frac{\gamma - \frac{\gamma-\alpha\beta}{1-\beta}}{\alpha - \frac{\gamma-\alpha\beta}{1-\beta}} = \frac{\frac{\alpha\beta-\gamma\beta}{1-\beta}}{\frac{\alpha-\gamma}{1-\beta}} =\beta.$$
It now follows that $\phi'(\phi(x))=x$ for any $x \in \mathbb{H}_5$.
Similarly $\phi(\phi'(x))=x$ for any $x \in \mathbb{U}_3$.
Hence $\phi$ is a bijection with inverse $\phi'$, so the partial fields $\mathbb{H}_5$ and $\mathbb{U}_3$ are isomorphic. \end{proof}
The next lemma is a consequence of \cite[Lemmas~5.7 and~5.25]{OSV00} when $\mathbb{P} = \mathbb{U}_2$, and is proved for $\mathbb{P} = \mathbb{H}_5$ in \cite[Lemma~7.3.16]{vanZwam09}.
\begin{lemma}[{\cite{OSV00,vanZwam09}}]
\label{u2stabs}
For $\mathbb{P} \in \{\mathbb{H}_5,\mathbb{U}_2\}$,
the matroids $U_{2,5}$ and $U_{3,5}$ are
non-binary, $3$-connected,
strong $\mathbb{P}$-stabilizers. \end{lemma}
\subsection*{Certifying non-representability}
Let $\mathbb{P}$ be a partial field. Let $M$ be a matroid and let $E(M)=X \cup Y$ where $X$ and $Y$ are disjoint. Let $A$ be an $X \times Y$ matrix with entries in $\mathbb{P}$ such that, for some distinct $a, b \in Y$, both $A-a$ and $A-b$ are $\mathbb{P}$-matrices, $M \backslash a=M[I|A-a]$, and $M \backslash b=M[I|A-b]$. Then we say $A$ is an $X \times Y$ \emph{companion $\mathbb{P}$-matrix} for $M$.
Let $B$ be a basis of $M$. We write $B^*$ to denote $E(M)-B$. Let $A$ be a $B\times B^*$ matrix with entries in $\mathbb{P}$. A subset~$Z$ of $E(M)$ \textit{incriminates} the pair $(M, A)$ if $A[Z]$ is square and one of the following holds: \begin{enumerate}
\item $\det(A[Z])\notin \mathbb{P}$,
\item $\det(A[Z])=0$ but $B\triangle Z$ is a basis of $M$, or
\item $\det(A[Z])\neq 0$ but $B\triangle Z$ is dependent in $M$. \end{enumerate}
The next lemma follows immediately.
\begin{lemma}
Let $M$ be a matroid, let $A$ be an $X\times Y$ matrix with entries in $\mathbb{P}$, where $X$ and $Y$ are disjoint, and $X\cup Y=E(M)$. Exactly one of the following statements is true: \begin{enumerate}
\item $A$ is a $\mathbb{P}$-matrix and $M=M[I | A]$, or
\item there is some $Z\subseteq X\cup Y$ that incriminates $(M, A)$. \end{enumerate} \end{lemma}
Let $M$ be an excluded minor for the class of $\mathbb{P}$-representable matroids. We will obtain a $B \times B^*$ companion $\mathbb{P}$-matrix $A$ for $M$ such that $\{x,y,a,b\}$ incriminates $(M,A)$ for some distinct $x,y \in B$ and $a,b \in B^*$. In this setting, for $p \in B$ and $q \in B^*$ where $A_{pq} \neq 0$, we say that the pivot $A^{pq}$ is \emph{allowable} if $\{p,q\} \cap \{x,y,a,b\} \neq \emptyset$ and $\{x,y,a,b\} \triangle \{p,q\}$ incriminates $(M,A^{pq})$, or $\{p,q\} \cap \{x,y,a,b\} = \emptyset$ and $\{x,y,a,b\}$ incriminates $(M,A^{pq})$. The next two lemmas describe situations where a pivot is allowable.
\begin{lemma}[{\cite[Lemma 5.10]{MvZW10}}]
\label{allowablexyrow2}
Let $A$ be a $B\times B^{*}$ companion $\mathbb{P}$-matrix for $M$. Suppose that $\{x,y,a,b\}$ incriminates $(M,A)$, for pairs $\{x,y\}\subseteq B$ and $\{a,b\}\subseteq B^{*}$. If $p\in \{x,y\}$, $q\in B^{*}-\{a,b\}$, and $A_{pq}\neq 0$, then
$A^{pq}$ is an allowable pivot. \end{lemma}
\begin{lemma}[{\cite[Lemma 5.11]{MvZW10}}]
\label{allowablenonxy2}
Let $A$ be a $B\times B^{*}$ companion $\mathbb{P}$-matrix for $M$. Suppose that $\{x,y,a,b\}$ incriminates $(M,A)$, for pairs $\{x,y\}\subseteq B$ and $\{a,b\}\subseteq B^{*}$. If $p\in B-\{x,y\}$, $q\in B^{*}-\{a,b\}$, $A_{pq}\neq 0$, and either $A_{pa}=A_{pb}=0$ or $A_{xq}=A_{yq}=0$, then
$A^{pq}$ is an allowable pivot. \end{lemma}
\subsection*{Delta-wye exchange}
Let $M$ be a matroid with a triangle $T=\{a,b,c\}$. Consider a copy of $M(K_4)$ having $T$ as a triangle with $\{a',b',c'\}$ as the complementary triad labelled such that $\{a,b',c'\}$, $\{a',b,c'\}$ and $\{a',b',c\}$ are triangles. Let $P_{T}(M,M(K_4))$ denote the generalised parallel connection of $M$ with this copy of $M(K_4)$ along the triangle $T$. Let $M'$ be the matroid $P_{T}(M,M(K_4))\backslash T$ where the elements $a'$, $b'$ and $c'$ are relabelled as $a$, $b$ and $c$ respectively. The matroid $M'$ is said to be obtained from $M$ by a \emph{$\Delta$\nobreakdash-$Y$\ exchange} on the triangle~$T$, and is denoted $\Delta_T(M)$. Dually, $M''$ is obtained from $M$ by a \emph{$Y$\nobreakdash-$\Delta$\ exchange} on the triad $T^*=\{a,b,c\}$ if $(M'')^*$ is obtained from $M^*$ by a $\Delta$\nobreakdash-$Y$\ exchange on $T^*$. The matroid $M''$ is denoted $\nabla_{T^*}(M)$.
We say that a matroid $M_1$ is \emph{$\Delta$\nobreakdash-$Y$-equivalent} to a matroid $M_0$ if $M_1$ can be obtained from $M_0$ by a sequence of $\Delta$\nobreakdash-$Y$\ and $Y$\nobreakdash-$\Delta$\ exchanges on coindependent triangles and independent triads, respectively.
We let $\Delta^*(M)$ denote the set of matroids that are $\Delta$\nobreakdash-$Y$-equivalent to $M$ or $M^*$.
Oxley, Semple, and Vertigan proved that the set of excluded minors for $\mathbb{P}$-representability is closed under $\Delta$\nobreakdash-$Y$\ exchange. \begin{proposition}[{\cite[Theorem~1.1]{OSV00}}]
\label{osvdelta}
Let $\mathbb{P}$ be a partial field, and let $M$ be an excluded minor for the class of $\mathbb{P}$-representable matroids.
If $M' \in \Delta^*(M)$, then $M'$ is an excluded minor for the class of $\mathbb{P}$-representable matroids. \end{proposition}
\subsection*{Robust and strong elements, and bolstered bases}
Let $M$ be a $3$-connected matroid, let $B$ be a basis of $M$, and let $N$ be a $3$-connected minor of $M$. Recall that we write $B^*$ to denote $E(M)-B$. An element $e \in E(M)$ is \textit{$(N,B)$-robust} if either \begin{enumerate}
\item $e\in B$ and $M/e$ has an $N$-minor, or
\item $e\in B^*$ and $M\backslash e$ has an $N$-minor. \end{enumerate}
\noindent Note that an $N$-flexible element of $M$ is clearly $(N,B)$-robust for any basis~$B$ of $M$.
An element $e \in E(M)$ is \textit{$(N,B)$-strong} if either \begin{enumerate}
\item $e\in B$ and $\si(M/e)$ is $3$-connected and has an $N$-minor, or
\item $e\in B^*$ and $\co(M\backslash e)$ is $3$-connected and has an $N$-minor. \end{enumerate}
Now let $\{a,b\}$ be a pair of elements of $M$ such that $M \backslash a,b$ is $3$-connected with an $N$-minor. Let $B$ be a basis of a matroid $M \backslash a,b$, and let $A$ be a $B\times B^{*}$ companion $\mathbb{P}$-matrix of $M$ such that $\{x,y,a,b\}$ incriminates $(M,A)$, for some $\{x,y\}\subseteq B$.
If either
\begin{itemize}
\item [(i)] $M \backslash a,b$ has exactly one $(N,B)$-strong element $u$ outside of $\{x,y\}$, and $\{u,x,y\}$ is a triad of $M \backslash a,b$; or
\item [(ii)] $M \backslash a,b$ has no $(N,B')$-strong elements outside of $\{x',y'\}$ for every choice of basis~$B'$ with a $B'\times (B')^{*}$ companion $\mathbb{P}$-matrix $A'$ of $M$ such that $\{x',y',a,b\}$ incriminates $(M,A')$, for some $\{x',y'\}\subseteq B'$;
\end{itemize}
then $B$ is a \emph{strengthened} basis.
In other words, a basis~$B$ is strengthened if $B$ is chosen such that either there is one $(N,B)$-strong element $u$ of $M \backslash a,b$ outside of $\{x,y\}$, and $\{u,x,y\}$ is a triad; or there are no $(N,B)$-strong elements outside of $\{x,y\}$, and, moreover, there are no $(N,B')$-strong elements outside of $\{x',y'\}$ for any choice of basis~$B'$ with an incriminating set $\{x',y',a,b\}$ where $\{x',y'\} \subseteq B'$.
In particular, for a strengthened basis~$B$ with no $(N,B)$-strong elements, an allowable pivot cannot introduce an $(N,B)$-strong element.
Now suppose $B$ is strengthened. We say that $B$ is \emph{bolstered} if \begin{enumerate}
\item when $M \backslash a,b$ has no $(N,B)$-strong elements outside of $\{x,y\}$, then
for any $B_1\times B_1^{*}$ companion $\mathbb{P}$-matrix~$A_1$ where $\{x_1,y_1,a,b\}$ incriminates $(M,A_1)$, with $\{x_1,y_1\}\subseteq B_1$ and $\{a,b\}\subseteq B_1^{*}$, the number of $(N,B)$-robust elements of $M \backslash a,b$ outside of $\{x,y\}$ is at least the number of $(N,B_1)$-robust elements of $M \backslash a,b$ outside of $\{x_1,y_1\}$; or
\item when $M \backslash a,b$ has an $(N,B)$-strong element $u$ of $M \backslash a,b$ outside of $\{x,y\}$, then for any $B_1\times B_1^{*}$ companion $\mathbb{P}$-matrix~$A_1$ such that
\begin{enumerate}[label=\rm(\Roman*)]
\item $\{x,y,a,b\}$ incriminates $(M,A_1)$, with $\{x,y\}\subseteq B_1$ and $\{a,b\}\subseteq B_1^{*}$, and
\item $u$ is the only $(N,B_1)$-strong element of $M \backslash a,b$, with $u \in B_1^*$,
\end{enumerate}
the number of $(N,B)$-robust elements of $M \backslash a,b$ is at least the number of $(N,B_1)$-robust elements of $M \backslash a,b$. \end{enumerate}
Loosely speaking, a strengthened basis~$B$ is bolstered if no allowable pivot increases the number of elements that are robust but not strong.
\section{Excluded minors are almost fragile} \label{almostfragsec}
We now recap results that we require from \cite{BCOSW20,paper1}. All of these results, appearing in the remainder of this section, are under the following hypotheses: Let $\mathbb{P}$ be a partial field. Let $M$ be an excluded minor for the class of $\mathbb{P}$-representable matroids, and let $N$ be a non-binary $3$-connected strong stabilizer for the class of $\mathbb{P}$-representable matroids, where $M$ has an $N$-minor.
The first result implies that we can essentially restrict attention to an excluded minor with no triads. A proof appears in \cite{paper1}, but it is essentially a consequence of the main theorem proved in \cite{BWW20,BWW21,BWW22}.
\begin{lemma}[{\cite[Lemma~3.1]{paper1}}]
\label{notriads2}
Suppose that $|E(M)| \ge |E(N)| + 10$.
Then there exists a matroid~$M_1 \in \Delta^*(M)$ such that $M_1$ has a pair of elements $\{a,b\}$ for which $M_1 \backslash a,b$ is $3$-connected and has a $\Delta^*(N)$-minor, and $M_1$ has no triads. \end{lemma}
The following theorem addresses the case where $M \backslash a,b$ is not $N$-fragile, and is extracted from {\cite[Theorem~6.7]{BCOSW20}}. For item~\ref{easywin2}, the fact that the triangle is closed is established as \cite[Lemma~3.4]{paper1}.
\begin{theorem}[{\cite[Theorem~6.7(ii)(b)]{BCOSW20}}]
\label{bcosw-thm}
Let $a,b \in E(M)$ be a pair of elements for which $M \backslash a,b$ is $3$-connected with an $N$-minor.
Suppose $|E(M)| \ge |E(N)| + 10$, and $M \backslash a,b$ is not $N$-fragile.
Then
\begin{enumerate}
\item $M$ has a bolstered basis~$B$, and a $B\times B^{*}$ companion $\mathbb{P}$-matrix $A$ for which $\{x,y,a,b\}$ incriminates $(M,A)$, where $\{x,y\}\subseteq B$ and $\{a,b\}\subseteq B^{*}$, and there is an element $u \in B^*-\{a,b\}$ that is $(N,B)$-strong in $M \backslash a,b$;
\item either\label{bcoswii}
\begin{enumerate}[label=\rm(\Roman*)]
\item the $N$-flexible, and $(N,B)$-robust, elements of $M\backslash a,b$ are contained in $\{u,x,y\}$, or\label{bcoswii1}
\item the $N$-flexible, and $(N,B)$-robust, elements of $M\backslash a,b$ are contained in $\{u,x,y,z\}$, where $z \in B$, and $(z,u,x,y)$ is a maximal fan of $M \backslash a,b$, or\label{bcoswii2}
\item the $N$-flexible, and $(N,B)$-robust, elements of $M\backslash a,b$ are contained in $\{u,x,y,z,w\}$, where $z \in B$, $w \in B^*$, and $(w,z,x,u,y)$ is a maximal fan of $M \backslash a,b$;\label{bcoswii3}
\end{enumerate}
\item the unique triad in $M \backslash a,b$ containing $u$ is $\{u,x,y\}$;\label{bcoswiii}
\item $M$ has a cocircuit $\{x,y,u,a,b\}$; and
\item for some $d\in \{a,b\}$, the set $\{d,x,y\}$ is a closed triangle.\label{easywin2}
\end{enumerate} \end{theorem}
A consequence of this theorem is that $M \backslash a,b$ has no $M(K_4)$ restriction or co-restriction, as shown below.
\begin{lemma}
\label{gadgetnotMK4}
Let $a,b \in E(M)$ be a pair of elements for which $M \backslash a,b$ is $3$-connected with an $N$-minor.
Suppose $|E(M)| \ge |E(N)| + 10$, and $M \backslash a,b$ is not $N$-fragile, so \cref{bcosw-thm} holds.
Let $Z$ be the set of $N$-flexible elements of $M \backslash a,b$.
Then there is no set $Z'$ containing $Z$ such that either $(M\backslash a,b)|Z' \cong M(K_4)$ or $(M\backslash a,b)^*|Z' \cong M(K_4)$. \end{lemma} \begin{proof}
Towards a contradiction, let $Z'$ be a set containing $Z$ such that $M'|Z' \cong M(K_4)$ for some $M' \in \{M\backslash a,b,(M\backslash a,b)^*\}$.
Note that every element of $Z'$ is in at least $2$ triangles $T_1$ and $T_2$ of $M'$, where $r_{M'}(T_1 \cup T_2)=3$.
Since $u \in Z \subseteq Z'$ and $\{u,x,y\}$ is the unique triad containing $u$, we have $M' = M \backslash a,b$.
Now $u$ is in two triangles of $(M \backslash a,b)|Z'$ that are not both contained in a common segment.
By orthogonality, one of these two triangles contains $\{u,x\}$, and the other contains $\{u,y\}$ (and, in particular, $x,y \in Z'$).
Since $\{u,y\}$ is contained in a triangle, neither \ref{bcoswii2} nor \ref{bcoswii3} of \cref{bcosw-thm}\ref{bcoswii} holds.
So \cref{bcosw-thm}\ref{bcoswii}\ref{bcoswii1} holds.
Then there exist elements $e,f,g \in E(M \backslash a,b) - \{x,y,u\}$ such that $\{u,x,f\}$, $\{u,y,g\}$, $\{e,f,g\}$ and $\{x,y,e\}$ are triangles.
But the triangle $\{x,y,e\}$ contradicts \cref{bcosw-thm}\ref{easywin2}. \end{proof}
The next three results are from \cite{paper1}. \begin{lemma}[{\cite[Lemma~3.3]{paper1}}]
\label{wmatype1again}
Suppose $M$ has a pair of elements $\{a,b\}$ such that $M\backslash a,b$ is $3$-connected with an $N$-minor, $|E(M)| \ge |E(N)| + 10$, and $M \backslash a,b$ is not $N$-fragile, so \cref{bcosw-thm} holds.
Assume \cref{bcosw-thm}\ref{easywin2} holds with $d=b$.
Then, either
\begin{enumerate}
\item the $N$-flexible elements of $M \backslash a,b$ are contained in $\{u,x,y\}$, or
\item $M \backslash a,x$ is $3$-connected with an $N$-minor, but is not $N$-fragile, and there are at most three $N$-flexible elements in $M \backslash a,x$.
\end{enumerate} \end{lemma}
\begin{theorem}[{\cite[Theorem~3.5]{paper1}}]
\label{thegrandfantasy}
Suppose $M$ has a pair of elements $\{a,b\}$ such that $M\backslash a,b$ is $3$-connected with an $N$-minor, $M$ has no triads, $|E(M)| \ge |E(N)| + 11$, and $M \backslash a,b$ is not $N$-fragile, so \cref{bcosw-thm} holds.
If the $N$-flexible elements of $M \backslash a,b$ are contained in $\{u,x,y\}$, then, for every $b' \in B-\{x,y\}$, the element $b'$ is $N$-essential in at least one of $M \backslash a,b\backslash u$ and $M \backslash a,b/u$. \end{theorem}
\begin{lemma}[{\cite[Lemma~4.1]{paper1}}]
\label{subfrag3conn}
Suppose $M$ has a pair of elements $\{a,b\}$ such that $M\backslash a,b$ is $3$-connected with an $N$-minor, $|E(M)| \ge |E(N)| + 10$, and $M \backslash a,b$ is not $N$-fragile, so \cref{bcosw-thm} holds.
Suppose the $N$-flexible elements of $M \backslash a,b$ are contained in $\{u,x,y\}$,
and $M \backslash a,b,u/x$ is not $N$-fragile.
Then either $M \backslash a,b,u/x/y$ or $M \backslash a,b,u/x \backslash y$ is $3$-connected and $N$-fragile. \end{lemma}
The next theorem is the counterpart to \cref{bcosw-thm} that addresses the case where $M \backslash a,b$ is $N$-fragile. Item~\ref{nostronginbasis} was established as \cite[Lemma~3.1]{BCOSW20}.
\begin{theorem}[{\cite[Theorem~6.7(ii)(a)]{BCOSW20}}]
\label{fragilecase}
Let $a,b \in E(M)$ be a pair of elements for which $M \backslash a,b$ is $3$-connected with an $N$-minor.
Suppose $|E(M)| \ge |E(N)| + 10$, and $M \backslash a,b$ is $N$-fragile.
Then
\begin{enumerate}
\item $M$ has a bolstered basis~$B$, and a $B\times B^{*}$ companion $\mathbb{P}$-matrix $A$ for which $\{x,y,a,b\}$ incriminates $(M,A)$, where $\{x,y\}\subseteq B$ and $\{a,b\}\subseteq B^{*}$; and
\item $M \backslash a,b$ has at most one $(N,B)$-robust element outside of $\{x,y\}$, where if such an element $u$ exists, then $u \in B^*-\{a,b\}$ is an $(N,B)$-strong element of $M \backslash a,b$, and $\{u,x,y\}$ is a coclosed triad of $M\backslash a,b$.
\item if $v$ is an $(N,B_1)$-strong element of $M \backslash a,b$, for some basis $B_1$ such that there exists a $B_1 \times B_1^*$ companion $\mathbb{P}$-matrix~$A_1$ of $M$ where $\{x_1, y_1, a, b\}$ incriminates $(M, A_1)$, and $\{x_1,y_1\} \subseteq B_1$ and $\{a,b\} \subseteq B_1^*$, then $v \notin B_1 - \{x_1,y_1\}$.\label{nostronginbasis}
\end{enumerate} \end{theorem}
The last theorem implies that $M \backslash a,b$ cannot have arbitrarily large fans, as proved below.
\begin{corollary}
\label{fragilefanscase}
Assume that $N$ has no $4$-element fans, and let $a,b \in E(M)$ be a pair of elements for which $M \backslash a,b$ is $3$-connected with an $N$-minor.
Suppose that $|E(M)| \ge |E(N)| + 10$ and $M \backslash a,b$ is $N$-fragile, so \cref{fragilecase} holds.
Then $M \backslash a,b$ has no fan with more than five elements.
Moreover, if $(f_1,f_2,f_3,f_4,f_5)$ is a fan
in $M \backslash a,b$, then either
\begin{enumerate}
\item there is a triad $\{u,x,y\} \in \{\{f_1,f_2,f_3\},\{f_3,f_4,f_5\}\}$, where $u$ is the unique $(N,B)$-robust element outside of $\{x,y\}$, and $u \in \{f_2,f_4\}$,\label{ffcc1}
\item $\{f_1,f_2,f_3\}$ is a triad, and $\{f_2,f_4\} = \{x,y\}$,\label{ffcc2}
\item each element in $\{f_2,f_3,f_4\}$ is $N$-essential, or\label{ffcc3}
\item $\{f_1,f_2,f_3\}$ is a triad, and $\si(M/f_3)$ is not $3$-connected.\label{ffcc4}
\end{enumerate} \end{corollary} \begin{proof}
By \cref{fragilecase}, $M \backslash a,b$ has at most one $(N,B)$-robust element outside of $\{x,y\}$ and if such an element $u$ exists, then $u$ is an $(N,B)$-strong element of $M \backslash a,b$ that is in $B^* - \{a,b\}$, and $\{u,x,y\}$ is a coclosed triad of $M \backslash a,b$.
Let $F$ be a maximal fan of $M \backslash a,b$ of size at least~$5$.
By \cref{fragilefanelements}, if $s$ is a spoke element of $F$ that is not $N$-essential, then $M \backslash a,b \backslash s$ has an $N$-minor;
whereas if $t$ is a rim element of $F$ that is not $N$-essential, then $M \backslash a,b / t$ has an $N$-minor.
The only $(N,B)$-robust elements are in $\{x,y\}$ or in $\{x,y,u\}$ for an element $u \in B^*-\{a,b\}$.
We deduce that $s \in B \cup u$ for any such spoke $s$ of $F$, and
$t \in B^* \cup \{x,y\}$ for any such rim $t$ of $F$.
\begin{claim}
If $M \backslash a,b$ has a triangle $\{t_1,t_2,t_3\}$ where $\{t_1,t_3\} \subseteq B$, the element $t_2$ is $N$-contractible in $M \backslash a,b$, and $\si(M/t_2)$ is $3$-connected, then $\{t_1,t_3\} = \{x,y\}$.
\label{ffcclaim}
\end{claim}
\begin{subproof}
Assume $M \backslash a,b$ has such a triangle $\{t_1,t_2,t_3\}$.
As $M \backslash a,b$ is $N$-fragile, $t_2$ is not $N$-deletable and, in particular, $t_2 \neq u$.
Moreover, $t_2 \in B^*$.
Suppose $\{t_1,t_3\}$ avoids $\{x,y\}$.
Then, by \cref{allowablenonxy2}, a pivot on $A_{t_1t_2}$ is allowable.
Let $B' = B \triangle \{t_1,t_2\}$.
Now $t_2 \in B'-\{x,y\}$ and $t_2$ is an $(N, B')$-strong element, contradicting \cref{fragilecase}\ref{nostronginbasis}.
Next suppose that $t_1 \in \{x,y\}$ but $t_3 \notin \{x,y\}$.
Without loss of generality, let $t_1 = x$.
First, observe that if the element $u$ exists, then by orthogonality between the triad $\{u,x,y\}$ and the triangle $\{t_1,t_2,t_3\}$, we have $t_2=u$, a contradiction.
So $M \backslash a,b$ has no $(N,B)$-strong elements.
By \cref{allowablexyrow2}, the pivot on $A_{xt_2}$ is allowable.
Let $B' = B \triangle \{x,t_2\}$.
Since $t_2$ is $N$-contractible, and $\{x,t_3\}$ is a parallel pair in $M \backslash a,b / t_2$, the element $x$ is $N$-deletable.
Then $x$ is $(N, B')$-robust, whereas $t_2$ is not $(N,B)$-robust, so the number of $(N,B')$-robust elements outside of $\{t_2,y\}$ is greater than the number of $(N,B)$-robust elements outside of $\{x,y\}$, contradicting that $B$ is a bolstered basis.
We deduce that $\{t_1,t_3\} = \{x,y\}$, as required.
\end{subproof}
Let $(f_1,f_2,\dotsc,f_\ell)$ be a fan ordering of $F$.
Suppose first that $M \backslash a,b$ has an $(N,B)$-robust element $u$, where $u \in F$.
If $u$ is a rim element of $F$, then $u$ is not $N$-deletable by \cref{fragilefanelements}, contradicting that $u$ is $(N,B)$-robust. So we may
assume that $u$ is a spoke element $f_i$ of $F$.
Suppose $3 \le i \le \ell-2$.
Then $f_{i-2}$ and $f_{i+2}$ are spokes, so they are $N$-deletable by \cref{fragilefanelements,fragilefans}.
Since $u$ is the only $(N,B)$-robust element of $M \backslash a,b$ in $B^*$, we have $\{f_{i-2},f_{i+2}\} \subseteq B$.
Let $F' = \{f_{i-2},f_{i-1},f_{i},f_{i+1},f_{i+2}\}$.
Observe that $\{x,y\} \neq \{f_{i-1},f_{i+1}\}$, for otherwise the rank-$3$ fan $F'$ contains four elements of the basis~$B$.
By orthogonality between the triad $\{u,x,y\}$ and the triangles $\{f_{i-2},f_{i-1},u\}$ and $\{u,f_{i+1},f_{i+2}\}$, we have $\{x,y\} \subseteq F'$.
Now $F'$ contains distinct triads $\{f_{i-1},u,f_{i+1}\}$ and $\{u,x,y\}$, so $r^*_{M \backslash a,b}(F') \le 3$.
But $r(F') = 3$, so $\lambda_{M \backslash a,b}(F') \le 1$, a contradiction.
Next, let $i = 2$.
Suppose $\{f_1,f_2,f_3\} = \{u,x,y\}$.
In the case that $|F| \ge 6$, the set $\{f_4,f_5,f_6\}$ is a triangle.
Then, by \cref{fragilefanelements,fragilefans}, $f_5$ is $N$-contractible, so $f_4$ and $f_6$ are $N$-deletable.
Moreover, $\si(M/f_5)$ is $3$-connected by \cref{fanends}.
So $\{f_4,f_6\} = \{x,y\}$ by \cref{ffcclaim}, a contradiction.
Thus $|F| = 5$, and \ref{ffcc1} holds in this case.
Now we may assume that $\{f_1,f_2,f_3\} \neq \{u,x,y\}$.
Observe that $\{f_1,f_3\}$ does not meet $\{x,y\}$,
for otherwise $\{u,x,y\}$ is not coclosed in $M \backslash a,b$.
In particular, $f_3 \notin \{x,y\}$.
So $f_4 \in \{x,y\}$, by orthogonality with the triangle $\{u,f_3,f_4\}$.
Since $u$ is $N$-deletable in $M\backslash a,b$, and $\{f_1,f_3\}$ is a parallel pair in $M \backslash a,b \backslash u$, the element $f_3$ is $N$-contractible.
So $f_3 \in B^*$.
Then $f_1 \in B$, since the triad $\{f_1,f_2,f_3\}$ cannot be contained in $B^*$.
But then $f_1$ is $(N,B)$-robust by \cref{fragilefanelements,fragilefans}, and $f_1 \notin \{x,y\}$, a contradiction.
Now let $i=1$, so $u$ is a spoke end of $F$.
By orthogonality, $\{x,y\}$ meets $\{f_2,f_3\}$.
If $f_3 \in \{x,y\}$, then $f_3$ is in distinct triads $\{u,x,y\}$ and $\{f_2,f_3,f_4\}$, which contradicts \cref{fanunique}.
So $f_3 \notin \{x,y\}$.
Without loss of generality, say $f_2 =x$.
If $y \notin F$, then $F \cup y$ is a fan, contradicting that $F$ is maximal.
So $y \in F$.
Then, by orthogonality, $y=f_\ell$ is a rim end.
Moreover, $u \in \cl(F-u)$ due to the triangle $\{u,x,f_3\}$, and $u \in \cl^*_{M \backslash a,b}(F-u)$ due to the triad $\{u,x,y\}$.
But $\lambda_{M \backslash a,b}(F-u) \le 2$ and so $\lambda_{M \backslash a,b}(F) \le 1$, implying that $|E(M \backslash a,b)-F| \le 1$.
If $F=E(M \backslash a,b)$, then it follows that $M \backslash a,b$ is a rank-$\frac{\ell}{2}$ wheel or whirl.
But $f_i \in B$ for each $i \in \{3,5,\dotsc,\ell-1\}$, and $x,y \in B$, so $|B| = \ell/2 + 1$, a contradiction.
Now $|E(M \backslash a,b)-F| = 1$, so let $E(M \backslash a,b)-F = \{q\}$.
Then $(\{u,x\}, f_3, f_4, \dotsc, f_{\ell-1},\{y,q\})$ is a path of $3$-separations, so $\{f_{\ell-1},y,q\}$ is a triangle or a triad. But it is not a triangle, by orthogonality with the triad $\{u,x,y\}$, and it is not a triad, since $F$ is maximal, a contradiction.
Finally, we may assume that there are no $(N,B)$-robust elements contained in $F - \{x,y\}$.
Suppose $F$ contains a $5$-element fan $F'$ with fan ordering $(f_1',f_2',\dotsc,f_5')$ such that $\{f'_1,f'_2,f'_3\}$ is a triangle.
Then $\{f'_3,f'_4,f'_5\}$ is also a triangle, and $f'_1$, $f'_3$, and $f'_5$ are the spoke elements.
\Cref{fragilefanelements,fragilefans} imply that $f'_1$ and $f'_5$ are $N$-deletable in $M \backslash a,b$.
Moreover, $\si(M/f'_2)$ and $\si(M/f'_4)$ are $3$-connected by \cref{fanends}.
If $f'_3$ is $N$-deletable, then $\{f'_1,f'_3\} = \{x,y\} = \{f'_3,f'_5\}$ by \cref{ffcclaim}, a contradiction.
Thus $f'_3$ is $N$-essential and, in particular, $|F|=5$, by \cref{fragilefans}.
Due to the triangles $\{f'_1,f'_2,f'_3\}$ and $\{f'_3,f'_4,f'_5\}$, and by \cref{fragilefanelements}, it follows that $f'_2$ and $f'_4$ are also $N$-essential.
So \ref{ffcc3} holds in this case.
We may now assume that $|F| = 5$, and when $(f_1,f_2,\dotsc,f_5)$ is a fan ordering of $F$, the set $\{f_2,f_3,f_4\}$ is a triangle.
Next we claim that if $\{f_2,f_3,f_4\}$ contains an element that is not $N$-essential, then no element of $F$ is $N$-essential.
Suppose $f_3$ is not $N$-essential.
Then $f_3$ is $N$-contractible, by \cref{fragilefanelements,fragilefans}.
Since $\{f_2,f_4\}$ is a parallel pair in $M \backslash a,b / f_3$, the elements $f_2$ and $f_4$ are $N$-deletable, so no element of $F$ is $N$-essential.
Similarly, if $f_2$ (or $f_4$) is not $N$-essential, then no element in $F$ is $N$-essential.
This proves the claim.
We may now assume $\{f_2,f_3,f_4\}$ contains an element that is not $N$-essential, otherwise \ref{ffcc3} holds.
Then, by the foregoing and \cref{fragilefanelements}, $f_2$ and $f_4$ are $N$-deletable, and $f_3$ is $N$-contractible.
So $\{f_2,f_4\} \subseteq B$, and hence $f_3 \in B^*$.
If $\si(M/f_3)$ is not $3$-connected, then \ref{ffcc4} holds.
Otherwise, $\si(M/f_3)$ is $3$-connected, in which case $\{f_2,f_4\} = \{x,y\}$, by \cref{ffcclaim}. So \ref{ffcc2} holds. \end{proof}
\section{\texorpdfstring{$\{U_{2,5},U_{3,5}\}$}{\{U(2,5),U(3,5)\}}-fragile matroids} \label{utfutfsec}
In this section, we recap some known properties of $\{U_{2,5},U_{3,5}\}$-fragile matroids \cite{CMvZW16}, and prove some further structural properties of this class that have not previously been explicitly stated. Recall that, by definition, when we say a matroid is $\{U_{2,5},U_{3,5}\}$-fragile, it has an $\{U_{2,5},U_{3,5}\}$-minor.
Throughout this section, we focus on $\{U_{2,5},U_{3,5}\}$-fragile matroids, rather than $U_{2,5}$-fragile or $U_{3,5}$-fragile matroids.
\Cref{utfutfequiv}, which follows from the following well-known lemma, connects these classes of fragile matroids.
\begin{lemma}[see {\cite[Proposition~12.2.15]{Oxley11}}, for example]
\label{utfutfprop}
Let $M$ be a $3$-connected matroid with rank and corank at least~$3$.
Then $M$ has a $U_{2,5}$-minor if and only if $M$ has a $U_{3,5}$-minor. \end{lemma}
\begin{corollary}
\label{utfutfequiv}
Let $M$ be a $3$-connected matroid with rank and corank at least~$3$, and $|E(M)| \ge 7$.
Then $M$ is $U_{2,5}$-fragile and $U_{3,5}$-fragile if and only if $M$ is $\{U_{2,5},U_{3,5}\}$-fragile. \end{corollary} \begin{proof}
Suppose $M$ is $\{U_{2,5},U_{3,5}\}$-fragile.
Then $M$ has a $\{U_{2,5},U_{3,5}\}$-minor, so, by \cref{utfutfprop}, $M$ has both a $U_{2,5}$- and a $U_{3,5}$-minor.
So clearly $M$ is $U_{2,5}$-fragile and $U_{3,5}$-fragile.
Now let $M$ be $U_{2,5}$-fragile and $U_{3,5}$-fragile and, towards a contradiction, suppose $M$ is not $\{U_{2,5},U_{3,5}\}$-fragile.
Clearly $M$ has a $\{U_{2,5},U_{3,5}\}$-minor.
So, by duality, we may assume that, for some $e \in E(M)$, the matroid $M \backslash e$ has a $U_{3,5}$-minor (but $M /e$ does not), and $M/e$ has a $U_{2,5}$-minor (but $M \backslash e$ does not).
Since $U_{2,5}$ and $U_{3,5}$ are $3$-connected, $\co(M \backslash e)$ has a $U_{3,5}$-minor and $\si(M/e)$ has a $U_{2,5}$-minor.
In particular, $\co(M \backslash e)$ has rank at least~$3$, and $\si(M/e)$ has corank at least~$3$.
Since $|E(M)| \ge 7$, the rank or corank of $M$ is at least~$4$.
Assume without loss of generality that $M$ has corank at least~$4$.
Then $M \backslash e$ and $\co(M \backslash e)$ have corank at least~$3$.
Since $M \backslash e$ has a $U_{3,5}$-minor, it is $U_{3,5}$-fragile.
As $M$ is $3$-connected, and by \cref{genfragileconn}, $M \backslash e$ is $3$-connected up to series classes.
Now $\co(M \backslash e)$ is $3$-connected and has rank and corank at least~$3$.
Thus $\co(M \backslash e)$, and hence $M \backslash e$, has a $U_{2,5}$-minor by \cref{utfutfprop}.
But $M / e$ has a $U_{2,5}$-minor, so $e$ is $U_{2,5}$-flexible in $M$, and hence $M$ is not $U_{2,5}$-fragile, a contradiction.
We deduce that $M$ is $\{U_{2,5},U_{3,5}\}$-fragile, as required. \end{proof}
Let $(x_1,x_2,x_3)$ be an ordered subset of elements of a matroid $M$ in which $\{x_1,x_2,x_3\}$ is a triangle~$T$. Let $W$ be a copy of the rank-$r$ wheel $M(\mathcal{W}_r)$ having a triangle $\{x_1,x_2,x_3\}$ where $x_1$ and $x_3$ are spoke elements. Let $X \subseteq \{x_1,x_2,x_3\}$ such that $x_2 \in X$. We say that \emph{gluing an $r$-wheel onto $(x_1,x_2,x_3)$} (with \emph{remove set} $X$) is the operation by which we obtain the matroid $P_T(M,W) \backslash X$, where $P_T(M,W)$ is the generalized parallel connection of $M$ and $W$ across the triangle $T$.
The following was proved by Chun et al.~\cite{CCCMWvZ13,CCMvZ15}.
Geometric representations of the matroids $M_{9,9}$, $X_8$ and $Y_8$ are given in \cref{pathdescminors-fig}.
\begin{theorem}[{\cite[Theorem~1.3 and Corollary~1.4]{CCCMWvZ13}}]
\label{ccmwvz-result}
Let $\mathbb{P} \in \{\mathbb{H}_5, \mathbb{U}_2\}$, and
let $M$ be a $3$-connected $\{U_{2,5},U_{3,5}\}$-fragile $\mathbb{P}$-representable matroid.
Then either
\begin{enumerate}
\item $M$ has an $\{X_8,Y_8,Y_8^*\}$-minor;\label{pathlikecase}
\item $M$ is isomorphic to a matroid in $\{U_{2,6},U_{4,6},P_6,M_{9,9},M_{9,9}^*\}$;
\item $M$ or $M^*$ can be obtained from $Y_8 \backslash 4$
by gluing a wheel to $(1,5,7)$;
\item $M$ or $M^*$ can be obtained from $U_{2,5}$, with $E(U_{2,5})=\{e_1,e_2,e_3,e_4,e_5\}$, by gluing up to two wheels to $(e_1,e_2,e_3)$ and $(e_3,e_4,e_5)$; or
\item $M$ or $M^*$ can be obtained from $U_{2,5}$, with $E(U_{2,5})=\{e_1,e_2,e_3,e_4,e_5\}$, by gluing up to three wheels to $(e_1,e_3,e_2)$, $(e_1,e_4,e_2)$, and $(e_1, e_5, e_2)$.
\end{enumerate} \end{theorem}
In the case that \ref{pathlikecase} holds, Clark et al.~\cite{CMvZW16} proved the following:
\begin{theorem}[\cite{CMvZW16,Clark15}]
\label{nicepathdescription}
Let $\mathbb{P} \in \{\mathbb{H}_5, \mathbb{U}_2\}$, and
let $M$ be a $3$-connected $\{U_{2,5},U_{3,5}\}$-fragile $\mathbb{P}$-representable matroid with an $\{X_8,Y_8,Y_8^*\}$-minor.
Then $M$ has path width three.
Moreover, $M$ has a guts-coguts path $(P_1,P_2,\dotsc,P_m)$ such that \begin{enumerate}
\item for $\{i,i'\} = \{1,m\}$, the set $P_i$ is path generating, and is either a triangle, triad, $4$-segment, $4$-cosegment, or fan of size at least~$4$;\label{npd1}
\item for $\{i,i'\} = \{1,m\}$, the set $P_i$ is maximal in the sense that there is no $P'$ with $P_i \subsetneqq P' \subseteq E(M) - P_{i'}$ such that $P'$ is a segment, cosegment or fan;
\item for $i \in \{1,m\}$, if $P_i$ is not a fan of size at least~$4$, then either $P_i$ is a segment containing an element that is not $\{U_{2,5},U_{3,5}\}$-deletable, or $P_i$ is a cosegment containing an element that is not $\{U_{2,5},U_{3,5}\}$-contractible; and\label{nicepathdesciii}
\item $|P_i| \le 3$ for each $i \in \{2,3,\dotsc,m-1\}$. \end{enumerate} \end{theorem}
\noindent Note that the result stated here is essentially a stronger version of \cite[Lemma~4.1 and Theorem 4.2]{CMvZW16} that follows from \cite[Lemmas~2.21 and 2.22]{CMvZW16} (see also \cite[Lemma~3.3.1]{Clark15}).
We say that a guts-coguts path $(P_1,P_2,\dotsc,P_m)$ as described in \cref{nicepathdescription} is a \emph{nice path description} for $M$.
Note that a nice path description is not necessarily unique, even up to reversal. However, a nice path description $(P_1,P_2,\dotsc,P_m)$ can be refined to a sequential ordering~$\sigma$. By \cref{welldefinedends}, $M$ has a well-defined pair of ends $\{L(M),R(M)\} = \{L(\sigma),R(\sigma)\}$. If both ends of $M$ are triangle or triad ends, then, by \cref{endslipperiness}\ref{esi}, $\{L(M),R(M)\} = \{P_1,P_m\}$. If $M$ has a segment or cosegment end, $L(M)$ say, then, by \cref{endslipperiness}\ref{esii}, $P_i \subseteq L(M)$ and $|P_i| \in \{3,4\}$ for some $i \in \{1,m\}$. In the case that $M$ has a fan end, the outcome from \cref{endslipperiness}\ref{esiii} is more complicated, partly due to the fact an $M(K_4)$ restriction has three distinct maximal $5$-element fans (see \cite[Theorem 1.6]{OW00}); however, we will see, as \cref{noMK4}, that a $3$-connected $\{U_{2,5},U_{3,5}\}$-fragile matroid has no $M(K_4)$ restriction or co-restriction.
\begin{figure}
\caption{$Y_8$.}
\caption{$X_8$.}
\caption{$M_{9,9}$.}
\caption{
Geometric representations of matroids appearing in \cref{ccmwvz-result}.}
\label{pathdescminors-fig}
\end{figure}
We now prove some more properties of $\{U_{2,5},U_{3,5}\}$-fragile matroids with nice path descriptions.
\begin{lemma}
\label{noessential}
Let $M$ be a matroid with an $\{X_8,Y_8,Y_8^*\}$-minor.
Then $M$ has no $\{U_{2,5},U_{3,5}\}$-essential elements. \end{lemma} \begin{proof}
Observe that, as $M/C \backslash D$ is isomorphic to a matroid in $\{X_8,Y_8,Y_8^*\}$ for some disjoint sets $C,D \subseteq E(M)$, it suffices to show that at least one of $N \backslash z$ and $N / z$ has a $\{U_{2,5},U_{3,5}\}$-minor for all $N \in \{X_8,Y_8,Y_8^*\}$ and all $z \in E(N)$.
We show this for $N \in \{X_8,Y_8\}$; the result then follows by duality.
Using the labelling given in \cref{pathdescminors-fig}, observe that $Y_8/3\backslash \{y_1,y_2\} \cong U_{2,5}$ for every $2$-element subset $\{y_1,y_2\}$ of $\{1,2,4\}$.
Since $Y_8 / 5 \backslash \{7,8\} \cong U_{2,5}$, it follows, by symmetry, that for all $z \in E(Y_8)$, at least one of $Y_8 \backslash z$ and $Y_8/z$ has a $\{U_{2,5},U_{3,5}\}$-minor.
Now $X_8 / \{5,7\} \backslash y \cong U_{2,5}$ for $y \in \{2,6\}$ and $X_8 / \{5,8\} \backslash 1 \cong U_{2,5}$.
Also $X_8 / 3 \backslash \{y_1,y_2\} \cong U_{3,5}$ for every $2$-element subset $\{y_1,y_2\}$ of $\{1,2,4\}$.
Thus $X_8 \backslash z$ or $X_8 / z$ has a $\{U_{2,5},U_{3,5}\}$-minor, for all $z \in E(X_8)$.
\end{proof}
\begin{lemma}
\label{pathdesctris}
Let $M$ be a $3$-connected $\{U_{2,5},U_{3,5}\}$-fragile $\mathbb{P}$-representable matroid with $|E(M)| \ge 10$, for $\mathbb{P} \in \{\mathbb{H}_5, \mathbb{U}_2\}$, such that $M$ has an $\{X_8,Y_8,Y_8^*\}$ minor.
\begin{enumerate}
\item If $T$ is a triangle of $M$, then at least two elements of $T$ are $\{U_{2,5},U_{3,5}\}$-deletable.
\item If $T^*$ is a triad of $M$, then at least two elements of $T^*$ are $\{U_{2,5},U_{3,5}\}$-contractible.
\end{enumerate} \end{lemma} \begin{proof}
Let $T=\{a,b,c\}$ be a triangle of $M$.
By \cref{noessential}, $M$ has no $\{U_{2,5},U_{3,5}\}$-essential elements.
If $c \in T$ is $\{U_{2,5},U_{3,5}\}$-contractible, say, then $a$ and $b$ are $\{U_{2,5},U_{3,5}\}$-deletable, since $\{a,b\}$ is a parallel pair in $M/c$, and $U_{2,5}$ and $U_{3,5}$ are $3$-connected.
Since $M$ is $\{U_{2,5},U_{3,5}\}$-fragile, neither $a$ nor $b$ is $\{U_{2,5},U_{3,5}\}$-contractible.
So $T$ contains at most one $\{U_{2,5},U_{3,5}\}$-contractible element. The result follows by duality. \end{proof}
\begin{lemma}
\label{pathdescprops}
Let $M$ be a $3$-connected $\{U_{2,5},U_{3,5}\}$-fragile $\mathbb{P}$-representable matroid with $|E(M)| \ge 10$, for $\mathbb{P} \in \{\mathbb{H}_5, \mathbb{U}_2\}$, having a nice path description $(P_1,P_2,\dotsc,P_m)$.
For each $i \in \{2,3,\dotsc,m-1\}$,
\begin{enumerate}
\item if $P_i$ is a guts set and $e \in P_i$, then $e$ is
$\{U_{2,5},U_{3,5}\}$-deletable, and
$\co(M \backslash e)$ is $3$-connected; and
\item if $P_i$ is a coguts set and $e \in P_i$, then $e$ is
$\{U_{2,5},U_{3,5}\}$-contractible, and
$\si(M / e)$ is $3$-connected.
\end{enumerate} \end{lemma} \begin{proof}
For some such $i$, let $e \in P_i$.
Suppose $P_i$ is a guts set.
Note that if $i=2$, then $P_1$ is not a triangle or $4$-segment.
It follows that $r(P_1 \cup \dotsm \cup P_{i-1}) \ge 3$.
By symmetry, $r(P_{i+1} \cup \dotsm \cup P_{m}) \ge 3$.
If $e \in P_i$ is $N$-contractible, for $N \in \{U_{2,5},U_{3,5}\}$,
then it follows from \cref{CPL} that $M$ has an element that is $N$-flexible, a contradiction.
So each $e \in P_i$ is not $\{U_{2,5},U_{3,5}\}$-contractible.
By \cref{noessential}, each $e \in P_i$ is $\{U_{2,5},U_{3,5}\}$-deletable.
Moreover, $\si(M/e)$ is not $3$-connected, by \cref{vert3sep}, so $\co(M\backslash e)$ is $3$-connected by Bixby's Lemma.
By a dual argument, if $P_i$ is a coguts set then each $e \in P_i$ is $\{U_{2,5},U_{3,5}\}$-contractible, and $\si(M/e)$ is $3$-connected. \end{proof}
We next consider fans appearing in $\{U_{2,5},U_{3,5}\}$-fragile matroids.
\begin{lemma}[{\cite[Lemma~2.22]{CMvZW16}}]
\label{cmvzw-fans}
Let $\mathbb{P} \in \{\mathbb{U}_2, \mathbb{H}_5\}$, and let $M$ be a $\{U_{2,5},U_{3,5}\}$-fragile $\mathbb{P}$-representable matroid.
Let $A = \{a, b, c\}$ be a coindependent triangle of $M$ such that $b$ is not $\{U_{2,5},U_{3,5}\}$-deletable.
Let $M'$ be obtained from $M$ by gluing an $r$-wheel $W$ onto $(a, b, c)$ with remove set $X \subseteq \{a, b, c\}$ such that $b \in X$.
If $M'$ is $3$-connected, then $M'$ is a $\{U_{2,5},U_{3,5}\}$-fragile $\mathbb{P}$-representable matroid.
Moreover, $F = E(W) - X$ is a fan. \end{lemma}
\noindent For simplicity, when gluing a wheel $W$ with remove set $X$ as in the last lemma, we refer to $F = E(W)-X$ as the \emph{resulting fan}.
We now strengthen \cref{fragilefans} in the case that $M$ is a $\{U_{2,5},U_{3,5}\}$-fragile matroid (that is, when $\mathcal{N} = \{U_{2,5},U_{3,5}\}$).
\begin{lemma}
\label{fragilefans2}
Let $M$ be a $3$-connected $\{U_{2,5},U_{3,5}\}$-fragile matroid, and let $F$ be a fan of $M$.
\begin{enumerate}
\item If $|F| = 4$ and $e$ is an end of $F$, then $e$ is not $\{U_{2,5},U_{3,5}\}$-essential.
\item If $|F| \ge 5$ and $e \in F$, then $e$ is not $\{U_{2,5},U_{3,5}\}$-essential.
\end{enumerate} \end{lemma} \begin{proof}
If $|F| \ge 6$, or $|F|=5$ and $e$ is an end of $F$, then the result follows from \cref{fragilefans}.
Suppose $|F| = 4$ and let $e$ be an end of $F$.
Since $M$ is $3$-connected and has a $\{U_{2,5},U_{3,5}\}$-minor, $r(M) \ge 3$ and $r^*(M) \ge 3$.
By \cref{utfutfprop}, $M$ has a $U_{2,5}$-minor and a $U_{3,5}$-minor.
Let $(f_1,f_2,f_3,e)$ be a fan ordering of $F$.
Suppose $e$ is a spoke of $F$, so $\{f_2,f_3,e\}$ is a triangle.
By \cref{fragilefanelements}, $e$ is not $U_{3,5}$-contractible.
Suppose $e$ is not $U_{3,5}$-deletable, so $e$ is $U_{3,5}$-essential.
Also by \cref{fragilefanelements}, $f_2$ is not $U_{3,5}$-contractible and $f_3$ is not $U_{3,5}$-deletable.
Moreover, $f_3$ is not $U_{3,5}$-contractible, for otherwise $e$ is $U_{3,5}$-deletable; and $f_2$ is not $U_{3,5}$-deletable, for otherwise $f_3$ is $U_{3,5}$-contractible.
So all elements in the triangle $\{f_2,f_3,e\}$ are $U_{3,5}$-essential.
Let $C,D \subseteq E(M)$ such that $M / C \backslash D \cong U_{3,5}$.
By the foregoing, $\{f_2,f_3,e\} \cap (C \cup D) = \emptyset$.
But $r_{M/C \backslash D}(\{f_2,f_3,e\}) \le 2$, a contradiction.
We deduce that $e$ is $U_{3,5}$-deletable, so it is $\{U_{2,5},U_{3,5}\}$-deletable.
By a dual argument, if $e$ is a rim of $F$, then e is not $U_{2,5}$-contractible, so it is $\{U_{2,5},U_{3,5}\}$-contractible.
This proves that for an end $e$ of $F$, the element~$e$ is not $\{U_{2,5},U_{3,5}\}$-essential.
Finally, suppose $F$ is a maximal fan with $|F|=5$ where $(f_1,f_2,f_3,f_4,f_5)$ is a fan ordering of $F$.
We use a similar argument to show that $f_3$ is not $\{U_{2,5},U_{3,5}\}$-essential.
By \cref{utfutfprop}, $M$ has a $U_{2,5}$-minor and a $U_{3,5}$-minor.
By duality, we may assume that $\{f_2,f_3,f_4\}$ is a triad.
By \cref{fragilefanelements}, $f_3$ is not $U_{2,5}$-contractible, and $f_2$ and $f_4$ are not $U_{2,5}$-deletable.
Suppose $f_3$ is not $U_{2,5}$-deletable.
Then $f_2$ and $f_4$ are not $U_{2,5}$-contractible, so they are $U_{2,5}$-essential.
Let $C,D \subseteq E(M)$ such that $M / C \backslash D \cong U_{2,5}$.
Now $r^*_{M/C \backslash D}(\{f_2,f_3,f_4\}) \le 2$, a contradiction.
This proves that $f_3$ is not $\{U_{2,5},U_{3,5}\}$-essential. \end{proof}
\begin{lemma}
\label{noMK4}
Let $M$ be a $3$-connected $\{U_{2,5},U_{3,5}\}$-fragile matroid.
Then there is no set $X \subseteq E(M)$ such that $M|X \cong M(K_4)$ or $M^*|X \cong M(K_4)$. \end{lemma} \begin{proof}
Suppose that $M|X \cong M(K_4)$ for some $X \subseteq E(M)$.
Then $M$ has three $5$-element fans $F_1=(f_1,f_2,f_3,f_4,f_5)$, $F_2=(g,f_2,f_4,f_3,f_5)$, and $F_3=(g,f_4,f_2,f_3,f_1)$, where $X = \{f_1,f_2,f_3,f_4,f_5,g\}$.
By \cref{fragilefans2}, no element in $X$ is $\{U_{2,5},U_{3,5}\}$-essential.
By \cref{fragilefanelements}, each spoke of $F_1$, $F_2$, or $F_3$ is $\{U_{2,5},U_{3,5}\}$-deletable, and each rim of $F_1$, $F_2$, or $F_3$ is $\{U_{2,5},U_{3,5}\}$-contractible.
But $f_4$ is a rim of $F_1$, and a spoke of $F_2$, so it is $\{U_{2,5},U_{3,5}\}$-flexible, a contradiction. \end{proof}
For a fan with at least six elements, in a $3$-connected matroid, \cref{fanends} tells us that we can retain $3$-connectivity up to series pairs or parallel pairs when a spoke is deleted or a rim is contracted, whereas we lose connectivity if a spoke is contracted or a rim is deleted. For the middle element of a $5$-element fan, no such guarantee can be made in general.
However, we can guarantee this for $5$-element fans appearing in $\{U_{2,5},U_{3,5}\}$-fragile matroids, as shown in the next lemma.
\begin{lemma}
\label{noMK4conn}
Let $M$ be a $3$-connected $\{U_{2,5},U_{3,5}\}$-fragile matroid, and suppose $M$ has a $5$-element fan with ordering $(f_1,f_2,\dotsc,f_5)$, where $\{f_2,f_3,f_4\}$ is a triangle. Then $\si(M/f_3)$ is $3$-connected. \end{lemma} \begin{proof}
Suppose that $\si(M/f_3)$ is not $3$-connected.
Then, by \cref{fanmiddle}, there exists some element $f_6$ such that $M^*|\{f_1,f_2,\dotsc,f_6\} \cong M(K_4)$, contradicting \cref{noMK4}. \end{proof}
We return to $\{U_{2,5},U_{3,5}\}$-fragile matroids with nice path descriptions: we next consider properties of the ends.
\begin{lemma}
\label{pathdescendconn}
Let $M$ be a $3$-connected $\{U_{2,5},U_{3,5}\}$-fragile $\mathbb{P}$-representable matroid with $|E(M)| \ge 10$, for $\mathbb{P} \in \{\mathbb{H}_5, \mathbb{U}_2\}$, having a nice path description $(P_1,P_2,\dotsc,P_m)$.
Let $i \in \{1,m\}$.
\begin{enumerate}
\item If $P_i$ is a triangle or $4$-segment, then $\si(M / e)$ is $3$-connected for each $e \in P_i$.
\item If $P_i$ is a triad or $4$-cosegment, then $\co(M \backslash e)$ is $3$-connected for each $e \in P_i$.
\end{enumerate} \end{lemma} \begin{proof}
Suppose $P_1$ is a triad or $4$-cosegment, and $\co(M\backslash e)$ is not $3$-connected for some $e \in P_1$.
By \cref{fclnontrivialsep}, $M \backslash e$ has a $2$-separation $(X,Y)$ for which $\fcl_{M \backslash e}(X) \neq E(M)$ and $\fcl_{M \backslash e}(Y) \neq E(M)$, and neither $X$ nor $Y$ is a series class of $M \backslash e$.
By the definition of a nice path description, $P_m$ contains a path-generating set $P_m'$ of size $3$.
Without loss of generality, $|P_m' \cap Y| \ge 2$.
Since $\fcl_{M \backslash e}(Y) \neq E(M)$, we may assume that $Y$ is fully closed, implying $E(M)-P_1 \subseteq Y$.
But then $X \subseteq P_1 - e$, where $P_1 - e$ is a series class in $M \backslash e$, a contradiction. \end{proof}
\begin{lemma}
\label{pathdescends}
Let $M$ be a $3$-connected $\{U_{2,5},U_{3,5}\}$-fragile $\mathbb{P}$-representable matroid with $|E(M)| \ge 10$, for $\mathbb{P} \in \{\mathbb{H}_5, \mathbb{U}_2\}$, having a nice path description $(P_1,P_2,\dotsc,P_m)$.
Let $i \in \{1,m\}$.
\begin{enumerate}
\item If $P_i$ is a segment of $M$, then $|P_i|-1$ elements of $P_i$ are $\{U_{2,5},U_{3,5}\}$-deletable, and the other element is $\{U_{2,5},U_{3,5}\}$-contractible.\label{pde1}
\item If $P_i$ is a cosegment of $M$, then $|P_i|-1$ elements of $P_i$ are $\{U_{2,5},U_{3,5}\}$-contractible, and the other element is $\{U_{2,5},U_{3,5}\}$-deletable.\label{pde2}
\item If $P_i$ is a fan of size at least~$4$, then each spoke of $P_i$ is $\{U_{2,5},U_{3,5}\}$-deletable, and each rim of $P_i$ is $\{U_{2,5},U_{3,5}\}$-contractible.\label{pde3}
\end{enumerate} \end{lemma} \begin{proof}
Suppose $P_i$ is a $k$-cosegment.
By \cref{nicepathdescription}\ref{nicepathdesciii}, $P_i$ has one element that is $\{U_{2,5},U_{3,5}\}$-deletable, so, by \cref{pathdesctris}, the other $k-1$ elements are $\{U_{2,5},U_{3,5}\}$-contractible as required.
A similar argument applies when $P_i$ is a segment.
When $P_i$ is a fan, the result follows from \cref{fragilefanelements,noessential}. \end{proof}
The next property of $\{U_{2,5},U_{3,5}\}$-fragile matroids with nice path descriptions builds on earlier results of the section.
\begin{lemma}
\label{pathdescrank}
Let $M$ be a $3$-connected $\{U_{2,5},U_{3,5}\}$-fragile $\mathbb{P}$-representable matroid with $|E(M)| \ge 10$, for $\mathbb{P} \in \{\mathbb{H}_5, \mathbb{U}_2\}$,
with an $\{X_8,Y_8,Y_8^*\}$-minor.
Let $C$ be the set of $\{U_{2,5},U_{3,5}\}$-contractible elements and let $D$ be the set of $\{U_{2,5},U_{3,5}\}$-deletable elements of $M$. Then $|C| = r(M)$ and $|D| = r^*(M)$. \end{lemma} \begin{proof}
By \cref{nicepathdescription}, $M$ has a nice path description $(P_1,P_2,\dotsc,P_m)$.
Since this is a path of $3$-separations, it is easily seen that if $M$ has $h$ coguts elements in $P_2 \cup \dotsm \cup P_{m-1}$, then $r(M) = r(P_1) +h+ r(P_m)-2$. Each of the $h$ coguts elements are $\{U_{2,5},U_{3,5}\}$-contractible, by \cref{pathdescprops}, whereas each guts element is $\{U_{2,5},U_{3,5}\}$-deletable.
It remains to show that an end, $P_1$ say, has exactly $r(P_1)-1$ elements that are $\{U_{2,5},U_{3,5}\}$-contractible.
If $P_1$ is a $k$-cosegment, then $k \in \{3,4\}$ and $r(P_1)=k$, and this follows from \cref{pathdescends}\ref{pde2}.
On the other hand, if $P_1$ is a segment, then $r(P_1)=2$ and $P_1$ has exactly one $\{U_{2,5},U_{3,5}\}$-contractible element, by \cref{pathdescends}\ref{pde1}.
If $P_1$ is a fan of size at least~$5$ having $t$ rim elements, then $r(P_1) = t+1$.
By \cref{pathdescends}\ref{pde3}, each rim element is $\{U_{2,5},U_{3,5}\}$-contractible and each spoke element is $\{U_{2,5},U_{3,5}\}$-deletable.
It is also easily checked that if $P_1$ is a $4$-element fan then $r(P_1)=3$ and $P_1$ has exactly two $\{U_{2,5},U_{3,5}\}$-contractible elements.
We deduce that there are exactly $r(M)=r(P_1)+h+r(P_m)-2$ elements that are $\{U_{2,5},U_{3,5}\}$-contractible. Since $M$ has no $\{U_{2,5},U_{3,5}\}$-essential elements, by \cref{noessential}, the result follows. \end{proof}
We also require the following \lcnamecref{keepfragilelabels} that ensures elements can be removed while retaining a nice path description.
\begin{lemma}
\label{keepfragilelabels}
Let $M$ be a $3$-connected $\{U_{2,5},U_{3,5}\}$-fragile $\mathbb{P}$-representable matroid with $|E(M)| \ge 10$,
for $\mathbb{P} \in \{\mathbb{H}_5, \mathbb{U}_2\}$, having a nice path description $(P_1,P_2,\dotsc,P_m)$.
\begin{enumerate}
\item If $a \in P_1$ and $b \in P_m$ are $\{U_{2,5},U_{3,5}\}$-deletable, then $M \backslash a,b$ is $\{U_{2,5},U_{3,5}\}$-fragile and has no $\{U_{2,5},U_{3,5}\}$-essential elements.\label{kfl1}
\item If $e \in P_1 \cup P_m$ is $\{U_{2,5},U_{3,5}\}$-contractible, then $M / e$ is $\{U_{2,5},U_{3,5}\}$-fragile and has no $\{U_{2,5},U_{3,5}\}$-essential elements.\label{kfl2}
\end{enumerate} \end{lemma} \begin{proof}
We sketch the proof only. Consider \ref{kfl1}.
Using the terminology of \cite{CMvZW16}, $M$ has a path sequence from which we can obtain a path sequence for $M \backslash a,b$, since $a$ and $b$ are at the ends.
This latter path sequence certifies that $M \backslash a,b$ is $\{U_{2,5},U_{3,5}\}$-fragile, and that $M \backslash a,b$ has an $\{X_8,Y_8,Y_8^*,M_{8,6}\}$-minor, by \cite[Lemma~6.1]{CMvZW16}.
By \cref{noessential}, if $M\backslash a,b$ has an $\{X_8,Y_8,Y_8^*\}$-minor, then $M \backslash a,b$ has no $\{U_{2,5},U_{3,5}\}$-essential elements.
Using a similar approach as in the proof of \cref{noessential}, it is easily checked that $M \backslash a,b$ has no $\{U_{2,5},U_{3,5}\}$-essential elements when $M \backslash a,b$ has only an $M_{8,6}$-minor.
A similar argument also applies for \ref{kfl2}. \end{proof}
Note that the previous \lcnamecref{keepfragilelabels} implies that after deleting the pair $\{a,b\}$ or contracting $e$, each element in the resulting matroid is $\{U_{2,5},U_{3,5}\}$-deletable (or $\{U_{2,5},U_{3,5}\}$-contractible) if and only if it was $\{U_{2,5},U_{3,5}\}$-deletable (or $\{U_{2,5},U_{3,5}\}$-contractible) in $M$; otherwise, $M$ would have $\{U_{2,5},U_{3,5}\}$-flexible elements.
\section{Excluded minors are almost \texorpdfstring{$\{U_{2,5},U_{3,5}\}$}{\{U(2,5),U(3,5)\}}-fragile} \label{utfutffragilesec}
Suppose that $M$ is an excluded minor for the class of $\mathbb{P}$-representable matroids, where $\mathbb{P} \in \{\mathbb{H}_5,\mathbb{U}_2\}$, and $M \backslash a,b$ is a $3$-connected
matroid with a $\{U_{2,5},U_{3,5}\}$-minor, for some distinct $a,b \in E(M)$. In this section, we show that if $|E(M)| \ge 16$, then $M \backslash a,b$ is $\{U_{2,5},U_{3,5}\}$-fragile.
\begin{lemma}
\label{noessential2}
For $\mathbb{P} \in \{\mathbb{H}_5,\mathbb{U}_2\}$ and $N \in \{U_{2,5},U_{3,5}\}$,
let $M$ be a $3$-connected $\{U_{2,5},U_{3,5}\}$-fragile $\mathbb{P}$-representable matroid.
If $M$ has an $\{X_8,Y_8,Y_8^*\}$-minor and $|E(M)| \ge 9$,
then $M$ has no $N$-essential elements. \end{lemma} \begin{proof}
Note that for any $M$ satisfying the hypotheses of the lemma, $M$ has a $\{U_{2,5},U_{3,5}\}$-minor, so $M$ is not a wheel or a whirl.
Towards a contradiction, suppose that $M$ has an $N$-essential element.
If $|E(M)| > 9$, then, by Seymour's Splitter Theorem, $M$ has a $3$-connected $\mathbb{P}$-representable $\{U_{2,5},U_{3,5}\}$-fragile minor $M'$, with $|E(M')| = |E(M)|-1$, such that $M'$ has an $\{X_8,Y_8,Y_8^*\}$-minor. Note that $M'$ also has an $N$-essential element.
It follows that there exists a $3$-connected $\mathbb{P}$-representable $\{U_{2,5},U_{3,5}\}$-fragile matroid $M''$, having an $\{X_8,Y_8,Y_8^*\}$-minor, such that $|E(M'')|=9$ and $M''$ has an $N$-essential element.
The $3$-connected $\mathbb{P}$-representable $\{U_{2,5},U_{3,5}\}$-fragile matroids on nine elements are given in \cite[Figure~8]{CCCMWvZ13}.
It can be readily checked that for each such matroid having an $\{X_8,Y_8,Y_8^*\}$-minor, the matroid has no $N$-essential elements.
So no such matroid $M''$ exists, a contradiction. \end{proof}
\begin{lemma}
\label{utfutfessentialrank4}
For $\mathbb{P} \in \{\mathbb{H}_5,\mathbb{U}_2\}$ and $N \in \{U_{2,5},U_{3,5}\}$,
let $M$ be a $3$-connected $\{U_{2,5},U_{3,5}\}$-fragile $\mathbb{P}$-representable matroid with
an $N$-essential
element and rank at most four. Then $|E(M)| \le 9$.
\end{lemma} \begin{proof}
By \cref{noessential2}, either $|E(M)| \le 8$ or $M$ has no $\{X_8,Y_8,Y_8^*\}$-minor. So we may assume $M$ has no $\{X_8,Y_8,Y_8^*\}$-minor.
By \cref{ccmwvz-result}, we may also assume that $M$ or $M^*$ can be obtained by gluing wheels to $U_{2,5}$ or $Y_8 \backslash 4$.
In this case, the fact that $r(M) \le 4$ forces $|E(M)| \le 9$; we omit the details.
\end{proof}
\begin{lemma}
\label{hopeful}
For $\mathbb{P} \in \{\mathbb{H}_5,\mathbb{U}_2\}$ and $N \in \{U_{2,5},U_{3,5}\}$,
let $M$ be a $3$-connected $\{U_{2,5},U_{3,5}\}$-fragile $\mathbb{P}$-representable matroid with at least three $N$-essential elements.
Then either $|E(M)| \le 7$ or $M \in \{Y_8,Y_8^*\}$. \end{lemma}
\begin{proof}
First, assume that $M$ has no $\{X_8,Y_8,Y_8^*\}$-minor.
Then, by \cref{ccmwvz-result}, either $M \in \{M_{9,9},M_{9,9}^*\}$, $M$ or $M^*$ can be obtained by gluing wheels to $U_{2,5}$ or $Y_8 \backslash 4$, or $|E(M)| \le 7$.
It is easily checked that $M_{9,9}$ has no $U_{2,5}$- or $U_{3,5}$-essential elements, so the former is not possible.
Assume that $M$ or $M^*$
can be obtained by gluing wheels to $U_{2,5}$ or $M_{7,1}$.
We claim that $|E(M)| \le 7$ in this case.
Suppose not; then there exists some minor-minimal matroid $M'$ such that $M'$ can be obtained by gluing wheels to $U_{2,5}$ or $M_{7,1}$, $|E(M')| \ge 8$, and $M'$ is a $3$-connected $\{U_{2,5},U_{3,5}\}$-fragile $\mathbb{P}$-representable matroid with at least three $N$-essential elements.
First, we observe that if $|E(M')|=8$, then $M'$ or $(M')^*$ is isomorphic to one of the four matroids referred to in \cite{CCCMWvZ13,CMvZW16} as $M_{8,1}$, $M_{8,3}$, $M_{8,5}$, and $M_{8,6}$ (in particular, $M' \cong M_{8,6}$ in the case that a wheel was glued to $M_{7,1}$), but these matroids have no $U_{2,5}$- or $U_{3,5}$-essential elements.
So $|E(M')| \ge 9$.
Suppose that $M'$ has a maximal fan~$F$ of length at least~$4$.
By contracting a rim end, or deleting a spoke end, of $F$, we obtain a $3$-connected minor~$M''$ of $M'$, by \cref{fanendsstrong}, with $|E(M'')| \ge 8$.
By \cref{fragilefanelements,fragilefans2}, $M''$ has a $\{U_{2,5},U_{3,5}\}$-minor, so this matroid is still $\{U_{2,5},U_{3,5}\}$-fragile, and $M''$ has at least as many $U_{2,5}$-fragile elements as $M'$.
But this shows $M'$ is not minor-minimal, a contradiction.
So $M'$ has no fans of length at least~$4$.
It follows that $M'$ can be obtained from $U_{2,5}$ by gluing three wheels so that the resulting fans each have length three: this matroid is referred to as $M_{9,18}$ in \cite{CCCMWvZ13,CMvZW16}.
But it is easily checked that $M_{9,18}$ has no $U_{2,5}$- or $U_{3,5}$-essential elements.
We deduce that $|E(M)| \le 7$ in the case that $M$ or $M^*$ can be obtained by gluing wheels to $U_{2,5}$ or $M_{7,1}$.
We may now assume that $M$ has an $\{X_8,Y_8,Y_8^*\}$-minor.
Then $M \in \{X_8,Y_8,Y_8^*\}$, for otherwise $M$ has no $N$-essential elements, by \cref{noessential2}.
It is readily checked that $X_8$ has exactly one $U_{2,5}$-essential element, and exactly one $U_{3,5}$-essential element, whereas $Y_8$ has three $U_{2,5}$-essential elements. So $M \in \{Y_8,Y_8^*\}$.
\end{proof}
\begin{lemma}
\label{utfutf3essentialv2}
For $\mathbb{P} \in \{\mathbb{H}_5,\mathbb{U}_2\}$ and $N \in \{U_{2,5},U_{3,5}\}$,
let $M$ be a $3$-connected $\{U_{2,5},U_{3,5}\}$-fragile $\mathbb{P}$-representable matroid with two $N$-essential elements.
Then either $|E(M)| \le 8$, or $M$ or $M^*$ can be obtained from $U_{2,5}$ by gluing a single wheel such that the resulting fan has at least five elements. \end{lemma} \begin{proof}
By \cref{noessential2}, either $|E(M)| \le 8$ or $M$ has no $\{X_8,Y_8,Y_8^*\}$-minor; so we may assume that $M$ has no $\{X_8,Y_8,Y_8^*\}$-minor.
We first apply \cref{ccmwvz-result}, deducing that
either $M \in \{M_{9,9},M_{9,9}^*\}$, $M$ or $M^*$ can be obtained by gluing wheels to $U_{2,5}$ or $Y_8 \backslash 4$, or $|E(M)| \le 8$.
It is easy to check that if $M \in \{M_{9,9},M_{9,9}^*\}$, then $M$ has no $N$-essential elements, so the former is not possible.
Assume that $M$ or $M^*$ can be obtained by gluing a wheel to $M_{7,1}$, or gluing at least two wheels to $U_{2,5}$.
We claim that $|E(M)| \le 8$.
Suppose not; then there exists some minor-minimal matroid $M'$ such that $M'$ can be obtained by gluing a wheel to $M_{7,1}$, or gluing at least two wheels to $U_{2,5}$; $|E(M')| \ge 9$; and $M'$ is a $3$-connected $\{U_{2,5},U_{3,5}\}$-fragile $\mathbb{P}$-representable matroid with at least two $N$-essential elements.
First, we observe that if $|E(M')|=9$, then $M'$ or $(M')^*$ is isomorphic to one of the matroids referred to in \cite{CCCMWvZ13,CMvZW16} as
$M_{9,1}$, $M_{9,2}$, $M_{9,7}$, $M_{9,15}$, and $M_{9,18}$ (in particular, $M' \cong M_{9,7}$ in the case that a wheel was glued to $M_{7,1}$).
But these matroids have at most one $N$-essential element (if such an element exists, it is $\{U_{2,5},U_{3,5}\}$-essential).
So $|E(M')| \ge 10$.
Suppose that $M'$ has a maximal fan~$F$ of length at least~$4$.
By contracting a rim end, or deleting a spoke end, of $F$, we obtain a $3$-connected minor~$M''$ of $M'$, by \cref{fanendsstrong}, with $|E(M'')| \ge 9$.
By \cref{fragilefanelements,fragilefans2}, $M''$ has a $\{U_{2,5},U_{3,5}\}$-minor, so this matroid is still $\{U_{2,5},U_{3,5}\}$-fragile, and $M''$ has at least as many $N$-fragile elements as $M'$.
But this shows $M'$ is not minor-minimal, a contradiction.
So $M'$ has no fans of length at least~$4$.
But then $|E(M')| \le 9$, a contradiction.
We deduce that $|E(M)| \le 8$ in the case that $M$ or $M^*$ can be obtained by gluing a wheel to $M_{7,1}$, or gluing at least two wheels to $U_{2,5}$.
Now, the only remaining possibility, when $|E(M)| \ge 9$, is that $M$ or $M^*$ can be obtained by gluing a single wheel to $U_{2,5}$, as required. \end{proof}
Next we work towards \cref{jail}, which describes some properties of matroids that are $N$-fragile, for $N \in \{U_{2,5},U_{3,5}\}$, but not $\{U_{2,5},U_{3,5}\}$-fragile.
We require a definition. Let $M$ be a $3$-connected matroid having the $3$-connected matroid $N$ as a minor. For $e \in E(M)$, we say $e$ is \emph{$N$-elastic} if $e$ is $N$-flexible and both $\si(M/e)$ and $\co(M \backslash e)$ are $3$-connected.
To prove \cref{jail}, we consider two cases: first, when $M$ has a $U_{2,5}$-elastic element, in \cref{jail-elastic}; and then when $M$ has no $U_{2,5}$-elastic elements, in \cref{jail-nonelastic}.
\begin{lemma}
\label{jail-elastic}
For $\mathbb{P} \in \{\mathbb{H}_5,\mathbb{U}_2\}$,
let $M$ be a $3$-connected $\mathbb{P}$-representable $U_{3,5}$-fragile matroid that is not $\{U_{2,5},U_{3,5}\}$-fragile, where $r(M) \ge 4$ and $r^*(M) \ge 4$.
If $M$ has a $U_{2,5}$-elastic element,
then $|E(M)| \le 9$, and $M$ has at most two $U_{3,5}$-essential elements. \end{lemma} \begin{proof}
Let $e$ be a $U_{2,5}$-elastic element of $M$, so $e$ is $U_{2,5}$-flexible, and both $\si(M/e)$ and $\co(M \backslash e)$ are $3$-connected. \begin{claim}
\label{coranktwo}
$r(\co(M \backslash e))=2$. \end{claim} \begin{subproof}
If $r(\co(M \backslash e)) \ge 3$, then, as $\co(M \backslash e)$ has a $U_{2,5}$-minor, \cref{utfutfprop} implies it also has a $U_{3,5}$-minor.
Moreover, $\si(M / e)$ has a $U_{2,5}$-minor, and this matroid has rank at least~$3$ (since $r(M) \ge 4$) and corank at least~$3$ (since it has a $U_{2,5}$-minor). So, by \cref{utfutfprop}, $M/e$ has a $U_{3,5}$-minor.
But then $e$ is $U_{3,5}$-flexible, a contradiction.
\end{subproof}
Now, by \cref{coranktwo}, and since $M \backslash e$ has a $U_{2,5}$-minor, $\co(M \backslash e) \cong U_{2,t}$ for some $t \ge 5$.
Therefore, the union of three series classes of $M \backslash e$ is a circuit.
Since $M$ is $\mathbb{P}$-representable, $t \le 6$.
We work towards showing, in \cref{simpcase,cosimpcase}, that when $f$ is in a non-trivial series class of $M \backslash e$, then, except in some particular situations, both $\si(M/f)$ and $\co(M \backslash f)$ are $3$-connected and have rank and corank at least~$3$.
\begin{claim}
\label{presimpcase1}
Let $f \in E(M\backslash e)$ where $f$ is in a non-trivial series class~$S$ of $M \backslash e$ with $|S| \ge 3$. Then $\si(M/f)$ is $3$-connected.
\end{claim}
\begin{subproof}
Suppose $\si(M/f)$ is not $3$-connected.
Then $M$ has a vertical $3$-separation $(X,f,Y)$.
In particular, $f \notin \cl^*(X)$ and $f \notin \cl^*(Y)$.
Let $f'$ and $f''$ be distinct elements in $S-f$.
Since $M$ is $3$-connected, $\{e,f',f''\}$ is a triad of $M$.
We may assume that at least two elements of this triad are contained in $X$, say.
But then $f \in \cl^*(X)$, a contradiction.
\end{subproof}
\begin{claim}
\label{presimpcase2}
Suppose $M \backslash e$ has at least two non-trivial series classes, and $\{f,f'\}$ is a series class of $M \backslash e$.
If $\si(M/f)$ is not $3$-connected, then $M$ has a vertical $3$-separation $(X,f,Y)$ such that $e \in X$ and $f' \in Y$, and either
\begin{enumerate}[label=\rm(\Roman*)]
\item every non-trivial series class of $M \backslash e$ distinct from $\{f,f'\}$ is contained in $X$, or\label{psc2i}
\item $M \backslash e$ has precisely two non-trivial series classes, $\{f,f'\}$ and $G$, and $X = (G-g) \cup e$ for some $g \in G$.\label{psc2ii}
\end{enumerate}
\end{claim}
\begin{subproof}
Suppose $\si(M/f)$ is not $3$-connected, so $M$ has a vertical $3$-separation $(X,f,Y)$.
Since $f \notin \cl^*(X)$ and $f \notin \cl^*(Y)$, we may assume that $e \in X$ and $f' \in Y$.
Let $G$ be a non-trivial series class of $M \backslash e$ distinct from $\{f,f'\}$, with distinct elements $g,g' \in G$.
If $\{g,g'\} \subseteq Y$, then $e \in \cl^*(Y)$, so $f \in \cl^*(Y)$, a contradiction.
Now suppose $g \in Y$ and $G-g \subseteq X$.
It suffices to prove that when \ref{psc2ii} does not hold, then $(X \cup g, f, Y-g)$ is a vertical $3$-separation.
Observe that $g \in \cl^*(X)-X$ and $f \in \cl(X)-X$.
By \cref{gutsandcoguts}, $\cl(X) = X \cup f$ and $\cl^*(X) = X \cup g$.
Hence, $G$ and $\{f,f'\}$ are the only non-trivial series classes of $M \backslash e$ not contained in $X$.
Recall that the union of three series classes of $M \backslash e$ is a circuit.
If $X$ contains two series classes of $M \backslash e$, then, as $G-g \subseteq X$, we have $g \in \cl(X)$, a contradiction.
In particular, if $M \backslash e$ has at least four non-trivial series classes, then two are contained in $X$, a contradiction.
So $M \backslash e$ has at most three non-trivial series classes,
and $Y$ contains at least two trivial series classes.
In particular, $|Y| \ge 4$.
If $X$ does not contain any series class of $M \backslash e$, then $X = (G-g) \cup e$ and \ref{psc2ii} holds.
So assume $X$ contains a series class $S$ of $M \backslash e$.
Let $C^*$ be a cocircuit contained in $Y$, and suppose $g \in C^*$.
If $f' \notin C^*$, then the circuit $S \cup G \cup \{f,f'\}$ intersects $C^*$ in a single element $g$, contradicting orthogonality.
Similarly, if $C^*$ avoids some element $y \in Y-\{f',g\}$, this violates orthogonality with the circuit $S \cup G \cup y$.
So $C^*=Y$.
Now $r(Y)=4$ and $r^*(Y)=|Y|-1$, so $\lambda(Y) = 3$, a contradiction.
Thus $g \notin C^*$.
It now follows that $(X \cup g, f, Y-g)$ is a cyclic $3$-separation, as required.
\end{subproof}
\begin{claim}
\label{simpcase}
Let $f \in E(M\backslash e)$ where $f$ is in a non-trivial series class~$S$ of $M \backslash e$.
Suppose either $|S| \ge 3$, or $M \backslash e$ has at least three non-trivial series classes.
Then $\si(M/f)$ is $3$-connected and has rank and corank at least~$3$.
\end{claim}
\begin{subproof}
Suppose $\si(M/f)$ is not $3$-connected.
By \cref{presimpcase1}, $|S| = 2$, so $M \backslash e$ has at least three non-trivial series classes.
Let $S = \{f,f'\}$.
By \cref{presimpcase2}, $M$ has a vertical $3$-separation $(X,f,Y)$ such that $f' \in Y$, and $X$ contains $e$ and every non-trivial series class of $M \backslash e$ distinct from $S$.
Note, in particular, that there are at least two non-trivial series classes contained in $X$.
The set $Y$ consists of $f'$ and a subset of the elements in trivial series classes of $M \backslash e$, with $|Y| \ge 3$.
Hence $X \cup f$ spans $E(M)$, contradicting that $(X,f,Y)$ is a vertical $3$-separation.
We deduce that $\si(M/f)$ is $3$-connected.
Clearly
$r(\si(M/f)) \ge 3$, since $r(M) \ge 4$.
We claim that $r^*(\si(M/f)) \ge 3$.
Suppose $f$ is in a triangle~$T$ of $M$.
If $T$ is also a triangle of $M \backslash e$, then $T$ contains $S$ by orthogonality, implying $\co(M \backslash e)$ is not $3$-connected, a contradiction.
So $e \in T$.
But then, for every series pair $\{g,g'\}$ of $M \backslash e$, the set $\{e,g,g'\}$ is a triad that meets $T$.
By orthogonality, $T$ contains an element of each series pair of $M \backslash e$.
In the case that $M \backslash e$ has at least three non-trivial series classes, we deduce that $f$ is not in a triangle, so $r^*(M/f) \ge 4$.
Otherwise, when $|S| \ge 3$, it follows that $f$ is in at most one triangle, so $r^*(\si(M/f)) \ge r^*(M/f) - 1 \ge 3$.
\end{subproof}
Recall that $\co(M \backslash e) \cong U_{2,t}$ for $t \in \{5,6\}$.
\begin{claim}
\label{precosimpcase}
Let $f \in E(M \backslash e)$ where $f$ is in a non-trivial series class~$S$ of $M \backslash e$.
Suppose $\co(M \backslash f)$ is not $3$-connected.
Then $M$ has a cyclic $3$-separation $(X,f,Y)$ such that
\begin{enumerate}[label=\rm(\Roman*)]
\item $e \in X$,
\item $X \cup f$ is coclosed,
\item $Y$ is the union of at least $t-2$ trivial series classes of $M \backslash e$, and\label{pcsc3}
\item there is a circuit~$C$ such that $(S-f) \cup e \subseteq C \subseteq X$.\label{pcsc4}
\end{enumerate}
\end{claim}
\begin{subproof}
Clearly $M$ has a cyclic $3$-separation $(X,f,Y)$ where, without loss of generality, $e \in X$ and $X \cup f$ is coclosed. It remains to show \ref{pcsc3} and \ref{pcsc4} hold.
First, observe that as $e \in X$ and $X \cup f$ is coclosed, for a series class $S'$ of $M \backslash e$ distinct from $S$, either $S' \subseteq X$ or $S' \subseteq Y$.
Similarly, either $S -f \subseteq X$ or $S -f \subseteq Y$.
The set $Y$ contains a circuit~$C_Y$.
Since $C_Y$ is also a circuit of $M \backslash e$, if $C_Y$ meets a series class~$S'$ of $M \backslash e$, then, by orthogonality, $C_Y$ contains $S'$.
Recall that the union of three series classes of $M \backslash e$ is a circuit.
It follows that $C_Y$ is the union of three series classes of $M \backslash e$; in particular, $C_Y$ avoids $S$.
If $S -f \subseteq Y$, then $f \in \cl(Y)$ so $f \notin \cl^*(X)$, a contradiction.
So $S -f \subseteq X$.
If there are series classes $S'$ and $S''$ of $M \backslash e$ contained in $X$, such that $S$, $S'$ and $S''$ are distinct, then $f \in \cl(X)$ and thus $f \notin \cl^*(Y)$, a contradiction.
So either $X-e = S -f$, or $X-e = S' \cup (S - f)$ for some series class $S'$ of $M \backslash e$.
There is also a circuit~$C_X$ contained in $X$.
If $e \notin C_X$, then $C_X$ is properly contained in the union of three series classes of $M \backslash e$, a contradiction.
So $e \in C_X$.
Then $e \notin \cl^*(Y)$, but $e$ blocks each non-trivial series class of $M \backslash e$.
Thus $Y$ is the union of trivial serial classes of $M \backslash e$, of which there are at least $t-2$, so \ref{pcsc3} holds.
Moreover, $S-f \subseteq C_X$ by orthogonality. So \ref{pcsc4} holds, thus proving the claim.
\end{subproof}
\begin{claim}
\label{cosimpcase}
Let $S$ be a non-trivial series class of $M \backslash e$.
\begin{enumerate}[label=\rm(\Roman*)]
\item If $M \backslash e$ has at least three non-trivial series classes, then, for each $f \in S$, the matroid $\co(M\backslash f)$ is $3$-connected.\label{csc1}
\item If $M \backslash e$ has precisely two non-trivial series classes, then there exists some $S' \subseteq S$ with $|S'| \ge |S|-1$ such that, for each $f \in S'$, the matroid $\co(M\backslash f)$ is $3$-connected.\label{csc2}
\end{enumerate}
Moreover, if $\co(M \backslash f)$ is $3$-connected for some $f \in S$, then $r(\co(M \backslash f)) \ge 3$ and $r^*(\co(M \backslash f)) \ge 3$.
\end{claim}
\begin{subproof}
Let $f \in S$.
Suppose $\co(M \backslash f)$ is not $3$-connected.
Then $M$ has a cyclic $3$-separation $(X,f,Y)$ as described in \cref{precosimpcase}.
In particular, \cref{precosimpcase}\ref{pcsc3} implies that when $M \backslash e$ has at least three non-trivial series classes, $\co(M \backslash f)$ is $3$-connected for each $f \in S$, proving \ref{csc1}.
For \ref{csc2}, assume $M \backslash e$ has precisely two non-trivial series classes.
By the foregoing, $Y$ is the union of the $t-2$ trivial series classes of $M \backslash e$, and there is a circuit~$C$ such that $(S-f) \cup e \subseteq C \subseteq X$.
Let $f' \in S-f$ and suppose $\co(M \backslash f')$ is not $3$-connected.
Then, by another application of \cref{precosimpcase}, $M$ has a cyclic $3$-separation $(X', f', Y')$ where $Y'=Y$.
Since $Y' = Y$, we have $C \subseteq X \subseteq X' \cup f'$.
But $f' \in C$, so $f' \in \cl(X')$, and hence $f' \notin \cl^*(Y')$, a contradiction.
This proves that for $S' = S-f$, each $f' \in S'$ has the property that $\co(M \backslash f')$ is $3$-connected.
Henceforth, let $f \in S$ such that $\co(M \backslash f)$ is $3$-connected, and suppose $M \backslash e$ has at least two non-trivial series classes.
Clearly $r^*(\co(M \backslash f)) \ge 3$, since $r^*(M) \ge 4$.
It remains to show that $r(\co(M \backslash f)) \ge 3$.
Let $(S_1,S_2,\dotsc,S_t)$ be a partition of $E(M \backslash e)$ into series classes, with $f \in S_1$.
Recall that $t \ge 5$ and, for distinct $i,j,k \in [t]$, the set $S_i \cup S_j \cup S_k$ is a circuit.
Suppose $f$ is in a triad $T^*$ of $M$ that is not contained in $S_1 \cup e$.
Without loss of generality, $S_2 \cap T^* \neq \emptyset$.
Let $h \in \{1,2\}$ and $i \in \{3,4,5\}$, so $C_{h,i} = (S_h \cup S_3 \cup S_4 \cup S_5) - S_i$ is a circuit.
By orthogonality with $C_{2,i}$ for each such $i$, we have $|S_2 \cap T^*| = 2$.
Similarly, orthogonality with $C_{1,i}$ for each $i$ implies that $|S_1 \cap T^*| = 2$, a contradiction.
So the only triads of $M$ containing $f$ are contained in $S_1 \cup e$.
Since $M \backslash e$ has at least one non-trivial series classes other than $S_1$, we have $r(\co(M \backslash f)) \ge r(\co(M \backslash e)) + 1 = 3$.
\end{subproof}
Recall that $\co(M \backslash e) \cong U_{2,t}$ for $t \in \{5,6\}$.
Assume first that $M \backslash e$ has precisely one non-trivial series class $S$.
Then $S \cup e$ is a cosegment of $M$.
Let $G = S \cup e$ and $L = E(M) - G$.
Note that $M / L \cong U_{|G|-2,|G|}$ and $M|L \cong U_{2,|L|}$; moreover, $\co(M \backslash e) \cong U_{2,|L|+1}$, so $|L| \in \{4,5\}$.
Suppose that $|G| \ge 5$.
Then each element $\ell \in L$ is $U_{3,5}$-contractible.
Moreover, $L-\ell$ is a non-trivial parallel class in $M / \ell$, implying each $\ell \in L$ is $U_{3,5}$-deletable.
So each $\ell \in L$ is $U_{3,5}$-flexible, a contradiction.
Hence $|G| \le 4$.
Now $|E(M)| = |L| + |G| \le 9$.
Suppose $|E(M)|=9$. Then $|L|=5$ and $|G|=4$, in which case $r(M)=4$ and $r^*(M) = 5$.
Each $s \in S$ is $U_{2,5}$-contractible so, by \cref{simpcase,utfutfprop}, $s$ is also $U_{3,5}$-contractible; and each $\ell \in L$ is $U_{2,5}$-deletable so, by \cref{lineconn}, $M \backslash \ell$ is $3$-connected, and hence $\ell$ is $U_{3,5}$-deletable.
So $M$ has at most one $U_{3,5}$-essential element, and thus the \lcnamecref{jail-elastic} holds in this case.
Now suppose $|E(M)|=8$. Then, since $r^*(M) \ge 4$, we have $|L|=|G|=4$.
Consider $M/e$.
Since $M/e$ has a $U_{2,5}$-minor, there exist distinct $s,s' \in S$ such that $\{s,s',\ell\}$ is independent for each $\ell \in L$.
Thus, $\{s,s',e,\ell\}$ is independent in $M$ for each $\ell \in L$.
Let $\{s''\} = S-\{s,s'\}$ and note that $S \cup \ell$ is independent in $M$ for each $\ell \in L$.
Choose $\ell' \in L$ so that $\{s,s'',e,\ell'\}$ is a circuit, or if no such circuit exists, then choose $\ell' \in L-\ell$ arbitrarily.
Then $M /s \backslash \ell' \cong P_6$ where $L-\ell'$ is the unique triangle in this matroid.
It follows that each $\ell \in L$ is $U_{3,5}$-deletable.
By orthogonality, each circuit~$C$ that meets $G$ has $|C \cap G| \ge 3$.
Thus, by the foregoing, there exists $\ell \in L$ such that $G' \cup \ell$ is independent for all $G' \subseteq G$ with $|G'|=3$.
Then $\si(M/\ell) \cong U_{3,5}$.
So $\ell$ is $U_{3,5}$-flexible, a contradiction.
Assume next that $M \backslash e$ has precisely two non-trivial series classes, each of size two.
Then $|E(M)| = t + 3 \le 9$.
Let $S_1$ and $S_2$ be the two series pairs of $M \backslash e$.
Suppose $|E(M)|=9$, so
$r(M) = 4$ and $r^*(M) = 5$.
Since $\co(M \backslash e) \cong U_{2,6}$, each element in $E(M)-e$ is $U_{2,5}$-deletable.
By \cref{cosimpcase,utfutfprop}, for each $i \in \{1,2\}$ there exists $s_i \in S_i$ such that $M \backslash s_i$ has a $U_{3,5}$-minor.
Now $M / s_1$, say, also has a $U_{2,5}$-minor.
If $s_1$ is in a triangle of $M$, then, by orthogonality, this triangle is contained in $S_1 \cup S_2 \cup e$, in which case $S_1 \cup S_2 \cup e$ is a $5$-element fan. It follows that $s_1$ is in at most one triangle, so
$\si(M/s_1)$ has rank and corank at least~$3$.
If $\si(M/s_1)$ is $3$-connected, then, by \cref{utfutfprop}, $M / s_1$ has a $U_{3,5}$-minor, so $s_1$ is $U_{3,5}$-flexible, a contradiction.
So $\si(M / s_1)$ is not $3$-connected.
Then, by \cref{presimpcase2}, $(S_2 \cup e, s_1, Y)$ is a vertical $3$-separation.
Similarly, $(S_1 \cup e, s_2, Y)$ is a vertical $3$-separation.
So there is a circuit contained in $S_1 \cup \{e,s_2\}$, and a circuit contained in $S_2 \cup \{e,s_1\}$. If these circuits are distinct, then, by circuit elimination, there is a circuit contained in $S_1 \cup S_2$, a contradiction.
So $\{s_1,e,s_2\}$ is a triangle and $S_1 \cup S_2 \cup e$ is a $5$-element fan.
Now, it is easily verified that $M$ has no $U_{3,5}$-essential elements, and thus the \lcnamecref{jail-elastic} holds in this case.
Now suppose $|E(M)| = 8$.
Let $E(M)-(S_1 \cup S_2 \cup e) = \{s_3,s_4,s_5\}$.
Each $s \in S_1 \cup S_2$ is $U_{2,5}$-contractible.
First, suppose there is a circuit $\{s_1,e,s_2\}$, where $S_i = \{s_i,s_i'\}$ for $i \in \{1,2\}$.
Then $S_1 \cup S_2 \cup e$ is a $5$-element fan, and, by \cref{fragilefanelements,fragilefans}, $s_i'$ is $U_{3,5}$-contractible, for $i \in \{1,2\}$.
By \cref{fanmiddle}, $\si(M/e)$ is $3$-connected, since $\{s_1,s_2,s_\ell\}$ is not a triad for any $\ell \in \{3,4,5\}$, by orthogonality.
Note that $\{s_1,e,s_2\}$ is the unique triangle containing $e$, so $\si(M/e)$ has rank and corank three.
Since $e$ is $U_{2,5}$-contractible, $e$ is also $U_{3,5}$-contractible by \cref{utfutfprop}.
Since $s_i$ is in a parallel pair in $M / e$, it follows that $s_i$ is $U_{3,5}$-deletable, for $i \in \{1,2\}$.
Similarly, $s_i'$ is $U_{3,5}$-contractible for $i \in \{1,2\}$.
Now $\co(M \backslash e) \cong M \backslash e / s_1',s_2' \cong U_{2,5}$.
If $M / s_1',s_2' \cong U_{2,6}$, then each element in $\{s_3,s_4,s_5\}$ is $U_{3,5}$-deletable, so the \lcnamecref{jail-elastic} holds.
Otherwise, it follows that there is a circuit $\{e,s_1',s_2',s_\ell\}$, for some $\ell \in \{3,4,5\}$.
But then $s_\ell$ is $U_{3,5}$-deletable, and again the \lcnamecref{jail-elastic} holds.
Now suppose there is no circuit of the form $\{s_1,e,s_2\}$ for $s_1 \in S_1$ and $s_2 \in S_2$.
Let $S_1 = \{s_1,s_1'\}$ and $S_2 = \{s_2,s_2'\}$, and let $\{i,j\} = \{1,2\}$.
First, observe that if $\{s_i,e\} \cup S_j$ is a circuit, then it is readily checked that $M / s_j' \backslash s_j \cong M / s_j \backslash s_j' \cong U_{3,5}$, so $s_j$ is $U_{3,5}$-flexible, a contradiction.
By \cref{presimpcase2}, either $\si(M/s_i)$ is $3$-connected, or there is a vertical $3$-separation $(X, s_i, Y)$ such that $s_i' \in Y$ and $S_{j} \cup e \subseteq X$.
In the latter case $r(Y)=3$, so, by closing $Y \cup s_i$, we may assume that $X = S_j \cup e$ and $Y = \{s_i',s_3,s_4,s_5\}$.
Then $s_i$ is in a circuit contained in $\{s_i,e\} \cup S_j$.
Such a circuit must contain $e$, so $\{s_i,e\} \cup S_j$ is a circuit, a contradiction.
So $\si(M/s_i)$ is $3$-connected.
Similarly, $\si(M/s)$ is $3$-connected for all $s \in S_1 \cup S_2$.
Moreover, $\si(M/s)$ has rank and corank at least~$3$, so, by \cref{utfutfprop}, $s$ is $U_{3,5}$-contractible.
If there exists some $\ell \in \{3,4,5\}$ such that $s_\ell$ is not in a $4$-element circuit with $e$ that meets $S_1$ and $S_2$, then $M/s_\ell \cong Q_6$, in which case $s_\ell$ is $U_{3,5}$-contractible, and $s_{\ell'}$ is $U_{3,5}$-deletable for each $\ell' \in \{3,4,5\} - \ell$.
Then $M$ has no $U_{3,5}$-essential elements, so the \lcnamecref{jail-elastic} holds.
So for each $\ell \in \{3,4,5\}$, there is a $4$-element circuit containing $\{s_\ell,e\}$ that meets $S_1$ and $S_2$.
Note that no two of these three circuits intersects $S_1 \cup S_2$ in the same pair of elements, for otherwise $r(M)=3$.
So, without loss of generality, $\{s_1,s_2,s_3,e\}$, $\{s_1,s_2',s_4,e\}$, and $\{s_1',s_2,s_5,e\}$ are circuits.
Now $M/e$ has triangles $\{s_1,s_2,s_3\}$, $\{s_1,s_2',s_4\}$, $\{s_1',s_2,s_5\}$, and $\{s_3,s_4,s_5\}$.
Since $M/e$ is a $7$-element rank-$3$ matroid with a $U_{2,5}$-minor, it has some element that is not in two distinct triangles.
It follows that $M/e$ has precisely the four aforementioned triangles.
But now $M/e \backslash s_1 \backslash s_5 \cong U_{3,5}$, so $s_1$ is $U_{3,5}$-flexible, a contradiction.
We may now assume that $M \backslash e$ has at least two non-trivial series classes where, if there are precisely two non-trivial series classes, then one has size at least~$3$.
First, assume that $\co(M \backslash e) \cong U_{2,6}$.
Let $S$ be a non-trivial series class of $M \backslash e$ where, if there are only two such series classes, then $|S| \ge 3$.
By \cref{cosimpcase}, there exists some $f \in S$ such that $\co(M \backslash f)$ is $3$-connected and has rank and corank at least~$3$.
Now $\co(M \backslash (S \cup e)) \cong U_{2,5}$, so $\co(M \backslash f)$ has a $U_{2,5}$-minor.
By \cref{utfutfprop}, $\co(M \backslash f)$, and hence $M \backslash f$, has a $U_{3,5}$-minor.
Moreover, since $\co(M \backslash e)$ has a $U_{2,5}$-minor, $M/f$ has a $U_{2,5}$-minor.
By \cref{simpcase,utfutfprop}, $M/f$ has a $U_{3,5}$-minor, so $f$ is $U_{3,5}$-flexible, a contradiction.
We may now assume that $\co(M \backslash e) \cong U_{2,5}$.
Let $(S_1,S_2,\dotsc,S_5)$ be a partition of $E(M \backslash e)$ into series classes where, for some $h \in \{2,3,4,5\}$, we have $|S_i| \ge 2$ if and only if $i \in [h]$, and, in the case that $h=2$, we have $|S_2| \ge 3$.
For $i \in [5]-[h]$, let $S_i = \{s_i\}$.
\begin{claim}
\label{noparallelcases}
Let $s_i \in S_i$ for $i \in [h]$.
Then $M / (\bigcup_{i \in [h]} S_i-s_i)$ is loopless and has a single parallel pair, which contains $e$.
\end{claim}
\begin{subproof}
The matroid $M' = M / (\bigcup_{i \in [h]} S_i - s_i)$ has rank two, and $M' \backslash e \cong U_{2,5}$, so either $M' \cong U_{2,6}$, or $e$ is a loop in $M'$, or $M'$ has a single parallel pair, which contains $e$.
For $i \in [h]$, let $S_i^- = S_i-s_i$.
Firstly, suppose that $M' \cong U_{2,6}$.
Then $e \notin \cl_M(\bigcup_{i \in [h]} S_i^-)$, and it follows that $M / (\bigcup_{i \in [h-1]} S_i^-) \backslash S_h \cong U_{2,5}$.
By \cref{cosimpcase}, there exists $f \in S_h$ such that $\co(M \backslash f)$ is $3$-connected with rank and corank at least~$3$.
Since $M \backslash f$ has a $U_{2,5}$-minor, \cref{utfutfprop} implies that $M \backslash f$ has a $U_{3,5}$-minor.
Moreover, by \cref{simpcase,utfutfprop}, $M/f$ has a $U_{3,5}$-minor, so $f$ is $U_{3,5}$-flexible, a contradiction.
Now we may assume that $e$ is a loop in $M'$.
Let $T = \bigcup_{i \in [h]} S_i^-$.
Then there is a circuit~$C$ contained in $T \cup e$.
Note that, by orthogonality, if $C$ meets $S_i^-$ for some $i \in [h]$, then $S_i^- \subseteq C$.
Since $|C| \ge 3$, either $C$ meets some $S_i^-$ where $|S_i^-| \ge 2$, or $C$ meets $S_i^-$ and $S_j^-$ for distinct $i,j \in [h]$.
Thus, in the case that
$h=2$, we have $S_2^- \subseteq C$.
For any $c \in C$, the matroid $M \backslash c$ has a $U_{2,5}$-minor, since $\co(M \backslash e) \cong M \backslash e / T \cong M \backslash c / e / (T-c) \cong U_{2,5}$.
If $h \ge 3$, then, by \cref{cosimpcase}\ref{csc1}, $\co(M \backslash c)$ is $3$-connected and has rank and corank at least~$3$, for each $c \in C$.
Otherwise, when $h=2$, the circuit $C$ contains $S_2^-$ with $|S_2^-| \ge 2$, and by \cref{cosimpcase}\ref{csc2} there exists some $c \in S_2^-$ such that $\co(M \backslash c)$ is $3$-connected and has rank and corank at least~$3$.
In either case, $M \backslash c$ has a $U_{3,5}$-minor by \cref{utfutfprop}.
Moreover, by \cref{simpcase,utfutfprop}, $M / c$ has a $U_{3,5}$-minor.
So $c$ is $U_{3,5}$-flexible, a contradiction. \end{subproof}
By \cref{noparallelcases}, we may now assume, for every choice of $s_i$'s, that $\{e,s_j\}$ is a parallel pair in $M / (\bigcup_{i \in [h]} S_i - s_i)$, for some $j \in [5]$.
So $M \backslash s_j$ has a $U_{2,5}$-minor.
Assume next that $M \backslash e$ has precisely two non-trivial series classes $S_1$ and $S_2$, with $|S_1| = 2$ and $|S_2|=3$.
We will show that this case is contradictory.
Let $S_1 = \{s_1,s_1'\}$ and $S_2 = \{s_2,s_2',s_2''\}$.
Since $|S_1 \cup S_2 \cup e|=6$ and $r(M)=5$, the set $S_1 \cup S_2 \cup e$ contains a circuit~$C$.
Since $\co(M \backslash e) \cong U_{2,5}$, it follows that $e \in C$.
Then, by orthogonality, $|C| \ge 4$.
By \cref{noparallelcases}, $|C| \neq 4$.
Suppose $S_1 \cup S_2 \cup e$ is a circuit.
It now follows from \cref{noparallelcases} and circuit elimination that, without loss of generality, $\{s_1,s_2,s_2',e,s_3\}$, $\{s_1,s_2,s_2'',e,s_4\}$, and $\{s_1,s_2',s_2'',e,s_5\}$ are circuits; and $\{s_1',s_2,s_2',e,s_5\}$, $\{s_1',s_2,s_2'',e,s_3\}$, and $\{s_1',s_2',s_2'',e,s_4\}$ are circuits.
But then $M/e$ has no $U_{2,5}$-minor, a contradiction.
Next suppose $S_2 \cup \{s_1,e\}$ is a circuit.
Then each element in $S_2$ is $U_{2,5}$-flexible.
By \cref{simpcase,cosimpcase,utfutfprop}, there is some $f \in S_2$ that is $U_{3,5}$-flexible, a contradiction.
Now, up to labels, we may assume that $S_1 \cup \{s_2',s_2'',e\}$ is a circuit.
Then each element in $S_1$ is $U_{2,5}$-flexible.
By \cref{cosimpcase,utfutfprop}, we may assume, up to labels, that $s_1$ is $U_{3,5}$-deletable.
We claim that $s_1$ is also $U_{3,5}$-contractible.
Consider $M / s_1 / s_2'$.
This is a rank-$3$ matroid with rank-$2$ sets $\{s_1',s_3,s_4,s_5\}$ and $\{s_1',s_2'',e\}$.
Moreover, by \cref{noparallelcases}, $M$ has a circuit contained in $\{s_1,s_2,s_2',e,q\}$, where $q \in \{s_1',s_2'',s_3,s_4,s_5\}$.
This circuit is distinct from the circuit $\{s_1,s_1',s_2',s_2'',e\}$.
By circuit elimination, and since no circuit is contained in $S_1 \cup S_2$, it follows that $q \in \{s_3,s_4,s_5\}$.
Without loss of generality, $\{s_2,e,s_3\}$ is a circuit of $M / s_1/s_2'$.
It now follows that $M / s_1 / s_2' \backslash s_1' \backslash s_3 \cong U_{3,5}$.
So $s_1$ is $U_{3,5}$-flexible, a contradiction.
Recall that when $h=2$, we may assume that $|S_2| \ge 3$.
By the foregoing, we may now also assume, in this case, that $|S_1|+|S_2| \ge 6$.
\begin{claim}
\label{parallelgoodcase}
Let $s_i \in S_i$ for $i \in [h]$.
Then $M / (\bigcup_{i \in [h]} S_i-s_i)$ has a parallel pair $\{e,s_j\}$, for some $j \in [5]-[h]$.
\end{claim}
\begin{subproof}
Suppose $M / (\bigcup_{i \in [h]} S_i-s_i)$ has a parallel pair $\{e,s_j\}$ for $j \in [h]$.
To begin with, assume also that $h \ge 3$.
Then \cref{cosimpcase}\ref{csc1} implies that $\co(M \backslash s_j)$ is $3$-connected with rank and corank at least~$3$.
Thus, by \cref{utfutfprop}, $M \backslash s_j$ has a $U_{3,5}$-minor.
Moreover, by \cref{simpcase,utfutfprop}, $M/s_j$ has a $U_{3,5}$-minor, so $s_j$ is $U_{3,5}$-flexible, a contradiction.
So $h=2$. In particular, $j \in \{1,2\}$.
Suppose that $|S_j| \ge 3$.
Let $S_i^- = S_i - s_i$ for $i \in \{1,2\}$, and let $\{j,j'\} = \{1,2\}$.
Then $S_{j'} \cup S_j \cup e$ contains a circuit~$C$ of $M$, with $\{e,s_j\} \subseteq C$.
Suppose $\co(M \backslash s_j)$ is not $3$-connected.
Then, by \cref{precosimpcase}, there exists a circuit~$C'$ such that $S_j^- \cup e \subseteq C' \subseteq S_{j'} \cup S_j^- \cup e$.
Note that $s_j \in C-C'$, so $C \neq C'$, and $e \in C \cap C'$.
By circuit elimination, there is a circuit contained in $(C \cup C') - e \subseteq S_{j'} \cup S_j$.
By orthogonality, $S_{j'} \cup S_j$ is a circuit.
But then $\co(M \backslash e)$ contains a parallel pair, a contradiction.
So $\co(M \backslash s_j)$ is $3$-connected; moreover, this matroid has rank and corank at least~$3$, by \cref{cosimpcase}.
Now $s_j$ is $U_{2,5}$-deletable, so, by \cref{utfutfprop}, $M \backslash s_j$ has a $U_{3,5}$-minor.
Moreover, by \cref{simpcase,utfutfprop}, $M/s_j$ has a $U_{3,5}$-minor, so $s_j$ is $U_{3,5}$-flexible, a contradiction.
We deduce that $|S_j| = 2$.
In particular, $j=1$, as $|S_2| \ge 3$. Since $|S_1| + |S_2| \ge 6$, we have $|S_2| \ge 4$.
Now, as $S_2 \cup e$ is a coclosed cosegment, $M.(S_2 \cup e) \cong U_{|S_2|-1,|S_2|+1}$ and $f \notin \cl^*(S_2 \cup e)$. Since $|S_2| \ge 4$, the element $f$ is $U_{3,5}$-flexible, a contradiction. \end{subproof}
By \cref{parallelgoodcase} we may now assume, for every choice of $s_i$'s, that $\{e,s_j\}$ is a parallel pair in $M / (\bigcup_{i \in [h]} S_i-s_i)$ for some $j \in [5] - [h]$ (in other words, for some $j$ such that $\{s_j\}$ is a series class of $M \backslash e$).
Then there is a circuit~$C$ such that $\{e,s_j\} \subseteq C \subseteq \{e,s_j\} \cup (\bigcup_{i \in [h]} S_i-s_i)$.
By orthogonality, $C= \{e,s_j\} \cup (\bigcup_{i \in [h]} S_i-s_i)$.
Note that $r(M) = 2 +\Sigma_{i \in [h]} (|S_i| - 1) = |C|$.
So $r(C) = r(M)-1$.
Moreover, any proper superset $D$ of $C$ contains at least two series classes of $M \backslash e$, in which case $D$ spans $E(M)$.
So $C$ is a circuit-hyperplane and $E(M)-C$ is a cocircuit.
Suppose that $M \backslash e$ has precisely one trivial series class, so $j=5$.
Then $\{e,s_5\} \cup \left(\bigcup_{i \in [4]} S_i - s_i\right)$ is a circuit and $C^*=\{s_1,s_2,s_3,s_4\}$ is a cocircuit.
But $\{e,s_5\} \cup \left(\bigcup_{i \in [3]} S_i-s_i\right) \cup (S_4-s_4')$ is also a circuit for $s_4' \in S_4-s_4$, and this circuit intersects $C^*$ in a single element, $s_4$, contradicting orthogonality.
Next suppose that $M \backslash e$ has precisely two trivial series classes, $\{s_4\}$ and $\{s_5\}$.
Suppose $|S_1|=|S_2|=|S_3|=2$, so $|E(M)| = 9$.
Then it is readily checked that $M / S_i \backslash s_4,s_5 \cong U_{3,5}$, for $i \in [3]$.
So $M$ has at most one $U_{3,5}$-essential element, and thus the \lcnamecref{jail-elastic} holds in this case.
We may now assume that $|S_3| \ge 3$, say.
Let $s_3,s_3',s_3''$ be distinct elements in $S_3$.
The set $C_1=\{e,s_j\} \cup \left(\bigcup_{i \in [3]} S_i - s_i\right)$ is a circuit and $C_1^*=\{s_1,s_2,s_3,s_{j'}\}$ is a cocircuit, for some $\{j,j'\} = \{4,5\}$.
Also, $C_2=\{e,s_k\} \cup \left(\bigcup_{i \in [2]} S_i - s_i\right) \cup (S_3-s_3')$ is a circuit and $C^*_2=\{s_1,s_2,s_3',s_{k'}\}$ is a cocircuit, for some $\{k,k'\} = \{4,5\}$.
By orthogonality between $C_1$ and $C^*_2$, we have $j=k'$, so $j'=k$.
Furthermore, $C_3=\{e,s_\ell\} \cup \left(\bigcup_{i \in [2]} S_i - s_i\right) \cup (S_3-s_3'')$ is a circuit and $C^*_3=\{s_1,s_2,s_3'',s_{\ell'}\}$ is a cocircuit, for some $\{\ell,\ell'\} = \{4,5\}$.
By orthogonality between $C_1$ and $C^*_3$, we have $j=\ell'$,
but by orthogonality between $C_2$ and $C^*_3$, we have $k=\ell'$, so $j=k$, a contradiction.
Now suppose that $M \backslash e$ has three trivial series classes, so $M \backslash e$ has precisely two non-trivial series classes, $S_1$ and $S_2$, and $|S_2| \ge 3$.
If $S_2$, say, has size at least~$4$, then $S_2 \cup e$ is a coclosed cosegment of size at least~$5$, and it follows that any $f \in S_1$ is $U_{3,5}$-flexible.
Recall also that $|S_1| + |S_2| \ge 6$.
So we may assume that $|S_1| = 3$ and $|S_2| = 3$.
Let $S_1 = \{t_1,t_2,t_3\}$ and $S_2 = \{u_1,u_2,u_3\}$.
As before, for $i,j \in [3]$, the set $C_{i,j}=\{e,w_{i,j}\} \cup (S_1 - t_i) \cup (S_2 - u_j)$ is a circuit and $C_{i,j}^*=\{t_i,u_j\} \cup (\{s_3,s_4,s_5\}-w_{i,j})$ is a cocircuit, with $\{w_{i,1},w_{i,2},w_{i,3}\} = \{3,4,5\}$ for $i \in [3]$ and $\{w_{1,j},w_{2,j},w_{3,j}\} = \{3,4,5\}$ for $j \in [3]$.
Without loss of generality,
$w_{i,j}=s_{((i+j) \bmod 3)+3}$.
As $M / e$ has a $U_{2,5}$-minor and $r(M)=6$, there exists a $3$-element independent set $C \subseteq E(M/e)$ such that $M/(C \cup e)$ has a $U_{2,5}$-minor.
Suppose that $C \subseteq E(M/e) - S_2$.
Then $(E(M/e)-S_2) - C$ is a parallel class of size three in $M/(C \cup e)$, in which case $|E(\co(M/(C \cup e)))| \le 4$, implying $M / (C \cup e)$ has no $U_{2,5}$-minor.
So $C$ meets $S_2$ and, similarly, $C$ meets $S_1$.
Let $C = (S_1 - t_i) \cup u_j$.
Then $M / (C \cup e)$ has two distinct parallel pairs, due to the circuits $C_{i,j'}$ for $j' \in [3]-j$, so again $M / (C \cup e)$ has no $U_{2,5}$-minor.
Now we may assume that $|C \cap S_1| = 1$ and $|C \cap S_2| = 1$.
Without loss of generality let $C=\{t_1,u_1,s_4\}$.
Then $\{s_3,s_5\}$ and $C_{2,2}-C=\{t_3,u_3\}$ are parallel pairs in $M/(C \cup e)$, so again $M / (C \cup e)$ has no $U_{2,5}$-minor.
We deduce that $M/e$ has no $U_{2,5}$-minor, a contradiction. \end{proof}
\begin{lemma}
\label{jail-nonelastic}
For $\mathbb{P} \in \{\mathbb{H}_5,\mathbb{U}_2\}$,
let $M$ be a $3$-connected $\mathbb{P}$-representable $U_{3,5}$-fragile matroid that is not $\{U_{2,5},U_{3,5}\}$-fragile, where $r(M) \ge 4$ and $r^*(M) \ge 4$.
Suppose $M$ has no $U_{2,5}$-elastic elements, and let $F$ be the set of $U_{2,5}$-flexible elements of $M$.
Then one of the following holds:
\begin{enumerate}
\item $|F| \in \{3,4,5\}$, the set $F$ is a fan that is contained in a $5$-element fan $F'$, and there exists an element $g$ such that either $M|(F' \cup g)$ or $M^*|(F' \cup g)$ is isomorphic to $M(K_4)$.
Moreover, if $|F|=3$, then $F$ is the set of internal elements of $F'$.
\item $|F| \in \{4,5\}$ but there is no $4$- or $5$-element maximal fan that contains $F$.
\item $|F|\ge 6$.
\end{enumerate} \end{lemma} \begin{proof}
We start by proving the following claim:
\begin{claim}
\label{flexstart}
If $e$ is a $\{U_{2,5},U_{3,5}\}$-flexible element of $M$, then $e$ is $U_{2,5}$-flexible.
\end{claim}
\begin{subproof}
Let $e$ be a $\{U_{2,5},U_{3,5}\}$-flexible element of $M$.
Clearly the claim holds if $e$ is $U_{3,5}$-essential, so assume otherwise.
Suppose $e$ is $U_{3,5}$-deletable
and $U_{2,5}$-contractible.
Then $\co(M \backslash e)$ has a $U_{3,5}$-minor, so $r(\co(M \backslash e)) \ge 3$.
Since $r^*(M) \ge 4$, we have $r^*(\co(M \backslash e)) \ge 3$.
Now $\co(M \backslash e)$ is a $3$-connected matroid with rank and corank at least~$3$, and having a $U_{3,5}$-minor.
Hence, by \cref{utfutfprop}, $\co(M \backslash e)$ has a $U_{2,5}$-minor.
Then $e$ is $U_{2,5}$-flexible, as claimed.
Suppose now that $e$ is $U_{3,5}$-contractible
and $U_{2,5}$-deletable.
Since $r(M) \ge 4$, we have $r(\si(M / e)) \ge 3$.
If $r^*(\si(M / e)) \ge 3$, then $\si(M / e)$ has both a $U_{2,5}$- and a $U_{3,5}$-minor, by \cref{utfutfprop}, in which case $e$ is $U_{2,5}$-flexible, as required.
Similarly, as $r^*(M \backslash e) \ge 3$, we have $r(\co(M \backslash e)) = 2$, otherwise, by \cref{utfutfprop}, $e$ is $U_{3,5}$-flexible, a contradiction.
So $r(\co(M \backslash e)) = 2$ and we may assume that $r^*(\si(M / e)) = 2$.
In particular, in $M$, the element~$e$ is in at least two distinct triangles, and at least two distinct triads.
If $e$ is in a $4$-element segment $L$ of $M$, then each triad containing $e$ is contained in $L$, by orthogonality. But then $M$ has a triangle-triad, contradicting that $M$ is $3$-connected.
It follows that $e$ is in triangles $T_1$ and $T_2$, with $r(T_1 \cup T_2) = 3$; and, similarly, $e$ is in triads $T_1^*$ and $T_2^*$ with $r^*(T_1^* \cup T_2^*) = 3$. By orthogonality, $T_1 \cup T_2 = T_1^* \cup T_2^*$.
But then $\lambda(T_1 \cup T_2) = 1$, so, as $M$ is $3$-connected, $|E(M)| \le 6$, a contradiction.
Hence $e$ is $U_{2,5}$-flexible, as claimed. \end{subproof}
Since $M$ is not $\{U_{2,5},U_{3,5}\}$-fragile, there exists an element~$e$ that is $\{U_{2,5},U_{3,5}\}$-flexible.
By \cref{flexstart}, $e$ is $U_{2,5}$-flexible.
\begin{claim}
\label{f0}
For $(M_0,N_0) \in \{(M,U_{3,5}), (M^*,U_{2,5})\}$, the matroid $M_0$ has at least three $N_0^*$-flexible elements. \end{claim} \begin{subproof}
By hypothesis, $e$ is not $U_{2,5}$-elastic in $M$.
Now, for some $(M_0,N_0) \in \{(M,U_{3,5}), (M^*,U_{2,5})\}$, the matroid $M_0$ is $N_0$-fragile and has an $N_0^*$-flexible element $e$ such that $\si(M/e)$ is not $3$-connected.
By \cref{vert3sep}, $M_0$ has a vertical $3$-separation $(X,e,Y)$.
By \cref{niceVertSep} we may assume that $|X \cap E(N_0^*)| \le 1$ and $Y \cup e$ is closed.
By \cref{CPL}, at most one element of $X$ is not $N_0^*$-flexible so, as $|X| \ge 3$, the set $X$ contains at least two $N_0^*$-flexible elements.
So $M_0$ has at least three $N_0^*$-flexible elements, as $e$ is also $N_0^*$-flexible.
The claim follows by duality. \end{subproof}
Now, for some $(M_1,N_1) \in
\{(M,U_{3,5}), (M^*,U_{2,5})\}$, the matroid $M_1$ is $N_1$-fragile and has two $N_1^*$-flexible elements $e_1$ and $e_2$ such that $\si(M/e_i)$ is not $3$-connected for $i \in \{1,2\}$.
By \cref{vert3sep}, $M_1$ has vertical $3$-separations $(X_i,e_i,Y_i)$ for $i \in \{1,2\}$, with $Y_i \cup e_i$ closed and $|X_i \cap E(N_1^*)| \le 1$.
\begin{claim}
\label{f3}
If $Z$ is the set of $N_1^*$-flexible elements of $M_1$, and $|Z|=3$, then $Z$ is a triangle that is contained in a $5$-element fan $F'$, and there exists an element $g$ such that $M_1^*|(F' \cup g) \cong M(K_4)$.
\end{claim}
\begin{subproof}
Suppose that $M_1$ has precisely three $N_1^*$-flexible elements.
Let $i \in \{1,2\}$.
Then, as $|X_i \cup e_i| \ge 4$ and at most one element in $X_i \cup e_i$ is not $N_1^*$-flexible, by \cref{CPL},
$|X_i|=3$.
Moreover, there exists $f_i \in X_i \cap \cl^*(Y_i)$ that is not $N_1^*$-flexible, and $e_i \in \cl(X_i-f_i)$.
Then $(X_i-f_i) \cup e_i$ is a triangle and $X_i$ is a triad, so $X_i \cup e_i$ is a $4$-element fan where $e_i$ is a spoke end and $f_i$ is a rim end.
Now the $N_1^*$-flexible elements form a triangle $\{e_1,e_2,e_3\}$, for some element $e_3$.
Let $F'$ be the fan with ordering $(f_2,e_1,e_3,e_2,f_1)$, noting that $f_1 \neq f_2$ follows from the fact that $M_1$ is $3$-connected.
Since $e_3$ is $N_1^*$-flexible but $M_1$ has no $N_1^*$-elastic elements, at least one of $\si(M / e_3)$ or $\co(M \backslash e_3)$ is not $3$-connected.
Suppose $\co(M \backslash e_3)$ is not $3$-connected.
Thus, there exists a cyclic $3$-separation $(X_3,e_3,Y_3)$ with $Y_3 \cup e_3$ coclosed and $|X_3 \cap E(N_1^*)| \le 1$.
By \cref{CPL}, at most one element of $X_3$ is not $N_1^*$-flexible.
As $M_1$ has three $N_1^*$-flexible elements, $|X_3| = 3$, so $X_3$ is a triangle.
But $\{e_1,e_2\} \subseteq X_3$, so $\{e_1,e_2\}$ is contained in a triangle distinct from $\{e_1,e_2,e_3\}$, contradicting orthogonality with the triads $\{f_2,e_1,e_3\}$ and $\{e_3,e_2,f_1\}$.
Thus $\si(M / e_3)$ is not $3$-connected.
Then, by \cref{fanmiddle}, there exists an element $g \in E(M_1)-F'$ such that $M_1^*|(F' \cup g) \cong M(K_4)$.
Letting $F = F' \cup g$, the claim follows.
\end{subproof}
\begin{claim}
\label{f4}
Let $Z$ be the set of $N_1^*$-flexible elements of $M_1$, and suppose $|Z|=4$.
Then either $Z$ is a fan contained in a set $F$ such that $M_1^*|F$ is isomorphic to $M(K_4)$, or
there is no $4$- or $5$-element maximal fan that contains $Z$.
\end{claim}
\begin{subproof}
Let $i \in \{1,2\}$.
Then, as at most one element in $X_i \cup e_i$ is not $N_1^*$-flexible, by \cref{CPL}, we have $|X_i| \in \{3,4\}$.
First, assume $X_1$ and $X_2$ both contain only two $N_1^*$-flexible elements.
Then, by \cref{CPL}, for each $i \in \{1,2\}$, we have $|X_i| = 3$, there exists $f_i \in X_i \cap \cl^*(Y_i)$ that is not $N_1^*$-flexible, and $e_i \in \cl(X_i-f_i)$.
Thus $(X_i-f_i) \cup e_i$ is a triangle and $X_i$ is a triad, so $X_i \cup e_i$ is a $4$-element fan where $e_i$ is a spoke end and $f_i$ is a rim end.
Let $X_i' = (X_i-f_i) \cup e_i$ for $i \in \{1,2\}$.
Since $X'_1 \cup X'_2 \subseteq Z$ and $|Z|=4$, we have $|X'_1 \cap X'_2| \ge 2$.
But if $|X'_1 \cap X'_2|=2$, then $X'_1 \cup X'_2$ is a $4$-element segment, contradicting orthogonality with the triad $X_1$.
So $X'_1 = X'_2$, and this set is a triangle of $N_1^*$-flexible elements.
Now $X'_1 = \{e_1,e_2,e_3\}$ for some element $e_3$.
Let $F'$ be the fan with ordering $(f_2,e_1,e_3,e_2,f_1)$,
where $f_1 \neq f_2$ follows from the fact that $M_1$ is $3$-connected.
Let $Z-\{e_1,e_2,e_3\} = \{e_4\}$.
Since $e_3$ is $N_1^*$-flexible but $M_1$ has no $N_1^*$-elastic elements, at least one of $\si(M / e_3)$ or $\co(M \backslash e_3)$ is not $3$-connected.
Suppose
$\co(M \backslash e_3)$ is not $3$-connected.
Thus, there exists a cyclic $3$-separation $(X_3,e_3,Y_3)$ with $Y_3 \cup e_3$ coclosed and $|X_3 \cap E(N_1^*)| \le 1$.
As at most one element of $X_3 \cup e_3$ is not $N_1^*$-flexible, by \cref{CPL}, we have $|X_3 \cap Z| \in \{2,3\}$, so $|X_3| \in \{3,4\}$.
If $|X_3| = 3$, then $X_3$ is a triangle that contains at least two elements of $\{e_1,e_2,e_4\}$, in which case it follows from orthogonality that either $X_3=\{e_1,f_2,e_4\}$ or $X_3=\{f_1,e_2,e_4\}$, so $Z$ is contained in a $6$-element fan and the claim holds.
So we may assume that $|X_3| = 4$, in which case $X_3 = \{e_1,e_2,e_4,p\}$ for some element $p$.
Since $p$ is not $N_1^*$-flexible, we have
$p \in \cl(Y_3)$ and $e_3 \in \cl^*(\{e_1,e_2,e_4\})$ by \cref{CPL}.
Now $\{e_1,e_2,e_4\}$ is $3$-separating, and $e_3$ is in the closure and coclosure of this set, so $\lambda(\{e_1,e_2,e_3,e_4\})=1$, a contradiction.
So we may assume that $\si(M / e_3)$ is not $3$-connected.
Then, by \cref{fanmiddle}, there exists an element $g \in E(M_1)-F'$ such that $M_1^*|(F' \cup g) \cong M(K_4)$.
If $g = e_4$, then the claim holds with $F = F' \cup e_4$.
So we may assume that $g \neq e_4$.
Suppose, for a contradiction, that $Z$ is contained in a fan $F$ with $|F|\le 5$.
By \cref{fanunique}, the unique triangle containing $e_3$ is $\{e_1,e_2,e_3\}$, and $\{e_1,e_2,g\}$ is the unique triad containing $\{e_1,e_2\}$ by orthogonality.
Thus, if $e_3$ is a spoke end of $F$, then $(e_3,e_i,e_j,g,e_4)$ is a fan ordering of $F$ for some $\{i,j\}=\{1,2\}$.
But then $\{e_j,g,e_4\}$ is a triangle, contradicting orthogonality.
So $e_3$ is not a spoke end of $F$.
As the unique triads containing $e_3$ are $\{e_3,e_1,f_2\}$ and $\{e_3,e_2,f_1\}$, by \cref{fanunique}, this implies that $f_i \in F$ for some $i \in \{1,2\}$.
Then, without loss of generality, $F=\{e_1,e_2,e_3,f_1,e_4\}$, so either $e_4$ is in a triangle with $f_1$,
or $e_4$ is in a triad with $e_1$.
But $\{e_4,f_1\}$ is not contained in a triangle, by orthogonality.
So, by orthogonality again, either $\{e_4,e_1,e_2\}$ or $\{e_4,e_1,e_3\}$ is a triad.
In the former case, $\{e_1,e_2,g,e_4\}$ is a cosegment, and in the latter case $\{e_1,f_2,e_3,e_4\}$ is a cosegment; both contradict orthogonality with the triangle $\{e_1,e_2,e_3\}$.
Now we may assume that $X_1$ contains precisely three $N_1^*$-flexible elements.
Suppose $|X_1|=4$. Then, by \cref{CPL}, there is a unique element $f_1 \in X_1$ that is not $N_1^*$-flexible, and, letting $X_1' = X_1 - f_1$ and $Y_1' = Y_1 \cup f_1$, there is a path of $3$-separations $(X_1', e_1, Y_1')$.
If $r(X_1') = 2$, then $Z=X_1' \cup e_1$ is a segment, so there is no fan that contains $Z$. So we may assume that $r(X_1') = 3$, in which case $(X_1', e_1, Y_1')$ is a vertical $3$-separation such that $Y_1' \cup e_1$ is closed, and $|X_1' \cap E(N_1^*)| \le 1$.
Thus, by replacing $(X_1,e_1,Y_1)$ with $(X_1',e_1,Y_1')$ if necessary, we may assume that $|X_1| = 3$.
By a similar argument, we may assume that $|X_2| = 3$.
If each element of $X_2$ is $N_1^*$-flexible, then $X_1 \cup e_1 = X_2 \cup e_2$. But then, as $X_1$ and $X_2$ are distinct triads, $X_1 \cup e_1$ is a cosegment, contradicting that $e_1$ is a guts element.
So $X_2$ has two $N_1^*$-flexible elements, in which case, as before, there is a unique element $f_2 \in X_2$ that is not $N_1^*$-flexible, and $X_2 \cup e_2$ is a $4$-element fan where $e_2$ is a spoke end and $f_2$ is a rim end.
Thus $e_2$ is in a triangle that contains $e_1$ and is contained in $X_1 \cup e_1$; we choose $e_3$ and $e_4$ so that this triangle is $\{e_1,e_2,e_3\}$, and $X_1 = \{e_2,e_3,e_4\}$.
Note that
$X_1$ is an independent triad,
$X_1 \cup e_1$ is a $4$-element fan where $e_1$ is a spoke end,
and $X_2 = \{f_2,e_1,e_3\}$.
Thus $(e_4,e_2,e_3,e_1,f_2)$ is an ordering of a $5$-element fan~$F'$, where
$\{e_1,e_2,e_3,e_4\}$ is the set of $N_1^*$-flexible elements in $F'$.
As $e_3$ is $N_1^*$-flexible but not $N_1^*$-elastic, either $\co(M_1 \backslash e_3)$ or $\si(M_1/e_3)$ is not $3$-connected.
If $\si(M_1 / e_3)$ is not $3$-connected, then, by \cref{fanmiddle}, there exists an element $g$ such that $M_1^*|(F' \cup g) \cong M(K_4)$, so \ref{f4} holds.
So we may assume that $\co(M_1 \backslash e_3)$ is not $3$-connected.
Then there exists a cyclic $3$-separation $(X_3,e_3,Y_3)$ with $Y_3 \cup e_3$ coclosed and $|X_3 \cap E(N_1^*)| \le 1$.
By \cref{CPL}, at most one element of $X_3$ is not $N_1^*$-flexible, so $|X_3| \in \{3,4\}$.
We may assume that $|X_3|=3$ (by the same argument used earlier for $X_1$ and $X_2$),
in which case $X_3$ is a triangle that contains at least two elements of $\{e_1,e_2,e_4\}$.
By orthogonality, $X_3 = \{e_4,e_2,x\}$ for some $x \notin F'$.
But then $F' \cup x$ is a $6$-element fan, so \ref{f4} holds. \end{subproof}
\begin{claim}
\label{f5}
Let $F$ be the set of $N_1^*$-flexible elements of $M_1$, and suppose $|F|=5$.
If $F$ is a maximal fan, then there exists an element $g \in E(M_1)-F$ such that either $M_1|(F \cup g)$ or $M_1^*|(F \cup g)$ is isomorphic to $M(K_4)$.
\end{claim}
\begin{subproof}
Suppose $F$ forms a maximal fan with fan ordering $(f_1,f_2,\dotsc,f_5)$.
For some $(M_2,N_2) \in \{(M_1,N_1), (M_1^*,N_1^*)\} = \{(M,U_{3,5}), (M^*,U_{2,5})\}$, the elements $f_1$ and $f_5$ are spoke ends of $F$, the matroid $M_2$ is $N_2$-fragile, and $F$ is the set of $N_2^*$-flexible elements in $M_2$.
As $f_3$ is $N_2^*$-flexible but not $N_2^*$-elastic, at least one of $\si(M_2/f_3)$ or $\co(M_2 \backslash f_3)$ is not $3$-connected.
Suppose $\si(M_2/f_3)$ is not $3$-connected.
Then there exists a vertical $3$-separation $(X,f_3,Y)$ with with $Y \cup f_3$ closed and $|X \cap E(N_2^*)| \le 1$.
By \cref{CPL}, at most one element of $X$ is not $N_2^*$-flexible, so $|X| \in \{3,4\}$.
Suppose $|X|=3$, in which case $X$ is a triad that contains at least two elements of $F - f_3$.
If $X \subseteq F-f_3$, then $X$ intersects one of the triangles $\{f_1,f_2,f_3\}$ or $\{f_3,f_4,f_5\}$ in a single element, contradicting orthogonality.
So, by orthogonality, $X \cap F$ is either $\{f_1,f_2\}$ or $\{f_4,f_5\}$.
But then $F \cup X$ is a $6$-element fan, contradicting that the fan $F$ is maximal.
Now $|X| = 4$.
If each element of $X$ is $N_2^*$-flexible, then $X = F-f_3$.
But then $f_3 \in \cl^*(X)$, so $f_3 \notin \cl(Y)$, a contradiction.
So there is an element $x \in X-F$ that is not $N_2^*$-flexible.
Then, by \cref{CPL}, $x \in \cl^*(Y)$.
It follows that $X-x$ is $3$-separating.
But $X-x \subseteq F-f_3$, so $X-x$ is not a triad, by orthogonality, and $X - x$ is not a triangle, as $r(F)=3$, a contradiction.
We deduce that $\co(M_2 \backslash f_3)$ is not $3$-connected.
Then, by \cref{fanmiddle}, there exists an element $g$ such that $M_2|(F \cup g) \cong M(K_4)$. \end{subproof}
It is easily seen that if there is a fan $F \subseteq X \subsetneqq E(M_1)$ such that $M_1|X \cong M(K_4)$, then $|F| \le 5$. The \lcnamecref{jail-nonelastic} now follows from this fact and \cref{f0,f3,f4,f5}. \end{proof}
\begin{proposition}
\label{jail}
Let $\mathbb{P} \in \{\mathbb{H}_5,\mathbb{U}_2\}$.
Suppose $M$ is a $3$-connected $\mathbb{P}$-representable $U_{3,5}$-fragile matroid that is not $\{U_{2,5},U_{3,5}\}$-fragile, where $r(M) \ge 4$ and $r^*(M) \ge 4$.
Let $F$ be the set of $U_{2,5}$-flexible elements of $M$.
Then one of the following holds:
\begin{enumerate}
\item $|E(M)| \le 9$, and $M$ has at most two $U_{3,5}$-essential elements.\label{9eltfewess}
\item $|F| \ge 4$ and $F$ is not contained in a maximal fan of size at most five.\label{othercase}
\item $|F| \in \{3,4,5\}$, the set $F$ is a fan that is contained in a $5$-element fan $F'$, and there exists an element $g$ such that either $M|(F' \cup g)$ or $M^*|(F' \cup g)$ is isomorphic to $M(K_4)$. Moreover, $F$ is the set of internal elements of $F'$ when $|F|=3$.\label{annoyingextracase}
\end{enumerate} \end{proposition} \begin{proof}
If $M$ has a $U_{2,5}$-elastic element, then \ref{9eltfewess} holds by \cref{jail-elastic}.
Otherwise, $M$ has no $U_{2,5}$-elastic elements, and \ref{othercase} or \ref{annoyingextracase} holds by \cref{jail-nonelastic}. \end{proof}
The next lemma was verified by computer. Note that computational techniques for efficiently enumerating $3$-connected $\mathbb{P}$-representable matroids, for a partial field $\mathbb{P}$, are described in \cite{BP20}.
\begin{lemma}
\label{maxsized}
For $\mathbb{P} \in \{\mathbb{H}_5,\mathbb{U}_2\}$,
suppose $M$ is a $3$-connected $\mathbb{P}$-representable matroid with $r(M) \le 3$. Then $|E(M)| \le 12$.
Moreover,
\begin{enumerate}
\item if $|E(M)| =9$, then $M$ has no $U_{2,5}$-essential elements, at most three $U_{3,5}$-essential elements, and at least six $U_{3,5}$-deletable elements; and\label{ms1}
\item if $|E(M)| \in \{10,11,12\}$, then $M$ has at least six $U_{2,5}$-flexible elements.\label{ms2}
\end{enumerate} \end{lemma}
\begin{theorem}
\label{utfutffragile}
Let $M$ be an excluded minor for the class of $\mathbb{P}$-representable matroids where $\mathbb{P} \in \{\mathbb{H}_5,\mathbb{U}_2\}$.
Suppose $|E(M)| \ge 16$, and there are distinct elements $a,b \in E(M)$ such that $M \backslash a,b$ is $3$-connected and has a $\{U_{2,5},U_{3,5}\}$-minor.
Then $M \backslash a,b$ is a $\{U_{2,5},U_{3,5}\}$-fragile matroid with rank and corank at least~$4$. \end{theorem} \begin{proof}
Clearly $r(M) \ge 3$ and $r^*(M \backslash a,b) \ge 3$, for otherwise $M \backslash a,b$ has a $U_{2,7}$- or $U_{5,7}$-minor, so is not $\mathbb{P}$-representable.
So, by \cref{utfutfprop}, $M \backslash a,b$ has both a $U_{2,5}$-minor and a $U_{3,5}$-minor.
Moreover, if $M \backslash a,b$ has rank or corank three, then it has at most 12 elements, by \cref{maxsized}, so $|E(M)| \le 14$, a contradiction.
So $r(M) \ge 4$ and $r^*(M \backslash a,b) \ge 4$.
Towards a contradiction, assume that $M \backslash a,b$ is not $\{U_{2,5},U_{3,5}\}$-fragile.
By \cref{utfutfequiv}, $M \backslash a,b$ is not $N$-fragile for some $N \in \{U_{2,5},U_{3,5}\}$.
Suppose $M \backslash a,b$ is $N^*$-fragile but not $N$-fragile.
Let $F$ be the set of $N$-flexible elements of $M \backslash a,b$.
Then, by \cref{jail}, either $|F| \ge 4$ and $F$ is not contained in a maximal fan of size at most five, or $F$ is contained in a set $F'$ such that either $M|F'$ or $M^*|F'$ is isomorphic to $M(K_4)$.
By \cref{bcosw-thm}\ref{bcoswii}, if $|F| \ge 4$, then $F$ is contained in a maximal fan of size at most five, so the former does not hold.
Moreover, $F$ is not contained in an $M(K_4)$ restriction or co-restriction,
by \cref{gadgetnotMK4}, so the latter does not hold.
We deduce that $M \backslash a,b$ is neither $U_{2,5}$-fragile nor $U_{3,5}$-fragile.
Let $N \in \{U_{2,5},U_{3,5}\}$.
By \cref{bcosw-thm}, $M$ has a basis~$B$ with $x,y \in B$, where $\{b,x,y\}$ is a triangle up to switching the labels of $a$ and $b$, and there is an $(N,B)$-strong element $u \in B^* - \{a,b\}$.
By \cref{wmatype1again}, we may assume, up to switching the labels of $b$ and $x$, that the $N$-flexible
elements of $M \backslash a,b$ are contained in the set $\{u,x,y\}$.
(We note that due to ``switching labels'' in this way, in order to avoid cumbersome notation, the elements henceforth referred to as $a$ and $b$ may not be the same as those given in the statement of the theorem, as we work towards a contradiction.)
Recall, by \cref{bcosw-thm}\ref{bcoswiii}, that $\{u,x,y\}$ is the unique triad containing $u$ in $M \backslash a,b$.
Thus $\{x,y\}$ is a series pair in $M \backslash a,b,u$.
Note also that $u$ is $N$-flexible in $M \backslash a,b$, for otherwise $M \backslash a,b$ is $N$-fragile.
Moreover, by applying \cref{bcosw-thm} with the minor~$N^*$, the matroid $M \backslash a,b$ has at most five $N^*$-flexible elements and, in the case that $M \backslash a,b$ has five $N^*$-flexible elements, they form a $5$-element maximal fan of $M \backslash a,b$.
\begin{claim}
\label{wmautfutffrag}
For some $M'' \in \{M \backslash a,b \backslash u / x,\, M \backslash a,b \backslash u / x \backslash y,\, M \backslash a,b \backslash u / x / y \}$, the matroid $M''$ is $3$-connected, $\{U_{2,5},U_{3,5}\}$-fragile, and has rank and corank at least~$4$.
\end{claim}
\begin{subproof}
Let $M' = M \backslash a,b \backslash u$.
By \cref{subfrag3conn}, we can choose $M'' \in \{M' / x,\, M' / x \backslash y,\, M' / x / y \}$ such that $M''$ is $3$-connected and $N$-fragile.
Since $|E(M)| \ge 15$, we have $|E(M'')| \ge 10$.
If $M''$ has rank or corank at most three, then, by \cref{maxsized}\ref{ms2}, $M''$ has at least six $N'$-flexible elements for some $N' \in \{U_{2,5},U_{3,5}\}$. But then $M \backslash a,b$ has six $N'$-flexible elements, a contradiction.
So $M''$ has rank and corank at least~$4$.
It remains to show that $M''$ is $\{U_{2,5},U_{3,5}\}$-fragile.
Suppose not.
Then, by \cref{utfutfequiv}, $M''$ is not $N^*$-fragile.
Let $F$ be the $N^*$-flexible elements of $M''$.
By \cref{jail} and since $|E(M'')| \ge 10$,
either $|F| \ge 4$,
or $F$ is a triangle or triad of $M''$ that are the internal elements of a $5$-element fan~$F'$ such that either $M''|(F' \cup g)$ or $(M'')^*|(F' \cup g)$ is isomorphic to $M(K_4)$ for some element $g \in E(M'')-F'$.
Note that $F \subseteq E(M \backslash a,b)-\{u,x\}$, and the elements in $F$ are also $N^*$-flexible in $M \backslash a,b$.
By \cref{bcosw-thm}\ref{bcoswii}, there are at most three $N^*$-flexible elements in $E(M \backslash a,b)-\{u,x\}$, where in the case there are precisely three,
$(y,u,x,z,w)$ is a maximal fan of $M \backslash a,b$, and the $N^*$-flexible elements of $M''$ are $\{y,z,w\}$.
It follows that $F=\{y,z,w\}$ is a triad in $M''$.
Now \cref{jail}\ref{annoyingextracase} holds, and $F$ is contained in a $5$-element fan~$F'$ such that $M''|(F' \cup g) \cong M(K_4)$ for some element $g$.
It follows that $\{z,w\}$ is contained in a triangle in $M''$, which, by orthogonality, is also a triangle in $M \backslash a,b$, contradicting that the fan $(y,u,x,z,w)$ in $M \backslash a,b$ is maximal.
From this contradiction we deduce that $M''$ is $\{U_{2,5},U_{3,5}\}$-fragile.
\end{subproof}
\begin{claim}
\label{maylemma6}
Let $M' \in \{M \backslash a,b \backslash u, M \backslash a,b / u\}$.
Then $M'$ has at most two $N$-essential elements.
\end{claim}
\begin{subproof}
Let $M' = M \backslash a,b \backslash u$ and suppose $M'$ has at least three $N$-essential elements.
By \cref{wmautfutffrag}, we can choose $M'' \in \{M' / x,\, M' / x \backslash y,\, M' / x / y \}$ such that $M''$ is $3$-connected and $\{U_{2,5},U_{3,5}\}$-fragile.
Note that $M''$ also has at least three $N$-essential elements.
By \cref{hopeful}, $|E(M'')| \le 8$, so $|E(M)| \le 13$, a contradiction.
Henceforth, we may assume that $M \backslash a,b \backslash u$ has at most two $N$-essential elements.
Let $M' = M \backslash a,b / u$ and suppose $M'$ has at least three $N$-essential elements.
We claim that
for some $M'' \in \{M',\, M' \backslash x,\, M' \backslash y,\, M' \backslash x,y\}$, the matroid $M''$ is $N$-fragile and $3$-connected up to series classes.
Since the $N$-flexible elements of $M \backslash a,b$ are contained in $\{u,x,y\}$, there is certainly some $M_0'' \in \{M',\, M' \backslash x,\, M' \backslash x,y\}$ that is $N$-fragile and $3$-connected up to series and parallel classes, by \cref{genfragileconn}.
Suppose $M_0''$ has a parallel pair.
Then $u$ is in a triangle of $M \backslash a,b$.
By orthogonality, this triangle meets $\{x,y\}$, so each parallel pair of $M_0''$ meets $\{x,y\}$.
In particular, if $M_0''$ has a parallel pair, then $M_0'' \neq M' \backslash x,y$.
If $y$ is in a parallel pair of $M_0''$, then $y$ is $N$-deletable, and so $M_0'' \backslash y$ is $N$-fragile.
If $x$ is in a parallel pair of $M_0''$, then $M_0'' = M'$ and $M_0'' \backslash x$ is $N$-fragile.
Now, for some $M'' \in \{M_0'', M_0'' \backslash x,\, M_0'' \backslash y\}$, the matroid $M''$ is $N$-fragile and $3$-connected up to series classes.
Since $\{u,x,y\}$ is the unique triad of $M \backslash a,b$ containing $u$, there is no triad of $M'$ containing $\{x,y\}$, and hence $x,y \notin S$ for each series class $S$ of $M''$.
By orthogonality with the triad $\{u,x,y\}$ of $M \backslash a,b$, either $x$ or $y$ is in the fundamental circuit $C(u,B)$ of $u$ with respect to $B$.
Without loss of generality, say $x \in C(u,B)$; then $B-x$ is a basis of $M\backslash a,b /u$.
Moreover, in this matroid $y$ is not in a series pair, and $\{x,y\}$ is not contained in a triad, so there exists some $q \in B^* - \{a,b,u\}$ such that $B' = (B-\{x,y\}) \cup q$ is a basis of $M \backslash a,b / u$, and also of $M''$.
Suppose either $M''$ has two non-trivial series classes, or a series class of size at least~$3$. Since $|S \cap B'| \ge |S|-1$ for each series class of $M''$, there exists some $b'_1 \in B'-q = B-\{x,y\}$ that is $N$-contractible in $M''$.
But then $b'_1 \in B-\{x,y\}$ is also $N$-contractible in $M \backslash a,b$, contradicting that all the $(N,B)$-robust elements are in $\{u,x,y\}$.
So $M''$ has at most one non-trivial series class, and this series class has size two.
In particular, $|E(M'')| \le |E(\co(M''))| +1$.
Suppose $\co(M'')$ is not $\{U_{2,5},U_{3,5}\}$-fragile.
Then, by \cref{utfutfequiv}, $\co(M'')$ is not $N^*$-fragile.
Let $F$ be the set of $N^*$-flexible elements of $\co(M'')$.
Then, by \cref{jail}, either $|E(\co(M''))| \le 9$; the matroid $\co(M'')$ has rank or corank three;
$|F| \ge 4$ and $F$ is not contained in a maximal fan of size at most five; or $|F|=3$ and $F$ is the set of internal elements of a $5$-element fan~$F'$ such that either $\co(M'')|(F' \cup g)$ or $(\co(M''))^*|(F' \cup g)$ is isomorphic to $M(K_4)$ for some element $g$.
Consider the latter two cases.
Note that $F \subseteq E(M \backslash a,b)-\{u\}$, and the elements in $F$ are also $N^*$-flexible in $M \backslash a,b$.
By \cref{bcosw-thm}\ref{bcoswii}, there are at most four $N^*$-flexible elements in $E(M \backslash a,b)-\{u\}$, and these elements are contained in a maximal fan of size four or five, with fan ordering $(y,x,u,z)$ or $(y,u,x,z,w)$ respectively.
In particular, $u$ is in a triangle $\{u,x,z\}$ in $M \backslash a,b$.
Since $M''$ is simple, it follows that $M'' \in \{M' \backslash x, M' \backslash x,y\}$.
Then $F \subseteq E(M \backslash a,b)-\{u,x\}$, and there are at most three $N^*$-flexible elements in $E(M \backslash a,b)-\{u,x\}$ (by \cref{bcosw-thm}\ref{bcoswii} again), so $F=\{y,z,w\}$ and $M'' = M'\backslash x$.
But then $\{z,w\}$ is a series pair in $M''$, so $F \nsubseteq E(\co(M''))$, a contradiction.
Now suppose $r(\co(M'')) = 3$ or $r^*(\co(M'')) = 3$.
If $|E(\co(M''))| \le 8$, then $|E(M)| \le 8 + 1 + 5 = 14$, a contradiction.
By \cref{maxsized}\ref{ms2}, $|E(\co(M''))| \le 12$, and if $|E(\co(M''))| \in \{10,11,12\}$, then $\co(M'')$ has at least six $N'$-flexible elements for some $N' \in \{U_{2,5},U_{3,5}\}$, in which case $M \backslash a,b$ has six $N'$-flexible elements, a contradiction.
So $|E(\co(M''))| =9$.
By \cref{maxsized}\ref{ms1}, and since $M''$ has at least three $N$-essential elements, either $\co(M'')$ has rank three, three $N$-essential elements, and the other six elements are $N$-deletable, where $N \cong U_{3,5}$; or $\co(M'')$ has corank three, three $N$-essential elements, and the other six elements are $N$-contractible, where $N \cong U_{2,5}$.
If $M''$ has no series pairs, then $|E(M)| \le 14$, a contradiction.
So let $\{s,s'\}$ be the unique series pair of $M''$, and, without loss of generality, $\co(M'') = M'' / s'$.
Consider the case where $r(\co(M''))=3$ and $N \cong U_{3,5}$.
If $s$ is $N$-deletable in $\co(M'')$, then $s$ is $N$-flexible in $M''$, contradicting that $M''$ is $N$-fragile.
Otherwise, $s$ is $N$-essential in $\co(M'')$, but $s$ and $s'$ are $N$-contractible in $M''$, in which case $M''$ has at most two $N$-essential elements, a contradiction.
Now consider the case where $r^*(\co(M''))=3$ and $N \cong U_{2,5}$.
Then $M''$ has rank~$7$, and has $B' = (B-\{x,y\}) \cup q$ as a basis, so there are at least five elements of $\co(M'')$ in $B-\{x,y\}$.
As $\co(M'')$ has six $N$-contractible elements, and $|E(\co(M''))| = 9$, there is some $b_1' \in B-\{x,y\}$ that is $N$-contractible in $\co(M'')$.
But then $b_1'$ is also $N$-contractible in $M \backslash a,b$, contradicting that all the $(N,B)$-robust elements are in $\{u,x,y\}$.
Finally, suppose $\co(M'')$ has rank and corank at least~$4$, but $|E(\co(M''))| \le 9$.
Then \cref{jail}\ref{9eltfewess} holds, so $\co(M'')$ has at most two $N$-essential elements.
But, as $M'$ has at least three $N$-essential elements, so does $\co(M'')$, a contradiction.
We deduce that $\co(M'')$ is $\{U_{2,5},U_{3,5}\}$-fragile.
Then, as $\co(M'')$ has at least three $N$-essential elements, \cref{hopeful} implies that $|E(\co(M''))| \le 8$.
But now $|E(M)| \le |E(\co(M''))| + 1 + 5 = 14$, a contradiction.
\end{subproof}
\begin{claim}
\label{mayclaim}
$M \backslash a,b \backslash u$ has at most one $N$-essential element.
\end{claim}
\begin{subproof}
By \cref{wmautfutffrag}, we can choose some $$M'' \in \{M \backslash a,b,u / x,\, M \backslash a,b,u / x \backslash y,\, M \backslash a,b,u / x / y \}$$ such that $M''$ is $3$-connected and $\{U_{2,5},U_{3,5}\}$-fragile.
Towards a contradiction, suppose $M \backslash a,b \backslash u$ has two $N$-essential elements.
By \cref{utfutf3essentialv2}, either $M''$ or $(M'')^*$ can be obtained from $U_{2,5}$ by gluing a wheel.
In either case, the resulting fan~$F$ has $|F| \ge 8$, since $|E(M'')| \ge 10$.
By \cref{fragilefanelements,fragilefans}, $F$ has at least four elements that are $N$-deletable in $M''$.
Let $d$ be an $N$-deletable element of $M''$.
If $d \notin B$, then $d$ is $(N,B)$-robust in $M''$ and hence also in $M \backslash a,b$.
But $u$ is the only $(N,B)$-robust element of $M\backslash a,b$ that is not in $B$, and $u \notin E(M'')$, so this is contradictory.
So each $N$-deletable element of $M''$ is in $B$.
In particular, $F \cap B$ has at least four elements that are $N$-deletable in $M''$.
As $x \notin E(M'')$, at least three of these are in $B - \{x,y\}$.
So $M''$, and hence $M \backslash a,b \backslash u$, has at least three elements in $B-\{x,y\}$ that are not $N$-essential.
Therefore, $M \backslash a,b / u$ has at least three $N$-essential elements, contradicting \cref{maylemma6}.
We deduce that $M \backslash a,b \backslash u$ has at most one $N$-essential element. \end{subproof}
By \cref{maylemma6}, $M \backslash a,b / u$ has at most two $N$-essential elements, and, by \cref{mayclaim}, $M \backslash a,b \backslash u$ has at most one $N$-essential element.
By \cref{thegrandfantasy}, for every $b' \in B-\{x,y\}$, the element $b'$ is $N$-essential in either $M \backslash a,b \backslash u$ or $M \backslash a,b / u$.
Suppose $r(M) \ge 6$.
Then $|B-\{x,y\}| \ge 4$, so either $M \backslash a,b \backslash u$ has at least two $N$-essential elements, or $M \backslash a,b / u$ has at least three $N$-essential elements, a contradiction.
So $r(M) \le 5$.
Moreover, \cref{wmautfutffrag} implies that $r(M) \ge 5$.
So $r(M)=5$, $M \backslash a,b / u$ has precisely two $N$-essential elements, and $M \backslash a,b \backslash u$ has precisely one $N$-essential element.
Let $M''$ be the matroid given by \cref{wmautfutffrag}.
Then $r(M'') = 4$ and $M''$ has an $N$-essential element, so $|E(M'')| \le 9$ by \cref{utfutfessentialrank4}, implying $|E(M)| \le 14$, a contradiction. \end{proof}
\section{Fragile matroids appearing in an excluded minor} \label{fragilepropssec}
Suppose that $M$ is an excluded minor for the class of $\mathbb{P}$-representable matroids, where $\mathbb{P} \in \{\mathbb{H}_5,\mathbb{U}_2\}$, and $M \backslash a,b$ is a $3$-connected
matroid with a $\{U_{2,5},U_{3,5}\}$-minor, for some distinct $a,b \in E(M)$. By \cref{utfutffragile}, if $|E(M)| \ge 16$, then $M \backslash a,b$ is $\{U_{2,5},U_{3,5}\}$-fragile. In this section we consider further properties of such a $\{U_{2,5},U_{3,5}\}$-fragile matroid $M \backslash a,b$.
We work under the following hypotheses throughout this section. Let $M$ be an excluded minor for the class of $\mathbb{P}$-representable matroids where $\mathbb{P} \in \{\mathbb{H}_5,\mathbb{U}_2\}$.
Let $M\backslash a,b$ be a $3$-connected $\{U_{2,5},U_{3,5}\}$-fragile matroid with rank and corank at least~$4$, for distinct $a,b \in E(M)$.
Let $N \in \{U_{2,5},U_{3,5}\}$; then $N$ is a non-binary $3$-connected strong $\mathbb{P}$-stabilizer by \cref{u2stabs}, and $M \backslash a,b$ is $N$-fragile by \cref{utfutfequiv}. By \cref{fragilecase}, there exists a bolstered basis~$B$ for $M$ and a $B \times B^*$ companion $\mathbb{P}$-matrix~$A$ for which $\{x,y,a,b\}$ incriminates $(M,A)$ where $\{x,y\} \subseteq B$ and $\{a,b\} \subseteq B^*$, and $M\backslash a,b$ has at most one $(N,B)$-robust element outside of $\{x,y\}$, where if such an element $u$ exists, then $u\in B^{*}-\{a,b\}$ is an $(N,B)$-strong element of $M\backslash a,b$, and $\{u,x,y\}$ is a coclosed triad of $M\backslash a,b$.
\begin{lemma}
\label{lemmaC}
Suppose that $|E(M)| \ge 15$.
Then
\begin{enumerate}[label=\rm(\Roman*)]
\item $M\backslash a,b$ has an $\{X_8,Y_8,Y_8^*\}$-minor,
\item $M\backslash a,b$ has a nice path description, and
\item for all $e \in E(M \backslash a,b)$, exactly one of $M \backslash a,b\backslash e$ and $M \backslash a,b/e$ has a $\{U_{2,5},U_{3,5}\}$-minor.
\end{enumerate} \end{lemma} \begin{proof}
We can begin by applying \cref{ccmwvz-result}.
If (i) holds, then the \lcnamecref{lemmaC} holds by \cref{nicepathdescription,noessential}.
So one of (ii)--(iv) holds, and $M$ or $M^*$ can be obtained by gluing up to three wheels to $U_{2,5}$ or $Y_8 \backslash 4$.
Each glued wheel corresponds to a fan of $M \backslash a,b$, by \cref{cmvzw-fans}.
Each of these fans has at most five elements, by \cref{fragilefanscase}.
So if (ii) holds, then $|E(M \backslash a,b)| \le 9$, a contradiction.
Similarly, if (iii) holds, then $|E(M \backslash a,b)| \le 10$, a contradiction.
So we may assume that (iv) of \cref{ccmwvz-result} holds.
Then $M \backslash a,b$ or its dual can be obtained from $U_{2,5}$ with ground set $\{x_1,x_2,x_3,x_4,x_5\}$ by gluing wheels to $(x_1,x_3,x_2)$, $(x_1,x_4,x_2)$, and $(x_1,x_5,x_2)$, and each of the resulting fans has size at most five.
Let $F$ be one of these fans of $M \backslash a,b$ with $|F| = 5$.
By \cref{noMK4conn}, if $(f_1,f_2,\dotsc,f_5)$ is a fan ordering of $F$, then $\si(M/f_3)$ is $3$-connected.
Now suppose $F = (f_1,f_2,f_3,f_4,f_5)$ and $F' = (f'_1,f'_2,f'_3,f'_4,f'_5)$ are distinct $5$-element fans obtained by gluing wheels. We claim that
$\{f_2,f_3,f_4\}$ and $\{f'_2,f'_3,f'_4\}$ each contain at least one element that is not $N$-essential.
Without loss of generality, $F$ is the fan obtained by gluing a wheel to $(x_1,x_3,x_2)$, and $F'$ is the fan obtained by gluing a wheel to $(x_1,x_4,x_2)$.
Let $F'' = E(M) -(F \cup F')$.
There is a $\{U_{2,5},U_{3,5}\}$-fragile minor~$M'$ of $M\backslash a,b$, obtained by deleting or contracting elements of $F''$, such that $M'$ can be obtained from $U_{2,5}$, with ground set $\{x_1,x_2,\dotsc,x_5\}$, by gluing one wheel to $(x_1,x_3,x_2)$ and gluing a second wheel to $(x_1,x_4,x_2)$.
If at most one of $x_1$ and $x_2$ is in the remove set when gluing these two wheels, then each fan has at most two $\{U_{2,5},U_{3,5}\}$-essential elements.
So we may assume that both $x_1$ and $x_2$ are removed as part of the operation of gluing these two wheels.
By contracting $f_1$ and $f_5$ from $M'$, we obtain a $\{U_{2,5},U_{3,5}\}$-fragile matroid where each element of $F'$ is not $\{U_{2,5},U_{3,5}\}$-essential; whereas by contracting $f'_1$ and $f'_5$ we obtain a $\{U_{2,5},U_{3,5}\}$-fragile matroid where each element of $F$ is not $\{U_{2,5},U_{3,5}\}$-essential.
This proves the claim.
Now, by \cref{fragilefanscase}, each fan of $M \backslash a,b$ has at most five elements, and if a fan has size five, then it contains $\{x,y\}$.
So at most one of the three fans has size five.
Observe that the size of each of these three fans of $M \backslash a,b$ has the same parity, due to how the wheels are glued to $U_{2,5}$.
Thus, if $M \backslash a,b$ has a $5$-element fan, then $|E(M \backslash a,b)| \le 11$; whereas if each fan of $M \backslash a,b$ has size at most four, then $|E(M \backslash a,b)| \le 12$.
Either case is contradictory, so this completes the proof. \end{proof}
\begin{lemma}
\label{atmostonetri}
Suppose that $|E(M)| \ge 15$.
Then $M \backslash a,b$ has at most one triangle, and if such a triangle~$T$ exists, then
\begin{enumerate}
\item $M \backslash a,b$ has an $(N,B)$-robust element $u \in B^*$ that is in a coclosed triad $\{u,x,y\}$,
\item $T$ contains $u$ and either $x$ or $y$, and
\item $T \cup \{x,y\}$ is a $4$-element fan.
\end{enumerate} \end{lemma} \begin{proof}
Let $T$ be a triangle of $M'=M \backslash a,b$.
By \cref{lemmaC,pathdesctris},
the triangle~$T$ either consists of three $N$-deletable elements, or two $N$-deletable elements and an $N$-contractible element.
In the former case, there exists an element in $T \cap B^*$, as $r(T) = 2$, and this element is $(N,B)$-robust.
So $M'$ has an $(N,B)$-robust element $u$ outside of $\{x,y\}$, with $u \in T$.
Now $u$ is in a triad $\{u,x,y\}$.
By orthogonality, one of $x$ and $y$ is in $T$, and the other is not, since $M'$ is $3$-connected.
In particular, $T \cup \{x,y\}$ is a $4$-element fan, as required.
We may now assume that $T$ consists of two elements that are $N$-deletable in $M'$, and one that is $N$-contractible.
First, suppose $\{x,y\} \subseteq T$.
Let $T= \{x,y,p\}$. Then $p \in B^*$.
Consider the case when $M$ has an $(N,B)$-robust element~$u$.
Note that $p \neq u$, since $\{u,x,y\}$ is a triad and $M$ is $3$-connected.
Moreover, $\co(M' \backslash u)$ is $3$-connected, but this matroid is isomorphic to $\co(M' \backslash u/x)$, which has a parallel pair $\{y,p\}$, a contradiction.
So $M'$ has no $(N,B)$-robust elements.
Now, by the definition of a bolstered basis, no allowable pivot can introduce an $(N,B)$-robust element.
If $p$ is $N$-deletable, then it is $(N,B)$-robust, a contradiction.
So without loss of generality we may assume that $p$ is the $N$-contractible element of $T$; thus $x$ and $y$ are $N$-deletable.
But a pivot on $A_{xp}$
is allowable, by \cref{allowablexyrow2}, where $\{p,y,a,b\}$ incriminates $(M,A^{xp})$, and $x$ is an $(N,B\triangle \{x,p\})$-robust element outside of $\{p,y\}$, contradicting that $B$ is bolstered.
Next, suppose $|\{x,y\} \cap T| = 1$.
Without loss of generality, $x \in T$ and $y \notin T$.
Suppose $M'$ has no $(N,B)$-robust elements.
There is at least one $N$-deletable element in $T-x$. Let $q$ be such an element; then $q \in B$. Now $T=\{x,p,q\}$ where $p \in B^*$ and $p$ is $N$-contractible, since $M'$ has no $(N,B)$-robust elements, so $x$ is $N$-deletable.
By \cref{allowablexyrow2}, a pivot on $A_{xp}$ is allowable, where $\{p,y,a,b\}$ incriminates $(M,A^{xp})$.
But now $x$ is an $(N,B \triangle \{x,p\})$-robust element outside of $\{p,y\}$, which contradicts that $B$ is a bolstered basis.
So we may assume that $M'$ has an $(N,B)$-robust element~$u$, in which case $\{u,x,y\}$ is a triad.
By orthogonality, $u \in T$, so let $T = \{x,u,q\}$.
Then $\{x,y,u,q\}$ is a $4$-element fan as required.
Finally, suppose $x,y \notin T$.
Recall that if $M'$ has an $(N,B)$-robust element~$u$ outside of $\{x,y\}$, then $\{u,x,y\}$ is a triad.
Thus, if such a $u$ exists, then by orthogonality $u \notin T$.
So $T$ does not contain an $(N,B)$-robust element.
Thus, the two $N$-deletable elements of $T$ are in $B$ and the $N$-contractible element is in $B^*$.
Let $\alpha,\beta \in B$ be the $N$-deletable elements of $T$ and let $\gamma \in B^*$ be the $N$-contractible element of $T$.
By \cref{lemmaC}, $M'$ has a nice path description $(P_1,P_2,\dotsc,P_m)$.
We claim that either $P_1$ or $P_1 \cup T$ is a maximal $4$-element fan with $\gamma$ as an internal element.
By \cref{allowablenonxy2}, a pivot on $A_{\alpha\gamma}$ is allowable, after which $\alpha$ and $\gamma$ become $(N,B')$-robust elements, where $B' =B\triangle \{\alpha,\gamma\}$, with $\alpha \in (B')^*$ and $\gamma \in B'$.
Now $\gamma \in P_i$, for some $i \in \seq{m}$, where,
by \cref{pathdescprops}, either $P_i$ is a coguts set, or $i \in \{1,m\}$.
In the former case, $\si(M' / \gamma)$ is $3$-connected, again by \cref{pathdescprops}, in which case $\gamma$ is an $(N,B')$-strong element in $B'-\{x,y\}$, contradicting \cref{fragilecase}\ref{nostronginbasis}.
So, without loss of generality, $\gamma \in P_1$.
If $P_1$ is a triangle or $4$-segment, then $\si(M' / \gamma)$ is $3$-connected by \cref{pathdescendconn}, so again $\gamma$ is an $(N,B')$-strong element in $B'-\{x,y\}$, a contradiction to \cref{fragilecase}\ref{nostronginbasis}.
If $P_1$ is a $4$-cosegment, then by orthogonality $T \subseteq P_1$, so $T$ is a triangle-triad, contradicting that $M'$ is $3$-connected.
Suppose $P_1$ is a fan of size at least~$4$.
Since $\gamma$ is $N$-contractible, \cref{fragilefanelements} implies that $\gamma$ is not a spoke element of the fan~$P_1$.
If $\gamma$ is a rim element, then $\si(M' / \gamma)$ is $3$-connected by \cref{fanends,noMK4conn}, a contradiction to \cref{fragilecase}\ref{nostronginbasis}.
So $\gamma$ is an internal element of $P_1$, where $P_1$ is a maximal $4$-element fan, as claimed.
Finally, if $P_1$ is a triad, then $F=P_1 \cup T$ is a $4$-element fan, by orthogonality.
As in the previous case, $\gamma$ is not a spoke or a rim element of $F$, so $F$ is a maximal $4$-element fan with $\gamma$ as an internal element.
Now let $F = P_1 \cup T$ if $P_1$ is a triad, otherwise let $F = P_1$; in either case, $F$ is a maximal $4$-element fan with $\gamma$ as an internal element.
By orthogonality and the maximality of $F$, we have $T \subseteq F$.
So, without loss of generality, $F$ has a fan ordering $(\alpha, \gamma, \beta, \delta)$, where $P_1 - T = \{\delta\}$.
Note that the only triangle containing $\gamma$ is $T$, and the only triad containing $\beta$ is $\{\gamma,\beta,\delta\}$, by orthogonality and the maximality of $F$.
Thus $\co(M \backslash \beta) \cong M \backslash \beta / \gamma \cong \si(M / \gamma)$.
By \cref{allowablenonxy2}, a pivot on $A_{\beta\gamma}$ is allowable, after which $\beta$ and $\gamma$ become $(N,B'')$-robust elements, where $B'' =B\triangle \{\beta,\gamma\}$, with $\beta \in (B'')^*$ and $\gamma \in B''$.
Recall that $\si(M/\gamma)$ is not $3$-connected, by \cref{fragilecase}\ref{nostronginbasis}, so $\co(M \backslash \beta)$ is not $3$-connected.
Thus both $\beta$ and $\gamma$ are $(N,B'')$-robust but not $(N,B'')$-strong.
As neither $\beta$ nor $\gamma$ is $(N,B)$-robust, this contradicts that $B$ is a bolstered basis. \end{proof}
\begin{lemma}
\label{rkcorkbounds}
Suppose that $|E(M)| \ge 15$. Then
\begin{enumerate}
\item $r(M\backslash a,b) \le r^*(M \backslash a,b) + 2$ and
\item $r^*(M\backslash a,b) \le r(M\backslash a,b) + 1$.
\end{enumerate}
Moreover, if $M \backslash a,b$ has an $(N,B)$-robust element outside of $\{x,y\}$, then $r(M\backslash a,b) \le r^*(M \backslash a,b) + 1$. \end{lemma} \begin{proof}
By \cref{lemmaC}, every element of $M \backslash a,b$ is $N$-deletable or $N$-contractible (but not both).
Let $r = r(M \backslash a,b)$ and $r^* = r^*(M \backslash a,b)$, and let $C$ and $D$ be the set of $N$-contractible and $N$-deletable elements of $M \backslash a,b$ respectively.
Recall that $M \backslash a,b$ has at most one $(N,B)$-robust element outside of $\{x,y\}$, and if this element exists it is in $B^*$.
So each of the $r-2$ elements of $B-\{x,y\}$ are $N$-deletable, $x$ and $y$ might be $N$-deletable, and at most one element in $B^*$ is $N$-deletable. In total, $|D| \le r+1$.
On the other hand, all of the $r^*$ elements of $B^*$ are $N$-contractible when $M\backslash a,b$ has no $(N,B)$-robust elements outside of $\{x,y\}$, but none of the elements in $B-\{x,y\}$ are $N$-contractible. So $|C| \le r^*+2$.
If $M \backslash a,b$ has an $(N,B)$-robust element outside of $\{x,y\}$, then $|C| \le r^*+1$.
By \cref{pathdescrank}, $r(M\backslash a,b)=|C|$, so $r = |C| \le r^*+2$; and $r^*(M\backslash a,b) =|D|$, so $r^* = |D| \le r+1$, as required.
\end{proof}
The next two lemmas are used to simplify the arguments in \cref{ndtsec}.
\begin{lemma}
\label{oneendisatriad}
Suppose that $|E(M)| \ge 13$ and
$M \backslash a,b$
has a nice path description $(P_1,P_2,\dotsc,P_m)$.
Then either $P_1$ or $P_m$ is a coclosed triad.
\end{lemma} \begin{proof}
Let $i \in \{1,m\}$ and suppose that $P_i$ is a $4$-cosegment.
We claim that $P_i \cap \{x,y\} \neq \emptyset$.
By \cref{pathdescends}, there is some $e \in P_i$ that is $\{U_{2,5},U_{3,5}\}$-deletable, and each element in $P_i-e$ is $\{U_{2,5},U_{3,5}\}$-contractible.
Recall that $M \backslash a,b$ has at most one $(N,B)$-robust element outside of $\{x,y\}$, where if such an element $u$ exists, then $u \in B^*$
and $\{u,x,y\}$ is a
triad of $M \backslash a,b$.
Since $r^*_{M \backslash a,b}(P_i-e)=2$, we have $|(P_i-e) \cap B^*| \le 2$, so there is an $(N,B)$-robust element in $(P_i-e) \cap B$.
So $P_i \cap \{x,y\} \neq \emptyset$ as claimed.
Towards a contradiction, suppose that $P_1$ and $P_m$ are both $4$-cosegments.
Then, without loss of generality, $x \in P_1$ and $y \in P_m$.
Let $e_1 \in P_1$ and $e_m \in P_m$ be $\{U_{2,5},U_{3,5}\}$-deletable elements, so,
letting $T_1^* = P_1-e_1$ and $T_m^* = P_m-e_m$, each element in $T_1^* \cup T_m^*$ is $\{U_{2,5},U_{3,5}\}$-contractible.
Note that $x \in T^*_1$ and $y \in T^*_m$, and $(T_1^* \cup T_m^*)-\{x,y\} \subseteq B^*$.
Let $Z = E(M \backslash a,b)-(T_1^* \cup T_m^*)$.
Then $Z \cap B = B-\{x,y\}$ and, as $r(Z) \leq r(M \backslash a,b)-2$, the set $B-\{x,y\}$ spans $Z$.
Suppose $M \backslash a,b$ has an $(N,B)$-robust element $u \in Z$. Then $\{u,x,y\}$ is a triad, so $r^*(T_1^* \cup T_2^* \cup u) \le 4$. But $|(T_1^* \cup T_2^* \cup u) \cap B^*| \ge 5$, a contradiction.
So $M \backslash a,b$ has no $(N,B)$-robust elements.
If $r^*(M \backslash a,b)\le 4$, then, by \cref{rkcorkbounds}, $r(M) \le 6$, so $|E(M)| \le 12$, a contradiction.
So we may assume $r^*(M \backslash a,b) > 4$.
Thus, there exists an element $q \in B^* \cap Z$ such that $q$ is not $(N,B)$-robust, and $A_{xq} = A_{yq} = 0$.
Since $q$ is not a loop, there exists an element $p \in B-\{x,y\}$ such that $A_{pq} \neq 0$.
Now $A^{pq}$ is an allowable pivot, by \cref{allowablenonxy2}, and $q$ is $N$-contractible in $B' = B \triangle \{p,q\}$.
So $q$ is $(N,B')$-robust, but $M \backslash a,b$ has no $(N,B)$-robust elements, contradicting that $B$ is a bolstered basis.
We deduce that $P_1$ and $P_m$ are not both $4$-cosegments.
Next, suppose that $P_1$ is a $4$-cosegment, and $P_m$ is a triad that is not closed. Then, by definition, there is an element $p_1 \in P_1$ such that $P_m \cup p_1$ is a $4$-cosegment.
The $4$-cosegments $P_1$ and $P_m \cup p_1$ each have a unique $\{U_{2,5},U_{3,5}\}$-deletable element, whereas the other elements are $\{U_{2,5},U_{3,5}\}$-contractible; and contain at most two elements in $B^*$, so at least two elements in $B$.
Since $M \backslash a,b$ has at most one $(N,B)$-robust element, it follows that $p_1$ is $\{U_{2,5},U_{3,5}\}$-contractible.
Moreover, $r^*_{M \backslash a,b}(P_1 \cup P_m) =3$, so $|(P_1 \cup P_m) \cap B^*| \le 3$ and $|(P_1 \cup P_m) \cap B| \ge 4$, implying that $p_1 \in B^*$.
Now, $P_1 - p_1$ and $P_m$ each contain two elements of $B$, at least one of which is $N$-contractible, and therefore $(N,B)$-robust. So we may assume that $x \in P_1-p_1$ and $y \in P_m$.
Note that $P_2$ and $P_{m-1}$ are guts sets and $m$ is odd.
Let $i \in \{2,4,\dotsc,m-1\}$, so that $P_i$ is a guts set.
Since $\sqcap^*(P_1,P_m)=1$, it follows from the duals of \cref{growpi,pflancoguts} that $|P_i| = 1$.
Hence $m \ge 5$.
Now consider the coguts set $P_3$. By \cref{pathdescprops}, each $e \in P_3$ is $\{U_{2,5},U_{3,5}\}$-contractible, so, as $e \notin \{x,y\}$, we have $e \in B^*$.
Thus, if $|P_3| \ge 2$, then $r^*_{M \backslash a,b}(P_1 \cup P_2 \cup P_3) = 3$ but $|(P_1 \cup P_2 \cup P_3) \cap B^*| \ge 4$, a contradiction.
So $|P_3| = 1$.
Now $P_2 \cup P_3 \cup P_4$ is a triangle, by the duals of \cref{growpi,pflantriad}.
But as $\{x,y\} \subseteq P_1 \cup P_m$, this contradicts \cref{atmostonetri}.
By symmetry, we deduce that if $P_i$ is a $4$-cosegment and $P_{j}$ is a triad for $\{i,j\} = \{1,m\}$, then $P_{j}$ is coclosed.
By \cref{atmostonetri} and \cref{nicepathdescription}\ref{npd1},
neither $P_1$ nor $P_m$ is a triangle.
Let $\{i,j\}=\{1,m\}$ and suppose that $P_i$ is a fan of size at least~$4$; then, by \cref{atmostonetri} again, $\{x,y\} \subseteq P_i$,
so $P_j$ is a (coclosed) triad. \end{proof}
\begin{lemma}
\label{niceends}
Suppose that $|E(M)| \ge 13$ and $M \backslash a,b$ has a nice path description $(P_1,P_2,\dotsc,P_m)$. Let $i \in \{1,m\}$. Then $P_i$ is either a coclosed cosegment, or a maximal fan. \end{lemma} \begin{proof}
By \cref{oneendisatriad}, we may assume that $P_1$ is a coclosed triad.
Then $P_2$ is a guts set. Moreover, for any $p_1 \in P_1$, we have $p_1 \notin \cl(P_m)$, so $P_m$ is closed.
Suppose $P_m$ is a cosegment that is not coclosed, or $P_m$ is a fan that is not maximal.
In either case, there is some $p_1 \in P_1 \cap \cl^*(P_m)$.
It follows that $(P_1-p_1,P_2,\dotsc,P_{m-1},\{p_1\},P_m)$ is a path of $3$-separations.
Hence $p_2 \in \cl(P_1-p_1)$ for each $p_2 \in P_2$, implying $P_1 \cup p_2$ is a fan.
This contradicts the definition of a nice path description, so we deduce that if $P_m$ is a cosegment, then it is coclosed, and if $P_m$ is a fan, then it is maximal. \end{proof}
\section{The delete-triple case} \label{dtsec}
We work under the following assumptions throughout this section.
Let $M$ be an excluded minor for the class of $\mathbb{P}$-representable matroids where $\mathbb{P} \in \{\mathbb{H}_5,\mathbb{U}_2\}$, and $M$ has no triads.
Suppose also that $|E(M)| \ge 16$.
Note that, by \cref{utfutffragile}, for any pair $\{a,b\} \subseteq E(M)$ such that $M \backslash a,b$ is $3$-connected with a $\{U_{2,5},U_{3,5}\}$-minor, the matroid $M \backslash a,b$ is $\{U_{2,5},U_{3,5}\}$-fragile.
We say that a triple $\{a,b,c\} \subseteq E(M)$ is a \emph{delete triple} for $M$ if $M \backslash a,b,c$ is $3$-connected with a $\{U_{2,5},U_{3,5}\}$-minor. In this section, we prove \cref{deltripleprop}, which says that, under the above assumptions, $M$ has no delete triples.
\begin{lemma}
\label{specialdeletetriples}
If $M$ has a delete triple, then it has some delete triple $\{a,b,c\}$ such that $M \backslash a,b,c$ has no triangles. \end{lemma} \begin{proof}
Suppose $\{a,b,e\}$ is a delete triple for $M$ but $M \backslash a,b,e$ has at least one triangle.
Observe that $M \backslash a,b$ is $\{U_{2,5},U_{3,5}\}$-fragile and has rank and corank at least~$4$, by \cref{utfutffragile}, and this matroid has at least one triangle.
Since $M \backslash a,b,e$ has a $\{U_{2,5},U_{3,5}\}$-minor, $M \backslash a,b,e$ is also $\{U_{2,5},U_{3,5}\}$-fragile.
Let $N \in \{U_{2,5},U_{3,5}\}$ such that $M\backslash a,b$ has an $N$-minor.
By \cref{fragilecase}, there exists a basis~$B$ for $M$ and a $B \times B^*$ companion $\mathbb{P}$-matrix~$A$ for which $\{x,y,a,b\}$ incriminates $(M,A)$ where $\{x,y\} \subseteq B$ and $\{a,b\} \subseteq B^*$.
By \cref{atmostonetri}, $M \backslash a,b$ has exactly one triangle~$T$, there is a unique $(N,B)$-robust element $u \in T \cap B^*$, and $T \cup \{x,y\}$ is a $4$-element fan~$F$.
If the fan~$F$ is maximal, then there is a spoke end $c$ of $F$.
Now $c$ is not $\{U_{2,5},U_{3,5}\}$-contractible in $M \backslash a,b$, by \cref{fragilefanelements}, so it is $\{U_{2,5},U_{3,5}\}$-deletable, by \cref{lemmaC},
and $M \backslash a,b,c$ is $3$-connected, by \cref{fanendsstrong}.
So $\{a,b,c\}$ is a delete triple such that $M \backslash a,b,c$ has no triangles, as required.
Now we may assume that $F$ is properly contained in a fan $F'$. By \cref{fragilefanscase}, $|F'| =5$, and, by \cref{atmostonetri}, we may assume up to swapping $x$ and $y$ that $F'$ has an ordering $(x,u,y,f_4,f_5)$ where $\{x,u,y\}$ is a triad.
Note that $e \notin F'$, since each element of $F'$ is in a triad of $M \backslash a,b$ but $M \backslash a,b,e$ is $3$-connected.
So the triangle $\{u,y,f_4\}$ of $M \backslash a,b$ is also a triangle of $M \backslash a,b,e$.
Moreover, by \cref{fanunique}, $e$ is not in the coclosure of either of the triads of $F'$, so $F'$ is also a fan of $M \backslash a,b,e$.
Again by \cref{fanunique}, the only triads containing $y$ in $M \backslash a,b,e$ are $\{u,x,y\}$ and $\{y,f_4,f_5\}$.
Since $M$ has no triads, either $a$ or $b$ blocks the triad $\{u,x,y\}$ of $M \backslash a,b$.
Without loss of generality, say $a$ blocks $\{u,x,y\}$.
Then, $\{u,x,y\}$ is not a triad in $M \backslash b,e$.
Furthermore, as $\{u,y,f_4\}$ is a triangle in $M \backslash a,b,e$, it is also a triangle in $M \backslash b,e$.
By \cref{utfutffragile,atmostonetri}, this is the unique triangle in $M \backslash b,e$, and it is contained in a $4$-element fan.
Applying the argument from the first paragraph, if this $4$-element fan is maximal, then there is a delete triple, $\{b,e,c'\}$ say, such that $M \backslash b,e,c'$ has no triangles, as required.
So we may assume that $M \backslash b,e$ has a $5$-element fan $F_2'$ whose internal elements are $\{u,y,f_4\}$.
Now $y$ is in at least one triad of $M \backslash b,e$.
Since $y$ is in exactly two triads of $M \backslash a,b,e$, at least one of which is blocked by $a$, the unique triad of $M \backslash b,e$ containing $y$ is $\{y,f_4,f_5\}$.
So
$f_4$ is a rim element of $F_2'$, and
$\{u,f_4\}$ is contained in a triad $T^*$ of $M \backslash b,e$.
Since $M \backslash a,b,e$ is $3$-connected, $T^*$ is also a triad of $M \backslash a,b,e$.
Then it follows that $T^* = \{u,f_4,q\}$ for some $q \in E(M \backslash a,b) - \{e,u,x,f_4,f_5\}$.
Now $(f_5,y,f_4,u,q)$ is a fan ordering of $F_2'$, which is a fan in $M \backslash b,e$ and $M \backslash a,b,e$.
Note that $x,q,f_5 \in \cl^*_{M \backslash a,b,c}(\{u,y,f_4\})$, and it follows that $\{x,q,f_5\}$ is also a triad of $M \backslash a,b,c$.
Thus $(M\backslash a,b,c)^*|(F' \cup q) \cong M(K_4)$.
But $M \backslash a,b,c$ is $\{U_{2,5},U_{3,5}\}$-fragile, so this contradicts \cref{noMK4}. \end{proof}
We say that a delete triple $\{a,b,c\}$ for $M$ is \emph{special} if $M \backslash a,b,c$ has no triangles.
The next lemma is straightforward, but important for the arguments that follow. \begin{lemma}
\label{pathbasics}
Let $M'$ be a $3$-connected matroid with $x \in E(M')$.
Suppose $M' \backslash x$ is $3$-connected, and both $M'$ and $M'\backslash x$ have path width three.
Let $(e_1,e_2,\dotsc,e_n)$ be a sequential ordering of $M'$, with $x = e_i$ for some $i \in \seq{n}$.
Then $\sigma=(e_1,\dotsc,e_{i-1},e_{i+1},\dotsc,e_n)$ is a sequential ordering of $M' \backslash x$.
Moreover, any triad of $M' \backslash x$ contained in either $\{e_1,\dotsc,e_{i-1}\}$ or $\{e_{i+1},\dotsc,e_n\}$ is not blocked by $x$. \end{lemma} \begin{proof}
For $j \in \seq{n-1}$, let $X_j = \{e_1,e_2,\dotsc,e_j\}$ and $Y_j = \{e_{j+1},e_{j+2},\dotsc,e_n\}$, and $X'_j = X_j - x$ and $Y'_j = Y_j - x$.
Suppose $|X'_j|,|Y'_j|\ge 2$ for some $j \in \seq{n-1}$.
To show that $\sigma$ is a sequential ordering of $M' \backslash x$, it suffices to show that $\lambda_{M' \backslash x}(X'_j) = 2$.
Since $M' \backslash x$ is $3$-connected, $\lambda_{M' \backslash x}(X'_j) \ge 2$.
Moreover, $\lambda_{M' \backslash x}(X'_j) = r(X'_j) + r(Y'_j) - r(M' \backslash x) \le r(X_j) + r(Y_j) - r(M') = \lambda_{M'}(X_j) = 2$, as required.
Now suppose that $\{e_1,\dotsc,e_{i-1}\}$ contains a triad $T^*$ of $M' \backslash x$.
It remains to prove that $T^*$ is not blocked by $x$.
First, assume that $r^*_{M'}(\{e_1,\dotsc,e_{i-1}\}) \ge 3$ and $r^*_{M'}(\{e_{i+1},\dotsc,e_n\}) \ge 3$.
Then, by the dual of \cref{vert3sep}, $x$ is not a coguts element, since $M' \backslash x$ is $3$-connected.
So $x$ is a guts element, in which case $x \in \cl(\{e_{i+1},\dotsc,e_n\}) \subseteq \cl(E(M' \backslash x) - T^*)$, implying that $x$ does not block $T^*$.
So we may assume that $r^*_{M'}(\{e_1,\dotsc,e_{i-1}\}) \le 2$ or $r^*_{M'}(\{e_{i+1},\dotsc,e_n\}) \le 2$.
In the former case, $x \notin \cl^*_{M'}(\{e_1,\dotsc,e_{i-1}\})$, for otherwise $M \backslash x$ is not $3$-connected, so $x \in \cl(\{e_{i+1},\dotsc,e_n\})$ by orthogonality.
In the latter case, $x \notin \cl^*_{M'}(\{e_{i+1},\dotsc,e_n\})$, similarly, and thus, as $(e_1,\dotsc,e_n)$ is a sequential ordering of $M'$, we must have $x \in \cl(\{e_{i+1},\dotsc,e_n\})$.
Since $x \in \cl(\{e_{i+1},\dotsc,e_n\})$ in either case, $x$ does not block $T^*$. \end{proof}
We come to the main result of this section. For ease of reference, we restate the section assumptions. \begin{theorem}
\label{deltripleprop}
Let $M$ be an excluded minor for the class of $\mathbb{P}$-representable matroids where $\mathbb{P} \in \{\mathbb{H}_5,\mathbb{U}_2\}$, and $M$ has no triads.
Suppose $|E(M)| \ge 16$.
Then $M$ has no delete triples. \end{theorem} \begin{proof}
Towards a contradiction, suppose $M$ has a delete triple. By \cref{specialdeletetriples}, we may assume that $M$ has a special delete triple $\{a,b,c\}$. Then, by \cref{utfutffragile}, each of $M \backslash a,b$, $M \backslash a,c$, and $M \backslash b,c$ is $\{U_{2,5},U_{3,5}\}$-fragile. By \cref{lemmaC}, each of these matroids has a nice path description, and no $\{U_{2,5},U_{3,5}\}$-essential elements. Since $M \backslash a,b$ is $\{U_{2,5},U_{3,5}\}$-fragile, and $c$ is $\{U_{2,5},U_{3,5}\}$-deletable in this matroid, $M \backslash a,b,c$ is also $\{U_{2,5},U_{3,5}\}$-fragile.
\begin{claim}
\label{mabcfragile}
$M \backslash a,b,c$ has path width three and no $\{U_{2,5},U_{3,5}\}$-essential elements. \end{claim} \begin{proof}
The matroid $M \backslash a,b,c$ is a $3$-connected $\{U_{2,5},U_{3,5}\}$-fragile $\mathbb{P}$-representable matroid, with $|E(M \backslash a,b,c)| \ge 12$.
Moreover, $M \backslash a,b,c$ has no triangles, so it has no fans of size at least~$4$.
Thus, by \cref{ccmwvz-result}, $M \backslash a,b,c$ has an $\{X_8,Y_8,Y_8^*\}$-minor.
Now $M \backslash a,b,c$ has path width three, by \cref{nicepathdescription}, and no $\{U_{2,5},U_{3,5}\}$-essential elements, by \cref{noessential}. \end{proof}
Let $L$ and $R$ be the ends of a sequential ordering of $M \backslash a,b,c$.
Note that, by \cref{welldefinedends}, for every sequential ordering $\sigma$ of $M \backslash a,b,c$, we have $\{L,R\} = \{L(\sigma),R(\sigma)\}$. Since $M \backslash a,b,c$ has no triangles, each of $L$ and $R$ is either a triad or a $4$-cosegment.
\begin{claim}
\label{only4cosegs}
$L$ and $R$ are $4$-cosegments. \end{claim} \begin{subproof}
Say $L$
is a triad.
Since $M$ has no triads, $L$ is blocked by at least one of $a$, $b$, and $c$.
Without loss of generality we may assume that $a$ blocks $L$.
Consider a sequential ordering $\sigma_a=(p_1,p_2,p_3,\dotsc,p_n)$ for $M \backslash b,c$.
Now $a=p_i$ for some $i \in \seq{n}$.
So $\sigma_a^-=(p_1,p_2,\dotsc,p_{i-1},p_{i+1},\ldots,p_n)$ is a sequential ordering for $M \backslash a,b,c$ by \cref{pathbasics}.
By \cref{welldefinedends}, we may assume (up to reversing the order of $\sigma_a^-$) that $L=L(\sigma_a^-)$ and $R=R(\sigma_a^-)$.
Suppose $i > 3$. Then $L=\{p_1,p_2,p_3\}$ is a triad of $M \backslash a,b,c$ by \cref{endslipperiness}\ref{esi}, and, as $\sigma_a$ is a sequential ordering for $M \backslash b,c$, the set $\{p_1,p_2,p_3\}$ is either a triangle or a triad of $M \backslash b,c$.
If $\{p_1,p_2,p_3\}$ is a triangle of $M \backslash b,c$, then it is also a triangle of $M \backslash a,b,c$, so $M \backslash a,b,c$ is not $3$-connected, a contradiction.
So $\{p_1,p_2,p_3\}$ is a triad of $M \backslash b,c$ and $M \backslash a,b,c$, in which case $a$ does not block $L$, a contradiction.
We deduce that $a = p_i$ for $i \in \{1,2,3\}$.
Now, if $\{p_1,p_2,p_3\}$ is a triad of $M \backslash b,c$, then $M \backslash a,b,c$ is not $3$-connected, a contradiction.
So $\{p_1,p_2,p_3\}$ is a triangle of $M \backslash b,c$.
As $L = L(\sigma_a^-)$ is a triad of $M \backslash a,b,c$, \cref{endslipperiness}\ref{esi} implies that this triad is $\{p_1,p_2,p_3,p_4\}-p_i$.
As the triad $L$ is blocked by $a$, we have that $\{p_1,p_2,p_3,p_4\}$ is a cocircuit of $M \backslash b,c$.
By \cref{atmostonetri}, the triangle $\{p_1,p_2,p_3\}$ is contained in a $4$-element fan $F$ of $M \backslash b,c$.
Since $M \backslash a,b,c$ is $3$-connected, $a$ is not contained in the triad of $F$.
So, for some element $z$ and $\{i,j,k\} = \{1,2,3\}$, the fan $F$ has ordering $(a,p_j,p_k,z)$ where $\{p_j,p_k,z\}$ is a triad.
Note that $p_4 \neq z$, since in $M \backslash b,c$ the set $\{p_j,p_k,p_4\}$ is properly contained in a cocircuit, whereas $\{p_j,p_k,z\}$ is a triad.
But then $\{p_j,p_k,p_4,z\}$ is a $4$-cosegment of $M \backslash a,b,c$ containing $L$, so $L$ is not a triad end of $M \backslash a,b,c$, a contradiction.
So $L$ is a cosegment of size at least~$4$. The fact that $|L| = 4$ follows from the fact that $M \backslash a,b,c$ is $\{U_{2,5},U_{3,5}\}$-fragile.
The result then follows by symmetry. \end{subproof}
By \cref{only4cosegs}, we may now assume that $|L|=4$ and $|R|=4$.
\begin{claim}
\label{unblockedtriad}
For each $x \in \{a,b,c\}$ and $X \in \{L,R\}$, the element $x$ does not block every triad contained in $X$. \end{claim} \begin{subproof}
It suffices to show that $a$ does not block every triad contained in $L$.
Consider a sequential ordering $\sigma_a=(p_1,p_2,p_3,\dotsc,p_n)$ for $M \backslash b,c$.
We have $a=p_i$ for some $i \in \seq{n}$, and $$\sigma_a^-=(p_1,p_2,\dotsc,p_{i-1},p_{i+1},\ldots,p_n)$$ is a sequential ordering for $M \backslash a,b,c$ by \cref{pathbasics}.
By reversing $\sigma_a^-$, if necessary, we may assume that $L(\sigma_a^-) = L$ and $R(\sigma_a^-) = R$, due to \cref{welldefinedends}.
Suppose $i > 3$.
Then $\{p_1,p_2,p_3\}$ is a triad of $M \backslash a,b,c$, by \cref{endslipperiness}\ref{esii}, and $\{p_1,p_2,p_3\}$ is either a triangle or a triad of $M \backslash b,c$.
However, if $\{p_1,p_2,p_3\}$ is a triangle of $M \backslash b,c$, then it is a triangle-triad in $M \backslash a,b,c$, contradicting $3$-connectivity.
So $\{p_1,p_2,p_3\}$ is a triad of $M \backslash b,c$ and $M \backslash a,b,c$.
Then $\{p_1,p_2,p_3\} \subseteq L$ and $\{p_1,p_2,p_3\}$ is a triad that is not blocked by $a$, as required.
So we may assume $a=p_i$ for some $i \le 3$.
Now, if $\{p_1,p_2,p_3\}$ is a triad of $M \backslash b,c$, then $M \backslash a,b,c$ is not $3$-connected, a contradiction.
So $\{p_1,p_2,p_3\}$ is a triangle.
As $L = L(\sigma_a^-)$ is a $4$-cosegment of $M \backslash a,b,c$, \cref{endslipperiness}\ref{esii} implies that $\{p_1,p_2,p_3,p_4\}-p_i$ is a triad~$T^*$ contained in $L$.
We may assume that $\{p_1,p_2,p_3,p_4\}$ is a cocircuit of $M \backslash b,c$, for otherwise $T^*$ is a triad contained in $L$ that is not blocked by $a$, as required.
By \cref{atmostonetri}, the triangle $\{p_1,p_2,p_3\}$ is contained in a $4$-element fan $F$ of $M \backslash b,c$.
Since $M \backslash a,b,c$ is $3$-connected, $a$ is not contained in the triad of $F$.
So, for some element $z$ and $\{i,j,k\} = \{1,2,3\}$, the fan $F$ has ordering $(a,p_j,p_k,z)$ where $\{p_j,p_k,z\}$ is a triad.
Note that $p_4 \neq z$, since, in $M \backslash b,c$ the set $\{p_j,p_k,p_4\}$ is properly contained in a cocircuit, whereas $\{p_j,p_k,z\}$ is a triad.
But then $\{p_j,p_k,p_4,z\}$ is a $4$-cosegment of $M \backslash a,b,c$, and it follows that $L = \{p_j,p_k,p_4,z\}$.
But then $\{p_j,p_k,z\}$ is a triad contained in $L$ that is not blocked by $a$, as required. \end{subproof}
\begin{claim}
\label{applyphp}
Some $x \in \{a,b,c\}$ blocks a triad in $L$ and a triad in $R$. \end{claim} \begin{subproof}
Every triad of $L$, and every triad of $R$, is blocked by at least one of $a$, $b$, and $c$, since $M$ has no triads.
Without loss of generality, $a$ blocks one of the four triads of $L$.
By \cref{unblockedtriad}, one of the other three triads of $L$ is not blocked by $a$; without loss of generality, there is a triad of $L$ blocked by $b$.
By \cref{unblockedtriad}, at least one triad of $R$ is not blocked by $c$.
So some triad of $R$ is blocked by $a$ or $b$, and hence \cref{applyphp} holds. \end{subproof}
By \cref{applyphp}, we may assume that $c$ blocks a triad in $L$ and a triad in $R$. Consider a sequential ordering $\sigma_c=(p_1,p_2,\dotsc,p_n)$ for $M \backslash a,b$, where $c=p_{i_c}$ for some $i_c \in \seq{n}$. Then $\sigma_c^-=(p_1,p_2,\dotsc,p_{i_c-1},p_{i_c+1},\ldots,p_n)$ is a sequential ordering for $M \backslash a,b,c$ by \cref{pathbasics}.
We now break into two cases depending on whether or not $L$ and $R$ are disjoint.
We first consider the case where $L$ meets $R$.
\begin{claim}
\label{singlecoflanoutcome}
The ends $L$ and $R$ are disjoint. \end{claim} \begin{subproof}
Towards a contradiction, suppose $L$ meets $R$.
If $|L \cap R| \ge 2$, then $L \cup R$ is a cosegment of $M \backslash a,b,c$, and it follows that $r^*(M \backslash a,b,c) = 2$.
But then, as $M \backslash a,b,c$ is $\mathbb{P}$-representable, $M \backslash a,b,c$ is isomorphic to a minor of $U_{4,6}$, implying $|E(M)| \le 9$, a contradiction.
So we may assume that $|L \cap R| = 1$. Let $L \cap R = \{s\}$ and $s = p_{i_s}$.
Suppose that $i_c \le 3$ up to reversing the ordering of $\sigma_c$.
Without loss of generality, $i_c = 3$.
Then $\{p_1,p_2,p_4\} \subseteq L$, by \cref{endslipperiness}\ref{esii}.
If $i_s > i_c$, then by \cref{pathbasics} $c$ does not block any triad contained in $R$, a contradiction.
So $i_s \in \{1,2\}$.
Now $L = \left\{p_1,p_2,p_4,p_{i_L}\right\}$ for some $i_L \ge 5$.
Since $p_{i_L} \in \cl^*_{M \backslash a,b}(\{p_1,p_2,p_4\})$, we may also assume that $i_L = 5$.
Let $(P_1,\dotsc,P_m)$ be the guts-coguts concatenation of $\sigma_c$ with ends $P_1 = \{p_1,p_2,c\}$ and $P_m = \{p_{n-2},p_{n-1},p_n\}$, where $s \in \{p_1,p_2\}$.
Then $P_1$ is a triangle, $P_m$ is a triad, $P_2$ is a coguts set containing $\{p_4,p_{i_L}\}$, and $P_{m-1}$ is a guts set.
Since $p_{i_s} \in \cl^*_{M \backslash a,b}(\{c,p_{n-2},p_{n-1},p_n\})$, we have $r^*_{M \backslash a,b}(P_m \cup \{c,p_{i_s}\})=3$.
Thus $\sqcap^*(\{c,s\},P_m) = 1$, and $\sqcap^*(P_1,P_m) \ge 1$ by the dual of \cref{growpi}.
Now, also using the dual of \cref{pflancoguts}, if $P_j$ is a guts set for $3 \le j \le m-1$, then $|P_j| = 1$.
Moreover, by the dual of \cref{pflantriad}, if $P_j$ is a coguts set with $|P_j|=1$ and $3 < j < m-1$, then $P_{j-1} \cup P_j \cup P_{j+1}$ is a triangle of $M \backslash a,b$.
As this triangle avoids $c$, it is also a triangle of $M \backslash a,b,c$, contradicting that $\{a,b,c\}$ is a special delete triple.
So any coguts set $P_j$ with $3 < j < m-1$ has $|P_j| \ge 2$.
Observe that $m$ is even, $P_j$ is a guts set for each odd $j>1$, and $P_j$ is a coguts set for each even $j < m$.
It follows that $r(M \backslash a,b) = 3+q$ where $q=|P_2|+|P_4|+|P_6|+\dotsb+|P_{m-2}|$.
Let $g = m/2 - 1$.
There are $g$ guts sets, each of size one, so $|E(M \backslash a,b)| = 6+q+g$, and thus $r^*(M \backslash a,b) = 3+g$.
By \cref{rkcorkbounds}, $3+q = r(M \backslash a,b) \le r^*(M \backslash a,b) + 2 = 5+g$, so $q \le g+2$.
On the other hand, there are $g$ coguts sets (excluding ends), and all have size at least~$2$.
So $q \ge 2g$.
Now $2g \le q \le g+2$, so $g \le 2$.
Moreover, $q \le g+2$, so $q \le 4$.
So $|E(M\backslash a,b)| = 6+q+g \le 12$ in the case that $i_c \le 3$, a contradiction.
Now we assume that $3 < i_c < n-2$.
If $3 < i_s < n-2$, then $L=\{p_1,p_2,p_3,p_s\}$ and $R=\{p_s,p_{n-2},p_{n-1},p_n\}$, by \cref{endslipperiness}\ref{esii}, and either no triad of $L$ is blocked by $p_c$ when $i_s < i_c$, or no triad of $R$ is blocked by $p_c$ when $i_c < i_s$.
So up to reversing the ordering we may assume that $i_s \le 3$.
By \cref{endslipperiness}\ref{esii}, $\{p_1,p_2,p_3\} \subseteq L$ and $\{p_{n-2},p_{n-1},p_n\} \subseteq R$.
Since $i_s \in \{1,2,3\}$, we have $R=\{p_{i_s},p_{n-2},p_{n-1},p_n\}$.
Let $L = \left\{p_1,p_2,p_3,p_{i_L}\right\}$ and observe that $i_L > i_c$, for otherwise $c$ does not block any triad of $L$ by \cref{pathbasics}.
As $p_{i_s} \in \cl^*_{M \backslash a,b}(\{c,p_{n-2},p_{n-1},p_n\})$, we may assume that $i_s = i_c-1$.
Using \cref{endslipperiness}\ref{esii}, we deduce that $i_s = 3$ and $i_c = 4$.
As $p_{i_L} \in \cl^*_{M \backslash a,b}(\{p_1,p_2,p_3,c\})$, we may also assume that $i_L = i_c+1=5$.
Let $(P_1,\dotsc,P_m)$ be the guts-coguts concatenation of $\sigma_c$ with ends $P_1 = \{p_1,p_2,s\}$ and $P_m = \{p_{n-2},p_{n-1},p_n\}$.
Note that $P_2 = \{c\}$ and $p_{i_L} \in P_3$.
Observe that $r^*_{M \backslash a,b}(P_m) = 2$ and $p_{i_s} \in \cl^*_{M \backslash a,b,c}(P_m)$, so $r^*_{M \backslash a,b}(P_m \cup \{c,p_{i_s}\})=3$.
It follows that $\sqcap^*(P_m,\{c,p_{i_s}\})=2+2-3=1$.
By the dual of \cref{growpi}, $\sqcap^*(P_1 \cup P_2, P_m) \ge 1$, and, by the dual of \cref{pflancoguts}, for every guts set $P_j$ with $j > 2$ we have $|P_j|=1$. We have also seen that $|P_2|=1$.
Suppose $P_j$ is a coguts set with
$j \neq 3$.
Then $5 \le j \le m-2$.
Then, by the duals of \cref{growpi,pflantriad}, $P_{j-1} \cup P_j \cup P_{j+1}$ is a triangle of $M \backslash a,b$.
As this triangle avoids $c$, it is also a triangle of $M \backslash a,b,c$, contradicting that $\{a,b,c\}$ is a special delete triple.
So for each coguts set $P_j$ with $j \neq 3$, we have $|P_j| \ge 2$.
Observe that $m$ is odd, $P_j$ is a guts set for each even $j$, and $P_j$ is a coguts set for each odd $j \notin \{1,m\}$.
It follows that $r(M \backslash a,b) = 4+q$ where $q=|P_3|+|P_5|+|P_7|+\dotsb+|P_{m-2}|$.
Let $g = (m-1)/2$.
There are $g$ guts sets, each of size one, so $|E(M \backslash a,b)| = 6+q+g$, and thus $r^*(M \backslash a,b) = 2+g$.
By \cref{rkcorkbounds}, $4+q = r(M \backslash a,b) \le r^*(M \backslash a,b) + 2 = 4+g$, so $q \le g$.
On the other hand, there are $g - 1$ coguts sets (excluding ends), and all except possibly $P_3$ has size at least~$2$.
So $q \ge 2(g - 1) - 1 = 2g - 3$.
Now $2g - 3 \le q \le g$, so $g \le 3$.
Moreover, $q \le g$, so $q \le 3$.
So $|E(M\backslash a,b)| = 6+q+g \le 12$, a contradiction. \end{subproof}
By \cref{singlecoflanoutcome}, we may now assume that $L$ and $R$ are disjoint. We may also assume that $\sigma_c=(p_1,p_2,\dotsc,p_n)$ is a sequential ordering for $M \backslash a,b$ such that some initial segment and some terminal segment of $\sigma_c$ are ends of a nice path description for $M \backslash a,b$.
Suppose that $i_c \le 3$. Then $\{p_1,p_2,p_3,p_4\}-p_{i_c} \subseteq L$, by \cref{endslipperiness}\ref{esii}. Since $L$ and $R$ are disjoint, $R \subseteq \{p_{i_c+1},\ldots,p_n\}$. By \cref{pathbasics}, no triad contained in $R$ is blocked by $c$, a contradiction. By symmetry, we deduce that $3 < i_c < n-2$.
By \cref{endslipperiness}\ref{esii}, $\{p_1,p_2,p_3\} \subseteq L$ and $\{p_{n-2},p_{n-1},p_n\} \subseteq R$. In particular, $\{p_1,p_2,p_3\}$ and $\{p_{n-2},p_{n-1},p_n\}$ are triads of $M \backslash a,b,c$. Let $L = \left\{p_1,p_2,p_3,p_{i_L}\right\}$ and $R = \left\{p_{i_R},p_{n-2},p_{n-1},p_n\right\}$. If $i_L < i_c$, then, by \cref{pathbasics}, $c$ does not block any triad of $L$, a contradiction. So $i_L > i_c$ and, similarly, $i_R < i_c$. Moreover, since $p_{i_R} \in \cl^*_{M \backslash a,b,c}(\{p_{n-2},p_{n-1},p_n\})$, we have $p_{i_R} \in \cl^*_{M\backslash a,b}(\{p_{i_c},\dotsc,p_n\})$, so we may assume that $i_R = i_c-1$. Similarly, we may assume that $i_L = i_c+1$.
Let $(P_1,\dotsc,P_m)$ be the guts-coguts concatenation of $\sigma_c$ with ends $P_1=\{p_1,p_2,p_3\}$ and $P_m=\{p_{n-2},p_{n-1},p_n\}$.
Choose $j_c \in \{2,3,\dotsc,m-1\}$ such that $c \in P_{j_c}$. Since $i_L-1 = i_c = i_R+1$, where $p_{i_c}$ is a guts element but $p_{i_L}$ and $p_{i_R}$ are coguts elements, we have $|P_{j_c}| = 1$.
Observe that $r^*_{M \backslash a,b}(P_1) = 2$ and $p_{i_L} \in \cl^*_{M \backslash a,b,c}(P_1)$, so $r^*_{M \backslash a,b}(P_1 \cup \{c,p_{i_L}\})=3$. It follows that $\sqcap_{M \backslash a,b}^*(P_1,\{c,p_{i_L}\})=2+2-3=1$. By the dual of \cref{growpi}, $\sqcap^*(P_1,P_{j_c} \cup \dotsb \cup P_m) \ge 1$, so, by the dual of \cref{pflancoguts}, for every guts set $P_j$ such that $j < j_c$ we have $|P_j| = 1$. We have also seen that $|P_{j_c}| = 1$. By symmetry, every guts set $P_j$ has size one.
Now suppose $P_j$ is a coguts set with $|P_j|=1$ and $j \notin \{j_c-1,j_c+1\}$.
Then, by the duals of \cref{growpi,pflantriad} and symmetry, $P_{j-1} \cup P_j \cup P_{j+1}$ is a triangle of $M \backslash a,b$. As this triangle avoids $c$, it is also a triangle of $M \backslash a,b,c$, contradicting that $\{a,b,c\}$ is a special delete triple. So for each coguts set $P_j$, where $j \notin \{j_c-1,j_c+1\}$, we have $|P_j| \ge 2$.
Observe that $m$ is odd, $P_j$ is a guts set for each even $j$, and $P_j$ is a coguts set for each odd $j \notin \{1,m\}$. It follows that $r(M \backslash a,b) = 4+q$ where $q=|P_3|+|P_5|+|P_7|+\dotsb+|P_{m-2}|$.
Let $g = (m-1)/2$. There are $g$ guts sets, each of size one, so $|E(M \backslash a,b)| = 6+q+g$, and thus $r^*(M \backslash a,b) = 2+g$.
By \cref{rkcorkbounds}, $4+q = r(M \backslash a,b) \le r^*(M \backslash a,b) + 2 = 4+g$, so $q \le g$.
On the other hand, there are $g - 1$ coguts sets (excluding ends), at most two of which have size one. So $q \ge 2(g - 1) - 2 = 2g - 4$. Now $2g - 4 \le q \le g$, so $g \le 4$.
Moreover, $q \le g$, so $q \le 4$.
So far, we have shown that $|E(M\backslash a,b)| =6+q+g \le 14$. If $q \le 3$, then, as $2g-4 \le q \le 3$, we have $g \le 3$, and $|E(M\backslash a,b)| =6+q+g \le 12$, as required. So it remains only to rule out the possibility that $q=4$.
Assume $q=4$. Then $g=4$, $m=9$, $r(M \backslash a,b) = 8$, and $|E(M \backslash a,b)| = 14$, and there are three coguts sets: two are singletons, and one has size two. Recall that for each coguts set $P_j$ with $j \notin \{j_c-1,j_c+1\}$ we have $|P_j| \ge 2$. So we may assume, without loss of generality, that $|P_3|=2$ and $|P_5|=|P_7| = 1$, where $j_c=6$, thus $(P_1,\dotsc,P_m) =$ $$(\{p_1,p_2,p_3\}, \{p_4\}, \{p_5,p_6\}, \{p_7\}, \{p_{i_R}\}, \{c\}, \{p_{i_L}\}, \{p_{11}\}, \{p_{12},p_{13},p_{14}\}).$$
Recall that, by \cref{mabcfragile}, $M \backslash a,b,c$ has no $\{U_{2,5},U_{3,5}\}$-essential elements. Moreover, if $e$ is $\{U_{2,5},U_{3,5}\}$-deletable (or $\{U_{2,5},U_{3,5}\}$-contractible) in $M \backslash a,b,c$, then it is $\{U_{2,5},U_{3,5}\}$-deletable (or $\{U_{2,5},U_{3,5}\}$-contractible respectively) in $M \backslash a,b$; and, since $M \backslash a,b$ is $\{U_{2,5},U_{3,5}\}$-fragile, the converse also holds. Thus, in what follows, when we say an element is $\{U_{2,5},U_{3,5}\}$-deletable (or $\{U_{2,5},U_{3,5}\}$-contractible), it is $\{U_{2,5},U_{3,5}\}$-deletable (or $\{U_{2,5},U_{3,5}\}$-contractible, respectively) in each of the matroids $M \backslash a,b,c$, $M \backslash a,b$, $M \backslash a,c$, and $M \backslash b,c$.
\begin{claim}
\label{revealtriads}
Up to labels,
\begin{enumerate}[label=\rm(\Roman*)]
\item $\{p_1,p_4,p_5\}$ and $\{p_2,p_4,p_6\}$ are triads of $M \backslash a,b,c$ not blocked by $c$, and $p_3$ is the unique $\{U_{2,5},U_{3,5}\}$-deletable element in $L$; and
\item $\{p_{i_L},p_{11},p_{14}\}$ is a triad of $M \backslash a,b,c$ not blocked by $c$, and $p_{12}$ is the unique $\{U_{2,5},U_{3,5}\}$-deletable element in $R$.
\end{enumerate} \end{claim} \begin{subproof}
If $p_4$ is $\{U_{2,5},U_{3,5}\}$-contractible, then some element in $\{p_1,p_2,p_3\}$ is $\{U_{2,5},U_{3,5}\}$-flexible by \cref{minor3conn}, a contradiction.
So $p_4$ is $\{U_{2,5},U_{3,5}\}$-deletable.
Similarly, $p_5$ and $p_6$ are $\{U_{2,5},U_{3,5}\}$-contractible and not $\{U_{2,5},U_{3,5}\}$-deletable.
Consider $M \backslash a,b \backslash p_5$.
Observe that $r^*_{M \backslash a,b \backslash p_5}(\{p_1,p_2,p_3,p_4,p_6\}) = 2$.
Thus, if $\{p_4,p_5\}$ does not cospan an element of $\{p_1,p_2,p_3\}$ in $M \backslash a,b$, then $(M \backslash a,b \backslash p_5)^*|\{p_1,p_2,p_3,p_4,p_6\} \cong U_{2,5}$, so $p_5$ is $\{U_{2,5},U_{3,5}\}$-deletable, a contradiction.
So $\{p_4,p_5\}$ cospans an element of $\{p_1,p_2,p_3\}$ in $M \backslash a,b$, and,
similarly $\{p_4,p_6\}$ cospans an element of $\{p_1,p_2,p_3\}$.
Without loss of generality, $\{p_1,p_4,p_5\}$ and $\{p_2,p_4,p_6\}$ are triads of $M \backslash a,b$, and hence also of $M \backslash a,b,c$.
It now follows that $p_1$ and $p_2$ are $\{U_{2,5},U_{3,5}\}$-contractible.
Since some initial segment of $\sigma_c$ is an end of a nice path description $\mathbf{P}_c$ for $M \backslash a,b$, and
$\{p_1,p_2,p_3\}$ is a coclosed triad in $M \backslash a,b$ not contained in a $4$-element fan, this triad is an end of $\mathbf{P}_c$.
By \cref{nicepathdescription}\ref{nicepathdesciii}, $p_3$ is $\{U_{2,5},U_{3,5}\}$-deletable.
In a similar manner, $r^*_{M \backslash a,b,c}\left(\left\{p_{i_R},p_{i_L},p_{11},p_{12},p_{13},p_{14}\right\}\right) = 3$ and $R=\left\{p_{i_R},p_{12},p_{13},p_{14}\right\}$ is a $4$-cosegment of $M \backslash a,b,c$, so if $\left\{p_{i_L},p_{11}\right\}$ does not cospan an element of $R$ in $M \backslash a,b,c$, then $M \backslash a,b,c \backslash p_{i_L}$ has a $5$-cosegment, implying $M \backslash a,b,c$ is not $\{U_{2,5},U_{3,5}\}$-fragile, a contradiction.
So we may assume that $\{p_{i_L},p_{11},p_{14}\}$ is a triad of $M \backslash a,b,c$.
As $c$ is a guts element, we know from $\sigma_c$ that $c \notin \cl^*_{M \backslash a,b}(\{p_{i_L},p_{11},p_{14}\})$, so $\{p_{i_L},p_{11},p_{14}\}$ is also a triad of $M \backslash a,b$.
Moreover, $p_{11}$ is $\{U_{2,5},U_{3,5}\}$-deletable, and hence $p_{i_L}$ and $p_{14}$ are $\{U_{2,5},U_{3,5}\}$-contractible.
Since some terminal segment of $\sigma_c$ is an end of a nice path description for $M \backslash a,b$, and by \cref{nicepathdescription}\ref{nicepathdesciii},
$\{p_{12},p_{13},p_{14}\}$ has a unique element that is $\{U_{2,5},U_{3,5}\}$-deletable;
we may assume that $p_{12}$ is this element, whereas $p_{13}$ is $\{U_{2,5},U_{3,5}\}$-contractible. \end{subproof}
Recall that each triad of $M \backslash a,b,c$ is blocked by at least one of $a$, $b$, or $c$, and observe that neither $\{p_1,p_2,p_3\}$ nor $\{p_{12},p_{13},p_{14}\}$ is blocked by $c$. By \cref{revealtriads}, we may assume that $\{p_1,p_4,p_5\}$, $\{p_2,p_4,p_6\}$, and $\{p_{i_L},p_{11},p_{14}\}$ are also triads of $M \backslash a,b,c$ not blocked by $c$.
Without loss of generality, assume $\{p_1,p_2,p_3\}$ is blocked by $a$.
We next consider what other triads can be blocked by $a$.
\begin{claim}
\label{whatablocks}
If a triad $T^*$ of $M \backslash a,b,c$ is blocked by $a$, then $T^* \cap L \neq \emptyset$.
Moreover, at most one of $\{p_1,p_4,p_5\}$ and $\{p_2,p_4,p_6\}$ is blocked by $a$. \end{claim} \begin{subproof}
Let $\sigma_a = (p'_1,p'_2,\dotsc,p'_{14})$ be a sequential ordering that is a refinement of a nice path description $\mathbf{P}_a$ of $M \backslash b,c$.
If the left end (or right end) of $\mathbf{P}_a$ is a fan of size at least~$4$, then we choose $\{p'_1,p'_2,p'_3\}$ (or $\{p'_{12},p'_{13},p'_{14}\}$, respectively) to be a triad.
Let $\sigma_a^-$ be the sequential ordering of $M \backslash a,b,c$ obtained from $\sigma_a$ by removing $a$, as described in \cref{pathbasics}.
By reversing these orderings, if necessary, we may assume that $L(\sigma_a^-) = L$ and $R(\sigma_a^-) = R$, due to \cref{welldefinedends}.
First, suppose $a \notin \{p'_1,p'_2,p'_3\}$.
Then $\{p'_1,p'_2,p'_3\} \subseteq L$ by \cref{endslipperiness}\ref{esii}, and $a$ does not block the triad $\{p'_1,p'_2,p'_3\}$.
But $a$ blocks $\{p_1,p_2,p_3\}$, so $\{p_1,p_2,p_3\} \neq \{p_1',p_2',p'_3\}$, implying $p_{i_L} \in \{p'_1,p'_2,p'_3\}$.
Now $\{p'_1,p'_2,p'_3\}$ is a triad in $M \backslash b,c$, and this triad is contained in the left end of the nice path description $\mathbf{P}_a$ for $M \backslash b,c$.
If this end is a $4$-cosegment, then it is $\{p'_1,p'_2,p'_3,p'_4\}$, in which case $L=\{p'_1,p'_2,p'_3,p'_4\}$ and $a$ does not block the triad $\{p_1,p_2,p_3\}$, a contradiction.
So the left end of $\mathbf{P}_a$ is either a triad, or a $4$- or $5$-element fan where $a$ forms a triangle with elements of the triad $\{p'_1,p'_2,p'_3\}$ (since $M \backslash a,b,c$ has no triangles).
We show, in any case, that $p_3 \in \{p'_1,p'_2,p'_3\}$.
Recall that $p_3$ is the unique $\{U_{2,5},U_{3,5}\}$-deletable element
in $L$.
Suppose $\{p'_1,p'_2,p'_3,a\}$ is a $4$-element fan in $M \backslash b,c$ that is contained in the left end of $\mathbf{P}_a$.
Since $\{p'_1,p'_2,p'_3\}$ is contained in a $4$-element fan with no $\{U_{2,5},U_{3,5}\}$-essential elements, this set contains a $\{U_{2,5},U_{3,5}\}$-deletable element.
So $p_3 \in \{p'_1,p'_2,p'_3\}$ as claimed.
Now suppose the left end of $\mathbf{P}_a$ is a triad.
Then this end is $\{p_1',p_2',p_3'\}$, and this set contains a $\{U_{2,5},U_{3,5}\}$-deletable element by \cref{nicepathdescription}\ref{nicepathdesciii}.
Since $p_3$ is the unique $\{U_{2,5},U_{3,5}\}$-deletable element
in $L$, we again have $p_3 \in \{p'_1,p'_2,p'_3\}$.
Now $\{p'_1,p'_2,p'_3\}=\{p_h,p_3,p_{i_L}\}$ for some $h \in \{1,2\}$.
We claim that $\{p_h,p_3,p_{i_L}\}$ is closed in $M \backslash a,b,c$.
Suppose not.
Assume $h=1$ and say $p_k \in \cl_{M \backslash a,b,c}(\{p_1,p_3,p_{i_L}\})$ for some $k \in \seq{14} - \{1,3,i_L\}$.
Then $\{p_1,p_3,p_{i_L},p_k\}$ is a circuit and, since $p_{i_L}$ is a coguts element in $\sigma_c^-$, we have $k=11$,
contradicting orthogonality with the triad $\{p_1,p_4,p_5\}$.
The argument is essentially the same when $h=2$, but with $(p_1,p_5)$ and $(p_2,p_6)$ swapped.
So $\{p_1',p_2',p_3'\}$ is closed in $M \backslash a,b,c$.
Now, since the left end of $\mathbf{P}_a$ is not a $4$-cosegment, $p'_4 \in \cl_{M \backslash b,c}(\{p'_1,p'_2,p'_3\})$, which implies that $p'_4=a$.
Thus, by \cref{pathbasics}, if $a$ blocks a triad, then this triad meets $\{p_h,p_3,p_{i_L}\} \subseteq L$. Moreover, $a$ does not block both $\{p_1,p_4,p_5\}$ and $\{p_2,p_4,p_6\}$.
Now suppose $a \in \{p'_1,p'_2,p'_3\}$.
We may assume, without loss of generality, that $a=p'_3$. Then $\{p'_1,p'_2,p'_4\} \subseteq L$, by \cref{endslipperiness}\ref{esii}, so $\{p'_1,p'_2,p'_4\}$ is a triad of $M \backslash a,b,c$.
Since $M \backslash a,b,c$ is $3$-connected, $\{p'_1,p'_2,p'_3\}$ is a triangle of $M \backslash b,c$.
Note that $\{p'_1,p'_2,p'_3,p'_4\}$ is not a $4$-element fan of $M \backslash b,c$, for otherwise we would have chosen $\sigma_a$ so that $\{p'_1,p'_2,p'_3\}$ is a triad.
So $\{p'_1,p'_2,a,p'_4\}$ is a cocircuit of $M \backslash b,c$, and the triangle $\{p'_1,p'_2,a\}$ is the left end of $\mathbf{P}_a$.
By \cref{nicepathdescription}\ref{nicepathdesciii} and \cref{pathdesctris}, the set $\{p'_1,p'_2,a\}$ contains one $\{U_{2,5},U_{3,5}\}$-contractible element and two $\{U_{2,5},U_{3,5}\}$-deletable elements.
Hence either $p'_1$ or $p'_2$ is $\{U_{2,5},U_{3,5}\}$-deletable.
Since $p_3$ is the unique $\{U_{2,5},U_{3,5}\}$-deletable element
in $L$, we have $p_3 \in \{p'_1,p'_2\}$.
Now $\{p'_1,p'_2\} \in \left\{\{p_1,p_3\}, \{p_2,p_3\}, \{p_{i_L},p_3\} \right\}$.
Thus, if $a$ blocks a triad, then the triad meets $\{p'_1,p'_2\} \subseteq L$.
Moreover, $a$ does not block both $\{p_1,p_4,p_5\}$ and $\{p_2,p_4,p_6\}$. \end{subproof}
By \cref{whatablocks}, we may now assume that $b$ blocks $\{p_{12},p_{13},p_{14}\}$.
\begin{claim}
\label{whatbblocks}
Either
\begin{enumerate}[label=\rm(\Roman*)]
\item $b$ blocks neither $\{p_1,p_4,p_5\}$ nor $\{p_2,p_4,p_6\}$; or
\item $\{p_5,p_7,p_{i_R}\}$ is a triad of $M \backslash a,b,c$ that is not blocked by $b$, up to swapping $(p_1,p_5)$ and $(p_2,p_6)$.
\end{enumerate} \end{claim} \begin{subproof}
Let $\sigma_b = (p''_1,p''_2,\dotsc,p''_{14})$ be a sequential ordering for $M \backslash a,c$ such that some initial segment and some terminal segment of $\sigma_b$ are ends of a nice path description $\mathbf{P}_b$ for $M \backslash a,c$, where if the left (or right) end of $\mathbf{P}_b$ is a fan of size at least~$4$, then we choose $\{p''_{1},p''_{2},p''_{3}\}$ (or $\{p''_{12},p''_{13},p''_{14}\}$, respectively) to be a triad.
Let $\sigma_b^-$ be the sequential ordering of $M \backslash a,b,c$ obtained from $\sigma_b$ by removing $b$, as described in \cref{pathbasics}.
By reversing these orderings, if necessary, we may assume that $L(\sigma_b^-) = L$ and $R(\sigma_b^-) = R$, due to \cref{welldefinedends}.
First we assume that $b \in \{p''_{12},p''_{13},p''_{14}\}$.
Without loss of generality, $b=p''_{12}$.
Then $\{p''_{11},p''_{13},p''_{14}\} \subseteq R$, by \cref{endslipperiness}\ref{esii}, so $\{p''_{11},p''_{13},p''_{14}\}$ is a triad of $M \backslash a,b,c$.
Since $M \backslash a,b,c$ is $3$-connected, $\{b,p''_{13},p''_{14}\}$ is a triangle of $M \backslash a,c$.
Note that $\{p''_{11},b,p''_{13},p''_{14}\}$ is not a $4$-element fan of $M \backslash a,c$, for otherwise we would have chosen $\sigma_b$ so that $\{p''_{12},p''_{13},p''_{14}\}$ is a triad.
So $\{p''_{11},b,p''_{13},p''_{14}\}$ is a cocircuit of $M \backslash a,c$ and the triangle $\{b,p''_{13},p''_{14}\}$ is the right end of $\mathbf{P}_b$.
By \cref{nicepathdescription}\ref{nicepathdesciii} and \cref{pathdesctris}, $\{b,p''_{13},p''_{14}\}$ contains one $\{U_{2,5},U_{3,5}\}$-contractible element, and two $\{U_{2,5},U_{3,5}\}$-deletable elements.
Hence either $p''_{13}$ or $p''_{14}$ is $\{U_{2,5},U_{3,5}\}$-deletable.
Since $p_{12}$ is the unique $\{U_{2,5},U_{3,5}\}$-deletable element in $R$, we have $p_{12} \in \{p''_{13},p''_{14}\}$.
So $\{p''_{13},p''_{14}\} \in \{\{p_{12},p_{14}\},\{p_{12},p_{13}\},\{p_{12},p_{i_R}\}\}$.
In any case, $b$ blocks neither $\{p_1,p_4,p_5\}$ nor $\{p_2,p_4,p_6\}$, as required.
Now we may assume that $b \notin \{p''_{12},p''_{13},p''_{14}\}$.
Then $\{p''_{12},p''_{13},p''_{14}\} \subseteq R$ by \cref{endslipperiness}\ref{esii}, and $b$ does not block the triad $\{p''_{12},p''_{13},p''_{14}\}$.
But $b$ blocks $\{p_{12},p_{13},p_{14}\}$, so $\{p_{12},p_{13},p_{14}\} \neq \{p''_{12},p''_{13},p''_{14}\}$, implying $p_{i_R} \in \{p''_{12},p''_{13},p''_{14}\}$.
Now $\{p''_{12},p''_{13},p''_{14}\}$ is a triad in $M \backslash a,c$, and this triad is contained in the right end of $\mathbf{P}_b$.
If this end is a $4$-cosegment, then it is $\{p''_{11},p''_{12},p''_{13},p''_{14}\}$, in which case $R = \{p''_{11},p''_{12},p''_{13},p''_{14}\}$ and $b$ does not block the triad $\{p_{12},p_{13},p_{14}\}$, a contradiction.
So the right end of $\mathbf{P}_b$ is either a triad, or a $4$- or $5$-element fan where $b$ forms a triangle with elements of the triad $\{p''_{12},p''_{13},p''_{14}\}$.
We show, in any case, that $p_{12} \in \{p''_{12},p''_{13},p''_{14}\}$.
Recall that $p_{12}$ is the unique $\{U_{2,5},U_{3,5}\}$-deletable element
in $R$.
Suppose $\{p''_{12},p''_{13},p''_{14},b\}$ is a $4$-element fan in $M \backslash a,c$ contained in the right end of $\mathbf{P}_b$.
Since $\{p''_{12},p''_{13},p''_{14}\}$ is contained in a $4$-element fan with no $\{U_{2,5},U_{3,5}\}$-essential elements, this set contains a $\{U_{2,5},U_{3,5}\}$-deletable element.
So $p_{12} \in \{p''_{12},p''_{13},p''_{14}\}$ as claimed.
Now suppose the right end of $\mathbf{P}_b$ is a triad.
Then this end is $\{p''_{12},p''_{13},p''_{14}\}$, and it contains a $\{U_{2,5},U_{3,5}\}$-deletable element, by \cref{nicepathdescription}\ref{nicepathdesciii}.
Since $p_{12}$ is the unique $\{U_{2,5},U_{3,5}\}$-deletable element
in $R$, we again have $p_{12} \in \{p''_{12},p''_{13},p''_{14}\}$.
Now $\{p''_{12},p''_{13},p''_{14}\} = \{p_{i_R},p_{12},p_{g}\}$ for some $g \in \{13,14\}$.
We claim that either $\{p_{i_R},p_{12},p_{g}\}$ is closed in $M \backslash a,b,c$, or $g=13$ and
$\cl_{M \backslash a,b,c}(\{p_{i_R},p_{12},p_{13}\}) = \{p_7,p_{i_R},p_{12},p_{13}\}$.
Suppose $p_k \in \cl_{M \backslash a,b,c}(\{p_{i_R},p_{12},p_{g}\})$ for some $k \in \seq{14} - \{i_R,12,g\}$.
Then $\{p_k,p_{i_R},p_{12},p_{g}\}$ is a circuit and,
since $p_{i_R}$ is a coguts element in $\sigma_c^-$, we have $k \in \{4,7\}$.
By orthogonality with the triads $\{p_1,p_4,p_5\}$ and $\{p_{i_L},p_{11},p_{14}\}$, we have $k=7$ and $g = 13$.
So either $\{p''_{12},p''_{13},p''_{14}\}$ is closed in $M \backslash a,b,c$, or $\cl_{M \backslash a,b,c}(\{p''_{12},p''_{13},p''_{14}\}) = \{p_7,p_{i_R},p_{12},p_{13}\}$, as claimed.
Now, since the right end of $\mathbf{P}_b$ is not a $4$-cosegment, $p''_{11} \in \cl_{M \backslash a,c}(\{p''_{12},p''_{13},p''_{14}\})$, which implies that either $p''_{11}=b$, or $p''_{11} = p_7$.
But in the former case, $b$ does not block either of the triads $\{p_1,p_4,p_5\}$ or $\{p_2,p_4,p_6\}$, as required.
So we may assume that $p''_{11} = p_7$.
Consider $p''_{10}$.
If $p''_{10}=b$, then neither $\{p_1,p_4,p_5\}$ nor $\{p_2,p_4,p_6\}$ is blocked by $b$, as required; so may assume that $p''_{10} \neq b$.
Let $Q=E(M \backslash a,b,c) - \{p''_{11},p''_{12},p''_{13},p''_{14}\} = \{p_1,p_2,p_3,p_4,p_5,p_6,p_{i_L},p_{11},p_{14}\}$, so $p''_{10} \in Q$.
Observe that each element in $Q$ is in a triad of $M \backslash a,b,c$ that is contained in $Q$.
Hence $p''_{10}$ is not a guts element, so $p''_{10} \in \cl^*_{M \backslash a,b,c}(\{p''_{11},p''_{12},p''_{13},p''_{14}\}) = \cl^*_{M \backslash a,b,c}(\{p_7,p_{i_R},p_{12},p_{13}\})$.
Note also that $p''_{10} \neq p_{14}$, for otherwise $b$ does not block $\{p_{12},p_{13},p_{14}\}$.
Since $C=\{p_1,p_2,p_3,p_4\}$ is a circuit, $p''_{10} \notin C$.
If $p''_{10} \in \{p_{11},p_{i_L}\}$, then $\{p_{14},p_{11},p_{i_L}\} \subseteq \cl^*_{M \backslash a,b,c}(\{p_7,p_{i_R},p_{12},p_{13}\})$, in which case $\{p_5,p_6\} \subseteq \cl^*_{M \backslash a,b,c}(\{p_7,p_{i_R},p_{12},p_{13}\})$ since $p_5$ and $p_6$ are coguts elements in $\sigma_c$.
It follows that $r^*(M \backslash a,b,c) \le 4$, so
$r(M \backslash a,b) \ge 9$, a contradiction.
Thus $p''_{10} \in \{p_5,p_6\}$.
Up to possibly swapping the labels on $(p_1,p_5)$ and $(p_2,p_6)$, we may now assume that $p''_{10} = p_5$.
We claim that $\{p_5,p_7,p_{i_R}\}$ is a triad that is not blocked by $b$.
As $\{p_{i_R},p_{12},p_{13}\}$ is a triad in $M \backslash a,b,c$ and $p_5 \in \cl^*_{M \backslash a,b,c}(\{p_7,p_{i_R},p_{12},p_{13}\})$,
we have that $\{p_5,p_7\}$ is contained in a $3$- or $4$-element cocircuit~$C^*$ that is contained in $\{p_5,p_7,p_{i_R},p_{12},p_{13}\}$.
By orthogonality with the circuit $\{p_{11},p_{12},p_{13},p_{14}\}$, either $C^* = \{p_5,p_7,p_{i_R}\}$ or $C^* = \{p_5,p_7,p_{12},p_{13}\}$, so we may assume the latter.
But then, by cocircuit elimination with $\{p_{i_R},p_{12},p_{13}\}$, there is a cocircuit contained in $\{p_5,p_7,p_{i_R},p_{12}\}$, which, again by orthogonality, is the triad $\{p_5,p_7,p_{i_R}\}$.
By \cref{pathbasics}, this triad is not blocked by $b$, as required. \end{subproof}
By \cref{whatablocks}, $a$ blocks at most one of $\{p_1,p_4,p_5\}$ and $\{p_2,p_4,p_6\}$. As neither of these triads is blocked by $c$, at least one of $\{p_1,p_4,p_5\}$ and $\{p_2,p_4,p_6\}$ is blocked by $b$. Now, by \cref{whatbblocks}, we may assume that $\{p_5,p_7,p_{i_R}\}$ is a triad of $M \backslash a,b,c$ that is not blocked by $b$. But $\{p_5,p_7,p_{i_R}\}$ is not blocked by $a$, by \cref{whatablocks}, and, recalling that $i_c > i_R > 7$, it is also not blocked by $c$. From this contradiction, we deduce that $M$ has no delete triples, thus completing the proof. \end{proof}
\section{The no-delete-triples case} \label{ndtsec}
In this section we prove the following:
\begin{theorem}
\label{nodeltriplethm}
Let $M$ be an excluded minor for the class of $\mathbb{P}$-representable matroids where $\mathbb{P} \in \{\mathbb{H}_5,\mathbb{U}_2\}$, and $M$ has no triads.
Suppose there is a pair $\{a,b\} \subseteq E(M)$ such that $M \backslash a,b$ is $3$-connected with a $\{U_{2,5},U_{3,5}\}$-minor.
If $M$ has no delete triples, then $|E(M)| \le 15$. \end{theorem}
The bulk of the work in proving this \lcnamecref{nodeltriplethm} is accomplished by \cref{nodeltripleprop}, which proves the result except when $M$ has 16 elements and specific structure. In \cref{nodeltriplematrices}, we show that the specific structure implies that $M \backslash a,b$ is, up to isomorphism, one of three particular $2$-regular matroids. We performed a computer search to show that, in fact, when $M$ satisfies the hypotheses of the theorem and contains one of these three matroids as a minor, then $M$ has a delete triple.
We first require some definitions.
For a guts-coguts path $\mathbf{P}=(P_1,P_2,\dotsc,P_m)$, we say that $\mathbf{P}$ is \emph{left-justified} if for all $i \in \{2,3,\dotsc,m-1\}$, \begin{enumerate}[label=\rm(\Roman*)]
\item if $P_i$ is a guts set, then $\cl\bigl(\bigcup_{j \in [i]} P_j\bigr) - \bigl(\bigcup_{j \in [i]}P_{j}\bigr) \subseteq P_m$; and
\item if $P_i$ is a coguts set, then $\cl^*\bigl(\bigcup_{j \in [i]} P_j\bigr) - \bigl(\bigcup_{j \in [i]} P_{j}\bigr) \subseteq P_m$. \end{enumerate} Similarly, we say that $\mathbf{P}$ is \emph{right-justified} if $(P_m,P_{m-1},\dotsc,P_1)$ is left-justified.
Given a guts-coguts path $\mathbf{P}$, one can easily obtain a left-justified guts-coguts path $\mathbf{P}' = (P'_1,P'_2,\dotsc,P'_{m'})$ with $P_1 = P'_1$ and $P_m = P'_{m'}$; we call $\mathbf{P}'$
the \emph{left-justification} of $\mathbf{P}$.
Suppose that $\mathbf{P} = (P_1,P_2,\dotsc,P_m)$ is a nice path description. Recall that a nice path description is a guts-coguts path. Note that the left-justification of $\mathbf{P}$, and the left-justification of $(P_m,P_{m-1},\dotsc,P_1)$, are also nice path descriptions. We say that the \emph{reversal} of $(P_1,P_2,\dotsc,P_m)$ is the nice path description $\mathbf{P'}$ obtained from the left-justification of $(P_m,P_{m-1},\dotsc,P_1)$. By \cref{niceends}, the ends of $\mathbf{P'}$ are $P_m$ and $P_1$.
For the remainder of this section we let $M$ be an excluded minor for the class of $\mathbb{P}$-representable matroids where $\mathbb{P} \in \{\mathbb{H}_5,\mathbb{U}_2\}$, and $M$ has no triads.
\begin{proposition}
\label{nodeltripleprop}
Suppose there is a pair $\{a,b\} \subseteq E(M)$ such that $M \backslash a,b$ is $3$-connected with a $\{U_{2,5},U_{3,5}\}$-minor.
If $M$ has no delete triples, then either
\begin{enumerate}
\item $|E(M)| \le 15$; or
\item $|E(M)| = 16$ and\label{ndtendgameoutcomes16}
\begin{enumerate}[label=\rm(\alph*)]
\item $M \backslash a,b$ has a nice path description \[(\{a',p_1',p_1\},\{p_2\},\{p_3\},\{p_4\},\{p_5,p_5'\},\{p_6\},\{p_7\},\{p_8\},\{p_9,p_9',b'\})\] where, for some $\{q,q'\} = \{p_5,p_5'\}$,
\begin{itemize}
\item $\{a',p_1',p_1\}$, $\{p_1,p_2,p_3\}$, $\{p_3,p_4,p_5\}$, $\{q,p_6,p_7\}$, $\{p_7,p_8,p_9\}$, and $\{p_9,p_9',b'\}$ are triads of $M \backslash a,b$,
\item $\{a',p_1,p_4,p_5'\}$ and $\{q',p_6,p_9,b'\}$ are cocircuits of $M \backslash a,b$,
\end{itemize}
and $M \backslash a,b$ has no triangles;\label{ndtendgameoutcome16a}
\item $M \backslash a,b$ has a nice path description \[(\{a',p_1'',p_1',p_1\},\{p_2,p_2'\},\{p_3,p_3'\},\{p_4\},\{p_5\},\{p_6\},\{p_7,p_7',b'\})\] where
\begin{itemize}
\item $\{a',p_1'',p_1',p_1\}$ is a cosegment of $M \backslash a,b$,
\item $\{p_1',p_2',p_3'\}$, $\{p_1,p_2,p_3\}$, $\{p_3,p_4,p_5\}$, $\{p_5,p_6,p_7\}$, and $\{p_7,p_7',b'\}$ are triads of $M \backslash a,b$, and
\item $\{p_3',p_4,p_7,b'\}$ is a cocircuit of $M \backslash a,b$;
\end{itemize}\label{ndtendgameoutcome16b}
and $M \backslash a,b$ has no triangles.
\end{enumerate}
\end{enumerate} \end{proposition} \begin{proof}
Suppose that $M$ has no delete triples and $|E(M)| \ge 16$.
By \cref{utfutffragile}, $M \backslash a,b$ is $\{U_{2,5},U_{3,5}\}$-fragile.
Loosely speaking, our strategy is to use the structure of this matroid to find a triple that, once we add back $a$ and $b$, is a delete triple in $M$; if we cannot find such a triple, then \labelcref{ndtendgameoutcomes16} holds.
The crux to this approach is the following:
\begin{claim}
\label{ndtcrux}
Suppose there are distinct elements $a',b',c' \in E(M\backslash a,b)$ such that the matroid $M \backslash \{a,b,a',b',c'\}$
\begin{itemize}
\item is $3$-connected up to series classes of size at most three,
\item has at least three distinct series classes, and
\item has a $\{U_{2,5},U_{3,5}\}$-minor,
\end{itemize}
and, in $M \backslash a,b$,
\begin{itemize}
\item $\{a',b',c'\}$ is not contained in a $5$-element cocircuit,
\item no pair of elements of $\{a',b',c'\}$ is contained in a $4$-element cocircuit, and
\item $\{a',b',c'\}$ is coindependent.
\end{itemize}
Then $\{a',b',c'\}$ is a delete triple for $M$. \end{claim} \begin{subproof}
First, we claim that if $S$ is a series pair of $M \backslash \{a,b,a',b',c'\}$, then it is blocked by $a$ or $b$ in $M \backslash a',b',c'$.
Let $S$ be a series pair of $M \backslash \{a,b,a',b',c'\}$.
Then $S \cup \{a',b',c'\}$ contains a cocircuit in $M \backslash a,b$.
If this cocircuit has size four or five, then it intersects $\{a',b',c'\}$ in two or three elements, respectively, a contradiction.
So $S \cup e$ is a triad of $M \backslash a,b$ for some $e \in \{a',b',c'\}$.
This triad is blocked by $a$ or $b$ in $M$, since $M$ has no triads.
Without loss of generality, say $a$ is the element that blocks the triad $S \cup e$ of $M \backslash a,b$.
Then
$a \in \cl^*_{M \backslash b}(S \cup e)$,
so $a \in \cl^*_{M \backslash b,a',b',c'}(S)$, and hence
the series pair $S$ of $M \backslash \{a,b,a',b',c'\}$ is blocked by $a$, thus proving the first claim.
Suppose $M \backslash a',b',c'$ has a coloop $e$.
Since $M \backslash \{a,b,a',b',c'\}$ has no coloops, $e \in \{a,b\}$.
Then $\{e,a',b',c'\}$ is a cocircuit of $M$, since $M$ is $3$-connected and has no triads.
But then $\{a',b',c'\}$ is not coindependent in $M \backslash a,b$, a contradiction.
Now suppose $M \backslash a',b',c'$ has a series pair $\{e,f\}$.
Then $\{e,f,a',b',c'\}$ contains a cocircuit of $M$.
If $\{a,b\} \cap \{e,f\} = \emptyset$, then $r^*_{M \backslash a,b,a',b',c'}(\{e,f\}) \le 1$.
Note that $\{e,f\}$ is not a series pair of $M \backslash \{a,b,a',b',c'\}$, by the first claim. So $e$ or $f$ is a coloop of $M \backslash \{a,b,a',b',c'\}$.
But this contradicts that $M \backslash \{a,b,a',b',c'\}$ is $3$-connected up to series classes.
Similarly, if $|\{a,b\} \cap \{e,f\}| = 1$, then $e$ or $f$ is a coloop of $M \backslash \{a,b,a',b',c'\}$, again contradicting that $M \backslash \{a,b,a',b',c'\}$ is $3$-connected up to series classes.
So $\{a,b\} = \{e,f\}$.
Then $\{a,b,a',b',c'\}$ contains a cocircuit of $M$, which meets $\{a',b',c'\}$ since $M$ is $3$-connected, so $\{a',b',c'\}$ is not coindependent in $M \backslash a,b$, a contradiction.
Next, we work towards proving that if $(U,V)$ is a $2$-separation of $M \backslash a',b',c'$, then we may assume, up to swapping $U$ and $V$, that $U$ is a cosegment such that for some $\{e,e'\} = \{a,b\}$, we have $e \in U \cap \cl(U-e)$ and $e' \in V$.
Let $(U,V)$ be a $2$-separation of $M \backslash a',b',c'$ with $a \in U$.
Since $M \backslash a',b',c'$ has no loops, coloops, parallel pairs or series pairs, $|U|,|V| \ge 3$.
Note that $\{a,b\}$ is coindependent in $M \backslash a',b',c'$, since $M$ is $3$-connected and $\{a',b',c'\}$ is coindependent in $M \backslash a,b$.
Now $(U-a,V)$ is a $2$-separation in $M \backslash \{a, a',b',c'\}$.
Since $a$ does not block the $2$-separating set $V$, we have $a \in \cl(U-a)$.
Suppose $U-a$ is contained in a series class of $M \backslash \{a, a',b',c'\}$.
Then $b \in V$, for otherwise the non-empty set $U-\{a,b\}$ consists of coloops of $M \backslash \{a,b,a',b',c'\}$, contradicting that $M \backslash \{a,b,a',b',c'\}$ is $3$-connected up to series classes. Now each pair of $U-a$ is a series pair of $M \backslash \{a,b,a',b',c'\}$ not blocked by $b$.
So $a$ blocks $U-a$, implying $a \in \cl^*_{M \backslash a',b',c'}(U-a)$.
Recalling that $a \in \cl(U-a)$, we see that $U$ is a cosegment of $M \backslash a',b',c'$ with $a \in \cl(U-a)$ as required.
So we may assume that $U-a$ is not contained in a series class of $M \backslash \{a,a',b',c'\}$.
Suppose $b \in U$.
Then $(U-\{a,b\},V)$ is a $2$-separation of $M \backslash \{a,b,a',b',c'\}$.
As neither $a$ nor $b$ blocks the $2$-separating set $V$, we have $\{a,b\} \subseteq \cl(U-\{a,b\})$.
If $V$ is contained in a series class of $M \backslash \{a,b,a',b',c'\}$, then it is also contained in a series class of $M \backslash a',b',c'$, a contradiction.
Since $M \backslash \{a, b, a',b',c'\}$ is $3$-connected up to series classes, $U-\{a,b\}$ is contained in a series class $S$ say.
Now $M \backslash \{a,b,a',b',c'\}$ contains some series class $S'$ distinct from $S$.
Since $a,b \in \cl(U-\{a,b\})$, neither $a$ nor $b$ blocks $S'$, a contradiction.
We deduce that $b \in V$.
Now, by symmetry,
$b \in \cl(V-b)$.
Since $M \backslash \{a, b, a',b',c'\}$ is $3$-connected up to series classes, either $U-a$ or $V-b$ is contained in a series class of $M \backslash \{a,b,a',b',c'\}$; without loss of generality, say it is $V-b$.
Since $a \in \cl(U-a)$, the element $a$ does not block $V-b$.
So $b$ blocks $V-b$, that is, $b \in \cl^*_{M \backslash a',b',c'}(V-b)$.
Now $V$ is a cosegment in $M \backslash a',b',c'$ with $b \in V \cap \cl(V-b)$ and $a \notin V$, as required.
Now we may assume that $M \backslash a',b',c'$ has a cosegment $G$, with $a \in G \cap \cl(G-a)$ and $b \notin G$, up to swapping $a$ and $b$, for otherwise $M$ has a delete triple, $\{a',b',c'\}$, as required.
Without loss of generality, $G$ is coclosed in $M \backslash a',b',c'$,
and it follows that $G-a$ is a series class in $M \backslash \{a, b, a',b',c'\}$.
Let $G-a$, $S'$ and $S''$ be distinct series classes of $M \backslash \{a, b, a',b',c'\}$.
Since $a \in \cl(G-a)$, it follows that $a$ blocks neither $S'$ nor $S''$.
So $b$ blocks both $S'$ and $S''$. Note that $b \notin \cl(S')$, for otherwise $b$ does not block $S''$; and similarly $b \notin \cl(S'')$.
We deduce that the only $2$-separations of $M \backslash a',b',c'$ are of the form $(G', E(M \backslash a',b',c')-G')$ where $G' \subseteq G$ is a cosegment and $a \in \cl(G'-a)$, and $b \notin G'$.
Now $G-a$ is a series class of $M \backslash \{a,b,a',b',c'\}$ that is blocked in $M \backslash a,b$.
Since $M \backslash a,b$ has no $4$-element cocircuits containing a pair of elements in $\{a',b',c'\}$, and no $5$-element cocircuit containing $\{a',b',c'\}$, each series pair of $M \backslash \{a,b,a',b',c'\}$ contained in $G-a$ is blocked by exactly one of $a'$, $b'$, and $c'$.
Observe that $|G-a| \in \{2,3\}$.
We claim that there is some $e \in \{a',b',c'\}$ that blocks every pair contained in $G-a$.
Clearly this is the case when $|G-a| =2$, so let $G-a=\{s,t,q\}$, and suppose $a'$ blocks $\{s,t\}$ but $a'$ does not block $\{s,q\}$.
Then, we may assume that $b'$ blocks $\{s,q\}$, so $\{s,t,a'\}$ and $\{s,q,b'\}$ are triads of $M \backslash a,b$.
By cocircuit elimination, $\{t,q,a',b'\}$ contains a cocircuit; so either $\{t,q,a'\}$ or $\{t,q,b'\}$ is a triad of $M \backslash a,b$.
In the former case, $\{s,q,a'\}$ is also a triad of $M \backslash a,b$, by cocircuit elimination, so $a'$ blocks $\{s,q\}$, a contradiction.
So $\{t,q,b'\}$ is a triad of $M \backslash a,b$.
But then, by cocircuit elimination with $\{s,q,b'\}$, so is $\{s,t,b'\}$, so $b'$ blocks each pair in $G-a$.
We may now assume that $a'$ blocks every pair contained in $G-a$.
Then $(G-a) \cup a'$ is a cosegment of $M \backslash a,b$.
Suppose $M \backslash b',c'$ is not $3$-connected. Let $(U,V)$ be a $2$-separation in $M \backslash b',c'$ with $a' \in U$.
Note that $|U| \ge 3$, since $a'$ is not in a parallel or series pair of $M \backslash b',c'$.
Since $a'$ is not a coloop in $M \backslash b',c'$, we have $\lambda_{M \backslash a',b',c'}(U-a') \le \lambda_{M \backslash b',c'}(U) \le 1$.
So $(U-a',V)$ is a $2$-separation in $M \backslash a',b',c'$.
Thus either $U-a'$ or $V$ is a cosegment $G' \subseteq G$, with $a \in G' \cap \cl(G'-a)$ and $b \notin G'$.
As $V$ is $2$-separating in both $M \backslash a',b',c'$ and $M \backslash b',c'$, we see that $a'$ does not block $V$, so
$a' \in \cl_{M\backslash b',c'}(U-a')$.
If $U-a' = G'$, then $a' \in \cl(G') = \cl(G'-a)$, and so in $M \backslash a,b$, the set $(G'-a) \cup a'$ is a dependent cosegment and is therefore $2$-separating, a contradiction.
So $V = G'$, and $a'$ does not block $G'$, with $a \in V$ and $b \in U$.
Note that $|V-a| \ge 2$,
so $a'$ does not block each series pair of $M \backslash a,b,a',b',c'$ contained in $G-a$, a contradiction.
So $M \backslash b',c'$ is $3$-connected.
As $M \backslash a',b',c'$ has a $\{U_{2,5},U_{3,5}\}$-minor, $M \backslash a',b',c'$ is $\{U_{2,5},U_{3,5}\}$-fragile, by \cref{utfutffragile}.
But then $M \backslash a',b',c'$ is $3$-connected up to series and parallel classes, by \cref{genfragileconn}, contradicting that $G$ is $2$-separating in $M \backslash a',b',c'$. \end{subproof}
Now, if $M \backslash a,b$ has a triple $\{a',b',c'\}$ as described in \cref{ndtcrux}, then $M$ has a contradictory delete triple. Our strategy is to attempt to find such a triple $\{a',b',c'\}$; when we cannot, we have the structure described in \ref{ndtendgameoutcomes16}.
Recall that $M \backslash a,b$ is $3$-connected and $\{U_{2,5},U_{3,5}\}$-fragile, and, due to \cref{lemmaC}, $M \backslash a,b$ has a nice path description $\mathbf{P} = (P_1,P_2,\dotsc,P_m)$ and every element of $M \backslash a,b$ is either $\{U_{2,5},U_{3,5}\}$-deletable or $\{U_{2,5},U_{3,5}\}$-contractible.
Let $N \in \{U_{2,5},U_{3,5}\}$ such that $M\backslash a,b$ has an $N$-minor.
By \cref{fragilecase}, there exists a basis~$B$ for $M$ and a $B \times B^*$ companion $\mathbb{P}$-matrix~$A$ for which $\{x,y,a,b\}$ incriminates $(M,A)$ where $\{x,y\} \subseteq B$ and $\{a,b\} \subseteq B^*$.
By \cref{atmostonetri}, $M \backslash a,b$ has at most one triangle, and if such a triangle $T$ exists, then
$T \cup \{x,y\}$ is a $4$-element fan containing $u$.
In what follows, we work in the matroid $M \backslash a,b$ unless explicitly specified otherwise; for example, when we say $P_1$ is a triad, we mean it is a triad of $M \backslash a,b$.
\begin{claim}
\label{ndtends}
Let $i \in \{1,m\}$. Then $P_i$ is either a cosegment or a $5$-element fan whose ends are rim elements. \end{claim} \begin{subproof}
The end $P_i$ contains either a triangle or a triad.
If $P_i$ does not contain a triangle, then it is a cosegment.
On the other hand, if $P_i$ contains a triangle, then, by \cref{atmostonetri}, $P_i$ is a fan of size at least~$4$.
If $P_i$ is a fan of size at least~$6$, then it contains at least $2$ triangles, a contradiction.
If the fan $P_i$ has a spoke end $d$, then $M \backslash a,b,d$ is $3$-connected and has a $\{U_{2,5},U_{3,5}\}$-minor, by \cref{fanends,pathdescends}, contradicting that $M$ has no delete triples.
It follows that $P_i$ has size five and both its ends are rim elements, as required. \end{subproof}
Suppose neither $P_1$ nor $P_m$ is a $5$-element fan. Then both $P_1$ and $P_m$ are cosegments, by \cref{ndtends}. By \cref{niceends}, $P_1$ and $P_m$ are coclosed. Hence $P_2$ and $P_{m-1}$ are guts sets. Moreover, $m$ is odd.
On the other hand, if $P_1$ is a $5$-element fan, then $P_2$ could be either a guts set (in which case $m$ is odd) or a coguts set (in which case $m$ is even).
For ease of notation, for any $i \in \{2,3,\dotsc,m-1\}$ we let $P_i^- = P_1 \cup \dotsm \cup P_{i-1}$ and $P_i^+ = P_{i+1} \cup \dotsm \cup P_{m}$.
\begin{claim}
\label{ndttriadseq}
Let $i \in \{2,3,\dotsc,m-1\}$ such that $P_i$ is a guts set.
Then $|P_i| \le 2$ and for each $e \in P_i$, there is a triad of $M \backslash a,b$ containing $e$ that meets $P_i^-$ and $P_i^+$. \end{claim} \begin{subproof}
First, observe that if some $e \in P_i$ is in a triad~$T^*$, then it follows from orthogonality that $T^*$ meets both $P_i^-$ and $P_i^+$.
If $|P_i| = 3$, then $P_i$ is a triangle, so $P_i$ is contained in a $4$-element fan by \cref{atmostonetri}.
But then there is a triad containing two elements of $P_i$, so it does not meet both $P_i^-$ and $P_i^+$, a contradiction.
So $|P_i| \le 2$.
Now let $e \in P_i$.
By \cref{pathdescprops}, $e$ is $\{U_{2,5},U_{3,5}\}$-deletable in $M \backslash a,b$, and $\co(M \backslash a,b \backslash e)$ is $3$-connected. Thus, if $e$ is not in a triad, then $\{a,b,e\}$ is a delete triple, a contradiction.
So $e$ is in a triad which, by the foregoing, meets both $P_i^-$ and $P_i^+$. \end{subproof}
We say that $T^*$ is an \emph{internal triad} if $T^*$ is a triad that contains an element in some guts set $P_i$, for $i \in \{2,3,\dotsc,m-1\}$.
\begin{claim}
\label{ndtdisjtriads}
Let $\{\ell,e,r\}$ and $\{\ell',e',r'\}$ be distinct internal triads, where $e \in P_i$ and $e' \in P_{i'}$ for guts sets $P_i$ and $P_{i'}$, and $\ell \in P_i^-$, $r \in P_i^+$, $\ell' \in P_{i'}^-$, and $r' \in P_{i'}^+$.
Then $\ell \neq \ell'$ and $r \neq r'$.
In particular, if for some guts set $P_j$ we have $|P_j|=2$, say $P_j = \{e,e'\}$, then the triads containing $e$ and $e'$ are disjoint. \end{claim} \begin{subproof}
Suppose $\ell = \ell'$. Then, by cocircuit elimination, there is a cocircuit~$C^*$ contained in $\{e,e',r,r'\}$.
First suppose that $i = i'$.
Since $e$ and $e'$ are in the guts set $P_i$, and $P_i^+ \cap C^* = \{r,r'\}$,
it follows from orthogonality that neither $e$ nor $e'$ is in the cocircuit~$C^*$. But then $\{r,r'\}$ is a series pair, a contradiction.
Now suppose that $i \neq i'$. Without loss of generality, let $i < i'$.
Then, it follows from orthogonality that $e \notin C^*$, so $\{r,e',r'\}$ is a triad, where $r \in P_i^+ \cap P_{i'}^-$.
Now $\{\ell,r,e',r'\}$ is a $4$-cosegment.
But then $\{\ell,r,e'\}$ is a triad that avoids $P_{i'}^+$, contradicting orthogonality.
So $\ell \neq \ell'$ and, similarly, $r \neq r'$.
\end{subproof}
We now assume that $\mathbf{P} = (P_1, \dotsc, P_m)$ is a nice path description for $M \backslash a,b$ such that $P_m$ is a triad, using \cref{oneendisatriad} and up to the reversal of $\mathbf{P}$. Recall also that the reversal of $\mathbf{P}$ is, by definition, left-justified.
\begin{claim}
\label{ndttriads}
$|P_{m-1}| = 1$, and if an internal triad meets $P_m$, then this triad contains $P_{m-1}$.
Moreover,
\begin{enumerate}[label=\rm(\Roman*)]
\item if $P_1$ is a triad, then $|P_2| = 1$, and each internal triad that meets $P_1$ contains $P_2$;\label{ndttriadsi}
\item if $P_1$ is a $4$-cosegment or a $5$-element fan and $P_2$ is a guts set, and $|P_i|=2$ for some guts set $P_i$ with $i > 2$, then for some $e \in P_i$ each triad containing $e$ is disjoint from $P_1$.\label{ndttriadsii}
\end{enumerate} \end{claim} \begin{subproof}
Let $P_1$ be a cosegment.
For each $p_2 \in P_2$, the set $P_1 \cup p_2$ contains a circuit.
If this circuit is a triangle, then, by orthogonality, $|P_1|=3$ and $P_1$ is contained in a $4$-element fan, violating the definition of a nice path description.
So the circuit contained in $P_1 \cup p_2$ has size at least~$4$.
In particular, if $P_1$ is a triad, then $P_1 \cup p_2$ is a $4$-element circuit for each $p_2 \in P_2$.
Suppose $P_1$ is a triad and $|P_2| \ge 2$.
Let $P_2 = \{p_2,p_2'\}$.
By \cref{ndttriadseq}, $p_2$ is in a triad $T^*$ that contains an element of $P_1$, and an element of $P_2^+$.
But then $|T^* \cap (P_1 \cup p_2')| = 1$, contradicting orthogonality.
So if $P_1$ is a triad, then $|P_2| = 1$.
Similarly, since $P_m$ is a triad, $|P_{m-1}| = 1$.
Let $P_1$ be a cosegment,
let $p_2 \in P_2$, and let $T^*_2 = \{\ell_2,p_2,r_2\}$ be the internal triad containing $p_2$, with $\ell_2 \in P_1$.
Suppose there is some internal triad $\{\ell_i,p_i,r_i\}$ with $\ell_i \in P_1$, and $p_i \in P_i$ for some guts set $P_i$ with $i > 2$.
Then $r_i \in P_i^+$, so $|\{\ell_i,p_i,r_i\} \cap (P_1 \cup p_2)| = 1$.
By orthogonality, $P_1 \cup p_2$ is not a circuit.
In particular, we deduce that if $P_1$ is a triad, then no such internal triad $\{\ell_i,p_i,r_i\}$ exists, and \cref{ndttriads}\ref{ndttriadsi} follows.
(By symmetry, if an internal triad meets $P_m$ then it contains $P_{m-1}$.)
If $P_1$ is a $4$-cosegment, then we deduce that $(P_1 - \ell_i) \cup p_2$ is a $4$-element circuit.
Now if there is some $e \in P_i-p_i$, then any internal triad containing $e$ does not contain $\ell_i$, by \cref{ndtdisjtriads}, and does not meet $P_1 - \ell_i$, by orthogonality.
So \cref{ndttriads}\ref{ndttriadsii} holds in the case that $P_1$ is a $4$-cosegment.
Finally, suppose $P_1$ is a $5$-element fan with ordering $(f_1,f_2,f_3,f_4,f_5)$ and $P_2$ is a guts set.
Then $\{f_2,f_3,f_4\}$ is a triangle.
For each internal triad $\{\ell_i,p_i,r_i\}$ with $p_i \in P_i$, $\ell_i \in P_i^-$ and $r_i \in P_i^+$, we have $|P_1 \cap \{\ell_i,p_i,r_i\}| \le 1$, so, by orthogonality, either $\ell_i \in \{f_1,f_5\}$ or $\ell_i \notin P_1$.
By \cref{ndtdisjtriads}, at most two internal triads meet $P_1$.
There is an internal triad containing $p_{2}$ that meets $P_1$.
Thus for any guts set $P_i$ with $i > 2$ and $|P_i| =2$, there is some $e \in P_i$ such that any internal triad containing $e$ avoids $P_1$, as required. \end{subproof}
Let $G$ and $Q$ be the guts and coguts elements in $P_2 \cup \dotsm \cup P_{m-1}$, respectively.
\begin{claim}
\label{ndtcogutsupper}
$|Q| \le |G|$.
Moreover,
\begin{enumerate}[label=\rm(\Roman*)]
\item if $P_1$ is a $4$-cosegment, or $M \backslash a,b$ has a triangle, then $|Q| \le |G|-1$; and
\item if $P_1$ is a $4$-cosegment and $M \backslash a,b$ has a triangle, then $|Q| \le |G|-2$.
\end{enumerate} \end{claim} \begin{subproof}
Observe that for each coguts element $q \in P_i$, we have \[q \notin \cl(P_1 \cup \dotsm \cup P_{i-1} \cup (P_i - q)).\]
It follows that $r(M \backslash a,b) = r(P_1) + |Q| + r(P_m) - 2$.
Suppose $P_1$ is a $5$-element fan.
Then, by \cref{atmostonetri}, $M \backslash a,b$ has an $(N,B)$-robust element outside of $\{x,y\}$, so $r(M \backslash a,b) \le r^*(M \backslash a,b)+1$ by \cref{rkcorkbounds}.
By \cref{pathdescends}\ref{pde3}, an element of $P_1$ is $\{U_{2,5},U_{3,5}\}$-deletable if and only if it is a spoke, so $P_1$ has precisely two elements that are $\{U_{2,5},U_{3,5}\}$-deletable.
On the other hand, the triad $P_m$ has precisely one $\{U_{2,5},U_{3,5}\}$-deletable element, also by \cref{pathdescends}\ref{pde2}.
So $M \backslash a,b$ has precisely $|G|+3$ elements that are $\{U_{2,5},U_{3,5}\}$-deletable, by \cref{pathdescprops}, and hence $r^*(M \backslash a,b) = |G|+3$, by \cref{pathdescrank}.
Since $r(P_1) = 4$ and $r(P_m) = 3$,
\begin{align*}
|Q| &= r(M \backslash a,b) + 2 - r(P_1) - r(P_m) \\
&\le r^*(M \backslash a,b) + 3 - 4 - 3\\
&= (|G|+3) -4 = |G|-1,
\end{align*}
as required.
Now, by \cref{ndtends}, we may assume that $P_1$ is a cosegment.
Then $M \backslash a,b$ has $|G| + 2$ elements that are $\{U_{2,5},U_{3,5}\}$-deletable, by \cref{pathdescprops,pathdescends}.
If $M \backslash a,b$ has a triangle, then $r(M \backslash a,b) \le r^*(M \backslash a,b)+1 = |G|+3$ by \cref{atmostonetri,rkcorkbounds}.
Otherwise, by \cref{rkcorkbounds,pathdescrank}, $r(M \backslash a,b) \le r^*(M \backslash a,b)+2 = |G| + 4$.
Observe that $r(P_1)\ge 3$ and $r(P_m)=3$.
In the case that $M \backslash a,b$ does not have a triangle,
\begin{align*}
|Q| &= r(M \backslash a,b) - r(P_1) - r(P_m) + 2 \\
&\le (|G|+4) -r(P_1) -3 + 2 \\
&= |G| -r(P_1) +3 \le |G|,
\end{align*}
and, if $P_1$ is a $4$-cosegment, then $r(P_1) = 4$, in which case $|Q| \le |G|-1$.
Similarly, if $M \backslash a,b$ has a triangle, then
\begin{align*}
|Q| &\le
|G| -r(P_1) +2 \le |G|-1,
\end{align*}
and, if $P_1$ is a $4$-cosegment, then $r(P_1) = 4$, in which case $|Q| \le |G|-2$. \end{subproof}
\begin{claim}
\label{ndt5eltfan}
If $P_1$ is a $5$-element fan, then $P_2$ is a guts set and $|P_2|=1$. \end{claim} \begin{subproof} Let $P_1$ be a $5$-element fan. First, suppose $P_2$ is a coguts set. Then $m$ is even.
Let $P_i$ be a guts set of $\mathbf{P}$ with $i \neq m-1$. If $|P_i| = 1$, then clearly $|P_i| \le |P_{i+1}|$. Otherwise, $|P_i| = 2$, by \cref{ndttriadseq}, in which case, by \cref{ndtdisjtriads,ndttriads}, there are two disjoint internal triads that meet $P_i$ and avoid $P_m$. Since $\mathbf{P}$ is left-justified, $|P_i| = 2 \le |P_{i+1}|$. Finally, observe that $1=|P_{m-1}| \le |P_2|$, by \cref{ndttriads}. Since $m$ is even, it follows that
$|G| \le |Q|$, but this contradicts \cref{ndtcogutsupper}. We deduce that $P_2$ is a guts set.
Now suppose $|P_2| \ge 2$. Then $|P_2| = 2$, by \cref{ndttriadseq}, so let $P_2 = \{e,e'\}$. By \cref{ndtdisjtriads} there are distinct elements $\ell$ and $\ell'$ such that $\{\ell,e\}$ and $\{\ell',e'\}$ are contained in triads where, for each triad, the final element is in $P_2^+$. By orthogonality, $\ell$ and $\ell'$ are the rim ends of the fan $P_1$. Let $(\ell,f_2,f_3,f_4,\ell')$ be an ordering of $P_1$. As $r(\{\ell,f_2,f_4,\ell',e,e'\}) = 4$, and $\{\ell,f_2,f_3\}$ is a triad, $r(\{f_4,\ell',e,e'\}) \le 3$. But $M \backslash a,b$ has at most one triangle, $\{f_2,f_3,f_4\}$, so $\{f_4,\ell',e,e'\}$ is a circuit. This circuit intersects the triad containing $\{\ell,e\}$ in a single element, contradicting orthogonality. We deduce that $|P_2| = 1$. \end{subproof}
By \cref{ndt5eltfan}, we may now assume that $P_2$ is a guts set, so $m$ is odd, and if $P_1$ is not a $4$-cosegment, then $|P_2|=1$.
We may also assume that $m \ge 5$, for otherwise $|E(M)| \le 11$. Suppose $m = 5$. As $|P_1|+|P_2| \le 6$ and $|P_{4}| + |P_5| = 4$, we may assume that $|P_3|=3$ and $|P_1|+|P_2| = 6$. But the latter implies that $P_1$ is either a $5$-element fan or $4$-cosegment, so $|Q| \le |G|-1$ by \cref{ndtcogutsupper}, in which case $|P_3| \le 2$. So $m \ge 7$.
\begin{claim}
\label{ndtendcandidates}
There exist elements $a' \in P_1$ and $b' \in P_m$ such that $M \backslash a,b \backslash a',b'$ is $\{U_{2,5},U_{3,5}\}$-fragile, where each $\{U_{2,5},U_{3,5}\}$-deletable (or $\{U_{2,5},U_{3,5}\}$-contractible) element of $M \backslash a,b$ not in $\{a',b'\}$ remains $\{U_{2,5},U_{3,5}\}$-deletable (or $\{U_{2,5},U_{3,5}\}$-contractible, respectively) in $M \backslash a,b \backslash a',b'$.
Moreover,
\begin{enumerate}[label=\rm(\Roman*)]
\item neither $a'$ nor $b'$ is in an internal triad of $M \backslash a,b$;
\item $a'$ is in a circuit contained in $P_1 \cup p_2$ for each $p_2 \in P_2$, and $P_{m-1} \cup P_m$ is a $4$-element circuit containing $b'$; and
\item if $P_1$ is a $5$-element fan, then $a'$ is a spoke of $P_1$.
\end{enumerate} \end{claim} \begin{subproof}
Let $i \in \{1,m\}$.
When $P_i$ is not a $5$-element fan, then, using \cref{pathdescends}\ref{pde2}, we choose $e_i$ to be the unique element in $P_i$ that is $\{U_{2,5},U_{3,5}\}$-deletable.
When $P_1$ is a $5$-element fan, we choose $e_1$ to be a spoke of $P_1$;
then $e_1$ is $\{U_{2,5},U_{3,5}\}$-deletable, by \cref{pathdescends}\ref{pde3}.
Let $a' = e_1$ and $b'=e_m$.
By \cref{keepfragilelabels}\ref{kfl1}, $M \backslash a,b \backslash a',b'$ is $\{U_{2,5},U_{3,5}\}$-fragile and has no $\{U_{2,5},U_{3,5}\}$-essential elements. Since $M \backslash a,b$ has no $\{U_{2,5},U_{3,5}\}$-flexible elements, each element of $M \backslash a,b \backslash a',b'$ is $\{U_{2,5},U_{3,5}\}$-deletable (or $\{U_{2,5},U_{3,5}\}$-contractible) if and only if it was $\{U_{2,5},U_{3,5}\}$-deletable (or $\{U_{2,5},U_{3,5}\}$-contractible, respectively) in $M \backslash a,b$.
This proves the first part of \cref{ndtendcandidates}.
Suppose $a'$ is in an internal triad $\{a',e,r\}$, where $e$ is the guts element.
Then $M \backslash a,b \backslash a'$ has a $\{U_{2,5},U_{3,5}\}$-minor, and $\{e,r\}$ is a series pair in this matroid, so $e$ is $\{U_{2,5},U_{3,5}\}$-contractible in $M \backslash a,b \backslash a'$ and thus also in $M \backslash a,b$.
But this contradicts \cref{pathdescprops}.
So $a'$ is not in an internal triad.
Similarly, neither is $b'$.
By \cref{ndttriads}, $|P_{m-1}| = 1$, so $P_m \cup P_{m-1}$ is a circuit containing $b'$.
Similarly, if $P_1$ is a triad, then $|P_2| = 1$ and $P_1 \cup P_2$ is a circuit containing $a'$.
If $P_1$ is a $5$-element fan, then there is a unique triangle $T \subseteq P_1$, and $a' \in T$, since $a'$ was chosen to be a spoke of $P_1$.
Finally, suppose that $P_1$ is a $4$-cosegment.
For each $p_2 \in P_2$, the set $P_1 \cup p_2$ contains a circuit.
It remains to show that this circuit contains $a'$.
Clearly this is the case if $P_1 \cup p_2$ is a circuit, so suppose otherwise.
By orthogonality, the circuit is not a triangle, so $(P_1 - a') \cup p_2$ is a $4$-element circuit.
Let $P_1 = \{p_1,p_1',p_1'',a'\}$, so $\{p_1,p_1',p_1'',p_2\}$ is a circuit.
By \cref{pathdescends}\ref{pde3}, $p_1$, $p_1'$, and $p_1''$ are $\{U_{2,5},U_{3,5}\}$-contractible. By \cref{keepfragilelabels}\ref{kfl2}, $M \backslash a,b / p_1$ is $\{U_{2,5},U_{3,5}\}$-fragile and has no $\{U_{2,5},U_{3,5}\}$-essential elements, so $p_1'$ is $\{U_{2,5},U_{3,5}\}$-contractible in $M \backslash a,b / p_1$.
Thus $M \backslash a,b / p_1,p_1'$ has a $\{U_{2,5},U_{3,5}\}$-minor, and $\{p_1'',p_2\}$ is a parallel pair in this matroid.
Thus $p_1''$ is $\{U_{2,5},U_{3,5}\}$-deletable in $M \backslash a,b / p_1,p_1'$, and hence in $M \backslash a,b$, a contradiction.
We deduce that, for each $p_2 \in P_2$, there is a circuit contained in $P_1 \cup p_2$ that contains $a'$, as required.
\end{subproof}
We work towards applying \cref{ndtcrux} using $a'$ and $b'$ as given in \cref{ndtendcandidates}.
First we require the following.
\begin{claim}
\label{localcoconn}
$\sqcap^*_{M \backslash a,b}(P_1,P_m) = 0$. \end{claim} \begin{subproof}
Suppose $\sqcap^*_{M \backslash a,b}(P_1,P_m) \ge 1$.
Let $P_i$ be a guts set for some $i \in \{2,3,\dotsc,m-1\}$.
Then, by the duals of \cref{growpi,pflancoguts}, $\sqcap^*_{M \backslash a,b}(P_i^-,P_i^+) = 1$, so $|P_i|=1$.
So every guts set has size one.
First, assume that $P_1$ is a cosegment.
Then, by \cref{ndtcogutsupper}, $|Q| \le |G|$.
Since each guts set has size one, and the number of guts sets is one more than the number of coguts sets, there is at most one coguts set of size two.
Recall that $m \ge 7$, so there exists a coguts set $P_j$ with $|P_j|=1$, for some $\{j,j'\} = \{3,5\}$.
By the dual of \cref{pflancoguts}, $P_{j-1} \cup P_j \cup P_{j+1}$ is a triangle.
But this implies that $|Q| \le |G|-1$ by \cref{ndtcogutsupper}, so every coguts set has size one.
In particular, $|P_{j'}| = 1$, so $P_{j'-1} \cup P_{j'} \cup P_{j'+1}$ is also a triangle, contradicting that $M \backslash a,b$ has at most one triangle.
Now assume $P_1$ is a $5$-element fan.
By \cref{ndtcogutsupper}, $|Q| \le |G|-1$, so every guts and coguts set has size one.
By the dual of \cref{pflancoguts}, $P_{2} \cup P_3 \cup P_{4}$ is a triangle, so $|Q| \le |G|-2$, by \cref{ndtcogutsupper}, a contradiction. \end{subproof}
\begin{claim}
\label{ndtgoodtrip}
Let $a'$ and $b'$ be as given in \cref{ndtendcandidates}, and let $c'$ be a guts element in $P_k$ for some $k \in \{4,6,\dotsc,m-3\}$.
Suppose $C^*$ is a cocircuit of $M \backslash a,b$.
\begin{enumerate}
\item $C^* \nsubseteq \{a',b',c'\}$.\label{ndtgt1}
\item If $\{a',c',b'\} \subseteq C^*$ and $|C^*|=5$, then $C^* = \{a'',a',c',b',b''\}$ for some $a'' \in P_1-a'$ and $b'' \in P_m-b'$.\label{ndtgt2}
\item If $\{a',c'\} \subseteq C^*$ and $|C^*|=4$, then $C^* = \{a'',a',c',r\}$ for some $a'' \in P_1-a'$ and $r \in P_k^+$.\label{ndtgt3}
\item If $\{c',b'\} \subseteq C^*$ and $|C^*|=4$, then $C^* = \{\ell,c',b',b''\}$ for some $\ell \in P_k^-$ and $b'' \in P_m-b'$.\label{ndtgt4}
\item If $\{a',b'\} \subseteq C^*$, then $|C^*| \neq 4$.\label{ndtgt5}
\end{enumerate} \end{claim} \begin{subproof}
Since $P_1 \cup P_2$ contains a circuit containing $a'$, by \cref{ndtendcandidates}, any cocircuit of $M \backslash a,b$ containing $a'$ meets $(P_1 -a')\cup P_2$, by orthogonality.
Similarly, $P_{m-1} \cup P_m$ contains a circuit containing $b'$, so any cocircuit of $M \backslash a,b$ containing $b'$ meets $P_{m-1} \cup (P_m - b')$.
It follows that $\{a',b',c'\}$ is coindependent in $M \backslash a,b$, thus proving \ref{ndtgt1}.
Observe that $M \backslash \{a,b,a',b',c'\}$ has a $\{U_{2,5},U_{3,5}\}$-minor, by \cref{ndtendcandidates} and since $c'$ is $\{U_{2,5},U_{3,5}\}$-deletable in $M \backslash a,b$.
Suppose $\{a',b',c'\}$ is contained in a $5$-element cocircuit $\{a',b',c',a'',b''\}$.
Since $M \backslash \{a,b,a',b',c'\}$ has a $\{U_{2,5},U_{3,5}\}$-minor and $\{a'',b''\}$ is a series pair in this matroid, $a''$ and $b''$ are $\{U_{2,5},U_{3,5}\}$-contractible in $M \backslash a,b$.
By \cref{pathdescprops}, $a''$ and $b''$ are not guts elements, so $a'' \in P_1-a$ and $b'' \in P_m-b'$, thus proving \ref{ndtgt2}.
Similarly, if some pair of elements in $\{a',b',c'\}$ is contained in a $4$-element cocircuit, then the other two elements in this cocircuit are $\{U_{2,5},U_{3,5}\}$-contractible in $M \backslash a,b$.
Cases~\ref{ndtgt3} and~\ref{ndtgt4} of the claim then follow from orthogonality.
For case~\ref{ndtgt5}, if $\{a',b'\}$ is contained in a $4$-element cocircuit $\{a'',a',b',b''\}$ say,
then $a'' \in P_1-a'$ and $b'' \in P_m-b'$, by orthogonality.
But this contradicts \cref{localcoconn} and the dual of \cref{picircuits}.
\end{subproof}
\begin{claim}
\label{ndtgoodtripsupplement}
Let $a'$ and $b'$ be as given in \cref{ndtendcandidates}, and let $c'$ be a guts element in $P_i$, for some $i \in \{4,6,\dotsc,m-3\}$, such that
\begin{enumerate}[label=\rm(\Roman*)]
\item neither $\{a',c'\}$ nor $\{c',b'\}$ is contained in a $4$-element cocircuit of $M \backslash a,b$, and\label{ndtgts1}
\item there is a unique triad containing $c'$, and this triad avoids $P_1 \cup P_m$.\label{ndtgts2}
\end{enumerate}
Then $\{a',b',c'\}$ is a delete triple. \end{claim} \begin{subproof}
Observe that $M \backslash \{a,b,a',b',c'\}$ has a $\{U_{2,5},U_{3,5}\}$-minor, by \cref{ndtendcandidates} and since $c'$ is $\{U_{2,5},U_{3,5}\}$-deletable in $M \backslash a,b$; hence this matroid is $\{U_{2,5},U_{3,5}\}$-fragile.
By \cref{genfragileconn}, it follows that $M\backslash \{a,b,a',b',c'\}$ is $3$-connected up to series classes.
We work towards an application of \cref{ndtcrux}.
First, observe that in $M \backslash a,b$, the set $\{a',b',c'\}$ is coindependent by \cref{ndtgoodtrip}\ref{ndtgt1}, and $\{a',b'\}$ is not contained in a $4$-element cocircuit by \cref{ndtgoodtrip}\ref{ndtgt5}.
Suppose $\{a',b',c'\}$ is contained in a $5$-element cocircuit of $M \backslash a,b$.
Then, by \cref{ndtgoodtrip}\ref{ndtgt2}, this cocircuit is $\{a'',a',c',b',b''\}$ for some $a'' \in P_1 - a'$ and $b'' \in P_m$.
Let $\{\ell,c',r\}$ be an internal triad containing $c'$, with $\ell \in P_i^-$ and $r \in P_i^+$.
Then, by \ref{ndtgts2}, $\ell \notin P_1$, and $r \notin P_m$.
By cocircuit elimination, there is a cocircuit~$C^*$ contained in $\{a'',a',\ell,r,b',b''\}$.
By the dual of \cref{pflancoguts}, $\sqcap^*_{M \backslash a,b}(P_i^-,P_i^+)=0$.
But $C^* \subseteq P_i^- \cup P_i^+$, so, by the dual of \cref{picircuits}, either $C^* \subseteq P_i^-$ or $C^* \subseteq P_i^+$.
Thus either $\{a'',a',\ell\}$ or $\{r,b',b''\}$ is a triad.
But $\ell \notin P_1$ and $r \notin P_m$, so this contradicts that $P_1$ and $P_m$ are ends of a nice path description.
So $\{a',b',c'\}$ is not contained in a $5$-element cocircuit.
It remains only to show that $M \backslash \{a,b,a',b',c'\}$ has at least three non-trivial series classes and
is $3$-connected up to series classes of size at most three.
Suppose $S'$ is a series pair of $M \backslash \{a,b,a',b',c'\}$.
Then $S' \cup \{a',b',c'\}$ contains a cocircuit $C^*$ in $M \backslash a,b$.
By \cref{ndtgoodtrip}, either $C^*=5$ and $\{a',b',c'\} \subseteq C^*$, or $C^*=4$ with $c' \in C^*$ and $C^* \cap \{a',b'\} = 1$, or $|C^*|=3$ and $|C^* \cap \{a',b',c'\}| = 1$.
But by the foregoing, and \ref{ndtgts2}, only the latter is possible; that is, every element in a non-trivial series class of $M \backslash \{a,b,a',b',c'\}$ is in a triad of $M \backslash a,b$ that contains one of $a'$, $b'$, or $c'$.
Let $S_a$, $S_b$ and $S_c$ be the set of elements in $M\backslash a,b$ that are in a triad with $a'$, $b'$, and $c'$, respectively.
Then, each of $S_a$, $S_b$ and $S_c$ is contained in a series class of $M \backslash \{a,b,a',b',c'\}$, and each element in a non-trivial series class of $M \backslash \{a,b,a',b',c'\}$ is in $S_a \cup S_b \cup S_c$.
Observe that $a'$, $b'$, and $c'$ are each in at least one triad,
so the sets $S_a$, $S_b$, and $S_c$ are non-empty.
We claim that these three sets have size at most three, and are pairwise disjoint.
Suppose $P_1$ is a cosegment, so $P_1-a' \subseteq S_a$.
Since $a'$ is in a circuit contained in $P_1 \cup p_2$ for any $p_2 \in P_2$, any triad containing $a'$ is either contained in $P_1$, or is an internal triad containing a guts element $p_2 \in P_2$, by orthogonality.
But $a'$ is not in an internal triad, so $S_a = P_1-a'$ when $P_1$ is a cosegment.
Similarly, $S_b = P_m-b'$.
Now suppose $P_1$ is a $5$-element fan.
We may assume that $(f_1,f_2,f_3,a',f_5)$ is a fan ordering of $P_1$, where $\{f_2,f_3,a'\}$ is a triangle.
Suppose $a'$ is in a triad that also contains some $z \notin P_1$.
Then, by orthogonality, this triad is $\{a',z,f_2\}$. But then $(M \backslash a,b)^*|(P_1 \cup z) \cong M(K_4)$, contradicting \cref{noMK4}.
So $S_a = \{f_3,f_5\} \subseteq P_1$ when $P_1$ is a $5$-element fan.
Now $|S_a| \le 3$ and $|S_b| = 2$.
By \ref{ndtgts2}, $|S_c| = 2$ and $S_c \cap (P_1 \cap P_m) = \emptyset$, so the sets $S_a$, $S_b$, and $S_c$ are pairwise disjoint.
It remains to show that the series classes of $M \backslash \{a,b,a',b',c'\}$ containing $S_a$, $S_b$, and $S_c$ are distinct.
We first show that $c' \notin \cl^*_{M \backslash a,b,a',b'}(S_a \cup S_b)$.
Suppose that $c' \in \cl^*_{M \backslash a,b,a',b'}(S_a \cup S_b)$.
Then $c'$ is in a cocircuit~$D_1$ of $M \backslash a,b$ contained in $S_a \cup S_b \cup \{a',b',c'\}$.
Note that $r^*_{M \backslash a,b}(S_a \cup S_b \cup \{a',b'\}) \le 4$, so $|D_1| \le 5$.
If $D_1$ contains at most two elements in $S_a \cup S_b$, then $S'=D_1-\{a',b',c'\}$ is a series pair in $M \backslash \{a,b,a',b',c'\}$, in which case $S' \cup c'$ is a triad.
But then $S' \subseteq S_c$, a contradiction.
So $|D_1 \cap (S_a \cup S_b)| \in \{3,4\}$.
Since $S_a \cup a' \subseteq P_1$ and $S_b \cup b' \subseteq P_m$, and $c'$ is a guts element,
$D_1 \cap P_m \neq \emptyset$.
By orthogonality, $|D_1 \cap P_m| \neq 1$, so $|D_1 \cap P_m| = 2$.
We claim that there is a cocircuit $D_2$ with $|D_2| \in \{4,5\}$ and $\{c',b'\} \subseteq D_2 \subseteq P_1 \cup \{a',c',b'\} \cup P_m$.
If $b' \in D_1$, then we can just let $D_2 = D_1$; so suppose that $b' \notin D_1$.
Let $s_b \in S_b$.
By cocircuit elimination, there is a cocircuit $D_2$ contained in $(D_1 \cup b') - s_b$.
By \cref{localcoconn}, this cocircuit contains $c'$.
Thus, arguing as for $D_1$, we have that $|D_2 \cap P_m| \ge 2$.
As $c'$ is a guts element, $D_2 \cap P_1 \neq \emptyset$.
Now $D_2$ has the claimed properties; in particular, $|D_2| \in \{4,5\}$.
By \ref{ndtgts1}, $|D_2| = 5$.
Then $a' \notin D_2$, since no $5$-element cocircuit contains $\{a',c',b'\}$.
Let $s_a \in S_a$.
By cocircuit elimination, there is a cocircuit $D_3$ contained in $(D_2 - a') \cup s_a$.
Arguing as before, $|D_3 \cap P_1|=|D_3 \cap P_m| \ge 2$, so $D_3 = D_2 \triangle \{a',s_1\}$.
Thus $\{a',b',c'\}$ is contained in a $5$-element cocircuit, a contradiction.
This proves that $c' \notin \cl^*_{M \backslash a,b,a',b'}(S_a \cup S_b)$.
By \cref{localcoconn}, $r^*_{M \backslash a,b}(S_a \cup S_b \cup \{a',b'\})=4$, so $r^*_{M \backslash a,b,a',b'}(S_a \cup S_b)=2$.
As $c' \notin \cl^*_{M \backslash a,b,a',b'}(S_a \cup S_b)$, we have $r^*_{M \backslash a,b,a',b',c'}(S_a \cup S_b) = 2$, so the series classes containing $S_a$ and $S_b$ are distinct.
Now we claim that the series classes containing $S_a$ and $S_c$ are distinct.
Pick $s_a \in S_a$ and $s_c \in S_c$, and observe that $r^*_{M \backslash a,b}(S_a \cup S_c \cup \{a',c'\}) = r^*_{M \backslash a,b}(\{a',s_a,c',s_c\})$.
Suppose $r^*_{M \backslash a,b}(S_a \cup S_c \cup \{a',c'\}) \le 3$.
Then $\{a',s_a,c',s_c\}$ is dependent in $(M \backslash a,b)^*$.
But $\{a',s_a,c',s_c\}$ does not contain a triad of $M \backslash a,b$, so $\{a',s_a,c',s_c\}$ is a cocircuit, contradicting \ref{ndtgts1}.
Thus $r^*_{M \backslash a,b}(S_a \cup S_c \cup \{a',c'\}) = 4$.
Suppose $b' \in \cl^*_{M \backslash a,b,a',c'}(S_a \cup S_c)$.
Then there is a cocircuit contained in $S_a \cup S_c \cup \{a',b',c'\}$ and containing $b'$.
But this cocircuit intersects the circuit $P_{m-1} \cup P_m$ in a single element, a contradiction.
So $b' \notin \cl^*_{M \backslash a,b,a',c'}(S_a \cup S_c)$.
Now $r^*_{M \backslash a,b,a',b',c'}(S_a \cup S_c) = 2$, so the series classes of $M \backslash \{a,b,a',b',c'\}$ containing $S_a$ and $S_c$ are distinct.
By a similar argument, the series classes containing $S_c$ and $S_b$ are distinct. This completes the proof. \end{subproof}
Next, we argue that each guts set of $\mathbf{P}=(P_1,P_2,\dotsc,P_m)$ has size one, except perhaps $P_2$ when $P_1$ is a $4$-cosegment.
\begin{claim}
\label{nodoubleguts}
Let $P_i$ be a guts set for some $i \in \{2, \dotsc, m-1\}$ such that if $P_1$ is a $4$-cosegment then $i \neq 2$.
Then $|P_i| = 1$. \end{claim} \begin{subproof}
Recall that $|P_{m-1}| = 1$ and if $P_1$ is not a $4$-cosegment then $|P_2| = 1$, so \cref{nodoubleguts} holds when $i = m-1$ or $i=2$.
So we may assume that $3 \le i < m-2$.
Let $a'$ and $b'$ be as given in \cref{ndtendcandidates}.
Let $P_i = \{c',c''\}$ such that each triad containing $c'$ is disjoint from $P_1$, where such a $c' \in P_i$ exists by \cref{ndttriads}\ref{ndttriadsii}.
Towards an application of \cref{ndtgoodtripsupplement}, it remains to show that there is a unique triad containing $c'$, which avoids $P_1 \cup P_m$, and neither $\{a',c'\}$ nor $\{c',b'\}$ is contained in a $4$-element cocircuit of $M \backslash a,b$.
Suppose $T_1^*$ and $T_2^*$ are distinct triads containing $c'$.
Then, by \cref{ndttriadseq,ndtdisjtriads}, $T_1^* = \{\ell_1,c',r_1\}$ and $T_2^* = \{\ell_2,c',r_2\}$ for distinct $\ell_1,\ell_2 \in P_i^-$ and distinct $r_1,r_2 \in P_i^+$.
By cocircuit elimination, there is a cocircuit contained in $\{\ell_1,\ell_2,r_1,r_2\}$, which has size at least~$3$, since $M \backslash a,b$ is $3$-connected. But this contradicts the dual of \cref{picircuits}.
We deduce that there is a unique triad containing $c'$.
Suppose $\{a',c'\}$ is contained in a $4$-element cocircuit.
Then this cocircuit is $\{a',a'',c',r'\}$ where $a'' \in P_1 - a'$ and $r' \in P_i^+$, by \cref{ndtgoodtrip}\ref{ndtgt3}.
Let $\{\ell,c',r\}$ be the internal triad containing $c'$, with $\ell \in P_i^- - P_1$ and $r \in P_i^+ - P_m$.
Then, by cocircuit elimination, there is a cocircuit~$C^*$ contained in $\{a',a'',\ell,r,r'\}$.
By the dual of \cref{pflancoguts}, $\sqcap^*_{M \backslash a,b}(P_i^-,P_i^+)=0$, so, by the dual of \cref{picircuits}, either $C^* \subseteq P_i^-$ or $C^* \subseteq P_i^+$.
Thus $\{a'',a',\ell\}$ is a triad.
But then $P_1 \cup \ell$ is a $4$-cosegment, contradicting that $P_1$ is an end of a nice path description.
By a symmetric argument, $\{c',b'\}$ is not contained in a $4$-element cocircuit.
Now, by \cref{ndtgoodtripsupplement}, $M$ has a delete triple, a contradiction.
This proves \cref{nodoubleguts}. \end{subproof}
\begin{claim}
\label{onedoublecoguts}
At most one coguts set of $\mathbf{P}$ has size more than one, and if a coguts set of size more than one exists, then it has size two.
Moreover,
\begin{enumerate}
\item if $P_1$ is a $4$-cosegment, then $|P_2|=|P_3|$ and $M \backslash a,b$ has no triangles; and\label{odcg1}
\item if $M \backslash a,b$ has a triangle, then $|P_i|=1$ for each $i \in \{2,3,\dotsc,m-1\}$.\label{odcg2}
\end{enumerate} \end{claim} \begin{subproof}
Suppose $P_1$ is not a $4$-cosegment.
By \cref{nodoubleguts}, every guts set has size one.
By \cref{ndtcogutsupper}, $|Q| \le |G|$.
So either every coguts set has size one, in which case $|Q| =|G|-1$,
or all but one coguts set has size one, and this coguts set has size two, in which case $|Q| =|G|$.
If $M \backslash a,b$ has a triangle,
then $|Q| \le|G|-1$ by \cref{ndtcogutsupper}, so every coguts set
also
has size one.
Now suppose $P_1$ is a $4$-cosegment.
Then $|Q| \le|G|-1$, by \cref{ndtcogutsupper}.
Again by \cref{nodoubleguts}, every guts set except perhaps $P_2$ has size one.
Thus, if $|P_2| = 1$, then every coguts set has size one and $|Q| = |G|-1$; in particular, $|P_2|=|P_3|=1$ and $M \backslash a,b$ has no triangles, the latter by \cref{ndtcogutsupper}.
On the other hand, if $|P_2| = 2$, then at most one coguts set has size two; by \cref{ndttriadseq,ndtdisjtriads} and since $\mathbf{P}$ is left-justified,
we have $|P_3|=2$,
so again $|Q| = |G|-1$, and $M \backslash a,b$ has no triangles by \cref{ndtcogutsupper}. \end{subproof}
By \cref{onedoublecoguts}, there is at most one coguts set with size two. If such a coguts set exists, let $j \in \{3,5,\dotsc,m-2\}$ such that $|P_j|=2$; otherwise, let $j=0$.
We work towards applying \cref{ndtgoodtripsupplement}, first when $P_1$ is a triad, and then when it is not. First we prove one more claim that holds in either case.
\begin{claim}
\label{ndtinternaltris}
Let $p_k \in P_k$ be a guts element, for some $k \in \{2,4,\dotsc,m-1\}$, and let $T^*$ be an internal triad containing $p_k$.
Then $T^* = \{\ell_k,p_k,r_k\}$ for some $\ell_k \in P_k^-$ and $r_k \in P_{k+1}$.
Moreover, if
\begin{enumerate}[label=\rm(\Roman*)]
\item $P_1$ is a triad, and $j=0$ or $k \le j+1$;\label{ndtit1}
\item $P_1$ is a $4$-cosegment, $|P_2|=|P_3|=2$, and $k \in \{2,4\}$; or\label{ndtit2}
\item $P_1$ is a $5$-element fan;\label{ndtit3}
\end{enumerate}
then $\ell_k \in P_{k-1}$. \end{claim} \begin{subproof}
By \cref{ndttriadseq}, we may assume that $T^* = \{\ell_k,p_k,r_k\}$ for some $\ell_k \in P_k^-$ and $r_k \in P_k^+$.
Since $\mathbf{P}$ is left-justified, either $r_k \in P_{k+1}$ or $r_k \in P_m$.
If $k < m-1$, then $r_k \notin P_m$ by \cref{ndttriads}.
Thus $r_k \in P_{k+1}$ for each even $k$.
We first consider when \ref{ndtit3} holds.
Let $P_1$ be a $5$-element fan $(f_1,f_2,f_3,f_4,f_5)$ and let $\{\ell_{i},p_{i},r_{i}\}$ is an internal triad for each guts element~$p_{i}$.
Suppose $\ell_t \in P_1$ for some even $t \ge 4$.
By orthogonality and \cref{ndtdisjtriads}, we may assume that $f_1 = \ell_2$ and $f_5 = \ell_t$.
By \cref{onedoublecoguts}\ref{odcg2}, we have $P_{i} = \{p_{i}\}$ for each $i \in \{2,3,\dotsc,m-1\}$.
Observe that $r^*_{M \backslash a,b}(P_1 \cup p_2) = 4$ and $r^*_{M \backslash a,b}(P_2^+) = r(M \backslash a,b)-2$.
Now $f_5 \in \cl^*_{M \backslash a,b}(P_2^+)$, so $\cl^*_{M \backslash a,b}(P_2^+ \cup \{f_4,f_5\})$ is contained in a cohyperplane. As $f_3 \in \cl^*_{M \backslash a,b}(P_2^+ \cup \{f_4,f_5\})$, the set $\{f_1,f_2,p_2\}$ is a triangle.
But then $P_1 \cup p_2$ is a $6$-element fan, contradicting that $P_1$ is an end of a nice path description.
Now suppose \ref{ndtit1} or \ref{ndtit2} holds. It remains to show that $\ell_k \in P_{k-1}$.
This is clear if $k = 2$.
Suppose $P_1$ is a $4$-cosegment, $|P_2|=2$, and $k=4$.
By the dual of \cref{pflancoguts}, $\sqcap^*_{M \backslash a,b}(P_1,P_2^+) = 0$.
Thus, by the dual of \cref{picircuits}, $\ell_4 \in P_2^+$, so $\ell_4 \in P_3$.
Now we may assume that $P_1$ is a triad and $j=0$ or $k \le j+1$.
Let $i$ be even, with $2 < i \le k$, and suppose for all $i'$ such that $2 \le i' < i$, we have $\ell_{i'} \in P_{i'-1}$.
Now $\ell_{i} \notin P_1$, by \cref{ndttriads}\ref{ndttriadsi}.
Observe, for each even $i'$ with $2 < i' < i$, we have $i' \le j-1$, since $i' < i \le k \le j+1$ where $i'$ and $k$ are even.
So, for such an $i'$, we have $P_{i'-1}=\{\ell_{i'}\}$, and thus $\ell_{i} \notin P_{i'-1}$, by \cref{ndtdisjtriads}.
By orthogonality, $\ell_{i}$ is not a guts element, so $\ell_{i} \in P_{i-1}$.
The claim follows by induction. \end{subproof}
\begin{claim}
\label{ndtendgame}
Suppose $P_1$ is a triad.
Then \labelcref{ndtendgameoutcomes16}\labelcref{ndtendgameoutcome16a} holds. \end{claim} \begin{subproof}
Recall that $\mathbf{P}=(P_1,P_2,\dotsc,P_m)$ is a nice path description of $M \backslash a,b$ with $m$ odd, where $P_2$ and $P_{m-1}$ are guts sets,
and $|P_i|=1$ for every $i \in \{2,3,\dotsc,m-1\} - j$.
Let $P_i = \{p_i\}$ for all $i \in \{2,3,\dotsc,m-1\} - j$ and, if $j \neq 0$, let $P_j = \{p_j,p_j'\}$.
Recall also that $M \backslash a,b$ has at most one triangle.
If $M \backslash a,b$ has a triangle, then, by \cref{onedoublecoguts}\ref{odcg2}, $j=0$, and, up to replacing $(P_1,P_2,\dotsc,P_m)$ with its reversal, $\{p_{m-3},p_{m-2},p_{m-1}\}$ is not a triangle.
So we may assume that $\{p_{m-3},p_{m-2},p_{m-1}\}$ is independent.
Since $13 \le |E(M \backslash a,b)| \le m+5$, and $m$ is odd, we have $m \ge 9$.
We distinguish the following cases:
\begin{enumerate}[label=\rm(\Roman*)]
\item $m=9$, and $j=5$.\label{ndteg1}
\item $m \ge 11$ and $j=5$.\label{ndteg2}
\item $m=9$ and $j=7$.\label{ndteg3}
\item None of \ref{ndteg1} to \ref{ndteg3} holds; that is, $j \notin \{5,7\}$, or $m \ge 11$ and $j =7$.\label{ndteg4}
\end{enumerate}
Note that $M \backslash a,b$ has no triangles in cases~\ref{ndteg1} to \ref{ndteg3}; in the case that $M \backslash a,b$ has a triangle, case~\ref{ndteg4} holds.
We first handle cases~\ref{ndteg2} to \ref{ndteg4}, before returning to case~\ref{ndteg1}.
Let $a'$ and $b'$ be as given in \cref{ndtendcandidates}.
In cases~\ref{ndteg2} and \ref{ndteg3} we let $c' = p_6$; in case~\ref{ndteg4} we let $c'=p_4$; whereas in case~\ref{ndteg1}, $c' \in \{p_4,p_6\}$ as appropriate.
Choose $k \in \{4,6\}$ so that $c' = p_k$.
We work towards an application of \cref{ndtgoodtripsupplement} with the elements $a',b',c'$; it remains to show that neither $\{a',c'\}$ nor $\{c',b'\}$ is contained in a $4$-element cocircuit of $M \backslash a,b$, and there is a unique triad containing $c'$, which avoids $P_1 \cup P_m$.
Suppose case~\ref{ndteg3} holds, so $m=9$, $j=7$, and $k=6$.
Note that, by \cref{ndtdisjtriads,ndtinternaltris}, $\{p_5,c',p_7\}$ is the unique triad containing $c'$, up to swapping the labels on $p_7$ and $p_7'$.
As $M \backslash a,b$ has no triangles, $\{p_2,p_3,p_4\}$ is independent, so $\sqcap^*_{M \backslash a,b}(P_1,P_4^+) = 0$ by the dual of \cref{pflantriad}.
By \cref{ndtgoodtrip}\ref{ndtgt3}, if there is a $4$-element cocircuit containing $\{a',c'\}$, then it avoids $\{p_2,p_3,p_4\}$; hence, by the dual of \cref{picircuits}, no such cocircuit exists.
Suppose $\{c',b'\}$ is contained in a $4$-element cocircuit.
By \cref{ndtgoodtrip}\ref{ndtgt4} and orthogonality, this cocircuit is $\{\ell,c',b',b''\}$ for some $b'' \in P_m -b'$ and $\ell \in \{p_3,p_5\}$.
If $\ell = p_3$, then, by cocircuit elimination with the triad $\{p_3,p_4,p_5\}$ there is also a cocircuit contained in $\{p_4,p_5,c',b',b''\}$, which (again by orthogonality) does not contain $p_4$.
So we may assume that $\ell=p_5$.
Recall that $\{p_5,c',p_7\}$ is a triad.
By cocircuit elimination, $\{c',p_7,b',b''\}$ contains a cocircuit.
As $c'$ is a guts element, $c' \notin \cl^*_{M \backslash a,b}(\{p_7,b',b''\})$.
Thus $\{p_7,b',b''\}$ is a triad.
But $p_7 \notin \cl^*_{M \backslash a,b}(P_m)$, since $P_m$ is an end of a nice path description, so this is contradictory.
So $\{c',b'\}$ is not contained in a $4$-element cocircuit.
Thus, by \cref{ndtgoodtripsupplement}, $M$ has a delete triple, a contradiction.
Now assume we are in case~\ref{ndteg2} or~\ref{ndteg4}.
Observe that $j=0$ or $k \le j+1$ in either case, so, by \cref{ndtdisjtriads,ndtinternaltris}, there is a unique triad containing $c'$, which we may assume is $\{p_{k-1},c',p_{k+1}\}$, up to switching the labels on $p_{k-1}$ and $p_{k-1}'$ when $j=k-1$.
We claim that $\{a',c'\}$ is not contained in a $4$-element cocircuit.
Towards a contradiction, suppose $\{a',c'\}$ is contained in a $4$-element cocircuit~$C^*$.
Then $C^* = \{a'',a',c',r\}$ with $a'' \in P_1 - a'$ and $r \in P_k^+$, by \cref{ndtgoodtrip}\ref{ndtgt3}.
Since $\mathbf{P}$ is left-justified, either $r \in P_{k+1}$ or $r \in P_m$.
But if $r \in P_m$, then $C^*$ intersects the circuit $P_m \cup p_{m-1}$ in a single element, contradicting orthogonality.
So $r \in P_{k+1}$.
Note that, in either case, $j \neq k+1$, so $|P_{k+1}|=1$ and $p_{k+1} = r$.
Recall that $\{p_{k-1},c',p_{k+1}\}$ is a triad.
By cocircuit elimination with $C^*$, there is a cocircuit contained in $\{a'',a',p_{k-1},c'\}$.
But $c' \notin \cl^*_{M \backslash a,b}(\{a'',a',p_{k-1}\})$, since $\{a'',a',p_{k-1}\} \subseteq P_k^-$ and $P_k=\{c'\}$ is a guts set, so $\{a'',a',p_{k-1}\}$ is a triad of $M \backslash a,b$.
Since $a'' \in P_1 - a'$,
the set $P_1 \cup p_{k-1}$ is a $4$-cosegment, contradicting that $P_1$ is an end of a nice path description.
We deduce that $\{a',c'\}$ is not contained in a $4$-element cocircuit.
Suppose $\{c',b'\}$ is contained in a $4$-element cocircuit~$C^*$. Then $C^*=\{c'',c',b',b''\}$ for some $b'' \in P_m -b'$ and $c'' \in P_k^-$, by \cref{ndtgoodtrip}\ref{ndtgt4}.
By the dual of \cref{picircuits}, the existence of $C^*$ implies that $\sqcap^*_{M \backslash a,b}(P_{k+1}^-,P_m) \ge 1$.
Recall that $\{p_{m-3},p_{m-2},p_{m-1}\}$ is not a triangle, so $\sqcap^*_{M \backslash a,b}(P_{m-3}^-,P_m) =0$, by the dual of \cref{pflantriad}.
But then, as $k \le m-5$ and by the dual of \cref{growpi}, $\sqcap^*_{M \backslash a,b}(P_{k+1}^-,P_m) \le \sqcap^*_{M \backslash a,b}(P_{m-3}^-,P_m) =0$, a contradiction.
We deduce that $\{c',b'\}$ is not contained in a $4$-element cocircuit.
Now, by \cref{ndtgoodtripsupplement}, $M$ has a delete triple, a contradiction.
It remains only to consider case~\ref{ndteg1}, where $m=9$, $j=5$ and $M \backslash a,b$ has no triangles.
Let $\mathbf{P}' = (P_1',P_2'\dotsc,P_m')$ be the (left-justified) reversal of $\mathbf{P} = (P_1,P_2,\dotsc,P_m)$, where $P_1' = P_m$ and $P_1 = P_m'$.
We may assume $|P_5'|=2$, for otherwise case~\ref{ndteg4} applies for $\mathbf{P}'$.
Thus $\mathbf{P}$ is both left- and right-justified; in particular, $\{p_1,p_2,p_3\}$ and $\{p_7,p_8,p_9\}$ are triads for some $p_1 \in P_1$ and $p_9 \in P_9$.
Up to swapping the labels on $p_5$ and $p_5'$, we may assume that $\{p_3,p_4,p_5\}$ is a triad, and this is the unique triad containing $p_4$, by \cref{ndtdisjtriads,ndtinternaltris}.
Let $q \in \{p_5,p_5'\}$ such that $\{q,p_6,p_7\}$ is a triad, and note that this is the unique triad containing $p_6$, by \cref{ndtdisjtriads,ndtinternaltris}.
Since $\sqcap^*_{M \backslash a,b}(P_1,P_4^+) = 0$ and $\sqcap^*_{M \backslash a,b}(P_6^-,P_9) = 0$, by the dual of \cref{pflantriad}, it follows from \cref{picircuits,ndtgoodtrip} that there are no $4$-element cocircuits containing $\{a',p_6\}$ or $\{p_4,b'\}$.
Thus, if $\{a',p_4\}$ is not contained in a $4$-element cocircuit, or $\{p_6,b'\}$ is not contained in a $4$-element cocircuit, then we can apply \cref{ndtgoodtripsupplement}, with $c' = p_4$ or $c' = p_6$ respectively, to deduce that $M$ has a contradictory delete triple.
So we may assume that $\{a',p_4\}$ and $\{p_6,b'\}$ are contained in $4$-element cocircuits.
Let the former cocircuit be $\{a'',a',p_4,r\}$.
Then $a'' \in P_1-a'$ and $r \in \{p_5,p_5'\}$, due to the left-justification of $\mathbf{P}$ and \cref{ndtgoodtrip}\ref{ndtgt3}.
If $r = p_5$, then, by cocircuit elimination with the triad $\{p_3,p_4,p_5\}$, the set $\{a'',a',p_3,p_4\}$ contains a cocircuit.
Since $p_4$ is a guts element, $\{a'',a',p_3\}$ is a triad, a contradiction.
So $r = p_5'$.
Now $\{a'',a',p_4,p_5'\}$ is a cocircuit.
By cocircuit elimination with the triad $P_1$, the set $\{p_1',p_1'',p_4,p_5'\}$ is a cocircuit for any pair $\{p_1',p_1''\} \subseteq P_1$.
By a symmetric argument, after letting $P_5 = \{q,q'\}$ such that the internal triad containing $p_6$ is $\{q,p_6,p_7\}$, the set $\{q',p_6,p_9',p_9''\}$ is a cocircuit for any pair $\{p_9',p_9''\} \subseteq P_9$.
So \labelcref{ndtendgameoutcomes16}\labelcref{ndtendgameoutcome16a} holds, thus completing the proof of \cref{ndtendgame}. \end{subproof}
Finally, we handle the case where one end of $\mathbf{P}$ is a $4$-cosegment or a $5$-element fan.
\begin{claim}
\label{ndtendgame2}
If $P_1$ is not a triad, then \labelcref{ndtendgameoutcomes16}\labelcref{ndtendgameoutcome16b} holds. \end{claim} \begin{subproof}
Suppose $P_1$ is not a triad, so $P_1$ is a $5$-element fan or a $4$-cosegment.
Recall that $\mathbf{P}=(P_1,P_2,\dotsc,P_m)$ is a nice path description of $M \backslash a,b$ with $m$ odd, where $P_2$ and $P_{m-1}$ are guts sets, and $m \ge 7$.
By \cref{nodoubleguts,onedoublecoguts}, if $P_1$ is a $5$-element fan, then every guts and coguts set has size one; whereas if $P_1$ is a $4$-cosegment, then every guts and coguts set except perhaps $P_2$ and $P_3$ has size one, and $|P_2|=|P_3| \in \{1,2\}$.
For all $i \in \{2,3,\dotsc,m-1\}$, let $P_i = \{p_i\}$ if $|P_i|=1$, otherwise let $P_i = \{p_i,p_i'\}$.
We distinguish the following cases:
\begin{enumerate}[label=\rm(\Roman*)]
\item $|P_2| = |P_3| = 2$, and $m \ge 9$.\label{ndtegg1}
\item $|P_2| = |P_3| = 2$, and $m = 7$.\label{ndtegg2}
\item $|P_2| = |P_3| = 1$ and $P_1$ is a $5$-element fan.\label{ndtegg3}
\item $|P_2| = |P_3| = 1$, $P_1$ is a $4$-cosegment, and $m \ge 11$.\label{ndtegg4}
\item $|P_2| = |P_3| = 1$, $P_1$ is a $4$-cosegment, and $m = 9$.\label{ndtegg5}
\end{enumerate}
Note that if $P_1$ is a $5$-element fan, then $|P_2| = |P_3| = 1$, by \cref{ndt5eltfan}.
So only in case~\ref{ndtegg3} is $P_1$ a $5$-element fan.
Moreover, by \cref{onedoublecoguts}\ref{odcg1}, if $M \backslash a,b$ has a triangle, then it is a triangle of $P_1$, where $P_1$ is a $5$-element fan; so only in case~\ref{ndtegg3} does $M \backslash a,b$ have any triangles.
Observe also that when cases~\ref{ndtegg1} to \ref{ndtegg3} do not hold, then, since $13 \le |E(M \backslash a,b)| \le m+5$ and $m$ is odd, we have $m \ge 9$.
So these five cases are exhaustive.
Let $a'$ and $b'$ be as given in \cref{ndtendcandidates}.
In case~\ref{ndtegg1} we let $c'=p_{m-5}$;
in case~\ref{ndtegg2} and~\ref{ndtegg3} we let $c'=p_4$;
in case~\ref{ndtegg4} we let $c'=p_6$;
while in case~\ref{ndtegg5}, $c' \in \{p_4,p_6\}$ as appropriate.
We work towards an application of \cref{ndtgoodtripsupplement} with the elements $a',b',c'$; it remains to show that neither $\{a',c'\}$ nor $\{c',b'\}$ is contained in a $4$-element cocircuit of $M \backslash a,b$, and there is a unique triad containing $c'$, which avoids $P_1 \cup P_m$.
Firstly we address cases~\ref{ndtegg1} and~\ref{ndtegg2}.
Recall that $c' = p_{m-5}$ in case~\ref{ndtegg1}, and $c'=p_4$ in case~\ref{ndtegg2}.
In either case, $c' \in P_3^+$.
Since $|P_2|=2$, we have $\sqcap^*_{M \backslash a,b}(P_1,P_2^+)=0$, by the dual of \cref{pflancoguts}.
By \cref{ndtgoodtrip}\ref{ndtgt3}, any $4$-element cocircuit containing $\{a',c'\}$ avoids $P_2$; so, by the dual of \cref{picircuits}, no such cocircuit exists.
Note also that, by \cref{ndtdisjtriads,ndtinternaltris}, there is a unique triad $\{\ell_c,c',r_c\}$ containing $c'$ where $\ell_c \in P_{m-5}^-$ and $r_c=p_{m-4}$ in case~\ref{ndtegg1}, and $\ell_c \in P_4^-$ and $r_c=p_5$ in case~\ref{ndtegg2}.
In either case, $\ell_c \notin P_1$, by \cref{picircuits}, since $\sqcap^*_{M \backslash a,b}(P_1,P_2^+)=0$.
Consider case~\ref{ndtegg1}.
It remains only to show that $\{c',b'\}$ is not contained in a $4$-element cocircuit.
As $\{p_{m-3},p_{m-2},p_{m-1}\}$ is independent, $\sqcap^*_{M \backslash a,b}(P_{m-3}^-,P_m) = 0$ by the dual of \cref{pflantriad}.
As $c' \in P_{m-3}^-$ and $b' \in P_m$, and by \cref{ndtgoodtrip}\ref{ndtgt4}, there is no $4$-element cocircuit containing $\{c',b'\}$, and hence $M$ has a delete triple, by \cref{ndtgoodtripsupplement}, a contradiction.
Now consider case~\ref{ndtegg2}, where $c'=p_4$ and $m=7$.
We will show that \labelcref{ndtendgameoutcomes16}\labelcref{ndtendgameoutcome16b} holds.
If $\{c',b'\}$ is not contained in a $4$-element cocircuit, then $M$ has a contradictory delete triple, by the foregoing and \cref{ndtgoodtripsupplement}.
So let $C^*$ be a $4$-element cocircuit containing $\{c',b'\}$.
Then $C^* = \{\ell,c',b',b''\}$ with $b'' \in P_m - b'$ and $\ell \in P_{4}^-$, by \cref{ndtgoodtrip}\ref{ndtgt4}.
Recall that there is a triad $\{\ell_c,p_4,p_5\}$ with $\ell_c \in P_4^- -P_1$.
By orthogonality, $\ell_c \notin P_2$, so we may assume that $\{p_3,p_4,p_5\}$ is a triad, up to swapping $p_3$ and $p'_3$.
By \cref{ndtinternaltris,ndtdisjtriads,ndtendcandidates}, we may assume that $\{p_1,p_2,p_3\}$ and $\{p_1',p_2',p_3'\}$ are internal triads, where $P_1 = \{a',a'',p_1,p_1'\}$, up to swapping the labels on $p_2$ and $p_2'$.
Observe that $p_2$ is in a circuit contained in $P_1 \cup p_2$, and $p_2'$ is in a circuit contained in $P_1 \cup p_2'$, where neither of these circuits is a triangle.
If there is some element of $P_1$ that both of these circuits avoid, then, by circuit elimination, there is a circuit contained in $P_1 \cup P_2$ that avoids two elements of $P_1$, contradicting orthogonality.
So $P_1 \cup P_2$ is the union of two circuits.
Now, by orthogonality, $\ell \notin P_1 \cup P_2$.
So $\ell \in \{p_3,p_3'\}$.
If $\ell = p_3$, then, by cocircuit elimination with the triad $\{p_3,p_4,p_5\}$, there is a cocircuit contained in $\{p_4,p_5,b',b''\}$.
But $p_4 \notin \cl^*_{M \backslash a,b}(\{p_5,b',b''\})$, since $\{p_5,b',b''\} \subseteq P_4^+$ and $p_4$ is a guts element, so $\{p_5,b',b''\}$ is a triad of $M \backslash a,b$, in which case $P_m \cup p_5$ is a $4$-cosegment, contradicting that $P_m$ is an end of a nice path description.
So $\ell = p_3'$ and $\{p_3',p_4,b',b''\}$ is a cocircuit.
By cocircuit elimination with the triad $P_7$, orthogonality, and the fact that $P_7$ is an end of a nice path description, we deduce $\{p_3',p_4,p_7,p_7'\}$ is a cocircuit for any pair $\{p_7,p_7'\} \subseteq P_7$.
It remains to show that $\{p_5,p_6,p_7\}$ is a triad.
If $\{p_3',p_6,p_7\}$ is a triad, then the corank-$5$ set $P_m \cup \{p_6, p_2',a'\}$ cospans $H^* = E(M \backslash a,b) - \{p_2,p_3,p_4\}$, so $H^*$ is contained in a cohyperplane. But then $\{p_2,p_3,p_4\}$ is a circuit,
a contradiction.
By \cref{ndtdisjtriads} and orthogonality, $\{p_5,p_6,p_7\}$ is a triad.
So \labelcref{ndtendgameoutcomes16}\labelcref{ndtendgameoutcome16b} holds.
Now consider case~\ref{ndtegg3}, where $P_1$ is a $5$-element fan.
Recall that $c' = p_4$, and the only triangle of $M \backslash a,b$ is contained in $P_1$.
Let $(f_1,f_2,f_3,f_4,f_5)$ be a fan ordering of $P_1$.
First, observe that $\{p_3,c',p_5\}$ is the unique triad containing $c'$, by \cref{ndtinternaltris}.
Next we show that $\{a',c'\}$ is not contained in a $4$-element cocircuit.
Towards a contradiction, let $C^*$ be a $4$-element cocircuit containing $\{a',c'\}$.
Then $C^* = \{a'',a',c',p_5\}$ with $a'' \in P_1 - a'$, by \cref{ndtgoodtrip}\ref{ndtgt3} and since $\mathbf{P}$ is left-justified.
By cocircuit elimination with the triad $\{p_3,c',p_5\}$, the set $\{a'',a',p_3,c'\}$ contains a cocircuit.
By orthogonality, this cocircuit is the triad $\{a'',a',p_3\}$.
Since $a'$ is a spoke of the fan $P_1$, by \cref{ndtendcandidates}, we may assume that $a' =f_2$.
Note that $a'' \neq f_3$, for otherwise $\{a'',a',p_3,f_1\}$ is a cosegment and $\{a'',a',f_4\}$ is a triangle, a contradiction.
So, by orthogonality, $a'' = f_4$.
Now $(M \backslash a,b)^*|(P_1 \cup p_3) \cong M(K_4)$, contradicting \cref{noMK4}.
We deduce that $\{a',c'\}$ is not contained in a $4$-element cocircuit.
We next show that $\{c',b'\}$ is not contained in a $4$-element cocircuit.
Assume that $m \ge 9$.
Then $c' \in P_{m-3}^-$.
Since $\{p_{m-3},p_{m-2},p_{m-1}\}$ is independent, $\sqcap^*_{M \backslash a,b}(P_{m-3}^-,P_m) = 0$, by the dual of \cref{pflantriad}.
Since $b' \in P_m$, it follows, by \cref{ndtgoodtrip}\ref{ndtgt4} and the dual of \cref{picircuits}, that $\{c',b'\}$ is not contained in a $4$-element cocircuit.
Now assume that $m=7$ and suppose that $\{c',b'\}$ is contained in a $4$-element cocircuit.
Then this cocircuit is $\{\ell,c',b',b''\}$ with $b'' \in P_m - b'$ and $\ell \in P_4^-$, by \cref{ndtgoodtrip}\ref{ndtgt4}.
Recall that $\{p_3,c',p_5\}$ is a triad.
If $\ell = p_3$, then, by cocircuit elimination and orthogonality, $\{p_5,b',b''\}$ is a triad, so $P_m$ is not coclosed, a contradiction.
So $\ell \neq p_3$.
Without loss of generality,
$\{f_1,p_2,p_3\}$ is a triad.
By \cref{ndtendcandidates}, $a' \in \{f_2,f_4\}$.
Now, by orthogonality, \cref{ndtdisjtriads}, and the foregoing, $\ell =f_5$.
Then $f_5 \in \cl^*_{M \backslash a,b}(P_2^+)$, where $r^*_{M \backslash a,b}(P_2^+) = r^*(M \backslash a,b)-2$.
Thus, $\cl^*_{M \backslash a,b}(P_2^+ \cup \{f_4,f_5\})$ is contained in a cohyperplane. As $f_3 \in \cl^*_{M \backslash a,b}(P_2^+ \cup \{f_4,f_5\})$, the set $\{f_1,f_2,p_2\}$ is a triangle, a contradiction.
So $\{c',b'\}$ is not contained in a $4$-element cocircuit.
By \cref{ndtgoodtripsupplement}, $M$ has a delete triple, a contradiction.
Next we assume that case~\ref{ndtegg4} or \ref{ndtegg5} holds, so $m \ge 9$.
In case~\ref{ndtegg5}, for now let $c' = p_6$.
Since $\{p_2,p_3,p_4\}$ is independent, $\sqcap^*_{M \backslash a,b}(P_1,P_{4}^+) =0$ by the dual of \cref{pflantriad}.
Observe that, by \cref{ndtdisjtriads,ndtinternaltris}, there is a unique triad containing $p_6$, which also contains $p_7$, and avoids $P_1$, by the dual of \cref{picircuits}.
As $a' \in P_1$ and $c' \in P_4^+$, the dual of \cref{picircuits} and \cref{ndtgoodtrip}\ref{ndtgt3} imply that $\{a',c'\}$ is not contained in a $4$-element cocircuit.
We first consider case~\ref{ndtegg4}; it remains to show that $\{c',b'\}$ is not contained in a $4$-element cocircuit.
Since $\{p_{m-3},p_{m-2},p_{m-1}\}$ is independent, $\sqcap^*_{M \backslash a,b}(P_{m-3}^-,P_{m}) =0$ by the dual of \cref{pflantriad}, with $b' \in P_m$.
As $c' \in P_{m-3}^-$, we have that $\{c',b'\}$ is not contained in a $4$-element cocircuit, by \cref{ndtgoodtrip}\ref{ndtgt4} and the dual of \cref{picircuits}.
By \cref{ndtgoodtripsupplement}, $M$ has a delete triple, a contradiction.
Now consider case~\ref{ndtegg5}, where $m=9$.
We may assume that $\{p_6,b'\}$ is in a $4$-element cocircuit, for otherwise, by the foregoing, we can apply \cref{ndtgoodtripsupplement} with $c' = p_6$ to obtain a contradictory delete triple.
By the dual of \cref{picircuits} and orthogonality, $C_1^* = \{\ell,p_6,b',b''\}$ is a cocircuit for $\ell \in \{p_3,p_5\}$ and $b'' \in P_m-b'$.
Observe also that, by \cref{ndtdisjtriads,ndtinternaltris}, there is a unique triad containing $p_4$, which also contains $p_5$.
It follows that this triad is either $\{p_3,p_4,p_5\}$, or $\{p_1,p_4,p_5\}$ for some $p_1 \in P_1-a'$.
Suppose $\{p_3,p_4,p_5\}$ is a triad.
As $\sqcap^*_{M \backslash a,b}(P_{6}^-,P_{m}) = 0$, by the dual of \cref{pflantriad}, with $p_4 \in P_{6}^-$ and $b' \in P_m$, the pair $\{p_4,b'\}$ is not contained in a $4$-element cocircuit, by \cref{ndtgoodtrip}\ref{ndtgt4} and the dual of \cref{picircuits}.
So we may assume that $\{a',p_4\}$ is in a $4$-element cocircuit, for otherwise we can apply \cref{ndtgoodtripsupplement} with $c'=p_4$.
Let $C_2^*$ be the $4$-element cocircuit containing $\{a',p_4\}$.
Then, by \cref{ndtgoodtrip} and since $\mathbf{P}$ is left-justified, $C_2^*=\{a'',a',p_4,p_5\}$ for $a'' \in P_1 - a'$.
By cocircuit elimination with $\{p_3,p_4,p_5\}$, there is a cocircuit contained in $\{a'',a',p_3,p_4\}$.
But then, since $p_4$ is a guts element, $\{a'',a',p_3\}$ is a triad, contradicting that $P_1$ is an end of a nice path description.
So we may assume that $\{p_1,p_4,p_5\}$ is a triad, where $P_1 = \{a',a'',p_1,p_1'\}$.
By \cref{ndtdisjtriads} and the left-justification of $\mathbf{P}$, the internal triad containing $p_2$ is, without loss of generality, $\{p_1',p_2,p_3\}$.
Now $\{p_1',p_2,p_3\}$ and $\{p_1,p_4,p_5\}$ are triads, and $C_1^*=\{\ell,p_6,b',b''\}$ is a cocircuit.
Since $\mathbf{P}$ is left-justified, $\{p_3,p_4\}$ is contained in a circuit which is, in turn, contained in $P_3^- \cup \{p_3,p_4\}$.
Hence, by orthogonality, $\ell = p_5$, so $C_1^*=\{p_5,p_6,b',b''\}$.
If $\{p_5,p_6,p_7\}$ is a triad, then, by cocircuit elimination, there is a cocircuit contained in $\{p_6,p_7,b',b''\}$.
Then, by orthogonality, $\{p_7,b',b''\}$ is a triad, so $P_m$ is not an end of a nice path description, a contradiction.
So there is an internal triad $\{\ell',p_6,p_7\}$ with $\ell' \in P_5^-$.
Since $\{p_2,p_3,p_4\}$ is independent, the dual of \cref{picircuits,pflantriad}, and orthogonality, imply that $\ell' =p_3$, so $\{p_3,p_6,p_7\}$ is a triad.
But this contradicts orthogonality, since $p_3$ is in a circuit contained in $P_5^-$. \end{subproof}
The \lcnamecref{nodeltripleprop} now follows from \cref{ndtendgame,ndtendgame2}. \end{proof}
\begin{corollary}
\label{nodeltriplematrices}
Suppose there is a pair $\{a,b\} \subseteq E(M)$ such that $M \backslash a,b$ is $3$-connected with a $\{U_{2,5},U_{3,5}\}$-minor,
$M$ has no delete triples, and $|E(M)| \ge 16$.
Then $M \backslash a,b \cong M[I|A_i]$, for some $i \in \{1,2,3\}$, where each $A_i$ is a $\mathbb{U}_2$-matrix as follows:
$$A_1= \kbordermatrix{
& a' & p_1 & p_5 & p_5' & p_9 & b' \\
p_1' & 1 & \alpha_1 & 0 & 0 & 0 & 0 \\
p_2 & 1 & \alpha_1 & 1 & 1 & 0 & 0 \\
p_3 & 1 & 1 & 1 & 1 & 0 & 0 \\
p_4 & 1 & 1 & 0 & 1 & 0 & 0 \\
p_6 & 0 & 0 & 0 & 1 & 1 & 1 \\
p_7 & 0 & 0 & 1 & 1 & 1 & 1 \\
p_8 & 0 & 0 & 1 & 1 & \alpha_2 & 1 \\
p_9' & 0 & 0 & 0 & 0 & \alpha_2 & 1 \\ },$$
$$A_2= \kbordermatrix{
& a' & p_1 & p_5 & p_5' & p_9 & b' \\
p_1' & 1 & \alpha_1 & 0 & 0 & 0 & 0 \\
p_2 & 1 & \alpha_1 & 1 & 1 & 0 & 0 \\
p_3 & 1 & 1 & 1 & 1 & 0 & 0 \\
p_4 & 1 & 1 & 0 & 1 & 0 & 0 \\
p_6 & 0 & 0 & 1 & 0 & 1 & 1 \\
p_7 & 0 & 0 & 1 & 1 & 1 & 1 \\
p_8 & 0 & 0 & 1 & 1 & \alpha_2 & 1 \\
p_9' & 0 & 0 & 0 & 0 & \alpha_2 & 1 \\ },$$
$$A_3= \kbordermatrix{
& p_1'' & p_2 & p_2' & p_3 & p_7 & b' \\
a' & 1 & \alpha_1 & 0 & \alpha_1 & 0 & 0 \\ p_1 & 0 & 1 & 0 & 1 & 0 & 0 \\ p_1' & 1 & 1 & 0 & 1 & 0 & 0 \\ p_3' & 1 & 1 & 1 & 1 & 0 & 0 \\ p_4 & 1 & 1 & 1 & 1 & \alpha_2-1 & 1 \\ p_5 & 1 & 1 & 1 & 0 & \alpha_2-1 & 1 \\ p_6 & 1 & 1 & 1 & 0 & \alpha_2 & 1 \\ p_7' & 0 & 0 & 0 & 0 & \alpha_2 & 1 \\
}.$$ \end{corollary}
\begin{proof}
We apply \cref{nodeltripleprop} and observe that, since $|E(M)| \ge 16$, \cref{nodeltripleprop}\ref{ndtendgameoutcomes16} holds.
It remains to find the matroids satisfying \ref{ndtendgameoutcomes16}\ref{ndtendgameoutcome16a} or \ref{ndtendgameoutcomes16}\ref{ndtendgameoutcome16b}, and $\mathbb{P}$-representations for these matroids.
These can be found by hand; here we do not give all the details, but observe a few key points.
Let $M' = M \backslash a,b$.
Observe that both ends of the nice path descriptions for $M'$ are cosegments, in either case~\ref{ndtendgameoutcomes16}\ref{ndtendgameoutcome16a} or case~\ref{ndtendgameoutcomes16}\ref{ndtendgameoutcome16b}.
By \cref{pathdescends}\ref{pde2}, each of these ends has a unique $\{U_{2,5},U_{3,5}\}$-deletable element.
Up to labels we assume that $a'$ is the unique $\{U_{2,5},U_{3,5}\}$-deletable at one end; similarly $b'$ is the unique $\{U_{2,5},U_{3,5}\}$-deletable element at the other end.
Consider when \ref{ndtendgameoutcomes16}\ref{ndtendgameoutcome16a} holds.
Recall that $M'$ has no triangles, and thus $\{a',p_1',p_1,p_2\}$, and $\{p_8,p_9,p_9',b'\}$ are circuits.
Since $M'$ is $\{U_{2,5},U_{3,5}\}$-fragile, it can be argued that $\{a',p_1,p_3,p_4\}$, $\{p_6,p_7,p_9,b'\}$, and $\{p_4,p_5,p_5',p_6\}$ are circuits,
and $\{p_1',p_2,p_8,p_{9}'\}$ is a cocircuit.
We obtain $M_1$ if $(q,q')=(p_5,p_5')$ and $M_2$ if $(q,q')=(p_5',p_5)$.
When \ref{ndtendgameoutcomes16}\ref{ndtendgameoutcome16b} holds, it can be argued that $\{p_2,p_3,p_5,p_6\}$, $\{p_4,p_5,p_7,b'\}$, and $\{p_6,p_7,p_7',b'\}$ are circuits and $\{p_1'',p_2,p_2',p_6,p_7'\}$ is a cocircuit; we obtain $M_3$ in this case.
Alternatively, these matroids and representations can be found by a computer search on all $\mathbb{P}$-representable matroids on 14 elements, for $\mathbb{P} \in \{\mathbb{U}_2, \mathbb{H}_5\}$ (recall that all $3$-connected $\mathbb{U}_2$-representable matroids with a $\{U_{2,5},U_{3,5}\}$-minor, and at most 15 elements, were enumerated in \cite{BP20}).
This approach was used to verify the correctness of the representations found by hand. \end{proof}
\begin{proof}[Proof of \cref{nodeltriplethm}]
By \cref{nodeltriplematrices}, if $M$ has no delete triples and $|E(M)| \ge 16$, then $|E(M)|=16$ and $M \backslash a,b$ is $2$-regular and isomorphic to $M[I|A_i]$, for some $i \in \{1,2,3\}$, with $A_i$ as described in \cref{nodeltriplematrices}.
By a computer search, we found all $\mathbb{H}_5$-representable matroids that are single-element extensions of these three matroids.
Fix some $i \in \{1,2,3\}$.
For each (not necessarily distinct) pair of extensions of $M[I|A_i]$, say $N_1$ and $N_2$, we found each matroid~$M$ with a pair $\{a,b\}$ such that $M \backslash a \cong N_1$ and $M \backslash b \cong N_2$, using the splicing techniques described in \cite{BP20}.
We then discarded any such matroid $M$ with at least one triad.
For each of the
matroids, we verified the matroid indeed has a delete triple.
For example, for $i=1$ there were $56$ pairwise non-isomorphic single-element extensions that were $2$-regular, and a further $7$ pairwise non-isomorphic single-element extensions that were only $\mathbb{H}_5$-representable; after splicing a pair of these matroids, $368$ matroids were obtained that had no triads. \end{proof}
\section{Proof of \texorpdfstring{\cref{intro1}}{Theorem 1.1}} \label{concsec}
Combining \cref{deltripleprop,nodeltriplethm}, we prove our main result.
\begin{proof}[Proof of \cref{intro1}]
Observe that $U_{2,5}$ is a non-binary $3$-connected strong stabilizer for the class of $\mathbb{P}$-representable matroids, by \cref{u2stabs}.
We may assume that $M$ has a $U_{2,5}$-minor, for otherwise $M$ is an excluded minor
for the class of near-regular matroids, in which case $|E(M)| \le 8$ \cite{HMvZ11}.
Assume that $|E(M)| \ge |E(U_{2,5})| + 11 = 16$.
By \cref{notriads2}, there exists a matroid~$M_1 \in \Delta^*(M)$ such that $M_1$ has a pair of elements $\{a,b\}$ for which $M_1 \backslash a,b$ is $3$-connected and has a $\{U_{2,5},U_{3,5}\}$-minor, and $M_1$ has no triads.
By \cref{osvdelta}, $M_1$ is an excluded minor for the class of $\mathbb{P}$-representable matroids.
By \cref{utfutffragile}, $M_1 \backslash a,b$ is $\{U_{2,5},U_{3,5}\}$-fragile.
If $M_1$ has a delete triple, then, by \cref{deltripleprop}, $|E(M_1)| \le 15$; whereas if $M_1$ has no delete triples, then, by \cref{nodeltriplethm}, $|E(M_1)| \le 15$.
But $|E(M)| = |E(M_1)|$, so this is contradictory.
We deduce that $|E(M)| \le 15$, as required. \end{proof}
\appendix
\section*{Appendix: matroids appearing as excluded minors}
The matroids $P_6$, $F_7$, $F_7^-$, $P_8$, and $P_8^=$ are well known (see Oxley~\cite[Appendix]{Oxley11}, for example), as is the rank-$r$ uniform matroid on $n$ elements, $U_{r,n}$.
We now provide representations for other matroids appearing as excluded minors in this paper. Note that we provide \emph{reduced} representations: that is, we provide a matrix $A$ such that $M \cong M[I|A]$. For maximum generality, we provide a representation for a matroid $M$ over $\mathbb{P}_M$, the universal partial field of $M$, but in each case, we describe how one can obtain a finite field representation of $M$.
The following are reduced $\mathbb{K}_2$-representations for $F_7^=$, $\mathit{TQ}_8$, and $P_8^-$, respectively. The partial field $\mathbb{K}_2$ is formally defined in \cite{PvZ10b}, but note that a $\textrm{GF}(4)$-representation can be obtained by setting $\alpha = \omega$, where $\omega$ is a generator of $\textrm{GF}(4)$. Alternatively, two inequivalent $\textrm{GF}(5)$-representations can be obtained by setting $\alpha \in \{2,3\}$.
The matroid $F_7^=$ can be obtained by relaxing a circuit-hyperplane of $F_7^-$, and it has the following reduced representation:
$$\begin{bmatrix}
1 & 1 & 0 & 1 \\
1 & 0 & 1 & 1 \\
0 & 1 & 1 & \alpha \end{bmatrix}$$
The matroid $\mathit{TQ}_8$ was introduced in \cite{BP20}, and it has the following reduced representation:
$$\begin{bmatrix}
0 & 1 & 1 & 1\\
1 & \alpha & 1 & \alpha + 1\\
1 & 1 & 1 & \alpha \\
1 & \alpha + 1 & \alpha & \alpha \end{bmatrix}$$
Finally, $P_8^-$ can be obtained by relaxing one of the pair of disjoint circuit-hyperplanes of $P_8$, and it has the following reduced representation:
$$\begin{bmatrix}
1 & \alpha+1 & 0 & 1\\
\alpha+1 & \alpha+1 & \alpha & 1\\
0 & \alpha & \alpha & 1\\
1 & 1 & 1 & 0 \end{bmatrix}$$
There is, up to isomorphism, a unique matroid that can be obtained by deleting an element from the affine geometry $\mathit{AG}(2,3)$; following \cite{HMvZ11}, we denote this matroid $\mathit{AG}(2,3)\backslash e$. We use $(\mathit{AG}(2,3)\backslash e)^{\Delta Y}$ to denote the self-dual matroid that can be obtained by performing a single $\Delta$\nobreakdash-$Y$\ exchange on a triangle of $\mathit{AG}(2,3)\backslash e$.
For a matroid $M$, let $M+e$ denote the free single-element extension of $M$. Consider the matroids that can be obtained by relaxing zero or more circuit-hyperplanes starting from the Fano matroid $F_7$. We obtain the sequence $$F_7, F_7^-, F_7^=, \{H_7,M(K_4)+e\}, \{\mathcal{W}^3+e,\Lambda_3\}, Q_6+e, P_6+e, U_{3,7}$$ where $\Lambda_3$ denotes the rank-$3$ tipped free spike. Note that $H_7$ is the dual of the matroid (unique up to isomorphism) that can be obtained by performing a $\Delta$\nobreakdash-$Y$\ exchange on a triangle of $M(K_4)+e$.
We first provide reduced $\mathbb{H}_4$-representations of $M(K_4)+e$ and $\Lambda_3$ (see \cite{PvZ10b} for a definition of the partial field $\mathbb{H}_4$). Note that four inequivalent $\textrm{GF}(5)$-representations can be obtained by substituting $(\alpha,\beta) \in \{(2,2), (3,3), (3,4), (4,3)\}$.
\noindent \begin{tabular}{p{.45\textwidth} p{0.45\textwidth}} $M(K_4)+e$: $\begin{bmatrix}
1 & \alpha & \alpha & 1 \\
0 & 1 & 1 & 1 \\
1 & 0 & \alpha & \frac{\beta(\alpha-1)}{1-\beta} \\ \end{bmatrix}$ & $\Lambda_3$: $\begin{bmatrix}
1 & 1 & \alpha+\beta-2\alpha\beta & \alpha\beta-1 \\
1 & \alpha & 0 & \alpha(\beta-1) \\
1 & 0 & \alpha(1-\beta) & \alpha(\beta-1) \\ \end{bmatrix}$ \end{tabular}
Finally, we provide reduced $\mathbb{H}_2$-representations of $\mathcal{W}^3+e$ and $Q_6+e$ (see \cite{PvZ10b} for a definition of the partial field $\mathbb{H}_2$). Note that two inequivalent $\textrm{GF}(5)$-representations can be obtained by substituting $i \in \{2,3\}$.
\noindent \begin{tabular}{p{.45\textwidth} p{0.45\textwidth}} $\mathcal{W}^3+e$: $\begin{bmatrix}
1 & 0 & i & 1 \\
i & 1 & 0 & 1 \\
0 & i & 1 & 1 \end{bmatrix}$ & $Q_6+e$: $\begin{bmatrix}
\frac{i+1}{2} & 0 & i & 1 \\
1 & 1 & 1 & 1 \\
0 & \frac{1-i}{2} & -i & 1 \end{bmatrix}$ \end{tabular}
\end{document} | arXiv |
Does the category of locally ringed spaces have products?
The category of schemes has all fibered products, but the proof uses affine schemes in a crucial way. I want to understand whether this is true for the category of locally ringed spaces. The standard sources for categorical properties (nLab and Stacks project) do not say or contradict this fact explicitly. Though they say that the fiber product of schemes is automatically fiber product in the category of locally ringed spaces (remark 16.2 in the Schemes chapter of Stacks project).
The simplest locally ringed space which is not a scheme is a single point with a local ring (which is not a field) as a structure sheaf. The tensor product of local rings need not be a local ring, but it seems possible that this tensor product is always the direct product of local rings. For fields this is true, and I was unable to verify it for the general case. Even if this holds (which I highly doubt), it's still possible that more complicated fibered product do not exist. Maybe there is an obvious counterexample which I missed?
algebraic-geometry ringed-spaces
DmitryDmitry
$\begingroup$ I don't know the immediate answer, but this mathoverflow.net/questions/13616/… overflow post has a comment by @MartinBrandenburg which indicates that it is true. $\endgroup$
– Alex Youcis
$\begingroup$ I've seen that comment, but unfortunately the link to the file (probably containing proofs) doesn't work. $\endgroup$
I don't know whether the following is available in published form somewhere, I learned it from Jens Franke. I have written notes in german, and Martin has them in english, I think?
Suppose $f: X\to Z$ and $g: Y\to Z$ are morphisms of locally ringed spaces. The fiber product $X\times_Z Y$ of $f$ and $g$ in the category $\textbf{LRS}$ can be described as follows:
Underlying set: The set underlying of $X\times_Z Y$ is given by $$X\times_Z Y := \{(x,y,{\mathfrak p})\ |\ x\in X, y\in Y, f(x)=g(y)=:z,\\\quad\quad\quad\quad\quad\quad{\mathfrak p}\in\text{Spec}({\mathcal O}_{X,x}\otimes_{\mathcal O_{Z,z}}{\mathcal O}_{Y,y}),\\ \quad\quad\quad\quad\quad\quad\quad\quad\ \iota_{x,y,X}^{-1}({\mathfrak p})={\mathfrak m}_{X,x}, \iota^{-1}_{x,y,Y}({\mathfrak p}) = {\mathfrak m}_{Y,y}\}$$ Here, $\iota_{x,y,X}: {\mathcal O}_{X,x}\to{\mathcal O}_{X,x}\otimes_{{\mathcal O}_{Z,z}}{\mathcal O}_{Y,y}$ and $\iota_{x,y,Y}: {\mathcal O}_{Y,y}\to{\mathcal O}_{X,x}\otimes_{{\mathcal O}_{Z,z}}{\mathcal O}_{Y,y}$ are the canonical maps.
Topology: For $U\subset X$ and $V\subset Y$ open and $f\in{\mathcal O}_X(U)\otimes_{{\mathcal O}_Z(Z)}{\mathcal O}_Y(V)$ put $${\mathcal U}(U,V,f)\ :=\ \{(x,y,{\mathfrak p})\in X\times_Z Y\ |\ x\in U, y\in V, (\text{im. of } f\text{ in } {\mathcal O}_{X,x}\otimes_{\mathcal O_{Z,z}}{\mathcal O}_{Y,y})\notin {\mathfrak p}\}.$$ This defines the base for a topology on $X\times_Z Y$.
Structure sheaf: For $(x,y,{\mathfrak p})$ denote ${\mathcal O}_{X\times_ ZY,(x,y,{\mathfrak p})} := ({\mathcal O}_{X,x}\otimes_{{\mathcal O}_{Z,z}}{\mathcal O}_{Y,y})_{\mathfrak p}$. For $W\subset X\times_Z Y$ put $${\mathcal O}_{X\times_Z Y}(W) := \{(\lambda_{x,y,{\mathfrak p}})\in\prod\limits_{(x,y,{\mathfrak p})\in W} {\mathcal O}_{X\times_Z Y,(x,y,{\mathfrak p})}\ |\ \text{for every } (x,y,{\mathfrak p})\in W\text{ there ex. } \\ \text{std. open }{\mathcal U}(U,V,f)\subset W\text{ cont. } (x,y,{\mathfrak p})\text{ and }\mu\in({\mathcal O}_X(U)\otimes_{{\mathcal O}_Z(Z)}{\mathcal O}_{Y}(V))_f\\ \text{s.t. for all }(x^{\prime},y^{\prime},{\mathfrak p}^{\prime})\in{\mathcal U}(U,V,f)\text{ we have } \lambda_{(x^{\prime},y^{\prime},{\mathfrak p}^{\prime})}=\mu_{(x^{\prime},y^{\prime},{\mathfrak p}^{\prime})}\}$$ (The stalk of ${\mathcal O}_{X\times_Z Y}$ at $(x,y,{\mathfrak p})$ it then indeed $({\mathcal O}_{X,x}\otimes_{{\mathcal O}_{Z,z}}{\mathcal O}_{Y,y})_{\mathfrak p})$
Structure morphisms: One has canonical projections $X\leftarrow X\times_Z Y\to Y$, details ommitted for now.
There are many things to be checked, but maybe you want to think about them yourself to familiarize with the definitions?
HannoHanno
$\begingroup$ That's.....gross. +1 $\endgroup$
$\begingroup$ The topology is not quite correct, you also need open subsets of $Z$ to enter there. $\endgroup$
– Martin Brandenburg
$\begingroup$ @MartinBrandenburg Are you sure? What to change? $\endgroup$
– Hanno
$\begingroup$ @Hanno: It's been a few years, but I feel like I'm gonna re-read this answer myself. I think what Martin Brandenburg meant is that you need an arbitrary open subset $W$ of $Z$ and consider only those pairs $(U,V)$ where $U \subseteq f^{-1}(W)$ and $V \subseteq g^{-1}(W)$. Then you have morphisms $\mathcal O_Z(W) \to \mathcal O_X(f^{-1}(W)) \to \mathcal O_X(U)$ and $\mathcal O_Z(W) \to \mathcal O_Y(g^{-1}(W)) \to \mathcal O_Y(V)$ that you can use to tensor both rings over $\mathcal O_Z(W)$. $\endgroup$
– Patrick Da Silva
$\begingroup$ Hey @PatrickDaSilva: I may well be missing and misremembering stuff - it is a very long time ago - but I don't think it matters whether we consider $f$ in the 'refined' tensor product you suggest, or the one used in the answer, because it only enters the definition through its images in the local tensor products. $\endgroup$
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What exactly is $\mathbb{P}_\mathbb{Z}^n$?
What are the "correct" modules over locally ringed spaces?
locally ringed space $(X,\mathcal{O}_X)$ isomorphic as Ringed Spaces to $Spec(A)$ but not isomorphic as Locally Ringed Spaces.
Is there a notion of "schemeification" analogous to that of sheafification of a presheaf?
introducing locally ringed space definition necessary at all?
Schemes and locally ringed spaces
Non-isomorphic locally ringed spaces which represent isomorphic functors $\mathsf{CommRing} \to \mathsf{Set}$.
Ringed space but not locally ringed space | CommonCrawl |
Principal ideal theorem
In mathematics, the principal ideal theorem of class field theory, a branch of algebraic number theory, says that extending ideals gives a mapping on the class group of an algebraic number field to the class group of its Hilbert class field, which sends all ideal classes to the class of a principal ideal. The phenomenon has also been called principalization, or sometimes capitulation.
This article is about the Hauptidealsatz of class field theory. For the theorem about Noetherian rings, see Krull's principal ideal theorem.
Formal statement
For any algebraic number field K and any ideal I of the ring of integers of K, if L is the Hilbert class field of K, then
$IO_{L}\ $
is a principal ideal αOL, for OL the ring of integers of L and some element α in it.
History
The principal ideal theorem was conjectured by David Hilbert (1902), and was the last remaining aspect of his program on class fields to be completed, in 1929.
Emil Artin (1927, 1929) reduced the principal ideal theorem to a question about finite abelian groups: he showed that it would follow if the transfer from a finite group to its derived subgroup is trivial. This result was proved by Philipp Furtwängler (1929).
References
• Artin, Emil (1927), "Beweis des allgemeinen Reziprozitätsgesetzes", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 5 (1): 353–363, doi:10.1007/BF02952531, S2CID 123050778
• Artin, Emil (1929), "Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 7 (1): 46–51, doi:10.1007/BF02941159, S2CID 121475651
• Furtwängler, Philipp (1929). "Beweis des Hauptidealsatzes fur Klassenkörper algebraischer Zahlkörper". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 7: 14–36. doi:10.1007/BF02941157. JFM 55.0699.02. S2CID 123544263.
• Gras, Georges (2003). Class field theory. From theory to practice. Springer Monographs in Mathematics. Berlin: Springer-Verlag. ISBN 3-540-44133-6. Zbl 1019.11032.
• Hilbert, David (1902) [1898], "Über die Theorie der relativ-Abel'schen Zahlkörper", Acta Mathematica, 26 (1): 99–131, doi:10.1007/BF02415486
• Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. Vol. 62 (2nd printing of 1st ed.). Springer-Verlag. p. 104. ISBN 3-540-63003-1. Zbl 0819.11044.
• Serre, Jean-Pierre (1979). Local Fields. Graduate Texts in Mathematics. Vol. 67. Translated by Greenberg, Marvin Jay. Springer-Verlag. pp. 120–122. ISBN 0-387-90424-7. Zbl 0423.12016.
| Wikipedia |
Structure Of Cyclohexane
This complex displays remarkable selectivity for the hydrolytic kinetic resolution of epoxides after activation with a suitable acid. Pickett Journal of the American Chemical Society 1970, 92 (25), 7281-7290 DOI: 1021/ja00728a009 This paper describes a theoretical method for setting up calculations for ring inversion. For example, in the tri-substituted cyclohexane below, the methyl group is cis to the ethyl group, and also trans to the chlorine. Alkenes are isomeric with cycloalkanes e. For unsubstituted cyclohexane, there are no cis-trans isomers. Some Conformations of Cyclohexane Rings. Two orthorhombic forms (Vm values are 2. Cyclohexane is a cycloalkane which is an alicyclic hydrocarbon. The structures of cyclohexene and cyclohexane are usually simplified in the same way that the Kekulé structure for benzene is simplified - by leaving out all the carbons and hydrogens. The molecular formula of cyclohexane is C6H12. The structure links the U. 2% with cyclohexanol and cylohexanone (KA oil) selectivity of 97. C6H12 Is the molecular formulaDot Structure is attached The objects will sinks if the buoyant force How has Kashmir been influenced by global warming Which, element isotpe is used in cure of cancer Tick the correct answer The resistance of a resistor is reduced to half of its initial value. AU - Leuwerink, F. Search text. THIERRYLUANGRATH. It is a colorless low-melting solid used in the production of polyester resins. Cyclohexane View entire compound with free spectra: 24 NMR, 9 FTIR, and 3 Raman. Silane-Based Poly(ethylene glycol) as a Primer for Surface. Surface Tension of Cyclohexane. Not affected by aqueous solutions of acids, alkalis, most oxidizing agents, and most reducing agents. A cyclohexane molecule can take the form of the conformations that it has. The structure has been solved by beta synthesis and refined three-dimensionally to an R factor of 10·9% with 817 observed re-flexions. Formula: C 6 H 1 2 Molecular mass: 84. With no torsional strain (and no angle strain), cyclohexane is the most stable of all the small rings of. A few soluble transition metals were widely used as homogeneous catalysts in the industrial oxidation of cyclohexane. In linguistic literature the term is used for the expressions where the meaning of one element is dependent on the other, irrespective of the structure and properties of the unit (V. Now consider another molecule somewhat similar to stearic acid, called a phoshpolipid. the other 2 bonds that each carbon can make, go to the hydrogen atoms. The classification is based on the combined structural-semantic principle and it also considers the quotient of stability of phraseological units. Market Analysis and Insights: Global Cyclohexane Dimethanol (CHDM) Market. Antibacterial activity against test bacteria of the novel cyclohexane-1,3-dione ligands and their metal complexes. Like all cyclohexanes, it can interconvert rapidly between two chair conformers. Ah, but it doesn't, because the equation describes a mole ratio. We have investigated the gaseous structure of cyclohexane- 1,4-dione via electron diffraction. Aside from composition, there are other important differences in the internal structure of the asteroids. Define skeletal structure. The oppositional structure of the category of gender can be shown schematically on the following The constant categorial feature "quantitative structure" (see Ch. More exactly, you need 2 substitutents at 2 different carbon centers of the cyclohexane ring for isomers to exist. Substituted Cyclohexanes. The two structures shown here are the result of chair-chair interconversions, and they also happen to be identical (ie. Sold by Odette & Marche - Locally Owned from the Midwest and ships from Amazon Fulfillment. The bond angles would necessarily be 120º, 10. The RCSB PDB also provides a variety of tools and resources. A cyclohexane conformation is any of several three-dimensional shapes adopted by a cyclohexane molecule. Molecular Weight: 194. Consumer products. Determining degrees of unsaturation from a structure: Are you ready for some good news? It turns out that determining IHD from a structure is actually even easier than doing so for a molecular formula. No Definition of 'cyclohexane' Found - It's still good as a Scrabble word though! Cyclohexane is worth 28 points in Scrabble, and 30 points in Words with Friends. This is due to the fact that cyclohexane has 2 hydrogens less compared to hexane. so if you draw and 6 carbons that form a ring (all single bonds) each carbon uses 2 of its potential 4 bonds. a catalyst G. 15948 grams. However, conventional separation methods suffer from cumbersome operation, huge energy expenditure, or use of entrainers. Water: Chime in new window. Draw the structure of methylcyclohexane. Peak 1 refers to cyclohexane and peak 2 refers to toluene. (Note that while you defined the bond midpoint, the angle will be the same regardless of whether it's the midpoint of the bond or the neighboring carbon atom itself. the other 2 bonds that each carbon can make, go to the hydrogen atoms. Search text. The cyclohexane ring adopts a chair conformation in which the two methine hydrogen atoms are in a trans configuration. You can also browse global suppliers,vendor,prices,Price,manufacturers of Cyclohexane(110-82-7). The bond angles would necessarily be 120º, 10. PubChem CID: 16275. Melt the Cyclohexane and carefully remove any that is adhering to the walls of the tube. when n is odd, the presence of S n axis implies the presence of 2n elements, in which a plane of symmetry (σ) makes an independent appearance. A tooth that is visible in your mouth is only a part of the entire tooth. Description. Properties of Organic Solvents. Optical constants of C 6 H 12 (Cyclohexane) Kerl and Varchmin 1995: 313 K; n 0. The effect of different reaction parameters/additive was optimized. Structure and physical data for. Atomic weights: C=12 H=1 O=16 Cyclohexanol=C6H12O=100 Cyclohexene=C6H12=84. Formula : KI 1. The carbon atoms in the chemical structure of 2-BUTYNE are implied to be located at the corner(s) and hydrogen atoms attached to carbon atoms are not indicated – each carbon atom is considered to be associated with enough hydrogen atoms to provide the carbon atom. The Cyclohexane Molecule -- Chemical and Physical Properties. Introduction. The lowest energy form of this monosubstituted methylcyclohexane occurs when the methyl group occupies an equatorial rather than an axial position. Organizational structures define the hierarchy or an organization, and determine the way However, the initial establishment of the structure does not end the discussion surrounding them. most stable structure: The most stable conformation of cyclohexane is the chair form shown to the right. It has no color and is flammable. In this instance, cyclohexane combustion, like with other fuels, is Electronic structure methods generally approximate electronic orbitals as linear combinations of the electronic orbital functions. 2 Structural Organization of the Human Body. 1 displays a mechanism for the reaction of cyclohexane on Pt111. There is a second nucleic acid in all cells called ribonucleic acid, or RNA. …compound possessing this structure is cyclohexane (not an isoprenoid), represented by structural formula 1, by a condensed version 2, or simply by the hexagon 3. No Definition of 'cyclohexane' Found - It's still good as a Scrabble word though! Cyclohexane is worth 28 points in Scrabble, and 30 points in Words with Friends. I found this too: Should the 802 cm-1 peak be higher than the 2853 cm-1 peak based on the molecular structure of cyclohexane?. Transport sources. ( i think it would be higher) Chemistry. Cyclohexane and the Chair Structure: The chair structure of cyclohexane is considered to be the "perfect" conformation. The molecular formula for ethanol is C 2 H 6 O and its molar mass is 46. an organic compound. The process according to claim 1 wherein benzene or a mixture of benzene and cyclohexane is used as a feed. Still in HS and I can no longer get. Communicative structure of the english and russian sentence. The structural formula is CH2CH2CH2CH2CH2CH2. ЦИКЛОГЕКСАН ТЕХНИЧЕСКИЙ. Benzene is built from hydrogen atoms (1s 1) and carbon atoms (1s 2 2s 2 2p x 1 2p y 1). nominative 2. According to the kinetic formula of kekule, benzene is an equilibrium mixture of the following two structures, in which these two structures rapidly change into each other. Draw the structure of cyclohexane. DNA Structure, Replication, and Repair. • cyclohexane gas. Cyclohexane (data page) • Cyclohexane conformation • Cyclohexane stereochemistry • Cyclohexane-1,2-diol dehydrogenase • Cyclohexane-1,3-dione hydrolase. In compounds of this kind, the six ring atoms are not coplanar, but the ring usually is puckered, as shown in 4 and 5. Also, every carbon-carbon bond in such a structure would be eclipsed. A wide variety of synthesis of cyclohexane options are available to you, such as classification, grade standard, and usage. THIERRYLUANGRATH. 5: Sequence Rules for Specifying Configuration). Structure metonymic concepts - is not only the language, but also our thoughts, attitudes and actions. Conformational Analysis of Cyclohexane. Based on structure of cyclohexane. cis-1,3-dimethylcyclohexane. The large amplitude motion, as suggested from spectroscopic, dipole moment and earlier electron diffraction data was studied. 049 and Rw = 0. This allows us to investigate energy differences between different conformations. Display References Display Compounds. In the chair form its 12 extracyclic bonds fall into two classes: six lie parallel to the main axis of symmetry and are designated "axial", while six extend radially outward at ±109. Organic Chemistry. You will be doing tests to determine how these three classes of hydrocarbons compare with each one in certain properties. Gas-phase structure of 1,3,5-trisilacyclohexane and comparison with cyclohexane and cyclohexasilane. Dicyclohexyl ether then is a probable side product of the dehydration of cyclohexanol. Reasonably accurate experimental data (T, the temperature vs. Cyclohexane Cyclohexane is a cycloalkane with the molecular formula C6H12. The structure of Ice-I consists of a layer of puckered hexagonal rings of water molecules that have the conformation of the chair form of cyclohexane. The cyclohexane ring is very important because it is virtually strain free. The most stable conformation of cyclohexane is the chair form shown to the right. Alkenes are isomeric with cycloalkanes e. Recherche par n° CAS des substances susceptibles d'avoir un effet sur la reproduction des hommes et des femmes exposés durant leur travail. This database provides structural information on all of the Zeolite Framework Types that have been approved by the Structure Commission of the International Zeolite Association (IZA-SC). Each of the nucleotides in RNA is made up of a nitrogenous base, a five-carbon sugar, and a phosphate group. shortened words correlated with w o r d s , e. Boat conformation interactions four sets of eclipsed C-H interactions, one flagpole interaction. We wish to report the synthesis of a dirhodium complex with the novel trans-P,P'-diphenyl-1,4-diphospha-cyclohexane (dpdpc) ligand. Hexane has a linear carbon chain whereas cyclohexane is a cyclic molecule. Visit ChemicalBook To find more ALKOXYLATED CYCLOHEXANE DIMETHANOL DIACRYLATE() information like chemical properties,Structure,melting point,boiling point,density,molecular formula,molecular weight, physical properties,toxicity information,customs codes. Formula: C 6 H 1 2 Molecular mass: 84. Oppositions in Morphology. 5 moles of dioxygen, which is perfectly reasonable. The structure of cyclohexane is a hexagon ring of bonded carbon atoms (six bonded carbons) encased by 12 hydrogen atoms (each carbon atom being bonded with two hydrogens). The experimental structure is compared to the result of quantum chemical (QC) calculations (HF, MP2 and B3LYP) using the 6-31G∗ basis set. Condensed structural formula cyclohexane keyword after analyzing the system lists the list of keywords related and the list of websites with related content, in addition you can see which keywords most interested customers on the this website. Some of the advantages of using. Esters are compounds formed by the reaction of carboxylic acids with alcohols, and they have a general structural formula of:. Cyclohexane substituents can be found in either axial or equatorial positions. Canvas uses Structure Sensor to capture 3D models of rooms in seconds, which can then be converted into professional-grade CAD files. The water molecule is so common that it is wise to just memorize that water is a BENT molecule. Toluene is a liquid, which is colourless, water-insoluble and smells like paint thinners. DCM, THF, hexanes, heptane, toluene, and cyclohexane). The internal angles of a flat, regular hexagon are 120˚, as you can see in the image below. Reaction scale could be reduced for a greener process using a small vigreux column. 4-Hydroxyphenylpyruvate dioxygenase (EC 1. 0 references. The root of the tooth is completely buried into the jaw bone. Cyclohexane is a colourless, flammable liquid with a distinctive detergent-like odor, reminiscent of cleaning products (in which it is sometimes used). Isocyanates and thioisocyanates, such as 1,4-CYCLOHEXANE DIISOCYANATE, are incompatible with many classes of compounds, reacting exothermically to release toxic gases. Using the Kekule structure for Benzene, what is the predicted ratio of the hydrogenation enthalpy for benzene to that for cyclohexene?. ЦИКЛОГЕКСАН ТЕХНИЧЕСКИЙ. The chair structure consists of a six-membered ring where every C-C bond exists in a staggered conformation. Sold by Odette & Marche - Locally Owned from the Midwest and ships from Amazon Fulfillment. 0) which comprises isopropenylaromatic units (A) contained in an amount of 5 to 95 wt. These include mobility, stability, posture, circulation, digestion, and more. Morphological structure of English words. Cyclohexane; Benzene, hexahydro-; Hexahydrobenzene; Hexamethylene; Hexanaphthene; Cicloesano. Home Tables for Chemistry Compound classes. Cyclohexane is also used as a solvent in chemical and industrial processes and recently has been substituted for benzene in many applications. Page 4 of 16 MSDS - Toluene immediate area). Because of the cyclic nature, the number of structures cyclohexane can take is much fewer than its acyclic analog hexane. Users can perform simple and advanced searches based on annotations relating to sequence, structure and function. The new structure for cyclohexane was combined with ab initio calculations to redetermine the key structural features of F-, CI-, Br- and I-cyclohexane, and it is found that while the equatorial CX bond lengths are comparable with those for secondary substitution in propane, the axial bond lengths are all longer and halfway between those for. The experimental data shown in these pages are freely available and have been published already in the DDB Explorer Edition. The author uses ussues of "The Guardian" from 01. Reference:. The crystal and molecular structure of the pseudo-sugar DL-(1,3,5/2,4)-1,2,3,4-tetraacetoxy-5-(acetoxymethyl)cyclohexane ("pseudo-beta-DL-glucopyranose pentaacetate") has been determined by X-ray diffraction and statistical-phasing procedures. GHS Hazard and Precautionary Statements. 4 g) were loaded to a self-designed double-walled three-necked batch reactor. cyclohexane C. skeletal structure synonyms, skeletal structure pronunciation, skeletal structure translation, English dictionary definition of skeletal. 5% cyclohexane and 61. Draw the two chair conformations for each of the following di-substituted cyclohexanes. Cycles And Motorcycles Iron Or Steel Roller Chain. Block or report user. Frontiers in Chemistry. Cyclohexane; Benzene, hexahydro-; Hexahydrobenzene; Hexamethylene; Hexanaphthene; Cicloesano. They have long chemical structure and susceptible. The light microscope uses light as a source of radiation, whereas the electron microscope uses electrons. Cyclohexene produces more colour intensity and more soot is given off. Herein, mesoporous silicas using a chiral cationic low-molecular-weight amphiphile and organic solvents such as toluene, cyclohexane, tetrachlorocarbon, and tetrahydrofuran through a dual-templating approach were described. In addition to the dominant C=O-mode, the C-C-double bond (example 2 -4)is also located in this area (1600-1660 cm-1). Molecular Weight 114. Cyclohexane Rings. Another postulate of Chomsky's language acquisition theory is the process of selecting the best grammar that matches with the data available. Reasonably accurate experimental data (T, the temperature vs. The data represent a small sub list of all available data in the Dortmund Data Bank. Others are 'rubble piles'. Cyclohexane is non-polar. The molecular formula of cyclohexane is C6H12. Water: Chime in new window. a catalyst G. Classification. 'Chloroform was removed from measured amounts of C6PS, C6PG, and C6PC stocks using a stream of nitrogen, and the lipid samples were dissolved in cyclohexane and lyophilized overnight. The arrows in the figure below are meant to show how the structure physically moves to get from one conformation to the other. Cyclohexane View entire compound with free spectra: 24 NMR, 9 FTIR, and 3 Raman. ОКП 24 1641. The molecular formula of cyclohexane is C6H12. 0 - 100 (6) 201 - 300 (1) 301 - 400 (2) Boiling Point (°C) cyclohexane for analysis. Vray & standart versions. The uptake of CO2 was three times and for cyclohexane ten times higher in activated carbon black than in the non-activated one. The simplest way to draw. Cyclohexane is the most stable alicyclic ring system. (Adapted from Sigma-Aldrich) Here are the steps to find the chiral centres. Cyclohexane is a colourless, flammable liquid with a distinctive detergent-like odor, reminiscent of cleaning products (in which it is sometimes used). Due to the cyclic arrangement in its molecular structure, cyclohexane possesses a lesser number of hydrogen atoms compared to hexane. The internal angles for a hexagon are found to be 120°. CYCLIC ALKANES: Substituents on a cyclic alkane can be either cis or trans to each other. In compounds of this kind, the six ring atoms are not coplanar, but the ring usually is puckered, as shown in 4 and 5. These include mobility, stability, posture, circulation, digestion, and more. 19 November 2016. Composition, Information on Ingredients CAS# Chemical Name Percent EINECS/ELINCS 110-82-7 Cyclohexane >99% 203-806-2 Hazard Symbols: XN F N Risk Phrases: 11 38 50/53 65 67 Hazards Identification EMERGENCY OVERVIEW Appearance: colourless. In this work, we employ all-atom molecular dynamics simulations to examine the hydration response of phospholipid reverse micelles in cyclohexane. Phonation is the creation of sound by structures in the upper respiratory tract of the respiratory system. 1,1-Diethoxycyclohexane. Human Metabolome Database ID. The bonds in. Answer to Problem 12. The synthesis and structure of cyclohexane are a very important type of prototype because of a wide range of compounds. Noun forming suffixes. Cyclohexanone diethyl ketal. A ring with alternating single and double bonds is also possible, and is known as a. The morphological structure of the English word. Download Clker's Cyclohexane Lewis Structure clip art and related images now. Cyclohexane is a cycloalkane with the molecular formula C6H12 The 6 vertexed ring does not conform to the shape of a perfect hexagon. Due to these structural differences, both hexane and cyclohexane own unique properties. The key to understanding trends in ring strain is that the atoms We begin by studying the most stable conformation of cyclohexane, which has completely staggered. Cyclohexane : C6H12 The figure above is just for formula description. This allows you to copy/paste/search a long list of Abs in a single search. Conformations of Cyclohexane. Linear and Planar Densities. The structure links the U. It also occurs naturally as a plant volatile and can be released from volcanoes. ; Bhattacharjee, Apurba K. For example, in the tri-substituted cyclohexane below, the methyl group is cis to the ethyl group, and also trans to the chlorine. A planar structure for cyclohexane is clearly improbable. Cyclohexane is a cycloalkane with the molecular formula C 6 H 12. The chair conformation […]. Cyclohexane is the most stable alicyclic ring system. Draw the structure of cyclohexane. Other articles where Cyclobutane is discussed: hydrocarbon: Cycloalkanes: Cyclobutane (C4H8) and higher cycloalkanes adopt nonplanar conformations in order to minimize the eclipsing of bonds on adjacent atoms. Spectroscopic and structural data of the complex will be discussed with a view towards understanding the steric and electronic. The results of research on lexics, syntax, structure of newspaper articles and headlines are provided. Molecular Weight: 194. Exoskeletons: Robotic Structures Making Paralyzed People Walk Again. Therefore, to reduce torsional strain, cyclohexane adopts a three-dimensional structure known as the chair conformation. The experimental data shown in these pages are freely available and have been published already in the DDB Explorer Edition. I found this too: Should the 802 cm-1 peak be higher than the 2853 cm-1 peak based on the molecular structure of cyclohexane?. …compound possessing this structure is cyclohexane (not an isoprenoid), represented by structural formula 1, by a condensed version 2, or simply by the hexagon 3. Cyclohexane. Conformations of Cyclohexane. Firstly, draw cyclohexane and label the carbon atoms in it :. CYCLIC ALKANES: Substituents on a cyclic alkane can be either cis or trans to each other. A cyclohexane conformation is any of several three-dimensional shapes adopted by a cyclohexane molecule. 5º larger than the ideal tetrahedral angle. • Membrane-bound spherical structure that houses genetic material of eukaryotic cell. You can also browse global suppliers,vendor,prices,Price,manufacturers of Cyclohexane(110-82-7). Cholesterol • Cholesterol is essential to life. 1-Bromocyclohexane;CYCLOHEXYL BROMIDE;Cyelohexyl bromide;Cyclohexane,bromo-;Bromocyclohexane>. Whats is Tetrahydrofuran (THF)? Physical and Chemical Properties, Formules and Structures, Polar Nature, Uses, and More of this High-purity Lab Solvent. Chemical properties of toluene Sulfonation : Forms p-toluenesulfonic acid on sulfonation. The two structures shown here are the result of chair-chair interconversions, and they also happen to be identical (ie. Write structural formula of a cycloalkane with molecular. Market Analysis and Insights: Global Cyclohexane Dimethanol (CHDM) Market. puckered conformation of cyclohexane ring in which carbons 1 & 4 are bend toward each other. Structure of Hydrocarbon. Presentation folder structure. A planar structure for cyclohexane is clearly improbable. Others are 'rubble piles'. hydrogenating benzene to cyclohexane in a gas phase reaction by passing said benzene over said catalyst. Sign up for free!. All nuclear carbons do not lie in one plane. Structural isomerism is not a form of stereoisomerism, and is dealt with on a separate page. Structure, properties, spectra, suppliers and links for: Cyclohexane, 110-82-7. Benzene is built from hydrogen atoms (1s 1) and carbon atoms (1s 2 2s 2 2p x 1 2p y 1). 4-Hydroxyphenylpyruvate dioxygenase (EC 1. Doc Brown's GCSE/IGCSE/O Level KS4 the isomeric cyclohexane does not have a double bond and is a saturated hydrocarbon, and does not react with. Computational Geometry; Computer Science and Game Theory; Computer Vision and Pattern Recognition; Computers and Society; Cryptography and Security; Data Structures and Algorithms. Cyclohexane is an organic compound having the chemical formula C 6 H 12 and is a cyclic structure. Actually these are functional group isomers because cyclohexane is a cycloalkane and hex-1-ene is an alkene. Although humans contain a thousand times more DNA than do bacteria, the best estimates are that humans have only about 20. When bromine is added to cyclohexane (at right), the color persists because no reaction occurs. The synthesis and structure of cyclohexane are a very important type of prototype because of a wide range of compounds are formed from cyclohexane. Both you and Dr Dave P have misunderstood the problem. The general rule for solubility in water is the factor of polarity. Read More; structural formula. Technical cyclohexane. Presentation folder structure. The resulting angle and eclipsing strains would severely destabilize this structure. The basic unit of life, the cell, can be seen clearly only with the aid of microscopes. Benzene, hexahydro-. This deviation in bond angle from the ideal bond angle 109. This energy diagram shows. The complexes catalysed conversion of cyclohexane with appreciable yields. Reaction Intermediates. This article discusses seven types of organizational structures and reasons to use each. It is also used as a chemical building block for downstream derivatives like adipic acid and caprolactam , both of which are used as chemical intermediates in the production of nylon. 18 Isomers of Octane - C 8 H 18. Toluene is a liquid, which is colourless, water-insoluble and smells like paint thinners. ALKENES - structure & chemical properties. 110-82-7 - XDTMQSROBMDMFD-UHFFFAOYSA-N - Cyclohexane - Similar structures search, synonyms, formulas, resource links, and other chemical information. Still in HS and I can no longer get. The chair conformation is more stable than the boat conformation. 5: Sequence Rules for Specifying Configuration). Cyclohexane is a colourless, flammable liquid with a distinctive detergent-like odor, reminiscent of cleaning products (in which it is sometimes used). Also, every carbon-carbon bond in such a structure would be eclipsed. 13 as research data. Emerging core-shell nanostructured catalysts of transition metal. Step by step tutorial for drawing the cyclohexane hexagon, chair conformations, and cyclohexane ring flip as required in your standard organic chemistry course. Nazwa analitu. Cyclohexane; Benzene, hexahydro-; Hexahydrobenzene; Hexamethylene; Hexanaphthene; Cicloesano. Because many compounds feature structurally similar six-membered rings, the structure and dynamics of cyclohexane are important prototypes of a wide range of compounds. Cyclohexane and bromine reaction observations Cyclohexane and bromine reaction observations. Ecological Information Environmental Fate: When released into the soil, this material may biodegrade to a moderate e xtent. Chemicalize is a powerful online platform for chemical calculations and predictions (like logP, logD, pKa, NMR), search, and name-structure conversion. This problem has been solved! See the answer. Eukaryotic Gene Structure. Transport sources. Màrius Ramírez Cardona 1, Iván Barajas Rosales 2,Carlos Gómez Aldapa 2. , cabbie (cabman), nightie (nightdress), teeny (teenager) ; shortened words correlated with p h r a s e s , e. It has a mild odour and is insoluble in water but soluble in alcohol, ether, acetone, benzene, and ligroin. Cyclohexane is a cycloalkane with the molecular formula C 6 H 12. Radiolytic decomposition of methanesulfonyl chloride in liquid cyclohexane. Provied information about Cyclohexane, nonyl-(Molecular Formula: C15H30, CAS Registry Number:2883-02-5 ) ,Boiling Point,Melting Point,Flash Point,Density,Molecular Structure,Risk Codes,Synthesis Route at guidechem. Molecular Weight: 194. Some Conformations of Cyclohexane Rings A planar structure for cyclohexane is clearly improbable. It contains an amine-reactive N-hydroxysuccinimide (NHS ester) and a sulfhydryl-reactive maleimide group. Half of the hydrogens are in the plane of the ring (equatorial) while the other half are perpendicular to the plane (axial). Dichlorocyclohexanes: an introduction D. Melting Point of Cyclohexane. Structure of Hydrocarbon. Reasonably accurate experimental data (T, the temperature vs. DCM, THF, hexanes, heptane, toluene, and cyclohexane). com/iz6sE1Luba. The di-axial conformer is very unstable due. Hence, there's a higher number of cyclohexane molecules in a given molecule when compared with hexane. The i-mutation was significant for the phonemic structure of OE. Benzene is an unsaturated molecule, but cyclohexane is saturated. CYCLOHEXANE. Technical cyclohexane. EINECS:203-806-2. Boiling Points of Alkanes Reminder about Alkanes:. The structure of the lungs includes the bronchial tree - air tubes branching off from the bronchi into smaller and smaller air tubes, each one ending in a pulmonary alveolus. Lewis Structure For Ethanol. Synthesis, crystal structure and catalytic property of a copper(II) complex derived from N,N'-Bis(4-methylsalicylidene)cyclohexane-1,2-diamine. Optical constants of C 6 H 12 (Cyclohexane) Kerl and Varchmin 1995: 293 K; n 0. Which of them should be higher? Ignore the other peaks which I wasn't concerned. OH CH OH 41. Cyclohexane is a cycloalkane with the molecular formula C6H12 The 6 vertexed ring does not conform to the shape of a perfect hexagon. Phraseological units are classified into: 1. Illustrated Glossary of Organic Chemistry - Cyclohexane. If you look carefully at the structure above or use your model, you should be able to see that a parallel relationship exists for all C-C bonds across the ring from one another. Structure and the Functions of the Teeth. Cyclohexane View entire compound with free spectra: 24 NMR, 9 FTIR, and 3 Raman. Cyclohexane. This article discusses seven types of organizational structures and reasons to use each. If the formula used in calculating molar mass is the molecular formula, the formula weight computed is the molecular weight. The most general meanings rendered by language and expressed by systemic correlations of. Cyclohexane. kekule introduced the kinetic formula of benzene in 1872 to correct this defect of benzene's structure III. The gas-phase structure of 1,3,5-trisilacyclohexane 1 has been determined by gas electron diffraction (GED). Cyclohexane is a colourless, flammable liquid with a distinctive detergent-like odor, reminiscent of cleaning products (in which it is sometimes used). Properties of Organic Solvents. Due to water's bent, tetrahedral structure, it contains a dipole, which makes it asymmetrical (since the electrons are denser around the central oxygen, it is partially-negative o. The chair is the most important form of cyclohexane, but not the only one. Let's go through the seven common types of org structures and reasons why you might consider each of them. De-esterified pectin can form so-called "egg-box structures" in the presence of Ca2+ ions (Mohnen, 2008; Harholt et al. Page 4 of 16 MSDS - Toluene immediate area). cyclohexane rus. Although humans contain a thousand times more DNA than do bacteria, the best estimates are that humans have only about 20. Cyclohexane has a non-polar structure that makes it almost free from ring strain. Eukaryotic Gene Structure. Mitochondria are enclosed by two membranes—a smooth outer membrane and a markedly folded or tubular inner mitochondrial membrane, which has a large surface and. Display References Display Compounds. Condensed structural formula cyclohexane keyword after analyzing the system lists the list of keywords related and the list of websites with related content, in addition you can see which keywords most interested customers on the this website. Ground state products. PRESENT PERFECT CONTINUOUS/PROGRESSIVE TENSE This post includes detailed expressions about Present Perfect Continuous tense and its structures in english. Cyclopentane. Some Conformations of Cyclohexane Rings. Ruscic, Uncertainty Quantification in Thermochemistry, Benchmarking Electronic Structure Computations, and Active Thermochemical Tables. Others are 'rubble piles'. Structure of Benzene. The C-C-C bonds are very close to 109. An organizational structure defines how jobs and tasks are formally divided, grouped and coordinated. In 1543, he published Dialecticae partitiones (The Structure of Dialectic), which in its second edition was called Dialecticae institutiones (Training in Dialectic), and Aristotelicae animadversions. The angle strain in cyclobutane is less than in cyclopropane, whereas cyclopentane and higher cycloalkanes are virtually free of angle strain. The invention relates to a preparation method of C-glucoside containing a saturated cyclohexane structure. n The chair conformation has two kinds of positions for substituents on the ring: axial positions and equatorial positions. Reasonably accurate experimental data (T, the temperature vs. The internal angles for a hexagon are found to be 120°. Conformations of silicon-containing rings. If the formula used in calculating molar mass is the molecular formula, the formula weight computed is the molecular weight. Some Conformations of Cyclohexane Rings. We have investigated the gaseous structure of cyclohexane- 1,4-dione via electron diffraction. If you draw a structural formula instead of using models, you have to bear in mind the possibility of this free. Conformational Analysis of Cyclohexane. 4 g) were loaded to a self-designed double-walled three-necked batch reactor. For estrogenic activity, structure activity relationship studies have led to the conclusion that the presence of two hydroxyl groups about 11-12 Å apart are sufficient. Structure Search. cyclohexane 1,2-dichloroethane dichloromethane dimethylformamide dimethyl sulfoxide dioxane ethanol ethyl acetate diethyl ether heptane hexane methanol methyl-t -butyl ether 2-butanone pentane n -propanol isopropanol diisopropyl ether tetrahydrofuran toluene trichloroethylene water xylene id e e e nl tate e rm e e e e e e l tate r e e l l-t r e. Molecular Formula: C7H12O. We are going to look at the difference in energy between conformations of 1,4-dichloro-cyclohexane. The synthesis and structure of cyclohexane are a very important type of prototype because of a wide range of compounds. Technical cyclohexane. Standard atmospheric pressure is 760 mm Hg. You can find other items of this collection by using the following keyword — 3DR199. cis-1,3-Dimethylcyclohexane | C8H16 | CID 252361 - structure, chemical names, physical and chemical properties, classification, patents, literature, biological. Cyclohexane View entire compound with free spectra: 24 NMR, 9 FTIR, and 3 Raman. Molecular Weight 114. On the syntagmatic level, the semantic structure of the word is analyzed in its linear relationships with neighboring words in connected speech. Hexane is linear while cyclohexane is cyclic. Cyclohexane C. It has a role as a non-polar solvent. It is simply cyclohexane and there are two hydrogens on each carbon atom. It may be written in a condensed form as (CH2)6. Glossary: Beat. Molecular Structure. A series of novel poly(1,4‐cyclohexanedimthylene terephthalate‐co‐1,4‐cyclohxylenedimethylene 2,6‐naphthalenedicarboxylate) (PCTN) copolyesters were successfully melt polymerized using different content of trans‐ or cis‐isomers. Read on to know about the structure and functions of the. The point groups C nh, D nh, and D nd. The chemical ATP, adenosine triphosphate, is the fuel that powers all life. Compound Cyclohexane with free spectra: 24 NMR, 9 FTIR, and 3 Raman. The energy zero is the heat of formation of the ele-. Cyclohexane-1,3-dione and its derivatives are important building blocks. Cyclohexane is non-polar. The volume cyclohexane occupies is lower than hexane due to its cyclic structure. EINECS:203-806-2. The second componentof the phonetic system of English is syllabic structure of its words in isolation and in phrases and sentences. Structures of the limbic system involved in memory formation. Each of these chains contains a compound known as heme, which in turn contains iron, which is what transports oxygen in. cis-1,3-dimethylcyclohexane. Doc Brown's GCSE/IGCSE/O Level KS4 the isomeric cyclohexane does not have a double bond and is a saturated hydrocarbon, and does not react with. Cyclohexane and the Chair Structure: The chair structure of cyclohexane is considered to be the "perfect" conformation. The molecular structure of hexakis(ethylidene)-cyclohexane has been studied experimentally by the gas electron diffraction method and theoretically by ab initio. A combined electron paramagnetic resonance (EPR) and density functional theory (DFT) investigation of the N,N′-bis(3,5-di-tert-butylsalicylidene)-1,2-cyclohexane-diamino cobalt(II) complex was undertaken. Deep structures. Graph the temperature of the cyclohexane as it cools versus the time in minutes. The chemical formula of toluene can be written as C 6 H 5 CH 3. 238000007792 addition Methods 0. Quadridentate nitrogen donor ligands acting as bridging di-bidentates: X-ray crystal and molecular structure of [Ag2{m-(R)(S)-1,2-[(2-C5H4N)C(H)=N]2-cyclohexane}]2] [O3SCF3]2 and the observation of 3J(107,109Ag-1H) in the 1H NMR spectrum of the dinuclear [Ag2L2]2+ cation. Benzene, on the other hand, has a delocalised system of pi bonded electrons that cause an even charge distribution over the entire molecule, therefore it is possible for the molecule to exist as a planar hexagonal shape. This model is a part of Molecule structure collection. 'Chloroform was removed from measured amounts of C6PS, C6PG, and C6PC stocks using a stream of nitrogen, and the lipid samples were dissolved in cyclohexane and lyophilized overnight. Structure in solution (1H and INEPT 109Ag NMR) and in the solid (X-ray) of [Ag{m-(R),(S)-1,2-(thiophene-2-CH=N)2-cyclohexane}2] (O3SCF3)2. Propane, being a GAS but an aliphatic hydrocarbon is soluble in cyclohexane, which is a liquid and a cycloaliphatic hydrocarbon. Advanced Search | Structure Search. N-heterocyclic carbenes catalyze intramolecular β-alkylations of α,β-unsaturated esters, amides, and nitriles that bear pendant leaving groups to form a variety of ring sizes. With such capabilities, they can form more complex molecules like cyclohexane and in rare instances aromatic hydrocarbons like benzene. Cyclohexane (110- 82 -7) No No None 12. ycloalkanes, structure and nomenclature ycloalkanes are alkanes. The publication first offers information on the theory of slow chain oxidations and the products of liquid-phase cyclohexane oxidation. Bone, or osseous tissue, is a hard, dense connective tissue that forms most of the adult skeleton, the support structure of the body. Numer serii. This is one of the reasons why compounds containing six-membered rings are very common. Hiding of all overloaded methods with same name in base class. Classification. Structure Search Login or Create account 0. Doc Brown's GCSE/IGCSE/O Level KS4 the isomeric cyclohexane does not have a double bond and is a saturated hydrocarbon, and does not react with. Emerging core-shell nanostructured catalysts of transition metal. Stereoisomers: Cyclohexane, 1,3-dimethyl-, trans-Cyclohexane, 1,3-dimethyl-(1R-trans)-1,3-dimethylcyclohexane. Structure, properties, spectra, suppliers and links for: Cyclohexane, 110-82-7. cyclohexane - PowerPoint PPT Presentation. Text lists sorted by. Any 'set' could be used to draw two parallel Newman projections. Isotopologues: Cyclohexane, d12; Cyclohexane, d12; Cyclohexane, d12; Cyclohexane-d12; Cyclohexane-1,2,3-d6; Hephane-d16. Циклогексан. More exactly, you need 2 substitutents at 2 different carbon centers of the cyclohexane ring for isomers to exist. CYCLIC ALKANES: Substituents on a cyclic alkane can be either cis or trans to each other. Helpful Hints D. Isotopologues: Cyclohexane, d12; Cyclohexane, d12; Cyclohexane, d12; Cyclohexane-d12; Cyclohexane-1,2,3-d6; Hephane-d16. How to convert 1 milliliter of cyclohexane to grams. Read on to know about the structure and functions of the. The boiling point of cyclohexane at 760 mm is 80. Numer serii. cis-1,3-Dimethylcyclohexane | C8H16 | CID 252361 - structure, chemical names, physical and chemical properties, classification, patents, literature, biological. Physical Chemistry Chemical Physics, 14, 16400-16408. Dichlorocyclohexanes: an introduction D. Each of the nucleotides in RNA is made up of a nitrogenous base, a five-carbon sugar, and a phosphate group. Benzene is one of the most fundamental compounds used in the manufacturing of various plastics. I do agree with Tom, you could use a pencil. Silane-Based Poly(ethylene glycol) as a Primer for Surface. Eukaryotic Gene Structure. 049 and Rw = 0. Both aliphatic (linear or branched from short C-2 chain to fatty length ) and aromatic diamines are used as a monomer to form copolymers like nylons, polyesters and polyurethanes for characteristic properties. Pathways called white matter tracts connect areas of the cortex to each other. Two orthorhombic forms (Vm values are 2. The Van Der Waals forces between molecules are inversely proportional to the distance between molecules. Cyclohexanol is converted to cyclohexene, not cyclohexane. A typical phospholipid structure is shown below. Define skeletal structure. to two different entities, on possessing one common characteristic, on linguistic semantic nearness, on a common component in their semantic structures. Introduction I Thematic principle of classification II Classification based on the semantic principle III The structural principle of classifying phraseological units. The data represent a small sub list of all available data in the Dortmund Data Bank. They have long chemical structure and susceptible. Glossary: Beat. Fraction A still consisted of 21. structure of the substance, it will have different solubility in different solvents depending on the nature of the solvent. Reactions with amines, aldehydes, alcohols, alkali metals, ketones, mercaptans, strong oxidizers, hydrides, phenols, and peroxides can cause vigorous releases of heat. cyclohexane C. Cyclohexane doesn't exist in hexagonal form. The di-axial conformer is very unstable due. Esophageal Structure. Presentation folder structure. The author uses ussues of "The Guardian" from 01. skeletal structure synonyms, skeletal structure pronunciation, skeletal structure translation, English dictionary definition of skeletal. As a continuous work to discover novel crop selective HPPD inhibitor, a series of 2-(aryloxyacetyl)cyclohexane-1,3-diones were rationally designed and synthesized by an efficient one-pot procedure using N,N′-carbonyldiimidazole (CDI), triethylamine, and acetone cyanohydrin. This is easy to mistake when hurrying, so be careful when you are intepreting any structural formulas which include hexagons. The reaction of cyclohexane combustion is: 9O2+ C6H12. Padding and Alignment of Structure Members. In the cyclohexane case, for example, there is a carbon atom at each corner, and enough hydrogens to make the total bonds on each carbon atom up to four. In this case, we're talking about Cyclohexane, so we're only dealing with hydrogens, so first, we're gonna put in the hydrogens that we call axial or axial groups here, and axial groups, you can think about the earth, so here is the earth, you can think about this as being a globe and this would be the axis going straight up and down, so. The bond angles would necessarily be 120º, 10. 049 and Rw = 0. Canvas uses Structure Sensor to capture 3D models of rooms in seconds, which can then be converted into professional-grade CAD files. com offers 846 synthesis of cyclohexane products. If you add cyclohexane to chlorine/bromine water or iodine solution and shake it, the halogen dissolves in the cyclohexane layer because the halogen and the cyclohexane have the same intermolecular forces (instantaneous dipole-induced dipole). Organizational structures can use. …compound possessing this structure is cyclohexane (not an isoprenoid), represented by structural formula 1, by a condensed version 2, or simply by the hexagon 3. While Hexene also has structural formula C6H12. Catalytic Test. Cyclohexane–methanol mixtures separate into two liquids below an upper critical solution temperature (UCST). The new structure for cyclohexane was combined with ab initio calculations to redetermine the key structural features of F-, CI-, Br- and I-cyclohexane, and it is found that while the equatorial CX bond lengths are comparable with those for secondary substitution in propane, the axial bond lengths are all longer and halfway between those for. HiTape® combines the benefits of automated processing, the cost-effectiveness of. Half of the hydrogens are in the plane of the ring (equatorial) while the other half are perpendicular to the plane (axial). N-heterocyclic carbenes catalyze intramolecular β-alkylations of α,β-unsaturated esters, amides, and nitriles that bear pendant leaving groups to form a variety of ring sizes. Due to the cyclic arrangement in its molecular structure, cyclohexane possesses a lesser number of hydrogen atoms compared to hexane. 5º larger than the ideal tetrahedral angle. Chemischer Informationsdienst 1975 , 6 (8) , no-no. This component may be viewed from two points - its syllable formation. Cyclohexane is not planar but puckered into a 3-D conformation that relieves all strain. Cyclohexane Cyclohexane (also known as CYX, hexamethylene, hexahydrobenzene, hexanaphthene, and benzenehexahydride) is a colourless, volatile, and flammable liquid with the formula C6H12. Terpinolene is a cyclic monoterpene compound found in some Labiatae herbs. Discussion : 1. The large amplitude motion, as suggested from spectroscopic, dipole moment and earlier electron diffraction data was studied. Get 1:1 help now from expert Chemistry tutors. The new structure for cyclohexane was combined with ab initio calculations to redetermine the key structural features of F-, CI-, Br- and I-cyclohexane, and it is found that while the equatorial CX bond lengths are comparable with those for secondary substitution in propane, the axial bond lengths are all longer and halfway between those for. Dispersion formula $$n^2-1=\frac{0. The crystal structure of pure cyclohexane has now been determined and it indeed confirms the chair form. See full list on study. 5 mL) and AC (0. Benzene, hexahydro-. Toluene is a naturally occurring compound derived primarily from petroleum or petrochemical processes. It is a styrene copolymer obtained by living polymerization (M w /M n =1. Synthesis Of Cyclohexene Via Dehydration Of Cyclohexanol Lab Report. Types of Organizational Structures. Octane; 2-Methylheptane; 3-Methylheptane; 4-Methylheptane; 2,2-Dimethylhexane; 2,3-Dimethylhexane. 2% with cyclohexanol and cylohexanone (KA oil) selectivity of 97. Hiding of all overloaded methods with same name in base class. To draw the structure of trans-1,4-dimethyl cyclohexane, we can follow the below steps. cyclohexane chair practice problems answers cyclohexane. In other settings mostly unreactive. When re leased into the soil, this. 2 Boiling point: 81°C Melting point: 7°C See Also: Toxicological Abbreviations CYCLOHEXANE. The mixture was stirred in the presence of molecular oxygen (40 mL/min) at 75[degrees]C for 14 hours (Scheme 1). Based on structure of cyclohexane. Structure vs class in C++. Evaporation. 3 Product. The C-C-C bonds are very close to 109. 110-82-7 Formula. Draw two isomeric alcohols with the formula C 4 H 10 O. Market Structure - Domestic Supply vs. The experimental data shown in these pages are freely available and have been published already in the DDB Explorer Edition. Cyclohexane structural formula keyword after analyzing the system lists the list of keywords related and the list of websites with related content, in addition you can see which keywords most interested customers on the this website. 18 Isomers of Octane - C 8 H 18. Terpinolene is a cyclic monoterpene compound found in some Labiatae herbs. cyclohexane. Module : Stereochemistry Conformational Analysis of Cyclohexane Author : Dr. cheminform abstract: carbon monofluoride, evidence for a structure containing an infinite array of cyclohexane boats. Due to the cyclic arrangement in its molecular structure, cyclohexane possesses a lesser number of hydrogen atoms compared to hexane. Cycloh = ? do you mean Cycloakenes with the formula CnH2n? Thalidomide is C13H10N2O4. Draw the structure of cyclohexane. A planar structure for cyclohexane is clearly improbable. Cyclohexane is used as a nonpolar solvent for the chemical industry, and also as a raw material for the industrial production of adipic acid and caprolactam, both of which are intermediates used in the production of nylon. QYQADNCHXSEGJT-UHFFFAOYSA-N cyclohexane-1,1-dicarboxylate;hydron Chemical class data:image/svg+xml;base64. Half-Sandwich and Triangular-Sandwich Supramolecular Solid State Structures of C 60 with Ir(ttp)Me. Cyclohexane is non-polar. The chemical formula for ethanol is CH 3 CH 2 OH or C 2 H 5 OH (condensed structural formulas). Collins English Dictionary defines metaphor as a figure of speech in which a word or phrase is applied to an. Crystal Structure of the elements. The molecular formula for ethanol is C 2 H 6 O and its molar mass is 46. The stable structure is the one that puts the lone pairs in equatorial locations, giving a T-shaped molecular structure. Stop leak if you can do it without risk. The chemical formula of cyclohexane is C 6 H 12 which is a non-polar solvent. A process for the production of cyclohexane by the hydrogenation of benzene is provided wherein the reactor is operated at a pressure wherein the reaction mixture is boiling under low hydrogen partial pressure in the range of about 0. The structure has been solved by beta synthesis and refined three-dimensionally to an R factor of 10·9% with 817 observed re-flexions. Structure of nonionic surfactant diglycerol monomyristate (C14G2) micelles in cyclohexane has been investigated by small-angle X-ray scattering (SAXS) technique. The uptake of CO2 was three times and for cyclohexane ten times higher in activated carbon black than in the non-activated one. Cyclohexane View entire compound with free spectra: 24 NMR, 9 FTIR, and 3 Raman. Benzene is one of the most fundamental compounds used in the manufacturing of various plastics. Get tutorials, Flutter news and other exclusive content delivered to Having the foundational structure of the Number Trivia App's architecture in place, we will begin. Introduction. Structure and physical data for. Structural elements generally include: (a) the words expression, air, attitude, and others which describe behaviour or Two step epithets have a fixed structure of Adv + Adj model. Posted by 14. C6H12 Is the molecular formulaDot Structure is attached The objects will sinks if the buoyant force How has Kashmir been influenced by global warming Which, element isotpe is used in cure of cancer Tick the correct answer The resistance of a resistor is reduced to half of its initial value. The rings exist in chair conformations and are mostly trans configured. The Au clusters encapsulated in the MCM-22 zeolite are highly active and selective for the oxidation of cyclohexane to KA-oil, which is superior to. If cyclohexane has two substituents and one has to be placed axial and one equatorial (as is the case in trans-1,2-disubstituted cyclohexanes), the lowest-energy conformation will be the one in which the bigger group goes in the equatorial position and the smaller group goes in the axial position. The simplest method of preparation is the Fischer method, in which an alcohol and an acid are reacted in an acidic medium. Atoms can be hybridized. The molecular formula of cyclohexane is C6H12. James Richard Fromm. Get tutorials, Flutter news and other exclusive content delivered to Having the foundational structure of the Number Trivia App's architecture in place, we will begin. Cyclohexane is used as polymerization thinner,Paint from film; Detergent; Adipic acid extraction. Atomic weights: C=12 H=1 O=16 Cyclohexanol=C6H12O=100 Cyclohexene=C6H12=84. China Jdch999 (1, 2-Cyclohexane dicarboxylic acid, di-isononyl ester), Find details about China Dinch, Car Wire Plasticizer from Jdch999 (1, 2-Cyclohexane dicarboxylic acid, di-isononyl ester) - Zhejiang Jiaao Enprotech Stock Co. Cyclohexane is non-polar. The separation of benzene and cyclohexane is one of the most challenging tasks in the petrochemical field. Structural elements generally include: (a) the words expression, air, attitude, and others which describe behaviour or Two step epithets have a fixed structure of Adv + Adj model. Kahoot! is a game-based learning platform that brings engagement and fun to 1+ billion players every year at school, at work, and at home. Computational Geometry; Computer Science and Game Theory; Computer Vision and Pattern Recognition; Computers and Society; Cryptography and Security; Data Structures and Algorithms. Cyclohexane is a ring of single-bonded carbon atoms each single-bonded to 2 hydrogen atoms. Others are the boat (a high-energy form; rarely seen), a half chair (very high energy) and a twist-boat (an intermediate in the ring-flip isomerism between two chair forms). Cyclohexane is the cycloalkane molecule with the formula C 6 H 12, with 6 carbon atoms connected to each other in a ring shape, and each connected to two hydrogen atoms. cyclohexane (English) retrieved. Cyclohexene | C6H10 | CID 8079 - structure, chemical names, physical and chemical properties, classification, patents, literature, biological activities, safety. Linguostylistics studies the nature, functioning and structure of stylistic devices and the styles of a Modern English lexicology investigates the problems of word structure and word formation; it also. 0 references. | CommonCrawl |
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Thursday April 2, 2020
Alice Pellet-Mary (COSIC team, KU Leuven)
B013, 10:30
Thursday February 6, 2020
Jean Kieffer (LFANT team, Université de Bordeaux)
Computing isogenies from modular equations in genus 2
Given two l-isogenous elliptic curves over a field k of large characteristic, an algorithm by Elkies uses modular polynomials to compute this isogeny explicitly. It is an essential primitive in the well-known SEA point counting algorithm.
In this talk, we generalize Elkies's algorithm to compute isogenies between Jacobians of genus 2 curves: we compute either l-isogenies or, in the RM case, cyclic beta-isogenies. The algorithm uses modular equations of Siegel or Hilbert type respectively. As in genus 1, there are two main steps:
Use derivatives of modular equations to compute the action of the isogeny on differential forms;
Solve a differential system to find the isogeny.
This algorithm has implications for point counting in genus 2: Elkies's method is now available, and could improve on the Schoof-Pila approach.
Baptiste Lambin (Ruhr Universitat Bochum)
Making (near) Optimal Choices for the Design of Block Ciphers
When designing block ciphers, we need to make decisions on which specific components to use such as which S-box, which linear layer etc. These decisions are made by taking into account the security of the resulting block cipher, but also the underlying cost in term of performances. In this talk, I will present two different works aiming at finding optimal components w.r.t a given criteria, while also trying to keep a very cheap implementation cost. I will show how the use of automatic tools (either already existing or specifically designed) can help us to make (near) optimal decisions while the initial search space is very large.
The first part of my talk will be focused on a block cipher construction called Generalized Feistel Networks, which is a very efficient construction in general. The idea of this construction is to split the plaintext into an even number of blocks, apply a Feistel transformation in parallel to those blocks, and then permute them. In a recent paper accepted at FSE 2020, we showed how to choose this permutation to be optimal w.r.t a certain criterion (called "diffusion round"), which also allowed us to solve a 10-year old problem.
The second part of my talk will be based on a paper accepted at SAC 2018, which aims at replacing the key-schedule of AES by a more efficient one (namely a permutation). AES is the current symmetric encryption standard, and its key-schedule (i.e. the algorithm allowing to transform the master key into several round keys) is quite intricate and costly. We thus studied what would happen if we replace this key-schedule by a permutation, and especially we show that we can get better security arguments even though we have a much simpler key-schedule.
Claire Delaplace (Ruhr Universitat Bochum)
Improved Low-Memory Subset Sum and LPN Algorithms via Multiple Collisions
For enabling post-quantum cryptanalytic experiments on a meaningful scale, there is a strong need for low-memory algorithms. We show that the combination of techniques from representations, multiple collisions finding, and the Schroeppel-Shamir algorithm leads to improved low-memory algorithms. For random subset sum instances (a1, ..., an, t) defined modulo 2n, our algorithms improve over the Dissection technique for small memory M < 20.02n and in the mid-memory regime 20.13n < M < 20.2n. An application of our technique to LPN of dimension k and constant error p yields significant time complexity improvements over the Dissection-BKW algorithm from Crypto 2018 for all memory parameters M < 20.35 k/log k. This is joint work with Andre Esser and Alexander May, published at IMACC 2019.
Rémi Clarisse (IRMAR, Université de Rennes 1, Orange Labs)
A group signature scheme in the generic group model
Group signatures, introduced by Chaum and van Heyst in 1991, enable members of a group to sign on behalf of the group, thanks to a certificate delivered by a group manager. The point is that the signature is anonymous: it cannot be traced back to its issuer, except for a specific entity, the opening authority, which can "open" any valid group signature.
Bellare, Micciancio and Warinschi provided the first security model where they defined the requirements for a group signature: anonymity, traceability and non-frameability.
Combining seemingly contradictory properties such as anonymity and traceability can be done through the Sign-Encrypt-Prove (SEP) framework: after signing a message, a group member encrypts both the signature and the certificate then proves in a non-interactive zero-knowledge fashion that everything is well formed (the opening authority still has a way to rescind the anonymity).
In this talk, we will construct a group signature that uses the nice interaction between Pointcheval and Sanders signatures and Fuchsbauer, Hanser and Slamanig equivalence-class signatures. Group members then remain anonymous by simply randomizing their signatures and certificates: there is no longer the need for encryption and proving.
Xavier Bonnetain (Inria Paris)
Quantum cryptanalysis using hidden structures
slides: pdf (1308 kb)
Quantum cryptanalysis studies how to break cryptographic primitives using a quantum computer. Multiple quantum algorithms for the hidden shift and hidden period problems have been proposed, which are vastly more efficient than their classical counterpart. In this talk, I will present some of them, with their applications to cryptanalyze multiple symmetric constructions and some isogeny-based key exchanges.
In more details, I will present Simon's algorithm and its use to attack multiple symmetric constructions using quantum queries, the offline approach which allows to apply it even if we restrict to classical queries, and some abelian hidden shift algorithms and their applications against some symmetric constructions and the key exchange CSIDH.
Thursday July 4, 2019
Simon Masson (Thales)
Verifiable delay functions from supersingular isogenies and pairings
Amphi C, 10:15
Franz Brausse (University of Trier)
Applying Exact Real Arithmetic to Solving Non-linear Constraints
slides: pdf (595 kb)
Exact Real Arithmetic describes implementations of Computable Analysis (TTE), which can be considered as a foundation of reliable computations on continuous data. TTE is a framework which allows to express and reason about real number computations on digital computers or Turing machines in a rigorous manner.
In this talk we report on recent applications of ERA to check satisfiability of non-linear constraints over real numbers in the setting of SMT. In particular, we present a CDCL style calculus and a prototypical implementation for this concrete problem. The procedure is based on symbolical methods to resolve linear conflicts and on exact real computations to construct linearisations of non-linear constraints local to conflicts. This approach allows the treatment of a large number of computable non-linear real functions such as polynomials, rational or trigonometrical functions and beyond. The implementation has been evaluated on several non-linear SMT-Lib examples and the results have been compared with state-of-the-art SMT solvers.
Diego Aranha (University of Aarhus, Denmark)
Implement all the pairings in software!
Since their first introduction as a cryptanalytic tool, bilinear pairings have moved to a very useful building block for cryptography. Current real-world applications of pairings include remote attestation, privacy-preserving blockchains and identity-based cryptography. However, recent advances in the discrete log computation have reduced the efficiency of the most popular pairing constructions and previous candidates for standardization. This talk discusses ongoing work regarding the efficient implementation of new sets of parameters for bilinear groups, to adjust security against these recent attacks, and techniques for their realization in software.
Robin Larrieu (LIX, Ecole Polytechnique)
Fast polynomial reduction for generic bivariate ideals
A008, 10:30
Let A, B in K[X,Y] be two bivariate polynomials over an effective field K, and let G be the reduced Gröbner basis of the ideal I := ⟨A,B⟩ generated by A and B, with respect to some weighted-degree lexicographic order. Under some genericity assumptions on A,B, we will see how to reduce a polynomial with respect to G with quasi-optimal complexity, despite the fact that G is much larger than the intrinsic complexity of the problem. For instance, if A,B have total degree n, that is O(n2) coefficients, then G has O(n3) coefficients but reduction can be done in time O(n2).
We will consider two situations where these new ideas apply, leading to different algorithms:
First, there is a class called "vanilla Gröbner bases" for which there is a so-called terse representation that, once precomputed, allows to reduce any polynomial P in time O(n2). In this setting, assuming suitable precomputation, multiplication and change of basis can therefore be done in time O(n2) in the quotient algebra K[X,Y] / ⟨A,B⟩.
Then, we assume that A and B are given in total degree and we consider the usual degree lexicographic order. Although the bases are not vanilla in this case, they admit a so-called concise representation with similar properties. Actually, the precomputation can also be done efficiently in this particular setting: from the input A,B, one can compute a Gröbner basis in concise representation in time O(n2). As a consequence, multiplication in K[X,Y] / ⟨A,B⟩ can be done in time O(n2) including the cost of precomputation.
Ben Smith (Inria Saclay, Ecole Polytechnique)
Progress and hard problems in commutative isogeny-based cryptography
In this talk we consider CSIDH, a new post-quantum key-exchange algorithm based on the hardness of constructing an unknown isogeny between two given elliptic curves. We will describe some algorithmic improvements, and also some important relations between the underlying "hard" problems in both the quantum and classical worlds.
Tuesday April 2, 2019
Matías Bender (Polsys team, LIP6, Sorbonne Universités/Inria/CNRS)
Gröbner bases and sparse polynomial systems
Gröbner bases are one the most powerful tools in algorithmic non-linear algebra. Their computation is an intrinsically hard problem with complexity at least single exponential in the number of variables. However, in most of the cases, the polynomial systems coming from applications have some kind of structure. For example, several problems in computer-aided design, robotics, vision, biology, kinematics, cryptography, and optimization involve sparse systems where the input polynomials have a few non-zero terms.
An approach to exploit sparsity is to embed the systems in a semigroup algebra and to compute Gröbner bases over this algebra. Prior to our work, the algorithms that follow this approach benefited from the sparsity only in the case where all the polynomials have the same sparsity structure, that is the same Newton polytope. In this talk, I will present the first algorithm that overcomes this restriction. Under regularity assumptions, it performs no redundant computations. Further, I will use it to compute Gröbner basis in the standard algebra and solve sparse polynomials systems over the torus. The complexity of the algorithm depends on the Newton polytopes and it is similar to the complexity of the solving techniques involving the sparse resultant. Hence, this algorithm closes a 25-years gap between the strategies to solve systems using resultants and Gröbner bases. Additionally, for particular families of sparse systems, I will use the multigraded Castelnuovo-Mumford regularity to improve the complexity bounds.
This is joint work with Jean-Charles Faugère and Elias Tsigaridas.
Thursday February 14, 2019
Sébastien Duval (Université catholique de Louvain, Belgique)
MDS and almost-MDS matrices with lightweight circuits
MDS matrices are essential in symmetric-key cryptography since they provide optimal diffusion in block ciphers. Many MDS matrices are known, but a problem remains which gathers a lot of attention: finding lightweight MDS matrices.
Several approaches exist, namely reducing the cost of already-known MDS matrices (to improve existing ciphers) and finding new MDS matrices lighter than the known ones (to make new ciphers).
We focus on the second case and, contrarily to the usual approach, we will not look for matrices whose coefficients are lightweight to implement. Rather than this local optimization, we will prefer a global optimization of the whole matrix, which allows reusing of intermediate values.
We propose an algorithm to search for lightweight formal MDS matrices on a polynomial ring, by enumerating circuits until we reach an MDS matrix. This approach allows us to get much better results than previous works. We also adapt this algorithm to look for almost-MDS matrices, which offer a trade-off between cost and security.
Yann Rotella (Radboud University, Nijmegen, Pays-Bas)
On the Security of Goldreich's Pseudorandom Generator
Almost two decades ago, Goldreich proposed a one-way function construction where output bits depends in a constant number of input bits. Afterwards, this one-way function construction was extended to Pseudo Random Generators.
In the polynomial regime, where the seed is of size n and the output of size ns for s > 1, the only known solution, commonly known as Goldreich's PRG, proceeds by applying a simple d-ary predicate (Boolean function of d variables) to public random size-d subsets of the bits of the seed.
The security of this PRG has been widely investigated, and was proven to be secure against a large class of attacks. However, by applying symmetric cryptanalysis techniques such as Guess-and-determine and Gröbner Basis, we were able to find sub-exponential time algorithms that recover the seed of this PRG. Moreover, as the predicate can be identified as a Boolean function, we also derive new security criteria that rely on the predicate that is used in this PRG. Finally, the complexity of our algorithms is counted in a non-asymptotic way, leading to a better understanding of the security of this Pseudo-Random Generator.
The results of this work were published at ASIACRYPT 2018, and a full version of the paper can be found at this link.
Joint work with Geoffroy Couteau and Aurélien Dupin and Pierrick Méaux and Mélissa Rossi.
Léo Perrin (Inria Paris, Equipe Secret)
S-Box Decompositions and some Applications
S-boxes are components used by many symmetric cryptographic algorithms, most prominently the AES. They are small non-linear functions whose properties are crucial in ensuring the security of the whole cipher against some attacks.
In this talk, I will present my work on the decomposition of S-boxes. Intuitively, a decomposition is a "simple" algorithm evaluating the S-box. The knowledge of a decomposition can allow a significantly improved implementation but, on the other hand, it can also reveal cryptographic flaws as particular decompositions can allow attacks. In fact, we could imagine that a malevolent designer would purposefully hide such a "backdooring" decomposition in their S-box. As an illustration, I will present some results on the S-box used by the last two Russian standards in symmetric cryptography. It turns out to exhibit very strong algebraic properties that warrant further investigation into the security of these algorithms.
Then, I will present a more surprising application of these techniques to a more theoretical problem. We were able to find a decomposition of the only known solution of a long-standing open problem in the design of S-boxes, namely the big APN problem (it deals with constructing S-boxes with optimal resistance against differential attack). This allowed us to find the first generalization of this function---though we could not use them to solve the big APN problem. On the bright side, the type of decomposition we found in this permutation has a deeper meaning than we first expected as it turns out to play a key role in "CCZ-equivalence" (a form of equivalence between S-boxes).
I will conclude with several open problems that have been opened by this line of research.
Sudarshan SHINDE (Inria Paris, Equipe Ouragan)
A complete classification of ECM-friendly families using modular curves
The elliptic curve method (ECM) is a factorization algorithm widely used in cryptography. It was proposed in 1985 by Lenstra and improved a couple of months later by Montgomery using well-chosen families of curves.
Algorithm succeeds in factoring if the elliptic curves it uses have desirable smoothness properties. These smoothness properties can be quantified using the mean valuation of a prime l in the cardinality of an elliptic curve E modulo random primes.
In this joint work with Razvan Barbulescu, we present a brief survey of previous works in this regard and then present two approaches to find these ECM-friendly elliptic curves. Combined with recent works of [1] and [2], we obtain a complete list of 1540 such families.
Cyril Bouvier (Laboratoire d'Informatique, de Robotique et de Microélectronique de Montpellier (LIRMM))
Faster cofactorization with ECM using mixed representations
The Elliptic Curve Method (ECM) is the asymptotically fastest known method for finding medium-size prime factors of large integers. It is also a core ingredient of the Number Field Sieve (NFS) for integer factorization and its variants for computing discrete logarithms. In NFS and its variants, ECM is extensively used in the cofactorization step (a subroutine of the sieving phase) which consists of breaking into primes billions of composite integers of a hundred-ish bits.
In this talk, we propose a novel implementation of ECM in the context of NFS that requires fewer modular multiplications than any other publicly available implementation. The main ingredients are: Dixon and Lenstra's idea on primes combination, fast point tripling, optimal double-base decompositions and Lucas chains, and a good mix of Edwards and Montgomery representations.
Simon Abelard (Caramba team)
Point-counting on hyperelliptic curves defined over finite fields of large characteristic: algorithms and complexities (PhD defense)
C005, 14:30
Counting points on algebraic curves has drawn a lot of attention due to its many applications from number theory and arithmetic geometry to cryptography and coding theory. In this thesis, we focus on counting points on hyperelliptic curves over finite fields of large characteristic p. In this setting, the most suitable algorithms are currently those of Schoof and Pila, because their complexities are polynomial in log q. However, their dependency in the genus g of the curve is exponential, and this is already painful even in genus 3.
Our contributions mainly consist of establishing new complexity bounds with a smaller dependency in g of the exponent of log p. For hyperelliptic curves, previous work showed that it was quasi-quadratic, and we reduced it to a linear dependency. Restricting to more special families of hyperelliptic curves with explicit real multiplication (RM), we obtained a constant bound for this exponent.
In genus 3, we proposed an algorithm based on those of Schoof and Gaudry-Harley-Schost whose complexity is prohibitive in general, but turns out to be reasonable when the input curves have explicit RM. In this more favorable case, we were able to count points on a hyperelliptic curve defined over a 64-bit prime field.
Aurore Guillevic (Caramba team)
A New Family of Pairing-Friendly Elliptic Curves
There have been recent advances in solving the finite extension field discrete logarithm problem as it arises in the context of pairing-friendly elliptic curves. This has lead to the abandonment of approaches based on supersingular curves of small characteristic, and to the reconsideration of the field sizes required for implementation based on non-supersingular curves of large characteristic. This has resulted in a revision of recommendations for suitable curves, particularly at a higher level of security. Indeed for AES-256 levels of security the BLS48 curves have been suggested, and demonstrated to be superior to other candidates. These curves have an embedding degree of 48. The well known taxonomy of Freeman, Scott and Teske only considered curves with embedding degrees up to 50. Given some uncertainty around the constants that apply to the best discrete logarithm algorithm, it would seem to be prudent to push a little beyond 50. In this note we announce the discovery of a new family of pairing friendly elliptic curves which includes a new construction for a curve with an embedding degree of 54.
Joint work with Michael Scott (Miracl).
Svyatoslav Covanov (Caramba team)
Multiplication algorithms: bilinear complexity and fast asymptotic methods (PhD defense)
Since 1960 and the result of Karatsuba, we know that the complexity of the multiplication (of integers or polynomials) is sub-quadratic: given a ring R, the product in R[X] of polynomials a0 + a1 X and b0 + b1 X, for any a0, a1, b0 and b1 in R, can be computed with three and not four multiplications over R: (a0 + a1 X) (b0 + b1 X) = m0 + (m2 - m0 - m1)X + m1 X2, with the three multiplications m0 = a0b0, m1 = a1b1, and m2 = (a0 + a1)(b0 + b1). In the same manner, Strassen's algorithm allows one to multiply two matrices 2nx2n with only seven products of matrices nxn.
The two previous examples fall in the category of bilinear maps: these are functions of the form Phi : Km x Kn -> Kl, given a field K, linear in each variable. Among the most classical bilinear maps, we have the multiplication of polynomials, matrices, or even elements of algebraic extension of finite fields. Given a bilinear map Phi, computing the minimal number of multiplications necessary to the evaluation of this map is a NP-hard problem. The purpose of this thesis is to propose algorithms minimizing this number of multiplications. Two angles of attack have been studied.
The first aspect of this thesis is to study the problem of the computation of the bilinear complexity under the angle of the reformulation of this problem in terms of research of matrix subspaces of a given rank. This work led to an algorithm taking into account intrinsic properties of the considered products such as matrix or polynomial products over finite fields. This algorithm allows one to find all the possible decompositions, over F2, for the product of polynomials modulo X5 and the product of matrices 3x2 by 2x3.
Another aspect of this thesis was the development of fast asymptotic methods for the integer multiplication. There is a particular family of algorithms that has been proposed after an article by Fürer published in 2007. This article proposed a first algorithm, relying on fast Fourier transform (FFT), allowing one to multiply n-bit integers in O(n log n 2O(log*n)), where log* is the iterated logarithm function. In this thesis, an algorithm, relying on a number theoretical conjecture, has been proposed, involving the use of FFT and generalized Fermat primes. With a careful complexity analysis of this algorithm, we obtain a complexity in O(n log n 4log*n).
Monday June 4, 2018
Chitchanok Chuengsatiansup (ARIC Team, LIP, ENS Lyon)
Optimizing multiplications with vector instructions
In this talk, I will explain techniques to achieve fast and secure implementations. I will introduce a vector unit, which is a part of a CPU, and ways to utilize it. I will also briefly emphasize the importance of and ways to prevent software side-channel attacks. Then, I will explain how to optimize scalar multiplication in Curve41417 and polynomial multiplication in Streamlined NTRU Prime 4591761. Karatsuba's method plays an important role in the former case, while combinations of Karatsuba's method and Toom--Cook's method are crucial in the latter case. Both implementations utilize the CPU's vector unit.
Friday June 1, 2018
Virginie Lallemand (Ruhr Universität Bochum)
cryptanalyse de systèmes de chiffrement symétrique
Alexandre Wallet (ARIC Team, LIP, ENS Lyon)
On the harvesting phase in Index-calculus over algebraic curves
This talk focuses on the relation collection phase (or harvesting) in Index-Calculus algorithms to compute discrete logs in an algebraic curve. I will present results from my thesis, improving two different types of harvesting. First, in the general case, we propose a sieving approach, which is a time/memory trade-off that can improve the practical running time of the algorithms. Second, when the target curve is defined over an extension of a finite field, the harvesting can be presented as the resolutions of multivariate systems. When the curve is hyperelliptic and the field has characteristic two, we show that several equations in the involved systems are "squares". We then exploit this property to improve the asymptotic complexity of the harvesting phase. The impact on practical computations allows to estimate the running time of a complete Index-calculus on a curve whose associated group has ~2184 elements. In term of practical difficulty, it compares to recent records over finite fields.
Joint work with Vanessa Vitse and Jean-Charles Faugère.
Benjamin Wesolowski (EPFL)
Mildly short vectors in cyclotomic ideal lattices in quantum polynomial time
We study the geometry of units and ideals of cyclotomic rings, and derive an algorithm to find a mildly short vector in any given cyclotomic ideal lattice in quantum polynomial time, under some plausible number-theoretic assumptions. More precisely, it finds an approximation of the shortest vector by a subexponential factor. This result exposes an unexpected hardness gap between these structured lattices and general lattices: the best known polynomial time generic lattice algorithms can only reach an exponential approximation factor. Following a recent series of attacks, these results call into question the hardness of various problems over structured lattices, such as Ideal-SVP and Ring-LWE, upon which relies the security of a number of cryptographic schemes.
This is a joint work with Ronald Cramer, Léo Ducas, Chris Peikert, and Oded Regev.
Matthieu Rambaud (Inria Saclay)
Dense families of curves over prime fields with many points after field extension
A folklore conjecture (Cascudo-Cramer-Xing-Yang 2012, Lemma IV.4 and Ballet-Pieltant-R.-Sijsling 2017 §3 for counterexamples) states that, for all p prime number and 2t an even integer, there exists a family of function fields over the prime field Fp such that:
(i) the genera tend to infinity;
(ii) the ratio of two successive genera tends to 1 (density condition) and
(iii) after field extension to Fp2t, the asymptotic number of points reaches the Ihara bound.
The only cases known so far are for t=1, with the classical modular curves X0(N) over Fp. We present here an explicit family of Shimura curves solving the case p=3 and 2t=6.
The talk will recall facts on the descent of covering maps, and illustrate recent numerical methods of Sijsling-Voight.
Tuesday November 7, 2017
Léo Ducas (CWI)
Enumeration in Lattices for the Number Field Sieve
summary in french
Laurent Imbert (ECO team, LIRMM)
Randomized Mixed-Radix Scalar Multiplication
Chloe Martindale (Technical University of Eindhoven, Netherlands)
Isogeny graphs, modular polynomials, and point counting for higher genus curves.
We recap the theory of modular polynomials and isogeny graphs of (ordinary) elliptic curves and give a natural generalisation to abelian g-folds (e.g. the Jacobian of a genus g curve). This is joint work with Marco Streng.
We then recap the algorithm of Schoof, Elkies, and Atkin for efficient point counting on elliptic curves over finite fields, and show how to generalise the algorithm to genus 2 curves over finite fields (with maximal real multiplication), using our generalisation of modular polynomials to genus 2. Under some heuristic assumptions, this is the fastest known algorithm to count points on genus 2 curves over large prime fields. This is joint work with Sean Ballentine, Aurore Guillevic, Elisa Lorenzo-Garcia, Maike Massierer, Ben Smith, and Jaap Top.
Vincent Neiger (AriC team, LIP, ÉNS Lyon -- Univ. of Waterloo (ON, CA))
Fast computation of normal forms of polynomial matrices
In this talk, we present recent results about fast algorithms for fundamental operations for matrices over K[X] (univariate polynomials). We mainly focus on the computation of normal forms, obtained by transforming the input matrix via elementary row operations. Depending on a degree measure specified by a "shift", these normal forms range from the Hermite form, which is triangular, to the Popov form, which minimizes the degrees of the rows.
For Popov or Hermite forms of m x m nonsingular matrices with entries of degree < d, the previous fastest algorithms use O~(m^w d) operations, where w is the exponent of K-linear algebra. We will discuss improvements in three directions: (1) improving the cost bound to O~(m^w D/m), where D/m is a type of average degree of the input matrix; (2) derandomizing Hermite form computation [*]; (3) achieving fast computation for arbitrary shifts. The last point will lead us to consider the problem of solving systems of linear modular equations over K[X], which generalizes Hermite-Pade approximation and rational reconstruction.
[*] joint work with George Labahn and Wei Zhou (U. Waterloo, ON, Canada).
Cécile Pierrot (Almasty team, LIP6, UPMC, Paris)
Génération d'aléa public et Bitcoin
Karim Bigou (Équipe CAIRN, IRISA, Lannion)
Calcul modulaire en RNS pour des implantations de cryptographie à clef publique
Monday November 23, 2015
Chenqi Mou (School of Mathematics and Systems Science, Beihang University, China)
Sparse FGLM algorithms for solving polynomial systems
Groebner basis is an important tool in computational ideal theory, and the term ordering plays an important role in the theory of Groebner bases. In particular, the common strategy to solve a polynomial system is to first compute the basis of the ideal defined by the system w.r.t. DRL, change its ordering to LEX, and perhaps further convert the LEX Groebner basis to triangular sets.
Given a zero-dimensional ideal I ⊂ 𝕂[x1,...,xn] of degree D, the transformation of the ordering of its Groebner basis from DRL to LEX turns out to be the bottleneck of the whole solving process. Thus it is of crucial importance to design efficient algorithms to perform the change of ordering.
In this talk we present several efficient methods for the change of ordering which take advantage of the sparsity of multiplication matrices in the classical FGLM algorithm. Combining all these methods, we propose a deterministic top-level algorithm that automatically detects which method to use depending on the input. As a by-product, we have a fast implementation that is able to handle ideals of degree over 60000. Such an implementation outperforms the Magma and Singular ones, as shown by our experiments.
First for the shape position case, two methods are designed based on the Wiedemann algorithm: the first is probabilistic and its complexity to complete the change of ordering is O(D(N1+n log(D)2)), where N1 is the number of nonzero entries of a multiplication matrix; the other is deterministic and computes the LEX Groebner basis of $√I$ via Chinese Remainder Theorem. Then for the general case, the designed method is characterized by the Berlekamp-Massey-Sakata algorithm from Coding Theory to handle the multi-dimensional linearly recurring relations. Complexity analyses of all proposed methods are also provided.
Furthermore, for generic polynomial systems, we present an explicit formula for the estimation of the sparsity of one main multiplication matrix, and prove that its construction is free. With the asymptotic analysis of such sparsity, we are able to show that for generic systems the complexity above becomes O(√(6/n π)D2+(n-1)/n).
This talk is based on joint work with Jean-Charles Faugere.
Frédéric Bihan (Université de Savoie Mont Blanc)
Polynomial systems with many positive solutions from bipartite triangulations
We use a version of Viro's method to construct polynomial systems with many positive solutions. We show that if a polytope admits an unimodular regular triangulation whose dual graph is bipartite, then there exists an unmixed polynomial system with this polytope as Newton polytope and which is maximally positive in that all its toric complex solutions are in fact real positive solutions. We present classical families of polytopes which admit such triangulations. These examples give evidence in favor of a conjecture due to Bihan which characterizes affine relations inside the support of a maximally polynomial system. We also use our construction to get polynomial systems with many positive solutions considering a simplicial complex contained in a regular triangulation of the cyclic polytope. This is joint work with Pierre-Jean Spaenlehauer (INRIA Nancy).
Jan Tuitman (Department of Mathematics, KU Leuven, Belgium)
Counting points on curves: the general case
Kedlaya's algorithm computes the zeta function of a hyperelliptic curve over a finite field using the theory of p-adic cohomology. We have recently dev eloped and implemented a generalisation of this algorithm that works for (almost) any curve. First, we will outline the theory involved. Then we will describe our algorithm and illustrate the main ideas by giving some examples. Finally, if time permits, we will talk about some current and future work of ours with various coauthors on improving the algorithm and applying it in other settings.
Roland Wen (Univ. of New South Wales, Sydney)
Engineering Cryptographic Applications: Leveraging Recent E-Voting Experiences in Australia to Build Failure-Critical Systems
Advanced, bespoke cryptographic applications are emerging for large-scale use by the general population. A good example is cryptographic electronic voting systems, which make extensive use of sophisticated cryptographic techniques to help attain strong security properties (such as secrecy and verifiability) that are required due to the failure-critical nature of public elections. Recently in Australia, cryptographic e-voting systems were used in two state elections: iVote, an Internet voting system, was used in New South Wales, and vVote, a polling place voting system, was used in Victoria.
However developing and deploying such complex and critical cryptographic applications involves a range of engineering challenges that have yet to be addressed in practice by industry and the research community. As with any complex, large-scale system, there were barriers to applying appropriately rigorous engineering practices in the two Australian e-voting systems. But since these e-voting systems are critical national infrastructure, such engineering practices are needed to provide high assurance of the systems and their required properties.
In this talk I will discuss some of the engineering challenges, practical barriers and issues, and what can be learned from the two recent Australian e-voting experiences.
Matthieu Rambaud (Télécom ParisTech)
Comment trouver de bons algorithmes de multiplication par interpolation ?
Sebastian Kochinke (Universität Leipzig)
The Discrete Logarithm Problem on non-hyperelliptic Curves of Genus g > 3.
We consider the discrete logarithm problem in the degree-0 Picard group of non-hyperelliptic curves of genus g > 3. A well known method to tackle this problem is by index calculus. In this talk we will present two algorithms based on the index calculus method. In both of them linear systems of small degree are generated and then used to produce relations. Analyzing these algorithms leads to several geometric questions. We will discuss some of them in more detail and state further open problems. At the end we will show some experimental results.
Enea Milio (LFANT team, Institut de Mathématiques de Bordeaux)
Calcul des polynômes modulaires en genre 2
Thomas Richard (CARAMEL team, LORIA)
Cofactorization strategies for NFS
Nowadays, when we want to factor large numbers, we use sieve algorithms like the number field sieve or the quadratic sieve. In these algorithms, there is an expensive part called the relation collection step. In this step, one searches a wide set of numbers to identify those which are smooth, i.e. integers where all prime divisors are less than a particular bound. In this search, a first step finds the smallest prime divisors with a sieving procedure and a second step tries to factor the remaining integers which are no longer divisible by small primes, using factoring methods like P-1, P+1 and ECM. This talk will introduce a procedure, following Kleinjung, to optimize the second step!
Andrea Miele (LACAL, EPFL, Lausanne)
Post-sieving on GPUs
The number field sieve (NFS) is the fastest publicly known algorithm for factoring RSA moduli. We show how the post sieving step, a compute-intensive part of the relation collection phase of NFS, can be farmed out to a graphics processing unit. Our implementation on a GTX 580 GPU, which is integrated with a state-of-the-art NFS implementation, can serve as a cryptanalytic co-processor for several Intel i7-3770K quad-core CPUs simultaneously. This allows those processors to focus on the memory-intensive sieving and results in more useful NFS-relations found in less time.
Christian Eder (Department of Mathematics, University of Kaiserslautern)
Computing Groebner Bases
In 1965 Buchberger introduced a first algorithmic approach to the computation of Groebner Bases. Over the last decades optimizations to this basic attempt were found. In this talk we discuss two main aspects of the computation of Groebner Bases: Predicting zero reductions is essential to keep the computational overhead and memory usage at a low. We show how Faugère's idea, initially presented in the F5 algorithm, can be generalized and further improved. The 2nd part of this talk is dedicated to the exploitation of the algebraic structures of a Groebner Basis. Thus we are not only able to replace polynomial reduction by linear algebra (Macaulay matrices, Faugère's F4 algorithm), but we can also specialize the Gaussian Elimination process for our purposes.
Nicholas Coxon (CARAMEL team, LORIA)
Nonlinear polynomials for NFS factorisation
To help minimise the running time of the number field sieve, it is desirable to select polynomials with balanced degrees in the polynomial selection phase. I will discuss algorithms for generating polynomial pairs with this property: those based on Montgomery's approach, which reduce the problem to the construction of small modular geometric progressions; and an algorithm which draws on ideas from coding theory.
Irene Márquez-Corbella (GRACE team, LIX, École Polytechnique)
Une attaque polynomiale du schéma de McEliece basé sur les codes géométriques
Guillaume Moroz (Vegas team, LORIA)
Évaluation et composition rapide de polynômes
Pierre-Jean Spaenlehauer (CARAMEL team, LORIA)
A Newton-like iteration and algebraic methods for Structured Low-Rank Approximation
Given an linear/affine space of matrices E with real entries, a data matrix U ∈ E and a target rank r, the Structured Low-Rank Approximation Problem consists in computing a matrix M ∈ E which is close to U (with respect to the Frobenius norm) and has rank at most r. This problem appears with different flavors in a wide range of applications in Engineering Sciences and symbolic/numeric computations.
We propose an SVD-based numerical iterative method which converges locally towards such a matrix M. This iteration combines features of the alternating projections algorithm and of Newton's method, leading to a proven local quadratic rate of convergence under mild tranversality assumptions. We also present experimental results which indicate that, for some range of parameters, this general algorithm is competitive with numerical methods for approximate univariate GCDs and low-rank matrix completion (which are instances of Structured Low-Rank Approximation).
In a second part of the talk, we focus on the algebraic structure and on exact methods to compute symbolically the nearest structured low-rank matrix M to a given matrix U ∈ E with rational entries. We propose several ways to compute the algebraic degree of the problem and to recast it as a system of polynomial equations in order to solve it with algebraic methods.
The first part of the talk is a joint work with Eric Schost, the second part is a joint work with Giorgio Ottaviani and Bernd Sturmfels.
Armand Lachand (Équipe Théorie des Nombres, IECL)
Quelques perspectives mathématiques sur la sélection polynomiale dans le crible algébrique NFS
Luca De Feo (CRYPTO, PRiSM, Univ. Versailles Saint-Quentin)
Algorithms for Fp
Realizing in software the algebraic closure of a finite field Fp is equivalent to construct so called "compatible lattices of finite fields", i.e. a collection of finite extensions of Fp together with embeddings Fpm ⊂ Fpn whenever m | n.
There are known algorithms to construct compatible lattices in deterministic polynomial time, but the status of the most practically efficient algorithms is still unclear. This talk will review the classical tools available, then present some new ideas towards the efficient construction of compatible lattices, possibly in quasi-optimal time.
Friday December 6, 2013
Cécile Pierrot (PolSys team, LIP6)
Crible Spécial sur Corps de Nombres (SNFS) – Application aux courbes elliptiques bien couplées.
Clément Pernet (AriC team, LIP, ENS Lyon)
Calcul de formes echelonnées et des profils de rang
Thursday November 7, 2013
Julia Pieltant (GRACE team, LIX, École Polytechnique)
Algorithme de type Chudnovsky pour la multiplication dans les extensions finies de Fq.
Bastien Vialla (LIRMM)
Un peu d'algèbre linéaire
Alice Pellet-Mary (CARAMEL team, LORIA)
Test rapide de cubicité modulaire
Svyatoslav Covanov (CARAMEL team, LORIA)
Implémentation efficace d'un algorithme de multiplication de grands nombres
Hamza Jeljeli (CARAMEL team, LORIA)
RNS Arithmetic for Linear Algebra of FFS and NFS-DL algorithms
Computing discrete logarithms in large cyclic groups using index-calculus-based methods, such as the number field sieve or the function field sieve, requires solving large sparse systems of linear equations modulo the group order. In this talk, we present how we use the Residue Number System (RNS) arithmetic to accelerate modular operations. The first part deals with the FFS case, where the matrix contains only small values. The second part discusses how we treat for NFS-DL, the particular dense columns corresponding to Schirokauer's maps, where the values are large.
Mourad Gouicem (PEQUAN team, LIP6)
Fractions continues et systèmes de numérations : applications à l'implémentation de fonctions élémentaires et à l'arithmétique modulaire
Friday April 5, 2013
François Morain (GRACE team, LIX)
ECM using number fields
Antoine Joux (CryptoExperts / CRYPTO, PRiSM, Univ. Versailles Saint-Quentin)
Logarithmes discrets dans les corps finis. Application en caractéristique "moyenne".
Adeline Langlois (AriC, LIP, ENS Lyon)
Classical Hardness of Learning with Errors
The decision Learning With Errors (LWE) problem, introduced by Regev in 2005 has proven an invaluable tool for designing provably secure cryptographic protocols. We show that LWE is classically at least as hard as standard worst-case lattice problems, even with polynomial modulus. Previously this was only known under quantum reductions.
Our techniques capture the tradeoff between the dimension and the modulus of LWE instances, leading to a much better understanding of the landscape of the problem. The proof is inspired by techniques from several recent cryptographic constructions, most notably fully homomorphic encryption schemes.
This work has been done with Z. Brakerski, C. Peikert, O. Regev and D. Stehlé.
Friday February 22, 2013
Jérémy Parriaux (CRAN, Univ. de Lorraine)
Contrôle, synchronisation et chiffrement
Emmanuel Jeandel (Équipe CARTE, LORIA)
Quête du plus petit jeu de tuiles apériodique
Maike Massierer (Mathematisches Institut, Universität Basel)
An Efficient Representation for the Trace Zero Variety
The hardness of the (hyper)elliptic curve discrete logarithm problem over extension fields lies in the trace zero variety. A compact representation of the points of this abelian variety is needed in order to accurately assess the hardness of the discrete logarithm problem there. Such representations have been proposed by Lange for genus 2 curves and by Silverberg for elliptic curves. We present a new approach for elliptic curves. It is made possible by a new equation for the variety derived from Semaev's summation polynomials. The new representation is optimal in the sense that it reflects the size of the group, it is compatible with the structure of the variety, and it can be computed efficiently.
Pierre-Jean Spaenlehauer (ORCCA, University of Western Ontario)
Résolution de systèmes polynomiaux structurés et applications en Cryptologie
Alin Bostan (Projet SpecFun, INRIA Saclay)
Computer algebra for the enumeration of lattice walks
Classifying lattice walks in restricted lattices is an important problem in enumerative combinatorics. Recently, computer algebra methods have been used to explore and solve a number of difficult questions related to lattice walks. In this talk, we will give an overview of recent results on structural properties and explicit formulas for generating functions of walks in the quarter plane, with an emphasis on the algorithmic methodology.
Mohab Safey El Din (Projet PolSys, LIP6, UPMC-IUF-INRIA)
Computer Algebra Algorithms for Real Solving Polynomial Systems: the Role of Structures
Real solving polynomial systems is a topical issue for many applications. Exact algorithms, using computer algebra techniques, have been deployed to answer various specifications such that deciding the existence of solutions, answer connectivity queries or one block-real quantifier elimination. In this talk, we will review some recent on-going works whose aims are to exploit algebraic and geometric properties in order to provide faster algorithms in theory and in practice.
Paul Zimmermann (Projet CARAMEL, LORIA)
Sélection polynomiale dans CADO-NFS
Alexandre Benoit (Lycée Alexandre Dumas, Saint Cloud)
Multiplication quasi-optimale d'opérateurs différentiels
Christophe Petit (Crypto Group, Université Catholique de Louvain)
On polynomial systems arising from a Weil descent
Polynomial systems of equations appearing in cryptography tend to have special structures that simplify their resolution. In this talk, we discuss a class of polynomial systems arising after deploying a multivariate polynomial equation over an extension field into a system of polynomial equations over the ground prime field (a technique commonly called Weil descent).
We provide theoretical and experimental evidence that the degrees of regularity of these systems are very low, in fact only slightly larger than the maximal degrees of the equations.
We then discuss cryptographic applications of (particular instances of) these systems to the hidden field equation (HFE) cryptosystem, to the factorization problem in SL(2, 2n) and to the elliptic curve discrete logarithm over binary fields. In particular, we show (under a classical heuristic assumption in algebraic cryptanalysis) that an elliptic curve index calculus algorithm due to Claus Diem has subexponential time complexity O(2c n2/3 log n) over the binary field GF(2n), where c is a constant smaller than 2.
Based on joint work with Jean-Charles Faugère, Ludovic Perret, Jean-Jacques Quisquater and Guénaël Renault.
Hugo Labrande (ENS de Lyon)
Accélération de l'arithmétique des corps CM quartiques
Laura Grigori (Projet Grand Large, INRIA Saclay-Île de France, LRI)
Recent advances in numerical linear algebra and communication avoiding algorithms
Numerical linear algebra operations are ubiquitous in many challenging academic and industrial applications. This talk will give an overview of the evolution of numerical linear algebra algorithms and software, and the major changes such algorithms have undergone following the breakthroughs in the hardware of high performance computers. A specific focus of this talk will be on communication avoiding algorithms. This is a new and particularly promising class of algorithms, which has been introduced in the late 2008 as an attempt to address the exponentially increasing gap between computation time and communication time - one of the major challenges faced today by the high performance computing community. I will also discuss novel preconditioning techniques for accelerating the convergence of iterative methods.
Francisco Rodríguez-Henríquez (CINVESTAV, IPN, México)
Computing square roots over prime extension fields
Taking square roots over finite fields is a classical number theoretical problem that has capture the attention of researchers across the centuries. Nowadays, the computation of square roots is especially relevant for elliptic curve cryptosystems, where hashing an arbitrary message to a random point that belongs to a given elliptic curve, point compression and point counting over elliptic curves, are among its most relevant cryptographic applications.
In this talk, we present two new algorithms for computing square roots over finite fields of the form Fq, with q = pm and where p is a large odd prime and m an even integer. The first algorithm is devoted to the case when q ≡ 3 (mod 4), whereas the second handles the complementary case when q ≡ 1 (mod 4). We include numerical comparisons showing the efficiency of our algorithms over the ones previously published in the open literature.
Răzvan Bărbulescu (Projet CARAMEL, LORIA)
Finding Optimal Formulae for Bilinear Maps
We describe a unified framework to search for optimal formulae evaluating bilinear or quadratic maps. This framework applies to polynomial multiplication and squaring, finite field arithmetic, matrix multiplication, etc. We then propose a new algorithm to solve problems in this unified framework. With an implementation of this algorithm, we prove the optimality of various published upper bounds, and find improved upper bounds.
Cyril Bouvier (Projet CARAMEL, LORIA)
Finding ECM-Friendly Curves through a Study of Galois Properties
In this paper we prove some divisibility properties of the cardinality of elliptic curves modulo primes. These proofs explain the good behavior of certain parameters when using Montgomery or Edwards curves in the setting of the elliptic curve method (ECM) for integer factorization. The ideas of the proofs help us to find new families of elliptic curves with good division properties which increase the success probability of ECM.
Aurore Guillevic (Équipe Crypto, ENS / Laboratoire Chiffre, Thales)
Familles de courbes hyperelliptiques de genre 2, calcul explicite de l'ordre de la jacobienne et constructions pour les couplages
Olivier Levillain (ANSSI)
SSL/TLS: état des lieux et recommandations
Luc Sanselme (Lycée Henri Poincaré, Nancy)
Algorithmique des groupes en calcul quantique
Jean-Charles Faugère (Projet PolSys, LIP6)
Gröbner Bases and Linear Algebra
There is a strong interplay between computing efficiently Gröbner bases and linear algebra. In this talk, we focus on several aspects of this convergence:
Algorithmic point of view: algorithms for computing efficiently Gröbner bases (F4, F5, FGLM, ...) rely heavily on efficient linear algebra.
The matrices generated by these algorithms have unusual properties: sparse, almost block triangular. We present a dedicated algorithm for computing Gaussian elimination of Gröbner bases matrices.
By taking advantage of the sparsity of multiplication matrices in the classical FGLM algorithm we can design an efficient algorithm to change the ordering of a Gröbner basis. The algorithm is a related to multivariate generalization of the Wiedemann algorithm. When the matrices are not sparse, for generic systems, the complexity is Õ(Dω) where D is the number of solutions and ω ≤ 2.3727 is the linear algebra constant.
Mixing Gröbner bases methods and linear algebra technique for solving sparse linear systems leads to an efficient algorithm to solve Boolean quadratic equations over F2; this algorithm is faster than exhaustive search by an exponential factor
Application point of view: for instance, a generalization of the eigenvalue problem to several matrices – the MinRank problem – is at the heart of the security of many multivariate public key cryptosystems.
Design of C library: we present a multi core implementation of these new algorithms; the library contains specific algorithms to compute Gaussian elimination as well as specific internal representation of matrices.The efficiency of the new software is demonstrated by showing computational results for well known benchmarks as well as some crypto applications.
Joint works with S. Lachartre, C. Mou, P. Gaudry, L. Huot, G. Renault, M. Bardet, B. Salvy, P.J. Spaenlehauer and M. Safey El Din.
Peter Schwabe (Academia Sinica, Taiwan)
EdDSA signatures and Ed25519
One of the most widely used applications of elliptic-curve cryptography is digital signatures. In this talk I will present the EdDSA signature scheme that improves upon previous elliptic-curve-based signature schemes such as ECDSA or Schnorr signatures in several ways. In particular it makes use of fast and secure arithmetic on Edwards curves, it is resilient against hash-function collisions and it supports fast batch verification of signatures. I will furthermore present performance results of Ed25519, EdDSA with a particular set of parameters for the 128-bit security level.
Karim Khalfallah (ANSSI)
Les canaux auxiliaires, approche sous l'angle du rapport signal-à-bruit
Wednesday March 7, 2012
Jérémie Detrey (Projet CARAMEL, LORIA)
Implémentation efficace de la recherche de formules pour applications bilinéaires
Marion Videau (Projets CARAMEL, LORIA)
Codes d'authentification de message (suite et fin)
Stéphane Glondu (Projets CASSIS/CARAMEL, LORIA)
Tutoriel Coq (suite et fin)
Charles Bouillaguet (CRYPTO, PRiSM, Univ. Versailles Saint-Quentin)
Presumably hard problems in multivariate cryptography
Public-key cryptography relies on the existence of computationaly hard problems. It is not widely accepted that a public-key scheme is worthless without a "security proof", i.e., a proof that if an attacker breaks the scheme, then she solves in passing an instance of an intractable computational problem. As long as the hard problem is intractable, then the scheme is secure. The most well-known hardness assumptions of this kind are probably the hardness of integer factoring, or that of taking logarithms in certain groups.
In this talk we focus on multivariate cryptography, a label covering all the (mostly public-key) schemes explicitly relying on the hardness of solving systems of polynomial equations in several variables over a finite field. The problem, even when restricted to quadratic polynomials, is well-known to be NP-complete. In the quadratic case, it is called MQ. Interestingly, most schemes in this area are not "provably secure", and a lot of them have been broken because they relied on another, less well-known, computational assumption, the hardness of Polynomial Linear Equivalence (PLE), which is a higher-degree generalization the problem testing whether two matrices are equivalent.
In this talk I will present the algorithms I designed to tackle these two hard problems. I will show that 80 bits of security are not enough for MQ to be practically intractable, and I will present faster-than-before, sometimes practical algorithms for various flavors of PLE.
ECM sur GPU
In this talk, I will present a new implementation of the Elliptic Curve Method algorithm (ECM) for graphic cards (GPU). This implementation uses Montgomery's curve, like GMP-ECM, but uses a different implementation and a different algorithm for the scalar multiplication. As there is no modular arithmetic library (like GMP) available for GPU, it was necessary to implement a modular arithmetic using array of unsigned integers from scratch, while taking into account constraints of GPU programming. The code, written for NVIDIA GPUs using CUDA, was optimized for factoring 1024 bits integers.
Sorina Ionica (Projet CARAMEL, LORIA)
Pairing-based algorithms for Jacobians of genus 2 curves with maximal endomorphism ring
Using Galois cohomology, Schmoyer characterizes cryptographic non-trivial self-pairings of the ℓ-Tate pairing in terms of the action of the Frobenius on the ℓ-torsion of the Jacobian of a genus 2 curve. We apply similar techniques to study the non-degeneracy of the ℓ-Tate pairing restrained to subgroups of the ℓ-torsion which are maximal isotropic with respect to the Weil pairing. First, we deduce a criterion to verify whether the jacobian of a genus 2 curve has maximal endomorphism ring. Secondly, we derive a method to construct horizontal (ℓ, ℓ)-isogenies starting from a jacobian with maximal endomorphism ring. This is joint work with Ben Smith.
Thursday October 6, 2011
Short Division of Long Integers (with David Harvey)
We consider the problem of short division — i.e., approximate quotient — of multiple-precision integers. We present ready-to-implement algorithms that yield an approximation of the quotient, with tight and rigorous error bounds. We exhibit speedups of up to 30% with respect to GMP division with remainder, and up to 10% with respect to GMP short division, with room for further improvements. This work enables one to implement fast correctly rounded division routines in multiple-precision software tools.
Diego F. Aranha (Universidade de Brasília)
Efficient Software Implementation of Binary Field Arithmetic Using Vector Instruction Sets
In this talk, we will describe an efficient software implementation of characteristic 2 fields making extensive use of vector instruction sets commonly found in desktop processors. Field elements are represented in a split form so performance-critical field operations can be formulated in terms of simple operations over 4-bit sets. In particular, we detail techniques for implementing field multiplication, squaring, square root extraction, half-trace and inversion and present a constant-memory lookup-based multiplication strategy. We illustrate performance with timings for scalar multiplication on a 251-bit curve and compare our results with publicly available benchmarking data.
Benoît Gaudel (Projet CARAMEL, LORIA)
Étude de stratégies de cofactorisation pour l'algorithme Function Field Sieve
Thursday June 9, 2011
Alain Couvreur (Projet COCQ, Institut de Mathématiques de Bordeaux)
Une nouvelle construction géométrique de codes sur de petits corps
Christophe Mouilleron (Projet Arénaire, LIP, ENS Lyon)
Génération de schémas d'évaluation avec contraintes pour des expressions arithmétiques
Marion Videau (ANSSI)
Cryptanalyse d'ARMADILLO2
Hamza Jeljeli (LACAL, EPFL, Lausanne)
RNS on Graphic Processing Units
Benjamin Smith (Projet TANC, LIX, École Polytechnique)
Middlebrow Methods for Low-Degree Isogenies in Genus 2
In 2008, Dolgachev and Lehavi published a method for constructing (ℓ,ℓ)-isogenies of Jacobians of genus 2 curves. The heart of their construction requires only elementary projective geometry and some basic facts about abelian varieties. In this talk, we put their method into practice, and consider what needs to be done to transform their method into a practical algorithm for curves over finite fields.
Thursday May 5, 2011
Xavier Pujol (Projet Arénaire, LIP, ENS Lyon)
Analyse de BKZ
Strong lattice reduction is the key element for most attacks against lattice-based cryptosystems. Between the strongest but impractical HKZ reduction and the weak but fast LLL reduction, there have been several attempts to find efficient trade-offs. Among them, the BKZ algorithm introduced by Schnorr and Euchner in 1991 seems to achieve the best time/quality compromise in practice. However, no reasonable time complexity upper bound was known so far for BKZ. We give a proof that after Õ(n3/k2) calls to a k-dimensional HKZ reduction subroutine, BKZk returns a basis such that the norm of the first vector is at most ≈ γkn/2(k-1) × det(L)1/n. The main ingredient of the proof is the analysis of a linear dynamic system related to the algorithm.
Améliorations au problème du logarithme discret dans Fp*
Alin Bostan (Projet ALGORITHMS, INRIA Paris-Rocquencourt)
Algébricité de la série génératrice complète des chemins de Gessel
Martin Albrecht (Projet SALSA, LIP6)
The M4RI & M4RIE libraries for linear algebra over GF(2) and small extensions
In this talk we will give an overview of the M4RI and M4RIE libraries. These open-source libraries are dedicated to efficient linear algebra over small finite fields with characteristic two. We will present and discuss implemented algorithms, implementation issues that arise for these fields and also some ongoing and future work. We will also demonstrate the viability of our approach by comparing the performance of our libraries with the implementation in Magma.
Bogdan Pasca (Projet Arénaire, LIP, ENS Lyon)
FPGA-specific arithmetic pipeline design using FloPoCo
In the last years the capacity of modern FPGAs devices has increased to the point that they can be used with success for accelerating various floating-point computations. However, ease of programmability has never rhymed with FPGAs. Obtaining good accelerations was most of the time a laborious and error-prone process.
This talk is addressing the programmability of arithmetic circuits on FPGAs. We present FloPoCo, a framework facilitating the design of custom arithmetic datapaths for FPGAs. Some of the features provided by FloPoCo are: an important basis of highly optimized arithmetic operators, a unique methodology for separating arithmetic operator design from frequency-directed pipelining the designed circuits and a flexible test-bench generation suite for numerically validating the designs.
The framework is reaching maturity, so far being tested with success for designing several complex arithmetic operators including the floating-point square-root, exponential and logarithm functions. Synthesis results capture the designed operator's flexibility: automatically optimized for several Altera and Xilinx FPGAs, wide range of target frequencies and several precisions ranging from single to quadruple precision.
Christophe Arène (Institut de Mathématiques de Luminy)
Complétude des lois d'addition sur une variété abélienne
Frederik Vercauteren (ESAT/COSIC, KU Leuven)
Fully homomorphic encryption via ideals in number rings
In this talk, I will review the concept of fully homomorphic encryption, describe its possibilities and then give a construction based on principal ideals in number rings. This is joint work with Nigel Smart.
Junfeng Fan (ESAT/COSIC, KU Leuven)
ECC on small devices
The embedded security community has been looking at the ECC ever since it was introduced. Hardware designers are now challenged by limited area (<15 kGates), low power budget (<100 μW) and sophisticated physical attacks. This talk will report the state-of-the-art ECC implementations for ultra-constrained devices. We take a passive RFID tag as our potential target. We will discuss the known techniques to realize ECC on such kind of devices, and what are the challenges we face now and in the near future.
Sylvain Collange (Arénaire, LIP, ENS Lyon)
Enjeux de conception des architectures GPGPU : unités arithmétiques spécialisées et exploitation de la régularité
Guillaume Batog (VEGAS, LORIA)
Sur le type d'intersection de deux quadriques de P3(R)
Vanessa Vitse (CRYPTO, PRiSM, Univ. Versailles Saint-Quentin)
Calcul de traces de l'algorithme F4 et application aux attaques par décomposition sur courbes elliptiques
Monday November 8, 2010
Mehdi Tibouchi (Équipe Cryptographie, Laboratoire d'Informatique de l'ENS)
Hachage vers les courbes elliptiques et hyperelliptiques
Marcelo Kaihara (LACAL, EPFL, Lausanne)
Implementation of RSA 2048 on GPUs
Following the NIST recommendations for Key Management (SP800-57) and the DRAFT recommendation for the Transitioning of Cryptographic Algorithms and Key Sizes (SP 800-131), the use of RSA-1024 will be deprecated from January 1, 2011. Major vendors and enterprises will start the transition to the next minimum RSA key size of 2048 bits, which is computationally much more expensive. This talk presents an implementation of RSA-2048 on GPUs and explores the possibility to alleviate the computationally overhead by offloading the operations on GPUs.
Louise Huot (Équipe SALSA, LIP6)
Étude des systèmes polynomiaux intervenant dans le calcul d'indice pour la résolution du problème du logarithme discret sur les courbes
Thomas Prest (Équipe CARAMEL, LORIA)
Sélection polynomiale pour le crible NFS
Peter Montgomery (Microsoft Research & CWI)
Attempting to Run NFS with Many Linear Homogeneous Polynomials
Julie Feltin (Équipe CARAMEL, LORIA)
Implantation de l'algorithme ECM sur GPU
Fabrice Rouillier (Projet SALSA, LIP6)
Quelques astuces pour résoudre les systèmes polynomiaux dépendant de 2 variables
Xavier Goaoc (Projet VEGAS, LORIA)
Influence du bruit sur le nombre de points extrêmes
1,82 ?
Francesco Sica (Department of Mathematics and Statistics, University of Calgary)
Une approche analytique au problème de la factorisation d'entiers
Jean-François Biasse (Projet TANC, LIX, École Polytechnique)
Calcul de groupe de classes d'idéaux de corps de nombres.
Mioara Joldeş (Projet Arénaire, LIP, ENS Lyon)
Chebyshev Interpolation Polynomial-based Tools for Rigorous Computing
Performing numerical computations, yet being able to provide rigorous mathematical statements about the obtained result, is required in many domains like global optimization, ODE solving or integration. Taylor models, which associate to a function a pair made of a Taylor approximation polynomial and a rigorous remainder bound, are a widely used rigorous computation tool. This approach benefits from the advantages of numerical methods, but also gives the ability to make reliable statements about the approximated function.
A natural idea is to try to replace Taylor polynomials with better approximations such as minimax approximation, Chebyshev truncated series or interpolation polynomials. Despite their features, an analogous to Taylor models, based on such polynomials, has not been yet well-established in the field of validated numerics. In this talk we propose two approaches for computing such models based on interpolation polynomials at Chebyshev nodes.
We compare the quality of the obtained remainders and the performance of the approaches to the ones provided by Taylor models. We also present two practical examples where this tool can be used: supremum norm computation of approximation errors and rigorous quadrature.
This talk is based on a joint work with N. Brisebarre.
Peter Schwabe (Department of Mathematics and Computer Science, TU Eindhoven)
Breaking ECC2K-130
In order to "increase the cryptographic community's understanding and appreciation of the difficulty of the ECDLP", Certicom issued several elliptic-curve discrete-logarithm challenges. After these challenges were published in 1997 the easy ones of less than 100 bits were soon solved — the last one in 1999.
In 2004 also all challenges of size 109 bits were solved but the 131-bit challenges have so far all not been successfully targeted.
Since end of 2009 a group of several institutions is trying to solve the challenge ECC2K-130, a discrete-logarithm problem on a Koblitz curve over the field F2131. In my talk I will describe the approach taken to solve this challenge and give details of the Pollard rho iteration function. Furthermore I will give implementation details on two different platforms, namely the Cell Broadband Engine and NVIDIA GPUs.
Romain Cosset (Projet CARAMEL, LORIA)
Formules de Thomae et isogénies
Luca De Feo (Projet TANC, LIX, École Polytechnique)
Isogeny computation in small characteristic
Isogenies are an important tool in the study of elliptic curves. As such their applications in Elliptic Curve Cryptography are numerous, ranging from point counting to new cryptographic schemes.
The problem of finding explicit formulae expressing an isogeny between two elliptic curves has been studied by many. Vélu gave formulae for the case where the curves are defined over C; these formulae have been extended in works by Morain, Atkin and Charlap, Coley & Robbins to compute isogenies in the case where the characteristic of the field is larger than the degree of the isogeny.
The small characteristic case requires another treatment. Algorithms by Couveignes, Lercier, Joux & Lercier, Lercier & Sirvent give solutions to different instances of the problem. We review these strategies, then we present an improved algorithm based over Couveignes' ideas and we compare its performance to the other ones.
Friday March 5, 2010
Wouter Castryck (Department of Mathematics, KU Leuven)
The probability that a genus 2 curve has a Jacobian of prime order
To generate a genus 2 curve that is suitable for use in cryptography, one approach is to repeatedly pick a curve at random until its Jacobian has prime (or almost prime) order. Naively, one would expect that the probability of success is comparable to the probability that a randomly chosen integer in the according Weil interval is prime (or almost prime). However, in the elliptic curve case it was observed by Galbraith and McKee that large prime factors are disfavoured. They gave a conjectural formula that quantifies this effect, along with a heuristic proof, based on the Hurwitz–Kronecker class number formula. In this talk, I will provide alternative heuristics in favour of the Galbraith–McKee formula, that seem better-suited for generalizations to curves of higher genus. I will then elaborate this for genus 2. This is joint (and ongoing) research with Hendrik Hubrechts and Alessandra Rigato.
Tuesday March 2, 2010
Osmanbey Uzunkol (KANT Group, Institute of Mathematics, TU Berlin)
Shimura's reciprocity law, Thetanullwerte and class invariants
In the first part of my talk I am going to introduce the classical class invariants of Weber, and their generalizations, as quotients of values of "Thetanullwerte", which enables to compute them more efficiently than as quotients of values of the Dedekind η-function. Moreover, the proof that most of the invariants introduced by Weber are actually units in the corresponding ring class fields will be given, which allows to obtain better class invariants in some cases, and to give an algorithm that computes the unit group of corresponding ring class fields.
In the second part, using higher degree reciprocity law I am going to introduce the possibility of generalizing the algorithmic approach of determining class invariants for elliptic curves with CM, to determining alternative class invariant systems for principally polarized simple abelian surfaces with CM.
Fabien Laguillaumie (Algorithmique, GREYC, Univ. Caen Basse-Normandie)
Factorisation des entiers N = pq2 et applications cryptographiques
Monday February 1, 2010
Pascal Molin (Institut de Mathématiques de Bordeaux)
Intégration numérique rapide et prouvée — Application au calcul des périodes de courbes hyperelliptiques
Éric Brier (Ingenico)
Familles de courbes pour factorisation par ECM des nombres de Cunningham
Iram Chelli (CACAO)
Fully Deterministic ECM
We present a FDECM algorithm allowing to remove — if they exist — all prime factors less than 232 from a composite input number n. Trying to remove those factors naively either by trial-division or by multiplying together all primes less than 232, then doing a GCD with this product both prove extremely slow and are unpractical. We will show in this article that with FDECM it costs about a hundred well-chosen elliptic curves, which can be very fast in an optimized ECM implementation with optimized B1 and B2 smoothness bounds. The speed varies with the size of the input number n. Special attention has also been paid so that our FDECM be the most implementation-independent possible by choosing a widespread elliptic-curve parametrization and carefully checking all results for smoothness with Magma. Finally, we have considered possible optimizations to FDECM first by using a rational family of parameters for ECM and then by determining when it is best to switch from ECM to GCD depending on the size of the input number n. To the best of our knowledge, this is the first detailed description of a fully deterministic ECM algorithm.
Răzvan Bărbulescu (CACAO)
Familles de courbes elliptiques adaptées à la factorisation des entiers
Tadanori Teruya (LCIS, University of Tsukuba, Japan)
Generating elliptic curves with endomorphisms suitable for fast pairing computation
This presentation is about a kind of ordinary elliptic curves introduced by Scott at INDOCRYPT 2005. These curves' CM discriminant is -3 or -1, then they have endomorphism for reducing Miller's loop length to half. These curves are also restricted in terms of the form of group order. Therefore, these are generated by Cocks-Pinch method. Cocks-Pinch method is a general method to obtain elliptic curve parameters with rho-value approximately 2. This method enables to fix group order, CM disciminant and embedding degree in advance as long as they meet the requirements. Elliptic curves introduced by Scott with CM discriminant -3, they were investigated by Scott and Takashima but CM discriminant -1 are not. In this presentation, we show the result of generating curve parameters with CM discriminant -1 and what amount of parameters meet the requirements.
Andy Novocin (ANR LareDa, LIRMM, Montpellier)
Gradual Sub-Lattice Reduction and Applications
One of the primary uses of lattice reduction algorithms is to approximate short vectors in a lattice. I present a new algorithm which produces approximations of short vectors in certain lattices. It does this by generating a reduced basis of a sub-lattice which is guaranteed to contain all short vectors in the given lattice. This algorithm has a complexity which is less dependent on the size of the input basis vectors and more dependent on the size of the output vectors.
To illustrate the usefulness of the new algorithm I will show how it can be used to give new complexity bounds for factoring polynomials in Z[x] and reconstructing algebraic numbers from approximations.
Nicolas Guillermin (Centre d'électronique de l'armement (CELAR), DGA)
Architecture matérielle pour la cryptographie sur courbes elliptiques et RNS
Judy-anne Osborn (Australian National University)
On Hadamard's Maximal Determinant Problem
The Maximal Determinant Problem was first posed around 1898. It asks for a square matrix of largest possible determinant, with the entries of the matrix restricted to be drawn from the set {0, 1}, or equivalently {+1, -1}.
Emperical investigations show an intriguing amount of structure in this problem, both in the numerical sequence of maximal determinants, and in the corresponding maximal determinant matrices themselves. But naive brute force search becomes infeasible beyond very small orders, due to the exponential nature of the search space.
High and maximal determinant matrices are useful in applications, particularly in statistics, which is one reason why it is desirable to have at hand a means of constructing these matrices. For certain sparse infinite subsequences of orders, constructive algorithms have been found - some relating to finite fields. However progress over the last one hundred years has been distinctly patchy, depending on elementary number theoretic properties of the matrix order: particularly its remainder upon division by four.
We discuss ways of setting up computations which may be feasible with current computing power and yet still yield new maximal determinant matrices that would not be accessible to a naive search.
Jérémie Detrey (CACAO)
Hardware Operators for Pairing-Based Cryptography – Part II: Because speed also matters –
Originally introduced in cryptography by Menezes, Okamoto and Vanstone (1993) then Frey and Rück (1994) to attack the discrete logarithm problem over a particular class of elliptic curves, pairings have since then been put to a constructive use in various useful cryptographic protocols such as short digital signature or identity-based encryption. However, evaluating these pairings relies heavily on finite field arithmetic, and their computation in software is still expensive. Developing hardware accelerators is therefore crucial.
In the second part of this double-talk, we will focus on the other end of the hardware design spectrum. While the first part (given by Jean-Luc Beuchat) presented a co-processor which, although quite slow, would strive to minimize the amount of hardware resources required to compute the Tate pairing, in this second part we will describe another co-processor architecture, designed to achieve much lower computation times, at the expense of hardware resources.
Jean-Luc Beuchat (Tsukuba)
Hardware Operators for Pairing-Based Cryptography – Part I: Because size matters –
In this talk, we will then present a hardware co-processor designed to accelerate the computation of the Tate pairing in characteristics 2 and 3. As the title suggests, this talk will emphasize on reducing the silicon footprint (or in our case the usage of FPGA resources) of the circuit to ensure scalability, while trying to minimize the impact on the overall performances.
Nicolas Estibals (ENS Lyon)
Multiplieurs parallèles et pipelinés pour le calcul de couplage en caractéristiques 2 et 3
Marc Mezzarobba (Projet Algo)
Suites et fonctions holonomes : évaluation numérique et calcul automatique de bornes
Éric Schost (University of Western Ontario.)
Deformation techniques for triangular arithmetic
Triangular representations are a versatile data structure; however, even basic arithmetic operations raise difficult questions with such objects. I will present an algorithm for multiplication modulo a triangular set that relies on deformation techniques and ultimately evaluation and interpolation. It features a quasi-linear running time (without hidden exponential factor), at least in some nice cases. More or less successful applications include polynomial multiplication, operations on algebraic numbers and arithmetic in Artin-Schreier extensions.
Joerg Arndt (Australian National University)
arctan relations for computing pi.
slides: pdf (89 kb)
http://www.jjj.de/arctan/arctanpage.html
Binary polynomial irreducibility tests avoiding GCDs.
Mathieu Cluzeau (INRIA Rocquencourt, équipe SECRET)
Reconnaissance d'un code linéaire en bloc
Nicolas Meloni (Université de Toulon)
Chaines d'additions différentielles et multiplication de point sur les courbes elliptiques
Laurent Imbert (LIRMM)
Quelques systèmes de numération exotiques (et applications)
Guillaume Melquiond (MSR-INRIA)
L'arithmétique flottante comme outil de preuve formelle
Aurélie Bauer (Université de Versailles Saint-Quentin-en-Yvelines, Laboratoire PRISM,)
Toward a Rigorous Variation of Coppersmith's Algorithm on Three Variables
In 1996, Coppersmith introduced two lattice reduction based techniques to find small roots in polynomial equations. One technique works for modular univariate polynomials, the other for bivariate polynomials over the integers. Since then, these methods have been used in a huge variety of cryptanalytic applications. Some applications also use extensions of Coppersmith's techniques on more variables. However, these extensions are heuristic methods.
In this presentation, we present and analyze a new variation of Coppersmith's algorithm on three variables over the integers. We also study the applicability of our method to short RSA exponents attacks. In addition to lattice reduction techniques, our method also uses Gröbner bases computations. Moreover, at least in principle, it can be generalized to four or more variables.
Nicolas Julien (Project-team Marelle, INRIA Sophia Antipolis.)
Arithmétique réelle exacte certifiée
Wednesday December 5, 2007
Christophe Doche
DBNS et cryptographie sur courbes elliptiques
Clément Pernet
Algèbre linéaire dense dans des petits corps finis: théorie et pratique.
B13, 10:30
Thomas Sirvent
Schéma de diffusion efficace basé sur des attributs
Jean-Luc Beuchat
Arithmetic Operators for Pairing-Based Cryptography
Since their introduction in constructive cryptographic applications, pairings over (hyper)elliptic curves are at the heart of an ever increasing number of protocols. Software implementations being rather slow, the study of hardware architectures became an active research area. In this talk, we first describe an accelerator for the $\eta_T$ pairing over $\mathbb{F}_3[x]/(x^{97}+x^{12}+2)$. Our architecture is based on a unified arithmetic operator which performs addition, multiplication, and cubing over $\mathbb{F}_{3^{97}}$. This design methodology allows us to design a compact coprocessor ($1888$ slices on a Virtex-II Pro~$4$ FPGA) which compares favorably with other solutions described in the open literature. We then describe ways to extend our approach to any characteristic and any extension field.
The talk will be based on the following research reports:
http://eprint.iacr.org/2007/091
Ley Wilson
Quaternion Algebras and Q-curves
Let K be an imaginary quadratic field with Hilbert class field H and maximal order OK. We consider elliptic curves E defined over H with the properties that the endomorphism ring of E is isomorphic to OK and E is isogenous to E over H for all \sigma\in Gal(H/K). Taking the Weil restriction W_{H/K} of such an E from H to K, one obtains an abelian variety whose endomorphism ring will be either a field or a quaternion algebra. The question of which quaternion algebras may be obtained in this way is one of our motivations.
For quaternion algebras to occur, the class group of K must have non-cyclic 2-Sylow subgroup, the simplest possible examples occuring when K has class number 4. In this case, investigating when W_{H/K}(E) has a non-abelian endomorphism algebra is closely related to finding extensions L/H such that Gal(L/K) is either the dihedral or quaternion group of order 8.
Jeremie Detrey
Évaluation en virgule flottante de la fonction exponentielle sur FPGA
David Kohel (Université de Sydney et UHP-Nancy 1)
Complex multiplication and canonical lifts
The $j$-invariant of an elliptic curve with complex multiplication by $K$ is well-known to generate the Hilbert class field of $K$. Such $j$-invariants, or rather their minimal polynomials in $\ZZ[x]$, can be determined by means of complex analytic methods from a given CM lattice in $\CC$. A construction of CM moduli by $p$-adic lifting techniques was introduced by Couveignes and Henocq. Efficient versions of one-dimensional $p$-adic lifting were developed by Br\"oker. These methods provide an alternative application of $p$-adic canonical lifts, as introduced by Satoh for determining the zeta function of an elliptic curves $E/\FF_{p^n}$.
Construction of such defining polynomials for CM curves is an area of active interest for use in cryptographic constructions. Together with Gaudry, Houtmann, Ritzenthaler, and Weng, we generalised the elliptic curve CM construction to genus 2 curves using $2$-adic canonical lifts. The output of this algorithm is data specifying a defining ideal for the CM Igusa invariants $(j_1,j_2,j_3)$ in $\ZZ[x_1,x_2,x_3]$. In contrast to Mestre's AGM algorithm for determining zeta functions of genus 2 curves $C/\FF_{2^n}$, this construction pursues the alternative application of canonical lifts to CM constructions. With Carls and Lubicz, I developed an analogous $3$-adic CM construction using theta functions. In this talk I will report on recent progress and challenges in extending and improving these algorithms.
David Lubicz
Relèvement canonique en caractéristique impaire.
Paul Zimmermann
Fast Multiplication over GF(2)[x]
Schönhage proposed in the paper "Schnelle Multiplikation von Polynomen über Körpern der Charakteristik 2" (Acta Informatica, 1977) an O(n log(n) log(log(n))) algorithm to multiply polynomials over GF(2)[x]. We describe that algorithm and report on its implementation in NTL.
Richard Brent
A Multi-level Blocking Algorithm for Distinct-Degree Factorization of Polynomials over GF(2).
Abstract: We describe a new multi-level blocking algorithm for distinct-degree factorization of polynomials over GF(2). The idea of the algorithm is to use one level of blocking to replace most GCD computations by multiplications, and a finer level of blocking to replace most multiplications by squarings (which are much faster than multiplications over GF(2)). As an application we give an algorithm that searches for all irreducible trinomials of given degree. Under plausible assumptions, the expected running time of this algorithm is much less than that of the classical algorithm. For example, our implementation gives a speedup of more than 60 over the classical algorithm for trinomials of degree 6972593 (a Mersenne exponent). [Joint work with Paul Zimmermann.]
Explicit isogenies of hyperelliptic Jacobians
Isogenies — surjective homomorphisms of algebraic groups with finite kernel — are of great interest in number theory and cryptography. Algorithms for computing with isogenies of elliptic curves are well-known, but in higher dimensions, the situation is more complicated, and few explicit examples of non-trivial isogenies are known. We will discuss some of the computational issues, and describe some examples and applications of isogenies of Jacobians of hyperelliptic curves.
Guillaume Hanrot
Problème du vecteur le plus court dans un réseau : analyse de l'algorithme de Kannan (travail commun avec D. Stehlé).
Guillaume Melquiond (Project-team Arénaire, INRIA Rhône-alpes.)
De l'arithmétique d'intervalles à la certification de programmes
Sylvain Chevillard (speaker), Christoph Lauter (Project-team Arénaire, INRIA Rhône-alpes.)
Une norme infinie certifiée pour la validation d'algorithmes numériques
Christoph Lauter (Project-team Arénaire, INRIA Rhône-alpes.)
Automatisation du contrôle de précision et de la preuve pour les formats double-double et triple-double
Sylvain Chevillard (Project-team Arénaire, INRIA Rhône-alpes.)
Approximation polynomiale de fonctions continues et nombres flottants
Cédric Lauradoux (Project-team Codes, INRIA Rocquencourt.)
Shift Registers Synthesis
Shift registers are very common hardware devices. They are always associated with a combinatorial/sequential feedback. Linear Feedback Shift Registers (LFSRs) are certainly the most famous setup amongst those circuits. LFSRs are used everywhere in communication systems: scramblers, stream-ciphers, spread spectrum, Built-in Self Test (BIST)... Despite their popularity, the impact of LFSRs characteristics has never been clearly studied. I have studied the LFSRs synthesis on Xilinx Spartan2E FPGA with several goals (area, critical path, throughput). Studying high throughtput synthesis is particularly interesting since it is a circuitous way to study software synthesis. I will describe which properties can be observed for both synthesis. In conclusion, other shift registers setups will be considered like Feedback with Carry Shift Registers (FCSRs) or Non Linear Feedback Shift Registers (NLFSRs).
Marcus Wagner (Technische Universität Berlin)
On Deuring correspondences of algebraic function fields
Using Deurings theory of correspondences we are able to construct homomorphisms between the degree zero classgroups of function fields. Correspondences are divisors of function fields with transcendent constant fields of degree one. They form an entire ring which is for example in the case of elliptic function fields isomorphic to an order of an imaginary quadratic number field. In this talk we show how to compute endomporphisms of elliptic and hyperelliptic curves using correspondences.
Michael Quisquater
Cryptanalyse lineaire des algorithmes de chiffrement par bloc.
Stef Graillat
Évaluation précise de polynômes en précision finie
Christopher Wolf
Division without Multiplication in Factor Rings
In a factor ring, i.e., in a polynomial ring F[z]/(m) or the integer ring Z_n, the conventional way of performing division a/b consists of two steps: first, the inverse b^{-1} is computed and then the product ab^{-1}. In this talk we describe a technique called "direct division" which computes the division a/b for given a,b directly, only using addition and multiplication in the underlying structure, i.e., finite field operations in F for the polynomial ring F[z]/(m) and addition and multiplication by 2 in the integer ring Z_n. This technique requires that the module m is not divisible by z, and the module n is odd.
Benoît Daireaux
Analyse dynamique des algorithmes euclidiens
Frederik Vercauteren
The Number Field Sieve in the Medium Prime Case
Thomas Plantard
Arithmétique modulaire pour la cryptographie.
Thursday January 5, 2006
Marion Videau (Project-team Codes, INRIA Rocquencourt.)
Propriétés cryptographiques des fonctions booléennes symétriques.
Alexander Kruppa (Technische Universität München)
Optimising the enhanced standard continuation of P-1, P+1 and ECM
The enhanced standard continuation of the P-1, P+1 and ECM factoring methods chooses pairs (a,b), where a is a multiple of a suitably chosen d, so that every prime in the desired stage 2 interval can be written as a-b. Montgomery [1] showed how to include more than one prime per (a,b) pair by instead evaluating f(a)-f(b) so that this bivariate polynomial has algebraic factors. However, he restricts his analysis to the algebraic factors a-b and a+b, and considers only prime values taken by these factors. We present a framework for generalising Montgomery's ideas by choosing (a,b) pairs as nodes in a partial cover of a bipartite graph, which allows utilising large prime factors of composite values, and algebraic factors of higher degree.
[1] P. L. Montgomery, Speeding the Pollard and elliptic Curve Methods of factorization, Math. Comp. 48 (177), 1987.
Wednesday November 23, 2005, informal workgroup
Sebastian Pauli (Technische Universität Berlin)
Construction Class Fields of Local Fields
Let K be a p-adic field. We give an explicit characterization of the abelian extensions of K of degree p by relating the coefficients of the generating polynomials of extensions L/K of degree p to the norm group N_{L/K}(L^*). This is applied in the construction of class fields of degree p^m.
Tuesday November 8, 2005, informal workgroup
Andreas Enge ( project-team TANC, INRIA Futurs, LIX)
Computing Hilbert class polynomials by floating-point approximations
Florian Hess (Technische Universität Berlin)
Arithmetic on general curves
The talk surveys various algorithms for computing in the divisor class groups of general non singular curves and gives a running time discussion.
Thursday September 29, 2005, informal workgroup
On finit l'exposé d'il y a deux semaines.
On finit l'exposé de la semaine dernière.
Calcul de fonctions holonomes en O(M(n) log(n)3)
Fonctions thétas et formules efficaces pour loi de groupe en genre 2.
Benjamin Werner ( LogiCal project-team, INRIA Futurs, LIX)
A propos de la preuve formelle du théorème des quatre couleurs
Roland Zumkeller (Projet LogiCal, INRIA Futurs, LIX)
Traitement d'inégalités réelles en Coq
Friday June 3, 2005, informal workgroup
Julien Cochet
à préciser
Damien Vergnaud (LMNO, CNRS / Université de Caen.)
On the decisional xyz-Diffie Hellman problem.
Digital signatures have the sometimes unwanted property of being universally verifiable by anybody having access to the signer's public key. In recent work with F. Laguillaumie and P. Pailler, we have proposed a signature scheme where the verification requires interaction with the signer. Its security relies on the « xyz » variant of the classical Diffie-Hellman problem. We present in this talk the underlying algorithmical problem within its cryptographical context, and give some assessment of its difficulty
Thursday April 7, 2005, informal workgroup
Jean-Yves Degos
Study of Basiri-Enge-Faugère-Gurel paper on arithmetic of C3,4 curves.
Wednesday March 23, 2005, informal workgroup
Study of P. L. Montgomery's paper: "Five, Six, and Seven-Term Karatsuba-Like Formulae".
Thursday March 10, 2005, informal workgroup
Study of the security proofs of OAEP and OAEP+
Régis Dupont ( project-team TANC, INRIA Futurs, LIX)
Theta constants and Borchardt mean, applications.
A curve of genus g defined over the complex field C is isomorphic to a torus with g holes, or equivalently to a quotient of the form Cg/(Zg.1⊕Zg.τ), τ being a g×g matrix called a Riemann matrix.
When the genus g equals one, the computation of τ from the equation of an elliptic curve is one of the classical applications of the arithmetico-geometric mean (AGM). The AGM can be interpreted using functions called theta constants.
We show how this special case extends to higher genus, using a generalization of the AGM known as the Borchardt mean.
In particular, we develop an algorithm for computing genus 2 Riemann matrices in almost linear time. This algorithm can be implemented easily.
As we also show, this technique allows for rapid computation of modular forms and functions, and we discuss the applications thereof (construction of CM curves, explicit computation of isogenies, …).
Last modification: Fri 17 Jan 2020 09:48:30 PM CET
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\begin{document}
\title{Stieltjes
moment properties and continued fractions from combinatorial
triangles
\thanks{Supported partially by the National Natural
Science Foundation of China (Nos. 11971206, 12022105), the Natural
Science Fund for Distinguished Young Scholars of Jiangsu Province
(No. BK20200048) and the Young Talents Invitation Program of
Shandong Province.
\newline\hspace*{5mm}
\begin{abstract} Many combinatorial numbers can be placed in the following generalized triangular array $[T_{n,k}]_{n,k\ge 0}$ satisfying the recurrence relation: \begin{equation*} T_{n,k}=\lambda(a_0n+a_1k+a_2)T_{n-1,k}+(b_0n+b_1k+b_2)T_{n-1,k-1}+\frac{d(da_1-b_1)}{\lambda}(n-k+1)T_{n-1,k-2} \end{equation*} with $T_{0,0}=1$ and $T_{n,k}=0$ unless $0\le k\le n$ for suitable $a_0,a_1,a_2,b_0,b_1,b_2,d$ and $\lambda$. For $n\geq0$, denote by $T_n(q)$ the generating function of the $n$-th row. In this paper, we develop various criteria for $\textbf{x}$-Stieltjes moment property and $3$-$\textbf{x}$-log-convexity of $T_n(q)$ based on the Jacobi continued fraction expression of $\sum_{n\geq0}T_n(q)t^n$, where $\textbf{x}$ is a set of indeterminates consisting of $q$ and those parameters occurring in the recurrence relation. With the help of a criterion of Wang and Zhu [Adv. in Appl. Math. (2016)], we show that the corresponding linear transformation of $T_{n,k}$ preserves Stieltjes moment properties of sequences. Finally, we present some related examples including factorial numbers, Whitney numbers, Stirling permutations, minimax trees and peak statistics.
\\ {\sl \textbf{MSC}:}\quad 05A20; 05A15; 11A55; 11B83; 15B48; 30B70; 44A60
\\ {\sl \textbf{Keywords}:}\quad Recurrence relations; Jacobi continued fractions; Stieltjes continued fractions; Total positivity; Hankel matrices; $\textbf{x}$-Stieltjes moment sequences; Binomial transformations; Convolutions; Row-generating functions; $\textbf{x}$-log-convexity \end{abstract}
\tableofcontents
\section{Introduction}
\subsection{Total positivity and Stieltjes moment sequences} Total positivity of matrices is an important and powerful concept that arises often in various branches of mathematics, such as classical analysis \cite{Sc30}, representation theory \cite{Lus94,Rie03}, network analysis \cite{Pos06}, cluster algebras \cite{BFZ96,FZ99}, combinatorics \cite{Bre95,GV85}, positive Grassmannians and integrable systems \cite{KW14}. We refer the reader to the monograph \cite{Kar68} for more details about total positivity.
Let $A=[a_{n,k}]_{n,k\ge 0}$ be a matrix of real numbers. It is called {\it totally positive} ({\it TP} for short) if all its minors are nonnegative. It is called {\it TP$_r$} if all minors of order $k\le r$ are nonnegative. For a sequence, the total positivity of its Hankel matrix plays an important role in different fields. Let us recall the definition. Given a sequence $\alpha=(a_k)_{k\ge 0}$, define its {\it Hankel matrix} $H(\alpha)$ by $$H(\alpha)=[a_{i+j}]_{i,j\ge 0}= \left[
\begin{array}{ccccc}
a_0 & a_1 & a_2 & a_3 & \cdots \\
a_1 & a_2 & a_3 & a_4 & \\
a_2 & a_3 & a_4 & a_5 & \\
a_3 & a_4 & a_5 & a_6 & \\
\vdots & & & & \ddots \\
\end{array} \right].$$
We say that $\alpha$ is a {\it Stieltjes moment} ({\it SM} for short) sequence if it has the form \begin{equation}\label{i-e} a_k=\int_0^{+\infty}x^kd\mu(x), \end{equation} where $\mu$ is a non-negative measure on $[0,+\infty)$ (see \cite[Theorem 4.4]{Pin10} for instance). Stieltjes proved that $\alpha$ is an SM sequence if and only if there exist nonnegative numbers $\alpha_0,\alpha_1,\ldots$ such that $$\sum_{n\geq0}a_nz^n=\frac{1}{1-\cfrac{\alpha_0z}{1-\cfrac{\alpha_1z}{1-\cdots}}}$$ in the sense of formal power series. It is well known that $\alpha$ is a Stieltjes moment sequence if and only if its Hankel matrix $H(\alpha)$ is TP. Stieltjes moment problem is one of classical moment problems and arises naturally in many branches of mathematics \cite{ST43,Wid41}. Indeed, SM sequences are closely related to log-convexity of sequences. The sequence $\alpha$ is called {\it log-convex} if $a_{k-1}a_{k+1}\ge a_k^2$ for all $k\ge 1$. Clearly, a sequence of positive numbers is log-convex if and only if its Hankel matrix is TP$_2$. As a result, SM property implies log-convexity. In addition, many log-convex sequences in combinatorics have SM property. We refer the reader to Liu and Wang \cite{LW07} and Zhu \cite{Zhu13} for log-convexity and Wang and Zhu \cite{WZ16} and Zhu\cite{Zhu19,Zhu191} for SM property.
In what follows, concepts for log-convexity, SM property and total positivity will be strengthened in a natural manner.
Let $\mathbb{P}$ denote the set of all positive integers and $\mathbb{N}=\mathbb{P}\cup\{0\}$. Let $\textbf{x}=\{x_i\}_{i\in{I}}$ be a set of indeterminates. A matrix $M$ with entries in $\mathbb{R}[\textbf{x}]$ is called \textbf{x-totally positive} (\textbf{x}-TP for short) if all its minors are polynomials with nonnegative coefficients in the indeterminates $\textbf{x}$ and is called \textbf{x-totally positive of order $r$} ( \textbf{x}-TP$_r$ for short)if all its minors of order $k\le r$ are polynomials with nonnegative coefficients in the indeterminates $\textbf{x}$. A sequence $(\alpha_n(\textbf{x}))_{n\geq0}$ with values in $\mathbb{R}[\textbf{x}]$ is called an \textbf{x-Stieltjes moment } ($\textbf{x}$-SM for short) sequence if its associated infinite Hankel matrix is $\textbf{x}$-totally positive. We use $f(x)\geq_{\textbf{x}}0$ to represent that all coefficients of the polynomial $f(\textbf{x})$ are nonnegative. It is called \textbf{x-log-convex } ($\textbf{x}$-LCX for short) if $$\alpha_{n+1}(\textbf{x})\alpha_{n-1}(\textbf{x})-\alpha_n(\textbf{x})^2\geq_{\textbf{x}}0$$ for all $n\in \mathbb{P}$ and is called \textbf{strongly x-log-convex } ($\textbf{x}$-SLCX for short) if $$\alpha_{n+1}(\textbf{x})\alpha_{m-1}(\textbf{x})-\alpha_n(\textbf{x})\alpha_m(\textbf{x})\geq_{\textbf{x}}0$$ for all $n\geq m\geq1$. Clearly, an $\textbf{x}$-SM sequence is both $\textbf{x}$-SLCX and $\textbf{x}$-LCX. Define an operator $\mathcal {L}$ by $$\mathcal {L}[\alpha_i(\textbf{x})]:=\alpha_{i-1}(\textbf{x})\alpha_{i+1}(\textbf{x})-\alpha_i(\textbf{x})^2$$ for $i\in \mathbb{P}$. Then the $\textbf{x}$-log-convexity of $(\alpha_i(\textbf{x}))_{i\geq 0}$ is equivalent to $\mathcal {L}[\alpha_i(\textbf{x})]\geq_{\textbf{x}}0$ for all $i\in \mathbb{P}$. In general, we say that $(\alpha_i(\textbf{x}))_{i\geq 0}$ is {\it $\textbf{k}$-\textbf{x-log-convex}} if the coefficients of $\mathcal {L}^m[\alpha_i(\textbf{x})]$ are nonnegative for all $m\leq k$, where $\mathcal {L}^m=\mathcal {L}(\mathcal {L}^{m-1})$. It is called {\it \textbf{infinitely x-log-convex}} if $(\alpha_i(\textbf{x}))_{i\geq 0}$ is $k$-$\textbf{x}$-log-convex for every $k \in \mathbb{N}$.
If $\textbf{x}$ contains a unique indeterminate $q$, then they reduce to $q$-LCX \cite{LW07}, $q$-SLCX \cite{CTWY10,CWY11,Zhu13,Zhu14,Zhu182} and $q$-SM \cite{WZ16,Zhu19,Zhu20}, respectively.
Finally, for brevity, let $\rm \textbf{SCF}[\alpha_{2i},\alpha_{2i+1};z]_{i\geq0}$ denote the {\it Stieltjes continued fraction} expansion $$\frac{1}{1-\cfrac{\alpha_0z}{1-\cfrac{\alpha_1z}{1-\cdots}}}.$$ The Stieltjes continued fraction expansion is closely related to the {\it Jacobi continued fraction} expansion, denoted by $\rm \textbf{JCF}[s_i,r_{i+1};z]_{i\geq0}$, $$\frac{1}{1-s_0z-\cfrac{r_1z^2}{1-s_1z-\cfrac{r_2z^2}{1-\cdots}}}$$ by the famous contraction formulae \begin{eqnarray} \DF{1}{1-\DF{t_1z}{1-\DF{t_2z}{1-\ldots}}} &=&\DF{1}{1- t_1z-\DF{t_1t_2z^2}{1- (t_2+t_3)z-\DF{t_3t_4z^2}{1- (t_4+t_5)z-\ldots}}}\label{contraction}\\ &=&1+\DF{t_1z}{1- (t_1+t_2)z-\DF{t_2t_3z^2}{1- (t_3+t_4)z-\DF{t_4t_5z^2}{1-\ldots}}}.\label{contraction+decom} \end{eqnarray} Thus we can write \begin{eqnarray} \rm \textbf{SCF}[\alpha_{2n},\alpha_{2n+1};z]_{n\geq0}=\rm \textbf{JCF}[\alpha_{2n-1}+\alpha_{2n},\alpha_{2n}\alpha_{2n+1};z]_{n\geq0}. \end{eqnarray}
We refer the reader to the monograph \cite{JT80} about continued fractions.
\subsection{Motivations} It is well-known that many classical combinatorial arrays satisfy certain recurrence relations. The following are some
examples:
\begin{ex}\label{basic-qSM} \begin{itemize}
\item [\rm (i)]
$\binom{n}{k}=\binom{n-1}{k}+\binom{n-1}{k-1}$, where $\binom{n}{k}$ is the binomial
coefficients;
\item [\rm (ii)] $\left[
\begin{array}{ccccc}
n \\
k\\
\end{array} \right]=(n-1)\left[
\begin{array}{ccccc}
n-1 \\
k\\
\end{array} \right]+\left[
\begin{array}{ccccc}
n-1 \\
k-1\\
\end{array} \right],$ where $\left[
\begin{array}{ccccc}
n \\
k\\
\end{array} \right]$ is the signless Stirling number counting the number of permutations of $n$ elements which are the product of $k$ disjoint cycles;
\item [\rm (iii)]
$\left\{
\begin{array}{ccccc}
n \\
k\\
\end{array} \right\}=k\left\{
\begin{array}{ccccc}
n-1 \\
k\\
\end{array} \right\}+\left\{
\begin{array}{ccccc}
n-1 \\
k-1\\
\end{array} \right\},$ where $\left\{
\begin{array}{ccccc}
n \\
k\\
\end{array} \right\}$ is the Stirling number of the second kind enumerating the number of partitions of an $n$-element set consisting of $k$ disjoint nonempty blocks;
\item [\rm (iv)] $\left\langle
\begin{array}{ccccc}
n \\
k\\
\end{array} \right\rangle=k\left\langle
\begin{array}{ccccc}
n-1 \\
k\\
\end{array} \right\rangle+(n-k+1)\left\langle
\begin{array}{ccccc}
n-1 \\
k-1\\
\end{array} \right\rangle$, where $\left\langle
\begin{array}{ccccc}
n \\
k\\
\end{array} \right\rangle$ is the classical Eulerian number counting the number of permutations of $n$ elements having $k-1$ descents; \item [\rm (v)] $\mathscr{B}_{n, k}=(k+1) \mathscr{B}_{n-1,k}+n \mathscr{B}_{n-1,k-1}+(n-k+1)\mathscr{B}_{n-1,k-2},$ where $\mathscr{B}_{n, k}$ is the number of symmetric tableaux of size $2n+1$ with $k+1$ diagonal cells \cite{ABN13}; \item [\rm (vi)] $\mathscr{T}_{n,k}=(k+1)\mathscr{T}_{n-1,k}+(n+1)\mathscr{T}_{n-1,k-1}+(n-k+1)\mathscr{T}_{n-1,k-2},$ where $\mathscr{T}_{n,k}$ is the number of staircase tableaux of size $n$ with $k$ labels $\alpha$ or $\delta$ in the diagonal \cite{ABD13}. \end{itemize} \end{ex}
These examples can be placed in a common framework. Let $\mathbb{R}$ (resp. $\mathbb{R^{+}}$, $\mathbb{R^{\geq}}$) be the set of all (resp., positive, nonnegative) real numbers. Let $\{a_1,b_1\}\in \mathbb{R}$, $\lambda\in\mathbb{R^{+}}$ and $\{a_0,a_2,b_0,b_2,d\}\subseteq \mathbb{R^{\geq}}$. Define a generalized triangular array $[T_{n,k}]_{n,k\ge 0}$ by the recurrence relation: \begin{equation}\label{Recurece+TT} T_{n,k}=\lambda(a_0n+a_1k+a_2)T_{n-1,k}+(b_0n+b_1k+b_2)T_{n-1,k-1}+\frac{d(da_1-b_1)}{\lambda}(n-k+1)T_{n-1,k-2} \end{equation} with $T_{0,0}=1$ and $T_{n,k}=0$ unless $0\le k\le n$. We also denote its row-generating function by $T_n(q)=\sum_{k\geq0}T_{n,k}q^k$ for $n\geq0$.
Many positivity properties of $[T_{n,k}]_{n,k}$ have been derived
for $d=0$, see Kurtz \cite{Kur72} for log-concavity of each row sequence,
Wang and Yeh~\cite{WYjcta05} for P\'olya Frequency property of each row
sequence, Liu and Wang \cite{LW07} for the $q$-log-convexity of $(T_n(q))_{n\geq0}$, Chen {\it et al.}~\cite{CWY11} for the strong $q$-log-convexity of $(T_n(q))_{n\geq0}$ and Zhu \cite{Zhu182} for linear transformations of $T_{n,k}$ preserving the strong $q$-log-convexity. It was proved that row-generating functions $T_n(q)$ for Stirling triangle of the second kind and Eulerian triangle form a $q$-Stieltjes moment sequence in $q$, respectively, see Wang and Zhu \cite{WZ16} for instance. For $d\neq0$, recently in \cite{Zhu201}, we proved for certain special case that $(T_n(q))_{n\geq0}$ is a $q$-Stieltjes moment sequence.
The aim of this paper is to consider the $q$-Stieltjes moment property of $(T_n(q))_{n\geq0}$ for the general case. In addition, in view of (\ref{Recurece+TT}), clearly, all elements $T_{n,k}$ are polynomials in the eight parameters $a_0,a_1,a_2,b_0,b_1$, $b_2$, $d$ and $\lambda$, and $T_n(q)$ can be regarded as a polynomial in nine indeterminates $a_0,a_1,a_2,b_0,$ $b_1$, $b_2$, $d$, $\lambda$ and $q$. It is natural to consider the following multi-variable question.
\begin{ques}\label{Que} Assume that the array $[T_{n,k}]_{n,k}$ is defined in (\ref{Recurece+TT}). When is $(T_n(q))_{n\geq0}$ an $\textbf{x}$-Stieltjes moment sequence with $\textbf{x}=(a_0,a_1,a_2,b_0,b_1,b_2,d,\lambda,q)$ ? \end{ques}
In order to answer Question \ref{Que}, our main tool is to use continued fraction expressions in Sections $2$ and $3$. For the array $[T_{n,k}]_{n,k}$ in (\ref{Recurece+TT}) without the term $T_{n-1,k-2}$, Sokal also conjectured that $(T_n(q))_{n\geq0}$ is $(a_0,a_1,a_2,b_0,b_1,b_2,q)$-Stieltjes moment \cite{SS,Sok}. We prove the next result.
\begin{thm}\label{thm+Ring+PSE+SM} If a triangle $[T_{n,k}]_{n,k}$ satisfies any of the following: \begin{itemize}
\item [\rm (i)]
$T_{n,k}=[a_0(n-1)+a_2]T_{n-1,k}+[b_0(n-1)+b_2]T_{n-1,k-1};$
\item [\rm (ii)] $T_{n,k}=[a_0(b_0+ b_1)(n-1)+a_0 b_1k+a_0 b_2]T_{n-1,k}+[b_0(n-1)+b_1(k-1)+b_2]T_{n-1,k-1}$;
\item [\rm (iii)]
$T_{n,k}=(a_1k+a_2)T_{n-1,k}+[b_1(k-1)+b_2]T_{n-1,k-1};$
\item [\rm (iv)] $T_{n,k}=[a_0(n-k-1)+a_2]T_{n-1,k}+[b_0(n-k)+b_2]T_{n-1,k-1};$
\item [\rm (v)]
$T_{n,k}=(a_1k+a_2)T_{n-1,k}+(b_0n-b_0k+b_2)T_{n-1,k-1}$ \text{for}
$0\in\{a_2,b_2,a_1-a_2,b_0-b_2\}$;
\item [\rm (vi)]
$T_{n,k}=b_0\left(n-2k+\frac{2a_2-a_1}{a_1}\right)T_{n-1,k}+[a_1\,(n-k)+a_2]T_{n-1,k-1}$;
\item [\rm (vii)]
$T_{n,k}=(a_1\,k+a_2)T_{n-1,k}+b_0\left(2k-n+\frac{2a_2-a_1}{a_1}\right)T_{n-1,k-1}$, \end{itemize} where $T_{0,0}=1$ and $T_{n,k}=0$ unless $0\le k\le n$, then $(T_n(q))_{n\geq0}$ is an $\textbf{x}$-Stieltjes moment and $3$-$\textbf{x}$-log-convex sequence with $\textbf{x}=(a_0,a_1,a_2,b_0,b_1,b_2,q)$. \end{thm}
Note that the next relationship was proved in \cite{Zhu20}. \begin{thm}\label{thm+m+tran}
Let $[T_{n,k}]_{n,k\ge 0}$ be defined in (\ref{Recurece+TT}). Then there exists an array $[A_{n,k}]_{n,k\geq0}$ satisfying the recurrence relation
\begin{eqnarray*} A_{n,k} &=&[[b_0+d(a_1-a_0)]n+(b_1-2da_1)k+b_2+d(a_1-a_2)]A_{n-1,k-1}+\\ &&(a_0n+a_1k+a_2)A_{n-1,k} \end{eqnarray*} with $A_{0,0}=1$ and $A_{n,k}=0$ unless $0\le k\le n$ such that their row-generating functions satisfy \begin{eqnarray}\label{rel+T+A} T_n(q)=(\lambda+dq)^nA_n(\frac{q}{\lambda+dq}) \end{eqnarray} for $n\geq0$. \end{thm} Thus, for the array $[T_{n,k}]_{n,k}$ satisfying a four term recurrence relation in (\ref{Recurece+TT}), we can get the corresponding results by Theorem \ref{thm+Ring+PSE+SM} and Theorem \ref{thm+m+tran}. For instance, we list three concise cases as follows.
\begin{thm}\label{thm+three+PSE+SM} If a triangle $[T_{n,k}]_{n,k}$ satisfies any of the following: \begin{itemize} \item [\rm (i)]
$T_{n,k}=\lambda(a_1k+a_2)T_{n-1,k}+[-da_1n+(b_1+2da_1)k+b_2-b_1-d(a_1-a_2)]T_{n-1,k-1}-\\\frac{d(da_1+b_1)}{\lambda}(n-k+1)T_{n-1,k-2};$
\item [\rm (ii)]
$T_{n,k}=\lambda(a_0n-a_0k+a_2-a_0)T_{n-1,k}+[(b_0+2da_0)(n-k)+b_2+da_2]T_{n-1,k-1}+\\
\frac{d(b_0+da_0)}{\lambda}(n-k+1)T_{n-1,k-2};$
\item [\rm (iii)]\footnote{This case was also proved in \cite{Zhu201}. We will give a different proof.}
$T_{n,k}=\lambda(a_1k+a_2)T_{n-1,k}+[(b_0-da_1)n-(b_0-2da_1)k+b_2-d(a_1-a_2)]T_{n-1,k-1}+\\ \frac{d(b_0-da_1)}{\lambda}(n-k+1)T_{n-1,k-2},$ \end{itemize} where $T_{0,0}=1$ and $T_{n,k}=0$ unless $0\le k\le n$, then $(T_n(q))_{n\geq0}$ is an $\textbf{x}$-Stieltjes moment and $3$-$\textbf{x}$-log-convex sequence with $\textbf{x}=(a_0,a_1,a_2,b_0,b_1,b_2,d,\lambda,q)$. \end{thm}
Let $M=[M_{n,k}]_{n,k\ge 0}$ be an infinite matrix. For $n\geq0$, define the $M$-convolution \begin{eqnarray}\label{a-c} z_n=\sum_{k=0}^{n}M_{nk}x_ky_{n-k}. \end{eqnarray} We say that \eqref{a-c} preserves the Stieltjes moment property: if both $(x_n)_{n\ge 0}$ and $(y_n)_{n\ge 0}$ are Stieltjes moment sequences, then so is $(z_n)_{n\ge 0}$.
Using positive definiteness of the quadratic form, P\'olya and Szeg\"o~\cite[Part VII, Theorem 42]{PS64} proved that the binomial convolution $$z_n=\sum_{k=0}^{n}\binom{n}{k}x_ky_{n-k}$$ preserves the Stieltjes moment property for real numbers. Recently, more and more triangular convolutions preserving the Stieltjes moment property for real numbers, see Wang and Zhu \cite{WZ16} and Zhu \cite{Zhu201}. In addition, the next generalized result was proved.
\begin{lem}\label{lem+conv}\emph{\cite{WZ16}} Let $M_n(q)=\sum_{k=0}^{n}M_{n,k}q^k$ be the $n$-th row generating function of a matrix $M$. Assume that $(M_n(q))_{n\ge 0}$ is a Stieltjes moment sequence for any fixed $q\ge 0$. Then the $M$-convolution \eqref{a-c} preserves the Stieltjes moment property for real numbers. \end{lem} Combining Theorems \ref{thm+Ring+PSE+SM} and \ref{thm+three+PSE+SM} and Lemma \ref{lem+conv}, we immediately have the next result. \begin{thm}\label{thm+T+conv+SM} Let $\{a_0,a_1,a_2,b_0,b_1,b_2,d\}\subseteq\mathbb{R^{\geq}}$ and $\lambda>0$. If the triangular array $[T_{n,k}]_{n,k}$ satisfies any recurrence relation in Theorem \ref{thm+Ring+PSE+SM} and Theorem \ref{thm+three+PSE+SM}, then its triangle-convolution $$z_n=\sum_{k=0}^{n}T_{n,k} x_ky_{n-k},\quad n=0,1,2,\ldots$$ preserves the Stieltjes moment property for real numbers. \end{thm}
For the row-generating function $T_n(q)$ of $[T_{n,k}]_{n,k}$, we also give a result for the $\textbf{x}$-Stieltjes moment property of $T_n(q)$ by taking $q$ to be a fixed $\mu$. \begin{thm}\label{thm+sequence+SM} If an array $[T_{n,k}]_{n,k}$ satisfies the recurrence relation:
$$T_{n,k}=(a_0n-\mu b_1k+a_2)T_{n-1,k}+(b_0n+b_1k+b_2)T_{n-1,k-1},$$ where $T_{n,k}=0$ unless $0\le k\le n$ and $T_{0,0}=1$, then $(T_n(\mu))_{n\geq0}$ is an $\textbf{x}$-Stieltjes moment sequence with $\textbf{x}=(a_0,a_2,b_0,b_1,b_2,\mu)$. \end{thm}
\section{Total positivity and continued fractions} In this section, we will present some criteria for total positivity from combinatorial arrays and continued fraction expansions. \begin{thm}\label{thm+Hakel+Jacbi+main} Let $\{r_n(\textbf{x}), s_n(\textbf{x}),t_n(\textbf{x})\}\subseteq\mathbb{R}[\textbf{x}]$ for $n\in \mathbb{N}$. Assume that an array $[D_{n,k}]_{n,k}$ satisfies the recurrence relation:
\begin{equation}\label{recurr+PJS} D_{n,k}=r_{k-1}(\textbf{x})D_{n-1,k-1}+s_k(\textbf{x})D_{n-1,k}+t_{k+1}(\textbf{x})D_{n-1,k+1},
\end{equation} were $D_{n,k}=0$ unless $0\le k\le n$ and $D_{0,0}=1$. Then we have the following results: \begin{itemize} \item [\rm (i)] The ordinary generating function has the Jacobi continued fraction expression \begin{eqnarray*} \sum\limits_{n=0}^{\infty}D_{n,0} z^n=\DF{1}{1- s_0(\textbf{x})z-\DF{r_0(\textbf{x})t_1(\textbf{x})z^2}{1- s_1(\textbf{x})z-\DF{r_1(\textbf{x})t_2(\textbf{x})z^2}{1- s_2(\textbf{x})z-\ldots}}}. \end{eqnarray*} \item [\rm (ii)] If the tridiagonal matrix \begin{equation*}\label{J-eq} J(r(\textbf{x}),s(\textbf{x}),t(\textbf{x}))=\left[ \begin{array}{ccccc} s_0(\textbf{x}) & r_0(\textbf{x}) & & &\\ t_1(\textbf{x}) & s_1(\textbf{x}) & r_1(\textbf{x}) &\\
& t_2(\textbf{x}) & s_2(\textbf{x}) &r_2(\textbf{x}) &\\ & & \ddots&\ddots & \ddots \\ \end{array}\right] \end{equation*} is $\textbf{x}$-TP$_r$ (resp., $\textbf{x}$-TP), then the Hankel matrix $[D_{i+j,0}]_{i,j\geq0}$ is $\textbf{x}$-TP$_r$ (resp., $\textbf{x}$-TP). \item [\rm (iii)] If $J(r(\textbf{x}),s(\textbf{x}),t(\textbf{x}))$ is $\textbf{x}$-TP$_{k+1}$ for $1\leq k\leq 3$, then $(D_{n,0})_{n\geq0}$ is $k$-$\textbf{x}$-log-convex. \end{itemize} \end{thm} \begin{proof} (i)
Let $h_k(z)=\sum_{n\geq k}D_{n,k}z^n$ for $k\geq0$. It follows from the recurrence relation (\ref{recurr+PJS}) that we have \begin{eqnarray*} h_0(z)&=&1+s_0zh_0(z)+t_1zh_1(z), \\ h_k(z)&=&r_{k-1}zh_{k-1}(z)+s_{k}zh_k(z)+t_{k+1}zh_{k+1}(z) \end{eqnarray*} for $k\geq1$, which imply \begin{eqnarray*} \frac{h_0(z)}{1}&=&\frac{1}{1-s_0z-t_1z\frac{h_1(z)}{h_0(z)}}, \\ \frac{h_1(z)}{h_0(z)}&=&\frac{r_0z}{1-s_1z-t_2z\frac{h_2(z)}{h_1(z)}},\\ &\vdots&\\ \frac{h_k(z)}{h_{k-1}(z)}&=&\frac{r_{k-1}z}{1-s_{k}z-t_{k+1}z\frac{h_{k+1}(z)}{h_k(z)}}. \end{eqnarray*} Thus we get \begin{eqnarray*} \sum\limits_{n=0}^{\infty}D_{n,0} z^n=h_0(z)=\DF{1}{1- s_0z-\DF{r_0t_1z^2}{1- s_1z-\DF{r_1t_2z^2}{1- s_2z-\ldots}}}. \end{eqnarray*}
(ii) Let $\alpha=(D_{n,0})_{n\geq0}$ and $D=[D_{n,k}]_{n,k\geq0}$. We will show the following fundamental expression. \begin{cl}\label{Cl} We have the fundamental formula $$DVD^T=H(\alpha),$$ where \[V=\left[\begin{array}{ccccc} V_{0}&&&&\\ &V_{1}&&&\\ &&V_{2}&&\\ &&&\ddots\\ \end{array}\right]\] with $V_0=1,V_n=\prod_{i=1}^{n}t_ir_{i-1}^{-1}$ for $n\geq1$. \end{cl} \begin{proof}
In order to show $DVD^T=H(\alpha)$, it suffices to prove that \begin{eqnarray} \sum_{k\geq0}D_{n,k}D_{m,k}V_k=D_{n+m,0} \end{eqnarray} for any nonnegative integers $n$ and $m$. It is obvious that $$\sum_{k\geq0}D_{n,k}D_{0,k}V_k=D_{0,0}D_{n,0}V_0=D_{n,0}.$$ Assume that the assertion is true for all $i\leq m-1$ and all $n$. Then \begin{eqnarray*} \sum_{k\geq0}D_{n,k}D_{m,k}V_k &=&\sum_{k\geq0}D_{n,k}\left[r_{k-1}D_{m-1,k-1}+s_kD_{m-1,k}+t_{k+1}D_{m-1,k+1}\right]V_k\\ &=&\sum_{k\geq0}D_{m-1,k}\left[r_{k}V_{k+1}D_{n,k+1}+s_kV_kD_{n,k}+t_{k}V_{k-1}D_{n,k-1}\right]\\ &=&\sum_{k\geq0}D_{m-1,k}\left[t_{k+1}D_{n,k+1}+s_kD_{n,k}+r_{k-1}D_{n,k-1}\right]V_k\\ &=&\sum_{k\geq0}D_{m-1,k}D_{n+1,k}V_k\\ &=&D_{n+m,0}, \end{eqnarray*} where the third equality is obtained by using $r_kV_{k+1}=t_{k+1}V_k$. \end{proof}
For the triangular array $[D_{n,k}]_{n,k}$, we can construct an associated triangular array $[D^*_{n,k}]_{n,k}$ as follows: \begin{equation}\label{rr+three} D^*_{n,k}=D^*_{n-1,k-1}+s_k\,D^*_{n-1,k}+r_{k}t_{k+1}\,D^*_{n-1,k+1} \end{equation} with $D^*_{n,k}=0$ unless $0\le k\le n$ and $D^*_{0,0}=1$. Clearly, $D^*_{n,0}=D_{n,0}$ because its ordinary generating function has the same Jacobi continued fraction expansion by (i). Let $D^*$ denote the matrix $[D^*_{n,k}]_{n,k\geq0}$. By the above Claim \ref{Cl}, we immediately have \begin{equation}\label{Hankel+decom} D^*V^*{(D^*)}^T=H(\alpha), \end{equation}
where \[V^*=\left[\begin{array}{ccccc} V^*_{0}&&&&\\ &V^*_{1}&&&\\ &&V^*_{2}&&\\ &&&\ddots\\ \end{array}\right]\] with $V^*_0=1,V^*_n=\prod_{i=1}^{n}t_ir_{i-1}$ for $n\geq1$.
Thus, applying the classical Cauchy-Binet formula to (\ref{Hankel+decom}), in order to prove that $H(\alpha)$ is $\textbf{x}$-TP$_r$ (resp., $\textbf{x}$-TP), it suffices to demonstrate that the matrix $D^*$ is $\textbf{x}$-TP$_r$ (resp., $\textbf{x}$-TP). This follows from the next two claims.
\begin{cl}\label{cl-equal+tridilog} The matrix \[J=\left[\begin{array}{ccccc} s_0&r_0&&&\\ t_1&s_1&r_1&&\\ &t_2&s_2&r_2&\\ &&\ddots&\ddots&\ddots \end{array}\right]\] is $\textbf{x}$-TP$_r$ (resp., $\textbf{x}$-TP) in $\mathbb{R}[\textbf{x}]$ if and only if the matrix \[J^*=\left[\begin{array}{ccccc} s_0&1&&&\\ r_0t_1&s_1&1&&\\ &r_1t_2&s_2&1&\\ &&\ddots&\ddots&\ddots \end{array}\right] \] is $\textbf{x}$-TP$_r$ (resp., $\textbf{x}$-TP) in $\mathbb{R}[\textbf{x}]$. \end{cl}
\begin{proof} It suffices to prove the corresponding result for their $n$-th leading principal submatrices, see Theorem 4.3 of Pinkus \cite{Pin10}. That is the matrix \[J_n=\left[\begin{array}{ccccc} s_0&r_0&&&\\ t_1&s_1&r_1&&\\ &t_2&s_2&r_2&\\ &&\ddots&\ddots&\ddots\\ &&&t_n&s_n\\ \end{array}\right]\] is $\textbf{x}$-TP$_r$ (resp., $\textbf{x}$-TP) in $\mathbb{R}[\textbf{x}]$ if and only if the matrix \[J^*_n=\left[\begin{array}{ccccc} s_0&1&&&\\ r_0t_1&s_1&1&&\\ &r_1t_2&s_2&1&\\ &&\ddots&\ddots&\ddots\\ &&&r_{n-1}t_n&s_n\\ \end{array}\right] \] is $\textbf{x}$-TP$_r$ (resp., $\textbf{x}$-TP) in $\mathbb{R}[\textbf{x}]$. This follows by induction on $n$ since $\det{J_n}=\det{J^*_n}$ from the following identities \begin{eqnarray*} \det{J_n}&=&s_n\det{J_{n-1}}-r_{n-1}t_n\det{J_{n-2}},\\ \det{J^*_n}&=&s_n\det{J^*_{n-1}}-r_{n-1}t_n\det{J^*_{n-2}}. \end{eqnarray*} \end{proof}
\begin{cl} If the matrix \[J=\left[\begin{array}{ccccc} s_0&r_0&&&\\ t_1&s_1&r_1&&\\ &t_2&s_2&r_2&\\ &&\ddots&\ddots&\ddots \end{array}\right]\] is $\textbf{x}$-TP$_r$ (resp., $\textbf{x}$-TP) in $\mathbb{R}[\textbf{x}]$, then so is the triangular matrix $D^*$. \end{cl} \begin{proof} Let $\overline{D^*}$ denote the matrix obtained from $D^*$ by deleting its first row. Also let $\overline{D_n^*}$ and $D_n^*$ denote the $(n+1)$-th leading principal submatrices of $\overline{D^*}$ and $D^*$, respectively. In order to prove that the triangular matrix $D^*$ is $\textbf{x}$-TP$_r$ (resp., $\textbf{x}$-TP), it suffices to prove that $D_n^*$ is $\textbf{x}$-TP$_r$ (resp., $\textbf{x}$-TP). It follows from Claim \ref{cl-equal+tridilog} that $J^*_n$ is $\textbf{x}$-TP$_r$ (resp., $\textbf{x}$-TP). By (\ref{rr+three}), we have $\overline{D_n^*}=D^*_nJ^*_n$. By induction on $n$, we immediately get that $D^*_n$ for $n\in N$ is $\textbf{x}$-TP$_r$ (resp., $\textbf{x}$-TP). \end{proof}
(iii) For brevity, we write $D_{n,0}$ for $D_n$. It follows from (ii) that the Hankel matrix $[D_{i+j}]_{i,j\geq0}$ is $\textbf{x}$-TP$_{r+1}$ for $1\leq r\leq3$. We will use some identities (see \cite{Zhu18} for instance): \begin{eqnarray*}
\mathcal {L}(D_k)&=&\left|\begin{array}{cc} D_{k-1}&D_{k}\\ D_{k}&D_{k+1}
\end{array}\right|,\\ \mathcal {L}^2(D_k)&=&\mathcal {L}(D_{k-1})\mathcal {L}(D_{k+1})-\left[\mathcal {L}(D_k)\right]^2\nonumber\\ &=&\left(D_{k+2}D_{k}-D_{k+1}^2\right)\left(D_{k}D_{k-2}-D_{k-1}^2\right)-\left(D_{k+1}D_{k-1}-D_k^2\right)^2\nonumber\\
&=&D_k\left|\begin{array}{ccc} D_{k-2}&D_{k-1}&D_{k}\\ D_{k-1}&D_k&D_{k+1}\\ D_k&D_{k+1}&D_{k+2}
\end{array}\right|,\\ \mathcal {L}^3(D_k)&=&\mathcal {L}^2(D_{k-1})\mathcal {L}^2(D_{k+1})-\left[\mathcal {L}^2(D_k)\right]^2\\
&=&(D_{k+1}D_{k-1}-D^2_k) D^2_k\left|\begin{array}{cccc} D_{k-3}&D_{k-2}&D_{k-1}&D_{k}\\ D_{k-2}&D_{k-1}&D_{k}&D_{k+1}\\ D_{k-1}&R_{k}&D_{k+1}&D_{k+2}\\ D_k&D_{k+1}&D_{k+2}&D_{k+3}
\end{array}\right|+\\
&& (D_{k+1}D_{k-1}-D^2_k)\left|\begin{array}{ccc} D_{k-3}&D_{k-2}&D_{k-1}\\ D_{k-2}&D_{k-1}&D_{k}\\ D_{k-1}&D_{k}&D_{k+1}
\end{array}\right|\left|\begin{array}{ccc} D_{k-1}&D_{k}&D_{k+1}\\ D_{k}&D_{k+1}&D_{k+2}\\ D_{k+1}&D_{k+2}&D_{k+3}
\end{array}\right|. \end{eqnarray*} Obviously, $(D_n)_{n\geq0}$ is strongly \textbf{x}-log-convex if $[D_{i+j}]_{i,j\geq0}$ is \textbf{x}-TP$_{2}$. If $[D_{i+j}]_{i,j\geq0}$ is \textbf{x}-TP$_{r+1}$ for $r=2,3$, then $(D_n)_{n\geq0}$ is $k$-\textbf{x}-log-convex. We complete the proof. \end{proof} \begin{rem} When $r_n=1$ for $n\geq0$, the array $[D_{n,k}]_{n,k}$ for real numbers was called the recursive triangle by Aigner \cite{Aig01}. Aigner also gave a determinant method to get the corresponding continued fraction and Flajolet \cite{Fla80} also presented a combinatorial interpretation. \end{rem}
\begin{rem} We refer the reader to \cite{Zhu13,Zhu14} for the original idea of total positivity in Theorem \ref{thm+Hakel+Jacbi+main}. Note that we can not directly get Hankel-total positivity using $V$ because $V_n$ may be a rational function not a polynomial. Our proof indicates the relation between arrays $[D_{n,k}]$ and $[D^*_{n,k}]$. \end{rem}
\begin{rem} Generally speaking, it is much easier to deal with total positivity of the tridiagonal matrix $J$ than that of $J^*$. In fact, under keeping the product of $\textbf{r}$ and $\textbf{t}$, we can choose different pairs of $\textbf{r}$ and $\textbf{t}$. \end{rem}
Recall the concept of $\gamma$-binomial transformation. Given a sequence $(a_n)_{n\geq0}$, its $\gamma$-binomial transformation is defined to be \begin{eqnarray} a^\circ_n=\sum_{k=0}^n\binom{n}{k}a_k\gamma^{n-k} \end{eqnarray} for $n\geq0$. For $\gamma=1$, it
reduces to the famous binomial transformation. More generally, for an array $[A_{n,k}]_{n,k}$, define its {\it $\gamma$-binomial transformation} $[A^\circ_{n,k}]_{n,k}$ by \begin{eqnarray} A^\circ_{n,k}=\sum_{i=0}^n\binom{n}{i}A_{i,k}\gamma^{n-i}. \end{eqnarray}
\begin{prop}\label{prop+x+binomial+array} Assume that $[A^\circ_{n,k}]_{n,k}$ is the $\gamma$-binomial transformation of $[A_{n,k}]_{n,k}$. \begin{itemize}
\item [\rm (i)]
If $[A_{n,k}]_{n,k}$ is $\textbf{x}$-TP$_r$ in $\mathbb{R}[\textbf{x}]$, then
$[A^\circ_{n,k}]_{n,k}$ is $(\textbf{x},\gamma)$-TP$_r$.
\item [\rm (ii)]
We have \begin{eqnarray} A^\circ_n(q):=\sum_{k\geq0}A^\circ_{n,k}q^k=\sum_{i=0}^n\binom{n}{i}A_i(q)\gamma^{n-i}. \end{eqnarray}
\item [\rm (iii)] If the array $[A_{n,k}]_{n,k}$ in $\mathbb{R}[\textbf{x}]$ satisfies the following recurrence relation:
\begin{eqnarray}
A_{n,k}=r_{k-1}(\textbf{x})A_{n-1,k-1}+s_k(\textbf{x})A_{n-1,k}+t_{k+1}(\textbf{x})A_{n-1,k+1}
\end{eqnarray} with $A_{n,k}=0$ unless $0\le k\le n$ and $A_{0,0}=1$, then $[A^\circ_{n,k}]_{n,k}$ satisfies the following recurrence relation: \begin{eqnarray} A^\circ_{n,k}=r_{k-1}(\textbf{x})A^\circ_{n-1,k-1}+(\gamma+s_k(\textbf{x}))A^\circ_{n-1,k}+t_{k+1}(\textbf{x})A^\circ_{n-1,k+1}
\end{eqnarray} with $A^\circ_{n,k}=0$ unless $0\le k\le n$ and $A^\circ_{0,0}=1$.
\item [\rm (iv)] If $(A_{n,0})_{n\geq0}$ is an $\textbf{x}$-Stieltjes moment sequence, then $(A^\circ_{n,0})_{n\geq0}$ is an $(\textbf{x},\gamma)$-Stieltjes moment sequence. \end{itemize} \end{prop}
\begin{proof} (i) Let $$\binom{n}{k}\gamma^{n-k}=B_{n,k}(\gamma)$$ for $n\geq k\geq0$. Clearly, $[B_{n,k}(\gamma)]_{n,k\geq0}$ is an array satisfying the recurrence relation \begin{equation}\label{recurrence+Binom+matrx} B_{n,k}(\gamma)=\gamma B_{n-1,k}(\gamma)+B_{n-1,k-1}(\gamma), \end{equation} where $B_{n,k}(\gamma)=0$ unless $0\le k\le n$ and $B_{0,0}(\gamma)=1$. Thus the $\gamma$-binomial transformation of $[A_{n,k}]_{n,k}$ is equivalent to the decomposition \begin{equation}\label{Id+x+binom+array} [A^\circ_{n,k}]_{n,k}=[B_{n,k}(\gamma)]_{n,k}[A_{n,k}]_{n,k}. \end{equation} Obviously, total positivity of the Pascal triangle implies that $[B_{n,k}(\gamma)]_{n,k\geq0}$ is $\gamma$-TP. Then applying the classical Cauchy-Binet formula to (\ref{Id+x+binom+array}), we immediately get (i).
(ii) Clearly, for $n\geq0$, we have \begin{eqnarray*} A^\circ_n(q)=\sum_{i=0}^nA^\circ_{n,i}q^i =\sum_{i=0}^n\sum_{k=0}^n\binom{n}{k}A_{k,i}\gamma^{n-k}q^i =\sum_{k=0}^n \binom{n}{k} A_k(q)\gamma^{n-k}. \end{eqnarray*}
(iii) We have \begin{eqnarray*} A^\circ_{n,k}&=&\sum_{i=0}^n\binom{n}{i}A_{i,k}\gamma^{n-i}=\sum_{i=0}^nB_{n,i}(\gamma)A_{i,k}=\sum_{i=0}^n[\gamma B_{n-1,i}(\gamma)+B_{n-1,i-1}(\gamma)]A_{i,k}\\ &=&\gamma A^\circ_{n-1,k}+\sum_{i=0}^{n-1}B_{n-1,i}(\gamma)A_{i+1,k}\\ &=&r_{k-1}(\textbf{x})A^\circ_{n-1,k-1}+(\gamma+s_k(\textbf{x}))A^\circ_{n-1,k}+t_{k+1}(\textbf{x})A^\circ_{n-1,k+1}. \end{eqnarray*} It is obvious that $A^\circ_{n,k}=0$ unless $0\le k\le n$ and $A^\circ_{0,0}=1$.
(iv) It follows from the decomposition (\ref{Id+x+binom+array}) and Claim \ref{Cl} in Theorem \ref{thm+Hakel+Jacbi+main} (ii) that \begin{eqnarray*} [A^\circ_{i+j,0}]_{i,j}&=&[A^\circ_{n,k}]_{n,k}V [A^\circ_{n,k}]_{n,k}^T\\ &=&[B_{n,k}(\gamma)]_{n,k}[A_{n,k}]_{n,k}V [A_{n,k}]_{n,k}^T[B_{n,k}(\gamma)]_{n,k}^T\\ &=&[B_{n,k}(\gamma)]_{n,k}[A_{i+j,0}]_{i,j}[B_{n,k}(\gamma)]_{n,k}^T. \end{eqnarray*} Note that $[B_{n,k}(\gamma)]_{n,k}$ is $\gamma$-TP. Then, using the classical Cauchy-Binet formula, we immediately deduce that $[A^\circ_{i+j,0}]_{i,j}$ is $(\textbf{x},\gamma)$-TP when $[A_{i+j,0}]_{i,j}$ is $\textbf{x}$-TP. Thus, if $(A_{n,0})_{n\geq0}$ is $\textbf{x}$-SM, then is $(A^\circ_{n,0})_{n\geq0}$ is $(\textbf{x},\gamma)$-SM. This completes the proof. \end{proof}
\begin{rem} The (iv) of Proposition \ref{prop+x+binomial+array} can also be proved by a different method in \cite{Zhu19}. But it seems proof here is more natural. \end{rem}
\section{Tridiagonal matrices and $\textbf{x}$-Stieltjes moment sequences}
The total positivity of the tridiagonal matrix \begin{equation*}\label{J-eq} J(\textbf{r},\textbf{s},\textbf{t})=\left[ \begin{array}{ccccc} s_0 & r_0 & & &\\ t_1 & s_1 & r_1 &\\
& t_2 & s_2 &r_2 &\\ & & \ddots&\ddots & \ddots \\ \end{array}\right] \end{equation*} plays an important role in Theorem \ref{thm+Hakel+Jacbi+main}. Thus, we will present some criteria for its total positivity.
By the Laplace expansion, if each element of $J(\textbf{r},\textbf{s},\textbf{t})$ is nonnegative, then it is not hard to get that the total positivity of $J(\textbf{r},\textbf{s},\textbf{t})$ is equivalent to that all submatrices with the contiguous rows and the same columns are totally positive. The following positivity result about perturbation for tridiagonal matrices is interesting and very important. \begin{prop}\label{prop+addition+tri} Assume for $n\in \mathbb{P}$ that $\{a_n, b_n, c_n, a'_n, b'_n, c'_n,b_n-b'_n,c_n-c'_n\}\subseteq \mathbb{R}^{\geq}[\textbf{x}]$. If the matrix \begin{eqnarray*}\left[
\begin{array}{cccc}
a_1 & b_1 & & \\
c_1 & a_2 & b_2 & \\
& c_2& a_3 & \ddots \\
& & \ddots & \ddots \\
\end{array}
\right] \end{eqnarray*} is $\textbf{x}$-TP$_r$ in $\mathbb{R}[\textbf{x}]$, then so is the tridiagonal matrix \begin{eqnarray*} \left[
\begin{array}{cccc}
a_1+a'_1 & b_1-b'_1 & & \\
c_1-c'_1 & a_2+a'_2 & b_2-b'_2 & \\
& c_2-c'_2& a_3+a'_3 & \ddots \\
& & \ddots & \ddots \\
\end{array}
\right]. \end{eqnarray*} \end{prop} \begin{proof} Assume that \begin{eqnarray*} T_n=\left[
\begin{array}{ccccc}
a_1+a'_1 & b_1-b'_1 & & \\
c_1-c'_1 & a_2+a'_2 & b_2-b'_2 & \\
& \ddots & \ddots & \ddots \\
&&c_{n-2}-c'_{n-2}&a_{n-1}+a'_{n-1}&b_{n-1}-b'_{n-1}\\
& &&c_{n-1}-c'_{n-1}&a_n+a'_n
\end{array}
\right]. \end{eqnarray*} In order to prove that the tridiagonal matrix \begin{eqnarray*} \left[
\begin{array}{cccc}
a_1+a'_1 & b_1-b'_1 & & \\
c_1-c'_1 & a_2+a'_2 & b_2-b'_2 & \\
& c_2-c'_2& a_3+a'_3 & \ddots \\
& & \ddots & \ddots \\
\end{array}
\right] \end{eqnarray*} is $\textbf{x}$-TP$_r$, it suffices to prove that $T_n$ is $\textbf{x}$-TP$_r$ for all $n\geq1$. In what follows, we proceed by induction on the number
$$|\{i:a'_i\not\equiv0\}\bigcup \{i:b'_i\not\equiv0\}\bigcup
\{i:c'_i\not\equiv0\}|.$$ Since the matrix \begin{eqnarray*}\left[
\begin{array}{cccc}
a_1 & b_1 & & \\
c_1 & a_2 & b_2 & \\
& c_2& a_3 & \ddots \\
& & \ddots & \ddots \\
\end{array}
\right] \end{eqnarray*} is $\textbf{x}$-TP$_r$, its minors are nonnegative. Thus it is true for
$$|\{i:a'_i\not\equiv0\}\bigcup \{i:b'_i\not\equiv0\}\bigcup
\{i:c'_i\not\equiv0\}|=0.$$ Suppose that it holds for
$|\{i:a'_i\not\equiv0\}\bigcup \{i:b'_i\not\equiv0\}\bigcup
\{i:c'_i\not\equiv0\}|=k-1$. Let us consider the next step for
$|\{i:a'_i\not\equiv0\}\bigcup \{i:b'_i\not\equiv0\}\bigcup
\{i:c'_i\not\equiv0\}|=k$. Without loss of generality, denote the new positive element by $a'_m$, $b'_m$ or $c'_m$. If one minor of order $\leq r$ of $T_n$ does not contain the new element, then it belongs to $\mathbb{R}^{\geq}[\textbf{x}]$ by the inductive hypothesis. Thus it suffices to prove the minors of order $\leq r$ containing the new element belong to $\mathbb{R}^{\geq}[\textbf{x}]$. Then it suffices to consider submatrices with contiguous rows and the same column in the following three cases.
Case $1$. Assume that the new positive element is $a'_m$. By dividing the row containing $a_m+a'_m$ into two rows, we have \begin{eqnarray*} \det\left[
\begin{array}{cccc}
\ddots & \ddots & & \\
\ddots & a_m+a'_m & \ddots & \\
& \ddots& \ddots & \ddots \\
\end{array}
\right]
&=&\det\left[
\begin{array}{cccc}
\ddots & \ddots & & \\
\ddots & a_m & \ddots & \\
& \ddots& \ddots & \ddots \\
\end{array}
\right]
+
\det\left[
\begin{array}{cccc}
\ddots & \ddots & & \\
0 & a'_m & 0 & \\
& \ddots& \ddots & \ddots \\
\end{array}
\right]\\
&=&M_1+a'_mM_2\\
&\geq_{\textbf{x}}&0 \end{eqnarray*} where both $M_1$ and $M_2$ are minors in the inductive hypothesis.
Case $2$. Assume that the new positive element is $b'_m$. By dividing the row containing $b_m-b'_m$ into two rows, we have \begin{eqnarray*} &&\det\left[
\begin{array}{cccccc}
\ddots&\ddots& & & & \\
\ddots& \ddots & \ddots & & \\
& b_m-b'_m & \ddots & \ddots \\
& & \ddots & \ddots \\
\end{array}
\right]\\
&=&\det\left[
\begin{array}{cccccc}
\ddots&\ddots& & & & \\
\ddots& \ddots & \ddots & & \\
& b_m & \ddots & \ddots \\
& & \ddots & \ddots \\
\end{array}
\right]
+
\det\left[
\begin{array}{cccccc}
\ddots&\ddots& & & & \\
\ddots& \ddots & \ddots & & \\
& -b'_m & 0 & \\
& & \ddots & \ddots \\
\end{array}
\right]\\
&=&M_1+b'_mM_2\\
&\geq_{\textbf{x}}&0 \end{eqnarray*} where both $M_1$ and $M_2$ are minors in the inductive hypothesis.
Case $3$. Assume that the new positive element is $c'_m$, which is similar to the Case $2$.
The proof is complete. \end{proof}
\begin{rem} The result in Proposition \ref{prop+addition+tri} is very useful for proving the total positivity of tridiagonal matrices because it transforms a totally positive matrix to more. \end{rem}
The next result for $\textbf{x}$ being a unique indeterminate was proved by Chen, Liang and Wang \cite{CLW15} using diagonally dominant matrices. Now, we give a new unified proof by Proposition \ref{prop+addition+tri}. \begin{prop}\label{prop+tri+TP} Assume for $n\in \mathbb{N}$ that $\{r_n, s_n,t_n\}\subseteq\mathbb{R}^{\geq}[\textbf{x}]$. Then the matrix $J(\textbf{r},\textbf{s},\textbf{t})$ is $\textbf{x}$-TP under any of the following conditions: \begin{itemize} \item [\rm (i)] $s_0\geq_{\textbf{x}} r_0$ and $s_n \geq_{\textbf{x}} r_n + t_n$ for $n\geq 1$; \item [\rm (ii)] $s_0\geq_{\textbf{x}} t_1$ and $s_n \geq_{\textbf{x}} r_{n-1} + t_{n+1}$ for $n\geq 1$; \item [\rm (iii)] $s_0\geq_{\textbf{x}} 1$ and $s_n \geq_{\textbf{x}} r_{n-1} t_{n}+1$ for $n\geq 1$; \item [\rm (iv)] $s_0\geq_{\textbf{x}} r_0t_1$ and $s_n \geq_{\textbf{x}} r_{n} t_{n+1}+1$ for $n\geq 1$. \end{itemize} \end{prop} \begin{proof} (i) By Claim \ref{cl-equal+tridilog} in the proof of Theorem \ref{thm+Hakel+Jacbi+main} (ii), it suffices to prove that the matrix \[J'_n=\left[\begin{array}{ccccc} s_0&1&&&\\ r_0t_1&s_1&1&&\\ &r_1t_2&s_2&1&\\ &&\ddots&\ddots&\ddots\\ &&&r_{n-1}t_n&s_n\\ \end{array}\right] \] is $\textbf{x}$-TP in $\mathbb{R}[\textbf{x}]$.
It follows from the decomposition \begin{eqnarray*} \left[
\begin{array}{cccc}
r_0 & 1 & & \\
t_1r_0 & t_1+r_1 & 1 & \\
& t_2r_1 & t_2+r_2 & \ddots \\
& & \ddots & \ddots \\
\end{array}
\right] =\left[\begin{array}{cccc} 0 & 1 & & \\
& t_1 & 1 & \\
& & t_2 & \ddots\\
& & & \ddots\\ \end{array}\right] \left[\begin{array}{cccc} 1 & & &\\ r_0 & 1 & &\\
& r_1 & 1 &\\
& & \ddots & \ddots\\ \end{array}\right] \end{eqnarray*} that \begin{eqnarray*}\left[
\begin{array}{cccc}
r_0 & 1 & & \\
t_1r_0 & t_1+r_1 & 1 & \\
& t_2r_1 & t_2+r_2 & \ddots \\
& & \ddots & \ddots \\
\end{array}
\right] \end{eqnarray*} is $\textbf{x}$-TP since $(r_n)_{n\geq0}$ and $(t_n)_{n\geq1}$ belong to $\mathbb{R}^{\geq}[\textbf{x}]$. Thus \[\left[\begin{array}{ccccc} s_0&1&&&\\ r_0t_1&s_1&1&&\\ &r_1t_2&s_2&1&\\ &&\ddots&\ddots&\ddots\\ &&&r_{n-1}t_n&s_n\\ \end{array}\right] \] is $\textbf{x}$-TP by taking $a_1'=s_0-r_0,a_{n+1}'=s_n-r_n-t_n, b_m'=c_n'=0$ for $n\geq1$ in Proposition \ref{prop+addition+tri}.
Similarly, (ii), (iii) and (iv) can respectively be proved by the next decompositions \begin{eqnarray*} \left[
\begin{array}{cccc}
t_1 & 1 & & \\
t_1r_0 & t_2+r_0 & 1 & \\
& t_2r_1 & t_3+r_1 & \ddots \\
& & \ddots & \ddots \\
\end{array}
\right] &=& \left[\begin{array}{cccc} 1 & & &\\ r_0 & 1 & &\\
& r_1 & 1 &\\
& & \ddots & \ddots\\ \end{array}\right] \left[\begin{array}{cccc} t_1 & 1 & & \\
& t_2 & 1 & \\
& & t_3 & \ddots\\
& & & \ddots\\ \end{array}\right], \end{eqnarray*} \begin{eqnarray*} \left[
\begin{array}{cccc}
1 & 1 & & \\
t_1r_0 & t_1r_0+1 & 1 & \\
& t_2r_1 & t_2r_1+ & \ddots \\
& & \ddots & \ddots \\
\end{array}
\right] &=&\left[\begin{array}{cccc} 0 & 1 & & \\
& t_1r_0 & 1 & \\
& & t_2r_1 & \ddots\\
& & & \ddots\\ \end{array}\right] \left[\begin{array}{cccc} 1 & & &\\ 1 & 1 & &\\
& 1 & 1 &\\
& & \ddots & \ddots\\ \end{array}\right],\\ \left[
\begin{array}{cccc}
t_1r_0 & 1 & & \\
t_1r_0 & 1+t_2r_1 & 1 & \\
& t_2r_1 & 1+t_3r_2 & \ddots \\
& & \ddots & \ddots \\
\end{array}
\right] &=&\left[\begin{array}{cccc} 0 & 1 & & \\
& 1 & 1 & \\
& & 1 & \ddots\\
& & & \ddots\\ \end{array}\right] \left[\begin{array}{cccc} 1 & & &\\ t_1r_0 & 1 & &\\
& t_2r_1 & 1 &\\
& & \ddots & \ddots\\ \end{array}\right]. \end{eqnarray*} \end{proof}
Based on Theorem \ref{thm+Hakel+Jacbi+main} and Proposition \ref{prop+tri+TP} (i), we obtain the following result which uses Jacobi continued fractions to prove x-Stieltjes moment property
\begin{thm}\label{thm+S+C} Let $\{\lambda_n(\textbf{x}), \mu_{n}(\textbf{x}),T_n(\textbf{x})\}\subseteq \mathbb{R}^{\geq}[\textbf{x}]$ for $n\in \mathbb{N}$ and $$\sum_{n\geq0} T_n(\textbf{x})z^n=\rm \textbf{JCF}[\lambda_n(\textbf{x}),\mu_{n+1}(\textbf{x});z]_{n\geq0}.$$ If there exists polynomials $\alpha_n(\textbf{x})$ and $\beta_n(\textbf{x})$ in $\mathbb{R}^{\geq}[\textbf{x}]$ such that $$\lambda_n(\textbf{x})=\beta_n(\textbf{x})+\alpha_{2n}(\textbf{x})+\alpha_{2n-1}(\textbf{x}), \mu_{n+1}(\textbf{x})=\alpha_{2n}(\textbf{x})\alpha_{2n+1}(\textbf{x}),$$ then $T_n(\textbf{x})$ form an $\textbf{x}$-SM and $3$-$\textbf{x}$-LCX sequence for $n\in \mathbb{N}$. \end{thm}
\begin{proof} In order to prove that $T_n(\textbf{x})$ form an $\textbf{x}$-SM and $3$-$\textbf{x}$-LCX sequence, by Theorem \ref{thm+Hakel+Jacbi+main}, it suffices to prove the corresponding tridiagonal matrix \[\left[\begin{array}{ccccc} \lambda_0(\textbf{x})&\alpha_{0}(\textbf{x})&&&\\ \alpha_{1}(\textbf{x})&\lambda_1(\textbf{x})&\alpha_{2}(\textbf{x})&&\\ &\alpha_{3}(\textbf{x})&\lambda_2(\textbf{x})&\alpha_{4}(\textbf{x})&\\ &&\ddots&\ddots&\ddots\\ &&&\alpha_{2n-1}(\textbf{x})&\lambda_n(\textbf{x})\\ \end{array}\right] \] is $\textbf{x}$-TP in $\mathbb{R}[\textbf{x}]$. Clearly, \begin{eqnarray*} \lambda_0(\textbf{x})=\beta_0(\textbf{x})+\alpha_{0}(\textbf{x})&\geq_{x}&\alpha_{0}(\textbf{x}),\\ \lambda_n(\textbf{x})=\beta_n(\textbf{x})+\alpha_{2n}(\textbf{x})+\alpha_{2n-1}(\textbf{x})&\geq_{x}&\alpha_{2n}(\textbf{x})+\alpha_{2n-1}(\textbf{x})\quad\text{for}\quad n\geq1. \end{eqnarray*} It follows from Proposition \ref{prop+tri+TP} (i) that we get the desired total positivity. \end{proof} The following result will play an important role in the proof of Theorem \ref{thm+Ring+PSE+SM}.
\begin{thm}\label{thm+Gass} Assume \begin{eqnarray*} \sum_{n\geq0}G_n(a,b,c)z^n=1+\sum_{n\geq1}z^n\prod_{k=0}^{n-1}\frac{a+bk}{1-c(k+1)z}. \end{eqnarray*} Then we have continued fraction expansions \begin{eqnarray*} \sum_{n\geq0}G_n(a,b,c)z^n&=&\rm \textbf{SCF}[\alpha_{2n},\alpha_{2n+1};z]_{n\geq0}\\ &=&\rm \textbf{JCF}[\alpha_{2n}+\alpha_{2n-1},\alpha_{2n}\alpha_{2n+1};z]_{n\geq0}, \end{eqnarray*} where $\alpha_{2n}(a)=a+nb$ and $\alpha_{2n+1}(a)=(c+b)(1+n)$ for $n\geq0$. In addition, $G_n(a,b,c)$ form an $\textbf{x}$-SM and $3$-$\textbf{x}$-LCX sequence with $\textbf{x}=(a,b,c)$ for $n\in \mathbb{N}$. \end{thm} \begin{proof} By Theorem \ref{thm+S+C}, it suffices to prove \begin{eqnarray*} \sum_{n\geq0}G_n(a,b,c)z^n &=&\rm \textbf{JCF}[\alpha_{2n}+\alpha_{2n-1},\alpha_{2n}\alpha_{2n+1};z]_{n\geq0} \end{eqnarray*} with $\alpha_{2n}(a)=a+nb$ and $\alpha_{2n+1}(a)=(c+b)(1+n)$ for $n\geq0$.
Define the function \begin{eqnarray*} F(a,z):=1+\sum_{n\geq1}z^n\prod_{k=0}^{n-1}\frac{a+bk}{1-c(k+1)z}, \end{eqnarray*} which implies
\begin{eqnarray}\label{eq+F}
F(a,z)=1+\frac{az}{1-cz}\,F(a+b,\frac{z}{1-cz}). \end{eqnarray} Assume that \begin{eqnarray*} F(a,z)&=&\frac{1}{1-\frac{\alpha_0(a)z}{1-\frac{\alpha_1(a)z}{1-\frac{\alpha_2(a)z}{1-\ldots}}}}. \end{eqnarray*} It follows from the contraction formula (\ref{contraction}) that \begin{eqnarray} F(a,z) &=&\DF{1}{1- \alpha_0(a)z-\DF{\alpha_0(a)\alpha_1(a)z^2}{1- (\alpha_1(a)+\alpha_2(a))z-\DF{\alpha_2(a)\alpha_3(a)z^2}{1- (\alpha_3(a)+\alpha_4(a))z-\ldots}}}\label{Eq+S2}. \end{eqnarray} Combining (\ref{eq+F}) and (\ref{Eq+S2}), we have \begin{eqnarray*} &&F(a,z)\\ &=&1+\frac{az}{1-cz}\,F(a+b,\frac{z}{1-cz})\\ &=&1+\DF{az}{1- (c+\alpha_0(a+b))z-\DF{\alpha_0(a+b)\alpha_1(a+b)z^2}{1- (c+\alpha_1(a+b)+\alpha_2(a+b))z-\DF{\alpha_2(a+b)\alpha_3(a+b)z^2}{1-\ldots}}}. \end{eqnarray*} On the other hand, by the contraction formula (\ref{contraction+decom}), we also have \begin{eqnarray*} F(a,z) &=&1+\DF{\alpha_0(a)z}{1- (\alpha_0(a)+\alpha_1(a))z-\DF{\alpha_1(a)\alpha_2(a)z^2}{1- (\alpha_2(a)+\alpha_3(a))z-\DF{\alpha_3(a)\alpha_4(a)z^2}{1- (\alpha_4(a)+\alpha_5(a))z-\ldots}}}. \end{eqnarray*} Thus, we get equations \begin{eqnarray*} \alpha_0(a)&=&a\\ \alpha_0(a)+\alpha_1(a)&=&c+\alpha_0(a+b)\\ \alpha_1(a)\alpha_2(a)&=&\alpha_0(a+b)\alpha_1(a+b)\\ \alpha_2(a)+\alpha_3(a)&=&c+\alpha_1(a+b)+\alpha_2(a+b)\\ &\cdots&. \end{eqnarray*} Solving equations, we get $\alpha_{2n}(a)=a+nb$ and $\alpha_{2n+1}(a)=(c+b)(1+n)$ for $n\geq0$. \end{proof}
\section{Proof of Theorem \ref{thm+Ring+PSE+SM}}
In this section, in order to apply previous results to prove Theorem \ref{thm+Ring+PSE+SM}, the key is to obtain the continued fraction expansion for the generating function $\sum_{n\geq0}T_n(q)t^n$.
We first present an important relationship between two different arrays. \begin{lem}\label{lem+shit} Let $\{a_0,a_2,b_0,b_1,b_2,\lambda\}\subseteq \mathbb{R}$. Assume that an array $[A_{n,k}]_{n,k\geq0}$ satisfies the recurrence relation: \begin{equation}\label{recurrence relation+A} A_{n,k}=(a_0n-\lambda b_1k+a_2)A_{n-1,k}+(b_0n+b_1k+b_2)A_{n-1,k-1}, \end{equation} where $A_{n,k}=0$ unless $0\le k\le n$ and $A_{0,0}=1$. If a polynomial $B_n(q)=A_n(q+\lambda)$, then the coefficient array $[B_{n,k}]_{n,k\geq0}$ of $B_n(q)$ satisfies the recurrence relation: \begin{equation*} B_{n,k}=[(a_0+\lambda b_0)n+\lambda b_1k+a_2+\lambda(b_1+b_2)]B_{n-1,k}+(b_0n+b_1k+b_2)B_{n-1,k-1}, \end{equation*} where $B_{0,0}=1$ and $B_{n,k}=0$ unless $0\le k\le n$. \end{lem} \begin{proof} By the recurrence relation (\ref{recurrence relation+A}), we have \begin{eqnarray*} A_n(q) &=& [a_0n+a_2+(b_0n+b_1+b_2)q]A_{n-1}(q)+b_1(-\lambda +q)qA'_{n-1}(q). \end{eqnarray*} Then using $B_n(q)=A_n(q+\lambda)$, we get \begin{eqnarray*} B_n(q) &=& [(a_0+\lambda b_0)n+a_2+\lambda(b_1+b_2)+(b_0n+b_1+b_2)q]B_{n-1}(q)+b_1(\lambda +q)qB'_{n-1}(q) \end{eqnarray*} for $n\geq1$. This implies that $[B_{n,k}]_{n,k\geq0}$ satisfies the recurrence relation: \begin{equation*} B_{n,k}=[(a_0+\lambda b_0)n+\lambda b_1k+a_2+\lambda(b_1+b_2)]B_{n-1,k}+(b_0n+b_1k+b_2)B_{n-1,k-1}, \end{equation*} where $B_{n,k}=0$ unless $0\le k\le n$ and $B_{0,0}=1$. \end{proof}
\begin{rem} For a triangle $[T_{n,k}]_{n,k}$, define its {\it reciprocal triangle} $[T^*_{n,k}]_{n,k}$ by $$T^*_{n,k}=T_{n,n-k},\quad 0\leq k \leq n.$$ In addition, we have $T^*_n(q)=q^nT_n(\frac{1}{q})$. \end{rem}
We present \textbf{the proof of Theorem \ref{thm+Ring+PSE+SM} as follows:}
\begin{proof} (i) From the recurrence relation in (i), we get \begin{eqnarray}\label{eq+fst} T_n(q)&=&(a_0n+a_2-a_0)T_{n-1}(q)+(b_0n+b_2-b_0)qT_{n-1}(q)\nonumber\\ &=&[(a_0n+a_2-a_0)+(b_0n+b_2-b_0)q]T_{n-1}(q)\nonumber\\ &=&\prod_{k=1}^n[(a_0+b_0q)k+a_2-a_0+(b_2-b_0)q] \end{eqnarray} for $n\geq1$. In Theorem \ref{thm+Gass}, if we take $$ a=a_2+b_2q,\quad \ b=a_0+b_0q\quad\text{and}\, c=0,$$ then we immediately have
\begin{eqnarray}\label{eq+fst+CF}
\sum\limits_{n=0}^{\infty}T_{n}(q) z^n
&=&\rm \textbf{SCF}[\alpha_{2n},\alpha_{2n+1};z]_{n\geq0}\nonumber\\
&=&\rm \textbf{JCF}[\alpha_{2n-1}+\alpha_{2n},\alpha_{2n}\alpha_{2n+1};z]_{n\geq0} \end{eqnarray} and $(T_n(q))_{n\geq0}$ is an $\textbf{x}$-SM sequence with $\textbf{x}=(a_0,a_2,b_0,b_2,q)$, where $\alpha_{2n}=a_2+b_2q+n(a_0+b_0q)$ and $\alpha_{2n+1}=(a_0+b_0q)(1+n)$.
(ii) First let us consider a degenerated array $[A_{n,k}]_{n,k\geq0}$ satisfying the recurrence relation $$A_{n,k}=a_0b_1(n-k-1)A_{n-1,k}+[b_0(n-1)+b_1(k-1)+b_2]A_{n-1,k-1}$$ for $n\geq1$ with $A_{0,0}=1$. It is easy to find $A_{n,k}=0$ for $n\neq k$. Then by induction on $n$, we immediately get for $n\geq1$ that \begin{eqnarray*} A_{n,k}&=&0\quad \text{for} \quad n\neq k,\\ A_{n,n}&=&\prod_{k=0}^{n-1}[b_2+(b_0+b_1)k]. \end{eqnarray*} Clearly, this triangle $[A_{n,k}]_{n,k}$ is degenerated to a diagonal sequence. So for $n\geq1$ the generating function $$A_n(q)=q^nA_{n,n}=q^n\prod_{k=0}^{n-1}[b_2+(b_0+b_1)k].$$ For the array $[T_{n,k}]_{n,k\geq0}$ in (ii), by taking $\lambda=-a_0$ in Lemma \ref{lem+shit}, we get $$T_n(q)=A_n(q+a_0)$$ for $n\geq1$. So $$T_n(q)=(q+a_0)^n\prod_{k=0}^{n-1}[b_2+(b_0+b_1)k]$$ for $n\geq1$. It follows from Theorem \ref{thm+Gass} that \begin{eqnarray*} \sum_{n\geq0}T_n(q)z^n&=&\rm \textbf{SCF}[\alpha_{2n},\alpha_{2n+1};z]_{n\geq0}\\ &=&\rm \textbf{JCF}[\alpha_{2n-1}+\alpha_{2n},\alpha_{2n}\alpha_{2n+1};z]_{n\geq0} \end{eqnarray*} with $\alpha_{2n}=(b_2+n(b_0+b_1))(q+a_0)$ and $\alpha_{2n+1}=(b_0+b_1)(q+a_0)(1+n)$, and $(T_n(q))_{n\geq0}$ is an $\textbf{x}$-SM sequence with $\textbf{x}=(a_0,b_0,b_1,b_2,q)$.
(iii) By Proposition \ref{prop+x+binomial+array} and \cite[Theorem 1]{Zhu19}, $[T_{n,k}]_{n,k\geq0}$ can be considered as the $a_2$-binomial transformation of $[\widetilde{T}_{n,k}]_{n,k\geq0}$, where $[\widetilde{T}_{n,k}]_{n,k\geq0}$ satisfies the recurrence relation \begin{equation*} \widetilde{T}_{n,k}=a_1k\widetilde{T}_{n-1,k}+(b_1k+b_2-b_1)\widetilde{T}_{n-1,k-1} \end{equation*} for $n\geq1$ and $\widetilde{T}_{0,0}=1$. Let $f_k=\sum_{n\geq0}\widetilde{T}_{n,k}z^n$ for $k\geq0$ and $F(z)=\sum_{k\geq0}f_kq^k$. By the recurrence relation, we get $$f_k=a_1kz f_k+(b_1k+b_2-b_1)zf_{k-1},$$ which implies $$f_k=\prod_{i=1}^k\frac{(b_1i+b_2-b_1) z}{1-a_1i z}$$ for $k\geq1$. Thus, we have \begin{eqnarray*} F(z)&=&1+\frac{b_2qz}{1-a_1z}+\frac{b_2(b_1+b_2)q^2z^2}{(1-a_1z)(1-2a_1z)}+\cdots\\ &=&1+\sum_{k\geq1}z^k\prod_{i=1}^k\frac{(b_1i+b_2-b_1)q}{1-a_1iz}. \end{eqnarray*} It follows from Theorem \ref{thm+Gass} that \begin{eqnarray*}
\sum\limits_{n=0}^{\infty}\widetilde{T}_{n}(q) z^n&=&\rm \textbf{JCF}[\alpha_{2n-1}+\alpha_{2n},\alpha_{2n}\alpha_{2n+1};z]_{n\geq0}\\
&=&\rm \textbf{SCF}[\alpha_{2n},\alpha_{2n+1};z]_{n\geq0}, \end{eqnarray*} where $\alpha_{2n}=(nb_1+b_2)q$ and $\alpha_{2n+1}=(n+1)(a_1+b_1q)$ for $n\geq0$. In addition, $(\widetilde{T}_n(q))_{n\geq0}$ is an $\textbf{x}$-SM sequence. Thus by Proposition \ref{prop+x+binomial+array}, we have \begin{eqnarray}\label{Stir+Sec+CF}
\sum\limits_{n=0}^{\infty}T_{n}(q) z^n&=&\rm \textbf{JCF}[a_2+\alpha_{2n-1}+\alpha_{2n},\alpha_{2n}\alpha_{2n+1};z]_{n\geq0}, \end{eqnarray} and $(T_n(q))_{n\geq0}$ is an $\textbf{x}$-SM sequence with $\textbf{x}=(a_1,a_2,b_1,b_2,q)$.
(iv) Clearly, the array $[T_{n,k}]_{n,k\geq0}$ in (iv) is the reciprocal array of that in (iii). By (\ref{Stir+Sec+CF}), we have \begin{eqnarray}\label{rec+ST}
\sum\limits_{n=0}^{\infty}T_{n}(q) z^n
&=&\rm \textbf{JCF}[b_2q+\alpha_{2n-1}+\alpha_{2n},\alpha_{2n}\alpha_{2n+1};z]_{n\geq0}, \end{eqnarray} where $\alpha_{2n}=na_0+a_2$ and $\alpha_{2n+1}=(n+1)(a_0+b_0q)$ for $n\geq0$. By Theorem \ref{thm+S+C}, $(T_n(q))_{n\geq0}$ is an $\textbf{x}$-SM sequence with $\textbf{x}=(a_0,a_2,b_0,b_2,q)$
(v) For the array $[T_{n,k}]_{n,k}$, let $B_n(q)=T_n(q+\frac{a_1}{b_0})$. By Lemma \ref{lem+shit}, we have the coefficient array $[B_{n,k}]_{n,k\geq0}$ of $B_n(q)$ satisfying the recurrence relation $$B_{n,k}=\left[a_1n-a_1k+a_2+\frac{a_1(b_2-b_0)}{b_0}\right]B_{n-1,k}+(b_0n-b_0k+b_2)B_{n-1,k-1}$$ for $n\geq1$, which is a special case in (iv). It follows from the continued fraction expansion in (\ref{rec+ST}) that
\begin{eqnarray}\label{CF+EF}
&&\sum\limits_{n=0}^{\infty}T_{n}(q) z^n=\sum\limits_{n=0}^{\infty}B_{n}(q-\frac{a_1}{b_0})
z^n\nonumber\\
&=&\rm \textbf{JCF}[n(a_1+b_0q)+a_2+b_2q,(n+1)(na_1b_0+a_2b_0+a_1b_2)q;z]_{n\geq0}\\ &=&\begin{cases}\rm \textbf{SCF}[na_1+a_2,(n+1)b_0q;z]_{n\geq0} &\text{for}\quad b_2=0\\ \rm \textbf{SCF}[(nb_0+b_2)q,(n+1)a_1;z]_{n\geq0}&\text{for}\quad a_2=0\\ \rm \textbf{SCF}[(n+1)b_0q,na_1+a_1+a_2;z]_{n\geq0}&\text{for}\quad b_2=b_0\\ \rm \textbf{SCF}[(n+1)a_1,(nb_0+b_0+b_2)q;z]_{n\geq0}&\text{for} \quad a_2=a_1.\end{cases}\nonumber \end{eqnarray} By Theorem \ref{thm+S+C}, $(T_n(q))_{n\geq0}$ is an $\textbf{x}$-SM sequence with $\textbf{x}=(a_1,a_2,b_0,b_2,q)$ for
$0\in\{a_2,b_2,a_1-a_2,b_0-b_2\}$.
(vi) For the array $[T_{n,k}]_{n,k}$ in (vi), in what follows, we will prove \begin{eqnarray}\label{CF++} \sum\limits_{n=0}^{\infty}T_{n}(q) z^n=\rm \textbf{JCF}[(a_1n+a_2)(q+\frac{2b_0}{a_1}),\frac{b_0(a_1n+2a_2)(n+1)(q+\frac{2b_0}{a_1})}{2};z]_{n\geq0}, \end{eqnarray} which immediately implies that $(T_n(q))_{n\geq0}$ is an $\textbf{x}$-SM sequence with $\textbf{x}=(a_1,a_2,b_0,q)$ by taking $s_n=(a_1 n+a_2)(q+\frac{2b_0}{a_1})$, $r_n=(\frac{na_1}{2}+a_2)(q+\frac{2b_0}{a_1})$ and $t_n=b_0n$ for $n\geq0$ in Theorem \ref{thm+Hakel+Jacbi+main} (ii) and Proposition \ref{prop+tri+TP} (i).
Let $S^*_n(q+\frac{2b_0}{a_1})=T_n(q)$ and $S^*_n(q)=\sum_{k\geq0}S^*_{n,k}q^k$ for $n\geq0$. Then for (\ref{CF++}) it suffices to prove \begin{eqnarray}\label{CF+++} \sum\limits_{n=0}^{\infty}S^*_{n}(q) z^n=\rm \textbf{JCF}[(a_1n+a_2)q,\frac{b_0(a_1n+2a_2)(n+1)q}{2};z]_{n\geq0}. \end{eqnarray} Moreover, by Lemma \ref{lem+shit}, the array $[S^*_{n,k}]_{n,k\geq0}$ satisfies the recurrence relation: \begin{equation} S^*_{n,k}=b_0\left(2k-n+1\right)S^*_{n-1,k}+[a_1\,(n-k)+a_2]S^*_{n-1,k-1} \end{equation} for $0\leq k\leq n$ with $S^*_{0,0}=1$ (In addition, it is easy to prove $S^*_{n,k}=0$ for $k< (n-1)/2$). In view of its reciprocal array, for (\ref{CF+++}), we will demonstrate \begin{eqnarray}\label{CF++++} \sum\limits_{n=0}^{\infty}S_{n}(q) z^n=\rm \textbf{JCF}[a_1n+a_2,\frac{b_0(a_1n+2a_2)(n+1)q}{2};z]_{n\geq0}, \end{eqnarray} where $S_n(q)$ is the row-generating function of the reciprocal array $[S_{n,k}]_{n,k\geq0}$ satisfying the recurrence relation: \begin{equation}\label{S+recurrence relation} S_{n,k}=(a_1k+a_2)S_{n-1,k}+b_0(n-2k+1)S_{n-1,k-1} \end{equation} for $0\leq k\leq n$, where $S_{0,0}=1$. It is not hard to get (\ref{CF++++}) from the next claim and the continued fraction expression (\ref{CF+EF}) in (v).
\begin{cl}\footnote{This result was proved in \cite{Zhu20}. For convenience of the reader, we cite its proof to be self-contained.} For the array $[S_{n,k}]_{n,k}$ in (\ref{S+recurrence relation}), there exists an array $[E_{n,k}]_{n,k}$ satisfying the recurrence relation: \begin{equation}\label{two+recurrence relation} E_{n,k}=(a_1k+a_2)E_{n-1,k}+[a_1(n-k)+a_2]E_{n-1,k-1} \end{equation} with $E_{n,k}=0$ unless $0\le k\le n$ and $E_{0,0}=1$ such that \begin{eqnarray*} E_n(x)=(1+x)^nS_n\left(\frac{\frac{2a_1}{b_0} x}{(1+x)^{2}}\right) \end{eqnarray*} for $n\geq1$. \end{cl} \begin{proof} We will prove this result by in induction on $n$. Let $\frac{2a_1}{b_0}=\lambda$. It is obvious for $n=0$. For $n=1$, $E_1(x)=a_2+a_2x$ and $S_1(x)=a_2$. Therefore we have $$E_1(x)=(1+x)S_1\left(\frac{\lambda x}{(1+x)^{2}}\right).$$ In the following we assume $n\geq2$. By the inductive hypothesis, we have \begin{eqnarray} E'_{n-1}(x)=(n-1)(1+x)^{n-2}S_{n-1}(\frac{\lambda x}{(1+x)^{2}})+\frac{(1+x)^{n-1}\lambda (1-x)}{(1+x)^3}S'_{n-1}(\frac{\lambda x}{(1+x)^{2}}). \end{eqnarray} On the other hand, by the recurrence relations (\ref{S+recurrence relation}) and (\ref{two+recurrence relation}), we have \begin{eqnarray*} S_n(x)&=&[a_2+b_0(n-1)x]S_{n-1}(x)+x(a_1-2b_0x)S'_{n-1}(x),\\ E_n(x)&=&[a_2(x+1)+a_1x(n-1)]E_{n-1}(x)+a_1x(1-x)E'_{n-1}(x). \end{eqnarray*} Then we get \begin{eqnarray*} E_n(x) &=&[a_2(x+1)+a_1x(n-1)](1+x)^{n-1}S_{n-1}\left(\frac{\lambda x}{(1+x)^{2}}\right)+a_1x(1-x)\\ &&\times\left[(n-1)(1+x)^{n-2}S_{n-1}\left(\frac{\lambda x}{(1+x)^{2}}\right)+\frac{(1+x)^{n-1}\lambda (1-x)}{(1+x)^3}S'_{n-1}\left(\frac{\lambda x}{(1+x)^{2}}\right)\right]\\ &=&(1+x)^n\left[a_2+\frac{a_1(n-1)x}{1+x}\right]S_{n-1}\left(\frac{\lambda x}{(1+x)^{2}}\right)+a_1x(1-x)(1+x)^n\\ &&\times\left[\frac{(n-1)}{(1+x)^{2}}S_{n-1}\left(\frac{\lambda x}{(1+x)^{2}}\right)+\frac{\lambda (1-x)}{(1+x)^4}S'_{n-1}\left(\frac{\lambda x}{(1+x)^{2}}\right)\right]\\ &=&(1+x)^n\left[a_2+b_0(n-1)\times\frac{\lambda x}{(1+x)^2}\right]S_{n-1}\left(\frac{\lambda x}{(1+x)^{2}}\right)+(1+x)^n\times \frac{\lambda x}{(1+x)^{2}}\times\\ &&\left[a_1-2b_0\times\frac{\lambda x}{(1+x)^{2}}\right]\times S'_{n-1}\left(\frac{\lambda x}{(1+x)^{2}}\right)\\ &=&(1+x)^nS_{n}\left(\frac{\lambda x}{(1+x)^{2}}\right). \end{eqnarray*} This proves the claim. \end{proof} (vii) Obviously, the array in (vii) is the reciprocal array of that in (vi). By (\ref{CF++}), we have the Jacobi continued fraction expansion \begin{eqnarray*} \sum\limits_{n=0}^{\infty}T_{n}(q) z^n=\rm \textbf{JCF}[(a_1n+a_2)(1+\frac{2b_0}{a_1}q),\frac{b_0(a_1n+2a_2)(n+1)(1+\frac{2b_0q}{a_1})q}{2};z]_{n\geq0} \end{eqnarray*} and $(T_n(q))_{n\geq0}$ is an $\textbf{x}$-SM sequence with $\textbf{x}=(a_1,a_2,b_0,q)$.
The proof is complete. \end{proof}
\section{Proof of Theorem \ref{thm+three+PSE+SM}} \begin{proof} (i) By Theorem \ref{thm+m+tran}, there exists an array $[A_{n,k}]_{n,k\geq0}$ satisfying the recurrence relation
\begin{eqnarray} A_{n,k}=(a_1k+a_2)A_{n-1,k}+[b_1(k-1)+b_2]A_{n-1,k-1} \end{eqnarray} with $A_{0,0}=1$ and $A_{n,k}=0$ unless $0\le k\le n$ such that their row-generating functions satisfy \begin{eqnarray}\label{rel+T+A} T_n(q)=(\lambda+dq)^nA_n(\frac{q}{\lambda+dq}) \end{eqnarray} for $n\geq0$. It follows from (\ref{Stir+Sec+CF}) that \begin{eqnarray}
\sum\limits_{n=0}^{\infty}T_{n}(q) z^n&=&\rm \textbf{JCF}[a_2(\lambda+dq)+\alpha_{2n-1}+\alpha_{2n},\alpha_{2n}\alpha_{2n+1};z]_{n\geq0}, \end{eqnarray} where $\alpha_{2n}=(nb_1+b_2)q$ and $\alpha_{2n+1}=(n+1)[(a_1d+b_1)q+\lambda a_1]$ for $n\geq0$. Thus by Theorem \ref{thm+S+C}, we get that $(T_n(q))_{n\geq0}$ is an $\textbf{x}$-SM sequence with $\textbf{x}=(a_1,a_2,b_1,b_2,\lambda,q)$.
(ii) By Theorem \ref{thm+m+tran}, there exists an array $[A_{n,k}]_{n,k\geq0}$ satisfying the recurrence relation
\begin{eqnarray} A_{n,k}=(a_0n-a_0k-a_0+a_2)A_{n-1,k}+[b_0(n-k)+b_2]A_{n-1,k-1} \end{eqnarray} with $A_{0,0}=1$ and $A_{n,k}=0$ unless $0\le k\le n$ such that their row-generating functions satisfy \begin{eqnarray}\label{rel+T+A} T_n(q)=(\lambda+dq)^nA_n(\frac{q}{\lambda+dq}) \end{eqnarray} for $n\geq0$. It follows from (\ref{rec+ST}) that \begin{eqnarray}
\sum\limits_{n=0}^{\infty}T_{n}(q) z^n
&=&\rm \textbf{JCF}[b_2q+\alpha_{2n-1}+\alpha_{2n},\alpha_{2n}\alpha_{2n+1};z]_{n\geq0}, \end{eqnarray} where $\alpha_{2n}=(na_0+a_2)(\lambda+dq)$ and $\alpha_{2n+1}=(n+1)[(a_0d+b_0)q+\lambda a_0]$ for $n\geq0$. By Theorem \ref{thm+S+C}, $(T_n(q))_{n\geq0}$ is an $\textbf{x}$-SM sequence with $\textbf{x}=(a_0,a_2,b_0,b_2,d,\lambda,q)$.
(iii) By Theorem \ref{thm+m+tran}, there exists an array $[A_{n,k}]_{n,k\geq0}$ satisfying the recurrence relation
\begin{eqnarray} A_{n,k}=(a_1k+a_2)A_{n-1,k}+[b_0(n-k)+b_2]A_{n-1,k-1} \end{eqnarray} with $A_{0,0}=1$ and $A_{n,k}=0$ unless $0\le k\le n$ such that their row-generating functions satisfy \begin{eqnarray}\label{rel+T+A} T_n(q)=(\lambda+dq)^nA_n(\frac{q}{\lambda+dq}) \end{eqnarray} for $n\geq0$. It follows from (\ref{CF+EF}) that \begin{eqnarray*}
&&\sum\limits_{n=0}^{\infty}T_{n}(q) z^n\nonumber\\
&=&\rm \textbf{JCF}[n(a_1(\lambda+dq)+b_0q)+a_2(\lambda+dq)+b_2q,(n+1)(na_1b_0+a_2b_0+a_1b_2)q(\lambda+dq);z]_{n\geq0}\nonumber\\ &=&\begin{cases}\rm \textbf{SCF}[(na_1+a_2)(\lambda+dq),(n+1)b_0q;z]_{n\geq0} &\text{for}\quad b_2=0\\ \rm \textbf{SCF}[(nb_0+b_2)q,(n+1)a_1(\lambda+dq);z]_{n\geq0}&\text{for}\quad a_2=0\\ \rm \textbf{SCF}[(n+1)b_0q,(na_1+a_1+a_2)(\lambda+dq);z]_{n\geq0}&\text{for}\quad b_2=b_0\\ \rm \textbf{SCF}[(n+1)a_1q,(nb_0+b_0+b_2)(\lambda+dq);z]_{n\geq0}&\text{for} \quad a_2=a_1.\end{cases} \end{eqnarray*} Then by Theorem \ref{thm+S+C}, $(T_n(q))_{n\geq0}$ is an $\textbf{x}$-SM sequence with $\textbf{x}=(a_1,a_2,b_0,b_2,d,\lambda,q)$ for
$0\in\{a_2,b_2,a_1-a_2,b_0-b_2\}$.
\end{proof}
\section{Proof of Theorem \ref{thm+sequence+SM}}
\begin{proof}
It follows from the recurrence relation
$$T_{n,k}=(a_0n-\mu
b_1k+a_2)T_{n-1,k}+(b_0n+b_1k+b_2)T_{n-1,k-1}$$ that
$$T_n(q)=[a_0n+a_2+q(b_0n+b_1+b_2)]T_{n-1}(q)+q(-\mu b_1+b_1q)T'_{n-1}(q).$$
Setting $q=\mu$, we have
\begin{eqnarray*}
T_n(\mu) &=&\prod_{k=1}^n[(a_0+\mu b_0)k+a_2+\mu(b_1+b_2)]
\end{eqnarray*} for
$n\geq1$.
By Theorem \ref{thm+Gass}, we immediately get that
\begin{eqnarray*} \sum_{n\geq0}T_n(\mu)z^n &=&\rm \textbf{JCF}[\alpha_{2n}+\alpha_{2n-1},\alpha_{2n}\alpha_{2n+1};z]_{n\geq0} \end{eqnarray*} with $\alpha_{2n}(a)=a_0+a_2+\mu(b_0+b_1+b_2)+n(a_0+\mu b_0)$ and $\mu_{2n+1}(a)=(a_0+\mu b_0)(1+n)$ for $n\geq0$ and $(T_n(\mu))_{n\geq0}$ is an $\textbf{x}$-SM sequence with $\textbf{x}=(a_0,a_2,b_0,b_1,b_2,\mu)$.
\end{proof}
\section{Examples} In this section, we present some examples related to continued fractions. \begin{ex}[\textbf{Factorial numbers}] By (\ref{eq+fst+CF}), if $a_0=b_2=q=1$ and $b_0=a_2=0$, then we immediately have \begin{eqnarray*} \sum_{n\geq0} n!z^n&=&\rm \textbf{SCF}[n+1,n+1;z]_{n\geq0} \end{eqnarray*} and similarly if $a_0=a_2=b_0=q=1$ and $b_2=0$, then we have \begin{eqnarray*} \sum_{n\geq0}(2n-1)!!z^n&=&\rm \textbf{SCF}[1+2n,2(1+n);z]_{n\geq0}, \end{eqnarray*} which were proved by Euler \cite{Eul76}. Thus the continued fraction expansion in (\ref{eq+fst+CF}) can be looked at a generalization of above two results and the corresponding row-generating function $T_n(a_0,a_2,b_0,b_2,q)$ in (\ref{eq+fst}) can be viewed as a five-variable refinement of $n!$ and $(2n-1)!!$. \end{ex}
\begin{ex}[\textbf{Whitney numbers of the first kind}] In \cite{CJ12}, the $r$-Whitney numbers of the first kind, denoted by $w_{m,r}(n, k)$, satisfy the recurrence relation \begin{equation*} w_{m,r}(n, k)=[(n-1)m+r]w_{m,r}(n-1, k)+w_{m,r}(n-1, k-1) \end{equation*} with $w_{m,r}(0, 0)=1$. It reduces to the signless Stirling number of the first kind for $m=r=1$. Let the row-generating functions $w_{n}(m,r,q)=\sum_{k\geq0}w_{m,r}(n, k)q^k$ for $n\geq0$. It follows from Theorem \ref{thm+Ring+PSE+SM} (i) and (\ref{eq+fst+CF}) that we have \begin{eqnarray*} \sum_{n\geq0}w_{n}(m,r,q)z^n&=&\rm \textbf{JCF}[r+q+2mn,(r+q+mn)m(n+1);z]_{n\geq0}\\ &=&\rm \textbf{SCF}[r+q+mn,m(n+1);z]_{n\geq0} \end{eqnarray*} and $(w_{n}(m,r,q))_{n\geq0}$ is an $\textbf{x}$-SM and $3$-$\textbf{x}$-LCX sequence with $\textbf{x}=(m,r,q)$. Thus by Theorem \ref{thm+Ring+PSE+SM} (i), the corresponding row-generating function $T_n(a_0,a_2,b_0,b_2,q)$ in (\ref{eq+fst}) can be considered as a five-variable refinement of $w_{n}(m,r,q)$. \end{ex}
\begin{ex}[\textbf{Stirling permutations}] Stirling permutations were introduced by Gessel and Stanley \cite{GS78}. A Stirling permutation of order $n$ is a permutation of the multiset $\{1^2,2^2,3^2, \ldots, n^2\}$ such that every element between the two occurrences of $i$ are greater than $i$ for each $i\in [n]$, where $[n] = \{1, 2, \ldots, n\}$. Denote by $Q_n$ the set of Stirling permutations of order $n$. For $\sigma =\sigma_1\sigma_2\ldots\sigma_{2n}\in Q_n$, an index $i\in[2n-1]$ is an ascent plateau if $\sigma_{i-1} < \sigma_i = \sigma_{i+1}$. Let $ap(\sigma)$ be the number of the ascent plateaus of $\sigma$. Let
$N_{n,k}=|\{\sigma\in Q_n: ap(\sigma)=k\}$. It is known that the array $[N_{n,k}]_{n,k\geq0}$ satisfies the recurrence relation \begin{eqnarray} N_{n+1,k}&=&2kN_{n,k}+(2n-2k+3)N_{n,k-1} \end{eqnarray} with initial conditions $N_{1,1}=1$ and $N_{1,k}=0$ for $k\geq2$ or $k\leq0$, see \cite{MY15}. Additionally, it is also related to the perfect matching. A perfect matching of $[2n]$ is a set partition of $[2n]$ with blocks (disjoint nonempty subsets) of size exactly $2$. Let $M_{2n}$ be the set of matchings of $[2n]$, and let $M\in M_{2n}$. Then $N_{n,k}$ is also the number of perfect matchings in $M_{2n}$ with the restriction that only $k$ matching pairs have odd smaller entries, see \cite{MY15}. For brevity, we let $N_{0,0}=1$ and $N_{0,k}=0$ for $k>1$ or $k<0$. It is clear that $[N_{n,k}]_{n,k\geq0}$ satisfies the recurrence relation \begin{eqnarray} N_{n,k}&=&2kN_{n-1,k}+(2n-2k+1)N_{n-1,k-1} \end{eqnarray} with $N_{0,0}=1$ and $N_{0,k}=0$ for $k\geq1$ or $k<0$. Let $N_n(q)=\sum_{k\geq0}N_{n,k}q^k$ for $n\in \mathbb{N}$. By Theorem \ref{thm+Ring+PSE+SM} (v) and (\ref{CF+EF}), we have the continued fraction expansion \begin{eqnarray*} \sum_{n\geq0}N_n(q)z^n&=&\rm \textbf{SCF}[(2n+1)q,2(n+1);z]_{n\geq0}\\ &=&\rm \textbf{JCF}[(2n+1)q+2n,2(n+1)(2n+1)q;z]_{n\geq0} \end{eqnarray*} and $(N_n(q))_{n\geq0}$ is $q$-SM and $3$-$q$-log-convex. \end{ex}
\begin{ex}[\textbf{Minimax trees}] Let $\mathcal {M}_{n,k}$ denote the set of all trees with $n$ vertices and $k$ leaves. Denote by $hr(T)$ the number of inner vertices of the second kind of a minimax tree $T$. Denote by $cr(T)$ the number of inner vertices $s$ of a minimax tree $T$ which have maximum label (it means that the label is maximum in the subtree $T(s)$ of root $s$) so that an increasing tree $T$ is a tree satisfying $cr(T)=0$. Let $\overrightarrow{m}_{n,k}(p,q)=m_{n+1,k+1}(p,q)$, where $$m_{n,k}(p,q):=\sum_{T\in \mathcal {M}_{n,k}}p^{hr(T)}q^{cr(T)}$$ for $n,k\geq0$ and $\overrightarrow{m}_{0,0}(p,q)=1$. Then \begin{eqnarray}\label{ex+FH} \overrightarrow{m}_{n,k}(p,q)=(1+p)(q+1)(k+1)\overrightarrow{m}_{n-1,k}(p,q)+(n-2k+1)\overrightarrow{m}_{n-1,k-1}(p,q) \end{eqnarray} for $0\leq k\leq (n+1)/2$, see Foata and Han \cite[Propositon 3.1]{FH01}. Let its row-generating function $\mathscr{M}_n(x)=\sum_{k\geq0}\overrightarrow{m}_{n,k}(p,q)x^k$. Then by (\ref{CF++++}), we have \begin{eqnarray} \sum\limits_{n=0}^{\infty}\mathscr{M}_n(x) z^n=\rm \textbf{JCF}[(1+p)(q+1)(n+1),\frac{(1+p)(q+1)(n+2)(n+1)x}{2};z]_{n\geq0}. \end{eqnarray} It is easy to check that $(\mathscr{M}_n(x))_{n\geq0}$ is not $x$-log-convex. But its reciprocal polynomial $\mathscr{M}^*_n(x)$ has the following property by the proof of (vi) in Theorem \ref{thm+Ring+PSE+SM}.
\begin{prop} For $n\geq0$, $\mathscr{M}^*_n\left(x+\frac{2}{(p+1)(q+1)}\right)$ form an $\textbf{x}$-SM and $3$-$\textbf{x}$-LCX sequence with $\textbf{x}=(x,p,q)$. \end{prop} \end{ex}
\begin{ex}[\textbf{Peak Statistics}] Let $pk(\pi)$ and $lpk(\pi)$ denote the numbers of interior peaks and the numbers of left peaks of $\pi\in S_n$, respectively. Let \begin{eqnarray*} W_{n}(q)&=&\sum_{\pi\in S_n}q^{pk(\pi)}=\sum_{k\geq0}W_{n,k}q^{k},\\ \widetilde{W}_{n}(q)&=&\sum_{\pi\in S_n}q^{lpk(\pi)}=\sum_{k\geq0}\widetilde{W}_{n,k}q^{k}. \end{eqnarray*} It is known that \begin{eqnarray*} W_{n,k}&=&(2k+2)W_{n-1,k}+(n-2k)W_{n-1,k-1},\\ \widetilde{W}_{n,k}&=&(2k+1)\widetilde{W}_{n-1,k}+(n-2k+1)\widetilde{W}_{n-1,k-1}, \end{eqnarray*} where $W_{1,0}=1$ and $\widetilde{W}_{0,0}=1$, see Stembridge \cite{St97}, Petersen \cite{Pet07} and \cite[A008303, A008971]{Slo} for instance. Then by (\ref{CF++++}), we have \begin{eqnarray} \sum\limits_{n=0}^{\infty}W_{n+1}(q) t^n&=&\rm \textbf{JCF}[2(n+1),(n+2)(n+1)q;t]_{n\geq0},\\ \sum\limits_{n=0}^{\infty}\widetilde{W}_{n}(q) t^n&=&\rm \textbf{JCF}[(2n+1),(n+1)^2q;t]_{n\geq0}. \end{eqnarray} It is easy to check that their $q$-log-convexity does not hold. Thus, both $(W_{n+1}(q))_{n\geq0}$ and $(\widetilde{W}_{n}(q))_{n\geq0}$ are not $q$-SM sequences. But their reciprocal polynomials $W^*_{n+1}(q)$ and $\widetilde{W}^*_{n}(q)$ have the following property by (vi) in Theorem \ref{thm+Ring+PSE+SM}. \begin{prop} Both $(W^*_{n+1}(q+1))_{n\geq0}$ and $(\widetilde{W}^*_{n}(q+1))_{n\geq0}$ are $q$-SM and $3$-$q$-LCX sequences. \end{prop} \end{ex}
\begin{rem} If we change the $\mathbb{R}[x]$ to an ordered commutative ring, then we can extend results of this paper to those in the commutative ring, see \cite{Zhu2018}. \end{rem} \section{Acknowledgments} The author is extremely grateful to the anonymous referee for his/her many valuable remarks and suggestions to improve the original manuscript.
\end{document} | arXiv |
\begin{document}
\title{Subsystems and time in quantum mechanics}
\author{Bradley A. Foreman} \affiliation{Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China}
\begin{abstract} This paper investigates the relationship between subsystems and time in a closed nonrelativistic system of interacting bosons and fermions. It is possible to write any state vector in such a system as an unentangled tensor product of subsystem vectors, and to do so in infinitely many ways. This requires the superposition of different numbers of particles, but the theory can describe in full the equivalence relation that leads to a particle-number superselection rule in conventionally defined subsystems. Time is defined as a functional of subsystem changes, thus eliminating the need for any reference to an external time variable. The dynamics of the unentangled subsystem decomposition is derived from a variational principle of dynamical stability, which requires the decomposition to change as little as possible in any given infinitesimal time interval, subject to the constraint that the state of the total system satisfy the Schr\"odinger equation. The resulting subsystem dynamics is deterministic. This determinism is regarded as a conceptual tool that observers can use to make inferences about the outside world, not as a law of nature. The experiences of each observer define some properties of that observer's subsystem during an infinitesimal interval of time (i.e., the present moment); everything else must be inferred from this information. The overall structure of the theory has some features in common with quantum Bayesianism, the Everett interpretation, and dynamical reduction models, but it differs significantly from all of these. The theory of information described here is largely qualitative, as the most important equations have not yet been solved. The quantitative level of agreement between theory and experiment thus remains an open question. \end{abstract}
\pacs{03.65.Ta}
\maketitle
\tableofcontents
\section{Introduction}
\begin{quote} \textit{It seems to me that for a systematic foundation of quantum mechanics one needs to begin with the composition and decomposition of quantum systems.} \\ \phantom{x}
--- W. Pauli \cite{[] [{, p.\ 3.}] AtmanspacherFuchs2014, Pauli1985, [{Another translation of Ref.\ \cite{Pauli1985} is provided by Howard: ``Quite independently of \emph{Einstein}, it appears to me that, in providing a systematic foundation for quantum mechanics, one should \emph{start} more from the composition and separation of systems than has until now (with Dirac, e.g.) been the case. --- This is indeed --- as Einstein has \emph{correctly} felt --- a very fundamental point in quantum mechanics, which has, moreover, a direct connection with your reflections about the \emph{cut} and the possibility of its being shifted to an arbitrary place.'' See }] [] Howard1997b} \end{quote}
In physics, as in everyday life, we must divide the world conceptually into subsystems before we can say anything about it. We cannot comprehend the undivided world; we can only talk about how its parts differ from or change relative to each other. In quantum mechanics the role of subsystem decompositions is, if anything, even more important. As Wheeler has said, ``we are first able to play the game when with chalk we have drawn a line across the empty courtyard'' \cite{Wheeler1981}. If we decline to draw such a line, ``then physics has vanished, and only a mathematical scheme remains'' \cite{[{}] [{, p.\ 58.}] Heisenberg1930}.
The ``mathematical scheme'' that Heisenberg refers to is the Schr\"odinger equation for the time evolution of a closed system. If the world is not divided into subsystems, time may pass on a sheet of paper, but nothing can be said to \emph{happen}, as there is no point of contact between theory and experiment. Indeed it is not even clear what the time variable in the Schr\"odinger equation \emph{means} if one cannot talk about relative changes between different parts of the system.
This paper explores the relationship between subsystems and time in nonrelativistic quantum mechanics. It springs from an examination of four interrelated questions: (1)~how to construct subsystem decompositions without entanglement; (2)~how to define time in a closed system; (3)~how to define a dynamics of interacting subsystems without entanglement; and (4)~how to extract information from these subsystems.
Cursory answers to these four questions are as follows: (1)~Unentangled subsystem decompositions can be constructed in Fock space, using superpositions of different numbers of particles. (2)~Time can be defined as a functional of subsystem changes. (3)~An entanglement-free dynamics can be derived by maximizing the stability of the subsystem decomposition. (4)~An observer obtains information from only one subsystem, and only during the present moment of time. All else must be inferred.
A brief introduction to each of these four subject areas is given below. It must be emphasized at the outset that this paper is exploratory in nature; the theory of information presented here has not yet reached a stage of development that would permit a direct comparison with experiment.
\subsection{Definition of subsystems}
\label{sec:intro_define_subsystems}
In quantum mechanics, subsystems are traditionally defined using a tensor product of vector spaces \cite{CohTan1977}. Subsystems defined in this way inevitably become entangled by interactions \cite{Schrodinger1935b}. Although entanglement is now commonly regarded as a resource \cite{Horodecki2009}, its consequences still leave many people uneasy with the foundations of quantum mechanics \cite{[] [{, p.\ 102.}] Weinberg2015, Leggett2005, [] [{, p.\ 185.}] Isham1995, Laloe2012, Bell2004}.
But is this definition necessary? Another approach has been advocated by Primas and Amann \cite{Primas1983, Primas1987, Primas1990a, *Primas1990b, Primas1991, Primas1993, Primas1994a, Primas1994b, Primas2000, Amann1988, Amann1991a, Amann1991b, *Amann1991c, Amann1993, Amann1995, AmannPrimas1997b, AmannAtmanspacher1999}. It is based on the concept of a \emph{quantum object}, which Primas defines as ``an open quantum system which in spite of its interaction with the environment is not [entangled] with the environment'' \cite{Primas1987}. Objects may change their properties but they are required to ``keep their identity in the course of time'' \cite{Primas1987}. The concept of an object is similar to that of a quasiparticle \cite{Nozieres1997, Kaxiras2003, Jain2007}. However, quasiparticles are designed to minimize interactions, whereas objects are designed to minimize entanglement.
Algebraic ``dressing'' techniques have been developed for the construction of quantum objects in some simple models, but only in the context of certain approximations and asymptotic limits \cite{Primas1983, Primas1987, Primas1990a, *Primas1990b, Primas1991, Primas1993, Primas1994a, Primas1994b, Primas2000, Amann1988, Amann1991a, Amann1991b, Amann1991c, Amann1993, Amann1995, AmannPrimas1997b, AmannAtmanspacher1999}. One approximation considered essential by Primas is neglecting the Pauli exclusion principle at the level of the interactions between subsystems. Thus, electrons in different subsystems are treated as distinguishable. However, this contradicts a basic principle of elementary-particle physics.
Is it possible to construct quantum objects without neglecting the Pauli exclusion principle? The answer is yes, if we define a subsystem decomposition as a tensor product of \emph{vectors} rather than a tensor product of vector \emph{spaces}. Of course, the answer also depends on how we define a system.
The system studied in this paper is a closed nonrelativistic system of interacting fermions and bosons. The dimensions of the underlying single-particle Hilbert spaces are assumed to be finite; the dimension of the resulting fermion Fock space is then finite too, but that of the boson Fock space is infinite. For example, one might consider a finite volume in coordinate space and cut off wavelengths shorter than some given value. The use of such a cutoff is congruent with the current understanding of the standard model of elementary-particle physics as an effective field theory \cite{WeinbergVol1}. Bell has called this type of system a ``serious part of quantum mechanics'' that covers a ``substantial fragment of physics'' \cite{Bell1990}, thus making it (in his opinion) a worthy object of study in the field of quantum foundations \cite{Bell1987, Bell1990}.
It is shown in Secs.\ \ref{sec:tensor_prod_indistinguishable} and \ref{sec:mathematics} that any state vector $\ket{\psi}$ in such a system can be written \emph{exactly} as an unentangled tensor product of an arbitrary number of subsystem vectors $\ket{u_k}$. Furthermore, for any given number of subsystems, this can be done in infinitely many ways. Despite the lack of entanglement, knowledge of the whole ($\ket{\psi}$) does not imply knowledge of the parts ($\ket{u_k}$), because the subsystem vectors are not confined to subspaces. On the contrary, $\ket{u_k}$ occupies the same Fock space as $\ket{\psi}$.
\subsection{Definition of time}
\label{sec:intro_define_time}
Such a decomposition is only meaningful if it persists over time. A crucial question that must be addressed is how to \emph{define} time in a closed system without making reference to an external time variable \cite{BarbourBertotti1982, PageWootters1983, Wootters1984, UnruhWald1989, Pegg1991, Isham1992, Page1994, Barbour1994a, Barbour1994b, GambiniPortoPullin2004a, GambiniPortoPullin2004b, GambiniPortoPullin2009, Poulin2006, MilburnPoulin2006, AlbrechtIglesias2008, Arce2012, Moreva2014}. The given subsystem decomposition is very useful in this regard. It allows time to be defined as a \emph{functional} that organizes information about subsystem changes.
Section \ref{sec:kinematics} begins the preparatory work for this definition by developing ways to quantify differences between subsystem decompositions. The main focus is on geometric concepts related to the Fubini--Study metric in Hilbert space. Section \ref{sec:time_functional} then defines time as a functional of two infinitesimally different (but otherwise arbitrary) subsystem decompositions. This functional is defined so as to maximize the amount of change that can be expressed in the form of Schr\"odinger dynamics. It is a functional of the entire subsystem decomposition; this eliminates the need to single out any particular subsystem as a clock.
\subsection{Subsystem dynamics}
\label{sec:intro_dynamics}
The time functional can be used to formulate a dynamics of subsystems by means of a variational principle of \emph{dynamical stability}. This principle requires the subsystem decomposition to change as little as possible in any given infinitesimal time interval, subject to the constraint that the total state vector $\ket{\psi}$ of the closed system satisfy the Schr\"odinger equation. However, the interacting subsystems derived from this principle do \emph{not} satisfy the Schr\"odinger equation.
The concept of \emph{subsystem} dynamics does not even exist in the usual tensor-product-of-vector-spaces formulation of subsystems. The tensor-product decomposition is just given arbitrarily, with no connection (in principle) between different times (although it is often taken to be time independent). The subsystem dynamics developed here (in Sec.\ \ref{sec:dynamical_stability}) therefore has no parallel in orthodox quantum mechanics.
Dynamically stable subsystems are \emph{quantum objects} in the sense defined above. Does this mean that their observable properties can be regarded as elements of a ``free-standing reality'' \cite{FuchsPeres2000, Fuchs2003}? Deriving such a description was in fact a large part of the original motivation for this study. However, the answer turns out to be an emphatic \emph{no}. Dynamically stable subsystems still have an unavoidable element of subjectivity.
There are two fundamental reasons for this. One is that the subjective choice of a \emph{number} of subsystems is essential to the dynamics. The results depend explicitly on this number, and it has to be put in by hand. Its value cannot be derived from the principle of dynamical stability itself.
The second reason is that the resulting dynamics is \emph{deterministic}. This investigation began with a vague expectation that the dynamical stability problem might not have unique solutions, thereby necessitating the introduction of ``objective'' probabilities at a fundamental level. This could be regarded as a new type of decoherence mechanism that is inherent in the dynamics of quantum objects.
However, this expectation proved to be false. The variation problem has a unique solution, so the resulting subsystem dynamics is deterministic. This means that the principle of dynamical stability cannot explain the lack of determinism exemplified by the ``quantum jumps'' of orthodox quantum mechanics \cite{[] [{, p.\ 36.}] Dirac1958, vonNeumann1955}.
If the principle of dynamical stability cannot be regarded as the foundation for a law of nature, what is it good for? It is essentially just a tool for observers to use. They can use it to infer something about the properties of subsystems in the past or the future from whatever information they have about those subsystems now. There is no guarantee that these inferences will agree with their experiences, because the subsystem dynamics is not viewed as a law governing the behavior of anything ``real.'' It is instead viewed as an instantiation of Wheeler's aphorism that ``the only law is the law that there is no law'' \cite{Wheeler1973}. That is, nature is not governed by laws; laws are just useful conceptual tools.
How are these results affected by the inclusion of superselection rules \cite{WickWightmanWigner1952, HegerfeldtKrausWigner1968, WickWightmanWigner1970, Wightman1995}? One must be careful in answering this question, because the standard rules are derived from the tensor-product-of-subspaces definition of subsystems, which is not relevant in the present context. (The standard rules are also highly controversial even in the proper context \cite{AharonovSusskind1967a, AharonovSusskind1967b, AharonovRohrlich2005, Mirman1969, Mirman1970, Mirman1979, Lubkin1970, Zeh1970, Zurek1982, GiuliniKieferZeh1995, Giulini2003a, Giulini2009a, WeinbergVol1, DowlingBartlettRudolphSpekkens2006, BartlettRudolphSpekkens2007, Earman2008}.) One must therefore go back to the underlying \emph{cause} of a superselection rule (i.e., the lack of an external reference frame \cite{BartlettRudolphSpekkens2007}) and reexamine its consequences for the definition of subsystems used here.
In the absence of an external reference frame, certain subsystem decompositions become observationally indistinguishable from one another. One can account for this indistinguishability by introducing \emph{equivalence classes} \cite{Jauch1964} of subsystem decompositions. For the reference frame that gives rise to a particle-number superselection rule, the subsystem dynamics with or without such equivalence classes is qualitatively the same, as shown in Sec.\ \ref{sec:superselection}. That is, the dynamics remains fully deterministic. However, the quantitative dynamics can be quite different in these two cases.
An intriguing consequence of the Pauli exclusion principle is that the subsystem dynamics depends on the \emph{order} in which the subsystem states $\ket{u_k}$ are multiplied. Ignoring this ordering would give rise to an apparent decoherence effect. However, when the ordering of subsystems is accounted for, the subsystem dynamics remains deterministic, as shown in Sec.\ \ref{sec:subsystem_permutations}.
\subsection{Information about subsystems}
\label{sec:intro_information}
Section \ref{sec:information_theory} addresses the question of precisely how inferences are to be drawn from information about the properties of subsystems. Because the subsystems are not entangled, we can use the methods of Bayesian inference familiar from \emph{classical} probability theory \cite{Jaynes1989, Jaynes2003, BernardoSmith2000, Appleby2005a, Appleby2005b, Jeffrey2004}. This automatically ensures compliance with Bohr's injunction that ``however far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms'' \cite{Bohr1949, BohrVol2}.
These subsystems are nonclassical, but due to their lack of entanglement, their properties can be regarded as ``beables'' rather than ``observables'' (in the sense of Bell \cite{Bell2004}). That is, the subsystems can be considered to \emph{have} certain properties, independently of whether they are ``measured.'' However, it should be emphasized that these properties are in general only \emph{inferred} to exist, and that they are \emph{contextual} in the sense that they depend on the subjective choice of a number of subsystems. As noted above, they cannot be regarded as elements of a free-standing reality.
How are we to reconcile the determinism of the subsystem dynamics with the ``quantum jumps'' of textbook quantum mechanics? This can be done by imposing limitations on the information an observer has access to. An observer is assumed to experience directly only those beables associated with \emph{one} subsystem, and only for an infinitesimal interval of time. This interval is called the \emph{present moment} of time. The properties of all other subsystems, as well as those of the observer's own subsystem in the past and future of the present moment, must be inferred from this information.
These limitations are not due to an observer's ignorance, in the literal sense of ignoring possibly accessible information. The observer simply has no access to anything other than her own experiences. These experiences are not predicted by anything in the theory. The degree to which an observer's inferences of an outside world match her experiences in successive present moments determines how useful her image of this outside world is. The degree of mismatch can be regarded as an effective ``quantum jump.''
The conceptual structure of this theory of information draws heavily upon the ideas of quantum Bayesianism (or QBism) developed by Caves, Fuchs, Schack, and Mermin \cite{CavesFuchsSchack2002, CavesFuchsSchack2007, Fuchs2003, Fuchs2010, Mermin2012a, *Mermin2012b, FuchsSchack2013, FuchsMerminSchack2014, Mermin2014b, Mermin2014a, *Mermin2014c, Mermin2014d, Mermin2016}. The main difference is that an observer is treated not as a black box but as a subsystem like any other. This allows multiple observers to be treated as equals, thus providing a framework for meaningful descriptions of intersubjective agreement. ``Objective'' properties can then be defined as those features of the common worldview that survive when more observers are added to the framework.
The theory of information is developed only to the level of a bare skeleton, however, because I have not solved the resulting system of equations. The discussion of this part of the theory is therefore mostly qualitative, and no proof can be offered that it leads to predictions in agreement with experiment. The challenge of adding flesh to this skeleton nevertheless opens up many promising avenues for future research.
\subsection{A guide for the reader}
As an aid to navigation, this subsection indicates which parts of the paper are most essential on a first reading. Overall, the most difficult parts are those dealing with technical details of boson vector spaces; these are largely relegated to a series of appendices. On a first reading, one should focus attention primarily on the properties of systems containing only fermions.
The beginning of Sec.\ \ref{sec:tensor_prod_indistinguishable} shows that, contrary to statements in the literature, systems of indistinguishable particles do have a tensor product structure. Readers who are willing to take this for granted can go straight to Eqs.\ (\ref{eq:uU0})--(\ref{eq:uvwUVW}), referring back to earlier material as needed for definitions of notation. The paragraph below Eq.\ (\ref{eq:U_fermi}) establishes when a fermion creator is invertible. Equation (\ref{eq:U_bf}) shows that for systems containing bosons, the complex numbers in the fermion matrix (\ref{eq:U_fermi}) are replaced by commuting operators.
Section \ref{sec:mathematics} explains why invertibility of subsystems is important and develops some mathematical tools for inversion. Readers who are willing to take invertibility for granted can go straight to the exponential representation of Eqs.\ (\ref{eq:U_exp_X}) and (\ref{eq:delta_eA}). The definition of the word ``quasiclassical'' and its relation to classical probability theory in Sec.\ \ref{sec:quasiclassical} are also worth noting.
Section \ref{sec:kinematics} deals with various kinematical aspects of the definition of subsystems. The definition of observable quantities in Secs.\ \ref{sec:observables} and \ref{sec:relational_properties} should be included in a first reading, but Secs.\ \ref{sec:permutations} and \ref{sec:subsystem_differences} can be glossed over and referred back to when needed. The geometric properties defined in Sec.\ \ref{sec:subsystem_geometry} are essential as they are used throughout the rest of the paper.
Section \ref{sec:time_functional} develops various connections between subsystems and time. Those willing to accept that interacting subsystems never satisfy the Schr\"odinger equation can skip Sec.\ \ref{sec:interacting_not_Schroedinger}. The remaining subsections all contain essential material. The key ideas of Sec.\ \ref{sec:time_functional_defn} are the conceptual foundation for the time functional in Eqs.\ (\ref{eq:Schr_ideal}) and (\ref{eq:lambda_defn}) and the resulting functional in Eq.\ (\ref{eq:Delta_t}). The central result of Sec.\ \ref{sec:properties_time_functional} is the inequality (\ref{eq:time_energy}).
The topic of Sec.\ \ref{sec:dynamical_stability} is subsystem dynamics. The concept of dynamical stability is quantified in Eq.\ (\ref{eq:chi_definition}) of Sec.\ \ref{sec:dynamical_stability_functional}. The simple case of a time-independent total system state $\ket{\psi}$ is considered first in Sec.\ \ref{sec:time_independent_psi}. The solution for the unique maximum of the dynamical stability functional is given in Eq.\ (\ref{eq:Dx_soln}). The extension to time-dependent $\ket{\psi}$ in Secs.\ \ref{sec:time_dependent_psi}, \ref{sec:real_matrix}, and \ref{sec:solve_dynamical_stability} can be glossed over on a first reading. The main result, shown in Eq.\ (\ref{eq:Delta_x_real}), is qualitatively the same as the previous solution (\ref{eq:Dx_soln}), apart from a necessary change of notation. Section \ref{sec:model_calculations} can be skipped on a first reading, but the results of Sec.\ \ref{sec:arbitrary_number} are essential for understanding why the subsystem decompositions used here cannot be considered objective.
Section \ref{sec:superselection} discusses a system lacking a phase reference, which would lead to a particle-number superselection rule in textbook quantum mechanics. In such a system, the time functional (\ref{eq:Delta_t}) can be replaced with the renormalized functional (\ref{eq:Delta_t_K}). After a similar renormalization of other variables, the overall solution for the dynamically stable subsystem change has the same form as before [i.e., Eq.\ (\ref{eq:Delta_x_real})]. This is an important result, but the details of the derivation are not needed in any later sections of the paper. The same is true for Sec.\ \ref{sec:subsystem_permutations}, which deals with the effect of subsystem permutations on subsystem dynamics.
Section \ref{sec:information_theory} describes how information is extracted from the preceding theory. It contains many key concepts but only one equation, so it can be read in its entirety the first time around. The same is true for the conclusions in Sec.\ \ref{sec:conclusions}.
\section{Tensor products and indistinguishable particles}
\label{sec:tensor_prod_indistinguishable}
The purpose of this section is to define clearly what is meant by the tensor product of many-particle quantum states. Only a small part of the material presented here is entirely new, but establishing a clear notation at the outset helps to simplify calculations in subsequent sections of the paper.
The most convenient tensor product for systems of many indistinguishable particles has an additional algebraic structure that accounts for symmetry or antisymmetry with respect to particle permutations. The precise form of this algebra is defined uniquely by the geometry of Hilbert space, in the form of a cluster decomposition property for the inner product in Fock space. The resulting tensor algebra is the same as that of particle creation operators in Fock space. Special care is needed to ensure closure of the algebra for systems containing bosons.
\subsection{Definition of a tensor product}
\label{sec:tensor}
Given two vector spaces $\mathcal{V}_1$ and $\mathcal{V}_2$, the tensor product of vectors $\ket{u} \in \mathcal{V}_1$ and $\ket{v} \in \mathcal{V}_2$ is written as $\ket{u} \otimes \ket{v}$. The tensor product is bilinear but not commutative \cite{Szekeres2004}. That is, it must be distributive over vector addition: \begin{subequations} \label{eq:distributive} \begin{align} \ket{u} \otimes (\ket{v} + \ket{w}) & = \ket{u} \otimes \ket{v} + \ket{u} \otimes \ket{w} , \\ (\ket{u} + \ket{z}) \otimes \ket{v} & = \ket{u} \otimes \ket{v} + \ket{z} \otimes \ket{v} , \end{align} \end{subequations} as well as bilinear with respect to scalar multiplication: \begin{equation} \alpha (\ket{u} \otimes \ket{v}) = (\alpha \ket{u}) \otimes \ket{v} = \ket{u} \otimes (\alpha \ket{v}) . \label{eq:bilinear} \end{equation} The set of linear combinations of such tensor products defines the vector space $\mathcal{V}_1 \otimes \mathcal{V}_2$. The tensor product of three or more vectors is associative: \begin{equation} (\ket{u} \otimes \ket{v}) \otimes \ket{w} = \ket{u} \otimes (\ket{v} \otimes \ket{w}) , \label{eq:associative} \end{equation} thus defining the vector space $\mathcal{V}_1 \otimes \mathcal{V}_2 \otimes \mathcal{V}_3$ uniquely.
\subsection{Indistinguishable particles}
\label{sec:indistinguishable}
The vector space for a system of $n$ identical particles is \begin{equation} \mathcal{H}^{n} = \underbrace{\mathcal{H} \otimes \mathcal{H} \otimes \cdots \otimes \mathcal{H}}_{n \text{ times}} , \end{equation} in which $\mathcal{H}$ is the Hilbert space of a single particle. The Fock space $\mathcal{F} (\mathcal{H})$ for a system with an indefinite number of identical particles is defined as the direct sum \cite{NegeleOrland1998, ReedSimonVol1, Geroch1985} \begin{equation} \mathcal{F} (\mathcal{H}) = \bigoplus_{n=0}^{\infty} \mathcal{H}^{n} = \mathcal{H}^{0} \oplus \mathcal{H} \oplus \mathcal{H}^{2} \oplus \cdots , \end{equation} where the zero-particle space $\mathcal{H}^{0}$ consists of all scalar multiples of the vacuum state $\ket{0}$ (the null vector is written as $0$). In the tensor algebra of Fock space, $\ket{0}$ is the multiplicative identity: \begin{equation} \ket{0} \otimes \ket{\psi} = \ket{\psi} \otimes \ket{0} = \ket{\psi} \qquad \forall \ket{\psi} \in \mathcal{F} (\mathcal{H}) . \label{eq:mult_ident} \end{equation} By construction, $\mathcal{F} (\mathcal{H})$ is therefore closed under tensor multiplication \footnote{Here we are considering tensor multiplication only at the formal level. See Appendix \ref{app:rigged} for a discussion of the subtleties that arise when normalization is considered.}: \begin{equation} \mathcal{F} (\mathcal{H}) \otimes \mathcal{F} (\mathcal{H}) = \mathcal{F} (\mathcal{H}) . \end{equation}
According to the symmetrization postulate \cite{Messiah1962, MessiahGreenberg1964},
the only physically meaningful states in $\mathcal{F} (\mathcal{H})$ are those satisfying the symmetry condition \begin{equation} S \ket{\psi} = \ket{\psi} , \label{eq:symmetry} \end{equation} where $S$ is a projector for states of appropriate symmetry (i.e., totally symmetric for bosons and totally antisymmetric for fermions). It is defined by \begin{equation} S = \sum_{n=0}^{\infty} \Pi_n S_n \Pi_n , \qquad S_n = \frac{1}{n!} \sum_{\sigma} \varepsilon (\sigma) \sigma , \label{eq:symmetrizer} \end{equation} in which $\Pi_n$ is the projector for the $n$-particle subspace $\mathcal{H}^n$ and $S_n$ is the projector for the symmetric or antisymmetric states of $\mathcal{H}^n$ (with $S_0 = S_1 = 1$). The sum over $\sigma$ covers the $n!$ permutation operators in $\mathcal{H}^n$. For fermions, $\varepsilon (\sigma)$ is the sign of the permutation $\sigma$: \begin{equation} \varepsilon (\sigma) = \begin{cases} +1 & \text{if } \sigma \text{ is even} , \\ -1 & \text{if } \sigma \text{ is odd} . \\ \end{cases} \end{equation} For bosons, $\varepsilon (\sigma) = 1$. The subspace of vectors in $\mathcal{F} (\mathcal{H})$ that satisfy the symmetry constraint (\ref{eq:symmetry}) is denoted $\mathcal{F}_{s} (\mathcal{H})$, while the corresponding subspace of $\mathcal{H}^{n}$ is denoted $S(\mathcal{H}^{n})$.
\subsection{The \texorpdfstring{$\psi$}{psi} product}
\label{sec:psi_product}
It is convenient at this point to introduce another tensor product that automatically accounts for all symmetry requirements. This binary operator is defined by \begin{equation} \ket{u^{(p)}} \odot \ket{v^{(q)}} = c(p, q) S (\ket{u^{(p)}} \otimes \ket{v^{(q)}}) , \label{eq:psi_product} \end{equation} in which $\ket{u^{(p)}} \in S(\mathcal{H}^{p})$, $\ket{v^{(q)}} \in S(\mathcal{H}^{q})$, and $c(p, q)$ is a numerical coefficient to be defined below. The linearity of $S$ and bilinearity of $\ket{u} \otimes \ket{v}$ then give a unique extension of $\ket{u} \odot \ket{v}$ to the case of general $\ket{u}, \ket{v} \in \mathcal{F}_{s} (\mathcal{H})$.
For fermions, $\ket{u} \odot \ket{v}$ is the same as the exterior or wedge product $\ket{u} \wedge \ket{v}$ \cite{Szekeres2004, Misner1973, LoomisSternberg1990, Abraham1988, KostrikinManin1989, Hassani2013}, while for bosons, $\ket{u} \odot \ket{v}$ is known as the symmetric product \cite{Szekeres2004, Abraham1988, KostrikinManin1989, Hassani2013}. For brevity, the name ``$\psi$ product'' is used here as an umbrella term covering both cases in the context of many-particle quantum states of generic symmetry. Note that the $\psi$ product is commutative only for bosons, since \cite{Szekeres2004, LoomisSternberg1990, Abraham1988, KostrikinManin1989, Hassani2013} \begin{equation} \ket{u^{(p)}} \odot \ket{v^{(q)}} = \zeta^{pq} \, \ket{v^{(q)}} \odot \ket{u^{(p)}} , \label{eq:signed_product} \end{equation} where $\zeta = +1$ for bosons and $\zeta = -1$ for fermions.
The coefficient $c(p, q)$ is partially defined by requiring the $\psi$ product to be associative. As shown in Appendix \ref{app:associative}, this requires $c(p, q)$ to have the form \cite{Abraham1988} \begin{equation} c(p, q) = \frac{f(p + q)}{f(p) f(q)} , \label{eq:cf} \end{equation} in which $f(1) = 1$ but $f(n)$ is otherwise arbitrary. Most authors choose either $f(n) = 1$ \cite{Szekeres2004, KostrikinManin1989} or $f(n) = n!$ \cite{Misner1973, LoomisSternberg1990, Abraham1988, Hassani2013}.
However, for applications in quantum mechanics, it is more convenient to choose $f(n) = \sqrt{n!}$ \footnote{See pp.\ 603 and 622--623 of Messiah \cite{Messiah1962}.}, due to the following cluster decomposition theorem. The theorem refers to a case in which $\mathcal{H} = \mathcal{H}_1 \oplus \mathcal{H}_2$, where the subspaces $\mathcal{H}_1$ and $\mathcal{H}_2$ are orthogonal. The Fock space therefore factors as $\mathcal{F}_{s} (\mathcal{H}) = \mathcal{F}_{s} (\mathcal{H}_1) \odot \mathcal{F}_{s} (\mathcal{H}_2)$.
\begin{theorem}[Cluster decomposition] \label{thm:cluster} Let $\ket{s t} = \ket{s} \odot \ket{t}$ and $\ket{u v} = \ket{u} \odot \ket{v}$ be vectors in $\mathcal{F}_{s} (\mathcal{H})$, where $\ket{s}, \ket{u} \in \mathcal{F}_{s} (\mathcal{H}_1)$ and $\ket{t}, \ket{v} \in \mathcal{F}_{s} (\mathcal{H}_2)$. Then the inner product $\inprod{s t}{u v}$ factors as \begin{equation} \inprod{s t}{u v} = \inprod{s}{u} \inprod{t}{v} \label{eq:cluster_decomp} \end{equation} for all such vectors if and only if $\abs{f(n)} = \sqrt{n!}$. \end{theorem} The factorization (\ref{eq:cluster_decomp}) is called a cluster decomposition \cite{Peres1995} because the subspaces $\mathcal{H}_1$ and $\mathcal{H}_2$ typically refer to different regions in coordinate space. Theorem \ref{thm:cluster} is proved in Appendix \ref{app:cluster_decomposition}.
The phase of $f(n)$ is not determined by this theorem. However, choosing $f(n)$ to be real and positive means that the coefficient of $\ket{u^{(p)}} \otimes \ket{v^{(q)}}$ in $\ket{u^{(p)}} \odot \ket{v^{(q)}}$ is also real and positive.
The $\psi$ product is required here to have the cluster decomposition property and to satisfy this phase convention. The coefficient in equation (\ref{eq:psi_product}) is thus given uniquely by \begin{equation} c(p, q) = \sqrt{\frac{(p+q)!}{p! q!}} . \label{eq:cpqf} \end{equation}
It is sometimes said that ``a tensor-product structure is not present'' in systems of indistinguishable particles \cite{Zanardi2002}. This is misleading, however, because the $\psi$ product has all of the essential properties of a tensor product (see Sec.\ \ref{sec:tensor}). Only the physically meaningless tensor-product structure of a system of \emph{distinguishable} particles is lacking.
\subsection{Subsystem creators}
\label{sec:creation}
The algebra of the $\psi$ product defined above is the same as the familiar algebra of particle creation operators in Fock space. To see this, one can start by defining a general $n$-particle $\psi$ product \begin{equation} \ket{\alpha_1 \alpha_2 \cdots \alpha_n} = \ket{\alpha_1} \odot \ket{\alpha_2} \odot \cdots \odot \ket{\alpha_n} , \label{eq:ket_general} \end{equation} in which $\ket{\alpha_k} \in \mathcal{H}$, and the set $\{ \ket{\alpha_k} \}$ is not assumed to be linearly independent or normalized. This vector is related to the corresponding unsymmetrized $n$-particle tensor product by \begin{equation} \ket{\alpha_1 \alpha_2 \cdots \alpha_n} = \sqrt{n!} \, S (\ket{\alpha_1} \otimes \ket{\alpha_2} \otimes \cdots \otimes \ket{\alpha_n}) , \label{eq:ket_relation} \end{equation} which follows from the definition of $\ket{u} \odot \ket{v}$ in Eqs.\ (\ref{eq:psi_product}) and (\ref{eq:cpqf}). The product of two such states has the composition property (due to associativity) \begin{equation} \ket{\alpha_1 \cdots \alpha_p} \odot \ket{\beta_1 \cdots \beta_q} = \ket{\alpha_1 \cdots \alpha_p \beta_1 \cdots \beta_q} . \end{equation}
Consider now the case in which the set $\{ \ket{\alpha_k} \}$ is orthonormal, with repetition of the same state permitted. Let $n_{\alpha}$ be the number of times a particular state $\ket{\alpha} \in \mathcal{H}$ occurs in equation (\ref{eq:ket_general}), with $\sum_{\alpha} n_{\alpha} = n$. For fermions, $\ket{\alpha} \odot \ket{\alpha} = 0$, so values of $n_{\alpha} > 1$ merely give rise to the null vector. Excluding such cases, the normalization of the vector (\ref{eq:ket_general}) is given by \cite{NegeleOrland1998} \begin{equation} \inprod{\alpha_1 \cdots \alpha_n}{\alpha_1 \cdots \alpha_n} = \prod_{\alpha} n_{\alpha} ! \; . \label{eq:normalization} \end{equation} That is, nonzero fermion states are normalized to unity, but this is true for bosons only if no single-particle state is repeated. Unit vectors are useful in some contexts (e.g., Appendix \ref{app:rigged}), but here the normalization (\ref{eq:normalization}) is more convenient.
The particle creation operator $a_{\lambda}^{\dagger}$ can now be defined as \cite{NegeleOrland1998} \begin{equation} a_{\lambda}^{\dagger} \ket{\lambda_1 \cdots \lambda_n} = \ket{\lambda} \odot \ket{\lambda_1 \cdots \lambda_n} = \ket{\lambda \lambda_1 \cdots \lambda_n} , \label{eq:creation_operator} \end{equation} which maps $S(\mathcal{H}^{n})$ into $S(\mathcal{H}^{n+1})$. The vectors $\{ \ket{\lambda_{i}} \}$ need not be orthonormal, although the operator commutation relations are simpler if they are \cite{NegeleOrland1998}. A product of creation operators can thus be used to generate any simple product state from the vacuum: \begin{equation} \ket{\lambda_1 \cdots \lambda_n} = a_{\lambda_1}^{\dagger} \cdots a_{\lambda_n}^{\dagger} \ket{0} . \label{eq:create_vacuum} \end{equation} The absence of numerical factors in this equation is due to the fact that no normalization convention is imposed on $\ket{\lambda_1 \cdots \lambda_n}$ \cite{NegeleOrland1998}.
Any vector $\ket{u} \in \mathcal{F}_{s} (\mathcal{H})$ can therefore be generated from the vacuum by \begin{equation} \ket{u} = U \ket{0} , \label{eq:uU0} \end{equation} in which $U$ is a linear combination of products of creation operators, including (in general) a scalar term for the creation of no particles. For convenience, the operator $U$ is called the \emph{creator} of the state $\ket{u}$. The lengthier phrase ``creation operator'' is reserved for the creator $a_{\lambda}^{\dagger}$ of a single-particle state $\ket{\lambda}$; thus, the set of creation operators is a proper subset of the creators.
Given another such state $\ket{v} = V \ket{0}$, the $\psi$ product of $\ket{u}$ and $\ket{v}$ is \begin{subequations} \label{eq:uvUV} \begin{align} \ket{u} \odot \ket{v} & = (U \ket{0}) \odot \ket{v} \\ & = U (\ket{0} \odot \ket{v}) \\ & = U \ket{v} \\ & = UV \ket{0} . \end{align} \end{subequations} The algebra of the vectors $\ket{u}$ and $\ket{v}$ is therefore the same as the algebra of the creators $U$ and $V$. This can be seen even more clearly when a redundant vacuum state is appended to $\ket{v}$: \begin{equation} \ket{u} \odot \ket{v} \odot \ket{0} = UV \ket{0} . \end{equation} This result can be extended to any number of $\psi$ products; for example, if $\ket{w} = W \ket{0}$, then \begin{equation} \ket{u} \odot \ket{v} \odot \ket{w} \odot \ket{0} = UVW \ket{0} . \label{eq:uvwUVW} \end{equation}
Of course, it is only meaningful to write such equations if the vector defined by this product is normalizable. No difficulty arises for fermions, because the dimension of the fermion Fock space is finite; fermion creation operators are therefore bounded. However, boson creators are unbounded (with respect to the topology defined by the usual Hilbert space norm); the $\psi$ product of two normalized boson vectors could therefore be unnormalizable. One must take care to choose a boson vector space that is closed under the $\psi$ product.
The construction of such a vector space is described in Appendix \ref{app:rigged}. The result, denoted $\mathcal{F}_{\psi} (\mathcal{H}_{\mathrm{b}})$, is a dense subspace of $\mathcal{F}_{s} (\mathcal{H}_{\mathrm{b}})$, where $\mathcal{H}_{\mathrm{b}}$ is the Hilbert space of a single boson. It should be noted that $\mathcal{F}_{\psi} (\mathcal{H}_{\mathrm{b}})$ is a Fr\'echet space \cite{ReedSimonVol1, Horvath2012} rather than a Hilbert space. This distinction can be ignored for many purposes, because the inner product from $\mathcal{F}_{s} (\mathcal{H}_{\mathrm{b}})$ remains well defined in the subspace $\mathcal{F}_{\psi} (\mathcal{H}_{\mathrm{b}})$; the only change is that the Fr\'echet-space topology is not defined by this inner product.
\subsection{Matrix notation for fermions}
\label{sec:fermion_matrix}
In a fermion system, the Fock space has a finite dimension $2^{d}$, in which $d$ is the dimension of the single-fermion Hilbert space $\mathcal{H}_{\mathrm{f}}$. It is therefore convenient to introduce matrix representations for the (always bounded) fermion creators $U$ and $V$.
Let $\mathcal{H}_{\mathrm{f}}$ be spanned by an orthonormal basis $\{ \ket{e_{k}} \}$, in which $k \in \{ 0, 1, 2, \ldots, d-1 \}$. The Fock space $\mathcal{F}_{s} (\mathcal{H}_{\mathrm{f}})$ is then spanned by the basis $\{ \ket{f_i} \}$, where the integer $i \in \{ 0, 1, 2, \ldots, 2^{d} - 1 \}$ has the binary representation \begin{equation} i = \sum_{k=0}^{d-1} i_{k} 2^{k} \qquad (i_k \in \{ 0, 1 \}) , \label{eq:binary_notation} \end{equation} in which $i_k$ is the $k$th binary digit of $i$. If $i_k = 1$, the state $\ket{e_k}$ is occupied in $\ket{f_i}$; otherwise, it is unoccupied. Thus, for example, when $d = 4$, the basis vector $\ket{f_5}$ can be written in various ways as \begin{equation} \ket{f_5} = \ket{0101} = \ket{e_2} \odot \ket{e_0} . \end{equation} In this notation, $\ket{f_0}$ is just the vacuum state $\ket{0}$.
As a simple example, consider the case $d = 2$, for which a general vector $\ket{u} \in \mathcal{F}_{s} (\mathcal{H}_{\mathrm{f}})$ can be written as \begin{equation} \ket{u} = \sum_{i=0}^{3} c_{i} \ket{f_i} , \qquad c_{i} \equiv \inprod{f_i}{u} . \label{eq:u_Fermi} \end{equation} The $\psi$ products of $\ket{u}$ with the basis vectors of $\mathcal{F}_{s} (\mathcal{H}_{\mathrm{f}})$ are then given by $\ket{u} \odot \ket{f_0} = \ket{u}$, $\ket{u} \odot \ket{f_{1}} = c_{0} \ket{f_{1}} + c_{2} \ket{f_{3}}$, $\ket{u} \odot \ket{f_{2}} = c_{0} \ket{f_{2}} - c_{1} \ket{f_{3}}$, and $\ket{u} \odot \ket{f_{3}} = c_{0} \ket{f_{3}}$. The matrix representing $U$ in this basis is therefore \begin{equation} U = \begin{pmatrix} c_{0} & 0 & 0 & 0 \\ c_{1} & c_{0} & 0 & 0 \\ c_{2} & 0 & c_{0} & 0 \\ c_{3} & c_{2} & -c_{1} & c_{0} \end{pmatrix} . \label{eq:U_fermi} \end{equation}
In general, $U$ has the form of a lower triangular matrix whenever the basis $\{ \ket{f_{i}} \}$ is arranged in order of nondecreasing particle number $\abs{i} \equiv \sum_{k=0}^{d-1} i_k$. (Such an arrangement is called graded lexicographic ordering \cite{Cox2007}.) Hence, $U$ is invertible if and only if $c_0 \ne 0$, since the determinant of a triangular matrix is just the product of its diagonal elements.
\subsection{Different types of particles}
\label{sec:different_types}
Up to this point it has tacitly been assumed that the system under consideration contains only one type of particle. To combine different types of either bosons or fermions, one can simply take a direct sum of the single-particle Hilbert spaces before constructing the Fock space \cite{[] [{, pp.\ 166--167 and 227--237.}] Sudbery1986}. All boson (fermion) creation operators then commute (anticommute) with each other. Such a description is possible because the operators for different fermions can be chosen arbitrarily to either commute or anticommute \cite{[] [{, pp.\ 250 and 475.}] LandauLifshitz1977, [] [{, p.\ 94.}] LandauLifshitz1980}. (These two choices are related by a Jordan--Wigner transformation \cite{JordanWigner1928, LiebSchultzMattis1961, SchultzMattisLieb1964, Araki1961, [] [{, Sec.\ 4-4.}] StreaterWightman1989}.)
In this mode of description, which is commonly used in the treatment of isospin \cite{Messiah1962, Sudbery1986, LandauLifshitz1977}, there are only two types of particles: bosons and fermions. Different types of bosons or different types of fermions are treated as formally indistinguishable; the distinction is maintained only at the level of quantum numbers within the single-particle Hilbert spaces $\mathcal{H}_{\mathrm{b}}$ and $\mathcal{H}_{\mathrm{f}}$ \footnote{In an alternative mode of description, different types of bosons and fermions are distinguished using an unsymmetrized tensor product (see Appendix \ref{app:different}). Formulas such as Eq.\ (\ref{eq:u_not_Schmidt}) then become much more cumbersome, but the physics is the same. In particular, the vector space $\mathcal{E}$ of Eq.\ (\ref{eq:total_E}) is not affected by this change of convention; only the labeling and phase of the basis vectors is altered \cite{Sudbery1986}.}.
Bosons and fermions are combined together into one system by means of an ordinary (unsymmetrized) tensor product. A general vector in the tensor-product space \begin{equation} \mathcal{E} = \mathcal{F}_{\psi} (\mathcal{H}_{\mathrm{b}}) \otimes \mathcal{F}_{s} (\mathcal{H}_{\mathrm{f}}) \label{eq:total_E} \end{equation} is thus of the form \begin{equation} \ket{u} = \sum_{j=0}^{\infty} \sum_{i=0}^{2^{d}-1} c_{ji} \ket{b_j} \otimes \ket{f_i} , \label{eq:u_expansion} \end{equation} in which $\{ \ket{b_j} \}$ and $\{ \ket{f_i} \}$ are fixed orthonormal bases for the boson and fermion Fock spaces (see Sec.\ \ref{sec:fermion_matrix} for the definition of $\{ \ket{f_i} \}$). This can also be written as \begin{equation} \ket{u} = \sum_{i=0}^{2^{d}-1} \ket{u_i} \otimes \ket{f_i} , \label{eq:u_not_Schmidt} \end{equation} in which $\ket{u_i} = \sum_{j} c_{ji} \ket{b_j}$. This is not a Schmidt decomposition, since the basis $\{ \ket{f_i} \}$ is independent of $\ket{u}$; the set $\{ \ket{u_i} \}$ is therefore generally not orthonormal.
In the matrix notation of Sec.\ \ref{sec:fermion_matrix}, $\ket{u}$ can be written as a $2^{d}$-component ``spinor,'' given here explicitly (in decimal and binary) for the case $d = 2$: \begin{equation} \ket{u} = \begin{pmatrix} \ket{u_{0}} \\ \ket{u_{1}} \\ \ket{u_{2}} \\ \ket{u_{3}} \end{pmatrix} = \begin{pmatrix} \ket{u_{00}} \\ \ket{u_{01}} \\ \ket{u_{10}} \\ \ket{u_{11}} \end{pmatrix} . \label{eq:u_spinor} \end{equation} The boson-fermion creator $U$ then has the form \begin{equation} U = \begin{pmatrix} U_{0} & 0 & 0 & 0 \\ U_{1} & U_{0} & 0 & 0 \\ U_{2} & 0 & U_{0} & 0 \\ U_{3} & U_{2} & -U_{1} & U_{0} \end{pmatrix} , \label{eq:U_bf} \end{equation} in which the constants $c_i$ of Eq.\ (\ref{eq:U_fermi}) are replaced by boson creators $U_i$ (defined by $\ket{u_i} = U_i \ket{0}_{\mathrm{b}}$).
As shown in Appendix \ref{app:different}, the definition of the $\psi$ product $\ket{u} \odot \ket{v}$ can easily be extended to the vector space $\mathcal{E}$. All of the main conclusions of Secs.\ \ref{sec:psi_product} and \ref{sec:creation}, including the cluster decomposition property and the algebraic equivalence shown in Eq.\ (\ref{eq:uvwUVW}), remain valid in $\mathcal{E}$.
\section{Basic tools for subsystem analysis}
\label{sec:mathematics}
Let us turn now to an investigation of the conditions under which a general ket $\ket{\psi} \in \mathcal{E}$ can be written as a $\psi$ product of the form $\ket{\psi} = \ket{u} \odot \ket{v}$. This equation is equivalent to the operator product \begin{equation} \Psi = U V , \label{eq:Psi_U_V} \end{equation} in which $\Psi$, $U$, and $V$ are the creators of the states $\ket{\psi}$, $\ket{u}$, and $\ket{v}$, respectively.
\subsection{Invertible subsystem creators}
\label{sec:invertible}
Such an equation can clearly be written for any \emph{invertible} subsystem creator $U$, since one can then take \begin{equation} V = U^{-1} \Psi \label{eq:V_Uinv_Psi} \end{equation} for arbitrary $\Psi$. Likewise, if $V$ is invertible, one can always let $U = \Psi V^{-1}$. This approach can easily be extended to more general products such as $\Psi = UVW$, since (for example) if $U$ and $W$ are invertible one can always let $V = U^{-1} \Psi W^{-1}$.
Under what conditions is a creator invertible? This question has already been answered for fermion systems; in Sec.\ \ref{sec:fermion_matrix}, it was noted that a fermion creator $U$ is invertible if and only if $c_0 \ne 0$, where $c_0 = \inprod{f_0}{u} = \inprod{0}{u}$ is the vacuum component of $\ket{u}$. Invertibility is thus a weak constraint: the set of invertible fermion creators is uncountable and forms a smooth manifold.
A similar condition can be derived for systems containing both bosons and fermions. The main result is expressed here in the form of a theorem (proved in Appendix \ref{app:invertibility_theorem}), using the notation of Sec.\ \ref{sec:different_types}. \begin{theorem}[Invertibility] \label{thm:invertible} For a boson-fermion creator $U$, the following statements are equivalent: (a)~The linear map $U : \mathcal{E} \to \mathcal{E}$ is invertible. (b)~The associated boson creator $U_0 : \mathcal{F}_{\psi} \to \mathcal{F}_{\psi}$ is invertible. (c)~The corresponding boson state $\ket{u_0} = U_{0} \ket{0}_{\mathrm{b}}$ is a coherent state. \end{theorem} The coherent states mentioned in part (c) of the theorem are just the familiar Glauber states \cite{Glauber1963c, KlauderSkagerstam1985, BlaizotRipka1986, NegeleOrland1998, Perelomov1986, ZhangFengGilmore1990, Gazeau2009}, as defined in Eq.\ (\ref{eq:coherent}) of Appendix \ref{app:rigged}.
The constraint imposed by Theorem \ref{thm:invertible} remains very weak. For $U$ to be invertible, the boson state $\ket{u_0}$ associated with the fermion vacuum $\ket{f_0} = \ket{0}_{\mathrm{f}}$ must be a coherent state, but all of the other kets $\ket{u_i}$ in Eq.\ (\ref{eq:u_not_Schmidt}) (i.e., those with $i \ne 0$) are completely arbitrary.
\subsection{Why invertibility is important}
\label{sec:invertible_important}
Equation (\ref{eq:V_Uinv_Psi}) is not the most general solution of Eq.\ (\ref{eq:Psi_U_V}). The most general solution has the form \cite{CampbellMeyer2009, BenIsrael2003, StewartSun1990, Nashed1976} \begin{equation} V = U^{-} \Psi + (1 - U^{-} U) Y , \label{eq:gen_inv} \end{equation} in which $U^{-}$ is a generalized inverse of $U$ \cite{CampbellMeyer2009, BenIsrael2003, StewartSun1990, Nashed1976} and the creator $Y$ is arbitrary. For finite-dimensional matrices, such a solution exists if and only if $\Psi \in \im U$, where $\im U$ denotes the image (or range) of $U$. When $U$ is invertible, Eq.\ (\ref{eq:gen_inv}) reduces to Eq.\ (\ref{eq:V_Uinv_Psi}), but invertibility of $U$ is not necessary for the existence of the solution (\ref{eq:gen_inv}).
Nevertheless, all subsequent analysis in this paper is based on the study of \emph{invertible} subsystem creators. This is done for two reasons. The first is simply the pragmatic reason that the algebra of the subsystem creators is much simpler when invertibility is assumed at the outset.
The second reason is more technical but also more compelling. For simplicity, consider the Moore--Penrose generalized inverse $A^{-}$ of a finite-dimensional square matrix $A$ \cite{CampbellMeyer2009, BenIsrael2003, StewartSun1990}. In this case, $(A + \delta A)^{-}$ is well known to be continuous at $\delta A = 0$ if and only if the rank of $(A + \delta A)$ is the same as the rank of $A$ for all perturbations $\delta A$ in some finite neighborhood of $\delta A = 0$ \cite{CampbellMeyer2009, BenIsrael2003, StewartSun1990}. But if $A$ does not have full rank, there exist matrices $A + \delta A$ with $\rank (A + \delta A) > \rank A$ for arbitrarily small $\norm{\delta A} > 0$. Hence, $(A + \delta A)^{-}$ is continuous at $\delta A = 0$ if and only if $A$ has full rank---i.e., if and only if $A$ is invertible.
Continuity of the subsystem (\ref{eq:gen_inv}) is essential because the subsystem dynamics in Sec.\ \ref{sec:dynamical_stability} is derived from a variational principle. In order for subsystems to be stationary states of the dynamical stability functional, they must first be continuous with respect to small variations. Subsystems derived from generalized inverses are consequently not used in this paper.
A simple corollary of Theorem \ref{thm:invertible} is that all invertible subsystems must have a vacuum component. This implies that all nontrivial (i.e., not purely vacuum) invertible subsystems must have an indefinite number of particles. A pragmatic reason for allowing this type of subsystem was pointed out by Bell \cite{Bell1976}: \begin{quote} The real world is made of electrons and protons and so on, and as a result the boundaries of natural objects are fuzzy, and some particles in the boundary can only doubtfully be assigned to either object or environment. I think that fundamental physical theory should be so formulated that such artificial divisions are manifestly inessential. \end{quote} The superposition of different numbers of particles is therefore essential to the definition of the subsystems considered here. This does not imply that the physical limitations leading to particle-number superselection rules must be ignored in this theory. A detailed discussion of this topic is, however, postponed until Sec.\ \ref{sec:superselection}.
\subsection{Functions of subsystem creators}
Before writing down any formulas for the inverse of a creator, it is helpful to start by establishing some general properties of creators and functions of creators. Creators are actually easier to work with than generic operators. The reason for this is that any creator $A$ can be decomposed into even and odd parts: \begin{equation} A = A_{+} + A_{-} , \end{equation} in which $A_{+}$ ($A_{-}$) comprises all terms with an even (odd) number of fermion creation operators. From Eq.\ (\ref{eq:signed_product}) we see that an even creator commutes with any creator, whereas two odd creators anticommute: \begin{equation} A_{+} B_{\pm} = B_{\pm} A_{+} , \qquad A_{-} B_{-} = - B_{-} A_{-} . \label{eq:creator_commute_anti} \end{equation} The commutator $[ A, B ] \equiv AB - BA$ of two creators is therefore given by \begin{equation} [ A, B ] = 2 A_{-} B_{-} , \label{eq:creator_commutator} \end{equation} which commutes with any other creator $C$, since $A_{-} B_{-}$ is even. Equation (\ref{eq:creator_commute_anti}) also implies that all odd creators are nilpotent: \begin{equation} A_{-}^2 = 0 . \label{eq:A_minus_nilpotent} \end{equation} The binomial expansion of any integer power of a creator thus contains only two terms: \begin{equation} A^{n} = (A_{+} + A_{-})^{n} = A_{+}^{n} + n A_{+}^{n-1} A_{-} . \label{eq:An_binomial} \end{equation} All functions $f(A)$ defined by a power series can be expanded likewise as \begin{equation} f(A) = f(A_{+}) + A_{-} f' (A_{+}) , \end{equation} in which $f'(x) = \mathrm{d} f / \mathrm{d} x$.
\subsection{Inversion formula}
\label{sec:inversion_formula}
Let us now return to the case of an invertible creator $U$. Given that $U_0$ is invertible, we can write $U$ as \begin{equation} U = U_0 (1 + Z) = (1 + Z) U_0 , \label{eq:Z} \end{equation} in which $Z \equiv U_0^{-1} U - 1 = U U_0^{-1} - 1$. Calculating $U_0^{-1}$ is trivial, because the operator $U_0$ for a coherent state is an exponential function [see Eq.\ (\ref{eq:coherent})]. In the matrix notation of Eq.\ (\ref{eq:U_bf}), $Z$ has the same lower-triangular form as $U$: \begin{equation} Z = \begin{pmatrix} Z_{0} & 0 & 0 & 0 \\ Z_{1} & Z_{0} & 0 & 0 \\ Z_{2} & 0 & Z_{0} & 0 \\ Z_{3} & Z_{2} & -Z_{1} & Z_{0} \end{pmatrix} \qquad (\text{for } d = 2) . \label{eq:Z_bf} \end{equation} The key difference is that $Z_0 = 0$, by definition of $Z$. The operator $Z$ is therefore nilpotent---i.e., $Z^{k} = 0$ for some finite integer $k$. Since the matrix size is $2^d \times 2^d$, we see immediately that $k \le 2^{d}$.
However, we can obtain a stronger bound on $k$ by noting from Eq.\ (\ref{eq:A_minus_nilpotent}) that the square of any \emph{monomial} function of the fermion creation operators is also zero. Since the maximum degree of any such monomial is $d$, inspection of Eq.\ (\ref{eq:An_binomial}) shows that \begin{equation} Z^k = 0 \qquad \forall k > \lceil (d/2) \rceil , \label{eq:Z_nilpotent} \end{equation} in which $\lceil x \rceil$ is the ceiling function. One can then use the geometric series to obtain the inversion formula \begin{equation} U^{-1} = U_0^{-1} \sum_{n=0}^{\lceil (d/2) \rceil} (-1)^{n} Z^{n} . \label{eq:U_inverse_series} \end{equation}
\subsection{The exponential representation}
In a similar fashion, we can use Eq.\ (\ref{eq:Z}) to calculate the logarithm of $U$: \begin{equation} \ln U = \ln U_0 + \ln (1 + Z) , \end{equation} in which $\ln (1 + Z)$ is given by the power series \begin{equation} \ln (1 + Z) = \sum_{n=1}^{\lceil (d/2) \rceil} \frac{(-1)^{n+1}}{n} Z^n . \end{equation} Every invertible creator can therefore be represented as an exponential function: \begin{align} U & = \exp X = e^{X_0} e^{(X - X_0)} \nonumber \\ & = e^{X_0} \sum_{n=0}^{\lceil (d/2) \rceil} \frac{(X - X_0)^n}{n!} , \label{eq:U_exp_X} \end{align} in which $X = \ln U$. The intermediate steps in Eq.\ (\ref{eq:U_exp_X}) used the facts that $X_0 = \ln U_0$ is even and that $X - X_0$ is nilpotent.
The exponential representation (\ref{eq:U_exp_X}) plays a crucial role in the remainder of this paper. For both computation and analysis, it is preferable to take the creator $X$ as fundamental and \emph{define} the subsystem $U$ as $U = \exp X$. This guarantees that regardless of any changes in $X$, $U$ always remains invertible, its inverse being given simply by $U^{-1} = \exp (-X)$. In this approach, the inversion formula (\ref{eq:U_inverse_series}) becomes redundant.
The Campbell--Baker--Hausdorff formula for two creators $A$ and $B$ can now be used to show that \begin{align} e^{A} e^{B} & = e^{A + B + [A, B] / 2} = e^{A + B + A_{-} B_{-}} \nonumber \\ & = e^{A + B} e^{A_{-} B_{-}} = e^{A + B} (1 + A_{-} B_{-}) , \end{align} in which Eq.\ (\ref{eq:creator_commutator}) and $(A_{-} B_{-})^2 = 0$ were used. Adding $e^{A} e^{B}$ and $e^{B} e^{A}$ then yields the very useful formula \begin{equation} e^{A+B} = \{ e^{A} , e^{B} \} , \label{eq:expAplusB} \end{equation} in which $\{ A, B \} \equiv \frac12 (AB + BA)$ denotes the symmetrized product. As a special case of Eq.\ (\ref{eq:expAplusB}), note that if $A$ is varied by $\delta A$, the corresponding first-order variation in $e^{A}$ is given by \begin{align} \delta (e^{A}) & \equiv e^{A + \delta A} - e^{A} \nonumber \\ & = \{ e^{\delta A} - 1 , e^{A} \} \nonumber \\ & = \{ \delta A , e^{A} \} , \label{eq:delta_eA} \end{align} in which terms of second and higher order in $\delta A$ were discarded in the final step. This yields a simple expression for the derivative of $e^{A}$ with respect to a parameter $s$: \begin{equation} \frac{\partial}{\partial s} (e^{A}) = \left\{ \frac{\partial A}{\partial s} , e^{A} \right\} . \end{equation} This result is much simpler than the corresponding formulas for a general operator $A$ \cite{Wilcox1967}. Several other identities for the symmetrized product of creators are collected together in Appendix \ref{app:creator_identities}.
\subsection{Quasiclassical subsystems}
\label{sec:quasiclassical}
The representation of quantum states by exponential functions has a long history, dating back at least to the WKB approximation of 1926. The modern theory of generalized coherent states \cite{KlauderSkagerstam1985, Perelomov1986, ZhangFengGilmore1990, Gazeau2009} also relies heavily on exponential representations. In the latter approach, the coherent states are tied to a particular Lie group (namely, the dynamical group associated with the Hamiltonian of the total system), and the exponential functions used to construct the coherent states are \emph{unitary} operators involving both creation and annihilation operators. Such unitary operators have the advantage of convenient normalization properties, but they are not useful in the present context because their algebra is not isomorphic to the algebra of the $\psi$ product.
The exponential representation (\ref{eq:U_exp_X}) can be viewed as a further generalization of the coherent-state concept, in that the creator $X$ is not tied to any Lie group. Indeed, $X$ is almost entirely arbitrary, the only restrictions being the algebraic closure constraint of Appendix \ref{app:rigged} (which requires all $\psi$ products of subsystems to be normalizable) and the invertibility constraint requiring $U_0 = \exp X_0$ to be a Glauber coherent state (\ref{eq:coherent}) (which implies that $X_0$ is a \emph{linear} function of the boson creation operators). But the latter constraint is not independent of the former, because the algebraic closure condition of Appendix \ref{app:rigged} was used in Appendix \ref{app:invertibility_theorem} to derive the linearity of $X_0$.
Subsystems permitting an exponential representation $U = \exp X$ will be referred to as \emph{quasiclassical} subsystems in this paper. However, since the word ``quasiclassical'' has various other connotations (including the WKB approximation and the Glauber coherent states \cite{CohTan1977}), it is important to be clear about the sense in which this label is being used here. In this paper, ``quasiclassical'' is just a synonym for ``invertible.''
This sense of the word quasiclassical does not imply the use of any approximation. As emphasized in Sec.\ \ref{sec:invertible}, the product $\Psi = UV$ provides an exact representation for arbitrary states $\Psi$. As discussed in Sec.\ \ref{sec:invertible_important}, invertibility of $U$ is the minimal restriction needed to ensure continuity of $V$ when $U$ is varied and $\Psi$ is held constant. Note that when $U$ is taken to be quasiclassical, $V$ is quasiclassical if and only if $\Psi$ is quasiclassical---but $\Psi$ need not be quasiclassical.
The fact that all coherent states are invertible is one possible reason for designating the latter as quasiclassical. A more significant reason is that the existence of an unentangled product $\ket{\psi} = \ket{u} \odot \ket{v}$ permits the use of \emph{classical} probability theory. Since different subsystem decompositions $\ket{u} \odot \ket{v}$ are mutually exclusive alternatives for the representation of $\ket{\psi}$, one's state of knowledge or belief about the suitability of various decompositions can be described using ordinary Bayesian probability theory. The implications of this idea are developed further in the next section.
\section{Kinematics of subsystems without subspaces}
\label{sec:kinematics}
This section explores several topics that are independent of and logically prior to any concept of subsystem dynamics, thus falling into the category of kinematics. These include the definition of observable quantities, the description of differences between two subsystem decompositions, and subsystem geometry.
For generality, let $\ket{\psi}$ be decomposed into a product of $m$ subsystems, where $m \ge 2$: \begin{equation} \ket{\psi} = \ket{u_1} \odot \ket{u_2} \odot \cdots \odot \ket{u_{m}} . \label{eq:psi_u_decomp} \end{equation} Here and below, the subscript $k$ in $\ket{u_k}$ is just a label for the $m$ different subsystems ($k \in \{ 1, 2, \ldots, m \}$). The subscripts introduced previously in Eq.\ (\ref{eq:u_not_Schmidt}) are henceforth retired as they have no further use.
\subsection{Observables and beables}
\label{sec:observables}
Given such a subsystem decomposition, how are we to define the observable quantities of the theory?
A fundamental hypothesis of the present paper is that \emph{all} observables are calculated from the subsystem vectors $\ket{u_k}$. In other words, no observable quantity is calculated directly from the total system vector $\ket{\psi}$. This is not to say that $\ket{\psi}$ is meaningless; the subsystem dynamics depends on $\ket{\psi}$. However, $\ket{\psi}$ does not appear anywhere in the definition of observables.
Apart from this change, the mathematical apparatus used to define observables is nearly the same as that in ordinary many-particle quantum mechanics \cite{NegeleOrland1998, BlaizotRipka1986, Merzbacher1998}. That is, observables are represented by hermitian operators $A$ that are totally symmetric with respect to permutations of identical particles. For each such operator, one can calculate a number \begin{equation} \expect{A}_k \equiv \frac{\matelm{u_k}{A}{u_k}}{\inprod{u_k}{u_k}} \label{eq:mean_value} \end{equation} for each subsystem $\ket{u_k}$. In ordinary quantum mechanics, only the eigenvalues of $A$ are observable; the numbers $\expect{A}_k$ are therefore interpreted as mean values.
Here, however, the numbers $\expect{A}_k$ are taken to be directly observable. This means that observables are defined using what Bell has called the ``density of stuff'' interpretation \cite{Bell1990}, first introduced by Schr\"odinger \cite{[] [{; English translation in Ref.\ \cite{Schrodinger1982}.}] Schrodinger1926c, [] [{, pp.\ 41--44.}] Schrodinger1982}. A similar mass-density interpretation has been used in the dynamical reduction theory of Ghirardi \emph{et al}.\ \cite{GhirardiPearleRimini1990, GhirardiGrassiBenatti1995, BassiGhirardi2003}, but here $\expect{A}_k$ can describe properties other than mass density.
According to Bell \cite{Bell2004}, the numbers $\expect{A}_k$ can then be classified as ``beables'' rather than observables. That is, subsystem $\ket{u_k}$ is taken to \emph{possess} the property $\expect{A}_k$ independently of any measurement (the concept of ``measurement'' having no place in the fundamental postulates of the theory).
Such an interpretation of $\expect{A}_k$ is possible for the reason already explained in Sec.\ \ref{sec:quasiclassical}---namely, that different subsystem decompositions can be described using classical probability theory. Without entanglement, there is no need for the properties of the subsystems to be described as \emph{potential} rather than \emph{actual}.
Of course, it remains to be seen whether this interpretation of $\expect{A}_k$ can give rise to experimental predictions similar to those obtained from ordinary quantum mechanics. This is a difficult problem for which only qualitative results are obtained in this paper. Further discussion of this topic is presented in Sec.\ \ref{sec:information_theory}.
\subsection{Relational properties}
\label{sec:relational_properties}
One aspect of observables that deserves special attention is that not all quantities $\expect{A}_k$ are necessarily observable. For example, since the total system is taken to be closed (i.e., not interacting with anything else), quantities such as absolute position or orientation in space have no meaning within the theory. One can of course calculate numbers for these quantities within a given model, but such numbers are meaningless due to the lack of any external reference frame.
The only meaningful properties are therefore relational properties. For example, although the absolute position of the center of mass of a given subsystem is meaningless, it is meaningful to talk about the relative distance between the centers of mass of two subsystems. Another meaningful quantity would be the mass density of a subsystem relative to the position of its own center of mass.
This type of restriction has received much attention lately in regard to the connection between reference frames (or their lack) and superselection rules \cite{BartlettRudolphSpekkens2007}. The consequences of such restrictions within the present theory will be investigated in detail (for a specific example) in Sec.\ \ref{sec:superselection}. Until then, however, it will be assumed that there are no restrictions (in principle) on the observable quantities $\expect{A}_k$.
\subsection{Subsystem permutations}
\label{sec:permutations}
Let us now return to the subsystem decomposition (\ref{eq:psi_u_decomp}), expressed in terms of creators: \begin{equation} \Psi = U_{1} U_{2} \cdots U_{m} . \end{equation} If the only observables are the numbers (\ref{eq:mean_value}), no observable quantity depends on the \emph{order} in which the subsystems are multiplied. Any permutation \begin{equation} \Psi_{\pi} = U_{\pi (1)} U_{\pi (2)} \cdots U_{\pi (m)} \label{eq:psi_pi} \end{equation} yields the same observables, where $\pi$ denotes one of the $m!$ permutations of the integers $(1, 2, \ldots, m)$. The value of the product does, however, depend on this order, as indicated by the subscript on $\Psi_{\pi}$.
According to the results of Sec.\ \ref{sec:quasiclassical}, all but one of the subsystems $U_k$ are taken to be quasiclassical. For definiteness, let $U_1$ be the one subsystem that need not be quasiclassical. In the permutation (\ref{eq:psi_pi}), $U_1$ is at position $k = k_v$, where \begin{equation} \pi (k_v) = 1 , \qquad k_v = \pi^{-1} (1) . \end{equation} We can then solve Eq.\ (\ref{eq:psi_pi}) for $U_1$, obtaining \begin{equation} V \equiv U_1 = U_{\pi (k_v - 1)}^{-1} \cdots U_{\pi (1)}^{-1} \Psi_{\pi} U_{\pi (m)}^{-1} \cdots U_{\pi (k_v + 1)}^{-1} . \end{equation} As noted here, the symbol $V$ or $\ket{v} = V \ket{0}$ will often be used to refer to this non-quasiclassical subsystem.
\subsection{Subsystem differences}
\label{sec:subsystem_differences}
The next topic is how to describe small differences between two subsystem decompositions. Consider two sets of subsystems, $\{ U_k \}$ and $\{ U_k' \}$, whose products are $\Psi$ and $\Psi'$, respectively. The quasiclassical subsystems can be given an exponential representation (\ref{eq:U_exp_X}), for which the difference $\Delta X_k \equiv X_k' - X_k$ is assumed to be small. The corresponding difference $\Delta U_k \equiv U_k' - U_k$ is then given (for $k \ne 1$) by Eq.\ (\ref{eq:delta_eA}): \begin{align} \Delta U_k & = \exp (X_k + \Delta X_k) - \exp X_k \nonumber \\ &= \{ \Delta X_k, U_k \} , \label{eq:Delta_Uk} \end{align} in which terms beyond the first order in $\Delta X_k$ have been neglected. Likewise, the first-order difference between $(U_k')^{-1}$ and $U_k^{-1}$ is (for $k \ne 1$) \begin{align} \Delta U_k^{-1} & = \exp (-X_k - \Delta X_k) - \exp (-X_k ) \nonumber \\ &= - \{ \Delta X_k, U_k^{-1} \} . \end{align} The difference $\Delta V = V' - V = U_1' - U_1$ is taken to be determined by the values of $\{ \Delta U_k^{-1} \}$ and $\Delta \Psi = \Psi'- \Psi$. To define $\Delta V$, it is helpful to introduce linear functionals $V_{X}$ and $\tilde{V}_k [Y]$ such that when $X = \Psi$ and $Y = U_{k}^{-1}$ we have \begin{equation} V_{\Psi} = V = \tilde{V}_k [U_{k}^{-1}] \qquad (k \ne 1) . \end{equation} To first order in small quantities, $\Delta V$ is then given by \begin{equation} \Delta V = V_{\Delta \Psi} + \sum_{k=2}^{m} \tilde{V}_k [\Delta U_k^{-1}] . \label{eq:Delta_V} \end{equation} Note that $\Delta V$ depends (implicitly) on the choice of permutation $\pi$ in Eq.\ (\ref{eq:psi_pi}).
For practical calculations, the exponents $\ket{x_k} = X_k \ket{0}$ are usually expanded in some orthonormal basis $\{ \ket{e_{ki}} \}$: \begin{equation} \ket{x_k} = \sum_{i} c_{ki} \ket{e_{ki}} \qquad (k \ne 1) , \label{eq:xk_cki} \end{equation} in which $c_{ki} = \inprod{e_{ki}}{x_k}$. The corresponding change $\ket{\Delta x_k}$ is thus \begin{equation} \ket{\Delta x_k} = \sum_{i} \Delta c_{ki} \ket{e_{ki}} , \quad \Delta c_{ki} = \inprod{e_{ki}}{\Delta x_k} . \label{eq:Dxk} \end{equation} Combining this expression with Eq.\ (\ref{eq:Delta_Uk}) then gives \begin{equation} \ket{\Delta u_k} = \sum_{i} \Delta c_{ki} \ket{f_{ki}} , \label{eq:Delta_uk} \end{equation} in which the creator of $\ket{f_{ki}}$ is defined to be \begin{equation} f_{ki} = \{ e_{ki}, U_k \} . \label{eq:fki} \end{equation} For a given value of $k$, the set $\{ \ket{f_{ki}} \}$ is generally not orthonormal, but it is linearly independent if and only if the set $\{ \ket{e_{ki}} \}$ is. [This can be shown easily using Eq.\ (\ref{eq:switch_basis}).] From Eq.\ (\ref{eq:Delta_uk}), we now see that $\ket{\Delta u_k}$ and $\ket{\Delta x_k}$ are related by \begin{equation} \ket{\Delta u_k} = \biggl( \sum_{i} \outprod{f_{ki}}{e_{ki}} \biggr) \ket{\Delta x_k} . \label{eq:DukDxk} \end{equation} A similar expression for $\ket{\Delta v} = \Delta V \ket{0}$ can be derived from Eq.\ (\ref{eq:Delta_V}): \begin{subequations} \label{eq:DvDxk} \begin{align} \ket{\Delta v} & = \ket{v_{\Delta \Psi}} + \sum_{k=2}^{m} \sum_{i} \Delta c_{ki} \ket{g_{ki}} \\ & = \ket{v_{\Delta \Psi}} + \sum_{k=2}^{m} \biggl( \sum_{i} \outprod{g_{ki}}{e_{ki}} \biggr) \ket{\Delta x_k} , \end{align} \end{subequations} in which the creators of $\ket{g_{ki}}$ are defined as \begin{equation} g_{ki} = - \tilde{V}_k [ \{ e_{ki}, U_k^{-1} \} ] . \end{equation}
\subsection{Subsystem geometry}
\label{sec:subsystem_geometry}
The preceding expressions for subsystem differences are useful primarily in the context of \emph{geometric} structures that allow us to measure the \emph{distance} between neighboring subsystem decompositions. Such a metric is essential for both the definition of the time functional in Sec.\ \ref{sec:time_functional} and the variational formulation of dynamical stability in Sec.\ \ref{sec:dynamical_stability}.
There are many ways to define such a distance, but the most suitable measure for the present purposes is the Hilbert--Schmidt distance \cite{Bengtsson2006}. This distance is based on the Hilbert--Schmidt inner product and norm \begin{equation} (A, B) = \frac12 \tr (A^{\dag} B) , \qquad \norm{A} = \sqrt{(A, A)} , \label{eq:HS_inprod} \end{equation} in which $A$ and $B$ are operators. The Hilbert--Schmidt distance $D$ is then defined as \begin{equation} D (A, B) = \norm{A - B} . \label{eq:HS_distance} \end{equation} The operators of interest are the subsystem projectors \begin{equation} \rho_{k} = \frac{\outprod{u_{k}}{u_{k}}}{\inprod{u_{k}}{u_{k}}} , \qquad \rho_{k}' = \frac{\outprod{u_{k}'}{u_{k}'}}{\inprod{u_{k}'}{u_{k}'}} . \end{equation} The square of the Hilbert--Schmidt distance between the subsystem states $\ket{u_k}$ and $\ket{u_k'}$ is thus given by \begin{subequations} \label{eq:D2k} \begin{align} D^2 (\rho_k, \rho_{k}') & = 1 - \tr (\rho_k \rho_{k}') \\ & = \frac{\matelm{u_{k}'}{(1 - \rho_{k})}{u_{k}'}}{\inprod{u_{k}'}{u_{k}'}} . \end{align} \end{subequations} This satisfies $0 \le D^2 (\rho_k, \rho_{k}') \le 1$, which is the reason for introducing the factor of $1/2$ in Eq.\ (\ref{eq:HS_inprod}) \cite{Bengtsson2006}.
Our primary interest is in the value of $D^2 (\rho_k, \rho_{k}')$ for small subsystem differences $\ket{\Delta u_{k}} = \ket{u_{k}'} - \ket{u_{k}}$. Noting that $(1 - \rho_{k}) \ket{u_{k}} = 0$, we can rewrite Eq.\ (\ref{eq:D2k}) as \begin{equation} D^2 (\rho_k, \rho_{k}') = \frac{\matelm{\Delta u_{k}}{(1 - \rho_{k})}{\Delta u_{k}}}{\inprod{u_{k}'}{u_{k}'}} , \end{equation} which shows that $D^2 (\rho_k, \rho_{k}')$ is of order $\norm{\Delta u_{k}}^2 \equiv \inprod{\Delta u_{k}}{\Delta u_{k}}$. Indeed, since $\inprod{u_{k}'}{u_{k}'} = \inprod{u_{k}}{u_{k}} + O (\norm{\Delta u_{k}})$, we have \begin{equation} D^2 (\rho_k, \rho_{k}') = \frac{\matelm{\Delta u_{k}}{(1 - \rho_{k})}{\Delta u_{k}}}{\inprod{u_{k}}{u_{k}}} + O (\norm{\Delta u_{k}}^3) . \end{equation} The leading term in this expression is the familiar Fubini--Study metric \cite{ProvostVallee1980, Wootters1981, Page1987, Berry1989, AnandanAharonov1990, Anandan1991, Bengtsson2006}. The Hilbert--Schmidt distance is only one of several large-scale measures of distance that lead to the Fubini--Study metric in the limit of infinitesimal $\norm{\Delta u_{k}}$ \cite{ProvostVallee1980, Bengtsson2006}, but it is generally the easiest of these to work with.
The next step is to extend this measure of distance to the subsystem decomposition (\ref{eq:psi_pi}) as a whole. The simplest way to do this is to construct a direct sum of the projectors $\rho_k$: \begin{equation} \rho \equiv \bigoplus_{k=1}^{m} \rho_{k} = \rho_1 \oplus \rho_2 \oplus \cdots \oplus \rho_m . \end{equation} The resulting operator $\rho$ is also a projector, since $\rho_k^2 = \rho_k$ implies $\rho^2 = \rho$. The Hilbert--Schmidt distance (\ref{eq:HS_distance}) between $\rho$ and $\rho'$ is then \begin{subequations} \label{eq:HS_total} \begin{align} D^2 (\rho, \rho') & = m - \tr (\rho \rho') \\ & = \sum_{k=1}^{m} D^2 (\rho_k, \rho_{k}') , \end{align} \end{subequations} which is clearly independent of the choice of permutation $\pi$ in Eq.\ (\ref{eq:psi_pi}). In the limit of infinitesimal $\norm{\Delta u_{k}}$, this reduces to \begin{equation} D^2 (\rho, \rho') = \sum_{k=1}^{m} \frac{\matelm{\Delta u_{k}}{(1 - \rho_{k})}{\Delta u_{k}}}{\inprod{u_{k}}{u_{k}}} , \label{eq:D2_rho_FS} \end{equation} which is the Fubini--Study metric for the entire subsystem decomposition.
This result can be expressed more concisely by using a direct-sum representation of vectors. For arbitrary subsystem kets $\ket{\varphi_k}$ and $\ket{\chi_k}$, let their direct sum be denoted by the same symbol without the subscript $k$: \begin{equation} \ket{\varphi} \equiv \bigoplus_{k=1}^{m} \ket{\varphi_k} , \qquad \ket{\chi} \equiv \bigoplus_{k=1}^{m} \ket{\chi_k} . \label{eq:direct_sum_ket} \end{equation} It is convenient also to bury the normalization factor $\inprod{u_{k}}{u_{k}}$ inside the definition of the inner product: \begin{equation} \inprod{\varphi}{\chi} \equiv \sum_{k=1}^{m} \frac{\inprod{\varphi_{k}}{\chi_{k}}}{\inprod{u_{k}}{u_{k}}} . \label{eq:direct_sum_inprod} \end{equation} This allows Eq.\ (\ref{eq:D2_rho_FS}) to be written simply as \begin{equation} \eta \equiv D^2 (\rho, \rho') = \matelm{\Delta u}{(1 - \rho)}{\Delta u} ,
\label{eq:D2_rho_FS_simple} \end{equation} in which the letter $\eta$ is introduced as a concise symbol for this functional of $\rho$ and $\ket{\Delta u}$.
\section{Time as a functional}
\label{sec:time_functional}
With this measure of distance in hand, we can now turn to the topic of subsystem dynamics, beginning with the concept of \emph{time}. This section starts by explaining why, in a system of interacting particles, it is impossible for both $\ket{\psi}$ and the subsystems $\ket{u_k}$ to satisfy the Schr\"odinger equation. Next, it is argued that in a closed system, the external time parameter $t$ of conventional quantum mechanics is meaningless. Instead, time should be defined internally via the relations between changes in subsystems. This is then used to construct a time \emph{functional}, which will be applied to calculations of subsystem dynamics in Sec.\ \ref{sec:dynamical_stability}.
\subsection{Why interacting subsystems cannot satisfy the Schr\"odinger equation}
\label{sec:interacting_not_Schroedinger}
Let us start by examining whether it is possible for a closed system and its subsystems to satisfy the Schr\"odinger equation. It is sufficient for this purpose to study a two-subsystem decomposition $\ket{\psi} = \ket{u} \odot \ket{v} = U \ket{v}$. The Schr\"odinger equation for the total system is \begin{equation} i \partial_{t} \ket{\psi} = H \ket{\psi} = HU \ket{v} , \label{eq:Schr_psi} \end{equation} in which $H$ is the Hamiltonian. But this is the same as \begin{subequations} \begin{align} i \partial_{t} \ket{\psi} & = [H, U] \ket{v} + U H \ket{v} \\ & = [H, U] \ket{v} + \ket{u} \odot (H\ket{v}) . \label{eq:Schr_b} \end{align} \end{subequations} If the operator $[H, U]$ is a creator, then $[H, U] \ket{v} = ([H, U] \ket{0}) \odot \ket{v}$, in which \begin{equation} [H, U] \ket{0} = HU \ket{0} - UH \ket{0} = (H - E_0) \ket{u} , \end{equation} where $E_0$ is the energy of the vacuum. Assuming that $E_0 = 0$ (which is necessary if $\ket{0}$ is to act as a time-independent multiplicative identity), Eq.\ (\ref{eq:Schr_b}) can thus be written as \begin{equation} i \partial_{t} \ket{\psi} = (H \ket{u}) \odot \ket{v} + \ket{u} \odot (H \ket{v}) . \end{equation} A comparison with the differential identity \begin{equation} \partial_{t} \ket{\psi} = (\partial_{t} \ket{u}) \odot \ket{v} + \ket{u} \odot (\partial_{t} \ket{v}) \end{equation} then shows that both $\ket{u}$ and $\ket{v}$ can satisfy the Schr\"odinger equation.
But when is it true that $[H,U]$ is a creator? If $H$ conserves particle number, it can be written as a polynomial (usually quadratic) function of the hopping operators $a_{i}^{\dag} a_{j}$ (which reduce to number operators $N_i = a_{i}^{\dag} a_{i}$ when $i = j$). Now the commutator of a hopping operator and a creation operator is just another creation operator: \begin{equation} [ a_{i}^{\dag} a_{j} , a_{k}^{\dag} ] = \delta_{jk} a_{i}^{\dag} . \label{eq:hop_commutator} \end{equation} Assuming that $H$ is \emph{linear} in the hopping operators---as would be the case for a system of \emph{noninteracting} particles---this shows that the commutator of $H$ with a creator always generates another creator. The subsystems $\ket{u}$ and $\ket{v}$ can therefore satisfy the Schr\"odinger equation in this case.
However, if pairs of particles interact, then $H$ also includes the pair distribution operator \cite{Merzbacher1998} \begin{equation} P_{ij} = N_{i} N_{j} - \delta_{ij} N_{i} , \end{equation} for which \begin{equation} [ P_{ij}, a_{k}^{\dag} ] = \delta_{ik} a_{k}^{\dag} N_{j} + \delta_{jk} a_{k}^{\dag} N_{i} . \end{equation} This contains both creation and annihilation operators. Hence, in a system of interacting particles, $[H,U]$ is not a creator, and it is impossible in general for all of $\ket{\psi}$, $\ket{u}$, and $\ket{v}$ to satisfy the Schr\"odinger equation.
\subsection{Relational time in quantum mechanics}
An extensive literature on the topic of \emph{relational time} in quantum mechanics also casts serious doubt on whether the time parameter $t$ in Eq.\ (\ref{eq:Schr_psi}) can have any meaning in a closed system (see, e.g., Refs.\ \cite{BarbourBertotti1982, PageWootters1983, Wootters1984, UnruhWald1989, Pegg1991, Isham1992, Page1994, Barbour1994a, Barbour1994b, GambiniPortoPullin2004a, GambiniPortoPullin2004b, GambiniPortoPullin2009, Poulin2006, MilburnPoulin2006, AlbrechtIglesias2008, Arce2012, Moreva2014}). The argument is very similar to that already given in Sec.\ \ref{sec:relational_properties}. Namely, within a closed system, one can observe only changes in the relations between various subsystems; one does not have access to any hypothetical absolute external time variable.
Page and Wootters \cite{PageWootters1983, Wootters1984, Page1994} have argued that this gives rise to an effective energy superselection rule in which a coherent superposition of different energy eigenstates is experimentally indistinguishable from a statistical mixture. Page \cite{Page1994}, Poulin \cite{Poulin2006}, and Milburn and Poulin \cite{MilburnPoulin2006} have extended this approach by using group averaging of density operators to eliminate the external time parameter $t$, thereby reducing a general unmixed state to a statistical mixture of energy eigenstates.
A common strategy in this type of approach is to identify one subsystem as a \emph{clock} and measure time via correlations between the clock subsystem and other subsystems. This gives rise to an effective decoherence mechanism if the clock is of finite size \cite{GambiniPortoPullin2004a, GambiniPortoPullin2004b, Poulin2006, MilburnPoulin2006, BartlettRudolphSpekkens2007}.
Barbour \cite{Barbour1994a} has argued that such approaches do not agree with how time is defined operationally. In practice, we define time not by looking at a single clock, but by using the time parameter $t$ to achieve the best fit to all of the experimental information at our disposal. This \emph{ephemeris time} concept was developed long ago by astronomers, but even today it is how time is defined from a network of atomic clocks, all of which operate in different environments and run at slightly different rates. From this perspective, ``ultimately the universe is the only clock'' \cite{Barbour1994a}.
Time is defined in the present paper by using the concept of ephemeris time in the context of the geometric approach to quantum mechanics developed by Anandan and Aharonov \cite{AnandanAharonov1990, Anandan1991}. These authors have alluded to this concept themselves, even going so far as to say that ``The parameter $t$ represents correlation between the Fubini--Study distances determined by different clocks'' \cite{AnandanAharonov1990}.
However, the correlation between different clocks is found only in these words, not in any of their equations. Their equations establish only a relationship between external time and the Fubini--Study distance traveled by a system evolving according to Schr\"odinger's equation \cite{AnandanAharonov1990, Anandan1991}. During a ``measurement,'' however, the system can move a finite distance in the projective Hilbert space during a time interval of zero \cite{AnandanAharonov1988}. It is not clear how these disparate Hilbert-space transport mechanisms are to be reconciled. But this is of course just the old conundrum posed by von Neumann's axioms of time evolution \cite{vonNeumann1955}.
This paper implements Anandan and Aharonov's idea mathematically by introducing a \emph{time functional} that is optimized to achieve the best fit between Schr\"odinger dynamics and the changes that occur in all subsystems. The actual value of these changes is not determined by this functional; that task is left to the principle of dynamical stability, to be discussed below in Sec.\ \ref{sec:dynamical_stability}.
\subsection{Definition of the time functional}
\label{sec:time_functional_defn}
The time functional can be defined using a simple extension of the geometric concepts introduced previously in Sec.\ \ref{sec:subsystem_geometry}. Consider two slightly different subsystem decompositions, $\rho$ and $\rho'$. If $\rho'$ differs from $\rho$ only by a Schr\"odinger time evolution, the two decompositions must be related by \begin{equation} \rho' \overset{?}{=} e^{-i \hat{H} \Delta \tau} \rho e^{i \hat{H} \Delta \tau} \label{eq:Schr_ideal} \end{equation} for some time interval $\Delta \tau$, in which \begin{equation} \hat{H} \equiv \bigoplus_{k=1}^{m} H \label{eq:H_hat} \end{equation} is the Hamiltonian in the direct-sum formalism. Of course, for arbitrary $\rho$ and $\rho'$, Eq.\ (\ref{eq:Schr_ideal}) will not be true, but we can try to get as close as possible to such a description by minimizing the Hilbert--Schmidt distance between the two sides of the equation. In other words, we can define a function \begin{subequations} \label{eq:lambda_defn} \begin{align} \lambda (\Delta \tau) & \equiv D^2 (e^{-i \hat{H} \Delta \tau} \rho e^{i \hat{H} \Delta \tau}, \rho') \\ & = D^2 (\rho, e^{i \hat{H} \Delta \tau} \rho' e^{-i \hat{H} \Delta \tau}) \end{align} \end{subequations} and seek the value of $\Delta \tau$ that minimizes this function. This special value, denoted $\Delta \tau = \Delta t$, provides the best fit between $\rho$ and $\rho'$ that can be expressed in the language of Schr\"odinger dynamics.
Our only concern is the case of infinitesimal differences $\norm{\Delta u_k}$, for which $\Delta t$ is also infinitesimal. We can therefore use the Fubini--Study metric of Eqs.\ (\ref{eq:D2_rho_FS}) and (\ref{eq:D2_rho_FS_simple}) to write \begin{subequations} \begin{align} \lambda (\Delta \tau) & = \matelm{u'}{e^{-i \hat{H} \Delta \tau} (1 - \rho) e^{i \hat{H} \Delta \tau}}{u'} \\ & = \sum_{k=1}^{m} \frac{\matelm{u_{k}'}{e^{-i H \Delta \tau} (1 - \rho_{k}) e^{i H \Delta \tau}}{u_{k}'}}{\inprod{u_{k}}{u_{k}}} . \end{align} \end{subequations} Here the exponentials can be expanded in the usual way:
\begin{multline} e^{-i \hat{H} \Delta \tau} (1 - \rho) e^{i \hat{H} \Delta \tau} = (1 - \rho) -i \Delta \tau [\hat{H}, 1-\rho] \\ - \tfrac12 \Delta \tau^2 [\hat{H}, [\hat{H}, 1 - \rho ]] + \cdots . \end{multline}
If we treat $\Delta \tau$ and $\norm{\Delta u}$ as of the same order and work to second order overall, the final result can be written as \begin{equation} \lambda (\Delta \tau) = \eta - 2 \Delta \tau \imag \inprod{\Delta u}{H} + \Delta \tau^2 \inprod{H}{H} , \label{eq:lambda_quad1} \end{equation} in which $\lambda (0) = \eta$ was already defined in Eq.\ (\ref{eq:D2_rho_FS_simple}). The vector $\ket{H}$ in this expression is defined as \begin{subequations} \label{eq:H_ket_defn} \begin{align} \ket{H} & \equiv (1 - \rho) \hat{H} \ket{u} \\ & = \bigoplus_{k=1}^{m} (1 - \rho_k) H \ket{u_k} . \end{align} \end{subequations} The minimum of the quadratic function (\ref{eq:lambda_quad1}) occurs at $\Delta \tau = \Delta t$, in which \begin{equation} \Delta t = \frac{\imag \inprod{\Delta u}{H}}{\inprod{H}{H}} . \label{eq:Delta_t} \end{equation} This is the desired expression giving the optimal time interval $\Delta t$ as a functional of the subsystem change $\ket{\Delta u}$.
\subsection{Properties of the time functional}
\label{sec:properties_time_functional}
Note that the solution (\ref{eq:Delta_t}) can be used to rewrite Eq.\ (\ref{eq:lambda_quad1}) as \begin{equation} \lambda (\Delta \tau) = \eta + \Delta \tau (\Delta \tau - 2 \Delta t) \inprod{H}{H} . \label{eq:lambda_quad2} \end{equation} When $\Delta \tau = \Delta t$, this function attains its minimum value \begin{subequations} \label{eq:lambda_min} \begin{align} \lambda (\Delta t) & = \eta - \Delta t^2 \inprod{H}{H} \label{eq:lambda_min_a} \\ & = \eta - \frac{(\imag \inprod{\Delta u}{H})^{2}}{\inprod{H}{H}} . \label{eq:lambda_min_b} \end{align} \end{subequations} Given the definition (\ref{eq:lambda_defn}) of $\lambda (\Delta \tau)$ as the square of a distance, it seems obvious that this minimum value must satisfy $\lambda (\Delta t) \ge 0$. However, it is not immediately clear from Eq.\ (\ref{eq:lambda_min}) that this is in fact the case.
To see explicitly that $\lambda (\Delta t)$ is indeed nonnegative, note that \begin{equation} (\imag \inprod{\Delta u}{H})^{2} \le \abs{\inprod{\Delta u}{H}}^{2} . \label{eq:imag_inequality} \end{equation} This inequality in conjunction with Eq.\ (\ref{eq:lambda_min_b}) implies that \begin{subequations} \begin{align} \lambda (\Delta t) & \ge \eta - \frac{\abs{\inprod{\Delta u}{H}}^{2}}{\inprod{H}{H}} \\ & = \matelm{\Delta u}{(1 - \rho - \Pi_{H})}{\Delta u} , \end{align} \end{subequations} in which $\Pi_{H}$ is the projector \begin{equation} \Pi_{H} \equiv \frac{\outprod{H}{H}}{\inprod{H}{H}} . \end{equation} Because $\rho$ and $\Pi_{H}$ are orthogonal, the operator $(1 - \rho - \Pi_{H})$ is also a projector. This can be used to write \begin{equation} \matelm{\Delta u}{(1 - \rho - \Pi_{H})}{\Delta u} = \inprod{w}{w} \ge 0 , \end{equation} in which \begin{equation} \ket{w} \equiv (1 - \rho - \Pi_{H}) \ket{\Delta u} . \end{equation} This proves that $\lambda (\Delta t) \ge 0$, and furthermore that a necessary condition for $\lambda (\Delta t) = 0$ is $\ket{w} = 0$ or \begin{equation} (1 - \rho) \ket{\Delta u} = \Pi_{H} \ket{\Delta u} . \end{equation} However, this condition is not sufficient. Tracing back to the previous inequality (\ref{eq:imag_inequality}), we see that $\real \inprod{\Delta u}{H} = 0$ is also required. Hence, in order to achieve $\lambda (\Delta t) = 0$, it is necessary and sufficient that \begin{equation} (1 - \rho) \ket{\Delta u} = i C \ket{H} , \label{eq:minimum_condition} \end{equation} in which $C$ is a real constant. In other words, the component of $\ket{\Delta u}$ that is orthogonal to $\ket{u}$ must be proportional to $\ket{H}$, with an imaginary coefficient.
What is the significance of this? According to the definition (\ref{eq:H_ket_defn}), the vector $\ket{H}$ is just the component of $\hat{H} \ket{u}$ that is orthogonal to $\ket{u}$. Hence, the condition (\ref{eq:minimum_condition}) says that in order to achieve $\lambda (\Delta t) = 0$, all subsystems must satisfy the Schr\"odinger equation, but only insofar as the component of $\ket{\Delta u}$ orthogonal to $\ket{u}$ is concerned. [The component of $\ket{\Delta u}$ that is parallel to $\ket{u}$ does not contribute to the distance (\ref{eq:D2_rho_FS_simple}).]
Another way of expressing $\ket{H}$ is \begin{equation} \ket{H} = \bigoplus_{k=1}^{m} (H - \expect{H}_k) \ket{u_k} , \end{equation} in which $\expect{H}_k$ is the mean value [cf.\ Eq.\ (\ref{eq:mean_value})] \begin{equation} \expect{H}_k \equiv \frac{\matelm{u_k}{H}{u_k}}{\inprod{u_k}{u_k}} . \end{equation} Hence, the inner product $\inprod{H}{H}$ can be written as \begin{equation} \inprod{H}{H} = \sum_{k=1}^{m} \frac{\matelm{u_k}{(H - \expect{H}_k)^{2}}{u_k}}{\inprod{u_k}{u_k}} . \end{equation} This provides a simple physical interpretation of $\inprod{H}{H}$: it is the combined energy variance of all subsystems. The corresponding standard deviation is denoted $\Delta E \equiv \sqrt{\inprod{H}{H}}$.
The inequality $\lambda (\Delta t) \ge 0$ can thus be written in the alternative form [cf.\ Eq.\ (\ref{eq:lambda_min_a})] \begin{equation} \Delta E^2 \Delta t^2 \le \eta . \label{eq:time_energy} \end{equation} This looks vaguely like a time--energy uncertainty relation, except that the inequality is pointing in the wrong direction---so actually it is nothing of the kind. It simply says that, for a given squared distance $\eta = D^2 (\rho, \rho')$, there is an upper bound on the optimal value of $\Delta \tau$ that can be fitted to $\rho$ and $\rho'$ using Eq.\ (\ref{eq:Schr_ideal}). Furthermore, because the optimal value $\Delta \tau = \Delta t$ is intimately related to Schr\"odinger dynamics, the numerical value of $\Delta t$ depends on the energy scale $\Delta E$ determined by the subsystem decomposition $\rho$.
To conclude this section, we may note that the time functional (\ref{eq:Delta_t}) offers a very simple way of implementing the idea that ``ultimately the universe is the only clock.'' But of course, as mentioned previously, the definition of such a clock tells us nothing about how the subsystems evolve in time. Finding a way to define this time evolution is the subject of the next section.
\section{Dynamical stability of subsystems}
\label{sec:dynamical_stability}
The most obvious criterion for defining subsystem dynamics is to maximize the stability of the subsystem decomposition. In other words, we should choose the dynamics such that the decomposition ``hops about the least'' \cite{[] [{, p.\ 518.}] Fuchs2011}. This concept of \emph{dynamical stability} has a long history. In the early days of quantum mechanics, Schr\"odinger used it in an attempt to interpret particles as stable wave packets \cite{[] [{; English translation in Ref.\ \cite{Schrodinger1982}.}] Schrodinger1926c, [] [{, pp.\ 41--44.}] Schrodinger1982}. More recently, its importance for decoherence theory has been repeatedly emphasized by Zeh \cite{Zeh1970, Zeh1971b, KublerZeh1973, Zeh1973, Zeh1979, JoosZeh1985, Zeh2000, Zeh2003ch2, Zeh2006}, and the basic idea has been developed extensively by Zurek under such names as the predictability sieve \cite{Zurek1993a, ZurekHabibPaz1993}, einselection \cite{Zurek1982, Zurek1998, Zurek2003}, the existential interpretation \cite{Zurek1993a, Zurek1998, Zurek2003}, and quantum Darwinism \cite{Zurek2003, Zurek2014}.
In this section, the concept of dynamical stability is defined for the subsystem decomposition (\ref{eq:psi_pi}) in terms of a dynamical stability functional $\chi$. The time evolution of the subsystems is then determined by maximizing $\chi$. For simplicity, the total system state $\ket{\psi}$ is initially assumed to be independent of time. This analysis is then extended to the case of time-dependent $\ket{\psi}$.
\subsection{Dynamical stability functional}
\label{sec:dynamical_stability_functional}
To simplify the description of dynamical stability, it is convenient to introduce the dimensionless variable \begin{equation} \sigma \equiv \Delta E \Delta t = \frac{\imag \inprod{\Delta u}{H}}{\Delta E} . \label{eq:sigma_defn} \end{equation} The dynamical stability functional $\chi$ is then defined as \begin{equation} \chi \equiv \frac{\sigma^2}{\eta} = \frac{\Delta E^2 \Delta t^2}{D^2 (\rho, \rho')} \qquad (\eta \ne 0) . \label{eq:chi_definition} \end{equation} The foundation for this definition is the inequality (\ref{eq:time_energy}), which says that $0 \le \chi \le 1$. The \emph{principle of dynamical stability} is implemented by holding $\rho$ fixed and varying $\rho'$ so as to maximize the value of $\chi$. The decompositions $\rho$ and $\rho'$ could thus be regarded as ``initial'' and ``final,'' although this has the potential to be misleading because it has nothing to do with the sign of $\Delta t$.
Maximizing $\chi$ with respect to variations in $\rho'$ simply requires that the subsystems change as little as possible (as measured by the Fubini--Study metric) in a given infinitesimal time interval $\Delta t$. According to the results of Sec.\ \ref{sec:interacting_not_Schroedinger}, if $\ket{\psi}$ is assumed to satisfy the Schr\"odinger equation, the upper limit $\chi = 1$ is generally unattainable in a system of interacting particles.
Suppose now that $\rho'$ is varied by a small amount $\delta \rho$. (This should perhaps be written as $\delta \rho'$, but the prime symbol can be omitted because $\rho$ itself is not varied.) This will give rise to corresponding variations $\delta \sigma$, $\delta \eta$, and $\delta \chi$, which are related by \begin{equation} \delta \chi = \frac{(\sigma + \delta \sigma)^2}{\eta + \delta \eta} - \frac{\sigma^2}{\eta} = \frac{2 \sigma \delta \sigma - \chi \delta \eta + \delta \sigma^2}{\eta + \delta \eta} . \end{equation} To first order in small quantities, this reduces to \begin{equation} \delta \chi = \eta^{-1} (2 \sigma \delta \sigma - \chi \delta \eta) , \end{equation} in which all variations are evaluated to first order in $\delta \rho$. The stationary states of the dynamical stability functional are then given by $\delta \chi = 0$ or \begin{equation} \frac{\delta \eta}{\eta} = 2 \frac{\delta \sigma}{\sigma} \qquad (\sigma \ne 0) , \label{eq:eta_sigma_stationary} \end{equation} in which $\sigma \ne 0$ can always be assumed because we have no interest in the minima of $\chi$.
\subsection{Time-independent \texorpdfstring{$\Psi$}{Psi}}
\label{sec:time_independent_psi}
Let us now apply the principle of dynamical stability to the special case in which the total system state $\Psi$ is assumed to be independent of time, so that $\Delta \Psi \equiv \Psi' - \Psi = 0$. This implies that $V_{\Delta \Psi} = 0$ in the general expression (\ref{eq:Delta_V}) for $\Delta V \equiv \Delta U_1$, which simplifies the analysis considerably.
The quantities to be varied are the ``final'' subsystem exponents $\ket{x_k'} = X_k' \ket{0}$ for the quasiclassical subsystems ($k \ne 1$). As before, it is convenient to use a direct-sum representation for the differences $\ket{\Delta x_k} = \ket{x_k'} - \ket{x_k}$: \begin{equation} \ket{\Delta x} = \bigoplus_{k=2}^{m} \ket{\Delta x_k} . \label{eq:Delta_x_sum} \end{equation} In contrast to the definition of $\ket{\Delta u}$ [see Eq.\ (\ref{eq:direct_sum_ket})], the value $k = 1$ is \emph{not} included in the definition of $\ket{\Delta x}$. There are two reasons for this. First, as noted in Sec.\ \ref{sec:subsystem_differences}, only the subsystems with $k \ne 1$ are treated as independently variable; the subsystem $k = 1$ is entirely determined by $\Psi$ and the other subsystems. Second, it is not generally even possible to define $X_1 = \ln U_1$, since $\Psi$ need not be quasiclassical.
The dimensionless time interval (\ref{eq:sigma_defn}) can now be expressed in terms of the \emph{independent} variables $\ket{\Delta x}$ as \begin{equation} \sigma = \frac{\imag \inprod{\Delta u}{H}}{\Delta E} \equiv \imag \inprod{\Delta x}{\sigma} , \label{eq:sigma_def} \end{equation} in which the components of the vector $\ket{\sigma}$ are given by [cf.\ Eqs.\ (\ref{eq:DukDxk}), (\ref{eq:DvDxk}), (\ref{eq:H_ket_defn})]
\begin{multline} \inprod{e_{ki}}{\sigma} = \frac{1}{\Delta E} \biggl[ \frac{\matelm{f_{ki}}{(1 - \rho_k)H}{u_k}}{\inprod{u_k}{u_k}} \\ + \frac{\matelm{g_{ki}}{(1 - \rho_v)H}{v}}{\inprod{v}{v}} \biggr] . \label{eq:sigma_ket} \end{multline}
In this expression, $\rho_v \equiv \rho_1$ and $\ket{v} \equiv \ket{u_1}$. The squared distance $\eta$ can be written likewise as \begin{equation} \eta = \matelm{\Delta x}{\hat{\eta}}{\Delta x} , \label{eq:eta_operator} \end{equation} in which the matrix elements of the operator $\hat{\eta}$ are given by [cf.\ Eq.\ (\ref{eq:D2_rho_FS_simple})]
\begin{multline} \matelm{e_{ki}}{\hat{\eta}}{e_{k'i'}} = \delta_{kk'} \frac{\matelm{f_{ki}}{(1 - \rho_k)}{f_{ki'}}}{\inprod{u_k}{u_k}} \\ + \frac{\matelm{g_{ki}}{(1 - \rho_v)}{g_{k'i'}}}{\inprod{v}{v}} . \label{eq:eta_hat} \end{multline}
The operator $\hat{\eta}$ is clearly positive ($\hat{\eta} \ge 0$), and it can be made positive definite ($\hat{\eta} > 0$) if we agree to exclude the vacuum state \footnote{The vacuum state can be excluded from the subsystem exponents $\ket{x_k}$ because its inclusion has no effect other than to change the normalization of $\ket{u_k}$. If this choice is made, one can readily verify using Eqs.\ (\ref{eq:fki}) and (\ref{eq:switch_basis}) that $\hat{\eta} > 0$.} from the orthonormal basis $\{ \ket{e_{ki}} \}$ used to define the subsystem exponents in Eq.\ (\ref{eq:xk_cki}).
If we now vary $\ket{x'}$ by $\ket{\delta x}$ (holding $\ket{x}$ fixed), $\ket{\Delta x}$ also varies by $\ket{\delta x}$. The resulting first-order variations in $\sigma$ and $\eta$ are \begin{subequations} \begin{align} \delta \sigma & = \imag \inprod{\delta x}{\sigma} = \frac{\inprod{\delta x}{\sigma} - \inprod{\sigma}{\delta x}}{2i} , \\ \delta \eta & = \matelm{\delta x}{\hat{\eta}}{\Delta x} + \matelm{\Delta x}{\hat{\eta}}{\delta x} . \end{align} \end{subequations} Substituting these expressions into the stationary-state condition (\ref{eq:eta_sigma_stationary}) gives \begin{equation} \matelm{\delta x}{\hat{\eta}}{\Delta x} + \matelm{\Delta x}{\hat{\eta}}{\delta x} = -i C (\inprod{\delta x}{\sigma} - \inprod{\sigma}{\delta x}) , \end{equation} in which $C \equiv \eta / \sigma$ is real. Because the variation $\ket{\delta x}$ is arbitrary, we can partition this equation in the usual way \cite[pp.\ 764--765]{Messiah1962} to obtain \begin{equation} \matelm{\delta x}{\hat{\eta}}{\Delta x} = -i C \inprod{\delta x}{\sigma} , \label{eq:eta_sigma_linear} \end{equation} together with its complex conjugate. Removing the arbitrary vector $\bra{\delta x}$ gives the linear algebraic equation \begin{equation} \hat{\eta} \ket{\Delta x} = -i C \ket{\sigma} , \end{equation} in which $\hat{\eta}$ is positive definite and therefore invertible. All stationary states of the dynamical stability functional $\chi$ with $\chi > 0$ are thus given explicitly by \begin{equation} \ket{\Delta x} = -i C \hat{\eta}^{-1} \ket{\sigma} . \label{eq:Dx_soln} \end{equation}
Upon substituting this result back into the definitions of $\sigma$, $\eta$, and $\chi$, we find \begin{subequations} \begin{align} \sigma & = C \matelm{\sigma}{\hat{\eta}^{-1}}{\sigma} , \\ \eta & = C^2 \matelm{\sigma}{\hat{\eta}^{-1}}{\sigma} , \\ \chi & = \matelm{\sigma}{\hat{\eta}^{-1}}{\sigma} . \label{eq:chi_solution} \end{align} \end{subequations} The only degree of freedom in the solution (\ref{eq:Dx_soln}) is the value of the real constant $C = \eta / \sigma = \sigma / \chi$. Since $\chi$ is independent of $C$, one can find the value of $C$ from a given time interval $\Delta t$ simply by calculating $C = \Delta E \Delta t / \chi$. The sign of $\Delta t$ can be positive or negative, but its magnitude should always be chosen small enough that $\eta \ll 1$ (or else the approximations used in deriving the basic equations are no longer valid).
Thus, for a given sign and magnitude of $\Delta t$, there is only \emph{one} stationary state of the dynamical stability functional with $\chi > 0$. This strongly suggests that this stationary state is the unique global maximum of $\chi$. A proof of this conjecture is given in Appendix \ref{app:chi_maximum}.
The solution (\ref{eq:Dx_soln}) can be viewed as a differential equation for $\ket{x}$, since \begin{equation} \frac{\partial \ket{x}}{\partial t} = \lim_{\Delta t \to 0} \frac{\ket{\Delta x}}{\Delta t} = -i \Delta E \frac{\hat{\eta}^{-1} \ket{\sigma}}{\matelm{\sigma}{\hat{\eta}^{-1}}{\sigma}} , \label{eq:dxdt} \end{equation} in which the limit $\Delta t \to 0$ is somewhat redundant because it has been assumed throughout the derivation. This equation can be integrated to obtain $\ket{x}$ as a function of $t$; in practice, this is done by using Eq.\ (\ref{eq:Dx_soln}) repeatedly for small but finite intervals $\Delta t$. Numerical calculations on simple models (see Sec.\ \ref{sec:model_calculations}) show that the change in $\ket{x}$ over a fixed time interval $T$ does indeed converge (quadratically) to a definite value in the limit $\Delta t \to 0$.
The time evolution generated by Eq.\ (\ref{eq:dxdt}) is deterministic. That is, the final subsystem decomposition is uniquely determined by the initial one, and if the subsystems are propagated forward and backward over a finite interval (by changing the sign of $\Delta t$ at the far end), the returning solution always converges to its initial value. Hence, even though the differential equation (\ref{eq:dxdt}) is nonlinear, it does not exhibit any of the lack of determinism so characteristic of standard quantum mechanics.
\subsection{Time-dependent \texorpdfstring{$\Psi$}{Psi}}
\label{sec:time_dependent_psi}
The next step is to lift the restriction $\Delta \Psi = 0$ that was imposed in Sec.\ \ref{sec:time_independent_psi}. Because the total system is closed, $\ket{\Delta \psi}$ is assumed to follow the Schr\"odinger equation (to first order in the time functional $\Delta t$): \begin{equation} \ket{\Delta \psi} = -i \Delta t H \ket{\psi} = -i \Delta t H \Psi \ket{0} . \label{eq:Delta_psi_Schr} \end{equation} This does not imply that $\Delta \Psi = -i \Delta t H \Psi$, because $H \Psi$ is not a creator. Rather, we have \begin{equation} \Delta \Psi = -i \Delta t (H \cdot \Psi) , \label{eq:Delta_Psi_Schr} \end{equation} in which $H \cdot \Psi$ denotes the creator defined by $(H \cdot \Psi) \ket{0} \equiv H \ket{\psi}$. Upon substituting this result into Eq.\ (\ref{eq:DvDxk}), we obtain \begin{equation} \ket{\Delta v} = -i \Delta t \ket{v_{H \cdot \Psi}} + \sum_{k=2}^{m} \biggl( \sum_{i} \outprod{g_{ki}}{e_{ki}} \biggr) \ket{\Delta x_k} . \label{eq:Delta_v_mod} \end{equation} The dimensionless interval $\sigma$ then takes the form \begin{align} \sigma & = \Delta E \Delta t = \frac{\imag \inprod{\Delta u}{H}}{\Delta E} \nonumber \\ & = \imag \inprod{\Delta x}{\sigma_0} + \frac{\Delta t}{\Delta E} \real \biggl[ \frac{\matelm{v_{H \cdot \Psi}}{(1 - \rho_v)H}{v}}{\inprod{v}{v}} \biggr] , \end{align} in which $\ket{\sigma_0}$ relabels the vector introduced previously in Eq.\ (\ref{eq:sigma_ket}). Noting that $\Delta t$ now appears on both sides of the equation, we can combine these terms to obtain \begin{equation} \omega \Delta E \Delta t = \imag \inprod{\Delta x}{\sigma_0} , \end{equation} in which $\omega$ is the real constant \begin{equation} \omega \equiv 1 - \frac{1}{\Delta E^2} \real \biggl[ \frac{\matelm{v_{H \cdot \Psi}}{(1 - \rho_v)H}{v}}{\inprod{v}{v}} \biggr] . \label{eq:omega_E} \end{equation} At this point it is convenient to redefine the vector $\ket{\sigma}$ so as to obtain the same outward appearance as Eq.\ (\ref{eq:sigma_def}): \begin{equation} \sigma = \Delta E \Delta t = \frac{\imag \inprod{\Delta x}{\sigma_0}}{\omega} \equiv \imag \inprod{\Delta x}{\sigma} , \label{eq:sigma_redef} \end{equation} in which $\ket{\sigma}$ is just a renormalized version of Eq.\ (\ref{eq:sigma_ket}):
\begin{multline} \inprod{e_{ki}}{\sigma} = \frac{1}{\omega \Delta E} \biggl[ \frac{\matelm{f_{ki}}{(1 - \rho_k)H}{u_k}}{\inprod{u_k}{u_k}} \\ + \frac{\matelm{g_{ki}}{(1 - \rho_v)H}{v}}{\inprod{v}{v}} \biggr] . \label{eq:sigma_ket_redef} \end{multline}
Hence, the time evolution of $\Psi$ affects the variable $\sigma$ only through the renormalization factor $\omega$.
However, its effect on $\eta$ is more profound. Because $\ket{\Delta v}$ is now linear in $\Delta t$, $\eta$ in Eq.\ (\ref{eq:D2_rho_FS_simple}) becomes a quadratic function of $\Delta t$. When expressed in terms of $\sigma$, this quadratic dependence takes the form \begin{equation} \eta (\Delta t) = \eta_0 + 2 \beta \sigma + \kappa \sigma^2 , \label{eq:eta_Delta_t} \end{equation} in which $\eta (\Delta t) = D^2 (\rho, \rho') = \matelm{\Delta u}{(1 - \rho)}{\Delta u}$ is the function defined in Eq.\ (\ref{eq:D2_rho_FS_simple}) and $\eta_0 \equiv \eta(0) = \matelm{\Delta x}{\hat{\eta}}{\Delta x}$ relabels the quantity defined earlier in Eq.\ (\ref{eq:eta_operator}). Although $\eta$ now depends on $\Delta t$, one should note carefully that $\eta (\Delta t) \ne \lambda (\Delta t)$ [see Eq.\ (\ref{eq:lambda_min})].
The new functional $\beta$ in Eq.\ (\ref{eq:eta_Delta_t}) is defined by \begin{equation} \beta = \imag \inprod{\Delta x}{\beta} , \end{equation} in which $\ket{\beta}$ is the vector \begin{equation} \inprod{e_{ki}}{\beta} = \frac{1}{\Delta E} \frac{\matelm{g_{ki}}{(1 - \rho_v)}{v_{H \cdot \Psi}}}{\inprod{v}{v}} . \label{eq:beta_ket_E} \end{equation} The dimensionless constant $\kappa$ in Eq.\ (\ref{eq:eta_Delta_t}) is defined by \begin{equation} \kappa = \frac{1}{\Delta E^2} \frac{\matelm{v_{H \cdot \Psi}}{(1 - \rho_v)}{v_{H \cdot \Psi}}}{\inprod{v}{v}} . \label{eq:kappa_E} \end{equation} With these results, we can now construct the dynamical stability functional $\chi = \sigma^2 / \eta$ just as before, in which $\eta \equiv \eta (\Delta t)$.
\subsection{Real matrix representation}
\label{sec:real_matrix}
There are several ways to solve the variation problem for $\chi$. The method used here has the advantage of requiring very little modification when the definition of distance is changed below in Sec.\ \ref{sec:superselection}.
When $\eta$ in Eq.\ (\ref{eq:eta_Delta_t}) is expressed as a function of the expansion coefficients $\Delta c_{ki} = \inprod{e_{ki}}{\Delta x}$ introduced in Eq.\ (\ref{eq:Dxk}), the result can be written as \begin{equation} \eta = \mu + \nu , \end{equation} in which $\mu$ has the same form as $\eta_0 = \matelm{\Delta x}{\hat{\eta}}{\Delta x}$: \begin{equation} \mu = \sum_{ki} \sum_{k' i'} \Delta c_{ki}^{*} \Delta c_{k'i'} \mu_{ki,k'i'} . \label{mu_def} \end{equation} However, $\nu$ is qualitatively different: \begin{equation} \nu = \real \left( \sum_{ki} \sum_{k' i'} \Delta c_{ki}^{*} \Delta c_{k'i'}^{*} \nu_{ki,k'i'} \right) . \label{nu_def} \end{equation} Here the matrix elements $\mu_{ki,k'i'}$ and $\nu_{ki,k'i'}$ are given by
\begin{multline} \mu_{ki,k'i'} = \delta_{kk'} \frac{\matelm{f_{ki}}{(1 - \rho_k)}{f_{ki'}}}{\inprod{u_k}{u_k}} + \frac{\matelm{g_{ki}}{(1 - \rho_v)}{g_{k'i'}}}{\inprod{v}{v}} \\ + \frac{1}{2} (\beta_{ki} \sigma_{k'i'}^{*} + \sigma_{ki} \beta_{k'i'}^{*} + \kappa \sigma_{ki} \sigma_{k'i'}^{*}) \label{eq:mu_matrix} \end{multline}
and \begin{equation} \nu_{ki,k'i'} = - \frac{1}{2} (\beta_{ki} \sigma_{k'i'} + \sigma_{ki} \beta_{k'i'} + \kappa \sigma_{ki} \sigma_{k'i'}) , \label{eq:nu_matrix} \end{equation} in which $\beta_{ki} = \inprod{e_{ki}}{\beta}$ and $\sigma_{ki} = \inprod{e_{ki}}{\sigma}$. Because $\nu$, unlike $\mu$ and $\eta_0$, is not a sesquilinear form, the stationary-state equation (\ref{eq:eta_sigma_stationary}) no longer reduces to a linear algebraic equation for the complex coefficients $\Delta c_{ki}$.
However, one can put Eq.\ (\ref{eq:eta_sigma_stationary}) into the form of a linear algebraic equation simply by separating \begin{equation} \Delta c_{ki} = \Delta c_{ki}' + i \Delta c_{ki}'' , \end{equation} and working with the real variables $\Delta c_{ki}'$ and $\Delta c_{ki}''$. If the real and imaginary parts of $\mu_{ki,k'i'}$ and $\nu_{ki,k'i'}$ are likewise separated, $\eta$ can be expressed in block matrix notation as \begin{equation} \eta = \begin{pmatrix} \Delta c' & \Delta c'' \end{pmatrix} \begin{pmatrix} \mu' + \nu' & -\mu'' + \nu'' \\ \mu'' + \nu'' & \mu' - \nu' \end{pmatrix} \begin{pmatrix} \Delta c' \\ \Delta c'' \end{pmatrix} , \end{equation} in which all matrix elements are real.
It is convenient to write this equation in a quasi-Dirac notation: \begin{equation} \eta = \pmatelm{\Delta x}{\tilde{\eta}}{\Delta x} , \end{equation} in which the rounded ket vector is represented by the real column matrix \begin{equation} \pket{\Delta x} = \begin{pmatrix} \Delta c' \\ \Delta c'' \end{pmatrix} , \end{equation} and the operator $\tilde{\eta}$ is represented by the real symmetric matrix \begin{equation} \tilde{\eta} = \begin{pmatrix} \mu' + \nu' & -\mu'' + \nu'' \\ \mu'' + \nu'' & \mu' - \nu' \end{pmatrix} . \label{eq:eta_tilde} \end{equation} This matrix is symmetric because $\mu'$, $\nu'$, and $\nu''$ are symmetric, whereas $\mu''$ is antisymmetric.
A similar representation can be introduced for the dimensionless time interval \begin{equation} \sigma = \imag \inprod{\Delta x}{\sigma} = \imag \left( \sum_{ki} \Delta c_{ki}^{*} \sigma_{ki} \right) , \end{equation} if we separate $\sigma_{ki} = \sigma_{ki}' + i \sigma_{ki}''$ just as for $\Delta c_{ki}$. This is written in quasi-Dirac notation as \begin{equation} \sigma = \pinprod{\Delta x}{\sigma} = \pinprod{\sigma}{\Delta x} , \end{equation} in which the matrix representation for $\pket{\sigma}$ is \begin{equation} \pket{\sigma} = \begin{pmatrix} \sigma'' \\ -\sigma' \end{pmatrix} . \label{eq:pket_sigma} \end{equation}
\subsection{Dynamically stable subsystem changes}
\label{sec:solve_dynamical_stability}
The dynamical stability functional $\chi$ now has a form very similar to that found in Sec.\ \ref{sec:time_independent_psi}: \begin{equation} \chi = \frac{\sigma^2}{\eta} = \frac{\pinprod{\Delta x}{\sigma}^2}{\pmatelm{\Delta x}{\tilde{\eta}}{\Delta x}} . \end{equation} When $\pket{\Delta x}$ is varied by $\pket{\delta x}$, the stationary states are determined by $\delta \chi = 0$ or [cf.\ Eq.\ (\ref{eq:eta_sigma_linear})] \begin{equation} \pmatelm{\delta x}{\tilde{\eta}}{\Delta x} = C \pinprod{\delta x}{\sigma} , \label{eq:eta_sigma_linear_real} \end{equation} in which $C \equiv \sigma / \chi = \eta / \sigma$. Because $\pket{\delta x}$ can range over the whole vector space, this is equivalent to \begin{equation} \tilde{\eta} \pket{\Delta x} = C \pket{\sigma} , \label{eq:eta_sigma_linear_real_ket} \end{equation} which can be solved as before to obtain the dynamically stable subsystem change \begin{equation} \pket{\Delta x} = C \tilde{\eta}^{-1} \pket{\sigma} . \label{eq:Delta_x_real} \end{equation} Note that the factor of $-i$ in Eq.\ (\ref{eq:Dx_soln}) is absent here because it is embedded into the definition (\ref{eq:pket_sigma}) of $\pket{\sigma}$.
The qualitative properties of this solution are again very similar to the solution (\ref{eq:Dx_soln}) found in Sec.\ \ref{sec:time_independent_psi}. In particular, the time evolution generated by Eq.\ (\ref{eq:Delta_x_real}) remains deterministic.
If $\Psi$ happens to be an energy eigenstate with energy $E$, we have $H \cdot \Psi = E \Psi$ and thus \begin{equation} \ket{v_{H \cdot \Psi}} = E \ket{v_{\Psi}} = E \ket{v} . \end{equation} In this case \begin{equation} (1 - \rho_v) \ket{v_{H \cdot \Psi}} = E (1 - \rho_v) \ket{v} = 0 , \end{equation} which implies that $\omega = 1$, $\ket{\beta} = 0$, and $\kappa = 0$. Consequently $\sigma$ and $\eta$ are exactly the same as when $\Delta \Psi = 0$, and there is no difference between the present results and those of Sec.\ \ref{sec:time_independent_psi}. This is reassuring because it is precisely what we would expect when time evolution does not change the ray that $\ket{\psi}$ belongs to.
\subsection{Model calculations and special cases}
\label{sec:model_calculations}
As a tool for developing insight, it is helpful to run some numerical calculations on simple models and see how well the general principles of the theory hold up in practice. The model used here was the extended Hubbard model \cite{[] [{, pp.\ 22 and 403.}] Mahan2000} for small one-dimensional lattices of interacting fermions. Tests were run on both spinless fermions (with nearest-neighbor interactions) and spin $1/2$ fermions (with on-site and nearest-neighbor interactions). For fermions, the algebra of the $\psi$ product can easily be implemented using bitwise operations in the binary representation of Eq.\ (\ref{eq:binary_notation}).
As noted already at the end of Sec.\ \ref{sec:time_independent_psi}, convergence tests of evolution over finite time intervals show that the subsystem dynamics is indeed deterministic, with the solutions converging quadratically in $\Delta t$. This remains true for the case of time-dependent $\Psi$.
An interesting test case is obtained by setting all terms derived from $\ket{v}$ equal to zero in Eqs.\ (\ref{eq:eta_hat}), (\ref{eq:omega_E}), (\ref{eq:sigma_ket_redef}), (\ref{eq:beta_ket_E}), (\ref{eq:kappa_E}), (\ref{eq:mu_matrix}), and (\ref{eq:nu_matrix}). This eliminates all constraints on $\Delta \Psi$, thereby converting the constrained variation problem to an unconstrained variation. With no constraints on the quasiclassical subsystems, one would expect the solutions of Eq.\ (\ref{eq:Delta_x_real}) to have the absolute maximum value of $\chi = 1$, corresponding to the limiting case of Schr\"odinger dynamics for all subsystems [cf.\ Eq.\ (\ref{eq:minimum_condition})]. This is precisely what happens.
A similar result is obtained if one keeps all terms derived from $\ket{v}$ but sets the particle interaction potential to zero. This again yields $\chi = 1$ and Schr\"odinger dynamics for all subsystems. As noted by Wiseman \cite{Wiseman2004}, such a limit is physically uninteresting because it turns each particle into an isolated universe having no connection with anything else. However, it is a crucial test for the logical coherence of the theory, in that it establishes the consistency of assuming Schr\"odinger dynamics for the state $\ket{\psi}$ of a closed system.
\subsection{The number of subsystems is dynamically essential}
\label{sec:arbitrary_number}
The number of subsystems $m$ has so far been treated as an arbitrary parameter. But what is the significance of this number? Does it play an active part in determining the subsystem dynamics, or is its role more passive?
This question addresses the distinction between \emph{trivial} and \emph{nontrivial} subsystems. A trivial subsystem is one that remains in a pure vacuum state ($\ket{u_k} = \ket{0}$) as time evolves. A trivial subsystem has no observable properties (see Sec.\ \ref{sec:observables}), so it makes no difference whether it is included in the subsystem decomposition. The value of $\chi$ is also unchanged by the addition of trivial subsystems. If trivial subsystems are allowed by the principle of dynamical stability, then the number $m$ plays no essential role in the dynamics, because one can add trivial subsystems to any given subsystem decomposition without changing any observable property.
However, vacuum subsystems are \emph{not} dynamically stable in systems of interacting particles. All quasiclassical subsystems, including vacuum subsystems, are coupled to each other by the terms derived from $\ket{v}$ in the matrix (\ref{eq:eta_hat}). A vacuum subsystem satisfies the time-dependent Schr\"odinger equation, and we know already from Secs.\ \ref{sec:interacting_not_Schroedinger} and \ref{sec:dynamical_stability_functional} that such a time dependence is not dynamically stable in a system of interacting particles. Because $\chi < 1$ (see Sec.\ \ref{sec:dynamical_stability_functional}), there is always room to increase $\chi$ by allowing an initial vacuum subsystem to evolve into a nonvacuum final subsystem. Hence, in a system of interacting particles, the number $m$ plays an essential role in the subsystem dynamics, because there are no trivial subsystems.
On the other hand, for noninteracting particles, dynamically stable subsystem decompositions always have $\chi = 1$. The most general such decomposition can be obtained by choosing an independent solution of the Schr\"odinger equation for each subsystem. Because the vacuum state satisfies the Schr\"odinger equation, trivial subsystems are dynamically stable. Hence, for noninteracting particles, the number $m$ need not be the same as the number of nontrivial subsystems. (However, this case is physically uninteresting, as noted in Sec.\ \ref{sec:model_calculations}.)
In conclusion, for the physically interesting case of interacting particles, the number of subsystems $m$ is an essential determining factor for the subsystem dynamics. The value of $m$ is arbitrary, but some number must be chosen in order to apply the principle of dynamical stability.
\section{Reference frames and superselection rules}
\label{sec:superselection}
Thus far, we have seen no sign of any deviation from strict determinism in the subsystem dynamics. This result seems to be in tension with the lack of determinism exhibited by ordinary quantum mechanics. However, up to this point it has also been assumed that there are in principle no restrictions on observable quantities. It is therefore of interest to consider the effect of restrictions arising from the lack of an external reference frame, which were discussed briefly in Sec.\ \ref{sec:relational_properties}. In standard quantum mechanics, it is well known that the lack of a reference frame generally gives rise to a superselection rule \cite{BartlettRudolphSpekkens2007} together with associated classical variables.
\subsection{Lack of phase reference}
\label{sec:phase_reference}
Rather than discussing reference frames in general, this paper focuses on a particular example relevant to nonrelativistic fermions, namely the number superselection rule arising from the lack of a phase reference \cite{BartlettRudolphSpekkens2007}. Lack of a phase reference simply means that the phase transformation \begin{equation} \ket{\psi} \to e^{i N \phi} \ket{\psi} \label{eq:phase_psi} \end{equation} has no observable consequences, where $N$ is the operator for the total number of particles and $\phi$ is any real number. If particle number is conserved ($[H, N] = 0$), then this symmetry is maintained over time, since \begin{equation} e^{-i H t} \ket{\psi} = e^{-i N \phi} e^{-i H t} e^{i N \phi} \ket{\psi} . \end{equation} Of course, in the present theory, observables are associated with the subsystems rather than $\ket{\psi}$ (Sec.\ \ref{sec:observables}). To see the effect on the subsystems, note that the transformation (\ref{eq:phase_psi}) is equivalent to \begin{equation} \Psi \to e^{i N \phi} \Psi e^{-i N \phi} , \label{eq:phase_Psi} \end{equation} because $N \ket{0} = 0$. But this is equivalent to applying the same phase transformation to every subsystem: \begin{equation} U_k \to e^{i N \phi} U_k e^{-i N \phi} \qquad (k = 1, 2, \ldots, m) . \label{eq:phase_Uk} \end{equation}
A crucial difference between this phase shift and the Schr\"odinger dynamics problem studied in Sec.\ \ref{sec:interacting_not_Schroedinger} is that both sides of the mapping (\ref{eq:phase_Uk}) are creators [see Eq.\ (\ref{eq:hop_commutator})]. Hence, a phase shift applied to the total state $\Psi$ propagates directly to all of the subsystems. This is analogous to Lubkin's description of superselection rules in standard quantum mechanics \cite{Lubkin1970}.
\subsection{Equivalence classes of subsystem decompositions}
In the direct-sum formalism, applying the phase shift $\ket{u_k} \to e^{i N \phi} \ket{u_k}$ to all subsystems $k$ is the same as applying the phase shift $\ket{u} \to e^{i \hat{N} \phi} \ket{u}$ to the direct sum of subsystems, in which [cf.\ Eq.\ (\ref{eq:H_hat})] \begin{equation} \hat{N} \equiv \bigoplus_{k=1}^{m} N . \end{equation} If this phase shift has no observable consequences, the relevant mathematical object is not the individual subsystem decomposition $\ket{u}$ but rather the \emph{equivalence class} \begin{equation} [u] \equiv \{ \exp (i \hat{N} \phi) \ket{u} : 0 \le \phi < 2\pi \} . \end{equation} The corresponding equivalence class for a subsystem projector $\rho$ is \begin{equation} [\rho] \equiv \{ \exp (i \hat{N} \phi) \rho \exp (-i \hat{N} \phi) : 0 \le \phi < 2\pi \} . \label{eq:rho_phase_orbit} \end{equation} Such equivalence classes are also referred to as \emph{phase orbits} \footnote{The name ``orbit'' is commonly used in this context; see, e.g., Refs.\ \cite{Bengtsson2006} and \cite{Isham1999}.}. In a system without a phase reference, all of the preceding theory must be reformulated in terms of orbits rather than individual subsystem decompositions.
\subsection{Distance between phase orbits}
\label{sec:phase_orbit_distance}
The first step is to define a suitable measure of distance between phase orbits. The distance between $[\rho]$ and $[\rho']$ can be defined simply as the minimum distance \footnote{It is interesting to note that the Fubini--Study metric can also be derived from such a minimum principle \cite{ProvostVallee1980}.} between any two elements of these orbits: \begin{equation} D^2 ([\rho], [\rho']) \equiv \min_{\theta, \phi} D^2 (e^{i \hat{N} \theta} \rho e^{-i \hat{N} \theta}, e^{i \hat{N} \phi} \rho' e^{-i \hat{N} \phi}) . \end{equation} One of these phase shifts is redundant, so we can write this definition more simply as \begin{equation} D^2 ([\rho], [\rho']) = \min_{\phi} \lambda (\phi) , \label{eq:D2_lambda} \end{equation} in which \begin{equation} \lambda (\phi) = D^2 (\rho, e^{i \hat{N} \phi} \rho' e^{-i \hat{N} \phi}) . \label{eq:lambda_phi} \end{equation} Here it is worthwhile to pause and note the similarity between $\lambda (\phi)$ and the function $\lambda (\Delta \tau)$ introduced previously in Eq.\ (\ref{eq:lambda_defn}). This similarity means that much of the following derivation will be almost identical to that given in Sec.\ \ref{sec:time_functional_defn}. Consequently, only a brief outline of the results is presented.
For small changes $\norm{\Delta u}$, Eq.\ (\ref{eq:lambda_phi}) reduces to \begin{equation} \lambda (\phi) = \matelm{u'}{e^{-i \hat{N} \phi} (1 - \rho) e^{i \hat{N} \phi}}{u'} . \end{equation} After expanding the right-hand side to second order in small quantities, we obtain the quadratic function \begin{equation} \lambda (\phi) = \eta_0 - 2 \phi \imag \inprod{\Delta u}{N} + \phi^2 \inprod{N}{N} , \label{eq:lambda_phi_quad} \end{equation} in which $\eta_0 = \lambda (0)$ and $\ket{N} \equiv (1 - \rho) \hat{N} \ket{u}$. The function (\ref{eq:lambda_phi_quad}) has a minimum at $\phi = \varphi$, in which \begin{equation} \varphi = \frac{\inprod{\Delta u}{N}}{\inprod{N}{N}} . \end{equation} The value of $\lambda (\phi)$ at the minimum is \begin{equation} \lambda (\varphi) = \eta_0 - \frac{(\imag \inprod{\Delta u}{N})^{2}}{\inprod{N}{N}} . \label{eq:lambda_phi_min} \end{equation} But this minimum value is just the desired distance (\ref{eq:D2_lambda}) between the two orbits: \begin{equation} D^2 ([\rho], [\rho']) = \matelm{\Delta u}{(1 - \rho)}{\Delta u} - \frac{(\imag \inprod{\Delta u}{N})^{2}}{\inprod{N}{N}} . \label{eq:D2_phase_orbit} \end{equation}
As before, the symbol $\eta$ is used to refer to the square of the basic measure of distance: \begin{equation} \eta \equiv D^2 ([\rho], [\rho']) = \lambda (\varphi) . \end{equation} It is convenient to write this more concisely as \begin{equation} \eta = \eta_0 - \xi^2 , \end{equation} in which $\xi$ is the functional \begin{equation} \xi \equiv \frac{\imag \inprod{\Delta u}{N}}{\Delta N} = \imag \inprod{\Delta x}{\xi} , \quad \Delta N \equiv \sqrt{\inprod{N}{N}} , \label{eq:xi_def} \end{equation} and $\ket{\xi}$ is the vector
\begin{multline} \inprod{e_{ki}}{\xi} = \frac{1}{\Delta N} \biggl[ \frac{\matelm{f_{ki}}{(1 - \rho_k)N}{u_k}}{\inprod{u_k}{u_k}} \\ + \frac{\matelm{g_{ki}}{(1 - \rho_v)N}{v}}{\inprod{v}{v}} \biggr] . \label{eq:xi_ket} \end{multline}
Again, it is worth noting the close similarity between these quantities and those defined in Secs.\ \ref{sec:time_functional} and \ref{sec:dynamical_stability}.
Sometimes it is necessary to calculate $D^2 ([\rho], [\rho'])$ in situations where $\norm{\Delta u}$ is not small. (See, for example, the last paragraph in Sec.\ \ref{sec:dynamical_stability_phase_orbits}.) This case is considered in Appendix \ref{app:phase_orbit_distance}.
\subsection{Time functional for phase orbits}
A time functional suitable for phase orbits can now be derived by minimizing the function [cf.\ Eqs.\ (\ref{eq:lambda_defn}), (\ref{eq:lambda_phi})] \begin{equation} \lambda (\phi, \Delta \tau) \equiv D^2 (\rho, e^{i \hat{N} \phi} e^{i \hat{H} \Delta \tau} \rho' e^{-i \hat{H} \Delta \tau} e^{-i \hat{N} \phi}) \end{equation} with respect to both $\phi$ and $\Delta \tau$. Given that $[H, N] = 0$, this reduces in the case of small $\norm{\Delta u}$ to \begin{equation} \lambda (\phi, \Delta \tau) = \matelm{u'}{e^{-i \hat{A}} (1 - \rho) e^{i \hat{A}}}{u'} , \label{eq:lambda_A} \end{equation} in which the operator $\hat{A}$ is defined by \begin{equation} \hat{A} (\phi, \Delta \tau) \equiv \hat{H} \Delta \tau + \hat{N} \phi . \end{equation} When $\norm{\Delta u}$ is small, $\phi$ and $\Delta \tau$ can also be treated as small quantities of the same order, and Eq.\ (\ref{eq:lambda_A}) can be expanded as usual to obtain the quadratic approximation \begin{equation} \lambda (\phi, \Delta \tau) = \eta_0 - 2 \imag \inprod{\Delta u}{A} + \inprod{A}{A} , \end{equation} in which $\eta_0 = \lambda (0, 0)$ and $\ket{A} = (1 - \rho) \hat{A} \ket{u}$.
The minimum of this function occurs at $(\phi, \Delta \tau) = (\varphi, \Delta t)$, in which $\varphi$ and $\Delta t$ satisfy the system of equations \begin{equation} \begin{pmatrix} \inprod{N}{N} & \inprod{N}{H} \\ \inprod{H}{N} & \inprod{H}{H} \end{pmatrix} \begin{pmatrix} \varphi \\ \Delta t \end{pmatrix} = \begin{pmatrix} \imag \inprod{\Delta u}{N} \\ \imag \inprod{\Delta u}{H} \end{pmatrix} . \end{equation} The matrix on the left is real and symmetric, because $[H, N] = 0$ implies $\inprod{N}{H} = \inprod{H}{N}$. Upon inverting this matrix, we find the desired time functional \begin{equation} \Delta t = \frac{\inprod{N}{N} \imag \inprod{\Delta u}{H} - \inprod{H}{N} \imag \inprod{\Delta u}{N}}{\inprod{N}{N} \inprod{H}{H} - \inprod{N}{H} \inprod{H}{N}} . \label{eq:Delta_t_messy} \end{equation} This solution is well defined as long as $\inprod{N}{N} > 0$, $\inprod{H}{H} > 0$, and (by the Schwarz inequality) $\ket{H} \ne c \ket{N}$, where $c$ is any constant. If $\ket{H} = c \ket{N}$, this simply means that Schr\"odinger dynamics cannot move the subsystems out of the initial phase orbit, so $\Delta t$ is ill defined (at least to first order in small quantities).
Equation (\ref{eq:Delta_t_messy}) looks considerably more complicated than the previous functional (\ref{eq:Delta_t}), but it can be simplified by introducing the operator \begin{equation} K \equiv H - \frac{\inprod{H}{N}}{\inprod{N}{N}} N . \end{equation} Here $K$ is just the component of $H$ that is orthogonal to $N$, in the sense that $\inprod{K}{N} = 0$. In terms of $K$, the time functional (\ref{eq:Delta_t_messy}) is simply \begin{equation} \Delta t = \frac{\imag \inprod{\Delta u}{K}}{\inprod{K}{K}} . \label{eq:Delta_t_K} \end{equation} The denominator satisfies $\inprod{K}{K} \le \inprod{H}{H}$, equality occurring if and only if $\inprod{H}{N} = 0$. In geometric terms, $\Delta K \equiv \sqrt{\inprod{K}{K}}$ is the length of $\ket{K}$, which is the component of $\ket{H}$ orthogonal to $\ket{N}$. Physically, $\Delta K$ is a renormalized energy uncertainty, just what remains of $\Delta E$ after the unphysical part of $H$ is removed (``unphysical'' because it has no observable consequences for this particular $\ket{u}$).
The phase angle $\varphi$ at the minimum of $\lambda (\phi, \Delta \tau)$ can be written likewise as \begin{equation} \varphi = \frac{\imag \inprod{\Delta u}{L}}{\inprod{L}{L}} , \qquad L \equiv N - \frac{\inprod{N}{H}}{\inprod{H}{H}} H . \end{equation} The value of $\lambda (\phi, \Delta \tau)$ at the minimum is
\begin{multline} \lambda (\varphi, \Delta t) = \matelm{\Delta u}{(1 - \rho)}{\Delta u} \\ - \frac{(\imag \inprod{\Delta u}{N})^{2}}{\inprod{N}{N}} - \frac{(\imag \inprod{\Delta u}{K})^{2}}{\inprod{K}{K}} , \end{multline}
which is similar to Eqs.\ (\ref{eq:lambda_min}) and (\ref{eq:lambda_phi_min}). This can also be written as \begin{equation} \lambda (\varphi, \Delta t) = \lambda (\varphi, 0) - \Delta K^2 \Delta t^2 , \end{equation} in which $\lambda (\varphi, 0)$ is the same as Eq.\ (\ref{eq:lambda_phi_min}): \begin{equation} \lambda (\varphi, 0) = \matelm{\Delta u}{(1 - \rho)}{\Delta u} - \frac{(\imag \inprod{\Delta u}{N})^{2}}{\inprod{N}{N}} . \label{eq:lambda_phi_0} \end{equation}
If we follow the argument used in Sec.\ \ref{sec:properties_time_functional}, it is easy to prove that $\lambda (\varphi, \Delta t) \ge 0$, in which a necessary condition for equality is \begin{equation} (1 - \rho) \ket{\Delta u} = (\Pi_{K} + \Pi_{N}) \ket{\Delta u} , \end{equation} where $\Pi_{K}$ and $\Pi_{N}$ are the projectors for $\ket{K}$ and $\ket{N}$. A necessary and sufficient condition for $\lambda (\varphi, \Delta t) = 0$ is \begin{equation} (1 - \rho) \ket{\Delta u} = i C_{K} \ket{K} + i C_{N} \ket{N} , \end{equation} in which the numbers $C_{K}$ and $C_{N}$ are real.
\subsection{Dynamical stability of phase orbits}
\label{sec:dynamical_stability_phase_orbits}
With the modified time functional (\ref{eq:Delta_t_K}) in hand, we can now apply the principle of dynamical stability in much the same way as before (see Sec.\ \ref{sec:dynamical_stability}). Many parts of the previous analysis can be carried over to the case of phase orbits simply by replacing $H \to K$ and $\Delta E \to \Delta K$. Thus, for example, the dimensionless time interval $\sigma$ is redefined as [cf.\ Eqs.\ (\ref{eq:sigma_def}), (\ref{eq:sigma_redef})] \begin{equation} \sigma \equiv \Delta K \Delta t = \frac{\imag \inprod{\Delta u}{K}}{\Delta K} = \imag \inprod{\Delta x}{\sigma} , \end{equation} in which the components of $\ket{\sigma}$ are [cf.\ Eq.\ (\ref{eq:sigma_ket_redef})]
\begin{multline} \inprod{e_{ki}}{\sigma} = \frac{1}{\omega \Delta K} \biggl[ \frac{\matelm{f_{ki}}{(1 - \rho_k)K}{u_k}}{\inprod{u_k}{u_k}} \\ + \frac{\matelm{g_{ki}}{(1 - \rho_v)K}{v}}{\inprod{v}{v}} \biggr] , \end{multline}
and the renormalization factor $\omega$ is [cf.\ Eq.\ (\ref{eq:omega_E})] \begin{equation} \omega = 1 - \frac{1}{\Delta K^2} \real \biggl[ \frac{\matelm{v_{H \cdot \Psi}}{(1 - \rho_v)K}{v}}{\inprod{v}{v}} \biggr] . \end{equation} Likewise, $\ket{\beta}$ and $\kappa$ in Eqs.\ (\ref{eq:beta_ket_E}) and (\ref{eq:kappa_E}) are redefined as \begin{equation} \inprod{e_{ki}}{\beta} = \frac{1}{\Delta K} \frac{\matelm{g_{ki}}{(1 - \rho_v)}{v_{H \cdot \Psi}}}{\inprod{v}{v}} , \end{equation} \begin{equation} \kappa = \frac{1}{\Delta K^2} \frac{\matelm{v_{H \cdot \Psi}}{(1 - \rho_v)}{v_{H \cdot \Psi}}}{\inprod{v}{v}} . \end{equation} The only truly new parameter to arise is \begin{equation} \theta \equiv \frac{1}{\Delta N \Delta K} \real \biggl[ \frac{\matelm{v_{H \cdot \Psi}}{(1 - \rho_v) N}{v}}{\inprod{v}{v}} \biggr] . \end{equation}
Given these definitions, the squared distance between neighboring phase orbits is [cf.\ Eq.\ (\ref{eq:eta_Delta_t})] \begin{equation} \eta = \eta_0 - \xi^2 + 2 \sigma (\beta - \theta \xi) + \sigma^2 (\kappa - \theta^2) , \end{equation} in which $\eta_0 = \matelm{\Delta x}{\hat{\eta}}{\Delta x}$ and $\xi$ is defined in Eq.\ (\ref{eq:xi_def}). Here it should be noted that $\Psi$ is treated as time dependent (see Sec.\ \ref{sec:time_dependent_psi}) and $\eta$ refers to the quantity \begin{equation} \eta = \eta(\Delta t) \equiv \lambda (\varphi, 0) \ne \lambda (\varphi, \Delta t) , \end{equation} in which $\lambda (\varphi, 0)$ is given in Eq.\ (\ref{eq:lambda_phi_0}).
It is convenient now to follow the approach of Sec.\ \ref{sec:real_matrix} and write $\eta = \mu + \nu$, in which $\mu$ and $\nu$ are defined in Eqs.\ (\ref{mu_def}) and (\ref{nu_def}). The only difference is that the matrix elements $\mu_{ki,k'i'}$ and $\nu_{ki,k'i'}$ in Eqs.\ (\ref{eq:mu_matrix}) and (\ref{eq:nu_matrix}) are modified to become
\begin{multline} \mu_{ki,k'i'} = \delta_{kk'} \frac{\matelm{f_{ki}}{(1 - \rho_k)}{f_{ki'}}}{\inprod{u_k}{u_k}} + \frac{\matelm{g_{ki}}{(1 - \rho_v)}{g_{k'i'}}}{\inprod{v}{v}} \\ + \frac{1}{2} [ (\beta_{ki} \sigma_{k'i'}^{*} + \sigma_{ki} \beta_{k'i'}^{*}) + (\kappa - \theta^2) \sigma_{ki} \sigma_{k'i'}^{*} \\ - \xi_{ki} \xi_{k'i'}^{*} - \theta (\xi_{ki} \sigma_{k'i'}^{*} + \sigma_{ki} \xi_{k'i'}^{*}) ] , \end{multline}
\begin{multline} \nu_{ki,k'i'} = - \frac{1}{2} [ (\beta_{ki} \sigma_{k'i'} + \sigma_{ki} \beta_{k'i'}) + (\kappa - \theta^2) \sigma_{ki} \sigma_{k'i'} \\ - \xi_{ki} \xi_{k'i'} - \theta (\xi_{ki} \sigma_{k'i'} + \sigma_{ki} \xi_{k'i'}) ] . \end{multline}
Aside from this change of definition, all of the subsequent analysis in Secs.\ \ref{sec:real_matrix} and \ref{sec:solve_dynamical_stability} follows through in the same way as before. Because the solution (\ref{eq:Delta_x_real}) has the same mathematical structure as before, it gives rise to the same qualitative behavior too. That is, the dynamics of phase orbits is also deterministic.
Of course, this does not mean that the subsystem dynamics is totally unchanged. The main difference arises because $\Delta K$ is generally smaller than $\Delta E$. The subsystems therefore tend to evolve in time more slowly, because all physically unobservable changes (lying entirely within a given orbit) have been filtered out.
Test calculations show that the phase-orbit dynamics obtained by integrating Eq.\ (\ref{eq:Delta_x_real}) over a finite time interval has the correct limiting behavior (i.e., Schr\"odinger subsystem dynamics) for the special cases discussed in Sec.\ \ref{sec:model_calculations}. To demonstrate this, one must use the phase-orbit distance formulas derived in Appendix \ref{app:phase_orbit_distance} for the case of large $\norm{\Delta u}$.
\section{Subsystem permutations}
\label{sec:subsystem_permutations}
Thus far we have considered only one of the $m!$ possible permutations of the subsystems in Eq.\ (\ref{eq:psi_pi}). The next step is to consider the set of all permutations.
\subsection{Influence of permutations on dynamics}
\label{sec:permutations_influence_dynamics}
According to the definition of an observable in Sec.\ \ref{sec:observables}, a permutation of the subsystems merely rearranges the labels $k$ on the numbers $\expect{A}_k$ in Eq.\ (\ref{eq:mean_value}). But the subsystem labels $k$ are not themselves observable, so a permutation cannot affect the value of any observable quantity at any given time.
On the other hand, permutation of the subsystems does affect the value of the product $\Psi_{\pi}$ in Eq.\ (\ref{eq:psi_pi}). This has no \emph{direct} effect on any observable quantity, but $\Delta \Psi_{\pi} = \Psi_{\pi}' - \Psi_{\pi}$ determines the value of the subsystem change $\Delta V$ [see Eq.\ (\ref{eq:Delta_V})] generated by given changes $\Delta U_k$ in the quasiclassical subsystems ($k \ne 1$).
This means that subsystem permutations do alter the subsystem \emph{dynamics}. Starting from a given initial subsystem decomposition $\rho$ or phase orbit $[\rho]$, a single time step (\ref{eq:Delta_x_real}) will in general carry each permutation into a \emph{different} final state. In other words, the permutation symmetry of observables is \emph{broken} by the dynamics. Subsystem permutations are thus qualitatively different from the phase transformations considered in Sec.\ \ref{sec:phase_reference}.
But this suggests that it might be possible, at least in principle, to use the effect of permutations on dynamics to obtain information about subsystem permutations from the time evolution of observables. Given unlimited information about the observables of all subsystems (which is an unrealistic assumption, as will be discussed in Sec.\ \ref{sec:information_theory}), one might even be able to deduce which individual permutation was consistent with a given set of experimental data.
\subsection{Subsystem ordering in orthodox quantum mechanics}
\label{sec:orthodox_subsystem}
It is important to pause here and note that this effect of subsystem permutations on dynamics is not limited to the context of the present dynamical stability formalism. It is a general consequence of the noncommutative algebra of fermion creation operators that holds even in orthodox quantum mechanics, although this aspect of the theory has received little prior attention.
Consider, for example, the products $\Psi_a = U_1 U_2$ and $\Psi_b = U_2 U_1$ of two creators $U_1$ and $U_2$. Since $\Psi_a$ and $\Psi_b$ generally belong to different rays in projective Hilbert space, they will of course evolve in different ways under the Schr\"odinger equation. That is really all there is to it.
An obvious objection to this conclusion is that $\Psi_a$ and $\Psi_b$ do \emph{not} belong to different rays if $U_1$ and $U_2$ have a definite number of fermions. In that case, $\Psi_a$ differs from $\Psi_b$ by at most a physically meaningless overall sign [see Eq.\ (\ref{eq:signed_product})]. More generally, this is also true if every subsystem $U_k$ is even or odd [see Eq.\ (\ref{eq:creator_commute_anti})]---or, in other words, if each $U_k$ has a definite univalence, where the univalence $W$ is the number of fermions modulo 2: \begin{equation} W \equiv N_{\mathrm{f}} \pmod 2 . \label{eq:univalence} \end{equation} Therefore, if the subsystems in orthodox quantum mechanics are required to satisfy a fermion-number or univalence superselection rule \cite{WickWightmanWigner1952, HegerfeldtKrausWigner1968, WickWightmanWigner1970, Wightman1995}, the Schr\"odinger dynamics of $\Psi$ does not depend on the order in which the subsystems are multiplied.
An answer to this objection can be found in the growing consensus \cite{AharonovSusskind1967a, AharonovSusskind1967b, AharonovRohrlich2005, Mirman1969, Mirman1970, Mirman1979, Lubkin1970, Zeh1970, Zurek1982, GiuliniKieferZeh1995, Giulini2003a, Giulini2009a, WeinbergVol1, DowlingBartlettRudolphSpekkens2006, BartlettRudolphSpekkens2007, Earman2008} that most (if not all) superselection rules are pragmatic expressions of practical limitations on experimental capabilities rather than fundamental laws of nature. This suggests that, at the level of fundamental theory, the superposition principle should be taken seriously, not lightly brushed aside \cite{Zeh2003ch2}.
The practical limitation that gives rise to a particle-number superselection rule in orthodox quantum mechanics is just the lack of a phase reference discussed in Sec.\ \ref{sec:phase_reference} \cite{BartlettRudolphSpekkens2007}. Since this limitation was already taken into account in the analysis of Sec.\ \ref{sec:superselection} and the discussion of Sec.\ \ref{sec:permutations_influence_dynamics}, it does not alter the conclusion that subsystem permutations can have an observable effect on subsystem dynamics.
\subsection{Significance of a univalence superselection rule}
\label{sec:univalence_ssr}
Thus, if one wishes to eliminate the dependence of dynamics on subsystem permutations, one must introduce the univalence superselection rule as an independent axiom; it cannot be derived from the phase symmetry (\ref{eq:phase_Uk}). This rule would require all quasiclassical subsystems to be even, but subsystem $U_1$ could be even or odd. One can easily verify that the univalence of all subsystems is conserved by the time evolution (\ref{eq:Delta_x_real}) if $[H, W] = 0$.
Introducing such a rule makes it easier to combine two subsystems into one. Suppose, for example, we have two subsystems localized in adjacent regions of coordinate space. In such a case it seems natural to talk about using the $\psi$ product to merge these subsystems. However, this cannot be done if they are separated in the product (\ref{eq:psi_pi}) by a subsystem that commutes with neither of them, even if that subsystem is localized far away in coordinate space. The univalence superselection rule therefore provides a natural framework for discussing composition and decomposition of subsystems.
However, at present the question of whether to introduce such an axiom is simply left open. The basic structure of the theory that follows does not depend on this choice.
\section{A bare-bones theory of information}
\label{sec:information_theory}
At this stage the basic theory of subsystem dynamics is more or less complete. The next step is to construct a theory of information (or, in other words, a theory of measurement) that connects the subsystem dynamics to the experiences of observers. This paper develops such a theory only at a very rudimentary level, focusing mostly on qualitative questions such as ``Whose information?'' and ``Information about what?''~\cite{Bell1990} Detailed investigations of this theory of information are left for future work.
\subsection{Bayesian inference in the present moment}
\label{sec:Bayesian_inference}
The present theory of information is essentially just a theory of Bayesian inference for individual observers treated as subsystems of a closed system. The use of Bayesian inference means that this theory has much in common with QBism \cite{CavesFuchsSchack2002, CavesFuchsSchack2007, Fuchs2003, Fuchs2010, Mermin2012a, *Mermin2012b, FuchsSchack2013, FuchsMerminSchack2014, Mermin2014b, Mermin2014a, *Mermin2014c, Mermin2014d, Mermin2016}. However, it differs from QBism in its assignment of subsystem vectors $\ket{u_k}$ to all observers whose experiences are being described.
Here the implementation of Bayesian inference is controlled by two fundamental principles: (1)~Each observer experiences directly only the beables associated with one subsystem $\ket{u_k}$. All else must be inferred. In particular, the existence of other subsystems is inferred from the influence of those subsystems on the dynamics of $\ket{u_k}$. (2)~Each observer experiences directly only the beables associated with a single moment of time (a \emph{moment} being defined as an infinitesimal interval of time). All else must be inferred. In particular, the dynamics of all subsystems in the past and future of the present moment is purely an inference.
The meaning, significance, and historical context of these statements are elaborated in this subsection. Their mathematical implications are discussed in Sec.\ \ref{sec:backbone}.
Let us start with the case in which inferences are drawn from the experiences of only one observer. In this case, principle (1) says that the existence of other subsystems is inferred from their influence on the dynamics of the observer's subsystem $\ket{u_k}$. Interactions between particles are crucial in this regard. If the particles do not interact, the time evolution of $\ket{u_k}$ is independent of the other subsystems, so nothing at all can be inferred about the properties of other subsystems.
The idea that an observer can only ``measure'' the state of his own subsystem was proposed long ago by London and Bauer \cite{LondonBauer1939a, *LondonBauer1939b, WheelerZurek1983}. They described this capacity of an observer as a ``faculty of introspection.'' London and Bauer's concept of measurement is therefore very different from that of von Neumann \cite{vonNeumann1955}, in which the consciousness of the observer somehow directly perceives the state of the outside world, even though the observer is expressly excluded from this state.
In the present theory, the ``faculty of introspection'' does not lead to any collapse of the total state vector $\ket{\psi}$. Instead, the observer merely takes note of the beables for subsystem $\ket{u_k}$ (or some subset thereof) during the present moment of time. This information is then used by the observer to perform a Bayesian updating of the probabilities that he ascribes to the various possible subsystem decompositions of $\ket{\psi}$. Here Bayesian updating is just the usual process of replacing prior probabilities with posterior probabilities, in which the words ``prior,'' ``posterior,'' and ``updating'' refer to a direction of logical inference, not to a direction in time. The total state $\ket{\psi}$ is not assumed to be known, since all subsystems are treated as initially unknown.
The idea that all observations are fundamentally \emph{self}-observations may seem strange from the perspective of orthodox measurement theory \cite{vonNeumann1955}, which requires an observer to always be \emph{outside} the observed system \cite{Wheeler1977}. Indeed the orthodox description is ostensibly the most natural one, since we intuitively regard our sense of vision as a direct perception of the world around us. However, our susceptibility to optical illusions shows clearly that the three-dimensional world we see is actually an inference based on very incomplete two-dimensional information provided by the retinas \cite{Pinker1997}.
The fact that observers are treated as subsystems does not mean that the problem of consciousness must be solved before this theory can be used. The theory merely limits what can be \emph{known} by any observer to the properties of a single subsystem. However, the consequences of imposing such a limit can be evaluated by studying simple non-biological subsystems. According to Wheeler \cite{Wheeler1981}, it is ``not consciousness but the distinction between the probe and the probed [that is] central to the elemental quantum act of observation.'' A similar remark was made by Heisenberg \cite{[{}] [{, p.\ 58.}] Heisenberg1930}: ``The observing system need not always be a human being; it may also be an inanimate apparatus, such as a photographic plate.''
Principle (2) bridges the gap between the ``block universe'' concept of time \cite{Price1996, Price2013, Zeh2007} used in most physical theories and the subjective flow of time that each of us experiences. According to Carnap \cite{Carnap1963}, Bergson's criticism of the block universe picture \cite{Bergson1910, Bergson1999, Ridley2014} was deeply troubling to Einstein: \begin{quote} Once Einstein said that the problem of the Now worried him seriously. He explained that the experience of the Now means something special for man, something essentially different from the past and the future, but that this important difference does not and cannot occur within physics. That this experience cannot be grasped by science seemed to him a matter of painful but inevitable resignation. \ldots We both agreed that this was not a question of a defect for which science could be blamed, as Bergson thought. \end{quote} The idea that the present moment \emph{does} play a crucial role in physics was emphasized by Wheeler in connection with his ``delayed-choice'' experiments \cite{Wheeler1978, Wheeler1980b, Wheeler1983b}: \begin{quote} The ``past'' is theory. The past has no existence except as it is recorded in the present. \end{quote} Wheeler's central message is that everything we say about the past is necessarily an \emph{inference} based on records in the present. This is of course a platitude for historians and paleontologists, but it carries important lessons for experimental physicists as well \cite{Jacques2007}.
Everett \cite{Everett1957, Everett1973, DeWittGraham1973}, Bell \cite{Bell1971, Bell1976, Bell1981}, Barbour \cite{Barbour1994d, Barbour1994b, Barbour2000}, and Page \cite{Page1994} have suggested ways of taking this into account by restructuring quantum theory around the records contained in the present value of $\ket{\psi}$. Mermin \cite{Mermin1998b, Mermin2014b, Mermin2014a, *Mermin2014c}, Hartle \cite{Hartle2005}, and Smolin \cite{Smolin2013} have all recently called attention to the physical significance of the present. But Zeh has remarked that ``physics does not even offer any conceptual means for deriving the concept of a present that would objectively separate the past from the future'' \cite{Zeh2007}.
Here this concept is not \emph{derived}; rather, it is built into the theory of information as a fundamental axiom about the \emph{subjective} experiences of observers. The subjective nature of the Now was stressed by Zeh \cite{Zeh2007}, and its importance for the foundations of both classical and quantum physics has been emphasized especially by Mermin \cite{Mermin2014b, Mermin2014a, *Mermin2014c}.
It should be noted that the infinitesimal time interval referred to in principle (2) is different from the finite time interval (of ``perhaps a few tenths of a second'' \cite{Mermin1998b}) that is associated with our intuitive perception of the present. As discussed by Barbour \cite{Barbour1994d, Barbour1994b} and Hartle \cite{Hartle2005}, the \emph{perceived} duration of the present moment is probably related to the way in which information about the immediate past is stored and processed in the brain.
\subsection{Mathematical backbone for inferences}
\label{sec:backbone}
Let us now examine some mathematical implications of the fundamental principles discussed in Sec.\ \ref{sec:Bayesian_inference}. The first step is to consider \emph{ideal} observers, each of whom is fully aware of every detail of the state of her subsystem. This is of course unrealistic, as an observer's experience will in general tell her only the values of some subset of the beables associated with her subsystem. The consequences of this further restriction are discussed below in Sec.\ \ref{sec:rest_of_skeleton}.
To begin, let us review the way in which the principle of dynamical stability was used in Sec.\ \ref{sec:dynamical_stability}. The problem solved there was a \emph{time evolution problem}. The initial subsystems $\ket{u_k}$ were all assumed to be known; the unknown quantities were the final subsystems $\ket{u_k'}$. The initial and final subsystems were assumed to differ only infinitesimally. The known subsystems were held fixed and the unknown subsystems were varied so as to maximize the dynamical stability functional $\chi$, subject to the constraint that the total states $\ket{\psi}$ and $\ket{\psi'}$ satisfy the Schr\"odinger equation (\ref{eq:Delta_psi_Schr}).
The problem of concern now is an \emph{environmental stabilization problem}, in which environmental subsystems are used to stabilize the overall subsystem decomposition. The difference lies in the classification of known and unknown subsystem states. Let us write the total number of subsystems as \begin{equation} m = s + r , \end{equation} in which $s$ is the number of subsystems that are classified as observers; the remaining $r$ subsystems are regarded as parts of the environment. For the environmental stabilization problem, all observer states $\ket{u_k}$ and $\ket{u_k'}$ are treated as known quantities (defined by the experiences of the observers), whereas the environmental states $\ket{u_k}$ and $\ket{u_k'}$ are treated as unknown. As before, the initial and final subsystems are assumed to differ only infinitesimally. As before, the known subsystems are held fixed and the unknown subsystems are varied so as to maximize $\chi$, subject to the constraint that $\ket{\psi}$ and $\ket{\psi'}$ satisfy the Schr\"odinger equation.
The solutions of the environmental stabilization problem are pairs $(\ket{u}, \ket{u'})$ of subsystem decompositions. Each such pair has its own (infinitesimal) time interval $\Delta t$, which is the duration of the present moment associated with that pair. This is always an \emph{inferred} quantity because $\Delta t$ is defined only for the subsystem decomposition as a whole, not for individual subsystems. Once the environmental stabilization problem has been solved, we can use time evolution to extrapolate any given pair $(\ket{u}, \ket{u'})$ into the past and future of $\Delta t$.
From their definition as variation problems, the time evolution and environmental stabilization problems each lead to a set of equations in which the number of equations is equal to the number of unknown variables. However, there is a big difference in the difficulty of these sets of equations. The time evolution problem leads to a set of linear equations (see Sec.\ \ref{sec:solve_dynamical_stability}), whereas the environmental stabilization problem leads to a set of nonlinear equations.
The latter set of equations is not written out explicitly here, but its qualitative properties can easily be seen by returning to the model system introduced in Sec.\ \ref{sec:model_calculations} (i.e., a system of interacting fermions). In this model, each state $\ket{u_k}$ or $\ket{u_k'}$ can be regarded as a function of $(2^d - 1)$ independent complex variables or $2(2^d - 1)$ independent real variables, where $d = \dim \mathcal{H}_{\mathrm{f}}$. The dynamical stability functional $\chi$ is a rational function of these variables. The environmental stabilization problem therefore requires one to find the common zeros of a set of rational functions. This is a difficult but well defined problem in algebraic geometry \cite{Cox2007}.
However, since the author has no training in this field, the surest route to progress is to publicize the problem and invite experts in algebraic geometry to work on it. For this reason, the quest for explicit solutions shall not be pursued any further here. The existence of solutions is simply taken for granted. Indeed, since the environmental stabilization problem is nonlinear, it may have many solutions [in contrast to the time evolution problem, for which the set of linear equations (\ref{eq:eta_sigma_linear_real_ket}) has a unique solution (\ref{eq:Delta_x_real})]. If no univalence superselection rule is imposed (see Sec.\ \ref{sec:univalence_ssr}), there will in general be a different set of solutions for each possible permutation of the subsystems, thus further increasing the total number of solutions.
Assuming that the environmental stabilization problem can be solved, how do we use its solutions to perform Bayesian inference? Consider the set of pairs $(\ket{u}, \ket{u'})$ of subsystem decompositions that satisfy the environmental stabilization problem for all possible choices of observer states. In this set, the observer states are \emph{not} required to agree with the experiences of the observers. However, to fix the scale of the subsystem changes, the Fubini--Study distance between the sets of observer states in $\ket{u}$ and $\ket{u'}$ is required to be the same as that given by the experiences of the observers.
The first step in the Bayesian inference problem is to assign a prior probability to each pair $(\ket{u}, \ket{u'})$ in this set. The choice of prior probability is subjective \cite{Appleby2005a, Appleby2005b, Jeffrey2004}, although with some work it may be possible to reduce the degree of subjectivity to a choice of symmetry principle \cite{Jaynes1989, Jaynes2003}. Posterior probabilities are then obtained simply by setting to zero the probabilities of all pairs $(\ket{u}, \ket{u'})$ that have the wrong observer states (i.e., states inconsistent with the experiences of the observers) and renormalizing.
The outcome of this inference problem is reminiscent of Wheeler's definition of reality
\cite{Wheeler1981}: \begin{quote} The vision of what we call ``reality'' is in large measure of a pale and theoretic hue. It is framed by a few iron posts of true observation---themselves also resting on theory for their meaning---but most of the walls and towers in the vision are of papier-m{\^ a}ch{\' e}, plastered in between those posts by an immense labor of imagination and theory. \end{quote} Here the iron posts are the observer subsystems in the present moment; everything else is inferred. In general, an observer's experiences in the present moment have a definiteness that is lacking in her memory of her inferences about that moment \footnote{The observer's inferences about the present moment were made in the inferred past, when the present was regarded as part of the future. Sentences such as this one illustrate how difficult it is to talk consistently about the past and future as inferences. The difficulty is that our language takes the reality of the past and future for granted. For most of this paper, this problem is dealt with in the simplest possible way, by not striving for absolute consistency.}. This plays a role similar to the ``reduction of the wave packet'' in orthodox quantum mechanics \cite{CohTan1977}.
It must be emphasized that this is only an analogy. No actual ``jump'' is supposed to take place; there is only a contrast between expectations and experiences. The analogy with the orthodox description of wave-packet reduction \cite{CohTan1977} is actually rather distant. A closer analogy can be found in the dynamical reduction models of Ghirardi and others \cite{BassiGhirardi2003}. In these models the reduction process is continuous, and the beables are defined in terms of expectation values (rather than eigenvalues) of operators, just as in Sec.\ \ref{sec:observables}. In the present paper, however, the beables are not limited to mass density in coordinate space, and the effective ``reduction'' takes place only at the subsystem level, leaving the Schr\"odinger dynamics of the total system untouched.
\subsection{How many subsystems?}
\label{sec:how_many_subsystems}
Are there any general criteria for choosing the subsystem numbers $s$ and $r$? This question is easiest to answer for the number of environmental subsystems $r$. As argued below, the sharpness of Bayesian inferences can be maximized by choosing the smallest possible value: namely, $r = 1$ \footnote{The value $r=0$ must be excluded because the resulting state $\ket{\psi}$ would not satisfy the Schr{\" o}dinger equation.}. The choice of the number of observers $s$ requires a longer answer, as it occupies the borderland between subjectivity and objectivity.
The value $r = 1$ is favored by Occam's razor, which can be regarded as a corollary of Bayesian probability theory \cite{Nemenman2015}. The basic argument is that, for any given value of $s$, the number of solutions to the environmental stabilization problem can be expected to increase rapidly as a function of $r$. This statement is plausible because both the number of equations and the degree of the polynomials involved are increasing functions of $r$. It is, however, not possible to prove this claim without actually solving the environmental stabilization problem. The sharpest Bayesian inferences are thus expected to be given by $r = 1$.
The most obvious value to choose for the number of observers is also $s = 1$. This choice seems to be consistent with each person's intuitive view of the world. If we assume that Darwinian natural selection has instilled in us a rough facsimile of the above Bayesian inference process as the basis for our perception of the world around us (which is, admittedly, a big assumption), then it would be difficult to explain how natural selection could settle on any value other than $s = 1$.
This instinctive value---the one used for the automatic subconscious inferences that underlie our perception of the world---cannot be changed. We cannot separate these subconscious inferences from our raw sense impressions any more than we could measure the bare (unrenormalized) charge of an electron in quantum electrodynamics. However, the value of primary concern here is not this instinctive value but the value of $s$ used for conscious mathematical calculations in quantum mechanics.
The value most commonly chosen in this context is also $s = 1$. A division of the world into observer and observed is the foundation for the quantum theory of measurement used by Heisenberg \cite{[{}] [{, p.\ 58.}] Heisenberg1930}, von Neumann \cite{vonNeumann1955}, and many others. As Zeh has noted \cite{Zeh2003ch2, Zeh1999}, it is never strictly necessary to introduce any other subsystems.
However, this choice leaves one open to a charge of \emph{solipsism}, because each of us has experiences that include encounters with other human beings. Choosing $s = 1$ confines each observer to a hermitage, within which the experiences of others are treated only as inferences, never as primary data. Different observers will therefore always base their worldviews on mutually exclusive subsystem decompositions. This leads inevitably to Wheeler's question \cite{Wheeler1979, Wheeler1983b}: \begin{quote} What keeps these images of something ``out there'' from degenerating into separate and private universes: one observer, one universe; another observer, another universe? \end{quote} Wheeler's answers are cryptic but instructive: \begin{quote} That is prevented by the very solidity of those iron posts, the elementary acts of observership-participancy.
That is the importance of Bohr's point that no observation is an observation unless we can communicate the results of that observation to others in plain language. \cite{Bohr1958} \end{quote} Translated from Wheelerian poetry into the language of the present theory, the first answer says that although different observers always have different experiences, these experiences---the iron posts---are not affected by the choice of $s$. A subsystem decomposition is just a tool observers use to draw inferences from their experiences. The only thing affected by the choice of $s$ is the set of inferences. Choosing $s > 1$ facilitates intersubjective agreement by allowing multiple observers to have equal status within the theory.
The second answer says that if we are to believe that the image of an outside world depicted by these inferences is anything more than a mirage or a hallucination, the essential features of this image must be independent of the value of $s$ used to perform the inferences. Indeed, we can take the set of those features that are robust under increases of $s$ as the \emph{definition} of what is objective. Here the word ``objective'' is used in roughly the sense defined by Primas \cite[p.\ 352]{Primas1983}: \begin{quote} That is, objectivity can never mean anything else but \emph{conditional intersubjective agreement}, conditioned by a jointly accepted context. The criterion of objectivity is that the perceptions can be shared. \end{quote} The idea that ``objective'' properties are necessarily contextual dates back at least to Bohr's thoughts on complementarity \cite{Bohr1928, Bohr1958}. Here the ``jointly accepted context'' is the entire structure of the present theory of information. The ``perceptions [that] can be shared'' are the common features of the worldview that emerges in the limit of many observers. Communication between observers therefore plays a central role in establishing which elements of the theory are meaningful \cite{Wheeler1979, Wheeler1983b, Wheeler1986b, Wheeler1988}.
To flesh out these answers, it is necessary to explain how the theory works when $s > 1$. The most egalitarian approach would be to assign all probabilities as a team. A minimum requirement for forming such a team is to get all teammates to agree to assign unit probability to the hypothesis that ``you are a being very much like myself, with your own private experience'' \cite{Mermin2014d}. Prior probabilities are assigned by team consensus and can be regarded as an expression of gambling commitments by the team as a whole. Inferences drawn from such calculations can then be used as the basis for group decisions.
Of course, this egalitarian approach is somewhat in tension with the subconscious inferences that define each observer's intuitive worldview. Members of such a team might still be wise to make \emph{personal} decisions by assigning less than unit probability to the hypotheses that the other teammates always tell the truth and never make mistakes. If the reported experiences of these teammates are weighted accordingly, the resulting monarchical structure is no longer fully egalitarian. However, it is not solipsist either, as long as these weights are nonzero.
This type of monarchy is the closest point of approach between QBism and the present theory. QBism requires quantum mechanics to be a ``single-user theory'' \cite{Fuchs2010, FuchsSchack2013, FuchsMerminSchack2014, Mermin2014d} but also emphasizes that communications with others can (and should) form part of the basis for single-user quantum state assignments. The two theories are not directly comparable because the words ``user'' (in QBism) and ``observer'' (in this paper) have different mathematical implications. In particular, the single-user theory of the QBist is not equivalent to choosing $s = 1$ in the present theory, because according to the principles of Sec.\ \ref{sec:Bayesian_inference} (which are not part of QBism), the choice $s = 1$ does not allow the experiences of more than one observer to be included in any way.
It should also be noted that the QBist arguments for requiring quantum mechanics to be a single-user theory do not rule out the possibility of choosing $s > 1$ in the present theory. The basic argument is that ``my internal personal experience is inaccessible to you except insofar as I attempt to represent it to you verbally, and vice-versa''~\cite{Mermin2014d}. But this is irrelevant here, because neither classical nor quantum Bayesian theory has any direct contact with internal personal experience; it can only deal with what can be distilled out of that experience and represented \emph{mathematically} \footnote{Private experiences may influence the choice of prior probabilities in a way that cannot be described mathematically. However, it is assumed here that a precondition for forming a team is consensus on the method of defining prior probabilities.}. But what can be described mathematically can also be communicated to another person ``in plain language'' \cite{Bohr1958}. So there is no reason why these mathematical representations of personal experiences cannot be pooled and used as the basis for group inferences, in the manner described above.
\subsection{Strong dynamical stability}
\label{sec:strong_dynamical_stability}
As shown in Sec.\ \ref{sec:how_many_subsystems}, the sharpness of Bayesian inferences can be optimized with respect to the number of environmental subsystems by choosing $r = 1$. However, given that the environmental stabilization problem of Sec.\ \ref{sec:backbone} has not yet been solved, it is not possible at this stage to say whether the resulting inferences would be sharp enough to be comparable to the predictions of orthodox quantum mechanics. The situation could be similar to that pointed out by Zurek \cite{Zurek1993a} in connection with the consistent histories formulation of quantum mechanics \cite{Griffiths2002, Omnes1999, GellMannHartle2014, Hohenberg2010}, where the consistency conditions alone are not sufficiently selective to single out emergent classical behavior.
Given this possibility, it is of interest to consider whether there are any means available for further sharpening of inferences. The method discussed here is based on a strengthening of the principle of dynamical stability. The basic idea is similar to the concept of the ``predictability sieve'' introduced by Zurek \cite{Zurek1993a, ZurekHabibPaz1993}.
The environmental stabilization problem of Sec.\ \ref{sec:backbone} involves holding the observer states fixed and maximizing $\chi$ with respect to variations in the environmental states. The observer states can be chosen arbitrarily, subject only to the (implicit) requirement of leading to a mathematically well defined variation problem.
But is this complete freedom of choice necessary? Might it be possible to narrow down the choices by maximizing $\chi$ with respect to \emph{all} subsystem states, including those of the observers? The choice of observer states would then no longer be completely arbitrary, but if this variation problem has a sufficiently large number of solutions, it may still be possible to account for the actual experiences of observers.
For obvious reasons, the variational principle thus defined is called the \emph{strong} principle of dynamical stability. Subsystem decompositions derived from this principle are maximally stable, in the sense that they are constrained solely by the requirement that $\ket{\psi}$ and $\ket{\psi'}$ satisfy the Schr\"odinger equation. Given that our experience of the world does have a certain stability, it is at least plausible that this experience is congruent with strong dynamical stability.
The selective power of strong dynamical stability is most noticeable in a context where the observer's perception of her state is assumed not to be infinitely precise. Given that an observer's experience is only capable of selecting a certain continuous range of states, strong dynamical stability may be able to narrow the possibilities down to a much smaller (perhaps finite) subset. This would lead to sharper Bayesian inferences.
\subsection{The rest of the skeleton: Complementarity and ``phenomenon''}
\label{sec:rest_of_skeleton}
Let us now consider the effect of removing the restriction to ideal observers imposed in Sec.\ \ref{sec:backbone}. What an observer actually perceives is not the state of her subsystem but some subset of the beables for that subsystem. The perceived values of these beables may be consistent with more than one state $\ket{u_k}$, so the state must be \emph{inferred} from the beables. This imposes another layer of inference, thus making the inferences drawn by a real observer less sharp than those drawn by an ideal observer.
The reason why all beables are not perceived is probably Darwinian. Survival requires an efficient mechanism for drawing inferences about the outside world, and it is most efficient to focus attention on those beables with the greatest signal-to-noise ratio and ignore the rest. This signal-to-noise ratio can be greatly enhanced by sense organs, so the set of perceived beables may vary between organisms with a different evolutionary history. However, the degree of variation is also constrained by efficiency considerations, since slowly changing beables are easier to keep track of than rapidly changing ones. All beables change more slowly under the strong principle of dynamical stability (Sec.\ \ref{sec:strong_dynamical_stability}), but some still change more slowly than others. This limits the possible sets of beables that it is worth developing sense organs for.
The question of which beables are most stable---thus readily seized upon by Darwinian natural selection---is essentially just the ``preferred basis problem'' of decoherence theory \cite{Schlosshauer2007}. Given that particle interactions are typically functions of number operators in coordinate space, the expectation values of such operators (see Sec.\ \ref{sec:observables}) will play the most prominent role in a strongly interacting system. This yields a description similar to the mass-density beables of dynamical reduction theories \cite{GhirardiPearleRimini1990, GhirardiGrassiBenatti1995, BassiGhirardi2003}. However, in general one must consider the dynamics generated by the total Hamiltonian, not just the interaction Hamiltonian. This typically shifts the arena from coordinate space to phase space \cite{Schlosshauer2007}.
Bohr's principle of complementarity \cite{Bohr1928, Bohr1958} is an immediate corollary of these Darwinian restrictions on the subset of perceived beables. Given that all inferences are performed in the present moment, it is simply impossible for an observer to infer definite values for all beables of her environment from a limited set of her own beables. But the set of sharply inferred environmental beables may change with her experiences in the present moment---depending, for example, on which piece of measurement apparatus she is looking at.
The fact that all subsystem beables are defined in the present theory, but not all of them play an active role in the worldview of observers, is similar to ideas used previously in modal interpretations of quantum mechanics. As noted by Vink \cite{Vink1993} and Bub \cite{Bub1995a, Bub1995b}, in modal interpretations it is possible to assign definite values to all possible observables without violating the Bell--Kochen--Specker theorem \cite{Bell1966, KochenSpecker1967, Mermin1993} for the \emph{measured} values of observables. The axiomatic foundations of the theory can thus be simplified by allowing the process of measurement (or, in the present case, Bayesian inference from a Darwinian subset) to take up some of the burden that would otherwise be shouldered by a restriction of the allowed beables.
As discussed in Sec.\ \ref{sec:how_many_subsystems}, our conscious experiences are largely based on subconscious inferences about the structure of our environment. We seem to perceive a ``real'' three-dimensional butterfly rather than a pair of two-dimensional butterfly-shaped patterns on our retinas. In order to match our conscious experiences to the mathematical structure of the present theory of information, we have to infer the latter description from the former. This additional layer of (inverse) inference makes the theory even more complicated. However, such a description is arguably more reasonable than assuming that we directly perceive the three-dimensional butterfly, which is the approach used in orthodox quantum mechanics.
This type of automatic subconscious inference (i.e., an inference about the environment based on a Darwinian-restricted set of observer beables) is an example of what Bohr and Wheeler call a ``phenomenon'' \cite{Bohr1949, Wheeler1978, Wheeler1979, Wheeler1980b}. A phenomenon is subjective in the sense that it depends on the sensory apparatus of the observer. For example, although the inferred ``blackening of a grain of silver bromide'' constitutes a phenomenon for us \cite{Wheeler1980b}, it presumably would not be regarded as such by a Kentucky cave shrimp, which lives its entire life underground and has no eyes.
\subsection{Comparison with other quantum theories}
\label{sec:compare_other_quantum}
The essence of the theory of information outlined here is its reliance upon inferences from the properties of a limited number of subsystems in the present moment. In some ways this is similar to the Everett interpretation \cite{Everett1957, Everett1973}, which also uses the properties of one subsystem to make inferences about others and also relies on memories and records in the present to make inferences about the past. However, Everett defines subsystems using the traditional tensor product of vector spaces, and he defines the present to be an instant rather than a moment (i.e., a point on the real line rather than an interval).
Because Everett's subsystems are entangled, he introduces the relative-state concept as a new axiom \cite{vonNeumann1955} that allows him to infer the state of the outside world from the instantaneous state of the observer. However, this inference assumes knowledge of the total state $\ket{\psi}$. It is not clear how such an instantaneous inference scheme can get off the ground if the observer starts in a state of ignorance of $\ket{\psi}$. That is, it is not clear why an observer using such a scheme would have any justification for believing that knowledge of his own present state says anything about the state of his environment.
By contrast, the subsystems in the present theory are defined to be unentangled quantum objects, and inferences are based on an infinitesimal time interval rather than a single point in time. The information about dynamics contained in this interval then allows an observer to infer something about the state of the environment even when the observer has no prior information about the environment. Of course, it is not clear whether this scheme can get off the ground either; at this stage, it is only a conjecture that the solution of the environmental stabilization problem (in either the original form of Sec.\ \ref{sec:backbone} or the strong form of Sec.\ \ref{sec:strong_dynamical_stability}) will provide inferences sharp enough to say anything significant about the environment. However, it is reasonable to assume that inferences from a moment in time would be considerably sharper than instantaneous inferences.
Bell has said that the really novel element in Everett's theory is ``a repudiation of the concept of the `past','' although he acknowledges that this interpretation might not be accepted by Everett \cite{Bell1981}. Bell did not endorse this interpretation, which Kucha{\v r} has described as ``solipsism of an instant'' \footnote{See Kucha{\v r}'s discussion with Page on p.\ 296 of Ref.\ \cite{Page1994}.}. For the reasons described above, this label might indeed be warranted for a truly instantaneous inference scheme.
However, the momentary inference scheme adopted here does not require a wholesale repudiation of the concept of the past. Rather, it places limits on what can be said about the past. It remains to be seen whether these limits are in quantitative agreement with the experiences of observers.
This approach also places limits on what can be said about the future. As noted near the end of Sec.\ \ref{sec:backbone}, the contrast between an observer's inferences about the future and her experiences in successive present moments has an effect roughly comparable to the quantum jumps of orthodox quantum theory.
The definition of subsystems used by Everett is the same as that used in nearly all other formulations of quantum mechanics. The vector spaces entering into the tensor product have no dynamics and can be chosen arbitrarily at different times. Because the resulting subsystem states are not objects (see Sec.\ \ref{sec:intro_define_subsystems}), it is meaningless to compare their observable properties directly with those of the present theory. The subsystems in the two theories simply refer to different things. Given this qualitative difference, perhaps the only meaningful test of the present theory would be a direct comparison between theory and experiment.
\subsection{Is anything missing?}
When one is contemplating the possible outcomes of such a comparison, two other qualitative differences between this theory and standard quantum mechanics stand out. One is that the quasiclassical subsystems used here have indefinite particle numbers (see Sec.\ \ref{sec:invertible_important}). Therefore, they cannot directly replicate standard textbook problems for distinguishable subsystems with definite particle numbers. Instead, these subsystems seem to enforce a compliance with Bohr's emphasis on the ``whole experimental arrangement'' \cite{Bohr1949}. That is, they would only be able to describe a hydrogen atom as a part of a larger subsystem, not as a subsystem by itself.
Another difference is the way in which observable (or beable) quantities were defined in Sec.\ \ref{sec:observables}. This definition is based on what would conventionally be described as expectation values rather than eigenvalues. The physical picture is closest to the ``density of stuff'' interpretation \cite{Bell1990} used in the dynamical reduction models of Ghirardi and others \cite{GhirardiPearleRimini1990, GhirardiGrassiBenatti1995, BassiGhirardi2003}, but with the continuous ``reduction'' occurring in the subsystem states $\ket{u_k}$ rather than the total system state $\ket{\psi}$ (see Sec.\ \ref{sec:backbone}).
Taken together, these differences raise doubts as to whether the present theory would be able to replicate the innumerable successful predictions of orthodox quantum mechanics. A particular concern is whether there is any element of this theory comparable to the discrete eigenvalue spectra predicted for the results of ideal measurements in orthodox quantum mechanics.
However, the predictions of dynamical reduction models are well known to be experimentally indistinguishable (given the current state of experimental capabilities) from those of orthodox quantum mechanics \cite{BassiGhirardi2003}. This shows it is unnecessary to define beables as eigenvalues. Since the definition of beables used in Sec.\ \ref{sec:observables} is just a generalization of the one used in dynamical reduction models, it is not unreasonable to expect it to yield comparable results. Testing this conjecture is a key challenge for future research.
\section{Conclusions}
\label{sec:conclusions}
This paper arose from the observation that ordinary many-particle quantum mechanics has a hitherto unnoticed mathematical structure that can be interpreted as an unentangled subsystem decomposition. This structure relies on the superposition of different numbers of particles, but it also permits a full description of the equivalence relation that leads to a particle-number superselection rule in orthodox quantum mechanics. The goal of the paper was to take this structure seriously and see what it leads to. Can the elimination of entanglement between subsystems help to resolve some of the conceptual difficulties at the heart of quantum mechanics?
To build on this foundation, one must link the subsystem states to the experiences of observers. The first step is the definition of time as a functional of subsystem changes. This functional can then be embedded into a dynamical stability functional that describes the subsystem dynamics. The resulting subsystem decompositions change by the smallest amount consistent with Schr\"odinger dynamics for the total system. This change is deterministic.
The observable or ``beable'' properties of subsystems are defined as expectation values of the conventional operators in many-particle quantum mechanics. These beables include, but are not limited to, the mass density functions used in dynamical reduction theories. An observer could in principle experience any beable associated with her subsystem. However, for Darwinian reasons it is assumed that only those beables with the greatest signal-to-noise ratio are experienced.
An observer's experiences are also limited to the present moment of time. These experiences are the ``iron posts'' upon which our concept of reality is based. From them, an observer can infer the existence of an outside world together with a past and future of the present moment. These inferences are extremely useful tools that are indispensable for us to ``orient ourselves in the labyrinth of sense impressions,'' but they always remain an ``arbitrary creation of the human (or animal) mind'' \cite{Einstein1936, Einstein1950, Einstein1954}. The resulting image of the world is thus unavoidably subjective. Objectivity emerges only in the limit of many observers.
Much work remains to be done in order to fill in the details of the theory of information outlined here. The most important task is to solve the variation problem in which an environmental subsystem is used to stabilize the dynamics of the observer subsystems in the present moment. Only then will it be possible to see whether the inferences derived in this way are in sufficiently close agreement with experience to be useful.
\begin{acknowledgments} I wish to thank Mike Burt for many stimulating conversations over the years, as well as Yik Man Chiang and Avery Ching for guidance on the theory of several complex variables. I also thank Gerald Bastard, Shengwang Du, Brian Ridley, Ping Sheng, Henry Tye, and Yi Wang for helpful comments on an earlier version of the manuscript. \end{acknowledgments}
\appendix
\section{Associativity of the \texorpdfstring{$\psi$}{psi} product}
\label{app:associative}
The $\psi$ product (\ref{eq:psi_product}) is associative if \begin{equation} (\ket{u^{(k)}} \odot \ket{v^{(l)}}) \odot \ket{w^{(m)}} = \ket{u^{(k)}} \odot (\ket{v^{(l)}} \odot \ket{w^{(m)}}) \label{eq:associative1} \end{equation} for all $\ket{u^{(k)}} \in S (\mathcal{H}^{k})$, $\ket{v^{(l)}} \in S (\mathcal{H}^{l})$, and $\ket{w^{(m)}} \in S (\mathcal{H}^{m})$. The explicit form of the associativity condition given by the definition (\ref{eq:psi_product}) is \begin{multline} c(k, l) c(k+l, m) S [ S (\ket{u^{(k)}} \otimes \ket{v^{(l)}}) \otimes \ket{w^{(m)}} ] \\ = c(k, l+m) c(l, m) S [ \ket{u^{(k)}} \otimes S (\ket{v^{(l)}} \otimes \ket{w^{(m)}}) ] . \label{eq:associative2} \end{multline} This appendix examines the implications of this constraint for the function $c(k, l)$.
Now it is well known that \cite{Szekeres2004, Abraham1988, KostrikinManin1989, LoomisSternberg1990} \begin{equation} S ( \ket{u} \otimes \ket{v} ) = S ( \ket{u} \otimes S \ket{v} ) = S ( S \ket{u} \otimes \ket{v} ) \end{equation} for all $\ket{u} \in \mathcal{H}^{p}$ and $\ket{v} \in \mathcal{H}^{q}$. Equation (\ref{eq:associative2}) consequently reduce to the condition \begin{equation} c(k, l) c(k+l, m) = c(k, l+m) c(l, m) , \label{eq:associative3} \end{equation} which must be satisfied in order for the $\psi$ product to be associative.
To solve this equation, it is convenient to follow the procedure suggested on p.\ 401 of Ref.\ \cite{Abraham1988}. Define a function $f (l)$ recursively by the relation \begin{equation} f (l + 1) = c(1, l) f (l) . \label{eq:psi_def} \end{equation} In the special case $k = 1$, Eq.\ (\ref{eq:associative3}) gives \begin{equation} \frac{c(l+1, m)}{c(l, m)} = \frac{c(1, l+m)}{c(1, l)} = \frac{f (l+m+1) f (l)}{f (l+m) f (l+1)} , \label{eq:c_ratio_psi} \end{equation} in which $c(1, l) = f (l+1) / f (l)$ was used in the last step. But this is just a special case of a more general relation \begin{equation} \frac{c(l+k, m)}{c(l, m)} = \frac{f (l+m+k) f (l)}{f (l+m) f (l+k)} , \label{eq:induction} \end{equation} which can be proved by mathematical induction. In the identity \begin{equation} \frac{c(l+k+1, m)}{c(l, m)} = \frac{c(l+k+1, m)}{c(l+k, m)} \frac{c(l+k, m)}{c(l, m)}, \end{equation} one can replace the first term using Eq.\ (\ref{eq:c_ratio_psi}) (with $l \rightarrow l + k$) and the second term using Eq.\ (\ref{eq:induction}). This gives
\begin{align} \frac{c(l+k+1, m)}{c(l, m)} & = \frac{f (l+k+m+1) f (l+k)}{f (l+k+m) f (l+k+1)} \nonumber \\ & \quad \times \frac{f (l+m+k) f (l)}{f (l+m) f (l+k)} \nonumber \\ & = \frac{f (l+m+k+1) f (l)}{f (l+m) f (l+k+1)} , \end{align}
which shows that Eq.\ (\ref{eq:induction}) holds for $k+1$ whenever it holds for $k$. The initial condition (\ref{eq:c_ratio_psi}) then establishes that (\ref{eq:induction}) holds for all $k \ge 1$. (Of course, it is trivially valid when $k = 0$ too.)
Substituting $k = q - l$ in Eq.\ (\ref{eq:induction}) then gives \begin{equation} \frac{c(q, m)}{c(l, m)} = \frac{f (q+m) f (l)}{f (l+m) f (q)} , \end{equation} which becomes \begin{equation} \frac{c(q, m)}{c(1, m)} = \frac{f (q+m) f (1)}{f (m+1) f (q)} \end{equation} when $l = 1$. Replacing $c(1, m) = f (m+1) / f (m)$ then yields the desired solution \begin{equation} c(q, m) = \frac{f (q+m) f (1)}{f (q) f (m)} , \label{eq:cqm} \end{equation} showing that $c(q, m) = c(m, q)$.
When $m = 0$, this reduces to \begin{equation} c(q, 0) = c(0, q) = \frac{f (1)}{f (0)} , \end{equation} which is independent of $q$. The value of $c(q, 0)$ can be fixed by requiring the vacuum state $\ket{0}$ to serve as a multiplicative identity for the $\psi$ product [cf.\ Eq.\ (\ref{eq:mult_ident})]: \begin{equation} \ket{0} \odot \ket{\Psi} = \ket{\Psi} \odot \ket{0} = \ket{\Psi} \qquad \forall \ket{\Psi} \in \mathcal{F}_{s} (\mathcal{H}) . \end{equation} Imposing this condition when $\ket{\Psi} = \ket{u^{(q)}}$ gives $c(q, 0) = 1$ or $f (0) = f (1)$. The result $\abs{f (0)} = \abs{f (1)}$ can also be derived from cluster decomposition, as shown in Appendix \ref{app:cluster_decomposition}.
The recursive definition (\ref{eq:psi_def}) of $f (l)$ leaves one value of $f (l)$ that can be chosen arbitrarily. It is convenient to choose $f (1) = 1$, thus reducing Eq.\ (\ref{eq:cqm}) to Eq.\ (\ref{eq:cf}), which was to be proved.
\section{Cluster decomposition theorem}
\label{app:cluster_decomposition}
This appendix contains a proof of Theorem \ref{thm:cluster}, which is about the cluster decomposition property $\inprod{st}{uv} = \inprod{s}{u} \inprod{t}{v}$ of the inner product in $\mathcal{F}_{s} (\mathcal{H})$. This inner product is derived from the inner product in $\mathcal{F} (\mathcal{H})$ and the definition of the $\psi$ product in Eqs.\ (\ref{eq:psi_product}) and (\ref{eq:cf}).
To define the inner product in $\mathcal{F} (\mathcal{H})$, let \begin{equation} \pket{\alpha_1 \alpha_2 \cdots \alpha_n} = \ket{\alpha_1} \otimes \ket{\alpha_2} \otimes \cdots \otimes \ket{\alpha_n} \end{equation} denote a general tensor-product state in $\mathcal{H}^{n}$, where $\ket{\alpha_{k}} \in \mathcal{H}$. The set $\{ \ket{\alpha_{k}} \}$ is not assumed to be linearly independent or normalized. The rounded bracket on the ket $\pket{\alpha_1 \alpha_2 \cdots \alpha_n}$ distinguishes this unsymmetrized tensor product from the symmetrized product $\ket{\alpha_1 \alpha_2 \cdots \alpha_n}$ defined in Eqs.\ (\ref{eq:ket_general}) and (\ref{eq:ket_relation}).
The inner product of two such tensor products is defined in the usual way as \cite{CohTan1977} \begin{equation} \newinprod{\alpha_1 \cdots \alpha_n}{\beta_1 \cdots \beta_n} = \inprod{\alpha_1}{\beta_1} \cdots \inprod{\alpha_n}{\beta_n} , \end{equation} where $\inprod{\alpha_k}{\beta_k}$ is the inner product in $\mathcal{H}$. Now let $\{ \ket{e_i} \}$ be an orthonormal basis in $\mathcal{H}$. The corresponding tensor products \begin{equation} \pket{e_{i_1} \cdots e_{i_n}} = \ket{e_{i_1}} \otimes \cdots \otimes \ket{e_{i_n}} \label{eq:ebasis} \end{equation} therefore form an orthonormal basis in $\mathcal{H}^{n}$, since \begin{equation} \newinprod{e_{i_1} \cdots e_{i_n}}{e_{j_1} \cdots e_{j_n}} = \delta_{i_1 j_1} \cdots \delta_{i_n j_n} . \end{equation}
These elementary results can now be used to evaluate the inner product $\inprod{s t}{u v}$ of the vectors $\ket{s t} = \ket{s} \odot \ket{t}$ and $\ket{u v} = \ket{u} \odot \ket{v}$ in Theorem \ref{thm:cluster}. The kets $\ket{s} \in \mathcal{F}_{s} (\mathcal{H}_1)$ and $\ket{t} \in \mathcal{F}_{s} (\mathcal{H}_2)$ are expanded as \begin{equation} \ket{s} = \sum_{k} \ket{s^{(k)}} , \qquad \ket{t} = \sum_{l} \ket{t^{(l)}} , \end{equation} in which $\ket{s^{(k)}} \in S (\mathcal{H}^{k})$ and $\ket{t^{(l)}} \in S (\mathcal{H}^{l})$. Hence \begin{equation} \ket{s t} = \sum_{k} \sum_{l} \ket{s^{(k)}} \odot \ket{t^{(l)}} = \sum_{k} \sum_{l} \ket{s^{(k)} t^{(l)}} , \end{equation} in which $\ket{s^{(k)} t^{(l)}} \in S (\mathcal{H}^{k+l})$. With a similar expansion for $\ket{u v}$, we can write \begin{equation} \inprod{s t}{u v} = \sum_{kl} \sum_{mn} \inprod{s^{(k)} t^{(l)}}{u^{(m)} v^{(n)}} . \label{eq:stuvklmn} \end{equation} Here $\inprod{s^{(k)} t^{(l)}}{u^{(m)} v^{(n)}}$ is zero unless $k+l = m+n$, since states with different numbers of particles are orthogonal.
But due to the orthogonality of the subspaces $\mathcal{H}_1$ and $\mathcal{H}_2$, a stronger restriction is possible: $\inprod{s^{(k)} t^{(l)}}{u^{(m)} v^{(n)}}$ is zero unless $k = m$ and $l = n$. Thus \begin{equation} \inprod{s t}{u v} = \sum_{kl} \inprod{s^{(k)} t^{(l)}}{u^{(k)} v^{(l)}} . \label{eq:stuvkl} \end{equation} The definition (\ref{eq:psi_product}) of the $\psi$ product can now be used to write \begin{equation} \ket{u^{(k)} v^{(l)}} = c(k, l) S (\ket{u^{(k)}} \otimes \ket{v^{(l)}}) . \end{equation} Here associativity requires $c(k, l)$ to have the form derived in Appendix \ref{app:associative}: \begin{equation} c(k, l) = \frac{f(k+l)}{f(k) f(l)} , \qquad f(1) = 1 , \label{eq:ckl} \end{equation} but the function $f(k)$ has not yet been determined.
The inner product (\ref{eq:stuvkl}) is therefore \begin{equation} \inprod{s t}{u v} = \sum_{kl} \abs{c(k, l)}^2 (\bra{s^{(k)}} \otimes \bra{t^{(l)}}) S^{\dagger} S (\ket{u^{(k)}} \otimes \ket{v^{(l)}}) . \end{equation} Since $S$ is an orthogonal projector, $S^{\dagger} S = S^2 = S$, thus \begin{equation} \inprod{s t}{u v} = \sum_{kl} \abs{c(k, l)}^2 (\bra{s^{(k)}} \otimes \bra{t^{(l)}}) S (\ket{u^{(k)}} \otimes \ket{v^{(l)}}) . \label{eq:stuv_expand} \end{equation} Here $(\ket{u^{(k)}} \otimes \ket{v^{(l)}}) \in \mathcal{H}^{k+l}$, so the definition of $S$ gives \begin{equation} S (\ket{u^{(k)}} \otimes \ket{v^{(l)}}) = \frac{1}{(k+l)!} \sum_{\sigma} \varepsilon (\sigma) \sigma (\ket{u^{(k)}} \otimes \ket{v^{(l)}}) . \label{eq:Skl} \end{equation}
Now if the product $(\ket{u^{(k)}} \otimes \ket{v^{(l)}})$ is expanded in the unsymmetrized basis of Eq.\ (\ref{eq:ebasis}) (with $n = k + l$), the first $k$ vectors $\ket{e_i}$ will be from $\mathcal{H}_1$ and the last $l$ vectors will be from $\mathcal{H}_2$. The same is true for the product $(\ket{s^{(k)}} \otimes \ket{t^{(l)}})$. The only permutations $\sigma$ in equation (\ref{eq:Skl}) that contribute nonvanishing terms to equation (\ref{eq:stuv_expand}) are therefore those that do not interchange any of the first $k$ vectors with the last $l$ vectors. These permutations are of the form $\sigma = \sigma_1 \sigma_2$, where $\sigma_1$ is any permutation of the first $k$ vectors and $\sigma_2$ is any permutation of the last $l$ vectors.
The orthogonality of the basis vectors (\ref{eq:ebasis}) therefore reduces Eq.\ (\ref{eq:stuv_expand}) to
\begin{multline} \inprod{s t}{u v} = \sum_{kl} \frac{\abs{c(k, l)}^2}{(k + l)!} \sum_{\sigma_1} \varepsilon (\sigma_1) \matelm{s^{(k)}}{\sigma_1}{u^{(k)}} \\ \times \sum_{\sigma_2} \varepsilon (\sigma_2) \matelm{t^{(l)}}{\sigma_2}{v^{(l)}} , \label{eq:stuv_expand2} \end{multline}
in which $\varepsilon (\sigma_1 \sigma_2) = \varepsilon (\sigma_1) \varepsilon (\sigma_2)$ was used. Now $\ket{u^{(k)}} \in S(\mathcal{H}^{k})$, so $\sigma_1 \ket{u^{(k)}} = \varepsilon (\sigma_1) \ket{u^{(k)}}$, and likewise $\sigma_2 \ket{v^{(l)}} = \varepsilon (\sigma_2) \ket{v^{(l)}}$. But $[\varepsilon (\sigma)]^2 = 1$ for any $\sigma$, hence \begin{align} \inprod{s t}{u v} & = \sum_{kl} \frac{\abs{c(k, l)}^2}{(k + l)!} \sum_{\sigma_1} \inprod{s^{(k)}}{u^{(k)}} \sum_{\sigma_2} \inprod{t^{(l)}}{v^{(l)}} \\ & = \sum_{kl} \babs{\frac{f(k+l)}{f(k) f(l)}}^2 \frac{k! l!}{(k + l)!} \inprod{s^{(k)}}{u^{(k)}} \inprod{t^{(l)}}{v^{(l)}} . \label{eq:stuv_expand3} \end{align}
At this point, choosing $\abs{f(k)} = \sqrt{k!}$ eliminates all of the numerical factors, yielding \begin{equation} \inprod{s t}{u v} = \sum_{k} \inprod{s^{(k)}}{u^{(k)}} \sum_{l} \inprod{t^{(l)}}{v^{(l)}} . \end{equation} Each factor can be rewritten as \begin{equation} \sum_{k} \inprod{s^{(k)}}{u^{(k)}} = \sum_{k} \sum_{m} \inprod{s^{(k)}}{u^{(m)}} = \inprod{s}{u} , \end{equation} since $\inprod{s^{(k)}}{u^{(m)}} = 0$ when $k \ne m$. Therefore $\inprod{s t}{u v} = \inprod{s}{u} \inprod{t}{v}$, thus proving the ``if'' part of the cluster decomposition theorem \ref{thm:cluster}.
Conversely, suppose it is given that $\inprod{s t}{u v} = \inprod{s}{u} \inprod{t}{v}$ for all $\ket{s}, \ket{u} \in \mathcal{F}_{s} (\mathcal{H}_1)$ and all $\ket{t}, \ket{v} \in \mathcal{F}_{s} (\mathcal{H}_2)$, where $\mathcal{H}_1$ and $\mathcal{H}_2$ are orthogonal subspaces of $\mathcal{H}$. What does this tell us about $f(k)$?
Under the given conditions, we are free to choose $\ket{u} = \ket{u^{(k)}}$ and $\ket{v} = \ket{v^{(l)}}$ for any values of $k$ and $l$. Equation (\ref{eq:stuv_expand3}) then reduces to \begin{equation} \inprod{s t}{u v} = \babs{\frac{f(k+l)}{f(k) f(l)}}^2 \frac{k! l!}{(k + l)!} \inprod{s^{(k)}}{u^{(k)}} \inprod{t^{(l)}}{v^{(l)}} , \end{equation} with no summation on $k$ and $l$. Likewise \begin{equation} \inprod{s}{u} \inprod{t}{v} = \inprod{s^{(k)}}{u^{(k)}} \inprod{t^{(l)}}{v^{(l)}} . \end{equation} Hence, $\inprod{s t}{u v} = \inprod{s}{u} \inprod{t}{v}$ in all such cases only if \begin{equation} \babs{\frac{f(k+l)}{f(k) f(l)}}^2 = \frac{(k + l)!}{k! l!} \qquad \forall k, l \ge 0 . \end{equation} Setting $l = 1$ and using $f(1) = 1$ then gives \begin{equation} \abs{f(k+1)}^2 = (k + 1) \abs{f(k)}^2 . \end{equation} This defines $\abs{f(k)}^2$ recursively as \begin{equation} \abs{f(k)}^2 = k! \abs{f(1)}^2 = k! \qquad (k \ge 1) , \end{equation} while $k = 0$ gives $\abs{f(1)}^2 = \abs{f(0)}^2$. Hence, $\abs{f(k)} = \sqrt{k!}$ for all $k$, thus completing the proof of Theorem \ref{thm:cluster}.
\section{Algebraic closure conditions for bosons}
\label{app:rigged}
This appendix describes the construction of the boson vector space $\mathcal{F}_{\psi} (\mathcal{H}_{\mathrm{b}})$ mentioned at the end of Sec.\ \ref{sec:creation}. The objective is to identify a subspace of $\mathcal{F}_{s} (\mathcal{H}_{\mathrm{b}})$ that is a good match for the algebra of the boson $\psi$ product. This algebra is easiest to describe using the Segal--Bargmann representation of Fock space \cite{Bargmann1961, Schweber1962, Folland1989, Hall2000}, which is often used in the definition of coherent boson states \cite{Glauber1963c, KlauderSkagerstam1985, NegeleOrland1998}.
Let us start by establishing a concise notation. Standard basis kets in Fock space are written as $\ket{n}$, where $n = (n_1, n_2, \ldots, n_b) \in \mathbb{N}^b$ is a vector of nonnegative integers, $b$ is the dimension of the single-boson Hilbert space $\mathcal{H}_{\mathrm{b}}$, and $n_i$ is the number of bosons in the single-particle state $i$. In multi-index notation \cite{ReedSimonVol1, Bargmann1961}, powers of a complex vector $z = (z_1, \ldots, z_b) \in \mathbb{C}^b$ are written as $z^n = z_1^{n_1} \cdots z_b^{n_b}$, and likewise for powers of the vector $a^{\dagger} = (a_1^{\dagger}, \ldots, a_b^{\dagger})$ of boson creation operators. This allows the normalized basis ket $\ket{n}$ to be written simply as $\ket{n} = (n !)^{-1/2} (a^{\dagger})^n \ket{0}$ [see Eqs.\ (\ref{eq:normalization}) and (\ref{eq:create_vacuum})], where $n! = n_1 ! \cdots n_b !$.
A general vector in $\mathcal{F}_{s} (\mathcal{H}_{\mathrm{b}})$ has the form $\ket{f} = \sum_{n} c_n \ket{n}$, where $c_n = \inprod{n}{f}$. The Fock space $\mathcal{F}_{s} (\mathcal{H}_{\mathrm{b}})$ is also required to be a Hilbert space, the members of which must satisfy $\norm{f} < \infty$, in which $\norm{f}^2 = \inprod{f}{f} = \sum_{n} \abs{c_n}^2$. For reasons to be explained below, the Hilbert-Fock space defined in this way is also written as $\mathcal{H}_1$.
The Segal--Bargmann representation of $\ket{f}$ is defined by the expression [cf.\ Eq.\ (\ref{eq:uU0})] \begin{equation} \ket{f} = F (a^{\dagger}) \ket{0} , \end{equation} in which the function $F(z)$ is defined by the power series \begin{equation} F (z) = \sum_{n} \frac{c_n}{\sqrt{n !}} z^n . \end{equation} If $\norm{f} < \infty$, $F(z) \in \mathbb{C}$ is an entire holomorphic function of $z = x + i y$, where $x, y \in \mathbb{R}^b$. This representation of ket vectors by entire functions is a powerful advantage of the Segal--Bargmann theory.
In the Segal--Bargmann representation, the inner product of two vectors $\ket{f}$ and $\ket{g}$ is given by the integral \begin{equation} \inprod{f}{g} = \int F^*(z) G(z) \rho (z) \, \ensuremath{\rmd^{2b} z} , \label{eq:Bargmann_ip} \end{equation} in which $\rho (z) = \pi^{-b} \exp (- \abs{z}^2)$, $\abs{z}^2 = \abs{z_1}^2 + \cdots + \abs{z_b}^2$, and $\ensuremath{\rmd^{2b} z} = \mathrm{d} x_1 \, \mathrm{d} y_1 \cdots \mathrm{d} x_b \, \mathrm{d} y_b$. Functions of the form $F(z) = \exp (\frac12 \gamma z^2 + \alpha \cdot z)$ are of special interest, where $\gamma \in \mathbb{C}$, $\alpha \in \mathbb{C}^b$, $\alpha \cdot z = \alpha_1 z_1 + \cdots + \alpha_b z_b$, and $z^2 = z \cdot z$. This function is normalizable (i.e., $\norm{f} < \infty$) if and only if $\abs{\gamma}^2 < 1$ \cite{Bargmann1961}. A general bound on all normalizable states is given by the Schwarz inequality \cite{Bargmann1961}: \begin{equation} \abs{F(z)} \le \exp (\tfrac12 \abs{z}^2) \norm{f} \qquad \forall z \in \mathbb{C}^b . \label{eq:Schwarz_Fz} \end{equation} This implies that the Hilbert-Fock space $\mathcal{H}_1 = \mathcal{F}_{s} (\mathcal{H}_{\mathrm{b}})$ is a poor match for the algebra of the $\psi$ product, since $\ket{f} \odot \ket{g}$ is represented by the product $F(z) G(z)$ [cf.\ Eq.\ (\ref{eq:uvUV})]. For example, the product of $F(z) = \exp (\frac12 \gamma z^2 + \alpha \cdot z)$ and $G(z) = \exp (\frac12 \delta z^2 + \beta \cdot z)$ is normalizable if and only if $\abs{\gamma + \delta}^2 < 1$, but this condition is violated by many pairs of states with $\abs{\gamma}^2 < 1$ and $\abs{\delta}^2 < 1$.
To find a suitable vector space for the algebra of the $\psi$ product, it is helpful to consider the family of vectors $\ket{f_k}$ defined by \begin{equation} F_k (z) = F (k z) , \qquad \inprod{n}{f_k} = k^{\abs{n}} \inprod{n}{f} , \label{eq:Fk_def} \end{equation} in which $k$ is a positive integer (i.e., $k \in \mathbb{N}_{+}$) and $\abs{n} \equiv n_1 + \cdots + n_b$. One can easily see that $\inprod{f_k}{g_k} = \inprod{f}{g}_k$, in which $\inprod{f}{g}_k$ denotes the inner product \begin{equation} \inprod{f}{g}_k = \int F^*(z) G(z) \rho_k (z) \, \ensuremath{\rmd^{2b} z} , \end{equation} where $\rho_k (z) = (\pi k^2)^{-b} \exp (- \abs{z}^2 / k^2 )$. This gives rise to a countable family of norms $\norm{f}_k = (\inprod{f}{f}_k)^{1/2}$; note that $\norm{f}_1$ is the same as the norm $\norm{f}$ defined by the inner product (\ref{eq:Bargmann_ip}). The set of vectors with $\norm{f}_k < \infty$ forms a Hilbert space, which is denoted $\mathcal{H}_k$. The statement $\ket{f} \in \mathcal{H}_k$ is the same as $\ket{f_k} \in \mathcal{H}_1$.
According to Eqs.\ (\ref{eq:Schwarz_Fz}) and (\ref{eq:Fk_def}), all vectors in $\mathcal{H}_k$ must satisfy \begin{equation} \abs{F(z)} \le \exp (\abs{z}^2 / 2 k^2) \norm{f} \qquad \forall z \in \mathbb{C}^b . \label{eq:Schwarz_Fz_k} \end{equation} Conversely, to show that $\ket{f} \in \mathcal{H}_k$, it is sufficient to find numbers $0 \le A < \infty$ and $0 \le \lambda < 1$ such that \cite{Bargmann1961} \begin{equation} \abs{F (z)} \le A \exp (\lambda \abs{z}^2 / 2 k^2) \qquad \forall z \in \mathbb{C}^b . \label{eq:F_in_Hk} \end{equation} From these results it is easy to see that \begin{equation} \mathcal{H}_{k+1} \subset \mathcal{H}_k , \end{equation} since Eq.\ (\ref{eq:Schwarz_Fz_k}) with $k \to k + 1$ yields an inequality of the type (\ref{eq:F_in_Hk}), with $A = \norm{f}$ and $\lambda = k^2 / (k+1)^2$.
Let us now define a vector space $\mathcal{F}_{\psi} = \mathcal{F}_{\psi} (\mathcal{H}_{\mathrm{b}})$ as the intersection of the Hilbert spaces $\mathcal{H}_k$ for all $k \in \mathbb{N}_{+}$. $\mathcal{F}_{\psi}$ is defined by the countable family of norms $\{ \norm{f}_k \}$, but it cannot be defined by any single norm. This space is therefore a Fr\'echet space \cite{ReedSimonVol1, Horvath2012},
not a Hilbert space. Such vector spaces are familiar from the rigged Hilbert space formalism of quantum mechanics
\cite{Antoine1998, Antoine2009b1, Bohm1993, Bohm1978, Bogolubov1975, Ballentine2015}, which can be used to provide a rigorous justification for the Dirac bra-ket formalism.
The subscript on $\mathcal{F}_{\psi}$ is intended to suggest that this space is a suitable arena for the algebra of the $\psi$ product. To show this, we need to prove that $\ket{h} = \ket{f} \odot \ket{g}$ belongs to $\mathcal{F}_{\psi}$ whenever $\ket{f}$ and $\ket{g}$ do. In other words, we must show that $\ket{h} \in \mathcal{H}_k$ for all $k \in \mathbb{N}_{+}$ whenever $\ket{f} \in \mathcal{H}_q$ and $\ket{g} \in \mathcal{H}_q$ for all $q \in \mathbb{N}_{+}$. But this is easily done, since Eq.\ (\ref{eq:Schwarz_Fz_k}) gives inequalities $\abs{F(z)} \le \exp (\abs{z}^2 / 2 q^2) \norm{f}$ and $\abs{G(z)} \le \exp (\abs{z}^2 / 2 q^2) \norm{g}$; the product $H(z) = F(z) G(z)$ thus satisfies $\abs{H (z)} \le A \exp (\lambda \abs{z}^2 / 2 k^2)$, where $A = \norm{f} \norm{g}$ and $\lambda = 2 k^2 / q^2$. According to Eq.\ (\ref{eq:F_in_Hk}), this implies that $\ket{h} \in \mathcal{H}_k$ as long as we are free to choose $\lambda < 1$, i.e., $q > \sqrt{2} k$. But this can be done for any $k \in \mathbb{N}_{+}$, by the definition of $\mathcal{F}_{\psi}$.
It should be clear from the above derivation that the algebra of the $\psi$ product cannot be accommodated within any vector space defined by a finite number of norms. Hence, the move from Hilbert space to Fr\'echet space is necessary for boson systems.
What type of vectors belong to $\mathcal{F}_{\psi}$? It was noted above that $F(z) = \exp (\frac12 \gamma z^2 + \alpha \cdot z)$ is in $\mathcal{H}_1$ if and only if $\abs{\gamma}^2 < 1$. However, Eqs.\ (\ref{eq:Schwarz_Fz_k}) and (\ref{eq:F_in_Hk}) show that it belongs to $\mathcal{F}_{\psi}$ if and only if $\gamma = 0$. The only exponential functions in $\mathcal{F}_{\psi}$ are therefore those of the form $F(z) = \exp (\alpha \cdot z)$, for arbitrary $\alpha \in \mathbb{C}^b$. But these are just the coherent states \begin{equation} \ket{\alpha} = \exp (\alpha \cdot a^{\dagger}) \ket{0} , \label{eq:coherent} \end{equation} which can be defined as eigenvectors of the boson annihilation operators $a_i$ (i.e., $a_i \ket{\alpha} = \alpha_i \ket{\alpha}$) \cite{Glauber1963c}. Note that the operator $\exp (\alpha \cdot a^{\dagger})$ in Eq.\ (\ref{eq:coherent}) is easy to invert; its inverse is $\exp (-\alpha \cdot a^{\dagger})$.
Bargmann called the functions $F (z) = \exp (\alpha \cdot z)$ ``principal vectors'' and showed that they are complete (although not orthogonal), in the sense that finite linear combinations of them are dense in $\mathcal{H}_1$ \cite{Bargmann1961}. This completeness is usually expressed as the integral \cite{Klauder1960, Glauber1963c} \begin{equation} \frac{1}{\pi^b} \int \outprod{\alpha}{\alpha} \exp (- \abs{\alpha}^2) \, \mathrm{d}^{2b} \alpha = 1 . \end{equation} The monomials $F(z) = (n!)^{-1/2} z^n$ also form a complete orthonormal basis \cite{Bargmann1961}, corresponding to the original basis $\ket{n} = (n !)^{-1/2} (a^{\dagger})^n \ket{0}$ in Fock space.
Finally, note from Eq.\ (\ref{eq:Fk_def}) that if $\ket{f} \in \mathcal{F}_{\psi}$, then as $\abs{n} \to \infty$, $\inprod{n}{f}$ must decrease faster than $\exp (- \kappa \abs{n})$ for any positive value of $\kappa$. This rate of decrease is even faster than that of the sequences of rapid descent encountered in connection with Schwartz spaces $\mathcal{S}$ \cite{Gelfand1968, ReedSimonVol1, Simon1971, Bogolubov1975, Bohm1978}.
\section{Different types of particles}
\label{app:different}
Consider a system containing two types of particles, labeled $A$ and $B$. If the corresponding single-particle Hilbert spaces are $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$, the vector space of the whole system can be defined as the tensor-product space \begin{equation} \mathcal{G} = \mathcal{F}_{s} (\mathcal{H}_{A}) \otimes \mathcal{F}_{s} (\mathcal{H}_{B}) . \label{eq:combined_space} \end{equation} That is, a general vector $\ket{u} \in \mathcal{G}$ is a linear combination of tensor products $\ket{u_{A}} \otimes \ket{u_{B}}$, where $\ket{u_{A}} \in \mathcal{F}_{s} (\mathcal{H}_{A})$ and $\ket{u_{B}} \in \mathcal{F}_{s} (\mathcal{H}_{B})$.
One can define a $\psi$ product in $\mathcal{G}$ by letting the $\psi$ product of Sec.\ \ref{sec:psi_product} act in parallel on the subspaces $\mathcal{F}_{s} (\mathcal{H}_{A})$ and $\mathcal{F}_{s} (\mathcal{H}_{B})$. That is, the $\psi$ product of two simple tensor products $\ket{u} = \ket{u_{A}} \otimes \ket{u_{B}}$ and $\ket{v} = \ket{v_{A}} \otimes \ket{v_{B}}$ is defined to be
\begin{multline} (\ket{u_{A}} \otimes \ket{u_{B}}) \odot (\ket{v_{A}} \otimes \ket{v_{B}}) \\ = (\ket{u_{A}} \odot \ket{v_{A}}) \otimes (\ket{u_{B}} \odot \ket{v_{B}}) . \end{multline}
This is then extended to arbitrary vectors $\ket{u}, \ket{v} \in \mathcal{G}$ by multilinearity. The algebra thus defined is associative, which follows directly from the associativity of the $\psi$ product in $\mathcal{F}_{s} (\mathcal{H}_{A})$ and $\mathcal{F}_{s} (\mathcal{H}_{B})$.
From this definition, it is a simple exercise to show that the cluster decomposition property of Theorem \ref{thm:cluster} is valid in $\mathcal{G}$ if it holds in both $\mathcal{F}_{s} (\mathcal{H}_{A})$ and $\mathcal{F}_{s} (\mathcal{H}_{B})$. The equivalence between the algebra of the $\psi$ product and the algebra of creation operators discussed in Sec.\ \ref{sec:creation} likewise remains valid in systems with more than one type of particle.
\section{Invertibility theorem}
\label{app:invertibility_theorem}
The first step in the proof of Theorem \ref{thm:invertible} is to show that a boson-fermion creator $U : \mathcal{E} \to \mathcal{E}$ is invertible if and only if its associated boson creator $U_0 : \mathcal{F}_{\psi} (\mathcal{H}_{\mathrm{b}}) \to \mathcal{F}_{\psi} (\mathcal{H}_{\mathrm{b}})$ is invertible. The basic reason for this can be seen intuitively from the matrix of creators (\ref{eq:U_bf}). The determinant of this triangular matrix is $\det U = (U_0)^{2^d}$, in which $d = \dim \mathcal{H}_{\mathrm{f}}$. If we assume that the standard theorems of matrix algebra can be extended to this matrix of commuting operators, then $U^{-1}$ exists if and only if $U_0^{-1}$ exists.
A more explicit argument is as follows. Because $U_0$ is a boson creator, we can use the multi-index notation of Appendix \ref{app:rigged} to write $U_0 = F (a^{\dagger})$, in which $F(z)$ is an entire function of $z \in \mathbb{C}^b$ and $b = \dim \mathcal{H}_{\mathrm{b}}$. Invertibility of $U_0$ thus requires that $1 / F(z)$ is also an entire function, or that $F(z) \ne 0$ for finite $z$. But if $F(z) = 0$ at $z = \alpha^{*}$, the coherent state $\ket{\alpha}$ in Eq.\ (\ref{eq:coherent}) is orthogonal to every vector in the image of $U_0$, as shown below. Hence, $U_0$ cannot be surjective (or onto) in this case. This also implies that $U$ is not surjective, because $\ket{\alpha} \otimes \ket{0}_{\mathrm{f}}$ is orthogonal to the image of $U$. Invertibility of $U_0$ is therefore necessary for invertibility of $U$. Its sufficiency follows immediately from Eqs.\ (\ref{eq:Z}) and (\ref{eq:U_inverse_series}).
To clarify the orthogonality relation mentioned above, let us start by writing $U_0^{\dagger} = \tilde{F} (a)$, in which $a = (a_1, \ldots, a_b)$ is a vector of boson annihilation operators and $\tilde{F} (z^{*}) \equiv [F(z)]^{*}$ is an entire function of $z^{*}$. Given that $F(z) = 0$ at $z = \alpha^{*}$, we have $\tilde{F} (\alpha) = 0$ and thus $\ket{U_0^{\dagger} \alpha} \equiv U_0^{\dagger} \ket{\alpha} = \tilde{F} (a) \ket{\alpha} = \tilde{F} (\alpha) \ket{\alpha} = 0$, because $\ket{\alpha}$ is an eigenket of $a$. For any $\ket{x} \in \mathcal{F}_{\psi} (\mathcal{H}_{\mathrm{b}})$ we then have $\inprod{U_0^{\dagger} \alpha}{x} = \matelm{\alpha}{U_0}{x} = 0$. But $\matelm{\alpha}{U_0}{x} = 0$ for all $\ket{x} \in \mathcal{F}_{\psi} (\mathcal{H}_{\mathrm{b}})$ is precisely the statement that $\ket{\alpha}$ is orthogonal to every vector in the image of $U_0$.
The second step in the proof of Theorem \ref{thm:invertible} is to show that $U_0$ is invertible if and only if $\ket{u_0}$ is a coherent state. The starting point is the condition $F(z) \ne 0$ established above. Now it is well known in the theory of a single complex variable $z \in \mathbb{C}$ that every entire function $F(z)$ with no zeros can be written as $F(z) = \exp [G (z)]$, where $G (z)$ is another entire function \cite{[] [{, Part II, p.\ 3.}] Knopp1996, [] [{, pp.\ 123--124, 376.}] Lang1999, [] [{, pp.\ 27, 206.}] Forster1981}. This is equivalent to the existence of a global logarithm of such a function $F(z)$, which depends essentially on whether the domain of $F$ is simply connected. The single-variable proof given in Ref.\ \cite{Forster1981} can also be extended to the case of entire functions of several complex variables $z \in \mathbb{C}^b$ \cite{[] [{, p.\ 248.}] Kaup1983}. In order for $U_0$ to be invertible, it is therefore necessary that $F (z) = \exp [G (z)]$ for some entire function $G(z)$.
However, according to the results of Appendix \ref{app:rigged}, if $\ket{u_0} \in \mathcal{F}_{\psi} (\mathcal{H}_{\mathrm{b}})$, then $G(z)$ can only be a linear function of $z$. That is, $F (z)$ must be proportional to $\exp(\alpha \cdot z)$ for some $\alpha \in \mathbb{C}^b$, and $\ket{u_0}$ must be proportional to one of the coherent states $\ket{\alpha}$ defined in Eq.\ (\ref{eq:coherent}). The necessity of $\ket{u_0}$ being a coherent state is therefore established.
To demonstrate its sufficiency, we only need to note that the operator $\exp (\alpha \cdot a^{\dagger})$ appearing in Eq.\ (\ref{eq:coherent}) is invertible, its inverse being given by $\exp (-\alpha \cdot a^{\dagger})$. Thus, $U_0$ is invertible whenever $\ket{u_0}$ is a coherent state. This concludes the proof of Theorem \ref{thm:invertible}.
\section{Creator identities}
\label{app:creator_identities}
A useful identity for the symmetrized product of three creators $A$, $B$, and $C$ is \begin{equation} \{ A, \{ B, C \} \} = \{ \{ A, B \} , C \} . \label{eq:ABC_sym} \end{equation} This can be derived simply by writing out the definition of the symmetrized products, which leads to the general operator identity \begin{equation} \{ A, \{ B, C \} \} - \{ \{ A, B \} , C \} = \frac14 [[ A, C ] , B ] . \label{eq:ABC_op_iden} \end{equation} Given that $A$, $B$, and $C$ are creators, the right-hand side vanishes due to Eq.\ (\ref{eq:creator_commutator}), yielding the identity in Eq.\ (\ref{eq:ABC_sym}).
A useful corollary of this identity is the equivalence \begin{equation} B = \{ U, A \} \quad \Leftrightarrow \quad A = \{ U^{-1}, B \} , \label{eq:switch_basis} \end{equation} in which $A$ and $B$ are creators and $U$ is an invertible creator. For example, the leftward implication can be derived from \begin{equation} \{ U, A \} = \{ U, \{ U^{-1}, B \} \} = \{ \{ U, U^{-1} \} , B \} = B , \end{equation} since $\{ U, U^{-1} \} = 1$.
\section{Proof that \texorpdfstring{$\chi$}{chi} has a global maximum}
\label{app:chi_maximum}
In Sec.\ \ref{sec:time_independent_psi} it was shown that, for a given value of $\Delta t$, the dynamical stability functional has only one stationary state with $\chi > 0$. To prove that this is indeed the global maximum of $\chi$, we can follow the approach used in Eq.\ (\ref{eq:imag_inequality}) to obtain the inequality \begin{equation} \chi = \frac{(\imag \inprod{\Delta x}{\sigma})^2}{\matelm{\Delta x}{\hat{\eta}}{\Delta x}} \le \frac{\matelm{\Delta x}{\Sigma}{\Delta x}}{\matelm{\Delta x}{\hat{\eta}}{\Delta x}} \equiv \gamma , \label{eq:chi_gamma} \end{equation} in which $\Sigma \equiv \outprod{\sigma}{\sigma}$. Varying the functional $\gamma$ leads to the generalized eigenvalue equation \begin{equation} \Sigma \ket{\Delta x} = \gamma \hat{\eta} \ket{\Delta x} , \end{equation} which is well defined because $\hat{\eta} > 0$. However, because $\Sigma$ is a projector of rank one, it has only one eigenvector with eigenvalue $\gamma > 0$. This is just \begin{equation} \ket{\Delta x} = -i C \hat{\eta}^{-1} \ket{\sigma} , \label{eq:gamma_eigenvector} \end{equation} where $C$ is an arbitrary \emph{complex} number. Since the functional $\gamma$ is bounded from above by its maximum eigenvalue, this eigenvalue is the global maximum of $\gamma$.
Looking back now at Eq.\ (\ref{eq:chi_gamma}), we see that choosing $C$ to be \emph{real} turns the inequality $\chi \le \gamma$ into an equality, and also makes the eigenvector (\ref{eq:gamma_eigenvector}) identical to the stationary state (\ref{eq:Dx_soln}) of $\chi$. Hence, the global maximum of $\gamma$ is also the global maximum of $\chi$, and the conjecture is proved.
\section{Distance between phase orbits}
\label{app:phase_orbit_distance}
The calculation of $D^2 ([\rho], [\rho'])$ in Sec.\ \ref{sec:phase_orbit_distance} was based on the assumption that $\norm{\Delta u}$ is small. If this is not true, we must return to Eqs.\ (\ref{eq:D2_lambda}) and (\ref{eq:lambda_phi}) and calculate the function \begin{align} \lambda (\phi) & = m - \tr (\rho e^{i \hat{N} \phi} \rho' e^{-i \hat{N} \phi}) \\ & = m - \sum_{k=1}^{m} \tr (\rho_k e^{i N \phi} \rho_k' e^{-i N \phi}) \end{align} without any approximations. This can be done by using the resolution of the identity $\sum_{n} \Pi_{n} = 1$, in which $\Pi_{n}$ is the projector for the $n$-particle subspace [cf.\ Eq.\ (\ref{eq:symmetrizer})]. The result is \begin{equation} \lambda (\phi) = m - \sum_{n,n'} M_{n n'} e^{i (n - n') \phi} , \label{eq:lambda_phi_M} \end{equation} in which \begin{align} M_{n n'} & \equiv \sum_{k=1}^{m} \tr (\rho_k \Pi_{n} \rho_k' \Pi_{n'}) \label{eq:M_rho} \\ & = \sum_{k=1}^{m} \frac{\matelm{u_k}{\Pi_{n}}{u_k'} \matelm{u_k'}{\Pi_{n'}}{u_k}}{\inprod{u_k}{u_k} \inprod{u_k'}{u_k'}} . \end{align} This matrix is hermitian, as can be seen from Eq.\ (\ref{eq:M_rho}). The function (\ref{eq:lambda_phi_M}) can therefore be written as $\lambda (\phi) = m - G_0 + 2 g (\phi)$, in which $G_l \equiv \sum_{n} M_{n+l, n}$ and \begin{equation} g (\phi) = - \sum_{l>0} \real (G_l) \cos (l \phi) + \sum_{l>0} \imag (G_l) \sin (l \phi) . \end{equation} Hence, in Eq.\ (\ref{eq:D2_lambda}), minimizing $\lambda (\phi)$ is the same as minimizing $g (\phi)$. This is easy to do in fermion systems with small $d = \dim \mathcal{H}_{\mathrm{f}}$, because $l \le d$. The minimum of $g (\phi)$ can then be found quickly using a simple grid search and Newton's method.
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\cite{Miller1990},\ pp.\ \bibinfo {pages} {259--280}\BibitemShut {NoStop} \bibitem [{\citenamefont {Primas}(1991)}]{Primas1991}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
{Primas}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Necessary and
sufficient conditions for an individual description of the measurement
process},}\ }in\ \href@noop {} {\emph {\bibinfo {booktitle} {Symposium on the
Foundations of Modern Physics 1990: Quantum Theory of Measurement and Related
Philosophical Problems}}},\ \bibinfo {editor} {edited by\ \bibinfo {editor}
{\bibfnamefont {P.}~\bibnamefont {Lahti}}\ and\ \bibinfo {editor}
{\bibfnamefont {P.}~\bibnamefont {Mittelstaedt}}}\ (\bibinfo {publisher}
{World Scientific},\ \bibinfo {address} {Singapore},\ \bibinfo {year}
{1991})\ pp.\ \bibinfo {pages} {332--346}\BibitemShut {NoStop} \bibitem [{\citenamefont {Primas}(1993)}]{Primas1993}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
{Primas}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The {Cartesian}
cut, the {Heisenberg} cut, and disentangled observers},}\ }in\ \href@noop {}
{\emph {\bibinfo {booktitle} {Symposia on the Foundations of Modern Physics
1992: The Copenhagen Interpretation and Wolfgang Pauli}}},\ \bibinfo {editor}
{edited by\ \bibinfo {editor} {\bibfnamefont {K.~V.}\ \bibnamefont
{Laurikainen}}\ and\ \bibinfo {editor} {\bibfnamefont {C.}~\bibnamefont
{Montonen}}}\ (\bibinfo {publisher} {World Scientific},\ \bibinfo {address}
{Singapore},\ \bibinfo {year} {1993})\ pp.\ \bibinfo {pages}
{245--269}\BibitemShut {NoStop} \bibitem [{\citenamefont {Primas}(1994{\natexlab{a}})}]{Primas1994a}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
{Primas}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Realism and
quantum mechanics},}\ }in\ \href@noop {} {\emph {\bibinfo {booktitle} {Logic,
Methodology and Philosophy of Science IX}}},\ \bibinfo {editor} {edited by\
\bibinfo {editor} {\bibfnamefont {D.}~\bibnamefont {Prawitz}}, \bibinfo
{editor} {\bibfnamefont {B.}~\bibnamefont {Skyrms}}, \ and\ \bibinfo {editor}
{\bibfnamefont {D.}~\bibnamefont {Westerst{\aa}hl}}}\ (\bibinfo {publisher}
{Elsevier},\ \bibinfo {address} {Amsterdam},\ \bibinfo {year} {1994})\ pp.\
\bibinfo {pages} {609--631}\BibitemShut {NoStop} \bibitem [{\citenamefont {Primas}(1994{\natexlab{b}})}]{Primas1994b}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
{Primas}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Endo- and
exo-theories of matter},}\ }in\ \href {\doibase 10.1007/978-3-642-48647-0}
{\emph {\bibinfo {booktitle} {Inside Versus Outside}}},\ \bibinfo {editor}
{edited by\ \bibinfo {editor} {\bibfnamefont {H.}~\bibnamefont
{Atmanspacher}}\ and\ \bibinfo {editor} {\bibfnamefont {G.~J.}\ \bibnamefont
{Dalenoort}}}\ (\bibinfo {publisher} {Springer},\ \bibinfo {address}
{Berlin},\ \bibinfo {year} {1994})\ pp.\ \bibinfo {pages}
{163--193}\BibitemShut {NoStop} \bibitem [{\citenamefont {Primas}(2000)}]{Primas2000}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
{Primas}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Asymptotically
disjoint quantum states},}\ }in\ \href {\doibase 10.1007/3-540-46657-6}
{\emph {\bibinfo {booktitle} {Decoherence: Theoretical, Experimental, and
Conceptual Problems}}},\ \bibinfo {editor} {edited by\ \bibinfo {editor}
{\bibfnamefont {P.}~\bibnamefont {Blanchard}}, \bibinfo {editor}
{\bibfnamefont {E.}~\bibnamefont {Joos}}, \bibinfo {editor} {\bibfnamefont
{D.}~\bibnamefont {Giulini}}, \bibinfo {editor} {\bibfnamefont
{C.}~\bibnamefont {Kiefer}}, \ and\ \bibinfo {editor} {\bibfnamefont {I.-O.}\
\bibnamefont {Stamatescu}}}\ (\bibinfo {publisher} {Springer},\ \bibinfo
{address} {Berlin},\ \bibinfo {year} {2000})\ pp.\ \bibinfo {pages}
{161--178}\BibitemShut {NoStop} \bibitem [{\citenamefont {Amann}(1988)}]{Amann1988}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Amann}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Chirality as a
classical observable in algebraic quantum mechanics},}\ }in\ \href {\doibase
10.1007/978-94-009-3005-6} {\emph {\bibinfo {booktitle} {Fractals,
Quasicrystals, Chaos, Knots and Algebraic Quantum Mechanics}}},\ \bibinfo
{editor} {edited by\ \bibinfo {editor} {\bibfnamefont {A.}~\bibnamefont
{Amann}}, \bibinfo {editor} {\bibfnamefont {L.}~\bibnamefont {Cederbaum}}, \
and\ \bibinfo {editor} {\bibfnamefont {W.}~\bibnamefont {Gans}}}\ (\bibinfo
{publisher} {Kluwer},\ \bibinfo {address} {Dordrecht},\ \bibinfo {year}
{1988})\ pp.\ \bibinfo {pages} {305--325}\BibitemShut {NoStop} \bibitem [{\citenamefont {Amann}(1991{\natexlab{a}})}]{Amann1991a}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Amann}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Chirality: A
superselection rule generated by the molecular environment?}}\ }\href
{\doibase 10.1007/BF01192570} {\bibfield {journal} {\bibinfo {journal} {J.
Math. Chem.}\ }\textbf {\bibinfo {volume} {6}},\ \bibinfo {pages} {1--15}
(\bibinfo {year} {1991}{\natexlab{a}})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Amann}(1991{\natexlab{b}})}]{Amann1991b}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Amann}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Molecules coupled
to their environment},}\ }in\ \cite{GansBlumenAmann1991},\ pp.\ \bibinfo
{pages} {3--22}\BibitemShut {NoStop} \bibitem [{\citenamefont {Amann}(1991{\natexlab{c}})}]{Amann1991c}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Amann}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Theories of
molecular chirality: A short review},}\ }in\ \cite{GansBlumenAmann1991},\
pp.\ \bibinfo {pages} {23--32}\BibitemShut {NoStop} \bibitem [{\citenamefont {Amann}(1993)}]{Amann1993}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Amann}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The {Gestalt}
problem in quantum theory: Generation of molecular shape by the
environment},}\ }\href {\doibase 10.1007/BF01255834} {\bibfield {journal}
{\bibinfo {journal} {Synthese}\ }\textbf {\bibinfo {volume} {97}},\ \bibinfo
{pages} {125--156} (\bibinfo {year} {1993})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Amann}(1995)}]{Amann1995}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Amann}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Structure,
dynamics and spectroscopy of single molecules: A challenge to quantum
mechanics},}\ }\href {\doibase 10.1007/BF01164662} {\bibfield {journal}
{\bibinfo {journal} {J. Math. Chem.}\ }\textbf {\bibinfo {volume} {18}},\
\bibinfo {pages} {247--308} (\bibinfo {year} {1995})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Amann}\ and\ \citenamefont
{Primas}(1997)}]{AmannPrimas1997b}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Amann}}\ and\ \bibinfo {author} {\bibfnamefont {H.}~\bibnamefont {Primas}},\
}\bibfield {title} {\enquote {\bibinfo {title} {What is the referent of a
nonpure quantum state?}}\ }in\ \cite{CohenHorneStachel1997vol2},\ pp.\
\bibinfo {pages} {9--29}\BibitemShut {NoStop} \bibitem [{\citenamefont {Amann}\ and\ \citenamefont
{Atmanspacher}(1999)}]{AmannAtmanspacher1999}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Amann}}\ and\ \bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
{Atmanspacher}},\ }\bibfield {title} {\enquote {\bibinfo {title} {{$C^*$}-
and {$W^*$}-algebras of observables, their interpretations, and the problem
of measurement},}\ }in\ \href {\doibase 10.1007/978-94-011-4581-7} {\emph
{\bibinfo {booktitle} {On Quanta, Mind and Matter: Hans Primas in
Context}}},\ \bibinfo {editor} {edited by\ \bibinfo {editor} {\bibfnamefont
{H.}~\bibnamefont {Atmanspacher}}, \bibinfo {editor} {\bibfnamefont
{A.}~\bibnamefont {Amann}}, \ and\ \bibinfo {editor} {\bibfnamefont
{U.}~\bibnamefont {M\"uller-Herold}}}\ (\bibinfo {publisher} {Kluwer},\
\bibinfo {address} {Dordrecht},\ \bibinfo {year} {1999})\ pp.\ \bibinfo
{pages} {57--79}\BibitemShut {NoStop} \bibitem [{\citenamefont {Nozi{\` e}res}(1997)}]{Nozieres1997}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
{Nozi{\` e}res}},\ }\href@noop {} {\emph {\bibinfo {title} {Theory of
Interacting Fermi Systems}}}\ (\bibinfo {publisher} {Westview},\ \bibinfo
{address} {Boulder, Colorado},\ \bibinfo {year} {1997})\BibitemShut {NoStop} \bibitem [{\citenamefont {Kaxiras}(2003)}]{Kaxiras2003}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
{Kaxiras}},\ }\href {\doibase 10.1017/CBO9780511755545} {\emph {\bibinfo
{title} {Atomic and Electronic Structure of Solids}}}\ (\bibinfo {publisher}
{Cambridge University Press},\ \bibinfo {year} {2003})\BibitemShut {NoStop} \bibitem [{\citenamefont {Jain}(2007)}]{Jain2007}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~K.}\ \bibnamefont
{Jain}},\ }\href {\doibase 10.1017/CBO9780511607561} {\emph {\bibinfo {title}
{Composite Fermions}}}\ (\bibinfo {publisher} {Cambridge University Press},\
\bibinfo {year} {2007})\BibitemShut {NoStop} \bibitem [{\citenamefont {Weinberg}(2005)}]{WeinbergVol1}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
{Weinberg}},\ }\href {\doibase 10.1017/CBO9781139644167} {\emph {\bibinfo
{title} {The Quantum Theory of Fields}}},\ Vol.~\bibinfo {volume} {1}\
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\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
{Bell}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Against
`measurement'},}\ }\href
{http://physicsworldarchive.iop.org/index.cfm?action=summary&doc=3
{\bibfield {journal} {\bibinfo {journal} {Phys. World}\ }\textbf {\bibinfo
{volume} {3}},\ \bibinfo {pages} {33--40} (\bibinfo {year} {1990})},\
\bibinfo {note} {{A}ugust; reprinted in Ref.\ \cite{Bell2004}.}\BibitemShut
{Stop} \bibitem [{\citenamefont {Bell}(1987)}]{Bell1987}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~S.}\ \bibnamefont
{Bell}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Beables for
quantum field theory},}\ }in\ \href@noop {} {\emph {\bibinfo {booktitle}
{Quantum Implications: Essays in Honour of David Bohm}}},\ \bibinfo {editor}
{edited by\ \bibinfo {editor} {\bibfnamefont {B.~J.}\ \bibnamefont {Hiley}}\
and\ \bibinfo {editor} {\bibfnamefont {F.~D.}\ \bibnamefont {Peat}}}\
(\bibinfo {publisher} {Routledge},\ \bibinfo {address} {London},\ \bibinfo
{year} {1987})\ pp.\ \bibinfo {pages} {227--234},\ \bibinfo {note} {reprinted
in Ref.\ \cite{Bell2004}.}\BibitemShut {Stop} \bibitem [{\citenamefont {Barbour}\ and\ \citenamefont
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~B.}\ \bibnamefont
{Barbour}}\ and\ \bibinfo {author} {\bibfnamefont {B.}~\bibnamefont
{Bertotti}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Mach{\textquoteright}s principle and the structure of dynamical theories},}\
}\href {\doibase 10.1098/rspa.1982.0102} {\bibfield {journal} {\bibinfo
{journal} {Proc. R. Soc. A}\ }\textbf {\bibinfo {volume} {382}},\ \bibinfo
{pages} {295--306} (\bibinfo {year} {1982})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Page}\ and\ \citenamefont
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~N.}\ \bibnamefont
{Page}}\ and\ \bibinfo {author} {\bibfnamefont {W.~K.}\ \bibnamefont
{Wootters}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Evolution
without evolution: Dynamics described by stationary observables},}\ }\href
{\doibase 10.1103/PhysRevD.27.2885} {\bibfield {journal} {\bibinfo
{journal} {Phys. Rev. D}\ }\textbf {\bibinfo {volume} {27}},\ \bibinfo
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\BibitemOpen
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{Wootters}},\ }\bibfield {title} {\enquote {\bibinfo {title} {`{Time}'
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{Unruh}}\ and\ \bibinfo {author} {\bibfnamefont {R.~M.}\ \bibnamefont
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{Pegg}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Time in a quantum
mechanical world},}\ }\href {\doibase 10.1088/0305-4470/24/13/018} {\bibfield
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gravity and the problem of time},}\ } (\bibinfo {year} {1992}),\ \Eprint
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entropy},}\ }in\ \cite{Halliwell1994},\ pp.\ \bibinfo {pages}
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timelessness of quantum gravity: {I}. {The} evidence from the classical
theory},}\ }\href {\doibase 10.1088/0264-9381/11/12/005} {\bibfield
{journal} {\bibinfo {journal} {Class. Quantum Grav.}\ }\textbf {\bibinfo
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{1994}{\natexlab{a}})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Barbour}(1994{\natexlab{b}})}]{Barbour1994b}
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timelessness of quantum gravity: {II}. {The} appearance of dynamics in static
configurations},}\ }\href {\doibase 10.1088/0264-9381/11/12/006} {\bibfield
{journal} {\bibinfo {journal} {Class. Quantum Grav.}\ }\textbf {\bibinfo
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problem of time in quantum mechanics and quantum gravity: a fundamental
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Phys.}\ }\textbf {\bibinfo {volume} {6}},\ \bibinfo {pages} {45} (\bibinfo
{year} {2004}{\natexlab{a}})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Gambini}\ \emph
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{Gambini}}, \bibinfo {author} {\bibfnamefont {R.~A.}\ \bibnamefont {Porto}},
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}\bibfield {title} {\enquote {\bibinfo {title} {Realistic clocks, universal
decoherence, and the black hole information paradox},}\ }\href {\doibase
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{Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {93}},\ \bibinfo {pages}
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{Gambini}, \citenamefont {Porto}, \citenamefont {Pullin},\ and\ \citenamefont
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\BibitemOpen
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{Gambini}}, \bibinfo {author} {\bibfnamefont {R.~A.}\ \bibnamefont {Porto}},
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ambiguity and the emergence of physical laws},}\ }\href {\doibase
10.1103/PhysRevD.77.063506} {\bibfield {journal} {\bibinfo {journal} {Phys.
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conditional probability and semiclassical interpretations for the problem of
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\citenamefont {Brida}, \citenamefont {Gramegna}, \citenamefont {Giovannetti},
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{Moreva}}, \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Brida}},
\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Gramegna}}, \bibinfo
{author} {\bibfnamefont {V.}~\bibnamefont {Giovannetti}}, \bibinfo {author}
{\bibfnamefont {L.}~\bibnamefont {Maccone}}, \ and\ \bibinfo {author}
{\bibfnamefont {M.}~\bibnamefont {Genovese}},\ }\bibfield {title} {\enquote
{\bibinfo {title} {Time from quantum entanglement: An experimental
illustration},}\ }\href {\doibase 10.1103/PhysRevA.89.052122} {\bibfield
{journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume}
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{Fuchs}}\ and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Peres}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Quantum theory needs no
`interpretation'},}\ }\href {\doibase 10.1063/1.883004} {\bibfield {journal}
{\bibinfo {journal} {Phys. Today}\ }\textbf {\bibinfo {volume} {53}},\
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\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.~A.}\ \bibnamefont
{Fuchs}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Quantum mechanics
as quantum information, mostly},}\ }\href {\doibase
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Mod. Opt.}\ }\textbf {\bibinfo {volume} {50}},\ \bibinfo {pages} {987--1023}
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\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.~A.~M.}\
\bibnamefont {Dirac}},\ }\href@noop {} {\emph {\bibinfo {title} {The
Principles of Quantum Mechanics}}},\ \bibinfo {edition} {4th}\ ed.\ (\bibinfo
{publisher} {Oxford University Press},\ \bibinfo {year} {1958})\BibitemShut
{NoStop} \bibitem [{\citenamefont {von Neumann}(1955)}]{vonNeumann1955}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {von
Neumann}},\ }\href@noop {} {\emph {\bibinfo {title} {Mathematical Foundations
of Quantum Mechanics}}}\ (\bibinfo {publisher} {Princeton University
Press},\ \bibinfo {year} {1955})\BibitemShut {NoStop} \bibitem [{\citenamefont {Wheeler}(1973)}]{Wheeler1973}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~A.}\ \bibnamefont
{Wheeler}},\ }\bibfield {title} {\enquote {\bibinfo {title} {From relativity
to mutability},}\ }in\ \href {\doibase 10.1007/978-94-010-2602-4} {\emph
{\bibinfo {booktitle} {The Physicist's Conception of Nature}}},\ \bibinfo
{editor} {edited by\ \bibinfo {editor} {\bibfnamefont {J.}~\bibnamefont
{Mehra}}}\ (\bibinfo {publisher} {Reidel},\ \bibinfo {address} {Dordrecht},\
\bibinfo {year} {1973})\ pp.\ \bibinfo {pages} {202--247}\BibitemShut
{NoStop} \bibitem [{\citenamefont {Wick}\ \emph {et~al.}(1952)\citenamefont {Wick},
\citenamefont {Wightman},\ and\ \citenamefont
{Wigner}}]{WickWightmanWigner1952}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~C.}\ \bibnamefont
{Wick}}, \bibinfo {author} {\bibfnamefont {A.~S.}\ \bibnamefont {Wightman}},
\ and\ \bibinfo {author} {\bibfnamefont {E.~P.}\ \bibnamefont {Wigner}},\
}\bibfield {title} {\enquote {\bibinfo {title} {The intrinsic parity of
elementary particles},}\ }\href {\doibase 10.1103/PhysRev.88.101} {\bibfield
{journal} {\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume}
{88}},\ \bibinfo {pages} {101--105} (\bibinfo {year} {1952})}\BibitemShut
{NoStop} \bibitem [{\citenamefont {Hegerfeldt}\ \emph {et~al.}(1968)\citenamefont
{Hegerfeldt}, \citenamefont {Kraus},\ and\ \citenamefont
{Wigner}}]{HegerfeldtKrausWigner1968}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~C.}\ \bibnamefont
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}\bibfield {title} {\enquote {\bibinfo {title} {Proof of the fermion
superselection rule without the assumption of time-reversal invariance},}\
}\href {\doibase 10.1063/1.1664539} {\bibfield {journal} {\bibinfo
{journal} {J. Math. Phys.}\ }\textbf {\bibinfo {volume} {9}},\ \bibinfo
{pages} {2029--2031} (\bibinfo {year} {1968})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Wick}\ \emph {et~al.}(1970)\citenamefont {Wick},
\citenamefont {Wightman},\ and\ \citenamefont
{Wigner}}]{WickWightmanWigner1970}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~C.}\ \bibnamefont
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\ and\ \bibinfo {author} {\bibfnamefont {E.~P.}\ \bibnamefont {Wigner}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Superselection rule for
charge},}\ }\href {\doibase 10.1103/PhysRevD.1.3267} {\bibfield {journal}
{\bibinfo {journal} {Phys. Rev. D}\ }\textbf {\bibinfo {volume} {1}},\
\bibinfo {pages} {3267--3269} (\bibinfo {year} {1970})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Wightman}(1995)}]{Wightman1995}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~S.}\ \bibnamefont
{Wightman}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Superselection
rules; old and new},}\ }\href {\doibase 10.1007/BF02741478} {\bibfield
{journal} {\bibinfo {journal} {Nuovo Cimento B}\ }\textbf {\bibinfo {volume}
{110}},\ \bibinfo {pages} {751--769} (\bibinfo {year} {1995})}\BibitemShut
{NoStop} \bibitem [{\citenamefont {Aharonov}\ and\ \citenamefont
{Susskind}(1967{\natexlab{a}})}]{AharonovSusskind1967a}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont
{Aharonov}}\ and\ \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
{Susskind}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Charge
superselection rule},}\ }\href {\doibase 10.1103/PhysRev.155.1428} {\bibfield
{journal} {\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume}
{155}},\ \bibinfo {pages} {1428--1431} (\bibinfo {year}
{1967}{\natexlab{a}})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Aharonov}\ and\ \citenamefont
{Susskind}(1967{\natexlab{b}})}]{AharonovSusskind1967b}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont
{Aharonov}}\ and\ \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
{Susskind}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Observability
of the sign change of spinors under $2\ensuremath{\pi}$ rotations},}\ }\href
{\doibase 10.1103/PhysRev.158.1237} {\bibfield {journal} {\bibinfo
{journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {158}},\ \bibinfo {pages}
{1237--1238} (\bibinfo {year} {1967}{\natexlab{b}})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Aharonov}\ and\ \citenamefont
{Rohrlich}(2005)}]{AharonovRohrlich2005}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont
{Aharonov}}\ and\ \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
{Rohrlich}},\ }\href {\doibase 10.1002/9783527619115} {\emph {\bibinfo
{title} {Quantum Paradoxes: Quantum Theory for the Perplexed}}}\ (\bibinfo
{publisher} {Wiley-VCH},\ \bibinfo {address} {Weinheim},\ \bibinfo {year}
{2005})\BibitemShut {NoStop} \bibitem [{\citenamefont {Mirman}(1969)}]{Mirman1969}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
{Mirman}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Coherent
superposition of charge states},}\ }\href {\doibase 10.1103/PhysRev.186.1380}
{\bibfield {journal} {\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo
{volume} {186}},\ \bibinfo {pages} {1380--1383} (\bibinfo {year}
{1969})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Mirman}(1970)}]{Mirman1970}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
{Mirman}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Analysis of the
experimental meaning of coherent superposition and the nonexistence of
superselection rules},}\ }\href {\doibase 10.1103/PhysRevD.1.3349} {\bibfield
{journal} {\bibinfo {journal} {Phys. Rev. D}\ }\textbf {\bibinfo {volume}
{1}},\ \bibinfo {pages} {3349--3363} (\bibinfo {year} {1970})}\BibitemShut
{NoStop} \bibitem [{\citenamefont {Mirman}(1979)}]{Mirman1979}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
{Mirman}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Nonexistence of
superselection rules: Definition of term \emph{frame of reference}},}\ }\href
{\doibase 10.1007/BF00715184} {\bibfield {journal} {\bibinfo {journal}
{Found. Phys.}\ }\textbf {\bibinfo {volume} {9}},\ \bibinfo {pages}
{283--299} (\bibinfo {year} {1979})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Lubkin}(1970)}]{Lubkin1970}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
{Lubkin}},\ }\bibfield {title} {\enquote {\bibinfo {title} {On violation of
the superselection rules},}\ }\href {\doibase 10.1016/0003-4916(70)90005-9}
{\bibfield {journal} {\bibinfo {journal} {Ann. Phys. (N.Y.)}\ }\textbf
{\bibinfo {volume} {56}},\ \bibinfo {pages} {69--80} (\bibinfo {year}
{1970})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Zeh}(1970)}]{Zeh1970}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.~D.}\ \bibnamefont
{Zeh}},\ }\bibfield {title} {\enquote {\bibinfo {title} {On the
interpretation of measurement in quantum theory},}\ }\href {\doibase
10.1007/BF00708656} {\bibfield {journal} {\bibinfo {journal} {Found.
Phys.}\ }\textbf {\bibinfo {volume} {1}},\ \bibinfo {pages} {69--76}
(\bibinfo {year} {1970})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Zurek}(1982)}]{Zurek1982}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.~H.}\ \bibnamefont
{Zurek}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Environment-induced superselection rules},}\ }\href {\doibase
10.1103/PhysRevD.26.1862} {\bibfield {journal} {\bibinfo {journal} {Phys.
Rev. D}\ }\textbf {\bibinfo {volume} {26}},\ \bibinfo {pages} {1862--1880}
(\bibinfo {year} {1982})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Giulini}\ \emph {et~al.}(1995)\citenamefont
{Giulini}, \citenamefont {Kiefer},\ and\ \citenamefont
{Zeh}}]{GiuliniKieferZeh1995}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
{Giulini}}, \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Kiefer}}, \
and\ \bibinfo {author} {\bibfnamefont {H.~D.}\ \bibnamefont {Zeh}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Symmetries, superselection
rules, and decoherence},}\ }\href {\doibase 10.1016/0375-9601(95)00128-P}
{\bibfield {journal} {\bibinfo {journal} {Phys. Lett. A}\ }\textbf
{\bibinfo {volume} {199}},\ \bibinfo {pages} {291--298} (\bibinfo {year}
{1995})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Giulini}(2003)}]{Giulini2003a}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
{Giulini}},\ }\enquote {\bibinfo {title} {Superselection rules and
symmetries},}\ Chap.~\bibinfo {chapter} {6},\ in\ \cite{JoosZeh2003}
(\bibinfo {year} {2003})\BibitemShut {NoStop} \bibitem [{\citenamefont {Giulini}(2009)}]{Giulini2009a}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
{Giulini}},\ }\enquote {\bibinfo {title} {Superselection rules},}\ pp.\
\bibinfo {pages} {771--779},\ in\ \cite{Greenberger2009} (\bibinfo {year}
{2009})\BibitemShut {NoStop} \bibitem [{\citenamefont {Dowling}\ \emph {et~al.}(2006)\citenamefont
{Dowling}, \citenamefont {Bartlett}, \citenamefont {Rudolph},\ and\
\citenamefont {Spekkens}}]{DowlingBartlettRudolphSpekkens2006}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.~R.}\ \bibnamefont
{Dowling}}, \bibinfo {author} {\bibfnamefont {S.~D.}\ \bibnamefont
{Bartlett}}, \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {Rudolph}}, \
and\ \bibinfo {author} {\bibfnamefont {R.~W.}\ \bibnamefont {Spekkens}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Observing a coherent
superposition of an atom and a molecule},}\ }\href {\doibase
10.1103/PhysRevA.74.052113} {\bibfield {journal} {\bibinfo {journal} {Phys.
Rev. A}\ }\textbf {\bibinfo {volume} {74}},\ \bibinfo {pages} {052113}
(\bibinfo {year} {2006})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Bartlett}\ \emph {et~al.}(2007)\citenamefont
{Bartlett}, \citenamefont {Rudolph},\ and\ \citenamefont
{Spekkens}}]{BartlettRudolphSpekkens2007}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.~D.}\ \bibnamefont
{Bartlett}}, \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {Rudolph}}, \
and\ \bibinfo {author} {\bibfnamefont {R.~W.}\ \bibnamefont {Spekkens}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Reference frames,
superselection rules, and quantum information},}\ }\href {\doibase
10.1103/RevModPhys.79.555} {\bibfield {journal} {\bibinfo {journal} {Rev.
Mod. Phys.}\ }\textbf {\bibinfo {volume} {79}},\ \bibinfo {pages} {555--609}
(\bibinfo {year} {2007})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Earman}(2008)}]{Earman2008}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
{Earman}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Superselection
rules for philosophers},}\ }\href {\doibase 10.1007/s10670-008-9124-z}
{\bibfield {journal} {\bibinfo {journal} {Erkenntnis}\ }\textbf {\bibinfo
{volume} {69}},\ \bibinfo {pages} {377--414} (\bibinfo {year}
{2008})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Jauch}(1964)}]{Jauch1964}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~M.}\ \bibnamefont
{Jauch}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The problem of
measurement in quantum mechanics},}\ }\href {\doibase 10.5169/seals-113486}
{\bibfield {journal} {\bibinfo {journal} {Helv. Phys. Acta}\ }\textbf
{\bibinfo {volume} {37}},\ \bibinfo {pages} {293--316} (\bibinfo {year}
{1964})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Jaynes}(1989)}]{Jaynes1989}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.~T.}\ \bibnamefont
{Jaynes}},\ }\href {\doibase 10.1007/978-94-009-6581-2} {\emph {\bibinfo
{title} {Papers on Probability, Statistics and Statistical Physics}}}\
(\bibinfo {publisher} {Kluwer},\ \bibinfo {address} {Dordrecht},\ \bibinfo
{year} {1989})\BibitemShut {NoStop} \bibitem [{\citenamefont {Jaynes}(2003)}]{Jaynes2003}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.~T.}\ \bibnamefont
{Jaynes}},\ }\href {\doibase 10.1017/CBO9780511790423} {\emph {\bibinfo
{title} {Probability Theory: The Logic of Science}}}\ (\bibinfo {publisher}
{Cambridge University Press},\ \bibinfo {year} {2003})\BibitemShut {NoStop} \bibitem [{\citenamefont {Bernardo}\ and\ \citenamefont
{Smith}(2000)}]{BernardoSmith2000}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~M.}\ \bibnamefont
{Bernardo}}\ and\ \bibinfo {author} {\bibfnamefont {A.~F.~M.}\ \bibnamefont
{Smith}},\ }\href {\doibase 10.1002/9780470316870} {\emph {\bibinfo {title}
{Bayesian Theory}}}\ (\bibinfo {publisher} {Wiley},\ \bibinfo {address}
{Chichester},\ \bibinfo {year} {2000})\BibitemShut {NoStop} \bibitem [{\citenamefont {Appleby}(2005{\natexlab{a}})}]{Appleby2005a}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~M.}\ \bibnamefont
{Appleby}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Facts, values
and quanta},}\ }\href {\doibase 10.1007/s10701-004-2014-6} {\bibfield
{journal} {\bibinfo {journal} {Found. Phys.}\ }\textbf {\bibinfo {volume}
{35}},\ \bibinfo {pages} {627--668} (\bibinfo {year}
{2005}{\natexlab{a}})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Appleby}(2005{\natexlab{b}})}]{Appleby2005b}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~M.}\ \bibnamefont
{Appleby}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Probabilities
are single-case or nothing},}\ }\href {\doibase 10.1134/1.2055942} {\bibfield
{journal} {\bibinfo {journal} {Opt. Spectrosc.}\ }\textbf {\bibinfo
{volume} {99}},\ \bibinfo {pages} {447--456} (\bibinfo {year}
{2005}{\natexlab{b}})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Jeffrey}(2004)}]{Jeffrey2004}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
{Jeffrey}},\ }\href {\doibase 10.1017/CBO9780511816161} {\emph {\bibinfo
{title} {Subjective Probability: The Real Thing}}}\ (\bibinfo {publisher}
{Cambridge University Press},\ \bibinfo {year} {2004})\BibitemShut {NoStop} \bibitem [{\citenamefont {Bohr}(1970)}]{Bohr1949}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.}~\bibnamefont
{Bohr}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Discussion with
{Einstein} on epistemological problems in atomic physics},}\ }in\ \href@noop
{} {\emph {\bibinfo {booktitle} {Albert Einstein: Philosopher--Scientist}}},\
\bibinfo {series} {The Library of Living Philosophers}, Vol.~\bibinfo
{volume} {7},\ \bibinfo {editor} {edited by\ \bibinfo {editor} {\bibfnamefont
{P.~A.}\ \bibnamefont {Schilpp}}}\ (\bibinfo {publisher} {Open Court},\
\bibinfo {address} {La Salle, Illinois},\ \bibinfo {year} {1970})\ \bibinfo
{edition} {3rd}\ ed.,\ pp.\ \bibinfo {pages} {199--241},\ \bibinfo {note}
{reprinted in Ref.\ \cite{BohrVol2}, pp.\ 32--66.}\BibitemShut {Stop} \bibitem [{\citenamefont {Bohr}(1987{\natexlab{a}})}]{BohrVol2}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.}~\bibnamefont
{Bohr}},\ }\href@noop {} {\emph {\bibinfo {title} {Essays 1932--1957 on
Atomic Physics and Human Knowledge}}},\ \bibinfo {series} {The Philosophical
Writings of Niels Bohr}, Vol.~\bibinfo {volume} {2}\ (\bibinfo {publisher}
{Ox Bow},\ \bibinfo {address} {Woodbridge, Conn.},\ \bibinfo {year}
{1987})\BibitemShut {NoStop} \bibitem [{\citenamefont {Caves}\ \emph {et~al.}(2002)\citenamefont {Caves},
\citenamefont {Fuchs},\ and\ \citenamefont {Schack}}]{CavesFuchsSchack2002}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.~M.}\ \bibnamefont
{Caves}}, \bibinfo {author} {\bibfnamefont {C.~A.}\ \bibnamefont {Fuchs}}, \
and\ \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Schack}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Quantum probabilities as
{Bayesian} probabilities},}\ }\href {\doibase 10.1103/PhysRevA.65.022305}
{\bibfield {journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo
{volume} {65}},\ \bibinfo {pages} {022305} (\bibinfo {year}
{2002})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Caves}\ \emph {et~al.}(2007)\citenamefont {Caves},
\citenamefont {Fuchs},\ and\ \citenamefont {Schack}}]{CavesFuchsSchack2007}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.~M.}\ \bibnamefont
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and\ \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Schack}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Subjective probability and
quantum certainty},}\ }\href {\doibase 10.1016/j.shpsb.2006.10.007}
{\bibfield {journal} {\bibinfo {journal} {Stud. Hist. Philos. Mod. Phys.}\
}\textbf {\bibinfo {volume} {38}},\ \bibinfo {pages} {255--274} (\bibinfo
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\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.~A.}\ \bibnamefont
{Fuchs}},\ }\href@noop {} {\enquote {\bibinfo {title} {{{QBism}, the
Perimeter of Quantum {Bayesianism}}},}\ } (\bibinfo {year} {2010}),\ \Eprint
{http://arxiv.org/abs/1003.5209} {arXiv:1003.5209} \BibitemShut {NoStop} \bibitem [{\citenamefont {Mermin}(2012{\natexlab{a}})}]{Mermin2012a}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.~D.}\ \bibnamefont
{Mermin}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Quantum
mechanics: Fixing the shifty split},}\ }\href {\doibase 10.1063/PT.3.1618}
{\bibfield {journal} {\bibinfo {journal} {Phys. Today}\ }\textbf {\bibinfo
{volume} {65}},\ \bibinfo {pages} {8--10} (\bibinfo {year}
{2012}{\natexlab{a}})},\ \bibinfo {note} {{July}}\BibitemShut {NoStop} \bibitem [{\citenamefont {Mermin}(2012{\natexlab{b}})}]{Mermin2012b}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.~D.}\ \bibnamefont
{Mermin}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Measured
responses to quantum {Bayesianism}},}\ }\href {\doibase 10.1063/PT.3.1803}
{\bibfield {journal} {\bibinfo {journal} {Phys. Today}\ }\textbf {\bibinfo
{volume} {65}},\ \bibinfo {pages} {12--15} (\bibinfo {year}
{2012}{\natexlab{b}})},\ \bibinfo {note} {{December}}\BibitemShut {NoStop} \bibitem [{\citenamefont {Fuchs}\ and\ \citenamefont
{Schack}(2013)}]{FuchsSchack2013}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.~A.}\ \bibnamefont
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}\bibfield {title} {\enquote {\bibinfo {title} {Quantum-{Bayesian}
coherence},}\ }\href {\doibase 10.1103/RevModPhys.85.1693} {\bibfield
{journal} {\bibinfo {journal} {Rev. Mod. Phys.}\ }\textbf {\bibinfo {volume}
{85}},\ \bibinfo {pages} {1693--1715} (\bibinfo {year} {2013})}\BibitemShut
{NoStop} \bibitem [{\citenamefont {Fuchs}\ \emph {et~al.}(2014)\citenamefont {Fuchs},
\citenamefont {Mermin},\ and\ \citenamefont
{Schack}}]{FuchsMerminSchack2014}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.~A.}\ \bibnamefont
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and\ \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Schack}},\
}\bibfield {title} {\enquote {\bibinfo {title} {An introduction to {QBism}
with an application to the locality of quantum mechanics},}\ }\href {\doibase
10.1119/1.4874855} {\bibfield {journal} {\bibinfo {journal} {Am. J. Phys.}\
}\textbf {\bibinfo {volume} {82}},\ \bibinfo {pages} {749--754} (\bibinfo
{year} {2014})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Mermin}(2014{\natexlab{a}})}]{Mermin2014b}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.~D.}\ \bibnamefont
{Mermin}},\ }\bibfield {title} {\enquote {\bibinfo {title} {{QBism} puts the
scientist back into science},}\ }\href {\doibase 10.1038/507421a} {\bibfield
{journal} {\bibinfo {journal} {Nature}\ }\textbf {\bibinfo {volume} {507}},\
\bibinfo {pages} {421--423} (\bibinfo {year}
{2014}{\natexlab{a}})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Mermin}(2014{\natexlab{b}})}]{Mermin2014a}
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.~D.}\ \bibnamefont
{Mermin}},\ }\bibfield {title} {\enquote {\bibinfo {title} {What {I} think
about {Now}},}\ }\href {\doibase 10.1063/PT.3.2290} {\bibfield {journal}
{\bibinfo {journal} {Phys. Today}\ }\textbf {\bibinfo {volume} {67}},\
\bibinfo {pages} {8--9} (\bibinfo {year} {2014}{\natexlab{b}})},\ \bibinfo
{note} {{March}}\BibitemShut {NoStop} \bibitem [{\citenamefont {Mermin}(2014{\natexlab{c}})}]{Mermin2014c}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.~D.}\ \bibnamefont
{Mermin}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Classical and
quantum framing of the {Now}},}\ }\href {\doibase 10.1063/PT.3.2495}
{\bibfield {journal} {\bibinfo {journal} {Phys. Today}\ }\textbf {\bibinfo
{volume} {67}},\ \bibinfo {pages} {8--9} (\bibinfo {year}
{2014}{\natexlab{c}})},\ \bibinfo {note} {{September}}\BibitemShut {NoStop} \bibitem [{\citenamefont {Mermin}(2014{\natexlab{d}})}]{Mermin2014d}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.~D.}\ \bibnamefont
{Mermin}},\ }\href@noop {} {\enquote {\bibinfo {title} {{Why {QBism} is not
the {Copenhagen} interpretation and what {John Bell} might have thought of
it}},}\ } (\bibinfo {year} {2014}{\natexlab{d}}),\ \Eprint
{http://arxiv.org/abs/1409.2454} {arXiv:1409.2454} \BibitemShut {NoStop} \bibitem [{\citenamefont {Mermin}(2016)}]{Mermin2016}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.~D.}\ \bibnamefont
{Mermin}},\ }\href {\doibase 10.1017/CBO9781139162579} {\emph {\bibinfo
{title} {Why Quark Rhymes with Pork and other Scientific Diversions}}}\
(\bibinfo {publisher} {Cambridge University Press},\ \bibinfo {year}
{2016})\BibitemShut {NoStop} \bibitem [{\citenamefont {Szekeres}(2004)}]{Szekeres2004}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
{Szekeres}},\ }\href {\doibase 10.1017/CBO9780511607066} {\emph {\bibinfo
{title} {A Course in Modern Mathematical Physics}}}\ (\bibinfo {publisher}
{Cambridge University Press},\ \bibinfo {year} {2004})\BibitemShut {NoStop} \bibitem [{\citenamefont {Negele}\ and\ \citenamefont
{Orland}(1998)}]{NegeleOrland1998}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~W.}\ \bibnamefont
{Negele}}\ and\ \bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
{Orland}},\ }\href@noop {} {\emph {\bibinfo {title} {Quantum Many-Particle
Systems}}}\ (\bibinfo {publisher} {Westview},\ \bibinfo {address} {Boulder,
Colorado},\ \bibinfo {year} {1998})\BibitemShut {NoStop} \bibitem [{\citenamefont {Reed}\ and\ \citenamefont
{Simon}(1980)}]{ReedSimonVol1}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{Reed}}\ and\ \bibinfo {author} {\bibfnamefont {B.}~\bibnamefont {Simon}},\
}\href@noop {} {\emph {\bibinfo {title} {Functional Analysis}}},\ \bibinfo
{series} {Methods of Modern Mathematical Physics}, Vol.~\bibinfo {volume}
{1}\ (\bibinfo {publisher} {Academic},\ \bibinfo {address} {San Diego},\
\bibinfo {year} {1980})\BibitemShut {NoStop} \bibitem [{\citenamefont {Geroch}(1985)}]{Geroch1985}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
{Geroch}},\ }\href@noop {} {\emph {\bibinfo {title} {Mathematical Physics}}}\
(\bibinfo {publisher} {University of Chicago},\ \bibinfo {year}
{1985})\BibitemShut {NoStop} \bibitem [{Note1()}]{Note1}
\BibitemOpen
\bibinfo {note} {Here we are considering tensor multiplication only at the
formal level. See Appendix \ref {app:rigged} for a discussion of the
subtleties that arise when normalization is considered.}\BibitemShut {Stop} \bibitem [{\citenamefont {Messiah}(1962)}]{Messiah1962}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Messiah}},\ }\href@noop {} {\emph {\bibinfo {title} {Quantum Mechanics}}}\
(\bibinfo {publisher} {North-Holland},\ \bibinfo {address} {Amsterdam},\
\bibinfo {year} {1962})\BibitemShut {NoStop} \bibitem [{\citenamefont {Messiah}\ and\ \citenamefont
{Greenberg}(1964)}]{MessiahGreenberg1964}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~M.~L.}\
\bibnamefont {Messiah}}\ and\ \bibinfo {author} {\bibfnamefont {O.~W.}\
\bibnamefont {Greenberg}},\ }\bibfield {title} {\enquote {\bibinfo {title}
{Symmetrization postulate and its experimental foundation},}\ }\href
{\doibase 10.1103/PhysRev.136.B248} {\bibfield {journal} {\bibinfo
{journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {136}},\ \bibinfo {pages}
{B248--B267} (\bibinfo {year} {1964})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Misner}\ \emph {et~al.}(1973)\citenamefont {Misner},
\citenamefont {Thorne},\ and\ \citenamefont {Wheeler}}]{Misner1973}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.~W.}\ \bibnamefont
{Misner}}, \bibinfo {author} {\bibfnamefont {K.~S.}\ \bibnamefont {Thorne}},
\ and\ \bibinfo {author} {\bibfnamefont {J.~A.}\ \bibnamefont {Wheeler}},\
}\href@noop {} {\emph {\bibinfo {title} {Gravitation}}}\ (\bibinfo
{publisher} {Freeman},\ \bibinfo {address} {New York},\ \bibinfo {year}
{1973})\BibitemShut {NoStop} \bibitem [{\citenamefont {Loomis}\ and\ \citenamefont
{Sternberg}(1990)}]{LoomisSternberg1990}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {L.~H.}\ \bibnamefont
{Loomis}}\ and\ \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
{Sternberg}},\ }\href@noop {} {\emph {\bibinfo {title} {Advanced
Calculus}}},\ \bibinfo {edition} {revised}\ ed.\ (\bibinfo {publisher}
{Jones and Bartlett},\ \bibinfo {address} {Boston},\ \bibinfo {year}
{1990})\BibitemShut {NoStop} \bibitem [{\citenamefont {Abraham}\ \emph {et~al.}(1988)\citenamefont
{Abraham}, \citenamefont {Marsden},\ and\ \citenamefont
{Ratiu}}]{Abraham1988}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
{Abraham}}, \bibinfo {author} {\bibfnamefont {J.~E.}\ \bibnamefont
{Marsden}}, \ and\ \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
{Ratiu}},\ }\href {\doibase 10.1007/978-1-4612-1029-0} {\emph {\bibinfo
{title} {Manifolds, Tensor Analysis, and Applications}}},\ \bibinfo {edition}
{2nd}\ ed.\ (\bibinfo {publisher} {Springer},\ \bibinfo {address} {New
York},\ \bibinfo {year} {1988})\BibitemShut {NoStop} \bibitem [{\citenamefont {Kostrikin}\ and\ \citenamefont
{Manin}(1989)}]{KostrikinManin1989}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~I.}\ \bibnamefont
{Kostrikin}}\ and\ \bibinfo {author} {\bibfnamefont {{\relax Yu}.~I.}\
\bibnamefont {Manin}},\ }\href@noop {} {\emph {\bibinfo {title} {Linear
Algebra and Geometry}}}\ (\bibinfo {publisher} {Gordon and Breach},\
\bibinfo {address} {New York},\ \bibinfo {year} {1989})\BibitemShut {NoStop} \bibitem [{\citenamefont {Hassani}(2013)}]{Hassani2013}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
{Hassani}},\ }\href {\doibase 10.1007/978-3-319-01195-0} {\emph {\bibinfo
{title} {Mathematical Physics}}},\ \bibinfo {edition} {2nd}\ ed.\ (\bibinfo
{publisher} {Springer},\ \bibinfo {address} {Heidelberg},\ \bibinfo {year}
{2013})\BibitemShut {NoStop} \bibitem [{Note2()}]{Note2}
\BibitemOpen
\bibinfo {note} {See pp.\ 603 and 622--623 of Messiah \cite
{Messiah1962}.}\BibitemShut {Stop} \bibitem [{\citenamefont {Peres}(1995)}]{Peres1995}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Peres}},\ }\href {\doibase 10.1007/0-306-47120-5} {\emph {\bibinfo {title}
{Quantum Theory: Concepts and Methods}}}\ (\bibinfo {publisher} {Kluwer
Academic},\ \bibinfo {address} {Dordrecht},\ \bibinfo {year}
{1995})\BibitemShut {NoStop} \bibitem [{\citenamefont {Zanardi}(2002)}]{Zanardi2002}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
{Zanardi}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Quantum
entanglement in fermionic lattices},}\ }\href {\doibase
10.1103/PhysRevA.65.042101} {\bibfield {journal} {\bibinfo {journal} {Phys.
Rev. A}\ }\textbf {\bibinfo {volume} {65}},\ \bibinfo {pages} {042101}
(\bibinfo {year} {2002})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Horv\'ath}(2012)}]{Horvath2012}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
{Horv\'ath}},\ }\href@noop {} {\emph {\bibinfo {title} {Topological Vector
Spaces and Distributions}}}\ (\bibinfo {publisher} {Dover},\ \bibinfo
{address} {Mineola, N.Y.},\ \bibinfo {year} {2012})\BibitemShut {NoStop} \bibitem [{\citenamefont {Cox}\ \emph {et~al.}(2007)\citenamefont {Cox},
\citenamefont {Little},\ and\ \citenamefont {O'Shea}}]{Cox2007}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
{Cox}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Little}}, \ and\
\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {O'Shea}},\ }\href
{\doibase 10.1007/978-0-387-35651-8} {\emph {\bibinfo {title} {Ideals,
Varieties, and Algorithms: An Introduction to Computational Algebraic
Geometry and Commutative Algebra}}},\ \bibinfo {edition} {3rd}\ ed.\
(\bibinfo {publisher} {Springer},\ \bibinfo {address} {Berlin},\ \bibinfo
{year} {2007})\BibitemShut {NoStop} \bibitem [{\citenamefont {Sudbery}(1986)}]{Sudbery1986}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Sudbery}},\ }\href@noop {} {\emph {\bibinfo {title} {Quantum Mechanics and
the Particles of Nature}}}\ (\bibinfo {publisher} {Cambridge University
Press},\ \bibinfo {year} {1986})\BibitemShut {NoStop} \bibitem [{\citenamefont {Landau}\ and\ \citenamefont
{Lifshitz}(1977)}]{LandauLifshitz1977}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {L.~D.}\ \bibnamefont
{Landau}}\ and\ \bibinfo {author} {\bibfnamefont {E.~M.}\ \bibnamefont
{Lifshitz}},\ }\href@noop {} {\emph {\bibinfo {title} {Quantum Mechanics}}},\
\bibinfo {edition} {3rd}\ ed.\ (\bibinfo {publisher} {Pergamon},\ \bibinfo
{address} {Oxford},\ \bibinfo {year} {1977})\BibitemShut {NoStop} \bibitem [{\citenamefont {Berestetskii}\ \emph {et~al.}(1980)\citenamefont
{Berestetskii}, \citenamefont {Lifshitz},\ and\ \citenamefont
{Pitaevskii}}]{LandauLifshitz1980}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {V.~B.}\ \bibnamefont
{Berestetskii}}, \bibinfo {author} {\bibfnamefont {E.~M.}\ \bibnamefont
{Lifshitz}}, \ and\ \bibinfo {author} {\bibfnamefont {L.~P.}\ \bibnamefont
{Pitaevskii}},\ }\href@noop {} {\emph {\bibinfo {title} {Quantum
Electrodynamics}}},\ \bibinfo {edition} {2nd}\ ed.\ (\bibinfo {publisher}
{Pergamon},\ \bibinfo {address} {Oxford},\ \bibinfo {year}
{1980})\BibitemShut {NoStop} \bibitem [{\citenamefont {Jordan}\ and\ \citenamefont
{Wigner}(1928)}]{JordanWigner1928}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
{Jordan}}\ and\ \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
{Wigner}},\ }\bibfield {title} {\enquote {\bibinfo {title} {\"{Uber} das
{Paulische} \"{Aquivalenzverbot}},}\ }\href {\doibase 10.1007/BF01331938}
{\bibfield {journal} {\bibinfo {journal} {Z. Phys.}\ }\textbf {\bibinfo
{volume} {47}},\ \bibinfo {pages} {631--651} (\bibinfo {year}
{1928})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Lieb}\ \emph {et~al.}(1961)\citenamefont {Lieb},
\citenamefont {Schultz},\ and\ \citenamefont
{Mattis}}]{LiebSchultzMattis1961}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
{Lieb}}, \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {Schultz}}, \
and\ \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Mattis}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Two soluble models of an
antiferromagnetic chain},}\ }\href {\doibase 10.1016/0003-4916(61)90115-4}
{\bibfield {journal} {\bibinfo {journal} {Ann. Phys. (N.Y.)}\ }\textbf
{\bibinfo {volume} {16}},\ \bibinfo {pages} {407--466} (\bibinfo {year}
{1961})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Schultz}\ \emph {et~al.}(1964)\citenamefont
{Schultz}, \citenamefont {Mattis},\ and\ \citenamefont
{Lieb}}]{SchultzMattisLieb1964}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.~D.}\ \bibnamefont
{Schultz}}, \bibinfo {author} {\bibfnamefont {D.~C.}\ \bibnamefont {Mattis}},
\ and\ \bibinfo {author} {\bibfnamefont {E.~H.}\ \bibnamefont {Lieb}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Two-dimensional {Ising}
model as a soluble problem of many fermions},}\ }\href {\doibase
10.1103/RevModPhys.36.856} {\bibfield {journal} {\bibinfo {journal} {Rev.
Mod. Phys.}\ }\textbf {\bibinfo {volume} {36}},\ \bibinfo {pages} {856--871}
(\bibinfo {year} {1964})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Araki}(1961)}]{Araki1961}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
{Araki}},\ }\bibfield {title} {\enquote {\bibinfo {title} {On the connection
of spin and commutation relations between different fields},}\ }\href
{\doibase 10.1063/1.1703710} {\bibfield {journal} {\bibinfo {journal} {J.
Math. Phys.}\ }\textbf {\bibinfo {volume} {2}},\ \bibinfo {pages} {267--270}
(\bibinfo {year} {1961})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Streater}\ and\ \citenamefont
{Wightman}(1989)}]{StreaterWightman1989}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.~F.}\ \bibnamefont
{Streater}}\ and\ \bibinfo {author} {\bibfnamefont {A.~S.}\ \bibnamefont
{Wightman}},\ }\href@noop {} {\emph {\bibinfo {title} {PCT, Spin and
Statistics, and All That}}}\ (\bibinfo {publisher} {Addison-Wesley},\
\bibinfo {address} {New York},\ \bibinfo {year} {1989})\BibitemShut {NoStop} \bibitem [{Note3()}]{Note3}
\BibitemOpen
\bibinfo {note} {In an alternative mode of description, different types of
bosons and fermions are distinguished using an unsymmetrized tensor product
(see Appendix \ref {app:different}). Formulas such as Eq.\ (\ref
{eq:u_not_Schmidt}) then become much more cumbersome, but the physics is the
same. In particular, the vector space $\protect \mathcal {E}$ of Eq.\ (\ref
{eq:total_E}) is not affected by this change of convention; only the labeling
and phase of the basis vectors is altered \cite {Sudbery1986}.}\BibitemShut
{Stop} \bibitem [{\citenamefont {Glauber}(1963)}]{Glauber1963c}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.~J.}\ \bibnamefont
{Glauber}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Coherent and
incoherent states of the radiation field},}\ }\href {\doibase
10.1103/PhysRev.131.2766} {\bibfield {journal} {\bibinfo {journal} {Phys.
Rev.}\ }\textbf {\bibinfo {volume} {131}},\ \bibinfo {pages} {2766--2788}
(\bibinfo {year} {1963})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Klauder}\ and\ \citenamefont
{Skagerstam}(1985)}]{KlauderSkagerstam1985}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~R.}\ \bibnamefont
{Klauder}}\ and\ \bibinfo {author} {\bibfnamefont {B.-S.}\ \bibnamefont
{Skagerstam}},\ }\href {\doibase 10.1142/9789814415118_fmatter} {\emph
{\bibinfo {title} {Coherent States: Applications in Physics and Mathematical
Physics}}}\ (\bibinfo {publisher} {World Scientific},\ \bibinfo {address}
{Singapore},\ \bibinfo {year} {1985})\BibitemShut {NoStop} \bibitem [{\citenamefont {Blaizot}\ and\ \citenamefont
{Ripka}(1986)}]{BlaizotRipka1986}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.-P.}\ \bibnamefont
{Blaizot}}\ and\ \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
{Ripka}},\ }\href@noop {} {\emph {\bibinfo {title} {Quantum Theory of Finite
Systems}}}\ (\bibinfo {publisher} {MIT Press},\ \bibinfo {address}
{Cambridge, Mass.},\ \bibinfo {year} {1986})\BibitemShut {NoStop} \bibitem [{\citenamefont {Perelomov}(1986)}]{Perelomov1986}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Perelomov}},\ }\href {\doibase 10.1007/978-3-642-61629-7} {\emph {\bibinfo
{title} {Generalized Coherent States and Their Applications}}}\ (\bibinfo
{publisher} {Springer},\ \bibinfo {address} {Berlin},\ \bibinfo {year}
{1986})\BibitemShut {NoStop} \bibitem [{\citenamefont {Zhang}\ \emph {et~al.}(1990)\citenamefont {Zhang},
\citenamefont {Feng},\ and\ \citenamefont {Gilmore}}]{ZhangFengGilmore1990}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.-M.}\ \bibnamefont
{Zhang}}, \bibinfo {author} {\bibfnamefont {D.~H.}\ \bibnamefont {Feng}}, \
and\ \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Gilmore}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Coherent states: Theory and
some applications},}\ }\href {\doibase 10.1103/RevModPhys.62.867} {\bibfield
{journal} {\bibinfo {journal} {Rev. Mod. Phys.}\ }\textbf {\bibinfo {volume}
{62}},\ \bibinfo {pages} {867--927} (\bibinfo {year} {1990})}\BibitemShut
{NoStop} \bibitem [{\citenamefont {Gazeau}(2009)}]{Gazeau2009}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.-P.}\ \bibnamefont
{Gazeau}},\ }\href {\doibase 10.1002/9783527628285} {\emph {\bibinfo {title}
{Coherent States in Quantum Physics}}}\ (\bibinfo {publisher} {Wiley-VCH},\
\bibinfo {address} {Weinheim},\ \bibinfo {year} {2009})\BibitemShut {NoStop} \bibitem [{\citenamefont {Campbell}\ and\ \citenamefont
{Meyer}(2009)}]{CampbellMeyer2009}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.~L.}\ \bibnamefont
{Campbell}}\ and\ \bibinfo {author} {\bibfnamefont {C.~D.}\ \bibnamefont
{Meyer}},\ }\href {\doibase 10.1137/1.9780898719048} {\emph {\bibinfo {title}
{Generalized Inverses of Linear Transformations}}}\ (\bibinfo {publisher}
{SIAM},\ \bibinfo {address} {Philadelphia},\ \bibinfo {year}
{2009})\BibitemShut {NoStop} \bibitem [{\citenamefont {Ben-Israel}\ and\ \citenamefont
{Greville}(2003)}]{BenIsrael2003}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Ben-Israel}}\ and\ \bibinfo {author} {\bibfnamefont {T.~N.~E.}\ \bibnamefont
{Greville}},\ }\href {\doibase 10.1007/b97366} {\emph {\bibinfo {title}
{Generalized Inverses: Theory and Applications}}},\ \bibinfo {edition} {2nd}\
ed.\ (\bibinfo {publisher} {Springer},\ \bibinfo {address} {New York},\
\bibinfo {year} {2003})\BibitemShut {NoStop} \bibitem [{\citenamefont {Stewart}\ and\ \citenamefont
{Sun}(1990)}]{StewartSun1990}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~W.}\ \bibnamefont
{Stewart}}\ and\ \bibinfo {author} {\bibfnamefont {J.-G.}\ \bibnamefont
{Sun}},\ }\href@noop {} {\emph {\bibinfo {title} {Matrix Perturbation
Theory}}}\ (\bibinfo {publisher} {Academic},\ \bibinfo {address} {Boston},\
\bibinfo {year} {1990})\BibitemShut {NoStop} \bibitem [{\citenamefont {Nashed}(1976)}]{Nashed1976}
\BibitemOpen
\bibinfo {editor} {\bibfnamefont {M.~Z.}\ \bibnamefont {Nashed}},\ ed.,\
\href@noop {} {\emph {\bibinfo {title} {Generalized Inverses and
Applications}}}\ (\bibinfo {publisher} {Academic},\ \bibinfo {address} {New
York},\ \bibinfo {year} {1976})\BibitemShut {NoStop} \bibitem [{\citenamefont {Bell}(1976)}]{Bell1976}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~S.}\ \bibnamefont
{Bell}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The measurement
theory of {Everett} and de {Broglie}'s pilot wave},}\ }in\ \href {\doibase
10.1007/978-94-010-1440-3} {\emph {\bibinfo {booktitle} {Quantum Mechanics,
Determinism, Causality, and Particles}}},\ \bibinfo {editor} {edited by\
\bibinfo {editor} {\bibfnamefont {M.}~\bibnamefont {Flato}}, \bibinfo
{editor} {\bibfnamefont {Z.}~\bibnamefont {Maric}}, \bibinfo {editor}
{\bibfnamefont {A.}~\bibnamefont {Milojevic}}, \bibinfo {editor}
{\bibfnamefont {D.}~\bibnamefont {Sternheimer}}, \ and\ \bibinfo {editor}
{\bibfnamefont {J.~P.}\ \bibnamefont {Vigier}}}\ (\bibinfo {publisher}
{Reidel},\ \bibinfo {address} {Dordrecht},\ \bibinfo {year} {1976})\ pp.\
\bibinfo {pages} {11--17},\ \bibinfo {note} {reprinted in Ref.\
\cite{Bell2004}.}\BibitemShut {Stop} \bibitem [{\citenamefont {Wilcox}(1967)}]{Wilcox1967}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.~M.}\ \bibnamefont
{Wilcox}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Exponential
operators and parameter differentiation in quantum physics},}\ }\href
{\doibase 10.1063/1.1705306} {\bibfield {journal} {\bibinfo {journal} {J.
Math. Phys.}\ }\textbf {\bibinfo {volume} {8}},\ \bibinfo {pages} {962--982}
(\bibinfo {year} {1967})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Merzbacher}(1998)}]{Merzbacher1998}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
{Merzbacher}},\ }\href@noop {} {\emph {\bibinfo {title} {Quantum
Mechanics}}},\ \bibinfo {edition} {3rd}\ ed.\ (\bibinfo {publisher}
{Wiley},\ \bibinfo {address} {New York},\ \bibinfo {year} {1998})\BibitemShut
{NoStop} \bibitem [{\citenamefont {Schr\"odinger}(1926)}]{Schrodinger1926c}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
{Schr\"odinger}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Der
stetige \"{U}bergang von der {Mikro}- zur {Makromechanik}},}\ }\href
{\doibase 10.1007/BF01507634} {\bibfield {journal} {\bibinfo {journal}
{Naturwissenschaften}\ }\textbf {\bibinfo {volume} {14}},\ \bibinfo {pages}
{664--666} (\bibinfo {year} {1926})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Schr\"odinger}(1982)}]{Schrodinger1982}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
{Schr\"odinger}},\ }\href@noop {} {\emph {\bibinfo {title} {Collected Papers
on Wave Mechanics}}}\ (\bibinfo {publisher} {Chelsea},\ \bibinfo {address}
{New York},\ \bibinfo {year} {1982})\BibitemShut {NoStop} \bibitem [{\citenamefont {Ghirardi}\ \emph {et~al.}(1990)\citenamefont
{Ghirardi}, \citenamefont {Pearle},\ and\ \citenamefont
{Rimini}}]{GhirardiPearleRimini1990}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~C.}\ \bibnamefont
{Ghirardi}}, \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Pearle}}, \
and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Rimini}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Markov processes in
{Hilbert} space and continuous spontaneous localization of systems of
identical particles},}\ }\href {\doibase 10.1103/PhysRevA.42.78} {\bibfield
{journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume}
{42}},\ \bibinfo {pages} {78--89} (\bibinfo {year} {1990})}\BibitemShut
{NoStop} \bibitem [{\citenamefont {Ghirardi}\ \emph {et~al.}(1995)\citenamefont
{Ghirardi}, \citenamefont {Grassi},\ and\ \citenamefont
{Benatti}}]{GhirardiGrassiBenatti1995}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~C.}\ \bibnamefont
{Ghirardi}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Grassi}}, \
and\ \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Benatti}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Describing the macroscopic
world: Closing the circle within the dynamical reduction program},}\ }\href
{\doibase 10.1007/BF02054655} {\bibfield {journal} {\bibinfo {journal}
{Found. Phys.}\ }\textbf {\bibinfo {volume} {25}},\ \bibinfo {pages} {5--38}
(\bibinfo {year} {1995})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Bassi}\ and\ \citenamefont
{Ghirardi}(2003)}]{BassiGhirardi2003}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Bassi}}\ and\ \bibinfo {author} {\bibfnamefont {G.~C.}\ \bibnamefont
{Ghirardi}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Dynamical
reduction models},}\ }\href {\doibase 10.1016/S0370-1573(03)00103-0}
{\bibfield {journal} {\bibinfo {journal} {Phys. Rep.}\ }\textbf {\bibinfo
{volume} {379}},\ \bibinfo {pages} {257--426} (\bibinfo {year}
{2003})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Bengtsson}\ and\ \citenamefont
{{\.Z}yczkowski}(2006)}]{Bengtsson2006}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {I.}~\bibnamefont
{Bengtsson}}\ and\ \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont
{{\.Z}yczkowski}},\ }\href {\doibase 10.1017/CBO9780511535048} {\emph
{\bibinfo {title} {Geometry of Quantum States}}}\ (\bibinfo {publisher}
{Cambridge University Press},\ \bibinfo {year} {2006})\BibitemShut {NoStop} \bibitem [{\citenamefont {Provost}\ and\ \citenamefont
{Vallee}(1980)}]{ProvostVallee1980}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~P.}\ \bibnamefont
{Provost}}\ and\ \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
{Vallee}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Riemannian
structure on manifolds of quantum states},}\ }\href {\doibase
10.1007/BF02193559} {\bibfield {journal} {\bibinfo {journal} {Commun. Math.
Phys.}\ }\textbf {\bibinfo {volume} {76}},\ \bibinfo {pages} {289--301}
(\bibinfo {year} {1980})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Wootters}(1981)}]{Wootters1981}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.~K.}\ \bibnamefont
{Wootters}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Statistical
distance and {Hilbert} space},}\ }\href {\doibase 10.1103/PhysRevD.23.357}
{\bibfield {journal} {\bibinfo {journal} {Phys. Rev. D}\ }\textbf {\bibinfo
{volume} {23}},\ \bibinfo {pages} {357--362} (\bibinfo {year}
{1981})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Page}(1987)}]{Page1987}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~N.}\ \bibnamefont
{Page}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Geometrical
description of {Berry's} phase},}\ }\href {\doibase 10.1103/PhysRevA.36.3479}
{\bibfield {journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo
{volume} {36}},\ \bibinfo {pages} {3479--3481} (\bibinfo {year}
{1987})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Berry}(1989)}]{Berry1989}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.~V.}\ \bibnamefont
{Berry}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The quantum
phase, five years after},}\ }in\ \href {\doibase
10.1142/9789812798381_fmatter} {\emph {\bibinfo {booktitle} {Geometric Phases
in Physics}}},\ \bibinfo {editor} {edited by\ \bibinfo {editor}
{\bibfnamefont {A.}~\bibnamefont {Shapere}}\ and\ \bibinfo {editor}
{\bibfnamefont {F.}~\bibnamefont {Wilczek}}}\ (\bibinfo {publisher} {World
Scientific},\ \bibinfo {address} {Singapore},\ \bibinfo {year} {1989})\ pp.\
\bibinfo {pages} {7--28}\BibitemShut {NoStop} \bibitem [{\citenamefont {Anandan}\ and\ \citenamefont
{Aharonov}(1990)}]{AnandanAharonov1990}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
{Anandan}}\ and\ \bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont
{Aharonov}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Geometry of
quantum evolution},}\ }\href {\doibase 10.1103/PhysRevLett.65.1697}
{\bibfield {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf
{\bibinfo {volume} {65}},\ \bibinfo {pages} {1697--1700} (\bibinfo {year}
{1990})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Anandan}(1991)}]{Anandan1991}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
{Anandan}},\ }\bibfield {title} {\enquote {\bibinfo {title} {A geometric
approach to quantum mechanics},}\ }\href {\doibase 10.1007/BF00732829}
{\bibfield {journal} {\bibinfo {journal} {Found. Phys.}\ }\textbf {\bibinfo
{volume} {21}},\ \bibinfo {pages} {1265--1284} (\bibinfo {year}
{1991})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Anandan}\ and\ \citenamefont
{Aharonov}(1988)}]{AnandanAharonov1988}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
{Anandan}}\ and\ \bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont
{Aharonov}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Geometric
quantum phase and angles},}\ }\href {\doibase 10.1103/PhysRevD.38.1863}
{\bibfield {journal} {\bibinfo {journal} {Phys. Rev. D}\ }\textbf {\bibinfo
{volume} {38}},\ \bibinfo {pages} {1863--1870} (\bibinfo {year}
{1988})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Fuchs}(2011)}]{Fuchs2011}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.~A.}\ \bibnamefont
{Fuchs}},\ }\href {\doibase 10.1017/CBO9780511762789} {\emph {\bibinfo
{title} {Coming of Age with Quantum Information: Notes on a Paulian Idea}}}\
(\bibinfo {publisher} {Cambridge University Press},\ \bibinfo {year}
{2011})\BibitemShut {NoStop} \bibitem [{\citenamefont {Zeh}(1971)}]{Zeh1971b}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.~D.}\ \bibnamefont
{Zeh}},\ }\bibfield {title} {\enquote {\bibinfo {title} {On the
irreversibility of time and observation in quantum theory},}\ }in\ \href@noop
{} {\emph {\bibinfo {booktitle} {Foundations of Quantum Mechanics}}},\
\bibinfo {series and number} {Proc. 49th Enrico Fermi School of Physics},\
\bibinfo {editor} {edited by\ \bibinfo {editor} {\bibfnamefont
{B.}~\bibnamefont {d'Espagnat}}}\ (\bibinfo {publisher} {Academic},\
\bibinfo {address} {New York},\ \bibinfo {year} {1971})\ pp.\ \bibinfo
{pages} {263--273}\BibitemShut {NoStop} \bibitem [{\citenamefont {K{\"u}bler}\ and\ \citenamefont
{Zeh}(1973)}]{KublerZeh1973}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {O.}~\bibnamefont
{K{\"u}bler}}\ and\ \bibinfo {author} {\bibfnamefont {H.~D.}\ \bibnamefont
{Zeh}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Dynamics of quantum
correlations},}\ }\href {\doibase 10.1016/0003-4916(73)90040-7} {\bibfield
{journal} {\bibinfo {journal} {Ann. Phys. (N.Y.)}\ }\textbf {\bibinfo
{volume} {76}},\ \bibinfo {pages} {405--418} (\bibinfo {year}
{1973})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Zeh}(1973)}]{Zeh1973}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.~D.}\ \bibnamefont
{Zeh}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Toward a quantum
theory of observation},}\ }\href {\doibase 10.1007/BF00708603} {\bibfield
{journal} {\bibinfo {journal} {Found. Phys.}\ }\textbf {\bibinfo {volume}
{3}},\ \bibinfo {pages} {109--116} (\bibinfo {year} {1973})}\BibitemShut
{NoStop} \bibitem [{\citenamefont {Zeh}(1979)}]{Zeh1979}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.~D.}\ \bibnamefont
{Zeh}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Quantum theory and
time asymmetry},}\ }\href {\doibase 10.1007/BF00708694} {\bibfield {journal}
{\bibinfo {journal} {Found. Phys.}\ }\textbf {\bibinfo {volume} {9}},\
\bibinfo {pages} {803--818} (\bibinfo {year} {1979})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Joos}\ and\ \citenamefont {Zeh}(1985)}]{JoosZeh1985}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
{Joos}}\ and\ \bibinfo {author} {\bibfnamefont {H.~D.}\ \bibnamefont {Zeh}},\
}\bibfield {title} {\enquote {\bibinfo {title} {The emergence of classical
properties through interaction with the environment},}\ }\href {\doibase
10.1007/BF01725541} {\bibfield {journal} {\bibinfo {journal} {Z. Phys. B}\
}\textbf {\bibinfo {volume} {59}},\ \bibinfo {pages} {223--243} (\bibinfo
{year} {1985})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Zeh}(2000)}]{Zeh2000}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.~D.}\ \bibnamefont
{Zeh}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The meaning of
decoherence},}\ }in\ \href {\doibase 10.1007/3-540-46657-6} {\emph {\bibinfo
{booktitle} {Decoherence: Theoretical, Experimental, and Conceptual
Problems}}},\ \bibinfo {editor} {edited by\ \bibinfo {editor} {\bibfnamefont
{P.}~\bibnamefont {Blanchard}}, \bibinfo {editor} {\bibfnamefont
{E.}~\bibnamefont {Joos}}, \bibinfo {editor} {\bibfnamefont {D.}~\bibnamefont
{Giulini}}, \bibinfo {editor} {\bibfnamefont {C.}~\bibnamefont {Kiefer}}, \
and\ \bibinfo {editor} {\bibfnamefont {I.-O.}\ \bibnamefont {Stamatescu}}}\
(\bibinfo {publisher} {Springer},\ \bibinfo {address} {Berlin},\ \bibinfo
{year} {2000})\ pp.\ \bibinfo {pages} {19--42}\BibitemShut {NoStop} \bibitem [{\citenamefont {Zeh}(2003)}]{Zeh2003ch2}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.~D.}\ \bibnamefont
{Zeh}},\ }\enquote {\bibinfo {title} {Basic concepts and their
interpretation},}\ Chap.~\bibinfo {chapter} {2},\ in\ \cite{JoosZeh2003}
(\bibinfo {year} {2003})\BibitemShut {NoStop} \bibitem [{\citenamefont {Zeh}(2006)}]{Zeh2006}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.~D.}\ \bibnamefont
{Zeh}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Roots and fruits of
decoherence},}\ }in\ \href {\doibase 10.1007/978-3-7643-7808-0} {\emph
{\bibinfo {booktitle} {Quantum Decoherence}}},\ \bibinfo {editor} {edited by\
\bibinfo {editor} {\bibfnamefont {B.}~\bibnamefont {Duplantier}}, \bibinfo
{editor} {\bibfnamefont {J.-M.}\ \bibnamefont {Raimond}}, \ and\ \bibinfo
{editor} {\bibfnamefont {V.}~\bibnamefont {Rivasseau}}}\ (\bibinfo
{publisher} {Birkh{\"a}user},\ \bibinfo {address} {Basel},\ \bibinfo {year}
{2006})\ pp.\ \bibinfo {pages} {151--175}\BibitemShut {NoStop} \bibitem [{\citenamefont {Zurek}(1993)}]{Zurek1993a}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.~H.}\ \bibnamefont
{Zurek}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Preferred states,
predictability, classicality and the environment-induced decoherence},}\
}\href {\doibase 10.1143/ptp/89.2.281} {\bibfield {journal} {\bibinfo
{journal} {Prog. Theor. Phys.}\ }\textbf {\bibinfo {volume} {89}},\ \bibinfo
{pages} {281--312} (\bibinfo {year} {1993})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Zurek}\ \emph {et~al.}(1993)\citenamefont {Zurek},
\citenamefont {Habib},\ and\ \citenamefont {Paz}}]{ZurekHabibPaz1993}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.~H.}\ \bibnamefont
{Zurek}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Habib}}, \ and\
\bibinfo {author} {\bibfnamefont {J.~P.}\ \bibnamefont {Paz}},\ }\bibfield
{title} {\enquote {\bibinfo {title} {Coherent states via decoherence},}\
}\href {\doibase 10.1103/PhysRevLett.70.1187} {\bibfield {journal} {\bibinfo
{journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {70}},\ \bibinfo
{pages} {1187--1190} (\bibinfo {year} {1993})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Zurek}(1998)}]{Zurek1998}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.~H.}\ \bibnamefont
{Zurek}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Decoherence,
einselection and the existential interpretation (the rough guide)},}\ }\href
{\doibase 10.1098/rsta.1998.0250} {\bibfield {journal} {\bibinfo {journal}
{Philos. Trans. R. Soc. Lond. A}\ }\textbf {\bibinfo {volume} {356}},\
\bibinfo {pages} {1793--1821} (\bibinfo {year} {1998})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Zurek}(2003)}]{Zurek2003}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.~H.}\ \bibnamefont
{Zurek}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Decoherence,
einselection, and the quantum origins of the classical},}\ }\href {\doibase
10.1103/RevModPhys.75.715} {\bibfield {journal} {\bibinfo {journal} {Rev.
Mod. Phys.}\ }\textbf {\bibinfo {volume} {75}},\ \bibinfo {pages} {715--775}
(\bibinfo {year} {2003})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Zurek}(2014)}]{Zurek2014}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.~H.}\ \bibnamefont
{Zurek}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Quantum
{Darwinism}, classical reality, and the randomness of quantum jumps},}\
}\href {\doibase 10.1063/PT.3.2550} {\bibfield {journal} {\bibinfo
{journal} {Phys. Today}\ }\textbf {\bibinfo {volume} {67}},\ \bibinfo {pages}
{44--50} (\bibinfo {year} {2014})},\ \bibinfo {note} {{October}}\BibitemShut
{NoStop} \bibitem [{Note4()}]{Note4}
\BibitemOpen
\bibinfo {note} {The vacuum state can be excluded from the subsystem
exponents $\protect \ensuremath {\delimiter 69640972 x_k \delimiter "526930B
}$ because its inclusion has no effect other than to change the normalization
of $\protect \ensuremath {\delimiter 69640972 u_k \delimiter "526930B }$. If
this choice is made, one can readily verify using Eqs.\ (\ref {eq:fki}) and
(\ref {eq:switch_basis}) that $\protect \mathaccentV {hat}05E{\eta } >
0$.}\BibitemShut {Stop} \bibitem [{\citenamefont {Mahan}(2000)}]{Mahan2000}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~D.}\ \bibnamefont
{Mahan}},\ }\href {\doibase 10.1007/978-1-4757-5714-9} {\emph {\bibinfo
{title} {Many-Particle Physics}}},\ \bibinfo {edition} {3rd}\ ed.\ (\bibinfo
{publisher} {Kluwer},\ \bibinfo {address} {New York},\ \bibinfo {year}
{2000})\BibitemShut {NoStop} \bibitem [{\citenamefont {Wiseman}(2004)}]{Wiseman2004}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.~M.}\ \bibnamefont
{Wiseman}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Defending
continuous variable teleportation: why a laser is a clock, not a quantum
channel},}\ }\href {\doibase 10.1088/1464-4266/6/8/035} {\bibfield {journal}
{\bibinfo {journal} {J. Opt. B: Quantum Semiclass. Opt.}\ }\textbf {\bibinfo
{volume} {6}},\ \bibinfo {pages} {S849--S859} (\bibinfo {year}
{2004})}\BibitemShut {NoStop} \bibitem [{Note5()}]{Note5}
\BibitemOpen
\bibinfo {note} {The name ``orbit'' is commonly used in this context; see,
e.g., Refs.\ \cite {Bengtsson2006} and \cite {Isham1999}.}\BibitemShut
{Stop} \bibitem [{Note6()}]{Note6}
\BibitemOpen
\bibinfo {note} {It is interesting to note that the Fubini--Study metric can
also be derived from such a minimum principle \cite
{ProvostVallee1980}.}\BibitemShut {Stop} \bibitem [{\citenamefont {London}\ and\ \citenamefont
{Bauer}(1939)}]{LondonBauer1939a}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont
{London}}\ and\ \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Bauer}},\
}\href@noop {} {\emph {\bibinfo {title} {La Th\'eorie de l'Observation en
M\'ecanique Quantique}}}\ (\bibinfo {publisher} {Hermann},\ \bibinfo
{address} {Paris},\ \bibinfo {year} {1939})\BibitemShut {NoStop} \bibitem [{\citenamefont {London}\ and\ \citenamefont
{Bauer}(1983)}]{LondonBauer1939b}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont
{London}}\ and\ \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Bauer}},\
}\enquote {\bibinfo {title} {The theory of observation in quantum
mechanics},}\ pp.\ \bibinfo {pages} {217--259},\ in\ \cite{WheelerZurek1983}
(\bibinfo {year} {1983})\BibitemShut {NoStop} \bibitem [{\citenamefont {Wheeler}\ and\ \citenamefont
{Zurek}(1983)}]{WheelerZurek1983}
\BibitemOpen
\bibinfo {editor} {\bibfnamefont {J.~A.}\ \bibnamefont {Wheeler}}\ and\
\bibinfo {editor} {\bibfnamefont {W.~H.}\ \bibnamefont {Zurek}},\ eds.,\
\href@noop {} {\emph {\bibinfo {title} {Quantum Theory and Measurement}}}\
(\bibinfo {publisher} {Princeton University Press},\ \bibinfo {year}
{1983})\BibitemShut {NoStop} \bibitem [{\citenamefont {Wheeler}(1977)}]{Wheeler1977}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~A.}\ \bibnamefont
{Wheeler}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Include the
observer in the wave function?}}\ }in\ \href {\doibase
10.1007/978-94-010-1196-9_1} {\emph {\bibinfo {booktitle} {Quantum Mechanics,
A Half Century Later}}},\ \bibinfo {editor} {edited by\ \bibinfo {editor}
{\bibfnamefont {J.}~\bibnamefont {Leite~Lopes}}\ and\ \bibinfo {editor}
{\bibfnamefont {M.}~\bibnamefont {Paty}}}\ (\bibinfo {publisher} {Reidel},\
\bibinfo {address} {Dordrecht},\ \bibinfo {year} {1977})\ pp.\ \bibinfo
{pages} {1--18}\BibitemShut {NoStop} \bibitem [{\citenamefont {Pinker}(1997)}]{Pinker1997}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
{Pinker}},\ }\href@noop {} {\emph {\bibinfo {title} {How the Mind Works}}}\
(\bibinfo {publisher} {Norton},\ \bibinfo {address} {New York},\ \bibinfo
{year} {1997})\BibitemShut {NoStop} \bibitem [{\citenamefont {Price}(1996)}]{Price1996}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
{Price}},\ }\href {\doibase 10.1093/acprof:oso/9780195117981.001.0001} {\emph
{\bibinfo {title} {Time's Arrow and Archimedes' Point: New Directions for the
Physics of Time}}}\ (\bibinfo {publisher} {Oxford University Press},\
\bibinfo {year} {1996})\BibitemShut {NoStop} \bibitem [{\citenamefont {Price}(2013)}]{Price2013}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
{Price}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Rebirthing
pains},}\ }\href {\doibase 10.1126/science.1239717} {\bibfield {journal}
{\bibinfo {journal} {Science}\ }\textbf {\bibinfo {volume} {341}},\ \bibinfo
{pages} {960--961} (\bibinfo {year} {2013})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Zeh}(2007)}]{Zeh2007}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.~D.}\ \bibnamefont
{Zeh}},\ }\href {\doibase 10.1007/978-3-540-68001-7} {\emph {\bibinfo {title}
{The Physical Basis of the Direction of Time}}},\ \bibinfo {edition} {5th}\
ed.\ (\bibinfo {publisher} {Springer},\ \bibinfo {address} {Berlin},\
\bibinfo {year} {2007})\BibitemShut {NoStop} \bibitem [{\citenamefont {Carnap}(1963)}]{Carnap1963}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
{Carnap}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Intellectual
autobiography},}\ }in\ \href@noop {} {\emph {\bibinfo {booktitle} {The
Philosophy of Rudolf Carnap}}},\ \bibinfo {series} {The Library of Living
Philosophers}, Vol.~\bibinfo {volume} {11},\ \bibinfo {editor} {edited by\
\bibinfo {editor} {\bibfnamefont {P.~A.}\ \bibnamefont {Schilpp}}}\ (\bibinfo
{publisher} {Open Court},\ \bibinfo {address} {La Salle, Illinois},\
\bibinfo {year} {1963})\ pp.\ \bibinfo {pages} {37--38}\BibitemShut {NoStop} \bibitem [{\citenamefont {Bergson}(1910)}]{Bergson1910}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
{Bergson}},\ }\href {https://archive.org/details/cu31924014360410} {\emph
{\bibinfo {title} {Time and Free Will: An Essay on the Immediate Data of
Consciousness}}}\ (\bibinfo {publisher} {George Allen \& Unwin},\ \bibinfo
{address} {London},\ \bibinfo {year} {1910})\BibitemShut {NoStop} \bibitem [{\citenamefont {Bergson}(1999)}]{Bergson1999}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
{Bergson}},\ }\href@noop {} {\emph {\bibinfo {title} {Duration and
Simultaneity: Bergson and the Einsteinian Universe}}},\ \bibinfo {edition}
{2nd}\ ed.\ (\bibinfo {publisher} {Clinamen},\ \bibinfo {address}
{Manchester},\ \bibinfo {year} {1999})\BibitemShut {NoStop} \bibitem [{\citenamefont {Ridley}(2014)}]{Ridley2014}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {B.~K.}\ \bibnamefont
{Ridley}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Classical and
quantum framing of the {Now}},}\ }\href {\doibase 10.1063/PT.3.2492}
{\bibfield {journal} {\bibinfo {journal} {Phys. Today}\ }\textbf {\bibinfo
{volume} {67}},\ \bibinfo {pages} {8} (\bibinfo {year} {2014})},\ \bibinfo
{note} {{September}}\BibitemShut {NoStop} \bibitem [{\citenamefont {Wheeler}(1978)}]{Wheeler1978}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~A.}\ \bibnamefont
{Wheeler}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The `past' and
the `delayed-choice' double-slit experiment},}\ }in\ \href {\doibase
10.1016/B978-0-12-473250-6.50006-6} {\emph {\bibinfo {booktitle}
{Mathematical Foundations of Quantum Theory}}},\ \bibinfo {editor} {edited
by\ \bibinfo {editor} {\bibfnamefont {A.~R.}\ \bibnamefont {Marlow}}}\
(\bibinfo {publisher} {Academic},\ \bibinfo {address} {New York},\ \bibinfo
{year} {1978})\ pp.\ \bibinfo {pages} {9--48}\BibitemShut {NoStop} \bibitem [{\citenamefont {Wheeler}(1980)}]{Wheeler1980b}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~A.}\ \bibnamefont
{Wheeler}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Delayed-choice
experiments and the {Bohr}--{Einstein} dialog},}\ }in\ \href@noop {} {\emph
{\bibinfo {booktitle} {The American Philosophical Society and the Royal
Society: Papers Read at a Meeting, June 5, 1980}}}\ (\bibinfo {publisher}
{American Philosophical Society},\ \bibinfo {address} {Philadelphia},\
\bibinfo {year} {1980})\ pp.\ \bibinfo {pages} {9--40}\BibitemShut {NoStop} \bibitem [{\citenamefont {Wheeler}(1983)}]{Wheeler1983b}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~A.}\ \bibnamefont
{Wheeler}},\ }\enquote {\bibinfo {title} {Law without law},}\ pp.\ \bibinfo
{pages} {182--213},\ in\ \cite{WheelerZurek1983} (\bibinfo {year}
{1983})\BibitemShut {NoStop} \bibitem [{\citenamefont {Jacques}\ \emph {et~al.}(2007)\citenamefont
{Jacques}, \citenamefont {Wu}, \citenamefont {Grosshans}, \citenamefont
{Treussart}, \citenamefont {Grangier}, \citenamefont {Aspect},\ and\
\citenamefont {Roch}}]{Jacques2007}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {V.}~\bibnamefont
{Jacques}}, \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Wu}},
\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Grosshans}}, \bibinfo
{author} {\bibfnamefont {F.}~\bibnamefont {Treussart}}, \bibinfo {author}
{\bibfnamefont {P.}~\bibnamefont {Grangier}}, \bibinfo {author}
{\bibfnamefont {A.}~\bibnamefont {Aspect}}, \ and\ \bibinfo {author}
{\bibfnamefont {J.-F.}\ \bibnamefont {Roch}},\ }\bibfield {title} {\enquote
{\bibinfo {title} {Experimental realization of {Wheeler{\textquoteright}s}
delayed-choice gedanken experiment},}\ }\href {\doibase
10.1126/science.1136303} {\bibfield {journal} {\bibinfo {journal}
{Science}\ }\textbf {\bibinfo {volume} {315}},\ \bibinfo {pages} {966--968}
(\bibinfo {year} {2007})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Everett}(1957)}]{Everett1957}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
{Everett}},\ }\bibfield {title} {\enquote {\bibinfo {title} {`{Relative}
state' formulation of quantum mechanics},}\ }\href {\doibase
10.1103/RevModPhys.29.454} {\bibfield {journal} {\bibinfo {journal} {Rev.
Mod. Phys.}\ }\textbf {\bibinfo {volume} {29}},\ \bibinfo {pages} {454--462}
(\bibinfo {year} {1957})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Everett}(1973)}]{Everett1973}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
{Everett}},\ }\enquote {\bibinfo {title}
{\href{http://hdl.handle.net/10575/1302}{The theory of the universal wave
function}},}\ pp.\ \bibinfo {pages} {3--140},\ in\ \cite{DeWittGraham1973}
(\bibinfo {year} {1973})\BibitemShut {NoStop} \bibitem [{\citenamefont {DeWitt}\ and\ \citenamefont
{Graham}(1973)}]{DeWittGraham1973}
\BibitemOpen
\bibinfo {editor} {\bibfnamefont {B.~S.}\ \bibnamefont {DeWitt}}\ and\
\bibinfo {editor} {\bibfnamefont {N.}~\bibnamefont {Graham}},\ eds.,\
\href@noop {} {\emph {\bibinfo {title} {The Many-Worlds Interpretation of
Quantum Mechanics}}}\ (\bibinfo {publisher} {Princeton University Press},\
\bibinfo {year} {1973})\BibitemShut {NoStop} \bibitem [{\citenamefont {Bell}(1971)}]{Bell1971}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~S.}\ \bibnamefont
{Bell}},\ }\href {http://inspirehep.net/record/1106054/} {\enquote {\bibinfo
{title} {On the hypothesis that the {Schroedinger} equation is exact},}\ }
(\bibinfo {year} {1971}),\ \bibinfo {note} {{CERN-TH-1424}; revised version
published as Ref.\ \cite{Bell1981}.}\BibitemShut {Stop}
\bibitem [{\citenamefont {Bell}(1981)}]{Bell1981}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~S.}\ \bibnamefont
{Bell}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Quantum mechanics
for cosmologists},}\ }in\ \href@noop {} {\emph {\bibinfo {booktitle} {Quantum
Gravity 2}}},\ \bibinfo {editor} {edited by\ \bibinfo {editor} {\bibfnamefont
{C.~J.}\ \bibnamefont {Isham}}, \bibinfo {editor} {\bibfnamefont
{R.}~\bibnamefont {Penrose}}, \ and\ \bibinfo {editor} {\bibfnamefont
{D.~W.}\ \bibnamefont {Sciama}}}\ (\bibinfo {publisher} {Clarendon},\
\bibinfo {address} {Oxford},\ \bibinfo {year} {1981})\ pp.\ \bibinfo {pages}
{611--637},\ \bibinfo {note} {reprinted in Ref.\
\cite{Bell2004}.}\BibitemShut {Stop} \bibitem [{\citenamefont {Barbour}(1994{\natexlab{c}})}]{Barbour1994d}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~B.}\ \bibnamefont
{Barbour}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The emergence
of time and its arrow from timelessness},}\ }in\ \cite{Halliwell1994},\ pp.\
\bibinfo {pages} {405--414}\BibitemShut {NoStop} \bibitem [{\citenamefont {Barbour}(2000)}]{Barbour2000}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
{Barbour}},\ }\href@noop {} {\emph {\bibinfo {title} {The End of Time}}}\
(\bibinfo {publisher} {Oxford University Press},\ \bibinfo {year}
{2000})\BibitemShut {NoStop} \bibitem [{\citenamefont {Mermin}(1998)}]{Mermin1998b}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.~D.}\ \bibnamefont
{Mermin}},\ }\bibfield {title} {\enquote {\bibinfo {title} {What is quantum
mechanics trying to tell us?}}\ }\href {\doibase 10.1119/1.18955} {\bibfield
{journal} {\bibinfo {journal} {Am. J. Phys.}\ }\textbf {\bibinfo {volume}
{66}},\ \bibinfo {pages} {753--767} (\bibinfo {year} {1998})}\BibitemShut
{NoStop} \bibitem [{\citenamefont {Hartle}(2005)}]{Hartle2005}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~B.}\ \bibnamefont
{Hartle}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The physics of
now},}\ }\href {\doibase 10.1119/1.1783900} {\bibfield {journal} {\bibinfo
{journal} {Am. J. Phys.}\ }\textbf {\bibinfo {volume} {73}},\ \bibinfo
{pages} {101--109} (\bibinfo {year} {2005})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Smolin}(2013)}]{Smolin2013}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
{Smolin}},\ }\href@noop {} {\emph {\bibinfo {title} {Time Reborn}}}\
(\bibinfo {publisher} {Houghton Mifflin Harcourt},\ \bibinfo {address}
{Boston},\ \bibinfo {year} {2013})\BibitemShut {NoStop} \bibitem [{Note7()}]{Note7}
\BibitemOpen
\bibinfo {note} {The observer's inferences about the present moment were made
in the inferred past, when the present was regarded as part of the future.
Sentences such as this one illustrate how difficult it is to talk
consistently about the past and future as inferences. The difficulty is that
our language takes the reality of the past and future for granted. For most
of this paper, this problem is dealt with in the simplest possible way, by
not striving for absolute consistency.}\BibitemShut {Stop} \bibitem [{Note8()}]{Note8}
\BibitemOpen
\bibinfo {note} {The value $r=0$ must be excluded because the resulting state
$\protect \ensuremath {\delimiter 69640972 \psi \delimiter "526930B }$ would
not satisfy the Schr{\" o}dinger equation.}\BibitemShut {Stop} \bibitem [{\citenamefont {Nemenman}(2015)}]{Nemenman2015}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {I.}~\bibnamefont
{Nemenman}},\ }\bibfield {title} {\enquote {\bibinfo {title} {A mathematical
framework for falsifiability},}\ }\href {\doibase 10.1063/PT.3.2929}
{\bibfield {journal} {\bibinfo {journal} {Phys. Today}\ }\textbf {\bibinfo
{volume} {68}},\ \bibinfo {pages} {11--12} (\bibinfo {year} {2015})},\
\bibinfo {note} {{October}}\BibitemShut {NoStop} \bibitem [{\citenamefont {Zeh}(1999)}]{Zeh1999}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.~D.}\ \bibnamefont
{Zeh}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Why {Bohm}'s
quantum theory?}}\ }\href {\doibase 10.1023/A:1021669308832} {\bibfield
{journal} {\bibinfo {journal} {Found. Phys. Lett.}\ }\textbf {\bibinfo
{volume} {12}},\ \bibinfo {pages} {197--200} (\bibinfo {year}
{1999})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Wheeler}(1979)}]{Wheeler1979}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~A.}\ \bibnamefont
{Wheeler}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Frontiers of
time},}\ }in\ \href@noop {} {\emph {\bibinfo {booktitle} {Problems in the
Foundations of Physics}}},\ \bibinfo {series and number} {Proc. 72nd Enrico
Fermi School of Physics},\ \bibinfo {editor} {edited by\ \bibinfo {editor}
{\bibfnamefont {G.}~\bibnamefont {Toraldo~di Francia}}}\ (\bibinfo
{publisher} {North-Holland},\ \bibinfo {address} {Amsterdam},\ \bibinfo
{year} {1979})\ pp.\ \bibinfo {pages} {395--492}\BibitemShut {NoStop} \bibitem [{\citenamefont {Bohr}(1987{\natexlab{b}})}]{Bohr1958}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.}~\bibnamefont
{Bohr}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Quantum physics
and philosophy: Causality and complementarity},}\ }in\ \href@noop {} {\emph
{\bibinfo {booktitle} {Essays 1958--1962 on Atomic Physics and Human
Knowledge}}},\ \bibinfo {series} {The Philosophical Writings of Niels Bohr},
Vol.~\bibinfo {volume} {3}\ (\bibinfo {publisher} {Ox Bow},\ \bibinfo
{address} {Woodbridge, Conn.},\ \bibinfo {year} {1987})\ pp.\ \bibinfo
{pages} {1--7}\BibitemShut {NoStop} \bibitem [{\citenamefont {Bohr}(1928)}]{Bohr1928}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.}~\bibnamefont
{Bohr}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The quantum
postulate and the recent development of atomic theory},}\ }\href {\doibase
10.1038/121580a0} {\bibfield {journal} {\bibinfo {journal} {Nature}\
}\textbf {\bibinfo {volume} {121}},\ \bibinfo {pages} {580--590} (\bibinfo
{year} {1928})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Wheeler}(1986)}]{Wheeler1986b}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~A.}\ \bibnamefont
{Wheeler}},\ }\bibfield {title} {\enquote {\bibinfo {title} {How come the
quantum?}}\ }\href {\doibase 10.1111/j.1749-6632.1986.tb12434.x} {\bibfield
{journal} {\bibinfo {journal} {Ann. N.Y. Acad. Sci.}\ }\textbf {\bibinfo
{volume} {480}},\ \bibinfo {pages} {304--316} (\bibinfo {year}
{1986})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Wheeler}(1988)}]{Wheeler1988}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~A.}\ \bibnamefont
{Wheeler}},\ }\bibfield {title} {\enquote {\bibinfo {title} {World as system
self-synthesized by quantum networking},}\ }\href@noop {} {\bibfield
{journal} {\bibinfo {journal} {IBM J. Res. Develop.}\ }\textbf {\bibinfo
{volume} {32}},\ \bibinfo {pages} {4--15} (\bibinfo {year}
{1988})}\BibitemShut {NoStop} \bibitem [{Note9()}]{Note9}
\BibitemOpen
\bibinfo {note} {Private experiences may influence the choice of prior
probabilities in a way that cannot be described mathematically. However, it
is assumed here that a precondition for forming a team is consensus on the
method of defining prior probabilities.}\BibitemShut {Stop} \bibitem [{\citenamefont {Griffiths}(2002)}]{Griffiths2002}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.~B.}\ \bibnamefont
{Griffiths}},\ }\href {\doibase 10.1017/CBO9780511606052} {\emph {\bibinfo
{title} {Consistent Quantum Theory}}}\ (\bibinfo {publisher} {Cambridge
University Press},\ \bibinfo {year} {2002})\BibitemShut {NoStop} \bibitem [{\citenamefont {Omn\`es}(1999)}]{Omnes1999}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
{Omn\`es}},\ }\href@noop {} {\emph {\bibinfo {title} {Understanding Quantum
Mechanics}}}\ (\bibinfo {publisher} {Princeton University Press},\ \bibinfo
{year} {1999})\BibitemShut {NoStop} \bibitem [{\citenamefont {Gell-Mann}\ and\ \citenamefont
{Hartle}(2014)}]{GellMannHartle2014}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{Gell-Mann}}\ and\ \bibinfo {author} {\bibfnamefont {J.~B.}\ \bibnamefont
{Hartle}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Adaptive coarse
graining, environment, strong decoherence, and quasiclassical realms},}\
}\href {\doibase 10.1103/PhysRevA.89.052125} {\bibfield {journal} {\bibinfo
{journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {89}},\ \bibinfo
{pages} {052125} (\bibinfo {year} {2014})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Hohenberg}(2010)}]{Hohenberg2010}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.~C.}\ \bibnamefont
{Hohenberg}},\ }\bibfield {title} {\enquote {\bibinfo {title} {An
introduction to consistent quantum theory},}\ }\href {\doibase
10.1103/RevModPhys.82.2835} {\bibfield {journal} {\bibinfo {journal} {Rev.
Mod. Phys.}\ }\textbf {\bibinfo {volume} {82}},\ \bibinfo {pages}
{2835--2844} (\bibinfo {year} {2010})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Schlosshauer}(2007)}]{Schlosshauer2007}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{Schlosshauer}},\ }\href {\doibase 10.1007/978-3-540-35775-9} {\emph
{\bibinfo {title} {Decoherence and the Quantum-to-Classical Transition}}}\
(\bibinfo {publisher} {Springer},\ \bibinfo {address} {Berlin},\ \bibinfo
{year} {2007})\BibitemShut {NoStop} \bibitem [{\citenamefont {Vink}(1993)}]{Vink1993}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~C.}\ \bibnamefont
{Vink}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Quantum mechanics
in terms of discrete beables},}\ }\href {\doibase 10.1103/PhysRevA.48.1808}
{\bibfield {journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo
{volume} {48}},\ \bibinfo {pages} {1808--1818} (\bibinfo {year}
{1993})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Bub}(1995{\natexlab{a}})}]{Bub1995a}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
{Bub}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Interference,
noncommutativity, and determinateness in quantum mechanics},}\ }\href
{\doibase 10.1007/BF00763477} {\bibfield {journal} {\bibinfo {journal}
{Topoi}\ }\textbf {\bibinfo {volume} {14}},\ \bibinfo {pages} {39--43}
(\bibinfo {year} {1995}{\natexlab{a}})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Bub}(1995{\natexlab{b}})}]{Bub1995b}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
{Bub}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Why not take all
observables as beables?}}\ }\href {\doibase
10.1111/j.1749-6632.1995.tb39018.x} {\bibfield {journal} {\bibinfo
{journal} {Ann. N.Y. Acad. Sci.}\ }\textbf {\bibinfo {volume} {755}},\
\bibinfo {pages} {761--767} (\bibinfo {year}
{1995}{\natexlab{b}})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Bell}(1966)}]{Bell1966}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~S.}\ \bibnamefont
{Bell}},\ }\bibfield {title} {\enquote {\bibinfo {title} {On the problem of
hidden variables in quantum mechanics},}\ }\href {\doibase
10.1103/RevModPhys.38.447} {\bibfield {journal} {\bibinfo {journal} {Rev.
Mod. Phys.}\ }\textbf {\bibinfo {volume} {38}},\ \bibinfo {pages} {447--452}
(\bibinfo {year} {1966})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Kochen}\ and\ \citenamefont
{Specker}(1967)}]{KochenSpecker1967}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
{Kochen}}\ and\ \bibinfo {author} {\bibfnamefont {E.~P.}\ \bibnamefont
{Specker}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The problem of
hidden variables in quantum mechanics},}\ }\href {\doibase
10.1512/iumj.1968.17.17004} {\bibfield {journal} {\bibinfo {journal} {J.
Math. Mech.}\ }\textbf {\bibinfo {volume} {17}},\ \bibinfo {pages} {59--87}
(\bibinfo {year} {1967})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Mermin}(1993)}]{Mermin1993}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.~D.}\ \bibnamefont
{Mermin}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Hidden variables
and the two theorems of {John Bell}},}\ }\href {\doibase
10.1103/RevModPhys.65.803} {\bibfield {journal} {\bibinfo {journal} {Rev.
Mod. Phys.}\ }\textbf {\bibinfo {volume} {65}},\ \bibinfo {pages} {803--815}
(\bibinfo {year} {1993})}\BibitemShut {NoStop} \bibitem [{Note10()}]{Note10}
\BibitemOpen
\bibinfo {note} {See Kucha{\v r}'s discussion with Page on p.\ 296 of Ref.\
\cite {Page1994}.}\BibitemShut {Stop} \bibitem [{\citenamefont {Einstein}(1936)}]{Einstein1936}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Einstein}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Physics and
reality},}\ }\href {\doibase 10.1016/S0016-0032(36)91047-5} {\bibfield
{journal} {\bibinfo {journal} {J. Franklin Inst.}\ }\textbf {\bibinfo
{volume} {221}},\ \bibinfo {pages} {349--382} (\bibinfo {year} {1936})},\
\bibinfo {note} {reprinted in Refs.\ \cite{Einstein1950} and
\cite{Einstein1954}.}\BibitemShut {Stop} \bibitem [{\citenamefont {Einstein}(1950)}]{Einstein1950}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Einstein}},\ }\href@noop {} {\emph {\bibinfo {title} {Out of My Later
Years}}}\ (\bibinfo {publisher} {Philosophical Library},\ \bibinfo {address}
{New York},\ \bibinfo {year} {1950})\ pp.\ \bibinfo {pages}
{59--97}\BibitemShut {NoStop} \bibitem [{\citenamefont {Einstein}(1954)}]{Einstein1954}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Einstein}},\ }\href@noop {} {\emph {\bibinfo {title} {Ideas and Opinions}}}\
(\bibinfo {publisher} {Bonanza},\ \bibinfo {address} {New York},\ \bibinfo
{year} {1954})\ pp.\ \bibinfo {pages} {290--323}\BibitemShut {NoStop} \bibitem [{\citenamefont {Bargmann}(1961)}]{Bargmann1961}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {V.}~\bibnamefont
{Bargmann}},\ }\bibfield {title} {\enquote {\bibinfo {title} {On a {Hilbert}
space of analytic functions and an associated integral transform: Part
{I}},}\ }\href {\doibase 10.1002/cpa.3160140303} {\bibfield {journal}
{\bibinfo {journal} {Comm. Pure Appl. Math.}\ }\textbf {\bibinfo {volume}
{14}},\ \bibinfo {pages} {187--214} (\bibinfo {year} {1961})}\BibitemShut
{NoStop} \bibitem [{\citenamefont {Schweber}(1962)}]{Schweber1962}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.~S.}\ \bibnamefont
{Schweber}},\ }\bibfield {title} {\enquote {\bibinfo {title} {On {Feynman}
quantization},}\ }\href {\doibase 10.1063/1.1724296} {\bibfield {journal}
{\bibinfo {journal} {J. Math. Phys.}\ }\textbf {\bibinfo {volume} {3}},\
\bibinfo {pages} {831--842} (\bibinfo {year} {1962})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Folland}(1989)}]{Folland1989}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~B.}\ \bibnamefont
{Folland}},\ }\href@noop {} {\emph {\bibinfo {title} {Harmonic Analysis in
Phase Space}}}\ (\bibinfo {publisher} {Princeton University Press},\
\bibinfo {year} {1989})\BibitemShut {NoStop} \bibitem [{\citenamefont {Hall}(2000)}]{Hall2000}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {B.~C.}\ \bibnamefont
{Hall}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Holomorphic
methods in analysis and mathematical physics},}\ }in\ \href {\doibase
10.1090/conm/260} {\emph {\bibinfo {booktitle} {First Summer School in
Analysis and Mathematical Physics: Quantization, the Segal--Bargmann
Transform and Semiclassical Analysis}}},\ \bibinfo {editor} {edited by\
\bibinfo {editor} {\bibfnamefont {S.}~\bibnamefont {P\'erez-Esteva}}\ and\
\bibinfo {editor} {\bibfnamefont {C.}~\bibnamefont {Villegas-Blas}}}\
(\bibinfo {publisher} {Amer. Math. Soc.},\ \bibinfo {address} {Providence,
R.I.},\ \bibinfo {year} {2000})\ pp.\ \bibinfo {pages} {1--59}\BibitemShut
{NoStop} \bibitem [{\citenamefont {Antoine}(1998)}]{Antoine1998}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.-P.}\ \bibnamefont
{Antoine}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Quantum
mechanics beyond {Hilbert} space},}\ }in\ \href {\doibase 10.1007/BFb0106772}
{\emph {\bibinfo {booktitle} {Irreversibility and Causality: Semigroups and
Rigged Hilbert Spaces}}},\ \bibinfo {editor} {edited by\ \bibinfo {editor}
{\bibfnamefont {A.}~\bibnamefont {Bohm}}, \bibinfo {editor} {\bibfnamefont
{H.-D.}\ \bibnamefont {Doebner}}, \ and\ \bibinfo {editor} {\bibfnamefont
{P.}~\bibnamefont {Kielanowski}}}\ (\bibinfo {publisher} {Springer},\
\bibinfo {address} {Berlin},\ \bibinfo {year} {1998})\ Chap.~\bibinfo
{chapter} {1}\BibitemShut {NoStop} \bibitem [{\citenamefont {Antoine}\ \emph {et~al.}(2009)\citenamefont
{Antoine}, \citenamefont {Bohm},\ and\ \citenamefont
{Wickramasekara}}]{Antoine2009b1}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.-P.}\ \bibnamefont
{Antoine}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Bohm}}, \
and\ \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Wickramasekara}},\
}\enquote {\bibinfo {title} {Rigged {Hilbert} spaces for the {Dirac}
formalism of quantum mechanics},}\ pp.\ \bibinfo {pages} {651--660},\ in\
\cite{Greenberger2009} (\bibinfo {year} {2009})\BibitemShut {NoStop} \bibitem [{\citenamefont {Bohm}(1993)}]{Bohm1993}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Bohm}},\ }\href@noop {} {\emph {\bibinfo {title} {Quantum Mechanics:
Foundations and Applications}}},\ \bibinfo {edition} {3rd}\ ed.\ (\bibinfo
{publisher} {Springer},\ \bibinfo {address} {New York},\ \bibinfo {year}
{1993})\BibitemShut {NoStop} \bibitem [{\citenamefont {B{\"o}hm}(1978)}]{Bohm1978}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{B{\"o}hm}},\ }\href {\doibase 10.1007/3-540-088431-1} {\emph {\bibinfo
{title} {The Rigged {Hilbert} Space and Quantum Mechanics}}}\ (\bibinfo
{publisher} {Springer},\ \bibinfo {address} {Berlin},\ \bibinfo {year}
{1978})\BibitemShut {NoStop} \bibitem [{\citenamefont {Bogolubov}\ \emph {et~al.}(1975)\citenamefont
{Bogolubov}, \citenamefont {Logunov},\ and\ \citenamefont
{Todorov}}]{Bogolubov1975}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.~N.}\ \bibnamefont
{Bogolubov}}, \bibinfo {author} {\bibfnamefont {A.~A.}\ \bibnamefont
{Logunov}}, \ and\ \bibinfo {author} {\bibfnamefont {I.~T.}\ \bibnamefont
{Todorov}},\ }\href@noop {} {\emph {\bibinfo {title} {Introduction to
Axiomatic Quantum Field Theory}}}\ (\bibinfo {publisher} {W. A. Benjamin},\
\bibinfo {address} {Reading, Mass.},\ \bibinfo {year} {1975})\BibitemShut
{NoStop} \bibitem [{\citenamefont {Ballentine}(2015)}]{Ballentine2015}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
{Ballentine}},\ }\href {\doibase 10.1142/9789810248604_fmatter} {\emph
{\bibinfo {title} {Quantum Mechanics: A Modern Development}}},\ \bibinfo
{edition} {2nd}\ ed.\ (\bibinfo {publisher} {World Scientific},\ \bibinfo
{address} {Singapore},\ \bibinfo {year} {2015})\BibitemShut {NoStop} \bibitem [{\citenamefont {Klauder}(1960)}]{Klauder1960}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~R.}\ \bibnamefont
{Klauder}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The action
option and a {Feynman} quantization of spinor fields in terms of ordinary
$c$-numbers},}\ }\href {\doibase 10.1016/0003-4916(60)90131-7} {\bibfield
{journal} {\bibinfo {journal} {Ann. Phys. (N.Y.)}\ }\textbf {\bibinfo
{volume} {11}},\ \bibinfo {pages} {123--168} (\bibinfo {year}
{1960})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Gel'fand}\ and\ \citenamefont
{Shilov}(1968)}]{Gelfand1968}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {I.~M.}\ \bibnamefont
{Gel'fand}}\ and\ \bibinfo {author} {\bibfnamefont {G.~E.}\ \bibnamefont
{Shilov}},\ }\href@noop {} {\emph {\bibinfo {title} {Generalized
Functions}}},\ Vol.~\bibinfo {volume} {2}\ (\bibinfo {publisher}
{Academic},\ \bibinfo {address} {New York},\ \bibinfo {year}
{1968})\BibitemShut {NoStop} \bibitem [{\citenamefont {Simon}(1971)}]{Simon1971}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont
{Simon}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Distributions and
their {Hermite} expansions},}\ }\href {\doibase 10.1063/1.1665472} {\bibfield
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\end{document} | arXiv |
Assessment of postharvest loss along potato value chain: the case of Sheka Zone, southwest Ethiopia
Benyam Tadesse1,
Fayera Bakala2 &
Lamirot W. Mariam3
Ethiopia has possibly the highest potential for potato production than any country in Africa. Postharvest loss (20–25%) is one of the major problems in the potato production. Therefore, this study was conducted with the objective of assessing postharvest losses along potato value chain actors and identifying its determinants in the study area.
The descriptive result indicated that the quantity of postharvest losses at producer, local trader, wholesaler and retailer level was 21.724, 1.838, 3.406 and 4.07 kg/qt, respectively. The average gross margin with loss of producers, local traders, wholesalers and retailers was 6464.70, 282,169.89, 219,644.61 and 345,826.36 Birr, respectively, which is less than the average gross margin without loss (10,146.12, 284,015.83, 221,274.69 and 352,986.62 Birr, respectively). Distance to the nearest market, area allocated for potato and total output determine postharvest loss positively, and sex, experience, family size of working age, selling price and access to credit determine postharvest loss negatively.
In the study area, lack of storage facilities for potato was raised by farmers and other actors as a priority problem. Intervention of government from input supply until the end consumers is paramount and preparing storage mechanism is a must.
Achieving food security continues to be a challenge as it is affected by a complexity of factors [1,2,3]. Increasing the food availability is therefore not only increasing the productivity in agriculture, but also a need to lower the losses [4,5,6]. Food losses after harvest until the food reach the consumer are significant [7]. A large amount of food and products are not reaching the consumer particularly due to postharvest losses [4, 7] during harvesting, handling, transporting, storage, processing, packaging and distribution. Postharvest losses reduce the availability of food crops and income that could be generated by selling these products; thus in terms of quantity they are linked to food security [7,8,9]. According to FAO [10], postharvest losses in developing countries can range from 15 to 50%.
Potato (Solanum tuberosum L.) is the fourth most important food crop in the world on the basis of production after maize, rice and wheat with annual production accounts of nearly 300 million tonnes [11]. Out of these, over half of production occurs in developing countries [12]. During the production year of 2015/2016, it ranks first in area coverage and third in both total production and productivity among the root crops grown in Ethiopia [13]. It produces considerably more energy and protein than cereals [14]. It is also the fastest growing staple food crop and source of cash income for smallholder farmers in Ethiopia [15, 16]. It has a short cropping cycle and a large production per unit area in a given time. It provides more nutritious food per land unit in less time and often under more adverse condition than other food crops due to its efficient water use. It is one of the most efficient crops in converting natural resources, labor and capital into a high-quality food. Potato provides more food per unit area than any other major staple crop. They are the perfect food and one of the few that can actually sustain life on its own. Thus, it has significant impact on providing nutrition to families, increasing household income and providing surplus to the wider market [17].
Ethiopia has possibly the highest potential for potato production than any country in Africa with 70% of the 13.5 million ha of arable land suitable for potato cultivation. Over one million highland farmers could grow potatoes in Ethiopia. Two of the three known agroecologies Woyina Dega (1500–2300 masl) and Dega (above 2300 masl) exhibit the best out-grower potato production [18]. However, the potato is widely regarded as a secondary non-cereal crop in part because it has never reached the potential that it has in supporting food security. It is estimated that 296,577.59 ha is now planted and 36,576,382.69 qt production annually [13]. Hence, it contributes toward efforts of ensuring food and nutrition security. In Ethiopia, potato is becoming a prominent source of income since the crop is the most important cash crop for smallholder farmers in the mid-altitude and highland areas of the country [19, 20]. Ethiopia has a much higher potential to increase agricultural production of the crop through use of improved seeds and undertaking technological innovation that facilitate the management and reduce postharvest losses [17, 21].
Postharvest loss (20–25%) is one of the major problems in the potato production. This entails that significant loss is incurred to the small holders that could have helped in nutrition, food security and income generation [22]. Potato yield productivity has increased far more than 24 tons per hectare due to adoption of new varieties [17]. However, postharvest loss reduction efforts have not been tailored well. Harvesting loss reduction helps increasing income, achieve food security and subsequent storage lose reduction [23]. Thus, to reduce postharvest losses, appropriate technologies should be developed and promoted. With the reduction in postharvest losses by 50%, food availability would be increased by 20% without cultivating an additional hectare of land for increasing crop yield [24]. Until recently, knowledge of postharvest handling of fruits and vegetables such as improved storage, packaging, transport and handling techniques in developing regions like southwest Ethiopia was virtually nonexistent for perishable crops in most areas, thus allowing for considerable losses of produce. Postharvest losses have been highlighted as one of the determinants of the food problem [25]. According to Oyekanmi [26], postharvest loss prevention technology techniques becomes paramount as more produce is transported to non-producing areas to supply the growing population as well as storing for longer period to obtain a year-round supply. Despite the remarkable progress made in increasing world food production at the global level, approximately half of the population in the third world does not have access to adequate food supplies. There are many reasons for this; one of which is food loss occurring in the postharvest and marketing systems. Evidence suggests that these losses tend to be highest in countries where the need for food is greatest [25]. Food supply can be improved either by increase in production or more importantly, reduction in loss. Since many researches show that great effort is being made in the area of food production, especially in the developing countries, the decline in food production therefore can be traced to postharvest losses. Reduction in postharvest losses therefore will increase food availability, hence alleviation of food shortage problems. Managing the effect of postharvest losses has the potential tendency to reduce the effect of the efforts put into production and increase marketing efficiency [27]. Therefore, this study seeks to examine postharvest losses along potato value chain, underlying factors that contribute to the massive postharvest losses, taking into consideration the postharvest handling practices and how it affects the income of potato farmers in southwest Ethiopia.
Description of study area
This study was conducted in southwest Ethiopia, Masha District of Sheka Zone. Masha is one of the three districts in Sheka Zone of Southern Nation Nationalities and Peoples Region (SNNPR), which is located at 677 km to southwest of Addis Ababa. Sheka Zone is known by its dense forest coverage, and the agroecology is 70% mid-altitude, 20% high altitude and 10% low altitudes. The zone composes three districts, viz., Yeki, Andracha and Masha. Its altitude ranges from 950 to 3300 m above sea level. Mean rainfall level is more than 2000 mm. Regarding rainfall distribution, high rainfall occurs at June, July, August, September, medium at October, November, March, low at April and May and little/no at December and January. Major crops grown in the area are enset, sorghum, maize, coffee, potato, field pea, fava bean, wheat, barley, haricot bean and teff. Agricultural farming system is enset-production-based farming system. Total land coverage of the area is 217,527.15 hectare and from this 26% is for cultivation, 41.3% plantations, 2.24% pasture, 8.96% cultivable and 3.5% uncultivable land. The area was selected for this study due to exceptionally high potential for potato production in the southwestern part of Ethiopia. The district is composed of 19 rural kebeles; among these the study was conducted in 3 kebeles, viz., Gatimo, Atiso and Shibo, as indicated in Fig. 1.
Map of study area
Data source and instruments
In this study, both primary and secondary data sources were used. Different approaches of primary data collection methods were used. This includes survey, focus group discussions, key informant interviews, field observations and market assessments. Individuals from agricultural development and cooperative offices as well as the local people who have knowledge and experience about potato production and marketing were selected as key informants and interviewed on the issues related to potato production and marketing.
In this study, three-stage sampling technique was used. In the first stage, high-quantity potato-producing district was purposively selected. At the second stage, among the 19 rural kebeles of district, 3 kebeles were again purposively selected based on the intensity of production and marketing of potato. In the third stage, household heads producing potato were selected randomly from the total potato producers from three kebeles. The sample size was determined by rule of thumb suggested by Greene [28] N ≥ 50 + 8 m, where N is sample size and "m" is the number of explanatory variables. The sample size was determined by proportional to total population within three kebeles. Thus, using the total potato producers household list 193 producers were selected proportionally to total population of three kebeles (Tables 1, 2).
Table 1 Sample size of producers.
Table 2 Sample size of traders.
Regarding other value chain actors sample size, 7 local traders, 5 wholesalers and 8 retailers were selected by snowball techniques.
Data were analyzed by descriptive statistics, gross margin and ordinary least squares regression analysis. Descriptive statistics such as frequency distribution, percentages and mean were used in analyzing socioeconomic characteristics of respondents and quantity of potato lost at each value chain actor's level, while gross margin analysis was used to estimate the profit made by potato value chain actors in the study area. The gross profit of a business is estimated as the difference between the total sales and variable cost incurred.
$${\text{GM}} = {\text{TR}} - {\text{TVC,}}$$
where GM is gross margin and TR is total revenue [value of output (amount realized from the sale of potato)]. It was obtained by multiplying the quantity of potato sold by the unit selling price. TVC is total variable cost which includes cost of all inputs (preharvest and postharvest labor wage, transportation costs and other input costs).
Econometric model
Econometrically, multiple linear regression model was used to examine the relationship between postharvest loss of potato and explanatory variables. The general form of multiple linear regressions is:
$$\begin{aligned} Y & = \, f\left( {X_{1} , \, X_{2} , \, X_{3} , \, X_{4} \ldots X_{k} } \right) \\ Y & = \, \beta_{\text{o}} + \, \beta_{1} X_{1} + \, \beta_{2} X_{2} + \, \beta_{3} X_{3} + \cdots + \beta_{4} X_{k} + \varepsilon i \\ \end{aligned}$$
Y represents postharvest loss and X1, X2, X3, X4…X k represents independent or explanatory variables, and \(\varepsilon i\) is the disturbance factor. It is possible to write the function as:
Quantity of potato lost = f (Sex, Age, farming experience, Education_level, Distance to nearest market, FMSZ_working_age, Area allocated for potato production, total_output, selling price, local_seed, Improved seed, CREDITS and training). Based on this quantity lost function of potato, the econometric model for quantity of potato lost was written as:
$$\begin{aligned} {\text{Quantity of potato lost}} & = \, \beta_{\text{o}} + \beta_{1} {\text{Sex }} + \, \beta_{2} {\text{Age}} + \, \beta_{3} {\text{experience }} + \, \beta_{4} {\text{Education}} \\ & \quad + \beta_{5} {\text{Distance}} + \beta_{6} {\text{FMSZ}} + \beta_{7} {\text{AOLP}} + \beta_{8} {\text{total}}\_{\text{output}} \\ & \quad + \beta_{9} {\text{sellingprice}} + \beta_{10} {\text{local\_seed}} + \beta_{11} {\text{Improved seed}} \\ & \quad + \beta_{12} {\text{CREDITS}} + \beta_{13} {\text{DIUGETTR}} + \varepsilon i \\ \end{aligned}$$
where βo = intercept term.
Socio-demographic characteristics of respondents
The average age of interviewed respondents was 37.6632 years, while the average family size of the respondents was 6 in numbers. The education levels of respondents in the study areas were generally low (mean of 5.3990 years of schooling). On average, producer households had 7.0415 years of experiences in potato production and marketing. The mean distance from the nearest market center of producers in the study area was 5.5026 km. From the total land owned by the respondent, the land allocated for potato production is 0.55 ha/household, that means 27.94% of total.
From the total respondents, only 2% were Females and the rest were males. About 96.4% of studied respondents were married, while the remaining were not married (Table 3). Only half of producers had access to market information. From the total respondents, 2.6% of respondents had access to credit, while the rest 97.4% of respondents had no access of credit. Concerning other services, about 60% of producers had access to extension (Table 4).
Table 3 Demographic characteristics of respondents for continuous variables.
Table 4 Socioeconomic characteristics of producers (categorical variables).
Socioeconomic characteristics of traders
In this study, trader refers to local traders, retailers and wholesalers. As indicated in Table 3, the average age of local traders, retailers and wholesalers was 31.43, 29.13 and 35 years, respectively. Local traders, retailers and wholesalers averagely have 4 family members. Averagely, wholesalers were more experienced (4.2 years) as compared to local traders (2.71 years) and retailers (2 years). But in education level retailers were more educated than wholesalers. The initial working capital of local traders, retailers and wholesalers was 6500.14,Footnote 1 359.38 and 21,928 Birr, respectively, but the respective current working capital was 59,000, 3410.34 and 65,034 Birr. Table 6 shows all traders and wholesalers were male, while 63% of retailers were female and the rest were males. All traders, retailers and wholesalers had market information; however, none of traders had access to credit sources (Table 5).
Table 5 Socio-demographic characteristics of traders (continuous variables).
According to the survey result in Table 6, all traders responded that they have access to market information. All local traders and wholesalers were males only, but 38% of retailers were females. All local traders, wholesalers and retailers responded that they have no access for credit service in the study area.
Table 6 Socio-demographic characteristics of traders (categorical variables).
Assessment of postharvest loss along potato value chain
The descriptive result indicated that the mean value of the amount of potato postharvest loss at producer level was 9.31 qt per year per household which means 21.72%. When we estimate it in ETB, one household lose 3683.11 Br1 per year due to potato postharvest loss (Fig. 2).
Harvesting of potato by plowing the cultivated area
Next to producer, postharvest loss of potato was higher at retailer level. The postharvest loss assessment of local traders revealed that the quantity of potato lost in quintal per household was 3.34 qt which accounts 0.59%. It indicated that local traders lose 1324.64 ETB per household per year due to potato postharvest loss. Wholesalers lose 2.5 qt per household per year due to postharvest loss which estimated 0.65%. It indicated that one wholesaler lose 1630.1 ETB per year. Retail level losses were about 1.92 percent of the total produce of potato in the study area. The causes of loss were physical injury during harvest, rotting and disease infection, lack of storage area and poor handling area. The discarded potato fetched no economic value to the retailers. The aggregate postharvest loss from production (farm get level) to consumption level is 24.88% (Table 7).
Table 7 Assessment of postharvest loss along potato value chain.
Causes of postharvest losses at producer level
In Masha District, potato is harvested by different mechanisms. The first round potato is harvested by hand and suffers mechanical injury during harvest period because of digging tools, wooden sticks, hoes or forks. At the second round, producer collected remaining potato from the first round harvest by plowing potato cultivated area and/or digging via hoe. This kind of harvesting mechanisms leads the crop to damage due to injuries, cutting through or scraping away the outer skin of produce which provide entry points for molds and bacteria causing decay, increase water loss from the damaged area and cause an increase in respiration rate, and thus heat production finally hastens its senescence.
Mechanical injury
The high moisture content and soft texture of root crops like potato make them susceptible to mechanical injury, which occurred from production to retail marketing because of poor harvesting practices in the study area (Figs. 2, 3).
Harvesting, collecting and storage habit of potato in the study area
The producers in Masha Zone usually stayed their harvested potato without any shade at farm gate for 2–4 days due to low local market demand, and since the area is recognized by its heavy rainfall, the harvested potato was easily spoiled again. They did not cure their harvested product in the sided shade to get a quality product. One of the simplest and most effective ways to reduce water loss and decay during postharvest storage of root, tuber and bulb crops is curing after harvest. The type of wound also affects periderm formation: Abrasions result in the formation of deep, irregular periderm; cuts result in a thin periderm; and compressions and impacts may entirely prevent periderm formation. And also crops are most likely to be injured at harvest by the digging tools, which may be wooden sticks, machetes, hoes or forks. Researchers observed huge amount of potato thrown away or discarded at the farm gate; these all are neither consumed nor marketed in any form (Fig. 4).
Discarded potato at the farm gate
Selection and grading
All potatoes showing greening decay or severe damage owing to harvesting or pest attack should be discarded at harvest. Immature tubers and those showing minor damage or wetted by rain should be put aside for immediate consumption. Potatoes to be stored for food, or seed should be fully mature and free from any visible damage or decay. But in the study area the reality is different (Fig. 5).
Selection and grading of potato
The term "storage" as now applied to fresh produce is the holding of produces under controlled conditions. Usually in the study area, producers took the harvested potato to home if they had not get buyers and wait for sometimes until they get buyers and there is a chance to store potatoes for seed for subsequent production period. They place the produce directly on to the soil, especially wet soil, use dirty harvesting or field containers contaminated with soil, crop residues or decaying produce and dirty containers. There was paramount potato postharvest loss occurred due to improper and lack of storing area, insects and worms as well poor handling techniques. There are also situations where commission agents and traders collect potato together in open area around accessible load trucks from producers after bargained the selling price, but they promise to pay the price after it loaded. In this case, if traders face some hindering problems to load the collected potato, there is a probability to keep for 2–3 days while beaten by rainfall until it loaded. Significant amount of spoiled potato was observed due to these circumstances, and even sometimes traders refuse to take collected potato and free from losses. Commission agents are working for traders and/or wholesalers, and they never incur loss in potato marketing system (Fig. 6).
Storing system in the study area
The primary modes of transportation of potatoes in the study area to the markets were by loading it on back of animals like horses and donkeys. Due to over-packing or under-packing of field or marketing containers, careless handling, such as dropping or throwing or walking on produce and sitting on packed containers during the process of grading, transport or marketing, and the perish ability nature of potato, there is high postharvest loss (Fig. 7).
Potato loading and unloading system in the study area
Late blight caused by Phytophthora infestans was a major fungus disease in the study area causing huge yield losses. The climate is favorable for the development of late blight, particularly during the two rainy seasons. High costs of chemicals to control the disease limit the use of fungicides, particularly by small-scale farmers. Producers had weak habit of preventive applications of fungicides rather they often start spraying when the disease is already present in the field making control very difficult. Infested fields are then the source of inoculums (spores) for other fields, and in this way late blight can spread rapidly throughout the production areas. The situation is further aggravated by poor spraying techniques and the use of cheaper and less effective chemicals. Introducing varieties with some degree of resistance was important, but they still need fungicide applications to control the disease. It has been observed that late maturing varieties tend to have more resistance to late blight than early varieties, but farmers in the study area prefer early maturing varieties. It can be concluded that methods to control late blight are available, but they are not adequately applied by producers with limited resources for inputs and lack of knowledge (Fig. 8).
Sample of potato lost due to disease
Comparison between gross margin with loss and without loss in potato value chain
The average gross margin with loss (6464.70 Birr) was less than the average gross margin without loss (10,146.12 Birr). This shows that postharvest losses reduce the income of farmers in the study area. The percentage loss of income incurred by the farmers was 36.3%. The average gross margin with loss of local traders, wholesalers and retailers was 282,169.89, 219,644.61 and 345,826.36 Birr which was less than the average gross margin without loss which is indicated in Table 8.
Table 8 Gross margin with loss and without loss.
Determinants of postharvest loss
Multiple linear regression model was used to assess factors that determine postharvest losses of potato at producer level in the study area. To exclude heteroscedasticity problem, we regress the variables by using robust but before running the regression we checked multicollinearity problem, and the result showed that there was no serious multicollinearity problem. Model is generally significant as given by the F coefficient and its probability. The R-squared is high showing that the explanatory variables explain about 81.9% of the variation in the dependent variable.
Out of 13 hypothesized variables, eight variables were found to determine postharvest loss of potato in the study area. These are sex (SEX), distance to the nearest market (DISFM), Experience (EXPR), and family size of working age (FMSZ), area allocated for potato (AOLP), total output (TOTP), selling price (PRICE) and access to credit (ACTC). Access to extension service (ACCEXT), improved seed (IMPSD), local seed (LOSD), age (AGE) and education level (EDUCL) were not significantly associated with dependent variable (Table 9).
Table 9 Determinants of losses at producer level.
The regression result of sex indicated that the variable had negative relationship with postharvest loss and significant at less than 10% significance level. This implies that female-headed households are likely to experience postharvest losses for these crops as compared to the male-headed households. The coefficient of the variable indicated that potato producer being a male should reduce potato postharvest loss by 1.914101 quintal. As similar to the hypothesis, Experience had negative effect on potato postharvest loss and was significant at < 10% probability level. The negative relationship between quantity lost and farm experience indicated that the increase in farm experience resulted in a decrease in quantity lost. The coefficient of the variable indicated that the increase in farm experience by 1 year resulted in a decrease in quantity lost by 0.0957481 quintal. The better postharvest handling practices also reduce postharvest loss of potato. Family size of working age was statistically significant at < 1% significance level. As expected, the variable has a negative effect on quantity postharvest lost of potato. The negative and significant relationship indicates that as potato production was labor-intensive activity, larger number of working age family size provides higher labor force to undertake potato production and postharvest handling activities. The coefficient of the variable indicated that the increase in one number of working age family size reduced quantity postharvest by 2.249992 quintal or the decrease in one number of working age family size increased quantity postharvest loss by 2.249992 quintal. Distance to nearest market had positive effect on potato postharvest loss and found to be statistically significant at < 1% significance level. The positive relationship indicates that the farther is a household from the market, farmers forced to transport or store their product and it leads to potato postharvest loss. The coefficient of the variable indicated that the increase in distance of market by 1 km resulted in a decrease in potato postharvest loss by 0.6984369 quintal. Area allocated for potato production was significant at < 1% significance level, and as expected it had a positive relationship with quantity of potato postharvest loss. The coefficient of the variable illustrated that the increase in 1 ha of area allocated for potato production resulted in 1.118513 quintal of quantity of potato postharvest loss. Total output was statistically significant at < 1% significance level and had positive effect on quantity of potato postharvest loss. The coefficient of the variable showed that the increase in the total output of potato by 1 quintal increased quantity postharvest of potato by 0.1334658 quintal. Selling price of potato was significant at 5% significance level. As hypothesized, it had negative effect on quantity postharvest loss of potato. The coefficient of the variable indicated that the increase in selling price of potato resulted in the decrease in quantity of potato postharvest lost by 0.0130786 Birr. Access to credit was significant at 5% significance level and had negative effect on quantity of postharvest lose. The coefficient of the variable indicated that access for credit resulted in 2.845566 quintal of potato postharvest loses.
The average age of interviewed respondents was 37.6632 years. The result indicates a good supply of agile workforce in potato production in the study area. This result is in line with finding of Ayandiji et al. [24]. Babalola et al. [29] argued that age is a very important demographic characteristic because it determines the size and quality of the labor force. Older farmers are expected to use their farming experience to decide on appropriate postharvest handling practices and hence an overall reduction in postharvest losses [30].
The descriptive result indicated that the mean value of the amount of potato postharvest loss at producer level was 9.31 qt per year per household which means 21.72%. Postharvest losses in fruits and vegetables are about 25–40% [31]. 75% losses were occurred at field level [32]. 30% of fruits, 10% of vegetables, 50% of root crops and 60% of cash crops were lost [33]. The results are in line with reports of Admassu [34], Abebe and Bekele [35] and Humble and Reneby [36]. According to Hodges et al. [9], postharvest losses in developing countries can range from 15 to 50%. Horticultural crops are perishable products, and they are more prone to greater losses than for non-perishable crops [4]. Postharvest losses at producer level occur due to harvesting injuries; for example, for the first round potato is harvested by hand and some simple digging material and at this stage loss could occur. At the second round, producers collect potato output remained from the first round harvest; hence, they plow potato cultivated area and/or dig via hoe. These potato harvesting mechanisms trigger potato loss at farm gate level. In addition to this, after harvested usually potato stayed at farm gate for 2–4 days due to lack of potato market demand in the area without any shade, and since the area is recognized by its heavy rainfall, the harvested potatoes were easily spoiled again. Researchers observed huge amount of potatoes thrown away or discarded fruits at the farm gate; these all are neither consumed nor marketed in any form. Estimated losses at farm, market and consumption level were reported as 38.6, 35.9 and 25.5% of the total losses, and total postharvest losses were 31% of the total production [31]. Martey et al. [37] found that about 75% of the total postharvest losses occurred at the farm level and about 25% at the market level.
Next to producer, postharvest loss of potato was higher at retailer level in the study area. Losses at retailer level are obvious because he is unaware about the daily sales and he buys potato according to his experience and bears losses in shape of unsold quantity. In the study area, retailers lack shops for performing their activities and they sell potato in the market place which is exposed to sunlight. Bari [31], FFTC [38], Liu [39] also indicate that unsold quantity was lost daily and also retailers don't have enough resources to store their unsold commodity. These results are also similar to FFTC [38] and Gajanana et al. [40]. Postharvest losses reduce the income of farmers in Masha District of southwest Ethiopia. This result is in line with the finding of Ayandiji et al. [24]. The average gross margin with loss was less than the average gross margin without loss for all potato value chain actors. The factors that influence the postharvest losses significantly at the farm level were identified in the study area.
The regression result of sex indicated that the variable had negative relationship with postharvest loss and significant at < 10% significance level. Married household heads are thought to have an advantage with regards to labor availability for their production and postharvest handling activities, which in turn could minimize postharvest losses [41]. Takane [42] contented that usually single-headed households are female-headed households, and because of the absence of husbands, female-headed households have fewer economically active household members and are in a disadvantageous position relative to their male-headed counterparts in deploying family labor for farm activities. Smallholder farmers find themselves at a major disadvantage because many do not understand the market well, how it works and why prices fluctuate, they have little or no information about market conditions and prices, they are not organized collectively, and they have no experience of market negotiation [43]. There is therefore a need for these farmers to be made aware of various products they can grow in relation to their climatic conditions and market demand [44]. Male-headed households are likely to experience postharvest losses for these crops as compared to the female-headed households. This result contradicts the finding of Liu [39] who concluded that female farmers were found to be more prone to high levels of losses than their male counterparts. A study conducted by Ortmann and King [41] revealed a positive and statistically significant relationship between sex of household head and both cabbage and spinach postharvest losses. Considering the fact that more time and careful handling are required to minimize mechanical damages in leafy and fruit vegetables, females are likely to encounter minimal losses since they are careful handlers and are considered to be more patient as compared to their male counterparts.
The education levels of respondents in the study areas were generally low (mean of 5.3990 years of schooling). This finding is consistent with the finding of Ortmann and King [41]. Education is the key in understanding consumer quality expectations in niche markets which is necessary also because consumers may have different expectations and acceptance of the same food product produced using different technologies [45]. The more educated the farmer, the less will be potato postharvest loss. It has had positive effect on the adoption of appropriate agricultural technologies and skills to the farming population over the years. This agrees with the findings of Oduekun [45] that the level of education influences participation in agricultural productive activities, adoption, transfer and application of innovations. This could be a contributory factor to the high postharvest losses in potato production in the study areas because only farmers with good education often appreciate and use most postharvest technologies available. This result is consistent with the findings of Fawole and Fasina [46], Basavaraja et al. [47], Adesina and Baidu-Forson [48]. Education level of farmers also had significant impact on postharvest losses [37].
As potato production was labor-intensive activity, larger number of working age family size provides higher labor force to undertake potato production and postharvest handling activities. This is because farmers who had larger household sizes tended to have lower levels of postharvest losses because they have relatively high amount of family labor. Distance from the farm to the market was positively and significantly related to tomato postharvest losses. Similar results were reported by Liu [30].
On average, producer households had 7.0415 years of experiences in potato production and marketing. As similar to the hypothesis, Experience had negative effect on potato postharvest loss and was significant at < 10% probability level. More experienced farmers are expected to have minimal postharvest losses as compared to the inexperienced ones since they can utilize the gained experience to make important handling decisions as well as having market contacts to ensure that harvested produce is sold quickly [41]. The research result implied that as the farmer become more experienced, postharvest losses diminish. An experienced farmer population implies good knowledge and adoption of postharvest handling technology among the farmers [30]. As revealed by [41], older household heads have farming experience and adopted new technologies than young farmers; therefore, farming experience was expected to have a negative relationship with postharvest losses. Farming experience is thought to positively influence technology adoption [37, 49].
The mean distance from the nearest market center of producers in the study area was 5.5026 km. Distance to nearest market had positive effect on potato postharvest loss and found to be statistically significant at < 1% significance level. PHLs also varied depending upon the distance to the market. This pushes to higher quantity of potato postharvest loss. When they are marketed to medium-distance markets, PHLs were 5.15%, whereas for long-distance markets they were 8.17% [50]. According to Ortmann and King [41], the distance between the farm and the market had a positive and statistically significant relationship with cabbage and tomato postharvest losses. The farther the market is from the farm, the longer it takes for the produce to reach the market and hence an increase in postharvest losses due to heat build-up and in transit mechanical injuries [4]. According to Ayandiji et al. [24], increase in the distance from the farm to the market will increase the quantity of fruit loss; this is because the longer the distance of the farm to the market, the longer the time it will take for the produce to get to the market, and so the losses will increase because of the congestion and packaging of the tomato together for a long time. The longer the distance the more the time it will take for the produce to get to the market and so, the losses will increase because of congestion of the product and build up of heat [30].
From the total land owned by the respondent, the land allocated for potato production is 0.55 ha/household (i.e., 27.94% of total farmland). Area allocated for potato production was significant at < 1% significance level, and as expected it had a positive relationship with quantity of potato postharvest loss. When the area for production increases, total output will increase and also it will increase potato postharvest loss. Higher land holdings serve as an incentive to produce surplus for markets [42]. With respect to postharvest losses, large land holdings imply large volumes being produced, and the higher the production volumes, the higher the losses since farmers face the constraints of poor handling practices and limited storage facilities [30, 43]. The larger the area put into cultivation leads to higher production and greater chance of losses due to poor handling and lack of proper storage [41]. The larger the farm size, the higher the likelihood for postharvest losses. Similar results were reported by Gajanana [40] and Ortmann and King [41]. They reported that as production scale increases farmers will have to contend with the problem of storage and transportation; and where these facilities are not adequate, losses are imminent. The larger the area put into cultivation, the higher the quantity harvested and chances of losses due to poor handling and lack of proper storage [30]. Takane [42], however, argued that farm size may have indirect positive impacts on market participation by enabling farmers to generate production surpluses and overcome credit market, thus reducing postharvest losses. Overall, an increase in farm size tends to have a significant increase in tomato postharvest losses [41]. According to the same source instead of increasing farm size or land allocated to tomatoes, reducing the farm size or land allocated to tomatoes has an overall effect of tomato postharvest loss reduction. The larger the area put into cultivation the higher the quantity harvested and chances of higher losses due to poor handling and lack of proper storage. Good hygiene practices such as hand washing and postharvest handling equipment washing minimize chances of produce contamination, hence a reduction in postharvest losses [4]. This was in line with study of Begum et al. [51] which was studied with the title of Economic Analysis of Postharvest Losses in Food Grains for Strengthening Food Security in Northern Regions of Bangladesh. Increase in the number of baskets harvested of fruits also results in increase in the losses because there is no effective method of storage; hence, the more the quantity of harvested produce, the more the spoilage [24].
The descriptive result of assessment of postharvest losses along potato value chain actors indicated that 21.72, 0.59, 0.655 and 1.92% loss was estimated at producer, local trader, wholesaler and retailer levels, respectively. The average gross margin with loss of producers, local traders, wholesalers and retailers was 6464.70, 282,169.89, 219,644.61 and 345,826.36 Birr which was less than the average gross margin without loss (10,146.12, 284,015.83, 221,274.69 and 352,986.62 Birr). Out of suggeted variables sex, distance to the nearest market, experience, and family size of working age, area allocated for potato, total output, selling price and access to credit. Farmers in the study area lack skills of pre- and postharvest management and disease prevention of potato. Therefore, training on cultivation, disease prevention and pre- and postharvest management are important factors for enhancement of potato productivity and reducing postharvest losses in the study area. In the study areas, lack of storage facilities for potato was raised by farmers and other actors as a priority problem. There is the need for provision of good storage facilities to store the produce that are harvested before they are taken to the market. This will help to reduce the losses that occur at the farm level. Therefore, it is recommended to expand DLS in high potato-producing areas as per standard DLS design and construction. Through technical support to the farmers, cost effective mechanism of expanding DLS should be considered. Postharvest technology should be introduced to reduce the losses. In the potato market survey, it was observed that potato is transported over long distance either spread on floor of the truck, back of horse or put in congested sacks. During loading and unloading, there is mishandling of the products which lead to quick spoilage and high loss. Therefore, it is important to establish potato transportation standards and enforce it. There should be ready market for the produce. The markets must be well organized and also the road network must be improved in order to aid easy transportation of their produce. Roads linking farms to market should be improved to reduce transit losses. Establishment of farmers market and cooperative marketing should be encouraged to reduce losses related to marketing functions.
The basic unit of money in Ethiopia; equal to 100 cents.
PHLs:
postharvest losses
DLS:
diluted light storage
ETB:
SNNPR:
Southern Nation Nationalities and Peoples Region
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The corresponding author carried out the responsibilities of proposal writing, data collection and preparation of the manuscript. The co-authors FB and LWM were participated in activities like communicating with the donors, commenting on issues to be raised on the proposal and questioner preparation and participating in the write-up of the manuscript. All authors read and approved the final manuscript.
This study received financial support from Mizan-Tepi University. DAs and government office workers of Masha District are acknowledged for their cooperation during the data collection.
We declare that whatever data have been used in the manuscript, it can be made available to anyone who desires to see them from the corresponding author on request.
We agreed that the information given in the manuscript now can be published by the Publication House and Journal of "Agriculture and Food Security."
It is to declare that we have all the ethical approval and consent to take participate in research paper writing and submission to any relevant journal from our organization where we are working and posted.
This work was funded by Mizan-Tepi University from its research and community service budgets.
College of Agriculture and Natural Resources, Department of Agro Economics, Mizan-Tepi University, Mizan-Aman, Ethiopia
Benyam Tadesse
College of Agriculture and Natural Resources, Department of Natural Resource Economics, Mizan-Tepi University, Mizan-Aman, Ethiopia
Fayera Bakala
College of Agriculture and Natural Resources, Department of Horticulture, Mizan-Tepi University, Mizan-Aman, Ethiopia
Lamirot W. Mariam
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Correspondence to Benyam Tadesse.
Tadesse, B., Bakala, F. & Mariam, L.W. Assessment of postharvest loss along potato value chain: the case of Sheka Zone, southwest Ethiopia. Agric & Food Secur 7, 18 (2018) doi:10.1186/s40066-018-0158-4
Multiple linear regressions
Postharvest loss | CommonCrawl |
\begin{definition}[Definition:Set Union/General Definition]
Let $S$ be a collection, which could be either a set or a class.
The '''union of $S$''' is:
:$\ds \bigcup S := \set {x: \exists X \in S: x \in X}$
That is, the set of all elements of all elements of $S$ which are themselves sets.
\end{definition} | ProofWiki |
A question on the definition of the Riemann–Stieltjes integral
I have been reading Wikipedia's article on the Riemann–Stieltjes integral (https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral) and I don't understand why the Riemann–Stieltjes integral isn't equaivalent to the Generalized Riemann–Stieltjes integral. I mean, if instead of $g$ we take the identity function (i.e. we consider the case of the Riemann integral) then they are the same. Why does this change for other $g$'s? I am quite confused because my textbook uses the definition Generalized Riemann–Stieltjes integral as the definition for the Riemann–Stieltjes integral and this is why I want to understand the difference between them.
EDIT: I would also like to know which properties of the generalized integral transfer to the ungeneralized one and which do not to fully grasp them.
real-analysis integration definition
JustAnAmateur
JustAnAmateurJustAnAmateur
$\begingroup$ ${}\,+1.$ One who edits Wikipedia articles should understand the way I edited the punctuation in this question. $\endgroup$
– Michael Hardy
$\begingroup$ Pollard, Henry (1920). "The Stieltjes integral and its generalizations". The Quarterly Journal of Pure and Applied Mathematics. 49. babel.hathitrust.org/cgi/… $\endgroup$
$\begingroup$ Pollard writes: "The integral so obtained exists and is identical with the integral previously defined whenever this exists, and exists in certain cases where this does not." I don't yet know what those "certain cases" are. $\qquad$ $\endgroup$
Pollard, Henry (1920). "The Stieltjes integral and its generalizations". The Quarterly Journal of Pure and Applied Mathematics. 49.
Pollard's paper appears to be where the generalization was introduced. The paper begins on page 73 and that page has only the title, the author's name and affiliation, and the references. The link above is to page 74.
On page 80, Pollard defines two functions: $$ f(x) = \begin{cases} 0 & x < 1, \\ k & x\ge 1, \end{cases} $$ $$ \varphi(x) = \begin{cases} 0 & x\le 1, \\ 1 & x>1. \end{cases} $$ He seems to claim that as a generalized Riemann–Stieltjes integral, apparently defined as a Moore–Smith limit of a net indexed by partitions of the interval $[0,2],$ the integral $$ \int_0^2 f(x)\,d\varphi(x) $$ exists and is equal to $k(\varphi(2) - \varphi(1)),$ but that as a Riemann–Stieltjes integral, defined as a limit as the mesh of the partition approaches $0,$ that integral does not exist. Pollard calls the the generalized Riemann–Stieltjes integral "the modified Stieltjes integral."
But I haven't carefully sifted through all the details.
Michael HardyMichael Hardy
$\begingroup$ If the partition does not contain the point $1$, then by choosing the corresponding $c$ either $< 1$ or $\geqslant 1$ you can make the sum $0$ or $k$. But if the partition contains the point $1$, then for the $[x_{k-1},1]$ interval you always have $f(c)\cdot (0 - 0)$ and for the $[1, x_{k+1}]$ interval you always have $k\cdot(1-0)$. $\endgroup$
$\begingroup$ I seem to recall Baby Rudin saying the Riemann–Stieltjes integral is undefined in cases where $f$ and $\varphi$ have a point of discontinuity in common. So that's where the "generalized" version goes beyond the ungeneralized one. $\endgroup$
$\begingroup$ Yes. The generalised Riemann–Stieltjes integral can deal with some common discontinuities, but not with all. $\endgroup$
$\begingroup$ Thank you very much! I agree with @MichaelHardy that the generalized version deals with some cases when the function have a point of discontinuity in common. What I would also like to know is which properties of the generalized integral transfer to the ungeneralized one and which do not, I will add this in my question. $\endgroup$
– JustAnAmateur
$\begingroup$ @JustAnAmateur : I am inclined to doubt that any useful properties of the generalized version fail to apply to the version defined by Stieltjes except that the one defined by Stieltjes fails to be defined when $f$ and $\varphi$ have a common discontinuity. The paper by Pollard might be the first place I'd look for an answer to that. $\endgroup$
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Dual bundle
In mathematics, the dual bundle is an operation on vector bundles extending the operation of duality for vector spaces.
Definition
The dual bundle of a vector bundle $\pi :E\to X$ is the vector bundle $\pi ^{*}:E^{*}\to X$ whose fibers are the dual spaces to the fibers of $E$.
Equivalently, $E^{*}$ can be defined as the Hom bundle $\mathrm {Hom} (E,\mathbb {R} \times X),$ that is, the vector bundle of morphisms from $E$ to the trivial line bundle $\mathbb {R} \times X\to X.$
Constructions and examples
Given a local trivialization of $E$ with transition functions $t_{ij},$ a local trivialization of $E^{*}$ is given by the same open cover of $X$ with transition functions $t_{ij}^{*}=(t_{ij}^{T})^{-1}$ (the inverse of the transpose). The dual bundle $E^{*}$ is then constructed using the fiber bundle construction theorem. As particular cases:
• The dual bundle of an associated bundle is the bundle associated to the dual representation of the structure group.
• The dual bundle of the tangent bundle of a differentiable manifold is its cotangent bundle.
Properties
If the base space $X$ is paracompact and Hausdorff then a real, finite-rank vector bundle $E$ and its dual $E^{*}$ are isomorphic as vector bundles. However, just as for vector spaces, there is no natural choice of isomorphism unless $E$ is equipped with an inner product.
This is not true in the case of complex vector bundles: for example, the tautological line bundle over the Riemann sphere is not isomorphic to its dual. The dual $E^{*}$ of a complex vector bundle $E$ is indeed isomorphic to the conjugate bundle ${\overline {E}},$ but the choice of isomorphism is non-canonical unless $E$ is equipped with a hermitian product.
The Hom bundle $\mathrm {Hom} (E_{1},E_{2})$ of two vector bundles is canonically isomorphic to the tensor product bundle $E_{1}^{*}\otimes E_{2}.$
Given a morphism $f:E_{1}\to E_{2}$ of vector bundles over the same space, there is a morphism $f^{*}:E_{2}^{*}\to E_{1}^{*}$ between their dual bundles (in the converse order), defined fibrewise as the transpose of each linear map $f_{x}:(E_{1})_{x}\to (E_{2})_{x}.$ Accordingly, the dual bundle operation defines a contravariant functor from the category of vector bundles and their morphisms to itself.
References
• 今野, 宏 (2013). 微分幾何学. 〈現代数学への入門〉 (in Japanese). 東京: 東京大学出版会. ISBN 9784130629713.
Manifolds (Glossary)
Basic concepts
• Topological manifold
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• in Lie theory
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manifolds
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| Wikipedia |
Geolocation of covert communication entity on the Internet for post-steganalysis
Fan Zhang1,2,
Fenlin Liu1,2 &
Xiangyang Luo1,2
Geolocation of covert communication entity is significantly important for the forensics of the crime but has significant challenges when the steganalyst locks the guilty actor IP and wants to know the physical location of the actor. This kind of post-steganalysis involves not only the stegos transmitted on the Internet but the IP package head and content. This paper presents a geolocation method for the location of the covert communication entity based on hop-hot path coding. The method estimates the location of the covert communication entity by combining the path and delay between probes and the covert communication entity IP, which improves the deficiency that similar delays do not necessarily mean close geographical locations of the IPs. Moreover, the similarity between the IPs' paths can be measured by coding the paths between IPs and probes. The results of a series of experiments show that the median error of the proposed method is within 6.16 km using different thresholds.
Steganography embeds secret messages into unsuspicious carriers and transmits the messages through public channels for covert communication without attracting attention [1]. This kind of communication not only hides the secret messages but also the communication behavior as the carriers are accessible to anyone on the public channels. As the steganography techniques could be maliciously used for stealing confidential information, it is practically significant to carry on researches to forensics of the crime. For decades, researchers have proposed many techniques for the forensics of steganography, including the stego detection [2,3,4,5,6,7,8,9,10], the payload location [11,12,13,14,15], the embedding key restore [16, 17], the secret message extraction [17], and the steganographer detection [18,19,20]. In practice, the covert communication entity on the Internet usually acts as the user of social platforms, whose location is virtual. Even if the covert communication entity is successfully detected, the physical location of the covert communication entity is still unknown. To achieve the complete forensics of the crime, the post-steganalysis that investigates the physical location of the covert communication entity should be carried out.
As the transmission of the stegos on the Internet involves the IP (Internet Protocol) packages and the IP usually reflects the physical location of the client, the physical location of the covert communication entity can be located based on the IP addresses in the IP packages of the stegos. At present, street-level geolocation method is suitable for locating the covert communication entity, such as SLG (Street-Level Geolocation) method [21] IRLD (Identification Routers and Local Delay Distribution Similarity based Geolocation) method [22], and TNN method (IP Geolocation Algorithm based on Two-tiered Neural Networks) [23]. These methods are based on an important assumption that when probes measure IPs with similar geographic locations, the delays from probes to IPs are often similar. However, in actual Internet environment, IPs' locations with similar delays are not necessarily adjacent. Therefore, it is difficult to ensure the reliability of the results of these methods.
Aiming at the above problem, this paper presents a geolocation method for the location of the covert communication entity based on hop-hot path coding. It is worth noting that the geolocation of the entity is for the post-steganalysis where the stegos transmitted on the Internet are successfully and the entity IP is locked. With the knowledge of the covert communication entity IP, the proposed method firstly obtains landmarks around the covert communication entity (landmarks are not covert communication entities; they are IP addresses of known geographical locations), and use probes to measure the known landmarks to get the delay and path information from probes to landmarks. Then, the path is encoded to get the vector of delay and path, and probes are used to measure the covert communication entity to obtain the entity vector of delay and path. After that, the vector of delay and path of the landmarks whose network environment is similar to the entity is taken as the input, and the corresponding latitude and longitude of the landmarks are as the output to train the neural network. Finally, input the entity vector of delay and path neural network to geolocate the entity.
The rest of this paper is organized as follows. The existing typical entity geolocation methods are introduced in Section 2. Section 3 introduces the basic principle and main steps of the proposed method for the geolocation of the covert communication entity based on hop-hot path coding. The performance of the method is evaluated and discussed through the experiments in Section 4. Finally, Section 5 summarizes the work of this paper.
In this section, the existing typical network entity geolocation methods are briefly analyzed and the problems involved are pointed out.
Existing network entity geolocation methods usually attempt to describe the conversion or statistical relationship between delay and geographical location. The accuracy of most of these methods can only reach city level. Only a few methods, such as SLG and TNN, can geolocate the network entity at street-level granularity.
The SLG method [21] uses a three-tier geolocation process to locate the network entity. A schematic diagram of the geolocation process of SLG method is shown in Fig. 1. In tier 1, the method convert the delay between the probes and the network entity into geographical distance, and geolocate the entity into a coarse-grained region based on multilateration. In tier 2, the relative delay between the landmarks and the entity is converted into distance; then, the entity is geolocated into a fine-grained region via multilateration. In tier 3, the location of the landmark with the minimum relative delay of the entity is taken as the estimated location of the entity, for example, the landmark L3.
Schematic diagram of SLG method
The TNN method [23] locates the network entity by training neural network; it also utilizes the idea of approaching tier by tier to geolocalize the entity. A schematic diagram of the geolocation process of TNN method is shown in Fig. 2. The method uses RBF (radial basis function) neural network and MLP (multilayer perceptron) neural network to learn the mapping between the delay and the location of the landmarks to achieve entity geolocation. The TNN method uses RBF neural network as the first tier to locate a smaller region in which the network entity resides, and then uses MLP neural network as the second tier to estimate its location more accurately within that region.
Schematic diagram of TNN method
Under normal circumstances, the above methods based on delay can achieve street-level entity geolocation. However, the delays of probes measuring entity can only represent the distance between the probes and the entity. In actual network environment, due to indirect or circuitous routing, entities with similar delays may be far apart, which will cause unreliable geolocation result. In order to overcome the above problems and improve reliability in entity geolocation, an entity geolocation method based on delay and path is proposed in this paper. Different from the above typical methods, the proposed method not only estimates the location of entity by delay but also takes into account the paths between the probes and the entity.
The proposed method
A large amount of measurement data shows that the end-to-end distance in the Internet can be approximated by delay, and the direction is determined by the path. According to this basic fact, the geolocation method trains neural networks based on the combination of delay and path. Because of the ISPs (Internet service providers) of the covert communication entities are unknown and ISPs in some countries do not fully realize the interconnection within the city, when geolocating the covert communication entities, we need to use landmarks around the entities to ensure the consistency of training samples.
The method is divided into six parts: obtaining landmarks, vectors construction of landmarks, acquisition of training sets, training neural networks, vector construction of the covert communication entity, and entity geolocation. Figure 3 shows the frame diagram of the method.
Method frame diagram
The specific steps of the method are as follows:
Obtaining landmarks. With the knowledge of the covert communication entity IP, get landmarks around the entity.
Vector construction of landmarks. Deploying n probes P1, P2, …, Pn, acquiring the delay and path from the probes to landmarks. Then, encoding the path with hop-hot path code method to get the vectors of delay and path
$$ {V}_k=\left({d}_{k,1},{d}_{k,2},\dots, {d}_{k,n},{C}_k\right). $$
where Vk represents the vector of delay and path of the k th landmark, and dk, i represents the delay from the probe i to the landmark k. Ck represents the encoded path vector of the landmark k.Acquisition of training sets. Use (1) to cluster the landmarks, and then use the latitude and longitude of the landmarks to cluster the landmarks. Take the intersection of the two clustering results to obtain the training sets, denoted as
$$ F=\left\{{S}_1,{S}_2,\dots, {S}_q\right\}. $$
where Si is the i th training set.
Neural networks training. Train the neural network for each training set. Taking (1) in the training set Si as input, and the latitude and longitude thereof as output, obtaining a well-trained neural network.
Vector construction of the covert communication entity. Acquiring the delay and path from the probes to entity. Encoding the path to get the vector of delay and path of the covert communication entity
$$ {V}_T=\left({d}_1,{d}_2,\dots, {d}_n,{C}_T\right). $$
where VT represents the vector of delay and path of the covert communication entity, and di represents the delay from the probe i to the entity. CT represents the encoded path vector of the entity.
Geolocation of covert communication entity. Calculate the similarity simi from (3) to Si.Setting the threshold U, and let \( M=\underset{i=1,\dots, q}{\max}\left({sim}_i\right) \) , if M ≥ U, inputting (3) into the neural network constructed by Si to obtain its latitude and longitude; otherwise, ending the method.
Among them, hop-hot path coding method, acquisition of training sets, and geolocation of covert communication entity are the important parts of the method, which will be described in detail in the following subsections.
Hop-hot path coding method
The path from probe to the entity IP is composed of router sequence, such as <probe, router1, router2, …, routern, entity IP>. One-hot coding can be used to measure the similarity between paths by judging whether routers in the paths, but the paths are sequential, one-hot coding cannot express this sequential well, so it is not very reasonable to express the paths by one-hot encoding. In order to better measure the degree of similarity between paths, this paper proposes a path coding method: hop-hot path coding. It can make the coded path vector directly into the machine learning model as a feature or compare similarity.
The process of coding is as follows. Firstly, stable router paths are obtained from probes to all landmarks, and all router sets are obtained. Then, the one-hot coding is used to encode each stable router path to obtain the path vector. After that, the path vector is quantized by hop number. Finally, the entity's router path vector is quantized. The details are as follows:
Step 1. Building router path set. n probes are used to measure m landmarks, then, a stable router path set whose size is n × m obtained. The set is recorded as
$$ \mathbf{E}=\left\{\begin{array}{l}{p}_{1,1},{p}_{1,2},\dots, {p}_{1,n}\\ {}{p}_{2,1},{p}_{2,2},\dots, {p}_{2,n}\\ {}\dots \\ {}{p}_{m,1},{p}_{m,2},\dots, {p}_{m,n}\end{array}\right\}. $$
where pk, i is the measured router path from the ith probe to the kth landmark.
Step 2. Extracting routers. All routers in the router paths from the ith probe to m landmarks are extracted. The extracting result is
$$ {\mathbf{O}}_i=\left\{{r}_{i,1},{r}_{i,2},\dots, {r}_{i,{l}_i}\right\}. $$
where ri, j is the jth router in the measured paths from the ith probe to m landmarks, and the order is inessential. li is the number of routers appearing in the measured paths whose source is the ith probe. The feature space of path coding is consistent to all Oi, and the feature space is recorded as
$$ \mathbf{L}=\left\{{\mathbf{O}}_{\mathbf{1}},{\mathbf{O}}_{\mathbf{2}},\dots, {\mathbf{O}}_{\mathbf{n}}\right\}. $$
That is equivalent to
$$ \mathbf{L}=\left\{\left\{{r}_{1,1},{r}_{1,2},\dots, {r}_{1,{l}_1}\right\},\left\{{r}_{2,1},{r}_{2,2},\dots, {r}_{2,{l}_2}\right\},\dots, \left\{{r}_{n,1},{r}_{n,2},\dots, {r}_{n,{l}_n}\right\}\right\}. $$
where n is the number of probes.
Step 3. Building landmarks' router path vector. For landmark k, according to the router paths from each probe to landmark k, the landmark is coded in feature space L. The coding result is recorded as
$$ {\mathbf{C}}_{\mathbf{k}}=\left(\ {V}_{1,1,k},{V}_{i,j,k},\dots, {V}_{n,{l}_n,k}\right). $$
The value of Vi, j, k is donated as
$$ {V}_{i,j,k}=\left\{\begin{array}{l}\beta, \kern1.25em if\ {r}_{i,j}\ not\ in\ {p}_{i,k}\\ {}{H}_{i,j,k},\kern1.25em if\ {r}_{i,j}\ in\ {p}_{i,k}\end{array}\right.. $$
where Hi, j, k is the number of hops from the router ri, j to landmark k, and β is a control parameter whose value is greater than the length of px, y, (1 ≤ x ≤ m, 1 ≤ y ≤ n).Step 4. Building the router path vector of the covert communication entity. As the same of landmark, the coding result of entity in feature space L is recorded as
$$ {\mathbf{C}}_T=\left(\ {V}_{1,1,T},{V}_{i,j,T},\dots, {V}_{n,{l}_n,T}\right). $$
The value of Vi, j, T is donated as
$$ {V}_{i,j,T}=\left\{\begin{array}{l}\beta \kern1.5em if\ {r}_j\ not\ in\ {p}_{i,T}\\ {}{H}_{i,j,T}\kern1.5em if\ {r}_j\ in\ {p}_{i,T}\end{array}\right.. $$
where Hi, j, T is the number of hops from the router ri, j to entity T. Meanwhile, if a router is in the router path from probes to entity but not in the router paths from probes to the landmarks, this router would not be considered.
Acquisition of training sets
In the actual Internet environment, there are multiple ISPs in some countries and regions. Even if the landmarks' locations are close, there may also be large gaps in vectors of delay and path between landmarks. If all the landmarks are used as the training set to train the neural network, the mapping relationship learned by it will not be strong, and the geolocation reliability is hard to guarantee. Therefore, the training set needs to be filtered so that the delays, paths, latitude, and longitude of the landmarks in each training set are similar. The specific steps are as follows:
Input: Vectors of delay and path of landmarks, longitude, and latitude of landmarks
Output: Filtered training sets
Step 1. Using (1) to perform K means clustering on the landmarks, wherein k value is iterated from small to big, calculating the contour coefficients of the clustering, selecting the k value corresponding to the maximum contour coefficient, recording the clustering set as K = {D1, D2, …, Dk}.Step 2. Using the latitude and longitude in the landmark set to cluster all the landmarks, in terms of the number of clusters, also selecting the value corresponding to the maximum contour coefficient and recording it as h, and recording the clustering set as Q = {L1, L2, …, Lh}.Step 3. Calculating F = K ∩ Q and recording the final set of categories as F = {S1, S2, …, Sq}.At this time, the delay, path, latitude, and longitude of the landmarks in each training set are similar. The neural network is trained by using the landmarks in each training set, and the mapping between delay, path, and location will be more reliable.
Geolocation of covert communication entity
After training the neural network for each training set, when geolocating the covert communication entity, it is first necessary to judge the training set to which the entity belongs. Then, the vector of delay and path is input into the neural network trained by the training set to obtain the latitude and longitude of the entity. Specific steps are as follows:
Input: The vector of delay and path of the entity
Output: Longitude and latitude of the entity
Step 1. Calculate the cosine similarity between the center of Di and (3), and choose the Di with the highest cosine similarity between center and (3) as the Di to which the entity T belongs.
Step 2. Calculate the cosine similarity between landmarks in Di and the entity. Find the landmark whose vector of delay and path is most similar to the entities' vector. Record the training set to which the landmark belongs as Sj, and use Sj as the training set of the entity. The vector similarity between landmark and entity is recorded as M.Step 3. Setting the threshold U, and if M ≥ U, using the neural network formed by the training set Sj to geolocate the entity; otherwise, ending the method.
This paper focuses at the geolocation of the covert communication entity on the Internet with the knowledge of the entity IP, while the detection of the stegos on the Internet and the acquisition of the IP address from the stegos IP packages are beyond this paper. In this section, the rationality and effectiveness of the proposed method are verified by two experiments: verification on the geolocation effect of the method, and comparative verification. The experimental setups are shown in Table 1.
Table 1 Experimental setups
In this paper, 53,433 measurable street-level landmarks in Chinese Mainland, New York State (USA), and Hong Kong (China) have been measured for 14 days and 3000 times with nine probes located in Beijing, Chengdu, Shanghai, Wuhan, Washington, Silicon Valley, New York, Atlanta, and Seattle. A large amount of path and delay information has been obtained.
The path acquisition part of the method combines the method of merging router aliases such as Ally, Mercator [25, 26], and the anonymous route parsing method in [27]. Merge the routers in the path from each probe to the landmark and select the most frequently occurring path as the path information, then set β to 30 empirically when encoding the path.
In order to obtain more accurate delay information, during network measurement, the delay from the probes to the landmarks is repeatedly measured, and the minimum delay on the stable path is selected as the delay information. The delay information on the stable path often represents the network stability and less congestion. Therefore, the obtained delay information is closer to the true propagation delay.
When training neural networks, MLP neural networks [28] are used, with formula (1) as the input of the neural networks, and the latitude and longitude of the landmark as the output of the neural networks. A neural network is trained for each training set.
Experiment of entity geolocation with different parameters
Based on the experimental setups in Table 1, we verify the effect of the geolocation method for the location of the covert communication entity in this subsection. To verify the geolocation error, 80% of the landmarks are randomly selected from each region as the candidate set of the training set for training network, and the remaining 20% of the landmarks (a total of 10,686) are used as the covert communication entities for geolocation verification. The landmarks can be divided into 145 categories by using the landmarks clustering in the method. Table 2 shows the relationship between the size of the training set, the number of clusters and the geographical location thereof.
Table 2 Statistics of stable path ratio
Table 3 shows the geolocation effects of training sets in different training sizes and different geolocation thresholds on the corresponding entities.
Table 3 Relationship between different training set sizes, different thresholds and the quantity of the entities that can be geolocated and geolocation error
Figure 4 shows the geolocation error cumulative distribution of the entities that can be geolocated under different training set sizes and different threshold conditions.
The CDF (cumulative distribution function) of geolocation error under different training set sizes and thresholds. The red dashed line, blue dot line, and the black solid line indicate the cumulative error distribution of all neural networks formed by the training sets with the landmarks greater than 100,300 and 500 in each training set for the corresponding entities geolocation, respectively. a Threshold is 0.9. b Threshold is 0.8. c Threshold is 0.7
Table 3 and Fig. 4 show that as the training scale N increases, the number of samples in a single training set is increasing, but the total number of landmarks in the landmark set is decreasing, the number of locatable entities is also decreasing, and the geolocation error is decreasing. The reason is that the network trained by the training set that does not satisfy a certain sample number is not universal, which is statistically reasonable. It can also be seen that different geolocation thresholds have different degrees of impact on the number of localizable entities and geolocation error. From the experimental results, this method has certain advantages in street-level geolocation.
Comparative verification
In this subsection, we compare the geolocation effect of the proposed method with the typical geolocation methods under the situations of same entities and landmarks. Table 4 shows the statistical results of proposed method, SLG method, and TNN method when the geolocation threshold is 0.9, and the geolocation error is 10 km, 20 km, and 40 km. Fig. 5 shows the geolocation cumulative distribution of the proposed method in this paper, the SLG method and the TNN method.
Table 4 The proportion of geolocation error when the threshold is 0.9
Comparison on IP geolocation error under the same conditions. The black line, red dashed line, and blue dot line indicate the geolocation error cumulative distribution of the proposed method, SLG method, and TNN method, respectively. a The training set size > 100, and the threshold is 0.9. b The training set size > 100, and the threshold is 0.8. c The training set size > 100, and the threshold is 0.7. d The training set size > 300, and the threshold is 0.9. e The training set size > 300, and the threshold is 0.8. f The training set size > 300, and the threshold is 0.7. g The training set size > 500, and the threshold is 0.9. h The training set size > 500, and the threshold is 0.8. i The training set size > 500, and the threshold is 0.7
As can be seen from Table 4 and Fig. 5, the reliability of the street-level positioning method of this method at 10 km, 20 km, and 40 km is better than the existing typical street-level methods.
Traditional steganalysis mainly detect whether the investigated object carries secret messages, while a few works are reported for the payload location, the embedding key restore, the secret message extraction and the steganographer detection. The geolocation of the covert communication entity reveals the physical location of the steganography on the Internet based on the IP address in the IP packages of the stegos. This paper presents a method for the geolocation of the covert communication entity based on hop-hot path coding. We do a preliminary work on the geolocation of the covert communication entity and there are still some geolocation errors as it is difficult to locate the covert communication entity within the last kilometer. Nevertheless, our method is very helpful to geolocate the covert communication entity in a certain area, and the geolocation accuracy is improved compared with the existing geolocation methods. In addition, the hop-hot path coding method in this paper is just an attempt. Whether there is a better coding method that is worth further exploring.
The datasets used and analyzed during the current study are available from the corresponding author on reasonable request.
U :
Geolocation threshold
QCG:
Quantity of the entities that can be geolocated
QCNG:
Quantity of the entities that cannot be geolocated
MGE:
Median geolocation error of the entities that can be geolocated
PGE < X:
Proportion of the entities within geolocation error being X
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This work was supported by the National Natural Science Foundation of China (no.U1636219, 61602508, 61772549, U1736214, U1804263), the National Key R&D Program of China (no. 2016YFB0801303, 2016QY01W0105), and the Science and Technology Innovation Talent Project of Henan Province (no. 184200510018).
PLA Strategic Support Force Information Engineering University, Zhengzhou, 450001, China
Fan Zhang, Fenlin Liu & Xiangyang Luo
State Key Laboratory of Mathematical Engineering and Advanced Computing, Zhengzhou, 450001, China
Fan Zhang
Fenlin Liu
Xiangyang Luo
FZ and FL conceived the idea. XL designed the experiments. FZ performed the experiments. FZ, FL wrote the paper. All authors read and approved the final paper.
Correspondence to Fenlin Liu.
Zhang, F., Liu, F. & Luo, X. Geolocation of covert communication entity on the Internet for post-steganalysis. J Image Video Proc. 2020, 15 (2020). https://doi.org/10.1186/s13640-020-00504-8
Covert communication
Entity geolocation
Post-steganalysis
New Advances on Intelligent Multimedia Hiding and Forensics | CommonCrawl |
\begin{definition}[Definition:Uncountable Ordinal]
Let $\alpha$ be an ordinal.
Then $\alpha$ is said to be '''uncountable''' {{iff}} it is an uncountable set.
\end{definition} | ProofWiki |
\begin{document}
\setcounter{page}{1} \title{Isoperimetric Bounds for Lower Order Eigenvalues} \renewcommand{\thefootnote}{} \footnotetext{2000 {\it Mathematics Subject Classification }: 35P15; 53C40; 58C40, 53C42.
\hspace*{2ex}Key words and phrases: Isoperimetric inequalities, eigenvalues, Laplacian, biharmonic Steklov problems, Wentzell-Laplace operator.} \author{Fuquan Fang and Changyu Xia} \date{} \maketitle ~~~\\[-15mm]
\begin{abstract} New isoperimetric inequalities for lower order eigenvalues of the Laplacian on closed hypersurfaces, of the biharmonic Steklov problems and of the Wentzell-Laplace on bounded domains in a Euclidean space are proven. Some open questions for further study are also proposed.
\end{abstract}
\markright{
Fuquan Fang and Changyu Xia
}
\section{Introduction and the main results} \renewcommand{\arabic{section}}{\arabic{section}} \renewcommand{\thesection.\arabic{equation}}{\arabic{section}.\arabic{equation}} \setcounter{equation}{0} \label{intro}
Let $(M, g)$ be a closed Riemannian manifold of dimension $\geq 2$. The spectrum of the Laplace operator on $M$ provides a sequence of global Riemannian invariants \be\nonumber 0=\lambda_0(M)<\lambda_1(M)\leq \lambda_2(M)\leq\cdots\nearrow\infty. \end{eqnarray} We adopt the convention that each eigenvalue is repeated according to its multiplicity. An important issue in spectral geometry is to obtain good estimates for these and other eigenvalues in terms of the geometric data of the manifold $M$ such as the volume, the diameter, the curvature, the isoperimetric constants, etc. See \cite{be},\cite{ber},\cite{c2},\cite{ds},\cite{sy} for references.
On the other hand, after the seminal works of Bleecker-Weiner \cite{bw} and Reilly \cite{r1}, the following approach is developed: the manifold $(M, g)$ is immersed isometrically into another Riemannian manifold. One then gets good estimates for $\lambda_k(M)$, mostly for $\lambda_1(M)$, in termos of the extrinsic geometric quantities of $M$. See for example \cite{bw}, \cite{dmwx}, \cite{ei}, \cite{gr}, \cite{h}, \cite{wx1}, \cite{x1}. Especially relevant for us is the quoted work of
Reilly \cite{r1}, where he obtained the following remarkable isoperimetric inequality for the first positive eigenvalue $\lambda_1(M)$ in the case that $M$ is embedded as a hypersurface bounding a domain $\Omega$ in $\mathbb{R}^{n}$: \be\label{0.1}
\lambda_1(M)\leq\frac{n-1}{n^2}\cdot\frac{|M|^2}{|\Omega|^2}.
\end{eqnarray} Here $|M|$ and $|\Omega|$ denote the Riemannian $(n-1)$-volume of $M$ and the Riemannian $n$-volume of $\Omega$, respectively. Moreover, equality holds in (\ref{0.1}) if and only if $M$ is a round sphere. Our first result improves (\ref{0.1}) to the sum of the first $n$ non-zero eigenvalues of the Laplace operator on $M$. \begin{theorem} \label{th1} Let $M$ be a closed embedded hypersurface bounding a domain $\Omega$ in $\mathbb{R}^{n}$. Then the first $n$ non-zero eigenvalues of the Laplacian on $M$ satisfy \be\label{0.2}
\sum_{i=1}^{n}\lambda_i\leq \frac{n-1}{n}\cdot\frac{|M|^2}{|\Omega|^2} \end{eqnarray} and \be\label{0.3}
\sum_{i=1}^{n}\lambda_i\leq \frac{(n-1)\sqrt{|M|}}{|\Omega|}\left(\int_M H^2\right)^{1/2}, \end{eqnarray} where $H$ stands for the mean curvature of $M$. Moreover, equality holds in either of (\ref{0.2}) and (\ref{0.3}) if and only if $M$ is a sphere. \end{theorem}
In the second part of this paper we study eigenvalues of fourth order Steklov problems. Let $\Omega$ be an $n$-dimensional compact Riemannian manifold with boundary and $\Delta$ and $\overline{\Delta}$ be the Laplace operators on $\Omega$ and $\partial\Omega$, respectively. Consider the eigenvalue problem \be\label{0.4}
\left\{\begin{array}{l}
\Delta^2 u= 0 \ \ {\rm in \ \ } \Omega, \\ \partial_{\nu}u = \partial_{\nu}(\Delta u)+\xi u =0 \ \ {\rm on \ \ } \partial\Omega, \end{array}\right. \end{eqnarray} where $\partial_{\nu}$ denotes the outward unit normal derivative. This problem was first discussed by J. R. Kuttler and V. G. Sigillito \cite{ks} in the case where $\Omega$ is a bounded domain in $\mathbb{R}^n$. The eigenvalue problem (\ref{0.4}) is important in biharmonic analysis and elastic mechanics. In the two dimensional case, it describes the deformation $u$ of the linear elastic supported plate $\Omega$ under the action of the transversal exterior force $f(x) = 0, \ x\in\Omega $ with Neumann boundary condition
$\partial_{\nu}u|_{\partial \Omega}=0$ (see, \cite{tg},\cite{v},\cite{xw2}). In addition, the first nonzero eigenvalue $\xi_1$ arises as an optimal constant in an a priori inequality (see \cite{ks}). The eigenvalues of the problem (\ref{0.4}) form a discrete and increasing sequence (counted with multiplicity): \be 0=\xi_0<\xi_1\leq\xi_2\leq\cdots\nearrow +\infty. \end{eqnarray} Let $\mathcal{D}_k$ be the space of harmonic homogeneous polynomials in $\mathbb{R}^n$ of degree $k$ and denote by $\mu_k$ the dimension of $\mathcal{D}_k$, $k=0, 1,\cdots,$. For the $n$-dimensional Euclidean ball with radius $R$, the eigenvalues of (\ref{0.4}) are $\xi_k=k^2(n+2k)/R^3, k=0, 1, 2, \cdots, $ and the multiplicity of $\xi_k$ is $\mu_k$ (see \cite{xw2}, Theorem 1.5 ). When $\Omega$ has nonnegative Ricci curvature with strictly convex boundary, a lower bound for $\xi_1(\Omega)$ has been given in \cite{xw1}. On the other hand, an isoperimetric upper bound for $\xi_1(\Omega)$ has been proven for the case where $\Omega$ is a bounded domain in $\mathbb{R}^n$ (see \cite{xw2}, Theorem 1.6 ). We have an isoperimetric inequality for the sum of the reciprocals of the first $n$ nonzero eigenvalues of the problem (\ref{0.4}) on bounded domains in $\mathbb{R}^n$. \begin{theorem} \label{th3} Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^n$. Then the first $n$ nonzero eigenvalues of the problem (\ref{0.4}) satisfy \be\label{0.5}
\sum_{i=1}^n\frac 1{\xi_i}\geq \frac{n^2|\Omega|\left(\frac{|\Omega|}{\omega_n}\right)^{2/n}}{(n+2)|\partial\Omega|}, \end{eqnarray} with equality holding if and only if $\Omega$ is a ball, where $\omega_n$ denotes the volume of the unit ball in $\mathbb{R}^n$. \end{theorem} Now we come to another Steklov problem for the bi-harmonic operator. Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ and $\tau$ a positive constant. Denote by $\nabla^2$ and $\nabla$ the Hessian on $\mathbb{R}^n$ and the gradient operator on $\Omega$, respectively. Consider the following Steklov problem of fourth
order \begin{eqnarray}\label{0.6} \left\{\begin{array}{ccc} \Delta^2 u-\tau \Delta u =0,&&~\mbox{in} ~~ \Omega, \\[2mm] \frac{\partial^2u}{\partial\nu^2}=0, &&~~\mbox{on}~~\partial \Omega,\\[2mm] \tau\frac{\partial u}{\partial \nu}-\mathrm{div}_{\partial M}\(\nabla^2 u(\nu)\)-\frac{\partial\Delta u}{\partial\nu}=\lambda u, &&~~\mbox{on}~~\partial \Omega. \end{array}\right. \end{eqnarray} This problem has a discrete spectrum which can be listed as $$0=\lambda_{0,\tau}<\lambda_{1,\tau}\leq\lambda_{2,\tau}\leq\cdots\leq\lambda_{k,\tau}\nearrow +\infty.$$ The eigenvalue $0$ is simple and the corresponding eigenfunctions are constants. Let $u_0, u_1,..., u_{k},\cdots, $ be the eigenfunctions of problem (\ref{0.6}) corresponding to the eigenvalues $0=\lambda_{0,\tau},\ \lambda_{1,\tau},\cdots, \lambda_{k,\tau}, \cdots, $. For each $k=1,\cdots,$ we have the following variational characterization
\begin{small}
\begin{eqnarray}\label{0.7}
\lambda_{k,\tau}=\mathrm{min}\left\{\frac{\int_{\Omega}\(|\nabla^2u|^2+\tau|\nabla u|^2\)}{\int_{\partial\Omega} u^2}\Bigg| u\in H^2(\Omega), u\neq0, \int_{\partial\Omega} u u_j=0, j=0,\cdots,k-1\right\}. \end{eqnarray} \end{small}
The eigenvalues and eigenfunctions on the ball in $\mathbb{R}^n$ have been determined by Buoso-Provenzano in \cite{bp}. In particular, if $\mathbf{B}_R^n$ is the ball of radius $R$ centered at the origin in $\mathbb{R}^n$, then \be \lambda_{1,\tau}(\mathbf{B}_R^n)=\lambda_{2,\tau}(\mathbf{B}_R^n)=\cdots=\lambda_{n,\tau}(\mathbf{B}_R^n)=\frac{\tau}{R} \end{eqnarray}
and the corresponding eigenspace is generated by $\{x_1,\cdots, x_n\}$. Buoso and Provenzano \cite{bp} also proved the following
isoperimetric inequality for the sums of the reciprocals of the first $n$ non-zero eigenvalues: \be\label{bp}
\sum_{i=1}^n\frac 1{\lambda_{1,\tau}(\Omega)}\geq \frac n{\tau}\left(\frac{|\Omega|}{\omega_n}\right)^{1/n} \end{eqnarray} with equality holding if and only if $\Omega$ is a ball.
Further study for the eigenvalues of the problem (\ref{0.6}) has been made in \cite{bcp}, \cite{dmwx}, \cite{xw2}, etc. Our next result is an isoperimetric inequality for the sum of the first $n$ non-zero eigenvalues of the problem (\ref{0.6}).
\begin{theorem}\label{th4} Let $\Omega$ be a bounded domain with smooth boundary $\partial \Omega$ in $\mathbb{R}^n$. Denoting by $\lambda_{i,\tau}$ the $i$-th eigenvalue of the (\ref{0.6}), we have \begin{eqnarray}\label{th4.1} \sum_{j=1}^{n}\lambda_{j,\tau}\leq
\frac{\tau|\partial\Omega|}{|\Omega|}. \end{eqnarray} Equality holds in (\ref{th4.1}) if and only if $\Omega$ is a ball. \end{theorem}
The final part of the present paper concerns the eigenvalue problem with Wentzell boundary conditions: \be\label{int1}\left\{\begin{array}{l} \Delta u =0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\rm in \ }\ \Omega,\\ -\beta\overline{\Delta} u+\partial_{\nu} u= \lambda u\ \ \ \quad~ {\rm on \ } \partial \Omega, \end{array}\right. \end{eqnarray} where $\beta$ is a nonnegative constant, $\Omega$ is a compact Riemannian manifold of dimension $n\geq 2$ with non-empty boundary, $\Delta$ and $\overline{\Delta}$ denote the the Laplacian on $\Omega$ and $\partial\Omega$, respectively. When $\beta=0$, (\ref{int1}) becomes the Steklov problem: \be\label{int2}\left\{\begin{array}{l} \Delta u =0 \ \ \ \ \ \ \ \ \ {\rm in \ }\ \Omega,\\ \partial_{\nu} u= p u\ \ \ \quad~ {\rm on \ } \partial \Omega, \end{array}\right. \end{eqnarray} which has been studied extensively ( see \cite{b}, \cite{bpr},\cite{ceg},\cite{e1}-\cite{fs3},\cite{hps},\cite{ks},\cite{st}, \cite{wx2},\cite{xw2},\cite{xi} ). The spectrum of the problem (\ref{int1}) consists in an increasing sequence
\be \nonumber\lambda_{0, \beta}=0<\lambda_{1, \beta}\leq\lambda_{2, \beta}\leq\cdots\nearrow +\infty, \end{eqnarray} with corresponding real orthonormal (in $L^2(\partial \Omega)$ sense) eigenfunctions $u_0, u_1, u_2,\cdots.$ Consider the Hilbert space \be H(\Omega)=\{ u\in H^1(\Omega), {\rm Tr}_{\partial \Omega}(u)\in H^1(\partial \Omega) \}, \end{eqnarray} where ${\rm Tr}_{\partial \Omega}$ is the trace operator. We define on $H(\Omega)$ the two bilinear forms \be A_{\beta}(u, v)=\int_{\Omega}\nabla u\cdot\nabla v +\beta\int_{\partial \Omega} \overline{\nabla} u\cdot\overline{\nabla} v, \ B(u, v)=\int_{\partial \Omega}uv, \end{eqnarray} where, $\nabla $ and $\overline{\nabla}$ are the gradient operators on $\Omega$ and $\partial \Omega$, respectively. Since we assume that $\beta$ is nonnegative, the two bilinear forms are positive and the variational characterization for the $k$-th eigenvalue is \be \label{vc}\lambda_{k, \beta}=\min\left\{\frac{A_{\beta}(u, u)}{B(u, u)}, u\in H(\Omega), u\neq 0,\ \int_{\partial \Omega} u u_i=0, i=0,\cdots,k-1\right\}. \end{eqnarray} When $k=1$, the minimum is taken over the functions orthogonal to the eigenfunctions associated to $\lambda_{0, \beta}= 0,$ i.e., constant functions.
If $\Omega=\mathbf{B}_R^n$, then \cite{dkl} $$\lambda_{1, \beta}=\lambda_{2, \beta}=\cdots \lambda_{n, \beta}=\frac{(n-1)\beta + R}{R^2}$$ and the corresponding eigenspace is generated by $\{x_i, i=1,\cdots,n\}$. For the Steklov problem (\ref{int2}), Brock \cite{b} showed that if $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^n$, then the first $n$ nonzero eigenvalues of $\Omega$ satisfy \be\label{th0.0}
\sum_{i=1}^n\frac 1{p_i(\Omega)} \geq n\left(\frac{|\Omega|}{\omega_{n}}\right)^{\frac 1n}, \end{eqnarray} with equality holding if and only if $\Omega$ is a ball. Brock's theorem has been generalized to the eigenvalues of the problem (\ref{int1}) in \cite{dmwx}. We prove
\begin{theorem} \label{th7} Let $\beta\geq 0$ and $\Omega$ be a bounded domain with smooth boundary $\partial \Omega$ in
$\mathbb{R}^n$. Denote by $\lambda_{1, \beta}\leq\lambda_{2, \beta}\leq\cdots\leq \lambda_{n, \beta}$
the first $n$ non-zero
eigenvalues of the following problem with the Wentzell boundary condition. \be\label{th7.1}\left\{\begin{array}{l} \Delta u =0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\rm in \ }\ \Omega,\\ -\beta\overline{\Delta} u+\partial_{\nu} u= \lambda u\ \ \ \ \quad ~~ {\rm on \ } \partial \Omega, \end{array}\right. \end{eqnarray} Then we have \be\label{th7.2}
\sum_{i=1}^n \lambda_{i,\beta}\leq \frac{|\partial\Omega|}{|\Omega|}+\frac{(n-1)\beta}n\cdot\frac{|\partial\Omega|^2}{|\Omega|^2}.
\end{eqnarray} Furthermore, equality holds in (\ref{th7.2}) if and only if $\Omega$ is a ball.
\end{theorem} Taking $\beta=0$ in (\ref{th7.2}), we have a new isoperimetric inequality for the first $n$ nonzero Steklov eigenvalues of a bounded domain $\Omega\subset\mathbb{R}^n$: \be\label{th5.3}
\sum_{i=1}^n p_i(\Omega)\leq \frac{|\partial\Omega|}{|\Omega|} \end{eqnarray} with quality holding if and only if $\Omega$ is a ball.
It has been conjectured by Henrot \cite{he1} that the first $n$ nonzero Steklov eigenvalues of a bounded domain $\Omega\subset\mathbb{R}^n$ satisfy \be\label{th5.3}
\overset{n}{\underset{i=1}{\prod}} p_i(\Omega)\leq\frac{\omega_n}{|\Omega|} \end{eqnarray} which is stronger than Brock's inequality (\ref{th0.0}). If $n=2$, or $n\geq 3$ and $\Omega$ is convex, then (\ref{th5.3}) is true ( see \cite{hps}, \cite{he2} ). This result can be extended to eigenvalues of the problem (\ref{int1}). Namely, we have \begin{theorem} \label{th8} Let the notation be as in Theorem \ref{th7} and when $n\geq 3$, assume further that $\Omega$ is convex. Then \be\label{th8.1}
\overset{n}{\underset{i=1}{\prod}} \lambda_{i,\beta}\leq\left(1+\frac{(n-1)\beta|\partial\Omega|}{n|\Omega|}\right)^n\cdot\frac{\omega_n}{|\Omega|} \end{eqnarray}
with quality holding if and only if $\Omega$ is a ball. \end{theorem}
\section{A Proof of Theorem \ref{th1}} \setcounter{equation}{0} In this section, we give a \vskip0.3cm {\it Proof of Theorem \ref{th1}}.
Let $\Delta$ and $\overline{\Delta}$ be the Laplace operators on $\mathbb{R}^n$ and $M$, respectively, and let $\{u_i\}_{i=0}^{+\infty}$ be an orthonormal system of eigenfunctions corresponding to the eigenvalues \be\label{pth3.1} 0=\lambda_0<\lambda_1\leq \lambda_2\leq\cdots\rightarrow\infty \end{eqnarray} of the Laplacian of $M$, that is, \be\label{th3.2} \overline{\Delta} u_i = -\lambda_i u_i, \quad \int_M u_iu_j
=\delta_{ij}. \end{eqnarray} We have $u_0=1/\sqrt{|M|}$ and for each $i=1,\cdots,$ the Rayleigh-Ritz characterization for $\lambda_i$ is given by
\be \label{pth1.1} \lambda_i=\underset{u\neq 0, \int_M uu_j=0, j=0,\cdots i-1}{\min}\frac{\int_M |\overline{\nabla} u|^2}{\int_M u^2},\end{eqnarray} being $\overline{\nabla}$ the gradient operator on $M$.
In order to obtain good upper bound for $\lambda_i$, we need to choose nice trial functions $\phi_i$ for each of the eigenfunctions $u_i$ and insure that these are orthogonal to the preceding eigenfunctions $u_0,\cdots, u_{i-1}$. We note that the coordinate functions are eigenfunctions corresponding to the first eigenvalue of the hypersphere in $\mathbb{R}^n$. For the $n$ trial functions $\phi_1, \phi_2, \cdots, \phi_n,$ we simply choose the $n$ coordinate functions: \be \phi_i=x_i, \ \ {\rm for}\ \ i=1,\cdots, n, \end{eqnarray} but before we can use these we need to make adjustments so that $\phi_i \perp{\rm span}\{u_0,\cdots, u_{i-1}\}$ in $L^2(\partial\Omega)$. By translating the origin appropriately we can assume that \be \int_{M} x_i=0, \ i=1,\cdots,n, \end{eqnarray} that is, $x_i\perp u_0$.
Nextly we show that a rotation of the axes can be made so that \be\label{pth1.} \int_{M}\phi_j u_i=\int_{M} x_j u_i=0, \end{eqnarray} for $j=2,3,\cdots, n$ and $i=1,\cdots,j-1$. In fact,
let us define an $n \times n$ matrix $P=\(p_{ji}\),$ where $p_{ji}=\int_{M} x_j u_i$, for $i,j=1,2,\cdots,n.$ Using the orthogonalization of Gram and Schmidt (QR-factorization theorem), one can find an upper triangle matrix $T=(T_{ji})$ and an orthogonal matrix $U=(a_{ji})$ such that $T=UP$, that is, \begin{eqnarray*} T_{ji}=\sum_{k=1}^n a_{jk}p_{ki}=\int_M \sum_{k=1}^n a_{jk}x_k u_i =0,\ \ 1\leq i<j\leq n. \end{eqnarray*} Letting $y_j=\sum_{k=1}^n a_{jk}x_k$, we have \begin{eqnarray}\label{c2} \int_M y_j u_i =\int_M \sum_{k=1}^n a_{jk}x_k u_i =0,\ \ 1\leq i<j\leq n. \end{eqnarray} Since $U$ is an orthogonal matrix, $y_1, y_2, \cdots, y_n$ are also coordinate functions on $\mathbb{R}^n$. Thus, denoting these coordinate functions still by $x_1, x_2,\cdots, x_n$, one arrives at the condition (\ref{pth1.}). It follows from (\ref{pth1.1}) that
\be\label{pt1.4} \lambda_i\int_M x_i^2\leq \int_M |\overline{\nabla}
x_i|^2, \ i=1,\cdots, n, \end{eqnarray} with equality holding if and only if \be\label{pth1.5} \overline{\Delta} x_i= -\lambda_i x_i. \end{eqnarray}
Integrating the equality \be\label{pth1.2} \frac 12\Delta x_i^2=1 \end{eqnarray} on $\Omega$ and using the divergence theorem, one gets \be\label{pth1.3}
|\Omega|=\int_M x_i\partial_{\nu}x_i, \ i=1,\cdots, n, \end{eqnarray} where $\nu$ denotes the outward unit normal of $\partial\Omega=M$. Taking the square of (\ref{pth1.3}) and using the H\"older inequality, we infer \be\label{pth1.6}
|\Omega|^2\leq\left(\int_M x_i^2\right)\left(\int_M(\partial_{\nu}x_i)^2\right), \ i=1,\cdots, n. \end{eqnarray} Multiplying (\ref{pt1.4}) by $\int_M(\partial_{\nu}x_i)^2$ and using (\ref{pth1.6}), we have \be\label{pth1.7}
|\Omega|^2\lambda_i\leq \left(\int_M |\overline{\nabla}x_i|^2\right)\left(\int_M(\partial_{\nu}x_i)^2\right) , \ i=1,\cdots, n. \end{eqnarray} Observing that on $M$
\be 1=|\nabla x_i|^2=|\overline{\nabla}x_i|^2+(\partial_{\nu}x_i)^2, \end{eqnarray} one deduces from (\ref{pth1.7}) that \be\label{pth1.8}
|\Omega|^2\lambda_i\leq \left(|M|-\int_M (\partial_{\nu}x_i)^2\right)\left(\int_M(\partial_{\nu}x_i)^2\right), \ i=1,\cdots, n. \end{eqnarray} Summing over $i$ and using Cauchy-Schwarz inequality, we get \be\label{pth1.9}
|\Omega|^2\sum_{i=1}^{n}\lambda_i&\leq& |M|\sum_{i=1}^n\left(\int_M(\partial_{\nu}x_i)^2\right)-\sum_{i=1}^n\left(\int_M (\partial_{\nu}x_i)^2\right)^2\\ \nonumber &\leq& |M|^2-\frac 1n\left(\sum_{i=1}^n\int_M (\partial_{\nu}x_i)^2\right)^2\\ \nonumber
&=& \frac{(n-1)|M|^2}n. \end{eqnarray} This proves (\ref{0.2}).
To prove (\ref{0.3}), we use divergence theorem and H\"older inequality to get \be\label{pth1.10}
\int_M |\overline{\nabla}x_i|^2=-\int_M x_i\overline{\Delta}x_i\leq \left(\int_M x_i^2\right)^{1/2}\left(\int_M (\overline{\Delta}x_i)^2\right)^{1/2}, \end{eqnarray} which, combining with (\ref{pt1.4}), gives \be\label{pth1.11} \lambda_i\left(\int_M x_i^2\right)^{1/2}\leq\left(\int_M (\overline{\Delta}x_i)^2\right)^{1/2}. \end{eqnarray} Multiplying (\ref{pth1.11}) by $\left(\int_M (\partial_{\nu}x_i)^2\right)^{1/2}$ and using (\ref{pth1.6}), we have \be\label{pth1.12}
\lambda_i|\Omega|\leq \left(\int_M (\overline{\Delta}x_i)^2\right)^{1/2}\left(\int_M (\partial_{\nu}x_i)^2\right)^{1/2}, \ i=1,\cdots,n. \end{eqnarray} Summing over $i$ and using Cauchy-Schwarz inequaty, we infer \be\label{pth1.13}
|\Omega|\sum_{i=1}^n\lambda_i&\leq& \left(\sum_{i=1}^n\int_M (\overline{\Delta}x_i)^2\right)^{1/2}\left(\sum_{i=1}^n\int_M (\partial_{\nu}x_i)^2\right)^{1/2}\\ \nonumber &=&
(n-1)\left(\int_M H^2\right)^{1/2}|M|^{1/2}, \end{eqnarray} where, in the last equality, we have used the fact that \be\label{pth1.14} \overline{\Delta} x \equiv (\overline{\Delta} x_1,\cdots, \overline{\Delta} x_n) =(n-1){\mathbf H}, \end{eqnarray} being ${\mathbf H}$ the mean curvature vector of $M$ in ${\mathbb{R}}^n$. Hence, (\ref{0.3}) holds.
If the equality holds in (\ref{0.2}), then the inequalities (\ref{pt1.4}), (\ref{pth1.6}), (\ref{pth1.8}) and (\ref{pth1.9}) must take equality sign. It then follows that (\ref{pth1.5}) holds, \be \int_M (\partial_{\nu}x_1)^2=\int_M (\partial_{\nu}x_2)^2=\cdots=\int_M (\partial_{\nu}x_n)^2, \end{eqnarray} and so \be \lambda_1=\lambda_2=\cdots=\lambda_n. \end{eqnarray} Thus, the position vector $x=(x_1,\cdots, x_n)$ when restricted on $\partial\Omega$ satisfies \be\label{p1} \Delta x = -\lambda_1 (x_1,\cdots, x_n).\end{eqnarray} Combining (\ref{p1}) and (\ref{pth1.14}), we have \be\label{p3} x=-\frac{(n-1)}{\lambda_1}{\mathbf H}, \ \
\ \ {\rm on\ \ \ }M. \end{eqnarray} Consider the function $h = |x|^2 : M\rightarrow {\mathbb{R}}$. It is easy to see from (\ref{p3}) that \be\nonumber Z h =2\langle Z, x\rangle = 0, \ \ \ \forall Z\in {\mathfrak{X}}(M). \end{eqnarray} Thus $g$ is a constant function and so $M$ is a hypersphere. If the equality holds in (\ref{0.3}), one can use similar arguments to deduce that $M$ is a hypersphere in $\mathbb{R}^n$.
$\Box$ \vskip0.2cm {\bf Remark 2.1} Noting that the Reilly inequality (\ref{0.1}) has been strengthened to \cite{wx1} \be \label{r1}
\lambda_1\leq \frac{(n-1)|M|}{n|\Omega|}\left(\frac{\omega_n}{|\Omega|}\right)^{1/n} \end{eqnarray} with equality holding if and only $M$ is a hypersphere, we believe that a stronger form of (\ref{0.2}) is valid. \begin{conjecture} If the conditions are as in Theorem \ref{th1}, then \be\label{c1}
\sum_{i=1}^n \lambda_i\leq \frac{(n-1)|M|}{|\Omega|}\left(\frac{\omega_n}{|\Omega|}\right)^{1/n}. \end{eqnarray} Moreover, the equality holds in (\ref{c1}) if and only $M$ is a round sphere. \end{conjecture} \vskip0.2cm {\bf Remark 2.2} It is easy to see from (\ref{pth1.8}) that \be \label{r2.2}
\lambda_n\leq\frac{|\partial\Omega|^2}{4|\Omega|^2} \end{eqnarray} which is also new. It would be interesting to know the best possible upper bound for $\lambda_n$.
\section{Proofs of Theorems \ref{th3} and \ref{th4}} \setcounter{equation}{0} In this section, we shall prove Theorems \ref{th3} and \ref{th4}. Before doing this, let us recall some known facts. Let $\{\phi_i\}_{i=0}^{\infty}$ be orthonormal eigenfunctions corresponding to the eigenvalues $\{\xi_i\}_{i=0}^{\infty}$ of the problem (\ref{0.4}). That is, \be\label{.}
\left\{\begin{array}{l}
\Delta^2 \phi_i= 0 \ \ {\rm in \ \ } \Omega, \\ \partial_{\nu}\phi_i = \partial_{\nu}(\Delta\phi_i)+\xi_i\phi_i =0 \ \ {\rm on \ \ } \partial\Omega \\ \int_{\partial\Omega}\phi_i\phi_j=\delta_{ij}. \end{array}\right. \end{eqnarray} For each $k=1,\cdots,$ the variational characterization for $\xi_k$ is given by \be\label{t3.1}
\xi_k=\underset{\underset{\int_{\partial\Omega}\phi\phi_i=0, i=0, 1,\cdots, k-1}{\phi\in H^2(\Omega), \partial_{\nu}\phi|_{\partial\Omega}=0, \phi|_{\partial\Omega}\neq 0}}{\inf}\frac{\int_{\Omega}(\Delta\phi)^2}{\int_{\partial\Omega}\phi^2}.
\end{eqnarray} Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ and $\Omega^*$ the ball centered at the origin in $\mathbb{R}^n$ such that $|\Omega^*|=|\Omega|$. The moments of inertia of $\Omega$ with respect to the hyperplanes $x_k=0$, are defined as \be\label{3.1} J_k(\Omega)=\int_{\Omega}x_k^2 \ {\rm for \ all\ } k\in\{1,\cdots,n\}. \end{eqnarray} By summation over $k$, we obtain the polar moment of inertia of $\Omega$ with respect to the origin denoted by \be\label{3.2} J_0(\Omega)=\sum_{k=1}^n\int_{\Omega}x_k^2. \end{eqnarray} Note that $J_0(\Omega)$ depends on the position of the origin.
In fact, $J_0(\Omega)$ is smallest when the origin coincides with the center of mass of $\Omega$, i.e. when we have \be \int_{\Omega} x_k=0, \ k=1,\cdots,n. \end{eqnarray} We need the following well known isoperimetric property \cite{bf}, \cite{hps}: \begin{theorem}\label{t03.1} Among all domains $\Omega$ of prescribed $n$-volume, the ball $\Omega^*$ centered at the origin has the smallest polar moment of inertia, that is, \be J_0(\Omega)\geq J_0(\Omega^*), \ \ \Omega\in \mathcal{O},
\end{eqnarray} for all bounded domain $\Omega$ of prescribed $n$-volume $|\Omega|$, with equality if and only if $\Omega$ coincides with $\Omega^*$. \end{theorem} By multiplication over $k$ in (\ref{3.1}), we obtain a quantity denoted by $J(\Omega)$, \be J(\Omega)=\overset{n}{\underset{k=1}{\prod}}J_k(\Omega)
\end{eqnarray}
which satisfies the
following isoperimetric inequality \cite{bl}, \cite{he2}: \be\label{4.2}
J(\Omega)\geq J(\Omega^*)=\frac{|\Omega|^{n+2}}{(n+2)^n\omega_n^2} \end{eqnarray} with equality if and only if $\Omega$ is an ellipsoid symmetric with respect to the hyperplanes $x_k=0, k=1,\cdots, n.$ \vskip0.3cm {\it Proof of Theorem \ref{th3}.} By a translation of the origin in $\mathbb{R}^n$, we can assume that \be\label{t3.1} \int_{\Omega} x_i=0, \ i=1,\cdots,n. \end{eqnarray} For each $i\in \{1,\cdots, n\}$, let $g_i$ be the solution of the problem \be\label{pth3.3}
\left\{\begin{array}{l}
\Delta g_i= x_i\ \ \ \ \ \mbox{in}\ \ \ \Omega, \\
\partial_{\nu}g_i|_{\partial \Omega}=0,\\
\int_{\partial \Omega} g_i=0. \end{array}\right. \end{eqnarray} We {\it claim} that if the coordinate functions $x_1,\cdots,x_n$ are chosen properly, then \be g_i\bot{\rm span}\{\phi_0,\cdots, \phi_{i-1}\}, \ i=1,\cdots,n. \end{eqnarray} To see this, let us fix a set of coordinate functions $x_1,\cdots, x_n$ and the solutions $g_1,\cdots,g_n$ as above. Consider the $n\times n$ matrix $H=(h_{ji})$ with $h_{ji}=\int_{M} g_j\phi_i$, for $i,j=1,2,\cdots,n.$ One can find an upper triangle matrix $S=(s_{ji})$ and an orthogonal matrix $T=(t_{ji})$ such that $S=TH$, that is, \be\label{pth3.4} s_{ji}=\sum_{k=1}^n t_{jk}h_{ki}=\int_M \sum_{k=1}^n t_{jk}g_k \phi_i =0,\ \ 1\leq i<j\leq n. \end{eqnarray} Letting $y_j=\sum_{k=1}^n t_{jk}x_k$, $\tilde{g}_j=\sum_{k=1}^n t_{jk}g_k$, we have from (\ref{pth3.3}) and (\ref{pth3.4}) that \be\label{pth3.5} \left\{\begin{array}{l}
\Delta\tilde{g}_i= y_i\ \ \ \ \ \mbox{in}\ \ \ \Omega, \\
\partial_{\nu}\tilde{g}_i|_{\partial \Omega}=0,\\
\int_{\partial \Omega} \tilde{g}_i=0 \end{array}\right. \end{eqnarray} and \be \tilde{g}_i\bot{\rm span}\{\phi_0,\cdots, \phi_{i-1}\}, \ i=1,\cdots,n. \end{eqnarray} Since $T=(t_{ji})$ is an orthogonal matrix, $y_1,\cdots,y_n$ are also coordinate functions of $\mathbb{R}^n$. Thus, our {\it claim} is true. Denoting these coordinate functions and the solutions of (\ref{pth3.5}) still by $x_1, x_2,···, x_n,$ and $g_1,\cdots,g_n$, respectively, we conclude from (\ref{t3.1}) that \be\label{pth3.6} \xi_i\leq \frac{\int_{\Omega} x_i^2}{\int_{\partial \Omega} g_i^2}, \ \ i=1,\cdots,n. \end{eqnarray} From divergence theorem we know that \be \int_{\Omega} x_i^2 =\int_{\Omega} x_i \Delta g_i=-\int_{\Omega}\langle\nabla x_i, \nabla g_i\rangle=-\int_{\partial \Omega} g_i\partial_{\nu}x_i, \end{eqnarray} which gives \be\label{pth3.7} \left(\int_{\Omega} x_i^2\right)^2\leq\int_{\partial\Omega} (\partial_{\nu} x_i)^2\int_{\partial\Omega} g_i^2,\ i=1,\cdots,n. \end{eqnarray} Combining (\ref{pth3.6}) into (\ref{pth3.7}), we infer \be\label{pth3.8} \xi_i{\int_{\Omega} x_i^2}\leq\int_{\partial \Omega}(\partial_{\nu} x_i)^2,\ i=1,\cdots,n, \end{eqnarray} which implies that \be\label{pth3.9} \sum_{i=1}^n\frac 1{\xi_i}\geq\sum_{i=1}^n\frac{\int_{\Omega} x_i^2}{\int_{\partial \Omega}(\partial_{\nu} x_i)^2}. \end{eqnarray} Using the arithmetic-geometric mean inequality and the isoperimetric inequality (\ref{4.2}), we have \be\label{pth3.10}\nonumber \sum_{i=1}^n\frac{\int_{\Omega} x_i^2}{\int_{\partial \Omega}(\partial_{\nu} x_i)^2}&\geq&\frac{n\left(\overset{n}{\underset{j=1}{\prod}}\int_{\Omega} x_i^2\right)^{1/n}}{\left(\overset{n}{\underset{j=1}{\prod}}\int_{\partial \Omega}(\partial_{\nu} x_i)^2\right)^{1/n}}\\ \nonumber
&\geq&\frac{n|\Omega|^{1+\frac 2n}}{(n+2)\omega_n^{2/n}}\cdot\frac 1{\left(\overset{n}{\underset{j=1}{\prod}}\int_{\partial \Omega}(\partial_{\nu} x_i)^2\right)^{1/n}}\\ \nonumber
&\geq&\frac{n|\Omega|^{1+\frac 2n}}{(n+2)\omega_n^{2/n}}\cdot\frac 1{\frac 1n \sum_{i=1}^n\int_{\partial \Omega}(\partial_{\nu} x_i)^2}\\
&=&\frac{n^2|\Omega|^{1+\frac 2n}}{(n+2)|\partial\Omega|\omega_n^{2/n}} \end{eqnarray} with equality holding if and only if \be\label{pth3.11}& & \frac{\int_{\Omega} x_1^2}{\int_{\partial \Omega}(\partial_{\nu} x_1)^2}=\cdots=\frac{\int_{\Omega} x_n^2}{\int_{\partial \Omega}(\partial_{\nu} x_n)^2},\\ & & \label{pth3.12}
\overset{n}{\underset{j=1}{\prod}}\int_{\Omega} x_i^2=\frac{|\Omega|^{n+2}}{(n+2)^n\omega_n^2} \end{eqnarray} and \be\label{pth3.13} \int_{\partial \Omega}(\partial_{\nu} x_1)^2=\cdots=\int_{\partial \Omega}(\partial_{\nu} x_n)^2. \end{eqnarray} Combining (\ref{pth3.9}) and (\ref{pth3.10}), one gets (\ref{0.5}). If the equality holds in (\ref{0.5}), then (\ref{pth3.6}), (\ref{pth3.7}), (\ref{pth3.8}), (\ref{pth3.9}) and (\ref{pth3.10}) should take equality. It follows that \be\label{pth3.14}
\int_{\Omega} x_1^2=\int_{\Omega} x_2^2=\cdots \int_{\Omega} x_n^2=\frac{|\Omega|}{n+2}\cdot\left(\frac{|\Omega|}{\omega_n}\right)^{2/n} \end{eqnarray} and so \be\label{pth3.15}
\int_{\Omega}\sum_{i=1}^n x_i^2=\frac{n|\Omega|}{n+2}\cdot\left(\frac{|\Omega|}{\omega_n}\right)^{2/n}. \end{eqnarray} Consequently, we conclude from Theorem \ref{t03.1} that $\Omega$ is a ball. On the other hand, if $\Omega$ is a ball of radius $R$ in $\mathbb{R}^n$, then \be
\sum_{i=1}^n\frac 1{\xi_i}=\frac n{\xi_1}=\frac n{\frac{(n+2)}{R^3}}=\frac{n^2|\Omega|\left(\frac{|\Omega|}{\omega_n}\right)^{2/n}}{(n+2)|\partial\Omega|}. \end{eqnarray} This completes the proof of Theorem \ref{th3}.
$\Box$ \vskip0.3cm {\bf Remark 3.1.} Consider a more general eigenvalue problem : \be\label{r3.2}
\left\{\begin{array}{l}
\Delta^2 u= 0 \ \ {\rm in \ \ } \Omega, \\ \partial_{\nu}u = \partial_{\nu}(\Delta u)+\zeta\rho u =0 \ \ {\rm on \ \ } \partial\Omega, \end{array}\right. \end{eqnarray} where $\rho$ is a continuous positive function on $\partial\Omega$. The eigenvalues of this problem can be arranged as (counted with multiplicity): \be 0=\zeta_0<\zeta_1\leq\zeta_2\leq\cdots\nearrow +\infty. \end{eqnarray} When $\Omega $ is a bounded domain with smooth boundary in $\mathbb{R}^n$, one can use similar arguments as in the proof of (\ref{pth3.17}) to show that the first $n$ nonzero eigenvalues of the problem (\ref{r3.2}) satisfy \be\label{pth3.17} \overset{n}{\underset{j=1}{\prod}}\xi_j\leq
\left(\frac{\omega_n}{|\Omega|}\right)^2\cdot\left(\frac{(n+2)}{n|\Omega|}\int_{\partial\Omega}\frac 1{\rho}\right)^n, \end{eqnarray} with equality holding implies that $\Omega$ is an ellipsoid. To see this, let us take an orthonormal set of eigenfunctions $\{\psi_i\}_{i=0}^{\infty}$ corresponding to the eigenvalues $\{\zeta_i\}_{i=0}^{\infty}$, that is, \be\label{r3.4}
\left\{\begin{array}{l}
\Delta^2 \psi_i= 0 \ \ {\rm in \ \ } \Omega, \\ \partial_{\nu}\psi_i = \partial_{\nu}(\Delta\psi_i)+\zeta_i\rho\psi_i =0 \ \ {\rm on \ \ } \partial\Omega.\\ \int_{\partial\Omega}\rho\psi_i\psi_j=\delta_{ij} \end{array}\right. \end{eqnarray} The variational characterization for $\zeta_k$ is given by \be\label{4.3}
\zeta_k=\underset{\underset{\int_{\partial\Omega}\rho\psi\psi_i=0, i=0, 1,\cdots, k-1}{\psi\in H^2(\Omega), \partial_{\nu}\psi|_{\partial\Omega}=0, \psi|_{\partial\Omega}\neq 0}}{\inf}\frac{\int_{\Omega}(\Delta\psi)^2}{\int_{\partial\Omega}\rho\psi^2}, \ k=1,\cdots. \end{eqnarray} We choose the origin in $\mathbb{R}^n$ so that (\ref{t3.1}) holds. For each $i\in \{1,\cdots, n\}$, let $h_i$ be the solution of the problem \be\label{r3.3}
\left\{\begin{array}{l}
\Delta h_i= x_i\ \ \ \ \ \mbox{in}\ \ \ \Omega, \\
\partial_{\nu}h_i|_{\partial \Omega}=0,\\
\int_{\partial \Omega}\rho h_i=0. \end{array}\right. \end{eqnarray} As in the proof of Theorem \ref{th3}, we can assume that \be\label{r3.4} \int_{\partial\Omega}\rho x_i\psi_j=0, \ i=1, 2, \cdots,n, \ j<i. \end{eqnarray} It follows from (\ref{4.3}) that \be\label{r3.5} \zeta_i\int_{\partial\Omega}\rho h_i^2\leq \int_{\Omega}x_i^2, \ i=1,\cdots,n. \end{eqnarray} Since \be\label{r3.6} \int_{\Omega} x_i^2=\int_{\Omega}x_i\Delta h_i=-\int_{\partial\Omega}h_i\partial_{\nu}x_i\leq\left(\int_{\partial\Omega}\rho h_i^2\right)^{1/2}\left(\int_{\partial\Omega}\frac{(\partial_{\nu}x_i)^2}{\rho}\right)^{1/2}, \end{eqnarray} we infer from (\ref{r3.5}) that \be\label{r3.6} \zeta_i\int_{\Omega} x_i^2\leq \int_{\partial\Omega}\frac{(\partial_{\nu}x_i)^2}{\rho}, \ i=1,\cdots,n. \end{eqnarray} By multiplication over $i$, one gets \be\label{pth3.16}\nonumber \overset{n}{\underset{j=1}{\prod}}\zeta_j\cdot \overset{n}{\underset{i=1}{\prod}}\int_{\Omega}x_i^2 &\leq& \overset{n}{\underset{i=1}{\prod}}\int_{\partial \Omega}\frac{(\partial_{\nu} x_i)^2}{\rho}\\ \nonumber &\leq& \left(\frac 1n\sum_{i=1}^n \int_{\partial \Omega}\frac{(\partial_{\nu} x_i)^2}{\rho}\right)^n \\ &=&\left(\frac 1n\int_{\partial\Omega}\frac 1{\rho}\right)^n, \end{eqnarray} which, combining with (\ref{4.2}), yields (\ref{pth3.17}). Also, when the equality holds (\ref{pth3.17}), we must have the equality in (\ref{4.2}) and so $\Omega$ is an ellipsoid. \vskip0.3cm {\it Proof of Theorem \ref{th4}} Let $u_0, u_1, u_2, \cdots, $ be orthonormal eigenfunctions corresponding to the eigenvalues $0, \lambda_{1,\tau}, \lambda_{1,\tau}, \cdots,$ that is, \begin{eqnarray*} \left\{\begin{array}{ccc} \Delta^2 u_i-\tau\Delta u_i =0,&&~\mbox{in}~~\Omega, \\[2mm] \frac{\partial^2u_i}{\partial\nu^2}=0, &&~~\mbox{on}~~\partial \Omega,\\[2mm] \tau\frac{\partial u_i}{\partial \nu}-\mathrm{div}_{\partial \Omega}\(\nabla^2 u_i(\nu)\)-\frac{\partial\Delta u_i}{\partial\nu}=-\lambda_{i,\tau} u_i,&&~~\mbox{on}~~\partial \Omega,\\[2mm] \int_{\partial\Omega} u_i u_j =\delta_{ij}. \end{array}\right. \end{eqnarray*}
Note that $u_0= 1/\sqrt{|\partial \Omega|}$. Using the same discussions as in the proof of Theorem \ref{th1}, we can assume that \be \int_{\partial\Omega} x_i u_j=0, \ i=1,\cdots, n, \ j=0,\cdots, i-1.\end{eqnarray} Thus, we have from (\ref{0.7}) that \begin{eqnarray}\label{c4} \lambda_{i,\tau}\int_{\partial\Omega} x_i^2\leq \int_{\Omega}
\(|\nabla^2x_i|^2 +\tau|\nabla x_i|^2\) = \tau |\Omega|, \ i=1,\cdots,n. \end{eqnarray} As in the proof of Theorem \ref{th1}, we have for each $i\in\{1,\cdots,n\}$ that \be\label{pt4.1}
|\Omega|^2=\left(\int_{\partial\Omega} x_i\partial_{\nu}x_i\right)^2\leq \left(\int_{\partial\Omega} x_i^2\right)\left(\int_{\partial\Omega} (\partial_{\nu}x_i)^2\right) \end{eqnarray} with equality holding if and only $\partial_{\nu}x_i=\eta_i x_i$ for some constant $\eta_i\neq 0$.
Multiplying (\ref{c4}) by $\int_{\partial\Omega} (\partial_{\nu}x_i)^2$ and using (\ref{pt4.1}), we get \begin{eqnarray}\label{c5}
\lambda_{i,\tau}|\Omega|^2\leq \tau|\Omega|\int_{\partial\Omega} (\partial_{\nu}x_i)^2, \ 1,\cdots, n. \end{eqnarray}
Dividing by $|\Omega|^2$ and summing over $i$, one gets \begin{eqnarray}\label{pt4.2} \sum_{j=1}^{n}\lambda_{j,\tau}\leq
\frac{\tau}{|\Omega|}\int_{\partial\Omega}\sum_{i=1}^n (\partial_{\nu}x_i)^2=\frac{\tau|\partial\Omega|}{|\Omega|}. \end{eqnarray} This proves (\ref{th4.1}). Moreover, if equality holds in (\ref{th4.1}), then $\partial_{\nu}x_i=\eta_i x_i, $ for some nonzero constants $\eta_i\ i=1,\cdots, n.$ It follows that \be \sum_{i=1}^n \eta_i^2 x_i^2=1 \ \ {\rm on\ } \partial\Omega. \end{eqnarray} If $z=\sum_{i=1}^n \eta_i^2 x_i^2$, then the outward unit normal of $\partial\Omega$ is given by \be \label{pt4.3}
\nu=\frac{\nabla z}{|\nabla z|}. \end{eqnarray} Note that \be \label{pt4.4} \nu=(\partial_{\nu}x_1,\cdots, \partial_{\nu}x_n)=(\eta_1 x_1,\cdots, \eta_n x_n). \end{eqnarray} Comparing (\ref{pt4.3}) and (\ref{pt4.4}), we infer $\eta_1=\eta_2=\cdots=\eta_n$, which shows that $\partial\Omega$ is a hypersphere and so $\Omega$ is a ball. On the other hand, we have \be
\lambda_{1,\tau}(\mathbf{B}_R^n)=\cdots\lambda_{n,\tau}(\mathbf{B}_R^n)=\frac{\tau}{R}=\frac{\tau|\partial\mathbf{B}_R^n|}
{n|\mathbf{B}_R^n|}, \end{eqnarray} that is, the equality holds for balls in (\ref{th4.1}).
$\Box$ \vskip0.3cm \begin{conjecture}
Under the same assumptions of Theorem \ref{th4}, we have
\be\label{c3.1} \overset{n}{\underset{j=1}{\prod}}\lambda_{j,\tau}\leq\frac{\tau^n\omega_n}{|\Omega|} \end{eqnarray} with equality holding if and only if $\Omega$ is a ball. \end{conjecture} \vskip0.2cm It should be mentioned that if $\Omega$ is convex, then the above conjecture is true. To see this, it is enough to take the products of the $n$ inequalities in (\ref{c4}) and use Lemma \ref{l4.1}.
\section{Proofs of Theorems \ref{th7} and \ref{th8}} \setcounter{equation}{0} In this section, we prove Theorems \ref{th7} and \ref{th8}. We shall need the following result \cite{hps}. \vskip0.3cm \begin{lemma} \label{l4.1}
Let $\Omega$ be a bounded convex domain in $\mathbb{R}^n$. Assume that origin coincides with the center of mass of $\partial\Omega$, that is, \be\label{pth7.1} \int_{\partial\Omega}x_i ds=0,\ i=1,\cdots,n. \end{eqnarray} Then we have \be \label{pth7.2}
\overset{n}{\underset{i=1}{\prod}}\int_{\partial\Omega} x_i^2 ds\geq \overset{n}{\underset{i=1}{\prod}}\int_{\partial\Omega^*} x_i^2 ds=\frac{|\Omega|^{n+1}}{\omega_n}, \end{eqnarray} with equality if and only if $\Omega=\Omega^*$. \end{lemma} \vskip0.3cm We prove the following result from which Theorem \ref{th7} follows.
\begin{theorem} \label{t4.1} Let $\beta\geq 0$ and $\Omega$ be a bounded domain with smooth boundary $\partial \Omega$ in
$\mathbb{R}^n$. Let $\rho$ be a positive continuous function on $\partial\Omega$ and denote by $0<\eta_{1, \beta}\leq\eta_{2, \beta}\leq\cdots\leq \eta_{n, \beta}\leq\cdots $
the
eigenvalues of the problem : \be\label{t7.1}\left\{\begin{array}{l} \Delta u =0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\rm in \ }\ \Omega,\\ -\beta\overline{\Delta} u+\partial_{\nu} u= \eta\rho u\ \ \ \ \quad ~~ {\rm on \ } \partial \Omega, \end{array}\right. \end{eqnarray} Then we have \be\label{t7.2}
\sum_{i=1}^n \eta_{i,\beta}\leq \frac 1{|\Omega|^2}\left((|\Omega|+\beta|\partial\Omega|)\int_{\partial\Omega}\rho^{-1} -\frac{\beta}n\left(\int_{\partial\Omega}\frac 1{\sqrt{\rho}}\right)^2\right).
\end{eqnarray} Furthermore, if $\rho$ is constant, the equality holds in (\ref{t7.2}) if and only if $\Omega$ is a ball. \end{theorem}
{\it Proof of Theorem \ref{t4.1}.} Let $u_0, u_1, u_2,\cdots$ be
orthonormal eigenfunctions corresponding to the eigenvalues $0=\eta_{0,\beta}<\eta_{1,\beta}\leq \eta_{2,\beta}\leq \cdots, $ of the problem (\ref{t7.1}), that is, \be\label{t7.3}\left\{\begin{array}{l} \Delta u_i =0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\rm in \ }\ \Omega,\\ -\beta\overline{\Delta} u+\partial_{\nu} u_i= \eta_i\rho u_i\ \ \ \ \quad {\rm on \ } \partial \Omega,\\ \int_{\partial \Omega} \rho u_iu_j=\delta_{ij}. \end{array}\right. \end{eqnarray} Note that $u_0$ is a constant function $1/(\int_{\partial\Omega}\rho)^{1/2}$. The eigenvalues $\eta_{i,\beta}, i=1, 2,\cdots,$ are characterized by \be \label{t7.4} \eta_{i,\beta}=\underset{\underset{\int_{\partial\Omega}\rho u u_j=0, \ j=0,1,\cdots, i-1}{u\in H(\Omega)\setminus\{0\}}}{\min}
\frac{\int_{\Omega}|\nabla u|^2 +\beta\int_{\partial\Omega} |\overline{\nabla}
u|^2}{\int_{\partial\Omega}\rho u^2}. \end{eqnarray} As in the proof of Theorem \ref{th1}, we can choose the coordinate functions $x_1,\cdots,x_n$ of $\mathbb{R}^n$ so that \be \int_{\partial\Omega}\rho x_iu_j=0,\ j<i, i=1,\cdots,n. \end{eqnarray} Hence \be
\eta_{i,\beta}\int_{\partial\Omega} \rho x_i^2 &\leq& \int_{\Omega}|\nabla x_i|^2 +\beta\int_{\partial\Omega} |\overline{\nabla}
x_i|^2\\ \nonumber &=&|\Omega|+\beta\int_{\partial\Omega} |\overline{\nabla}
x_i|^2\\ \label{pth7.6} &=& |\Omega|+\beta\left(|\partial\Omega|-\int_{\partial\Omega}(\partial_{\nu}x_i)^2\right), \ i=1,\cdots,n. \end{eqnarray} We have from (\ref{pt4.1}) that \be\label{t7.5}
|\Omega|^2\leq \left(\int_{\partial\Omega}\rho x_i^2\right)\left(\int_{\partial\Omega}\rho^{-1}(\partial_{\nu}x_i)^2\right). \end{eqnarray} Multiplying (\ref{pth7.6}) by $\int_{\partial\Omega}\rho^{-1}(\partial_{\nu}x_i)^2$ and using (\ref{t7.5}), we get \be\nonumber
\eta_{i,\beta}|\Omega|^2&\leq& (|\Omega|+\beta|\partial\Omega|)\int_{\partial\Omega}\rho^{-1}(\partial_{\nu}x_i)^2 -\beta\left(\int_{\partial\Omega}(\partial_{\nu}x_i)^2\right)\left(\int_{\partial\Omega}\rho^{-1}(\partial_{\nu}x_i)^2\right)
\\ \label{pth7.7} &\leq& (|\Omega|+\beta|\partial\Omega|)\int_{\partial\Omega}\rho^{-1}(\partial_{\nu}x_i)^2 -\beta\left(\int_{\partial\Omega}\frac 1{\sqrt{\rho}}(\partial_{\nu}x_i)^2\right)^2. \end{eqnarray} Summing over $i$ and using Cauchy-Schwarz inequality, one has \be\nonumber
|\Omega|^2\sum_{i=1}^n\eta_{i,\beta}
&\leq& (|\Omega|+\beta|\partial\Omega|)\int_{\partial\Omega}\rho^{-1}-\frac{\beta}n\left(\sum_{i=1}^n\int_{\partial\Omega}\frac 1{\sqrt{\rho}}(\partial_{\nu}x_i)^2\right)^2
\\ \label{pth7.8} &=&(|\Omega|+\beta|\partial\Omega|)\int_{\partial\Omega}\rho^{-1} -\frac{\beta}n\left(\int_{\partial\Omega}\frac 1{\sqrt{\rho}}\right)^2.
\end{eqnarray} Dividing by $|\Omega|^2$, we get (\ref{th7.2}). Moreover, when $\rho$ is constant, the equality holds in (\ref{th7.2}) if and only if $\Omega$ is a ball.
$\Box$
\vskip0.3cm {\it Proof of Theorem \ref{th8}.} Let us choose the origin in $\mathbb{R}^n$ as the center of mass of $\partial\Omega$. Taking $\rho=1$ and using the same arguments as in the proof of Theorem \ref{th7}, we can get
\be \lambda_{i,\beta}\int_{\partial\Omega} x_i^2 \leq |\Omega|+\beta\left(|\partial\Omega|-\int_{\partial\Omega}(\partial_{\nu}x_i)^2\right), \ i=1,\cdots,n. \end{eqnarray} By multiplication of these inequalities, one infers \be\nonumber & & \overset{n}{\underset{i=1}{\prod}}\lambda_{i,\beta}\overset{n}{\underset{j=1}{\prod}}\int_{\partial\Omega} x_j^2\\ \nonumber &\leq&
\overset{n}{\underset{i=1}{\prod}}\left((|\Omega|+\beta|\partial\Omega|)-\beta\int_{\partial\Omega}(\partial_{\nu}x_i)^2\right)
\\ \nonumber &\leq&\left(\frac 1n\sum_{i=1}^n\left((|\Omega|+\beta|\partial\Omega|)-\beta\int_{\partial\Omega}(\partial_{\nu}x_i)^2\right)\right)^n
\\ \label{pth8.1} &=&\left(|\Omega|+\frac{(n-1)\beta}n|\partial\Omega|\right)^n \end{eqnarray} Substituting (\ref{pth7.2}) into (\ref{pth8.1}), we obtain (\ref{th8.1}). It is clear from the proof that equality holds in (\ref{th8.1}) if and only $\Omega$ is a ball.
$\Box$
\vskip0.2cm {\bf Remark 4.1} We believe that the convexity assumption in Theorem \ref{th8} is unnecessary.
\vskip0.4cm
Department of Mathematics, Southern University of Science and Technology
Shenzhen, 518055, GUANDONG, P. R. CHINA
fuquan\[email protected]
\vskip0.6cm
[email protected]
\end{document} | arXiv |
Truncated dodecahedral prism
In geometry, a truncated dodecahedral prism is a convex uniform polychoron (four-dimensional polytope).
Truncated dodecahedral prism
Schlegel diagram
Decagonal prisms hidden
TypePrismatic uniform polychoron
Uniform index60
Schläfli symbolt0,1,3{3,5,2} or t{3,5}×{}
Coxeter-Dynkin
Cells34 total:
2 t0,1{5,3}
12 {}x{10}
20 {}x{3}
Faces154 total:
40 {3}
90 {4}
24 {10}
Edges240
Vertices120
Vertex figure
Isosceles-triangular pyramid
Symmetry group[5,3,2], order 240
Propertiesconvex
It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes.
Alternative names
• Truncated-dodecahedral dyadic prism (Norman W. Johnson)
• Tiddip (Jonathan Bowers: for truncated-dodecahedral prism)
• Truncated-dodecahedral hyperprism
See also
• Truncated 120-cell,
External links
• 6. Convex uniform prismatic polychora - Model 60, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora) x o3x5x - tiddip".
| Wikipedia |
Energy, Sustainability and Society
Electricity market relationship between Great Britain and its neighbors: distributional effects of Brexit
Christopher Stephen Ball ORCID: orcid.org/0000-0003-3884-54901,
Kristina Govorukha2,
Wilhelm Kuckshinrichs1,
Philip Mayer2,
Dirk Rübbelke2 &
Stefan Vögele1
Energy, Sustainability and Society volume 12, Article number: 32 (2022) Cite this article
Beyond Great Britain, Brexit could also have ripple effects on the electricity systems of certain other EU member states. This paper investigates the possible effects of reduced growth in interconnectivity between Great Britain and mainland Europe by 2030 on the electricity system in GB and across other EU member states in addition to the effects of Pound depreciation. Effects are analyzed across a "Green Scenario" and "Blue Scenario" in 2030, based on the ENTSO-E (European Network of Transmission System Operators-Electricity) 10-year development plans. There is a greater expansion of nuclear and renewables in Green than in Blue and, in Blue, the British CO2 price is higher than in the EU. Within each scenario, there are four variants: full vs. reduced expansion of interconnection capacity, in combination with no devaluation and 10% depreciation of the British Pound. The EMME (Electricity Market Model for Europe) is used to model these impacts across the different scenario variants.
Interconnector utilization is more volatile in the Green Scenario variants, leading to concerns about investor incentives, especially given the increased uncertainty under Brexit. In terms of electricity prices, GB consumers lose out across both Blue and Green scenario variants, whereas EU and GB producers both gain and lose in different variants. Across the Green Scenario variants, EU neighbors' trade balances with GB deteriorate slightly, but the impact is far stronger in Blue due to a loss of opportunities to export power. GB sees significant increases in electricity costs across scenario variants. Green scenario variants offer potential for modest emission reductions in certain EU nations, whereas Blue Scenario variants lead to greater emission reductions in the EU neighbors which contrasts with a sharp rise in GB emissions.
There is a significant link between NTC expansion and wholesale prices. Delayed or cancelled NTC expansion could negatively affect the GB power system's low-carbon transition. Pound depreciation and reduced expansion of NTCs lead to shifts in generation-related CO2 emissions. A higher cost burden for electricity is a risk for GB, whereas, for EU neighbors, their trade position with the UK risks deteriorating.
The Brexit vote of 2016 represents a turning point in Britain's position in Europe, with wide-ranging economic and political ramifications. Since energy and climate policies of GB (Great Britain) and the EU are interlinked, it is important to study the effects of Brexit on the energy system and emissions in both GB and its EU neighbors. As regards the EU's objectives for energy and climate, these are based around a common energy policy encompassing 'solidarity between member states' and greater integration of energy systems, enabled by a European grid [1]. A target has been set to increase net transfer capacities (NTCs) to 15% of all installed capacity across member states by 2030 [2], with such cross-border transmission possibilities important in systems with increasing shares of renewable power [3]. The redesign of electricity systems has major implications for markets and infrastructure [4] and for economic, social and political structures [5]. Brexit runs counter to the trend towards greater integration, albeit principally concerning one country—GB—and may, through creating greater uncertainty, lead to lower than expected integration of the British grid with the grids of its EU neighbors [6,7,8,9]. In 2021, the EU–UK Trade and Cooperation Agreement was reached, setting out principles for post-Brexit cooperation between GB and the EU in a number of areas, including energy and climate. Within the agreement, it is reiterated that GB is leaving the Internal Energy Market (IEM), EU ETS (Emissions Trading) scheme and EURATOM (European Atomic Energy Community), but GB also commits to 'non-regression' on climate change action and carbon pricing [10]. In relation to electricity interconnectors, a specific market coupling mechanism has been established which will determine interconnector capacity allocation—this mechanism is based on a separate algorithm which is only used for trades between the grids of Great Britain and those of other EU bidding zones. This agreement represents efforts to continue cooperation in energy and climate matters, however, there is a clear implication that GB is an external partner as opposed to an integral part of the Energy Union and this leads to wider questions about future British divergence and participation within the IEM.
Adam [11] is interested in creating "a framework for thinking about the impacts of Brexit" (P.9) on the British economy. This article's goal is similar, seeking to explore implications of possible scenarios from Brexit in terms of the electricity system of GB and, crucially, its neighbors. Our interest is not the short-term effects of Brexit, but the long-term implications for the electricity systems based on uncertainties about (i) the trajectory that the British power system will follow to 2030; (ii) the continued success of joint projects between the UK and EU (i.e., interconnectors) and (iii) the evolution of the Euro vs. the GBP. We focus on two potential consequences of Brexit that may have a significant impact on electricity market developments: (i) a reduced expansion of Net Transfer Capacities (NTC) and (ii) a depreciation of the GBP against the Euro. In our conception, a soft Brexit implies no reduction to NTC expansion, whereas a hard Brexit implies an expansion of NTC capacity of 65% of the planned level to 2030. No depreciation constitutes the "default" exchange rate (def), whereas depreciation of 10% in the Pound constitutes inflation (infl). We identify possible effects of Brexit on electricity prices, electricity flow structures, the utilization rates of the relevant interconnectors, shifts in CO2 emissions, and a monetary assessment of potential costs for the electricity system. This study adds value to existing literature in that it considers the impact of exchange rate effects and the implications of Brexit for the electricity systems of GB's neighbors.
This analysis is organized as follows: initially, we provide a critical insight into the observable effects of Brexit to date on NTC expansion projects and on the Pound-to-Euro exchange rate. This is done, in particular, with a view to the two scenario assumptions regarding NTC expansion and the exchange rate of the British Pound. Subsequently, we deal with the relevance of Brexit for the electricity systems of neighboring countries. We then explain the methodological approach and present the employed bottom-up model of the European electricity market. Finally, results and conclusions arising from the analysis are discussed.
Current status and critical assessment of the advancing Brexit
Broadly, two possible outcomes of Brexit are described, namely a 'soft' Brexit, in which GB remains in close alignment with EU institutions, and a 'hard Brexit', corresponding to GB leaving the IEM [12]. Although Brexit has already been the subject of several studies [6, 7, 9], little attention has been paid so far to the general potential impact of Brexit not only on GB itself, but also on neighboring countries and thus on the entire European electricity market. The present study contributes to closing this gap. Newbery's [12] and Geske et al.'s [13] distinction between a hard and soft Brexit, in electricity terms, is based on whether or not the UK leaves the IEM. This is a good guide, although GB could still reach a robust and efficient trading relationship with its neighbors outside of the IEM.
We take a critical look at the two Brexit scenario assumptions regarding the development of the NTC and the exchange rates of the Pound. Brexit has led to complications with interconnector projects. Notably, the French Regulatory Commission's decision that it would have to suspend decisions as to whether to support future interconnector projects between Britain and France has led to delays according to the FAB (France–Alderney–Britain) interconnector project [14]. Likewise, according to a report by the French Regulatory Commission, a soft Brexit could lead to a 10% fall in the value of interconnection between Britain and France, whereas, a hard Brexit, leading to market decoupling, could lead to a fall in value of as much as 30% [15]. In the practitioner literature, there is the suggestion that growth in interconnection is likely despite Brexit, due to increasing opportunities in GB's balancing market [16, 17]. However, with the end of the transition period on January 1st 2021, GB left the Internal Energy Market (IEM) [18]. Thus, in terms of international electricity trade, GB is treated as a third party with trades no longer being aided by EU single market tools. As a result, electricity trade between the EU and GB became less efficient and reports predicted a drop in the value of interconnection [17] and, more specifically, complications arising from the loss of access to the current market coupling arrangements [19].
The development of interconnectors between GB the neighboring EU states are listed as Projects of Common Interest (PCI). On the one hand, these projects receive an accelerated permitting process and, on the other hand, they are eligible for public funding. Between 2014 and 2020, 21 projects directly related to GB received funding, according to the EC's PCI database [18, 21]. Of these projects, 15 involved research and development of NTCs. Three others relate to interconnection between France and Ireland. The maximum committed financial support for these projects is approximately €646 million. According to a British Government White Paper, published in 2020, GB plans to realize 18 GW of NTC by 2030, which is three times the current capacity [22]. In the event that some or all of the future interconnector projects are stripped of the PCI status, funding for these projects could be jeopardized [23]. Mathieu, Deane [20] argue that Norway as a full member of the IEM, for example, could take advantage of the situation in order to receive stronger support from the EU for their interconnector projects. In addition to the loss of EU funding, Brexit has also increased uncertainties regarding investments in the energy sector. This may have a negative impact on the financing costs of interconnector projects [20, 24].
In addition to the reduced expansion of the NTC, this study also looks at the consequences of a devaluation of the Pound. Various studies already identified a link between Brexit and devaluation [25,26,27,28]. Nabarro and Schulz [29] estimate, that the value of Sterling could fall between 5 and 10% in trade weighted terms. Plakandaras, Gupta and Wohar [27] argue that a major part of the depreciation of Sterling is due to the uncertainty associated with Brexit. Moreover, Stoupos and Kiohos [28] found that a further depreciation of the Pound is likely, which may lead to a decline in value especially against the Euro and the Dollar.
Although it is unclear how long this effect will persist, an examination of the time series of the GBP and Euro exchange rate shows that no recovery to a pre-Brexit level has taken place so far. Figure 1 displays the Pound-to-Euro exchange rate from January 2016 to March 2021. A sharp fall in the value of the Pound occurs following the Brexit vote in June 2016 and the value of the Pound has never regained its mid-2016 level, although there have been substantial fluctuations. This sustained lower value is reflective of perceptions about a more difficult trading relationship between GB and the EU [30]. Moreover, this depreciation of the GBP against the euro occurred during the Eurozone crisis, affecting Portugal, Greece, Italy and Spain, with the euro declining seriously against the dollar in April 2014 and only gaining significant ground in early 2018 [31]. The decline of GBP against the euro despite the Eurozone crisis indicates that the effect of Brexit on the GBP was substantial. The structural break in the exchange rates possibly falls in the period between November 2016 and July 2017, close to the date when the EU was notified about GB decision to withdraw. The structural change identified is the period after which there is a consistent change in the Pound-to-euro exchange rate.Footnote 1 For the purposes of this study, we assume that, in certain scenarios, this depreciation persists in 2030, whereas, in other scenarios, there is no depreciation. The notes on the estimation of the structural break are given in the Appendix (Fig. 1).
GBP to Euro Exchange Rate 2016–2021
The above analysis indicates that, in the short-term, Brexit has had significant structural effects on the Pound and that, in the longer-term, the smooth running of interconnector projects could be jeopardized by the uncertainty created.
Why Brexit is relevant to the power system of GB and its EU neighbors
Studies have been published about the implications of Brexit on energy and climate policy [33,34,35] while there has also been research relating to economic and financial market analysis around Brexit [36,37,38,39].
Geske, Green [6] analyze the economic consequences for GB and France if GB were to leave the EU's IEM. Lockwood, Froggatt [7] contrast advantages and disadvantages of Brexit by identifying and evaluating potential tradeoffs between market integration and political freedom of action. Mayer, Ball [8] investigate the impact at the actor level in GB by considering both a reduced NTC expansion and devaluation of the British Pound. By applying a model for the European Electricity System, MacIver, Bukhsh [40] examine the implications of increased interconnectivity of the GB electricity market with Europe and conclude, among other things, that unilateral CO2 taxation in GB can lead to local reductions in CO2 emissions, which are offset by additional emissions in the rest of Europe. Employing a coupled modeling approach [35], examine the sectoral implications of Brexit for the United Kingdom, Europe, and the rest of the world. They found that a positive or negative outcome of Brexit for GB depends heavily on its relationship with the rest of the world. In contrast, the picture for Europe was more pessimistic. In only one of a total of eight scenarios can Europe achieve positive gross value added as a result of Brexit. Other studies suggest more modest impacts from Brexit for the UK's electricity system, thanks to the TCA's (EU–UK Trade and Cooperation Agreement) focus on maintaining cooperation in the energy sector, highlighting minor increases in trade barriers [41] and a minimal value of an additional GW of interconnector capacity with France for UK consumers in 2025 [42]. Guo and Newbery [43] estimate the social costs of uncoupling to be substantially lower than other projections in the literature at €28 million per year. Furthermore, the marginal value of interconnector capacity with the UK could be more significant for French, Dutch and Belgian consumers [42], indicating that it is interesting to study the effects on the UK's neighbors. Costs from uncoupling the UK from the Single Electricity Market arise from increases in inefficient trading, with estimates by Gissey, Guo [44] of a 3% and 2% increase in the price differential between the UK and France and the Netherlands, respectively.
The EU's IEM involves markets clearing at the same time and transmission capacity to be allocated automatically and this minimizes the errors in electricity trading [12]. Geske, Green [6] argue that uncoupled markets may weaken incentives for investors to expand interconnection capacity between GB and EU grids, due to lower trading efficiency. Mathieu, Deane [20] argue that the welfare losses of an exclusion of GB would be all the greater the further European market integration progresses.
GB is a net importer of electricity, with France being the most significant trading partner, followed by Belgium and the Netherlands [22]. The commercial value of Britain's interconnectors with France and the Netherlands is estimated at €500 million annually, since gains can be made from trading electricity from low-cost to high-cost markets, with the social value from contributions to energy security adding an additional €25 million in social value [45]. The imposition of a unilateral carbon price in the GB market causes losses in welfare [46], as it has reversed the direction of electricity trade flows between GB and the continent, with GB importing from the Netherlands and France despite having lower generation costs and carbon intensity [45]. While there is high private and social value from interconnection, the asymmetric carbon price imposed by GB on its own generation is arguably harmful to the social value of interconnection.
Costs of British withdrawal from the IEM are estimated by Newbery, Gissey [45] at €300 million annually for Britain by 2030, whereas the welfare losses, according to Geske, Green [6] amount to €700 million per year by 2030 for both Britain and France. The economic effects are distributed very differently among stakeholders. Geske, Green [6] find that wholesale costs for British consumers increase by 4%, whereas they fall slightly for French consumers. In contrast, they estimate that British producers benefit, whereas French producers lose out from a loss of access to trading opportunities, through a stymied expansion of interconnection capacity between the British and French grids to 5 GW (rather than 10 GW). There will also be implications for GB's energy security, especially at peak times and the loss of access to the EU's shared electricity balancing system, currently in development, will entail very large costs [47].
Negative consequences may result not only from a possibly costly coupling process of the EU ETS (EU Emissions Trading System) with the GB ETS, introduced in January, 2021, but also from insufficient interconnection at the power sector level. This could have an impact, especially with regard to the integration of high shares of intermittent renewables, both on the expansion of these and on security of supply. The transition to a low-carbon system involves high investments (both in terms of generation facilities and grid expansion). Investments of up to €130 to €330 billion could be required by 2030 [48]. The political and regulatory uncertainties associated with Brexit could have a negative impact on willingness to invest, leading to delays in innovation and the transformation of the energy sector and ultimately to insufficient progress on climate protection [20, 34, 49, 50]. The UK established its own Emissions Trading Scheme upon leaving the EU in January 2021. While there are signs of convergence between the UK and the EU ETS prices [41], there has also been divergence, with the spread between the UK and the EU ETS price reaching a high of 50% (UK price of £90 over the EU price of £60 per ton) in September 2021 [51]. While there has been talk of linking the UK ETS with the EU ETS, this would make the UK a rule-taker [52] and the process could take time [53]. While the UK and the EU have both committed themselves to Net Zero targets, indicating a similar trajectory towards more ambitious decarbonization, the UK is experimenting with its ETS system—integrating provisions for carbon dioxide removal technologies, for example [54]. This indicates that a certain degree of uncertainty about convergence on UK and EU carbon prices persists.
A common oversight of existing literature in the field of electricity market analysis is the role of exchange rates. While these have been considered in empirical studies [26, 55] and more aggregate modeling approaches such as input–output [56], or CGE models [57], they have been mostly neglected in a bottom-up electricity market analysis. Mayer, Ball [8] consider a possible fall in the value of the Pound relative to the Euro, due to greater trade barriers between GB and the EU following Brexit. Yet, their study focuses on the implications for GB, largely omitting effects for the rest of Europe. Exchange rate effects could amplify the effect of reduced NTCs on consumers and producers in GB and connected countries while changing flows of electricity could influence carbon emissions across these countries. This paper will explore the possible implications of these fluctuations for the power system of both GB and its EU neighbors.
Bottom-up electricity market model
The Electricity Market Model for Europe (EMME) is a bottom-up electricity dispatch model [58], consisting of 28 European countries. It applies a linear optimization method to minimize total system costs \(Z\) under the transmission and operational constraints. Total system costs comprise electricity generation costs, imports and exports of electricity between countries as described by the objective function in Eq. 1:
$$\underset{}{\mathrm{min}}Z=\sum_{h,i,d}\left[\mathrm{Pr}\left(h,i,d\right)\bullet \mathrm{Cst}\left(i,d\right)\right]+\sum_{h,d,k}\mathrm{Im}\left(h,d,k\right)\bullet T$$
subject to:
$$\sum_{i}\mathrm{Pr}\left(h,i,d\right)+\sum_{k}\mathrm{Im}\left(h, d,k\right)-\sum_{k}\mathrm{Ex}\left(h,d,k\right)=\mathrm{Dm}\left(h,d\right) \forall h,d,$$
$$\frac{\mathrm{Pr}\left(h,i,d\right)}{y}\le \mathrm{Cp}\left(h,i,d\right),$$
$$\frac{\mathrm{Im}\left(h,d,k\right)}{y}\le \mathrm{NTC}\left(d,k\right),$$
with \(i\): index for generation technology type; \(h\): specific hour of the year [−]; \(d\), and \(k\): countries indexes [−]; \(\mathrm{Cst}\): variable generation costs [Euro/MWh]; \(\mathrm{Pr}\): electricity production [MWh]; \(\mathrm{Cp}\): generation capacity [MW]; \(\mathrm{Im}\): electricity imports from country k to country d [MWh]; \(\mathrm{Ex}\): electricity exports from country d to country k [MWh]; \(T:\) transport costs for imports and exports (Euro/MWh); \(\mathrm{Dm}\): electricity demand [MWh]; \(\mathrm{NTC}\): net transfer capacity between two markets [MW]; and \(y\): conversion factor (MWh to MW) [unit: hours].
Equation 2 is the central constraint that represents the energy balance. It ensures that the given hourly electricity demand is balanced by the supply side at every hour in each modeled country. The model comprises the detailed representation of the electricity generation mix: power plants, their capacities, vintage structure and respective variable costs for each country. Imports and exports between the neighboring countries are constrained by net transfer capacities (NTCs). Generation capacities, energy commodity prices, CO2 certificate prices, NTCs and power demand are exogenous model input parameters. Diverse sets of input parameters represent assumptions about the future of the system and are combined in the scenarios. The production, imports, exports and electricity prices in each country result from the modeled economic dispatch. Based on these model results, we estimate emissions, consumer and producer surpluses across various scenarios.
Geographic coverage includes the EU-28 countries, with each country treated as a single node, which are linked via NTCs. Figure 2 provides an overview of the geographical scope.
Geographical scope of the EMME model
The calculation of CO2 emissions takes into account the vintage structure of a power plant \(i\) and the type of fuel used:
$${\mathrm{CO}}_{2} \left(d\right)= \sum_{i}\left({\mathrm{sec}}_{i}\bullet \sum_{h}\mathrm{Pr}(h, i, d)\right),$$
with \({\mathrm{CO}}_{2} :\) emissions in country d\([t]\) and \({\mathrm{sec}}_{i}:\) specific emission coefficient \(\left[\frac{{{t}_{\mathrm{CO}}}_{2}}{{\mathrm{MWh}}_{\mathrm{el}}}\right]\).
Equation 6 below shows the trade balance comprising the difference between the value of imports and exports for each country:
$$\mathrm{TB} \left(d\right)= \sum_{h,k}\left(\mathrm{Im}\left(h,d,k\right)\bullet X(h,k\right))-\sum_{h,k}\left(\mathrm{Ex}\left(h,d,k\right)\bullet X(h,d\right)),$$
with \(X\): the electricity price in the country-importer \(k\) or -exporter \(d\) [Euro/MWh]—the marginal electricity price in the respective region (shadow price of the demand constraint in Eq. 2).
The total expenses for the provision of electricity in each region can give a general overview of the costs pertinent to the described electricity system. We focus on the variable costs and do not regard the overnight costs for present generation capacities, as we do not regard investments in the electricity generation fleet. The effects of the GBP devaluation on generation costs (through e.g., increasing the cost of imported natural gas) are built into the model. The total expenses for the provision of electricity are as follows:
$$\mathrm{TE} \left(d\right)=\sum_{h,i}(\mathrm{Pr}\left(h, i, d\right)\bullet X\left(h,d\right))+\sum_{h,k}\left(\mathrm{Im}\left(h,d,k\right)\bullet (X\left(h,k\right)+T(k,d)\right))-\sum_{h,k}\left(\mathrm{Ex}\left(h,d,k\right)\bullet (X(h,d\right)).$$
Scenario assumptions
The possible distributional impacts on EU countries are investigated under the two scenarios Blue and Green (see Table 1) and these are both based on the 10-year development plans found within the ENTSO-E documentation [59]. Brexit is a cause of uncertainty in the realm of multilateral agreements, such as NTC expansion and climate policy, especially since GB will no longer be subject to the EU Renewables Directive. Both Green and Blue scenarios describe GB following strict unilateral climate policies, represented by a high GB carbon price. The Blue scenario sheds light on the developments between GB and EU countries, in the context of a high GB carbon price and a lower EU ETS price. In contrast, under the Green Scenario, the ETS and GB carbon prices are equal, i.e., the EU ETS price is increased to match the high GB carbon price. Under the Green scenario, there is a far greater expansion of renewable and nuclear capacity than in the Blue scenario—this is based on the details of the development plans described by ENTSO-E. Details of the Blue and Green scenarios are given in Table 1 below. Effects from reduced NTC expansion and the depreciation of the Pound on electricity prices and CO2 emissions are differentiated across the Blue and Green scenarios.
Table 1 Scenarios and scenario assumptions
In this section, the changes in power flows arising from the different variants of Brexit alongside the impacts for the utilization of NTCs, economic effects and the impacts on emissions in GB and the EU are presented.
Changes in power flows and electricity prices
Examining the changes in power flows between GB and its European neighbors helps to understand the underlying impacts of Brexit on other European countries' power systems. The changes induced across the variants of the Blue scenario and Green scenario are discussed below. All changes are relative to the default variant of the Blue and Green scenarios in which there is no reduced expansion of NTCs and no devaluation of the Pound (Table 2).
Table 2 Total imports of electricity to GB and exports from GB in GWh to neighbors in 2030 in default Blue and Green Scenarios
Since, in the scenario variants of Blue, GB follows a policy of a unilaterally high carbon price combined with lower expansion of low-carbon generation alternatives, GB's domestic generation is less competitive than the domestic generation of its European counterparts. This means that GB is a net importer under the Blue scenario and the scenario variants overwhelmingly influence the imports of power to the GB grid from the EU; see Table 3. Effects on flows in the Blue scenarios can be differentiated between Belgium and the Netherlands, for whom the depreciation effect is stronger, and the others, where the NTC effect is dominant. The falls in imports to the GB grid from Belgium and the Netherlands indicate that, under depreciation of the Pound, these countries' power exports face a substantial loss of competitiveness in relation to Britain's domestic generation. It is possible that Belgian and Dutch generation provides GB with a relatively small amount of peaking power which is no longer worthwhile, as its relative cost increases through depreciation. Curiously, under a reduced expansion of NTCs, GB imports more Dutch power, suggesting these imports are displacing those from France and the other nations. Exports from France, Denmark, Ireland, and Norway to GB are minimally affected by the depreciation, but there are substantial drops under a reduced NTC expansion. This indicates that, under the Blue Scenario, imports to GB from Belgium and the Netherlands fulfill a different function than those from France, Denmark Norway, and Ireland. It is suggested that the power flows from France, Denmark, Norway, and Ireland to GB are more constant and stable, whereas the British grid only has recourse to Belgian and Dutch power occasionally to meet peak demand needs (Table 4).
Table 3 Changes in imports to GB from EU neighbors and exports from GB to EU neighbors in blue scenario variants compared to variant soft_def (GWh)
Table 4 Changes in imports to GB from EU neighbors and exports from GB to EU neighbors in green scenario variants compared to variant soft_def
Under the Green scenario, GB's position in the EU electricity market is different, in that it is now much more of an exporter of power to the continent thanks to its expansion of nuclear and renewable capacity. In this case, the effects are stronger on the export-side for the Netherlands, Belgium and France and stronger on the import side for Denmark and Norway (i.e., imports to GB). Across the Netherlands, Belgium, France and Denmark, exports from GB are strongly boosted by a depreciation of the Pound, as would be expected, given the enhanced competitiveness of British power exports. In contrast, a fall in NTC expansion leads to considerable reductions in exports from GB to these countries. In the case of France, imports to GB behave in the expected way, but they do not in the case of the Netherlands and Belgium, although the effects are very small in the latter cases. In the case of Denmark, imports to GB fall sharply in the case of depreciation, whereas exports from GB rise, as would be expected. Under a fall in NTC expansion, there are no effects on imports to GB, but exports from GB fall significantly. For Norway, imports to GB grid fall substantially under a reduced NTC expansion, but there is little impact from the depreciation of the Pound. In the case of Ireland, imports to GB are affected strongly by both effects. In summary, Belgium and the Netherlands experience substantial changes in terms of their exports from GB, whereas, in France, the picture is mixed and, in Denmark, Norway, and Ireland, their imports to GB are strongly affected.
The changes in the flows under the different scenarios and their variants help to understand the reasons underlying changes in the utilization of NTCs, the electricity-related costs of Brexit and the impacts of Brexit on emissions. Figure 3 shows NTC utilization as the ratio of electricity flows from and to GB in relation to the NTC. The calendar weeks of the scenario year 2030 are plotted on the Y-axis, the X-axis shows the hours of a day. Light areas indicate a high, dark areas a low utilization of the NTC (Fig. 3).
Utilization of NTCs across Blue Scenario Variants (weeks of year on y axis, hours of day on x axis)
A comparison of the pattern of GB's net electricity imports shows that there is a structural difference between the Green and Blue scenario groups. All scenario groups of Blue show a relatively constant high utilization. At the same time, a slight seasonal shift can be seen with regard to the hours with the highest utilization rates. Interestingly, it seems that the combination of both a reduction in NTC and a devaluation of the British Pound lead on average to a lower utilization rate of the transfer capacities, than the reduction of NTC alone.
Although the effect is less pronounced, it can be seen that, in the Green scenario variants, the exchange rate effect tends to show an opposite trend. Here, the devaluation of the Pound leads to an increase in the utilization rate. This can be seen in the top right hand picture of Fig. 4, where, under a soft Brexit (soft_infl), implying full NTC expansion, a devaluation leads to more utilization in the early weeks of the year (0–10) and the final weeks of the year (45–50) at night, shown by the greater amount of yellow at those periods (reflecting higher utilization). Devaluation has similar effects under a hard Brexit in Green, with greater amounts of yellow during those periods. In general, Green shows a more fragmented picture, with hourly rapid shifts between high and low rates of utilization of NTCs. With regard to international electricity trade and the expansion of NTCs, our analysis shows that the basic structure of imports and exports between GB and neighboring countries depends significantly on the assumed development path. While Blue is characterized by a high average utilization of lines, the picture is different for Green. Hours with high and low utilization often alternate hourly. With a view to the future development of line capacities, the question arises as to whether an import–export structure as shown in Green creates sufficient investment incentives for the expansion of NTCs. On the other hand, however, it must be taken into account that due to the volatile generation structure of renewable plants in the Green scenario, the expansion of NTCs can be significantly more important for system stability than is the case in the Blue scenario variants.
Utilization of NTCs across Green Scenario Variants (weeks of year on y axis, hours of day on x axis)
In Fig. 5, the price deviations from the reference scenario (soft Brexit, no Pound depreciation) are shown. It is clear that, in the Green scenario variants, the effects are more distributed and that the impact of depreciation is more important than the impact of reduced NTC expansion. This is intuitive given the role of GB as a net exporter in the Green scenarios, with the Pound depreciation leading to the greater attractiveness of British power on the continent. In contrast, in the Blue scenarios, the effects are concentrated in France, Norway, Sweden and Denmark and are driven by the reduction in NTC expansion rather than the Pound depreciation. This effect is caused by the fact that GB is a net importer under the Blue scenarios and a fall in NTC expansion capacities reduces its ability to import from its key partners. The distributional effects differ according to the scenario. Generally, in the Green scenario variants, the changes caused by the Pound depreciation are good for EU consumers, who benefit from lower wholesale prices, and for GB producers, who are better able to export to the EU markets. In contrast, the changes are bad for GB consumers, who see price increases, and for EU producers who must compete against cheaper British imports. Under the Blue variants, the changes are good for EU consumers and for GB producers, but negative for EU producers, not able to sell as much electricity as they would like to GB and for GB consumers, who face higher prices through the reduced import potential.
Price effects across Europe in Max in green (top) and blue (bottom). Sharper colors indicate stronger price increases
Costs of Brexit
The costs of Brexit in terms of GB leaving the IEM have been estimated at €300 million by Newbery, Gissey [45] and €700 million by Geske, Green [6], respectively. In this section, we conduct a monetary assessment of the different scenario variants. In doing so, we compare the monetary value of electricity flows between GB and the neighboring states on the one hand, and the total expenditure on electricity provision in GB on the other. All estimates are in 2019 euros.
Figure 6 shows the monetized power flows between GB and neighboring EU member states in reference to the soft_def variants—i.e., the variants without reduced expansion of NTCs and without Pound devaluation. The values shown represent the differences in trade balances, according to Eq. 6, of the respective countries with GB. Positive values correspond to additional spending by neighboring countries, negative values to reduced spending. The variants of the Green scenario are plotted on the left, those of the Blue scenario on the right.
Monetized power flows between GB and neighboring states in €million
An examination of the Green scenario variants shows that here the exchange rate effect exceeds that of the NTC reduction: with full NTC development and a permanent devaluation of the Pound against the Euro, the expenditures of neighboring European countries on British electricity increase by a total of almost €700 million per annum. In contrast, hard_def leads to a decrease in spending of around €135 million annually. The combination of both effects together leads to additional expenditures of around €454 million annually, with the reduced expansion of NTC capacity mitigating the exchange rate effect.
The Blue scenarios show a somewhat different picture. Here, the effect of a reduced NTC expansion clearly predominates, while the exchange rate effect plays only a minor role. In the event of a devaluation of the Pound (soft_infl), spending by EU member states on GB electricity will fall by €169 million annually. The other two sub-scenarios result in an increase of €1,688 per annum (hard_def), and €1,508 per annum (hard_infl) million, respectively, resulting from selling less power to the GB grid.
In summary, in the Green Scenarios, from the perspective of the European neighboring countries, a reduction in electricity trade expenditures with GB can only be achieved if NTC capacity is reduced. In the analysis presented, a devaluation of the British Pound and an expansion of capacities as planned leads to GB being able to sell higher volumes of green electricity to the EU member states at more favorable prices overall. In contrast, the unilateral introduction of a carbon price in GB results in electricity producers in surrounding states having a competitive advantage over GB producers. The devaluation of the Pound can only compensate for this to a very limited extent. Here, a reduction in planned NTC capacity means that neighboring countries can sell less electricity to GB, which negatively impacts trade balances.
Total GB spending on electricity supply, described by Eq. 7 in the methodology section, is €22.99 billion for Green and €25.77 billion for Blue.
The potential cost of Brexit can then be estimated using the difference in total spending in each scenario variant relative to the scenario without NTC capacity reduction and without permanent depreciation of the Pound, as shown in Table 5.
Table 5 GB's total expenditure for electricity supply compared to baseline scenarios in €million
Again, we see that total costs in the Green scenario are only marginally responsive to the reduction in NTC capacity. The hard_def variant responds with only a slight increase in expenditures. In contrast, the exchange rate effect leads to a more significant increase in soft_infl, and the combined effect of the exchange rate and reduced NTC capacity in hard_infl causes expenditures to increase the most. Although in this case, too, the exchange rate effect has a stronger impact than the reduction in NTC capacity, the difference in Blue's scenario variants is smaller. The isolated exchange rate effect results in an increase of 6.64%, the NTC effect in an increase of 5.20%. The combined effect results in an overall growth of 11.74%. Compared to the estimates of €300 million to €700 million annually [6, 45], our cost assessments show that they are significantly higher on average. At €269.19 million, Green_hard_def is the scenario with the lowest additional costs due to Brexit. At €1,430 million and €1,619 million, the other variants are significantly more pessimistic than the estimates by Newbery, Gissey [45] and Geske, Green [6]. In the case of Blue, the range between €1,341 million and €3,025 million is even higher.
Impacts of Brexit on emissions
The impacts of Brexit on emissions from power generation in GB and neighboring countries across the Blue and Green scenario variants are shown in Fig. 7. We are not concerned with the overall level of UK–EU CO2 emissions, but, rather, changes in the distribution of these emissions. Our goal is to identify shifts in the neighboring countries' power sector emissions across the different scenarios.
Percentage change in CO2 emissions by country in relation to soft-def scenario (below absolute values in Mt CO2)
The reduced expansion of NTC capacities is the trigger for emissions changes in the Blue variants, with changes being much more pronounced than in the Green scenario variants. Under Blue_soft_infl, there are limited rises in GB's and Austria's emissions contrasted with minimal falls in emissions in the Netherlands, Belgium and France. However, in Blue_hard_def, emissions rise substantially in GB (more than 30%) and this is accompanied by significant falls in France (8%) and Denmark (5%). Belgium and Spain also both experience falls in emissions of 3% under a reduced expansion of NTC capacity, with Austria seeing a rise of 3%. In Blue_hard_infl, the depreciation of the Pound, for the most part, amplifies the effect of the reduced NTC capacity expansion—of course, this is especially the case for GB, with power sector emissions increasing by 35% compared to the reference case (Blue_soft_def).
While there is a certain degree of movement in emissions in Blue_soft_infl, caused by the depreciation effect, this is relatively weak. GB, under the Blue scenario, is a net importer, so there are marginal reductions in emissions in the Netherlands, Belgium and France, as their exports of power are less attractive to the GB market. As a net importer, GB is far more reliant on imports from its neighbors and, under reduced expansion of NTC capacity, it must resort to less efficient natural gas plants to meet its needs. This drives up emissions considerably. For the exporters of power to GB, this leads to opportunities to reduce their power-related emissions, especially for France and Belgium.
Figure 7 shows that the depreciation effect is the dominant effect in the Green Scenario variants. This is due to GB's position as a net exporter of renewable power in the Green scenario; the depreciation of the Pound leads to cheaper imports of clean power for its EU neighbors. Emissions fall in Belgium and the Netherlands and there is a corridor to central and Eastern Europe, with emissions reductions in Germany, the Czech Republic and Poland in the Green_soft_infl and Green_hard_infl variants. It is proposed that the greater availability of British green electricity in France, the Netherlands and Belgium has a cascading impact on these central and Eastern European countries, namely they benefit indirectly through being able to purchase surplus French and German power. Poland, in particular, experiences a 16% drop in emissions and this can be partly attributed to the greater presence of fossil fuel generation in Poland in 2030 compared to other states. There is a southern corridor, with French emissions increasing and Spanish emissions decreasing. In fact, in Green_soft_infl, French production increases slightly whereas Spanish production decreases by a similar quantity, with the additional French power produced being passed through to Spain, shifting the emissions reduction to Spain. Denmark and Austria are outliers, seeing spikes in emissions of 25% and 27%, respectively, however, it must be said that Danish emissions are very minimal in the Green scenario and this could be caused by a small reduction in imports from GB. In the case of GB, there is a significant rise of 10% in Green_soft_infl and this is caused by increased production linked with exports which have become more competitive thanks to the Pound depreciation.
Under the reduced expansion of NTC, there is minimal impact on emissions—with only GB and Poland experiencing effects. In GB's case, more limited import capacity means that it has to resort to domestic fossil fuel production in times of power shortages, while the causes of the effect in Poland are not clear at this stage.
An analysis of this picture demonstrates that there are opportunities and risks arising out of Brexit for continental Europe. Under the Green Scenario variants, there are opportunities for modest reductions in emissions in certain EU nations, with some notable exceptions, including France and Austria. Across the Blue scenarios, there is, in general, potential for emission reductions across the continent. From GB's perspective, the risks are considerably higher and these are amplified in the Blue scenario variants with reduced expansion of interconnector capacity. In these variants, GB's power sector-related emissions rise considerably and this is a political concern for GB, especially given its recent commitments, including the introduction of its own emissions trading scheme and Net Zero pledge [60]. The British government must consider how it could respond to Brexit-related vulnerabilities related to currency and NTC capacities—e.g., whether it needs to invest in additional back-up capacity, in the form of greater amounts of hydrogen or batteries.
Conclusions and limitations
The developments arising from GB's exit from the EU may not only have far-reaching consequences for the electricity market in GB itself, but also in the neighboring EU member states and, thus, in the EU as a whole. We show that there is a significant link between the expansion of NTCs and wholesale electricity prices in the affected countries. This effect, in turn, has a direct impact on the flow of electricity between countries.
A closer look at the electricity flows over the course of the year shows that even adverse effects can occur with regard to the climate protection targets. Due to the higher price spread, a power plant fleet as assumed in the Blue scenario would use the existing NTCs much more intensively than in the case of the Green scenario. Higher price spreads and higher NTC utilization in Blue suggest that investment incentives in additional transmission capacity could be significantly higher than in Green. However, due to the volatile generation profiles of renewable energy plants, such as wind power or PV plants, sufficient interconnectors to neighboring countries could make a significant contribution to the stability of national power systems. Consequently, delayed or even cancelled NTC expansion could have a negative long-term effect on the power sector's transition to a low-carbon system.
Despite Brexit, national CO2 avoidance targets remain in place for both GB and European member states. Our results show that both the depreciation of the British Pound and a reduced expansion of NTCs can have a significant impact on shifts in electricity generation-related CO2 emissions. Due to the high share of renewable energy sources in Green, emissions from electricity generation react minimally to the scenario variants. In Blue, on the other hand, it can be seen that the variants can lead to a local shift in emissions. In order to successfully decarbonize both the European and GB electricity systems, it is important to account for such regional shifts so that emissions abatements in one region cannot simply be offset in the other.
Lastly, we assessed the potential costs of Brexit in terms of electricity supply in monetary terms. Other studies, which also determined the electricity sector-related annual costs of Brexit, valued them at €500 million and €700 million, respectively. Although one scenario variant in our study results in additional costs for GB of only €269 million, all other scenarios show a significantly higher burden of up to €3025 million Euro. In addition, an examination of the monetarily valued electricity flows of neighboring countries shows that their expenditures for electricity trade also increase in most scenario variants.
By looking at different scenarios, we were able to show the range of potential effects on the power systems of GB and its EU neighbors. This range reflects the great uncertainty associated with Brexit. The authors conclude that many potentially negative effects can be mitigated or even averted by appropriate policy measures. In order to ensure this, however, a suitable framework must be created for the stakeholders involved, which reduces or, at best, eliminates the existing uncertainties regarding the future partnership between GB and the EU. A future study could consider ways of doing this while preserving the integrity of the Single Market. For instance, ways and implications of linking the UK ETS with the EU ETS could be explored in more detail. Our study provides putative lessons from Brexit for the electricity systems of GB and its neighbors and could, therefore, stimulate debate about possible electricity system-related risks should other member states, such as Poland or Hungary, consider departing from the EU.
A limitation of this study is that it, unlike Pollitt and Chyong [42], does not consider the impact of reductions in GDP, arising from Brexit, on the British electricity demand and future studies would benefit from including GDP-impacts into their analysis. Furthermore, future studies could benefit from looking at the impact of progressive reductions in the expansion of interconnector capacity, i.e., from a modest 1GW to more substantial reductions in expansion. Future studies could also consider the impact on the energy markets in the context of wider economic effects from Brexit on supply chains, labor markets and the long-term position of markets for capital, services and goods.
The datasets used and analyzed in during the current study are available from the corresponding author on reasonable request.
The analysis for structural change is based on the normal linear regression model and capture parameter instabilities in both regression coefficients and error variance (see [32]).
GB:
ENTSO-E:
European Network of Transmission System Operators-Electricity
EMME:
Electricity Market Model for Europe
NTC:
Net Transfer Capacities
IEM:
Internal Energy Market
EURATOM:
European Atomic Energy Community
Def:
Default exchange rate
Infl:
Pound depreciation
FAB:
France–Alderney–Britain interconnector
EU ETS:
EU Emissions Trading Scheme
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Open Access funding enabled and organized by Projekt DEAL. This research did not receive any specific grant from funding agencies in the commercial, non-profit or public sector.
Forschungszentrum Jülich, Institute of Energy and Climate Research-Systems Analysis and Technology Evaluation (IEK-STE), 52425, Jülich, Germany
Christopher Stephen Ball, Wilhelm Kuckshinrichs & Stefan Vögele
TU Bergakademie Freiberg, 09599, Freiberg, Germany
Kristina Govorukha, Philip Mayer & Dirk Rübbelke
Christopher Stephen Ball
Kristina Govorukha
Wilhelm Kuckshinrichs
Philip Mayer
Dirk Rübbelke
Stefan Vögele
CB, KG, PM and SV worked on the conceptualization, writing and visualization of the results. PM worked on the methodology. SV developed the model. DR and WK undertook reviewing and editing. All authors read and approved the final manuscript.
Correspondence to Christopher Stephen Ball.
No human participants, human data or human tissue have been involved.
The structural brakes were defined with the methodology described in Zeileis, Shah [32]. The computations are carried out in the R system for statistical computing with packages fxregime 1.0–4 [61] and strucchange 1.5–2 [62].
The exchange rate model is a standard linear regression model (see Table
Table 6 Coefficients of the linear model applied in the analysis of structural changes
6 for the summary):
$${y}_{i}={x}_{i}^{t}\beta +{u}_{i} \left(i=1, \dots , n\right),$$
in which the \({y}_{i}\) are returns of the target currency and the \({x}_{i}\) is the vector of the monthly currency exchange rates. To identify structural change, we compare the natural logarithm of the SMI time series with the natural logarithm of the lagged (two and four month lags) time series:
$$\mathrm{log}\left({x}_{i}\right)-\mathrm{log}\left({x}_{i-2}\right)-\mathrm{log}\left({x}_{i-4}\right).$$
The fluctuation of GBP and Euro exchange rate (Fig. 1) a structural change that with some certainty can be identified within the period between October 2016 and July 2017. The Bayesian information criterion (BIC) criteria reach its minimum for the 1 breakpoint (see Fig.
Bayesian information criterion
8). The same is proved by the F statistics (LR/Wald) for all single break alternatives (see Fig.
F Statistics (LR/Wald) for all single break alternatives
9). They correspond to the breaks between October 2016 and July 2017. Visually, this approves changes in the mean, max and min values of the GBP and Euro exchange rate presented in Fig. 1.
Ball, C.S., Govorukha, K., Kuckshinrichs, W. et al. Electricity market relationship between Great Britain and its neighbors: distributional effects of Brexit. Energ Sustain Soc 12, 32 (2022). https://doi.org/10.1186/s13705-022-00358-0
Interconnectors
Electricity system
Distributional effects
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\begin{document}
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\def \prend{\vrule depth-1pt height7pt width6pt} \def \proof{\bigbreak\noindent{\bf Proof.\ \ }} \def \endpf{{\ \ \prend \medbreak}}
\def
{
}
\newtheorem{theorem}{T\/heorem}[section] \newtheorem{apptheo}{T\/heorem}[section] \newtheorem{corollary}{Corollary}[section] \newtheorem{definition}{Definition}[section] \newtheorem{lemma}{Lemma}[section] \newtheorem{applem}{Lemma}[section] \newtheorem{example}{Example}[section]
\newtheorem{appexample}{Example}[section] \newtheorem{fact}{Fact}[section] \newtheorem{claim}{Claim}[section] \newtheorem{proposition}{Proposition} \newtheorem{remark}{Remark}[section] \newcommand{\subset}{\subset} \newtheorem{problem}{Open problem}
\title{Operational State Complexity of\\
Deterministic Unranked Tree Automata}
\author{Xiaoxue Piao \qquad\qquad Kai Salomaa \institute{School of Computing, Queen's University\\ Kingston, Ontario K7L 3N6, Canada} \email{\{piao, ksalomaa\}@cs.queensu.ca} } \defUnranked Tree Automata{Unranked Tree Automata} \defX. Piao \& K. Salomaa{X. Piao \& K. Salomaa}
\maketitle
\begin{abstract}
We consider the state complexity of basic operations on tree languages recognized by deterministic unranked
tree automata.
For the operations of
union and intersection the upper and lower bounds of both weakly and strongly
deterministic
tree automata are obtained. For tree concatenation we establish a tight upper bound that is of a different order than the known state complexity of concatenation of regular string languages. We show that $(n+1) ( (m+1)2^n-2^{n-1} )-1$
vertical states are sufficient, and necessary in the worst case, to recognize the concatenation of tree languages recognized by (strongly or weakly) deterministic automata with, respectively,
$m$ and $n$ vertical states.\\ Keywords:
operational state complexity, tree automata,
unranked trees, tree operations \end{abstract}
\section{Introduction}\label{s:in}
As XML \cite{dtddef} has played increasingly important roles in data representation and exchange through the web, tree automata have gained renewed interest, particularly tree automata operating on unranked trees. XML documents can be abstracted as unranked trees, which makes unranked tree automata a natural and fundamental model for various XML processing tasks \cite{CDG,MaN,Sc}. Both deterministic and nondeterministic unranked tree automata have been studied.
One method to handle unranked trees is to encode them as ranked trees and then use the classical theory of ranked tree automata. However, the encoding may result in trees of unbounded height since there is no a priori restriction on the number of the children of a node in unranked trees. Also depending on various applications, it may be difficult to come up with a proper
choice of the encoding method.
Descriptional complexity of finite automata and related structures has been extensively studied in recent years~\cite{GH,HK2,HK,Yu,Yu2}. Here we consider operational state complexity of deterministic unranked tree automata. Operational state complexity describes how the size of an automaton varies under regularity preserving operations. The corresponding results for string languages are well known~\cite{galina,Yu,YuZhSa94}, however, very few results have been obtained for tree automata. While state complexity results for tree automata operating on ranked trees
are often similar to corresponding results on regular string automata \cite{Yu}, the situation becomes essentially different for automata operating on unranked trees. An unranked tree automaton has two different types of states, called horizontal and vertical states, respectively. There are also other automaton models that can be used to process unranked trees, such as nested word automata and stepwise tree automata. The state complexity of these models has been studied in~\cite{HS,mn,nestps}.
We study two different models of determinism for unranked tree automata. We call the usual deterministic unranked tree automaton \cite{CDG} model where the horizontal languages defining the transitions are specified by DFAs (deterministic finite automata),
a {\em weakly deterministic tree automaton} (or WDTA). For the other variant of determinism for unranked tree automata, we refer to the corresponding automaton model as a
{\em strongly deterministic unranked tree automaton} (or SDTA).
This model was introduced by Cristau, L\"oding and Thomas~\cite{CLT}, see also Raeymaekers and Bruynooghe~\cite{RB}. SDTAs can be minimized efficiently and the minimal automaton is unique~\cite{CLT}. On the other hand, the minimization problem for WDTAs is NP-complete and the minimal automaton need not be unique~\cite{mn}.
We give upper and lower bounds for the numbers of both vertical and horizontal states for the operations of union and intersection. The upper bounds for vertical states are tight for both SDTAs and WDTAs. We also get upper bounds which are almost tight for the number of the horizontal states of SDTAs. Obtaining a matching lower bound for the horizontal states of WDTAs turns out to be very problematic. This is mainly because the minimal WDTA may not be unique and the minimization of WDTAs is intractable~\cite{mn}. Also, the number of horizontal states of WDTAs can be reduced by adding vertical states, i.e., there can be trade-offs between the numbers of horizontal and vertical states, respectively.
The upper bounds for the number of
vertical states for union and intersection of WDTAs and SDTAs are, as expected, similar to the upper bound for the corresponding operation on ordinary string automata. Already in the case of union and intersection, the upper bounds for the numbers horizontal states are dramatically different for WDTAs and SDTAs, respectively.
In an SDTA, the horizontal language associated with label $\sigma$ is represented with a single DFA $H_\sigma$ augmented with an output function $\lambda$. The state assigned to a node labeled with $\sigma$ is determined by the final state reached in $H_\sigma$ and $\lambda$. On the other hand, in a WDTA, the horizontal languages associated with a given label $\sigma$ and different states are represented by distinct DFAs. The state assigned to a node labeled with $\sigma$ depends on the choice of the DFA.
We consider also the state complexity of
(tree) concatenation of SDTAs. It is well known that $m2^n-2^{n-1}$ states are sufficient to accept the concatenation of an $m$ state DFA and an $n$ state DFA~\cite{YuZhSa94}. However, the tight upper bound to accept the concatenation of unranked tree automata, with $m$ and $n$ vertical states respectively, turns out to be $(n+1) ( (m+1)2^n-2^{n-1} )-1$. The factor $(n+1)$ is necessary here because the automaton accepting the concatenation of two tree languages must keep track of the computations where no concatenation has been done. For string concatenation, there is only one path and the concatenation always takes place somewhere on that path.
For non-unary trees, there is no way that the automaton can foretell on which branch the concatenation is done and, consequently, the automaton for concatenation needs considerably more states.
It should be emphasized that this phenomenon is not caused by any particular construction used for the automaton to accept the concatenation of given tree languages, and we have a matching lower bound result.
Since complementation is an ``easy'' operation for both strongly and weakly deterministic tree automata, we do not investigate its state complexity in this paper. Note that we do not require the automaton models to be complete (i.e., some transitions may be undefined). A (strongly or weakly) deterministic automaton accepting the complement of a tree language recognized by the same type of automaton would need at most one additional vertical state and it is easy to see that this bound can be reached in the worst case.
The paper is organized as follows. Definitions of unranked tree automata and other notations are given in section~\ref{pre}. The upper bounds and corresponding lower bounds for union and intersection of SDTAs are presented in section~\ref{sdta}. In section~\ref{wdta}, the state complexity of union and intersection of WDTAs is discussed. The tight bound for the number of vertical states for tree concatenation of SDTAs is given in section~\ref{con}. The same construction works for WDTAs.
\section{Preliminaries}\label{pre}
Here we briefly recall some notations and definitions concerning trees and tree automata. A general reference on tree automata is~\cite{CDG}.
Let ${\rm I\!N}$ be the set of non-negative integers. A {\em tree domain} $D$ is a finite set of elements in ${\rm I\!N}^*$ with the following two properties: (i) If $w\in D$ and $u$ is a prefix of $w$ then $u\in D$. (ii) If $ui \in D$, $i\in {\rm I\!N}$ and $j<i$ then $uj \in D$. The nodes in an unranked tree $t$ can be denoted by a tree domain $dom(t)$, and $t$ is a mapping from $dom(t)$ to the set of labels $\Sigma$. The set of $\Sigma$-labeled trees is $T_\Sigma$.
For $t,t'\in T_\Sigma$ and $u \in dom(t')$, $t'(u\leftarrow t)$ denotes the tree obtained from $t'$ by replacing the subtree at node $u$ by $t$. The concatenation of trees $t$ and $t'$ is defined as $t\cdot t' = \{t'(u\leftarrow t)\mid u\in leaf(t')\}$. The concatenation operation is extended in the natural way to sets of trees $L_1$, $L_2$: $$L_1\cdot L_2 = \bigcup_{t\in L_1, t'\in L_2} t\cdot t'.$$
We denote a tree $t = b ( a_1, \ldots ,a_n )$, whose root is labeled by $b$ and leaves are labeled by $ a_1, \ldots ,a_n $, simply as $b ( a_1 \ldots a_n)$. When $a_1= \ldots =a_n=a$, write $t=b(a^n)$. By a slight abuse of notation, for a unary tree $t = a_1 ( a_2 ( \ldots (a_n)\ldots)) $, we write $t = a_1 a_2 \ldots a_n$ for abbreviation. When $a_1= \ldots =a_n=a$, we write $t=a^n$ for short. (In each case it should be clear from the context whether $a^n$ refers to a sequence of leaves or to a unary tree.)
Next we briefly recall the definitions of the two variants of deterministic bottom-up tree automata considered here. A {\em weakly deterministic unranked tree automaton} (WDTA) is a 4-tuple $A=(Q,\Sigma,\delta,F)$ where $Q$ is a finite set of states, $\Sigma$ is the alphabet, $F \subseteq Q$ is the set of final states, $\delta$ is a mapping from $Q\times\Sigma$ to the subsets of $(Q \cup \Sigma)^*$ which satisfies the condition that, for each $q \in Q, \sigma \in \Sigma, \delta(q,\sigma)$ is a regular language and for each label $\sigma$ and every two states $q_1\neq q_2$, $\delta(q_1,\sigma)\bigcap\delta(q_2,\sigma)=\emptyset$. The language $\delta(q,\sigma)$ is called the {\em horizontal language} associated with $q$ and $\sigma$ and it is specified by a DFA $H_{q,\sigma}^A$.
Roughly speaking, a WDTA operates as follows. If $A$ has reached the children of a $\sigma$-labelled node $u$ in states
$q_1$, $q_2$ ,..., $q_n$, the computation assigns state $q$ to node $u$ provided that $q_1q_2...q_n\in\delta(q,\sigma)$. In the sequence $q_1q_2...q_n$ an element $q_i \in \Sigma$ is interpreted to correspond to a leaf labeled by that symbol. A WDTA is a deterministic hedge automaton~\cite{CDG} where each horizontal language is specified using a DFA.
Note that in the usual definition of~\cite{CDG} the horizontal languages are subsets of $Q^*$. In order to simplify some constructions,
we allow also the use of symbols of the alphabet $\Sigma$ in the horizontal languages, where a symbol $\sigma \in \Sigma$ occurring in a word of a horizontal language is always interpreted to label a leaf of the tree. The convention does not change the state complexity bounds in any significant way because we use small constant size alphabets and we can think that the tree automaton assigns to each leaf labeled by $\sigma \in \Sigma$ a particular state that is not used anywhere else in the computation.
A {\em strongly deterministic unranked tree automaton} (SDTA) is a 4-tuple $A=(Q,\Sigma,F,\delta)$, where $Q, \Sigma, F$ are similarly defined as for WDTAs. For each $a \in \Sigma$, the horizontal languages $\delta(q, a)$, $q \in Q$, are defined by a single DFA augmented with an output function as follows. For $a \in \Sigma$ define $D_a=(S_a,Q \cup \Sigma,s_a^0,\gamma_a,E_a, \lambda_a)$ where $(S_a,Q \cup \Sigma,s_a^0,\gamma_a,E_a)$ is a DFA and $\lambda_a$ is a mapping $S_a\rightarrow Q$. For all $q \in Q$ and $a \in \Sigma$, the horizontal language $\delta(q,a)$ is specified by $D_a$ as the set $\{ w \in (Q \cup \Sigma)^* \mid \lambda_a(\gamma_a^*(s_a^0,w))=q\}$. Intuitively, when $A$ has reached the children of a node $u$ labelled by $a$ in states $q_1, \ldots, q_m$ (an element $q_i \in \Sigma$ is interpreted as a label of a leaf node), the state at $u$ is determined (via the function $\lambda_a$) by the state that the DFA $D_a$ reaches after reading the word $q_1 \cdots q_m$. More information on SDTA's can be found in~\cite{CLT}.
Given a tree automaton $A=(Q,\Sigma,F,\delta)$, the states in $Q$ are called {\em vertical states\/}. The DFAs recognizing the horizontal languages are called {\em horizontal DFAs\/} and their states
are called horizontal states. We define the {\em (state) size of $A$,} ${\rm size}(A)$, as a pair of integers $[ |Q|, n ]$, where $n$ is the sum of the sizes of all horizontal DFAs associated with $A$.
\section{Union and intersection}
We investigate the state complexity of union and intersection operations on unranked tree automata. The upper bounds on the numbers of vertical states are similar for SDTAs and WDTAs, however the upper bounds on the numbers of horizontal states differ between the two models.
\subsection{Strongly deterministic tree automata}\label{sdta}
The following result gives the upper bounds and the lower bounds for the operations of union and intersection for SDTAs.
\begin{theorem}\label{xxx} For any two arbitrary SDTAs $A_i=(Q_i,\Sigma,\delta_i,F_i)$, $i=1,2$, whose transition function associated with $\sigma$ is represented by a DFA $H_{\sigma}^{A_i}=(C_{\sigma}^i, Q_i \cup \Sigma, \gamma_{\sigma}^i, c_{\sigma,0}^i, E_{\sigma}^i)$, we have \begin{description}
\item[1] Any SDTA $B_\cup$ recognizing $L(A_1)\cup L(A_2)$ satisfies that $${\rm size}(B_{\cup}) \leq [ \; (|Q_1|+1)\times
(|Q_2|+1)-1; \; \sum_{\sigma \in \Sigma} ((|C_{\sigma}^{1}|+1)
\times (|C_{\sigma}^{2}|+ 1) -1) \; ].$$
\item[2] Any SDTA $B_\cap$ recognizing $L(A_1)\cap L(A_2)$ satisfies that $${\rm size}(B_{\cap}) \leq [ \; |Q_1|\times
|Q_2|; \; \sum_{\sigma \in \Sigma} |C_{\sigma}^{1}| \times
|C_{\sigma}^{2}| \; ].$$
\item[3] For integers $m, n\geq 1$ and relatively prime numbers
$k_1,k_2,\ldots,k_m,k_{m+1},\ldots,\\ k_{m+n}$, there exists
tree languages $T_1$ and $T_2$ such that $T_1$ and $T_2$, respectively, can be
recognized by SDTAs with $m$ and $n$ vertical states,
$\prod_{i=1}^m k_i+O(m)$ and $\prod_{i=1+m}^{m+n} k_i+O(n)$ horizontal states, and \begin{description}
\item[i] any SDTA recognizing $T_1\cup T_2$ has at least
$(m+1)(n+1)-1$ vertical
states and $\prod_{i=1}^{m+n} k_i$ horizontal states.
\item[ii] any SDTA recognizing $T_1\cap T_2$ has at least $mn$ vertical
states and $\prod_{i=1}^{m+n} k_i$ horizontal states. \end{description} \end{description} \end{theorem}
The upper bounds on vertical and horizontal states are obtained from product constructions, and Theorem~\ref{xxx} shows that for the operations of union and intersection on SDTAs the upper bounds
are tight for vertical states and almost tight for horizontal states.
\subsection{Weakly deterministic automata}\label{wdta}
In this section, the upper bounds on the numbers of vertical and horizontal states for the operations of union and intersection on WDTAs are investigated, and followed by matching lower bounds on the numbers of vertical states.
\begin{lemma}\label{union} Given two WDTAs $A_i=(Q_i,\Sigma,\delta_i,F_i)$, $i=1,2$, each horizontal language $\delta_i(q,\sigma)$ is represented by a DFA $D_{q,\sigma}^{A_i}=(C_{q,\sigma}^i, Q_i \cup \Sigma, \gamma_{q,\sigma}^i, c_{q,\sigma,0}^i, E_{q,\sigma}^i)$.
The language $L(A_1)\cup L(A_2)$ can be recognized by a WDTA $B_{\cup}$ with \begin{eqnarray*} \ & {\rm size}(B_{\cup}) \leq [
\; (|Q_1|+1)\times (|Q_2|+1)-1; &\ \\
\ & |\Sigma| \times (\sum_{q \in Q_1,p \in Q_2}|D_{q,\sigma}^{A_1}| \times |D_{p,\sigma}^{A_2}| + \sum_{q\in Q_1}|D_{q,\sigma}^{A_1}|\times \prod_{p \in Q_2}|D_{p,\sigma}^{A_2}| + \sum_{p\in Q_2}|D_{p,\sigma}^{A_2}| &\ \\
\ & \times \prod_{q \in Q_1}|D_{q,\sigma}^{A_1}|) \; ] & \end{eqnarray*}
The language $L(A_1)\cap L(A_2)$ can be recognized by a WDTA $B_{\cap}$ with \begin{eqnarray*}\ & {\rm size}(B_{\cap}) \leq [
\; |Q_1|\times |Q_2|; \; |\Sigma| \times \sum_{q \in Q_1,p\in Q_2}
|D_{q,\sigma}^{A_1}| \times |D_{p,\sigma}^{A_2}| \; ].&\ \end{eqnarray*} \end{lemma} The theorem below shows that the upper bounds for the vertical states are tight.
\begin{theorem}\label{dtadfa} For any two WDTAs $A_1$ and $A_2$ with $m$ and $n$ vertical states respectively, we have \begin{itemize}
\item[1] any WDTA recognizing $L(A_1)\cup L(A_2)$ needs at
most $(m+1)(n+1)-1$ vertical states,
\item[2] any WDTA recognizing $L(A_1)\cap L(A_2)$ needs at
most $mn$ vertical states,
\item[3] for any integers $m,n\geq 1$, there exist tree languages $T_1$
and $T_2$ such that $T_1$ and $T_2$ can be recognized by
WDTAs with $m$ and $n$ vertical states respectively, and any WDTA
recognizing $T_1 \cup T_2$ has at least $(m+1)(n+1)-1$ vertical
states, and any WDTA recognizing $T_1 \cap T_2$ has at least $mn$ vertical
states. \end{itemize} \end{theorem}
\begin{problem} Are the upper bounds for the numbers of horizontal states given in Lemma~\ref{union} tight? \end{problem} In the case of WDTAs we do not have a general method to establish lower bounds on the number of the horizontal states. It remains an open question to give (reasonably) tight lower bounds on the number of horizontal states needed to recognize the union or intersection of tree languages recognized by two WDTA's. \section{Concatenation of strongly deterministic tree automata}\label{con}
We begin by giving a construction of an SDTA recognizing the concatenation of two tree languages recognized by given
SDTAs.
\begin{lemma}\label{cons} Let $A_1$ and $A_2$ be two arbitrary SDTAs. $A_i=(Q_i,\Sigma,\delta_i,F_i)$, $i=1,2$, transition function for each $\sigma\in\Sigma$ is represented by a DFA $H_{\sigma}^{A_i}=(C_{\sigma}^i, Q_i \cup \Sigma, \gamma_{\sigma}^i, c_{\sigma,0}^i, E_{\sigma}^i)$ with an output function $\lambda_\sigma^i$.
The language $L(A_2)\cdot L(A_1)$ can be recognized by an SDTA $B$ with
$${\rm size}(B ) \leq [ \; (|Q_1| + 1)\times (2^{|Q_1|}
\times (|Q_2|+1)-2^{|Q_1|-1})-1; \; |\Sigma| (|C_{\sigma}^2|+1)
(|C_{\sigma}^1|+1)\times 2^{|C_{\sigma}^1| + 1} \; ].$$ \end{lemma}
\begin{proof} Choose $B = (Q_1' \times Q_1'' \times Q_2', \Sigma, \delta, F)$, where $Q_1'= Q_1 \cup \{dead\} $, $Q_1''= {\cal P} (Q_1) $, $Q_2' = Q_2 \cup \{ dead\}$. Let $P_2 \subseteq Q_1 $. $( p_1 , P_2 ,q)\in Q_1' \times Q_1'' \times Q_2'$ is final if there exists $p\in P_2$ such that $p\in F_1$.
The transition function $\delta$ associated with each $\sigma$ is represented by a DFA $H_{\sigma}^{B}=( S \times S'' \times S', (Q_1' \times Q_1'' \times Q_2') \cup \Sigma, \mu, ( c_{\sigma,0}^1 , ( \{ c_{\sigma,0}^1 \} , 0) ,c_{\sigma,0}^2), V)$ with an output function $\lambda_\sigma^{B}$, where $S = C_{\sigma}^1 \cup \{dead\} $, $S'' = {\cal P} ( C_{\sigma}^1 ) \times \{ 0 ,1 \} $, $S' = C_{\sigma}^2 \cup \{dead\} $. Let $C_2\subseteq C_{\sigma}^1$, $x=1,0$. $( c_1 ,( C_2 , x ) , c^2) \in S \times S'' \times S'$ is final if $c^2 \in E_{\sigma}^2$ or there exists $c \in c_1 \cup C_2$ such that $c \in E_{\sigma}^1$. $\mu$ is defined as below:
For any input $a\in \Sigma$, $$ \mu(( c_1 , ( C_2 , x) ,c^2), a) = ( \gamma_\sigma^1(c_1, a), (\bigcup_{c_2\in C_2} \gamma_\sigma^1(c_2, a), x), \gamma_\sigma^2(c^2, a)) $$ For any input $( p_1 , P_2 ,q)\in Q_1' \times Q_1'' \times Q_2' $, if $P_2 \neq \emptyset$, $$ \mu(( c_1 , ( C_2 , 0) , c^2), ( p_1 , P_2 ,q) ) = ( \gamma_\sigma^1(c_1, p_1), (\bigcup_{p_2\in P_2} \gamma_\sigma^1(c_1, p_2) , 1), \gamma_\sigma^2(c^2, q)) $$ $$ \mu(( c_1 , ( C_2 , 1) , c^2), ( p_1 , P_2 ,q) ) = ( \gamma_\sigma^1(c_1, p_1), ( \bigcup_{p_2\in P_2}\gamma_\sigma^1(c_1, p_2) \cup \bigcup_{c_2\in C_2} \gamma_\sigma^1(c_2, p_1) , 1), \gamma_\sigma^2(c^2, q)) $$ if $P_2 = \emptyset$, $$ \mu(( c_1 , ( C_2 , 0) , c^2), ( p_1 , \emptyset ,q) ) = ( \gamma_\sigma^1(c_1, p_1), ( \emptyset , 0), \gamma_\sigma^2(c^2, q)) $$ $$ \mu(( c_1 , ( C_2 , 1) , c^2), ( p_1 , \emptyset ,q) ) = ( \gamma_\sigma^1(c_1, p_1), (\bigcup_{c_2\in C_2} \gamma_\sigma^1(c_2, p_1) , 1), \gamma_\sigma^2(c^2, q)) $$ Write the computation above in an abbreviated form as $\mu(( c_1 , ( C_2 , x) , c^2), r ) = ( p_1' , P_2' ,q' )$, $r\in \Sigma\cup Q_1'\times Q_1''\times Q_2'$. When compute $p_1'$ and $q'$, if any $\gamma_\sigma^i (c, \alpha)$, $i=1,2$, $c=c_1,c^2$, $\alpha \in \Sigma \cup Q_i $, is not defined in $A_i$, assign $dead$ to $p_1'$ or $q'$. When compute $P_2'$, add nothing to $P_2'$ if any $\gamma_\sigma^i (c, \alpha)$ is not defined.
Let $p_{leaf}\in Q_1$ denote the state assigned to the leaf in $A_1$ substituted by a tree in $L(A_2)$. $\lambda_\sigma^{B}$ is defined as: for any final state $e = ( c_1 , ( C_2 , x) ,c^2)$, $x_1 = c_1 \cap E_\sigma^1$, $X_2 = C_2 \cap E_\sigma^1$, \begin{itemize}
\item[1] If $c^2 \in E_\sigma^2$
$$ \lambda_\sigma^{B}(e) = \left\{ \begin{array}{l}
( \lambda_\sigma^1 (x_1) , p_{leaf} \cup \bigcup_{x_2\in X_2}\lambda_\sigma^1 (x_2) , \lambda_\sigma^2 (c^2) ), \mbox{ if } \lambda_\sigma^2 (c^2) \in F_2 \mbox{ and } x=1
\\
( \lambda_\sigma^1 (x_1) , p_{leaf} , \lambda_\sigma^2 (c^2) ), \mbox{ if } \lambda_\sigma^2 (c^2) \in F_2 \mbox{ and } x=0
\\
(\lambda_\sigma^1 (x_1) , \bigcup_{x_2\in X_2} \lambda_\sigma^1 (x_2) ,\lambda_\sigma^2 (c^2) ), \mbox{ if } \lambda_\sigma^2 (c^2) \notin F_2 \mbox{ and } x=1
\\
( \lambda_\sigma^1 (x_1) , \emptyset , \lambda_\sigma^2 (c^2) ), \mbox{ if } \lambda_\sigma^2 (c^2) \notin F_2 \mbox{ and } x=0 \end{array} \right. $$ \item[2] If
$c^2 \notin E_{\sigma}^2$, $$ \lambda_\sigma^{B}(e) = \left\{ \begin{array}{l} ( \lambda_\sigma^1 (x_1) , \emptyset , dead ) \mbox{ if } x = 0
\\ ( \lambda_\sigma^1 (x_1) , \bigcup_{x_2\in X_2} \lambda_\sigma^1 (x_2) ,dead ) \mbox{ if } x=1 \end{array} \right. $$ If $x_1 = \emptyset$, define $ \lambda_\sigma^1 (x_1) = dead $. If $X_2 = \emptyset$, define $\bigcup_{x_2\in X_2} \lambda_\sigma^1 (x_2) = \emptyset$. \end{itemize} The state in $B$ has three components $( p_1 , P_2 ,q)$. $p_1$ is used to keep track of $A_1$'s computation where no concatenation is done. $p_1$ is computed by the first component $c_1$ in the state of $H_{\sigma}^{B}$. $P_2$ traces the computation where the concatenation takes place. In a state $( c_1 , ( C_2 , x) ,c^2)$ of $H_{\sigma}^{B}$, $x=1$ (or $x=0$) records there is (or is not) a concatenation in the computation. The third component $q$ keeps track of the computation of $A_2$. When a final state is reached in $A_2$, which means a concatenation might take place, an initial state $p_{leaf}$ is added to $P_2$, which is achieved by the $\lambda_\sigma^{B}$ function in $B$.
According to the definition of $\lambda_\sigma^{B}$, when $\lambda_\sigma^2 (c^2) \in F_2$, $p_{leaf}$ is always in the second component of the state. Exclude the cases when
$\lambda_\sigma^2 (c^2) \in F_2$, and $p_{leaf}$ is not in the second component of the state, and we do not require $B$ be complete. $B$ has $(|Q_1| + 1)\times (2^{|Q_1|} \times
(|Q_2|+1)-2^{|Q_1|-1})-1$ vertical states in worst case. \end{proof} \endpf
Lemma~\ref{cons} gives an upper bound on both the numbers of vertical and horizontal states recognizing the concatenation of $L(A_2)$ and $L(A_1)$. In the following we give a matching lower bound for the number of vertical states of any SDTA recognizing $L(A_2) \cdot L(A_1)$.
For our lower bound construction we define tree languages consisting of trees where, roughly speaking, each branch belongs to the worst-case languages used for string concatenation in~\cite{YuZhSa94} and, furthermore, the minimal DFA reaches the same state at an arbitrary node $u$ in computations starting from any two leaves below $u$. For technical reasons, all leaves of the trees are labeled by a fixed symbol and the strings used to define the tree language do not include the leaf symbols.
As shown in Figure~\ref{f:dfa}, $A$ and $B$ are the DFAs used in Theorem~1 of \cite{YuZhSa94} except that a self-loop labeled by an additional symbol $d$ is added to each state in $B$. We use the symbol $d$ as an identifier of DFA $B$, which always leads to a dead state in the computations of $A$. This will be useful for establishing that all vertical states of the SDTA constructed as in Lemma~\ref{cons} are needed to recognize the concatenation of tree languages defined below.
\begin{figure}
\caption{DFA $A$ and $B$}
\label{f:dfa}
\end{figure}
Based on the DFAs $A$ and $B$ we define the tree languages $T_A$ and $T_B$ used in our lower bound construction. The tree language $T_B$ consists of $\Sigma$-labeled trees $t$, $\Sigma = \{ a, b, c, d \}$, where: \begin{enumerate} \item All leaves are labeled by $a$ and if a node $u$ has a child that is a leaf, then all the children of $u$ are leaves. \item $B$ accepts the string of symbols labeling a path from any node of height one to the root. \item The following holds for any $u \in {\rm dom}(t)$ and any nodes
$v_1$ and $v_2$
of height one below $u$. If $w_i$ is the string of symbols labeling the path from $v_i$ to $u$, $i = 1, 2$, then $B$ reaches the same state after reading strings $w_1$ and $w_2$. \end{enumerate} Intuitively, the above condition means that when, on a tree of $T_B$, the DFA $B$ reads strings of symbols labeling paths starting from nodes of height one upwards, the computations corresponding to different paths ``agree'' at each node. This property is used in the construction of an SDTA $M_B$ for $T_B$ below.
Note that the computations of $B$ above are started from the nodes of height one and they ignore the leaf symbols. This is done for technical reasons because in tree concatenation a leaf symbol is replaced by a tree, i.e., the original symbol labeling the leaf will not appear in the resulting tree.
$T_B$ can be recognized by an SDTA $M_B=(Q_B,\{a,b,c,d\}, \delta_B, F_B)$ where $Q_B=\{0,1,\ldots,n-1\}$ and $F_B=\{n-1\}$. The transition function is defined as: \begin{itemize}
\item[(1)] $\delta_B(0,a)=\epsilon$,
\item[(2)] $\delta_B(i,a)=\bigcup_{0 \leq i \leq n-1} i^+$,
\item[(3)] $\delta_B(i,d)=\bigcup_{0 \leq i \leq n-1} i^+$,
\item[(4)] $\delta_B(j, b) = (j-1)^+, 1 \leq j \leq n-1$ and
$\delta_B(0, b) = (n-1)^+$,
\item[(5)] $\delta_B(1,c)=\{0, \ldots, n-1\}^+$. \end{itemize}
The tree language $T_A$ and an SDTA $M_A$ recognizing it are defined similarly based on the DFA $A$. Note that $T_A$ has no occurrences of the symbol $d$ and $M_A$ has no transitions defined on $d$.
The SDTAs $M_A$ and $M_B$ have $m$ and $n$ vertical states, respectively.
An SDTA $C$ recognizing tree language $T_A \cdot T_B$\ \ \footnote{Recall from section~\ref{pre} that $T_A \cdot T_B$ consists of trees where in some tree of $T_B$ a leaf is replaced by a tree of $T_A$.} is obtained from $M_A$ and $M_B$ using the construction given in Lemma~\ref{cons}. The vertical states in $C$ are of the following form \begin{equation}\label{state1} (q,S,p), 0\leq q\leq n, S\subseteq \{0,1,\ldots,n-1\}, 0\leq p\leq m, \end{equation} where if $p=m-1$ then $0\in S$, and if $S=\emptyset$ then $q=n$ and $p=m$ can not both be true. The number of states in~(\ref{state1}) is $(n+1) ((m+1)2^n-2^{n-1})-1$. State $q=n$ (or $p=m$) denotes $q=dead$ (or $p=dead$) in the construction of lemma~\ref{cons}. We will show that $C$ needs at least $(n+1) ((m+1)2^n-2^{n-1})-1$ vertical states. We prove this by showing that each state in~(\ref{state1}) is reachable and all states are pairwise inequivalent, or distinguishable. Here distinguishability means that for any distinct states $q_1$ and $q_2$ there exists $t \in T_\Sigma[x]$ such that the (unique deterministic) computation of $C$ on $t(x \leftarrow q_1)$ leads to acceptance if and only if the computation of $C$ on $t(x \leftarrow q_2)$ does not lead to acceptance.
\begin{lemma} All states of $C$ are reachable. \label{reach} \end{lemma}
\begin{proof} We introduce the following notation. For a unary tree \\$t=a_1(a_2(\ldots a_m(b)\ldots))$, we denote $word(t)=a_ma_{m-1}\ldots a_1 \in \Sigma^*$. Note that $word(t)$ consists of the sequence of labels of $t$ from the node of height one to the root, and the label of the leaf is not included.
We show that all the states in~(\ref{state1}) are reachable by using induction on $|S|$.
When $|S|=0$, $(i, \emptyset, j)$, $0\leq i\leq n-1$, $0\leq j\leq m-2$ is reachable from $(0,\emptyset,0)$ by reading tree $t$ where $word(t)=b^ia^j$. State $(n, \emptyset, j)$, $1\leq j\leq m-2$ is reachable from $(0,\emptyset,0)$ by reading tree $a(t_1,t_2)$ where $word(t_1)=ba^{j-1}$ and $word(t_2)=b^2a^{j-1}$. State $(n,\emptyset,0)$ is reachable by reading symbol $b$ from state $(n, \emptyset, j)$, $1\leq j\leq m-2$. State $(i, \emptyset, m)$, $0\leq i\leq n-1$ is reachable from $(0,\emptyset,0)$ by reading tree $b(t_1,t_2)$ where $word(t_1)=b^{i-1}a$ and $word(t_2)=b^{i-1}a^2$.
When $|S|=1$, $(i, \{0\}, m-1)$, $0\leq i\leq n-1$ is reachable from $(0,\emptyset,0)$ by reading tree $t$ where $word(t)=b^ia^{m-1}$.
State $(n, \{0\}, m-1)$, is reachable from $(0,\emptyset,0)$ by reading tree $a(t_1,t_2)$ where $word(t_1)=ba^{m-2}$ and $word(t_2)=b^2a^{m-2}$.
State $(i, \{0\}, j)$, $0\leq i\leq n$, $0\leq j\leq m-2$ is reachable from $(i, \{0\}, m-1)$ by reading a sequence of unary symbol $a^{1+j}$.
State $(i, \{0\}, m)$, $0\leq i\leq n-1$ is reachable from $(0,\emptyset,0)$ by reading tree $t$ where $word(t)=b^{i}a^{m-1}d$.
From $(0,\emptyset,0)$ by reading subtree $b(b(a),b(b(a)))$, state $(n,\emptyset,0)$ is reached. State $(n, \{0\}, m)$ is reached from $(n,\emptyset,0)$ by reading a sequence of unary symbols $a^{m-1}d$.
That is all the states $(i, \{0\}, j)$, $0\leq i\leq n$, $0\leq j\leq m$ are reachable.
Then state $(i, \{k\}, j)$, $0\leq i\leq n-1$, $0\leq j\leq m-1$, $1\leq k\leq n-1$ is reachable from $(\overline{i-1},\{k-1\},j)$ by reading a sequence of unary symbols $ba^j$. For any integer $x$, $$ \overline{x} = \left\{ \begin{array}{l} x \mbox{ if } x \geq 0
\\ n+x \mbox{ if } x< 0 \end{array} \right. $$ State $(n, \{k\}, j)$, $0\leq j\leq m-1$, $1\leq k\leq n-1$ is reachable from $(n,\{k-1\},j)$ by reading a sequence of unary symbols $ba^j$. State $(i, \{k\}, m)$, $0\leq i\leq n-1$, $1\leq k\leq n-1$ is reachable from $(\overline{i-1},\{k-1\},m)$ by reading a unary symbol $b$. State $(n, \{k\}, m)$, $1\leq k\leq n-1$ is reachable from $(n,\{k-1\},m)$ by reading a unary symbol $b$.
That is all the states $(i, \{k\}, j)$, $0\leq i\leq n$, $0\leq j\leq m$, $0\leq k\leq n-1$ are reachable.
Now assume that for $|S|\leq z$, all the states $(i, S, j)$, $0\leq i\leq n$, $0\leq j\leq m$, $S\subseteq \{0,\ldots,n-1\}$ are reachable. And this is the inductive assumption.
We will show that any state $(x, S', y)$, $0\leq x\leq n$, $0\leq y\leq m$, $|S'|=z+1$ is reachable.
First consider the case where $y\neq m-1$. Let $s_1 > s_2 > \ldots > s_z > s_{z+1}$ be the elements in $S'$. Let $P= \{s_1-s_{z+1} , s_2-s_{z+1} , \ldots , s_z-s_{z+1}\}$.
When $0\leq x\leq n-1$, according to the inductive assumption, state $(\overline{x-s_{z+1}}, P , 0)$, is reachable. Then state $(\overline{x-s_{z+1}}, P\cup \{0\}, m-1)$ is reachable from $(\overline{x-s_{z+1}}, P, 0)$ by reading a sequence of unary symbols $a^{m-1}$. State $(x, S', y)$, $0\leq y\leq m-2$ is reachable from $(\overline{x-s_{z+1}}, P\cup \{0\}, m-1)$ by reading a sequence of unary symbols $b^{s_{z+1}}a^{y}$. State $(x, S', m)$ is reachable from $(\overline{x-s_{z+1}}, P\cup \{0\}, m-1)$ by reading a sequence of unary symbols $b^{s_{z+1}}d$.
When $x=n$, according to the inductive assumption, state $(n, P , 0)$, is reachable. Then state $(n, P\cup \{0\}, m-1)$ is reachable from $(n, P, 0)$ by reading a sequence of unary symbols $a^{m-1}$. $(n, S', y)$, $0\leq y\leq m-2$ is reachable from $(n, P\cup \{0\}, m-1)$ by reading a sequence of unary symbols $b^{s_{z+1}}a^{y}$. State $(n, S', m)$ is reachable from $(n, P\cup \{0\}, m-1)$ by reading a sequence of unary symbols $b^{s_{z+1}}d$.
Now consider the case when $y=m-1$. According to the definition of (\ref{state1}), $0\in S'$. According to the inductive assumption, state $(x, S'-\{0\}, m-2)$ is reachable. Then state $(x, S', m-1)$ is reachable by reading a unary symbol $a$.
Since $(x, S', y)$ is an arbitrary state with $|S'|=z+1$, we have proved that all the states $(x, S', y)$, $0\leq x\leq n$, $0\leq y\leq m$, $|S'|=z+1$ is reachable.
Thus, all the states in (\ref{state1}) are reachable. \end{proof}\endpf
\begin{lemma} All states of $C$ are pairwise inequivalent. \footnote{Proof omitted due to length restriction.} \label{inequi} \end{lemma}
According to the upper bound in Lemma~\ref{cons} and Lemmas~\ref{reach} and~\ref{inequi}, we have proved the following theorem.
\begin{theorem}\label{ti} For arbitrary SDTAs $A_1$ and $A_2$, where $A_i=(Q_i,\Sigma,\delta_i,F_i)$, $i=1,2$, any SDTA $B=(Q,\Sigma,\delta,F)$ recognizing $L(A_2)\cdot L(A_1)$ satisfies
$|Q|\leq (|Q_1| + 1)\times (2^{|Q_1|} \times
(|Q_2|+1)-2^{|Q_1|-1})-1$.
For any integers $m,n\geq 1$, there exists tree languages $T_A$ and $T_B$, such that $T_A$ and $T_B$ can be recognized by SDTAs having $m$ and $n$ vertical states, respectively,
and any SDTA recognizing $T_A \cdot T_B$ needs at least $(n+1) ( (m+1)2^n-2^{n-1} )-1$ vertical states. \end{theorem}
We do not have a matching lower bound for the number of horizontal states given by Lemma~\ref{cons}. With regards to the number of vertical states, both the upper bound of Lemma~\ref{cons} and the lower bound of Theorem~\ref{ti} can be immediately modified for WDTAs. (The proof holds almost word for word.) In the case of WDTAs, getting a good lower bound for the number of horizontal states would likely be very hard.
\section{Conclusion}
We have studied the operational state complexity of two variants of deterministic unranked tree automata. For union and intersection, tight upper bounds on the number of vertical states were established for both strongly and weakly deterministic automata. An almost tight upper bound on the number of horizontal states was obtained in the case of strongly deterministic unranked tree automata. For weakly deterministic automata,
lower bounds on the numbers of horizontal states are hard to establish because there can be trade-offs between the numbers of vertical and horizontal states
. This is indicated also by the fact that minimization of weakly deterministic unranked tree automata is intractable and the minimal automaton need not be unique~\cite{mn}.
As ordinary strings can be viewed as unary trees, it is easy to predict that the state complexity of a given operation for tree automata should be greater or equal to the state complexity of the corresponding operation on string languages.
As our main result, we showed that for deterministic unranked tree automata, the
state complexity of concatenation of an $m$ state and an $n$ state automaton is at most $(n+1) ( (m+1)2^n-2^{n-1} )-1$ and that this bound can be reached in the worst case. The bound is of a different order than the known state complexity $m2^n-2^{n-1}$ of concatenation of regular string languages.
\end{document} | arXiv |
\begin{definition}[Definition:Bounded Ordered Set/Real Numbers/Definition 2]
Let $\R$ be the set of real numbers.
Let $T \subseteq \R$ be a subset of $\R$ such that:
:$\exists K \in \R: \forall x \in T: \size x \le K$
where $\size x$ denotes the absolute value of $x$.
Then $T$ is '''bounded in $\R$'''.
\end{definition} | ProofWiki |
\begin{document}
\title{An analogue of a van der Waerden's theorem\\ and its application to two-distance preserving mappings} \author{Victor Alexandrov} \date{ } \maketitle \begin{abstract} The van der Waerden's theorem reads that an equilateral pentagon in Euclidean 3-space $\Bbb E^3$ with all diagonals of the same length is necessarily planar and its vertex set coincides with the vertex set of some convex regular pentagon. We prove the following many-dimensional analogue of this theorem: for $n\geqslant 2$, every $n$-dimensional cross-polytope in $\Bbb E^{2n-2}$ with all diagonals of the same length and all edges of the same length necessarily lies in $\Bbb E^n$ and hence is a convex regular cross-polytope. We also apply our theorem to the study of two-distance preserving mappings of Euclidean spaces. \par \noindent\textit{Mathematics Subject Classification (2010)}: 52B11; 52B70; 52C25; 51K05. \par \noindent\textit{Key words}: Euclidean space, pentagon, cross-polytope, Cayley-Menger determinant, Beckman-Quarles theorem \end{abstract}
\textbf{1. van der Waerden's theorem and its many-dimensional analogue.} In 1970 B.L. van der Waerden has shown that an equilateral and isogonal pentagon in Euclidean 3-space $\Bbb E^3$ is necessarily planar and its vertex set coincides with the vertex set of some convex regular pentagon. For more details about this theorem we refer to \cite{Wa70} and \cite{Bo73} and references given there. More recent results related to this theorem may be found in \cite{BK11}, \cite{FS12}, and \cite{OH13}.
The van der Waerden's theorem may be reformulated as follows: \textit{an equilateral pentagon in Euclidean 3-space $\Bbb E^3$ with all diagonals of the same length is necessarily planar and its vertex set coincides with the vertex set of some convex regular pentagon.} The aim of this paper is to find a many-dimensional analogue of this statement by replacing a pentagon by a polyhedron which, under some conditions on the lengths of its edges and diagonals, is `surprisingly flat.' The latter means that we start with a polyhedron located in $\Bbb E^N$ and prove that it necessarily lies in $\Bbb E^n$ for some $n<N$.
Let $n\geqslant 1$ be an integer and $\{ {\bf e}_1, {\bf e}_2, \dots, {\bf e}_n \}$ be an orthonormal basis in Euclidean $n$-space $\Bbb E^n$. Denote by $V_n$ the set of the end-points of the vectors $\pm{\bf e}_1, \pm{\bf e}_2, \dots, \pm{\bf e}_n $. The convex hull of $V_n$ is called the {\it standard $n$-dimensional cross-polytope} in $\Bbb E^n$ and is denoted by $S_n$. Obviously, $V_n$ is the set of vertices of $S_n$.
Let $N$ and $n$ be positive integers. In this paper, every injective mapping $f:V_n\to\Bbb E^N$ is called an {\it $n$-dimensional cross-polytope} in $\Bbb E^N$. We prefer this definiton for the following two reasons:
(i) using $f$, the reader may easily reconstruct an abstract simplicial complex $C$ which is combinatorially equivalent to $S_n$ and whose vertex set is $f(V_n)$ (it suffice to put by definiton that a simplex $\Delta$ (of arbitrary dimension) with vertices $w_1,w_2,\dots,w_k\in f(V_n)$ belongs to $C$ if and only if the simplex with vertices $f^{-1}(w_1),f^{-1}(w_2),\dots,f^{-1}(w_k)$ is a face of $S_n$) and
(ii) from our definition the reader may easily see that the realization of the abstract simplicial complex $C$ described in (i) may be degenerate (e.g., the vertices $w_1,w_2,\dots,w_k\in f(V_n)$ may lay on a single line in $\Bbb E^N$).
For every two points $u,v\in V_n\subset S_n$ there are only two possibilities: either $u$ and $v$ are joint together by an edge of $S_n$ or $u+v=0$. In the first case the straight-line segment with the points $f(u)$ and $f(v)$ in $\Bbb E^N$ is called an {\it edge} of the $n$-dimensional cross-polytope $f:V_n\to\Bbb E^N$, while in the second case this segment is called a {\it diagonal} of $f$.
The main result of this paper is the following theorem.
\textbf{Theorem 1.} \textit{Let $n\geqslant 2$ be an integer, let $a$, $b$ be positive numbers, and let $f:V_n\to \Bbb E^{2n-2}$ be an $n$-dimensional cross-polytope such that the length of every edge of $f$ is equal to $a$ and the length of every diagonal of $f$ is equal to $b$. Then $f$ is isometric to a homothetic copy of $S_n$, the standard $n$-dimensional cross-polytope in $\Bbb E^n$. In particular, $b=\sqrt{2}a$. }
\textbf{Proof:} As soon as we know distances between every two vertices of $f$, we may treat $f$ a simplex with $2n$ vertices $$ f({\bf e}_1), f(-{\bf e}_1), f({\bf e}_2), f(-{\bf e}_2), \dots, f({\bf e}_n), f(-{\bf e}_n).\eqno(1) $$ In general, a simplex with $2n$ vertices and prescribed edge lengths is located in $\Bbb E^{2n-1}$. According to assumptions of Theorem 1, $f$ is located in $\Bbb E^{2n-2}$ and, thus, its $(2n-1)$-dimensional volume is equal to 0.
Let's make use of the Cayley--Menger formula for the $k$-dimensional volume, $\mbox{vol}_k$, of a simplex with $k+1$ vertices in $\Bbb E^k$ (see, e.g., \cite[p. 98]{Bl53}): $$ (-1)^{k+1}2^k(k!)^2(\mbox{vol}_k)^2=
\left| \begin{array}{cccccc} 0 & 1 & 1 & 1 & \dots & 1 \\ 1 & 0 & d_{12}^2 & d_{13}^2 & \dots & d_{1,k+1}^2 \\ 1 & d_{21}^2 & 0 & d_{23}^2 & \dots & d_{2,k+1}^2 \\ 1 & d_{31}^2 & d_{32}^2 & 0 & \dots & d_{3,k+1}^2 \\ \dots & \dots & \dots & \dots & \dots & \dots \\ 1 & d_{k+1,1}^2 & d_{k+1,2}^2 & d_{k+1,3}^2 & \dots & 0 \end{array}
\right|. \eqno(2) $$ Here we assume that the vertices are labeled with numbers from 1 to $k+1$ and $d_{ij}$ is the Euclidean distance between $i$-th and $j$-th vetrices.
Enumerating the vertices of $f$ according to (1) and taking into account that $\mbox{vol}_{2n-1}=0$, we obtain from (2)
$$\left| \begin{array}{ccccc} 0 & \begin{array}{cc}
1 & 1
\end{array} & \begin{array}{cc}
1 & 1
\end{array} &
\dots & \begin{array}{cc}
1 & 1
\end{array} \\ \begin{array}{cc}
1 \\ 1
\end{array} & \fbox{$\begin{array}{cc}
0 & b^2 \\
b^2 & 0
\end{array}$} & \begin{array}{cc}
a^2 & a^2 \\
a^2 & a^2
\end{array} & \begin{array}{c}
\dots \\
\dots
\end{array} & \begin{array}{cc}
a^2 & a^2 \\
a^2 & a^2
\end{array} \\ \begin{array}{cc}
1 \\ 1
\end{array} & \begin{array}{cc}
a^2 & a^2 \\
a^2 & a^2
\end{array} & \fbox{$\begin{array}{cc}
0 & b^2 \\
b^2 & 0
\end{array}$} & \begin{array}{c}
\dots \\
\dots
\end{array} & \begin{array}{cc}
a^2 & a^2 \\
a^2 & a^2
\end{array} \\ \cdots & \cdots\cdots & \cdots\cdots & \ddots & \cdots\cdots \\ \begin{array}{cc}
1 \\ 1
\end{array} & \begin{array}{cc}
a^2 & a^2 \\
a^2 & a^2
\end{array} & \begin{array}{cc}
a^2 & a^2 \\
a^2 & a^2
\end{array} & \begin{array}{c}
\dots \\
\dots
\end{array} & \fbox{$\begin{array}{cc}
0 & b^2 \\
b^2 & 0
\end{array}$} \end{array}
\right|=0. \eqno(3) $$ Note that the matrix in (3) contains $n$ blocks of the form $$ \left[ \begin{array}{cc} 0 & b^2 \\ b^2 & 0 \end{array} \right]. $$ In (3), these blocks are boxed for better visibility.
Denote the determinant in (3) by $D_n$. We are going to compute $D_n$ and, thus, replace (3) by an explicit relation involving $a$ and $b$.
Multiply the first row of $D_n$ by $(-a^2)$ and add the result to every other row. This yields $$
D_n=\left| \begin{array}{ccccc} 0 & \begin{array}{cc}
1\hphantom{b^2} & \hphantom{b^2}1
\end{array} & \begin{array}{cc}
1\hphantom{b^2} & \hphantom{b^2}1
\end{array} &
\dots & \begin{array}{cc}
1\hphantom{b^2} & \hphantom{b^2}1
\end{array} \\ \begin{array}{cc}
1 \\ 1
\end{array} & \fbox{$\begin{array}{cc}
-a^2 & b^2-a^2 \\
b^2-a^2 & -a^2
\end{array}$} & \begin{array}{cc}
0\hphantom{b^2} & \hphantom{b^2}0 \\
0\hphantom{b^2} & \hphantom{b^2}0
\end{array} & \begin{array}{c}
\dots \\
\dots
\end{array} & \begin{array}{cc}
0\hphantom{b^2} & \hphantom{b^2}0 \\
0\hphantom{b^2} & \hphantom{b^2}0
\end{array} \\ \begin{array}{cc}
1 \\ 1
\end{array} & \begin{array}{cc}
0\hphantom{b^2} & \hphantom{b^2}0 \\
0\hphantom{b^2} & \hphantom{b^2}0
\end{array} & \fbox{$\begin{array}{cc}
-a^2 & b^2-a^2 \\
b^2-a^2 & -a^2
\end{array}$} & \begin{array}{c}
\dots \\
\dots
\end{array} & \begin{array}{cc}
0\hphantom{b^2} & \hphantom{b^2}0 \\
0\hphantom{b^2} & \hphantom{b^2}0
\end{array} \\ \cdots & \cdots\cdots & \cdots\cdots & \ddots & \cdots\cdots \\ \begin{array}{cc}
1 \\ 1
\end{array} & \begin{array}{cc}
0\hphantom{b^2} & \hphantom{b^2}0 \\
0\hphantom{b^2} & \hphantom{b^2}0
\end{array} & \begin{array}{cc}
0\hphantom{b^2} & \hphantom{b^2}0 \\
0\hphantom{b^2} & \hphantom{b^2}0
\end{array} & \begin{array}{c}
\dots \\
\dots
\end{array} & \fbox{$\begin{array}{cc}
-a^2 & b^2-a^2 \\
b^2-a^2 & -a^2
\end{array}$} \end{array}
\right|.\eqno(4) $$ In (4), subtract the second row from the third, the forth from the fifth, \dots, and the $(2n)$th from the $(2n+1)$th. We obtain $$
D_n=\left| \begin{array}{ccccc} 0 & \begin{array}{cc}
1\hphantom{b^2} & \hphantom{b^2}1
\end{array} & \begin{array}{cc}
1\hphantom{b^2} & \hphantom{b^2}1
\end{array} &
\dots & \begin{array}{cc}
1\hphantom{b^2} & \hphantom{b^2}1
\end{array} \\ \begin{array}{cc}
1 \\ 1
\end{array} & \fbox{$\begin{array}{cc}
-a^2 & b^2-a^2 \\
\hphantom{1} b^2 & -b^2
\end{array}$} & \begin{array}{cc}
0\hphantom{b^2} & \hphantom{b^2}0 \\
0\hphantom{b^2} & \hphantom{b^2}0
\end{array} & \begin{array}{c}
\dots \\
\dots
\end{array} & \begin{array}{cc}
0\hphantom{b^2} & \hphantom{b^2}0 \\
0\hphantom{b^2} & \hphantom{b^2}0
\end{array} \\ \begin{array}{cc}
1 \\ 1
\end{array} & \begin{array}{cc}
0\hphantom{b^2} & \hphantom{b^2}0 \\
0\hphantom{b^2} & \hphantom{b^2}0
\end{array} & \fbox{$\begin{array}{cc}
-a^2 & b^2-a^2 \\
\hphantom{1}b^2 & -b^2
\end{array}$} & \begin{array}{c}
\dots \\
\dots
\end{array} & \begin{array}{cc}
0\hphantom{b^2} & \hphantom{b^2}0 \\
0\hphantom{b^2} & \hphantom{b^2}0
\end{array} \\ \cdots & \cdots\cdots & \cdots\cdots & \ddots & \cdots\cdots \\ \begin{array}{cc}
1 \\ 1
\end{array} & \begin{array}{cc}
0\hphantom{b^2} & \hphantom{b^2}0 \\
0\hphantom{b^2} & \hphantom{b^2}0
\end{array} & \begin{array}{cc}
0\hphantom{b^2} & \hphantom{b^2}0 \\
0\hphantom{b^2} & \hphantom{b^2}0
\end{array} & \begin{array}{c}
\dots \\
\dots
\end{array} & \fbox{$\begin{array}{cc}
-a^2 & b^2-a^2 \\
\hphantom{1}b^2 & -b^2
\end{array}$} \end{array}
\right|.\eqno(5) $$ In (5), add the second column to the third, the forth to the fifth, \dots, and the $(2n)$th to the $(2n+1)$th. We get $$
D_n=\left| \begin{array}{ccccc} 0 & \begin{array}{cc}
1\hphantom{b^2} & \hphantom{b^2}2
\end{array} & \begin{array}{cc}
1\hphantom{b^2} & \hphantom{b^2}2
\end{array} &
\dots & \begin{array}{cc}
1\hphantom{b^2} & \hphantom{b^2}2
\end{array} \\ \begin{array}{cc}
1 \\ 0
\end{array} & \fbox{$\begin{array}{cc}
-a^2 & b^2-2a^2 \\
\hphantom{1} b^2 & 0
\end{array}$} & \begin{array}{cc}
0\hphantom{b^2} & \hphantom{b^2}0 \\
0\hphantom{b^2} & \hphantom{b^2}0
\end{array} & \begin{array}{c}
\dots \\
\dots
\end{array} & \begin{array}{cc}
0\hphantom{b^2} & \hphantom{b^2}0 \\
0\hphantom{b^2} & \hphantom{b^2}0
\end{array} \\ \begin{array}{cc}
1 \\ 0
\end{array} & \begin{array}{cc}
0\hphantom{b^2} & \hphantom{b^2}0 \\
0\hphantom{b^2} & \hphantom{b^2}0
\end{array} & \fbox{$\begin{array}{cc}
-a^2 & b^2-2a^2 \\
\hphantom{1}b^2 & 0
\end{array}$} & \begin{array}{c}
\dots \\
\dots
\end{array} & \begin{array}{cc}
0\hphantom{b^2} & \hphantom{b^2}0 \\
0\hphantom{b^2} & \hphantom{b^2}0
\end{array} \\ \cdots & \cdots\cdots & \cdots\cdots & \ddots & \cdots\cdots \\ \begin{array}{cc}
1 \\ 0
\end{array} & \begin{array}{cc}
0\hphantom{b^2} & \hphantom{b^2}0 \\
0\hphantom{b^2} & \hphantom{b^2}0
\end{array} & \begin{array}{cc}
0\hphantom{b^2} & \hphantom{b^2}0 \\
0\hphantom{b^2} & \hphantom{b^2}0
\end{array} & \begin{array}{c}
\dots \\
\dots
\end{array} & \fbox{$\begin{array}{cc}
-a^2 & b^2-2a^2 \\
\hphantom{1}b^2 & 0
\end{array}$} \end{array}
\right|.\eqno(6) $$ Expand the determinant in (6) along the third row, the fifth row, \dots, and the $(2n+1)$th row. The result is $$
D_n=(-1)^n b^{2n}\left| \begin{array}{ccccc} 0 & 2 & 2 & \dots & 2 \\ 1 & \fbox{$b^2-2a^2$} & 0 & \dots & 0 \\ 1 & 0 & \fbox{$b^2-2a^2$} & \dots & 0 \\ \cdots & \cdots & \cdots & \ddots & \cdots \\ 1 & 0 & 0 & \dots & \fbox{$b^2-2a^2$} \end{array}
\right|. \eqno(7) $$ Note that the size of the matrix in (7) is reduced to $n+1$.
If $b^2-2a^2=0$, the determinant in (7) is equal to zero (e.g., because the second row is equal to the third). If $b^2-2a^2\neq 0$, rewrite (7) in the form $$
D_n=2\frac{(-1)^n b^{2n}}{b^2-2a^2}\left| \begin{array}{ccccc} 0 & 1 & 1 & \dots & 1 \\ b^2-2a^2 & \fbox{$b^2-2a^2$} & 0 & \dots & 0 \\ b^2-2a^2 & 0 & \fbox{$b^2-2a^2$} & \dots & 0 \\ \cdots & \cdots & \cdots & \ddots & \cdots \\ b^2-2a^2 & 0 & 0 & \dots & \fbox{$b^2-2a^2$} \end{array}
\right|.\eqno(8) $$ In (8), subtract the second, the third, \dots, and the $(n+1)$st column from the first column. This yields $$
D_n=2\frac{(-1)^n b^{2n}}{b^2-2a^2}\left| \begin{array}{ccccc} -n & 1 & 1 & \dots & 1 \\ 0 & \fbox{$b^2-2a^2$} & 0 & \dots & 0 \\ 0 & 0 & \fbox{$b^2-2a^2$} & \dots & 0 \\ \cdots & \cdots & \cdots & \ddots & \cdots \\ 0 & 0 & 0 & \dots & \fbox{$b^2-2a^2$} \end{array}
\right|= 2nb^{2n}(2a^2-b^2)^{n-1}. $$ Hence, (3) is equivalent to $b^2-2a^2=0$. This means that the cross-polytope $f$ is congruent to a homothetic copy of the standard $n$-dimensional cross-polytope $S_n$ in $\Bbb E^n$. Q.E.D.
\textbf{2. Two-distance preserving mappings.} A mapping $g:\Bbb E^n \to \Bbb E^m$ is said to be \textit{unit
distance preserving} if, for all $x,y\in \Bbb E^n$, the equality $|x-y|=1$ implies the equality $|g(x)-g(y)|=1$. Here by $|x|$ we denote the Euclidean norm of a vector $x\in\Bbb E^n$.
In 1953, F.S. Beckman and D.A. Quarles \cite{BQ53} proved that, \textit{for $n\geqslant 2$, every unit distance preserving mapping $g:\Bbb E^n \to \Bbb E^n$ is an isometry of $\Bbb E^n$} (i.e., $g$ preserves all distances). Since that time, the problem `does a unit distance preserving mapping is an isometry' was studied for spaces of various types (hyperbolic \cite{Ku80}, Banach \cite{Ra07}, $\Bbb Q^n$ \cite{Za05}, just to name a few).
In 1985, B.V. Dekster \cite{De85} found a mapping $g:\Bbb E^2 \to \Bbb E^6$, which is unit distance preserving but is not an isometry. That example motivated geometers to look for other conditions that make the statement `every unit distance preserving mapping $g:\Bbb E^n \to \Bbb E^m$ is an isometry' correct even if $m\neq n$. One of the possible sets of such conditions uses the notions of cable and strut defined as follows.
Given a mapping $g:\Bbb E^n \to \Bbb E^m$, a positive real number $c$ is called a \textit{cable} of $g$ if, for all
$x,y\in\Bbb E^n$, the equality $|x-y|=c$ implies
$|g(x)-g(y)|\leqslant c$ and a positive real number $s$ is called a \textit{strut} of $g$ if, for all
$x,y\in\Bbb E^n$, the equality $|x-y|=s$ implies
$|g(x)-g(y)|\geqslant s$.
In 1999, K. Bezdek and R. Connelly \cite{BC99} proved that \textit{if $n\geqslant 2$, $c$ is a cable of a mapping $g:\Bbb E^n \to \Bbb E^m$, $s$ is a strut of $g$ and $c/s<(\sqrt{5}-1)/2$, then $g$ is an isometry.}
As a corollary of Theorem 1, we prove the following theorem.
\textbf{Theorem 2.} \textit{Let $n\geqslant 6$ and $0\leqslant m\leqslant 2n-2$ be integers, let $A$ and $B$ be positive real numbers, and let
$g:\Bbb E^n \to \Bbb E^m$ be a mapping such that, for all $x,y\in\Bbb E^n$, the equality $|x-y|=A$ implies $|g(x)-g(y)|=A$ and the equality $|x-y|=\sqrt{2}A$ implies $|g(x)-g(y)|=B$. Then $g$ is an isometry.}
\textbf{Proof:} First, observe that \textit{$B$ is necessarily equal to $\sqrt{2}A$}.
In fact, given $x,y\in\Bbb E^n$ such that
$|x-y|=\sqrt{2}A$, find an $n$-dimensional cross-polytope $P$ in $\Bbb E^n$ with the following properties:
(i) $P$ is congruent to a homothetic copy of the standard $n$-dimensional cross-polytope $S_n$ in $\Bbb E^n$ with the scale factor $A/\sqrt{2}$;
(ii) $x$ and $y$ belong to the vertex set of $P$;
(iii) the straight line segment with the endpoints $x$ and $y$ is a diagonal of $P$.
Since $g:\Bbb E^n \to \Bbb E^m$ and $m\leqslant 2n-2$, Theorem 1 yields that the image of $P$ under the mapping $g$ is congruent to $P$. In particular, this means that
$|g(x)-g(y)|=\sqrt{2}A$ and, thus, $B=\sqrt{2}A$.
Now, let's prove that \textit{the real number $c=2A/\sqrt{n}$ is a cable of the mapping $g$,} i.e., let's prove that if $v_1,v_2\in\Bbb E^n$ are such that
$|v_1-v_2|=c=2A/\sqrt{n}$
then $|g(v_1)-g(v_2)|\leqslant c$.
Let $L$ be the $(n-1)$-dimensional plane in $\Bbb E^n$ that passes thought the point $(v_1+v_2)/2$ and is orthogonal to the vector $v_1-v_2$. Let $v_3, v_4, \dots , v_{n+2}$ be the vertices of an $(n-1)$-dimensional regular simplex in $L$ with edge lengths $\sqrt{2}A$ and circumcenter at the point $(v_1+v_2)/2$. The latter means that the point $(v_1+v_2)/2$ is the center of an $(n-2)$-dimensional sphere which passes through all the vertices $v_3, v_4, \dots , v_{n+2}$. Since the radius of this $(n-2)$-sphere is equal to $R=A\sqrt{1-1/n}$ (see, e.g., \cite[pp. 294--295]{Co48}), Pythagora's Theorem gives
$|v_i-v_j|=A$ for all $i=1,2$ and $j=3,4,\dots, n+2$.
It follows from conditions of Theorem 2 and the relation $B=\sqrt{2}A$ that, for every $i=1,2$, the $n$-simplex $\Delta_i$ with vertices $v_i, v_3, v_4, \dots , v_{n+2}$ is congruent to the $n$-simplex with vertices $g(v_i), g(v_3), g(v_4), \dots , g(v_{n+2})$. For short, denote the latter simplex by $g(\Delta_i)$. Denote by $\lambda$ the $(n-1)$-dimensional plane containing the points $g(v_3), g(v_4), \dots , g(v_{n+2})$.
Since $g(\Delta_1)$ is congruent to $\Delta_1$, it follows that $m\geqslant n$.
If $m=n$, the non-degenerate $n$-simplices $g(\Delta_1)$ and $g(\Delta_2)$ lie either in the same half-space of $\Bbb E^m$ determined by $\lambda$
(in this case $|g(v_1)-g(v_2)|=0<c$) or in the different half-spaces of $\Bbb E^m$ determined by $\lambda$ (in this case
$|g(v_1)-g(v_2)|= |v_1-v_2|=c$). In both cases,
$|g(v_1)-g(v_2)|\leqslant c$.
If $m>n$, the $n$-simplices $g(\Delta_1)$ and $g(\Delta_2)$ may be obtained from $\Delta_1$ and $\Delta_2$ in two steps: first, we apply to $\Delta_1$ and $\Delta_2$ such an isometry $h:\Bbb E^n\to\Bbb E^m$ that $h(v_j)=g(v_j)$ for all $j=1,3,4,\dots, n+2$ and then rotate the simplex $h(\Delta_2)$ around $\lambda$ in such a way that $h(v_2)$ coincides with $g(v_2)$. From this description we conclude that
$|g(v_1)-g(v_2)|\leqslant |h(v_1)-h(v_2)|=|v_1-v_2|=c$. Hence, $c=2A/\sqrt{n}$ is a cable.
Obviously, we may consider $s=\sqrt{2}A$ as a strut of $g$.
Since $n\geqslant 6$, $c/s=\sqrt{2/n}<(\sqrt{5}-1)/2$. Thus, according to the above-cited theorem by K. Bezdek and R. Connelly, $g$ is an isometry. Q.E.D.
\noindent{Victor Alexandrov}
\noindent\textit{Sobolev Institute of Mathematics}
\noindent\textit{Koptyug ave., 4}
\noindent\textit{Novosibirsk, 630090, Russia}
and
\noindent\textit{Department of Physics}
\noindent\textit{Novosibirsk State University}
\noindent\textit{Pirogov str., 2}
\noindent\textit{Novosibirsk, 630090, Russia}
\noindent\textit{e-mail: [email protected]}
\noindent{Submitted: April 17, 2015}
\end{document} | arXiv |
\begin{document}
\subjclass[2010]{Primary: 46B80, 46B20. Secondary: 20F65}
\keywords{Large scale, coarse and uniform geometry, Banach spaces, topological groups, cocycles} \thanks{The author was partially supported by a Simons Foundation Fellowship (Grant \#229959) and also recognises support from the NSF (DMS 1201295 \& DMS 1464974)}
\begin{abstract} We study uniform and coarse embeddings between Banach spaces and topological groups. A particular focus is put on equivariant embeddings, i.e., continuous cocycles associated to continuous affine isometric actions of topological groups on separable Banach spaces with varying geometry. \end{abstract}
\title{Equivariant geometry of Banach spaces and topological groups}
\tableofcontents
\section{Introduction} The present paper is a contribution to the study of large scale geometry of Banach spaces and topological groups and, in particular, to questions of embeddability between these objects. In a sense, our aim is somewhat wider than usual as we will be dealing with general Polish groups as opposed to only locally compact groups. This is achieved by using the recently developed framework of \cite{rosendal-coarse}, which allows us to treat Banach spaces and topological groups under one heading. Still our focus will be restricted as we are mainly interested in equivariant maps, that is, cocycles associated to affine isometric actions on Banach spaces.
To state the results of the paper, let us begin by fixing the basic terminology. Given a map $\sigma\colon (X,d)\rightarrow (Y,\partial)$ between metric spaces, we define the {\em compression modulus} by $$ \kappa_\sigma(t)=\inf\big(\partial(\sigma(a),\sigma(b))\mid d(a,b)\geqslant t\big) $$ and the {\em expansion modulus} by $$ \theta_\sigma(t)=\sup\big(\partial(\sigma(a),\sigma(b))\mid d(a,b)\leqslant t\big). $$ Thus $\sigma$ is {\em uniformly continuous} if $\lim_{t\rightarrow 0_+}\theta_\sigma(t)=0$ (in which case $\theta_\sigma$ becomes the modulus of uniform continuity) and a {\em uniform embedding} if, moreover, $\kappa_\sigma(t)>0$ for all $t>0$. Furthermore, $\sigma$ is {\em bornologous} if $\theta_\sigma(t)<\infty$ for all $t<\infty$ and {\em expanding} if $\lim_{t\rightarrow \infty}\kappa_\sigma(t)=\infty$. A bornologous expanding map is called a {\em coarse embedding}. We also define $\sigma$ to be {\em uncollapsed} if $\kappa_\sigma(t)>0$ for just some sufficiently large $t>0$.
As we will be studying uniform and coarse embeddability between Banach spaces and topological groups, we must extend the above concepts to the larger categories of uniform and coarse spaces and also show how every topological group is canonically equipped with both a uniform and a coarse structure.
Postponing for the moment this discussion, let consider the outcomes of our study. As pointed out by N. Kalton \cite{kalton2}, the concepts of uniform and coarse embeddability between Banach spaces seem very tightly related. Though, Kalton \cite{kalton3} eventually was able to give an example of two separable Banach spaces that are coarsely equivalent, but not uniformly homeomorphic, the following basic question of Kalton concerning embeddings remains open. \begin{quest} Does the following equivalence hold for all (separable) Banach spaces? $$ X \text{ is uniformly embeddable into }E\;\Longleftrightarrow\; X \text{ is coarsely embeddable into }E. $$ \end{quest}
Relying on entirely elementary techniques, our first result shows that in many settings we do have an implication from left to right. \begin{thm}\label{intro:ell p sum} Suppose $\sigma\colon X\rightarrow E$ is an uncollapsed uniformly continuous map between Banach spaces. Then, for any $1\leqslant p<\infty$, $X$ admits a simultaneously uniform and coarse embedding into $\ell^p(E)$. \end{thm} Since both uniform and coarse embeddings are uncollapsed, we see that if $X$ is uniformly embeddable into $E$, then $X$ is coarsely embeddable into $\ell^p(E)$. For the other direction, if $X$ admits a uniformly continuous coarse embedding into $E$, then $X$ is uniformly embeddable into $\ell^p(E)$. It is therefore natural to ask to which extent bornologous maps can be replaced by uniformly continuous maps. In particular, is every bornologous map $\sigma\colon X\rightarrow E$ between Banach spaces {\em close} to a uniformly continuous map, i.e., is there a uniformly continuous map $\tilde \sigma$ so that $\sup_{x\in X}\norm{\sigma(x)-\tilde\sigma(x)}<\infty$?
As it turns out, A. Naor \cite{naor-nets} was recently able to answer our question in the negative, namely, there are separable Banach spaces $X$ and $E$ and a bornologous map between them which is not close to any uniformly continuous map. Nevertheless, several weaker questions remain open.
Also, in case the passage from $E$ to $\ell^p(E)$ proves troublesome, we can get by with $E\oplus E$ under stronger assumptions. \begin{thm}\label{intro:E plus E} Suppose $\sigma\colon X\rightarrow B_E$ is an uncollapsed uniformly continuous map from a Banach space $X$ into the ball of a Banach space $E$. Then $X$ admits a uniformly continuous coarse embedding into $E\oplus E$. \end{thm} For example, if $X$ is a Banach space uniformly embeddable into its unit ball $B_X$, e.g., if $X=\ell^2$, then, whenever $X$ uniformly embeds into a Banach space $E$, it coarsely embeds into $E\oplus E$.
The main aim of the present paper however is to consider equivariant embeddings between topological groups and Banach spaces, i.e., continuous cocycles. So let $\pi\colon G\curvearrowright E$ be a strongly continuous isometric linear representation of a topological group $G$ on a Banach space $E$, i.e., each $\pi(g)$ is a linear isometry of $E$ and, for every $\xi\in E$, the map $g\in G\mapsto \pi(g)\xi$ is continuous. A continuous {\em cocycle} associated to $\pi$ is a continuous map $b\colon G\rightarrow E$ satisfying the cocycle equation $$ b(gf)=\pi(g)b(f)+b(g). $$ This corresponds to the requirement that $\alpha(g)\xi=\pi(g)\xi+b(g)$ defines a continuous action of $G$ by affine isometries on $E$. As $b$ is simply the orbit map $g\mapsto \alpha(g)0$, it follows that continuous cocycles are actually uniformly continuous and bornologous. We call a continuous cocycle $b\colon G\rightarrow E$ {\em coarsely proper} if it is a coarse embedding of $G$ into $E$. In this case, we also say that the associated affine isometric action $\alpha$ is {\em coarsely proper}.
For the next result, a topological group $G$ has the {\em Haagerup property} if it has a strongly continuous unitary representation with an associated coarsely proper cocycle. Though it is known that every locally compact second countable amenable group has the Haagerup property \cite{BCV}, this is very far from being true for general topological groups. Indeed, even for Polish groups, that is, separable completely metrisable topological groups, such as separable Banach spaces, this is a significant requirement.
Building on work of I. Aharoni, B. Maurey and B. S. Mityagin \cite{maurey}, we show the following equivalence. \begin{thm}\label{intro:haagerup equiv} The following conditions are equivalent for an amenable Polish group $G$, \begin{enumerate} \item $G$ coarsely embeds into a Hilbert space, \item $G$ has the Haagerup property. \end{enumerate} \end{thm}
Restricting to Banach spaces, we have a stronger result, where the equivalence of (1) and (2) is due to N. L. Randrianarivony \cite{randrianarivony2}. \begin{thm}\label{intro:haagerup banach} The following conditions are equivalent for a separable Banach space $X$, \begin{enumerate} \item $X$ coarsely embeds into a Hilbert space, \item $X$ uniformly embeds into a Hilbert space, \item $X$ admits an uncollapsed uniformly continuous map into a Hilbert space, \item $X$ has the Haagerup property. \end{enumerate} \end{thm}
Even from collapsed maps we may obtain information, provided that there is just some single distance not entirely collapsed. \begin{thm} Suppose $\sigma\colon X\rightarrow \ku H$ is a uniformly continuous map from a separable Banach space into Hilbert space so that, for some single $r>0$, $$ \inf_{\norm{x-y}=r}\norm{\sigma(x)-\sigma(y)}>0. $$ Then $B_X$ uniformly embeds into $B_\ku H$. \end{thm}
One of the motivations for studying cocycles, as opposed to general uniform or coarse embeddings, is that cocycles are also algebraic maps, i.e., reflect algebraic features of the acting group $G$. As such, they have a higher degree of regularity and permit us to carry geometric information from the phase space back to the acting group. For example, a continuous uncollapsed cocycle between Banach spaces is automatically a uniform embedding. Similarly, a cocycle $b\colon X\rightarrow E$ between Banach spaces associated to the trivial representation $\pi\equiv {\rm id}_E$ is simply a bounded linear operator, so a cocycle may be considered second best to a linear operator. We shall encounter and exploit many more instances of this added regularity throughout the paper.
Relaxing the geometric restrictions on the phase space, the next case to consider is that of super-reflexive spaces, i.e., spaces admitting a uniformly convex renorming. For this class, earlier work was done by V. Pestov \cite{pestov} and Naor--Y. Peres \cite{naor-peres} for discrete amenable groups $G$. For topological groups, several severe obstructions appear and it does not seem possible to get an exact analogue of Theorem \ref{intro:haagerup equiv}. Indeed, the very concept of amenability requires reexamination. We say that a Polish group $G$ is {\em F\o lner amenable} if there is either a continuous homomorphism $\phi\colon H\rightarrow G$ from a locally compact second countable amenable group $H$ with dense image or if $G$ admits a chain of compact subgroups with dense union. For example, a separable Banach space is F\o lner amenable.
\begin{thm}\label{intro:fin repr} Let $G$ be a F\o lner amenable Polish group admitting a uniformly continuous coarse embedding into a Banach space $E$. Then, for every $1\leqslant p<\infty$, $G$ admits a coarsely proper continuous affine isometric action on a Banach space $V$ that is finitely representable in $L^p(E)$. \end{thm}
We note that, if $E$ is super-reflexive, then so is every $V$ finitely representable in $L^2(E)$. Again, for Banach spaces, this leads to the following.
\begin{thm}\label{intro:pestov banach} Suppose $X$ is a separable Banach space admitting an uncollapsed uniformly continuous map into a super-reflexive space. Then $X$ admits a coarsely proper continuous cocycle with values in a super-reflexive space. \end{thm}
For our next results, we define a topological group to be {\em metrically stable} if it admits a compatible left-invariant stable metric. By results of \cite{ben yaacov, megrelishvili2, shtern}, a Polish group is metrically stable if and only if it is isomorphic to a subgroup if the linear isometry group of a separable reflexive Banach space under the strong operator topology. Also, by a result of Y. Raynaud \cite{raynaud}, a metrically stable Banach space contains a copy of $\ell^p$ for some $1\leqslant p<\infty$.
Using the construction underlying Theorem \ref{intro:fin repr}, we obtain information on spaces uniformly embeddable into balls of super-reflexive spaces. \begin{thm}\label{intro:embedding super-reflexive} Let $X$ be a Banach space admitting an uncollapsed uniformly continuous map into the unit ball $B_E$ of a super-reflexive Banach space $E$. Then $X$ is metrically stable and contains an isomorphic copy of $\ell^p$ for some $1\leqslant p<\infty$. \end{thm} As an application, this restricts the class of super-reflexive spaces uniformly embeddable into their own balls.
For a general non-amenable topological group $G$, we may also produce coarsely proper affine isometric actions on reflexive spaces starting directly from a stable metric or \'ecart. \begin{thm}\label{intro:stable metric refl} Suppose a topological group $G$ carries a continuous left-invariant coarsely proper stable \'ecart. Then $G$ admits a coarsely proper continuous affine isometric action on a reflexive Banach space. \end{thm}
Intended applications here are, for example, automorphisms groups of countable atomic models of countable stable first-order theories that, under the additional assumption of being locally (OB), satisfy the conditions of Theorem \ref{intro:stable metric refl}.
The final part of our investigations concern a fixed point property for affine isometric group actions. We identify a geometric incompatibility between a topological group $G$ and a Banach space $E$ strong enough to ensure that not only does $G$ have no coarsely proper affine isometric action on $E$, but every affine isometric action even has a fixed point.
The two main concepts here are solvent maps and geometric Gelfand pairs. First, a map $\phi\colon X\rightarrow Y$ between metric spaces is {\em solvent} if, for every $n$, there is an $R$ with $R\leqslant d(x,x')\leqslant R+n\Rightarrow d(\phi x, \phi x')\geqslant n$. Refining earlier results of Kalton, we show that every bornologous map from $c_0$ to a reflexive Banach space is insolvent. Also a coarsely proper continuous isometric action $G\curvearrowright X$ of a topological group $G$ on a metric space $X$ is said to be a {\em geometric Gelfand pair} if, for some $K$ and all $x,y,z,u\in X$ with $d(x,y)\leqslant d(z,u)$, there is $g\in G$ so that $d(g(x),z)\leqslant K$ and $d(z,g(y))+d(g(y),u)\leqslant d(z,u)+K$. This second condition is typically verified when $X$ is sufficiently geodesic and the action of $G$ is almost doubly transitive. For example, if ${\rm Aff}(X)$ denotes the group of affine isometries of a Banach space $X$, then ${\rm Aff}(X)\curvearrowright X$ is a geometric Gelfand pair when $X=L^p([0,1])$, $1\leqslant p<\infty$, and when $X$ is the Gurarii space. Similarly, if $X$ is the integral, $\Z\U$, or rational, $\Q\U$, Urysohn metric space, then ${\rm Isom}(X)\curvearrowright X$ is a geometric Gelfand pair.
\begin{thm}\label{intro:bdd orbits} Suppose $G\curvearrowright X$ is a geometric Gelfand pair and $Y$ is a metric space so that every bornologous map $X\rightarrow Y$ is insolvent. Then every continuous isometric action $G\curvearrowright Y$ has bounded orbits. \end{thm}
Combining Theorem \ref{intro:bdd orbits} with the observations above and the fixed point theorems of \cite{ryll} and \cite{monod}, we obtain the following corollary. \begin{cor}\label{intro:fixed point} Every continuous affine isometric action of ${\rm Isom}(\Q\U)$ on a reflexive Banach space or on $L^1([0,1])$ has a fixed point. \end{cor}
\begin{center}{ Acknowledgements}\end{center} I would like to thank a number of people for helpful conversations and other aid during the preparation of the paper. They include F. Baudier, B. Braga, J. Galindo, G. Godefroy, W. B. Johnson, N. Monod, A. Naor, Th. Schlumprecht and A. Thom.
\section{Uniform and coarse structures on topological groups}\label{coarse defi} As is well-known, the non-linear geometry of Banach spaces and large scale geometry of finitely generated groups share many common concepts and tools, while a priori dealing with distinct subject matters. However, as shown in \cite{rosendal-coarse}, both theories may be viewed as instances of the same overarching framework, namely, the coarse geometry of topological groups. Thus, many results or problems admitting analogous but separate treatments for Banach spaces and groups can in fact be entered into this unified framework.
Recall that, if $G$ is a topological group, the {\em left-uniform structure} $\ku U_L$ on $G$ is the uniform structure generated by the family of entourages $$ E_V=\{(x,y)\in G\times G\mid x^{-1} y\in V\}, $$ where $V$ varies over identity neighbourhoods in $G$. It is a fact due to A. Weil that the left-uniform structure on $G$ is given as the union $\ku U_L=\bigcup_d\ku U_d$ of the uniformities $\ku U_d$ induced by continuous left-invariant \'ecarts (aka. pseudometrics) $d$ on $G$. Thus, $E\in \ku U_L$ if it contains some $$ E_\alpha=\{(x,y)\in X\times X\del d(x,y)<\alpha\}, $$ with $\alpha>0$ and $d$ a continuous left-invariant \'ecart on $G$. Apart from the weak-uniformity on a Banach space, this is the only uniformity on $G$ that we will consider and, in the case of the additive group $(X,+)$ of a Banach space, is simply the uniformity given by the norm.
More recently, in \cite{rosendal-coarse} we have developed a theory of coarse geometry of topological groups the basic concepts of which are analogous to those of the left-uniformity. For this, we first need J. Roe's concept of a coarse space \cite{roe}. A {\em coarse structure} on a set $X$ is a family $\ku E$ of subsets $E\subseteq X\times X$ called {\em coarse entourages} satisfying \begin{itemize} \item[(i)] the diagonal $\Delta$ belongs to $\ku E$, \item[(ii)] $F\subseteq E\in \ku E\Rightarrow F\in \ku E$, \item[(iii)] $E\in \ku E\Rightarrow E^{-1}=\{(y,x)\del (x,y)\in E\}\in \ku E$, \item[(iv)] $E,F\in \ku E\Rightarrow E\cup F\in \ku E$, \item[(v)] $E,F\in \ku E\Rightarrow E\circ F=\{(x,y)\del \e z\; (x,z)\in E\;\&\; (z,y)\in F\}\in \ku E$. \end{itemize} Just as the prime example of a uniform space is a metric space, the motivating example of a coarse space $(X,\ku E)$ is that induced from a (pseudo) metric space $(X,d)$. Indeed, in this case, we let $E\in \ku E_d$ if $E$ is contained in some $$ E_\alpha=\{(x,y)\in X\times X\del d(x,y)<\alpha\}, $$ with $\alpha<\infty$.
Now, if $G$ is a topological group, we define the left-coarse structure $\ku E_L$ to be the coarse structure given by $$ \ku E_L=\bigcap_d\ku E_d, $$ where the intersection is taken over the family of coarse structures $\ku E_d$ given by continuous left-invariant \'ecarts on $G$. Again, this is the only coarse structure on a topological group we will consider and, in the case of a finitely generated or locally compact, compactly generated group, coincides with the coarse structure given by the (left-invariant) word metric. Similarly, for a Banach space, the coarse structure $\ku E_L$ is simply that given by the norm.
A subset $A$ of a topological group has {\em property (OB) relative to $G$} if $A$ has finite diameter with respect to every continuous left-invariant \'ecart on $G$. Also, $G$ has {\em property (OB)} if is has property (OB) relative to itself. A continuous left-invariant \'ecart $d$ on a topological group $G$ is said to be {\em coarsely proper} if it induces the coarse structure, i.e., if $\ku E_L=\ku E_d$. In the class of Polish groups, having a coarsely proper \'ecart is equivalent to the group being {\em locally (OB)}, that is, having a relatively (OB) identity neighbourhood. Finally, $G$ is {\em (OB) generated} if it is generated by a relatively (OB) set.
Recall that a map $\sigma\colon (X,\ku U)\rightarrow (Y,\ku V)$ between uniform spaces is {\em uniformly continuous} if, for every $F\in \ku V$, there is $E\in \ku U$ so that $(a,b)\in E\Rightarrow (\sigma(a),\sigma(b))\in F$. Also, $\sigma$ is a {\em uniform embedding} is, moreover, for every $E\in \ku U$, there is $F\in \ku V$ so that $(a,b)\notin E\Rightarrow (\sigma(a),\sigma(b))\notin F$. Similarly, a map $\sigma\colon (X,\ku E)\rightarrow (Y,\ku F)$ between coarse spaces is {\em bornologous} if, for every $E\in \ku E$, there is $F\in \ku F$ so that $(a,b)\in E\Rightarrow (\sigma(a),\sigma(b))\in F$. And $\sigma$ is {\em expanding} if, every $F\in \ku F$, there is $E\in \ku E$ so that $(a,b)\notin E\Rightarrow (\sigma(a),\sigma(b))\notin F$. An expanding bornologous map is called a {\em coarse embedding}.
A coarse embedding $\sigma$ is a {\em coarse equivalence} if, moreover, the image $\sigma[X]$ is {\em cobounded} in $Y$, that is, there is some $F\in \ku F$ so that $$ \a y\in Y\; \e x\in X\; (y,\sigma(x))\in F. $$
A map $\sigma\colon (X,\ku E)\rightarrow (Y,\ku U)$ from a coarse space $X$ to a uniform space $Y$ is {\em uncollapsed} if there are a coarse entourage $E\in \ku E$ and a uniform entourage $F\in \ku U$ so that $$ (a,b)\notin E\;\Rightarrow\; (\sigma(a),\sigma(b))\notin F. $$ These definitions all agree with those given for the specific case of metric spaces.
It turns out to be useful to introduce a finer modulus than the compression. For this, given a map $\sigma\colon (X,d)\rightarrow (Y,\partial)$ between metric spaces, define the {\em exact compression modulus} by $$ \tilde\kappa_\sigma(t)=\inf\big(\partial(\sigma(a),\sigma(b))\mid d(a,b)= t\big) $$ and observe that $ \kappa_\sigma(t)=\inf_{s\geqslant t}\tilde\kappa_\sigma(s)$.
\section{Uniform versus coarse embeddings between Banach spaces}\label{uniform vs coarse} Observe first that a map $\sigma\colon X\rightarrow M$ from a Banach space $X$ to a uniform space $(M, \ku U)$ is uncollapsed if there are $\Delta>0$ and an entourage $F\in \ku U$ so that $$ \norm {x-y}>\Delta\;\Rightarrow\; (\sigma(x),\sigma(y))\notin F. $$ For example, a uniform embedding is uncollapsed. Note also that, if $\sigma\colon X\rightarrow G$ is a uniformly continuous uncollapsed map into a topological group $G$, then $x\mapsto (\sigma(x), \sigma(2x), \sigma(3x), \ldots)$ defines a uniform embedding of $X$ into the infinite product $\prod_{n\in \N}G$.
Our main results about uncollapsed maps between Banach spaces are as follows. \begin{thm}\label{ell p sum} Suppose $\sigma\colon X\rightarrow E$ is an uncollapsed uniformly continuous map between Banach spaces. Then, for any $1\leqslant p<\infty$, $X$ admits a simultaneously uniform and coarse embedding into $\ell^p(E)$. \end{thm}
The next corollary then follows from observing that all of the classes of spaces listed are closed under the operation $E\mapsto \ell^p(E)$ for an appropriate $1\leqslant p<\infty$. \begin{cor}\label{cor to ell p sum} If a Banach space $X$ is uniformly embeddable into $\ell^p$, $L^p$ (for some $1\leqslant p<\infty$), a reflexive, super-reflexive, stable, super-stable, non-trivial type or cotype space, then $X$ admits a simultaneously uniform and coarse embedding into a space of the same kind. \end{cor}
If we wish to avoid the passage from $E$ to the infinite sum $\ell^p(E)$, we can get by with a direct sum $E\oplus E$, but only assuming that $\sigma$ maps into a bounded set. Also, the resulting map may no longer be a uniform embedding.
\begin{thm}\label{E plus E} Suppose $\sigma\colon X\rightarrow B_E$ is an uncollapsed uniformly continuous map from a Banach space $X$ into the ball of a Banach space $E$. Then $X$ admits a uniformly continuous coarse embedding into $E\oplus E$. \end{thm}
Both propositions will be consequences of somewhat finer and more detailed results with wider applicability. Indeed, Theorem \ref{ell p sum} is a direct corollary of Lemma \ref{unif saa coarse} below.
\begin{lemme}\label{unif saa coarse} Suppose $X$ and $E$ are Banach spaces and $P_n\colon E\rightarrow E$ is a sequence of bounded projections onto subspaces $E_n\subseteq E$ so that, $E_m\subseteq \ker P_n$ for all $m\neq n$. Assume also that $\sigma_n\colon X\rightarrow E_n$ are uncollapsed uniformly continuous maps. Then $X$ admits a simultaneously uniform and coarse embedding into $E$. \end{lemme} This lemma applies in particular to the case when $E$ is a Schauder sum of a sequence of subspaces $E_n$.
\begin{proof} By composing with a translation, we may suppose that $\sigma_n(0)=0$ for each $n$. Fix also $\Delta_n,\delta_n,\epsilon_n>0$ so that $$ \norm{x-y}\geqslant \Delta_n\;\Rightarrow\; \norm{\sigma_n(x)-\sigma_n(y)}\geqslant \delta_n $$ and $$ \norm{x-y}\leqslant \epsilon_n\;\Rightarrow\; \norm{\sigma_n(x)-\sigma_n(y)}\leqslant 2^{-n}. $$ Note that, if $\norm{x-y}\leqslant k\cdot\epsilon_n$ for some $k\in \N$, then there are $z_0=x, z_1, z_2, \ldots, z_k=y\in X$ so that $\norm{z_i-z_{i+1}}=\frac 1k\norm{x-y}\leqslant \epsilon_n$, whence $$ \norm{\sigma_n(x)-\sigma_n(y)}\leqslant \sum_{i=1}^k\norm{\sigma_n(z_{i-1})-\sigma_n(z_i)}\leqslant k\cdot 2^{-n}. $$ Thus, setting $\psi_n(x)=\frac{\sigma_n(n\Delta_n\cdot x)}{\lceil\frac{n^2\Delta_n}{\epsilon_n}\rceil}$, we have, for all $x,y\in X$, \[\begin{split} \norm{x-y}\leqslant n &\;\Rightarrow\; \norm{n\Delta_n\cdot x-n\Delta_n\cdot y}\leqslant n^2\Delta_n<\Big\lceil\frac{n^2\Delta_n}{\epsilon_n}\Big\rceil\cdot \epsilon_n\\ &\;\Rightarrow\; \norm{\sigma_n\big(n\Delta_n\cdot x\big)-\sigma_n\big(n\Delta_n\cdot y\big)}\leqslant \Big\lceil\frac{n^2\Delta_n}{\epsilon_n}\Big\rceil\cdot 2^{-n}\\ &\;\Rightarrow\; \norm{\psi_n(x)-\psi_n(y)}\leqslant 2^{-n}, \end{split}\] while \[\begin{split} \norm{x-y}\geqslant \frac 1n &\;\Rightarrow\; \norm{n\Delta_n\cdot x-n\Delta_n\cdot y}\geqslant \Delta_n\\ &\;\Rightarrow\; \norm{\sigma_n\big(n\Delta_n\cdot x\big)-\sigma_n\big(n\Delta_n\cdot y\big)}\geqslant \delta_n\\ &\;\Rightarrow\; \norm{\psi_n(x)-\psi_n(y)}\geqslant \frac {\delta_n}{\big\lceil\frac{n^2\Delta_n}{\epsilon_n}\big\rceil}. \end{split}\]
Now choose $\xi_n>0$ so that $$ \norm{x-y}\leqslant \xi_n\;\Rightarrow\; \norm{\sigma_n(x)-\sigma_n(y)}\leqslant \frac{\delta_n}{n\norm{P_n}}\cdot2^{-n} $$ and set $\phi_n(x)=\frac{n\norm{P_n}}{\delta_n}\cdot\sigma_n\big(\frac{\xi_n}n\cdot x\big)$. Then \[\begin{split} \norm{x-y}\leqslant n &\;\Rightarrow\; \norm{\frac {\xi_n}n\cdot x-\frac {\xi_n}n\cdot y}\leqslant \xi_n\\ &\;\Rightarrow\; \Norm{\sigma_n\big(\frac {\xi_n}n\cdot x\big)-\sigma_n\big(\frac {\xi_n}n\cdot y\big)}\leqslant \frac{\delta_n}{n\norm{P_n}}\cdot2^{-n}\\ &\;\Rightarrow\; \norm{\phi_n(x)-\phi_n(y)}\leqslant 2^{-n}, \end{split}\] while \[\begin{split} \norm{x-y}\geqslant \frac{n\Delta_n}{\xi_n} &\;\Rightarrow\; \norm{\frac {\xi_n}n\cdot x-\frac {\xi_n}n\cdot y}\geqslant \Delta_n\\ &\;\Rightarrow\;\Norm{\sigma_n\big(\frac {\xi_n}n\cdot x\big)-\sigma_n\big(\frac {\xi_n}n\cdot y\big)}\geqslant \delta_n\\ &\;\Rightarrow\; \norm{\phi_n(x)-\phi_n(y)}\geqslant n\norm{P_n}. \end{split}\]
In particular, if $\norm{x-y}\leqslant m$, then $\norm{x-y}\leqslant n$ for all $n\geqslant m$, whence $$ \sum_{n=1}^\infty\norm{\psi_{2n-1}(x)-\psi_{2n-1}(y)}\leqslant \sum_{n=1}^{m-1}\norm{\psi_{2n-1}(x)-\psi_{2n-1}(y)}+\sum_{n=m}^\infty2^{-2n+1}<\infty $$ and $$ \sum_{n=1}^\infty\norm{\phi_{2n}(x)-\phi_{2n}(y)}\leqslant \sum_{n=1}^{m-1}\norm{\phi_{2n}(x)-\phi_{2n}(y)}+\sum_{n=m}^\infty2^{-2n}<\infty. $$ Setting $y=0$, we see that both $\sum_{n=1}^\infty\psi_{2n-1}(x)$ and $\sum_{n=1}^\infty\phi_{2n}(x)$ are absolutely convergent in $E$, whence we may define $\omega\colon X\rightarrow E$ by $$ \omega(x)=\sum_{n=1}^\infty\psi_{2n-1}(x)+\sum_{n=1}^\infty\phi_{2n}(x). $$
First, to see that $\omega$ is uniformly continuous and thus bornologous, let $\epsilon>0$ and find $m$ large enough so that $2^{-2m+2}<\frac\eps3$. Since each of $\sigma_n$ is uniformly continuous, so are the $\psi_n$ and $\phi_n$. We may therefore choose $\eta>0$ so that $$ \NORM{\Big(\sum_{n=1}^{m-1}\psi_{2n-1}(x)+\sum_{n=1}^{m-1}\phi_{2n}(x)\Big)-\Big(\sum_{n=1}^{m-1}\psi_{2n-1}(y)+\sum_{n=1}^{m-1}\phi_{2n}(y)\Big)}<\frac \eps3 $$ whenever $\norm{x-y}<\eta$. Thus, if $\norm{x-y}<\min\{\eta, m\}$, we have \[\begin{split} \norm{\omega(x)-\omega(y)} \leqslant &\NORM{\Big(\sum_{n=1}^{m-1}\psi_{2n-1}(x)+\sum_{n=1}^{m-1}\phi_{2n}(x)\Big)-\Big(\sum_{n=1}^{m-1}\psi_{2n-1}(y)+\sum_{n=1}^{m-1}\phi_{2n}(y)\Big)}\\ &+\sum_{n=m}^\infty\Norm{\psi_{2n-1}(x)-\psi_{2n-1}(y)}+\sum_{n=m}^\infty\Norm{\phi_{2n}(x)-\phi_{2n}(y)}\\ <& \frac \eps3+\frac \eps3+\frac \eps3, \end{split}\] showing uniform continuity.
Secondly, to see that $\omega$ is a uniform embedding, suppose that $\norm{x-y}>\frac 1{2n-1}$ for some $n\geqslant 1$. Then, \[\begin{split} \norm{\omega(x)-\omega(y)} &\geqslant\frac1{ \norm{P_{2n-1}}}\norm{P_{2n-1}\omega(x)-P_{2n-1}\omega(y)}\\ &\geqslant\frac1{ \norm{P_{2n-1}}}\norm{\psi_{2n-1}(x)-\psi_{2n-1}(y)}\\ &\geqslant \frac 1 {\norm{P_{2n-1}}}\cdot \frac {\delta_{2n-1}}{\big\lceil\frac{(2n-1)^2\Delta_{2n-1}}{\epsilon_{2n-1}}\big\rceil}. \end{split}\]
Finally, to see that $\omega$ is a coarse embedding, observe that, if $\norm{x-y}\geqslant \frac{2n\Delta_{2n}}{\xi_{2n}}$, then \[\begin{split} \norm{\omega(x)-\omega(y)} &\geqslant\frac1{ \norm{P_{2n}}}\norm{P_{2n}\omega(x)-P_{2n}\omega(y)}\\ &\geqslant\frac1{ \norm{P_{2n}}}\norm{\phi_{2n}(x)-\phi_{2n}(y)}\\ &\geqslant 2n, \end{split}\] which finishes the proof. \end{proof}
We should mention here that B. Braga \cite{braga1} has been able to use our construction above coupled with a result of E. Odell and T. Schlumprecht \cite{distortion} to show that $\ell^2$ admits a simultaneously uniform and coarse embedding into every Banach space with an unconditional basis and finite cotype.
Our next result immediately implies theorem \ref{E plus E}. \begin{lemme}\label{unif-coarse} Suppose $\sigma\colon X\rightarrow B_E$ and $\omega\colon X\rightarrow B_F$ are uncollapsed uniformly continuous maps from a Banach space $X$ into the balls of Banach spaces $E$ and $F$. Then $X$ admits uniformly continuous coarse embedding into $E\oplus F$. \end{lemme}
\begin{proof} Since $\sigma$ and $\omega$ are uncollapsed, pick $\Delta\geqslant 2$ and $\delta>0$ so that $$ \norm {x-y}>\Delta\;\Rightarrow\; \norm{\sigma(x)-\sigma(y)}>\delta \;\;\&\;\; \norm{\omega(x)-\omega(y)}>\delta. $$ We will inductively define bounded uniformly continuous maps $$ \phi_1,\phi_2,\ldots\colon X\rightarrow E $$ and $$ \psi_1,\psi_2,\ldots\colon Y\rightarrow F $$ with $\phi_n(0)=\psi_n(0)=0$ and numbers $0=t_0<r_1<t_1<r_2<t_2<\ldots$ with $\lim_{n}r_n=\infty$ so that, for all $n\geqslant 1$, $$ \norm{x-y}\geqslant r_n\;\Rightarrow\; \Norm{\sum_{i=1}^n\phi_i(x)- \sum_{i=1}^n\phi_i(y)}\geqslant 2^n, $$ $$ \norm{x-y}\geqslant t_n\;\Rightarrow\; \Norm{\sum_{i=1}^n\psi_i(x)- \sum_{i=1}^n\psi_i(y)}\geqslant 2^n, $$ $$ \norm{x-y}\leqslant t_{n-1}\;\Rightarrow\; \Norm{\phi_n(x)- \phi_n(y)}\leqslant 2^{-n} $$ and $$ \norm{x-y}\leqslant r_n\;\Rightarrow\; \Norm{\psi_n(x)- \psi_n(y)}\leqslant 2^{-n}. $$
Suppose that this have been done. Then $$ \norm{x-y}\leqslant t_{n-1}\;\;\Rightarrow\;\; \sum_{i=n}^\infty\Norm{\phi_i(x)- \phi_i(y)}\leqslant \sum_{i=n}^\infty2^{-i}\leqslant 1. $$ In particular, setting $y=0$, we see that the series $\sum_{i=1}^\infty\phi_i(x)$ is absolutely convergent for all $x\in X$. Similarly, $$ \norm{x-y}\leqslant r_{n}\;\Rightarrow\; \sum_{i=n}^\infty\Norm{\psi_i(x)- \psi_i(y)}\leqslant \sum_{i=n}^\infty2^{-i}\leqslant 1, $$ showing that also $\sum_{i=1}^\infty\psi_i(x)$ is absolutely convergent for all $x\in X$. So define $\phi\colon X\rightarrow E$ and $\psi\colon X\rightarrow F$ by $\phi(x)=\sum_{i=1}^\infty\phi_i(x)$ and $\psi(x)=\sum_{i=1}^\infty\psi_i(x)$.
We claim that $\phi$ and $\psi$ are uniformly continuous. To see this, let $\alpha>0$ be given and pick $n\geqslant 2$ so that $2^{-n+2}<\alpha$. By uniform continuity of the $\phi_i$, we may choose $\beta>0$ small enough so that $\sum_{i=1}^{n-1}\Norm{\phi_i(x)- \phi_i(y)}<\frac \alpha2$ whenever $\norm{x-y}<\beta$. Thus, if $\norm{x-y}<\min\{\beta, t_{n-1}\}$, we have \[\begin{split} \norm{\phi(x)-\phi(y)} &\leqslant \sum_{i=1}^{n-1}\Norm{\phi_i(x)- \phi_i(y)}+ \sum_{i=n}^\infty\Norm{\phi_i(x)- \phi_i(y)}\\ &<\frac \alpha2+\sum_{i=n}^\infty2^{-i}\\ &<\alpha, \end{split}\] showing that $\phi$ is uniformly continuous. A similar argument works for $\psi$.
Now, suppose $\norm {x-y}\geqslant r_m$. Then either $r_n\leqslant \norm {x-y}\leqslant t_n$ or $t_n\leqslant \norm {x-y}\leqslant r_{n+1}$ for some $n\geqslant m$. In the first case, \[\begin{split} \Norm{\phi(x)-\phi(y)} &=\Norm{\sum_{i=1}^\infty\phi_i(x)- \sum_{i=1}^\infty\phi_i(y)}\\ &\geqslant \Norm{\sum_{i=1}^n\phi_i(x)- \sum_{i=1}^n\phi_i(y)}-\sum_{i=n+1}^\infty\Norm{\phi_i(x)- \phi_i(y)}\\ &\geqslant 2^n-1, \end{split}\] while, in the second case, \[\begin{split} \Norm{\psi(x)-\psi(y)} &=\Norm{\sum_{i=1}^\infty\psi_i(x)- \sum_{i=1}^\infty\psi_i(y)}\\ &\geqslant \Norm{\sum_{i=1}^n\psi_i(x)- \sum_{i=1}^n\psi_i(y)}-\sum_{i=n+1}^\infty\Norm{\psi_i(x)- \psi_i(y)}\\ &\geqslant 2^n-1. \end{split}\] Thus, $$ \norm {x-y}\geqslant r_m\;\;\Rightarrow\;\; \norm{\phi(x)-\phi(y)}+\norm{\psi(x)-\psi(y)}\geqslant 2^m-1, $$ showing that $\phi\oplus\psi\colon X\rightarrow E\oplus F$ is expanding. As each of $\phi$ and $\psi$ is uniformly continuous, so is $\phi\oplus\psi$ and therefore also bornologous. It follows that $\phi\oplus\psi$ is a coarse embedding of $X$ into $E\oplus F$.
Let us now return to the construction of $\phi_i, \psi_i, r_i$ and $t_i$. We begin by letting $t_0=0$, $r_1=\Delta$ and $\phi_1=\frac{2}\delta\sigma$. Now suppose that $\phi_1,\ldots, \phi_n$, $\psi_1,\ldots, \psi_{n-1}$ and $t_0<r_1<t_1<\ldots< r_{n}$ have been defined satisfying the required conditions. As the $\psi_i$ are bounded, let $$ S=\sup_{x,y\in X}\Norm{\sum_{i=1}^{n-1}\psi_i(x)-\sum_{i=1}^{n-1}\psi_i(y)}. $$ Also, as $\omega$ is uniformly continuous, pick $0<\epsilon<1$ so that $\norm{\omega(x)-\omega(y)}<\frac\delta{(S+2^{n})2^{n}}$ whenever $\norm{x-y}\leqslant \epsilon$ and let $\psi_n(x)=\frac{S+2^{n}}\delta\omega(\frac \epsilon{r_n}x)$. Then, if $\norm{x-y}\leqslant r_n$, we have $\norm{\frac \epsilon{r_n}x-\frac \epsilon{r_n}y}\leqslant \epsilon$ and so $$ \norm{\psi_n(x)-\psi_n(y)}=\frac{S+2^{n}}\delta\NORM{\omega\big(\frac \epsilon{r_n}x\big)-\omega\big(\frac \epsilon{r_n}y\big)}<2^{-n}. $$ On the other hand, if we let $t_n=\frac{r_n\Delta}\epsilon$, then $$ \norm{x-y}\geqslant t_n\;\Rightarrow\; \norm{\psi_n(x)-\psi_n(y)}\geqslant S+2^n\;\Rightarrow\; \Norm{\sum_{i=1}^n\psi_i(x)- \sum_{i=1}^n\psi_i(y)}\geqslant 2^n. $$ Note that, as $\epsilon<1$ and $\Delta\geqslant 2$, we have $t_n>2r_n$. A similar construction allows us to find $\phi_{n+1}$ and $r_{n+1}>2t_n$ given $\phi_1,\ldots, \phi_n$, $\psi_1,\ldots, \psi_n$ and $t_0<r_1<\ldots< t_n$. \end{proof}
In the light of the previous results, it would be very interesting to determine when a coarse embedding can be replaced by a uniformly continuous coarse embedding. As mentioned earlier, Naor \cite{naor-nets} was able to construct a bornologous map between two separable Banach spaces, which is not close to any uniformly continuous map. \begin{quest} Suppose $X$ is a separable Banach space coarsely embedding into a separable Banach space $E$. Is there a uniformly continuous coarse embeddding of $X$ into $E$? \end{quest}
\section{Cocycles and affine isometric representations}\label{affine actions} By the Mazur--Ulam Theorem, every surjective isometry $A$ of a Banach space $X$ is {\em affine}, that is, there are a unique invertible linear isometry $T\colon X\rightarrow X$ and a vector $\eta\in X$ so that $A$ is given by $A(\xi)=T(\xi)+\eta$ for all $\xi \in X$. It follows that, if $\alpha\colon G\curvearrowright X$ is an isometric action of a group $G$ on a Banach space $X$, there is an isometric linear representation $\pi\colon G\curvearrowright X$, called the {\em linear part} of $\alpha$, and a corresponding {\em cocycle} $b\colon G\rightarrow X$ so that $$ \alpha(g)\xi=\pi(g)\xi +b(g) $$ for all $g\in G$ and $\xi \in X$. In particular, $b$ is simply the orbit map $g\mapsto \alpha(g)0$. Moreover, the cocycle $b$ then satisfies the {\em cocycle equation} $$ b(gf)=\pi(g)b(f)+b(g) $$ for $g,f\in G$. Finally, as $\alpha$ is an action by isometries, we have \[\begin{split} \norm{ b(f)-b(g)}&=\norm{\alpha(f)0-\alpha(g)0}\\ &=\norm{\alpha(g^{-1} f)0-0}\\ &=\norm{b(g^{-1} f)}. \end{split}\]
Now, if $G$ is a topological group, the action $\alpha$ is continuous, i.e., continuous as a map $\alpha\colon G\times X\rightarrow X$, if and only if the linear part $\pi$ is {\em strongly continuous}, that is, $g\in G\mapsto \pi(g)\xi\in X$ is continuous for every $\xi \in X$, and $b\colon G\rightarrow X$ is continuous. Moreover, since $b$ is simply the orbit map $g\mapsto \alpha(g)0$, in this case, the cocycle $b\colon G\rightarrow X$ is both uniformly continuous and bornologous. We say that the action $\alpha$ is {\em coarsely proper} if and only if $b\colon G\rightarrow X$ is a coarse embedding.
If $\pi\colon G\curvearrowright X$ is a strongly continuous isometric linear representation, we let $Z^1(G,\pi)$ denote the vector space of continuous cocycles $b\colon G\rightarrow X$ assiciated to $\pi$. Also, let $B^1(G,\pi)$ denote the linear subspace of {\em coboundaries}, i.e., cocycles $b$ of the form $b(g)=\xi-\pi(g)\xi$ for some $\xi\in X$. Note that the cocycle $b$ has this form if and only if $\xi$ is fixed by the corresponding affine isometric action $\alpha$ induced by $\pi$ and $b$.
As noted above, continuous cocycles are actually uniformly continuous. But, in the case of cocycles between Banach spaces, we have stronger information available.
\begin{prop}\label{affine banach} Let $b\colon X\rightarrow E$ be a continuous uncollapsed cocycle between Banach spaces $X$ and $E$, i.e., there are $\Delta,\delta>0$ so that $$ \norm x>\Delta\;\Rightarrow\; \norm{b(x)}>\delta. $$ Then $b\colon X\rightarrow E$ is a uniform embedding. In fact, there are constants $c,C>0$ so that $$ c\cdot \min\{\norm{x-y},1\}\leqslant \norm{b(x)-b(y)}\leqslant C\norm{x-y}+C. $$ \end{prop}
\begin{proof} Suppose that $\pi\colon X\curvearrowright E$ is the strongly continuous isometric linear action for which $b\colon X\rightarrow E$ an uncollapsed continuous cocycle. As noted above, $b$ is uniformly continous and hence, by the Corson--Klee lemma (Proposition 1.11 \cite{lindenstrauss}), also Lipschitz for large distances. This shows the second inequality.
We now show the first inequality with $c=\min\{\delta, \frac \delta{2\Delta}\}$, which implies uniform continuity of $b^{-1}$. Since $\norm{b(x)-b(y)}=\norm{b(x-y)}$, it suffices to verify that $$ c\cdot \min\{\norm{x},1\}\leqslant \norm{b(x)} $$ for all $x\in X$. For $\norm{x}>\Delta$, this follows from our assumption and choice of $c$. So suppose instead that $x\in X\setminus \{0\}$ with $\norm x\leqslant \Delta$ and let $n$ be minimal so that $n\norm{x}> \Delta$. Then $\norm x\leqslant \frac {\Delta}{n-1}\leqslant \frac {2\Delta}{n}$ and \[\begin{split} \delta&< \norm{b(n\cdot x)}\\ &=\norm{\pi^{n-1}(x)b(x)+\pi^{n-2}(x)b(x)+\ldots +b(x)}\\ &\leqslant\norm{\pi^{n-1}(x)b(x)}+\norm{\pi^{n-2}(x)b(x)}+\ldots +\norm{b(x)}\\ &=n\cdot\norm{b(x)}, \end{split}\] i.e., $$ \norm{b(x)}\geqslant \frac \delta n \geqslant \frac \delta {2\Delta}\cdot \frac {2\Delta}{n}\geqslant \frac \delta{2\Delta}\cdot\norm x\geqslant c\norm x $$ as required. \end{proof} Since any continuous cocycle is both bornologous and uniformly continuous, we see that any coarsely proper continuous cocycle $b\colon X\rightarrow E$ between Banach spaces is simultaneously a uniform and coarse embedding.
The following variation is also of independent interest. \begin{prop}\label{affine ball} Let $b\colon X\rightarrow E$ be a continuous cocycle between Banach spaces satisfying $$ \inf_{\norm{x}=r}\norm{b(x)}>0 $$ for some $r> 0$. Then $B_X$ uniformly embeds into $B_E$. \end{prop}
\begin{proof} Set $\delta=\inf_{\norm{x}=r}\norm{b(x)}$ and find, by uniform continuity of $b$, some $m$ so that $$ \norm{b(x)}>\frac \delta2 $$ whenever $r\leqslant \norm x\leqslant r+\frac 2m$. Then, if $0<\norm x\leqslant \frac 1m$, fix $n$ minimal so that $r\leqslant \norm{ nx}\leqslant r+\frac 2m$. With this choice of $n$ we have as in the proof of Proposition \ref{affine banach} that $$ \norm {b(x)}\geqslant \frac \delta{2n}\geqslant \frac\delta{4r}\norm x, $$ showing that $b\colon \frac 1mB_X\rightarrow E$ is a uniform embedding. The proposition follows by rescaling $b$. \end{proof} Though we shall return to the issue later, let us just mention that uniform embeddings between balls of Banach spaces has received substantial attention. For example, Y. Raynaud \cite{raynaud} (see also Section 9.5 \cite{lindenstrauss}) has shown that $B_{c_0}$ does not uniformly embed into a stable metric space, e.g., into $L^p([0,1])$ with $1\leqslant p<\infty$.
Our next result replicates the construction from Section \ref{uniform vs coarse} within the context of cocycles.
\begin{prop}\label{cocycle amplification} Suppose $b\colon X\rightarrow E$ is an uncollapsed continuous cocycle between Banach spaces. Then, for every $1\leqslant p<\infty$, there is a coarsely proper continuous cocycle $\tilde b\colon X\curvearrowright \ell^p(E)$. \end{prop}
\begin{proof} Let $b$ be associated to the strongly continuous isometric linear representation $\pi\colon X\curvearrowright E$. As in the proof of Lemma \ref{ell p sum}, there are constants $\epsilon_n$ and $K_n$ so that $$ x\mapsto (K_1\cdot b(\epsilon_1 x), K_2\cdot b(\epsilon_2 x),\ldots) $$ defines a simultaneously uniform and coarse embedding of $X$ into $\ell^p(E)$. We claim that the map $\tilde b\colon X\rightarrow \ell^p(E)$ so defined is a cocycle for some strongly continuous isometric linear representation $\tilde\pi\colon X\curvearrowright \ell^p(E)$. Indeed, observe that each $x\mapsto K_n\cdot b(\epsilon_nx)$ is a cocycle for the isometric linear representation $x\mapsto \pi(\epsilon_n x)$ on $E$, so $\tilde b$ is a cocycle for the isometric linear representation $$ \tilde \pi(x)=\pi(\epsilon_1 x)\otimes \pi(\epsilon_2x)\otimes \ldots $$ of $X$ on $\ell^p(E)$. \end{proof}
\begin{lemme}\label{cocycle isometry group} Let $E$ be a separable Banach space and ${\rm Isom}(E)$ the group of linear isometries of $E$ equipped with the strong operator topology. If $\pi\colon {\rm Isom}(E)\curvearrowright \ell^p(E)$ denotes the diagonal isometric linear representation for $1\leqslant p<\infty$, there is a continuous cocycle $b\colon {\rm Isom}(E)\rightarrow \ell^p(E)$ associated to $\pi$ that is a uniform embedding of ${\rm Isom}(E)$ into $\ell^p(E)$. \end{lemme}
\begin{proof} Fix a dense subset $\{\xi_n\}_{n\in \N}$ of the sphere $S_E$ and let $$ b(g)=\Big(\frac{\xi_1-g(\xi_1)}{2^1},\frac{\xi_2-g(\xi_2)}{2^2},\frac{\xi_3-g(\xi_3)}{2^3},\ldots\Big). $$ Then $b$ is easily seen to be a cocycle for $\pi$. Moreover, by the definition of the strong operator topology, $b$ is a uniform embedding. \end{proof}
\begin{prop}\label{cocycle amplification banach} Supppose $\pi\colon X\rightarrow {\rm Isom}(E)$ is an uncollapsed strongly continuous isometric linear representation of a Banach space $X$ on a separable Banach space $E$. Then, for every $1\leqslant p<\infty$, $X$ admits coarsely proper continuous cocycle $b\colon X\rightarrow \ell^p(E)$. \end{prop}
\begin{proof} Let $c\colon {\rm Isom}(E)\rightarrow \ell^p(E)$ be the cocycle given by Lemma \ref{cocycle isometry group} associated to the diagonal isometric linear representation $\rho\colon {\rm Isom}(E)\curvearrowright \ell^p(E)$. It follows that $c\circ \pi\colon X\rightarrow \ell^p(E)$ is a continuous uncollapsed cocycle associated to the isometric linear representation $\rho\circ \pi\colon X\curvearrowright \ell^p(E)$. By Proposition \ref{cocycle amplification} and the fact that $\ell^p(\ell^p(E))=\ell^p(E)$, we obtain a coarsely proper continuous cocycle $b\colon X \rightarrow \ell^p(E)$. \end{proof}
Cocycles between Banach spaces are significantly more structured maps than general maps. For example, as shown in Proposition \ref{affine banach}, a coarsely proper continuous cocycle $b\colon X\rightarrow E$ between Banach spaces $X$ and $E$ is automatically both a uniform and coarse embedding, but, moreover, $b$ also preserves a certain amount of algebraic structure, depending on the isometric linear representation $\pi\colon X\curvearrowright E$ of which it is a cocycle.
To study these algebraic features, we must introduce a topology on the space of cocycles. So fix a strongly continuous isometric linear representation $\pi\colon G\curvearrowright E$ of a topological group $G$ on a Banach space $E$. Every compact set $K\subseteq G$ determines a seminorm $\norm\cdot_K$ on $Z^1(G,\pi)$ by $\norm{ b}_K=\sup_{g\in K}\norm{b(g)}$ and the family of seminorms thus obtained endows $Z^1(G,\pi)$ with a locally convex topology. With this topology, one sees that a cocycle $b$ belongs to the closure $\overline{B^1(G,\pi)}$ if and only if the corresponding affine action $\alpha=(\pi,b)$ {\em almost has fixed points}, that is, if for any compact set $K\subseteq G$ and $\epsilon>0$ there is some $\xi=\xi_{K,\epsilon}\in E$ verifying $$ \sup_{x\in K}\norm{\big(\pi(x)\xi+b(x)\big)-\xi}=\sup_{x\in K}\norm{b(x)-\big(\xi-\pi(x)\xi\big)}<\epsilon. $$ Elements of $\overline{B^1(G,\pi)}$ are called {\em almost coboundaries}.
Note that, if $b$ is a coboundary, then $b(G)$ is a bounded subset of $E$. Conversely, suppose $b(G)$ is a bounded set and $E$ is reflexive. Then any orbit $\ku O$ of the corresponding affine action is bounded and its closed convex hull $C=\overline{\rm conv}(\ku O)$ is a weakly compact convex set on which $G$ acts by affine isometries. It follows by the Ryll-Nardzewski fixed point theorem \cite{ryll} that $G$ fixes a point on $C$, meaning that $b$ must be a coboundary.
Now, if $b\in\overline{B^1(G,\pi)}$ and, for every compact $K\subseteq G$, we can choose $\xi=\xi_{K,1}$ above to have arbitrarily large norm, we see that the supremum $$ \sup_{x\in K}\Norm{\pi(x)\frac \xi{\norm \xi}-\frac\xi{\norm \xi}}<\frac{\sup_{x\in K}\norm{b(x)}+1}{\norm \xi} $$ can be made arbitrarily small, which means that the linear action $\pi$ {\em almost has invariant unit vectors}. If, on the other hand, for some $K$ the choice of $\xi_{K,1}$ is bounded (but non-empty), then the same bound holds for any compact $K'\supseteq K$, whereby we find that $b(G)\subseteq E$ is a bounded set and so, assuming $E$ is reflexive, that $b\in B^1(G,\pi)$. This shows that, if $E$ is reflexive and $\pi$ does not almost have invariant unit vectors, then $B^1(G,\pi)$ is closed in $Z^1(G,\pi)$.
We define the {\em first cohomology group} of $G$ with coefficients in $\pi$ to be the quotient space $H^1(G,\pi)=Z^1(G,\pi)/B^1(G,\pi)$, while the {\em reduced cohomology group} is $\overline{H^1}(G,\pi)=Z^1(G,\pi)/\overline{B^1(G,\pi)}$.
If $E$ is separable and reflexive, the Alaoglu--Birkhoff decomposition theorem \cite{alaoglu} implies that $E$ admits $\pi(G)$-invariant decomposition into closed linear subspaces $E=E^G\oplus E_G$, where $E^G$ is the set of $\pi(G)$-fixed vectors. We can therefore write $b=b^G\oplus b_G$, where $b^G\colon X\rightarrow E^G$ and $b_G\colon X\rightarrow E_G$ are cocycles for $\pi$. In particular, $$ b^G(xy)=\pi(x)b^G(y)+b^G(x)=b^G(y)+b^G(x)=b^G(x)+b^G(y), $$
i.e., $b^G$ is a continuous homomorphism from $G$ to $(E^G,+)$. Also, Theorem 2 of \cite{BRS}, implies that, if $G$ is abelian, then $\overline{H^1}(G, \pi|_{E_G})=0$ and so $b_G\in \overline{B^1(G,\pi)}$.
Now, suppose $\pi\colon X\curvearrowright E$ is a strongly continuous isometric linear representation of a separable Banach space $X$ on a separable reflexive Banach space $E$. Assume that $b\colon X\rightarrow E$ is a continuous cocycle and let $E=E^X\oplus E_X$ and $b^X\colon X\rightarrow E^X$ and $b_X\colon X\rightarrow E_X$ be the decompositions as above. Being a continuous additive homomorphism, $b^X$ is a bounded linear operator from $X$ to $E^X$. Also, if $b^X$ is coarsely proper or even just uncollapsed, then $b^X$ must be an isomorphic embedding of $X$ into $E^X$.
Now, since $X$ is abelian, $b_X\colon X\rightarrow E_X$ belongs to $\overline{B^1(X,\pi)}$, which means that, for every norm-compact subset $C\subseteq X$ and $\epsilon>0$, there is $\xi \in E_X$ so that $$ \norm{\xi-\pi(x)\xi-b_X(x)}<\epsilon $$ for all $ x\in C$.
We summarise the discussion so far in the following lemma. \begin{lemme}\label{banach on banach} Suppose $\pi\colon X\curvearrowright E$ is a strongly continuous isometric linear representation of a separable Banach space $X$ on a separable reflexive Banach space $E$ and assume that $b\colon X\rightarrow E$ is a continuous cocycle. Then there is a $\pi(X)$-invariant decomposition $E=E^X\oplus E_X$ and a decomposition $b=b^X\oplus b_X$ so that $b^X\colon X\rightarrow E^X$ is a bounded linear operator and $b_X\colon X\rightarrow E_X$ belongs to $\overline{B^1(X,\pi)}$. \end{lemme}
\section{Amenability}\label{amenability} Central in our investigation is the concept of amenability, which for general topological groups is defined as follows. \begin{defi} A topological group $G$ is {\em amenable} if every continous action $\alpha \colon G\curvearrowright C$ by affine homeohomorphisms on a compact convex subset $C$ of a locally convex topological vector space $V$ has a fixed point in $C$. \end{defi}
If $G$ is a topological group, we let ${\rm LUC}(G)$ be the vector space of bounded left-uniformly continuous functions $\phi\colon G\rightarrow \R$ and equip it with the supremum norm induced from the inclusion ${\rm LUC}(G)\subseteq \ell^\infty(G)$. It then follows that the {\em right-regular representation} $\rho\colon G\curvearrowright {\rm LUC}(G)$, $\rho(g)(\phi)=\phi(\cdot\, g)$, is continuous. Moreover, if $G$ is amenable, there exists a $\rho$-invariant mean, that is, a positive continuous linear functional $\go m\colon {\rm LUC}(G)\rightarrow \R$ with $\go m(\mathbb 1)=1$, where $\mathbb1 $ is the function with constant value $1$, and so that $\go m\big(\rho(g)(\phi)\big)=\go m(\phi)$.
While we shall use the existence of invariant means on ${\rm LUC}(G)$, at one point this does not seem to suffice. Instead, we shall rely on an appropriate generalisation of F\o lner sets present under additional assumptions.
\begin{defi} A topological group $G$ is said to be {\em approximately compact} if there is a countable chain $K_0\leqslant K_1\leqslant \ldots \leqslant G$ of compact subgroups whose union $\bigcup_nK_n$ is dense in $G$. \end{defi}
This turns out to be a fairly common phenomenon among non-locally compact Polish groups. For example, the unitary group $U(\ku H)$ of separable infinite-dimensional Hilbert space with the strong operator topology is approximately compact. Indeed, if $\ku H_1\subseteq \ku H_2\subseteq \ldots\subseteq \ku H$ is an increasing exhaustive sequence of finite-dimensional subspaces and $U(n)$ denotes the group of unitaries pointwise fixing the orthogonal complement $\ku H_n^\perp$, then each $U(n)$ is compact and the union $\bigcup_nU(n)$ is dense in $U(\ku H)$.
More generally, as shown by P. de la Harpe \cite{harpe}, if $M$ is an approximately finite-dimensional von Neumann algebra, i.e., there is an increasing sequence $A_1\subseteq A_2\subseteq\ldots\subseteq M$ of finite-dimensional matrix algebras whose union is dense in $M$ with respect the strong operator topology, then the unitary subgroup $U(M)$ is approximately compact with respect to the strong operator topology.
Similarly, if $G$ contains a locally finite dense subgroup, this will witness approximate compactness. Again this applies to, e.g., ${\rm Aut}([0,1],\lambda)$ with the weak topology, where the dyadic permutations are dense, and ${\rm Isom}(\U)$ with the pointwise convergence topology (this even holds for the dense subgroup ${\rm Isom}(\Q\U)$ by an unpublished result of S. Solecki; see \cite{RZ} for a proof).
Of particular interest to us is the case of non-Archimedean Polish groups. By general techniques, these may be represented as automorphism groups of countable locally finite (i.e., any finitely generated substructure is finite) ultrahomogeneous structures. And, in this setting, we have the following reformulation of approximate compactness. \begin{prop}[A.S. Kechris \& C. Rosendal \cite{turbulence}] Let $\bf M$ be a locally finite, countable, ultrahomogeneous structure. Then ${\rm Aut}(\bf M)$ is approximately compact if and only if, for every finite substructure $\bf A\subseteq \bf M$ and all {\em partial} automorphisms $\phi_1,\ldots, \phi_n$ of $\bf A$, there is a finite substructure $\bf A\subseteq \bf B\subseteq \bf M$ and {\em full} automorphisms $\psi_1,\ldots,\psi_n$ of $\bf B$ extending $\phi_1,\ldots,\phi_n$ respectively. \end{prop}
Whereas a locally compact group is amenable if and only if it admits a F\o lner sequence, there is no similar characterisation of general amenable groups. Nevertheless, one may sometimes get by with a little less, which we isolate in the following definition. \begin{defi}\label{folner} A Polish group $G$ is said to be {\em F\o lner amenable} if either \begin{enumerate} \item $G$ is approximately compact, or \item there is a continuous homomorphism $\phi\colon H\rightarrow G$ from a locally compact second countable amenable group $H$ so that $G=\overline{\phi[H]}$. \end{enumerate} \end{defi}
Apart from the approximately compact or locally compact amenable groups, easy examples of F\o lner amenable Polish groups are, e.g., Banach spaces or, more generally, abelian groups. Indeed, every abelian Polish group $G$ contains a countable dense subgroup $\Gamma$, which, viewed as a discrete group, is amenable and maps densely into $G$.
\section{Embeddability in Hilbert spaces} We shall now consider Hilbert valued cocycles, for which we need some background material on kernels conditionally of negative type. The well-known construction of inner products presented here originates in work of E. H. Moore \cite{moore}. A full treatment can be found, e.g., in \cite{bekka}, Appendix C. \begin{defi} A (real-valued) {\em kernel conditionally of negative type} on a set $X$ is a function $\Psi\colon X\times X\rightarrow \R$ so that \begin{enumerate} \item $\Psi(x,x)=0$ and $\Psi(x,y)=\Psi(y,x)$ for all $x,y\in X$, \item for all $x_1,\ldots, x_n\in X$ and $r_1,\ldots, r_n\in \R$ with $\sum_{i=1}^nr_i=0$, we have $$ \sum_{i=1}^n\sum_{j=1}^n r_ir_j\Psi(x_i,x_j)\leqslant 0. $$ \end{enumerate} \end{defi} For example, if $\sigma\colon X\rightarrow \ku H$ is any mapping from $X$ into a Hilbert space $\ku H$, then a simple calculation shows that $$ \sum_{i=1}^n\sum_{j=1}^n r_ir_j\norm{\sigma(x_i)-\sigma(x_j)}^2=-2\Norm{\sum_{i=1}^nr_i\sigma(x_i)}\leqslant 0, $$ whenever $\sum_{i=1}^nr_i=0$, which implies that $\Psi(x,y)=\norm{\sigma(x)-\sigma(y)}^2$ is a kernel conditionally of negative type.
Suppose that $\Psi$ is a kernel conditionally of negative type on a set $X$ and let $\M(X)$ denote the vector space of finitely supported real valued functions $\xi$ on $X$ of mean $0$, i.e., $\sum_{x\in X}\xi(x)=0$. We define a positive symmetric linear form $\langle\cdot\del \cdot\rangle_\Psi$ on $\M(X)$ by $$ \Big\langle\sum_{i=1}^nr_i\delta_{x_i}\Del \sum_{j=1}^ks_j\delta_{y_i}\Big\rangle_\Psi=-\frac 12\sum_{i=1}^n\sum_{j=1}^k r_is_j\Psi(x_i,y_j). $$ Also, if $N_\Psi$ denotes the null-space $$ N_\Psi=\{\xi\in \M(X)\del \langle\xi\del \xi\rangle_\Psi=0\}, $$ then $\langle\cdot\del \cdot\rangle_\Psi$ defines an inner product on the quotient $\M(X)/N_\Psi$ and we obtain a real Hilbert space $\ku K$ as the completion of $\M(X)/N_\Psi$ with respect to $\langle\cdot\del \cdot\rangle_\Psi$.
We remark that, if $\Psi$ is defined by a map $\sigma\colon X\rightarrow \ku H$ as above and $e\in X$ is any choice of base point, the map $\phi_e\colon X\rightarrow \ku K$ defined by $\phi_e(x)=\delta_x-\delta_e$ satisfies $\norm{\phi_e(x)-\phi_e(y)}_\ku K=\norm{\sigma(x)-\sigma(y)}_\ku H$. Indeed, \[\begin{split} \norm{\phi_e(x)-\phi_e(y)}^2_\ku K &=\langle \phi_e(x)-\phi_e(y)\del \phi_e(x)-\phi_e(y)\rangle\\ &=\langle \delta_x-\delta_y\del \delta_x-\delta_y\rangle\\ &=-\frac 12\big(\Psi(x,x)+\Psi(y,y)-\Psi(x,y)-\Psi(y,x)\big)\\ &=\Psi(x,y)\\ &=\norm{\sigma(x)-\sigma(y)}_\ku H^2. \end{split}\]
Also, if $G\curvearrowright X$ is an action of a group $G$ on $X$ and $\Psi$ is $G$-invariant, i.e., $\Psi(gx,gy)=\Psi(x,y)$, this action lifts to an action $\pi\colon G\curvearrowright\M(X)$ preserving the form $\langle\cdot\del\cdot\rangle_\Psi$ via $\pi(g)\xi=\xi(g^{-1}\,\cdot\,)$. It follows that $\pi$ factors through to an orthogonal (i.e., isometric linear) representation $G\curvearrowright\ku K$.
A version of Lemma \ref{pre-maurey} below is originally due to I. Aharoni, B. Maurey and B. S. Mityagin \cite{maurey} for the case of abelian groups and has been extended and refined several times recently in connection with the coarse geometry of Banach spaces and locally compact groups (see, e.g., \cite{tessera, randrianarivony, randrianarivony2}). Since more care is needed when dealing with general amenable as opposed to locally compact amenable or abelian groups, we include a full proof.
Let us first recall that, if $\sigma \colon X\rightarrow Y$ is a map between metric spaces, the exact compression modulus $\tilde \kappa$ of $\sigma$ is given by $\tilde\kappa(t)=\inf_{d(x,x')=t}d(\sigma(x), \sigma(x'))$.
\begin{lemme}\label{pre-maurey} Suppose $d$ is a compatible left-invariant metric on an amenable topological group $G$ and $\sigma\colon (G,d)\rightarrow \ku H$ is a uniformly continuous and bornologous map into a Hilbert space $\ku H$ with exact compression modulus $\tilde \kappa$ and expansion modulus $\theta$. Then there is a continuous $G$-invariant kernel conditionally of negative type $\Psi\colon G\times G\rightarrow \R_+$ satisfying $$ \tilde\kappa\big(d(g,f)\big)^2\leqslant \Psi(g,f)\leqslant \theta\big(d(g,f)\big)^2. $$ \end{lemme}
\begin{proof} For fixed $g,h\in G$, we define a function $\phi_{g,h}\colon G\rightarrow \R$ via $$ \phi_{g,h}(f)=\norm{\sigma(fg)-\sigma(fh)}^2. $$ Since, for all $g,h,f\in G$, we have $$ \tilde\kappa\big(d(g,h)\big)^2=\tilde\kappa\big(d(fg,fh)\big)^2\leqslant \norm{\sigma(fg)-\sigma(fh)}^2\leqslant \theta\big(d(g,h)\big)^2, $$ it follows that $$ \tilde\kappa\big(d(g,h)\big)^2\leqslant \phi_{g,h}\leqslant\theta\big(d(g,h)\big)^2 $$ and so, in particular, $\phi_{g,h}\in \ell^\infty(G)$.
We claim that $\phi_{g,h}$ is {\em left}-uniformly continuous, i.e., that for all $\epsilon>0$ there is $W\ni 1$ open so that $|\phi_{g,h}(f)-\phi_{g,h}(fw)|<\epsilon$, whenever $f\in G$ and $w\in W$. To see this, take some $\eta>0$ so that $4\eta\norm{\phi_{g,h}}_\infty+4\eta^2<\epsilon$ and find, by uniform continuity of $\sigma$, some open $V\ni1$ so that $\norm{\sigma(f)-\sigma(fv)}<\eta$ for all $f\in G$ and $v\in V$. Pick also $W\ni 1$ open so that $Wg\subseteq gV$ and $Wh\subseteq hV$. Then, if $f\in G$ and $w\in W$, there are $v_1,v_2\in V$ so that $wg=gv_1$ and $wh=hv_2$, whence \[\begin{split}
\big|\phi_{g,h}(f)-\phi_{g,h}(fw)\big|
&=\Big| \norm{\sigma(fg)-\sigma(fh)}^2- \norm{\sigma(fwg)-\sigma(fwh)}^2\Big| \\
&=\Big| \norm{\sigma(fg)-\sigma(fh)}^2- \norm{\sigma(fgv_1)-\sigma(fhv_2)}^2\Big|\\ &<4\eta\norm{\phi_{g,h}}_\infty+4\eta^2\\ &<\epsilon. \end{split}\] Thus, every $\phi_{g,h}$ belongs to the closed linear subspace ${\rm LUC}(G)\subseteq \ell^\infty(G)$ of left-uniformly continuous bounded real-valued functions on $G$ and a similar calculation shows that the map $(g,h)\in G\times G\mapsto \phi_{g,h}\in \ell^\infty(G)$ is continuous.
Now, since $G$ is amenable, there exists a mean $\go m$ on ${\rm LUC}(G)$ invariant under the {\em right}-regular representation $\rho\colon G\curvearrowright {\rm LUC}(G)$ given by $\rho(g)\big(\phi\big)=\phi(\,\cdot\, g)$. Using this, we can define a continuous kernel $\Psi\colon G\times G\rightarrow \R$ by $$ \Psi(g,h)=\go m(\phi_{g,h}) $$ and note that $\Psi(fg,fh)=\go m(\phi_{fg,fh})=\go m\big(\rho(f)\big(\phi_{g,h}\big)\big)=\go m(\phi_{g,h})=\Psi(g,h)$ for all $g,h,f\in G$.
We claim that $\Psi$ is a kernel conditionally of negative type. To verify this, let $g_1,\ldots, g_n\in G$ and $r_1,\ldots, r_n\in \R$ with $\sum_{i=1}^nr_i=0$. Then, for all $f\in G$, $$ \sum_{i=1}^n\sum_{j=1}^n r_ir_j\phi_{g_i,g_j}(f)=\sum_{i=1}^n\sum_{j=1}^n r_ir_j\norm{\sigma(fg_i)-\sigma(fg_j)}^2\leqslant 0, $$ since $(g,h)\mapsto \norm{\sigma(fg)-\sigma(fh)}^2$ is a kernel conditionally of negative type. As $\go m$ is positive, it follows that also $$ \sum_{i=1}^n\sum_{j=1}^n r_ir_j\Psi(g_i,g_j) = \go m\Big(\sum_{i=1}^n\sum_{j=1}^n r_ir_j\phi_{g_i,g_j}\Big) \leqslant 0. $$
Finally, as $\go m$ is a mean and $$ \tilde\kappa\big(d(g,h)\big)^2\leqslant \phi_{g,h}\leqslant \theta\big(d(g,h)\big)^2, $$ it follows that $$ \tilde\kappa\big(d(g,h)\big)^2\leqslant \Psi(g,h)\leqslant \theta\big(d(g,h)\big)^2 $$ as required. \end{proof}
\begin{thm}\label{maurey} Suppose $d$ is a compatible left-invariant metric on an amenable topological group $G$ and $\sigma\colon (G,d)\rightarrow \ku H$ is a uniformly continuous and bornologous map into a Hilbert space $\ku H$ with exact compression modulus $\tilde \kappa$ and expansion modulus $\theta$.
Then there is a continuous cocycle into a real Hilbert space $b\colon G\rightarrow \ku K$ so that $$ \tilde \kappa\big(d(g,f)\big)\leqslant \norm{b(g)-b(f)}\leqslant \theta\big(d(g,f)\big), $$ for all $g,f\in G$. \end{thm}
\begin{proof} Let $\Psi$ be the $G$-invariant kernel conditionally of negative type given by Lemma \ref{pre-maurey}. As above, we define a positive symmetric form $\langle\cdot\del \cdot\rangle_\Psi$ on $\M(G)$. Note that, since $\Psi$ is $G$-invariant, the form $\langle\cdot\del \cdot\rangle_\Psi$ is invariant under the left-regular representation $\lambda\colon G\curvearrowright \M(G)$ given by $\lambda(g)(\xi)=\xi(g^{-1}\,\cdot\,)$ and so $\lambda$ induces a strongly continuous orthogonal representation $\pi$ of $G$ on the Hilbert space completion $\ku K$ of $\M(G)/N_\Psi$.
Moreover, as is easily checked, the map $b\colon G\rightarrow \ku K$ given by $b(g)=(\delta_g-\delta_1)+N_\Psi$ is a cocycle for $\pi$. Now $$ \norm{b(g)}^2=\langle \delta_g-\delta_1 \del \delta_g-\delta_1\rangle_\Psi=\Psi(g,1), $$ whence $$ \tilde \kappa\big(d(g,1)\big)\leqslant \norm{b(g)}\leqslant \theta\big(d(g,1)\big). $$ Replacing $g$ with $f^{-1} g$, the theorem follows. \end{proof}
The first application concerns uniform embeddability of balls. Recall, for example, that a Banach space $X$ whose unit ball $B_X$ is uniformly embeddable into a Hilbert space must have finite cotype (Proposition 5.3 \cite{raynaud}). \begin{cor}\label{balls hilbert} Let $\sigma\colon X\rightarrow \ku H$ be a uniformly continuous map from a Banach space $X$ into a Hilbert space satisfying $$ \inf_{\norm{x-y}=r}\norm{\sigma(x)-\sigma(y)}>0 $$ for just some $r>0$. Then $B_X$ is uniformly embeddable into Hilbert space. \end{cor}
\begin{proof} Observe that the exact compression modulus of $\sigma$ satisfies $\tilde\kappa_\sigma(r)>0$. Thus, applying Theorem \ref{maurey}, we obtain a Hilbert valued continuous cocycle $b\colon X\rightarrow \ku K$ with $\tilde\kappa_b(r)>0$. Finally, an application of Proposition \ref{affine ball}, shows that $B_X$ uniformly embeds into $B_\ku K$. \end{proof}
The requirement that $\sigma$ be uniformly continuous in Theorem \ref{maurey} above is somewhat superfluous. Indeed, W. B. Johnson and N. L. Randrianarivony (Step 0 in \cite{randrianarivony}) have shown that any separable Banach space admitting a coarse embedding into a Hilbert space also has a uniformly continuous coarse embedding into a Hilbert space and the proof carries over directly to prove the following.
\begin{lemme}\cite{randrianarivony}\label{randrianarivony} Suppose $\sigma\colon (X,d) \rightarrow \ku H$ is a map from a metric space into a Hilbert space and assume that $\sigma$ is Lipschitz for large distances and expanding. Then $(X,d)$ admits a uniformly continuous coarse embedding into a Hilbert space. \end{lemme}
For a topological group, the question is then when may may work with Lipshcitz for large distance maps rather than bornologous maps. \begin{lemme}\label{lipschitz} Let $\sigma\colon G\rightarrow (X,d)$ be a bornologous map from a Polish group into a metric space. Then there is a compatible left-invariant metric $\partial$ on $G$ so that $$ \sigma\colon (G,\partial)\rightarrow (X,d) $$ is Lipschitz for large distances. \end{lemme}
\begin{proof} For $a\in G$, set $$ w(a)=\sup_{x^{-1} y=a}d(\sigma(x),\sigma(y)) $$ and observe that, as $\sigma$ is bornologous, $w(a)=w(a^{-1})<\infty$. We claim that there is some symmetric open identity neighbourhood $V$ so that $$ C=\sup_{a\in V}w(a)<\infty. $$ If not, there are $a_n\rightarrow 1$ so that $w(a_n)>n$. But then $K=\{a_n,1\}_n$ is a compact and thus relatively (OB) set and $$ E_K=\{(x,y)\in G\times G\del x^{-1} y\in K\} $$ a coarse entourage on $G$. As $\sigma$ is bornologous, we see that $(\sigma\times \sigma)E_K$ is a coarse entourage in $X$, i.e., $\sup_{x^{-1} y \in K}d(\sigma(x),\sigma(y))<\infty$, contradicting the assumptions on $K$.
So pick $V$ as claimed and let $D\leqslant 1$ be some compatible left-invariant metric on $G$. Fix also $\epsilon>0$ so that $V$ contains the ball $B_D(2\epsilon)$. Define \[\begin{split} \partial(x,y)=\inf\Big(\sum_{i\in B}D(a_i,1)+\sum_{i\notin B}\big(w(a_i)+1\big)\Del x=ya_1\cdots a_n\;\;\&\;\; a_i\in V\text{ for }i\in B\Big). \end{split}\] Since $D$ is a compatible metric and $V\ni 1$ is open, $\partial$ is a continuous left-invariant \'ecart. Moreover, as $\partial\geqslant D$, we see that $\partial$ is a compatible metric on $G$.
Now, suppose $x,y\in G$ are given and write $x=ya_1\cdots a_n$ for some $a_i\in G$ with $a_i\in V$ for $i\in B$ so that $\sum_{i\in B}D(a_i,1)+\sum_{i\notin B}\big(w(a_i)+1\big)\leqslant \partial(x,y)+\epsilon$. Observe that, if $D(a_i,1)<\epsilon$ and $D(a_{i+1}, 1)< \epsilon$ for some $i<n$, then $a_ia_{i+1}\in V$, so, by coalescing $a_i$ and $a_{i+1}$ into a single term $a_ia_{i+1}\in V$, we only decrease the final sum. We may therefore assume that, for every $i<n$, either $D(a_i,1)\geqslant \epsilon$ or $D(a_{i+1}, 1)\geqslant \epsilon$. It thus follows that $$
\partial(x,y)+\epsilon\geqslant \sum_{i\in B}D(a_i,1)+\sum_{i\notin B}\big(w(a_i)+1\big)\geqslant \sum_{i\leqslant n}D(a_i,1)\geqslant \frac{n-1}2\cdot \epsilon, $$ i.e., $n\leqslant \frac {2\partial(x,y)}\epsilon+3$.
Therefore, \[\begin{split} d(\sigma(x),\sigma(y)) \leqslant&
\sum_{i\in B}d\big(\sigma(ya_1\cdots a_{i}), \sigma(ya_1\cdots a_{i-1})\big)\\
&+\sum_{i\notin B}d\big(\sigma(ya_1\cdots a_{i}), \sigma(ya_1\cdots a_{i-1})\big)\\
\leqslant & C\cdot n+\sum_{i\notin B}w(a_{i})\\ \leqslant & \Big(\frac{2C}\epsilon+1\Big)\cdot\partial(x,y) +3C+\epsilon. \end{split}\] In other words, $\sigma\colon (G,\partial)\rightarrow (X,d)$ is Lipschitz for large distances. \end{proof}
We can now extend the definition of the Haage\-rup property (see, e.g., Definition 2.7.5 \cite{bekka} or \cite{CCJJV}) from locally compact groups to the category of all topological groups. \begin{defi} A topological group $G$ is said to have the {\em Haagerup property} if it admits a coarsely proper continuous affine isometric action on a Hilbert space. \end{defi}
Thus, based on Theorem \ref{maurey} and Lemmas \ref{randrianarivony} and \ref{lipschitz}, we have the following reformulation of the Haage\-rup property for amenable Polish groups.
\begin{thm}\label{haagerup equiv} The following conditions are equivalent for an amenable Polish group $G$, \begin{enumerate} \item $G$ coarsely embeds into a Hilbert space, \item $G$ has the Haagerup property. \end{enumerate} \end{thm}
\begin{proof}(2)$\Rightarrow$(1): If $\alpha\colon G\curvearrowright \ku H$ is a coarsely proper continuous affine isometric action, with corresponding cocycle $b\colon G\rightarrow \ku H$, then $b\colon G\rightarrow \ku H$ is a uniformly continuous coarse embedding.
(1)$\Rightarrow$(2): If $\eta\colon G\rightarrow \ku H$ is a coarse embedding, then by Lemma \ref{lipschitz} there is a compatible left-invariant metric $d$ on $G$ so that $\eta\colon (G,d)\rightarrow \ku H$ is Lipschitz for large distances. Since $\eta$ is a coarse embedding, it must be expanding with respect to the metric $d$. It follows from Lemma \ref{randrianarivony} that $\eta\colon (G,d)\rightarrow \ku H$ may also be assumed to be uniformly continuous. Thus, by Theorem \ref{maurey}, there is a continuous affine isometric action $\alpha\colon G\curvearrowright \ku K$ on a Hilbert space $\ku K$ with associated cocycle $b\colon G\rightarrow \ku K$ so that, for all $g\in G$, $$ \kappa_\eta\big(d(g,f)\big)\leqslant \norm{b(g)-b(f)}\leqslant\theta_\eta\big(d(g,f)\big). $$ Since $\eta$ is a coarse embedding, the cocycle $b\colon G\rightarrow \ku K$ is coarsely proper and so is the action $\alpha$. \end{proof}
U. Haagerup \cite{haagerup} initially showed that finitely generated free groups have the Haagerup property. It is also known that amenable locally compact groups \cite{BCV} (see also \cite{CCJJV}) have the Haagerup property. However, this is not the case for amenable Polish groups. For example, a separable Banach space not coarsely embedding into Hilbert space such as $c_0$ of course also fails the Haagerup property.
There is also a converse to this. Namely, E. Guentner and J. Kaminker \cite{GK} showed that, if a finitely generated discrete group $G$ admits a affine isometric action on a Hilbert space whose cocycle $b$ grows faster than the square root of the word length, then $G$ is amenable (see \cite{tessera} for the generalisation to the locally compact case). It is not clear what, if any, generalisation of this is possible to the setting of (OB) generated Polish groups. For example, by Theorem 7.2 \cite{OB}, the homeomorphism group of the $n$-sphere $\mathbb S^n$ has property (OB) and thus is quasi-isometric to a point. It therefore trivially fulfills the assumptions of the Guentner--Kaminker theorem, but is not amenable.
It is by now well-known that the are Polish groups admitting no non-trivial continuous unitary or, equivalently, orthogonal, representations (the first example seems to be due to J. P. R. Christensen and W. Herer \cite{christensen}). Also many of these examples are amenable. While the non-existence of unitary representations may look like a local condition on the group that can be detected in neighbourhoods of the identity, the following result shows that under extra assumptions this is also reflected in the large scale behaviour of the group.
\begin{prop} Suppose $G$ is an amenable Polish group with no non-trivial unitary representations. Then $G$ is either coarsely equivalent to a point or is not coarsely embeddable into Hilbert space. \end{prop}
\begin{proof}Suppose $G$ is not coarsely equivalent to a point. Then, by Theorem \ref{haagerup equiv}, if $G$ is coarsely embeddable into Hilbert space, there is a coarsely proper continuous affine isometric action of $G$ on Hilbert space. So either the linear part $\pi$ is a non-trivial orthogonal representation of $G$ or the corresponding cocycle $b\colon G\rightarrow \ku H$ is a coarsely proper continuous homomorphism. Thus, as $G$ is not coarsely equivalent to a point, in the second case, $b$ is unbounded and so composing with an appropriate linear functional, we get an unbounded homomorphism into $\R$. In either case, $G$ will admit a non-trivial orthogonal and thus also unitary representation contrary to our assumptions. \end{proof}
Finally, let us sum up the equivalences for Banach spaces. \begin{thm}\label{haagerup banach} The following conditions are equivalent for a separable Banach space $X$, \begin{enumerate} \item $X$ coarsely embeds into a Hilbert space, \item $X$ uniformly embeds into a Hilbert space, \item $X$ admits an uncollapsed uniformly continuous map into a Hilbert space, \item $X$ has the Haagerup property. \end{enumerate} \end{thm}
Here Condition (4) of Theorem \ref{haagerup banach} is a priori the strongest of the four since a continuous coarsely proper cocycle $b\colon X\rightarrow \ku H$ will be simultaneously a uniform and coarse embedding. The equivalence of (1) and (2) was proved in \cite{randrianarivony} (the implication from (3) to (1) and (2) of course also being a direct consequence of Theorem \ref{ell p sum}), while the implication from (2) to (4) following from Theorem \ref{haagerup equiv}. A seminal characterisation of uniform embeddability into Hilbert spaces utilising a version of Lemma \ref{pre-maurey} appears in \cite{maurey}.
\section{Preservation of local structure}\label{super-refl} Weakening the geometric restrictions on the phase space from euclidean to uniformly convex, we still have a result similar to Theorem \ref{maurey}. However, in this case, we do not know if amenability suffices, but must rely on strengthenings of this.
Before we state the next result, we recall that a Banach space $X$ is said to be {\em finitely representable} in a Banach space $Y$ if, for every finite-dimensional subspace $F\subseteq X$ and every $\epsilon>0$, there is a linear embedding $T\colon F\rightarrow Y$ so that $\norm T\cdot \norm{T^{-1}}<1+\epsilon$. We also say that $X$ is {\em crudely finitely representable} in $Y$ if there is a constant $K$ so that every finite-dimensional subspace of $X$ is $K$-isomorphic to a subspace of $Y$. Finally, if $E$ is a Banach space and $1\leqslant p<\infty$, we let $L^p(E)$ denote the Banach space of equivalence classes of measurable functions $f\colon [0,1]\rightarrow E$ so that the $p$-norm $$ \norm{f}_{L^p(E)}=\Big(\int_0^1 \norm{f}_E^p \;d\lambda\Big)^{1/p} $$ is finite.
The next theorem is the basic result for preservation of local structure. Earlier versions of this are due to A. Naor and Y. Peres \cite{naor-peres} and V. Pestov \cite{pestov} respectively for finitely generated amenable and locally finite discrete groups.
\begin{thm}\label{pestov} Suppose that $d$ is a continuous left-invariant \'ecart on a F\o lner amenable Polish group $G$ and $\sigma\colon (G,d)\rightarrow E$ is a uniformly continuous and bornologous map into a Banach space $E$ with exact compression modulus $\tilde\kappa$ and expansion modulus $\theta$.
Then, for every $1\leqslant p<\infty$, there is a continuous affine isometric action of $G$ on a Banach space $V$ finitely representable in $L^p(E)$ with corresponding cocycle $b$ satisfying $$ \tilde\kappa\big(d(x,y)\big)\leqslant \norm{b(x)-b(y)}_V\leqslant \theta\big(d(x,y)\big) $$ for all $x,y\in G$. \end{thm}
\begin{proof}
Let us first assume that $G$ satisfies the second assumption of Definition \ref{folner} and let $\phi \colon H\rightarrow G$ be the corresponding mapping. Then, replacing $H$ by the amenable group $H/\ker \phi$, we may suppose that $\phi$ is injective. Let also $|\cdot|$ denote a right-invariant Haar measure on $H$. Since $H$ is amenable and locally compact, there is a F\o lner sequence $\{F_n\}$, i.e., a sequence of Borel subsets $F_n\subseteq H$ of finite positive meaure so that $\lim_n\frac{|F_nx\triangle F_n|}{|F_n|}=0$ for all $x\in H$. Let also $L^p(F_n,E)$ denote the Banach space of $p$-integrable functions $f\colon F_n\rightarrow E$ with norm $$ \norm{f}_{L^p(F_n,E)}=\Big(\int_{F_n}\norm{f(x)}_E^p\Big)^{1/p}. $$
Now fix a non-principal ultrafilter $\ku U$ on $\N$ and let $\big(\prod_nL^p(F_n,E)\big)_\ku U$ denote the corresponding ultraproduct. That is, if we equip $$ W=\{(\xi_n)\in \prod_nL^p(F_n,E)\del \sup_n\norm{\xi_n}_{L^p(F_n,E)}<\infty\} $$ with the semi-norm $$ \norm{(\xi_n)}_\ku U=\lim_\ku U\norm{\xi_n}_{L^p(F_n,E)} $$ and let $N=\{(\xi_n)\in W\del \norm{(\xi_n)}_W=0\}$ denote the corresponding null-space, then $\big(\prod_n\ell^p(F_n,E)\big)_\ku U$ is the quotient $W/N$. For simplicity of notation, if $(\xi_n)\in W$, we denote the element $(\xi_n)+N\in \big(\prod_nL^p(F_n,E)\big)_\ku U$ by $(\xi_n)_\ku U$.
Consider the linear operator $\Theta\colon L^\infty(H, E)\rightarrow \big(\prod_nL^p(F_n,E)\big)_\ku U$ given by $$
\Theta(f)= \big(|F_n|^{-1/p}\cdot f\!\upharpoonright_{F_n}\big)_\ku U. $$ Then $\norm{f}_{\ku U,p}=\Norm{\Theta(f)}_\ku U$ defines a semi-norm on $L^\infty(H, E)$ satisfying \[\begin{split} \norm{f}_{\ku U, p}^p
&=\NORM{\big(|F_n|^{-1/p}\cdot f\!\upharpoonright_{F_n}\big)_\ku U}_\ku U^p\\
&=\Big(\lim_\ku U\Norm{|F_n|^{-1/p}\cdot f\!\upharpoonright_{F_n}}_{L^p(F_n,E)}\Big)^p\\
&=\lim_\ku U\int_{F_n}\Norm{|F_n|^{-1/p}\cdot f(x)}_{E}^p\\
&=\lim_\ku U\frac 1{|F_n|}\int_{F_n}\Norm{f(x)}_E^p. \end{split}\]
We claim that $\norm{\cdot }_{\ku U,p}$ is invariant under the right-regular representation $\rho \colon H\curvearrowright L^\infty(H,E)$. Indeed, for all $f\in L^\infty(H,E)$ and $y\in H$, \[\begin{split} \norm{\rho(y)f}_{\ku U, p}^p-\norm{f}_{\ku U, p}^p
&=\lim_\ku U\frac 1{|F_n|}\int_{x\in F_n}\Norm{\big(\rho(y)f\big)(x)}_E^p-\lim_\ku U\frac 1{|F_n|}\int_{x\in F_n}\Norm{f(x)}_E^p\\
&=\lim_\ku U\frac 1{|F_n|}\Big(\int_{x\in F_n}\Norm{f(xy)}_E^p-\int_{x\in F_n}\Norm{f(x)}_E^p\Big)\\
&=\lim_\ku U\frac 1{|F_n|}\Big(\int_{x\in F_ny}\Norm{f(x)}_E^p-\int_{x\in F_n}\Norm{f(x)}_E^p\Big)\\
&\leqslant \lim_\ku U\frac 1{|F_n|}\int_{x\in F_ny\triangle F_n}\Norm{f(x)}_E^p\\
&\leqslant \norm{f}_{L^\infty(H, E)}^p\cdot \lim_\ku U \frac {|F_ny\triangle F_n|}{|F_n|}\\ &=0. \end{split}\]
Since $\phi\colon H\rightarrow G$ is a homomorphism and $d$ is left-invariant, for $x,y,z\in H$ we have $$ d\big(\phi(x),\phi(y)\big)=d\big(\phi(z)\phi(x),\phi(z)\phi(y)\big)=d\big(\phi(zx),\phi(zy)\big) $$ and hence \[\begin{split} \tilde\kappa\big[d\big(\phi(x),\phi(y)\big)\big] &=\tilde\kappa\big[d\big(\phi(zx),\phi(zy)\big)\big]\\ &\leqslant \Norm{\sigma\phi(zx)-\sigma\phi(zy)}_E\\ &= \Norm{\big(\rho(x)\sigma\phi\big)(z)-\big(\rho(y)\sigma\phi\big)(z)}_E\\ &\leqslant \theta\big[d\big(\phi(x),\phi(y)\big)\big]. \end{split}\] This shows that, for $x,y\in H$, we have $\big(\rho(x)\sigma\phi\big)-\big(\rho(y)\sigma\phi\big)\in L^\infty(H,E)$ and $$ \tilde\kappa\big[d\big(\phi(x),\phi(y)\big)\big]^p \leqslant
\frac 1{|F_n|}\int_{F_n}\Norm{\big(\rho(x)\sigma\phi\big)-\big(\rho(y)\sigma\phi\big)}_E^p \leqslant \theta[d\big(\phi(x),\phi(y)\big)\big]^p $$ for all $n$. By the expression for $\norm\cdot_{\ku U,p}$, it follows that $$ \tilde\kappa\big[d\big(\phi(x),\phi(y)\big)\big] \leqslant \Norm{ \big(\rho(x)\sigma\phi\big)-\big(\rho(y)\sigma\phi\big) }_{\ku U, p}\leqslant \theta\big[d\big(\phi(x),\phi(y)\big)\big]. $$
Therefore, by setting $y=1$, we see that the mapping $b\colon H\rightarrow L^\infty(H,E)$ given by $$ b(x)=\big(\rho(x)\sigma\phi\big)-\sigma\phi $$ is well-defined. Also, as $b(x)-b(y)= \big(\rho(x)\sigma\phi\big)-\big(\rho(y)\sigma\phi\big)$, we see that $$ \tilde\kappa\big[d\big(\phi(x),\phi(y)\big)\big] \leqslant \Norm{b(x)-b(y)}_{\ku U,p}\leqslant \theta\big[d\big(\phi(x),\phi(y)\big)\big]. $$ Since $\sigma$ uniformly continuous and thus $\lim_{\epsilon\rightarrow 0_+}\theta(\epsilon)=0$, it follows that $b$ is uniformly continuous with respect to the metric $d\big(\phi(\cdot),\phi(\cdot)\big)$ on $H$ and the semi-norm $\norm\cdot_{\ku U,p}$ on $L^\infty(H,E)$.
Let now $M\subseteq L^\infty(H,E)$ denote the null space of the semi-norm $\norm\cdot_{\ku U,p}$ and $X$ be the completion of $L^\infty(H,E)/M$ with respect to $\norm\cdot_{\ku U,p}$. Clearly, $X$ is isometrically embeddable into $\big(\prod_nL^p(F_n,E)\big)_\ku U$ and the right-regular representation $\rho\colon H\curvearrowright L^\infty(H,E)$ induces a linear isometric representation of $H$ on $X$, which we continue denoting $\rho$. Similarly, we view $b$ as a map into $X$.
As is easy to see, $b\in Z^1(H,\rho)$, that is, $b$ satisfies the cocycle identity $b(xy)=\rho(x)b(y)+b(x)$ for $x,y\in H$. So, in particular, the linear span of $b[H]$ is $\rho[H]$-invariant. Moreover, since each $\rho(x)\in \rho[H]$ is an isometry, the same holds for the closed linear span $V\subseteq X$ of $b[H]$.
We claim that, for every $\xi\in V$, the map $x\mapsto \rho(x)\xi$ is uniformly continuous with respect to the metric $d\big(\phi(\cdot),\phi(\cdot)\big)$. Since the linear span of $b[H]$ is dense in $V$, it suffices to prove this for $\xi\in b[H]$. So fix some $z\in H$ and note that, for $x,y\in H$, we have \[\begin{split} \Norm{\rho(x)b(z)-\rho(y)b(z)}_{\ku U,p} &=\Norm{\big(b(xz)-b(x)\big)-\big(b(yz)-b(y)\big)}_{\ku U,p}\\ &\leqslant\Norm{b(xz)-b(yz)}_{\ku U,p}+\Norm{b(y)-b(x)}_{\ku U,p}\\ &\leqslant\theta\big[ d\big(\phi(xz), \phi(yz)\big) \big]+\theta\big[ d\big(\phi(x),\phi(y) \big)\big]. \end{split}\] Now, suppose that $\epsilon>0$ is given. Then, by uniform continuity of $\sigma$, there is $\delta>0$ so that $\theta(\delta)<\frac \eps2$. Also, since multiplication by $\phi(z)$ on the right is left-uniformly continuous on $G$, there is an $\eta>0$ so that $d\big(\phi(x),\phi(y) \big)<\eta$ implies that $d\big(\phi(xz),\phi(yz) \big)<\delta$. It follows that, provided $d\big(\phi(x),\phi(y) \big)<\min \{\eta,\delta\}$, we have $\Norm{\rho(x)b(z)-\rho(y)b(z)}_{\ku U,p}<\epsilon$, hence verifying uniform continuity.
To sum up, we have now an isometric linear representation $\rho\colon H\curvearrowright V$ with associated cocycle $b\colon H\rightarrow V$ so that, with respect to the metric $d\big(\phi(\cdot),\phi(\cdot)\big)$ on $H$, the mappings $b$ and $x\mapsto \rho(x)\xi$ are uniformly continuous for all $\xi\in V$.
Identifying $H$ with its image in $G$ via the continuous embedding $\phi$, it follows that there are unique continuous extensions of these mappings to all of $G$, which we continue denoting $b$ and $x\mapsto \rho(x)\xi$. Also, simple arguments using continuity and density show that, for all $x\in G$, the map $\xi\mapsto \rho(x)\xi$ defines a linear isometry $\rho(x)$ of $V$ so that $\rho(xy)=\rho(x)\rho(y)$ and $b(xy)=\rho(x)b(y)+b(x)$. In other words, $\rho$ is a continuous isometric linear representation and $b\in Z^1(G,\rho)$. Moreover, $$ \tilde\kappa\big[d(x,y)\big] \leqslant \Norm{b(x)-b(y) }_{\ku U, p}\leqslant \theta\big[d(x,y)\big] $$ for all $x,y\in G$.
To finish the proof, it now suffices to verify that $V$ is finitely representable in $L^p(E)$. To see this, note that, since each $L^p(F_n,E)$ is isometrically a subspace of $L^p(E)$, it follows from the properties of the ultraproduct that $\big(\prod_nL^p(F_n,E)\big)_\ku U$ is finitely representable in $L^p(E)$. Also, by construction, $X$ and a fortiori its subspace $V$ are isometrically embeddable into $\big(\prod_nL^p(F_n,E)\big)_\ku U$, which proves the theorem under the second assumption.
Consider now instead the case when $G$ is approximately compact. The proof is very similar to the second case, so we shall just indicate the changes needed. Thus, let $K_1\leqslant K_2\leqslant K_3\leqslant\ldots\leqslant G$ be a chain of compact subgroups with dense union in $G$. Instead of considering the sets $F_n$ with the Haar measure $|\cdot|$ from $H$, we now use the $K_n$ with their respective Haar measures and similarly $L^p(K_n,E)$ in place of $L^p(F_n,E)$. As before, the ultraproduct $\big(\prod_nL^p(K_n,E)\big)_\ku U$ is finitely representable in $L^p(E)$.
Also, if $x\in \bigcup_nK_n$, then the right-regular representation $\rho(x)$ defines a linear isometry of all but finitely many $L^p(K_n,E)$, which means that $\rho(x)$ induces a linear isometry of the ultraproduct $\big(\prod_nL^p(K_n,E)\big)_\ku U$ simply by letting $\rho(x)\big(\xi_n\big)_\ku U=\big(\rho(x)\xi_n\big)_\ku U$.
Also, observe that, for $x,y\in G$, we have $\rho(x)\sigma,\rho(y)\sigma\in L^p(K_n,E)$ for all $n$ and by computations similar to those above, we find that $$ \tilde\kappa\big[d\big(\phi(x),\phi(y)\big)\big] \leqslant \NORM{ \big(\rho(x)\sigma-\rho(y)\sigma \big)_\ku U }_{\ku U}\leqslant \theta\big[d\big(\phi(x),\phi(y)\big)\big]. $$ We may therefore define $b\colon G\rightarrow \big(\prod_nL^p(K_n,E)\big)_\ku U$ by letting $$ b(x)=\big(\rho(x)\sigma-\sigma\big)_\ku U $$ and note that $$ \tilde\kappa\big[d\big(\phi(x),\phi(y)\big)\big] \leqslant \Norm{ b(x)-b(y) }_{\ku U}\leqslant \theta\big[d\big(\phi(x),\phi(y)\big)\big]. $$ One easily sees that, when restricted to $\bigcup_nK_n$, $b$ is a cocycle associated to the representation $\rho\colon \bigcup_nK_n\curvearrowright \big(\prod_nL^p(K_n,E)\big)_\ku U$.
Moreover, as before, we verify that, for every $\xi \in V=\overline{b[\bigcup_nK_n]}$, the mapping $x\in \bigcup_nK_n\mapsto \rho(x)\xi\in V$ is left-uniformly continuous, so the representation $\rho$ extends to an isometric linear representation of $G$ on $V$ with associated cocycle $b$. \end{proof}
Naor and Peres \cite{naor-peres} showed the above result for finitely generated discrete amenable groups using F\o lner sequences and asked in this connection whether one might bypass the F\o lner sequence of the proof and instead proceed directly from an invariant mean on the group. Of course, for a finitely generated group, this question is a bit vague since having an invariant mean or a F\o lner sequence are both equivalent to amenability. However, for general Polish groups this is not so and we may therefore provide a precise statement capturing the essence of their question. \begin{prob}\label{naor} Does Theorem \ref{pestov} hold for a general amenable Polish group $G$? \end{prob}
\begin{rem} Subsequently to the appearence of the present paper in preprint, F. M. Schneider and A. Thom have developed a more general notion of F\o lner sets in topological groups \cite{thom} and were able to combine this with the mechanics of the above proof to solve Problem \ref{naor} in the affirmative. Moreover, they were also able to construct an example of an amenable Polish group, which is not F\o lner amenable, thereby showing that their result is indeed a proper strengthening of the above. \end{rem}
Our main concern here being the existence of coarsely proper affine isometric actions, let us first consider the application of Theorem \ref{pestov} to that problem. \begin{cor}\label{fin repr} Let $G$ be a F\o lner amenable Polish group admitting a uniformly continuous coarse embedding into a Banach space $E$. Then $G$ admits a coarsely proper continuous affine isometric action on a Banach space $V$ that is finitely representable in $L^2(E)$. \end{cor} Though we do not in general have tools permitting us to circumvent the assumption of uniform continuity as in Lemma \ref{randrianarivony}, for non-Archimedean groups we do. Indeed, assume $\sigma\colon G\rightarrow E$ is a coarse embedding of a non-Archimedean Polish group into a Banach space $E$. Then $G$ is locally (OB) and thus has an open subgroup $V\leqslant G$ with property (OB) relative to $G$. Since $\sigma$ is bornologous, there is a constant $K>0$ so that $\norm{\sigma(g)-\sigma(f)}\leqslant K$ whenever $f^{-1} g\in V$. Letting $X\subseteq G$ denote a set of left-coset representatives for $V$, we define $\eta(g)=\sigma(h)$, where $h\in X$ is the coset representative of $gV$. Then $\norm{\eta(g)-\sigma(g)}\leqslant K$ for all $g\in G$, so $\eta\colon G\rightarrow \ku H$ is a coarse embedding and clearly constant on left-cosets of $V$, whence also uniformly continuous on $G$. \begin{cor}\label{fin repr2} Let $G$ be a F\o lner amenable non-Archimedean Polish group admitting a coarse embedding into a Banach space $E$. Then $G$ admits a coarsely proper continuous affine isometric action on a Banach space $V$ that is finitely representable in $L^2(E)$. \end{cor}
Our second task is now to identify various properties of a Banach space $E$ that are inherited by any space $V$ finitely representable in $L^2(E)$ (or in other $L^p(E)$). Evidently, these must be {\em local properties} of Banach spaces, i.e., only dependent on the class of finite-dimensional subspaces of the space in question.
We recall that a Banach space $V$ is {\em super-reflexive} if every other space crudely finitely representable in $V$ is reflexive. Since evidently $V$ is finitely representable in itself, super-reflexive spaces are reflexive. Also, every space that is crudely finitely representable in a super-reflexive space must not only be reflexive but even super-reflexive. Moreover, super-reflexive spaces are exactly those all of whose ultrapowers are reflexive. By a result of P. Enflo \cite{enflo} (see also G. Pisier \cite{pisier} for an improved result or \cite{fabian} for a general exposition), the super-reflexive spaces can also be characterised as those admitting an equivalent uniformly convex renorming.
It follows essentially from the work of J. A. Clarkson \cite{clarkson} that, if $E$ is uniformly convex, then so is $L^p(E)$ for all $1<p<\infty$. In particular, if $E$ is super-reflexive, then so is every space finitely representable in $L^2(E)$.
For a second application, we shall note the preservation of Rademacher type and cotype in the above construction. For that we fix a {\em Rademacher sequence}, i.e., a sequence $(\epsilon_n)_{n=1}^\infty$ of mutually independent random variables $\epsilon_n\colon \Omega\rightarrow \{-1,1\}$, where $(\Omega, \mathbb P)$ is some probability space, so that $\mathbb P(\epsilon_n=-1)=\mathbb P(\epsilon_n=1)=\frac 12$. E.g., we could take $\Omega=\{-1,1\}^\N$ with the usual coin tossing measure and let $\epsilon_n(\omega)=\omega(n)$. \begin{defi} A Banach space $X$ is said to have {\em type} $p$ for some $1\leqslant p\leqslant 2$ if there is a constant $C$ so that $$ \Big(\mathbb E\NORM{\sum_{i=1}^n\epsilon_ix_i}^p\Big)^\frac 1p\leqslant C\cdot\Big(\sum_{i=1}^n\norm{x_i}^p\Big)^\frac 1p $$ for every finite sequence $x_1,\ldots, x_n\in X$.
Similarly, $X$ has {\em cotype} $q$ for some $2\leqslant q<\infty$ if there is a constant $K$ so that $$ \Big(\sum_{i=1}^n\norm{x_i}^q\Big)^\frac 1q\leqslant K\cdot \Big(\mathbb E\NORM{\sum_{i=1}^n\epsilon_ix_i}^q\Big)^\frac 1q $$ for every finite sequence $x_1,\ldots, x_n\in X$. \end{defi} We note that, by the triangle inequality, every Banach space has type $1$. Similarly, by stipulation, every Banach space is said to have cotype $q=\infty$.
Whereas the $p$ in the formula $\Big(\sum_{i=1}^n\norm{x_i}^p\Big)^\frac 1p$ is essential, this is not so with the $p$ in $\Big(\mathbb E\NORM{\sum_{i=1}^n\epsilon_ix_i}^p\Big)^\frac 1p$. Indeed, the Kahane--Khintchine inequality (see \cite{albiac}) states that, for all $1<p<\infty$, there is a constant $C_p$ so that, for every Banach space $X$ and $x_1,\ldots, x_n\in X$, we have $$ \mathbb E\NORM{\sum_{i=1}^n\epsilon_ix_i} \leqslant \Big(\mathbb E\NORM{\sum_{i=1}^n\epsilon_ix_i}^p\Big)^\frac 1p \leqslant C_p\cdot\mathbb E\NORM{\sum_{i=1}^n\epsilon_ix_i}. $$ In particular, for any $p,q\in [1,\infty[$, the two expressions $\Big(\mathbb E\NORM{\sum_{i=1}^n\epsilon_ix_i}^p\Big)^\frac 1p$ and $\Big(\mathbb E\NORM{\sum_{i=1}^n\epsilon_ix_i}^q\Big)^\frac 1q$ differ at most by some fixed multiplicative constant independent of the space $X$ and the vectors $x_i\in X$.
Clearly, if $E$ has type $p$ or cotype $q$, then so does every space crudely finitely representable in $X$. Also, by results of W. Orlicz and G. Nordlander (see \cite{albiac}), the space $L^p$ has type $p$ and cotype $2$, whenever $1\leqslant p\leqslant 2$, and type $2$ and cotype $p$, whenever $2\leqslant p<\infty$. Moreover, assuming again that $E$ has type $p$ or cotype $q$, then $L^2(E)$ has type $p$, respectively, cotype $q$. We refer the reader to \cite{albiac} for more information on Rademacher type and cotype.
By the above discussion, Corollary \ref{fin repr} gives us the following.
\begin{cor}\label{type} Let $G$ be a F\o lner amenable Polish group admitting a uniformly continuous coarse embedding into a Banach space $E$ that is either (a) super-reflexive, (b) has type $p$ or (c) cotype $q$. Then $G$ admits a coarsely proper continuous affine isometric action on a Banach space $V$ that is super-reflexive, has type $p$, respectively, has cotype $q$. \end{cor} In particular, this applies when $G$ is a Banach space. Also, in fact, any combination of properties (a), (b) and (c) verified by $E$ can be preserved by $V$.
Corollary \ref{type} applies, in particular, when $G$ admits a uniformly continuous coarse embedding into an $L^p$ space. In this connection, let us note that, by results of J. Bretagnolle, D. Dacunha-Castelle and J.-L. Krivine \cite{bretagnolle2, bretagnolle} and M. Mendel and Naor \cite{naor1}, for $1\leqslant q\leqslant p\leqslant \infty$, there is a map $\phi\colon L^q\rightarrow L^p$ which is simultaneously a uniform and coarse embedding. Thus, if $G$ admits a uniformly continuous coarse embedding into $L^p$, $p\leqslant 2$, then it also admits such an embedding into $L^2$. On the other hand, by results of Mendel and Naor \cite{naor2}, $L^q$ does not embed coarsely into $L^p$, whenever $\max \{2,p\}<q<\infty$.
Using Theorems \ref{ell p sum} and \ref{pestov}, we conclude the following. \begin{thm}\label{pestov banach} Let $\sigma\colon X\rightarrow E$ be an uncollapsed uniformly continuous map between separable Banach spaces. Then, for every $1\leqslant p<\infty$, $X$ admits a coarsely proper continuous affine isometric action on a Banach space $V$ finitely representable in $L^p(E)$.
\end{thm}
Also, similarly to the proof of Corollary \ref{balls hilbert}, we may conclude the following. \begin{cor}\label{balls local} Let $\sigma\colon X\rightarrow E$ be a uniformly continuous map from a Banach space $X$ into a super-reflexive or superstable space satisfying $$ \inf_{\norm{x-y}=r}\norm{\sigma(x)-\sigma(y)}>0 $$ for just some $r>0$. Then $B_X$ is uniformly embeddable into a super-reflexive, respectively, superstable space. \end{cor} Again, by Proposition 5.3 \cite{raynaud}, in the superstable case, we may further conclude that $X$ has finite cotype. In the super-reflexive case, we may combine this with a result of Kalton to conclude the following. \begin{cor}\label{kalton-ros} Let $\sigma\colon X\rightarrow E$ be a uniformly continuous map from a Banach space $X$ with non-trivial type into a super-reflexive space satisfying $$ \inf_{\norm{x-y}=r}\norm{\sigma(x)-\sigma(y)}>0 $$ for just some $r>0$. Then $X$ is super-reflexive. \end{cor} \begin{proof} Indeed, by Theorem 5.1 \cite{kalton}, if $B_X$ is uniformly embeddable into a uniformly convex space or, equivalently, into a super-reflexive space, and $X$ has non-trivial type, then $X$ is super-reflexive. The result now follows by applying Corollary \ref{balls local}. \end{proof}
\section{Stable metrics, wap functions and reflexive spaces} While it would be desirable to have a result along the lines of Theorems \ref{maurey} and \ref{pestov} for reflexive spaces, things are more complicated here and requires auxiliary concepts, namely, stability and weakly almost periodic functions.
\begin{defi} A function $\Phi\colon X\times X\rightarrow \R$ on a set $X$ is {\em stable} provided that, for all sequences $(x_n)$ and $(y_m)$ in $X$, we have $$ \lim_{n\rightarrow \infty}\lim_{m\rightarrow \infty}\Phi(x_n,y_m)=\lim_{m\rightarrow \infty}\lim_{n\rightarrow \infty}\Phi(x_n,y_m), $$ whenever both limits exist in $\R$. \end{defi} Observe that, if $\Phi$ is a metric or just an \'ecart on $X$, then, if either of the two limits exists, both $(x_n)$ and $(y_m)$ are bounded. Thus, for an \'ecart to be stable, it suffices to verify the criterion for bounded sequences $(x_n)$ and $(y_m)$. With this observation in mind, one can show that an \'ecart (respectively, a bounded function) $\Phi$ is stable provided that, for all bounded $(x_n)$, $(y_m)$ (respectively all $(x_n)$, $(y_m)$), we have $$ \lim_{n\rightarrow \ku U}\lim_{m\rightarrow \ku V}\Phi(x_n,y_m)=\lim_{m\rightarrow \ku V}\lim_{n\rightarrow \ku U}\Phi(x_n,y_m) $$ whenever $\ku U, \ku V$ are non-principal ultrafilters on $\N$.
We remark that a simple, but tedious, inspection shows that, if $d$ is a stable \'ecart on $X$, then so is the uniformly equivalent bounded \'ecart $D(x,y)=\min\{d(x,y),1\}$.
In keeping with the terminology above, a Banach space $(X, \norm\cdot)$ is {\em stable} if the norm metric is stable. This class of spaces was initially studied by J.-L. Krivine and Maurey \cite{KM} in which they showed that every stable Banach space contains a copy of $\ell^p$ for some $1\leqslant p<\infty$. As we shall see, for several purposes including the geometry of Banach spaces, the following more general class of groups plays a central role. \begin{defi} A topological group $G$ is {\em metrically stable} if $G$ admits a compatible left-invariant stable metric. \end{defi}
Recall that a bounded function $\phi\colon G\rightarrow \R$ on a group $G$ is said to be {\em weakly almost periodic (WAP)} provided that its orbit $\lambda(G)\phi=\{\phi(g^{-1}\,\cdot\, )\del g\in G\}$ under the left regular representation $\lambda\colon G\curvearrowright \ell^\infty(G)$ is a relatively weakly compact subset of $\ell^\infty(G)$. The connection between stability and weak almost periodicity is provided by the following well-known criterion of A. Grothendieck \cite{grothendieck}. \begin{thm}[A. Grothendieck \cite{grothendieck}] A bounded function $\phi\colon G\rightarrow \R$ on a group $G$ is weakly almost periodic if and only if the function $\Phi(x,y)=\phi(x^{-1} y)$ is stable. \end{thm} There is a tight connection between weakly almost periodic functions and representations on reflexive Banach space borne out by several classical results. Central among these is the fact that every continuous bounded weakly almost periodic function $\phi$ on a topological group $G$ is a {\em matrix coefficient} of a strongly continuous isometric linear representation $\pi\colon G\curvearrowright E$ on a reflexive Banach space $E$, that is, for some $\xi \in E$ and $\eta^*\in E^*$, $$ \phi(x)=\langle \pi(x)\xi, \eta^*\rangle. $$ Conversely, every such matrix coefficient is weakly almost periodic.
We shall need some more recent results clarifying reflexive representability of Polish groups. Here a topological group $G$ is said to {\em admit a topologically faithful isometric linear representation on a reflexive Banach space $E$} if $G$ is isomorphic to a subgroup of the linear isometry group ${\rm Isom}(E)$ equipped with the strong operator topology. The following theorem combines results due to A. Shtern \cite{shtern}, M. Megrelishvili \cite{megrelishvili2} and I. Ben Yaacov, A. Berenstein and S. Ferri \cite{ben yaacov}.
\begin{thm}\cite{shtern,megrelishvili2, ben yaacov}\label{smben} For a Polish group $G$, the following are equivalent, \begin{enumerate} \item $G$ is metrically stable, \item $G$ has a topologically faithful isometric linear representation on a separable reflexive Banach space, \item for every identity neighbourhood $V\subseteq G$, there is a continuous weakly almost periodic function $\phi\colon G\rightarrow [0,1]$ so that $\phi(1_G)=1$ and ${\rm supp}(\phi)\subseteq V$. \end{enumerate} \end{thm} In particular, the isometry group of a separable reflexive Banach space is metrically stable. Moreover, if $E$ is a stable Banach space, then ${\rm Isom}(E)$ is metrically stable. Indeed, by Lemma \ref{cocycle isometry group}, ${\rm Isom}(E)$ admits a cocycle $b\colon {\rm Isom}(E)\rightarrow \ell^2(E)$ which is a uniform embedding. As also $\ell^2(E)$ is stable, it follows that $d(x,y)=\norm{b(x)-b(y)}_{\ell^2(E)}$ is a compatible left-invariant stable metric on ${\rm Isom}(E)$.
Conversely, there are examples, such as the group ${\rm Homeo}_+[0,1]$ of increasing homeomorphisms of the unit interval \cite{megrelishvili1}, that admit no non-trivial continuous isometric linear actions on a reflexive space.
We now aim at extending earlier results of Y. Raynaud \cite{raynaud} on the existence of $\ell^p$ subspaces of Banach spaces. Observe first that every left-invariant compatible metric on a Banach space is in fact bi-invariant and uniformly equivalent to the norm metric. Thus, a Banach space is metrically stable exactly when it admits an invariant stable metric uniformly equivalent to the norm. Formulated in our terminology, Th\'eor\`eme 4.1 \cite{raynaud} states the following. \begin{thm}[Y. Raynaud]\label{raynaud} Every metrically stable Banach space contains an isomorphic copy of $\ell^p$ for some $1\leqslant p<\infty$. \end{thm}
Also, by Theorem \ref{smben} and Proposition \ref{cocycle amplification banach}, we see that a metrically stable separable Banach space admits a coarsely proper continuous affine isometric action on a separable reflexive Banach space.
Furthermore, in Th\'eor\`eme 0.2 \cite{raynaud}, Raynaud showed that any Banach space uniformly embeddable in a {\em superstable} Banach space, i.e., a space all of whose ultrapowers are stable, is metrically stable (in fact, that it has a compatible invariant superstable metric).
Note finally that, since by \cite{raynaud} the class of superstable Banach spaces is closed under passing from $E$ to $L^p(E)$, $1\leqslant p<\infty$, and under finite representability, Theorem \ref{pestov banach} implies that a separable Banach space $E$ uniformly embeddable into a superstable Banach space also admits a coarsely proper continuous cocycle on a superstable space.
\begin{thm}\label{embedding super-reflexive} Let $X$ be a separable Banach space admitting an uncollapsed uniformly continuous map into the unit ball $B_E$ of a super-reflexive Banach space $E$. Then $X$ is metrically stable and contains an isomorphic copy of $\ell^p$ for some $1\leqslant p<\infty$. \end{thm}
\begin{proof}Let $\sigma\colon X\rightarrow B_E$ be the map in question. By replacing $E$ by the closed linear span of the image of $Y$, we can suppose that $E$ is separable. Let $\kappa$ and $\theta$ be the compression and expansion moduli of $\sigma$. As $\sigma$ is uncollapsed, $\kappa(t)>0$ for some $t>0$ and, as $\sigma$ maps into $B_E$, we have $\theta(s)\leqslant 2$ for all $s$. By Theorem \ref{pestov}, there is a Banach space $V$ finitely representable in $L^2(E)$ and a strongly continuous isometric linear representation $\pi\colon X\curvearrowright V$ with a continuous cocycle $b\colon X\rightarrow V$ satisfying $$ \kappa\big(\norm{x-y}\big)\leqslant \norm{b(x)-b(y)}_V\leqslant \theta\big(\norm{x-y}\big). $$ Being finitely representable in the super-reflexive space $L^2(E)$, it follows that $V$ is super-reflexive itself. Also, $b$ is bounded and uncollapsed, whence, by Proposition \ref{affine banach}, $b$ is a uniform embedding of $X$ into $V$. Being a bounded cocycle in a super-reflexive space, we conclude by the Ryll-Nardzewski fixed point theorem that $b$ is a coboundary, i.e., that $b(x)=\xi-\pi(x)\xi$ for some $\xi\in V$ and all $x\in X$.
Now, since $x\mapsto \xi-\pi(x)\xi$ is a uniform embedding, so is $x\mapsto \pi(x)\xi$, which shows that $\pi\colon X\rightarrow {\rm Isom}(V)$ is a topologically faithful isometric linear representation on a reflexive Banach space. By Theorem \ref{smben}, we conclude that $X$ is metrically stable. Finally, by Theorem \ref{raynaud}, $X$ contains an isomorphic copy of $\ell^p$ for some $1\leqslant p<\infty$. \end{proof}
A question first raised by Aharoni \cite{aharoni2} (see also the discussion in Chapter 8 \cite{lindenstrauss}) is to determine the class of Banach spaces $X$ uniformly embeddable into their unit ball $B_X$. It follows from \cite{aharoni} that $c_0$ embeds uniformly into $B_{c_0}$ and, in fact, every separable Banach space $X$ containing $c_0$ embeds uniformly into its ball $B_X$. Similarly, for $1\leqslant p\leqslant 2$, $L^p([0,1])$ is uniformly embeddable into the unit ball of $L^2([0,1])$ \cite{maurey}. Since also, for all $1\leqslant p<\infty$, the unit balls of $L^p([0,1])$ and $L^2([0,1])$ are uniformly homeomorphic by a result of E. Odell and Th. Schlumprecht \cite{distortion}, it follows that $L^p([0,1])$ in uniformly embeddable into the ball $B_{L^p([0,1])}$ for $1\leqslant p\leqslant 2$. This fails for $p>2$ by results of \cite{maurey}.
\begin{cor} Let $E$ be a separable super-reflexive Banach space not containing $\ell^p$ for any $1\leqslant p<\infty$ or, more generally, which is not metrically stable. Then $E$ is not uniformly embeddable into $B_E$. \end{cor}
For example, this applies to the $2$-convexification $T_2$ of the Tsirelson space and to V. Ferenczi's uniformly convex HI space \cite{ferenczi}.
One may wonder whether Theorem \ref{embedding super-reflexive} has an analogue for uniform embeddings into super-reflexive spaces as opposed to their balls. I.e., if $X$ is an infinite-dimensional Banach space uniformly embeddable into a super-reflexive space, does it follow that $X$ contains an isomorphic copy of $\ell^1$ or an infinite-dimensional super-reflexive subspace? However, as shown by B. Braga (Corollary 4.15 \cite{braga3}), Tsirelson's space $T$ uniformly embeds into the super-reflexive space $(T_2)_{T_2}$ without, of course, containing $\ell^1$ or a super-reflexive subspace. The best one might hope for is thus some asymptotic regularity property of $X$.
We shall now consider the existence of coarsely proper continuous affine isometric actions on reflexive spaces by potentially non-amenable groups. For locally compact second countable groups, such actions always exist, since N. Brown and E. Guentner \cite{BG} showed that every countable discrete group admits a proper affine isometric action on a reflexive Banach space and U. Haagerup and A. Przybyszewska \cite{haagerup-affine} generalised this to locally compact second countable groups. Also, Kalton \cite{kalton} showed that every stable metric space may be coarsely embedded into a reflexive Banach space, while, on the contrary, the Banach space $c_0$ does not admit a coarse embedding into a reflexive Banach space.
Our goal here is to provide an equivariant counter-part of Kalton's theorem.
\begin{thm}\label{stable metric refl} Suppose a topological group $G$ carries a continuous left-invariant coarsely proper stable \'ecart. Then $G$ admits a coarsely proper continuous affine isometric action on a reflexive Banach space. \end{thm}
Theorem \ref{stable metric refl} is a corollary of the following more detailed result.
\begin{thm}\label{wap refl} Suppose $d$ is a continuous left-invariant \'ecart on a topological group $G$ and assume that, for all $\alpha>0$, there is a continuous weakly almost periodic function $\phi\in \ell^\infty(G)$ with $d$-bounded support so that $\phi\equiv 1$ on $D_\alpha=\{g\in G\del d(g,1)\leqslant \alpha\}$.
Then $G$ admits a continuous isometric action $\pi\colon G\curvearrowright X$ on a reflexive Banach space and a continuous cocycle $b\colon G\rightarrow X$ that is coarsely proper with respect to the \'ecart $d$. \end{thm}
\begin{proof}Under the given assumptions, we claim that, for every integer $n\geqslant 1$, there is a continuous weakly almost periodic function $0\leqslant \phi_n\leqslant 1$ on $G$ so that \begin{enumerate} \item $\norm{\phi_n}_\infty=\phi(1_G)=1$, \item$\norm{\phi_n-\lambda(g)\phi_n}_\infty\leqslant \frac 1{4^n} \text{ for all } g\in D_n$ and \item ${\rm supp}(\phi_n)$ is $d$-bounded. \end{enumerate}
To see this, we pick inductively sequences of continuous weakly almost periodic functions $(\psi_i)_{i=1}^{4^n}$ and radii $(r_i)_{i=0}^{4^n}$ so that \begin{enumerate} \item[(i)] $0=r_0<2n<r_1<r_1+2n<r_2<r_2+2n<r_3<\ldots<r_{4^n}$, \item[(ii)] $0\leqslant \psi_i\leqslant 1$, \item[(iii)] ${\psi}_{i}\equiv 1$ on $D_{r_{i-1}+n}$, \item[(iv)] ${\rm supp}(\psi_i)\subseteq D_{r_i}$. \end{enumerate} Note first that, by the choice of $r_i$, the sequence $$ D_{r_0+n}\setminus D_{r_0},\; D_{r_1}\setminus D_{r_0+n}, \;D_{r_1+n}\setminus D_{r_1},\; D_{r_2}\setminus D_{r_1+n}, \; \ldots\;, D_{r_{4^n}}\setminus D_{r_{4^n-1}+n}, \; G\setminus D_{r_{4^n}} $$ partitions $G$. Also, for all $1\leqslant i\leqslant 4^n$, $$ \psi_1\equiv \ldots\equiv \psi_i\equiv 0, \text{ while } \psi_{i+1}\equiv \ldots\equiv \psi_{4^n}\equiv 1\text{ on }D_{r_i+n}\setminus D_{r_i} $$ and $$ \psi_1\equiv \ldots\equiv \psi_{i-1}\equiv 0, \text{ while } \psi_{i+1}\equiv \ldots\equiv \psi_{4^n}\equiv 1\text{ on }D_{r_i}\setminus D_{r_{i-1}+n}. $$ Setting $\phi_n=\frac 1{4^n}\sum_{i=1}^{4^n}\psi_i$, we note that, for all $1\leqslant i\leqslant 4^n$, $$ \phi_n\equiv \frac {4^n-i}{4^n} \quad \text{ on }D_{r_i+n}\setminus D_{r_i} $$ and $$
\frac {4^n-i}{4^n}\leqslant \phi_n\leqslant \frac {4^n-i+1}{4^n} \quad \text{ on }D_{r_{i}}\setminus D_{r_{i-1}+n}. $$
Now, if $g\in D_n$ and $f\in G$, then $|d(g^{-1} f,1)-d(f,1)|=|d(f,g)-d(f,1)|\leqslant d(g,1)\leqslant n$. So, if $f$ belongs to some term in the above partition, then $g^{-1} f$ either belongs to the immediately preceding, the same or the immediately following term of the partition. By the above estimates on $\phi_n$, it follows that $|\phi_n(f)-\phi_n(g^{-1} f)|\leqslant \frac1{4^n}$. In other words, for $g\in D_n$, we have \[\begin{split}
\norm{\phi_n-\lambda(g)\phi_n}_\infty= \sup_{f\in G}|\phi_n(f)-\phi_n(g^{-1} f)|\leqslant \frac 1{4^n}, \end{split}\] which verifies condition (2). Conditions (1) and (3) easily follow from the construction.
Consider now a specific $\phi_n$ as above and define $$ W_n=\overline{\rm conv}\big(\lambda(G)\phi_n\cup -\lambda(G)\phi_n\big)\subseteq B_{\ell^\infty(G)} $$ and, for every $k\geqslant 1$, $$ U_{n,k}=2^kW_n+2^{-k}B_{\ell^\infty(G)}. $$ Let $\norm{\cdot}_{n,k}$ denote the gauge on ${\ell^\infty}(G)$ defined by $U_{n,k}$, i.e., $$ \norm{\psi}_{n,k}=\inf(\alpha>0\del \psi\in \alpha\cdot U_{n,k}). $$
If $g\in D_n$, then $\norm{\phi_n-\lambda(g)\phi_n}_\infty\leqslant \frac 1{4^n}$ and so, for $k\leqslant n$, $$ \phi_n-\lambda(g)\phi_n\in \frac 1{2^n}\cdot 2^{-k}B_{\ell^\infty(G)}\subseteq \frac 1{2^n}\cdot U_{n,k}. $$ In particular, \begin{equation}\label{a} \norm{\phi_n-\lambda(g)\phi_n}_{n,k}\leqslant \frac 1{2^n},\;\;\text{ for all } k\leqslant n \text{ and }g\in D_n. \end{equation}
On the other hand, for all $g\in G$ and $k$, we have $\phi_n-\lambda(g)\phi_n\in 2W_n\subseteq \frac1{2^{k-1}}U_{n,k}$. Therefore, \begin{equation}\label{b} \norm{\phi_n-\lambda(g)\phi_n}_{n,k}\leqslant \frac 1{2^{k-1} },\;\;\text{ for all } k\text{ and }g. \end{equation}
Finally, since $U_{n,1}\subseteq 2B_{\ell^\infty(G)}+\frac 12B_{\ell^\infty(G)}=\frac 52B_{\ell^\infty(G)}$, we have $\norm\cdot_\infty\leqslant \frac 52\norm\cdot_{n,1}$. So, if $g\notin ({\rm supp}\;\phi_n)^{-1}$, then \begin{equation}\label{c} \norm{\phi_n-\lambda(g)\phi_n}_{n,1}\geqslant \frac 25,\;\;\text{ for all } g\notin ({\rm supp}\;\phi_n)^{-1}. \end{equation}
It follows from (\ref{a}) and (\ref{b}) that, for $g\in D_n$, we have \[\begin{split} \sum_k\norm{\phi_n-\lambda(g)\phi_n}^2_{n,k} &\leqslant \underbrace{\Big(\frac 1{2^n}\Big)^2+\ldots+\Big(\frac 1{2^n}\Big)^2}_{n\text{ times}} + \Big(\frac 1{2^{(n+1)-1}}\Big)^2+ \Big(\frac 1{2^{(n+2)-1}}\Big)^2+\ldots\\ &\leqslant\frac 1{2^{n-2}}, \end{split}\] while using (\ref{c}) we have, for $g\notin ({\rm supp}\;\phi_n)^{-1}$, $$ \sum_k\norm{\phi_n-\lambda(g)\phi_n}^2_{n,k}\geqslant \frac4{25}. $$
Define $\triple{\cdot}_n$ on $\ell^\infty(G)$ by $\triple{\psi}_n=\big(\sum_k\norm{\psi}_{n,k}^2\big)^{\frac 12}$ and set $$ X_n=\{\psi\in \overline{\rm span}(\lambda(G)\phi_n)\subseteq \ell^\infty(G)\del \triple{\psi}_n<\infty\}\subseteq \ell^\infty(G). $$ By the main result of W. J. Davis, T. Figiel, W. B. Johnson and A. Pe\l czy\'nski \cite{dfjp}, the interpolation space $(X_n,\triple{\cdot}_n)$ is a reflexive Banach space. Moreover, since $W_n$ and $U_{n,k}$ are $\lambda(G)$-invariant subsets of $\ell^\infty(G)$, one sees that $\norm{\cdot}_{n,k}$ and $\triple{\cdot}_n$ are $\lambda(G)$-invariant and hence we have an isometric linear representation $\lambda\colon G\curvearrowright (X_n, \triple{\cdot}_n)$.
Note that, since $\phi_n\in W_n$, we have $\phi_n\in X_n$ and can therefore define a cocycle $b_n\colon G\rightarrow X_n$ associated to $\lambda$ by $b_n(g)=\phi_n-\lambda(g)\phi_n$. By the estimates above, we have $$ \triple{b_n(g)}_n=\triple{\phi_n-\lambda(g)\phi_n}_n\leqslant\big( \frac 1{{\sqrt 2}}\big)^{n-2} $$ for $g\in D_n$, while $$ \triple{b_n(g)}_n=\triple{\phi_n-\lambda(g)\phi_n}_n\geqslant\frac2{5} $$ for $g\notin ({\rm supp}\;\phi_n)^{-1}$.
Let now $Y=\big(\bigoplus_n(X_n,\triple{\cdot}_n)\big)_{\ell^2}$ denote the $\ell^2$-sum of the spaces $(X_n,\triple{\cdot}_n)$. Let also $\pi\colon G\curvearrowright Y$ be the diagonal action and $b=\bigoplus b_n$ the corresponding cocycle. To see that $b$ is well-defined, note that, for $g\in D_n$, we have \[\begin{split} \norm{b(g)}_Y &=\Big(\sum_{m=1}^\infty\triple{b_m(g)}_m^2\Big)^\frac 12\\ &=\Big(\text{finite}+\sum_{m=n}^\infty\triple{b_m(g)}_m^2\Big)^\frac 12\\ &\leqslant\Big(\text{finite}+\sum_{m=n}^\infty\frac 1{2^{m-2}}\Big)^\frac 12\\ &<\infty, \end{split}\] so $b(g)\in Y$.
Remark that, whenever $g\notin ({\rm supp}\;\phi_n)^{-1}$, we have $$ \norm{b(g)}_Y\geqslant \Big( \underbrace{\big(\frac 25\big)^2+\ldots+\big(\frac 25\big)^2}_{n\text{ times}} \Big)^\frac 12\geqslant \frac {\sqrt n}3. $$ As $({\rm supp}\;\phi_n)^{-1}$ is $d$-bounded, this shows that the cocycle $b\colon G\rightarrow Y$ is coarsely proper with respect to $d$. We leave the verification that the action is continuous to the reader. \end{proof}
Let us now see how to deduce Theorem \ref{stable metric refl} from Theorem \ref{wap refl}. So fix a continuous left-invariant coarsely proper stable \'ecart $d$ on $G$ with corresponding balls $D_\alpha$. Then, for every $\alpha>0$, we can define a continuous bounded weakly almost periodic function $\phi_\alpha\colon G\rightarrow \R$ by $$ \phi_\alpha(g)=2-\max\Big\{1, \min\big\{\frac{d(g,1)}\alpha, 2\big\}\Big\}. $$ We note that $\phi_\alpha$ has $d$-bounded support, while $\phi_\alpha\equiv 1$ on $D_\alpha$, thus verifying the conditions of Theorem \ref{wap refl}.
\section{A fixed point property for affine isometric actions}
\subsection{Kalton's theorem and solvent maps}\label{section solvent} The result of this final section has a different flavor from the preceding ones, in that our focus will be on the interplay between coarse geometry and harmonic analytic properties of groups as related to fixed points of affine isometric actions. As a first step we must consider a weakening of the concept of coarse embeddings and show how it relates to work of Kalton.
If $M\subseteq \N$ is an infinite set and $r\geqslant 1$, let $P_r(M)$ be the set of all $r$-tuples $n_1<\ldots<n_r$ with $n_i\in M$. We define a graph structure on $P_r(M)$ by letting two $r$-tuples $n_1<\ldots< n_r$ and $m_1<\ldots<m_r$ be related by an edge if $$ n_1\leqslant m_1\leqslant n_2\leqslant m_2\leqslant \ldots\leqslant n_r\leqslant m_r $$ or vice versa. When equipped with the induced path metric, one sees that $P_r(M)$ is a finite diameter metric space with the exact diameter depending on $r$.
\begin{thm}[N. J. Kalton \cite{kalton}]\label{kalton1} Suppose $r\in \N$ and let $E$ be a Banach space such that $E^{(2r)}$ is separable. Then given any uncountable family $\{f_i\}_{i\in I}$ of bounded maps $f_i\colon P_r(\N)\rightarrow E$ and any $\epsilon>0$, there exist $i\neq j$ and an infinite subset $M\subseteq \N$ so that $$ \norm{f_i(\sigma)-f_j(\sigma)}<\theta_{f_i}(1)+\theta_{f_j}(1)+\epsilon $$ for all $\sigma\in P_r(M)$. \end{thm}
Kalton uses this result to show that $c_0$ neither embeds uniformly nor coarsely into a reflexive Banach space. However a close inspection of the proof reveals a stronger result in the coarse setting.
\begin{defi} A map $\phi\colon (X,d)\rightarrow (M,\ku E)$ from a metric space $X$ to a coarse space $M$ is said to be {\em solvent} if, for every coarse entourage $E\in \ku E$ and $n\geqslant 1$, there is a constant $R$ so that $$ R\leqslant d(x,y)\leqslant R+n\;\;\Rightarrow\;\; \big(\phi(x),\phi(y)\big)\notin E. $$ \end{defi} In the case $(M,\partial)$ is a metric space, our definition becomes more transparent. Indeed $\phi$ is solvent if there are constants $R_n$ for all $n\geqslant 1$ so that $$ R_n\leqslant d(x,y)\leqslant R_n+n\;\;\Rightarrow\;\; \partial\big(\phi(x),\phi(y)\big)\geqslant n. $$ Also, in case $X$ is actually a geodesic metric space, we have the following easy reformulation. \begin{lemme}\label{solvent reform} Let $(X,d)$ be a geodesic metric space of infinite diameter and suppose that $\phi\colon (X,d)\rightarrow (M,\partial)$ is a bornologous map into a metric space $(M,\partial)$. Then $\phi$ is solvent if and only if $$ \sup_{t}\tilde\kappa_\phi(t)=\sup_{t}\inf_{d(x,y)=t}\partial\big(\phi(x),\phi(y)\big)=\infty $$ \end{lemme}
\begin{proof} Suppose that the second condition holds and find constants $t_n$ with $$ \inf_{d(x,y)=t_n}\partial\big(\phi(x),\phi(y)\big)\geqslant n. $$ Since $\phi$ is bornologous and $X$ geodesic, $\phi$ is Lipschitz for large distances and hence $$ \partial\big(\phi(x),\phi(y)\big)\leqslant K\cdot d(x,y)+K $$ for some constant $K$ and all $x,y\in X$.
Now, assume $t_{n^2}\leqslant d(x,y)\leqslant t_{n^2}+n$. Then, as $X$ is geodesic, there is some $z\in X$ with $d(x,z)=t_{n^2}$, while $d(z,y)\leqslant n$. It follows that $\partial\big(\phi(x),\phi(z)\big)\geqslant n^2$, while $\partial\big(\phi(z),\phi(y)\big)\leqslant Kn+K$, i.e., $$ \partial\big(\phi(x),\phi(y)\big)\geqslant n^2-Kn-K\geqslant n $$ provided $n$ is sufficiently large. Setting $R_n=t_{n^2}$, we see that $\phi$ is solvent.
The converse is proved by noting that every distance is attained in $X$. \end{proof}
While a solvent map $\phi\colon X\rightarrow M$ remains solvent when post-composed with a coarse embedding of $M$ into another coarse space, then dependence on $X$ is somewhat more delicate. For this and ulterior purposes, we need the notion of near isometries. \begin{defi} A map $\phi\colon X\rightarrow Y$ between metric spaces $X$ and $Y$ is said to be a {\em near isometry} if there is a constant $K$ so that, for all $x,x'\in X$, $$ d(x,x')-K\leqslant d(\phi(x),\phi(x'))\leqslant d(x,x')+K. $$ \end{defi} It is fairly easy to find a near isometry $\phi\colon X\rightarrow X$ of a metric space $X$ that is not close to any isometry of $X$, i.e., so that $\sup_xd(\phi(x),\psi(x))=\infty$ for any isometry $\psi$ of $X$. However, in the case of Banach spaces, positive results do exist. Indeed, by the work of several authors, in particular, P. M. Gruber \cite{gruber} and J. Gervirtz \cite{gervirtz}, if $\phi\colon X\rightarrow Y$ is a surjective near isometry between Banach spaces $X$ and $Y$ with defect $K$ as above and $\phi(0)=0$, then there is surjective linear isometry $U\colon X\rightarrow Y$ with $\sup_{x\in X}\norm{\phi(x)-U(x)}\leqslant 4K$ (see Theorem 15.2 \cite{lindenstrauss}).
Now, suppose instead that $\phi$ neither surjective nor is $\phi(0)=0$, but only that $\phi$ $K$-cobounded, i.e., $\sup_{y\in Y}\inf_{x\in X}\norm{y-\phi(x)}\leqslant K$. Then $Y$ has density character at most that of $X$ and hence cardinality at most that of $X$, whence a short argument shows that there is a surjective map $\psi\colon X\rightarrow Y$ with $\psi(0)=0$ so that $\sup_{x\in X}\norm{\psi(x)+\phi(0)-\phi(x)}\leqslant 42K$. In particular, $\psi$ is a near isometry with defect $85K$, whence there is a surjective linear isometry $U\colon X\rightarrow Y$ with $\sup_{x\in X}\norm{\psi(x)-U(x)}\leqslant 340K$. All in all, we find that $$ \sup_{x\in X}\norm{A(x)-\phi(x)}\leqslant 382K, $$ where $A\colon X\rightarrow Y$ is the surjective affine isometry $A=U+\phi(0)$. In other words, every cobounded near isometry $\phi\colon X\rightarrow Y$ is close to a surjective affine isometry $A\colon X\rightarrow Y$.
Let us also note the following straightforward fact. \begin{lemme}\label{coarse dependence} Let $$ X\overset{\sigma}\longrightarrow Y\overset{\phi}\longrightarrow Z \overset{\psi}\longrightarrow W $$ be maps between metric spaces $X$, $Y$ and coarse spaces $Z$, $W$ so that $\sigma$ is a near isometry, $\phi$ is solvent and $\psi$ is a coarse embedding. Then the composition $\psi\phi\sigma\colon X\rightarrow W$ is also solvent. \end{lemme}
To gauge of how weak the existence of solvable maps is, we may reutilse the ideas of Section \ref{uniform vs coarse}. \begin{prop}\label{triviality} Supppose $X$ and $E$ are Banach spaces and that there is no uniformly continuous solvent map $\psi\colon X\rightarrow \ell^2(E)$. Then, for every uniformly continuous map $\phi\colon X\rightarrow E$, we have $$ \sup_r\inf_{\norm{x-y}=r}\Norm{\phi(x)-\phi(y)}=0. $$ \end{prop}
\begin{proof} Assume for a contradiction that $\phi\colon X\rightarrow E$ is a uniformly continuous map and that $r>0$ satisfies $\inf_{\norm{x-y}=r}\Norm{\phi(x)-\phi(y)}=\delta>0$. Without loss of generality, we may assume that $\phi(0)=0$.
Now fix $n\geqslant 1$ and choose $\epsilon_n>0$ small enough that $\theta_\phi(\epsilon_n)\leqslant \frac \delta{n2^n}$, where $\theta_\phi$ is the expansion modulus of $\phi$. Set also $\psi_n(x)=\frac n\delta\phi\big(\frac {\epsilon_n} n x\big)$. Then \[\begin{split} \norm{x-y}\leqslant n &\;\Rightarrow\; \NORM{\phi\Big(\frac {\epsilon_n} nx\Big)-\phi\Big(\frac {\epsilon_n} ny\Big)}\leqslant \frac \delta{n2^n}\\ &\;\Rightarrow\; \norm{\psi_n(x)-\psi_n(y)}\leqslant \frac 1{2^n}, \end{split}\] while \[\begin{split} \norm{x-y}=\frac {rn}{\epsilon_n} &\;\Rightarrow\; \norm{\psi_n(x)-\psi_n(y)}\geqslant n. \end{split}\]
Finally, let $\psi\colon X\rightarrow \ell^2(E)$ be defined by $\psi(x)=(\psi_1(x), \psi_2(x),\ldots)$. Then the above inequalities show that $\psi$ is well-defined and uniformly continuous and also that, for every $n\geqslant 1$, there is some $R$ with $$ \norm{x-y}=R \;\Rightarrow\; \norm{\psi_n(x)-\psi_n(y)}_E\geqslant n\;\Rightarrow\; \norm{\psi(x)-\psi(y)}_{\ell^2(E)}\geqslant n. $$ Applying Lemma \ref{solvent reform}, we conclude that $\psi$ is solvent, contradicting our assumptions. \end{proof}
We now come to the improved statement of Kalton's theorem. \begin{thm}\label{kalton2} Every bornologous map $\phi\colon c_0\rightarrow E$ into a reflexive Banach space $E$ is insolvent. \end{thm}
\begin{proof} Let $(e_k)_{k=1}^\infty$ denote the canonical unit vector basis for $c_0$. If $a\subseteq \N$ is a finite subset, we let $\chi_a=\sum_{k\in a}e_k$. Now, for $r\geqslant 1$ and $A\subseteq \N$ infinite, define $$ f_{r,A}(n_1,\ldots, n_r)=\sum_{i=1}^r\chi_{A\cap[1,n_i]}. $$ Note that, if $n_1\leqslant m_1\leqslant n_2\leqslant m_2\leqslant \ldots\leqslant n_r\leqslant m_r$ with $n_1<n_2\ldots< n_r$ and $m_1<m_2<\ldots<m_r$, then $\norm{f_{r,A}(n_1,\ldots, n_r)-f_{r,A}(m_1,\ldots, m_r)}\leqslant 1$, that is, $\theta_{f_{r,A}}(1)\leqslant 1$ and thus also $\theta_{\phi f_{r,A}}(1)\leqslant \theta_\phi(1)$.
Now fix $r$. Then by Theorem \ref{kalton1} there are infinite subsets $A,B, M\subseteq \N$ so that $$ \norm{\phi f_{r,A}(\sigma)-\phi f_{r,B}(\sigma)}<\theta_{\phi f_{r,A}}(1)+\theta_{\phi f_{r,B}}(1)+1\leqslant 2\,\theta_\phi(1)+1. $$ for all $\sigma\in P_r(M)$. On the other hand, as $A\neq B$, there is some $\sigma\in P_r(M)$ so that $\norm{f_{r,A}(\sigma)-f_{r,B}(\sigma)}=r$, whereby $$ \inf_{\norm{x-y}=r}\norm{\phi(x)-\phi(y)} \leqslant\norm{\phi f_{r,A}(\sigma)-\phi f_{r,B}(\sigma)}\leqslant 2\,\theta_\phi(1)+1. $$ As any interval of the form $[R,R+n]$, for $n\geqslant 1$, contains an integral point $r$, we see that $\phi$ is insolvent. \end{proof}
\begin{cor}\label{kalton3} Every bornologous map $\phi\colon c_0\rightarrow L^1([0,1])$ is insolvent. \end{cor}
\begin{proof} By Lemma \ref{coarse dependence}, it suffices to observe that $L^1([0,1])$ coarsely embeds into the reflexive space $L^2([0,1])$, which follows from \cite{bretagnolle2}. \end{proof}
Applying Proposition \ref{triviality} and the fact that $E\mapsto \ell^2(E)$ preserves reflexivity, we also obtain the following corollary. \begin{cor} Every uniformly continuous map $\phi\colon c_0\rightarrow E$ into a reflexive Banach space satisfies $$ \sup_r\inf_{\norm{x-y}=r}\Norm{\phi(x)-\phi(y)}=0. $$ \end{cor}
For a thorough study of regularity properties of Banach spaces preserved under solvent maps, the reader may consult \cite{braga2}.
\subsection{Geometric Gelfand pairs} Our next step is to identify a class of topological groups whose large scale geometry is partially preserved through images by continuous homomorphisms. This is closely related to the classical notion of Gelfand pairs.
\begin{defi}\label{geom gelfand} A {\em geometric Gelfand pair} consists of a coarsely proper continuous isometric action $G\curvearrowright X$ of a topological group $G$ on a metric space $X$ so that, for some constant $K$ and all $x,y,z,u\in X$ with $d(x,y)\leqslant d(z,u)$, there is $g\in G$ with $d\big(g(x),z\big)<K$ and $d\big(z,g(y)\big)+d\big(g(y),u\big)<d(z,u)+K$. \end{defi}
An immediate observation is that, if $G\curvearrowright X$ is a geometric Gelfand pair and $x,z\in X$, we may set $x=y$ and $z=u$ and thus find some $g\in G$ so that $d\big(g(x),z)<K$. In other words, the action is also cobounded, whereby the orbit map $g\in G\mapsto g(x)\in X$ is a coarse equivalence between $G$ and $X$ for any choice of $x\in X$.
To better understand the notion of geometric Gelfand pairs, suppose that $G\curvearrowright X$ is a coarsely proper continuous and {\em doubly transitive} isometric action on a geodesic metric space, i.e., so that, for all $x,y,z,v\in X$ with $d(x,y)=d(z,v)$, we have $g(x)=z$ and $g(y)=v$ for some $g\in G$. Then $G\curvearrowright X$ is a geometric Gelfand pair. Indeed, if $d(x,y)\leqslant d(z,u)$, pick by geodecity some $v\in X$ so that $d(z,v)+d(v,u)=d(z,u)$ and $d(z,v)=d(x,y)$. Then, by double transitivity, there is $g\in G$ with $g(x)=z$ and $g(y)=v$, verifying that this is a geometric Gelfand pair.
Observe here that it suffices that $X$ is geodesic with respect to its distance set $S=d(X\times X)$, i.e., that, for all $x,y\in X$ and $s\in S$ with $s\leqslant d(x,y)$, there is $z\in X$ with $d(x,z)+d(z,y)=d(x,y)$ and $d(x,z)=s$.
If $K$ is a compact subgroup of a locally compact group $G$, then $(G,K)$ is said to be a {\em Gelfand pair} if the convolution algebra of compactly supported $K$-bi-invariant continuous functions on $G$ is commutative (see Section 3.3 \cite{bekka} or Section 24.8 \cite{simonnet}). A basic result due to I. M. Gelfand (Proposition 24.8.1 \cite{simonnet}) states that, if $G$ admits a involutory automorphism $\alpha$ so that $g^{-1} \in K\alpha(g)K$ for all $g\in G$, then $(G,K)$ is a Gelfand pair.
In the case above of a coarsely proper continuous and doubly transitive isometric action $G\curvearrowright X$ on a geodesic metric space, let $K=\{g\in G\del g(x)=x\}$, which is a closed subgroup of $G$. Then, if $g,f\in G$ and $f\in KgK$, there is some $h\in K$ so that $f(x)=hg(x)$ and so $$ d(f(x),x)=d(hg(x),x)=d(g(x), h^{-1}(x))=d(g(x),x). $$ Conversely, if $g,f\in G$ and $d(f(x),x)=d(g(x),x)$, then, by double transitivity, there is some $h\in G$ with $hf(x)=g(x)$ and $h(x)=x$, i.e., $g^{-1} hf\in K$ and $h\in K$ whereby $f\in KgK$. In other words, $$ f\in KgK\;\;\Longleftrightarrow\;\; d(f(x), x)=d(g(x),x). $$ As $d(g^{-1}(x), x)=d(x,g(x))$, this shows that $g^{-1}\in KgK$ for all $g\in G$. So $(G,K)$ fulfils the condition of Gelfand's result, except for the fact that $G$ and $K$ may not be locally compact, respectively, compact. This motivates the terminology of Definition \ref{geom gelfand}.
\begin{exa} Probably the simplest non-trivial example of a geometric Gelfand pair is that of the canonical action $D_\infty \curvearrowright \Z$ of the infinite dihedral group $D_\infty$ on $\Z$ with the euclidean metric. This action is doubly transitive and proper. As moreover $\Z$ is geodesic with respect to its distance set $\N$, we see that $D_\infty \curvearrowright \Z$ is a geometric Gelfand pair. \end{exa} \begin{exa} Similarly, if ${\rm Aff}(\R^n)=O(n)\ltimes \R^n$ denotes the group of (necessarily affine) isometries of the $n$-dimensional euclidean space, then ${\rm Aff}(\R^n)\curvearrowright\R^n$ is a geometric Gelfand pair. \end{exa}
Another class of examples are constructed as above from Banach spaces. For this, let $X$ be a separable Banach space and ${\rm Isom}(X)$ the group of linear isometries of $X$ equipped with the strong operator topology. As every isometry of $X$ is affine, the group of all isometries of $X$ decomposes as the semi-direct product ${\rm Aff}(X)={\rm Isom}(X)\ltimes (X,+)$. Also, $(X,\norm\cdot)$ is said to be {\em almost transitive} if the action ${\rm Isom}(X)\curvearrowright X$ induces a dense orbit on the unit sphere $S_X$ of $X$. \begin{prop} Let $X$ be a separable Banach space. Then ${\rm Aff}(X)\curvearrowright X$ is a geometric Gelfand pair if and only if ${\rm Isom}(X)$ has property (OB) and $(X,\norm\cdot)$ is almost transitive. \end{prop} \begin{proof} First, as shown in \cite{rosendal-coarse}, the tautological action ${\rm Aff}(X)\curvearrowright X$ is coarsely proper if and only if ${\rm Isom}(X)$ has property (OB). So assume that ${\rm Isom}(X)$ has property (OB) and consider the transitivity condition.
We observe that, if $(X,\norm\cdot)$ is almost transitive, then ${\rm Aff}(X)\curvearrowright X$ is almost doubly transitive in the sense that, for all $x,y,z,u\in X$ with $\norm{x-y}=\norm{z-u}$ and $\epsilon>0$ there is $g\in {\rm Aff}(X)$ so that $g(x)=z$ and $\norm{g(y)-u}<\epsilon$. In this case, ${\rm Aff}(X)\curvearrowright X$ is a geometric Gelfand pair. Conversely, suppose ${\rm Aff}(X)\curvearrowright X$ is a geometric Gelfand pair with a defect $K$. Then, for $y,z\in S_X$ and $\epsilon>0$, there is some $g\in {\rm Aff}(X)$ so that $\norm{g(0)-0}<K$ and $\norm{g(\frac{2K}{\epsilon}y)-\frac {2K}{\epsilon}z}<K$. Letting $f\in {\rm Isom}(X)$ be the linear isometry $f(x)=g(x)-g(0)$, we find that $\norm{f(\frac{2K}{\epsilon}y)-\frac {2K}{\epsilon}z}<2K$, i.e., $\norm{f(y)-z}<\epsilon$, showing that $(X,\norm\cdot)$ is almost transitive. \end{proof} Examples of almost transitive Banach spaces include $L^p([0,1])$ for all $1\leqslant p<\infty$ \cite{rolewicz} and the Gurarii space $\mathbb G$ \cite{lusky}. Moreover, as all of these spaces are separably categorical in the sense of continuous model theory, it follows from Theorem 5.2 \cite{OB} that their linear isometry group has property (OB). We therefore conclude that ${\rm Aff}(L^p([0,1]))\curvearrowright L^p([0,1])$ with $1\leqslant p<\infty$ and ${\rm Aff}(\mathbb G)\curvearrowright\mathbb G$ are geometric Gelfand pairs.
\begin{exa} Suppose $\Gamma$ is a countable {\em metrically homogeneous} connected graph, that is, so that any isometry $f\colon A\rightarrow B$ with respect to the path metric on $\Gamma$ between two finite subsets extends to a full automorphism of $\Gamma$. Then the action ${\rm Aut}(\Gamma)\curvearrowright \Gamma$ is coarsely proper \cite{rosendal-coarse} and metrically doubly transitive. Since also the path metric is $\Z$-geodesic, it follows that ${\rm Aut}(\Gamma)\curvearrowright \Gamma$ is a geometric Gelfand pair.
Examples of metrically homogeneous countable graphs include the $n$-regular trees $T_n$ for all $1\leqslant n\leqslant \aleph_0$ and the integral Urysohn metric space $\Z\U$. \end{exa}
This latter may be described as the Fra\"iss\'e limit of all finite $\Z$-metric spaces (i.e., with integral distances) and plays the role of a universal object in the category of $\Z$-metric spaces. Alternatively, $\Z\U$ is the unique universal countable $\Z$-metric space so that any isometry between two finite subspaces extends to a full isometry of $\Z\U$. The rational Urysohn metric space $\Q\U$ is described similarly with $\Q$ in place of $\Z$.
Let us also recall that, since $\Q\U$ is countable, ${\rm Isom}(\Q\U)$ is a Polish group when equipped with the {\em permutation group topology}, which is that obtained by declaring pointwise stabilisers to be open.
\begin{exa} As shown in \cite{autom}, the tautological action ${\rm Isom}(\Q\U)\curvearrowright \Q\U$ is coarsely proper. Since it is also doubly transitive and $\Q\U$ is $\Q$-geodesic, it follows that ${\rm Isom}(\Q\U)\curvearrowright \Q\U$ is a geometric Gelfand pair. \end{exa}
\subsection{Nearly isometric actions and quasi-cocycles} Having identified the class of geometric Gelfand pairs, we will now show a strong geometric rigidity property for their actions.
\begin{defi} Let $G$ be a group and $M$ a metric space. A {\em nearly isometric action} of $G$ on $M$ is a map $\alpha\colon G\times M\rightarrow M$ so that, for some constant $C$ and all $g,f\in G$ and $x,y\in M$, we have $$ d\big(\alpha(g,x), \alpha(g, y)\big) \leqslant d(x,y)+C $$ and $$ d\big(\alpha(f, \alpha(g,x)), \alpha(fg, x)\big) \leqslant C. $$ \end{defi} Thus, if for $g\in G$ we let $\alpha(g)$ denote the map $\alpha(g, \cdot)\colon M\rightarrow M$, we see firstly that each $\alpha(g)$ is a contraction $M\rightarrow M$ up to an additive defect $C$ and secondly that $\alpha(f)\alpha(g)$ and $\alpha(fg)$ agree as maps on $M$ up to the same additive defect $C$. Thus, a nearly isometric action needs neither be a true action nor do the maps need to be exact isometries.
Nearly isometric actions can be constructed from Banach space valued quasi-cocycles. \begin{defi} A map $b\colon G\rightarrow E$ from a group $G$ to a Banach space $E$ is said to be a {\em quasi-cocycle} provided that there is an isometric linear representation $\pi\colon G\curvearrowright E$ so that $$ \sup_{g,f\in G}\Norm{\pi(g)b(f)+b(g)-b(gf)}<\infty. $$ \end{defi} Observe that if $b\colon G\rightarrow E$ is a quasi-cocycle associated to an isometric linear representation $\pi\colon G\curvearrowright E$, then the formula $$ \alpha(g)x= \pi(g)x+b(g) $$ defines a nearly isometric action of $G$ on $E$. In fact, in this case, each map $\alpha(g)$ is an actual isometry. We also have the following converse to this.
\begin{lemme} Suppose that $\alpha\colon G\times E\rightarrow E$ is a nearly isometric action of a group $G$ on a Banach space $E$ so that the $\alpha(g)\colon E\rightarrow E$ are uniformly cobounded. Then $b\colon G\rightarrow E$ defined by $b(g)=\alpha(g)0$ is a quasi-cocycle associated to an isometric linear representation $\pi\colon G\curvearrowright E$ so that $$ \sup_{g\in G}\sup_{x\in E}\norm{\pi(g)x+b(g)-\alpha(g)x}<\infty. $$ \end{lemme}
\begin{proof} Since the maps $\alpha(g)$ are uniformly cobounded, there is a constant $C$ larger than the defect of $\alpha$ so that $\inf_{y\in E}\norm{x-\alpha(g)y}< C$ for all $g\in G$ and $x\in E$.
Given any $x\in E$, find $y\in E$ with $\norm{x-\alpha(1)y}\leqslant C$ and observe that \[\begin{split} \norm{\alpha(1)x-x} &\leqslant \norm{\alpha(1)x-\alpha(1)y}+C\\ &\leqslant \norm{\alpha(1)x-\alpha(1)\alpha(1)y}+2C\\ &\leqslant \norm{x-\alpha(1)y}+3C\\ &\leqslant 4C. \end{split}\] Thus, for all $x,y\in E$ and $g\in G$, \[\begin{split} \norm{x-y} &\leqslant \norm{\alpha(1)x-\alpha(1)y}+8C\\ &\leqslant \norm{\alpha(g^{-1})\alpha(g)x-\alpha(g^{-1})\alpha(g)y}+10C\\ &\leqslant \norm{\alpha(g)x-\alpha(g)y}+11C\\ &\leqslant \norm{x-y}+12C, \end{split}\] showing that $\alpha(g)$ is a near isometry of $E$ with defect $11C$. Hence, as observed in Section \ref{section solvent}, there is a linear isometry $\pi(g)$ of $E$ so that, for $b(g)=\alpha(g)0$ and $K=5000C$, we have $$ \sup_{x\in E}\norm{\pi(g)x+b(g)-\alpha(g)x}\leqslant K. $$ It follows that the map $A\colon G\rightarrow {\rm Aff}(E)$ given by $A(g)x=\pi(g)x+b(g)$ defines a {\em rough action} of $G$ on $E$ in the sense of Section 13.1 \cite{monod2}, that is, each $A(g)$ is an affine isometry of $E$ and $\sup_{x\in E}\norm{A(g)A(f)x-A(gf)x}\leqslant 4K<\infty$. In particular, $\pi\colon G\rightarrow {\rm Isom}(E)$ is an isometric linear representation of $G$ and $b\colon G\rightarrow E$ an associated quasi-cocycle (Lemma 13.1.2 \cite{monod2}). Indeed $\norm{\pi(g)b(f)+b(g)-b(gf)}=\norm{A(g)A(f)0-A(gf)0}\leqslant 4K$, whereby \[\begin{split} \norm{\pi(g)\pi(f)x-&\pi(gf)x}\\ &\leqslant\Norm{\pi(g)\big(\pi(f)x+b(f)\big)+b(g)-\big(\pi(gf)x+b(gf)\big)}+4K\\ &\leqslant\Norm{A(g)A(f)x-A(gf)x}+4K\\ &\leqslant 8K \end{split}\] for all $x\in E$. As $\norm{\pi(g)\pi(f)x-\pi(gf)x}$ is positive homogeneous in $x$, we see that $\pi(g)\pi(f)=\pi(gf)$ for all $g,f\in G$. \end{proof}
As our applications deal with topological groups, it is natural to demand that a nearly isometric action $\alpha$ respects some of the topological group structure. As continuity may be too restrictive, a weaker assumption is that some orbit map is bornologous.
\begin{lemme}\label{baire orbit map} Let $\alpha\colon G\times M\rightarrow M$ be a nearly isometric action of a Polish group. Then, if some orbit map $g\in G\mapsto \alpha(g)x\in M$ is Baire measurable, it is also bornologous. \end{lemme}
\begin{proof} For every $n$, let $B_n=\{g\in G\del d\big(\alpha(g)x,x\big)<n\;\&\; d\big(\alpha(g^{-1})x,x\big)<n\}$. Then $G=\bigcup_nB_n$ is a covering of $G$ by countable many Baire measurable symmetric subsets, whence by the Baire category theorem some $B_n$ must be nonmeagre. Applying Pettis' theorem, it follows that $B_nB_n$ is an identity neighbourhood in $G$.
Now suppose that $A\subseteq G$ is relatively (OB) and find a finite set $F\subseteq G$ and $m\geqslant 1$ so that $A\subseteq (FB_nB_n)^m$. Let also $K$ be a number larger than all of $n$, $\max_{f\in F}d\big(\alpha(f)x,x\big)$ and the defect of $\alpha$. Then, if $h,g\in G$ and $h^{-1} g\in A$, there are $k_1,\ldots, k_{2m}\in B_n$ and $f_1,\ldots, f_m\in F$ with $g=hf_1k_1k_2h_2k_3k_4\cdots f_mk_{2m-1}k_{2m}$. Thus, \[\begin{split} d\big(\alpha(g)x,\alpha(h)x\big) = & d\big(\alpha(hf_1k_1k_2\cdots f_mk_{2m-1}k_{2m})x,\alpha(h)x\big)\\ \leqslant & d\big(\alpha(hf_1k_1k_2\cdots f_mk_{2m-1}k_{2m})x,\alpha(h)\alpha(f_1k_1k_2\cdots f_mk_{2m-1}k_{2m})x\big)\\ &+d\big(\alpha(h)\alpha(f_1k_1k_2\cdots f_mk_{2m-1}k_{2m})x,\alpha(h)x\big)\\ \leqslant & d\big(\alpha(f_1k_1k_2\cdots f_mk_{2m-1}k_{2m})x,x\big)+2K\\ \leqslant & d\big(\alpha(f_1k_1k_2\cdots f_mk_{2m-1}k_{2m})x,\alpha(f_1)\alpha(k_1k_2\cdots f_mk_{2m-1}k_{2m})x\big)\\ &+ d\big(\alpha(f_1)\alpha(k_1k_2\cdots f_mk_{2m-1}k_{2m})x,\alpha(f)x\big)+ d\big(\alpha(f)x,x\big)+2K\\ \leqslant & d\big(\alpha(k_1k_2\cdots f_mk_{2m-1}k_{2m})x,x\big)+5K\\ \leqslant &\ldots\\ \leqslant & (2+3m)K. \end{split}\] I.e., $$ h^{-1} g \in A\;\Rightarrow \; d\big(\alpha(g)x,\alpha(h)x\big)\leqslant (2+3m)K, $$ showing that the orbit map $g\in G\mapsto \alpha(g)x\in M$ is bornologous. \end{proof}
In particular, since a quasi-cocycle $b\colon G\rightarrow E$ associated to an isometric linear representation $\pi\colon G\curvearrowright E$ is simply the orbit map $g\mapsto \alpha(g)0$ of the associated rough action $\alpha(g)x=\pi(g)x+b(g)$, we have the following corollary.
\begin{lemme}\label{baire cocycle} Let $b\colon G\rightarrow E$ be a Baire measurable quasi-cocycle on a Polish group $G$. Then $b$ is bornologous. \end{lemme}
\begin{exa}\label{automatic borno} In a great number of mainly topological transformation groups $G$, we can remove the condition of Baire measurability from Lemmas \ref{baire orbit map} and \ref{baire cocycle}. Indeed, in the above proof, we need only that, whenever $G=\bigcup_nB_n$ is a countable increasing covering by symmetric subsets, then there are $n$ and $m$ so that ${\rm int}(B_n^m)\neq \emptyset$. One general family of groups satisfying this criterion are Polish groups with {\em ample generics}, i.e., so that the diagonal conjugacy action $G\curvearrowright G^n$ has a comeagre orbit for every $n\geqslant 1$ \cite{turbulence}. For example, both ${\rm Isom}(\Q\U)$ and, by the same proof, ${\rm Isom}(\Z\Q)$ have ample generics, while ${\rm Aut}(T_{\aleph_0})$ has an open subgroup with ample generics \cite{turbulence}. Thus, in all three cases, every quasi-cocycle defined on the group is bornologous. \end{exa}
\begin{prop}\label{near isom} Suppose $G\curvearrowright X$ is a geometric Gelfand pair and that $\alpha$ is a nearly isometric action of $G$ on a metric space $M$ so that every bornologous map $X\rightarrow M$ is insolvent. If some orbit map $g\in G\mapsto \alpha(g)\xi\in M$ is bornologous, then every orbit $\alpha[G]\zeta$ is bounded. \end{prop}
\begin{proof} Fix $\xi \in M$ so that the orbit map $g\in G\mapsto \alpha(g)\xi$ is bornologous and let $C$ be the defect of $\alpha$, i.e., for all $g,f\in G$ and $\zeta, \eta\in M$, $$ d_X\big(\alpha(g)\zeta, \alpha(g) \eta\big) \leqslant d_X(\zeta,\eta)+C \;\;\;\;\text{ and }\;\;\;\; d_X\big(\alpha(f)\alpha(g)\zeta, \alpha(fg)\zeta\big) \leqslant C. $$
Let also $K$ be a constant witnessing that $G\curvearrowright X$ is a geometric Gelfand pair. Fix $x\in X$ and choose for each $y\in X$ some $\gamma_y\in G$ such that $d_X\big(\gamma_y(x),y\big)<K$. Define then $\phi\colon X\rightarrow M$ by $\phi(y)=\alpha(\gamma_y)\xi$. We set $V=\{g\in G\del d_X(gx,x)<3K\}$, which is a relatively (OB) symmetric identity neighbourhood in $G$ and let $$ L=\sup_{f^{-1} g\in V}d_M\big(\alpha(g)\xi,\alpha(f)\xi\big). $$
We claim that $\phi$ is bornologous. Indeed, observe that the composition $y\mapsto \gamma_y(x)$ of $\gamma\colon X\rightarrow G$ with the orbit map $g\in G\mapsto g(x)\in X$ will be close to the identity map on $X$. So, as the orbit map $g\mapsto g(x)$ is coarsely proper and hence expanding, this implies that $\gamma$ and thus also $\phi$ are bornologous
Now, suppose that $m\geqslant 1$ and that $$ d_M\big(\phi(x),\phi(y)\big)\geqslant m+3C+2L+\theta_\phi(m+2K) $$ for some $y\in X$. We claim that, for all $z,u\in X$, $$ d_X(x,y)\leqslant d_X(z,u)\leqslant d_X(x,y)+m \;\;\Rightarrow\;\; d_M\big(\phi(z),\phi(u)\big)\geqslant m. $$
To see this, suppose that $d_X(x,y)\leqslant d_X(z,u)\leqslant d_X(x,y)+m$ and find some $g\in G$ so that $d_X\big(g(x),z\big)<K$ and $d_X\big(z,g(y)\big)+d_X\big(g(y),u\big)<d_X(z,u)+K$, i.e., \[\begin{split} d_X\big(g(y),u\big) &\leqslant d_X(z,u)+K-d_X\big(z,g(y)\big)\\ &\leqslant d_X(x,y)+m+K-d_X\big(g(x),g(y)\big)+K\\ &=m+2K. \end{split}\] In particular, $d_M\big(\phi(gy),\phi(u)\big)\leqslant \theta_\phi(m+2K)$.
Now, $d_X(g\gamma_x(x),\gamma_z(x))\leqslant d_X(g(x),z)+2K<3K$, so $\gamma_z^{-1} g\gamma_x\in V$, whence $$ d_M\big(\alpha(g\gamma_x)\xi, \phi(z)\big)=d_M\big(\alpha(g\gamma_x)\xi, \alpha(\gamma_z)\xi\big)\leqslant L. $$ Similarly, $d_X\big(g\gamma_y(x),\gamma_{gy}(x)\big)\leqslant d_X\big(g\gamma_y(x),g(y)\big)+d_X\big(g(y),\gamma_{gy}(x)\big)<2K$, so $\gamma_{gy}^{-1} g\gamma_y\in V$, whence $$ d_M\big(\alpha(g\gamma_y)\xi, \phi(gy)\big)=d_M\big(\alpha(g\gamma_x)\xi, \alpha(\gamma_{gy})\xi\big)\leqslant L. $$ Thus \[\begin{split} d_M\big(\phi(x),\phi(y)\big) &=d_M\big(\alpha(\gamma_x)\xi, \alpha(\gamma_y)\xi\big)\\ &\leqslant d_M\big(\alpha(g^{-1} )\alpha(g\gamma_x)\xi, \alpha(g^{-1} )\alpha(g\gamma_y)\xi\big)+2C\\ &\leqslant d_M\big(\alpha(g\gamma_x)\xi, \alpha(g\gamma_y)\xi\big)+3C\\ &\leqslant d_M\big(\phi(z),\phi(gy)\big)+3C+2L\\ &\leqslant d_M\big(\phi(z),\phi(u)\big)+3C+2L+\theta_\phi(m+2K), \end{split}\] whence $d_M\big(\phi(z),\phi(u)\big)\geqslant m$, proving the claim.
It follows from our claim that, if $\phi$ is unbounded, then, for every $m$, there is some $R_m$ so that $$ R_m\leqslant d_X(z,u)\leqslant R_m+m \;\;\Rightarrow\;\; d_M\big(\phi(z),\phi(u)\big)\geqslant m $$ i.e., $\phi$ is solvent, which is absurd. So $\phi$ is a bounded map, which easily implies that every orbit $\alpha[G]\zeta$ is bounded in $M$. \end{proof}
\begin{prop} Suppose $G\curvearrowright X$ is a geometric Gelfand pair and $H$ a topological group so that every bornologous map $X\rightarrow H$ is insolvent. Then, if $\pi\colon G\rightarrow H$ is a continuous homomorphism, $\pi[G]$ is relatively (OB) in $H$. \end{prop}
\begin{proof} Pick a constant $K$ witnessing that $G\curvearrowright X$ is a geometric Gelfand pair, fix $x\in X$ and choose for every $y\in X$ some $\gamma_y\in G$ so that $d(\gamma_y(x), y)<K$. As in the proof of Proposition \ref{near isom}, $\gamma\colon X\rightarrow G$ is bornologous. Finally, let $V=\{g\in G\del d_X(gx,x)<3K\}$, which is a symmetric relatively (OB) identity neighbourhood in $G$.
Now assume that $A\subseteq H$ is relatively (OB), $m\geqslant 1$ and fix a symmetric relatively (OB) set $D\subseteq G$ so that $\gamma_v\in \gamma_wD$ whenever $d(w,v)\leqslant m+2K$. Suppose that $\gamma_y\notin \gamma_xV\pi^{-1}(A)DV$ for some $y\in X$.
Suppose that $d_X(x,y)\leqslant d_X(z,u)\leqslant d_X(x,y)+m$ for some $z,u\in Z$ and find some $g\in G$ so that $d_X\big(g(x),z\big)<K$ and $d_X\big(z,g(y)\big)+d_X\big(g(y),u\big)<d_X(z,u)+K$, i.e., $d_X\big(g(y),u\big)\leqslant m+2K$. In particular, $\gamma_{gy}\in \gamma_uD$. Also $d_X(g\gamma_x(x),\gamma_z(x))\leqslant d_X(g(x),z)+2K<3K$ and $$ d_X\big(g\gamma_y(x),\gamma_{gy}(x)\big)\leqslant d_X\big(g\gamma_y(x),g(y)\big)+d_X\big(g(y),\gamma_{gy}(x)\big)<2K, $$ so $g\gamma_x\in \gamma_zV$ and $g\gamma_y\in \gamma_{gy}V$. It follows that \[\begin{split} \gamma_x^{-1} \gamma_y=(g\gamma_x)^{-1} g\gamma_y\in V\gamma_{z}^{-1}\gamma_{gy}V\subseteq V\gamma_{z}^{-1} \gamma_uDV \end{split}\] and so $\pi(\gamma_{z})^{-1} \pi(\gamma_u)\notin A$. In other words, for all $z,u\in X$, $$ d_X(x,y)\leqslant d_X(z,u)\leqslant d_X(x,y)+m \;\; \Rightarrow\;\; \pi(\gamma_{z})^{-1} \pi(\gamma_u)\notin A. $$
This shows that, if, for all relatively (OB) sets $A\subseteq H$ and $D\subseteq G$, there is some $\gamma_y\notin \gamma_xV\pi^{-1} (A)DV$, then the map $z\in X\mapsto \pi(\gamma_z)\in H$ is both bornologous and solvent, which is impossible. So choose some $A$ and $D$ for which it fails. As ${\rm im}(\gamma)$ is cobounded in $G$, it follows that $G=V\pi^{-1}(A)U$ for some relatively (OB) set $U\subseteq G$ and hence $\pi[G]$ is included in the relatively (OB) set $\pi[V]A\pi[U]$. In particular, $\pi[G]$ is relatively (OB) in $H$. \end{proof}
We are now in a position of deducing the main application of this section. Note first that ${\rm Isom}(\Q\U)$ is isomorphic to a closed subgroup of the group $S_\infty$ of all permutations of a countable set and so, in particular, ${\rm Isom}(\Q\U)$ admits continuous unitary representations given by permutations of an othogonal basis $(e_x)_{x\in \Q\U}$. Moreover, by results of \cite{solecki} and \cite{turbulence}, ${\rm Isom}(\Q\U)$ is approximately compact and thus amenable. As shown in \cite{autom}, ${\rm Isom}(\Q\U)$ is coarsely equivalent to $\Q\U$ itself. So, as $c_0$ nearly isometrically embeds into $\Q\U$, this shows that ${\rm Isom}(\Q\U)$ cannot have a coarsely proper affine isometric action on a reflexive space. This is in opposition to the result of Brown--Guentner \cite{BG} and Haagerup--Przybyszewska \cite{haagerup-affine} that every locally compact second countable group admits such an action. However, our result here indicates a much higher degree of geometric incompatibility.
\begin{thm}\label{fix} Let $E$ be either a reflexive Banach space or $E=L^1([0,1])$ and let $G$ be ${\rm Isom}(\Z\U)$ or ${\rm Isom}(\Q\U)$. Then every quasi-cocycle $b\colon G\rightarrow E$ is bounded. In particular, every affine isometric action $G\curvearrowright E$ has a fixed point. \end{thm}
\begin{proof} Observe that there are near isometries from $c_0$ into both $\Z\U$ and $\Q\U$. It thus follows from Theorem \ref{kalton2}, respectively Corollary \ref{kalton3}, that every bornologous map from $\Z\U$ or $\Q\U$ into a reflexive space or into $L^1([0,1])$ is insolvent. Now, by Example \ref{automatic borno}, every quasi-cocycle on $G$ is bornologous, so it follows from Proposition \ref{near isom} that every quasi-cocycle $b\colon G\rightarrow E$ is bounded.
Now, if $\alpha\colon G\curvearrowright E$ is an affine isometric action, then the associated cocycle in bounded and thus the affine action has a bounded orbit. Thus, by the Ryll-Nardzewski fixed point theorem \cite{ryll} for the reflexive case or the fixed point theorem of U. Bader, T. Gelander and N. Monod \cite{monod} for $E=L^1([0,1])$, there is a fixed point in $E$. \end{proof}
Theorem \ref{fix} is motivated by questions pertaining to the characterisation of property (OB) among non-Archimedean Polish groups. Every continuous affine isometric action of a Polish group with property (OB) on a reflexive space has a fixed point and one may ask if this characterises property (OB) for {\em non-Archimedean} Polish groups, i.e., closed subgroups of $S_\infty$. As ${\rm Isom}(\Q\U)$ acts transitively on the infinite-diameter metric space $\Q\U$, it fails property (OB) and so the answer to our question is no. However, the following question remains open. \begin{quest} Suppose $\alpha\colon {\rm Isom}(\Q\U)\curvearrowright E$ is a continuous affine action on a reflexive Banach space $E$. Does $\alpha$ necessarily have a fixed point? \end{quest} The difference with Theorem \ref{fix} here is that the action is not required to be isometric and thus the linear part $\pi$ could map onto an unbounded subgroup of the general linear group ${\rm GL}(E)$.
\end{document} | arXiv |
\begin{document}
\title{Supersingular $j$-invariants and the Class Number of $\mathbb{Q}(\sqrt{-p})$}
\author{Guanju Xiao}
\address{Key Laboratory of Mathematics Mechanization, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People's Republic of China} \address{School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \email{[email protected]}
\author{Lixia Luo} \address{Key Laboratory of Mathematics Mechanization, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People's Republic of China} \address{School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \email{[email protected]}
\author{Yingpu Deng} \address{Key Laboratory of Mathematics Mechanization, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People's Republic of China} \address{School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People's Republic of China}
\email{[email protected]}
\subjclass[2020]{Primary 14H52; Secondary 11R11, 11R29, 11Y16}
\keywords{Supersingular $j$-invariants, Class Polynomials, Class Number}
\begin{abstract}
For a prime $p>3$, let $D$ be the discriminant of an imaginary quadratic order with $|D|< \frac{4}{\sqrt{3}}\sqrt{p}$. We research the solutions of the class polynomial $H_D(X)$ mod $p$ in $\mathbb{F}_p$ if $D$ is not a quadratic residue in $\mathbb{F}_p$. We also discuss the common roots of different class polynomials in $\mathbb{F}_p$. As a result, we get a deterministic algorithm (Algorithm 3) for computing the class number of $\mathbb{Q}(\sqrt{-p})$. The time complexity of Algorithm 3 is $O(p^{3/4+\epsilon})$. \end{abstract}
\maketitle
\section{Introduction}
For a prime $p>3$, we know that $\mathbb{Q}(\sqrt{-p})$ is an imaginary quadratic field with discriminant $-p$ (resp. $-4p$) if $p \equiv 3 \pmod 4$ (resp. $p \equiv 1 \pmod 4$). We shall be concerned with the computational problem of calculating the class number of $\mathbb{Q}(\sqrt{-p})$.\par Currently, the best available rigorous methods for computing class number of an imaginary quadratic field of discriminants $D$ is of complexity $(|D|^{1/2+\epsilon})$(see \cite{MR1933052} and \cite[Proposition 9.7.15]{MR2300780}). Assuming the Generalized Riemann Hypothesis (GRH), Shanks's algorithm \cite{MR0316385,MR0371855} computes the class number $h(D)$ in $O(|D|^{1/5+\epsilon})$ operations, and can be used to compute the structure of the class group $C(D)$ in $O(|D|^{1/4+\epsilon})$ operations. Moreover, the values of $h(D)$ computed by Shanks's algorithm could not be guaranteed to be correct if the GRH is false. In 1989, Hafner and McCurley \cite{MR1002631} proposed a Las Vegas algorithm that compute the structure of the class group of $C(D)$ in an expected time of $L(D)^{\sqrt{2}+o(1)}$ bit operations under the assumption of GRH, where
$$L(D)=\exp(\sqrt{\log |D| \log \log |D|}).$$\par In this paper, we will propose algorithms to compute the class number of $\mathbb{Q}(\sqrt{-p})$ using the supersingular $j$-invariants in $\mathbb{F}_p$.\par For a prime $p>3$, we know that the number of supersingular $j$-invariants in $\mathbb{F}_p$ depends on the class number of $\mathbb{Q}(\sqrt{-p})$. If the discriminant $D$ of an imaginary quadratic order $O$ is not a quadratic residue in $\mathbb{F}_p$, then Deuring's theorem \cite[Theorem 13.12]{MR0409362} shows that the roots of $H_D(X)$ mod $p$ are supersingular $j$-invariants where $H_D(X)$ is the class polynomial. Moreover, Kaneko's theorem \cite[Theorem 1]{MR1040429} shows that every supersingular $j$-invariant contained in the prime field $\mathbb{F}_p$ is a roots of some $H_D(X)$ mod $p$ with $|D| < \frac{4}{\sqrt{3}}\sqrt{p}$. We can get the class number of $\mathbb{Q}(\sqrt{-p})$ by calculating the roots of $H_D(X)$ mod $p$ in $\mathbb{F}_p$ for all $|D|< \frac{4\sqrt{p}}{\sqrt{3}}$ with $\left ( \frac{D}{p} \right ) =-1$, however there exists no efficient algorithm to compute the class polynomials.\par To overcome this disadvantage, we consider the prime factorization of $pO_H$ in $H=\mathbb{Q}(j(O))$ where $O$ is an imaginary quadratic order with discriminant $D$. We will show that every prime ideal with norm $p$ in $H$ corresponds to a root of $H_D(X)$ mod $p$ in $\mathbb{F}_p$ if $|D| < \frac{4}{\sqrt{3}}\sqrt{p}$. The new problem is that it is difficult to construct the field $H$ without $j(O)$ in general, so we need a simpler subfield of $H$. Let $F$ be the genus field of $O$ and $L$ be the ring class field of $O$, it's easy to show that the maximal real subfield $E$ of $F$ is a subfield of $H$. What's more, we have the following diagram about the field extensions. \[ \xymatrix{ &L \ar@{-}[d] \ar@{-}[rd] &\\ &F \ar@{-}[d] \ar@{-}[rd] &H \ar@{-}[d] \\ &K \ar@{-}[rd] &E \ar@{-}[d] \\
&& \mathbb{Q} } \] \par It is easier to compute the prime factorization of $p$ in $E$, and we will prove that every prime ideal with norm $p$ in $E$ corresponds to a root of $H_D(X)$ mod $p$ in $\mathbb{F}_p$ with $|D|< \frac{4\sqrt{p}}{\sqrt{3}}$. We also discuss the common roots of different class polynomials $H_{D_1}(X)$ and $H_{D_2}(X)$ in $\mathbb{F}_p$. We propose two algorithms (Algorithm 2 and Algorithm 3) in Section 4. Compared with this algorithm, the theoretical results are more interesting. \par In this paper, $\text{N}(\mathfrak{p})$ is the absolute norm of $\mathfrak{p}$ if $\mathfrak{p}$ is an ideal in a number field, and $\text{Nrd}(\alpha)$ is the reduced norm of $\alpha$ if $\alpha$ is an element of a quaternion algebra. We always assume $p>3$. \par The remainder of this paper is organized as follows. In Section 2, we review some preliminaries on supersingular $j$-invariants, class polynomials and genus theory. Theoretical results are in Section 3, including Theorem 5, 8 and 9. We propose two algorithms (Algorithm 2 and Algorithm 3) for computing the class number of $\mathbb{Q}(\sqrt{-p})$ and analyse the time complexity of them in Section 4. Finally, we make a conclusion in Section 5. \par
\section{Preliminaries}
\subsection{Supersingular $j$-invariants} We will present some basic facts about the supersingular elliptic curves, and the reader can refer to \cite{MR2514094} for more details. For an elliptic curve $E:Y^2=X^3+aX+b$ over a finite field with characteristic $p>3$, the $j$-invariant of $E$ is $j(E)=1728\cdot 4a^3/(4a^3+27b^2)$. Different elliptic curves with the same $j$-invariant are isomorphic over the algebraic closed field $\overline{\mathbb{F}}_p$. Moreover, the $j$-invariant of every supersingular elliptic curve over $\overline{\mathbb{F}}_p$ is proved to be in $\mathbb{F}_{p^2}$ and it is called a supersingular $j$-invariant. For a supersingular elliptic curve $E$ over $\mathbb{F}_{p^2}$, the endomorphism ring $\text{End}(E)$ is isomorphic to a maximal order of $B_{p,\infty}$, where $B_{p,\infty}$ is a quaternion algebra defined over $\mathbb{Q}$ and ramified at $p$ and $\infty$. Moreover, $j(E)\in \mathbb{F}_p$ if and only if $\text{End}(E)$ contains a root of $x^2+p=0$ (see \cite{MR3451433}). \par Choose a prime integer $q$ such that $q \equiv 3 \pmod 8$ and $\left( \frac{p}{q} \right) =-1$, then $B_{p,\infty}$ can be written as $B_{p,\infty}=\mathbb{Q}+\mathbb{Q}\alpha+\mathbb{Q}\beta+\mathbb{Q}\alpha\beta$ where $\alpha^2=-p$, $\beta^2=-q$ and $\alpha\beta=-\beta\alpha$. Choosing an integer $r$ such that $r^2+p \equiv 0 \pmod q$, put $$\mathcal{O}(q,r)=\mathbb{Z} + \mathbb{Z}\frac{1+\beta}{2} + \mathbb{Z} \frac{\alpha(1+\beta)}{2} + \mathbb{Z}\frac{(r+\alpha)\beta}{q}.$$ When $p \equiv 3 \pmod 4$, we further choose an integer $r'$ such that $r'^2+p \equiv 0 \pmod {4q}$ and put $$\mathcal{O}'(q,r')=\mathbb{Z} + \mathbb{Z}\frac{1+\alpha}{2} + \mathbb{Z} \beta + \mathbb{Z}\frac{(r'+\alpha)\beta}{2q}.$$\par Ibukiyama's results \cite{MR683249} show that both $\mathcal{O}(q,r)$ and $\mathcal{O}'(q,r')$ are maximal orders of $B_{p,\infty}$ and the endomorphism ring $\text{End}(E)$ is isomorphic to $\mathcal{O}(q,r)$ or $\mathcal{O'}(q,r')$ with suitable choice of $q$ if $E$ is a supersingular elliptic curve over $\mathbb{F}_p$.\par For a prime $p > 3$, let $S_p$ be the set of all supersingular $j$-invariants in $\mathbb{F}_p$. Then (see \cite{MR3451433}) $$ \# S_{p}= \left\{ \begin{array}{lcl} \frac{1}{2}h(-4p) & &{\text{if} \ p\equiv 1 \pmod{4} ,} \\ h(-p) & &{\text{if} \ p\equiv 7 \pmod{8} ,}\\ 2h(-p) & &{\text{if} \ p\equiv 3 \pmod{8} ,} \end{array} \right. $$ where $h(D)$ is the class number of the imaginary quadratic order with discriminant $D$.
\subsection{Class Polynomials} We first recall some basic facts about the class polynomials, and the general references are \cite{MR3236783,MR0409362}. Given an order $O$ in an imaginary quadratic field $K$, the class polynomial $H_D(X)$ is the monic minimal polynomial of $j(O)$ over $\mathbb{Q}$ where $D$ is the discriminant of $O$. Note that $H_D(X)$ has integer coefficients. Let $\{ \mathfrak{a}_i \}$ be a complete set of coset representatives of the $h(D)$ proper ideal classes of $O$. We have that $$H_D(X)=\prod^{h(D)}_{i=1} (X-j(\mathfrak{a}_i))$$
is an irreducible polynomial in $\mathbb{Z}[X]$. The coefficients of $H_D(X)$ grow rapidly with the size of the discriminant $D$. Sutherland \cite{MR2728992} presented a space-efficient algorithm to compute the class polynomial $H_D(X)$ modulo a positive integer $p$, based on an explicit form of the Chinese Remainder Theorem. Under the Generalized Riemann Hypothesis, the algorithm uses $O(|D|^{1/2+\epsilon} \log p)$ space and has an expected running time of $O(|D|^{1+\epsilon})$. \par As we known \cite[Lemma 9.3]{MR3236783}, $L=K(j(O))$ is an abelian extension of $K$ and the Galois group $\text{Gal}(L/K)$ is isomorphic to the class group $C(D)$ of $O$. Moreover, $L/\mathbb{Q}$ is a generalized dihedral extension and the Galois group $\text{Gal}(L/\mathbb{Q})$ can be written as a semidirect product $$\text{Gal}(L/ \mathbb{Q})\simeq \text{Gal}(L/K) \rtimes (\mathbb{Z}/2\mathbb{Z})\simeq C(D)\rtimes (\mathbb{Z}/2\mathbb{Z}),$$ and the nontrivial element of $\mathbb{Z}/2\mathbb{Z}$ acts on $\text{Gal}(L/K)$ via conjugation by $\sigma$ which is a complex conjugation.\par We also need Deuring's reduction theorem \cite[Theorem 13.12]{MR0409362}. \begin{theorem} Let $\tilde{E}$ be an elliptic curve over a number field with $\text{End} (\tilde{E}) \simeq O$, where $O$ is an order of an imaginary quadratic field $K$. Let $\mathfrak{p}$ be a prime ideal of $\overline{\mathbb{Q}}$ over a prime number $p$, at which $\tilde{E}$ has non-degenerate reduction $E$. $E$ is supersingular if and only if $p$ does not split in $K$. \end{theorem} Kaneko proved two theorems in \cite{MR1040429}. \begin{theorem}\label{t2}
Every supersingular $j$-invariant contained in the prime field $\mathbb{F}_p$ is a root of some $H_D(X)$ mod $p$ with $| D | \le \frac{4}{\sqrt{3}} \sqrt{p}$. \end{theorem} \begin{theorem}\label{t3}
If two different discriminants $D_1$ and $D_2$ satisfy $D_1 D_2 < 4p$ (in particular $|D_1|, |D_2| < 2\sqrt{p}$), then two polynomials $H_{D_1}(X)$ mod $p$ and $H_{D_2}(X)$ mod $p$ in $\mathbb{F}_p[X]$ have no roots in common. In other words, every prime factor $p$ of the resultant of $H_{D_1}(X)$ and $H_{D_2}(X)$ satisfies $p \le \frac{D_1 D_2}{4}$.\par Furthermore, if $\mathbb{Q}(\sqrt{D_1})= \mathbb{Q}(\sqrt{D_2})$, the above inequality $D_1 D_2<4p$ (resp. $p \le \frac{D_1 D_2}{4}$) can be replaced by $D_1 D_2<p^2$ (resp. $p\le \sqrt{D_1 D_2}$). \end{theorem}
\subsection{Genus Fields} Let $K$ be an imaginary quadratic field. $O$ is an order of $K$ with discriminant $D$. We write $D=-4n$ if $D\equiv \ 0 \pmod 4$. The following proposition \cite[Proposition 3.11]{MR3236783} determines the number of elements of order $2$ in the class group $C(D)$. \begin{proposition}\label{p1}
Let $D\equiv 0,1 \pmod 4$ be negative, and let $t$ be the number of odd primes dividing $D$. Define the number $\mu$ as follows: if $D\equiv 1 \ \pmod 4$, then $\mu =t$, and if $D\equiv \ 0 \pmod 4$, then $D=-4n$, where $n>0$, and $\mu$ is determined by the following table:
\begin{table}[H]
\centering
\begin{tabular}{c|c}
\hline
$n$ & $\mu$ \\
\hline
$n \equiv \ 3 \pmod 4$ & $t$ \\
$n \equiv \ 1,\ 2 \pmod 4$ & $t+1$ \\
$n \equiv \ 4 \pmod 8$ & $t+1$ \\
$n \equiv \ 0 \pmod 8$ & $t+2$ \\
\hline
\end{tabular}
\end{table}
Then the class group $C(D)$ has exactly $2^{\mu-1}$ elements of order $\le 2$. \end{proposition} Let $p_1, \ldots, p_t$ be the odd prime divisors of $D$, then the genus field of $O$ \cite[Theorem 2.2.23]{MR1313719} is $K_0(\sqrt{p_1^*}, \ldots, \sqrt{p_t^*})$ where $p_i^*=(-1)^{\frac{p_i-1}{2}}p_i$ and $K_0$ satisfies the following conditions $$\left \{ \begin{array}{ll}
K_0=\mathbb{Q} & \text{if} \ D \ \text{odd or} \ n \equiv -1\pmod 4, \\
K_0=\mathbb{Q}(i) & \text{if} \ n \equiv 1,4,5\pmod 8, \\
K_0=\mathbb{Q}(\sqrt{-2}) & \text{if} \ n \equiv 2\pmod 8, \\
K_0=\mathbb{Q}(\sqrt{2}) & \text{if} \ n \equiv -2\pmod 8, \\
K_0=\mathbb{Q}(i,\sqrt{2}) & \text{if} \ n \equiv 0\pmod 8. \end{array} \right .$$
\section{The Number of Supersingular $j$-invariants over $\mathbb{F}_p$ } In the first subsection, we will compute the class number of $\mathbb{Q}(\sqrt{-p})$ with class polynomials. The algorithm is simple, however there exists no efficient algorithm to compute class polynomials. In the second subsection, We will research the solutions of the class polynomial $H_D(X)$ mod $p$ in $\mathbb{F}_p$ if $D$ is not a quadratic residue in $\mathbb{F}_p$. We also discuss the common roots of different class polynomials in $\mathbb{F}_p$. \subsection{Computing the Class Number with Class Polynomials}
Let $D\equiv 0,1 \pmod 4$ be the discriminant of an imaginary quadratic order. For a prime integer $p>3$, if the Legendre symbol $\left (\frac{D}{p} \right ) \neq 1$, then the roots of $H_D(X)$ mod $p$ in $\mathbb{F}_p$ are the supersingular $j$-invariants in $\mathbb{F}_p$ by Deuring's reducing theorem. To get the class number of $\mathbb{Q}(\sqrt{-p})$, we will compute the number of supersingular $j$-invariants over $\mathbb{F}_p$. Theorem \ref{t2} implies that we just need compute all the roots of $H_D(X)$ mod $p$ in $\mathbb{F}_p$ for $|D| < \frac{4}{\sqrt{3}}\sqrt{p}$, so we have Algorithm 1.\par In fact, the degree of $\text{gcd}(H_D(X),X^p-X)$ is the number of roots of $H_D(X)$ in $\mathbb{F}_p$. Since two class polynomials $H_{D_1}(X)$ mod $p$ and $H_{D_2}(X)$ mod $p$ may have common roots in $\mathbb{F}_p$, we compute explicit roots of $H_D(X)$ in $\mathbb{F}_p$ in Algorithm 1. We will get a sufficient and necessary condition under which we can determine whether $H_{D_1}(X)$ mod $p$ and $H_{D_2}(X)$ mod $p$ have common roots in $\mathbb{F}_p$ or not in the next subsection.\par The correctness of Algorithm 1 is obvious, but the time complexity is worse than other algorithms. Even though, we can get all the supersingular $j$-invariants over $\mathbb{F}_p$ by Algorithm 1.
\begin{algorithm}[H] \caption{ Computing the class number of $\mathbb{Q}(\sqrt{-p})$ with $H_D(X)$.} \label{alg:Framwork} \begin{algorithmic}[1] \Require The prime $p$; \Ensure The class number of $\mathbb{Q}(\sqrt{-p})$; \label{code:fram:extract} \State Let $S_p=\varnothing$; \label{code:fram:trainbase}
\State For every negative discriminant $D\equiv 0$ or $1 \pmod 4$ satisfying $|D| < \frac{4}{\sqrt{3}}\sqrt{p}$, do step 3; \label{code:fram:add} \State If Legendre symbol $\left (\frac{D}{p} \right )=-1$, then add all the roots of $H_D(X)$ in $\mathbb{F}_p$ to $S_p$; \label{code:fram:classify} \State $ h= \left\{ \begin{array}{ll} \frac{1}{2}\# S_{p} &{\text{if} \ p\equiv 3 \pmod{8} ,} \\ \# S_{p} &{\text{if} \ p\equiv 7 \pmod{8} ,}\\ 2\# S_{p} &{\text{if} \ p\equiv 1 \pmod{4} ;} \end{array} \right.$ \label{code:fram:select} \\ \Return $h$; \end{algorithmic} \end{algorithm} \subsection{Computing the Number of Supersingular $j$-invariants over $\mathbb{F}_p$ without Class Polynomials} In this subsection, we will calculate the roots of $H_D(X)$ mod $p$ in $\mathbb{F}_p$ by the prime factorization of $(p)$ in the field $E$ firstly. Secondly, we will consider the common roots of different class polynomials in $\mathbb{F}_p$. By this way, we can compute the number of supersingular $j$-invariants over $\mathbb{F}_p$ without class polynomials. \subsubsection{The Roots of $H_D(X)$ mod $p$ in $\mathbb{F}_p$} Let $K$ be an imaginary quadratic field. $O$ is an order of $K$ with discriminant $D$. Let $L$ be the ring class field of $O$ and $H=\mathbb{Q}(j(O))$. We have the following diagram about field extensions. \[ \xymatrix{ &&L \ar@{-}[ld] \ar@{-}[rd] &\\ &K \ar@{-}[rd] &&H \ar@{-}[ld] \\ &&\mathbb{Q} & } \] \par In general, we have the following theorem \cite[Theorem 3.3.5]{MR1313719} about the prime factorization of prime integer $p$ in number fields. \begin{theorem}
Let $H=\mathbb{Q}(\theta)$ be an extension field not necessarily normal but generated by an algebraic integer that satisfies the monic equation
$$f(x)=x^m+a_1x^{m-1}+ \cdots +a_m=0.$$
Then $\mathbb{Z}[\theta]$ is generally only a submodule of $O_H$, the integer ring of $H$, of index $i=[O_H:\mathbb{Z}[\theta]]$. Assume $p \nmid i$. Then if we factor
$$f(x)\equiv f_1(x)^{e_1} \cdots f_g(x)^{e_g} \ \text{mod} \ p$$
into polynomials $\{f_t(x)\}$ irreducible mod $p$ and distinct, this factorization simulates the prime factorization of $pO_H$,
$$pO_H=\mathfrak{p}_1^{e_1} \cdots \mathfrak{p}_g^{e_g}.$$
If the polynomial $f_t(x)$ is of degree $f_t$, then this is the degree of $\mathfrak{p}_t$, which can be written in the form $\mathfrak{p}_t=(p, f_t(\theta))$. \end{theorem}
In our case, $H=\mathbb{Q}(j(O))$. We assume $\left ( \frac{D}{p} \right ) =-1$ and $|D| < \frac{4}{\sqrt{3}}\sqrt{p}$, so $p$ is inert in $K$ and $pO_K$ splits completely in $L$ which implies that $p$ is not ramified in $H$. We claim that $p$ does not divide the discriminant of $H_D(X)$. If not, there is a multiple root of $H_D(X)$ mod $p$ which implies that $O_1=O$ and $O_2=O$ can be embedded in a maximal order $\mathcal{O}$ of $B_{p,\infty}$ with different images. We have $p\le |D|$ by the proof of Theorem \ref{t3} in \cite{MR1040429}, which contradicts $p>3$ and $p\ge \frac{3D^2}{16}$. We must have $p \nmid [O_H : \mathbb{Z}[j(O)]]$ and the above theorem holds in our case.\par By Theorem 4, every root of $H_D(X)$ mod $p$ in $\mathbb{F}_p$ corresponds to an ideal $\mathfrak{P}$ in $O_H$ with norm $p$. In other words, we transform the problem of computing the number of roots of $H_D(X)$ mod $p$ in $\mathbb{F}_p$ into the problem of computing the number of integral ideals with norm $p$ in $H$.\par How can we get the prime factorization of $p$ in $H$ without $H_D(X)$? If the genus field $F$ of $O$ is the same as the ring class field $L$ of $O$, then $H=L \cap \mathbb{R}$ is a composition of some real quadratic fields. In this case, $H$ is Galois over $\mathbb{Q}$, and it is easy to determine whether $p$ splits completely in $H$ or not. This special case enlightens us to consider the prime factorization of $p$ in the maximal real subfield of $F$. \par Let $p_1, \ldots, p_t$ be the odd prime divisors of $D$. Suppose that the first $m$ ($m \le t$) primes satisfying $p_i=p_i^*\equiv \ 1 \pmod 4$. For the other odd primes, we define $q_j=p_j^*$ for $j=1,\ldots,n=t-m$. We write $D=-4n$ if $D\equiv \ 0 \pmod 4$.\par
We define a field $E$ which is a subfield of $F$ as follows. $$E= \left \{ \begin{array}{ll}
\mathbb{Q}(\sqrt{p_1},\ldots, \sqrt{p_m}, \sqrt{\frac{D}{q_1}}, \ldots, \sqrt{\frac{D}{q_n}}) & \text{if} \ D \ \text{odd or} \ n \equiv -1 \pmod 4, \\
\mathbb{Q}(\sqrt{p_1},\ldots, \sqrt{p_t}) & \text{if} \ n \equiv 1,4,5 \pmod 8, \\
\mathbb{Q}(\sqrt{p_1},\ldots, \sqrt{p_m}, \sqrt{-2q_1},\ldots, \sqrt{-2q_n}) & \text{if} \ n \equiv 2 \pmod 8, \\
\mathbb{Q}(\sqrt{2},\sqrt{p_1},\ldots, \sqrt{p_m}, \sqrt{\frac{D}{q_1}}, \ldots, \sqrt{\frac{D}{q_n}}) & \text{if} \ n \equiv -2 \pmod 8, \\
\mathbb{Q}(\sqrt{2},\sqrt{p_1},\ldots, \sqrt{p_t}) & \text{if} \ n \equiv 0 \pmod 8. \end{array} \right .$$ \par In fact, $E$ is the maximal real subfield of $F$. Moreover, $E$ is a Galois extension over $\mathbb{Q}$ and $\text{Gal}(E/\mathbb{Q})\simeq (\mathbb{Z}/2\mathbb{Z})^{\mu-1}$ where $\mu$ is defined as Proposition \ref{p1}. It is easy to show that $E=F\cap H$. Furthermore, we have the following diagram about the field extensions. \[ \xymatrix{ &L \ar@{-}[d] \ar@{-}[rd] &\\ &F \ar@{-}[d] \ar@{-}[rd] &H \ar@{-}[d] \\ &K \ar@{-}[rd] &E \ar@{-}[d] \\ && \mathbb{Q} } \] \par As we know, $L/E$ is a Galois extension with Galois group $G'= \text{Gal}(L/E)$. Let $\text{Gal}(L/H)=\langle \sigma \rangle$. If there exists an ideal $\mathfrak{P}$ in $H$ with norm $p$, then $\mathfrak{P}$ is inert in $L$ and the Frobenius map $\left (\frac{\mathfrak{P}}{L/H} \right )=\sigma$ has degree $2$. Moreover, the norm of the prime ideal $\mathfrak{p}=\mathfrak{P}\cap E$ is $p$.
\begin{theorem}
Let $p>3$ be a prime integer and $D$ the discriminant of an imaginary quadratic order. Suppose that $|D|< \frac{4}{\sqrt{3}}\sqrt{p}$ and $\left( \frac{D}{p} \right )=-1$. If $p$ splits completely in $E$, then there exist $2^{\mu-1}$ roots of $H_D(X)$ mod $p$ in $\mathbb{F}_p$. Otherwise, there exists no root of $H_D(X)$ mod $p$ in $\mathbb{F}_p$. \end{theorem} \begin{proof}
As we know, the class polynomial $H_D(X)$ mod $p$ has roots in $\mathbb{F}_p$ if and only if there exist prime ideals in $H$ with norm $p$. Let $\mathfrak{P}$ be a prime ideal in $H$ with norm $p$, then the norm of $\mathfrak{p}=\mathfrak{P}\cap E$ is $p$ and $p$ splits completely in $E$. By the proof of Tchebotarev Density Theorem in Janusz's book \cite[Theorem 10.4 Chapter 5]{MR0366864}, the number of prime ideals with norm $p$ in $H$ is $d=[C_{G}(\sigma):\langle \sigma \rangle]$ with $G=\text{Gal}(L/\mathbb{Q})$. For any $\iota \in G$, we have $\iota \sigma \iota^{-1}= \sigma$ if and only if $\sigma \iota \sigma^{-1}=\iota$ if and only if $\iota^{-1}=\iota$ since $\sigma \iota \sigma^{-1}=\iota^{-1}$, so the order of $\iota$ is less than or equal to $2$. As we know, the number of elements with order less than or equal to $2$ in $G$ is $2^\mu$ where $\mu$ is defined in Proposition 2.1, so $d=2^{\mu-1}$. Moreover, we have that $p$ splits into $2^{\mu-1}$ prime ideals in $E$.\par Suppose that $p$ splits completely in $E$. If $\mathcal{P}$ is a prime ideal in $L$ with norm $p^2$, then the Frobenius map $\left ( \frac{\mathcal{P}}{L/\mathbb{Q}}\right) |_E$ is trivial. Notice that $G'=\text{Gal}(L/E)=C(D)^2 \rtimes \langle \sigma \rangle$, so we have $\left( \frac{\mathcal{P}}{L/\mathbb{Q}} \right) \in G'$. As we know, the order of $\left( \frac{\mathcal{P}}{L/\mathbb{Q}}\right)$ is $2$ and $\left( \frac{\mathcal{P}}{L/\mathbb{Q}} \right)$ is not the square of an element in $C(D)$ since $\left ( \frac{\mathcal{P}}{L/\mathbb{Q}}\right) |_K$ is nontrivial, so we have that $\left( \frac{\mathcal{P}}{L/\mathbb{Q}}\right)=\sigma$ or $\left( \frac{\mathcal{P}}{L/\mathbb{Q}}\right)=\tau \sigma $ with $\tau \in C(D)^2$. If $\left( \frac{\mathcal{P}}{L/\mathbb{Q}}\right)=\sigma$, then the class polynomial $H_D(X)$ mod $p$ has roots in $\mathbb{F}_p$. If $\left( \frac{\mathcal{P}}{L/\mathbb{Q}}\right)=\delta^2 \sigma $ with $\delta \in C(D)$, then $\left( \frac{\delta^{-1} \mathcal{P}}{L/\mathbb{Q}}\right)=\delta^{-1} (\delta^2 \sigma ) \delta=\sigma$, which means that the class polynomial $H_D(X)$ mod $p$ has roots in $\mathbb{F}_p$. In conclusion, if $p$ splits completely in $E$, then the class polynomial $H_D(X)$ mod $p$ has roots in $\mathbb{F}_p$.
\end{proof} \begin{remark}
Suppose $|D|< \frac{4}{\sqrt{3}}\sqrt{p}$ and $\left( \frac{D}{p} \right )=-1$. If the class number $h(D)$ is odd, then there exists exactly one root of $H_D(X)$ mod $p$ in $\mathbb{F}_p$. \end{remark}
\subsubsection{The Common Roots of Different Class Polynomials}
For an imaginary quadratic order $O_i$ with discriminant $D_i$, we denote $F_i$ its genus field and $E_i=F_i \cap \mathbb{R}$. In this subsection, we assume that $p$ is inert in $\mathbb{Q}(\sqrt{D_i})$ and $p$ splits completely in $E_i$.\par Let $D_1$ and $D_2$ be two different negative discriminants. If $\mathbb{Q}(\sqrt{D_1})=\mathbb{Q}(\sqrt{D_2})$, then every prime factor $p$ of the resultant of $H_{D_1}(X)$ and $H_{D_2}(X)$ satisfies $p^2\le D_1D_2$ by Theorem \ref{t3}. We assume $|D_i| < \frac{4}{\sqrt{3}}\sqrt{p}$, so $D_1D_2 < \frac{16}{3}p$. Let $p>5$ be a prime integer, then $H_{D_1}(X)$ mod $p$ and $H_{D_2}(X)$ mod $p$ in $\mathbb{F}_p$ have no roots in common. We assume $\mathbb{Q}(\sqrt{D_1})\neq \mathbb{Q}(\sqrt{D_2})$ in the following. \par Let $D_1$, $D_2$ be two distinct negative discriminants and write $$J(D_1,D_2)=\prod_{\substack{[\tau_1],[\tau_2] \\ \text{disc}(\tau_i)=D_i}}(j(\tau_1)-j(\tau_2)),$$ where $[\tau_i]$ runs over all elements of the upper half-plane with discriminant $D_i$ modulo $\text{SL}_2(\mathbb{Z})$. Let $w_i$ denote the number of roots of unity in the imaginary quadratic order of discriminant $D_i$. Let $f_i$ denote the conductor of $D_i$. Gross and Zagier \cite{MR772491} studied the prime factorizations of $J(D_1,D_2)$ where $D_1$ and $D_2$ are two fundamental discriminants which are relatively prime. Recently, Lauter and Viray \cite{MR3431591} studied the prime factorizations of $J(D_1,D_2)$ for arbitrary $D_1$ and $D_2$. We need two theorems in \cite{MR3431591}.
\begin{theorem} Let $D_1$, $D_2$ be any two distinct discriminants. Then there exists a function $F$ that takes non-negative integers of the form $\frac{D_1 D_2 - x^2}{4}$ to (possibly fractional) prime powers. This function satisfies $$J(D_1, D_2)^{\frac{8}{w_1 w_2}} = \pm \prod_{\substack{x^2 \le D_1D_2 \\ x^2 \equiv D_1 D_2 \pmod 4}} F \left (\frac{D_1 D_2 - x^2}{4} \right ),$$ Moreover, $F(m) = 1$ unless either (1) $m = 0$ and $D_2 = D_1 \ell^{2k}$ for some prime $\ell$ or (2) the Hilbert symbol $(D_1, -m)_{\ell} = -1$ at a unique finite prime $\ell$ and this prime divides $m$. In both of these cases, $F(m)$ is a (possibly fractional) power of $\ell$. \end{theorem}
\begin{theorem} Let $m$ be a non-negative integer of the form $\frac{D_1D_2-x^2}{4}$ and $\ell$ a fixed prime that is coprime to $f_1$. If $m > 0$ and either $\ell > 2$ or $2$ does not ramify in both $\mathbb{Q}(\sqrt{D_1})$ and $\mathbb{Q}(\sqrt{D_2})$, then $v_{\ell}(F(m))$ can be expressed as a weighted sum of the number of certain invertible integral ideals in $O_{D_1}$ of norm $m/\ell^r$ for $r > 0$. Moreover, if $m$ is coprime to the conductor of $D_1$, then $v_{\ell}(F(m))$ is an integer and the weights are easily computed and constant; more precisely, we have $$v_{\ell}(F(m))= \left \{ \begin{array}{ll}
\frac{1}{e}\rho(m)\sum_{r \geq 1} \mathfrak{A}(m/\ell^r) & \text{if} \ \ell \nmid f_2 , \\
\rho(m)\mathfrak{A}(m/\ell^{1+v(f_2)}) & \text{if} \ \ell \mid f_2 , \end{array} \right. $$ where $e$ is the ramification degree of $\ell$ in $\mathbb{Q}(\sqrt{D_1})$ and $$\rho(m)=\left \{ \begin{array}{ll}
0 & \text{if} \ (D_1,-m)_p=-1 \ \text{for} \ p\mid D_1, p\nmid f_1 \ell, \\
2^{\# \{p|(m,D_1):p\nmid f_2 \ \text{or} \ p=\ell \} } & \text{otherwise}, \end{array} \right. $$ $$\mathfrak{A}(N)=\# \left \{ \begin{array}{cl}
& N(\mathfrak{b})=N, \mathfrak{b} \ \text{invertible,} \\
\mathfrak{b} \subseteq O_{D_1} & p\nmid \mathfrak{b} \ \text{for all} \ p\mid(N,f_2),p\nmid \ell D_1 \\
& \mathfrak{p}^3 \nmid \mathfrak{b} \ \text{for all} \ \mathfrak{p}\mid p \mid (N,f_2,D_1), p\neq \ell \end{array} \right \}.$$ If $m=0$, then either $v_{\ell}(F(0))=0$ or $D_2=D_1\ell^{2k}$ and $$v_{\ell}(F(0))=\frac{2}{w_1}\cdot \#\text{Pic}(O_{D_1}).$$ \end{theorem}
We assume $|D_1|, |D_2| < \frac{4}{\sqrt{3}}\sqrt{p}$ and $\mathbb{Q}(\sqrt{D_1})\neq \mathbb{Q}(\sqrt{D_2})$, so $p \mid \frac{D_1D_2-x^2}{4}$ for an integer $x$ if and only if $4p=D_1D_2-x^2$. We will compute $v_{\ell}(F(m))$ for $m= \ell =p$.\par We assume that $p$ is inert in $\mathbb{Q}(\sqrt{D_1})$, so $e=1$. By definition, $\mathfrak{A}(p/p^{r})$ is $1$ (resp. $0$) if $r =1$ (resp. $r >1$), so $\sum_{r \ge 1} \mathfrak{A}(p/p^r)=1$. The cardinality of $\{q | (p,D_1):q\nmid f_2 \ \text{or} \ q=p \}$ is $1$. Moreover, we have that $(D_1,-p)_q=-1$ at a unique finite prime $q=p$ if $p$ splits completely in $E_1$. We illustrate this when $D_1$ is odd, and other cases can be proved similarly.\par First, we have $(D_1,-p)_p=\left ( \frac{D_1}{p} \right )=-1$. If $q \nmid pD_1$ or $p \mid f_1$, then $(D_1,-p)_q=1$ obviously. For an odd prime $q | D_1$ and $q \nmid f_1p$, we can write $D_1=q^kD_1'$ where $q \nmid D_1'$. If $2\mid k$, then $(D_1, -p)_q=1$. If $2\nmid k$, we have that $$(D_1, -p)_q=\left(\frac{-p}{q} \right)^k=\left(\frac{-p}{q} \right).$$ If $q \equiv 1 \pmod 4$, then $\left(\frac{-p}{q} \right)=\left(\frac{q}{p} \right)=1$. If $q \equiv 3 \pmod 4$, then $\left(\frac{-p}{q} \right)=-\left(\frac{-q}{p} \right)=1$ since $\left(\frac{D_1}{p} \right)=-1$ and $\left( \frac{D_1/-q}{p} \right)=1$. We have $\rho(p) =2$ and $v_p(F(p))=2$. \begin{theorem}
Let $D_1$ and $D_2$ be two distinct discriminants satisfying $\sqrt{3p} < |D_1| < |D_2| < \frac{4}{\sqrt{3}} \sqrt{p}$ and $\mathbb{Q}(\sqrt{D_1}) \neq \mathbb{Q}(\sqrt{D_2})$. Assume that the prime integer $p>5$ is inert in $\mathbb{Q}(\sqrt{D_1})$ and $\mathbb{Q}(\sqrt{D_2})$. If $p$ splits completely in $E_1$ and $E_2$, then the class polynomials $H_{D_1}(X)$ mod $p$ and $H_{D_2}(X)$ mod $p$ have one common root in $\mathbb{F}_p$ if and only if there exists an integer $x$ such that $D_1D_2-x^2=4p$. \end{theorem} \begin{proof}
If $p \ge 7$, then $|D_2|> |D_1| >4$ and $w_1=w_2=2$. If there exists an integer $x$ such that $D_1D_2-x^2=4p$, then $F(p)=p^2$ and $v_p(J(D_1,D_2))=1$. There exists one common root of $H_{D_1}(X)$ mod $p$ and $H_{D_2}(X)$ mod $p$ and the common root must be in $\mathbb{F}_p$. Otherwise, if $\alpha \in \mathbb{F}_{p^2} \setminus \mathbb{F}_p$ is the common root, then $\bar{\alpha}$ is also a common root and $p^2 | J(D_1,D_2)$. \par
On the contrary, if $H_{D_1}(X)$ mod $p$ and $H_{D_2}(X)$ mod $p$ have one common root in $\mathbb{F}_p$, then $p | J(D_1,D_2)$ and there exists an integer $x$ such that $D_1D_2-x^2=4p$. \end{proof} \begin{remark}
Kaneko proved in \cite{MR1040429} that each prime factor $p$ of the resultant of $H_{D_1}(X)$ and $H_{D_2}(X)$ divides a positive integer of the form $(D_1D_2-x^2)/4$. If $\sqrt{3p} < |D_1| < |D_2| < \frac{4}{\sqrt{3}} \sqrt{p}$ and $4p=D_1D_2-x^2$, Theorem 8 implies that $H_{D_1}(X)$ mod $p$ and $H_{D_2}(X)$ mod $p$ have a common root in $\mathbb{F}_p$. \end{remark}
For a discriminant $D_1$ satisfying $\sqrt{3p} < |D_1| < \frac{4}{\sqrt{3}} \sqrt{p}$, if there exists a discriminant $D_2$ with $\sqrt{3p} < |D_1| < |D_2| < \frac{4}{\sqrt{3}} \sqrt{p}$ and $D_1D_2 - x^2=4p$ where $x$ is a positive integer, then the form $f(X,Y)=-D_1X^2-2xXY-D_2Y^2$ is a positive definite quadratic form with discriminant $-16p$. We claim that $f(X,Y)$ is reduced.\par We just need to prove $2x < -D_1$. If not, $2x \ge -D_1$ implies that $16p = 4D_1D_2-4x^2 < 8x \cdot \frac{4\sqrt{p}}{\sqrt{3}}-4x^2$. This inequality holds if and only if $\frac{2\sqrt{p}}{\sqrt{3}} < x < \frac{6\sqrt{p}}{\sqrt{3}}$, which contradicts $x^2=D_1D_2-4p<\frac{4}{3}p$. We do not require that $f(X,Y)$ is primitive.\par In fact, we have the following theorem. \begin{theorem}
Let $D_1$, $D_2$ and $D_3$ be three distinct discriminants satisfying $|D_1| < |D_2| < |D_3| < \frac{4}{\sqrt{3}} \sqrt{p}$. Assume that the prime integer $p>5$ is inert in $\mathbb{Q}(\sqrt{D_i})$. If $p$ splits completely in $E_i$, then the class polynomials $H_{D_1}(X)$, $H_{D_2}(X)$ and $H_{D_3}(X)$ have no common root in $\mathbb{F}_p$. \end{theorem} \begin{proof}
We assume that $\mathbb{Q}(\sqrt{D_1})$, $\mathbb{Q}(\sqrt{D_2})$ and $\mathbb{Q}(\sqrt{D_3})$ are not equal to each other. Let $|D_2|$ be the smallest value such that $H_{D_1}(X)$ mod $p$ and $H_{D_2}(X)$ mod $p$ have a common root $j$ in $\mathbb{F}_p$. We must have $|D_1|>\sqrt{3p}$ since $D_1D_2>4p$. As we know, $f(X,Y)=-D_1X^2-2xXY-D_2Y^2$ is a reduced positive definite quadratic form with discriminant $-16p$. On the other hand, we have that $O_i=\mathbb{Z}[\frac{-D_i+\sqrt{D_i}}{2}]$ can be embedded into the endomorphism ring $\mathcal{O}$ of $E(j)$ for $i=1,2$. We assume $$\mathcal{O}=\mathcal{O}(q,r)= \mathbb{Z} + \mathbb{Z}\frac{1+\beta}{2} + \mathbb{Z} \frac{\alpha(1+\beta)}{2} + \mathbb{Z}\frac{(r+\alpha)\beta}{q},$$ where $q \equiv 3 \ \text{mod} \ 8$, $\left ( \frac{p}{q} \right )=-1$, $\alpha^2=-p$, $\beta^2=-q$ and $\alpha\beta=-\beta\alpha$. If $\mathcal{O}=\mathcal{O}'(q,r')$, the proof is similar.\par There exist $\gamma_i \in \mathcal{O}$ such that $\text{Trd}(\gamma_i)=-D_i$ and $\text{Nrd}(\gamma_i)=\frac{D_i^2-D_i}{4}$ for $i=1,2$. Let $\gamma_i= w_i + x_i \frac{1+\beta}{2} + y_i \frac{\alpha(1+\beta)}{2} + z_i \frac{(r+\alpha)\beta}{q}$, then we have the following Diophantine equations: $$ 2w_i + x_i =-D_i$$ and $$(w_i + \frac{x_i}{2})^2 + \frac{p}{4}y_i^2 + q(\frac{x_i}{2}+\frac{z_ir}{q})^2 + pq(\frac{y_i}{2}+ \frac{z_i}{q})^2 = \frac{D_i^2-D_i}{4}.$$
We must have $y_i=0$ since $|D_i| < p$ and $$qx_i^2 + 4rx_iz_i + \frac{4r^2+4p}{q}z_i^2=-D_i.$$ Let $g(X,Y)=qX^2+4rXY+\frac{4r^2+4p}{q}Y^2$. We know that $g(x,y) \equiv 0$ or $3 \pmod 4$ for any $x,y \in \mathbb{Z}$.\par We assume that the first (resp. second) successive minimum of $g(X,Y)$ is $\sqrt{-D_1}$ (resp. $\sqrt{-D_2}$) which coincides with $f(X,Y)$. By Proposition 5.7.3 and Theorem 5.7.6 of \cite{MR2300780}, $g(X,Y)$ can be reduced to $f(X,Y)$ or $f(X,-Y)$. It is easy to see that the minimal value of the function $f(X,Y)$ is $f(0,0)=0$. If we further assume $X\neq 0$ and $Y\neq 0$, then $f(X,Y) \ge f(1,1)$. Moreover, we have $$\begin{aligned} f(1,1)= -D_1-2x-D_2>&2\sqrt{D_1D_2}-2x=2\sqrt{4p+x^2}-2x \\ =&\frac{8p}{\sqrt{4p+x^2}+x} \\ >&\frac{8p}{\sqrt{4p+\frac{4p}{3}}+\frac{2\sqrt{p}}{\sqrt{3}}}\\ =&\frac{4}{\sqrt{3}}\sqrt{p}, \end{aligned}$$
where the second inequality holds since $0<x<\frac{2}{\sqrt{3}}\sqrt{p}$. Suppose that there exists an imaginary quadratic order $O_3$ which can be embedded in $\mathcal{O}(q,r)$, then the discriminant $D_3$ satisfies $|D_3|>\frac{4}{\sqrt{3}}\sqrt{p}$.\par If $\sqrt{-D_1}$ (resp. $\sqrt{-D_2}$) is not the first (resp. second) successive minimum of $g(X,Y)$, then $|D_2|>\frac{4}{\sqrt{3}}\sqrt{p}$ and this theorem holds. \par \end{proof}
\section{Computing the class number of $\mathbb{Q}(\sqrt{-p})$} It is time to present our algorithm (Algorithm 2) to compute the class number of $\mathbb{Q}(\sqrt{-p})$. \begin{algorithm}[H] \caption{ Computing the class number of $\mathbb{Q}(\sqrt{-p})$.} \label{alg:Framwork} \begin{algorithmic}[1] \Require A prime $p>5$; \Ensure The class number of $\mathbb{Q}(\sqrt{-p})$; \State Let $s_p=0$ and $T=\varnothing$; \label{code:fram:let}
\State For every negative discriminant $D\equiv 0$ or $1 \pmod4$ satisfying $|D| < \frac{4}{\sqrt{3}}\sqrt{p}$, if the Legendre symbol $\left (\frac{D}{p} \right )=-1$, do the following two steps; \label{code:fram:for} \State Factor $D$ and compute the generators of the field $E$; \label{code:fram:if}
\State If $p$ splits completely in $E$, then $s_p=s_p+2^{\mu-1}$; Furthermore, if $|D|>\sqrt{3p}$, then add $D$ to the set $T$; \label{code:fram:split} \State Calculate the number $t$ of pairings $(D_1,D_2)$ with $D_1,D_2 \in T$ where $D_1D_2-4p$ is a square; \State $ h= \left\{ \begin{array}{ll} \frac{1}{2} (s_{p}-t) &{\text{if} \ p\equiv 3 \pmod{8} ,} \\ s_{p}-t &{\text{if} \ p\equiv 7 \pmod{8} ,}\\ 2(s_{p}-t) &{\text{if} \ p\equiv 1 \pmod{4}; } \end{array} \right.$ \label{code:fram:equation} \\ \Return $h$. \end{algorithmic} \end{algorithm} The correctness of Algorithm 2 is guaranteed by Theorem 5, 8 and 9. We now analyze the complexity of Algorithm 2. There are about $\frac{2}{\sqrt{3}}\sqrt{p}$ many $D$ in step 2. We just need to compute less than $\log p$ many Legendre symbols in step 4 which can be calculated in polynomial time of $\log p$.\par Factoring large integers is difficult. Hittmeir \cite{MR3834692} presented a deterministic algorithm that provably computes the prime factorisation of a positive integer $N$ in $N^{2/9+\epsilon}$ bit operations. Recently, the exponent for deterministic factorization was improved to $1/5$ by Harvey in \cite{harvey2020exponent}.
What's more, Coppersmith algorithm uses lattice basis reduction techniques, and it has the advantage of requiring little memory space. There are several probabilistic algorithms for factoring integers. The quadratic sieve\cite{MR825590}, the number field sieve\cite{MR1321216} and the elliptic curve method \cite{MR916721} are subexponential algorithms for factoring integers.\par In step 5, it is easy to determine whether $D_1D_2-4p$ is a square or not. There are at most ($\frac{\sqrt{3p}}{3} \times \frac{\sqrt{3p}}{3}$) many pairings $(D_1,D_2)$ in $T$. The time complexity of step 5 is $O(p)$, and the constant is small by further analysis. In general, the time complexity of Algorithm 2 is $O(p)$.\par As we can see, the step 5 occupies the most time in Algorithm 2. If $D_1D_2-4p$ is a square, then the equation $x^2 \equiv -4p \pmod {|D_2|}$ is solvable. Moreover, if the solution $x$ satisfies $\sqrt{3p} < \frac{x^2+4p}{|D_2|} < |D_2|$ and $\frac{x^2+4p}{D_2}$ is a negative discriminant, then we get a pairing $(D_1=\frac{x^2+4p}{|D_2|},D_2)$ such that $D_1D_2-4p$ is a square. Moreover, we have Algorithm 3.\par The correctness of Algorithm 3 is obvious. We will analyze the time complexity of Algorithm 3.\par We will discuss the time complexity of step 5. Assume $|D|=p_1^{a_1}\cdots p_t^{a_t}$ with $a_i>0$, we have the following equations: $$\left \{ \begin{array}{c} x^2 \equiv -4p \pmod {p_1^{a_1}}; \\ \cdots\\ x^2 \equiv -4p \pmod {p_t^{a_t} }. \end{array} \right.$$\par If $p_i>2$, the solutions of $x^2 \equiv -4p \pmod {p_i^{a_i}}$ can be derived from the solutions of $x^2 \equiv -4p \pmod {p_1}$ by Theorem 9.3 in \cite[Chapter 2]{MR665428}. Moreover, one can solve the equation $x^2 \equiv -4p \pmod {p_1}$ in polynomial time if we assume the Generalized Riemann Hypothesis (GRH). If $p_i=2$, then $a_i \ge 2$ and we can solve the equation $x^2 \equiv -p \pmod {2^{a_i-2}}$ by Theorem 5.1 in \cite[Chapter 3]{MR665428}. By this way, it is easy to get the solutions of $x^2 \equiv -4p \pmod {2^{a_i}}$, and we omit the details. If we do not assume the GRH, Bourgain et al.\cite{MR3378857} researched the polynomial factorization problem when the polynomial $f$ fully splits over $\mathbb{F}_p$. They proved the following theorem. \begin{theorem}
There is a deterministic algorithm that, given a squarefree polynomial $f \in \mathbb{F}_p[X]$ of degree $n$ that fully splits over $\mathbb{F}_p$, finds in time $(n+p^{1/2})p^{\epsilon}$ a root of $f$. \end{theorem}
Next, we can get $x \pmod{|D|}$ by Chinese Remainder Theorem. Generally, there are about $2^t$ many solutions where $t$ is the number of distinct prime factors of $|D|$. As we know (see \cite[Chapter 5.10]{MR665428}), $t\sim \log\log(|D|)$ for almost all $|D|$, so there are $O(\log |D|)$ many solutions of $x^2 \equiv -4p \pmod{|D|}$ for almost all $|D|$. As we discussed above, step 5 can be done in polynomial time, and factoring $D$ in step 3 occupies the most time in Algorithm 3. \begin{algorithm}[H] \caption{ Computing the class number of $\mathbb{Q}(\sqrt{-p})$.} \label{alg:Framwork} \begin{algorithmic}[1] \Require A prime $p$; \Ensure The class number of $\mathbb{Q}(\sqrt{-p})$; \State Let $s_p=0$ and $t=0$; \label{code:fram:let}
\State For every negative discriminant $D\equiv 0$ or $1 \pmod4$ satisfying $|D| < \frac{4}{\sqrt{3}}\sqrt{p}$, if the Legendre symbol $\left (\frac{D}{p} \right )=-1$, do the following three steps; \label{code:fram:for} \State Factor $D$ and compute the generators of the field $E$; \label{code:fram:if}
\State If $p$ splits completely in $E$, then $s_p=s_p+2^{\mu-1}$; Furthermore, if $|D|>2\sqrt{p}$, then do the next step; \label{code:fram:split}
\State Solve the equation $x^2 \equiv -4p \pmod {|D|}$ and calculate the number $t_D$ of solutions satisfying $\sqrt{3p} < \frac{x^2+4p}{|D|} < |D|$ and $\frac{x^2+4p}{D} \equiv 0,1 \pmod 4$. Let $t=t+t_D$; \State $ h= \left\{ \begin{array}{ll} \frac{1}{2} (s_{p}-t) &{\text{if} \ p\equiv 3 \pmod{8} ,} \\ s_{p}-t &{\text{if} \ p\equiv 7 \pmod{8} ,}\\ 2(s_{p}-t) &{\text{if} \ p\equiv 1 \pmod{4}; } \end{array} \right.$ \label{code:fram:equation} \\ \Return $h$. \end{algorithmic} \end{algorithm}
\begin{theorem} The time complexity of Algorithm 3 is $O(p^{3/4+\epsilon})$. Moreover, if we factor $D$ by probabilistic algorithms and assume the Generalized Riemann Hypothesis (GRH), the time complexity can be reduced to $\tilde{O}(p^{1/2})$. \end{theorem} \begin{proof}
As discussed above, for every $D$ with $\left ( \frac{D}{p} \right )=-1$, we need factor $|D|$ in step 3. Moreover, for $|D|>2\sqrt{p}$, we also need solve the equation $x^2 \equiv -4p \pmod {|D|}$, which can be done in polynomial time under the Generalized Riemann Hypothesis (GRH)or in time $O(p_i^{1/2+\epsilon})(O(p^{1/4+\epsilon}))$ by Theorem 10. Other steps can be computed in polynomial time. There are about $\frac{2}{\sqrt{3}}\sqrt{p}$ many $D$, so this theorem holds. \end{proof} We present in the following table the results of applying Algorithm 3 to compute class numbers of $\mathbb{Q}(\sqrt{-p})$ with various size discriminants. The computations were all carried out on a PC microcomputer with Intel(R) Core(TM) i5-10210U by PARI/GP. Here $h$ is the class number of the quadratic field $\mathbb{Q}(\sqrt{-p})$, and $T$ is the time required to compute the class number of $\mathbb{Q}(\sqrt{-p})$ by Algorithm 3. All these $T$ are expressed in second. \begin{table}[H]
\centering
\setlength{\tabcolsep}{5mm}{
\begin{tabular}{l r r }
\toprule
$p$ & $h$ & $T$ \\
\midrule
$10^{11}+283$ & $88847$ & $14$ \\
$10^{12}+547$ & $240171$ & $50$ \\
$10^{13}+99$ & $670135$ & $182$ \\
$10^{14}+99$ & $1981515$ & $694$ \\
$10^{15}+9867$ & $4921593$ & $2605$\\
\bottomrule
\end{tabular}} \end{table} The memory space of Algorithm 3 can be neglected. For similar-sized $p$'s, the running time changes as the class number changes. We use the deterministic factorization algorithms in the program. The data in the table illustrate the correctness and effectiveness of Algorithm 3.
\section{Conclusion}
For a prime $p$, let $D_1$, $D_2$ be any two negative discriminants. Assume that $D_1$ and $D_2$ are not quadratic residues in $\mathbb{F}_p$ with $|D_1|< |D_2| < \frac{4\sqrt{p}}{3}$, then $H_{D_1}(X)$ and $H_{D_2}(X)$ have a common root in $\mathbb{F}_p$ if $D_1D_2-4p$ is a square. The case which $H_{D_1}(X)$ and $H_{D_2}(X)$ have a common root in $\mathbb{F}_{p^2} \setminus \mathbb{F}_p$ is also interesting, and it may be more difficult.\par The correctness of our algorithms, Algorithm 2 and Algorithm 3, are guaranteed by Theorem 5, Theorem 8 and Theorem 9. In fact, we use neither the supersingular $j$-invariants nor the class polynomials in the calculation process.
\section{Acknowledge} The authors would like to thank Shparlinski for suggestions on the complexity of Algorithm 3 and Jianing Li for comments on the proof of Theorem 5.
\end{document} | arXiv |
\begin{document}
\title{Consciousness and the Collapse of the Wave Function hanks{Forthcoming in (S. Gao, ed.) extit{Consciousness and Quantum Mechanics}
\begin{abstract}
Does consciousness collapse the quantum wave function? This idea was taken seriously by John von Neumann and Eugene Wigner but is now widely dismissed. We develop the idea by combining a mathematical theory of consciousness (integrated information theory) with an account of quantum collapse dynamics (continuous spontaneous localization). Simple versions of the theory are falsified by the quantum Zeno effect, but more complex versions remain compatible with empirical evidence. In principle, versions of the theory can be tested by experiments with quantum computers. The upshot is not that consciousness-collapse interpretations are clearly correct, but that there is a research program here worth exploring.
\end{abstract}
{\bf Keywords:} wave function collapse, consciousness, integrated information theory, continuous spontaneous localization
\tableofcontents
\section{Introduction}
One of the hardest philosophical problems arising from contemporary science is the problem of quantum reality. What is going on in the physical reality underlying the predictions of quantum mechanics? It is widely accepted that quantum-mechanical systems are describable by a wave function. The wave function need not assign definite position, momentum, and other definite properties to physical entities. Instead it may assign a superposition of multiple values for position, momentum, and other properties. When one measures these properties, however, one always obtains a definite result. On a common picture, the wave function is guided by two separate principles. First, there is a process of evolution according to the Schrödinger equation, which is linear, deterministic, and constantly ongoing. Second, there is a process of collapse into a definite state, which is nonlinear, nondeterministic, and happens only on certain occasions of measurement.
This picture is standardly accepted at least as a basis for empirical predictions, but it has been less popular as a story about the underlying physical reality. The biggest problem is the \textit{measurement problem} (see \textcite{Albert1992}; \textcite{Bell1990}). On this picture, a fundamental measurement-collapse principle says that collapses happen when and only when a measurement occurs. But on the face of it, the notion of “measurement” is vague and anthropocentric, and is inappropriate to play a role in a fundamental specification of reality. To make sense of quantum reality, one needs a much clearer specification of the underlying dynamic processes.
Another of the hardest philosophical problems arising from contemporary science is the mind-body problem. What is the relation between mind and body, or more specifically, between consciousness and physical processes? By consciousness, what is meant is phenomenal consciousness, or subjective experience. A system is conscious when there is something it is like to be that system, from the inside. A mental state is conscious when there is something it is like to be in that state.
There are many aspects to the problem of consciousness, including the core problem of why physical processes should give rise to consciousness at all. One central aspect of the problem is the \textit{consciousness-causation problem}: how does consciousness play a causal role in the physical world? It seems obvious that consciousness plays a causal role, but it is surprisingly hard to make sense of what this role is and how it can be played.
There is a long tradition of trying to solve the consciousness-causation problem and the quantum measurement problem at the same time, by saying that measurement is an act of consciousness, and that consciousness plays the role of bringing about wave function collapse. The locus classicus of this consciousness-collapse thesis is Eugene Wigner's 1961 article “Remarks on the mind-body question”. There are traces of the view in earlier work by \textcite{vonNeumann1955} and \textcite{LondonBauer1939}.\footnote{It is clear that \textcite{vonNeumann1955} endorses a measurement-collapse interpretation, and he says (p.418) that subjective perception is “related” to measurement, but he does not clearly identify measurement with conscious perception. In his discussion of observed systems (I), measuring instruments (II), and “actual observer” (III), he says “the boundary can just as well be drawn between I and II+III as between I+III and III”. This suggests neutrality on whether the collapse process is triggered by measuring devices or by conscious observers. He also says that the boundary is “arbitrary to a very large extent” (p.420), which is not easy to reconcile with the fact that different locations for collapse are empirically distinguishable in principle, as we discuss in section 6. \textcite[section 11]{LondonBauer1939} say more clearly: “We note the essential role played by the consciousness of the observer in this transition from the mixture to the pure case. Without his effective intervention, one would never obtain a new psi function” (although see \textcite{French2020} for an alternative reading).} In recent years the approach has been pursued by Henry Stapp (1993) and others.
The central motivations for the consciousness-collapse view come from the way it addresses these problems. Where the problem of quantum reality is concerned, the view provides one of the few interpretations of quantum mechanics that takes the standard measurement-collapse principle at face value. Other criteria for measurement may be possible, but understanding measurement in terms of consciousness has a number of motivations. First, it provides one of the few non-arbitrary criteria for when measurement occurs. Second, it is arguable that our core pretheoretical concept of measurement is that of measurement by a conscious observer. Third, the consciousness-collapse view is especially well-suited to save the central epistemological datum that ordinary conscious observations have definite results. Fourth, understanding measurement as consciousness provides a potential solution to the consciousness-causation problem: consciousness causes collapse.
Despite these motivations, the consciousness-collapse view has not been popular among contemporary researchers in the foundations of physics. Some of this unpopularity may stem from the popularity of the view in unscientific circles: for example, popular treatments by \textcite{Capra1975} and \textcite{Zukav1979}, who link the view to Eastern religious traditions. More substantively, the view is frequently set aside in the literature on the basis of \textit{imprecision} and on the basis of \textit{dualism}.
The objection from imprecision is stated succinctly by \textcite[pp.82--3]{Albert1992}
\begin{quote}
“How the physical state of a certain system evolves (on this proposal) depends on whether or not that system is conscious; and so in order to know precisely how things physically behave, we need to know precisely what is conscious and what isn't. What this “theory” predicts will hinge on the precise meaning of the word conscious; and that word simply doesn't have any absolutely precise meaning in ordinary language; and Wigner didn't make any attempt to make up a meaning for it; so all this doesn't end up amounting to a genuine physical theory either.” \end{quote}
We think that the force of this objection is limited. Of course it is true that ‘conscious’ in ordinary language is highly ambiguous and imprecise, but it is easy to disambiguate the term and make it more precise. Philosophers have distinguished a number of meanings for the term, the most important of which is phenomenal consciousness. As usually understood, a system is phenomenally conscious when there is something it is like to be that system: so if there is something it is like to be a bat, a bat is phenomenally conscious, and if there is nothing it is like to be a rock, a rock is not phenomenally conscious. One might question the precision of this concept in turn, but it is at least a common and widely defended view (see e.g. \textcite{Antony2006}; \textcite{Simon2017}) that it picks out a definite and precise property. On this view, phenomenal consciousness comes in a number of varieties, but it is either definitely present or definitely absent in a given system at a given time.
In recent years, theories that give precise mathematically-defined conditions for the presence or absence of consciousness have begun to be developed. The most well-known of these theories is Tononi’s integrated information theory (\cite{Tononi2008}), which specifies a mathematical structure for conscious states and quantifies them with a mathematical measure of integrated information. Of course it is early days in the science of consciousness, and current theories are unlikely to be final theories. Nevertheless, it is possible to envisage precise theories of consciousness, and to reason about they might be combined with a consciousness-collapse view to yield precise interpretations of quantum mechanics.
Crucially, when different precise theories of consciousness are combined with the consciousness-collapse view, these yield subtly different experimental predictions. As a result, we have a further motivation for taking consciousness-collapse interpretations seriously: they can be tested experimentally. As we discuss in section \ref{experiments}, there is a long-term research program of experimentally testing consciousness-collapse interpretations and eventually supporting a precise consciousness-collapse interpretation. The required experiments are difficult, but advances in quantum computing may already exclude certain simple consciousness-collapse interpretations. Because of these considerations, the underdetermination of conditions for consciousness does not reflect any fundamental imprecision in consciousness-collapse views. It simply reflects an experimentally testable degree of freedom.
The second common objection to the consciousness-collapse view is that it is committed to dualism: the view that the mental and the physical are fundamentally distinct. The consciousness-collapse view treats consciousness in a special way that seems to exempt it from the standard quantum-mechanical laws governing physical systems. This remark by Peter Lewis (this volume) reflects a common attitude:
\begin{quote}
“Wigner postulates a strong form of interactive dualism in order to justify a duality in the physical laws. Few will want to follow Wigner down this path: non-physical minds, especially causally active ones, are mysterious at best.” \end{quote}
Again, we think the force of this objection is limited.
First: the consciousness-collapse thesis need not lead to dualism. It is compatible with materialist views on which consciousness is a complex physical property. For example, let us suppose a materialist version of integrated information theory on which consciousness is identical to $\Phi^{*}$, the property of having integrated information above a certain threshold. Then the consciousness-collapse theory will say that $\Phi^{*}$ causes collapse. This interpretation of quantum mechanics will involve a fundamental physical law saying that under the conditions specified by $\Phi^{*}$, collapse is brought about according to the Born rule. A fundamental law involving a complex physical property may be unlike familiar physical laws, but it involves nothing nonphysical.
Second: where consciousness is concerned, there are reasons to take dualism seriously. There are familiar reasons to question whether any purely physical theory can explain consciousness. One common reason (\cite{Chalmers2003}) is that physical theories explain only structure and dynamics (the so-called “easy problems” of behavior and the like), and explaining consciousness (the so-called “hard problem”) requires explaining more than structure and dynamics. These reasons need not lead to substance dualism, on which consciousness involves a separate nonphysical entity akin to an ego or soul, but they have led many theorists to adopt a form of property dualism where consciousness is accepted as a fundamental property akin to spacetime, mass, and charge.
Where physical theories give fundamental physical laws that connect physical properties to each other, a property dualist theory of consciousness gives fundamental psychophysical laws that connect physical properties to consciousness. For example, on a property dualist construal of integrated information theory, there might be a fundamental physics-to-consciousness law saying that when a system has $\Phi$ above a certain threshold, the system will have a corresponding state of consciousness. Such a law has a structure akin to the Newtonian mass-to-gravitational-field law, saying that when a system has a certain mass, the system will have a corresponding gravitational field. On a consciousness-causes-collapse theory, there will be an additional consciousness-to-physics law saying that states of consciousness bring about wave function collapse in a certain way. Putting these theories together might yield a mathematically precise version of property dualism that specifies the conditions under which consciousness arises and the role that it plays.
Interestingly, the most common reason among philosophers for rejecting property dualist theories of consciousness is an argument from physics. This argument runs roughly as follows: (1) every physical effect has only physical causes, (2) consciousness causes physical effects, so (3) consciousness is physical. The key first premise is a causal closure thesis, supported by the observation that there are no causal gaps in standard physics that a nonphysical consciousness might fill. But wave function collapse in quantum mechanics appears to be precisely such a gap, and consciousness-collapse models are at least not obviously ruled out by known physics. The situation is that many physicists rule out consciousness-collapse models for philosophical reasons (they are dualistic), while philosophers rule out property dualist models for physics-based reasons (they violate causal closure).
The upshot is that a central reason to reject the consciousness-collapse thesis (it leads to dualism) and a central reason to reject interactionist property dualism (it violates the causal closure of physics) provide no reason to reject the two views when taken together. Perhaps there are other reasons to reject the consciousness-collapse thesis or to reject dualism, but these reasons must be found elsewhere.
A third common objection to the consciousness-collapse thesis is that it is not \textit{necessary} to invoke consciousness in an interpretation of quantum mechanics, as there are alternative interpretations that give it no special role. Even if we retain the measurement-collapse framework, it is possible to understand measurement independently of consciousness, so that nonconscious systems such as ordinary measuring devices can collapse the wave function. Going beyond this framework, a number of alternative interpretations have been developed that give no role to the notion of measurement. These include spontaneous-collapse interpretations (e.g. \textcite{Pearle1976}; \textcite{Ghirardi1986}) which retain a collapse process but dispense with the need for measurement as a trigger, and hidden-variable interpretations (\cite{Bohm1952}) and many-worlds interpretations (\cite{Everett1957}), which eliminate collapse entirely.
We agree that one is not forced to accept a role for consciousness in quantum mechanics. At the same time, the mere existence of alternative interpretations is not itself good reason to reject the consciousness-collapse thesis. If it were, we would have good reason to reject all interpretations. Perhaps the underlying thought is that the consciousness-collapse thesis is extravagant and has certain costs, such as dualism. For there to be a serious objection here, an opponent needs to articulate the costs as objections in their own right. As with every other interpretation of quantum mechanics, the consciousness-collapse interpretation has both serious costs (dualism) and serious benefits (taking the standard dynamics at face value, solving the consciousness-causation problem). To assess any interpretation, we need to weigh its costs against its benefits.
In this article, we are exploring consciousness-collapse models rather than endorsing them. In particular, we are not asserting that these interpretations are superior to other interpretations of quantum mechanics. Both of us have considerable sympathy with other interpretations and especially with many-worlds interpretations (see \textcite[ch.10]{Chalmers1996} and \textcite{McQueenVaidman2019}). But we think that consciousness-collapse interpretations deserve close attention. If it turns out that these interpretations have fatal flaws, they can be set aside. But if there are consciousness-collapse interpretations without clear fatal flaws, then these interpretations should be taken seriously as possible descriptions of quantum-mechanical reality.
In our view, by far the most important challenge to consciousness-collapse models is not the issue of imprecision or of dualism, but the question of \textit{dynamic principles}. Can we find a simple, coherent, and empirically viable set of dynamic principles governing how consciousness collapses the wave function? If we can find such principles, consciousness-collapse models should be placed alongside other dynamic models (including Bohmian hidden-variable models, Everettian many-worlds models, and Pearle-GRW style spontaneous collapse models) as serious contenders to be the correct interpretation of quantum mechanics. If we cannot, then consciousness-collapse models may remain an important speculative class of models, but they will stay on the second tier of interpretations until they are cashed out with dynamic principles.
In what follows, we will explore the prospects for consciousness-collapse interpretations of quantum mechanics. We will do this mainly by exploring and evaluating potential dynamic principles. We focus especially on what we call \textit{super-resistance models}, according to which there are special properties that resist superposition and trigger collapse. When these models are combined with the consciousness-collapse thesis, we obtain models in which consciousness or its physical correlates resist superposition and trigger collapse. We think super-resistance consciousness-collapse models are worth investigating, and in this article we investigate some of them.
In this article we are not trying to solve the hard problem of how physical processes give rise to consciousness. We are giving an account of the causal role of consciousness that can be combined with many different approaches to the hard problem. Our approach is consistent with both materialist views, on which consciousness is identified with a complex physical property, and dualist views, on which consciousness is a primitive property that correlates with physical properties. Our approach is also consistent with many different theories of consciousness that correlate consciousness with underlying physical processes. For concreteness we will often assume a Tononi-style theory of consciousness on which consciousness is identical to or correlated with integrated information, but much of what we say should translate straightforwardly to other theories of consciousness.
We will not be addressing problems that come up for collapse models of quantum mechanics quite generally. For example, collapse models face important challenges stemming from the theory of relativity (collapse seems to require a privileged reference frame (\cite{Maudlin2011})), and the tails problem (collapse leaves wave functions with tails (\cite{McQueen2015})). The collapse models we consider certainly face these challenges. These are important challenges, but for present purposes we will be happy if consciousness-collapse interpretations can be shown to be about as viable as widely discussed spontaneous-collapse interpretations. Interpretations in both classes will still face the general problems. A number of ideas about how to deal with them have been put forward, but this is a topic for another day.
Our aim is to set out the best consciousness-collapse model that we can and to assess it. Our discussion is speculative and our conclusions are mixed. We articulate both positive models and serious limitations. We first articulate a simple consciousness-collapse model on which consciousness is entirely superposition-resistant. This model is subject to a conclusive objection (distinct from those outlined above) arising from the quantum Zeno effect. We then articulate a model that is not subject to this objection, combining integrated information theory with Pearle’s continuous-collapse theory. We explore the prospects of empirically testing these models, and discuss some objections. The model is still subject to both empirical and philosophical objections, but there are some potential ways forward. The upshot is not that consciousness-collapse interpretations are clearly correct, but that there is a research program here worth exploring.
\section{Consciousness as super-resistant}
One can clarify the options for a consciousness-collapse theory by asking a crucial question for any collapse model of quantum mechanics: What is the locus of collapse? That is, which observable determines the definite states that the collapse process projects superposed states onto? Here there are two options: there can be a \textit{variable locus} (different observables serve as the locus on different occasions of collapse) or a \textit{fixed locus} (the same observable always serves as the locus of collapse).
A variable-locus model is closest to standard formulations of quantum mechanics. On a standard understanding, many different observable quantities (e.g. position, momentum, mass, and spin) can be measured and thereby serve as the locus of collapse. Every observable is associated with an operator. Upon measurement, the wave function collapses probabilistically into an eigenstate of that operator, and the measurement reveals the corresponding eigenvalue for the observable (such as a specific position for the particle), with probabilities determined by the prior quantum state according to the Born rule.
Henry Stapp’s consciousness-collapse model (\cite{Stapp1993}) is a variable-locus model, on which consciousness collapses whatever observable is being consciously observed at a given time. The variable-locus approach has some attractions, but it also faces some hard questions. Not least is the question: what determines which observable is being measured? This question is hard enough that Stapp's model postulates an entirely separate process that determines the locus of collapse. Stapp calls this process “asking a question of nature”, which is supposed to be something that takes place in the mind of an observer. Stapp takes this to be a third process distinct from von Neumann’s standard dual processes of collapse itself and Schrödinger evolution. Stapp takes this third process as primitive. There are options for analyzing it (perhaps via a precisely specified observation relation between observers and observables, for example, or by building awareness of observables into the structure of consciousness), but it is clear that such a theory will be complex.
One option for a variable-locus consciousness-collapse theory invokes the idea that consciousness represents certain objects and properties in its environment. For example, visual experiences typically represent the color, shape, and location of observed objects, while auditory experiences represent locations, pitches, and the like. A consciousness-collapse view may hold that when consciousness represents observable properties of an observed object, the object collapses into a definite state of those observables. For example, perceiving the location of a ball that was previously in a superposition will collapse the ball into a definite location. One trouble here is that on standard representationalist views, the represented properties are built into a state of consciousness but the represented objects are not. In some cases an experience as of a single object may be caused by no object or by multiple objects in reality, so there is still a difficult question about which object if any undergoes collapse. This approach may work better with relationist views where consciousness involves direct awareness of specific objects and properties, but there will still be many complications.\footnote{For representationalist views, see \textcite{Tye1995}. For relationist views, see \textcite{ByrneLogue2009}. These views may face a version of the Zeno problem in the next section, arising from whether the states of consciousness themselves can enter superpositions.}
Fixed-locus models are simpler in a number of respects, and we will focus on them. In a fixed-locus measurement-collapse model, there are special properties that serve as the locus of collapse. In a fixed locus consciousness-collapse model, consciousness itself (or perhaps its physical correlate) serves as the locus of collapse. It is this idea that we will develop in what follows.
One natural way to develop a fixed-locus collapse model is through the idea of superposition-resistance, which we will sometimes abbreviate as super-resistance. The idea is that there are special superposition-resistant observables, which as a matter of fundamental law resist superposition and cause the system to collapse onto eigenstates of these observables (with probabilities given by the Born rule). The corresponding class of models are \textit{super-resistance models} of quantum mechanics.\footnote{In earlier versions of this article we called superposition-resistant observables “m-properties” (short for “measurement properties”) and super-resistance models “m-property models”.} There are a number of different ways to make the dynamics of super-resistance precise, some of which we will explore in the following sections. A strong version of super-resistance invokes fundamental \textit{superselection rules} (\cite{WickWightmanWigner1952}), according to which certain observables are entirely forbidden from entering superpositions. A weaker version invokes principles according to which these superpositions are unstable and tend to collapse.
There are super-resistance models of collapse that give no special role to consciousness or measurement. One well-known super-resistance model is Penrose’s model (\cite{Penrose2014}) of quantum mechanics on which spacetime structure is superposition-resistant: when the structure of spacetime evolves into superpositions over a certain threshold, these superpositions collapse onto a definite structure. One can also see the GRW interpretation of quantum mechanics as an interpretation on which position is mildly superposition-resistant: superpositions of position tend to collapse, though with low probability for isolated particles.
Super-resistance models work well with measurement-collapse interpretations of quantum mechanics. In the context of these interpretations, we can think of a super-resistant property not as a \textit{measured} property (e.g. particle position) but as a \textit{measurement} property (e.g. a pointer position or a conscious experience). To sketch the idea intuitively: suppose there is a special class of measurement devices (e.g. oscilloscopes) which have special measurement properties (e.g. meter readings or pointer locations) that (as a matter of fundamental law) resist superposition and tend to collapse. When a measurement takes place, a measured property affects a measurement property. Suppose that we have a quantum system (e.g. a particle) in a superposition of locations $a$ and $b$, which we represent (simplifying by omitting amplitudes) as the quantum state $\ket{a} + \ket{b}$. The particle interacts with a measurement system such that if not for this principle, it would yield an entangled superposition
$\ket{a}\ket{M(a)} + \ket{b}\ket{M(b)}$, where $M(a)$ and $M(b)$ are the states of the measurement system. Because $M$ is superposition-resistant, the particle and measurement system will instead evolve into a collapsed state $\ket{a}\ket{M(a)}$ or $\ket{b}\ket{M(b)}$, with probabilities given by the Born rule. The effect will be much the same as if the measured property collapsed directly, but now the measurement properties serve as a single locus of collapse.
Superposition-resistance is an especially natural idea in the context of consciousness-collapse models of quantum mechanics. The idea that consciousness resists superposition is suggested in a brief passage in \textcite{Wigner1961}, and is later developed by \textcite{Albert1992}, and \textcite{Chalmers2003}.
Wigner writes:
\begin{quote}
``If the atom is replaced by a conscious being, the wave function $\alpha(\phi_1 \times \chi_1) + \beta (\phi_2 \times \chi_2)$ (which also follows from the linearity of the equations) appears absurd because it implies that my friend was in a state of suspended animation before he answered my question. It follows that the being with a consciousness must have a different role in quantum mechanics than the inanimate measuring device: the atom considered above. In particular, the quantum mechanical equations of motion cannot be linear.” (\cite[p.180]{Wigner1961})
\end{quote}
Wigner’s suggestion seems to be that a state of consciousness cannot be superposed because it would require being in a “state of suspended animation”. Wigner does not suggest a dynamic process for collapse here, but potential processes are fleshed out a little by Albert and Chalmers. Albert suggests a picture on which the physical correlates of consciousness immediately collapse once superposed:
\begin{quote}
All physical objects almost always evolve in strict accordance with the dynamical equations of motion. But every now and then, in the course of some such dynamical evolutions (in the course of measurements, for example), the brain of a sentient being may enter a state wherein (as we've seen) states connected with various different conscious experiences are superposed; and at such moments, the mind connected with that brain (as it were) opens its inner eye, and gazes on that brain, and that causes the entire system (brain, measuring instrument, measured system, everything) to collapse, with the usual quantum-mechanical probabilities, onto one or another of those states; and then the eye closes, and everything proceeds again in accordance with the dynamical equations of motion until the next such superposition arises, and then that mind's eye opens up again, and so on. (\cite[pp.81-2]{Albert1992})
\end{quote}
Albert is entertaining the view mainly for the sake of argument, and he almost immediately rejects it in the passage quoted earlier about the imprecision of consciousness. Chalmers writes more sympathetically:
\begin{quote}
Upon observation of a superposed system, Schrödinger evolution at the moment of observation would cause the observed system to become correlated with the brain, yielding a resulting superposition of brain states and so (by psychophysical correlation) a superposition of conscious states. But such a superposition cannot occur, so one of the potential resulting conscious states is somehow selected (presumably by a nondeterministic dynamic principle at the phenomenal level). The result is that (by psychophysical correlation) a definite brain state and state of the observed object are also selected. The same might apply to the connection between consciousness and non-conscious processes in the brain: when superposed non-conscious processes threaten to affect consciousness, there will be some sort of selection. In this way, there is a causal role for consciousness in the physical world. (\cite[pp.262-3]{Chalmers2003})
\end{quote}
Chalmers in effect combines Wigner’s suggestion that consciousness cannot superpose with Albert’s suggestion that consciousness collapses its physical correlates. The key idea here is that consciousness is a superposition-resistant property and that its physical correlates therefore resist superposition too. That is, it is difficult or impossible for a subject to be in a superposition of two different states of consciousness, and this results in the collapse of physical processes that interact with consciousness.\footnote{\textcite{Halvorson2011} also argues for a picture on which mental states cannot be superposed and therefore bring about collapse in the physical world.}
Here the relevant states are total conscious states of a subject at a time. The total conscious state of a subject is what it is like to be that subject: if what it is like to be subject A is the same as what it is like to be subject B, then A and B are in the same total conscious state. A subject’s total conscious state at a time may include many aspects: visual experience, auditory experience, the experience of thought, and so on. Like position or mass or color or shape, consciousness in this form can take on many specific values. Its specific values are the vast range of possible total conscious states of a subject at a time.
This view assumes that there is a physical correlate of consciousness (PCC): a set of physical states that correlate perfectly with a system's conscious states. For simplicity, we can start by assuming a materialist view where the total conscious state and its physical correlate are identical. Things work best if we also assume that the physical correlate of consciousness (PCC) can itself be represented as a quantum observable with an associated operator. This assumption is nontrivial, as not every physical property is an observable; we return to it later. A PCC observable will have many different eigenstates corresponding to distinct total states of consciousness. This makes it straightforward to treat consciousness as a super-resistant property.
To illustrate how this works, we can again suppose an electron in a superposition of locations (again omitting amplitudes for simplicity) $\ket{a} + \ket{b}$. The electron registers on a measurement device and then the result is perceived by a human subject. Assuming the measurement device is not conscious, then at the first stage the electron and the device will go into an entangled state $\ket{a}\ket{M(a)} + \ket{b}\ket{M(b)}$. When the human looks, this result will affect the eye (E), early areas of the nervous system and brain (B), and eventually the physical correlates of consciousness (PCC). Under Schrödinger evolution, we would expect the electron, device, and subject to go into an entangled state $\ket{a}\ket{M(a)}\ket{E(a)}\ket{B(a)}\ket{PCC(a)} + \ket{b}\ket{M(b)}\ket{E(b)}\ket{B(b)}\ket{PCC(b)}$. However, this superposed state would yield a superposition of states of consciousness. So at the point where the PCC is affected, the system will collapse. It collapses into $\ket{a} \ket{M(a)}\ket{E(a)}\ket{B(a)}\ket{PCC(a)}$ or $\ket{b} \ket{M(b)}\ket{E(b)}\ket{B(b)}\ket{PCC(b)}$, with Born rule probabilities. In effect, at the point where the measurement reaches consciousness, the electron, the measurement device, and the brain will collapse into a definite state.
On a dualist view on which consciousness merely correlates with physical properties, things are a little more complicated. We focus on forms of dualism where there are psychophysical laws correlating physical states of a system with states of consciousness. There will be a set of physical correlates of consciousness (which may be disjunctive if necessary) that are in one-to-one correspondence with total states of consciousness. A subject will be in a given state of consciousness if and only if it is in the corresponding PCC state. We can assume as before that the PCC is a quantum observable. Psychophysical laws connect unsuperposed PCC eigenstates to unsuperposed states of consciousness. They also connect superpositions of PCC states to the corresponding superpositions of states of consciousness. A given subject's PCC is in a superposition of PCC eigenstates with certain amplitudes if and only if the subject’s conscious experience is in a superposition of the corresponding total states of consciousness with the same distribution of amplitudes.
On a dualist view, a fundamental principle will say that consciousness resists superposition. Whenever Schrödinger evolution plus the psychophysical laws entail that a system enters or is about to enter a superposition of total states of consciousness, the system will collapse into a definite total state of consciousness. As a result, the PCC will also collapse into an eigenstate, and other physical entities that are entangled with the PCC will collapse as described above.
One motivation for the super-resistance consciousness-collapse model is given by Wigner’s suggestion that superpositions of consciousness are “absurd”. That is, something about the very nature of consciousness or the concept of consciousness rules out total states where consciousness is superposed. It is certainly at least very hard to imagine subjects who are in superposed states of consciousness (at least without these states becoming total states of consciousness in their own right). If something about the nature of consciousness explains why it cannot be superposed, then this might provide a possible explanation of why collapse comes about. This explanatory motivation might be seen as a further motivation for understanding consciousness as the trigger of collapse.
Taking Wigner’s motivation seriously leads to the idea that consciousness is \textit{absolutely superposition-resistant}: that is, that it can never enter superpositions, even brief and unstable ones. Invoking absolute superposition-resistance leads to a clean and simple dynamic model for collapse involving superselection rules. Unfortunately this model leads to a fatal problem for absolute super-resistance, which we explore in the next section.
\section{Superselection and the Zeno problem}
To develop super-resistance models in more detail, we can start by thinking of them independently of consciousness. In principle any observable could serve as a super-resistant observable, with distinct models of quantum mechanics arising from taking different observables to resist superposition. Later we can consider the special case where consciousness or its physical correlates serve as super-resistant observables.
The simplest (albeit fatally flawed) super-resistance model invokes superselection: the strong form of super-resistance where certain superpositions are ruled out entirely. In particular, it invokes the familiar concept of a superselection rule: a rule postulating that superpositions of a specified observable are forbidden.
Superselection rules are invoked for a number of purposes in quantum mechanics.\footnote{Superselection rules were introduced by \textcite{WickWightmanWigner1952}. There are many somewhat different definitions of superselection rules, analyzed thoroughly by \textcite{Earman2008}. Here we use a common informal definition. Superselection rules are invoked in analyses of the measurement process by \textcite{Bub1988}, \textcite{Hepp1972}, \textcite{MachidaNamiki1980}, and others. \textcite{Thalos1998} gives an excellent review. The most common strategy is to argue that superselection rules can emerge from the Schrödinger dynamics governing the interaction of a system with its environment. It is unclear to us whether anyone has explicitly proposed a superselection collapse interpretation, but we are open to pointers.} Sometimes they are postulated to analyze quantum-mechanical properties that are never found in superpositions, such as the difference in charge between a proton and a neutron. Sometimes they are used to help analyze quantum-mechanical symmetries. Sometimes they are used to help address measurement in quantum mechanics, most often through the idea that superselection can emerge through interaction with the environment by Schrödinger evolution alone.
Here we are exploring a somewhat different idea: the idea of a \textit{superselection collapse model}, with a fundamental superselection rule governing the collapse process. Such a model will specify a superselection observable, such that physical systems must always be in eigenstates of the operator corresponding to the observable. The associated collapse postulate says that whenever a system would otherwise enter a superposition of eigenstates of this operator (given Schrödinger dynamics alone), it instead enters a definite eigenstate, with probabilities given by the Born rule. In the special case where consciousness (or its physical correlate) is a superselection observable, then whenever consciousness would otherwise be about to enter a superposition, it must collapse to a definite state according to the Born probabilities.
To specify the dynamics better, we can first suppose that the collapse takes place at a time interval of $\Delta t$, so that if the system has evolved (according to the Schrödinger equation) in the preceding $\Delta t$ into a non-eigenstate of the superselection observable, it collapses probabilistically into an eigenstate of that operator, with probabilities given by the Born rule. This yields a well-defined stochastic process. For the absolute super-resistance model, the dynamics is the limiting case of this process as $\Delta t$ approaches zero.
The superselection collapse model has a dynamics that is already familiar in quantum mechanics: it is precisely the dynamics that would obtain (on a traditional measurement interpretation) if the resistant observable were being continuously measured by an outside observer. The current approach does not require that there are any outside observers, or that resistant properties themselves are ever measured, or that continuous measurement ever takes place (though to aid the imagination, one could metaphorically suppose that God is continuously measuring the resistant properties of the entire universe). All that it requires is the mathematical dynamics associated with continuous measurement of resistant properties, which is fairly straightforward.
Unfortunately, the dynamics of continuous measurement leads to a well known effect, the quantum Zeno effect, which renders any superselection collapse model empirically inadequate. The quantum Zeno effect is the effect whereby the more often one measures a quantum observable, the harder it is for the system to enter different states of that observable. In the extreme case where an observable is measured continuously, it cannot change at all.
The source of the quantum Zeno effect lies in the mathematical fact that for a system to evolve under Schrödinger evolution from some initial eigenstate of an operator to some other eigenstate of that operator, it must evolve through superpositions of eigenstates.\footnote{One could argue that this mathematical fact is the common explanation both of the Zeno effect and of the problem for superselection collapse models, rather than the Zeno effect explaining the problem. Still, the problem is still aptly called a Zeno problem, tied to the impossibility of motion.} Eigenstates are orthogonal to each other, so the continuous process of Schrodinger evolution cannot evolve directly from eigenstate to eigenstate. If a system governed by this process cannot pass through superpositions of these eigenstates, then the system cannot change from one eigenstate to another. Another way to put things is that if small superpositions are permitted, an initial superposition will assign probability 1-$\epsilon$ (where $\epsilon$ is negligible) to the initial eigenstate. So if there is a measurement of this observable in the first moment, the superposition will collapse to the initial eigenstate with probability 1-$\epsilon$. Continuous measurement will therefore force the system to remain in that initial eigenstate.
This leads to the Zeno problem for superselection collapse interpretations. If there is a superselection observable (one that can never enter superpositions), every system will remain forever in a single eigenstate of that observable. This consequence may be acceptable for standard superselection observables in physics (such as the charge difference between a proton and a neutron), but it is clearly unacceptable for observables tied to measurement that serve as triggers of the collapse process.\footnote{Mariam \textcite[p.538]{Thalos1998} raises a version of this problem for superselection-based accounts of measurement, arguing that if a classical quantity is governed by a superselection rule, it can never change its magnitude in evolution over time.} For example, if a superselection observable corresponds to the position of the pointer on a measurement device, then that pointer will be forever stuck in one location and unable to give useful measurement results.
We can illustrate the Zeno problem by taking the superselection observable to be consciousness (or its physical correlate). We know that systems have different conscious states at different times, and sometimes evolve from being unconscious to being conscious. If consciousness or its physical correlate was a superselection observable, it would obey the dynamics of continuous measurement so it could not change at all. If we started in an unconscious state, we could never become conscious. The unfortunate consequence would be that we could never wake up from a nap. Furthermore, if there is no consciousness in the early universe, then consciousness could never emerge later.\footnote{Barry \textcite{Loewer2002} raises a different early-universe problem for consciousness-collapse theories: if the first collapse requires the universe to be in a non-null eigenstate of consciousness, then this will never happen, while if collapse is triggered by any superposition of consciousness, then the first collapse will happen too early. The absolute super-resistance model takes the second horn. On this view, Loewer’s “collapse too early” problem can be minimized by having conditions for consciousness that are not satisfied in the early universe (so that in its early stages, the universe will be in a null eigenstate of consciousness), and also by noting that most initial collapses when they occur will be onto a null state of consciousness. The Zeno problem as it arises for the early universe is the distinct but related problem that all collapses will be onto a null state of consciousness.}
The Zeno problem is not just a problem for superselection collapse interpretations. In “Zeno Goes to Copenhagen”, we argue that the Zeno problem is a serious problem for almost any measurement-collapse interpretation of quantum mechanics. Any such interpretation faces the question of whether measurement itself can enter quantum superpositions. If measurement can enter superpositions, the standard dynamics of collapse upon measurement is ill-defined, and new dynamics is required. If measurement cannot enter superpositions, the quantum Zeno effect suggests that measurements can never start or finish, at least if measurement is an observable. One way out is to deny that measurement is an observable, but this option leads to further commitments (embracing a strong form of dualism or construing measurement as a special wave-function property) that themselves require a highly revisionary approach.
In this article, however, we are focusing on the Zeno problem as a problem for super-resistance interpretations. To handle the Zeno problem in this framework, the obvious move is to abandon superselection (on which superpositions of the relevant observable are entirely forbidden) for a weaker version of super-resistance. An {\em approximately super-resistant} observable is one that can enter superpositions but nevertheless resists superposition, at least in some circumstances. On a simple version of this view, superpositions of the observable in question are unstable and they probabilistically tend to collapse over time.
To make the idea of approximate super-resistance precise, we require nonstandard physics. Fortunately, there is a wealth of resources for developing such physics in the literature on modern dynamical collapse theories (\cite{Bassietal2013}). In section 6, we show how these theories can be adapted to yield a model on which consciousness is approximately super-resistant. The rough idea is that as a total state of consciousness (and/or its physical correlate) enters increasingly large superpositions (where a large superposition is roughly one that gives significant amplitude to distant states), this yields higher probabilities of collapse of consciousness onto a more definite state. Admittedly it is far from clear what a superposition of states of consciousness would amount to. We return to this matter in the final section.
\section{Integrated information theory}
There are many ways to spell out the details of a consciousness-collapse super-resistance model. We can combine the view with many different theories of consciousness, and with various different accounts of the collapse dynamics. In what follows we spell out one way of working out some details, by combining the theory with a specific theory of consciousness (integrated information theory, or IIT) and a specific model of approximate super-resistance dynamics (inspired by Pearle’s continuous spontaneous localization interpretation of quantum mechanics).
We focus on IIT for several reasons. First, it is one of the few mathematically precise theories of consciousness. Second, unlike many competitors it purports to be a fundamental theory of consciousness that offers basic and universal principles connecting consciousness to physical processes. Third, it offers a specific physical correlate for total states of consciousness, using its notion of a Q-shape (qualia shape). Fourth, it has a distance metric between total states of consciousness, which plays an important role in our framework. None of this means that we are endorsing IIT. Many objections have been made to IIT (e.g. \textcite{Aaronson2014}, \textcite{Bayne2018}, \textcite{BarrettMediano2019}, \textcite{Doerig2019}) and they raise important issues. Our approach could in principle be combined with any theory that has the four properties just listed.
IIT is a theory that associates systems with both quantitative amounts of consciousness and qualitative states of consciousness. Its systems are classical Markovian networks made up of interconnected units that interact with each other according to deterministic or probabilistic rules. Each unit can take on a number of states, and the state of the system is made up of the states of each of the units in the system.
One limitation of IIT as it stands (\textcite{BarrettMediano2019}) is that its assigns amounts and states of consciousness to discrete Markovian network systems but not to real physical systems. To apply it to real physical systems, we need to combine it with a mapping from physical systems to network structures. In what follows we will assume such a mapping (or some other generalization of IIT) so that IIT applies to physical systems.
IIT is derived from phenomenological axioms rather than from experimental evidence. Experimental support for it is somewhat limited to date, especially because it is impractical to measure and calculate its measures of consciousness in biological systems. However, some measurable approximations of its quantitative measures have been shown to correlate with level of consciousness, see \textcite{Massiminietal2005}, \textcite{Casarottoetal2016}, \textcite{Leungetal2020}, and \textcite{Afrasiabietal2021}. Additionally, spatiotemporal patterns of integrated information (approximating IIT's qualitative measures) have been derived from brain areas and correlated with the contents of conscious perceptions of faces and other objects (\textcite{Haunetal2017}). In any case, we will treat IIT as a potential empirical theory of consciousness. Much of our discussion should generalize to other theories.
IIT is built around the notions of information and integration. The information in a system is a measure of the extent to which the present state of a system constrains its potential past and future states. One centerpiece of IIT is its measure of integration, which it labels $\Phi$. $\Phi$ is a measure of the extent to which the information in a system is irreducible to the information of its components. It quantifies how much the causal powers of a system fail to be accounted for by any partitioned version of it.
The simplest system with nonzero $\Phi$ is a dyad: a network AB with two interacting nodes A and B that swap their states. If A is on or off, B turns on or off at the next time step, and vice versa. In this case, AB has causal powers that are not reducible to those of A and B taken alone, and $\Phi$(AB) = 1. (We spell out the mathematics in an appendix.) By contrast, if A and B are not interacting, then the causal powers of AB are reducible to those of A and B taken alone, so $\Phi$(AB) = 0.
IIT says that a system is conscious if and only if it is a maximum of $\Phi$: that is, if the system has higher $\Phi$ than any system nested within it and higher $\Phi$ than any system it is nested within. The amount of consciousness in a system is $\Phi^{max}$, which is equivalent to $\Phi$ if the system is a maximum and 0 if the system is not. In what follows we drop the superscript for simplicity.
One way to combine IIT with a super-resistance model is to say that $\Phi$ is super-resistant. That is, $\Phi$ resists superposition and superpositions of $\Phi$ trigger collapse. Unfortunately, this view faces a fatal problem. It fails to suppress superpositions of qualitatively distinct conscious states with the same value of $\Phi$. Consider a conscious subject and a screen in a dark isolated room. The screen can display green or blue. If it is put into a superposition of displaying both, then the subject will be put into a superposition of experiencing green and experiencing blue. There is no reason to assume that these experiences differ in their $\Phi$–value. But then there is no $\Phi$–superposition, and so no collapse. The subject remains in a superposition of qualitatively distinct total states of consciousness. Such a theory therefore will not yield determinate experiences for many crucial observations. The underlying problem is that $\Phi$ is not a genuine physical correlate of consciousness – that is, it is not a physical correlate of a total state of consciousness. It is merely a physical correlate of a scalar degree of consciousness, where the same degree can be present in many different conscious states.\footnote{We canvassed the idea of using $\Phi$ as an absolutely super-resistant property in an early version of this article that raised the Zeno problem for absolute super-resistance and suggested approximate super-resistance via continuous localization as a possible solution. In an article responding to our early presentation and building on the ideas there, \textcite{OkonSebastian2018} develop the idea that $\Phi$ could be an approximately super-resistant property using continuous localization. Okon and Sebastian respond to our current objection by saying that decoherence makes it extremely unlikely that there will be superposed conscious states with the same value of $\Phi$. The blue/green case seems a clear case of this sort of superposition, however, as does any ensuing state resulting from interactions with their environment that makes no difference to their total state of consciousness. The dyad system discussed in the main text and the appendix gives a simple illustration of a superposition of states with different Q-shapes but with the same value of $\Phi$. In addition, the Q-shape collapse model is much better suited for giving all aspects of consciousness a causal role, whereas the $\Phi$-collapse model gives degree of consciousness a causal role and leaves everything else epiphenomenal.}
Fortunately, IIT also postulates a physical correlate of total states of consciousness. The Q-shape (qualia shape) of a system is an entity that serves as an abstract representation of the structure of the integrated information in a system. IIT specifies a mathematical mapping from network structures to Q-shapes. If we assume (as above) that the total physical state of a system determines a network structure, then IIT will derivatively specify a mapping from total physical states to Q-shapes.
The Q-shape of $S$ is a set of weighted points, one for each mechanism in $S$. A mechanism is a subsystem $m$ of $S$ -- that is, a nonempty set of elements of $S$ -- with $\phi(m)>0$ (as defined in Appendix A). If $S$ has $n$ elements, then it has up to $N=2^n-1$ mechanisms. For example, in the dyad system AB, which has two elements A and B, the subsystems are A, B, and AB, and the mechanisms are A and B. The weight associated with a mechanism $m$ is $\phi(m)$, a non-negative real number representing the integrated information associated with $m$. The point associated with $m$ is given by two probability distributions over the $2^n$ states of $S$, the so-called maximally irreducible cause repertoire and maximally irreducible effect repertoire associated with $m$.
According to IIT, a system’s Q-shape determines (at least nomologically) the total state of consciousness associated with that system. A Q-shape is itself a mathematical entity, and it is not obvious just how a Q-shape determines a state of consciousness. What matters most for our purposes is that according to IIT, (i) having a given Q-shape is a physically definable property (we might call it physical Q-shape), (ii) Q-shape is a physical correlate of consciousness, in that any two physical systems with the same associated physical Q-shape will have the same state of consciousness. It will also be helpful to assume the stronger theses that (iii) the mathematical structure of a conscious state is given by a Q-shape (call this a system's phenomenal Q-shape) and (iv) as a matter of psychophysical law, a system has a given phenomenal Q-shape (that is, it has a conscious experience with a given structure) if and only if has the isomorphic physical Q-shape (that is, it has a physical state with the same structure as defined by IIT). These claims are far from obviously correct, but something like them seems to be intended by IIT.
As before, it does not matter too much for our purposes whether these claims of IIT are correct. It is plausible that a final mathematical theory of consciousness will specify \textit{some} mathematical structure for consciousness (though there may be more to consciousness than its mathematical structure, as inverted qualia cases suggest). And it is plausible that this mathematical structure should be realized in some way in the physical correlates of consciousness. If necessary, we can replace Q-shape by that mathematical structure. What matters most is that there is some precise theory of consciousness for which psychophysical isomorphism principles like this are correct.
Different states of the dyad system AB discussed earlier can be associated with different Q-shapes. Consider state 10, where A is on and B is off, and state 00, where both A and B are off. As we show in the appendix, both states have $\Phi=1$, but they are associated with distinct Q-shapes. In principle one can prepare a dyad system in a superposition of these two states 10 and 00: we might call this \textit{Schrödinger’s dyad}. If Q-shape is super-resistant, Schrödinger’s dyad will be unstable and will collapse into a state with a definite Q-shape. We discuss a framework for combining IIT with quantum mechanics along these lines in the next two sections. In section 7, we discuss possible experimental tests, which are likely to rule out the simple Q-shape collapse interpretation but which suggest a program for empirically refining collapse interpretations.
\section[Combining IIT with quantum mechanics]{Combining IIT with quantum mechanics\footnote{This section is co-authored with Johannes Kleiner (Münich Center for Mathematical Philosophy, Ludwig Maximilian University).}}
The standard IIT framework (\textcite{Oizumietal2014}) maps classical network states to Q-shapes. We have assumed a derivative mapping from classical physical states to Q-shapes. To combine IIT with quantum mechanics, we need to extend the IIT mapping so that it maps quantum physical states to Q-shapes or to superpositions of Q-shapes. The core idea of a Q-shape collapse model is that systems in superpositions of Q-shape always collapse toward having a determinate Q-shape.
To extend the IIT mapping to quantum physical states, the obvious way to proceed is to use IIT's physical definition of Q-shape to define a set of Q-shape collapse operators, one for each dimension of Q-shape. The joint eigenstates of these operators will be physical states with determinate Q-shapes.
A challenge to defining these Q-shape operators is that in the classical IIT framework, $\phi$ and Q-shape depend on probabilities of state-transitions in a network, which may depend on the position and momentum of the system’s parts. Position and momentum are noncommuting operators, so physical systems cannot be in joint eigenstates of them. High-mass systems may have precise enough position and momentum to determine $\phi$ and Q-shape, but these quantities may not be defined for low-mass entities such as electrons in quantum systems (\textcite[p97]{McQueen2019a}).
There are various options for addressing this challenge. We could redefine $\Phi$ and Q-shape so they depend only on positions or mass densities of elements of the system. We could also give special treatment for low-mass systems, for example modifying $\Phi$ to stipulate that $\Phi=0$ for systems with mass below a certain threshold, or we could invoke a coarse-grained or ``smeared" version of $\Phi$ and Q-shape observables, with significant smearing mainly required for systems with very low mass.
Alternatively, we can invoke newer versions of IIT that are defined over quantum states. One framework for an IIT-driven collapse model has been developed by \textcite{KremnizerRanchin2015}, who define a new measure of quantum integrated information QII for quantum systems. On their model, a system's QII determines the probability of collapses onto a position basis, so that systems with higher QII are more likely to collapse on to the position basis. However, Kremnizer and Ranchin’s interesting model is a super-resistance theory only in a weak sense: the properties that trigger collapse (QII) are quite distinct from the collapse basis (position), and position resists superposition only in certain contexts with high QII. Also, while Kremnizer and Ranchin speculate that their quantity QII may be a measure of consciousness, this will yield at best a limited causal role for consciousness, on which the scalar amount of consciousness determines probability of collapse but the specific conscious state of a subject plays no role.
\textcite{zanardi2018quantum} have developed a more thoroughgoing quantum-mechanical version of IIT, defining quantum mechanical operators for each IIT notion (including Q-shape as well as $\phi$) across a broad class of quantum-mechanical networks. (These are networks of finite-dimensional non-relativistic qudits, interacting via Markovian trace preserving completely positive maps.) Further generalizations have been given by \textcite{KleinerTull2020}. These models do not yet give a complete mapping from physical states to Q-shapes, but they come closer to doing this than standard IIT. In what follows, we will assume a fully developed model along these lines with a complete mapping from physical states to Q-shapes.
Quantum IIT specifies a mapping $E$ from states of quantum systems to {\em quantum Q-shapes}. Quantum Q-shapes are are quantum analogs of classical Q-shapes, the Q-shapes invoked in standard IIT. Classical Q-shapes for an $n$-element system $S$ can be represented as $N=2^n-1$ weighted points, one for each subsystem of $S$, where points are pairs of probability distributions and weights are non-negative real numbers (for a subsystem that is not a mechanism, the weight will be zero). Quantum Q-shapes likewise involve $N$ weighted points, where points are now pairs of density operators associated with the Hilbert space of $S$ and weights are non-negative reals. Where the space of classical Q-shapes is the Cartesian product of $N$ copies (one for each subsystem) of $Pr(S) \times Pr(S) \times R^+_0$, the space of quantum Q-shapes is the Cartesian product of $N$ copies of $D(S) \times D(S) \times R^+_0$. Here $Pr(S)$ is the space of probability distributions over $S$, whose quantum analog $D(S)$ is the space of density operators over $S$. $R^+_0$ is the set of non-negative real numbers.\footnote{If $S$ is a network of elements with binary states, each weighted point will have $2^{n+1}+1$ dimensions (two $2^n$-dimensional probability spaces plus a real number), so classical Q-space has $(2^n-1)(2^{n+1}+1)$ dimensions. In quantum IIT, the $2^n$ dimensional probability-spaces are replaced by $2^{2n}$-dimensional density spaces, so quantum Q-space has $(2^n-1)(2^{2n+1}+1)$ dimensions.}
There is a natural mapping from classical Q-shapes to a subclass of quantum Q-shapes, deriving from a mapping from $Pr(S)$ to $D(S)$, defined as follows: $(p(s_i)) \mapsto \sum_i p(s_i) \ket{s_i} \bra{s_i}$. We can call this distinguished subclass of quantum Q-shapes the {\em quasi-classical} Q-shapes. Any quantum Q-shape can be seen as a superposition of quasi-classical Q-shapes.
Quantum IIT as it stands does not say much about how quantum Q-shapes correspond to states of consciousness. For our purposes we can add the further claims that (i) quasi-classical Q-shapes correspond to determinate states of consciousness, exactly as the corresponding classical Q-shapes do in classical IIT, and (ii) other quantum Q-shapes are superpositions of quasi-classical Q-shapes and correspond to superpositions of the corresponding states of consciousness.
We can define the quasi-classical {\em states} of a quantum system as those quantum states that quantum IIT associates (via the mapping $E$) with a quasi-classical Q-shape. If $\mathcal C$ is the class of quasi-classical Q-shapes, the class of quasi-classical quantum states is $\mathds{E}^{-1}(\mathcal C)$, the preimage of $\mathcal C$ under $\mathds E$. Every state of a quantum system can then be represented as a superposition of quasi-classical states, and its associated Q-shape will be a superposition of the corresponding quasi-classical Q-shapes. We can then set up collapse operators so that quantum systems always collapse toward these quasi-classical states with quasi-classical Q-shapes.
One limitation of quantum IIT as it currently stands is that these quasi-classical states (picked out as those that quantum IIT associates with quasi-classical Q-shapes) may not closely correspond to what we usually think of as quasi-classical quantum states such as mass density eigenstates. As a result, the Q-shape collapse dynamics need not lead to collapse toward standard ``classical" states such as mass density eigenstates and may result in a superposition of these states (along with a relatively determinate state of consciousness). If we want to avoid these quantum superpositions as physical correlates of determinate consciousness, there is at least a research program of developing a version of quantum IIT on which quasi-classical Q-shapes and determinate states of consciousness are associated with more ``classical" quantum states. In what follows it may be helpful to assume such a version of the framework.
We can now define Q-shape collapse operators. Recall that a Q-shape is a point in the direct product of $N$ copies of the density operator space $D(S)$. Any density operator in $D(S)$ can be represented (in the quasi-classical basis $\ket{s_i}$) as
\begin{equation}
\rho = \sum_{i,j} c_{ij} \ket{s_i}\bra{s_j} \end{equation}
The Q-shape for any given quantum state $\psi$ consists of $2N$ density operators of this kind and $N$ non-negative real numbers. The Q-shape can therefore be represented by $2N$ sets of coefficients $c_{ij}^k$ which we denote as $c_{ij}^k(\psi)$ (for $k=1 \ldots 2N$), and $N$ non-negative real numbers which we denote $\varphi^k(\psi)$ (for $k=1\ldots N$). For notational simplicity, we duplicate each of the latter, so that for each $k=1, \, ... \, 2N$, we have a $c_{ij}^k(\psi)$ which describes the first or second factor in $D(S) \times D(S) \times R^+_0$ and a $\varphi^k(\psi)$ which describes the third factor.
We can then define an ensemble of orthogonal self-adjoint collapse operators as follows:
\begin{equation}\label{Kleiner}
\hat Q^k_{ij} := \sum_{\psi \in \mathds{E}^{-1}(\mathcal C)}
\varphi^k(\psi) ((c_{ij}(\psi)+(c_{ji}(\psi)) \ket{\psi} \bra{\psi}.
\end{equation}
The sum has been restricted so that it runs over the class $\mathds{E}^{-1}(\mathcal C)$ of quasi-classical quantum states, that is, those whose Q-shapes are quasi-classical. As $k$ ranges from 1 to $2N$ (where $N=2^n-1$) and $i$ and $j$ range from 1 to $2^n$, an $n$-element system will be associated with $2^{2n+1}(2^n-1)$ collapse operators.\footnote{For reasons tied to the role of weights within IIT, we have combined the real weights $\varphi^k$ and the density operator coefficients $c_{ij}$ in defining the Q-shape collapse operators. As a result there are $N$ fewer collapse operators than dimensions of Q-space. Alternatively one can define separate operators for the coefficients and real weights as follows: $\hat Q^k_{ij} := \sum_{\psi \in \mathds{E}^{-1}(\mathcal C)}
((c_{ij}(\psi)+(c_{ji}(\psi)) \ket{\psi} \bra{\psi}$ and $\hat B^l := \sum_{\psi \in \mathds{E}^{-1}(\mathcal C)}\varphi^l(\psi) \ket{\psi} \bra{\psi}$.}
\section{Continuous collapse dynamics}
To complete our picture of super-resistant consciousness-based collapse, we need an account of the dynamics of super-resistant collapse. Fortunately, there exist models of dynamic collapse (due to Philip Pearle and Lajos Di\'osi, among others) that can be generalized to model the continuous collapse of any observable. It is not difficult to adapt these models to model the continuous collapse of consciousness and its physical correlates such as Q-shape. We start by informally reviewing these models and the adaptation to consciousness-collapse models, before providing formal details.\footnote{Thanks to Maaneli Derakhshani, Philip Pearle, and Johannes Kleiner for their extensive help with the material in this section.}
We start with the continuous spontaneous localization (CSL) model due to Pearle (\cite*{Pearle1976}, \cite*{Pearle1999}, \cite*{Pearle2021}). Pearle's model is a continuous relative of the well known GRW model, on which the position of isolated particles undergo spontaneous localization of position with low probability at any given time. On CSL, wave functions undergo a gradual stochastic collapse process at all times. The model provides continuous collapse onto mass density: the amount of mass present at various locations. It provides a dynamics by which superpositions of mass density gradually collapse toward definite states of mass density, with faster collapse in high-mass systems. In effect, CSL is a model on which mass density is super-resistant.
Pearle’s model can be informally motivated by an analogy between gradual collapse and the gambler's ruin game in classical probability theory (\cite{Pearle1982}). In the gambler's ruin, a number of gamblers play against each other until all but one of them is “wiped out”. Consider two gamblers, $G_1$ and $G_2$, who have \$100 between them such that $G_1$ has \$60 and $G_2$ has \$40. They toss a coin: if heads $G_1$ gives a dollar to $G_2$, if tails $G_2$ gives a dollar to $G_1$. As they keep playing, their respective amounts fluctuate, but the total remains the same. Eventually, the game ends, as one player acquires \$100. It turns out that $G_1$ wins 60\% of the time while $G_2$ wins 40\% of the time. That is, the probability that a given gambler wins is determined by the initial stakes.
In CSL, the squared amplitudes in a superposition (in the preferred basis) play a continuous stochastic gambler's ruin game against each other, fluctuating up and down until one “wins”, thereby completing the collapse. The probability that a given state vector “wins” a collapse in the long run is determined by its initial squared amplitude according to the Born rule. Crucially, we may control the speed at which the games are played in terms of certain (experimentally bounded) parameters. This allows large superpositions to collapse quickly and small superpositions to collapse at a negligible rate.
Like the GRW theory, Pearle’s theory involves a weak sort of super-resistance. Mass density resists superposition weakly, in that an isolated particle will only gradually collapse toward a definite position and so a definite mass density. At the fundamental level, superpositions of mass density will be ubiquitous. However, when many particles are entangled in a macroscopic system, the mass density of the system as a whole will collapse extremely fast, so that we will never encounter macroscopic systems in large superpositions of mass density.
Continuous collapse models can be adapted to work with super-resistant properties other than position and mass density. Given any observable, we can postulate a continuous collapse process with a version of the Pearle dynamics applied to this observable. Squared amplitudes for eigenstates of the observable engage in a stochastic gambler's ruin process, so that systems in superpositions of the observable collapse quickly or slowly toward their eigenstates via a gamblers-ruin process.
A related collapse process is postulated in the \textcite{Penrose2014} model of gravitational collapse, where spacetime curvature is super-resistant. Superpositions of spacetime curvature collapse onto definite states. Unlike Pearle, Penrose does not give a fully defined dynamics for collapse. He defines a superposition lifetime, $\hbar/\Delta E_G$, where $\hbar$ is Planck’s constant and $\Delta E_G$ is the gravitational self-energy of the difference between the mass distributions belonging to the two states in the superposition. But the dynamics of collapse during this lifetime are not specified.\footnote{The \textcite{HameroffPenrose} ``Orch OR" model extends Penrose's model of collapse into a model of consciousness. The Penrose-Hameroff model is not a consciousness-collapse model either: Penrose and Hameroff hold that collapse is triggered by superpositions of spacetime curvature rather than by consciousness or measurement, and that collapse causes consciousness rather than vice versa. Our approach might be considered a distant cousin of the Penrose-Hameroff model, with the main differences on our approach being: (i) consciousness causes collapse rather than vice versa, (ii) collapse is onto Q-shape rather than onto spacetime curvature, (iii) the collapse dynamics corresponds somewhat more closely to Pearle's model rather than Di\'osi-Penrose's, and (iv) as discussed later, we make no claims about quantum coherence and quantum computation in the brain.}
An account of the dynamics of gravitational collapse has been independently provided by Lajos \textcite{Diosi1987}. Di\'osi sets out a stochastic version of the Schrodinger equation on which there is a continuous collapse process onto spacetime structure. Di\'osi's dynamic collapse process is closely related to Pearle's continuous spontaneous localization process, with some differences arising from the use of a collapse onto gravitational structure as opposed to mass density.
It turns out that the Di\'osi and Pearle dynamics are both instances of a general formulation of continuous collapse dynamics which can be applied to any collapse operator. Such a formulation has been presented by Angelo Bassi and coauthors (2017).\footnote{See also \textcite[eqn.10]{Pearle1999} and \textcite[eqn.14]{Bassietal2013}.} We will adapt this formulation to set out a dynamics for continuous collapse onto consciousness.
In the context of IIT, we can use this general dynamics to develop a view on which Q-shape is super-resistant. Informally: Suppose a system is in a superposition of two Q-shapes, each with an associated amplitude. We can stipulate a ``localization" dynamics for this superposition that works much like Pearle’s except that collapse is toward eigenstates of Q-shape. The amplitudes trade off probabilistically with each other over time, in effect playing gambler's ruin at a rate proportional to the distance between the two Q-shapes. In the long run, the system will collapse onto a specific Q-shape with probability given by its initial squared amplitude.
We can spell out the mathematical details as follows. The general framework for continuous collapse rests on using a modified version of the Schrodinger equation that includes a nonlinear and stochastic term for collapse as well as the standard linear deterministic evolution. To be consistent and compatible with constraints such as no superluminal signalling, nonlinear modifications to the Schrodinger equation must take a highly constrained stochastic form. This yields the following general form for continuous collapse models (\cite[p.27]{Bassietal2017}):
\begin{equation}\label{Bassi(74)} d\psi_t = [-i\hat{H}_0dt + \sqrt{\lambda}(\hat{A} - \langle\hat{A}\rangle_t)dW_t - \frac{\lambda}{2}(\hat{A}-\langle\hat{A}\rangle_t)^2dt ]\psi_t \end{equation}
Here $\psi_t$ is the wave function state at $t$, $\hat{H}_0$ is the Hamiltonian, $\lambda$ is a real-valued parameter governing collapse rate, $\hat{A}$ is a collapse operator, $\langle\hat{A}\rangle_t$ is its expected value at $t$, and $W_t$ is a noise function allowing for stochastic behavior. The equation allows continuous stochastic collapse toward an eigenstate of the operator $\hat{A}$ at a rate governed by $\lambda$ and $W$, with probabilities given by the Born rule.
It is straightforward to generalize this equation to multiple collapse operators.\footnote{\textcite[eqn.36]{Bassietal2013}} Using our Q-shape collapse operators defined in (\ref{Kleiner}), we can propose the following dynamics:
\begin{equation}\label{ourproposal1} \begin{split} d\psi_t = [-\frac{i}{\hbar}\hat{H}dt + \sqrt{\lambda}\sum_{\alpha}(\hat{Q}_{\alpha} - \langle\hat{Q}_{\alpha}\rangle_t)dW_{\alpha,t} \\ -\frac{\lambda}{2}\sum_{\alpha}(\hat{Q}_{\alpha} - \langle\hat{Q}_{\alpha}\rangle_t)^2dt]\psi_t.
\end{split} \end{equation}
Here $\alpha = i,j,k$ is a multi-index that comprises the indices in (\ref{Kleiner}). If there is little difference in the superposed Q-shapes, then the first term on the right hand side (representing Schrödinger evolution) dominates. Otherwise, the system collapses toward a joint eigenstate of the collapse operators, at a rate proportional to the sum of the difference between their eigenvalues.
The noise function $W_{\alpha,t}$ is responsible for the stochastic ``gambler's ruin" collapse behavior described earlier.\footnote{For a simple illustration of how this works, see \textcite[sec. 2.2]{Pearle1999}.} In CSL and other mass density collapse models, the collapse operators correspond to local mass densities $\hat{m}(x)$ (the amount of mass at location $x$). The CSL noise function is given by Wiener processes $W_t(x)$, representing Brownian motion through time at location $x$. The noise at different spatial locations $x$ and $y$ is correlated by a spatial correlation function $G(x-y)$, which in CSL is a Gaussian function of the distance between $x$ and $y$. This ensures that collapse rate depends on the distance between mass density distributions.
In our IIT-based collapse model, we can define the collapse rate so that it depends on the extended Earth movers distance EMD* between Q-shapes (see the appendix). To ensure this, we can stipulate that the spatial correlation function involved in the noise functions $W_t(x)$ is defined in terms of Earth movers' distance: specifically, $G(x,y)=1/EMD^*(x,y)$ (with an appropriate cut-off for when $EMD^*(x,y)$ is small or zero; we omit the details).
In CSL it is also standard to ``smear" the mass density operator with the same Gaussian $G(x-y)$, so that collapse is onto smeared mass density eigenstates rather than precise mass density eigenstates, thereby avoiding large violations of energy conservation.\footnote{Smearing the mass density operator results in the following equation: $d\psi_t = [-\frac{i}{\hbar}\hat{H}dt + \sqrt{\lambda}\int d^3x(\hat{m}(\textbf{x} - \langle\hat{m}(\textbf{x})\rangle_t)dW_t(\textbf{x}) -\frac{\lambda}{2}\int d^3x \int d^3y \boldsymbol{G}(\textbf{x - y})(\hat{m}(\textbf{x}) - \langle\hat{m}(\textbf{x})\rangle_t)(\hat{m}(\textbf{y}) - \langle\hat{m}(\textbf{y})\rangle_t)dt]\psi_t$.} Our equation is simpler because our collapse operators do not correspond to points (or smeared regions) in a continuous space but instead correspond to a discrete set of mechanisms. We therefore do not need to include a smearing function in our equation. In principle, however, it is straightforward to add such a smearing function as in mass density models.
Our equation (\ref{ourproposal1}) assumes that all superposed Q-shapes are Q-shapes of a single system (network of units) with a fixed number of units and a fixed causal structure. It does not address the case where we have a superposition involving Q-shapes of systems with different numbers of units or causal structures. Extending the current framework to handle those cases is a further project.
The overall theory may look complex, but the underlying principles are fairly simple. First, there is an IIT-style quasi-classical psychophysical theory linking physical Q-shape by a structural isomorphism to phenomenal Q-shape in states of consciousness. Second, there is a generalization of this theory to the quantum realm, so that superpositions of physical Q-shape are linked to superpositions of phenomenal Q-shape and so to superpositions of states of consciousness. Third, there is the key claim that consciousness is super-resistant. More specifically, phenomenal Q-shapes resist superposition via a Pearle-style principle of continuous collapse for Q-shapes, so that superpositions of consciousness rapidly become more determinate. Putting these elements together: superpositions in the environment lead to superpositions of Q-shape in the brain, which lead to superpositions of consciousness. These superpositions of consciousness will rapidly collapse, yielding collapse in the correlated Q-shapes and collapse in the brain states and the environmental states that are entangled with Q-shape.
\section{Experimental tests}\label{experiments}
Different super-resistant collapse models make different predictions. For any proposed super-resistant property, in principle it is possible (though usually extremely difficult) to test whether a system is in a superposition of that property. This means that in principle (although not yet in practice) it is possible to test which systems can collapse quantum wave functions, and in virtue of which of their properties. For example, in principle we can test whether atoms, molecules, cells, worms, mice, dogs, or humans, as well as oscilloscopes, computers, and other devices have the capacity to collapse a wave function.\footnote{It is occasionally suggested that we know from existing results that ordinary measuring devices collapse the wave function, perhaps because we always find them in definite states, or because their measurements do not lead to quantum interference. However, it is easy to see that these observations are all equally consistent with a view on which only humans (say) collapse wave functions, and measurement devices are observed by humans and entangled with their environment. Sophisticated variants of this objection are made by \textcite{KochHepp2006} and \textcite{CarpenterAnderson}. \textcite{OkonSebastian2016} explain what goes wrong in these objections.}
To test whether a given property supports superpositions, one can use an interferometer for this property, which detects interference between superposed quantities in much the same way that a double-slit experiment detects interference between superposed positions. In practice it is extraordinarily difficult to set up interferometers for complex properties instantiated by complex systems, because of the need to prepare the relevant system in complete isolation from environmental effects. To date, the most complex such measurements have detected interference in large molecules with around 2000 atoms (\cite{Feinetal2019}). Current limitations are practical rather than principled, and measurements for more complex properties are certainly possible in principle.
These tests have clear implications for super-resistance models. In absolute super-resistance models, superpositions of super-resistant observables are impossible. In approximate super-resistance models, these superpositions are unstable. So at least on a first approximation: if we detect widespread superpositions of an observable, that tends to disconfirm models on which that observable is super-resistant.
On a second approximation, all this depends on just how unstable the superpositions are. We can distinguish \textit{fast-collapse} models on which large superpositions of a super-resistant observable are rare, from \textit{slow-collapse} models on which large superpositions are common. Here a large superposition of an observable is a superposition of significantly different eigenstates of the observable with significant amplitudes for significant periods (where ``significant" is a placeholder for now). If we frequently detect large superpositions of an observable, this tends to disconfirm at least fast-collapse super-resistance models involving that observable. These results do not disconfirm slow-collapse models as easily. Still, where consciousness-collapse models are concerned, fast-collapse models are arguably preferable to slow-collapse models, as the latter allow that large superpositions of conscious states are common. So for now, we will focus on fast-collapse models, returning to slow-collapse models shortly.
We may already be in a position to test fast-collapse models in which Q-shape is super-resistant. This project is aided by the fact that even quite simple systems (such as a dyad) can have nonzero $\Phi$ and nontrivial Q-shapes, as we have seen. To test the hypothesis, we need only prepare a quantum computer to enter superpositions of Q-shape. The simplest example is Schrödinger’s dyad (from section 4): two units A and B in a superposition of connected and disconnected states with distinct Q-shapes. If we find the interference effects predicted by standard quantum mechanics (which assumes that simple systems do not perform measurements and evolve according to Schrödinger dynamics), this will falsify the hypothesis that Q-shape is super-resistant, at least on a fast-collapse model. If we do not find these effects, this will suggest that these superpositions are impossible or unstable and will tend to support the hypothesis that Q-shape is super-resistant.
Something along these lines could be done with a quantum version of a Fredkin crossover gate.\footnote{Thanks to Scott Aaronson for this suggestion.} A classical Fredkin gate involves three bits, a control bit and two other bits A and B. If the control bit is 1, bits A and B are swapped. If the control bit is 0, bits A and B are left as is. In a quantum version of the Fredkin gate, the control bit can be in superposition, and the AB system will then be in a superposition of bit-swapping and staying constant. As a result, IIT appears to suggest that the AB system will be in a Q-shape superposition. If Q-shape is super-resistant in a fast-collapse model, we should expect this superposition to collapse.
In fact, a quantum Fredkin gate has recently been constructed (\cite{Pateletal2016}), with results indicating a successful superposition. However, in this example, it does not seem that the conditions for $\Phi$(AB)=1 are met, because there is no two-way feedback interaction between gates A and B. In IIT, purely \textit{feedforward} networks typically have zero $\Phi$. A feedforward network can have nonzero $\Phi$ if it has overlapping inputs and overlapping outputs, but this does not appear to be happening in the quantum Fredkin gate.\footnote{This points to another test case that can be realized by a quantum computer. Perhaps the simplest feedforward system with nonzero $\Phi$ is a dyad system CD that forms a layer of a feedforward network, whereby a node from a previous layer gives input to both C and D, and both C and D give input to a node in a subsequent layer. For illustration, see \textcite[Fig. 7(B)]{Oizumietal2014}.}
How might we properly construct feedback systems such as AB using quantum computers? In the quantum computing literature, two primary types of quantum feedback are distinguished. The traditional type is \textit{measurement-based feedback}. Here, a quantum system performs some (usually feedforward) processing and is measured, and the measurement result is then fed back into the quantum system as input. This will not help for our purposes. A more recent development is \textit{coherent quantum feedback} (\cite{Lloyd2000}), where feedback connectivity obtains in the quantum system itself. Superpositions of coherent quantum feedback could be used to build our dyad system in a superposition of states.
For example, consider the ion-trap example discussed by \textcite[p4]{Lloyd2000}. The initial state of the system is $\ket{\psi}_s\ket{0}_m\ket{\phi}_c$, where $\ket{\psi}_s$ is the unknown state of the ``system" ion, $\ket{\phi}_c$ is the prepared state of the ``controller" ion and $\ket{0}_m$ is the vibrational mode cooled to its ground state. Lloyd explains how certain directed pulses can evolve the system from $\ket{\psi}_s\ket{0}_m\ket{\phi}_c$ to $\ket{\downarrow}_s\ket{\psi'}_m\ket{\phi}_c$, to $\ket{\downarrow}_s\ket{\phi'}_m\ket{\psi}_c$, and finally to $\ket{\phi}_s\ket{0}_m\ket{\psi}_c$. In effect, the initial unknown state of the system ion is swapped with the initial state of the controller ion. Schrödinger’s dyad may then be constructed by putting the input pulses into a superposition of implementing this swap and not implementing this swap, yielding: $\alpha\ket{\psi}_s\ket{0}_m\ket{\phi}_c + \beta\ket{\phi}_s\ket{0}_m\ket{\psi}_c$. If the two terms in the superposition yield distinct Q-shapes, then our model predicts that this superposition is unstable and will eventually collapse, even if the system remains isolated.
The issue is not entirely straightforward, as it might be denied that the full conditions for $\Phi$(AB)=1 are met (perhaps because of the role of the vibrational mode or the pulses). Still, it seems likely that some technologically feasible quantum computation involves a superposition of Q-shapes. If found, such a superposition will falsify the combination of standard IIT (on which Q-shape is the physical correlate of consciousness) and the fast-collapse consciousness-collapse thesis.
More generally, most proponents of quantum computing predict that superposed states in larger and larger systems will gradually be demonstrated. It would be foolhardy to bet against these predictions. In the face of these results, one could maintain an IIT-collapse view by modifying IIT somewhat: for example to say that a system is conscious (and has a Q-shape) only when $\Phi$ is above a certain threshold, or by adding other constraints to the definition of $\Phi$ so that the relevant simple systems have $\Phi$ = 0. Alternatively one could adopt a slow-collapse version of the model; one could reject IIT entirely for a different theory of consciousness; or one could reject the consciousness-collapse thesis. Still, this shows how even near-term experimental results from quantum mechanics can have some bearing on theories of consciousness.
All this brings out that the consciousness-collapse thesis in its fast-collapse version is not easy to combine with panpsychist theories of consciousness on which consciousness is found even in very simple systems. A strong panpsychist fast-collapse view on which position or mass or charge quickly collapses the wave function is straightforwardly refuted by standard experimental results showing interference effects. The more recent results of Fein et al demonstrating superpositions of position in 2000-atom systems tend to suggest that the threshold for collapse lies somewhere beyond that level. There are some quasi-panpsychist collapse views involving slightly more complex properties distinct from position that have not yet been tested, but we should easily enough be able to test them as above, and few would expect them to be supported. The consciousness-collapse thesis (in fast-collapse versions) tends to fit more comfortably with non-panpsychist views on which consciousness arises only in relatively complex systems. These views are consistent with existing and likely near-term-future observations, while still being subject to experimental test eventually.
There remains the possibility of slow-collapse models on which superpositions of consciousness tend to collapse slowly across long periods. If these models allow widespread large superpositions of human states of consciousness, these views are hard to reconcile with introspection, and it also becomes less clear why we should accept the consciousness-collapse view over an Everett-style view where one's consciousness is constantly in large superpositions. Perhaps there could be a CSL-style slow-collapse panpsychist model on which superpositions of consciousness are common but unstable at the microphysical level, in the way that superpositions of mass distribution are common but unstable at the microphysical level in CSL. In CSL, large superpositions of macroscopic mass distributions are nevertheless uncommon. Likewise, a panpsychist slow-collapse view might have the consequence that large superpositions of human consciousness are uncommon, especially on a constitutive panpsychist view on which human consciousness is constituted by patterns of microconsciousness. Such a view will face the notorious combination problem of how this constitution works, and it may also have less of an irreducible causal role for human consciousness than other collapse views. Still, there are various versions of a slow-collapse model worth exploring.
There are also empirical constraints on super-resistance models tied to energy conservation (collapses tend to produce excess energy, so they cannot be too frequent or too dramatic\footnote{The main difficulty in the experimental detection of such effects involves controlling all the possible ways of cooling. Thus, in their discussion of testing GRW and CSL, \textcite{FeldmannTumulka2012} consider the Kubacher Kristallhöhle, the largest natural cave in Germany, which is 9$^{\circ}$ C all year around. When surface temperatures are low, heat spontaneously created in the cave cannot be transported away, thereby suggesting a way of obtaining an empirical bound on the rate of spontaneous warming. It is much more difficult to see how we could find empirical bounds on spontaneous warming in conscious systems, but it may not be impossible.}) and to the quantum Zeno effect (a super-resistance model must allow superpositions to persist long enough to avoid Zeno effects, while not persisting so long that measurements do not have definite outcomes). All these phenomena impose constraints that narrow the class of available super-resistance models: super-resistant properties are not too simple and not too complex, while collapses are not too frequent and not too slow.
For a super-resistance model to be empirically supported, we will eventually have to find systems and properties that resist superposition. One key (if currently far-fetched) experiment would use an interferometer on a human isolated from their environment, preparing them to enter a superposition of conscious states and seeing if interference effects are observed. If interference effects are not observed, one will have experimental support for the claim that humans can collapse wave functions. As before this would not decisively demonstrate that consciousness is doing the work, but it would give reason to take that view seriously. If interference effects are not observed, one will have experimental support for the claim that humans cannot collapse wave functions. This will also tend to falsify any measurement-collapse formulation of quantum mechanics, and in particular will tend to falsify the view that consciousness collapses the wave function. In this way the framework of this article may ultimately be subject to empirical test.
Admittedly, it is not clear that it will ever be possible to isolate and test a conscious human brain in this way. Perhaps somewhat more feasible in the long term could be running a detailed simulation of a human brain on a quantum computer. If interference effects are not observed, one will have experimental support for the claim that the computational structure of the human brain can collapse wave functions. If they are not observed, one will have evidence against this claim. However, this result will leave open the hypothesis that other features of the human brain that are not replicated in a simulation, such as biological features, are responsible for wave-function collapse. It may be especially difficult to test biological collapse models, as many standard methods of isolating systems to test for superposition require low temperatures where the biology may break down. Still, these quantum computing experiments might at least give us evidence for or against a consciousness-collapse model where the correlates of consciousness are computational. In the long run, advances in quantum computing are likely to heavily constrain the prospects for consciousness-collapse models.
\section{The causal role of consciousness}
On the picture we have sketched, superpositions of physical Q-shape drive collapse. How does this yield a causal role for consciousness?
On a materialist view which identifies physical Q-shape (a physical property) with phenomenal Q-shape (a property of consciousness), the causal role is straightforward. Superpositions of consciousness involve superpositions of phenomenal Q-shapes, which trigger collapse onto more definite phenomenal Q-shapes, which are themselves more definite physical Q-shapes, leading to more definite physical consequences.
On a dualist view, physical Q-shape may be ontologically distinct from phenomenal Q-shape, so a causal role for the former is not yet a causal role for consciousness. The simplest way to derive a causal role for phenomenal Q-shape is to assume (i) that consciousness has a quantum structure whereby subjects are in superpositions of phenomenal Q-shapes iff they are in corresponding superpositions of physical Q-shapes, and (ii) a fundamental principle saying that phenomenal Q-shape is super-resistant and obeys the collapse dynamics we have developed. When subjects are in superpositions of phenomenal Q-shapes, these Q-shapes collapse according to the dynamics. Phenomenal Q-shapes are perfectly correlated with physical Q-shapes, so collapse of phenomenal Q-shapes leads to collapse of physical Q-shapes, and the standard ensuing physical effects of collapse.
Someone might object that we do not give a genuine causal role to nonphysical consciousness at all. Instead, all the causal work is done by the physical correlates of consciousness.
One version of this objection notes that on a dualist consciousness-collapse interpretation, there will be PCC states (e.g. physical Q-shapes, on the IIT framework) that correlate perfectly with consciousness. One can then develop a physicalist collapse interpretation on which the primary locus of superposition-resistance is the PCC states. Collapse of the PCC states does all the causal work, and collapse of consciousness is causally irrelevant. There will at least be a possible world (we might think of it as a quantum zombie world) where collapse works this way. In that world, the physical wave function will evolve just as in our world. So even in our world, consciousness may seem redundant.
In response: on the dualist interpretation spelled out above, it is consciousness that directly causes the wave function to collapse. There is a fundamental principle saying that consciousness resists superposition. (In the IIT framework, phenomenal Q-shapes resist superposition.) This leads to probabilistic collapse toward determinate states of consciousness. This collapse of consciousness brings about physical collapse to a more determinate PCC state, because of a psychophysical law ensuring that states of consciousness and their physical correlates (in the IIT framework, phenomenal Q-shapes and physical Q-shapes) are always in alignment. So consciousness is causally responsible for collapse in our world. There may be other models where physical correlates cause collapse directly, but that is not how things work on the dualist interpretation we have specified.
The quantum zombie scenario does suggest that there is a sort of structural/mathematical explanation that might be given for our actions without mentioning consciousness. Still (as is familiar from discussions of panpsychism and Russellian monism), this structural explanation would not provide a complete explanation of our actions, precisely because it leaves out the role of consciousness in grounding that structure. Like many structural explanations, it leaves out the actual causes. In the actual world consciousness is causing the relevant behavior, and consciousness may explain why it is that we behave determinately at all.
A related objection asks: in the actual world, how do we know that it is consciousness that triggers collapse, and not its physical correlates? As we discussed in the last section, if there is a perfect correlation between the two, these hypotheses cannot be distinguished experimentally. Still, insofar as we already have reason to believe that consciousness is a fundamental property, then the hypothesis that consciousness triggers collapse has at least two advantages. First, this way the fundamental law of collapse involves a fundamental property. Second, this way we have a causal role for consciousness, cohering with a strong pretheoretical desideratum. These virtues give reasons to favor the view over the alternative.
One might also object that even if our models give consciousness a causal role, they do not give consciousness the kind of causal role that we pretheoretically would expect it to have. One worry is that collapsing consciousness may affect the objects we perceive, but we want consciousness to affect action, producing intelligent behavior and verbal reports such as ‘I am conscious'.
One worry is that the most obvious effects of collapse point the wrong way: collapse of consciousness will collapse perceived objects such as measurement instruments, but what we want is for consciousness to affect action. In response, we can note that a collapse of consciousness will collapse an associated PCC state in the brain, and this brain state will be entangled with action states or will at least cause a corresponding action state, so a collapse of consciousness will help bring about a determinate action. For example, if consciousness probabilistically collapses into an experience of red rather than an experience of blue, this collapse will bring about a PCC state associated with experience of red, which will tend to lead to an utterance of 'I am experiencing red' rather than 'I am experiencing blue'.
Furthermore, consciousness also involves the experience of agency and action: say, the experience of choosing to lift one's left hand rather than one's right hand. Superpositions of these states will collapse into definite states, which will lead to actions such as raising one’s left hand.
This picture naturally raises issues about free will. On this view, the experience of choice plays a nondeterministic causal role in bringing about action. On some popular conceptions of ``free will", on which what matters for free will is nondeterminism and a role for consciousness, this picture may vindicate free will in the relevant sense. Others may object that the choices are themselves selected probabilistically, and that random choices are no better than deterministic choices when it comes to free will. We think the issues are far from straightforward, so we will set aside issues about free will here, but we note that a causal role for consciousness can be expected to have some bearing on those issues.
Another objection is that if consciousness always collapses via the Born role, then any effect of consciousness on action will at best be a sort of dice-rolling role. It will probabilistically select between different available outcomes, but it will not yield a qualitatively special outcome. Under a hypothesis where PCC states collapse the wave function, purely physical quantum zombies would have behaved the same way. So consciousness will not make outcomes on which humans behave intelligently or on which they say 'I am conscious' any more likely than they would have been if some other property had collapsed the wave function. One might even simulate the dynamics in a classical computer (with a pseudorandom number generator), with no role for consciousness, and the same patterns of behavior would ensue.
Most of what this objector says is correct. The quantum zombie scenario suggests that there is a sort of structural/mathematical explanation that might be given for our actions without mentioning consciousness. Still, this structural explanation would not provide a complete explanation of our actions, precisely because it leaves out the role of consciousness in grounding that structure. (Like many structural explanations, it leaves out the actual causes.) In the actual world consciousness is causing the relevant behavior, and consciousness may explain why it is that we behave determinately at all. One might have liked a stronger, more transformative causal role for consciousness that could not even in principle have been duplicated without consciousness, but it is not clear why such a role is essential.
If one does want a stronger role for consciousness, the most obvious move is to suggest that the role for consciousness in collapse is not entirely constrained by the Born probabilities. Perhaps perceptual consciousness obeys those constraints (thereby explaining our observations in quantum experiments), but agentive experience does not. For example, collapses due to agentive experience might be biased in such a way that more ``intelligent" choices that lead to more intelligent behavior tend to be favored than they would be according to the Born rule. This picture sacrifices the great simplicity of the original quantum dynamics, and it could perhaps be disconfirmed through the right sort of experiments and simulations, but it is arguable that our current evidence leaves room open for it. We do not find this picture especially attractive, but it is at least worth putting it onto the table.\\
\section{Philosophical objections}
We have already considered many objections to our account. Some are technical issues specific to the use of IIT: for example, whether IIT applies to real physical states, whether Q-shape operators can be defined, and whether a Q-shape/collapse theory has already been falsified by existing experimental results. These are serious issues that may require modifying IIT or moving to a different theory of the physical correlates of consciousness. Some are versions of objections that arise for many objective collapse theories: for example, consistency with relativity and the tails problem. These are also serious issues that we have set aside for now with the preliminary aim of getting consciousness-collapse models closer to the level of seriousness of existing objective collapse theories. A final technical issue is whether the parameters of a consciousness-collapse theory can be set to avoid the Zeno effect.
In this final section we consider a number of philosophical objections. We have already considered objections concerning the causal role of consciousness. The largest objection remaining concerns superposed states of consciousness.\\
\noindent \textbf{Objection 1: What is a superposed state of consciousness?}
As we saw earlier, Wigner said that it is “absurd” to suppose that a subject could be in a state of “suspended animation”, that is, in a superposition of multiple states of consciousness. However, the approximate super-resistance model we have developed requires that subjects can be in such superposed states. Large superpositions of consciousness (those between significantly different states with significant amplitude for significant periods) will be rare, at least on a fast-collapse model, but they will be possible. Small superpositions of consciousness (those that are like large superposition except that they are brief, or low-amplitude, or between closely related states) may be ubiquitous. In fact, on these models it may be that most or all conscious subjects are in small superpositions of consciousness most or all of the time. This raises the questions: are superpositions of consciousness possible, and if so how can we understand them?
There are a few different ways of trying to understand superposed states of consciousness. First, one could try to understand them as familiar states: for example, a superposition of seeing an object at positions A and B might be a state of double vision. However, double vision is an ordinary state of consciousness that can enter superpositions. It leads to reports such as “I see an object at A and at B”. The superposed state does not. It leads to reports such as “I see an object at A” (if the introspection and report process triggers collapse), or at worst a superposition of “I see an object at A” and “I see an object at B” (if no collapse is triggered). This brings out that the sort of superpositions we need are not introspectible or reportable and will be quite different from familiar states such as double vision.\footnote{\textcite{Shimony1963} reads \textcite{LondonBauer1939} as allowing superpositions of consciousness and critiques the idea in part by arguing that phenomena such as blurred vision and indecision do not really involve superpositions.}
A more radical alternative says that superposed states of consciousness involve multiple subjects having distinct total states of conscious experience. We will set aside this option as extravagant (do subjects pop into and out of existence in superposition and collapse?), though it is perhaps worth some attention.
A third option is to say that a superposition of states of consciousness is a state that the subject is in, but it is not itself a total state of consciousness. That is, when a subject is in a superposition of conscious states A and B, there is no subjective experience of being in this superposition. There is something it is like to be in A, and something it is like to be in B, but nothing it is like to be in A and B simultaneously. The subject has the experience of being in A and the experience of being in B, without having any conjoint experience of being in the superposition. This violates the Unity Thesis articulated by \textcite{BayneChalmers2003} holding that whenever a subject is in multiple conscious states, they are also in a single conscious state that subsumes and unifies them. Some theorists hold that the Unity Thesis is false, at least for split-brain patients and other fragmented subjects: these subjects do not have a single determinate total conscious state, but instead have multiple conscious states as fragments.\footnote{On split-brain cases, see for example \textcite{Nagel1971} who argues for indeterminacy here. \textcite{BayneChalmers2003} argue that in these cases there is a single subject with a single determinate state of consciousness, while \textcite{Schechter2017} argues that there are multiple subjects each with a determinate state of consciousness.} It is far from obvious what is really going on in these cases, and any analogy with superposed states seems fairly distant. Still, these cases at least bring out that the Unity Thesis and the corresponding assumption that every subject is in a single determinate total state of consciousness is not non-negotiable.
A fourth option is to say that a superposition of total states of consciousness is itself a total state of consciousness – albeit one quite unlike the ordinary total states of consciousness that we are introspectively familiar with. On this view, when a subject is in a superposition of conscious states A and B, there is something it is like to be in this superposition. It presumably involves some combination of the experience of being in A and the experience of being in B, combined by some novel phenomenal mode of combination. This mode of combination is not something we could introspect or report for the reasons discussed above, so it would have to be something that we have no introspective familiarity with. The phenomenological role of amplitudes is also not clear. Perhaps amplitudes give the ordinary states of consciousness relative weights in the combined states. As a result, it is far from clear what the phenomenology of a superposed state would be like. Still, it is far from obvious that a mode of combination like this is impossible.
We think that the fourth option is perhaps the most worthy of consideration, followed by the third. On the fourth option, we can no longer say that total states of consciousness correspond one-to-one with PCC eigenstates. Instead, ordinary non-superposed total states of consciousness will correspond to PCC eigenstates, and superposed total states of consciousness will correspond to superpositions of these eigenstates.
There is precedent to the thought that there are states of consciousness that we cannot introspect or report. Theorists (e.g. Block) who believe in an “overflow” of consciousness outside attention often postulate such aspects: if introspecting and reporting a state always involve attending to it, unattended states cannot be introspected or reported. One can perhaps make unnoticed superpositions more palatable by noting that on a fast-collapse model they will usually be small superpositions, involving very similar states of consciousness, very low amplitudes, and/or very brief periods of time. As a result, the superpositions may largely fall below the grain of our ordinary introspective access.
Still, the fact that our super-resistance model has to postulate superposed states of consciousness is a significant cost of the view. Is it possible to develop a super-resistance consciousness-collapse model that avoids superpositions of consciousness while also avoiding the Zeno problem? Such a model would need to give up on the tight connection between definite conscious states and PCC eigenstates, in order that never-superposed conscious states do not lead to never-superposed PCC states and so to the Zeno effect. At the same time, it would need to retain enough of a connection between consciousness and physical states that the definiteness of consciousness leads to collapse in its physical basis. It is not easy to meet both demands at once. One path invokes a looser connection between consciousness and PCC eigenstates, whereby superposed PCC states can coexist with definite states of consciousness at least briefly. For example, one might hold that superposed PCC states determine a definite state of consciousness probabilistically according to the Born rule, and that this definite state of consciousness leads to collapse onto a corresponding PCC state but only after a time delay. Perhaps this view and others in the neighborhood are at least worth developing.
In any case: in ordinary quantum mechanics, many theorists say that they cannot really imagine what it is for a physical state to be in a superposition. At the same time, they adopt the idea and run with it, and the idea seems to be theoretically fruitful. Our suggestion is that we do something like this for superpositions of states of consciousness, at least for now. We should simply adopt the idea and see whether it is fruitful. If it is, we can later return to the question of just what superposed states of consciousness involve.\\
\noindent \textbf{Objection 2: How do quantum effects make a difference to macroscopic brain processes?}
Quantum theories of brain processes are sometimes criticized on the grounds that it is hard to see how low-level quantum processes can affect high-level processing in neurons. A more specific version of this objection is that on some accounts (e.g. Hameroff and Penrose), quantum coherence at the neural level is required for distinctively quantum effects in neural processing, but the high temperatures in the brain are likely to lead to decoherence below the neural level. These objections do not apply to our approach, which does not involve any special effects of low-level quantum processes on neural processes and is entirely consistent with decoherence at relatively low levels. In fact, in our central illustrations, we have treated brain states as superpositions of numerous decoherent eigenstates, which themselves may involve relatively classical processing in neurons. The only high-level quantum process that plays an essential role in our framework is the collapse process, which selects one or more of these eigenstates as outlined above. Our picture is consistent with further macroscopic quantum effects, but they are not required.\\
\noindent \textbf{Objection 3: What about macroscopic superpositions?}
One might worry that on a consciousness-collapse view ordinary macroscopic objects such as measurement devices will exist in states of superposition until they are observed. Our view does not necessarily lead to this consequence. For a start, if a correct theory of consciousness associates these devices with some amount of consciousness (as may be the case for IIT), then the devices will collapse wave functions much as humans do. Even if these devices are not conscious, it is likely that typical measuring devices will be entangled with humans and other conscious systems, so that they will typically be in a collapsed state too. Still, in special cases where such a device is entirely isolated from conscious systems and records a quantum interaction, it will enter a macroscopic superposition. Of course we will never observe such a superposition, as our observation will collapse the state of the system. But we might in principle get empirical evidence of this superposition if we can eventually measure associated interference effects. Perhaps the existence of macroscopic superpositions is counterintuitive, but many cosmological theories already allow macroscopic objects to be in superposition in the early universe where there are no observers. It is unclear why allowing this in the current universe is any worse. \\
\noindent \textbf{Objection 4: What about the first appearance of consciousness in the universe?}
As we saw earlier, if consciousness is absolutely super-resistant, the quantum Zeno effect entails that it can never emerge for the first time in the development of the universe. On an approximate super-resistance model, there is less of a problem. For eons, the universe can persist in a wholly unconscious superposed state without any collapses. At some point, a physical correlate of consciousness may emerge in some branch of the wave function, yielding a superposition of consciousness and unconsciousness (or their physical correlates) with low amplitude for consciousness. With high probability the universe will collapse back toward an unconscious state. As this happens repeatedly in many branches of the wave function, there will eventually be a low probability collapse toward a state of consciousness, and consciousness will be in a position to take hold.
\section{Conclusion}
The results of our analysis are mixed. We have developed a consciousness-collapse model with a reasonably clear and precise dynamics. But it must be admitted that the model we have developed is not as simple and powerful as the original (simple if imprecise) measurement-collapse framework.
Our initial superselection collapse model was simple, but it leads to the Zeno problem. Avoiding the Zeno problem has led to a number of complications. First, we have had to countenance superpositions in states of consciousness, and it is not at all clear that this is possible. Second, we have had to introduce Pearle-style collapse dynamics along with parameters for the rate of collapse, and these parameters have to be constrained carefully in order to yield empirically acceptable results. We have also had to invoke a complex theory of consciousness -- though this is less of a cost, since a theory of consciousness is needed even in the absence of the quantum measurement problem.
Is this consciousness-collapse model the best that we can do? We have seen that to avoid countenancing superposed states of consciousness while also avoiding the Zeno problem, a consciousness-collapse model will need to break the strong link between definite states of consciousness and eigenstates of a PCC observable. Perhaps there are alternative models on which the physical correlates of consciousness involve a more complex wave-function property, or on which consciousness can vary independently of any physical properties. There also remain the possibility of variable-locus models, though these may also need to break the strong link between consciousness and its physical correlates to avoid the Zeno problem. In any case, models along these lines are certainly worth exploring.
Overall: the model we have developed is perhaps not as simple or powerful as some of the leading interpretations of quantum mechanics. If it is the best we can do, then the upshot may be that consciousness-collapse models are subject to principled limitations. Nevertheless, it at least serves as an existence proof for a relatively precise consciousness-collapse model. The model is open to empirical test, and it is not out of the question that a more powerful model along these lines could be developed. In the meantime, the research program of consciousness-collapse models deserves attention.
\appendix
\section{Appendix: Calculating Q-shape for a dyad system in IIT 3.0 and quantum IIT}
In this appendix, we illustrate some mathematical details of standard IIT (IIT3.0) and quantum IIT (QIIT), by showing how $\Phi$ and Q-shape are determined in simple dyad systems with two elements. The IIT formalisms are complex, but dyads avoid some complications. We will also define a distance measure between Q-shapes which is important for the collapse dynamics.
We begin with IIT3.0.\footnote{Thanks to Nao Tsuchiya and Leo Barbosa. Our calculations follow the supporting information in \textcite{Mayneretal2018} especially S1: Calculating $\Phi$. See also \textcite{Oizumietal2014} and \textcite{Tononietal2016}. For the earlier, simpler IIT formalism for calculating $\Phi(AB)$, see \textcite[fig. 5]{Tononi2004}, \textcite{Tsuchiya2017}, and \textcite{McQueen2019b}. The reader can experiment with calculating $\Phi$ for various systems including the dyad AB at http://integratedinformationtheory.org/calculate.html. Details of the underlying software can be found in \textcite{Mayneretal2018}.} We assume a dyad system with two elements A and B, each of which can be in one of two states: [1] or [0]. The composite system AB can be in one of four possible states: [11], [00], [10], or [01]. The transition rules are a simple swap: the state of A at one time is determined by copying the state of B at the previous time and vice versa. We can stipulate that in the system under consideration, the current state of AB is [10]. The next state is thereby determined to be [01]. Subsystems of AB are the nonempty sets of elements of the system: \{A\}, \{B\}, and \{A, B\}, which we will abbreviate as A, B, and AB when there is no chance of confusion. Mechanisms are subsystems with nonzero weight.
The Q-shape of a system consists of a location $L(m)$ for each mechanism $m$ in the system, weighted by the measure $\phi(m)$. $L(m)$ is a point in a $2^{n+1}$-dimensional space with two dimensions for each of the $2^n$ possible states of the system, where $n$ is the number of elements. $L(m)$ is determined by conjoining two probability distributions over the states $S$ of the system: $p_m(S)$ and $p'_m(S)$, where the former is defined in terms of the effects of $m$ and the latter is defined in terms of the causes of $m$. Each distribution is associated with a $\phi$ value. The weight $\phi(m)$ is the minimum of these two values. The Q-shape of AB lives in an 8-dimensional space, as AB has four possible states. As we will see, of the three subsystems of AB, only A and B yield mechanisms with nonzero weight. Hence, The Q-shape of AB consists in two weighted points located in an 8-dimensional space.
IIT3.0 distinguishes two notions of integrated information: $\phi$ (small phi), which applies to individual mechanisms, and $\Phi$ (big phi), which applies to the total system. To know $\Phi(AB)$ we must first calculate AB's Q-shape. To know AB's Q-shape we must first calculate $\phi$ for AB’s mechanisms. To begin with, we illustrate how the probability distribution $p_m(S)$ and $\phi(m)$ are calculated and then used to define Q-shape and $\Phi$.
The distribution $p_m(S)$ is a distribution over future states $S$ of the system, reflecting their probability of occurrence given that the elements of $m$ are fixed to their current state (while any other elements are allowed to vary). For the candidate mechanism AB, both elements will be fixed to their current value [10]. $p_{AB}(S)$ is the probability that the following state will be S, given the current state [10]. The following state is guaranteed to be [01], so $p_{AB}$ assigns probability 1 to [01] and probability 0 to the other three states.
Recall that $L(m)$ is determined by conjoining two probability distributions over the states $S$ of the system: $p_m(S)$ and $p'_m(S)$. If we consider just $p_m(S)$, then the location $L(AB)$ can be seen as a point in 4-dimensional space corresponding to the distribution $p_{AB}(S)$. Let us say the four dimensions are ordered as [00], [01], [10], [11]. Then L(AB)=[0,1,0,0], which assigns 1 to [01] and 0 to the other states. While $p_m(S)$ is a distribution over possible future states, $p'_m(S)$ is a distribution over possible preceding states (i.e. the probabilities that the preceding state was $S$, given the current state). For our system AB the two distributions are the same, so the 8-dimensional location will be a repeated version of the 4-dimensional location: that is, L(AB)=[0,1,0,0,0,1,0,0].
For candidate mechanism A, $p_A(S)$ is the probability of the following state being $S$ given that element A is fixed to its current value [1], while the other element B can vary with probability 0.5 for each value [0] or [1]. Under these conditions, the following state may be either [01] or [11], and $p_A$ will assign these two states probability 0.5 each. Likewise, $p_B$ will assign probability 0.5 each to states [00] and [01], the two states that can follow a state where B is fixed to 0. As with AB, $p_m(S)$ = $p'_m(S)$ for A and for B. As a result, $L(A)=[0,0.5,0,0.5,0,0.5,0,0.5]$ and $L(B)=[0.5,0.5,0,0,0.5,0.5,0,0]$.
The integrated information [small phi] $\phi(AB)$ is determined by considering the difference between the probabilistic effects of the subsystem AB with the effects of a partitioned subsystem A-B where we consider only the effects of A and B taken separately on each other. We can define a probability distribution $p_{A-B}$ as the tensor product of two distributions: a distribution $p_{A|B}$ over states of A given that B is fixed to its current value 0 (so A=[1] has probability 1) and a distribution $p_{B|A}$ over states of B given that A is fixed to its current value 1 (so B=[0] has probability 0). The product distribution $p_{A-B}$ assigns 1 to [10] and 0 to every other state.
We can then define $\phi(AB) = EMD(p_{AB}, p_{A-B})$. For two probability distributions $p_1$ and $p_2$ over the same state-space, EMD($p_1,p_2$) is the Earth mover’s distance between $p_1$ and $p_2$. This can be defined as the minimal amount of work required to turn $p_1$ into $p_2$ by moving the ``Earth" of probability from some points in the $2^n$-dimensional space to other points, where work is measured by the amount of probability moved multiplied by the Hamming distance between the points. In the case just described, $p_{AB}$ and $p_{A-B}$ are exactly the same distribution, so the Earth mover’s distance between them is 0. So $\phi(AB) = 0$.
The quantity $\phi(A)$ can be defined as a related Earth-mover’s distance over states of B, comparing the distribution over those states with A fixed to its current value of [1] (resulting in probability 1 to B=[0]) to a distribution that ignores the value of A (resulting in probability 0.5 each to B=[0] or B=[1]). In this case, $\phi(A) = 0.5$. Likewise, $\phi(B)=0.5$.
As a result, we can fully specify the Q-shape $Q_{AB}$ of the system AB. It consists of location $L(AB)=[0,1,0,0,0,1,0,0]$ with associated weight $\phi(AB) = 0$, location $L(A)= [0,0.5,0,0.5,0,0.5,0,0.5]$ with associated weight $\phi(A)=0.5$, and location $L(B)=[0.5,0.5,0,0,0.5,0.5,0,0]$ with associated weight $\phi(B)=0.5$.
In the above calculations we took a shortcut that should now be made explicit. For each candidate mechanism, we chose to consider the probability distribution assigned to the future (or past) states \textit{of a particular subsystem}. For candidate mechanism AB we chose subsystem AB. For candidate mechanism A we chose subsystem B. And for B we chose A. These choices are not arbitrary, but are the result of an optimization procedure. For each candidate mechanism, we in fact consider all possible subsystems and choose the subsystem that maximizes $\phi$. For example, when considering candidate mechanism A, it turns out that A has more integrated information about B than about AB. After all, there are three possible ways of disconnecting A from AB: disconnect A to A, A to B, A to AB. Nothing happens by disconnecting A to A (there was no connection there to begin with!). But then that is the minimal information partition, implying that A has zero $\phi$ about AB. On the other hand, there is only one way to disconnect A from B, and that disconnection does make a difference, giving nonzero $\phi$. For details see \textcite{Barbosaetal2021}.
We can define the distance between two Q-shapes $Q_1$ and $Q_2$ (defined over the same states $S$, with associated probability distributions $p_{m,1}$ and $p_{m,2}$ and weights $\phi_1(m)$ and $\phi_2(m)$) as an extended Earth mover’s distance $EMD^*(Q_1, Q_2)$:
\begin{equation}\label{EMD*} \begin{split} EMD^*(Q_1, Q_2) = \\
\sum_i (|\phi_1(m_i)-\phi_2(m_i)|\times (EMD(p_{m_i,1} , p_{m_i,2})) + EMD(p'_{m_i,1} , p'_{m_i,2}))) \end{split} \end{equation}
This distance is the minimal amount of work required to transform the $\phi_1$ distribution over mechanisms $m$ into $\phi_2$ by repeatedly moving the ``Earth" of $\phi$ from one mechanism $m_1$ in $Q_1$ to another mechanism $m_2$ in $Q_2$. (A complication is that in some cases (where $Q_1$ has more total $\phi$ than $Q_2$), we need to send the excess to an unconstrained distribution $p_{uc}$ associated with $Q_2$.)
We can then define $\Phi(AB)$ as the minimal value of EMD*($Q_{AB}, Q_{AB^*})$, across all partitions $AB^*$ of $AB$. A partition of a system requires cutting one or more causal connections between its units. For system AB, a partition cuts the connection from A to B or from B to A or both. In this case, either cut reduces $\phi$ to zero for both mechanisms A and B, and their probability distributions are flattened. The reason why cutting just one of these two connections destroys both mechanisms is tied to the fact that $\phi(m)$ is defined as the minimum of two $\phi$ values, the one that pertains to the future state and the one that pertains to the past state. Each cut will send one of these $\phi$ values to zero.
Recall that $Q_{AB}$ assigns $\phi(A) = \phi(B)=0.5$, where these serve as weights for $L(A)=[0,0.5,0,0.5,0,0.5,0,0.5]$ and $L(B)=[0.5,0.5,0,0,0.5,0.5,0,0]$. $Q_{AB^*}$ instead assigns zero weights to both $L(A^*)$ and $L(B^*)$, where $L(A^*) = L(B^*) = [0.25,0.25,0.25,0.25,0.25,0.25,0.25,0.25]$. We thus have:
$EMD^*(Q_{AB}, Q_{AB^*}) = |\phi(A)-\phi(A^*)|\times (EMD(p_A, p_{A^*}) + EMD(p'_A, p'_{A^*})) + |\phi(B)-\phi(B^*)|\times (EMD(p_B, p_{B^*}) + EMD(p'_B, p'_{B^*}))$ = $(0.5\times (0.5 + 0.5)) + (0.5\times (0.5 + 0.5)) = 1$.
The crucial quantity $\Phi^{max} (AB)$ is defined as $\Phi(AB)$ if AB is a maximum of $\Phi$, and 0 otherwise. Here AB is a maximum of $\Phi$ if $\Phi(AB) > \Phi(S)$ for all systems $S$ such that $S$ has elements in common with AB. In our case, we can stipulate that AB is isolated from its environment so that no other system containing A or B has higher $\Phi$. In this case, AB is a maximum of $\Phi$, so $\Phi^{max}(AB) = 1$. According to IIT, $\Phi^{max}$ is a measure of consciousness, so system AB has one unit of consciousness.
In section 4 we noted that if AB is in a different state (either 01, 00, or 11), than the calculation for $\Phi$ is the same, but the Q-shape is different. This can now be seen by the fact that changing the initial state changes the locations but not their weights. Thus, if the initial state is instead 00, then we still have two mechanisms A and B, each with weight 0.5, but their locations become $L(A)=[0.5,0.5,0,0,0.5,0.5,0,0]$ and $L(B)=[0.5,0,0.5,0,0.5,0,0.5,0]$. This is not enough to change $\Phi$, but it is enough to change the Q-shape. We can thus define Schroedinger's dyad as AB in a superposition of 10 and 00. A collapse model base only on $\Phi$ would fail to collapse this superposition, despite it being a superposition of conscious states.
We now move to quantum IIT (QIIT).\footnote{Thanks to Johannes Kleiner. Our calculations are intended to follow \textcite{zanardi2018quantum} and \textcite{KleinerTull2020}.} To simplify the calculations of the dyad, it is easier to start A and B in the same initial state ($\ket{00}$ or $\ket{11}$) so that they remain stationary. We add the further stipulation that A (B) maintains its own state over time. We may now consider A and B to be AND gates that each take two inputs as depicted.
\begin{center} \includegraphics[scale=.7]{drawing.pdf}\\[.5em] \end{center}
State $|00\rangle$ has zero $\Phi$ and Q-shape, that is, $\Phi(|00\rangle) = Q(|00\rangle) = 0$. For if we partition the system by replacing one of the directed edges with random input, the inputs are still only either 00 or 01, whereas the AND gates require an input of 11 to change state. Partitioning does not make a difference.
Partitioning makes a difference if the system is instead in state $|11\rangle$: If any of the edges are removed and replaced by random input, at least in half the cases it will feed a $0$ to its target, so that in light of the AND gate the state of the target will change from $1$ to $0$. This implies that the system in that state has non-zero $\Phi$ value, and its Q-shape isn't null.
We can therefore introduce collapse operators for the Q-shapes of these two states, and then use them to define a small consciousness superposition.
Our new dyad still has three subsystems (AB, A, and B). For each we consider the integrated information $\phi$ of both future and past states. So for the collapse operators $Q^k_{ij}$, the $k$ index runs from 1 to 6. Since the Hilbert space of the system in this case is $4$ dimensional, the indices $i$ and $j$ run from $1$ to $4$ each.
The $c_{ij}^k(\psi)$ in (\ref{Kleiner}) are the coefficients of the operator $\rho$ which is the $k$th component of the Q-shape of $\psi$. Because $Q(00) = 0$, it follows that $c_{ij}^k(\psi_0)= 0$. Since $\ket{00}$ and $\ket{11}$ are wave functions with classical Q-shapes, they are contained in $\mathds{E}^{-1}(\mathcal C)$ and are summed over in (\ref{Kleiner}). It follows that
\begin{equation}
\hat Q^k_{ij} \ket{00} = \sum_{\psi \in \mathds{E}^{-1}(\mathcal C)} \varphi^k(\psi) c_{ij}^k(\psi) \ket{\psi} \bra\psi\ket{00} = \varphi^k(00) c_{ij}^k(00)\ket{00} = 0\ket{00} \end{equation}
We have assumed the wave functions with classical Q-shapes are orthogonal. Thus $\ket{00}$ is an eigenvector of every operator $\hat Q^k_{ij}$ with eigenvalue $0$. We also have
\begin{equation}
\hat Q^k_{ij} \ket{11} = \sum_{\psi \in \mathds{E}^{-1}(\mathcal C)} \varphi^k(\psi) c_{ij}^k(\psi) \ket{\psi} \bra\psi\ket{11} = \varphi^k(11) c_{ij}^k(11)\ket{11} \end{equation}
so that $\ket{11}$ is an eigenvector of $Q^k_{ij}$ with eigenvalue $c_{ij}^k(\ket{11})$.
Letting $\ket{11}$ be the alive (conscious) state and $\ket{00}$ be the dead (unconscious) state, we can provide (in addition to the section 4 example) another example of Schroedinger's dyad:
\begin{equation}
\Ket{\Psi}_{AB} = \alpha\Ket{00} + \beta\Ket{11} \end{equation}
Our dynamics (in section 6) predicts that this state is not completely stable, but continuously collapses towards one of the two Q-shape eigenstates, in accord with the Born rule.
\printbibliography
\end{document} | arXiv |
Multi-level evidence of an allelic hierarchy of USH2A variants in hearing, auditory processing and speech/language outcomes
Peter A. Perrino ORCID: orcid.org/0000-0002-6849-67571,2,
Lidiya Talbot3,
Rose Kirkland4,5,
Amanda Hill6,
Amanda R. Rendall1,2,
Hayley S. Mountford3,
Jenny Taylor4,7,
WGS500 Consortium,
Alexzandrea N. Buscarello1,2,
Nayana Lahiri8,
Anand Saggar8,
R. Holly Fitch ORCID: orcid.org/0000-0002-1748-95061,2 &
Dianne F. Newbury ORCID: orcid.org/0000-0002-9557-268X3
Language development builds upon a complex network of interacting subservient systems. It therefore follows that variations in, and subclinical disruptions of, these systems may have secondary effects on emergent language. In this paper, we consider the relationship between genetic variants, hearing, auditory processing and language development. We employ whole genome sequencing in a discovery family to target association and gene x environment interaction analyses in two large population cohorts; the Avon Longitudinal Study of Parents and Children (ALSPAC) and UK10K. These investigations indicate that USH2A variants are associated with altered low-frequency sound perception which, in turn, increases the risk of developmental language disorder. We further show that Ush2a heterozygote mice have low-level hearing impairments, persistent higher-order acoustic processing deficits and altered vocalizations. These findings provide new insights into the complexity of genetic mechanisms serving language development and disorders and the relationships between developmental auditory and neural systems.
Learning to use and understand language requires the coordinated development of a whole host of underlying skills and processes. Assuming an initial bottom-up framework as posited by Tallal & others1, the first step in typical language development requires hearing, which allows infants to identify and parse meaningful signals from background noise and environmental distractors (auditory perception). These sounds are then mapped onto speech units (phonemes) enabling the extraction of meaningful linguistic constructs (comprehension). Even newborn babies turn their heads towards familiar voices and show a preference for speech over white noise2. These inclinations are fined-tuned over early life such that, at 3 months, infants show a preference for their native tongue over foreign languages and other vocal sounds (e.g. laughter)3. These behavioural distinctions are mirrored at the cognitive level; many studies show that speech evokes distinct neural responses, indicating that language is subject to privileged processing4. Early speech processing skills correlate with later language milestones5 suggesting that they modulate successful language acquisition.
While hearing is not strictly essential to successful language development (see ref. 6), overt disturbance of the contributory processes can directly disrupt language development; congenital hearing loss is associated with widespread cognitive deficits in domains including attention, memory and language7. Less obvious disturbances of audition can also indirectly impact spoken language development, especially if they occur during critical time periods or in addition to other insults. For example, auditory processing disorder (APD), characterized by poor central sound processing despite apparently normal hearing, can lead to difficulties understanding speech in noisy environments8 and, when persistent, is associated with an increased risk of Developmental Language Disorder (DLD) and Attention Deficit, Hyperactivity Disorder (ADHD)9. In the current study we test this framework by assessing whether typical variations across genes that function in auditory pathways may form part of a complex risk mechanism in the emergence of language disorders. In support of our proposed framework, subtle disturbances of auditory processing have been described across many neurodevelopmental disorders where hearing is unaffected (e.g. autistic disorder (as reviewed by ref. 10, ADHD11 and dyslexia12). Similarly, declines in later-life auditory processing skills are correlated with altered language function13.
The observed overlaps and comorbidities between neurodevelopmental and language disorders illustrate the complexities of contributory networks and the inter-reliance of developmental psychopathologies. In the long-term, all of these disorders are associated with academic limitation and reduced quality of life8,14 and represent an economic burden upon health-care and education systems15.
Although APD and DLD are both common childhood conditions (as high as 10%)16,17, our understanding of the relationships between hearing, auditory processing and language are confounded by these comorbidities alongside a lack of consistent diagnostic guidelines and failure to identify causal mechanisms18. As for many neurodevelopmental disorders, genetic contributions are poorly understood6,19. While monogenic forms of language disorder have been identified20, the majority of genetic risk factors are expected to function within complex models where each variant has only a small effect size and is influenced by additional genetic and environmental interactors19. Genome-wide screens indicate that large sample sets will be required to map variants that contribute to language disorder21 and acquisition22 and recent studies indicate shared genetic effects across behavioural subsets and between disorder and typical development19.
Current research suggests that genetic effects can overlap between syndromes that have traditionally been considered as clinically distinct. Similarly, genetic overlaps are reported between Mendelian forms of disease and more complex forms of disorder23,24,25. These changes in the field led us to apply an alternative approach to the investigation of molecular mechanisms underlying hearing and language within the current study. More specifically, we investigate the molecular overlaps between hearing, auditory processing and language through the targeted study of a gene that has an established role in audition—USH2A. This gene encodes the Usherin protein, which acts as a lateral link between stereocilium, providing structural organization for hair cell bundle development26. Homozygous pathogenic changes in USH2A, and therefore complete absence of the usherin protein, result in disorganization or loss of cochlear outer hair cells27, leading to congenital hearing loss clinically described as Usher Syndrome (OMIM#276901)28,29. This syndrome is a monogenic recessive disorder (4 in 100,000 births) associated with hearing loss specific to the high-frequency ranges, often accompanied by retinitis pigmentosa. The disorder splits into three clinical categories (Types I–III), relating to severity and age of onset28,30 and homozygous USH2A mutations result in the majority of Type-II cases in which diagnosed individuals are born with hearing loss and develop retinitis pigmentosa at the onset of puberty, yet do not experience the vestibular dysfunction (i.e. difficulties with balance and coordination) found in other types of Usher31. In accordance with currently accepted genetic models, individuals with a heterozygous USH2A mutations are considered to be unaffected carriers.
In the current study, we employ whole-genome sequencing in a discovery family to target association and gene x environment interaction analyses in two large population cohorts; the Avon Longitudinal Study of Parents and Children (ALSPAC) and UK10K. We complement these statistical methodologies with the characterization of a mouse model that shows behavioural effects of a heterozygous knockout on hearing, complex acoustic processing, and communicative vocalization. We propose the existence of an "alleleic hierarchy" of variation within which different variant types have divergent effects upon auditory thresholds. We show that USH2A variants exert direct effects upon low-frequency hearing and indirectly affect the risk of language disorder through the modulation of subsequent auditory perception and language development. These findings suggest a shared genetic etiology between hearing mechanisms, central auditory processing and language development and support the targeted investigation of sub-serving mechanisms in relation to language development.
AnUSH2A variant cosegregates with language disorder
A multi-generational family was ascertained for genetic investigation (Fig. 1). The family included seven Non-Founder individuals, all of whom were affected by a severe expressive language disorder. Affected individuals presented with slow and dysfluent speech and difficulties characteristic of APD, namely processing speech and following instructions, particularly in the presence of background noise or absence of visual cues (e.g. on the phone). The proband (IV.I, Fig. 1) and her sister (IV.2, Fig. 1) attended a special school for children with speech and language difficulties. Hearing assessments in the proband and sister were normal. Pure-tone audiometry tests in the Grandfather (II.I, Fig. 1) indicated normal hearing thresholds but deficits were noted across three tests of central auditory processing (dichotic digits, frequency pattern and duration pattern). Further details of the phenotype are provided in the Methods. Genome sequencing of two individuals (II.2 and IV.1) identified a novel stop-gain mutation (a change which results in a truncated protein) in the USH2A gene (NP_996816:p.Gln4541*) shared by all affected family members (Fig. 1, Supplementary Dataset 1). This variant (rs765476745) has previously been documented in cases of Usher syndrome and retinitis pigmentosa in whom it was documented to occur with a secondary pathogenic variant in a compound heterozygote mechanism32,33,34. The hearing of heterozygotes was not assessed within these previous studies32,33,34. rs765476745 has a Combined Annotation Dependent Depletion (CADD) score of 40, placing it in the top 0.01% of deleterious variants in the genome35, and meets clinical guidelines from the American College of Medical Genetics (ACMG) for a "pathogenic variant"36; it confers a stop mutation (Very strong evidence of pathogenicity—PVS1), has previously been described as pathogenic (Strong evidence of pathogenicity—PS1), is absent from population databases (Moderate evidence of pathogenicity—PM2) and cosegregates with disease (Supporting evidence of pathogenicity—PP1)36. No secondary putative pathogenic changes were found in other genes related to Usher syndrome or hearing loss (Supplementary Dataset 1). Given the inheritance pattern and complete cosegregation, in the absence of a second pathogenic variant, we hypothesized that the observed heterozygous loss of USH2A could account for the observed language disorder in this family.
Fig. 1: Discovery Family and cosegregation of USH2A variant.
Individuals affected by APD are coloured in black. Unaffected individuals are shown in white. Line through symbol indicates that individual is deceased. All descendants of individual I.1 were affected by a severe expressive language disorder. The only unaffected individuals in the family were incoming Founder members (I.2, II.2, II.3, III.4). Chromatograms show validation of USH2A variant in wider family members. The variant (position indicated by a blue highlight bar) was observed in a heterozygous form in all affected individuals.
Heterozygous Ush2a KOs have altered auditory perception
To elucidate the behavioural effects of heterozygous Ush2a loss, we generated heterozygous (HT) and full knockout (KO) mice. Hearing and complex acoustic discrimination thresholds were compared against WT controls on a series of prepulse inhibition (PPI) tasks37. Initially, using a simple single-tone detection task at 40 kHz (high-frequency), individual ANOVAs revealed that KO mice trended to the expected hearing impairment typical of Usher syndrome, while HT mice performed similarly to WT controls [(Overall): F(2,32) = 1.995, p = 0.153, one-tail; (WT vs. HT): F(1, 22) = 0.619, p = 0.440; (WT vs. KO): F(1, 21) = 3.125, p = 0.092; (HT vs. KO): F(1, 21) = 1.517, p = 0.232 (Fig. 2a). In contrast, using this same task at 15 kHz (low-frequency), HT mice performed significantly worse than KOs and trended to worse than WTs [(Overall): F(2,32) = 3.697, p = 0.36; (WT vs. HT): F(1, 22) = 3.201, p = 0.087; (WT vs. KO): F(1, 21) = 1.054, p = 0.316; (HT vs. KO): F(1, 21) = 5.016, p = 0.036], while WT and KO did not differ (Fig. 2a). In order to assess possible higher-order processing deficits, individual scores on the above single-tone task were used as covariates (frequency-matched) to analyze more complex PPI measures (thus eliminating variance due to hearing impairments). In repeated measures ANCOVAs with Genotype as the between-subjects variable and Day and Cue as the within-subject variables, deficits were again evident for HT mice on complex low-frequency tasks [Embedded Tone 100: 10.5 kHz: F(2, 31) = 3.691, p = 0.036; Embedded Tone 10: 10.5 kHz: F(2, 31) = 4.635, p = 0.017)], and for KO mice on higher frequency tasks [Pitch Discrimination: 40.5 kHz (WT vs. KO); F(1, 20) = 9.232, p = 0.006] (Fig. 2b–d). These findings indicate that Ush2a-mediated perceptual deficits include higher-order dysfunction, even when variance due to hearing loss was removed.
Fig. 2: Ush2a HT mice show low-frequency auditory impairments.
Relative performance on various prepulse inhibition paradigms (lower scores equal better performance). a Normal Single Tone at 15 kHz and 40 kHz displayed. b Embedded Tone 0-100 at 10.5 kHz analyzed across days using NST 15 kHz as a covariate. c Embedded Tone 0-10 at 10.5 kHz analyzed across days using NST 15 kHz as a covariate. d Pitch Discrimination at 40 kHz analyzed across days using NST 40 kHz as covariate. Data shown are mean ± SEM for each Genotype. *p < 0.05; #p < 0.15. White diamond indicates Genotype mean. All panels included 35 biologically independent animals (12 WT, 12 HT, 11 KO). Data underlying these figures are provided in Supplementary Dataset 3.
Heterozygous Ush2a KOs have altered vocalizations
Given the reported comorbidity between auditory processing and language impairments and the presence of dysarthria in the discovery family, we investigated whether Ush2a knockout in mice altered the properties of their ultrasonic vocalizations. Results showed that Ush2a HT mice vocalized at significantly higher frequencies (pitch) across most syllable types [F(2, 16100) = 83.476, p = 0.000] and produced calls that were shorter and louder than WTs [duration: F(2, 16100) = 26.70, p = 0.000; volume: F(2, 16100) = 142.54, p = 0.000] (Fig. 3). (See Supplementary Fig. 1 for images and coding technology for eight primary call types assessed). Interestingly, Ush2a KO mice also produced higher pitched calls suggesting that disruption to auditory processing ability (regardless of the frequency of the stimuli) results in impaired expressive communication ability. This could reflect the importance of intact auditory feedback for vocal development (e.g. anomalous song production in deafened birds), and is consistent with the higher vocal pitch observed in in profoundly deaf speakers38. The putative social impact of any vocalization anomalies will require additional study. Our behavioural investigations of the mouse model indicate that heterozygous disruption of Ush2a leads to altered low-frequency hearing thresholds, a phenotype distinct from the high-frequency hearing loss of Usher syndrome (and replicated in null Ush2a mice). Moreover, these altered low-frequency thresholds were further associated with higher-order auditory processing deficits, as well as disrupted vocalizations.
Fig. 3: Ush2a HT mice produce altered ultrasonic vocalizations (USVs).
a Syllable Frequency per Genotype collapsed across syllable category. b Syllable Duration per Genotype collapsed across syllable category. c Syllable Volume per Genotype collapsed across syllable category. Data shown are mean ± SEM for each Genotype. ***p < 0.001. White diamond indicates Genotype mean. All panels included 34 biologically independent animals (12 WT, 11 HT, 11 KO). Data underlying these figures are provided in Supplementary Dataset 4.
Pathogenic USH2A carriers recapitulate the mouse model
The description of a single case family does not warrant a claim of causation, even when supported by an animal model. We therefore sought to characterize the developmental profiles of other individuals with heterozygous USH2A knockout. The effects of pathogenic USH2A variants were explored through the investigation of developmental profiles of UK10K children39. Fourteen UK10K individuals (12 M:2 F, from 1646 individuals with sequence and phenotypic data available, 0.85%, Table 1) were identified as carriers of USH2A changes that were designated as "pathogenic" in ClinVar. These consisted of five distinct variants which were always detected in a heterozygous form (Table 1). The variant found in the discovery family (rs765476745) was not present in the UK10K samples (Table 1). Analyses of developmental behavioural data showed that USH2A carriers scored below expected on measures of early vocabulary (Cohen's d = 0.7237, 95% CI = 0.18–1.27) and word combinations (Cohen's d = 1.09, 95% CI = 0.54–1.63) (Table 2). Parents of carriers were twice as likely to be concerned about their child's speech at 3 years of age (RR = 2.07, 95% CI = 0.57–7.49) and reported a higher incidence of stuttering (RR = 2.92, 95% CI = 1.05–8.08) and dyslexia (RR = 1.88, 95% CI = 0.28–12.45) at age 8 (Table 3). In support of our mouse model, we also observed a low-frequency-specific hearing phenotype in heterozygous individuals; carriers had average low-frequency (500 Hz) hearing thresholds 1.2 dB HL above those of non-carriers (Table 2) but did not display overt hearing loss (Table 3) or differences at higher frequencies (Table 2). Thus across mouse and human data, we find evidence that heterozygous USH2A variants affect higher-order auditory processing and increase the risk of delayed language milestones. In line with current literature, these variants alone do not result in a discernible carrier phenotype (as would be expected in a monogenic model)40. Instead, we propose that they form part of a genetic risk factor within a complex genetic model.
Table 1 Clinically relevant mutations observed in the UK10K dataset.
Table 2 Quantitative measures of language, reading and cognition in carriers of USH2A compared to non-carriers in UK10K dataset.
Table 3 Discrete measures of educational support, neurodevelopmental disorders and hearing in carriers of USH2A compared to non-carriers in UK10K dataset.
USH2A variants are associated with low-frequency hearing
Targeted association analyses were performed to investigate the wider effects of USH2A polymorphisms on hearing and language within typical development. SNPs tagging common USH2A variants were analysed for allelic association in the ALSPAC cohort41 (N = 7691 individuals, N = 127 variants). Three measures of language (early vocabulary size (vocab), nonword repetition (NWR) and developmental language disorder (DLD) status), and two hearing measures (low-frequency hearing (minimum air conduction threshold at 0.5 kHz; MinLow) and mid-frequency hearing (minimum air conduction threshold at 1, 2 and 4 kHz; MinMid)) were assessed. In support of the carrier and mouse findings described above, association was observed for a cluster of SNPs located between exons 4 and 12 of USH2A, specifically with low-frequency hearing (Table 4). The top-associated SNP (rs10864237, minP = 6.9 × 10−5) explained 0.3% of variance in low-frequency hearing thresholds (βSE = 0.13) representing a 1 dB HL difference between risk (TT genotype) and non-risk (CC genotype) individuals. These analyses therefore extend our previous findings to encompass common variants in USH2A and low-frequency hearing thresholds within the typical range.
Table 4 Association analyses of variants across USH2A in relation to language and hearing outcomes.
Gene-environment interactions associate with language
We then explored a hypothesis that USH2A modulates low-frequency hearing thresholds through indirect modulatory effects on subsequent auditory perception and language development. In this model, common USH2A variants impact hearing but also exert secondary impacts on speech and language development, presumably as a result of less effective higher-order auditory perception. Specifically, we again assessed common variants in USH2A for association to language outcomes in ALSPAC, but this time included low-frequency hearing thresholds as an interaction factor42. Association was now observed with early vocabulary (rs7532570) (Table 4). Within this interactive model, rs7532570 had a p-value of 8.6 × 10−5 compared to P = 0.15 in the additive model. When combined with the findings of the direct association analyses, these data suggest that common variants in USH2A can modify low-frequency hearing thresholds and that, when thresholds are altered USH2A can, in turn, modulate the risk of disrupted language development. This mirrors the relationship observed between low-frequency hearing and vocalization in Ush2a HT mice, suggesting a parallel modulatory gene-environment (GxE) interaction in mice and together confirming that auditory perception represents a building block for language development.
Genetic variants in USH2A exert distinct effects
To gain a more complete picture of USH2A variation, gene-based analyses were performed using UK10K genome sequence data (N = 1646 individuals), enabling the combined consideration of rare and common variants across coding and non-coding regions within a single test. This large sequence dataset allows the detection of variants with expected frequency as low at 0.03%. Three variant selection thresholds were considered (all variants (N = 7619), rare variants (MAF ≤ 1%, N = 5424) common variants (MAF ≥ 5%, N = 1335)) in relation to the same three language and two hearing measures described above. Results showed dichotomous effects between variant frequency classes; association to hearing measures was driven by common variants while DLD status was marginally associated with rare variants (Table 4).
GxE effects implicate hearing-modulated language pathways
To explore similar genetic effects at a genome-wide level, a GxE interaction study (GWIs) was completed. These exploratory analyses enabled the identification of common variants that influence language through low-frequency hearing and, additionally allowed the evaluation of genes implicated in hearing within the model identified41. Taking direction from the GxE analyses above, a linear regression model was employed with early vocabulary as the dependent variable and low-frequency hearing thresholds as an interaction term (Supplementary Fig. 2). Eight SNPs reached genome-wide significance (P < 5 × 10−8), while 450 SNPs across 139 HGNC transcripts were nominally associated (P ≤ 10−5) (Supplementary Dataset 2). Pathway analyses did not indicate an enrichment of genes previously related to hearing or language (Supplementary Table 2) but instead revealed an enrichment of protein-binding factors involved in cell adhesion and cellular movement (Table 5). Cellular components of cell projections, lamellipodia and synapses were also over-represented (Table 5). These genome-wide analyses therefore implicate cell migration and connectivity as potential mechanisms for the underlying effect of auditory perception upon speech and language development.
Table 5 Pathway analyses of genes implicated in GxE interaction effects of low-frequency hearing upon vocabulary development.
Language development is a multifaceted trait that relies on interactions between many sub-servant mechanisms each subject to genetic, cognitive and environmental effects, including auditory processing and hearing. In this study, we consider developmental links between a specific candidate gene (USH2A), hearing, auditory perception, communicative mouse vocalization and human vocabulary. Together, our data provide evidence that auditory perception represents a building block for communicative and language development. The identification of a USH2A stop-gain variant in the discovery family was substantiated by behavioural investigation of heterozygous Ush2a knockout mice (Ush2a+/−). In contrast to full knockouts, these mice presented with a distinctive low-frequency hearing loss (p < 0.05 at 15 Hz), accompanied by impairments in complex sound processing that was present even after variance due to hearing loss was removed, and also altered ultrasonic vocalizations. Population data corroborated the functional effects of USH2A in audition and early language development; children in the UK10K cohort39 who carried pathogenic variants had increased low-frequency hearing thresholds (+1.2 dB HL at 500 Hz) and showed reduced early vocabulary when compared to non-carriers. In a cohort of typically developing individuals41, variants at the 5′ end of the gene were directly associated with increased low-frequency hearing thresholds (minP = 6.9 × 10−5). Within an interactive genetic model, individuals carrying risk variants in the presence of altered low-frequency hearing thresholds were found to have a smaller vocabulary than those who carried only one of these risk factors in isolation (minP = 8.6 × 10−5). Together, these data demonstrate that allelic variations in USH2A are associated with altered low-level hearing thresholds that, in turn, impact speech and language development through the modulation of higher-order acoustic processing. As such, even a subtle degradation in hearing and subsequent complex acoustic processing (as seen in heterozygous Ush2a mice) could developmentally derail language processing in humans.
Our findings are consistent with emerging evidence that different variant types can associate with variable outcomes, forming an "allelic hierarchy" of disease-causing and complex risk variants, representing a shift from Mendelian genetic models43. We extend this hypothesis by demonstrating a multifaceted allelic hierarchy in which rare and common variants within the same gene can form reciprocal influences upon gene functions under different environmental influences. The finding that heterozygous disruption of USH2A led to altered hearing thresholds in the low-frequency ranges was unexpected, as complete loss of this gene results in Type-II Usher Syndrome characterized by congenital high-frequency hearing loss40. While one previous study suggested that carrier individuals may experience slight hearing disturbances44, heterozygotes are generally considered aphenotypic and do not show obvious deficits in clinical hearing tests40. Our findings provide a molecular explanation for this; heterozygous gene disruptions are typified by subtle changes in the processing of low-frequency sounds that may be incidental to routine audiologic assessment. Such changes would not necessarily be detected in a clinical setting where the focus would be on Usher-related high-frequency hearing loss. Notably, we found that the low-frequency thresholds of carrier individuals were consistently (marginally) below those of non-carriers, though still within typical range. While it is unlikely that such subtle changes in hearing thresholds (1–2 dB) at these frequencies would directly lead to language disorder, we propose that mild changes in low-level hearing may exert a snow-ball effect that derails higher-order communicative processing. This model is akin to that described for persistent otitis media with effusion which, in itself, does not cause language disorder but may represent a risk factor when persistent45. Our findings are of clinical importance given that heterozygous loss of the USH2A gene is relatively common–we found that carriers of heterozygous pathogenic variants constitute 0.85% of the UK10K cohort studied here. This figure aligns well with gnomAD European samples, of whom 1.1% are carriers46. Thus although behavioural effects are likely to be subtle, and may exert indirect effects within a more complex genetic model as indicated by our gene-environment analyses, the fact that 1 in 100 worldwide may be at risk calls for universal updates to screening protocols.
Beyond this, our study highlights a directionality of effects in which genetically-mediated differences in hearing (directly or indirectly) affect the neuronal development of central auditory processing systems and consequently influence language acquisition. These observations generate two distinct temporal models; (1), the feedback model, in which altered auditory input directly affects neuronal development leading to perceptual deficits that, in turn, increase the risk of speech and language disorders, or (2), the double-hit model, in which altered hearing thresholds combine with existing genetic factors to moderate the risk of speech and language disorders. Exploratory network analyses implicate synaptic connections and cell growth as important processes in hearing-mediated language pathways perhaps suggesting the importance of feedback mechanisms. Importantly, the mouse strains employed here (I129) have a homogeneous background that lacks overt risk mutations. This combines with low Ush2a brain-expression47 and a lack of reported neuronal anomalies in Ush2a knockouts27 to further substantiate the hypothesis that patterns of emergent cochlear output can be shaped by primary stereocilia activity48,49. The feedback model fits with a recent single-cell sequencing study, which showed that auditory input during early life can shape gene expression patterns in spiral ganglion neurons (the primary tract between the cochlea and brainstem)49. Under this model, early variations in hearing thresholds can have long-lasting and complex downstream effects, presumably through the modification of central mechanisms. The double-hit model aligns with emerging knowledge from high-throughput genomic studies, which indicate the existence of complex shared mechanisms between disorders and further suggest that a "one-gene, one-disorder" expectation represents a gross simplification of genetic mechanisms both in disease and typical development50,51. The exact mechanisms by which low-frequency hearing may influence language development remain unclear but given the findings presented here, we propose that differences in auditory input can alter perception of speech. When these occur at critical time-points or are combined with other (as yet unidentified) risk factors, we hypothesise that this may have repercussions for the development of expressive language. The current study considers only air conductance thresholds but future studies may consider other aspects of hearing, for example through the addition of conductive hearing tests or measures of auditory brain responses and exploration of the effects of otitis media. Future investigations are needed to delineate the temporal effects reported here. Such studies will allow us to distinguish between impaired input at the synaptic interface between hair cells and the brain, versus altered linguistic circuitry or feedback, as well as to investigate genetic modifiers and define critical developmental windows for these interactions. The current study directly advances our understanding of the behavioural effects of changes in the USH2A gene and indicates that different levels of disruption can target different sound frequencies. Our mouse models further suggests that Ush2a-mediated alterations of sound perception can lead to behavioural deficits that extend to vocalization.
The discovery pedigree consisted of 12 members (Fig. 1). Eight individuals and all descendants of individual I.1 (Fig. 1) were affected by expressive language disorder characterized by acute auditory processing difficulties and speech dysarthria. All descendants of individual I.1 were affected indicating an autosomal dominant inheritance pattern. The family was ascertained through proband IV.1.
Language phenotype
The proband (IV.1) was born at full-term by normal delivery following an uneventful pregnancy. There were no early developmental concerns and all gross motor milestones were achieved. However, early language milestones were delayed. First word was reported at 18 months and she was referred to a speech and language therapist at 2 years of age. A diagnosis of Specific Language Impairment (SLI) was given at age 4 years and 8 months. In this assessment, she showed particular difficulties understanding abstract language and linguistic concepts and often failed to follow conversations when no visual cues were given. She had an extensive vocabulary but her language processing was slow and she often showed difficulties finding the word she needed. She showed grammatical difficulties such as sequencing errors, simplification of sentence structure and errors with word structure. On the Children's Communication Checklist (CCC-2), she scored below the 15th percentile on all four language scales (speech, syntax, semantics and coherence) but above this range in scales of inappropriate initiation (60th percentile), use of context (34th percentile), nonverbal communication (42nd percentile) and interests (36th percentile). At a clinical assessment at 58 months, she showed typical hand-eye coordination and performance. She did not present with dysmorphic features and hearing assessments were normal.
There were also concerns regarding the proband's younger Sister's (IV.2) language development. Her first words appeared around the age of 2 years. Motor development was normal. She had mild to moderate bilateral conductive hearing impairment due to recurrent ear infection and grommets were inserted at the age of 3. Following this, hearing assessments were normal but her speech and language difficulties continued and she was diagnosed as having a severe speech disorder in particular with expressive language and dysfluency with very good receptive language skills. The proband (IV.1) and her sister (IV.2) both attend special language units.
The proband's Mother (III.3), maternal Great-Uncle (II.2) and Grandfather (II.1) indicate that the deficits observed in the proband are typical across all family members. The Mother and maternal Great-Uncle have not had formal assessments but both struggled at school requiring speech and language therapy and have difficulties with expressive speech and processing. The similarities between their early difficulties and that of the proband and her sister are striking. Assessment of the Grandfather (II.1) at 62 years of age indicated poor performance across cognitive tasks (verbal and nonverbal) with particular difficulties in tests of recall memory, visual recognition, literacy, executive function and information processing (all below tenth percentile). In contrast, verbal recognition, object naming and auditory attention skills were within the expected range. Pure-tone audiometry showed normal hearing thresholds but deficits were noted across all three tests of central auditory processing (dichotic digits, frequency pattern and duration pattern).
SNP genotyping
Seven members of the discovery family (five affected individuals, II.1, II.2, III.3, IV.1 and IV.2, and two unaffected individuals, II.4, III.4, Fig. 1) were genotyped on Illumina HumanOmniExpress-12v1 Beadchips (San Diego, CA, USA; ~750,000 SNPs). SNPs were excluded if the gentrain (genotype clustering quality) score was <0.5 or genotyping success rate was <95%. All individuals had a genotype rate>95%. Genotype data were used to construct haplotype sharing patterns across the pedigree and to call copy number variants (CNVs) as described below.
Haplotype reconstruction
SNP genotype data from seven family members were used to construct haplotype sharing patterns within the Merlin package52. These data were employed to filter candidate variants from the whole-genome sequence data as described below.
Copy number calling
CNVs were called by two separate algorithms; PennCNV53 and QuantiSNP54. All samples had a log R ratio (LRR) SD < 0.35, a B-allele frequency (BAF) drift value <0.002 and a waviness factor between −0.04 and 0.04 in PennCNV and an average LRR SD < 0.3 and BAF SD < 0.15 in QuantiSNP. Any CNV that contained at least three consecutive SNPs, had a confidence value (PennCNV) or log Bayes Factor (QuantiSNP) of >10 and was predicted by both PennCNV and QuantiSNP, with a minimum intersection of 50% each way, was considered to be of 'high confidence'. The innermost boundaries of the two algorithm calls were used. CNVs were excluded if they spanned the centromere or telomeres.
Whole-genome sequencing
DNA from two members of the discovery family (II.2 and IV.1, Fig. 1) were subject to whole-genome sequencing enabling the identification of possibly pathogenic variants within shared chromosome regions across the wider pedigree. Sequencing was performed as part of the Oxford University-Illumina WGS500 collaboration (http://www.well.ox.ac.uk/wgs500)55. This project includes whole-genome sequences for 156 samples from clinical cases in whom standard genetic tests were negative or where no standard tests were available55.
Sequencing was completed on the Illumina HiSeq platform (Illumina Inc, San Diego, CA, USA) with 100nt, paired-end runs. Alignment was performed against the Human Reference genome (build 37d5, hg19) in Stampy56 and duplicate reads removed using Picard (http://broadinstitute.github.io/picard/). Variant sites (Single nucleotide variants and indels less than 50 bp) were called using Platypus (v0.1.8)57. The mean depth across all mapped sites was 28.03 and the transtition-transversion ratio across the two samples was 1.99.
Variants that were shared by the two family members and passed quality filters with PHRED quality scores≥20 were identified within vcftools58 (N = 17,767, Supplementary Dataset 1). These were subsequently filtered through a step-wise procedure to include variants which fell within chromosome regions shared only between affected family members (using haplotype reconstruction data from the wider pedigree as described above) (remaining N = 3743, Supplementary Dataset 1). Potential functional relevance of shared variants were annotated using SnpEff (v3.2)59. Variants that conferred a coding change (frameshift, non-synonymous, canonical splice-variant or stop/start-gain/loss) (remaining N = 1223, Supplementary Dataset 1) and were not described (or had a minor allele frequency of 0) in the 1000 Genomes Phase I (v2) data (Apr 2012)60 (remaining N = 36, Supplementary Dataset 1) and dbSNP (build 147)61 (remaining N = 6, Supplementary Dataset 1) were prioritized for follow-up. Variants and filter data are shown in Supplementary Dataset 1. Candidate variants were validated by Sanger sequencing using BigDye (v3.1) on a 3730XL DNA analyzer (Applied Biosystems, California) using standard protocols. Chromatograms were visualized within FinchTV (www.geospiza.com/finchtv).
Replication cohorts
Targeted analyses of the identified candidate gene (USH2A ± 10Kb - chr1:215786236-216606738, hg19) were performed in two large population cohorts; the Avon Longitudinal Study of Parents and Children (ALSPAC, 7,141 children, 3,615M:3,526F)41,62 and the UK10K dataset (1646 individuals, 785M:861F)39. The ALSPAC population cohort offers a wide range of neurodevelopmental phenotypes (including language, memory, hearing and neuropsychiatric measures) from children born to 14541 mothers from Avon in 199141. In addition to phenotype data, ALSPAC also provides genotype data (Illumina Human Hap 550-quad array) for 8365 children41 allowing SNP-based association analyses. A subset of ALSPAC children (1867 individuals) had whole-genome sequence data available as part of the UK10K project39 allowing gene-based association analyses of rare and common variants across the candidate gene.
Both replication cohorts were filtered to include only individuals with available phenotype data, of British ethnicity, born at more than 32 weeks gestation and a birth weight >1500 g. Additional filters were applied for the analysis of common variation in the ALSPAC cohort. These aimed to exclude children with overt pathology that may confound language development, namely nonverbal IQ < 65 and hearing loss (hearing thresholds above 40dbL). After these filters, the ALPSAC replication set included 7141 children (3615M:3526F) and the UK10K replication set included 1681 individuals (806M:875F). The UK10K cohort included fourteen children with heterozygous USH2A mutations (Table 2) allowing the consideration of developmental profiles including measures of early language development, later language and cognitive ability, hearing function and neurodevelopmental disorders across carrier children (30 measures in total, Tables 2 and 3).
Analyses targeted three measures of language (early vocabulary, Nonword repetition and Developmental Language Disorder (DLD)) and two measures of air conductance (mid- and low- frequency hearing thresholds) as directed by observations in the heterozygote knockout mice. Details of these measures are provided below and the ALSPAC website contains details of every available measure through a fully searchable data dictionary and variable search tool (http://www.bristol.ac.uk/alspac/researchers/our-data/).
Early vocabulary (vocab)
The vocabulary measure represents a sum of items that children could use and/or understand, from a list of 123 words, at age 3 (ALSPAC variable KG865). This measure was derived from a parental questionnaire. Data were available for 6165 genotyped children from the ALSPAC cohort and 1614 children from the UK10K cohort. Scores across both datasets ranged from 0 (child did not understand or use any of the 123 words) to 246 (child could use and understand all of the 123 words) (mean = 229.8, SD = 29.4).
Nonword repetition (NWR)
An adaptation of the Nonword memory test63 was used to assess short-term memory (ALSPAC variable F8SL105). This measure has been shown to provide an accurate biomarker of speech and language difficulties64,65. The tests were completed in clinic and consisted of 12 nonsense words of between 3 and 5 syllables which the child had to listen to and repeat. This test was completed at 8 years of age and data were available for 5229 genotyped children from the ALSPAC cohort and 1572 children from the UK10K cohort. Scores across both datasets ranged from 0 to 12 (mean = 7.3, SD = 2.5).
DLD status (DLD)
A binary measure of DLD status was defined in line with our previous publications65,66; cases performed at least 1 SD below mean on WOLD comprehension (ALSPAC variable F8SL040) OR had CCC verbal fluency AND syntax (ALSPAC variables KU503b and KU504b respectively) >1 SD below mean with no evidence for Autistic Spectrum Disorder (ASD) or hearing impairment. Typically developing controls were selected to perform above expected levels across all of the three language measures used to define cases (WOLD comprehension, CCC syntax and CCC verbal fluency) and had nonverbal IQ > 80 and presented without neurodevelopmental disorders or special educational needs. The ALSPAC cohort included 731 cases and 2114 controls and the UK10K cohort included 36 cases and 582 controls.
Mid-frequency hearing (MinMid)
Audiometry was performed as per British Society of Audiologists (BSA) standards—thresholds were taken as 2/3 presentations on the ascending scales. Both air- and bone-conduction were performed using either a GSI 61 clinical audiometer or a Kamplex AD12 audiometer. All hearing tests were carried out by audiologists and trained Staff in a room with minimal external noise (not exceeding 35 dB). Minimum air conduction thresholds were measured for the left and right ears at 0.5, 1, 2 and 4 kHz. Mid-range hearing was defined as the minimum air conduction thresholds across the right and left ears averaged across 1, 2 and 4 KHz (ALSPAC variables F7HS018 and F7HS028, respectively). This measure was available for 4645 genotyped children from the ALSPAC cohort and 1300 children from the UK10K cohort. Thresholds ranged from −8.3 to +40 across these samples.
Low-frequency hearing (MinLow)
Low-frequency hearing thresholds were defined as the minimum air conduction thresholds across the left and right ears at 0.5KHz. This measure was derived from ALSPAC variables F7HS017, F7HS018, F7HS027 and F7HS028). Data were available for 4563 genotyped children from the ALSPAC cohort and 1277 children from the UK10K cohort. Thresholds ranged from −10 to +40 across these samples.
Separate high-frequency threshold data were not available within the ALSPAC data release for this project.
Gene-based association analyses
The UK10K cohort offered genome sequence data, allowing characterization of developmental profiles in identified heterozygous carriers. These sequence data were also employed for gene-based analyses of common and rare variants within RVTESTS67. Gene-based testing employed SKAT; a kernel-based method that allows for variants with different directions of effects and can analyze both rare and common variants within a single model68. In total, 7691 variants were analyzed. All variants had an allele count of at least 1 in the sample set, affected only single nucleotides (i.e. SNVs), had a minimum mean quality score of 20 and a minimum mean depth of 3 across samples and HWEp > 1 × 10−5. The transtition-transversion ratio of the SNVs was 2.2.
Association analyses of common variants
SNP data were available for ALSPAC from Illumina 660 and Illumina 550 SNP arrays allowing allelic association analyses of common variants with the PLINK package42. Standard quality control procedures69 were completed on genome-wide SNP data prior to analyses; variants with a minor allele frequency <5%, a call rate of <5%, a Hardy-Weinberg equilibrium p < 5 × 10−7 or a heterozygosity rate more than three standard deviations from the mean were excluded. Per SNP genotype rates were compared between DLD cases and controls and any SNP with a differential missing rate was excluded. Individuals with a genotype rate <95%, discordant sex information or non-Caucasian genetic background were excluded. Following quality control, SNPs across the USH2A gene ±10Kb (chr1:215786236-216606738) were pruned using the Tagger algorithm within Haploview70,71 to derive a pairwise tagging SNP set with R2 < 0.8 consisting of 127 SNPs across 820Kb. Tagging SNPs were analyzed for allelic association within PLINK42 using a linear model of regression for quantitative traits and a logistic model for discrete traits.
Genetic interaction analyses
Gene-environment interaction effects were further modeled within ALSPAC at the gene and genome-wide level using PLINK in which the–interaction command can be used to model SNPxcovariate interactions within a linear regression model (Y = b0 + b1.ADD + b2.COV1 + b3.ADDxCOV1 + e)42. Gene-level analyses were performed for 12 SNPs across the 5' region of the USH2A gene and comprised of three language outcome measures (Early vocabulary (vocab), nonword repetition (NWR) and DLD status) and one interaction factor (Low-frequency hearing threshold (MinLow)). At the genome level, a single outcome measure (Early vocabulary (vocab)) was modelled with a single interaction factor (Low-frequency hearing threshold (MinLow)) for 488205 autosomal SNPs. Manhattan plots were generated using the qqman package72 within R (v3.4.4) (https://www.r-project.org/).
Pathway analyses
SNPs that had P values ≤ 10−5 in the genome-wide interaction analyses (N = 450, Supplementary Dataset 2) were positioned within UCSC (https://genome-euro.ucsc.edu/index.html, hg19) and those which mapped onto known HGNC transcripts (N = 139, Supplementary Dataset 2) were entered into pathway analyses to identify over-represented gene classes. Pathway analyses were performed within STRING (https://string-db.org)73. Gene ontology classes were analyzed for over-representations using a Fisher exact test with FDR multiple test correction (Table 5). Identified genes were further compared to a list of 37 candidate genes for speech and language-related phenotypes (Taken from ref. 23 and supplemented with a list from refs. 23,74,75) and a list of 197 candidate genes for hearing-related phenotypes (taken from http://hereditaryhearingloss.org/, supplemented with a list from the IMPC75) (Supplementary Table 2).
Ush2a mice—subjects
Six Ush2a knockout (KO) male mice27 were provided by Dr. Jun Yang (University of Utah), and were re-derived on an 129S4/SvJaeJ background strain at the Gene Targeting and Transgenic Facility (GTTF) at UConn Health. All subjects were single housed in standard Plexiglass mouse-tubs (12 h/12 h light-dark cycle), with food and water available ad libitum. F1 subjects were delivered to the University of Connecticut where they were crossed with six wild-type (WT) controls (129S4/SvJaeJ; stock number 009104) purchased from The Jackson Laboratory (Bar Harbor, ME). The resulting F2 offspring were heterozygous (HT) for the Ush2a gene, which shows 71% identity with its Human orthologue. Breeding pairs (HT × HT) were used to generate the experimental subjects, such that all genotypes (homozygous knockout, heterozygous, and wild-type) were represented within-litter (F3). F3 genotypes were determined via PCR of earpunch DNA using the following DNA primers: Common (5′-GTGAATACAGGCACCTCTGAATGTGAC-3′), WT (5′-GTCACGGCTGAATCCCGAAGC-3′), KO (5′-GAGATCAGCAGCCTCTGTTCCAC-3′). Twelve WT male mice, 12 HT male mice, and 11 Ush2a KO male mice from F3 were randomly selected for behavioural testing as outlined below (12 WT, 11 HT and 11 KO mice were used when recording ultrasonic vocalizations).
Ush2a mice—auditory processing
Following puberty, subjects were tested on a battery of auditory processing tasks using a modified prepulse inhibition paradigm which allows free movement during the presentation of sounds that include an unpredictable loud noise burst (see Fitch et al., 2008 for review)37. PPI provides a superior index of acoustic processing at higher levels of the central auditory system most relevant to receptive communication. In brief, PPI offers an index of stimulus parameters that are behaviourally detectable, and while simple PPI is brainstem and mid-brain mediated, the use of complex acoustic cues clearly engages auditory cortex76. The engagement of cortical/behavioural thresholds is crucial to an ethologically-relevant model of receptive communicative processing. The ability to suppress an acoustic startle response (ASR; an involuntary, reflexive response to an unexpected auditory stimulus [startle eliciting stimulus (SES); 105 dB, 50 ms, broadband white noise burst (1–10 kHz)]) was measured. Subjects were placed on cell-loaded platforms (Med Associates, St. Albans, VT), and presented with varying auditory stimuli generated via RPvdsEx software and a RZ6 multifunction processor (Tucker Davis Technologies, Alachua, FL). Subject motor reflex responses were recorded via a Biopac MP150 acquisition system and Acqknowledge 4.1 software (Biopac Systems, Goleta, CA) connected to the load cell platforms. Tasks are detailed below and included detection of simple tones (15 or 40 kHz) in silence; and of deviant tones (variable duration) in a pure-frequency background (Embedded Tone: 10.5 or 40 kHz background tone, 5.6 or 35 kHz cue tone; Pitch Discrimination: 10.5 or 40.5 kHz tone ± 75 Hz or 8 kHz cue tone). Tone frequencies were determined based on low and high-frequency bounds of the mouse audiogram (~2–50 KHz)77. Testing began at postnatal (P) day 65 and continued to P114. Normal Single Tone consisted of 104 trials conducted over one day, where Embedded Tone and Pitch Discrimination each consisted of 300 trials and were conducted over 5 consecutive days. During cued trials, subjects were presented with an auditory cue (prepulse) 50 ms before the presentation of the SES (no cue presentation occurred during uncued trials). If the subject was able to detect the auditory cue, an attenuation (or reduction) of their ASR was expected relative to their ASR during an uncued trial. If the auditory cue was not detected, the response was expected to equate to an uncued trial. Quantification of this phenomenon was termed the "attenuation score" (ATT), which compared the mean amplitude of cued ASR to that of the uncued ASR for each subject, for each session condition.
$$\frac{{{\mathrm{Mean}}\,{\mathrm{cued}}\,{\mathrm{ASR}}}}{{{\mathrm{Mean}}\,{\mathrm{uncued}}\,{\mathrm{ASR}}}}\times100$$
Normal single tone
Subjects were first tested on Normal Single Tone (NST) to measure baseline prepulse inhibition, general auditory ability, and to rule out any underlying auditory processing impairments that might impede performance on subsequent auditory processing tasks (i.e. impaired reflex mechanics). Subjects were required to detect a simple single tone (50 ms, 75 dB) against a silent background. This cue was presented 50 ms before the SES on half of the trials (104 cued and uncued trials each, pseudorandom and evenly distributed), at inter-trial intervals (ITI) ranging from 16 s–24 s. Two versions of this task were developed – a 15 kHz version (cue; 50 ms, 75 dB, 15,000 Hz tone) and a 40 kHz version (cue; 50 ms, 75 dB, 40,000 Hz tone). All subjects were able to perform both versions of the task (15 kHz – P65; 40 kHz – P104). The frequency-matched NST score for each subject was used as a covariate in the analysis of further tasks, specifically to eliminate individual differences in PPI or hearing from subsequent auditory processing analyses.
Embedded tone
The variable duration Embedded Tone Task (EBT) consisted of 300 pseudorandom trials with ITIs ranging from 16–24 s. Subject's ability to detect a change in tone frequency from a constant pure-tone background was measured, and ATT scored were calculated. During cued trials, a single cue was presented 100 ms before the SES; for uncued trials, the "cue" was presented 0 ms before the SES (i.e. no cue). Three versions of this task were used: (1) a long-duration EBT task, where the cue duration ranged from 0 ms to 100 ms (cue; 75 dB, 5600 Hz tone & pure-tone background; 75 dB, 10,500 Hz tone); (2) a short-duration EBT task, where the cue duration ranged from 0 ms to 10 ms (cue; 75 dB, 5600 Hz tone and pure-tone background; 75 dB, 10,500 Hz tone); (3) an ultrasonic long-duration EBT task where the cue duration ranged from 0 ms to 100 ms (cue; 75 dB, 35,000 Hz tone & pure-tone background; 75 dB, 40,000 Hz tone). This combination of frequencies and temporal durations was designed to capture the range of processing capacities, allowing us to test for genotype-specific differences in that range. Non-ultrasonic and ultrasonic versions of the task were necessary to determine any Genotype effects observed were frequency dependent. Both non-ultrasonic versions of the task were administered for five consecutive days, and the ultrasonic version of the task was administered for four consecutive days (P67–P78; P103–106).
Pitch discrimination
Pitch Discrimination (PD) testing assessed the subject's ability to detect subtle changes in pitch within a constant pure-tone background. Each testing session consisted of 300 pseudorandom trials, with an ITI ranging from 16 s to 24 s. During cued trials, the cue was presented for 300 ms, 100 ms before the SES. "Cues" presented during uncued trials were presented at the same frequency as the pure-tone background. Two versions of this task were used for this study: (1) PD task where the cue frequency deviated 5–75 Hz above or below a 10,500 Hz pure-tone background (cue: 300 ms, 75 dB tone and pure-tone background: 10,500 Hz tone); and (2) ultrasonic PD task where the cue frequency deviated 5–75 Hz above or below a 40,500 Hz pure-tone background (cue: 300 ms, 75 dB tone & pure-tone background: 40,500 Hz tone). A non-ultrasonic PD task was administered for five consecutive days, and an ultrasonic PD task was administered for three consecutive days.
Ultrasonic vocalizations (USVs; P115–P120)
Following assessment of auditory processing ability, ultrasonic vocalizations (USVs) were recorded and analyzed using methods adapted from Chabout et al.78. Using WT female homecage bedding and urine collected 5 days prior to testing, a single experimental male mouse was placed in a standard Plexiglass tub with a single novel WT female mouse and allowed to freely interact for 5 min. In this setting, a male mouse will vocalize while the female does not, such that recoded calls can be attributed to the male. A Brüel & Kjær Type 4954-B microphone (Brüel & Kjær, Nærum, Denmark), connected to a RME Fireface UC audio interface (RME Audio, Haimhausen, Germany), was placed 5 cm above the top of the Plexiglass tub. USVs were recorded at 192,000 Hz using DIGICheck 5.92 (RME Audio, Haimhausen, Germany) to ensure all USVs were captured. Following USV recording, sound files (.wav) were analyzed in MATLAB (MathWorks) using MUPET (Mouse Ultrasonic Profile ExTraction79). Syllables in the range of 35,000 Hz to 110,000 Hz, and duration between 8 ms to 200 ms, were analyzed. If syllables occurred less than 5 ms apart, they were excluded from analyses. Following these parameters, a syllable repertoire was generated, illustrating 40 unique syllables (Supplementary Fig. 1). These 40 unique syllables were then assigned to one of ten potential syllable groups, as defined by Heckman et al.80. Eight syllable categories were created; Short, Down-FM, Up-FM, Chevron, Flat, 1-Freq Step, Noisy, and Complex80. The mean frequency (kHz) of each syllable was exported from MUPET and used for statistical analyses. Since comparable Genotype effects were seen on all call types, only the mean frequency shift is reported in the text (Fig. 3)—these USV frequency means collapse across all eight call types.
Statistics and reproducibility (genomic analyses)
The ALPSAC replication set included 7141 children (3615M:3526F), providing 96% power to detect a variant that explains 0.5% of the trait variance at a Bonferroni-corrected alpha level of 7.87 × 10−5. The final UK10K replication set included 1681 individuals (806M:875F) providing 81% power to detect a variant that explains 1% of the trait variance at a Bonferroni-corrected alpha level of 0.0033.
Gene-based analyses were performed within the UK10K dataset. These analyses employed the SKAT test in RVTESTS67 and considered five traits (as detailed above) using three SNP selection thresholds (all variants, rare variants (MAF ≤ 1%) common variants (MAF ≥ 5%)), yielding a Bonferroni significance threshold of P = 0.0033 at an alpha level of 0.05.
SNP-based analyses were performed within the ALSPAC dataset. These analyses employed tests of allelic association within PLINK42 using a linear model of regression for quantitative traits and a logistic model for discrete traits. Five phenotypes were analyzed across 127 SNPs, yielding a Bonferroni significance threshold of P = 7.87 × 10−5 at an alpha level of 0.05.
Gene-environment interaction effects were modeled within the ALSPAC dataset at the gene and genome-wide level. These analyses used PLINK42, which employs a linear regression model. Gene-level analyses were performed for 12 SNPs across the 5′ region of the USH2A gene and comprised of three language outcome measures (Early vocabulary (vocab), nonword repetition (NWR) and DLD status) and one interaction factor (Low-frequency hearing threshold (MinLow)) yielding a Bonferroni significance threshold of P = 0.0014 at an alpha level of 0.05. At the genome level, a single outcome measure (Early vocabulary (vocab)) was modelled with a single interaction factor (Low-frequency hearing threshold (MinLow)) for 488205 autosomal SNPs yielding a Bonferroni significance threshold of P = 1.02 × 10−7 at an alpha level of 0.05.
Gene ontology classes were analyzed for over-representations using a Fisher exact test with FDR multiple test correction.
Statistics and reproducibility (mouse analyses)
Normal Single Tone attenuation scores were analyzed using a one-way analysis of variance (ANOVA) comparing WT, HT, and Ush2a KO performance. To account for individual variation in baseline prepulse inhibition and hearing, NST was used a covariate for subsequent statistical analyses (NST 15 kHz was used as a covariate for all non-ultrasonic auditory tasks; NST 40 kHz was used as a covariate for all ultrasonic auditory tasks). EBT and PD tasks were analyzed using a mixed factorial design. Differences in ATT scores for non-ultrasonic EBT 100 and EBT 10 were conducted using a 3 × 5 × 9 repeated measures ANCOVA, with Genotype (three levels; WT, HT, Ush2a KO) as the between-subjects variable, and Day (five levels) and cue Duration (nine levels) as the within-subjects variables. Ultrasonic EBT 100 data were analyzed using a 3 × 4 × 5 repeated measures ANCOVA with Genotype (three levels) as the between-subjects variable, and Day (four levels) and cue Duration (five levels) as the within-subjects variables. For Pitch Discrimination, a 3 × 5 × 9 and a 3 × 3 × 5 (for non-ultrasonic PD and ultrasonic PD, respectively) repeated measures ANCOVA was used to determine ATT differences, where Genotype (three levels) was the between-subject variables and Day (five levels, three levels) and Frequency (nine levels, five levels) were the within-subject effects. Statistical analyses were completed used SPSS 24 with an alpha criterion of 0.05.
For the analysis of ultrasonic vocalizations, the overall mean frequency (collapsed across syllable category) was analyzed using a one-way analysis of variance (ANOVA) comparing WT, HT and Ush2a KO scores (Fig. 3).
All animal behavioural testing were performed blind to genotype.
Ethical approval for the discovery family was provided by University of London & St George's University Hospitals. All members provided informed consent/assent of investigation. Ethical approval for ALSPAC was obtained from the ALSPAC Ethics and Law Committee and the Local Research Ethics Committees (http://www.bristol.ac.uk/alspac/researchers/research-ethics/). All animal procedures conformed to the Guide for the Care and Use of Laboratory Animals and were approved by the University of Connecticut Institute for Animal Care and Use Committee (IACUC). The current animal study design adheres to the ARRIVE guidelines81.
All shared variants found in the discovery family are provided in Supplementary Dataset 1. ALSPAC and UK10K SNP and sequence data are available upon application as outlined at http://www.bristol.ac.uk/alspac/researchers/access/. The ALSPAC website additionally contains details of all the data that is available through a fully searchable data dictionary and variable search tool (http://www.bristol.ac.uk/alspac/researchers/our-data/). Auditory processing and USV data from murine behavioral testing are provided in Supplementary Datasets 3 and 4, respectively.
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We thank Maria Bitner-Glindzicz for her suggestions in the early stages of this work. This manuscript is dedicated to her. We also thank Amanda Hall for her advice regarding hearing phenotypes. The work of the Newbury Lab is currently funded by Oxford Brookes University, the Leverhulme Trust and the Economic and Social Research Council. This work was, in part, completed while Dianne Newbury was at the Wellcome Trust Centre for Human Genetics, Oxford as an MRC Career Development Fellow (G1000569/1). We are extremely grateful to all the families who took part in this study, the midwives for their help in recruiting them, and the whole ALSPAC team, which includes interviewers, computer and laboratory technicians, clerical workers, research scientists, volunteers, managers, receptionists and nurses. The UK Medical Research Council and Wellcome (Grant ref: 102215/2/13/2) and the University of Bristol provide core support for ALSPAC. This publication is the work of the authors and Dianne Newbury will serve as a guarantor for the contents of this paper. A comprehensive list of grants funding is available on the ALSPAC website (http://www.bristol.ac.uk/alspac/external/documents/grant-acknowledgements.pdf). Whole-genome sequencing of the ALSPAC samples was performed as part of the UK10K consortium (a full list of investigators who contributed to the generation of the data is available from www.UK10K.org.uk). ALSPAC GWAS data was generated by Sample Logistics and Genotyping Facilities at Wellcome Sanger Institute and LabCorp (Laboratory Corporation of America) using support from 23andMe. We thank the WGS500 (see the Supplementary Information for a list of consortium members, including co-author Jenny Taylor) through the High-Throughput Genomics Group at the Wellcome Trust Centre for Human for the generation of the sequencing and genotyping data in the discovery family. The WGS500 project was funded by a Wellcome Trust Core Award (090532/Z/09/Z, Peter Donnelly) and a Medical Research Council Hub grant (G0900747 91070, Peter Donnelly), the NIHR Biomedical Research Centre Oxford, the UK Department of Health's NIHR Biomedical Research Centres funding scheme and Illumina. Animal work was supported by funding from the University of Connecticut Murine Behavioral Neurogenetics Facility, the Connecticut Institute for Brain and Cognitive Sciences (IBACS), and Science of Learning & Art of Communication (NSF Grant DGE-1747486).
Department of Psychological Science/Behavioral Neuroscience, University of Connecticut, Storrs, CT, USA
Peter A. Perrino, Amanda R. Rendall, Alexzandrea N. Buscarello & R. Holly Fitch
UConn Institute of Brain and Cognitive Sciences; UConn Murine Behavioral Neurogenetics Facility, Storrs, CT, USA
Faculty of Health and Life Sciences, Oxford Brookes University, Oxford, OX3 0BP, UK
Lidiya Talbot, Hayley S. Mountford & Dianne F. Newbury
Wellcome Trust Centre for Human Genetics, Roosevelt Drive, Headington, Oxford, OX3 7BN, UK
Rose Kirkland & Jenny Taylor
School of Veterinary Medicine and Science, University of Nottingham, Sutton Bonington Campus, Leicestershire, LE12 5RD, UK
Rose Kirkland
Population Health Sciences, Bristol Medical School, University of Bristol, Bristol, BS8 2BN, UK
NIHR Biomedical Research Centre, John Radcliffe Hospital, Headley Way, Headington, Oxford, OX3 9DU, UK
Institute of Molecular and Clinical Sciences, St George's, University of London & St George's University Hospitals NHS Foundation Trust, London, UK
Nayana Lahiri & Anand Saggar
Peter A. Perrino
Lidiya Talbot
Amanda R. Rendall
Hayley S. Mountford
Alexzandrea N. Buscarello
Nayana Lahiri
Anand Saggar
R. Holly Fitch
Dianne F. Newbury
WGS500 Consortium
N.L. and A.S. ascertained and assessed the discovery family. R.K., J.T. and D.F.N. generated and analyzed whole-genome and SNP data in the discovery family. A.H. was the ALSPAC data buddy for this project and compiled and verified all ALSPAC and UK10K datasets for analysis. L.T., H.S.M. and D.F.N. analyzed the ALSPAC and UK10K datasets. P.A.P., A.R.R., A.N.B. and R.H.F. were responsible for all mouse rearing, care, phenotyping and data analysis at the University of Connecticut Murine Behavioral Neurogenetics Facility. P.A.P., L.N., H.S.M., R.H.F. and D.F.N. were responsible for drafting this manuscript. All authors reviewed and agreed the manuscript content. R.H.F. and D.F.N. act as corresponding authors for the murine and Human work respectively.
Correspondence to R. Holly Fitch or Dianne F. Newbury.
Perrino, P.A., Talbot, L., Kirkland, R. et al. Multi-level evidence of an allelic hierarchy of USH2A variants in hearing, auditory processing and speech/language outcomes. Commun Biol 3, 180 (2020). https://doi.org/10.1038/s42003-020-0885-5
Quantitative genome-wide association analyses of receptive language in the Danish High Risk and Resilience Study
Ron Nudel
, Camilla A. J. Christiani
, Jessica Ohland
, Md Jamal Uddin
, Nicoline Hemager
, Ditte Ellersgaard
, Katrine S. Spang
, Birgitte K. Burton
, Aja N. Greve
, Ditte L. Gantriis
, Jonas Bybjerg-Grauholm
, Jens Richardt M. Jepsen
, Anne A. E. Thorup
, Ole Mors
, Thomas Werge
& Merete Nordentoft
BMC Neuroscience (2020) | CommonCrawl |
\begin{document}
\title{\Large On a subset sums problem of Chen and Wu }\author{\large Min Tang\thanks{Corresponding author. This work was supported by the National Natural Science Foundation of China(Grant No. 11971033) and top talents project of Anhui Department of Education(Grant No. gxbjZD05).} and Hongwei Xu} \date{} \maketitle
\vskip -3cm \begin{center} \vskip -1cm { \small \begin{center}
School of Mathematics and Statistics, Anhui Normal University \end{center} \begin{center} Wuhu 241002, PR China \end{center}} \end{center}
{\bf Abstract:} For a set $A$, let $P(A)$ be the set of all finite subset sums of $A$. We prove that if a sequence $B=\{11\leq b_1<b_2<\cdots\}$ satisfies $b_2=3b_1+5$, $b_3=3b_2+2$ and $b_{n+1}=3b_n+4b_{n-1}$ for all $n\geq 3$, then there is a sequence of positive integers $A=\{a_1<a_2<\cdots\}$ such that
$P(A)=\mathbb{N}\setminus B$. This result shows that the answer to the problem of Chen and Wu [`The inverse problem on subset sums', European. J. Combin. 34(2013), 841-845] is negative.
{\bf Keywords:} subsetsum; completement; representation problem
2020 Mathematics Subject Classification: 11B13\vskip8mm
\section{Introduction} Let $\mathbb{N}$ be the set of all nonnegative integers. For a sequence of integers $A=\{a_1<a_2<\cdots\}$, let $$P(A)=\left\{\sum \varepsilon_ia_i: a_i\in A, \varepsilon_i=0\text{ or }1, \sum \varepsilon_i<\infty\right\}.$$ Here $0\in P(A)$.
In 1970, S. A. Burr \cite{Burr} asked the following question: which subsets $S$ of $\mathbb{N}$ are equal to $P(A)$ for some $A$? Burr showed the following result (unpublished):
\noindent{\bf Theorem A} (\cite{Burr}). {\it Let $B=\{4\leq b_1<b_2<\cdots\}$ be a sequence of integers for which $b_{n+1}\geq b_n^2$ for $n=1,2,\ldots$. Then there exists $A=\{a_1<a_2<\cdots\}$ for which
$P(A)=\mathbb{N}\setminus B$.}
Burr \cite{Burr} ever mentioned that if $B$ grows sufficiently rapidly, then there exists a sequence $A$ such that $P(A)=\mathbb{N}\setminus B$.
It is natural to ask how slow can sequence $B$ grow. In 1996, Hegyv\'{a}ri \cite{Hegy} improved Burr's result:
\noindent{\bf Theorem B} (\cite{Hegy}, Theorem 1). {\it Let $B=\{7\leq b_1<b_2<\cdots\}$ be a sequence of integers. Suppose that for every $n$, $b_{n+1}\geq 5b_n$. Then there exists a sequence of integers $A=\{a_1<a_2<\cdots\}$ for which
$P(A)=\mathbb{N}\setminus B$.}
In 2012, Chen and Fang \cite{chen2012} precisely extended Hegyv\'{a}ri's result related to Burr¡¯s question:
\noindent{\bf Theorem C} (\cite{chen2012}, Theorem 1). {\it Let $B=\{b_1<b_2<\cdots\}$ be a sequence of integers with $b_1\in\{4,7,8\}\cup \{b: b\geq 11, b\in \mathbb{N}\}$ and $b_{n+1}\geq 3b_n+5$ for all $n\geq 1$. Then there exists a sequence of positive integers $A=\{a_1<a_2<\cdots\}$ for which
$P(A)=\mathbb{N}\setminus B$.}
\noindent{\bf Theorem D} (\cite{chen2012}, Theorem 2). {\it Let $B=\{b_1<b_2<\cdots\}$ be a sequence of positive integers with $b_1\in\{3,5,6,9,10\}$ or $b_2=3b_1+4$ or $b_1=1$, $b_2=9$ or $b_1=2$, $b_2=15$. Then there is no sequence of positive integers $A=\{a_1<a_2<\cdots\}$ for which
$P(A)=\mathbb{N}\setminus B$.}
In 2013, Chen and Wu \cite{chen2013} further improved Theorem C.
\noindent{\bf Theorem E} (\cite{chen2013}, Theorem 1). {\it If $B=\{b_1<b_2<\cdots\}$ is a sequence of integers with $b_1\in\{4,7,8\}\cup \{b: b\geq 11, b\in \mathbb{N}\}$, $b_2\geq 3b_1+5$, $b_3\geq 3b_2+3$ and $b_{n+1}> 3b_n-b_{n-2}$ for all $n\geq 3$, then there exists a sequence of positive integers $A=\{a_1<a_2<\cdots\}$ such that
$P(A)=\mathbb{N}\setminus B$ and
$$P(A_s)=[0,2b_s]\setminus \{b_1,\ldots, b_s, 2b_s-b_{s-1}, \ldots,2b_s-b_1\},$$
where $A_s=A\cap [0, b_s-b_{s-1}]$ for all $s\geq 2$.}
Moreover, Chen and Wu \cite{chen2013} posed the following problem:
\noindent{\bf Problem 1} (\cite{chen2013}, Problem 1). {\it Let $B=\{b_1<b_2<\cdots\}$ be a sequence of positive integers. Let $d_1=10$, $d_2=3b_1+4$, $d_3=3b_2+2$ and $d_{n+1}=3b_n-b_{n-2}(n\geq 3)$. If $b_m=d_m$ for some $m\geq 3$ and $b_n>d_n$ for all $n\neq m$. Is it true there is no sequence of positive integers $A=\{a_1<a_2<\cdots\}$ with $P(A)=\mathbb{N}\setminus B$?}
In 2013, Wu \cite{Wu2013} gave a segment version of Theorem E. Recently, Fang and Fang \cite{Fang2019} determined the critical value for $b_3$ such that there exists an infinite sequence of positive integers $A$ for which
$P(A)=\mathbb{N}\setminus B$(under the condition $b_1>1$ and $b_2=3b_1+5$).
In this paper, we obtain the following results:
\begin{thm}\label{thm0} Let $B=\{11\leq b_1<b_2<\cdots\}$ be a sequence of integers with $b_2=3b_1+5$, $b_3=3b_2+2$ and $b_{n+1}=3b_n+4b_{n-1}$ for all $n\geq 3$. Then there exists a sequence of positive integers $A=\{a_1<a_2<\cdots\}$ such that
$P(A)=\mathbb{N}\setminus B$. \end{thm}
\noindent{\bf Remark 1} Theorem \ref{thm0} shows that the answer to Problem 1 is negative for $m=3$.
\begin{thm}\label{thm1} Let $B=\{3\leq b_1<b_2<\cdots\}$ be a sequence of integers. If $b_2=3b_1+3$ or $b_2=3b_1+2$, then there is no sequence of positive integers $A=\{a_1<a_2<\cdots\}$ such that
$P(A)=\mathbb{N}\setminus B$. \end{thm}
\begin{thm}\label{thm2} Let $B=\{b_1<b_2<\cdots\}$ be an infinite arithmetic progression with common difference $d$ and $b_1\in\{4,7,8\}\cup \{b: b\geq 11, b\in \mathbb{N}\}$. If $b_1+2\leq d\leq 2b_1+1$, then there exists a sequence of positive integers $A=\{a_1<a_2<\cdots\}$ such that
$P(A)=\mathbb{N}\setminus B$. \end{thm} \noindent{\bf Remark 2} Theorem \ref{thm2} shows that there exists a sequence of positive integers $A=\{a_1<a_2<\cdots\}$ such that
$P(A)=\mathbb{N}\setminus B$ for a given special common difference sequence with $2b_1+2\leq b_2\leq 3b_1+1$. Theorem \ref{thm1} and Theorem \ref{thm2} further enrich our understanding of Burr's problem.
\section{Lemma}
\begin{lem}\label{lem2} Let $A=\{a_1<a_2<\cdots\}$ and $B=\{b_1<b_2<\cdots\}$ be two sequences of positive integers with $b_1>1$ such that $P(A)=\mathbb{N}\setminus B$. If $P(\{a_1,\ldots,a_k\})=[0,b_1-1]$ and $b_2\geq 2b_1+2$, then $a_{k+1}=b_1+1$, $a_{k+2}\leq 2b_1+1$ and $$P(\{a_1,\ldots,a_{k+1}\})=[0, 2b_1]\setminus \{b_1\},$$ $$P(\{a_1,\ldots,a_{k+2}\})=[0, a_{k+2}+2b_1] \setminus \{b_1, a_{k+2}+b_1\}.$$ \end{lem} \begin{proof} Since $$P(\{a_1,\ldots,a_k\})=[0, b_1-1],$$ $b_1+1\in P(A)$ and $b_1\not\in P(A)$, we have $a_{k+1}=b_1+1$. Hence $$P(\{a_1,\ldots,a_{k+1}\})=[0, b_1-1]\cup [a_{k+1},a_{k+1}+b_1-1]=[0, 2b_1]\setminus \{b_1\},$$ and $$a_{k+2}+P(\{a_1,\ldots,a_{k+1}\})=[a_{k+2}, a_{k+2}+2b_1]\setminus \{a_{k+2}+b_1\}.$$
If $a_{k+2}\geq 2b_1+2$, then $2b_1+1\not\in P(A)$ and $b_2=2b_1+1$, which contradicts with $b_2\geq 2b_1+2$. So $a_{k+2}\leq 2b_1+1$ and $$P(\{a_1,\ldots,a_{k+2}\})=[0, a_{k+2}+2b_1] \setminus \{b_1, a_{k+2}+b_1\}.$$
This completes the proof of Lemma \ref{lem2}. \end{proof}
\section{Proof of Theorem \ref{thm0}} First, we shall prove that the following result:
\noindent{\bf Fact I} {\it There exists a set sequence $\{A_k\}_{k=3}^{\infty}$ such that
\noindent(i) $A_k=A_{k-1}\cup \{b_{k-1}+2b_{k-3}, b_{k-1}+b_{k-2}-b_{k-3}, b_{k-1}+2b_{k-2}-b_{k-3}\}$ for $k\geq 4$;
\noindent(ii) $P(A_k)=[0, b_k+b_{k-1}]\setminus \{b_1,\ldots,b_k, b_k+b_{k-1}-b_i: \; i=1,\ldots,k-2\}$ for $k\geq 4$;
\noindent(iii) $b_{k}=3b_{k-1}+4b_{k-2}$ for $k\geq 4$.} \vskip 3mm By the proof of [2, Theorem 1], there exists $A_1=\{a_1<a_2<\ldots<a_k\}\subseteq [1,b_1-1]$ such that
$$P(A_1)=[0,b_1-1].$$ By Lemma \ref{lem2}, we have $a_{k+1}=b_1+1$ and $$P(A_1\cup \{b_1+1\})=[0, 2b_1]\setminus \{b_1\}.$$ Let $a_{k+2}=b_1+2,\; a_{k+3}=b_1+3.$ Then $$P(\{a_1,\ldots,a_{k+3}\})=[0, b_1+b_2] \setminus \{b_1, b_2\}.$$ Let $a_{k+4}=b_1+b_2$, $a_{k+5}=2b_2-2b_1+2$. Then $$b_1+b_2+P(\{a_1,\ldots,a_{k+3}\})=[b_1+b_2,2b_1+2b_2]\backslash \{2b_1+b_2, b_1+2b_2\},$$ thus by $b_3=3b_2+2$, we have $$P(\{a_1,\ldots,a_{k+4}\})=[0,2b_1+2b_2]\backslash \{b_1,b_2,2b_1+b_2, b_1+2b_2\},$$ $$a_{k+5}+P(\{a_1,\ldots,a_{k+4}\})=[2b_2-2b_1+2,b_3+b_2]\setminus \mathcal{B}_0,$$ where $\mathcal{B}_0=\{2b_2-b_1+2,3b_2-2b_1+2,b_3, b_3+b_2-b_1\}.$
Write $$A_3=A_1\cup\{b_1+1, b_1+2,b_1+3\}\cup \{b_1+b_2, 2b_2-2b_1+2\}.$$ Since $$2b_2-2b_1+2<2b_1+b_2<2b_2-b_1+2<b_1+2b_2<3b_2-2b_1+2<2b_1+2b_2,$$
we have$$P(A_3)=[0, b_3+b_2] \setminus \{b_1, b_2,b_3,b_3+b_2-b_1\}.$$
Noting that $$\max A_3=2b_2-2b_1+2<b_3+2b_1<b_3+b_2-b_1<b_3+2b_2-b_1,$$and $$ b_3+b_2-b_1+P(A_3)=[b_3+b_2-b_1, 2b_3+2b_2-b_1]\setminus \mathcal{B}_{3,1},$$ where $$\mathcal{B}_{3,1}=\Big\{b_3+b_2, b_3+2b_2-b_1, 2b_3+b_2-b_1,2b_3+2b_2-2b_1\Big\}.$$ Thus $$P(A_3\cup\{b_3+b_2-b_1\})=[0,2b_3+2b_2-b_1]\setminus \mathcal{B}_{3,2},$$ where $$\mathcal{B}_{3,2}=\Big\{b_1,b_2, b_3, b_3+2b_2-b_1, 2b_3+b_2-b_1,2b_3+2b_2-2b_1\Big\}.$$
Similarly, we have $$P(A_3\cup\{b_3+b_2-b_1, b_3+2b_2-b_1\})=[0,3b_3+4b_2-2b_1]\setminus \mathcal{B}_{3,4},$$ where $$\mathcal{B}_{3,4}=\Big\{b_1,b_2, b_3, 2b_3+4b_2-2b_1, 3b_3+3b_2-2b_1,3b_3+4b_2-3b_1\Big\}.$$
Noting that $$b_3+2b_1+P(A_3\cup\{b_3+b_2-b_1,b_3+2b_2-b_1\})=[b_3+2b_1, 4b_3+4b_2]\setminus \mathcal{B}_{3,5},$$ where $$\mathcal{B}_{3,5}=\Big\{b_3+3b_1, b_3+b_2+2b_1,2b_3+2b_1, 3b_3+4b_2,4b_3+3b_2, 4b_3+4b_2-b_1\Big\}.$$ Let \begin{equation}\label{eq1}b_4=3b_3+4b_2,\end{equation} \begin{equation}\label{eq2}A_4=A_3\cup \{b_3+b_2-b_1,b_3+2b_2-b_1,b_3+2b_1\}.\end{equation} Then \begin{equation}\label{eq3}P(A_4)=[0, b_4+b_3]\setminus \{b_1,b_2, b_3,b_4, b_4+b_3-b_2, b_4+b_3-b_1\}.\end{equation}
By (\ref{eq1}), (\ref{eq2}), (\ref{eq3}), we know that Fact I is true for $k=4$. Suppose that Fact I is true for $k(\geq 4)$. Now we consider the case $k+1$.
Since $$A_{k}=A_{k-1}\cup \{b_{k-1}+b_{k-2}-b_{k-3},b_{k-1}+2b_{k-2}-b_{k-3},b_{k-1}+2b_{k-3}\},$$ $$P(A_k)=[0, b_k+b_{k-1}]\setminus \{b_1,\ldots,b_k, b_k+b_{k-1}-b_i: \; i=1,\ldots,k-2\},$$ and $$\max A_k=b_{k-1}+2b_{k-2}-b_{k-3}<b_k+2b_{k-2}<b_k+b_{k-1}-b_{k-2}<b_k+2b_{k-1}-b_{k-2},$$ we have $$b_k+b_{k-1}-b_{k-2}+P(A_k)=[b_k+b_{k-1}-b_{k-2}, 2b_k+2b_{k-1}-b_{k-2}] \setminus \mathcal{B}_{k,1},$$ where $$\mathcal{B}_{k,1}=\Big\{b_k+b_{k-1}-b_{k-2}+b_i,2b_k+2b_{k-1}-b_{k-2}-b_i: \; i=1,\ldots,k-1\Big\}.$$ Noting that $$\begin{array}{ll}&{\bf b_k+b_{k-1}-b_{k-2}}<b_k+b_{k-1}-b_{k-2}+b_1<\cdots<b_k+b_{k-1}-b_{k-2}+b_{k-3}\\ <&{\bf b_k+b_{k-1}-b_{k-3}}<\cdots<b_k+b_{k-1}-b_1\\ <&{\bf b_k+b_{k-1}=b_k+b_{k-1}-b_{k-2}+b_{k-2}},\end{array}$$ we have $$P(A_k\cup \{b_k+b_{k-1}-b_{k-2}\})=[0, 2b_k+2b_{k-1}-b_{k-2}] \setminus \mathcal{B}_{k,2},$$ where $$\mathcal{B}_{k,2}=\Big\{b_1,\ldots,b_k,2b_k+2b_{k-1}-b_{k-2}-b_i: i=1,\ldots,k\Big\}.$$ Noting that $$\begin{array}{ll}&b_k+2b_{k-1}-b_{k-2}+P(A_k\cup \{b_k+b_{k-1}-b_{k-2}\})\\ =&[b_k+2b_{k-1}-b_{k-2}, 3b_k+4b_{k-1}-2b_{k-2}] \setminus \mathcal{B}_{k,3},\end{array}$$ where $$\mathcal{B}_{k,3}=\Big\{b_k+2b_{k-1}-b_{k-2}+b_i,3b_k+4b_{k-1}-2b_{k-2}-b_i: \; i=1,\ldots,k\Big\}.$$ Noting that $$\begin{array}{ll}&{\bf b_k+2b_{k-1}-b_{k-2}}<b_k+2b_{k-1}-b_{k-2}+b_1<\cdots<b_k+3b_{k-1}-b_{k-2}\\ <&{\bf 2b_k+b_{k-1}-b_{k-2}}<\cdots<2b_k+2b_{k-1}-b_{k-2}-b_1\\ <&{\bf 2b_k+2b_{k-1}-b_{k-2}},\end{array}$$ we have $$\begin{array}{ll}&P(A_k\cup \{b_k+b_{k-1}-b_{k-2},b_k+2b_{k-1}-b_{k-2}\})\\=&[0, 3b_k+4b_{k-1}-2b_{k-2}] \setminus \mathcal{B}_{k,4},\end{array}$$ where $$\mathcal{B}_{k,4}=\Big\{b_1,\ldots, b_k, 3b_k+4b_{k-1}-2b_{k-2}-b_i: i=1,\ldots,k\Big\}.$$ Noting that $$\begin{array}{ll}&b_k+2b_{k-2}+P(A_k\cup \{b_k+b_{k-1}-b_{k-2},b_k+2b_{k-1}-b_{k-2}\})\\ =&[b_k+2b_{k-2}, 4b_k+4b_{k-1}] \setminus \mathcal{B}_{k,5},\end{array}$$ where $$\mathcal{B}_{k,5}=\Big\{b_k+2b_{k-2}+b_i,4b_k+4b_{k-1}-b_i: \; i=1,\ldots,k\Big\}.$$ Noting that $$\begin{array}{ll}&{\bf b_k+2b_{k-2}}<b_k+2b_{k-2}+b_1<\cdots<2b_k+2b_{k-2}\\ <&{\bf 2b_k+4b_{k-1}-2b_{k-2}}<\cdots<{\bf 3b_k+4b_{k-1}-2b_{k-2}-b_1},\end{array}$$ we have $$P(A_k\cup \{b_k+b_{k-1}-b_{k-2},b_k+2b_{k-1}-b_{k-2},b_k+2b_{k-2}\})=[0, 4b_k+4b_{k-1}] \setminus \mathcal{B}_{k,6},$$ where $$\mathcal{B}_{k,6}=\Big\{b_1,\ldots, b_k, 4b_k+4b_{k-1}-b_i: \; i=1,\ldots,k\Big\}.$$
Write $$b_{k+1}=3b_k+4b_{k-1},$$ $$A_{k+1}=A_{k}\cup \{b_k+b_{k-1}-b_{k-2},b_k+2b_{k-1}-b_{k-2},b_k+2b_{k-2}\},$$ we have $$P(A_{k+1})=[0,b_{k+1}+b_k]\setminus \{b_1,\ldots,b_{k+1},b_{k+1}+b_k-b_i: \; i=1,\ldots,k-1\}.$$ Second, let $$A=\bigcup_{k=4}^{\infty}A_k.$$ If $n\in P(A)$, let $n<b_k+2b_{k-2}$, then noting that $$A\setminus A_i\subseteq [b_{k}+2b_{k-2},+\infty)$$ for all $i\geq k$, we have $n\in P(A_{k})$.
By Fact I (ii) we have \begin{equation}\label{eq4}n\not\in \{b_1,\ldots,b_k,b_k+b_{k-1}-b_i: \; i=1,\ldots,k-2\}.\end{equation} If $n\leq b_k$, then by (\ref{eq4}) we have $n\not\in B$. If $b_k<n<b_k+2b_{k-2}$, then by $b_k<n<b_{k+1}$, we have $n\not\in B$.
That is, $n\in \mathbb{N}\setminus B$.
Conversely, if $n'\in \mathbb{N}\setminus B$, then $n'\not\in B$, let $n'<b_{k'}$, we have $$n'\not\in \{b_1,\ldots,b_{k'}, b_{k'}+b_{k'-1}-b_i: \; i=1,\ldots,k'-2\}.$$ By Fact I (ii) we have $n'\in P(A_{k'})$. So $n'\in P(A)$.
Hence $P(A)=\mathbb{N}\setminus B$.
This completes the proof of Theorem \ref{thm0}.
\section{Proof of Theorem \ref{thm1}} By Theorem D, we know that if $b_1\in \{3,5,6,9,10\}$, then there is no sequence of positive integers $A=\{a_1<a_2<\cdots\}$ for which
$P(A)=\mathbb{N}\setminus B$.
Now, it is sufficient to consider positive integers sequence $B=\{1<b_1<b_2<\cdots\}$ with $b_1\in\{4,7,8\}\cup \{b: b\geq 11, b\in \mathbb{N}\}$.
Assume that there exists a sequence $A=\{a_1<a_2<\cdots\}$ of positive integers such that
$P(A)=\mathbb{N}\setminus B$. By the proof of [2, Theorem 1], there exists $A_1=\{a_1<a_2<\ldots<a_k\}\subseteq [1,b_1-1]$ such that
$$P(A_1)=[0,b_1-1].$$ By Lemma \ref{lem2}, we have $$P(\{a_1,\ldots,a_{k+2}\})=[0, a_{k+2}+2b_1] \setminus \{b_1, a_{k+2}+b_1\}.$$
We divide into two cases:
{\bf Case 1.} $b_2=3b_1+3$. If $a_{k+2}\geq b_1+3$, then $b_2\in [0, a_{k+2}+2b_1]$. Since $b_2\not\in P(\{a_1,\ldots,a_{k+2}\})$, we have $b_2=a_{k+2}+b_1$. Thus $$a_{k+2}=b_2-b_1=2b_1+3>2b_1+1.$$ By Lemma \ref{lem2}, it is impossible. Thus $a_{k+2}=b_1+2$ and $$P(\{a_1,\ldots,a_{k+2}\})=[0,3b_1+2]\setminus \{b_1,2b_1+2\}.$$ Hence $$a_{k+3}+P(\{a_1,\ldots,a_{k+2}\})=[a_{k+3}, a_{k+3}+3b_1+2]\setminus \{a_{k+3}+b_1, a_{k+3}+2b_1+2\}.$$
If $a_{k+3}\geq 2b_1+3$, then $2b_1+2\not\in P(A)$, thus $b_2=2b_1+2$, a contradiction. Hence $a_{k+3}\leq 2b_1+2$.
Since $a_{k+3}>a_{k+2}$, we have $a_{k+3}\geq b_1+3$, thus $b_1+a_{k+3}\neq 2b_1+2$ and $$P(\{a_1,\ldots,a_{k+3}\})=[0, a_{k+3}+3b_1+2]\setminus \{b_1, a_{k+3}+2b_1+2\}.$$ Since $b_2=3b_1+3\in [0, a_{k+3}+3b_1+2]$ and $b_2\not\in P(\{a_1,\ldots,a_{k+3}\})$, we have $$b_2=3b_1+3=a_{k+3}+2b_1+2\geq 3b_1+5,$$ a contradiction.
{\bf Case 2.} $b_2=3b_1+2$. Since $a_{k+2}\geq b_1+2$, then $b_2\in [0, a_{k+2}+2b_1]$. Since $b_2\not\in P(\{a_1,\ldots,a_{k+2}\})$, we have $b_2=a_{k+2}+b_1$. Thus $$a_{k+2}=b_2-b_1=2b_1+2>2b_1+1.$$ By Lemma \ref{lem2}, it is impossible.
This completes the proof of Theorem \ref{thm1}.
\section{Proof of Theorem \ref{thm2}}
First, we shall prove that the following result:
\noindent{\bf Fact II} {\it There exists a set sequence $\{A_k\}_{k=2}^{\infty}$ such that
\noindent(i) $A_2\subseteq A_3\subseteq\ldots$;
\noindent(ii) $P(A_k)=[0, 2b_1+(2^{k}-1)d]\setminus \{b_i: \; i=1,\ldots,2^k\}$.} \vskip 3mm For $b_1\in \{4,7,8\}\cup \{b: b\geq 11\}$, by the proof of [2, Theorem 1], there exists $A_1=\{a_1<a_2<\ldots<a_k\}\subseteq [1,b_1-1]$ such that
$$P(A_1)=[0,b_1-1].$$ By Lemma \ref{lem2}, we have $a_{k+2}\leq 2b_1+1$ and $$P(A_1\cup \{b_1+1\})=[0, 2b_1]\setminus \{b_1\}.$$
Since $a_{k+2}\leq 2b_1+1$ and $b_1+2\leq d\leq 2b_1+1$, we can choose $a_{k+2}=d$, thus $$P(A_1\cup \{b_1+1,d\})=[0, 2b_1+d]\setminus\{b_1,b_1+d\}.$$ Choose $a_{k+3}=2d$, then $$a_{k+3}+P(\{a_1,\ldots,a_{k+2}\})=[2d, 2b_1+3d]\setminus \{b_1+2d,b_1+3d\}.$$ Write $$A_2=A_1\cup \{b_1+1,d,2d\}.$$ Since $d\geq b_1+2>b_1$, we have $b_1+2d>2b_1+d$, thus $$P(A_2)=[0, 2b_1+3d]\setminus \{b_1,b_2,b_3,b_4\}.$$
We have proved that Fact II (ii) is true for $k=2$. Suppose that Fact II is true for $k(\geq 2)$. Now we consider the case $k+1$.
Since $$A_k=A_1\cup \{b_1+1,d, \ldots, 2^{k-1}d\},$$ $$P(A_k)=[0, 2b_1+(2^{k}-1)d]\setminus \{b_i: \; i=1,\ldots,2^k\},$$ we have $$2^{k}d+P(A_k)=[2^{k}d, 2b_1+(2^{k+1}-1)d]\setminus \{b_i+2^kd: \; i=1,\ldots,2^{k}\}.$$ Write $$A_{k+1}=A_{k}\cup \{2^{k}d\}.$$ Since $$b_i+2^{k}d=b_{2^k+i}, \quad i=1,\ldots,2^k,$$
we have $$P(A_{k+1})=[0, 2b_1+(2^{k+1}-1)d]\setminus \{b_i: \; i=1,\ldots,2^{k+1}\}.$$
Second, put $$A=\bigcup_{k=2}^{\infty}A_k.$$ If $n\in P(A)$, let $n\leq 2^{k-1}d$, then, by $$A\setminus A_i\subseteq [2^{k-1}d+1,+\infty)$$ for all $i\geq k$, we have $n\in P(A_{k})$. By Fact II (ii) we have $n\not\in \{b_1,\ldots,b_{2^k}\}$. Since $n\leq 2^{k-1}d<b_{2^k}$, we have $n\not\in B$. That is, $n\in \mathbb{N}\setminus B$.
Conversely, if $n'\in \mathbb{N}\setminus B$, then $n'\not\in B$, let $n'<b_{2^{k'}}$, we have $n'\not\in \{b_1,\ldots,b_{2^{k'}}\}$. By Fact II (ii) we have $n'\in P(A_{k'})$. So $n'\in P(A)$.
Hence $P(A)=\mathbb{N}\setminus B$.
This completes the proof of Theorem \ref{thm2}.
\end{document} | arXiv |
\begin{document}
\title{The Cauchy problem on large time for the Water Waves equations with large topography variations} \author{Mésognon-Gireau Benoît\footnote{UMR 8553 CNRS, Laboratoire de Mathématiques et Applications de l'Ecole Normale Supérieure, 75005 Paris, France. Email: [email protected]}} \date{} \maketitle
\begin{abstract} This paper shows that the long time existence of solutions to the Water Waves equations remains true for large amplitude topography variations in presence of surface tension. More precisely, the dimensionless equations depend strongly on three parameters $\epsilon,\mu,\beta$ measuring the amplitude of the waves, the shallowness and the amplitude of the bathymetric variations respectively. In \cite{alvarez}, the local existence of solutions to this problem is proved on a time interval of size $\frac{1}{\max (\beta,\epsilon)}$ and uniformly with respect to $\mu$. In presence of large bathymetric variations (typically $\beta \gg \epsilon$), the existence time is therefore considerably reduced. We remove here this restriction and prove the local existence on a time interval of size $\frac{1}{\epsilon}$ under the constraint that the surface tension parameter must be at the same order as the shallowness parameter $\mu$. We also show that the result of \cite{bresch_metivier} dealing with large bathymetric variations for the Shallow Water equations can be viewed as a particular endpoint case of our result.\end{abstract}
\section{Introduction} We recall here classical formulations of the Water Waves problem, with and without surface tension. We then shortly introduce the meaningful dimensionless parameters of this problem, and then state the local existence result proved by \cite{alvarez}. We discuss the dependance of the size of the time interval with respect to these parameters and then explain the strategy adopted in this paper to get an improved local existence result. \subsection{Formulations of the Water Waves problem} The Water Waves problem puts the motion of a fluid with a free surface into equations. We recall here two equivalent formulations of the Water Waves equations for an incompressible and irrotationnal fluid. We then introduce the surface tension, and recall a local existence result by \cite{alvarez}. \subsubsection{Free surface $d$-dimensional Euler equations} The motion, for an incompressible, inviscid and irrotationnal fluid occupying a domain $\Omega_t$ delimited below by a fixed bottom and above by a free surface is described by the following quantities : \\\\ - the velocity of the fluid $U=(V,w)$, where $V$ and $w$ are respectively the horizontal and vertical components \\ - the free top surface profile $\zeta$ \\ - the pressure $P.$ \\ All these functions depends on the time and space variables $t$ and $(X,z) \in\Omega_t$. There exists a function $b:\mathbb{R}^d\rightarrow \mathbb{R}$ such that the domain of the fluid at the time $t$ is given by $$\Omega_t = \lbrace (X,z)\in\mathbb{R}^{d+1},-H_0+ b(X) < z <\zeta(t,X)\rbrace,$$ where $H_0$ is the typical depth of the water. The unknowns $(U,\zeta,P)$ are governed by the Euler equations: \begin{align} \begin{cases} \partial_t V + (V\cdot\nabla + w\partial_z)V = - \nabla P \text{ in } \Omega_t\\ \mbox{\rm div}(U) = 0 \text{ in } \Omega_t\\ \mbox{\rm curl}(U) = 0 \text{ in } \Omega_t . \label{euler} \end{cases}\end{align}
In these equations, $\underline{V}$ and $\underline{w}$ are the horizontal and vertical components the velocity evaluated at the surface. We denote here $-ge_z$ the acceleration of gravity, where $e_z$ is the unit vector in the vertical direction. The notation $\nabla$ denotes here the gradient with respect to the horizontal variable $X$. \\
These equations are completed by boundary conditions :
\begin{align} \begin{cases} \partial_t \zeta +\underline{V}\cdot\nabla\zeta - \underline{w} = 0 \\ U\cdot n = 0 \text{ on } \lbrace z=-H_0+ b(X)\rbrace \\ P = P_{atm}\text{ on } \lbrace z=\zeta(X)\rbrace. \label{boundary_conditions} \end{cases} \end{align} The vector $n$ in the last equation stands for the normal upward vector at the bottom $(X,z=-H_0+b(X))$. We denote $P_{atm}$ the constant pressure of the atmosphere at the surface of the fluid. The first equation states the assumption that the fluid particles does not cross the surface, while the last equation states the assumption that they do not cross the bottom. The equations \eqref{euler} with boundary conditions \eqref{boundary_conditions} are commonly referred to the free surface Euler equations. \subsubsection{Craig-Sulem-Zakharov formulation} Since the fluid is by hypothesis irrotational, it derives from a scalar potential: $$U = \nabla_{X,z} \Phi.$$ Here, $\nabla_{X,z}$ denotes the three dimensional gradient with respect to both variables $X$ and $z$. Zakharov remarked in \cite{zakharov} that the free surface profile $\zeta$ and the potential at the surface $\psi = \Phi_{\vert z=\zeta}$ fully determine the motion of the fluid, and gave an Hamiltonian formulation of the problem. Later, Craig-Sulem, and Sulem (\cite{craigsulem1} and \cite{craigsulem2}) gave a formulation of the Water Waves equation involving the Dirichlet-Neumann operator. The following Hamiltonian system is equivalent (see \cite{david} and \cite{alazard} for more details) to the free surface Euler equations \eqref{euler} and \eqref{boundary_conditions}: \begin{align*}\begin{cases}
\displaystyle{\partial_t \zeta - G\psi = 0} \\
\displaystyle{\partial_t \psi + g\zeta + \vert\nabla\psi\vert^2 - \frac{(G\psi +\nabla\zeta\cdot\nabla\psi)^2}{2(1+\mid\nabla\zeta\mid^2)}=0 } \end{cases}\end{align*} where the unknown are $\zeta$ (free top profile) and $\psi$ (velocity potential at the surface) with $t$ as time variable and $X\in\mathbb{R}^d$ as space variable. The fixed bottom profile is $b$, and $G$ stands for the Dirichlet-Neumann operator, that is
\begin{equation*} G\psi = G[\zeta, b]\psi = \sqrt{1+ \modd{\nabla\zeta}} \partial_n \Phi_{\vert z=\zeta}, \end{equation*} where $\Phi$ stands for the potential, and solves Laplace equation with Neumann (at the bottom) and Dirichlet (at the surface) boundary conditions
\begin{align}\begin{cases} \Delta_{X,z} \Phi = 0 \quad \text{in } \mathbb{R}^d \times \lbrace -H_0+ b < z < \zeta\rbrace \\ \phi_{\vert z=\zeta} = \psi,\quad \partial_n \Phi_{\vert z=-H_0+ b} = 0 \label{dirichlet} \end{cases}\end{align} with the notation, for the normal derivative $$\partial_n \Phi_{\vert z=-H_0+b(X)} = \nabla_{X,z}\Phi(X,-H_0+b(X))\cdot n$$ where $n$ stands for the normal upward vector at the bottom $(X,-H_0+b(X))$. See also \cite{david} for more details. \\
\subsubsection{Water Waves with surface tension}
In the presence of surface tension, the only difference with the classical Euler problem \eqref{euler} is that the pressure condition at the surface is changed into $$P-P_{atm} = \sigma \kappa(\zeta) \text{ on } \lbrace z=\zeta(t,X)\rbrace,$$ where $\sigma$ denotes the surface tension coefficient, and $\kappa(\zeta)$ is the mean curvature at the surface $$\kappa(\zeta) = -\nabla\cdot(\frac{\nabla\zeta}{\sqrt{1+\vert\nabla\zeta\vert^2}}).$$ The Zakharov/Craig-Sulem formulation has therefore to be changed into
\begin{align}\begin{cases}
\displaystyle{\partial_t \zeta - G\psi = 0} \\
\displaystyle{\partial_t \psi + g\zeta + \vert\nabla\psi\vert^2 - \frac{(G\psi +\nabla\zeta\cdot\nabla\psi)^2}{2(1+\mid\nabla\zeta\mid^2)}=-\frac{\sigma}{\rho}\kappa(\zeta). } \label{ww_equation} \end{cases}\end{align}
\subsubsection{Dimensionless equations} Since the properties of the solutions depend strongly on the characteristics of the fluid, it is more convenient to non-dimensionalize the equations by introducing some characteristic lengths of the wave motion: \\\\ (1) The characteristic water depth $H_0$ \\ (2) The characteristic horizontal scale $L_x$ in the longitudinal direction \\ (3) The characteristic horizontal scale $L_y$ in the transverse direction (when $d=2$)\\ (4) The order of the free surface amplitude $a_{surf}$\\ (5) The order of bottom topography $a_{bott}$.\\
Let us then introduce the dimensionless variables: $$x'=\frac{x}{L_x},\quad y'=\frac{y}{L_y},\quad \zeta'=\frac{\zeta}{a_{surf}},\quad z'=\frac{z}{H_0},\quad b'=\frac{b}{a_{bott}},$$ and the dimensionless variables: $$t'=\frac{t}{t_0},\quad \Phi'=\frac{\Phi}{\Phi_0},$$ where $$t_0 = \frac{L_x}{\sqrt{gH_0 }},\quad \Phi_0 = \frac{a_{surf}}{H_0}L_x \sqrt{gH_0}.$$
After re scaling, four dimensionless parameters appear in the equation. They are
\begin{align*} \frac{a_{surf}}{H_0} = \epsilon, \quad \frac{H_0^2}{L_x^2} = \mu,\quad \frac{a_{bott}}{H_0} = \beta,\quad \frac{L_x}{L_y} = \gamma,\quad B_0=\frac{\rho gL_x^2}{\sigma}, \end{align*} where $\epsilon,\mu,\beta,\gamma$ are commonly referred respectively as "nonlinearity", "shallowness", "topography", "transversality" and "Bond" parameters.\\
For instance, the Zakharov-Craig-Sulem system (\ref{ww_equation}) becomes (see \cite{david} for more details) in dimensionless variables (we omit the "primes" for the sake of clarity):
\begin{align}\begin{cases}
\displaystyle{\partial_t \zeta - \frac{1}{\mu} G_{\mu,\gamma}[\epsilon\zeta,\beta b]\psi = 0 }\\
\displaystyle \partial_t \psi + \zeta + \frac{\epsilon}{2}\vert\nabla^{\gamma}\psi\vert^2 - \frac{\epsilon}{\mu}\frac{(G_{\mu,\gamma}[\epsilon\zeta,\beta b]\psi +\epsilon\mu\nabla^{\gamma}\zeta\cdot\nabla^{\gamma}\psi)^2}{2(1+\epsilon^2\mu\mid\nabla^{\gamma}\zeta\mid^2)}=-\frac{1}{B_0}\frac{\kappa_\gamma(\epsilon\sqrt{\mu}\zeta)}{\epsilon\sqrt{\mu}}, \label{ww_equation1} \end{cases}\end{align} where $G_{\mu,\gamma}[\epsilon\zeta,\beta b]\psi$ stands for the dimensionless Dirichlet-Neumann operator,
\begin{equation*} G_{\mu,\gamma}[\epsilon\zeta,\beta b]\psi = \sqrt{1+\epsilon^2 \modd{\nabla^{\gamma}\zeta}} \partial_n \Phi_{\vert z=\epsilon\zeta} = (\partial_z\Phi-\mu\nabla^{\gamma}(\epsilon \zeta)\cdot\nabla^{\gamma}\Phi)_{\vert z=\epsilon\zeta}, \end{equation*} where $\Phi$ solves the Laplace equation with Neumann (at the bottom) and Dirichlet (at the surface) boundary conditions
\begin{align*}\begin{cases} \Delta^{\mu,\gamma} \Phi = 0 \quad \text{in } \mathbb{R}^d \times \lbrace -1+\beta b < z < \epsilon\zeta\rbrace \\ \phi_{\vert z=\epsilon\zeta} = \psi,\quad \partial_n \Phi_{\vert z=-1+\beta b} = 0, \end{cases}\end{align*} \\ and where the surface tension term is $$\kappa_\gamma(\zeta) = -\nabla^{\gamma}\cdot(\frac{\nabla^{\gamma}\zeta}{\sqrt{1+\vert\nabla^{\gamma}\zeta\vert^2}}).$$ We used the following notations:
\begin{align*} &\nabla^{\gamma} = {}^t(\partial_x,\gamma\partial_y) \quad &\text{ if } d=2\quad &\text{ and } &\nabla^{\gamma} = \partial_x &\quad \text{ if } d=1 \\ &\Delta^{\mu,\gamma} = \mu\partial_x^2+\gamma^2\mu\partial_y^2+\partial_z^2 \quad &\text{ if } d=2\quad &\text{ and } &\Delta^{\mu,\gamma} = \mu\partial_x^2+\partial_z^2 &\quad \text{ if } d=1 \end{align*} and $$ \partial_n \Phi_{\vert z=-1+\beta b}=\frac{1}{\sqrt{1+\beta^2\vert\nabla^{\gamma} b\vert^2}} (\partial_z\Phi-\mu\nabla^{\gamma}(\beta b)\cdot\nabla^{\gamma}\Phi)_{\vert z=-1+\beta b}.$$
\subsection{Main result}
Alvarez-Lannes (\cite{alvarez}) proved the following local existence result. We use the notation $a\vee b = \max(a,b)$. \begin{theorem} Under reasonable assumptions on the initial conditions $(\zeta^0,\psi^0)$, there exists a unique solution $(\zeta,\psi)$ of the Water Waves equations $(\ref{ww_equation1})$ with initial condition $(\zeta^0,\psi^0)$ on a time interval $[0;\frac{T}{\epsilon\vee\beta}]$, where $T$ only depends on initial data.\label{david} \end{theorem} For a precise statement, see section \ref{statement} and see \cite{david} Theorem 9.6 and Chapter 4 for a complete proof. The fact that $T$ does not depend on $\mu$ allowed the authors to provide a rigorous justification of most of the Shallow Water models used in the literature for the description, among others, of coastal flows. In these models, one has $\beta = O(\epsilon)$ and therefore the time scale for the solution is $O(\frac{1}{\epsilon\vee\beta})=O(\frac{1}{\epsilon})$. There exists however some asymptotics models assuming small amplitude surface variation ($\epsilon = O(\mu)$) and large bottom variation $\beta = O(1)$. This is the case of the well-known Boussinesq-Peregrine model (\cite{boussinesq1},\cite{boussinesq2},\cite{peregrine}) that has been used a lot in applications. For such a regime, Theorem \eqref{david} provides an existence time of order $O(1)$ only. Our aim is here to improve this result in order to reach an $O(\frac{1}{\epsilon})$ existence time. We prove the following result.
\begin{theorem} The Water Waves equation with surface tension \eqref{ww_equation3} admits a solution on a time interval of the form $[0;\frac{1}{\epsilon}]$ where $T$ only depends on $B_0\mu$ and on the initial data. \label{theorem_flou} \end{theorem}
In order to prove Theorem \eqref{theorem_flou}, we use a method inspired by Bresch-Métivier \cite{bresch_metivier} and Métivier-Schochet \cite{schochetmetivier}. The only condition we need is that there is a small amount of surface tension. More precisely, we assume that the capillary parameter $\frac{1}{B_0}$ is at the same order as the shallowness parameter $\mu$. \\
In the context of the method used by \cite{bresch_metivier}, we start by rescaling the time by setting $t'=t\epsilon$. The Theorem \ref{david} now gives an existence time of order $\epsilon$ in these new scaled variables. The Craig-Sulem-Zakharov formulation of the Water Waves problem in the newly scaled variables is \begin{align}\begin{cases}
\displaystyle{ \partial_t \zeta - \frac{1}{\mu\epsilon} G\psi = 0} \\
\displaystyle\partial_t \psi + \frac{1}{\epsilon}\zeta + \frac{1}{2}\mid\nabla^{\gamma}\psi\mid^2 - \frac{1}{\mu}\frac{(G\psi +\epsilon\mu\nabla^{\gamma}\zeta\cdot\nabla^{\gamma}\psi)^2}{2(1+\epsilon^2\mu\mid\nabla^{\gamma}\zeta\mid^2)}=-\frac{1}{B_0\epsilon}\frac{\kappa_\gamma(\epsilon\sqrt{\mu}\zeta)}{\epsilon\sqrt{\mu}}. \label{ww_equation3} \end{cases}\end{align}
The difficulty is therefore to handle the singular $O(\frac{1}{\epsilon})$ terms of this equation. For the sake of clarity, let us sketch the method of \cite{bresch_metivier}, \cite{schochetmetivier} on the example of the Shallow-Water equations, implemented in \cite{bresch_metivier}. The Shallow-Water equations can be read in the present setting variables
\begin{align} \begin{cases} \displaystyle \partial_t\zeta +\frac{1}{\epsilon}\nabla^{\gamma}\cdot (h\overline{V}) = 0 \\ \displaystyle \partial_t\overline{V}+(\overline{V}\cdot\nabla^{\gamma})\overline{V} + \frac{1}{\epsilon}\nabla^{\gamma}\zeta = 0.\label{shallowater1} \end{cases} \end{align} We denoted $h$ the total height of water: $$h(t,X)=1+\epsilon\zeta(t,X)-\beta b(X)$$ and $$\overline{V}(t,X) = \frac{1}{h(t,X)}\int_{-1+\beta b(X)}^{\epsilon\zeta(t,X)} V(t,X,z)dz$$ the vertical mean of $V$, the horizontal component of the velocity. The natural energy associated to this equation is $$E(\zeta,\overline{V}) = \frac{1}{2}\vert\zeta\vert_2^2+\frac{1}{2}(h\overline{V},\overline{V})_2.$$ By derivating in time it is easy to check that $E\big((\epsilon\partial_t)^k\zeta, (\epsilon\partial_t)^k\overline{V}\big)$ is uniformly bounded with respect to $\epsilon$. It is however not the case for $H^s$ norms of these unknowns with $s\geq 1$, because of commutators terms with the bottom parametrization that are of order $\frac{\beta}{\epsilon}$ and therefore singular if $\beta =O(1)$. Now, to recover an energy uniformly bounded with respect to $\epsilon$ for the spatial derivatives, we use the equation \eqref{shallowater1} to write $$\nabla^{\gamma}\zeta = \epsilon\partial_t\overline{V} +\epsilon R,$$ where $\vert R\vert_2\leq C$ with $C$ independent of $\epsilon$. Thus $\nabla^{\gamma}\zeta$ is bounded in $L^2$ norm uniformly with respect to $\epsilon$. It allows us to recover a control of $\vert\zeta\vert_{H^1}$. For $\overline{V}$, the equation \eqref{shallowater1} gives that $\vert\nabla^{\gamma}\cdot(h\overline{V})\vert_2$ is bounded by $\vert(\epsilon\partial_t)\zeta\vert_2$ (bounded uniformly), and taking the rotational of the second equation, one has that $\mbox{rot}(\overline{V})$ satisfies a symmetric hyperbolic equation of the form $$\partial_t \mbox{\rm curl}\overline{V}+\overline{V}\cdot\nabla^{\gamma}\mbox{\rm curl}\overline{V} = R$$ with $$\vert R\vert_{L^2} \leq C,$$ and with $C$ independent of $\epsilon$, and thus $\mbox{\rm curl}\overline{V}$ is uniformly bounded in $L^2$ norm. With an ellipticity argument, one recovers an uniform bound for $\overline{V}$ in $H^1$ norm. By induction, one can recover the same bound for higher order space derivatives. \\
We propose here an adaptation of this method to the Water Waves problem. The structure of the equation is important for this method. For example, for the Water Waves equation \eqref{ww_equation3} without surface tension ($\displaystyle \frac{1}{B_0} = 0$), one can differentiate the energy $$E(\zeta,\psi) = \frac{1}{2}(G\psi,\psi)_2+\frac{1}{2}\vert\zeta\vert_2^2$$ with respect to time, and check that \begin{equation}E\big((\epsilon\partial_t)\zeta,(\epsilon\partial_t)\psi\big)\leq C,\label{doublestar}\end{equation} where $C$ does not depend on $\epsilon$. One has, for this energy, the equivalence \begin{equation}E(\zeta,\psi)\sim \frac{1}{2}\vert\mathfrak{P}(\epsilon\partial_t)\psi\vert^2_2+\frac{1}{2}\vert(\epsilon\partial_t)\zeta\vert^2_2\label{equivaenergy}\end{equation} where $\mathfrak{P}$ is equivalent to the square root of the Dirichlet-Neumann operator, and acts as an order $1/2$ operator (see later Section \ref{statement}): $$\mathfrak{P} = \frac{\vert D^\gamma\vert}{(1+\sqrt{\mu}\vert D^\gamma\vert)^{1/2}}.$$ The problem is then to recover higher order space derivatives estimates. The second equation of \eqref{ww_equation3} gives $$\zeta = \epsilon\partial_t\psi+\epsilon R$$ where $\vert\mathfrak{P} R\vert_2\leq C$ with $C$ independent of $\epsilon$. Using \eqref{equivaenergy}, this equation only provides a bound uniform with respect to $\epsilon$ for $\vert\mathfrak{P}\zeta\vert_2$, and one recover only $1/2$ space derivative. The same goes for the first equation: $\vert G\psi\vert_2$ is bounded uniformly with respect to $\epsilon$ by $\vert \epsilon\partial_t\zeta\vert_2$, and the ellipticity of $G$ allows us to recover a bound for $\psi$ in $H^1$ norm, and thus we only gained $1/2$ derivative with respect to the control provided by \eqref{doublestar}. We can overcome this problem by taking into account the surface tension term. Indeed, the energy for the equation \eqref{ww_equation3} is $$E(\zeta,\psi)=\vert\zeta\vert_2^2+\frac{1}{B_0}\vert \nabla^{\gamma} \zeta\vert_2^2+\vert\mathfrak{P}\psi\vert_2^2.$$ As explained above, it is easy to get a uniform bound of the energy for times derivatives. The first equation yields $$\frac{1}{\mu}G\psi = (\epsilon\partial_t)\zeta$$ and thus $\frac{1}{\mu}G\psi$ is bounded uniformly in $L^2$ and $\frac{1}{\sqrt{B_0}} H^1$ norm. One can then prove a uniform bound for $\vert\mathfrak{P}\nabla\psi\vert_2$, depending on $B_0\mu$, using the ellipticity of $G$. Note that we pay special attention to the dependence of the surface tension on the existence time; it is indeed important to get a dependence on $Bo\mu$ and not only on $B_0$ since this weaken our assumption on the size of the capillary effects.
\subsection{Notations}\label{notations} We introduce here all the notations used in this paper.
\subsubsection{Operators and quantities} Because of the use of dimensionless variables (see before the "dimensionless equations" paragraph), we use the following twisted partial operators: \begin{align*} &\nabla^{\gamma} = {}^t(\partial_x,\gamma\partial_y) \quad &\text{ if } d=2\quad &\text{ and } &\nabla^{\gamma} = \partial_x &\quad \text{ if } d=1 \\ &\Delta^{\mu,\gamma} = \mu\partial_x^2+\gamma^2\mu\partial_y^2+\partial_z^2 \quad &\text{ if } d=2\quad &\text{ and } &\Delta^{\mu,\gamma} = \mu\partial_x^2+\partial_z^2 &\quad \text{ if } d=1 \\ &\nabla^{\mu,\gamma} = {}^t(\sqrt{\mu}\partial_x,\gamma\sqrt{\mu}\partial_y,\partial_z)\quad &\text{ if } d=2\quad &\text{ and } &{}^t(\sqrt{\mu}\partial_x,\partial_z) &\quad \text{ if } d=1 \\ &\nabla^{\mu,\gamma}\cdot = \sqrt{\mu}\partial_x+\gamma\sqrt{\mu}\partial_y+\partial_z\quad &\text{ if } d=2\quad &\text{ and } &\sqrt{\mu}\partial_x+\partial_z &\quad \text{ if } d=1 \\ &\mbox{\rm curl}^{\mu,\gamma} = {}^t(\sqrt{\mu}\gamma\partial_y-\partial_z,\partial_z-\sqrt{\mu}\partial_x,\partial_x-\gamma\partial_y)&\text {if } d=2\quad \end{align*} \begin{remark}All the results proved in this paper do not need the assumption that the typical wave lengths are the same in both direction, ie $\gamma = 1$. However, if one is not interested in the dependance of $\gamma$, it is possible to take $\gamma = 1$ in all the following proofs. A typical situation where $\gamma\neq 1$ is for weakly transverse waves for which $\gamma=\sqrt{\mu}$; this leads to weakly transverse Boussinesq systems and the Kadomtsev–Petviashvili equation (see \cite{lannes_saut}). \end{remark}
We use the classical Fourier multiplier $$\Lambda^s = (1-\Delta)^{s/2} \text{ on } \mathbb{R}^d$$ defined by its Fourier transform as $$\mathcal{F}(\Lambda^s u)(\xi) = (1+\vert\xi\vert^2)^{s/2}(\mathcal{F}u)(\xi)$$ for all $u\in\mathcal{S}'(\mathbb{R}^d)$. The operator $\mathfrak{P}$ is defined as \begin{equation} \mathfrak{P} = \frac{\vert D^{\gamma}\vert}{(1+\sqrt{\mu}\vert D^{\gamma}\vert)^{1/2}}\label{defp} \end{equation} where $$\mathcal{F}(f(D)u)(\xi) = f(\xi)\mathcal{F}(u)(\xi)$$ is defined for any smooth function $f$ and $u\in\mathcal{S}'(\mathbb{R}^d)$. The pseudo-differential operator $\mathfrak{P}$ acts as the square root of the Dirichlet Neumann operator (see later \ref{equivanorme}). \\
We denote as before by $G_{\mu,\gamma}$ the Dirichlet-Neumann operator, which is defined as followed in the scaled variables:
\begin{equation*} G_{\mu,\gamma}\psi = G_{\mu,\gamma}[\epsilon\zeta,\beta b]\psi = \sqrt{1+\epsilon^2 \modd{\nabla^{\gamma}\zeta}} \partial_n \Phi_{\vert z=\epsilon\zeta} = (\partial_z\Phi-\mu\nabla^{\gamma}(\epsilon\zeta)\cdot\nabla^{\gamma}\Phi)_{\vert z=\epsilon\zeta}. \end{equation*} where $\Phi$ solves the Laplace equation
\begin{align*} \begin{cases} \Delta^{\gamma,\mu}\Phi = 0\\ \Phi_{\vert z=\epsilon\zeta} = \psi,\quad \partial_n \Phi_{\vert z=-1+\beta b} = 0 \end{cases} \end{align*}
For the sake of simplicity, we use the notation $G[\epsilon\zeta,\beta b]\psi$ or even $G\psi$ when no ambiguity is possible. \\
\subsubsection{The Dirichlet-Neumann problem} In order to study the Dirichlet-Neumann problem \eqref{dirichlet}, we need to map $\Omega_t$ into a fixed domain (and not on a moving subset). For this purpose, we introduce the following fixed strip: $$\mathcal{S} = \mathbb{R}^d\times (-1;0)$$ and the diffeomorphism \begin{displaymath} \Sigma_t^{\epsilon} : \left.
\begin{array}{rcl}
\mathcal{S} & \rightarrow &\Omega_t \\
(X,z) & \mapsto & (1+\epsilon\zeta(X)-\beta b(X))z+\epsilon\zeta(X) \\
\end{array} \right. \end{displaymath} It is quite easy to check that $\Phi$ is the variational solution of \eqref{dirichlet} if and only if $\phi = \Phi\circ\Sigma_t^{\epsilon}$ is the variational solution of the following problem: \begin{align}\begin{cases}
\nabla^{\mu,\gamma}\cdot P(\Sigma_t^{\epsilon})\nabla^{\mu,\gamma} \phi = 0 \label{dirichletneumann}\\
\phi_{z=0}=\psi,\quad \partial_n\phi_{z=-1} = 0, \end{cases} \end{align} and where $$P(\Sigma_t^{\epsilon}) = \vert \det J_{\Sigma_t^{\epsilon}}\vert J_{\Sigma_t^{\epsilon}}^{-1}~^t(J_{\Sigma_t^{\epsilon}}^{-1}),$$ where $J_{\Sigma_t^{\epsilon}}$ is the jacobian matrix of the diffeomorphism $\Sigma_t^{\epsilon}$. For a complete statement of the result, and a proof of existence and uniqueness of solutions to these problems, see \cite{david} Chapter 2.\\
We introduce here the notations for the shape derivatives of the Dirichlet-Neumann operator. More precisely, we define the open set $\bold{\Gamma}\subset H^{t_0+1}(\mathbb{R}^d)^2$ as:\\ $$\bold{\Gamma} =\lbrace \Gamma=(\zeta,b)\in H^{t_0+1}(\mathbb{R}^d)^2,\quad \exists h_0>0,\forall X\in\mathbb{R}^d, \epsilon\zeta(X) +1-\beta b(X) \geq h_0\rbrace$$ and, given a $\psi\in \overset{.}H{}^{s+1/2}(\mathbb{R}^d)$, the mapping: \begin{displaymath}G[\epsilon\cdot,\beta\cdot] : \left. \begin{array}{rcl} &\bold{\Gamma} &\longrightarrow H^{s-1/2}(\mathbb{R}^d) \\ &\Gamma=(\zeta,b) &\longmapsto G[\epsilon\zeta,\beta b]\psi. \end{array}\right.\end{displaymath} We can prove the differentiability of this mapping. See Appendix \ref{appendixA} for more details. We denote $d^jG(h,k)\psi$ the $j$-th derivative of the mapping at $(\zeta,b)$ in the direction $(h,k)$. When we only differentiate in one direction, and no ambiguity is possible, we simply denote $d^jG(h)\psi$ or $d^j G(k)\psi$.
\subsubsection{Functional spaces} The standard scalar product on $L^2(\mathbb{R}^d)$ is denoted by $(\quad,\quad)_2$ and the associated norm $\vert\cdot\vert_2$. We will denote the norm of the Sobolev spaces $H^s(\mathbb{R}^d)$ by $\vert \cdot\vert_{H^s}$.\\ \\We introduce the following functional Sobolev-type spaces, or Beppo-Levi spaces: \begin{definition} We denote $\dot{H}^{s+1}(\mathbb{R}^d)$ the topological vector space $$\dot{H}^{s+1}(\mathbb{R}^d) = \lbrace u\in L^2_{loc}(\mathbb{R}^d),\quad \nabla u\in H^s(\mathbb{R}^d)\rbrace$$ endowed with the (semi) norm $\vert u\vert_{\dot{H}^{s+1}(\mathbb{R}^d)} = \vert\nabla u\vert_{H^s(\mathbb{R}^d)} $. \end{definition} Just remark that $\dot{H}^{s+1}(\mathbb{R}^d)/\mathbb{R}^d$ is a Banach space (see for instance \cite{lions}). \\
The space variables $z\in\mathbb{R}$ and $X\in\mathbb{R}^d$ play different roles in the equations since the Euler formulation (\ref{euler}) is posed for $(X,z)\in \Omega_t$. Therefore, $X$ lives in the whole space $\mathbb{R}^d$ (which allows to take fractionary Sobolev type norms in space), while $z$ is actually bounded. For this reason, we need to introduce the following Banach spaces: \begin{definition} The Banach space $(H^{s,k}((-1,0) \times\mathbb{R}^d),\vert .\vert_{H^{s,k}})$ is defined by $$H^{s,k}((-1,0) \times\mathbb{R}^d) = \bigcap_{j=0}^{k} H^j((-1,0);H^{s-j}(\mathbb{R}^d)),\quad \vert u\vert_{H^{s,k}} = \sum_{j=0}^k \vert \Lambda^{s-j}\partial_z^j u\vert_2.$$ \end{definition}
\section{Main result} This section is dedicated to the proof of Theorem \ref{theorem_flou}. In \S \ref{energysection} we introduce the energy space $\mathcal{E}^N_\sigma$ used in the Water Waves equations. This energy plays an important role in the proof of the main result, since the key point consists in proving that this energy is uniformly bounded with respect to $\epsilon$. We also recall in this Subsection the method used to prove the local existence theorem for the Water Waves equation. The proof of the local existence relies on the important assumption that the Rayleigh-Taylor condition holds; this is discussed in \S \ref{rayleigh_section}. The following \S \ref{statement} states the main result of this paper, that is, the precise statement of Theorem \ref{theorem_flou}. The last \S \ref{proof_section} is dedicated to the proof of this result. \subsection{The energy space}\label{energysection} The purpose of this section is to introduce the energy space used in the proof of the local existence result for the Water Waves equation. To this purpose, we explain the strategy of this proof. We adapt here the approach of \cite{david} to the rescaled in time equations \eqref{ww_equation2}, pointing out where the singular terms are. We recall that we rescale the time variable for the equation \eqref{ww_equation1} by setting $$t'=t\epsilon.$$ The Water Waves equation with surface tension \eqref{ww_equation1} in the newly scaled variables is \begin{align}\begin{cases}
\displaystyle{ \partial_t \zeta - \frac{1}{\mu\epsilon} G\psi = 0} \\
\displaystyle\partial_t \psi + \frac{1}{\epsilon}\zeta + \frac{1}{2}\mid\nabla^{\gamma}\psi\mid^2 - \frac{1}{\mu}\frac{(G\psi +\epsilon\mu\nabla^{\gamma}\zeta\cdot\nabla^{\gamma}\psi)^2}{2(1+\epsilon^2\mu\mid\nabla^{\gamma}\zeta\mid^2)}=-\frac{1}{B_0\epsilon}\frac{\kappa_\gamma(\epsilon\sqrt{\mu}\zeta)}{\epsilon\sqrt{\mu}}. \label{ww_equation2} \end{cases}\end{align} \begin{remark} We recall that $$\kappa_\gamma(\zeta) = -\nabla^{\gamma}\cdot(\frac{\nabla^{\gamma}\zeta}{\sqrt{1+\vert\nabla^{\gamma}\zeta\vert^2}}),$$ so the surface tension term that appears on the right hand side of the second equation of \eqref{ww_equation2} is only of size $\displaystyle \frac{1}{\epsilon}$ (thought at first sight it seems of size $\displaystyle \frac{1}{\epsilon^2}$).\end{remark}
The purpose of the proof of an existence time uniform with respect to $\epsilon$ for this equation in the newly time scaled variables, is to get an uniform bound with respect to $\epsilon$ for a good quantity called the energy which controls Sobolev norms of the unknowns. By a continuity argument, one can deduce a time existence independent of $\epsilon$. For the Water Waves equation with surface tension \eqref{ww_equation2}, a natural quantity appears to act as an "energy". If we look at the linearized equation around the rest state $\zeta=0,\psi=0$, we find a system of evolution equations $$\partial_t U+\frac{1}{\epsilon}\mathcal{A}_\sigma U = 0,\quad \text{ with }\quad \mathcal{A}_\sigma=\begin{pmatrix} &0 &-\frac{1}{\mu}G[0,\beta b] \\ &1-\frac{1}{B_0}\Delta^\gamma &0\end{pmatrix}.$$ This system can be made symmetric if we multiply it by the symmetrizer
$$\begin{pmatrix} &1-\frac{1}{B_0}\Delta^\gamma & 0\\ &0 & \frac{1}{\mu}G[0,\beta b] \end{pmatrix},$$ where $U={}^t(\zeta,\psi)$. In \cite{alvarez},\cite{david}, $G[0,\beta b]$ is replaced by $G[0,0]$ in $\mathcal{A}_\sigma$. Here, we cannot perform this simplification because the error would be of size $O(\frac{\beta}{\epsilon})$ and therefore singular since $\beta = O(1)$. This suggests a natural energy of the form $$\vert\zeta\vert_2^2+\frac{1}{B_0}\vert\nabla^{\gamma}\zeta\vert_2^2+(\frac{1}{\mu}G[0,\beta b]\psi,\psi)_2.$$ The last term is uniformly equivalent to $\vert\mathfrak{P}\psi\vert_2^2$ where $\mathfrak{P}$ is defined in \eqref{defp}. See later Remark \ref{energy_size} for a precise statement. Thus, $\mathfrak{P}$ acts as the square root of the Dirichlet-Neumann operator and is of order $1/2$. \\
This energy has not the sufficient order of derivatives to have a real control of the unknowns. For instance, the product $\vert\nabla^{\gamma}\psi\vert^2$ in the second equation of $\eqref{ww_equation2}$ is not defined if $\psi$ is only $H^{1/2}(\mathbb{R}^d)$. To recover a control of the unknowns at a higher order, the classical scheme for this kind of method, is to differentiate the equation \eqref{ww_equation2} in order to get an evolution equation of the unknowns $\partial_{X_i}^k\zeta,\partial_{X_i}^k\psi$. For this purpose, we need to use an explicit shape derivative formula with respect to the surface for the Dirichlet Neumann operator. It is given in Appendix Theorem \ref{321}: $$dG(h)\psi = -\epsilon G(h\underline{w})-\epsilon\mu\nabla^{\gamma}\cdot(h\underline{V})$$ with $$\underline{w} = \frac{G\psi+\epsilon\mu\nabla^{\gamma}\zeta\cdot\nabla^{\gamma}\psi}{1+\epsilon^2\mu\modd{\nabla^{\gamma}\zeta}} \quad\text{ and }\quad \underline{V} = \nabla^{\gamma}\psi-\epsilon\underline{w}\nabla^{\gamma}\zeta.$$ See \cite{david} Chapter 3 for a full proof of this formula. By differentiating $N$ times the first equation of \eqref{ww_equation2}, one finds after some computation that, for $\vert(\alpha,k)\vert =N$, $$\partial_t\partial^{(\alpha,k)}\zeta +\nabla^\gamma\cdot (\underline{V}\partial^{(\alpha,k)}\zeta)-(\frac{1}{\mu\epsilon}G(\partial^{(\alpha,k)}\psi-\epsilon\underline{w}\partial^{(\alpha,k)} \zeta)+\frac{1}{\epsilon}\underset{j\in\mathbb{N^*},l_1+...+l_j+\delta = (\alpha,k)}{\sum} d^j G(\partial^{l_1}b,...,\partial^{l_j}b) \partial^{\delta}\psi )) = R,$$ with, for any $d+1$-uplet of integers $(\alpha,k) = (\alpha_1,...,\alpha_{d},k)$, $\vert(\alpha,k)\vert = \sum_{i=1}^{d} \alpha_i+k$, and with the notation $$\forall (\alpha,k)=(\alpha_1,...,\alpha_{d},k)\in\mathbb{N}^{d+1},\quad \partial^{(\alpha,k)} f = \partial_{X_1}^{\alpha_1}...\partial_{X_d}^{\alpha_d}(\epsilon\partial_t)^kf$$ and where, without entering technical details $$\vert R\vert_{H^{N-1}} \leq C$$ where $C$ does not depend on $\epsilon$. This evolution equation can be found by differentiating $N$ times the Dirichlet-Neumann operator $G[\epsilon\zeta,\beta b]\psi$ and thus $R$ contains derivatives of the form $$dG(\partial^{l_1}\zeta,...,\partial^{l_j}\zeta,\partial^{m_1}\beta b,...,\partial^{m_k}\beta b)\partial^\delta\psi$$ where the shape derivatives of $G$ with respect to $\zeta$ have an $\epsilon$ factor that cancel the $\displaystyle \frac{1}{\epsilon}$ singularity (see also the definition of $dG$ in Section \ref{notations}). Finally, the only singular terms comes from the derivatives of the Dirichlet Neumann operator with respect to the bottom. Up to some extra singular source terms coming from the shape derivatives with respect to the bottom, this equation has the same structure as \eqref{ww_equation2}, with $\partial^{(\alpha,k)} \zeta$ and $\partial^{(\alpha,k)}\psi-\epsilon\underline{w}\partial^{(\alpha,k)} \zeta$ playing the role of $\zeta$ and $\psi$ respectively, These quantities are the so called "Alinhac good unknowns" and will be denoted by:\begin{equation} \forall \vert(\alpha,k)\vert\geq 1\quad \zeta_{(\alpha,k)} = \partial^{(\alpha,k)} \zeta, \quad \psi_{(\alpha,k)} = \partial^{(\alpha,k)} \psi - \epsilon \underline{w} \partial^{(\alpha,k)}\zeta.\quad \label{defpsia}\end{equation} An evolution equation for the unknown $\psi_{(\alpha,k)}$ in term of the good unknowns can be also obtained (see later Section \ref{proof_section}). \\
In the surface tension case, the leading order operator is the surface tension term $\kappa_\gamma$. This leads to some technical complications. For instance, in order to control the time derivatives of $\kappa_\gamma$, one has to include the time derivatives of the unknowns in the energy. This method has been used by \cite{rousset}, \cite{david2} to study the Water Waves Problem with surface tension. Time derivatives and space derivatives play a different role in this proof, and we use the notation $$\forall k\in\mathbb{N},\qquad \zeta_{(k)} = (\epsilon\partial_t)^k \zeta,\quad\text{ and } \psi_{(k)} = (\epsilon\partial_t)^k\psi-\epsilon\underline{w}(\epsilon\partial_t)^k\zeta$$ for time derivatives, and $$\forall\alpha\in\mathbb{N}^d,\forall k\in\mathbb{N},\qquad \zeta_{(\alpha,k)} = (\epsilon\partial_t)^k\partial^\alpha \zeta\quad\text{ and }\quad \psi_{(\alpha,k)} = (\epsilon\partial_t)^k\partial^\alpha \psi-\epsilon\underline{w}(\epsilon\partial_t)^k\partial^\alpha\zeta$$ such that $f_{(\alpha,k)}$ denotes indeed the good unknown defined by \eqref{defpsia} with index the $d+1$-uplet $(\alpha,k)$. See \cite{david} Chapter 9 for more details about how to handle the time derivative in the energy.\\
All these considerations explain why we do not use, for the local existence result of the Water Waves equations, an energy involving terms of the form $\partial_{X_i}^k\psi$ but rather the following energy:
\begin{equation}\mathcal{E}_\sigma^N(U) = \modd{\zeta}_2+\modd{\mathfrak{P}\psi}_{H^{t_0+3/2}}+\underset{(\alpha,k)\in\mathbb{N}^{d+1},1\leq\mid(\alpha,k)\mid\leq N}{\sum} \modd{\zeta_{(\alpha,k)}}_{H^1_\sigma}+\modd{\mathfrak{P} \psi_{(\alpha,k)}}_2\label{energie_theoreme}\end{equation} where \begin{equation}\vert f\vert_{H^1_\sigma}^2 = \vert f\vert_2^2+\frac{1}{B_0}\vert \nabla^{\gamma} f\vert_2^2. \label{defh1}\end{equation}
The choice of $N$ is of course purely technical, and made in particular to have the different products of functions well-defined in the Sobolev Spaces used. Again, it is very important to note that the time derivatives of order less than $N$ appear in the equation. \\
We consider solutions $U=(\zeta,\psi)$ of the Water Waves equations in the following space:
\begin{equation*} E_{\sigma,T}^N = \lbrace U\in C(\left[ 0,T\right];H^{t_0+2}\times\overset{.}H{}^2(\mathbb{R}^d)), \mathcal{E}_\sigma^N(U(.))\in L^{\infty}(\left[ 0,T\right])\rbrace \end{equation*}
\subsection{The Rayleigh-Taylor condition}\label{rayleigh_section} We explained in Subsection \ref{energysection} that the Water Waves equations \eqref{ww_equation2} can be "quasilinearized". In these quasilinearized equations, a quantity appears to play an important role. It is called the "Rayleigh-Taylor coefficient" (see \cite{david} Chapter 4 and also \cite{taylor} for more details) and is defined by\begin{equation}\underline{\mathfrak{a}}(\zeta,\psi) = 1+\epsilon(\epsilon\partial_t+\epsilon \underline{V}\cdot\nabla^{\gamma})\underline{w} = -\epsilon\frac{P_0}{\rho a g}(\partial_z P)_{\vert z = \epsilon\zeta}\label{rtdef}\end{equation} where $\underline{w} = (\partial_z\Phi)_{\vert z=\epsilon\zeta}$ and $\underline{V} = (\nabla^{\gamma}\Phi)_{\vert z=\epsilon\zeta}$ are respectively the horizontal and vertical component of the velocity $U=\nabla_{X,z}\Phi$ evaluated at the surface.
The condition for strict hyperbolicity of the Water Waves system appears to be the following "Rayleigh-Taylor condition": $\underline{\mathfrak{a}}>0$. This makes the link with the classical Rayleigh-Taylor criterion $$\underset{\mathbb{R}^d}{\inf}(-\partial_z P)_{\vert z=\epsilon\zeta}>0$$ where $P$ is the dimensionless pressure. See \cite{raileigh} for more details. Ebin (\cite{ebin}) showed that the Water Waves problem (without surface tension) is ill-posed if the Rayleigh-Taylor condition is not satisfied. Wu \cite{wu1} proved that this condition is satisfied by any solution of the Water Waves problem in infinite depth. It is proved also in \cite{david} Chapter 4 that this is also true in finite depth for the case of flat bottom. For the surface tension case, the Rayleigh-Taylor condition does not need to be satisfied in order to have well-posedness, but the existence time depends too strongly on the surface tension coefficient and is then too small for most applications to oceanography.
\subsection{Statement of the result}\label{statement} We now state the main result.
\begin{theorem}\label{uniform_result} Let $t_0>d/2$,$N\geq t_0+t_0\vee 2+3/2$. Let $U^0 = (\zeta^0,\psi^0)\in E_0^N,b\in H^{N+1\vee t_0+1}(\mathbb{R}^d)$. Let $\epsilon,\gamma,\beta$ be such that $$0\leq \epsilon,\beta,\gamma\leq 1,$$ and moreover assume that: \begin{equation}\exists h_{min}>0,\exists a_0>0,\qquad 1+\epsilon\zeta^0-\beta b\geq h_{min}\quad\text{ and } \quad \underline{\mathfrak{a} }(U^0)\geq a_0\label{rayleigh}\end{equation} Then, there exists $T>0$ and a unique solution $U^\epsilon\in E_{\sigma,T}^N$ to \eqref{ww_equation2} with initial data $U^0$. Moreover, $$\frac{1}{T}= C_1,\quad\text{ and }\quad \underset{t\in [0;T]}{\sup} \mathcal{E}_\sigma^N(U^\epsilon(t)) = C_2$$ with $\displaystyle C_i=C(\mathcal{E}_\sigma^N(U^0),\frac{1}{h_{min}},\frac{1}{a_0},\vert b\vert_{H^{N+1\vee t_0+1}},\mu B_0)$ for $i=1,2$. \end{theorem}
It is very important to note that $T$ does not depend on $\epsilon$ for small values of $\epsilon$. The Theorem \ref{theorem_flou} gives an existence time of order $\epsilon$ as $\epsilon$ goes to zero, if $\beta$ is of order $1$. We prove here that it is in fact, of order $1$. Note that the topography parameter $\beta$ is fixed in all this study.\\
Let us now give a result for the Water Waves equation with surface tension \eqref{ww_equation1} in the initial time variable. The local existence Theorem \ref{david} provides an existence time of order $\frac{1}{\epsilon\vee\beta}$ to the initial Water Waves equation \eqref{ww_equation1} with surface tension. After the rescaling in time $t'=t\epsilon$, the Theorem \ref{david} provides an existence time of order $\frac{\epsilon}{\epsilon\vee\beta}\sim \frac{\epsilon}{\beta}$ as $\epsilon$ goes to $0$. Now, the main result Theorem \ref{uniform_result} claims that the existence time is in fact of order $1$ in this variable. It gives then the following result:
\begin{theorem} Under the assumptions of Theorem \ref{uniform_result}, there exists a unique solution $(\zeta,\psi)$ of the Water Waves equation \eqref{ww_equation3} with initial condition $(\zeta^0,\psi^0)$ on a time interval $[0;\frac{ T}{\epsilon}]$ where $$\frac{1}{T}= C_1,\quad\text{ and }\quad \underset{t\in [0;T]}{\sup} \mathcal{E}_\sigma^N(U^\epsilon(t)) = C_2$$ with $\displaystyle C_i=C(\mathcal{E}_\sigma^N(U^0),\frac{1}{h_{min}},\frac{1}{a_0},\vert b\vert_{H^{N+1\vee t_0+1}},\mu B_0)$ for $i=1,2$. \end{theorem} This last result gives a gain of an order $\displaystyle \frac{1}{\epsilon}$ with respect to the time existence provided by Theorem \ref{david}. It is very important to note that the existence time given by Theorem \ref{uniform_result} depends on the constant $B_0\mu$. It implies for instance that in the shallow water limit ($\mu \ll 0$), less on less capillary effects are required (recall that the capillary effects are of order $\frac{1}{B_0}$). This is the reason why in the limit case $\mu = 0$ corresponding to the shallow water equations and investigated in \cite{bresch_metivier}, no surface tension is needed. We discuss about the shallow water regime in the Section \ref{shallow_section}.
\subsection{Proof of Theorem \ref{uniform_result}}\label{proof_section} The key point of the proof of Theorem \ref{uniform_result} is the following Proposition: \begin{proposition} \label{keypoint} Let $U^{\epsilon}=(\zeta^{\epsilon},\psi^{\epsilon})$ be the unique solution of the equation \eqref{ww_equation2} on the time interval $[0;T^\epsilon]$ and $$K^\epsilon = \underset{t\in [0,T^\epsilon]}{\sup} \mathcal{E}_\sigma^N(U^\epsilon(t)).$$ Then we have, with the previous notations: \begin{equation} \forall t\in [0;T^{\epsilon}], \qquad\mathcal{E}_\sigma^N(U^{\epsilon})(t) \leq C_0 + C_1(K^\epsilon) (t+\epsilon)\label{desired} \end{equation} where $C_0=C_0(\mathcal{E}_\sigma^N(U^\epsilon_{\vert t=0}))$ and $\displaystyle C_1(K^\epsilon) = C_1(K^\epsilon,\frac{1}{h_{min}},\frac{1}{a_0},\vert b\vert_{H^{N+1\vee t_0+1}},B_0\mu)$ are non decreasing functions of their arguments. \label{main_result} \end{proposition} \begin{proof}[Proof of Proposition \ref{keypoint}]
The quantity $\epsilon$ is fixed throughout the proof. We consider a solution $U^\epsilon = (\zeta^\epsilon,\psi^\epsilon)$ on a time interval $[0;T^\epsilon]$ of \eqref{ww_equation1} given by the standard local existence Theorem \ref{david}. To alleviate the notations, we omit, when no ambiguity is possible, the $\epsilon$ in the notation $U^{\epsilon}$ in the following estimates. Moreover, $C$ will stand for any non decreasing continuous positive function. Let us first sketch the proof. \\ (i) The evolution equation for time derivatives of the unknowns is "skew symmetric" with respect to $\displaystyle \frac{1}{\epsilon}$ terms: these large terms cancel one another in energy estimates. This allows us to get the improved estimate \eqref{desired} for time derivatives: \begin{equation}\vert (\epsilon\partial_t)^k\zeta\vert_{H^1_\sigma} + \vert \mathfrak{P}(\epsilon\partial_t)^k\psi\vert_2\leq C_1(K)t+C_0;\quad k=0..N\label{time_estimate}\end{equation} This is proved in Lemma \ref{lemma_control}. \\ (ii) To get higher order estimates (with respect to space variables), we use the equation \eqref{ww_equation2} to get $$\frac{1}{\mu}G((\epsilon\partial_t)^k\psi) = (\epsilon\partial_t)^{k+1}\zeta + \epsilon R $$ where $\vert R\vert_{H^1_\sigma} \leq C_1(K)$. By the first step of the proof, the term $(\epsilon\partial_t)^{k+1}\zeta $ satisfies the "good" control \eqref{time_estimate} in $H^1_\sigma$ norm. Since $G$ is of order one and elliptic, this should allow us to recover one space derivative for $\mathfrak{P}(\epsilon\partial_t)^k\psi$, with the desired control \eqref{desired}. But there is a little constraint, due to the factor $\frac{1}{B_0}$ in the definition of the $H^1_\sigma$ norm \eqref{defh1}: $$\vert f\vert_{H^1_\sigma}^2 = \vert f\vert_2^2+\frac{1}{B_0}\vert\nabla^{\gamma} f\vert_2^2.$$ One has to use precisely the definition of the operator $\mathfrak{P}$ given in \eqref{defp} by $$\mathfrak{P} = \frac{\vert D^\gamma\vert}{(1+\sqrt{\mu}\vert D^\gamma\vert)^{1/2}}$$ and the inequality $$\vert \mathfrak{P}\psi\vert_2 \leq \frac{M}{\mu} (G\psi,\psi)_2$$ (see below Proposition \ref{equivanorme}) to get that $\vert D^\gamma\vert\mathfrak{P}\psi$ is bounded in $H^1$ norm by a constant depending on $\mu B_0$. One can use the same technique to control $\Delta^\gamma\zeta$ with $(\epsilon\partial_t)^{k+1}\psi$ in $H^1_\sigma$ norm, and recover again one space derivative for $\zeta$. By finite induction, one recovers the control of the form \eqref{desired} for $\zeta_{(k)}$ and $\psi_{(k)}$. This is done in Lemma \ref{lemma_recover}. \\
For this proof, we choose (taking smaller time existence if necessary) $T^\epsilon$ such that
\begin{equation}\forall t\in [0;T^\epsilon],\quad \underline{\mathfrak{a} }(t) \geq \frac{a_0}{2}\quad\text{ and }\quad h(t)=1+\epsilon\zeta(t)-\beta b \geq \frac{h_{min}}{2}.\label{condition_time} \end{equation} The first condition may be satisfied given the continuity in time of $\underline{\mathfrak{a} }$ (see the definition of $\underline{\mathfrak{a} }$ in \eqref{rtdef}) on the time interval $[0;T^\epsilon]$, provided that $\partial_t \underline{\mathfrak{a} }\in L^\infty([0;T^\epsilon];\mathbb{R}^d)$ (this is proved in the control of $A_3$ below). The second condition is satisfied by the fact the solution $\zeta$ lives in the space $C([0;T^\epsilon];H^{t_0+2}(\mathbb{R}^d))$, and the continuous embedding $H^{t_0}\subset L^\infty(\mathbb{R}^d)$ given $t_0 >d/2$. This gives the continuity in time of $h$ (note that $b$ is also in $L^\infty(\mathbb{R}^d)$).\\
Now let us prove the desired estimate \eqref{desired} for time derivatives of $\psi$ and $\zeta$. Because of the energy space introduced in \eqref{energie_theoreme}, we want to control quantities like \begin{equation*} \mathcal{E}_{(\alpha,k)} = \modd{\zeta_{(\alpha,k)}}_{H^1_\sigma}+\modd{\mathfrak{P} \psi_{(\alpha,k)}}_2, \qquad \vert (\alpha,k)\vert \leq N \end{equation*}
with $\alpha = 0$. Let $k$ be fixed. We look for the equations for the unknown $U^k=(\zeta_{(k)},\psi_{(k)})$. We denote the Rayleigh-Taylor coefficient by $$\underline{\mathfrak{a}} = 1+\epsilon(\epsilon\partial_t+\epsilon \underline{V}\cdot\nabla^{\gamma})\underline{w}.$$ By differentiating $k$ times the equations \eqref{ww_equation2} with $(\epsilon\partial_t)^k$, one find after some computations the following result:
\begin{lemma}\label{lemma_quasilinear} The unknown $U_{(k)}$ satisfies the following equation: \begin{equation} \partial_t U_{(k)}+\frac{1}{\epsilon}\mathcal{A}_\sigma[U]U_{(k)}+B[U]U_{(k)}+C_k[U]U_{(k-1)} = {}^t(R_k,S_k)\label{quasilinear} \end{equation} with the operators $$\displaystyle \mathcal{A}_\sigma[U] = \begin{pmatrix} &0& -\frac{1}{\mu}G \\ &\underline{\mathfrak{a} }-\frac{1}{B_0}\nabla^{\gamma}\cdot \mathcal{K}(\sqrt{\mu}\epsilon\nabla^{\gamma}\zeta)\nabla^{\gamma} &0 \end{pmatrix},$$ $$\displaystyle \mathcal{B}[U] = \begin{pmatrix} &\underline{V}\cdot\nabla^{\gamma} & 0 \\ &0 &\underline{V}\cdot\nabla^{\gamma} \end{pmatrix},$$ and $$\displaystyle \mathcal{C}_k[U] = \begin{pmatrix} &0& -\frac{1}{\mu}dG(\epsilon\partial_t\zeta) \\ &\frac{1}{B_0\epsilon}\nabla^{\gamma}\cdot \mathcal{K}_{(k)}[\sqrt{\mu}\nabla^{\gamma}\zeta] &0 \end{pmatrix},$$
and where $$\mathcal{K}(\nabla^{\gamma}\zeta) = \frac{(1+\vert\nabla^{\gamma}\zeta\vert^2)I_d-\nabla^{\gamma}\zeta\otimes\nabla^{\gamma}\zeta}{(1+\vert\nabla^{\gamma}\zeta\vert^2)^{3/2}},$$ and $$\mathcal{K}_{(k)}[\nabla^{\gamma}\zeta]F = -\nabla^{\gamma}\cdot \Big[ d\mathcal{K}(\nabla^{\gamma}\partial_t\zeta)\nabla^{\gamma} F+d\mathcal{K}(\nabla^{\gamma} F)\nabla^{\gamma} \partial_t\zeta\Big].$$ The residual ${}^t(R_k,S_k)$ satisfies the following control : \begin{equation}\vert R_k\vert_{H^1_\sigma}+\vert\mathfrak{P}S_k\vert_2\leq C_1(K).\label{reste}\end{equation}\end{lemma} \begin{remark} Let us explain why the residual has to satisfy an estimate of the form \eqref{reste}. The energy for $\zeta_{(k)},\psi_{(k)}$ is of the form $$\vert\mathfrak{P} \psi_{(k)}\vert_2^2+\vert\zeta_{(k)}\vert_2^2+\frac{1}{B_0}\vert\nabla^{\gamma}\zeta_{(k)}\vert_2^2.$$ In order to get energy estimate, we differentiate this energy with respect to time, which leads to the control of terms of the form $$\frac{1}{\mu}(\partial_t \psi_{(k)},G\psi_{(k)})_2,\qquad (\partial_t\zeta_{(k)},\zeta_{(k)})_{H^1_\sigma}.$$ In order to control these quantities, we replace $\partial_t(\zeta_{(k)},\psi_{(k)})$ by their expressions given by the equation \eqref{quasilinear}. All terms satisfying a control of the form \eqref{reste} are harmless for the energy estimate, since they lead to the control of terms such as $$\frac{1}{\mu} (G\psi_{(k)},S_k),\qquad (\zeta_{(k)},R_k)_{H^1_\sigma}$$ which is easily done. \end{remark} \begin{proof}[Proof of Lemma \ref{lemma_quasilinear}] The differentiation of the first equation of $\eqref{ww_equation2}$ takes the form (recall that $G\psi$ stands for $G[\epsilon\zeta,\beta b]\psi$ and thus any derivative of $G$ with respect to $\zeta$ involves an $\epsilon$ factor) \begin{align*} (\epsilon\partial_t)^k\partial_t\zeta &= \frac{1}{\mu\epsilon} G(\epsilon\partial_t)^k\psi + \frac{1}{\mu\epsilon}dG((\epsilon\partial_t)^k\zeta)\psi+\frac{1}{\mu\epsilon} dG(\epsilon\partial_t\zeta)\psi_{(k-1)}\\ &+\frac{1}{\mu\epsilon}\sum_{1\leq j_1+...+j_m+l\leq k\atop 1\leq l} \epsilon^{j_1+...+j_m}dG((\epsilon\partial_t)^{j_1}\zeta,...,(\epsilon\partial_t)^{j_m}\zeta)(\epsilon\partial_t)^l\psi.\end{align*}Using the explicit shape derivative formula with respect to the surface for $G$ given by Proposition \ref{321}, we get that $$dG((\epsilon\partial_t)^k\zeta)\psi = -\epsilon G((\epsilon\partial_t)^k\zeta \underline{w})-\epsilon\mu\nabla^{\gamma}\cdot((\epsilon\partial_t)^k\zeta\underline{V}),$$ and thus using the definition of $\zeta_{(k)},\psi_{(k)}$ given by \eqref{defpsia}, one gets the following evolution equation: $$(\epsilon\partial_t)^k\partial_t\zeta +\nabla^{\gamma}\cdot(\underline{V}\zeta_{(k)})- \frac{1}{\mu\epsilon} G\psi_{(k)}-\frac{1}{\mu\epsilon} dG(\epsilon\partial_t\zeta)\psi_{(k-1)}=\frac{1}{\mu\epsilon}\sum_{1\leq j_1+...+j_m+l\leq k\atop 1\leq l} \epsilon^{j_1+...+j_m}dG((\epsilon\partial_t)^{j_1}\zeta,...,(\epsilon\partial_t)^{j_m}\zeta)(\epsilon\partial_t)^l\psi.$$The term $dG(\epsilon\partial_t\zeta)\psi_{(k-1)}$ is controlled in $L^2$ norm, but not in $H^1_\sigma$ norm. The terms of the right hand side involve derivatives of $\psi$ of order less than $N-2$ and then can be put in a residual $R_k$ with a control \eqref{reste}, using Proposition \ref{328}. The differentiation of the second equation of \eqref{ww_equation2} involves the linearization of the surface tension term $$ \frac{1}{\epsilon\sqrt{\mu}}(\epsilon\partial_t)^k \kappa_\gamma (\epsilon\sqrt{\mu}\zeta) = -\nabla^{\gamma}\cdot\mathcal{K}(\epsilon\sqrt{\mu}\nabla^{\gamma}\zeta)\nabla^{\gamma} (\epsilon\partial_t)^k\zeta + K_{(k)}[\epsilon\sqrt{\mu}\nabla^{\gamma}\zeta](\epsilon\partial_t)^{k-1}\zeta+...$$ The second order operator $\mathcal{K}_{(k)}$ is not controlled in $H^{1/2}$ norm (or $\vert \mathfrak{P}\cdot\vert_2$ norm). The other terms can be put in the residual $S_k$ with a control \eqref{reste}. See \cite{david} Chapter 9 for a complete proof of the evolution equation in terms of unknowns $\psi_{(k)}$,$\zeta_{(k)}$.\end{proof}
$\qquad \Box $
We now show that the singular terms of size $O(\frac{1}{\epsilon})$ are transparent in the energy estimates for the evolution equation \eqref{quasilinear}. This yields the bounds announced in \eqref{time_estimate}.
\begin{proposition} One has the following estimate for all $0\leq k\leq N$: $$ \vert \zeta_{(k)}\vert_{H^1_\sigma}+\vert\mathfrak{P}\psi_{(k)}\vert_2\leq C_2(K)t+C_0,$$ where $C_0=C(\mathcal{E}_\sigma^N(U_{\vert t=0}))$ and $C_2=\displaystyle C(\frac{1}{h_{\min}},\frac{1}{a_0},\vert b\vert_{H^{N+1\vee t_0+1}})$ are non decreasing function of their arguments. \label{lemma_control} \end{proposition}
\begin{remark} An evolution equation can also be obtained for space derivatives, and then takes the form \begin{equation} \partial_t U_{(\alpha,k)}+\frac{1}{\epsilon}\tilde{\mathcal{A}}_\sigma[U]U_{(\alpha,k)}+B[U]U_{\widecheck{(\alpha,k)}}+C_{(\alpha,k)} = {}^t(R_{(\alpha,k)},S_{(\alpha,k)}) \end{equation} with $$\displaystyle \tilde{\mathcal{A}}_\sigma[U] = \begin{pmatrix} &0& -\frac{1}{\mu}G \\ &\underline{\mathfrak{a} }-\frac{1}{B_0}\nabla^{\gamma}\cdot \mathcal{K}(\sqrt{\mu}\epsilon\nabla^{\gamma}\zeta)+\sum_{\vert\alpha\vert +\vert\delta\vert\leq N} dG(\partial^{\alpha_1} b,...,\partial^{\alpha_j}b)\partial^\delta \psi &0 \end{pmatrix}.$$ and $$U_{\widecheck{(\alpha,k)}} = \sum_{j=1}^d U_{(\alpha-e_j),k} + U_{(\alpha,k-1)}$$ with $e_j$ the unit vector in the $j-th$ direction. This system is then non symmetrizable with respect to $\displaystyle \frac{1}{\epsilon}$ terms, and the controls are not uniform with respect to $\epsilon$, due to spatial derivatives of the bottom. This is the reason why we have to control the time derivatives first, and then use the structure of the equation to recover higher order derivatives. In the case of a flat bottom $\beta=0$, or almost flat bottom $\beta = O(\epsilon)$, the terms involving space derivatives of $b$ in $\tilde{\mathcal{A}}_\sigma[U]$ can be put in the residual and are easy to control. The proof of Theorem \ref{david} as considered in \cite{alvarez} gives then a time existence of order $\frac{1}{\epsilon}$.
\end{remark} \begin{proof}[Proof of Lemma \ref{lemma_control}] The system \eqref{quasilinear} can be symmetrized with respect to main order terms of size $\displaystyle \frac{1}{\epsilon}$ if we multiply it by the operator \begin{equation} \mathcal{S}^1[U]=\begin{pmatrix} &\underline{\mathfrak{a} }-\frac{1}{B_0}\nabla^{\gamma}\cdot \mathcal{K}(\epsilon\sqrt{\mu}\nabla^{\gamma}\zeta)\nabla^{\gamma}&0\\ &0&\frac{1}{\mu}G \end{pmatrix}.\label{symmetrizer1}\end{equation} This suggests to introduce \begin{align}&E_0=\frac{1}{2}\vert\zeta\vert_{H^1_\sigma}^2+\frac{1}{2\mu}(G\psi,\psi)_2+\frac{1}{2B_0}(\frac{\nabla^{\gamma}\zeta}{\sqrt{1+\epsilon^2\mu\vert\nabla^{\gamma}\zeta\vert^2}},\nabla^{\gamma}\zeta)_2,\qquad &k=0\nonumber \\ &E_k = (\mathcal{S}^1[U] U_{(k)},U_{(k)})_2 .\qquad &k\neq 0\label{energie_twisted} \end{align} The quantity $\displaystyle\sum_{k=0}^N E_k$ is uniformly equivalent to the energy $\mathcal{E}_\sigma^N$ introduced in \eqref{energie_theoreme}: \begin{lemma}\label{energy_size} There exists $M=C(\frac{1}{h_{\min}},\vert\zeta\vert_{H^{t_0+1}},\vert b\vert_{H_{t_0+1}},\frac{1}{a_0},K)$ where $C$ is a non decreasing function of its arguments, such that $$\frac{1}{M}\mathcal{E}_\sigma^N \leq \displaystyle\sum_{k=0}^N E_k\leq M\mathcal{E}_\sigma^N.$$ \end{lemma} \begin{proof}[Proof of Lemma \ref{energy_size}] This is proved in \cite{david}; we give here the main steps of the proof for the sake of completeness. \\
(i) We start to use the following inequalities (see \cite{david} Chapter 3):
\begin{equation*} (\psi,\frac{1}{\mu}G\psi)_2 \leq M_0\vert\mathfrak{P}\psi\vert_2^2\quad\text{ and }\quad \vert\mathfrak{P}\psi\vert_2^2 \leq M_0(\psi,\frac{1}{\mu}G\psi)_2 \label{equivanorme} \end{equation*} for all $\psi\in \overset{.}H{}^{1/2}(\mathbb{R}^d)$, where $M_0$ is a constant of the form $C(\frac{1}{h_{\min}},\vert\zeta\vert_{H^{t_0+1}},\vert b\vert_{H_{t_0+1}})$. The same inequality stands for space and time derivatives.
(ii)Thanks to the Rayleigh-Taylor condition \eqref{condition_time}, we have also:
$$\frac{1}{M}\frac{1}{2}\modd{\zeta}\leq \frac{1}{2}(\zeta,\underline{\mathfrak{a} }\zeta)_2 \leq M \frac{1}{2}\modd{\zeta},$$with $M$ a constant of the form $C(\frac{1}{a_0},K)$. The same inequality stands for space and time derivatives.
(iii) At last, $\mathcal{K}(\epsilon\sqrt{\mu}\nabla^{\gamma}\zeta)$ is a $d\times d$ symmetric matrix uniformly bounded with respect to time and $\epsilon$,
$$\frac{1}{M}\vert\zeta_{(k)}\vert_{H^1_\sigma}\leq (\mathcal{K}(\epsilon\sqrt{\mu}\nabla^{\gamma}\zeta)\nabla^{\gamma}\zeta_{(k)},\nabla^{\gamma}\zeta_{(k)})_2\leq M\vert\zeta_{(k)}\vert_{H^1_\sigma}$$ where $M$ is a constant of the form $C_1(K)$.
$\qquad \Box $ \end{proof}
Because the term $\mathcal{C}_k[U]U^{(k-1)}$ which appear in the equation \eqref{quasilinear} contains order two derivatives with respect to $\zeta_{(k-1)}$, the time derivative of the energies $E_k$ are actually not controlled by the energy $\mathcal{E}^N_\sigma$. To overcome this problem, we slightly adjust the twisted energy $E_k$ for $k=N$ by defining \begin{align*} F_k &= \epsilon(\mathcal{S}^2[U] U_{(k-1)},U_{(k)})_2 &\text{ if } k=N,\\ &=0&\text{ if }k\neq N, \end{align*} where \begin{equation*} \mathcal{S}^2[U] = \begin{pmatrix} &\frac{1}{B_0}\mathcal{K}_{(k)}(\epsilon\sqrt{\mu}\nablag\zeta) &0 \\ &0 &\frac{1}{\mu}dG(\epsilon\partial_t\zeta) \end{pmatrix}. \end{equation*}
The presence of the $\epsilon$ in $F_k$ is a consequence of the first time scaling $t'=t\epsilon$. The matrix operator $\mathcal{S}^2[U]$ symmetrizes the subprincipal term $\mathcal{C}_k$. One derives with respect to time this "energy". Our goal is to have, for all $0\leq k\leq N$ \begin{equation*} \frac{d}{dt}(E_k+F_k) \leq C_1(K). \end{equation*} We will at the end recover a similar estimate for the energy $E_k$ by a Young inequality in the control of $F_k$ by the $E_j$.\\
\textbf{Control of $\frac{d}{dt}E_0$}\\
One get, using the symmetry of $G$: $$\frac{dE_0}{dt} = (\partial_t\zeta,\zeta)_2+\frac{1}{B_0}(\frac{\nabla^{\gamma}\zeta}{\sqrt{1+\epsilon^2\mu\vert\nabla^{\gamma}\zeta\vert^2}},\nabla^{\gamma}\partial_t\zeta)_2+\frac{1}{\mu}(G\psi,\partial_t\psi)_2+A_1,$$ where, the commutator terms $[G,\partial_t]$ and $[\frac{1}{\sqrt{1+\epsilon^2\mu\vert\nabla^{\gamma}\zeta\vert^2}},\partial_t]$ are $$A_1 = -\big(\frac{1}{2B_0}\frac{(\epsilon^2\mu\nabla^{\gamma}\zeta\cdot\partial_t\nabla^{\gamma}\zeta) \nabla^{\gamma}\zeta}{(1+\epsilon^2\mu\vert\nabla^{\gamma}\zeta\vert^2)^{3/2}},\nabla^{\gamma}\zeta\big)_2 +\frac{1}{2\mu}(dG(\epsilon\partial_t\zeta)\psi,\psi)_2.$$ Using the equations $\eqref{ww_equation2}$ to replace $\partial_t\zeta$ and $\partial_t\psi$ in this equality, one can write
\begin{align*}\frac{dE_0}{dt} &= \frac{1}{\mu\epsilon}(G\psi,\zeta)_2-\frac{1}{\mu\epsilon}(G\psi,\zeta)_2 +\frac{1}{\mu\epsilon}\frac{1}{B_0}(\frac{\nabla^{\gamma}\zeta}{\sqrt{1+\epsilon^2\mu\vert\nabla^{\gamma}\zeta\vert^2}},\nabla^{\gamma} G\psi)_2-\frac{1}{\mu\epsilon}\frac{1}{B_0}(\frac{\nabla^{\gamma}\zeta}{\sqrt{1+\epsilon^2\mu\vert\nabla^{\gamma}\zeta\vert^2}},\nabla^{\gamma} G\psi)_2 \\&+ A_1+B_1+B_2,\end{align*} where
$$B_1 = -\frac{1}{2\mu}(\vert\nabla^{\gamma}\psi\vert^2,G\psi)_2,$$ $$B_2 = \frac{1}{\mu}\big(\frac{(G\psi+\mu\nabla^{\gamma}(\epsilon\zeta)\cdot\nabla^{\gamma}\psi)^2}{2(1+\epsilon^2\mu\vert\nabla^{\gamma} \zeta\vert^2)},\frac{1}{\mu}G\psi\big)_2.$$ The large terms of order $\displaystyle\frac{1}{\epsilon}$ cancel one another, thanks to the symmetry of the equation. One must now control $A_1,B_1$ and $B_2$ in order to get the desired estimate for $E_0$.\\
\textit{- Control of $A_1$} Let us start with the first term of $A_1$. We use the fact that $\vert\epsilon\partial_t\zeta\vert_{H^1_\sigma}$ and $\vert\zeta\vert_{H^1_\sigma}$ are bounded by $C_2(K)$, since $N\geq 2$. Moreover, since $N\geq t_0+1$, one has that $\nabla^{\gamma}\zeta\in L^\infty(\mathbb{R}^d)$ with a control by $C_2(K)$, and thus one can write, using Cauchy-Schwartz inequality, \begin{align*}\vert -\big(\frac{1}{2B_0}\frac{(\epsilon^2\mu\nabla^{\gamma}\zeta\cdot\partial_t\nabla^{\gamma}\zeta) \nabla^{\gamma}\zeta}{(1+\epsilon^2\mu\vert\nabla^{\gamma}\zeta\vert^2)^{3/2}},\nabla^{\gamma}\zeta\big)_2\vert &\leq \frac{1}{2} \vert\epsilon\partial_t\zeta\vert_{H^1_\sigma}\vert \zeta\vert_{H^1_\sigma}\vert \nabla^{\gamma}\zeta\vert_{L^\infty(\mathbb{R}^d)}^2 \\ &\leq C_2(K). \end{align*}
To control the second term of $A_1$, we use the Proposition \ref{328} in the Appendix with $s=0$ to write \begin{align*} \vert \frac{1}{2\mu}(dG(\epsilon\partial_t\zeta)\psi,\psi)_2\vert & \leq M_0\vert \epsilon\partial_t\zeta\vert_{H^{t_0+1}}\vert\mathfrak{P}\psi\vert_2\vert\mathfrak{P}\psi\vert_2, \end{align*} where $M_0$ is a constant of the form $C(\frac{1}{h_{\min}},\vert\zeta\vert_{H^{t_0+1}},\vert b\vert_{H^{t_0+1}}).$
Moreover, $\epsilon\partial_t\zeta$ is controlled in $H^{t_0+1}$ norm by $C_2(K)$, since $N\geq t_0+2$ and thus, one has \begin{align*} \vert \frac{1}{2\mu}(dG(\epsilon\partial_t\zeta)\psi,\psi)_2\vert & \leq C_2(K). \end{align*} \textit{- Control of $B_1$} One has to remark that $\nabla^{\gamma}\psi$ is $L^\infty(\mathbb{R}^d)$. Indeed, one has $$\vert\Lambda^{t_0}\nabla^{\gamma}\psi\vert_2\leq C \vert\mathfrak{P}\psi\vert_{H^{t_0+3/2}},$$ where $C$ does not depends on $\mu$ nor $\psi$. Since this last term is controlled by the energy, one has that $\nabla^{\gamma}\psi\in L^\infty(\mathbb{R}^d)$ with a control by $C_2(K)$. Thus, one can write \begin{align*} \vert\frac{1}{2\mu}(\vert\nabla^{\gamma}\psi\vert^2,G\psi)_2\vert&\leq \vert\nabla^{\gamma}\psi\vert_{L^\infty(\mathbb{R}^d)}\frac{1}{\mu}\vert G\psi\vert_2\vert\nabla^{\gamma}\psi\vert_2\\ &\leq C_2(K). \end{align*} Now, using the second point of Proposition \ref{314} with $s=1/2$, one gets that
\begin{align*}\vert\frac{1}{2\mu}(\vert\nabla^{\gamma}\psi\vert^2,G\psi)_2\vert\leq C_2(K)M(3/2)\vert\mathfrak{P}\psi\vert_{H^{1}}\end{align*} where $M(3/2)$ is a constant of the form $\displaystyle C(\frac{1}{h_0},\vert\zeta\vert_{H^{t_0+1}},\vert b\vert_{H^{t_0+1}},\vert\zeta\vert_{H^{3/2}},\vert b\vert_{H^{3/2}})$. Just note that the first point of Proposition \ref{314} does not suffice here, since we want a control of $\frac{1}{\mu}G\psi$ and not only of $\frac{1}{\mu^{3/4}}G\psi$. This is the interest of the second point of this Proposition: the Dirichlet-Neumann operator $G$ has to be seen as a $3/2$ order operator in order to be controlled uniformly with respect to $\mu$. \\
\textit{- Control of $B_2$} We already noticed in the previous controls that $\nabla^{\gamma}\zeta$ and $\nabla^{\gamma}\psi$ were in $L^\infty(\mathbb{R}^d)$ with a $C_2(K)$ control. Moreover, by Proposition \ref{314} with $s=t_0+1/2$, one gets \begin{align*}\frac{1}{\mu}\vert \Lambda^{t_0} G\psi\vert_2 &\leq M(t_0+3/2)\vert\mathfrak{P}\psi\vert_{H^{t_0+1/2}} \\ &\leq C_2(K)\end{align*} and thus $\frac{1}{\mu}G\psi$ is $L^\infty(\mathbb{R}^d)$ with a $C_2(K)$ control. Then, we write, using Cauchy-Schwartz inequality,
\begin{align*} B_2&\leq\frac{1}{\mu} \vert G\psi\vert_{L^\infty(\mathbb{R}^d)}+\epsilon\mu\vert\nabla^{\gamma}\zeta\vert_{L^\infty(\mathbb{R}^d)}\vert\nabla^{\gamma}\psi\vert_{L^\infty(\mathbb{R}^d)})\bigg( \vert\nabla^{\gamma}\psi\vert_{L^\infty(\mathbb{R}^d)}\epsilon\vert\nabla^{\gamma}\zeta\vert_2+\vert G\psi\vert_2\bigg)\frac{1}{\mu}\vert G\psi\vert_2 \\ &\leq C_2(K) \end{align*} where we used again Proposition \ref{314} to control $\frac{1}{\mu}G\psi$ in $L^2$ norm. \\
\textit{- Synthesis} To conclude, we proved that $$\frac{dE_0}{dt}\leq C_2(K),$$ which gives, by integrating in time, the following inequality: $$\forall t\in[0;T^\epsilon],\qquad E_0(t)\leq C_2(K)t+C_0$$ where $C_0$ only depends on the norm of the initial data.
\textbf{Control of $\frac{d}{dt}(E_k+F_k)$ for $k\neq 0$} \\
We deal here with the case $k=N$ (if $k<N$, there is no term of order more than $N$ that appears in the derivative of the energy). Recall that for $k=N$, we have $$E_k+F_k = (\mathcal{S}^1[U] U_{(k)},U_{(k)})_2+\epsilon(\mathcal{S}^2[U] U_{(k-1)},U_{(k)})_2.$$ Therefore, by derivating in time, and using the symmetry of $\mathcal{S}^1[U]$ we get
\begin{align*} \frac{d}{dt}(E_k+F_k) &= (\SyU_{(k)},\partial_t U_{(k)})_2+\epsilon(\mathcal{S}^2[U] U_{(k-1)},\partial_tU_{(k)})_2\\ &+(\big[\partial_t,\mathcal{S}^1[U]\big]U_{(k)},U_{(k)})_2+(\partial_t(\mathcal{S}^2[U] U_{(k-1)}),U_{(k)})_2. \end{align*} We replace $\partial_tU_{(k)}$ by its expression given in the quasilinear system \eqref{quasilinear}. One gets
\begin{align*} \frac{d}{dt}(E_k+F_k) &= (\SyU_{(k)},-\frac{1}{\epsilon}\mathcal{A}_\sigma[U]U_{(k)}-B[U]U_{(k)}-\mathcal{C}_k[U]U_{(k-1)})_2 \\ & + \epsilon(\mathcal{S}^2[U] U_{(k-1)},-\frac{1}{\epsilon}\mathcal{A}_\sigma[U]U_{(k)}-B[U]U_{(k)}-\mathcal{C}_k[U]U_{(k-1)})_2 \\ &+(\big[\partial_t,\mathcal{S}^1[U]\big]U_{(k)},U_{(k)})_2+(\partial_t(\mathcal{S}^2[U] U_{(k-1)}),U_{(k)})_2 \\ &+(\SyU_{(k)},{}^t(S_k,R_k))_2+\epsilon(\mathcal{S}^2[U] U_{(k-1)},{}^t(S_k,R_k))_2. \end{align*} Thanks to the symmetry, the large terms of size $\displaystyle \frac{1}{\epsilon}$ cancel one another, ie $$\frac{1}{\epsilon}(\SyU_{(k)},\mathcal{A}_\sigma[U]U_{(k)})_2 = 0.$$ This is fundamental and based on the fact that the evolution equation for the unknown $U_{(k)}$ is still symmetrizable with respect to $\frac{1}{\epsilon}$ terms. Again, it is not the case for the evolution equation in term of spatial derivatives $ U_{(\alpha,k)}$. Then, the commutators between $U_{(k)}$ and subprincipal terms of $\mathcal{C}_k$, which are not controlled by the energy (mainly because of the order two operators) also cancel one another, because of the choice of $F_k$. More precisely, one gets $$-(\SyU_{(k)},\mathcal{C}_k[U]U_{(k-1)})_2-(\mathcal{S}^2[U] U_{(k-1)},\mathcal{A}_\sigma[U]U_{(k)})_2=0.$$ One can also check that $$\epsilon(\mathcal{S}^2[U] U_{(k-1)},\mathcal{C}_k[U]U_{(k-1)})_2 = 0. $$To conclude, we proved that
$$\frac{d}{dt}(E_k+F_k) = A_1+A_2+A_3+A_4+B_1+B_2+B_3+B_4+B_5+B_6,$$ where $(\big[\partial_t,\mathcal{S}^1[U]\big]U_{(k)},U_{(k)})_2=A_1+A_2+A_3$ with
$$A_1 = \frac{1}{2\mu}(dG(\epsilon\partial_t\zeta)\psi_{(k)},\psi_{(k)})_2,$$ $$A_2 = \frac{1}{2B_0}(\partial_t\ \big(K(\epsilon\sqrt{\mu}\nabla^{\gamma}\zeta)\big)\nabla^{\gamma}\zeta_{(k)},\nabla^{\gamma}\zeta_{(k)}\bigg)_2,$$ $$A_3 =\frac{1}{2} ((\partial_t\underline{\mathfrak{a} })\zeta_{(k)},\zeta_{(k)})_2.$$ The term $(\partial_t(\mathcal{S}^2[U] U_{(k-1)}),U_{(k)})_2=A_4$ is given by $$A_4 = \frac{1}{\mu}(\epsilon\partial_t\big(dG(\epsilon\partial_t\zeta)(\psi_{(k-1)})\big),\psi_{(k)})_2+\frac{1}{B_0}(\epsilon\partial_t\bigg(\mathcal{K}_{(k)}[\sqrt{\mu}\nabla^{\gamma}\zeta]\zeta_{(k-1)}\bigg),\zeta_{(k)})_2.$$ We denote the terms coming from the evolution equation contained in $-(\SyU_{(k)}-B[U]U_{(k)})_2$ by
$$B_1 = (-\underline{V}\nabla^{\gamma}\cdot\zeta_{(k)},\underline{\mathfrak{a} }\zeta_{(k)})_2-\frac{1}{B_0}(\nabla^{\gamma} (\underline{V}\cdot\nabla^{\gamma}\zeta),\mathcal{K}(\sqrt{\mu}\epsilon\nabla^{\gamma}\zeta)\nabla^{\gamma}\zeta_{(k)})_2$$ and $$B_5 = -(\underline{V}\cdot\nabla^{\gamma}\psi_{(k)},\frac{1}{\mu}G\psi_k)$$ and the term $-\epsilon(\mathcal{S}^2[U] U_{(k-1)},B[U]U_{(k)})_2$ is $$B_2 = -\epsilon\frac{1}{\mu}(dG(\partial_t\zeta)\psi_{(k-1)},\underline{V}\cdot\nabla^{\gamma}\psi_{(k)})_2-\frac{1}{B_0}\epsilon(\mathcal{K}_{(k)}[\sqrt{\mu}\nabla^{\gamma}\zeta]\zeta_{(k-1)},\underline{V}\cdot\nabla^{\gamma}\zeta_{(k)})_2.$$ Finally, the residual terms $(\SyU_{(k)},{}^t(S_k,R_k))_2+\epsilon(\mathcal{S}^2[U] U_{(k-1)},{}^t(S_k,R_k))_2$ which are the most easy terms to control are denoted by $$B_3 = (R_k,\underline{\mathfrak{a} }\zeta_{(k)})_2,$$ $$B_4 = +\epsilon\frac{1}{\mu}(dG(\partial_t\zeta)\psi_{(k-1)},S_k)_2+\epsilon(\mathcal{K}_{(k)}[\sqrt{\mu}\nabla^{\gamma}\zeta]\zeta_{(k-1)},R_k)_2$$ and $$B_6 = (S_k,\frac{1}{\mu}G\psi_{(k)})_2.$$ We then need to control all these terms by a constant of the form $C_2(K)$ in order to get the desired estimate. These controls requires bounds for quantities such as $dG(h,k)\psi$ or $G\psi$, which can be obtained by the use of Propositions \ref{314}, \ref{318} and \ref{328}.
\textit{-Control of $A_1$} Using Proposition \ref{328} with $s=0$, one gets \begin{align*} \vert \frac{1}{2\mu}(dG(\epsilon\partial_t\zeta)\psi_{(k)},\psi_{(k)})_2\vert &\leq M_0\vert\epsilon\partial_t\zeta\vert_{H^{t_0+1}}\vert\mathfrak{P}\psi_{(k)}\vert_2^2\\ &\leq C_2(K) \end{align*} since $N\geq t_0+2$. \\
\textit{-Control of $A_2$} We start to check that $\partial_t(K(\epsilon\sqrt{\mu}\nabla^{\gamma}\zeta))$ is bounded in $L^\infty(\mathbb{R}^d)$ by $C_2(K)$:
\begin{align*}\partial_t(K(\epsilon\sqrt{\mu}\nabla^{\gamma}\zeta)) &= \frac{2\epsilon^2\mu\nabla^{\gamma}\zeta\cdot\partial_t\nabla^{\gamma}\zeta Id-\epsilon^2\mu(\nabla^{\gamma}\zeta\otimes\nabla^{\gamma}\partial_t\zeta+\nabla^{\gamma}\partial_t\zeta\otimes\nabla^{\gamma}\zeta)}{(1+\epsilon^2\mu\vert\nabla^{\gamma}\vert^2)^{3/2}}\\ &-\frac{3}{2}\frac{(1+\epsilon^2\mu\vert\nabla^{\gamma}\zeta\vert^2Id-\epsilon^2\mu\nabla^{\gamma}\zeta\otimes\nabla^{\gamma}\zeta)(2\epsilon^2\mu\partial_t\nabla^{\gamma}\zeta\cdot\nabla^{\gamma}\zeta)}{(1+\epsilon^2\mu\vert\nabla^{\gamma}\zeta\vert^2)^{5/2}}.\end{align*}
Because of the energy expression given by \eqref{energie_theoreme}, one has that $\nabla^{\gamma}\zeta\in L^\infty(\mathbb{R}^d)$ (since $N\geq t_0+1$), $\epsilon\partial_t\nabla^{\gamma}\zeta \in L^\infty(\mathbb{R}^d)$ (since $N\geq t_0+2$) and thus we get $$\vert \partial_t(K(\epsilon\sqrt{\mu}\nabla^{\gamma}\zeta))\vert_{L^\infty(\mathbb{R}^d)}\leq C_2(K).$$\\
Thus, one can write \begin{align*} \left\vert \frac{1}{2B_0}\bigg(\partial_t\ \big(K(\epsilon\sqrt{\mu}\nabla^{\gamma}\zeta)\big)\nabla^{\gamma}\zeta_{(k)},\nabla^{\gamma}\zeta_{(k)}\bigg)_2\right\vert &\leq C_2(K) \frac{1}{B_0}\vert\nabla^{\gamma}\zeta_{(k)}\vert^2 \\ &\leq C_2(K). \end{align*} \textit{-Control of $A_3$} We prove that $$\vert\partial_t\underline{\mathfrak{a} }\vert_{L^\infty(\mathbb{R}^d)}\leq C_2(K).$$ Recall that $$\underline{\mathfrak{a}} = 1+\epsilon(\epsilon\partial_t+\epsilon \underline{V}\cdot\nabla^{\gamma})\underline{w}.$$
By derivating with respect to time, one get $$\partial_t\underline{\mathfrak{a} } = (\epsilon\partial_t)^2\underline{w}+(\epsilon\partial_t)\underline{V}\cdot\nabla^{\gamma}\underline{w}+\underline{V}\cdot\nabla^{\gamma}(\epsilon\partial_t)\underline{w}.$$
We need to use an explicit expression of the horizontal and vertical component of the velocity at the surface $\underline{V}$ and $\underline{w}$ here:
$$\underline{w} = \frac{G\psi+\epsilon\mu\nabla^{\gamma}\zeta\cdot\nabla^{\gamma}\psi}{1+\epsilon^2\mu\vert\nabla^{\gamma}\zeta\vert^2},\quad\text{ and }\quad \underline{V} = \nabla^{\gamma}\psi-\epsilon\underline{w}\nabla^{\gamma}\zeta. $$ If one takes brutally the $\epsilon\partial_t$ and $(\epsilon\partial_t)^2$ derivatives of these expressions, one has to deal with terms of the form $$d^2G((\epsilon\partial_t)^2\zeta)\psi,\quad d^2G(\epsilon\partial_t\zeta,\epsilon\partial_t\zeta)\psi,\quad dG(\epsilon\partial_t\zeta)(\epsilon\partial_t\psi),\quad G(\epsilon\partial_t)^2\psi.$$
For the first and second terms, we use Proposition \ref{328b} with $s=t_0+1/2$ to get \begin{align*} \vert d^2G(\epsilon\partial_t\zeta,\epsilon\partial_t\zeta)\psi\vert_{H^{t_0}} &\leq M_0\mu^{3/4}\vert (\epsilon\partial_t)^2\zeta\vert_{H^{t_0+1}}^2\vert\mathfrak{P}\psi\vert_{H^{t_0+1/2}} \\ &\leq C_2(K) \end{align*} since $N\geq t_0+3$. For the third term, we apply the same result to get
\begin{align*} \vert dG(\epsilon\partial_t\zeta)(\epsilon\partial_t\psi)\vert_{H^{t_0}}\leq M_0\mu^{3/4}\vert\epsilon\partial_t\zeta\vert_{H^{t_0+1}}\vert\mathfrak{P}\epsilon\partial_t\psi\vert_{H^{t_0+1/2}}. \end{align*} There is a bit more work to achieve in order to control the term $\vert\mathfrak{P}\epsilon\partial_t\psi\vert_{H^{t_0+1/2}}$. We write, noticing that $N\geq t_0+3/2,$ \begin{align*} \vert\mathfrak{P}\epsilon\partial_t\psi\vert_{H^{t_0+1/2}} &\leq \sum_{\beta\in\mathbb{N}^d,\vert\beta\vert\leq N-1}\vert\mathfrak{P}\epsilon\partial_t\partial^\beta\psi\vert_2 \\ &\leq \sum_{\beta\in\mathbb{N}^d,\vert\beta\vert\leq N-1} \vert \mathfrak{P}\psi_{(\beta,1)}+\mathfrak{P}\epsilon\underline{w}\zeta_{(\beta,1)}\vert_2 \end{align*} using the definition of $\psi_{(\alpha,k)}$ given by \eqref{defpsia}. Using the fact that $\vert\mathfrak{P} f\vert_2\leq C\mu^{-1/4}\vert f\vert_{H^{1/2}}$, we get
\begin{align*} \vert\mathfrak{P}\epsilon\partial_t\psi\vert_{H^{t_0+1/2}} &\leq \sum_{\beta\in\mathbb{N}^d,\vert\beta\vert\leq N-1}\vert \mathfrak{P}\psi_{(\beta,1)}\vert_2+\mu^{-1/4}\epsilon\vert \underline{w}\zeta_{(\beta,1)}\vert_{H^{1/2}} \\ &\leq \sum_{\beta\in\mathbb{N}^d,\vert\beta\vert\leq N-1}\vert \mathfrak{P}\psi_{(\beta,1)}\vert_2+\mu^{-1/4}\epsilon\vert \underline{w}\vert_{H^{t_0}}C_2(K) \\ \end{align*} where we used the Sobolev estimate $\vert f g\vert_{1/2} \leq C \vert f\vert_{H^{t_0}} \vert g\vert_{H^{1/2}}$ to derive the last inequality. At last, we use Proposition \ref{314} with $\underline{w}$ (see the remark at the end of this Proposition) and with $s=t_0+1/2$ to get \begin{align*} \vert\mathfrak{P}\epsilon\partial_t\psi\vert_{H^{t_0+1/2}} &\leq \sum_{\beta\in\mathbb{N}^d,\vert\beta\vert\leq N-1}\vert \mathfrak{P}\psi_{(\beta,1)}\vert_2+\epsilon\mu^{1/2}M(t_0+1)\vert\mathfrak{P}\psi\vert_{H^{t_0+1}}C_2(K) \\ &\leq C_2(K) \end{align*} and finally we proved $$\vert dG(\epsilon\partial_t\zeta)(\epsilon\partial_t\psi)\vert_{H^{t_0}}\leq C_2(K).$$
It remains $\vert G(\epsilon\partial_t)^2\psi\vert_{H^{t_0}}$ to be controlled. We use Proposition \ref{314} to write, with $s=t_0+1/2$, $$\vert G(\epsilon\partial_t)^2\psi\vert_{H^{t_0}} \leq \mu^{3/4} M(t_0+1)\vert\mathfrak{P}(\epsilon\partial_t)^2\psi\vert_{H^{t_0+1}}$$ and we use the previous technique to prove that $\vert\mathfrak{P}(\epsilon\partial_t)^2\psi\vert_{H^{t_0+1}} \leq C_2(K)$.\\
Combining all these results, one can control all term of $\partial_t\underline{\mathfrak{a} }$ in $L^\infty$ norm by $C_2(K)$, and one get the desired result: $$\vert \partial_t\underline{\mathfrak{a} }\vert_{L^\infty(\mathbb{R}^d)}\leq C_2(K).$$ It is now easy to get \begin{align*}\vert \frac{1}{2} ((\partial_t\underline{\mathfrak{a} })\zeta_{(k)},\zeta_{(k)})_2\vert &\leq C_2(K)\vert\zeta_{(k)}\vert_2^2\\ &\leq C_2(K). \end{align*}
\textit{- Control of $A_4$} The fact that we get an $\epsilon\partial_t$ derivative here, and not only a $\partial_t$ derivative is essential here. The first term involves terms of the form $$(dG(\zeta_{(j)})\psi_{(l)},\psi_{(k)})_2$$ with $j,l\leq N$ and the Proposition \ref{328} with $s=0$ allows to control them by $C_2(K)$. The second term of $A_4$ involves $\frac{1}{B_0}\nabla^{\gamma}\zeta_{(k)}$ and $\zeta_{(k)}$ terms, which are controlled in $L^2$ norm, and other $L^{\infty}$ terms (see the control of $A_2$ for example). There is no other difficulty than computation, to control $A_2$ by $C_2(K)$. \\
\textit{-Control of $B_1$} The control of the first term requires a classical symmetry trick. We write, by integrating by parts,
\begin{align*}(\underline{V}\cdot\nabla^\gamma\zeta_{(\alpha)},\underline{\mathfrak{a} }\zeta_{(\alpha)})_2 &= -(\zeta_{(\alpha)},\nabla^\gamma\cdot(\underline{\mathfrak{a} }\zeta_{(\alpha)}\underline{V}))_2 \\ &=-(\zeta_{(\alpha)},\nabla^\gamma\cdot(\underline{\mathfrak{a} }\underline{V})\zeta_{(\alpha)})_2-(\zeta_{(\alpha)},\underline{\mathfrak{a} }\underline{V}\cdot\nabla^\gamma\zeta_{(\alpha)})_2\end{align*} and thus one gets
$$(\underline{V}\cdot\nabla^\gamma\zeta_{(\alpha)},\underline{\mathfrak{a} }\zeta_{(\alpha)})_2 = -\frac{1}{2}(\zeta_{(\alpha)},\nabla^\gamma\cdot(\underline{\mathfrak{a} }\underline{V})\zeta_{(\alpha)})_2.$$ We can use the same technique as in the control of $A_3$ to get $$\vert \nabla^{\gamma}\cdot(\underline{\mathfrak{a} }\underline{V})\vert_{L^\infty(\mathbb{R}^d)}\leq C_2(K)$$ and we then get that
\begin{align*}\vert (\underline{V}\cdot\nabla^\gamma\zeta_{(\alpha)},\underline{\mathfrak{a} }\zeta_{(\alpha)})_2\vert &\leq \vert \nabla^{\gamma}\cdot(\underline{\mathfrak{a} }\underline{V})\vert_{L^\infty(\mathbb{R}^d)} \vert \zeta_{(k)}\vert_2^2 \\ &\leq C_2(K). \end{align*}
It is the same trick for the second term. Using the symmetry of $\mathcal{K}$, we write \begin{align*} \frac{1}{B_0}(\underline{V}\nabla^{\gamma}\cdot(\nabla^{\gamma}\zeta_{(k)}),\mathcal{K}(\epsilon\sqrt{\mu}\nablag\zeta)\nabla^{\gamma}\zeta_{(k)})_2 &= \frac{1}{B_0}(\nabla^{\gamma}\cdot(\nabla^{\gamma}\zeta_{(k)}),\mathcal{K}(\epsilon\sqrt{\mu}\nablag\zeta)\underline{V}\cdot\nabla^{\gamma}\zeta_{(k)})_2 \end{align*} and, by integrating by parts, \begin{align*} \frac{1}{B_0}(\underline{V}\nabla^{\gamma}\cdot(\nabla^{\gamma}\zeta_{(k)}),\mathcal{K}(\epsilon\sqrt{\mu}\nablag\zeta)\nabla^{\gamma}\zeta_{(k)})_2 &= - \frac{1}{B_0}(\nabla^{\gamma}\zeta_{(k)},\nabla^{\gamma}\cdot(\mathcal{K}(\epsilon\sqrt{\mu}\nablag\zeta)\underline{V})\nabla^{\gamma}\zeta_{(k)})_2\\& - \frac{1}{B_0}(\nabla^{\gamma}\zeta_{(k)},\mathcal{K}(\epsilon\sqrt{\mu}\nablag\zeta)\underline{V}\nabla^{\gamma}\cdot(\nabla^{\gamma}\zeta_{(k)}))_2 \\ &=- \frac{1}{B_0}(\nabla^{\gamma}\zeta_{(k)},\nabla^{\gamma}\cdot(\mathcal{K}(\epsilon\sqrt{\mu}\nablag\zeta)\underline{V})\nabla^{\gamma}\zeta_{(k)})_2\\&-\frac{1}{B_0}(\underline{V}\nabla^{\gamma}\cdot(\nabla^{\gamma}\zeta_{(k)}),\mathcal{K}(\epsilon\sqrt{\mu}\nablag\zeta)\nabla^{\gamma}\zeta_{(k)})_2 \end{align*} and thus $$\frac{1}{B_0}(\underline{V}\nabla^{\gamma}\cdot(\nabla^{\gamma}\zeta_{(k)}),\mathcal{K}(\epsilon\sqrt{\mu}\nablag\zeta)\nabla^{\gamma}\zeta_{(k)})_2 = -\frac{1}{2B_0}(\nabla^{\gamma}\zeta_{(k)},\nabla^{\gamma}\cdot(\mathcal{K}(\epsilon\sqrt{\mu}\nablag\zeta)\underline{V})\nabla^{\gamma}\zeta_{(k)})_2.$$ We can then use the same type of computation as for the control of $A_2$ to show that $$\nabla^{\gamma}\cdot(\mathcal{K}(\epsilon\sqrt{\mu}\nablag\zeta)\underline{V}) \in L^\infty(\mathbb{R}^d)$$ with a $C_2(K)$ bound, and finally we get, by Cauchy-Schwartz's inequality \begin{align*}\vert \frac{1}{B_0}(\underline{V}\nabla^{\gamma}\cdot(\nabla^{\gamma}\zeta_{(k)}),\mathcal{K}(\epsilon\sqrt{\mu}\nablag\zeta)\nabla^{\gamma}\zeta_{(k)})_2\vert &\leq C_2(K)\frac{1}{B_0}\vert\nabla^{\gamma}\zeta_{(k)}\vert^2 \\ &\leq C_2(K). \end{align*} \textit{- Control of $B_2$} For the first term, we write $$(\underline{V}\cdot\nabla^{\gamma}\psi_{(k)},\frac{1}{\mu}dG(\epsilon\partial_t\zeta)\psi_{(k-1)}) =(\frac{\nabla^{\gamma}}{(1+\sqrt{\mu}\vert D^\gamma\vert)^{1/2}}\psi_{(k)},(1+\sqrt{\mu}\vert D^\gamma\vert)^{1/2}(\underline{V}\frac{1}{\mu}dG(\epsilon\partial_t)\psi_{(k-1)})$$ and we deduce with Cauchy-Schwartz inequality that this quantity is bounded in absolute value by $$\vert\underline{V}\vert_{H^{t_0}}\vert\mathfrak{P}\psi_{(k)}\vert_2(\frac{1}{\mu}\vert dG(\epsilon\partial_t\zeta)\psi_{(k-1)}\vert_2+\mu^{-3/4}\vert dG(\epsilon\partial_t\zeta)\vert_{H^{1/2}}).$$ We now use Proposition \eqref{328b} to control this term by $C_2(K)$. The second term of $B_2$ is controlled by using Cauchy-Schwartz inequality. \\
\textit{-Control of $B_3$} We use the control $\vert\underline{\mathfrak{a} }\vert_{L^\infty(\mathbb{R}^d)}\leq C_2(K)$ and the control over $R_k$ given by \eqref{reste} to get, with Cauchy-Schwartz inequality \begin{align*}\vert( R_k,\underline{\mathfrak{a} }\zeta_{(k)})_2\vert &\leq \vert\underline{\mathfrak{a} }\vert_{L^\infty(\mathbb{R}^d)} \vert R_k\vert_2\vert\zeta_{(k)}\vert_2 \\ &\leq C_2(K). \end{align*}
\textit{ - Control of $B_4$} It is a direct use of Cauchy-Schwartz inequality and Proposition \ref{328}, and the control over $R_k$ and $S_k$ given by \eqref{reste}. \\
\textit{- Control of $B_5$} To control this term, we use a direct application of Proposition \ref{329} to write
\begin{align*} \vert(\underline{V}\cdot\nabla^{\gamma}\psi_{(k)},\frac{1}{\mu}G\psi_{(k)})_2\vert &\leq M\vert\underline{V}\vert_{W^{1,\infty}}\vert\mathfrak{P}\psi_{(k)}\vert_2^2 \\
&\leq C_2(K). \end{align*}
\textit{- Control of $B_6$} We use Proposition \ref{318} with $s=0$ to get
\begin{align*} \vert (S_k,\frac{1}{\mu}G\psi_{(k)})_2\vert &\leq \mu M_0 \vert\mathfrak{P} S_k\vert_2\vert\mathfrak{P}\psi_{(k)}\vert_2 \\ &\leq C_2(K) \end{align*} where we used the control over $S_k$ given by \eqref{reste} to derive the last inequality. \\
\textit{- Synthesis} We proved that $$\frac{d}{dt}(E_k+F_k) \leq C_2(K)$$ and thus we get by integrating in time: $$\forall t\in [0;T^\epsilon],\quad(E_k+F_k)(t)\leq C_2(K)t+C_0$$ where $C_0$ only depends on the initial energy. It is easy to get that $$\vert F_k\vert \leq C_2(K)\epsilon$$ and thus $$\forall t\in [0;T^\epsilon],\quad E_k(t) \leq C_2(K)(t+\epsilon)+C_0$$ Thanks to the equivalence between $E_k$ and the initial energy for this problem $\mathcal{E}^k_\sigma$, introduced for Theorem \ref{uniform_result} by \eqref{energie_theoreme}, proved in Remark \ref{energy_size}, we conclude to the desired result : $$\forall t\in [0;T^\epsilon],\forall 0\leq k\leq N,\quad\vert \zeta_{(k)}\vert_{H^1_\sigma}+\vert \mathfrak{P}\psi_{(k)}\vert_2 \leq C_2(K)(t+\epsilon)+C_0$$
$\qquad \Box $ \end{proof} The Proposition \ref{lemma_control} gives the desired estimate of the form \eqref{desired} for time derivatives (more precisely, for $\vert \zeta_{(k)}\vert_{H^1_\sigma}+\vert\mathfrak{P}\psi_{(k)}\vert_2$). We want to recover the same estimate for space derivatives, using directly the equation \eqref{ww_equation2}. To this purpose, one has to make precisely the link between the $H^1$ norm of $\mathfrak{P}\psi_{(k)}$ and the $H^1_\sigma$ norm of $\zeta_{(k+1)}$. This is the point of the following Lemma.
\begin{lemma}\label{lemma_recover} For all $0\leq\vert\alpha\vert\leq N$, one has: $$\vert \mathfrak{P}\psi_{(\alpha,k)}\vert_2 + \vert \zeta_{(\alpha,k)}\vert_{H^1_\sigma} \leq C_1(K)(t+\epsilon)+C_0$$ where $C_0$ only depends on the initial energy of the unknowns, for all $0\leq k\leq N-\vert\alpha\vert$. \end{lemma} \begin{proof}[Proof of Lemma \ref{lemma_recover}] The proof is done by induction on $\vert\alpha\vert$. For $\vert\alpha\vert=0$ it is the Lemma \ref{lemma_control}. \\
Assume that the result is true for $\vert\alpha\vert-1$, with $\vert\alpha\vert \geq 1$. Let $0\leq k\leq N-\vert\alpha\vert$. We start to give an evolution equation in term of unknowns $\zeta_{(\alpha,k)}$ and $\psi_{(\alpha,k)}$ :
\begin{align} \begin{cases} \displaystyle \partial_t\zeta_{(\alpha,k)}+\underline{V}\cdot\nabla^{\gamma}\zeta_{(\alpha,k)}-\frac{1}{\mu\epsilon}G\psi_{(\alpha,k)}-\frac{1}{\mu\epsilon}\sum dG(\partial^{l_1}\beta b,...,\partial^{l_i}\beta b,\partial^{m_1}\epsilon\zeta,...,\partial^{m_j}\epsilon\zeta) \partial^\delta \psi \\ \displaystyle \partial_t\psi_{(\alpha,k)}+\underline{V}\cdot\nabla^{\gamma}\psi_{(\alpha,k)}+\frac{1}{\epsilon}\underline{\mathfrak{a} } \zeta_{(\alpha,k)}-\frac{1}{\epsilon B_0}\nabla^{\gamma}\cdot\mathcal{K}(\sqrt{\mu}\nabla^{\gamma}\zeta)\nabla^{\gamma}\zeta_{(\alpha,k)} + \mathcal{K}_{(\alpha)}[\epsilon\sqrt{\mu}\nabla^{\gamma}\zeta]\zeta_{\langle\widecheck{(\alpha,k)}\rangle}=S_k \label{quasilinear2} \end{cases} \end{align} where the summation in the first equation is over the index $(i,j,l_1,...,l_j,m_1,...,m_j,\delta)$ satisfying $$1\leq i+j,\vert l_1+...+l_i+m_1+...+m_j\vert+\delta =\vert \alpha\vert+k.$$ We used the notation $$(\zeta_{\langle\widecheck{(\alpha,k)}\rangle} = (\zeta_{(\widecheck{\alpha}^1)},...,\zeta_{(\widecheck{\alpha}^d)},\zeta_{(\alpha,k-1)})$$ with $\widecheck{\alpha}^j=\alpha-e_j$, where $e_j$ denote the unit vector in the $j$-th direction of $\mathbb{R}^d$, and where $$\vert \mathfrak{P} S_k\vert_2\leq C_1(K).$$ This system is very similar to the evolution equation \eqref{quasilinear} in term of unknowns $\zeta_{(k)},\psi_{(k)}$, except that the space derivatives of the bottom $b$ appears from the derivation of $G\psi$, and the same goes for the space derivatives of $\zeta$ in the derivation of the surface tension term $\kappa_\gamma$. Now, using Proposition \ref{equivanorme}, one can write :
\begin{align*} \vert \mathfrak{P}\psi_{(\alpha,k)}\vert_2^2+\vert\zeta_{(\alpha,k)}\vert_{H^1_\sigma}^2 \leq C_1(K) \frac{1}{\mu} \big(G\psi_{(\alpha,k)},\psi_{(\alpha,k)})_2+ (\underline{\mathfrak{a} } \zeta_{(\alpha,k)}-\frac{1}{B_0}\nabla^{\gamma}\mathcal{K}(\sqrt{\mu}\epsilon\nabla^{\gamma}\zeta)\nabla^{\gamma}\zeta_{(\alpha,k)},\zeta_{(\alpha,k)})_2\big) \end{align*} We now use the evolution equation \eqref{quasilinear2} in order to express $\psi_{(\alpha,k)}$ and $\zeta_{(\alpha,k)}$ with respect to time derivatives plus over terms of size $\epsilon$:
\begin{align} \vert \mathfrak{P}\psi_{(\alpha,k)}\vert_2^2+\vert\zeta_{(\alpha,k)}\vert_{H^1_\sigma}^2 &\leq C_1(K) \bigg( \epsilon\partial_t\zeta_{(\alpha,k)}+\epsilon\underline{V}\cdot\nabla^{\gamma}\zeta_{(\alpha,k)}\nonumber\\ &-\frac{1}{\mu}\sum dG(\partial^{l_1}\beta b,...,\partial^{l_i}\beta b,\partial^{m_1}\epsilon\zeta,...,\partial^{m_j}\epsilon\zeta) \partial^\delta \psi,\psi_{(\alpha,k)} \bigg)_2\nonumber\\ &+\bigg(-\epsilon\partial_t\psi_{(\alpha,k)}-\epsilon\underline{V}\cdot\nabla^{\gamma}\psi_{(k)}-\frac{\epsilon}{B_0}\mathcal{K}(\alpha,k)[\sqrt{\mu}\nabla^{\gamma}\zeta]\zeta_{\langle\widecheck{(\alpha,k)}\rangle}+S_k,\zeta_{(\alpha,k)}\bigg)_2. \label{recovery_calcul} \end{align}
Let us control the first term of the r.h.s. of \eqref{recovery_calcul}. One has to express this term with respect to $\mathfrak{P}\psi_{(\alpha,k)}$. To this purpose, we assume for convenience that $\alpha^1\neq 0$ (recall that $\vert\alpha\vert\geq 1)$. One computes :
\begin{align*} \vert( \epsilon\partial_t\zeta_{(\alpha,k)},\psi_{(\alpha,k)})_2\vert &= (\partial^\alpha \zeta_{(k+1)},\psi_{(\alpha,k)})_2\vert \\ &= \vert (\partial^{\alpha-e_1}\zeta_{(k+1)},\partial^{e_1}\psi_{(\alpha,k)})_2\vert \\ &\leq (\vert\zeta_{(\alpha-e_1,k+1)}\vert,\vert\vert D^\gamma\vert\psi_{(\alpha,k)}\vert)_2 \\ &\leq \left\vert \frac{(1+\sqrt{\mu}\vert D^\gamma\vert)^{1/2}}{(1+\frac{1}{\sqrt{B_0}}\vert D^\gamma\vert)}(1+\frac{1}{\sqrt{B_0}}\vert D^\gamma\vert)\zeta_{(\alpha-e_1,k+1)}\right\vert_2 \left\vert \frac{\vert D^\gamma\vert}{(1+\sqrt{\mu}\vert D^\gamma\vert)^{1/2}}\psi_{(\alpha,k)}\right\vert_2 \end{align*}
Now, remark that $$ \frac{(1+\sqrt{\mu}\vert D^\gamma\vert)^{1/2}}{(1+\frac{1}{\sqrt{B_0}}\vert D^\gamma\vert)} \leq C(B_0\mu)$$ where $C(B_0\mu)$ is a constant that only depends on $B_0\mu$ (recall here that $0\leq \gamma \leq 1$ and the definition of $\vert D^\gamma\vert$ given in section \ref{notations}). It comes: \begin{align*} ( \epsilon\partial_t\zeta_{(\alpha,k)},\psi_{(\alpha,k)})_2 &\leq C_1(K) \vert \zeta_{(\alpha-e_1,k+1)}\vert_{H^1_\sigma}\vert \mathfrak{P}\psi_{(\alpha,k)}\vert_2 \\ &\leq (C_1(K)(t+\epsilon)+C_0)\vert \mathfrak{P}\psi_{(\alpha,k)}\vert_2 \end{align*} where we used the induction assumption to control $\vert \zeta_{(\alpha-e_1,k+1)}\vert_{H^1_\sigma}$, since $\vert\alpha-e_1\vert \leq \vert\alpha\vert-1$. \\
For the control of the third term of the r.h.s. of \eqref{recovery_calcul}, one can prove, using Proposition \ref{328} that for $j\neq 0$, one has (see also \cite{david} Chapter 4 for details): $$\frac{1}{\mu}(dG(\partial^{l_1}\beta b,...,\partial^{l_i}\beta b,\partial^{m_1}\epsilon\zeta,...,\partial^{m_j}\epsilon\zeta) \partial^\delta \psi,\psi_{(\alpha,k)})_2= \epsilon R$$ with $$R\leq C_1(K).$$ For $j=0$, we use again Proposition \ref{328} to write :
\begin{align*} \frac{1}{\mu}(dG(\partial^{l_1}\beta b,...,\partial^{l_i}\beta b) \partial^\delta \psi,\psi_{(\alpha,k)})_2 &\leq C_1(K) \vert\partial^{l_1}\beta b\vert_{H^{t_0}}...\vert\partial^{l_i}\beta b\vert_{H^{t_0}} \vert \mathfrak{P}\partial^\delta\psi\vert_2\vert\mathfrak{P} \psi_{(\alpha,k)}\vert_2 \end{align*} Now, recall that $b\in H^{N+1\vee t_0+1}$. In order to prove that this last term is controlled by $C_1(K)(t+\epsilon)+C_0$, one computes, using the definition of $\psi_{(\alpha)}$ given by \eqref{defpsia}:
\begin{align*} \vert\mathfrak{P}\partial^\delta\psi\vert_2 &\leq \vert \mathfrak{P}\psi_{(\delta)}\vert_2+\epsilon\vert\mathfrak{P} \underline{w}\zeta_{(\delta)}\vert_2 \\ &\leq \vert \mathfrak{P}\psi_{(\delta)}\vert_2+\epsilon\vert \underline{w}\vert_{H^{t_0}}\vert \zeta_{(\delta)}\vert_{H^1} \end{align*} where we used the identity $\vert \mathfrak{P} f\vert_2\leq \vert f\vert_{H^1}$. Now, the Proposition \ref{314} gives $\vert\underline{w}\vert \leq \mu^{3/4} C_1(K)\vert \mathfrak{P}\psi\vert_{H^{t_0+3/2}}$. One can use the induction assumption, because the term $\psi_{(\delta)}$ only contains spatial derivatives of $\psi$ of order less than $\vert\alpha\vert-1$ due to the fact that $i\geq 1$ (the only term with spatial derivative of order $\vert \alpha\vert$ of $\psi$ that appears in the system \eqref{quasilinear2} is $G\psi_{(\alpha,k)}$). One gets :
$$\vert\mathfrak{P}\partial^\delta\psi\vert_2 \leq C_1(K)(t+\epsilon)+C_0,$$ and finally
$$(dG(\partial^{l_1}\beta b,...,\partial^{l_i}\beta b,\partial^{m_1}\epsilon\zeta,...,\partial^{m_j}\epsilon\zeta) \partial^\delta \psi,\psi_{(\alpha,k)})_2 \leq \big(C_1(K)(t+\epsilon)+C_0\big)\vert\mathfrak{P}\psi_{(\alpha,k)}\vert.$$
Now, we focus on the most difficult remaining term of \eqref{recovery_calcul}, which is $(-\epsilon\partial_t\psi_{(\alpha,k)},\zeta_{(\alpha,k)})_2$. One uses the definition of $\psi_{(\alpha,k)}$ given by \eqref{defpsia} $\psi_{(\alpha,k)} = \partial^\alpha(\epsilon\partial_t)^k\psi-\epsilon\underline{w}\partial^\alpha(\epsilon\partial_t)^k\zeta$ to write :
\begin{align*} \vert(\epsilon\partial_t\psi_{(\alpha,k)},\zeta_{(\alpha,k)})_2 \vert&=\vert(\partial_1\psi_{(\alpha-e_1,k+1)},\zeta_{(\alpha,k)})_2 + (\epsilon(\partial_1\underline{w})\zeta_{(\alpha-e_1,k+1)}-(\epsilon\partial_t)(\epsilon\underline{w})\zeta_{(\alpha,k)},\zeta_{(\alpha,k)})_2\vert \\ &\leq \vert \mathfrak{P} \psi_{(\alpha-e_1,k+1)}\vert_2 \vert(\frac{1+\sqrt{\mu}\vert D^\gamma\vert)^{1/2}}{1+\frac{1}{\sqrt{B_0}}\vert D^\gamma\vert} (1+\frac{1}{\sqrt{B_0}}\vert D^\gamma\vert)\zeta_{(\alpha,k)}\vert_2 + \epsilon C_1(K) \end{align*}
We now use the induction assumption to control $ \vert \mathfrak{P} \psi_{(\alpha-e_1,k+1)}\vert_2$ by $C_1(K)(t+\epsilon)+C_0$, and the identity $$ \frac{(1+\sqrt{\mu}\vert D^\gamma\vert)^{1/2}}{(1+\frac{1}{\sqrt{B_0}}\vert D^\gamma\vert)} \leq C(B_0\mu)$$ where $C(B_0\mu)$ is a constant that only depends on $B_0\mu$. One gets:
\begin{align*} \vert(\epsilon\partial_t\psi_{(\alpha,k)},\zeta_{(\alpha,k)})_2 \vert &\leq (C_1(K)(t+\epsilon)+C_0)(1+\vert \zeta_{(\alpha,k)}\vert_{H^1_\sigma}) \end{align*}
To conclude, we proved :
$$\vert \psi_{(\alpha,k)}\vert^2+\vert \zeta_{(\alpha,k)}\vert_{H^1_\sigma}^2 \leq (C_1(K)(t+\epsilon)+C_0)(1+\vert\psi_{(\alpha,k)}\vert_2+\vert \zeta_{(\alpha,k)}\vert_{H^1_\sigma})$$ and one can use the Young's inequality to get the desired estimate, and the Lemma \ref{lemma_recover} is true at rank $\vert\alpha\vert$.
$\qquad \Box $\end{proof}
According to the definition of $\mathcal{E}_\sigma^N (U)$ given by \eqref{energie_theoreme}, there is a remaining term to be controlled: it is $\vert\mathfrak{P}\psi\vert_{H^{t_0+3/2}}$. The control can be done as follows: let $r_0$ be such that $t_0+3/2\leq r_0 \leq N-1$. We have, for all $r\in\mathbb{N}^{d+1}$, $\vert r\vert = r_0$,
$$\vert\mathfrak{P}\partial^r\psi\vert_2 \leq
\vert\mathfrak{P}\psi_{(r)}\vert_2+\epsilon\vert\mathfrak{P}\underline{w}\partial^r\zeta \vert_2 $$ with the definition of $\psi_{(r)}$. The first term of the right hand side is controlled by previous estimates. For the second term, we use again the same technique as in the control of $A_4$ to write \begin{align*}\vert\mathfrak{P}\partial^r\psi\vert_2&\leq C_1(K)t+C_0+\epsilon\mu^{-1/4}\vert\underline{w}\partial^r\zeta\vert_{H^{1/2}} \\ &\leq C_1(K)t+C_0+\epsilon\mu^{-1/4}\vert\underline{w}\vert_{H^{t_0+1/2}}\vert\zeta\vert_{H^N} \\ &\leq C_1(K)t+C_0+\epsilon\mu^{-1/4} \mu^{3/4}\vert\mathfrak{P}\psi\vert_{H^{t_0+1}} \\ &\leq C_1(K)t+C_0+\epsilon C_1(K). \end{align*}
The result then comes from the identity $$\vert\mathfrak{P}\psi\vert_{H^{t_0+3/2}} \leq \vert\mathfrak{P}\psi\vert_{H^{r_0}} \lesssim \vert\mathfrak{P}\psi\vert_2 + \underset{\vert r\vert = r_0}{\sum}\vert\mathfrak{P}\partial^r\psi\vert_2.$$
Finally, we prove $$\mathcal{E}^N(U)(t)\leq C_0+C_1(K)(t+\epsilon)$$ for all $t$ in $[0;T_\epsilon]$, which closes the proof of Proposition \ref{main_result}.
\end{proof}$\qquad \Box $
We can now prove Theorem \ref{uniform_result} by constructing an existence time for all solutions $U^\epsilon$ of the system $\eqref{ww_equation2}$, which does not depend on $\epsilon$. \begin{proof}[Proof of Theorem \ref{uniform_result}] We define $$\epsilon_0 = \frac{1}{2C_1(2C_0)}.$$ Let fix a $\epsilon \leq \epsilon_0$. Let us consider \begin{equation*} T_{\epsilon}^* = \underset{t>0}{\sup}\lbrace t,U^{\epsilon}\text{ exists on } \left[ 0,t\right]\text{ and } \mathcal{E}^N(U^{\epsilon})(t) \leq 2C_0,1+\epsilon\zeta(t)-\beta b\geq h_{\min}/2,\underline{\mathfrak{a} }(t)\geq a_0/2\text{ on }\left[ 0,t\right]\rbrace. \end{equation*} We know that $T_{\epsilon}^*$ exists and that $U^{\epsilon}$, solution to (\ref{ww_equation1}) exists on $\left[ 0,T_{\epsilon}^*\right]$. The Proposition \ref{main_result} gives the following estimate: \begin{equation*} \mathcal{E}_{\alpha}(U^{\epsilon})(t) \leq C_1(K)(t+\epsilon) +C_0\quad\forall t\in\left[ 0,T_{\epsilon}^* \right]. \end{equation*} We then consider \begin{equation*} T_0 = \frac{1}{2C_1(2C_0)}\inf\lbrace1,h_{\min}\rbrace. \end{equation*} Let us show that $T_0\leq T_{\epsilon}^*$. Suppose $T_{\epsilon}^* < T_0$. First of all, let us prove that the condition over the height $1+\epsilon\zeta-\beta b$ is satisfied. One can write for all $0\leq t\leq T_0$\begin{align*} (1+\epsilon\zeta-\beta b)(t) &= (1+\epsilon\zeta-\beta b)(0)+\int_0^t \partial_t (1+\epsilon\zeta-\beta b)(s)ds \\ &\geq (1+\epsilon\zeta-\beta b)(0)-t\underset{s\in[0;t]}{\sup} \vert\partial_t( 1+\epsilon\zeta-\beta b)\vert_{L^\infty(\mathbb{R}^d)} \\ &\geq (1+\epsilon\zeta-\beta b)(0)-tC_1(K) \\ &\geq h_{\min}-T_0C(2C_0) \\ &\geq \frac{h_{\min}}{2}. \end{align*}One can do the same for the Rayleigh-Taylor condition $\underline{\mathfrak{a} }(t)\geq \frac{a_0}{2}$. Now, for the energy condition, we would have for all $t\in\left[ 0,T_{\epsilon}^*\right]$: \begin{align*} \mathcal{E}_{\alpha}(U^{\epsilon})(t) &\leq C_0+C_1(K)(t+\epsilon) \\ &\leq C_0 + C_1(2C_0)(T_{\epsilon}^*+\frac{1}{2C_1(2C_0)}) \\&< C_0 + C_1(2C_0)(T_0+\frac{1}{2C_1(2C_0)}) \\&< 2C_0 \end{align*} using the monotony of $C$ and the definition of $T_{\epsilon}^*$. We can therefore continue the solution to an interval $\left[ 0,\widetilde{T_{\epsilon}^*}\right]$ such that
\begin{equation*} \mathcal{E}_{\alpha}(U^{\epsilon})(t) \leq 2C_0,\quad\quad\forall t\in \left[ 0,\widetilde{T_{\epsilon}^*} \right] \end{equation*} which contradicts the definition of $T_{\epsilon}^*$. We then have $T_0\leq T_{\epsilon}^*$. \\\\Conclusion: the solution $U^\epsilon$ exists on the time interval $[0;T_0]$. \end{proof}
$\qquad \Box $
\subsection{Shallow Water limit}\label{shallow_section} We discuss here the size of the existence time for solutions of the Water Waves equation \eqref{ww_equation3} when the shallowness parameter $\mu$ goes to zero. This regime corresponds to the Shallow Water model:
\begin{align}\begin{cases}\label{shallow_eq} \partial_t\zeta+\nabla^{\gamma}\cdot(h\overline{V})=0 \\ \partial_t\overline{V}+\nabla^{\gamma}\zeta+\epsilon(\overline{V}\cdot\nabla^{\gamma})\overline{V} = 0 \end{cases}\end{align} where $h=1+\epsilon\zeta-\beta b$ is the height of the Water. Using the existence time given by Theorem \ref{uniform_result}, one can deduce easily the following long time existence result for the Shallow Water problem, proved in \cite{bresch_metivier}:
\begin{theorem} Let $t_0>d/2$. Let $(\overline{V}_0,\zeta_0)\in H^{t_0+1}(\mathbb{R}^d)^{d+1}$. Then, there exists $T>0$ and a unique solution $(\overline{V},\zeta)\in C([0;\frac{T}{\epsilon}];H^{t_0+1}(\mathbb{R}^d)^{d+1})$ to the Shallow Water equation \eqref{shallow_eq} with initial condition $(\overline{V}_0,\zeta_0)$, with $$\frac{1}{T} = C_1,\quad\text{ and } \underset{t\in [0;\frac{T}{\epsilon}]}{\sup} \vert (\zeta , \overline{V})(t)\vert_{H^{t_0+1}(\mathbb{R}^d)^{d+1}}=C_2$$ where $C_i = C(\frac{1}{h_{\min}},\vert (\zeta_0,\overline{V}_0)\vert_{H^{t_0+1}(\mathbb{R}^d)^{d+1}})$ is a non decreasing function of its arguments. \end{theorem} The result of \cite{bresch_metivier} can therefore be understood as a particular endpoint of our main result. It is important to notice that the time existence provided here for the solutions of Shallow Water equations does not depend on the bathymetric parameter $\beta$.
\begin{proof}
Let us fix $\epsilon >0$ during all the proof. We also set $\mu B_0 = 1$. We recall that Theorem \ref{uniform_result} gives a solution $U^\mu = (\zeta^\mu,\psi^\mu)$ to the Water Waves equation \eqref{ww_equation3}, on a time interval $[0;\frac{T}{\epsilon}]$ with\begin{equation}\label{shallow_bound}\frac{1}{T}=C_1\quad\text{ and }\quad\underset{t\in [0;\frac{T}{\epsilon}]}{\sup} \sum_{\vert (\alpha,k)\vert\leq N} \vert \mathfrak{P}\psi_{(\alpha,k)}^\mu\vert_2+\vert \zeta_{(\alpha,k)}^\mu\vert_2 =C_2\end{equation} where $C_i = C(\mathcal{E}_\sigma^N(U^0),\frac{1}{h_{\min}},\frac{1}{a_0},\vert b\vert_{H^{N+1\vee t_0+1}},\mu B_0)$ is a non decreasing function of its arguments. \\
We now need an asymptotic expansion of $\nabla^{\gamma}\psi^\mu$ and $G\psi^\mu$ with respect to the vertical mean of the horizontal component of the velocity $V = \nabla^{\gamma}\Phi$ in shallow water:
\begin{equation}\label{asymptotics} G\psi^\mu = -\mu\nabla^{\gamma}\cdot(h^\mu\overline{V}^\mu)\quad \text{ and }\quad \nabla^{\gamma}\psi^\mu = \overline{V}^\mu +\mu R, \end{equation} with $\vert R\vert_{H^{t_0+1}} \leq C_2$ (recall that $N\geq t_0+t_0\vee 2+3/2$ with $t_0>d/2$) and $\displaystyle \overline{V}^\mu = \frac{1}{h^\mu} \int_{-1+\beta b}^{\epsilon\zeta} V^\mu (z)dz$. For a complete proof of this latter result, see \cite{david} Chapter 3. Now, we take the first equation of \eqref{ww_equation3}, and the gradient of the second equation of \eqref{ww_equation3} and we replace $G\psi^\mu$ and $\nabla^{\gamma}\psi^\mu$ by the expressions given by \eqref{asymptotics}. The surface tension term is of size $\mu$ since $\frac{1}{B_0}=\mu$. One can check that we get the following equation, satisfied in the distribution sense of $D'([0;\frac{T}{\epsilon}]\times\mathbb{R}^d)$:
\begin{align}\begin{cases}\label{shallow_reste} \partial_t\zeta^\mu +\nabla^{\gamma}\cdot(h^\mu\overline{V}^\mu) \\ \partial_t \overline{V}^\mu+\epsilon \overline{V}^\mu\cdot\nabla^{\gamma}\overline{V}^\mu +\nabla^{\gamma}\zeta^\mu = \mu R \end{cases} \end{align} with $\vert R\vert_{H^{t_0+1}} \leq C_2$. It is then easy to show that the sequences $(\overline{V}^\mu)_{\mu}$ and $(\zeta^\mu)_{\mu}$ are bounded in $W^{1,\infty}([0;\frac{T}{\epsilon}];H^{t_0+1}(\mathbb{R}^d))$, using the bound given by \eqref{shallow_bound}. Therefore, up to a subsequence we get the weak convergence of $(\overline{V}^\mu,\zeta^\mu)_{\mu}$ and $(\zeta^\mu)_{\mu}$ to an element $(\overline{V},\zeta)\in C([0;\frac{T}{\epsilon}];H^{t_0+1}(\mathbb{R}^d)^{d+1})$. The weak convergence of the linear terms of the equation \eqref{shallow_reste} does not raise any difficulty. Since $H^{t_0+1}(\mathbb{R}^d)$ is embedded in $C^1(\mathbb{R}^d)$, we also get the convergence of the non linear terms in the equation \eqref{shallow_reste}. Finally, the limit $(\overline{V},\zeta)$ satisfies the Shallow Water equations \eqref{shallow_eq} in the distribution sense of $D'([0;\frac{T}{\epsilon}]\times\mathbb{R}^d)$. The uniqueness is classical for this kind of symmetrizable quasi-linear hyperbolic system, and is done for instance in \cite{taylor3} Chapter XVI. $\qquad \Box $ \end{proof}
Let us give a qualitative explanation of this latter result. We recall that Proposition \ref{energy_size} claims that $$\frac{1}{M_0}\vert\mathfrak{P}\psi\vert_2\leq(\psi,\frac{1}{\mu}G\psi)_2 \leq M_0 \vert\mathfrak{P}\psi\vert_2^2$$ where $$\mathfrak{P} = \frac{\vert D^\gamma\vert}{(1+\sqrt{\mu}\vert D^\gamma\vert)^{1/2}}$$ and therefore $\frac{1}{\mu}G$ acts like an order one operator with respect to $\psi$. The idea of the proof for Theorem \ref{uniform_result} is to get a "good" energy estimate for time derivatives: $$\frac{1}{2}\vert\partial_t\mathfrak{P}\psi\vert_2+\frac{1}{2}\vert\partial_t\zeta\vert_2 \leq C(K)t\epsilon+C_0$$ and then use the equation to recover the same estimate for space derivatives. Using the first equation $\partial_t\zeta=\frac{1}{\mu}G\psi$, and the ellipticity of the order one operator $\frac{1}{\mu}G$ would only provide us a gain of half a space derivative for $\psi$ (we already have an estimate for $\mathfrak{P}\psi$ where $\mathfrak{P}$ is of order $1/2$). This is why we need surface tension which provides an additional estimate for $\frac{1}{B_0}\vert\partial_t\nabla^{\gamma}\zeta\vert_2$, which leads to recover exactly one space derivative in the estimate of $\psi$. In the shallow water regime, using \eqref{asymptotics}, one gets that $\frac{1}{\mu}G\psi \sim -\nabla^{\gamma}\cdot(h\nabla^{\gamma}\psi)$ as $\mu$ goes to zero. Therefore, when $\mu$ goes to zero, $\frac{1}{\mu}G\psi$ goes to an order one operator with respect to $\overline{V} = \nabla^{\gamma}\psi$. \footnote{Another way to see it is that in the limit $\mu\rightarrow 0$, $\mathfrak{P}$ must be seen as a first order operator ($\mathfrak{P}\sim \vert D^\gamma\vert$) and the control of $\mathfrak{P}\psi$ gives the control of a full derivative of $\psi$.} Therefore, if one has a "good" estimate for time derivatives like $$\frac{1}{2}\vert\partial_t\zeta\vert_2+\frac{1}{2}\vert\partial_t \overline{V}\vert_2\leq Ct\epsilon+C_0$$ with $C$ independent of $\epsilon$, one can recover, using the first equation, a "good" estimate for $\nabla^{\gamma}\cdot(\overline{V})$. Using the second equation to get the same good estimate for $\mbox{curl}\overline{V}$, one can recover exactly one space derivative of $\overline{V}$ in the estimates. And therefore, we does not need surface tension in this case. Moreover, the surface tension term of size $\frac{1}{B_0} = \mu$ vanishes as $\mu$ goes to zero.
\appendix
\section{The Dirichlet Neumann Operator}\label{appendixA} Here are for the sake of convenience some technical results about the Dirichlet Neumann operator, and its estimates in Sobolev norms. See \cite{david} Chapter 3 for complete proofs. The first two propositions give a control of the Dirichlet-Neumann operator.
\begin{proposition}\label{314} Let $t_0$>d/2, $0\leq s \leq t_0+3/2$ and $(\zeta,\beta)\in H^{t_0+1}\cap H^{s+1/2}(\mathbb{R}^d)$ such that \begin{equation*} \exists h_0>0,\forall X\in\mathbb{R}^d, \epsilon\zeta(X)-\beta b(X) +1 \geq h_0\end{equation*}
(1)\quad The operator $G$ maps continuously $\overset{.}H{}^{s+1/2}(\mathbb{R}^d)$ into $H{}^{s-1/2}(\mathbb{R}^d)$ and one has \begin{equation*} \vert G\psi\vert_{H^{s-1/2}} \leq \mu^{3/4} M(s+1/2) \vert\mathfrak{P}\psi\vert_{H^s}, \end{equation*} where $M(s+1/2)$ is a constant of the form $C(\frac{1}{h_0},\vert\zeta\vert_{H^{t_0+1}},\vert b\vert_{H^{t_0+1}},\vert\zeta\vert_{H^{s+1/2}},\vert b\vert_{H^{s+1/2}})$. \\
(2)\quad The operator $G$ maps continuously $\overset{.}H{}^{s+1}(\mathbb{R}^d)$ into $H{}^{s-1/2}(\mathbb{R}^d)$ and one has \begin{equation*} \vert G\psi\vert_{H^{s-1/2}} \leq \mu M(s+1) \vert\mathfrak{P}\psi\vert_{H^s+1/2}, \end{equation*} where $M(s+1)$ is a constant of the form $C(\frac{1}{h_0},\vert\zeta\vert_{H^{t_0+1}},\vert b\vert_{H^{t_0+1}},\vert\zeta\vert_{H^{s+1}},\vert b\vert_{H^{s+1}})$. \\
Moreover, it is possible to replace $G$ by $\underline{w}$ in the previous result, where $\underline{w} = \frac{G\psi+\epsilon\mu\nabla^{\gamma}\zeta\cdot\nabla^{\gamma}\psi}{1+\epsilon^2\mu\vert\nabla^{\gamma}\zeta\vert^2}$(vertical component of the velocity $U=\nabla_{X,z}\Phi$ at the surface). \end{proposition}
\begin{proposition}\label{318} Let $t_0>d/2$, and $0\leq s\leq t_0+1/2$. Let also $\zeta,b\in H^{t_0+1}(\mathbb{R}^d)$ be such that $$\exists h_0>0, \forall X\in\mathbb{R}^d, 1+\epsilon\zeta(X)-\beta b(X) \geq h_0$$ Then, for all $\psi_1$, $\psi_2\in \overset{.}H{}^{s+1/2}(\mathbb{R}^d)$, we have $$(\Lambda^sG\psi_1,\Lambda^s\psi_2)_2 \leq\mu M_0 \vert \mathfrak{P}\psi_1\vert_{H^s}\vert \mathfrak{P}\psi_2\vert_{H^s},$$ where $M_0$ is a constant of the form $C(\frac{1}{h_0},\vert\zeta\vert_{H^{t_0+1}},\vert b\vert_{H^{t_0+1}})$. \end{proposition}
The second result gives a control of the shape derivatives of the Dirichlet-Neumann operator. More precisely, we define the open set $\bold{\Gamma}\subset H^{t_0+1}(\mathbb{R}^d)^2$ as:\\ $$\bold{\Gamma} =\lbrace \Gamma=(\zeta,b)\in H^{t_0+1}(\mathbb{R}^d)^2,\quad \exists h_0>0,\forall X\in\mathbb{R}^d, \epsilon\zeta(X) +1-\beta b(X) \geq h_0\rbrace$$ and, given a $\psi\in \overset{.}H{}^{s+1/2}(\mathbb{R}^d)$, the mapping: \begin{equation}\label{mapping}G[\epsilon\cdot,\beta\cdot] : \left. \begin{array}{rcl} &\bold{\Gamma} &\longrightarrow H^{s-1/2}(\mathbb{R}^d) \\ &\Gamma=(\zeta,b) &\longmapsto G[\epsilon\zeta,\beta b]\psi. \end{array}\right.\end{equation} We can prove the differentiability of this mapping. The following Theorem gives a very important explicit formula for the first-order partial derivative of $G$ with respect to $\zeta$:
\begin{theorem}\label{321} Let $t_0>d/2$. Let $\Gamma = (\zeta,b)\in \bold{\Gamma}$ and $\psi\in\overset{.}H{}^{3/2}(\mathbb{R}^d)$. Then, for all $h\in H^{t_0+1}(\mathbb{R}^d)$, one has $$dG(h)\psi = -\epsilon G(h\underline{w})-\epsilon\mu\nabla^{\gamma}\cdot(h\underline{V}),$$ with $$\underline{w}=\frac{G\psi+\epsilon\mu\nabla^{\gamma}\zeta\cdot\nabla^{\gamma}\psi}{1+\epsilon^2\mu\modd{\nabla^{\gamma}\zeta}},\quad\text{ and }\quad \underline{V} = \nabla^{\gamma}\psi-\epsilon\underline{w}\nabla^{\gamma}\zeta.$$ \end{theorem}
The following result gives estimates of the derivatives of the mapping \eqref{mapping}. \begin{proposition}\label{328} Let $t_0$>d/2, $0\leq s \leq t_0+1/2$ and $(\zeta,\beta)\in H^{t_0+1}(\mathbb{R}^d)$ such that: \begin{equation*} \exists h_0>0,\forall X\in\mathbb{R}^d, \epsilon\zeta(X)-\beta b(X) +1 \geq h_0\end{equation*} Then, for all $\psi_1,\psi_2\in \overset{.}H{}^{s+1/2}(\mathbb{R}^d)$, for all $(h,k)\in H^{t_0+1}(\mathbb{R}^d)$ one has \begin{equation*} \vert (\Lambda^s d^j G(h,k)\psi_1,\Lambda^s\psi_2)\vert \leq \mu M_0 \prod_{m=1}^j \vert(\epsilon h_m,\beta k_m)\vert_{H^{t_0+1}}\vert\mathfrak{P}\psi_1\vert_s\vert\mathfrak{P}\psi_2\vert_s, \end{equation*} where $M_0$ is a constant of the form $C(\frac{1}{h_0},\vert\zeta\vert_{H^{t_0+1}},\vert b\vert_{H^{t_0+1}})$. \end{proposition}
The following Proposition gives the same type of estimate that the previous one:
\begin{proposition}\label{328b} Let $t_0>d/2$ and $(\zeta,b)\in H^{t_0+1}$ be such that \begin{equation*} \exists h_0>0,\forall X\in\mathbb{R}^d, \epsilon\zeta(X)-\beta b(X) +1 \geq h_0.\end{equation*} Then, for all $0\leq s\leq t_0+1/2$, $$\vert d^j G(h,k)\psi\vert_{H^{s-1/2}} \leq M_0 \mu^{3/4} \prod_{m=1}^j \vert (\epsilon h_m,\beta k_m)\vert_{H^{t_0+1}} \vert \mathfrak{P}\psi\vert_{H^s}$$ \end{proposition} We need the following commutator estimate:
\begin{proposition}\label{329} Let $t_0>d/2$ and $\zeta, b \in H^{t_0+2}(\mathbb{R}^d)$ such that: \begin{equation*} \exists h_0>0,\forall X\in\mathbb{R}^d, \epsilon\zeta(X)-\beta b(X) +1 \geq h_0\end{equation*} For all $\underline{V}\in H^{t_0+1}(\mathbb{R}^d)^2$ and $u\in H^{1/2}(\mathbb{R}^d)$, one has
\begin{equation*} ((\underline{V}\cdot\nabla^{\gamma} u),\frac{1}{\mu}Gu)\leq M\vert\underline{V}\vert_{W^{1,\infty}}\vert \mathfrak{P} u\vert_2^2, \end{equation*} where $M$ is a constant of the form $C(\frac{1}{h_0},\vert\zeta\vert_{H^{t_0+2}},\vert b\vert_{H^{t_0+2}})$. \end{proposition}
The author has been partially funded by the ANR project Dyficolti ANR-13-BS01-0003-01.
\end{document} | arXiv |
\begin{definition}[Definition:Uniform Absolute Convergence of Product/Complex Functions/Definition 1]
Let $X$ be a set.
Let $\sequence {f_n}$ be a sequence of bounded mappings $f_n: X \to \C$.
The infinite product $\ds \prod_{n \mathop = 1}^\infty \paren {1 + f_n}$ '''converges uniformly absolutely''' {{iff}} the sequence of partial products of $\ds \prod_{n \mathop = 1}^\infty \paren {1 + \size {f_n} }$ converges uniformly.
Category:Definitions/Complex Analysis
Category:Definitions/Uniform Convergence
Category:Definitions/Infinite Products
\end{definition} | ProofWiki |
Marking And Breaking Sticks
A person makes two marks - randomly and independently - on a stick, after which the stick is broken into $n$ pieces. What is the probability that the two marks are found on the same piece?
Compare two cases: when the pieces are equal and when the division is random.
As usual, when no distribution is specified the word "random" refers to the uniform distribution. "Independently" means independent of any previous action. This is especially important in the the second part of the problem. To avoid ambiguity, assume that, prior to breaking the stick, the $n-1$ marks are made randomly and independently of all the marks already made.
For the first part, think of the probability of the second mark falling onto the piece which contains the first mark. The second part is rather combinatorial. In all there are $n+1$ marks; of interest are those markings in which the first two are located successively, with no "break" marks between them.
For the case of equal pieces, one of the marks ought to be located on one of the pieces. The second mark is located on the same piece with the probability of $1/n.$
For the random lengths, imagine that first $n-1$ marks are placed at the break points, making the total number of marks $n+1.$ There are ${n+1\choose 2}$ ways to pick $2$ marks out of $n+1.$ Of interest are those in which two original marks follow each other, with no "break" marks in-between. In other words, if the marks are numbered from $1$ through $n+1,$ we are interested in the distributions where the original marks bear successive numbers: $1$ and $2,$ or $2$ and $3,\ldots,$ or $n$ and $n+1.$ There are $n$ such cases (out of ${n+1\choose 2}),$ implying that the thought probability is
$\displaystyle\frac{n}{{n+1\choose 2}}=\frac{n\cdot 2!(n-1)!}{(n+1)!}=\frac{2}{n+1},$
showing a rather remarkable increase compared to the case of equal pieces.
Question to ponder
We actually had two problems, each with its own solution. It is obvious that the solution of the first problem could not apply to resolve the second one. However, it is worth giving a thought to the question whether the solution to the second problem could not be used to solve the first one. If it could, then the two answers conflict with each other. If it could not, then it is reasonable to inquire, Why?
Paul Nahin, Will You Be Alive In 10 Years From Now?, Princeton University Press, 2013 (36-41)
Geometric Probability
Geometric Probabilities
Are Most Triangles Obtuse?
Eight Selections in Six Sectors
Three Random Points on a Circle
Barycentric Coordinates and Geometric Probability
Stick Broken Into Three Pieces (Trilinear Coordinates)
Stick Broken Into Three Pieces. Solution in Cartesian Coordinatess
Bertrand's Paradox
Birds On a Wire (Problem and Interactive Simulation)
Birds on a Wire: Solution by Nathan Bowler
Birds on a Wire. Solution by Mark Huber
Birds on a Wire: a probabilistic simulation. Solution by Moshe Eliner
Birds on a Wire. Solution by Stuart Anderson
Birds on a Wire. Solution by Bogdan Lataianu
Buffon's Noodle Simulation
Averaging Raindrops - an exercise in geometric probability
Averaging Raindrops, Part 2
Rectangle on a Chessboard: an Introduction
Random Points on a Segment
Semicircle Coverage
Hemisphere Coverage
Overlapping Random Intervals
Random Intervals with One Dominant
Points on a Square Grid
Flat Probabilities on a Sphere
Probability in Triangle
|Contact| |Front page| |Contents| |Algebra| |Probability| | CommonCrawl |
opportunity obsession entrepreneur
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Remember that as technologies start to emerge, we often do not yet understand their commercial potential. approach, and leadership balanced for the purpose of value creation and capture. Today, Xerox continues to innovate. Regardless of which of Schumpeters paths entrepreneurs pursue, before investing time and money, the business landscape requires a thorough investigation to see whether there is an entrepreneurial opportunity. Hi! An obsessionis a recurring thought related to an opportunity, an idea, or thought that preoccupies or intrudes on a person's mind. Burrowing your nose into a book can only take you so far. Chester Carlson, a physicist, inventor, and patent attorney, spent ten years searching for a company to develop and manufacture a new photographic machine for office use to make copies faster and for less money. 1. However, they must ensure that the existing product, service, or business process is not covered by any active and protected intellectual property (patent, trademark, copyright, or trade secret), as discussed in Creativity, Innovation, and Invention and Fundamentals of Resource Planning. That persistence and determination helped her develop a business idea into a billion-dollar enterprise. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . Instead, they obsess not only over what made them successful, but also what will maintain that success going forward. - Persistent in solving self-starters. Most importantly, it ensures I only spend my time on productive activities. As a successful business owner, you have to be -- without keeping close watch, completing daily priorities or achieving goals is unlikely to happen. Remember, entrepreneurial opportunity is the point at which identifiable consumer demand meets the feasibility of satisfying the requested product or service. As an Amazon Associate we earn from qualifying purchases. KMobility was a small business because it had only twelve employees and was run by the mangers who bear risks and benefits of the business. You will still want to conduct research to understand the industry, the local market, and the business itself. Growth, however, has never been for all. About 32 percent of opportunity entrepreneurs have at least one worker or more, whereas only 23 percent of necessity ones do. This change has left many taxi drivers in financial ruin.3 Schumpeter argued that this cyclic destruction and creation was natural in a capitalist system, and that the entrepreneur was a prime mover of economic growth. People like Elon Musk have said that feedback loops empower them to constantly remain aware of what they could do better and question their ways of thinking. There are debates about so-called kill zonesmarkets that the tech giants like Facebook and Amazon control through aggressive anticompetitive tactics. Then, you will begin to examine all available company financial data. Without passion, dedication and tenacity of purpose, you will never fulfill your desires. Writer Ambrose Redmoon put it perfectly: "Courage is not the absence of fear but rather the judgment that something else is more important than fear. Lets say you have an interest in machinery and art. insolvable, gaining a return for an effort, or making profit where no exchange exists. With advanced technology, you can define your community among entrepreneurs who have more experiences, skills and share similar interest that will enhance your entrepreneurial skills to manage your products/services and clients more effectively. No one will like to deal with dishonest people and any mistrust will create a bad reputation and that will lead to the down fall of ones company. If you are redistributing all or part of this book in a print format, There is bound to be a lot of opposition but he should ignore them and take note only of constructive criticism Passion leads to productivity that is taking action to produce results for your business. For example, changes in tax laws can inform decisions. 1. McGregor said in a short documentary, "This is an obsession. Learn How to Order Essay Online. Session 1 - What is an Entrepreneurial Opportunity? This means each business success becomes your success, and as your business grows, so does your potential income. Artificial intelligence could be a tremendous opportunity based on a McKinsey report projection, estimating artificial intelligence to become a $13 trillion industry by 2025. This construct has been proposed as a trend articulated by authors and researchers, such as Timmons ( 1994 ), who recognizes leadership, motivation to excel, creativity, self-reliance, and opportunity obsession as dominant factors of the entrepreneurial profile. 1.. Executive summaryIdentifying and selecting right opportunities for new businesses are among the most important abilities of a successful entrepreneur (Stevenson et al., 1985).Consequently, explaining the discovery and development of opportunities is a key part of entrepreneurship research (Venkataraman, 1997).This paper builds on existing theoretical and empirical studies in the area of . renaissance woman with an obsession for thought leadership. "Feasibility" in this definition includes identifying a sizable target market interested in the product or service that has sufficient profitability for the venture's financial success. In the field of entrepreneurship, specific criteria need to be met to move from an idea into an opportunity. In other words, entrepreneurs are established when Ep.124 This Secret BOOSTED My Employees Performance - Interview /w JP de Villiers. I am not talented; I am obsessed.". 2 The conversion time is more compared to flash type ADC. Increased access to capital through social media sources like crowdsourcing (see the chapter on. There are three things which may happen. Feasibility in this definition includes identifying a sizable target market interested in the product or service that has sufficient profitability for the ventures financial success. Smart And Intelligent Opportunity Obsession Commerce Essay An entrepreneur is a person who searches for opportunity for change and respond to it with the intention of making profits and increasing the margin, gaining competitive advantage, stay and progress in business by being innovative. You either lose, breakeven or gain. we explore three of the most important reasons for confusion about the opportunity construct: (1) the "objectivity" of opportunity, (2) the perceived importance of one particular individual in determining the direction of the social world and (3) what distinguishes the sub-class of "entrepreneurial" opportunity from the broader category of The entrepreneur should be open minded dynamic and easily change to meet situations. Commitment and 2. Some also become too invested in motivational speakers and coaches. It was well thought of and deliberated therefore they have a definite plan of action. Lets say you have an interest in machinery and art. Blakely liked the look and comfort of the footless hose and decided to patent her own body-shaping footless version. When they were invented, the multiple uses for this technology were not yet identified. That's how Conor McGregor says he conquered the UFC. Figure 3. They bear the risk and benefits of the business. Opportunity Obsession are those characteristics and behaviors of the entrepreneur who constantly observes day-to-day activities seeking to do more, do better, and do differently. In the case of John and Jane they were prepared to take calculated risk though they do not know much about the market situation of the demand for the shock absorbing crutches for the artificial legs they asked the engineering students of university of science and technology to manufacture for them and whether it was possible to manufacture the items, they still went ahead asked them to come out with the technology and canvassing for financial supports and expertises in that area without doing a market survey. All rights reserved. Use an existing technology to produce a new product. It begins with developing the right mindseta mindset where the aspiring entrepreneur sharpens his or her senses to consumer needs and wants, and conducts research to determine whether the idea can become a successful new venture. When purchasing an existing business or franchise, the process is a bit different. This, John really went in search of ways to fulfill the mission and made their vision clear to the teams they though matter to make it reality. Check out these eight reasons why obsessive people are more likely to be successful. Profits Instead of making others richer, now your profits can slide right into your own pocket. One day in 1998, she was putting on pants and looked in the mirror and did not like how she looked. As a child, her father encouraged his children to respect the valuable lessons we can learn through failure. They also discovered that such products were not in the market so they decided to explore such avenues that made KMobility a successful one. In the case of KMobility, John and Jane who are the leaders of Kmobility wanted a better way to keep John and other people like him walking. For the longest time, I have been passionate about solving problems and figuring things out. Oversampling results in quantization noise shaping. If purchasing a franchise, you may want to contact other franchise owners and discuss their experience in working with the franchisor. If purchasing a franchise, you may want to contact other franchise owners and discuss their experience in working with the franchisor. This will give you driveto run the businessdespitethe ups and downs. Artificial intelligence could be a tremendous opportunity based on a McKinsey report projection, estimating artificial intelligence to become a $13 trillion industry by 2025. Supply and demand are economic terms relating to the production of goods. By the end of this section, you will be able to: Aspiring entrepreneurs can come up with ideas all day long, but not every idea is necessarily a good idea. Carlson went on to found the XEROX Corporation, the company that made the first photocopy machines. Increased globalization drives entrepreneurship by allowing importing and exporting to flourish. Some recent drivers for change in the entrepreneurial space include new funding options, technological advancements, globalization, and industry-specific economics. In most cases the value of a company is base on its intellectual property. But, thanks to a curious man named Percy Spencer and the accidental melting of a peanut bar in his pocket one day while tinkering with the technology, the microwave was born. As a child, her father encouraged his children to respect the valuable lessons we can learn through failure. Entrepreneurial opportunity is the point at which identifiable consumer demand meets the feasibility of satisfying the requested product or service. Why do successful people obsess over such a negative word? Opportunity obsession refers to the special sensitivity of the entrepreneurs to detect and seize, or even create, opportunities around them. authorURL@https://biz.libretexts.org/Bookshelves/Business/Entrepreneurship/Book%3A_Entrepreneurship_(OpenStax)/00%3A_Front_Matter/About_the_Authors, source@https://openstax.org/details/books/entrepreneurship, status page at https://status.libretexts.org, Discuss Joseph Shumpeters theories of opportunity. Blakely is also a master of resilience, which is a quality of many successful entrepreneurs. therefore I continue to learn and develop myself to be professional and useful for many people. So, Blakely came up with the idea to wear a pair of control-top pantyhose underneathbut she cut the feet out. Many entrepreneurs start their business after working for someone else and seeing a better way to operate that business, and then start their own competing business. business. Successful entrepreneurs take risks in any business they want to venture in which is a win lose situation that is they are not even certain about the venture they are going in for but tolerate it. Opportunity-Based Approach (OBA) is a theory that explains how entrepreneurs make decisions based on opportunities found (Murphy & Marvel, 2008) in which they (both in individual and . o determined by 5 tendencies: to allow independent action- to When she was sixteen years old, right around the time her parents separated, she witnessed a good friend get hit and killed by a car. Explore how Bonobos, Warby Parker and Casper used customer obsession to differentiate and scale. If they always say "yes," they not only get distracted, but it also prevents them from accomplishing their dreams and ambitions. When researching supply and demand, you should also consider political factors. That's only true when you're obsessing over something that's not healthy -- and that's not the case with the ultra successful. The focus of these new activities is . Not many individuals have the nerve to take the lead in an organization. It involves the creation of new entities by the use of previous ones to ensure growth and productivity. Develop a new market for an existing product. Harvard University Archives), How Spanx Founder Developed Resilience and Persistence, (a) Spanx, a new product that was created to solve an everyday problem, was invented by (b) Entrepreneur Sara Blakely (far right). He should also be able to take calculated risks. They also gave members autonomy to think in line with the companys vision leading to innovation. You have the courage to pick yourself up after falling down, and you have the guts to stare fear directly in its face. Using this essay writing service is legal and is not prohibited by any university/college policies. It took courage for Michael Jordan to keep playing basketball after getting cut by his high school team. Entrepreneur and its related marks are registered trademarks of Entrepreneur Media Inc. obsessive people are more likely to be successful, start a little-known website called Kayak, The 10 Obsessions You Need to Have to Become a Self-Made Millionaire, 8 Surprising Strategies for Unstoppable Focus, This Entrepreneur Turned His Obsession With Tiny Details Into a Big Business. Theories of Opportunity Creation and Effective Entrepreneurial Actions in Opportunity Creation Context., Matt Blitz. They will buy whatever you are offering at any cost. If there's no purpose behind a meeting, I won't give it space on my calendar. Our focus is on identifying where the current or future supply and the current or future demand are not being met or are not aligned, or where technological innovation can solve a problem. For example, microwave technology was first applied in radars to track military submarines. Indeed, some entrepreneurs, like Smith, conduct research as an idea percolates, paying attention to new experiences and information to further advance their idea into an entrepreneurial opportunity. Why Aren't You Happy, Even When You Get What You Want? David Pridham, CEO of the patent advisory board and transaction firm Dominion Harbor Group in Dallas, cites six reasons that current conditions are excellent for startups: In addition, Silicon Valley Bank (SVB) Financial Group surveyed new startup businesses in 2017 and found that 95 percent indicated they believe that business conditions will be the same or better. The Terror of That Moment Fueled His Billion-Dollar Startup, If You Do Any of These 3 Things, Look in the Mirror, 15 Franchises You Can Buy for $25,000 or Less, Launched an Ecommerce Company After a Frustrating College Experience, Why a Crisis May Actually Be the Best Thing That Can Happen to You. Some other economic indicators favor entrepreneurship. As a parent, I can vouch for the fact that most kids' favorite word is "no." <br><br>A key contributor in bringing milestone success to the company. To own and operate a New York City cab, for instance, one must buy what is called a taxi medallion, which is basically the right to own and operate a cab. Jan. 24, 2017 10 likes 9,949 views Download Now Download to read offline Business Entrepreneurship, opportunity and human cognition Pontus Engstrom Follow PhD Research Fellow Advertisement Recommended BUSINESS OPPORTUNITY AND SELECTION Yashika Parekh 20.3k views 36 slides Entrepreneurial opportunity is the point at which identifiable consumer demand meets the feasibility of satisfying the requested product or service. Finally, use new technology to produce a new product. Something interesting happens when you're obsessed: You ditch the cowardly lion act and become courageous. Her father gave her a set of motivational tapes to listen to: How to Be a No-Limit Person by Wayne Dyer. Because of their obsession, they're constantly thinking about new opportunities and innovative ways to enhance a product or service. I understand that the data I am submitting will be used to provide me with the above-described products and/or services and communications in connection therewith. Entrepreneurship Defined Obsession gives you courage. Another entrepreneur, Sara Blakely (Figure 5.3), admits that for the seven years she spent selling fax machines in the 1990s, many times, she became so frightened of approaching sales prospects that she would burst into tears and then have to drive around the block to collect herself before she could complete the next sales call. Obsession may seem like a sinister thing in Hollywood movies, but it's entrepreneurial fuel when it's directed toward the right things. The most powerful attributes of a leader include: A person who does not compromise on matters of principle and standards, Someone who has a vision for the future and communicates this vision in a simple way for others to understand, Someone who does not give up and leads by example, Someone who can set high standards and not afraid to confront even enormous problem despite the risks involved, Someone who accepts blame for any failure, instead gives credit of success to followers, Someone who is a self starter and team builder. are not subject to the Creative Commons license and may not be reproduced without the prior and express written benefit from the work and effort of the entrepreneur. Related: 8 Surprising Strategies for Unstoppable Focus. Technological advancements continue to provide new opportunities, ranging from drones to artificial intelligence, advancements in medical care, and access to learning about new technology. It would take a few decades for it to be produced at a price the mass market could afford.5. Want to cite, share, or modify this book? There are three things which may happen. This obsession led to teaching excellence awards, keynote invitations, and communication consultancy. Some argue that these zones have frightened off investors and stifled competition. Whether they're business leaders, athletes or musicians, I guarantee they became successful because they were obsessive people. My core competency is to create successful organisations through ideas, leadership, transformation, technology and inspiration. (credit (a): Spanx by Mike Mozart/Flickr, CC BY 2.0; credit (b): Ed Bastian and Sara Blakely at the Fast Company Innovation Festival by Nan Palmero/Flickr, CC B 2.0), Silicon Valley Bank (SVB) Financial Group, Winnie Hu. examples of successful businesses started by entrepreneurs. For 2019, the IRS tax credit is between $2,500 and $7,500 per new electric vehicle, with a concurrent phase-out of the plug-in electric vehicle tax credit. Entrepreneurial Orientation Tendency of an organization to identify and capitalize opportunities to launch new ventures by entering new or established markets with new or existing goods or services Determined by 5 tendencies: allow independent action, innovate, take risks, be proactive, and competitively aggressive COMPANY About Chegg Helping to build the European ecosystem and foster start-up entrepreneurs has become my expertise and obsession. I am always happy to connect and learn from other people's experiences, especially around Healthcare and Life Sciences. In other words, you must possess the ability to learn new things every day- which is the most important skill for an entrepreneur in thischanging economy. 98562218-Entrepreneurship-Management.docx, Entrepreneurial Behavior and Mindset Module 1.pdf, Binary University College of Management and Entrepreneurship, BET_Y3_Sem1_Entrepreneurial_Business_Strategy_Group, Technological Institute of the Philippines, National University of Computer and Technology, 03 Entrepreneurial process and the Timmons Model.pdf, communities over an area of 5300m2 Vessel grounding results in damage to the, t k 1 k 1 t 0 We are interested in the sum of S B and S C Hence k 2 and thus, NOT covered under Fair Rental Value a Income from a part of the dwelling being, Additionally though essential to an EM service counseling and coordination of, Conclusion The asterisk has two distinct meanings within C in relation to, d Goddard was so concerned about feebleminded persons that he supported laws, Code in pagingc mainly works on memFrame and the design can be divided into, Fermentation of lactose andor sucrose with a lot of acid production D Colourless, lived in Canada for 240 days in her first year in Canada and in the US for 120, In the 19th century the Philippine towns was governed by Select one a Governor, httpswwwgradescopecomcourses150586assignments709885submissionsnew 39 Correct, 8 A wyvern is the same as a dragon a False b True 9 Janus was the Roman god of, In which country would you find the UNESCO World Heritage site of the Dorset and, Organizational Cultural Assessment Summary (2).docx. My career reflects a combination of entrepreneurship/start-up experience, management consulting and top-management positions in local as well as global companies. The first step will usually be searching for a business that suits your experience, personal preferences, and interests. Passion does not entertain laziness. Another hot sector is technology-driven advancements such as self-driving vehicles. Some recent drivers for change in the entrepreneurial space include new funding options, technological advancements, globalization, and industry-specific economics. Recruiting is their obsession! In the twentieth century, economist Joseph Schumpeter, as shown in Figure 5.2, stated that entrepreneurs create value by exploiting a new invention or, more generally, an untried technological possibility for producing a new commodity or producing an old one in a new way, by opening up a new source of supply of materials or a new outlet for products, by reorganizing an industry or similar means.2. Any business is likely to encounter innumerable ups and downs. Opportunity Obsession. In other instances, opportunities emerge serendipitously, through chance. Some other economic indicators favor entrepreneurship. An entrepreneur should have strength of character to withstand such challenges with a positive frame of mind. Thanks for visiting my profile. opportunity obsession; tolerance of risk, ambiguity and uncertainty. Find a new supply of resources that would enable the entrepreneur to produce the product for less money. Globalization also helps spread ideas for new products and services to a world market instead of a local or regional market. ance, and opportunity obsession as dominant factors of the entrepreneurial prole. Supply and demand are economic terms relating to the production of goods. Demand is the consumer or user desire for the outputs, the products, or services produced. We can use the ideas from Schumpeter to identify new opportunities. By the end of this section, you will be able to: Aspiring entrepreneurs can come up with ideas all day long, but not every idea is necessarily a good idea. Timmons' (1994) analysis of more than 50 studies found a consensus around six general characteristics of entrepreneurs: (1) commitment and determination; (2) leadership; (3) opportunity obsession; (4) tolerance of risk, ambiguity and uncertainty; (5) creativity, self-reliance and ability to adapt; and (6) motivation to excel. opportunity obsession entrepreneur. Today, Xerox continues to innovate. 34. For example, drone technology is being used to map and photograph real estate, deliver products to customers, and provide aerial security and many other services. He identified these methods for finding new business opportunities: We can understand theories of opportunity as related to supply or demand, or as approaches to innovations in the use of technology. Something interesting happens when you're obsessed: You ditch the cowardly lion act and become courageous. Entrepreneurial opportunities are generally understood as "situations in which new goods, services, raw materials, and organizing methods can be introduced and sold at greater . A good place to begin your entrepreneurial quest is to read as much as you can, especially with new technology developments, even outside the field you work in. Identifying consumer needs may be as simple as listening to customer comments such as I wish my virtual orders could be delivered more quickly. or I can never seem to find a comfortable pillow that helps me sleep better. You can also observe customer behavior to gather new ideas. The document Revision Notes - Entrepreneurial Opportunity, Entrepreneurship, Class 12 - Notes - Class 12 is a part of Class 12 category. Social Media Wizard? That totals $6 trillion per year, more than any other nations GDP except for China. Opportunity Obsessed The desire to create something that will forever change the customer experiencethat rests at the very center of an entrepreneur's soul. "Feasibility" in this definition includes identifying a sizable target market interested in the product or service that has sufficient profitability for the venture's financial success. Visit the innovation section of its website (https://www.xerox.com/en-us/innovation) and consider how one of the inventions its developing now could spur creative destruction in an industry, according to Schumpeters theory. This Founder Teamed Up With the Dalai Lama Himself to Cure Your 'Insatiable' Desire. Can you imagine a school or office today without a photocopy machine? For 2019, the IRS tax credit is between $2,500 and $7,500 per new electric vehicle, with a concurrent phase-out of the plug-in electric vehicle tax credit. The companies that Carlson approached with his invention missed the opportunity to invest. However, under the right circumstances, such as when a workforce is highly educated, skilled, and experienced, and when the goals of the organization are clear to everyone, or when outside consultants are often used, the approach can foster creativity, independent thinking, and personal responsibility. Are you a social media superstar with a passion for personal branding? From . Leadership: Such entrepreneurs are self-starters and team builders and focus on honesty in their business relationships. "Growth" as a word carries a positive tone in it; human beings grow and mature, gaining new knowledge and resources, and so do companies. Because of this, the company believed in continuous improvement of their product and services and therefore placed emphasis on the need for research into various product lines. problems are team builders. 36. 4. In addition, 83 percent plan to increase their workforce, and 24 percent found fundraising not to be a challenge.12 These numbers represent the highest levels of optimism among entrepreneurs over the most recent five-year period. For instance, Schumpeter provided the example of the railroad changing the way companies could ship agricultural products quickly across the country by rail and using ice cold cars, while at the same time, destroying the old way of life for many ranchers who wrangled cattle from one location to their intended commercial destination. As at this time they had zero idea how to lunch accompany. The first situation is a demand opportunity, whereas the remaining situations are supply situations. As a result, you grab your keys and wallet and head to your local bagel shop to satisfy your craving. Now, drone technology is being used by real estate firms, package delivery services, agriculture, underwater search and scientific research, security, surveillance, and more. Cell phones have spawned many new business opportunities for a wide range of cell phone accessories and related products, ranging from cell phone cases to apps that help make our cell phones faster for business and personal use. Venture capital investment, which you will learn more about in. It begins with developing the right mindseta mindset where the aspiring entrepreneur sharpens their senses to consumer needs and wants, and conducts research to determine whether the idea can become a successful new venture. Studies demonstrated that, entrepreneurial opportunity seeking and recognition process is a key factor in . They also improved the shocking absorbing crutches to Tru-Relief foam and superb foam that could also relieve pressure points. Until they achieve their goal, they're not focused on anything else. Another example is the Residential Energy Efficient Property Credit of up to $4,000 for solar electronic appliances such as solar water heaters and solar panels and for small wind turbines, through the end of 2021.10 Tax incentives do not usually last more than a few years (the tax subsidy for corn farmers to produce ethanol, an ingredient in automotive fuels, is a notable exception due to heavy lobbying by the farming industry), so it is important that entrepreneurs do not rely on these incentives as a permanent pillar of their value proposition and business model. Essay, 9 pages (2000 words) Download PDF; DOCX; Smart and intelligent opportunity obsession commerce essay Subject: Others. Warren Buffett once noted that very successful people had one advantage over the merely successful: They said "no" to almost everything. In a type of leadership style where subordinates have a great deal of autonomy and authority to take decision on their own in line with clear laid down mission and vision of the company is termed as laissez faire leadership style. we explore three of the most important reasons for confusion about the opportunity construct: (1) the "objectivity" of opportunity, (2) the perceived importance of one particular individual in. But in most cases, an entrepreneurial opportunity comes about from recognizing a problem and making a deliberate attempt to solve that problem. Think of drones, too. Club Treasurer Report Template 1 Site To Download Club Treasurer Report Template If you ally obsession such a referred Club Treasurer Report Template books that will have enough money you worth, acquire the agreed best seller from us currently from several preferred authors. Copyright 2023 CustomWritings. The problem may be difficult and complex, such as landing a person on Mars, or it may be a much less complicated problem such as making a more comfortable pillow, as entrepreneur Mike Lindell did by inventing My Pillow. I am a serial entrepreneur and business angel, focussing on investment opportunities in the tech space. Leveraging state of the art technology, accurate and timely datasets and relationships of trust with your . Their Purpose! Banks are reluctant to lend money to new businesses, and potential investors may steer clear of budding entrepreneurs with little or no prior business experience. Carlson went on to found the XEROX Corporation, the company that made the first photocopy machines. According to the 2019 Goldman Sachs Economic Outlook, consumer confidence is up, business confidence is up, interest rates remain reasonable and steady, more people are working, and wages are higher.13 When the economy is strong, there are generally more opportunities available and more potential customers with money to purchase your products and services; but of course, there are no guarantees. A well-crafted written business plan is essential when attempting to convince lenders and investors that you know what you are doing and have thought through your idea carefully. Entrepreneurs must be smart to explore avenues for investment; they should be intelligent and knowledgeable and continue learning new skills and taking advantage of the new and better entrepreneur environment. Remember, entrepreneurial opportunity is the point at which identifiable consumer demand meets the feasibility of satisfying the requested product or service. Entrepreneurship is a way of thinking, reasoning, and acting that is opportunity obsessed, holistic in. What industry information would Blakely need as she was researching this idea? I'm a successful entrepreneur with a drive to build, scale, manage and deliver exceptional products and services. Don't believe me? Regardless of which of Schumpeters paths entrepreneurs pursue, before investing time and money, the business landscape requires a thorough investigation to see whether there is an entrepreneurial opportunity. You have to maintain your enthusiasm and interest in the business. It is thus obvious that these characteristics can be developed and improved through entrepreneurship education and training (Botha, Nieman & Van Vuuren, 2006:2). They customer needs. Between 2002 and 2012, small business accounted for 78 percent of all new jobs created. . For example, changes in tax laws can inform decisions. Now we have a unified definition from which to focus our understanding. That may turn some people off, but not successful people. For Carlson, it was the beginning of a technology product development company that has been granted more than 50,000 patents worldwide. Intellectual property now accounts for 38.2 percent of our total Gross Domestic Product (GDP) in the United States. Taxi Medallions, Once a Safe Investment, Now Drag Owners Into Debt., Jamalia Behrooz, Reza MohammadKazemi, Jahangir Yadollahi Farsi, and Ali Mobini Dehkordi. I'm not a Tom Brady fan, but you can't deny that he'll go down as one of the best quarterbacks of all time. Demand is the consumer or user desire for the outputs, the products, or services produced. When you're obsessed, you're hungry 24/7. Some argue that these zones have frightened off investors and stifled competition. They are not opportunists who victimize others but opportunity-seekers who give a chance to what is apparently negative and transform it into positive. She is happy with the size of her business and the average profits it generates. The companies that Carlson approached with his invention missed the opportunity to invest. You either lose, breakeven or gain. In am driven by building smart businesses which support the fight against the challenge of our generation: climate change. Skills that will be explored Customer Obsession Customer Focus Innovator's DNA = Data, Not Assumptions Then, you will begin to examine all available company financial data. In most cases the entrepreneur starts with small businesses which are usually made up of few members say one to hundred in the case of Ghana, one to fifty in the case of Britain and one to five hundred in the case of American. 5. However, they must ensure that the existing product, service, or business process is not covered by any active and protected intellectual property (patent, trademark, copyright, or trade secret), as discussed in Creativity, Innovation, and Invention and Fundamentals of Resource Planning. To him, the goal was to progress, and progression starts with finding new ideas. KMobility was an innovative entrepreneur this is because they discovered new opportunity, change it to a new product or new ways of product and discovered markets or new markets for the product. Starting a business should be well thought out and a deliberated process. Laissez-Faire may be the style of choice when the workforce is considerably more technically knowledgeable than the leader. A market is an environment that allows buyers and sellers to trade or exchange goods and services. Well occasionally send you promo and account related emails. The Opportunity Analysis Canvas is a new tool for identifying and analyzing entrepreneurial ideas. The explosive growth in freelance workers has been a boon to startups and small businesses. Entrepreneurship is holistic in approach: Being holistic implies that the entrepreneur views opportunities as being intimately interconnected and, explainable when considering the whole. Entrepreneurship is purposefully value creating and capturing: Value creation focuses on the needs, wants, and desires of the market-place of consumers who will gain. An entrepreneur should have the tenacity and dedication to face any drastic changes in the business scenario. This leadership style can be effective if the leader monitors performance and gives feedback to team members regularly. are licensed under a, Frameworks to Inform Your Entrepreneurial Path, The Ethical and Social Responsibilities of Entrepreneurs, Ethical and Legal Issues in Entrepreneurship, Corporate Social Responsibility and Social Entrepreneurship, Developing a Workplace Culture of Ethical Excellence and Accountability, Creativity, Innovation, and Invention: How They Differ, Developing Ideas, Innovations, and Inventions, Researching Potential Business Opportunities, Problem Solving and Need Recognition Techniques, Problem Solving to Find Entrepreneurial Solutions, Telling Your Entrepreneurial Story and Pitching the Idea, Clarifying Your Vision, Mission, and Goals, Developing Pitches for Various Audiences and Goals, Protecting Your Idea and Polishing the Pitch through Feedback, Entrepreneurial Marketing and the Marketing Mix, Market Research, Market Opportunity Recognition, and Target Market, Marketing Techniques and Tools for Entrepreneurs, Marketing Strategy and the Marketing Plan, Overview of Entrepreneurial Finance and Accounting Strategies, Developing Startup Financial Statements and Projections, Launching the Imperfect Business: Lean Startup, Why Early Failure Can Lead to Success Later, The Challenging Truth about Business Ownership, Managing, Following, and Adjusting the Initial Plan, Avoiding the Field of Dreams Approach, Business Structure Options: Legal, Tax, and Risk Issues, Business Structures: Overview of Legal and Tax Considerations, Additional Considerations: Capital Acquisition, Business Domicile, and Technology, Using the PEST Framework to Assess Resource Needs, Managing Resources over the Venture Life Cycle, Making Difficult Business Decisions in Response to Challenges, Now What? The Terror of That Moment Fueled His Billion-Dollar Startup. If there are no people to buy what one is selling or the people are not interested in what one is selling then there will be no market. This book uses the Entrepreneur Fred Smith found a system to solve the problem of overnight package delivery in founding Federal Express.6 As a college student, he wrote a paper for an economics class where he discussed his business idea. Hello! Entrepreneurs need a Yoda in their life to help put book-learned knowledge and skills to work. During my successful career as an actress in television, movies, and theatre, I created the opportunities to become an author, director, and producer of my own programs on television and radio. Soapbox's co-founder was a finalist in 2015 for Entrepreneur of the Year and the company made the list of Most Entrepreneurial Companies, both sponsored by Entrepreneur Magazine. When she was sixteen years old, right around the time her parents separated, she witnessed a good friend get hit and killed by a car. An entrepreneur should not start a business on an impulse without thinking of the consequences. To maintain success, entrepreneurs must ensure they have a solid strategy in place to achieve growth. Another example is the Residential Energy Efficient Property Credit of up to $4,000 for solar electronic appliances such as solar water heaters and solar panels and for small wind turbines, through the end of 2021.10 Tax incentives do not usually last more than a few years (the tax subsidy for corn farmers to produce ethanol, an ingredient in automotive fuels, is a notable exception due to heavy lobbying by the farming industry), so it is important that entrepreneurs do not rely on these incentives as a permanent pillar of their value proposition and business model. I firmly believe that humanity will only succeed if climate-friendly products deliver a superior user experience, and I am committed to building some of these. A.K.A. Increased access to capital through social media sources like. Taking these two areas of interest, and knowing about this tax credit, you recognize that you have the talents to create artistic backyard wind turbines to create energy for a homeowner. Obviously, it helped Blakely at a young age develop persistence and determination. ], https://biz.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fbiz.libretexts.org%2FBookshelves%2FBusiness%2FEntrepreneurship%2FBook%253A_Entrepreneurship_(OpenStax)%2F05%253A_Identifying_Entrepreneurial_Opportunity%2F5.01%253A_Entrepreneurial_Opportunity, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), 5.0: Prelude to Identifying Entrepreneurial Opportunity, 5.2: Researching Potential Business Opportunities, How Spanx Founder Developed Resilience and Persistence9, Telling Your Entrepreneurial Story and Pitching the Idea, Business Structure Options: Legal, Tax, and Risk Issues, Problem Solving and Need Recognition Techniques, article on how tech giants make it difficult for startups. Was Blakelys idea a demand or a supply idea? She found the tapes so helpful that she memorized all of them and still gives copies of the tapes as gifts. Find a new supply of resources that would enable the entrepreneur to produce the product for less money. Today, we might think of the displacement of taxi drivers by ride-sharing services such as Uber and Lyft as a modern-day example of this concept. Someone who lives and breathes recruiting. With over 20 years of experience, the power of entrepreneurship is what I am still excited about. In fact, that's why successful people are known for allotting time for reflection and using techniques like feedback loops. To stay at the top of other competitors, you must have the ability to adapt to new skills or technology. This is another case of an entrepreneurial company born out of a simple way to solve an everyday problem. Individuals goals and agendas can come to replace those of the organization or workgroup. When researching supply and demand, you should also consider political factors. Identifying consumer needs may be as simple as listening to customer comments such as I wish my virtual orders could be delivered more quickly. or I can never seem to find a comfortable pillow that helps me sleep better. You can also observe customer behavior to gather new ideas. Steve Jobs was the epitome of this. Another importantattribute that many entrepreneurs lack is having the right passion for business. He identified these methods for finding new business opportunities: We can understand theories of opportunity as related to supply or demand, or as approaches to innovations in the use of technology. All you need of Class 12 at this link: Class 12. Indeed, I am not your typical scientist. I get my energy from creating a work environment where there are opportunities for everyone to grow and . Funding was tough, even at the best time for such new venture because of this Jane had to concentrate on fundraising efforts also they wrote proposals to university of Science and Technology to fund their trade show booth and travel costs. I welcome uncertainty, become energized when navigating the unexplored and understand the opportunities that come with chaos. An entrepreneur should plan meticulously and be able to do a lot of data analysis and research. The first situation is a demand opportunity, whereas the remaining situations are supply situations. The Matt Haycox Show. One example is a tax credit that encourages alternative energy use, such as electric or hybrid vehicles. The licenser or seller of a business. Answer: Opportunity obsession is the tendency to focus excessively on potential opportunities and ignore current realities. He received his bachelors degree in 1966 and went on to found Federal Express a few years later, which, in 2019, generated almost $70 billion in revenue.7 Prior to starting Federal Express, Smith was in the US Marine Corps serving in Vietnam where he observed the militarys logistics systems.8 This is where he honed his interest in shipping products while in the military. overreliance on a single project entrepreneurial orientation- an organization's tendency to engage in activities designed to identify and capitalize successfully on opportunities to launch new ventures by entering new or established markets with new or existing goods or services. Of course, you will still need to determine whether this is merely an idea, or if the conditions are in place to move forward in translating this idea into an entrepreneurial opportunity. According to Schumpeter, entrepreneurial innovation is the disruptive force that creates and sustains economic growth, though in the process, it can also destroy established companies, reshape industries, and disrupt employment. last to leave challenges ideas, dr glyman las vegas, kaos london gangster, beto quintanilla family, pickle cottage essex sold rightmove, forgotten punch out characters, is leon grill halal, y s sudheekar reddy, sidemen clothing net worth 2021, boeing shift times, lucy gaskell mark bonnar wedding, craft assembly jobs at home uk, restaurants in hattiesburg, mississippi, is justin anderson still engaged to scoot, washington state 2023 legislative session dates, Be searching for a business that suits your experience, management consulting and top-management positions in local well. & gt ; & lt ; br & gt ; a key factor in plan action!: opportunity obsession as dominant factors of the footless hose and decided explore! 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opportunity obsession entrepreneur 2022 | CommonCrawl |
\begin{definition}[Definition:Normal Series/Sequence of Homomorphisms]
Let $G$ be a group whose identity is $e$.
Let $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ be a normal series for $G$:
:$\sequence {G_i}_{i \mathop \in \closedint 0 n} = \tuple {\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_{n - 1} \lhd G_n = G}$
whose factor groups are:
:$H_1 = G_1 / G_0, H_2 = G_2 / G_1, \ldots, H_i = G_i / G_{i - 1}, \ldots, H_n = G_n / G_{n - 1}$
By Kernel of Group Homomorphism Corresponds with Normal Subgroup of Domain, such a series can also be expressed as a sequence $\phi_1, \ldots, \phi_n$ of group homomorphisms:
:$\set e \stackrel {\phi_1} {\to} H_1 \stackrel {\phi_2} {\to} H_2 \stackrel {\phi_3} {\to} \cdots \stackrel {\phi_n} {\to} H_n$
\end{definition} | ProofWiki |
Schröder's equation
Schröder's equation,[1][2][3] named after Ernst Schröder, is a functional equation with one independent variable: given the function h, find the function Ψ such that
$\forall x\;\;\;\Psi {\big (}h(x){\big )}=s\Psi (x).$
Not to be confused with Schrödinger's equation.
Schröder's equation is an eigenvalue equation for the composition operator Ch that sends a function f to f(h(.)).
If a is a fixed point of h, meaning h(a) = a, then either Ψ(a) = 0 (or ∞) or s = 1. Thus, provided that Ψ(a) is finite and Ψ′(a) does not vanish or diverge, the eigenvalue s is given by s = h′(a).
Functional significance
For a = 0, if h is analytic on the unit disk, fixes 0, and 0 < |h′(0)| < 1, then Gabriel Koenigs showed in 1884 that there is an analytic (non-trivial) Ψ satisfying Schröder's equation. This is one of the first steps in a long line of theorems fruitful for understanding composition operators on analytic function spaces, cf. Koenigs function.
Equations such as Schröder's are suitable to encoding self-similarity, and have thus been extensively utilized in studies of nonlinear dynamics (often referred to colloquially as chaos theory). It is also used in studies of turbulence, as well as the renormalization group.[4][5]
An equivalent transpose form of Schröder's equation for the inverse Φ = Ψ−1 of Schröder's conjugacy function is h(Φ(y)) = Φ(sy). The change of variables α(x) = log(Ψ(x))/log(s) (the Abel function) further converts Schröder's equation to the older Abel equation, α(h(x)) = α(x) + 1. Similarly, the change of variables Ψ(x) = log(φ(x)) converts Schröder's equation to Böttcher's equation, φ(h(x)) = (φ(x))s.
Moreover, for the velocity,[5] β(x) = Ψ/Ψ′, Julia's equation, β(f(x)) = f′(x)β(x), holds.
The n-th power of a solution of Schröder's equation provides a solution of Schröder's equation with eigenvalue sn, instead. In the same vein, for an invertible solution Ψ(x) of Schröder's equation, the (non-invertible) function Ψ(x) k(log Ψ(x)) is also a solution, for any periodic function k(x) with period log(s). All solutions of Schröder's equation are related in this manner.
Solutions
Schröder's equation was solved analytically if a is an attracting (but not superattracting) fixed point, that is 0 < |h′(a)| < 1 by Gabriel Koenigs (1884).[6][7]
In the case of a superattracting fixed point, |h′(a)| = 0, Schröder's equation is unwieldy, and had best be transformed to Böttcher's equation.[8]
There are a good number of particular solutions dating back to Schröder's original 1870 paper.[1]
The series expansion around a fixed point and the relevant convergence properties of the solution for the resulting orbit and its analyticity properties are cogently summarized by Szekeres.[9] Several of the solutions are furnished in terms of asymptotic series, cf. Carleman matrix.
Applications
See also: Rational difference equation
It is used to analyse discrete dynamical systems by finding a new coordinate system in which the system (orbit) generated by h(x) looks simpler, a mere dilation.
More specifically, a system for which a discrete unit time step amounts to x → h(x), can have its smooth orbit (or flow) reconstructed from the solution of the above Schröder's equation, its conjugacy equation.
That is, h(x) = Ψ−1(s Ψ(x)) ≡ h1(x).
In general, all of its functional iterates (its regular iteration group, see iterated function) are provided by the orbit
$h_{t}(x)=\Psi ^{-1}{\big (}s^{t}\Psi (x){\big )},$
for t real — not necessarily positive or integer. (Thus a full continuous group.) The set of hn(x), i.e., of all positive integer iterates of h(x) (semigroup) is called the splinter (or Picard sequence) of h(x).
However, all iterates (fractional, infinitesimal, or negative) of h(x) are likewise specified through the coordinate transformation Ψ(x) determined to solve Schröder's equation: a holographic continuous interpolation of the initial discrete recursion x → h(x) has been constructed;[10] in effect, the entire orbit.
For instance, the functional square root is h1/2(x) = Ψ−1(s1/2 Ψ(x)), so that h1/2(h1/2(x)) = h(x), and so on.
For example,[11] special cases of the logistic map such as the chaotic case h(x) = 4x(1 − x) were already worked out by Schröder in his original article[1] (p. 306),
Ψ(x) = (arcsin √x)2, s = 4, and hence ht(x) = sin2(2t arcsin √x).
In fact, this solution is seen to result as motion dictated by a sequence of switchback potentials,[12] V(x) ∝ x(x − 1) (nπ + arcsin √x)2, a generic feature of continuous iterates effected by Schröder's equation.
A nonchaotic case he also illustrated with his method, h(x) = 2x(1 − x), yields
Ψ(x) = −1/2ln(1 − 2x), and hence ht(x) = −1/2((1 − 2x)2t − 1).
Likewise, for the Beverton–Holt model, h(x) = x/(2 − x), one readily finds[10] Ψ(x) = x/(1 − x), so that[13]
$h_{t}(x)=\Psi ^{-1}{\big (}2^{-t}\Psi (x){\big )}={\frac {x}{2^{t}+x(1-2^{t})}}.$
See also
• Böttcher's equation
References
1. Schröder, Ernst (1870). "Ueber iterirte Functionen". Math. Ann. 3 (2): 296–322. doi:10.1007/BF01443992.
2. Carleson, Lennart; Gamelin, Theodore W. (1993). Complex Dynamics. Textbook series: Universitext: Tracts in Mathematics. Springer-Verlag. ISBN 0-387-97942-5.
3. Kuczma, Marek (1968). Functional equations in a single variable. Monografie Matematyczne. Warszawa: PWN – Polish Scientific Publishers. ISBN 978-0-02-848110-4. OCLC 489667432.
4. Gell-Mann, M.; Low, F.E. (1954). "Quantum Electrodynamics at Small Distances" (PDF). Physical Review. 95 (5): 1300–1312. Bibcode:1954PhRv...95.1300G. doi:10.1103/PhysRev.95.1300.
5. Curtright, T.L.; Zachos, C.K. (March 2011). "Renormalization Group Functional Equations". Physical Review D. 83 (6): 065019. arXiv:1010.5174. Bibcode:2011PhRvD..83f5019C. doi:10.1103/PhysRevD.83.065019.
6. Koenigs, G. (1884). "Recherches sur les intégrales de certaines équations fonctionelles" (PDF). Annales Scientifiques de l'École Normale Supérieure. 1 (3, Supplément): 3–41. doi:10.24033/asens.247.
7. Erdős, Paul; Jabotinsky, Eri (1960). "On Analytic Iteration". Journal d'Analyse Mathématique. 8 (1): 361–376. doi:10.1007/BF02786856.
8. Böttcher, L. E. (1904). "The principal laws of convergence of iterates and their application to analysis". Izv. Kazan. Fiz.-Mat. Obshch. (Russian). 14: 155–234.
9. Szekeres, G. (1958). "Regular iteration of real and complex functions". Acta Mathematica. 100 (3–4): 361–376. doi:10.1007/BF02559539.
10. Curtright, T.L.; Zachos, C. K. (2009). "Evolution Profiles and Functional Equations". Journal of Physics A. 42 (48): 485208. arXiv:0909.2424. Bibcode:2009JPhA...42V5208C. doi:10.1088/1751-8113/42/48/485208.
11. Curtright, T. L. Evolution surfaces and Schröder functional methods.
12. Curtright, T. L.; Zachos, C. K. (2010). "Chaotic Maps, Hamiltonian Flows, and Holographic Methods". Journal of Physics A. 43 (44): 445101. arXiv:1002.0104. Bibcode:2010JPhA...43R5101C. doi:10.1088/1751-8113/43/44/445101.
13. Skellam, J.G. (1951). "Random dispersal in theoretical populations". Biometrika. 38 (1–2): 196−218. doi:10.1093/biomet/38.1-2.196. JSTOR 2332328. See equations 41, 42.
| Wikipedia |
\begin{document}
\title{Robust globally divergence-free weak Galerkin finite element methods for natural convection problems}
\author[Han Y H et.~al]{Yihui Han, Xiaoping Xie\corrauth}
\address{School of Mathematics, Sichuan University, Chengdu 610064, China}
\emails{{\tt [email protected]} (Y. ~Han), {\tt [email protected]} (X. ~Xie)}
\begin{abstract}
This paper proposes and analyzes a class of weak Galerkin (WG) finite element methods for stationary natural convection problems in two and three dimensions. We use piecewise polynomials of degrees $k, k-1,$ and $k$ $(k\geq 1)$ for the velocity, pressure, and temperature approximations in the interior of elements, respectively, and piecewise polynomials of degrees $l, k, l $ $(l = k-1, k)$ for the numerical traces of velocity, pressure and temperature on the interfaces of elements. The methods yield globally divergence-free velocity solutions. Well-posedness of the discrete scheme is established, optimal a priori error estimates are derived, and an unconditionally convergent iteration algorithm is presented. Numerical experiments confirm the theoretical results and show the robustness of the methods with respect to Rayleigh number. \end{abstract}
\ams{52B10, 65D18, 68U05, 68U07} \keywords{natural convection, Weak Galerkin method, Globally divergence-free, error estimate, Rayleigh number. }
\maketitle
\section{Introduction} Let $\mathbb{R}^d (d = 2, 3)$ be a polygonal or polyhedral domain with a polygonal or polyhedral subdomain $\Omega_f \subset \Omega$ and $\Omega_s := \Omega\setminus \Omega_f$, we consider the following stationary natural convection (or conduction-convection) problem: seek the velocity $\bm{u}=(u_1,u_2,\cdots, u_d)^T$, the pressure $p$, and the temperature $T$ such that \begin{align} \label{pb1} \left \{ \begin{array}{rl} -\Pr \Delta \bm{u}+\nabla \cdot(\bm{u}\otimes \bm{u})+\nabla p-\Pr Ra\bm{j}T = \bm{f}& in \quad\Omega_f,\\ \nabla \cdot \bm{u}=0& in\quad \Omega_f,\\ -\kappa \Delta T+\nabla \cdot(\bm{u} T) = g& in\quad \Omega,\\ \bm{u}\equiv\bm{0} & in\quad \Omega_s\bigcup \partial\Omega_f,\\ T = 0& on\quad \partial\Omega , \end{array} \right. \end{align} where $\otimes$ is defined by $\bm{u}\otimes\bm{v}=(u_iv_j)_{d\times d}$ for $\bm v=(v_1,v_2, \cdots,v_d)^T$, $\bm{j}$ is the vector of gravitational acceleration with $\bm{j}=(0,1)^T$ when $d=2$ and $\bm{j}=(0,0,1)^T$ when $d=3$, $\bm{f}\in[L^2(\Omega_f)]^d$, $g\in L^2(\Omega)$ are the forcing functions, and $ \Pr $, $Ra$ denote the Prandtl and Rayleigh numbers, respectively,.
The model problem \eqref{pb1}, arising both in nature and in engineering applications, is a coupled system of fluid flow, governed by the incompressible Navier-Stokes equations, and heat transfer, governed by the energy equation. Due to its practical significance, the development of efficient numerical methods for natural convection has
attracted a great many of research efforts; see, e.g.
\cite{Ben2011A},\cite{Boland1990An},\cite{boland1990error},\cite{c2001new},\cite{Mayne2000h-adaptive},\cite{de1983natural},
\cite{H1994An},\cite{JOHANNES2012A},\cite{Luo2003A},\cite{Luo2010An},\cite{manzari1999explicit},\cite{massarotti1998characteristic},\cite{Shi2009Nonconforming},\cite{Shi2010A}, \cite{Si2011A},
\cite{Zhang2014Error},\cite{Zhang2014A}.
In \cite{Boland1990An, boland1990error}, error estimates for some finite element methods were derived in approximating stationary and non-stationary natural convection problems.
\cite{Luo2003A, Luo2010An} applied Petrov-Galerkin least squares mixed finite element methods to discretize the problems. \cite{Shi2009Nonconforming,Shi2010A} developed a nonconforming mixed element method and a Petrov-Galerkin least squares nonconforming mixed element method for the stationary problems. In \cite{zhang2015decoupled}, three kinds of decoupled two level finite element methods were presented. \cite{Zhang2014Error, Zhang2014A} applied the variational multiscale method to solve the stationary and non-stationary problems.
In this paper, we consider a weak Galerkin (WG) finite element discretization of the model problem \eqref{pb1}.
The WG method was first proposed and analyzed to solve second-order elliptic problems \cite{wang2013weak,Wang2014weakmixed}. It is designed by using a weakly defined gradient operator over functions with discontinuity, and then allows the use of totally discontinuous functions in the finite element procedure. Similar to the hybridized discontinuous Galerkin (HDG) method \cite{Cockburn2009Unified}, the WG method is of the property of local elimination of unknowns defined in the interior of elements.
We note that in some special cases the WG method and the HDG method are equivalent (cf. \cite{chen2016robust,chen-feng-xie2017, chen-xie2016}).
In \cite{chen2016robust}, a class of robust globally divergence-free weak Galerkin methods for Stokes equations were developed, and then were extended in \cite{Zheng2017A} to solve incompressible quasi-Newtonian Stokes equations. We also refer to \cite{Chen-W-W-Y2015,Li-X2015,Deka2018wginterface,Li-X2016,Mu-W-Y-Z2015, Wang-W-Z-Z2016, Wang-Y2016, Zheng-X2017,Chen-C-X2018,zhaiqilong2018hyperbolic,wangruishu2018interface,zhangjiachuan2018WGNS} for some other developments and applications of the WG method.
This paper aims to propose a class of WG methods for the natural convection problems. The methods include as unknowns the velocity, pressure, and temperature variables both in the interior of elements and on the interfaces of elements. In the interior of elements, we use piecewise polynomials of degrees $k, k-1,$ and $k$ $(k\geq 1)$ for the velocity, pressure, and temperature approximations, respectively. On the interfaces of elements, we use piecewise polynomials of degrees $l, k, l $ $(l = k-1, k)$ for the numerical traces of velocity, pressure and temperature. The methods are shown to yield globally divergence-free velocity approximations.
The rest of the paper is organized as follows. Section 2 introduces the WG finite element scheme. Section 3 shows the existence and uniqueness of the discrete solution. Section 4 derives a priori error estimates. Section 5 discusses the local elimination property and the convergence of an iteration method for the WG scheme. Finally, Section 6 provides numerical examples to verify the theoretical results.
Throughout this paper, we use $a\apprle b$ $(a\apprge b)$ to denote $a\leq Cb$ $(a\geq Cb)$, where the constant C is positive independent of mesh size $h, h_K, h_e$ and the $\Pr$, $\kappa$ and Rayleigh number.
\section{WG finite element scheme} \subsection{Notation} For any bounded domain $D\in \mathbb{R}^s (s= d, d-1)$, let $H^m(D)$ and $H_0^m(D)$ denote the usual $m^{th}$-order Sobolev spaces on D, and $\lVert \cdot\rVert_{m,D}, \lvert\cdot\rvert_{m,D}$ denote the norm and semi-norm on these spaces. We use $(\cdot,\cdot)_{m,D}$ to denote the inner product of $H^m(D)$, with $(\cdot,\cdot)_D := (\cdot,\cdot)_{0,D}$. When $D = \Omega$, we set $\lVert\cdot\rVert_m := \lVert\cdot\rVert_{m,\Omega},\lvert\cdot\rvert_m :=\lvert\cdot\rvert_{m,\Omega}, $ and $(\cdot,\cdot) := (\cdot,\cdot)_\Omega $. In particular, when $D \subset R^{d-1}$, we use $\langle\cdot,\cdot\rangle_D$ to replace $(\cdot,\cdot)_D$. For integer $k\geqslant0$, $P_k(D)$ denotes the set of all polynomials on D with degree no more than $k$. We also need the following spaces: \begin{center}
$L_0^2(\Omega) :=\{q\in L^2(\Omega) :(q,1)=0\}$,\\
$\bm{H}(div,D) :=\{\bm{v}\in \bm{L}^2(D) :\nabla\cdot\bm{v}\in L^2(D)\}.$ \end{center}
Let $\mathcal{T}_h^s$ and $\mathcal{T}_h^f$ be shape-regular simplicial decompositions of the subdomains $\Omega_s$ and $\Omega_f$, respectively. Then $\mathcal{T}_h := \mathcal{T}_h^s\cup \mathcal{T}_h^f=\bigcup \{K\}$ is a shape-regular simplicial decomposition of $\Omega$. Let $\varepsilon_h^s$ and $\varepsilon_h^f$ be the sets of all edges (faces) of all elements in $\mathcal{T}_h^s$ and $\mathcal{T}_h^f$, respectively, and set $\varepsilon_h: = \varepsilon_h^s\cup \varepsilon_h^f=\bigcup\{e\}$.
For any $K\in \mathcal{T}_h$, $e\in \varepsilon_h$, we denote by $h_K$ and $h_e$ the diameters of $K$ and $e$, respectively, and set $h: = \max\limits_{K\in \mathcal{T}_h}h_K$. Let $\bm{n}_K$ and $\bm{n}_e$ be the outward unit normal vectors along the boundary $\partial K$ and $e$. We denote by $\nabla_h$ and $\nabla_h\cdot$ the piecewise-defined gradient and divergence with respect to $\mathcal{T}_h$. We also introduce the mesh-dependent inner products and mesh-dependent norms: \begin{align*} \langle u,v\rangle_{\partial\mathcal{T}_h} :=&\sum_{K\in \mathcal{T}_h}\langle u,v\rangle_{\partial K},\quad \lVert u\rVert_{0,\partial\mathcal{T}_h} :=\left(\sum_{K\in \mathcal{T}_h}\lVert u\rVert_{0,\partial K}^2\right)^{1/2} \\ (u,v)_{\mathcal{T}_h} :=&\sum_{K\in \mathcal{T}_h}(u,v)_{K},\quad\quad \lVert u\rVert_{0,\mathcal{T}_h} :=\left(\sum_{K\in \mathcal{T}_h}\lVert u\rVert_{0,K}^2\right)^{1/2}. \end{align*}
\subsection{Weak problem} We first introduce the space \begin{align*} \bm{W} := \{\bm{v}\in [H_0^1(\Omega_f)]^d: \nabla\cdot \bm{v}=0\} \end{align*} and the following bilinear and trilinear forms: for any $ \bm{u},\bm{v}, \bm{w}\in H_0^1(\Omega_f)$, $q\in L_0^2(\Omega_f)$, and $T,s\in H_0^1(\Omega)$,
\begin{align*} a(\bm{u},\bm{v}) &:= \Pr (\nabla\bm{u},\nabla\bm{v}),\quad b(\bm{v},q) := (\nabla q,\bm{v}),\\ d(T,\bm{v}) &:= \Pr Ra(\bm{j}T,\bm{v}),\quad \overline{a}(T,s) := \kappa(\nabla T,\nabla s),\\ c(\bm{w};\bm{u},\bm{v}) &:= ((\bm{w}\cdot \nabla)\bm{u},\bm{v}), \quad \overline{c}(\bm{w};T,s) := ((\bm{w}\cdot \nabla)T,s). \end{align*} It is easy to see that, for $\bm u\in \bm W$, \begin{align*} c(\bm{u};\bm{u},\bm{v}) &= \frac{1}{2}(\nabla\cdot (\bm{u}\otimes\bm{u}),\bm{v})-\frac{1}{2}(\nabla\cdot (\bm{u}\otimes\bm{v}),\bm{u}),\\
\overline{c}(\bm{u};T,s) &= \frac{1}{2}(\nabla\cdot (\bm{u}T),s)-\frac{1}{2}(\nabla\cdot (\bm{u}s),T). \end{align*}
Then the variational problem of (\ref{pb1}) reads as follows: seek $(\bm{u},p,T)\in \bm{W}\times L_0^2(\Omega_f)\times H_0^1(\Omega)$ such that \begin{align} \label{pbv} \left \{ \begin{array}{rl} A(\bm{u};\bm{u},\bm{v})+b(\bm{v},p)-b(\bm{u},q)-d(T,\bm{v})&= (\bm{f},\bm{v}), \forall (\bm{v},q)\in \bm{W}\times L_0^2(\Omega_f),\\ \overline{A}(\bm{u};T,s)&= (g,s), \forall s\in H_0^1(\Omega), \end{array} \right. \end{align} where \begin{align*} A(\bm{u};\bm{u},\bm{v}) &:= a(\bm{u},\bm{v})+c(\bm{u};\bm{u},\bm{v}),\\
\overline{A}(\bm{u};T,s)&=\overline{a}(T,s)+\overline{c}(\bm{u};T,s).
\end{align*}
\begin{thm}\label{continuous exsit}
\cite{boland1990error}For $\bm{f}\in [H^{-1}(\Omega_f)]^d$ and $g\in H^{-1}(\Omega)$, the weak problem (\ref{pbv}) has at least one solution.
In addition, it admits a unique solution $(\bm{u},p,T)\in \bm{W}\times L_0^2(\Omega_f)\times H_0^1(\Omega)$ if
\begin{equation*}\label{assump1}
({\Pr}^{-1}\mathcal{N}Ra\kappa^{-1}+Ra\mathcal{M}\kappa^{-2})\lVert g\rVert_{-1}+\mathcal{N}{\Pr}^{-2}\lVert \bm{f}\rVert_{-1} < 1,
\end{equation*} where \begin{align*} \mathcal{N} := \sup\limits_{\bm{0}\neq\bm{w},\bm{u},\bm{v}\in \bm{W}}\frac{c(\bm{w};\bm{u},\bm{v})}{\lvert \bm{w}\rvert_1\lvert \bm{u}\rvert_1\lvert \bm{v}\rvert_1},\quad \mathcal{M} := \sup\limits_{\substack{\bm{0}\neq\bm{w}\in \bm{W},\\ 0\neq T,s\in H_0^1(\Omega)}}\frac{\overline{c}(\bm{w};T,s)}{\lvert \bm{w}\rvert_1\lvert T\rvert_1\lvert s\rvert_1}. \end{align*}
\end{thm}
In what follows, we assume that the solution $(\bm{u},p,T)$ is unique and, more precisely, there exists a fixed constant $\delta> 0$ such that \begin{equation*}\label{assump2} ({\Pr}^{-1}\mathcal{N}Ra\kappa^{-1}+Ra\mathcal{M}\kappa^{-2})\lVert g\rVert_{-1}+\mathcal{N}{\Pr}^{-2}\lVert \bm{f}\rVert_{-1} < 1-\delta. \end{equation*}
\subsection{Discrete weak operators} In order to design a WG finite element scheme for the problem (1.1), we introduce the discrete weak gradient operator $\nabla_{w,r}$ and the discrete weak divergence operator $\nabla_{w,r}\cdot$ as follows.
\begin{defi}
For any $K\in \mathcal{T}_h$ and $v\in \mathcal{V}(K):=\left\{\{v_0,v_b\}:v_0\in L^2(K),v_b\in H^{1/2}(\partial K)\right\}$, the discrete weak gradient $\nabla_{w,r,K}v\in [P_r(K)]^d$ on $K$ is determined by the equation
\begin{equation*}\label{weak gradient}
(\nabla_{w,r,K}v,\bm{\tau})_K= -(v_0,\nabla\cdot\bm{\tau})_K+\langle v_b,\bm{\tau} \cdot \bm{n}_K\rangle_{\partial K}\quad \forall \bm{\tau}\in [P_r(K)]^d.
\end{equation*} \end{defi} Then we define the global discrete weak gradient operator $\nabla_{w,r}$ by \begin{center}
$\nabla_{w,r}|_K=\nabla_{w,r,K},\quad \forall K\in \mathcal{T}_h$. \end{center}
For a vector $\bm{v}=(v_1,\cdots,v_d)^T\in [\mathcal{V}(K)]^d$, we define its discrete weak gradient $\nabla_{w,r}\bm{v}$ by $$\nabla_{w,r}\bm{v}:=(\nabla_{w,r}v_1,\cdots,\nabla_{w,r}v_d)^T.$$
\begin{defi}
For any $K\in \mathcal{T}_h$ and $\bm{v}\in \mathcal{W}(K):=\left\{\{\bm{v}_0,\bm{v}_b\}:\bm{v}_0\in [L^2(K)]^d,\bm{v}_b\cdot\bm{n}_K\in H^{-1/2}(\partial K)\right\}$, the discrete weak divergence $\nabla_{w,r,K}\cdot \bm{v}\in P_r(K)$ is determined by the equation
\begin{equation*}\label{weak divergence}
(\nabla_{w,r,K}\cdot \bm{v},\tau)_K= -(\bm{v}_0,\nabla\tau)_K+\langle \bm{v}_b\cdot \bm{n}_K,\tau \rangle_{\partial K}\quad \forall \tau\in P_r(K).
\end{equation*} \end{defi} Then we define the global discrete weak divergence operator $\nabla_{w,r}\cdot$ by
$$\nabla_{w,r}\cdot|_K=\nabla_{w,r,K}\cdot,\forall K\in \mathcal{T}_h.$$ For a tensor $\bm w=(\bm w_1,\cdots, \bm w_d)^T\in [\mathcal{W}(K)]^{d\times d}$ with $\bm w_i\in [\mathcal{W}(K)]^{d}$ for $i=1,\cdots,d$, we define its discrete weak divergence $\nabla_{w,r}\cdot \bm w$ by $$ \nabla_{w,r}\cdot \bm w=( \nabla_{w,r}\cdot \bm w_1,\cdots, \nabla_{w,r}\cdot \bm w_d)^T.$$
\subsection{WG finite element scheme} For any $K\in \mathcal{T}_h,e\in \varepsilon_h$ and any integer $j\geq 0$, let $Q_j^0:L^2(K)\rightarrow P_j(K)$ and $Q^b_j:L^2(e)\rightarrow P_j(e)$ be the usual $L^2$ projection operators. We shall use $\bm{Q}_j^b$ to denote $Q^b_j$ for vector spaces.
For any integer $k\geq 1$ and $l = k-1,k$, we introduce the following finite dimensional spaces: \begin{align*}
\bm{V}_h&=\{\bm{v}_h=\{\bm{v}_{h0},\bm{v}_{hb}\}:\bm{v}_{h0}|_K\in [P_k(K)]^d,\bm{v}_{hb}|_e\in [P_l(e)]^d,\forall K\in \mathcal{T}_h,\forall e\in \varepsilon_h\},\\
\bm{V}_h^0&=\{\bm{v}_h=\{\bm{v}_{h0},\bm{v}_{hb}\}\in \bm{V}_h:\bm{v}_{hb}|_{\partial \Omega_f}=0\},\\
Q_h&=\{q_h=\{q_{h0},q_{hb}\}:q_{h0}|_K\in P_{k-1}(K),q_{hb}|_e\in P_k(e),\forall K\in \mathcal{T}_h,\forall e\in \varepsilon_h\},\\ Q_h^0&=\{q_h=\{q_{h0},q_{hb}\}\in Q_h:q_{h0}\in L_0^2(\Omega_f)\},\\
S_h&=\{s_h=\{s_{h0},s_{hb}\}:s_{h0}|_K\in P_k(K),s_{hb}|_e\in P_l(e),\forall K\in \mathcal{T}_h,\forall e\in \varepsilon_h\},\\
S_h^0&=\{s_h=\{s_{h0},s_{hb}\}\in S_h:s_{hb}|_{\partial \Omega}=0\}. \end{align*}
For any $\bm{u}_h =\{\bm{u}_{h0},\bm{u}_{hb}\}, \bm{v}_h =\{\bm{v}_{h0},\bm{v}_{hb}\}\in \bm{V}_h^0$, $q_h = \{q_{h0},q_{hb}\}\in Q_h^0$, and $T_h = \{T_{h0},T_{hb}\}, s_h = \{s_{h0},s_{hb}\}\in S_h^0$, define the following bilinear and trilinear forms: \begin{align*} a_h(\bm{u}_h,\bm{v}_h) &:= \Pr (\nabla_{w,m}\bm{u}_h,\nabla_{w,m}\bm{v}_h)+\Pr \langle \tau(\bm{Q}_l^b\bm{u}_{h0}-\bm{u}_{hb}),\bm{Q}_l^b\bm{v}_{h0}-\bm{v}_{hb}\rangle_{\partial \mathcal{T}_h^f},\\ b_h(\bm{v}_h,q_h) &:= (\nabla_{w,k}q_h,\bm{v}_{h0}),\\ d_h(T_h,\bm{v}_h) &:= \Pr Ra(\bm{j}T_{h0},\bm{v}_{h0}),\\ \overline{a}_h(T_h,s_h) &:= \kappa(\nabla_{w,m}T_h,\nabla_{w,m}s_h)+\kappa\langle \tau(Q_l^bT_{h0}-T_{hb}),Q_l^bs_{h0}-s_{hb}\rangle_{\partial \mathcal{T}_h},\\ c_h(\bm{w}_h;\bm{u}_h,\bm{v}_h) &:= \frac{1}{2}(\nabla_{w,k}\cdot (\bm{u}_h\otimes\bm{w}_h),\bm{v}_{h0})-\frac{1}{2}(\nabla_{w,k}\cdot (\bm{v}_h\otimes\bm{w}_h),\bm{u}_{h0}),\\ \overline{c}_h(\bm{u}_h;T_h,s_h) &:= \frac{1}{2}(\nabla_{w,k}\cdot (\bm{u}_hT_h),s_{h0})-\frac{1}{2}(\nabla_{w,k}\cdot (\bm{u}_hs_h),T_{h0}). \end{align*} It is easy to see that \begin{equation}
c_h(\bm{w}_h;\bm{v}_h,\bm{v}_h) =0,\quad
\overline{c}_h(\bm{w}_h;s_h,s_h) =0.\label{3356} \end{equation}
The WG finite element scheme for (\ref{pb1}) is then given as follows: seek $\bm{u}_h = \{\bm{u}_{h0},\bm{u}_{hb}\}\in \bm{V}_h^0$, $p_h = \{p_{h0},p_{hb}\}\in Q_h^0$, and $T_h =\{T_{h0},T_{hb}\}\in S_h^0$ such that \begin{align} \label{pb2} \left \{ \begin{array}{rl} A_h(\bm{u}_h;\bm{u}_h,\bm{v}_h)+b_h(\bm{v}_h,p_h)-b_h(\bm{u}_h,q_h)-d_h(T_h,\bm{v}_h)&= (\bm{f},\bm{v}_{h0}), \forall (\bm{v}_h,q_h)\in \bm{V}_h^0\times Q_h^0,\\ \overline{A}_h(\bm{u}_h;T_h,s_h)&= (g,s_{h0}), \forall s_h\in S_h^0, \end{array} \right. \end{align} where \begin{align} A_h(\bm{w}_h;\bm{u}_h,\bm{v}_h) &:= a_h(\bm{u}_h,\bm{v}_h)+c_h(\bm{w}_h;\bm{u}_h,\bm{v}_h),\\
\overline{A}_h(\bm{u}_h;T_h,s_h)&:=\overline{a}_h(T_h,s_h)+\overline{c}_h(\bm{u}_h;T_h,s_h),\label{224}
\end{align}
$\tau|_{\partial T}=h_T^{-1}$, and m is an integer with $k-1\leq m\leq l$.
\begin{rem}
It's easy to show that the scheme (\ref{pb2}) yields globally divergence-free velocity approximation $\bm{u}_{h0}$. In fact, let $K_1,K_2\in \mathcal{T}_h$ be any two adjacent elements with a common face $e$, introduce a function $r_{hb}\in L^2(\varepsilon_h)$ with
\begin{equation*}
r_{hb}|_e=\left\{
\begin{array}{l}
-(\bm{u}_{h0}\cdot \bm{n}_e)|_{K_1\bigcap e}-(\bm{u}_{h0}\cdot \bm{n}_e)|_{K_2\bigcap e},\quad \forall e\in \varepsilon_h^f/\partial \Omega_f,\\
0,\quad \forall e\in \partial\Omega_f,
\end{array}
\right.
\end{equation*}
and set $c_0:=\frac{1}{|\Omega_f|} \int_{\Omega_f} \nabla_h\cdot \bm{u}_{h0}d\bm{x}$. Then, taking $(\bm{v}_{h0},\bm{v}_{hb},q_{h0},q_{hb},s_{h0},s_{hb})=(\bm{0},\bm{0},\nabla_h\cdot \bm{u}_{h0}-c_0,r_{hb}-c_0,0,0) $
in (\ref{pb2}) yields
\begin{equation*}
\lVert \nabla_h\cdot \bm{u}_{h0}\rVert_0^2 +\sum_{e\in \varepsilon_h^f/\partial \Omega_f}\lVert (\bm{u}_{h0}\cdot \bm{n}_e)|_{K_1}+(\bm{u}_{h0}\cdot \bm{n}_e)|_{K_2}\rVert_{0,e}^2=0.
\end{equation*}
This indicates $\bm{u}_{h0}\in\bm{H}(div,\Omega_f)$ and $\nabla_h\cdot \bm{u}_{h0}=\nabla\cdot \bm{u}_{h0}=0$, i.e. the velocity approximation $\bm{u}_{h0}$ is globally divergence-free in a pointwise sense. \end{rem}
\section{Well-posedness of the discrete scheme}
\subsection{Some basic results} For the projections $Q_j^0$ and $Q_j^b$ with $j\geq 0$, the following stability and approximation results are standard.
\begin{lem}(\cite{shi2013finite}) \label{ineq}
Let $s$ be an integer with $1\leq s\leq j+1$. Then we have, for any $K\in \mathcal{T}_h$ and $e\in \varepsilon_h$,
\begin{align*}
\lVert v-Q_j^0v\rVert_{0,K}+h_K\lvert v-Q_j^0v\rvert_{1,K}&\apprle h_K^s\lvert v\rvert_{s,K},\forall v\in H^s(K),\\
\lVert v-Q_j^0v\rVert_{0,\partial K}&\apprle h_K^{s-1/2}\lvert v\rvert_{s,K},\forall v\in H^s(K),\\
\lVert v-Q_j^bv\rVert_{0,\partial K}&\apprle h_K^{s-1/2}\lvert v\rvert_{s,K},\forall v\in H^s(K),\\
\lVert Q_j^0v\rVert_{0,K}&\leq \lVert v\rVert_{0,K},\forall v\in L^2(K),\\
\lVert Q_j^bv\rVert_{0,e}&\leq \lVert v\rVert_{0,e},\forall v\in L^2(e).
\end{align*}
\end{lem}
By using the trace theorem, the inverse inequality, and scaling arguments metioned in {\cite{shi2013finite}}, we can get the following lemma. \begin{lem}\label{trace inequality}
For all $K\in \mathcal{T}_h$, $w\in H^1(K)$, and $1\leq \tilde{q}\leq \infty$, we have
\begin{equation*}
\lVert w\rVert_{0,\tilde{q},\partial K}\apprle h_K^{-\frac{1}{\tilde{q}}}\lVert w\rVert_{0,\tilde{q},K} + h_K^{1-\frac{1}{\tilde{q}}}\lvert w\rvert_{1,\tilde{q},K},
\end{equation*}
In particular, for all $w\in P_k(K)$,
\begin{equation*}
\lVert w\rVert_{0,\tilde{q},\partial K}\apprle h_K^{-\frac{1}{\tilde{q}}}\lVert w\rVert_{0,\tilde{q},K}.
\end{equation*} \end{lem}
\begin{lem}(\cite{chen2016robust}) \label{lem3.3}
Let $0\leq k-1\leq m\leq l\leq k$. For all $K\in \mathcal{T}_h$ and $\bm{v}_h=\{\bm{v}_{h0},\bm{v}_{hb}\}\in [P_k(K)]^d\times [P_l(\partial K)]^d$, the following estimates hold:
\begin{align}
\lVert \nabla\bm{v}_{h0}\rVert_{0,K}&\apprle \lVert \nabla_{w,m}\bm{v}_h\rVert_{0,K}+h_K^{-1/2}\lVert \bm{Q}_l^b\bm{v}_{h0}-\bm{v}_{hb}\rVert_{0,\partial K},\label{equivalentu1}\\
\lVert \nabla_{w,m}\bm{v}_h\rVert_{0,K}&\apprle \lVert \nabla\bm{v}_{h0}\rVert_{0,K}+h_K^{-1/2}\lVert \bm{Q}_l^b\bm{v}_{h0}-\bm{v}_{hb}\rVert_{0,\partial K},\label{equivalentu2}
\end{align}
\end{lem}
We introduce the following semi-norms: for any $(\bm{v}_h,q_h,s_h)\in \bm{V}_h\times Q_h\times S_h$, \begin{align*} \interleave\bm{v}_h\interleave^2:=& \lVert\nabla_{w,m}\bm{v}_h\rVert_0^2+ \lVert\tau^{1/2}(\bm{Q}_l^b\bm{v}_{h0}-\bm{v}_{hb})\rVert_{0,\partial\mathcal{T}_h^f}^2,\\ \lVert q_h\rVert^2:=&\lVert q_{h0}\rVert_0^2+\sum_{K\in\mathcal{T}_h^f}\lVert\nabla_{w,k}q_h\rVert_{0,K}^2,\\ \interleave s_h\interleave^2:=& \lVert\nabla_{w,m} s_h\rVert_0^2+ \lVert\tau^{1/2}(Q_l^bs_{h0}-s_{hb})\rVert_{0,\partial\mathcal{T}_h}^2. \end{align*}
Here we recall that $\tau|_{\partial K}=h_K^{-1}$. It is easy to see that the above three semi-norms are norms on $\bm{V}_h^0$, $Q_h^0$ and $S_h^0$, respectively (cf. \cite{chen2016robust}). In addition, from the lemma above it follows
\begin{equation}\label{1seminorm v}
\lVert \nabla_h\bm{v}_{h0}\rVert_0\apprle \interleave \bm{v}_h\interleave,\quad \forall \bm{v}_h\in \bm{V}_h^0.
\end{equation}
\begin{rem} We note that the estimates \eqref{equivalentu1}, \eqref{equivalentu2}, and \eqref{1seminorm v} also hold for all $s_h\in S_h^0$ due to the fact that $s_h|_K=\{s_{h0},s_{hb}\}\in P_k(K)\times P_l(\partial K)$. \end{rem}
\begin{lem}(\cite{karakashian2007convergence})
For all $s_{h0}\in S_{h0} = \{s_{h0}:s_{h0}|_K\in P_k(K),\forall K\in \mathcal{T}_h\}$, there exists an interpolation
$I_ks_{h0}\in S_{h0}\cap H_0^1(\Omega)$ such that
\begin{align*}
\sum_{K\in \mathcal{T}_h}\lVert s_{h0}-I_ks_{h0}\rVert_{0,K}^2&\apprle \sum_{e\in \varepsilon_h} h_e\lVert [s_{h0}]\rVert_{0,e}^2,\\
\sum_{K\in \mathcal{T}_h}\lVert \nabla(s_{h0}-I_ks_{h0})\rVert_{0,K}^2&\apprle \sum_{e\in \varepsilon_h}h_e^{-1}\lVert [s_{h0}]\rVert_{0,e}^2.
\end{align*}
\end{lem}
From this lemma it follows that, for all $\bm{v}_{h0}\in \bm{V}_{h0} = \{\bm{v}_{h0}:\bm{v}_{h0}|_K\in P_k(K),\forall K\in \mathcal{T}_h^f\}$, there exists an interpolation $\bm{I}_k\bm{v}_{h0}\in \bm{V}_{h0}\cap [H_0^1(\Omega)]^d$ such that
\begin{align}
\sum_{K\in \mathcal{T}_h}\lVert \bm{v}_{h0}-\bm{I}_k\bm{v}_{h0}\rVert_{0,K}^2&\apprle \sum_{e\in \varepsilon_h} h_e\lVert [\bm{v}_{h0}]\rVert_{0,e}^2,\label{bmI_k}\\
\sum_{K\in \mathcal{T}_h}\lVert \nabla(\bm{v}_{h0}-\bm{I}_k\bm{v}_{h0})\rVert_{0,K}^2&\apprle \sum_{e\in \varepsilon_h}h_e^{-1}\lVert [\bm{v}_{h0}]\rVert_{0,e}^2.\label{bmI_ksemi}
\end{align}
\begin{lem}\label{discrete sobolev embedding}
For all $\bm{v}_h\in \bm{V}_h^0$ and $s_h\in S_h^0$, we have
\begin{align}
\lVert \bm{v}_{h0}\rVert_{0,\tilde{q}}&\leq C_{\tilde{q}1}\interleave \bm{v}_h\interleave,\label{ineq 11}\\
\lVert s_{h0}\rVert_{0,\tilde{q}}&\leq C_{\tilde{q}2}\interleave s_h\interleave,\label{ineq 12}
\end{align}
where $2\leq \tilde{q}< \infty$ when $d = 2$, $2 \leq \tilde{q}\leq 6$ when $d = 3$, and $C_{\tilde{q}1}$, $C_{\tilde{q}2}$ are positive constants only depending on $\tilde{q}$.
\end{lem} \begin{proof}
For all $\bm{v}_h\in \bm{V}_h^0$, we apply the Sobolev embedding theorem and Poinc\'{a}re inequality to get
\begin{equation}\label{318}
\lVert \bm{I}_k\bm{v}_{h0}\rVert_{0,\tilde{q}}\apprle \lVert \bm{I}_k\bm{v}_{h0}\rVert_1 \apprle \lVert \nabla\bm{I}_k\bm{v}_{h0}\rVert_0.
\end{equation}
From (\ref{bmI_ksemi}), (\ref{1seminorm v}),
the definition of $ \interleave\cdot \interleave$, and the projection property of $\bm{Q}_l^b$, it follows
\begin{equation}\label{319}
\begin{aligned}
\lVert \nabla\bm{I}_k\bm{v}_{h0}\rVert_0 &\apprle \lVert \nabla_h \bm{v}_{h0}\rVert_0 + \big(\sum_{e\in \varepsilon_h}h_e^{-1}\lVert [\bm{v}_{h0}]\rVert_{0,e}^2\big)^{\frac{1}{2}}\\
&\apprle \interleave \bm{v}_h\interleave + \big(\sum_{e\in \varepsilon_h}h_e^{-1}\lVert [\bm{v}_{h0}-\bm{v}_{hb}]\rVert_{0,e}^2\big)^{\frac{1}{2}}\\
&\apprle \interleave \bm{v}_h\interleave.
\end{aligned}
\end{equation}
Using the Sobolev embedding theorem and the inverse inequality once again, by the properties of the projection-mean operator (\cite{shi2013finite}) $\bm{\Pi_h}:\bm{V}_{h0} = \{\bm{v}_{h0}:\bm{v}_{h0}|_K\in P_k(K),\forall K\in \mathcal{T}_h^f\}\rightarrow W^{1,2}(\Omega_f)\cap W^{0,\tilde{q}}(\Omega_f)$ and the fact that $2\leq \tilde{q}< \infty$ when $d = 2$ and $2 \leq \tilde{q}\leq 6$ when $d = 3$, we have
\begin{equation*}
\begin{aligned}
\lVert \bm{v}_{h0}-\bm{I}_k\bm{v}_{h0}\rVert_{0,\tilde{q}} &\leq \lVert \bm{v}_{h0}-\bm{\Pi_h}\bm{v}_{h0}\rVert_{0,\tilde{q}}+\lVert \bm{\Pi_h}\bm{v}_{h0}-\bm{I}_k\bm{v}_{h0}\rVert_{0,\tilde{q}}\\
&\apprle h\lVert \nabla_h\bm{v}_{h0}\rVert_{0,\tilde{q}} +\lVert \bm{\Pi_h}\bm{v}_{h0}-\bm{I}_k\bm{v}_{h0}\rVert_{1,2} \\
&\apprle h^{1-(\frac{d}{2}-\frac{d}{\tilde{q}})}\lVert \nabla_h\bm{v}_{h0}\rVert_{0,2} + \lVert \bm{v}_{h0}-\bm{\Pi_h}\bm{v}_{h0}\rVert_{1,2}+\lVert \bm{v}_{h0}-\bm{I}_k\bm{v}_{h0}\rVert_{1,2}\\
&\apprle \lVert \nabla_h \bm{v}_{h0}\rVert_0+\lVert \nabla_h (\bm{v}_{h0}-\bm{I}_k\bm{v}_{h0})\rVert_0\\
&\apprle \interleave \bm{v}_h\interleave,
\end{aligned}
\end{equation*} which, together with \eqref{318} and \eqref{319}, yields the desired estimate (\ref{ineq 11}).
Similarly, we can obtain (\ref{ineq 12}). This finishes the proof. \end{proof}
For any nonnegative integer $j$ and any $K\in \mathcal{T}_h$, we introduce the local Raviart-Thomas(RT) element space \begin{equation*} \bm{RT}_j(K)=[P_j(K)]^d+\bm{x}P_j(K). \end{equation*}
Lemmas \ref{RT3.6}-\ref{RT38} show some properties of the $RT$ projection which can be founded in ({\cite{brezzi2008mixed}}.Page 9-10).
\begin{lem}\label{RT3.6}
For any $\bm{v}_{h0}\in \bm{RT}_j(K)$, $\nabla\cdot \bm{v}_{h0}|_K=0$ implies $\bm{v}_{h0}\in [P_j(K)]^d$. \end{lem}
\begin{lem}\label{RT}
For any $K\in \mathcal{T}_h$ and $\bm{v}\in [H^1(K)]^d$, there exists a unique $\bm{P}_j^{RT}\bm{v}\in \bm{RT}_j(K)$ such that
\begin{align}
\langle\bm{P}_j^{RT}\bm{v}\cdot \bm{n}_e,w_j\rangle_e&= \langle\bm{v}\cdot \bm{n}_e,w_j\rangle_e,\quad \forall w_j\in P_j(e),e\in \partial K,\label{RT1}\\
(\bm{P}_j^{RT}\bm{v},\bm{w}_{j-1})_K&= (\bm{v},\bm{w}_{j-1})_K,\quad \forall \bm{w}_{j-1}\in [P_{j-1}(K)]^d\label{RT2}.
\end{align}
If $j=0$, $\bm{P}_j^{RT}\bm{v}$ is determined only by (\ref{RT1}). Moreover, the following approximation holds:
\begin{equation*}\label{RT3}
\lVert \bm{v}-\bm{P}_j^{RT}\bm{v}\rVert_{0,K}\apprle h_K^r\lvert \bm{v}\rvert_{r,K},\quad \forall 1\leq r\leq j+1,\forall \bm{v}\in[H^r(K)]^d.
\end{equation*} \end{lem}
\begin{lem}\label{RT38}
The operator $\bm{P}_j^{RT} $ defined in Lemma \ref{RT} satisfies
\begin{equation*}\label{RTdiv}
(\nabla\cdot \bm{P}_j^{RT}\bm{v},q_h)_K= (\nabla\cdot \bm{v},q_h)_K,\quad \forall \bm{v}\in [H^1(K)]^d, q_h\in P_j(K),K\in \mathcal{T}_h.
\end{equation*} \end{lem}
\begin{lem}(\cite{chen2016robust})\label{commutativity}
It holds the following commutativity properties:
\begin{align}
\nabla_{w,m}\{\bm{P}_k^{RT}\bm{v},\bm{Q}_l^b\bm{v}\}&=\bm{Q}_m^0(\nabla\bm{v}),\quad \forall \bm{v}\in [H^1(\Omega_f)]^d. \label{com1}\\
\nabla_{w,k}\{Q_{k-1}^0q,Q_k^bq\}&=\bm{Q}_k^0(\nabla q),\quad \forall q\in H^1(\Omega_f)\label{com2},\\
\nabla_{w,m}\{Q_k^0 s,Q_l^b s\}&=\bm{Q}_m^0(\nabla s),\quad \forall s\in H^1(\Omega). \label{com3}
\end{align} \end{lem}
\subsection{Stability conditions} \begin{lem}\label{boundedness}
For any $\bm{u}_h,\bm{v}_h \in \bm{V}_h$, and $T_h,s_h \in S_h$, the following inequalities hold:
\begin{align}
a_h(\bm{u}_h,\bm{v}_h) &\apprle \Pr\interleave \bm{u}_h\interleave \cdot\interleave \bm{v}_h\interleave,\label{continuous result ah} \\
a_h(\bm{v}_h,\bm{v}_h) &= \Pr\interleave \bm{v}_h\interleave^2,\label{coercivity ah} \\
\overline{a}_h(T_h,s_h) &\apprle \kappa\interleave T_h\interleave \cdot\interleave s_h\interleave,\label{continuous result ahbar} \\
\overline{a}_h(s_h,s_h) &= \kappa\interleave s_h\interleave^2,\label{coercivity ahbar} \\
c_h(\bm{w}_h;\bm{u}_h,\bm{v}_h) &\apprle \interleave \bm{w}_h\interleave \cdot \interleave \bm{u}_h\interleave \cdot \interleave \bm{v}_h\interleave ,\label{continuous result ch} \\
\overline{c}_h(\bm{w}_h;T_h,s_h) &\apprle \interleave \bm{w}_h\interleave \cdot \interleave T_h\interleave \cdot \interleave s_h\interleave ,\label{continuous result chbar} \\
d_h(T_h,\bm{v}_h)&\apprle \Pr Ra \interleave T_h\interleave \cdot\interleave \bm{v}_h\interleave. \label{continuous result dh}
\end{align} \end{lem} \begin{proof}
From the definitions of $a_h(\cdot,\cdot), \overline{a}_h(\cdot,\cdot),c_h(\cdot;\cdot,\cdot),\overline{c}_h(\cdot;\cdot,\cdot), d_h(\cdot,\cdot)$, Cauchy-Schwarz inequality and Lemma \ref{discrete sobolev embedding}, we can easily get (\ref{continuous result ah}),(\ref{continuous result ahbar}), and (\ref{continuous result dh}).
For all $\bm{u}_h,\bm{v}_h \in \bm{V}_h$, by the definition of $\nabla_{w,k}\cdot$ we have
\begin{align*}
2 c_h(\bm{w}_h;\bm{u}_h,\bm{v}_h) = &(\bm{v}_{h0}\otimes\bm{w}_{h0},\nabla_h\bm{u}_{h0})-(\bm{u}_{h0}\otimes\bm{w}_{h0},\nabla_h\bm{v}_{h0}) \\
&\quad - \langle \bm{v}_{hb}\otimes\bm{w}_{hb}\bm{n},\bm{u}_{h0}\rangle_{\partial\mathcal{T}_h^f} +\langle \bm{u}_{hb}\otimes\bm{w}_{hb}\bm{n},\bm{v}_{h0}\rangle_{\partial\mathcal{T}_h^f}\\
= &\big((\bm{v}_{h0}\otimes\bm{w}_{h0},\nabla_h\bm{u}_{h0})-(\bm{u}_{h0}\otimes\bm{w}_{h0},\nabla_h\bm{v}_{h0})\big) \\
&+ \langle (\bm{u}_{h0}-\bm{u}_{hb})\otimes(\bm{w}_{h0}-\bm{w}_{hb})\bm{n},\bm{v}_{h0}\rangle \\
&- \langle (\bm{u}_{h0}-\bm{u}_{hb})\otimes\bm{w}_{h0}\bm{n},\bm{v}_{h0}\rangle \\
&- \langle (\bm{v}_{h0}-\bm{v}_{hb})\otimes(\bm{w}_{h0}-\bm{w}_{hb})\bm{n},\bm{u}_{h0}\rangle \\
&+ \langle (\bm{v}_{h0}-\bm{v}_{hb})\otimes\bm{w}_{h0}\bm{n},\bm{u}_{h0}\rangle \\
=:& \sum_{i=1}^{5}R_i.
\end{align*}
In light of H\"{o}lder's inequality and Lemma \ref{discrete sobolev embedding}, we obtain
\begin{align*}
\lvert R_1 \rvert &\leq \lVert \bm{v}_{h0}\rVert_{0,4}\lVert \bm{w}_{h0}\rVert_{0,4}\lVert \nabla_h\bm{u}_{h0}\rVert_{0,2} +\lVert \bm{u}_{h0}\rVert_{0,4}\lVert \bm{w}_{h0}\rVert_{0,4}\lVert \nabla_h\bm{v}_{h0}\rVert_{0,2} \\
&\apprle \interleave \bm{w}_h\interleave \cdot \interleave \bm{u}_h\interleave \cdot \interleave \bm{v}_h\interleave.
\end{align*}
From H\"{o}lder's inequality, Lemma \ref{trace inequality}, Lemma \ref{discrete sobolev embedding}, and the inverse inequality, it follows
\begin{align*}
|R_2| &\leq \sum_{K\in\mathcal{T}_h^f}\lVert \bm{w}_{h0}-\bm{w}_{hb}\rVert_{0,3,\partial K}\lVert \bm{u}_{h0}-\bm{u}_{hb}\rVert_{0,2,\partial K}\lVert \bm{v}_{h0}\rVert_{0,6,\partial K} \\
&\leq \sum_{K\in\mathcal{T}_h^f}h_K^{-\frac{d-1}{6}}\lVert \bm{w}_{h0}-\bm{w}_{hb}\rVert_{0,2,\partial K}\lVert \bm{u}_{h0}-\bm{u}_{hb}\rVert_{0,2,\partial K}h_K^{-\frac{1}{6}}\lVert \bm{v}_{h0}\rVert_{0,6, K} \\
&\leq \sum_{K\in\mathcal{T}_h^f}h_K^{-\frac{1}{2}}\lVert \bm{w}_{h0}-\bm{w}_{hb}\rVert_{0,2,\partial K}h_K^{-\frac{1}{2}}\lVert \bm{u}_{h0}-\bm{u}_{hb}\rVert_{0,2,\partial K}h_K^{1-\frac{d}{6}}\lVert \bm{v}_{h0}\rVert_{0,6, K} \\
&\apprle \interleave \bm{w}_h\interleave \cdot \interleave \bm{u}_h\interleave \cdot \interleave \bm{v}_h\interleave
\end{align*}
and
\begin{align*}
|R_3| &\leq \sum_{K\in\mathcal{T}_h^f}\lVert \bm{w}_{h0}\rVert_{0,4,\partial K}\lVert \bm{u}_{h0}-\bm{u}_{hb}\rVert_{0,2,\partial K}\lVert \bm{v}_{h0}\rVert_{0,4,\partial K} \\
&\leq \sum_{K\in\mathcal{T}_h^f}h_K^{-\frac{1}{4}}\lVert \bm{w}_{h0}\rVert_{0,4,K}\lVert \bm{u}_{h0}-\bm{u}_{hb}\rVert_{0,2,\partial K}h_K^{-\frac{1}{4}}\lVert \bm{v}_{h0}\rVert_{0,4,K} \\
&\leq \sum_{K\in\mathcal{T}_h^f}\lVert \bm{w}_{h0}\rVert_{0,4,K}h_K^{-\frac{1}{2}}\lVert \bm{u}_{h0}-\bm{u}_{hb}\rVert_{0,2,\partial K}\lVert \bm{v}_{h0}\rVert_{0,4,K} \\
&\apprle \interleave \bm{w}_h\interleave \cdot \interleave \bm{u}_h\interleave \cdot \interleave \bm{v}_h\interleave.
\end{align*}
Similarly, we can get
\begin{align*}
|R_4|+|R_5| &\apprle \interleave \bm{w}_h\interleave \cdot \interleave \bm{u}_h\interleave \cdot \interleave \bm{v}_h\interleave.
\end{align*}
As a result, the estimate (\ref{continuous result ch}) holds.
The estimate (\ref{continuous result chbar}) follows similarly. \end{proof}
By \eqref{3356}, Lemma \ref{boundedness}, and the definitions of the trilinear forms $A_h(\cdot;\cdot,\cdot)$ and $\overline A_h(\cdot;\cdot,\cdot)$, we easily get the following continuity and coercivity results.
\begin{lem}\label{lem31}
For any $\bm{w}_h,\bm{u}_h,\bm{v}_h \in \bm{V}_h, T_h,s_h \in S_h$, it holds
\begin{align}
A_h(\bm{w}_h;\bm{u}_h,\bm{v}_h) &\apprle (\Pr+\interleave \bm{w}_h\interleave)\interleave \bm{u}_h\interleave \cdot\interleave \bm{v}_h\interleave,\\
\overline{A}_h(\bm{w}_h;T_h,s_h) &\apprle (\kappa +\interleave \bm{w}_h\interleave)\interleave T_h\interleave \interleave s_h\interleave,\\
A_h(\bm{v}_h;\bm{v}_h,\bm{v}_h) &= \Pr \interleave \bm{v}_h\interleave^2,\\
\overline{A}_h(\bm{v}_h;s_h,s_h) &= \kappa \interleave s_h\interleave^2 .\label{Core-Bar-A}
\end{align} \end{lem}
By following the same routine as in the proof of (\cite[Theorem 3.1]{chen2016robust}), we can obtain the following inf-sup inequality. \begin{lem}\label{infsup}
For any $(\bm{v}_h,q_h)\in \bm{V}_h^0\times Q_h^0$, it holds
\begin{equation*}
\sup\limits_{\bm{v}_h\in \bm{V}_h^0} \frac{b_h(\bm{v}_h,q_h)}{\interleave \bm{v}_h\interleave} \apprge \lVert q_h\rVert.
\end{equation*} \end{lem}
\subsection{Existence and uniqueness results}
We define a space \begin{equation*} \bm{W}_h := \{\bm{w}_h\in \bm{V}_h^0:b_h(\bm{w}_h,q_h) = 0,\forall q_h\in Q_h^0\}, \end{equation*} and introduce the following discretization problem: seek $(\bm{u_h},T_h)\in \bm{W}_h\times S_h^0$ \begin{align}\label{pb3} \left \{ \begin{array}{rl} A_h(\bm{u}_h;\bm{u}_h,\bm{v}_h)-d_h(T_h,\bm{v}_h) &= (\bm{f},\bm{v}_{h0}), \forall \bm{v}_h\in \bm{W}_h,\\ \overline{A}_h(\bm{u}_h;T_h,s_h) &= (g,s_{h0}), \forall s_h\in S_h^0. \end{array} \right. \end{align} It is easy to see that, by Lemma \ref{infsup} and the theory of mixed finite element methods \cite{brezzi2008mixed}, the following conclusion holds. \begin{lem}\label{lem313}
The problems (\ref{pb2}) and (\ref{pb3}) are equivalent in the sense that (i) and (ii) hold:
(i) if $(\bm{u}_h,p_h,T_h)\in \bm{V}_h^0\times Q_h^0\times S_h^0$ is the solution to the problem (\ref{pb2}), then $(\bm{u_h},T_h)$
is the solution to the problem \eqref{pb3};
(ii) if $(\bm{u_h},T_h)\in \bm{W}_h\times S_h^0$ is the solution to the problem \eqref{pb3}, then $(\bm{u}_h,p_h,T_h)$
is the solution to the problem (\ref{pb2}), where $p_h\in Q_h^0$ is determined by
\begin{equation*}
b_h(\bm{v}_h,p_h)= (\bm{f},\bm{v}_{h0})-A_h(\bm{u}_h;\bm{u}_h,\bm{v}_h)+d_h(T_h,\bm{v}_h), \forall \bm{v}_h \in \bm{V}_h^0.
\end{equation*}
\end{lem}
In what follows we shall discuss the existence and uniqueness of the solution to the problem (\ref{pb3}). To this end, we set
\begin{align} \mathcal{N}_h &:= \sup\limits_{0\neq\bm{w}_h,\bm{u}_h,\bm{v}_h\in \bm{W}_h}\frac{c_h(\bm{w}_h;\bm{u}_h,\bm{v}_h)}{\interleave \bm{w}_h\interleave\cdot \interleave \bm{u}_h\interleave\cdot\interleave \bm{v}_h\interleave},\label{Nh}\\ \mathcal{M}_h &:= \sup\limits_{\substack{0\neq\bm{w}_h\in \bm{W}_h,\\ 0\neq T,s\in S_h^0}}\frac{\overline{c}_h(\bm{w}_h;T_h,s_h)}{\interleave \bm{w}_h\interleave\cdot \interleave T_h\interleave\cdot\interleave s_h\interleave},\label{Mh}\\
\lVert \bm{f}\rVert_h& := \sup\limits_{0\neq\bm{v}_h\in \bm{W}_h}\frac{(\bm{f},\bm{v}_{h0})}{\interleave \bm{v}_h\interleave},\label{f_h}\\ \lVert g\rVert_h &:= \sup\limits_{0\neq s_h\in S_h^0}\frac{(g,s_{h0})}{\interleave s_h\interleave}.\label{g_h} \end{align} From Lemma \ref{boundedness} we easily know that $\mathcal{N}_h, \mathcal{M}_h$ are bounded from above by a positive constant independent of the mesh size $h$.
\begin{thm} \label{exist} The problem (\ref{pb3}) admits
at least one solution $(\bm{u}_h,T_h)\in \bm{W}_h \times S_h^0$. \end{thm} \begin{proof}
First, by Lemma \ref{lem31} it is easy to see that, for a given $\bm{u}_h\in \bm{W}_h$, the bilinear form
$\overline{A}_h(\bm{u}_h;\cdot,\cdot)
$ is continuous and coercive on $S_h^0\times S_h^0$. Hence, by Lax-Milgram theorem there is a unique $T_h\in S_h^0$ such that the second equation of (\ref{pb3}) holds.
Define a mapping $F: \bm{W}_h\rightarrow S_h^0$ by $F(\bm{u}_h) = T_h$. Then the thing left is to show that there exists at least one $\bm{u}_h\in \bm{W}_h$ such that
\begin{equation}\label{pb4}
A_h(\bm{u}_h;\bm{u}_h,\bm{v}_h)=a_h(\bm{u}_h,\bm{v}_h) + c_h(\bm{u}_h;\bm{u}_h,\bm{v}_h) = d_h(F(\bm{u}_h),\bm{v}_h) + (\bm{f},\bm{v}_{h0}), \forall \bm{v}_h\in \bm{W}_h.
\end{equation}
Take $s_h = T_h$ in the second equation of (\ref{pb3}), and apply (\ref{g_h}) and \eqref{Core-Bar-A} to get
\begin{equation*}
\kappa \interleave T_h\interleave^2 = (g,T_{h0})\leq \lVert g\rVert_h\cdot \interleave T_h\interleave,
\end{equation*}
which yields
\begin {equation}\label{regular estimates Th}
\interleave F(\bm{u}_h)\interleave = \interleave T_h\interleave \leq \kappa^{-1}\lVert g\rVert_h.
\end{equation} Take $\bm{v}_h = \bm{u}_h$ in (\ref{pb4}), and we obtain
\begin{align*}
\Pr\interleave \bm{u}_h\interleave^2&= d_h(F(\bm{u}_h),\bm{u}_h) + (\bm{f},\bm{u}_{h0}) \\
&\leq (\Pr Ra\interleave F(\bm{u}_{h})\interleave + \lVert \bm{f}\rVert_h)\interleave \bm{u}_h\interleave \\
&\leq (\Pr Ra\kappa ^{-1}\lVert g\rVert_h + \lVert \bm{f}\rVert_h)\interleave \bm{u}_h\interleave.
\end{align*} This indicates \begin{align}\label{stabs}
\interleave \bm{u}_h\interleave\leq Ra\kappa ^{-1}\lVert g\rVert_h +{\Pr}^{-1} \lVert \bm{f}\rVert_h. \end{align} By Lemma \ref{boundedness} and (\ref{f_h}), we also have
\begin{equation*}
\lvert -c_h(\bm{u}_h;\bm{u}_h,\bm{v}_h) +d_h(F(\bm{u}_h),\bm{v}_h) + (\bm{f},\bm{v}_{h0})\rvert
\apprle \big( \interleave \bm{u}_h\interleave^2+\Pr Ra\interleave F(\bm{u}_h)\interleave + \lVert \bm{f}\rVert_h\big)\interleave \bm{v}_h\interleave.
\end{equation*}
Now we introduce another mapping, $\mathcal{A}: \bm{W}_h\rightarrow \bm{W}_h$, defined by $\mathcal{A}(\bm{u}_h) = \bm{w}$, where $\bm{w}\in \bm{W}_h$ is determined by
\begin{equation}\label{pbw}
a_h(\bm{w},\bm{v}_h) = -c_h(\bm{u}_h;\bm{u}_h,\bm{v}_h) + d_h(F(\bm{u}_h),\bm{v}_h) + (\bm{f},\bm{v}_{h0}), \forall \bm{v}_h\in \bm{W}_h.
\end{equation}
Clearly, $\bm{u}_h$ is a solution to (\ref{pb4}) if it is a solution to
\begin{equation*}
\mathcal{A}(\bm{u}_h) = \bm{u}_h.
\end{equation*}
To show this system has a solution, from the Leray-Schauder's principle it suffices to prove the following two assertions:
(i) $\mathcal{A}$ is a continuous and compact mapping;
(ii) for any $0\leq \lambda \leq 1$, the set $\bm{W}_{\lambda,h}:=\{\bm{v}_h\in \bm{W}_h: \bm{v}_h = \lambda \mathcal{A}\bm{v}_h\}$ is bounded.
Let $\bm{u}_{1h},\bm{u}_{2h}\in \bm{W}_h$, set $\bm{w}_2 = \mathcal{A}(\bm{u}_{2h})$ and $\bm{w}_1 = \mathcal{A}(\bm{u}_{1h})$, then we obtain
\begin{align}
a_h(\bm{w}_1,\bm{v}_h) = -c_h(\bm{u}_{1h};\bm{u}_{1h},\bm{v}_h) + d_h(F(\bm{u}_{1h}),\bm{v}_h) + (\bm{f},\bm{v}_{h0}),\label{pbw1}\\
a_h(\bm{w}_2,\bm{v}_h) = -c_h(\bm{u}_{2h};\bm{u}_{2h},\bm{v}_h) + d_h(F(\bm{u}_{2h}),\bm{v}_h) + (\bm{f},\bm{v}_{h0}).\label{pbw2}
\end{align}
Subtracting \eqref{pbw2} from \eqref{pbw1}, and taking $\bm{v}_h =\bm{w}: = \bm{w}_1-\bm{w}_2$, we get
\begin{equation}\label{355}
a_h(\bm{w},\bm{w}) = -c_h(\bm{u}_{1h}-\bm{u}_{2h};\bm{u}_{1h},\bm{w}) -c_h(\bm{u}_{2h};\bm{u}_{1h}-\bm{u}_{2h},\bm{w}) + d_h(F(\bm{u}_{1h})-F(\bm{u}_{2h}),\bm{w}).
\end{equation}
Substitute $T_h=F(\bm{u}_{1h})$ and $T_h=F(\bm{u}_{2h})$ into the second equation of (\ref{pb3}), respectively, and subtract the two resultant equations each other, then, in view of \eqref{224}, we have
\begin{equation*}
\overline{a}_h(F(\bm{u}_{1h})-F(\bm{u}_{2h}),s_h) = -\overline{c}_h(\bm{u}_{1h}-\bm{u}_{2h};F(\bm{u}_{1h}),s_h) -\overline{c}_h(\bm{u}_{2h};F(\bm{u}_{1h})-F(\bm{u}_{2h}),s_h) , \forall s_h\in S_h^0,
\end{equation*}
Taking $s_h = F(\bm{u}_{1h})-F(\bm{u}_{2h})$ in this equation, together with \eqref{3356}, \eqref{regular estimates Th}, and Lemma \ref{boundedness}, leads to
\begin{equation}\label{T1-T2}
\begin{aligned}
\kappa \interleave F(\bm{u}_{1h})-F(\bm{u}_{2h})\interleave &\apprle \interleave \bm{u}_{1h}-\bm{u}_{2h}\interleave\cdot \interleave F(\bm{u}_{1h})\interleave\\
&\leq \kappa^{-1} \interleave \bm{u}_{1h}-\bm{u}_{2h}\interleave\cdot \lVert g\rVert_h.
\end{aligned}
\end{equation}
As a result, from \eqref{355} and \eqref{stabs} it follows
\begin{align*}
\interleave \mathcal{A}(\bm{u}_{1h})-\mathcal{A}(\bm{u}_{2h})\interleave&=\interleave \bm{w}\interleave\apprle ({\Pr}^{-1}(\interleave \bm{u}_{1h}\interleave+\interleave \bm{u}_{2h}\interleave) + \kappa^{-2}Ra\lVert g\rVert_h)\interleave \bm{u}_{1h}-\bm{u}_{2h}\interleave\\
&\leq \big(2{\Pr}^{-1}(Ra\kappa ^{-1}\lVert g\rVert_h +{\Pr}^{-1} \lVert \bm{f}\rVert_h) + \kappa^{-2}Ra\lVert g\rVert_h\big)\interleave \bm{u}_{1h}-\bm{u}_{2h}\interleave,
\end{align*} which means that $\mathcal{A}$ is equicontinuous and uniformly bounded. Thus, $\mathcal{A}$ is compact by the Arzel\'{a}-Ascoli theorem\cite{brezis2010functional}.
It remains to show (ii). If $\lambda = 0$, then $\bm{W}_{\lambda,h}=\{0\}$. For $\lambda \in (0,1]$ and $\bm{v}_h\in \bm{W}_{\lambda,h}$, by \eqref{pbw} and \eqref{3356} we have
\begin{align*}
\lambda^{-1}a_h(\bm{v}_h,\bm{v}_h) =a_h(\mathcal{A}\bm{v}_h,\bm{v}_h) &= -c_h(\bm{v}_h;\bm{v}_h,\bm{v}_h) + d_h(F(\bm{v}_h),\bm{v}_h) + (\bm{f},\bm{v}_{h0})\\ &= d_h(F(\bm{v}_h),\bm{v}_h) + (\bm{f},\bm{v}_{h0}),
\end{align*} which implies
\begin{equation*}
\interleave \bm{v}_h\interleave \leq \lambda Ra\interleave F(\bm{v}_h)\interleave + \lambda {\Pr}^{-1}\lVert \bm{f}\rVert_h\\
\leq \lambda Ra\kappa^{-1}\lVert g\rVert_h + \lambda {\Pr}^{-1}\lVert \bm{f}\rVert_h.
\end{equation*}
This completes the proof.
\end{proof}
We now give a global uniqueness criteria for the case of small data (small Rayleigh number $Ra$). \begin{thm}\label{thm32}
Suppose
\begin{equation}\label{assu}
({\Pr}^{-1}\mathcal{N}_hRa\kappa^{-1}+\mathcal{M}_h Ra\kappa^{-2})\lVert g\rVert_h + \mathcal{N}_h{\Pr}^{-2}\lVert \bm{f}\rVert_h < 1 .
\end{equation}
Then the problem (\ref{pb3}) admits a unique solution $(\bm{u}_h,T_h)\in \bm{W}_h \times S_h^0$ with $T_h=F(\bm{u}_h)$. \end{thm} \begin{proof} By Theorem \ref{exist}, let
$\bm{u}_{1h}, \bm{u}_{2h}\in \bm{W}_h$ be two solutions to the problem (\ref{pb4}). Then it suffices to show $\bm{u}_{1h}=\bm{u}_{2h}$. In fact, we have
\begin{align*}
a_h(\bm{u}_{1h},\bm{v}_h) = -c_h(\bm{u}_{1h};\bm{u}_{1h},\bm{v}_h) + d_h(F(\bm{u}_{1h}),\bm{v}_h) + (\bm{f},\bm{v}_{h0}),\\
a_h(\bm{u}_{2h},\bm{v}_h) = -c_h(\bm{u}_{2h};\bm{u}_{2h},\bm{v}_h) + d_h(F(\bm{u}_{2h}),\bm{v}_h) + (\bm{f},\bm{v}_{h0}).
\end{align*}
Subtracting the above two equations each other with $\bm{v}_h = \bm{u}_{1h} - \bm{u}_{2h}$, and using \eqref{3356}, we obtain
\begin{equation*}
a_h(\bm{u}_{1h}-\bm{u}_{2h},\bm{u}_{1h}-\bm{u}_{2h}) = -c_h(\bm{u}_{1h}-\bm{u}_{2h};\bm{u}_{1h},\bm{u}_{1h}-\bm{u}_{2h}) + d_h(F(\bm{u}_{1h})-F(\bm{u}_{2h}),\bm{u}_{1h}-\bm{u}_{2h}),
\end{equation*} which, together with Lemma \ref{boundedness}, (\ref{T1-T2}) and \eqref{stabs}, yields
\begin{align*}
\Pr \interleave \bm{u}_{1h}-\bm{u}_{2h}\interleave^2 &\leq \mathcal{N}_h\interleave \bm{u}_{1h}-\bm{u}_{2h}\interleave^2\cdot \interleave \bm{u}_{1h}\interleave + \Pr Ra\interleave F(\bm{u}_{1h})-F(\bm{u}_{2h})\interleave\cdot \interleave \bm{u}_{1h}-\bm{u}_{2h}\interleave\\
&\leq \mathcal{N}_h\interleave \bm{u}_{1h}-\bm{u}_{2h}\interleave^2\cdot \interleave \bm{u}_{1h}\interleave + \mathcal{M}_h\Pr Ra\kappa^{-2}\cdot \interleave \bm{u}_{1h}-\bm{u}_{2h}\interleave^2\cdot\lVert g\rVert_h,\\
&\leq
\big((\mathcal{N}_hRa\kappa^{-1}+\mathcal{M}_h\Pr Ra\kappa^{-2})\lVert g\rVert_h + \mathcal{N}_h{\Pr}^{-1}\lVert \bm{f}\rVert_h\big)\interleave \bm{u}_{1h}-\bm{u}_{2h}\interleave^2.
\end{align*}
If $\bm{u}_{1h} \neq \bm{u}_{2h}$, then, by the assumption \eqref{assu}, we further have
\begin{align*}
\Pr \interleave \bm{u}_{1h}-\bm{u}_{2h}\interleave^2
< \Pr \interleave \bm{u}_{1h}-\bm{u}_{2h}\interleave^2,
\end{align*} which contradicts. Therefore $\bm{u}_{1h} = \bm{u}_{2h}$. \end{proof}
\section{A priori error estimates} This section is devoted to the error estimation of the WG scheme (\ref{pb2}). We set \begin{equation*} \bm{I}_h\bm{u}:=\{\bm{P}_k^{RT}\bm{u},\bm{Q}_l^b\bm{u}\},J_hp:=\{Q_{k-1}^0p,Q_k^bp\},H_h T:=\{Q_k^0T,Q_l^bT\}. \end{equation*} We recall that $k\geq 1$ and $l=k, k-1$.
\begin{lem}\label{E-N}
For any $\bm{w},\bm{u}\in \bm{W}$, $T\in H_0^1(\Omega)$, $\bm{v}_h\in \bm{V}_h^0$ and $s_h\in S_h^0$, it holds
\begin{align}
c_h(\bm{I}_h\bm{w};\bm{I}_h\bm{u},\bm{v}_h) =& (\nabla\cdot (\bm{u}\otimes\bm{w}),\bm{v}_{h0})+E_N(\bm{w};\bm{u},\bm{v}_h),\label{c_h}\\
\overline{c}_h(\bm{I}_h\bm{u};H_hT,s_h) =& (\nabla\cdot (\bm{u}T),s_{h0})+\overline{E}_N(\bm{u};T,s_h),\label{c_hhat}
\end{align}
where
\begin{equation*}
\begin{aligned}
E_N(\bm{w};\bm{u},\bm{v}_h) :=& \frac{1}{2}(\bm{u}\otimes\bm{w}-\bm{P}_k^{RT}\bm{u}\otimes\bm{P}_k^{RT}\bm{w},\nabla_h\bm{v}_{h0})-\frac{1}{2}\langle (\bm{u}\otimes\bm{w}-\bm{Q}_l^b\bm{u}\otimes\bm{Q}_l^b\bm{w})\cdot\bm{n},\bm{v}_{h0}\rangle_{\partial \mathcal{T}_h^f}\\
&\quad -\frac{1}{2}(\bm{w}\cdot\nabla\bm{u}-\bm{P}_k^{RT}\bm{w}\cdot\nabla_h\bm{P}_k^{RT}\bm{u},\bm{v}_{h0})-\frac{1}{2}\langle \bm{v}_{hb}\otimes\bm{Q}_l^b\bm{w}\cdot\bm{n},\bm{P}_k^{RT}\bm{u}\rangle_{\partial\mathcal{T}_h^f},\\
\overline{E}_N(\bm{u};T,s_h) :=& \frac{1}{2}(\bm{u}T-\bm{P}_k^{RT}\bm{u}Q_k^0T,\nabla_h s_{h0})-\frac{1}{2}\langle (\bm{u}T-\bm{Q}_l^b\bm{u}Q_l^b T)\cdot\bm{n},s_{h0}\rangle_{\partial \mathcal{T}_h}\\
&\quad -\frac{1}{2}(\bm{u}\cdot\nabla T-\bm{P}_K^{RT}\bm{u}\cdot\nabla_h Q_k^0T,s_{h0})-\frac{1}{2}\langle (\bm{Q}_l^b\bm{u}s_{hb})\cdot\bm{n},Q_k^0T\rangle_{\partial\mathcal{T}_h}.
\end{aligned}
\end{equation*} \end{lem} \begin{proof}
From the definition of weak divergence and Green's formula, we have
\begin{equation*}
\begin{aligned}
(\nabla_{w,k}\cdot(\bm{I}_h\bm{u}\otimes\bm{I}_h\bm{w}),\bm{v}_{h0}) =& (\nabla\cdot (\bm{u}\otimes\bm{w}),\bm{v}_{h0})+
(\bm{u}\otimes\bm{w}-\bm{P}_k^{RT}\bm{u}\otimes\bm{P}_k^{RT}\bm{w},\nabla_h\bm{v}_{h0})\\
&-\langle (\bm{u}\otimes\bm{w}-\bm{Q}_l^b\bm{u}\otimes\bm{Q}_l^b\bm{w})\cdot\bm{n},\bm{v}_{h0}\rangle_{\partial \mathcal{T}_h^f},\\
(\nabla_{w,k}\cdot(\bm{v}_h\otimes\bm{I}_h\bm{w}),\bm{P}_k^{RT}\bm{u}) =& (\nabla\cdot (\bm{u}\otimes\bm{w}),\bm{v}_{h0})+(\bm{w}\cdot\nabla\bm{u}-\bm{P}_K^{RT}\bm{w}\cdot\nabla_h\bm{P}_k^{RT}\bm{u},\bm{v}_{h0})\\
&+\langle \bm{v}_{hb}\otimes\bm{Q}_l^b\bm{w}\cdot\bm{n},\bm{P}_k^{RT}\bm{u}\rangle_{\partial\mathcal{T}_h^f},
\end{aligned}
\end{equation*} which, together with the definition of the trilinear form $c_h(\cdot;\cdot,\cdot) $, yields (\ref{c_h}).
Similarly, we can obtain (\ref{c_hhat}). \end{proof}
\begin{lem}\label{RTgeneral}
Let $j, r$ be nonnegative integers. For any $K\in \mathcal{T}_h$ and $ \bm{v}\in [H^r(K)]^d$, the following estimates hold for the RT projection operator:
\begin{align}
\lvert \bm{v}-\bm{P}_j^{RT}\bm{v}\rvert_{1,2,K}&\apprle h_K^{r-1}\lvert \bm{v}\rvert_{r,2,K},\forall 1\leq r\leq j+1,\label{RT12} \\
\lvert \bm{v}-\bm{P}_j^{RT}\bm{v}\rvert_{0,3,K}&\apprle h_K^{r-\frac{d}{6}}\lvert \bm{v}\rvert_{r,2,K},\forall 0\leq r\leq j+1,\label{RT03}\\
\lvert \bm{v}-\bm{P}_j^{RT}\bm{v}\rvert_{0,2,\partial K}&\apprle h_K^{r-\frac{1}{2}}\lvert \bm{v}\rvert_{r,2,K},\forall 1\leq r\leq j+1, \label{RT02edge} \\
\lvert \bm{v}-\bm{P}_j^{RT}\bm{v}\rvert_{0,3,\partial K}&\apprle h_K^{r-\frac{1}{3}-\frac{d}{6}}\lvert \bm{v}\rvert_{r,2,K},\forall 1\leq r\leq j+1. \label{RT03edge}
\end{align} \end{lem} \begin{proof}
We only prove (\ref{RT12}), since the estimates (\ref{RT03})-(\ref{RT03edge}) follow similarly.
For $ 1\leq r\leq j+1$, by the triangle inequality, the inverse inequality, Lemma \ref{ineq}, and Lemma \ref{RT}, we get
\begin{align*}
\lvert \bm{v}-\bm{P}_j^{RT}\bm{v}\rvert_{1,2,K} &\leq \lvert \bm{v}-\bm{Q}_{r-1}^0\bm{v}\rvert_{1,2,K} +\lvert \bm{Q}_{r-1}^0\bm{v}-\bm{P}_j^{RT}\bm{v}\rvert_{1,2,K} \\
&\apprle \lvert \bm{v}-\bm{Q}_{r-1}^0\bm{v}\rvert_{1,2,K} +h_K^{-1}\lvert \bm{Q}_{r-1}^0\bm{v}-\bm{P}_j^{RT}\bm{v}\rvert_{0,2,K} \\
&\leq \lvert \bm{v}-\bm{Q}_{r-1}^0\bm{v}\rvert_{1,2,K} +h_K^{-1}\lvert \bm{Q}_{r-1}^0\bm{v}-\bm{v}\rvert_{0,2,K}+h_K^{-1}\lvert \bm{v}-\bm{P}_j^{RT}\bm{v}\rvert_{0,2,K} \\
&\apprle h_K^{r-1}\lvert \bm{v}\rvert_{r,2,K},
\end{align*}
i.e. (\ref{RT12}) holds. \end{proof} \begin{lem}\label{lem4.3}
For $\bm{u}\in [H^{k+1}(\Omega_f)]^d$ with $\nabla\cdot\bm{u} = 0$ and $T\in H^{k+1}(\Omega)$, it holds
\begin{align*}
E_N(\bm{u};\bm{u},\bm{v}_h)&\apprle h^{k}\lVert \bm{u}\rVert_2\lVert \bm{u}\rVert_{k+1}\interleave \bm{v}_h\interleave, \forall \bm{v}_h\in \bm{V}_h^0,\\
\overline{E}_N(\bm{u};T,s_h)&\apprle h^{k}(\lVert \bm{u}\rVert_2\lVert T\rVert_{k+1}+\lVert T\rVert_2\lVert \bm{u}\rVert_{k+1})\interleave s_h\interleave, \forall s_h\in S_h^0,
\end{align*}
for $l=k$ when $d = 2,3$, and for $l = k-1$ when $d=2$. \end{lem} \begin{proof}
From the h\"{o}lder inequality, the sobolev inequality, and the projection properties, we have
\begin{align*}
&\lvert (\bm{u}\otimes \bm{u}-\bm{P}_k^{RT}\bm{u}\otimes\bm{P}_k^{RT}\bm{u},\nabla_h\bm{v}_{h0})\rvert \\
\leq& \lvert ((\bm{u}-\bm{P}_k^{RT}\bm{u})\otimes \bm{u},\nabla_h\bm{v}_{h0})\rvert + \lvert (\bm{P}_k^{RT}\bm{u}\otimes(\bm{u}-\bm{P}_k^{RT}\bm{u}),\nabla_h\bm{v}_{h0})\rvert \\
\apprle& \sum_{K\in \mathcal{T}_h}\lvert \bm{u}-\bm{P}_k^{RT}\bm{u}\rvert_{0,2,K}\lvert \bm{u}\rvert_{0,\infty,K}\lVert \nabla_h\bm{v}_{h0}\rVert_{0,2,K} + \sum_{K\in \mathcal{T}_h}\lvert \bm{u}-\bm{P}_k^{RT}\bm{u}\rvert_{0,2,K}\lvert \bm{P}_k^{RT}\bm{u}\rvert_{0,\infty,K}\lVert \nabla_h\bm{v}_{h0}\rVert_{0,2,K} \\
\apprle& h^{k+1}\lvert \bm{u}\rvert_{0,\infty}\lvert \bm{u}\rvert_{k+1}\interleave \bm{v}_h\interleave \\
\apprle& h^{k+1}\lVert \bm{u}\rVert_2\lVert \bm{u}\rVert_{k+1}\interleave \bm{v}_h\interleave.
\end{align*}
For $l=k$ when $d = 2,3$, and for $l = k-1$ when $d=2$, we have
\begin{align*}
&\lvert \langle (\bm{u}\otimes \bm{u}-\bm{Q}_l^b\bm{u}\otimes\bm{Q}_l^b\bm{u})\cdot \bm{n},\bm{v}_{h0}\rangle_{\partial\mathcal{T}_h^f}\rvert= \lvert \langle (\bm{u}\otimes \bm{u}-\bm{Q}_l^b\bm{u}\otimes\bm{Q}_l^b\bm{u})\cdot \bm{n},\bm{v}_{h0}-\bm{v}_{hb}\rangle_{\partial\mathcal{T}_h^f}\rvert \\
\leq& \lvert \langle (\bm{u}-\bm{Q}_l^b\bm{u})\otimes \bm{u}\cdot \bm{n},\bm{v}_{h0}-\bm{v}_{hb}\rangle_{\partial\mathcal{T}_h^f}\rvert + \lvert \langle \bm{Q}_l^b\bm{u}\otimes(\bm{u}-\bm{Q}_l^b\bm{u})\cdot \bm{n},\bm{v}_{h0}-\bm{v}_{hb}\rangle_{\partial\mathcal{T}_h^f}\rvert \\
\apprle& \lvert \langle (\bm{u}-\bm{Q}_l^b\bm{u})\otimes (\bm{u}-\bm{Q}_k^0\bm{u})\cdot \bm{n},\bm{v}_{h0}-\bm{v}_{hb}\rangle_{\partial\mathcal{T}_h^f}\rvert + \lvert \langle (\bm{u}-\bm{Q}_l^b\bm{u})\otimes \bm{Q}_k^0\bm{u}\cdot \bm{n},\bm{v}_{h0}-\bm{v}_{hb}\rangle_{\partial\mathcal{T}_h^f}\rvert \\
&+ \lvert \langle (\bm{Q}_l^0\bm{u}-\bm{Q}_l^b\bm{u})\otimes (\bm{u}-\bm{Q}_l^b\bm{u})\cdot \bm{n},\bm{v}_{h0}-\bm{v}_{hb}\rangle_{\partial\mathcal{T}_h^f}\rvert + \lvert \langle \bm{Q}_l^0\bm{u}\otimes(\bm{u}-\bm{Q}_l^b\bm{u}) \cdot \bm{n},\bm{v}_{h0}-\bm{v}_{hb}\rangle_{\partial\mathcal{T}_h^f}\rvert \\
\leq& \sum_{K\in\mathcal{T}_h}(\lvert \bm{u}-\bm{Q}_l^b\bm{u}\rvert_{0,2,\partial K}\lvert \bm{u}-\bm{Q}_k^0\bm{u}\rvert_{0,2,\partial K}\lvert \bm{v}_{h0}-\bm{v}_{hb}\rvert_{0,\infty,\partial K} +\lvert \bm{u}-\bm{Q}_l^b\bm{u}\rvert_{0,2,\partial K}\lvert \bm{Q}_k^0\bm{u}\rvert_{0,\infty,\partial K}\lvert \bm{v}_{h0}-\bm{v}_{hb}\rvert_{0,2,\partial K}) \\
&+\sum_{K\in\mathcal{T}_h}(\lvert \bm{Q}_l^0\bm{u}-\bm{Q}_l^b\bm{u}\rvert_{0,2,\partial K}\lvert \bm{u}-\bm{Q}_l^b\bm{u}\rvert_{0,2,\partial K}\lvert \bm{v}_{h0}-\bm{v}_{hb}\rvert_{0,\infty,\partial K} +\lvert \bm{u}-\bm{Q}_l^b\bm{u}\rvert_{0,2,\partial K}\lvert \bm{Q}_k^0\bm{u}\rvert_{0,\infty,\partial K}\lvert \bm{v}_{h0}-\bm{v}_{hb}\rvert_{0,2,\partial K}) \\
\apprle& h^{l+1/2}\lvert \bm{u}\rvert_{l+1}h^{1/2}\lvert \bm{u}\rvert_1h^{1-d/2}\interleave \bm{v}_h\interleave + h^{l+1/2}\lvert \bm{u}\rvert_{l+1}\lvert \bm{u}\rvert_{0,\infty}h^{1/2}\interleave \bm{v}_h\interleave \\
\apprle& h^{k}\lVert \bm{u}\rVert_{2}\lVert \bm{u}\rVert_{k+1}\interleave \bm{v}_h\interleave,
\end{align*}
\begin{align*}
&\lvert (\bm{u}\cdot \nabla\bm{u}-\bm{P}_k^{RT}\bm{u}\cdot\nabla_h\bm{P}_k^{RT}\bm{u},\bm{v}_{h0})\rvert \\
\leq& \lvert ((\bm{u}-\bm{P}_k^{RT}\bm{u})\cdot \nabla\bm{u},\bm{v}_{h0})\rvert + \lvert (\bm{P}_k^{RT}\bm{u}\cdot
(\nabla\bm{u}-\nabla_h\bm{P}_k^{RT}\bm{u}),\bm{v}_{h0})\rvert \\
\apprle& \sum_{K\in \mathcal{T}_h}\lvert \bm{u}-\bm{P}_k^{RT}\bm{u}\rvert_{0,3,K}\lvert \nabla\bm{u}\rvert_{0,2,K}\lVert \bm{v}_{h0}\rVert_{0,6,K} + \sum_{K\in \mathcal{T}_h}\lvert \nabla\bm{u}-\nabla\bm{P}_k^{RT}\bm{u}\rvert_{0,2,K}\lvert \bm{P}_k^{RT}\bm{u}\rvert_{0,\infty,K}\lVert \bm{v}_{h0}\rVert_{0,2,K} \\
\apprle& h^{k+1-d/6}\lvert \bm{u}\rvert_{k+1}\lvert \bm{u}\rvert_{1,2}\lvert \bm{v}_{h0}\rvert_{1,2} + \lvert \bm{u}\rvert_{0,\infty}h^k\lvert \bm{u}\rvert_{k+1}\lvert \bm{v}_{h0}\rvert_{0,2} \\
\apprle& h^{k}\lVert \bm{u}\rVert_2\lVert \bm{u}\rVert_{k+1}\interleave \bm{v}_h\interleave,
\end{align*} and
\begin{align*}
&\lvert \langle \bm{v}_{hb}\otimes \bm{Q}_l^b\bm{u}\cdot \bm{n},\bm{P}_k^{RT}\bm{u}\rangle_{\partial\mathcal{T}_h^f}\rvert = \lvert \langle \bm{v}_{hb}\otimes \bm{Q}_l^b\bm{u}\cdot \bm{n},\bm{P}_k^{RT}\bm{u}-\bm{Q}_k^b\bm{u}\rangle_{\partial\mathcal{T}_h^f}\rvert \\
\leq& \lvert \langle (\bm{v}_{h0}-\bm{v}_{hb})\otimes (\bm{Q}_l^b\bm{u}-\bm{Q}_l^0\bm{u})\cdot \bm{n},\bm{P}_k^{RT}\bm{u}-\bm{Q}_k^b\bm{u}\rangle_{\partial\mathcal{T}_h^f}\rvert + \lvert \langle \bm{v}_{h0}\otimes (\bm{Q}_l^b\bm{u}-\bm{Q}_l^0\bm{u})\cdot \bm{n},\bm{P}_k^{RT}\bm{u}-\bm{Q}_k^b\bm{u}\rangle_{\partial\mathcal{T}_h^f}\rvert \\
&+\lvert \langle (\bm{v}_{h0}-\bm{v}_{hb})\otimes \bm{Q}_l^0\bm{u}\cdot \bm{n},\bm{P}_k^{RT}\bm{u}-\bm{Q}_k^b\bm{u}\rangle_{\partial\mathcal{T}_h^f}\rvert + \lvert \langle \bm{v}_{h0}\otimes \bm{Q}_l^0\bm{u}\cdot \bm{n},\bm{P}_k^{RT}\bm{u}-\bm{Q}_k^b\bm{u}\rangle_{\partial\mathcal{T}_h^f}\rvert \\
\leq& \sum_{K\in\mathcal{T}_h}\lvert \bm{v}_{h0}-\bm{v}_{hb}\rvert_{0,\infty,\partial K}\lvert \bm{Q}_l^b\bm{u}-\bm{Q}_l^0\bm{u}\rvert_{0,2,\partial K}\lvert \bm{P}_k^{RT}\bm{u}-\bm{Q}_k^b\bm{u}\rvert_{0,2,\partial K} + \sum_{K\in\mathcal{T}_h}\lvert \bm{v}_{h0}\rvert_{0,\infty,\partial K}\lvert \bm{Q}_l^b\bm{u}-\bm{Q}_l^0\bm{u}\rvert_{0,2,\partial K}\lvert \bm{P}_k^{RT}\bm{u}-\bm{Q}_k^b\bm{u}\rvert_{0,2,\partial K} \\
&+\sum_{K\in\mathcal{T}_h}\lvert \bm{v}_{h0}-\bm{v}_{hb}\rvert_{0,2,\partial K}\lvert \bm{Q}_l^0\bm{u}\rvert_{0,6,\partial K}\lvert \bm{P}_k^{RT}\bm{u}-\bm{Q}_k^b\bm{u}\rvert_{0,3,\partial K} + \sum_{K\in\mathcal{T}_h}\lvert \bm{v}_{h0}\rvert_{0,3,\partial K}\lvert \bm{Q}_l^0\bm{u}\rvert_{0,6,\partial K}\lvert \bm{P}_k^{RT}\bm{u}-\bm{Q}_k^b\bm{u}\rvert_{0,2,\partial K} \\
\apprle& h^{1-d/2}\interleave \bm{v}_h\interleave h^{1/2}\lvert \bm{u}\rvert_1h^{k+1/2}\lvert \bm{u}\rvert_{k+1} + h^{-d/6}\lvert \bm{v}_{h0}\rvert_{0,6}h^{1/2}\lvert \bm{u}\rvert_1h^{k+1/2}\lvert \bm{u}\rvert_{k+1} \\
&+ h^{1/2}\interleave \bm{v}_h\interleave h^{-1/6}\lvert \bm{u}\rvert_{0,6}h^{k+1-1/3-d/6}\lvert \bm{u}\rvert_{k+1} + h^{-1/3}\lvert \bm{v}_{h0}\rvert_{0,3}h^{-1/6}\lvert \bm{u}\rvert_{0,6}h^{k+1/2}\lvert \bm{u}\rvert_{k+1} \\
\apprle& h^{k}\lVert \bm{u}\rVert_{2}\lVert \bm{u}\rVert_{k+1}\interleave \bm{v}_h\interleave.
\end{align*} Similarly, we can obtain \begin{align*} &\lvert (\bm{u}T-\bm{P}_k^{RT}\bm{u}Q_k^0T,\nabla_hs_{h0})\rvert \\ \leq& \lvert (\bm{u}(T-Q_k^0T),\nabla_hs_{h0})\rvert + \lvert ((\bm{u}-\bm{P}_k^{RT}\bm{u})Q_k^0T,\nabla_hs_{h0})\rvert \\ \apprle& \sum_{K\in \mathcal{T}_h}\lvert T-Q_k^0T\rvert_{0,2,K}\lvert \bm{u}\rvert_{0,\infty,K}\lVert \nabla_hs_{h0}\rVert_{0,2,K} + \sum_{K\in \mathcal{T}_h}\lvert \bm{u}-\bm{P}_k^{RT}\bm{u}\rvert_{0,2,K}\lvert Q_k^0T\rvert_{0,\infty,K}\lVert \nabla_hs_{h0}\rVert_{0,2,K} \\ \apprle& h^{k+1}\lvert \bm{u}\rvert_{0,\infty}\lvert T\rvert_{k+1}\interleave s_h\interleave +h^{k+1}\lvert T\rvert_{0,\infty}\lvert \bm{u}\rvert_{k+1}\interleave s_h\interleave \\ \apprle& h^{k+1}\lVert \bm{u}\rVert_2\lVert T\rVert_{k+1}\interleave s_h\interleave +h^{k+1}\lVert T\rVert_2\lVert \bm{u}\rVert_{k+1}\interleave s_h\interleave, \end{align*} \begin{align*} &\lvert (\bm{u}\cdot \nabla T-\bm{P}_k^{RT}\bm{u}\cdot\nabla_h Q_k^0T,s_{h0})\rvert \\ \leq& \lvert ((\bm{u}-\bm{P}_k^{RT}\bm{u})\cdot \nabla T,s_{h0})\rvert + \lvert (\bm{P}_k^{RT}\bm{u}\cdot (\nabla T-\nabla_h Q_k^0T),s_{h0})\rvert \\ \apprle& \sum_{K\in \mathcal{T}_h}\lvert \bm{u}-\bm{P}_k^{RT}\bm{u}\rvert_{0,3,K}\lvert \nabla T\rvert_{0,2,K}\lVert s_{h0}\rVert_{0,6,K} + \sum_{K\in \mathcal{T}_h}\lvert \nabla T-\nabla_h Q_k^0T\rvert_{0,2,K}\lvert \bm{P}_k^{RT}\bm{u}\rvert_{0,\infty,K}\lVert s_{h0}\rVert_{0,2,K} \\ \apprle& h^{k+1-d/6}\lvert \bm{u}\rvert_{k+1}\lvert T\rvert_{1,2}\lvert s_{h0}\rvert_{1,2} + \lvert \bm{u}\rvert_{0,\infty}h^k\lvert T\rvert_{k+1}\lvert s_{h0}\rvert_{0,2} \\ \apprle& h^{k}\lVert \bm{u}\rVert_2\lVert T\rVert_{k+1}\interleave s_h\interleave +h^{k}\lVert T\rVert_2\lVert \bm{u}\rVert_{k+1}\interleave s_h\interleave. \end{align*} For $l=k$ when $d = 2,3$, and for $l = k-1$ when $d=2$, we have \begin{align*} &\lvert \langle (\bm{u}T-\bm{Q}_l^b\bm{u}Q_l^bT)\cdot \bm{n},s_{h0}\rangle_{\partial\mathcal{T}_h}\rvert =
\lvert \langle (\bm{u}T-\bm{Q}_l^b\bm{u}Q_l^bT)\cdot \bm{n},s_{h0}-s_{hb}\rangle_{\partial\mathcal{T}_h}\rvert \\ \leq& \lvert \langle (\bm{u}(T-Q_l^bT))\cdot \bm{n},s_{h0}-s_{hb}\rangle_{\partial\mathcal{T}_h}\rvert + \lvert \langle ((\bm{u}-\bm{Q}_l^b\bm{u})Q_l^bT)\cdot \bm{n},s_{h0}-s_{hb}\rangle_{\partial\mathcal{T}_h}\rvert \\ \apprle& \lvert \langle (T-Q_l^bT)(\bm{u}-\bm{Q}_k^0\bm{u})\cdot \bm{n},s_{h0}-s_{hb}\rangle_{\partial\mathcal{T}_h}\rvert + \lvert \langle (T-Q_l^bT)\bm{Q}_k^0\bm{u}\cdot \bm{n},s_{h0}-s_{hb}\rangle_{\partial\mathcal{T}_h}\rvert \\ & +\lvert \langle (Q_l^bT-Q_l^0T)(\bm{u}-\bm{Q}_l^b\bm{u})\cdot \bm{n},s_{h0}-s_{hb}\rangle_{\partial\mathcal{T}_h}\rvert + \lvert \langle Q_l^0T(\bm{u}-\bm{Q}_l^b\bm{u})\cdot \bm{n},s_{h0}-s_{hb}\rangle_{\partial\mathcal{T}_h}\rvert \\ \leq& \sum_{K\in\mathcal{T}_h}(\lvert T-Q_l^bT\rvert_{0,2,\partial K}\lvert \bm{u}-\bm{Q}_k^0\bm{u}\rvert_{0,2,\partial K}\lvert s_{h0}-s_{hb}\rvert_{0,\infty,\partial K} +\lvert T-Q_l^bT\rvert_{0,2,\partial K}\lvert \bm{Q}_k^0\bm{u}\rvert_{0,\infty,\partial K}\lvert s_{h0}-s_{hb}\rvert_{0,2,\partial K}) \\ & +\sum_{K\in\mathcal{T}_h}(\lvert Q_l^bT-Q_l^0T\rvert_{0,2,\partial K}\lvert \bm{u}-\bm{Q}_l^b\bm{u}\rvert_{0,2,\partial K}\lvert s_{h0}-s_{hb}\rvert_{0,\infty,\partial K} +\lvert Q_l^0T\rvert_{0,\infty,\partial K}\lvert \bm{u}-\bm{Q}_l^b\bm{u}\rvert_{0,2,\partial K}\lvert s_{h0}-s_{hb}\rvert_{0,2,\partial K}) \\ \apprle& h^{l+1/2}\lvert T\rvert_{l+1}h^{1/2}\lvert \bm{u}\rvert_1h^{1-d/2}\interleave s_h\interleave + h^{l+1/2}\lvert T\rvert_{l+1}\lvert \bm{u}\rvert_{0,\infty}h^{1/2}\interleave s_h\interleave \\ &+h^{l+1/2}\lvert T\rvert_{l+1}h^{1/2}\lvert \bm{u}\rvert_1h^{1-d/2}\interleave s_h\interleave + \lvert T\rvert_{0,\infty}h^{l+1/2}\lvert \bm{u}\rvert_{l+1}h^{1/2}\interleave s_h\interleave \\ \apprle& h^{k}\lVert \bm{u}\rVert_{2}\lVert T\rVert_{k+1}\interleave s_h\interleave + h^{k}\lVert T\rVert_{2}\lVert \bm{u}\rVert_{k+1}\interleave s_h\interleave, \end{align*} \begin{align*} &\lvert \langle s_{hb}\bm{Q}_l^b\bm{u}\cdot \bm{n},Q_k^0T\rangle_{\partial\mathcal{T}_h}\rvert= \lvert \langle s_{hb}\bm{Q}_l^b\bm{u}\cdot \bm{n},Q_k^0T-Q_k^bT\rangle_{\partial\mathcal{T}_h}\rvert \\ \leq& \lvert \langle (s_{h0}-s_{hb})(\bm{Q}_l^b\bm{u}-\bm{Q}_l^0\bm{u})\cdot \bm{n},Q_k^0T-Q_k^bT\rangle_{\partial\mathcal{T}_h}\rvert + \lvert \langle s_{h0}(\bm{Q}_l^b\bm{u}-\bm{Q}_l^0\bm{u})\cdot \bm{n},Q_k^0T-Q_k^bT\rangle_{\partial\mathcal{T}_h}\rvert \\ &+\lvert \langle (s_{h0}-s_{hb})\bm{Q}_l^0\bm{u}\cdot \bm{n},Q_k^0T-Q_k^bT\rangle_{\partial\mathcal{T}_h}\rvert + \lvert \langle s_{h0}\bm{Q}_l^0\bm{u}\cdot \bm{n},Q_k^0T-Q_k^bT\rangle_{\partial\mathcal{T}_h}\rvert \\ \leq& \sum_{K\in\mathcal{T}_h}\lvert s_{h0}-s_{hb}\rvert_{0,\infty,\partial K}\lvert \bm{Q}_l^b\bm{u}-\bm{Q}_l^0\bm{u}\rvert_{0,2,\partial K}\lvert Q_k^0T-Q_k^bT\rvert_{0,2,\partial K} + \sum_{K\in\mathcal{T}_h}\lvert s_{h0}\rvert_{0,\infty,\partial K}\lvert \bm{Q}_l^b\bm{u}-\bm{Q}_l^0\bm{u}\rvert_{0,2,\partial K}\lvert Q_k^0T-Q_k^bT\rvert_{0,2,\partial K} \\ &+\sum_{K\in\mathcal{T}_h}\lvert s_{h0}-s_{hb}\rvert_{0,2,\partial K}\lvert \bm{Q}_l^0\bm{u}\rvert_{0,6,\partial K}\lvert Q_k^0T-Q_k^bT\rvert_{0,3,\partial K} + \sum_{K\in\mathcal{T}_h}\lvert s_{h0}\rvert_{0,3,\partial K}\lvert \bm{Q}_l^0\bm{u}\rvert_{0,6,\partial K}\lvert Q_k^0T-Q_k^bT\rvert_{0,2,\partial K} \\ \apprle& h^{1-d/2}\interleave s_h\interleave h^{1/2}\lvert \bm{u}\rvert_1h^{k+1/2}\lvert T\rvert_{k+1} + h^{-d/6}\lvert s_{h0}\rvert_{0,6}h^{1/2}\lvert \bm{u}\rvert_1h^{k+1/2}\lvert T\rvert_{k+1} \\ &+ h^{1/2}\interleave s_h\interleave h^{-1/6}\lvert \bm{u}\rvert_{0,6}h^{k+2/3-d/6}\lvert T\rvert_{k+1} + h^{-1/3}\lvert s_{h0}\rvert_{0,3}h^{-1/6}\lvert \bm{u}\rvert_{0,6}h^{k+1/2}\lvert T\rvert_{k+1} \\
\apprle& h^k\lVert \bm{u}\rVert_{2}\lVert T\rVert_{k+1}\interleave s_h\interleave. \end{align*} As a result, the two desired results follow from the definitions of $E_N(\bm{u};\bm{u},\bm{v}_h)$,
$\overline{E}_N(\bm{u};T,s_h) $ given in Lemma \ref{E-N}. \end{proof}
\begin{lem}\label{lem44}
Let $(\bm{u},p,T)$ be the solution to the problem (\ref{pb1}), then it holds
\begin{eqnarray}\label{Iu}
A_h(\bm{I}_h\bm{u};\bm{I}_h\bm{u},\bm{v}_h)&+&b_h(\bm{v}_h,J_hp)-b_h(\bm{u}_h,q_h)-d_h(H_hT,\bm{v}_h)\nonumber \\
&=& (\bm{f},\bm{v}_{h0})+E_L(\bm{u},\bm{v}_h)+E_N(\bm{u};\bm{u},\bm{v}_h), \forall (\bm{v}_h,q_h)\in \bm{V}_h^0\times Q_h^0, \label{HT}\\
\overline{A}_h(\bm{I}_h\bm{u};H_hT,s_h)&=&(g,s_{h0})+\overline{E}_L(T,s_h)+\overline{E}_N(\bm{u};T,s_h), \forall s_h\in S_h^0,\label{411}
\end{eqnarray}
where
\begin{align*}
E_L(\bm{u},\bm{v}_h) &:= \Pr \langle(\nabla\bm{u}-\bm{Q}_m^0\nabla\bm{u})\cdot \bm{n},\bm{v}_{h0}-\bm{v}_{hb}\rangle_{\partial \mathcal{T}_h^f}+\Pr \langle\tau(\bm{P}_k^{RT}\bm{u}-\bm{u}),\bm{v}_{h0}-\bm{v}_{hb}\rangle_{\partial \mathcal{T}_h^f},\\
\overline{E}_L(T,s_h) &:= \kappa\langle(\nabla T-\bm{Q}_m^0\nabla T)\cdot \bm{n},s_{h0}-s_{hb}\rangle_{\partial \mathcal{T}_h}+\kappa\langle\tau(Q_k^0 T-T),s_{h0}-s_{hb}\rangle_{\partial \mathcal{T}_h}.
\end{align*}
In addition, it holds
\begin{equation}\label{local RT}
\bm{P}_k^{RT}\bm{u}|_K\in [P_k(K)]^d,\quad \forall K\in \mathcal{T}_h^f.
\end{equation} \end{lem} \begin{proof} We first show \eqref{local RT}. In fact,
for all $K\in \mathcal{T}_h,\varphi \in P_k(K)$, by Lemma \ref{RT38} we get \begin{equation*}
(\nabla\cdot \bm{P}_k^{RT}\bm{u},\varphi)_K= (\nabla\cdot \bm{u},\varphi)_K=0,
\end{equation*}
which indicates
\begin{equation}\label{div415}
\nabla\cdot \bm{P}_k^{RT}\bm{u}=0.
\end{equation}
Thus, the result (\ref{local RT}) follows from Lemma \ref{RT3.6}.
By the definition of $a_0(\cdot,\cdot),A(\cdot,\cdot)$ and $d_h(\cdot,\cdot)$, we obtain
\begin{equation}\label{R}
\begin{aligned}
&A_h(\bm{I}_h\bm{u};\bm{I}_h\bm{u},\bm{v}_h)+b_h(\bm{v}_h,J_hp)-b_h(\bm{u}_h,q_h)-d_h(H_hT,\bm{v}_h)\\
=&\Pr (\nabla_{w,m}\bm{I}_h\bm{u},\nabla_{w,m}\bm{v}_h)\\
&+\Pr \langle \tau\bm{Q}_l^b(\bm{P}_k^{RT}\bm{u}-\bm{u}),\bm{Q}_l^b\bm{v}_{h0}-\bm{v}_{hb}\rangle_{\partial \mathcal{T}_h^f}\\
&+\frac{1}{2}(\nabla_{w,k}\cdot (\bm{I}_h\bm{u}\otimes \bm{I}_h\bm{u}),\bm{v}_{h0})-\frac{1}{2}(\nabla_{w,k}\cdot (\bm{v}_h\otimes \bm{I}_h\bm{u}),\bm{P}_k^{RT}\bm{u})\\
&+(\nabla_{w,k}\{Q_{k-1}^0p,Q_k^bp\},\bm{v}_{h0})\\
&-(\nabla_{w,k}q_h,\bm{P}_k^{RT}\bm{u})\\
&-\Pr Ra(\bm{j}Q_k^0T,\bm{v}_{h0})\\
:=&\sum_{i=1}^{6}R_i.
\end{aligned}
\end{equation}
From the commutativity property (\ref{com1}), the definition of weak gradient, Green's formula, the property of the projection $\bm{Q}_l^b$, and the relation $\langle \nabla\bm{u}\cdot \bm{n},\bm{v}_{hb}\rangle_{\partial \mathcal{T}_h}=0$, it follows
\begin{equation}\label{R1}
\begin{aligned}
R_1&=\Pr (\bm{Q}_m^0\nabla\bm{u},\nabla_{w,k}\bm{v}_h)\\
&=-\Pr (\nabla_h\cdot \bm{Q}_m^0\nabla\bm{u},\bm{v}_{h0})+\Pr\langle \bm{Q}_m^0\nabla\bm{u}\cdot \bm{n},\bm{v}_{hb}\rangle_{\partial \mathcal{T}_h^f}\\
&=\Pr (\bm{Q}_m^0\nabla\bm{u},\nabla_h\bm{v}_{h0})+\Pr\langle\bm{Q}_m^0\nabla\bm{u}\cdot \bm{n},\bm{v}_{hb}-\bm{v}_{h0}\rangle_{\partial \mathcal{T}_h^f}\\
&=-\Pr (\Delta\bm{u},\bm{v}_{h0})+\Pr\langle (\nabla\bm{u}-\bm{Q}_m^0\nabla\bm{u})\cdot \bm{n},\bm{v}_{hb}-\bm{v}_{h0}\rangle_{\partial \mathcal{T}_h^f}.
\end{aligned}
\end{equation}
By the definitions of the projections $\bm{Q}_l^b$ and $Q_k^0$, we have
\begin{equation}\label{R2}
R_2=\Pr \langle\tau(\bm{P}_k^{RT}\bm{u}-\bm{u}),\bm{Q}_l^b\bm{v}_{h0}-\bm{v}_{hb}\rangle_{\partial \mathcal{T}_h^f},\\
\end{equation}
\begin{equation}\label{R6}
R_6 = \Pr Ra(\bm{j}Q_k^0T,\bm{v}_{h0}) = \Pr Ra(\bm{j}T,\bm{v}_{h0}).
\end{equation} By (\ref{c_h}), we get
\begin{align}
R_3 =& (\nabla\cdot (\bm{u}\otimes\bm{u}),\bm{v}_{h0})+E_N(\bm{u};\bm{u},\bm{v}_h).
\end{align}
The commutativity property (\ref{com2}) gives
\begin{equation}\label{R4}
R_4=(\bm{Q}_k^0\nabla p,\bm{v}_{h0})=(\nabla p,\bm{v}_{h0}).
\end{equation}
In view of \eqref{div415}, (\ref{RT1}), and the definitions of $d_h(\cdot,\cdot)$ and weak gradient, we obtain
\begin{equation}\label{R5}
\begin{aligned}
R_5&=-(\bm{P}_k^{RT}\bm{u},\nabla_{w,k}q_h)\\
&=(\nabla\cdot \bm{P}_k^{RT}\bm{u},q_{h0})-\langle \bm{P}_k^{RT}\bm{u}\cdot \bm{n},q_{hb}\rangle_{\partial \mathcal{T}_h^f}\\
&=-\langle \bm{u}\cdot \bm{n},q_{hb}\rangle_{\partial \mathcal{T}_h^f}\\
&=0.
\end{aligned}
\end{equation}
Finally, the desired relation \eqref{HT} follows from the combination of (\ref{R})-(\ref{R5}) and the first equation of \eqref{pb1}.
Similarly, we can get the relation (\ref{411}). This completes the proof. \end{proof} \begin{lem}\label{lem4.5}
For $\bm{u}\in [H^{k+1}(\Omega_f)]^d$ and $T\in H^{k+1}(\Omega)$, it holds
\begin{align}
\lvert E_L(\bm{u},\bm{v}_h)\rvert &\apprle \Pr h^k\lvert \bm{u}\rvert_{k+1} \interleave \bm{v}_h \interleave,\quad \forall \bm{v}_h\in \bm{V}_h^0,\label{ELerroru}\\
\lvert \overline{E}_L(T,s_h)\rvert &\apprle \kappa h^k\lvert T\rvert_{k+1} \interleave s_h \interleave,\quad \forall s_h\in S_h^0.\label{ELerrorT}
\end{align} \end{lem} \begin{proof}
By Lemma \ref{ineq} and the definition of $\bm{Q}_m^0$, we have
\begin{equation*}
\begin{aligned}
\lvert E_L(\bm{u},\bm{v}_h)\rvert&=\lvert \Pr\langle(\nabla\bm{u}-\bm{Q}_m^0\nabla\bm{u})\cdot \bm{n},\bm{v}_{h0}-\bm{v}_{hb}\rangle_{\partial \mathcal{T}_h^f}\rvert+\lvert\Pr\langle\tau(\bm{P}_k^{RT}\bm{u}-\bm{u}),\bm{Q}_l^b\bm{v}_{h0}-\bm{v}_{hb}\rangle_{\partial \mathcal{T}_h^f}\rvert\\
&\leq \sum_{K\in\mathcal{T}_h^f}\Pr\lVert \nabla\bm{u}-\bm{Q}_m^0\nabla\bm{u}\rVert_{0,\partial K}(\lVert \bm{v}_{h0}-\bm{Q}_l^b\bm{v}_{h0}\rVert_{0,\partial K} +\lVert \bm{Q}_l^b\bm{v}_{h0}-\bm{v}_{hb}\rVert_{0,\partial K})\\
&\quad +\sum_{K\in\mathcal{T}_h^f}\Pr\lVert \tau^{1/2}(\bm{P}_k^{RT}\bm{u}-\bm{u})\rVert_{0,\partial K}\lVert \tau^{1/2}(\bm{Q}_l^b\bm{v}_{h0}-\bm{v}_{hb})\rVert_{0,\partial K}\\
&\apprle \Pr h^k\lvert \bm{u}\rvert_{k+1}(\lVert \nabla_h\bm{v}_{h0}\rVert_0 +\lVert \tau^{1/2}(\bm{v}_{h0}-\bm{v}_{hb})\rVert_{0,\partial\mathcal{T}_h^f}) +\Pr h^k\lvert \bm{u}\rvert_{k+1}\lVert\lvert \bm{v}_h\rvert\rVert \\
&\apprle \Pr h^k\lvert \bm{u}\rvert_{k+1}\lVert\lvert \bm{v}_h\rvert\rVert.
\end{aligned}
\end{equation*} i.e. \eqref{ELerroru} holds. The estimate (\ref{ELerrorT}) follows similarly. \end{proof}
\begin{thm}\label{error1}
Let $(\bm{u},p,T)\in [H^{k+1}(\Omega_f)]^d\times H^k(\Omega_f)\times H^{k+1}(\Omega)$ and $(\bm{u}_h,p_h,T_h)\in \bm{V}_h^0\times Q_h^0\times S_h^0$ be the solutions to the problem (\ref{pb1}) and the WG scheme (\ref{pb2}), respectively. Then, under the assumption \eqref{assu} with
\begin{equation}\label{c-0}
C_0:=1 -({\Pr}^{-1}\mathcal{N}_hRa\kappa^{-1}+\mathcal{M}_h Ra\kappa^{-2})\lVert g\rVert_h + \mathcal{N}_h{\Pr}^{-2}\lVert \bm{f}\rVert_h> 0 ,
\end{equation}
it holds the following estimates: for $l=k$ when $d = 2,3$, and for $l = k-1$ when $d=2$,
\begin{align}
&\interleave \bm{I}_h\bm{u}-\bm{u}_h\interleave \apprle C_1h^k\lVert \bm{u}\rVert_{k+1},\label{erroru1}\\
&\interleave H_h T-T_h\interleave \apprle (C_1\mathcal{M}_h\kappa^{-2}\lVert g\rVert_h +\kappa^{-1}\lVert T\rVert_2)h^k\lVert \bm{u}\rVert_{k+1}+(1 +\kappa^{-1}\lVert \bm{u}\rVert_2)h^k\lVert T\rVert_{k+1},\label{errorT1}\\
&\lVert J_hp-p_h\rVert \apprle \Pr (C_1+ Ra\kappa^{-1}\lVert T\rVert_2)h^k\lVert \bm{u}\rVert_{k+1}+\Pr Ra(1 +\kappa^{-1}\lVert \bm{u}\rVert_2)h^k\lVert T\rVert_{k+1},\label{errorp}
\end{align}
where $C_1 := \frac{1+{\Pr}^{-1}\lVert \bm{u}\rVert_2}{C_0+\mathcal{M}_hRa\kappa^{-2}\lVert g\rVert_h}$. \end{thm} \begin{proof}
From (\ref{pb2}) and Lemma \ref{lem44} we easily get the following error equations:
\begin{equation}\label{430}
\begin{aligned}
&a_h(\bm{I}_h\bm{u}-\bm{u}_h,\bm{v}_h)+c_h(\bm{I}_h\bm{u};\bm{I}_h\bm{u},\bm{v}_h)-c_h(\bm{u}_h;\bm{u}_h,\bm{v}_h)+b_h(\bm{v}_h,J_hp-p_h) \\
&\quad -b_h(\bm{I}_h\bm{u}-\bm{u}_h,q_h)
-d_h(H_hT-T_h,\bm{v}_h)=E_L(\bm{u},\bm{v}_h)+E_N(\bm{u};\bm{u},\bm{v}_h),\forall (\bm{v}_h,q_h)\in \bm{V}_h^0\times Q_h^0,\\
\end{aligned}
\end{equation}
\begin{equation}
\overline{a}_h(H_hT-T_h,s_h)+\overline{c}_h(\bm{I}_h\bm{u};H_hT,s_h)-\overline{c}_h(\bm{u}_h;T_h,s_h)=\overline{E}_L(T,s_h)+\overline{E}_N(\bm{u};T,s_h), \forall s_h\in S_h^0.
\end{equation}
Take $(\bm{v}_h,q_h,s_h) = (\bm{I}_h\bm{u}-\bm{u}_h,J_hp-p_h,H_hT-T_h)$ in the above two equations, then we have
\begin{equation}\label{431}
\begin{aligned}
\Pr \interleave \bm{I}_h\bm{u}-\bm{u}_h\interleave^2 &= E_L(\bm{u},\bm{I}_h\bm{u}-\bm{u}_h)+E_N(\bm{u};\bm{u},\bm{I}_h\bm{u}-\bm{u}_h)\\
&-c_h(\bm{I}_h\bm{u};\bm{I}_h\bm{u},\bm{I}_h\bm{u}-\bm{u}_h)+c_h(\bm{u}_h;\bm{u}_h,\bm{I}_h\bm{u}-\bm{u}_h),
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\kappa \interleave H_h T-T_h\interleave^2 &= \overline{E}_L(T,H_h T-T_h)+\overline{E}_N(\bm{u};T,H_h T-T_h)\\
&-\overline{c}_h(\bm{I}_h\bm{u};H_hT,H_h T-T_h)+\overline{c}_h(\bm{u}_h;T_h,H_h T-T_h)
\end{aligned}
\end{equation}
By \eqref{3356} and (\ref{stabs}), it holds
\begin{equation*}
\begin{aligned}
&c_h(\bm{I}_h\bm{u};\bm{I}_h\bm{u},\bm{I}_h\bm{u}-\bm{u}_h)-c_h(\bm{u}_h;\bm{u}_h,\bm{I}_h\bm{u}-\bm{u}_h) \\
=& c_h(\bm{I}_h\bm{u}-\bm{u}_h;\bm{u}_h,\bm{I}_h\bm{u}-\bm{u}_h)\\ \leq& \mathcal{N}_h\interleave \bm{u}_h\interleave \interleave \bm{I}_h\bm{u}-\bm{u}_h\interleave^2\\
\leq& \mathcal{N}_h(Ra\kappa^{-1}\lVert g\rVert_h+{\Pr}^{-1}\lVert \bm{f}\rVert_h)\interleave \bm{I}_h\bm{u}-\bm{u}_h\interleave^2,
\end{aligned}
\end{equation*} which, together with \eqref{431}, \eqref{c-0}, Lemma \ref{lem4.3}, and Lemma \ref{lem4.5}, leads to
\begin{equation*}
\begin{aligned}
(C_0+\mathcal{M}_h Ra\kappa^{-2}\lVert g\rVert_h)\interleave \bm{I}_h\bm{u}-\bm{u}_h\interleave
\apprle (1+{\Pr}^{-1}\lVert \bm{u}\rVert_2)h^k\lVert \bm{u}\rVert_{k+1},
\end{aligned}
\end{equation*} i.e. \eqref{erroru1} holds.
Similarly, we can obtain
\begin{align*}
\interleave H_h T-T_h\interleave &\leq \mathcal{M}_h\kappa^{-2}\lVert g\rVert_h\interleave \bm{I}_h\bm{u}-\bm{u}_h\interleave + \kappa^{-1}(\kappa +\lVert \bm{u}\rVert_2)h^k\lVert T\rVert_{k+1} + \kappa^{-1}h^k\lVert T\rVert_2\lVert \bm{u}\rVert_{k+1}\\
& \apprle (C_1\mathcal{M}_h\kappa^{-2}\lVert g\rVert_h +\kappa^{-1}\lVert T\rVert_2)h^k\lVert \bm{u}\rVert_{k+1}+(1 +\kappa^{-1}\lVert \bm{u}\rVert_2)h^k\lVert T\rVert_{k+1},
\end{align*}
i.e. \eqref{errorT1} holds.
Finally, let us estimate $\lVert J_hp-p_h\rVert$. In light of Theorem \ref{infsup}, \eqref{430}, Lemma \ref{boundedness}, Lemma \ref{lem4.3}, Lemma \ref{lem4.5}, \eqref{erroru1}, and \eqref{errorT1}, we have
\begin{align*}
&\lVert J_hp-p_h\rVert \apprle \sup\limits_{\bm{0}\neq\bm{v}_h\in \bm{V}_h^0} \frac{b_h(\bm{v}_h,J_hp-p_h)}{\interleave \bm{v}_h\interleave}\\
=& \sup\limits_{\bm{0}\neq\bm{v}_h\in \bm{V}_h^0} \frac{E_L(\bm{u},\bm{v}_h)+E_N(\bm{u};\bm{u},\bm{v}_h)-a_h(\bm{I}_h\bm{u}-\bm{u}_h,\bm{v}_h)-c_h(\bm{I}_h\bm{u};\bm{I}_h\bm{u},\bm{v}_h)+c_h(\bm{u}_h;\bm{u}_h,\bm{v}_h)+d_h(H_hT-T_h,\bm{v}_h)}{\interleave \bm{v}_h\interleave}\\
\apprle& (\Pr +\lVert \bm{u}\rVert_2)h^k\lVert \bm{u}\rVert_{k+1}+(\Pr + \mathcal{N}_h(Ra\kappa^{-1}\lVert g\rVert_h+{\Pr}^{-1}\lVert \bm{f}\rVert_h))\interleave \bm{I}_h\bm{u}-\bm{u}_h\interleave + \Pr Ra\interleave H_h T-T_h\interleave\\
\apprle& \Pr (C_1+ Ra\kappa^{-1}\lVert T\rVert_2)h^k\lVert \bm{u}\rVert_{k+1}+\Pr Ra(1 +\kappa^{-1}\lVert \bm{u}\rVert_2)h^k\lVert T\rVert_{k+1},
\end{align*}
i.e. \eqref{errorp} holds. \end{proof}
From Theorem \ref{error1}, Lemma \ref{lem3.3}, and the triangle inequality, it follows the following error estimates: \begin{thm}
Under the same conditions of Theorem \ref{error1}, it holds
\begin{align*}
&\lVert \nabla\bm{u}-\nabla_h\bm{u}_{h0}\rVert_0 \apprle (C_1+1)h^k\lVert \bm{u}\rVert_{k+1},\\
&\lVert \nabla\bm{u}-\nabla_{w,m}\bm{u}_h\rVert_0 \apprle (C_1+1)h^k\lVert \bm{u}\rVert_{k+1},\\
&\lVert \nabla T-\nabla_h T_{h0}\rVert_0 \apprle (\mathcal{M}_h\kappa^{-2}\lVert g\rVert_hC_1 +\kappa^{-1}\lVert T\rVert_2)h^k\lVert \bm{u}\rVert_{k+1}+(1 +\kappa^{-1}\lVert \bm{u}\rVert_2)h^k\lVert T\rVert_{k+1},\\
&\lVert \nabla T-\nabla_{w,m}T_h\rVert_0 \apprle (\mathcal{M}_h\kappa^{-2}\lVert g\rVert_hC_1 +\kappa^{-1}\lVert T\rVert_2)h^k\lVert \bm{u}\rVert_{k+1}+(1 +\kappa^{-1}\lVert \bm{u}\rVert_2)h^k\lVert T\rVert_{k+1},\\
&\lVert p-p_{h0}\rVert_0 \apprle \Pr (C_1+ Ra\kappa^{-1}\lVert T\rVert_2)h^k\lVert \bm{u}\rVert_{k+1}+\Pr Ra(1 +\kappa^{-1}\lVert \bm{u}\rVert_2)h^k\lVert T\rVert_{k+1} + h^k\lVert p\rVert_k.
\end{align*} \end{thm}
\section{Local elimination property and iteration scheme}
\subsection{Local elimination} In this subsection, we shall show that in the WG scheme (\ref{pb2}), the velocity, pressure, and temperature approximations, $(\bm{u}_{h0}, p_{h0}, T_{h0})$, defined in the interior of the elements, can be locally eliminated by using the numerical traces, $(\bm{u}_{hb}, p_{hb}, T_{hb})$, defined on the interface of the elements. Therefore, after the local elimination the resultant system only involves degrees of freedom of $(\bm{u}_{hb}, p_{hb}, T_{hb})$ as unknowns.
We rewrite the scheme (\ref{pb2}) as the following form: seek $\bm{u}_h = (\bm{u}_{h0},\bm{u}_{hb})\in \bm{V}_h^0$, $p_h = (p_{h0},p_{hb})\in Q_h^0$ and $T_h = (T_{h0},T_{hb})\in S_h^0$ such that \begin{align} \label{pb6} \left \{ \begin{array}{rl} A_h(\bm{u}_h;\bm{u}_h,\bm{v}_h)+b_h(\bm{v}_h,p_h)-b_h(\bm{u}_h,q_h)-d_h(T_h,\bm{v}_h)&= (\bm{f},\bm{v}_{h0}), \forall \bm{v}_h\in \bm{V}_h^0,\\ \overline{A}_h(\bm{u}_h;T_h,s_h)&= (g,s_{h0}), \forall s_h\in S_h^0. \end{array} \right. \end{align}
For all $K\in \mathcal{T}_h^f$, taking $\bm{v}_{h0}|_{\mathcal{T}_h^f/K} = \bm{0}, \bm{v}_{hb} = \bm{0}$, $q_{h0}|_{\mathcal{T}_h^f/K} = 0, q_{hb} = 0$ and $T_{h0}|_{\mathcal{T}_h^f/K} = 0, T_{hb} = 0$ in (\ref{pb6}), we can get the following local problem: seek $(\bm{u}_{h0},p_{h0},T_{h0})\in [P_k(K)]^d\times P_{k-1}(K)\times P_k(K)$ such that, for $\forall (\bm{v}_{h0},q_{h0})\in [P_k(K)]^d\times P_{k-1}(K), s_{h0}\in P_k(K)$, \begin{align} \label{pb7} \left \{ \begin{array}{rl} A_{h,K}(\bm{u}_{h0};\bm{u}_{h0},\bm{v}_{h0})+b_{h,K}(\bm{v}_{h0},p_{h0})-b_{h,K}(\bm{u}_{h0},q_{h0})-d_{h,K}(T_{h0},\bm{v}_{h0})&= F_{h,K}(\bm{v}_{h0}),\\ \overline{A}_{h,K}(\bm{u}_{h0},T_{h0};s_{h0})&= G_{h,K}(s_{h0}). \end{array} \right. \end{align} where \begin{align*} A_{h,K}(\bm{u}_{h0};\bm{u}_{h0},\bm{v}_{h0}) :=& a_{h,K}(\bm{u}_{h0},\bm{v}_{h0}) + c_{h,K}(\bm{u}_{h0};\bm{u}_{h0},\bm{v}_{h0});\\ a_{h,K}(\bm{u}_{h0},\bm{v}_{h0}) :=& \Pr (\nabla_{w,m}\{\bm{u}_{h0},\bm{0}\},\nabla_{w,m}\{\bm{v}_{h0},\bm{0}\}) + \Pr\langle \tau\bm{Q}_l^b\bm{u}_{h0},\bm{Q}_l^b\bm{v}_{h0}\rangle_{\partial K},\\ c_{h,K}(\bm{u}_{h0};\bm{u}_{h0},\bm{v}_{h0}) :=& \frac{1}{2}(\nabla_{w,k}\cdot \{\bm{u}_{h0}\otimes\bm{u}_{h0},\bm{0}\otimes \bm{0}\},\bm{v}_{h0})-\frac{1}{2}(\nabla_{w,k}\cdot \{\bm{v}_{h0}\otimes\bm{u}_{h0},\bm{0}\otimes\bm{0}\},\bm{u}_{h0}),\\ b_{h,K}(\bm{v}_{h0},p_{h0}) :=& (\nabla_{w,k}\{p_{h0},0\},\bm{v}_{h0}),\\ d_{h,K}(T_{h0},\bm{v}_{h0}) :=& \Pr Ra(\bm{j}T_{h0},\bm{v}_{h0}),\\ \overline{A}_{h,K}(\bm{u}_{h0},T_{h0};s_{h0}) :=& \overline{a}_{h,K}(T_{h0},s_{h0}) + \overline{c}_{h,K}(\bm{u}_{h0};T_{h0},s_{h0});\\ \overline{a}_{h,K}(T_{h0},s_{h0}) :=& \kappa (\nabla_{w,m}\{T_{h0},0\},\nabla_{w,m}\{s_{h0},0\}) + \kappa\langle \tau Q_l^b T_{h0},Q_l^bs_{h0}\rangle_{\partial K}\\ \overline{c}_{h,K}(\bm{u}_{h0};T_{h0},s_{h0}) :=& \frac{1}{2}(\nabla_{w,k}\cdot \{\bm{u}_{h0}T_{h0},\bm{0}\},s_{h0})-\frac{1}{2}(\nabla_{w,k}\cdot \{\bm{u}_{h0}s_{h0},\bm{0}\},T_{h0})\\ F_{h,K}(\bm{v}_{h0},q_{h0}) :=& (\bm{f},\bm{v}_{h0}) - \Pr (\nabla_{w,m}\{\bm{0},\bm{u}_{hb}\},\nabla_{w,m}\{\bm{v}_{h0},\bm{0}\}) +\Pr\langle \tau\bm{u}_{hb},\bm{Q}_l^b\bm{v}_{h0}\rangle_{\partial K} \\ & \quad -\frac{1}{2}(\nabla_{w,k}\cdot \{\bm{0}\otimes \bm{0},\bm{u}_{hb}\otimes\bm{u}_{hb}\},\bm{v}_{h0}) - (\nabla_{w,k}\{0,p_{hb}\},\bm{v}_{h0}),\\ G_{h,K}(s_{h0}): =& (g,s_{h0}) - \kappa (\nabla_{w,m}\{0,T_{hb}\},\nabla_{w,m}\{s_{h0},0\}) +\Pr\langle \tau T_{hb},\bm{Q}_l^bs_{h0}\rangle_{\partial K} \\ & \quad -\frac{1}{2}(\nabla_{w,k}\cdot \{\bm{0},\bm{u}_{hb}T_{hb}\},s_{h0}). \end{align*}
For any $K\in \mathcal{T}_h$, we define the following semi-norms: \begin{align*} \interleave \bm{v}_{h0}\interleave_K :=&\left( \lVert \nabla_{w,m}\{\bm{v}_{h0},\bm{0}\}\rVert_{0,K}^2 + \lVert \tau^{1/2}\bm{Q}_l^b\bm{v}_{h0}\rVert_{0,\partial K}^2\right)^{1/2},\\
\interleave s_{h0}\interleave_K: =&\left( \lVert \nabla_{w,m}\{s_{h0},0\}\rVert_{0,K}^2 + \lVert \tau^{1/2}Q_l^bs_{h0}\rVert_{0,\partial K}^2\right)^{1/2}. \end{align*} It is easy to see that the above semi-norms are norms on the local spaces $[P_k(K)]^d$ and $ P_k(K)$, respectively.
By following the same routine as in Section 3 for the global problem \eqref{pb2}, we can obtain the following existence and uniqueness results for the local problem \eqref{pb6}.
\begin{thm} For any given $\bm{u}_{hb}, p_{hb}$ and $T_{hb}$, and any $K\in \mathcal{T}_h$, the local problem (\ref{pb7}) admits at least one solution. In addition, it admits a unique solution if
\begin{equation*}
({\Pr}^{-1}\mathcal{N}_{h,K}Ra\kappa^{-1}+\mathcal{M}_{h,K}Ra\kappa^{-2})\lVert G_{h,K}\rVert_h + \mathcal{N}_{h,K}{\Pr}^{-2}\lVert F_{h,K}\rVert_h < 1,
\end{equation*}
where
\begin{align*} \mathcal{N}_{h,K} &:= \sup\limits_{0\neq\bm{w}_{h0},\bm{u}_{h0},\bm{v}_{h0}\in \bm{W}_{h,K}}\frac{c_{h,K}(\bm{w}_{h0};\bm{u}_{h0},\bm{v}_{h0})}{\interleave \bm{w}_{h0}\interleave_K\cdot \interleave \bm{u}_{h0}\interleave_K\cdot\interleave \bm{v}_{h0}\interleave_K},\\%\label{NhK}\\ \mathcal{M}_{h,K} &:= \sup\limits_{\substack{0\neq\bm{w}_{h0}\in \bm{W}_{h,K},\\ 0\neq T_{h0},s_{h0}\in P_k(K)}}\frac{\overline{c}_{h,K}(\bm{w}_{h0};T_{h0},s_{h0})}{\interleave \bm{w}_{h0}\interleave_K\cdot \interleave T_{h0}\interleave_K\cdot\interleave s_{h0}\interleave_K},\\%\label{MhK}\\ \lVert F_{h,K}\rVert_h &:= \sup\limits_{0\neq\bm{v}_{h0}\in \bm{W}_{h,K}}\frac{F_{h,K}(\bm{v}_{h0})}{\interleave \bm{v}_{h0}\interleave_K},\quad \lVert G_{h,K}\rVert_h := \sup\limits_{0\neq s_{h0}\in P_k(K)}\frac{G_{h,K}(s_{h0})}{\interleave s_{h0}\interleave_K}, \end{align*} and $ \bm{W}_{h,K} := \{\bm{w}_{h0}\in [P_k(K)]^d:b_{h,K}(\bm{w}_{h0},q_{h0}) = 0,\forall q_{h0}\in P_{k-1}(K)\}.$
\end{thm}
\subsection{Iteration scheme}
Since the WG scheme \eqref{pb2} is nonlinear, we introduce the following Oseen's iteration scheme: given $\bm{u}_h^0$, for $n=1,2,\cdots,$ and $\forall (\bm{v}_h,q_h,s_h)\in \bm{V}_h^0\times Q_h^0 \times S_h^0$, \begin{align} \label{pboseen} \left \{ \begin{array}{rl} &a_h(\bm{u}_h^n,\bm{v}_h)+c_h(\bm{u}_h^{n-1};\bm{u}_h^n,\bm{v}_h)+b_h(\bm{v}_h,p_h^n)-b_h(\bm{u}_h^n,q_h)=(\bm{f},\bm{v}_{h0})+d_h(T_h^n,\bm{v}_h),\\ &\overline{a}_h(T_h^n,s_h)+\overline{c}_h(\bm{u}_h^{n-1};T_h^n,s_h)= (g,s_{h0}). \end{array} \right. \end{align} We have the following convergence theorem. \begin{thm}
Let $(\bm{u}_h,p_h,T_h)\in \bm{V}_h^0\times Q_h^0 \times S_h^0$ be the solution to the WG scheme (\ref{pb2}), and assume that (\ref{c-0}) holds. Then the Oseen's iteration scheme (\ref{pboseen}) is convergent in the following sense:
\begin{align*}
\lim\limits_{n\longrightarrow \infty}\interleave \bm{u}_h^n- \bm{u}_h\interleave = 0, \lim\limits_{n\longrightarrow \infty}\lVert p_h^n- p_h\rVert = 0, \lim\limits_{n\longrightarrow \infty}\interleave T_h^n- T_h\interleave = 0.
\end{align*} \end{thm} \begin{proof}
Set $\bm{e}_u^n := \bm{u}_h^n-\bm{u}_h, e_p^n: = p_h^n-p_h, e_T^n: = T_h^n-T_h$, then, from (\ref{pb2}) and (\ref{pboseen}), we have, for $\forall (\bm{v}_h,q_h,s_h)\in \bm{V}_h^0\times Q_h^0\times S_h^0$,
\begin{align}\label{pbiteration}
\left \{
\begin{array}{rl}
a_h(\bm{e}_u^n,\bm{v}_h)&=-b_h(\bm{v}_h,e_p^n)+b_h(e_u^n,q_h)+c_h(\bm{u}_h;\bm{u}_h,\bm{v}_h)-c_h(\bm{u}_h^{n-1};\bm{u}_h^n,\bm{v}_h)\\
\overline{a}_h(e_T^n,s_h)&=\overline{c}_h(\bm{u}_h;T_h,s_h)-\overline{c}_h(\bm{u}_h^{n-1};T_h^n,s_h).
\end{array}
\right.
\end{align}
Taking $\bm{v}_h = \bm{e}_u^n, q_h = e_p^n, s_h = e_T^n$ in (\ref{pbiteration}), in view of \eqref{3356} and Lemma \ref{boundedness}, we get
\begin{equation}\label{509}
\begin{aligned}
\Pr\interleave \bm{e}_u^n\interleave^2
=&c_h(\bm{u}_h;\bm{u}_h,\bm{e}_u^n)-c_h(\bm{u}_h^{n-1};\bm{u}_h^n,\bm{e}_u^n) + d_h(e_T^n,\bm{e}_u^n)\\
=&-c_h(\bm{e}_u^{n-1};\bm{u}_h,\bm{e}_u^n)-c_h(\bm{u}_h^{n-1};\bm{e}_u^{n},\bm{e}_u^n)+d_h(e_T^n,\bm{e}_u^n)\\
\leq&\mathcal{N}_h\interleave \bm{e}_u^{n-1}\interleave \interleave \bm{u}_h\interleave \interleave \bm{e}_u^n\interleave+\Pr Ra\interleave e_T^n\interleave\interleave \bm{e}_u^n\interleave,
\end{aligned}
\end{equation}
\begin{equation}\label{510}
\begin{aligned}
\kappa\interleave e_T^n\interleave^2=&\overline{c}_h(\bm{u}_h;T_h,e_T^n)-\overline{c}_h(\bm{u}_h^{n-1};T_h^n,e_T^n)\\
= &-\overline{c}_h(\bm{e}_u^{n-1};T_h,e_T^n)-\overline{c}_h(\bm{u}_h^{n-1};e_T^{n},e_T^n) \\
\leq& \mathcal{M}_h\interleave \bm{e}_u^{n-1}\interleave \interleave T_h\interleave\interleave \bm{e}_T^n\interleave,
\end{aligned}
\end{equation}
which, together with \eqref{regular estimates Th}, \eqref{stabs}, and \eqref{c-0}, implies
\begin{equation*}
\begin{aligned}
\interleave \bm{e}_u^n\interleave &\leq {\Pr}^{-1}\mathcal{N}_h\interleave \bm{e}_u^{n-1}\interleave \interleave \bm{u}_h\interleave +Ra\interleave e_T^n\interleave \\
&\leq ( {\Pr}^{-1}\mathcal{N}_h \interleave \bm{u}_h\interleave +\kappa^{-1}\mathcal{M}_h\interleave T_h\interleave)\interleave \bm{e}_u^{n-1}\interleave \\
&\leq (1-C_0)\interleave \bm{e}_u^{n-1}\interleave \leq \cdots
\leq (1-C_0)^n\interleave \bm{e}_u^{0}\interleave .
\end{aligned}
\end{equation*}
Since $0<C_0<1$, the above inequality leads to the conclusion
\begin{align}\label{ulim}
\lim\limits_{n\longrightarrow \infty}\interleave \bm{u}_h^n- \bm{u}_h\interleave =\lim\limits_{n\longrightarrow \infty}\interleave\bm{e}_u^n\interleave= 0.
\end{align}
Thus, from \eqref{510} and \eqref{regular estimates Th} it follows
\begin{align}\label{Tlim}
\lim\limits_{n\longrightarrow \infty}\interleave T_h^n- T_h\interleave =\lim\limits_{n\longrightarrow \infty}\interleave e_T^n\interleave= 0.
\end{align}
Finally, in light of Lemma \ref{infsup} and the first equation of \eqref{pbiteration}, we obtain \begin{align*} \lVert e_p^n\rVert \leq &\sup\limits_{\bm{v}_h\in \bm{V}_h^0}\frac{b_h(\bm{v}_h,e_p^n)}{\interleave\bm{v}_h \interleave} \\ = &\sup\limits_{\bm{v}_h\in \bm{V}_h^0}\frac{1}{\interleave\bm{v}_h \interleave}(-a_h(\bm{e}_u^n,\bm{v}_h)+c_h(\bm{u}_h;\bm{u}_h,\bm{v}_h)-c_h(\bm{u}_h^{n-1};\bm{u}_h^n,\bm{v}_h)+d_h(e_T^n,\bm{v}_h))\\
= &\sup\limits_{\bm{v}_h\in \bm{V}_h^0}\frac{1}{\interleave\bm{v}_h \interleave}(-a_h(\bm{e}_u^n,\bm{v}_h)-c_h(\bm{e}_u^{n-1};\bm{u}_h,\bm{v}_h)-c_h(\bm{e}_u^{n-1};\bm{e}_u^n,\bm{v}_h)-c_h(\bm{e}_u^{n};\bm{u}_h,\bm{v}_h)+d_h(e_T^n,\bm{v}_h))\\ \leq &\Pr\interleave \bm{e}_u^{n}\interleave +\mathcal{N}_h(\interleave \bm{u}_h\interleave\interleave \bm{e}_u^{n-1}\interleave+\interleave \bm{e}_u^{n}\interleave\interleave \bm{e}_u^{n-1}\interleave+\interleave \bm{u}_h\interleave\interleave \bm{e}_u^{n}\interleave) +\Pr Ra\interleave e_T^n \interleave, \end{align*} which, together with (\ref{ulim}) and (\ref{Tlim}), yields
$\lim\limits_{n\longrightarrow \infty}\lVert p_h^n- p_h\rVert = 0$. \end{proof}
\section{Numerical experiments} In this section, we shall show some numerical results to examine the performance of the proposed WG methods for the natural convection equations. The Oseen's iteration scheme \eqref{pboseen} with initial guess $\bm{u}_h^0=0$ is used in all the numerical experiments.
We consider three cases of our WG methods with $k=1,2$: \begin{eqnarray*} \begin{aligned} WG-I&:& l &= k,&m& = k,\\ WG-II&:& l &= k,&m& = k-1,\\ WG-III&:& l &= k-1,&m& = k-1. \end{aligned} \end{eqnarray*}
\begin{exmp} Take $\Omega = [-1,1]\times[0,1]$ and $\Omega_f = [0,1]\times[0,1]$.
The exact solution to the problem \eqref{pb1} is given by
\begin{align*}
\left \{
\begin{array}{rll}
u_1 &= -x^2(x-1)^2y(y-1)(2y-1) & \text{in} \quad\Omega_f,\\
u_2 &= y^2(y-1)^2x(x-1)(2x-1) & \text{in}\quad\Omega_f,\\
p &= x^6-y^6 & \text{in} \quad\Omega_f,\\
T &= (x-1)(x+1)y(y-1) & \text{in} \quad\Omega.
\end{array}
\right.
\end{align*}
with $\Pr = 1,\kappa = 1, Ra = 10$. Regular triangular meshes are used for the computation (see Figure 1). \end{exmp}
\begin{figure}
\caption{Regular triangular meshes: $4\times 2$ mesh (Left) and $ 8\times 4$ mesh (Right)}
\label{division}
\end{figure}
Tables 1 and 2 show the history of convergence for the velocity $\bm{u}_{h0} $, pressure $p_{h0}$, and temperature $T_{h0}$. Results of $div_hU_h =: \max\limits_{K\in \mathcal{T}_h^f}h_K^{-1}\lVert \nabla\cdot \bm{u}_{h0}\rVert_{0,K}$
are also listed.
From the numerical results we have the following observations:
\begin{itemize}
\item The convergence rates of $\lVert \nabla\bm{u}-\nabla_h\bm{u}_{h0} \rVert_0,$ $ \lVert p-p_{h0} \rVert_0$, and $\lVert \nabla T-\nabla_hT_{h0} \rVert_0$ for the proposed WG methods with $k = 1,2$ are of $k^{\text{th}}$ orders, as is consistent with the theoretical results. In addition, the convergence rates of $\lVert \bm{u}-\bm{u}_{h0}\rVert_0$ and $\lVert T-T_{h0} \rVert_0$ are of $(k+1)^{\text{th}}$ orders.
\item Since $\lVert \nabla_h\cdot \bm{u}_{h0}\rVert_{0,\infty}\apprle \max\limits_{K\in \mathcal{T}_h^f}h_K^{-1}\lVert \nabla\cdot \bm{u}_{h0}\rVert_{0,K}$, the velocity approximations obtained by our methods are globally divergence-free, which are conformable to the conclusion in Remark 2.1.
\end{itemize}
\begin{table}[H]
\normalsize
\caption{Results for different methods with $k = 1$ }\label{KK1}
\centering
\tiny
\subtable[Method:WG-I]{
\begin{tabular}{p{0.75cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}}
\hline
\multirow{2}{*}{mesh}&
\multicolumn{2}{c|}{ $\frac{\lVert \nabla\bm{u}-\nabla_h\bm{u}_{h0} \rVert_0}{\lVert \nabla\bm{u}\rVert_0}$}&\multicolumn{2}{c|}{$\frac{\lVert \bm{u}-\bm{u}_{h0}\rVert_0}{\lVert \bm{u}\rVert_0}$}&\multicolumn{2}{c|}{ $\frac{\lVert p-p_{h0} \rVert_0}{\lVert p\rVert_0}$}&\multicolumn{2}{c|}{ $\frac{\lVert \nabla T-\nabla_hT_{h0} \rVert_0}{\lVert \nabla T\rVert_0}$}&\multicolumn{2}{c|}{ $\frac{\lVert T-T_{h0} \rVert_0}{\lVert T\rVert_0}$}
&\multirow{2}{*}{$div_hU_h$}\cr\cline{2-11}
&error&order&error&order&error&order&error&order&error&order&\cr
\cline{1-12}
$8\times 4$ &5.9412E-01 & &1.6959E-01 & &4.4819E-01 & &2.4656E-01 & &2.7341E-02 & &3.9988E-16\\
\hline
$16\times 8$ &3.1494E-01 &0.92 &4.7778E-02 &1.83 &2.3637E-01 &0.92 &1.2464E-01 &0.98 &6.8747E-03 &1.99 &1.9062E-15\\
\hline
$32\times 16$ &1.5988E-01 &0.98 &1.2396E-02 &1.95 &1.1983E-01 &0.98 &6.2498E-02 &0.99 &1.7191E-03 &2.00 &3.0715E-15\\
\hline
$64\times 32$ &8.0247E-02 &0.99 &3.1249E-03 &1.99 &6.0122E-02 &0.99 &3.1272E-02 &1.00 &4.2894E-04 &2.00 &3.1834E-14\\
\hline
$128\times 64$ &4.0162E-02 &1.00 &7.8018E-04 &2.00 &3.0087E-02 &1.00 &1.5639E-02 &1.00 &1.0704E-04 &2.00 &4.6475E-14\\
\hline
\end{tabular} } \subtable[Method:WG-II]{
\begin{tabular}{p{0.75cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}}
\hline
\multirow{2}{*}{mesh}&
\multicolumn{2}{c|}{ $\frac{\lVert \nabla\bm{u}-\nabla_h\bm{u}_{h0} \rVert_0}{\lVert \nabla\bm{u}\rVert_0}$}&\multicolumn{2}{c|}{$\frac{\lVert \bm{u}-\bm{u}_{h0}\rVert_0}{\lVert \bm{u}\rVert_0}$}&\multicolumn{2}{c|}{ $\frac{\lVert p-p_{h0} \rVert_0}{\lVert p\rVert_0}$}&\multicolumn{2}{c|}{ $\frac{\lVert \nabla T-\nabla_hT_{h0} \rVert_0}{\lVert \nabla T\rVert_0}$}&\multicolumn{2}{c|}{ $\frac{\lVert T-T_{h0} \rVert_0}{\lVert T\rVert_0}$}
&\multirow{2}{*}{$div_hU_h$}\cr\cline{2-11}
&error&order&error&order&error&order&error&order&error&order&\cr
\cline{1-12}
$8\times 4$ &7.0486E-01 & &7.8104E-01 & &4.7353E-01 & &2.6104E-01 & &1.3922E-01 & &1.5492E-15\\
\hline
$16\times 8$ &3.2996E-01 &1.10 &1.8899E-01 &2.05 &2.3962E-01 &0.98 &1.2868E-01 &1.02 &3.5017E-02 &1.99 &5.5321E-16\\
\hline
$32\times 16$ &1.6192E-01 &1.03 &4.8031E-02 &1.98 &1.2025E-01 &0.99 &6.4066E-02 &1.01 &8.7749E-03 &2.00 &6.8348E-15\\
\hline
$64\times 32$ &8.0518E-02 &1.01 &1.2196E-02 &1.98 &6.0178E-02 &1.00 &3.1996E-02 &1.00 &2.1989E-03 &2.00 &9.5579E-15\\
\hline
$128\times 64$ &4.0158E-02 &1.00 &3.0774E-03 &1.99 &3.0095E-02 &1.00 &1.5993E-02 &1.00 &5.5203E-04 &2.00 &2.0390E-14\\
\hline
\end{tabular} } \subtable[Method:WG-III]{
\begin{tabular}{p{0.75cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}}
\hline
\multirow{2}{*}{mesh}&
\multicolumn{2}{c|}{ $\frac{\lVert \nabla\bm{u}-\nabla_h\bm{u}_{h0} \rVert_0}{\lVert \nabla\bm{u}\rVert_0}$}&\multicolumn{2}{c|}{$\frac{\lVert \bm{u}-\bm{u}_{h0}\rVert_0}{\lVert \bm{u}\rVert_0}$}&\multicolumn{2}{c|}{ $\frac{\lVert p-p_{h0} \rVert_0}{\lVert p\rVert_0}$}&\multicolumn{2}{c|}{ $\frac{\lVert \nabla T-\nabla_hT_{h0} \rVert_0}{\lVert \nabla T\rVert_0}$}&\multicolumn{2}{c|}{ $\frac{\lVert T-T_{h0} \rVert_0}{\lVert T\rVert_0}$}
&\multirow{2}{*}{$div_hU_h$}\cr\cline{2-11}
&error&order&error&order&error&order&error&order&error&order&\cr
\cline{1-12}
$8\times 4$ &7.4774E-01 & &8.3792E-01 & &4.7910E-01 & &3.1663E-01 & &1.6162E-01 & &1.0304E-16\\
\hline
$16\times 8$ &3.3583E-01 &1.02 &1.9985E-01 &2.07 &2.4031E-01 &0.99 &1.5503E-01 &1.03 &4.0626E-02 &1.99 &2.0466E-16\\
\hline
$32\times 16$ &1.6272E-01 &1.01 &5.0551E-02 &1.98 &1.2033E-01 &1.00 &7.7080E-02 &1.01 &1.0178E-02 &2.00 &1.7369E-15\\
\hline
$64\times 32$ &8.0623E-02 &1.00 &1.2810E-02 &1.98 &6.0183E-02 &1.00 &3.8485E-02 &1.00 &2.5494E-03 &2.00 &2.3551E-15\\
\hline
$128\times 64$ &4.0212E-02 &1.00 &3.2282E-03 &1.99 &3.0093E-02 &1.00 &1.9235E-02 &1.00 &6.3946E-04 &2.00 &6.5944E-15\\
\hline
\end{tabular} } \end{table}
\renewcommand\arraystretch{1.1} \begin{table}[H]
\normalsize
\caption{Results for different methods with $k = 2$ }\label{KK2}
\centering
\tiny
\subtable[Method:WG-I]{
\begin{tabular}{p{0.75cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}}
\hline
\multirow{2}{*}{mesh}&
\multicolumn{2}{c|}{ $\frac{\lVert \nabla\bm{u}-\nabla_h\bm{u}_{h0} \rVert_0}{\lVert \nabla\bm{u}\rVert_0}$}&\multicolumn{2}{c|}{$\frac{\lVert \bm{u}-\bm{u}_{h0}\rVert_0}{\lVert \bm{u}\rVert_0}$}&\multicolumn{2}{c|}{ $\frac{\lVert p-p_{h0} \rVert_0}{\lVert p\rVert_0}$}&\multicolumn{2}{c|}{ $\frac{\lVert \nabla T-\nabla_hT_{h0} \rVert_0}{\lVert \nabla T\rVert_0}$}&\multicolumn{2}{c|}{ $\frac{\lVert T-T_{h0} \rVert_0}{\lVert T\rVert_0}$}
&\multirow{2}{*}{$div_hU_h$}\cr\cline{2-11}
&error&order&error&order&error&order&error&order&error&order&\cr
\cline{1-12}
$8\times 4$ &1.6192E-01 & &2.8177E-02 & &6.6611E-02 & &2.3814E-02 & &1.5210E-03 & &1.5852E-15\\
\hline
$16\times 8$ &4.2800E-02 &1.92 &3.6801E-03 &2.94 &1.7476E-02 &1.92 &5.9899E-03 &1.98 &1.9029E-04 &2.99 &1.3501E-14\\
\hline
$32\times 16$ &1.0767E-02 &1.99 &4.6124E-04 &2.99 &4.4430E-03 &1.98 &1.4995E-03 &1.99 &2.3790E-05 &3.00 &6.8867E-14\\
\hline
$64\times 32$ &2.6808E-03 &2.01 &5.7386E-05 &3.01 &1.1115E-03 &1.99 &3.7495E-04 &2.00 &2.9736E-06 &3.00 &3.8064E-14\\
\hline
$128\times 64$ &6.7021E-04 &2.00 &7.1513E-06 &3.00 &2.7795E-04 &2.00 &9.3738E-05 &2.00 &3.7173E-07 &3.00 &7.2047E-14\\
\hline
\end{tabular}
}
\subtable[Method:WG-II]{
\begin{tabular}{p{0.75cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}}
\hline
\multirow{2}{*}{mesh}&
\multicolumn{2}{c|}{ $\frac{\lVert \nabla\bm{u}-\nabla_h\bm{u}_{h0} \rVert_0}{\lVert \nabla\bm{u}\rVert_0}$}&\multicolumn{2}{c|}{$\frac{\lVert \bm{u}-\bm{u}_{h0}\rVert_0}{\lVert \bm{u}\rVert_0}$}&\multicolumn{2}{c|}{ $\frac{\lVert p-p_{h0} \rVert_0}{\lVert p\rVert_0}$}&\multicolumn{2}{c|}{ $\frac{\lVert \nabla T-\nabla_hT_{h0} \rVert_0}{\lVert \nabla T\rVert_0}$}&\multicolumn{2}{c|}{ $\frac{\lVert T-T_{h0} \rVert_0}{\lVert T\rVert_0}$}&\multirow{2}{*}{$div_hU_h$}\cr\cline{2-11}
&error&order&error&order&error&order&error&order&error&order&\cr
\cline{1-12}
$8\times 4$ &2.5023E-01 & &5.9209E-02 & &6.6212E-02 & &4.1197E-02 & &4.9610E-03 & &6.5550E-16\\
\hline
$16\times 8$ &6.3163E-02 &1.98 &7.4474E-03 &2.99 &1.7485E-02 &1.92 &1.0276E-02 &1.99 &6.1111E-04 &3.02 &7.4872E-15\\
\hline
$32\times 16$ &1.5659E-02 &2.01 &9.3395E-04 &2.99 &4.4432E-03 &1.98 &2.5691E-03 &2.00 &7.5883E-05 &3.01 &5.6488E-15\\
\hline
$64\times 32$ &3.8820E-03 &2.01 &1.1720E-04 &3.00 &1.1117E-03 &2.00 &6.4257E-04 &2.00 &9.4569E-06 &3.00 &2.4648E-14\\
\hline
$128\times 64$ &9.6547E-04 &2.00 &1.4685E-05 &3.00 &2.7815E-04 &2.00 &1.6070E-04 &2.00 &1.1804E-06 &3.00 &2.1412E-13\\
\hline
\end{tabular}
}
\subtable[Method:WG-III]{
\begin{tabular}{p{0.75cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}|p{0.3cm}<{\centering}|p{1.24cm}<{\centering}}
\hline
\multirow{2}{*}{mesh}&
\multicolumn{2}{c|}{ $\frac{\lVert \nabla\bm{u}-\nabla_h\bm{u}_{h0} \rVert_0}{\lVert \nabla\bm{u}\rVert_0}$}&\multicolumn{2}{c|}{$\frac{\lVert \bm{u}-\bm{u}_{h0}\rVert_0}{\lVert \bm{u}\rVert_0}$}&\multicolumn{2}{c|}{ $\frac{\lVert p-p_{h0} \rVert_0}{\lVert p\rVert_0}$}&\multicolumn{2}{c|}{ $\frac{\lVert \nabla T-\nabla_hT_{h0} \rVert_0}{\lVert \nabla T\rVert_0}$}&\multicolumn{2}{c|}{ $\frac{\lVert T-T_{h0} \rVert_0}{\lVert T\rVert_0}$}
&\multirow{2}{*}{$div_hU_h$}\cr\cline{2-11}
&error&order&error&order&error&order&error&order&error&order&\cr
\cline{1-12}
$8\times 4$ &1.3075E-01 & &6.2217E-02 & &6.6237E-02 & &2.1605E-02 & &5.3332E-03 & &1.6739E-15\\
\hline
$16\times 8$ &3.4979E-02 &1.90 &7.6750E-03 &3.02 &1.7492E-02 &1.92 &5.4667E-03 &1.98 &6.6033E-04 &3.02 &3.5685E-15\\
\hline
$32\times 16$ &8.9627E-03 &1.96 &9.4948E-04 &3.01 &4.4333E-03 &1.98 &1.3734E-03 &1.99 &8.2232E-05 &3.01 &3.2603E-14\\
\hline
$64\times 32$ &2.2617E-03 &1.99 &1.1834E-04 &3.00 &1.1121E-03 &2.00 &3.4409E-04 &2.00 &1.0263E-05 &3.00 &7.3185E-14\\
\hline
$128\times 64$ &5.6761E-04 &2.00 &1.4791E-05 &3.00 &2.7826E-04 &2.00 &8.6112E-05 &2.00 &1.2821E-06 &3.00 &2.6642E-13\\
\hline
\end{tabular}
} \end{table}
\begin{figure}
\caption{The physical domain with its boundary conditions: $4\times4$ mesh}
\label{buoyancy}
\end{figure}
\begin{exmp}
We consider the well-known test cave for the natural convection codes which is called buoyancy-driven cavity problem. This problem describes the two-dimensional flow of a Boussinesq fluid in an upright square cavity of side $L=1$. Fig.\ref{buoyancy} shows the physical domain with the boundary conditions. The velocity is zero on all the boundaries. The horizontal walls are insulated with $\frac{\partial T}{\partial \bm{n}} = 0$, and the vertical sides are at temperatures $T_H=1$ and $T_C=0$. We take $\Omega=\Omega_f=[0,1]\times [0,1]$, $\kappa = 1, \Pr = 0.71, \bm{f} = 0$, and $g = 0$.
\end{exmp}
For different Rayleigh numbers, i.e. $Ra=10^3, 10^4,10^5, 10^6,10^7$, we use the WG-I method with $k=1,2$ to compute the following quantities at different mesh sizes:
\begin{table}[H]
\normalsize
\centering{
\begin{tabular}{c|c p{5cm}|}
\hline
$u1_{max}$ &the maximum horizontal velocity on the vertical mid-plane of the cavity \\
\hline
$u2_{max}$ &the maximum vertical velocity on the horizontal mid-plane of the cavity \\
\hline
$\overline{Nu}$ &the average Nusselt number throughout the cavity \\
\hline
$Nu_{max}$ &the maximum value of the local Nusselt number on the boundary at x=0 \\
\hline
$Nu_{min}$ &the minimum value of the local Nusselt number on the boundary at x=0 \\
\hline
\end{tabular}
} \end{table}
The results are listed in Table \ref{num2} and compared with the famous benchmark solutions of de Vahl Davis \cite{de1983natural} and of some other authors such as Manzari \cite{manzari1999explicit}, Massarotti et al \cite{massarotti1998characteristic}, Wan et al \cite{c2001new}, and Zhang et al \cite{Zhang2014Error}. Figure \ref{streamline1} and Figure \ref{isotherms1} show the contour maps of the stream function and the isotherms of the flow. We have the following observations:
\begin{itemize} \item From Table \ref{num2} we can see that the WG-I method gives good results for all the quantities for different Rayleigh numbers. In particular, the method with $k=2$ behaves very well at the coarsest mesh $40\times40$.
\item Figure \ref{streamline1} demonstrates that, as Rayleigh number $Ra $ increases, the circular vortex at the cavity center begins to deform into an ellipse and then breaks up into two vortices, and then there's a big vortex in the center.
\item Figure \ref{isotherms1} shows that, when Rayleigh number $Ra $ is small, the heat transfer mainly depends on heat conduction (isotherms almost vertical), with the increasing of $Ra$, the heat transfer pattern gradually turns to heat convection and boundary layers appear around the two walls (isotherms almost horizontal at the center). \end{itemize} \begin{figure}
\caption{Contour maps of stream function (left to right) with Ra = $10^3, 10^4, 10^5, 10^6, 10^7$.}
\label{streamline1}
\end{figure}
\begin{figure}
\caption{Isotherms(left to right) with Ra = $10^3, 10^4, 10^5, 10^6, 10^7$}
\label{isotherms1}
\end{figure}
\renewcommand\arraystretch{1.1} \begin{table}[H]
\normalsize
\centering
\caption{Natural convection in a square cavity: comparison with the benchmark solutions }\label{num2}
\centering
\scriptsize
\begin{threeparttable}
\centering{
\begin{tabular}{p{0.25cm}<{\centering}|p{0.75cm}<{\centering}|p{1.17cm}<{\centering}|p{1.17cm}<{\centering}|p{1.17cm}<{\centering}|p{1.17cm}<{\centering}|p{0.8cm}<{\centering}|p{0.9cm}<{\centering}|p{0.83cm}<{\centering}|p{0.83cm}<{\centering}|p{1.23cm}<{\centering}}
\hline
\multirow{2}{*}{ Ra}&{}&{WG-I,k=1}&{WG-I,k=1}&{WG-I,k=2}&{WG-I,k=2}&{Ref.\cite{de1983natural}}&{ Ref.\cite{Zhang2014Error}}&{Ref.\cite{manzari1999explicit}}&{ Ref.\cite{massarotti1998characteristic}}&{ Ref.\cite{c2001new}}\cr
&{}&{$40\times 40$}&{$70\times 70$}&{$40\times 40$}&{$50\times 50$}&{}&{ $64\times 64$}&{$70\times 70$}&{$70\times 70$}&{$100\times 100$}\cr
\cline{1-11}
\multirow{5}{*}{$10^3$}&$u1_{max}$ &3.653 &3.654 &3.640 &3.646 &3.649 &- &3.68 &- &3.489 \\
&$u2_{max}$ &3.711 &3.698 &3.697 &3.697 &3.697 &- &3.73 &3.686 &3.69 \\
&$\overline{Nu}$ &1.118 &1.118 &1.118 &1.118 &1.118 &- &1.074 &1.117 &1.117 \\
&$Nu_{max}$ &1.506 &1.506 &1.560 &1.506 &1.505 &- &1.47 &- &1.501 \\
&$Nu_{min}$ &0.691 &0.691 &0.691 &0.691 &0.692 &- &0.623 &- &0.691 \\
\hline
\multirow{5}{*}{$10^4$}&$u1_{max}$ &16.227 &16.188 &16.183 &16.180 &16.178 &16.19 &16.10 &- &16.122 \\
&$u2_{max}$ &19.744 &19.611 &19.600 &16.628 &19.617 &19.63 &19.90 &19.63 &19.79 \\
&$\overline{Nu}$ &2.243 &2.244 &2.245 &2.245 &2.243 &- &2.084 &2.243 &2.254 \\
&$Nu_{max}$ &3.528 &3.530 &3.531 &3.531 &3.528 &- &3.47 &- &3.579 \\
&$Nu_{min}$ &0.585 &0.585 &0.585 &0.585 &0.586 &- &0.4968 &- &0.577 \\
\hline
\multirow{5}{*}{$10^5$}&$u1_{max}$ &34.829 &34.771 &34.715 &34.702 &34.81 &34.74 &34.00 &- &34.00 \\
&$u2_{max}$ &69.049 &68.736 &67.875 &68.290 &68.22 &68.48 &70.00 &68.85 &70.63 \\
&$\overline{Nu}$ &4.515 &4.519 &4.522 &4.522 &4.519 &- &4.30 &4.521 &4.598 \\
&$Nu_{max}$ &7.701 &7.713 &7.716 &7.720 &7.717 &- &7.71 &- &7.945 \\
&$Nu_{min}$ &0.726 &0.727 &0.728 &0.728 &0.729 &- &0.614 &- &0.698 \\
\hline
\multirow{5}{*}{$10^6$}&$u1_{max}$ &64.977 &64.710 &64.835 &64.541 &64.63 &64.81 &65.40 &- &65.40 \\
&$u2_{max}$ &217.307 &221.534 &208.237 &220.609 &219.36 &220.46 &228 &221.6 &227.11 \\
&$\overline{Nu}$ &8.797 &8.813 &8.825 &8.825 &8.800 &- &8.743 &8.806 &8.976 \\
&$Nu_{max}$ &17.676 &17.511 &17.462 &17.536 &17.925 &- &17.46 &- &17.86 \\
&$Nu_{min}$ &0.970 &0.976 &0.980 &0.979 &0.989 &- &0.716 &- &0.913 \\
\hline
\multirow{5}{*}{$10^7$}&$u1_{max}$ &154.770 &148.802 &148.454 &148.596 &145.267*&148.40 &139.7 &- &143.56 \\
&$u2_{max}$ &819.329 &695.512 &703.702 &707.696 &703.253*&694.14 &698 &702.3 &714.48 \\
&$\overline{Nu}$ &16.564 &16.484 &16.522 &16.521 &- &- &13.99 &16.40 &16.656 \\
&$Nu_{max}$ &47.155 &40.374 &40.935 &40.329 &41.025* &- &30.46 &- &38.6 \\
&$Nu_{min}$ &1.359 &1.353 &1.363 &1.367 &1.380* &- &0.787 &- &1.298 \\
\hline
\end{tabular}
\begin{tablenotes}
\footnotesize
\item[1] The benchmark solutions with * were mentioned in \cite{Mayne2000h-adaptive} when $Ra=10^7$.
\end{tablenotes}
} \end{threeparttable} \end{table}
\section{Conclusions} In this paper, we have developed a class of weak Galerkin finite element methods with globally divergence-free velocity approximation for the steady-state natural convection problems. Well-posedness of the discrete scheme is analyzed, and optimal error estimates for the velocity, temperature and pressure approximations are derived. The proposed Oseen's iteration algorithm is unconditionally convergent. Numerical experiments verify the theoretical results.
\section*{Acknowledgments} This work was supported by National Natural Science Foundation of China (11771312) and Major Research Plan of National Natural Science Foundation of China (91430105).
\end{document} | arXiv |
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