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\begin{document}

\begin{abstract}
	Flows are a topological model of concurrency which enables to encode the notion of refinement of observation and to understand the homological properties of branchings and mergings of execution paths. Roughly speaking, they are Grandis' $d$-spaces without an underlying topological space. They just have an underlying homotopy type. This note is twofold. First, we give a new construction of the model category structure of flows which is more conceptual thanks to Isaev's results. It avoids the use of difficult topological arguments. Secondly, we prove that this model category is left determined by adapting an argument due to Olschok. The introduction contains some speculations about what we expect to find out by localizing this minimal model category structure.
	\linebreak[4]\linebreak[4]
	Les flots sont un mod\`ele topologique de la concurrence qui permet d'encoder la notion de raffinement de l'observation et de comprendre les propri\'et\'es homologiques des branchements et des confluences des chemins d'ex\'ecution. Intuitivement, ce sont des $d$-espaces au sens de Grandis sans espace topologique sous-jacent. Ils ont seulement un type d'homotopie sous-jacent. Cette note a deux objectifs. Premi\`erement de donner une nouvelle construction de la cat\'egorie de mod\`eles des flots plus conceptuelle gr\^ace au travail d'Isaev. Cela permet d'\'eviter des arguments topologiques difficiles. Deuxi\`emement nous prouvons que cette cat\'egorie de mod\`eles est d\'etermin\'ee à gauche en adaptant un argument de Olschok. L'introduction contient quelques sp\'eculations sur ce qu'on s'attend \`a trouver en localisant cette cat\'egorie de mod\`eles minimale.
\end{abstract}

\maketitle
\tableofcontents

\section{Introduction}

\subsection{Topological models of concurrency}
There is a multitude of topological models of concurrency: flows which are the subject of this paper and which are introduced in \cite{model3}, but also $d$-spaces \cite{mg}, streams \cite{MR2545830}, inequilogical spaces \cite{equilogical}, spaces with distinguished cu\-bes \cite{distinguishedcube1} \cite{distinguishedcube2}, multipointed $d$-spaces \cite{mdtop} etc... All these mathematical devices contain the same basic examples coming from concurrency (e.g. the geometric realizations of precubical sets), a local ordering modeling the direction of time and its irreversibility, execution paths, a set or a topological space of states and a notion of homotopy between execution paths to model concurrency. Grandis' $d$-spaces give rise to a vast literature studying the directed fundamental category and the directed components of directed spaces whatever the definition we give to this notion of \textit{directed space}.

This paper belongs to the sequence of papers \cite{model3} \cite{hocont} \cite{1eme} \cite{2eme} \cite{3eme} \cite{4eme} \cite{nonexistence} \cite{exbranch}. The main feature of the model category of flows is to enable the formalization and the study of the notion of refinement of observation (the cofibrant replacement functor plays a crucial role in the formalization indeed). The model category of flows also enables the study of the homological properties of the branching areas and merging areas of execution paths in concurrent systems, in particular a long exact sequence, and their interaction with the refinement of observation, actually their invariance with respect to them. Since flows have also a labeled version (see \cite[Section~6]{ccsprecub}), they can be used for modeling the path spaces of process algebras for any synchronization algebra \cite{ccsprecub} \cite{symcub}.

\subsection{Speculative digression}

In our line of research, the objects are multipointed, i.e. they are equipped with a distinguished set of states. The set of states provided by the description of a concurrent process is not forgotten. It is exactly the same phenomenon as in the formalism of simplicial set. A simplicial set is equipped with a set of vertices which comes from the description of the space. This family of topological models of concurrency differs from other topological models of concurrency like Grandis' $d$-spaces or streams which are not multipointed.

The interest of our approach is that it is already possible to build model category structures such that the weak equivalences preserve the causal stru\-cture of a process. Indeed, all flows are fibrant. By Ken Brown's lemma, two flows are therefore weakly equivalent if and only if they can be related by a zig-zag of trivial fibrations. Every trivial fibration satisfies the right lifting property with respect to any cofibration, and therefore with respect to any reasonable notion of extension of paths. Thus two weakly equivalent flows are bisimilar in Joyal-Nielsen-Winskel's sense \cite{0856.68067} for any reasonable notion of extension of paths.

The main drawback of a lot of, and to the best of our knowledge, actually all other model categories introduced in directed homotopy is that their weak equivalences destroy the causal structure for example by identifying the directed segment up to weak equivalence with a point. Indeed, the directed segment should not be contractible (in a directed sense) in directed homotopy. To understand the reason, consider the well-known example of the Swiss Flag example (cf. Figure~\ref{ex3}). It consists of two processes concurrently executing the instructions $PA.PB.VA.VB$ and $PB.PA.VB.VA$ where $Px$ means taking the control of a shared ressource $x$ and $Vx$ means releasing it. We suppose that at most one process can take the control of a given shared ressource. There are in our example two shared ressources $A$ and $B$ which can be for example buffers to temporarily store data. The execution paths start from the bottom left corner, ends to the top right corner, and are supposed to be nondecreasing with respect to each axis of coordinates. Then the point of coordinates $(PB,PA)$ is a deadlock because it is impossible from it to reach the desired final state. As soon as the directed segment becomes weakly equivalent to a point, the deadlock $(PB,PA)$ could disappear up to weak equivalence (the picture on the right is an object having the same causal structure). It means that a weak equivalence could break the causal structure as soon as the directed segment is contractible in a directed sense and, in this case, if left properness is assumed.

\begin{figure}
	\begin{center}
		\includegraphics[width=8cm]{swiss1.pdf}
	\end{center}
	\caption{Swiss Flag example}
	\label{ex3}
\end{figure}

The weak equivalences of the model category of flows do preserve the causal structure. However, they are too restrictive. It is not even possible up to weak equivalence to replace in a flow a directed segment by a more refined one (cf. Figure~\ref{ex2}), by adding additional points in the middle of the segment in the distinguished set of states. This annoying behaviour can be overcome by adding weak equivalences by homotopical localization. One of the challenges of our line of research is precisely to understand the homotopical localization of the model category of (labeled) flows with respect to this kind of maps: they are called T-homotopy equivalences in \cite{3eme}.

\begin{figure}
	\begin{center}
		\includegraphics[width=8cm]{generalizedT.pdf}
	\end{center}
	\caption{Replacement of a directed segment by a more refined one}
	\label{ex2}
\end{figure}

The paper \cite{nonexistence} proves not only that the class of weak equivalences of this homotopical localization contains more equivalences than the dihomotopy equivalences of flows as defined in \cite{hocont}, but also that there is no hope to obtain a model category structure on flows such that the weak equivalences are exactly these dihomotopy equivalences, even if a notion of fibrant object (the homotopy continuous flows \cite[Definition~4.3]{hocont}) with the associated Whitehead theorem exists for dihomotopy equivalences \cite[Theorem~4.6]{hocont}.

By now, we only know by studying examples that the weak equivalences of this homotopical localization seem to be, in the labeled case, dihomotopy equivalences in the sense of \cite{hocont} up to a kind of bisimulation. In particular, it means that the weak equivalences of this homotopical localization likely preserve causality, but not the branching and merging homologies, and not the underlying homotopy type of a flow as defined in \cite{4eme} which is, roughly speaking, the homotopy type of the space obtained after removing the directed structure of a flow.

It is actually possible to homotopically localize the model category of labeled flows by the whole class of bisimulations in Joyal-Nielsen-Winskel's sense since this class of maps is accessible. Then another strange phenomenon occurs. We would then have to deal in the localization with weak equivalences breaking the causal structure. The latter phenomenon is explained in \cite[Theorem~12.4]{pastsim} within the combinatorial framework of Catta\-ni-Sassone higher dimensional transition systems but it can be easily adapted and generalized to many other frameworks of directed homotopy, including the one of flows.

\subsection{Purpose of this note}

The construction of the model category of flows as carried out in \cite{model3} is quite long and tricky. It makes use of rather complicated topological lemmas, in particular because colimits of flows are difficult to understand. Indeed, flows are roughly speaking \textit{small categories without identities (precategories ? pseudocategories ?) enriched over topological spaces}~\footnote{There is no known relation between the model category of flows and the model category of topologically enriched small categories of \cite{MR3094501}: the obvious adjunction is not even a Quillen adjunction; moreover the terminology of ``flow'' must not mislead the reader, it has nothing to do with a similar terminology in Morse theory.} Therefore new paths are created as soon as states are identified in a colimit, which may generate complicated modifications of the topology of the path space (e.g. see the proof of \cite[Theorem~15.2]{model3} which is not only complicated but also contains a flaw which will be fixed in a subsequent paper). The first purpose of this note is to drastically simplify this construction using \cite{Isaev}. We then explain in a second part why the model category of flows is left determined in the sense of \cite{rotho} by adapting an argument due to Marc Olschok for the model category of topological spaces. The latter fact is a new result (it was mentioned without proof in \cite[Section~12]{pastsim}). In the (complicated) quest of finding out better model categories with as much weak equivalences as possible preserving the causal structure, this result means that it is not possible in the framework of flows to remove weak equivalences without changing the set of generating cofibrations.

\subsection{Organization}

Section~\ref{section1} recalls what we need to use from Isaev's paper.  Section~\ref{section2} explains the new construction of the model category structure of flows (Theorem~\ref{simplification}). Section~\ref{section3} recalls the notion of left determined model category and proves that the category of flows is left determined (Theorem~\ref{leftdetflow}). Section~\ref{conc} makes some final comments.

\subsection{Notations}

All categories are locally small. The category of sets is denoted by $\set$. The set of maps in a category $\K$ from $X$ to $Y$ is denoted by $\K(X,Y)$.  The initial (final resp.) object, if it exists, is always denoted by $\varnothing$ ($\mathbf{1}$ resp.). The identity of an object $X$ is denoted by $\id_X$.  The composite of two maps $f:A\to B$ and $g:B\to C$ is denoted by $g.f$. A subcategory is always isomorphism-closed (replete). Let $f$ and $g$ be two maps of a category $\K$. Denote by $f\boxslash g$ when $f$ satisfies the \emph{left lifting property} (LLP) with respect to $g$, or equivalently $g$ satisfies the \emph{right lifting property} (RLP) with respect to $f$. Let $\C$ be a class of maps. Let us introduce the notations $\inj(\C) = \{g \in \K, \forall f \in \C, f\boxslash g\}$ and $\cof(\C)=\{f\mid \forall g\in \inj(\C), f\boxslash g\}$. The class of morphisms of $\K$ that are transfinite compositions of pushouts of elements of $\C$ is denoted by $\cell(\C)$. We refer to \cite{MR95j:18001} for locally presentable categories, to \cite{MR2506258} for combinatorial model categories.  We refer to \cite{MR99h:55031} and to \cite{ref_model2} for more general model categories.

\section{Isaev approach for constructing model categories}
\label{section1}

Let $\K$ be a locally presentable category. A combinatorial model category structure is characterized by its set of generating cofibrations and by its class of fibrant objects by \cite[Proposition E.1.10]{quasicat}. Therefore, for a given set of maps $I$, there exists at most one combinatorial model category structure on $\K$ such that the set of generating cofibrations is $I$ and such that all objects are fibrant. In \cite{Isaev}, several methods are expounded to obtain model category structures such that all objects are fibrant. We summarize in the next theorem what we are going to need in this note.

\bth \cite[Theorem~4.3, Proposition~4.4, Proposition~4.5 and Corollary~4.6]{Isaev} \label{Isa}
Let $\K$ be a locally presentable category. Let $I$ be a set of maps of $\K$ such that the domains of the maps of $I$ are $I$-cofibrant (i.e. belong to $\cof(I)$). Suppose that for every map $i:U\to V \in I$, the relative codiagonal map $V\sqcup_U V \to V$ factors as a composite $V\sqcup_U V \stackrel{\gamma_0\sqcup \gamma_1}\to C_U(V)\to V$ such that the left-hand map belongs to $\cof(I)$. Let $J_I = \{\gamma_0:V\to C_U(V)\mid U\to V \in I\}$. Suppose that there exists a path functor $\cocyl:\K \to \K$, i.e. an endofunctor of $\K$ equipped with two natural transformations $\tau:\id\Rightarrow\cocyl$ and $\pi:\cocyl\Rightarrow \id\p \id$ such that the composite $\pi.\tau$ is the diagonal. Moreover we suppose that the path functor satisfies the following hypotheses:
\begin{enumerate}
	\item With $\pi=(\pi_0,\pi_1)$, $\pi_0:\cocyl(X)\to X$ and $\pi_1:\cocyl(X)\to X$ have the RLP with respect to $I$.
	\item The map $\pi:\cocyl(X)\to X\p X$ has the RLP with respect to the maps of $J_I$.
\end{enumerate}
Then there exists a unique model category structure on $\K$ such that the set of generating cofibrations is $I$ and such that the set of generating trivial cofibrations is $J_I$. Moreover, all objects are fibrant.
\eth

Unlike in \cite{Isaev}, we can drop the hypothesis about the smallness of the domains and the codomains of the maps of $I$ with respect to $I$ by \cite[Proposition~1.3]{MR1780498} because the ambient category is supposed to be locally presentable . Note that every map of $J_I$ is a split monomorphism since the composite $V\sqcup_U V \to C_U(V)\to V$ is the relative codiagonal. Therefore every object is fibrant indeed.

\section{The model category of flows}
\label{section2}

\begin{nota} The category $\top$ denotes a bicomplete locally presentable cartesian closed full subcategory of the category of general topological spa\-ces containing all CW-complexes. \end{nota}

The category of $\Delta$-generated spaces, i.e. the colimits of simplices, or equivalently the colimits of the segment $[0,1]$ by \cite[Proposition~3.17]{LisDicov}, satisfies these hypotheses \cite{FR}. It is also possible to add weak separability hypotheses like this one: for every continuous map $g:\Delta^n\to X$ where $\Delta^n$ is the topological $n$-simplex with $n\geq 0$, $g(\Delta^n)$ is closed in $X$. For a tutorial about these topological spaces, see for example \cite[Section~2]{mdtop}. We take $\top$ to be equipped with the standard Quillen model category structure.

\begin{nota}
	The internal hom functor is denoted by $\ttop(-,-)$.
\end{nota}

\bd \cite{model3} A {\rm flow} $X$ consists of a topological space $\P X$ of execution paths, a discrete space $X^0$ of states, two continuous maps $s$ and $t$ from $\P X$ to $X^0$ called the source and target map respectively, and a continuous and associative map \[*:\{(x,y)\in \P X\p \P X; t(x)=s(y)\}\longrightarrow \P X\] such that $s(x*y)=s(x)$ and $t(x*y)=t(y)$.  A morphism of flows $f:X\longrightarrow Y$ consists of a set map $f^0:X^0\longrightarrow Y^0$ together with a continuous map $\P f:\P X\longrightarrow \P Y$ such that $f(s(x))=s(f(x))$, $f(t(x))=t(f(x))$ and $f(x*y)=f(x)*f(y)$. The corresponding category is denoted by $\dtop$. \ed

\begin{nota}
	For a topological space $X$, let $\glob(X)$ be the flow defined by $\glob(X)^0=\{0,1\}$ and $\P \glob(X)=X$ with $s$ and $t$ being the constant functions $s=0$ and $t=1$. The Glob mapping induces a functor from the category $\top$ of topological spaces to the category $\dtop$ of flows.
\end{nota}

We need to recall the two following easy propositions:

\bp \label{ortho3} \cite[Proposition~13.2]{model3} A morphism of flows
$f:X\longrightarrow Y$ satisfies the RLP with respect to
$\glob(U)\longrightarrow \glob(V)$ if and only if for any
$\alpha,\beta\in X^0$, $\P_{\alpha,\beta}X\longrightarrow
\P_{f(\alpha),f(\beta)}Y$ satisfies the RLP with respect to
$U\longrightarrow V$. \ep

\bp\label{ortho2} \cite[Proposition~16.2]{model3} Let $f$ be a morphism of flows. Then the
following conditions are equivalent:
\begin{enumerate}
	\item $f$ is bijective on states
	\item $f$ satisfies the RLP
	with respect to $R:\{0,1\}\longrightarrow \{0\}$ and
	$C:\varnothing\subset \{0\}$.
\end{enumerate}
\ep

We will also need this new proposition which does not seem to be proved in one of our previous papers about flows:

\bp \label{connected-colim-glob}
The globe functor $\glob:\top\to \dtop$ preserves connected colimits (i.e. colimits such that the underlying small category is connected).
\ep

Note that the connectedness hypothesis is necessary. Indeed, $V$ and $W$ being two topological spaces, the flow $\glob(V\sqcup W)$ has two states whereas the flow $\glob(V) \sqcup \glob(W)$ has four states.

\bpf Let $V$ be a topological space. Giving a map from the flow $\glob(V)$ to a flow $X$ is equivalent to choosing two states $\alpha$ and $\beta$ of $X$ (the image of the states $0$ and $1$ of $\glob(V)$) and a continuous map from $V$ to $\P_{\alpha,\beta}X$. Thus the following natural bijection of sets holds \begin{equation}
\label{truc}
\dtop(\glob(V),X)\iso \bigsqcup_{(\alpha,\beta)\in X^0\p X^0} \top(V,\P_{\alpha,\beta}X).
\end{equation}
We obtain the sequence of natural bijections ($\liminj V_i$ being a connected colimit of topological spaces)
\begin{align*}
\dtop(\glob(\liminj V_i),X) & \iso \bigsqcup_{(\alpha,\beta)\in X^0\p X^0} \top(\liminj V_i,\P_{\alpha,\beta}X) \\
&\iso \bigsqcup_{(\alpha,\beta)\in X^0\p X^0} \limproj\top(V_i,\P_{\alpha,\beta}X)\\
&\iso \limproj \bigsqcup_{(\alpha,\beta)\in X^0\p X^0} \top(V_i,\P_{\alpha,\beta}X) \\
&\iso \limproj \dtop(\glob(V_i),X) \\
&\iso \dtop(\liminj \glob(V_i),X),
\end{align*}
the first and the fourth isomorphisms by (\ref{truc}), the second and the fifth isomorphisms by definition of a (co)limit and the third isomorphism by the connectedness of the limit.
The proof is complete using the Yoneda lemma.
\epf

\begin{nota} \cite[Notation~7.6]{model3} Let $U$ be a topological space. Let $X$ be a flow. The flow $\{U,X\}_S$ is defined as follows:
	\begin{enumerate}
		\item The set of states of $\{U,X\}_S$ is $X^0$.
		\item For $\alpha,\beta\in X^0$, let $\P_{\alpha,\beta}\{U,X\}_S = \ttop(U,\P_{\alpha,\beta}X)$.
		\item For $\alpha,\beta,\gamma\in X^0$, the composition law
		\[*:\P_{\alpha,\beta}\{U,X\}_S\p
		\P_{\beta,\gamma}\{U,X\}_S\longrightarrow\P_{\alpha,\gamma}\{U,X\}_S\]
		is the composite
		\[\P_{\alpha,\beta}\{U,X\}_S\p \P_{\beta,\gamma}\{U,X\}_S\iso
		\ttop\left(U,\P_{\alpha,\beta}X\p
		\P_{\beta,\gamma}X\right)\longrightarrow
		\ttop\left(U,\P_{\alpha,\gamma}X\right)\]
		induced by the
		composition law of $X$.
	\end{enumerate}
\end{nota}

The flow $\{U,X\}_S$ is functorial with respect to $U$ and $X$ (contravariant with respect to $U$ and covariant with respect to $X$). The flow $\{\varnothing,Y\}_S$ is the flow having the same set of states as $Y$ and exactly one non-constant execution path between two points of $Y^0$. The flow $\{\{0\},X\}_S$ is canonically isomorphic to $X$ for all flows $X$. Maybe the latter assertion deserves a little explanation because it is precisely why cartesian closedness matters (the fact that $\{0\}$ is exponentiable is actually sufficient). For all topological spaces $U$, we have the natural bijections \[\top(U,\P_{\alpha,\beta}X) \iso\top(U\p \{0\},\P_{\alpha,\beta}X) \iso \top(U,\ttop(\{0\},\P_{\alpha,\beta}X)).\] Thus by Yoneda, we obtain the homeomorphism \[\P_{\alpha,\beta}X\iso\ttop(\{0\},\P_{\alpha,\beta}X)).\]

\begin{nota}  Let $n\geq 1$. Denote by $\mathbf{D}^n = \{b\in \mathbb{R}^n, |b| \leq 1\}$ the $n$-dimensional disk, and by $\mathbf{S}^{n-1}$ the $(n-1)$-dimensional sphere. By convention, let $\mathbf{D}^{0}=\{0\}$ and $\mathbf{S}^{-1}=\varnothing$. Let $I^{gl}_+=\{\glob(\mathbf{S}^{n-1}) \subset \glob(\mathbf{D}^{n})\mid n\geq 0\} \cup \{C:\varnothing \to \{0\},R:\{0,1\} \to \{0\}\}$. \end{nota}

We recall an elementary lemma about the model category $\top$ which is a straightforward consequence of the fact that the Quillen model structure is Cartesian monoidal:

\begin{lem} \label{fib}
	Let $i:U\to V$ be a cofibration of $\top$. Then for all topological spaces $X$, the map $i^*:\ttop(V,X)\to \ttop(U,X)$ is a fibration. If moreover $i$ is a weak equivalence, then the map $i^*:\ttop(V,X)\to \ttop(U,X)$ is a trivial fibration.
\end{lem}

We can now easily carry out the construction of the model category structure.

\bth \label{simplification} There exists a unique model category structure such that $I^{gl}_+$ is the set of generating cofibrations and such that all objects are fibrant. \eth

\bpf We have to check the hypotheses of Theorem~\ref{Isa}. The category $\dtop$ is locally presentable since $\top$ is locally presentable (see for example the proof of \cite[Proposition~6.11]{nonexistence}). That all $\glob(\mathbf{S}^{n-1})$ for all $n\geq 0$ are $I^{gl}_+$-cofibrant comes from the fact that the $(n-1)$-sphere is cofibrant in $\top$. We can factor the relative codiagonal map $\mathbf{D}^{n} \sqcup_{\mathbf{S}^{n-1}}\mathbf{D}^{n} \to \mathbf{D}^{n}$ as a composite $\mathbf{D}^{n} \sqcup_{\mathbf{S}^{n-1}}\mathbf{D}^{n} \subset \mathbf{D}^{n+1} \to \mathbf{D}^{n}$ for all $n\geq 0$. Thus for $U\to V$ being one of the maps $\glob(\mathbf{S}^{n-1}) \subset \glob(\mathbf{D}^{n})$ for $n\geq 0$, we set $C_U(V) = \glob(\mathbf{D}^{n+1})$. We have a pushout diagram of topological spaces
\[\xymatrix
{
	\mathbf{S}^{n}\fr{}\fd{} & \mathbf{D}^{n} \sqcup_{\mathbf{S}^{n-1}}\mathbf{D}^{n}\fd{}\\
	\mathbf{D}^{n+1}\fr{} & \cocartesien\mathbf{D}^{n+1}
}\]
which gives rise to the pushout diagram of flows
\[\xymatrix
{
	\glob(\mathbf{S}^{n})\fd{}\fr{} & \glob(\mathbf{D}^{n}) \sqcup_{\glob(\mathbf{S}^{n-1})}\glob(\mathbf{D}^{n})\fd{}\\
	\glob(\mathbf{D}^{n+1})\fr{} & \cocartesien \glob(\mathbf{D}^{n+1})
}\]
for all $n\geq 0$ by Proposition~\ref{connected-colim-glob}. This implies that for $U\to V$ being one of the maps $\glob(\mathbf{S}^{n-1}) \subset \glob(\mathbf{D}^{n})$ for $n\geq 0$, the map $V\sqcup_U V\to C_U(V)$ belongs to $\cell(I^{gl}_+)$. The map $C:\varnothing\to\{0\}$ gives rise to the relative codiagonal map $\{0\}\sqcup \{0\}\to \{0\}$. Thus we set $C_\varnothing(\{0\})=\{0\}$. In this case, the map $V\sqcup_U V\to C_U(V)$ is $R:\{0,1\}\to\{0\}$ which belongs to $\cell(I^{gl}_+)$. The map $R:\{0,1\}\to \{0\}$ gives rise to the relative codiagonal map $\id_{\{0\}}$. Thus we set $C_{\{0,1\}}(\{0\})=\{0\}$. In this case, the map $V\sqcup_U V\to C_U(V)$ is $\id_{\{0\}}$ which belongs to $\cell(I^{gl}_+)$. The set of generating trivial cofibrations will be therefore the set of maps  $\glob(\mathbf{D}^{n}) \subset \glob(\mathbf{D}^{n+1})$ for $n\geq 0$. Let $\cocyl(X)=\{[0,1],X\}_S$ for all flows $X$. The composite map $\{0,1\}\subset [0,1] \to \{0\}$ yields a natural composite map of flows $X\iso \{\{0\},X\}_S \to \cocyl(X) \to \{\{0,1\},X\}_S$ which is constant on states and which gives rise to the composite continuous map $\P_{\alpha,\beta}X \to \ttop([0,1],\P_{\alpha,\beta}X) \to \P_{\alpha,\beta}X \p\P_{\alpha,\beta}X$ on the spaces of paths for all $(\alpha,\beta)\in X^0\p X^0$. We obtain a natural composite map of flows $X\stackrel{\tau}\longrightarrow \cocyl(X) \stackrel{\pi}\longrightarrow X\p X$ since the set of states of $X\p X$ is $X^0 \p X^0$ and the space of paths from $(\alpha,\alpha')$ to $(\beta,\beta')$ is $\P_{\alpha,\beta}X \p \P_{\alpha',\beta'}X$ by \cite[Theorem~4.17]{model3}. We have obtained a path object in the sense of Theorem~\ref{Isa}. Since the maps $\pi_0$ and $\pi_1$ are bijective on states, they satisfy the RLP with respect to $\{C:\varnothing \to \{0\},R:\{0,1\} \to \{0\}\}$ by Proposition~\ref{ortho2}. By Proposition~\ref{ortho3}, the maps $\pi_0$ and $\pi_1$ satisfy the RLP with respect to $\glob(\mathbf{S}^{n-1}) \subset \glob(\mathbf{D}^{n})$ for $n\geq 0$ if and only if the evaluation maps $\ttop([0,1],\P_{\alpha,\beta}X) \rightrightarrows \P_{\alpha,\beta}X$  on $0$ and $1$ satisfy the RLP with respect to the inclusion $\mathbf{S}^{n-1}\subset \mathbf{D}^{n}$ for $n\geq 0$ and for all $(\alpha,\beta)\in X^0\p X^0$, i.e. if and only if the evaluation maps $\ttop([0,1],\P_{\alpha,\beta}X) \rightrightarrows \P_{\alpha,\beta}X$ are trivial fibrations for all $(\alpha,\beta)\in X^0\p X^0$. The latter fact is a consequence of Lemma~\ref{fib} and from the fact that the inclusions $\{0\}\subset [0,1]$ and $\{1\}\subset [0,1]$ are trivial cofibrations of $\top$. Finally we have to check that the map $\pi:\cocyl(X)\to X\p X$ satisfies the RLP with respect to the maps $\glob(\mathbf{D}^{n}) \subset \glob(\mathbf{D}^{n+1})$ for $n\geq 0$. By Proposition~\ref{ortho3} again, it suffices to prove that the map $\ttop([0,1],\P_{\alpha,\beta} X) \to \ttop(\{0,1\},\P_{\alpha,\beta} X) = \P_{\alpha,\beta}X\p \P_{\alpha,\beta}X$ is a fibration of topological spaces for all $(\alpha,\beta)\in X^0\p X^0$. By Lemma~\ref{fib} again, this comes from the fact that the inclusion $\{0,1\}\subset [0,1]$ is a cofibration of $\top$.
\epf

This model category structure coincides with the one of \cite{model3}.

\section{Left determinedness of the model category of flows}
\label{section3}

Let us now recall the definition of a left determined model category:

\bd \label{localizer}
Let $I$ be a set of maps of a locally presentable category $\K$. A class of maps $\W$ is a {\rm localizer (with respect to $I$)} or an {\rm $I$-localizer} if $\W$ satisfies:
\begin{itemize}
	\item Every map satisfying the RLP with respect to the maps of $I$ belongs to $\W$.
	\item $\W$ is closed under retract and satisfies the $2$-out-of-$3$ property.
	\item The class of maps $\cof(I)\cap \W$ is closed under pushout and transfinite composition.
\end{itemize}
The class of all maps is an $I$-localizer. The class of $I$-localizers is closed under arbitrarily large intersection. Therefore there exists a smallest $I$-localizer denoted by $\W_I$.
\ed

\bd \cite{rotho} A combinatorial model category $\K$ with the set of generating cofibrations $I$ is {\rm left determined} if the class of weak equivalences is $\W_I$. \ed

Consider a combinatorial model category $\K$ such that all objects are fibrant with a class of weak equivalences $\W$ and a set of generating cofibrations $I$. The localizer $\W_I$ could be strictly smaller than $\W$. If $\W_I$ is the class of weak equivalences of a model category structure on $\K$, then all objects of this model category structure are fibrant, and therefore $\W=\W_I$. To the best of our knowledge, we can only say, using \cite[Theorem~2.2]{rotho}, that every combinatorial model category such that all objects are fibrant is left determined if we assume Vop\v{e}nka's principle. Since in any model category, two fibrant objects are weakly equivalent if and only if they are related by a span of trivial fibrations, and since all trivial fibrations belong to the smallest localizer, it is also true that there is an equivalence of categories $\K[\W_I^{-1}]\simeq \K[\W^{-1}]$ between the categorical localizations of $\K$ with respect to $\W_I$ and $\W$ if all objects of the combinatorial model category $\K$ are fibrant. Note that in \cite{Isaev}, a localizer is just the class of weak equivalences of a model category structure. In the latter sense, a model category of fibrant objects has a minimal localizer indeed.

In our case, it is possible to conclude that the model category is left determined without assuming Vop\v{e}nka's principle by adapting a technique we learned from Marc Olschok for the model category of topological spaces.

\bth \label{leftdetflow} The model category of flows is left determined. \eth

\bpf Let $f:X\to Y$ be a weak equivalence of flows. Then $f$ factors as a composite $f=f_2.f_1$ where $f_1$ is a trivial cofibration, i.e.  $f_1\in \cof(\{\glob(\mathbf{D}^{n}) \subset \glob(\mathbf{D}^{n+1})\mid n\geq 0\})$ and where $f_2$ is a trivial fibration. In particular $f_2$ satisfies the RLP with respect to $C:\varnothing\to \{0\}$ and $R:\{0,1\}\to \{0\}$. Thus $f_2$ is bijective on states by Proposition~\ref{ortho2}. The functor $X\mapsto X^0$ from $\dtop$ to $\set$ is colimit-preserving since it has a right adjoint (the functor taking a set $S$ to the flow with the set of states $S$ and exactly one path between each pair of states). Therefore $f_1$ is bijective on states since the maps $\glob(\mathbf{D}^{n}) \subset \glob(\mathbf{D}^{n+1})$ for all $n\geq 0$ are bijective on states. We deduce that $f$ is bijective on states. Consider the commutative diagram of flows
\[
\xymatrix
{
	X \fR{\exists !\overline{f}} \fD{f} && N_f \cartesien \fR{p'} \fD{f'} && X \fD{f} \\
	&& && \\
	Y \fR{\tau} && \{[0,1],Y\}_S \fR{\pi_0} \fD{\pi_1} && Y \\
	&& && \\
	&& Y &&
}
\]
where the existence of $\overline{f}$ comes from the universal property of the pullback. All arrows are bijective on states. Using the fact that the functor $\P:\dtop\to\top$ is limit-preserving by \cite[Theorem~4.17]{model3}, one obtains the commutative diagram  of topological spaces
\[
\xymatrix
{
	\P X \fR{\P \overline{f}} \fD{\P f} && \P N_f \cartesien \fR{\P p'} \fD{\P f'} && \P X \fD{\P f} \\
	&& && \\
	\P Y \fR{\P\tau} && \ttop([0,1],\P Y) \fR{\P(\pi_0)} \fD{\P(\pi_1)} && \P Y \\
	&& && \\
	&& \P Y &&
}
\]
By \cite[Proposition~4.64]{MR1867354}, the map $\P(\pi_1).\P(f')$ is a Hurewicz fibration, and therefore a fibration of the model category of $\top$. By Proposition~\ref{ortho3}, $\pi_1.f'$ satisfies the RLP with respect to all trivial cofibrations of flows, i.e. $\pi_1.f'$ is a fibration of flows. The maps $\P(\pi_0)$ and $\P(\pi_1)$ are trivial fibrations of the model category of $\top$ by Lemma~\ref{fib}.  By Proposition~\ref{ortho3} and Proposition~\ref{ortho2}, $\pi_0$ and $\pi_1$ satisfy the RLP with respect to all cofibrations of $\dtop$, i.e. $\pi_0$ and $\pi_1$ are trivial fibrations of flows. Thus $p'$ is a trivial fibration of $\dtop$ since it is a pullback of a trivial fibration. Since $f$ is a weak equivalence of flows by hypothesis, we deduce by the $2$-out-of-$3$ property that $f'$ is a weak equivalence of $\dtop$. Thus $\pi_1.f'$ is a trivial fibration of flows as well. We have $p'.\overline{f}=\id_X$. Since $p'$ is a trivial fibration, it belongs to the smallest localizer. Therefore by the $2$-out-of-$3$ property, $\overline{f}$ belongs to the smallest localizer. Since $\pi_1.f'$ is a trivial fibration, it belongs to the smallest localizer as well. Since $\pi_1.f'.\overline{f} = \pi_1.\tau.f=f$, we deduce that $f$ belongs to the smallest localizer.
\epf

\section{Concluding remarks}
\label{conc}

The hypothesis that $\top$ is locally presentable can be removed. Theorem~\ref{simplification} and Theorem~\ref{leftdetflow} hold by working in any bicomplete cartesian closed full subcategory of the general category of topological spaces containing all CW-complexes. But then, we have to check that all domains and all codomains of the maps of $I_{gl}^+$ are small relative to $\cell(I_{gl}^+)$. This is done in \cite[Section~11]{model3} and there is no known way to avoid the use of some difficult topological arguments. However, the model category of flows is left proper but not cellular because of the presence of $R:\{0,1\}\to \{0\}$ in the generating cofibrations. So outside the framework of locally presentable categories, we have no tools to prove the existence of any homotopical localization and to study the homotopical localization of $\dtop$ with respect to the refinement of observation.


TITLE: Matrix Exponent
QUESTION [3 upvotes]: I know that the exponential function for an $n\times n$ non-degenerate square matrix $A$ is:
$$e^{At}=S \operatorname{diag}(e^{\lambda_1t},\dots,e^{\lambda_nt}) \space S^{-1}, $$
where $S$ is the eigenvector matrix and $\lambda_i$ are its eigenvalues.
What is $e^{A}$? Is it
$$e^{A}=S \operatorname{diag}(e^{\lambda_1},\dots,e^{\lambda_n}) \space S^{-1}?$$
Also, what is $\cos(A)$?

REPLY [4 votes]: With the greatest possible generality, if $A$ is a square matrix and $p(x) = \sum_{k=0}^n a_k x^k$ is a polynomial, then you can define
$$
 p(A) := \sum_{k=0}^n a_k A^k, \quad A^0 := I.
$$
More generally, if $f(x)$ is an analytic function whose Taylor series $f(x) = \sum_{k=0}^\infty a_k x^k$ has radius of convergence $R > \|A\|$, where $\|A\|$ denotes the operator norm of $A$, then you can define
$$
 p(A) := \sum_{k=0}^\infty a_k A^k,
$$
with convergence in the operator norm, and hence, in particular, entry-wise convergence. Thus, since
$$
 e^x = \sum_{k=0}^\infty \frac{1}{k!} x^k, \quad \cos x = \sum_{k=0}^\infty \frac{(-1)^k}{(2k)!}x^{2k}
$$
with infinite radii of convergence, you can, in complete generality, write
$$
 e^A := \sum_{k=0}^\infty \frac{1}{k!}A^k, \quad \cos A = \sum_{k=0}^\infty \frac{(-1)^k}{(2k)!}A^{2k}.
$$
Now, something very nice happens when $A$ is diagonalisable, i.e., when
$$
 A = S \operatorname{diag}(\lambda_1,\dotsc,\lambda_n)S^{-1}
$$
for $S$ invertible and $\lambda_k$ the eigenvalues of $A$. Then, one can check—and you do need to check this on partial sums—that for $f(x) = \sum_{k=0}^\infty a_k x^k$ analytic with sufficiently large radius of convergence,
$$
 f(A) = \sum_{k=0}^\infty a_k A^k = \sum_{k=0}^\infty a_k (S \operatorname{diag}(\lambda_1,\dotsc,\lambda_n)S^{-1})^k\\ = \sum_{k=0}^\infty a_k S \operatorname{diag}(\lambda_1^k,\dotsc,\lambda_n^k)S^{-1}\\ = S \operatorname{diag}\left(\sum_{k=0}^\infty a_k \lambda_1^k,\dotsc, \sum_{k=0}^\infty a_k \lambda_n^k\right) S^{-1}\\ = S\operatorname{diag}(f(\lambda_1),\dotsc,f(\lambda_n))S^{-1}.
$$
Indeed, this motivates, in particular, the definition of $f(A)$ for $f(x)$ arbitrary as
$$
 f(A) := S\operatorname{diag}(f(\lambda_1),\dotsc,f(\lambda_n))S^{-1},
$$
which turns out to be independent of the choice of diagonalisation—this is a baby example of what functional analysts call a functional calculus. Hence, in your case,
$$
 e^A = S\operatorname{diag}(e^{\lambda_1},\dotsc,e^{\lambda_n})S^{-1}, \quad \cos A = S \operatorname{diag}(\cos \lambda_1,\dotsc,\cos \lambda_n)S^{-1}.
$$


text:
\begin{document}
\maketitle

\section{Introduction}
Aneta Stefanovska expressed a vision ``to build a self-consistent theory of non-autonom\-ous oscillators" (June 2014).  In this direction she introduced the class of ``chronotaxic" systems \cite{SCS}, defined as ``oscillatory systems with time-varying, but stable, amplitudes and frequencies''.

This chapter presents a view of a non-autonomous oscillator as a mapping from input functions of time to a circle of possible solutions (state functions of time).  It indicates how this view encompasses chronotaxic systems and enables one, at least conceptually, to understand the extent of synchronisation in networks of oscillators, whether autonomous or not.  For the latter a hierarchical aggregation scheme is introduced.

The approach is based on the theory of normal hyperbolicity \cite{F,HPS}.  This theory is the mathematical expression of Haken's slaving principle \cite{Ha}, the idea that some variables for a dynamical system might contract relatively rapidly onto some invariant submanifold in the state space, and then it suffices to study the dynamics on the submanifold.  Two key results of normal hyperbolicity theory are:~(i) conditions guaranteeing existence of such a submanifold, and (ii) smooth persistence of normally hyperbolic (NH) submanifolds as parameters are varied smoothly.  It was developed before Haken's slaving principle and deserves to be better known in the physics community.  It is a generalisation of centre manifold theory, which is the main mathematical tool Haken used, but has much wider scope.  An obstacle is that it demands considerable technical expertise in mathematical analysis.  Yet the obstacles are genuine:~it turns out that NH submanifolds are differentiable some number $r$ times, depending on the ratio between normal and tangential contraction rates, but typically not more than $r$ times.  This is important to recognise, as there is a tendency in physics to consider such functions as pathologies (though physicists do understand that there can be fractal functions).

It is a project on which I have been working for many years, notably with PhD student Stephen Gin (2006--13).  It was prompted initially by Mohammad Ghaffari Saadat in 2003, who had formulated a limit-cycle model for a bipedal robot walking down a slope \cite{TGN} and asked me how much non-uniformity of slope it could cope with.  I proposed to tackle this problem by fitting it into the framework of the non-autonomous version of the theory of NH submanifolds, where the result of a not too large forcing function on an oscillator is a circle of possible trajectories.   Gin and I attempted to develop good versions of the proofs of normal hyperbolicity results to produce realistic conditions guaranteeing the outcome \cite{G}.  Our approach is still incomplete, but I present here the key ideas.

In the world of conservative dynamics, an oscillator is considered to be a Hamiltonian system with an elliptic equilibrium point; this view has fundamental importance but is not the appropriate one for present purposes.

Outside the world of conservative dynamics, an oscillator is usually considered to be an autonomous dynamical system with an attracting periodic orbit.  The concept has been extended to cater for chaotic oscillators, but I will postpone treating that extension until near the end of this chapter.

This concept of oscillator as a system with an attracting limit-cycle, however, fails to include the many situations where it is subject to time-dependent forcing.  Also, in a network of oscillators, each is subject to input from others, in general time-dependent, so even if the network is autonomous it is useful to consider time-dependent forcing on each of its oscillators.

So I propose a view of an oscillator as a mapping from input functions $f$ of time to a circle's worth of solutions for its state $x$ as a function of time.  Each input function $f$ (possibly with more than one component) causes a response $x_\theta$ (a function of time) with a phase $\theta \in S^1$ labelling the different possible responses.
This view is justified by the theory of normal hyperbolicity, at least for not too strong forcing.  It is also my interpretation of chronotaxic systems.

The idea is to consider a non-autonomous system $\dot{x} = v(x,t)$ on a state space $X$ as an autonomous system in the extended state space $X \times \R$, with the real line $\R$ representing time.  The dynamics has the form
\begin{eqnarray}
\dot{x} &=& v(x,s) \label{eq:sys}\\
\dot{s} &=& 1. \nonumber
\end{eqnarray}
First suppose the vector field $v = v_0$ is independent of $s$ and $\dot{x}=v_0(x)$ has a limit cycle $\gamma$ (in the strong sense of a periodic orbit with no Floquet multipliers\footnote{The Floquet multipliers of a periodic orbit are the eigenvalues of the derivative of the return map to a transverse section.} on the unit circle).  The most relevant case for applications might be the attracting case (all Floquet multipliers inside the unit circle), but one can allow the more general situation.
Then in $X\times \R$, the extended system (\ref{eq:sys}) has an extended verson of $\gamma$, namely an invariant cylinder $\gamma \times \R$.  The trajectories form helices on the cylinder, representing the same periodic solution but shifted in $s$.

This cylinder is an example of a NH submanifold.  In general, a {\em NH submanifold} for a $C^1$ dynamical system is an invariant $C^1$ submanifold for which the linearised normal dynamics decomposes into components which contract exponentially in forward or backward time respectively, and faster than the linearised tangential dynamics.  Note that the use of the word ``normal'' might suggest perpendicular, but actually, a normal vector to a submanifold is defined to be an equivalence class of vectors at a point modulo vectors tangent to the submanifold at that point.
In the above case, the linearised tangential dynamics neither contracts nor expands on average, because the phase difference between any pair of the helices remains constant.  The linearised normal dynamics decomposes into exponentially contracting components in forward and backward time, corresponding to the Floquet multipliers inside and outside the unit circle, respectively.

Now allow $v$ to depend weakly on $s$.  The key result for NH submanifolds is that they persist under $C^1$-small  perturbation.  Thus the perturbed system has a $C^1$-nearby invariant cylinder, no longer in general of product form but diffeomorphic to $S^1\times\R$.  Furthermore, the vector field on it is close to that on the unperturbed cylinder, and the normal dynamics is close to that for the unperturbed case.  The solutions on the perturbed cylinder are not in general just a family of periodic solutions differing by phase.   In particular, there may be solutions on the cylinder to which all nearby ones converge in forward time.  There may also be solutions to which all nearby ones converge in backward time.  Or neither may happen.  In any case, there is a circle's worth of solutions on the cylinder, which one could label by the intersections of the cylinder with $s=0$ for example.

In particular, if $v(x,t) = v_0(x) + f(t)$ then the forcing function $f$ produces a circle's worth of state functions $x$ of time on the cylinder.  In general a forcing function $f$ should be allowed to depend on the state $x$ too, so $v=v_0(x)+f(x,t)$, and by normal hyperbolicity theory, the same conclusion holds.

As an illustration, consider a model of a quasiperiodically forced limit-cycle oscillator from \cite{CS}:
\begin{eqnarray}
\dot{x}&=&-qx-\omega y \\
\dot{y}&=&\omega x - q y + \gamma f(t), \nonumber
\end{eqnarray}
with $q= \alpha(\sqrt{x^2+y^2}-a)$, $f(t) = \sin 2\pi t + \sin 4t$, $\alpha, a >0, \gamma \ge 0$ (more natural would be $q=\alpha(x^2+y^2-a^2)$ because it makes the dynamics smooth at the origin, but the interest is in the behaviour for $r=\sqrt{x^2+y^2}$ near $a$).
In polar coordinates $(r,\theta)$ and extended state-space, this is
\begin{eqnarray}
\dot{r}&=&-\alpha r(r-a) + \gamma f(s) \sin \theta \\
\dot{\theta}&=& \omega-\frac{\gamma}{r}f(s)\cos\theta \nonumber \\
\dot{s}&=&1. \nonumber
\end{eqnarray}
For $\gamma=0$ there is an invariant cylinder $r=a$.  It attracts exponentially with exponent $-\alpha a$ and the motion on the cylinder is $\dot{\theta}=\omega$, $\dot{s}=1$, which has Lyapunov exponents 0.  So the cylinder is NH and persists to a deformed invariant cylinder for small enough $\gamma$.  A rough estimate of the range of $\gamma$ for which persistence is guaranteed is given by the range for which tangential contraction is weaker than normal contraction on the unperturbed cylinder.  The normal contraction rate (onto the unperturbed cylinder) is still $\alpha a$.  The tangential contraction (or expansion if negative) $-\frac{\partial \dot{\theta}}{\partial \theta} = -\frac{\gamma}{r}f(s)\sin\theta$.  This is smaller than $\alpha a$ for all $s, \theta$ iff $2\gamma < \alpha a^2$.  Thus one can expect the NH cylinder to persist for $\gamma$ up to something of the order of $\alpha a^2/2$.

When $\gamma$ exceeds $\alpha a^2/2$ one can not expect the invariant cylinder to persist.  It is shown numerically in \cite{CS} that the cylinder is replaced by a (non-autonomous) chaotic attractor with one unstable Lyapunov exponent (coming from the $s,\theta$ for which the tangential dynamics is expanding).  For a class of examples where a NH submanifold (in fact two 2-tori) can be proved to break up, see \cite{BaM1}.  In this chapter, however, I will concentrate on regimes of weak enough coupling that NH submanifolds persist.

As an aside, this view of an oscillator fits in Willems' ``behavioural approach'' to systems and control \cite{W}.  His view was that the description of a dynamical system should be considered to be the restrictions on the set of possible functions of time for all variables.  Normal hyperbolicity strikes me a key tool for delivering his approach.
On the other hand, he also proposed that one should go beyond the idealisation of inputs and outputs by treating all coupling as two-way, a line that I shall not follow consistently.

In this chapter I will explain how this view of an oscillator illuminates the phenomena of phase-locking, synchronisation and chimera \cite{AS}, allows to extend the concept of coupling, and allows a hierarchical reduction treatment of synchronisation in networks of oscillators.  I will extend the results to allow excitable oscillators and chaotic oscillators.  I will outline how the theory of normal hyperbolicity underlies the results.  There is a huge literature on synchronisation, e.g.~\cite{PRK}, and much of what I will say will be familiar but the important emphasis here is on synchronisation in aperiodically forced systems, which has been treated much less.

Perhaps this direction is not what Aneta had in mind, but I believe it provides a self-consistent theory for non-autonomous oscillators and I hope that it will be useful.

\section{Phase-locking}
\label{sec:pl}
It is well-known that an oscillator may phase-lock to some features of its inputs.  Indeed, this is the principle of phase-locked loops in electronic engineering \cite{Br} and of synchronous generators and motors in AC electrical networks.

My definition of phase-locking of an oscillator to forcing is that the NH cylinder (assumed attracting) has an attracting trajectory on it and the initial condition is in its basin of attraction.

Any discussion of attractors for non-autonomous systems requires care because the dynamics is unbounded in the time direction of extended state-space, so there are inequivalent choices of neighbourhoods of a trajectory.  For example, for the 2D system $\dot{x}=x,\dot{s}=1$, any trajectory has a neighbourhood of attraction, despite looking unstable, e.g.~for the solution $x=0$ just take neighbourhood of the form $|x|<\eps e^{2s}$.
So I make precise here that by ``attracting trajectory'' I mean the case with zero unstable space of a uniformly hyperbolic trajectory in the non-autonomous sense.  To explain what this means would take some space, so I refer the reader to \cite{BiM} (with my PhD student Zahir Bishnani), but the important feature is to choose a notion of distance in extended state-space that is uniform in time (so that one does not allow neighbourhoods like that in the above example).  There might be a bundle of trajectories which all converge together in forward time, but in general there is only one trajectory in the bundle that has a uniform-in-time neighbourhood of attraction.  It is a pullback attractor (for this concept, see the contribution by Kloeden in this volume).  My concept of attracting trajectory is distinct, however, from that of pullback attractor, because it can also occur that a pullback attractor is not uniformly hyperbolic (it may be repelling after some time).

An alternative way to describe phase-locking is that the oscillator is synchronised to its inputs.  I use ``synchronise" in a weak sense: that to a given input function of time there is a locally unique forwards asymptotic solution (the strong sense applies to systems of identical oscillators with a symmetry of the coupling that maps any oscillator to any other, and consists in all oscillators doing the same; for an example, see \cite{YM}).  Note that a forced oscillator may have more than one such attracting trajectory; this would allow different synchronisations to the same input.

This is in contrast to non-synchronisation, where there is a circle's worth of solutions that do not converge asymptotically to a discrete subset.  The strongest version of non-synchronisation is when there is a time-dependent choice of $C^1$ coordinate $\phi$ around the cylinder, replacing an initial coordinate $\theta$, such that $\dot{\phi}=\omega(t)$, a positive function of $t$ only, and $\frac{\partial \phi}{\partial \theta}$ and its inverse are bounded.  Then with a new time $\tau$ defined by $d\tau/dt=\omega(t)$, we obtain $d\phi/d\tau = 1$.  It would be interesting to investigate the probability of this case with respect to a distribution of oscillator frequencies for given weak forcing, perhaps obtaining a sort of non-autonomous KAM result\footnote{The original KAM theory gives a set of invariant tori for near-integrable Hamiltonian systems, the measure of whose complement goes to zero as the perturbation from integrability goes to zero.}, extending the theory of reducibility of cocycles (see \cite{DS} for an early example).

The main conclusion of this section is that synchronisation of an oscillator to its inputs is dimension-reduction.  In particular, if there is no immediate feedback from the oscillator to any of its inputs, then one could delete that oscillator, replacing its outputs by some modifications of the outputs from its inputs.

\section{Synchronisation of two oscillators}
\label{sec:2osc}
Let us start with two autonomous oscillators $x_i = v_i(x_i)$, $i=1,2$, meaning each has a limit cycle $\gamma_i$, and couple them in the standard sense of a modification to the vector field of the product system, depending on the state of each but not too strongly, so
\begin{equation}
\dot{x}_i = v_i(x_i) + g_i(x_1,x_2),
\end{equation}
with $g_i$ $C^1$-small.
Then the product system has a NH 2-torus, being a small perturbation of $\gamma_1 \times \gamma_2$.

If the difference of the frequencies of the uncoupled limit cycles is smaller in a suitable dimensionless sense than the coupling then the NH torus has an attracting limit cycle on it, which makes one turn in the $\gamma_2$ direction for each turn in the $\gamma_1$ direction.  I say the two oscillators have gone into $1:1$ synchronisation.  Recall Huygens' clocks.  The torus may have more than one attracting limit cycle on it, in which case several synchronised solutions are possible.  It may also have unstable limit cycles on it.

Similarly, if the frequencies are close to being in (coprime) integer ratio $m:n$ then coupling might produce an attracting $m:n$ limit cycle on the NH torus, which makes $m$ revolutions in the $\gamma_1$ direction and $n$ in the $\gamma_2$ direction per period.  On the other hand, for weak coupling and smooth enough dynamics, the non-synchronised situation occurs with high probability.  More precisely, if one adds a free parameter varying the unperturbed frequency ratio, then KAM theory gives a set of parameter values of nearly full measure for which the dynamics is conjugate to a constant vector field on a 2-torus with irrational frequency ratio (e.g.~\cite{LD} for a version by my PhD student Jo\~ao Lopes Dias).  Thus synchronisation does not always result.

Now consider the non-autonomous situation, where one or both of the oscillators is subject to external forcing.  If the forcing is not too strong then the resulting system has a NH submanifold in extended state space, diffeomorphic to $\gamma_1 \times \gamma_2 \times \R$, which I call a torus-cylinder.  More generally, for any manifold $M$ I define an $M$-cylinder to be a manifold diffeomorphic to $M\times\R$.  Thus an ordinary cylinder can be called a circle-cylinder.
If the unperturbed frequencies are close to integer ratio $m:n$ then the NH submanifold might contain a NH attracting submanifold diffeomorphic to a circle cross time, being a perturbation of the product of a $m:n$ synchronised limit cycle for the autonomous system and time.  In this situation the non-autonomous pair of oscillators can be replaced by a single one.

So again, synchronisation of two oscillators is a dimension-reduction.

\section{What is coupling?}
In the previous section I used the standard dynamical systems notion for coupling as a perturbation of the product of two vector fields.  One might want, however, to allow more general forms of coupling, for example incorporating time-delays or coupling via an intermediate dynamical system.  Furthermore, suppose one achieved a dimension-reduction as in section~\ref{sec:pl} or \ref{sec:2osc} and then wants to consider how the new effective oscillator is coupled to others that originally were coupled to one or both of the pair of oscillators.  This is no longer describable as a standard perturbation of the product of vector fields.

So I generalise the notion of coupling of two non-autonomous oscillators.  As already defined, a non-autonomous oscillator is a non-autonomous system with NH cylinder on which the dynamics can be described by one phase $\theta$ with $\dot{\theta} = f(\theta,t)$.  A coupling of two non-autonomous oscillators is a non-autonomous system with a NH torus-cylinder on which the dynamics can be described by two phases $\theta = (\theta_1,\theta_2)$ with $\dot{\theta}_i=\tilde{f}_i(\theta,t)$ and $\tilde{f}_i(\theta,t)$ close to $f_i(\theta_i,t)$ for some $f_i$.

Then the dynamics on the NH torus-cylinder may contain a NH attracting circle-cylinder, as in the more restricted case of the previous section.  If the trajectory is in its basin of attraction, I say the two oscillators synchronise.

\section{Synchronisation of $N$ oscillators}
Not too strong coupling of $N$ non-autonomous oscillators produces a NH $N$-torus-cylinder.  The dynamics on it might contain an attracting NH $d$-torus-cylinder for some $d<N$.  If $d=1$ the whole group is synchronised and can be replaced by a single effective non-autonomous oscillator.  If $d=0$ the whole group is phase-locked to its inputs and can be eliminated.

Once again, synchronisation, whether partial or complete, means dimension-reduction.

\section{Hierarchical aggregation}
In a network of oscillators, the above dimension-reductions can in principle be iterated.  First one identifies groups of oscillators which synchronise or phase-lock to their inputs.  One reduces to a new network of effective oscillators.  Then one repeats, if possible.  The end result is a decomposition into synchronised clusters.

Although I did not find out about his work until after I'd proposed this, it is a direct example of Willems' ``tearing, zooming, linking'' approach \cite{W}.

One should note that the end result is not necessarily complete synchronisation.  Indeed, it could well be a chimera \cite{AS}, meaning a system in which some of the oscillators are synchronised but others behave chaotically.  The chaotic ones force the synchronised ones and the synchronised ones force the chaotic ones, but our approach of non-autonomous oscillators caters for both of these.  There is now a huge literature on chimera.  To me the phenomenon was not a surprise because it fits in my framework, but without the framework it can admittedly be considered surprising.

\section{Normal hyperbolicity estimates}
To achieve the above dimension-reductions requires good normal hyperbolicity estimates, i.e.~results guaranteeing existence of NH submanifolds.

The easiest case, namely, 1D submanifolds, which are just uniformly hyperbolic trajectories of non-autonomous systems, was already treated in \cite{BiM} (incidentally, it was formulated with attracting trajectories in mind, but another application would be to the unstable trajectories of geophysical flows that form boundaries between trajectories of different classes, e.g.~\cite{FPET}).  So that takes care of the case of phase-locking.

Higher-dimensional NH submanifolds, however, require more theory.  The classic references are \cite{F,HPS}.  They are not particularly well adapted to producing practical estimates.  Thus I set Stephen Gin onto developing a better way.  His PhD thesis \cite{G} gives the outcome, but it is not a complete treatment.  So here, I sketch an approach to NH estimates that I believe will be useful.  It is in the classic dynamical systems setting of a vector field on the product of state space and time, but hopefully could be extended to take care of the more general forms of coupling that I have described here.

I restrict attention to submanifolds that are torus-cylinders, but of arbitrary dimension $m+1$.
So suppose
\begin{eqnarray}
\dot{\theta} &=& \Theta(\theta,r,t) \\
\dot{r} &=& R(\theta,r,t), \nonumber
\end{eqnarray}
for $\theta \in \T^m$, $r \in U$, a neighbourhood of $0\in \R^p$.
I suppose that the product $|R_\theta| |\Theta_r|$ is small (where subscript denotes derivatives), the $r$-dynamics is hyperbolic, and the Green function for linearised normal dynamics decays faster than any contraction that may occur in $\theta$-dynamics.

Given a Lipschitz graph $r=\rho(\theta,t)$, a candidate for an invariant submanifold, construct a new one, $T\rho$, by the following steps:
\begin{enumerate}
\item For all $(\theta_0,t_0)$, let $\theta()$ be the trajectory of $\dot{\theta}(t)=\Theta(\theta,\rho(\theta,t),t)$ from $\theta(t_0)=\theta_0$.
\item Solve $\dot{r}(t) = R(\theta(t),r(t),t)$ for the unique function $r()$ such that $r(t)$ is near $\rho(\theta(t),t)$ for all $t$.
\item Set $(T\rho)(\theta_0,t_0) = r(t_0)$.
\end{enumerate}
To achieve the second step, I assume that $L:C^1(\R,\R^{p})\to C^0(\R,\R^{p})$ defined by
$$L[x](t) = \dot{x}(t) - R_r(\theta(t),r(t),t) x(t)$$
on infinitesimal displacements $x$ in $r$ has bounded inverse.  This is equivalent to the first part of the NH condition, namely a splitting of the normal bundle into exponentially contracting backwards and forwards subspaces.

Having thus constructed the ``graph transform'' $T$, I want to prove that it is a contraction on a suitable space of graphs and hence has a unique fixed point there, which will be an invariant graph.  In the direction of achieving this,
define a {\em slope} to be a linear map $\sigma$ from displacements in $\theta$ to displacements in $r$.
For an approximation $\tilde{\sigma}$ to the expected derivative $\rho_\theta$, define
$M_{\tilde{\sigma}}: W^{1,\infty}(\R,\R^{mp}) \to W^{0,\infty}(\R,\R^{mp})$ by
$$M_{\tilde{\sigma}}[\sigma] = \dot{\sigma} - R_r\sigma+\sigma(\Theta_\theta + \Theta_r\tilde{\sigma})$$
on slope functions $\sigma$ of $t$, where $W^{s,\infty}$ are the spaces of functions with essentially bounded $s^{th}$ derivative.
Suppose that $M_{\tilde{\sigma}}$ has bounded inverse.  This is the second part of the NH condition, namely faster normal contraction than tangential contraction.

Then $T$ should be a contraction in the space of $C^0$ functions with an a priori Lipschitz constant.  So it would have a unique fixed point $\rho$.  Any fixed point is invariant and actually $C^1$ with slope $\rho_\theta$ being the fixed point of the contraction map $\sigma \mapsto M_\sigma^{-1}[R_\theta]$.

To complete this programme requires detailed estimates.  Formulated in terms of contraction maps as here, it should be possible to obtain excellent estimates, along the lines of the uniformly hyperbolic case in \cite{BiM}.  We might do best to follow the approach of \cite{H} (cf.~\cite{E}), but replacing their exponential hypotheses by our hypotheses of invertibility of $L$ and $M$ and modifying their exponentially weighted norm to use the linearised tangential flow.  I would like to finish this one day.

\section{Extension to class 1 neurons}
So far, I have considered the simplest type of oscillator, namely limit cycles, but the treatment can be extended to class I neurons (or excitable oscillators).  These are dynamical systems with an attracting invariant cylinder in the autonomous case and dynamics on it in simplest form given by
\begin{eqnarray}
\dot{\theta} &=& \mu + 1-\cos\theta \\
\dot{\mu}&=&0 . \nonumber
\end{eqnarray}
Since $\mu$ is constant, one could think of it as an external parameter, but I wish to consider it as a state variable because coupling from another neuron can make $\mu$ change in time.
It is best to think of $\mu$ as bounded, so the attracting cylinder can be considered an invariant annulus.

They arise in modelling of ``excitable neurons'' whose frequency goes to zero as a parameter ($\mu$) is varied and then settle at a $\mu$-dependent resting state, or in reverse go from a resting state to large amplitude periodic spiking.
An example is the Morris-Lecar model \cite{ML}, but it was \cite{EK} who identified the phenomenon as the unfolding of a saddle-node on a cycle (I proposed this independently to physiologist H.Barlow in the same year and then in 1991 proposed to C.Koch the extension to allow crossover at a ``saddle-node loop'' \cite{S} to the unfolding of a homoclinic orbit to a saddle).
Thus the non-autonomous version has an attracting NH annulus-cylinder.

I had an undergraduate student study networks of such neurons in 1989/90, with the state $\mu$ of each neuron driven by the spiking of some others (with time-delay kernels), which produced periodic bursting \cite{M2}.

Two class I neurons coupled not too strongly have a NH attracting annulus$\times$annulus-cylinder.  Generic bifurcation diagrams in the autonomous case were given in \cite{BaM}.  The dynamics on it has attracting submanifolds of various types.  The non-autonomous case has non-autonomous versions of them.

The theory of this paper applies just as well to class I neurons as to ordinary oscillators, with the addition of the $\mu$-direction for each class I neuron.

\section{Extension to chaotic oscillators}
The approach can also be extended to chaotic oscillators if they have an attracting NH submanifold containing the attractor.  For example, think of a R\"ossler attractor \cite{R}, which is contained in a solid torus in $\R^3$.  Then the non-autonomous system has a solid-torus-cylinder.  A R\"ossler attractor can be phase-locked to forcing, meaning that the dynamics is attracted onto a disk-cylinder (a solid torus is the product of a disk and a circle).  This should be quite easy because the R\"ossler attractor was observed to be nearly phase-coherent.  I interpret that as meaning that there is a cross-section with nearly constant return time (equivalently, for a given cross-section $\Sigma$ there is a constant $c>0$ and a function $b:\Sigma \to \R$ such that the return time $\tau(x) = c + b(f(x))-b(x)$, where $f:\Sigma \to \Sigma$ is the return map).

Synchronisation of chaotic attractors with NH cylinders of dimensions $N_1+1, N_2+1$ means there is a NH cylinder for the coupled system with dimension less than $N_1+N_2+1$.

Even better, the theory of NH submanifolds extends to NH laminations \cite{HPS}.  A lamination is a topological space in which each point has a neighbourhood homeomorphic to the product of a Euclidean space with a general topological space.  It decomposes into leaves, which are locally submanifolds but in general only injectively immersed, so a leaf may accumulate onto itself.  The theory of NH laminations requires a $C^1$-structure in addition, but is basically the same as for NH submanifolds.  In particular, a NH lamination persists under $C^1$-small perturbation.

This means one can treat some chaotic attractors in greater detail.  In particular, imagine we start with a non-trivial uniformly hyperbolic attractor of an autonomous system, for example a suspension of a Plykin attractor \cite{P}.  This is perhaps less familiar than R\"ossler's attractor but deserves to be better known, as the simplest uniformly hyperbolic attractor after equilibria and periodic orbits.  The Plykin attractor was constructed for a discrete-time system, but the map is isotopic to the identity so one can realise it as the first return map of an associated continuous-time system.  My PhD student Tim Hunt showed an explicit way to realise it in a system of three ODEs, extended by another PhD student Linling Ru, and less cumbersome ways have been proposed (though not yet with rigorous justification) \cite{K}.  It is a NH lamination, whose leaves are its unstable manifolds (of dimension two:~one expanding dimension and one time dimension) and they form a Cantor set transversally.  Under time-dependent forcing, it persists to a Cantor set of 3D leaves whose tangent space is spanned by one expanding dimension and two near neutral dimensions.  The persistence is highly robust, requiring only that any tangential contraction be slower than any transverse contraction.  Then one can ask what happens on the leaves.  The dynamics might collapse onto a 2D subleaf with the same expanding dimension one neutral dimension.  I would say the attractor has synchronised to the forcing.

Similarly, one could couple a suspended Plykin attractor to a limit-cycle oscillator.  It produces an attractor with a Cantor set of 3D leaves (the product of the 2D leaves of the chaotic attractor with the limit cycle).  The dynamics of each leaf might collapse onto 2D subleaves.  I would say the Plykin attractor and limit cycle synchronise together.

More generally, one could couple a continuous-time autonomous uniformly hyperbolic attractor with $M$ unstable dimensions to $N$ limit cycle oscillators and obtain an attractor with a mixture of chaos and nearly quasiperiodic behaviour.  It would have $M$ unstable dimensions, $N$ nearly quasiperiodic dimensions, and the flow dimension, with the remaining dimensions contracting onto the leaves.  By the theory of NH laminations, such attractors persist for small smooth perturbations, though the dynamics in the quasiperiodic dimensions cannot be expected to remain quasiperiodic.  Nonetheless, it will have small Lyapunov exponents for those dimensions and perhaps there is a non-autonomous KAM theory that would even give truly quasiperiodic motion for a set of nearly full measure of parameters.  I propose this as an explanation of the scenario reported recently by \cite{YK}.

As a final note, one might ask about physical realisation of attractors like R\"ossler's.  I designed an electronic oscillator back in 1981, principally to demonstrate period-doubling sequences \cite{M1}, but moving the parameter further it exhibited a R\"ossler type of attractor.  Model equations for the voltages at three points have the form
\begin{eqnarray}
\dot{x}&=&ax-by \\
\dot{y}&=& cx-ez \nonumber \\
\dot{z} &=& -fy-g(z) , \nonumber
\end{eqnarray}
with $a,b,c,e,f$ positive constants of which $a$ was adjustable by a 10-turn potentiometer, and $g$ an approximately odd cubic nonlinearity produced with a pair of transistors.  Interestingly, as I increased $a$ further, the R\"ossler attractor turned into what Chua later called a double-scroll attractor \cite{MCK}.  Indeed, Chua's equations turn out to be equivalent to mine after minor changes of variable.

\section{Conclusion}
I have shown that the behaviour of networks of oscillators, autonomous or not, can be aided by identifying normally hyperbolic submanifolds.  This allows a deeper understanding of synchronisation of oscillators to forcing and to each other, especially in the aperiodic case.  There are many studies on synchronisation in autonomous or periodically forced systems (for one example, see \cite{SST}) but relatively few on the aperiodically forced case.  The fundamental feature of synchronisation is dimension-reduction of an associated normally hyperbolic submanifold.  In a network of oscillators, even if autonomous, the inputs that an individual oscillator sees are in general aperiodic.  This motivates a hierarchical aggregation scheme for understanding the dynamics of a network of oscillators:~oscillators that synchronise to their inputs can be eliminated, groups of oscillators that synchronise together can be replaced by a single effective oscillator.  All this depends on generalising the notion of oscillator from a limit cycle of an autonomous dynamical systems to a mapping from input functions of time to solutions and generalising the notion of coupling.  Finally, I extended the treatment from limit-cycle oscillators to excitable oscillators and chaotic oscillators.

config: arxiv


TITLE: Why can the neutron not be measured in this fixed-target experiment?
QUESTION [1 upvotes]: Consider a fixed-target-experiment, where negatively charged pions are shot at protons (the latter being at rest). The kinetic energy of the pions shall be known. One possible reaction is
$$
\pi^- + p \to n + \pi^+ + \pi^-
$$
creating a neutron. The neutron apparently remains undetected by measuring devices, and only one of the pions can be measured.
I have two conceptual questions:
(1) Why is this? (I mean, both pions carry charge...)
(2) If the neutron cannot be measured, then how can we be sure that it is there?

REPLY [2 votes]: Exclusive & semi-inclusive reactions are commonly as written target(beam, detected)undetected:
$$p(\pi^-, \pi^+)n\pi^-$$
To understand the detection of the final state, we really need more information. While there are detectors that can handle multiple particle final states (https://www.jlab.org/physics/hall-b/clas), this experiment wasn't done there.
More commonly, fixed target experiments put a beam on target and look at reaction products with a spectrometer (e.g., SLAC's 8 GeV spectrometer: https://www.slac.stanford.edu/pubs/slacpubs/5750/slac-pub-5753.pdf). This grabs a small bite of both angular space and momentum space. It uses (a) dipole magnet(s) to select a central momentum, and quadruple magnets to focus particle tracks so that momentum and lab-angle (regardless of out-of-plane angle) can be determined.
The spectrometer can only look a one charged final state, but that can be reversed by reversing the current. Since the beam is $\pi^-$, the $\pi^+$ final state is much cleaner.
The way to separate the reaction from:
$$p(\pi^-, \pi^+)X$$
is the invariant mass of $X$. The initial state's 4-momentum is:
$$p_i^{\mu} = p_p^{\mu} + p_p^{\pi^-} $$
$$p_i^{\mu} = (M_p, 0,0,0) + (E_{\pi^-}, 0,0,p_{\pi^-}) $$
while the final state is:
$$p_f^{\mu} = p^{\mu}_{\pi^+} + p_X^{\mu}   $$
Of course, $p_i^{\mu}= p_f^{\mu}$ so that:
$$p_X^{\mu} = p_i^{\mu} - p^{\mu}_{\pi^+} $$
$$p_X^{\mu} = (M_p+E_{\pi^-}, 0,0, p_{\pi^-}) - (E_{\pi^+}, p_{\pi^+}\sin{\theta},0, p_{\pi^+}\cos{\theta}) $$
If you work that all out at a fixed angle, there should be a momentum region that ensures $X$ is a nucleon plus a pion, and since it has charge -1, it has to be a neutron and $\pi^-$. (Another reason to detect the positive pion, as a negative pion in the final state does NOT distinguish between an undetected $n\pi^+$ and $p\pi^-$). That also rules out a lot of other final states, so by detecting  the $\pi^+$, there is a kinematic region where $X=n\pi^-$ is the only viable option.


TITLE: Would it make sense to do a two-factor decomposition similar to singular-value decomposition?
QUESTION [3 upvotes]: A finite-dimensional matrix $M$ can be decomposed as
$$
M = U \Sigma V^*
$$
where $U$ and $V^*$ are unitary matrices whose geometric interpretation is rotation and $\Sigma$ scaling along the orthonormal basis.   Would it be possible to decompose $M$ as just one rotation followed by one scaling
$$
M = \Sigma R
$$
?

REPLY [1 votes]: Suppose the singular-value decomposition (SVD) of the finite dimensional matrix $M$ is
$$
M = U \Sigma V^*,
$$
one can insert identity represented as the product of Hermitian matrices $I = V V^*$ in the decomposition and regroup the factors as a product of a unitary matrix and a normal matrix
$$
M = (U V^*) (V \Sigma V^*) = W N,
$$
where $W$ is a unitary matrix representing a rotation, and $N$ a normal matrix representing contraction and expansion along axes in an appropriate orthogonal basis that can be obtained by the rotation represented by the unitary matrix $V^*$.  Note the normal matrix $N$ is unitarily similar to the diagonal matrix $\Sigma$.
So instead of $M = \Sigma R$, one can always get decomposition of form $M = W N$ which is polar decomposition.
If $M$ is invertible, the the decomposition is unique.  This is analogues to that non-zero complex number has unique decomposition $z = r e^{i \theta}$ up to the periodicity in $\theta$.


TITLE: Polarizable variations of (mixed) Hodge structures
QUESTION [3 upvotes]: I am trying to come to grips with Saito's theory of mixed Hodge modules (slightly) beyond just the basic axiomatic formalism. I will take my Hodge structures and sheaves to be rational, but I would be grateful if the experts would point out any subtleties with my questions if rational is replaced by real. Let me at the moment focus on questions related to polarizability:

Let $\mathcal{L}$ be a local system underlying a polarizable variation of Hodge structure on a smooth variety. Does the polarizability imply $\mathcal{L}$ is self-dual? If yes, then does every direct summand of this local system also have to be self-dual (I am guessing no to the latter)?
More generally, let $M$ be a polarizable mixed Hodge module on some variety (not necessarily smooth). Is polarizability equivalent to Verdier self-duality (up to Tate twist) in the derived category of mixed Hodge modules? If not equivalent, does it at least imply Verdier self-duality?

REPLY [6 votes]: You have my sympathies for trying come to grips with this stuff. Fortunately the answer, which is both yes and no, can be explained
in the simplest case of a point.  A polarization on a pure Hodge structure $H$ of weight $w$
is a pairing
 $$\langle, \rangle: H\otimes H\to \mathbb{Q}(-w)$$
such that $(2\pi i)^n\langle -, C -\rangle$ is positive definite and symmetric, where $C$ is the Weil operator which acts by $i^{p-q}$ on $H^{pq}$. These conditions known as the Hodge-Riemann bilinear relations imply that $H\cong H^*(-w)$ as Hodge structures, so in this sense $H$ self dual. However, just having such an isomorphism would not give everything else.
If $H$ is replaced by a variation of Hodge structures, then a polarization is flat pairing
as above satisfying the Hodge-Riemann conditions on the fibres. For  Hodge modules
the story is similar but more delicate. Since the whole theory is constructed by induction on dimension of support, a polarization is also defined in this manner.
Given Hodge module
$H$ of weight $w$ on $X$, a polarization is a pairing $H\otimes H\to \mathbb{Q}(\dim X-w)[2\dim X]$ satisfying the inductive conditions (0.8)-(0.10) of Saito's "Modules Hodge Polarizables". Such a pairing should induce an isomorphism $H\cong DH(?)$ up to  twist,
but it's not equivalent. (Added I don't have a precise reference or proof for this,
just a feeling that one could prove it as follows: Buried in Saito's second paper,
"Mixed Hodge modules", is a proof that any simple polarizable Hodge module is
$j_{!*}V$, where $V$ is  a VHS. This effectively reduces the duality statement to the previous case.)


TITLE: Question about the zeros of $x^2+1 \in \Bbb Z_7.$
QUESTION [1 upvotes]: So obviously $6$ isn't a quadratic residue $mod(7)$ thus there are no zeros in $\Bbb Z_7$. So what I did next is considered considered the field $\Bbb Z_7[x]/<x^2+1>$ obviously $x+<x^2+1>$ is a zero for the polynomial so we will denote that by $i$, keep in mind that $i^2=6$. My question is really about the field $\Bbb Z_7(i)$, that is the smallest field containing all of $\Bbb Z_7$ and $i$. Well since $i^2=6$ this field has all its elements of the form $a+bi$ where $a,b \in \Bbb Z_7$ so that field contains only 14 elements which isn't a power of a prime. I must be doing something wrong since I am a bit rusty on algebra, can someone please point out my mistake.

REPLY [0 votes]: $(a+bi)^2 = a^2 - b^2 + 2abi = -1$ only iff $2ab = 0$, which gives us that either $a = 0$ or $b=0$. You have excluded the options where $b=0$, which leaves us with $a = 0$. Here we get the solutions $b = 6$ and $b = 1$.