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\section{Local Search over Graphs}
In this section we provide communication lower bounds on the communication complexity of local search over several families of graphs.
\begin{theorem}\label{theo:opt}
The following bound holds for the randomized communication complexity of $\textsc{SumLS}$:
\begin{enumerate}
\item\label{theo:opt-bounded} $CC(\textsc{SumLS}(G))= \Omega(\sqrt{N})$, when $G$ is a specific constant-degree (36) graph with $N$ vertices.
\item\label{theo:opt-hypercube} $CC(\textsc{SumLS}(\textsf{Hyp}_n)) = \Omega(\sqrt{N}) = \Omega(2^{n/2})$ where $N=2^n$ is the number of vertices.
\item\label{theo:opt-grid} $CC(\textsc{SumLS}(H)= \Omega(\sqrt{N})$, when $H$ is a grid with a constant dimension (119) grid with $N$ vertices.
\item\label{theo:odd} $CC(\textsc{SumLS}(\textsf{Odd}_n)) = \Omega(2^{n/2})$.
\end{enumerate}
\end{theorem}
We note that results \ref{theo:opt-bounded}, \ref{theo:opt-hypercube}, and \ref{theo:opt-grid} are optimal since \cite{Ald} provides a randomized algorithm that finds a local maximum in these graph using $O(\sqrt N)$ value queries.
Result \ref{theo:odd}, on the other hand, is not necessarily optimal because the odd graph has $N\approx 4^{n}$ vertices, so in terms of the number of vertices our lower bound is $\Omega(\sqrt[4]{N})$.
Result \ref{theo:opt-grid} proves an optimal bound for a grid with a constant dimension, but this dimension is quite large (119). We are able to show that finding a local optimum in the three-dimensional grid is hard, but our lower bound in this case is only $\Omega(N^c)$, for some constant $c>0$ (in contrast to an optimal bound of $\Omega(\sqrt N)$ for the $119$-dimensional grid). To prove this, we first show that finding a local maximum is hard even for degree $4$ graphs.
\begin{theorem}\label{theo:grid}
There exists a constant $c>0$ such that the randomized communication complexity of $\textsc{SumLS}$ satisfies:
\begin{enumerate}
\item\label{theo:deg-bounded} $CC(\textsc{SumLS}(G))\geq N^c$, when $G$ is a specific degree 4 graph with $N$ vertices.
\item\label{theo:dim-grid} $CC(\textsc{SumLS}(\textsf{Grid}_{N\times N \times [2]})= N^c$.
\end{enumerate}
\end{theorem}
The overall structure of the proofs of Theorems \ref{theo:opt} and \ref{theo:grid} is similar, but the proofs use different techniques.
The proof of Theorem \ref{theo:opt} appears in Section \ref{sec:main-pr} and the proof of Theorem \ref{theo:grid} appears in Section \ref{sec:grid-pr}.
\section{Proof of Theorem \ref{theo:opt}}\label{sec:main-pr}
Our starting point is a communication variant of a pebbling game. In this problem,
$D=(V,E)$ is a known directed acyclic graph. The input is a boolean assignment for the vertices $b:V \rightarrow \{0,1\}$ such that every source is true ($b(v)=1$) and every sink is false ($b(v)=0$). The output is a false vertex whose all predecessors are true (i.e., $v\in V$ such that $b(v)=0$ and $b(u)=1$ for all $u\in V$, $(u,v)\in E$). Note that the problem is total.
The communication variant of the pebbling game $\textsc{Pebb(D)}$ is defined by distributing the information $b(v)\in \{0,1\}$ of every vertex by a
constant size index-gadget $\{0,1\}^3\times [3] \rightarrow \{0,1\}$.
In \citep{GP14} it is shown that there exists a constant degree graph $D$ with $N$ vertices where both the randomized communication complexity of the problem is $\Theta(\sqrt{N})$. The proof is done in three steps.
\paragraph{Step 1} We introduce an intermediate communication problem $\textsc{VetoLS}(G)$ where Alice holds the potential function and Bob holds a subset of valid vertices (equivalently, Bob vetoes the vertices that are not in the set that he holds). The goal is to find a valid local maximum: a valid vertex whose \emph{valid} neighbours have (weakly) lower potential. Given a graph $D$ as above, we construct a constant degree graph $G$ with $O(N)$ vertices and reduce $\textsc{Peb(D)}$ to $\textsc{VetoLS}(G)$. This gives us an optimal communication lower bound for $\textsc{VetoLS}$ for a concrete graph $G$.
\paragraph{Step 2} We define a certain notion of ``embedding'' of one graph into the other. We show that if $G'$ can be embedded in $G$, then $CC(\textsc{VetoLS}(G'))\geq CC(\textsc{VetoLS}(G))$. We use this observation to prove optimal hardness bound of $\textsc{VetoLS}$ over the hypercube by embedding $G$ into an hypercube of dimension $\log(N)+c$ for a constant $c$.
\paragraph{Step 3} For every graph $G$, we show that $CC(\textsc{VetoLS}(G))\approx CC(\textsc{SumLS}(G))$.
\vspace{0.1in}\noindent We now provide a detailed description of each step.
\subsection{Starting Point: Pebbling Games}\label{sec:step0}
In Section \ref{sec:emb} we use the concrete structure of the constant degree graph $D$ for which the hardness of the pebbling game is proved. Hence, we start with providing an explicit description of the graph $D$.
The vertices of $D$ are given by $V=[M^3]\times [M]\times [M] \times [M]$. Each vertex $v=(k_1,k_2,k_3,k_4)$ has six successors:
$$\{v+(1,\pm 1,0,0),v+(1,0,\pm 1,0),v+(1,0,0,\pm 1)\}$$
where the $\pm 1$ addition in the last three coordinates is done modulo $M$. The addition in the first coordinate is the standard addition. Thus, each vertex has six predecessors:
$$\{v+(-1,\pm 1,0,0),v+(-1,0,\pm 1,0),v+(-1,0,0,\pm 1)\}$$
The sources of the graph are $\{(1,\cdot,\cdot,\cdot)\}$ and its sinks are $\{(M^3,\cdot,\cdot,\cdot)\}$.
In \citep{GP14} an optimal bound on the communication complexity is proved the following optimal bound on the communication of the following variant of pebbling games. In the communication problem $\textsc{Pebb}(D)$, Alice's input is an assignment $b:V\times [3] \rightarrow \{0,1\}$. Bob's input is an index for each vertex $I:V \rightarrow [3]$. The input satisfies $b(v,I(v))=1$ for every source $v$, and $b(v,I(v))=0$ for every sink $v$. The output is a vertex $v\in S$ such that $b(v,I(v))=0$ and for every predecessor $u$ of $v$ holds $b(u,I(u))=1$.
\begin{theorem}[\citep{GP14}]\label{theo:gp}
$CC(\textsc{Pebb}(D))=\Omega(M^3)$. This bound also holds for randomized protocols.
\end{theorem}
\subsection{Step 1: From Pebbling to $\textsc{VetoLS}$}\label{sec:veto}
Given a graph $G$, the communication problem $\textsc{VetoLS}(G)$ is defined as follows. Alice's input is a function $f:V\rightarrow [W]$. Bob's input is a non-empty subset $S\subset V$. The output is a vertex $v\in S$ such that $f(v)\geq f(w)$ for every $w\in S$ such that $\{v,w\}\in E$ (i.e., for every valid neighbour). We show that the communication complexity of $\textsc{VetoLS}(G)$ is at least that of $\textsc{Pebb}(D)$, for some $G$ that is related to $D$. Next we show how to obtain the graph $G$ from $D$.
We construct the graph $G$ in two stages. First, given a graph $D$ of the pebbling game, let $G'$ be an undirected version of $D$ which additionally has an edge from every source of $D$ to some sink of $D$. Let $G$ be the graph that is obtained from $G'$ by replacing each vertex in $G$ with three new vertices and duplicating the edges so that each new vertex is connected to all the copies of its neighbors in $G'$. We call the graph $G$ the \emph{replication graph} of $D$.
\begin{proposition}\label{pro:peb-ls}
Let $D$ be a graph and $G$ be its replication graph. The communication complexity of $\textsc{VetoLS}(G)$ is at least that of $\textsc{Pebb}(D)$.
\end{proposition}
\begin{proof}
Let $V$ denote that set of vertices of $D$ and $V'=V\times [3]$ be the set of vertices of $G$. Let $t:V'\rightarrow \mathbb R$ be a topological numbering of the vertices of $G$. I.e., $t((u,i))>t((v,j))$ if there exists a directed edge $(u,v)$ in $D$.
Alice's input in $\textsc{Pebb}(D)$ is an assignment $b:V\times [3] \rightarrow \{0,1\}$. We use this to define the potential function $f$ that Alice holds in $\textsc{VetoLS}(G)$: for vertex $v\in V'$ let $f(v)=t(v)+6N\mathds{1}_{b(v)=0}$. Bob's input in $\textsc{Pebb}(D)$ is the function $I:V \rightarrow [3]$. Bob defines the set of valid vertices in $\textsc{VetoLS}(G)$ to be $S=\{(v,I(v)):v\in V\}$. Namely, among the three copies of $v$ only the one with correct index is valid. This choice of valid vertices has the desirable property that the subgraph of \emph{valid} vertices is precisely $G'$ and the assignment $b(v,i)$ over the vertices of $G'$ is precisely the decomposed assignment $b(v,I(v))$.
We argue that the local maxima of $f$ are precisely all false vertices whose all incoming neighbours are true. Those are indeed local maxima, because their ``predecessors" do not have the bonus of $6N$ and their ``successors" have lower topological number. The source cannot be a local maximum because it is a ``true'' vertex and it is connected to a sink that is a ``false'' vertex. A true vertex (other than source) is not local maximum because its predecessor has higher topological number. Similarly, a false vertex with false predecessor is not local maximum. This leaves us only with false vertices whose predecessors are true.
\end{proof}
\subsection{Step 2: Embedding the Bounded Degree Graph}\label{sec:emb}
In this step we define a certain notion of embedding of one graph into another. We will see that if a graph $G$ can be embedded into $H$ then the communication complexity of local search on $H$ is essentially at least as large as the communication complexity of local search on $G$. We will then see how to embed the graph $G$ of the previous steps into the three dimensional grid, the hypercube, and the odd graph.
\begin{definition}\label{def:vied}
A \emph{vertex-isolated edge-disjoint (VIED) embedding} of a graph $G=(V_G,E_G)$ in a graph $H=(V_H,E_H)$ is a pair of mappings $\varphi : V_G \rightarrow V_H$ and $\chi: E_G \rightarrow P(H)$, where $P(H)$ is the set of \emph{simple paths} on $H$, such that:
\begin{itemize}
\item $\varphi$ is injective.
\item For every edge $\{v,w\}\in E_G$, the path $\chi(\{v,w\})$ connects $\varphi(v)$ to $\varphi(w)$.
\item The interior vertices of the paths $\chi(\{v,w\})$ and $\chi(\{v',w'\})$ are disjoint (edge disjointness).
\item For every $v\in V_G$ and every $\{w,w'\}\in E_G$ such that $v\neq w,w'$ holds $d(\varphi(v),\chi(\{w,w'\}))\geq 2$, where $d$ denotes the distance in $H$ of the vertex from the path (vertex isolation).
\end{itemize}
\end{definition}
That is, in a VIED embedding every edge of $G$ is replaced by a path in $H$ that connects the corresponding vertices such that these paths do not share a vertex. Moreover, for every $v\in V_G$, $\varphi(v)$ is isolated in the sense that no path passes through the neighbours of $\varphi(v)$.
\begin{lemma}\label{lem:emb}
Let $G$ be a graph and suppose it can be VIED embedded into some other graph $H$. Then $CC(\textsc{VetoLS}(G))\leq CC(\textsc{VetoLS}(H))$.
\end{lemma}
\begin{proof}
Alice's potential is defined as follows. For vertices $w\in \varphi(V_G)$ we define $f_H(\varphi(v))=f_G(v)$. Consider a vertex $w\in \chi(E_G)$ that belongs to an edge $\{u,v\}\in E_G $. Suppose that $w$ is the $k$'th element in the path $\chi(\{u,v\})$ and $l$ is the total length of this path. Define:
\begin{align}\label{eq:edge}
f_H(w)=\frac{k}{l} f_G(u) + \frac{l-k}{l}f_G(v)
\end{align}
In all other vertices Alice's potential will not play a role because these vertices will not be valid, thus we can simply set $f_H(w)\equiv 0$ for all other vertices.
We recall that Bob's input in $\textsc{VetoLS}(G)$ is $S_G\subset V_G$.
We denote by $E_G(S_G)\subset E_G$ the set of internal edges of $S_G$.
Bob's subset of valid vertices in $H$ is defined by\footnote{By $\chi(E_G(S_G))$ we obviously mean the corresponding \emph{vertices} in these paths.} $S_H=\varphi(S_G) \cup \chi(E_G(S_G))$.
If $v\in V_G$ is a valid local maximum, then $\varphi(v)\in V_H$ is a valid local maximum because all its valid neighbours are valid edges in which $v$ participates (here we use the isolation property), and the value along these edges is a weighted average of $f_G(v)$ and $f_G(u)\leq f_G(v)$, where $u$ is a valid neighbour of $v$.
We argue that there are no additional valid local maxima in $H$. Indeed, if $v\in V_G$ is not a local maximum then $\varphi(v)\in V_H$ is not a local maximum because there is a valid edge where the potential increases. If $w\in \chi(E_G(S_G))$, by distinctness, $f_G(u)\neq f_G(v)$ therefore in one of the directions of the path $\chi(\{u,v\})$ the potential increases. All other vertices are invalid.
\end{proof}
\subsubsection{An Explicit Description of the Graph G}
In the embeddings we use the specifics of the DAG $D$ for which the hardness of pebbling games is proved in \cite{GP14}. We now explicitly describe the replication graph $G$ that is obtained from $D$ so that Proposition \ref{pro:peb-ls} can be applied.
Let $G'$ be the undirected version of the DAG $D$ for which the hardness of pebbling games is proved with additional edges that connect the sources and sinks of $D$ in a same way other vertices in $D$ are connected. Formally, the vertices of $G'$ are $V=[M^3]\times [M]\times [M] \times [M]$, and the edges are:
$$E=\{(u,v):u-v\in \{(\pm 1,\pm 1,0,0),(\pm 1,0,\pm 1,0),(\pm 1,0,0,\pm 1)\}\}$$
Let $G$ be the graph that is obtained from $G'$ by replacing each vertex in $G$ with three new vertices and duplicating the edges so that each new vertex is connected to all the copies of its neighbors in $G'$. Formally, the vertices of $G$ are $\{(v,i):v\in V,i\in [3]\}$ and the edges are $\{((u,i),(v,j)):(u,v)\in E, i,j\in [3]\}$. Note that $G$ is a graph with $3M^6$ vertices and (constant) degree $d=36$.
\subsubsection{Embedding into the Hypercube}
In this section we show how to embed the replication graph $G$ obtained in the previous step into the hypercube. Moreover, the embedding is such that the number of vertices in the hypercube increases only by a constant factor. This small blowup is crucial for obtaining an optimal $2^{n/2}$ bound.
\begin{lemma}\label{lem:hyp}
The graph $G$ (with $3M^6$ vertices) can be VIED-embedded into the $n$'th-dimensional hypercube $\textsf{Hyp}_n$ for $n=6\lceil \log M \rceil + 111$. As a corollary, $CC(\textsc{VetoLS}(\textsf{Hyp}_n))=\Omega(2^{n/2})$.
\end{lemma}
\begin{proof}
For clarity of exposition we assume that $M=2^c$ is a power of 2.
We start with some notations and properties of the graph $G$. Recall that the vertices of $G$ are $V=[M^3]\times [M] \times [M] \times [M]\times [3]$.
For a vertex $v=(k_1,k_2,k_3,k_4,i)$, $k_1$ is called the \emph{layer of $v$}. Note that all edges connect $k$ layer vertices to $k+1$ layer vertices. $k_1+k_2+k_3+k_4 \mod 2$ is called the \emph{parity of $v$}. $i$ is called the \emph{replication index of $v$}.
We present an edge coloring of $G$ with $108$ colors in which no two adjacent edges are colored the same (a ``valid'' coloring).
We first color all edges from layer 1 to layer 2 with $54$ colors. Given a vertex $v=(k_2,k_3,k_4)$, edges are specified by a displacement $d\in \{\pm 1,0,0),(0,\pm 1,0),(0,0,\pm 1)\}$ that operates on $(k_2,k_3,k_4)$ and pair of replication indices $i,j\in [3]$ ($i$ is the replication index of the vertex at layer $1$ and $j$ is the replication index of the vertex at layer $2$). Note that we have $6\cdot 9=54$ such specifications. It is easy to verify that coloring these edges in $54$ different colors is a valid edge coloring.
We proceed by coloring all edges between layers $2$ and $3$ with \emph{different} $54$ colors using a similar coloring method. Similarly, all edges from layer $2k-1$ to layer $2k$ are colored as edges between layers $1$ and $2$ and all edges from layer $2k$ to layer $2k+1$ are colored as edges between layers $2$ and $3$.
This defines an edge coloring of $G$.
Now we present some notation. The vertices of the hypercube are partitioned into blocks as follows:
\begin{itemize}
\item For $i=1,...,5$ the \emph{$i$'th index} block consists of bits that represent the $i$'th index. The sizes of the blocks are $(3c,c,c,c,2)$ for $i=1,2,3,4,5$ correspondingly.
\item A \emph{parity bit} memorizes the parity of a vertex.
\item The \emph{edge} block consists of $108$ bits.
\item The \emph{counter} block consists of $3$ bits that serves as a counter to keep track of the block on which we currently apply the changes along the embedding path (see below).
\end{itemize}
\paragraph{Embedding the vertices.}
Let $(h_1,...,h_{M^3})$ be a Hamiltonian path of the $3c$-dimensional hypercube. Let $(h'_1,...,h'_{M})$ be a Hamiltonian path of the $c$-dimensional hypercube and $(h''_1,...,h''_{4})$ be a Hamiltonian path of the $2$-dimensional hypercube. To define $\phi(v)$, we embed a vertex $v=(k_1,k_2,k_3,k_4,i)$ into the vertex of the hypercube whose first block is the bits of $h_{k_1}$, the second block is $h'_{k_2}$, then $h'_{k_3}$, $h'_{k_4}$ and $h''_{k_5}$. We set the parity bit to be the parity of $v$, the edge block to $\textbf{0}$, and the counter block to $\textbf{0}$.
\paragraph{Embedding the edges.}
Note that the coloring of $G$ in $108$ colors naturally induces an order on the edges. Every vertex has at most one $m$'th edge, and two adjacent vertices agree on the index of this edge.
The $m$'th edge of $v$, from $v$ in layer $k_1$ to $u$ in layer $k_1+1$, is defined by the following sequence of bit flipping.
\begin{enumerate}
\item The $m$'th bit in the edge block is flipped to 1.
\item A single bit in the counter block is flipped to encode the integer 1.
\item A single bit in the first index block is flipped to encode the integer $k_1+1$.
\item A single bit in the counter block is flipped to encode the integer 2.
\item If the displacement of the edge is $(\pm 1,0,0)$, a single bit in the second index block is flipped to encode the integer $k_2 \pm 1$. If the displacement of the edge is $(0,\pm 1,0)$, a single bit in the third index block is flipped to encode the integer $k_3 \pm 1$. If the displacement of the edge is $(0,0,\pm 1)$, a single bit in the fourth index block is flipped to encode the integer $k_3 \pm 1$.
\item A single bit in the counter block is flipped to encode the integer 3.
\item The two bits of the fifth index block are flipped (one by one in a fixed order) to encode the integer $j$ (the replication index of $u$).
\item The counter block returns back to $\textbf{0}$.
\item The $m$'th bit in the edge block is flipped back to $\textbf{0}$.
\end{enumerate}
It is easy to see that this path ends up at $\phi(u)$ (note that the parity of $v$ and $u$ is the same, and indeed we did not flip the parity bit). We argue that the defined paths are disjoint. It is sufficient to prove that given a node on the path one can recover the previous node. Given the color of the edge and the counter, it is immediate to recover the previous node in all intermediate steps excluding steps (3) and (5). In steps (3) and (5) it is unclear whether we should flip the corresponding index block or the counter block. To determine this we use the parity bit: In step (3), if the parity bit is equal to the parity of the encoded vertices, then it means that we did not flip yet a bit, and to get the previous vertex we set the counter block to encode 0. If the parity bit differs from the parity of the encoded indices, then it means that we have flip a bit, and to get the previous vertex we should flip a bit in the index block. In step (5) we do the opposite. If the parity bit differs from the parity of the encoded indices, then we flip the counter. If the parity bit is equal to the parity of the encoded indices, then we flip the index block.
It is easy to check that the embedding is vertex isolated because of the parity bit.
\end{proof}
\subsubsection{Embedding into the Grid}
\begin{lemma}\label{lem:grid1}
The graph $G$ (with $3M^6$ vertices) can be VIED-embedded in a constant-dimension grid with $O(M^6)$ vertices. As a corollary, $CC(\textsc{VetoLS}(\textsf{Grid}_d))=\Omega(\sqrt{N})$ for some constant-dimension grid with $N$ vertices.
\end{lemma}
\begin{proof}(sketch)
The embedding is very similar to the one we presented in Lemma \ref{lem:hyp} for embedding into the hypercube. In the proof of Lemma \ref{lem:hyp} we only used the fact that the hypercube has an Hamiltonian cycle. For the grid, we will take advantage of the observation that the two-dimensional grid has an Hamiltonian cycle.
Specifically, a vertex $v=(k_1,k_2,k_3,k_4,i)$ is embedded into the vertex of the grid whose first block is the bits of that correspond to a Hamiltonian cycle on $\textsf{Grid}_{M^{1.5}\times M^{1.5}}$, blocks $2-4$ are specified using the Hamiltonian cycle on $\textsf{Grid}_{M^{0.5}\times M^{0.5}}$, and block $5$ using the Hamiltonian cycle on $\textsf{Grid}_{2}\times \textsf{Grid}_2$. We set the parity bit to be the parity of $v$, the edge block to $\textbf{0}$, and the counter block to $\textbf{0}$.
Applying very similar arguments to the proof of Lemma \ref{lem:hyp} we establish the embedding of $G$ into the grid $[M^{1.5}]^2\times [M^{0.5}]^6 \times [2]^{111}$.
\end{proof}
\subsubsection{Embedding into the Odd Graph}
\begin{lemma}\label{lem:odd}
There exists a VIED embedding of $\textsf{Hyp}_n$ in $\textsf{Odd}_{n+2}$. As a corollary,
$CC(\textsc{VetoLS}(\textsf{Odd}_n))=\Omega(2^{n/2})$.
\end{lemma}
\begin{proof}
We first embed $\textsf{Hyp}_n$ in $\textsf{Hyp}_{n+1}$ simply by $\phi_1(v)=(v,0)$ and $\chi_1(\{v,w\})=\{(v,0),(w,0)\}.$ Obviously this embedding is edge disjoint (but not vertex isolated).
We now embed $\textsf{Hyp}_{n+1}$ in $\textsf{Odd}_{n+2}$. We refer to each vertex of $\textsf{Hyp}_{n+1}$ as a subset $S\subset [n+1]$. We denote $S+n+1=\{i+n+1:i\in S\}$. We denote $T^c=[n+1]\setminus T$ (this notation will be relevant for subsets of $[n+1]$ rather than subsets of $[2n+3]$ as the vertices of $\textsf{Odd}_{n+2}$).
The embedding is defined by
\begin{align*}
\phi_2(S)=&S\cup (S^c + n+1). \\
\chi_2(S,S\cup \{i\})=&S\cup (S^c + n+1) \rightarrow (S^c\setminus \{i\}) \cup (S+n+1) \cup \{2n+3\} \rightarrow \\
& S \cup \{i\} \cup ((S\cup \{i\})^c + n+1)
\end{align*}
It is easy to check that this indeed defines a valid path on $\textsf{Odd}_{n+2}$. All the defined paths are disjoint because given a vertex on a path $T\cup (T'+n)\cup \{2n+3\}$ we can identify the edge: $S=T'$ and $i$ is the unique element that is missing from both sets $T$ and $T'$.
Now we define the embedding of $\textsf{Hyp}_n$ in $\textsf{Odd}_{n+2}$ to be the decomposition of these two embeddings; I.e., $\phi(v)=\phi_2(\phi_1(v))$ and $\chi(e)=\chi_2(\chi_1(e))$. The embedding $(\phi,\chi)$ is edge disjoint because both embeddings $(\phi_1,\chi_1)$ and $(\phi_2,\chi_2)$ are edge disjoint. Now we prove that $(\phi,\chi)$ is vertex isolated. A vertex $\phi_2(\phi_1(v))=S\cup (S^c + 2n)$ has $n+2$ neighbours in $\textsf{Odd}_{n+2}$. Among these neighbours, $n+1$ participate in an embedding of the outgoing edges of $S\in \textsf{Hyp}_{n+1}$. So there is a single neighbour, $S^c \cup (S+n+1)$, who is suspected to belong to an embedding of an independent edge. Note that $S^c \cup (S+n+1)=\phi_2(S^c)$ and $S^c\in \textsf{Hyp}_{n+1}$ \emph{does not} belong to the embedding of $\textsf{Hyp}_n$ in $\textsf{Hyp}_{n+1}$: indeed, for every vertex $v\in \textsf{Hyp}_n$ the complementary vertex $\overline{(v,0)}=(\overline{v},1)\in \textsf{Hyp}_{n+1}$ does not belong to the embedding of $\textsf{Hyp}_n$ in $\textsf{Hyp}_{n+1}$ (neither to $\phi_1(V_{\textsf{Hyp}_n})$ nor to $\chi_1(E_{\textsf{Hyp}_n})$).
\end{proof}
\subsection{Step 3: From $\textsc{VetoLS}$ to $\textsc{SumLS}$}\label{sec:veto-sum}
First, recall that the potential function gets values in $[W]$. We reduce the problem $\textsc{VetoLS}(G)$ to $\textsc{SumLS}(G)$.
Alice's potential remains unchanged (i.e., $f_A(v):=f_G(v)$). Bob fixes some valid vertex $v^*\in S$ and sets his potential as follows: $f_B(v):=0$ if $v\in S$, otherwise he sets $f_B(v)=-d(v,v^*)\cdot (W+1)$, where $d$ is the distance in $G$. Indeed every valid local maximum $v$ is a local maximum of the sum because all the valid neighbours have lower sum of potentials $f_A(v)+f_B(v)=f_G(v)\geq f_G(w)=f_A(v)+f_B(v)$ and all invalid neighbours have negative sum of potentials $f_A(w)+f_B(w)\leq W - (W+1)<0$. It is easy to check that every valid vertex that is not a local maximum is not a local maximum of the sum. Finally, every invalid vertex $v$ is not a local maximum of the sum because the neighbour $w$ in the direction of the shortest path to $v^*$ has higher sum of potentials:
\begin{align*}
f_A(v)+f_B(v) &\leq W - d(v,v^*)(W+1) < -(d(v,v^*)-1)(W+1) \\ &= -(d(w,v^*)-1)(W+1)\leq f_A(w)+f_B(w).
\end{align*}
We apply this reduction on the graphs considered in Lemmas \ref{lem:grid1}, \ref{lem:hyp} and \ref{lem:odd} to deduce the theorem.
\section{Proof of Theorem \ref{theo:grid}}\label{sec:grid-pr}
The overall structure of the proof is similar to that of Theorem \ref{theo:opt}.
\paragraph{Step 0} We start with a local-search-related communicationally-hard problem over some graph $H$.
\paragraph{Step 1} We use the intermediate problem $\textsc{VetoLS}(G)$, where $G$ is constructed from $H$.
\paragraph{Step 2} We embed $G$ in the three-dimensional grid.
\paragraph{Step 3} We reduce $\textsc{VetoLS}(\textsf{Grid})$ to $\textsc{SumLS}(\textsf{Grid})$.
\vspace{0.1in} \noindent However, in order to be able to embed $G$ in the three-dimensional grid, the degree of $G$ should be very low; at most 6.
The pebbling game result of \citep{GP14} does not serve our purposes because the degree of the graph $G$ is 36.
Hence, our starting point is some different local-search-related communicationally hard problem over some degree 3 graph $H$. In Step 1, we carefully modify $H$ to $G$ by increasing the degree only by 1; i.e., $G$ is degree 4 graph. Now, in Step 2 we are able to embed $G$ in the three-dimensional grid. Step 3 is identical to that in the proof of Theorem \ref{theo:opt}.
\subsection{Step 0: The Query Complexity of Local Search and its Simulated Variant}
In the problem $\textsc{QuLS}(H)$ there is a graph $H$ and a function $h$ that gives a value $h(v)$ for every vertex. The function $h$ can only be accessed via queries $h(v)$. Furthermore, for each two vertices $v,u$ are distinct: $h(v)\neq h(u)$. The goal is to find a local maximum of $h$ while minimizing the number of queries.
Santha and Szegedy \citep{SS} introduced a general connection between the query complexity of the local search problem and the expansion of a graph. Since random $3$-regular graphs are expanders with high probability, we have that there exists a degree 3 graph $H$ with $N$ vertices for which finding a local maximum requires $\textsf{poly}(N)$ queries. However, their construction does not assume that $h(v)\neq h(u)$ for every two vertices $v$ and $u$. This is easy to fix: let $h'(v)=2N\cdot h(v) +v$ (where $v\in [N]$ denotes the index of $v$). Observe that each local maximum of $h'$ is also a local maximum of $h$ and that the query $h'(v)$ can be computed by one query $h(v)$, so the number of queries required to find a local maximum of $h'$ is at least the number of queries required to find a local maximum of $h$. We therefore have:
\begin{lemma}[essentially \citep{SS}]
There exists a degree 3 graph $H$ with $N$ vertices and a function $h'$ such that every vertex has a distinct value for which finding a local maximum requires $\textsf{poly}(N)$ queries.
\end{lemma}
The simulation theorems provides us a recipe to produce problems with high communication complexity, given a problem with high query complexity. In particular, \citep{GPW,AGJKM} suggest the \emph{index-gadget} recipe, which starting from $\textsc{QuLS}(H)$ is translated to the following communication problem $\textsc{SimLS}(H)$: for each vertex $v\in H$, Alice holds an array of valuations $(f(v,i))_{i\in [M]}$ where $f(v,I(v))=h(v)$ and\footnote{E.g., $M=N^{256}$ in \citep{GPW}.} $M=\textsf{poly}(N)$. Bob holds the correct index $I(v)\in [M]$. Their goal is to compute a local maximum of the function $f(v,I(v))$. Direct application of the simulation theorems to our setting gives that:
$$CC(\textsc{SimLS}(H))=\Theta(\log N)QC(\textsc{QuLS}(H))=\textsf{poly}(N)$$
\subsection{Step 1: The Communication Complexity of $\textsc{VetoLS}$}
In this step we prove the communication hardness of $\textsc{VetoLS}$ on a certain bounded degree graph. We recall the definition of $\textsc{VetoLS}(G)$. Alice's input is a function $f_G:V\rightarrow [W]$. Bob's input is a non-empty subset $S\subset V$. The output is a vertex $v\in S$ such that $f_G(v)\geq f_G(w)$ for every $w\in S$ such that $\{v,w\}\in E$ (i.e., for every valid neighbour).
Unlike the communication pebbling game problem that uses index gadgets of size 3, the simulated $\textsc{QuLS}$ problem uses gadgets of size $M=\textsf{poly} N$ ($N$ is the number of vertices of $G$). The idea in the proof of Theorem \ref{theo:opt} is to replicate each vertex according to the gadget size, and connect every vertex with all its replicated neighbours. This idea is impractical here, because the degree of the resulting graph will be huge. Instead, we replace each replicated vertex with degree $3M$ by a carefully chosen binary tree structure in order to reduce the degree.
\begin{figure}[h]
\caption{The graph $G$. The replacement of a vertex by $M$ pairs of binary trees, and the neighbours of the leaves of $T^{out}$.}\label{fig:g}
\centering
\vspace*{3mm}
\begin{tikzpicture}
\node[circle, draw, minimum size=0.8cm] (v) at (-1,0) {$v$};
\node[circle, draw] (w1) at (-2,1) {$w_1$};
\node[circle, draw] (w2) at (-1,1) {$w_2$};
\node[circle, draw] (w3) at (0,1) {$w_3$};
\draw (v) -- (w1);
\draw (v) -- (w2);
\draw (v) -- (w3);
\draw[<->,ultra thick] (0.5,0) -- (1.5,0);
\filldraw[black] (3,0) circle (0.1);
\draw (3,0) -- (1.7,1.3) -- (4.3,1.3) -- (3,0);
\draw (3,0) -- (1.7,-1.3) -- (4.3,-1.3) -- (3,0);
\node at (3,1) {$T^{out}(v,1)$};
\node at (3,-1) {$T^{in}(v,1)$};
\filldraw[black] (6,0) circle (0.1);
\draw (6,0) -- (4.7,1.3) -- (7.3,1.3) -- (6,0);
\draw (6,0) -- (4.7,-1.3) -- (7.3,-1.3) -- (6,0);
\node at (6,1) {$T^{out}(v,2)$};
\node at (6,-1) {$T^{in}(v,2)$};
\filldraw[black] (7.8,0) circle (0.05);
\filldraw[black] (8,0) circle (0.05);
\filldraw[black] (8.2,0) circle (0.05);
\filldraw[black] (10,0) circle (0.1);
\draw (10,0) -- (8.7,1.3) -- (11.3,1.3) -- (10,0);
\draw (10,0) -- (8.7,-1.3) -- (11.3,-1.3) -- (10,0);
\node at (10,1) {$T^{out}(v,M)$};
\node at (10,-1) {$T^{in}(v,M)$};
\draw[decorate,decoration={brace,amplitude=6pt,mirror,raise=4pt},yshift=0pt]
(11.3,0.1) -- (11.3,1.3) node [black,midway,xshift=0.7cm] {
$3a$};
\draw[decorate,decoration={brace,amplitude=6pt,mirror,raise=4pt},yshift=0pt]
(11.3,-1.3) -- (11.3,-0.1) node [black,midway,xshift=0.7cm] {
$3a$};
\draw[decorate,decoration={brace,amplitude=6pt,mirror,raise=4pt},yshift=0pt]
(1.7,-1.3) -- (4.3,-1.3) node [black,midway,yshift=-0.6cm] {
$M^3$};
\filldraw[black] (3.8,1.3) circle (0.1);
\node at (3.8,1.6) {$t_{(i,j,k)}(v,1)$};
\filldraw[black] (3.5,4.3) circle (0.1);
\draw (3.5,4.3) -- (2.2,3) -- (4.8,3) -- (3.5,4.3);
\filldraw[black] (6.5,4.3) circle (0.1);
\draw (6.5,4.3) -- (5.2,3) -- (7.8,3) -- (6.5,4.3);
\filldraw[black] (9.5,4.3) circle (0.1);
\draw (9.5,4.3) -- (8.2,3) -- (10.8,3) -- (9.5,4.3);
\node at (3.5,3.3) {$T^{in}(w_1,i)$};
\node at (6.5,3.3) {$T^{in}(w_2,j)$};
\node at (9.5,3.3) {$T^{in}(w_3,k)$};
\draw (3.8,1.9) -- (2.5,3);
\draw (3.8,1.9) -- (5.5,3);
\draw (3.8,1.9) -- (8.5,3);
\end{tikzpicture}
\end{figure}
\paragraph{The Graph $G$.} Without loss of generality we assume that $M=2^a$ is a power of 2.
We obtain our graph $G$ by replacing every vertex $v\in H$ by a tuple of $M$ graphs $(T^{out}(v,i)\cup T^{in}(v,i))_{i\in M}$,
where $T^{out}(v,i)\cup T^{in}(v,i)$ denotes two binary trees with an overlapping root, both of depth $\log (M^3)=3a$ (see Figure \ref{fig:g}).
Roughly speaking, the role of $T^{out}(v,i)$ is to decode the correct indices of the three neighbours,
and in parallel to split the outgoing edges from $v_i$.
The role of $T^{in}(v,i)$ is simply to gather the incoming edges into $v_i$.
More formally, the vertices of $T^{out}(v,i)$ at depth $d$ are denoted by $(t_s(v,i))_{s\in \{0,1\}^d}$.
The vertices of $T^{in}(v,i)$ at depth $d$ are denoted by $(t'_s(v,i))_{s\in \{0,1\}^d}$. The vertices at depth $3a$ will be called \emph{leaves}\footnote{Note that they are leaves only with respect to the tree. In the graph $G$ they will not be leaves.}.
As was mentioned above, the vertex at depth 0 of these two trees coincides (i.e., $t_\emptyset (v,i) = t'_\emptyset (v,i)$).
Now we describe how the leaves of $T^{out}(v,i)$ connect to the leaves of $T^{in}(w,j)$ for $w\neq v$.
For a leaf $t_s(v,i)\in T^{out}(v,i)$ we denote $s=(j_1,j_2,j_3)$ where $j_1,j_2,j_3\in [M]$ are the indices of the three neighbors of $v$, $w_1,w_2,w_3$. The leaf $t_s(v,i)\in G$ has a single edge to the tree $T^{in}(w_1,j_1)$, a single edge to the tree $T^{in}(w_2,j_2)$, and a single edge to the tree $T^{in}(w_3,j_3)$ (see Figure \ref{fig:g}).
In principle, we should specify which leaf exactly in $T^{in}(w_1,j_1)$ is connected to $t_s(v,i)$. However,
since it will not play any role in our arguments, we just implement a counting argument to ensure that the
number of neighbours from other trees of every leaf $t'_{s'}(w,j)$ is at most 3. If $w$ has a neighbour $v$, then for every $i\in [M]$
exactly $M^2$ vertices $t_s(v,i)$ will encode the index $j$. So from $T^{out}(v,i)$ we have $M\cdot M^2=M^3$ incoming edges.
Summing over the 3 neighbours we get $3M^3$ incoming edges. If we distribute them equally among the $M^3$ vertices, we get 3 neighbours for each.
\paragraph{Alice's Potential. }
Alice's potential function is defined by $f_G(t'_s(v,i))=7af(v,i)+3a-|s|$ and $f_G(t_s(v,i))=7af(v,i)+3a+|s|$. Namely the potential in the tree $T'_s(v,i)$ starts at a value of $7af(v,i)$ in the leaves of $T^{in}(v,i)$. It increases by $1$ after every edge
until it gets to the root. At the root we move to the tree $T^{out}(v,i)$ where it proceeds to increase by $1$ until it gets to the leaves of $T^{out}(v,i)$ where the value of the potential is $7af(v,i)+6a$.
\paragraph{Bob's valid Vertices. }Now we define the subset of valid vertices $S$ held by Bob.
Let $bin(i)\in \{0,1\}^a$ denote the binary representation of an index $i\in [M]$.
We denote by $nbin(v)=(bin(I(w_i)))_{i=1,2,3}$ the binary
representation of the triple of $v$'s neighbours. For a binary string $b$ we denote by $b_{[k]}$ its first $k$ elements.
A vertex $t_s(v,i)\in S$ iff $i=I(v)$ and $s=nbin(v)_{[|s|]}$ (recall that $I(v)$ is Bob's input in $\textsc{SimLS}$). Informally speaking the valid vertices are those where the tree $T^{out}(v,i)$ (or $T^{in}(v,i)$) has the correct index, and if the vertex is in $T^{out}(v,i)$ we require, in addition, that the prefix of the encoding of the neighbours' indices will be correct.
\paragraph{Local Maxima in $G$.} Since the potential of Alice increases starting from the leaves of $T^{in}(v,i)$ and ending at the leaves of $T^{out}(v,i)$, and in addition for every valid vertex there exists a valid neighbour with higher (lower) depth in $T^{out}(v,i)$ (in $T^{in}(v,i)$) the valid local maxima appear only on the leaves of $T^{out}(v,i)$. Every valid leaf of $T^{out}(v,i)$ has a potential of $7af(v,I(v))+6a$ (i.e., the correct potential) and is connected to leaves of $T^{in}(w_j,I(w_j))$ for $j=1,2,3$ with a potential of $7af(w_j,I(w_j))$ (i.e., the correct potential of the neighbours). Note that the potential values are integers. Therefore, $7af(v,I(v))+6a \geq 7af(w,I(w))$ if and only if $f(v,I(v))\geq f(w,I(w))$. Hence, there is a one-to-one correspondence between valid local maxima of $f_G$ with respects to the set if valid vertices $S$ and local maxima of $h$ over $H$.
This completes the proof item \ref{theo:deg-bounded} of the Theorem.
\subsection{Step 2: Embedding the Degree 4 Graph Into the Grid}
We VIED embed (see Definition \ref{def:vied}) the degree 4 graph $G$ obtained in the previous step into the grid. We use Lemma \ref{lem:emb} to deduce hardness of $\textsc{VetoLS}$ over the grid.
\begin{lemma}\label{lem:grid}
Every degree 4 graph $G$ with $N$ vertices can be VIED-embedded in $\textsf{Grid}_{4N\times (2N+2) \times 2}$. As a corollary, $CC(\textsc{VetoLS}(\textsf{Grid}_{N\times N \times 2}))=\textsf{poly}(N)$.
\end{lemma}
\begin{proof}
We embed the graph $G$ in the grid whose vertices are $\{3,4,...,4N+2\}\times \{-1,0,...,2N\} \times \{0,1\}$. We denote the vertices of $G$ by $\{v_i\}_{i\in [N]}$ and we embed $\phi(v_i)=(4i,0,0)$. We use (for instance) the structure of Figure \ref{fig:paths} to place the four outgoing edges of $(4i,0,0)$ at the points $(4i-1,1,0),(4i,1,0),(4i+1,1,0)$ and $(4i+2,1,0)$.
\begin{figure}[h]
\caption{The outgoing edges of the embedded vertices.}\label{fig:paths}
\centering
\begin{tikzpicture}[scale=0.7]
\draw[step=1,gray,thin] (0,0) grid (16,3);
\draw[line width=3] (1,2) -- (1,0);
\draw[line width=3] (2,1) -- (0,1);
\draw[line width=3] (0,1) -- (0,2);
\draw[line width=3] (2,1) -- (2,2);
\draw[line width=3] (1,0) -- (3,0);
\draw[line width=3] (3,2) -- (3,0);
\draw[line width=3] (5,2) -- (5,0);
\draw[line width=3] (6,1) -- (4,1);
\draw[line width=3] (4,1) -- (4,2);
\draw[line width=3] (6,1) -- (6,2);
\draw[line width=3] (5,0) -- (7,0);
\draw[line width=3] (7,2) -- (7,0);
\draw[line width=3] (14,2) -- (14,0);
\draw[line width=3] (15,1) -- (13,1);
\draw[line width=3] (13,1) -- (13,2);
\draw[line width=3] (15,1) -- (15,2);
\draw[line width=3] (14,0) -- (16,0);
\draw[line width=3] (16,2) -- (16,0);
\filldraw (1,1) circle (0.3);
\filldraw (5,1) circle (0.3);
\filldraw (14,1) circle (0.3);
\node at (0,-0.5) {3};
\node at (1,-0.5) {4};
\node at (5,-0.5) {8};
\node at (14,-0.5) {$4N$};
\node at (-0.5,0) {-1};
\node at (-0.5,1) {0};
\node at (-0.5,2) {1};
\end{tikzpicture}
\end{figure}
We denote by $\{e_i\}_{i\in [m]}$ the edges in the graph $G$. Note that $m\leq 4N/2=2N$ because the graph degree is 4. The embedding of the edges is by an increasing order $e_1,...,e_m$. For an edge $e_i=(v_j,v_k)$ let $r_j\in \{-1,0,1,2\}$ be the minimal index such that the vertex $(4j+r_j,1,0)$ is not yet used by previous edges $\{e_{i'}\}_{i'<i}$. Similarly we define $r_k$. The edge $e_i=(v_j,v_k)$ is embedded to the path:
$$(4j+r_j,1,0)\leftrightsquigarrow (4j+r_j,i,0) \leftrightarrow (4j+r_j,i,1) \leftrightsquigarrow (4k+r_k,i,1) \leftrightarrow (4k+r_k,i,0) \leftrightsquigarrow (4k+r_k,1,0)$$
where $(x,y,z) \leftrightsquigarrow (x,y',z)$ denotes a straight line that consistently changes the second coordinate (similarly for $(x,y,z) \leftrightsquigarrow (x',y,z)$).
The embedding is VIED because all horizontal lines appear at $(\cdot,\cdot, 1)$ while all vertical lines appear at $(\cdot,\cdot, 0)$. The embedding is vertex isolated by the construction of Figure \ref{fig:paths}.
\end{proof}
Finally Step 3 is identical to Section \ref{sec:veto-sum}. We use the reduction from $\textsc{VetoLS}$ to $\textsc{SumLS}$ to deduce the Theorem.
\section{Identifying Ordinal Potential Games}\label{ap:ident-ord}
We will prove two results, one for two-player $N$-action games and one for $n$-player $2$ action games. In both we use the following two-player two-action game for $x,y\in \{0,2\}$:
\begin{center}
\begin{tabular}{|l|l|}
\hline
$2,1$ & $1,2$ \\ \hline
$1,y$ & $x,1$ \\ \hline
\end{tabular}
\end{center}
\noindent
This game has a better-reply cycle if and only if $x=y=2$.
\begin{proposition}\label{pro:2-ord}
Recognizing whether a two-player $N$-action game is an ordinal potential game requires $\textsf{poly}(N)$ bits of communication, even for randomized protocols.
\end{proposition}
\begin{proof}
Denote by $u'$ the two-player $2N\times 2N$ table that contains $N\times N$ copies of this game with the parameters $(x_{i,j},y_{i,j})_{i,j\in [N]}$. We denote by $u''$ the two-player $2N\times 2N$ game with the payoffs $u''(a,b)=(3\lceil \frac{a}{2}\rceil,3\lceil \frac{b}{2}\rceil)$. And we denote $u=u'+u''$. The game $u$ has a better-reply cycle if and only if there exist $i,j\in [N]$ such that $x_{i,j}=y_{i,j}=2$. Indeed if $x_{i,j}=y_{i,j}=2$, since we have added a constant payoff of $3i$ to player $1$ ($3j$ to player 2) to the $(i,j)$ copy of the game, the better reply cycle remains a better reply cycle in $u$. If $(x_{i,j},y_{i,j})\neq (2,2)$ for all $i,j$ then we have no better reply cycle within the copies of the $2\times 2$ games, and we have no better reply cycles across the $2\times 2$ games because at least one player has dominant strategy. Therefore the determination of ordinal potential property is as hard as disjointness, which requires $\textsf{poly}(N)$ communication, even with randomized communication.
\end{proof}
\begin{proposition}\label{pro:n-ord}
Recognizing whether an $n$-player $2$-action game is an ordinal potential game requires $2^{\Omega(n)}$ bits of communication, even for randomized protocols.
\end{proposition}
\begin{proof}
Consider an $(n+2)$-player game where for each profile $a\in \{0,1\}^n$ the last two players are playing the above $2\times 2$ game with parameters $x_a,y_a$. For the first $n$ players we set the utilities such that $1$ is dominant strategy (e.g., $u_i(a_i,a_{-i})=a_i$ for $i\in [n]$). Similarly to the previous arguments, the game contains a better reply cycle if and only if $x_a=y_a=2$ for some $a\in \{0,1\}^n$. Again, we obtain a reduction to disjointness.
\end{proof}
\section{Introduction}
This paper deals with the communication complexity of local search problems. The general problem involves a search over some ``universe'' $V$, for an element $v^* \in V$ that maximizes, {\em at least ``locally''}, some objective function $f:V\rightarrow \mathbb{R}$. The notion of ``locality'' is formalized by putting a fixed, known, neighbourhood structure $E$ on the set of elements, so the requirement of local optimality is that for all $u \in V$ such that $(v^*,u) \in E$ we have that $f(v^*) \ge f(u)$. The notion of local optimality is interesting from two points of view: first, it captures the outcome of a wide range of ``gradual-improvement'' heuristics where the neighbourhood structure represents the types of gradual improvements allowed, and second, locally-optimal solutions provide a notion of stability, where the neighborhood structure models the possible ``deviations'' from stability.
In the context of computational complexity, local search problems are captured by the complexity class PLS \citep{JPY} which is a subset of the well studied class TFNP (defined in \citep{MP} and studied, e.g., in \citep{PSY, BCEI, DGP, hubacek2017journey}): search problems for which a witness always exists (``total search problems'') and can by efficiently verified (``in NP''). The problem has also been widely studied in the model of {\em query complexity} where the cost of an algorithm is the number of black-box queries to the objective function $f$, from the pioneering work of \citep{Ald} on the Boolean hypercube, to a rather complete characterization of not only the deterministic query complexity but also the randomized and even quantum complexities on any graph \citep{SS,Aar,SY}.
The interest in analyzing local search from a communication complexity point of view is clear: in essentially any application, the objective function $f$ is not really given as a ``black box'' but is somehow determined by the problem structure. When this structure has any element of distributed content then communication may become an important bottleneck. The question of how the information is distributed is key: in the simplest imaginable scenario, the search space $V$ is split somehow between the (say, two) parties, where each party holds the values $f(v)$ for its subset of $v \in V$ (the fixed commonly known neighbourhood structure still involves all of $V$). However, in this scenario even a global maximum (which is certainly also a local one) can be easily found with a small amount of communication by each player finding the maximum among his subset, and only communicating and comparing the maxima of the parties. Thus, for the problem to be interesting we must split the information $f(v)$ of each vertex between the parties. There are various ways to do this and the most natural one, conceptually and in terms of applications, is probably to split $f$ as the {\em sum} of two functions $f_A:V\rightarrow \mathbb{R}$ and $f_B:V\rightarrow \mathbb{R}$ held by Alice and Bob. So we consider the following problem:
\vspace{0.1in}
\noindent
{\bf Definition:} For a fixed, commonly known graph $G=(V,E)$, the $\textsc{SumLS}(G)$ communication problem is the following:
Alice holds a function $f_A:V\rightarrow \{1,...,W\}$,
Bob holds a function $f_B:V\rightarrow \{1,...,W\}$, and their goal is to find a vertex $v^* \in V$ such that $f_A(v^*)+f_B(v^*) \ge f_A(u)+f_B(u)$ for all $u \in V$ with $(v^*,u) \in E$.
\vspace{0.1in}
\noindent Determining the communication complexity of $\textsc{SumLS}$ on certain families of graphs is easy. For example, a simple reduction from disjointness shows that the communication complexity of $\textsc{SumLS}$ on the clique with $n$ vertices is $\Omega(n)$. Our main theorem proves optimal lower bounds for several important families of graphs, all have small degree. The technical challenge is that the non-deterministic communication complexity of the problem on small degree graphs is clearly low: to {\em verify} that $v^*$ is a local optimum, Alice and Bob need only communicate the values $f(u)$ and $g(u)$ for the small number of $v^*$'s neighbours in the graph (note that the degree of all graphs that we consider is indeed small: $\log N$ or even constant). There are only a few results in the communication complexity literature that manage to prove good lower bounds for total problems where verification is easy, most notably for Karchmer-Wigderson games \citep{KW,KRW,RM} and for PPAD-like communication problems \citep{BR,GoosRub}.
\begin{maintheo*}
\
\begin{enumerate}
\item The communication complexity of local search on the $n$-dimensional hypercube with $N=2^n$ vertices is $\Omega(\sqrt{N})$.
\item The communication complexity of local search on a constant-dimension grid with $N$ vertices is $\Omega(\sqrt{N})$.
\item The communication complexity of local search on a specific family of constant degree graphs with $N$ vertices is $\Omega(\sqrt{N})$.
\item The communication complexity of local search on the odd graph with $N$ vertices is $\Omega(\sqrt[4]{N})$.
\end{enumerate}
\end{maintheo*}
We note that all our bounds hold for \emph{randomized} communication complexity. Interestingly, the first three bounds are optimal: first, since for these families of graphs an algorithm by \cite{Ald} finds a local optimum with $O(\sqrt N)$ queries in expectation, which clearly implies an analogous communication algorithm with the same efficiency.
Our proof starts from considering the communication variant of a pebbling game \cite{GP14}. $D=(V,E)$ is a known directed acyclic graph. The input is a boolean assignment for the vertices $b:V \rightarrow \{0,1\}$ such that every source is true ($b(v)=1$) and every sink is false ($b(v)=0$). The output is a false vertex whose all predecessors are true (i.e., $v\in V$ such that $b(v)=0$ and $b(u)=1$ for all $u\in V$, $(u,v)\in E$). \citep{GP14} consider the communication variant of the game which is obtained by distributing the information $b(v)\in \{0,1\}$ of every vertex by a constant size index-gadget $\{0,1\}^3\times [3] \rightarrow \{0,1\}$. They show that for some constant-degree graph $D$ with $N$ vertices the communication complexity of the problem is $\Theta(\sqrt{N})$, which is optimal.
Our proof is composed of three steps. The first step shows how to reduce the pebbling game to a variant of local search on a graph $G$ ($\textsc{VetoLS}$) where Alice holds
the function $f$ and Bob holds a set of valid vertices. The goal is to find a local maximum in the subgraph that is composed of the valid vertices.
The second step is the most technically challenging one. We first define a notion of embedding one graph to the other, and show that if a graph $G$ can be embedded into $H$ then the communication of $\textsc{VetoLS}(H)$ is at least that of $\textsc{VetoLS}(H)$. We then show that the graph $G$ obtained in the previous step can be embedded into each of the families considered in the theorem. This embedding is quite delicate and uses specifics properties of the graph $G$, since the number of vertices of $G$ and $H$ must be almost the same, in order to obtain an optimal bound of $\Omega(\sqrt N)$ for $\textsc{VetoLS}(H)$, where $N$ is the number of vertices of $H$.
Finally, in the third step we show that the communication complexity of $\textsc{VetoLS}$ on any graph is at least that of local search, thus establishing the theorem.
The constants that are obtained in our theorem are quite big (the dimension of the grid has to be at least $119$, and the degree of the constant degree graph is $36$). Thus, we also provide an alternative proof that obtains better constants, at the cost of a worse communication bound. Specifically, we show that there exists a specific family of $4$-degree graphs for which the communication complexity of local search is $\Omega(N^c)$ for some constant $c>0$. We also show a lower bound of the form $\Omega(N^c)$ for the three dimensional grid $N\times N\times 2$. The alternative proof uses the more recent and more generic ``simulation'' lemmas that ``lift'' lower bounds from the query complexity setting to the communication complexity setting \citep{GPW,GPW15,RM}, instead of the ``simulation'' lemma of \citep{GP14} that was developed for specific settings like the pebbling game. The main technical difficulty that we overcome is that the ``combination gadgets'' used in these lemmas (specifically the index function) are very different from the simple sum that we desire.
We now describe two applications of our basic lower bound. In both applications we study communication variants of problems that are known to be PLS complete, have low non-deterministic complexity and, as we show, high communication complexity.
\subsection{Potential Games}
The communication requirements for reaching various types of equilibria in different types of games have received a significant amount of recent interest (\cite{BR,GoosRub}) as they essentially capture the convergence time of arbitrary dynamics in scenarios where each player only knows his own utilities (``uncoupled dynamics'' \citep{HMas,HMan}) and must ``learn'' information about the others. Of particular importance here is the class of potential games \citep{MS}.
\vspace{0.1in}
\noindent
{\bf Definition:} An $n$-player game with strategy sets $A_1,...,A_n$ and utility functions $u_1,...,u_n$
is an \emph{exact potential game} if there exists a single potential function
$\phi : A_1 \times \cdots \times A_n \rightarrow \mathbb{R}$ so that for every player $i$, every two strategies $a_i,a'_i \in A_i$ and every tuple of strategies $a_{-i} \in A_{-i}$ we have that $u_i(a_i,a_{-i})-u_i(a'_i,a_{-i})=\phi(a_i,a_{-i})-\phi(a'_i,a_{-i})$.
The game is an \emph{ordinal potential function} if there exists a single potential function
$\phi : A_1 \times \cdots \times A_n \rightarrow \mathbb{R}$ so that for every player $i$, every two strategies $a_i,a'_i \in A_i$ and every tuple of strategies $a_{-i} \in A_{-i}$ we have that $sign(u_i(a_i,a_{-i})-u_i(a'_i,a_{-i}))=sign(\phi(a_i,a_{-i})-\phi(a'_i,a_{-i}))$, i.e., the value of the potential function increases if and only if the player improves his utility.
\vspace{0.1in}
The class of exact potential games includes, in particular, all congestion games. A key property of potential games (exact or ordinal) is that every sequence of better responses converges to an equilibrium and therefore every potential game always has a pure Nash equilibrium.
\citep{HMan} study the communication complexity of pure Nash equilibrium in \emph{ordinal} potential games. They consider $n$-player games where each player has four actions and show (by a reduction from disjointness) that exponential communication is required to distinguish between the case where the game is an ordinal potential game (and thus has a Nash equilibrium) and the case where the game is not a potential game and does not admit any Nash equilibrium. This immediately implies that finding an equilibrium in games that are guaranteed to have one takes $exp(n)$ bits of communication.
Does finding an equilibrium become any easier for \emph{exact} potential games? In \citep{Noam-blog-2} it was shown that exponentially many \emph{queries} are needed to find an equilibrium, but maybe in the communication model the problem becomes much easier. The technical challenge is again that the non-deterministic communication complexity of the problem is low, i.e, verifying that a certain profile is a Nash equilibrium does not require much communication (each player only has to make sure that he plays his best response). Nevertheless, we provide a ray of hope and show that in contrast to ordinal potential games, there is a randomized protocol that uses only $\textsf{polylog}(|A|)$ (when $|A|=|A_1|\cdot ... \cdot |A_n|$ is the game size) bits of communication and determines whether the game is an exact potential game or not.
We then show that although it is easy to recognize whether a game is an exact potential game or not, finding an equilibrium requires polynomial (in the size of the game) communication (and in particular exponential in the number of players). These results provide a negative answer to an open question posed in \citep{Noam-blog}.
\begin{theo}
For some constant $c>0$, the following problem requires at least $N^c$ communication (even randomized): Alice gets an $N \times N$ matrix $u_A$ and Bob gets an
$N \times N$ matrix $u_B$, they are promised that the game defined by these matrices is an (exact) potential game and they must output a pure Nash equilibrium of the game.
\end{theo}
\begin{theo}
For some constant $c>0$, the following problem requires at least $2^{cn}$ communication (even randomized): Alice gets the utility functions of the first $n$ players in a $2n$-player $2$-action game. Bob gets the utility functions of the last $n$ players. They are promised that the game defined by these matrices is an (exact) potential game and they must output a pure Nash equilibrium of the game.
\end{theo}
\noindent Our proofs are via reductions from local search on (certain) degree 4 graphs
in the two-player $N$-action case, and from local search on the hypercube in
the $2n$-player 2-action case. While the relation between equilibria of potential games and local maxima is well known and very simple, the reduction is actually quite subtle.
First the neighbourhood structures do not naturally match (in the two-player case), but more crucially
the input to the players here is very limited: only very specifically related matrices $u_A$ and $u_B$ give an (exact) potential game, while the lower bounds for local search were for arbitrary inputs.
We also show that the search for a pure Nash equilibrium in exact potential games can be formulated as a \emph{total search problem}: Either find a pure Nash equilibrium (that is guaranteed to exist in exact potential games) or provide a succinct evidence that the game is not an exact potential game. Interestingly such a succinct evidence of violation of exact potential property is guaranteed to exist by \citep{MS}. As an immediate corollary from our results we deduce hardness of this total search problem.
\subsection{Local Optima in Combinatorial Auctions}
Our second application concerns attempts to weaken the global optimality constraints in market allocations. Consider a combinatorial auction of $m$ indivisible items among $n$ players, each with his own valuation function $v_i$ that gives a real value to every subset of the items. The usual goal of optimizing social welfare aims to globally maximize $\sum_i v_i(S_i)$ over all allocations $(S_1,...,S_n)$ of the items.
A corresponding notion of equilibrium is the Walrasian equilibrium, which includes also a vector of prices $p_1,...,p_m$ such that every player receives his globally-optimal set of items at these prices.
While these notions provide very strong guarantees, they are usually ``too good to be true'': Walreasian equilibria only rarely exist and optimizing social welfare is
usually infeasible, in essentially any sense of the word, and in particular in the sense of requiring exponential communication \citep{nisan2006communication}.
Several papers have tried to relax the notion of a Walrasian equilibrium or similarly view the allocation problem as a game and analyze the equilibria in this game. In particular, in the model of simultaneous second price auctions \citep{christodoulou2008bayesian} it is easy to see that when the valuations are submodular every allocation that is \emph{locally optimal} can be part of an equilibrium in the game, and the same goes for the endowed equilibrium of \citep{BDO18}. Recall that a locally optimal allocation in a combinatorial auction is an allocation of the items $(S_1,\ldots, S_n)$ such that transferring any single item $j\in S_i$ to some other player $i'$ does not improve the welfare.
Since local optima play a central role in various relaxed notions of equilibria, an obvious question is whether they are easy to find. In \citep{BDO18} it is shown that for some succinctly represented submodular valuations it is PLS hard to compute a locally optimal allocation in combinatorial auction. Furthermore, in the query model it is shown that finding a locally optimal allocation is as hard as finding a local maximum in the odd graph. Combining the same reduction with our communication hardness of local search on the odd graph, we get that:
\begin{theo}
The communication complexity of finding a locally optimal allocation between two players with submodular valuations is $2^{\Omega(n)}$.
\end{theo}
\section{The Communication Complexity of Exact Potential Games}
Recall that a game is an exact potential game if there exists a potential function $\phi:A^n\rightarrow \mathbb R$, such that $\phi(a_i,a_{-i})-\phi(a'_i,a_{-i})=u_i(a_i,a_{-i})-u_i(a'_i,a_{-i})$ for every player $i$, every pair of actions $a_i,a'_i\in A_i$, and every profile of the opponents $a_{-i}\in A_{-i}$.
In this section we study the communication complexity of exact potential games. We assume that each of the players knows only his own utility function and the goal is to compute a pure Nash equilibrium in the game. In game theoretic settings this form of information distribution is called \emph{uncoupledness} \citep{HMas,HMan}. It is known that the communication complexity of computing an equilibrium captures (up to a logarithmic factor) the rate of convergence of uncoupled dynamics to equilibrium \citep{CS,HMan}.
As a preliminary result, we demonstrate that \emph{determining} whether a game is an exact potential games (under the uncoupled distribution of information) requires low communication. This result is in contrast to ordinal potential games (see Appendix \ref{ap:ident-ord}).
\begin{proposition}\label{pro:epd}
Consider a game with $n$ players and $N$ actions. There exists a randomized communication protocol that determines whether the game is an exact potential game or not that uses only $\textsf{poly}(\log(N),n)$ bits of communication.
\end{proposition}
The proof is quite simple, and we demonstrate it here for $2$-player games. Monderer and Shapley \citep{MS} show that a two-player game $(A,B,u_A,u_B)$ is an exact potential game if and only if for every four actions $a,a'\in A$, and $b,b'\in B$ we have
\begin{align}\label{eq:cycle}
\begin{split}
&(u_A(a',b)-u_A(a,b))+(u_B(a',b')-u_B(a',b)) \\
&+(u_A(a,b')-u_A(a',b'))+(u_B(a,b)-u_B(a,b'))=0
\end{split}
\end{align}
Namely, the sum of gains/losses from unilateral divinations over every cycle of size four should sum up to zero.
Now each player checks, for every possible four-action cycle, whether the sum of changes in his utility equals the negative of the change in utility of the other player for the same cycle. Verifying this simultaneously for all cycles can be done by applying any efficient protocol for the equality problem (we recall that we focus on \emph{randomized} communication protocols). For a general number of players, a similar characterization exists and we have to use protocols based on the ``equal sum'' problem as demonstrated below.
\begin{proof}[Proof of Proposition \ref{pro:epd}]
By \citep{MS}, an $n$-player game $(A,u)$ is an exact potential game if and only if for every pair of permutations $\overline{\pi},\underline{\pi}$ over $[n]$ and for every pair of action profiles $a,b\in A$ we have
\begin{align}\label{eq:n-cyc}
\begin{split}
&\sum_{k=1}^n u_{\overline{\pi}(k)}(b,a,\overline{\pi}([k]))-u_{\overline{\pi}(k)}(b,a,\overline{\pi}([k-1]))+ \\
&\sum_{k=1}^n u_{\underline{\pi}(k)}(a,b,\underline{\pi}([k]))-u_{\overline{\pi}(k)}(a,b,\underline{\pi}([k-1]))=0
\end{split}
\end{align}
Simply speaking, for every sequence of unilateral deviations that starts at $a$ goes back and forth to $b$, where each player changes his strategy from $a_i$ to $b_i$ once and from $b_i$ to $a_i$ once, the sum in the gains/losses of all players from the unilateral divinations should sum up to 0.
The players should check whether Equation \eqref{eq:n-cyc} holds for all possible pairs of profiles $a,b\in [N]^n$ and pairs of permutations $\overline{\pi},\underline{\pi}$ over $[n]$. The number of these equations is $c=m^{2n} (n!)^2$. Each player can generate from his private input a vector in $\{-2W,...,0,...,2W\}^c$ which captures the sum of changes in his utility for each one of the tuples $(a,b,\overline{\pi},\underline{\pi})$. So the problem can be reduced to the following:
Each player $i$ holds a vector $v_i\in \{-2W,...,0,...,2W\}^c$ and the goal of the players is to determine whether $\sum_{i\in [n]} v_i = \textbf{0}_c$. This variant of the equality problem has a $\textsf{poly}(\log W,\log c)=\textsf{poly}(n,\log N)$ randomized communication protocol \citep{nisan1993communication,viola2015communication}.
\end{proof}
In contrast, identifying whether a game is an \emph{ordinal} potential game is hard, even for randomized communication protocols. Identification of the ordinal potential property has a reduction to the disjointness problem. We relegate these reductions (for two-player and for $n$-player games) to Appendix \ref{ap:ident-ord}.
The contrast between the hardness of identifying whether a game is an ordinal potential game and the easiness of identifying whether a game is an exact potential game might give some hope that computing an equilibrium in exact potential games is much easier than in ordinal potential games. Unfortunately, our main results for this section show that finding a Nash equilibrium remains hard even for exact potential games.
\begin{theorem}\label{theo:2pot}
Consider the two-party promise communication problem where Alice holds the utility $u_A:[N]\times [N] \rightarrow \mathbb{R}$, and Bob holds the utility $u_B$ of an exact potential game. The goal is to output a pure Nash equilibrium of the game. The problem requires $\textsf{poly}(N)$ bits of communication, even for randomized protocols.
\end{theorem}
We can also show hardness for the $2n$-player $2$-action case.
\begin{theorem}\label{theo:n-pot}
Consider the two-party promise communication problem where Alice holds the utilities of $(u_i)_{i\in [n]}$ and Bob holds the utilities $(u_i)_{i\in [2n]\setminus [n]}$ of an exact potential game, and they should output a pure Nash equilibrium of the game.
The problem requires $2^{\Omega(n)}$ communication, even for randomized protocols.
\end{theorem}
This problem is obviously requires at least as much communication as the $2n$-party communication problem where each player holds his own utility function.
In both theorems, we reduce from the problem of finding a local maximum (on a bounded degree graph in the two player case and on the hypercube in the $n$ player case) and show that the set of pure Nash equilibria corresponds exactly to the set of local maxima. The proofs of the Theorems appear in Sections \ref{sec:pr-2} and \ref{sec:pr-n}.
\subsection{Total variants of Pure Nash Equilibrium Search}\label{sec:tot}
In Theorems \ref{theo:2pot} and \ref{theo:n-pot} we have demonstrated communicational hardness of two \emph{promise} problems. Such hardness results are not rare in the literature. For instance, finding a pure Nash equilibrium in a game when it is promised that such an equilibrium exists.
To appreciate the novelty of our results we focus on a \emph{total} variant of equilibrium search problem $\textsc{TotExPot}$: either find a Nash equilibrium or provide a succinct evidence that the game is not an exact potential game. By \citep{MS} such a succinct evidence, in the form of a violating cycle (see Equations \eqref{eq:cycle},\eqref{eq:n-cyc}), necessarily exists.
More formally, in the problem $\textsc{TotExPot}(2,N)$ Alice holds the utility $u_A$, Bob holds a utility $u_B$ of an $N\times N$ game, and the output is either a pure Nash equilibrium or a cycle of actions of size 4 that violates Equation \eqref{eq:cycle}. Similarly in the problem $\textsc{TotExPot}(2n,2)$ Alice holds the utilities $(u_i)_{i\in n}$, Bob holds the utilities $(u_i)_{i\in [2n]\setminus [n]}$ of an $2n$-player 2-action game, and the output is either a pure Nash equilibrium or a cycle of actions of size $4n$ that violates Equation \eqref{eq:n-cyc}.
In Proposition \ref{pro:epd} we showed that low communication is needed to determine whether a game is an exact potential game or not (accompanied with an evidence in case it is not). From these observation along with Theorem \ref{theo:2pot} we deduce that
\begin{corollary}
The total search problem $\textsc{TotExPot}(2,N)$ requires $\textsf{poly}(N)$ communication.
\end{corollary}
Similarly for the $2n$-player 2-action case we have
\begin{corollary}
The total search problem $\textsc{TotExPot}(2n,2)$ requires $2^{\Omega(n)}$ communication.
\end{corollary}
Note that the non-deterministic complexity of $\textsc{TotExPot}(2,N)$ is $\log(N)$. Indeed a Nash equilibrium can be described by single action profile ($\Theta(\log N)$ bits), and a violating cycle can be described by $4$ action profiles. Each player can verify his best-reply condition and communicate a single bit to the opponent. Also verification of violating cycle can be done by communicating 4 valuations of utility.
Similarly, we can show that the non-deterministic complexity of $\textsc{TotExPot}(2n,n)$ is $\textsf{poly}(n)$. Thus again, our results demonstrate an exponential separation between the non-deterministic and the randomized communication complexity of a total search problem.
\section{Proof of Theorem \ref{theo:2pot}}\label{sec:pr-2}
We reduce the problem of finding a local maximum on a graph $G$ with degree $4$ to finding a Nash equilibrium in an exact potential game with two players and $N$ actions. We then apply Theorem \ref{theo:grid}(\ref{theo:deg-bounded}) to get our communication bound.
We construct the following exact potential game. For a vertex $v\in V$ we denote by $n_i(v)$ the $i$'th neighbour of $v$ for $i=1,2,3,4$. The strategy set of both players is $A=B=V\times [W]^5$ (recall that the potentials in $\textsc{SumLS}(G)$ get values in $[W]$ and that $W=\textsf{poly}(N)$). The interpretation of a strategy $(v,x)\in A$ where $\overrightarrow x=(x_0,x_1,...,x_4)\in [W]^5$ is (Alice's reported) potential for $v$ and its four neighbours. This report induces a valuation for all vertices $w\in V$ by
\begin{align*}\label{eq:rep-val}
val^{(v,\overrightarrow x)}(w)=
\begin{cases}
x_0 & \text{ if } w=v; \\
x_i & \text{ if } w=n_i(v); \\
0 & \text{ otherwise.}
\end{cases}
\end{align*}
A strategy $(v,\overrightarrow x)$ is \emph{truthful} if and only if $x_0=f_A(v)$ and $x_i=f_A(n_i(v))$ for all neighbours of $v$ (in short, $\overrightarrow x=n(v)$). Similarly Bob's strategy $(w,\overrightarrow y)$ induces a valuation $val^{(w,\overrightarrow y)}(v)$ on all vertices $v\in V$, and a truthful report is similarly defined.
The utilities of Alice and Bob are given by (recall that $d(v,w)$ is the distance in the graph between two vertices $v$ and $w$):
\begin{equation*}
\begin{split}
& u_A((v,\overrightarrow x),(w,\overrightarrow y))= 4W\cdot \mathds{1}_{d(v,w)\leq 1}+4W\cdot \mathds{1}_{x=n(v)} + val^{(w,\overrightarrow y)}(v) + val^{(v,\overrightarrow x)}(w)+ f_A(v) \\
& u_B((v,\overrightarrow x),(w,\overrightarrow y))= 4W\cdot \mathds{1}_{d(v,w)\leq 1}+4W\cdot \mathds{1}_{y=n(w)}
+ val^{(w,\overrightarrow y)}(v) + val^{(v,\overrightarrow x)}(w)+ f_B(w)
\end{split}
\end{equation*}
Namely, both players get large reward of $4W$ if they choose adjacent vertices, or the same vertex. Both players get large reward of $4W$ if they report truthfully their own valuations in the neighbourhood of their vertex. Both players get the sum of valuations of the two chosen vertices $v,w$ according to the report of the opponent. In addition Alice gets the (partial) potential of her vertex according to $f_A$, and Bob gets the potential of his vertex according to $f_B$.
\begin{lemma}\label{lem:exact}
The game is an exact potential game.
\end{lemma}
\begin{proof}
We will see that the game can be ``decomposed'' to two exact potential games, and will use this ``decomposition'' to provide a potential function for our game. We will use the following basic properties of potential games. We recall the notation of $(A_1,A_2,u_1,u_2)=(A,u)$ for a two-player game, where each $A_i$ is the action space of player $i$ and $u_i$ is the utility function of player $i$.
\begin{itemize}
\item An \emph{identical interest game} $(A,u)$ is a game in which $u_1=u_2$. An identical interest game is an exact potential game with potential function $\varphi=u_1$.
\item An \emph{opponent independent game} is a game in which the utility of each player $i$ depends only on his own actions: $u_i(a_1,a_2)=u_i(a_i)$ for every $(a_1,a_2)\in A$. Every opponent independent game is an exact potential game where the potential function is simply the sum of the utilities of the players.
\item For every pair of exact potential games $(A,u'),(A,u'')$ with potentials $\varphi',\varphi''$, the game $(A,u'+u'')$ is an exact potential game with potential $\varphi=\varphi'+\varphi''$.
\end{itemize}
Note that our game can be written as a sum of an identical interest game:
$$u'_A=u'_B= 4W\cdot \mathds{1}_{d(v,w)\leq 1} + val^{(w,\overrightarrow y)}(v) + val^{(v,\overrightarrow x)}(w)$$
and an opponent independent game:
$$u''_A=4W\cdot \mathds{1}_{\overrightarrow x=n(v)} + f_A(v), \ u''_B=4W\cdot \mathds{1}_{\overrightarrow y=n(w)} + f_B(w)$$
Therefore their sum is a potential game with potential:
\begin{align}\label{eq:pot}
\begin{split}
\phi((v,\overrightarrow x),(y,\overrightarrow w))= & 4W\cdot \mathds{1}_{d(v,w)\leq 1}+4W\cdot \mathds{1}_{\overrightarrow x=n(v)}+4W\cdot \mathds{1}_{\overrightarrow y=n(w)} \\
& + val^{(w,\overrightarrow y)}(v) + val^{(v,\overrightarrow x)}(w)+ f_A(v) +f_B(w)
\end{split}
\end{align}
\end{proof}
\begin{lemma}\label{lem:pne}
The pure Nash equilibria of the game are precisely $((v,\overrightarrow x),(v,\overrightarrow x'))$ such that $v$ is a local maximum of $f_A+f_B$ and $\overrightarrow x$ and $\overrightarrow x'$ are truth reports of the values of $v$ and its neighbours according to $f_A$ and $f_B$, respectively.
\end{lemma}
\begin{proof}
Pure Nash equilibria are the local maxima (with respect to a unilateral deviation) of the potential. It is easy to check that in a local maximum $x$ and $y$ are truth reports, because the gain in a truthful report is $4W$ whereas if the players do not report truthfully they lose this reward. However, Alice can gain at most $val^{(v,\overrightarrow x)}(w)+ f_A(v)\leq 2W$ from misreporting the value, and Bob's loss is similar. Similarly, in a local maximum $v$ and $w$ are neighbours (or the same vertex), because the gain of $4W$ is lost if $v$ and $w$ are not neighbors, in which case Alice's gain from the terms $val^{(w,\overrightarrow y)}(v) + val^{(v,\overrightarrow x)}(w)+ f_A(v)$ is at most $3W$. A similar argument holds for Bob. For a profile of strategies that satisfies the above the potential is equal to (see Equation \eqref{eq:pot}):
\begin{align}\label{eq:tpot}
\phi((v,\overrightarrow x),(w,\overrightarrow y))=12W+f_B(v)+f_A(w) +f_A(v)+ f_A(w)
\end{align}
A profile where $v\neq w$ is not a Nash equilibrium because by the distinctness assumption, $f_A(v)+ f_B(v) \neq f_A(w)+ f_B(w)$, so if $f_A(v)+ f_B(v) < f_A(w)+ f_B(w)$ Alice can deviate to $(w,\overrightarrow y)$ and increase the potential; Otherwise Bob can deviate to $(v,\overrightarrow x)$ and increase the potential. Finally, a profile $((v,\overrightarrow x),(v,\overrightarrow x'))$ with truth reporting is clearly a Nash equilibrium if it is a local maximum of $f_A+f_B$. If $v$ is not a local maximum of $f_A+f_B$, then Alice will increase the potential (given in Equation \eqref{eq:tpot}) if she deviates to the action $(w,n(w))$ where $w$ is a neighbour of $v$ with $f_A(w)+f_B(w)>f_A(v)+f_B(v)$.
\end{proof}
Lemmas \ref{lem:exact} and \ref{lem:pne} complete the proof of the theorem.
\section{Proof of Theorem \ref{theo:n-pot}}\label{sec:pr-n}
The proof of Theorem \ref{theo:n-pot} is done in two steps. First, we show a $2^{\Omega(\sqrt[3]{n})}$ bound. This is the significant part, in terms of the deduced result and also in terms of the techniques. Thereafter, in Section \ref{sec:2n} we improve the bound to $2^{\Omega(n)}$ building upon the arguments of this Section.
We start with proving the $2^{\Omega(\sqrt[3]{n})}$ bound. Our starting point is the proof of the hardness of $2$-player $n$-actions exact potential games (Theorem \ref{theo:2pot}). However, since we consider $n$-player binary-action games, it is convenient to reduce the problem $\textsc{SumLS}(\textsf{Hyp}_n)$ (local search on the $n$-th hypercube). We will get an exact potential game with $\Theta(n^3)$ players, where each player has only two actions.
\paragraph{A naive approach and an obstacle. } The simplest idea that comes to mind is to consider a \emph{group} of $n$-players who will choose $v\in \textsf{Hyp}_n$, and a \emph{group} of $(n+1) \lceil\log W \rceil$ players who will report the valuation vector $\overrightarrow x$ of the vertex itself and its $n$ neighbours, and similarly for Bob. We would like to set the group of Alice's players an \emph{identical utility} that is similar to the utility of Alice in the two-player game. An obstacle that arises with this approach is that if the groups of Alice's and Bob's players are playing two adjacent vertices $v,w\in \textsf{Hyp}_n$ with truthful valuations, none of them will want to switch to the opponent's vertex, even if at the adjacent vertex the sum of $f_A+f_B$ is higher. This follows from the fact if $(v,\overrightarrow x)$ is a truthful valuation, then $(w,\overrightarrow x)$ is not necessarily a truthful valuation (because the relevant vertices and their order is different with respect to $v$ and with respect to $w$). Thus, players in Alice's group will gain the difference in the potentials (at most $3W$) but lose $4W$ because now the group report is not truthful. Note that the same obstacle does not arise in the two-player case. In the two-player case Alice could change the vertex $v$ and the report $\overrightarrow x$ \emph{simultaneously}. In the $n$-player case we consider unilateral deviations that correspond to changes of single bits and thus such simultaneous deviations are impossible.
\paragraph{The solution to the obstacle. } To resolve the above problematic issue, we modify the form of the report $\overrightarrow x$ in the game.
\begin{itemize}
\item Instead of reporting the values in the ball of radius 1 around $v$ (i.e., the neighbors of $v$), each player reports the values in the ball of radius 2 around $v$. In the hypercube, this means that the report consists of $m=1+n+\frac{n(n-1)}{2}$ valuations.
\item Instead of reporting the values in a fixed order (namely $(v,n_1(v),...,n_4(v))$), the players jointly report pairs, where each pair consists of an index of a vertex $v$ and $f_A(v)$ (or $f_B(v)$).
\end{itemize}
\paragraph{The construction. }More formally, for Alice, we have a group of $n$ players with binary actions who jointly choose the vertex $v\in \{0,1\}^n$. In other words, the action of the $i$'th player in the group corresponds to the $i$'th bit in the index of the vertex. We have a group of $mn$ players with binary actions who jointly choose a list of $m$ vertices $\overrightarrow{xv}= (xv_1,...,xv_m)\in (\{0,1\}^n)^m$. Finally, we have a group of $m b:=m\lceil \log W \rceil$ players with binary actions who jointly choose a list of $m$ valuations $\overrightarrow{xf}=(xf_1,...,xf_m)\in (\{0,1\}^b)^m$. We denote $\overrightarrow{x}=(\overrightarrow{xv},\overrightarrow{xf})$.
Similarly to the two-player case, a report $\overrightarrow{x}=(\overrightarrow{xv},\overrightarrow{xf})$ defines a valuation function over all vertices.
For a list $\overrightarrow{xv}$ we denote $I_{\min}(\overrightarrow{xv}):=\{i\in [m]: xv_i \neq xv_j \text{ for all } j<i\}$ the set of indices with \emph{first} appearance of a vertex. The valuation is defined by
\begin{align*}
val^{\overrightarrow{x}}(w)=\begin{cases}
val(xf_i) &\text{if } w=xv_i \text{ for } i\in I_{\min}(\overrightarrow{xv}); \\
0 &\text{otherwise.}
\end{cases}
\end{align*}
where $val(\cdot)\in [W]$ denotes the numerical value of the binary string. Note that in case of multiple appearances of $w$ in the list we choose the value at the first appearance.
Similarly for Bob, we have three groups who jointly choose $w$, $\overrightarrow{yw}$, and $\overrightarrow{yf}$. The report $\overrightarrow{y}$ defines a valuation function $val^{\overrightarrow{y}}$ over all vertices. Note that the total number of players in the game is $2(n+m(n+b))=O(n^3)$.
Before we present the actual utilities we informally describe the prioritization according to which we set the utilities. In the two-player case there were only two levels of prioritization: the top level priority included the distance $d(v,w)$ (the $\mathds{1}_{d(v,w)\leq 1}$ term in the utility functions) and the truthfulness of the report (the $\mathds{1}_{x=n(v)}$ term in the utility functions). The bottom level priority included the remaining potential related terms ($val^{(w,\overrightarrow y)}(v), val^{(v,\overrightarrow x)}(w), f_A(v)$). More formally by \emph{prioritization} we mean that improving the higher priority term by $1$ should increase the utility \emph{irrespective} of how the lower priority terms change. Indeed the multiplier $4W$ was set in such a way. In the current construction, the prioritization levels are more involved, and we sketch them here from the highest priority to the lowest.
\begin{enumerate}
\item The distance $d(v,w)$.
\item The list $\overrightarrow{xv}$ should contain $v$ and its neighbours.
\item The valuations $\overrightarrow{xf}$ should be correct for $v$ and its neighbours.
\item The potential related terms (the core of the proof).
\item The list $\overrightarrow{xv}$ should contain the vertices within a distance 2 from $v$.
\item The valuations $\overrightarrow{xf}$ should be correct for vertices within a distance 2 from $v$.
\end{enumerate}
Now we describe what is the analogue of each one of these priorities in the $n$-player case. Hereafter, $d(\cdot,\cdot)$ will denote the \emph{hamming distance} (in the corresponding dimension). We denote by $B_r(v)$ the ball of radius $r$ around $v$ with respect to the hamming distance.
\begin{enumerate}
\item $\mathds{1}_{d(v,w)\leq 1}$ is translated to $-d(v,w)\cdot \mathds{1}_{d(v,w)\geq 2}$. Namely the loss is 0 in case the players choose the same vertex or adjacent vertices. Otherwise the loss increases with the distance.
\item Given $v$, we denote by $N_1(v):=\{(v_1,...,v_m): \{v_1,...,v_m\} \supset B_1(v)\}\subset \{0,1\}^{mn}$. Namely, $N_1(v)$ specifies $v$ and its neighbours. At the second priority we have $-d(\overrightarrow{xv},N_1(v))$.
\item Given $v$ and $\overrightarrow{xv}$, for an index $i\in I_{\min}(\overrightarrow{xv})$ such that $xv_i\in B_1(v)$ we have at the third priority the term $-d(xf,bin(f_A(xv_i))$ when we recall that $bin(z)\in \{0,1\}^b$ represents the binary representation of the potential value $z\in [W]$. Note that this definition takes into account only the \emph{first} appearance of every neighbour, which is consistent with the definition of $val^{\overrightarrow{x}}$. For other indices $i\in [m]$ the term will be identical but it will appear at the lowest sixth priority.
\item The profile $(v,\overrightarrow{x}),(w,\overrightarrow{y})$ defines a natural analogue of the two-player potential terms: $val^{\overrightarrow y}(v), val^{\overrightarrow y}(w), f_A(v), f_B(w)$. These terms are at the forth priority.
\item Given $v$, we denote by $N_2(v):=\{(v_1,...,v_m): \{v_1,...,v_m\} = B_2(v)\}\subset \{0,1\}^{mn}$ the lists that include precisely the set of all vertices within a radius 2 from $v$. At the fifth priority we have $-d(\overrightarrow{xv},N_2(v))$.
\item Finally, similarly to item 3, given $v$ and $\overrightarrow{xv}$, for every index $i\in [m]$ we have at the sixth priority the term $-d(xf,bin(f_A(xv_i))$.
\end{enumerate}
Now we are ready to define the utilities. As was mentioned above all the players in Alice's groups have identical utilities which is equal to:
\begin{align*}
u^A_i(v,\overrightarrow{x},w,\overrightarrow{y})=
& - k_1 \cdot d(v,w)\mathds{1}_{d(v,w)\geq 2} \\
& - k_2 \cdot d(\overrightarrow{xv},N_1(v)) \\
& - k_3 \cdot \sum_{i\in I_{\min}(\overrightarrow{xv}) \text{ s.t. } xv_i\in B_1(v)} d(xf,bin(f_A(xv_i)) \\
& + k_4 [val^{\overrightarrow y}(v) + val^{\overrightarrow x}(w) + f_A(v)] \\
& - k_5 \cdot d(\overrightarrow{xv},N_2(v)) \\
& - k_6 \cdot \sum_{i\in [m]} d(xf,bin(f_A(xv_i)),
\end{align*}
when we set $k_1,...,k_6$ as follows. We set $k_6=1$. Now we set $k_5$ to be greater than the maximal difference of sixth priority terms, e.g., $k_5=2n^2 b>mb$. Now we set $k_4$ to be the greater than the maximal total difference of sixth and fifth priority terms, e.g., $k_4=2n^3 b>mb+k_5(nm)$. Similarly we may proceed with $k_3 = 8W n^3 b$, $k_2 = 8Wn^5 b^2$, and $k_1=8Wn^8 b^2$.
Similarly we define each member in Bob's group to have the following identical utility function:
\begin{align*}
u^B_i(v,\overrightarrow{x},w,\overrightarrow{y})=
& - k_1 \cdot d(v,w)\mathds{1}_{d(v,w)\geq 2} \\
& - k_2 \cdot d(\overrightarrow{yw},N_1(w)) \\
& - k_3 \cdot \sum_{i\in I_{\min}(\overrightarrow{yw}) \text{ s.t. } yw_i\in B_1(w)} d(yf,bin(f_A(yw_i)) \\
& + k_4 [val^{\overrightarrow y}(v) + val^{\overrightarrow x}(w) + f_B(w)] \\
& - k_5 \cdot d(\overrightarrow{wy},N_2(w))\\
& - k_6 \cdot \sum_{i\in [m]} d(yf,bin(f_A(yw_i)).
\end{align*}
\begin{lemma}\label{lem:potential}
The defined $(2n+2m(n+b))$-player binary action game is an exact potential game.
\end{lemma}
\begin{proof}
If we view the game as a \emph{two}-player game where Alice chooses $(s,\hat{x})$ and Bob chooses $(r,\hat{y})$ the game is an exact potential game by similar arguments to those in Lemma \ref{lem:exact}. Namely it is the sum of two games where one is identical interest game and the other is opponent independent game. The potential function of the game is given by:
\begin{align*}
\phi(v,\overrightarrow{x},
& w,\overrightarrow{y})= - k_1 \cdot d(v,w)\mathds{1}_{d(v,w)\geq 2} \\
& - k_2 [d(\overrightarrow{xv},N_1(v))+d(\overrightarrow{yw},N_1(w))] \\
& - k_3 [\sum_{i\in I_{\min}(\overrightarrow{xv}) \text{ s.t. } xv_i\in B_1(v)} d(xf,bin(f_A(xv_i)) + \sum_{i\in I_{\min}(\overrightarrow{yw}) \text{ s.t. } yw_i\in B_1(w)} d(yf,bin(f_A(yw_i))] \\
& + k_4 [val^{\overrightarrow y}(v) + val^{\overrightarrow x}(w) + f_A(v)+f_B(w)] \\
& - k_5 [d(\overrightarrow{xv},N_2(v))+d(\overrightarrow{yw},N_1(w))] \\
& - k_6 [\sum_{i\in [m]} d(xf,bin(f_A(xv_i))+\sum_{i\in [m]} d(yf,bin(f_B(yw_i))]
\end{align*}
Note that by replacing Alice (Bob) by a group of $n+m(n+b)$ players all with the same utility we only reduced the set of possible unilateral deviations. For each one of these unilateral deviation by the two-player result the change is the utility is equal to the change in the potential.
\end{proof}
\begin{lemma}\label{lem:npne}
Every pure Nash equilibrium of the defined $(2n+2m(n+b))$-player binary action game is of the form $(v,\overrightarrow{x},v,\overrightarrow{y})$ where $v$ is a local maximum of $f_A+f_B$ over the hypercube.
\end{lemma}
\begin{proof}
The proof proceeds by narrowing the set of equilibria candidates according to the prioritization levels, with a twist at the fourth priority level.
First, in every equilibrium $d(v,w)\leq 1$ because otherwise there exists a player in Alice's $v$ group who can switch his strategy and decrease the distance by 1. Such a switch increases the first term in the utility of the group by $k_1$. By the choice of $k_1$, any change in the other terms of utilities is smaller.
Second, in every equilibrium $\overrightarrow{xv}\in N_1(v)$, because otherwise there exists a player in Alice's $\overrightarrow{xv}$ group who can switch his strategy and decrease the distance by 1. Such a switch does not effect the first term of the utility, and it increases the second term by $k_2$. By the choice of $k_2$, any change in the other terms of utilities is smaller. Similarly for Bob we have $\overrightarrow{yw}\in N_1(w)$.
Third, in every equilibrium for every $i\in I_{\min}(\overrightarrow{xv})$ such that $xv_i\in B_1(v)$ we have $xf_i=bin(f_A(xv_i))$. Simply speaking, all first appearances of elements in $B_1(v)$ (which indeed appear by the argument regarding the second priority level) have correct valuation. If it wasn't so, then there exists a player in Alice's $\overrightarrow{xf_i}$ group who can switch his strategy and decrease the distance by 1. Such a switch does not affect the first two terms of the utility, and it increases the third term by $k_3$. By the choice of $k_3$, any change in the other terms of utilities is smaller. Similarly for Bob, all first appearances of elements in $B_1(w)$ have correct valuation.
Now we jump to the fifth and the sixth priority levels. Given that $v,w$ are neighbours (or the same vertex) and their values already appear in the report $\overrightarrow{x}$ the terms of the utility in the fourth priority level are not affected by the vertices $xv_i$ such that $i\notin I_{\min}(\overrightarrow{xv})$ or $xv_i \notin B_1(v)$. Therefore, we can deduce that necessarily in equilibrium we have $\overrightarrow{xv}\in N_2(v)$ because otherwise some player in the $\overrightarrow{xv}$ group can decrease the distance by 1 without affecting any of the first four terms, and increase the fifth term by $k_5$. Any change in the last terms is smaller.
Similarly we can argue for the sixth priority level, that the values of $xf_i$ for the corresponding indices do not affect any other term.
From these arguments it follows that in any equilibrium both Alice (and Bob) report a list $\overrightarrow{xv}$ ($\overrightarrow{yw}$) that contains exactly all the vertices in the ball of radius 2 around $v$ ($w$), moreover all valuations of all these vertices are correct.
Now we go back to the fourth priority. Assume by way of contradiction that $v\neq w$. Similarly to the two-player case, the fourth term in the \emph{potential function} of the game is $val^{\overrightarrow{x}}(w)+val^{\overrightarrow{y}}(v)+f_A(v)+f_B(w)$, which under all the above restrictions of equilibria is equal to $f_A(w)+f_B(v)+f_A(v)+f_B(w)$.
Assume by way of contradiction that $v\neq w$, then we may assume w.l.o.g. that $f_A(w)+f_B(w)\geq f_A(v)+f_B(v)+1$ (we recall that we may assume that the sum defers at adjacent vertices and has integer values), then there exists a player in Alice's $v$ group who can switch his bit and turn the vertex $v$ into $w$. Let us examine the effect of this change on the potential. The first priority level term remains 0. The key observation is that the second and third priority level terms also remain 0. Note that the list $\overrightarrow{xv}$ includes all the vertices within radius 2 from $v$, and in particular all the vertices within radius 1 from $w$. Similarly the valuations $\overrightarrow{xf}$ of these vertices remain correct. Therefore the potential increases by at least $k_4$ in the first four terms, and any change in the fifth and sixth terms is smaller.
Finally for the case of $v=w$ where $v$ is not a local maximum we apply very similar arguments: There exists a player in Alice's group who can increase the potential of the game by $k_3$ and change only the fifth and sixth terms of the potential.
\end{proof}
Lemmas \ref{lem:potential}, and \ref{lem:npne} complete the proof of the $2^{\Omega(\sqrt[3]{n})}$ bound.
\subsection{Proof of Theorem \ref{theo:n-pot}: Improving the Bound to $2^{\Omega(n)}$}\label{sec:2n}
The presented above reduction has $\Theta(n^3)$ players, which yields a lower bound of $2^{\Omega(\sqrt[3]{n})}$ on the problem of finding a pure Nash equilibrium. Here we modify the reduction to have $\Theta(n)$ players, which implies a lower bound of $2^{\Omega(n)}$ on the problem of finding a pure Nash equilibrium. The idea is to reduce the unnecessary ``wasting" of players in the reduction. In the presented reduction Alice reports to Bob the valuations of \emph{all} vertices within radius 2 around $v$ (there are $\Theta(n^2)$ such vertices). However, the arguments of the proof of Theorem \ref{theo:grid} can be modified to show the existence of hard instances over the hypercube where for most of the neighbours within radius 2 from $v$, Alice and Bob \emph{know} the valuations of each other over these vertices. In fact, for these hard instances there exist only a \emph{constant} number of neighbours for which Alice does not know Bob's valuation, and Bob does not know Alice's. In the modified reduction, Alice's group will report only the valuation of the unknown vertices, which will require only $O(n)$ players for her group.
We start with a modification of Lemma \ref{lem:hyp}, which embeds the constant degree graph $G$ in $\textsf{Hyp}_n$. We present an embedding of $G$ in $\textsf{Hyp}_n$ with the additional property that every ball of radius 2 in $\textsf{Hyp}_n$ contains at most \emph{constant} number of vertices of the embedding's image. Formally, given an embedding $(\varphi,\chi)$ where $\varphi:V_G \rightarrow \{0,1\}^n$, $\chi:E_G \rightarrow P(\textsf{Hyp}_n)$, we denote the \emph{image of the embedding} by $Im(G)=\{w\in \{0,1\}^n: w\in \varphi(V_G) \cup \chi(E_G)\}$.
\begin{lemma}\label{lem:hyp-const}
Let $G$ be the graph with $N$ vertices that is defined in Section \ref{sec:veto} (the constant degree graph for which Theorem \ref{theo:opt}(\ref{theo:opt-bounded}) holds).
The graph $G$ can be VIED-embedded in $\textsf{Hyp}_n$ for $n=O(\log N)$, such that for every $w\in\{0,1\}^n$ we have\footnote{More concretely, $n=3\log N+333$ and $|B_2(w)\cap Im(G)|\leq 73$.} $|B_2(w)\cap Im(G)|=O(1)$.
\end{lemma}
\begin{proof}
We ``sparse" the embedding of Lemma \ref{lem:hyp} to reach a situation where every pair of independent edges are embedded to paths that are within a distance of at least 3 one from the other. This can be done, for instance, by embedding $G$ in a hypercube of dimension $n=3(\log N+111)$ rather than dimension $n'=\log N +111$, when we replace every vertex in $\textsf{Hyp}_{n'}$ by three copies of itself. Such a change multiplies the hamming distance by a factor of 3. For such an embedding the maximal number of vertices of $Im(G)$ in a ball or radius 2 is obtained at a vertex $w\in \phi(V_G)$ and is equal to $1+2\cdot 32$; the vertex and two vertices of every one of the 36 embedded edges.
\end{proof}
We proceed with a short presentation of the arguments that prove Theorem \ref{theo:opt}(\ref{theo:opt-hypercube}) from Theorem \ref{theo:opt}(\ref{theo:opt-bounded}), followed by a Corollary that will be essential in our reduction. The arguments below are very similar, but yet slightly defer from the proof that is presented in Section \ref{sec:main-pr}.
\paragraph{Proof of Theorem \ref{theo:opt}(\ref{theo:opt-hypercube}) from Theorem \ref{theo:opt}(\ref{theo:opt-bounded}). } We reduce $\textsc{SumLS}(G)$ to $\textsc{SumLS}(\textsf{Hyp}_n)$ using the VIED embedding $(\varphi,\chi)$ of Lemma \ref{lem:hyp-const}. Let $Im(G)\subset \textsf{Hyp}_n$ be the image of the embedding and let $w^*\in Im(G)$ be some fixed vertex. Given an instance $(f_A,f_B)$ of $\textsc{SumLS}(G)$ we define an instance $(f'_A,f'_B)$ of $\textsc{SumLS}(\textsf{Hyp}_n)$ by
\begin{align*}
f'_A(w)=
\begin{cases}
f_A(v) &\text{if } w=\varphi(v)\in \varphi(V_G) \\
\frac{k}{l} f_A(v) + \frac{l-k}{l}f_A(v') &\text{if } w\in \chi(\{v,v'\})\subset \chi(E_G) \\
-d(w,w^*) &\text{otherwise.}
\end{cases}\\
f'_B(w)=
\begin{cases}
f_B(v) &\text{if } w=\varphi(v)\in \varphi(V_G) \\
\frac{k}{l} f_B(v) + \frac{l-k}{l}f_B(v') &\text{if } w\in \chi(\{v,v'\})\subset \chi(E_G) \\
-d(w,w^*) &\text{otherwise.}
\end{cases}
\end{align*}
where in the case $w\in \chi(\{v,v'\})$ we assume that $w$ is the $k$'th element in the path $\chi(\{v,v'\})$ and $l$ is the total length of this path. Simply speaking, we set the functions $f'_A,f'_B$ to have the values $f_A(v)$ on the embedded vertices $\varphi(v)$. On intermediate vertices along a path that embeds an edge we set the value to be a weighted average of the two extreme valuations. For vertices out of $Im(G)$ we set $f'_A(w)=f'_B(w)$ to be a negative constant that does not depend on the instance $(f_A,f_B)$.
It can be easily checked that the local maxima of $f'_A+f'_B$ over $\textsf{Hyp}_n$ are precisely $\{\varphi(v): v \text{ is a local maximum of } f_A+f_B \text{ over } G\}$.
\begin{corollary}\label{cor:img-fixed}
Finding local maximum in $\textsf{Hyp}_n$ with the promise of $f'_A(w)=f'_B(w)=-d(w,w^*)$ for all $w\notin Im(G)$, requires $2^{\Omega(n)}$ communication.
\end{corollary}
Now we construct a potential game with $\Theta(n)$-players that solves the \emph{promise} $\textsc{SumLS}$ problem of Corollary \ref{cor:img-fixed}.
We mimic the arguments of the previous $2^{\sqrt[3]{n}}$ bound with one change: the reports $\overrightarrow{x}$ and $\overrightarrow{y}$ are done on vertices in $B_2(v)\cap Im(G)$ rather than $B_2(v)$. By Lemma \ref{lem:hyp-const} it is sufficient to report $73$ vertices (rather than $\Theta(n^2)$). A report consists of $\overrightarrow{x}=(\overrightarrow{xv},\overrightarrow{xf})$ where $\overrightarrow{xv}$ is a 73-tuple of vertices, and $\overrightarrow{xf}$ is a 73-tuple of valuations. The valuation function $val^{\overrightarrow{x}}(w)$ is modified to be $val^{\overrightarrow{x}}(w) = val(xf_i)$ if $w=xv_i$ for $i\in I_{\min}(\overrightarrow{xv})$; Otherwise, if $w\notin Im(G)$ we set $val^{\overrightarrow{x}}(w)=-d(w,w^*)$; Otherwise, we set $val^{\overrightarrow{x}}(w)=0$.
Similarly for Bob.
We also modify the definition of the neighbour vertices of $v\in \textsf{Hyp}_n$:
\begin{align*}
N_1(v)&:=\{(v_1,...,v_73): \{v_1,...,v_73\} \supset B_1(v)\cap Im(G)\}\subset \{0,1\}^{73n} \\
N_2(v)&:=\{(v_1,...,v_9): \{v_1,...,v_73\} \supset B_2(v) \cap Im(G)\}\subset \{0,1\}^{73n}.
\end{align*}
Note that by the Lemma \ref{lem:hyp-const} $N_1(v),N_2(v)\neq \emptyset$ for all vertices $v$.
From here, we apply similar arguments to those in Section \ref{sec:npot-pr} to prove a reduction from the local search promise problem of Corollary \ref{cor:img-fixed} to pure Nash equilibrium in potential games. The only additional argument that is needed is that $val^{\overrightarrow{x}},val^{\overrightarrow{y}}$ have the correct valuation for all vertices $w\notin Im(G)$ (in particular those within radius 2).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,081 |
Rodney M Bliss
Thoughts on building highly effective teams and individuals (That's original sounding, right?)
dreams, field of dreams, ghosts, invention, movies, star trek, Star Wars
Star Trek, Baseball and Ghosts
The most famous ship in the Star Trek universe is the USS Enterprise. In fact, in the series and the movies there have been multiple Enterprise ship.
U.S.S. Enterprise NX-01
U.S.S. Enterprise 1701
U.S.S. Enterprise 1701-A
U.S.S. Enterprise 1701-B
U.S.S. Enterprise 1701-C
U.S.S. Enterprise 1701-D
U.S.S. Enterprise 1701-E
I.S.S. Enterprise
There have probably been a few more that I'm missing. There's a famous scene in Star Trek IV "The Voyage Home" where the crew travels back in time to 1986 to procure some radioactive material. LT Chekov communicates to Admiral Kirk
Admiral. We have found the nuclear vessel.
Well done, Team two.
And Admiral. . .It is the Enterprise.
They were stealing material from the U.S.S. Enterprise aircraft carrier. It was a funny scene. Especially since Chekov pronounces it "wessel."
When NASA was searching for a name for it's original space shuttle, an online petition was started that suggested the name Enterprise in honor of the movie star ship.
And yet, in the Star Trek canon, the Star Ship Enterprise is actually named after the original space shuttle. In a sense, The Enterprise is named after itself. I have no doubt that if we ever do build a faster than light spaceship, it will be named Enterprise.
If you build it he will come.
It's hard to believe that Star Trek was made in teh late 1960s. It's been over 50 years.
One of my kids has a "dumb" phone. No email. No apps. Just phone and text. It's slightly smaller than a deck of playing cards. One of the biggest differences between this modern "dumb" phone and the early phones are that the earlier ones were "flip" phones. You had to physically open them to talk and physcially close them to hang up.
They were designed that way because on Star Trek to use the original communicators you opened them to talk and you closed them to hang up.
Gene Roddenberry, the creator of Star Trek built it. . .and they came.
Here are some similar things that Star Trek showed us and then we invented them.
Star Trek replicators became 3d Printers
Universal translators are a thing
Tricorders are handheld scanners
Holodecks are now allowing dead celebrities to go on tour
Communicator badges are now powered by Bluetooth
Saying "Computer" now sounds like "Hey Siri" or "Hello Google"
Phasers are still theoretical but direct energy weapons exist
Hyposprays are used to administer flu shots
Androids meet robots
And even Star Wars light sabers are possible.
The movie Back To The Future suggested we'd have flying cars by 2016. We don't have flying DeLoreans. But, just a few years later, drones are being turned into personal flying taxis. Back To The Future also predicted that after 107 years the Chicago Cubs would win the World Series in 2016. They missed that prediction too. But, only by a year. The Cubs won it all in 2017.
I'm not a great programmer. In fact, when it comes down to the details of coding, I'm not very patient. My buddy CK is much better at the details of coding. He spent years at Microsoft. He's been a Dev-Test lead for years.
But, here's the funny thing. I'm better with a blank page than CK is. I can start from nothing and lay out the classes, the procedures, the methods. If CK can see what is being created he can build off of that.
It's one of the unique and nearly magical things about us as a species and as a society. We think. We think. We imagine. We dream. And after someone imagines it. . .we build it.
Field of Dreams, one of the best baseball movies of all time, sees a slightly confused Kevin Costner build a baseball stadium in the middle of a corn field. He imagines it. Then he builds it. And then the magic starts to happen. (More than the normal baseball magic.) Oh and the ghosts show up.
Traveling faster than light is impossible. Science explains that. But, it makes the movies better.
I have no doubt that someday we'll manage it. And I have no doubt that ship will be named the U.S.S. Enterprise.
Rodney M Bliss is an author, columnist and IT Consultant. His blog updates every weekday. He lives in Pleasant Grove, UT with his lovely wife, thirteen children and grandchildren.
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What My Dysfunctional Family Taught Me About Business Communication
Sure We'll Test. It Won't Make Any Difference, Of Course
Training My Replacement
A Problem Not Seen In 101 Years. . .Safe For Another 101
Agile Cars Headed Over The Waterfall
Help And Help. . .It's Not Always the Same
Breakfast (And Dinner) Of Champions
The Sum of Their Fears. . .And Eating Dolphin
Freezing In Florida
The New York Times Bestseller And The Corporate Trainer | {
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{"url":"http:\/\/math.stackexchange.com\/questions\/144294\/complex-and-real-representations-their-differences-by-decomposition","text":"# Complex and Real Representations, their differences by decomposition\n\n1. The problem statement, all variables and given\/known data\n\nDecompose $\\mathbb{C}^{5}$, the 5 dimensional complex Euclidean space) into invariant subspaces irreducible with respect to the group $C_{5} \\cong \\mathbb{Z}_{5}$ of cyclic permutations of the basis vectors $e_{1}$ through $e_{5}$.\n\nHint: The group is Abelian, so all the irreps are one-dimensional. Therefore, you can use the simplified form of the projection operators, with characters.\n\nFurther, try to do the same for $\\mathbb{R}^{5}$, insisting that the basis vectors can only be combined with real coefficients. What is the difference between real and complex reps?\n\n2. Relevant equations\n\nThis may be the right projection operator, unsure: $$P^{\\alpha}=\\frac{d_{\\alpha}}{|G|} \\sum_{g} \\chi^{(\\alpha)}(g)*O_{g}$$\n\n3. The attempt at a solution\n\nI am confused by the term decompose, so my attempts have been floundering. I tried to write out the character table for $\\mathbb{Z}_5$ and I think I succeeded in that, but am unsure if it is needed. The hint about the projection operators served to confuse me more, although I readily understand the part about 1D irreps and Abelian. Is this asking me to construct reps (matrices) using cyclic permutations of $C_{5}$? If so, how am I supposed to use projection operators in this case to get them; This seems right however.\n\nAny help would be wonderful.\n\n-\nThe \"decompose\" part is asking you to write $V=\\mathbb{C}^5$ as $V_1\\oplus\\cdots\\oplus V_5$ where the $V_i$ are subspaces of $V$ that are fixed by the action of the group, i.e. for all $v \\in V_i$ and all $g \\in \\mathbb{Z}\/5$ you have $gv \\in V_i$. Each of those supspaces is one-dimensional of course, so you are really looking for five non-zero vectors $v_1,\\ldots,v_5$ such that $gv_i$ is a scalar multiple of $v_i$. \u2013\u00a0mt_ May 12 '12 at 19:03\nThe character table will help you, because it tells you how this group acts on its one-dimensional reps. Let $g$ be a generator of $\\mathbb{Z}_5$. Then if $V_1=\\langle v_1 \\rangle$ has character $\\chi$, you'll have $gv_1 = \\chi(g)v_1$. You know how $g$ acts on every vector in $V$ because you know how it acts on the basis $\\{e_i\\}$. So to find $v_1$, write it as a linear combination of $e_i$s, and solve the equation $gv_1 = \\chi(g)v_1$. Then repeat with a different character.... \u2013\u00a0mt_ May 12 '12 at 19:16\nI TeXified your post and attempted to recover the numbering by \"hard-coding\" the numbers. If something went wrong, I apologize. Please check that it looks all right. \u2013\u00a0Jyrki Lahtonen May 12 '12 at 19:21\nFor complex case you can just use the projections. In the real case you need to combine \"conjugate\" projections. As a hint I disclose that you should find a 2-dimensional real subspace, where the given generator acts by a 72-degree rotation, and another 2-dimensional real subspace, where the same generator acts by a 144-degree rotation. \u2013\u00a0Jyrki Lahtonen May 12 '12 at 19:23\n\nThe generator of your group acts via the linear transformation $T:\\mathbb{C}^5\\rightarrow\\mathbb{C}^5, (x_1,x_2,x_3,x_4,x_5)\\mapsto(x_5,x_1,x_2,x_3,x_4)$. Write $\\zeta=e^{2\\pi i\/5}$. Then the vector $$\\vec{u}_j=(1,\\zeta^{4j},\\zeta^{3j},\\zeta^{2j},\\zeta^{j})$$ is an eigenvector belonging to the eigenvalue $\\zeta^j$ for $j=0,1,2,3,4$. Therefore they each generate an invariant 1-dimensional (complex) subspace as described in mt_'s comments. They can also be gotten by applying the projection operators $$P^{-j}=\\frac15\\sum_{k=0}^5\\chi_j^*(g^k)T^k$$ to the vector $(5,0,0,0,0)$. Here $\\chi_j$ is the character $\\chi_j(g^k)=\\zeta^{jk}$ for all $k=0,1,2,3,4$ and $g$ is the generator of the group.[\/Edit]\nThe real case requires a bit more work. Observe that $\\zeta$ and $\\zeta^4$ as well as $\\zeta^2$ and $\\zeta^3$ are complex conjugates of each other. Thus the complex vector spaces $U_1=\\mathbb{C}\\vec{u}_1\\oplus\\mathbb{C}\\vec{u}_4$ and $U_1=\\mathbb{C}\\vec{u}_2\\oplus\\mathbb{C}\\vec{u}_3$ are stable under componentwise complex conjugation, because componentwise complex conjugation simply swaps the eigenvectors. It follows that the real vector spaces $$V_1=U_1\\cap \\mathbb{R}^5\\qquad\\text{and}\\qquad V_2=U_2\\cap \\mathbb{R}^5$$ are 2-dimensional (they are the eigenspaces belonging to eigenvalue 1 of the componentwise complex conjugation acting on $U_1$ and $U_2$ respectively), and also stable under the action of $T$. A calculation (see also answers to this question, in particular the link to Keith Conrad's lecture notes) shows that the matrix of $T$ with respect to the basis (over the reals) $\\{\\vec{u}_1+\\vec{u}_4, i(\\vec{u}_1-\\vec{u}_4)\\}$ of $V_1$ is the familiar rotation matrix. Therefore $T$ must act on $U_1$ as a rotation by 72 degrees.\nSimilarly the restriction of $T$ on $V_2$ is a rotation by 144 degrees.\nA rotation of a real plane does not have any invariant subspaces, so we cannot refine the direct sum decomposition $$\\mathbb{R}^5=\\langle(1,1,1,1,1)\\rangle\\oplus V_1\\oplus V_2$$ any further. Here $(1,1,1,1,1)=\\vec{u}_0$.","date":"2013-05-20 03:30:31","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9143328070640564, \"perplexity\": 157.17967997272524}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-20\/segments\/1368698222543\/warc\/CC-MAIN-20130516095702-00009-ip-10-60-113-184.ec2.internal.warc.gz\"}"} | null | null |
To compare the Bankers' perception about the profitability determinants of islamic and conventional banks.
For decades, North Dakota produced millions of barrels from wells drilled into conventional oil accumulations found in the Madison and Lodgepole formations.
010 excess material that must be finished with conventional polishing or EDM.
The differences between conventional drying and through-air-drying (TAD).
Within numerous scientific studies, such as the one referenced above, reviewed by the Organic Center for an upcoming State of Science Review, there were fifteen direct comparisons of antioxidant levels in organic versus conventional fruit and vegetables.
Throughout Operation Enduring Freedom, assets from conventional forces that SOF traditionally would have played a supporting role to regularly supported SOE Army forces were used to secure SOF bases, and a Navy aircraft carrier served in direct support of SOF operations.
By 2010, the feds hope, fleet mandates will help reduce conventional gasoline use by 30 percent.
Marrone contends that for organic and conventional farmers to enjoy widespread biopesticide use, such products must equal or even exceed performance of their competition.
This growing interest may be related to dissatisfaction with conventional western medicine (also known as biomedicine) which is perceived as high-cost technology driven, associated with significant morbidity, and focused on the disease rather than the whole patient.
The market shares of the conventional mortgage system are not only small relative to the amount borne by government institutions; they are also broadly distributed across the major types of institutions in the system.
He explains that because environmental regulations vary from location to location, some companies with large capital investments in conventional imagesetters won't be motivated to embrace the new technology immediately.
To prevent the almost inevitable pesticide "drift"--contamination carried by the wind from adjoining conventional fields--NOSB could have refused organic certification to so-called "split operations" (like, for instance, board member Eppley's), with both organic and conventional farming. | {
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Lothar Fischer (* 21. Juni 1942 in Homburg; † 3. Februar 2013 ebenda) war ein deutscher Politiker (SPD) und ehemaliger Bundestagsabgeordneter.
Leben
Fischer machte von 1956 bis 1957 eine Lehre bei einer Zollagentur. Auf dem Aufbaugymnasium machte er 1963 das Abitur und begann mit einem Studium der Mathematik. Im Jahr 1968 machte er sein Diplomhauptexamen im Hauptfach Mathematik, mit dem Nebenfach Physik, und arbeitete anschließend bis 1973 als Assistent am Mathematischen Institut der Universität des Saarlandes. Von 1969 bis 1980 arbeitete er zudem noch als Mathematik- und Physiklehrer eines Gymnasiums, zuletzt am Homburger Christian von Mannlich-Gymnasium. Er beendete seine Tätigkeit, um sich der Politik zu widmen.
Politik
Fischer war 1966 der SPD beigetreten und etablierte sich schnell als Vorsitzender verschiedener Juso-Ämter. So war er beispielsweise Unterbezirksvorsitzender und stellvertretender Stadtverbandsvorsitzender. Von 1981 bis 1990 war Fischer stellvertretender Unterbezirksvorsitzender der SPD, dem Bundestag gehörte er von 1980 bis 2002 an.
Weblinks
Einzelnachweise
Bundestagsabgeordneter (Saarland)
SPD-Mitglied
Deutscher
Geboren 1942
Gestorben 2013
Mann
Politiker (20. Jahrhundert)
Politiker (21. Jahrhundert) | {
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{"url":"http:\/\/fparena.blogspot.com\/2012_03_01_archive.html","text":"## Saturday, March 31, 2012\n\nFewer links, more commentary than usual. \u00a0Almost a regular post, but not quite. \u00a0I'm mostly just going to focus on the latest debate(s) about the state of IR. \u00a0I've come across a lot of other stuff worth commenting on since I last posted one of these, but I'm short on time, and only decided to post because I felt I needed to comment on the recent kerfuffle.\n\n## Saturday, March 24, 2012\n\n### Breaking Down Fearon 1995\n\nThis is the first in a semi-regular series on prominent applications of game theory to the study of international conflict.$$^1$$ My goal is to clarify the main contribution of the piece. I'll do so first by offering a brief synopsis before going through the key claims in more detail. Along the way, I'll try to note some of the important implications and points of common confusion. Assuming that format makes sense to you all, I'll do the same with future articles.\n\nWe begin with what I consider to be the single most important contribution to the study of international conflict in the last 25 years: Fearon's Rationalist Explanations for War.\n\n## Tuesday, March 20, 2012\n\n### My American Foreign Policy Class, Part III\n\nIn this part of the class, I discussed the role different actors within the US play in shaping what policies are actually chosen, and why \"the national interest\" is so hard to define.","date":"2015-08-28 05:22:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.29811468720436096, \"perplexity\": 1088.6062092751915}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-35\/segments\/1440644060413.1\/warc\/CC-MAIN-20150827025420-00294-ip-10-171-96-226.ec2.internal.warc.gz\"}"} | null | null |
Solange Theodoro (Bauru, 22 de abril de 1954) é uma atriz brasileira.
Carreira
Na televisão
No cinema
No Teatro
Ligações externas
Página oficial
Atrizes de São Paulo (estado)
Naturais de Bauru | {
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Diane Bardwell : Healing Music, Psychospiritual Therapy
Integral Psychotherapy Workshop Facilitator
O Breathe Us Deep
Psychospiritual Work
Sacred Hymn Download
Singer/Songwriter Psychospiritual Work
Diane Bardwell
Bio Music Events Store Links Contact Listen to the Music of Diane Bardwell Listen to the Music of Diane Bardwell
My Work With Diane
— Robert Augustus Masters, Ph.D.
"Diane is my wife, my beloved, my deepest friend, and my partner in all things, including the psychospiritual work we do together. What she brings to our work is a deeply compassionate, strongly grounded presence, as well as considerable skill in guiding others through their pain and obstacles into a deeper, more authentic and integrated life. Like me, she works not from behind a preset methodology, but from an intuitive sensing of what's needed moment to moment.
When we are working together, we not only bring our individual abilities to the work that needs to be done, but also our relationship, the presence of which contributes greatly to the work being done, according to many group participants. There is no effort in this, for it's simply a matter of us being with each other in the presence of others; we are not holding ourselves apart as an example of the far reaches of relational intimacy, but rather remain simply present in deeply connected mutuality and love for whatever work is needed.
Without Diane, I would have very likely burned out from all the work I do and feel so compellingly drawn to do. Through her, I am deepened and softened and reinforced in my commitment to being of authentic service to others. If you work with us, you may see me outfront more, but you will also see Diane right there, just as involved as I am in the work being done, contributing deeply not only through her heartful presence, but also through her insights, intuitions, and loving openness.
I am grateful, day after day, that she and I are together, and that we get to do this work together; we didn't consciously plan it thus, but it has felt utterly natural to evolve into working together. There is no retiring from this; as long as we can function, I'm sure we'll be doing our work, however much it evolves, with others."
For a list of currently available workshops, please see the Events page.
Read more about the Masters Center here
Interviews with Robert and Diane Masters about Transformation Through Intimacy
– the Journey Towards Awakened Monogamy
Life Coach Mary Allen: http://www.lifecoachmary.com/robertanddianemasters.htm (register free)
Terry Patten: http://beyondawakeningseries.com/blog/general/archive/
(register free and scroll down to our interview)
Ken Wilber: http://integrallife.com/node/112685 (after you register, you get one month's membership free)
"Robert and Diane are truly unique facilitators in creating a change in your emotional, mental, and physical states. Individually, they each bring their own awareness and loving compassion to each moment as it unfolds. Robert's ability to filter our story while deeply listening to the underlying wounds under the surface is unlike any professional I have worked with. His intuitive movements are quick and very effective to get us out of our head and into our hearts. ??Diane's loving voice and clear guidance allows us to trust even deeper that she has our back and gives us the reassurance to go beyond what we thought possible. Between her healing music and unconditional love, Diane opens new doors in our psyche that were previously blocked.??Together Diane and Robert create a space so profoundly safe that even the most horrible experiences we may have faced can be melted into light and love. These two are a remarkable force and can help anyone in search of better life."
— Terry Rauh, Massage Therapist/Energy Worker, Ashland, OR
Find Us On Facebook Bio | Music | Events | Store | Links | Contact | NEW!Sign up for our mailing list!Copyright © 2012 Diane Bardwell. All Rights Reserved
" I have found that her music and her voice also have a huge, yet gentle impact to help carry me through the difficulties I might be working through, or inspire and touch me in a way that takes me even deeper into where I need to go. Her music is a beautiful vessel for compassion, love, joy, grace, and beauty. For me, at its best, music was made to achieve what Diane's music does: through it, I experience being reconnected to humanity in my joy as well as in my pain.
— Dr. Laura Calderon de la Barca
"O Breathe Us Deep is the most powerful and beautiful music I have listened to in a very long time. It is deeply spiritual and transcendant, yet roars and weeps and laughs and sighs. These songs resonate in my bones and in my soul and invite me to journey with them, and be opened more fully by them. A truly sacred invitation to be fully alive and awake in Mystery. There is something very special being offered in this collaboration between Diane and her husband Robert (who wrote the lyrics). I feel deeply blessed to be receiving it."
Saleem Berryman, Goias, Brazil/San Francisco
"The music in each of these beautiful songs takes my breath away. The soul and emotional depth expressed through both the lyrics and Diane's magical voice moves me to tears. I relate to every feeling in every song. Thank you for this amazing and heart-felt gift."
— Pamala Oslie, author of LOVE COLORS and MAKE YOUR DREAMS COME TRUE
"Diane's CD is therapy in of itself. I was listening on my way to work and it blew me away. Best CD I have heard in a very long time. In fact, I have never heard anything like it. Totally amazing. If you can't make one of Robert Augustus Masters' sessions, get this CD. All I can say is that its having a healing effect on me that I have never experienced from any music before. This music is loaded with powerful stuff. I have heard her perform live also, her voice fills a room (without the need of amplification), she's a natural. These two make an impressive couple."
— Daniel Smith, Florida
"I was struck by the astonishing beauty of Diane's new album Emegence, both aesthetically and lyrically. I love how the music plays with tribal rhythms, conventional pop/R&B song structures, and sweet transcendent melodies. Diane's voice sounds absolutely gorgeous--I can feel a real transmission coming through her, evoking a subtle sense of being 'pulled up' out of mere gross-body awareness, even as the music keeps me grounded in the sensual. Highly recommended!"
— Corey deVos, Boulder, CO
"Emergence calls our authenticity out of hiding, heartfully reminding us of the chord of truth reverberating in the deep within. With depthful simplicity and a divinely inspired voice, Diane Bardwell sings us home. There is nowhere else to go- just home. I love this disc."
— Jeff Brown, Author
Soulshaping: A Journey of Self-Creation
"Diane's voice is irresistible! She projects a clarity and openness that I have not heard since Karen Carpenter or Barbara Streisand. This recording is a powerful blend of beautiful music and profound messages which gently invites you to leave off thinking about spirituality and enter into an experience of authentic and felt spirituality that is already present within each of us, awaiting our recognition."
— Ronald Radford, Flamenco Guitarist www.ronaldradford.com
"I am finding listening to Emergence very nourishing. It's a remedy, a comfort and an inspiration to me. I love the richness of the arrangements and am loving how on deeper and deeper listening new elements are always popping out at me.
— Frances Andre, Music Teacher, London
Photo by: Sequoia Pettengell
"Diane has a truly healing presence. From her voice to her gaze, she emanates grace, warmth and strength. Her empathy, compassion, and intuition are multi-layered, deep, and wholly genuine. What Diane brings to the group work is invaluable. She is truly there with you, truly present. Her openness and warmth guide you to the truth that lies within you. There is no pretense, only 100% embodied authenticity. She is the real deal."
— Jessica, Sarasota, Florida
"In my experience of the work that Robert and Diane do together, Diane's presence is both gentle and powerful. Her insights during the session are spot on, and her delivery is kind and caring; she can say things in a way that makes them digestible, even if they are confronting me with something difficult to be with. I always feel her care and her commitment to my wellbeing in her contributions. I always feel respected, and this increases my capacity to trust the process and the facilitators. She also has a special gift with mothering energy. For those of us that felt at times a lack of care and love from our mothers, her gaze, full of kindness, love and compassion, can dissolve pain as old as one's journey on Earth." — Dr. Laura Calderon de la Barca
Diane's Work with
Robert Masters
"I have attended a number of groups with Robert, and with Robert and Diane, attended both couples and individual sessions, and continue to be amazed and surprised at the depth attained by further work. This most recent week long group was truly incredible. I touched more deeply into what exists in me again. My life has changed dramatically since that first group I did in 2007. I am stronger, able to speak from a deeper place within myself, have more integrity, and walk through each day with more dignity. Thank you so much for bringing a dozen of us together for this week. The mix of people created something unexpected each day. What you create in that group is so grounded and present that it somehow seems wrong to call it magic. But it was. We all stepped more deeply into the mystery, and found ourselves more clear about life and love and pain and fear. I will be back. Thank you again." — Alistair Moes, Therapist, Vancouver, BC
"Diane Bardwell Masters shares her radiant energy in the group work that she does with her husband Robert. She is a wonderful singer who brings the ineffable sense of the Beloved to the room with her songs. Her depth, intelligence, and compassion shine through her eyes. Her astute sense of process adds to the flux and flow of the rhythm in the work. With grace and love, she implores all of us to seek what really matters — not to settle, but to open fully to all that is."
— Sarah Jane McGillivray, M.D. Labrador | {
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\section{Introduction}
Living cells carry some re-occurring genetic patterns called \textit{motifs} which are responsible for some key functions of cell physiology \cite{milo2002network,shen2002network}. Focusing on these motifs, synthetic biologists have designed models, first mathematically and then experimentally, to understand the nature of the dynamics in whole-cell or organism scenario. The synthetically constructed gene arrangements, often referred as the \textit{circuits}, are implemented in living cell, or the \textit{host} for operating with particular feature tasks. Interestingly the dynamics of the host and circuits often get coupled, most of the time in a nonlinear fashion. Functionality of the implemented synthetic circuit depends solely upon its host's physiology, host's growth and replication, operating temperature, pH, binding specificities and several other factors. Recent studies, provides sufficient information that host's physiology can affect, modulate or modify the circuit response not only in a low scale but also in a high scale. The output response can be moderated in a substantial scale or deviated from expected output in a low scale change, and in a high scale modulation, the output response is completely out of sink, giving rise to a new response or even a complete failure of the experiment \cite{weisse2015mechanistic,tan2009emergent,towbin2017optimality}. Limited knowledge about inter-cellular dynamics may restrict us in explaining these emergent responses, arising mainly because of the nonlinear couplings between the host and the circuit. From last decade, researchers have started focusing on this host-circuit coupling dynamics and gaining insightful results which in long term helps in designing robust and complex synthetic circuit also.
\\ One of the most important reasons of this influence of cellular context, is the dependency of the circuit on the host for the resources required for its gene expression. Here, by resource, we mean the cellular ingredients like RNAP, ribosome, ATP, protein degradation machinery \cite{borkowski2016overloaded, mcbride2017analyzing}, transcription factors etc. that are supplied by the host cell for the gene expression process. Any gene circuit, endogenous or synthetic, uses the resources from the host cell for their respective gene expression. Experiments prove that these resources are present in a limited manner inside cell \cite{Vind1993SynthesisOP,babu2003functional}. Recent experiments establish that the amount of available functional RNAP in the cell limits the transcription process majorly \cite{churchward1982transcription}. The effect of transcription factor sharing and its copy number in gene expression process is shown in \cite{zabet2013effects} and it has been established that in isogenic population of cells, this resource sharing enhances noise in the process of mRNA distribution \cite{das2017effect}. These works indicate that as long as the demand of resource is low and supply exceeds the demand, a gene circuit functions as expected. But for a complex network where the resource demand is high, or the circuit is functioning at a higher production regime, the resource supply might be insufficient compared to the demand. Under this circumstances, the circuit can develop unprecedented competition with other genes that are gathering resources from the same local pool.
\\Recently, to capture cellular resource sharing mathematically, models have been developed by researchers considering different approaches (e.g., resources as variable source, resource availability as a function of cell growth rate etc.) \cite{qian2017resource,carbonell2016dealing, weisse2015mechanistic,darlington2018dynamic}. Georgy et al. in their study provides an experimental evidence of cellular economy and proteins showing isocost like expression, while operating in a tight budget of ribosomes\cite{gyorgy2015isocost}. Recent study in this field reported major changes in toggle functionality as a consequence of resource competition \cite{chakraborty2021emergent,chakraborty2022Resource}. Theoretical models on competition of canonical and alternative sigma factors for RNAP in the steps of transcription initiation \cite{Grigorova} and transcription elongation \cite{Mauri2014AMF} shows bacterial responses on environmental fluctuations. Resource competition, specially ribosome competition effects in the protein production curve and insightful results in amplification and sensitivity modulation of proteins has been also reported \cite{9691638} using mathematical models. The effect of sharing degradation machinery and protease class of proteins is shown in \cite{mcbride2017analyzing} where the system shows emergent responses as a consequence of competition.
\\ In this work, we attempt to establish that resource distribution and competition has the capability to regulate the local and global dynamics of the system. When not taken into consideration, the outcome seems unpredictable and drastically different from the expected response. We consider a motif that apparently, does not bear any resemblance with the well-known motif, Feed-Forward Loop (FFL); however, due to resource competition interesting responses can be observed similar to the FFL motif. Here, we show that the inherent structure of the FFL circuit are compensated by a limited pool of availability, the resource competition plays similar role like an indirect repression. We report that the selected motifs behave like conventional FFL motifs and the unique responses of FFL motifs like response delay, pulse formation etc. are found to be carried out by resource driven motifs as well. In section \ref{ffl concept} we have briefly discussed about conventional basic structure of FFL motifs. In section \ref{model} we have discussed about the selected three gene motifs regulated by resource competition. In section \ref{result} we discuss the output responses giving rise to FFL-like behavior. Finally we conclude with some relevant discussion in section \ref{diss} respectively.
\section{Feed-forward loop motif.}\label{ffl concept}
In bacterial physiology, presence of the three gene motif known as Feed-Forward Loop (FFL) is found to modulate cellular dynamics of \textit{Yeast} and \textit{E.coli} very prominently. FFL motif is one of the most abundantly found motif in nature where three genes having their unique pattern of regulation (activation or repression) gives rise to coherent and incoherent motifs as shown in Fig. \ref{motif ffl}. In a three gene motif (say $X$, $Y$ and $Z$), one regulating the next in series (i.e $X$ regulating $Y$, $Y$ is regulating $Z$) and also the first gene (say $X$) is regulating the third gene (say $Z$) in a direct fashion. Distinctive resultant behaviour in protein synthesis, like response acceleration, response delay, pulse formation etc. can be achieved by these regulatory motifs.\\
Depending upon these mode of regulations (activation or repression) FFL motifs are conventionally classified in two groups, each containing four motifs, namely coherent FFL and incoherent FFL motifs as shown in Fig. \ref{motif ffl}. In coherent type FFL motifs, the direct regulation hand of $X \rightarrow Z$ is in harmony with the indirect regulation hand ($X$ regulating $Z$ via $Y$); these two are of opposite regulation in case of incoherent motifs. Some of these motifs arise abundantly in bacterial physiology while some of these are arising less frequently.
Presence of further AND gate logic and OR gate logic specifies either both the direct and indirect regulation in a combined way regulates $Z$ production (AND logic) or any one of this regulation is sufficient to initiate $Z$ regulation (OR logic).
\\In convention, an activator say $S_x$ and $S_y$ activates the proteins.
Presence of activator, that is when $S_x=1$, the first protein is in active state $X$, and in absence of activator, $S_x=0$, implies $X=0$ and similar for $Y$ (In the experimentally verified ara system, $X$ = CRP, $Y$ = araC, $Z$ = araBAD, $S_x$ = cAMP and $S_y$ = L-arabinose) \cite{mangan2003coherent}. Concentration of $Y$ and $Z$ can be represented by the set of equation, in a constitutive production of $X$:
\begin{equation*}
\frac{dY}{dt}=B_y+\beta_y\:\textit{f}(X,K_{xy})-Y\:\delta_y
\end{equation*}
\begin{equation}
\frac{dZ}{dt}=B_z+\beta_z\;\textit{G}(X,K_{xz},Y,K_{yz})-Z\:\delta_z
\label{FFL}
\end{equation}
Where $\beta_i$, $(i \in \{y,z\})$ is the maximum production rate and $K_{ij}$ $(i,j \in \{x,y,z\},i \neq j)$ is the activation or repression coefficient, signifies the activation on gene $i$ by transcription factor $j$. The AND gate function is represented by,
\begin{equation*}
\textit{G}=\textit{f}(X,K_{xz})\;\;\textit{f}(Y,K_{yz})
\end{equation*}
where the activator function is given by $\textit{f}(u,k)=\frac{(\frac{u}{k})^n}{1+(\frac{u}{k})^n}$ and repression function is represented by $\textit{f}(u,k)=\frac{1}{1+(\frac{u}{k})^n}$. $B_y$ and $\delta_y$ are the basal transcription rate and total degradation rate of $Y$ respectively, which includes the total dilution and degradation rates of $Y$ in cell. $B_z$ and $\delta_z$ represents same for $Z$. $n$ is co-operativity which accounts for the multimer formation of proteins.\\
Existence of this motif in yeast \cite{shen2002network}, \textit{C. elegence} \cite{mangan2003coherent}, \textit{B. subtilis} \cite{milo2004superfamilies,eichenberger2004program}, Sea urchin \cite{milo2004superfamilies}, \textit{E. coli} \cite{milo2004superfamilies,mangan2003coherent}, fruit fly \cite{milo2004superfamilies}, human \cite{odom2004control} and in many more diverse organisms are already seen. Instead of having this diversity, all two gene input circuits are not a FFL motif. As example, in \textit{E. coli} nearly $40\%$ two input operons are found to participate as FFL \cite{shen2002network}. In search of the reason for this, it is observed that the mutation in the binding sites of promoters can change the regulation arrows, even to the extent of removing it completely, resulting no more regulatory linkage in them. Thus $X$ to $Y$ regulation arrow is absent in some motifs which are referred as simple motifs (Fig. \ref{res motif}c) \cite{mangan2003structure}, and compared with FFL motif (Fig. \ref{motif ffl}) to explain its unique regulatory behavior. Conventionally $X$ $\rightarrow$ $Z$ and $Y$ $\rightarrow$ $Z$ these two regulations are considered as essential to maintain AND gate regulation and it can be said that the presence or absence of $X$ $\rightarrow$ $Y$ is the key factor that differentiates FFL like motif and simple regulation motif. \\
Mutation in gene dynamics is sometime biased by the preferences of bio-chemical reactions but majorly it is a random process \cite{loewe2008genetic}. It is possible that in some mutation the $X$ $\rightarrow$ $Z$ disappears leaving the rest of the wiring pattern intact. In this scenario, if the two proteins develop a resource competition with asymmetric resource affinity, an effective repression will come into picture. In this paper, we have carefully chosen some three gene motifs where certain conventional direct regulation, more precisely repression (i.e. Hill function type repression which includes co-operativity with $n=2$ via dimer formation, as shown in Fig. \ref{ffl concept}a), can be replaced by this resource competition scenario as shown in Fig. \ref{res motif}b. We have also considered resource driven simple regulation model in Fig. \ref{res motif}d which will be further used for evaluating the performance of resource driven FFL motif. \\
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{1.png}
\end{center}
\caption{Feed-Forward Loop model motifs. Arrow head symbols represents activation, hammer-head symbol represents repression. There are $4$ coherent motifs and $4$ incoherent motifs conventionally.}
\label{motif ffl}
\end{figure}
\section{Model Formulation}\label{model}
In our resource sharing model we focus especially on ribosome competition in the step of translation process. To illustrate this process, we can take the example of \textit{yeast} where approximately 60000 mRNA molecule starts translating parallelly \cite{warner1999economics, zenklusen2008single}, while available ribosome (which is limited, nearly 240000 in \textit{yeast}) possibly scans the same transcript simultaneously. Now, if any one mRNA starts accommodating ribosome with higher binding affinity, others translation initiation will be delayed (as the total supplier pool is getting affected) and suppressed as a result. \\
Considering this circumstances, let us consider a simple pathway where $X$ $\dashv$ $Y$ $\rightarrow$ $Z$ ($X$ represses $Y$, $Y$ activates $Z$). Suppose mRNAs for $X$ and $Z$ are involved in such a resource competition here.
If $X$ recruits ribosomes more efficiently for its production, $Z$ will suffer a deficiency, resulting into a repression in terms of resource. The intention behind choosing this particular architecture is to erase the $X$ to $Z$ direct regulation (repression, for coherent type 2 FFL) and introduce resource competition instead. This will complete the resource driven FFL (rFFL) motif where $X$ represses $Z$ via resource competition. Before implementing mathematical equations, let us elaborate our considerations below:
\begin{figure}
\centering
\includegraphics[width=\textwidth]{2.png}
\caption{(a). Selected FFL motif among all motifs where $X$ to $Z$ regulation is of repression type. $X$ to $Z$ dotted line represents these repressions can be replaced by resource competition for emergent resource driven repression in these motifs. Arrowhead symbols represents activation and hammerhead symbols represents repression. Activations will be replaced by repression according to motif requirement in (b), (c), (d) figures. (b). Representing schematic diagram of proposed resource competition model. $X$ and $Z$ are collecting resource from the same pool $T$, with affinities respectively $res_x$ and $res_z$. (c). Simple regulation motif of conventional FFL motif structure. $X$ and $Y$ are regulating $Z$ but there is no regulation of $X$ to $Y$. (d). Resource competition regulated simple motif structure. }
\label{res motif}
\end{figure}
\begin{itemize}
\item We consider ribosome is distributed over small several cytoplasmic compartments in cell. The limited presence of this translational resource in protein production is verified experimentally in some recent work \cite{gyorgy2015isocost}. We focus in the local resource pool here, present in the immediate vicinity of circuit of interest, which captures the circuit dynamics in a realistic way. Let $T$ represents the pool of ribosome, available for translation for its neighbourhood genes.
\item The pool of mRNA, as a result of transcription are respectively $g_x$ and $g_z$, which are ready to be translated into proteins $X$ and $Z$. Being expressed in the local field of cytoplasm, we consider both $X$ and $Z$ are collecting resource ribosomes from the same pool $T$.
\item The mRNA copies, which are ready for translation makes a ribosome bound complex in a step and get translated in next step. The small sub-unit of ribosome binds to three initiation factors IF1, IF2, IF3 along with a methionine-carrying tRNA first, then binds to mRNA and forms the complex. Let, $c_x$ and $c_z$ represents ribosome bound complex of $X$ and $Z$ respectively. Now from available total free pool $T$, $c_x$ and $c_z$ represents the bound complex, further free ribosome thus available for translation is given by ($T-c_x-c_z$).
\item mRNA binds with ribosomes with certain affinity. Let us consider $res_x$ and $res_z$ represents the resource affinity for $X$ and $Z$ mRNA respectively. This affinity for resource allocation depends upon various factors. The accessibility of the ribosome binding site on the mRNA significantly determines the basal translation level \cite{gualerzi2015initiation, cifuentes2019domains}, while Polycistronic mRNA pool contains a multiple ribosome binding sites (RBS) in most of the bacterial organisms \cite{burkhardt2017operon}. The recruitment of ribosome to this RBS are temperature dependent. Temperature fluctuation induces re-folding of the mRNA which interacts with proteins and regulates the synthesis level in selective cases \cite{breaker2018riboswitches}. The nature of the environmental legands also modulates this ribosome recruitment. Thus resource affinity $res_x$ and $res_z$ can be taken as different taking care of all these biological factors.
\item Protein is produced from respective complex at a certain rate, $\epsilon_x$ and $\epsilon_z$ respectively for $X$ and $Z$.
\item $\delta_x$, $\delta_y$, $\delta_z$, $\delta c_x$, $\delta c_z$ represents the overall degradation rates which accounts for the dilution and degradation inside cell for respectively protein $X$, $Y$, $Z$ and complex $c_x$ and $c_z$.
\item We achieve conventional AND gate logic of FFL, where both the direct regulation of $X$ to $Z$ hand, along with the indirect regulation hand of $Z$ regulation via $Y$ acts combinely as electronic AND gate logic, by multiplying $c_z \epsilon_z$ ( $Z$ production term from its respective complex) with $Y$ regulatory term.
\item In \cite{mangan2003structure} a step like behavior of $X$ induced by $S_x$ was considered. The same is achieved by allowing $X$ to produce from its complex for a time period say $t=0$ to $10$ when $S_x=1$. For our model, we consider $S_x=0$ blocks the complex formation $c_x$ at $t=10$, thus the protein $X$ is allowed to decay sharply, giving a nearly step like production of $X$ wrt. time $t$.
\item Respective resource driven simple regulation model is represented by the same equation of $Z$ with $Y$ constitutively expressed, $Y=1$. As mentioned before a schematic diagram of resource driven simple regulation model is shown in Fig. \ref{ffl concept}d.
\end{itemize}
Mathematical modelling for the type 2 coherent FFL motif by our proposed resource competition model where $X$ to $Z$ repression arises due to resource competition is given by Eq. \ref{chr2}.
\begin{eqnarray}\label{chr2}
\frac{dc_x}{dt}&=&res_x\:(T-c_x-c_z)\:g_x-c_x\:\delta c_x\\
\frac{dc_z}{dt}&=&res_z\:(T-c_x-c_z)\:g_z-c_z\:\delta c_z \nonumber\\
\frac{dX}{dt}&=&c_x\:\epsilon_x-X\:\delta_x\nonumber\\
\frac{dY}{dt}&=&B_y+\beta_y\:\frac{1}{1+(\frac{X}{k_{xy}})^n}-Y\:\delta_y\nonumber\\
\frac{dZ}{dt}&=&B_z+\beta_z\:\frac{c_z\:\epsilon_z\:(\frac{Y}{k_{yz}})^n}{1+(\frac{Y}{k_{yz}})^n}-Z\:\delta_z\nonumber
\end{eqnarray}
Following similar arguments, we re-create resource driven FFL (rFFL) capable of mimicking coherent type $3$, incoherent type $2$ and incoherent type $3$ FFL motif. The reason behind choosing these four motifs is that in all these architectures $X$ to $Z$ competition is a repression.
\section{Results}\label{result}
\subsection{Behavior of Coherent type 2 rFFL motif:}
We analyse the response of $Z$ upon step-like addition of inducer $S_x$, in presence of $Y$ in our AND logic rFFL coherent type $2$ motif. At time $t=0$, $X$ starts producing from its complex $c_x$ by collecting resource ribosome with affinity $res_x$ from the pool $T$. At the same time $Z$ also started to produce its complex $c_z$ and thus allocating resource at a rate of $res_z$. With the higher value of $res_x$ ($res_x> res_z$), $X$ is allocating more resources from fixed pool $T$ than $Z$. Thus availability of resource for $Z$ production decreases, causing lower production of $Z$. This puts an effective repression on $Z$ as $X$ gets produced at the cost of $Z$. Moreover, $X$ also repress $Y$, the activator of $Z$ production. Thus $Z$ is low throughout the active $X$ state. At $t=10$, following the conventional FFL motifs we make $S_x$, the inducer of $X$ production zero. This blocks the complex of $X$ production, $c_x = 0$. As soon as $c_x$ production stops $X$ drastically falls to zero and we investigate the pattern of $Z$ formation here. For reference, we put the nature of AND logic conventional FFL (cFFL) of coherent type $2$ behavior in Fig. \ref{chrnt2}c.
\\ Now we compare rFFL coherent type $2$ with corresponding simple motif (Fig. \ref{ffl concept}d). With respect to the resource driven simple motif, where $X$ to $Z$ resource competition and $Y$ to $Z$ regulation works separately, we find the rFFL like structure shows delay in reaching $Z$ steady state as shown in Fig. \ref{chrnt2}a. The response time of protein $Z$, defined as the time to reach $50\%$ of its final concentration \cite{rosenfeld2003response, savageau1976biochemical} is greater in rFFL motif than that of corresponding simple motif. Also the variation in resource allocation rates, which determines the efficiency of collecting resource for the production of the protein significantly modulates the delay here as shown in Fig. \ref{chrnt2}b. The red line of $X$ shows the behavior of protein production kinetics with respect to $S_x$ on step at $t = 0$, and $S_x$ off state at $t = 10$. $res_x = 1, res_z = 0.05$ the green representative of protein $Z$ is delayed wrt. $res_x = res_z =1$ the blue line.
\\ Similar to the cFFL motifs \cite{mangan2003structure} $Z$ goes on with $X$ off, that is output is inverted in nature. This is easily explainable as $X$ goes off, repression upon $Z$ via resource and repression upon activator of $Z$, that is repression on $Y$ both vanishes. Thus $Y$ activates $Z$ and $Z$ production increases making it on for $S_x$ off state.
\\Steady state logic of $Z$ is both sensitive to $S_x$ and $S_y$ as similar to cFFL motif. Putting $Y$ in off state by $S_y = 0$, $Z$ steady state goes off.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{3.png}
\caption{Comparative analysis of Kinetic behavior of coherent type 2 rFFL motif and cFFL motif in AND gate logic. (a). Resource driven motif. Red line represents $X$. The green and the blue line represents $Z$ in rFFL and resource driven simple logic respectively. Note that wrt. blue line (the resource driven simple logic), the green curve (rFFL logic) shows delay in reaching the steady state. (b). Variation in resource affinity affects the delay in reaching the steady state of $Z$. Red line is for $X$. Blue line is for $Z$, $res_x = 1, res_z = 1$. Green line is for $Z$, when $res_x = 1, res_z = 0.05$. (c). Kinetics of coherent type $2$ cFFL AND gate motif \cite{mangan2003structure}. $k_{xy} = 0.1, k_{yz} = 0.1, k_{xz} =1, n = 2$. Red line is for $X$, Green line shows nature of $Z$ in FFL motif response, blue line is for corresponding simple regulation model. $\beta_y = \beta_z= 1$, $\delta_x = \delta_y = \delta_z = 1, \delta c_x = 1, \delta c_z = 1, \epsilon_x = 1, \epsilon_z = 1, g_x = 5, g_z = 5, T=10, B_y = B_z =0$ for both (a) and (b).}
\label{chrnt2}
\end{figure}
\subsection{Key characteristics features of conventional FFL demonstrated by rFFL:}
\begin{itemize}
\item{\textit{$S_x$ off state delay in Coherent rFFL motif:}}
$Z$ steady state state is delayed in coherent type $2$ and type $3$ rFFL motif wrt. the corresponding resource driven simple regulation motif in $S_x$ off state. The behavior is similar to the conventional motif responses \cite{mangan2003structure}. Additionally, here the variation in resource affinity value regulates the delay response significantly (Fig. \ref{chrnt2}b, Fig. \ref{chrnt3}b).
\\ As explained earlier, from $t=0$ to $t=10$ the resource driven repression of $Z$ via $X$ is on; along with for coherent type 2, $X$ repress the activator ($Y$) of $Z$ (for coherent type 3, $X$ activates the repressor of $Z$). So with respect to resource driven simple motif where $X$ to $Z$ is repression and $Y$ activates $Z$, in rFFL motif $Y$ level is low due to presence of repression on $Y$ via $X$. At $t=10$, when the repression is withdrawn from $Z$, the level rises and becomes steady in a higher level. Also a delay is seen in reaching the steady state of $Z$ for coherent type $2$ rFFL logic as the activator $Y$ level is low here thus further $Z$ production is delayed (similar logic applicable for rFFL coherent type $3$ motif) when compared with resource driven simple motif. These behaviors are similar to the cFFL responses.
\item{\textit{Steady state logic of $Z$ is dependent on $S_x$ in Coherent rFFL:}}
Steady state logic of $Z$ is inverted with $S_x$ in coherent type $2$ and coherent type $3$ rFFL motif, that is $Z$ goes on in $S_x$ off step. As mentioned earlier, $S_x$ off state releases the repression on $Z$ both in terms of resource and indirect regulatory repression via $Y$ in both coherent type $2$ and $3$ rFFL motif, thus $Z$ production increases making $Z$ response inverted with $S_x$.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{4.png}
\caption{Comparative analysis of Kinetic behavior of coherent type 3 rFFL motif and cFFL motif in AND gate logic. (a). Resource driven motif. Red line represents $X$, green line is for rFFL $Z$ response and blue line is resource driven simple logic. Note that wrt. blue line (the resource driven simple logic), green curve (the rFFL logic) shows delay in reaching the steady state. (b). Variation in resource affinity affects the delay in reaching the steady state of $Z$. Red line is for $X$. Blue line is for $Z$, $res_x = 1, res_z = 1$. Green line is for $Z$, when $res_x = 1, res_z = 0.01$. (c). Kinetics of coherent type $3$ cFFL AND gate motif \cite{mangan2003structure}. Red line is for $X$, Green line shows nature of $Z$ in FFL motif response, blue line is for corresponding simple regulation model. Parameter values are $k_{xy} = k_{yz} = k_{xz} = 1, n = 2,$. For (a), (b) and (c) rest parameters are $\beta_y = \beta_z= 1$, $\delta_x = \delta_y = \delta_z = 1$ $B_y = B_z = 0$.}
\label{chrnt3}
\end{figure}
\item{\textit{Steady state of $Z$ is dependent on $S_y$ in Coherent type 2 rFFL motif but not in Coherent type 3 rFFL :}}
The steady state of $Z$ responds strongly to $S_y$ in case of coherent type $2$ rFFL and the inverted output nature is lost when $S_y=0$ (at $S_x = 0$, $S_y = 1$ the $Z$ steady state is inverted in nature). But the steady state of $Z$ is not dependent on $S_y$ in case of coherent type $3$ rFFL motif. Both these behaviors are similar to the conventional motif responses.
\item{\textit{Pulse generation in $S_x$ off state of Incoherent rFFL motifs:}}
Incoherent type $2$ and $3$ cFFL model motif shows pulse formation in $S_x$ off state for AND gate logic. Our AND logic rFFL model shows similar results in output as shown in Fig. \ref{pulse}a and \ref{pulse}c. $X$ and $Z$ are collecting resource from the pool $T$, and thus $X$ is putting a repression in $Z$ production, strength depending upon resource affinity. Now for incoherent type $3$ motif, at $t=10$, the complex formation of $X$, that is in a straight forward way the production of $X$ is blocked and no resource demands for $X$ is valid as well. Thus the available resource pool is now open for $Z$ and the repression in terms of resource is no more. Thus $Z$ production suddenly increases at $S_x$ off state. The wiring pattern shows $X$ activates $Y$, which is also an activator of $Z$, thus as $X=0$ now this $X$ can't activate $Y$, eventually $Y$ production decreases and further $Z$ production decreases as $Y$ is not produced enough so it can't activate $Z$, thus $Z$ decays eventually. Thus $Z$ shows a pulse in output as the sudden increase in production and eventually dies out. Similarly the pulse formation in Incoherent type $2$ rFFL motif can be explained.
\item{\textit{No pulse is created in $S_x$ on state of Incoherent rFFL motifs:}}
The incoherent type $2$ and $3$ cFFL do not generate pulse in response to $S_x$ on step. The resource driven motifs are showing similar results (Fig. \ref{pulse}a and \ref{pulse}c).
\item{\textit{Steady state behavior of Incoherent rFFL with no basal activity, $S_y$ effect:}}
Steady state logic of incoherent type 2 rFFL motif is found to depend on $S_y$. In presence of $S_y$, $Z$ creates a pulse and then comes down to low state eventually, while in absence of $S_y$ the steady state is high and no pulse is created. But type $3$ incoherent rFFL motif has a constant steady state $0$, which do not depends upon $S_y$. These behaviors are similar to the cFFL behaviors as well.
\item{\textit{$S_y$ effect in pulse generation in rFFL Incoherent motifs:}}
Similar to the cFFL motif our resource driven motif also gives similar result in case of incoherent type $2$ rFFL motif. Here, $Z$ shows pulse in output when $S_y$ is on, but $Z$ is high and steady when $S_y$ is off. In incoherent type $3$ rFFL motif, $Z$ shows no pulse in $S_y$ off state.
\end{itemize}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{5.png}
\caption{Pulse formation in AND logic rFFL incoherent motifs and comparative cFFL motif output response. Red line is for $X$, blue is for $Z$ in simple regulation system, green is for $Z$ in FFL motif response output. (a) Resource driven incoherent type $3$ motif. $ k_{xy} = 0.1, k_{yz} = 0.1, res_x = 1, res_z = 0.5$. A pulse is seen in the $S_x$ off state for $Z$ response. (b). Incoherent type $3$ cFFL motif. $ k_{xy} = 1, k_{yz} = 0.5, k_{xz} = 0.5$. (c). Pulse formation in Incoherent type $2$ rFFL motif. Parameter values are $k_{xy} = 0.1, k_{yz} = 0.1, res_x = 1, res_z = 0.5$. For both (a) and (c) $n = 2, T = 10, \delta c_x = 1, \delta c_z = 1, \epsilon_x = 1, \epsilon_z = 1, g_x = 5, g_z = 5, B_y = B_z = 0, \beta_y = \beta_z = 1$.}
\label{pulse}
\end{figure}
\section{Conclusion and Future directions}\label{diss}
Emergent responses in biological circuits as a consequence of context dependency is drawing the attention of research community in recent past \cite{tan2009emergent,nikolados2019growth, chakraborty2021emergent,9691638}. Among these several context dependencies, ribosome limitation is a major controlling factor in gene expression dynamics. Different processes, which are apparently not connected, gets coupled implicitely due to limited presence of this essential translational resource in the neighboring cellular environment. In this work, we have taken some commonly occurring motifs, specifically three gene patterns, which are quite different from conventional FFL structures as a repressive regulation is absent. This absences of this regulatory arm can arise from mutation in the system, which is a very random stochastic yet unavoidable change in genome structure. In human, average 175 mutations per diploid genome per generation (i.e., average mutation rate of $2.5\times 10^{-8}$) is noted \cite{nachman2000estimate}, while in E. coli the average mutation rate is $2.1\times 10^{-7}$ per gene per generation \cite{chen2013no}. The selected motifs shows FFL-like response when driven by resource competition, in a resource-limited cellular environment. An emergent repression arising from the context dependency, more precisely two mRNAs competing for translational ribosome, fulfills the repression condition in the chosen motifs, and the fundamental functional responses of conventional FFLs, like response delay, pulse generation, dependency of steady sates upon inducers etc., are achieved in proposed rFFL architectures. Acceleration or delay in response is depicted in FFL motif due to its unique construction, and the same is achieved in rFFL, solely caused by sharing resource from common ribosome pool. This nonlinear coupling between the host and the circuit can modulate the dynamics of the entire system significantly. Our work not only depicts the possibilities of vast modification in gene circuit response due to resource limitation, but also proposes an emergent architecture for one of the most common genetic motifs, Feed-forward loops.
\\It is important to note that our considerations are only valid for a resource limited, low growth system. Growth of the system is directly linked with the number of active ribosomes participated in translation and thus with the biomass of the system. Cellular
macromolecular composition could be highly correlated with cell growth \cite{klumpp2009growth}, and further investigations can be planned considering this factor in a future work. Moreover, biological processes mostly take place in a noisy environment. In a recent work on the noise characteristics of FFL \cite{ghosh2005noise}, the relation between functionality and abundance has been suggested, keeping the noise factor in mind. Deterministic and stochastic characteristics of functionality, dynamics and response of the proposed rFFL motifs can also elaborately studied, to develop further understanding on complex synthetic circuit operation in diverse host cells.
\section*{Acknowledgement}
\noindent PC and SG acknowledge the support by DST-INSPIRE, India, vide sanction Letter No. DST/INSPIRE/04/2017/002765 dated- 13.03.2019.
\section*{References}
\bibliographystyle{iop}
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RNC plans in jeopardy as Jacksonville council president opposes city bill
Andrew Pantazi
Florida Times-Union
The 2020 Republican National Convention's plans for coming to Jacksonville next month will face a major roadblock after Mayor Lenny Curry filed a bill Wednesday evening: a city council president who said he now opposes the effort.
The bill, which still leaves gaps about how the city plans to handle testing, treatment and the potential spread of coronavirus, grants Curry extraordinary power to spend $33 million in federal security funds how he thinks is necessary.
City Council President Tommy Hazouri told the Times-Union he opposes the bill, though it's possible he could be persuaded to change his vote depending on the outcome of a Friday council meeting where he's asked the sheriff and other officials to testify.
The fact that he hasn't gotten the mayor to commit to be there, he said, is a strike against the bill's chances of passing. If the bill fails a two-thirds majority vote next Tuesday, Hazouri said, he doesn't see how the convention could come.
Sheriff Mike Williams, on the other hand, told Hazouri he would attend. Williams said this week that "we are simply past the point of no return," and he didn't see how he could keep the event safe with such little time to plan for it.
The sheriff's Monday comments has sent plans for the convention in a tailspin. Curry held a news conference Tuesday, ostensibly about new COVID-19 testing sites, but he didn't get into any specifics when asked repeatedly what changes were necessary to make the convention a reality.
Curry said he agreed with the sheriff's comments even as he contradicted the sheriff by saying the convention could still happen and be safe.
Hazouri said the sheriff's opinion likely has turned enough of the council against the effort to kill the bill. "That [opinion] in itself stops it in its tracks with some of the council members and it would with me if he doesn't have the ability to keep people safe downtown."
The mayor's bill is necessary to spend the federal funds necessary for safety and security. Other funds are coming from fundraising for the convention's host committee.
However, it's possible the convention could continue without the council approving funding.
The Republican National Convention is scheduled to take place downtown at several venues from Aug. 24 through Aug. 27. It was initially scheduled for Charlotte, but President Trump moved it after the North Carolina governor indicated masks may be mandatory during the event.
A majority of Jacksonville residents oppose the convention coming here, a poll has found. And the day after the mayor announced the city had secured the Republican convention, community and business leaders as well as local Democratic Party leaders protested the planned event over health, crime and crowd concerns.
Brian Hughes, the city's chief administrative officer, has said in emails to Hazouri that the city wouldn't be on the hook for any spending, writing that "budgets for the events are the responsibility of the Host Committee and their partners."
He also wrote that he didn't believe any contracts or agreements necessary required City Council approval.
The bill does much more than approve funding. It also details insurance requirements, approves a mutual-aid plan for bringing in law-enforcement officers from other agencies and waives alcohol restrictions.
The bill also sets the perimeter for marching routes, free-speech zones and for the convention's celebrations, along with permitting processes for those wanting to participate or protest.
The bill left any plans for dealing with COVID-19 vague at best, saying that the city would assist "in providing resources to implement and administer the COVID-19 pandemic health protocols plan developed by the RNC and Host Committee in consultation with the City."
Hazouri has been unable to get answers about what that means. A spokeswoman for the host committee wouldn't answer questions.
The City Council is made up of two-thirds Republicans. Hazouri, a Democrat, however said he believed some Republicans on council are likely as skeptical as he, and he expects they'll ask many questions at Friday's meeting.
"I'm proud of them and they too will put the public ahead of politics. Maybe that's wishful thinking, but I think they will."
Councilman Matt Carlucci, who called himself a "Jeb Bush Republican," said he's concerned by how little details about the convention he had so close to the convention's go-date. As of now, he opposes the effort.
After Carlucci met with the mayor's staff, he said, "there were more questions than there were answers," he said. "How much money has the host committee raised? Well, they can't say."
He added, "I won't support it unless our sheriff says he can keep our community safe and secure. That's one box, but there are other boxes as well, such as the ever-changing COVID problem."
Hazouri said it's still possible the bill could pass, if every concern is sufficiently satisfied by Friday.
"This bill right now is on the 40-yard line, and it's fourth down, and they've got to get it across the goal line," he said. "They've got to do everything they can to make it acceptable, and depending on the answers and the questions, that's where the council has to make tough decisions."
Without the sheriff's endorsement, Hazouri said that he won't support the bill.
"Until I feel the health, safety and welfare are solidly protected, I can't support this bill or any bill that could potentially implode with the virus and our safety and the welfare of our community and those that are coming here."
He said he's concerned by how much energy the convention is taking up while the city also deals with a federal investigation into the JEA privatization debacle, near-record highs in terms of homicides and civil unrest in the wake of police violence, not to mention a global pandemic.
"Honestly, they should not have been moving on this with the pandemic," he said. "The pandemic itself should've been the [end] for the convention." | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,371 |
\section{Introduction}
The sequence of Fibonacci numbers $\left(F_{n}\right)_{n\in\NN}$ is given by the recurrence equation:
\begin{align*}
F_0&=0\\
F_1&=1\\
F_n&=F_{n-1}+F_{n-2}\\
\left(F_n\right)_{n\in\NN}&=0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946\ldots
\end{align*}
In this paper, we investigate whether this famous sequence, when concatenated behind a decimal, creates a \textit{normal number}. A number in a base $\beta$ is $\textit{normal}$ if every string of length $k$ occurs in the limit as often as every other string of length $k$, namely with a frequency of $1/\beta^k$. To prove that the Fibonacci concatenation is normal in a base $\beta$, we use a technique that measures the frequency of each digit in every place value of the Fibonacci numbers in base $\beta$. A major tool we use is the well-known fact that the Fibonacci sequence is purely periodic modulo every integer base $\beta$. The length of one period of the Fibonacci sequence modulo a base $\beta$ is known as the \textit{Pisano period}. The aim of this paper is to connect the notion of normal numbers and Pisano periods, to prove that the Fibonacci concatenation is normal in Base $10$, and to provide computations and heuristic evidence that the Fibonacci concatenation \textit{absolutely normal}.
\begin{theorem}[Main Theorem]
The Fibonacci sequence, when concatenated behind a decimal, is normal in every base of the form $5^x2^y$ for nonnegative integers $x$ and $y$.
\end{theorem}
For the convenience of the reader, we provide here an outline of the paper and a summary of techniques used:
\begin{description}
\item [Section 2] Background information on normal numbers and Pisano periods.
\item [Section 3] Connecting the relationship between periodicity of the Fibonacci sequence modulo an integer $\beta$ with the sequences of digits in each place value of the Fibonacci numbers in base $\beta$.
\item [Section 4] Proof that the Fibonacci concatenation is normal in every base of the form $\beta=5^x$.
\item [Section 5] Proof that the Fibonacci concatenation is normal in every base of the form $\beta=2^y$.
\item [Section 6] Combining the results from the previous two sections to show that the Fibonacci concatenation is normal in every base of the form $\beta=5^x2^y$.
\item [Section 7] Computational evidence that the Fibonacci concatenation is in fact normal in every base, thus is \textit{absolutely normal}.
\end{description}
\section{Preliminaries}
There are many definitions of a normal number. The first was given by Borel in 1909 \cite{[Borel]}, but the definition has been refined during the last century. We use the definition given by Davenport and Erdös \cite{[Davenport]}.
\begin{definition}
Let $x$ be a real number in a base $\beta$. Let $a_1a_2\cdots a_k$ be any string of length $k$ written in base $\beta$. Let $N(t)$ denote the number of times that this string occurs among the first $t$ digits of $x$. Then, $x$ is \textit{normal} in $\beta$ if
\[\lim_{t\to\infty}\frac{N(t)}{t}=\frac{1}{\beta^k}.\]
In the case where $k=1$, we say $x$ is \textit{simply normal} in base $\beta$ if
\[\lim_{t\to\infty}\frac{N(t)}{t}=\frac{1}{\beta}.\]
Finally, a number is \textit{absolutely normal} if it is normal in every base.
\end{definition}
More plainly, a number is \textit{normal} in a base $\beta$ if every string of length $k$ occurs with a limiting relative frequency of $1/\beta^k$ and is \textit{simply normal} if each digit occurs with a limiting relative frequency of $1/\beta$. The equivalent definition of \textit{normal} that was given by Borel \cite{[Borel]} in 1909 was further refined by Pillai \cite{[Pillai]} in 1940:
\begin{lemma}[Pillai]\label{lemma:Pillai}
A number is \textit{normal} in base $\beta$ if and only if it is simply normal in base $\beta^k$ for all positive integers $k$.
\end{lemma}
Borel \cite{[Borel]} showed that in every base, almost every number is normal, in the sense that non-normal numbers have Lebesgue measure $0$. Despite their overwhelming population on the real line, mathematicians have yet to prove that any of the commonplace mathematical constants like $\pi$, $e$, $\sqrt{2}$, $\phi$ etc. are normal. Because of this, the earliest examples of normal numbers were concatenations of well known sequences. Champernowne \cite{[Champernowne]} showed the concatenation of the natural numbers is normal in base $10$. Other early examples of normal numbers in base $10$ include the concatenation of the square numbers (Besicovitch \cite{[Besicovitch]}) and the concatenation of the primes (Copeland and Erd\H{o}s \cite{[Copeland]}).
\begin{align}
\nonumber &\text{Champernowne}&.123456789101112131415161718192021222324252\ldots\\
\nonumber &\text{Besicovitch}&.149162536496481100121144169196225256289324\ldots\\
\nonumber &\text{Copeland \& Erd\H{o}s}&.235711131719232931374143475359616771737983\ldots
\end{align}
We consider here the concatenation of the Fibonacci sequence behind a decimal. Let $\left(F_{n,\beta}\right)$ denote the Fibonacci sequence in base $\beta$ and let $\left<\left(F_{n,\beta}\right)\right>$ denote its concatenation behind a decimal.
\begin{example*}
In bases $2,\ldots,10$, the concatenation of the Fibonacci sequence is given by the following:
\begin{align}
\nonumber \left<\left(F_{n,2}\right)\right>=&.01110111011000110110101100010110111101100110010000111\ldots\\
\nonumber \left<\left(F_{n,3}\right)\right>=&.01121012221112101021200110022121002212211122221112111\ldots\\
\nonumber \left<\left(F_{n,4}\right)\right>=&.01123112031111202313112121003221113212120233123120331\ldots\\
\nonumber \left<\left(F_{n,5}\right)\right>=&.01123101323411142103241034141330024420124222234240314\ldots\\
\nonumber \left<\left(F_{n,6}\right)\right>=&.01123512213354131225400102514252454432311221155443120\ldots\\
\nonumber \left<\left(F_{n,7}\right)\right>=&.01123511163046106155264452104615312610444110351151222\ldots\\
\nonumber \left<\left(F_{n,8}\right)\right>=&.01123510152542671312203515711142173330755030101251515\ldots\\
\nonumber \left<\left(F_{n,9}\right)\right>=&.01123581423376110817027845874713162164348156551024616\ldots\\
\nonumber \left<\left(F_{n,10}\right)\right>=&.01123581321345589144233377610987159725844181676510946\ldots
\end{align}
\end{example*}
There are periodic patterns in the Fibonacci sequence for every integer modulus. In 1877, Lagrange \cite{[Lagrange]} noted that the Fibonacci sequence modulo $10$ repeats every $60$ terms. Several specific results for the periodicity of the Fibonacci sequence modulo various integers followed, culminating in the following theorem of Wall from 1960~\cite{[Wall]}.
\begin{theorem}[Wall]
For every positive integer $m$, the Fibonacci sequence is periodic modulo $m$.
\end{theorem}
The length of this period is known as the \textit{Pisano period} and is denoted $\pi(m)$. Pisano periods have been generalized for Lucas numbers, Pell numbers, $(a,b)$-Fibonacci numbers, and $n$-step Fibonacci numbers. For our purpose, we are only concerned with the Fibonacci sequence, and so will use the following definition.
\begin{definition}
The \textit{Pisano period} of a positive integer $m$, denoted $\pi(m)$, is the smallest number $k>0$ such that $F_k\equiv0\mod{m}$ and $F_{k+1}\equiv1\mod{m}$.
\[\text{-Equivalently-}\]
The Pisano period $\pi(m)$ is the length of one (shortest) period of the Fibonacci sequence modulo~$m$.
\end{definition}
\begin{example*} The sequence of Pisano periods can be found at OEIS: A001175 \cite{[A001175]}. Below are the first few Pisano periods up to $20$.
\begin{table}[H]
\centering
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}
$m$&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19&20\\
\hline
$\pi(m)$&3&8&6&20&24&16&12&24&60&10&24&28&48&40&24&36&24&18&60\\
\end{tabular}
\caption{Pisano period of the first $20$ integers.}
\end{table}
\end{example*}
Notice that the only odd integer that is a Pisano period is $\pi(2)=3$. Combining this with the results of Stanley \cite{[Stanley]} and Ehrlich \cite{[Ehrlich]}, Renault \cite{[Renault]} showed the following:
\begin{proposition}[Renault]
For every even integer $n>4$, there is an integer $m$ such that $\pi(m)=n$.
\end{proposition}
A property that we will use frequently is the \textit{nesting property} of Pisano periods. For every positive integer $n$, the Pisano period of $n$ divides the Pisano period of every multiple of $n$.
\begin{lemma}[Wall \cite{[Wall]}]\label{nesting}
For any two positive integers $m,n$, if $m\mid n$ then $\pi(m)\mid\pi(n)$.
\end{lemma}
For our purpose, we only need the nesting property for powers of the base we are considering. Given an integer $m$, $\pi(\beta^i)\mid\pi(\beta^{i+1})$.
\begin{example}
Following the investigation of Legrange, Geller \cite{[Geller]} observes the nesting of powers of 10:
\[
\pi(10)=60, \qquad \pi(100)=300, \qquad \pi(1000)=1500, \qquad \pi(10000)=15000, \ldots
\]
\end{example}
This pattern was later generalized by Jarden~\cite{[Jarden]} into the following proposition:
\begin{proposition}[Jarden]
The Fibonacci sequence is periodic modulo powers of ten with the following periods:
\[\pi(10^k)=\begin{cases}
60&k=1\\
300&k=2\\
15\cdot 10^{k-1}&k\vargeq3\qquad .
\end{cases}\]
\end{proposition}
From the Lemma~\ref{nesting} it follows that for a positive prime $p$, $\pi(p^i)\varleq\pi(p^{i+1})$. We can refine this expression to strict inequality, provided the prime $p$ is not a Wall-Sun-Sun prime.
\begin{definition}[Wall, Sun, Sun]
A Wall-Sun-Sun prime is a prime number $p$ such that $\pi(p)=\pi(p^2)$.
\end{definition}
Although there are conjectured to be infinitely many Wall-Sun-Sun primes \cite{[Klaska]}, none have been found. If it can be shown that Wall-Sun-Sun primes do not exist, that would imply Fermat's last theorem \cite{[Sun-Sun]}. Wall \cite{[Wall]} proposed the following lemma that tests whether a candidate integer can be a Wall-Sun-Sun prime.
\begin{lemma}[Wall]\label{lemma:WallCriteria}
Given a prime $p$, let $t$ be the largest integer such that $\pi(p^t)=\pi(p)$. Then for all $e\vargeq t$, $\pi(p^e)=p^{e-t}\pi(p)$.
\end{lemma}
The online research community \underline{PrimeGrid} is currently leading a computer search for Wall-Sun-Sun primes with its leading edge over $3\times10^{17}$. Because we have still not found a Wall-Sun-Sun prime, we can say that for every known prime that has been tested, $t=1$ and $\pi(p^e)=p^{e-1}\pi(p)$.
\begin{corollary}\label{conj:PiGrowth1}
For every prime $p$ that is not a Wall-Sun-Sun prime, $\pi(p^i)<\pi(p^{i+1})$ for $i \in \ZZ^+$.
\end{corollary}
\begin{proof}
Because we are supposing that $p$ is not a Wall-Sun-Sun prime, Wall's lemma \ref{lemma:WallCriteria} tells us that for all $e\vargeq1$, $\pi(p^e)=p^{e-1}\pi(p)$. Then $\pi(p^i)=p^{i-1}\pi(p)<p^{(i+1)-1}\pi(p)=\pi(p^{i+1})$.
\end{proof}
Leveraging the following Lemma by Wall \cite{[Wall]}, we can extend Corollary \ref{conj:PiGrowth1} to composite integers.
\begin{lemma}[Wall]\label{lemma:lcm}
If $m$ has the prime factorization $m=p_1^{e_1}p_2^{e_2}\cdots p_n^{e_n}$, then
\[
\pi(m)=\text{LCM}\left(\pi(p_1^{e_1}),\pi(p_2^{e_2}),\cdots ,\pi(p_n^{e_n})\right).
\]
\end{lemma}
Then we have the following result:
\begin{proposition}\label{conj:PiGrowth2}
For every integer $m$ that has no Wall-Sun-Sun primes in its prime factorization, $\pi(m^i)<\pi(m^{i+1})$ for $i \in \ZZ^+$.
\end{proposition}
\begin{proof}
Suppose $m$ has the prime factorization $m=p_1^{e_1}p_2^{e_2}\ldots p_n^{e_n}$, where no $p$ is a Wall-Sun-Sun prime. Then we have the following:
\[
\pi(m^i)=\text{LCM}\left(\pi(p_1^{{ie_1}})\pi(p_2^{{ie_2}})\cdots\pi(p_n^{{ie_n}})\right).
\]
And by Theorem \ref{conj:PiGrowth1} we have that for all $j=1,2,\ldots,n$, $\pi(p_j^{ie_j})<\pi(p_j^{(i+1)e_j})$. Then
\begin{align*}
\pi(m^i)&=\text{LCM}\left(\pi(p_1^{{ie_1}})\pi(p_2^{{ie_2}})\cdots\pi(p_n^{i(e_n)})\right)\\
&<\text{LCM}\left(\pi(p_1^{{(i+1)e_1}})\pi(p_2^{{(i+1)e_2}})\cdots\pi(p_n^{{(i+1)e_n}})\right)\\
&=\pi(m^{i+1}).\qedhere
\end{align*}
\end{proof}
\section{Bases and Moluli}
We prove the Main Theorem by exploiting the connection between the digits in the $\beta^k$'s place of the Fibonacci sequence in base $\beta$ and the Fibonacci sequence modulo $\beta^{k+1}$. To extract the digits $0,1,2,\ldots,\beta-1$ in the $\beta^k$'s place, we introduce the following function:
\begin{definition}
Denote by $\left(\Phi_{\beta^k}(n)\right)_{n\in\mathbb{N}}$ the sequence of $\beta^k$'s place digits in the Fibonacci sequence in base $\beta$. To obtain this sequence, we can use the following function:
\begin{align*}
\Phi_{\beta^k}:\mathbb{N}&\rightarrow\mathbb{Z}/\beta\mathbb{Z} \\
n&\mapsto\left\lfloor \frac{F_n}{\beta^k} \right\rfloor \mod{\beta} .
\end{align*}
\end{definition}
Let $\left(\Phi_{\beta^k}(n)\right)_{n=1}^{N_k}$ denote one (shortest) period of the $\beta^k$'s place digits of the Fibonacci sequence in base $\beta$ where $N_k$ is the length of a shortest period of the sequence $\left(\Phi_{\beta^k}(n)\right)_{n\in\mathbb{N}}$. This notation is justified in the following Lemma.
\begin{lemma}
\label{thm:PiDigits}
The sequence $\left(\Phi_{\beta^k}(n)\right)_{n\in\mathbb{N}}$ is periodic. Furthermore, if the base $\beta$ is not divisible by any Wall-Sun-Sun primes, the length $N_k$ of its period is equal to $\pi(\beta^{k+1})$.
\end{lemma}
\begin{proof}
Periodicity of the sequence $\left(\Phi_{\beta^k}(n)\right)_{n\in\mathbb{N}}$ follows immediately from the periodicity of the Fibonacci sequence modulo $\beta^{k+1}$; all of the digits must repeat after $\pi(\beta^{k+1})$ terms, so the $k^\text{th}$ digit (in base $\beta$) certainly repeats every $\pi(\beta^{k+1})$ terms. It follows that $N_k \mid \pi(\beta^{k+1})$.
To show equality, we proceed by contradiction. Suppose that we do not have equality, then $N_k < \pi(\beta^{k+1})$. That says that the $k^\text{th}$ digit (in base $\beta$) repeats on a shorter period than the Fibonacci number modulo $\beta^{k+1}$. So the sequence modulo $\beta^{k+1}$, ignoring the $k^\text{th}$ digit, would have the same period. In other words, $\pi(\beta^k ) = \pi(\beta^{k+1})$. But this contradicts Proposition~\ref{conj:PiGrowth2}.
\end{proof}
\begin{example*}
Consider the Fibonacci sequence in ternary (base $3$). The $3^0$'s place digits are given by the sequence:
\begin{align}
\nonumber \left(\Phi_{3^0}(n)\right)_{n=1}^{8}=0, 1, 1, 2, 0, 2, 2, 1\circlearrowleft .
\end{align}
This sequence repeats every $8$ terms. The length exactly coincides with the Pisano period $\pi(3)=\pi(\beta^{1})$ (this sequence is indeed the Fibonacci sequence modulo $3$). Consider now the sequence of $3^1$'s place digits in base $3$:
\begin{align}
\nonumber \left(\Phi_{3^1}(n)\right)_{n=1}^{24}=0, 0, 0, 0, 1, 1, 2, 1, 1, 2, 0, 2, 0, 2, 2, 2, 2, 1, 0, 1, 2, 0, 2, 0\circlearrowleft .
\end{align}
This sequence repeats every $24$ terms. The length exactly coincides with the Pisano period $\pi(9)=\pi(\beta^{2})$. Again, consider the sequence of $3^2$'s place digits in base $3$:
\begin{align}
\nonumber \left(\Phi_{3^2}(n)\right)_{n=1}^{72}=&0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 2, 1, 1, 0, 2, 2, 1, 1, 2, 1, \\
\nonumber &1, 2, 0, 2, 2, 1, 0, 2, 0, 2, 0, 2, 0, 2, 2, 2, 2, 2, 2, 1, 0, 2, 2, 2, \\
\nonumber &2, 1, 0, 1, 1, 2, 0, 0, 1, 1, 0, 1, 2, 0, 2, 0, 0, 1, 2, 0, 2, 0, 2, 0\circlearrowleft .
\end{align}
This sequence repeats every $72$ terms. The length exactly coincides with the Pisano period of $\pi(27)=\pi(\beta^3)$.
\end{example*}
It is well known \cite{[Gupta], [Renault]} that the integer bases, with respect to the Fibonacci sequence, can be evenly divided into three categories according to the number of zeros in their Pisano period.
\begin{definition}
For a base $\beta$, denote by $\omega(\beta)$ the number of zeros in one Pisano period of $\beta$.
\end{definition}
The sequence produced by $\omega(n)$ for $n=1,2,3,\ldots$ can be found at OEIS A001176 \cite{[A001176]}.
\begin{table}[H]
\centering
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}
$\beta$&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19&20\\
\hline
$\pi(\beta)$&3&8&6&20&24&16&12&24&60&10&24&28&48&40&24&36&24&18&60\\
\hline
$\omega(\beta)$&1&2&1&4&2&2&2&2&4&1&2&4&2&2&2&4&2&1&2
\end{tabular}
\caption{Table of $\beta$, $\pi(\beta)$, and $\omega(\beta)$ for $2\varleq\beta\varleq20$.}
\end{table}
It has been proven by Gupta et.~al.~\cite{[Gupta]} that the only possible values for $\omega(n)$ are $1, 2,$ and $4$. In the table below, we categorize the integer bases according to the number of zeros in their Pisano period.
\begin{table}[H]
\centering
\begin{tabular}{c|c|c}
&&\\
$\omega(\beta)=1$&$\omega(\beta)=2$&$\omega(\beta)=4$\\
&&\\
\hline
1, 2, 4, 11, 19, 22, 29, 31,&3, 6, 7, 8, 9, 12, 14, 15,&5, 10, 13, 17, 25, 26, 34, 37,\\
38, 44, 58, 59, 62, 71, 76, 79,&16, 18, 20, 21, 23, 24, 27, 28,&50, 53, 61, 65, 73, 74, 85, 89,\\
101, 116, 118, 121, 124, 131,&30, 32, 33, 35, 36, 39, 40,&97, 106, 109, 113, 122, 125,\\
139,142, 151, 158, 179, 181,\ldots&41, 42, 43, 45, 46, 47, 48, 49,\ldots&130, 137, 146, 149, 157\ldots\\
&&\\
OEIS: A053031 \cite{[A053031]}&OEIS: A053030 \cite{[A053030]}&OEIS: A053029 \cite{[A053029]}
\end{tabular}
\caption{Splitting of bases $\beta$ according to the number of zeros in one Pisano period}
\end{table}
Although there are countably many integer bases in each category, the rate at which each category grows is not balanced. Of the first ten thousand bases, $1013$ bases have one zero in their Pisano period, $7917$ bases have two zeros in their Pisano period, and $1070$ bases have four zeros in their Pisano period. Evidence of this uneven distribution in found in a series of conjectures on the OEIS: A053029, A053030, and A053031 \cite{[A053029], [A053030], [A053031]} concerning the number of zeros in an integer's Pisano period. We have summarized the three conjectures from the OEIS into the following:
\begin{conjecture}\label{conj:oeis}[OEIS: A053029, A053030, A053031]\\
\\
\romannumeral1.\relax \ An integer $m$ has four zeros in its Pisano period if and only if $m$ is an odd number, all of whose factors have four zeros in their Pisano period, or if $m$ is twice such a number.\\
\\
\romannumeral2.\relax \ An integer $m$ has one zero in its Pisano period if and only if $m$ is an odd number, all of whose factors have one zero in their Pisano period, or if $m$ is twice or four times such a number.\\
\\
\romannumeral3.\relax \ Every other integer has one zero in its Pisano period.
\end{conjecture}
There is a table of relations compiled by Vinson \cite{[Vinson]} that was distilled into a theorem of Renault \cite{[Renault]} categorizing when a number has exactly $1$, $2$, or $4$ zeros in its Pisano period.
\begin{theorem}[Renault]\label{thm:Renault}
Let $m$ and $n$ be integer bases, then:
\begin{table}[H]
\centering
\begin{tabular}{cc|ccc}
&&&$\omega(m)$&\\
&&&&\\
&&1&2&4\\
\hline
&&&&\\
&1&1&2&4 if $m=2$, else 2\\
&&&&\\
$\omega(n)$&2&2&2&2\\
&&&&\\
&4&4 if $n=2$, else 2&2&4
\end{tabular}
\caption{Table of $\omega\left(\text{LCM}(m,n)\right)$.}
\end{table}
\end{theorem}
We will primarily concern ourselves in this paper with bases $\beta$ where $\omega(\beta)=4$. There is a stability to these bases that allows us to determine if the Fibonacci concatenation is normal. First we will show that in the smallest such base, base $5$, the Fibonacci concatenation is \textit{normal}.
\section{Bases of the form $5^x$}
The technique to prove the Main Theorem relies on measuring the distribution of the digits of the Fibonacci sequence. We are particularly interested in the case where the digits are \textit{uniformly distributed}.
\begin{definition}
Given a number in base $\beta$, the digits $0,1,2,\ldots,\beta-1$ are \textit{uniformly distributed} if every digit occurs with a frequency of $1/\beta$.
\end{definition}
Consider the Fibonacci sequence in base $5$. The digits in base $5$ are $0,1,2,3,4$. Then the Fibonacci sequence in base 5 begins:
\begin{align}
\nonumber (F_{n,5})=0, 1, 1, 2, 3, 10, 13, 23, 41, 114, 210, 324, 1034, 1413, 3002, 4420, 12422, 22342, 40314, \ldots
\end{align}
There is a curious fact \cite{[Niederreiter]} about the distribution of the digits in the Fibonacci sequence that only holds modulo a power of $5$:
\begin{lemma}[Niederreiter \cite{[Niederreiter]}]\label{lemma:Niederreiter}
The digits $0,1,2,\ldots,5^k-1$ in the Fibonacci sequence modulo $5^k$ are uniformly distributed.
\end{lemma}
\begin{proposition}\label{prop:base5unif}
For every positive integer $k$, the digits $0,1,2,3,4$ are uniformly distributed in the sequences $\left(\Phi_{5^k}(n)\right)_{n=1}^{N_k}$.
\end{proposition}
\begin{proof}
We proceed by induction on $k$.
For $k=0$, Lemma~\ref{lemma:Niederreiter} says that each digit is uniformly distributed in $\left(\Phi_{5^0}(n)\right)_{n=1}^{N_0}$. Now, suppose for all $0\varleq i< j$ that the sequences $\left(\Phi_{5^i}(n)\right)_{n=1}^{N_i}$ are uniformly distributed and consider the sequence $\left(\Phi_{5^{j}}(n)\right)_{n=1}^{N_{j}}$. By Lemma~\ref{lemma:Niederreiter}, we know that the Fibonacci sequence modulo $5^{j+1}$ is uniformly distributed. Further, the digits in each place $5^i$ for all $0\varleq i<j$ are uniformly distributed. So the digits of the Fibonacci sequence in the $5^{j}$'s place must be uniformly distributed. It follows that the digits in the sequence $\left(\Phi_{5^{j}}(n)\right)_{n=1}^{N_{j}}$ are uniformly distributed.
\end{proof}
\begin{example*}
Consider the function $\Phi_{5^0}(n)$. A single period of the $5^0$'s place is given below:
\begin{align}
\nonumber \left(\Phi_{5^0}(n)\right)_{n=1}^{20}=0,1,1,2,3,0,3,3,1,4,0,4,4,3,2,0,2,2,4,1\circlearrowleft .
\end{align}
Notice that each digit $0,1,2,3,4$ appears \textit{exactly} four times. After each period of twenty terms of the Fibonacci sequence in base $5$, each digit in the $5^0$'s place has a frequency of $1/5$; the digits are uniformly distributed. Next, consider the $5^1$'s place. The length of the period is given by $\pi(5^2)=100$. A single period of the $5^1$'s place is given below:
\begin{align}
\nonumber \left(\Phi_{5^1}(n)\right)_{n=1}^{100}=&0,0,0,0,0,1,1,2,4,1,1,2,3,1,0,2,2,4,1,1,3,4,2,1,3,\\
\nonumber &0,3,3,2,0,3,3,1,0,2,3,0,3,3,2,1,3,4,2,1,4,0,4,0,4,\\
\nonumber &0,4,4,4,4,4,3,2,0,3,4,2,1,3,4,3,2,0,3,3,2,0,2,3,1,\\
\nonumber &0,1,1,2,4,2,1,3,4,2,2,4,1,1,2,4,1,0,2,3,1,4,0,4,0\circlearrowleft .
\end{align}
Notice again that each digit $0,1,2,3,4$ occurs \textit{exactly} twenty times; the digits are uniformly distributed. Further, there are exactly five periods of $5^0$'s place digits nested in every one period of $5^1$'s place digits. We can continue this pattern for every place value of the Fibonacci sequence in base $5$.
\end{example*}
\begin{theorem}\label{thm:coloring}
The concatenation of the Fibonacci sequence in base $5$ is \textit{simply normal}.
\end{theorem}
\begin{proof}
Because the digits in every place value of the Fibonacci sequence in base $5$ are uniformly distributed, we can construct its concatenation in a particular way that demonstrates that it is simply normal. Suppose we color the digits Fibonacci sequence so that every digit that occurs in the $k$'s place value is colored the same:
\begin{align}
\nonumber {\color{BrickRed}{\left(\Phi_{5^0}(n)\right)_{n\in\mathbb{N}}}}\ \ \ \ \ {\color{NavyBlue}{\left(\Phi_{5^1}(n)\right)_{n\in\mathbb{N}}}}\ \ \ \ \ {\color{OliveGreen}\left(\Phi_{5^2}(n)\right)_{n\in\mathbb{N}}}\ \ \ \ \ {\color{Plum}{\left(\Phi_{5^3}(n)\right)_{n\in\mathbb{N}}}}\ \ \ \ \ {\color{Dandelion}{\left(\Phi_{5^4}(n)\right)_{n\in\mathbb{N}}}}
\end{align}
\begin{align}
\nonumber (F_{n,5})= {\color{BrickRed}{0}}, {\color{BrickRed}{1}}, {\color{BrickRed}{1}}, {\color{BrickRed}{2}}, {\color{BrickRed}{3}}, {\color{NavyBlue}{1}}{\color{BrickRed}{0}}, {\color{NavyBlue}{1}}{\color{BrickRed}{3}}, {\color{NavyBlue}{2}}{\color{BrickRed}{3}}, {\color{NavyBlue}{4}}{\color{BrickRed}{1}}, {\color{OliveGreen}{1}}{\color{NavyBlue}{1}}{\color{BrickRed}{4}}, {\color{OliveGreen}{2}}{\color{NavyBlue}{1}}{\color{BrickRed}{0}}, {\color{OliveGreen}{3}}{\color{NavyBlue}{2}}{\color{BrickRed}{4}}, {\color{Plum}{1}}{\color{OliveGreen}{0}}{\color{NavyBlue}{3}}{\color{BrickRed}{4}}, {\color{Plum}{1}}{\color{OliveGreen}{4}}{\color{NavyBlue}{1}}{\color{BrickRed}{3}}, {\color{Plum}{3}}{\color{OliveGreen}{0}}{\color{NavyBlue}{0}}{\color{BrickRed}{2}}, {\color{Plum}{4}}{\color{OliveGreen}{4}}{\color{NavyBlue}{2}}{\color{BrickRed}{0}}, {\color{Dandelion}{1}}{\color{Plum}{2}}{\color{OliveGreen}{4}}{\color{NavyBlue}{2}}{\color{BrickRed}{2}}, {\color{Dandelion}{2}}{\color{Plum}{2}}{\color{OliveGreen}{3}}{\color{NavyBlue}{4}}{\color{BrickRed}{2}}, {\color{Dandelion}{4}}{\color{Plum}{0}}{\color{OliveGreen}{3}}{\color{NavyBlue}{1}}{\color{BrickRed}{4}}, \ldots
\end{align}
Then the concatenation of the Fibonacci sequence in base 5 is given by:
\begin{align}
\nonumber \left<(F_{n,5})\right>= {\color{BrickRed}{0}}{\color{BrickRed}{1}}{\color{BrickRed}{1}}{\color{BrickRed}{2}}{\color{BrickRed}{3}}{\color{NavyBlue}{1}}{\color{BrickRed}{0}}{\color{NavyBlue}{1}}{\color{BrickRed}{3}}{\color{NavyBlue}{2}}{\color{BrickRed}{3}}{\color{NavyBlue}{4}}{\color{BrickRed}{1}}{\color{OliveGreen}{1}}{\color{NavyBlue}{1}}{\color{BrickRed}{4}}{\color{OliveGreen}{2}}{\color{NavyBlue}{1}}{\color{BrickRed}{0}}{\color{OliveGreen}{3}}{\color{NavyBlue}{2}}{\color{BrickRed}{4}}{\color{Plum}{1}}{\color{OliveGreen}{0}}{\color{NavyBlue}{3}}{\color{BrickRed}{4}}{\color{Plum}{1}}{\color{OliveGreen}{4}}{\color{NavyBlue}{1}}{\color{BrickRed}{3}}{\color{Plum}{3}}{\color{OliveGreen}{0}}{\color{NavyBlue}{0}}{\color{BrickRed}{2}}{\color{Plum}{4}}{\color{OliveGreen}{4}}{\color{NavyBlue}{2}}{\color{BrickRed}{0}}{\color{Dandelion}{1}}{\color{Plum}{2}}{\color{OliveGreen}{4}}{\color{NavyBlue}{2}}{\color{BrickRed}{2}}{\color{Dandelion}{2}}{\color{Plum}{2}}{\color{OliveGreen}{3}}{\color{NavyBlue}{4}}{\color{BrickRed}{2}}{\color{Dandelion}{4}}{\color{Plum}{0}}{\color{OliveGreen}{3}}{\color{NavyBlue}{1}}{\color{BrickRed}{4}}\ldots
\end{align}
We can imagine the process of concatenating the Fibonacci sequence in base $5$ through this coloring technique by taking the digits $0,1,2,3,4$ in a piecemeal fashion extracted from each sequence $\left(\Phi_{5^k}(n)\right)_{n\in\mathbb{N}}$ and threading them into the appropriate place. Every digit in the Fibonacci concatenation in base $5$ belongs to a particular $\left(\Phi_{5^k}(n)\right)_{n\in\mathbb{N}}$. Further, all of the digits in ${\color{BrickRed}{\left(\Phi_{5^0}(n)\right)_{n\in\mathbb{N}}}}$ occur with a frequency of $1/5$, and all of the digits in ${\color{NavyBlue}{\left(\Phi_{5^1}(n)\right)_{n\in\mathbb{N}}}}$ occur with a frequency of $1/5$, and all of the digits in ${\color{OliveGreen}{\left(\Phi_{5^2}(n)\right)_{n\in\mathbb{N}}}}$ occur with a frequency of $1/5$, etc. That is, the limiting relative frequency of every digit $0,1,2,3,4$ in each place value of the Fibonacci concatenation base $5$ is $1/5$. Then the limiting relative frequency of each digit $0,1,2,3,4$ in the Fibonacci concatenation base $5$ is $1/5$. It follows that the concatenation of the Fibonacci sequence in base $5$ is \textit{simply normal}.
\end{proof}
Note that the result by Niederreiter \ref{lemma:Niederreiter} shows that the Fibonacci sequence is uniformly distributed modulo \textit{every} power of $5$. It immediately follows that the same coloring technique used in Theorem~\ref{thm:coloring} can be used to show that the Fibonacci concatenation is simply normal in every base $\beta=5^x$.
\begin{proposition}\label{thm:simplynormal5x}
In every base of the form $5^x$, the concatenation of the Fibonacci sequence is \textit{simply normal}.
\end{proposition}
\begin{proof}
From Lemma~\ref{lemma:Niederreiter} we know that in every base of the form $5^x$, the digits $0,1,2,\ldots,5^x-1$ in the Fibonacci sequence base $5^x$ are uniformly distributed. By induction on the result of Proposition \ref{prop:base5unif}, the digits in the sequences $\left(\Phi_{{(5^x)}^k}(n)\right)_{n\in\mathbb{N}}$ are uniformly distributed. We can apply the same coloring technique that was used in Theorem \ref{thm:coloring} to every base that is a power of $5$ because the digits are always uniformly distributed. It follows that the Fibonacci concatenation in every base of the form $\beta=5^x$ is \textit{simply normal}.
\end{proof}
\begin{corollary}
In every base of the form $5^x$, the Fibonacci concatenation is \textit{normal}.
\end{corollary}
\begin{proof}
This follows immediately from Proposition~\ref{thm:simplynormal5x} and Lemma \ref{lemma:Pillai}.
\end{proof}
\section{Bases of the form $2^y$}
Consider the Fibonacci sequence in base 2 where the only digits are $0$ and $1$. We are looking at the sequence:
\[ F_{n,2}= 0, 1, 1, 10, 11, 101, 1000, 1101, 10101, 100010, 110111, 1011001, 10010000, 11101001,\ldots\]
We would like to apply the same coloringing technique we used in base $5$ to the Fibonacci concatenation in base $2$. Unfortunately, the frequency of $0$'s and $1$'s is not equal in the first five place values.
\begin{table}[H]
\centering
\begin{tabular}{c|c|c|c|c}
$k$'s place&$\pi(2^{k+1})$&$\left(\Phi_{2^k}(n)\right)$&$0$'s&$1$'s\\
\hline
$2^0$'s place&3&$0,1,1\circlearrowleft$&$1$&$2$\\
\hline
$2^1$'s place&$6$&$0, 0, 0, 1, 1, 0\circlearrowleft$&$4$&$2$\\
\hline
$2^2$'s place&$12$&$0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0\circlearrowleft$&$8$&$4$\\
\hline
$2^3$'s place&$24$&$0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0\circlearrowleft$&$10$&$14$\\
\hline
$2^4$'s place&$48$&$0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1,$&&\\
&&$0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0\circlearrowleft$&$28$&$20$
\end{tabular}
\caption{Distribution of $0$'s and $1$'s in the first five place values of the Fibonacci sequence in base $2$.}
\end{table}
However, from Theorem~\ref{Jacobson} below, the sequences $\left(\Phi_{2^k}(n)\right)_{n=1}^{N_k}$ are uniformly distributed for all $k\vargeq5$.
For example, in the $2^5$'s place, there are exactly forty eight $0$'s and forty eight $1$'s.
\begin{align}
\nonumber \left(\Phi_{2^5}(n)\right)_{n=1}^{96}=&0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1,\\
\nonumber &1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1,\\
\nonumber &0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0,\\
\nonumber &1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0\circlearrowleft .
\end{align}
The proof technique that we will use to demonstrate normality of the Fibonacci concatenation in bases of the form $2^y$ is very similar to the coloring technique, except that we will ignore the first five place values.
\begin{theorem}\label{thm:LimitingFreq}
For a base $\beta$, if there exists a $K$ such that for all $k\vargeq K$, the digits $0, 1, 2,\ldots, \beta-1$ in the $\beta^k$'s place of the Fibonacci sequence base $\beta$ occur with a frequency of $1/\beta$, then the Fibonacci concatenation in base $\beta$ is \textit{simply normal}.
\end{theorem}
\begin{proof}
Suppose in base $\beta$ that the Fibonacci concatentaion is not uniformly distributed for the first $K$ places, but for all $k\vargeq K$, the digits in the $\beta^k$'s place are uniformly distributed. The digits in the $\beta^0$'s place, $\beta^1$'s place, $\ldots, \beta^K$'s place do not effect the limiting frequency of each digit, in the sense that the digits in these first place values contribute a Lebesgue measure $0$ number of digits to the Fibonacci concatenation base $\beta$. There are a finite number of place values that are not uniformly distributed, and an infinite number of place values that are uniformly distributed. It follows that the measure of the digits in the $\beta^0$'s place, $\beta^1$'s place, $\ldots, \beta^K$'s place is zero.
\end{proof}
Another way to say this is that the digits in the $\beta^0$'s place, $\beta^1$'s place, $\ldots, \beta^K$'s place get overwhelmed in the limit. Suppose we choose a digit at random from the Fibonacci concatenation in such a base; the odds of choosing a digit from a place value that is not uniformly distributed has probability $0$. With this theorem in mind, we turn our attention back to the Fibonacci sequence in base $2$.
The distribution of digits of the Fibonacci sequence modulo $2^k$ is well established. Jacobson \cite{[Jacobson]} completely described the number of occurrences of $0$'s and $1$'s. He observed that for the Fibonacci sequence modulo $2^k$, there is a ``type of stability" that occurs when $k\vargeq 5$.
\begin{theorem}[Jacobson]\label{Jacobson}
Denote the number of occurrences of $z$ as a residue in one (shortest) period of the Fibonacci sequence modulo $m$ by $v(m,z)$. Then for the Fibonacci sequence modulo~$2^k$ and for $k\vargeq5$, we have:
\begin{align*}
v(2^k,z)=&\begin{cases}
1 & \text{if}\ z\equiv3\mod4\\
2 & \text{if}\ z\equiv0\mod8\\
3 & \text{if}\ z\equiv1\mod4\\
8 & \text{if}\ z\equiv2\mod32\\
0 & \text{for all other residues.}
\end{cases}
\end{align*}
\end{theorem}
Jacobson recognized that after the $2^4$'s place, there is a regular pattern that emerges. This stability of the Fibonacci sequence modulo $2$ allows us to make observations about the Fibonacci concatenation in base $2$.
\begin{theorem}\label{thm:UniDistBase2}
For all $k\vargeq 5$, the digits $0$ and $1$ in $\left(\Phi_{2^k}(n)\right)_{n\in\mathbb{N}}$ are uniformly distributed.
\end{theorem}
\begin{proof}
Consider the Fibonacci sequence in base $2$ and let $k\vargeq5 $. We will represent numbers written in binary modulo $2^{k+1}$ in the following way:
\[
1\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}} a_4a_3a_2a_1a_0 \quad \text{ and } \quad 0\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}}a_4a_3a_2a_1a_0.
\]
Consider the occurence of numbers congruent to $3$ modulo $4$ in the Fibonacci sequence modulo $2^{k+1}$. Every number that has a residue of $3$ modulo $4$ occurs exactly once in one (shortest) period of the Fibonacci sequence modulo $2^{k+1}$. That is, the numbers
\[
1\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}} a_4a_3a_211 \quad \text{ and } \quad 0\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}}a_4a_3a_211.
\]
each occur exactly once in $\left(\Phi_{2^k}(n)\right)_{n=1}^{N_k}$. Next, consider the occurence of numbers congruent to $0$ modulo $8$ in the Fibonacci sequence modulo $2^{k+1}$. Every number that has a residue of $0$ modulo $8$ occurs exactly twice in one (shortest) period of the Fibonacci sequence modulo $2^{k+1}$. That is, the numbers
\[
1\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}} a_4a_3000 \quad \text{ and } \quad 0\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}}a_4a_3000.
\]
each occur exactly twice in $\left(\Phi_{2^k}(n)\right)_{n=1}^{N_k}$. Next, consider the occurence of numbers congruent to $1$ modulo $4$ in the Fibonacci sequence modulo $2^{k+1}$. Every number that has a residue of $1$ modulo $4$ occurs exactly three times in one (shortest) period of the Fibonacci sequence modulo $2^{k+1}$. That is, the numbers
\[
1\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}} a_4a_3a_201 \quad \text{ and } \quad 0\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}}a_4a_3a_201.
\]
each occur exactly three times in $\left(\Phi_{2^k}(n)\right)_{n=1}^{N_k}$. Finally, consider the occurence of numbers congruent to $2$ modulo $32$ in the Fibonacci sequence modulo $2^{k+1}$. Every number that has a residue of $2$ modulo $32$ occurs exactly eight times in one (shortest) period of the Fibonacci sequence modulo $2^{k+1}$. That is, the numbers
\[
1\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}}00010 \quad \text{ and } \quad 0\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}}00010.
\]
each occur exactly eight times in $\left(\Phi_{2^k}(n)\right)_{n=1}^{N_k}$. All other residues occur exactly zero times in the Fibonacci sequence modulo $2^{k+1}$ for $k\vargeq 5$. Then there is a one-to-one correspondence between numbers in the smallest period with a 1 in the $2^k$ digit and numbers with a 0 in the $2^k$ digit. Hence, the digits in the Fibonacci sequence are uniformly distributed in every $2^k$'s place for $k\vargeq 5$.
\end{proof}
Now that we know that the digits $0$ and $1$ are uniformly distributed for sufficiently large place values, we have the following result:
\begin{corollary}\label{SimplyNormalPowers2}
The Fibonacci concatenation is \textit{simply normal} in every base of the form $\beta=2^k$.
\end{corollary}
\begin{proof}
This immediately follows from Theorem~\ref{thm:LimitingFreq} and Theorem ~\ref{thm:UniDistBase2}.
\end{proof}
Now that we know every base that is a power of $2$ is simply normal, we can invoke Borel's definition of normality and achieve a stronger result:
\begin{theorem}
The Fibonacci concatenation is \textit{normal} in every base of the form $\beta=2^y$.
\end{theorem}
\begin{proof}
This follows immediately from Corollary~\ref{SimplyNormalPowers2} and Lemma~\ref{lemma:Pillai}
\end{proof}
\section{Bases of the form $5^x2^y$}\label{sec:MainThm}
Consider now bases of the form $5^x2^y$. Because $5$ and $2$ are not Wall-Sun-Sun primes, by Lemma~\ref{lemma:WallCriteria} we know that $\pi(5^x)=20(5^{x-1})$ and $\pi(2^y)=3(2^{y-1})$. Further, because $2^y$ and $5^x$ will always be coprime, by Lemma \ref{lemma:lcm} we know that
\begin{align*}
\pi(5^x2^y)=\text{LCM}\left(\pi(5^x),\pi(2^y)\right)=\begin{cases}
12\times5^x & \text{if}\ 1\varleq y\varleq3\\
12\times5^x\times2^{y-3} & \text{if}\ y>3.
\end{cases}
\end{align*}
For example, if $x=y=1$, then $\pi(2\times5)=12(5^1)=60=\pi(10)$. Jacobson \cite{[Jacobson]} considered the Fibonacci sequence modulo integers of the form $5^x2^y$ and obtained the following result:
\begin{theorem}[Jacobson]\label{Jacobson2}
Denote the number of occurrences of $z$ as a residue in one (shortest) period of the Fibonacci sequence modulo $5^x2^y$ by $v(5^x2^y,z)$. Then for $x\vargeq0$ and $y\vargeq5$ we have that
\[
v(5^x2^y,z)=\begin{cases}
1 & \text{if}\ z\equiv3\mod4\\
2 & \text{if}\ z\equiv0\mod8\\
3 & \text{if}\ z\equiv1\mod4\\
8 & \text{if}\ z\equiv2\mod32\\
0 & \text{for all other residues.}
\end{cases}
\]
\end{theorem}
Jacobson showed here that because the Fibonacci sequence is uniformly distributed modulo every power of $5$, the residues in the Fibonacci sequence modulo any power of $5^x2^y$ will be the same as the residues modulo any power of $2^y$ provided, $y\vargeq5$.
\begin{theorem}\label{thm:UniDistBase5x2y}
For $\beta=5^x2^y$, and for all $k\vargeq 5$, the digits $0, 1, \ldots, \beta-1$ in $\left(\Phi_{\beta^k}(n)\right)_{n\in\mathbb{N}}$ are uniformly distributed.
\end{theorem}
\begin{proof}
Let $\beta = 5^x 2^y$. The case $x=0$ is exactly the statement of Theorem~\ref{Jacobson}, and the case $y=0$ is exactly the statement of Proposition~\ref{prop:base5unif}.
Assume now that $x\geq1$ and $y\geq1$, so in particular $10 \mid \beta$. The proof follows the same structure as Theorem~\ref{Jacobson}, but the bookkeeping is a bit more complicated.
Since $10 \mid \beta$, we have $4 \mid 100_\beta$, $8 \mid 1000_\beta$, and $32 \mid 100000_\beta$, meaning a number is 3 (resp.~1) modulo 4 exactly when the last two digits are 3 (resp.~1) modulo 4, a number is 2 modulo 8 exactly when the last three digits are 2 modulo 8, and a number is 8 modulo 32 exactly when the last five digits are 8 modulo 32.
Let $a_1a_0$ be a two-digit number in base $\beta$ that is congruent to 3 modulo 4. Then from~\ref{Jacobson2}, each of these numbers (written in base $\beta$) appears exactly once in one (shortest) period of the Fibonacci sequence modulo $\beta^{k+1}$:
\begin{align*}
\beta-1&\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}} a_4a_3a_2a_1a_0\\
&\vdots\\
1&\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}} a_4a_3a_2a_1a_0\\
0&\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}}a_4a_3a_2a_1a_0.
\end{align*}
Similarly, if $a_1a_0$ is a two-digit number in base $\beta$ that is congruent to 1 modulo 4, then each of these numbers (written in base $\beta$) appears exactly three times in one (shortest) period of the Fibonacci sequence modulo~$\beta^{k+1}$:
\begin{align*}
\beta-1&\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}} a_4a_3a_2a_1a_0\\
&\vdots\\
1&\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}} a_4a_3a_2a_1a_0\\
0&\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}}a_4a_3a_2a_1a_0.
\end{align*}
If $a_2a_1a_0$ represents a three-digit number in base $\beta$ that is congruent to 0 modulo 8, then each of these numbers appears exactly twice in one (shortest) period of the Fibonacci sequence modulo~$\beta^{k+1}$:
\begin{align*}
\beta-1&\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}} a_4a_3a_2a_1a_0\\
&\vdots\\
1&\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}} a_4a_3a_2a_1a_0\\
0&\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}}a_4a_3a_2a_1a_0.
\end{align*}
And if $a_4a_3a_2a_1a_0$ represents a five-digit number that is congruent to 2 modulo 32, then each of these numbers appears exactly eight times in one (shortest) period of the Fibonacci sequence modulo $\beta^{k+1}$:
\begin{align*}
\beta-1&\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}} a_4a_3a_2a_1a_0\\
&\vdots\\
1&\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}} a_4a_3a_2a_1a_0\\
0&\underbrace{\ldots\ldots\ldots}_\text{$k-5$ \text{ digits}}a_4a_3a_2a_1a_0.
\end{align*}
No other numbers appear in the Fibonacci sequence modulo $\beta^{k+1}$. This one-to-one correspondence shows that each first digit $0, 1, \dots, \beta-1$ appears exactly as often as every other digit in the Fibonacci sequence modulo $\beta^{k+1}$. It follows that the digits $0, 1, \dots, \beta-1$ in $\left(\Phi_{\beta^k}(n)\right)_{n\in\mathbb{N}}$ are uniformly distributed for $k\vargeq 5$.
\end{proof}
\begin{proposition}\label{SimplyNormalPowers5x2y}
The Fibonacci concatenation is \textit{simply normal} in every base of the form $\beta=5^x2^y$.
\end{proposition}
\begin{proof}
This immediately follows from Theorem~\ref{thm:UniDistBase5x2y} and Theorem~\ref{thm:UniDistBase5x2y}.
\end{proof}
As before, now that we know every base that is a power of $5^x2^y$ is simply normal, we can invoke Borel's definition of normality and achieve a stronger result.
\begin{proof}[Proof of Main Theorem]
The proof that the Fibonacci concatenation is \textit{normal} in every base of the form $\beta=5^x2^y$ immediately follows from Proposition~\ref{SimplyNormalPowers5x2y} and Lemma~\ref{lemma:Pillai}.
\end{proof}
\section{Other bases}
In this section we provide computational evidence to suggest that our Main Theorem holds for infinitely many bases, perhaps in every base. We begin with bases $\beta$ such that $\omega(\beta)=4$~\cite{[A053029]}:
\[
5, 10, 13, 17, 25, 26, 34, 37, 50, 53, 61, 65, 73, 74, 85, 89, 97, 106, 109, 113, 122, 125, 130, 137, 146, 149,\ldots
\]
We have seen that in the smallest entry on this list, base $5$, the Fibonacci concatenation is normal. There is computational and heuristic evidence to believe the Fibonacci concatenation is normal in \textit{every} base on this list. We begin with a definition.
\begin{definition}\label{def:Upsilon}
For an integer base $\beta$, let $\Upsilon(\beta)$ denote the smallest $K$ such that for all $k\vargeq K$, the digits $0,1,2,\ldots,\beta-1$ are uniformly distributed in $\left(\Phi_{\beta^k}(n)\right)_{n=1}^{N_k}$.
\end{definition}
\begin{conjecture}\label{conj:omega4UD}
For every base $\beta$ where $\omega(\beta)=4$, there exists a $K$ such that for all $k\vargeq K$, the digits in the sequences $\left(\Phi_{\beta^k}(n)\right)_{n\in\mathbb{N}}$ are uniformly distributed.
\end{conjecture}
In fact, the data collected in Table 6\footnote{Computations done in Mathematica.} below suggests that for all \textit{prime} bases $p>5$ where $\omega(p)=4$, conjecture \ref{conj:omega4UD} holds for $K=1$. In other words, the only time $\left(\Phi_{p^k}(n)\right)_{n\in\mathbb{N}}$ is not uniformly distributed is in the ones place.
\begin{table}[H]\label{table6}
\begin{tabular}{c|c|c}
Base $\beta$&$\Upsilon(\beta)$&Searched to $\beta^k$'s place\\
\hline
5&0&Proven normal\\
\hline
13&1&$13^4$'s place\\
\hline
17&1&$17^4$'s place\\
\hline
37&1&$37^3$'s place\\
\hline
53&1&$53^2$'s place\\
\hline
61&1&$61^2$'s place\\
\end{tabular}
\caption{The first few prime bases with four zeros in their Pisano period.}
\end{table}
\begin{conjecture}\label{conj:omega4normal}
In every base $\beta$ where $\omega(\beta)=4$, and for all nonnegative $x$, the Fibonacci concatenation is normal in every base of the form $2^y\beta$.
\end{conjecture}
First, computations below for small $\beta$ on the list indicate that the digits in each place beyond $\beta^0$ are uniformly distributed. If Conjecture \ref{conj:oeis} holds, then whenever $\omega(\beta)=4$ then $\omega(\beta^i)=4$ for all $i\vargeq1$. This suggests
applying Theorem~\ref{thm:LimitingFreq} to show that the Fibonacci concatenation is simply normal in these bases. Furthermore, from Theorem~\ref{thm:Renault}, if $\beta_1$ and $\beta_2$ are relatively prime such that $\omega(\beta_1) = \omega(\beta_2) = 4$, then $\omega(\beta_1\beta_2) = 4$ as well. Similarly, if $\omega(\beta)=4$, then $\omega(2^y\beta)=4$. All of this suggests that one could mimic the proofs in Section~\ref{sec:MainThm} in these cases.
We could then determine what happens when a base has a prime factorization $\beta=2^yp_1^{e_1}p_2^{e_2}\cdots p_j^{e_j}$ where for every odd prime $p$, $\omega(p)=4$.
Jacobson \cite{[Jacobson]} demonstrated how to combine information about distribution of Fibonacci numbers modulo relatively prime bases $5^x$ and $2^y$ into a result modulo $5^x2^y$. We suggest a generalization of his approach phrased in terms of uniform distribution of digits:
\begin{conjecture}\label{UpsilonMAX}
Given two coprime bases $\beta_1$ and $\beta_2$ such that $\Upsilon(\beta_1)=u$ and $\Upsilon(\beta_2)=v$ then $\Upsilon(\beta_1\beta_2)=\text{max}(uv)$.
Inductively, it would follow for any finite set of coprime bases $\beta_1, \beta_2, \ldots, \beta_n$ where $\Upsilon(\beta_1)=u_1$, $\Upsilon(\beta_2)=u_2$, $\ldots$, $\Upsilon(\beta_n)=u_n$ then $\Upsilon(\beta_1, \beta_2, \ldots, \beta_n)=\text{max}(u_1,u_2,\ldots,u_n)$.
\end{conjecture}
If we are able to determine that the sequences $\left(\Phi_{\beta^k}(n)\right)_{n\in\mathbb{N}}$ are uniformly distributed for each base of the form $2^y\beta$ where $\omega(\beta)=4$ and $y\vargeq0$, we could proceed as above. Applying Theorem \ref{thm:LimitingFreq} we know that every base of the form $\beta\times2^y$ (including powers of these bases from Conjecture~\ref{conj:oeis}) is \textit{simply normal}, and by Lemma~\ref{lemma:Pillai} we would conclude that every base where $\omega(\beta)=4$ is \textit{normal}.
We now turn our attention to bases not yet covered by Conjecture~\ref{conj:omega4normal}. For a base $\gamma$ that falls outside the criteria of Conjecture \ref{conj:omega4normal}, the sequences $\left(\Phi_{\gamma^k}(n)\right)_{n\in\mathbb{N}}$ do not obviously become uniformly dirstibuted based on our computer search. It is possible that there is a $K$ such that for all $k\vargeq K$ the sequences $\left(\Phi_{\gamma^k}(n)\right)_{n\in\mathbb{N}}$ are all uniformly distributed, but if this is the case, the smallest such $K$ is beyond our computational power. Nevertheless, we provide below computational evidence to believe that the Fibonacci concatenation is \textit{normal} in every base.
\begin{conjecture}\label{conj:absolutelynormal}
The Fibonacci concatenation is \textit{absolutely normal}.
\end{conjecture}
We will build the argument to support Conjecture \ref{conj:absolutelynormal} like before, by showing that the Fibonacci concatenation is \textit{simply normal} in every base. It follows that the Fibonacci concatenation is simply normal in every \emph{power} of every base. Then, by Lemma~\ref{lemma:Pillai}, the Fibonacci concatenation is \textit{normal} in every base.
\begin{conjecture}\label{conj:gammasimplynormal}
The Fibonacci concatenation is simply normal in \textit{every} base.
\end{conjecture}
\begin{example*}
Consider again the Fibonacci sequence in base 3. Below is a table of the first few sequences of $\left(\Phi_{3^k}(n)\right)_{n}^{N_k}$ along with a running total of the frequency of the digits $0, 1, 2$ and a running percentage showing the distribution at finite levels. The running total and running percentages are calculated using the nesting property (Lemma \ref{nesting}): in one period of $3^k$'s place digits there are three periods of $3^{k-1}$'s place digits, nine periods of $3^{k-2}$'s place digits, twenty seven periods of $3^{k-3}$'s place digits etc.
\begin{table}[H]
\centering
\begin{tabular}{c|c|c|c|c|c}
$3^k$'s&$0$'s&$1$'s&$2$'s&Running total of $0$'s:$1$'s:$2$'s&Running Percentage of $0$'s:$1$'s:$2$'s\\
\hline
$3^0$'s&2&3&3&2 : 3 : 3&25.0000\% : 37.5000\% : 37.5000\%\\
\hline
$3^1$'s&9&6&9&15 : 15 : 18&31.2500\% : 31.2500\% : 37.5000\%\\
\hline
$3^2$'s&27&18&27&72 : 63 : 81&$33.\overline{3333}$\% : $29.1\overline{666}$\% : 37.5000\%\\
\hline
$3^3$'s&75&66&75&291 : 255 : 318&$33.680\overline{5}\%$ : $29.513\overline{8}$\% : $36.80\overline{55}\%$\\
\hline
$3^4$'s&216&216&216&1089 : 981 : 1170&$33.6\overline{111}$\% : $30.2\overline{777}$\% : $36.\overline{1111}$\%\\
\hline
$3^5$'s&630&684&630&3897 : 3627 : 4140&33.4105\% : 31.0957\% : 35.4938\%\\
\hline
$3^6$'s&1971&1890&1971&13662 : 12771 : 14391&33.4656\% : 31.2831\% : 35.2513\%\\
\hline
$3^7$'s&5859&5778&5859&46845 : 44091 : 49032&33.4684\% : 31.5008\% : 35.0309\%\\
\hline
$3^8$'s&17577&17334&17577&158112 : 149607 : 164673&33.4705\% : 31.6701\% : 34.8594\%\\
\hline
$3^9$'s&52326&52812&52326&5266621 : 501633 : 546345&33.4465\% : 31.8570\% : 34.6964\%\\
\hline
$3^{10}$'s&157707&156978&157707&1737693 : 1661877 : 1796742&33.4409\% : 31.9818\% : 34.4577\%\\
\hline
$3^{11}$'s&472635&471906&472635&5685714 : 5457537 : 5862861&33.4334\% : 32.0916\% : 34.4750\%
\end{tabular}
\caption{Table of the number of times each digit appears in $\left(\Phi_{3^k}(n)\right)_{n=1}^{N_k}$ for $k\varleq11$.}
\end{table}
\end{example*}
Notice in the table that the limiting frequency of each digit seems to approach $1/3$. If this running percentage of the number of times each digit $0, 1, 2$ appears in $\left(\Phi_{3^k}(n)\right)_{n=1}^{N_k}$ does in fact tend to $1/3$, then each digit will appear equally often in the limit of the Fibonacci concatenation. This is exactly what we would expect from a number that is \textit{simply normal}. From this, we could stitch together the Fibonacci concatenation in base $3$ exactly like we did before with the coloring technique. Note that base $3$ is not the only base $\gamma$ in which the running percentage of digits appears to converge to $1/\gamma$. In every base we have studied, the running percentage of digits in base $\gamma$ converges to $1/\gamma$. Below, we graph the running percentage of the first few such bases.
\begin{figure}[H]
\minipage{0.32\textwidth}
\caption*{Base 3}
\includegraphics[width=\linewidth]{Base3Convergence.png}
\caption*{$3^0$'s$\ldots3^{11}$'s}
\endminipage\hfill
\minipage{0.32\textwidth}
\caption*{Base 6}
\includegraphics[width=\linewidth]{Base6Convergence.png}
\caption*{$6^0$'s$\ldots6^{8}$'s}
\endminipage\hfill
\minipage{0.32\textwidth}
\caption*{Base 7}
\includegraphics[width=\linewidth]{Base7Convergence.png}
\caption*{$7^0$'s$\ldots7^{6}$'s}
\endminipage\hfill
\minipage{0.32\textwidth}
\caption*{Base 9}
\includegraphics[width=\linewidth]{Base9Convergence.png}
\caption*{$9^0$'s$\ldots9^{5}$'s}
\endminipage\hfill
\minipage{0.32\textwidth}
\caption*{Base 11}
\includegraphics[width=\linewidth]{Base11Convergence.png}
\caption*{$11^0$'s$\ldots11^{5}$'s}
\endminipage\hfill
\minipage{0.32\textwidth}
\caption*{Base 12}
\includegraphics[width=\linewidth]{Base12Convergence.png}
\caption*{$12^0$'s$\ldots12^{5}$'s}
\endminipage\hfill
\minipage{0.32\textwidth}
\caption*{Base 14}
\includegraphics[width=\linewidth]{Base14Convergence.png}
\caption*{$14^0$'s$\ldots14^{5}$'s}
\endminipage\hfill
\minipage{0.32\textwidth}
\caption*{Base 15}
\includegraphics[width=\linewidth]{Base15Convergence.png}
\caption*{$15^0$'s$\ldots15^{4}$'s}
\endminipage\hfill
\minipage{0.32\textwidth}
\caption*{Base 18}
\includegraphics[width=\linewidth]{Base18Convergence.png}
\caption*{$18^0$'s$\ldots18^{4}$'s}
\endminipage\hfill
\caption{Running Percentage of digits $0, 1, 2, \ldots, 1-\gamma$ with line $1/\gamma$}
\end{figure}
These computations provide evidence that Conjecture \ref{conj:gammasimplynormal} holds. If Conjecture \ref{conj:gammasimplynormal} holds for a base $\gamma$, then it necessarily holds for every base $\gamma^i$ for all $i$. We could then invoke Lemma \ref{lemma:Pillai} one more time and claim that the Fibonacce concatenation is \textit{absolutely normal}.
\section*{Acknowledgements}
The authors would like to extend their sincerest gratitude to Will Brian for his early encouragement on this project, to Marc Renault, and Joseph Vandehey for their correspondence and expertise, and to Michael De Vlieger for his brilliant Mathematica code.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,528 |
SYNONYM
#### According to
The Catalogue of Life, 3rd January 2011
#### Published in
Nat. Arr. Brit. Pl. (London) 1: 633 (1821)
#### Original name
Agaricus ciliaris Bolton, 1788
### Remarks
null | {
"redpajama_set_name": "RedPajamaGithub"
} | 2,963 |
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\section{Introduction}
The flow of a two-phase or multicomponent incompressible mixture is nowadays one of the most attractive theoretical and numerical problems in Fluid Mechanics (see, for instance, \cite{AMW1998,GKL2018,GZ2018,LIN2012,PS2016} and the references therein). This is mainly due to the interplay between the motion of the interface separating the two fluids (or phases) and the surrounding fluids. A natural description of this phenomenon is based on a free-boundary formulation. Let $\Omega$ be a bounded domain in $\mathbb{R}^d$ with $d=2,3$, and $T>0$. We assume that $\Omega$ is filled by two incompressible fluids (e.g. two liquids or a liquid and a gas), and we denote by $\Omega_1=\Omega_1(t)$ and $\Omega_2=\Omega_2(t)$ the subsets of $\Omega$ containing, respectively, the first and the second fluid portions for any time $t\geq 0$. The equations of motion are
\begin{equation}
\label{FB}
\begin{cases}
\rho_1 \big( \partial_t \textbf{\textit{u}}_1 + \textbf{\textit{u}}_1 \cdot \nabla \textbf{\textit{u}}_1 \big) - \nu_1 \mathrm{div}\, D \textbf{\textit{u}}_1 +\nabla p_1 =0, \quad &\mathrm{div}\, \textbf{\textit{u}}_1 =0, \quad \text{ in } \Omega_1\times (0,T),\\
\rho_2 \big( \partial_t \textbf{\textit{u}}_2 + \textbf{\textit{u}}_2 \cdot \nabla \textbf{\textit{u}}_2 \big) - \nu_2 \mathrm{div}\, D \textbf{\textit{u}}_2 +\nabla p_2 =0, \quad &\mathrm{div}\, \textbf{\textit{u}}_2 =0, \quad \text{ in } \Omega_2 \times (0,T).
\end{cases}
\end{equation}
Here, $\textbf{\textit{u}}_1$, $\textbf{\textit{u}}_2$ and $p_1$ and $p_2$ are, respectively, the velocities and pressures of the two fluids, while $\rho_1, \rho_2$ and $\nu_1,\nu_2$ are the (constant) densities and viscosities of the two fluids, respectively. The symmetric gradient is $D=\frac12 (\nabla +\nabla^t)$. The effect of the gravity are neglected for simplicity. Denoting by $\Gamma=\Gamma(t)$ the (moving) interface between $\Omega_1$ and $\Omega_2$, system \eqref{FB} can be equipped with the classical free boundary conditions
\begin{equation}
\label{Y-L}
\textbf{\textit{u}}_1=\textbf{\textit{u}}_2, \quad \big( \nu_1 D \textbf{\textit{u}}_1 -\nu_2 D \textbf{\textit{u}}_2 \big) \cdot \textbf{\textit{n}}_\Gamma = (p_1-p_2+\sigma H)\textbf{\textit{n}}_\Gamma \quad \text{ on }\Gamma \times (0,T),
\end{equation}
together with the no-slip boundary condition
\begin{equation}
\label{FB-ns}
\textbf{\textit{u}}_1=\mathbf{0}, \quad \textbf{\textit{u}}_2=\mathbf{0} \quad \text{ on }\partial \Omega \times (0,T).
\end{equation}
The vector $\textbf{\textit{n}}_\Gamma$ in \eqref{Y-L} is the unit normal vector of the interface from $\partial \Omega_1(t)$, $H$ is the mean curvature of the interface ($H= - \mathrm{div}\, \textbf{\textit{n}}_\Gamma$). In this setting, $\Gamma(t)$ is assumed to move with the velocity given by
\begin{equation}
\label{inter-vel}
V_{\Gamma(t)}= \textbf{\textit{u}} \cdot \textbf{\textit{n}}_{\Gamma(t)}.
\end{equation}
The coefficient $\sigma>0$ is the surface tension, which introduces a discontinuity in the normal stress proportional to the mean curvature of the surface. Since
$\frac{\d}{\d t} \mathcal{H}^{d-1}(\Gamma(t))= - \int_{\Gamma(t)} H V_\Gamma \, {\rm d} \mathcal{H}^{d-1}$, where $\mathcal{H}^{d-1}$ is the $d-1$-dimensional Hausdorff measure,
the (formal) energy identity for system \eqref{FB}-\eqref{Y-L} is
\begin{equation}
\label{FB-energy}
\frac{\d}{\d t} \Big\lbrace \sum_{i=1,2} \int_{\Omega_i(t)} \frac{\rho_i}{2}|\textbf{\textit{u}}_i|^2 \, {\rm d} x + \sigma \mathcal{H}^{d-1}(\Gamma(t)) \Big\rbrace
+ \sum_{i=1,2} \int_{\Omega_i(t)} \nu_i |D \textbf{\textit{u}}_i|^2 \, {\rm d} x
=0.
\end{equation}
We refer the reader to \cite{A2007,DS1995,Plo1993,PS2010-1,PS2016,T1995, TT1995} for the analysis of classical and varifold solutions to the system \eqref{FB}-\eqref{FB-ns}.
The twofold Lagrangian and Eulerian nature of system \eqref{FB}-\eqref{FB-ns} has led to the breakthrough idea (mainly from numerical analysts, see the review \cite{SS2003}) to reformulate the above system in the Eulerian description by interpreting the effect of the surface tension as a singular force term localized at the interface. Let us introduce the so-called level set function $\phi: \Omega\times(0,T) \rightarrow \mathbb{R}$ such that
$$
\phi>0 \quad \text{ in } \Omega_1\times (0,T), \quad \phi<0 \quad \text{ in }\Omega_2 \times (0,T), \quad \phi=0 \quad \text {on } \Gamma \times (0,T),
$$
namely the interface is the zero level set of $\phi$. We consider the Heaviside type function
\begin{equation}
\label{Heav}
K(\phi)=
\begin{cases}
1 \quad &\phi>0,\\
0 \quad &\phi=0,\\
-1 \quad &\phi<0,
\end{cases}
\end{equation}
and we denote by $\textbf{\textit{u}}$ the velocity such that $\textbf{\textit{u}}=\textbf{\textit{u}}_1$ in $\Omega_1\times (0,T)$ and $\textbf{\textit{u}}=\textbf{\textit{u}}_2$ in $\Omega_2\times (0,T)$.
It was shown in \cite[Section 2]{CHMS1994} that the system \eqref{FB}-\eqref{FB-ns} is formally equivalent to
\begin{equation}
\label{FB2}
\begin{cases}
\rho(\phi) \big( \partial_t \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}\big) - \mathrm{div}\, (\nu(\phi)D\textbf{\textit{u}}) + \nabla P=\sigma H(\phi) \nabla \phi \delta (\phi),\\
\mathrm{div}\, \textbf{\textit{u}}=0,\\
\partial_t \phi +\textbf{\textit{u}}\cdot \nabla \phi = 0,
\end{cases}
\quad \text{in }\Omega\times(0,T),
\end{equation}
together with the boundary condition \eqref{FB-ns}. Here
$$
\rho(\phi)= \rho_1 \frac{1+K(\phi)}{2}+ \rho_2 \frac{1-K(\phi)}{2}, \
\nu(\phi)=\nu_1 \frac{1+K(\phi)}{2}+ \nu_2 \frac{1-K(\phi)}{2}, \
H(\phi)= \mathrm{div}\, \left( \frac{\nabla \phi}{|\nabla \phi|} \right).
$$
Here, $\delta$ is the Dirac distribution, and $\nabla \phi$ is oriented as $\textbf{\textit{n}}_\Gamma$. The equation \eqref{FB2}$_3$ represents the motion of the interface $\Gamma$ that is simply transported by the flow.
This follows from the immiscibility condition, which translates into
$(\textbf{\textit{u}}, 1) \in \text{Tan} \lbrace (x,t)\in \Omega \times (0,T): x\in \Gamma(t) \rbrace$.
Although \eqref{FB2} seems to be more amenable than \eqref{FB}-\eqref{Y-L}, the presence of the Dirac mass still makes the analysis challenging. In the literature, two different approaches have been used to overcome the singular nature of the right-hand side of \eqref{FB2}$_1$, which both rely on the idea of continuous transition at the interface. The first approach is the Level Set method developed in the seminal works
\cite{CHMS1994,OK2005,OS1988,SSO1994} (see also the review \cite{SS2003}).
This approach consists in approximating the Heaviside function $K(\phi)$ by a smoothing regularization $K_\varepsilon(\phi)$.
More precisely, for a given $\varepsilon>0$, we introduce the function
\begin{equation}
\label{Heav-e}
K_\varepsilon(\phi)=
\begin{cases}
1 \quad &\phi>\varepsilon,\\
\frac12 \Big[ \frac{\phi}{\varepsilon} + \frac{1}{\pi} \sin \big( \frac{\pi \phi}{\varepsilon}\big) \Big]\quad &|\phi| \leq \varepsilon,\\
-1 \quad &\phi<-\varepsilon.
\end{cases}
\end{equation}
The resulting approximating system reads as follows
\begin{equation}
\label{LS1}
\begin{cases}
\rho_\varepsilon(\phi) \big( \partial_t \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}\big) - \mathrm{div}\, (\nu_\varepsilon(\phi)D\textbf{\textit{u}}) + \nabla P=\sigma H(\phi) \nabla \phi \delta_\varepsilon (\phi),\\
\mathrm{div}\, \textbf{\textit{u}}=0,\\
\partial_t \phi +\textbf{\textit{u}}\cdot \nabla \phi = 0,
\end{cases}
\quad \text{in } \Omega \times (0,T),
\end{equation}
where
$$
\rho_\varepsilon(\phi)= \rho_1 \frac{1+K_\varepsilon(\phi)}{2}+ \rho_2 \frac{1-K_\varepsilon(\phi)}{2}, \quad
\nu_\varepsilon(\phi)=\nu_1 \frac{1+K_\varepsilon(\phi)}{2}+ \nu_2 \frac{1-K_\varepsilon(\phi)}{2}, \quad \delta_\varepsilon= \frac{\mathrm{d} K_\varepsilon(\phi)}{\mathrm{d} \phi}.
$$
As a consequence of the approximation \eqref{Heav-e}, the thickness of the interface is approximately $\frac{2\varepsilon}{|\nabla \phi|}$. This necessarily requires that $|\nabla \phi|=1$ when $|\phi|\leq \varepsilon$, namely $\phi$ is a signed-distance function near the interface. However, even though the initial condition is suitably chosen, the evolution under the transport equation \eqref{LS1}$_3$ does not guarantee that this property remains true for all time. This fact had led to different numerical algorithms aiming to avoid the expansion of the interface (see \cite{SS2003} and the references therein).
In addition, as pointed out in \cite{LT1998}, another drawback of this approach is that the dynamics is sensitive to the particular choice of the approximation for the surface stress tensor.
The second approach is the so-called Diffuse Interface method (see \cite{AMW1998,FLSY2005,GKL2018}). This is based on the postulate that the interface is a layer with positive volume, whose thickness is determined by the interactions of particles occurring at small scales. In this context, the auxiliary function $\phi$ represents the difference between the fluids concentrations (or rescaled density/volume fraction). This function may exhibit a smooth transition at the interface, which is identified as intermediate level sets between the two values $1$ and $-1$.
The evolution equations for the state variables (density, velocity, concentration) are derived by combining the theory of binary mixtures and the energy-based formalism from thermodynamics and statistical mechanics.
In this framework, the surface stress tensor is replaced by a diffuse stress tensor whose action is essentially localized in the regions of high gradients, namely, $-\sigma \mathrm{div}\,(\nabla \phi \otimes \nabla \phi)$. This tensor is known as (Korteweg) capillary tensor (cf., e.g., \cite{AMW1998}). The resulting Diffuse Interface system, also called ``complex fluid" model (see, e.g., \cite[Sec.5]{LIN2012}), is the following
\begin{equation}
\label{Complex}
\begin{cases}
\rho(\phi) \big( \partial_t \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}} \big) - \mathrm{div}\, (\nu(\phi)D\textbf{\textit{u}}) + \nabla P= -\sigma \mathrm{div}\,(\nabla \phi \otimes \nabla \phi),\\
\mathrm{div}\, \textbf{\textit{u}}=0,\\
\partial_t \phi +\textbf{\textit{u}}\cdot \nabla \phi = 0,
\end{cases}
\quad \text{in } \Omega \times(0,T),
\end{equation}
equipped with the no-slip boundary condition
\begin{equation}
\label{Complex-bc}
\textbf{\textit{u}}=\mathbf{0} \quad \text{on } \partial \Omega \times (0,T).
\end{equation}
Here
\begin{equation}
\label{rhonu}
\rho(\phi)= \rho_1 \frac{1+\phi}{2}+ \rho_2 \frac{1-\phi}{2}, \quad
\nu(\phi)=\nu_1 \frac{1+\phi}{2}+ \nu_2 \frac{1-\phi}{2}.
\end{equation}
The energy associated to system \eqref{Complex} is defined as
$$
E(\textbf{\textit{u}},\phi)= \int_{\Omega} \frac12 \rho(\phi) |\textbf{\textit{u}}|^2+ \sigma \Big( \frac12 |\nabla \phi|^2 + \Psi(\phi) \Big) \, {\rm d} x,
$$
where $\Psi$ is a double-well potential from $[-1,1] \rightarrow \mathbb{R}$,
and the corresponding energy identity is
\begin{equation}
\label{Complex-EE}
\frac{\d}{\d t} E(\textbf{\textit{u}}, \phi) +\int_{\Omega} \nu(\phi) |D \textbf{\textit{u}}|^2 \, {\rm d} x=0.
\end{equation}
This model dissipates energy due to viscosity, but there are no regularization effects for $\phi$. It is worth noting that \eqref{Complex} is also related to the models for viscoelastic fluids (see, for instance, \cite{LLZ2005}) or to the two-dimensional incompressible MHD system without magnetic diffusion (cf., e.g., \cite{RWXZ2014} and the references therein).
Notice that, after rescaling the capillary tensor and the free energy by a parameter $\varepsilon$, it is possible to recognize the connection between \eqref{FB}-\eqref{Y-L} and \eqref{Complex}. Indeed, we have formally the convergences of the stress tensor (see, for instance, \cite{AL2018,LIN2012,PS2016} for further details on the sharp interface limit)
$$
-\int_{\Omega} \sigma \varepsilon \mathrm{div}\,(\nabla \phi \otimes \nabla \phi) \cdot \textbf{\textit{v}} \, {\rm d} x
\xrightarrow{\varepsilon\rightarrow 0} \int_{\Gamma} \sigma H \textbf{\textit{n}}_\Gamma \cdot \textbf{\textit{v}} \, {\rm d} \mathcal{H}^{d-1},
$$
where the limit integral corresponds to the weak formulation of \eqref{Y-L}, and of the (Helmholtz) free energy $\int_{\Omega} \left(\frac{\varepsilon}{2} |\nabla \phi|^2 + \frac{1}{\varepsilon}\Psi(\phi)\right) \, {\rm d} x$ to the area functional $\mathcal{H}^{d-1}(\Gamma)$ (see \cite{Modica}).
Before proceeding with the introduction of diffusive relaxations of the transport equation and their physical motivations, it is important to point out two main properties of \eqref{Complex}$_2$-\eqref{Complex}$_3$:
\begin{itemize}
\item[1.] Conservation of mass:
\begin{equation}
\label{CM}
\int_{\Omega} \phi(t)\, {\rm d} x=\int_{\Omega} \phi_0\, {\rm d} x, \quad \forall \, t \in [0,T].
\end{equation}
\item[2.] Conservation of $L^\infty(\Omega)$-norm:
\begin{equation}
\label{CLinf}
\| \phi(t)\|_{L^\infty(\Omega)}=\| \phi_0\|_{L^\infty(\Omega)}, \quad \forall \, t\in [0,T],
\end{equation}
which implies that
\begin{equation}
\label{Crange}
-1\leq \phi_0(x)\leq 1 \quad \text{a.e. in} \ \Omega \quad \Rightarrow
\quad -1\leq \phi(x,t)\leq 1 \quad \text{a.e. in} \ \Omega \times (0,T).
\end{equation}
\end{itemize}
The theory of binary mixtures takes into accounts dissipative mechanisms occurring at the interfaces. The molecules of two fluids interact at a miscoscopic scale, and their disposition is the result of a competition between the diffusion of molecules and the attraction of molecules of the same fluid (mixing vs demixing or ``philic" vs ``phobic" effects). This liquid-liquid phase separation phenomenon, though already well-known in Materials Science, has recently become a sort of paradigm in Cell Biology (see, for instance, \cite{AD2019,BTP2015,HWF2014,ShB17}). This competition is
described in the Helmholtz free energy of the system $\mathcal{E}(\phi)$ defined by
$$
\mathcal{E}(\phi)= \int_{\Omega} \frac12 |\nabla \phi|^2 + \Psi(\phi) \, {\rm d} x.
$$
The first term describes weakly non-local interactions (see \cite{CH1958}, cf. also \cite{E1989}). The potential $\Psi$ is the Flory-Huggins free energy density\footnote{For a system of finite number of molecules $A$ and $B$ occupying a lattice with $M$ sites, the thermodynamic properties of the system of molecules are derived from the partition function
\begin{equation}
\label{pf}
Z=\sum_{\Omega} \mathrm{e}^{\Big( \frac{H(\sigma_1,\dots,\sigma_M)}{k_B T}\Big)}
\end{equation}
where the Hamiltonian $H(\sigma_1,\dots,\sigma_M)$ denotes the energy of the arrangement $\sigma_1,\dots, \sigma_M$ ($\sigma_n=1$ if the lattice is occupied by molecule $A$, $\sigma_n=0$ otherwise), and $\Omega$ is the set of all possible arrangements. Here $k_B$ is the Boltzmann constant and $T$ is the temperature. It is common to describe only nearest neighbor interactions between particles, which lead to the particular Hamiltonian
\begin{equation}
H(\lbrace \sigma\rbrace)=\frac12 \sum_{m,n} \Big( e_{AA} \sigma_m \sigma_n +e_{BB} (1-\sigma_m)(1-\sigma_n)+e_{AB}\big(\sigma_m(1-\sigma_n)+\sigma_n(1-\sigma_m)\big)\Big),
\end{equation}
where $e_{AA}$, $e_{BB}$, and $e_{AB}$ are coefficients. In the Mean Field approximation the arrangements $\sigma_n$ and $1-\sigma_n$ are approximated by the probability (average) that a site is occupied by a molecule $A$ and $B$, namely $\phi_A=\frac{N_A}{M}$ and $\phi_B=\frac{N_B}{M}$($N_A$ and $N_B$ are the number of molecules of type $A$ and $B$, and $M=N_A+N_B$). Then, the partition function is given by
$$
Z= \frac{M!}{N_A! N_B !} \rm e^{- \frac{H(\phi_A,\phi_B)}{k_B T }}, \quad H(\phi_A,\phi_B)= \frac{z M}{2} \big(e_{AA} \phi_A^2+2e_{AB} \phi_A \phi_B+e_{BB} \phi_B^2 \big),
$$
where $z$ is the number of neighbors in a lattice. By using the Stirling approximation, the free energy density reads as
$$
f(\phi_A,\phi_B)= \frac{-k_B T \ln Z}{M} \approx \frac{k_B T}{\nu} \Big[ \phi_A \ln \phi_A +\phi_B \ln \phi_B\Big]+ \frac{z}{2\nu} \Big[ e_{AA} \phi_A^2+2e_{AB}\phi_A\phi_B+e_{BB}\phi_B^2\Big],
$$
where $\nu$ is the volume of molecules. By defining $\phi =\phi_A-\phi_B$ (with range $[-1,1]$), and setting appropriately the constants $\theta$ and $\theta_0$, the Flory-Huggins potential \eqref{Log} immediately follows. As usual, the function $\Psi$ is meant as the continuous extension at the values $s=\pm 1$.
Roughly speaking, the logarithmic term accounts for the entropy after mixing and the quadratic perturbation represents the internal energy after mixing. For more details, we refer the reader to \cite{LL2013}.
}
\begin{equation} \label{Log}
\Psi(s)=\frac{\theta}{2}\left[ (1+s)\log(1+s)+(1-s)\log(1-s)\right]-\frac{%
\theta_0}{2} s^2, \quad s \in [-1,1].
\end{equation}
We consider hereafter the case $0<\theta<\theta_0$, which implies,
in particular, that $\Psi$ is a non-convex potential\footnote{In the case, $\theta\geq \theta_0$, mixing prevails over demixing, and no separation takes place.}. It is worth mentioning that the Landau theory that leads to the well-known Ginzburg-Landau free energy is just an approximation of the above $\mathcal{E}(\phi)$ obtained through a Taylor expansion of the logarithmic potential $\Psi$. This choice is very common in the related literature (see, for instance, \cite{CM1995} and \cite{ES1986}). However, it has the main
drawback that the solution does not belong in general to the physical interval $[-1,1]$ (cf. \eqref{Crange}).
In order to include dissipative mechanisms in the dynamics of the concentration, we define the first variation of the Helmholtz free energy. This is called chemical potential and it is given by
$$
\mu= \frac{\delta \mathcal{E}(\phi)}{\delta \phi}= -\Delta \phi+ \Psi'(\phi).
$$
Two fundamental relaxation models proposed in the Diffuse Interface theory for binary mixture are the following modifications of the transport equation \eqref{Complex}$_3$:
\begin{itemize}
\item[$1.$] \textbf{Mass-conserving Allen-Cahn dynamics}\footnote{This equation differs from the classical Allen-Cahn equation due to the presence of term $\overline{\mu}$ (see \cite{RS1992,YFLS2006}, cf. also \cite{AC1979, BS1997,CHL2010,GO1997,MHVB2018,VRC2014}).} (\cite{RS1992,YFLS2006})
$$
\partial_t \phi +\textbf{\textit{u}}\cdot \nabla \phi + \gamma \big(\mu- \overline{\mu}\big)=0 \quad \text{in} \ \Omega \times (0,T), \quad \partial_\textbf{\textit{n}} \phi=0 \quad \text{on} \ \partial \Omega \times (0,T);
$$
\item[$2.$] \textbf{Cahn-Hilliard dynamics} (\cite{CH1958,CH1961})
$$
\partial_t \phi +\textbf{\textit{u}}\cdot \nabla \phi - \gamma \Delta \mu=0 \quad \text{in} \ \Omega \times (0,T), \quad \partial_\textbf{\textit{n}} \phi= \partial_\textbf{\textit{n}} \mu=0 \quad \text{on} \ \partial \Omega \times (0,T).
$$
\end{itemize}
Here $\overline{\mu}$ is the spatial average defined by $$
\overline{\mu}= \frac{1}{|\Omega|}\int_{\Omega} \mu \, {\rm d} x,
$$
and $\gamma$ is the elastic relaxation time. We point out that from the thermodynamic viewpoint the relaxation terms describe dissipative diffusional flux at the interface (cf. \cite{HMR2012,LT1998}).
As for the transport equation, both the mass-conserving Allen-Cahn and Cahn-Hilliard equations satisfy the conservation properties \eqref{CM} and \eqref{Crange}. In addition, their dynamics maintain the integrity of the interface: the mixing-demixing mechanism (which also translates into $\mu$) allows a balance which avoids uncontrolled expansion or shrinkage of the interface layer (cf. \cite{FLSY2005}).
In this work, we study a Diffuse Interface model that has been recently derived in \cite[Part I, Chap.2, 4.2.1]{GKL2018}. It accounts for unmatched densities and viscosities of the fluids, as well as dissipation due to interface mixing. The dynamics of $\phi$ is described through the following modification of the transport equation
$$
\partial_t \phi +\textbf{\textit{u}}\cdot \nabla \phi + \gamma \Big( \mu + \rho'(\phi)\frac{|\textbf{\textit{u}}|^2}{2} - \xi \Big)=0, \quad \text{ in} \ \Omega \times (0,T),
$$
where $\textbf{\textit{u}}$ denotes the volume averaged fluid velocity and
$$
\xi(t)= \frac{1}{|\Omega|}\int_\Omega \mu+ \rho'(\phi)\frac{|\textbf{\textit{u}}|^2}{2} \, {\rm d} x, \quad \text{ in} \ (0,T).
$$
Here the dissipation mechanism is similar to that of the mass-conserving Allen-Cahn dynamics, but it also includes an extra term due to the difference of densities. We thus have the nonhomogeneous Navier-Stokes-Allen-Cahn system
\begin{equation}
\label{Complex2}
\begin{cases}
\rho(\phi) \big( \partial_t \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}} \big) - \mathrm{div}\, (\nu(\phi)D\textbf{\textit{u}}) + \nabla P= -\sigma \mathrm{div}\,(\nabla \phi \otimes \nabla \phi),\\
\mathrm{div}\, \textbf{\textit{u}}=0,\\
\partial_t \phi +\textbf{\textit{u}}\cdot \nabla \phi + \gamma \big(\mu + \rho'(\phi)\frac{|\textbf{\textit{u}}|^2}{2} - \xi\big) =0,
\end{cases}
\quad \text{in } \Omega \times (0,T).
\end{equation}
This system is usually subject to a no-slip boundary condition for $\textbf{\textit{u}}$ and a homogeneous Neumann boundary condition for $\phi$, namely
\begin{equation}
\label{bc-C2}
\textbf{\textit{u}}=\mathbf{0},\quad \partial_{\textbf{\textit{n}}} \phi =0 \quad \text{ on } \partial\Omega
\times (0,T).
\end{equation}
In the last part of this work, we will consider the mass-conserving Euler-Allen-Cahn system
\begin{equation}
\label{Euler}
\begin{cases}
\partial_t \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}} + \nabla P= - \sigma \mathrm{div}\,(\nabla \phi \otimes \nabla \phi),\\
\mathrm{div}\, \textbf{\textit{u}}=0,\\
\partial_t \phi +\textbf{\textit{u}}\cdot \nabla \phi + \gamma \big(\mu- \overline{\mu}\big)=0,
\end{cases}
\quad \text{ in } \Omega \times (0,T),
\end{equation}
endowed with the boundary conditions
\begin{equation}
\label{bc-EAC}
\textbf{\textit{u}}\cdot \textbf{\textit{n}} =0,\quad \partial_{\textbf{\textit{n}}} \phi =0 \quad \text{ on } \partial\Omega
\times (0,T).
\end{equation}
The above model is obtained from \eqref{Complex2} in the case of inviscid flow and matched densities. We observe that other boundary conditions can be considered, for instance, periodic (cf. \cite[Part I, Chap.2, 4.2.3]{GKL2018}) and also \cite{MCYZ2017} for moving contact lines.
The mathematical literature concerning systems similar to the Navier-Stokes-Allen-Cahn system \eqref{Complex2}-\eqref{bc-C2} has been widely developed in last years, in terms of both physical modeling and
well-posedness analysis.
First, we report that there are different ways of accounting for the unmatched densities for incompressible binary mixtures. Among the existing literature, we mention \cite{AGG2012,B2001,HMR2012,HMR2012-2,FK2017,LT1998}. The model herein studied is derived via an energetic variational approach in \cite{JLL2017} (see also \cite{GKL2018} and, for the Navier-Stokes-Cahn-Hilliard system \cite{LSY2015}).
The system \eqref{Complex2}-\eqref{bc-C2} has been investigated in \cite{JLL2017} in the case of constant viscosity and standard Allen-Cahn equation with regular Landau potential $\Psi_0(s)=\frac14(s^2-1)^2$ and no mass conservation. The authors prove the existence of a global weak solution in three dimensions and the existence as well as uniqueness of the global strong solution in two dimensions. In the latter case, they also show the convergence of a weak solution to a single stationary state and they establish the existence of a global attractor. Thanks to their choice of potential and the absence of mass constraint, the authors can easily ensure that $\phi$ takes values in the physical range $[-1,1]$. This fact is crucial for their proofs. However, the mass constraint would not allow to establish a comparison principle even if the double-well potential is smooth. We also mention the previous contributions \cite{GG2010,GG2010-2,HM2019,W17,WX13,XZL2010} for the case with constant density, and \cite{AL2018} for the sharp interface limit in the Stokes case.
Additionally, there are works devoted to Navier-Stokes-Allen-Cahn models in which density is regarded as an independent variable (see, for instance, \cite{FL2019,LDH2016,LH2018}). In these works the potential is the classical Landau double-well and there is no mass conservation. The (non-conserved) compressible case (see \cite{Bl1999,FK2017} for modeling issues) has been analyzed, for instance, in \cite{DLL2013,FRPS2010,K2012,YZ2019} (see also \cite{W2011} for sharp interface limits).
On the other hand, in comparison with the viscous case above-mentioned, only few works have been addressed with the Euler-Allen-Cahn system \eqref{Euler}-\eqref{bc-EAC}. In this respect, we mention
\cite{ZGH2011} (see also \cite{Gal2016} for a nonlocal model), where the authors prove local existence of smooth solutions for the Euler-Allen-Cahn in the case of no-mass conservation and Landau potential.
The aim of this paper is to address the existence, uniqueness and (possibly) regularity of the solutions to the aforementioned Diffuse Interface systems\footnote{Without loss of generality, we consider the values of the parameters $\sigma=\gamma=1$ in our analysis.}: the complex fluid model \eqref{Complex}-\eqref{Complex-bc}, the Navier-Stokes-Allen-Cahn system \eqref{Complex2}-\eqref{bc-C2}, and the Euler-Allen-Cahn system \eqref{Euler}-\eqref{bc-EAC}.
On one hand, the purpose of our analysis is to stay as close as possible to a thermodynamically grounded framework by keeping densities and viscosities to be dependent on $\phi$, and the physically relevant Flory-Huggins potential \eqref{Log}. Although this choice requires some technical efforts, it provides results which are physically more reasonable.
On the other hand, by working in this general setting, we demonstrate that the dynamics originating from a general initial condition become global (in time) when the mass-conserving Allen-Cahn relaxation is taken into account. The latter is achieved in three dimensions for finite energy (weak) solutions, and in two dimensions even for more regular solutions in the case of non-constant density and viscosity and of constant density and zero viscosity.
Before concluding this introduction, we make some more precise comments on the analysis and on the main novelties of our techniques.
First, we recall that the existence and uniqueness of local (in time) regular solutions to the complex fluid system \eqref{Complex}-\eqref{Complex-bc} has been proven in \cite{LLZ2005,LZ2008} for constant density and viscosity. Here we generalize this result by allowing $\rho$ and $\nu$ to depend on $\phi$ and taking a more general initial datum
$(\textbf{\textit{u}}_0,\phi_0) \in ({\mathbf{V}}_\sigma\cap \mathbf{H}^2(\Omega)) \times W^{2,p}(\Omega)$, with $p>2$ in two dimensions and $p>3$ in three dimensions (see Theorem \ref{CF-T}).
Next, we study the Navier-Stokes-Allen-Cahn system \eqref{Complex2}-\eqref{bc-C2}. We prove the existence of a global weak solution with $(\textbf{\textit{u}}_0,\phi_0)\in \mathbf{H}_\sigma \times H^1(\Omega)$ (see Theorems \ref{weak-D} and \ref{W-S}), and the existence of a global strong solution with $(\textbf{\textit{u}}_0,\phi_0)\in {\mathbf{V}}_\sigma \times H^2(\Omega)$ such that $\Psi'(\phi_0)\in L^2(\Omega)$ (see Theorem \ref{strong-D}).
For the latter, we combine a classical energy approach, a new end-point estimate of the product of two functions in $L^2(\Omega)$ (see Lemma \ref{result1} below), and a new estimate for the Stokes system with non-constant viscosity. The proof is concluded with a logarithmic Gronwall argument that leads to double-exponential control. However, in light of the singularity of the Flory-Huggins potential, the uniqueness of these strong solutions seem to be a hard task. To overcome this issue, we then establish global estimates on the derivatives of the entropy\footnote{We define the (mixing) entropy as $F(s)=\frac{\theta}{2}\left[ (1+s)\log(1+s)+(1-s)\log(1-s)\right]$, for $s \in [-1,1]$. This corresponds to the convex part of \eqref{Log}.} (entropy estimates) provided that $\Vert\rho^\prime\Vert_{L^\infty(-1,1)}$ is small enough and $F''(\phi_0)\in L^1(\Omega)$. These estimates allow us to prove that $F''(\phi)^2 \log (1+F''(\phi)) \in L^1(\Omega \times (0,T))$, and, in turn, the uniqueness of such strong solutions. As a consequence of these entropy estimates,
we achieve the so-called uniform separation property. The latter says that $\phi$ stays uniformly away from the pure states in finite time\footnote{It is worth pointing out that the initial concentration $\phi_0$ for strong solutions is not separated from the pure phases. Indeed, the imposed conditions $F'(\phi_0)\in L^2(\Omega)$ or $F''(\phi_0)\in L^1(\Omega)$ allow $\phi_0$ being arbitrarily close to $+1$ and $-1$.}. This fact, besides being physically relevant, entails further regularity properties of the solution.
Note that in the case of a smooth potential and no mass conservation (cf. \cite{JLL2017}) this issue is trivial since the potential is smooth and a comparison principle holds. Finally, we consider the inviscid case, namely the Euler-Allen-Cahn system \eqref{Euler}-\eqref{bc-EAC}. Although this system turns out to be similar to the MHD equations with magnetic diffusion and without viscosity, the classical argument in the literature (see, e.g., \cite{CW2011}) does not apply because of the logarithmic potential. In our proof, it is crucial to make use of the structure of the incompressible Euler equations \eqref{Euler}$_1$-\eqref{Euler}$_2$, and the end-point estimate of the product (Lemma \ref{result1}). This gives the existence of global solutions with $(\textbf{\textit{u}}_0,\phi_0) \in (\mathbf{H}_\sigma\cap \mathbf{H}^1(\Omega)) \times H^2(\Omega)$ in two dimensions. Next, in light of the entropy estimates, we also prove the existence of smoother global solutions originating from $(\textbf{\textit{u}}_0,\phi_0) \in (\mathbf{H}_\sigma\cap \mathbf{W}^{1,p}(\Omega)) \times H^2(\Omega)$ provided that $p>2$ and $\nabla \mu_0:= \nabla (-\Delta \phi_0+\Psi'(\phi_0))\in \mathbf{L}^2(\Omega)$.
\smallskip
\noindent
\textbf{Plan of the paper.} In Section \ref{2} we introduce the notation, some functional inequalities and then prove an estimate for the product of two functions. In Section \ref{S-Complex} we show the local well-posedness of system \eqref{Complex}-\eqref{Complex-bc}. Section \ref{S-WEAK} is devoted to the existence of global weak solutions for the Navier-Stokes-Allen-Cahn system \eqref{Complex2}-\eqref{bc-C2}.
In Section \ref{S-STRONG} we study the existence and uniqueness of strong solutions to the Navier-Stokes-Allen-Cahn system \eqref{Complex2}-\eqref{bc-C2}.
Section \ref{EAC-sec} is devoted to the global existence of solutions to the Euler-Allen-Cahn system \eqref{Euler}-\eqref{bc-EAC}. In Appendix \ref{App-0} we prove a result on the Stokes problem with variable viscosity, and in Appendix \ref{App} we recall the Osgood lemma and two logarithmic versions of the Gronwall lemma.
\section{Preliminaries}
\label{2}
\setcounter{equation}{0}
\subsection{Notation}
For a real Banach space $X$, its norm is denoted by $\|\cdot\|_{X}$.
The symbol $\langle\cdot, \cdot\rangle_{X',X}$ stands for the duality pairing between $X$ and its dual space $X'$. The boldface letter $\bm{X}$ denotes the vectorial space endowed with the product structure.
We assume that $\Omega\subset \mathbb{R}^d$, $d=2,3$, is a bounded domain with smooth boundary $\partial \Omega$, $\textbf{\textit{n}}$ is the unit outward normal vector on $\partial \Omega$, and
$\partial_\textbf{\textit{n}}$ denotes the outer normal derivative on $\partial \Omega$.
We denote the Lebesgue spaces by $L^p(\Omega)$ $(p\geq 1)$ with norms $\|\cdot\|_{L^p(\Omega)}$. When $p=2$, the inner product in the Hilbert space $L^2(\Omega)$ is denoted by
$(\cdot, \cdot)$.
For $s\in \mathbb{R}$, $p\geq 1$, $W^{s,p}(\Omega)$
is the Sobolev space with corresponding norm $\|\cdot\|_{W^{s,p}(\Omega)}$. If $p=2$, we use the notation $W^{s,p}(\Omega)=H^s(\Omega)$. For every $f\in (H^1(\Omega))'$, we denote by $\overline{f}$ the generalized mean value over $\Omega$ defined by
$\overline{f}=|\Omega|^{-1}\langle f,1\rangle_{(H^1(\Omega))',H^1(\Omega)}$. If $f\in L^1(\Omega)$, then $\overline{f}=|\Omega|^{-1}\int_\Omega f \, {\rm d} x$.
Thanks to the generalized Poincar\'{e} inequality, there exists a positive constant $C=C(\Omega)$ such that
\begin{equation}
\label{normH1-2}
\| f\|_{H^1(\Omega)}\leq C \Big(\| \nabla f\|_{L^2(\Omega)}^2+ \Big|\int_{\Omega} f \, {\rm d} x\Big|^2\Big)^\frac12, \quad \forall \, f \in H^1(\Omega).
\end{equation}
We introduce the Hilbert space of solenoidal vector-valued functions
\begin{align*}
&\mathbf{H}_\sigma=\{ \textbf{\textit{u}}\in \mathbf{L}^2(\Omega): \mathrm{div}\, \textbf{\textit{u}}=0,\ \textbf{\textit{u}}\cdot \textbf{\textit{n}} =0\quad \text{on}\ \partial \Omega\} = \overline{C_{0,\sigma}^\infty(\Omega)}^{\mathbf{L}^2(\Omega)},\\
& {\mathbf{V}}_\sigma =\{ \textbf{\textit{u}}\in \mathbf{H}^1(\Omega): \mathrm{div}\, \textbf{\textit{u}}=0,\ \textbf{\textit{u}}=\mathbf{0}\quad \text{on}\ \partial \Omega\}=\overline{C_{0,\sigma}^\infty(\Omega)}^{\mathbf{H}^1(\Omega)},
\end{align*}
where $C_{0,\sigma}^\infty(\Omega)$ is the space of divergence free vector fields in $C_{0}^\infty(\Omega)$.
We also use $( \cdot ,\cdot )$ and
$\| \cdot \|_{L^2(\Omega)}$ for
the inner product and the norm in $\mathbf{H}_\sigma$. The space ${\mathbf{V}}_\sigma$ is endowed with the inner product and norm
$( \textbf{\textit{u}},\textbf{\textit{v}} )_{{\mathbf{V}}_\sigma}=
( \nabla \textbf{\textit{u}},\nabla \textbf{\textit{v}} )$ and $\|\textbf{\textit{u}}\|_{{\mathbf{V}}_\sigma}=\| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}$, respectively.
We denote by ${\mathbf{V}}_\sigma'$ its dual space.
We recall the Korn inequality
\begin{equation}
\label{KORN}
\|\nabla\textbf{\textit{u}}\|_{L^2(\Omega)} \leq \sqrt2\|D\textbf{\textit{u}}\|_{L^2(\Omega)} \leq \sqrt2 \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)},
\quad \forall \, \textbf{\textit{u}} \in {\mathbf{V}}_\sigma,
\end{equation}
where $D\textbf{\textit{u}} = \frac12\big(\nabla \textbf{\textit{u}}+ (\nabla \textbf{\textit{u}})^t\big)$.
We define the Hilbert space
$\mathbf{W}_\sigma= \mathbf{H}^2(\Omega)\cap {\mathbf{V}}_\sigma$
with inner product and norm
$ ( \textbf{\textit{u}},\textbf{\textit{v}})_{\mathbf{W}_\sigma}=( \mathbf{A}\textbf{\textit{u}}, \mathbf{A} \textbf{\textit{v}} )$ and $\| \textbf{\textit{u}}\|_{\mathbf{W}_\sigma}=\|\mathbf{A} \textbf{\textit{u}} \|$, where $\mathbf{A}$ is the Stokes operator.
We recall that there exists $C>0$ such that
\begin{equation}
\label{H2equiv}
\| \textbf{\textit{u}}\|_{H^2(\Omega)}\leq C\| \textbf{\textit{u}}\|_{\mathbf{W}_\sigma}, \quad \forall \, \textbf{\textit{u}}\in \mathbf{W}_\sigma.
\end{equation}
\subsection{Analytic tools}
We recall the Ladyzhenskaya, Agmon, Gagliardo-Nirenberg, Brezis-Gallouet-Wainger and trace interpolation inequalities:
\begin{align}
\label{LADY}
&\| f\|_{L^4(\Omega)}\leq C \|f\|_{L^2(\Omega)}^{\frac12}\|f\|_{H^1(\Omega)}^{\frac12}, &&\forall \, f \in H^1(\Omega), \ d=2,\\
\label{GN2}
&\| f\|_{L^p(\Omega)}\leq C p^\frac12 \| f\|_{L^2(\Omega)}^{\frac{2}{p}} \| f\|_{H^1(\Omega)}^{1-\frac{2}{p}}, &&\forall \, f \in H^1(\Omega),\ \ 2\leq p<\infty, \ d=2,\\
\label{GN3}
&\| f\|_{L^p(\Omega)}\leq C(p) \| f\|_{L^2(\Omega)}^{\frac{6-p}{2p}} \| f\|_{H^1(\Omega)}^{\frac{3(p-2)}{2p}}, &&\forall \, f \in H^1(\Omega),\ \ 2\leq p\leq 6, \ d=3,\\
\label{Agmon2d}
&\| f\|_{L^\infty(\Omega)}\leq C \|f\|_{L^2(\Omega)}^{\frac12}\|f\|_{H^2(\Omega)}^{\frac12}, && \forall \, f \in H^2(\Omega),\ d=2,\\
\label{GN-L4}
&\| \nabla f\|_{W^{1,4}(\Omega)}\leq C\| f \|_{H^2(\Omega)}^\frac12
\| f \|_{L^\infty(\Omega)}^\frac12, && \forall \, f \in H^2(\Omega),\ d=3,\\
\label{BGI}
&\| f\|_{L^\infty(\Omega)}\leq C \| f\|_{H^1(\Omega)} \log^\frac12 \Big({e}\frac{\| f\|_{H^2(\Omega)}}{\| f\|_{H^1(\Omega)}} \Big), &&\forall \, f \in H^2(\Omega), \ d=2,\\
\label{BGW}
&\| f\|_{L^\infty(\Omega)}\leq C(p) \| f\|_{H^1(\Omega)} \log^\frac12 \Big( C(p) \frac{\| f\|_{W^{1,p}(\Omega)}}{\| f\|_{H^1(\Omega)}} \Big) , &&\forall \, f \in W^{1,p}(\Omega), \ p>2, \ d=2,\\
\label{trace}
&\| f\|_{L^2(\partial \Omega)} \leq C \| f\|_{L^2(\Omega)}^\frac12 \| f\|_{H^1(\Omega)}^\frac12, &&\forall \, f \in H^1(\Omega), \ d=2.
\end{align}
Here, the constant $C$ depends only on $\Omega$, whereas the constant $C(p)$ depends on $\Omega$ and $p$.
\smallskip
We now prove the following end-point estimate for the product of two functions, which will play an important role in the subsequent analysis. This is a generalization of \cite[Proposition C.1]{GMT2019}.
\begin{lemma}
\label{result1}
Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary. Assume that $f\in H^1(\Omega)$ and $g\in L^p(\Omega)$ for some $p>2$, $g$ is not identical to $0$. Then, we have
\begin{equation}
\label{logprod}
\| f g\|_{L^2(\Omega)}\leq C \Big(\frac{p}{p-2} \Big)^\frac12 \| f\|_{H^1(\Omega)}
\| g\|_{L^2(\Omega)} \log^{\frac12} \Big( {e} |\Omega|^{\frac{p-2}{2p}} \frac{\| g\|_{L^p(\Omega)}}{\| g\|_{L^2(\Omega)}} \Big),
\end{equation}
for some positive constant $C$ depending only on $\Omega$.
\end{lemma}
\begin{proof}
Let us consider the Neumann operator $A=-\Delta + I$ on $L^2(\Omega)$ with domain $D(A)=\lbrace u\in H^2(\Omega): \partial_{\textbf{\textit{n}}}u=0$ on $\partial \Omega\rbrace$. By the classical spectral theory, there exists a sequence of positive eigenvalues $\lambda_k$ ($k\in \mathbb{N}$) associated with $A$ such that $\lambda_1=1$, $\lambda_{k}\leq \lambda_{k+1}$ and $\lambda_{k}\rightarrow +\infty$ as $k$ goes to $+\infty$. The sequence of eigenfunctions $w_k\in D(A)$ satisfying $A w_k=\lambda_k w_k$ forms an orthonormal basis in $L^2(\Omega)$ and an orthogonal basis in $H^1(\Omega)$.
Let us fix $N \in \mathbb{N}_0$ whose value will be chosen later. We write $f$ as follows
\begin{equation}
\label{decompsition}
f=\sum_{n=0}^N f_n + f_N^{\bot},
\end{equation}
where
$$
f_n=\sum_{k:\, {e}^n\leq \sqrt{\lambda_k}<{e}^{n+1}}
(f,w_k) w_k, \quad f_{N}^{\bot}= \sum_{k:\,\sqrt{\lambda_k} \geq {e}^{N+1}} (f,w_k) w_k.
$$
By using the above decomposition and H\"{o}lder's inequality, we have
\begin{align*}
\| f g\|_{L^2(\Omega)} \leq
\sum_{n=0}^N \| f_n g \|_{L^2(\Omega)}+
\| f_{N}^{\bot} g \|_{L^2(\Omega)} \leq \sum_{n=0}^N \| f_n\|_{L^\infty(\Omega)} \| g\|_{L^2(\Omega)}
+ \| f_N^{\bot}\|_{L^{p'}(\Omega)}
\| g\|_{L^p(\Omega)},
\end{align*}
where $p>2$ and $\frac{1}{p}+\frac{1}{p'}=\frac12$. By using \eqref{GN2} and \eqref{Agmon2d}, we obtain
\begin{align*}
\| fg\|_{L^2(\Omega)}
&\leq C \sum_{n=0}^N \| f_n\|_{L^2(\Omega)}^\frac12
\| f_n\|_{H^2(\Omega)}^\frac12 \| g\|_{L^2(\Omega)}+ C \Big(\frac{2p}{p-2}\Big)^\frac12 \| f_N^{\bot}\|_{L^2(\Omega)}^\frac{2}{p'}
\| f_N^{\bot}\|_{H^1(\Omega)}^{1-\frac{2}{p'}}
\| g\|_{L^p(\Omega)},
\end{align*}
for some $C$ independent of $p$. We recall that
$$
\| f_n\|_{L^2(\Omega)}^2= \sum_{k:\, {e}^n\leq \sqrt{\lambda_k}<{e}^{n+1}} |(f,w_k)|^2 \leq \frac{1}{{e}^{2n}} \sum_{k:\, {e}^n\leq \sqrt{\lambda_k}<{e}^{n+1}} \lambda_k |(f,w_k)|^2 = \frac{1}{{e}^{2n}} \| f_n\|_{H^1(\Omega)}^2,
$$
where we have used the fact $D(A^\frac12)=H^1(\Omega)$.
Observing that $\partial_\textbf{\textit{n}} f_n=0$ on $\partial \Omega$ ($f_n$ is a finite sum of $w_k$'s), by the regularity theory of the Neumann problem, we have
\begin{align*}
\| f_n\|_{H^2(\Omega)}^2 &\leq C \| A f_n\|_{L^2(\Omega)}^2= C \sum_{k:\, {e}^n\leq \sqrt{\lambda_k}<{e}^{n+1}} \lambda_k^2 |(f,w_k)|^2\\
&\leq C \sum_{k:\, {e}^n\leq \sqrt{\lambda_k}<{e}^{n+1}} {e}^{2(n+1)} \lambda_k |(f,w_k)|^2\\
&\leq C {e}^{2(n+1)} \| f_n\|_{H^1(\Omega)}^2.
\end{align*}
Thus, we infer that
$$
\| f_n\|_{L^2(\Omega)}^\frac12
\| f_n\|_{H^2(\Omega)}^\frac12 \leq C {e}^\frac12 \| f_n\|_{H^1(\Omega)},
$$
where the constant is independent of $n$. On the other hand, reasoning as above, we deduce that
$$
\| f_{N}^{\bot}\|_{L^2(\Omega)}^2\leq \frac{1}{{e}^{2(N+1)}} \| f_{N}^{\bot}\|_{H^1(\Omega)}^2.
$$
Combining the above inequalities and applying the Cauchy-Schwarz inequality, we get
\begin{align}
\| fg\|_{L^2(\Omega)} &\leq
C \sum_{n=0}^N {e}^\frac12 \| f_n\|_{H^1(\Omega)} \| g\|_{L^2(\Omega)} + C \frac{\Big(\frac{2p}{p-2}\Big)^\frac12}{{e}^{\frac{2(N+1)}{p'}}} \| f_N^{\bot}\|_{H^1(\Omega)} \| g\|_{L^p(\Omega)} \notag\\
&\leq C \| g\|_{L^2(\Omega)}
\Bigg( \sum_{n=0}^N {e}^\frac12 \| f_n\|_{H^1(\Omega)}
+ \frac{\Big(\frac{2p}{p-2}\Big)^\frac12}{{e}^{\frac{(p-2)(N+1)}{p}}}
\frac{\| g\|_{L^p(\Omega)}}{\| g\|_{L^2(\Omega)}} \| f_N^{\bot}\|_{H^1(\Omega)}\Bigg) \notag\\
&\leq C \| g\|_{L^2(\Omega)}
\Bigg( {e} (N+1) + \frac{\Big(\frac{2p}{p-2}\Big)}{{e}^{\frac{2(p-2)(N+1)}{p}}}
\frac{\| g\|^2_{L^p(\Omega)}}{\| g\|^2_{L^2(\Omega)}} \Bigg)^\frac12 \Bigg( \sum_{n=0}^N
\| f_n\|_{H^1(\Omega)}^2 +\| f_N^{\bot}\|^2_{H^1(\Omega)}
\Bigg)^\frac12 \notag \\
&\leq C \| g\|_{L^2(\Omega)}
\Bigg( {e} (N+1) + \frac{ \Big(\frac{2p}{p-2}\Big)}{{e}^{\frac{2(p-2)(N+1)}{p}}}
\frac{\| g\|_{L^p(\Omega)}^2}{\| g\|_{L^2(\Omega)}^2} \Bigg)^\frac12 \| f\|_{H^1(\Omega)}, \label{est2}
\end{align}
where we have used the fact $p'=\frac{2p}{p-2}$ and the constant $C$ is independent of $N$. Now, we choose the non-negative integer $N \in \mathbb{N}_0$ such that
$$
\frac{p}{2(p-2)} \log \Bigg( {e} |\Omega|^{\frac{p-2}{p}}\frac{\| g\|^2_{L^{p}(\Omega)}}{\| g\|^2_{L^2(\Omega)}}\Bigg) \leq N+1 < 1+ \frac{p}{2(p-2)}\log \Bigg({e} |\Omega|^{\frac{p-2}{p}}\frac{\| g\|^2_{L^p(\Omega)}}{\| g\|^2_{L^2(\Omega)}}\Bigg).
$$
We observe that the logarithm term in the above relations is greater than $1$ for any function $g\in L^p(\Omega)$ with $p>2$, $g\neq 0$.
Then by using the choice of $N$ in \eqref{est2}, we infer that
\begin{align*}
\| f g\|_{L^2(\Omega)}
&\leq
C \| f\|_{H^1(\Omega)} \| g\|_{L^2(\Omega)}
\Bigg( {e} \Bigg[ 1+\frac{p}{2(p-2)} \log \Big( {e} |\Omega|^{\frac{p-2}{p}} \frac{\| g\|^2_{L^p(\Omega)}}{\| g\|^2_{L^2(\Omega)}}\Big) \Bigg] +\frac{2p}{{e} (p-2) |\Omega|^{\frac{p-2}{p}}} \Bigg)^\frac12\\
&\leq C \| f\|_{H^1(\Omega)} \| g\|_{L^2(\Omega)}
\Bigg( \frac{3{e}}{2} \frac{p}{(p-2)} \log \Big( {e}^2 |\Omega|^{\frac{p-2}{p}} \frac{\| g\|^2_{L^p(\Omega)}}{\| g\|^2_{L^2(\Omega)}}\Big) +\frac{2p}{{e} (p-2) |\Omega|^{\frac{p-2}{p}}} \Bigg)^\frac12,
\end{align*}
which implies the desired conclusion.
\end{proof}
\begin{remark}
The conclusion of Lemma \ref{result1} holds as well in $\mathbb{T}^2$.
\end{remark}
\begin{remark}
It is well-known that $H^1(\Omega)$ is not an algebra in two dimensions. An interesting application of Lemma \ref{result1} together with the Brezis-Gallouet-Wainger inequality \eqref{BGW} is that
$$
\| f g\|_{H^1(\Omega)}\leq C_1 \| f\|_{H^1(\Omega)} \| g\|_{H^1(\Omega)} \log^{\frac12} \Big( C_2 \frac{\| g\|_{W^{1,p}(\Omega)}}{\| g\|_{H^1(\Omega)}}\Big),
$$
for any $f\in H^1(\Omega)$, $g \in W^{1,p}(\Omega)$ for some $p>2$, where $C_1$ and $C_2$ are two positive constants depending only on $\Omega$ and $p$.
\end{remark}
\begin{remark}
Lemma \ref{result1} can be regarded as a generalization of H\"{o}lder and Young inequalities. This inequality is sharp since the product between $f$ and $g$ is not defined in $L^2(\Omega)$ if $f\in H^1(\Omega)$ and $g\in L^2(\Omega)$. Indeed, we have the following counterexample in $\mathbb{R}^2$:
$$
g(x)= \frac{1}{|x|\log^{\frac34} \big(\frac{1}{|x|}\big)}, \quad f(x)=\log^{\frac12-\frac{1}{100}} \Big( \frac{1}{|x|}\Big),
\quad 0< x\leq 1.
$$
We notice that $g \in L^2(B_{\mathbb{R}^2}(0,1))$ since
$$
\int_{B_{\mathbb{R}^2}(0,1)} |g(x)|^2 \, {\rm d} x= 2 \pi \int_0^1 \frac{1}{r \log^{\frac32}(\frac{1}{r} \big) } \, {\rm d} r= 2\pi \int_1^{+\infty} \frac{1}{s\log^{\frac32}(s)} \, {\rm d} s <+\infty.
$$
However, $g \notin L^p(B_{\mathbb{R}^2}(0,1))$ for any $p>2$ because
$$
\int_{B_{\mathbb{R}^2}(0,1)} |g(x)|^p \, {\rm d} x= 2 \pi \int_0^1 \frac{1}{r^{p-1} \log^{\frac{3p}{4}}(\frac{1}{r} \big) } \, {\rm d} r = 2 \pi \int_1^{+\infty} \frac{1}{s^{3-p}\log^{ \frac{3p}{4}}(s)} \, {\rm d} s =+\infty.
$$
We easily observe that $f \in L^2(B_{\mathbb{R}^2}(0,1))$, but $f \notin L^\infty(B_{\mathbb{R}^2}(0,1))$ since
$\lim_{|x|\rightarrow 0} f(x)=+\infty$.
This, in turn, implies that $f \notin W^{1,p}(B_{\mathbb{R}^2}(0,1))$ for any $p>2$, due to the Sobolev embedding theorem. Nonetheless, we have
$$
\partial_{x_i}f(x)= \Big(\frac12 -\frac{1}{100}\Big) \frac{x_i}{|x|^2} \frac{1}{\log^{\frac{1}{2}+\frac{1}{100}} \big(\frac{1}{|x|} \big)}, \quad i=1,2,
$$
such that
\begin{align*}
\int_{B_{\mathbb{R}^2}(0,1)} |\partial_{x_i}f(x)|^2 \, {\rm d} x &\leq 2\pi \Big(\frac12 -\frac{1}{100}\Big)^2 \int_0^1
\frac{1}{r \log^{2(\frac12+\frac{1}{100})} \big( \frac{1}{r}\big)} \, {\rm d} r\\
&\leq C \int_0^1 \frac{1}{r \log^{1+\frac{1}{50}} \big( \frac{1}{r}\big)} < + \infty.
\end{align*}
Thus, we have $f \in W^{1,2}(B_{\mathbb{R}^2}(0,1))$.
Finally, we observe that
\begin{align*}
\int_{B_{\mathbb{R}^2}(0,1)} |g(x)f(x)|^2 \, {\rm d} x &
= \int_{B_{\mathbb{R}^2}(0,1)} \frac{\log^{1-\frac{1}{50}} \big( \frac{1}{|x|}\big)}{|x|^2\log^{\frac32} \big(\frac{1}{|x|}\big) } \, {\rm d} x\\
&= 2 \pi \int_0^1 \frac{1}{r \log^{\frac12 + \frac{1}{50}} \big( \frac{1}{r} \big) } \, {\rm d} r =+\infty,
\end{align*}
namely, the product $fg \notin L^2(B_{\mathbb{R}^2}(0,1))$.
The above counterexample can be generalized to any pair of functions
$$
g(x)= \frac{1}{|x|\log^\alpha \big(\frac{1}{|x|}\big)}, \quad f(x)=\log^\beta \big( \frac{1}{|x|}\big),
\quad x\in B_\mathbb{R}^2(0,1),
$$
where $\frac12 <\alpha<1$ and $\beta< \frac12 $ such that $\alpha-\beta<\frac{1}{2}$.
\end{remark}
\section{Complex Fluids Model: Local Well-posedness}
\label{S-Complex}
\setcounter{equation}{0}
\noindent
In this section we consider the complex fluids system
\begin{equation}
\label{CF}
\begin{cases}
\rho(\phi)(\partial_t \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}) - \mathrm{div}\, (\nu(\phi)D\textbf{\textit{u}}) + \nabla P= - \mathrm{div}\,(\nabla \phi \otimes \nabla \phi),\\
\mathrm{div}\, \textbf{\textit{u}}=0,\\
\partial_t \phi +\textbf{\textit{u}}\cdot \nabla \phi = 0,
\end{cases}
\quad \text{ in } \Omega \times (0,T),
\end{equation}
subject to the boundary condition
\begin{equation} \label{boundary-P}
\textbf{\textit{u}}=\mathbf{0} \quad \text{ on } \partial\Omega
\times (0,T),
\end{equation}
and to the initial conditions
\begin{equation}
\label{IC-P}
\textbf{\textit{u}}(\cdot, 0)= \textbf{\textit{u}}_0, \quad \phi(\cdot, 0)=\phi_0 \quad \text{ in } \Omega.
\end{equation}
We recall that $\textbf{\textit{u}}$ is the (volume) averaged velocity of the binary mixture, $P$ is the pressure, and $\phi$ denotes the difference of the concentrations (volume fraction) of the two fluids. The coefficients $\rho(\cdot)$ and $\nu(\cdot)$ represent the density and the viscosity of the mixture depending on $\phi$.
Throughout this work, motivated by the linear interpolation density and viscosity functions in \eqref{rhonu}, we assume that
\begin{equation}
\label{Hp-rn}
\rho, \nu \in C^2([-1,1]): \quad \rho(s)\in [\rho_1, \rho_2], \quad \nu(s)\in [\nu_1,\nu_2] \ \ \text{for}\ \ s\in[-1,1],
\end{equation}
where $\rho_1$, $\rho_2$ and $\nu_1$, $\nu_2$ are, respectively, the (positive) densities and viscosities of two homogeneous (different) fluids. In addition, we will use the notation
$$\rho_\ast=\min \lbrace \rho_1,\rho_2\rbrace>0,\qquad \nu_\ast =\min \lbrace \nu_1,\nu_2 \rbrace>0.$$
The aim of this section is to prove the local existence and uniqueness of regular solutions to problem \eqref{CF}-\eqref{IC-P} with general initial data. This generalizes \cite[Theorem 1.1]{LZ2008} to the case with unmatched densities and viscosities depending on the concentration, and to initial data $\phi_0$ belonging to $W^{2,p}(\Omega)$, instead of $\phi_0 \in H^3(\Omega)$ (see also \cite[Theorem 2.2]{LLZ2005} for the Cauchy problem in $\mathbb{R}^2$).
\begin{theorem}[Local well-posedness in 2D]
\label{CF-T}
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^2$.
For any initial datum $(\textbf{\textit{u}}_0, \phi_0)$ such that
$\textbf{\textit{u}}_0 \in {\mathbf{V}}_\sigma\cap \mathbf{H}^2(\Omega)$, $\phi_0 \in W^{2,p}(\Omega)$ for some $p>2$, with $|\phi_0(x)|\leq 1$, for all $x\in \Omega$, there exists a positive time $T_0$, which depends only on the norms of the initial data, and a unique solution $(\textbf{\textit{u}},\phi)$ to problem \eqref{CF}-\eqref{IC-P} on $[0,T_0]$ such that
\begin{align*}
& \textbf{\textit{u}} \in L^\infty(0,T_0; \mathbf{V}_\sigma\cap \mathbf{H}^2(\Omega)) \cap L^\frac{2p}{p-2}(0,T_0;\mathbf{W}^{2,p}(\Omega))\cap W^{1,2}(0,T_0; {\mathbf{V}}_\sigma)
\cap W^{1,\infty}(0,T_0; \mathbf{H}_\sigma(\Omega)), \\
& \phi \in L^\infty(0,T_0;W^{2,p}(\Omega))\cap W^{1,\infty}(0,T_0; H^1(\Omega)\cap L^\infty(\Omega)),\ \ \ |\phi(x,t)|\leq 1\ \ \mathrm{in}\ \ \Omega\times[0,T_0].
\end{align*}
\end{theorem}
\smallskip
\begin{proof}
We perform some \textit{a priori} estimates for the solutions to problem \eqref{CF}-\eqref{IC-P}, and then we prove the uniqueness. With these arguments, the existence of local solutions to \eqref{CF}-\eqref{IC-P} follows from the method of successive approximations (Picard's method). This relies on the definition of a suitable sequence $(\textbf{\textit{u}}_k,\phi_k)$ via an iteration scheme, \textit{a priori} bounds on $(\textbf{\textit{u}}_k,\phi_k)$ in terms of $(\textbf{\textit{u}}_{k-1},\phi_{k-1})$, and uniform estimates of $(\textbf{\textit{u}}_k-\textbf{\textit{u}}_{k-1},\phi_k-\phi_{k-1})$ (by arguing as in the uniqueness proof reported below).
We refer to \cite{LS1975} for the details of this type of argument in the case of the nonhomogeneous Navier-Stokes equations (see also, e.g., \cite{LH2018} for the Navier-Stokes-Allen-Cahn system).
\smallskip
\textbf{First estimate.}
Multiplying \eqref{CF}$_1$ by $\textbf{\textit{u}}$ and integrating over $\Omega$, we find
$$
\frac12\int_{\Omega} \rho(\phi) \partial_t |\textbf{\textit{u}}|^2 \, {\rm d} x +\frac12
\int_{\Omega} \rho(\phi) \textbf{\textit{u}} \cdot \nabla \big( |\textbf{\textit{u}}|^2\big) \, {\rm d} x + \int_{\Omega} \nu(\phi) |D\textbf{\textit{u}}|^2 \, {\rm d} x= -\int_{\Omega} \Delta \phi \nabla \phi \cdot \textbf{\textit{u}} \, {\rm d} x.
$$
Taking the gradient of \eqref{CF}$_3$, we have
$$
\nabla \partial_t \phi + \nabla (\textbf{\textit{u}}\cdot \nabla \phi)=0.
$$
Multiplying the above identity by $\nabla \phi$, integrating over $\Omega$ and using the no-slip boundary condition of $\textbf{\textit{u}}$, we obtain
$$
\frac12\frac{\d}{\d t} \int_\Omega |\nabla \phi|^2 \, {\rm d} x - \int_{\Omega} (\textbf{\textit{u}} \cdot \nabla \phi ) \Delta \phi \, {\rm d} x=0.
$$
By adding the two obtained equations, and using the identity
$$
\partial_t \rho(\phi) + \mathrm{div}\, (\rho(\phi)\textbf{\textit{u}})= \rho'(\phi) \big(\partial_t \phi + \textbf{\textit{u}} \cdot \nabla \phi \big)=0,
$$
we find the basic energy law
$$
\frac12 \frac{\d}{\d t} \int_{\Omega} \Big( \rho(\phi) |\textbf{\textit{u}}|^2 + |\nabla \phi|^2 \Big) \, {\rm d} x + \int_{\Omega} \nu(\phi) |D \textbf{\textit{u}}|^2 \, {\rm d} x =0.
$$
Integrating over $[0,t]$, we obtain
$$
E_0(\textbf{\textit{u}}(t),\phi(t))+ \int_0^t \int_{\Omega} \nu(\phi) |D \textbf{\textit{u}}|^2 \, {\rm d} x =
E_0(\textbf{\textit{u}}_0,\phi_0), \quad \forall \, t \geq 0.
$$
where
$$
E_0(\textbf{\textit{u}},\phi)= \frac12 \int_{\Omega} \rho(\phi) |\textbf{\textit{u}}|^2 + |\nabla \phi|^2 \, {\rm d} x.
$$
In addition, the transport equation yields that, for all $p\in [2, \infty]$, it holds
\begin{equation}
\label{cons-Lp}
\| \phi(t)\|_{L^p(\Omega)}= \| \phi_0\|_{L^p(\Omega)}, \quad \forall \, t \geq 0.
\end{equation}
Thus, we infer that
\begin{equation}
\label{CF-1}
\textbf{\textit{u}} \in L^\infty(0,T;\mathbf{H}_\sigma)\cap L^2(0,T;{\mathbf{V}}_\sigma), \quad \phi \in L^\infty(0,T;H^1(\Omega)\cap L^\infty(\Omega)).
\end{equation}
\medskip
\textbf{Second estimate.} We multiply \eqref{CF}$_1$ by $\partial_t \textbf{\textit{u}}$ and integrate over $\Omega$. After integrating by parts and using the fact that $\partial_t \textbf{\textit{u}} =0$ on $\partial \Omega$, we obtain
\begin{align*}
& \frac12 \frac{\d}{\d t} \int_{\Omega} \nu(\phi)|D \textbf{\textit{u}}|^2 \, {\rm d} x+ \int_{\Omega} \rho(\phi) |\partial_t \textbf{\textit{u}}|^2 \, {\rm d} x \\
&\quad = \frac12 \int_{\Omega} \nu'(\phi)\partial_t \phi |D \textbf{\textit{u}}|^2 \, {\rm d} x - \int_{\Omega} \rho(\phi) (\textbf{\textit{u}} \cdot \nabla) \textbf{\textit{u}} \cdot \partial_t \textbf{\textit{u}} \, {\rm d} x + \int_{\Omega} (\nabla \phi \otimes \nabla \phi) : \nabla \partial_t \textbf{\textit{u}} \, {\rm d} x.
\end{align*}
Combining \eqref{LADY}, \eqref{Agmon2d} and \eqref{CF-1}, together with the relation $\partial_t \phi= - \textbf{\textit{u}} \cdot \nabla \phi$, we have
\begin{align}
&\frac12 \frac{\d}{\d t} \int_{\Omega} \nu(\phi)|D \textbf{\textit{u}}|^2 \, {\rm d} x+ \int_{\Omega} \rho(\phi) |\partial_t \textbf{\textit{u}}|^2 \, {\rm d} x \notag \\
&\quad \leq C \| \partial_t \phi\|_{L^2(\Omega)} \| D \textbf{\textit{u}}\|_{L^4(\Omega)}^2 +
C \| \textbf{\textit{u}}\|_{L^\infty(\Omega)} \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)} \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}+ \| \nabla \phi\|_{L^4(\Omega)}^2 \| \nabla \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)} \notag \\
&\quad \leq C \| \textbf{\textit{u}} \cdot \nabla \phi\|_{L^2(\Omega)} \| D \textbf{\textit{u}}\|_{L^2(\Omega)} \| \textbf{\textit{u}}\|_{H^2(\Omega)} + C \| \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac12 \| \textbf{\textit{u}}\|_{H^2(\Omega)}^\frac12 \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)} \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)} \notag \\
&\qquad + C \| \nabla \phi\|_{L^2(\Omega)} \| \phi\|_{H^2(\Omega)} \| \nabla \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)} \notag \\
&\quad \leq \frac{\rho_\ast}{4} \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2+ \frac{\nu_\ast}{4} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2
+ C \| \textbf{\textit{u}}\|_{L^\infty(\Omega)}\| \nabla \phi\|_{L^2(\Omega)} \| D \textbf{\textit{u}}\|_{L^2(\Omega)} \| \textbf{\textit{u}}\|_{H^2(\Omega)} \notag \\
&\qquad +C \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}^2 \| \textbf{\textit{u}}\|_{H^2(\Omega)} + C
\| \phi\|_{H^2(\Omega)}^2 \notag \\
&\quad \leq \frac{\rho_\ast}{4} \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2+ \frac{\nu_\ast}{4} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2+ C \| D \textbf{\textit{u}}\|_{L^2(\Omega)} \| \textbf{\textit{u}}\|_{H^2(\Omega)}^\frac32 \notag \\
&\qquad+C \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2 \| \textbf{\textit{u}}\|_{H^2(\Omega)} + C
\| \phi\|_{H^2(\Omega)}^2.
\label{CF-2}
\end{align}
Here we have also used that $\| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}$ is equivalent to $\| D \textbf{\textit{u}}\|_{L^2(\Omega)}$ thanks to \eqref{KORN}. Next, we rewrite \eqref{CF}$_1$-\eqref{CF}$_2$ as a Stokes problem with non-constant viscosity
$$
\begin{cases}
-\mathrm{div}\, (\nu(\phi)D \textbf{\textit{u}})+\nabla P= \textbf{\textit{f}}, & \text{ in } \Omega \times (0,T),\\
\mathrm{div}\, \textbf{\textit{u}}=0,& \text{ in } \Omega \times (0,T),\\
\textbf{\textit{u}}=\mathbf{0}, & \text{ on } \partial \Omega \times (0,T),
\end{cases}
$$
where $\textbf{\textit{f}}= -\rho(\phi)(\partial_t \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}) - \mathrm{div}(\nabla \phi \otimes \nabla \phi)$.
By exploiting Theorem \ref{Stokes-e} with $p=2$, $s=2$, $r=\infty$, we infer that
\begin{align*}
\| \textbf{\textit{u}}\|_{H^2(\Omega)}
&\leq C \| \rho(\phi) \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}+ C\| \rho(\phi) (\textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}})\|_{L^2(\Omega)} \\
&\quad + C\| \mathrm{div}(\nabla \phi \otimes \nabla \phi)\|_{L^2(\Omega)}
+ C \|\nabla \phi \|_{L^\infty(\Omega)}\| D \textbf{\textit{u}}\|_{L^2(\Omega)} \\
&\leq C \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)} + C \| \textbf{\textit{u}}\|_{L^\infty(\Omega)} \|\nabla \textbf{\textit{u}} \|_{L^2(\Omega)} + C \| \nabla \phi\|_{L^\infty(\Omega)} \big( \|\phi \|_{H^2(\Omega)} + \| D \textbf{\textit{u}}\|_{L^2(\Omega)}\big)\\
&\leq C\| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)} + C\| \textbf{\textit{u}}\|_{H^2(\Omega)}^\frac12 \|D \textbf{\textit{u}} \|_{L^2(\Omega)} + C \| \nabla \phi\|_{L^\infty(\Omega)} \big( \|\phi \|_{H^2(\Omega)} + \| D \textbf{\textit{u}}\|_{L^2(\Omega)}\big).
\end{align*}
Here we have used \eqref{Agmon2d} and \eqref{CF-1}.
Thus, by Young's inequality we find
\begin{align}
\| \textbf{\textit{u}}\|_{H^2(\Omega)}
&\leq C\| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)} + C \|D \textbf{\textit{u}} \|_{L^2(\Omega)}^2 + C \| \nabla \phi\|_{L^\infty(\Omega)} \big( \|\phi \|_{H^2(\Omega)} + \| D \textbf{\textit{u}}\|_{L^2(\Omega)}\big).
\label{CF-uH2}
\end{align}
Inserting \eqref{CF-uH2} into \eqref{CF-2}, and using again Young's inequality, we get
\begin{align}
& \frac12 \frac{\d}{\d t} \int_{\Omega} \nu(\phi)|D \textbf{\textit{u}}|^2 \, {\rm d} x
+ \frac{\rho_\ast}{2}\int_{\Omega} |\partial_t \textbf{\textit{u}}|^2 \, {\rm d} x \notag \\
& \leq \frac{\nu_\ast}{4} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2
+ C \|D \textbf{\textit{u}} \|_{L^2(\Omega)}^4+ C \|\nabla \phi \|_{L^\infty(\Omega)}^2
(\| \phi\|_{H^2(\Omega)}^2+\| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2)+ C \|\phi\|_{H^2(\Omega)}^2.
\label{CF-2b}
\end{align}
\medskip
\textbf{Third estimate.}
We differentiate \eqref{CF} with respect to the time to obtain
\begin{align*}
\rho(\phi) \partial_{tt} \textbf{\textit{u}} &+ \rho(\phi) \big( \partial_t \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \partial_t \textbf{\textit{u}}\big) + \rho'(\phi)\partial_t \phi (\partial_t \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}) - \mathrm{div}\, ( \nu(\phi) D \partial_t \textbf{\textit{u}}) \\
& \quad - \mathrm{div}\,(\nu'(\phi)\partial_t \phi D \textbf{\textit{u}} ) + \nabla \partial_t P= - \mathrm{div}\,(\nabla \phi \otimes \nabla \partial_t \phi+ \nabla \partial_t \phi \otimes \nabla \phi).
\end{align*}
Multiplying the above equation by $\partial_t \textbf{\textit{u}}$ and integrating over $\Omega$, we are led to
\begin{align*}
& \frac12 \int_{\Omega} \rho(\phi) \partial_t |\partial_t \textbf{\textit{u}}|^2 \, {\rm d} x
+\int_{\Omega} \rho(\phi) \big( \partial_t \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \partial_t \textbf{\textit{u}}\big)\cdot \partial_t \textbf{\textit{u}} \, {\rm d} x + \int_{\Omega} \rho'(\phi)\partial_t \phi (\partial_t \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}) \cdot \partial_t \textbf{\textit{u}} \, {\rm d} x\\
& + \int_{\Omega} \nu(\phi) |D \partial_t \textbf{\textit{u}}|^2 \, {\rm d} x+ \int_{\Omega} \nu'(\phi) \partial_t \phi D \textbf{\textit{u}} : D \partial_t \textbf{\textit{u}} \, {\rm d} x= \int_{\Omega} \big( \nabla \phi \otimes \nabla \partial_t \phi + \nabla \partial_t \phi \otimes \nabla \phi \big) : \nabla \partial_t \textbf{\textit{u}} \, {\rm d} x.
\end{align*}
Since
\begin{align*}
&\frac12 \int_{\Omega} \rho(\phi) \partial_t |\partial_t \textbf{\textit{u}}|^2 \, {\rm d} x
+ \frac12 \int_{\Omega} \rho(\phi) \textbf{\textit{u}} \cdot \nabla|\partial_t \textbf{\textit{u}}|^2 \,{\rm d} x\\
&\quad = \frac12 \frac{\d}{\d t} \int_{\Omega} \rho(\phi)|\partial_t \textbf{\textit{u}}|^2 \,{\rm d} x - \frac12 \int_{\Omega} \underbrace{ \big( \partial_t \rho(\phi)+ \mathrm{div}\,( \rho(\phi) \textbf{\textit{u}} ) \big)}_{=0} |\partial_t \textbf{\textit{u}}|^2\, {\rm d} x
= \frac12 \frac{\d}{\d t} \int_{\Omega} \rho(\phi)|\partial_t \textbf{\textit{u}}|^2 \,{\rm d} x,
\end{align*}
we have
\begin{align}
&\frac12 \frac{\d}{\d t} \int_{\Omega} \rho(\phi)|\partial_t \textbf{\textit{u}}|^2 \,{\rm d} x
+ \int_{\Omega} \nu(\phi) |D \partial_t \textbf{\textit{u}}|^2 \, {\rm d} x\notag\\
&\quad = -\int_{\Omega} \rho(\phi) (\partial_t \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}) \cdot \partial_t \textbf{\textit{u}} \, {\rm d} x - \int_{\Omega} \rho'(\phi)\partial_t \phi (\partial_t \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}) \cdot \partial_t \textbf{\textit{u}} \, {\rm d} x\notag\\
&\qquad - \int_{\Omega} \nu'(\phi) \partial_t \phi D \textbf{\textit{u}} : D \partial_t \textbf{\textit{u}} \, {\rm d} x + \int_{\Omega} \big( \nabla \phi \otimes \nabla \partial_t \phi + \nabla \partial_t \phi \otimes \nabla \phi \big) : \nabla \partial_t \textbf{\textit{u}} \, {\rm d} x.
\label{CF-3}
\end{align}
We now estimate the terms on the right-hand side of the above equality.
By using \eqref{KORN}, \eqref{LADY}, and the equation \eqref{CF}$_3$, we find
\begin{align}
-\int_{\Omega} \rho(\phi) (\partial_t \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}) \cdot \partial_t \textbf{\textit{u}} \, {\rm d} x &\leq C \| \partial_t \textbf{\textit{u}}\|_{L^4(\Omega)}^2 \|\nabla \textbf{\textit{u}}\|_{L^2(\Omega)} \notag \\
&\leq \frac{\nu_\ast}{16} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2+ C\| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2 \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2,
\label{CF-4}
\end{align}
and
\begin{align}
& -\int_{\Omega} \rho'(\phi)\partial_t \phi (\partial_t \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}) \cdot \partial_t \textbf{\textit{u}} \, {\rm d} x\notag\\
&\quad \leq C \| \partial_t \phi \|_{L^2(\Omega)} \| \partial_t \textbf{\textit{u}}\|_{L^4(\Omega)}^2 + C \| \partial_t \phi\|_{L^2(\Omega)} \| \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}\|_{L^4(\Omega)}\| \partial_t \textbf{\textit{u}}\|_{L^4(\Omega)}
\notag \\
&\quad \leq C \| \textbf{\textit{u}} \cdot \nabla \phi\|_{L^2(\Omega)} \|\partial_t \textbf{\textit{u}} \|_{L^2(\Omega)} \| \nabla \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}\notag \\
& \qquad + C \| \textbf{\textit{u}} \cdot \nabla \phi\|_{L^2(\Omega)} \| \textbf{\textit{u}}\|_{L^\infty(\Omega)} \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac12 \|\textbf{\textit{u}} \|_{H^2(\Omega)}^\frac12 \|\partial_t \textbf{\textit{u}} \|_{L^2(\Omega)}^\frac12 \| \nabla \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac12 \notag \\
&\quad \leq \frac{\nu_\ast}{16} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2 +
C \| \textbf{\textit{u}}\|_{L^\infty(\Omega)}^2 \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2 +
C \| \textbf{\textit{u}}\|_{L^\infty(\Omega)}^\frac83 \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac23 \| \textbf{\textit{u}}\|_{H^2(\Omega)}^\frac23 \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac23 \notag \\
&\quad \leq \frac{\nu_\ast}{16} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2 +
C \| \textbf{\textit{u}}\|_{H^2(\Omega)} \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2 +
C \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac23 \| \textbf{\textit{u}}\|_{H^2(\Omega)}^2 \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac23 \notag \\
&\quad \leq \frac{\nu_\ast}{16} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2 +
C \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^3 + C \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2 \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2\notag \\
&\qquad + C \| \nabla \phi\|_{L^\infty(\Omega)} (\|\phi \|_{H^2(\Omega)}+\| D \textbf{\textit{u}}\|_{L^2(\Omega)}) \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2+C \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac23 \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac83 \notag \\
&\qquad +
C \|D \textbf{\textit{u}} \|_{L^2(\Omega)}^\frac{2}{3}
\big( \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^4+ \| \nabla \phi \|_{L^\infty(\Omega)}^2( \|\phi \|_{H^2(\Omega)}^2+ \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2) \big)
\| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac23,
\label{CF-5}
\end{align}
where we have also used \eqref{CF-1} and \eqref{CF-uH2}. Moreover, we obtain
\begin{align}
&-\int_{\Omega} \nu'(\phi) \partial_t \phi D \textbf{\textit{u}} : D \partial_t \textbf{\textit{u}} \, {\rm d} x \notag \\
&\quad \leq C \| \partial_t \phi\|_{L^4(\Omega)} \| D \textbf{\textit{u}}\|_{L^4(\Omega)} \| D \partial_t \textbf{\textit{u}} \|_{L^2(\Omega)} \notag \\
&\quad \leq \frac{\nu_\ast}{16} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2 +C \| \textbf{\textit{u}} \cdot \nabla \phi\|_{L^2(\Omega)} \| \nabla \partial_t \phi\|_{L^2(\Omega)} \| D \textbf{\textit{u}}\|_{L^2(\Omega)}\| \textbf{\textit{u}}\|_{H^2(\Omega)} \notag \\
&\quad \leq \frac{\nu_\ast}{16} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2 +C \| \nabla \partial_t \phi\|_{L^2(\Omega)} \| D \textbf{\textit{u}}\|_{L^2(\Omega)} \| \textbf{\textit{u}}\|_{H^2(\Omega)}^\frac32,
\label{CF-6}
\end{align}
and
\begin{align}
&\int_{\Omega} \big( \nabla \phi \otimes \nabla \partial_t \phi + \nabla \partial_t \phi \otimes \nabla \phi \big) : \nabla \partial_t \textbf{\textit{u}} \, {\rm d} x\notag \\
&\quad \leq C \| \nabla \phi\|_{L^\infty(\Omega)} \|\nabla \partial_t \phi \|_{L^2(\Omega)} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)} \notag \\
&\quad \leq \frac{\nu_\ast}{16} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2 +C\| \nabla \phi\|_{L^\infty(\Omega)}^2 \|\nabla \partial_t \phi \|_{L^2(\Omega)}^2.
\label{CF-7}
\end{align}
It is clear that an estimate of $\nabla \partial_t \phi$ is needed in order to control of the last two terms in \eqref{CF-6} and \eqref{CF-7}. For this purpose, we have
$$
\nabla \partial_t \phi= (\nabla \textbf{\textit{u}})^t \nabla \phi + \nabla^2 \phi \, \textbf{\textit{u}}.
$$
Then, we easily deduce that
\begin{align}
\|\nabla \partial_t \phi \|_{L^2(\Omega)}
&\leq \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}\| \nabla \phi\|_{L^\infty(\Omega)}+ \| \phi\|_{W^{2,p}(\Omega)} \| \textbf{\textit{u}}\|_{L^{\frac{2p}{p-2}}(\Omega)} \notag \\
& \leq C \| D \textbf{\textit{u}}\|_{L^2(\Omega)} \big( \| \nabla \phi\|_{L^\infty(\Omega)}+ \| \phi\|_{W^{2,p}(\Omega)}\big),
\label{CF-nphit}
\end{align}
for $p>2$.
Combining \eqref{CF-4}-\eqref{CF-7} and \eqref{CF-nphit} with \eqref{CF-uH2}-\eqref{CF-3}, we arrive at
\begin{align}
& \frac12 \frac{\d}{\d t} \int_{\Omega} \rho(\phi)|\partial_t \textbf{\textit{u}}|^2 \,{\rm d} x
+ \frac{3\nu_\ast}{4}\int_{\Omega} |D \partial_t \textbf{\textit{u}}|^2 \, {\rm d} x\notag\\
&\quad
\leq C\| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2 \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2+ \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^3 \notag \\
&\qquad + C \| \nabla \phi\|_{L^\infty(\Omega)} (\|\phi \|_{H^2(\Omega)}+\| D \textbf{\textit{u}}\|_{L^2(\Omega)}) \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2+C \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac23 \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac83 \notag \\
&\qquad +
C \|D \textbf{\textit{u}} \|_{L^2(\Omega)}^\frac{2}{3}
\big( \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^4+ \| \nabla \phi \|_{L^\infty(\Omega)}^2( \|\phi \|_{H^2(\Omega)}^2+ \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2) \big)
\| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac23\notag \\
&\qquad + C \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2 (\| \nabla \phi\|_{L^\infty(\Omega)}+\| \phi\|_{W^{2,p}(\Omega)}) (\| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac32 +\| D \textbf{\textit{u}}\|_{L^2(\Omega)}^3 )\notag \\
&\qquad + C \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2 (\| \nabla \phi\|_{L^\infty(\Omega)}+\| \phi\|_{W^{2,p}(\Omega)}) \| \nabla \phi\|_{L^\infty}^\frac32 \big( \|\phi \|_{H^2(\Omega)}^\frac32+ \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac32 \big) \notag \\
& \qquad +C \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2 \big( \| \nabla \phi\|_{L^\infty(\Omega)}^2+ \| \phi\|_{W^{2,p}(\Omega)}^2\big) \| \nabla \phi \|_{L^\infty(\Omega)}^2.
\label{CF-8}
\end{align}
\medskip
\textbf{Fourth estimate.}
In light of \eqref{CF-2b} and \eqref{CF-8}, we are left to control the $W^{2,p}(\Omega)$-norm of $\phi$. To this aim, we make use of the following equivalent norm
$$
\| f\|_{W^{2,p}(\Omega)}=\Big( \| f\|_{L^p(\Omega)}^p + \sum_{|\alpha|=2} \| \partial^\alpha f\|_{L^p(\Omega)}^p \Big)^\frac{1}{p},
$$
where $\alpha$ is a multi-index.
Next, we apply $\partial^\alpha$ to the transport equation \eqref{CF}$_3$
$$
\partial_t \partial^\alpha \phi + \partial^\alpha \big( \textbf{\textit{u}} \cdot \nabla \phi \big)=0.
$$
Multiplying the above equation by $|\partial^\alpha \phi|^{p-2}\partial^\alpha \phi$ and integrating over $\Omega$, we get
\begin{align}
\frac{1}{p} \frac{\d}{\d t} \int_{\Omega} |\partial^\alpha \phi|^p \, {\rm d} x&+
\int_{\Omega} \partial^\alpha \big( \textbf{\textit{u}} \cdot \nabla \phi \big) |\partial^\alpha \phi |^{p-2} \partial^\alpha \phi \, {\rm d} x=0.
\label{CF-9}
\end{align}
Since $\textbf{\textit{u}}$ is divergence free, the above can be rewritten as
\begin{align}
\frac{1}{p}\frac{\d}{\d t} \int_{\Omega} |\partial^\alpha \phi|^p \, {\rm d} x&+
\int_{\Omega} \Big( \partial^\alpha \big( \textbf{\textit{u}} \cdot \nabla \phi \big)- \textbf{\textit{u}} \cdot \nabla \partial^\alpha \phi\Big) |\partial^\alpha \phi |^{p-2} \partial^\alpha \phi \, {\rm d} x=0.
\label{CF-10}
\end{align}
By summing over all multi-inder of order $2$, and using \eqref{cons-Lp},
we find
\begin{align}
\frac{1}{p}\frac{\d}{\d t} \| \phi\|_{W^{2,p}(\Omega)}^p=
-\sum_{|\alpha|=2} \int_{\Omega} \Big( \partial^\alpha \big( \textbf{\textit{u}} \cdot \nabla \phi \big)- \textbf{\textit{u}} \cdot \nabla \partial^\alpha \phi\Big) |\partial^\alpha \phi |^{p-2} \partial^\alpha \phi \, {\rm d} x.
\label{CF-11}
\end{align}
It is easily seen that the right-hand side can be written as
\begin{align}
&\sum_{|\alpha|=2} \int_{\Omega} \Big( \partial^\alpha \big( \textbf{\textit{u}} \cdot \nabla \phi \big)- \textbf{\textit{u}} \cdot \nabla \partial^\alpha \phi\Big) |\partial^\alpha \phi |^{p-2} \partial^\alpha \phi \, {\rm d} x \notag \\
&=
\sum_{|\alpha|=2} \int_{\Omega} \big( \partial^\alpha \textbf{\textit{u}} \cdot \nabla \phi \big) |\partial^\alpha \phi |^{p-2} \partial^\alpha \phi \, {\rm d} x+ \sum_{|\beta|=1, |\gamma|=1, \beta+\gamma=\alpha} \int_{\Omega} \big( \partial^\beta \textbf{\textit{u}} \cdot \nabla \partial^\gamma \phi) |\partial^\alpha \phi |^{p-2} \partial^\alpha \phi \, {\rm d} x.\label{CF-12}
\end{align}
Observe that
\begin{align}
\sum_{|\alpha|=2} \int_{\Omega} \big( \partial^\alpha \textbf{\textit{u}} \cdot \nabla \phi \big) |\partial^\alpha \phi |^{p-2} \partial^\alpha \phi \, {\rm d} x
\leq C \| \textbf{\textit{u}}\|_{W^{2,p}(\Omega)} \| \nabla \phi\|_{L^\infty(\Omega)} \| \phi\|_{W^{2,p}(\Omega)}^{p-1},
\label{CF-13}
\end{align}
and
\begin{align}
\sum_{|\beta|=1, |\gamma|=1, \beta+\gamma=\alpha} \int_{\Omega} \big( \partial^\beta \textbf{\textit{u}} \cdot \nabla \partial^\gamma \phi) |\partial^\alpha \phi |^{p-2} \partial^\alpha \phi \, {\rm d} x\leq C \| \textbf{\textit{u}} \|_{W^{1,\infty}(\Omega)} \| \phi\|_{W^{2,p}(\Omega)}^{p}.
\label{CF-14}
\end{align}
Collecting \eqref{CF-11}-\eqref{CF-14} together, and using the Sobolev embedding $W^{2,p}(\Omega) \hookrightarrow W^{1,\infty}(\Omega)$ (with $p>2$), we obtain
$$
\frac{1}{p} \frac{\d}{\d t} \| \phi\|_{W^{2,p}(\Omega)}^p \leq C \| \textbf{\textit{u}}\|_{W^{2,p}(\Omega)} \| \phi\|_{W^{2,p}(\Omega)}^p.
$$
Notice that the above inequality is equivalent to
\begin{align}
\frac{1}{2} \frac{\d}{\d t} \| \phi\|_{W^{2,p}(\Omega)}^2 \leq C \| \textbf{\textit{u}}\|_{W^{2,p}(\Omega)} \| \phi\|_{W^{2,p}(\Omega)}^2.
\label{CF-15}
\end{align}
Next, by exploiting Theorem \ref{Stokes-e} with $s=p>2$ and $r=\infty$, we deduce that
\begin{align}
& \| \textbf{\textit{u}}\|_{W^{2,p}(\Omega)} \notag\\
&\quad \leq
C \|\rho(\phi) \partial_t \textbf{\textit{u}} + \rho(\phi) (\textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}) + \mathrm{div}(\nabla \phi \otimes \nabla \phi)\|_{L^p(\Omega)}+
C \|\nabla \phi \|_{L^\infty(\Omega)} \| D \textbf{\textit{u}}\|_{L^p(\Omega)} \notag \\
&\quad \leq C\big(\| \partial_t \textbf{\textit{u}}\|_{L^p(\Omega)}+ \|\textbf{\textit{u}}\|_{L^\infty(\Omega)} \| D \textbf{\textit{u}}\|_{L^p(\Omega)} +
\| \phi \|_{W^{2,p}(\Omega)} \| \nabla \phi\|_{L^\infty(\Omega)}+ \|\nabla \phi \|_{L^\infty(\Omega)} \| D \textbf{\textit{u}}\|_{L^p(\Omega)}\big) \notag \\
&\quad \leq C \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac{2}{p} \| D \partial_t \textbf{\textit{u}} \|_{L^2(\Omega)}^\frac{p-2}{p}
+C \| \textbf{\textit{u}}\|_{H^2(\Omega)}^\frac12 \| \textbf{\textit{u}}\|_{W^{2,p}(\Omega)}^\frac12 \| \textbf{\textit{u}}\|_{L^p(\Omega)}^\frac12 \notag \\
&\qquad
+C\| \nabla \phi\|_{L^\infty(\Omega)} \big( \| \phi\|_{W^{2,p}(\Omega)}+ \| \textbf{\textit{u}}\|_{W^{2,p}(\Omega)}^\frac12
\| \textbf{\textit{u}}\|_{L^p(\Omega)}^\frac12 \big). \notag
\end{align}
Thus, by Young's inequality we find
\begin{align}
\| \textbf{\textit{u}}\|_{W^{2,p}(\Omega)} &\leq C \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac{2}{p} \| D \partial_t \textbf{\textit{u}} \|_{L^2(\Omega)}^\frac{p-2}{p}+C \| \phi\|_{W^{2,p}(\Omega)}^2\notag\\
&\quad +C (\| \textbf{\textit{u}}\|_{H^2(\Omega)}+\| \nabla \phi\|_{L^\infty(\Omega)}^2) \| D \textbf{\textit{u}}\|_{L^2(\Omega)}.
\label{CF-uW2p}
\end{align}
Inserting \eqref{CF-uH2} and \eqref{CF-uW2p} into \eqref{CF-15}, we are led to
\begin{align}
\frac{1}{2} \frac{\d}{\d t} \| \phi\|_{W^{2,p}(\Omega)}^2
&\leq C \big( \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac{2}{p} \| D \partial_t \textbf{\textit{u}} \|_{L^2(\Omega)}^\frac{p-2}{p}
+ \| \phi\|_{W^{2,p}(\Omega)}^2 \big) \| \phi\|_{W^{2,p}(\Omega)}^2 \notag \\
& \quad + C ( \| \textbf{\textit{u}}\|_{H^2(\Omega)}+ \| \nabla \phi\|_{L^\infty(\Omega)}^2) \|D \textbf{\textit{u}}\|_{L^2(\Omega)} \| \phi\|_{W^{2,p}(\Omega)}^2 \notag\\
&\leq \frac{\nu_\ast}{4} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^{2} +
C \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^{\frac{4}{p+2}} \|\phi\|_{W^{2,p}(\Omega)}^\frac{4p}{p+2} + C \| \phi\|_{W^{2,p}(\Omega)}^4 \notag \\
&\quad + C \big(\| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)} + \| \phi\|_{H^2(\Omega)} \| \nabla \phi\|_{L^\infty(\Omega)} + \| \nabla \phi\|_{L^\infty(\Omega)}^2 \big) \| D \textbf{\textit{u}}\|_{L^2(\Omega)} \| \phi\|_{W^{2,p}(\Omega)}^2\notag \\
&\quad + C (\| D \textbf{\textit{u}}\|_{L^2(\Omega)}^3+ \| \nabla \phi\|_{L^\infty(\Omega)} \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2 ) \| \phi\|_{W^{2,p}(\Omega)}^2.
\label{CF-16}
\end{align}
\medskip
\textbf{Final estimate.} By adding \eqref{CF-2b}, \eqref{CF-8} and \eqref{CF-16} together, and using the embeddings $W^{2,p}(\Omega)\hookrightarrow W^{1,\infty}(\Omega)$, $W^{2,p}(\Omega)\hookrightarrow H^2(\Omega)$ for $p>2$, we deduce that
\begin{align}
\frac{\d}{\d t} Y(t) + \rho_\ast\int_{\Omega} |\partial_t \textbf{\textit{u}}|^2 \, {\rm d} x
+ \nu_\ast\int_{\Omega} |D \partial_t \textbf{\textit{u}}|^2 \, {\rm d} x \leq C(1+ Y^3(t)),
\label{Y}
\end{align}
where
$$
Y(t)= \int_{\Omega} \nu(\phi(t))|D \textbf{\textit{u}}(t)|^2 \, {\rm d} x
+ \int_{\Omega} \rho(\phi(t))|\partial_t \textbf{\textit{u}}(t)|^2 \,{\rm d} x+ \| \phi(t)\|_{W^{2,p}(\Omega)}^2.
$$
Concerning the initial data, we observe from \eqref{CF} that
\begin{align*}
\int_{\Omega} \rho(\phi(0))|\partial_t \textbf{\textit{u}}(0)|^2 \,{\rm d} x \leq
C \big( \| \textbf{\textit{u}}_0\|_{L^\infty(\Omega)}^2 + \|\phi_0 \|_{W^{2,p}(\Omega)} ^2\big) \| \nabla \textbf{\textit{u}}_0\|_{L^2(\Omega)}^2+C \| \textbf{\textit{u}}_0\|_{H^2(\Omega)}^2+ C \|\phi_0\|_{W^{2,p}(\Omega)}^4,
\end{align*}
which, in turn, implies
$$
Y(0)\leq Q (\| \textbf{\textit{u}}_0\|_{H^2(\Omega)}, \| \phi_0\|_{W^{2,p}(\Omega)}),
$$
where $Q$ is a positive continuous and increasing function of its arguments.
Finally, we deduce from \eqref{Y} that there exists a positive time $T_0<\frac{1}{2C(1+Y(0))^2}$, which depends on the parameters of the system and on the norms of the initial data $\| \textbf{\textit{u}}_0\|_{H^2(\Omega)}$ and $\| \phi_0\|_{W^{2,p}(\Omega)}$, such that
\begin{align}
& \int_{\Omega} |D \textbf{\textit{u}}(t)|^2 \, {\rm d} x + \int_{\Omega} |\partial_t \textbf{\textit{u}}(t)|^2 \,{\rm d} x+ \| \phi (t)\|_{W^{2,p}(\Omega)}^2 +\int_0^t \|\partial_t \textbf{\textit{u}}(\tau)\|_{H^1(\Omega)}^2 \, {\rm d} \tau \leq C_0,\label{CF-est}
\end{align}
for all $t \in [0,T_0]$, where $C_0$ is a positive constant depending on $T_0$, $\| \textbf{\textit{u}}_0\|_{H^2(\Omega)}$, $\| \phi_0\|_{W^{2,p}(\Omega)}$. In addition, we learn from \eqref{CF-uW2p} and \eqref{CF-est} that
$$
\int_0^t \| \textbf{\textit{u}}(\tau)\|_{W^{2,p}(\Omega)}^{\frac{2p}{p-2}} \, {\rm d} \tau \leq C,\quad \forall\, t\in [0,T_0].
$$
Similarly, we also deduce from \eqref{CF-uH2}, \eqref{CF-nphit}, and \eqref{CF-est} that
$$
\|\textbf{\textit{u}}(t)\|_{H^2(\Omega)}+\|\partial_t\phi(t)\|_{H^1(\Omega)}+\|\partial_t\phi(t)\|_{L^\infty(\Omega)}\leq C, \quad \forall\, t\in [0,T_0].
$$
We have obtained all the necessary \textit{a priori} estimates. Then the existence result follows as outlined at the beginning of the proof.
\medskip
\textbf{Uniqueness.} Let $(\textbf{\textit{u}}_1,\phi_1)$ and $(\textbf{\textit{u}}_2,\phi_2)$ be two solutions to problem \eqref{CF}-\eqref{IC-P} originating from the same initial datum. The difference of solutions $(\textbf{\textit{u}},\phi, P):=(\textbf{\textit{u}}_1-\textbf{\textit{u}}_2, \phi_1-\phi_2, P_1-P_2)$ solves the system
\begin{align}
&\rho(\phi_1)\big( \partial_t \textbf{\textit{u}} + \textbf{\textit{u}}_1 \cdot \nabla \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}_2 \big)- \mathrm{div}\, \big( \nu(\phi_1)D\textbf{\textit{u}}\big)+ \nabla P \notag\\
&\quad = - \mathrm{div}\,(\nabla \phi_1 \otimes \nabla \phi) -\mathrm{div}\,(\nabla \phi \otimes \nabla \phi_2) - (\rho(\phi_1)-\rho(\phi_2)) (\partial_t \textbf{\textit{u}}_2 + \textbf{\textit{u}}_2 \cdot \nabla \textbf{\textit{u}}_2) \notag\\
&\qquad + \mathrm{div}\, \big( (\nu(\phi_1)-\nu(\phi_2))D\textbf{\textit{u}}_2\big),
\label{CF-Diff1}\\
& \partial_t \phi +\textbf{\textit{u}}_1\cdot \nabla \phi +\textbf{\textit{u}} \cdot \nabla \phi_2=0,\label{CF-Diff2}
\end{align}
for almost every $(x,t) \in \Omega \times (0,T)$, together with the incompressibility constraint $\mathrm{div}\, \textbf{\textit{u}}=0$.
Multiplying \eqref{CF-Diff1} by $\textbf{\textit{u}}$ and integrating over $\Omega$, we find
\begin{align}
&\frac12 \frac{\d}{\d t} \int_{\Omega} \rho(\phi_1) |\textbf{\textit{u}}|^2 \, {\rm d} x +
\int_{\Omega} \rho(\phi_1) (\textbf{\textit{u}}_1 \cdot \nabla) \textbf{\textit{u}} \cdot \textbf{\textit{u}} \, {\rm d} x
+\int_{\Omega} \rho(\phi_1) (\textbf{\textit{u}}\cdot \nabla )\textbf{\textit{u}}_2 \cdot \textbf{\textit{u}} \, {\rm d} x
+\int_{\Omega} \nu(\phi_1)|D \textbf{\textit{u}}|^2 \, {\rm d} x \notag\\
&=\int_{\Omega} (\nabla \phi_1\otimes \nabla \phi+ \nabla \phi\otimes \nabla \phi_2) : \nabla \textbf{\textit{u}} \, {\rm d} x
- \int_{\Omega} (\rho(\phi_1)-\rho(\phi_2)) (\partial_t \textbf{\textit{u}}_2 + \textbf{\textit{u}}_2 \cdot \nabla \textbf{\textit{u}}_2) \cdot \textbf{\textit{u}} \, {\rm d} x \notag \\
&\quad - \int_{\Omega} (\nu(\phi_1)-\nu(\phi_2))D\textbf{\textit{u}}_2 : D \textbf{\textit{u}} \, {\rm d} x + \frac12 \int_{\Omega} |\textbf{\textit{u}}|^2 \rho'(\phi_1) \partial_t \phi_1 \,{\rm d} x.
\label{U1}
\end{align}
Noticing the identity
$$
\int_{\Omega} \rho(\phi_1) ( \textbf{\textit{u}}_1 \cdot \nabla)\textbf{\textit{u}} \cdot \textbf{\textit{u}} \, {\rm d} x=
\int_{\Omega} \rho(\phi_1) \textbf{\textit{u}}_1 \cdot \nabla \Big( \frac12 |\textbf{\textit{u}}|^2 \Big) \, {\rm d} x=
- \frac12\int_{\Omega} \rho'(\phi_1) (\nabla \phi_1 \cdot \textbf{\textit{u}}_1) |\textbf{\textit{u}}|^2 \, {\rm d} x,
$$
since $\phi_1$ solves the transport equation \eqref{CF}$_3$, we can rewrite \eqref{U1} as follows
\begin{align}
&\frac12 \frac{\d}{\d t} \int_{\Omega} \rho(\phi_1) |\textbf{\textit{u}}|^2 \, {\rm d} x
+\int_{\Omega} \nu(\phi_1)|D \textbf{\textit{u}}|^2 \, {\rm d} x \notag\\
&=\int_{\Omega} (\nabla \phi_1\otimes \nabla \phi+ \nabla \phi\otimes \nabla \phi_2) : \nabla \textbf{\textit{u}} \, {\rm d} x
- \int_{\Omega} (\rho(\phi_1)-\rho(\phi_2)) (\partial_t \textbf{\textit{u}}_2 + \textbf{\textit{u}}_2 \cdot \nabla \textbf{\textit{u}}_2) \cdot \textbf{\textit{u}} \, {\rm d} x \notag \\
&\quad - \int_{\Omega} (\nu(\phi_1)-\nu(\phi_2))D\textbf{\textit{u}}_2 : D \textbf{\textit{u}} \, {\rm d} x- \int_{\Omega} \rho(\phi_1) (\textbf{\textit{u}}\cdot \nabla )\textbf{\textit{u}}_2 \cdot \textbf{\textit{u}} \, {\rm d} x.
\label{U2}
\end{align}
By using the embedding $W^{2,p}(\Omega)\hookrightarrow W^{1,\infty}(\Omega)$ for $p>2$, we find that
\begin{align*}
\int_{\Omega} (\nabla \phi_1\otimes \nabla \phi+ \nabla \phi\otimes \nabla \phi_2) : \nabla \textbf{\textit{u}} \, {\rm d} x
&\leq \big( \| \nabla \phi_1\|_{L^\infty(\Omega)}+ \| \nabla \phi_2\|_{L^\infty(\Omega)}\big) \| \nabla \phi\|_{L^2(\Omega)} \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}\\
&\leq \frac{\nu_\ast}{8}\|D \textbf{\textit{u}} \|_{L^2(\Omega)}^2 +C \| \nabla \phi\|_{L^2(\Omega)}^2.
\end{align*}
Next, by H\"{o}lder's inequality, we have
\begin{align*}
&- \int_{\Omega} (\rho(\phi_1)-\rho(\phi_2)) (\partial_t \textbf{\textit{u}}_2 + \textbf{\textit{u}}_2 \cdot \nabla \textbf{\textit{u}}_2) \cdot \textbf{\textit{u}} \, {\rm d} x\\
&\quad \leq C \| \phi\|_{L^6(\Omega)} \|\partial_t \textbf{\textit{u}}_2 + \textbf{\textit{u}}_2 \cdot \nabla \textbf{\textit{u}}_2 \|_{L^3(\Omega)} \| \textbf{\textit{u}} \|_{L^2(\Omega)} \\
&\quad \leq C \big( \| \partial_t \textbf{\textit{u}}_2\|_{L^3(\Omega)}+ \| \textbf{\textit{u}}_2\|_{L^\infty(\Omega)}\| \nabla \textbf{\textit{u}}_2\|_{L^3(\Omega)} \big)\| \phi\|_{H^1(\Omega)} \| \textbf{\textit{u}}\|_{L^2(\Omega)},
\end{align*}
\begin{align*}
- \int_{\Omega} (\nu(\phi_1)-\nu(\phi_2))D\textbf{\textit{u}}_2 : D \textbf{\textit{u}} \, {\rm d} x
& \leq C \| \phi\|_{L^6(\Omega)} \| D \textbf{\textit{u}}_2\|_{L^3(\Omega)} \| D \textbf{\textit{u}}\|_{L^2(\Omega)}\\
&\leq \frac{\nu_\ast}{8}\|D \textbf{\textit{u}} \|_{L^2(\Omega)}^2 +C \| D \textbf{\textit{u}}_2\|_{L^3(\Omega)}^2 \| \phi\|_{H^1(\Omega)}^2,
\end{align*}
and
\begin{align*}
- \int_{\Omega} \rho(\phi_1) (\textbf{\textit{u}}\cdot \nabla )\textbf{\textit{u}}_2 \cdot \textbf{\textit{u}} \, {\rm d} x
\leq C \| \nabla \textbf{\textit{u}}_2\|_{L^\infty(\Omega)} \| \textbf{\textit{u}}\|_{L^2(\Omega)}^2.
\end{align*}
Collecting the above estimates together, we deduce from \eqref{U2} that
\begin{align}
&\frac{\d}{\d t} \int_{\Omega} \rho(\phi_1) |\textbf{\textit{u}}|^2 \, {\rm d} x
+\frac{3\nu_\ast}{2}\int_{\Omega} |D \textbf{\textit{u}}|^2 \, {\rm d} x \notag\\
&\quad \leq C (1+ \| \partial_t \textbf{\textit{u}}_2\|_{L^3(\Omega)}+ \| \textbf{\textit{u}}_2\|_{L^\infty(\Omega)}\| \nabla \textbf{\textit{u}}_2\|_{L^3(\Omega)} + \|D \textbf{\textit{u}}_2 \|_{L^3(\Omega)}^2+ \| \nabla \textbf{\textit{u}}_2\|_{L^\infty(\Omega)}) \notag \\
&\qquad \times \big(\| \textbf{\textit{u}}\|_{L^2(\Omega)}^2 + \| \phi\|_{H^1(\Omega)}^2 \big).
\label{U3}
\end{align}
Next, we multiply \eqref{CF-Diff2} by $\phi$ and get
$$
\frac12 \frac{\d}{\d t} \| \phi\|_{L^2(\Omega)}^2 + \int_{\Omega} (\textbf{\textit{u}} \cdot \nabla \phi_2) \phi \, {\rm d} x=0.
$$
Then taking the gradient of \eqref{CF-Diff2} and multiplying the resulting equation by $\nabla \phi$, we find
$$
\frac12 \frac{\d}{\d t} \| \nabla \phi\|_{L^2(\Omega)}^2 + \int_{\Omega} \nabla \big( \textbf{\textit{u}}_1 \cdot \nabla \phi \big) \cdot \nabla \phi \, {\rm d} x+ \int_{\Omega} \nabla \big( \textbf{\textit{u}} \cdot \nabla \phi_2 \big) \cdot \nabla \phi \, {\rm d} x=0.
$$
By adding the last two equations, we obtain
\begin{align*}
\frac12 \frac{\d}{\d t} \|\phi\|_{H^1(\Omega)}^2 &+ \int_{\Omega} (\nabla \textbf{\textit{u}}_1 \nabla \phi) \cdot \nabla \phi \, {\rm d} x+ \int_{\Omega} (\nabla \textbf{\textit{u}} \nabla \phi_2) \cdot \nabla \phi \, {\rm d} x\\
&+ \int_{\Omega} (\nabla^2 \phi_2 \textbf{\textit{u}}) \cdot \nabla \phi \, {\rm d} x+ \int_{\Omega} (\textbf{\textit{u}} \cdot \nabla \phi_2) \phi \, {\rm d} x=0.
\end{align*}
We have
\begin{align*}
-\int_{\Omega} (\nabla \textbf{\textit{u}}_1 \nabla \phi) \cdot \nabla \phi \, {\rm d} x \leq
\| \nabla \textbf{\textit{u}}_1 \|_{L^\infty(\Omega)} \| \nabla \phi\|_{L^2(\Omega)}^2,
\end{align*}
and by $W^{2,p}(\Omega)\hookrightarrow W^{1,\infty}(\Omega)$, $p>2$, we get
\begin{align*}
-\int_{\Omega} (\nabla \textbf{\textit{u}} \nabla \phi_2) \cdot \nabla \phi \, {\rm d} x
& \leq \|\nabla \textbf{\textit{u}} \|_{L^2(\Omega)} \| \nabla \phi_2\|_{L^\infty(\Omega)} \| \nabla \phi\|_{L^2(\Omega)}\\
& \leq \frac{\nu_\ast}{8} \|D \textbf{\textit{u}} \|_{L^2(\Omega)}^2+ C \| \nabla \phi\|_{L^2(\Omega)}^2,
\end{align*}
\begin{align*}
-\int_{\Omega} (\textbf{\textit{u}} \cdot \nabla \phi_2) \phi \, {\rm d} x \leq \| \textbf{\textit{u}}\|_{L^2(\Omega)} \| \nabla \phi_2\|_{L^\infty(\Omega)} \| \phi\|_{L^2(\Omega)}
\leq C \| \textbf{\textit{u}}\|_{L^2(\Omega)}^2 +C\| \phi\|_{L^2(\Omega)}^2.
\end{align*}
Using \eqref{GN2}, we obtain
\begin{align*}
-\int_{\Omega} (\nabla^2 \phi_2 \textbf{\textit{u}}) \cdot \nabla \phi \, {\rm d} x
&\leq \| \nabla^2 \phi_2\|_{L^p(\Omega)} \| \textbf{\textit{u}}\|_{L^{\frac{2p}{p-2}}(\Omega)} \| \nabla \phi\|_{L^2(\Omega)}\\
&\leq C \| \phi_2\|_{W^{2,p}(\Omega)} \| \textbf{\textit{u}}\|_{L^2(\Omega)}^{\frac{p-2}{p}}\| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}^{\frac{2}{p}} \| \nabla \phi\|_{L^2(\Omega)}\\
&\leq C \| \textbf{\textit{u}}\|_{L^2(\Omega)}^{\frac{p-2}{p}}\| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}^{\frac{2}{p}} \| \nabla \phi\|_{L^2(\Omega)}\\
&\leq \frac{\nu_\ast}{8} \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2+
C \| \textbf{\textit{u}} \|_{L^2(\Omega)}^2 + C \| \nabla \phi\|_{L^2(\Omega)}^2.
\end{align*}
Collecting the above estimates, we are led to
\begin{align}
\frac{\d}{\d t} \|\phi\|_{H^1(\Omega)}^2 &\leq \frac{\nu_\ast}{2} \| D \textbf{\textit{u}}\|^2+
C (1+ \| \nabla \textbf{\textit{u}}_1\|_{L^\infty(\Omega)} ) \big( \| \textbf{\textit{u}}\|_{L^2(\Omega)}^2+ \|\phi\|_{H^1(\Omega)}^2 \big).
\label{U4}
\end{align}
By adding \eqref{U3} and \eqref{U4}, we end up with the differential inequality
\begin{align}
\frac{\d}{\d t} \big( \| \textbf{\textit{u}}\|_{L^2(\Omega)}^2+ \|\phi\|_{H^1(\Omega)}^2 \big) + \nu_\ast \| D \textbf{\textit{u}}\|^2
\leq C R(t) \big( \| \textbf{\textit{u}}\|_{L^2(\Omega)}^2+ \|\phi\|_{H^1(\Omega)}^2\big),
\label{U4a}
\end{align}
where
\begin{align*}
R= 1+ \| \partial_t \textbf{\textit{u}}_2\|_{L^3(\Omega)}+ \| \textbf{\textit{u}}_2\|_{L^\infty(\Omega)}\| \nabla \textbf{\textit{u}}_2\|_{L^3(\Omega)} + \|D \textbf{\textit{u}}_2 \|_{L^3(\Omega)}^2+ \| \nabla \textbf{\textit{u}}_2\|_{L^\infty(\Omega)}+ \|\nabla \textbf{\textit{u}}_1 \|_{L^\infty(\Omega)}.
\end{align*}
Since $R\in L^1(0,T_0)$, then the uniqueness of strong solutions follows from Gronwall's lemma.
\end{proof}
\begin{remark}
The local well-posedness result stated in Theorem \ref{CF-T} is also valid in three dimensional case, provided that the initial condition $\phi \in W^{2,p}(\Omega)$ for some $p>3$. The strategy used in the above proof can be adapted to the this case by using the corresponding Sobolev inequalities in three dimensions.
\end{remark}
\section{Mass-Conserving Navier-Stokes-Allen-Cahn System: Weak Solutions}
\setcounter{equation}{0}
\label{S-WEAK}
In this section, we consider the Navier-Stokes-Allen-Cahn system for a binary mixture of two incompressible fluids with different densities. This model was proposed in \cite[Section 4.2.2]{GKL2018} and derived through an energetic variational approach (see also \cite{JLL2017} for the case with no mass constraint). The system reads as follows
\begin{equation}
\label{NSAC-D}
\begin{cases}
\rho(\phi)\big( \partial_t \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}} \big)- \mathrm{div}\, \big( \nu(\phi)D\textbf{\textit{u}}\big)+ \nabla P
= - \mathrm{div}\,(\nabla \phi \otimes \nabla \phi),\\
\mathrm{div}\, \textbf{\textit{u}}=0,\\
\partial_t \phi +\textbf{\textit{u}}\cdot \nabla \phi + \mu + \displaystyle{\rho'(\phi)\frac{|\textbf{\textit{u}}|^2}{2}} = \xi, \smallskip\\
\mu= -\Delta \phi + \Psi' (\phi), \quad \xi= \displaystyle{\overline{\mu+ \rho'(\phi)\frac{|\textbf{\textit{u}}|^2}{2}}},
\end{cases}
\quad \text{ in } \Omega \times (0,T),
\end{equation}
subject to the boundary conditions
\begin{equation} \label{boundary-D}
\textbf{\textit{u}}=\mathbf{0},\quad \partial_{\textbf{\textit{n}}} \phi =0 \quad \text{ on } \partial\Omega
\times (0,T),
\end{equation}
and to the initial conditions
\begin{equation}
\label{IC-D}
\textbf{\textit{u}}(\cdot, 0)= \textbf{\textit{u}}_0, \quad \phi(\cdot, 0)=\phi_0 \quad \text{ in } \Omega.
\end{equation}
Here, $\rho(\phi)$ and $\nu(\phi)$ are, respectively, the density and the viscosity of the mixture, which satisfy the assumptions \eqref{Hp-rn}. The nonlinear function $\Psi$ is the Flory-Huggins potential \eqref{Log}.
The total energy of system \eqref{NSAC-D}-\eqref{boundary-D} is given by
\begin{equation}
E(\textbf{\textit{u}},\phi)= \int_{\Omega} \frac12 \rho(\phi) |\textbf{\textit{u}}|^2+ \frac12 |\nabla \phi|^2 + \Psi(\phi) \, {\rm d} x.
\label{NSAC-Denergy}
\end{equation}
The main results of this section concern with the existence of global weak solutions.
\begin{theorem}[Global weak solution]
\label{weak-D}
Let $\Omega$ be a bounded domain in $\mathbb{R}^d$ with smooth boundary, $d=2,3$. Assume that the initial datum $(\textbf{\textit{u}}_0,\phi_0)$ satisfies
$\textbf{\textit{u}}_0 \in \mathbf{H}_\sigma, \phi_0\in H^1(\Omega)\cap L^\infty(\Omega)$ with $
\| \phi_0\|_{L^\infty(\Omega)}\leq 1$ and $ |\overline{\phi}_0|<1$. Then, there exists a global weak solution $(\textbf{\textit{u}},\phi)$ to system \eqref{NSAC-D}-\eqref{IC-D} in the following sense:
\begin{itemize}
\item[(i)] For all $T>0$, the pair $(\textbf{\textit{u}},\phi)$ satisfies
\begin{align*}
&\textbf{\textit{u}} \in L^\infty(0,T;\mathbf{H}_\sigma)\cap L^2(0,T;{\mathbf{V}}_\sigma),\\
&\phi \in L^\infty(0,T; H^1(\Omega))\cap L^q(0,T;H^2(\Omega)),\quad \partial_t \phi \in L^q(0,T;L^2(\Omega)), \\
&\phi \in L^\infty(\Omega\times (0,T)) : |\phi(x,t)|<1 \ \text{a.e. in} \ \Omega\times(0,T),\\
&\mu \in L^q(0,T;L^2(\Omega)),
\end{align*}
with $q=2$ if $d=2$, $q=\frac{4}{3}$ if $d=3$.
\item[(ii)] For all $T>0$, the system \eqref{NSAC-D} is solved as follows
\begin{align*}
&-\int_0^T\!\int_\Omega (\rho'(\phi) \partial_t \phi \eta(t)+ \rho(\phi)\eta'(t)) \textbf{\textit{u}}\cdot \textbf{\textit{v}} \, {\rm d} x {\rm d} t+ \int_0^T\!\int_\Omega \big(\rho(\phi)\textbf{\textit{u}}\cdot\nabla\textbf{\textit{u}} \big) \cdot \textbf{\textit{v}} \eta(t) \, {\rm d} x{\rm d} t \\
&\quad + \int_0^T\!\int_\Omega \nu(\phi)(D\textbf{\textit{u}}: D \textbf{\textit{v}})\eta(t) \, {\rm d} x{\rm d} t= \int_\Omega \rho(\phi_0)\textbf{\textit{u}}_0\textbf{\textit{v}}\eta(0) \, {\rm d} x+
\int_0^T\!\int_\Omega \big((\nabla \phi \otimes \nabla \phi): \nabla \textbf{\textit{v}} \big)\eta(t) \, {\rm d} x{\rm d} t,
\end{align*}
for $\textbf{\textit{v}} \in {\mathbf{V}}_\sigma$, $\eta\in C^1([0,T])$ with $\eta(T)=0$, and
\begin{align*}
&\partial_t \phi+ \textbf{\textit{u}}\cdot \nabla \phi -\Delta \phi+ \Psi'(\phi)+ \displaystyle{\rho'(\phi)\frac{|\textbf{\textit{u}}|^2}{2}}=\overline{\Psi'(\phi)+ \rho'(\phi)\frac{|\textbf{\textit{u}}|^2}{2}}, \quad \text{a.e. in} \ \Omega \times (0,T).
\end{align*}
\item[(iii)] The pair $(\textbf{\textit{u}},\phi)$ fulfills the regularity $\textbf{\textit{u}} \in C([0,T];(\mathbf{H}_\sigma)_w)$ and $\phi \in C([0,T];(H^1(\Omega))_w)$, for all $T>0$,
and $\textbf{\textit{u}}|_{t=0}=\textbf{\textit{u}}_0$,
$\phi|_{t=0}=\phi_0$ in $\Omega$.
In addition, $ \partial_{\textbf{\textit{n}}}\phi=0$ on $\partial\Omega\times(0,T)$ for all $T>0$.
\item[(iv)] The energy inequality
\begin{align*}
E(\textbf{\textit{u}}(t), \phi(t))+\int_0^t \int_{\Omega} \nu(\phi) |D \textbf{\textit{u}}|^2 \, {\rm d} x {\rm d}\tau + \int_0^t \|\partial_t \phi + \textbf{\textit{u}} \cdot \nabla \phi \|_{L^2(\Omega)}^2 \, {\rm d} \tau \leq E(\textbf{\textit{u}}_0, \phi_0)
\end{align*}
holds for all $t \geq 0$.
\end{itemize}
\end{theorem}\smallskip
Next, we investigate the special case with matched densities (i.e. $\rho_1=\rho_2$, so that $\rho\equiv 1$). The resulting model is the
homogeneous mass-conserving Navier-Stokes-Allen-Cahn system
\begin{equation}
\label{NSAC}
\begin{cases}
\partial_t \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}} - \mathrm{div}\, (\nu(\phi)D\textbf{\textit{u}}) + \nabla p= - \mathrm{div}\,(\nabla \phi \otimes \nabla \phi),\\
\mathrm{div}\, \textbf{\textit{u}}=0,\\
\partial_t \phi +\textbf{\textit{u}}\cdot \nabla \phi + \mu= \overline{\mu}, \\
\mu= -\Delta \phi + \Psi' (\phi),
\end{cases}
\quad \text{ in } \Omega \times (0,T).
\end{equation}
This system is associated with the boundary and the initial conditions
\begin{equation} \label{bic}
\textbf{\textit{u}}=\mathbf{0},\quad \partial_{\textbf{\textit{n}}} \phi =0 \quad \text{ on } \partial\Omega
\times (0,T), \quad \textbf{\textit{u}}(\cdot, 0)= \textbf{\textit{u}}_0, \quad \phi(\cdot, 0)=\phi_0 \quad \text{ in } \Omega.
\end{equation}
We first state the existence of global weak solutions, whose proof follows from similar {\it a priori} estimates as the ones obtained for the nonhomogeneous case in the proof of Theorem \ref{weak-D} below.
\begin{theorem}[Global weak solution]
\label{W-S}
Let $\Omega$ be a bounded domain in $\mathbb{R}^d$, $d=2,3$, with smooth boundary. Assume that the initial datum $(\textbf{\textit{u}}_0,\phi_0)$ satisfies
$\textbf{\textit{u}}_0 \in \mathbf{H}_\sigma, \phi_0\in H^1(\Omega)\cap L^\infty(\Omega)$ with $
\| \phi_0\|_{L^\infty(\Omega)}\leq 1$ and $ |\overline{\phi}_0|<1$. Then there exists a global weak solution $(\textbf{\textit{u}},\phi)$ to problem \eqref{NSAC}-\eqref{bic}. This is, the solution $(\textbf{\textit{u}},\phi)$ satisfies, for all $T>0$,
\begin{align*}
&\textbf{\textit{u}} \in L^\infty(0,T;\mathbf{H}_\sigma)\cap L^2(0,T;{\mathbf{V}}_\sigma),\\
&\partial_t \textbf{\textit{u}} \in L^2(0,T;{\mathbf{V}}'_\sigma) \ \text{if} \ d=2, \quad
\partial_t \textbf{\textit{u}} \in L^\frac43(0,T;{\mathbf{V}}'_\sigma) \ \text{if} \ d=3,\\
&\phi \in L^\infty(0,T; H^1(\Omega))\cap L^2(0,T;H^2(\Omega)), \\
&\phi \in L^\infty(\Omega\times (0,T)) : |\phi(x,t)|<1 \ \text{a.e. in } \ \Omega\times(0,T),\\
&\partial_t \phi \in L^2(0,T;L^2(\Omega)) \ \text{if} \ d=2, \quad
\partial_t \phi \in L^\frac43(0,T;L^2(\Omega)) \ \text{if} \ d=3,
\end{align*}
and
\begin{align*}
&\langle \partial_t \textbf{\textit{u}}, \textbf{\textit{v}}\rangle + (\textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}, \textbf{\textit{v}})+ (\nu(\phi)D\textbf{\textit{u}},\nabla \textbf{\textit{v}})
= (\nabla \phi \otimes \nabla \phi, \nabla \textbf{\textit{v}}), &&\forall \, \textbf{\textit{v}} \in {\mathbf{V}}_\sigma, \ \text{a.e.} \ t \in (0,T),\\
&\partial_t \phi+ \textbf{\textit{u}}\cdot \nabla \phi -\Delta \phi+ \Psi'(\phi)=\overline{\Psi'(\phi)}, && \text{a.e.} \ (x,t) \in \Omega \times (0,T).
\end{align*}
Moreover, the initial and boundary conditions and the energy inequality hold as in Theorem \ref{weak-D}.
\end{theorem}
Furthermore, due to the particular form of the density function, we are able to prove a uniqueness result in dimension two.
\begin{theorem}[Uniqueness of weak solutions in 2D]
\label{uni2d}
Assume $d=2$. Let
$(\textbf{\textit{u}}_1,\phi_1)$ and $(\textbf{\textit{u}}_2,\phi_2)$ be two weak solutions to problem \eqref{NSAC}-\eqref{bic} on $[0,T]$ subject to the same initial condition $(\textbf{\textit{u}}_0, \phi_0)$ which satisfies the assumptions of Theorem \ref{W-S}. Moreover, we assume that $\phi_1$ satisfies the additional regularity $L^\gamma(0,T;H^2(\Omega))$ with $\gamma>\frac{12}{5}$. Then $(\textbf{\textit{u}}_1,\phi_1)=(\textbf{\textit{u}}_2,\phi_2)$ on $[0,T]$.
\end{theorem}
\begin{remark}
The existence of strong solutions obtained in Section \ref{S-STRONG} (cf. Remark \ref{strong-hom}), which yields the regularity $\phi\in L^\gamma(0,T;H^2(\Omega))$, where $\gamma>\frac{12}{5}$, entails that
the Theorem \ref{uni2d} can be regarded as a weak-strong uniqueness result for problem \eqref{NSAC}-\eqref{bic} in two dimensions. That is, the weak solution originating from an initial condition $(\textbf{\textit{u}}_0,\phi_0)$ such that $\textbf{\textit{u}}_0\in {\mathbf{V}}_\sigma$ and $\phi_0\in H^2(\Omega)$ with $\Psi'(\phi_0)\in L^2(\Omega)$ coincides with the (unique) strong solution departing from the same initial datum.
\end{remark}
\subsection{Proof of Theorem \ref{weak-D}}
First, we derive \textit{a priori} estimates of problem \eqref{NSAC-D}-\eqref{IC-D} that will be crucial to prove the existence of global weak solutions. \medskip
\textbf{Mass conservation and energy dissipation.}
First, integrating the equation \eqref{NSAC-D}$_3$ over $\Omega$ and using the definition of $\xi$, we observe that
$$
\frac{1}{|\Omega|}\int_{\Omega} \phi (t) \, {\rm d} x=\frac{1}{|\Omega|} \int_{\Omega} \phi_0 \, {\rm d} x, \quad \forall \, t \geq 0.
$$
Next, we derive the energy equation associated with \eqref{NSAC-D}.
Multiplying \eqref{NSAC-D}$_1$ by $\textbf{\textit{u}}$ and integrating over $\Omega$, we obtain
\begin{equation}
\label{NSAC-D1}
\int_{\Omega} \frac12 \rho(\phi) \partial_t |\textbf{\textit{u}}|^2 \, {\rm d} x+ \int_{\Omega} \rho(\phi) (\textbf{\textit{u}} \cdot \nabla) \textbf{\textit{u}} \cdot \textbf{\textit{u}} \, {\rm d} x+ \int_{\Omega} \nu(\phi) |D \textbf{\textit{u}}|^2 \, {\rm d} x= -\int_{\Omega} \Delta \phi \nabla \phi \cdot \textbf{\textit{u}} \, {\rm d} x.
\end{equation}
Here we have used the relation $-\Delta \phi \nabla \phi= \frac12 \nabla |\nabla \phi|^2 -{\rm div}(\nabla \phi \otimes \nabla \phi)$ and the incompressibility condition \eqref{NSAC-D}$_2$. Thanks to the no-slip boundary condition for $\textbf{\textit{u}}$, we observe that
\begin{align*}
\int_{\Omega} \rho(\phi) (\textbf{\textit{u}} \cdot \nabla) \textbf{\textit{u}} \cdot \textbf{\textit{u}} \, {\rm d} x &=
\int_{\Omega} \rho(\phi) \textbf{\textit{u}} \cdot \nabla \Big( \frac12 |\textbf{\textit{u}}|^2 \Big) \, {\rm d} x
\\
&= - \frac12 \int_{\Omega} \mathrm{div}\, ( \rho(\phi) \textbf{\textit{u}} ) |\textbf{\textit{u}}|^2 \, {\rm d} x
= - \int_{\Omega} \rho'(\phi)\nabla \phi \cdot \textbf{\textit{u}} \ \frac{|\textbf{\textit{u}}|^2}{2} \, {\rm d} x.
\end{align*}
Next, we multiply \eqref{NSAC-D}$_3$ by $\partial_t \phi+ \textbf{\textit{u}} \cdot \nabla \phi$ and integrate over $\Omega$. Noticing that $\overline{\partial_t \phi + \textbf{\textit{u}}\cdot \nabla \phi}=0$, we get
\begin{equation}
\label{NSAC-D2}
\| \partial_t \phi+ \textbf{\textit{u}} \cdot \nabla \phi \|_{L^2(\Omega)}^2+ \int_{\Omega} \mu \, \big( \partial_t \phi+ \textbf{\textit{u}} \cdot \nabla \phi\big) \, {\rm d} x + \int_{\Omega} \rho'(\phi) \frac{|\textbf{\textit{u}}|^2}{2}\big(\partial_t \phi+ \textbf{\textit{u}} \cdot \nabla \phi\big)\, {\rm d} x=0.
\end{equation}
On the other hand, the following equalities hold
\begin{align*}
&\int_{\Omega} \mu \, \partial_t \phi \, {\rm d} x= \frac{\d}{\d t} \int_{\Omega} \frac12 |\nabla \phi|^2 + \Psi(\phi) \, {\rm d} x,
\\
&\int_{\Omega} \mu \, \textbf{\textit{u}} \cdot \nabla \phi \, {\rm d} x=
\int_{\Omega} -\Delta \phi \nabla \phi \cdot \textbf{\textit{u}} \, {\rm d} x
+ \int_{\Omega} \textbf{\textit{u}} \cdot \nabla \Psi(\phi) \, {\rm d} x= \int_{\Omega} -\Delta \phi \nabla \phi \cdot \textbf{\textit{u}} \, {\rm d} x,\\
&
\int_{\Omega} \rho'(\phi) \frac{|\textbf{\textit{u}}|^2}{2} \partial_t \phi \, {\rm d} x=
\int_{\Omega} \partial_t (\rho(\phi)) \frac{|\textbf{\textit{u}}|^2}{2} \, {\rm d} x.
\end{align*}
Thus, by adding \eqref{NSAC-D1} and \eqref{NSAC-D2}, and using the above identities, we obtain the energy equation
\begin{align}
\frac{\d}{\d t} E(\textbf{\textit{u}}, \phi)+\int_{\Omega} \nu(\phi) |D \textbf{\textit{u}}|^2 \, {\rm d} x + \|\partial_t \phi + \textbf{\textit{u}} \cdot \nabla \phi \|_{L^2(\Omega)}^2=0.\label{BEL-D}
\end{align}
\medskip
\textbf{Lower-order estimates.} We assume that $\phi \in L^\infty(\Omega\times (0,T))$ such that $ |\phi(x,t)|<1$ almost everywhere in $\Omega\times(0,T)$ (cf. Existence of weak solutions below).
Since $\rho$ is strictly positive, it is immediately seen from \eqref{BEL-D} that
\begin{equation}
\label{E-bound}
E(\textbf{\textit{u}}(t),\phi(t))+ \int_0^t \int_{\Omega} \nu(\phi) |D \textbf{\textit{u}}|^2 \, {\rm d} x {\rm d} \tau + \int_0^t \|\partial_t \phi + \textbf{\textit{u}} \cdot \nabla \phi \|_{L^2(\Omega)}^2 \, {\rm d} \tau\leq E(\textbf{\textit{u}}_0, \phi_0), \quad \forall \, t \geq 0.
\end{equation}
Therefore, we deduce
\begin{equation}
\label{B1-D}
\textbf{\textit{u}} \in L^\infty(0,T; \mathbf{H}_\sigma)\cap L^2(0,T;{\mathbf{V}}_\sigma), \quad \phi\in L^\infty(0,T;H^1(\Omega))
\end{equation}
and
\begin{equation}
\label{B2-D}
\partial_t \phi + \textbf{\textit{u}} \cdot \nabla \phi \in L^2(0,T; L^2(\Omega)).
\end{equation}
In light of \eqref{B1-D} and \eqref{B2-D}, when $d=2$, we have
$$
\Big\| - \rho'(\phi) |\textbf{\textit{u}}|^2 \Big\|_{L^2(\Omega)}\leq C \| \textbf{\textit{u}}\|_{L^4(\Omega)}^2 \leq C \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)},
$$
which entails that $\rho'(\phi) |\textbf{\textit{u}}|^2\in L^2(0,T;L^2(\Omega))$.
Instead, when $d=3$, we have
$$
\Big\| - \rho'(\phi) |\textbf{\textit{u}}|^2 \Big\|_{L^2(\Omega)}\leq C \| \textbf{\textit{u}}\|_{L^4(\Omega)}^2 \leq C \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac32,
$$
thus
$\rho'(\phi) |\textbf{\textit{u}}|^2\in L^\frac43(0,T;L^2(\Omega))$.
Since $\overline{\rho'(\phi) |\textbf{\textit{u}}|^2} \in L^\infty(0,T)$,
we also learn that
\begin{equation}
\label{B3-D}
\mu - \overline{\mu}\in L^q(0,T;L^2(\Omega)),
\end{equation}
for $q=2$ if $d=2$, $q=\frac{4}{3}$ if $d=3$.
Thanks to the boundary condition for $\phi$, we see that $\overline{\Delta \phi}=0$. Then, multiplying \eqref{NSAC-D}$_4$ by $-\Delta \phi$ and integrating by parts, we have
$$
\int_{\Omega} |\Delta \phi|^2 + F''(\phi) |\nabla \phi|^2 \, {\rm d} x= \theta_0
\|\nabla \phi\|_{L^2(\Omega)}^2 - \int_{\Omega} (\mu-\overline{\mu})\Delta \phi \, {\rm d} x,
$$
where $F$ is the convex part of the potential $\Psi$, i.e. $F(s)=\frac{\theta}{2}\left[ (1+s)\log(1+s)+(1-s)\log(1-s)\right].$
By \eqref{B1-D} and \eqref{B3-D}, we obtain
\begin{equation}
\label{H2-D}
\| \Delta \phi \|_{L^2(\Omega)} \leq C(1+ \| \mu-\overline{\mu}\|_{L^2(\Omega)}).
\end{equation}
Then, from the regularity theory of the Neumann problem, we infer that
\begin{equation}
\label{B4-D}
\phi \in L^q(0,T;H^2(\Omega)).
\end{equation}
From \eqref{LADY}, \eqref{GN3} and the above bounds, we have
\begin{align*}
\| \textbf{\textit{u}} \cdot \nabla \phi\|_{L^2(\Omega)}
&\leq C \| \textbf{\textit{u}} \|_{L^4(\Omega)}
\|\nabla \phi\|_{L^4(\Omega)}\notag\\
&\leq C \| \textbf{\textit{u}} \|_{L^2(\Omega)}^\frac12\| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac12 \|\nabla \phi\|_{L^2(\Omega)}^\frac12 \| \phi\|_{H^2(\Omega)}^\frac12\notag\\
& \leq C \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac12 \| \phi\|_{H^2(\Omega)}^\frac12,\quad \text{if}\ d=2,
\end{align*}
and
\begin{align*}
\| \textbf{\textit{u}} \cdot \nabla \phi\|_{L^2(\Omega)}
&\leq C \| \textbf{\textit{u}} \|_{L^4(\Omega)}
\|\nabla \phi\|_{L^4(\Omega)}\notag\\
&\leq C \| \textbf{\textit{u}} \|_{L^2(\Omega)}^\frac14\| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac34 \|\phi\|_{L^\infty(\Omega)}^\frac12 \| \phi\|_{H^2(\Omega)}^\frac12\notag\\
& \leq C \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac34 \| \phi\|_{H^2(\Omega)}^\frac12,\quad \text{if}\ d=3,
\end{align*}
which implies $\textbf{\textit{u}} \cdot \nabla \phi\in L^q(0,T;L^2(\Omega))$. Thus
\begin{equation}
\label{B5-D}
\partial_t \phi \in L^q(0,T;L^2(\Omega)).
\end{equation}
Moreover, we observe that
\begin{align}
\| \mu-\overline{\mu}\|_{L^2(\Omega)}&\leq \| \partial_t \phi\|_{L^2(\Omega)}+ \| \textbf{\textit{u}} \cdot \nabla \phi\|_{L^2(\Omega)}
+ \Big\|-\rho'(\phi)\frac{|\textbf{\textit{u}}|^2}{2}\Big\|_{L^2(\Omega)} + |\Omega|^{-\frac12} \Big\|-\rho'(\phi)\frac{|\textbf{\textit{u}}|^2}{2}\Big\|_{L^1(\Omega)} \notag\\
&\leq \| \partial_t \phi\|_{L^2(\Omega)}+ C \| \textbf{\textit{u}} \|_{L^4(\Omega)}
\|\nabla \phi\|_{L^4(\Omega)}+C \| \textbf{\textit{u}}\|_{L^4(\Omega)}^2+ C \| \textbf{\textit{u}}\|_{L^2(\Omega)}^2\notag \\
&\leq \| \partial_t \phi\|_{L^2(\Omega)}+ C \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac12 \| \phi\|_{H^2(\Omega)}^\frac12 +C \| \textbf{\textit{u}}\|_{L^2(\Omega)}\| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)} + C \| \textbf{\textit{u}}\|_{L^2(\Omega)}^2\notag \\
&\leq \| \partial_t \phi\|_{L^2(\Omega)}+ C \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac12 \| \phi\|_{H^2(\Omega)}^\frac12
+C \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)} + C,\quad \text{if}\ d=2,
\label{mu-L2-2}
\end{align}
and
\begin{align}
\| \mu-\overline{\mu}\|_{L^2(\Omega)}&\leq
\| \partial_t \phi\|_{L^2(\Omega)}+ C \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac34 \| \phi\|_{H^2(\Omega)}^\frac12
+C \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac32 + C,\quad \text{if}\ d=3.
\label{mu-L2-3}
\end{align}
Recalling \eqref{B1-D} and \eqref{H2-D}, and using Young's inequality, we find that
\begin{equation}
\label{estH2-D}
\|\phi \|_{H^2(\Omega)}\leq C(1+ \| \partial_t \phi\|_{L^2(\Omega)}+ \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}), \quad \text{if}\ d=2,
\end{equation}
and
\begin{equation}
\label{estH3-D}
\|\phi \|_{H^2(\Omega)}\leq C(1+ \| \partial_t \phi\|_{L^2(\Omega)}+ \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac32), \quad \text{if}\ d=3.
\end{equation}
In order to recover the full $L^2$-norm of $\mu$, we observe that
$$
\overline{\mu}=\overline{F'(\phi)}- \theta_0 \overline{\phi}.
$$
Since $|\overline{\phi}(t)|=|\overline{\phi_0}|<1$, it is well-known that
$$
\int_{\Omega} |F'(\phi)| \, {\rm d} x \leq C \int_{\Omega} F'(\phi) (\phi-\overline{\phi}) \, {\rm d} x+ C
$$
for some positive constant $C$ depending on $F$ and $\overline{\phi}$.
Multiplying \eqref{NSAC-D}$_4$ by $\phi - \overline{\phi}$ and using the boundary condition on $\phi$, we obtain
$$
\|\nabla \phi\|_{L^2(\Omega)}^2+ \int_{\Omega} F'(\phi) (\phi-\overline{\phi})= \int_{\Omega} \mu (\phi-\overline{\phi}) \, {\rm d} x+ \int_{\Omega} \theta_0 \phi (\phi-\overline{\phi}) \, {\rm d} x.
$$
Combining the above two relations and exploiting the energy bounds \eqref{B1-D}, we reach
\begin{equation}
\label{mubar}
\| F'(\phi) \|_{L^1(\Omega)} \leq C (1+ \| \mu-\overline{\mu}\|_{L^2(\Omega)}).
\end{equation}
This actually implies that
\begin{equation}
\mu \in L^q(0,T;L^2(\Omega))
\label{muq2}
\end{equation} and, in view of \eqref{B4-D}, we also have
\begin{equation}
F'(\phi)\in L^q(0,T;L^2(\Omega)),\label{fpq2}
\end{equation}
where $q=2$ if $d=2$, $q=\frac{4}{3}$ if $d=3$.
\smallskip
Besides, we have the following estimate for the time translations of $\textbf{\textit{u}}$:
\begin{lemma}\label{est-tran}
For any $\delta\in(0,T)$, the following bound holds
\begin{align}
\int_0^{T-\delta}\|\textbf{\textit{u}}(t+\delta)-\textbf{\textit{u}}(t)\|_{L^2(\Omega)}^2 \, {\rm d} t\leq C\delta^\frac14.\label{est-tr}
\end{align}
\end{lemma}
\begin{proof}
We only present the proof for the case $d=3$.
The case $d=2$ follows along the same lines.
It follows from \eqref{B1-D} and the interpolation \eqref{GN3} with $p=3$ that $\textbf{\textit{u}}\in L^4(0,T;L^3(\Omega))$. Similar to \cite{Lions} (see also \cite[Lemma 3.5]{JLL2017}), we have
\begin{align*}
&\|\sqrt{\rho(\phi(t+\delta))}(\textbf{\textit{u}}(t+\delta)-\textbf{\textit{u}}(t))\|_{L^2(\Omega)}^2 \nonumber\\
&\quad \leq -\int_\Omega(\rho(\phi(t+\delta))-\rho(\phi(t)))\textbf{\textit{u}}(t)\cdot (\textbf{\textit{u}}(t+\delta)-\textbf{\textit{u}}(t)) \, {\rm d} x\nonumber\\
&\qquad -\int_{t}^{t+\delta} \!\int_\Omega \rho(\phi(\tau))(\textbf{\textit{u}}(\tau)\cdot\nabla)\textbf{\textit{u}}(\tau)\cdot (\textbf{\textit{u}}(t+\delta)-\textbf{\textit{u}}(t))\, {\rm d} x{\rm d} \tau\nonumber\\
&\qquad -\int_t^{t+\delta}\!\int_\Omega \nu(\phi(\tau))D\textbf{\textit{u}}(\tau):D(\textbf{\textit{u}}(t+\delta)-\textbf{\textit{u}}(t))\, {\rm d} x{\rm d} \tau\nonumber\\
&\qquad +\int_t^{t+\delta}\!\int_\Omega (\nabla \phi(\tau)\otimes\nabla \phi(\tau)): \nabla (\textbf{\textit{u}}(t+\delta)-\textbf{\textit{u}}(t)) \, {\rm d} x{\rm d} \tau\nonumber\\
&\qquad +\int_t^{t+\delta}\!\int_\Omega \rho'(\phi)\partial_\tau \phi(\tau)\textbf{\textit{u}}(\tau)\cdot (\textbf{\textit{u}}(t+\delta)-\textbf{\textit{u}}(t))\, {\rm d} x{\rm d} \tau\nonumber := \sum_{i=1}^{5}J_i.\nonumber
\end{align*}
Observe now
\begin{align}
\int_0^{T-\delta} J_1(t)\, {\rm d} t
&\leq \int_0^{T-\delta}\!\int_{t}^{t+\delta}\!\int_\Omega
|\rho'(\phi)||\partial_\tau\phi(\tau)||\textbf{\textit{u}}(t)| (|\textbf{\textit{u}}(t+\delta)|+|\textbf{\textit{u}}(t)|)\, {\rm d} x{\rm d}\tau{\rm d} t\nonumber\\
&\leq \int_0^{T-\delta}(\|\textbf{\textit{u}}(t+\delta)\|_{L^3(\Omega)}+\|\textbf{\textit{u}}(t)\|_{L^3(\Omega)})
\|\textbf{\textit{u}}(t)\|_{L^6(\Omega)}\int_t^{t+\delta}\| \partial_\tau\phi(\tau)\|_{L^2(\Omega)} \, {\rm d} \tau {\rm d} t\nonumber\\
&\leq C\delta^\frac14\left(\int_0^T \|\nabla \textbf{\textit{u}}(t)\|_{L^2(\Omega)} \, {\rm d} t\right) \left(\int_0^{T}\| \partial_t\phi(t)\|_{L^2(\Omega)}^\frac43{\rm d} t\right)^\frac34 \leq C\delta^\frac14,\nonumber
\end{align}
and, in a similar manner,
\begin{align}
\int_0^{T-\delta} J_5(t) \, {\rm d} t
&\leq \int_0^{T-\delta}(\|\textbf{\textit{u}}(t+\delta)\|_{L^3(\Omega)}+\|\textbf{\textit{u}}(t)\|_{L^3(\Omega)})
\int_t^{t+\delta}\|\textbf{\textit{u}}(\tau)\|_{L^6(\Omega)}\| \partial_\tau\phi(\tau)\|_{L^2(\Omega)} \, {\rm d} \tau {\rm d} t\nonumber\\
&\leq C\delta^\frac14\left(\int_0^T \|\nabla \textbf{\textit{u}}(t)\|_{L^2(\Omega)}\, {\rm d} t\right) \left(\int_0^{T}\| \partial_t\phi(t)\|_{L^2(\Omega)}^\frac43{\rm d} t\right)^\frac34
\leq C\delta^\frac14.\nonumber
\end{align}
Next, we have
\begin{align}
&\int_0^{T-\delta} J_2(t) \, {\rm d} t\nonumber\\
&\quad \leq \int_0^{T-\delta}\!\int_{t}^{t+\delta} \| \rho(\phi(\tau))\|_{L^\infty(\Omega)}\|\textbf{\textit{u}}(\tau)\|_{L^6(\Omega)}
\|\nabla\textbf{\textit{u}}(\tau)\|_{L^2(\Omega)} \, {\rm d} \tau (\|\textbf{\textit{u}}(t+\delta)\|_{L^3(\Omega)}+\|\textbf{\textit{u}}(t)\|_{L^3(\Omega)})\, {\rm d} t\nonumber\\
&\quad \leq C\delta^\frac12\int_0^{T-\delta}\left(\int_t^{t+\delta}
\|\nabla\textbf{\textit{u}}(\tau)\|_{L^2(\Omega)}^2 \, {\rm d} \tau\right)^\frac12(\|\textbf{\textit{u}}(t+\delta)\|_{L^3(\Omega)}+\|\textbf{\textit{u}}(t)\|_{L^3(\Omega)}) \, {\rm d} t\nonumber\\
&\quad \leq C\delta^\frac12\left(\int_0^T
\|\nabla\textbf{\textit{u}}(t)\|_{L^2(\Omega)}^2 \, {\rm d} t\right)^\frac12\int_0^T\|\textbf{\textit{u}}(t)\|_{L^3(\Omega)} \, {\rm d} t \leq C\delta^\frac12,\nonumber
\end{align}
and
\begin{align}
&\int_0^{T-\delta} J_3(t) \, {\rm d} t\nonumber\\
&\quad \leq
\int_0^{T-\delta} \int_t^{t+\delta} \|\nu(\phi(\tau))\|_{L^\infty(\Omega)}\|D\textbf{\textit{u}}(\tau)\|_{L^2(\Omega)} \, {\rm d} \tau
(\|D\textbf{\textit{u}}(t+\delta)\|_{L^2(\Omega)}+\|D\textbf{\textit{u}}(t)\|_{L^2(\Omega)})\, {\rm d} t\nonumber\\
&\quad \leq C\delta^\frac12 \int_0^{T-\delta}
\left(\int_t^{t+\delta}
\|\nabla \textbf{\textit{u}}(\tau)\|_{L^2(\Omega)}^2 \, {\rm d} \tau\right)^\frac12 (\|D\textbf{\textit{u}}(t+\delta)\|_{L^2(\Omega)}+\|D\textbf{\textit{u}}(t)\|_{L^2(\Omega)}) \, {\rm d} t\nonumber\\
&\quad \leq C\delta^\frac12
\left(\int_0^{T}
\|\nabla \textbf{\textit{u}}(t)\|_{L^2(\Omega)}^2 \, {\rm d} t\right)^\frac12 \int_0^{T} \|\nabla \textbf{\textit{u}}(t)\|_{L^2(\Omega)}\, {\rm d} t \leq C\delta^\frac12,\nonumber
\end{align}
Finally, by using \eqref{GN-L4} we get
\begin{align*}
\int_0^{T-\delta} J_4(t) \, {\rm d} t &\leq \int_0^{T-\delta}\! \int_t^{t+\delta} \|\nabla \phi(\tau)\|_{L^4(\Omega)}^2 \, {\rm d} \tau (\|\nabla \textbf{\textit{u}}(t+\delta)\|_{L^2(\Omega)} +\|\nabla \textbf{\textit{u}}(t)\|_{L^2(\Omega)}) \, {\rm d} t\nonumber\\
&\quad \leq C\delta^\frac14 \int_0^{T-\delta} \left(\int_t^{t+\delta} \| \phi(\tau)\|_{H^2(\Omega)}^\frac43 \, {\rm d} \tau\right)^\frac34 (\|\nabla \textbf{\textit{u}}(t+\delta)\|_{L^2(\Omega)} +\|\nabla \textbf{\textit{u}}(t)\|_{L^2(\Omega)})\, {\rm d} t\nonumber\\
&\quad \leq C\delta^\frac14\left(\int_0^{T} \| \phi(t)\|_{H^2(\Omega)}^\frac43 \, {\rm d} t\right)^\frac34 \int_0^T \|\nabla \textbf{\textit{u}}(t)\|_{L^2(\Omega)} \, {\rm d} t \leq C\delta^\frac14.
\end{align*}
From the above estimate and the fact that $\rho$ is strictly bounded from below, we obtain the conclusion \eqref{est-tr}. The proof is complete.
\end{proof}
\medskip
\textbf{Existence of weak solutions.} With the above \textit{a priori} estimates, we are able to prove the existence of a global weak solution by using a semi-Galerkin scheme similar to \cite{JLL2017}. More precisely, for any $n\in \mathbb{N}$, we find a local-in-time approximating solution $(\textbf{\textit{u}}_n, \phi_n)$ where $\textbf{\textit{u}}_n$ solves \eqref{NSAC-D}$_1$ as in the classical Galerkin approximation and $\phi_n$ is the (non-discrete) solution to the Allen-Cahn equations \eqref{NSAC-D}$_3$-\eqref{NSAC-D}$_4$ with the velocity $\textbf{\textit{u}}_n$, the singular potential and the nonlocal term. This is achieved via a Schauder fixed point argument. For this approach, it is needed to solve separately a convective nonlocal Allen-Cahn equation. This can be done by introducing a family of regular potentials
$\lbrace \Psi_\varepsilon \rbrace$ that approximates the original singular potential
$\Psi$ by setting (see, e.g., \cite{GGW2018})
\begin{equation}
\Psi_\varepsilon(s)=F_\varepsilon(s)-\frac{\theta_0}{2}s^2,\quad \forall\, s\in \mathbb{R},\nonumber
\end{equation}
where
\begin{equation}
F_\varepsilon(s)=
\begin{cases}
\displaystyle{\sum_{j=0}^2 \frac{1}{j!}}
F^{(j)}(1-\varepsilon) \left[s-(1-\varepsilon)\right]^j,
\qquad\qquad \!\! \forall\,s\geq 1-\varepsilon,\\
F(s), \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad
\forall\, s\in[-1+\varepsilon, 1-\varepsilon],\\
\displaystyle{\sum_{j=0}^2
\frac{1}{j!}} F^{(j)}(-1+\varepsilon)\left[ s-(-1+\varepsilon)\right]^j,
\qquad\ \forall\, s\leq -1+\varepsilon.
\end{cases}
\nonumber
\end{equation}
Substituting the regular potential $\Psi_\varepsilon$ into the original Allen-Cahn equation, we are able to prove the existence of an approximating solution $\phi_\varepsilon$ to the resulting regularized equation using the semigroup approach like in \cite[Lemma 3.2]{JLL2017} or simply by the Galerkin method. For the approximating solution $\phi_{\varepsilon}$, we can derive estimates that are uniform in $\varepsilon$ and then pass to the limit as $\varepsilon\to 0$ to recover the case with singular potential. Here, we would like to remark that, thanks to the singular potential, we can show that the phase field takes values in $[-1,1]$ (using a similar argument like in \cite{GGW2018}), without the additional assumption $s\rho'(s)\geq 0$ for $|s|>1$ that was required in \cite{JLL2017}.
Next, thanks to the {\it a priori} estimates showed above, it follows that the existence time interval of any solution $(\textbf{\textit{u}}_n, \phi_n)$ is independent to $n$. From the same argument, we deduce uniform estimates that allows compactness for the phase field $\phi$.
Then, the key issue is to obtain uniform estimates of translations $\int_0^{T-\delta} \|\textbf{\textit{u}}(t+\delta)-\textbf{\textit{u}}(t)\|_{L^2(\Omega)}^2\, {\rm d} t$ (see Lemma \ref{est-tran}) that yields compactness of the velocity field in the case of unmatched densities (cf. \cite{Lions}). The above two-level approximating procedure is standard and we omit the details here.
\medskip
\textbf{Time continuity and initial condition.} We first observe that the regularity properties \eqref{B1-D} and \eqref{B5-D}, together with the global bound $\| \phi\|_{L^\infty(0,T;L^\infty(\Omega))}\leq 1$, entail that $\phi \in C([0,T]; L^p(\Omega))$, for any $2\leq p<\infty$ if $d=2,3$. In addition, since $\phi \in L^\infty(0,T;H^1(\Omega))$, we also infer from \cite[Theorem 2.1]{STRAUSS} that $\phi \in C([0,T];(H^1(\Omega))_w)$. If $d=2$, since $\phi \in L^2(0,T;H^2(\Omega))\cap W^{1,2}(0,T;L^2(\Omega))$, we deduce that $\phi \in C([0,T];H^1(\Omega))$. Next, the weak formulation of \eqref{NSAC-D}$_1$-\eqref{NSAC-D}$_2$ can be written as
\begin{align*}
\frac{\d}{\d t} \langle \mathbb{P}(\rho(\phi)\textbf{\textit{u}}), \textbf{\textit{v}} \rangle=\langle \widetilde{\textbf{\textit{f}}}, \textbf{\textit{v}}\rangle,
\end{align*}
for all $\textbf{\textit{v}} \in {\mathbf{V}}_\sigma$, in the sense of distribution on $(0,T)$, where $\mathbb{P}$ is the Leray projection onto $\mathbf{H}_\sigma$ and
$$
\langle \widetilde{\textbf{\textit{f}}}, \textbf{\textit{v}}\rangle= (\rho'(\phi)\partial_t \phi \, \textbf{\textit{u}}, \textbf{\textit{v}})-(\rho(\phi)(\textbf{\textit{u}} \cdot\nabla) \textbf{\textit{u}}, \textbf{\textit{v}})-
(\nu(\phi) D \textbf{\textit{u}}, \nabla \textbf{\textit{v}})+ (\nabla \phi \otimes \nabla \phi, \nabla \textbf{\textit{v}}).
$$
Arguing similarly to the proof of Lemma \ref{est-tran}, we observe that
\begin{align*}
\| \widetilde{\textbf{\textit{f}}}\|_{{\mathbf{V}}_\sigma'}
&\leq C \| \partial_t \phi\|_{L^2(\Omega)} \| \textbf{\textit{u}}\|_{L^3(\Omega)}+ C \| \textbf{\textit{u}}\|_{L^3(\Omega)} \| D \textbf{\textit{u}}\|_{L^2(\Omega)}+ C \| D \textbf{\textit{u}}\|_{L^2(\Omega)}+ C \| \nabla \phi\|_{L^4(\Omega)}^2\\
&\leq C \| \partial_t \phi\|_{L^2(\Omega)} \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac12 + C \| D\textbf{\textit{u}}\|_{L^2(\Omega)}^{\frac32}+ \| D \textbf{\textit{u}}\|_{L^2(\Omega)} + C \| \phi\|_{H^2(\Omega)}\\
&\leq C \big( 1+ \| \partial_t \phi\|_{L^2(\Omega)}^\frac43 + \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2+C \| \phi\|_{H^2(\Omega)}^\frac43 \big).
\end{align*}
In light of the regularity of the weak solution, we infer that $\widetilde{\textbf{\textit{f}}} \in L^1(0,T;{\mathbf{V}}_\sigma')$. By definition of the weak time derivative, this implies that $\partial_t \mathbb{P}(\rho(\phi)\textbf{\textit{u}})\in L^1(0,T;{\mathbf{V}}_\sigma')$. Observing that $\mathbb{P} (\rho(\phi)\textbf{\textit{u}}) \in L^\infty(0,T;\mathbf{H}_\sigma)$, we have $\mathbb{P} (\rho(\phi)\textbf{\textit{u}}) \in C([0,T];{\mathbf{V}}_\sigma')$. As a consequence, we deduce from \cite[Theorem 2.1]{STRAUSS} that $\mathbb{P} (\rho(\phi)\textbf{\textit{u}}) \in C([0,T]; (\mathbf{H}_\sigma)_w)$. It easily follows from the properties of the Leray operator $\mathbb{P}$ that $\mathbb{P} (\rho(\phi)\textbf{\textit{u}}) \in C([0,T]; (\mathbf{L}^2(\Omega))_w)$. Now, repeating the argument in \cite[Section 5.2]{ADG2013}, we deduce that $\rho(\phi)\textbf{\textit{u}} \in C([0,T]; (\mathbf{L}^2(\Omega))_w)$. Therefore, since $\rho(\phi) \in C([0,T];L^2(\Omega))$ and $\rho(\phi)\geq \rho_\ast>0$, we conclude that $\textbf{\textit{u}} \in C([0,T];(\mathbf{L}^2(\Omega))_w)$. Finally, thanks to the time continuity of $\textbf{\textit{u}}$ and $\phi$, a standard argument ensures that $\textbf{\textit{u}}|_{t=0}=\textbf{\textit{u}}_0$, $\phi|_{t=0}=\phi_0$ in $\Omega$.
\hfill$\square$
\smallskip
\subsection{Proof of Theorem \ref{uni2d}}
Let us consider two global weak solutions $(\textbf{\textit{u}}_1,\phi_1)$ and $(\textbf{\textit{u}}_2,\phi_2)$ to problem \eqref{NSAC}-\eqref{bic} given by Theorem \ref{W-S}. Denote the differences of solutions by $\textbf{\textit{u}}=\textbf{\textit{u}}_1-\textbf{\textit{u}}_2$, $\phi=\phi_1-\phi_2$. Then we have
\begin{align}
\langle \partial_t \textbf{\textit{u}}, \textbf{\textit{v}}\rangle + (\textbf{\textit{u}}_1 \cdot \nabla \textbf{\textit{u}}, \textbf{\textit{v}})&+ (\textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}_2, \textbf{\textit{v}})+ (\nu(\phi_1)D\textbf{\textit{u}},\nabla \textbf{\textit{v}}) + ((\nu(\phi_1)-\nu(\phi_2))D\textbf{\textit{u}}_2,\nabla \textbf{\textit{v}})
\notag \\
&= (\nabla \phi_1 \otimes \nabla \phi, \nabla \textbf{\textit{v}})+
(\nabla \phi \otimes \nabla \phi_2, \nabla \textbf{\textit{v}}) \label{NS-diff}
\end{align}
for all $ \textbf{\textit{v}} \in {\mathbf{V}}_\sigma$, almost every $t \in (0,T)$, and
\begin{equation}
\partial_t \phi+ \textbf{\textit{u}}_1\cdot \nabla \phi + \textbf{\textit{u}} \cdot \nabla \phi_2-\Delta \phi+ \Psi'(\phi_1)-\Psi'(\phi_2)=\overline{\Psi'(\phi_1)}-\overline{\Psi'(\phi_2)} \label{AC-diff}
\end{equation}
almost every $(x,t) \in \Omega \times (0,T)$.
Following the same strategy as in \cite{GMT2019}, we take $\textbf{\textit{v}}= \mathbf{A}^{-1}\textbf{\textit{u}}$, where $\mathbf{A}$ is the Stokes operator, and we find
\begin{align*}
\frac12 \frac{\d}{\d t} \| \textbf{\textit{u}}\|_{\ast}^2 &+ (\nu(\phi_1) D\textbf{\textit{u}}, \nabla \mathbf{A}^{-1}\textbf{\textit{u}}) =
(\textbf{\textit{u}}\otimes \textbf{\textit{u}}_1, \nabla \mathbf{A}^{-1}\textbf{\textit{u}})+ (\textbf{\textit{u}}_2\otimes \textbf{\textit{u}}, \nabla \mathbf{A}^{-1}\textbf{\textit{u}})\\
&- ((\nu(\phi_1)-\nu(\phi_2))D\textbf{\textit{u}}_2,\nabla \mathbf{A}^{-1}\textbf{\textit{u}})
+ (\nabla \phi_1 \otimes \nabla \phi, \nabla \mathbf{A}^{-1}\textbf{\textit{u}})
+(\nabla \phi \otimes \nabla \phi_2, \nabla \mathbf{A}^{-1}\textbf{\textit{u}}),
\end{align*}
where $\| \textbf{\textit{u}}\|_{\ast}= \| \nabla \mathbf{A}^{-1} \textbf{\textit{u}}\|_{L^2(\Omega)}$, which is a norm on ${\mathbf{V}}'_\sigma$ equivalent to the usual\ dual norm. Here, we have used the equality $\textbf{\textit{u}}_i \cdot \nabla \textbf{\textit{u}}= \mathrm{div}\, ( \textbf{\textit{u}}\otimes \textbf{\textit{u}}_i)$, $i=1,2$,
due to the incompressibility condition.
Multiplying \eqref{AC-diff} by $\phi$, integrating over $\Omega$ and observing that
$$
\int_{\Omega} (\textbf{\textit{u}}_1 \cdot \nabla \phi) \, \phi \,{\rm d} x= \int_{\Omega} \textbf{\textit{u}}_1 \cdot \frac12 \nabla \phi^2 \,{\rm d} x=0, \quad \int_{\Omega} (\overline{\Psi'(\phi_1)}-\overline{\Psi'(\phi_2)} ) \phi \, {\rm d} x= (\overline{\Psi'(\phi_1)}-\overline{\Psi'(\phi_2)} ) \overline{\phi} =0,
$$
we obtain
$$
\frac12 \frac{\d}{\d t} \|\phi\|_{L^2(\Omega)}^2+ \| \nabla \phi\|_{L^2(\Omega)}^2+ \int_{\Omega} (\textbf{\textit{u}} \cdot \nabla \phi_2) \, \phi \, {\rm d} x+ \int_{\Omega} (F'(\phi_1)-F'(\phi_2)) \, \phi \, {\rm d} x=
\theta_0 \| \phi\|_{L^2(\Omega)}^2.
$$
By adding the above two equations and using the convexity of $F$, we deduce that
\begin{align}
\label{u-est1}
& \frac{\d}{\d t} G(t) + (\nu(\phi_1) D\textbf{\textit{u}}, \nabla \mathbf{A}^{-1}\textbf{\textit{u}}) + \| \nabla \phi\|_{L^2(\Omega)}^2 \notag \\
&\quad \leq (\textbf{\textit{u}}\otimes \textbf{\textit{u}}_1, \nabla \mathbf{A}^{-1}\textbf{\textit{u}})+ (\textbf{\textit{u}}_2\otimes \textbf{\textit{u}}, \nabla \mathbf{A}^{-1}\textbf{\textit{u}})
- ((\nu(\phi_1)-\nu(\phi_2))D\textbf{\textit{u}}_2,\nabla \mathbf{A}^{-1}\textbf{\textit{u}}) \notag \\
&\qquad + (\nabla \phi_1 \otimes \nabla \phi, \nabla \mathbf{A}^{-1}\textbf{\textit{u}})
+(\nabla \phi \otimes \nabla \phi_2, \nabla \mathbf{A}^{-1}\textbf{\textit{u}})- (\textbf{\textit{u}} \cdot \nabla \phi_2 , \phi) + \theta_0 \| \phi\|_{L^2(\Omega)}^2,
\end{align}
where
$$
G(t)= \frac12 \| \textbf{\textit{u}}(t)\|_{\ast}^2+ \frac12 \|\phi(t)\|_{L^2(\Omega)}^2.
$$
In order to recover a $L^2(\Omega)$-norm of $\textbf{\textit{u}}$, which is a key term to control the nonlinear terms on the right-hand side, we obtain by integration by parts (see \cite[(3.9)]{GMT2019})
\begin{align*}
(\nu(\phi_1) D\textbf{\textit{u}}, \nabla \mathbf{A}^{-1}\textbf{\textit{u}})&= (\nabla \textbf{\textit{u}}, \nu(\phi_1)D \mathbf{A}^{-1}\textbf{\textit{u}})\\
&=- (\textbf{\textit{u}}, \mathrm{div}\, (\nu(\phi_1)D \mathbf{A}^{-1}\textbf{\textit{u}} )\\
&=- (\textbf{\textit{u}}, \nu'(\phi_1) D \mathbf{A}^{-1}\textbf{\textit{u}} \nabla \phi_1) - \frac12 (\textbf{\textit{u}}, \nu(\phi_1) \Delta \mathbf{A}^{-1}\textbf{\textit{u}}).
\end{align*}
Here we have used that $\mathrm{div}\, \nabla^t \textbf{\textit{v}}= \nabla \mathrm{div}\, \textbf{\textit{v}}$.
Notice that, by definition of Stokes operator,
there exists a scalar function $p\in L^\infty(0,T;H^1(\Omega))\cap L^2(0,T;H^2(\Omega))$ (unique up to a constant) such that
$ -\Delta \mathbf{A}^{-1} \textbf{\textit{u}}+ \nabla p= \textbf{\textit{u}}$ for almost any
$(x,t)\in \Omega \times (0,T)$. Moreover, we have the following estimates from \cite{Galdi} and \cite[Appendix B]{GMT2019}
\begin{equation}
\label{p}
\| p\|_{L^2(\Omega)}\leq C \| \nabla \mathbf{A}^{-1} \textbf{\textit{u}} \|_{L^2(\Omega)}^{\frac12} \| \textbf{\textit{u}}\|_{L^2(\Omega)}^{\frac12},\
\| p \|_{H^1(\Omega)}\leq C \|\textbf{\textit{u}} \|_{L^2(\Omega)}, \ \| p\|_{H^2(\Omega)}\leq C \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}.
\end{equation}
Then, we can write
\begin{align*}
- \frac12 (\textbf{\textit{u}}, \nu(\phi_1) \Delta A^{-1}\textbf{\textit{u}})&= \frac12 ( \textbf{\textit{u}}, \nu(\phi_1) \textbf{\textit{u}} ) -\frac12 (\textbf{\textit{u}}, \nu(\phi_1) \nabla p)\\
&= \frac12 ( \textbf{\textit{u}}, \nu(\phi_1) \textbf{\textit{u}} ) + \frac12 (\mathrm{div}\, (\nu(\phi_1)\textbf{\textit{u}} ), p)\\
&= \frac12 ( \textbf{\textit{u}}, \nu(\phi_1) \textbf{\textit{u}} ) + \frac12 (\nu'(\phi_1) \nabla \phi_1 \cdot \textbf{\textit{u}}, p).
\end{align*}
Hence, recalling that $\nu(\cdot)\geq \nu_\ast>0$, we find
\begin{align*}
(\nu(\phi_1) D\textbf{\textit{u}}, \nabla \mathbf{A}^{-1}\textbf{\textit{u}}) \geq \frac{\nu_\ast}{2} \| \textbf{\textit{u}}\|_{L^2(\Omega)}^2+
\frac12 (\nu'(\phi_1) \nabla \phi_1 \cdot \textbf{\textit{u}}, p)- (\textbf{\textit{u}}, \nu'(\phi_1) D \mathbf{A}^{-1}\textbf{\textit{u}} \nabla \phi_1).
\end{align*}
Owing to the above estimate, we rewrite \eqref{u-est1} as follows
\begin{align}
\label{u-est2}
\frac{\d}{\d t} G(t) &+ \frac{\nu_\ast}{2} \| \textbf{\textit{u}}\|_{L^2(\Omega)}^2 + \| \nabla \phi\|_{L^2(\Omega)}^2 \notag \\
&=(\textbf{\textit{u}}\otimes \textbf{\textit{u}}_1, \nabla \mathbf{A}^{-1}\textbf{\textit{u}})+ (\textbf{\textit{u}}_2\otimes \textbf{\textit{u}}, \nabla \mathbf{A}^{-1}\textbf{\textit{u}})
+ ((\nu(\phi_1)-\nu(\phi_2))D\textbf{\textit{u}}_2,\nabla \mathbf{A}^{-1}\textbf{\textit{u}}) \notag \\
&\quad + (\nabla \phi_1 \otimes \nabla \phi, \nabla \mathbf{A}^{-1}\textbf{\textit{u}})
+(\nabla \phi \otimes \nabla \phi_2, \nabla \mathbf{A}^{-1}\textbf{\textit{u}})- (\textbf{\textit{u}} \cdot \nabla \phi_2 , \phi) \notag \\
&\quad + \theta_0 \| \phi\|_{L^2(\Omega)}^2
+ (\textbf{\textit{u}}, \nu'(\phi_1) D \mathbf{A}^{-1}\textbf{\textit{u}} \nabla \phi_1)-\frac12 (\nu'(\phi_1) \nabla \phi_1 \cdot \textbf{\textit{u}}, p).
\end{align}
By the Ladyzhenskaya inequality \eqref{LADY}, together with \eqref{H2equiv} and the bounds for weak solutions, we have
\begin{align*}
& (\textbf{\textit{u}}\otimes \textbf{\textit{u}}_1, \nabla \mathbf{A}^{-1}\textbf{\textit{u}})+ (\textbf{\textit{u}}_2\otimes \textbf{\textit{u}}, \nabla \mathbf{A}^{-1}\textbf{\textit{u}})\\
&\quad \leq \| \textbf{\textit{u}}\|_{L^2(\Omega)} \big( \| \textbf{\textit{u}}_1\|_{L^4(\Omega)}+ \| \textbf{\textit{u}}_2\|_{L^4(\Omega)}\big)
\| \nabla \mathbf{A}^{-1}\textbf{\textit{u}}\|_{L^4(\Omega)}\\
&\quad \leq C \big(\| \textbf{\textit{u}}_1\|_{H^1(\Omega)}^\frac12+ \| \textbf{\textit{u}}_2\|_{H^1(\Omega)}^\frac12) \| \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac32 \|\textbf{\textit{u}} \|_{\ast}^\frac12\\
&\quad \leq \frac{\nu_\ast}{20} \| \textbf{\textit{u}}\|_{L^2(\Omega)}^2 + C \big(\| \textbf{\textit{u}}_1\|_{H^1(\Omega)}^2+ \| \textbf{\textit{u}}_2\|_{H^1(\Omega)}^2) \|\textbf{\textit{u}} \|_{\ast}^2.
\end{align*}
Similarly, we obtain
\begin{align*}
&(\nabla \phi_1 \otimes \nabla \phi, \nabla \mathbf{A}^{-1}\textbf{\textit{u}})
+(\nabla \phi \otimes \nabla \phi_2, \nabla \mathbf{A}^{-1}\textbf{\textit{u}})
\\
&\quad \leq \big( \| \nabla \phi_1\|_{L^4(\Omega)}+ \| \nabla \phi_2\|_{L^4(\Omega)}\big)
\| \nabla \phi\|_{L^2(\Omega)}
\| \nabla \mathbf{A}^{-1}\textbf{\textit{u}}\|_{L^4(\Omega)}\\
&\quad \leq C \big(\| \phi_1\|_{H^2(\Omega)}^\frac12+ \| \phi_2\|_{H^2(\Omega)}^\frac12) \| \nabla \phi\|_{L^2(\Omega)} \|\textbf{\textit{u}} \|_{L^2(\Omega)}^\frac12 \|\textbf{\textit{u}} \|_{\ast}^\frac12\\
&\quad \leq \frac{\nu_\ast}{20} \| \textbf{\textit{u}}\|_{L^2(\Omega)}^2 + \frac{1}{12} \| \nabla \phi\|_{L^2(\Omega)}^2+ C \big(\| \phi_1\|_{H^2(\Omega)}^2+ \| \phi_2\|_{H^2(\Omega)}^2) \|\textbf{\textit{u}} \|_{\ast}^2,
\end{align*}
and
\begin{align*}
(\textbf{\textit{u}} \cdot \nabla \phi_2, \phi)
&\leq \|\textbf{\textit{u}} \|_{L^2(\Omega)} \| \nabla \phi_2 \|_{L^4(\Omega)} \|\phi \|_{L^4(\Omega)}\\
&\leq C \|\textbf{\textit{u}} \|_{L^2(\Omega)} \| \nabla \phi_2 \|_{H^1(\Omega)}^\frac12 \|\phi \|_{L^2(\Omega)}^\frac12 \|\nabla \phi \|_{L^2(\Omega)}^\frac12\\
&\leq \frac{\nu_\ast}{20} \| \textbf{\textit{u}}\|_{L^2(\Omega)}^2+ \frac{1}{12} \| \nabla \phi\|_{L^2(\Omega)}^2
+C \|\phi_2 \|_{H^2(\Omega)}^2 \| \phi\|_{L^2(\Omega)}^2,
\end{align*}
where we have also used the inequality \eqref{normH1-2} and the conservation of mass $\overline{\phi}=0$.
Next, since $\nu'$ is bounded, by exploiting \eqref{LADY} we have
\begin{align*}
(\textbf{\textit{u}}, \nu'(\phi_1) D \mathbf{A}^{-1}\textbf{\textit{u}} \nabla \phi_1)
&\leq C \| \textbf{\textit{u}}\|_{L^2(\Omega)} \| D \mathbf{A}^{-1}\textbf{\textit{u}}\|_{L^4(\Omega)} \| \nabla \phi_1\|_{L^4(\Omega)}\\
&\leq C \| \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac32 \| \textbf{\textit{u}}\|_{\ast}^\frac12 \| \nabla \phi_1\|_{H^1(\Omega)}^\frac12\\
&\leq \frac{\nu_\ast}{20} \| \textbf{\textit{u}}\|_{L^2(\Omega)}^2+ C \| \phi_1\|_{H^2(\Omega)}^2\| \textbf{\textit{u}}\|_{\ast}^2.
\end{align*}
By using the Stokes operator (i.e. $\mathbf{A}=\mathbb{P}(-\Delta)$) and the integration by parts, we infer that
\begin{align*}
-\frac12 (\nu'(\phi_1) \nabla \phi_1 \cdot \textbf{\textit{u}}, p)
&= \frac12 \big( \Delta \mathbf{A}^{-1} \textbf{\textit{u}} , \mathbb{P} ( \nu'(\phi_1)\nabla \phi_1 p ) \big)\\
&=-\frac12 \int_{\Omega} (\nabla \mathbf{A}^{-1} \textbf{\textit{u}})^t : \nabla \mathbb{P} ( \nu'(\phi_1)\nabla \phi_1 p ) \, {\rm d} x \\
&\quad +\frac12 \int_{\partial \Omega} \big( (\nabla \mathbf{A}^{-1} \textbf{\textit{u}})^t \mathbb{P}( \nu'(\phi_1)\nabla \phi_1 p ) \big) \cdot \textbf{\textit{n}} \, {\rm d} \sigma.
\end{align*}
Thanks to \eqref{H2equiv}, \eqref{trace}, and the properties of the Leray projection, we find
\begin{align}
&-\frac12 (\nu'(\phi_1) \nabla \phi_1 \cdot \textbf{\textit{u}}, p) \notag \\
&\quad \leq C \| \nabla \mathbf{A}^{-1}\textbf{\textit{u}}\|_{L^2(\Omega)} \| \nabla \mathbb{P} ( \nu'(\phi_1)\nabla \phi_1 p ) \|_{L^2(\Omega)}
+ C \| \nabla \mathbf{A}^{-1}\textbf{\textit{u}}\|_{L^2(\partial \Omega)} \| \mathbb{P}( \nu'(\phi_1)\nabla \phi_1 p )\|_{L^2(\partial \Omega)} \notag \\
&\quad \leq C \| \textbf{\textit{u}}\|_{\ast} \| \nu'(\phi_1)\nabla \phi_1 p \|_{H^1(\Omega)}
+C \|\textbf{\textit{u}}\|_{\ast}^\frac12 \| \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac12
\| \mathbb{P}( \nu'(\phi_1)\nabla \phi_1 p )\|_{L^2(\Omega)}^\frac12
\| \mathbb{P}( \nu'(\phi_1)\nabla \phi_1 p )\|_{H^1(\Omega)}^\frac12
\notag \\
&\quad \leq C \| \textbf{\textit{u}}\|_{\ast} \| \nu'(\phi_1)\nabla \phi_1 p \|_{H^1(\Omega)} +C \| \textbf{\textit{u}}\|_{\ast}^\frac12 \| \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac12
\| \nu'(\phi_1)\nabla \phi_1 p \|_{L^2(\Omega)}^\frac12
\| \nu'(\phi_1)\nabla \phi_1 p \|_{H^1(\Omega)}^\frac12.
\label{pterm}
\end{align}
Owing to \eqref{LADY}, \eqref{BGI}, Lemma \ref{result1} and \eqref{p}, we observe that
\begin{align*}
\| \nu'(\phi_1)\nabla \phi_1 p \|_{L^2(\Omega)}
&\leq C \|\nabla \phi_1 \|_{L^4(\Omega)} \| p\|_{L^4(\Omega)}\\
&\leq C \| \phi_1\|_{H^2(\Omega)}^\frac12 \| p\|_{L^2(\Omega)}^\frac12 \| p\|_{H^1(\Omega)}^\frac12 \\
&\leq C \| \phi_1\|_{H^2(\Omega)}^\frac12 \| \nabla \mathbf{A}^{-1} \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac14 \| \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac34,
\end{align*}
and
\begin{align*}
\| \nu'(\phi_1)\nabla \phi_1 p \|_{H^1(\Omega)}
&\leq \| \nu'(\phi_1)\nabla \phi_1 p \|_{L^2(\Omega)}
+\| \nu''(\phi_1)\nabla \phi_1 \otimes \nabla \phi_1 p \|_{L^2(\Omega)} \\
&\quad + \| \nu'(\phi_1) \nabla^2 \phi_1 p\|_{L^2(\Omega)}
+ \| \nu'(\phi_1) \nabla \phi_1 \otimes \nabla p\|_{L^2(\Omega)}\\
&\leq C \| \textbf{\textit{u}}\|_{L^2(\Omega)} \log^\frac12 \Big( C \frac{\|\nabla \textbf{\textit{u}}\|_{L^2(\Omega)} }{\| \textbf{\textit{u}}\|_{L^2(\Omega)}}\Big) + C \| \nabla \phi_1\|_{L^4(\Omega)}^2 \| p\|_{L^\infty(\Omega)}\\
&\quad + C \| \phi_1\|_{H^2(\Omega)} \| p\|_{L^\infty(\Omega)} +
C \| \phi_1\|_{H^2(\Omega)} \| p\|_{H^1(\Omega)} \log^\frac12 \Big( C \frac{\| p\|_{H^2(\Omega)}}{\|p \|_{H^1(\Omega)}}\Big)\\
&\leq C \big( 1+ \| \phi_1\|_{H^2(\Omega)}\big) \| \textbf{\textit{u}}\|_{L^2(\Omega)} \log^\frac12 \Big( C \frac{\|\nabla \textbf{\textit{u}}\|_{L^2(\Omega)} }{\| \textbf{\textit{u}}\|_{L^2(\Omega)}}\Big).
\end{align*}
Combining the above estimates with \eqref{pterm}, we are led to
\begin{align*}
-\frac12 (\nu'(\phi_1) \nabla \phi_1 \cdot \textbf{\textit{u}}, p)
& \leq C \big( 1+ \| \phi_1\|_{H^2(\Omega)}\big) \| \textbf{\textit{u}}\|_{\ast} \| \textbf{\textit{u}}\|_{L^2(\Omega)} \log^\frac12 \Big( C \frac{\|\nabla \textbf{\textit{u}}\|_{L^2(\Omega)} }{\| \textbf{\textit{u}}\|_{L^2(\Omega)}}\Big)\\
&\quad + C \big( 1+ \| \phi_1\|_{H^2(\Omega)}^\frac34 \big) \| \textbf{\textit{u}}\|_{\ast}^\frac58 \| \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac{11}{8} \log^\frac14 \Big( C \frac{\|\nabla \textbf{\textit{u}}\|_{L^2(\Omega)} }{\| \textbf{\textit{u}}\|_{L^2(\Omega)}}\Big)\\
&\leq \frac{\nu_\ast}{20} \| \textbf{\textit{u}}\|_{L^2(\Omega)}^2+
C \big( 1+ \| \phi_1\|_{H^2(\Omega)}^2 \big) \| \textbf{\textit{u}}\|_{\ast}^2 \log \Big( C \frac{\|\nabla \textbf{\textit{u}}\|_{L^2(\Omega)} }{\| \textbf{\textit{u}}\|_{L^2(\Omega)}}\Big)\\
&\quad +C \big( 1+ \| \phi_1\|_{H^2(\Omega)}^\frac{12}{5} \big) \| \textbf{\textit{u}}\|_{\ast}^2 \log^\frac45 \Big( C \frac{\|\nabla \textbf{\textit{u}}\|_{L^2(\Omega)} }{\| \textbf{\textit{u}}\|_{L^2(\Omega)}}\Big).
\end{align*}
In order to handle the logarithmic terms, we recall that $\frac{ C \|\nabla \textbf{\textit{u}}\|_{L^2(\Omega)} }{\| \textbf{\textit{u}}\|_{L^2(\Omega)}}>1$. Since $\frac{C' \| \textbf{\textit{u}}\|_{L^2(\Omega)}}{\|\textbf{\textit{u}}\|_{\ast}}>1$, for some $C'>0$ depending on $\Omega$, we have
\begin{align*}
\log^\frac45 \Big( C \frac{\|\nabla \textbf{\textit{u}}\|_{L^2(\Omega)} }{\| \textbf{\textit{u}}\|_{L^2(\Omega)}}\Big)
& \leq 1+ \log \Big( C \frac{\|\nabla \textbf{\textit{u}}\|_{L^2(\Omega)} }{\| \textbf{\textit{u}}\|_{L^2(\Omega)}}\Big)\\
& \leq 1+ \log \Big( C \frac{ C' \|\nabla \textbf{\textit{u}}\|_{L^2(\Omega)} }{\| \textbf{\textit{u}}\|_{\ast}}\Big)\\
&\leq C+ \log \big(1+\| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)} \big) + \log \Big( \frac{\widetilde{C} }{\| \textbf{\textit{u}}\|_{\ast}}\Big),
\end{align*}
where $\widetilde{C}>0$ is a sufficiently large constant such that
$\log \Big( \frac{\widetilde{C} }{\| \textbf{\textit{u}}\|_{\ast}}\Big)>1$, which holds true in light of \eqref{B1-D}. Thus, we obtain
\begin{align*}
-\frac12 (\nu'(\phi_1) \nabla \phi_1 \cdot \textbf{\textit{u}}, p)
& \leq \frac{\nu_\ast}{20} \| \textbf{\textit{u}}\|_{L^2(\Omega)}^2+
C \big( 1+ \| \phi_1\|_{H^2(\Omega)}^\frac{12}{5} \big)
\log \big(1+\| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)} \big)
\| \textbf{\textit{u}}\|_{\ast}^2\\
&\quad +
C \big( 1+ \| \phi_1\|_{H^2(\Omega)}^\frac{12}{5} \big) \|\textbf{\textit{u}}\|_{\ast}^2 \log \Big( \frac{\widetilde{C}}{\| \textbf{\textit{u}}\|_{\ast}}\Big).
\end{align*}
Netx, by using Lemma \ref{result1}, we infer that
\begin{align*}
&-((\nu(\phi_1)-\nu(\phi_2))D\textbf{\textit{u}}_2,\nabla \mathbf{A}^{-1}\textbf{\textit{u}})\\
&\quad =-\int_{\Omega} \int_0^1 \nu'(\tau \phi_1+ (1-\tau)\phi_2) \, {\rm d} \tau \, \phi D \textbf{\textit{u}}_2 : \nabla \mathbf{A}^{-1}\textbf{\textit{u}} \, {\rm d} x\\
&\quad \leq C \| D\textbf{\textit{u}}_2\|_{L^2(\Omega)} \| \phi \nabla \mathbf{A}^{-1}\textbf{\textit{u}}\|_{L^2(\Omega)}\\
&\quad \leq C \| \textbf{\textit{u}}_2 \|_{H^1(\Omega)} \| \nabla \phi\|_{L^2(\Omega)} \| \textbf{\textit{u}}\|_{\ast}
\log^\frac12 \Big(C\frac{ \| \textbf{\textit{u}}\|_{L^2(\Omega)}}{\| \textbf{\textit{u}}\|_{\ast}} \Big)\\
&\quad \leq \frac{1}{12} \|\nabla \phi \|_{L^2(\Omega)}^2
+C \| \textbf{\textit{u}}_2 \|_{H^1(\Omega)}^2 \| \textbf{\textit{u}}\|_{\ast}^2
\log \Big(\frac{ \widetilde{C}}{\| \textbf{\textit{u}}\|_{\ast}} \Big),
\end{align*}
where $\widetilde{C}$ is chosen sufficiently large as above.
Summing up, we arrive at the differential inequality
\begin{equation}
\label{u-est3}
\frac{\d}{\d t} G(t) + \frac{\nu_\ast}{4} \| \textbf{\textit{u}}\|_{L^2(\Omega)}^2 + \frac12 \| \nabla \phi\|_{L^2(\Omega)}^2
\leq C S(t) G(t) \log \Big( \frac{\widetilde{C}}{G(t)}\Big),
\end{equation}
where
$$
S(t)= \Big(1+ \| \textbf{\textit{u}}_1\|_{H^1(\Omega)}^2+ \| \textbf{\textit{u}}_2\|_{H^1(\Omega)}^2
+ \|\phi_2 \|_{H^2(\Omega)}^2 +\| \phi_1\|_{H^2(\Omega)}^2 + \|\phi_1 \|_{H^2(\Omega)}^\frac{12}{5} \big( 1+ \log \big(1+\| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)} \big) \big)\Big).
$$
Here we have used that the function $s \log \Big(\frac{\widetilde{C}}{s} \Big)$ is increasing on $(0, \frac{\widetilde{C}}{e})$.
We observe that $S\in L^1(0,T)$ provided that $\phi_1 \in L^{\gamma}(0,T;H^2(\Omega))$ with $\gamma>\frac{12}{5}$. Indeed, we recall that
$\log(1+s)\leq C(\kappa) (1+s)^\kappa$, for any $\kappa>0$. Taking $\kappa= \frac{2(5\gamma -12)}{5\gamma}$, we have
\begin{align*}
\int_0^T \|\phi_1(\tau) \|_{H^2(\Omega)}^\frac{12}{5}& \log \big(1+\| \nabla \textbf{\textit{u}} (\tau)\|_{L^2(\Omega)} \big) \, {\rm d} \tau\\
&\leq C \int_0^T \| \phi_1 (\tau)\|_{H^2(\Omega)}^\frac{12}{5} \big(1+\| \nabla \textbf{\textit{u}}(\tau)\|_{L^2(\Omega)} \big)^\frac{2(5\gamma -12)}{5\gamma} \, {\rm d} \tau \\
&\leq C \int_0^T \| \phi_1(\tau)\|_{H^2(\Omega)}^\gamma + \| \nabla \textbf{\textit{u}}_1(\tau)\|_{L^2(\Omega)}^2+ \| \nabla \textbf{\textit{u}}_2 (\tau)\|_{L^2(\Omega)}^2 \, {\rm d} \tau.
\end{align*}
Throughout the rest of the proof, we will assume that $S\in L^1(0,T)$.
Integrating \eqref{u-est3} on the time interval $[0,t]$, we find
$$
G(t) \leq G(0)+C \int_0^t S(s) G(s) \log \Big( \frac{\widetilde{C}}{G(s)} \Big) \, {\rm d} s,
$$
for almost every $t \in [0,T]$. We observe that $\int_0^1 \frac{1}{s\log(\frac{C}{s})} \, {\rm d} s= \infty$. Thus, if $G(0)=0$, applying the Osgood lemma \ref{Osgood}, we deduce that $G(t)=0$ for all $t\in [0,T]$, namely $\textbf{\textit{u}}_1(t)=\textbf{\textit{u}}_2(t)$ and $\phi_1(t)=\phi_2(t)$. This demonstrates the uniqueness of solutions in the class of weak solutions satisfying the additional regularity $\phi_1 \in L^\gamma(0,T;H^2(\Omega))$ with $\gamma>\frac{12}{5}$. Indeed, we are able to deduce a continuous dependence estimate with respect to the initial datum. To this end, we define
$\mathcal{M}(s)=\log( \log(\frac{\widetilde{C}}{s}) )$. by the Osgood lemma, for $G(0)>0$, we are led to
\begin{equation}
\label{u-est4}
-\log \Big(\log \Big(\frac{\widetilde{C}}{G(t)}\Big)\Big)+\log \Big(\log \Big(\frac{\widetilde{C}}{G(0)}\Big)\Big)\leq C\int_0^t S(s)\, {\rm d} s
\end{equation}
for almost every $t \in [0,T]$.
Taking the double exponential of \eqref{u-est4}, we eventually infer the control
\begin{equation}
\label{CD}
G(t)\leq \widetilde{C} \Big(\frac{G(0)}{\widetilde{C}}\Big)^{ \exp(-C\int_{0}^t S(s)\, \mathrm{d}s)} \quad \forall \, t \in [0,T_0],
\end{equation}
where $T_0>0$ is defined by
$$
\log \Big(\log \Big(\frac{\widetilde{C}}{G(0)}\Big)\Big)\geq C \int_0^{T_0} S(s)\, {\rm d} s.
$$
The proof is complete.
\hfill $\square$
\medskip
\begin{remark}
We note that the same existence result as in Theorem \ref{W-S} holds for $\Omega=\mathbb{T}^d$, $d=2,3$.
In the particular case $\Omega=\mathbb{T}^2$, the uniqueness of weak solutions can be achieved, without the additional regularity $\phi\in L^\gamma(0,T;H^2(\Omega))$ as in Theorem \ref{uni2d}.
Indeed, in this case the solutions of the Stokes operator $\mathbf{A}^{-1}\textbf{\textit{u}}$ and $p$ are given by (see \cite[Chapter 2.2]{Temam})
$$
\mathbf{A}^{-1}\textbf{\textit{u}}= \sum_{k\in \mathbb{Z}^2} g_k {e}^{\frac{2i\pi k \cdot x}{L}},
\quad p= \sum_{k\in \mathbb{Z}^2} p_k {e}^{\frac{2i\pi k \cdot x}{L}},
$$
where
$$
g_k=-\frac{L^2}{4\pi^2|k|^2} \Big( \textbf{\textit{u}}_k-\frac{(k\cdot \textbf{\textit{u}}_k)k}{|k|^2}\Big), \quad
p_k= \frac{L k \cdot \textbf{\textit{u}}_k}{2i\pi |k|^2}, \quad
k \in \mathbb{Z}^2, \ k\neq 0,
$$
$L>0$ is the cell size. Here $\textbf{\textit{u}}_k$ is the $k$-mode of $\textbf{\textit{u}}$. We observe that we only consider the case $k \neq 0$ since $\overline{\textbf{\textit{u}}}$ is conserved for \eqref{NSAC}$_1$ on $\mathbb{T}^2$, and so we can choose $\overline{\textbf{\textit{u}}}=0$. Moreover, since $\textbf{\textit{u}}\in \mathbf{H}_\sigma$, we have that $k \cdot \textbf{\textit{u}}_k=0$ for any $k \in \mathbb{Z}^2$, which implies that $p_k=0$ for any $k \in \mathbb{Z}^2$. Thus, following the above proof, we are led to the differential inequality \eqref{u-est2} without the last term on the right-hand side, i.e. $-\frac12 (\nu'(\phi_1)\nabla \phi_1\cdot \textbf{\textit{u}},p)$.
Hence, we eventually end up with
$$
\frac{\d}{\d t} G(t) + \frac{\nu_\ast}{4} \| \textbf{\textit{u}}\|_{L^2(\Omega)}^2 + \frac12 \| \nabla \phi\|_{L^2(\Omega)}^2
\leq C \widetilde{S}(t) G(t) \log \Big( \frac{\widetilde{C}}{G(t)}\Big),
$$
where
$$
\widetilde{S}(t)= \Big(1+ \| \textbf{\textit{u}}_1\|_{H^1(\Omega)}^2+ \| \textbf{\textit{u}}_2\|_{H^1(\Omega)}^2
+ \|\phi_1 \|_{H^2(\Omega)}^2 +\| \phi_2\|_{H^2(\Omega)}^2 \Big).
$$
Since $\widetilde{S}(t)\in L^1(0,T)$ for any couple of weak solutions, an application of the Osgood lemma as above entails the uniqueness of weak solutions (without additional regularity) and a continuous dependence estimate with respect to the initial data,
i.e. \eqref{CD}.
\end{remark}
\section{Mass-conserving Navier-Stokes-Allen-Cahn System: Strong Solutions}
\setcounter{equation}{0}
\label{S-STRONG}
This section is devoted to the analysis of global strong solutions to the nonhomogeneous Navier-Stokes-Allen-Cahn system \eqref{NSAC-D}-\eqref{IC-D} in two dimensions. The main results are as follows.
\begin{theorem}[Global strong solution in 2D]
\label{strong-D}
Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^2$.
Assume that $\textbf{\textit{u}}_0 \in {\mathbf{V}}_\sigma$, $\phi_0 \in H^2(\Omega)$ such that $\partial_{\textbf{\textit{n}}} \phi_0=0$ on $\partial \Omega$, $F'(\phi_0)\in L^2(\Omega)$, $\|\phi_0 \|_{L^\infty(\Omega)}\leq 1$ and $|\overline{\phi}_0|<1$.
\smallskip
\begin{itemize}
\item[(1)]
There exists a global strong solution $(\textbf{\textit{u}},\phi)$ to problem \eqref{NSAC-D}-\eqref{IC-D} satisfying, for all $T>0$,
\begin{align*}
&\textbf{\textit{u}} \in L^\infty(0,T;{\mathbf{V}}_\sigma)\cap L^2(0,T;\mathbf{H}^2(\Omega))\cap H^1(0,T;\mathbf{H}_\sigma),\\
&\phi \in L^\infty(0,T;H^2(\Omega))\cap L^2(0,T;W^{2,p}(\Omega)), \\
&\partial_t \phi \in L^\infty(0,T;L^2(\Omega))\cap L^2(0,T;H^1(\Omega)),\\
&F'(\phi) \in L^\infty(0,T;L^2(\Omega))\cap L^2(0,T;L^p(\Omega))
\end{align*}
where $p \in (2,\infty)$. The strong solution satisfies the system \eqref{NSAC-D} almost everywhere in $\Omega \times (0,\infty)$.
Besides, $|\phi(x,t)|<1$ for almost any $(x,t)\in \Omega\times(0,\infty)$ and $\partial_{\textbf{\textit{n}}} \phi=0$ on $\partial \Omega\times(0,\infty)$.
\smallskip
\item[(2)] There exists $\eta_1>0$ depending only on the norms of the initial data and on the parameters of the system:
$$
\eta_1=\eta_1(E(\textbf{\textit{u}}_0,\phi_0), \| \textbf{\textit{u}}_0\|_{{\mathbf{V}}_\sigma}, \| \phi_0\|_{H^2(\Omega)},\|F'(\phi_0)\|_{L^2(\Omega)},\theta,\theta_0).
$$
If, in addition, $\|\rho'\|_{L^\infty(-1,1)}\leq \eta_1$ and $F''(\phi_0)\in L^1(\Omega)$, then, for any $T>0$, we have
\begin{align}
F''(\phi)\in L^\infty(0,T;L^1(\Omega)),\quad F'' \in L^q(0,T;L^p(\Omega)),
\end{align}
where $\frac{1}{p}+\frac{1}{q}=1$, $p \in (1,\infty)$, and
\begin{align}
\label{F''log}
(F''(\phi))^2 \log (1+ F''(\phi)) \in L^1(\Omega\times(0,T)).
\end{align}
In particular, the strong solution satisfying \eqref{F''log} is unique.
\end{itemize}
\end{theorem}
\smallskip
\begin{theorem}[Propagation of regularity in 2D]
\label{Proreg-D}
Let the assumptions in Theorem \ref{strong-D}-(1) be satisfied.
Assume that $\| \rho'\|_{L^\infty(-1,1)}\leq \eta_1$. Given a strong solution from Theorem \ref{strong-D}-(1), for any $\sigma>0$, there holds
$$
(F''(\phi))^2 \log (1+ F''(\phi)) \in L^1(\Omega\times(\sigma,T)),
$$
and
$$
\partial_t\textbf{\textit{u}}\in L^\infty(\sigma, T; \mathbf{H}_\sigma)\cap L^2(\sigma, T; {\mathbf{V}}_\sigma),\quad \partial_t \phi\in L^\infty(\sigma, T; H^1(\Omega))\cap L^2(\sigma, T; H^2(\Omega)).
$$
In addition, for any $\sigma>0$, there exists $\delta=\delta(\sigma)>0$ such that
$$
-1+\delta \leq \phi(x,t) \leq 1-\delta, \quad \forall \, x \in \overline{\Omega}, \ t \geq \sigma.
$$
\end{theorem}
\begin{remark}
The smallness assumption on $\rho'$ (see \eqref{Hyp} below for the explicit form) can be reformulated in terms of the difference of the (constant) densities of the two fluids when $\rho$ is a linear interpolation function. In this case, we have
$$
\rho(s)= \rho_1\frac{1+s}{2}+ \rho_2 \frac{1-s}{2}, \quad \rho'(s)= \frac{\rho_1-\rho_2}{2} \quad \forall \, s \in [-1,1].
$$
Roughly speaking, the results given by Theorem \ref{strong-D} and Theorem \ref{Proreg-D} imply that uniqueness and further regularity of strong solutions to the nonhomogeneous system hold provided that the two fluids have similar densities ($\rho_1 \approx \rho_2$).
\end{remark}
\begin{remark}[Matched densities]
\label{strong-hom}
It is worth noticing that Theorem \ref{strong-D} and Theorem \ref{Proreg-D} hold true in the case of constant density $\rho\equiv 1$ (i.e. $\rho_1=\rho_2$) without any smallness assumption.
\end{remark}
\subsection{Proof of Theorem \ref{strong-D}}
We perform higher-order \textit{a priori} estimates that are necessary for the existence of global strong solutions.\smallskip
\textbf{Higher-order estimates.}
Multiplying \eqref{NSAC-D}$_1$ by $\partial_t \textbf{\textit{u}}$, integrating over $\Omega$, and observing that
$$
\int_{\Omega} \nu( \phi)D\textbf{\textit{u}} \cdot D \partial_t \textbf{\textit{u}} \, {\rm d} x= \frac12 \frac{\d}{\d t} \int_{\Omega} \nu(\phi) |D\textbf{\textit{u}}|^2 \, {\rm d} x - \frac12 \int_{\Omega} \nu'(\phi) \partial_t \phi |D \textbf{\textit{u}}|^2 \, {\rm d} x,
$$
we obtain
\begin{align}
\frac12 &\frac{\d}{\d t} \int_{\Omega} \nu(\phi) |D\textbf{\textit{u}}|^2 \, {\rm d} x
+ \int_{\Omega} \rho(\phi) |\partial_t \textbf{\textit{u}}|^2 \, {\rm d} x
\notag \\
&= - ( \rho(\phi) \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}, \partial_t \textbf{\textit{u}}) - \int_{\Omega}\Delta \phi \, \nabla \phi \cdot \partial_t \textbf{\textit{u}} \, {\rm d} x+ \frac12 \int_{\Omega} \nu'(\phi) \partial_t \phi |D \textbf{\textit{u}}|^2 \, {\rm d} x.
\label{NS1-D}
\end{align}
Next, differentiating \eqref{NSAC-D}$_3$ in time, multiplying the resultant by $\partial_t \phi$ and integrating over $\Omega$, we obtain
\begin{align}
\frac12 &\frac{\d}{\d t} \| \partial_t \phi\|_{L^2(\Omega)}^2+ \int_{\Omega} \partial_t \textbf{\textit{u}} \cdot \nabla \phi \, \partial_t \phi \, {\rm d} x + \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2
+\int_{\Omega} F''(\phi) |\partial_t \phi|^2 \, {\rm d} x \notag \\
&= \theta_0 \| \partial_t \phi\|_{L^2(\Omega)}^2
- \int_{\Omega} \rho''(\phi) |\partial_t \phi|^2 \frac{|\textbf{\textit{u}}|^2}{2}\, {\rm d} x
- \int_{\Omega} \rho'(\phi) \textbf{\textit{u}} \cdot \partial_t \textbf{\textit{u}} \, \partial_t \phi \, {\rm d} x + \partial_t \xi \int_{\Omega} \partial_t \phi \, {\rm d} x.
\label{AC1-D}
\end{align}
Since $\overline{\partial_t \phi} =0$,
by adding the equations \eqref{NS1-D} and \eqref{AC1-D}, we find that
\begin{align}
&\frac{\d}{\d t} H(t) +\rho_\ast \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2 + \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2+
\int_{\Omega} F''(\phi)|\partial_t \phi|^2 \, {\rm d} x \notag \\
&\quad \leq - (\rho(\phi) \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}, \partial_t \textbf{\textit{u}}) - \int_{\Omega}\Delta \phi \, \nabla \phi \cdot \partial_t \textbf{\textit{u}} \, {\rm d} x + \frac12 \int_{\Omega} \nu'(\phi) \partial_t \phi |D \textbf{\textit{u}}|^2 \, {\rm d} x+ \theta_0 \| \partial_t \phi\|_{L^2(\Omega)}^2\notag \\
&\qquad - \int_{\Omega} \partial_t \textbf{\textit{u}} \cdot \nabla \phi \, \partial_t \phi \, {\rm d} x - \int_{\Omega} \rho''(\phi) |\partial_t \phi|^2 \frac{|\textbf{\textit{u}}|^2}{2}\, {\rm d} x
- \int_{\Omega} \rho'(\phi) \textbf{\textit{u}} \cdot \partial_t \textbf{\textit{u}} \, \partial_t \phi \, {\rm d} x,
\label{NSAC1-D}
\end{align}
where
\begin{equation}
\label{H-D}
H(t)= \frac12 \int_{\Omega} \nu(\phi) |D\textbf{\textit{u}}|^2 \, {\rm d} x + \frac12 \| \partial_t \phi\|_{L^2(\Omega)}^2.
\end{equation}
In \eqref{NSAC1-D}, we have used that $\rho$ is strictly positive ($\rho(s)\geq \rho_\ast>0$). In addition, we simply infer from \eqref{NSAC-D} that
$$
\|\partial_t \phi\|_{L^2(\Omega)}\leq C\big( 1+ \| \textbf{\textit{u}}\|_{H^1(\Omega)}\big) \| \phi\|_{H^2(\Omega)}+C \| F'(\phi)\|_{L^2(\Omega)}+ C \| \textbf{\textit{u}}\|_{H^1(\Omega)}^2.
$$
Therefore, it follows from the assumptions on the initial data that $H(0)<+\infty$.
We proceed to estimate the right-hand side of \eqref{NSAC1-D}.
By using \eqref{KORN} and \eqref{BGW}, we have
\begin{align*}
-( \rho(\phi) \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}, \partial_t \textbf{\textit{u}})
&\leq\| \rho(\phi)\|_{L^\infty(\Omega)} \| \textbf{\textit{u}}\|_{L^\infty(\Omega)} \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)} \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}\\
&\leq C \| D\textbf{\textit{u}}\|_{L^2(\Omega)}^2 \log^\frac12 \Big( C \frac{\| \textbf{\textit{u}}\|_{W^{1,p}(\Omega)}}{\| D \textbf{\textit{u}}\|_{L^2(\Omega)}}\Big) \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}\\
&\leq \frac{\rho_\ast}{8} \|\partial_t \textbf{\textit{u}} \|_{L^2(\Omega)}^2+ C \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^4 \log \Big( C \frac{\| \textbf{\textit{u}}\|_{W^{1,p}(\Omega)}}{\| D \textbf{\textit{u}}\|_{L^2(\Omega)}}\Big),
\end{align*}
for some $p>2$. Moreover, it holds
\begin{align*}
- \int_{\Omega}\Delta \phi \, \nabla \phi \cdot \partial_t \textbf{\textit{u}} \, {\rm d} x
&\leq \| \Delta \phi \|_{L^2(\Omega)} \| \nabla \phi\|_{L^\infty(\Omega)}
\| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}\\
&\leq C \|\Delta \phi \|_{L^2(\Omega)} \| \nabla \phi\|_{H^1(\Omega)} \log^\frac12 \Big( C \frac{\| \nabla \phi\|_{W^{1,p}(\Omega)}}{\| \nabla \phi\|_{H^1(\Omega)}}\Big) \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}\\
&\leq \frac{\rho_\ast}{8} \|\partial_t \textbf{\textit{u}} \|_{L^2(\Omega)}^2+ C \|\Delta \phi \|_{L^2(\Omega)}^2 \| \nabla \phi\|_{H^1(\Omega)}^2\log \Big( C \frac{\| \nabla \phi\|_{W^{1,p}(\Omega)}}{\| \nabla \phi\|_{H^1(\Omega)}}\Big).
\end{align*}
Next, by exploiting Lemma \ref{result1}, together with \eqref{normH1-2} and $\overline{\partial_t \phi}=0$, we obtain
\begin{align*}
\frac12 \int_{\Omega} \nu'(\phi) \partial_t \phi |D \textbf{\textit{u}}|^2 \, {\rm d} x
&\leq \| \nu'(\phi)\|_{L^\infty(\Omega)} \| \partial_t \phi |D \textbf{\textit{u}}|\|_{L^2(\Omega)} \| D \textbf{\textit{u}}\|_{L^2(\Omega)}\\
& \leq C \| \nabla \partial_t \phi\|_{L^2(\Omega)} \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2 \log^\frac12 \Big( C \frac{\|D \textbf{\textit{u}}\|_{L^p(\Omega)}}{\| D \textbf{\textit{u}}\|_{L^2(\Omega)}}\Big)\\
&\leq \frac18 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2+ C \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^4 \log \Big( C \frac{\|D \textbf{\textit{u}}\|_{L^p(\Omega)}}{\| D \textbf{\textit{u}}\|_{L^2(\Omega)}}\Big).
\end{align*}
It remains to control the last three terms on the right-hand side of \eqref{NSAC1-D}. By using \eqref{LADY} and \eqref{B1-D}, we obtain
\begin{align*}
- \int_{\Omega} \partial_t \textbf{\textit{u}} \cdot \nabla \phi \, \partial_t \phi \, {\rm d} x
&\leq \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)} \| \nabla \phi\|_{L^4(\Omega)} \| \partial_t \phi\|_{L^4(\Omega)}\\
&\leq \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)} \| \nabla \phi\|_{L^2(\Omega)}^\frac12 \| \phi\|_{H^2(\Omega)}^\frac12 \| \partial_t \phi\|_{L^2(\Omega)}^\frac12 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^\frac12 \\
&\leq \frac{\rho_\ast}{8} \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2
+\frac18 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2+ C \| \phi\|_{H^2(\Omega)}^2
\| \partial_t \phi\|_{L^2(\Omega)}^2,
\end{align*}
\begin{align*}
- \int_{\Omega} \rho''(\phi) |\partial_t \phi|^2 \frac{|\textbf{\textit{u}}|^2}{2}\, {\rm d} x
&\leq C \| \rho''(\phi)\|_{L^\infty(\Omega)} \| \partial_t \phi\|_{L^4(\Omega)}^2
\|\textbf{\textit{u}}\|_{L^4(\Omega)}^2\\
&\leq C \|\partial_t \phi \|_{L^2(\Omega)} \| \nabla \partial_t \phi\|_{L^2(\Omega)} \| \textbf{\textit{u}}\|_{L^2(\Omega)}\| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}\\
&\leq \frac18 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2
+C \|\partial_t \phi \|_{L^2(\Omega)}^2\| D\textbf{\textit{u}}\|_{L^2(\Omega)}^2,
\end{align*}
and
\begin{align*}
- \int_{\Omega} \rho'(\phi) \textbf{\textit{u}} \cdot \partial_t \textbf{\textit{u}} \, \partial_t \phi \, {\rm d} x
&\leq C \| \rho'(\phi)\|_{L^\infty(\Omega)}
\| \textbf{\textit{u}}\|_{L^4(\Omega)}\| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)} \| \partial_t \phi\|_{L^4(\Omega)}\\
&\leq C \| \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac12 \|\nabla \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac12
\| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)} \| \partial_t \phi\|_{L^2(\Omega)}^\frac12 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^\frac12\\
&\leq \frac{\rho_\ast}{8} \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2+ \frac18 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2
+C \| \partial_t \phi\|_{L^2(\Omega)}^2\| D\textbf{\textit{u}}\|_{L^2(\Omega)}^2.
\end{align*}
Combining \eqref{NSAC1-D} and the above inequalities, we deduce that
\begin{align*}
& \frac{\d}{\d t} H(t) +\frac{\rho_\ast}{2} \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2 + \frac12 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2 \\
&\quad \leq C \| \partial_t \phi\|_{L^2(\Omega)}^2
+C \big(\| D\textbf{\textit{u}}\|_{L^2(\Omega)}^2+\|\phi \|_{H^2(\Omega)}^2\big)
\|\partial_t \phi \|_{L^2(\Omega)}^2
+ C\| D \textbf{\textit{u}}\|_{L^2(\Omega)}^4 \log \Big( C \frac{\| \textbf{\textit{u}}\|_{W^{1,p}(\Omega)}}{\|D \textbf{\textit{u}}\|_{L^2(\Omega)}}\Big)\\
&\qquad + C \| \Delta \phi\|_{L^2(\Omega)}^2 \| \nabla \phi\|_{H^1(\Omega)}^2 \log \Big( C \frac{\| \nabla \phi\|_{W^{1,p}(\Omega)}}{\| \nabla \phi\|_{H^1(\Omega)}}\Big).
\end{align*}
From the inequalities
\begin{align}
\label{ineq0}
&x^2\log \Big(\frac{C y}{x}\Big)\leq x^2 \log (Cy) +1,\quad \forall\,x,\,y>0,\\
& \frac{\nu_\ast}{2} \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2 +\frac12 \| \partial_t \phi\|_{L^2(\Omega)}^2
\leq H(t)\leq C \Big( \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2 +\| \partial_t \phi\|_{L^2(\Omega)}^2\Big),
\label{Hbb}
\end{align}
and the estimate \eqref{estH2-D}, we can rewrite the above differential inequality as follows
\begin{align}
& \frac{\d}{\d t} H(t) +\frac{\rho_\ast}{2} \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2 + \frac12 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2 \notag \\
&\quad \leq C\big(1+ H(t) + H^2(t)\big)
+C H^2(t) \log \big( C \| \textbf{\textit{u}}\|_{W^{1,p}(\Omega)}\big) \nonumber \\
&\qquad + C \big( 1+H^2(t) \big)
\log \big( C \| \phi\|_{W^{2,p}(\Omega)}\big).
\label{NSAC2-D}
\end{align}
Let us now estimate the argument of the logarithmic terms on the right-hand side of
\eqref{NSAC2-D}. First, we rewrite \eqref{NSAC-D}$_1$ as a Stokes problem with non-constant viscosity
$$
\begin{cases}
-\mathrm{div}\, (\nu(\phi)D \textbf{\textit{u}})+\nabla P= \textbf{\textit{f}}, & \text{ in } \Omega\times (0,T),\\
\mathrm{div}\, \textbf{\textit{u}}=0, & \text{ in } \Omega\times (0,T),\\
\textbf{\textit{u}}=\mathbf{0}, & \text{ on } \partial \Omega\times (0,T),
\end{cases}
$$
where $\textbf{\textit{f}}= -\rho(\phi) \big( \partial_t \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}} \big) - \Delta \phi \nabla \phi$.
We now apply Theorem \ref{Stokes-e} with the following choice of parameters $p=1+\varepsilon$, $\varepsilon \in (0,1)$, and $r\in (2,\infty)$ such that $\frac{1}{r}=\frac{1}{1+\varepsilon}-\frac12$. We infer that
\begin{align*}
\| \textbf{\textit{u}}\|_{W^{2,1+\varepsilon}(\Omega)}
&\leq C \big( \| \partial_t \textbf{\textit{u}}\|_{L^{1+\varepsilon}(\Omega)}+ \| \textbf{\textit{u}}\cdot \nabla \textbf{\textit{u}}\|_{L^{1+\varepsilon}(\Omega)}
+\|\Delta \phi \nabla \phi\|_{L^{1+\varepsilon}(\Omega)} \big)+
C \| D \textbf{\textit{u}}\|_{L^2(\Omega)} \| \nabla \phi\|_{L^r(\Omega)}\\
&\leq C \big( \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}+ \| \textbf{\textit{u}}\|_{L^\frac{2(1+\varepsilon)}{1-\varepsilon}(\Omega)}\|\nabla \textbf{\textit{u}} \|_{L^2(\Omega)}+
\| \nabla \phi\|_{L^\frac{2(1+\varepsilon)}{1-\varepsilon}(\Omega)} \| \Delta \phi\|_{L^2(\Omega)} \big)\\
&\quad + C \| D \textbf{\textit{u}}\|_{L^2(\Omega)} \| \phi\|_{H^2(\Omega)}\\
&\leq C \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}+ C \|D \textbf{\textit{u}} \|_{L^2(\Omega)}^2 +
\| \phi\|_{H^2(\Omega)}^2\\
&\leq C \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}+ C(1+H(t)),
\end{align*}
where the constant $C$ depends on $\varepsilon$.
We recall the Sobolev embedding $W^{2,1+\varepsilon}(\Omega)\hookrightarrow W^{1,p}(\Omega)$ where $\frac{1}{p}=\frac{1}{1+\varepsilon}- \frac12$. Therefore, for any $p\in (2,\infty)$ there exists a constant $C$ depending on $p$ such that
\begin{align}
\|\textbf{\textit{u}}\|_{W^{1,p}(\Omega)}\leq C \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)} + C(1+H(t)).
\label{est-uw1p}
\end{align}
Next, by reformulating the equation \eqref{NSAC-D}$_4$ as the elliptic problem
\begin{equation}
\begin{cases}
-\Delta \phi+ F'(\phi)=\mu+\theta_0\phi, &\quad \text{in}\ \Omega\times(0,T),\\
\partial_\textbf{\textit{n}} \phi=0, &\quad \text{on}\ \partial \Omega\times (0,T).
\end{cases}
\end{equation}
We infer from the elliptic regularity theory (see, e.g., \cite[Lemma 2]{A2009} and \cite{GMT2019}) that
\begin{align}
\| \phi\|_{W^{2,p}(\Omega)} +\|F'(\phi)\|_{L^p(\Omega)}
&\leq C (1+\|\phi\|_{L^2(\Omega)}+\|\mu+\theta_0\phi\|_{L^p(\Omega)})\nonumber\\
&\leq C (1+\|\phi\|_{L^p(\Omega)}+\|\mu\|_{L^p(\Omega)}),
\label{pw2p}
\end{align}
for $p\in (2,\infty)$. On the other hand, from the equation \eqref{NSAC-D}$_3$, we see that
$$
\mu= -\partial_t \phi-\textbf{\textit{u}}\cdot \nabla \phi- \rho'(\phi) \frac{|\textbf{\textit{u}}|^2}{2}+\displaystyle{\overline{\mu+ \rho'(\phi)\frac{|\textbf{\textit{u}}|^2}{2}}}.
$$
Observe now that
$$
\Big\| -\rho'(\phi) \frac{|\textbf{\textit{u}}|^2}{2}\Big\|_{L^p(\Omega)}\leq C \| \textbf{\textit{u}}\|_{L^{2p}(\Omega)}^2
\leq C \|\nabla \textbf{\textit{u}} \|_{L^2(\Omega)}^2.
$$
Then, owing to Sobolev embedding and \eqref{normH1-2}, we have
\begin{align*}
\|\mu-\overline{\mu} \|_{L^p(\Omega)}
&\leq \| \partial_t \phi\|_{L^p(\Omega)} + \| \textbf{\textit{u}} \cdot \nabla \phi\|_{L^p(\Omega)}+\left\| \rho'(\phi) \frac{|\textbf{\textit{u}}|^2}{2} -
\displaystyle{\overline{\rho'(\phi)\frac{|\textbf{\textit{u}}|^2}{2}}}\right\|_{L^p(\Omega)}\\
&\leq C \| \nabla \partial_t \phi\|_{L^2(\Omega)}
+C \| \textbf{\textit{u}}\|_{H^1(\Omega)} \| \phi\|_{H^2(\Omega)}+C \|\nabla \textbf{\textit{u}} \|_{L^2(\Omega)}^2.
\end{align*}
In light of \eqref{mu-L2-2} and \eqref{mubar}, the above inequality yields
\begin{align}
\| \mu\|_{L^p(\Omega)}
&\leq C \|\mu-\overline{\mu} \|_{L^p(\Omega)} +C |\overline{\mu}| \notag\\
&\leq C \|\mu-\overline{\mu} \|_{L^p(\Omega)}+ C(1+\| \mu-\overline{\mu} \notag \|_{L^2(\Omega)})\\
&\leq C (1+\| \nabla \partial_t \phi\|_{L^2(\Omega)}
+H(t)).
\label{mu-Lp}
\end{align}
Thus, for any $p>2$, we deduce from the above estimate and \eqref{pw2p} that
\begin{equation}
\label{estW2p-D}
\| \phi\|_{W^{2,p}(\Omega)} \leq C(1+\| \nabla \partial_t \phi\|_{L^2(\Omega)} +H(t)),
\end{equation}
for some positive constant $C$ depending on $p$.
We recall the generalized Young inequality
\begin{align}
xy \leq \Phi(x)+ \Upsilon(y), \quad \forall \, x,\,y >0,
\label{Young0}
\end{align}
where
$$
\Phi(s)= s \log s -s+1, \quad \Upsilon(s)= {e}^{s}-1.
$$
Then we have
\begin{align*}
H(t)\log(1+\| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)})
&\leq H(t)\log H(t)+ 1 + \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}.
\end{align*}
Thus, using the above estimate and the elementary inequality
$$ \log(x+y)<\log(1+x)+\log(1+y),\quad x,y>0,$$
we can estimate the second term on the right-hand side of \eqref{NSAC2-D} as follows
\begin{align}
&CH^2(t)\log(C\|\textbf{\textit{u}}\|_{W^{1,p}(\Omega)})\nonumber\\
&\quad \leq CH^2(t)\log \big(C \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)} + C(1+H(t)) \big)\nonumber\\
&\quad \leq CH^2(t)\big(1+\log(1+\| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)})+\log(1+H(t))\big)\nonumber\\
&\quad \leq CH^2(t)+ CH(t)\big(H(t)\log H(t)+ 1\big) + CH(t)\| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}+ CH^2(t)\log(1+H(t))\nonumber\\
&\quad \leq \frac{\rho_\ast}{4} \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2+ C\big(1+H^2(t)\big)+ CH^2(t)\log(e+H(t)).
\label{RHD3-D}
\end{align}
In a similar manner, we have
\begin{align*}
H(t)\log(1+\| \nabla \partial_t \phi\|_{L^2(\Omega)})
&\leq H(t)\log H(t)+ 1 + \| \nabla \partial_t \phi\|_{L^2(\Omega)}.
\end{align*}
Then, using \eqref{estW2p-D}, the third term on the right-hand side of \eqref{NSAC2-D} can be estimated as follows
\begin{align}
&C \big( 1+H^2(t) \big)
\log \big( C \| \phi\|_{W^{2,p}(\Omega)}\big)\nonumber\\
&\quad \leq C \big( 1+H^2(t) \big) \log \big(C(1+\| \nabla \partial_t \phi\|_{L^2(\Omega)} +H(t)) \big) \nonumber\\
&\quad \leq C\big(1+H^2(t)\big) +C\log\big(1+\|\nabla \partial_t \phi\|_{L^2(\Omega)}+H(t)\big)
+H^2(t)\log\big(1+\| \nabla \partial_t \phi\|_{L^2(\Omega)}+ H(t)\big)\nonumber\\
&\quad \leq C \big( 1+H^2(t) \big) +C \big( 1+\| \nabla \partial_t \phi\|_{L^2(\Omega)}+H(t) \big)
+ C\| \nabla \partial_t \phi\|_{L^2(\Omega)} H(t)\nonumber\\
&\qquad
+ H^2(t) \log(1+H(t))\nonumber\\
&\quad \leq \frac{1}{8}\| \nabla \partial_t \phi\|_{L^2(\Omega)}^2+ C\big(1+H^2(t)\big)+ C H(t)\big( e+H(t) \big)\log (e+H(t)).
\label{RHD4-D}
\end{align}
Hence, by \eqref{RHD3-D} and \eqref{RHD4-D}, we easily deduce from \eqref{NSAC2-D} that
\begin{align}
\frac{\d}{\d t} (e+H(t)) &+\frac{\rho_\ast}{4} \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2 + \frac14 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2\leq C+CH(t) (e+H(t)) \log (e+H(t)).
\label{NSAC3-D}
\end{align}
Thanks to \eqref{E-bound}, \eqref{estH2-D}, and \eqref{Hbb}, we obtain
\begin{equation}
\label{intH}
\int_t^{t+1} H(\tau) \, {\rm d} \tau \leq Q(E_0), \quad \forall \, t \geq 0,
\end{equation}
where $Q$ is independent of $t$, and $E_0=E(\textbf{\textit{u}}_0,\phi_0)$.
We now apply the generalized Gronwall lemma \ref{GL2} to \eqref{NSAC3-D} and find the estimate
$$
\sup_{t \in [0,1]} H(t)\leq C \big(e+H(0)\big)^{{e}^{Q(E_0)}}.
$$
Moreover, by using the generalized uniform Gronwall lemma \ref{UGL2} together with \eqref{intH}, we infer that
$$
\sup_{t\geq 1} H(t)\leq C e^{(e+Q(E_0) ) e^{(1+Q(E_0))}}.
$$
By combining the above inequalities, we get
\begin{align}
\sup_{t \geq 0} H(t)\leq Q(E_0, \| \textbf{\textit{u}}_0\|_{{\mathbf{V}}_\sigma}, \| \phi_0\|_{H^2(\Omega)}, \| F'(\phi_0)\|_{L^2(\Omega)}).
\label{NSAC3h-D}
\end{align}
In addition, integrating \eqref{NSAC3-D} on the time interval $[t,t+1]$, we have, for all $t\geq 0$,
\begin{align}
\int_t^{t+1} \| \partial_t \textbf{\textit{u}}(\tau)\|_{L^2(\Omega)}^2 + \| \nabla \partial_t \phi(\tau)\|_{L^2(\Omega)}^2\, {\rm d} \tau \leq Q(E_0, \| \textbf{\textit{u}}_0\|_{{\mathbf{V}}_\sigma}, \| \phi_0\|_{H^2(\Omega)}, \| F'(\phi_0)\|_{L^2(\Omega)}).
\label{NSAC4}
\end{align}
Then we can deduce that
\begin{equation}
\label{str-1}
\textbf{\textit{u}} \in L^\infty(0,T; {\mathbf{V}}_\sigma)\cap H^1(0,T; \mathbf{H}_\sigma) \quad
\partial_t \phi \in L^\infty(0,T; L^2(\Omega))\cap L^2(0,T;H^1(\Omega)).
\end{equation}
Thanks to \eqref{estH2-D} and \eqref{estW2p-D}, we also get,
\begin{equation}
\label{str-2}
\sup_{t\geq 0} \| \phi(t)\|_{H^2(\Omega))}
\leq Q(E_0, \| \textbf{\textit{u}}_0\|_{{\mathbf{V}}_\sigma}, \| \phi_0\|_{H^2(\Omega)}, \| F'(\phi_0)\|_{L^2(\Omega)}),
\end{equation}
and, for all $t\geq 0$,
\begin{equation}
\label{str-2'}
\int_t^{t+1} \| \phi(\tau)\|_{W^{2,p}(\Omega))}^2 \, {\rm d} \tau \leq Q(E_0, \| \textbf{\textit{u}}_0\|_{{\mathbf{V}}_\sigma}, \| \phi_0\|_{H^2(\Omega)}, \| F'(\phi_0)\|_{L^2(\Omega)}),
\end{equation}
for any $p \in (2,\infty)$.
This entails that $\phi \in L^\infty(0,T;H^2(\Omega))\cap L^2(0,T;W^{2,p}(\Omega))$.
According to \eqref{mu-L2-2}, \eqref{mubar} and \eqref{mu-Lp}, it follows that $\mu\in L^\infty(0,T;L^2(\Omega))\cap L^2(0,T;L^{p}(\Omega))$
and, as a consequence,
$$
F'(\phi) \in L^\infty(0,T;L^2(\Omega))\cap L^2(0,T;L^p(\Omega)).
$$
Finally, by exploiting Theorem \ref{Stokes-e} with $p=2$ and $r=\infty$, together with the regularity of $\phi$ obtained above, we have, for all $t\geq 0$,
\begin{equation}
\label{str-3}
\int_t^{t+1} \|\textbf{\textit{u}}(\tau)\|_{H^2(\Omega)}^2 \, {\rm d} \tau \leq Q(E_0, \| \textbf{\textit{u}}_0\|_{{\mathbf{V}}_\sigma}, \| \phi_0\|_{H^2(\Omega)}, \| F'(\phi_0)\|_{L^2(\Omega)}),
\end{equation}
which yields that $\textbf{\textit{u}} \in L^2(0,T;\mathbf{H}^2(\Omega))$.
\smallskip
\textbf{Entropy bound in $L^\infty(0,T;L^1(\Omega))$.}
First of all, we observe that, for all $s\in (-1,1)$,
\begin{equation}
\label{Fder1}
F'(s)= \frac{\theta}{2} \log \Big( \frac{1+s}{1-s}\Big), \quad F''(s)= \frac{\theta}{1-s^2}, \quad F'''(s)= \frac{2\theta s}{(1-s)^2(1+s)^2}
\end{equation}
and
\begin{equation}
\label{Fder2}
F^{(4)}(s)= \frac{2 \theta(1+3s^2)}{(1-s)^3(1+s)^3}>0.
\end{equation}
Next, we compute
\begin{align*}
\frac{\d}{\d t} \int_{\Omega} F''(\phi) \, {\rm d} x&= \int_{\Omega} F'''(\phi) \partial_t \phi \, {\rm d} x\\
&=\int_{\Omega} F'''(\phi) \Big( \Delta \phi - \textbf{\textit{u}} \cdot \nabla \phi -F'(\phi)+ \theta_0 \phi - \rho'(\phi) \frac{|\textbf{\textit{u}}|^2}{2}+ \xi\Big) \, {\rm d} x.
\end{align*}
Since
$$
\int_{\Omega} F'''(\phi) \textbf{\textit{u}} \cdot \nabla \phi \, {\rm d} x=
\int_{\Omega} \textbf{\textit{u}} \cdot \nabla ( F''(\phi)) \, {\rm d} x=0,
$$
and exploiting the integration by parts, we rewrite the above equality as follows
\begin{align}
&
\frac{\d}{\d t} \int_{\Omega} F''(\phi) \, {\rm d} x + \int_{\Omega} F^{(4)}(\phi) |\nabla \phi|^2 \, {\rm d} x + \int_{\Omega} F'''(\phi) F'(\phi) \, {\rm d} x\nonumber\\
&\quad = \int_{\Omega} F'''(\phi) \Big( \theta_0 \phi - \rho'(\phi) \frac{|\textbf{\textit{u}}|^2}{2}+\xi\Big) \, {\rm d} x.
\label{EntE-}
\end{align}
In particular, by using \eqref{Fder2}, we have
\begin{equation}
\label{EntE}
\frac{\d}{\d t} \int_{\Omega} F''(\phi) \, {\rm d} x + \int_{\Omega} F'''(\phi) F'(\phi) \, {\rm d} x
\leq \int_{\Omega} F'''(\phi) \Big( \theta_0 \phi - \rho'(\phi) \frac{|\textbf{\textit{u}}|^2}{2} + \xi\Big) \, {\rm d} x.
\end{equation}
It follows from \eqref{Young0} that
\begin{equation}
\label{Young}
xy \leq \varepsilon x \log x + {e}^{\frac{y}{\varepsilon}},\quad \forall \, x>0, y>0,\, \varepsilon \in (0,1).
\end{equation}
which implies
\begin{align}
\int_{\Omega} -F'''(\phi) \rho'(\phi) \frac{|\textbf{\textit{u}}|^2}{2} \, {\rm d} x&\leq
\int_{\Omega} |F'''(\phi)| |\rho'(\phi)| \frac{|\textbf{\textit{u}}|^2}{2} \, {\rm d} x \notag \\
& \leq \varepsilon \int_{\Omega} |F'''(\phi)| \log ( |F'''(\phi)| ) \, {\rm d} x+
\int_{\Omega} {e}^{\frac{|\rho'(\phi)|}{\varepsilon} \frac{|\textbf{\textit{u}}|^2}{2}}\, {\rm d} x.
\label{EE1}
\end{align}
We observe that, for all $s\in [0,1)$, it holds
\begin{align*}
\log (|F'''(s)|) &= \log( F'''(s))= \log \Big( \frac{2\theta s}{(1-s)^2(1+s)^2}\Big)\notag \\
&= 2 \log \Big( \frac{1+s}{1-s} \frac{\sqrt{2\theta s}}{(1+s)^2}\Big) \leq 2 \log \Big( \sqrt{2\theta} \frac{1+s}{1-s}\Big)= \log(2\theta) + \frac{4}{\theta} F'(s).
\end{align*}
Since both $F'(s)$ and $F'''(s)$ are odd, we easily deduce that
$$
\log (|F'''(s)|) \leq C_0+ \frac{4}{\theta} |F'(s)|, \quad \forall \, s \in (-1,1),
$$
where $C_0=\log(2\theta)$ (without loss of generality, we assume in the sequel that $C_0>0$).
Then, using the fact that $F'''(s)F'(s)\geq 0$ for all $s\in (-1,1)$, we obtain
\begin{align*}
|F'''(s)|\log(|F'''(s)|)\leq C_0|F'''(s)|+ \frac{4}{\theta} F'''(s) F'(s), \quad \forall \, s \in (-1,1).
\end{align*}
Fix the constant $\alpha \in (0,1)$ such that $F'(\alpha)=1$. We infer that
\begin{equation}
\label{estF'''}
|F'''(s)|\log(|F'''(s)|)\leq C_1+ C_2F'''(s) F'(s), \quad \forall \, s \in (-1,1).
\end{equation}
where
$$
C_1= C_0F'''(\alpha), \quad C_2=\frac{4}{\theta} +C_0.
$$
Taking $\varepsilon=\frac{1}{2C_2}$ in \eqref{EE1}, we arrive at
\begin{align}
\int_{\Omega} -F'''(\phi) \rho'(\phi) \frac{|\textbf{\textit{u}}|^2}{2} \, {\rm d} x
& \leq \frac{C_1 |\Omega|}{2C_2} + \frac12 \int_{\Omega} F'''(\phi) F'(\phi) \, {\rm d} x+
\int_{\Omega} {e}^{C_2 |\rho'(\phi)| |\textbf{\textit{u}}|^2}\, {\rm d} x.
\label{EntE2}
\end{align}
Arguing in a similar way ($\varepsilon= \frac{1}{4C_2}$), we obtain
$$
\int_{\Omega} F'''(\phi) \, (\theta_0 \phi+ \xi) \, {\rm d} x \leq \frac{C_1 |\Omega|}{4C_2} + \frac14 \int_{\Omega} F'''(\phi) F'(\phi) \, {\rm d} x+
\int_{\Omega} {e}^{4C_2 |\theta_0\phi +\xi |}\, {\rm d} x.
$$
Since $\phi$ is globally bounded ($\|\phi \|_{L^\infty(\Omega \times (0,T))}\leq 1$) and $\|\xi\|_{L^\infty(0,T)}\leq C^\ast_2$, we get
\begin{equation}
\label{EntE3}
\int_{\Omega} F'''(\phi) \, (\theta_0+\xi) \phi \, {\rm d} x \leq \frac14 \int_{\Omega} F'''(\phi) F'(\phi) \, {\rm d} x+ \frac{C_1 |\Omega|}{4C_2} +
{e}^{4 C_2 (\theta_0+C_2^\ast)} |\Omega|.
\end{equation}
Combining \eqref{EntE} with \eqref{EntE2} and \eqref{EntE3}, we deduce that
\begin{align}
\frac{\d}{\d t} \int_{\Omega} F''(\phi) \, {\rm d} x &+ \frac14 \int_{\Omega} F'''(\phi) F'(\phi) \, {\rm d} x\notag \\
&\leq\frac{3C_1 |\Omega|}{4C_2} +
{e}^{4 C_2 (\theta_0+C_2^\ast)} |\Omega|+ \int_{\Omega} {e}^{ C_2 |\rho'(\phi)| \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}^2 \left(\frac{|\textbf{\textit{u}}|^2}{\|\nabla \textbf{\textit{u}}\|_{L^2(\Omega)}^2}\right)}\, {\rm d} x.
\label{EntE4}
\end{align}
In order to control the last term on the right-hand side of \eqref{EntE4}, we shall use the Trudinger-Moser inequality (see, e.g., \cite{Moser}). Namely, let $f\in H_0^1(\Omega)$ ($d=2$) such that
$\int_{\Omega} |\nabla f|^2 \,{\rm d} x\leq 1$. Then, there exists a constant $C_{TM}=C_{TM}(\Omega)$ (which depends only on the domain $\Omega$) such that
\begin{align}
\int_{\Omega} {e}^{4\pi |f|^2} \, {\rm d} x \leq C_{TM}(\Omega).\label{TrM}
\end{align}
Next, as a consequence of \eqref{NSAC3h-D}, we have the following uniform estimate
\begin{equation}
\sup_{t\geq 0 }\| \nabla \textbf{\textit{u}}(t)\|_{L^2(\Omega)}\leq Q(E(\textbf{\textit{u}}_0,\phi_0) ,H(0))=:R_0,
\end{equation}
where $R_0$ is independent of time. The exact value of $R_0$ can be estimated in terms of the norm of the initial conditions.
Now we make the following assumptions:
\begin{equation}
\label{Hyp} |\rho'(s)|_{L^\infty(-1,1)}\leq \frac{4 \pi}{C_2 R_0^2}.
\end{equation}
Thanks to \eqref{Hyp}, we conclude that
\begin{equation}
\label{EntE5}
\frac{\d}{\d t} \int_{\Omega} F''(\phi) \, {\rm d} x + \frac14 \int_{\Omega} F'''(\phi) F'(\phi) \, {\rm d} x
\leq\frac{3C_1 |\Omega|}{4C_2} +
{e}^{4 C_2 (\theta_0+C_2^\ast)} |\Omega|
+ C_{TM}(\Omega).
\end{equation}
Observe now that, for $s\in \big[\frac12,1)$,
\begin{align*}
F''(s)=\frac{\theta}{1-s^2}= \frac{(1-s)(1+s)}{2s} F'''(s)\leq \frac{3}{4F'(\frac12)} F'''(s)F'(s).
\end{align*}
This gives
\begin{equation}
\label{estF''}
F''(s)\leq C_3 + C_4 F'''(s)F'(s), \quad \forall \, s \in (-1,1),
\end{equation}
where
$$
C_3= F''\Big(\frac12\Big), \quad C_4=\frac{3}{4F'(\frac12)}.
$$
Hence, we are led to
$$
\frac{\d}{\d t} \int_{\Omega} F''(\phi) \, {\rm d} x + \frac{1}{4 C_4} \int_{\Omega} F''(\phi) \, {\rm d} x
\leq C_5,
$$
where
$$
C_5=\frac{3C_1 |\Omega|}{4C_2} +
{e}^{4 C_2 (\theta_0+C_2^\ast)} |\Omega|+ C_{TM}(\Omega)+ \frac{C_3|\Omega|}{4C_4}.
$$
We recall that $F''(\phi_0)\in L^1(\Omega)$. Then, an application of the Gronwall lemma entails that
\begin{equation}
\label{EB1}
\int_{\Omega} F''(\phi(t)) \, {\rm d} x \leq \| F''(\phi_0)\|_{L^1(\Omega)} {e}^{-\frac{t}{4C_4}} + 4 C_4 C_5, \quad \forall \, t \geq 0.
\end{equation}
In addition, integrating \eqref{EntE5} on the time interval $[t,t+1]$, we find
\begin{equation}
\label{EB2}
\int_t^{t+1}\! \int_{\Omega} F'''(\phi) F'(\phi) \,{\rm d} x {\rm d} \tau \leq 4 \| F''(\phi_0)\|_{L^1(\Omega)} + C_6, \quad \forall \, t \geq 0,
\end{equation}
where $$C_6= 4 C_5- \frac{C_3|\Omega|}{C_4}.$$ This allows us to improve the integrability of $F''(\phi)$. Indeed, arguing similarly to \eqref{estF''}, we have for $s\in \big[ \frac12 , 1)$
\begin{align*}
(F''(s))^2 \log(1+F''(s))&= \frac{\theta^2}{(1-s)^2(1+s)^2} \log \Big( 1+\frac{\theta}{1-s^2} \Big) \\
&\leq \theta F'''(s) \log \Big( \frac{1+s}{1-s} \frac{1-s^2 +\theta}{(1+s)^2}\Big)\\
&\leq 2F'''(s)F'(s) + \theta F'''(s) \log \Big( \frac12 + \frac{2\theta}{3} \Big) \\
&\leq C_7 F'''(s)F'(s).
\end{align*}
Hence, we infer that
$$
(F''(s))^2 \log(1+F''(s))\leq C_7 F'''(s)F'(s)+ C_8, \quad \forall \, s\in (-1,1).
$$
In light of \eqref{EB2}, we deduce \eqref{F''log}. Indeed, we have
\begin{equation}
\label{EB3}
\int_t^{t+1}\! \int_{\Omega} (F''(\phi))^2 \log ( 1+F''(\phi)) \,{\rm d} x {\rm d} \tau \leq
4 C_7 \| F''(\phi_0)\|_{L^1(\Omega)} + C_6 C_7 +C_8, \quad \forall \, t \geq 0.
\end{equation}
We notice that, by keeping the (non-negative) term $F^{(4)}(\phi)|\nabla \phi|^2$ (cf. \eqref{EntE-}) on the left-hand side of \eqref{EntE5} in the above argument, we can also deduce that
$$
\int_{t}^{t+1}\! \int_{\Omega} F^{(4)}(\phi)|\nabla \phi|^2 \, {\rm d} x {\rm d} \tau
\leq C_9, \quad \forall \, t \geq 0,
$$
where $C_9$ depends on $\|F''(\phi_0)\|_{L^1(\Omega)}$, $R_0$, $\theta$, $\theta_0$ and $\Omega$.
Since $ \left(\frac{s}{\sqrt{1-s^2}}\right)'=(1-s^2)^{-\frac32}$, we infer that
$$
\int_{t}^{t+1}\! \int_{\Omega} \Big| \nabla \Big( \frac{\phi}{\sqrt{1-\phi^2}}\Big)\Big|^2 \, {\rm d} x {\rm d} \tau \leq \frac{C_9}{2\theta}, \quad \forall \, t \geq 0.
$$
Setting $\psi= \frac{\phi}{\sqrt{1-\phi^2}}$, and observing that $F''(s)= \theta \Big[ \big( \frac{s}{\sqrt{1-s^2}}\big)^2 +1\Big]$, we have (cf. \eqref{EntE5})
$$
\| \psi(t)\|_{L^2(\Omega)}^2+ \int_t^{t+1}\! \| \nabla \psi(\tau)\|_{L^2(\Omega)}^2 \, {\rm d} \tau \leq C_{10}, \quad \forall \, t \geq 0.
$$
This implies that $\psi \in L^\infty(0,T;L^2(\Omega))\cap L^2(0,T;H^1(\Omega))$. By Sobolev embedding, we also have that $\psi \in L^q(0,T;L^p(\Omega))$ where $\frac12=\frac{1}{p}+\frac{1}{q}$, $p \in (2,\infty)$. As a consequence, we conclude that
\begin{equation}
\label{F''Lp}
\int_t^{t+1}\! \| F''(\phi(\tau))\|_{L^p(\Omega)}^q \, {\rm d} \tau \leq C_{11}, \quad \forall \, t \geq 0,
\end{equation}
where $1=\frac{1}{p}+\frac{1}{q}$, $p\in (1,\infty)$.
\medskip
\textbf{Uniqueness of strong solutions.}
Let us consider two strong solutions $(\textbf{\textit{u}}_1,\phi_1,P_1)$ and $(\textbf{\textit{u}}_2,\phi_2.P_2)$ to system \eqref{NSAC-D}-\eqref{IC-D} satisfying the entropy bound \eqref{F''log} and originating from the same initial datum.
The solutions difference $(\textbf{\textit{u}},\phi, P):=(\textbf{\textit{u}}_1-\textbf{\textit{u}}_2, \phi_1-\phi_2, P_1-P_2)$ solves
\begin{align}
\label{D-Diff1}
&\rho(\phi_1)\big( \partial_t \textbf{\textit{u}} + \textbf{\textit{u}}_1 \cdot \nabla \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}_2 \big)- \mathrm{div}\, \big( \nu(\phi_1)D\textbf{\textit{u}}\big)+ \nabla P \notag\\
& = - \Delta \phi_1 \nabla \phi -\Delta \phi \nabla \phi_2 - (\rho(\phi_1)-\rho(\phi_2)) (\partial_t \textbf{\textit{u}}_2 + \textbf{\textit{u}}_2 \cdot \nabla \textbf{\textit{u}}_2) + \mathrm{div}\, \big( (\nu(\phi_1)-\nu(\phi_2))D\textbf{\textit{u}}_2\big)
\end{align}
and
\begin{align}
\label{D-Diff2}
&\partial_t \phi +\textbf{\textit{u}}_1\cdot \nabla \phi +\textbf{\textit{u}} \cdot \nabla \phi_2
-\Delta \phi + \Psi' (\phi_1)-\Psi'(\phi_2)
\notag\\
&\quad = - \rho'(\phi_1)\frac{|\textbf{\textit{u}}_1|^2}{2}+ \rho'(\phi_2)\frac{|\textbf{\textit{u}}_2|^2}{2}
+\xi_1-\xi_2,
\end{align}
for almost every $(x,t) \in \Omega \times (0,T)$, together with the incompressibility constraint $\mathrm{div}\, \textbf{\textit{u}}=0$.
It follows that $\overline{\phi}(t)= 0$.
Multiplying \eqref{D-Diff1} by $\textbf{\textit{u}}$ and integrating over $\Omega$, we obtain
\begin{align}
&\frac{\d}{\d t} \int_{\Omega} \frac{\rho(\phi_1)}{2} |\textbf{\textit{u}}|^2 \, {\rm d} x +
\int_{\Omega} \rho(\phi_1) (\textbf{\textit{u}}_1 \cdot \nabla) \textbf{\textit{u}} \cdot \textbf{\textit{u}} \, {\rm d} x
+\int_{\Omega} \rho(\phi_1) (\textbf{\textit{u}}\cdot \nabla )\textbf{\textit{u}}_2 \cdot \textbf{\textit{u}} \, {\rm d} x
+\int_{\Omega} \nu(\phi_1)|D \textbf{\textit{u}}|^2 \, {\rm d} x \notag\\
&\quad =-\int_{\Omega} \Delta \phi_1 \nabla \phi \cdot \textbf{\textit{u}} \, {\rm d} x
- \int_{\Omega} \Delta \phi\nabla \phi_2 \cdot \textbf{\textit{u}} \, {\rm d} x
- \int_{\Omega} (\rho(\phi_1)-\rho(\phi_2)) (\partial_t \textbf{\textit{u}}_2 + \textbf{\textit{u}}_2 \cdot \nabla \textbf{\textit{u}}_2) \cdot \textbf{\textit{u}} \, {\rm d} x \notag \\
&\qquad - \int_{\Omega} (\nu(\phi_1)-\nu(\phi_2))D\textbf{\textit{u}}_2 : D \textbf{\textit{u}} \, {\rm d} x + \int_{\Omega} \frac12 |\textbf{\textit{u}}|^2 \rho'(\phi_1) \partial_t \phi_1 \,{\rm d} x.
\label{D1}
\end{align}
Next, multiplying \eqref{D-Diff2} by $-\Delta \phi$ and integrating over $\Omega$, we find
\begin{align}
&\frac{\d}{\d t} \int_{\Omega} \frac12 |\nabla \phi|^2 \,{\rm d} x + \|\Delta \phi \|_{L^2(\Omega)}^2
= \int_{\Omega} (\textbf{\textit{u}}_1 \cdot \nabla \phi) \, \Delta \phi \, {\rm d} x
+ \int_{\Omega} (\textbf{\textit{u}}\cdot \nabla \phi_2) \, \Delta \phi \,{\rm d} x \notag\\
& + \int_{\Omega} (F'(\phi_1)-F'(\phi_2)) \Delta \phi \, {\rm d} x
+ \theta_0 \|\nabla \phi\|_{L^2(\Omega)}^2 + \int_{\Omega}
\Big( \rho'(\phi_1)\frac{|\textbf{\textit{u}}_1|^2}{2}- \rho'(\phi_2)\frac{|\textbf{\textit{u}}_2|^2}{2} \Big) \Delta \phi \, {\rm d} x.
\label{D2}
\end{align}
Here we have used the fact that $\overline{\Delta \phi}=0$ which implies that $\int_{\Omega} (\xi_1-\xi_2) \Delta \phi \, {\rm d} x=0$.
Adding \eqref{D1} and \eqref{D2}, together with the bound from below of the viscosity, we have
\begin{align}
&\frac{\d}{\d t} \Big( \int_{\Omega} \frac{\rho(\phi_1)}{2} |\textbf{\textit{u}}|^2 \, {\rm d} x +
\int_{\Omega} \frac12 |\nabla \phi|^2 \,{\rm d} x \Big)
+ \nu_\ast \|D \textbf{\textit{u}}\|_{L^2(\Omega)}^2
+ \|\Delta \phi \|_{L^2(\Omega)}^2 \notag\\
&\leq -\int_{\Omega} \rho(\phi_1) (\textbf{\textit{u}}_1 \cdot \nabla) \textbf{\textit{u}} \cdot \textbf{\textit{u}} \, {\rm d} x-\int_{\Omega} \rho(\phi_1) (\textbf{\textit{u}}\cdot \nabla )\textbf{\textit{u}}_2 \cdot \textbf{\textit{u}} \, {\rm d} x
-\int_{\Omega} \Delta \phi_1 \nabla \phi \cdot \textbf{\textit{u}} \, {\rm d} x\notag \\
&\quad - \int_{\Omega} (\rho(\phi_1)-\rho(\phi_2)) (\partial_t \textbf{\textit{u}}_2 + \textbf{\textit{u}}_2 \cdot \nabla \textbf{\textit{u}}_2) \cdot \textbf{\textit{u}} \, {\rm d} x - \int_{\Omega} (\nu(\phi_1)-\nu(\phi_2))D\textbf{\textit{u}}_2 : D \textbf{\textit{u}} \, {\rm d} x \notag \\
&\quad + \int_{\Omega} \frac12 |\textbf{\textit{u}}|^2 \rho'(\phi_1) \partial_t \phi_1 \,{\rm d} x +
\int_{\Omega} (\textbf{\textit{u}}_1 \cdot \nabla \phi) \, \Delta \phi \, {\rm d} x + \int_{\Omega} (F'(\phi_1)-F'(\phi_2)) \Delta \phi \, {\rm d} x \notag\\
&\quad
+ \theta_0 \|\nabla \phi\|_{L^2(\Omega)}^2+ \int_{\Omega}
\Big( \rho'(\phi_1)\frac{|\textbf{\textit{u}}_1|^2}{2}- \rho'(\phi_2)\frac{|\textbf{\textit{u}}_2|^2}{2} \Big) \Delta \phi \, {\rm d} x.
\end{align}
We now proceed by estimating the terms on the right hand side of the above differential equality. We would like to mention that most of the bounds obtained below are easy applications of the Sobolev embedding theorem and interpolation inequalities in view of the estimates for global strong solutions that have been obtained before. Nevertheless, less standard is the estimate of the term involving the difference of the nonlinear terms ($F'(\phi_1)-F'(\phi_2)$) which makes use of the entropy bound \eqref{EB3}.
By using the regularity of strong solutions, \eqref{KORN} and \eqref{LADY}, we have
\begin{align*}
-\int_{\Omega} \rho(\phi_1) (\textbf{\textit{u}}_1 \cdot \nabla) \textbf{\textit{u}} \cdot \textbf{\textit{u}} \, {\rm d} x
&\leq C \| \textbf{\textit{u}}_1\|_{L^\infty(\Omega)}\| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)} \| \textbf{\textit{u}}\|_{L^2(\Omega)}\\
&\leq \frac{\nu_\ast}{12} \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2 +C \| \textbf{\textit{u}}_1\|_{L^\infty(\Omega)}^2\| \textbf{\textit{u}}\|_{L^2(\Omega)}^2,
\end{align*}
\begin{align*}
-\int_{\Omega} \rho(\phi_1) (\textbf{\textit{u}}\cdot \nabla )\textbf{\textit{u}}_2 \cdot \textbf{\textit{u}} \, {\rm d} x
&\leq C \| \nabla \textbf{\textit{u}}_2\|_{L^2(\Omega)} \| \textbf{\textit{u}}\|_{L^4(\Omega)}^2\\
& \leq \frac{\nu_\ast}{12} \|D \textbf{\textit{u}} \|_{L^2(\Omega)}^2+ C\| \textbf{\textit{u}}\|_{L^2(\Omega)}^2,
\end{align*}
\begin{align*}
-\int_{\Omega} \Delta \phi_1 \nabla \phi \cdot \textbf{\textit{u}} \, {\rm d} x
&\leq C \|\Delta \phi_1 \|_{L^4(\Omega)} \| \nabla \phi\|_{L^2(\Omega)} \| \textbf{\textit{u}}\|_{L^4(\Omega)}\\
&\leq \frac{\nu_\ast}{12} \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2+ C \|\Delta \phi_1 \|_{L^4(\Omega)}^2 \| \nabla \phi\|_{L^2(\Omega)}^2,
\end{align*}
\begin{align*}
&- \int_{\Omega} (\rho(\phi_1)-\rho(\phi_2)) (\partial_t \textbf{\textit{u}}_2 + \textbf{\textit{u}}_2 \cdot \nabla \textbf{\textit{u}}_2) \cdot \textbf{\textit{u}} \, {\rm d} x \notag\\
&\quad \leq C \| \phi\|_{L^4(\Omega)} \| \partial_t \textbf{\textit{u}}_2 + \textbf{\textit{u}}_2 \cdot \nabla \textbf{\textit{u}}_2\|_{L^2(\Omega)} \| \textbf{\textit{u}}\|_{L^4(\Omega)}\\
&\quad \leq C \| \nabla \phi\|_{L^2(\Omega)} \| \partial_t \textbf{\textit{u}}_2 + \textbf{\textit{u}}_2 \cdot \nabla \textbf{\textit{u}}_2\|_{L^2(\Omega)} \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}\\
&\quad \leq \frac{\nu_\ast}{12} \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2 + C \| \partial_t \textbf{\textit{u}}_2 + \textbf{\textit{u}}_2 \cdot \nabla \textbf{\textit{u}}_2\|_{L^2(\Omega)}^2 \| \nabla \phi\|_{L^2(\Omega)}^2,
\end{align*}
\begin{align*}
- \int_{\Omega} (\nu(\phi_1)-\nu(\phi_2))D\textbf{\textit{u}}_2 : D \textbf{\textit{u}} \, {\rm d} x
&\leq C \|\phi \|_{L^4(\Omega)} \| D \textbf{\textit{u}}_2\|_{L^4(\Omega)} \| D \textbf{\textit{u}}\|_{L^2(\Omega)}\\
&\leq \frac{\nu_\ast}{12} \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2 +
C \| D \textbf{\textit{u}}_2\|_{L^4(\Omega)}^2 \| \nabla \phi\|_{L^2(\Omega)}^2,
\end{align*}
\begin{align*}
\int_{\Omega} \frac12 |\textbf{\textit{u}}|^2 \rho'(\phi_1) \partial_t \phi_1 \,{\rm d} x
&\leq C \| \textbf{\textit{u}}\|_{L^4(\Omega)}^2 \| \partial_t \phi_1 \|_{L^2(\Omega)}\\
&\leq \frac{\nu_\ast}{12} \| D \textbf{\textit{u}}\|_{L^2(\Omega)}^2+ C \| \textbf{\textit{u}}\|_{L^2(\Omega)}^2,
\end{align*}
\begin{align*}
\int_{\Omega} (\textbf{\textit{u}}_1 \cdot \nabla \phi) \, \Delta \phi \, {\rm d} x
&\leq \|\textbf{\textit{u}}_1 \|_{L^\infty(\Omega)} \| \nabla \phi\|_{L^2(\Omega)} \| \Delta \phi\|_{L^2(\Omega)}\\
&\leq \frac{1}{6} \| \Delta \phi\|_{L^2(\Omega)}^2+ C \|\textbf{\textit{u}}_1 \|_{L^\infty(\Omega)}^2 \| \nabla \phi\|_{L^2(\Omega)}^2,
\end{align*}
\begin{align*}
\int_{\Omega}
&\Big( \rho'(\phi_1)\frac{|\textbf{\textit{u}}_1|^2}{2}- \rho'(\phi_2)\frac{|\textbf{\textit{u}}_2|^2}{2} \Big) \Delta \phi \, {\rm d} x\\
&= \int_{\Omega} \Big( \rho'(\phi_1)-\rho'(\phi_2) \Big) \frac{|\textbf{\textit{u}}_1|^2}{2} \Delta \phi \, {\rm d} x + \int_{\Omega} \frac{\rho'(\phi_2)}{2} \Big( \textbf{\textit{u}}_1\cdot \textbf{\textit{u}}+ \textbf{\textit{u}}\cdot \textbf{\textit{u}}_2 \Big) \Delta \phi \, {\rm d} x\\
&\leq C \| \phi\|_{L^4(\Omega)}\| \textbf{\textit{u}}_1\|_{L^8(\Omega)}^2 \| \Delta \phi\|_{L^2(\Omega)} + C \| \textbf{\textit{u}}\|_{L^2(\Omega)} (\| \textbf{\textit{u}}_1\|_{L^\infty(\Omega)}+\| \textbf{\textit{u}}_2\|_{L^\infty(\Omega)}) \| \Delta \phi\|_{L^2(\Omega)}\\
&\leq \frac{1}{6} \| \Delta \phi\|^2_{L^2(\Omega)} + C\| \nabla \phi\|_{L^2(\Omega)}^2+
C (\| \textbf{\textit{u}}_1\|_{L^\infty(\Omega)}^2+\| \textbf{\textit{u}}_2\|_{L^\infty(\Omega)}^2) \| \textbf{\textit{u}}\|_{L^2(\Omega)}^2.
\end{align*}
Using the generalized Young inequality \eqref{Young0} and the standard Young inequality, for $x>0$, $y>0$, $z>0$ with $Cz>y$, we obtain
\begin{align}
x^2 y^2 \log \Big( \frac{Cz}{y} \Big)
&\leq xy^2\left(x\log x+\frac{Cz}{y}\right)\nonumber\\
& \leq \varepsilon z^2+ x^2 y^2 \log x +C^2\varepsilon ^{-1}x^2y^2, \quad \forall \, \varepsilon>0.
\label{ineqy}
\end{align}
By making use of \eqref{BGI} and \eqref{ineqy}, we obtain that
\begin{align*}
& \int_{\Omega} (F'(\phi_1)-F'(\phi_2)) \Delta \phi \, {\rm d} x\\
&\quad = \int_{\Omega} \int_0^1 F''(\tau\phi_1+(1-\tau)\phi_2) \, {\rm d} \tau\, \phi \Delta \phi \, {\rm d} x\\
&\quad \leq C\big( \| F''(\phi_1)\|_{L^2(\Omega)}+ \| F''(\phi_2)\|_{L^2(\Omega)} \big) \| \phi\|_{L^\infty(\Omega)}\| \Delta \phi\|_{L^2(\Omega)}\\
&\quad \leq C\big( \| F''(\phi_1)\|_{L^2(\Omega)}+ \| F''(\phi_2)\|_{L^2(\Omega)} \big) \| \nabla \phi\|_{L^2(\Omega)} \log^\frac12 \Big( C \frac{\|\Delta \phi \|_{L^2(\Omega)}}{\| \nabla \phi\|_{L^2(\Omega)}} \Big) \| \Delta \phi\|_{L^2(\Omega)}\\
&\quad \leq \frac{1}{12} \| \Delta \phi\|_{L^2(\Omega)}^2 +C
\big( \| F''(\phi_1)\|_{L^2(\Omega)}^2+ \| F''(\phi_2)\|_{L^2(\Omega)}^2 \big) \| \nabla \phi\|_{L^2(\Omega)}^2 \log \Big( C \frac{\|\Delta \phi \|_{L^2(\Omega)}}{\| \nabla \phi\|_{L^2(\Omega)}} \Big)\\
&\quad \leq \frac{1}{6} \| \Delta \phi\|_{L^2(\Omega)}^2 +
C \| F''(\phi_1)\|_{L^2(\Omega)}^2 \big( 1+ \log \big( \| F''(\phi_1)\|_{L^2(\Omega)} \big) \big)
\| \nabla \phi\|_{L^2(\Omega)}^2 \\
&\qquad + C \| F''(\phi_2)\|_{L^2(\Omega)}^2 \big( 1+ \log\big( \| F''(\phi_2)\|_{L^2(\Omega)}\big) \big) \| \nabla \phi\|_{L^2(\Omega)}^2 .
\end{align*}
Collecting the above bounds, we find the differential inequality
\begin{align}
&\frac{\d}{\d t} \Big( \int_{\Omega} \frac{\rho(\phi_1)}{2} |\textbf{\textit{u}}|^2 \, {\rm d} x +
\int_{\Omega} \frac12 |\nabla \phi|^2 \,{\rm d} x \Big)
+ \frac{\nu_\ast}{2} \int_{\Omega} |D \textbf{\textit{u}}|^2 \, {\rm d} x +\frac12 \|\Delta \phi \|_{L^2(\Omega)}^2 \notag \\
&\quad \leq W_1(t) \int_{\Omega} \frac{\rho(\phi_1)}{2} |\textbf{\textit{u}}|^2 \, {\rm d} x+ W_2(t) \| \nabla \phi\|_{L^2(\Omega)}^2,
\label{D-DI}
\end{align}
where
\begin{align*}
W_1(t)= C\big( 1+ \| \textbf{\textit{u}}_1\|_{L^\infty(\Omega)}^2+ \| \textbf{\textit{u}}_2\|_{L^\infty(\Omega)}^2+ \| \partial_t \textbf{\textit{u}}_2 + \textbf{\textit{u}}_2 \cdot \nabla \textbf{\textit{u}}_2\|_{L^2(\Omega)}^2\big),
\end{align*}
and
\begin{align*}
W_2(t)&= C \Big( 1+ \| \Delta \phi_1\|_{L^4(\Omega)}^2+\| \partial_t \textbf{\textit{u}}_2 + \textbf{\textit{u}}_2 \cdot \nabla \textbf{\textit{u}}_2\|_{L^2(\Omega)}^2+ \| D\textbf{\textit{u}}_2\|_{L^4(\Omega)}^2+
\| \textbf{\textit{u}}_1\|_{L^\infty(\Omega)}^2
\Big)\\
&\quad + C \| F''(\phi_1)\|_{L^2(\Omega)}^2 \log \big( \| F''(\phi_1)\|_{L^2(\Omega)} \big) +
C \| F''(\phi_2)\|_{L^2(\Omega)}^2 \log \big( \| F''(\phi_2)\|_{L^2(\Omega)} \big).
\end{align*}
Here we have used that $\rho(s)\geq \rho_\ast$ for all $s \in (-1,1)$. In order to apply the Gronwall lemma, we are left to show that
\begin{equation}
\label{F''2log}
\int_0^T \| F''(\phi_i)\|_{L^2(\Omega)}^2 \log \big( \| F''(\phi_i)\|_{L^2(\Omega)} \big) \, {\rm d} \tau\leq C(T), \quad i=1,2.
\end{equation}
To this aim, we introduce the function
$$
g(s)= s \log ( C^\ast s), \quad \forall \, s\in (0,\infty),
$$
where $C^\ast$ is a positive constant. It is easily seen that $g$ is continuous and convex ($g''(s)= \frac{1}{s}>0$). By applying Jensen's inequality, we have
$$
g \Big( \frac{1}{|\Omega|}\int_{\Omega} |F''(\phi)|^2 \, {\rm d} x \Big)
\leq \frac{1}{|\Omega|} \int_{\Omega} g(|F''(\phi)|^2) \, {\rm d} x.
$$
Using the explicit form of $g$, this is equivalent to
\begin{align*}
\frac{1}{|\Omega|}\|F''(\phi)\|_{L^2(\Omega)}^2 \log \Big( \frac{C^\ast}{|\Omega|}
\|F''(\phi)\|_{L^2(\Omega)}^2 \Big) \leq \frac{1}{|\Omega|}
\int_{\Omega} |F''(\phi)|^2 \log (C^\ast |F''(\phi)|^2) \, {\rm d} x.
\end{align*}
Taking $C^\ast= |\Omega|$ and integrating the above inequality over $[0,T]$, we find
\begin{equation}
\label{jensen}
\int_0^T \|F''(\phi)\|_{L^2(\Omega)}^2 \log \big( \|F''(\phi)\|_{L^2(\Omega)} \big) \, {\rm d} \tau \leq \int_{0}^T\! \int_{\Omega} |F''(\phi)|^2 \log ( |\Omega| |F''(\phi)|^2) \, {\rm d} x {\rm d} \tau.
\end{equation}
Then, \eqref{F''2log} immediately follows from the entropy bounds \eqref{EB3} and \eqref{jensen}. As a consequence, both $W_1$ and $W_2$ belong to $L^1(0,T)$.
Finally, an application of the Gronwall lemma entails the uniqueness of strong solutions.\hfill$\square$
\medskip
\begin{remark}[Entropy Estimates in $L^p$, $p>1$]
Notice that the entropy estimate in $L^1(\Omega)$ proved in Theorem \ref{strong-D}-(2) can be generalized to the $L^p(\Omega)$ case with $p>1$. More precisely, for any $p\in \mathbb{N}$, there exists $\eta_p>0$ with the latter depending on the norms of the initial data and on the parameters of the system
$$
\eta_p=\eta_p(E(\textbf{\textit{u}}_0,\phi_0), \| \textbf{\textit{u}}_0\|_{{\mathbf{V}}_\sigma}, \| \phi_0\|_{H^2(\Omega)},\| F'(\phi_0)\|_{L^2(\Omega)},\theta,\theta_0)
$$
such that, if $\|\rho'\|_{L^\infty(-1,1)}\leq \eta_p$ and $F''(\phi_0)\in L^p(\Omega)$, then, for any $T>\sigma$, we have
\begin{align*}
F''(\phi)\in L^\infty(0,T;L^p(\Omega)),\quad |F''(\phi)|^{p-1}F'''(\phi) F'(\phi)\in L^1(\Omega\times (0,T)).
\end{align*}
Such result follows from the above proof by replacing $\frac{\d}{\d t} \int_{\Omega} F''(\phi) \, {\rm d} x$ by $\frac{\d}{\d t} \int_{\Omega} (F''(\phi))^p \, {\rm d} x$, and the observation that, for any $p>2$, there exist two positive constants $C^1_p$ and $C^2_p$ such that
$$
|(F''(s))^{p-1}F'''(s)| \log \big( |(F''(s))^{p-1}F'''(s)| \big) \leq
C^1_p+C^2_p (F''(s))^{p-1}F'''(s) F'(s), \quad \forall \, s \in (-1,1).
$$
\end{remark}
\smallskip
\subsection{Proof of Theorem \ref{Proreg-D}}
We now prove the propagation of entropy bound as stated
in Theorem \ref{Proreg-D}.
For every strong solution given by Theorem \ref{strong-D}-(1), we have
\begin{align*}
\|\textbf{\textit{u}}\cdot \nabla \phi\|_{H^1(\Omega)}&\leq \|\textbf{\textit{u}}\|_{L^4(\Omega)}\|\nabla \phi\|_{L^4}+\|\nabla \textbf{\textit{u}}\|_{L^4(\Omega)}\|\nabla \phi\|_{L^4(\Omega)}+\|\textbf{\textit{u}}\|_{L^\infty(\Omega)}\|\phi\|_{H^2(\Omega)}\nonumber\\
&\leq C+C\|\textbf{\textit{u}}\|_{H^2(\Omega)}^\frac12\|\phi\|_{H^2(\Omega)}^\frac12+C\|\phi\|_{H^2(\Omega)},
\end{align*}
and
\begin{align*}
\|\rho'(\phi)|\textbf{\textit{u}}|^2\|_{H^1(\Omega)}
&\leq C\|\textbf{\textit{u}}\|_{L^4(\Omega)}^2+C\|\nabla \phi\|_{L^\infty(\Omega)}\|\textbf{\textit{u}}\|_{L^4(\Omega)}^2+C\|\nabla \textbf{\textit{u}}\|_{L^4(\Omega)}\|\textbf{\textit{u}}\|_{L^4(\Omega)}\nonumber\\
&\leq C+C\|\phi\|_{W^{2,3}(\Omega)}+C\|\textbf{\textit{u}}\|_{H^2(\Omega)},
\end{align*}
which imply that
$$
\int_t^{t+1} \| \textbf{\textit{u}}(\tau)\cdot \nabla \phi(\tau)\|_{H^1(\Omega)}^2+ \|\rho'(\phi(\tau))\frac{|\textbf{\textit{u}}(\tau)|^2}{2}\|_{H^1(\Omega)}^2 \, {\rm d} \tau \leq C, \quad \forall \, t \geq 0,
$$
for some $C$ independent of $t$. In light of \eqref{NSAC4}, it follows that $$
\int_t^{t+1} \|-\Delta \phi(\tau) +F'(\phi(\tau))\|_{H^1(\Omega)}^2 \, {\rm d} \tau \leq C, \quad \forall \, t \geq 0.
$$
By using \cite[Lemma 7.4]{GGW2018}, we infer that, for any $p\geq 1$, there exists $C=C(p)$ such that
\begin{equation}
\label{propF''}
\| F''(\phi)\|_{L^p(\Omega)}\leq C\Big(1+{e}^{C\| -\Delta \phi +F'(\phi)\|_{H^1(\Omega)}^2}\Big)\quad \text{a.e. in}\ (0,T).
\end{equation}
Notice that we are not able to conclude that the right hand side of \eqref{propF''} is $L^1(0,T)$. Nevertheless, since integrable function are finite almost everywhere, the above inequality entails that there exists some $\sigma \in (0,1)$ (actually $\sigma$ can be taken arbitrarily small but positive) such that
\begin{align}
F''(\phi(\sigma))\in L^p(\Omega) \quad\text{with}\quad \| F''(\phi(\sigma))\|_{L^p(\Omega)}\leq C(p,\sigma),\quad \forall\,p \in [1,\infty).\label{FttLp}
\end{align}
Then, under the condition \eqref{Hyp} but without the additional assumption $F''(\phi_0)\in L^1(\Omega)$ on the initial datum, we are able to deduce that the previous estimates \eqref{EB1}-\eqref{EB3} hold for $t\geq \sigma>0$. More precisely, we have
\begin{equation}
\label{EB-sig}
\int_t^{t+1}\! \int_{\Omega} (F''(\phi))^2 \log ( 1+F''(\phi)) \,{\rm d} x {\rm d} \tau \leq C(\sigma), \quad \forall \, t \geq 0.
\end{equation}
Differentiating \eqref{NSAC-D}$_1$ with respect to time and testing the resultant by $\partial_t\textbf{\textit{u}}$, integrating over $\Omega$, we have
\begin{align*}
& \frac12 \int_{\Omega} \rho(\phi) \partial_t |\partial_t \textbf{\textit{u}}|^2 \, {\rm d} x
+\int_{\Omega} \rho(\phi) \big( \partial_t \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \partial_t \textbf{\textit{u}}\big)\cdot \partial_t \textbf{\textit{u}} \, {\rm d} x + \int_{\Omega} \rho'(\phi)\partial_t \phi (\partial_t \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}) \cdot \partial_t \textbf{\textit{u}} \, {\rm d} x\\
& + \int_{\Omega} \nu(\phi) |D \partial_t \textbf{\textit{u}}|^2 \, {\rm d} x+ \int_{\Omega} \nu'(\phi) \partial_t \phi D \textbf{\textit{u}} : D \partial_t \textbf{\textit{u}} \, {\rm d} x= \int_\Omega \partial_t(\nabla \phi\otimes \nabla \phi):\nabla \partial_t\textbf{\textit{u}} \, {\rm d} x.
\end{align*}
Since
\begin{align*}
&\frac12 \int_{\Omega} \rho(\phi) \partial_t |\partial_t \textbf{\textit{u}}|^2 \, {\rm d} x
= \frac12 \frac{\d}{\d t} \int_{\Omega} \rho(\phi)|\partial_t \textbf{\textit{u}}|^2 \,{\rm d} x - \frac12 \int_{\Omega} \rho'(\phi)\partial_t \phi |\partial_t \textbf{\textit{u}}|^2\, {\rm d} x,
\end{align*}
we find
\begin{align}
&\frac12\frac{{\rm d}}{{\rm d} t} \int_{\Omega} \rho(\phi)|\partial_t \textbf{\textit{u}}|^2 \, {\rm d} x
+\int_\Omega \nu(\phi)|D\partial_t \textbf{\textit{u}}|^2 \, {\rm d} x\nonumber\\
&=-\int_\Omega \rho(\phi)(\partial_t\textbf{\textit{u}}\cdot \nabla \textbf{\textit{u}}+ \textbf{\textit{u}}\cdot \nabla \partial_t \textbf{\textit{u}})\cdot \partial_t\textbf{\textit{u}} \, {\rm d} x
- \frac12 \int_{\Omega} \rho'(\phi)\partial_t \phi |\partial_t \textbf{\textit{u}}|^2\, {\rm d} x \notag \\
&\quad -\int_{\Omega} \rho'(\phi)\partial_t \phi (\textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}) \cdot \partial_t \textbf{\textit{u}} \, {\rm d} x - \int_\Omega \nu'(\phi)\partial_t \phi D\textbf{\textit{u}}: \nabla \partial_t\textbf{\textit{u}} + \int_\Omega \partial_t(\nabla \phi\otimes \nabla \phi):\nabla \partial_t\textbf{\textit{u}} \, {\rm d} x. \notag
\end{align}
In view of \eqref{str-1}, by using \eqref{LADY}, we have
\begin{align}
-\int_{\Omega} \rho(\phi)(\partial_t \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}) \cdot \partial_t \textbf{\textit{u}} \, {\rm d} x &\leq C \| \partial_t \textbf{\textit{u}}\|_{L^4(\Omega)}^2 \|\nabla \textbf{\textit{u}}\|_{L^2(\Omega)} \notag \\
&\leq \frac{\nu_\ast}{16} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2+ C\| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2,
\nonumber
\end{align}
and
\begin{align*}
-\int_{\Omega} \rho(\phi)(\textbf{\textit{u}} \cdot \nabla \partial_t \textbf{\textit{u}}) \cdot \partial_t \textbf{\textit{u}} \, {\rm d} x &\leq C \| \textbf{\textit{u}}\|_{L^4(\Omega)} \| \nabla \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)} \| \partial_t \textbf{\textit{u}}\|_{L^4(\Omega)}\\
&\leq \frac{\nu_\ast}{16} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2+ C\| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2.
\end{align*}
Similarly, we obtain
\begin{align*}
-\frac12 \int_{\Omega} \rho'(\phi)\partial_t \phi |\partial_t \textbf{\textit{u}}|^2\, {\rm d} x
&\leq C \| \partial_t \phi\|_{L^2(\Omega)} \| \partial_t \textbf{\textit{u}} \|_{L^4(\Omega)}^2\\
&\leq \frac{\nu_\ast}{16} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2+ C\| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2,
\end{align*}
and
\begin{align*}
-\int_{\Omega} \rho'(\phi)\partial_t \phi (\textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}) \cdot \partial_t \textbf{\textit{u}} \, {\rm d} x
&\leq C \| \partial_t \phi\|_{L^4(\Omega)} \| \textbf{\textit{u}} \|_{L^4(\Omega)}
\| \nabla \textbf{\textit{u}}\|_{L^4(\Omega)} \| \partial_t \textbf{\textit{u}}\|_{L^4(\Omega)}\\
&\leq C \| \nabla \partial_t \phi\|_{L^2(\Omega)}^\frac12 \| \textbf{\textit{u}}\|_{H^2(\Omega)}^\frac12 \| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac12 \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^\frac12\\
&\leq \frac{\nu_\ast}{16} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2
+ C\| \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2 +C \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2+C\| \textbf{\textit{u}}\|_{H^2(\Omega)}^2.
\end{align*}
Besides, by means of \eqref{Agmon2d}, we deduce that
\begin{align}
&-\int_{\Omega} \nu'(\phi) \partial_t \phi D \textbf{\textit{u}} : D \partial_t \textbf{\textit{u}} \, {\rm d} x\notag\\
&\quad \leq C \| \partial_t \phi\|_{L^\infty(\Omega)} \| D \textbf{\textit{u}}\|_{L^2(\Omega)} \| D \partial_t \textbf{\textit{u}} \|_{L^2(\Omega)} \notag \\
&\quad \leq \frac{\nu_\ast}{16} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2 +C \| \partial_t \phi\|_{L^2(\Omega)} \| \partial_t \phi\|_{H^2(\Omega)} \notag \\
&\quad \leq \frac{\nu_\ast}{16} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2 + \frac{1}{14} \| \Delta \partial_t \phi\|_{L^2(\Omega)}^2 +C \| \nabla \partial_t \phi\|_{L^2(\Omega)} ^2,
\nonumber
\end{align}
and
\begin{align}
&\int_\Omega \partial_t(\nabla \phi\otimes \nabla \phi):\nabla \partial_t\textbf{\textit{u}} {\rm d} x\nonumber\\
&\quad \leq
\|\nabla \phi\|_{L^4(\Omega)}\|\nabla \partial_t \phi\|_{L^4(\Omega)}\|D \partial_t\textbf{\textit{u}}\|_{L^2(\Omega)}\nonumber\\
&\quad \leq \frac{\nu_\ast}{16} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2
+C\|\nabla \partial_t\phi\|_{L^2(\Omega)}\|\nabla \partial_t \phi\|_{H^1(\Omega)}\nonumber\\
&\quad \leq \frac{\nu_\ast}{16} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2
+ \frac{1}{14} \| \Delta \partial_t \phi\|_{L^2(\Omega)}^2 +C \| \nabla \partial_t \phi\|_{L^2(\Omega)} ^2 \notag.
\end{align}
Next, we differentiate \eqref{NSAC-D}$_3$ with respect to time, multiply the resultant by $-\Delta \partial_t \phi$, and integrate over $\Omega$ to obtain
\begin{align*}
& \frac12 \frac{\d}{\d t} \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2+ \| \Delta \partial_t \phi\|_{L^2(\Omega)}^2
\notag \\
&= \theta_0 \|\nabla \partial_t \phi\|_{L^2(\Omega)}^2+ \int_{\Omega} F''(\phi) \partial_t \phi \Delta \partial_t \phi \, {\rm d} x+\int_{\Omega} (\partial_t \textbf{\textit{u}} \cdot \nabla \phi) \Delta \partial_t \phi\, {\rm d} x \\
&\quad + \int_{\Omega} (\textbf{\textit{u}} \cdot \nabla \partial_t \phi) \Delta \partial_t \phi\, {\rm d} x +\frac12 \int_{\Omega} \rho''(\phi) \partial_t \phi |\textbf{\textit{u}}|^2 \Delta \partial_t \phi \, {\rm d} x
+ \int_{\Omega} \rho'(\phi)( \textbf{\textit{u}} \cdot \partial_t \textbf{\textit{u}}) \Delta \partial_t \phi \, {\rm d} x.
\end{align*}
Here we have used that $\overline{\Delta \partial_t \phi}=0$ since $\partial_\textbf{\textit{n}} \partial_t \phi=0$ on the boundary $\partial \Omega$.
Exploiting \eqref{BGI}, we get
\begin{align*}
\int_{\Omega} F''(\phi) \partial_t \phi \Delta \partial_t \phi \, {\rm d} x
&\leq \| F''(\phi)\|_{L^2(\Omega)} \| \partial_t \phi\|_{L^\infty(\Omega)} \| \Delta \partial_t \phi \|_{L^2(\Omega)}\\
&\leq \| F''(\phi)\|_{L^2(\Omega)} \| \nabla \partial_t \phi\|_{L^2(\Omega)} \log^\frac12 \Big( C\frac{\| \Delta \partial_t \phi\|_{L^2(\Omega)}}{\| \nabla \partial_t \phi\|_{L^2(\Omega)}}\Big)\| \Delta \partial_t \phi \|_{L^2(\Omega)}\\
&\leq \frac{1}{28} \| \Delta \partial_t \phi \|_{L^2(\Omega)}^2 +C \| F''(\phi)\|_{L^2(\Omega)}^2 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2 \log \Big( C\frac{\| \Delta \partial_t \phi\|_{L^2(\Omega)}}{\| \nabla \partial_t \phi\|_{L^2(\Omega)}}\Big).
\end{align*}
Recalling \eqref{ineqy}, we obtain
\begin{align}
&\int_{\Omega} F''(\phi) \partial_t \phi \Delta \partial_t \phi \, {\rm d} x \notag\\
&\quad \leq \frac{1}{14} \| \Delta \partial_t \phi \|_{L^2(\Omega)}^2 +C \| F''(\phi)\|_{L^2(\Omega)}^2 \log \big( C\| F''(\phi)\|_{L^2(\Omega)} \big)\| \nabla \partial_t \phi\|_{L^2(\Omega)}^2.\label{FFF}
\end{align}
Next, using \eqref{LADY} and \eqref{str-1}, we see that
\begin{align}
\int_{\Omega} (\partial_t \textbf{\textit{u}} \cdot \nabla \phi) \Delta \partial_t \phi\, {\rm d} x
&\leq \|\partial_t \textbf{\textit{u}} \|_{L^4(\Omega)}\|\nabla \phi\|_{L^4(\Omega)}\|\Delta \partial_t \phi\|_{L^2(\Omega)}\nonumber\\
&\leq \frac{\nu_\ast}{16} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2
+ \frac{1}{14}\| \Delta \partial_t \phi\|_{L^2(\Omega)}^2 + C \|\partial_t \textbf{\textit{u}} \|_{L^2(\Omega)}^2, \nonumber
\end{align}
and
\begin{align}
\int_{\Omega} (\textbf{\textit{u}} \cdot \nabla \partial_t \phi) \Delta \partial_t \phi\, {\rm d} x& \leq \| \textbf{\textit{u}}\|_{L^4(\Omega)} \| \nabla \partial_t \phi\|_{L^4(\Omega)}
\| \Delta \partial_t \phi\|_{L^2(\Omega)} \nonumber\\
&\leq \frac{1}{14} \| \Delta \partial_t \phi\|_{L^2(\Omega)}^2 + C \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2.
\label{unpt}
\end{align}
Finally, in a similar manner we find that
\begin{align*}
\frac12 \int_{\Omega} \rho''(\phi) \partial_t \phi |\textbf{\textit{u}}|^2 \Delta \partial_t \phi \, {\rm d} x
&\leq C \| \partial_t \phi\|_{L^4(\Omega)} \| \textbf{\textit{u}}\|_{L^8(\Omega)}^2 \| \Delta \partial_t \phi\|_{L^2(\Omega)}\\
&\leq \frac{1}{14} \| \Delta \partial_t \phi\|_{L^2(\Omega)}^2 + C \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2,
\end{align*}
and
\begin{align*}
\int_{\Omega} \rho'(\phi)( \textbf{\textit{u}} \cdot \partial_t \textbf{\textit{u}}) \Delta \partial_t \phi \, {\rm d} x
&\leq C \| \textbf{\textit{u}}\|_{L^4(\Omega)} \| \partial_t \textbf{\textit{u}} \|_{L^4(\Omega)} \| \Delta \partial_t \phi\|_{L^2(\Omega)}\\
&\leq \frac{\nu_\ast}{16} \| D \partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2
+ \frac{1}{14}\| \Delta \partial_t \phi\|_{L^2(\Omega)}^2 + C \|\partial_t \textbf{\textit{u}} \|_{L^2(\Omega)}^2.
\end{align*}
From the above estimates, we deduce that
\begin{align}
\frac{{\rm d}}{{\rm d} t}L(t)+ \frac{\nu_*}{2}\|D\partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2
+\frac{1}{2}\| \Delta \partial_t \phi\|_{L^2(\Omega)}^2 \leq CK(t)L(t)+ C \| \textbf{\textit{u}}\|_{H^2(\Omega)}^2,
\end{align}
where
\begin{align*}
&L(t)=\frac12 \int_{\Omega} \rho(\phi) |\partial_t\textbf{\textit{u}}(t)|^2 \, {\rm d} x+\frac12\|\nabla \partial_t \phi(t)\|_{L^2(\Omega)}^2,\\
&K(t)=1+\| F''(\phi)\|_{L^2(\Omega)}^2 \log \big( C\| F''(\phi)\|_{L^2(\Omega)} \big).
\end{align*}
Recalling estimates \eqref{NSAC4} and \eqref{str-3}, we have
$$
\int_{t}^{t+1} L(\tau) + \| \textbf{\textit{u}}(\tau)\|_{H^2(\Omega)}^2 \, {\rm d} \tau \leq C, \quad \forall \, t \geq 0,
$$
where $C$ is independent of $t$.
As a consequence, there exists $\sigma \in (0,1)$ ($\sigma$ can be chosen arbitrary small but positive) such that
\begin{align}
L(\sigma)\leq C(\sigma).\label{Lini}
\end{align}
Notice that, without loss of generality, this value of $\sigma$ can be chosen equal to the one in \eqref{FttLp}.
Then, by exploiting \eqref{EB-sig} and the Jensen inequality (cf. \eqref{jensen}), we obtain
$$
\int_t^{t+1} K(\tau) \, {\rm d} \tau \leq C, \quad \forall \, t \geq \sigma,
$$
where $C$ depends on $\sigma$, but is independent of $t$. Thus, by using the Gronwall lemma on the time interval $[\sigma,1]$ and the uniform Gronwall lemma for $t\geq 1$, we deduce that
$$
L(t)+ \int_t ^{t+1} \|D\partial_t \textbf{\textit{u}}\|_{L^2(\Omega)}^2
+\| \Delta \partial_t \phi\|_{L^2(\Omega)}^2 \, {\rm d} \tau \leq C(\sigma),\quad \forall\, t\geq \sigma.
$$
Hence we have
$$
\partial_t\textbf{\textit{u}}\in L^\infty(\sigma, T; \mathbf{H}_\sigma)\cap L^2(\sigma, T; {\mathbf{V}}_\sigma),\quad \partial_t \phi\in L^\infty(\sigma, T; H^1(\Omega))\cap L^2(\sigma, T; H^2(\Omega)).
$$
In light of \eqref{est-uw1p} and \eqref{NSAC3h-D}, we infer that
$$
\textbf{\textit{u}} \in L^\infty(\sigma,T;\mathbf{W}^{1,p}(\Omega)), \quad \forall \, p \in (2,\infty).
$$
An immediate consequence of the above regularity results is that $$\widetilde{\mu}=-\Delta \phi+F'(\phi) \in L^2(\sigma,T;L^\infty(\Omega)).$$ Thanks to \cite[Lemma 7.2]{GGW2018}, we deduce that $F'(\phi)\in L^2(\sigma,T;L^\infty(\Omega))$. This property entails that there exists $\sigma' \in (\sigma,\sigma+1)$ such that
\begin{equation}
\label{F'sigma}
\|F'(\phi(\sigma'))\|_{L^\infty(\Omega)}\leq C(\sigma).
\end{equation}
Note that $\sigma'$ can also be chosen arbitrarily close to $\sigma$.
Now, we rewrite \eqref{NSAC-D} as follows
$$
\partial_t \phi + \textbf{\textit{u}} \cdot \nabla \phi - \Delta \phi +F'(\phi) = U(x,t),
$$
where $U=\theta_0 \phi- \rho'(\phi) \frac{|\textbf{\textit{u}}|^2}{2}+\xi$.
Thanks to the above regularity, it easily seen that $U \in L^\infty(0,T;L^\infty(\Omega))$. In particular,
$\sup_{t\geq \sigma} \| U(t)\|_{L^\infty(\Omega)}\leq C(\sigma)$.
For any $p\geq 2$, we compute
\begin{align*}
\frac{1}{p} \frac{\d}{\d t} \int_{\Omega} |F'(\phi)|^p \, {\rm d} x
&= \int_{\Omega} |F'(\phi)|^{p-2} F'(\phi) F''(\phi) \partial_t \phi \, {\rm d} x\\
&= \int_{\Omega} |F'(\phi)|^{p-2} F'(\phi) F''(\phi)
\Big( - \textbf{\textit{u}}\cdot \nabla \phi + \Delta \phi -F'(\phi) + U \Big) \, {\rm d} x.
\end{align*}
Since
$$
\int_{\Omega} |F'(\phi)|^{p-2} F'(\phi) F''(\phi) \textbf{\textit{u}} \cdot \nabla \phi \, {\rm d} x=
\int_{\Omega} \textbf{\textit{u}} \cdot \nabla \Big( \frac{1}{p} |F'(\phi)|^p \Big) \, {\rm d} x=0,
$$
we deduce that
\begin{align*}
& \frac{1}{p} \frac{\d}{\d t} \int_{\Omega} |F'(\phi)|^p \, {\rm d} x
+ \int_{\Omega} \Big( (p-1) |F'(\phi)|^{p-2} F''(\phi)^2 +
|F'(\phi)|^{p-1} F'(\phi) F'''(\phi) \Big) |\nabla \phi|^2 \, {\rm d} x\\
&\quad +\int_{\Omega} |F'(\phi)|^{p} F''(\phi) \, {\rm d} x
= \int_{\Omega} |F'(\phi)|^{p-2} F'(\phi) F''(\phi)
U \, {\rm d} x.
\end{align*}
We notice that the second term on the left-hand side is non-negative.
Next, we observe that
$$
F''(s)\leq \theta \mathrm{e}^{\frac{2}{\theta} |F'(s)|}, \quad \forall \, s \in (-1,1).
$$
Owing to the above inequality, and using the fact that $s \leq \mathrm{e}^s$ for $s\geq 0$, we deduce that
\begin{align*}
\log \Big(|F'(s)|^{p-1} F''(s) \Big) \leq \log (\theta)+ \Big(1+\frac{2}{\theta} \Big) (p-1) |F'(s)|, \quad \forall \, s \in (-1,1).
\end{align*}
Thus, we get
\begin{equation}
|F'(s)|^{p-1} F''(s) \log \Big(|F'(s)|^{p-1} F''(s) \Big) \leq C_1 p |F'(s)|^p F''(s) + C_2, \quad \forall \, s \in (-1,1),
\end{equation}
for some $C_1,C_2>0$ independent of $p$. Recalling
\begin{equation*}
xy \leq \varepsilon x \log x + {e}^{\frac{y}{\varepsilon}},\quad \forall \, x>0, y>0,\quad \varepsilon \in (0,1),
\end{equation*}
and taking $\varepsilon = \frac{1}{2C_1 p}$, we arrive at
\begin{align*}
\frac{1}{p} \frac{\d}{\d t} \int_{\Omega} |F'(\phi)|^p \, {\rm d} x
+ \frac12 \int_{\Omega} |F'(\phi)|^{p} F''(\phi) \, {\rm d} x
\leq \frac{C_2 |\Omega|}{2C_1}+ \int_{\Omega} {e}^{2C_1 p |U| }\, {\rm d} x.
\end{align*}
Since $U$ is globally bounded, we obtain
$$
\frac{C_2 |\Omega|}{2C_1}+ \int_{\Omega} {e}^{2C_1 p |U| }\, {\rm d} x
\leq \frac{C_2 |\Omega|}{2C_1} + |\Omega| {e}^{2C_3 p} \leq C_4
{e}^{C_5 p},
$$
for some $C_4,C_5>0$ independent of $p$ and $t$.
Observing that $ F''(s)\geq \theta$ for all $s\in (-1,1)$, we rewrite the above differential inequality for $p\geq 2$ as follows
\begin{equation}
\frac{\d}{\d t} \int_{\Omega} |F'(\phi)|^p \, {\rm d} x
+\theta \int_{\Omega} |F'(\phi)|^{p} \, {\rm d} x
\leq C_4 p {e}^{C_5 p}.\nonumber
\end{equation}
By applying the Gronwall lemma on the time interval $[\sigma', \infty)$, we infer that
\begin{equation}
\| F'(\phi(t))\|_{L^p(\Omega)}^p \leq \| F'(\phi (\sigma'))\|_{L^p(\Omega)}^p e^{-\theta (t-\sigma')} + \frac{C_4 p {e}^{C_5 p}}{\theta}, \quad \forall \, t \geq \sigma'.
\end{equation}
We recall the elementary inequality for $q<1$
$$
(x+y)^q\leq x^q+y^q, \quad \forall \, x>0,y>0.
$$
Choosing $q=\frac{1}{p}$, with $p\geq 2$, we find
\begin{equation}
\label{di-sp}
\| F'(\phi(t))\|_{L^p(\Omega)} \leq \| F'(\phi(\sigma'))\|_{L^p(\Omega)} {e}^{-\frac{\theta (t-\sigma')}{p}} +\Big( \frac{C_4 p}{\theta}\Big)^{\frac{1}{p}} {e}^{C_5}, \quad \forall \, t \geq \sigma'.\nonumber
\end{equation}
Recalling \eqref{F'sigma} and taking the limit as $p\rightarrow +\infty$, we deduce that
\begin{equation}
\label{sp1}
\| F'(\phi(t))\|_{L^\infty(\Omega)} \leq \| F'(\phi (\sigma'))\|_{L^\infty(\Omega)} + {e}^{C_5}, \quad \forall \, t \geq \sigma'. \nonumber
\end{equation}
As a result, there exists $\delta=\delta(\sigma)>0$ such that
$$
-1+\delta \leq \phi(x,t) \leq 1-\delta, \quad \forall \, x \in \overline{\Omega}, \ t \geq \sigma'.
$$
The proof is complete. \hfill $\square$
\section{Mass-conserving Euler-Allen-Cahn System in Two Dimensions}
\label{EAC-sec}
\setcounter{equation}{0}
In this section, we study the dynamics of ideal two-phase flows in a bounded domain $\Omega \subset \mathbb{R}^2$ with smooth boundary, which is described by the mass-conserving Euler-Allen-Cahn system:
\begin{equation}
\label{EAC}
\begin{cases}
\partial_t \textbf{\textit{u}} + \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}} + \nabla P= - \mathrm{div}\,(\nabla \phi \otimes \nabla \phi),\\
\mathrm{div}\, \textbf{\textit{u}}=0,\\
\partial_t \phi +\textbf{\textit{u}}\cdot \nabla \phi + \mu= \overline{\mu}, \\
\mu= -\Delta \phi + \Psi' (\phi),
\end{cases}
\quad \text{ in } \Omega \times (0,T).
\end{equation}
The above system corresponds to the inviscid NS-AC system \eqref{NSAC-D} (i.e. $\nu\equiv 0$) with matched densities (i.e. $\rho \equiv 1$). The system is subject to the following boundary conditions
\begin{equation} \label{boundaryE}
\textbf{\textit{u}}\cdot \textbf{\textit{n}} =0,\quad \partial_{\textbf{\textit{n}}} \phi =0 \quad \text{ on } \partial\Omega
\times (0,T),
\end{equation}
and initial conditions
\begin{equation}
\label{ICE}
\textbf{\textit{u}}(\cdot, 0)= \textbf{\textit{u}}_0, \quad \phi(\cdot, 0)=\phi_0 \quad \text{ in } \Omega.
\end{equation}
The main result of this section is as follows:
\begin{theorem}
\label{Th-EAC}
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^2$.
\begin{itemize}
\item[1.] Assume that $\textbf{\textit{u}}_0 \in \mathbf{H}_\sigma\cap \mathbf{H}^1(\Omega)$, $\phi_0\in H^2(\Omega)$ such that
$F'(\phi_0)\in L^2(\Omega)$, $\| \phi_0\|_{L^\infty(\Omega)}\leq 1$,
$|\overline{\phi}_0|<1$ and $\partial_\textbf{\textit{n}} \phi_0=0$ on $\partial \Omega$. Then, there exists a global solution $(\textbf{\textit{u}},\phi)$
which satisfies the problem \eqref{EAC}-\eqref{ICE} in the sense of distribution on $\Omega \times (0,\infty)$ and, for all $T>0$,
\begin{align*}
&\textbf{\textit{u}} \in L^\infty(0,T;\mathbf{H}_\sigma \cap \mathbf{H}^1(\Omega)), \quad \phi \in L^\infty(0,T; H^2(\Omega))\cap L^2(0,T; W^{2,p}(\Omega)),\\
&\partial_t \phi \in L^\infty(0,T;L^2(\Omega)) \cap L^2(0,T;H^1(\Omega)),\\
&\phi \in L^\infty(\Omega\times (0,T)) : |\phi(x,t)|<1 \ \text{a.e. in} \ \Omega\times(0,T),
\end{align*}
where $p\in (2,\infty)$. Moreover, $\partial_\textbf{\textit{n}} \phi=0$ on $\partial\Omega\times(0,\infty)$.
\medskip
\item[2.] Assume that $
\textbf{\textit{u}}_0 \in \mathbf{H}_\sigma\cap \mathbf{W}^{1,p}(\Omega)$, $p \in (2,\infty)$, $\phi_0\in H^2(\Omega)
$ such that $F'(\phi_0)\in L^2(\Omega)$,
$F''(\phi_0) \in L^1(\Omega)$, $\| \phi_0\|_{L^\infty(\Omega)}\leq 1$, $ |\overline{\phi}_0|<1$, $\partial_\textbf{\textit{n}} \phi_0=0$ on $\partial \Omega$, and in addition $\nabla \mu_0= \nabla ( -\Delta \phi_0+F'(\phi_0)) \in \mathbf{L}^2(\Omega)$. Then, there exists a global solution $(\textbf{\textit{u}},\phi)$ which satisfies the problem \eqref{EAC}-\eqref{ICE} almost everywhere in $\Omega \times (0,\infty)$ and, for all $T>0$,
\begin{align*}
&\textbf{\textit{u}} \in L^\infty(0,T;\mathbf{H}_\sigma \cap \mathbf{W}^{1,p}(\Omega)), \quad \phi \in L^\infty(0,T; W^{2,p}(\Omega)),\\
&\partial_t \phi \in L^\infty(0,T;H^1(\Omega)) \cap L^2(0,T;H^2(\Omega)),\\
&\phi \in L^\infty(\Omega\times (0,T)) : |\phi(x,t)|<1 \ \text{a.e. in} \ \Omega\times(0,T).
\end{align*}
In addition, for any $\sigma>0$, there exists $\delta=\delta(\sigma)>0$ such that
$$
-1+\delta \leq \phi(x,t) \leq 1-\delta, \quad \forall \, x \in \overline{\Omega}, \ t \geq \sigma.
$$
\end{itemize}
\end{theorem}
To prove Theorem \ref{Th-EAC}, we first derive formal estimates leading to the required estimates of solutions.
Then the existence results can be proved by a suitable approximation scheme with fixed point arguments and then passing to the limit, which is standard owing to
uniform estimates obtained in the first step. Hence, here below we only focus on the \textit{a priori} estimates and omit further details.
\medskip
\subsection{Case 1}
Let us first consider initial datum $(\textbf{\textit{u}}_0,\phi_0)$ such that
$$
\textbf{\textit{u}}_0 \in \mathbf{H}_\sigma\cap \mathbf{H}^1(\Omega), \quad \phi_0\in H^2(\Omega), \quad \partial_\textbf{\textit{n}} \phi_0=0\ \ \text{on}\ \partial\Omega,
$$
with
$$
\| \phi_0\|_{L^\infty(\Omega)}\leq 1, \quad |\overline{\phi}_0|<1 \quad \text{and} \quad F'(\phi_0)\in L^2(\Omega).
$$
\textbf{Lower-order estimate.} As in the previous section, we have the conservation of mass
$$
\overline{\phi}(t)= \overline{\phi}_0, \quad \forall \, t \geq 0.
$$
By the same argument for \eqref{BEL-D}, we deduce the energy balance
\begin{equation}
\label{EE2}
\frac{\d}{\d t} E(\textbf{\textit{u}}, \phi) + \|\partial_t \phi + \textbf{\textit{u}} \cdot \nabla \phi\|_{L^2(\Omega)}^2=0.
\end{equation}
Integrating the above relation on $[0,t]$, we find
$$
E(\textbf{\textit{u}}(t),\phi(t))+ \int_0^t \| \partial_t \phi + \textbf{\textit{u}} \cdot \nabla \phi\|_{L^2(\Omega)}^2 \, {\rm d} \tau = E(\textbf{\textit{u}}_0,\phi_0), \quad \forall \, t \geq 0.
$$
This implies that
\begin{equation}
\label{E1}
\textbf{\textit{u}} \in L^\infty(0,T; \mathbf{H}_\sigma), \quad \phi\in L^\infty(0,T;H^1(\Omega)), \quad \partial_t \phi + \textbf{\textit{u}} \cdot \nabla \phi \in L^2(0,T;L^2(\Omega)),
\end{equation}
where the last property also implies $\mu-\overline{\mu}\in L^2(0,T;L^2(\Omega))$. In addition, it follows from the estimates \eqref{H2-D} and \eqref{mubar} that
$$
\phi \in L^2(0,T;H^2(\Omega)),\ \ \mu\in L^2(0,T; L^2(\Omega))\ \ \text{and}\ \ F'(\phi)\in L^2(0,T;L^2(\Omega)).
$$
The latter entails that $\phi \in L^\infty(\Omega \times (0,T))$ such that
$|\phi(x,t)|<1$ almost everywhere in $\Omega\times(0,T)$.
We remark that in comparison with the viscous case, it is not possible at this stage to prove that $\partial_t \phi\in L^2(\Omega\times (0,T))$.
\medskip
\textbf{Higher-order estimates.}
In the two dimensional case, it is convenient to consider the equation for the vorticity $\omega= \frac{\partial u_2}{\partial x_1}-\frac{\partial u_1}{\partial x_2}$ that reads as follows
\begin{equation}
\label{vort-eq}
\partial_t \omega+ \textbf{\textit{u}} \cdot \nabla \omega = \nabla \mu \cdot (\nabla \phi)^\perp,
\end{equation}
where $\textbf{\textit{v}}^\perp= (v_2,-v_1)$ for any $\textbf{\textit{v}}=(v_1,v_2)$. Multiplying \eqref{vort-eq} by $\omega$ and integrating over $\Omega$, we obtain
\begin{equation}
\label{Vor1}
\frac12 \frac{\d}{\d t} \| \omega\|_{L^2(\Omega)}^2= \int_{\Omega} \nabla \mu \cdot (\nabla \phi)^\perp \, \omega \, {\rm d} x.
\end{equation}
On the other hand, differentiating \eqref{EAC}$_3$ with respect to time, multiplying by $\partial_t \phi$ and integrating over $\Omega$, we find
\begin{align}
\frac12 \frac{\d}{\d t} \| \partial_t \phi\|_{L^2(\Omega)}^2+ \int_{\Omega} \partial_t \textbf{\textit{u}} \cdot \nabla \phi \, \partial_t \phi \, {\rm d} x + \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2
+\int_{\Omega} F''(\phi) |\partial_t \phi|^2 \, {\rm d} x= \theta_0 \| \partial_t \phi\|_{L^2(\Omega)}^2.
\label{TestAC2}
\end{align}
Here we have used the following equalities
$$
\int_{\Omega} \textbf{\textit{u}}\cdot \nabla \partial_t \phi \, \partial_t \phi \, {\rm d} x=
\int_{\Omega} \textbf{\textit{u}}\cdot \nabla \Big( \frac12 |\partial_t \phi|^2 \Big) \, {\rm d} x=0 \quad
\text{and}\quad
\int_{\Omega} \partial_t \phi \, {\rm d} x=0.
$$
We now define
$$
H(t) = \frac12 \| \omega\|_{L^2(\Omega)}^2+ \frac12 \| \partial_t \phi\|_{L^2(\Omega)}^2.
$$
By adding together \eqref{Vor1} and \eqref{TestAC2}, we infer from the convexity of $F$ (i.e. $F''>0$) that
\begin{equation}
\label{In-EAC}
\frac{\d}{\d t} H(t) + \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2 \leq \int_{\Omega} \nabla \mu \cdot (\nabla \phi)^\perp \, \omega \, {\rm d} x - \int_{\Omega} \partial_t \textbf{\textit{u}} \cdot \nabla \phi \, \partial_t \phi \, {\rm d} x + \theta_0 \| \partial_t \phi\|_{L^2(\Omega)}^2.
\end{equation}
Before proceeding to control the terms on the right-hand side of \eqref{In-EAC}, we rewrite the second one using the Euler equation.
We first observe that
$$
\partial_t \textbf{\textit{u}} = \mathbb{P} \big( \mu \nabla \phi - \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}} \big),
$$
where $\mathbb{P}$ is the Leray projection operator.
Thus, we write
\begin{align*}
\int_{\Omega} \partial_t \textbf{\textit{u}} \cdot \nabla \phi \, \partial_t \phi \, {\rm d} x
&= \int_{\Omega} \mathbb{P} \big( \mu \nabla \phi - \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}} \big) \cdot \nabla \phi \, \partial_t \phi \, {\rm d} x \\
&= \int_{\Omega} \mu \nabla \phi \cdot \mathbb{P} \big( \nabla \phi \, \partial_t \phi\big) \, {\rm d} x - \int_{\Omega} (\textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}) \cdot \mathbb{P} \big( \nabla \phi \, \partial_t \phi\big) \, {\rm d} x \\
&= -\int_{\Omega} \mu \nabla \phi \cdot \mathbb{P} \big( \phi \,\nabla \partial_t \phi\big) \, {\rm d} x - \int_{\Omega} \mathrm{div}\, ( \textbf{\textit{u}} \otimes \textbf{\textit{u}} ) \cdot \mathbb{P} \big( \nabla \phi \, \partial_t \phi\big) \, {\rm d} x \\
&= -\int_{\Omega} \mu \nabla \phi \cdot \mathbb{P} \big( \phi \,\nabla \partial_t \phi\big) \, {\rm d} x + \int_{\Omega} (\textbf{\textit{u}} \otimes \textbf{\textit{u}}) : \nabla \mathbb{P} \big( \nabla \phi \, \partial_t \phi\big) \, {\rm d} x \\
&\quad - \int_{\partial \Omega} \textbf{\textit{u}}\otimes \textbf{\textit{u}} \mathbb{P}\big( \nabla \phi \, \partial_t \phi\big) \cdot \textbf{\textit{n}} \, {\rm d} \sigma \\
&= -\int_{\Omega} \mu \nabla \phi \cdot \mathbb{P} \big( \phi \,\nabla \partial_t \phi\big) \, {\rm d} x + \int_{\Omega} (\textbf{\textit{u}} \otimes \textbf{\textit{u}}) : \nabla \mathbb{P} \big( \nabla \phi \, \partial_t \phi\big) \, {\rm d} x \\
&\quad - \int_{\partial \Omega} (\textbf{\textit{u}} \cdot \textbf{\textit{n}}) \big(\textbf{\textit{u}} \cdot \mathbb{P}\big( \nabla \phi \, \partial_t \phi\big) \big) \, {\rm d} \sigma \\
&= -\int_{\Omega} \mu \nabla \phi \cdot \mathbb{P} \big( \phi \,\nabla \partial_t \phi\big) \, {\rm d} x + \int_{\Omega} (\textbf{\textit{u}} \otimes \textbf{\textit{u}}) : \nabla \mathbb{P} \big( \nabla \phi \, \partial_t \phi\big) \, {\rm d} x.
\end{align*}
Here we have used that $\mathbb{P}( \nabla v)=0$ for any $v \in H^1(\Omega)$, the relation $\mathrm{div}\, (S^t \textbf{\textit{v}})= S^t : \nabla \textbf{\textit{v}}+ \mathrm{div}\, S \cdot \textbf{\textit{v}}$ for any $d \times d$ tensor $S$ and vector $\textbf{\textit{v}}$, and the no-normal flow condition $\textbf{\textit{u}} \cdot \textbf{\textit{n}} =0$ at the boundary.
As a consequence, we rewrite \eqref{In-EAC} as follows
\begin{align}
\frac{\d}{\d t} H(t) + \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2
& \leq \int_{\Omega} \nabla \mu \cdot (\nabla \phi)^\perp \, \omega \, {\rm d} x + \int_{\Omega} \mu \nabla \phi \cdot \mathbb{P} \big( \phi \, \nabla \partial_t \phi\big) \, {\rm d} x \notag \\
&\quad - \int_{\Omega} (\textbf{\textit{u}} \otimes \textbf{\textit{u}}) : \nabla \mathbb{P} \big( \nabla \phi \, \partial_t \phi\big) \, {\rm d} x + \theta_0 \| \partial_t \phi\|_{L^2(\Omega)}^2.
\label{In-EAC2}
\end{align}
We now turn to estimate the right-hand side of \eqref{In-EAC2}.
By H\"{o}lder's inequality, we have
\begin{equation}
\label{I1}
\int_{\Omega} \nabla \mu \cdot (\nabla \phi)^\perp \, \omega \, {\rm d} x
\leq \| \nabla \mu\|_{L^2(\Omega)} \| \nabla \phi\|_{L^\infty(\Omega)} \| \omega\|_{L^2(\Omega)}.
\end{equation}
By taking the gradient of \eqref{EAC}$_3$, we observe that
$$
\| \nabla \mu\|_{L^2(\Omega)}
\leq \| \nabla \partial_t \phi\|_{L^2(\Omega)}+ \|\nabla^2 \phi\, \textbf{\textit{u}} \|_{L^2(\Omega)} + \| \nabla \textbf{\textit{u}} \, \nabla \phi\|_{L^2(\Omega)}.
$$
Recalling the elementary inequality
$$
\| \textbf{\textit{v}}\|_{H^1(\Omega)}\leq C \Big(\| \textbf{\textit{v}}\|_{L^2(\Omega)} +
\| \mathrm{div}\, \textbf{\textit{v}}\|_{L^2(\Omega)} + \| \mathrm{curl}\, \textbf{\textit{v}}\|_{L^2(\Omega)}+ \| \textbf{\textit{v}}\cdot \textbf{\textit{n}}\|_{H^\frac12(\partial \Omega)}\Big), \quad \forall \, \textbf{\textit{v}} \in \mathbf{H}^1(\Omega),
$$
and exploiting Lemma \ref{result1} as well as \eqref{BGW}, we find that
\begin{align*}
\| \nabla \mu\|_{L^2(\Omega)}
&\leq \| \nabla \partial_t \phi\|_{L^2(\Omega)}
+ C \| \textbf{\textit{u}}\|_{H^1(\Omega)} \| \nabla^2 \phi\|_{L^2(\Omega)} \log^\frac12
\Big( C \frac{\| \nabla^2 \phi\|_{L^{p}(\Omega)}}{\| \nabla^2 \phi\|_{L^2(\Omega)}} \Big) \\
&\quad
+ \| \nabla \textbf{\textit{u}} \|_{L^2(\Omega)} \|\nabla \phi\|_{L^\infty(\Omega)}\\
&\leq \| \nabla \partial_t \phi\|_{L^2(\Omega)}
+C (1+ \| \omega\|_{L^2(\Omega)}) \| \nabla^2 \phi\|_{L^2(\Omega)} \log^\frac12
\Big( C \frac{\| \nabla^2 \phi\|_{L^{p}(\Omega)}}{\| \nabla^2 \phi\|_{L^2(\Omega)}} \Big)\\
&\quad + C (1+ \| \omega\|_{L^2(\Omega)})
\|\nabla \phi\|_{H^1(\Omega)}\log^\frac12
\Big( C \frac{\| \nabla \phi\|_{W^{1,p}(\Omega)}}{\| \nabla \phi\|_{H^1(\Omega)}} \Big),
\end{align*}
for some $p>2$. Using \eqref{ineq0}, we rewrite the above estimate as follows
\begin{align*}
\| \nabla \mu\|_{L^2(\Omega)}
&\leq \| \nabla \partial_t \phi\|_{L^2(\Omega)}
+C (1+ \| \omega\|_{L^2(\Omega)}) \Big( \| \nabla \phi\|_{H^1(\Omega)}
\log^\frac12 \big( C \|\nabla \phi\|_{W^{1,p}(\Omega)} \big) + 1 \Big).
\end{align*}
Then, using again the inequality \eqref{BGW}, \eqref{I1} can be controlled as follows
\begin{align*}
\int_{\Omega} \nabla \mu \cdot (\nabla \phi)^\perp \, \omega \, {\rm d} x
&\leq \| \nabla \partial_t \phi\|_{L^2(\Omega)} \| \omega\|_{L^2(\Omega)} \|\nabla \phi\|_{H^1(\Omega)}\log^\frac12
\Big( C \frac{\| \nabla \phi\|_{W^{1,p}(\Omega)}}{\| \nabla \phi\|_{H^1(\Omega)}} \Big) \\
&\quad + C \| \omega\|_{L^2(\Omega)}(1+ \| \omega\|_{L^2(\Omega)}) \Big( \|\nabla \phi\|_{H^1(\Omega)}
\log^\frac12 \big( C \| \nabla \phi\|_{W^{1,p}(\Omega)} \big) + 1 \Big)\\
&\quad \times
\|\nabla \phi\|_{H^1(\Omega)}\log^\frac12
\Big( C \frac{\| \nabla \phi\|_{W^{1,p}(\Omega)}}{\| \nabla \phi\|_{H^1(\Omega)}} \Big)\\
&\leq \| \nabla \partial_t \phi\|_{L^2(\Omega)} \| \omega\|_{L^2(\Omega)} \Big( \|\nabla \phi\|_{H^1(\Omega)}
\log^\frac12 \big( C \| \nabla \phi\|_{W^{1,p}(\Omega)}\big) +1\Big)\\
&\quad + C (1+\| \omega\|_{L^2(\Omega)}^2) \Big( \| \nabla \phi\|_{H^1(\Omega)}^2
\log \big( C \| \nabla \phi\|_{W^{1,p}(\Omega)} \big) + 1 \Big)\\
&\leq \frac16 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2+
C (1+\| \omega\|_{L^2(\Omega)}^2) \Big( \| \phi\|_{H^2(\Omega)}^2
\log \big( C \| \phi\|_{W^{2,p}(\Omega)} \big) + 1 \Big),
\end{align*}
for some $p>2$.
Next, since $\phi$ is globally bounded, we have
\begin{align*}
\int_{\Omega} \mu \nabla \phi \cdot \mathbb{P} \big( \phi \, \nabla \partial_t \phi\big) \, {\rm d} x
&\leq C \| \mu\|_{L^2(\Omega)} \| \nabla \phi\|_{L^\infty(\Omega)} \| \phi \nabla \partial_t \phi \|_{L^2(\Omega)}\\
&\leq C \| \mu\|_{L^2(\Omega)} \|\nabla \phi\|_{H^1(\Omega)}\log^\frac12
\Big( C \frac{\| \nabla \phi\|_{W^{1,p}(\Omega)}}{\| \nabla \phi\|_{H^1(\Omega)}} \Big) \| \phi\|_{L^\infty(\Omega)} \| \nabla \partial_t \phi\|_{L^2(\Omega)}\\
&\leq C \| \mu\|_{L^2(\Omega)} \Big( \| \phi\|_{H^2(\Omega)}
\log^\frac12 \big( C \| \phi\|_{W^{2,p}(\Omega)} \big) + 1 \Big) \| \nabla \partial_t \phi\|_{L^2(\Omega)},
\end{align*}
for some $p>2$. In order to estimate the $L^2$-norm of $\mu$, we notice that
\begin{align*}
\| \mu-\overline{\mu}\|_{L^2(\Omega)}
&\leq \| \partial_t \phi\|_{L^2(\Omega)}+ \| \textbf{\textit{u}}\cdot \nabla \phi\|_{L^2(\Omega)}\\
&\leq \| \partial_t \phi\|_{L^2(\Omega)}+ \|\textbf{\textit{u}} \|_{L^4(\Omega)} \| \nabla \phi\|_{L^4(\Omega)} \\
&\leq \| \partial_t \phi\|_{L^2(\Omega)}+ C \|\textbf{\textit{u}} \|_{L^2(\Omega)}^\frac12 \| \textbf{\textit{u}}\|_{H^1(\Omega)}^\frac12 \| \nabla \phi\|_{L^2(\Omega)}^\frac12 \| \phi\|_{H^2(\Omega)}^\frac12 \\
&\leq \| \partial_t \phi\|_{L^2(\Omega)}+ C(1+\| \omega\|_{L^2(\Omega)})^\frac12 (1+\| \mu-\overline{\mu}\|_{L^2(\Omega)})^\frac12 \nonumber\\
&\leq \| \partial_t \phi\|_{L^2(\Omega)}+ C(1+\| \omega\|_{L^2(\Omega)}) +\frac12 \| \mu-\overline{\mu}\|_{L^2(\Omega)}.
\end{align*}
Here we have used the equation \eqref{EAC}$_3$, the Ladyzhenskaya inequality, and the estimates \eqref{H2-D}, \eqref{E1}. Since $\|\mu\|_{L^2(\Omega)}\leq C(1+\| \mu-\overline{\mu}\|_{L^2(\Omega)})$ (recalling \eqref{mubar}), we then infer that
$$
\| \mu\|_{L^2(\Omega)}\leq C(1+ \| \partial_t \phi\|_{L^2(\Omega)}+ \| \omega\|_{L^2(\Omega)} ).
$$
Thus, we can deduce that
\begin{align*}
\int_{\Omega} &\mu \nabla \phi \cdot \mathbb{P} \big( \phi \, \nabla \partial_t \phi\big) \, {\rm d} x\\
&\leq \frac16 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2 + C \big(1+ \| \partial_t \phi\|_{L^2(\Omega)}^2 + \| \omega\|_{L^2(\Omega)}^2 \big)\Big( \| \phi\|_{H^2(\Omega)}^2
\log \big( C \| \phi\|_{W^{2,p}(\Omega)} \big) + 1 \Big).
\end{align*}
Recalling that $\mathbb{P}$ is a bounded operator from $\mathbf{H}^1(\Omega)$ to $\mathbf{H}_\sigma \cap \mathbf{H}^1(\Omega)$, and using the inequalities \eqref{LADY}, \eqref{BGW}, Poincar\'{e}'s inequality and Lemma \ref{result1}, we have
\begin{align*}
&-\int_{\Omega} (\textbf{\textit{u}} \otimes \textbf{\textit{u}}) : \nabla \mathbb{P} \big( \nabla \phi \, \partial_t \phi\big) \, {\rm d} x \\
&\quad \leq \| \textbf{\textit{u}}\|_{L^4(\Omega)}^2 \|\mathbb{P} (\nabla \phi \, \partial_t \phi) \|_{H^1(\Omega)}\\
&\quad \leq C \| \textbf{\textit{u}}\|_{L^2(\Omega)} \| \textbf{\textit{u}}\|_{H^1(\Omega)} \| \nabla \phi \, \partial_t \phi\|_{H^1(\Omega)}\\
&\quad \leq C (1+\| \omega\|_{L^2(\Omega)}) \Big( \| \nabla \phi \, \partial_t \phi\|_{L^2(\Omega)} +
\| \nabla^2 \phi \,\partial_t \phi \|_{L^2(\Omega)}+ \| \nabla \phi \, \nabla \partial_t \phi\|_{L^2(\Omega)} \Big) \\
&\quad \leq C (1+\| \omega\|_{L^2(\Omega)}) \Big[ \| \nabla \phi\|_{L^\infty(\Omega)} \| \nabla \partial_t \phi\|_{L^2(\Omega)}\\
&\qquad + \| \nabla \partial_t \phi\|_{L^2(\Omega)} \| \nabla^2 \phi\|_{L^2(\Omega)} \log^\frac12
\Big( C \frac{\| \nabla^2 \phi\|_{L^{p}(\Omega)}}{\| \nabla^2 \phi\|_{L^2(\Omega)}} \Big) \Big] \\
&\quad \leq C (1+\| \omega\|_{L^2(\Omega)}) \| \nabla \partial_t \phi\|_{L^2(\Omega)}
\Big( \|\nabla \phi\|_{H^1(\Omega)}\log^\frac12
\Big( C \frac{\| \nabla \phi\|_{W^{1,p}(\Omega)}}{\| \nabla \phi\|_{H^1(\Omega)}} \Big)\\
&\qquad + \| \nabla^2 \phi\|_{L^2(\Omega)} \log^\frac12
\Big( C \frac{\| \nabla^2 \phi\|_{L^{p}(\Omega)}}{\| \nabla^2 \phi\|_{L^2(\Omega)}} \Big) \Big)\\
&\quad \leq C (1+\| \omega\|_{L^2(\Omega)}) \| \nabla \partial_t \phi\|_{L^2(\Omega)}
\Big( \| \phi\|_{H^2(\Omega)} \log^\frac12\big( C\| \phi\|_{W^{2,p}(\Omega)}\big)+1 \Big)\\
&\quad \leq \frac16 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2+
C\big(1+\|\omega \|_{L^2(\Omega)}^2\big) \Big( \| \phi\|_{H^2(\Omega)}^2 \log\big( C\| \phi\|_{W^{2,p}(\Omega)}\big)+1 \Big),
\end{align*}
for some $p>2$.
Combining the above estimates together with \eqref{In-EAC2}, we arrive at the differential inequality
\begin{align}
\label{In-EAC3}
\frac{\d}{\d t} H(t) + \frac12 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2
\leq C (1+H(t)) \Big( \| \phi\|_{H^2(\Omega)}^2 \log\big( C \| \phi\|_{W^{2,p}(\Omega)}+1 \Big).
\end{align}
In order to close the estimate, we are left to absorb the logarithmic term on the right-hand side of the above differential inequality. To this aim, we first multiply $\mu=-\Delta \phi+\Psi'(\phi)$ by $|F'(\phi)|^{p-2}F'(\phi)$, for some $p>2$, and integrate over $\Omega$. After integrating by parts and using the boundary condition for $\phi$, we obtain
$$
\int_{\Omega} (p-1)|F'(\phi)|^{p-2} F''(\phi) |\nabla \phi|^2 \, {\rm d} x+
\| F'(\phi)\|_{L^p(\Omega)}^p= \int_{\Omega} (\mu +\theta_0 \phi)|F'(\phi)|^{p-2}F'(\phi) \, {\rm d} x.
$$
By Young's inequality and the fact that $F''>0$, we deduce
$$
\| F'(\phi)\|_{L^p(\Omega)} \leq C(1+ \|\mu\|_{L^p(\Omega)} ).
$$
Using a well-known elliptic regularity result, together with the above inequality and \eqref{E1}, we obtain that (cf. \eqref{pw2p})
$$
\| \phi\|_{W^{2,p}(\Omega)} \leq C (1+\| \mu\|_{L^p(\Omega)}).
$$
On the other hand, we infer from equation \eqref{EAC}$_3$ that
\begin{align*}
\|\mu-\overline{\mu} \|_{L^p(\Omega)}
\leq \| \partial_t \phi\|_{L^p(\Omega)} + \| \textbf{\textit{u}} \cdot \nabla \phi\|_{L^p(\Omega)}.
\end{align*}
Then by Poincar\'{e}'s inequality and a Sobolev embedding theorem, we find
\begin{align*}
\| \mu\|_{L^p(\Omega)}
&\leq C \|\mu-\overline{\mu} \|_{L^p(\Omega)} +C |\overline{\mu}|\\
&\leq C \| \nabla \partial_t \phi\|_{L^2(\Omega)}
+C \| \textbf{\textit{u}}\|_{H^1(\Omega)} \| \phi\|_{H^2(\Omega)} + C(1+\| \mu-\overline{\mu}\|_{L^2(\Omega)})\\
&\leq C \| \nabla \partial_t \phi\|_{L^2(\Omega)}
+C (1+\| \omega\|_{L^2(\Omega)}) (1+ \| \mu-\overline{\mu}\|_{L^2(\Omega)})\\
&\leq C \| \nabla \partial_t \phi\|_{L^2(\Omega)}
+C (1+\| \omega\|_{L^2(\Omega)}) (1+ \| \partial_t \phi\|_{L^2(\Omega)}+ \|\omega\|_{L^2(\Omega)}).
\end{align*}
Thus, for $p>2$, we reach
$$
\| \phi\|_{W^{2,p}(\Omega)} \leq C ( 1+ \| \nabla \partial_t \phi\|_{L^2(\Omega)} +
H(t) ),
$$
which, in turn, allows us to rewrite \eqref{In-EAC3} as
\begin{equation}
\label{In-EAC4}
\frac{\d}{\d t} H(t) + \frac12 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2
\leq C (1+H(t)) \Big( \| \phi\|_{H^2(\Omega)}^2 \log\Big( C\big( 1+ \| \nabla \partial_t \phi\|_{L^2(\Omega)} +
H(t)\big)\Big)+1 \Big).
\end{equation}
We now observe that, for any $\varepsilon>0$, the following inequality holds
$$
x\log(C y)\leq \varepsilon y+ x\log\Big(\frac{ C x}{\varepsilon}\Big) \quad \forall \, x, y >0.
$$
By using the above inequality with $x=1+H(t)$, $y= 1+ \| \nabla \partial_t \phi\|_{L^2(\Omega)} +H(t) $ and $\varepsilon=1$, we deduce that
\begin{align*}
& \frac{\d}{\d t} H(t) + \frac12 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2\nonumber\\
&\quad \leq \| \nabla \partial_t \phi\|_{L^2(\Omega)} \| \phi\|_{H^2(\Omega)}^2
+ C ( 1+\| \phi\|_{H^2(\Omega)}^2 ) (1+H(t)) \log \big( C (1+H(t))\big).
\end{align*}
By Young's inequality, we obtain
\begin{align*}
\frac{\d}{\d t} H(t) + \frac14 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2
\leq \| \phi\|_{H^2(\Omega)}^4
+ C ( 1+\| \phi\|_{H^2(\Omega)}^2 ) (1+H(t)) \log \big( C (1+H(t))\big).
\end{align*}
Recalling that $\| \phi\|_{H^2(\Omega)}^2 \leq C(1+H(t))$, we are finally led to the differential inequality
\begin{equation}
\label{In-EAC5}
\frac{\d}{\d t} H(t) + \frac14 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2
\leq C ( 1+\| \phi\|_{H^2(\Omega)}^2 ) (1+H(t)) \log \big( C (1+H(t))\big).
\end{equation}
Since $\phi \in L^2(0,T;H^2(\Omega))$, then applying the generalized Gronwall lemma \ref{GL2}, we find the double exponential bound
\begin{align*}
\sup_{t \in [0,T]} &\Big( \|\partial_t \phi (t)\|_{L^2(\Omega)}^2+ \|\omega(t) \|_{L^2(\Omega)}^2 \Big)\\
&\leq C \big(1+ \| \textbf{\textit{u}}_0\|_{H^1(\Omega)}^2 \| \phi_0\|_{H^2(\Omega)}^2
+\| \phi_0\|_{H^2(\Omega)}^2+ \| \Psi'(\phi_0)\|_{L^2(\Omega)}^2+ \| \textbf{\textit{u}}_0\|_{H^1(\Omega)}^2 \big)^{{e}^{\int_0^T 1+\| \phi(s)\|_{H^2(\Omega)}^2 \, {\rm d} s}},
\end{align*}
for some constant $C>0$.
Here we have used that
$$
\| \partial_t \phi (0)\|_{L^2(\Omega)} \leq C \| \textbf{\textit{u}}_0\|_{H^1(\Omega)} \| \phi_0\|_{H^2(\Omega)}
+C \| \phi_0\|_{H^2(\Omega)}+C \| \Psi'(\phi_0)\|_{L^2(\Omega)}.
$$
Hence, we get
\begin{equation}
\label{Reg-E1}
\partial_t \phi \in L^\infty(0,T;L^2(\Omega)) \cap L^2(0,T;H^1(\Omega)), \quad
\omega \in L^\infty(0,T;L^2(\Omega)),
\end{equation}
which, in turn, entail that
\begin{equation}
\label{Reg-E2}
\textbf{\textit{u}} \in L^\infty(0,T;\mathbf{H}^1(\Omega)), \quad \phi \in L^\infty(0,T; H^2(\Omega))\cap L^2(0,T; W^{2,p}(\Omega)),
\end{equation}
for any $p \in [2,\infty)$. \medskip
\subsection{Case 2}
We now consider an initial condition $(\textbf{\textit{u}}_0,\phi_0)$ such that
$$
\textbf{\textit{u}}_0 \in \mathbf{H}_\sigma\cap \mathbf{W}^{1,p}(\Omega) , \quad \phi_0\in H^2(\Omega),\quad\partial_\textbf{\textit{n}} \phi_0=0\ \ \text{on}\ \partial\Omega,
$$
for $p \in (2,\infty)$, with $\| \phi_0\|_{L^\infty(\Omega)}\leq 1$, $|\overline{\phi}_0|<1$ and
$$
F'(\phi_0)\in L^2(\Omega), \quad F''(\phi_0)\in L^1(\Omega), \quad \nabla \mu_0= \nabla ( -\Delta \phi_0+F'(\phi_0)) \in L^2(\Omega).
$$
Thanks to the first part of Theorem \ref{Th-EAC}, we have a solution $(\textbf{\textit{u}},\phi)$ satisfying \eqref{Reg-E1} and \eqref{Reg-E2}. Moreover, repeating the same argument performed in Section \ref{S-STRONG}, we have (cf. \eqref{EntE4})
$$
\frac{\d}{\d t} \int_{\Omega} F''(\phi) \, {\rm d} x + \frac14 \int_{\Omega} F'''(\phi) F'(\phi) \, {\rm d} x
\leq C,
$$
for some positive constant $C$ only depending on $\Omega$ and the parameters of the system. Since $F''(\phi_0)\in L^1(\Omega)$, we learn, in particular, that (cf. \eqref{EB3})
\begin{equation}
\int_t^{t+1}\! \int_{\Omega} |F''(\phi)|^2 \log ( 1+F''(\phi)) \,{\rm d} x {\rm d} \tau \leq
C, \quad \forall t \geq 0.
\end{equation}
Multiplying \eqref{vort-eq} by $|\omega|^{p-2}\omega$ ($p>2$) and integrating over $\Omega$, we obtain
$$
\frac{1}{p} \frac{\d}{\d t} \|\omega\|_{L^p(\Omega)}^p = \int_{\Omega} \nabla \mu \cdot (\nabla \phi)^\perp |\omega|^{p-2}\omega \, {\rm d} x.
$$
By H\"{o}lder's inequality, we easily get
$$
\frac{1}{p} \frac{\d}{\d t} \|\omega\|_{L^p(\Omega)}^p \leq \| \nabla \mu \cdot (\nabla \phi)^\perp\|_{L^p(\Omega)} \|\omega\|_{L^p(\Omega)}^{p-1},
$$
which, in turn, implies
$$
\frac12 \frac{\d}{\d t} \|\omega\|_{L^p(\Omega)}^2 \leq \| \nabla \mu \cdot (\nabla \phi)^\perp\|_{L^p(\Omega)} \| \omega\|_{L^p(\Omega)}.
$$
Next, differentiating \eqref{EAC}$_3$ with respect time, then multiplying the resultant by $-\Delta \partial_t \phi$ and integrating over $\Omega$, we obtain
\begin{align*}
& \frac{\d}{\d t} \frac12 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2+ \| \Delta \partial_t \phi\|_{L^2(\Omega)}^2
\notag \\
&= \theta_0 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2+ \int_{\Omega} F''(\phi) \partial_t \phi \Delta \partial_t \phi \, {\rm d} x+\int_{\Omega} (\partial_t \textbf{\textit{u}} \cdot \nabla \phi) \Delta \partial_t \phi\, {\rm d} x + \int_{\Omega} (\textbf{\textit{u}} \cdot \nabla \partial_t \phi) \Delta \partial_t \phi\, {\rm d} x.
\end{align*}
Here we have used the fact that $\overline{\Delta \partial_t \phi}=0$ since $\partial_\textbf{\textit{n}} \partial_t \phi=0$ on $\partial \Omega$. Collecting the above two estimates, we find that
\begin{align*}
&\frac{\d}{\d t} \Big( \frac12 \|\omega\|_{L^p(\Omega)}^2+\frac12 \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2 \Big) + \| \Delta \partial_t \phi\|_{L^2(\Omega)}^2 \\
&\quad
\leq \| \nabla \mu \cdot (\nabla \phi)^\perp\|_{L^p(\Omega)} \| \omega\|_{L^p(\Omega)} + \theta_0 \|\nabla \partial_t \phi\|_{L^2(\Omega)}^2+ \int_{\Omega} F''(\phi) \partial_t \phi \Delta \partial_t \phi \, {\rm d} x\\
&\qquad
+\int_{\Omega} (\partial_t \textbf{\textit{u}} \cdot \nabla \phi) \Delta \partial_t \phi\, {\rm d} x + \int_{\Omega} (\textbf{\textit{u}} \cdot \nabla \partial_t \phi) \Delta \partial_t \phi\, {\rm d} x.
\end{align*}
Notice that, by \eqref{EAC}$_3$, we have the relation $\nabla \mu= \nabla \partial_t \phi + (\nabla \textbf{\textit{u}})^t \nabla \phi + (\textbf{\textit{u}}\cdot \nabla ) \nabla \phi$. By exploiting this identity, we obtain
\begin{align*}
&\| \nabla \mu \cdot (\nabla \phi)^\perp\|_{L^p(\Omega)} \| \omega\|_{L^p(\Omega)}\\
&\quad \leq \big( \|\nabla \partial_t \phi\|_{L^p(\Omega)} + \|(\nabla \textbf{\textit{u}})^t \nabla \phi\|_{L^p(\Omega)} + \| (\textbf{\textit{u}}\cdot \nabla ) \nabla \phi\|_{L^p(\Omega)} \big) \|\nabla \phi\|_{L^\infty(\Omega)} \| \omega\|_{L^p(\Omega)}.
\end{align*}
Using the Gagliardo-Nirenberg inequality \eqref{GN2} and the following inequality for divergence free vector fields satisfying the boundary condition \eqref{boundaryE}$_1$
\begin{equation}
\label{u-v}
\| \nabla \textbf{\textit{u}}\|_{L^p(\Omega)}\leq C(p) \| \omega\|_{L^p(\Omega)}, \quad p \in [2,\infty),
\end{equation}
we deduce that
\begin{align*}
&\| \nabla \mu \cdot (\nabla \phi)^\perp\|_{L^p(\Omega)} \| \omega\|_{L^p(\Omega)}\\
&\leq C \| \nabla \partial_t \phi\|_{L^2(\Omega)}^\frac{2}{p} \| \Delta \partial_t \phi\|_{L^2(\Omega)}^{1-\frac{2}{p}} \| \nabla \phi\|_{L^\infty(\Omega)} \| \omega\|_{L^p(\Omega)} + C \| \nabla \textbf{\textit{u}}\|_{L^p(\Omega)} \| \nabla \phi\|_{L^\infty(\Omega)}^2 \| \omega\|_{L^p(\Omega)}\\
&\quad + \| \textbf{\textit{u}}\|_{L^\infty(\Omega)} \| \phi\|_{W^{2,p}(\Omega)}
\| \nabla \phi\|_{L^\infty(\Omega)} \| \omega\|_{L^p(\Omega)}\\
&\leq \frac18 \| \Delta \partial_t \phi\|_{L^2(\Omega)}^2+ C \| \nabla \phi\|_{L^\infty(\Omega)}^{\frac{2p}{p+2}} \| \nabla \partial_t \phi\|_{L^2(\Omega)}^{\frac{4}{p+2}} \| \omega\|_{L^p(\Omega)}^\frac{2p}{p+2}\\
&\quad + C\big( \| \nabla \phi\|_{L^\infty(\Omega)}^2 + \| \phi\|_{W^{2,p}(\Omega)} \| \nabla \phi\|_{L^\infty(\Omega)} \big) \big(1+ \| \omega\|_{L^p(\Omega)}^2\big).
\end{align*}
Next, using \eqref{EAC}$_1$ together with the bounds \eqref{Reg-E1}, we have
\begin{align*}
&\int_{\Omega} \partial_t \textbf{\textit{u}} \cdot \nabla \phi \Delta \partial_t \phi\, {\rm d} x\\
&\quad \leq \int_{\Omega} \mathbb{P} \big( -\textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}} -\Delta \phi \nabla \phi \big) \cdot \nabla \phi \Delta \partial_t \phi \, {\rm d} x\\
&\quad \leq C \| \mathbb{P}\big( \textbf{\textit{u}} \cdot \nabla \textbf{\textit{u}}\big)\|_{L^2(\Omega)} \|\nabla \phi \Delta \partial_t \phi \|_{L^2(\Omega)} + C \| \mathbb{P}\big( \Delta \phi \nabla \phi\big)\|_{L^2(\Omega)} \| \nabla \phi \Delta \partial_t \phi\|_{L^2(\Omega)}\\
&\quad \leq \frac{1}{8} \| \Delta\partial_t \phi \|_{L^2(\Omega)}^2
+C \| \textbf{\textit{u}}\|_{L^\infty(\Omega)}^2 \| \nabla \textbf{\textit{u}}\|_{L^2(\Omega)}^2 \| \nabla \phi\|_{L^\infty(\Omega)}^2+
C\| \Delta \phi \|_{L^2(\Omega)}^2 \| \nabla \phi\|_{L^\infty(\Omega)}^4\\
&\quad \leq \frac{1}{8} \| \Delta\partial_t \phi \|_{L^2(\Omega)}^2
+C (1+\| \omega\|_{L^p(\Omega)}^2) \| \nabla \phi\|_{L^\infty(\Omega)}^2+C\| \nabla \phi\|_{L^\infty(\Omega)}^4.
\end{align*}
Arguing as for \eqref{FFF} and \eqref{unpt}, we have
\begin{align}
& \int_{\Omega} F''(\phi) \partial_t \phi \Delta \partial_t \phi \, {\rm d} x
\leq \frac18 \| \Delta \partial_t \phi \|_{L^2(\Omega)}^2 +C \| F''(\phi)\|_{L^2(\Omega)}^2 \log \big( C\| F''(\phi)\|_{L^2(\Omega)} \big)\| \nabla \partial_t \phi\|_{L^2(\Omega)}^2,\nonumber
\end{align}
\begin{align}
&\int_{\Omega} (\textbf{\textit{u}} \cdot \nabla \partial_t \phi) \Delta \partial_t \phi\, {\rm d} x
\leq \frac{1}{8} \| \Delta \partial_t \phi\|_{L^2(\Omega)}^2 + C \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2.
\nonumber
\end{align}
Collecting the above estimates and using Young's inequality, we arrive at the differential inequality
\begin{align*}
&\frac{\d}{\d t} \Big( \|\omega\|_{L^p(\Omega)}^2+ \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2 \Big) + \| \Delta \partial_t \phi\|_{L^2(\Omega)}^2 \leq R_1(t) \Big( \|\omega\|_{L^p(\Omega)}^2+ \| \nabla \partial_t \phi\|_{L^2(\Omega)}^2 \Big) +R_2(t),
\end{align*}
where
$$
R_1= C \Big( 1+
\| \nabla \phi\|_{L^\infty(\Omega)}^2 + \| F''(\phi)\|_{L^2(\Omega)}^2 \log \big( C\| F''(\phi)\|_{L^2(\Omega)}\big) \Big)
$$
and
$$
R_2 = C \| \phi\|_{W^{2,p}(\Omega)}^2 + C \big(\| \nabla \phi\|_{L^\infty(\Omega)}^4+1).
$$
By using \eqref{BGW}, and recalling \eqref{ineq0}, we see that
\begin{align*}
\| \nabla \phi\|_{L^\infty(\Omega)}^4 \leq C \| \nabla^2 \phi\|_{L^2(\Omega)}^4 \log^2 \big( \| \nabla^2 \phi\|_{L^p(\Omega)} \big) +1 \leq C \log^2 \big( \| \phi\|_{W^{2,p}(\Omega)} \big) +1,
\end{align*}
for $p>2$. In light of \eqref{Reg-E2}, we infer that both $R_1$ and $R_2$ belong to $L^1(0,T)$. Thanks to Gronwall's lemma, we obtain
$$
\|\omega(t)\|_{L^p(\Omega)}^2+ \| \nabla \partial_t \phi(t)\|_{L^2(\Omega)}^2
\leq \Big( \|\omega(0)\|_{L^p(\Omega)}^2+ \| \nabla \partial_t \phi(0)\|_{L^2(\Omega)}^2 + \int_0^T R_2(\tau)\, {\rm d} \tau\Big) {e}^{\int_0^T R_1(\tau)\, {\rm d} \tau},
$$
for any $t\in [0,T]$. Since $\| \omega (0)\|_{L^p(\Omega)}\leq \| \nabla \textbf{\textit{u}}_0\|_{L^p(\Omega)}$ and
\begin{align*}
\| \nabla \partial_t \phi(0)\|_{L^2(\Omega)}
&\leq \| (\nabla \textbf{\textit{u}}_0)^t \nabla \phi_0\|_{L^2(\Omega)} + \| (\textbf{\textit{u}}_0\cdot \nabla) \nabla \phi_0\|_{L^2(\Omega)} + \|\nabla \mu_0 \|_{L^2(\Omega)}\\
&\leq C \| \nabla \textbf{\textit{u}}_0\|_{L^p(\Omega)} \|\phi_0 \|_{H^2(\Omega)}+
C \| \textbf{\textit{u}}_0\|_{L^\infty(\Omega)} \|\phi_0 \|_{H^2(\Omega)}+\|\nabla \mu_0 \|_{L^2(\Omega)}\\
&\leq C\| \textbf{\textit{u}}_0\|_{W^{1,p}(\Omega)} \|\phi_0 \|_{H^2(\Omega)}+\|\nabla \mu_0 \|_{L^2(\Omega)},
\end{align*}
we deduce that for any $p\in (2,\infty)$
$$
\omega \in L^\infty(0,T;L^p(\Omega)),\quad \partial_t \phi \in L^\infty(0,T; H^1(\Omega))\cap L^2(0,T;H^2(\Omega)).
$$
This, in turn, implies that
$$
\textbf{\textit{u}} \in L^\infty(0,T;W^{1,p}(\Omega)), \quad \phi \in L^\infty(0,T;W^{2,p}(\Omega)).
$$
As a consequence, the above estimates yield that $$\widetilde{\mu}=-\Delta \phi+F'(\phi) \in L^2(0,T;L^\infty(\Omega)).$$
The rest part of the proof is the same as the proof of Theorem \ref{Proreg-D} with the choice $\sigma>0$.
The proof of Theorem \ref{Th-EAC} is complete.
\section{Conclusions and Future Developments}
In this paper we present mathematical analysis of some Diffuse Interface models that describe the evolution of incompressible binary mixture having (possibly) different densities and viscosities. We focus on the mass-conserving Allen-Cahn relaxation of the transport equation with the physically relevant Flory-Huggins potential. We show the existence of global weak solution in three dimension and of global strong solutions in two dimensions. For the latter, we discuss additional properties, such as uniqueness, regularity and the separation property. On the other hand, several still unsolved questions concern the analysis of the complex fluid, Navier-Stokes-Allen-Cahn and Euler-Allen-Cahn systems in the three dimensional case, which will be the subject of future investigations. We conclude by mentioning some interesting open problems related to the results proved in this work:
\medskip
$\bullet$ An important possible development of this work is to show the existence of global solutions to the complex fluids system \eqref{CF}-\eqref{IC-P} originating from small perturbation of some particular equilibrium states. We mention that some remarkable results in this direction have been achieved in \cite{LLZ2005,LZ2008,RWXZ2014} (see also \cite{LIN2012} and the references therein). In addition, it would be interesting to study the global existence of weak solutions as in \cite{HL2016} and to generalize Theorem \ref{CF-T} to the case with zero viscosity (cf. \cite[Theorem 3.1]{LLZ2005}).
\smallskip
$\bullet$ Two possible improvements of this work concern the Navier-Stokes-Allen-Cahn system \eqref{NSAC-D}-\eqref{IC-D}. The first question is whether the entropy estimates in Theorem \ref{strong-D} can be achieved for strong solutions with small initial data, but without restrictions on the parameters of the system, or even without any condition on the initial data. The second issue is to show the uniqueness of strong solutions given from Theorem \ref{strong-D}-(1), without relying on the entropy estimates in Theorem \ref{strong-D}-(2). Also, we mention the possibility of considering moving contact lines for the Navier-Stokes-Allen-Cahn system (see \cite{MCYZ2017} for numerical).
\smallskip
$\bullet$ Interesting open issues regarding the Euler-Allen-Cahn system \eqref{EAC}-\eqref{ICE} are the existence and the uniqueness of solutions corresponding to an initial datum $\omega_0 \in L^\infty(\Omega)$ as well as the study of the inviscid limit on arbitrary time intervals (cf. \cite{ZGH2011} for short times).
\smallskip
\section*{Acknowledgments}
\noindent Part of this work was carried out during the first and second authors' visit to School of Mathematical Sciences of Fudan University whose hospitality is gratefully acknowledged. M.~Grasselli is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). H.~Wu is partially supported by NNSFC grant No. 11631011 and the Shanghai Center for Mathematical Sciences at Fudan University.
\section*{Compliance with Ethical Standards}
\noindent The authors declare that they have no conflict of interest. The authors also confirm that the manuscript has not been submitted to more than one journal for simultaneous consideration and the manuscript has not been published previously (partly or in full).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,645 |
{"url":"https:\/\/mathoverflow.net\/questions\/236204\/hessians-on-kahler-manifolds","text":"# Hessians on Kahler Manifolds\n\nThis is primarily a linear algebra question, but for motivation I want to state this question in its natural, global context. Whenever we have a non-relativistic quantum field theory (renormalized, of course), we can take the classical phase space with polarization as a high-dimensional Kahler manifold $K$ (A priori, one would get $K=\\mathbb C^N$, but physical considerations can yield more interesting parameter spaces, e.g. modding-out by gauge transformations).\n\nI am interested in seeing the local picture in all of this: take the classical field Hamiltonian $H$, which is a real-valued $C^\\infty$-function on $K$. The canonical quantization of the Hessian of this function at a point $x\\in K$ (roughly) represents the mean-field approximation for the statistical dynamics of a field configuration localized near $x\\in K$. How do I provide the following?:\n\n1. A proof that $d^2H$ is pointwise diagonalizable by a symplectic transformation.\n2. An algorithm\/formula for said symplectic diagonalization.\n3. A formula for the resulting change in complex structure on the tangent space.\n\nAny help would be appreciated, including references (Kahler geometry is new to me).","date":"2021-03-07 21:40:02","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8894816637039185, \"perplexity\": 469.5084973945959}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178378872.82\/warc\/CC-MAIN-20210307200746-20210307230746-00446.warc.gz\"}"} | null | null |
There are an increasing number of homes being broken into because simple home protection measures were not taken.
Try to make your home look occupied while you are not there. Use timer switches on lamps or your radio.
If the system is breached, a monitoring station informs the key holders and police within minutes.
If activated, an alarm sound will alert your neighbours. These alarms are less expensive than monitored systems and are a good deterrent. However, you cannot guarantee anyone in the area will react to it and alert you or the police.
If the alarm is activated, the sound will alert your neighbours and the system will dial a series of telephone numbers. This allows the receiver to respond to the alarm.
If you are considering buying an intruder alarm you could use the Trading Standards Website to check for vetted suppliers in your area. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,112 |
package org.wso2.carbon.device.mgt.mobile.windows.api.services.enrollment.beans;
import org.wso2.carbon.device.mgt.mobile.windows.api.common.PluginConstants;
import javax.xml.bind.annotation.*;
@XmlRootElement
@XmlAccessorType(XmlAccessType.FIELD)
@XmlType(name = "ContextItem", namespace = PluginConstants.SOAP_AUTHORIZATION_TARGET_NAMESPACE,
propOrder = {"Value"})
public class ContextItem {
@XmlAttribute(name = "Name")
protected String Name;
@XmlElement(name = "Value", required = true,
namespace = PluginConstants.SOAP_AUTHORIZATION_TARGET_NAMESPACE)
protected String Value;
public String getValue() {
return Value;
}
public void setValue(String value) {
Value = value;
}
public String getName() {
return Name;
}
public void setName(String name) {
Name = name;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,035 |
class IncreaseLimitFtaFundingTypesCode < ActiveRecord::Migration[4.2]
def change
change_column :fta_funding_types, :code, :string, limit: 6
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,148 |
package org.apache.felix.cm.impl;
import java.util.LinkedList;
import org.osgi.service.log.LogService;
/**
* The <code>UpdateThread</code> is the thread used to update managed services
* and managed service factories as well as to send configuration events.
*/
public class UpdateThread implements Runnable
{
// the configuration manager on whose behalf this thread is started
// (this is mainly used for logging)
private final ConfigurationManager configurationManager;
// the thread group into which the worker thread will be placed
private final ThreadGroup workerThreadGroup;
// the thread's base name
private final String workerBaseName;
// the queue of Runnable instances to be run
private final LinkedList updateTasks;
// the actual thread
private Thread worker;
public UpdateThread( final ConfigurationManager configurationManager, final ThreadGroup tg, final String name )
{
this.configurationManager = configurationManager;
this.workerThreadGroup = tg;
this.workerBaseName = name;
this.updateTasks = new LinkedList();
}
// waits on Runnable instances coming into the queue. As instances come
// in, this method calls the Runnable.run method, logs any exception
// happening and keeps on waiting for the next Runnable. If the Runnable
// taken from the queue is this thread instance itself, the thread
// terminates.
public void run()
{
for ( ;; )
{
Runnable task;
synchronized ( updateTasks )
{
while ( updateTasks.isEmpty() )
{
try
{
updateTasks.wait();
}
catch ( InterruptedException ie )
{
// don't care
}
}
task = ( Runnable ) updateTasks.removeFirst();
}
// return if the task is this thread itself
if ( task == this )
{
return;
}
// otherwise execute the task, log any issues
try
{
// set the thread name indicating the current task
Thread.currentThread().setName( workerBaseName + " (" + task + ")" );
configurationManager.log( LogService.LOG_DEBUG, "Running task {0}", new Object[]
{ task } );
task.run();
}
catch ( Throwable t )
{
configurationManager.log( LogService.LOG_ERROR, "Unexpected problem executing task", t );
}
finally
{
// reset the thread name to "idle"
Thread.currentThread().setName( workerBaseName );
}
}
}
/**
* Starts processing the queued tasks. This method does nothing if the
* worker has already been started.
*/
synchronized void start()
{
if ( this.worker == null )
{
Thread workerThread = new Thread( workerThreadGroup, this, workerBaseName );
workerThread.setDaemon( true );
workerThread.start();
this.worker = workerThread;
}
}
/**
* Terminates the worker thread and waits for the thread to have processed
* all outstanding events up to and including the termination job. All
* jobs {@link #schedule(Runnable) scheduled} after termination has been
* initiated will not be processed any more. This method does nothing if
* the worker thread is not currently active.
* <p>
* If the worker thread does not terminate within 5 seconds it is killed
* by calling the (deprecated) <code>Thread.stop()</code> method. It may
* be that the worker thread may be blocked by a deadlock (it should not,
* though). In this case hope is that <code>Thread.stop()</code> will be
* able to released that deadlock at the expense of one or more tasks to
* not be executed any longer.... In any case an ERROR message is logged
* with the LogService in this situation.
*/
synchronized void terminate()
{
if ( this.worker != null )
{
Thread workerThread = this.worker;
this.worker = null;
schedule( this );
// wait for all updates to terminate (<= 10 seconds !)
try
{
workerThread.join( 5000 );
}
catch ( InterruptedException ie )
{
// don't really care
}
if ( workerThread.isAlive() )
{
this.configurationManager.log( LogService.LOG_ERROR,
"Worker thread {0} did not terminate within 5 seconds; trying to kill", new Object[]
{ workerBaseName } );
workerThread.stop();
}
}
}
// queue the given runnable to be run as soon as possible
void schedule( Runnable update )
{
synchronized ( updateTasks )
{
configurationManager.log( LogService.LOG_DEBUG, "Scheduling task {0}", new Object[]
{ update } );
// append to the task queue
updateTasks.add( update );
// notify the waiting thread
updateTasks.notifyAll();
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,304 |
Keep knives and other frequently used tools and utensils conveniently within reach with this wall-mounted Magnetic Knife Holder. Made of top-class stainless steel, the seamless square bar is anti-rust and durable. Inserted with super strong magnetic material with double layer magnet protection. Stronger adsorption force for safe tableware holding and storage. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,254 |
{"url":"https:\/\/www.physicsforums.com\/threads\/general-relativity-and-tidal-forces.636994\/","text":"# General relativity and tidal forces\n\n1. Sep 18, 2012\n\n### altergnostic\n\nTides on earth are described with newton's theory of gravitation. Relativistic effects on tides theoretically become measurable on very strong gravitational fields, possibly becoming twice as strong as tides predicted by newtonian gravity: http:\/\/adsabs.harvard.edu\/abs\/1983ApJ...264..620N\n\nTides are presumably outcomes of gravitational forces. Einstein ditched forces and the concept of inertia in GR (http:\/\/archive.org\/stream\/TheBornEinsteinLetters\/Born-TheBornEinsteinLetters_djvu.txt). So how is GR used to calculate tidal forces? If different parts of the body travel different geodesics, this would cause the body to tear apart over time. How can tides be described with the geometry of space-time?\n\n2. Sep 18, 2012\n\n### Staff: Mentor\n\nIf the forces holding the body together are strong enough, they will accelerate the different parts of the body off their geodesics and onto non-geodesic worldlines that stay close enough that the body doesn't tear apart. The center of mass of the object follows a geodesic and the other parts of the body experience fictitious forces that tend to pull the body apart and are resisted by whatever forces hold the body together.\n\nThese fictitious forces are tides.\n\n3. Sep 18, 2012\n\n### pervect\n\nStaff Emeritus\nTidal forces, when suitably defined, can be identified as being components of the Riemann curvature tensor.\n\nUnder most circumstances, taking the tidal force as one would measure it via Newtonian means ( a couple of accelerometers separated by a rigid rod) is an excellent approximation to (one of the) geometric definitions, which is related to the apparent relative acceleration of nearby geodesics which are initially parallel.\n\nIn fact, you write earlier (this is a very good insight)\n\nThe point is that when you measure the forces needed to hold a rigid body together, to keep it rigid, you are indirectly measuring \"how fast\" the geodesics would expand (accelerate away from each other) if said restoring forces did not exist.\n\nMTW's textbook \"Gravitation\", and a number of other textbooks, take this approach, though MTW is perhaps the textbook which invites the reader to take it most seriously.\n\nThe full Riemann curvature can, given a local description of time (a frame of reference, for instance, more formally a timelike congruence of worldlines) be decomposed by the Bel Decomposition http:\/\/en.wikipedia.org\/w\/index.php?title=Bel_decomposition&oldid=512613685 into three parts. One part, called the electrogravitic tensor, describes static gravity, and givenby the above \"tidal forces\", so the Newtonian tidal tensor can be pretty much directly be linked to the electrogravitic part of the Riemann tensor.\n\nAnother part, called the magnetogravitic tensor, described frame-dragging effects (which affect moving bodies, but don't directly affect static bodies). A third part, the topogravitic tensor, describes spatial curvature.\n\nThe Bel decomposition, unfortunately, is usually given short shrift in textbooks, so it may be hard to find a formal treatment I was introduced to it rather informally here on PF, for instance.\n\n4. Sep 19, 2012\n\n### altergnostic\n\nAll right, so I think I understand the basis of how GR creates tides, even though I have a minor issue with fictitious forces counteracted by real forces in this particular case, but that's a subject for another thread, I don't want to get into this here.\nMy problem with this analysis is that different parts of the body are not only trying to travel different geodesics, but also, in order to stay rigid, the outer parts of the rotating orbiter must have a faster tangential component than the inner parts. This is a bigger problem when you consider a body in tidal lock, such as the moon. Different geodesics would tend to rip the body apart radially, but different velocities would tend to shear te body in the line of orbit. In all gravitational theories, the tangential component of the velocity can't be caused by the gravitational field and is a constant. Newton called it the body's innate velocity.\n\nThink of it this way: a body is at a constant linear velocity, so all parts of the body travel at the same linear speed. Then it is captured by a gravitational field. It starts to orbit that planet and the linear velocity is not changed, but the body is accelerated into a curved motion (or continues to have a constant velocity in curved spacetime). The tangential component must still be the same for all parts of the body since gravity imparts no (fictitious) forces tangentially. This would tend to cause shearing, especially in a body in tidal lock. If it doesn't start to shear, presumably the tangential velocities are different on different parts of the body, with the far side faster then the near side (this would be true for all bodies, not only the ones in tidal lock), but how can that be? What forces act on the body to change the tangential (innate) velocities if gravity has no way to do so neither with Newton nor with GR? Are tides capable of changing tangential velocities in either GR or Newton's theory?\n\n5. Sep 19, 2012\n\n### pervect\n\nStaff Emeritus\nWhile \"centrifugal\" forces, i.e. forces due to rotation, do contribute to the strain on a rigid bar, they do not contribute to the Riemann curvature tensor, which is ultimately based on how fast geodesics separate (or converge). The \"force-on-a-bar\" idea is very useful, but it can only be used if\/when the bar isn't rotating.\n\nSo in your orbiter example, either you'd need to imagine that your spacecraft was not rotating (in which case in 1\/2 an orbit the outer side would be the inner side, assuming no frame dragging effects), or if your space-craft is tide-locked, you'd have to manually subtract the forces due to its absolute rotation (once per orbit, again assuming no frame dragging) from the measured strain on the bar to get the tensor components.\n\nThe Electrogravitic component of the Riemann tensor must be traceless. The centrifugal forces on a rotating sphere are not traceless, this is one way you can tell if a system is rotating.\n\nAccurate measurements of the gravity tensor, typically using rather exotic means such as superconductors and SQUID's for the detectors, are an expensive, but semi-routine, part of modern prospecting. Some interesting references are http:\/\/www.physics.umd.edu, http:\/\/www.bellgeo.com\/tech\/technology_theory_of_FTG.html [Broken], and http:\/\/www.dtic.mil\/cgi-bin\/GetTRDoc?AD=ADA496707\/GRE\/NASA_SGG.pdf [Broken]. These papers describe some of the modern techniques that are actually used to measure the gravity tensor. The last has some discussion of the physics as well, though it's oriented mostly towards Newtonian gravity.\n\nThe Wiki article is also mildly helpful, http:\/\/en.wikipedia.org\/w\/index.php?title=Gravity_gradiometry&oldid=508813691, giving a list of some of the basic systems that have been implemented.\n\nLast edited by a moderator: May 6, 2017\n6. Sep 19, 2012\n\n### altergnostic\n\nI'm not sure i follow. Forces due to rotation cause strain, but can only be used if the bar is not rotating??????\n\nIf the orbiter does not rotate on its axis, as in your first example, everything is fine, because there's no variation in tangential speeds. But if the body is in tidal lock, then how does the tangential velocities adjust to maintain the body in position? Or, if they don't adjust, how come there's no shearing?\n\n7. Sep 19, 2012\n\n### Staff: Mentor\n\nWhy do you say that? Sounds like an issue with geometry to me, like you think that different velocities mean different parts of the object are moving apart. What you're missing is that the object is rotating. Spin a pencil on your desk and you'll see that different parts travel at different speeds, but there is no shear.\nA torque is produced if a body's tidal bulge is not aligned with the source of the bulge: http:\/\/en.wikipedia.org\/wiki\/Tidal_locking\n\n8. Sep 19, 2012\n\n### altergnostic\n\nSorry, I meant to say that equal tangential velocities would tend to cause shearing, not different velocities. I'm making a diagram to clarify.\n\n9. Sep 19, 2012\n\n### altergnostic\n\nOk, I'm having a hard time putting the problem into words so I drew a diagram that may help to clarify the whole thing:\nhttp:\/\/www.pictureshoster.com\/files\/aix43ezq7zp99daihgzu.jpg\nhttp:\/\/www.pictureshoster.com\/files\/aix43ezq7zp99daihgzu.jpg\n\nIf different parts of the body have different velocities, there's no shearing. But if all parts of the body have the same linear velocity, then a body in tidal lock should exhibit shearing.\n\nIf there's no shearing, it follows that different parts of the body have different tangential components. Suppose that a body is traveling in a straight line at a constant velocity. It passes near a second body and starts to orbit it. If nothing else happens, the body wouldn't start to rotate on it's own axis and in \"1\/2 an orbit the outer side would be the inner side\" just like pervect said. For a body in tidal lock, something seems to affect linear velocities on the near and far side so that the far side orbits faster than the near side.\n\nMy question is what causes the changes in these tangential velocities. Or alternatively, why is there no shearing. Since gravity has no tangential component neither in Newton nor in GR, I'm lost.\n\nAt first I thought maybe it had something to do with the rigidity of the body and tidal forces. I thought that gravity would accelerate the near and far sides differently and force the body in tidal lock, accelerating the far side more than the near side, but this turned out to be a dead end.\n\nI think the make up of the problem is clearer now with my diagram, and I believe my logic is consistent. A body in this situation must either shear or achieve different tangential velocities on the near and on the far sides. If this is the case, what is the cause of these changes?\n\nLast edited: Sep 20, 2012\n10. Sep 20, 2012\n\n### D H\n\nStaff Emeritus\n\nYou are looking in the wrong directions. First look at a spec of mass on the orbiting body at the point furthest from the central mass. The velocity of that spec of mass is a function of the radius of the orbiting body and the velocity of the orbiting body's center of mass. Now imagine what would happen if that orbiting body wasn't there; all you have is the spec of mass as an orbiting body. That free particle will follow a different path than would our spec of mass. In particular, it would move outward. The tidal force at this point is radially outward away from the central mass, not tangential. Similarly, the tidal force on the point closest to the central mass is radially inward, toward the central mass. In both cases, the tidal force is away from the center of mass of the orbiting body. There is also a lesser effect (about half as much) for a particle at the leading and trailing points on the orbiting body. A free particle at those leading and trailing points would follow different paths than will a particle fixed to the orbiting body, but now the tidal force is directed toward the center of mass of the orbiting body. The end result is that the tidal forces act to pull the object apart radially, squeeze it together tangentially.\n\nNewtonian mechanics and general relativity agree on the above description so long as the central mass isn't particular massive or the distance to the central mass is sufficiently large. The reason for this agreement is that space-time is locally flat. \"Locally\" is a fairly large volume in these weak field circumstances, at least as far as physicists are concerned. (Mathematicians will disagree; local means infinitesimally small to them.) Make the central body massive enough or close enough and those weak field approximations become invalid. You'll start seeing effects that result from the curvature of space-time. Newtonian mechanics and relativity diverge at this point. Newtonian mechanics does not properly describe the extreme spaghettification that results from close proximity to extremely massive objects.\n\n11. Sep 20, 2012\n\n### pervect\n\nStaff Emeritus\nIf you connect the \"red x\"s on the same point on a non-rotating body, you'll get an elliptical orbit more like the one I'll attach, the picture you drew isn't right.\n\nIf you go through the math (using just Newtonian theory), you'll get the results DW quoted.\n\nIn particular, if the radial Newtonian force is -GM\/r^2, differentiating this with respect to r gives the well-known result for the radial tidal force 2GM\/r^3. See also the wiki article, http:\/\/en.wikipedia.org\/w\/index.php?title=Tidal_tensor&oldid=332450104.\n\nIt takes more work to go through the math to get the compressive tidal forces, the results dW quotes are correct however.\n\nThe results for GR are formally similar to the Newtonian results in a local frame-field, if you replace the radial distance \"r\" with the radial coordinate \"r\" for the Schwarzschild metric. Local frame fields seem to confuse more readers than they should, the math to compute them is somewhat involved, but the end result is just the forces\/fields\/tensors that a local observer would measure with local clocks and local rulers.\n\n#### Attached Files:\n\n\u2022 ###### tidal.png\nFile size:\n15.5 KB\nViews:\n139\nLast edited: Sep 20, 2012\n12. Sep 20, 2012\n\n### altergnostic\n\nI completely agree. But this does not address the problem. In the diagram, I hid the effects of tides, I want to focus on the tangential components of the velocities. I am aware that the specs would move apart, I actually said that explicitly in my first post.\n\nWhat keeps the near and far side from moving apart, which was answered in the beginning of the post, are the forces holding the body together, so it is clear that tidal forces tend to elongate the body radially, and it is clear that neither tides nor gravity have mechanisms to act tangentially on the near and far sides to change these velocities.\n\nPlease be patient since my question has not been answered. Leave tidal forces out for a moment and consider this:\n\n- A body Y moves in a straight line at a constant velocity V.\n- All parts of the body have the same linear velocity V.\n- This body starts to orbit a central mass M.\n- M imparts a centripetal acceleration due to gravity on Y.\n- Y feels no tangential forces from M, and no drag.\n- Y achieves tidal lock: the far side has a faster tangential velocity than the inner side.\n\nNow, if Y did not achieve tidal lock, and no forces other then gravity from M act on it, we may assume that there is no change in the tangential velocities and there would be no fixed near and far sides, the tides would travel, the body would appear to rotate as seen from M, the body would not be rotating in it's own axis locally (relative to it's own orbit) and everything is fine.\n\nBut if Y orbits in tidal lock, the far side orbits faster than the near side, and we must assume a change in the tangential velocities of each sides, namely, a positive acceleration on the far side and a negative one on the near side.\n\nGravity can't cause these tangential accelerations, so there's no way M causes it. The body Y also cannot impart forces on itself, so again, it cannot be the cause of these changes.\n\nMy question is: what causes the changes in the tangential velocities of the near and far sides?\n\n13. Sep 20, 2012\n\n### Staff: Mentor\n\nThe body Y cannot apply forces to itself, but one part of the body can apply forces to another part. Replace the body with a cloud of dust, such that each dust particle is connected to its neighbors with a spring. Now if two of the particles are on diverging geodesics, the spring between them will be stretched, applying a force to both particles that will accelerate them off their inertial path and onto other geodesics. Obviously this process can change the shape of the body\/cloud; but it can also change the velocity of one part of the cloud relative to another, as in tidal locking.\n\n14. Sep 20, 2012\n\n### Staff: Mentor\n\nEven though it acts only in a radial direction, gravity can produce tangential accelerations. Consider, for example, dropping a horizontally oriented bar towards the surface of the earth. Each end of the bar will experience a gravitational force pointing towards the center of the earth. These two vectors are not parallel; if I extend them far enough they will intersect at the center of the earth, so they are ever so slightly converging. Thus, both vectors have a small tangential component that is pushing the ends of the rod towards the middle and is resisted by the rigidity of the rod.\n\nIt's true that these forces are balanced so they cannot affect the center-of-mass movement of the rod - but that's true of tidal locking of a rotating body as well.\n\n15. Sep 20, 2012\n\n### altergnostic\n\nThat's a good point, but isn't that one of the reasons the body should resist changes on tangential velocities on the near and far sides? For example, if I try to push only the far side, these internal bonds (rigidity) would tend to transfer forces to the near side (and everywhere else). But the force is only transfered in this case, isn't it? That means that there must be an external force acting on the tangent in this case, and this force would be resisted by rigidity.\n\nYou mention tangential accelerations caused by gravity in a rigid rod falling horizontally towards the central mass. But in this case, the body is not in orbit, and if it was (even if slowly falling), the forces would be on the trailing and leading sides, wouldn't they? And this would only help the radial bulging anyway. These tangential components are not capable of altering tangential velocities on the near and far sides. A good analogy would be to hold the same rod vertically wrt earth and throw it straight towards the horizon (ignore drag). Would the far side of the rod start to speed up relative to the near side? Of course not. But that would be a rod in tidal lock.\n\nSince there's no tangential forces in gravity, I don't see where these changes in tangential velocities come from. The stretching you mentioned with the springs, caused by gravity, is only radial, it would cause radial stretching (tidal bulges). Exactly how can this radial force change the tangential velocities of the near and far sides?\n\nLast edited: Sep 20, 2012\n16. Sep 20, 2012\n\n### Mentz114\n\nIt might be constructive to look at some actual figures. For a test particle in the Schwarzschild vacuum the proper acceleration felt by an observer in a circular path u is\n\n$$\\frac{du}{d\\tau} = \\frac{m\\,{\\cos\\left( \\theta\\right) }^{2}}{{r}^{2}-2\\,m\\,r}\\ dr -\\frac{m\\,\\cos\\left( \\theta\\right) \\,\\sin\\left( \\theta\\right) }{r-3\\,m}\\ d\\theta$$\nIf the particle is entirely in the equatorial plane (EP, \u03b8 = \u03c0\/2) both components are zero, and the orbit is a geodesic. Considering a small sphere in orbit, with its centre following the geodesic we can make some deductions about the forces required to keep the whole body together.\n\nThe acceleration in the \u03b8-direction ( the one pointing up and down from the equatorial plane) will change sign between the two halves so it always acts towards the EP causing compressive stress. In the radial direction, leaving the EP in either direction will cause an increase in the acceleration in the +ve r-direction. I don't know what deformation this causes. Both components get smaller as the radius increases so it looks like there is also some stretching in the r and \u03b8-directions.\n\nThe acceleration above is calculated from the Schwarzschild circular orbit geodesic in the coordinate basis with 4-velocity\n$$u=\\frac{\\sqrt{r}\\,\\sqrt{m\\,{\\sin\\left( \\theta\\right) }^{2}+r-3\\,m}}{\\sqrt{r-3\\,m}\\,\\sqrt{r-2\\,m}}\\ \\partial_t -\\frac{\\sqrt{m}}{r\\,\\sqrt{r-3\\,m}}\\ \\partial_\\phi$$\nThis is only a geodesic if sin(\u03b8) = 1. So any circular path off the EP must apply forces to offset the accelerations engendered.\n\n17. Sep 20, 2012\n\n### zonde\n\nLet's say that this body in tidal lock does not have perfect spherical symmetry. Then as it rotates it's gravity affects large mass M differently. Now let's say that body M is not entirely rigid so that there are slightly different responses to small body in different rotation angle positions.\nAnother possibility. Body in orbit itself is not entirely rigid and then radial stretch elongates the body in different directions as it rotates. If this response to stretch in different directions is different then shouldn't it be possible to dump angular momentum using that radial force?\n\n18. Sep 21, 2012\n\n### Staff: Mentor\n\nWhen you're holding the rod vertically, it's exactly lined up with the radial gravitational force; both ends of the rod and the center of the earth lie in the same line, and the gravitational force acts along that line. But when you throw the rod it moves off that vertical line. Now the forces acting on the two ends are no longer exactly parallel (extend the vectors and they meet at the center of the earth, therefore are very slightly converging). Furthermore, the forces have very slightly different magnitudes, because the distance to the center of the earth is different for the two ends.\n\nSo we have forces of different magnitude acting in different directions on the two ends of the rod. Why shouldn't one end speed up relative to the other?\n\n19. Sep 21, 2012\n\n### altergnostic\n\nMentz114, Thanks for this. Correct me if i'm wrong, i take this to mean that anything above the equatorial plane suffers an accel towards the equatorial plane. That would be the flattening at the poles of tidal theory. As you also pointed out, there should also be stretching in the equatorial plane (i assume only on the near and far sides) which would be the tidal bulges. Still, i see no component that would cause different tangential forces on the near and far sides that could adjust velocities to cause a body to achieve tidal lock, is this correct?\n\n20. Sep 21, 2012\n\n### altergnostic\n\nI have considered this, and i believe that would tend to cause shearing. If the tangential velocities on the near and far sides remain unchenged, these forces that you mention would work against this stretching, but not overcome it. If a body has almost no rigidity, it could not hold itself together and it would shear (near side ahead of the far side), stretch radially and tear apart.\n\nLast edited: Sep 21, 2012","date":"2017-10-22 23:09:20","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7062787413597107, \"perplexity\": 581.7124530140475}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-43\/segments\/1508187825473.61\/warc\/CC-MAIN-20171022222745-20171023002745-00478.warc.gz\"}"} | null | null |
STEVENSON, Md., Feb. 21, 2014 (GLOBE NEWSWIRE) -- The securities litigation firm of Brower Piven, A Professional Corporation, announces that a class action lawsuit has been commenced in Illinois state court on behalf of all common stockholders of AMCOL International Corporation ("AMCOL" or the "Company") (NYSE:ACO).
According to the complaint, under the terms of the proposed transaction, AMCOL shareholders will receive $41 for each share of AMCOL common stock they own. The claims concern whether the AMCOL Board of Directors breached their fiduciary duties to stockholders by failing to maximize shareholder value before agreeing to enter into this transaction, and whether Imerys S.A. is underpaying for AMCOL shares.
If you currently own common stock of AMCOL International Corporation and would like to learn more about the lawsuit, you may email or call Brower Piven, who will, without obligation or cost to you, attempt to answer your questions. You may contact Brower Piven by email at hoffman@browerpiven.com, by calling (410) 415-6616, or at Brower Piven, A Professional Corporation, 1925 Old Valley Road, Stevenson, Maryland 21153. Attorneys at Brower Piven have combined experience litigating securities and other class action cases of over 60 years. | {
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\section*{}
Pooling design is a mathematical tool to reduce the number of tests
in DNA library screening \cite{CD,CD2,DH}. A pooling design is
usually represented by a binary matrix with columns indexed with
items and rows
indexed with pools. A cell $(i,j)$ contains a 1-entry if and
only if the $i$th pool contains the $j$th item. Biological
experiments are notorious for producing erronous outcomes.
Therefore, it would be wise for pooling designs to allow some
outcomes to be affected by errors. A binary matrix $M$ is called
$s^e$-{\em disjunct} if given any $s+1$ columns of $M$ with one
designated, there are $e+1$ rows with a 1 in the designated column
and 0 in each of the other $s$ columns. A $s^0$-disjunct matrix is
also called $s$-{\em disjunct}. An $s^e$-disjunct matrix is called
{\it fully $s^e$-disjunct} if it is not $s_1^{e_1}$-disjunct
whenever $s_1>s$ or $e_1>e$. An $s^e$-disjunct matrix is $\lfloor
e/2\rfloor$-error-correcting (see \cite{DF}).
For positive integers $k\leq n$, let $[n]=\{1,2,\ldots,n\}$ and
$\left([n]\atop k\right)$ be the set of all $k$-subsets of
$[n]$.
Macula \cite{Macula,Macula2} proposed a novel way of constructing
disjunct matrices by the containment relation of subsets in a finite
set.
\begin{defin}{\rm(\cite{Macula})}\, For positive integers $1\leq d<k< n$,
let $M(d,k,n)$ be the binary
matrix with rows indexed with $\left([n]\atop
d\right)$ and columns indexed with $\left([n]\atop k\right)$ such that $M(A,B)=1$ if
and only if $A\subseteq B$.
\end{defin}
D'yachkov et al. \cite{DF2} discussed the error-correcting property
of $M(d,k,n)$.
\begin{thm}{\rm (\cite{DF2})}\label{thmdy} For positive integers $1\leq d<k< n$ and $s\leq d$, $M(d,k,n)$ is fully
$s^{e_1}$-disjunct, where $e_1=\left(k-s\atop d-s\right)-1$.
\end{thm}
Ngo and Du \cite{Ngo} constructed disjunct matrices by the
containment relation of subspaces in a finite vector space.
D'yachkov et al. \cite{DF} discussed the error-tolerant property of
Ngo and Du's construction. Huang and Weng \cite{HW} introduced the
comprehensive concept of pooling spaces, which is a significant
addition to the general theory. Recently,
many pooling designs have been constructed using the ``containment
matrix" method, see e.g. \cite{bai, hhw, hww}.
Next we shall introduce our construction.
\begin{defin}
Given integers $1\leq d< k<n $ and $0\leq i\leq
d$. Let $M(i;d,k,n)$ be the binary matrix with rows indexed
$\left([n]\atop d\right)$ and columns indexed with
$\left([n]\atop k\right)$ such that $M(A,B)=1$ if and only if
$|A\cap B|=i$.
\end{defin}
Note that $M(i;d,k,n)$ and $M(d,k,n)$ have the same size, and
$M(i;d,k,n)$ is an $\left(n\atop d\right)\times \left(n\atop
k\right)$ matrix with row weight $\left(d\atop
i\right)\left(n-d\atop k-i\right)$ and column weight $\left(k\atop
i\right)\left(n-k\atop d-i\right)$. Since $M(d;d,k,n)=M(d,k,n)$,
our construction is a generalization of Macula's matrix.
Let $B\in\left([n]\atop k\right)$ and $C=[n]\backslash B$. Then, for any
$D\in\left([n]\atop d\right)$, $|D\cap B|=i$ if and only if $|D\cap
C|=d-i$. Therefore,
$M(i;d,k,n)=M(d-i;d,n-k,n)$ when $n>k+d-i$. Since $i\leq\lfloor d/2\rfloor$ if
and only if $d-i\geq\lfloor(d+1)/2\rfloor$, we always assume that
$i\geq\lfloor(d+1)/2\rfloor$ in this case.
\begin{thm}\label{thm2.1}
Let $1\leq s\leq i,\lfloor(d+1)/2\rfloor\leq i\leq d<k$ and
$n-k-s(k+d-2i)\geq d-i$. Then
\begin{itemize}
\item[\rm(i)] $M(i;d,k,n)$ is an $s^{e_2}$-disjunct matrix, where
$e_2=\left(k-s\atop i-s\right)\left(n-k-s(k+d-2i)\atop
d-i\right)-1$;
\item[\rm(ii)] For a given $k$, if $i<d,$ then $\lim\limits_{n\rightarrow\infty}
\frac{e_2+1}{e_1+1}=\infty.$
\end{itemize}
\end{thm}
\begin{pf} (i)
Let $B_{0},B_{1},\ldots,B_{s}\in\left([n]\atop k\right)$ be any
$s+1$ distinct columns of $M(i;d,k,n)$. Then, for each $j\in[s]$,
there exists an $x_j$ such that $x_j\in B_0\backslash B_j.$ Suppose
$X_0=\{x_j\mid 1\leq j\leq s\}.$ Then $X_0\subseteq B_0$, and
$X_0\not\subseteq B_j$ for each $j\in[s]$. Note that the number of
$i$-subsets of $B_0$ containing $X_0$ is $\left(k-|X_0|\atop
i-|X_0|\right)= \left(k-|X_0|\atop k-i\right)$. Since $
\left(k-|X_0|\atop k-i\right)$ is decreasing for $1\leq |X_0|\leq s$
and gets its minimum at $|X_0|=s$, the number of $i$-subsets of
$B_0$ containing $X_0$ is at least $\left(k-s\atop k-i\right)$.
Let $A_0$ be an $i$-subset of $B_0$ containing $X_0$. Then
$|A_0\cap B_j|<i$ for each $j\in[s]$.
Let $D\in\left([n]\atop
d\right)$ satisfying $|D\cap B_0|=i$. If there exists $j\in [s]$
such that $|D\cap B_j|=i$, then $|B_0\cap B_j|\geq|D\cap B_0\cap
B_j|\geq 2i-d$. Suppose $|B_0\cap B_j|\geq 2i-d$ for each
$j\in[s]$. Since $|\bigcup_{0\leq j\leq s}B_j|\leq k+s(k+d-2i)$, the
number of $d$-subsets $D$ of $[n]$ containing $A_0$ satisfying
$|D\cap B_{0}|=i$ and $|D\cap B_{j}|\not=i$ for each $j\in[s]$ is at
least $\left(n-k-s(k+d-2i)\atop d-i\right)$. Then the number of
$d$-subsets $D$ containing $X_0$ in $\left([n]\atop d\right)$
satisfying $|D\cap B_{0}|=i$ and $|D\cap B_{j}|\not=i$ for each
$j\in[s]$ is at least $\left(k-s\atop
i-s\right)\left(n-k-s(k+d-2i)\atop d-i\right)$. Therefore, (i)
holds.
(ii) is straightforward by (i) and Theorem~\ref{thmdy}.\qed
\end{pf}
\begin{example}
$M(5,7,50)$ is fully $1^{14},2^{9}$ and $3^5$-disjunct, but
$M(3;5,7,50)$ is $1^{9989},2^{2324}$ and $3^{299}$-disjunct;
$M(4,5,13)$ is fully $1^3$ and $2^2$-disjunct, but $M(3;4,5,13)$ is
$1^{29}$ and $2^5$-disjunct.
\end{example}
\section*{Concluding remarks}
(i) For given integers $d<k$ the following limit holds: $\lim\limits_{n\rightarrow
\infty} \frac{\left(n\atop d\right)}{\left(n\atop k\right)}=0$.
This shows that the test-to-item of $M(i;d,k,n)$ is small enough
when $n$ is large enough. By Theorem~\ref{thm2.1}, our pooling
design are better than Macula's designs when $n$ is large enough.
(ii) It seems to be interesting to compute $e$ such that
$M(i;d,k,n)$ is fully $s^e$-disjunct.
(iii) In \cite{NG}, Nan and the first author discussed the similar
construction of $s^e$-disjunct matrices in a finite vector space,
but the number $e$ is not well expressed. By the method of this
paper, $e$ may be larger. We will study this problem in a separate
paper.
(iv) For positive integers $1\leq d<k< n$, let $I$ be a nonempty
proper subset of $\{0,1,\ldots, d\}$, and let $M(I;d,k,n)$ be the
binary matrix with rows indexed with
$\left([n]\atop d\right)$ and columns indexed with $\left([n]\atop k\right)$ such
that $M(A,B)=1$ if and only if $|A\cap B|\in I$. How
about the error-tolerant property of $M(I;d,k,n)$?
\section*{Acknowledgment}
We would like thank the referees for their valuable suggestions.
This research is partially supported by NSF of China,
NCET-08-0052, Langfang Teachers' College (LSZB201005), and the
Fundamental Research Funds for the Central Universities of China.
| {
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Q: getApplicationContext returns null I have the following fragment.The problem is that every time I launch this the app crashes with the following exception : attempt to invoke virtual method android.content.Context android.support.v4.app.FragmentActivity.getApplicationContext() on a null object reference. From what I understand getActivity is called correctly, however the FragmentActivityTesting does not return any context, I'm not sure why this is the case:
public class FragmentATesting extends Fragment {
@Override
public void onAttach(Activity activity) {
super.onAttach(activity);
}
@Override
public void onCreate(Bundle SavedInstanceState) {
super.onCreate(SavedInstanceState);
}
@Override
public View onCreateView(LayoutInflater inflater,ViewGroup container,Bundle savedInstanceState) {
return inflater.inflate(R.layout.fragment_a,container,false);
}
@Override
public void onActivityCreated(Bundle SavedInstanceState){
super.onActivityCreated(SavedInstanceState);
}
@Override
public void onStart() {
super.onStart();
final SessionManager session = new SessionManager(getActivity().getApplicationContext());
TextView username = (TextView) getView().findViewById(R.id.username_info_txt);
HashMap<String,String> userString = session.getUserDetails();
String usernamee = userString.get("username");
username.setText(usernamee);
vote(usernamee);
}
@Override
public void onResume() {
super.onResume();
final SessionManager session = new SessionManager(getActivity().getApplicationContext());
TextView username = (TextView) getView().findViewById(R.id.username_info_txt);
HashMap<String,String> userString = session.getUserDetails();
String usernamee = userString.get("username");
username.setText(usernamee);
vote(usernamee);
}
@Override
public void onPause() {
super.onPause();
}
@Override
public void onStop() {
super.onStop();
}
@Override
public void onDestroyView() {
super.onDestroyView();
}
@Override
public void onDestroy() {
super.onDestroy();
}
private void vote(final String userId) {
class UploadImage extends AsyncTask<String, Void, String> {
RequestHandler rh = new RequestHandler();
@Override
protected String doInBackground(String... params) {
HashMap<String, String> data = new HashMap<>();
data.put("username",userId);
String result = rh.sendPostRequest(SL_URL2, data);
return result;
}
@Override
protected void onPreExecute() {
super.onPreExecute();
}
@Override
protected void onPostExecute(String s) {
super.onPostExecute(s);
JSONObject jsonObject = null;
String likess = "";
try {
jsonObject = new JSONObject(s);
} catch (JSONException e) {
e.printStackTrace();
}
try {
likess = jsonObject.getString("amount");
} catch (JSONException e) {
e.printStackTrace();
}
TextView amount = (TextView) getView().findViewById(R.id.likes_info_txt);
amount.setText(likess + " Likes");
}}
UploadImage ui = new UploadImage();
ui.execute();
}`
It is inside my ViewPager defined as follows:
public class FragmentActivityTesting extends FragmentActivity {
ViewPager viewPager = null;
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.activity_fragment_activity_testing);
viewPager = (ViewPager) findViewById(R.id.pic_pager);
setStatusBarColor();
FragmentManager fragmentManager = getSupportFragmentManager();
viewPager.setAdapter(new MyAdapter(fragmentManager));
viewPager.setCurrentItem(1);
IntentFilter intentFilter = new IntentFilter();
intentFilter.addAction("CLOSE_ALL");
BroadcastReceiver broadcastReceiver = new BroadcastReceiver() {
@Override
public void onReceive(Context context, Intent intent) {
finish();
}
};
registerReceiver(broadcastReceiver, intentFilter);
}
@TargetApi(21)
public void setStatusBarColor() {
Window window = this.getWindow();
// clear FLAG_TRANSLUCENT_STATUS flag:
window.clearFlags(WindowManager.LayoutParams.FLAG_TRANSLUCENT_STATUS);
// add FLAG_DRAWS_SYSTEM_BAR_BACKGROUNDS flag to the window
window.addFlags(WindowManager.LayoutParams.FLAG_DRAWS_SYSTEM_BAR_BACKGROUNDS);
if (Integer.valueOf(android.os.Build.VERSION.SDK) >= 21) {
window.setStatusBarColor(this.getResources().getColor(R.color.green));
}
}}
class MyAdapter extends FragmentStatePagerAdapter {
public MyAdapter(FragmentManager fm) {
super(fm);
}
@Override
public Fragment getItem(int position) {
Fragment fragment = null;
if (position ==0) {
fragment = new FragmentATesting();
}
else if (position == 1) {
fragment = new FragmentBTesting();
}
else if (position == 2) {
fragment = new FragmentCTesting();
}
return fragment;
}
@Override
public int getCount() {
return 3;
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,030 |
// Copyright (c) Microsoft. All rights reserved.
// Licensed under the MIT license. See LICENSE file in the project root for full license information.
using System;
using System.Diagnostics;
using System.Globalization;
using System.Linq;
using System.Text.RegularExpressions;
using System.Threading;
using System.Windows.Interop;
using EnvDTE;
using liblinux;
using liblinux.Persistence;
using Microsoft.SSHDebugPS.Docker;
using Microsoft.SSHDebugPS.SSH;
using Microsoft.SSHDebugPS.UI;
using Microsoft.SSHDebugPS.Utilities;
using Microsoft.VisualStudio.Linux.ConnectionManager;
using Microsoft.VisualStudio.Shell;
using Microsoft.VisualStudio.Shell.Interop;
namespace Microsoft.SSHDebugPS
{
internal class ConnectionManager
{
public static DockerConnection GetDockerConnection(string name, bool supportSSHConnections)
{
if (string.IsNullOrWhiteSpace(name))
return null;
DockerContainerTransportSettings settings;
Connection remoteConnection;
if (!DockerConnection.TryConvertConnectionStringToSettings(name, out settings, out remoteConnection) || settings == null)
{
string connectionString;
bool success = ShowContainerPickerWindow(IntPtr.Zero, supportSSHConnections, out connectionString);
if (success)
{
success = DockerConnection.TryConvertConnectionStringToSettings(connectionString, out settings, out remoteConnection);
}
if (!success || settings == null)
{
VSMessageBoxHelper.PostErrorMessage(StringResources.Error_ContainerConnectionStringInvalidTitle, StringResources.Error_ContainerConnectionStringInvalidMessage);
return null;
}
}
string displayName = DockerConnection.CreateConnectionString(settings.ContainerName, remoteConnection?.Name, settings.HostName);
if (DockerHelper.IsContainerRunning(settings.HostName, settings.ContainerName, remoteConnection))
{
return new DockerConnection(settings, remoteConnection, displayName);
}
else
{
VSMessageBoxHelper.PostErrorMessage(
StringResources.Error_ContainerUnavailableTitle,
StringResources.Error_ContainerUnavailableMessage.FormatCurrentCultureWithArgs(settings.ContainerName));
return null;
}
}
public static SSHConnection GetSSHConnection(string name)
{
ConnectionInfoStore store = new ConnectionInfoStore();
ConnectionInfo connectionInfo = null;
StoredConnectionInfo storedConnectionInfo = store.Connections.FirstOrDefault(connection =>
{
return string.Equals(name, SSHPortSupplier.GetFormattedSSHConnectionName((ConnectionInfo)connection), StringComparison.OrdinalIgnoreCase);
});
if (storedConnectionInfo != null)
connectionInfo = (ConnectionInfo)storedConnectionInfo;
if (connectionInfo == null)
{
IVsConnectionManager connectionManager = (IVsConnectionManager)ServiceProvider.GlobalProvider.GetService(typeof(IVsConnectionManager));
IConnectionManagerResult result;
if (string.IsNullOrWhiteSpace(name))
{
result = connectionManager.ShowDialog();
}
else
{
ParseSSHConnectionString(name, out string userName, out string hostName, out int port);
result = connectionManager.ShowDialog(new PasswordConnectionInfo(hostName, port, Timeout.InfiniteTimeSpan, userName, new System.Security.SecureString()));
}
if ((result.DialogResult & ConnectionManagerDialogResult.Succeeded) == ConnectionManagerDialogResult.Succeeded)
{
// Retrieve the newly added connection
store.Load();
connectionInfo = store.Connections.First(info => info.Id == result.StoredConnectionId);
}
}
return SSHHelper.CreateSSHConnectionFromConnectionInfo(connectionInfo);
}
/// <summary>
/// Open the ContainerPickerDialog
/// </summary>
/// <param name="hwnd">Parent hwnd or IntPtr.Zero</param>
/// <param name="supportSSHConnections">SSHConnections are supported</param>
/// <param name="connectionString">[out] connection string obtained by the dialog</param>
public static bool ShowContainerPickerWindow(IntPtr hwnd, bool supportSSHConnections, out string connectionString)
{
ThreadHelper.ThrowIfNotOnUIThread("Microsoft.SSHDebugPS.ShowContainerPickerWindow");
ContainerPickerDialogWindow dialog = new ContainerPickerDialogWindow(supportSSHConnections);
if (hwnd == IntPtr.Zero) // get the VS main window hwnd
{
try
{
// parent to the global VS window
DTE dte = (DTE)Package.GetGlobalService(typeof(SDTE));
hwnd = new IntPtr(dte?.MainWindow?.HWnd ?? 0);
}
catch // No DTE?
{
Debug.Fail("No DTE?");
}
}
if (hwnd != IntPtr.Zero)
{
WindowInteropHelper helper = new WindowInteropHelper(dialog);
helper.Owner = hwnd;
}
bool? dialogResult = dialog.ShowModal();
if (dialogResult.GetValueOrDefault(false))
{
connectionString = dialog.SelectedContainerConnectionString;
return true;
}
connectionString = string.Empty;
return false;
}
// Default SSH port is 22
internal const int DefaultSSHPort = 22;
/// <summary>
/// Parses the SSH connection string. Expected format is some permutation of username@hostname:portnumber.
/// If not defined, will provide default values.
/// </summary>
internal static void ParseSSHConnectionString(string connectionString, out string userName, out string hostName, out int port)
{
userName = StringResources.UserName_PlaceHolder;
hostName = StringResources.HostName_PlaceHolder;
port = DefaultSSHPort;
const string TempUriPrefix = "ssh://";
try
{
// In order for Uri to parse, it needs to have a protocol in front.
Uri connectionUri = new Uri(TempUriPrefix + connectionString);
if (!string.IsNullOrWhiteSpace(connectionUri.UserInfo))
userName = connectionUri.UserInfo;
if (!string.IsNullOrWhiteSpace(connectionUri.Host))
hostName = connectionUri.Host;
if (!connectionUri.IsDefaultPort)
port = connectionUri.Port;
}
catch (UriFormatException)
{ }
// If Uri sets anything to empty string, set it back to the placeholder
if (string.IsNullOrWhiteSpace(userName))
{
userName = StringResources.UserName_PlaceHolder;
}
if (string.IsNullOrWhiteSpace(hostName))
{
// handle case that its just a colon by replacing it with the default string
hostName = StringResources.HostName_PlaceHolder;
}
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,794 |
PEDIATRIC HEALTH CHOICE is a family of companies carefully engineered and dedicated to Pediatric Health Care. We are the premier provider of alternative-site health care services for medically complex, technology-dependent and behaviorally challenged children. Over twenty five years ago, we pioneered an innovative, child-focused and family-oriented service delivery model for pediatric health care that has subsequently been adopted throughout the United States today. Presently, we are the largest operator of Ambulatory Medical Day Health Centers in the country. Within our family of companies, we are certified by Medicaid and fully licensed by state regulatory agencies.
Ingrained in our philosophy is: commitment to the highest level of child and family state of wellness/health; dedication to individualized, family-centered care; assurance of a safe, nurturing and progressive environment; provision of qualified, caring professional staff; enhanced utilization of health care dollars; collaborative team effort; a multicultural approach and parent/professional partnership in care.
Currently based in Florida, Pennsylvania, Louisiana, Mississippi and Delaware, we offer a wide range of quality-oriented, comprehensive, specialized and cost-effective community-based pediatric health care disciplines. Our integrated, multi-disciplinary delivery model affords children efficient physician-prescribed services whether acute, chronic or intermittent in nature including: Ambulatory Medical Day Health Center Services; and Physical, Occupational and Oral-Motor/Speech/Language Services. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,485 |
exports.setEnvironment = function() {
process.env['AZURE_BATCH_ACCOUNT'] = 'lchency4';
process.env['AZURE_BATCH_ENDPOINT'] = 'https://lchency4.westcentralus.batch.azure.com';
process.env['AZURE_SUBSCRIPTION_ID'] = '3ee7eaf5-6a2f-49fd-953f-d760b5ac2e05';
};
exports.scopes = [[function (nock) {
var result =
nock('http://lchency4.westcentralus.batch.azure.com:443')
.filteringRequestBody(function (path) { return '*';})
.post('/pools?api-version=2018-03-01.6.1', '*')
.reply(201, "", { 'transfer-encoding': 'chunked',
'last-modified': 'Fri, 09 Mar 2018 22:44:04 GMT',
etag: '0x8D5860F42C979DB',
location: 'https://lchency4.westcentralus.batch.azure.com/pools/nodesdktestpool1',
server: 'Microsoft-HTTPAPI/2.0',
'request-id': 'd409329b-fca1-4288-9136-68b9e1f429fc',
'strict-transport-security': 'max-age=31536000; includeSubDomains',
'x-content-type-options': 'nosniff',
dataserviceversion: '3.0',
dataserviceid: 'https://lchency4.westcentralus.batch.azure.com/pools/nodesdktestpool1',
date: 'Fri, 09 Mar 2018 22:44:04 GMT',
connection: 'close' });
return result; },
function (nock) {
var result =
nock('https://lchency4.westcentralus.batch.azure.com:443')
.filteringRequestBody(function (path) { return '*';})
.post('/pools?api-version=2018-03-01.6.1', '*')
.reply(201, "", { 'transfer-encoding': 'chunked',
'last-modified': 'Fri, 09 Mar 2018 22:44:04 GMT',
etag: '0x8D5860F42C979DB',
location: 'https://lchency4.westcentralus.batch.azure.com/pools/nodesdktestpool1',
server: 'Microsoft-HTTPAPI/2.0',
'request-id': 'd409329b-fca1-4288-9136-68b9e1f429fc',
'strict-transport-security': 'max-age=31536000; includeSubDomains',
'x-content-type-options': 'nosniff',
dataserviceversion: '3.0',
dataserviceid: 'https://lchency4.westcentralus.batch.azure.com/pools/nodesdktestpool1',
date: 'Fri, 09 Mar 2018 22:44:04 GMT',
connection: 'close' });
return result; }]]; | {
"redpajama_set_name": "RedPajamaGithub"
} | 315 |
Bård Torstensen (born 13 September 1961) is a Norwegian guitarist and record producer, best known as a founder and longtime member of the rap metal band Clawfinger.
Torstensen also plays guitar in the ice skate-country band Melkesyra and the jazz/metal band Okavango, both of which are from his home town. Before founding Clawfinger in 1990 with Zak Tell (lead vocals), Jocke Skog (keyboards) and Erlend Ottem (guitar), he and Ottem played in a local band named Theo. This band made only one single and one LP, The Good, the Bad, and the Ugly (1988).
Torstensen is a known activist in his home town Arendal, and is engaged in preserving his community Barbu by working to stop new building plans.
Equipment and gear
During his time in Clawfinger, he can mostly be seen playing various Gibson Les Pauls, most mostly relied on Gibson Les Paul Studios. He could also be seen playing Les Paul Customs and Standards, as well as an Epiphone Les Paul Standard Baritone which was used for songs that required Drop A tuning. He could also be seen using guitars from other brands, such as a Schecter Celloblaster C5-X Baritone and an Ibanez Xiphos 7-string, both of which were also use in drop A tuning, although the Schecter was sometimes tuned to an alternate tuning where the lowest 3 strings are tuned to G in 2 octaves, and the remaining strings were tuned to D, and G above an octave (G-G-G-D-G). This tuning can be heard on songs like "Nothing Going On".
For amplification, the band claims to have never relied on traditional amplifiers; solely on pre-amps, speaker cabinet emulation, multi-effect processors, and amp modeling. During the 1990s and early 2000s, Torstensen mostly relied SansAmp preamps, starting with the original pedal Sansamp in the early-mid 1990s. A Boss ME-5 processor was used along with it. During the late-1990s and early 2000s, he relied on a small rack unit that contained a Tech21 SansAmp PSA-1 analog pre-amp, Digitech 2112 multi-effects processor for speaker emulations, and a wireless unit. From 2001 to 2004, he began using a Boss GX700 pre-amp and multi-FX processor that ran into the Digitech or a Matchbox MB10 for speaker simulations. Sometime after the release of Zeroes and Heroes, he switched to using Line6 processors, relying on the PODxt series. This change is apparent in the band's next album Hate Yourself With Style, due to the guitars sounding less "processed" and more "metallic". In the studio, the POD would be run into a PA system power amp which fed into a Line6 4x12, which would be picked up via microphone. Live, the POD would run straight into the venue's PA system. Sometime during 2009, the band began using Fractal Axe-FX processors, which he would use until the end of the band.
Discography (as guitarist)
Theo – The Good, the Bad, and the Ugly (1988)
Tone Norum – Don't Turn Around(1992)
Clawfinger – Deaf Dumb Blind (1993)
Clawfinger – Use Your Brain (1995)
Clawfinger – Clawfinger (1997)
Clawfinger – A Whole Lot of Nothing (2001)
Clawfinger – Zeros & Heroes (2003)
Clawfinger – Hate Yourself With Style (2005)
Melkesyra – Hurtigløpskøntri (2006)
Clawfinger – Life Will Kill You (2007)
Melkesyra – Melkesyra går allround (2009)
Okavango – Phonogene (2011)
Neonato – The End of Music (2015)
Melkesyra – Rett fram og til venstre (2018)
References
External links
Clawfinger webpage
Melkesyra webpage
Okavango webpage
Norwegian heavy metal guitarists
Norwegian record producers
Musicians from Arendal
Living people
1961 births | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,447 |
A former NorthShore University HealthSystem/Swedish Covenant obstetrics and gynecology doctor - Fabio Ortega, 75 – has pleaded guilty to sexually assaulting one of our clients, and another woman, during closed-door exams, in 2016 and 2017. In exchange for his plea of guilty, to two counts of felony Aggravated Criminal Sexual Abuse, Ortega was sentenced to three-years in the Illinois Department of Corrections.
After his plea, our client made a Victim Impact Statement to Ortega and NorthShore's lawyers, saying, "You have broken me. You haunt my dreams and creep into my most intimate moments. I am a shell of my former self. I'm not living, I am barely surviving day to day... No matter what I do, how much medicine I take, how many times a week I see my therapist, how many people I surround myself with, this will forever be my new identity. I will always be a victim but I am hoping that one day I will learn to be a survivor." Ms. Doe 3 is available for further comment.
The NorthShore/Swedish female patients allege that Ortega engaged in the following perverted conduct behind closed-doors:
Ortega penetrated his fingers into female patients' vaginas;
Ortega grabbed female patients' breasts;
Ortega asked female patients' sexually perverted questions;
Ortega touched female patients' anuses;
Ortega called female patients late at night, in an effort to groom them.
The lawsuits also allege that NorthShore's top officials - including its head of the Gynecology and Obstetrics Department - allowed Ortega to continue to work despite knowing he was under criminal investigation for sexual assault. Then, NorthShore allowed Ortega to quietly retire rather than fire him. The women's claims span over the years of 1996-2017.
"Ortega's guilty plea would not have occurred without the help of many people outside of the NorthShore system. We are so thankful for the investigators at Skokie and Lincolnwood police departments, the prosecutors with the Cook County State's Attorney's office. These are the people who believed Ortega's victims. NorthShore did not. Instead, NorthShore and Swedish choose profits over patient safety. They ignored the cries of their female patients. They harbored a sexual predator.
And still, even after Ortega's guilty plea, they refuse to apologize to the female patients who were abused under their roof. NorthShore and Swedish continue to gaslight Ortega's victims and say they are not liable for his abuses. In court filings, NorthShore/Swedish have made a litany of defenses, including that Ortega's conduct was proper; that the women are confused or lying; and, that NorthShore/Swedish are not liable because the women's claims are time-barred by the statute of limitations. We are determined to end the institutional abuse of women who put their lives into the hands of their healthcare providers," says Chicago-based attorney Tamara Holder.
To date, Ms. Holder and her co-counsel Johanna J. Raimond have filed a combined 14 lawsuits against NorthShore and Swedish. Currently, there are seven pending lawsuits against NorthShore, and two pending lawsuits against Swedish Covenant, where Ortega worked before NorthShore. NorthShore has settled at least 6 lawsuits since 2019.
If you have additional information about NorthShore or Ortega, please contact us at: 312-818-3850 or email: contact@tamaraholder.com.
NorthShore University HealthSystem has settled with three more of our clients who claim they were sexually assaulted by Dr. Fabio Ortega, an OB/GYN who practiced at NorthShore's Skokie, Evanston and Lincolnwood offices, and Evanston Hospital.
One NorthShore University HealthSystem patient alleged she was sexually assaulted by Dr. Ortega at Evanston Hospital just house after she gave birth to her daughter. That patient, and another patient, claimed that NorthShore knew about Ortega's sexually perverted conduct and allowed him to see patients even while he was under investigation for sexual assault. In fact, Dr. Richard Silver who heads the Department of Obstetrics and Gynecology met with Ortega and continued to allow him to see women without a chaperone. Allegations include touching women's vaginas, grabbing their breasts, and asking them sexually perverted questions under the guise of performing proper pelvic exams.
Ortega is facing two charges of sexual assault in Cook County. NorthShore continues to stand-by gynecologist Dr. Ortega and is defending itself, and him, in legal proceedings. Swedish Covenant Health where Ortega worked before going to NorthShore is also defending Ortega is two lawsuits.
If you have information about Fabio Ortega, please contact us: contact@tamaraholder.com
The Law Firm of Tamara N. Holder, LLC 312-818-3850
THE LAW FIRM OF TAMARA N. HOLDER
MICHAEL REINSDORF (CHICAGO BULLS), TOM RICKETTS (CHICAGO CUBS) AMONG BIG CHICAGO NAMES ON NORTHSHORE UNIVERSITY HEALTHSYSTEM BOARD OF DIRECTORS WHILE HOSPITAL GROUP HARBORED SEXUAL PREDATOR GYNECOLOGIST
NorthShore's top officials knew that Fabio Ortega was under criminal investigation for sexual assault, had history of complaints, yet still allowed him to work unsupervised, concealed investigation from female patients.
This week, Jane Doe 17 filed a 10-count complaint against Dr. Fabio Ortega and NorthShore University HealthSystem alleging:
Michael Reinsdorf, Tom Ricketts, A. Steven Crown were on NorthShore's Board of Directors in 2017 when NorthShore knew Dr. Fabio Ortega was under active criminal investigation for sexually assaulting female patients. The Board is responsible for credentialing of its doctors.
Despite being a multi-billion-dollar charitable institution, NorthShore applied for, and received, $85 million in CARES Act" grants – the statute was intended to protect financially insecure hospitals during the COVID-19 crisis.
NorthShore falsely claims it provides patients with "the right to be free from mental, physical, sexual, and verbal abuse…neglect…and exploitation."
NorthShore harbored, for thirteen years, a now-twice-indicted male gynecologist – Dr. Fabio Ortega – who sexually assaulted multiple women under the guise of providing legitimate medical care.
For years, female patients complained that Ortega touched their anuses, penetrated his fingers into their vaginas, grabbed their breasts, rubbed their thighs, and asked them sexually perverted questions. Rather than believe the women, NorthShore ignored them, and continued to steer trusting women to Ortega. Women complained when Ortega was at Swedish Hospital, too, before NorthShore hired him.
In February 2017, police notified NorthShore that Ortega was under criminal investigation for sexually assaulting a woman at one of its clinics. NorthShore immediately hired defense attorneys to protect itself, while continuing to steer women to Ortega and concealing that he was under active criminal investigation.
In July 2017, NorthShore scheduled Jane Doe 17 who was then just 21-years old, for a gynecological exam with Ortega. NorthShore not only concealed from her that Ortega was under criminal investigation for sexual assault, it failed to even supervise him during his closed-door "exam." Ortega then sexually assaulted her.
The 10-count complaint demands a trial by jury and seeks unspecified damages
In other lawsuits, NorthShore defends that it is not responsible for Ortega and that his conduct was legitimate – essentially stating that the women are lying.
In December 2019, our firm filed two lawsuits against Swedish Covenant Lawsuit on behalf of former female patients who allege they were sexually abused by gynecologist Fabio Ortega. Ortega has since been indicted for sexually abusing two other patients while he was an OBGYN at NorthShore University HealthSystem.
According to one woman's complaint:
During every appointment, he engaged in the following behavior, disguised as a medically necessary exam, in ways including but not limited to:
a. Ortega touched her vagina;
b. He inserted his fingers into her vagina;
c. Ortega would cause her extreme pain when he inserted his fingers inside of her.
d. When she told Ortega that he was hurting her, he would compare his fingers to her husband's penis with comments like, "I am sure you've had bigger down there," and then proceed to shove his fingers deeper into her vagina;
e. After he removed his fingers from her vagina, he would linger over her, then remove his gloves and touch the inside of her thigh with his bare hand;
f. During most appointments, he would grope her breasts "to check for lumps," even though she never complained to about her breasts;
g. During most appointments, she would ask to hear her baby's heartbeat but he would dismiss her request and say, "Everything is fine."
According to a second woman's complaint:
During every single visit, Ortega would engage in the following behavior disguised as medically necessary exams:
a. Ortega would first instruct her lay down and open her legs;
b. Next, He would touch her clitoris with his ungloved fingers;
c. Then, would he would tell her to "just relax," as he inserted his ungloved fingers into her vagina;
d. Then, he would instruct her to "squeeze my fingers;"
e. Then, he would stand closely over her, between her open legs, as he took notes, in an effort to disguise his sexual assault as a medically necessary exam.
NorthShore has also been sued by six (6) women who allege Ortega sexually assaulted them under the guise of providing proper medical care at Evanston Hospital, and his clinics in Skokie and Lincolnwood, Illinois.
These two patients will return to court once the courthouse reopens following the coronavirus pandemic. NorthShore has claimed in recent filings that it is not liable for Ortega's behavior, despite having knowledge of patient complaints and continuing to allow him to practice without unfettered access to trusting female patients. /em>
If you or someone you know has information about Fabio Ortega, please contact Tamara Holder 847-651-7222 or contact@tamaraholder.com | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,630 |
namespace apl
{
namespace dnp
{
MockFrameSink::MockFrameSink() : mNumFrames(0), mLowerOnline(false)
{}
void MockFrameSink::OnLowerLayerUp()
{
mLowerOnline = true;
}
void MockFrameSink::OnLowerLayerDown()
{
mLowerOnline = false;
}
void MockFrameSink::Reset()
{
this->ClearBuffer();
mNumFrames = 0;
}
bool MockFrameSink::CheckLast(FuncCodes aCode, bool aIsMaster, boost::uint16_t aDest, boost::uint16_t aSrc)
{
return (mCode == aCode) && (aIsMaster == mIsMaster) && (mSrc == aSrc) && (mDest == aDest);
}
bool MockFrameSink::CheckLastWithFCB(FuncCodes aCode, bool aIsMaster, bool aFcb, boost::uint16_t aDest, boost::uint16_t aSrc)
{
return (mFcb == aFcb) && CheckLast(aCode, aIsMaster, aDest, aSrc);
}
bool MockFrameSink::CheckLastWithDFC(FuncCodes aCode, bool aIsMaster, bool aIsRcvBuffFull, boost::uint16_t aDest, boost::uint16_t aSrc)
{
return (mIsRcvBuffFull == aIsRcvBuffFull) && CheckLast(aCode, aIsMaster, aDest, aSrc);
}
// Sec to Pri
void MockFrameSink::Ack(bool aIsMaster, bool aIsRcvBuffFull, boost::uint16_t aDest, boost::uint16_t aSrc)
{
mIsRcvBuffFull = aIsRcvBuffFull;
this->Update(FC_SEC_ACK, aIsMaster, aDest, aSrc);
}
void MockFrameSink::Nack(bool aIsMaster, bool aIsRcvBuffFull, boost::uint16_t aDest, boost::uint16_t aSrc)
{
mIsRcvBuffFull = aIsRcvBuffFull;
this->Update(FC_SEC_NACK, aIsMaster, aDest, aSrc);
}
void MockFrameSink::LinkStatus(bool aIsMaster, bool aIsRcvBuffFull, boost::uint16_t aDest, boost::uint16_t aSrc)
{
mIsRcvBuffFull = aIsRcvBuffFull;
this->Update(FC_SEC_LINK_STATUS, aIsMaster, aDest, aSrc);
}
void MockFrameSink::NotSupported (bool aIsMaster, bool aIsRcvBuffFull, boost::uint16_t aDest, boost::uint16_t aSrc)
{
mIsRcvBuffFull = aIsRcvBuffFull;
this->Update(FC_SEC_NOT_SUPPORTED, aIsMaster, aDest, aSrc);
}
// Pri to Sec
void MockFrameSink::TestLinkStatus(bool aIsMaster, bool aFcb, boost::uint16_t aDest, boost::uint16_t aSrc)
{
mFcb = aFcb;
this->Update(FC_PRI_TEST_LINK_STATES, aIsMaster, aDest, aSrc);
}
void MockFrameSink::ResetLinkStates(bool aIsMaster, boost::uint16_t aDest, boost::uint16_t aSrc)
{
this->Update(FC_PRI_RESET_LINK_STATES, aIsMaster, aDest, aSrc);
}
void MockFrameSink::RequestLinkStatus(bool aIsMaster, boost::uint16_t aDest, boost::uint16_t aSrc)
{
this->Update(FC_PRI_REQUEST_LINK_STATUS, aIsMaster, aDest, aSrc);
}
void MockFrameSink::ConfirmedUserData(bool aIsMaster, bool aFcb, boost::uint16_t aDest, boost::uint16_t aSrc, const boost::uint8_t* apData, size_t aDataLength)
{
mFcb = aFcb;
this->WriteToBuffer(apData, aDataLength);
this->Update(FC_PRI_CONFIRMED_USER_DATA, aIsMaster, aDest, aSrc);
}
void MockFrameSink::UnconfirmedUserData(bool aIsMaster, boost::uint16_t aDest, boost::uint16_t aSrc, const boost::uint8_t* apData, size_t aDataLength)
{
this->WriteToBuffer(apData, aDataLength);
this->Update(FC_PRI_UNCONFIRMED_USER_DATA, aIsMaster, aDest, aSrc);
}
void MockFrameSink::AddAction(boost::function<void ()> aFunc)
{
mActions.push_back(aFunc);
}
void MockFrameSink::ExecuteAction()
{
if(mActions.size() > 0) {
boost::function<void ()> f = mActions.front();
mActions.pop_front();
f();
}
}
void MockFrameSink::Update(FuncCodes aCode, bool aIsMaster, boost::uint16_t aDest, boost::uint16_t aSrc)
{
++mNumFrames;
mCode = aCode;
mIsMaster = aIsMaster;
mDest = aDest;
mSrc = aSrc;
this->ExecuteAction();
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,078 |
using System;
using System.Linq;
using Microsoft.Azure.Management.Sql.LegacySdk.Models;
namespace Microsoft.Azure.Management.Sql.LegacySdk.Models
{
/// <summary>
/// Represent the properties of a sync group.
/// </summary>
public partial class SyncGroupProperties
{
private Microsoft.Azure.Management.Sql.LegacySdk.Models.ConflictResolutionPolicyType? _conflictResolutionPolicy;
/// <summary>
/// Optional. The policy of resolving confliction between hub and
/// member database in the sync group.The possible values: 'HubWin'
/// and 'MemberWin'.
/// </summary>
public Microsoft.Azure.Management.Sql.LegacySdk.Models.ConflictResolutionPolicyType? ConflictResolutionPolicy
{
get { return this._conflictResolutionPolicy; }
set { this._conflictResolutionPolicy = value; }
}
private string _hubDatabaseUserName;
/// <summary>
/// Optional. The user name of the hub database.
/// </summary>
public string HubDatabaseUserName
{
get { return this._hubDatabaseUserName; }
set { this._hubDatabaseUserName = value; }
}
private int? _interval;
/// <summary>
/// Optional. Specifies the interval time of doing data synchronization.
/// </summary>
public int? Interval
{
get { return this._interval; }
set { this._interval = value; }
}
private System.DateTime? _lastSyncTime;
/// <summary>
/// Optional. The last sync time of a sync group.
/// </summary>
public System.DateTime? LastSyncTime
{
get { return this._lastSyncTime; }
set { this._lastSyncTime = value; }
}
private SyncGroupSchema _schema;
/// <summary>
/// Optional. The simple schema of sync group.
/// </summary>
public SyncGroupSchema Schema
{
get { return this._schema; }
set { this._schema = value; }
}
private string _syncDatabaseId;
/// <summary>
/// Optional. The database used to store sync meta data.
/// </summary>
public string SyncDatabaseId
{
get { return this._syncDatabaseId; }
set { this._syncDatabaseId = value; }
}
private string _syncState;
/// <summary>
/// Optional. The sync state of a sync group. The possible values: Enum
/// ('NotReady', 'Error', 'Warning', 'Processing', 'Good').
/// </summary>
public string SyncState
{
get { return this._syncState; }
set { this._syncState = value; }
}
/// <summary>
/// Initializes a new instance of the SyncGroupProperties class.
/// </summary>
public SyncGroupProperties()
{
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,978 |
'Available only in participating outlets' notice inadmissible?
Franchising Germany
According to Section 5a(3)(2) of the Act against Unfair Competition, an advertiser must display its identity and address on advertisements. If a company advertises not only its brand (image advertising), but also actual products and their prices, it must identify in print where the offer originates from and where the company is based.
This obligation presents considerable problems for franchise systems and groups, as well as for distributor systems. Most advertisements do not have the space to include the identities and addresses of potentially hundreds of franchisees. Whether a reference to a website of the company providing that information is adequate (Ie, the change from one medium to another) is disputed. Judgments have thus far rejected such references on different grounds. However, associations of competitors have sometimes accepted this solution in their claims and concluded settlements accordingly.
The Dusseldorf Higher Regional Court has to a certain extent added to this problem.
Cartel law provides that a retailer can, in principle, determine its own sale prices. Franchisees, groups of stores or distributors can therefore decide themselves how much to demand from customers. This must be taken into account in the franchisor's supra-regional advertising. Such advertising should not put any pressure for uniformity on the franchisee, to the effect that the franchisee is de facto compelled to charge the prices in the advertisement.
For this reason, supra-regional advertising by franchisors (or distributor systems) usually carries a footnote saying "non-binding price recommendation" or "only at participating dealers". According to the Federal Cartel Office, a franchisor's or manufacturer's enquiry as to whether a sales partner has complied with or is participating in the price recommendation carries the risk of being classified as an inadmissible pricing measure under cartel law.
In view of the August 5 2014 Dusseldorf Higher Regional Court decision, this footnote or notice could breach Section 5a(3)(2) of the Act against Unfair Competition, because the advertising does not contain the name and address of all participating dealers.
A retail franchisor issued an advertising supplement of approximately 20 pages listing more than 100 products and prices nationwide. All prices were offered as "non-binding price recommendations" and "obtainable only in participating outlets".
A consumer association commenced proceedings, complaining that the advertisement did not specify the outlets in which the offers were actually available. It is presumed that a customer who had enquired about the offer in a franchise outlet which did not provide the offer or non-binding price recommendation had complained to the consumer association, which then sued the franchisor for cessation of the advertisement.
While the Krefeld District Court correctly dismissed the claim (June 17 2011, 11 O 12/11), the Dusseldorf Higher Regional Court banned the franchisor from this form of advertising.
The Dusseldorf Higher Regional Court first rejected the need to refer the case – because of the cartel law issue – to the relevant cartel court (the senate of the higher district court hearing the case is not responsible for cartel matters). Cartel law was not decisive to the case. The higher regional court based this incorrect view on the general reference that the franchisor could avoid the cartel law problem by refraining in future from using this form of special offer or advertisement. The court held that the franchisor "could restrict itself in accordance with its role as franchisor, which does not sell the products to the consumer, to image advertising for the franchise system".
This demonstrates the court's lack of understanding of franchising distribution methods.
In addition, the court stated that it could not be assumed that the average consumer would know how a franchise company differs from a branch and, to a lesser extent, what legal significance this has for advertising. The reference that "all offers are exclusively non-binding price recommendations and only obtainable in participating outlets" did not change this.
The court also commented that consumers are accustomed to non-binding price recommendations originating from the manufacturer and not, for example, from wholesalers, or – in this case – from franchisors. When buying a vehicle, a consumer is accustomed to vehicles being offered by independent distributors at differing prices; but a non-binding price recommendation makes no sense in the case of a trading company (eg, the franchisor). A consumer would not be aware or would not perceive a non-binding price recommendation as a reservation of price changes of the trading company actually advertising the offer.
A consumer is also not required to consider a notice that "offers are available only in participating outlets". This is all the more relevant if the consumer is also unaware that a franchise system is involved, and of the legal consequences. A customer would see the advertisement as typical retailer advertising and assume the offer to be available in all of the retailer's outlets.
Advertising by a franchisor (which operates no outlets of its own) for its franchisees must state the franchisees' names and addresses. If a company, such as a franchisor, advertises the offers of other companies, the identities of the advertising companies must be stated.
The franchisor could do this in a brochure, as all franchisees providing the offer need not be included. According to the court, this "must however be those which actually provide the advertised offer".
If the franchisor is unable to legally determine the participating franchisees, "it cannot advertise with offers and must leave this form of advertising with concrete offers to its franchisees".
According to the court, if a franchisor observes cartel law and does not ask franchisees whether they are following the price recommendation, it may not issue any supra-regional advertising. This is because the franchisor, without making such an enquiry, cannot say which franchisees or dealers are participating in a campaign offer and what the associated price recommendation is. It is unrealistic for franchisees and unusual for most franchise systems to issue supra-regional advertising.
In this case the court forced the franchisor either to enquire about the participation of franchisees in the price recommendation and thereby risk a penalty under cartel law, or to dispense with such risky inquiry and forgo nationwide advertising by the system base or franchisor.
Even if a franchisor takes the risky option under cartel law and asks about participation, the problem of what form the names and addresses of participating outlets should be provided remains (ie, whether displaying this on a website is sufficient or whether the printed advertisement itself must contain the names and addresses of participating outlets).
It remains to be seen whether in the pending appeal (I ZR 194/14), the Federal Court of Justice will put an end to this and confirm a practicable, reasonable way for franchisors to advertise supra-regionally in compliance with the law.
For further information on this topic please contact Karsten Metzlaff or Tom Billing at Noerr LLP's Berlin office by telephone (+49 30 20 94 20 00), fax (+49 30 20 94 20 94) or email (karsten.metzlaff@noerr.com or tom.billing@noerr.com). Alternatively, contact Karl Rauser at Noerr's Munich office by telephone (+49 89 28 62 80), fax (+49 89 28 0110) or email (karl.rauser@noerr.com). The Noerr LLP website can be accessed at www.noerr.com.
Karsten Metzlaff
Tom Billing
Karl Rauser
Bogus self-employment and unethical franchise agreements
For want of an asterisk: regional court scrutinises franchisor's TV advertising
Admissibility of lawsuit despite effective mediation clause in franchise agreement
Requirements for interim injunction to enforce obligation to operate business
No compensation claim for franchisee where franchisor must block customer data when agreement terminated | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,263 |
{"url":"http:\/\/www.mathnet.ru\/php\/archive.phtml?jrnid=sm&wshow=issue&year=1974&volume=136&volume_alt=94&issue=4&issue_alt=8&option_lang=eng","text":"RUS\u00a0 ENG JOURNALS \u00a0 PEOPLE \u00a0 ORGANISATIONS \u00a0 CONFERENCES \u00a0 SEMINARS \u00a0 VIDEO LIBRARY \u00a0 PACKAGE AMSBIB\n General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS\n\n Mat. Sb.: Year: Volume: Issue: Page: Find\n\n 1974, Volume\u00a094(136), Number\u00a04(8)\n\n Holomorphic functions with positive imaginary part in the future tube.\u00a0IIV.\u00a0S.\u00a0Vladimirov 499 On the semiregularity of boundary points for nonlinear equationsE.\u00a0B.\u00a0Frid 516 On unconditional convergence in the space\u00a0$L_1$B.\u00a0S.\u00a0Kashin 540 Weakly special and special radicals in semigroupsA.\u00a0V.\u00a0Tishchenko 551 On the theory of the discrete spectrum of the three-particle Schro\u0308dinger operatorD.\u00a0R.\u00a0Yafaev 567 Elliptic modulesV.\u00a0G.\u00a0Drinfeld 594 Graduated formations of groupsL.\u00a0A.\u00a0Shemetkov 628 On the Galois action on rational cohomology classes of type $(p,p)$ of Abelian varietiesM.\u00a0V.\u00a0Borovoi 649","date":"2020-06-01 18:43:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2153434157371521, \"perplexity\": 5273.014278478349}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590347419593.76\/warc\/CC-MAIN-20200601180335-20200601210335-00201.warc.gz\"}"} | null | null |
{"url":"https:\/\/educationalresearchtechniques.com\/category\/latex\/","text":"# Basics of LateX\n\nIn this post, we will explore more concepts about Latex the typesetting language.\n\nOptional Commands\n\nOptional commands appear in brackets [\u00a0 \u00a0]\u00a0 when you are using Latex. In the example below, we will set the font size to 20pt in the preamble of the document. The code is as follows.\n\n\\documentclass[4paper,12pt]{article} \\begin{document}\nBehold the power of \\LaTeX\n\\end{document}\n\nHere is what it looks like\n\nInside the brackets, we set the paper size to A4 and the font size to 12pt. Many if not most commands have optional commands that can be used to customize the behavior of the document.\n\nLike most coding languages Latex allows you to make comments. To do this you need to place a % sign in front of your comment. As shown below\n\n\\documentclass[4paper,12pt]{article} \\begin{document}\nBehold the power of \\LaTeX\n%This will not print\n\\end{document}\n\nEverything after the % did not print. To stop this action simply press enter to move to the next line and you can continue with your document.\n\nFun with Fonts\n\nThere are many different ways to set the fonts. Generally, you can use the\u00a0\\text**{\u00a0 } code. Where the asterisks are is where you can specify the behavior you want of the text. Below is a simple example of the use of several different formats to the font.\n\n\\documentclass[4paper,12pt]{article}\n\\begin{document}\nYou can \\textit{italicized}\nText can be \\textsl{slanted}\nOff course, you can \\textbf{bold} text\nYou can also make text in \\textsc{small caps}\nIt is also possible to use several commands at the same \\textit{\\textbf{time}}\nBehold the power of \\LaTeX\n\\end{document}\n\nNotice how you put the command in front of the word that you want to format. This might seem cumbersome. However, once you get comfortable with this it is much faster to format documents then the point and click style of Word.\n\nEnvironments\n\nIf you want a certain effect to last for awhile you can use an environment. An environment is a space you declare in your document in which a center behavior takes place. Generally, environments are used to improve the readability of your code. Below is an example.\n\n\\documentclass[4paper,12pt]{article}\n\\begin{document}\n\\begin{bfseries}\nEverything is bold here\n\\end{bfseries}\n\\begin{itshape}\nEverything is bold here\n\\end{itshape}\nBehold the power of \\LaTeX\n\\end{document}\n\nAn environment always begins with the \\begin command and ends with the \\end command. In the curly braces, you type whatever is required for your formatting goals. There are scores of commands you can place inside the curly braces.\n\nConclusion\n\nThere is so much more to learn but this is just a beginning. One of the main benefits of learning Latex is the fixed nature of the formatting and the speed at which you can produce content once you are familiar with how to use this language.\n\n# Intro to LaTex\n\nHistory\n\nLaTex is an open-sourced typesetting document developed about 30 years ago by Leslie Lamport and based on the Tex\u00a0typesetting of Donald Knuth. It is commonly used in the domains of physics and math for producing mathematical\u00a0equations and other technical\u00a0documents. Below is a simple example of an equation developed using LaTex\n\nLaTex is a document markup language, which means that you indicate the commands and then it is processed to produce the desired effect. This is in contrast to Microsoft Word which utilizes a WYSIWYG\u00a0(What you see is what you get) approach.\n\nBenefits\n\nUsing LaTex provides several benefits. Cross-referencing\u00a0is easily accomplish especially with the help of BibTex. It is also multi-lingual and able to make glossaries, indexes, and figures\/tables with ease. In addition, LaTex is highly portable and opening a file on any computer is not a problem. Sometimes moving to another computer using Microsoft Word can cause issues with formatting.\n\nAnother benefit is psychology, using LaTex allows the author to focus on content and not appearance when writing. It is easy to get distracted when using Word to try and make something work through the point and click mechanism we are so used to when writing.\n\nCons\n\nIt takes extensive time to use LaTex. It looks similar to coding which is intimidating for many. However, once a certain mastery is achieved. Producing documents can be faster as everything is text-based and not point click based using a mouse.\n\nUsing LaTex\n\nTo use LaTex you need to install TexLive and TexWorks. TextLive is a LaTex distribution and TexWorks is one of many LaTex editors. The editor allows you to manipulate the LaTex code that you generate.\n\nOnce you have installed both programs you can type the following into TexWorks. Make sure the typeset is set to pdfLaTex. This allows the output to be a pdf file.\n\n\\documentclass{article}\n\\begin{document}\nThis is an example of what LaTex does\n\\end{document}\n\nWhat happened is as follows\n\n1. We entered the command \\documentclass{article}. All commands begin with a slash followed by the name. The curly braces are required\u00a0arguments. In this case, we are using the article template which is one of many templates available in LaTex.\n2. The next command is \\begin and this command indicates the beginning of the actual text of the document. Everything above the \\begin command is part of what we call the preamble.\n3. Next is the actual text that we want to appear in the pdf.\n4. Lastly, we have the \\end command which tells LaTex that the document is finished. Everything between \\begin and \\end command is part of the environment.\n\nConclusion\n\nThere is so much more that can be accomplished with this typesetting software. The possibilities will be explored in the near future.","date":"2018-03-17 22:19:11","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8798686861991882, \"perplexity\": 1038.310107058167}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-13\/segments\/1521257645362.1\/warc\/CC-MAIN-20180317214032-20180317234032-00339.warc.gz\"}"} | null | null |
Mária Mázorová (rodným jménem Marie Panczaková, 8. února 1928, Praha – 12. února 2016, Zvolen) byla slovenská folkloristka, pedagožka, choreografka a zakladatelka dvou známých slovenských folklorních souborů Zornička a Marína.
Založila taneční obor Lidové školy umění ve Zvolenu, vytvořila osnovy, učební plány a metodiky pro Lidové školy umění, napsala knihu Slovenské ľudové tance a iniciovala a vedla Mezinárodní dětský folklorní festival ve Zvolenu.
V roce 2006 získala Cenu města Zvolen a v roce 2008 ocenění Kvet kultúry a umenia.
Reference
Externí odkazy
Slovenští folkloristé
Slovenští tanečníci
Slovenští choreografové
Narození v roce 1928
Narození 8. února
Narození v Praze
Úmrtí v roce 2016
Úmrtí 12. února
Úmrtí ve Zvolenu
Ženy | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 103 |
\section{Introduction}
The advent of large galaxy surveys has transformed the study of large scale structure, allowing high-precision measurements of galaxy clustering statistics. Imaging and spectroscopic surveys, such as the Sloan Digital Sky Survey (SDSS, \citealt{York2000}), the Dark Energy Survey (DES, \citealt{Abbott2016}), the Dark Energy Spectroscopic Instrument (DESI, \citealt{collaboration2016a}), and the upcoming Legacy Survey of Space and Time (LSST, \citealt{LSST2009,Ivezic2019}), provide extraordinary opportunities to utilize such clustering measurements to study both galaxy formation and cosmology. However, it is difficult to model these directly since they depend on complex baryonic processes that are not fully understood. In the standard framework of $\Lambda$CDM cosmology, galaxies form and evolve in dark matter haloes \citep{White1978}, and therefore galaxy clustering can be modelled through halo clustering and galaxy-halo connection.
The formation and evolution of the dark matter haloes are dominated by gravity and their abundance and clustering can be well predicted by analytic models \citep{Press74,Bond91,Mo96,Sheth99,Paranjape2013} and by using high-resolution cosmological numerical simulations \citep{Springel2005,Prada2012,Villaescusa-Navarro2019,Wang2020}. Numerical $N$-body simulations track the evolution of dark matter particles under the influence of gravity and are able to accurately reproduce non-linear clustering on small scales. Haloes or subhaloes can be identified \citep{Springel2001b,Behroozi2013} and merger tree can then be constructed by linking the haloes or subhaloes to their progenitors and descendants at each snapshot in the simulation.
A useful approach for incorporating the predictions of galaxy formation physics is with semi-analytic modelling (SAM), in which the simulated dark matter haloes are populated with galaxies and evolved according to specified prescriptions for gas cooling, galaxy formation, feedback processes, and merging \citep{DeLucia2007,Guo2011,Guo2013,Croton2016,Stevens2016,Cora2018}. Such models have been successful in reproducing several measured properties of galaxy populations and have become a popular method to explore the galaxy-halo connection.
An alternative approach to model galaxy formation is provided by cosmological hydrodynamic simulations \citep{Schaye2015,Nelson2019}, which simulate both the dark matter particles and the stellar and gas components. The baryonic processes are tracked by a combination of fluid equations and subgrid prescriptions. Cosmological hydrodynamical simulations are starting to play a major role in studying galaxy formation, but are computationally expensive for the large volumes involved.
Empirical models such as halo occupation distribution (HOD) modelling \citealt{Berlind2002,Cooray2002,Zheng2005,Zehavi2005,Zehavi2011}) and subhalo abundance matching (SHAM, \citealt{Conroy2006,Behroozi2010,Reddick2013,Guo2016,Chaves-Montero16,Contreras2020}) are also used to model galaxy clustering by characterizing the relation between galaxies and their host haloes. In the HOD approach, one fits or utilizes a model for the halo occupation function, the average number of central and satellite galaxies in the host halo as a function of the halo mass. In contrast, the SHAM methodology connects galaxies to dark matter (sub)haloes using a monotonic relation between the galaxy's luminosity (or stellar mass) and the subhalo mass (or maximum circular velocity). Compared to SAM and hydrodynamic simulations, HOD and SHAM are practical and faster ways to generate realistic galaxy mock catalogues, increasingly important for the planning and analysis of galaxy surveys.
In the standard HOD or SHAM approaches, the galaxy content only depends on the halo or subhalo mass (or related mass indicators). However, halo clustering has been shown to depend on secondary halo properties or more generally on the assembly history or large-scale environment of the haloes, a phenomenon termed (halo) assembly bias \citep{Sheth2004,Gao2005,Wechsler2006,Gao2007,Paranjape2018,Ramakrishnan2019}. The dependences on these secondary parameters manifest themselves in different ways and are not trivially described \citep{Mao2018,Salcedo2018,Xu2018,Han2019}. Halo assembly bias might impact large scale galaxy clustering as well, if the formation of galaxy is correlated to that of the host halo, an effect commonly referred to as galaxy assembly bias (GAB hereafter; e.g., \citealt{Croton2007,Zu2008,Chaves-Montero16,Contreras2019,Xu2020,Xu2021}). In such a case, the halo occupation by galaxies will no longer depend solely on halo mass, but will vary with these secondary halo and environmental properties. These expected occupancy variations have recently been studied in SAM and hydrodynamical simulations \citep{Zehavi2018,Zehavi2019,Artale2018,Bose2019,Xu2021}).
If the GAB is significant in the real universe, neglecting it would have direct implications for interpreting galaxy clustering and the inferred galaxy-halo connection and cosmological constraints \citep{Zentner2014,McEwen2018,McCarthy2019,Lange2019}. Some extensions to include environment or other halo properties have been suggested (e.g., \citealt{Hearin2016,McEwen2018,Contreras2021,Xu2021}). However, given the complexities involved, it is very hard to develop a scheme which will simultaneously incorporate the occupancy variation (hereafter OV) of all relevant halo properties. Moreover, as demonstrated in \citet{Xu2021}, each halo property on its own contributes only a small fraction of the GAB signal, such that a mix of multiple properties will likely be required. this makes first principles predictions for assembly bias challenging. Alternative approaches to predict galaxy properties based on halo assembly history have been proposed \citep{Moster2018,Behroozi2019}, however, the full galaxy-halo connection could be high-dimensional and non-linear, which is difficult to capture by these models.
Machine learning (ML) provides a potentially powerful approach to study the galaxy-halo connection, inferring intricate relations from the complex multi-dimensional data in order to accurately connect the galaxies to the dark matter haloes. In recent years, ML techniques have become a versatile tool with a range of applications in large-scale structure and cosmology \citep{Aragon-Calvo2019,Berger2019,Lucie-Smith2020,deOliveira2020,Arjona2020,Ntampaka2020}. It is also helpful for processing observational data and performing classification \citep{DeLaCalleja2004,Sanchez2014,Tanaka2018,Cheng2020,Wu2020,Mucesh2021,Zhou2021}. In the context of halo modelling, ML can be implemented to predict galaxy properties based on input halo information \citep{Xu2013,Kamdar2016a,Kamdar2016b, Agarwal2018,Wadekar2020,Lovell2021,Moews2021}, and also applied in the reverse sense, predicting halo properties based on galaxy information \citep{Armitage2019,Calderon2019}. More specifically, \citet{Xu2013} make a first attempt to predict the number of galaxies given the halo's properties that can be utilized to create mock catalogues, matching the large scale correlation function to $5\%-10\%$. \citet{Agarwal2018} predict central galaxy properties based on halo properties and environment and find that the average relations of these properties with halo mass are accurately recovered. In \citet{Kamdar2016a,Kamdar2016b}, several galaxy properties such as gas mass, stellar mass, star formation rate, and colour are predicted based on subhalo information. Recently, \citet{Lovell2021} also present a study reproducing several galaxy properties based on subhalo properties in the EAGLE set of hydrodynamic simulations \citep{Schaye2015}.
In this paper, we aim to train a ML model to learn the relation between halo properties and the occupation numbers of galaxies from a galaxy formation simulation. This invariably includes the complex set of effects related to GAB (such as the preferential occupation of galaxies in early-formed haloes as one example). We utilize here Random Forest (RF) classification and regression, one of the most effective ML models for predictive analytics \citep{Breiman2001}. RF is an ensemble supervised learning method that works by combining decisions from a sequence of base models (decision trees). We use for this purpose stellar mass selected galaxy samples from the \citet{Guo2011} SAM applied to the Millennium Run Simulation \citep{Springel2005}. The input is the halo catalogue including an exhaustive set of halo properties and environment measures and the output will be the occupation numbers of central and satellite galaxies. The RF model will then be used to create mock galaxy catalogues and compared to the true levels of galaxy clustering and large-scale GAB.
We begin with a RF model that uses all internal and environmental halo properties as input and find an excellent agreement between the predicted HOD, galaxy clustering, and GAB and those measured in the SAM. The RF also provides feature importance which enables us to select the top properties for predicting occupations. Interestingly, the environment properties are found to be important for the satellites occupation but not for central one. We find that using only the top four input features can still recover the full level of GAB. We perform additional tests where we build RF models based on only mass and environment, and alternatively, using the internal halo properties alone.
This methodology can be applied to other galaxy formation models as well, and serve as the basis for an efficient way to populate galaxies in dark matter only simulations, capturing the pertinent information of the galaxy-halo relation and recovering the right level of galaxy clustering including the detailed effects of assembly bias. Additionally, evaluating the relative feature importance can provide valuable insight regarding the contributors to assembly bias and the importance of halo and environmental properties to galaxy formation and evolution.
Compared to other related ML works which predict the stellar mass of central galaxies (e.g., \citealt{Xu2013,Wadekar2020}; C.\ Cuesta, in prep.), our work utilizes the occupation numbers, more directly probing assembly bias, and allows to naturally incorporate both central and satellite galaxies.
In contrast to \citet{Xu2021} which evaluated the individual contributions to GAB and produced mock catalogues that recover the full level of GAB and OV with respect to specific environment measures, here we use the full ensemble of properties and are able to reproduce the OV with multiple properties simultaneously. This latter property allows for more realistic and complete mock catalogues, which may be important for certain cosmological applications.
The paper is organized as follows. In Section~\ref{simu}, we briefly describe the $N$-body simulation, the halo and environmental properties, and the SAM galaxy formation model. Section~\ref{ml_measure} provides an introduction to the RF algorithm and the performance measures used to evaluate our models.
In Section~\ref{predictions}, we present our results for the halo occupation, galaxy clustering, and GAB with different combinations of halo and environmental properties. We conclude in Section~\ref{summary}. Appendices~A and B present further results of our analysis.
\section{Dark matter halo and galaxy samples}
\label{simu}
\subsection{$N$-body simulation and halo properties}
\label{simu_halo}
We use in this work the dark matter halo sample from the Millennium $N$-body simulation \citep{Springel2005}. The simulation was run using the GADGET-2 code \citep{Springel2001a}, and adopts the first-year WMAP $\Lambda$CDM cosmology \citep{Spergel2003}, corresponding to the following cosmological parameters: $\Omega_{\rm m}=0.25$, $\Omega_{\rm b}=0.045$, $h=0.73$, $\sigma_8=0.9$, and $n_s$=1. The simulation is in a periodic box with a length of 500 $h^{-1}{\rm Mpc}$ on a side, with $2160^3$ total number of dark matter particles of mass $8.6\times10^8 \,h^{-1}\,{\rm M}_\odot$. The simulation outputs 64 snapshots spanning $z=127$ to $z=0$. At each redshift, the distinct haloes are identified by a friends-of-friends (FoF) group finding algorithm \citep{Davis1985}, and the subhaloes are identified by the \texttt{SUBFIND} algorithm \citep{Springel2001b}. Finally, a halo merger tree is constructed by linking each subhalo to its progenitor and descendant \citep{Springel2005}.
We utilize a set of internal halo properties as well as environmental measures, similar to those used in \citet{Xu2021}, as the input features for the RF models. These halo properties characterise halo structure and assembly history, and the environmental ones measure the density and tidal field at the position of the halo. We list and define all properties used in Table~\ref{table:haloprops}. The halo properties are separated into two categories. The first one are properties that can be obtained from the information from a single snapshot, here the one corresponding to $z=0$, such as $M_{\rm vir}$, $V_{\rm max}$, halo concentration $c$ defined as $V_{\rm max}/V_{\rm vir}$, and specific angular momentum $j$. The second category of halo properties pertains to the assembly history of the haloes and can be calculated from the merger tree. These include $V_{\rm peak}$, $a_{\rm 0.5}$, $a_{\rm 0.8}$, $a_{\rm vpeak}$, the mass accretion rate $\dot M$, $\dot M /M$, $z_{\rm first}$, $z_{\rm last}$, and $N_{\rm merge}$. The environmental properties we use are the mass densities on different smoothing scales, $\delta_{\rm 1.25}$, $\delta_{\rm 2.5}$, $\delta_{\rm 5}$, $\delta_{\rm 10}$, and the tidal anisotropy $\alpha_{1,5}$ \citep{Xu2021}.
\begin{table*}
\caption{Halo properties and environmental measures used as input features for the RF models. The top part correspond to properties obtained directly from the $z=0$ snapshot in the Millennium database. The middle part are properties computed using the merger tree of the simulation, and the bottom part corresponds to the environmental properties.}
\centering
\begin{tabular}{p{0.1\textwidth}p{0.75\textwidth}}
\hline
Properties & Definition\\
\hline\hline
$M_{\rm vir}$ & Halo mass enclosed by the virial radius, defined by 200 times the critical density \\ \hline
$V_{\rm max}$ & Maximum circular velocity of particles in the halo \\ \hline
$c$ & Halo concentration, defined as $V_{\rm max}/V_{\rm vir}$ \\ \hline
$j$ & Specific angular momentum, the angular momentum of the halo normalized by halo mass \\ \hline \hline
$V_{\rm peak}$ & Peak circular velocity, the peak value of maximum circular velocity in the history of the halo \\ \hline
$a_{\rm 0.5}$ & Scale factor when the halo first reaches 0.5 of its final mass, often referred as the halo formation time (age)\\ \hline
$a_{\rm 0.8}$ & Scale factor when the halo first reaches 0.8 of its final mass \\ \hline
$a_{\rm vpeak}$ & Scale factor corresponding to the peak circular velocity \\ \hline
$\dot M$ & Halo mass accretion rate \\ \hline
$\dot M /M$ & Specific mass accretion rate \\ \hline
$z_{\rm first}$ & Redshift of the first major merger, defined by a 1:3 mass ratio \\ \hline
$z_{\rm last}$ & Redshift of the last major merger \\ \hline
$N_{\rm merge}$ & Total number of the major mergers in the main branch of the merger tree \\ \hline \hline
$\delta_{\rm 1.25}$ & Matter density field at the halo position with a Gaussian smoothing scale of 1.25 $h^{-1}{\rm Mpc}$ \\ \hline
$\delta_{\rm 2.5}$ & Matter density field at the halo position with a Gaussian smoothing scale of 2.5 $h^{-1}{\rm Mpc}$ \\ \hline
$\delta_{\rm 5}$ & Matter density field at the halo position with a Gaussian smoothing scale of 5 $h^{-1}{\rm Mpc}$ \\ \hline
$\delta_{\rm 10}$ & Matter density field at the halo position with a Gaussian smoothing scale of 10 $h^{-1}{\rm Mpc}$ \\ \hline
$\alpha_{1,5}$ & Tidal anisotropy parameter, defined as $\sqrt{q^2_R}/(1+\delta_{\rm 5})$ where $q^2_R$ is the tidal torque \citep{Paranjape2018}, measured with a $5 \,h^{-1}\,{\rm Mpc}$ smoothing scale
\\ [0.5ex]
\hline
\end{tabular}
\label{table:haloprops}
\end{table*}
\subsection{Galaxy formation model}
\label{sam}
We use the galaxy sample corresponding to the \citet{Guo2011} galaxy formation SAM implemented on the Millennium simulation. It models the main physical processes involved in galaxy formation in a cosmological context. These processes include reionization, gas cooling, star formation, angular momentum evolution, black hole growth, galaxy merger and disruption, and AGN and supernova feedback. The \citep{Guo2011} is a version of L-galaxies, the SAM code of the Munich group\citep{DeLucia2004,Croton2006,Guo2013,Henriques2015,Henriques2020}, and uses the subhalo merger tree of the simulation to trace and evolve the galaxies through cosmic time. The prescription parameters in the model are tuned to luminosity, colour, abundance, and clustering of observed galaxies. The \citet{Guo2011} SAM model is widely used in literature (e.g., \citealt{Wang2013,Lu2015,Lin2016,Zehavi2018,Xu2021}), and it is publicly available at the
Millennium database \footnote{\url{http://gavo.mpa-garching.mpg.de/Millennium/}}.
When constructing our galaxy samples, we first apply a halo mass cut of $10^{10.7} \,h^{-1}\,{\rm M}_\odot$, below which the number of dark matter particles is too low to reliably host galaxies. We define stellar mass selected samples with different number densities. For our main analysis we focus on a sample with a stellar-mass threshold of $1.42\times10^{10} \,h^{-1}\,{\rm M}_\odot$, corresponding to a number density of $n=0.01 \,h^{3}\,{\rm Mpc}^{-3}$. This sample includes a total of $745,027$ central galaxies and $505,784$ satellite galaxies. For some of our analysis, we use two additional samples with stellar-mass thresholds of $3.88\times10^{10} \,h^{-1}\,{\rm M}_\odot$ and $0.185\times10^{10} \,h^{-1}\,{\rm M}_\odot$, corresponding to $n=0.00316 \,h^{3}\,{\rm Mpc}^{-3}$ and $n=0.0316 \,h^{3}\,{\rm Mpc}^{-3}$, respectively. These three samples are approximately evenly spaced in logarithmic number density and follow the choices made in \citet{Zehavi2018} and \citet{Xu2021}. While the results presented in this paper are limited to the \citet{Guo2011} SAM at z=0, the developed methodology can be applied to any SAM sample and redshift.
\section{Machine learning methodology}
\label{ml_measure}
\subsection {Random forest classification and regression}
\label{MLmethod}
We first briefly discuss the choice of the machine learning model. Linear regression and classification models are the simplest ML models to learn the relation between the input features and the output. However, linear models are limited since even the simplest non-linear transformation (e.g., a polynomial) can lead to a large increase in the number of features, thereby slowing down the learning process. Support vector machines (SVM) are powerful ML algorithms which can transform the input features into higher dimensions without explicitly transforming the features \citep{Aizerman1964,Boser1992}. However, they suffer from increased training time complexity with the size of training data. In contrast, ensemble methods such as Random Forest \citep{Breiman2001} are suitable for our purpose of learning the relation between halo properties and halo occupation because of their ability of dealing with large and high-dimensional datasets.
The Random Forest algorithm combines the output of multiple randomly created Decision Trees to generate the final output. It uses bootstrap aggregation to create random subsets of the training data with replacement on which the decision trees are trained. The decision tree is a flow-like structure in which each internal node represents a ``test'' of an attribute, each branch represents the outcome, and each terminal node or leaf represents the output (the decision taken after computing all attributes). Combining a large number of decision trees, the prediction of RF is the class that is predicted by the majority of the decision trees in the case of RF classification. For RF regression, the prediction is the average prediction from all decision trees. Thus, for our purpose here, training the RF on a subset of the Millennium halo catalogues and the corresponding SAM galaxy occupations, allows to take into account all the halo properties and predict whether a given halo has a central galaxy or not (classification) and the expected number of satellite galaxies (regression).
The main advantage of decision trees is that they perform well with non-linear problems and are computationally cheap since the decision trees can be trained in parallel. One of the major concerns about decision trees is that they can be unstable due to the hierarchical nature of trees: a small change in the training set can result in a difference in the root split which is propagated down to subsequent splits. However, this is mitigated in RF by averaging the predictions over many uncorrelated trees. Decision trees also tend to be strong learners, meaning that individual trees tend to overfit the data. Overfitting is addressed by aggregating the results over many high-variance and low-bias trees. Another important feature of the RF algorithm is that it provides the relative feature importance, i.e the contribution of each input property in making the predictions which we will examine in Section~\ref{predictions}. For a more rigorous discussion of the RF algorithm, we refer the reader to Chapters 9 and 15 of \citet{hastie01} and Chapters 6 and 7 of \citet{Geronml}.
\subsection{Performance measures}
\label{scores}
The RF model includes several `hyper-parameters' which characterize the ensemble of decision trees. In this work we focus on three of them, the total number of the trees in RF, the maximum depth of each tree, and the minimum number of samples in the leaf node of the tree. As common in machine learning analyses, we optimize the performance of the RF algorithm by doing a grid search over these parameters and finding the best fit values. The grid search is performed over 80\% of the full halo catalogue in the simulation, using the so-called 4-fold cross-validation technique (see, e.g., Chapter 5 of \citealt{CVref}). For each choice of hyper parameters, this data is split into four subsets; three are used for training and the remaining one is used for validation and obtaining the ``performance scores''. This is repeated four times so that each of the four subsets is used for validation, and the performance scores are averaged. This process is repeated for each choice on the hyper parameters grid, resulting in the grid point with the highest score.
\begin{figure}
\centering
\includegraphics[width=0.7\hsize, height =0.7\hsize]{confusion_all-revised.pdf}
\caption{Confusion matrix for central galaxy predictions for the $n=0.01 \,h^{3}\,{\rm Mpc}^{-3}$ galaxy sample, with all the halo internal and environmental properties used as input. The predictions are obtained from the full sample, with the rows corresponding to the ML predicted values and the columns showing the values in the SAM (see text).
}
\label{confusion_mat}
\end{figure}
For classification, a useful way to evaluate its performance is to look at the confusion matrix. To illustrate this we show in Figure~\ref{confusion_mat} the confusion matrix trained using the $n=0.01 \,h^{3}\,{\rm Mpc}^{-3}$ galaxy sample, using all halo and environmental features. Each row represents the RF predicted class (0 or 1), whereas each column represents the true class in the SAM (0 or 1). In our case, 1 refers to haloes containing a central galaxy and 0 otherwise. Haloes containing central galaxies and predicted as such are referred to as true positives (TP) whereas those predicted as 0 are referred to as false negatives (FN). Haloes without a central galaxy and predicted as such are referred to as true negatives (TN) while those predicted as 1 are false positives (FP). A perfect classifier would have only TN and TP and zero off-diagonal values. The confusion matrix shows the fraction of haloes in each category. We see that, in our case, the fractions of TP and FN are 0.91 and 0.09, respectively, where the predictions are normalized by the total number of haloes containing a central galaxy. The fractions of TN and FN are 0.98 and 0.02, respectively, normalized in this case by the total number of haloes not containing a central galaxy.
A more concise metric utilizing the confusion matrix is the $F_1$ score defined as:
\begin{equation}
F_1 = 2PR/[P + R],
\end{equation}
where $P$ and $R$ are the Precision and Recall. Precision measures the accuracy rate,
\begin{equation}
P = \text{TP}/[\text{TP} + \text{FP}],
\end{equation}
while the recall, also known as sensitivity or true positive rate, is
\begin{equation}
R = \text{TP}/[\text{TP} + \text{FN}].
\end{equation}
Since precision and recall measure different aspects of the success of the predictions, they are usually combined to evaluate a classifier. We use the $F_1$ score, conveying the balance of precision and recall, to optimize the choice of hyper parameters for the RF classification of central galaxies.
For regression, we use the $R^2$ score or the coefficient of determination defined as:
\begin{equation}
R^2 = 1 - S_{\text{res}}/S_{\text{tot}},
\end{equation}
where $S_{\text{res}}$ is the residual sum of squares,
\begin{equation}
S_{\text{res}} = \sum_i (p_i - y_i)^2 ,
\end{equation}
where $p_i$ is the prediction for each input data and $y_i$ the true value. This sum is normalized by the underlying total sum of squares relative to the mean $\bar{y}$:
\begin{equation}
S_{\text{tot}} = \sum_i (y_i - \bar{y})^2.
\end{equation}
Even though we explored other performance measures, we chose the $R^2$ score to set the hyper parameters for the RF regression predictions of the number of satellite galaxies for the cases we explore.
We utilize the Python package \texttt{sklearn} for performing all grid searches and RF training. We use 80\% of the full halo catalogue in the Millennium simulation as the training set. For each application, we first set the RF hyper parameters to those that give the highest scores in the grid search. We then proceed to train the RF classification and regression models to predict the number of central and satellite galaxies in each halo. In practice, when estimating the clustering and GAB, we average the predictions of 10 training sets (each containing 80\% of the total haloes) drawn randomly out of 90\% of the full catalogue. This allows to reduce the sensitivity to the specifics of the training set (though the sets clearly still have a large overlap). The remainder 10\% of the haloes are left as an independent test set, not used for either the training or cross-validation.
\section{Machine learning results}
\label{predictions}
In this section, we present the results of our RF models. For the main analysis described here, we use the stellar-mass selected $n=0.01 \,h^{3}\,{\rm Mpc}^{-3}$ sample as mentioned in Section~\ref{sam}. The direct predictions output of the ML model are the numbers of central and satellite galaxies in each halo. We comprehensively compare them with the 'true' distribution of the SAM galaxy sample in multiple ways. We first directly compare the galaxy numbers on a halo-by-halo basis. We then compare the halo occupation functions, namely the average number of galaxies as a function of halo mass, as well as the variations in these halo occupation functions with secondary properties (referred here as the OV; e.g., \citealt{Zehavi2018}). We then proceed to populate the halo sample with the predicted number of galaxies to create a mock galaxy catalogue based on the ML predictions. We calculate the clustering of the ML galaxy sample and compare to that of the SAM sample. Finally, we examine and compare the impact of GAB on the large-scale clustering signal. We describe all these in detail below.
We show the results using the full halo catalogue of the Millennium simulation, which includes the training sets, used to build the ML model, and the smaller (10\% of the haloes) test sample. We have repeated our main analysis using only the test sample, finding similar results to the ones shown here.
\subsection{All features}
\label{allfeatures}
Here we present the ML results when using all available features, namely all the internal halo properties and environmental measures specified in Table~\ref{table:haloprops}.
The accuracy of the ML predictions for hosting a central galaxy with stellar mass larger than our sample's threshold in the individual haloes has already been presented in Figure~\ref{confusion_mat}. Again, we find that for haloes which host a central galaxy above the stellar-mass threshold in the SAM, 91\% of them are predicted to host a central galaxy by our ML model. For haloes that do not host a central galaxy, 98\% of them are accurately predicted as such in our model. The difference in the relative values likely simply reflects the larger number of haloes with no central galaxy for this stellar-mass threshold, such that the number of misclassified haloes is roughly comparable. Note that we do not expect the ML algorithm to provide an accurate prediction for every single halo, due to the stochasticity involved, for example in the scatter between stellar mass and halo mass (and such a case would indicate extreme overfitting in the least). We view this agreement as very good.
\begin{figure}
\centering
\begin{subfigure}[h]{0.48\textwidth}
\includegraphics[width=\textwidth]{scatterplot-hoderror.pdf}
\end{subfigure}
\hfill
\caption{Comparison between the RF predicted number of satellite in each halo and the actual number from the SAM. The blue dots show these values for each individual halo, for the ML model applied to the $n=0.01 \,h^{3}\,{\rm Mpc}^{-3}$ galaxy sample, using all halo features. The diagonal grey line indicates the idealized case where the number is identical, and the shaded region represents the Poisson error often assumed in HOD models.
}
\label{scatter}
\end{figure}
The `raw' predicted numbers of satellite galaxies from the RF regression model are not required to have an integer value a-priori. We assign it to the nearest integers following a Bernoulli distribution with this mean. In practice, this amounts to assigning, e.g., 4.3 satellites to 3 with a 70\% probability or to 4 with 30\% probability. The relation between these discrete (integer) predictions for the number of satellites and the SAM number of satellites in each halo is presented in Figure~\ref{scatter}. Each point represents the satellite occupation in a single halo, showing the scatter of the RF predictions along the y-axis. The grey shaded area shows, for comparison, a simple Poisson scatter as is often assumed in HOD modelling (the shaded area appears to increase at low numbers, just due to the log scale plotted). The scatter in the ML prediction is larger than the Poisson scatter, due to the more complex model and limitations of the RF regression. This also suggests that we are not overfitting the data here. Though not shown here, for clarity, we also perform a linear fit of the points to examine any bias in the predictions. For a fully unbiased prediction, the slope of the linear fit would be one. However, we find a slope of 0.96 which indicates a slight underprediction. This is likely caused by the lower ML prediction relative to the SAM at the largest occupation numbers (high halo mass). This underprediction is also found in \citet{Xu2013} and is considered a result of the small number of the most massive haloes in the simulation. Since the level of the underprediction is low, it should not impact the results in this paper.
\begin{figure}
\centering
\begin{subfigure}[h]{0.48\textwidth}
\includegraphics[width=\textwidth]{hod-all.pdf}
\end{subfigure}
\hfill
\begin{subfigure}[h]{0.48\textwidth}
\includegraphics[width=\textwidth]{logxi-csall.pdf}
\end{subfigure}
\hfill
\caption{\textbf{Top}: The halo occupation function for the SAM $n=0.01 \,h^{3}\,{\rm Mpc}^{-3}$ sample (black) and ML prediction (blue) using all the halo and environmental properties. The individual contributions from central and satellite galaxies are shown as dotted and dashed lines, respectively. \textbf{Bottom}: The galaxy two-point auto-correlation function of the ML prediction (blue) compared to the SAM (black). The small difference on small scales is due to the galaxy profile in the SAM slightly deviating from the NFW profile assumed for the ML prediction.
}
\label{hodall}
\end{figure}
Moving away from the comparisons on an individual halo basis, we now shift to comparing the central and satellite galaxy numbers averaged in mass bins, namely the halo occupation functions commonly used in the HOD framework. The top panel of Figure~\ref{hodall} compares the halo occupation function corresponding to the ML predictions (blue) with that of the SAM (black) for the $n=0.01 \,h^{3}\,{\rm Mpc}^{-3}$ galaxy sample. We find that the predictions are in excellent agreement with the halo occupation of the SAM galaxies, as can be seen from the indistinguishable lines.
With the predicted number of central and satellite galaxies in each halo, we populate the haloes and create a mock galaxy catalogue to measure the clustering. For each halo, we place the central galaxy at the halo center and populate satellites with an NFW profile, going out to twice the virial radius. The bottom panel of Figure~\ref{hodall} shows the resulting two-point auto-correlation function relative to that measured from the SAM. Again, we find excellent agreement between the ML predictions and the SAM. On small scales, the prediction deviates from the SAM since an NFW profile is adopted in the mock catalogue, which is slightly different from the radial distribution of the SAM satellites (e.g., \citealt{Jimenez2019}). Since we are focused here on modelling GAB, we will only show our predicted clustering results on large scales (larger than $\sim 7 h^{-1}{\rm Mpc}$) from here on.
\begin{figure*}
\centering
\begin{subfigure}[h]{0.8\textwidth}
\includegraphics[width=\textwidth]{OV-all1.pdf}
\end{subfigure}
\hfill
\caption{The occupancy variations in the predicted halo occupation functions, when using all halo and environmental properties as input features. Each panel corresponds to a different secondary property, $c$, $a_{\rm 0.5}$, $\delta_{\rm 1.25}$, and $\alpha_{\rm 0.3,1.25}$, as labelled. In all panels, red and blue and lines represent the SAM occupations in the 10\% of haloes with the highest and lowest values, respectively, of the secondary properties in fixed mass bins. Pink and cyan lines show the corresponding cases for the ML predictions. The numbers of centrals, satellites, and all galaxies are shown by dotted, dashed, and solid lines, respectively.
}
\label{ov_all}
\end{figure*}
In addition to halo occupation as function of mass, we also examine in detail the variations of the halo occupations with secondary properties. Since halo clustering also depends on such properties (halo assembly bias), together with the OV, galaxy clustering would also be impacted. An HOD model that captures the OV dependence on a specific halo property would thus also capture the GAB caused by this halo property \citep{Xu2021}. These OVs are shown in Figure~\ref{ov_all} for some representative cases of the internal halo properties (concentration, $c$, and halo formation time, $a_{\rm 0.5}$, shown in the top panels) and the environmental measures ($\delta_{\rm 1.25}$ and $\alpha_{\rm 0.3, 1.25}$, shown on the bottom). Similar to $\alpha_{\rm 1,5}$, $\alpha_{\rm 0.3, 1.25}$ is defined as a measurement of tidal anisotropy on the smoothing scale of $1.25 \,h^{-1}\,{\rm Mpc}$:
\begin{equation}
\label{alpha03g125}
\alpha_{\rm 0.3,1.25}=\sqrt{q^2_R}/(1+\delta_{\rm 1.25})^{0.3},
\end{equation}
where $q^2_R$ is the tidal torque measured with the same smoothing scale and the normalization is modified by a 0.3 power \citep{Xu2021}. The red and blue curves in each panel show the occupations for the 10\% of the halo population in each mass bin with the highest and lowest values of the secondary property in the SAM sample, whereas cyan and pink show those predicted by the ML models. Dotted, dashed, and solid curves indicate the central, satellite, and total occupation number. We note that we use $a_{\rm 0.5}$, the scale factor when the halo accretes half of its halo mass, as a proxy for halo age. Highest $a_{\rm 0.5}$ values thus correspond to later formation times and the youngest ages, and vice versa, the earliest formation times correspond to the oldest age (and are colour coded accordingly).
The OVs shown in Figure~\ref{ov_all} generally follow the trends already examined in detail in previous works \citep{Zehavi2018,Contreras2019,Xu2021}. E.g., older haloes (higher formation time, smaller $a_{\rm 0.5}$ values) tend to start occupying central galaxies at lower halo masses. In contrast, such haloes, host on average less satellites than later-forming haloes. The striking result in this work is the excellent agreement between the ML predictions and the SAM ones, for all secondary properties. That implies that the RF algorithm is able to accurately learn and reproduce the different secondary trends.
Note that while $\alpha_{\rm 1,5}$ is one of the input features, $\alpha_{\rm 0.3, 1.25}$ is not, and while they may be correlated to some extent, they play different roles in GAB. \citet{Xu2021} show that $\alpha_{\rm 1,5}$ accounts for a small fraction of GAB, whereas $\alpha_{\rm 0.3, 1.25}$ captures the full effect on galaxy clustering. The tidal anisotropy parameter $\alpha_{\rm 0.3, 1.25}$ is also partially correlated with $\delta_{\rm 1.25}$, but include additional information on the tidal shear. So it is interesting to see that the OV dependence on $\alpha_{\rm 0.3, 1.25}$ can be well reproduced by the ML algorithm, without serving as input for it. More generally, since GAB is a result of halo assembly bias combined with the OV, and the individual OVs are accurately reproduced, we expect that the GAB signal can be well recovered as well.
\begin{figure*}
\centering
\begin{subfigure}[h]{0.48\textwidth}
\includegraphics[width=\textwidth]{full-cenxi-all-D80-rand10.pdf}
\end{subfigure}
\hfill
\begin{subfigure}[h]{0.48\textwidth}
\includegraphics[width=\textwidth]{plot3-csall-D80-rand10.pdf}
\end{subfigure}
\hfill
\caption{Comparison of the measured correlation functions and GAB of the SAM and the ML predicted mock catalogue, when using all features. The left-hand side shows these clustering results for central galaxies only, while the right-hand side shows the same for all (central and satellite) galaxies corresponding to the $n=0.01 \,h^{3}\,{\rm Mpc}^{-3}$ sample. For both these cases, the top panels show the ratios of the correlation function of the ML-predicted mock catalogues relative to that of the SAM. Ratios of the original (unshuffled) correlation functions are shown in black, while the ratios of the shuffled samples of each are shown in red. The bottom panels (on both sides), show the measured GAB signal, namely the ratio of the original correlation function to that of the shuffled sample. Here, the SAM GAB measurement is shown in black while the ML GAB is shown in blue. The shaded areas, in all panels, indicate the error bar measured from 10 different shuffled samples of the SAM galaxies and the 10 different realizations of the RF model.
}
\label{cluster_gab_all}
\end{figure*}
The GAB signature is usually measured as the ratio between the correlation function of the galaxy sample and that of a shuffled sample, created by randomly reassigning the galaxies among haloes of the same mass \citep{Croton2007}. The shuffling process effectively removes the connection of the galaxies to the assembly history of the haloes and eliminates the dependence on any secondary property other than halo mass (i.e it erases all OVs). Comparison between the clustering of the shuffled sample and the original thus reveals the overall effect of GAB, typically seen as an increased clustering amplitude on large scales. Following standard practice \citep{Croton2007,Zehavi2018,Contreras2019,Xu2021}, we shuffle the central galaxies and then move the satellites together with their associated central galaxy. This results in the shuffled sample having the same clustering as the original sample on small (one-halo) scales.
These results are examined in detail in Figure~\ref{cluster_gab_all}, showing the different large-scale clustering measurements separately for the central galaxies only on the left-hand side and for the full (central and satellite galaxies) sample on the right. We already saw in Figure~\ref{hodall} that the overall clustering of the ML mock sample is highly consistent with that of the SAM on large scales. This is presented more clearly in the top panels of Figure~\ref{cluster_gab_all}, where the black line shows the ratio of the ML predicted clustering to that of the SAM. The shaded regions hereafter indicate the uncertainty associated with the 10 different training sets (see \S~\ref{scores}). In both cases, we see that the SAM clustering is accurately reproduced.
Our results are a vast improvement compared to \citet{Xu2013} who recover the amplitude of galaxy clustering to 5\%-10\% using the halo occupations as well. We reproduce the clustering to sub-percent precision, perhaps due to both using a larger training sample and including also environmental properties. The latter is in line with recent studies that demonstrate the important role of environment in accurately capturing the level of galaxy clustering \citep{Hadzhiyska2020,Xu2021}.
We then proceed to examine the results of the shuffled samples. We shuffle each of the SAM sample and the ML mock sample in bins of fixed halo mass, as described above. The ratios of the shuffled ML predicted clustering to that of the shuffled SAM clustering are presented as the red lines in the top panels of Figure~\ref{cluster_gab_all}. Once again, these ratios are extremely close to unity, indicating an excellent agreement between the shuffled ML clustering and the shuffled SAM clustering.
We examine directly the GAB signature in the bottom panels of Figure~\ref{cluster_gab_all}. Namely, we present ratios of the large-scales correlation function of the original sample to that of the shuffled sample, $\xi/\xi_{\rm shuffled}$. Black lines represent this ratio, i.e the GAB signal, in the SAM while the blue lines represent the ML-predicted GAB signal. The error bar on the SAM measurement is the scatter from 10 different shuffled samples, while the error bar on the ML predictions arises from the 10 different training sets (each with its own shuffled sample). Again, this is shown for the central galaxies only on the left-hand side and for the full samples, including satellites, on the right.
These ratios have already been studied with this specific SAM sample \citep{Zehavi2018,Zehavi2019,Xu2021}. The roughly 15\% increase of clustering in the original SAM sample versus the shuffled one arises from the differentiated occupation of haloes with galaxies according to secondary halo properties which exhibit halo assembly bias. For example, galaxies tend to preferentially occupy older haloes which exhibit stronger clustering, resulting in an increased large-scale galaxy clustering (GAB). We note, again, that the excess clustering shown here is the overall combined effect from all secondary properties.
The remarkable result clearly shown in the bottom panels of Figure~\ref{cluster_gab_all} is the excellent agreement between the GAB signal measured by the ML-predicted sample and that of the original SAM galaxy sample. This is exhibited by the nearly perfect agreement between the blue and black lines in each panel, for central galaxies only (left) and for the full sample (right). The RF model applied trained on the individual halo occupations is thus able to accurately reproduce the GAB effect in the large-scale galaxy clustering. Together with the recovered OVs, we see that the ML model is highly successful in reproducing all aspects of the complex phenomena of assembly bias.
A simple measure of the agreement between the GAB signals, beyond the striking agreement by eye, is provided by
\begin{equation}
\label{eq:fab}
f_{\rm AB}=\langle (\xi_{\rm ML}/\xi_{\rm shuffled,ML} - 1) / (\xi_{\rm SAM}/\xi_{\rm shuffled,SAM} - 1) \rangle,
\end{equation}
which represents the recovered fraction of GAB. The averaging is done over the clustering ratio values measured on large scales of $9 \sim 30 h^{-1}{\rm Mpc}$. For the cases shown in the bottom panels of Figure~\ref{cluster_gab_all}, namely the $n=0.01 \,h^{3}\,{\rm Mpc}^{-3}$ sample using all the available features in the ML model, we obtain nearly perfect recovery with $f_{\rm AB}=0.99$ for the central galaxies only case and $f_{\rm AB}=0.98$ for the full sample (i.e they recover the full GAB signal to 1-2\%). The recovered level of the correlation function can be similarly estimated as $\langle \xi_{\rm ML}/\xi_{\rm SAM} \rangle$, returning a value of 1.00 for both these cases (to the level of accuracy quoted).
These values are summarized in Table~\ref{table:clustering}, for all the cases explored in this paper, and include also the values of the $F_1$ and $R^2$ performance scores of the RF predictions (\S~\ref{scores}).
The results of the RF models with all features are listed in the top two lines of Table~\ref{table:clustering}. The following lines in the table are the results of other RF models with different sets of input features as labelled, for which we provide more details and discussion in the following subsections. Table~\ref{table:clustering} also includes the values obtained using all features for two additional stellar-mass selected galaxy samples corresponding to $n=0.00316 \,h^{3}\,{\rm Mpc}^{-3}$ and $n=0.0316 \,h^{3}\,{\rm Mpc}^{-3}$. The clustering and GAB results for these two samples are presented in Appendix~\ref{num_den}.
\begin{table*}
\caption{Prediction results for RF models with different input features. The first two columns indicate the input features for the central and satellite galaxies. The centrals-only cases are indicated by a ``$-$'' in the second (satellite) column. The performance scores $F_{1}$ and $R^2$ for the centrals and satellites are listed in the third and fourth columns, respectively. The next column shows the recovered fraction of the correlation function, $\langle \xi_{\rm ML}/\xi_{\rm SAM} \rangle$, averaged over scales of $9 \sim 30 h^{-1}{\rm Mpc}$. We do not include a separate column for this property measured for the shuffled samples, since its accuracy is 1.00 (within the significance quoted) for {\it all} cases shown. The final column represents the accuracy of recovering the GAB signal using the $f_{\rm AB}$ measure. The main predictions are all based on the galaxy sample of number density $n=0.01 \,h^{3}\,{\rm Mpc}^{-3}$ and are listed in top 10 lines. The predictions with all features for two other number densities $n=0.00316 \,h^{3}\,{\rm Mpc}^{-3}$ and $n=0.0316 \,h^{3}\,{\rm Mpc}^{-3}$ are listed at the bottom. }
\centering
\setlength{\tabcolsep}{6pt}
\begin{tabular}{c c c c c c}
\hline
input (cen) & input (sat) & $F_1$ score & $R^2$ score & recovered $\xi$ & recovered $f_{\rm AB}$\\
\hline\hline
all & -- & 0.89 & -- & 1.00 & 0.99 \\ \hline
all & all & 0.89 & 0.94 & 1.00 & 0.98 \\ \hline
top 4 & -- & 0.88 & -- & 1.00 & 0.97 \\ \hline
top 4 & top 4 & 0.88 & 0.93 & 1.00 & 1.00 \\ \hline
$M_{\rm vir}$+$\delta_{\rm 1.25}$ & -- & 0.79 & -- & 0.99 & 0.86 \\ \hline
$M_{\rm vir}$+$\delta_{\rm 1.25}$ & $M_{\rm vir}$+$\delta_{\rm 1.25}$ & 0.79 & 0.91 & 0.99 & 0.92 \\ \hline
internal & -- & 0.89 & -- & 1.00 & 0.99 \\ \hline
internal & internal & 0.89 & 0.91 & 0.97 & 0.70 \\ \hline
single-epoch & -- & 0.85 & -- & 1.00 & 1.00 \\ \hline
single-epoch & single-epoch & 0.85 & 0.91 & 0.99 & 0.95 \\
\hline \hline
all (n=0.00316) & -- & 0.74 & -- & 0.98 & 0.83 \\ \hline
all (n=0.00316) & all (n=0.00316) & 0.74 & 0.87 & 0.99 & 0.96 \\ \hline
all (n=0.0316) & -- & 0.96 & -- & 1.00 & 1.00 \\ \hline
all (n=0.0316) & all (n=0.0316) & 0.96 & 0.95 & 1.00 & 0.99 \\ [0.5ex]
\hline
\end{tabular}
\label{table:clustering}
\end{table*}
\subsection{Feature importance }
\label{feature_imp}
\begin{figure*}
\centering
\begin{subfigure}[h]{0.46\textwidth}
\includegraphics[width=\textwidth]{featureimp_cen.pdf}
\end{subfigure}
\hfill
\begin{subfigure}[h]{0.52\textwidth}
\includegraphics[width=\textwidth]{corrcoef-cen.pdf}
\end{subfigure}
\hfill
\caption{
\textbf{Left:} Relative feature importance for the top 10 features of the RF predictions for central galaxies. \textbf{Right:} The correlation matrix of these top 10 features. The numbers shown are Pearson correlation coefficients between each pair of features.
}
\label{feature_imp_cent}
\end{figure*}
\begin{figure*}
\centering
\begin{subfigure}[h]{0.46\textwidth}
\includegraphics[width=\textwidth]{featureimp_sat.pdf}
\end{subfigure}
\hfill
\begin{subfigure}[h]{0.52\textwidth}
\includegraphics[width=\textwidth]{corrcoef-sat.pdf}
\end{subfigure}
\hfill
\caption{
Relative feature importance and correlation matrix for the top 10 features of the RF predictions for satellite galaxies.
}
\label{feature_imp_sat}
\end{figure*}
The above results show that the RF models are capable of accurately reproducing galaxy clustering and GAB. However, the number of input features is large which increases the complexity and running time of RF models. In this section, we aim to build simpler RF models with fewer input features that can achieve the same purpose. In addition to the prediction of galaxy numbers per halo, the RF algorithm also provides an estimate of the relative importance of the input features (i.e., all the secondary halo and environmental properties). It is evaluated based on the contribution of the input features to the construction of the RF decision trees. We show the top 10 properties ranked by feature importance in the left-hand side panels of Figure~\ref{feature_imp_cent} and Figure~\ref{feature_imp_sat}, for the central galaxies and satellites predictions, respectively.
For the central galaxies, we find that $V_{\rm max}$, the haloes' maximum circular velocity, is the most important feature followed by $z_{\rm last}$, $V_{\rm peak}$, and $a_{\rm 0.5}$. $V_{\rm max}$ can be considered as a halo mass indicator (e.g., \citealt{Zehavi2019}), and the other properties characterise the formation history of a halo. $V_{\rm peak}$, the peak value of $V_{\rm max}$ over the history of the halo, is a special case among them since it highly correlates with $V_{\rm max}$ (with a 0.99 Pearson correlation coefficient, as noted in the right panel of Figure~\ref{feature_imp_cent}). We perform a simple test that runs the RF prediction inputting the same feature twice (for example the halo mass), to mimic the situation of two highly correlated features). We find that it tends to maintain the importance of one feature and lower the importance of the other one. So it is likely that the roles of $V_{\rm max}$ and $V_{\rm peak}$ are comparable for the central galaxies prediction. Given the extreme correlation between the two, once $V_{\rm max}$ is utilized, $V_{\rm peak}$ does not really add any new information and thus it is not necessary to keep them both.
The importance of $V_{\rm max}$ is consistent with the finding by \citet{Zehavi2019} that $V_{\rm max}$ or $V_{\rm peak}$ better correlates with the central galaxies occupation than $M_{\rm vir}$ in the SAM sample, such that using the former reduces significantly the central galaxies OV with other secondary properties and the related trends in the stellar mass - halo mass relation. \citet{Xu2020} reach a similar conclusion with the Illustris simulation, namely that the stellar mass of central galaxies in fixed $V_{\rm peak}$ bins exhibits a weaker dependence on halo age or concentration than that in $M_{\rm vir}$ bins. This is not surprising since $V_{\rm max}$ or $V_{\rm peak}$ contains more internal structure information than $M_{\rm vir}$ alone, and in particular is also related to the concentration. Recently, \citet{Lovell2021} provide a ML approach to predict several galaxy properties from subhalo properties based on hydrodynamic simulations, also finding that $V_{\rm max}$ is the most important property for the prediction.
The next two properties in order of feature importance are $z_{\rm last}$ and $a_{\rm 0.5}$. Both are specific epochs in the formation history of the host halo. The halo formation time, $a_{\rm 0.5}$, is defined as the scale factor at the time when the host halo first reached half of its present mass, so is indicative of the halo age and is widely explored in assembly bias studies. At fixed halo mass, early-formed haloes (smaller $a_{\rm 0.5}$) tend to host more massive central galaxies than late-formed haloes (larger $a_{\rm 0.5}$), and thus are more likely to host central galaxies above a given stellar-mass threshold \citep{Zehavi2018}. The other parameter, $z_{\rm last}$, is the redshift of the last major merger of the host halo. It is another important epoch in the mass assembly history that could relate to the formation of the central galaxy. So it is reasonable that it is important for the central galaxies ML prediction.
Interestingly, we find that no environmental properties appear in the top 10 features for central galaxies. This may be supported by the fact that the OV with environment is much smaller than with internal halo properties like age \citep{Zehavi2018}, as also demonstrated in Figure~\ref{ov_all}. However, recent studies have shown that environment is the most informative property for describing GAB \citep{Hadzhiyska2020,Xu2021}. We will provide tests in the following subsections investigating the importance and necessity of the environment for reproducing the central galaxies and full GAB.
The left panel of Figure~\ref{feature_imp_sat} shows the feature importance for the satellites prediction. Halo mass, $M_{\rm vir}$, is the most important feature followed by the environment features $\delta_{\rm 2.5}$, $\delta_{\rm 1.25}$, and $\delta_{\rm 5}$. As expected, these three environmental measures are strongly correlated with each other, as can be seen in the right panel of Figure~\ref{feature_imp_sat}. In contrast to the central galaxies prediction, we note that here the environment is more important than secondary internal halo properties for predicting the number of satellites. Halo concentration is next and $V_{\rm max}$ and $V_{\rm peak}$ follow but with lower importance, which is again consistent with \citet{Zehavi2019} who showed that using $V_{\rm max}$ (or $V_{\rm peak}$) is detrimental to encapsulating the satellites OV relative to using $M_{\rm vir}$. These differences of feature importance between the central galaxies and satellites occupations highlight again the complexities of assembly bias. They imply that the formation and evolution of central and satellite galaxies may follow different paths and are impacted by different internal or environmental halo properties, and it is reasonable to model them separately with machine learning.
While the RF model estimates the input features importance, we should keep in mind that the features are correlated with each other. We take this into consideration when attempting to select fewer features for a less complex model. To illustrate that, in the right panel of Figure~\ref{feature_imp_cent} and Figure~\ref{feature_imp_sat}, we plot the correlation matrix which shows the Pearson correlation coefficients between each pair of the top 10 features included in the left panels. A correlation coefficient of 1 (shown by dark blue) indicates a positive maximal one-to-one correlation between the two properties, and a correlation coefficient of -1 (shown by dark orange) indicates a maximal anti-correlation. A correlation coefficient close to 0 indicates no correlation, with the two properties largely independent of each other. Values between 0 and 1 (-1) represent then a positive (negative) correlation with scatter, and the scatter is smaller for larger absolute values indicating a tighter correlation. In selecting a subset of top features, it is more effective to select a few such features that are important and yet less correlated with each other, in order to represent most of the information. For central galaxies, since $V_{\rm max}$ and $V_{\rm peak}$ are tightly correlated, we select $V_{\rm max}$, $z_{\rm last}$, $a_{\rm 0.5}$, and $M_{\rm vir}$ as the top features. For the satellite galaxies, we select $M_{\rm vir}$, $\delta_{\rm 2.5}$, $\delta_{\rm 1.25}$, and concentration $c$ as the top features. We show in the next section the RF prediction results with the selected top four features.
\subsection{Top Features}
\label{topfeatures}
\begin{figure*}
\centering
\begin{subfigure}[h]{0.8\textwidth}
\includegraphics[width=\textwidth]{OV-ctop-stop1.pdf}
\end{subfigure}
\hfill
\caption{Similar to Figure~\ref{ov_all}, the predicted OV with $c$, $a_{\rm 0.5}$, $\delta_{\rm 1.25}$, and $\alpha_{\rm 0.3,1.25}$, but now when using only the top four features in the RF algorithm. The four features for the central galaxies are $V_{\rm max}$, $a_{\rm lastmerg}$, $a_{\rm 0.5}$, and $M_{\rm vir}$. The four features for the satellite galaxies are $M_{\rm vir}$, $\delta_{\rm 2.5}$, $\delta_{\rm 1.25}$, and $c$.
}
\label{ov_top}
\end{figure*}
\begin{figure*}
\centering
\begin{subfigure}[h]{0.48\textwidth}
\includegraphics[width=\textwidth]{full-cenxi-top-D80-rand10.pdf}
\end{subfigure}
\hfill
\begin{subfigure}[h]{0.48\textwidth}
\includegraphics[width=\textwidth]{plot3-ctop-stop-D80-rand10.pdf}
\end{subfigure}
\hfill
\caption{
Similar to Figure~\ref{cluster_gab_all}, the predicted galaxy clustering and GAB measurement for centrals only (left) and all (central and satellite) galaxies (right), now obtained using only the top four features for central galaxies and satellites in the ML.
}
\label{cluster_gab_top}
\end{figure*}
In this section, we predict the number of central and satellite with the top four features selected separately for central galaxies and satellites in Section~\ref{feature_imp}. We first perform new grid searches for the two sets of top features to tune the RF classification and regression models for centrals and satellites, respectively. The $F_1$ and $R^2$ scores are listed in the third and fourth lines of Table~\ref{table:clustering}, which are very similar to those from the all features models. Figure~\ref{ov_top} presents the ML predicted OVs compared to those from the SAM. Similar to the OV prediction with all features shown in Figure~\ref{ov_all}, the considered OVs are all accurately reproduced. This is even more impressive in this case, when using only four features for each centrals or satellites. It is worth noting that other than $M_{\rm vir}$ which is common to both, no secondary property is present in both the centrals and satellites top features. Thus in all panels of Figure~\ref{ov_top}, showing the OV with $c$, $a_{\rm 0.5}$, $\delta_{\rm 1.25}$, and $\alpha_{\rm 0.3,1.25}$, these properties are not involved in all predictions. We therefore conclude that the top four features for centrals and satellites are highly efficient in capturing the information needed for reproducing the halo occupation numbers.
With the predicted occupations from the top features, we again populate the haloes to create a mock galaxy catalogue and measure galaxy clustering and the GAB signal. The results are presented in Figure~\ref{cluster_gab_top} and summarized in Table~\ref{table:clustering}. For the centrals-only sample, the predicted original clustering, shuffled clustering, and the GAB are highly consistent with those of the SAM (left panels), with recovered fractions of 1.00, 1.00, and 0.97, respectively. These results are very similar to those from the prediction using all features, and the RF classification with the top four features works equally well as the one with all features. It is worth noting again that the top four features for central galaxies are all halo internal properties without explicitly including environment. This seems to imply that environment measures are not necessary for recovering the centrals GAB. However, other works have shown that environment is crucial for capturing GAB (\citealt{Hadzhiyska2020,Xu2021}; C.\ Cuesta, in prep.). To gain more insight on the role of environment in recovering GAB, we examine in Section~\ref{mass_env} obtaining ML predictions based on only mass and environment, and in Section~\ref{internal} the predictions based solely on internal halo properties.
The right-hand side panels of Figure~\ref{cluster_gab_top} provide the predicted clustering and GAB for all galaxies including satellites. The satellite occupation is predicted with the top four features specific for satellites selected in Section~\ref{feature_imp} (which are different than the top four features for centrals) and include environmental properties. The recovered original clustering, shuffled clustering, and GAB fraction are all in excellent agreement with the SAM measurements (with recovered fractions of 1.00 for all). These fractions are in fact slightly higher than those for the all features model, but we consider them to be equally good due to the randomness associated with the prediction, populating galaxies, and shuffling. Combining the results from the centrals and satellites predictions, we find that the ML mock with only the top four features for each can well capture the galaxy-halo connection in the SAM and reproduce the expected galaxy clustering and GAB.
\subsection{Halo mass and one environmental feature}
\label{mass_env}
\begin{figure*}
\centering
\begin{subfigure}[h]{0.48\textwidth}
\includegraphics[width=\textwidth]{full-cenxi-ME-D80-rand10.pdf}
\end{subfigure}
\hfill
\begin{subfigure}[h]{0.48\textwidth}
\includegraphics[width=\textwidth]{plot3-csME-D80-rand10.pdf}
\end{subfigure}
\hfill
\caption{
Similar to Figure~\ref{cluster_gab_top}, the predicted galaxy clustering and GAB measurement for centrals only (left) and all galaxies (right), using only halo mass and environment $\delta_{\rm 1.25}$ for both centrals and satellites.
}
\label{cluster_gab_mass_env}
\end{figure*}
In Section~\ref{feature_imp} and Section~\ref{topfeatures}, we saw that environmental properties are listed in the top features for the satellite galaxies occupation. However, they are not included in the top 10 features for the central galaxies prediction, and the top four features for centrals (without environment) can well reproduce the centrals GAB. This seems to suggest that environment is not necessary for a recovery of the centrals GAB. We clarify that the internal halo properties (e.g., age $a_{\rm 0.5}$) are surely dependent on environment to some degree, since they produce assembly bias, but it is of interest to know whether an environmental measure is needed to be explicitly included. Traditional (non-ML) analyses show that environment is the most informative property for GAB, more significant than any other single secondary property in either SAM or hydrodynamic galaxy samples \citep{Hadzhiyska2020,Hadzhiyska2021,Xu2021}. In particular, \citet{Xu2021} demonstrated that $\delta_{\rm 1.25}$ can capture the full level of GAB in the SAM. To further examine the role of environment in GAB, we repeat our analysis but now only use the halo mass $M_{\rm vir}$ and $\delta_{\rm 1.25}$ as input features to the RF algorithm models.
The OVs predicted by the ML models based on $M_{\rm vir}$ and $\delta_{\rm 1.25}$ are shown in Appendix~\ref{ov_ME}. We find that the models are less successful in reproducing the OVs compared to the models with all features and the top four features. The OV dependence on $\delta_{\rm 1.25}$ is recovered as expected, as well as the ones for $\alpha_{\rm 0.3,1.25}$ to a large extent. However, the variations with halo properties such as concentration and age are poorly recovered, especially for the satellites. We note that these results are in agreement with those by \citet{Xu2021}. While they were able to mimic the full level of GAB with only halo mass and $\delta_{\rm 1.25}$, they were similarly unable to recover the OVs with other secondary properties. In our ML analysis, the weaker recovery is also reflected by the somewhat lower $F_1$ and $R^2$ performance scores of the RF models in this case (lines 5-6 in Table~\ref{table:clustering}). These scores reflect the halo-by-halo prediction accuracy, such that a lower value will lead to less accurate recovery of the OVs.
We then populate haloes with the predicted occupations and measure galaxy clustering and GAB. The results for these are shown in Figure~\ref{cluster_gab_mass_env} and summarized in Table~\ref{table:clustering} as well. For both centrals-only and all galaxies, the predicted shuffled clustering is in perfect agreement with the SAM results (the red solid lines in the top panels), indicating that the halo mass dependence of clustering is reproduced. However, for the original (unshuffled, including assembly bias) SAM clustering the ML recovery for both these cases is slightly lower (the black solid lines in the top panels). It is still reasonably good with a recovery fraction of 0.99, but stands out in contrast to the excellent agreement of the predictions with all features and top features explored earlier. This leads to a reduced ability to recover the GAB signal, denoted by the solid blue lines in the bottom panels of Figure~\ref{cluster_gab_mass_env}. These correspond to $f_{\rm AB}$ values of 0.86 and 0.92 for the centrals-only GAB and the all-galaxies one, respectively. This result is consistent with \citet{Xu2021} who show that shuffling galaxies among haloes with fixed mass and $\delta_{\rm 1.25}$ (which can also be considered as populating haloes according to only mass and $\delta_{\rm 1.25}$) reproduces $\sim$90\% of the full GAB signal. The performance of the RF models based on only mass and environment is also similar to that of the modified HOD model provided by \citet{Xu2021}, while in the latter the GAB parameters are tunable to reproduce the full effect.
Our analysis suggests that mass and environment are efficient in capturing most of the GAB signal and are useful for reproducing galaxy clustering within 1\% if halo internal properties are unavailable. Combined with the results from Section~\ref{topfeatures}, we find that the central GAB can be recovered with either a few internal halo properties or the environment. The former achieves the purpose by capturing most of the assembly bias effects in halo occupation, whereas the latter achieves this by ``mimicking'' the effect on the clustering. The satellites assembly bias effects can be largely recovered by environment alone, but including information on internal properties improves the OV. Would internal properties alone be able to reproduce both the centrals and satellites GAB? Is the environment required for reproducing the full GAB? We answer these questions in Section~\ref{internal} by testing the RF models using now only the internal halo properties.
\subsection{Internal Features}
\label{internal}
\begin{figure*}
\centering
\begin{subfigure}[h]{0.48\textwidth}
\includegraphics[width=\textwidth]{full-cenxi-in-D80-rand10.pdf}
\end{subfigure}
\hfill
\begin{subfigure}[h]{0.48\textwidth}
\includegraphics[width=\textwidth]{plot3-cin-sin-D80-rand10.pdf}
\end{subfigure}
\hfill
\caption{
Similar to Figure~\ref{cluster_gab_all}, the predicted galaxy clustering and GAB for centrals only (left) and all galaxies (right) when using all internal properties (and no environment measures) for the ML predictions.
}
\label{cluster_gab_internal}
\end{figure*}
In this section, we explore the performance of the ML predictions when using only internal halo properties, commonly associated with halo assembly bias, rather than environment measures directly. We include all the halo properties listed in lines 1-13 of Table~\ref{table:haloprops}. In contrast to the previous case with halo mass and environment, the models with internal properties accurately recover the OV with concentration and $a_{\rm 0.5}$ accurately, as shown in Figure~\ref{ov_in} in Appendix~\ref{more_ov}. The OV with $\delta_{\rm 1.25}$ and $\alpha_{\rm 0.3,1.5}$ are partially recovered, with the centrals OV well reproduced but smaller OV for the satellite galaxies. As before, we proceed to create mock galaxy catalogues with the ML predicted occupations, to study the impact on clustering and GAB.
The clustering and GAB of the RF mock are shown in Figure~\ref{cluster_gab_internal}. For the central galaxies only (left-hand side), we find that the original clustering, shuffled clustering, and GAB are all well reproduced at sub percent accuracy. These results are similar to those with only the top four properties shown in Section~\ref{topfeatures}, which for the central galaxies were comprised of only internal properties ($V_{\rm max}$, $z_{\rm last}$, $a_{\rm 0.5}$, and $M_{\rm vir}$). These top properties appear to include most of the information needed to reproduce the centrals clustering and GAB, such that now including all internal properties does not change the results.
The situation for the satellites, however, is different since environment measures have a prominent role in the top features. Consequently, we find that when adding the satellite galaxies, the clustering and GAB are not well with only the internal properties. The recovered clustering is lower than that of the SAM by 3\%, and only 70\% of the GAB is reproduced.
We conclude that while the environment is not necessary for centrals clustering, it is required for an accurate representation of the satellites clustering. This is consistent with the feature importance provided by the RF models. In summary, secondary halo properties include enough information to recover in full the centrals OV, clustering and GAB, but the environment is needed for accurately predicting the satellites OV and the full level of clustering and GAB, and cannot be replaced with internal properties alone.
\subsection{Single-Epoch Features}
\label{nontree}
\begin{figure*}
\centering
\begin{subfigure}[h]{0.48\textwidth}
\includegraphics[width=\textwidth]{full-cenxi-non-D80-rand10.pdf}
\end{subfigure}
\hfill
\begin{subfigure}[h]{0.48\textwidth}
\includegraphics[width=\textwidth]{plot3-csnon-D80-rand10.pdf}
\end{subfigure}
\hfill
\caption{
Similar to Figure~\ref{cluster_gab_internal}, the predicted galaxy clustering and GAB for centrals only (left) and all galaxies (right), now using only the following single-epoch (i.e not involving the halo merger tree) features $M_{\rm vir}$, $V_{\rm max}$, concentration $c$, angular momentum $j$, and $\delta_{\rm 1.25}$for both centrals and satellites.
}
\label{cluster_gab_nontree}
\end{figure*}
The main purpose of this paper is to explore the possibility of creating realistic mock galaxy catalogues from halo catalogues of $N$-body simulations using ML to capture the detailed galaxy-halo connection. However, for some low-resolution $N$-body simulations, the halo merger tree which follows the haloes' evolution is unavailable. In such cases, one will not be able to obtain halo properties that rely on the merger tree, such as $a_{\rm 0.5}$, $V_{\rm peak}$, and $z_{\rm last}$. The only available properties will be single-epoch properties typically obtained from the final snapshot of the simulation. These include $M_{\rm vir}$, $V_{\rm max}$, concentration $c$, angular momentum $j$, and the environment measures. In this section we test the performance of ML models based on these single-epoch properties. We include, for both centrals and satellites, the above four internal halo properties and $\delta_{\rm 1.25}$.
We find that the OVs in this case are mostly well reproduced, as shown in Figure~\ref{ov_non}. The OV with $c$ and $\delta_{\rm 1.25}$ are particularly well reproduced, as expected, since they are part of the input features. The only notable deviation is for the centrals OV with $a_{\rm 0.5}$ where the ML prediction is slightly smaller than in the SAM. The predicted clustering and GAB signal are shown in Figure~\ref{cluster_gab_nontree}. For the centrals-only prediction, both galaxy clustering and GAB are extremely well reproduced. Adding the satellites, the SAM clustering is recovered to within 1\% and the GAB is recovered to within 5\%. These are better than the ML with only internal properties or $M_{\rm vir}$ and $\delta_{\rm 1.25}$ alone, and slightly worse than the models with all features or the top four features. We suspect that including additional available (single-epoch) environment measures, such as $\delta_{\rm 2.5}$, would have improved this result.
Overall, the analysis illustrates that when the halo formation history is not available (for example, in low-resolution $N$-body simulations), ML models can still reproduce the clustering and GAB to reasonable accuracy. Using ML to predict the halo occupation and populate haloes with galaxies accordingly thus provide a viable practical approach to creating realistic mock galaxy catalogues, even in such cases.
\section{Summary and Discussion}
\label{summary}
In this paper, we describe a machine learning approach to predict the number of galaxies above a stellar-mass threshold in dark matter haloes using halo and environment properties as input. We use the halo catalogue from the Millennium simulation and the galaxy sample from the \citet{Guo2011} SAM model to train and test our ML method. We use random forest classification and regression for the central galaxies and satellites, respectively, and adopt commonly-used $F_1$ and $R^2$ scores to evaluate the performance of the models. We test different combinations of input properties. For each set of the input properties, we tune the hyper-parameters of the RF models to maximize the performance scores. With the predicted number of central and satellite galaxies in each halo, we then populate the Millennium simulation haloes to create a mock galaxy catalogue and measure the galaxy clustering and galaxy assembly bias signal to compare with those of the SAM.
We start by using all the available internal and environmental halo properties, listed in Table~\ref{table:haloprops}, as input features. The predicted HOD and occupancy variations are consistent with those measured from the SAM. The ML mock catalogue matches well the galaxy clustering, shuffled sample clustering, and GAB as that of the original SAM sample. The clustering is recovered to sub percent accuracy and GAB is recovered at the two percent level. Our results show that machine learning is capable of capturing the complex high-dimensional relations between halo properties and the galaxy occupation in the SAM model and reproduce the expected galaxy clustering accurately, including the intricate effects of assembly bias.
The RF models also provide an estimate of the relative importance of the different features. We find that $V_{\rm max}$ is the most important feature for central galaxies, followed by formation history (internal) properties and halo mass. Environmental properties are not included in the top 10 features. On the other hand, the satellite galaxies prediction relies the most on halo mass and environmental properties. We construct simpler RF models with the top four halo properties for the centrals and satellite galaxies separately, based on the feature importance and correlation matrix between them. We select $V_{\rm max}$, $z_{\rm last}$, $a_{\rm 0.5}$, and $M_{\rm vir}$ for central galaxies prediction and $M_{\rm vir}$, $\delta_{\rm 1.25}$, $\delta_{\rm 2.5}$, and concentration $c$ for the satellites prediction. The OVs, clustering and GAB are again well reproduced. This demonstrates that the ML methodology is powerful enough such that, with only a few halo properties, it can achieve similar performance as when using all the available information.
We perform two additional tests to further explore the role of the environment in reproducing galaxy clustering and GAB. We first use only halo mass and one environmental property ($\delta_{\rm 1.25}$) as input for the RF models. With the ML-constructed mock, we still recover the SAM (original and shuffled) galaxy clustering to within 1\% and about 92\% of the full GAB signal (Figure~\ref{cluster_gab_mass_env}). We conclude that $\delta_{1.25}$ along with the host halo mass is enough to reproduce GAB to $\sim 10\%$ accuracy. This is in agreement with previous works \citep{Hadzhiyska2020,Contreras2021,Xu2021} that showed that using environmental properties can realistically incorporate assembly bias into empirical models such as the HOD or SHAM. However, these methods do not recover the full occupancy variation for halo properties with inherent halo assembly bias, such as concentration or age (see Appendix~\ref{more_ov} for more details). This puts a limitation on such approaches when using statistics that need a more detailed modelling of the galaxy-halo connection (like galaxy lensing). Other approaches that add assembly bias to mock catalogues using a single secondary property like the halo concentration will also necessarily fail to reproduce the galaxy-halo connection, since such properties are not able to capture on their own the full GAB of a semi-analytic galaxy sample \citep{Croton2007,Xu2021}. To our knowledge, the approach presented in this paper is the most efficient model capable of populating galaxies in N-Body simulations, while taking into account the correlations between the halo occupation and the secondary halo properties, and recovering a realistic GAB signal.
The second test employs all secondary assembly bias properties as input, excluding the environment. The clustering and GAB for central galaxies alone are recovered at sub percent accuracy, at the same level as those with all or top four properties. However, after adding satellite galaxies, the predicted ML mock catalogue only recovers about 70\% of the GAB signal. This clearly indicates that internal properties alone are not able to fully capture the relation between the satellite occupations and the host haloes. Perhaps further information can be introduced by including additional internal properties not included in this work, however using readily-available environment measures seems the more practical approach here. Combining the results from the two tests, we find that both internal properties and environmental properties can reproduce the centrals clustering and GAB, but that environment is necessary for reproducing the full clustering and GAB. Furthermore, environment alone (together with halo mass) goes a long way toward mimicking the correct level of assembly bias, however including assembly bias properties in needed to recover the OV with such properties and reproduce GAB to percent level accuracy.
Finally, to explore a potential application of our ML method in cases where the halo merger tree might not be available in low-resolution $N$-body simulations, we limit the input properties to single-epoch ones which can be obtained from the present-day simulation. We therefore use $M_{\rm vir}$, $V_{\rm max}$, concentration, angular momentum, and $\delta_{\rm 1.25}$ as input for the RF models. The OVs in this case are reasonably reproduced, galaxy clustering is matched at sub percent level, and the GAB signal is recovered to 5\%. An improvement in the GAB level may be reached if including additional environment parameters. Utilizing such a model can be a practical approach for populating large dark-matter-only simulations, like the Millennium XXL Simulation \citep{Angulo2012} and others, where the resolution of the halo merger trees is insufficient for use in a SAM. Instead, one can train and fine-tune a ML model on a smaller volume high-resolution galaxy formation simulation. Once the model is determined, it is straight-forward to apply it to the larger simulation to create mock galaxy catalogues with all the required attributes.
Overall, our results demonstrate the ability of machine learning to successfully capture the high-dimensional relationship between the halo occupation and multiple halo properties. Our tests here are with a SAM, but we expect similar performance when matching hydrodynamical simulations, which we leave for future work. As just mentioned, it is particularly advantageous to learn these relations from existing SAM or hydrodynamic galaxy samples in order to create realistic mock galaxy catalogues with haloes in larger cosmological volumes. This has the advantage of reproducing the detailed galaxy-halo connection of state-of-the-art galaxy formation models, which might be computationally-prohibitive otherwise. Additionally, with the single-epoch test, we show that ML can also be used to reproduce galaxy clustering and assembly bias in low-resolution $N$-body simulations for emulators, which are becoming benchmarks for cosmological studies. In this work, we focus on predicting the occupation of galaxies in halo for stellar-mass selected samples, but it can be extended to other types of galaxy samples, for example, star formation rate selected samples and colour selected sample which are also frequently used in observations, as well as galaxy samples at higher redshifts. We leave these as well for future studies.
Different studies in the literature have focused on predicting galaxy properties from haloes with ML techniques. \citet{Xu2013} predict the number of galaxies based on six halo properties and reproduce the galaxy clustering to a 5\%-10\%, which is similar to our internal properties predictions without using environment. Our extended work now reaches sub percent accuracy. Other works based on ML techniques predict properties of central galaxies such as stellar mass, star formation rate, and gas mass to mimic galaxy formation in hydrodynamic simulations (e.g., \citealt{Kamdar2016b,Agarwal2018,Wadekar2020}). In contrast, our study using the occupation number more directly probes galaxy clustering and assembly bias and allows to naturally predict both central and satellite galaxies.
For the purpose of modelling the halo occupation, our work can be considered as a ML alternative to the HOD approach. The standard HOD framework models the number of galaxies in a halo as a function of only halo mass. Different extensions of the HOD (e.g., \citealt{Hearin2016,Xu2021,Yuan2021}) include an additional dependence on one or two secondary halo properties, but the galaxy-halo relations obtained are still limited. With ML-based methods, the non-linear dependence of the halo occupation on multiple halo and environment properties can be maximally reproduced, without assuming an analytic relation between them or fixing the parameters. Similarly, compared to empirical SHAM models, ML methods can capture and reproduce more complex multivariate dependencies between the galaxy and halo properties. This advantage makes ML a powerful approach for studying the galaxy-halo connection and for creating realistic mock galaxy catalogues which will be useful for upcoming large galaxy surveys.
\section*{Acknowledgements}
XX, SK and IZ acknowledge support by NSF grant AST-1612085. SC acknowledges the support of the ``Juan de la Cierva Formaci\'on'' fellowship (FJCI-2017-33816).
\section*{Data Availability}
The data underlying this article are available in GitHub at \url{https://github.com/xiaojux2020/RFmodels}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,459 |
Fontana di Piazza delle Cinque Scole, även benämnd Fontana del Pianto, är en fontän på Piazza delle Cinque Scole i Rione Regola i Rom. Fontänen designades av skulptören Giacomo della Porta år 1591 och stod ursprungligen på den närbelägna Piazza Giudea. Fontänen förses med vatten från Acqua Paola.
Beskrivning
Påve Gregorius XIII gav Giacomo della Porta i uppdrag att formge fontänen men själva stenhuggeriarbetet utfördes inte förrän 1591–1593 av Pietro Gucci med vit marmor från Serapis tempel på Quirinalen.
Fontänen har två brunnskar; det övre har gorgonansikten, vilka sprutar vatten.
Beteckningen "Cinque Scole" syftar på Roms tidigare fem synagogor: den kastilianska, den katalanska, den sicilianska, den nya och den italienska. Tillnamnet "Pianto" åsyftar den närbelägna kyrkan Santa Maria del Pianto.
I samband med saneringen av Roms getto i slutet av 1800-talet nedmonterades fontänen och återuppställdes på sin nuvarande plats år 1930.
Bilder
Källor
Noter
Webbkällor
Tryckta källor
Externa länkar
Fontäner i Rom
Rione Regola
Verk av Giacomo della Porta
Skulpturer från 1500-talet
Arkitekturåret 1591 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,195 |
Q: Is there a defined ordering between dragend and drop events? According to the documentation on the HTML5 drag and drop API, two events are fired when an element is dropped:
*
*A drop event is fired from the drop target
*A dragend event is fired from the source of the drag
In doing a simple test (see snippet), the drop event always fires just before the dragend event (at least in Chrome) but I can't find anything about the ordering of these events in the spec.
Is the ordering of these events defined, or are they free to fire in either order?
function allowDrop(ev) {
ev.preventDefault();
}
function drag(ev) {
ev.dataTransfer.setData("text", ev.target.id);
}
function drop(ev) {
console.log("drop at " + Date.now());
ev.preventDefault();
var data = ev.dataTransfer.getData("text");
ev.target.appendChild(document.getElementById(data));
}
function dragend(ev) {
console.log("dragend at " + Date.now());
}
#div1 {
background-color: red;
height: 100px;
width: 100px;
}
#drag1 {
background-color: green;
height: 50px;
width: 50px;
}
<div>Drag the green square in to the red one</div>
<div id="div1" ondrop="drop(event)" ondragover="allowDrop(event)" width="100px" height="100px"></div>
<div id="drag1" draggable="true" ondragstart="drag(event)" ondragend="dragend(event)" width="50px" height="50px">
A: According to the drag-and-drop processing model specified in the current (updated June 8, 2021) HTML specification, the drop() event must fire before the dragend() event.
The corresponding information is deeply nested in the document, but the section describing the end of the drag operation looks as follows (omissions and emphasis mine):
Otherwise, if the user ended the drag-and-drop operation (e.g. by
releasing the mouse button in a mouse-driven drag-and-drop interface),
or if the drag event was canceled, then this will be the last
iteration. Run the following steps, then stop the drag-and-drop
operation:
*
*[...]
Otherwise, the drag operation might be a success; run these substeps:
*
*Let dropped be true.
*If the current target element is a DOM element, fire a DND event named drop at it; otherwise, use platform-specific
conventions for indicating a drop.
*[...]
*Fire a DND event named dragend at the source node.
*[...]
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,199 |
Q: Passing custom log4j.properties file from s3 I'm trying to set custom logging configurations. If I add the log file to the cluster and reference it in my spark submit, the configurations take effect. But if I try to access the file using --files s3://... then it doesn't work.
Works (assuming I placed the file in the home dir):
spark-submit \
--master yarn \
--conf spark.driver.extraJavaOptions=-Dlog4j.configuration=file:log4j.properties \
--conf spark.executor.extraJavaOptions=-Dlog4j.configuration=file:log4j.properties \
Doesn't work:
spark-submit \
--master yarn \
--files s3://my_path/log4j.properties \
--conf spark.driver.extraJavaOptions=-Dlog4j.configuration=log4j.properties \
--conf spark.executor.extraJavaOptions=-Dlog4j.configuration=log4j.properties \
How can I use a config file in s3 to set the logging configuration?
A: You can't directly Log4J loads its files from the local filesystem, always.
You can use configs inside a JAR, and as spark will download JARs with your job, you should be able to get it indirectly. Create a JAR containing only the log4j.properties file, tell spark to load it with the job
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,351 |
Q: Backup MS SQL Server database through Java program? My question is simple. There's any possible that I follow the process through to completion.
A command like that
backup database master to disk='\\csd0201wnt29f\backup\master.bak' with stats=1
I mean the stats=1 clause.
The method
public void executar(Backup bkp, String comando) throws Exception {
Connection conexao = ConnectionFactory.connectBase(bkp.getServidor(), bkp.getUsuario(), bkp.getSenha());
PreparedStatement cmd = conexao.prepareCall(comando, ResultSet.TYPE_FORWARD_ONLY, ResultSet.CONCUR_READ_ONLY);
PreparedStatement cmdQuery = conexao.prepareCall("select @@spid");
ResultSet set = cmdQuery.executeQuery();
int spid=0;
if (set.next()) {
spid=set.getInt(1);
System.out.println("SPID" + set.getInt(1));
}
SQLMonitor s = new SQLMonitor(bkp, spid);
s.start();
cmd.execute();
cmdQuery.close();
cmd.close();
conexao.close();
}
The call
public void RealizaBackup(Backup bkp) throws Exception {
SQLComando cmd = new SQLComando();
System.out.println("backup database " + bkp.getBase() + " to disk='" + bkp.getCaminho() + "\\" + bkp.getBase() + ".bak' with stats=1");
cmd.executar(bkp, "backup database " + bkp.getBase() + " to disk='" + bkp.getCaminho() + "\\" + bkp.getBase() + ".bak' with stats=1");
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,523 |
Q: Python bvp solver with array like arguements I am trying to solve a 4th order bvp using the bvp solver in Python. The differential equation is of the form given below :-
d4w(x)/dx4=t/(t-w(x))
with boundary conditions w(0)=w(10)=w'(0)=w'(10)=0 . If t in the differential equation is a simple constant, the code is simple and is given by
import numpy as np
from scipy.integrate import odeint
import matplotlib.pyplot as plt
from scipy.integrate import solve_bvp
t=1
def fun(x, y):
return np.vstack([y[1], y[2], y[3], t/(t-y[0])])
def bc(ya, yb) :
return np.array([ya[0], ya[1], yb[0], yb[1]])
xmesh = np.linspace(0, 10, 100)
y = np.zeros((4, len(xmesh)))
sol = solve_bvp(fun, bc, xmesh, y)
plt.figure()
plt.plot(xmesh, sol.sol(xmesh)[0])
plt.legend()
But in my case, t is a pre-defined constant 1-D array. I am facing some issues in writing the code for this case. Any help regarding this will be highly appreciated.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,606 |
Purschenstein Castle () in Neuhausen/Erzgeb. in East Germany was built in the late 12th century, around 1200, probably by Boresch I (Borso). The toll and escort castle protected a salt road running from Central Germany to Bohemia. This long-distance trading route, also called the Old Bohemian Track (Alter Böhmischer Steig), ran from Leipzig past present-day Neuhausen and over the Deutscheinsiedler Saddle towards Prague.
In 2005, the castle was bought by a Dutch businessman. Since then it has been renovated and houses a hotel, the Schlosshotel Purschenstein.
Gallery
See also
List of castles in Saxony
External links
The von Schönberg at Purschenstein
Home page of Purschenstein Castle
Photo gallery of Purschenstein Castle
detailed information on Purschenstein as part of Project "Alte Salzstraße"
Castles in Saxony
Buildings and structures in Mittelsachsen
Neuhausen, Saxony | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,853 |
\section*{Introduction}
Let $\Omega$ be a bounded strongly pseudoconvex open subset of $\mathbb{C}^n$.
Given $\phi \in {\mathcal C}^0(\partial \Omega)$ and $f \in L^p(\Omega)$,
we consider the Dirichlet problem
$$
MA(\Omega,\phi,f): \left\{ \begin{array}{l}
u \in PSH(\Omega) \cap {\mathcal C}^0(\overline{\Omega}) \\
u=\phi \text{ on } \partial \Omega \\
(dd^c u)^n=f \beta_n \text{ in } \Omega.
\end{array} \right.
$$
Here $\beta_n=dV$ denotes the euclidean volume form in $\mathbb{C}^n$, $d=\partial+\overline{\partial}$,
$d^c=i(\overline{\partial}-\partial)$, $PSH(\Omega)$ is the set of plurisubharmonic
(psh for short) functions
in $\Omega$ (the set of locally integrable functions $u$ such that $dd^c u \geq 0$
in the sense of currents), and $( dd^c \; \cdot )^n$ denotes the complex Monge-Amp\`ere operator:
this operator is well defined on the subset of bounded (in particular continuous) psh functions,
as follows from the work of E.Bedford and A.Taylor [BT 2]. We refer the reader to
[K 4] for a recent survey on its properties.
The equation $MA(\Omega,\phi,f)$ has been studied intensively during the last decades.
Let ${\mathcal B}(\Omega,\phi,f)$ denote the family of subsolutions to
$MA(\Omega,\phi,f)$, i.e. the set of bounded functions $v$ that are psh in $\Omega$ with
$v \leq \phi$ on $\partial \Omega$ (i.e. $\limsup_{\zeta \rightarrow z} v(\zeta) \leq \phi(z)$
when $z \in \partial \Omega$) and $(dd^c v)^n \geq f \beta_n$
in $\Omega$. It follows from the comparison principle [BT 2] that if a solution exists,
it must coincide with the Perron-Bremermann envelope,
$$
u(z)=u(\Omega,\phi,f)(z):=\sup \{ v(z) \, / \, v \in {\mathcal B}(\Omega,\phi,f) \}.
$$
It follows from the works of H.J.Bremermann [Br], J.Walsh [W]
and E.Bedford-A.Taylor [BT 1]
that $u(\Omega,\phi,f)$ is indeed
a solution to $MA(\Omega,\phi,f)$ when $f \in {\mathcal C}^0(\overline{\Omega})$.
S.Kolodziej has further shown in [K 2]
that the solution $u(\Omega,\phi,f)$
is still continuous when $f \in L^p(\Omega)$, for some
$p>1$ (it is easy to check that $u(\Omega,\phi,f)$ is not necessarily
locally bounded when the density is merely in $L^1(\Omega)$).
Higher regularity results have also been provided. E.Bedford and A.Taylor showed in
[BT 1] that if $\phi \in Lip_{2\alpha}(\Omega)$ and $f^{1/n} \in Lip_{\alpha}(\overline{\Omega})$,
then $u(\Omega,\phi,f) \in Lip_{\alpha}(\overline{\Omega})$.
The smoothness of the solution (assuming smoothness of $\phi$ and $f>0$) is established in [CKNS].
Our aim here is to establish H\"older-continuity of $u(\Omega,\phi,f)$ in the spirit
of [BT 1], only assuming the density $f$ belongs to $L^p(\Omega)$, $p >1$,
as in [K 2]. Our main result is the following:
\vskip.2cm \noindent
{\bf Theorem A.}
{\it Assume $f \in L^p(\Omega)$, for some $p>1$, and $\phi \in Lip_{2\alpha}(\partial \Omega)$,
with $\nabla u(\Omega,\phi,0) \in L^2(\Omega)$. Then
$$
u(\Omega,\phi,f) \in Lip_{\alpha'}(\overline{\Omega}),
\; \;
\text{ for all } \alpha'<\min(\alpha,2/[qn+2]),
$$
where $1/p+1/q=1$.}
\vskip.2cm
The condition $\nabla u(\Omega,\phi,0) \in L^2(\Omega)$ is automatically
satisfied if $\phi \in {\mathcal C}^{1,1}(\partial \Omega)$: in this
case $u(\Omega,\phi,0) \in Lip_1(\overline{\Omega})$, hence
$\nabla u(\Omega,\phi,0)$ is actually bounded in $\Omega$ (see [BT 1]).
What really matters here is that there should exist a subsolution
$v \in {\mathcal B}(\Omega,\phi,0)$ such that $\nabla v \in L^2(\Omega)$.
This implies (see Lemma 3.1), that
$u(\Omega,\phi,0)$ and $u(\Omega,\phi,f)$
both have gradient in $L^2(\Omega)$.
We could not avoid the use of this extra (technical ?) hypothesis on
the homogenous solution $u(\Omega,\phi,0)$.
Also the exponent $\alpha'$ is probably not optimal. We can get a better
exponent by assuming that $\Delta u(\Omega,\phi,0)$ has finite mass in $\Omega$
(this is automatically satisfied when $\phi \in {\mathcal C}^2 (\partial \Omega)$).
\vskip.2cm \noindent
{\bf Theorem B.}
{\it Assume $f \in L^p(\Omega)$, for some $p>1$, and
$\phi \in Lip_{2\alpha}(\partial \Omega)$ is such that
that $\Delta u(\Omega,\phi,0)$ has finite mass in $\Omega$.
Then
$$
u(\Omega,\phi,f) \in Lip_{\alpha''}(\overline{\Omega}),
\; \;
\text{ for all } \alpha''<\min(\alpha,2/[qn+1]),
$$
where $1/p+1/q=1$.}
\vskip.2cm
The exponent $\alpha''$ is not far from being optimal as Example 4.2 shows.
\section{The stability estimate}
Our main tool is the following estimate which is proved in [EGZ]
in a compact setting (under growth -- but no boundary -- conditions, see
Proposition 3.3 in [EGZ]).
A similar -- but weaker -- estimate was established by S.Kolodziej in [K 3].
\begin{thm}
Fix $0 \leq f \in L^p(\Omega)$, $p>1$.
Let $\varphi,\psi$ be two bounded psh functions in $\Omega$ such that
$
(dd^c \varphi)^n=f \beta_n \text{ in } \Omega,
\text{ and } \varphi \geq \psi \text{ on } \partial \Omega.
$
Fix $r \geq 1$ and $0 \leq \gamma <r/[nq+r]$, $1/p+1/q=1$. Then
$$
\sup_{\Omega} (\psi-\varphi) \leq C ||(\max(\psi-\varphi,0)||_{L^r(\Omega)}^{\gamma},
$$
for some uniform constant $C=C(\gamma,||f||_{L^p(\Omega)},||\psi||_{L^{\infty}})>0$.
\end{thm}
The proof follows closely the one given in [EGZ], so we only
recall the main ingredients, for the reader's convenience.
The estimate is a simple consequence of the following result.
\begin{pro}
Fix $f \in L^p(\Omega)$, $p>1$, and let
$\varphi,\psi$ be bounded psh functions in $\Omega$ such that
$\varphi \geq \psi$ on $\partial \Omega$.
If $(dd^c \varphi)^n=f \beta_n$, then for all $\varepsilon>0,\tau>0$,
$$
\sup_{\Omega} (\psi-\varphi) \leq \varepsilon+C \left[ \text{Cap}(\varphi-\psi<-\varepsilon ) \right]^{\tau},
$$
for some uniform constant $C=C(\tau, ||f||_{L^p(\Omega)})$.
\end{pro}
Here $\text{Cap}(\cdot)$ denotes the Monge-Amp\`ere capacity introduced and studied
by E.Bedford and A.Taylor in [BT 2].
Recall that for $K \subset \Omega$,
$$
\text{Cap}(K):=\sup \left\{ \int_K (dd^c v)^n \, / \, v \in PSH(\Omega)
\text{ with } -1 \leq v \leq 0 \right\}.
$$
The proposition is a direct consequence of the following three lemmas:
\begin{lem}
Fix $\varphi,\psi \in PSH(\Omega) \cap L^{\infty}(\Omega)$ such that
$\overline{\lim}_{\zeta \rightarrow \partial \Omega} (\varphi-\psi) \leq 0$.
Then for all $t,s>0$,
$$
t^n \text{Cap}(\varphi-\psi<-s-t) \leq \int_{(\varphi-\psi<-s)} (dd^c \varphi)^n.
$$
\end{lem}
\begin{lem}
Assume $0 \leq f \in L^p(\Omega)$, $p>1$.
Then for all $\tau>1$, there exists $C_{\tau}>0$ such that
for all $K \subset \Omega$,
$$
0 \leq \int_K f dV \leq C_{\tau} \left[ \text{Cap}(K) \right]^{\tau}.
$$
\end{lem}
When $(dd^c \varphi)^n=f dV$, one can
combine Lemma 1.3 and Lemma 1.4 to control
$\text{Cap}(\varphi-\psi<-s)$ by $\text{Cap}(\varphi-\psi<-s-t)]^{\tau}$.
This has strong consequences since $\tau>1$, as the following
result shows, when applied to $g(t)=\text{Cap}(\varphi-\psi<-t-\varepsilon)^{1/n}$:
\begin{lem}
Let $g:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ be a decreasing right-continuous function.
Assume there exists $\tau,B>1$ such that
$g$ satisfies
$$
H(\alpha,B) \hskip1cm
tg(s+t) \leq B [ g(s) ]^{\tau}, \;
\forall s,t>0.
$$
Then there exists $S_{\infty}=S_{\infty}(\tau,B) \in \mathbb{R}^+$ such that
$g(s)=0$ for all $s \geq S_{\infty}$.
\end{lem}
One word about the proofs. Lemma 1.3 is a direct consequence
of the ``comparison principle'' of E.Bedford and A.Taylor [BT 2]
(see [K 4] p.32 for a detailed proof).
Using H\"older's inequality, one reduces the proof of Lemma 1.4 to
showing that the euclidean volume is bounded from above by the
Monge-Amp\`ere capacity. One can actually show that
$$
Vol(K) \lesssim \exp[-Cap(K)^{-1/n}] \; \;
(\text{see Theorem 7.1 in [Z]}),
$$
which is a much better control than what we actually need.
The last lemma is an elementary exercise, whose proof is given
in [EGZ], Lemma 2.3.
\section{H\"older continuous barriers}
For fixed $\delta >0$ we consider
$\Omega_{\delta}:=\{ z \in \Omega \, / \, \text{dist}(z,\partial \Omega) >\delta \}$ and set
$$
u_{\delta}(z):=\sup_{||\zeta|| \leq \delta } u(z+\zeta), \, z \in \Omega_{\delta}.
$$
This is a psh function in $\Omega_{\delta}$, when $u$ is psh in $\Omega$,
which measures the modulus of continuity
of $u$. We would like to use Theorem 1.1 applied with $\psi=u_{\delta}$.
However $u_{\delta}$ is not globally defined in $\Omega$, so we need
to extend it with control on the boundary values.
This is the contents of our next result which makes heavy use of the
pseudoconvexity assumption.
\begin{pro} Let $u \in PSH (\Omega) \cap L^{\infty} (\Omega)$ be a psh
function such that $u_{|\partial \Omega}=\phi \in Lip_{2 \alpha}(\partial \Omega)$.
Then there exists a family $(\tilde{u}_{\delta})_{0 < \delta < \delta_0}$
of bounded psh functions on $\Omega$
such that $\tilde{u}_{\delta} \searrow u$ in $\Omega$ as $\delta \searrow 0$, with
$$
\tilde{u}_{\delta} = \left\{ \begin{array}{l}
\max (u_{\delta} - C \delta^{\alpha}, u) \text{ in } \Omega_{\delta} \\
u \text{ in } \Omega \setminus \Omega_{\delta}
\end{array} \right.
$$
\noindent In particular $\sup_{\Omega_{\delta}} \vert \tilde{u}_{\delta} - u_{\delta}\vert \leq C \delta^{\alpha}$
for $0 < \delta < \delta_0$.
\end{pro}
The proof relies on the construction of H\"older continuous plurisubharmonic
and plurisuperhamonic barriers for the Dirichlet problem $MA (\Omega,\phi,f)$:
\begin{lem} Fix $\phi \in Lip_{2 \alpha} (\partial \Omega)$, $f \in L^p(\Omega)$,
$p>1$, and set $u := u(\Omega,\phi,f)$.
Then there exists $v, w \in PSH (\Omega) \cap Lip_{\alpha} (\overline \Omega)$ such that
\begin{enumerate}
\item $ v (\zeta) = \phi (\zeta) = - w (\zeta), \forall \zeta \in \partial \Omega$,\\
\item $ v (z) \leq u (z) \leq - w (z), \forall z \in \Omega$.
\end{enumerate}
\end{lem}
\begin{proof}
Assume first that $\phi \equiv 0$. We are going to show that
there exists a weak barrier $b_f \in PSH (\Omega) \cap Lip_1 (\Omega)$ for the Dirichlet problem
$MA (0,f,\Omega)$, i.e. a psh function which satisfies
\begin{itemize}
\item $(i)$ $ b_f (\zeta) = 0, \ \forall \zeta \in \partial \Omega,$\\
\item $(ii)$ $ b_f \leq u (\Omega,0,f), $ in $\Omega$,\\
\item $(iii)$ $ \vert b_f (z) - b_f (\zeta) \vert \leq C_1 \vert z - \zeta \vert, \ \forall z \in \Omega, \ \forall \zeta \in \Omega$,
\end{itemize}
for some uniform constant $C_1 > 0$.
In order to construct $b_f$, we set $u_0 := u (\Omega,0,f)$ and assume first that
the density $f$ is bounded near $\partial \Omega$: there
exists a compact subset $K \subset \Omega$ such that $0 \leq f \leq M$ on $\Omega \setminus K$.
Let $\rho$ be a ${\mathcal C}^2$ strictly plurisubharmonic defining function for $\Omega$.
Then for $A > 0$ large enough the function $b_f := A \rho$ satisfies the condition
$(dd^c b_f)^n \geq M \beta_n \geq f \beta_n$ on $\Omega \setminus K$.
Moreover taking $A$ large enough we also have $ A \rho \leq m \leq u_0$ on a neighborhood of $K$,
where $m := \min_{\Omega} u_0.$
Therefore the function $b_f$ is a ${\mathcal C}^2$ plurisubharmonic function on $\Omega$ satisfying the
conditions $(dd^c b_f)^n \geq (dd^c u_0)^n$ on $\Omega \setminus K$ and
$b_f \leq u_0$ on $\partial (\Omega \setminus K)$.
This implies, by the comparison principle [BT 2],
that $b_f \leq u_0$ in $\Omega \setminus K$, hence in $\Omega$.
When $f$ is not bounded near $\partial \Omega$, we can proceed as follows.
Fix a large ball $\mathbb{B} \subset \mathbb{C}^n$ so that $\Omega \Subset \mathbb{B} \subset \mathbb{C}^n$.
Define $\tilde f := f$ in $\Omega$ and $\tilde f = 0$ in $\mathbb{B} \setminus \Omega$.
We can use our previous construction to find a barrier function
$b_{\tilde f} \in PSH (\mathbb{B}) \cap {\mathcal C}^2 (\mathbb{B})$ for the Dirichlet problem
$MA (\mathbb{B},0, \tilde f)$ for the ball $\mathbb{B}$.
Let $h= u (\Omega,-b_{\tilde f},0)$ denote the Bremermann function
in $\Omega$ with boundary values $-b_{\tilde f}$, for the zero density.
Since $-b_{\tilde f} \in {\mathcal C}^2(\partial \Omega)$,
the psh function $h$ is Lipschitz in $\Omega$ (see [BT 1]),
therefore $b_f :=h + b_{\tilde f} \in PSH (\Omega) \cap Lip_1 (\Omega)$
is a barrier function for the Dirichlet problem $MA(\Omega,0,f)$.
It remains to construct the functions $v,w$ satisfying (1),(2) above.
It follows from [BT 1] that the psh
functions $u (\Omega,\pm \phi,0)$ are H\"older continuous of order $\alpha$.
We let the reader check that the functions $v := u (\Omega,\phi,0) + b_f$
and $w := u (\Omega,-\phi,0) + b_f$ do the job.
\end{proof}
We are now ready for the proof of the proposition.
\begin{proof}
It follows from Lemma 2.2 that
$$
|u(z) - u (\zeta)| \leq C |z - \zeta|^{\alpha},
\; \forall \zeta \in \partial \Omega, \forall z \in \Omega.
$$
The functions $u_{\delta}(z):=\sup_{||\zeta|| \leq \delta} u(z+\zeta)$
are psh in $\Omega_{\delta}$.
Observe that if $z \in \partial \Omega_{\delta}$ and $\zeta \in \mathbb{C}^n$ with
$||\zeta||\leq \delta$ then $z + \zeta \in \partial \Omega$, hence
$u_{\delta} - C \delta^{\alpha} \leq u (z)$.
Thus the functions
$$
\tilde{u}_{\delta} (z) := \left\{ \begin{array}{l}
\sup \{u_{\delta} (z) - C \delta^{\alpha}, u (z)\} \text{ in } \Omega_{\delta} \\
u \text{ in } \Omega \setminus \Omega_{\delta} \end{array} \right.
$$
are psh and bounded in $\Omega$
and decrease to $u$ as $\delta$ decreases to $0$.
\end{proof}
Our construction of barriers allows us to control the total
mass of the Laplacian of solutions to $MA(\Omega,\phi,f)$.
This will be important in section 4.
\begin{pro}
Fix $0 \leq f \in L^p(\Omega)$, $p>1$, and $\phi \in Lip_{2 \alpha}(\partial \Omega)$.
Then $\Delta u(\Omega,0,f)$ has finite
mass in $\Omega$.
In particular, if $\Delta u(\Omega,\phi,0)$ has finite mass in $\Omega$,
then $\Delta u(\Omega,\phi,f)$ has finite mass in $\Omega$.
\end{pro}
Note that $\Delta u(\Omega,\phi,0)$ has finite mass in $\Omega$ when
$\phi \in {\mathcal C}^2(\Omega)$,
as explained in the proof below.
\begin{proof}
Assume first that $\phi \in {\mathcal C}^2(\Omega)$.
Consider any smooth extension of $\phi$ and correct it by adding $A \rho$, $A>>1$,
in order to obtain a smooth plurisubharmonic extension $\hat{\phi}$
which is defined in
a neighborhood of $\overline{\Omega}$.
Since $\hat{\phi}$ is a subsolution to $MA(\Omega,\phi,0)$ whose
Laplacian has finite mass in $\Omega$, it follows from the comparison
principle that $\Delta u(\Omega,\phi,0)$ also has finite mass
in $\Omega$.
Let $\tilde{f}$ be the trivial extension of $f$ to a large ball $\mathbb{B}$
containing $\Omega$. Let $b_{\tilde{f}} \in {\mathcal C}^2(\mathbb{B})$
be a plurisubharmonic barrier for $MA(\mathbb{B},0,\tilde{f})$ (see the proof of Lemma 2.2).
Then $b_f:=u(\Omega,-b_{\tilde{f}},0)+b_{\tilde{f}}$ is
a plurisubharmonic barrier for $MA(\Omega,0,f)$. Its Laplacian has finite mass
in $\Omega$ since $b_{\tilde{f}}$ is smooth, so it follows from the comparison
principle that $\Delta u(\Omega,0,f)$ has finite mass in $\Omega$.
Set now $v:=u(\Omega,0,f)+u(\Omega,\phi,0)$.
This is a plurisubharmonic function in $\Omega$ such that
$v=\phi$ on $\partial \Omega$ and $(dd^c v)^n \geq f dV$ in $\Omega$.
If $\Delta u(\Omega,\phi,0)$ has finite mass in $\Omega$, then
$\Delta v$ has finite mass in $\Omega$, hence
$\Delta u(\Omega,\phi,f)$ also has finite mass in $\Omega$.
\end{proof}
\section{Gradient estimates}
This section is devoted to the proof of Theorem A.
Let $u=u(\Omega,\phi,f)$ be the unique solution to the
complex Monge-Amp\`ere equation
$$
(dd^c u)^n=f \beta_n \text{ in } \Omega,
$$
with boundary values $u=\phi \in Lip_{2\alpha}(\partial \Omega)$.
Since $f \in L^p(\Omega)$, $p>1$, it follows from [K 2] that
$u$ is a continuous plurisubharmonic function.
Our aim is to show that $u$ is H\"older continuous.
Let $\tilde{u}_{\delta}$ be the functions given by Proposition 3.1.
The stability estimate (Theorem 1.1) applied with $r=2$ yields
$$
\sup_{\Omega_{\delta}} (u_{\delta}-u) \leq
C_1 \delta^{\alpha}+\sup_{\Omega} (\tilde{u}_{\delta}-u)
\leq C_1 \delta^{\alpha}+ C_2 ||u_{\delta}-u||_{L^2(\Omega_{\delta})}^{\gamma},
$$
for $\gamma<2/(nq+2)$, $1/p+1/q=1$.
It remains to
show that $||u_{\delta}-u||_{L^2(\Omega_{\delta})} \leq C_4 \delta$ to conclude the proof.
It will be a consequence of Lemma 3.1 below that $\nabla u \in L^2(\Omega)$.
Assuming this for the moment, we derive the desired upper-bound
on $||u_{\delta}-u||_{L^2(\Omega_{\delta})}$.
Averaging the gradient of u on balls of radius $\delta$ yields, for all $||\zeta||<\delta$,
$$
\left( \int_{\Omega_{\delta}} |u(z+\zeta)-u(z)|^2 dV(z) \right)^{1/2}
\leq \delta \; || \nabla u||_{L^2(\Omega)}.
$$
By Choquet's lemma, there exists a sequence $\zeta_j$, $||\zeta_j|| <\delta$, such that
$u_{\delta}=( \sup_j u_j )^*$, in $\Omega_{\delta}$,
where $u_j(z):=u(z+\zeta_j)$.
Since $u_j-u \leq u_{\delta}-u$,
it follows from Lebesgue's dominated convergence theorem that
$$
\left( \int_{\Omega_{\delta}} |u_{\delta}(z)-u(z)|^2 dV(z) \right)^{1/2}
\leq \delta \; || \nabla u||_{L^2(\Omega)}.
$$
This ends the proof of Theorem A up to the fact, to be established now, that
$u$ has gradient in $L^2(\Omega)$.
\vskip.2cm
Since $u$ is plurisubharmonic and continuous, $\nabla u \in L_{loc}^2(\Omega)$.
It follows from Lemma 3.1 below that $\nabla u \in L^2(\Omega)$ as soon as
$u$ is bounded from below by a continuous psh function $v$ such that
$v \leq u \text{ in } \Omega$, $v=u=\phi$ on $\partial \Omega$, and $\nabla v \in L^2(\Omega)$.
Our extra assumption in Theorem A precisely yields such a function $v$.
Indeed set $v:=u(\Omega,\phi,0)+b_f$, where $b_f$ is the psh barrier
constructed in the proof of Lemma 2.2: this is a psh function such that
\begin{itemize}
\item $v=\phi+0=u$ on $\partial \Omega$;
\item $(dd^c v)^n \geq (dd^c b_f)^n \geq f \beta_n$ in $\Omega$, thus $v \leq u$ in $\Omega$;
\item $\nabla u(\Omega,\phi,0) \in L^2(\Omega)$
and $\nabla b_f \in L^{\infty}(\Omega)$, hence $\nabla v \in L^2(\Omega)$.
\end{itemize}
It is easy to check that $\nabla u(\Omega,\phi,0) \in L^{\infty}(\Omega) \subset L^2(\Omega)$
when $\phi \in {\mathcal C}^2(\partial \Omega)$.
We refer the reader to [BT 1] for a proof of the more delicate result that this still holds
when $\phi \in {\mathcal C}^{1,1}(\partial \Omega)$.
\begin{lem} Let $u, v \in PSH (\Omega) \cap C^{0} (\overline{\Omega})$ such that
$v \leq u$ on $ \Omega$ and $v = u$ on $\partial \Omega$. Then
$ \int_{\Omega} d u \wedge d^c u \wedge \omega^{n - 1}
\leq \int_{\Omega} d v \wedge d^c v \wedge \omega^{n - 1}.$
\end{lem}
\begin{proof}
First assume that $u = v$ near the boundary $\partial \Omega$.
Then we can approximate by smooth plurisubharmonic functions $u_{\delta}\downarrow u$ and
$v_{\delta} \downarrow v$ with $u_{\delta} \leq v_{\delta}$ on $\Omega_{\delta}$ and
$u_{\delta} = v_{\delta}$
in a neighborhood of $\partial \Omega_{\varepsilon},$ for $0 < \delta < \varepsilon$ and $\varepsilon$ small enough.
Integrating by parts, we get
$$
\int_{\Omega_{\varepsilon}} d v_{\delta} \wedge d^c v_{\delta} \wedge \omega^{n - 1} =
\int_{\partial \Omega_{\varepsilon}} v_{\delta} d^c v_{\delta} \wedge \omega^{n - 1} -
\int_{\Omega_{\varepsilon}} v_{\delta} dd^c v_{\delta} \wedge \omega^{n - 1} .
$$
Since $v_{\delta} = u_{\delta}$ on a neighbourhood of $\partial \Omega_{\varepsilon}$ we conclude that
$v_{\delta} d^c v_{\delta} = u_{\delta} d^c u_{\delta}$ on this neighbourhood and
then using again integration by parts we get
$$
\int_{\Omega_{\varepsilon}} d v_{\delta} \wedge d^c v_{\delta} \wedge \omega^{n - 1} =
\int_{\Omega_{\varepsilon}} d u_{\delta} \wedge d^c u_{\delta} \wedge \omega^{n - 1}
+ \int_{\Omega_{\varepsilon}} (u_{\delta} dd^c u_{\delta} - v_{\delta} dd^c v_{\delta}) \wedge \omega^{n - 1}.
$$
On the other hand, since $v_{\delta} = u_{\delta}$ on a neighbourhood of
$\partial \Omega_{\varepsilon}$ we conclude that
$v_{\delta} \wedge d^c u_{\delta} = u_{\delta} \wedge d^c v_{\delta}$ on this neighbourhood and then
\begin{eqnarray*}
\int_{\Omega_{\varepsilon}} u_{\delta} dd^c v_{\delta} - v_{\delta} dd^c u_{\delta}) \wedge \omega^{n - 1}
& = &\int_{\partial \Omega_{\varepsilon}} (u_{\delta} d^c v_{\delta} - v_{\delta} d^c u_{\delta}) \wedge \omega^{n - 1} \\
& - & \int_{\Omega_{\varepsilon}} (d u_{\delta} \wedge d^c v_{\delta} - d v_{\delta}
\wedge d^c u_{\delta}) \wedge \omega^{n - 1} = 0.
\end{eqnarray*}
Therefore
$$
\int_{\Omega_{\varepsilon}} d v_{\delta} \wedge d^c v_{\delta} \wedge \omega^{n - 1}
= \int_{\partial \Omega_{\varepsilon}} d u_{\delta} \wedge d^c u_{\delta} \wedge \omega^{n - 1}
+ \int_{\Omega_{\varepsilon}} (u_{\delta} - v_{\delta}) (dd^c u_{\delta} + dd^c v_{\delta}) \wedge \omega^{n - 1}.
$$
Since $v_{\delta} \leq u_{\delta}$ on $\Omega_{\delta},$ we obtain
$$
\int_{\Omega_{\varepsilon}} d v_{\delta} \wedge d^c v_{\delta} \wedge \omega^{n - 1}
\geq \int_{\partial \Omega_{\varepsilon}} d u_{\delta} \wedge d^c u_{\delta} \wedge \omega^{n - 1}.
$$
We know by Bedford and Taylor's convergence theorem that
$d v_{\delta} \wedge d^c v_{\delta} \wedge \omega^{n - 1}
\rightarrow d v \wedge d^c v \wedge \omega^{n - 1}.$
Taking the limit when $\delta \searrow 0$ we get
$$
\int_{\overline{\Omega}_{\varepsilon}} d v \wedge d^c v \wedge \omega^{n - 1}
\geq \int_{\Omega_{\varepsilon}} d u \wedge d^c u \wedge \omega^{n - 1}.
$$
Taking the limit when $\varepsilon \downarrow 0$ we get the required inequality.
Now if we only know that $u = v,$ we can define for $\varepsilon > 0$
$u_{\varepsilon} := \sup \{u - \varepsilon , v\}.$ Then $v \leq u_{\varepsilon}$ on $\Omega$ and $u_{\varepsilon} = v$
near the boundary of $\Omega$. Therefore we have for $\delta > 0$ small enough
$$
\int_{\Omega} d v \wedge d^c v \wedge \omega^{n - 1} \geq \int_{\Omega} d u_{\varepsilon}
\wedge d^c v_{\varepsilon} \wedge \omega^{n - 1}.
$$
Now by Bedford and Taylor's convergence theorem, we know that
$d u_{\varepsilon} \wedge d^c u_{\varepsilon} \wedge \omega^{n - 1} \to d u \wedge d^c u \wedge \omega^{n - 1}$
as $\varepsilon \downarrow 0$. Therefore we have
$$
\int_{\Omega} d v \wedge d^c v \wedge \omega^{n - 1} \geq \int_{\Omega}
d u \wedge d^c u \wedge \omega^{n - 1},
$$
which proves the required inequality.
\end{proof}
\section{Laplacian estimates}
This section is devoted to the proof of Theorem B.
We use the same method as above. The finiteness of the
total mass of $\Delta u(\Omega,\phi,0)$
allows a good control (see Lemma 4.2)
on the terms $\hat u_{\delta} - u$,
where
$$
\hat u_{\delta} (z) := \frac{1}{\tau_{n} \delta^{2 n}} \int_{\vert
\zeta - z\vert \leq \delta} u (\zeta) d V (\zeta), z \in \Omega_{\delta},
$$
where $\tau_n$ denotes the volume of the unit ball in $\mathbb{C}^n$.
We shall compare $\hat u_{\delta}$ to $u_{\delta}$ in Lemma 4.1 below.
It follows from the construction of plurisubharmonic H\"older continuous barriers
that the solution $u = u (\Omega,\phi,f)$ is H\"older continuous near the boundary,
i.e. for $\delta > 0$ small enough, we have
\begin{equation}
u (z) - u (\zeta) \leq c_0 \delta^{\alpha},
\end{equation}
for $z, \zeta \in \overline{\Omega}$ with
$\hbox{dist} (z, \partial \Omega) \leq \delta, \hbox{dist} (\zeta, \partial \Omega) \leq \delta $
and $ \vert z - \zeta \vert \leq \delta.$
\vskip.1cm
The link between $u_{\delta}$ and $\hat{u}_{\delta}$ is made by the
following lemma.
\begin{lem} Given $\alpha \in ]0, 1[$, the following two conditions are equivalent.
$(i)$ There exists $\delta_0, A > 0$ such that for any
$0 < \delta \leq \delta_0$,
$$
u_{\delta} - u \leq A \delta^{\alpha} \, \hbox{ on } \, \Omega_{\delta}.
$$
$(ii)$ There exists $\delta_1, B > 0$ such that for any $0 <\delta< \delta_1$,
$$
\hat u_{\delta} - u \leq B \delta^{\alpha} \, \hbox{ on } \Omega_{\delta}.
$$
\end{lem}
\begin{proof}
Observe that $\hat u_{\delta} \leq u_{\delta}$ in $\Omega_{\delta}$, hence
$(i) \Longrightarrow (ii)$ follows immediately.
We now prove that $(ii) \Longrightarrow (i)$.
We need to show that there exists $A,\delta_0>0$ such that
for $0 < \delta \leq \delta_0$,
$$
\omega (\delta) := \sup_{z \in \Omega_{\delta}} [u_{\delta} (z) - u (z)] \leq A \delta^{\alpha}.
$$
Fix $\delta_{\Omega}>0$ small enough so that $\Omega_{\delta} \neq \emptyset$ for
$\delta \leq 3 \delta_{\Omega}$.
Since $u$ is uniformly continuous, for any fixed
$0< \delta <\delta_{\Omega}$,
$$
\nu (\delta) := \sup_{\delta < t \leq \delta_{\Omega}}
\omega (t) t^{- \alpha} < + \infty.
$$
We claim that there exists $\delta_0 > 0$ small enough so that for any
$0 < \delta \leq \delta_0$,
$$
\omega (\delta) \leq A \delta^{\alpha},
\text{ with }
A= (1+4^{\alpha}) c_0+2^{\alpha} 4^n B+\nu(\delta_{\Omega}),
$$
where $c_0$ is the constant arising in inequality (1), while
$B$ is the constant from condition (ii).
Assume this is not the case.
Then there exists $0 < \delta < \delta_{\Omega}$ such that
\begin{equation}
\omega (\delta)> A \delta^{\alpha}.
\end{equation}
Set $\delta:=\sup \{ t <\delta_{\Omega} \, / \, \omega(t)> A t^{\alpha} \}$.
Then
\begin{equation}
\frac{\omega(\delta)}{\delta^{\alpha}} \geq A \geq \frac{\omega(t)}{t^{\alpha}}
\text{ for all } t \in [\delta,\delta_{\Omega}].
\end{equation}
Since $u$ is continuous, we can find
$z_0 \in \overline{\Omega_{\delta}}$, $ \zeta_0 \in \overline{\Omega}$ with
$ \vert z_0 - \zeta_0 \vert \leq \delta$ s.t.
$$
\omega(\delta) = \sup_{z \in \Omega_{\delta}} \left[ \sup_{w \in B(z,\delta)} u(w) -u(z)
\right]=u (\zeta_0) - u (z_0).
$$
\vskip.2cm
We first derive a contradiction if $z_0$
is close enough to the boundary of $\Omega$.
Assume that $\hbox{dist} (z_0,\partial \Omega) \leq 3 \delta$.
Take $z_1 \in \partial \Omega$ such that
$\hbox{dist} (z_0,\partial \Omega) = \hbox{dist} (z_0, z_1) \leq 4 \delta$.
It follows from $(1)$ that
$$
\omega(\delta) = u (\zeta_0) - u (z_0) =
[u (\zeta_0) - u (z_1)] + [u (z_1) - u (z_0)] \leq [1+4^{\alpha}] c_0 \delta^{\alpha}.
$$
This contradicts $(3)$.
Thus we can assume that $\hbox{dist} (z_0,\partial \Omega) > 3 \delta$.
Fix $b>1$ so that $\hbox{dist} (z_0,\partial \Omega) > (2b+1) \delta$.
Thus any $z \in \mathbb{B}(\zeta_0,b\delta)$ satisfies
$z \in \mathbb{B}(z_0,[b+1]\delta)$, hence $z \in \Omega_{b \delta}$.
By using inequality (3) with $t=b\delta$, we get
$u(\zeta_0)-u(z) \leq b^{\alpha} \omega(\delta)$, hence
\begin{equation}
u(z) \geq u(\zeta_0)-b^{\alpha} \omega(\delta),
\, \text{ for all } z \in \mathbb{B}(\zeta_0,b\delta).
\end{equation}
Observe now that $\mathbb{B}(\zeta_0,\delta) \subset \mathbb{B}(z_0,[b+1]\delta)$, hence
\begin{eqnarray*}
\hat u_{(b+1) \delta} (z_0)
& = & \left(\frac{b}{b+1} \right)^{2n} \hat u_{b \delta} (\zeta_0) +
\frac{1}{\tau_n (b+1)^{2n} \delta^{2 n} }
\int_{ \mathbb{B} (z_0, (b+1) \delta) \setminus \mathbb{B} (\zeta_0, b \delta)} u d V \\
& \geq &
\left(\frac{b}{b+1} \right)^{2n} u(\zeta_0) +
\left[(1 - \frac{b^{2n}}{(b+1)^{2 n}}\right]
[u (\zeta_0) - b ^{\alpha} \omega (\delta)] \\
&=& u(\zeta_0) -b^{\alpha} \left[1 - \frac{b^{2n}}{(b+1)^{2 n}}\right] \omega(\delta),
\end{eqnarray*}
where we have used the subharmonicity of $u$ together with inequality (4).
Since $u(\zeta_0)=u(z_0)+\omega(\delta)$, we infer, letting $b \rightarrow 1$,
$$
\hat u_{2 \delta} (z_0) \geq u(z_0)
+4^{-n}\omega(\delta).
$$
We now use assumption (ii), only considering small enough values of $\delta>0$:
since $\hat{u}_{2 \delta}(z_0) \leq u(z_0)+B 2^{\alpha} \delta^{\alpha}$, we get
$$
\omega(\delta) \leq 4^n 2^{\alpha} B \delta^{\alpha} < A \delta^{\alpha}.
$$
This contradicts the definition of $\delta$, hence we have proved that
$(ii) \Rightarrow (i)$.
\end{proof}
It is straightforward to check that if (i) is satisfied, then
$u$ belongs to $Lip_{\alpha}(\overline{\Omega})$.
Thus Theorem B will be proved if we can establish (ii).
It follows from Theorem 1.1 that is suffices to get control
on the $L^1$-average of $\hat{u}_{\delta}-u$.
This is the contents of our next result.
\begin{lem}
For $\delta > 0$ small enough, we have
$$
\int_{\Omega_{\delta}} [\hat u_{\delta} (z) - u (z)] d V_{2 n} (z) \leq c_n
\Vert \Delta u\Vert \delta^2,
$$
where $c_n > 0$ is a uniform constant.
\end{lem}
\begin{proof}
It follows from Jensen's formula that for $z \in \Omega_{\delta}$ and $0 < r < \delta$,
$$
\frac{1}{\sigma_{2 n - 1}} \int_{\vert \xi\vert = 1} u (z + r \xi) d S_{2 n - 1} = u (z) +
\int_0^{r} t^{1 - 2 n} (\int_{\vert \zeta\vert \leq t} dd^c u \wedge \beta_{n - 1}) d t.
$$
Using polar coordinates we get, for $z \in \Omega_{\delta}$,
$$
\hat u_{\delta} (z) - u (z) = \frac{1}{\sigma_{2 n - 1}
\delta^{2 n}} \int_{0}^{\delta} r^{2 n - 1} d r \int_0^{r} t^{1 - 2 n}
(\int_{\vert \zeta - z\vert \leq t} dd^c u \wedge \beta_{n - 1}) d t.
$$
Finally Fubini's theorem yields
\begin{eqnarray*}
\int_{\Omega_{\delta}} (\hat u_{\delta} - u) d V_{2 n} &\leq&
a_n \delta^{- 2 n}\int_{0}^{\delta} r^{2 n - 1} d r
\int_0^{r} t^{1 - 2 n} (\int_{\vert \zeta\vert \leq t} (\int_{\Omega} \Delta u) d t \\
&\leq& c_n \delta^2 \Vert \Delta u\Vert.
\end{eqnarray*}
\end{proof}
This ends the proof of Theorem B since by proposition 2.3,
$\Delta u=\Delta u(\Omega,\phi,f)$ has finite mass in $\Omega$.
\vskip.2cm
We now give a simple example which shows that one can not expect a better
exponent than $\alpha=2/nq$, for $1/p+1/q=1$.
\begin{exa}
Consider $u(z_1,\ldots,z_n):=|z_1|^{\alpha} \cdot ||(z_2,\ldots,z_n)||^2$.
This is a plurisubharmonic function in $\mathbb{C}^n$ which is H\"older-continuous
of exponent $\alpha \in ]0,1[$. We let the reader check that
$$
(dd^c u)^n=f dV, \;
\text{ with } f(z)=\frac{1}{|z_1|^{2-n\alpha}} g(z_2,\ldots z_n),
$$
where $g>0$ is a smooth density.
Given $p>1$, $f$ belongs to $L_{loc}^p(\mathbb{C}^n)$ whenever
$\alpha=\varepsilon+2/nq$, for some $\varepsilon>0$.
This shows that we cannot get a better exponent than
$2/nq$ in Theorems A,B.
\end{exa}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,254 |
\section{Introduction and Summary}
\label{sec:intro}
Eigensystems of Dirac operator in equilibrium gauge backgrounds carry the
information on fermionic aspects of quark--gluon dynamics. As an important
example, inspection of Dirac spectral representation for scalar fermionic
density immediately reveals that spontaneous chiral symmetry breaking (SChSB)
is equivalent to mode condensation of massless Dirac operator. Ever since
this association has been pointed out~\cite{Ban80A}, it became popular
to think about SChSB in terms of quark near--zeromodes.
Contrary to SChSB, the mode condensation property is a well--defined notion
in generic quark--gluon system, i.e. even when all quarks are massive.
While there is no chiral symmetry of physical degrees of freedom to break
in this case, condensing dynamics can still be described via chiral symmetry
considerations. Indeed, one can introduce e.g. a pair of fictitious fermionic
fields (``valence quarks'') of degenerate mass $m_v$, and cancel their contribution
to the action by also adding the associated bosonic partners~\cite{Mor87A}.
This keeps the dynamics of physical quarks and gluons unchanged, but makes it
meaningful to consider chiral rotations of valence fields in such extended system,
and to inquire about ``valence SChSB'' in the $m_v \to 0$ limit. In this
language,
\begin{equation}
\mbox{\bf vSChSB} \quad \Longleftrightarrow \quad \mbox{\bf QMC}
\label{eq:005}
\end{equation}
i.e. quark mode condensation (QMC) in arbitrary quark--gluon system is equivalent
to valence spontaneous chiral symmetry breaking (vSChSB): dynamics supports
condensing modes if and only if it supports valence chiral condensate and valence
Goldstone pions.
It is useful to think about SChSB in the above more general sense, especially
when inquiring about the mechanism underlying the phenomenon~\cite{Ale12D}.
Indeed, the response of massless valence quarks to gauge backgrounds of various
quark--gluon systems provides a relevant point of dynamical distinction for
associated theories: they either support ``broken'' or ``symmetric'' dynamics
of the external massless probe. Moreover, valence SChSB is readily observed
in lattice simulations with physically relevant flavor arrangements, and the
associated dynamical characteristics change smoothly in the light quark
regime~\cite{Ale12D}. Valuable lessons on SChSB can thus be learned by
studying its valence version with massive dynamical quarks: vSChSB becomes
SChSB as dynamical chiral limit is approached.
Unfortunately, the equivalence of quark mode condensation and valence SChSB
does not provide
window into specifics of broken quark dynamics. Indeed, the mode condensation
property is merely a restatement of symmetry breakdown condition in Dirac
spectral representation. However, it was recently proposed that another
relation may hold, possibly with similar scope of validity, but with
non--trivial dynamical connection to inner workings of the breaking
phenomenon~\cite{Ale12D}. In particular, it was suggested that
\begin{equation}
\mbox{\bf vSChSB} \quad \Longleftrightarrow \quad \mbox{\bf DChC}
\label{eq:015}
\end{equation}
i.e. that valence SChSB is equivalent to {\em dynamical chirality condensation}
(DChC). This offers an intuitively appealing notion that the vacuum effect of
chiral symmetry breaking is in fact the phenomenon of chirality condensation.
In light of Eq.~(1), the above relation carries the same information
as QMC--DChC equivalence, which may be preferable for explicit checks.
While entities involved in the above relations will all be defined in
Sec.~\ref{sec:background}, it should be pointed out now that DChC
relates to {\em dynamical} notion of local chirality in modes~\cite{Ale10A}:
it expresses the tendency for asymmetry in magnitudes of left--right components
(local chiral polarization), measured with respect to the baseline of statistical
independence. The associated quantifier, the correlation coefficient of polarization
$C_A \in [-1,1]$, is invariant with respect to the choice of parametrization for
the asymmetry. It thus provides the information on the quark--gluon system that
is inherently dynamical. DChC occurs when near--zeromodes are chirally
polarized ($C_A>0$) and sufficiently abundant, namely when {\em chiral polarization
density} $\rho_{ch}(\lambda) \equiv \rho(\lambda) \,C_A(\lambda)$ is positive
at $\lambda=0$ in infinite volume.
Important aspect of DChC is that it manifests itself in chiral spectral properties
away from strictly infrared limit. Indeed, it was shown~\cite{Ale12D,Ale10A} that
mode--condensing theories of physical interest exhibit {\em chiral polarization scale}
$\chps$, marking the spectral point where functions
$C_A(\lambda)$, $\rho_{ch}(\lambda)$ change sign and modes become anti--polarized
(Fig.~\ref{fig:illus}(left)). The existence of $\chps$
in chirally broken asymptotically free theories can be intuitively understood from
the fact that free modes are strictly anti--chiral and $C_A(\lambda)$ is thus
expected to assume negative values at sufficiently high $\lambda$.
The simultaneous occurrence of DChC and $\chps$ is thus not viewed as
accidental but rather generic. This is, in fact, an integral part of the vSChSB--DChC
conjecture as formulated in Ref.~\cite{Ale12D}, so that
\begin{equation}
\mbox{\bf vSChSB} \quad \Longleftrightarrow \quad
\mathbf{\chps > 0}
\label{eq:025}
\end{equation}
with suitably general definition of $\chps$. Chiral polarization scale can
thus be viewed as a non--standard ``order parameter'' of the breaking phenomenon,
and is expected to be naturally tied to the mechanism of SChSB~\cite{Ale12D}.
\begin{figure}[t]
\begin{center}
\centerline{
\hskip 0.00in
\includegraphics[width=18.0truecm,angle=0]{f_illus_0.pdf}
}
\vskip -0.10in
\caption{Layer of chirally polarized modes around $\lambda=0$
is visible e.g. in the low--energy behavior of $\rho_{ch}(\lambda)$. Standard
situation (left) and two singular ones (middle, right) are shown in infinite
volume. The dashed segment indicates that infinite volume limit
was approached from chirally polarized side. See Sec.~\ref{ssec:chidefs} for
definition of $\Lambda_{ch}^\infty$.}
\label{fig:illus}
\vskip -0.45in
\end{center}
\end{figure}
Relations (\ref{eq:015},\ref{eq:025}) acquire their full meaning only when
the corresponding range of theories is specified. This is relevant since the larger
the scope of theories conforming to vSChSB--DChC correspondence, the deeper the connection
of local chirality to the symmetry breaking phenomenon. Indeed, if the association
is generic, then it must be ascribed to the very nature of quark--gluon interaction.
The original vSChSB--DChC conjecture was formulated in the context of SU(3) gauge theories
with arbitrary number of fundamental quark flavors of arbitrary masses, and at arbitrary
temperature. Thus both broken and symmetric theories are included in this landscape
with corresponding transitions providing for most interesting tests of the conjecture.
While the maximal range of validity may be significantly
larger,\footnote{For example, extension to general SU(N) gauge groups may hold as well.}
this is a physically relevant setup currently associated with the above statements.
\medskip
\noindent The focus of this work involves two main aspects.
\medskip
{\em (I)} In Sec.~\ref{sec:background} we provide more complete description of chiral
polarization phenomenon, and formulate the proposed connections to vSChSB within such
wider context. Refined description reflects the premise that the vacuum feature we
are associating with vSChSB is the layer of chirally polarized modes around
$\lambda\!=\!0$ (``surface of the Dirac sea''). Apart from its width ($\Lambda_{ch}$),
global characterization of this structure also includes volume density of modes
involved ($\Omega$), and the volume density of total chirality generated by
them ($\Omega_{ch}$).
Incorporating these characteristics allows us to distinguish and discuss various
special/singular cases that might arise. For example, the layer could
approach zero width in the infinite volume limit, but acquire sufficiently singular
abundance of modes so as to generate positive $\Omega_{ch}$ and $\Omega$
(see Fig.~\ref{fig:illus}(middle)). Another possibility is that polarization
density asymptotically vanishes on the layer, due to modes being concentrated
on a subset of space--time with measure zero, but $\Omega$ remains positive in
the infinite volume limit. This is schematically shown in
Fig.~\ref{fig:illus}(right). Even if behaviors of the above type won't appear
in target continuum theories, they are likely to show up at low lattice cutoffs.
Working within a framework capturing such cases is beneficial to make the proposed
connections as universal as possible.
\medskip
\noindent Related to the above is a suggestion, described in Sec.~\ref{ssec:finvol},
whose conceptual content goes beyond that in Ref.~\cite{Ale12D}. To convey this,
note that our discussion implicitly proceeded in infinite volume where the notions
of condensate and symmetry breaking are defined. For example,
the generic form of relation \eqref{eq:025} in the extended framework
(see {\sl Conjecture 2''}) is
\begin{equation}
\mbox{\bf vSChSB} \quad \Longleftrightarrow \quad
\mathbf{\Omega > 0}
\tag{{\ref{eq:025}}'}
\label{eq:035}
\end{equation}
implying that appearance of chirally polarized layer around the surface of Dirac
sea is considered to be physically significant if
$\Omega \equiv \lim_{V\to\infty} \Omega(V) > 0$. Theories without polarized low
modes in large finite volumes ($\Omega(V)$ identically zero) are then predicted
to be unbroken, but theories with polarized low modes ($\Omega(V)>0$) could also be
unbroken if $\lim_{V\to\infty} \Omega(V)\!=\!0$. Consistently with available data, we
propose in {\sl Conjecture 3} that the latter possibility does not occur:
if large--volume dynamics generates chiral polarization in low end of the spectrum,
then valence chiral condensate appears in the infinite volume limit and vice versa.
This provides for the closest possible relationship between vSChSB and chiral
polarization (ChP), and it is in this sense that we refer to $\Lambda_{ch}$,
$\Omega_{ch}$ and $\Omega$ as {\em finite--volume ``order parameters''}.\footnote{As
discussed in Sec.~\ref{ssec:finvol}, $\Lambda_{ch}$, $\Omega_{ch}$ and $\Omega$ are
zero/non--zero simultaneously in finite volume: they effectively represent a single
finite--volume order parameter.}
In this context, the correspondence analogous to \eqref{eq:015} is expressed as
\begin{equation}
\mbox{\bf vSChSB} \quad \Longleftrightarrow \quad \mbox{\bf ChP} \;\,
\mbox{\rm in large finite volumes}
\label{eq:045}
\tag{{\ref{eq:015}}'}
\end{equation}
and the right-hand side of relation \eqref{eq:035} modifies to:
$\,\Omega(V)>0$ for $V_0<V<\infty$.
\medskip
{\em (II)} We present new lattice data supporting the above ideas. It is known
that, at zero temperature, N$_f$=0 theory~\cite{Ale12D,Ale10A} as well as
N$_f$=2+1 theory at physical point~\cite{Ale12D} exhibit chiral polarization,
in accordance with the presence of vSChSB, and thus with the proposed equivalence.
There is also an initial evidence that subjecting
N$_f$=0 QCD to thermal agitation, chiral polarization and vSChSB cease to exist
at common temperature T$_{ch}$ \cite{Ale12D}. When contemplating the validity of
the vSChSB--ChP relationship over the vast theory landscape considered, it is useful
to think of N$_f$=0, T=0 theory as a reference point~\cite{Ale12D}. Indeed, this
dynamics produces maximal breaking of valence chiral symmetry, with thermal effects
and the effects of light dynamical quarks providing two possible routes to symmetry
restoration. Thus, in pilot investigations, it is natural to examine these two
deformations of quenched theory independently of one another,
both to ascertain conjecture's validity in such instances, as well as to learn
about specific features associated with the two qualitatively different
effects. This work is the first step in that direction with finite--temperature
aspect examined in Sec.~\ref{sec:fintemp}, and many--flavor dynamics
in Sec.~\ref{sec:light}. All of our results are consistent with the vSChSB--ChP
equivalence.
\medskip
Few noteworthy byproducts of our main inquiry are also discussed here.
{\em (i)} Performing a volume analysis at the lattice
cutoff studied, we show the existence of the (lattice) phase $T_c < T < T_{ch}$
in N$_f$=0, simultaneously exhibiting vSChSB and
deconfinement~\cite{Edw99A,Ale12A,Ale12B}. This chiral polarization dynamics
appears to be of the singular type
$(\Lambda_{ch}\!=\!0,\, \Omega_{ch}\!>\!0,\, \Omega\!>\!0)$.
{\em (ii)} One of the characteristic
features of the above ``mixed phase'' is the appearance of very inhomogeneous
near--zeromodes, well distinguished from the bulk of the spectrum. We find
an intermediate phase (in quark mass) with such properties also in N$_f$=12.
The relevance of these phases for continuum physics remains an open issue in both
cases. {\em (iii)} Elementary analysis of the detailed chiral polarization
characteristic, namely absolute \Xg--distribution, is also performed. Among
other things, our data in N$_f$=0 indicate that deconfinement is characterized by
the appearance of distributions with indefinite convexity.
\section{The Background and the Conjectures}
\label{sec:background}
We start by reviewing the relevant background, and formulating the connections we
aim to test. Our discussion will be fairly detailed and self--contained, in part
to sufficiently extend pertinent parts of Letter~\cite{Ale12D} which were necessarily
rather brief. In addition, we build an extended formalism for description of chiral
polarization, which allows us to include special/singular behaviors and has some
benefits for continuum--limit considerations.
\subsection{Valence Chiral Symmetry and Mode Condensation}
\label{ssec:qmc}
Implicitly assumed in what follows is the setup involving SU(3) gluons interacting
with N$_f$ fundamental quarks of masses $M=(m_1,m_2,\ldots,m_{N_f})$.
To formally include valence chiral symmetry considerations, the system is
augmented by a pair of degenerate valence quarks of mass $m_v$, and a pair of
complex commuting fields (pseudofermions) compensating for the dynamical effect
of these fictitious particles~\cite{Mor87A}. This schematically corresponds to the action,
\begin{equation}
S \,=\, S_g \,+\,
\sum_{f=1}^{N_f} \psibar_f \Bigl( D_{(1)} + m_f \Bigr) \psi_f \,+\,
\sum_{i=1}^2 \bar{\eta}_i \Bigl( D_{(2)} + m_v \Bigr) \eta_i \,+\,
\sum_{i=1}^2 \phi^{\dagger}_i \Bigl( D_{(2)} + m_v \Bigr) \phi_i
\label{eq:2.015}
\end{equation}
where $S_g$ is the pure glue contribution.
When viewing the above as expression in the continuum, then $D_{(1)}=D_{(2)}=D$,
namely the continuum Dirac operator. However, on the lattice it is possible, and
sometimes desirable, to consider different discretizations for dynamical and
valence quarks. In particular, the role of $D_{(2)}$ in our case is to describe
the response of physical vacuum to external chiral probe. It is thus desirable
that it provides for exact lattice chiral symmetry, even though some numerically
cheaper $D_{(1)}$ could have been used to simulate physical quarks
and to define the theory.
Flavored chiral rotations of valence quark fields in the above extended system
become the symmetry of the action in the $m_v \to 0$ limit, and one can
meaningfully ask whether this symmetry is broken by the vacuum. If so, we speak
of {\em valence chiral symmetry breaking} (vSChSB). It has the usual consequence
of being associated with the triplet of massless valence pions. While these are
not physical states, they express the ability of the physical vacuum
to support a specific type of long range order: the same kind of order that is
required for physical chiral symmetry to be broken in the dynamical massless
limit. vSChSB is thus a relevant vacuum characteristic of QCD--like theories.
Following the steps involved in derivation of the standard Banks--Casher
relation~\cite{Ban80A}, one can inspect that the valence chiral condensate
in theory (\ref{eq:2.015}) is given by
\begin{equation}
\Sigma(M) \, \equiv \,
\lim_{m_v\to 0} \, \lim_{V \to \infty}
\langle\, \bar{\eta} \eta \, \rangle_{M,m_v,V} \;=\;
\pi \lim_{\epsilon \to 0^+} \, \frac{1}{\epsilon} \,
\lim_{V \to \infty} \sigma_{(2)}(\epsilon,M,V)
\label{eq:2.025}
\end{equation}
where $\eta$ is one (arbitrary) of the valence flavors,
$\sigma_{(2)}=\sigma_{(2)}(\lambda,M,V)$ the cumulative eigenmode density of
$D_{(2)}$ and $V$ the 4--volume.
In the continuum, $\sigma_{(2)}\to \sigma$ is the cumulative eigenmode density of
continuum Dirac operator. On the lattice, we implicitly assume that $D_{(2)}$
is the overlap Dirac operator~\cite{Neu98BA} which is our discretization of choice
in this study, and for which the relation such as (\ref{eq:2.025}) can be
straightforwardly derived~\cite{Cha98A}.
The notion of {\em cumulative eigenmode density}, used above, is defined as
\begin{equation}
\sigma(\lambda,M,V) \equiv \frac{1}{V} \,
\langle \, \sum_{0 \le \lambda_k < \lambda} \,1\;\rangle_{M,V}
\label{eq:2.030}
\end{equation}
where $\lambda$ (real number) represents the imaginary part of Dirac eigenvalue.
In case of overlap Dirac operator one can also take it to be the magnitude of
the eigenvalue multiplied by the sign of the imaginary part. $\lambda_k$ are
the values associated with given gauge background and ordered appropriately.
Note that $\sigma(\lambda,M,V)\equiv 0$ for $\lambda \le 0$. The differential
version of $\sigma$, referred to as {\em eigenmode density}, is generally
available and frequently useful, namely
\begin{equation}
\rho(\lambda,M,V) \equiv \lim_{\epsilon \to 0^+} \,
\frac{\sigma(\lambda+\epsilon,M,V) - \sigma(\lambda,M,V)}{\epsilon}
\,=\, \frac{\partial}{\partial_+ \lambda} \, \sigma(\lambda,M,V)
\label{eq:2.035}
\end{equation}
Note that we chose to distinguish the above eigenmode density from the usual
\begin{equation}
\bar{\rho}(\lambda,M,V) \equiv \frac{1}{V} \,
\sum\limits_k \langle \, \delta(\lambda - \lambda_k) \,\rangle_{M,V}
\label{eq:2.040}
\end{equation}
which, in some singular cases, has to be represented by a generalized function
with at most countably many ``atoms'' ($\delta$--functions), while $\rho$ is
always an ordinary function which simply takes the value ``$\infty$''
at the position of the atoms.
The infinite volume limit of $\rho(\lambda,M,V)$ will be defined as
\begin{equation}
\rho(\lambda,M) \equiv \frac{\partial}{\partial_+ \lambda}
\lim_{V \to \infty} \sigma(\lambda,M,V)
\label{eq:2.045}
\end{equation}
rather than as point--wise limit of $\rho(\lambda,M,V)$, but the two can only
differ in certain singular points of the spectrum. The theory is said to exhibit
{\em quark mode condensation} (QMC) if
\begin{equation}
\lim_{\epsilon \to 0^+} \, \frac{1}{\epsilon} \,
\lim_{V \to \infty} \,\sigma(\epsilon,M,V)
\;=\; \rho(0,M) > 0
\label{eq:2.050}
\end{equation}
i.e. when the abundance of ``infinitely infrared'' modes scales as the total
number of modes, namely with space--time volume. Relation \eqref{eq:2.025} then
implies
\begin{equation}
\Sigma(M) > 0 \quad \Longleftrightarrow \quad \rho(0,M) > 0
\tag{{\ref{eq:005}}'}
\label{eq:2.055}
\end{equation}
which is an explicit representation of equivalence (\ref{eq:005}) between vSChSB
and QMC.
\medskip
\noindent Few remarks regarding the above should now be made.
\medskip
\noindent {\em (i)} The discussion has been carried out at a rather general
level mainly to accommodate the possibility of most singular behavior at the origin
in the infinite volume limit. If $\lambda=0$ is the only ``atom'' in that situation,
then $\bar{\rho}(\lambda,M) = C(M) \delta(\lambda) + \hat{\rho}(\lambda,M)$, with
$\hat{\rho}$ being an ordinary function, and
$C(M) = \lim_{\epsilon\to 0} \int_0^\epsilon \bar{\rho}(\lambda,M)\, d\lambda$.
We then have
\begin{equation}
\rho(0,M) \; = \;
\lim_{\epsilon\to 0^+} \frac{C(M)}{\epsilon}
\;+\; \lim_{\lambda \to 0^+} \rho(\lambda,M)
\label{eq:2.060}
\end{equation}
since $\lim_{\lambda \to 0^+} \hat{\rho}(\lambda,M) =
\lim_{\lambda \to 0^+} \rho(\lambda,M)$. Therefore, in addition to the usual
second term, mode condensate acquires an infinite value if there is an atom
of spectral mode density at the origin.\footnote{Note that $\rho(0,M)$ can in
principle be infinite even when $C(M)=0$ since $\hat{\rho}(\lambda)$ could still
have an ordinary integrable divergence at $\lambda=0$.} This is in accordance
with diverging chiral condensate in such instance, namely
\begin{equation}
\Sigma(M) \; = \;
\lim_{m_v \to 0} \, \frac{2 C(M)}{m_v}
\;+\; \pi \lim_{\lambda \to 0^+} \rho(\lambda,M)
\label{eq:2.065}
\end{equation}
We emphasize that the singular accumulation of modes discussed above is not due
to exact zero--modes of finite volumes: their contribution is well--known to
vanish in the infinite volume limit. Rather, $C(M)>0$ would have to be generated
by modes that are ``squeezed'' into near--zeromodes at arbitrary finite volume
but become ``infinitely infrared'' as volume is taken to infinity. It is not known
whether there are continuum theories generating such dynamics, but we will discuss
the behavior that might be of this type at finite cutoff.
\medskip
\noindent {\em (ii)} It should be noted that, at finite cutoff, it is in principle
possible to obtain contradictory answers on vSChSB (QMC) with different choices of
$D_{(2)}$. However, such discrepancies, if any, are expected to disappear
sufficiently close to the continuum limit.
\medskip
\noindent {\em (iii)} In the above considerations we assumed $T=0$ for simplicity.
Incorporating theories on 3--volume $V_3$ at finite temperature $T$ is
straightforward and simply involves replacing $V \rightarrow (T,V_3)$ in labels,
and $V \rightarrow V_3/T$ in volume factors.
\subsection{Spectral Measures of Dynamical Chirality}
\label{ssec:chidefs}
The term {\em ``dynamical local chirality''} refers to characterization of local
asymmetry in left--right values of Dirac eigenmodes using absolute polarization
method of Ref.~\cite{Ale10A}. Dynamical nature of this approach mainly stems
from the fact that it quantifies polarization relative to the population of
statistically independent left--right components. The basic characteristic
is the correlation coefficient of polarization $\cop_A \in [-1,1]$,
assigned to a given eigenmode $\psi_\lambda$. It is linearly related to
the probability that the local value of $\psi_\lambda$ is more polarized
than value chosen from associated distribution of statistically independent
left--right components. In case of correlation ($\cop_A>0$), dynamics enhances
polarization and the mode is referred to as chirally polarized, while
anti--correlation ($\cop_A<0$) indicates that dynamics suppresses polarization
and the mode is chirally anti--polarized. Such comparison of polarization can
also be performed in a detailed differential manner, resulting in absolute
$\Xg$--distribution $\xd_A(\Xg)$, with $\Xg \in [-1,1]$. A concise
introduction to these concepts together with precise definitions can be found
in Appendix~\ref{app:chirality}.
Spectral characteristics of a given theory based on dynamical chirality measures
$\cop_A$ and $\xd_A(\Xg)$ can be defined as follows~\cite{Ale12D}. The cumulative
dynamical chirality per unit volume ({\em cumulative chiral polarization density}),
is given by
\begin{equation}
\sigma_{ch}(\lambda,M,V) \equiv \frac{1}{V} \,
\langle \, \sum_{0 \le \lambda_k < \lambda} \,\cop_{A,k}\;\rangle_{M,V}
\label{eq:4.005}
\end{equation}
where $\cop_{A,k}$ is chiral polarization (correlation) in the k--th mode. The associated
differential contribution due to modes at scale $\lambda$, namely
\begin{equation}
\rho_{ch}(\lambda,M,V) \,\equiv\, \lim_{\epsilon \to 0^+} \,
\frac{\sigma_{ch}(\lambda+\epsilon,M,V) - \sigma_{ch}(\lambda,M,V)}{\epsilon}
\;=\; \frac{\partial}{\partial_+ \lambda} \, \sigma_{ch}(\lambda,M,V)
\label{eq:4.015}
\end{equation}
is referred to as {\em chiral polarization density} as is its formal companion
\begin{equation}
\bar{\rho}_{ch}(\lambda,M,V) \equiv \frac{1}{V} \,
\sum\limits_k \langle \, \delta(\lambda - \lambda_k) \, \cop_{A,k} \,\rangle_{M,V}
\label{eq:4.025}
\end{equation}
which has to be represented by a generalized function in certain singular cases.
The average chiral polarization at scale $\lambda$ is given by
\begin{equation}
\cop_A(\lambda,M,V) \,\equiv\, \lim_{\epsilon \to 0^+}
\frac{\sigma_{ch}(\lambda+\epsilon,M,V) - \sigma_{ch}(\lambda,M,V)}
{\sigma(\lambda+\epsilon,M,V) - \sigma(\lambda,M,V)}
\;=\;
\frac{\rho_{ch}(\lambda,M,V)}{\rho(\lambda,M,V)}
\label{eq:4.035}
\end{equation}
where the second equation only applies when $0 < \rho(\lambda,M,V) < \infty$. We
similarly define the average absolute $\Xg$--distribution at scale $\lambda$ which,
written without shorthands, reads
\begin{equation}
\xd_A(\Xg,\lambda,M,V) \,\equiv\, \lim_{\epsilon \to 0^+}
\frac{\langle \; \sum\limits_{\lambda \le \lambda_k < \lambda+\epsilon}
\xd_{A,k}(\Xg) \; \rangle_{M,V}}
{\langle \; \sum\limits_{\lambda \le \lambda_k < \lambda+\epsilon}
1 \; \rangle_{M,V}}
\label{eq:4.045}
\end{equation}
where $\xd_{A,k}(\Xg)$ is the absolute $\Xg$--distribution of the eigenmode
associated with $\lambda_k$.
The infinite volume limit of $\rho_{ch}(\lambda,M,V)$ is defined to be
\begin{equation}
\rho_{ch}(\lambda,M) \equiv \frac{\partial}{\partial_+ \lambda} \,
\lim_{V \to \infty} \sigma_{ch}(\lambda,M,V)
\label{eq:4.055}
\end{equation}
We say that the theory exhibits {\em dynamical chirality condensation}~\cite{Ale12D} if
\begin{equation}
\lim_{\epsilon \to 0^+} \, \frac{1}{\epsilon} \,
\lim_{V \to \infty} \,\sigma_{ch}(\epsilon,M,V)
\;=\; \rho_{ch}(0,M) > 0
\label{eq:4.065}
\end{equation}
i.e. if its ``infinitely infrared'' Dirac modes are chirally polarized and their
contribution to dynamical chirality scales with space--time volume. Similarly,
dynamical anti--chirality condensation occurs when $\rho_{ch}(0,M)<0$. Note that
(anti-)chirality condensation implies mode condensation but not vice--versa.
Since $\rho_{ch}(\lambda)$ is a real--valued function of indefinite sign, we can assign
{\em chiral polarization scale} $\chps \ge 0$ to it as the largest $\Lambda$ such
that $\rho_{ch}(\lambda)>0$ on $[0,\Lambda)$ except for isolated zeros.
The provision for ``isolated zeros'' has two rationales. First, even with such zeros
present, $\chps$ retains its intended meaning as a spectral range of dynamical
chirality around the surface of Dirac sea. Second, it ensures that defining chiral
polarization scale via $\cop_A(\lambda)$ leads to the same scale in finite volume.
Indeed, if $\Lambda'_{ch} \ge 0$ is the largest $\Lambda$ such that
$\cop_A(\lambda)>0$ on $[0,\Lambda)$ except for isolated zeros, then
$\chps = \Lambda'_{ch}$ in finite volume.\footnote{What is relevant here
is that a zero of $\cop_A$ is also a zero of $\rho_{ch}$ but not necessarily
vice-versa.} Note that when positive $\chps$ doesn't exist, the above definition is
vacuously true for $\chps=0$, which is then the assigned chiral polarization
scale. Also, $\chps = \infty$ is associated with $\rho_{ch}(\lambda)$ that is positive
on $[0,\infty)$ except for possible isolated zeros. Given $\chps=\chps(M,V)$,
the {\em total dynamical chirality} of Dirac spectrum at low energy
$\Omega_{ch}=\Omega_{ch}(M,V)$ is
\begin{equation}
\Omega_{ch} \,\equiv\,
\max \,\{\, \sigma_{ch} (\chps),\;
\lim_{\epsilon \to 0^+} \sigma_{ch}(\chps+\epsilon) \,\}
\label{eq:4.060}
\end{equation}
where the possibility of discontinuity at $\chps$ has been taken into account.
The associated {\em total number of chirally polarized modes} $\Omega=\Omega(M,V)$ is
\begin{equation}
\Omega \,\equiv\, \begin{cases}
\sigma(\chps) &
\text{if $\Omega_{ch}=\sigma_{ch} (\chps)$}\\
\lim\limits_{\epsilon \to 0^+} \sigma(\chps+\epsilon) &
\text{if $\Omega_{ch}=\lim\limits_{\epsilon \to 0^+} \sigma_{ch}(\chps+\epsilon)$}
\end{cases}
\label{eq:4.062}
\end{equation}
Note that the characteristics $\Omega \ge 0$, $\Omega_{ch} \ge 0$ are volume densities
while the condensates $\rho(0)$, $\rho_{ch}(0)$ are both volume and spectral densities.
\begin{figure}[tbp]
\begin{center}
\centerline{
\hskip -0.20in
\includegraphics[width=14.5truecm,angle=0]{f_illus_1.pdf}
}
\vskip -0.60in
\caption{Examples of assigning $\Lambda_{ch}$ and $\Omega_{ch}$ to
$\sigma_{ch}(\lambda)$. The associated $\rho_{ch}(\lambda)$ is shown in
the lower figure for each case. For theory in {\em infinite volume},
{\sl Conjecture 2'} identifies the first/second pair of rows as options
that can only occur in broken/symmetric vacuum, while the behavior
in the third pair of rows is predicted to be impossible for theories
in ${\mathscr T}$. The possibilities in each category are not meant
to be exhaustive or guaranteed to occur.}
\label{fig:sigch_vs_lam_illus}
\end{center}
\end{figure}
\smallskip
Infinite volume limits of the above objects require some attention since the order
of operations might be relevant in certain cases. In discussion of the condensation
phenomena, we emphasized the primary role of cumulative densities, with
their infinite volume limits
\begin{equation}
\sigma(\lambda,M)\equiv \lim_{V\to \infty} \sigma(\lambda,M,V) \qquad\quad
\sigma_{ch}(\lambda,M)\equiv \lim_{V\to \infty} \sigma_{ch}(\lambda,M,V)
\label{eq:4.064}
\end{equation}
being the basis for computation of the condensates. However, one virtue of chiral
polarization framework is that such insistence on the order of operations
ceases to be crucial. In fact, our default view of infinite volume limits
for $\chps$, $\Omega$ and $\Omega_{ch}$ is just the direct limit of their
finite--volume versions, rather than corresponding functionals of
$\sigma_{ch}(\lambda,M)$ and $\sigma(\lambda,M)$. Thus, while we explicitly
distinguish the two options, for example
\begin{equation}
\Omega_{ch}(M) \,\equiv\, \lim_{V\to \infty} \Omega_{ch}(M,V)
\qquad\quad
\Omega_{ch}^\infty(M) \,\equiv\, \Omega_{ch}[\sigma_{ch}(\lambda,M)]
\label{eq:4.070}
\end{equation}
it is the first definition that is implicitly understood if not stated otherwise.
We use analogous notational convention also in case of $\chps$ and $\Omega$.
\vfill\eject
\noindent Few remarks regarding these concepts should be made.
\medskip
\noindent {\em (i)} The above definitions of $\chps$ and $\Omega_{ch}$ assume
certain analytic properties of $\sigma_{ch}(\lambda)$, such as existence
of $\rho_{ch}(\lambda)$ or the limit in Eq.~\eqref{eq:4.060}. In case of
cumulative mode density $\sigma(\lambda)$, such properties are, for most
part, inherent in its definition, i.e. it is continuous except possibly at
countably many finite jumps, and differentiable almost everywhere. While these
properties are expected to hold also for $\sigma_{ch}(\lambda)$ in any theory,
it is comforting that $\chps$ and $\Omega_{ch}$ can be assigned to
$\sigma_{ch}(\lambda)$ that is completely generic. The corresponding definition
and related considerations are discussed in Appendix~\ref{app:spectral}.
\medskip
\noindent {\em (ii)} In terms of $\sigma_{ch}(\lambda)$, chiral polarization
scale corresponds to the largest $\Lambda$ such that $\sigma_{ch}(\lambda)$ is
strictly increasing on $[0,\Lambda]$, and $\Omega_{ch}$ its associated maximal
value. Fig.~\ref{fig:sigch_vs_lam_illus} shows various behaviors of
$\sigma_{ch}(\lambda)$, illustrating how $\chps$ and $\Omega_{ch}$ are assigned
via above definitions. For the current purpose, one should only view
$\sigma_{ch}(\lambda)$, $\rho_{ch}(\lambda)$ shown as admissible pairs of
functions: many of these situations are not expected to occur in theories
of interest.
\medskip
\noindent {\em (iii)} In the above considerations we assumed $T=0$ for
notational simplicity. Extension of all definitions to finite temperatures
is straightforward (see remark {\em (iii)} of Sec.~\ref{ssec:qmc}).
\subsection{Conjecture Formulations}
\label{ssec:conjectures}
We will consider and extend {\em Conjecture 2} of Ref.~\cite{Ale12D}, which ties
the phenomenon of valence spontaneous chiral symmetry breaking to that of
dynamical local chirality in low--lying modes. Such vSChSB--DChC correspondence
was proposed to hold over the set ${\mathscr T}$ containing SU(3) gauge theories
in 3+1 space--time dimensions with any number ($N_f$) of fundamental quarks of
arbitrary masses $M\equiv (m_1,m_2,\ldots,m_{N_f})$, and at arbitrary
temperature $T$. Different continuum theories are thus labeled by $(M,T)$.
Statements of this section are formulated in infinite volume (infinite extent of
all spatial dimensions) so that spontaneous symmetry breaking and the condensation
concepts have definite meaning. To begin with, we formulate {\em Conjecture 2}
using more precise language and concepts developed here.
By ``chiral polarization characteristics'' we mean the parameters $\Lambda_{ch}$,
$\Lambda_{ch}^\infty$, $\Omega_{ch}$, $\Omega_{ch}^\infty$, $\Omega$,
$\Omega^\infty$.
\medskip
\noindent {\bf Conjecture 2}
\smallskip
\noindent {\sl The following holds in every lattice--regularized $(M,T) \in {\mathscr T}$
at sufficiently large ultraviolet cutoffs $\Lambda_{lat}$ and infinite volume.}
\begin{description}
\item[{\sl (I)}] {\sl Chiral polarization characteristics exist and are zero or
non--zero simultaneously. Moreover, $\chps = \Lambda_{ch}^\infty$ and
$0 \le \chps \ll \Lambda_{lat}$.
If $\Lambda_{ch}>0$ then $\rho_{ch}(\lambda)$ is positive on $[0,\Lambda_{ch})$.}
\item[{\sl(II)}] {\sl The property of valence spontaneous chiral symmetry breaking
is characterized by}
\begin{equation}
\Sigma > 0 \quad \Longleftrightarrow \quad
\rho(\lambda=0) > 0 \quad \Longleftrightarrow \quad \Lambda_{ch} > 0
\label{eq:5.020}
\end{equation}
\item[{\sl(III)}] {\sl $\rho_{ch}(\lambda)$ is non--positive for $\lambda > \Lambda_{ch}$,
except possibly in the vicinity of $\Lambda_{lat}$.}
\end{description}
\vfill\eject
\noindent The statement is thus split into three parts which we now discuss.
\bigskip
\noindent {\sl (I)} $\Lambda_{ch}\!=\!\Lambda_{ch}^\infty$ implies that
$\Omega_{ch}=\Omega_{ch}^\infty$ and $\Omega=\Omega^\infty$. Neither this pairwise
equality nor the simultaneous positivity of all six parameters follows from definitions
alone. Rather, they represent anticipated constraints on actual dynamical behavior in
theories under consideration close to the continuum. Note that the statement rules
out the possibility of chiral polarization over the whole Dirac spectrum. Also, while
$\Lambda_{ch}>0$ implies positivity of $\rho_{ch}(\lambda)$ on $[0,\Lambda_{ch})$ only
up to isolated points, such points are not expected to be present in infinite volume,
as explicitly stated.
\medskip
\noindent {\sl (II)} Relation (\ref{eq:5.020}) asserts that chiral polarization
scale $\Lambda_{ch}$ is an unconventional order parameter of vSChSB on ${\mathscr T}$:
chiral symmetry is broken if and only if this dynamical scale is generated.
Note that from {\sl (I)} and {\sl (II)} it follows that
$\rho(0) > 0 \, \Leftrightarrow \, \rho_{ch}(0) > 0$, which is the mathematical
representation of relation (\ref{eq:015}): quark mode condensation (hence vSChSB)
is equivalent to dynamical chirality condensation. Anti--chirality doesn't
condense~\cite{Ale12D}.
\medskip
\noindent {\sl (III)} This part reflects the expectation that dynamical chirality can
only occur in the low end of the Dirac spectrum, characterized by $\chps$. In lattice
theory, chiral polarization could exist in the vicinity of the cutoff (due to lattice
artifacts) but will scale out in the continuum limit. While generic anti--chirality
in the ultraviolet is supported by asymptotic freedom, the assertion
that chiral polarization cannot dominate at intermediate scales, i.e. in a spectral band
separated from origin, is not easy to verify directly. Indeed, reaching such
intermediate parts of Dirac spectra via numerical lattice QCD sufficiently close to
the continuum limit can be computationally demanding. Nevertheless, at least at zero
temperature, there is little doubt that the above scenario holds since running of the gauge
coupling, well understood, is monotonic across scales. The situation at finite temperature
is more involved though. The effects of asymmetry between magnetic and electric
couplings~\cite{Com97A} and the influence of infrared fixed point in dimensionally
reduced (3-d) theory~\cite{Bra05A} could improve prospects for more complicated behavior
at sufficiently high temperatures. However, the available data is not hinting
the existence of intermediate--scale chirality, as reflected in the statement.
\medskip
The meaning of {\sl Conjecture 2} is directly tied to the lattice definition
of the theory: following some line of constant physics in the parameter space of
a regularized setup, it is claimed that the statement becomes valid in associated lattice
theories with sufficiently large cutoff.\footnote{In the continuum language, masses
$M \equiv (m_1,m_2,\ldots,m_{N_f})$ label different continuum theories (different lines
of constant physics) and should thus be viewed as renormalized quark masses in some
fixed scheme.} It is thus useful to attempt a formulation of the
vSChSB $\leftrightarrow$ DChC correspondence valid for the widest range of cutoffs
possible. Similarly to vSChSB $\leftrightarrow$ QMC connection, which holds at
arbitrary cutoff, the range of validity may include even symmetry breaking instances
due to lattice artifacts. To formulate the alternative version of the conjecture,
we were mostly guided by results of the numerical study at finite temperature, described
in Sec.~\ref{sec:fintemp}.
These results suggest the viability of the scenario in which
chirally broken dynamics generates chiral polarization only via discrete contribution from
strictly infrared modes. To incorporate these cases, it is necessary to abandon the notion
that chiral polarization characteristics are non--zero simultaneously. {\sl Conjecture 2'}
stated below provides for the minimal extension of this type.
\bigskip
\noindent {\bf Conjecture 2'}
\smallskip
\noindent {\sl The following holds in every lattice--regularized $(M,T) \in {\mathscr T}$
at sufficiently large ultraviolet cutoffs $\Lambda_{lat}$ and infinite volume.}
\begin{description}
\item[{\sl (I)}] {\sl Chiral polarization characteristics exist,
$\Lambda_{ch} = \Lambda_{ch}^\infty$ and $0 \le \chps \ll \Lambda_{lat}$.
If $\Lambda_{ch}>0$ then $\rho_{ch}(\lambda)$ is positive on $[0,\Lambda_{ch})$.}
\item[{\sl(II)}] {\sl The property of valence spontaneous chiral symmetry breaking
is characterized by}
\begin{equation}
\Sigma > 0 \quad \Longleftrightarrow \quad
\rho(\lambda=0) > 0 \quad \Longleftrightarrow \quad \Omega_{ch} > 0
\tag{{\ref{eq:5.020}}'}
\label{eq:5.020a}
\end{equation}
\item[{\sl(III)}] {\sl $\rho_{ch}(\lambda)$ is non--positive for $\lambda > \Lambda_{ch}$,
except possibly in the vicinity of $\Lambda_{lat}$.}
\end{description}
\medskip
\noindent Discussion following {\em Conjecture 2} mostly applies here as well but
it is $\Omega_{ch}$ that serves as the order parameter of vSChSB on ${\mathscr T}$:
chiral symmetry is broken if and only if total low energy chirality
per unit volume is nonzero. Singular cases motivating this extension arise when
positive core in $\rho_{ch}(\lambda,V)$ becomes proportional to $\delta(\lambda)$ in
the infinite volume limit, leading to $\Lambda_{ch}=0$, $\Omega_{ch}>0$ as shown in
Fig.~\ref{fig:illus}(middle).\footnote{The possibility of transition
from $\Lambda_{ch}$ to $\Omega_{ch}$ has been discussed in Ref.~\cite{Ale12D} already.
However, $\Omega_{ch}$ was denoted as $\Omega$ in that work.} Note that
vSChSB $\leftrightarrow$ DChC correspondence still follows from {\sl (I)} and {\sl (II)}.
Part {\sl (III)} implies that $\Omega_{ch}$ is not just the total low--energy chirality
but the total chirality of the entire Dirac spectrum. The essential content of
{\sl Conjecture 2'} can then be summarized by saying that quark--gluon dynamics breaks
chiral symmetry if and only if it generates volume density of dynamical chirality.
It is interesting to look back at Fig.~2 in light of the above statement. Indeed,
assuming that various $\sigma_{ch}(\lambda)$, $\rho_{ch}(\lambda)$ shown represent
infinite volume limits, the first pair of rows corresponds to options for chirally
broken vacuum, the second pair is associated with symmetric vacuum, and the cases in
the third pair do not occur on ${\mathscr T}$ if {\sl Conjecture 2'} is valid.
With the same rationale that motivated {\sl Conjecture 2'}, we now put forward yet
more generic association of chiral polarization and vSChSB. In particular, it is
possible to drop the notion that DChC is a necessary companion of vSChSB, while
still maintaining the connection to chiral polarization. This can arise when, in
addition to simultaneous positivity, the pairwise equality of chiral polarization
parameters is abandoned as well. The prototypical situation we have in mind is when
$\Lambda_{ch}(V)>0$ converges to $\Lambda_{ch}>0$ in the infinite volume limit, but
$\rho_{ch}(\lambda, V)$ scales to zero on interval $[0,\Lambda_{ch})$. This results
in $\Lambda_{ch}^\infty\!=\!0$ and $\Omega_{ch}\!=\!0$
(see Fig.~\ref{fig:illus}(right)). Note that
$\rho_{ch} = \rho\, \cop_A$ can approach zero due to {\sl (i)} $\cop_A \to 0$ or
{\sl (ii)} $\rho \to 0$ or {\sl (iii)} both.
Option {\sl (i)} is quite interesting since it can occur when
low--lying Dirac modes are dimensionally reduced.\footnote{Loosely speaking,
eigenmodes are {\em dimensionally reduced} when their effective support comprises
a vanishing fraction of the associated domain in the infinite--volume limit.}
Indeed, the ``active'' part of the eigenmode can be chirally polarized and
induce vSChSB, but its contribution to total polarization gets overwhelmed by
the uncorrelated bulk in the infinite volume limit. While $\Omega_{ch}=0$ in this
situation, $\Omega>0$ still signals the association of chiral symmetry breaking
and chiral polarization. Thus, in the form below, the conjecture states that
vSChSB proceeds if and only if there is a volume density of chirally polarized
modes around the surface of the Dirac sea.
\medskip
\noindent {\bf Conjecture 2''}
\smallskip
\noindent {\sl The following holds in every lattice--regularized $(M,T) \in {\mathscr T}$
at sufficiently large ultraviolet cutoffs $\Lambda_{lat}$ and infinite volume.}
\begin{description}
\item[{\sl (I)}] {\sl Chiral polarization characteristics exist and
$0 \le \Lambda_{ch} + \Lambda_{ch}^\infty \ll \Lambda_{lat}$.}
\item[{\sl(II)}] {\sl The property of valence spontaneous chiral symmetry breaking
is characterized by}
\begin{equation}
\Sigma > 0 \quad \Longleftrightarrow \quad
\rho(\lambda=0) > 0 \quad \Longleftrightarrow \quad \Omega > 0
\tag{{\ref{eq:5.020}}''}
\label{eq:5.020b}
\end{equation}
\item[{\sl(III)}] {\sl $\rho_{ch}(\lambda)$ is non--positive for
$\lambda > \Lambda_{ch}^\infty$, except possibly in the vicinity of $\Lambda_{lat}$.}
\end{description}
\medskip
It should be emphasized that all three versions of the conjecture may be valid
simultaneously. Indeed, they do not necessarily exclude one another, but rather
express varied degrees of detail in which chiral polarization could manifest itself
in vSChSB. In fact, it is entirely feasible that the differences are only relevant
at sufficiently low lattice cutoffs.
\subsection{Finite Volume}
\label{ssec:finvol}
Discussion in the previous section has been carried out in the infinite volume
which is a native setting for vSChSB and various condensates. However, it is
both relevant practically and interesting conceptually, to examine how chiral
polarization concepts enter the finite volume considerations.
To begin with, it is useful to fix a convention regarding exact zero--modes.
Indeed, the default lattice setup in this discussion involves standard
(anti--)periodic boundary conditions and the overlap Dirac operator as
a chiral probe. Consequently, exact zero--modes can appear in finite volume
whether infinite--volume theory breaks chiral symmetry or not. In either case
though, their abundance doesn't scale with volume. This renders them inessential
for valence condensate and there is a choice whether to include them in
specific considerations. For our purposes it is more convenient to leave
zero--modes out which is what will be assumed from now on in this article.
In fact, one useful advantage of using overlap operator is that it cleanly
separates out topological modes, ensuring that their {\em a priori} local
chirality doesn't contaminate that of near--zeromodes which are of actual
interest.
The definition of chiral symmetry breaking in Dirac eigenmode representation
involves a strictly infrared condition: the existence of mode condensate (QMC).
This is of course not surprising: the strictly infrared nature of relevant mass
($m_v=0$) gets translated into strictly infrared corner of the Dirac spectrum
($\lambda=0$). However, applying QMC condition to detect vSChSB in finite volume
is futile since it is never satisfied. Indeed, $\rho(\lambda \!\to\! 0,V)$ is
zero identically. This is symptomatic of the fact that
vSChSB $\leftrightarrow$ QMC correspondence is kinematic in nature: since
the definition of vSChSB demands infinite volume, QMC condition, being its
equivalent, reacts to avoid conflict with the presence of infrared cutoff.
However, if the mechanism of vSChSB was known, it would be reasonable to expect
that other properties of quark modes, those suggested by the mechanism,
could be used to signal vSChSB even in given finite volume. After all, broken
and symmetric dynamics should be distinguishable in any volume. In fact, if
infrared cutoff is smaller than relevant {\em finite} scales in the theory,
the distinction from single volume should be essentially unambiguous.
While the mechanism of vSChSB has not been satisfactorily clarified yet, one of
the driving motivations for developing chiral polarization framework was to
provide a possible indicator of the above type. It is important in that regard
that, unlike $\rho(0)$, chiral polarization parameters $\chps$, $\Omega_{ch}$
and $\Omega$ can be readily non--zero in finite volume. If forming chirally
polarized layer around the surface of Dirac sea provides sufficient and necessary
condition for producing vSChSB, as conjectures of the previous section propose,
then chiral polarization characteristics represent viable candidates for such
{\em finite--volume ``order parameters''} indeed.
It is interesting that the situation in finite volume is in fact simpler than
what we dealt with in the previous section. There are just three
characteristics which, taking into account the convention on exact zero--modes,
can only be non--zero simultaneously, effectively yielding a single
finite--volume order parameter. Theory $(M,T,V)$ is said to be in
{\em chirally polarized phase} if chiral polarization characteristics are
positive, say $\Omega(M,T,V)>0$. We emphasize again that this provides
a well--defined dynamical distinction between theories in finite volume based
on their chiral behavior. However, in addition to this and the fact that
$\Omega(M,T)$ is expected to be a valid order parameter in traditional sense,
our notion of finite--volume order parameter for vSChSB involves another
feature contained in the following statement consistent with available data.
\medskip
\noindent {\bf Conjecture 3}
\smallskip
\noindent {\sl The following holds in every lattice--regularized $(M,T) \in {\mathscr T}$
at sufficiently large ultraviolet cutoffs $\Lambda_{lat}$.}
\medskip
\noindent{\sl (I)} {\sl Chiral polarization characteristics exist and
$0 \le \Lambda_{ch}(V) \ll \Lambda_{lat}$ for sufficiently large $V$.}
\medskip
\noindent{\sl(II)} {\sl The property of valence spontaneous chiral symmetry breaking
is characterized by}
\begin{equation}
\Sigma > 0 \quad \Longleftrightarrow \quad \Omega(V) > 0
\quad \mbox{\rm for} \quad V_0 < V < \infty
\label{eq:5.030}
\end{equation}
\noindent {\sl i.e. it occurs if and only if the theory is in chirally polarized phase
in large finite volumes.}
\medskip
\noindent{\sl(III)} {\sl $\rho_{ch}(\lambda,V)$ is non--positive for
$\lambda > \Lambda_{ch}(V)$, except possibly in the vicinity of $\Lambda_{lat}$.}
\bigskip
The conceptual novelty in the above statement is that the right--hand side
of Eq.~\eqref{eq:5.030} does not involve explicit infinite volume limit.
This is different e.g. from {\em Conjecture 2''} wherein the infinite volume
limit of $\Omega(M,T,V)$, while significantly easier to deal with than QMC,
still has to be investigated: the theory could stay in chirally polarized
phase for arbitrary large volumes, but with $\Omega(M,T,V)$ scaling to zero.
However, {\em Conjecture 3} proposes that it is impossible to approach chirally
symmetric physics in infinite volume via finite--volume physics in chirally
polarized phase.\footnote{Note that, for discussion in paragraph
preceding and motivating {\em Conjecture 2''}, this implies that options
{\sl (ii)} and {\sl (iii)} don't occur.}
In other words, it suggests that, at least sufficiently close to the continuum
limit, vSChSB and dynamical chirality are inextricable.
We emphasize that the above is not to say that it is impossible to have
a ``finite volume correction'' to prediction on vSChSB based on chiral
polarization in single volume. However, in the realm of {\sl Conjecture 3},
the false positive corresponds to missing out on the whole dynamically--defined
phase. In other words, it is expected that only in very small volumes, when
other aspects of physics are severely mutilated as well, is such occurrence likely.
Finally, we remark that while $\rho(\lambda \!=\!0,V)$ and
$\rho_{ch}(\lambda \!=\!0,V)$ always vanish and cannot be utilized as
indicators of vSChSB, this is not necessarily true for all strictly infrared
constructs in finite volume. For example, in chirally broken case,
$\lambda \!=\! 0$ is typically an isolated zero of $\rho(\lambda,V)$, but
$\cop_A(\lambda\!=\!0,V) \equiv \lim_{\lambda \to 0} \rho_{ch}(\lambda,V)/\rho(\lambda,V)$
is expected to be well defined and positive. In fact, the strictly infrared
version of {\sl Conjecture 3} with $\Omega(M,T,V)$ on the right--hand side of
Eq.~\eqref{eq:5.030} replaced by
$\cop_A(\lambda \!=\! 0,M,T,V)$ is an equally viable representation of
the correspondence, albeit less appealing from practical standpoint.
\begin{figure}[t]
\begin{center}
\centerline{
\hskip 0.00in
\includegraphics[width=18.0truecm,angle=0]{f_illus_2.pdf}
}
\vskip -0.10in
\caption{Possible behaviors of $\sigma_{ch}(\sigma)$ in the vicinity of
$\sigma=0$. In finite volume, case $(\alpha)$ predicts vSChSB of infinite
volume theory, while $(\beta)$ and $(\gamma)$ entail valence
chiral symmetry.}
\label{fig:sigch_vs_sig_illus}
\vskip -0.45in
\end{center}
\end{figure}
\subsection{$\sigma$--Parametrization and the Universal Scale of vSChSB}
\label{ssec:universal}
For certain purposes, some of which are discussed below, it is useful to
parametrize chiral polarization properties by cumulative eigenmode density
$\sigma$, rather than Dirac spectral parameter $\lambda$. Recall that
the description discussed so far is based on two cumulative densities,
namely $\sigma(\lambda)$ and $\sigma_{ch}(\lambda)$: the behavior of
$\sigma_{ch}(\lambda)$ defines polarization parameters
$\Lambda_{ch}$ and $\Omega_{ch}$, while $\Omega$ emerges ``a posteriori''
from $\sigma(\lambda)$ and $\Lambda_{ch}$.
We wish to eliminate $\lambda$ from the pair $\sigma(\lambda)$,
$\sigma_{ch}(\lambda)$ and consider $\sigma_{ch}=\sigma_{ch}(\sigma)$.
The latter is certainly a well-defined function when $\sigma(\lambda)$ is
one-to-one, but this is also true in general case. Indeed, since
$\sigma(\lambda)$ is non--decreasing and can have finite jumps, there are only
two special circumstances to examine.
({\em a}) If $\sigma(\lambda)$ is constant on interval $[\lambda_1,\lambda_2]$
then multiple values of $\sigma_{ch}$ could in principle be associated with
$\sigma_1=\sigma(\lambda_1)$. However, it follows from its definition that
$\sigma_{ch}(\lambda)$ is then also constant on $[\lambda_1,\lambda_2]$,
and $\sigma_{ch}(\sigma_1)$ is thus unique.
({\em b}) If $\sigma(\lambda)$ has a jump at $\bar{\lambda}$ with $\sigma^l$,
$\sigma^r$ being the left and right values respectively, then $\sigma_{ch}(\sigma)$
is a priori undefined on the interval $(\sigma^l,\sigma^r)$. However, since
the population of modes associated with interval $(\sigma^l,\sigma^r)$ is assigned
a common value of chiral correlation, namely $\cop_A(\bar{\lambda})$, there is
a natural unique definition of $\sigma_{ch}(\sigma)$ on $(\sigma^l,\sigma^r)$:
the linear dependence whose graph connects the points $(\sigma^l, \sigma_{ch}^l)$
and $(\sigma^r, \sigma_{ch}^r)$. Here $\sigma_{ch}^l$, $\sigma_{ch}^r$ are the left
and right values of $\sigma_{ch}(\lambda)$ at $\bar{\lambda}$. Note that the slope
of $\sigma_{ch}(\sigma)$ is $\cop_A(\sigma)$ for every $\sigma$.
\medskip
\noindent There are few points we wish to emphasize regarding the utility of
the above.
\medskip
\noindent {\em (i)} Since $\sigma_{ch}(\lambda)$ can only have jumps at arguments
where $\sigma(\lambda)$ does (see Appendix~\ref{app:spectral}), the function
$\sigma_{ch}(\sigma)$ is not only well--defined but always continuous on its
domain.\footnote{In lattice units, this domain is in fact the interval $[0,12]$
since $N_c=3$ for theories in ${\mathscr T}$. This is irrespective of volume,
cutoff or lattice Dirac operator used.}
Given that $\sigma_{ch}(\sigma\!=\!0)\!=\!0$, there are then three possible behaviors
of $\sigma_{ch}(\sigma)$ in the vicinity of zero: it can $(\alpha)$ turn positive,
$(\beta)$ remain identically zero or $(\gamma)$ turn negative. While it simply
classifies theories as chirally polarized, unpolarized or anti--polarized at lower
spectral end, this dynamical distinction acquires deeper meaning in light of our
conjectures. Indeed, in finite volume, the first case indicates that the theory is
in chirally polarized phase, and is thus predicted to be chirally broken, as opposed
to the remaining two options (see Fig.~\ref{fig:sigch_vs_sig_illus}).
In this way, the ``low $\sigma$'' behavior of $\sigma_{ch}(\sigma)$ provides for
rather succinct and elegant representation of the ideas discussed here.
\medskip
\noindent One noteworthy point in this regard is that when presented with
$\sigma_{ch}(\sigma)$ in infinite volume only, the behavior $(\beta)$ in itself
is indefinite with regard to vSChSB. Indeed, it could correspond to the situation
$\Omega_{ch}=0$, $\Omega>0$ with $\Omega$ undetermined by $\sigma_{ch}(\sigma)$
alone. In such case the information on large--volume behavior is needed to resolve
the ambiguity.
\medskip
\noindent {\em (ii)} In our discussion we paid little attention to the issues
related to precise behavior of various new constructs in the continuum limit.
In fact, this is not essential for our purposes. Indeed, our goal is to understand
the dynamics of vSChSB, which means gaining insight in the regime of lattice theory
where it becomes insensitive to the ultraviolet cutoff. Various ``order parameters''
we discussed are well--defined at arbitrary finite cutoff, and serve as indicators
of vSChSB which itself is well--defined at arbitrary finite cutoff if chirally
symmetric Dirac operator (overlap operator) is used.
\medskip
\noindent Nevertheless, it is appealing to characterize chiral polarization via
parameters with well--defined continuum values. As mentioned in~\cite{Ale12D}
already, $\Lambda_{ch}$ is expected to require a normalization factor to define
its unique continuum limit e.g. in N$_f$=2+1 zero temperature QCD at physical
point.\footnote{Note that in this discussion we implicitly assume that
$D_{(1)}=D_{(2)}=D_{overlap}$ in Eq.~\eqref{eq:2.015}, but we expect our points
to apply whenever $D_{(2)}=D_{overlap}$.}
However, the renormalization properties of mode density described
in~\cite{Del05A,Giu07A} imply that $\sigma_{ren}(\lambda_{ren})=\sigma(\lambda)$
for renormalized/bare cumulative density. Hence, $\Omega$ is expected to be free
of renormalization factors and have a universal continuum limit. The same would hold
also for $\Omega_{ch}$ if the renormalization of $\rho(\lambda)$ and
$\rho_{ch}(\lambda)$ proceeded via the same factor. This, however, would have
to be established. Note that, in such case, the whole function $\sigma_{ch}(\sigma)$
would be universal.
\medskip
\noindent {\em (iii)} Removing the reference to spectral parameter $\lambda$
underscores the dynamical nature of the vSChSB $\leftrightarrow$ ChP equivalence.
Indeed, in this form any explicit connection to ``strictly infrared'' scales, which
has its roots in kinematic considerations, is eliminated. Rather, the correspondence
relies exclusively on local dynamical behavior of lowest modes, irrespective of
precise Dirac eigenvalues they are labeled with. Given that $\sigma_{ch}(\sigma)$
can be conveniently computed directly, without invoking the spectral representation,
it is worthwhile to formulate the proposed connection directly in this language.
In fact, the formulation becomes somewhat more concise: $\Omega$ is defined
as the maximal $\bar{\sigma}$ such that $\sigma_{ch}(\sigma)$ is strictly
increasing on $[0,\bar{\sigma}]$, while $\Omega_{ch}$ is always simply
$\sigma_{ch}(\sigma=\Omega)$ due to continuity. The analog of {\sl Conjecture 3}
is then as follows, with $\Omega(V)$ in Eq.~\eqref{eq:5.040} replaceable by
$\Omega_{ch}(V)$ if so desired.
\medskip
\noindent {\bf Conjecture 3'}
\smallskip
\noindent {\sl The following holds in every lattice--regularized $(M,T) \in {\mathscr T}$
at sufficiently large ultraviolet cutoffs $\Lambda_{lat}$.}
\medskip
\noindent{\sl (I)} {\sl Function $\sigma_{ch}(\sigma,V)$ is continuous on its
domain $\sigma \in [0,12 \Lambda_{lat}^4]$, and $\Omega(V) \ll 12\Lambda_{lat}^4$ for
sufficiently large $V$.}
\medskip
\noindent{\sl(II)} {\sl The property of valence spontaneous chiral symmetry breaking
is characterized by}
\begin{equation}
\Sigma > 0 \quad \Longleftrightarrow \quad \Omega(V) > 0
\quad \mbox{\rm for} \quad V_0 < V < \infty
\label{eq:5.040}
\end{equation}
\medskip
\noindent{\sl(III)} {\sl $\sigma_{ch}(\sigma,V)$ is non--increasing for
$\sigma > \Omega(V)$, except possibly in the vicinity of $\sigma=12 \Lambda_{lat}^4$.}
\medskip
\noindent {\em (iv)} Relative to discussion in Ref.~\cite{Ale12D}, our description
of chiral polarization around the surface of Dirac sea became more detailed.
In addition to the width of the polarized layer ($\Lambda_{ch}$), the phenomenon
is also characterized by the volume density of total number of modes involved
($\Omega$) and the associated total chirality ($\Omega_{ch}$). In the spirit
of Refs.~\cite{Ale12D,Ale10A} and the conjectures discussed here, we consider
the corresponding scales, namely $\Lambda_{ch}$, $\Omega^{1/4}$ and
$\Omega_{ch}^{1/4}$, to be the dynamical scales associated with the phenomenon of
vSChSB. In the massless light--quark limit of ``real--world QCD'' (i.e. N$_f$=2+1),
they become the scales of SChSB. In Ref.~\cite{Ale12D} we estimated the
(unrenormalized) value of chiral polarization scale in N$_f$=2+1 QCD at physical
point to be $\Lambda_{ch} \approx 80$ MeV. The estimated values of the other
two parameters in this case are $\Omega^{1/4} \approx 150$ MeV (expected to be
universal) and $\Omega_{ch}^{1/4} \approx 60$ MeV.
\section{Temperature Effects}
\label{sec:fintemp}
In this section, we present results of the finite--temperature study in
N$_f$=0 QCD. This theory has a well--established deconfinement transition
temperature $T_c$ defined via expectation value of the Polyakov loop.
According to the standard scenario, vSChSB disappears in close vicinity
of $T_c$, but in general at some other temperature $T_{ch}$. We find that
mode condensation and chiral polarization of valence overlap quarks exactly
follow each other, in accordance with vSChSB--ChP correspondence.
The data also shows that, at the fixed lattice cutoff used
($\Lambda_{lat} \simeq 2.3\,$GeV), chiral transition temperature $T_{ch}$ is
strictly larger than $T_c$.
\subsection{Lattice Setup and Polyakov Loop Sectors}
We simulate pure--glue SU(3) theory with Wilson action at $\beta=6.054$.
The non--perturbative parametrization of Ref.~\cite{Gua98A} is used to set
the lattice scale, resulting in $a/r_0=0.170$ at the aforementioned gauge
coupling. Using the standard value $r_0\!=\!0.5\,$fm for reference scale, this
translates into $a=0.085\,$fm. To perform a basic temperature scan, we vary
the ``time'' extent of the lattice between $N_t\!=\!4$ and $N_t\!=\!20$ which
corresponds to temperatures $T=1/(N_ta)$ in the range 116--579 MeV. The spatial
extent of the lattice is kept fixed at $N\!=\!20$, corresponding to volume
$V_3=(N a)^3=(1.7\,$fm $\!\!)^3$. The information about these ensembles is
summarized in Table~\ref{tab:fint_ensemb} with some relevant explanations
provided below.
\begin{table}[t]
\centering
\begin{tabular}{@{} ccccccccc @{}}
\toprule
Ensemble & $N_t$ & $T/T_c$ & $T$[MeV] & $N_{cfg}$ &
$|\lambda|_{min}^{av}$ & $|\lambda|_{min}$ &
$|\lambda|_{max}^{av}$ & $|\lambda|_{max}$\\
\midrule
$E_{1}$ & 20 & 0.42 & 116 & 100 & 0.0204 & 0.0027 & 0.6128 & 0.6160 \\
$E_{2}$ & 12 & 0.70 & 193 & 200 & 0.0320 & 0.0065 & 0.7241 & 0.7270 \\
$E_{3}$ & 10 & 0.84 & 232 & 200 & 0.0379 & 0.0018 & 0.7658 & 0.7701 \\
$E_{4}$ & 9 & 0.93 & 258 & 200 & 0.0402 & 0.0039 & 0.7912 & 0.7944 \\
$E_{5}$ & 8 & 1.05 & 290 & 400 & 0.0859 & 0.0011 & 0.8208 & 0.8246 \\
$E_{6}$ & 7 & 1.20 & 331 & 400 & 0.2473 & 0.0006 & 0.8631 & 0.8675 \\
$E_{7}$ & 6 & 1.39 & 386 & 100 & 0.4038 & 0.0498 & 0.9233 & 0.9283 \\
$E_{8}$ & 4 & 2.09 & 579 & 100 & 0.7868 & 0.7129 & 1.1608 & 1.1673 \\
\bottomrule
\end{tabular}
\caption{20$^3 \times N_t$ ensembles of N$_f$=0 theory with Wilson gauge
action ($\beta=6.054$), used in the overlap eigenmode calculations.
$|\lambda|_{min}^{av}$ is the average magnitude of smallest non--zero
eigenvalue in a configuration, while $|\lambda|_{max}^{av}$ that of
the largest one. The magnitudes of all computed non--zero eigenvalues in
an ensemble satisfy $|\lambda|_{min} \le |\lambda| \le |\lambda|_{max}$.}
\label{tab:fint_ensemb}
\vskip -0.06in
\end{table}
Distinctive aspect of studying chiral issues in N$_f$=0 theory relates
to the fact that, while the deconfinement transition is associated
with spontaneous breakdown of Z$_3$ symmetry signaled by the expectation
value of Polyakov loop~\cite{Pol78A,Sve82A}, Dirac spectral properties in
the deconfined phase depend on which vacuum broken theory happens
to visit. In particular, it was pointed out that valence chiral symmetry
in the ``real sector'' might be restored at lower temperature than in
the ``complex sectors''~\cite{Cha95A}, with the former transition
being the one occurring close to $T_c$. While this conclusion remained somewhat
controversial~\cite{Gat02A}, it became common to study the real sector in
connection with chiral symmetry restoration in N$_f$=0 QCD. This is also tied
to the fact that dynamical fermions tend to bias gauge fields correspondingly
(see e.g.~\cite{Kov08A}), bringing up the expectation that the dynamics of
the real--sector vacuum better resembles the behavior in real--world QCD.
\begin{table}[b]
\centering
\begin{tabular}{@{} cccccccc @{}}
\toprule
Ensemble & $N$ & $L$[fm] & $N_{cfg}$ &
$|\lambda|_{min}^{av}$ & $|\lambda|_{min}$ &
$|\lambda|_{max}^{av}$ & $|\lambda|_{max}$\\
\midrule
$G_{1}$ & 16 & 1.36 & 400 & 0.301513 & 0.005507 & 1.029540 & 1.034761 \\
$G_{2}$ & 24 & 2.04 & 400 & 0.200300 & 0.000089 & 0.746919 & 0.753360 \\
$G_{3}$ & 32 & 2.72 & 200 & 0.067461 & 0.000046 & 0.598446 & 0.607929 \\
\bottomrule
\end{tabular}
\caption{$N^3\times$7 ensembles of N$_f$=0 theory with Wilson gauge plaquette action
($\beta=6.054$), used to study finite volume effects in the $N_t=7$ system ($E_6$).}
\label{tab:fint_ensemb2}
\end{table}
Here we adopt the above point of view and
present results for the real Polyakov loop sector. While in infinite volume the Z$_3$
distinction is only relevant above $T_c$, it is reasonable to keep the separation
on both sides of the transition for finite system due to tunneling. To that
effect, we rotated each generated configuration into complementary Z$_3$ sectors, but
$N_{cfg}$ of Table~\ref{tab:fint_ensemb} refers to independent unrotated
configurations. Except for ensembles $E_1$ and $E_2$, Dirac eigensystems were
calculated in all Z$_3$ sectors for each configuration, and the statistics for real--phase
results is thus $N_{cfg}$. For $E_1$ and $E_2$, only configurations originally
generated in real sector, 31 and 61 of them respectively, were included.
Overlap Dirac operator with Wilson kernel ($r\!=\!1$) and $\rho\!=\!26/19$ was used
in these valence quark calculations, and the quoted spectral
bounds refer to the real Polyakov--loop sector. For all ensembles used in this work,
200 lowest eigenmodes with non--negative imaginary part were computed.
We use $T_c/\sqrt{\sigma}=0.631$, quoted in Ref.~\cite{Kar97A}, as a reference value
for infinite--volume continuum--limit transition temperature to label our ensembles.
With string tension value $\sigma=(440\, \mbox{\rm MeV})^2$ this translates into
$T_c=277\,$MeV.
Since the volume and lattice cutoff are finite, and there is also a small uncertainty
in the determination of the lattice scale, it is prudent to check whether Z$_3$
transition in our system occurs in the expected range of temperatures. To do so, we
show the scatter plot of the Polyakov loop in Fig.\ref{fig:z3} (left). As can be seen
quite clearly, the expected symmetric distribution below $T_c$ is contrasted with
Z$_3$--concentrated population above $T_c$. This is also confirmed by the behavior
of Polyakov--loop susceptibility shown in Fig.\ref{fig:z3} (right). There is thus little
doubt that ensembles $E_1$--$E_4$ and $E_5$--$E_8$ represent the system in confined and
deconfined phases respectively.
\begin{figure}[t]
\begin{center}
\centerline{
\hskip 0.00in
\includegraphics[width=9.5truecm,angle=0]{f_polyloop_allT.pdf}
\hskip 0.0in
\includegraphics[width=7.5truecm,angle=0]{f_polyloop_suscept_vs_T.pdf}
}
\vskip 0.00in
\caption{Left: scatter plots of Polyakov loop for ensembles $E_1$--$E_8$. Each point
corresponds to a configuration with Z$_3$ symmetrization included. Sliding scale has
been adjusted in each case so as to contain all points. Right: Polyakov--loop
susceptibilities.}
\label{fig:z3}
\vskip -0.45in
\end{center}
\end{figure}
In the course of our analysis it will turn out that, while clearly in the deconfined
phase, the $N_t\!=\!7$ system still exhibits chiral polarization and vital signs of
vSChSB. To ascertain this, we study the finite--volume behavior on additional
lattices with details given in Table~\ref{tab:fint_ensemb2}.
The ensemble $G_3$ with $V_3=(2.72\,$fm $\!\!)^3$ corresponds to the largest system
studied. All the above notes concerning $E$--ensembles also apply to $G$--ensembles.
\subsection{Raw Data}
One useful way to obtain a quick overview of the situation at hand is to examine
the scatter plots of $\cop_A$ versus $\lambda$ since they provide a simultaneous
qualitative picture of spectral abundance and chiral polarization. Thus, each
eigenmode associated with given ensemble contributes a point to the plot, specified
by the magnitude of the eigenvalue ($\lambda$) and its correlation coefficient of
polarization ($\cop_A$). In Fig.~\ref{fig:T_raw_CA} we show these plots for
$E$--ensembles. Note that the temperature increases in lexicographic order.
Regarding the vSChSB$\leftrightarrow$ChP correspondence, the main aspect to examine
is whether presence of near--zero modes is always associated with tendency for chiral
polarization at low energy, and that chiral polarization is always absent in
accessible spectrum if near--zero modes are not produced. Such qualitative
correlation is clearly observed in Fig.~\ref{fig:T_raw_CA}.
Quick look at scatter plots also suggests three kinds of qualitative behavior as the
temperature is increased. First, the ``low--temperature dynamics'', exemplified by
$E_1$ ($T/T_c=0.42$) and discussed extensively in \cite{Ale12D}, appears to apply
throughout the confined phase. Second, the ``transition dynamics'', exemplified
by $E_6$ ($T/T_c=1.20$) extends approximately from $T_c$ to chiral transition point
$T_{ch}$. It is characterized by the spectral separation of near--zero modes from
the bulk and the creation of mode--depleted region between them. Finally,
the "high--temperature dynamics'' turns on above $T_{ch}$ as near--zeromodes can
no longer be supported in sufficient numbers, and the anti--polarization of the bulk
takes over.
\begin{figure}[t]
\begin{center}
\centerline{
\hskip 0.00in
\includegraphics[width=17.0truecm,angle=0]{f_CA_vs_lambda_ns20_alltemp.png}
}
\vskip -0.10in
\caption{Scatter plots of chiral polarization correlation ($\cop_A$) versus
eigenvalue magnitude ($\lambda$) for $E$--ensembles of
Table~\ref{tab:fint_ensemb}. Note the change of scale for $E_7$, $E_8$ relative
to $E_1$--$E_6$.}
\label{fig:T_raw_CA}
\vskip -0.45in
\end{center}
\end{figure}
In the above temperature scan, the system associated with ensemble
$E_6$ ($T/T_c=1.20$) is of prime interest: not only does it suggest itself as
a lattice example of vSChSB with deconfined gauge fields~\cite{Edw99A} but, more
importantly, it is the most borderline case where the vSChSB$\leftrightarrow$ChP
association should be ascertained. To do this at the level of raw data, we show
in Fig.~\ref{fig:T_raw_CA_nt7} the $\lambda$--$\cop_A$ scatter plots for this system
in increasing 3--volumes. The abundance of near--zero modes clearly rises with volume,
with the layer of high concentration increasingly focused toward the origin. Thus,
mode condensation appears all but inevitable. At the same time, chiral polarization
persists as predicted. Decreasing width of the polarized layer suggests possible
singular behavior in the vein of our discussion in Sec.~\ref{ssec:conjectures}. Note
that we also show the associated scatter plots of Polyakov loop, clearing any suspicion
that the system could be in confined phase.
\begin{figure}[t]
\begin{center}
\centerline{
\hskip 0.00in
\includegraphics[width=17.5truecm,angle=0]{f_CA_vs_lambda_nt7_alln_scatter_poly.png}
}
\vskip -0.10in
\caption{Scatter plots ($\cop_A$ vs $\lambda$) for $N_t\!=\!7$ system ($T/T_c=1.20$)
in varying spatial volume. The associated raw data for Polyakov loop is also shown.}
\label{fig:T_raw_CA_nt7}
\vskip -0.45in
\end{center}
\end{figure}
\subsection{Chiral Polarization Transition}
We now start putting the observations of the previous section into more formal terms.
The first task in this process is to determine the transition point (in temperature) for
chiral polarization. It should be emphasized that whether the system is in the polarized
phase or not is a well--defined question in any finite volume. The temperature scan
will be performed for $E$--ensembles sharing the same 3--volume. Resulting transition
point is thus associated with the corresponding ultraviolet cutoff and volume.
\begin{figure}[]
\begin{center}
\centerline{
\hskip 0.00in
\includegraphics[width=16.5truecm,angle=0]{f_sigmach_vs_lambda_alltemp_dl40.pdf}
}
\vskip -0.00in
\caption{Cumulative chiral polarization density for all $E$--ensembles. Ranges
are fixed.}
\label{fig:T_sigmach_lam_all}
\vskip -0.45in
\end{center}
\end{figure}
\begin{figure}[]
\begin{center}
\centerline{
\hskip 0.00in
\includegraphics[width=16.5truecm,angle=0]{f_sigmach_vs_lambda_and_sigma_sometemp_v2.pdf}
}
\vskip -0.00in
\caption{Low--energy closeup of $\sigma_{ch}(\lambda)$ (left column) and
$\sigma_{ch}(\sigma)$ (right column) for ensembles $E_4$--$E_7$. Theory is clearly
chirally polarized at $T/T_c=1.20$. Note the sharp transition in the latter
representation when changing to $T/T_c=1.39$.}
\label{fig:T_sigmach_lam_sig}
\vskip -0.45in
\end{center}
\end{figure}
To do this in a straightforward way, we monitor the behavior of cumulative chiral
polarization density $\sigma_{ch}(\lambda)$. The result is shown in
Fig.~\ref{fig:T_sigmach_lam_all} with temperature increasing in lexicographic order.
The characteristic positive bump (see discussion for Fig.~\ref{fig:sigch_vs_lam_illus})
appears at low temperatures signaling the creation of chirally polarized layer around
the surface of the Dirac sea. Data is shown on identical scales for all ensembles
to see the changing position of the maximum ($\Lambda_{ch}$) as well as its value
($\Omega_{ch}$). The polarization feature is clearly absent at $T/T_c=1.39$, while it
appears to be present at the borderline case ($T/T_c=1.20$) which, however would benefit
from better resolution.
In Fig.~\ref{fig:T_sigmach_lam_sig} we show closeups of the low part of the spectrum for
ensembles $E_4$--$E_7$. Note that the scales are no longer fixed in order to properly
resolve the polarization feature.
Temperature grows from top to bottom with left column displaying
$\sigma_{ch}(\lambda)$ and the right column the associated $\sigma_{ch}(\sigma)$.
As advertised, and suggested by the raw data, at $T/T_c=1.20$ the system is still
in chirally polarized phase. Both $\sigma_{ch}(\lambda)$ and $\sigma_{ch}(\sigma)$
tell the same story, but note how the latter effectively removes the depleted regions
of the Dirac spectrum (ensembles $E_6$ and $E_7$) from consideration, focusing on
polarization properties of existing modes.
Finally, in Fig.~\ref{fig:params_vs_T} we show the temperature dependence for the three
global characteristics of the chirally polarized layer, namely $\Lambda_{ch}$,
$\Omega_{ch}$ and $\Omega$. As discussed in Sec.~\ref{ssec:finvol}, they are all
equivalent indicators of chiral polarization in finite volume, and potentially
{\em finite--volume order parameters} of vSChSB, as proposed by {\em Conjecture 3}.
\begin{figure}[h]
\begin{center}
\centerline{
\hskip 0.12in
\includegraphics[height=3.98truecm,angle=0]{f_Lamch_vs_temp_real_dl10MeV.pdf}
\hskip -0.16in
\includegraphics[height=3.98truecm,angle=0]{f_Omegach_vs_temp_real_dl10MeV.pdf}
\hskip -0.16in
\includegraphics[height=3.98truecm,angle=0]{f_Omega_vs_temp_real_dl10MeV.pdf}
}
\vskip -0.05in
\caption{Global characteristics of chiral polarization in $E$--ensembles as functions
of temperature. They indicate the transition temperature $1.2\,T_c < T_{ch} < 1.39\,T_c$
for the cutoff and volume in question. $\delta\lambda$ refers to coarse--graining
parameter used in determination of $\Lambda_{ch}$.}
\label{fig:params_vs_T}
\vskip -0.25in
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\centerline{
\hskip 0.04in
\includegraphics[width=5.76truecm,angle=0]{f_rho_vs_lam_nt7_n32_dl40MeV_v2.pdf}
\hskip -0.14in
\includegraphics[width=5.76truecm,angle=0]{f_CA_vs_lam_nt7_n32_dl40MeV_v2.pdf}
\hskip -0.14in
\includegraphics[width=5.76truecm,angle=0]{f_rhoch_vs_lam_nt7_n32_dl40MeV_v2.pdf}
}
\vskip -0.1in
\centerline{
\hskip 0.05in
\includegraphics[width=5.8truecm,angle=0]{f_rho_vs_lam_nt7_n32_dl4MeV_closeup.pdf}
\hskip -0.14in
\includegraphics[width=5.8truecm,angle=0]{f_CA_vs_lam_nt7_n32_dl4MeV_closeup.pdf}
\hskip -0.14in
\includegraphics[width=5.8truecm,angle=0]{f_rhoch_vs_lam_nt7_n32_dl4MeV_closeup.pdf}
}
\vskip -0.1in
\caption{Large view (top) and closeup (bottom) of $\rho(\lambda)$, $\cop_A(\lambda)$,
$\rho_{ch}(\lambda)$ for ensemble $G_3$.}
\label{fig:n32_nt7}
\vskip -0.25in
\end{center}
\end{figure}
\subsection{Infinite Volume}
To test the proposed conjectures, it is necessary to deal with infinite volume
considerations since the definition of vSChSB explicitly relies on it.
In more concrete terms,
it is important to check which forms of {\em Conjecture 2} (if any) apply
to the finite--cutoff situation at hand. Also, assessing possible merits in the notion
of finite--volume order parameter, associated with {\em Conjecture 3}, depends on
the behavior of polarization observables in large volumes.
As is clear from our discussion in preceding sections, the relevant situation
to study in this regard is $T/T_c=1.2$, represented by $E_6$ and the $G$--ensembles.
Indeed, this is the borderline case, providing the most sensitive test for vSChSB--ChP
correspondence. To overview the situation directly in the largest volume available
($N\!=\!32$), we show in Fig.~\ref{fig:n32_nt7} (top) the triplet of characteristics
$\rho(\lambda)$, $\cop_A(\lambda)$, $\rho_{ch}(\lambda)$ over a large spectral range,
fully covering the depleted region observed in the raw data. While the increased
density near the origin is clearly visible, chiral polarization feature is not
recognizable at this resolution. However, closeup to the vicinity of the origin
in Fig.~\ref{fig:n32_nt7} (bottom) reveals a clear polarized layer as anticipated.
Note that the feature became quite thin in this largest volume.
To examine the situation in detail, we first focus on the ChP side of the correspondence.
In practical terms, this means determining the form of the chiral polarization layer
for large and infinite volume. To that effect, we show in Fig.~\ref{fig:nt7_sigmach_sig}
the behavior of $\sigma_{ch}(\sigma)$ for the four volumes available. As pointed out in
the previous section, this is the most robust way of visualizing the layer in case of
depleted spectra such as those we are dealing with in the transition region. Indeed,
even though Fig.~\ref{fig:T_raw_CA_nt7} indicates severe depletion (even close to
the spectral origin) for the $N\!=\!16$ system, the polarization layer is still quite
clearly visible in $\sigma_{ch}(\sigma)$. In larger volumes, the characteristic positive
bump tends to grow both in $\sigma$ and $\sigma_{ch}$--directions, strongly suggesting
that the ChP layer remains the dynamical feature of the system in the infinite
volume limit.
There is, however, a difference in how infinite--volume ChP is realized at $T/T_c \!=\! 1.2$
and at zero or small temperatures. This is revealed in Fig.~\ref{fig:params_vs_V}
where we show the dependence of chiral polarization parameters on the infrared cutoff.
As foretold by Fig.~\ref{fig:nt7_sigmach_sig}, $\Omega_{ch}$ and $\Omega$ grow as
the cutoff is removed, but $\Lambda_{ch}$ decreases, likely toward zero.
Continuation of these trends in larger volumes is only possible when $\rho(\lambda,V)$
and $\rho_{ch}(\lambda,V)$ develop a $\delta(\lambda)$--core in $V \to \infty$ limit,
to keep the observed volume density of polarized modes and volume density of dynamical
chirality finite. We may thus be dealing with a singular case of the type
($\Lambda_{ch}\!=\!0$, $\Omega_{ch} \!>\! 0$, $\Omega \!> \!0$) discussed repeatedly
in Sec.~\ref{sec:background}. Including such ChP behaviors was the main motivation behind
extending our formalism to current form.
\begin{figure}[t]
\begin{center}
\centerline{
\hskip -0.1in
\includegraphics[width=15.5truecm,angle=0]{f_sigmach_vs_sigma_all_nt7_ds0008.pdf}
}
\vskip -0.00in
\caption{Polarization characteristic $\sigma_{ch}(\sigma)$ at $T/T_c=1.2$
for increasing 3--volumes.}
\label{fig:nt7_sigmach_sig}
\vskip -0.45in
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\centerline{
\hskip 0.03in
\includegraphics[height=4.0truecm,angle=0]{f_Lamch_vs_L_nt7_dl4MeV_v2.pdf}
\hskip -0.15in
\includegraphics[height=4.0truecm,angle=0]{f_Omegach_vs_L_nt7_dl4MeV.pdf}
\hskip -0.15in
\includegraphics[height=4.0truecm,angle=0]{f_Omega_vs_L_nt7_dl4MeV.pdf}
}
\vskip -0.05in
\caption{Global characteristics of chiral polarization at $T/T_c=1.2$ against
infrared cutoff. Simple power--law fit was included in case of
$\Lambda_{ch}(1/L)$ to guide the eye.}
\label{fig:params_vs_V}
\vskip -0.20in
\end{center}
\end{figure}
Let's now examine the vSChSB side of the correspondence. This might seem unnecessary from
strictly logical standpoint since the result of our ChP analysis is such that it already
carries with it the implication of vSChSB (with divergent condensate) at $T/T_c=1.2$.
However, it certainly helps the case if the same conclusion can also be reached on its own,
independently of chiral polarization. We thus wish to check directly whether dynamics
at $T/T_c=1.2$ is consistent with the definition of mode--condensing theory. To begin with,
note that the behavior of $\rho(\lambda)$ for largest volume available, shown in the
lower--left plot of Fig.~\ref{fig:n32_nt7}, is in itself strongly suggestive of mode
condensation. Indeed, contrary to monotonically increasing function typical of zero and
low temperatures, this $\rho(\lambda)$ is monotonically decreasing in the very infrared
range shown. One thus naively expects non--zero mode density to survive at the spectral
origin.
However, the definition \eqref{eq:2.050} of mode condensation demands that volume
trends be examined at fixed infrared spectral windows to make meaningful conclusions.
In line with this definition, we consider the coarse--grained version of $\rho(0)$, namely
\begin{equation}
\rho(\lambda\!=\!0, \Delta, V) \,\equiv\, \frac{1}{\Delta} \,
\sigma(\lambda\!=\!\Delta,V)
\end{equation}
to see the associated volume tendencies for various values of $\Delta$.
The result of such calculation for a range of infrared windows down to 4 MeV is shown
in Fig.~\ref{fig:rho0_Delta_all}. Note that for any fixed $V$, function
$\rho(0,\Delta,V)$ will approach zero for $\Delta \to 0$. This is explicitly seen in
$N\!=\!16$ and $20$ cases but not for the two larger 3--volumes where the downward bend
occurs at yet smaller values of $\Delta$. Important feature of these results is
that $\rho(0,\Delta,V)$ grows with volume for all $\Delta$ shown. In fact, the rate of
growth increases at small $\Delta$, and even before the occurrence of the bend, leaving
little room for the possibility that
$\lim_{\Delta\to 0} \lim_{V\to\infty} \rho(0,\Delta,V)$ vanishes. Rather, the data
is consistent with diverging mode condensate as expected from ChP analysis.
To summarize, we presented evidence that the layer of chirally polarized modes around
the surface of the Dirac sea remains the feature of N$_f$=0 QCD at $T/T_c=1.2$ even in
the infinite--volume limit. At the same time, the system was found to be mode--condensing,
in accordance with the general vSChSB--ChP correspondence. The specific form of ChP layer
conforms to the types described by {\em Conjectures 2',2''}, but most likely doesn't fall
into the realm of {\em Conjecture 2}. We emphasize that this doesn't mean that
{\em Conjecture 2} is invalid: this would only transpire if the concluded type of ChP
behavior persisted at arbitrarily large ultraviolet cutoff. Note also that our
analysis is in agreement with {\em Conjectures 3,3'}, thus lending support to the concept
of finite--volume order parameter.
\begin{figure}[t]
\begin{center}
\centerline{
\hskip -0.0in
\includegraphics[height=8.0truecm,angle=0]{f_rho0_vs_dlam_nt7_alln_dl4MeV.pdf}
}
\vskip -0.05in
\caption{Coarse--grained mode condensate $\rho(0,\Delta)$ at $T/T_c=1.2$
for increasing 3--volumes.}
\label{fig:rho0_Delta_all}
\vskip -0.15in
\end{center}
\end{figure}
\subsection{Absolute \Xg--Distributions at Finite Temperature}
\label{ssec:Xd}
In the previous analysis of QCD phase transition, we focused on spectral properties
based on the correlation coefficient of chiral polarization $\cop_A$. Indeed, this
is sufficient for formulating and verifying the vSChSB--ChP correspondence which is
our main focus here. However, for other purposes involving vacuum structure, more
detailed information contained in absolute \Xg--distributions $\xd_A(\Xg)$ might be
valuable. In Ref.~\cite{Ale10A} ({\em Proposition 1}) it was concluded that,
in N$_f$=0 QCD at zero temperature, $\xd_A(\Xg)$ has a simple behavior for
low--energy Dirac eigenmodes: it is either purely convex or purely concave, with
the former being associated with polarization while the latter with anti--polarization.
It is thus a natural question to ask whether something different happens in this
regard due to thermal agitation.
To start such inquiry, we wish to establish how \Xg--distribution of lowest modes
changes with temperature and, in particular, whether a qualitative shift occurs when
crossing the chiral transition point $T_{ch}$. To formalize this question, it is
preferable to think in terms of $\xd_A(\Xg,\sigma)$ rather than the canonical
$\xd_A(\Xg,\lambda)$ of Eq.~\eqref{eq:4.045}. Indeed, this puts the low--temperature
systems with abundance of small eigenvalues, and the high--temperature systems with
low--energy depletion, on the same footing. We are then interested in
$\xd_A(\Xg,\sigma \!\to\! 0)$ which in practice needs to be coarse--grained with
respect to $\sigma$. The latter is accomplished by considering
$\xd_A(\Xg,\sigma\!=\!0,\Delta)$, which represents $\xd_A(\Xg,\sigma)$ averaged
over $\sigma \in [0,\Delta]$.
\begin{figure}[t]
\begin{center}
\centerline{
\hskip 0.0in
\includegraphics[height=7.0truecm,angle=0]{f_PA_vs_temp_real_25modes.pdf}
}
\vskip -0.10in
\caption{The absolute \Xg--distribution $\xd_A(\Xg,\sigma\!\to\!0)$, i.e. for
lowest modes in $E$--ensembles.}
\label{fig:PA0_E_all}
\vskip -0.35in
\end{center}
\end{figure}
The result of such calculation for $E$--ensembles is shown in Fig.~\ref{fig:PA0_E_all}.
We chose to fix the (total) number of included modes to 25 for each system, which
corresponds to small coarse--graining parameter $\Delta$ in the range 0.02--0.15 fm$^{-4}$.
This statistics is also sufficient to verify that further lowering this cut doesn't
change the behavior of absolute $\Xg$--distribution. As one can see, chiral transition
is dramatically reflected in $\xd_A(\Xg,\sigma\!=\!0)$: we observe a new type of functional
behavior above $T_{ch}$ (ensembles $E_7$ and $E_8$), namely that of indefinite convexity.
To specify the convexity properties more precisely, we will implicitly view $\xd_A(\Xg)$
as defined on a positive chiral branch, namely $\Xg \in [0,1]$, in what follows. The new
behaviors we found are then characterized by presence of a single inflection point
$\Xg_0 \in (0,1)$ of the type concave-to-convex, i.e. $\xd_A(\Xg)$ is concave on
$[0,\Xg_0]$ and convex on $[\Xg_0,1]$.
\medskip
\noindent {\bf Conjecture 4}
\smallskip
\noindent {\sl Consider lattice--regularized N$_f$=0 theory at arbitrary temperature
$T$. Let $T_{ch}\!=\!T_{ch}(\Lambda_{lat},V_3)$ be the temperature of chiral polarization
transition at sufficiently large ultraviolet cutoff $\Lambda_{lat}$, and sufficiently
large 3--volume $V_3$. Then absolute \Xg--distribution $\xd_A(\Xg,\sigma=0)$ is convex
for $T<T_{ch}$, and has a single inflection point of type concave-to-convex for
$T>T_{ch}$.}
\medskip
\noindent Note that in the fixed--scale approach, utilized in our numerical work,
the set of accessible temperatures is discrete. Thus, rather than a unique $T_{ch}$
at fixed finite $\Lambda_{lat}$, there is a range associated with the two successive
values of $N_t$ across which the transition occurs. This is implicitly understood in
the above. A fine--grained question in this regard is whether there could be a brief
phase above $T_{ch}$ where $\xd_A(\Xg,\sigma\!=\!0)$ is concave, and which cannot be
resolved at the lattice cutoff we are using.
The above finding naturally raises questions about the prevalence of modes
with indefinite convexity in the bulk of the finite--temperature spectrum.
First of all, we have not found any modes of indefinite convexity for $T<T_c$.
The absence of such modes at zero temperature was the the main part of
{\em Propositions 1,3} in Ref.~\cite{Ale10A}, and it carries over to this wider
regime. The proposed statement in the language used here is as follows.
\medskip
\noindent {\bf Conjecture 5a}
\smallskip
\noindent {\sl Consider lattice--regularized N$_f$=0 theory at temperature
$T\!<\!T_c \!=\!T_c(\Lambda_{lat})$. If $\,\Omega\!=\!\Omega(\Lambda_{lat},V_3)$ is
the density of chirally polarized modes, then the following holds at sufficiently
large $\Lambda_{lat}$ and $V_3$. Absolute \Xg--distribution $\xd_A(\Xg,\sigma)$ is
(i) convex in $\Xg$ for $\sigma \in (0,\Omega)$, (ii) uniform for $\sigma=\Omega$,
and (iii) concave at least for some band $\sigma > \Omega$.}
\medskip
\begin{figure}[t]
\begin{center}
\centerline{
\hskip 0.0in
\includegraphics[width=12.0truecm,angle=0]{f_PA_real_nt6_manysigma_ds02_v3.pdf}
}
\vskip -0.15in
\caption{The absolute \Xg--distribution $\xd_A(\Xg,\sigma,\delta=0.2)$ for
ensemble $E_7$ ($T/T_c\!=\!1.39$) with increasing values of $\sigma$.
Value $\delta=0.2$ entails averaging over 46 eigenmodes.}
\label{fig:PAsig_E_7}
\vskip -0.40in
\end{center}
\end{figure}
\noindent It turns out that the situation is in fact analogous for $T>T_{ch}$ except
that the role of chirally polarized (convex) modes is assumed by modes with
indefinite convexity of $\xd_A(\Xg)$. To support this, we show in
Fig.~\ref{fig:PAsig_E_7} the sequence of absolute \Xg--distributions with increasing
$\sigma$ for ensemble $E_7$. Note that for non--zero values of $\sigma$ we use
symmetric coarse--graining, averaging over the interval
$(\sigma - \delta/2, \sigma + \delta/2)$. The data suggests the existence of
a point $\sigma = \Omega_1$ where the behavior changes from convex--indefinite
to concave. At this particular temperature the transition occurs in the vicinity of
$\,\Omega_1 \approx 3.0\,$fm$^{-4}$ and a precise determination can be performed if
desired. We are thus led to formulate the following statement.
\medskip
\noindent {\bf Conjecture 5b}
\smallskip
\noindent {\sl Consider lattice--regularized N$_f$=0 theory at temperature
$T\!>\!T_{ch} \!=\!T_{ch}(\Lambda_{lat},V_3)$. At sufficiently large
$\Lambda_{lat}$ and $V_3$ there exists $\,\Omega_1\!=\!\Omega_1(\Lambda_{lat},V_3,T)$
such that the following holds. Absolute \Xg--distribution $\xd_A(\Xg,\sigma)$
(i) has a single inflection point of type concave-to-convex for $0<\sigma<\Omega_1$
and (ii) is concave at least for some band $\sigma > \Omega_1$.}
\medskip
We finally turn to the ``mixed phase'' ($T_c < T < T_{ch}$). This dynamics
exhibits chiral polarization ($\Omega>0$), which in case of confined vacuum happens
to be synonymous with convexity of $\xd_A(\Xg)$ for $\sigma < \Omega$.
However, {\em Conjecture 5b} suggests that
deconfinement is tied to indefinite convexity of absolute \Xg--distributions.
How then is the coexistence of vSChSB and deconfinement, and thus of chiral
polarization and indefinite convexity, realized in the mixed phase? The specific
arrangement we found is exemplified via ensemble $G_3$ in Fig.~\ref{fig:pa_n32_nt7}.
The spectrum starts with a layer of convex modes (top left), like at low
temperatures, but the distribution loses its definite convexity for
$\Omega' < \sigma < \Omega_1$, after which it becomes concave (bottom right).
There is a band $\Omega_0 < \sigma < \Omega_1$ within the convex--indefinite regime,
where modes are of the type found at $T>T_{ch}$, i.e. their $\xd_A(\Xg)$ has one
concave-to-convex inflection point (bottom left). We thus propose the following.
\medskip
\noindent {\bf Conjecture 5c}
\smallskip
\noindent {\sl Consider lattice--regularized N$_f$=0 theory at finite temperature
and overlap valence quarks. There exist lattice cutoffs $\Lambda_{lat}$ such
that $T_c(\Lambda_{lat}) < T_{ch}(\Lambda_{lat},V_3)$ for sufficiently large $V_3$.
At temperatures $T_c < T < T_{ch}$ there are $\Omega' < \Omega_0 < \Omega_1$
such that $\xd_A(\Xg,\sigma)$ is (i) convex for $0 < \sigma < \Omega'$,
(ii) convex--indefinite but not of type (iii) for $\Omega' < \sigma < \Omega_0$,
(iii) has one inflection point of concave-to-convex type for $\Omega_0 < \sigma < \Omega_1$,
and (iv) is concave at least for some band $\sigma > \Omega_1$.
Moreover, $\Omega' < \Omega < \Omega_0$.}
\medskip
\noindent It should be pointed out that the convex--indefinite spectral band of
{\sl (ii)} may just be the ``reversed'' version of {\sl (iii)}, namely that
the associated $\xd_A(\Xg)$ has a single inflection point of convex-to-concave type.
However, our statistics is not large enough to support this aspect with sufficient
certainty. Note also that the above formulation doesn't explicitly exclude
the possibility that $\Omega'=0$, neither in finite volume nor in the infinite volume
limit.\footnote{This doesn't necessarily contradict {\em Conjecture 4} which is only
concerned with the limit $\xd_A(\Xg,\sigma\to 0)$.} However, $\Omega_0$ is predicted
to be positive in both cases since $\Omega$ is.
\begin{figure}[t]
\begin{center}
\centerline{
\hskip 0.00in
\includegraphics[width=8.0truecm,angle=0]{f_PA_real_nt7_ns32_sigma0005_ds001_v2.pdf}
\hskip -0.00in
\includegraphics[width=8.0truecm,angle=0]{f_PA_real_nt7_ns32_sigma0045_ds001_v2.pdf}
}
\vskip -0.15in
\centerline{
\hskip 0.00in
\includegraphics[width=8.0truecm,angle=0]{f_PA_real_nt7_ns32_sigma1_ds02.pdf}
\hskip -0.00in
\includegraphics[width=8.0truecm,angle=0]{f_PA_real_nt7_ns32_sigma4_ds02.pdf}
}
\vskip -0.1in
\caption{Absolute \Xg--distributions in ensemble $G_3$ for increasing $\sigma$
representing four kinds of behavior found. The upper two results were obtained
with coarse--graining parameter $\delta=0.01$ (23 modes),
while $\delta=0.2$ (460 modes) was used for the lower two.}
\label{fig:pa_n32_nt7}
\vskip -0.35in
\end{center}
\end{figure}
\subsection{Dirac Mode Landscape at Finite Temperature}
\label{ssec:convexity}
According to vSChSB--ChP correspondence, chiral symmetry restoration at finite
temperature is the process of chiral depolarization (positive $\cop_A$ becoming
negative) in the Dirac spectrum. However, results of the previous section suggest
that more detailed polarization characteristic -- absolute \Xg--distribution -- encodes
{\em both} major effects of thermal agitation: valence chiral symmetry restoration and
deconfinement. This information is stored in convexity
properties of $\xd_A(\Xg)$ which can in this case even be invoked on their own, without
explicit reference to $\cop_A$. The resulting eigenmode ``convexity landscape''
is schematically shown in Fig.~\ref{fig:illus2} with blue and red color marking
purely convex and purely concave behavior of $\xd_A(\Xg)$ respectively. The two shades
of green represent convex--indefinite $\xd_A(\Xg)$, with darker version signifying
the presence of a single concave-to-convex inflection point.
\begin{figure}[t]
\begin{center}
\centerline{
\hskip 0.0in
\includegraphics[width=16.0truecm,angle=0]{f_plot-4.pdf}
}
\vskip -0.00in
\caption{Schematic view of Dirac eigenmodes at finite temperature according to
their absolute \Xg--distributions with blue representing convexity, red concavity,
and shades of green indefinite convexity. The left side corresponds to continuum--limit
scenario with $T_c<T_{ch}$ when the mixed phase (middle bar) exists, while the right
side to scenario with $T_c=T_{ch}$.}
\label{fig:illus2}
\vskip -0.35in
\end{center}
\end{figure}
Notice that Fig.~\ref{fig:illus2} offers two scenarios for possible behavior
in the vicinity of QCD phase transition. Indeed, whether {\em continuum} N$_f$=0 QCD
exhibits the mixed phase or not remains an open issue, and the formulation of
{\em Conjecture 5c} reflects this status. Thus, the option with mixed phase is shown
on the left, and the one without it on the right. Regardless of which possibility
is realized in the continuum however, the results provide for definite and
somewhat unexpected new characterization of confinement via chirality properties.
\medskip
\noindent {\em In N$_f$=0 QCD at finite temperature, confined vacuum supports
$\xd_A(\Xg)$ of definite convexity in overlap Dirac modes, while deconfined
vacuum produces a band of modes with convex--indefinite $\xd_A(\Xg)$ at
low $\sigma$.}
\medskip
With regard to the situation at high temperature ($T>T_{ch}$), one should mention
a possible connection of these findings to recent works, proposing that QCD phase
transition can be viewed as a version of Anderson localization, with increasing
temperature controlling the degree of randomness~\cite{Gar06A,Kov10A,Kov12A}.
In this scenario, there
is an analogue of Anderson's mobility edge, below which Dirac modes are localized.
Given that absolute \Xg--distribution is a fully dynamical characteristic of
the modes, it is reasonable to put forward the hypothesis that the suggested mobility
edge is associated with ``concavity edge'' $\Omega_1$ from the above analysis.
Denoting by $\lambda=\Lambda_{ch1}$ the spectral scale corresponding
to $\Omega_1$, it is natural to expect that the mobility edge in fact coincides
with $\Lambda_{ch1}$. The merits of this hypothesis need to be examined in a detailed
dedicated study.
\vfill\eject
\section{Effects of Many Light Flavors}
\label{sec:light}
The second and qualitatively different route toward chiral symmetry restoration
within ${\mathscr T}$ proceeds via including increasing number of dynamical quark flavors.
While the general tendency is significantly more general, the usual setup deals with
N$_f$ massless flavors at zero temperature. Owing to numerous lattice studies, as well
as other considerations, there is very little doubt that the canonical N$_f$=2 case
exhibits SChSB. However, massless fermions weaken the running of the gauge coupling,
and it is expected that there is a critical number of flavors N$_{f,cr}$, beyond which
chiral symmetry remains unbroken. The existence of N$_{f,cr}$ is connected to a larger
issue, namely the existence of a ``conformal window'' in flavor, containing theories
with infrared fixed point~\cite{Ban82A}. Indeed, this is expected to occur at
N$_{f,cr}<\;$N$_f<16.5$ for SU(3), and the reliable determination of N$_{f,cr}$ is of
an ongoing interest. The specific issue whether N$_f$=12 theory belongs to the conformal
window gained a particular attention recently, as discussed e.g. in
reviews~\cite{Nei12A,Gie12A,Ito13A}.
Rather than entering the discussion of unresolved problems such as the above, our aim
is to check the plausibility of vSChSB--ChP correspondence in this important corner of
quark--gluon dynamics. It should be kept in mind that the validity of the proposed
relation is to be examined for any given lattice regularization of any given
theory from ${\mathscr T}$: it either holds or not for the regularized system at hand.
We will thus not be much concerned with extrapolating quark mass to zero, or detailed
issues of continuum limit. Instead, our goal is to check the correspondence in the situation
where the effect of many light fermions is apparent, and the possibility for valence
chiral restoration exists.
\subsection{Lattice Setup}
For purposes of this pilot inquiry, we obtained some of the previously generated N$_f$=12
staggered fermion ensembles described in Ref.~\cite{Has12B}. More specifically,
the regularization in question uses a negative adjoint plaquette term (coupling $\beta_A$)
in addition to fundamental plaquette (coupling $\beta_F$), and nHYP--smeared staggered
fermions. This arrangement helps with ameliorating the problems of spurious UV fixed
points and the unphysical lattice phases encountered in theories with many light flavors.
Such issues have been carefully studied in this particular setting~\cite{Has12A}, and
are not expected to arise for ensembles listed in Table~\ref{tab:stag_ensemb}.
\begin{table}[b]
\centering
\begin{tabular}{@{} cccccccccc @{}}
\toprule
Ensemble & Size & $\beta_F$ & $\beta_A/\beta_F$ & am & N$_{conf}$ &
$|\lambda|_{min}^{av}$ & $|\lambda|_{min}$ &
$|\lambda|_{max}^{av}$ & $|\lambda|_{max}$\\
\midrule
$S_{1}$ & $16^{3}\times 32$ & 2.8 & -0.25 & 0.0200 & 100 &
0.0098 & 0.0007 & 0.5190 & 0.5259\\
$S_{2}$ & $32^{3}\times 64$ & 2.8 & -0.25 & 0.0025 & 30 &
0.0007 & 0.0001 & 0.2016 & 0.2035\\
$S_{3}$ & $24^{3}\times 48$ & 2.8 & -0.25 & 0.0025 & 50 &
0.0193 & 0.0001 & 0.3046 & 0.3075\\
\bottomrule
\end{tabular}
\caption{Ensembles of N$_f$=12 lattice QCD with nHYP smeared staggered fermions and
fundamental--adjoint ($\beta_F\,$--$\,\beta_A$) gauge plaquette action~\cite{Has12B}.}
\label{tab:stag_ensemb}
\end{table}
Simulations of many--light--flavor systems have to deal with the fact that the equilibrium
gauge fields are quite rough at currently accessible lattice cutoffs. Incorporating
smearing into definition of lattice Dirac operator helps to make such calculations
feasible. To define our overlap chiral probe, we use the same smearing procedure that
was used in Monte Carlo generation of the ensembles. Nevertheless, it is prudent
to exercise some care when using overlap operator even on fields that are moderately
rough. Indeed, the physical branch of the Wilson--Dirac spectrum can shift significantly
away from the origin on such backgrounds, and the mass parameter $\rho \in (0,2)$
in overlap construction needs to be chosen sufficiently large to contain it.
To avoid potential issues of this kind, we set $\rho=1.55$, which is somewhat larger
than $\rho=26/19 \approx 1.37$ used in our ``real world QCD'' simulations.
Performing small statistics calculations with ensemble $S_1$, we verified that overlap
low--mode abundances are reasonably stable in the vicinity of this value.
At fixed N$_f$, the mass $m$ of degenerate quarks provides for the only parameter
distinguishing various physical behaviors in this set of theories. At N$_f$=2,
there is valence spontaneous chiral symmetry breaking at arbitrary $m$, while
at N$_f$=12, there could be a transition to chirally symmetric vacuum when $m$ is
sufficiently small. The logic behind the choice of ensembles in
Table~\ref{tab:stag_ensemb} is that system $S_1$, characterized by larger mass, was
found in Ref.~\cite{Has12B} to be mode condensing (vSChSB) with respect to staggered
Dirac operator, while the system represented by $S_2$ and $S_3$ appeared non--condensing
(valence chiral symmetry) at this cutoff. Note that $S_2$ and $S_3$ only differ by volume
to give a sense of finite--volume effects.
\begin{figure}[t]
\begin{center}
\centerline{
\hskip 0.00in
\includegraphics[width=8.0truecm,angle=0]{f_CA_vs_lambda_nf12_1632.png}
\hskip -0.00in
\includegraphics[width=8.0truecm,angle=0]{f_sigmach_vs_sigma_nf12_1632_ds0153e-4.pdf}
}
\vskip -0.05in
\centerline{
\hskip 0.00in
\includegraphics[width=8.0truecm,angle=0]{f_rho_vs_lambda_nf12_1632_dl0015_v2.pdf}
\hskip -0.00in
\includegraphics[width=8.0truecm,angle=0]{f_PA_real_nf12_1632_low15_perensemble.pdf}
}
\vskip -0.1in
\caption{Chiral polarization characteristics for ensemble $S_1$. See the discussion
in the text.}
\label{fig:S1}
\vskip -0.35in
\end{center}
\end{figure}
\subsection{Selected Results}
We now proceed to discuss the results from the perspective of
vSChSB--ChP correspondence. Our extensive treatment of formalism in
Sec.~\ref{sec:background} and analysis of finite--temperature data in
Sec.~\ref{sec:fintemp} served in part to identify effective ways
to perform a study of this type. Here we show the raw data for $\cop_A$
vs $\lambda$, together with the plots of $\sigma_{ch}(\sigma)$,
$\rho(\lambda)$, and $\xd_A(\Xg,\sigma \to 0)$. The former three
are designed to reveal whether vSChSB and ChP are tied together,
while the latter serves as a first step to explore the newly proposed
convexity connections in this particular corner of quark--gluon dynamics.
Fig.~\ref{fig:S1} shows the above set of characteristics for the system
at larger of the two quark masses, represented by ensemble $S_1$. A mere
glance at the raw data (top left) reveals the presence of chiral
polarization at low energy without noticeable depletion of eigenvalues
near the origin. The system thus shows simultaneous signs of chiral
polarization and mode condensation in accordance with vSChSB--ChP
correspondence. This is confirmed by the behavior of $\sigma_{ch}(\sigma)$
(top right) and $\rho(\lambda)$ (bottom left). Indeed, the former shows
the positive bump at low $\sigma$, characteristic of chiral polarization,
while the behavior in the latter is typical of mode--condensing theory.
Focusing now on the situation at smaller mass, the same set of characteristics
are shown in Fig.~\ref{fig:S2S3}, with the smaller volume ($S_3$) in the left
column and the larger volume ($S_2$) in the right column. From the raw data
alone (top row) one can immediately see that a qualitative change in the Dirac
spectrum indeed occurred. However, it is not a simple depletion of eigenvalues in
the vicinity of the origin as one would naively expect. Rather, there is a break
in the spectrum, characterized by significant depletion, together with the
accumulation of modes very close to zero. Given that, vSChSB--ChP correspondence
predicts chiral polarization at the low end of the spectrum. This is indeed
featured in the raw data qualitatively, and is properly quantified via the behavior
of $\sigma_{ch}(\sigma)$ (second row).
While the options are limited for finite--volume scaling with only two volumes
available, there is little doubt that the theory in question is overlap mode
condensing. Indeed, as one can see in the third row of Fig.~\ref{fig:S2S3},
the peak in the density near the origin is actually growing as the volume
is increased. On the chiral polarization side, the positive maximum in
$\sigma_{ch}(\sigma)$, namely $\Omega_{ch}$, visibly shrinks at larger volume.
Thus, a detailed finite volume study is required to decide whether
the infinite--volume correspondence in the form of {\sl Conjecture 2} or
{\sl Conjecture 2'} holds true.
However, the position of the maximum in $\sigma_{ch}(\sigma)$, namely $\Omega$,
in fact mildly increases. We thus expect that the correspondence in the form
of {\sl Conjecture 2''} certainly holds in this case. Needless to say, our results
are also in agreement with chiral polarization characteristics being
the finite--volume order parameters of vSChSB, and thus concur with
{\sl Conjectures 3,3'}.
Lastly, we comment on the behavior of absolute \Xg--distributions for lowest
modes, i.e. $\xd_A(\Xg,\sigma\to 0)$. These results are shown in the lower right
plot of Fig.~\ref{fig:S1} and the last row of Fig.~\ref{fig:S2S3}. Only the lowest
15 modes from each ensemble were used in the computation, leading to
coarse--graining parameter $\Delta \ll \Omega$ in each case. All three
systems exhibit convex behavior, thus following the same pattern observed in
case of N$_f$=0 at finite temperature for chirally polarized theories ($T<T_{ch}$).
It should be mentioned here that the same holds for the N$_f$=2+1 systems close
to ``real world QCD'' studied in Ref.~\cite{Ale12D}. It is thus reasonable
to expect that vSChSB is equivalent to convexity of $\xd_A(\Xg,\sigma\to 0)$
over the whole base set ${\mathscr T}$.
\begin{figure}
\begin{center}
\centerline{
\hskip -0.05in
\includegraphics[width=8.0truecm,angle=0]{f_CA_vs_lambda_nf12_2448.png}
\hskip -0.00in
\includegraphics[width=8.0truecm,angle=0]{f_CA_vs_lambda_nf12_3264_v3.png}
}
\vskip -0.15in
\centerline{
\hskip 0.14in
\includegraphics[width=8.0truecm,angle=0]{f_sigmach_vs_sigma_nf12_2448_ds0002e-4.pdf}
\hskip -0.00in
\includegraphics[width=8.0truecm,angle=0]{f_sigmach_vs_sigma_nf12_3264_ds0002e-4.pdf}
}
\vskip -0.02in
\centerline{
\hskip 0.00in
\includegraphics[width=8.0truecm,angle=0]{f_rho_vs_lambda_nf12_2448_dl0005_v2.pdf}
\hskip -0.00in
\includegraphics[width=8.0truecm,angle=0]{f_rho_vs_lambda_nf12_3264_dl0005_v2.pdf}
}
\vskip -0.15in
\centerline{
\hskip 0.20in
\includegraphics[width=8.0truecm,angle=0]{f_PA_real_nf12_2448_low15_perensemble.pdf}
\hskip 0.05in
\includegraphics[width=8.0truecm,angle=0]{f_PA_real_nf12_3264_low15_perensemble.pdf}
}
\vskip -0.1in
\caption{Chiral polarization characteristics for ensemble $S_2$ (right column)
and $S_3$ (left column). See the discussion in the text.}
\label{fig:S2S3}
\vskip -0.15in
\end{center}
\end{figure}
\subsection{Intermediate Phases}
The above discussion of finite--temperature and many--flavor results evokes
certain analogy between the mixed phase in N$_f$=0 QCD and the situation
found in N$_f$=12 at lighter mass (ensembles $S_2$ and $S_3$). It appears
that decreasing quark mass for N$_f$=12 has a dynamical effect similar to
increasing temperature for N$_f$=0 in that a phase with narrow band of
near--zeromodes separated from the bulk is generated, at least for certain
range of cutoffs. In thermal case, the onset of this behavior appears
to coincide with deconfinement, and thus $T_c$. In many--flavor case,
where confinement is non--trivial to define, there is presumably a mass
$m_{in}$ below which this starts occurring.\footnote{If this behavior survives
the continuum limit, then some universal parametrization of the transition
point e.g. in terms of ratios of certain hadron masses, will be more
appropriate. The same applies to $m_{ch}$.}
Regardless of whether the existence of $m_{in}$ is a lattice artifact,
we can define an analog of $T_{ch}$, namely $m_{ch}$, below which valence chiral
symmetry gets restored in N$_f$=12. The aforementioned analogy would then apply
to the regime $T_c<T<T_{ch}$ of N$_f$=0 and $m_{ch}<m<m_{in}$ of N$_f$=12,
which we refer to as ``intermediate phases'' in what follows. It should be
emphasized that whether $m_{ch}$ is zero or non--zero is currently an open issue.
In what follows we summarize few observations on the two intermediate phases
and compare them. Given that the available data on the many--flavor side is
rather limited, this should be considered an initial assessment which can
serve as a starting point for more detailed investigation.
\medskip
\noindent {\bf (i) Separation from the bulk.} Both intermediate phases are characterized
by anomalous behavior of spectral density $\rho(\lambda)$, wherein the narrow
peak forms near the origin, and is clearly distinguished from the rest of the spectrum.
This is ``anomalous'' in the sense that, by virtue of the above, $\rho(\lambda)$
becomes non--monotonic. Note that in the thermal case, data indicates that the peak
of near--zeromodes becomes of $\delta$--function type in the infinite volume limit
while it is currently not clear what happens in N$_f$=12.
\medskip
\noindent {\bf (ii) Inhomogeneity.} The anomalous near--zeromodes are highly
inhomogeneous in both intermediate phases. By this we mean that the bulk of their
norm is carried by very small fraction of space--time points. Here we will not
focus on quantifying this feature, but it will certainly become a characteristic
of interest if either one of the intermediate phases turns out to be the reality of
the continuum limit.
\medskip
\noindent {\bf (iii) Indefinite convexity.} As discussed in
Secs.~\ref{ssec:Xd},\ref{ssec:convexity}, thermal transition to intermediate (mixed)
phase in N$_f$=0 is characterized by the appearance of modes with indefinite convexity
of absolute \Xg--distributions. In case of N$_f$=12 we do not find a clear evidence
of this happening. In fact, the spectral transition from chirally polarized to chirally
anti--polarized regime looks more like a direct transition from strictly convex to strictly
concave distribution. In Fig.~\ref{fig:pa_nf12_trans} (left column) we show
$\xd_A(\Xg,\sigma)$
for $\sigma \approx \Omega$. As one can see both for heavier mass (top) and the lighter
mass in intermediate phase (bottom), the absolute \Xg--distribution is flat which is
characteristic of the direct transition. The right column in the figure illustrates the
concave behavior at larger values of $\sigma$. Thus, to the extent that indefinite
convexity of $\xd_A$ reflects deconfinement even in the situation with light dynamical
quarks, the intermediate phase in N$_f$=12 exhibits not only valence chiral symmetry
breaking but also signs of confinement. Detailed inquiry at yet lower mass should clarify
this further.
\begin{figure}[t]
\begin{center}
\centerline{
\hskip 0.00in
\includegraphics[width=8.0truecm,angle=0]{f_PA_real_nf12_1632_sigma5e-5_ds1e-5_v2.pdf}
\hskip -0.00in
\includegraphics[width=8.0truecm,angle=0]{f_PA_real_nf12_1632_sigma2e-4_ds1e-5_v2.pdf}
}
\vskip -0.15in
\centerline{
\hskip 0.00in
\includegraphics[width=8.0truecm,angle=0]{f_PA_real_nf12_3264_sigma1e-6_ds2e-7_v2.pdf}
\hskip -0.00in
\includegraphics[width=8.0truecm,angle=0]{f_PA_real_nf12_3264_sigma4e-6_ds2e-7_v2.pdf}
}
\vskip -0.1in
\caption{Absolute \Xg--distributions $\xd_A(\Xg,\sigma)$ for ensemble $S_1$ (top)
and $S_2$ (bottom). In the left column $\sigma\approx \Omega$ while in the right
column $\sigma > \Omega$. Coarse--graining parameter $\delta=10^{-5}$ was
used for $S_1$ and $\delta = 2 \times 10^{-7}$ for $S_2$.}
\label{fig:pa_nf12_trans}
\vskip -0.35in
\end{center}
\end{figure}
\section{Discussion}
The main purpose of this work is to test the idea that, for quark--gluon interactions
of QCD type, generating chiral condensate is the same thing as generating the layer
of chirally polarized Dirac eigenmodes at low end of the spectrum~\cite{Ale12D}.
When viewed as a feature characterizing the state of vacuum correlations, this
connection is not limited to theories containing massless quarks, but applies
generically, even when all quarks are massive. In such generalized context,
vacuum correlations are probed by a pair of external (valence) massless quarks
and the role of chiral symmetry breaking is assumed by its valence counterpart.
Examining the temperature effects in N$_f$=0 QCD, and effects of many light flavors
at $T\!=\!0$ (for N$_f$=12), we find a complete agreement with this vSChSB--ChP
correspondence.
One relevant aspect of this connection, both practically and conceptually,
is that it is perfectly well defined even at lattice--regularized level. Indeed, if
lattice fermions with exact chiral symmetry provide the massless probe, then vSChSB
has a full-fledged lattice representation.\footnote{ChP is in fact well--defined
even if probing fermion is not exactly chiral away from continuum limit.}
The associated conjectures can then be formulated directly for lattice theories,
as done here, but with compliance only expected sufficiently close to the continuum
limit. Nevertheless, we found the agreement for every lattice system studied, even in
corners where relevance for continuum physics hasn't yet been established, e.g. in
the ``mixed phase'' of N$_f=0$ QCD. It is thus conceivable that the connection is
even more robust than originally expected.
Important motivation to identify the correspondence of vSChSB--ChP type is to narrow
down options in searches for specific mechanism of the breaking phenomenon. Indeed,
the two aspects involved are not required to be locked together by general
principles and are in fact quite different in nature. Like that of any broken symmetry,
the definition of vSChSB is intimately tied to thermodynamic limit (infinite volume).
On the other hand, ChP is defined in any fixed finite volume which is quite natural
given its role to characterize and detect chiral symmetry breaking nature of
the interaction at hand. Indeed, such indicator is not expected to turn itself on
in infinite volume only, but rather when volume is sufficiently large to contain
relevant finite scales of the theory.
Above considerations lead to the notion of {\em finite--volume ``order parameter''}
introduced in Sec.~\ref{ssec:finvol}. This is intended to be a dynamical quantifier
assuming non--zero value in sufficiently large {\em finite} volumes if and only if
the symmetry in question is broken (in infinite volume). The width of chiral
polarization layer ($\Lambda_{ch}$), the volume density of total chirality
($\Omega_{ch}$), and the volume density of polarized modes ($\Omega$), are viewed
as prototypes of such objects associated with vSChSB (see {\sl Conjectures 3,3'}$\,$),
and all available data is consistent with this proposition. In fact, we have not yet
encountered a reversal of dynamical tendency for chiral polarization due to the volume
being too small. In this vein, it is instructive to think of chiral polarization
framework as an attempt to construct a ``predictor'' of valence chiral symmetry
breaking for QCD--like theories. Such predictor, say $\Omega$, will be most efficient
if it is also a finite--volume order parameter. Indeed, a well--founded wager on
vSChSB can then be made based on the value of $\Omega$ in a single otherwise
acceptable volume.
Conceptual simplicity in the notion of finite--volume order parameter is further
underlined by the fact that all three ChP--based characteristics are (non)zero
simultaneously in finite volume. Thus, although the state of chiral polarization
is characterized by the triplet
\begin{equation}
\mbox{\rm ChP}_{V<\infty} \quad \longleftrightarrow \quad
\bigl(\,\Lambda_{ch}(V),\, \Omega_{ch}(V),\, \Omega(V) \,\bigr)
\end{equation}
the associated vSChSB inquiry in fact involves a single object, namely
\begin{equation}
\chi_p(V) \,\equiv\, \sgn \bigl( \Lambda_{ch} (V) \bigr)
\,=\, \sgn \bigl( \Omega_{ch} (V) \bigr)
\,=\, \sgn \bigl( \Omega(V) \bigr) \; \in \{0,1\}
\end{equation}
Note that, as finite--volume order parameters, elements of ChP$_{V<\infty}$ are not
required to have non--zero infinite--volume limits when symmetry is broken, but
$\chi_p \!\equiv\! \lim_{V\to\infty} \chi_p(V)=1$.
If, contrary to initial evidence, the notion of finite--volume order parameter on
$\euT$ doesn't materialize, vSChSB--ChP correspondence can still be based on order
parameters of more traditional variety: those indicating the symmetry breakdown
by positive {\em infinite--volume} value. The situation becomes more structured
though, since the classification of possible ChP behaviors in infinite volume is
given by the heptaplet of non--negative parameters
\begin{equation}
\mbox{\rm ChP}_{V=\infty} \quad \longleftrightarrow \quad
(\Lambda_{ch}, \Lambda_{ch}^\infty, \Omega_{ch}, \Omega_{ch}^\infty,
\Omega, \Omega^\infty,\chi_p)
\label{eq:5.010}
\end{equation}
with variety of mixed--positivity scenarios allowed in principle. In this language,
the hypothetical lack of finite--volume order parameter implies that $\chi_p$ is
not a good infinite--volume indicator of vSChSB. Nevertheless, in the vSChSB--ChP
correspondence being constructed, $\chi_p\!=\!0$ still indicates symmetric vacuum
since it implies vanishing of all elements in ChP$_{V=\infty}$ and thus
absence of any polarized behavior. However, to formulate the equivalence, one needs
to specify which forms of ChP$_{V=\infty}$ with $\chi_p\!=\!1$ are associated with
vSChSB. This heavily depends on the scope of polarized behaviors generated
by theories in $\euT$.
To this effect, we formulated three versions of such infinite--volume correspondence
that are not mutually exclusive, but include an increasingly large variety of chiral
polarization. The most restrictive form, {\sl Conjecture 2}, assumes that no singular
cases occur sufficiently close to the continuum limit~\cite{Ale12D}. Here
``singular'' refers to any combination of the first six elements in ChP$_{V=\infty}$
with mixed positivity, or with paired characteristics not matching
(e.g. $\Lambda_{ch} \ne \Lambda_{ch}^\infty$). In this case, each element of
ChP$_{V=\infty}$ (except $\chi_p$) is individually a valid infinite--volume order
parameter of vSChSB. Such scenario was only found to be violated in the narrow mixed
phase of N$_f$=0 QCD, but complete agreement may hold closer to continuum limit,
which is sufficient. In {\sl Conjecture 2'} we minimally expanded the range of
anticipated ChP behaviors in order to include the singular option encountered above,
namely $\Lambda_{ch}\!=\!\Lambda_{ch}^\infty\!=\!0$ and $0\!<\!\Omega_{ch}\!<\!\Omega$.
Here $\Omega_{ch}$, $\Omega$ and their partners are each an infinite--volume
order parameter of vSChSB. Most generally, {\sl Conjecture 2''} admits all singular
ChP behaviors, in which case $\Omega$ becomes the sole reliable indicator of
vSChSB: the layer of chirally polarized modes is expected to be physically relevant
only when total volume density of participating modes remains positive in
the infinite--volume limit.
We emphasize that having differently focused versions of vSChSB--ChP correspondence
simply reflects differently aimed benefits of the underlying relationship. Indeed,
the more restrictive the form of correspondence turns out to be valid, the more
information on the continuum behavior of vSChSB it conveys. On the other hand,
the more generic the formulation becomes, the wider its applicability becomes
in terms of cutoff theories. In an extreme case, the relationship might turn out
to be as generic as the vSChSB--QMC correspondence, i.e. valid at arbitrary
non--zero cutoff.
The analysis in this paper (and the discussion above) focuses attention to $\Omega$
as a central characteristic of ChP in relation to vSChSB. Indeed, not only
does $\Omega$ provide for the simple and most generic way to express vSChSB--ChP
correspondence, it is also expected to be a universal quantity characterizing QCD
vacuum (see Sec.~\ref{ssec:universal}). Moreover, our numerical experiments show that
the most practical scheme for detecting chiral polarization proceeds via computing
the dependence of $\sigma_{ch}$ on $\sigma$, from which $\Omega$ (and $\Omega_{ch}$)
is directly determined.
During the course of our main inquiry we encountered few results that are noteworthy
in their own right. The first one relates to the issue of thermal mixed phase
($T_c < T < T_{ch}$) in N$_f$=0 QCD, i.e. the existence of deconfined system in real
Polyakov line vacuum but with broken valence chiral symmetry~\cite{Edw99A}. This is
a well--posed and interesting question even at given cutoff, but its resolution
requires a careful volume study. Our data involves several volumes and the results
indicate that the mixed phase with overlap valence quarks does indeed exist, at least
for some range of cutoffs. The associated vSChSB proceeds via band of spatially
inhomogeneous and chirally polarized near--zeromodes, well separated from the rest of
the spectrum. The width of the band appears to vanish in the infinite--volume limit,
possibly involving $\delta(\lambda)$ singularity in spectral density. We found that
a similar phase also exists in N$_f$=12 theory at light quark mass, which deserves
a dedicated study.
The last side result we wish to discuss suggests novel characterization of
(de)confinement. While at $T<T_c$ only convex or concave absolute
$\Xg$--distributions of Dirac modes are found in N$_f$=0 QCD, the band of
convex--indefinite modes appears at $T>T_c$. Thus, at least in this setting,
the existence of such band seems to play the same role for deconfinement as
the existence of chirally polarized band plays for vSChSB. Both layers are present
in the mixed phase as they should. The situation at high temperatures brings up
an interesting question, namely how does the ``concavity edge'', marking the transition
from convex--indefinite to concave behavior in Dirac spectra at $T>T_c$, relate to
``mobility edge'' feature discussed in Refs.~\cite{Gar06A,Kov10A,Kov12A}?
With natural expectation being the coincidence of the associated scales, computations
needed to explore this issue are straightforward to set up.
\bigskip
\noindent{\bf Acknowledgments:}
We are indebted to Anna Hasenfratz and David Schaich for sharing their N$_f$=12
staggered fermion configurations for the purposes of this work. Thanks to Terry Draper
for discussions, and to Mingyang Sun for help with graphics. A. A. is supported by U.S.
National Science Foundation under CAREER grant PHY-1151648. I.H. acknowledges
the support by Department of Anesthesiology at the University of Kentucky.
\bigskip\medskip
\begin{appendix}
\section{Absolute Polarization and Dynamical Chirality}
\label{app:chirality}
In this Appendix we briefly describe the absolute (dynamical) polarization method
of Ref.~\cite{Ale10A}. In the present context the elementary object of study is
a chirally decomposed eigenmode $\psi=\psi_L+\psi_R$. Given that we are only
interested in local (on--site) relationship between left and right, the sufficient
information is stored in the probability distribution $\df(\psi_L,\psi_R)$ of its
values $(\psi_L(x),\psi_R(x))$. This setup coincides with the starting point of
a general approach that considers arbitrary stochastic quantity $Q$ with values
in a vector space decomposed into a pair of equivalent orthogonal subspaces
($Q = Q_1 + Q_2$, $Q_1 \cdot Q_2 = 0$), and whose ``dynamics'' is described by
symmetric probability distribution $\df(Q_1,Q_2)=\df(Q_2,Q_1)$.
Given the goal of characterizing the (normalized) asymmetry between the two subspaces
in favored values of $Q$ (polarization), the method proceeds by first marginalizing
the full distribution $\df(Q_1,Q_2)$ to the distribution of magnitudes $\db(q_1,q_2)$.
Indeed, the weight of a given subspace in sample $Q$ can be assessed via magnitude
$q_i \equiv |Q_i|$ of its component. One possibility for a normalized variable
expressing the desired relationship is~\cite{Hor01A}
\begin{equation}
\rpc \,=\, \frac{4}{\pi} \, \tan^{-1}\Bigl(\frac{q_2}{q_1}\Bigr) \,-\,1
\equiv \Fgr(q_1,q_2)
\label{eq:3.100}
\end{equation}
namely the {\em reference polarization coordinate}. Note that $\rpc \in [-1,1]$
with extremal values taken by samples strictly polarized into one of the subspaces,
and zero value assigned to strictly unpolarized samples ($q_1=q_2$). Probability
distribution of $\rpc$ in $\db(q_1,q_2)$, namely
\begin{equation}
\dop(\rpc) \,=\,
\int_0^\infty d q_1 \int_0^\infty d q_2 \,
\db(q_1,q_2) \; \delta\Bigl( \rpc - \Fg_r(q_1,q_2) \Bigr)
\label{eq:3.110}
\end{equation}
is called the {\em reference \Xg--distribution}, and represents detailed polarization
characteristic of dynamics $\df(Q_1,Q_2)$ with respect to polarization measure $\Fgr$.
The large freedom in choosing the polarization measure makes the above characteristic
highly non--unique and thus kinematic. Various ``reference frames'' of polarization
can be represented by suitably constructed {\em polarization functions} $\Fg(\rpc)$.
In this language, the reference \Xg--distribution is associated with polarization
function $\Fg_r(\rpc)=\rpc$. The main idea driving the absolute polarization method
is to adjust the polarization function characterizing $\db(q_1,q_2)$ so that it
measures polarization relative to its ``own statistical independence'', namely
relative to the stochastic dynamics described by
\begin{equation}
\db^u(q_1,q_2) \,\equiv\, p(q_1)\, p(q_2) \qquad\qquad
p(q) \equiv \int_0^\infty d q_2 \, \db(q,q_2) =
\int_0^\infty d q_1 \, \db(q_1,q)
\label{eq:3.120}
\end{equation}
One can show~\cite{Ale10A} that this is accomplished by utilizing the polarization function
\begin{equation}
\Fg_A(\rpc) \equiv 2 \int_{-1}^\rpc d y \, \dop^u(y) - 1
\label{eq:3.130}
\end{equation}
where reference \Xg--distribution $\dop^u(\rpc)$ is associated with uncorrelated
dynamics $\db^u(q_1,q_2)$. The corresponding distribution of polarization values, namely
\begin{equation}
\xd_A(\Xg) \,\equiv\,
\int_{-1}^1 d \rpc \,
\dop(\rpc) \; \delta\Bigl( \Xg - \Fg_A(\rpc) \Bigr)
\,=\, \frac{1}{2} \,
\frac{\dop \Bigl( \Fg_A^{-1}\,(\,\Xg\,) \Bigr)}
{\dop^u \Bigl( \Fg_A^{-1}\,(\,\Xg\,) \Bigr)}
\label{eq:3.140}
\end{equation}
is called the {\em absolute \Xg--distribution}. By construction, and as seen from
the above explicit form, $\xd_A(\Xg)$ is a differential measure quantifying polarization
tendencies relative to statistical independence. Moreover, it is unique: arbitrary
choice of the reference polarization coordinate (function) leads to the same
absolute \Xg--distribution~\cite{Ale10A}.\footnote{This uniqueness of the ``correlational''
approach is in fact why the method is referred to as {\em absolute}.}
Consequently, absolute \Xg--distribution is viewed as a genuinely dynamical concept.
Differential information contained in $\xd_A(\Xg)$ can be integrated into the
{\em correlation coefficient of polarization} $\cop_A \in [-1,1]$, namely
\begin{equation}
\cop_A \,\equiv \,
2\,\int_{-1}^{1} d \Xg \, |\Xg| \, \xd_A(\Xg) \,-\,1
\label{eq:3.150}
\end{equation}
Statistical meaning of $\cop_A$ is clarified by noting that the integral in the above
expression is the probability for sample drawn from $\db(q_1,q_2)$ to be more polarized
than sample drawn from $\db^u(q_1,q_2)$. Consequently, positive correlation means that
stochastic dynamics enhances polarization relative to statistical independence, while
negative correlation (anti--correlation) implies its suppression. $\df(Q_1,Q_2)$ is
said to support {\em dynamical polarization} in the former case and
{\em dynamical anti--polarization} in the latter.
In the context of Dirac eigenmodes and their chiral decomposition,
expressions like ``mode is chirally polarized'' vs ``anti--polarized'',
``mode is chirally correlated'' vs ``anti--correlated'', or
``mode supports dynamical chirality'' vs ``anti--chirality'',
are all verbal descriptions of $\cop_A>0$ vs $\cop_A<0$.
\section{Generalities on Spectral Definitions}
\label{app:spectral}
The primary entities involved in spectral definitions are the cumulative
densities $\sigma(\lambda)$ and $\sigma_{ch}(\lambda)$. At the regularized level,
$\sigma$ is proportional to certain cumulative probability function of $\lambda$
(non--decreasing, bounded) and can thus have at most countably many finite
discontinuities. All of its possible behaviors are then contained in the form
\begin{equation}
\sigma(\lambda,M,V) \,=\,
\sum_j A_j\, H(\lambda - \alpha_j) \;+\; \hat{\sigma}(\lambda,M,V)
\label{eq:a.010}
\end{equation}
where $\alpha_j=\alpha_j(M,V) \ge 0$ are the points of discontinuity,
$A_j=A_j(M,V) \ge 0$, and $H(x)$ is the left--continuous version of the Heaviside
step function ($H(x)=0$ for $x \le 0$ and $H(x)=1$ for $x>0$). $\hat{\sigma}$ is
a continuous non--decreasing function of $\lambda$ and, as such, it can only be
non--differentiable on the set of Lebesgue measure zero. However, the exotic
``Cantor function''--like cases, where the subset of non--differentiability is
uncountable, are very unlikely to appear in this physical context. Differentiable
functions producing derivatives that are discontinuous on uncountable subsets can
be quite safely omitted for the same reason. Thus, the ``standard model'' of
cumulative mode density, expected to cover all theories considered,
is \eqref{eq:a.010} with $\hat{\sigma}(\lambda)$ being a continuous non--decreasing
function that is continuously differentiable except for countably many (thus isolated)
points. The differential representation (generalized function) then exists
and is given by
\begin{equation}
\bar{\rho}(\lambda,M,V) \;=\;
\sum_j A_j\, \delta(\lambda - \alpha_j) \;+\; \hat{\rho}(\lambda,M,V)
\label{eq:a.020}
\end{equation}
where the ordinary function $\hat{\rho}(\lambda) = d \hat{\sigma}(\lambda)/d_+\lambda$
exists and is continuous except for points $a_k=a_k(M,V) \ge 0$ where it can have
an integrable divergence or simple discontinuity. Note that there is no a priori
relation between sets $\{\alpha_j\}$ and $\{a_k\}$.
The definition of cumulative chirality density is a priori less constraining on
its behavior. Indeed, while $\sigma_{ch}(\lambda)$ is bounded in absolute value
by $\sigma(\lambda)$, it is not necessarily monotonic,
and the most general form analogous to \eqref{eq:a.010} is thus not strictly
guaranteed. On the other hand, more singular behavior of $\sigma_{ch}(\lambda)$ would
require $\cop_A(\lambda)$ -- a dynamical property -- to be discontinuous on uncountable
subsets which is highly unlikely in this physical setting. Thus, the most general
form of $\sigma_{ch}(\lambda)$ expected to occur is
\begin{equation}
\sigma_{ch}(\lambda,M,V) \,=\,
\sum_j B_j\, H(\lambda - \beta_j) \;+\; \hat{\sigma}_{ch}(\lambda,M,V)
\label{eq:a.030}
\end{equation}
where $B_j=B_j(M,V)$ has indefinite sign, $\beta_j=\beta_j(M,V) \ge 0$,
and $\hat{\sigma}_{ch}$ is a continuous function of $\lambda$. In slightly
more restrictive ``standard model'', guaranteeing differential representation,
$\hat{\sigma}_{ch}(\lambda)$ is also continuously differentiable except for
countably many points, i.e.
\begin{equation}
\bar{\rho}_{ch}(\lambda,M,V) \;=\;
\sum_j B_j\, \delta(\lambda - \beta_j) \;+\; \hat{\rho}_{ch}(\lambda,M,V)
\label{eq:a.040}
\end{equation}
The function $\hat{\rho}_{ch}(\lambda) = d \hat{\sigma}_{ch}(\lambda)/d_+\lambda$
exists and is continuous everywhere except for points $b_k=b_k(M,V) \ge 0$
where it can have an integrable divergence or a simple discontinuity. Note that
the sets $\{\beta_j\} \subseteq \{\alpha_j\}$ need not be identical, and neither
do the sets $\{a_k\}$, $\{b_k\}$.
There is very little doubt that the behavior of $\sigma(\lambda)$ and
$\sigma_{ch}(\lambda)$ in all theories considered falls under the ``standard model''
description specified above. In fact, it appears very likely that only more
restricted forms will actually appear. Nevertheless, it is interesting to note
that the key concepts utilized in our discussion, namely that of chiral polarization
scale $\chps$ and low--energy chirality $\Omega_{ch}$, are well defined without
any assumptions placed on $\sigma_{ch}(\lambda)$ beyond its existence. More
precisely, below we put forward definitions that assign definite characteristics
to any real--valued function $\sigma_{ch}(\lambda)$ such that
$\sigma_{ch}(\lambda) \equiv 0$ for $\lambda \le 0$, and bounded on any
$(-\infty,\Lambda]$.
We first define $\chps$ as the ``largest'' $\Lambda$, such that
$\sigma_{ch}(\lambda)$ is strictly increasing on $[0,\Lambda]$, i.e.
\begin{equation}
\chps[\sigma_{ch}] \,\equiv\, \sup \, \{\, \Lambda \, | \,
\sigma_{ch}(\lambda_1) < \sigma_{ch}(\lambda_2)
\;\, \mbox{\rm for all} \;\,
0 \le \lambda_1 < \lambda_2 \le \Lambda \,\}
\label{eq:a.050}
\end{equation}
when $\sigma_{ch}$ is not strictly increasing on $[0,\infty)$, and $\chps=\infty$
otherwise. Note that when positive $\chps$ doesn't exist, the defining condition
\eqref{eq:a.050} is vacuously satisfied by $\chps=0$ which is then its assigned
value. Thus, $\chps[\sigma_{ch}]$ always exists and is non--negative.
Next, associate with $\sigma_{ch}(\lambda)$ its ``running maximum function''
\begin{equation}
\sigma_{ch}^m(\lambda) \,\equiv\,
\sup \, \{ \sigma_{ch}(\lambda') \,|\, \lambda' \le \lambda \,\}
\label{eq:a.060}
\end{equation}
which is a non--negative non--decreasing function bounded on any $(-\infty,\Lambda]$.
Thus, it can only be discontinuous via countably many finite jumps, and one--sided
limits exist everywhere. The low energy chiral polarization is then defined as
\begin{equation}
\Omega_{ch}[\sigma_{ch}] \,\equiv\,
\lim_{\lambda \to \chps^+} \sigma_{ch}^m(\lambda)
\label{eq:a.070}
\end{equation}
We emphasize that the above definitions of $\chps$ and $\Omega_{ch}$ coincide with those
given in the main text when $\sigma_{ch}(\lambda)$ is of ``standard form''.
\end{appendix}
\bigskip
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,422 |
\section{Introduction}
\textit{Stochastic resonance} (SR) is one of the most interesting
\textit{noise-induced phenomena} that arises from the interplay
between \textit{deterministic} and \textit{random} dynamics in a
\textit{nonlinear} system \cite{RMP}. A large number of examples
showing SR occur in \textit{extended} systems: for example, diverse
experiments were carried out to explore the role of SR in sensory
and other biological functions \cite{biol} or in chemical systems
\cite{sch}. These, together with the possible technological
applications, motivated many recent studies showing the possibility
of achieving an enhancement of the system response by means of the
coupling of several units in what conforms an \textit{extended
medium} \cite{extend1,extend2,extend3}.
In previous works \cite{extend2,extend3} we have studied the
stochastic resonant phenomenon in extended systems, when transitions
between two different spatial patterns occurs, exploiting the
concept of the \textit{non-equilibrium potential} (NEP)
\cite{GR,I0}: a Lyapunov functional of the associated deterministic
system that, for non-equilibrium systems, plays a role similar to
that of a thermodynamic potential in equilibrium thermodynamics.
Such NEP characterizes the global properties of the dynamics:
attractors, relative (or nonlinear) stability of these attractors,
height of the barriers separating attraction basins and, in
addition, allowing us to evaluate the transition rates among the
different attractors. In another work \cite{extend3c} we have also
shown that, for a scalar reaction-diffusion system with a
density-dependent diffusion and a known form of the NEP, the
non-homogeneous spatial coupling changes the effective dynamics of
the system and contributes to enhance the SR phenomenon.
Here we report on a study of SR in an extended system: an array of
FitzHugh-Nagumo \cite{FHN} units, with a density-dependent
(diffusive-like) coupling. The NEP for this system was found within
the excitable regime and for particular values of the coupling
strength \cite{extend3}. In the general case, however, the form of
the NEP has not been found yet. Nevertheless, the idea of the
existence of such a NEP is always \textit{underlying} our study.
Hence, we have resorted to an study based on numerical simulations,
analyzing the influence of different parameters on the system
response. The results show that the enhancement of the
signal-to-noise ratio found for a scalar system \cite{extend3c} is
robust, and that the indicated non-homogeneous coupling could
clearly contribute to enhance the SR phenomenon in more general
situations.
\section{Theoretical Framework}
\subsection{The Model}
For the sake of concreteness, we consider a simplified version of
the FitzHugh-Nagumo \cite{extend3,I0,FHN} model. This model has been
useful for gaining qualitative insight into the excitable and
oscillatory dynamics in neural and chemical systems \cite{LGN04}. It
consist of two variables, in one hand $u$, a (fast) activator field
that in the case of neural systems represents the voltage variable,
while in chemical systems represents a concentration of a
self-catalytic species. On the other hand $v$, the inhibitor field,
associated with the concentration of potassium ions in the medium
(within a neural context), that inhibits the generation of the $u$
species (in a chemical reaction). Instead of considering the usual
cubic like nonlinear form, we use a piece-wise linear version
\begin{eqnarray}
\label{eq:ucont}\epsilon \, \frac{\partial u(x,t)}{\partial t} &=&
\frac{\partial }{\partial x} \left( D_u(u) \, \frac{\partial
u}{\partial x} \right) + f(u) - v + \xi(x,t) \\
\frac{\partial v(x,t)}{\partial t} &=&
\label{eq:vcont}\frac{\partial }{\partial x} \left( D_v(v) \,
\frac{\partial u}{\partial x}\right) + \beta\,u - \alpha\, v,
\end{eqnarray}
where $f(u) = -u + \Theta(u-\phi_c) $, and $\xi(x,t)$ is a
$\delta$-correlated white Gaussian noise, that is $\left< \xi(x,t)
\right> = 0$ and $ \left< \xi(x,t) \xi(x',t') \right> = 2 \gamma
\delta(x-x') \delta(t-t')$. Here $\gamma$ indicates the noise
intensity and $\phi_c $ is the ``discontinuity" point, at which the
piece-wise linearized function $f(u)$ presents a jump. In what
follows, the parameters $\alpha $ and $\beta $ are fixed as $\alpha
=0.3$ and $\beta = 0.4$. Finally, $\epsilon$ is the parameter that
indicates the time-scale ratio between activator and inhibitor
variables, and is set as $\epsilon=0.03$. We consider Dirichlet
boundary conditions at $x=\pm L$. Although the results are
qualitatively the same as those that could appear considering the
usual FitzHugh-Nagumo equations, this simplified version allows us
to compare directly with the previous analytical results for this
system \cite{extend3}.
\begin{figure}
\centering \label{fig:pattern}
\includegraphics[width=7cm,angle=-90]{patterns.ps}
\caption{ We show the stable patterns that arise in the system.
There is one stable pattern that is identically zero, i.e.
$P^u_0(x)=P^v_0(x)=0$ and another which is non-zero ($P^u_1(x)$,
$P^v_1(x)$). The patterns for the fields $u(x)$ and $v(x)$ are
plotted in dashed and solid lines, respectively. The parameters, are
$D_u= 0.3$, $D_v= 1$, $h=2$.}
\end{figure}
As in \cite{extend3c}, we assume that the diffusion coefficient
$D_u(u)$ is not constant, but depends on the field $u$ according to
$D_u(u) = \, D_u \left[ 1 + h \, \Theta(u-\phi_c) \right] $. This
form implies that the value of $D_u(u)$ depends ``selectively" on
whether the field $u$ fulfills $u > \phi_c$ or $u < \phi_c$. $D_u$
is the value of the diffusion constant without such ``selective"
term, and $h$ indicates the size of the difference between the
diffusion constants in both regions (clearly, if $h=0$ then
$D_u(u)=D_u$ constant). $D_v(v)$ is the diffusion for the inhibitor
$v$, that here we assume to be homogeneously constant.
It is worthwhile noting that when the parameter $h$ is large enough,
under some circumstances the coupling term might become negative.
This is what is known as ``inhibitory coupling" \cite{Dayan}. This
is a very interesting kind of coupling that has attracted much
attention in the last years, both in neural and chemical context,
that we will not discuss here.
This system is known to exhibit two stable stationary patterns. One
of them is $u(x)=0$, $v(x)=0$, while the other is one with non-zero
values and can be seen in Fig. 1. We will denote with $P^{u,v}_0(x)$
and $P^{u,v}_1(x)$, the patterns for $u$ and $v$ fields. Further, we
consider that an external, periodic, signal enters into the system
through the value of the threshold $\phi_c$,
\begin{equation}
\phi_c(t) = \phi_c + \delta\phi \, \cos (\omega t),
\end{equation}
where $\omega$ is the signal frequency, and $\delta\phi $ its
intensity.
All the results shown in this paper were obtained through numerical
simulations of the system. The second order spatially discrete
version of the system indicated in Eqs. (\ref{eq:ucont},
\ref{eq:vcont}) reads
\begin{eqnarray}
\dot{u_i} &=& D_{u,i} ( u_{i-1} + u_{i+1}-2 u_{i} ) +
(D_{u,i+1}-D_{u,i-1}) ( u_{i+1} + u_{i-1} ) \nonumber \\
& & \label{eq:disc1} \hspace{8cm}+ f(u_i) - v_i + \xi _{i}(t) \\
\label{eq:disc2}\dot{v_i} &=& D_{v} ( v_{i-1} + v_{i+1}-2 v_{i} ) +
\beta\, u_i - \alpha\,v_i.
\end{eqnarray}
We have performed extensive numerical simulations of this set of
equations exploiting the Heun's algorithm \cite{1999_nises}.
\subsection{Response's Measures}
Since the discovery of the stochastic resonance phenomenon, several
different forms of characterizing it have been introduced in the
literature. Some examples are: (i) output signal-to-noise ratio
(SNR) \cite{RMP,McNM}, (ii) the spectral amplification factor (SAF)
\cite{jung1,jung2}, (iii) the residence time distribution
\cite{haenggi1,gamma1}, and, more recently, (iv) information theory
based tools \cite{bulsladri,neiman,noso}. Along this paper, we will
use the output SNR at the driving frequency $\omega$.
In this spatially-extended system, there are different ways of
measuring the overall system response to the external signal. In
particular, we evaluated the output SNR in three different ways (the
units being given in dB)
\begin{itemize}
\item SNR for the element $N/4$ of the chain evaluated over the
dynamical evolution of $u_{N/4}$, that we call $\textit{SNR}_1$.
\item SNR for the middle element of the chain evaluated over the
dynamical evolution of $u_{N/2}$, that we call $\textit{SNR}_2$.
Having Dirichlet boundary conditions, the local response depends on
the distance to the boundaries.
\item In order to measure the overall response of the system to
the external signal, we computed the SNR as follows: We digitized
the system dynamics to a dichotomic process $s(t)$: At time $t$ the
system has an associated value of $s(t)=1\, (0)$ if the Hilbert
distance to pattern $1\, (0)$ is lower than to the other pattern.
Stated in mathematical terms, we computed the distance
$\mathcal{D}_2[\cdot,\cdot]$ defined by
$$\mathcal{D}_2[f,g]=\left( \int_{-L}^L dx\, \left( f(x)-g(x)
\right)^2 \right)^{1/2}$$ in the Hilbert space of the real-valued
functions in the interval $[-L,L]$, i.e. $\mathcal{L}_2$. At time
$t$, a digitized process is computed by means of
\begin{equation}
s (t) = \left\{ %
\begin{array}{c}
1 \qquad \hbox{if } \mathcal{D}_2
\left[P^u_1(x),u(x,t)\right]<\mathcal{D}_2 \left[
P^u_0(x),u(x,t)\right]\\ 0 \qquad \hbox{if } \mathcal{D}_2
\left[P^u_1(x),u(x,t)\right]\geq\mathcal{D}_2 \left[ P^u_0(x),u(x,t)
\right]
\end{array} \right. ,
\end{equation}
We call this measure $SNR_{p}$.
\end{itemize}
\section{Results}
As indicated above, Eqs. (\ref{eq:disc1}) and (\ref{eq:disc2}) have
been integrated by means of the Heun method \cite{TSM}. We have
fixed the parameters $\epsilon=0.03$, $\phi_c=0.52$ and adopted an
integration step of $\Delta t=10^{-3}$. For the signal frequency we
adopted $\omega=2 \pi / 3.2 = 1.9634295\ldots$. The simulation was
repeated 250 times for each parameter set, and the SNR was computed
by recourse of the average power spectral density.
Figure \ref{SNR-gamma} depicts the results for the different SNR's
measures we have previously defined as function of the noise
intensity $\gamma$. We adopted the following values: $\delta \phi =
0.4$, $D_v = 1.$ and $N = 51$. In all three cases it is apparent
that there is an enhancement of the response for $h > 0$, when
compared with the $h = 0$ case, while for $h < 0$ the response is
smaller.
\begin{figure}
\centering
\includegraphics[width=4.5cm,angle=-90]{snr-gamma1.ps}
\includegraphics[width=4.5cm,angle=-90]{snr-gamma2.ps}
\includegraphics[width=4.5cm,angle=-90]{snr-gammap.ps}
\caption{ SNR vs. $\gamma$, the noise intensity, for the three
different measures we use. The parameters are $\delta\phi= 0.4$,
$D_v= 1.$, $\omega=2\pi/3.2$, $N=51$. The different curves represent
different values of $h$, showing an enhancement of the response to
the external signal for $\gamma>0$. In particular it is shown:
$h=-2$ ($+$), $h=-1$ ($\triangle$), $h=0$ ($\diamond$), $h=1$
($\square$) and $h=2$ ($\bigcirc$). }\label{SNR-gamma}
\end{figure}
In Fig. \ref{SNR-h} we show the same three response's measures, but
now as a function of $h$. We have plotted the maximum of each SNR
curve, for three different values of the noise intensity, and for
$\delta\phi= 0.4$, $D_v= 1.$, $\gamma = 0.01, 0.1, 0.3$, $D_u=0.3$,
and $N=51$. It is clear that there exists an optimal value of
$\gamma$ such that, for such a value, the phenomenon is stronger
(that is, the response is larger). It is apparent the rapid fall in
the response for $h<0$.
\begin{figure}
\centering
\includegraphics[width=4.5cm,angle=-90]{snr-h1.ps}
\includegraphics[width=4.5cm,angle=-90]{snr-h2.ps}
\includegraphics[width=4.5cm,angle=-90]{snr-hp.ps}
\caption{SNR vs. $h$, the selectiveness of coupling, for the three
different measures we use. The parameters are $\delta\phi= 0.4$,
$D_v= 1.$, $\gamma = 0.032~(\bigcirc), 0.32~(\square),
0.6~(\diamond), 1.2~(\square), 3.2~(\times)$, $D_u=0.3$,
$N=51$.}\label{SNR-h}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=4.5cm,angle=-90]{snr-h1_dX.ps}
\includegraphics[width=4.5cm,angle=-90]{snr-h2_dX.ps}
\includegraphics[width=4.5cm,angle=-90]{snr-hp_dX.ps}
\caption{SNR vs. $h$, the selectiveness of coupling, for different
values of $D_u$. The parameters are $\delta\phi= 0.4$, $D_v= 1.$,
$\gamma = 0.32$, $D_u=0.0~(\bigcirc), 0.1~(\square),
0.3~(\diamond)$, $N=51$.}\label{SNR-h_dX}
\end{figure}
In figure \ref{SNR-h_dX} we show the dependance of SNR on $h$, for
different values of the diffusion which depends on the activator
density $D_u$. It is apparent that the response becomes larger when
the value of $D_u$ is larger. However, as was discussed in
\cite{extend2,extend3}, it is clear that for still larger values of
$D_u$, the symmetry of the underlying potential (that is the
relative stability between the attractors) is broken and the
response finally falls-down.
Figure \ref{SNR-Du} shows the results of the SNR, but now as
function of $D_u$, the activator diffusivity, for different values
of $\gamma$, and for $\delta\phi= 0.4$, $D_v= 1.$ and $N=51$. It can
be seen that, independently from the coupling strength $D_u$, the
response to the external signal grows with the selectiveness of the
coupling, showing the robustness of the phenomenon.
\begin{figure}
\centering
\includegraphics[width=4.5cm,angle=-90]{snr-Du1.ps}
\includegraphics[width=4.5cm,angle=-90]{snr-Du2.ps}
\includegraphics[width=4.5cm,angle=-90]{snr-Dup.ps}
\caption{SNR vs. $D_u$, the diffusiveness in activator variable $u$,
for the three different measures we use. The parameters are
$\delta\phi= 0.4$, $D_v= 1.$, $\gamma = 0.1~(\bigcirc),
0.32~(\square), 1.0~(\diamond)$, while the white symbols represent
$h=2$ and the black ones, $h=0$. The system size is $N=51$.
}\label{SNR-Du}
\end{figure}
Next, in figure \ref{SNR-Dv}, we present the results for the SNR as
function of $D_v$, the activator diffusivity, for different values
of $\gamma$, and for $\delta\phi= 0.4$, $D_u= 0.3$, and $N=51$. We
see that for $h \geq 0$ the response is more or less flat, however,
it is again apparent the SNR's enhancement for $h>0$. For $h<0$ we
see that the system's response decays very fast with increasing
$D_v$. This effect could be associated to the fact (as found in
those cases where the NEP is known \cite{extend2,extend3}) that in
the underlying NEP the bistability is lost as a consequence of the
disappearance of some of the attractors \cite{I0}.
\begin{figure}
\centering
\includegraphics[width=4.5cm,angle=-90]{snr-Dv1.ps}
\includegraphics[width=4.5cm,angle=-90]{snr-Dv2.ps}
\includegraphics[width=4.5cm,angle=-90]{snr-Dvp.ps}
\caption{SNR vs. $D_v$, the diffusiveness in the inhibitor variable
$v$. The parameters are $\delta\phi= 0.4$, $D_u= 0.3$, $\omega =
2\,\pi / 3.2$, $N=51$. $\gamma = 0.1~(\bigcirc), 0.32~(\square),
1.0~(\diamond)$, while the white symbols represent $h=2$ and the
black ones, $h=0$.}\label{SNR-Dv}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=4.5cm,angle=-90]{snr-N1.ps}
\includegraphics[width=4.5cm,angle=-90]{snr-N2.ps}
\includegraphics[width=4.5cm,angle=-90]{snr-Np.ps}
\caption{SNR vs. $N$, the system size, for the three different
measures we use. The parameters are $\delta\phi= 0.4$, $D_u= 0.3$,
$D_v= 1.$, $\gamma = 0.32$, $h=-2~(\bigcirc), 0~\hbox{(grey
squares)}, 2~(\blacklozenge)$.}\label{SNR-N}
\end{figure}
Finally, in figure \ref{SNR-N} we depict the same three SNR's
measures but as a function of $N$, the system size. For the two
measures $SNR_1$ and $SNR_2$, we see that, for different values of
$h$ and $\gamma$, the response is very flat, and do not seems to be
too much dependent on $N$. It is clear that there is an increase of
the response when $h$ increases. At variance, for $SNR_p$, the
dependence to $N$ is apparent: the SNR decays to zero, in a fast or
slow way, depending of $h=0$ or $h>0$. Here $\delta\phi= 0.4$, $D_u=
0.3$, $D_v= 1.$, $\gamma = 0.01, 0.1, 0.3$.
\section{Conclusions}
We have analyzed a simplified version of the FitzHugh-Nagumo model
\cite{extend3,I0,FHN}, where the activator's diffusion is
density-dependent. Such a system, when both diffusions are constant
(that is: $D_u >0$ and $D_v = 0$), has a known form of the NEP
\cite{extend3}. However, in the general case we have not been able
to find the form of the NEP (but the idea of such a NEP is always
\textit{underlying} our analysis) and we have to resort to an
analysis based on numerical simulations.
Through the numerical approach we have studied the influence of the
different parameters on the system response. From the results it is
apparent the enhancement of the output SNR as $h$, the selectivity
parameter, is increased. This is seen through three different ways
of characterizing the system's response. We can conclude that the
phenomenon of enhancement of the SNR, due to a selectivity in the
coupling, initially found for a scalar system \cite{extend3c} is
robust, and that the indicated nonhomogeneous coupling could clearly
contribute to enhance the SR phenomenon in very general systems.
This phenomenon is also robust to variations of the parameter that
controls the selectiveness of the coupling, up to a point that even
in the case of inhibitory coupling the phenomenon holds.
An aspect worth to be studied in detail is the dependence of the SNR
on $N$, the number of coupled units. In this way we could analyze
the dependence of the so called \textit{system size stochastic
resonance} \cite{haenggi2,SSSR} on the \textit{selective coupling}.
The thorough study of this problem will be the subject of further
work.
\vspace{0.25cm}
{\bf Acknowledgments:} HSW thanks to the European Commission for the
award of a {\it Marie Curie Chair} at the Universidad de Cantabria,
Spain.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,452 |
Владимир Романович Эйгес (12 января 1877, Суджа, Курская губерния — 28 октября 1949, Москва) — русский и советский философ и математик.
Биография
Родился 31 декабря 1876 года (по старому стилю) в семье земского врача Рувима Манасиевича Эйгеса (1840—1926) и переводчицы Софии Иосифовны Эйгес (1846—1910). Вырос и окончил гимназию в Брянске; после окончания университета вновь работал в Брянске и в Рыбинске (где родилась его младшая дочь).
Заведовал кафедрой математики Московского автомобильно-дорожного института имени В. М. Молотова (МАДИ), до середины 1930-х годов доцент, затем профессор.
До 1917 года опубликовал несколько философских работ. Основные научные труды в области геометрии.
В напечатанной в газете «Правда» от 3 июля 1936 года статье «О врагах в советской маске», продолжавшей травлю математика Н. Н. Лузина, ему среди прочего вменялась незаслуженная характеристика научной деятельности В. Р. Эйгеса: "Ещё более блестящие и столь же незаслуженные отзывы давал академик Лузин и о работах В. Эйгеса, В. Депутатова, П. Бессонова и многих других. Так, беспардонно захваливая В. Эйгеса, он писал: «В. Эйгес является автором весьма глубокого, ценного и интересного исследования по основам геометрии. Эти исследования ещё важны и тем, что они совершенно оригинальны, замыслены и выполнены их автором самостоятельно в порядке одинокого исследователя, вне всяких влияний со стороны». Проверка этой работы проф. Хинчиным показала, что «как этот доклад, так и несколько представленных автором рукописей, посвящённых различным вопросам геометрии, носят вполне ученический характер и не содержат элементов сколько-нибудь серьёзного научного исследования». В этой связи публикации о деле Лузина часто путают В. Р. Эйгеса с его младшим братом Александром, также математиком и доцентом Московского института стали и сплавов, который не занимался геометрией.
Семья
Дочери — Тамара Владимировна Эйгес (1913—1981), художник; Нелли (1909—?).
Сёстры — Екатерина Романовна Эйгес (1890—1958), поэтесса и библиотечный работник, была замужем за математиком П. С. Александровым; Анна Романовна Эйгес (1873/1874—1966), переводчица (известен её перевод «Страданий молодого Вертера» Гёте, 1893 и 1937); Надежда Романовна Эйгес (1883—1975), педагог, основательница первых в России яслей.
Братья — Константин Эйгес, композитор, философ и теоретик музыки; Иосиф Эйгес (1887—1953), литературовед и музыковед; Александр Эйгес (1880—1944), математик и литературовед; Вениамин Эйгес (1888—1956), художник; Евгений Эйгес (1878—1957), врач.
Публикации
Критика феноменализма. Брянск: Типография А. Итина, 1905. — 95 с.
Критика феноменализма (О мироотношениях). М.: Типография О. Л. Сомовой, 1914. — 95 с.
Философские этюды: I. Сознание и бытие. Настоящее и временность. Трансцендентные бездны. II. О философии Вл. Соловьева, Бергсона и Лосского. М.: Труд, 1917. — 178 с.
Сборник задач по аналитической геометрии в пространстве с подробными решениями (Из задач, дававшихся на упражнениях в 1922—1923 уч. г. в Моск. высш. техн. училище). / Сборник сост. А. С. Некрасовым и А. А. Шубиным, под ред. В. М. Эйгес. М.: Макиз, 1923. — 32 с.
Мнимый сегмент и мнимый пояс евклидовой сферы как субстрат неевклидовых геометрий. Труды Всероссийского съезда математиков в Москве (27 апреля — 4 мая 1927). Под редакцией проф. И. И. Привалова. М.—Л.: ОГИЗ, 1928. — 280 с.
Дифференциальная геометрия в пространстве. М.: МАДИ, 1934. — 44 с.
Строительная индустрия: справочное руководство по гражданскому и промышленному строительству. Т. 2: Математика (раздел «Аналитическая геометрия»). М.—Л.: ОНТИ НКТП СССР, 1935. — 559 с.
Примечания
Философы Российской империи
Математики Российской империи
Математики СССР
Преподаватели МАДИ
Персоналии:Брянск | {
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{"url":"https:\/\/mathematica.stackexchange.com\/questions\/151344\/how-to-define-the-scope-of-subscript-in-a-iteration","text":"# How to define the scope of subscript in a iteration?\n\nNewbee to MMA, and completely have no clue about it. I need do a calculation of some iteration, seen in the picture below.\n\n$n=3,4,5,..., \\begin{cases} L_n=c_n\\\\ L_{n-1}=c_{n-1}\\\\ L_{n-p+1}=\\displaystyle\\sum_{r=1}^{p-2} (p-r-1)\\;c_{p-r-1}\\;L_{n-r+1}+c_{n-p+1}\\;(p=3,4,5,...,n)\\\\ \\end{cases}$\n\nAnd for convenience and necessity,n equals 8. And the code is editted as\n\nSubscript[L, 8] = Subscript[c, 8]\nSubscript[L, 7] = Subscript[c, 7]\nSubscript[L, 9 - p] =\nSum[(p - r - 1)*Subscript[c, p - r - 1]*Subscript[L, 9 - r] +\nSubscript[c, 9 - p], {r, 1, p - 2}](*p=3,4,5,6,7,8*)\n\n\n2 problems are encountered here.\n\nProblem 1:\n\n$c_1$ to $c_8$ is considered as constants with no certain values. Which code is needed to make it happen?\n\nProblem 2:\n\nAs can be seen in the last part of my code, p has a range from $3$ to $8$. And again, which code is needed?\n\nAfter searching so much info on the Internet, no solution is acquired....\n\n\u2022 Being a newbee, avoid Subscript. It is made for formatting, not calculations. For your indexed symbols, simply use c[8], L[7] etc. You can use a Do loop over p to get all the definitions done. \u2013\u00a0Marius Ladeg\u00e5rd Meyer Jul 13 '17 at 13:56\n\u2022 @MariusLadeg\u00e5rdMeyer thank you for your advice about Subscript, but I do need use it for further programming, for the iteration presented here is just a small step in my work. And I will try Do loop. Thank you again. \u2013\u00a0Robin_Lyn Jul 14 '17 at 1:37\n\nI agree in part with Marius Ladeg\u00e5rd Meyer concerning Subscript : never use it in the left-hand side of a definition because you cannot easily remove it after. But I like using it on the right hand side. So,\n\nL[8] := Subscript[c, 8]\nL[7] := Subscript[c, 7]\nL[k_] := Sum[((9 - k) - r - 1)*Subscript[c, (9 - k) - r - 1]*\nL[9 - r] + Subscript[c, k], {r, 1, (9 - k) - 2}]\n\n\nremark the use of := instead of = : delayed assignment\nAlso remark the substitution 9-p -> k in defining the left hand side of the sum : read up on 'pattern recognition' to learn why you need a single argument there (like ' k ' and not a subtraction like ' 9-p ' )\nand last but not least,L[k_]:= read up on 'Patterns' and argument naming.\nYou'll get the hang of it soon enough. ;-)\n\n\u2022 Many thankssssss for your help. And the problem 1 is still no solved, that can Subscript[c, i](i=1,2,3,...,8) be considered as constants and how to do it? And after that, whether is it possible to get mathematical expressions of L[k_] you mentioned above? @Wouter \u2013\u00a0Robin_Lyn Jul 14 '17 at 1:50\n\u2022 @Robin_Lyn : have you tried to use the definitions in my answer to express the values of L[k] using Table[L[k], {k, 8}]? it produces {6 Subscript[c, 1] + 5 Subscript[c, 5] Subscript[c, 7] + 6 Subscript[c, 6] Subscript[c, 8] + 4 Subscript[c, 4] (Subscript[c, 6] + Subscript[c, 1] Subscript[c, 8]) +.... \u2013\u00a0Wouter Jul 14 '17 at 10:36\n\u2022 yes, I tried and it works~~~And the calculations followed are solved all the way down, for they are similar iterations. A huge stone in my heart is taken away. Greaaaat jooooooy! \u2013\u00a0Robin_Lyn Jul 14 '17 at 11:05","date":"2020-01-25 17:34:37","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6312240362167358, \"perplexity\": 2212.510300279524}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579251678287.60\/warc\/CC-MAIN-20200125161753-20200125190753-00307.warc.gz\"}"} | null | null |
class ConditionIsBuyabox < ConditionSimple
def match?(card)
card.buyabox
end
def to_s
"is:buyabox"
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,704 |
Q: Счетчик для сканера штрих-кодов Подскажите, в каком направлении двигаться. Нужна программа, которая будет отнимать -1 от числа заранее заданного после каждого считывания сканера штрих-кодов и как только число будет равно нулю - блокировать программу на Windows. Не могу найти нормального счетчика для сканера. Программа должна работать в фоновом режиме.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,286 |
Q: When does a dense matrix have a sparse inverse? Many common systems in PDEs (finite element etc.) result in sparse matrices that have dense inverses. I was wondering, are there ever matrices which go the other way? In that the matrix itself is dense, but the inverse is sparse?
I.e. are there any problems were you might encounter
\begin{align}
A x = b
\end{align}
where $A$ is dense, and $A^{-1}$ is sparse? What would the significance of this be?
A: Matrices that come from finite elements tend to be sparse because they are discretizing a PDE that consists of local differential operators. Non-local problems (integral equations e.g.) would yield dense matrices. I'm not aware of any cases where a seemingly dense problem later turned out, for deep reasons, to have a sparse inverse; oftentimes it is obvious at the PDE level that you are discretizing the inverse of a local operator, so you refactor your equations to avoid this (as Omnom says in the comments).
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,140 |
I'm going to be honest. The past 18 months of my life's story have been hard. Hard down to the deepest parts of my soul. My heart has grieved as my dream of life and family have been shifted and refined. There have been days that waking up and moving through the simple rhythms of the day has taken all my energy and focus.
In the summer of 2012, my life turned upside down. My marriage of over ten years suddenly and dramatically fell apart and I found myself waking up to the reality that I was drowning in the busyness of life. It had been my desire to live with purpose and intention for many years but I had allowed small and seemingly harmless things to take up residence in my life. The constant checking of Facebook, the obsessive following of blogs and the pursuit of a "perfect" body had robbed me of living present and purposeful for each day. Fear had taken up residence and was driving my choices and my beliefs about myself as a woman, wife, and mother. I have known from a young age that I was created for relationship with God and that I am loved by Jesus. But there is a difference between knowing and living. And there is no fear when I choose to live by faith.
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Practically, I looked to the future and wondered how I was going to support myself and my two young boys. My thought and desire had been to stay home with my boys until they were in school full time. But with the ending of my marriage that was no longer financially possible. Late one night I was roaming the web looking for ideas. As I researched different options, I stumbled across the Parent Coaching Institute. As I read the description of the program, I knew it fit my giftings, knowledge and my desire to partner with others to experience an engaged and wholehearted life. Parent Coaches team up with parents in a unique relationship that equips parents to engage in life from their strengths and giftings as a parent and person to move them towards their dream for their family.
I graduated this month as a Certified PCI Parent Coach® and have started my own coaching practice called Foundations Parent and Life Coaching. My heart is to work with parents and individuals who want to build a solid foundation for a thriving life. Parenting and just living in general require being purposeful; otherwise busyness and distractions will crowd out the things that are truly important. Sometimes we come to a season of life where we need to be purposeful in establishing healthy, sustainable practices for ourselves and our families. That is when working with a coach can be beneficial. The coaching relationship is all about partnership. We work together to establish the rhythms that will help you and your family be your best selves. People come to coaching for a variety of reasons. For some it is when they have a newborn and are learning about who they are as a parent and what they want for their family. For others, it is when their child is school aged and they are struggling to find balance between screen time and physical activity. Some parents finding coaching helps them navigate a diagnosis. No matter the reason, what I know to be true is that when we are working towards being our best self, we give a gift to our children that will impact their future in the best way possible.
Hannah is a teacher with a background in early childhood development and is now certified as a parent and life coach. You can find more about the exceptional Parent and Life Coaching services she offers on her website Foundations Parent and Life Coaching or read her thoughts about thriving daily rhythms at her blog The Daily Rhythms of Life.
Well written post! I'm glad you found an outlet for income that was also satisfying to you- God is good! Thank you for linking up on "A Group Look" at A look at The Book!
What a great post! As co-host of the "A Group Look" Link up from A Look At the Book and Women of Worship – we thank you for coming by! I hope you will stop by and join us again next week on Friday.
I most certainly will! Thank you for visiting M2M! | {
"redpajama_set_name": "RedPajamaC4"
} | 2,754 |
\section{Introduction.}
Low-dimensional magnets are actively studied during last decades
both theoretically and experimentally. Spin-ladder is one of the
simplest models of the field, that is just one step more
complicated than the Heisenberg spin chain, the keystone of the
low-dimensional magnetism. Such a system consists of two chains
forming the "legs" of the spin ladder, which are coupled by simple
interchain coupling forming "rungs" of the ladder. The Hamiltonian
of the single spin ladder with the equivalent positions along the
ladder is
\begin{eqnarray}\label{eqn:ham}
{\cal H}&=&J_{leg}\sum_i (\vect{S}_{1,i}\vect{S}_{1,i+1}+\vect{S}_{2,i}\vect{S}_{2,i+1})+\nonumber\\
&&J_{rung}\sum_i \vect{S}_{1,i}\vect{S}_{2,i}+\mu_B \vect{H} \hat{g}\vect{S}+{\cal H}_{anis}
\end{eqnarray}
\noindent it includes Heisenberg exchange couplings $J_{leg}$ and
$J_{rung}$, Zeeman interaction (with usually anisotropic
$g$-tensor) and weak anisotropic spin-spin interactions ${\cal H}_{anis}$.
Independently on the ratio between the $J_{leg}$ and $J_{rung}$,
the excitation spectrum of the spin ladder is gapped, ground state
is non-magnetic and excited states are $S=1$
quasiparticles.\cite{kolezhukmikeska} However, most of the
experimentally available examples of the spin ladder systems are
the so-called strong-rung ladders with the dominating in-rung
interaction $J_{rung}$. Strong-leg ladders remains a rarity in
this family. Additional complication of the real systems is a
presence of the anisotropic spin-spin interactions breaking the
ideal symmetry of the Heisenberg model. Such interactions limit
excitations lifetime (to the point of total damping in some
extreme cases\cite{ntenp}) and could lift degeneracy of the $S=1$
states. Thus, estimation of such interactions strength and,
ideally, search for the systems with negligible anisotropic part
of the Hamiltonian is an important quest when comparing real
magnets with model predictions.
Adequate accounting for the effect of anisotropic interactions in
a spin-gap magnet is also a challenge. This problem was addressed
in a 1D field theory models via bosonic \cite{Affleck} and
fermionic \cite{tsvelik} approaches and in an independently
developed macroscopic model.\cite{farmar} However reliable
microscopic models remains a rarity (see, e.g. Ref.
\onlinecite{Kolezhuk}): most of the real spin-gap magnets have a
complicated network of the exchange couplings allowing far too
numerous possibilities of the anisotropic interactions parameters.
The adequate microscopic approaches are of particular interest in
connection with a particular case of the effect of a uniform
Dzyaloshinskii-Moria interaction on the properties of a quantum
magnet. \cite{povarov}
Recently found organometallic compound
(C$_7$H$_{10}$N$_2$)$_2$CuBr$_4$ , abbreviated DIMPY for short, is
an example of the strong-leg ladder with very weak anisotropic
interactions.\cite{dimpy1,dimpy2,dimpy-dave-prb,dimpy-dave-prl}
Presence of the energy gap in the excitation spectrum was revealed
by magnetic susceptibility \cite{dimpy1}, specific heat
\cite{dimpy2} and magnetization \cite{dimpy-dave-prl,dimpy-magn}
bulk measurements as well as by inelastic neutron scattering.
\cite{dimpy2,dimpy-dave-prb} The energy gap was found\cite{dimpy2}
to be 0.33 meV, it can be closed by the magnetic field $\mu_o
H_{c1}=3.0$~T, while the saturation field is much higher
\cite{dimpy-magn} $\mu_0 H_{sat} \approx 30$~T. The values of the
exchange constants were determined from the DMRG fit of the
measured inelastic neutron scattering spectra
\cite{dimpy-dave-prl} and were found to be $J_{leg}=1.42$~meV and
$J_{rung}=0.82$~meV. The magnetic field induced ordering is
observed at very low temperatures ($~T_N^{(max)} \approx 300$~mK
at $\mu_0 H \sim 15$~T).\cite{dimpy-dave-prl}
Electron spin resonance (ESR) spectroscopy is a powerful tool to
probe for the weak anisotropic interactions in the magnetic
systems. Inelastic neutron scattering experiments
\cite{dimpy-dave-prb} have shown that DIMPY is an almost perfect
realisation of the Heisenberg spin ladder. ESR technique allows
much higher energy resolution (routinely resolved ESR linewidth of
100 Oe corresponds approximately to the energy resolution of 1
$\mu$eV) and thus allows to probe possible effects of anisotropic
interactions with high accuracy.
In the present manuscript we report results of the ESR study of
low-energy spin dynamics in DIMPY in the temperature range from
400~mK to 300~K. We observe angular and temperature dependences of
the ESR linewidth at high temperatures which can be described as an
effect of the uniform Dzyaloshinskii-Moria (DM) interaction, which
is allowed by the lattice symmetry. At low temperatures we observe
slitting of the ESR absorption line due to lifting of the triplet
state degeneracy, which is also possibly due to the same DM
interaction. Additionally we observe well resolved ESR absorption
lines from the inequivalent ladders which allowed an upper estimate
of interladder exchange interaction.
\section{Samples and experimental details.}
Single crystals of non-deuterated DIMPY were grown from the solution
by slow diffusion in a temperature gradient. Samples quality was
checked by X-ray diffraction and magnetization measurements.
Concentration of the paramagnetic defects estimated from the 500~mK
magnetization curve is below 0.05\%.
DIMPY belongs to the monoclinic space group P2(1)/n with lattice
parameters $a=7.504$~\AA{}, $b=31.613$~\AA{}, $c=8.206$~\AA{} and
the angle $\beta=98.972^{\circ}$.\cite{dimpy1} As-grown crystals
have a well developed plane orthogonal to the $b$-axis and are
elongated along the $a$ direction.
ESR experiments were performed using set of the home-made
transmission-type ESR spectrometers at the frequencies 18-38~GHz.
Lowest available temperature of 400~mK was obtained by He-3 vapours
pumping cryostat. At the measurements below 77~K magnetic field was
created by compact superconducting magnets, typical nonuniformity of
the magnetic field at the resonance conditions in our experiments is
estimated as $<5\ldots20$~Oe depending on the magnet used.
High-temperature experiments where done with a resistive
water-cooled magnet with the field nonuniformity about 5 Oe.
\section{Lattice symmetry and possible anisotropic interactions.\label{sect:sym}}
\begin{figure}
\centering
\epsfig{file=struct2.eps, width=\figwidth, clip=}
\caption{(color online) Crystallographic structure of DIMPY with two
magnetically nonequivalent spin ladders. Only Cu and Br ions are
shown along with the main exchange bonds $J_{leg}$ and
$J_{rung}$. Solid arrows (red) indicate directions of $g$-tensor
main axes for inequivalent ladders,
dashed arrows (blue) indicate directions of the Dzyaloshinskii-Moriya
vectors for inequivalent ladders, as found from the data fit (see
text). Broad double-headed arrows links DM vector and $g$-tensor
axis corresponding to the same ladder.}\label{fig:struct}
\end{figure}
Monoclinic unit cell of DIMPY includes four magnetic Cu$^{2+}$
ions that belongs to two spin ladders: two pairs of copper ions
form rungs of the spin-ladders, which are then reproduced by
translations along the $a$-axis. This results in the formation of
two ladders differently oriented with respect to the crystal
\cite{dimpy-dave-prb} (see Figure \ref{fig:struct}).
Space symmetry of the DIMPY lattice includes inversion center in
the middle of each rung and a second order screw axis parallel to
the crystallographic $b$ direction that links different ladders.
These symmetries place strong restrictions on the possible
microscopic anisotropic interactions in DIMPY despite the low
crystallographic symmetry. First, all anisotropic interactions
along the legs of the ladders should be uniform because of
translational symmetry. Second, Dzyaloshinskii-Moria antisymmetric
interaction ${\cal H}_{DM}=\vect D \cdot \left[ {\vect S}_1 \times
{\vect S}_2 \right]$ on the rungs is forbidden by the inversion
symmetry. The same inversion symmetry requires that direction of
the Dzyaloshinskii-Moria vector $\vect D$ have to be exactly
opposite on the legs of the same ladder. Third, inversion center
on the rungs of the ladder ensures that $g$-tensor is always the
same for the given spin ladder so there are no complications of
anisotropic Zeeman splitting.
Second order axis establish relations between the $g$-tensor
components and Dzyaloshinskii-Moria vector direction in different
ladders. In particular, because of this second order axis the
effective $g$-factor values are the same for both ladders for the
field applied parallel or orthogonally to this axis.
This analysis neglects anisotropic interladder couplings and
possible symmetric anisotropic exchange (SAE) coupling ${\cal
H}_{SAE} = \sum _{\mu,\tau} J_{\mu\tau} { S}_1^\mu {S}_2^\tau$,
where $J_{\mu\tau}$ are components of a symmetric exchange tensor
$\hat{\mathbf{A}}$, which is usually constrained by condition $Tr
J_{\mu,\tau}=0$. Symmetric interaction is allowed both on rungs and
legs of the ladder and is also constrained by symmetry operations.
However, as we will demonstrate below, our observations point that
Dzyaloshinskii-Moria interaction is dominating anisotropic
interaction in the case of DIMPY.
As for the anisotropic interladder couplings, there is a
possibility that anisotropic couplings between the equivalent
ladders stacked in $c$ direction could be important as well: Cu-Cu
distance in this direction is even less then the distance on the
rungs ($8.2$~\AA{} against $8.9$~\AA{}) and suppression of the
Heisenberg exchange interaction in this direction is most likely
due to unfavorable mutual orientation of the electron orbitals of
bromine ions mediating this superexchange route which could be
less important for the anisotropic spin-spin interactions arising
through involvement of differently oriented excited electron
orbitals mixed with the ground state by spin-orbital
interaction.\cite{eremins} In the present work we neglect this
possibility.
Thus, the main anisotropic interactions in DIMPY are really simple
to analyse. They include anisotropic $g$-tensor, which is the same
for all magnetic ions of the given spin-ladder and a
Dzyaloshinskii-Moria interaction, which is uniform along the leg
of the ladder and the Dzyaloshinskii-Moria vectors are exactly
opposite on the legs of the given ladder.
\section{Experimental results.}
\subsection{Angular dependence of the ESR absorption at 77~K.}
\begin{figure}
\centering
\epsfig{file=g-angular.eps, width=\figwidth, clip=}
\caption{(color online) Main panel: angular dependence of the $g$-factor at
77~K. Symbols - experimental data, curves - uniaxial $g$-tensor
best fit (see text). Experimental error is about of 0.1\% and is
within symbol size. Inset: example of ESR absorption spectra at
representative orientations. Symbols - experimental data, curves
- best fit with Lorentzian lineshape, vertical lines corresponds
to certain $g$-factor values. Narrow line with $g=2.00$ (at 6.13
kOe) is a DPPH marker. }\label{fig:g-angular}
\end{figure}
\begin{figure}
\centering
\epsfig{file=width-angular3.eps, width=\figwidth, clip=}
\caption{(color online) Angular dependence of the ESR linewidth at f=17.2~GHz,
T=77~K (half-width at half-height). Vertical bar at the left panel shows typical
errorbar size (double error). Squares: width of the high-$g$ component.
Circles: width of the low-$g$ component. Curves: model
description (see text), solid lines show full linewidth, dashed
and dotted lines show contributions due to DM and SAE
interactions respectively. Marks "1" and "2" indicate
contributions corresponding to the same
ladder.}\label{fig:width-angular}
\end{figure}
We have taken rotational patterns of ESR absorption for the
magnetic field applied in different crystallographic planes.
Because of monoclinic lattice symmetry care should be taken with
consistent determination of the field direction with respect to
the lattice axes. We will use a cartesian basis with $X||a$,
$Y||b$ and $Z||c^{{}*{}}$ for the direction description. Rotation
patterns were taken for the field confined to $(XY)$ and $(XZ)$
planes and to the plane containing $Z$ axis and an $(Y-X)$
direction. All rotation patterns were taken for more then $180^0$
angular sweeps.
Examples of absorption spectra and angular dependences of the
$g$-factor are shown of the Figure \ref{fig:g-angular}. We observe
one or two Lorentzian absorption lines. These absorption lines are
clearly due to the different spin ladders: the ladders are
equivalent with respect to the magnetic field for $\vect{H}||Y$
and $\vect{H} \perp Y$, and we observe single component absorption
at these orientations. Anisotropy of $g$-factor is typical for the
Cu$^{2+}$ ion, $g$-factor varies from about 2.03 to 2.30 in
agreements with powder ESR measurements of
Ref.\onlinecite{dimpy-highfieldESR}
Angular dependence of the ESR linewidth was determined by fitting
observed absorption spectra with a single lorentzian line or with
a sum of two lorentzian lines (Fig.\ref{fig:width-angular}).
Typical half-width at half-height at 77~K was around 50 Oe. Field
inhomogeneity in the used magnet and uncertainties of the fit
procedure limit accuracy of the linewidth determination to about
5~Oe, however angular dependence is clearly present. We were able
to cross-check our results at certain selected orientations on a
commercial Bruker X-band spectrometer and we have found that
X-band data are in agreement with our results.
\subsection{Low-temperature ESR.}
\begin{figure}
\centering
\epsfig{file=scans_t.eps, width=\figwidth, clip=}
\caption{(color online) ESR absorption spectra at low
temperatures, $\vect{H}||(X+Y)$. Vertical
dashed lines mark resonance fields corresponding to the shown
$g$-factor values. Horizontal dashed line at 0.45~K curve is a
guide to the eye at zero-absorption level. Narrow absorption line at $g=2.00$ is a DPPH marker.}\label{fig:scans-lt}
\end{figure}
\begin{figure}
\centering
\epsfig{file=max_split.eps, width=\figwidth, clip=}
\caption{(color online) ESR absorption spectra at the temperature $~T \approx 1$~K
at different frequencies, $\vect{H}||(X+Y)$. All spectra are shifted along the field
axis to fit positions of the main absorption subcomponents. Left
panel: left absorption component ($g \approx 2.28$), weak
absorption subcomponent is magnified by the factor of 5 or 10
for better presentation. Right panel:
right absorption component ($g \approx 2.05$). Vertical dashed
lines mark positions of the absorption subcomponent at lowest
frequency. Triangles on the right panel mark position of the
DPPH marker absorption ($g=2.00$). }\label{fig: max-split}
\end{figure}
\begin{figure}
\centering
\epsfig{file=int-final.eps, width=\figwidth, clip=}
\caption{(color online) Left panel: temperature dependence of the ESR intensity
below 1~K at f=34.6~GHz, $\vect{H}||(X+Y)$. Inset: examples of ESR absorption and ESR
components and subcomponents notations. Symbols - experimental
data, dashed lines - fits with thermoactivation law
$I\propto \exp(-\Delta/T)$. Right panel: dependence of the determined
activation gaps for different spectral subcomponents. Filled
symbols: intense A1 and B1 subcomponents, open symbols - weak
A2 and B2 subcomponents. Lines: parameters-free model
dependence calculated with the zero-field gap value
$\Delta_{INS}$ known from the inelastic neutron scattering
experiments.\cite{dimpy2,dimpy-dave-prb}
Inset: scheme of the energy levels of a spin-gap magnet in a
magnetic field. Solid vertical arrows show transitions
corresponding to the observed ESR absorption, dashed vertical
arrows mark activation gaps for these transitions.
}\label{fig:int-final}
\end{figure}
Low temperature (below 77~K) ESR absorption was measured at certain
fixed field directions: for the field applied parallel to the
symmetry axis $\vect H||Y$ and for the field canted by approximately
$45^0$ towards $X$ axis. In the first case both ladders are
equivalently oriented with respect to the magnetic field, while the
later case corresponds to the maximal difference of the ladders'
effective $g$-factor, as evidenced by 77~K measurements.
As expected, we observe single-component ESR absorption for $\vect
{H}||Y$ and two resolved ESR signals from different ladders for the
canted sample. Temperature evolution of the ESR absorption spectra
is qualitatively similar in both cases (Figure \ref{fig:scans-lt}).
Below 10~K the ESR absorption intensity freeze down due to the
presence of the energy gap. ESR signal continue to loose intensity
down to 450~mK and almost vanishes at this temperature. Lowest
temperature (450 mK) ESR absorption includes broad powder-like
absorption spectrum probably related to the distorted surface of the
sample.
We did not observed any additional absorption signals which could be
related to the formation of the field induced ordered phase above
the critical field to appear at the lowest temperature of 450 mK in
the fields up to 10~T at the frequencies of 26...35~GHz. This is in
agreement with the known phase diagram of DIMPY
\cite{dimpy-dave-prl} demonstrating that highest temperature of the
transition into the ordered state is about 300 mK.
Additional splitting of the ESR absorption lines was observed around
1~K (Figure \ref{fig: max-split}), resonance fields of the split
sub-components differ by approximately 150 Oe. This splitting was
observed at various frequencies, it was most pronounced on the
high-field component of the canted sample ESR absorption spectra.
One of the split sub-components is much weaker then the other and
freeze out faster on cooling. Remarkably, mutual orientation of the
weaker and stronger sub-components is different for the low-field
and high-field components. We did not observe resolved splitting for
the $\vect{ H}||Y$ orientation, instead a weak peak of the linewidth
was observed around the same temperature of 1~K probably indicating
unresolved splitting.
At low temperatures intensities of all components follow
exponential law $I\propto\exp(-\Delta/T)$ (Figure
\ref{fig:int-final}). Energy gap for the weaker sub-components is
larger then that for the main subcomponents. By taking temperature
dependences of the ESR absorption at different frequencies we were
able to determine the values of the energy gaps at several
frequencies revealing dependence of the activation energy from the
resonance field (Figure \ref{fig:int-final}).
\subsection{ESR linewidth evolution from 300~K to 400~mK.}
\begin{figure}
\centering
\epsfig{file=width_t.eps, width=\figwidth, clip=}
\caption{(color online) Temperature dependence of the ESR
linewidth. Upper panel: $\vect{H}||(X+Y)$. Circles: low-field
(high-$g$) component, squares: high field (low-$g$) component.
All data are measured on the samples from the same batch.
Lower panel: $\vect{H}||Y$. Filled and open symbols corresponds to the data
measured on samples from different batches.
Experimental data were collected in different experimental setups operating at
different frequencies, vertical lines mark approximate temperature
boundaries for different experiments, microwave frequencies of
each experiment are given. Typical errorbar size is around
symbol size. Curves on both panels show empirical fit equations
(see text and Table \ref{tab:fits}). }\label{fig:width_t}
\end{figure}
Temperature evolution of the ESR linewidth was measured from room
temperature down to 400~mK (Figure \ref{fig:width_t}). The
temperature dependence is qualitatively similar in all
orientations and demonstrate strongly non-monotonous behavior. At
high temperatures (above 90~K) linewidth strongly increases with
heating rising from about 50 Oe at 77~K to about 300 Oe at 300~K.
On cooling below 77~K linewidth again increases reaching maximum
at temperature $~T_{max}=9.0\pm 0.2$~K. Temperature of the maximum
is the same for all orientations, while linewidth value at the
maximum varies from 90~Oe to 140~Oe, both of the extreme values
being observed in the orientation of maximal splitting
$\vect{H}||(X\pm Y)$ for different components of ESR absorption.
Below $~T_{max}$ linewidth again decreases reaching its minimal
value of about 10~Oe (observed at $\vect H||Y$) at 2~K, which is
most likely limited by the field inhomogeneity in our setup. On
cooling below 2~K a peak in the linewidth is observed around 1~K.
The peak is most pronounced for the high-field component in the
orientation of maximal splitting of the ESR absorption components,
peak position coincides with the temperature of subcomponents
appearance. Similar but less pronounced peak is observed for
$\vect{H}||Y$. High-$g$ component in the $\vect{H}||(X+Y)$
orientation does not demonstrate such a peak, which is probably
related to the very low intensity of the appearing weaker
subcomponent which cause fitting procedure to lock on the main
spectral subcomponent. Finally, on cooling below 1~K linewidth of
both components in the $\vect{H}||(X+Y)$ orientation increases
again.
\section{Discussion.}
\subsection{Recovery of the $g$-tensor.\label{sec:g-tensor}}
Observed angular dependences of the $g$-factor can be fitted
assuming uniaxial $g$-tensor. As was described in the Section
\ref{sect:sym} $g$-tensor is the same for the given ladder and
orientations of the $g$-tensors in inequivalent ladders are bound
by 2-nd order axis. Hence, directions of the main axis can be
expressed via polar angles as $\vect {n}_{g1,2}=(\pm \sin \Theta
\cos\phi; \cos\Theta;\pm \sin\Theta \sin\phi)$, here we count
polar angle $\Theta$ from the 2-nd order axis $Y||b$, different
signs corresponds to the different ladders.
Least squares fit of our data (see Figure \ref{fig:g-angular})
yields $g$-tensor components $g_{\parallel}=2.296\pm0.010$ and
$g_{\perp}=2.040\pm0.006$ and angles $\Theta=(34.8\pm 1.5)^\circ$
and $\phi=(178\pm 4)^\circ$. Fit quality can be improved by
assuming general form of the $g$-tensor. However, this results
only in minor planar anisotropy (with principal $g$-factor values
of $2.038\pm 0.010$ and $2.058\pm 0.010$) which is on the edge of
experimental error.
Main axis of the $g$-tensor within accuracy of our experiment lies
in the $(XY)$-plane of the crystal. This seems to be accidental as
there is no symmetry reasons to choose this plane in the
monoclinic crystal. Orientation of the $g$-tensor main axes
$\vect{n}_{g1,2}$ with respect to the crystal structure is shown
at the Figure \ref{fig:struct}. We can not decipher which of the
orientations corresponds to different ladders.
Found values of the main $g$-tensor components coincide with the
values found in earlier powder high-field ESR experiment.
\cite{dimpy-highfieldESR} The value of the $g$-factor for the
$\vect{H}||a$ case $g_a=2.17$ was found in
Ref.\onlinecite{dimpy-dave-prl} by magnetization fit, this value
disagree by 2\% with the value $g_a^{(ESR)}=2.130\pm0.005$ found
in our experiment.
\subsection{Interladder coupling estimation.}
Anisotropy of the $g$-tensor opens a direct way to the estimation
of interladder coupling. If the coupling between the ladders with
different g-tensor orientation would be strong enough, then a
common spin precession mode would be observed, a well known
exchange narrowing phenomenon.\cite{anderson-stat,exnar2}
Instead, we observe well resolved ESR absorption signals from the
inequivalent ladders. Thus, upper limit on the interladder coupling
can be estimated from the minimal splitting $\Delta H$ observed,
which is around 40 Oe at 17~GHz experiment (corresponds to
components $g$-factor difference $\Delta g=0.015$ at Figure
\ref{fig:g-angular}).
As it is known from the exchange narrowing theory, equally intense
components with splitting $\Delta\omega$ will form a common
precession mode if the coupling strength is above $J_c \simeq
\hbar\Delta\omega$. Hence we obtain an estimate $J_{inter}<{h\nu}
\frac{\Delta H}{H_{res}}\simeq 5$~mK or about 0.5 $\mu$eV.
This value is in reasonable agreement with the earlier estimate of
the interladder exchange coupling from the ordered phase boundary
calculated in mean-field approximation\cite{dimpy-dave-prl} as $n
J'_{MF}=6.3~\mu$eV (here $n$ is the number of coupled ladders,
coupling considered to be equal in all directions). Note that our
observation provides a direct estimate of the coupling between the
unequivalently oriented ladders only.
\subsection{Linewidth temperature dependence.\label{sec:width-t}}
\begin{table*}
\caption{ESR linewidth empirical fit equations parameters in
different temperature ranges and at different orientations. For
the orientations with two ESR components resolved (LF) and (HF)
marks fit results for the low-field and high-field components
correspondingly, (HF+LF) marks cases where linewidths of both
components are close and their temperature dependences are fitted
jointly. \label{tab:fits}}
\begin {ruledtabular}
\begin{tabular}{lccccc}
{Temperature range}&Parameters & Typical fit &$\vect {H}||Y$&$\vect{H}||(x+ y)$&$\vect{H}||(y + z)$\\
{and fit eqns.} & &accuracy & & &\\
\hline
$~T>80$~K& $\Delta H_0$, Oe& $\pm 5$~Oe & 46& 45 (HF); 56 (LF) & \\
$\Delta H=\Delta H_0+A \exp(-E_a/T)$& $A$, $\times 10^4$~Oe&$\pm 50$\% & $5.9 $&$1.7 $ (HF); $1.8$ (LF) & \\
& $E_a$, K&$\pm 150$~K &1510 & 1240 (HF); 1310 (LF)& \\
\hline
15K$<T<80$~K& $\Delta H_\infty$, Oe& $\pm 5$~Oe &35 &35 (HF); 44(LF) & 40 (HF+LF)\\
$\Delta H=\Delta H_\infty(1+\Theta/T)$&$\Theta$, K&$\pm 3$~K & 24& 16 (HF); 22(LF) & 14 (HF+LF)\\
\hline
1.5K$<T<7$~K&$\Delta H_0$, Oe&$\pm 2$~Oe & 10 &15 (HF);15 (LF)& 15(HF+LF)\\
$\Delta H=\Delta H_0+A \exp(-E_a/T)$& $A$, $\times 10^2$~Oe&$\pm 30$\% & 6.4 &1.7 (HF);3.9 (LF) & 5.3 (HF+LF)\\
&$E_a$, K&$\pm 1.5$ & 14 (9.6\footnote{Value of $E_a=9.6~K$ corresponds to the best fit with fixed parameter $\Delta H_0=0$ (see text).}) & 6.4 (HF);8.4 (LF) & 10.5(HF+LF)\\
\end{tabular}
\end{ruledtabular}
\end{table*}
Non-monotonous temperature dependence
of the linewidth indicates that spin precession relaxation is
governed by different processes in the different parts of the
studied temperature range. We fit this temperature dependence by
set of empirical equations as shown on Figure \ref{fig:width_t}
and discussed below. Values of the fit coefficients for the
empiric equations used are gathered in the Table \ref{tab:fits}.
High temperature increase of the ESR linewidth (above 77~K) is
naturally related to the spin-lattice relaxation: increase of the
phonons population numbers leads to the increase of the relaxation
rate. Linewidth dependences can be fitted by the sum of the
constant contribution describing the high-temperature spin-spin
relaxation and an empirical activation law $\Delta H=\Delta H_0+A
\exp(-E_a/T)$. Activation energy is $E_a=(1400\pm150)K$. Similar
behavior with activation energy of the same order of magnitude was
reported for other cuprates \cite{marsel,monika} and it was
discussed \cite{monika} as a relaxation via excited state with a
competing Jan-Teller distortion. However, detailed analysis of the
lattice relaxation is beyond the scopes of the present paper.
Lattice contribution vanishes with cooling. Shallow minimum of the
linewidth at 70-100~K indicates that phonon relaxation channel is
practically frozen down here. Hence, we assume that linewidth
measured at 77~K is mostly due to spin-spin relaxation.
It is well known \cite{Abragam, AltKoz} that anisotropic spin-spin
interactions are responsible for the spin-spin relaxation. Thus,
ESR linewidth provides access to determine these interactions
strength. For the concentrated magnets Dzyaloshinskii-Moria
interaction (which is allowed by the lattice symmetry of DIMPY)
and symmetric anisotropic exchange interaction are the main
contributions. Temperature dependence of the ESR linewidth is one
of the physical effects to find which of theses anisotropic
interactions dominates the linewidth.
Detailed description of the ESR linewidth in a quantum spin-ladder
is only emerging now: case of spin ladder with symmetric
anisotropic spin-spin coupling was considered by recently by
Furuya and Sato \cite{Sato}, a theory accounting for the uniform
Dzyaloshinskii-Moriya interaction is still to be constructed.
Theory of an ESR linewidth for a quantum $S=1/2$ chain was
developed by Oshikawa and Affleck more then decade
ago.\cite{oshikawa} We will apply their results to understand
qualitatively temperature dependence of the ESR linewidth at high
temperatures $~T\gg J_{leg,rung}$. Oshikawa and Affleck have
demonstrated that contribution of the symmetric anisotropic
exchange interaction to the ESR linewidth (exchange anisotropy in
their terms) decreases with cooling. They also considered
contribution of the staggered Dzyaloshinskii-Moria interaction and
have found that in this case ESR linewidth is increasing as
$1/T^2$ at low temperatures. Oshikawa and Affleck demonstrated
that at high temperature limit staggered Dzyaloshinskii-Moria
interaction results in the linewidth increasing with cooling as
$1/T$. As the high-temperature linewidth is determined by the pair
spin correlations, this result should be actually the same for the
staggered and uniform Dzyaloshinskii-Moria interaction. This
conclusion is in agreement with the results of
Ref.\onlinecite{marsel} for the uniform Dzyaloshinskii-Moria
interaction in quasi one dimensional antiferromagnet
Cs$_2$CuCl$_4$. Thus, increase of the linewidth with cooling below
80~K is a direct indication of the dominating role of the
Dzyaloshinskii-Moria interaction for the spin relaxation processes
in DIMPY. To model first order of the $1/T$ expansion we fit our
data by the law $\Delta H=\Delta H_\infty (1+ \Theta/T)$.
Characteristic temperature $\Theta$ is anisotropic and varies from
15 to 25~K in the orientations presented on the Figure
\ref{fig:width_t}. This temperature scale is close to the
exchange integral value in agreement with the results of
Refs.\onlinecite{oshikawa},\onlinecite{marsel}.
The crude estimate of this interaction strength can be obtained
from the linewidth at 80~K, which is approximately 50 Oe. As this
temperature far exceeds the exchange integral scale
high-temperature approximation can be used. We will discuss exact
calculations below while describing angular dependence, but as an
estimate one can write $\hbar \Delta\omega \sim \frac {D^2}{J}$ or
$D \sim \sqrt{g\mu_B \Delta H J} \sim 0.3$~K.
As the temperature decreases below approximately 10~K linewidth
start to decrease. This decrease is naturally related to the
gapped spectrum of the spin ladder. At low temperatures magnetic
properties of a spin ladder can be described on the triplet
quasiparticles language and linewidth is then interpreted as an
inverse lifetime of these quasiparticles, which is partially
determined by their interaction. As temperature approaches scale
of the energy gap, quasiparticles population numbers decreases,
gas of the quasiparticles became diluted and quasiparticles
interaction contribution froze out. This results in the narrowing
of the ESR absorption line with cooling. The linewidth temperature
dependence indeed follows thermoactivation law $\Delta H =\Delta
H_0+A \exp(-E_a'/T)$ with activation energy $E_a'=6.4...14$~K in
different orientations.
If the relaxation processes would be due to the pair interaction
of quasiparticles, the relaxation rate would be proportional to
the quasiparticles concentration squared and the activation energy
would be about $2\Delta_0$, where $\Delta_0=0.33$ meV (or 3.8~K)
is a zero-field gap. The activation energies for $\vect{H}||(X+Y)$
orientation are close to this expectation. However, for the field
applied along the second-order axis $\vect{H}||Y$ the best fit
activation energy ($14\pm1.5$~K) far exceeds the doubled
zero-field gap value. Most likely this result is an artefact due
to the effects of field inhomogeneity in our experimental setup:
the low temperature linewidth is minimal for $\vect{H}||Y$ and
could be limited by experimental resolution. This leads to
overestimation of $\Delta H_0$ parameter which in turn results in
overestimation of the activation energy. Tentative fit of the
$\vect{H}||Y$ data with $\Delta H_0$ value fixed to zero yields
activation energy of 9.6~K which is much closer to twice
zero-field gap value. Similar $\Delta H_0=0$ fits in other
orientations lead to the smaller corrections of the determined
activation energy.
Thus, within accuracy of our experiment, which is mostly limited
by field inhomogeneity of the magnetic field in the
superconducting coil used, we can conclude that pair
quasiparticles interactions dominates spin-spin relaxation
processes of the spin-ladder in low-temperature regime.
The peak of the linewidth around 1~K is related to the splitting of
the ESR lines into subcomponent, its origin is related to the
classical exchange narrowing phenomenon.\cite{anderson-stat,exnar2}
The exchange frequency became temperature dependent being related to
the quasiparticles concentration. At low temperatures (low
quasiparticles concentration) split ESR line is observed, at higher
temperatures (higher quasiparticles concentration) effective
exchange interaction between the quasiparticles gain efficiency and
a common precession mode is formed. Crossover between these regimes
results in the broadening of ESR line. Similar effect is observed in
other spin-gap magnets \cite{glazkov-tlcucl3,glazkov-phcc} and in
other systems. \cite{chestnut}
Finally, definitive increase of the linewidth below 700~mK is
probably indicative of the critical regime in the vicinity of the
field induced phase transition.
\subsection{Angular dependence of the ESR linewidth.}\label{sec:width-ang}
According to the theory of the exchange narrowed resonance spectra
\cite{Anderson1953,Castner1971}, the half width at half maximum
for a single Lorentzian shaped line is given by
\begin{equation}
\Delta H = C\left[\frac{M_{2}^{3}}{M_{4}}\right]^{1/2},
\label{linewidth}
\end{equation}
\noindent where $C$ is dimensionless constant of order unity,
depending on how the wings of Lorentzian profile drop at fields of
the order of exchange field ($J/g\mu_{\rm B} \ll \Delta
H$)\cite{Castner1971}; $M_2$ and $M_4$ are the second and fourth
moments of resonance line, firstly introduced by Van Vleck
\cite{VanVleck1948}
\begin{equation}
M_2 =\frac{\langle [{\cal H}_{anis},S^+][S^-,{\cal H}_{anis}]\rangle}{h^2\langle S^+S^-\rangle}, \label{M2}
\end{equation}
\begin{equation}
M_4 = \frac{\langle [{\cal H}_{ex},[{\cal H}_{anis},S^+]][[S^-,{\cal H}_{anis}],{\cal H}_{ex}] \rangle}{h^4\langle S^+S^-\rangle}. \label{M4}
\end{equation}
where $S^{\pm}$ denote left/right circular components of the total
spin summed up over the whole sample, ${\cal H}_{ex}$ is isotropic
exchange Hamiltonian, ${\cal H}_{anis}$ is anisotropic one that
doesn't commute with ${\cal H}_{ex}$ hence causing broadening of the
resonance line.
Analysis of the ESR linewidth based on calculation of the spectral
moments is a well developed method which allows to identify nature
of spin-spin interactions and estimate their magnitudes in
magnetically concentrated systems.\cite{Zakharov2008} Its benefit
is that in high temperature limit ($T\to\infty$) an exact
expression for linewidth can be found out for an arbitrary spin
system, whatever space dimension and exchange
couplings.\cite{AltKoz,Huber1999}
In the present paper we apply the "method of moments" to a
strong-leg spin ladder system, described by Hamiltonian
(\ref{eqn:ham}) with uniform Dzyaloshinskii-Moriya interaction
\begin{equation}
{\cal H}_{DM}=\sum_i \sum_{l=1,2}\vect{D}_l
[\vect{S}_{l,i}\times\vect{S}_{l,i+1}],\label{eqn:Hanis}
\end{equation}
\noindent here $i$ enumerates rungs of the ladder and $l$ enumerates
legs of the ladder, DM vectors on the legs are considered arbitrary
for the moment ($\vect{D}_1\neq\vect{D}_2$). Substituting Eq.
(\ref{eqn:Hanis}) into Eqs.(\ref{M2}), (\ref{M4}) and using the
corresponding commutation relations for $S=1/2$ spin operators, for
the linewidth Eq. (\ref{linewidth}) in high temperature limit we
have
\begin{equation}
\Delta H_{\infty}^{DM}(Oe)=C\sum_{l=1,2}\frac{[D_x^2+D_y^2+2D_z^2]_l}{4\sqrt{2}\mu_{B}\tilde{J}_Dg(\theta,\phi)}
\label{eqn:dH}
\end{equation}
\noindent where $\tilde{J}_D=\sqrt{J_{leg}^2+2J_{rung}^2}$ mean an
average exchange integral, \cite{Huber1999} and angular dependence
is determined by the transformation for DM vector
\begin{eqnarray}
D_x &=& D_X\cos{\beta}\cos{\alpha}+D_Y\cos{\beta}\sin{\alpha}-D_Z\sin{\beta},\nonumber \\
D_y &=& D_Y\cos{\alpha}-D_X\sin{\alpha}, \label{Dtr} \\
D_z &=& D_X\sin{\beta}\cos{\alpha}+D_Y\sin{\beta}\sin{\alpha}+D_Z\cos{\beta}. \nonumber
\end{eqnarray}
\noindent Here angles $\alpha$ and $\beta$ define orientation of
the local coordinate system $(x,y,z)$ where Zeeman term in Eq.
(\ref{eqn:ham}) takes diagonal form $g\mu_{B}HS^z$ with
\begin{equation}
g=\sqrt{A^2+B^2+C^2}, \label{geff}
\end{equation}
where
\begin{eqnarray}
A&=&g_{XX}\sin\Theta\cos\phi+g_{XY}\cos\Theta+g_{XZ}\sin\Theta\sin\phi, \nonumber \\
B&=&g_{YX}\sin\Theta\cos\phi+g_{YY}\cos\Theta+g_{YZ}\sin\Theta\sin\phi, \nonumber \\
C&=&g_{ZX}\sin\Theta\cos\phi+g_{ZY}\cos\Theta+g_{ZZ}\sin\Theta\sin\phi \nonumber \\
\text{and} \nonumber \\
&\cos\alpha&=\frac{A}{\sqrt{A^2+B^2}}, \cos\beta=\frac{C}{\sqrt{A^2+B^2+C^2}}, \nonumber
\end{eqnarray}
here polar $\Theta$ and azimuthal $\phi$ angles define the direction
of external magnetic field, so that $\Theta$ and $\phi$ are counted
from $Y$ and $X$ axes, respectively, as during the $g$-tensor
recovery procedure.
Note that by setting $J_{rung}=0$ and $\vect{D}_1=\vect{D}_2$ in
Eqn. (\ref{eqn:dH}) we immediately arrive to the known result for
1D Heisenberg chain with uniform DM interaction [see formula (16)
in Ref. \onlinecite{marsel}].
DIMPY has two inequivalent ladders with different $g$-tensors and
DM vectors. In accordance with crystal symmetry of the DIMPY (see
Sec. \ref{sect:sym}), the legs within same ladder are linked by
inversion, so that
\begin{equation}
\hat{\mathbf{g}}_{1}^{(k)}=\hat{\mathbf{g}}_{2}^{(k)},\;
\vect{D}_1^{(k)}=-\vect{D}_2^{(k)},\;(k=1\;\text{or}\;2),
\label{symrel1}
\end{equation}
\noindent here upper index ($k=1,2$) denotes the inequivalent
ladders and lower index enumerate legs od the ladder. The legs of
inequivalent ladders are linked by screw rotation along the second
order axis, hence
\begin{equation} \label{symrel2}
\begin{split}
\vect{D}_{l}^{(2)}=&C_{2}(Y)\vect{D}_{l}^{(1)},\\
{\hat{\mathbf{g}}}_{l}^{(2)}=C_2(Y){\hat{\mathbf{g}}}_{l}^{(1)}&C_2(Y)^{-1},\;(l=1,2),
\end{split}
\end{equation}
Orientation of the local axes is essentially different for inequivalent
spin ladders. Having known directions of the main axes of $g$-tensors (see Sec.
\ref{sec:g-tensor}), it's easy to find their components referred to
crystallographic axes
\begin{eqnarray}
\hat{g}^{(1,2)}=\begin{pmatrix} 2.128 & \mp0.12 & -0.008 \\
\mp0.12 & 2.214 & \pm0.005 \\
-0.008 & \pm0.005 & 2.038 \end{pmatrix},
\label{gten}
\end{eqnarray}
\noindent where upper(down) sign corresponds to the ladder with
upper(down) sign of $\vect{n}_g$.
Simulation of experimental data on the linewidth angular
dependence by Eq. (\ref{eqn:dH}) showed that the
Dzyaloshinskii-Moria interaction describes the angular variation
of the linewidth in DIMPY well enough (within experimental error).
However model including Dzyaloshinskii-Moria interaction only
predicts value of linewidth which is systematically less then the
experimental values by about 12 Oe. This fact indicates there is
an additional (small compared to DM interaction) source of the
line broadening in DIMPY. The modelled values of the linewidth can
be reconciled with the experimental ones by adding isotopic
contribution $\Delta H_0=12$~Oe, which can be probably ascribed to
the residual spin-lattice relaxation, or by considering other
anisotropic spin-spin couplings. Contribution to the linewidth
from dipole-dipole interaction is quite small for DIMPY and at the
shortest distance between Cu ions ($r=a\approx7.5$~\AA{})
following conventional estimation\cite{Yamada1996} it does not
exceed $\sim~0.5$~Oe. Additional broadening in DIMPY can be
related to SAE interaction along legs and rungs of the spin
ladders which usually appear as further sources of ESR line
broadening beyond the dominant DM interaction.
\cite{marsel,Eremin2008} The contribution to the linewidth due to
SAE interaction is derived in Appendix \ref{app:SAE}.
Taking into account symmetry relations (Eqs. (\ref{symrel1}),
(\ref{symrel2}), (\ref{gten}), (\ref{symrel3})) and, for
definiteness setting $\vect{D}_{1}^{(1)}=\vect{D}$,
$\hat{\mathbf{A}}_1^{(1)}=\hat{\mathbf{A}}$, the fitting of
linewidth angular dependence yields $D_X=0.15$, $D_Y=-0.14$,
$D_Z=0.075$~K and almost diagonal exchange-tensor with components
$J_{XX}=0.08$, $J_{YY}=-0.03$, $J_{ZZ}=-0.05$, $J_{XY}=-0.015$~K
and $J_{XZ}=J_{YZ}=0$. During simulation the Lorentzian profile
with exponential wings was assumed, which implies $C=\pi\sqrt{2}$.
Directions of the found $\vect{D}$ vectors are shown on the Figure
\ref{fig:struct}. Components of the DM vector and SAE tensor given
above correspond to the ladder with the $g$-tensor main axis
orientation $\vect{n}_{g1}=(\sin\Theta\cos\phi;
\cos\Theta;\sin\Theta\sin\phi)\approx(-0.57; 0.82;0)$ (when
comparing with Fig.\ref{fig:struct} note that $\vect{n}_g$ and
$-\vect{n}_g$ are physically equivalent), angle between
$\vect{D}$ and $\vect{n}_g$ vectors (reduced to
$(-\frac{\pi}{2};\frac{\pi}{2})$ range for convenience) is
approximately 23$^\circ$. Magnitude of the obtained
Dzyaloshinskii-Moria vector $|\vect{D}|=0.22$~K agrees well with
the crude estimation above (section \ref{sec:width-t}).
As it is seen from Fig. \ref{fig:width-angular}, taking into
consideration only an exchange mechanisms of spin anisotropy
within the legs of ladders gives a good compliance with
experiment. However, it is necessary to stress out, that without
DM interaction SAE coupling only (see Appendix \ref{app:SAE})
totally failed to give a correct description of the angular
dependence of linewidth in DIMPY.
Our simulation shows that absolute value as well as angular
anisotropy of the linewidth are predominantly attributed by DM
interaction, while contribution to the linewidth due to SAE
interaction is relatively small compared to DM one and gives a weak
angular variation. Similar behavior of ESR linewidth with coexistent
contributions from DM and SAE interactions within $S=1/2$
antiferromagnetic chains was observed in hight symmetry crystal
structure KCuF$_3$ \cite{Eremin2008}.
The found DM vector have not only transverse but also nonzero
\textit{longitudinal} (with respect to the Cu-Cu exchange bond)
component within the legs. Such result does't contradict with the
general rules for DM vector, established by Moriya
\cite{Moriya1960} based on general symmetry grounds for a pair of
exchange interacting ions. Moreover, a simple analysis of the
recovered $g$-tensors (see Sec. \ref{sec:g-tensor}) leads to the
same conclusion about direction of the DM vector. Since the axial
component of the $g$-tensor has a maximal value, then the ground
state orbitals of Cu$^{2+}$ ions (typically
"$\tilde{x}^2-\tilde{y}^2$"-like symmetry and $\tilde{z} ||
\vect{n}_{g}$) should predominantly lie within the plane
perpendicular to the main axis of a $g$-tensor, because the
maximal matrix element ($<\tilde{x}^2-\tilde{y}^2|
l_g|\tilde{x}\tilde{y}>=-2\imath$) relevant to spin-orbital
coupling appears only in the case when an external magnetic field
is applied parallel to the main axis of $g$-tensor. For the same
reason an effective DM vector predominantly should lie along the
main axis of the $g$-tensor. It should be noted that conventional
rule determining DM vector as
$\vect{D}\propto \left[ n_{1} \times n_{2} \right]$,
\cite{Keffer1962, Moskvin1977} where $n_1$ and
$n_2$ are unit vectors connecting a exchange interacting ions with
bridging ion, is not applicable in present case. Possible failure
of this rule was mentioned before in Ref. \onlinecite{Eremin1965},
referring to the features of exchange process through a two
bridging ions, which is also the case of DIMPY (see Fig.
\ref{fig:struct}).
Thus, analysis of ESR linewidth allowed us to conclude that the
DIMPY is a rare case of compound in which DM vector has a component
along the line connecting the pair of exchange interacting ions.
This is a consequence of low crystal symmetry of DIMPY and
nontrivial orbital ordering.
\subsection{Low-temperature sub-components appearance.}
First, we recall main observations on the subcomponent appearance.
ESR components splits around 1~K into two sub-components, one of
which is much weaker. The splitting is best observed at
$\vect{H}||(X+Y)$ orientation. Position of the weaker
sub-component with respect to the stronger sub-component is
different for both ESR absorption components. Maximal splitting is
about 150~Oe and it decreases as the resonance field approaches
critical field, weaker subcomponent became unresolvable at the
fields above 2/3 of the critical field. Activation energies for
the stronger and weaker subcomponents are different.
All these observations can be explained as an effect of the zero
field splitting of triplet sublevels. This effect was already
observed for various spin-gap magnets, e.g. TlCuCl$_3$,
\cite{glazkov-tlcucl3} or PHCC. \cite{glazkov-phcc} Anisotropic
interactions lift degeneracy of the $S=1$ triplet state and
frequencies of the dipolar transitions $|S^z=+1 \rangle
\leftrightarrow |S^z=0 \rangle$ and $|S^z=-1 \rangle
\leftrightarrow |S^z=0 \rangle$ would become different. Here we
assume, which is perfectly valid for the case of DIMPY, that the
anisotropy is very small and spin projection on the field
direction $S^z$ can be considered as a good quantum number.
Therefore, in the presence of such an anisotropy the resonance
fields for $|S^z=+1 \rangle \leftrightarrow |S^z=0 \rangle$ and
$|S^z=-1 \rangle \leftrightarrow |S^z=0 \rangle$ transitions in
the constant frequency ESR experiment would differ and ESR
absorption split into two sub-components.
Observed difference of the activation energies for the absorption
sub-components and dependence of the activation energy on the
microwave frequency used in the experiment is a direct consequence
of this explanation. The ESR intensity at low temperature is
determined by the population of the lowest sublevel. Hence, for
the $|S^z=+1 \rangle \leftrightarrow |S^z=0 \rangle$ transition
the activation energy is $\Delta \approx \Delta_0-g\mu_B
H_{res}=\Delta_0-h\nu$, being determined by the population of the
$|S^z=+1 \rangle$ sublevel (energy of this sublevel decrease with
field, see inset on Figure \ref{fig:int-final}). In the same time
the activation energy for the $|S^z=-1 \rangle \leftrightarrow
|S^z=0 \rangle$ remains constant (and equal to $\Delta_0$) since
the energy of $|S^z=0 \rangle$ sublevel is field independent. The
dependences of the activation energy on the microwave frequency of
the ESR experiment are described by this model parameter-free
using the zero-field gap value of 0.33~meV from the inelastic
neutron scattering experiment. \cite{dimpy2,dimpy-dave-prb}
Behaviour of the sublevels of the spin-gap magnet in the vicinity
of the critical field is a long-discussed problem. There is a
general macroscopic (or bosonic) approach of Refs.
\onlinecite{Affleck}, \onlinecite{farmar} and a 1D fermionic
approach of Tsvelik\cite{tsvelik} developed for the spin-chains.
Fermionic model of Tsvelik yields results formally equivalent to
the results of perturbation treatment of anisotropic
interactions.\cite{zaliznyak} Thus, within these approaches the
sublevels behave linearly in the vicinity of the critical field
and the splitting of the ESR subcomponent should be then field
independent. Bosonic model, on the contrary, predicts
square-root-like approach to the critical field for the low-energy
sublevel, while field dependence of the high-energy sublevel
remains linear in the vicinity of the critical field. Therefore,
sub-components splitting will change close to the critical field.
However, this nonlinearity of the bosonic model extends only in
the small vicinity of the critical field $(H_c-H) \sim \Delta
E/\mu_B \sim \Delta H$, here $\Delta E$ is the zero field triplet
sublevels splitting and $\Delta H \simeq 150$~Oe is the observed
sub-components splitting. We observe (Figure \ref{fig: max-split})
that the observed splitting is halved (compare 50.18~GHz and 26.30
~GHz curves at the Figure \ref{fig: max-split}) in the field of
about 2/3 of the critical field ($H_c \simeq 30$ kOe, zero-field
gap of 0.33~meV corresponds to the frequency of 80~GHz), i.e. well
below this nonlinearity range. This probably indicates that field
evolution of the split sub-components follows some other laws on
approaching the critical field. Similar behaviour of the ESR line
split by the uniform Dzyaloshinskii-Moria interaction was recently
reported for a quasi-1D antiferromagnet
Cs$_2$CuCl$_4$.\cite{timofey}
Under an assumption that the uniform Dzyaloshinskii-Moria
interaction along the legs of the ladder is responsible for the
observed splitting, anisotropy axis have to be aligned along the
$\vect{D}$ vector. We calculated effects of the DM coupling
perturbatively for the limiting case of strong-rung ladder (see
Appendix \ref{app:perturb}). Interdimer DM interaction mixes one
and two-particle excited $S^z=\pm 1$ states which results in the
triplet sublevels splitting by $\delta E=\frac{D^2}{2 J}$,
$S^z=\pm 1$ sublevels being shifted down. This corresponds to the
easy-axis anisotropy for the triplet excitations, $\vect{D}$
direction being the easy axis direction. Taking the magnitude of
the DM vector $D\approx 0.20$~K as estimated from the
high-temperature ESR linewidth analysis and substituting energy
gap of 0.33~meV as an exchange parameter of the perturbative model
we obtain an estimate of the sublevels splitting $\delta E \simeq
5$~mK which corresponds to sub-components splitting of about
40~Oe, factor of four less then the experimentally observed value.
However, perturbative treatment starting from the uninteracting
dimers is at best a qualitative model for a strong-leg ladder and
a detailed description of a strong-leg spin ladder with uniform
Dzyaloshinskii-Moria interaction needs a separate theoretical
effort.
We can not unambiguously determine type of the anisotropy from our
experimental observation since our setup does not allow to take an
angular dependence at He-3 temperature range. However, as it is
known from the formally similar problem of $S=1$ ion in a crystal
field \cite{Abragam,AltKoz} the effective anisotropy constant
changes monotonously with field rotating away from the anisotropy
axis $C_{eff}=\frac{C}{2}\left(3 \cos^2\xi-1\right)$, where $\xi$
is an angle counted from the anisotropy axis $z$ and anisotropy
$C$ enters spin Hamiltonian as $C ({S}^z)^2$. It is maximal at the
field parallel to the anisotropy axis, it change sign and
decreases by the factor of two at the orthogonal orientation of
the magnetic field and it turns to zero at a magic angle. Thus, as
splitting observed for the high-field component is larger then
that from the low-field component (approximately 150 Oe vs. 110
Oe, see Figure \ref{fig: max-split}) and weaker sub-component is
located on different side from the main sub-component, we find it
more likely that the high-field component corresponds to the
ladder with the magnetic field close to the true anisotropy axis.
In this case, as for the field applied close to anisotropy axis
the weaker subcomponent is located to the right from the stronger
subcomponent, the splitting of the triplet sublevels follows
easy-axis type of anisotropy energy of $S^z=\pm1$ states being
lower then energy of $S^z=0$ state in zero field.
However, this tentative identification of the anisotropy axis
deviates from the simple model of DM interaction only: as the
vectors $\vect{D}$ and $\vect{n}_g$ are quite close for the given
ladder the low-field component (corresponding to the higher
longitudinal $g$-factor) should then be closer to the anisotropy
axis. Possible reason for this deviation is the effect of
symmetric anisotropic exchange on the rungs of the ladder (see
Appendix \ref{app:perturb}). SAE coupling is smaller in magnitude,
but it enters to the triplet sublevels splitting linearly, why DM
contribution is quadratic. This is contrary to the linewidth
calculations where both couplings enter quadratically. Thus,
description of the subcomponents splitting probably lies beyond
the simple model with DM interaction only and requires accounting
for other anisotropic interactions.
\section{Conclusions.}
The strong-leg spin ladder system DIMPY is an established test
example of the Heisenberg spin ladder. However, anisotropic
spin-spin interactions, and in particular Dzyaloshinskii-Moria
interaction of intriguing geometry: uniform along the leg of the
ladder and exactly opposite on the other leg, give rise to a
family of interesting phenomena.
We have estimated parameters of Dzyaloshinskii-Moria interaction
from high-temperature data. We observe splitting of the ESR line at
low temperatures which is related to the zero-field splitting of the
triplet sublevels by the same interaction. Finally, we observe
series of crossovers between different regimes of relaxation of
spin precession on cooling from room temperature to 400~mK.
We present qualitative explanations of our observations. Simple
geometry of the exchange couplings and anisotropic spin-spin
interactions makes DIMPY one of the few candidates for the
model-free microscopic description of the effects of anisotropic
interactions on the properties of a spin-gap magnet, which is
still awaiting for a theoretical effort.
\acknowledgements
We thank A.B.Drovosekov (Kapitza Institute) for the assistance with
ESR experiment above 77~K and Prof.A.K.Vorobiev (M.Lomonosov Moscow
State University) for the possibility to perform reference
measurements with X-band spectrometer. We thank K.Povarov, Prof.
M.V.Eremin and Prof. A.I.Smirnov for valuable and stimulating
discussions. Authors acknowledge usage of "Balls \& Sticks" software
to build crystal structure images.
The work was supported by Russian Foundation for Basic Research
Grant No.15-02-05918, Russian Presidential Grant for the Support
of the Leading Scientific Schools No.5517.2014.2. M .F. work was
supported by the Russian Government Program of Competitive Growth
of Kazan Federal University. This work was partially supported by
the Swiss National Science Foundation, Division 2.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,388 |
At the Dance Theater Workshop, in Chelsea, early the following Saturday night, the lines on the staircase included college kids, agents, middle-aged black men in tweed sport coats, old women in minks, and sisters in cornrows. To a dignified experimental theater like D.T.W., The Spook Show was obviously more than just a mild sensation. Twenty minutes before curtain, the only seats were on the floor.
Next, this funky Don Quixote ran down a European trip. His spiel was accompanied by a series of eye-popping physical transformations: into a stewardess steering a quivering beverage cart; a microwaved airplane string bean; a German burgher ogling "the Schwarze" making his way down narrow Amsterdam streets and thanking God for legal hashish.
Next, Whoopi Goldberg distorted her voice and contorted her body into a character so handicapped one wondered whether she could make it work without veering into bathos, or bad taste. Like Lily Tomlin, who has walked a similar tightrope, she brought it off.
The last character, the little girl, looked so young that one observer swore Whoopi Goldberg was thirteen. She wore an old white skirt draped over her head and called it "long blond hair." She said her ambition was to sail on the Love Boat, but with a headful of pigtails that "don't do nuffin', don't blow in the wind," she wouldn't be welcomed aboard. She had tried other ways not to be black, but bathing in Clorox didn't work. This night the little girl approached a black woman in the audience. "Do you go out with a skirt on your head?" The answer was no, but the little girl pressed on, asking, "Can I touch your hair?" She was so clever and charming, her victim couldn't refuse.
I had not seen Whoopi Goldberg since 1980, when she and I and a group of Second City actors were working for a La Jolla, California, shrink tank on a state-funded PBS show about loneliness and isolation. Unlike the rest of us, Whoopi had been hired locally. She was not just the token black member of the cast. Like a seltzer bottle in the desert, she was the only black comedienne in all of San Diego County.
She became the toast as well as the scourge of San Diego. When she introduced her piece on teenage abortion, the right-to-lifers picketed her act. "I had Nazis on my back," says Whoopi. She got threatening letters. "Thanks for the free publicity," she said.
Touring Europe with Schein, she was referred to by the Germans as "die schwarze Schauspielerin," or "the black actress." When they returned to Berkeley, Schein recalls, Fontaine's European monologue was dashed off the top of her head in one inspired night.
"Girl, I've never seen anything like this before," said New York Afro-American psychotherapist Constance Carr-Shepherd after Whoopi Goldberg wiped imaginary snot from her finger and ran it through Connie Carr–from–Philadelphia's hair.
Unlike the characters of Lily Tomlin, Richard Pryor, and even Lenny Bruce, who deliver interior monologues essentially uninfluenced by audience reaction, Whoopi's "spooks" talk directly to the audience, and in these conversations the characters will bend dramatically to meet the audience's reactions. No two shows are the same. I saw seven. The soul of the Whoopi Goldberg variations is improvisation, the illumination of the moment, as in a revival meeting or in jazz.
On the word of a San Francisco theater colleague, the D.T.W. had booked Whoopi Goldberg sight unseen. Suddenly, her show was successful beyond an avant-garde theater's most surrealistic dreams. Its star had been reviewed twice in the Village Voice, a full-page story about her was about to be published in Newsweek, and after a week of Liz Smith's promos, she was finally seen chatting behind dark glasses with Jack Cafferty on Live at Five.
Since the D.T.W. was helpless to capitalize on the bonanza—its season was already booked—the journalist-tummeler Bob Weiner took it upon himself to convince a bunch of club owners and show-business honchos that a trip downtown to see The Spook Show was better than watching movie stars eat blinis at the Russian Tea Room.
Following Weiner's tip, Greg Dawson promptly offered her a space at the Ballroom, on Twenty-eighth Street, for an extended run. Joe Cates offered to produce her in a "regular situation," where she could "build as a theater artist" in a "classy Off Broadway venue." Larry Josephson, WBAI's sour gourmand, asked her to write material for the Radio Foundation, which produces The Bob and Ray Show.
It is possible there were even more spectacular offers: strangers pressed their phone numbers on Whoopi Goldberg until her pockets were stuffed. Her backstage visitors included Bette Midler, Burt Bacharach, Carole Bayer Sager, Jerry Stiller, Anne Meara, Warner Bros. vice president Diane Sokolow, Mrs. Oscar Hammerstein II, Norma Kamali, supermanager Sandy Gallin, and superagent Sam Cohn. After the penultimate performance, one of her idols, Mike Nichols, stood before her for a speechless five minutes. Mike Nichols did not become famous for sentimentality, but when he embraced Whoopi Goldberg and called her "a true artist," he was in tears.
No celebrities were present at Whoopi Goldberg's two great shows at Manhattan Community College, on Chambers Street. In front of a student body of which 70 percent belonged to a minority, Whoopi really let it rip. Among the smart and sophisticated middle-class crowd at D.T.W., she was the dangerous one, with her provocative language, her forays into the audience, and her threats to steal rings. Here, the most galvanic laugh of the afternoon was at the dope fiend Fontaine's discovery that a French bathroom attendant wants money for the toilet paper. Here, when she sassed the audience, the audience sassed back.
Describing his trip into the Schwarzwald, Fontaine observed, "There was me and it. The last black person these folks seen was Hannibal." Instantly, Whoopi figured out why the kids didn't laugh. "You don't know who Hannibal is!" Fontaine gasped. "Is this a college? Hannibal crossed the Alps on an elephant! He was a very dark-skinned man!" Horrified, Fontaine scratched his crotch and adjusted his shades. "Check out the libraries, y'all," he said.
At the Odeon that night, she ordered, as Fontaine would have, a filet mignon "done-done." She explained to the waiter that she got freaked eating blood.
The steak arrived oozing red. Conversations with show-business mavens that week, she was saying, had yielded this advice: "Get a good writer"; "Don't do a sitcom right away"; "Develop a strong New York cult following"; and "Say you're twenty-nine." Her flicker of hesitation gave way to defiance, and Whoopi pushed away the twenty-one-dollar dish. She looked blankly at her steak and said, "Save it for the dog." One wondered how anyone, including Moms Mabley, could tell her how to direct this sort of success. Over lemon dacquoise, her options sounded less freewheeling than choosing a hairdo for a corpse.
In New York, Sandy Gallin had "dragged everyone from dress designers to high-school friends" to see Whoopi's show. And his partners in Hollywood heard reports that he had seen "the greatest black actress of our lifetime." At the Russian Tea Room, Gallin asked Whoopi to consider signing with Katz-Gallin-Morey & Addis, promised to read The Color Purple, and then had her driven to the projects in a limousine.
Now, after Mac Davis's Vegas show at the MGM Grand, Whoopi grilled Davis about signing with Sandy Gallin, while Gallin grilled David Schein, who said, "Just let her do what she wants. She's very surprising when she does what she wants." She wanted to sign. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,245 |
Q: Going from setTimeout AJAX call to Long Polling with timeout I have created a site which auto refreshes a DIV every 15 seconds using a timeOut.
This works pretty solid, but it can take up to 5 minutes before the DIV is actually updated, which means the script will do the call 20 times before a change happens. Pretty good waste of costly bandwidth and server performance :)
I have been reading a lot about doing Long Polling instead and I've been giving it a go.
My AJAX call was:
intval = window.setTimeout(function() {
$.ajax({
type: 'GET',
cache: false,
url: 'url',
beforeSend: function() { $('#timerimg').attr('src', 'img/icons/loading.gif'); },
success: function(data) { $('#ajaxcontent').html(data); },
complete: function() { $('#timerimg').attr('src', 'img/icons/stop.gif'); }
});
}, 15000);
This function was placed inside the page being refreshed all the time, which made the timeout-function to being kept repeated.
Now I have tried to follow http://techoctave.com/c7/posts/60-simple-long-polling-example-with-javascript-and-jquery to create a simple Long Polling function
(function poll() {
$.ajax({
type: 'GET',
cache: false,
url: 'url',
success: function(data) { $('#ajaxcontent').html(data); },
complete: poll,
timeout: 30000
});
})();
If I am setting the url to hi.txt and making that write Hello World! then it finished constantly and doing a new poll. Which obviously means A LOT of simultaneous polls.
How do I correct this?
AND:
The autorefreshing DIV is a large chart with calculations from a huge SQL-Server query. So maybe it should just check if there has been the slightest change in the .getRows() (compared to the original) after the SQL query?
I am doing the whole thing in jQuery and ASP-Classic.
A: Long polling is not a client only solution. Long polling requires the server to hold the call until something changes. This means that you will be using up one of your concurrent connections for EVERY client logged in. This can be a problem with some service providers that limit concurrency, so check your ISP policy before you get a 503.
Also you are addressing more than one issue here. The server should keep track of whether or not the last request needs to be updated, or to just send you back the cached response. If you are using long polling, then the server will hold your connection until it has changed.
Previously you were using interval polling. I would suggest continuing that route (with server side caching) instead of dedicating half of your IE bandwidth (2 concurrent max) for an infrequent update. Just have the server kick back a "false" response to your inquiry if things haven't changed.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,287 |
Did Kanye West Attend the 2021 Met Gala With Kim Kardashian? The Truth About Her Masked Date
The 2021 Met Gala had a few fashionable surprises in store—but did you expect Kanye West to be one of them?
By Kisha Forde • Published September 13, 2021
Kimye fans were sent into a frenzy after a mystery man in a masked ensemble, who looked eerily similar to Kanye West, accompanied Kim Kardashian at the 2021 Met Gala on Monday night.
Not to mention, the "Keeping Up With the Kardashians" star's outfit alone created much chatter online after she made a dramatic red carpet entrance in a very polarizing look by Balenciaga.
For the momentous occasion, Kardashian donned a head-to-toe, all-black custom Balenciaga look, complete with a face covering and train.
While many believed that West made a surprise appearance at the fashion extravaganza, a source exclusively tells E! News he didn't attend the Met Ball.
Instead, the person standing by the KKW Beauty founder's side in an incognito look was none other than designer Demna Gvasalia, who is the creative director of Balenciaga.
See All the Celeb Couples at the 2021 Met Gala
The 2021 Met Gala in Photos
entertainment news 8 hours ago
Guitarist Tom Verlaine, Co-Founder of Punk-Era Band Television, Dies at 73
Bruce Springsteen 10 hours ago
Despite West's no-show at this year's Met Gala, the insider tells E! News, "His presence will be felt on the carpet."
"It was him who introduced Kim to Demna and was instrumental in the newly formed relationship between her and Balenciaga," the source explains, adding that it's a new fashion era for Kim, "This look on Kim is like a new subculture and fashion statement. No logo, no face, but everyone knows it's her."
The insider continues, "Kanye gave her the courage to push creativity and people's imagination through art. It's the ultimate confidence."
In fact, the Skims founder has donned many BDSM-inspired ensembles in recent weeks.
Just this past weekend, Kardashian stepped out in a jaw-dropping head-to-toe leather look -- complete with a matching face covering -- for New York Fashion Week. To no one's surprise, she sizzled in the bondage-inspired Balenciaga design.
Both Kardashian and West, who she filed for divorce from in February, have occasionally worn similar face coverings by the luxury brand. Last month, the reality TV star turned heads when she looked unrecognizable in a masked bodysuit during the rapper's "Donda" album listening party in Atlanta.
See some of the top looks from the 2021 Met Gala.
At the final "Donda" listening party in late August, Kardashian switched up her style and stunned in a striking haute couture wedding dress by Balenciaga. She even joined West on stage to show her support for him.
"Kanye asked her to do something and Kim was happy to do it," a source close to the beauty mogul told E! News. "She has always supported his work and will continue to do that in the future. She enjoys collaborating with Kanye. They have a bond for life and she wants to be there for him."
A second insider echoed similar sentiments, adding, "She knew how much this meant to him and was honored to be involved. The wedding dress was symbolic of their relationship and for the song. It was not a vow renewal. There is still so much love between the two, but they aren't reconciling."
It seems that West and Kardashian's final Met Gala appearance together was in 2019.
Copyright E! Online
Met GalaKim Kardashian | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,485 |
Q: How to send keystore in 3rd party desktop application using C# Windows application I am facing problem this work only for 2 or 3 days after this send error is access denied but forground windows is focused. and activated ....
namespace WindowsFormsApplication8
{
public partial class Form1 : Form
{
public Form1()
{
InitializeComponent();
}
[DllImport("User32.dll", SetLastError = true)]
public static extern IntPtr FindWindow(String lpClassName, String lpWindowName);
private void button1_Click(object sender, EventArgs e)
{
try
{
Process myproc= Process.GetProcessesByName("name").FirstOrDefault();
if (myproc!= null)
{
var ffCondition = new PropertyCondition(AutomationElement.ProcessIdProperty, myproc.Id);
var ffWindows = AutomationElement.RootElement.FindAll(TreeScope.Children, ffCondition);
var soWindow = ffWindows.Cast<AutomationElement>().FirstOrDefault(w => w.Current.Name.Contains("title of window "));
if (soWindow != null)
{
var soWindowPattern = soWindow.GetCurrentPattern(WindowPattern.Pattern) as WindowPattern;
if (soWindowPattern != null)
{
// Restore window (activating it).
soWindowPattern.SetWindowVisualState(WindowVisualState.Normal);
Thread.Sleep(100);
// Close window.
//soWindowPattern.Close();
SendKeys.Send("{f1}");
//System.Windows.Forms.SendKeys.SendWait("{f1}");
//string send = System.Windows.Input.Key.F1.ToString();
//SendKeys.SendWait(send);
var soWindow1 = ffWindows.Cast<AutomationElement>().FirstOrDefault(w => w.Current.Name.Contains("another child window "));
}
}
}
}
catch (Exception ex)
{
MessageBox.Show(ex.ToString());
}
}
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,326 |
Q: CREATE UNIQUE creates duplicate relation My Cypher query:
START reference=node(0)
CREATE UNIQUE
reference
-[:REFERENCES]->
(categories {name: 'categories'})
-[:CATEGORY]->
category_user{name : 'user_categorie'}),
reference
-[:REFERENCES]->
(categories {name: 'categories'})
-[:CATEGORY]->
category_project{name : 'project_categorie'})
The problem I am having is that it creates two REFERENCES relationships between the reference node and the categories node, where you'd expect only one.
A: It wouldn't know that the node with name "categories" is unique by name so it simply creates the whole pattern.
You could do something like:
START reference=node(0)
CREATE UNIQUE reference-[:REFERENCES]->(categories {name: 'categories'})
CREATE UNIQUE (categories)-[:CATEGORY]->(category_user {name : 'user_categorie'})
CREATE UNIQUE (categories)-[:CATEGORY]->(category_project {name : 'project_categorie'})
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,898 |
Introduced in 1997, the Gulfstream G-V was the first ultra-long range business jet and continues to set the standard for the class today. With a maximum range of about 6,500 miles, it is capable of completing nearly any flight you could ask for. The plane lives up to the standards of performance and reliability that Gulfstream is renowned for.
Modern business jets still have trouble competing with the level of performance that the Gulfstream G-V set at the time of its introduction. Even at maximum load capacity the plane can still travel 6,230 miles, well within the range of nearly all transoceanic flights. The plane usually holds 15 passengers, but can be configured to hold more if necessary. Standard amenities include a galley, fold-out work tables, a sink with running water, power outlets, and separate lavatories for passengers and crew. The plane can also be upgraded to include satellite TV and satellite phones at every seat as well. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,617 |
JAMMU, Feb 13: Jugal Kishore Sharma, BJP Member of Parliament from Jammu Poonch Lok Sabha constituency, today visited village Raipur and Bhalwal in Raipur-Domana constituency and sanctioned Rs 15 lakh for development works.
During the visit, Jugal also paid obeisance at the holy Temple of Lord Shiva at Raipur Mandi and prayed for the peace and development of the whole community.
While addressing the gathering, he highlighted the policies and programmes of Prime Minister Narendra Modi who has been taking the country forward in every arena. The MP also listened to the problems and demands of the people of the area and assured that all their demands would be re-addressed and fulfilled.
As the gathering demanded a community hall in the area, the MP sanctioned Rs 10 lakh for the same from his MPLAD scheme.
Jugal Kishore also visited village Bhalwal and redressed the demand of inhabitants by sanctioning Rs 5 lakh for the development of cremation ground.
Pandit Prabhakar Khajuria, Bakshi Singh Jamwal and locals of village Raipur and Bhalwal accompanied the MP. They said that sanctioning the amount from MPLAD scheme for various development works would greatly help the people.
District president Rural BJP, Omi Khajuria, Yash Pal Seth, ex-Sarpanch Bhupinder Singh ,ex-Sarpanch Narinder Singh Jamwal, Booby , Raju Singh, Shakti Dutt, Sheesh Pal, Harsh Vardhan Singh, Pt Ravi Sharma, Nittu Verma, Joginder Kumar, Sunil Rakka, Gulshan Bhagat and others accompanied the MP. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,432 |
12/28/2020 - Mercer County Health Officer Association Distribution of Moderna Vaccine Began in Hamilton Township
HAMILTON, NJ – Hamilton Township in collaboration with the Mercer County Health Officer Association (MCHOA) held the first COVID-19 vaccine clinic earlier today to Phase 1A healthcare workers.
In order to maximize COVID vaccination efforts, the Mercer County Health Officers Association has joined together to serve all communities within Mercer County throughout the four phases of the COVID-19 vaccination distribution. Under CDC and State health guidelines, the Moderna doses will first be distributed to healthcare workers who qualify under Phase 1A and who have not been vaccinated for COVID-19 through their employer or the federal Pharmacy Partnership for Long Term Care (LTC) Program administered through CVS and Walgreens.
This MCHOA is currently planning a series of COVID-19 vaccination clinics to support ongoing efforts to vaccinate healthcare workers which include emergency medical services. The MCHOA will administer the Moderna COVID-19 vaccinations at clinics throughout Mercer County municipalities point of dispensing (POD) locations. The clinics will be held twice a week on a rotating schedule and have the capacity to handle 500 vaccines per week. The COVID-19 vaccine clinics will be by appointment only and subject to the availability of vaccine doses on hand or accessible within the supply chain.
"Hamilton Township is proud to partner with the Mercer County Health Officer Association in order to ensure that those on the frontlines in our fight against this virus receive the vaccine as quickly as possible," said Mayor Jeff Martin. "The arrival of the Moderna COVID-19 vaccine is a continued step forward to provide protection to more of our community's critical healthcare workforce and eventually the general adult population."
"Vaccination is a critical component to protecting our residents," stated Hamilton Township Health Officer Christopher Hellwig. "Working together to safeguard the citizens of Mercer County is exactly what the founding members of the MCHOA had in mind when they formed in 1972. Our vaccination clinics will continue that ideal and work to protect the public's health particularly those that have been most impacted by COVID-19, while giving us a clear end to this pandemic."
Local Health Departments are one piece of the puzzle to vaccinate the State goal of 70% of the adult population in 6 months. This collective effort will ensure that our residents are provided with the opportunity to receive their vaccination in a timely manner and in a safe medical setting. COVID-19 vaccines will continue to be rolled out in phases determined by the State. The Mercer County Health Officers Association collaboration will continue to work closely with federal, state, and local partners. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,072 |
Q: Classify $I(a,b)=\int_{0}^{\frac{\pi}{2}} \frac{dx}{\sin(x)^a\cos(x)^b}$ with $a,b\in \mathbb{R}$ Classify $I(a,b)=\int_{0}^{\frac{\pi}{2}} \frac{dx}{\sin(x)^a\cos(x)^b}$ with $a,b\in \mathbb{R}$.
I'm really lost, this excercise is so different from the others and I can't find any similar questions online, the integral seems to converge if -1< a+b <1 with some exceptions that I can't figure out.
Edit: I solved it using user Greg Martin advice but I can't mark it as if he solved my problem so I'll credit him here.
A: HINT:
The function $f$, defined as
$$f(x)=\begin{cases}
\frac{x}{\sin(x)}&,x\ne 0\\\\1&, x=0\end{cases}$$
is analytic on $[0,\pi,2]$.
Now, write $\frac{1}{(\sin(x))^a}=\left(\frac{x}{\sin(x)}\right)^a\,\frac1{x^a}$.
Similarly, $\cos(x)=\sin(\pi/2-x)$
Can you proceed?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,604 |
void ofSoundStopAll(){
#ifdef OF_SOUND_PLAYER_FMOD
ofFmodSoundStopAll();
#endif
}
//--------------------
void ofSoundSetVolume(float vol){
#ifdef OF_SOUND_PLAYER_FMOD
ofFmodSoundSetVolume(vol);
#endif
}
//--------------------
void ofSoundUpdate(){
#ifdef OF_SOUND_PLAYER_FMOD
ofFmodSoundUpdate();
#endif
}
#ifndef TARGET_ANDROID
//--------------------
void ofSoundShutdown(){
#ifdef OF_SOUND_PLAYER_FMOD
ofFmodSoundPlayer::closeFmod();
#endif
}
#endif
//--------------------
float * ofSoundGetSpectrum(int nBands){
#ifdef OF_SOUND_PLAYER_FMOD
return ofFmodSoundGetSpectrum(nBands);
#elif defined(OF_SOUND_PLAYER_OPENAL)
return ofOpenALSoundPlayer::getSystemSpectrum(nBands);
#else
ofLog(OF_LOG_ERROR, "ofSoundGetSpectrum returning NULL - no implementation!");
return NULL;
#endif
}
#include "ofSoundPlayer.h"
//---------------------------------------------------------------------------
ofSoundPlayer::ofSoundPlayer (){
player = ofPtr<OF_SOUND_PLAYER_TYPE>(new OF_SOUND_PLAYER_TYPE);
}
//---------------------------------------------------------------------------
void ofSoundPlayer::setPlayer(ofPtr<ofBaseSoundPlayer> newPlayer){
player = newPlayer;
}
//--------------------------------------------------------------------
ofPtr<ofBaseSoundPlayer> ofSoundPlayer::getPlayer(){
return player;
}
//--------------------------------------------------------------------
bool ofSoundPlayer::loadSound(string fileName, bool stream){
if( player != NULL ){
return player->loadSound(fileName, stream);
}
return false;
}
//--------------------------------------------------------------------
void ofSoundPlayer::unloadSound(){
if( player != NULL ){
player->unloadSound();
}
}
//--------------------------------------------------------------------
void ofSoundPlayer::play(){
if( player != NULL ){
player->play();
}
}
//--------------------------------------------------------------------
void ofSoundPlayer::stop(){
if( player != NULL ){
player->stop();
}
}
//--------------------------------------------------------------------
void ofSoundPlayer::setVolume(float vol){
if( player != NULL ){
player->setVolume(vol);
}
}
//--------------------------------------------------------------------
void ofSoundPlayer::setPan(float pan){
if( player != NULL ){
player->setPan(CLAMP(pan,-1.0f,1.0f));
}
}
//--------------------------------------------------------------------
void ofSoundPlayer::setSpeed(float spd){
if( player != NULL ){
player->setSpeed(spd);
}
}
//--------------------------------------------------------------------
void ofSoundPlayer::setPaused(bool bP){
if( player != NULL ){
player->setPaused(bP);
}
}
//--------------------------------------------------------------------
void ofSoundPlayer::setLoop(bool bLp){
if( player != NULL ){
player->setLoop(bLp);
}
}
//--------------------------------------------------------------------
void ofSoundPlayer::setMultiPlay(bool bMp){
if( player != NULL ){
player->setMultiPlay(bMp);
}
}
//--------------------------------------------------------------------
void ofSoundPlayer::setPosition(float pct){
if( player != NULL ){
player->setPosition(pct);
}
}
//--------------------------------------------------------------------
void ofSoundPlayer::setPositionMS(int ms){
if( player != NULL ){
player->setPositionMS(ms);
}
}
//--------------------------------------------------------------------
float ofSoundPlayer::getPosition(){
if( player != NULL ){
return player->getPosition();
} else {
return 0;
}
}
//--------------------------------------------------------------------
int ofSoundPlayer::getPositionMS(){
if( player != NULL ){
return player->getPositionMS();
} else {
return 0;
}
}
//--------------------------------------------------------------------
bool ofSoundPlayer::getIsPlaying(){
if( player != NULL ){
return player->getIsPlaying();
} else {
return false;
}
}
//--------------------------------------------------------------------
bool ofSoundPlayer::isLoaded(){
if( player != NULL ){
return player->isLoaded();
} else {
return false;
}
}
//--------------------------------------------------------------------
float ofSoundPlayer::getSpeed(){
if( player != NULL ){
return player->getSpeed();
} else {
return 0;
}
}
//--------------------------------------------------------------------
float ofSoundPlayer::getPan(){
if( player != NULL ){
return player->getPan();
} else {
return 0;
}
}
//--------------------------------------------------------------------
float ofSoundPlayer::getVolume(){
if( player != NULL ){
return player->getVolume();
} else {
return 0;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 286 |
Q: Finding expected value when conditional distribution is known If the distribution of $Y$ conditional on $X=x$ is known, and the distribution of $X$ is known, what would be the general process for finding the expected value $\Bbb E[Y]$? IS there a general process, or does one need to know the exact distributions?
A: Use the fact that:
$$E[Y] = E[E[Y|X]]$$
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,822 |
Q: SelectNodes in HtmlAgilityPack always return null I'm trying to get data from this site: https://www.hltv.org/results, and it doesnt work. SelectNodes always returns null, I've tried using xpath, full xpath and so on and different things other than HtmlAgilityPack. Maybe its not the HtmlAgilityPack maybe the problem is in the tags.
Please take a look at the code and view also the tags to see if if i copied it right:
HtmlWeb web = new HtmlWeb();
var doc = web.Load("https://www.hltv.org/results");
var teams = doc.DocumentNode.SelectNodes("//*[@class='team team-won']");
Please Help, Thanks!
A: Use load for only files. In this example, you ar loading an url,
So you should use web.LoahHtml
İf you do this. It will work
A: HtmlAgilityPack does not handle javascript, so you can use selenium to store the html source code via webDriver.
IWebDriver webDriver = new ChromeDriver();
webDriver.Url = "https://www.hltv.org/results";
var pageSource = webDriver.PageSource;
var doc = new HtmlDocument();
doc.LoadHtml(pageSource);
var xPath = @"//*[@class='team team-won']";
var node = doc.DocumentNode.SelectSingleNode(xPath);
Console.WriteLine(node.InnerText);
Make sure that you have already downloaded and stored in same directory the webDriver.(i used chrome driver downloaded from here https://chromedriver.chromium.org/downloads)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 577 |
{"url":"https:\/\/gmat.la\/question\/OG2017Q-DS-226","text":"A total of 20 amounts are entered on a spreadsheet that has 5 rows and 4 columns; each of the 20 positions in the spreadsheet contains one amount. The average (arithmetic mean) of the amounts in row i is Ri (1 \u2264 i <\u22645). The average of the amounts in column j is Cj (1 \u2264 j \u2264 4). What is the average of all 20 amounts on the spreadsheet?\n\n(1) R1 + R2 + R3 + R4 + R5 = 550\n\n(2) C1 + C2 + C3 + C4 = 440\n\nStatement (1) ALONE is sufficient, but statement (2) alone is not sufficient.\n\nStatement (2) ALONE is sufficient, but statement (1) alone is not sufficient.\n\nBOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.\n\nEACH statement ALONE is sufficient.\n\nStatements (1) and (2) TOGETHER are NOT sufficient.\n\n##### \u8003\u9898\u8bb2\u89e3\n\n\uff081\uff09\u00a0 \u6bcf\u4e00\u884c\u76844\u4e2a\u6570\u7684\u548c\u4e3a4Ri\uff0c\u6240\u4ee55\u884c\u517120\u4e2a\u6570\u7684\u548c\u4e3a4~$(R_1+R_2+R_3+R_4+R_5)=4*550=2200$~\uff0c20\u4e2a\u6570\u7684\u5e73\u5747\u6570\u4e3a2200\/20=110\uff0c\u9898\u76ee\u5f97\u89e3\u3002\n\n\uff082\uff09\u00a0 \u6bcf\u4e00\u52175\u4e2a\u6570\u7684\u548c\u4e3a5Cj\uff0c\u6240\u4ee54\u5217\u517120\u4e2a\u6570\u7684\u548c\u4e3a~$5(C_1+C_2+C_3+C_4)=5*440=2200$~\uff0c20\u4e2a\u6570\u7684\u5e73\u5747\u6570\u4e3a2200\/20=110\uff0c\u9898\u76ee\u5f97\u89e3\u3002","date":"2021-12-08 19:32:33","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.37995538115501404, \"perplexity\": 780.6942171450636}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964363520.30\/warc\/CC-MAIN-20211208175210-20211208205210-00324.warc.gz\"}"} | null | null |
{"url":"https:\/\/wumbo.net\/notation\/","text":"# Notation\n\nA collection of math syntax and notation. The pages demonstrate the usage of symbols, expressions, and operators to represent mathematical concepts.\n\n## Arithmetic\n\nIn mathematics the plus symbol represents addition between two numbers. Addition is the process of combining two things together.\n\nSubtraction | Notation\n\nIn mathematics the minus symbol represents the subtraction operator. The expression on the right is subtracted from the expression on the left.\n\nMultiplication | Notation\n\nA dot between two numbers denotes multiplication in mathematics.\n\nMultiplication Parentheses | Notation\n\nTwo expressions placed next to eachother and separated by parenthesis are assumed to be multiplied together.\n\nMultiplication Variables | Notation\n\nIn mathematics when two variables are next to each other the implied operation is multiplication. This makes many equations and formulas more simple.\n\nDivision | Notation\n\nIn mathematics there are multiple different ways to represent division. The obelus is often used in elementary mathematics, then later the horizontal bar is adopted for convenience, and the forward slash is used because of the prevelence of computers and calculators.\n\nFraction | Notation\n\nA fraction is represented using a horizontal bar between two expressions. The expression on top is called the numerator and the expression on bottom is called the denominator.\n\nGrouping | Notation\n\nIn mathematics parentheses are used to group expressions to be evaluated together and organize the order of operations. Everything within the parantheses is evaluated, then the rest of the calculation is carried out.\n\nDecimal Point | Notation\n\nThe decimal point is used to separate the whole part of a number from the decimal part. The whole part is represented using number digits to the left of the decimal point. The decimal part is also represented using digits to the right of the decimal point.\n\nRatio | Notation\n\nThe syntax for a ratio is two numbers separated by the colon symbol.\n\n## Algebra\n\nVariable | Notation\n\nA variable is a letter or symbol that represents a value that either changes or is unknown.\n\nFunction | Notation\n\nIn mathematics f(x) represents the notation of a function that takes in a number x and returns a number as output.\n\nSlope | Notation\n\nSlope is denoted as the change in y over the change in x. The capital greek letter delta (\u0394) is used to represent the change in a variable.\n\nAbsolute Value | Notation\n\nThe notation for taking the absolute value is two vertical lines on either side of the expression being evaluated.\n\nSquare Root | Notation\n\nThe radical symbol by itself is used to denote taking the square root of a number.\n\nA radical is used to represent fractional exponents. By itself it is used to represent the square root of an expression, but it also is used to represent higher roots as well.\n\nExponent | Notation\n\nIn mathematics superscript text is used to indicate the exponent of some number.\n\nLogarithm | Notation\n\nLogarithms are often abreviated as \"log\" followed by a subscript number representing the base and the number that the logarithm is being applied to.\n\nFactorial | Notation\n\nThe exclamation mark is used to represent the factorial of a number in mathematics. A factorial is a unary operator.\n\nSummation | Notation\n\nThe capital Greek letter Sigma is used to represent the summation operator in mathematics.\n\nProduct | Notation\n\nThe capital Greek letter Pi is used to represent the product operator in mathematics. The operator has three parts: an initial value, an end value, and the expression being evaluated.\n\n## Logic\n\nEqual | Notation\n\nTwo stacked horizontal lines respresents the equals symbol in mathematics. The two expressions on either side are equal, or the same, when the equal sign is placed in between them.\n\nGreater Than | Notation\n\nThe symbol > represents the logical expression that the left side is greater than the right side.\n\nGreater Than or Equal | Notation\n\nThe symbol for a less than or equal to is \u2265. It combines the greater than symbol > and the equals symbol = together.\n\nLess Than | Notation\n\nThe symbol < represents the logic expression that the left side is less than the right side.\n\nLess Than or Equal | Notation\n\nThe symbol for a less than or equal to is \u2264. It combines the less than symbol < and the equals symbol = together.\n\nNot Equal | Notation\n\nThe symbol for not equal is a equal sign with a diagonal line through it.\n\nApproximately | Notation\n\nThe symbol for approximately equal is a squiggly equals sign. It is used to show that two numbers are roughly equal, but not exactly equal.\n\n## Geometry\n\nAngle | Notation\n\nThe notation for a symbol is a small symbol written in text, sometimes followed by three letters that correspond to a figure.\n\nCircle | Notation\n\nA circle in text is denoted using the circle symbol with a dot in the center followed by a letter that corresponds to the center point of the circle.\n\nComplementary Angles | Notation\n\nComplementary angles can visually be denoted as two angles who sum to a perpendicular or square angle.\n\nCongruent Angles | Notation\n\nCongruent angles are denoted with tick marks across the angle.\n\nLine | Notation\n\nA line is denoted by two letters representing the start and end of the line with a line over top.\n\nParallel Lines | Notation\n\nParallel lines are denoted by the parallel symbol placed betwen the notation of the two lines. A line is denoted by the start and end letter with a line over top.\n\nPerpendicular Angle | Notation\n\nA perpendicular angle is visually denoted by drawing a square at the vertex of the angle. The measured angle is equal to \u03c0\/2 radians or 90\u00b0.\n\nPerpendicular Lines | Notation\n\nThe symbol for two perpendicular lines is a horizontal line with another line drawn perpendicular to it.\n\nSupplementary Angles | Notation\n\nSupplementary angles can visually be denoted as two angles who sum to 180 degrees or PI degrees.\n\nTriangle | Notation\n\nA triangle is denoted using the triangle symbol followed by three letters that represent the points of the triangle.\n\nTheta | Notation\n\nThe symbol theta is often used as a variable to represent an angle in illustrations, functions, and equations.\n\nSlope | Notation\n\nSlope is denoted as the change in y over the change in x. The capital greek letter delta (\u0394) is used to represent the change in a variable.\n\nCartesian Coordinate System | Notation\n\nA point in the cartesian coordinate system is denoted by two numbers in parentheses separated by a comma. The first number represents the distance from origin in the x-direction and teh second number represents the distance from the origin in the y-direction.\n\nPolar Coordinate System | Notation\n\nA point in the polar coordinate system is denoted by two numbers in parentheses separated by a comma. The first number represents the radius r (distance from the origin) and an angle \u03b8 (the greek letter theta) relative to the origin.\n\n## Trigonometry\n\nAngle | Notation\n\nThe notation for a symbol is a small symbol written in text, sometimes followed by three letters that correspond to a figure.\n\nLine | Notation\n\nA line is denoted by two letters representing the start and end of the line with a line over top.\n\nPerpendicular Angle | Notation\n\nA perpendicular angle is visually denoted by drawing a square at the vertex of the angle. The measured angle is equal to \u03c0\/2 radians or 90\u00b0.\n\nPerpendicular Lines | Notation\n\nThe symbol for two perpendicular lines is a horizontal line with another line drawn perpendicular to it.\n\nTriangle | Notation\n\nA triangle is denoted using the triangle symbol followed by three letters that represent the points of the triangle.\n\n## Probability\n\nRandom Variable | Notation\n\nIn probability a random variable is denoted using capital latin letters usually X, Y, and Z or A, and B respectively.\n\nProbability Distribution | Notation\n\nA probability distribution is denoted like a function. Often a capital P is used for the function name, and the capital letter X is used for the argument to the function.\n\nConditional Probability | Notation\n\nConditional probability is denoted using a vertical bar between the two variables\n\nJoint Distribution | Notation\n\nA joint probability distribution is denoted like a function often using P as the function name. The capital letters X and Y are often used to represent the random variables of the distribution and are arguments to the function.\n\nExpected Value | Notation\n\nThe expected value of a probability distribution is denoted as the function E(X).\n\nArithmetic Mean | Notation\n\nThe arithmetic mean of a data set is denoted by a horizontal bar over the variable x.\n\nN Choose R Combination | Notation\n\nThe number of possible ways to choose r combinations from n total items is denoted using two parentheses with the n value above the r value. A subscript p or c is used to denote whether it is a combination or permutation.\n\nN Choose R Permutation | Notation\n\nThe number of possible ways to choose r permutations from n total items is denoted using two parentheses with the n value above the r value. A subscript p or c is used to denote whether it is a combination or permutation.\n\nPopulation Mean | Notation\n\nMeaningful description here.\n\nStandard Deviation | Notation\n\nMeaningful description here.\n\nVariance | Notation\n\nMeaningful description here.\n\n## Numbers\n\nEulers Number | Notation\n\nEuler's Number is represented in mathematics with the letter e and has some unique exponential properties.\n\nPi | Notation\n\nThe greek letter pi is a number in mathematics that describes the ratio a circle's circumference divided by its diamater.\n\nTau | Notation\n\nThe symbol tau is used in mathematics to represent the ratio of the circumeference of the circle divided by its radius.\n\nSet of Natural Numbers | Notation\n\nSet of Integers | Notation\n\nSet of Rational Numbers | Notation\n\nSet of Real Numbers | Notation\n\n2D Space | Notation\n\nTwo dimensional space is denoted using the symbol for the set of real numbers followed by a superscript two.\n\n3D Space | Notation\n\nThree dimensional space is denoted using the symbol for the set of real numbers followed by a superscript three.\n\nComplex Number | Notation\n\nA complex number is represented in two parts. The first is the real number part and the second is the imaginary number part.\n\nDecimal Point | Notation\n\nThe decimal point is used to separate the whole part of a number from the decimal part. The whole part is represented using number digits to the left of the decimal point. The decimal part is also represented using digits to the right of the decimal point.\n\nSet of Complex Numbers | Notation\n\nThe set of complex numbers is denoted using the Latin capital letter C most often presented with a double-struck typeface.\n\nScientific Notation | Notation\n\nScientific notation is a notation that scientists use to keep track of the number of significant digits in a number as well as easily recognize the magnitude of a number.\n\n## Set Theory\n\nEmpty Set | Notation\n\nAn empty set is respresented as a zero with a diagonal line through it, or as an empty pair of curly braces.\n\nSet | Notation\n\nThe notation for a set is two curly braces containing the elements of the set separated by commas.\n\nExists | Notation\n\nThe syntax for \"there exists\" is a backwards captial E. It is often used in conjuction with a variable with certain properties.\n\nFor All | Notation\n\nAn upside down capital \"A\" represents the for all symbol in mathematics.\n\nElement Of | Notation\n\nThe element of symbol describes membership to a set. When reading an equation the symbol can be read as \"in\" or \"belongs to\".\n\nProper Subset | Notation\n\nA proper subset is denoted by the subset symbol which looks like a U rotated ninety degrees to the right.\n\nProper Superset | Notation\n\nA proper superset is denoted by the superset symbol which looks like a U rotated ninety degrees to the left.\n\nSuch That | Notation\n\nThe colon symbol is used in math to denote a condition for a statement. The symbol can be read as \"such that\" in a math expression.\n\nSuperset | Notation\n\nThe superset operator in set theory is denoted using the superset symbol which looks like a U turned ninety degrees counter clockwise with a horizontal line underneath.\n\nSubset | Notation\n\nThe subset operator is denoted using a U shapes symbol rotated ninety degrees to the righ with a horizontal line underneath.\n\nIntersection | Notation\n\nThe cap symbol is used in mathematics to represent the intersect operation for two sets.\n\nSet Difference | Notation\n\nThe minus symbol is used in set theory to represent the difference operator for two sets. The operation removes all elements found in one set from another and returns the resulting set.\n\nUnion | Notation\n\nThe cup symbol is used in mathematics to represent the union operation for two sets. The union operator returns a set containing the elements from both sets.\n\n## Boolean Logic\n\nAnd | Notation\n\nThe \"and\" operator is denoted with the carrot symbol, which looks like a equilateral triangle with no bottom side. The operator evaluates to true if both the left and right expressions are true, otherwise evaluates to false.\n\nEqual | Notation\n\nTwo stacked horizontal lines respresents the equals symbol in mathematics. The two expressions on either side are equal, or the same, when the equal sign is placed in between them.\n\nOr | Notation\n\nThe logical or symbol describes when either one thing is true, the other thing is true, or both is true.\n\nEquivalent | Notation\n\nThe equivalent operator is used to express that two expressions are equivalent, although not necassarily equal.\n\nImplies | Notation\n\nThe boolean operator for implication is denoted using a double arrow pointing to the right.\n\nNegation | Notation\n\nThe negation symbol is used to reperesent the unary operator for negation, which inverts the value of the expression it is applied to.\n\nXor | Notation\n\nXor is the boolean operator that describes the operation of exclusive or. For example, when a waiter asks whether you want orange juice or coffee, they are really asking an exclusive or: you can have one or the other, but not both.\n\n## Linear Algebra\n\nVector | Notation\n\nA variable that represents a vector often has an arrow over the top to indicate that it is a vector.\n\nMagnitude | Notation\n\nTo vertical bars on either side of a vector represent taking the magnitude or length of that vector\n\nCross Product | Notation\n\nIn linear algebra the cross product operator is the times operator that is used in elementary mathematics.\n\nMatrix | Notation\n\nIn Linear Algebra, a m by n matrix is denoted as a grid of numbers with two brackets on either side. The variable m corresponds to the number of rows, and the variable n corresponds to the number of columns.\n\nDeterminant | Notation\n\nThe syntax for a determinant is to vertical bars on either side of the matrix.\n\nScalar Product | Notation\n\nThe syntax for a scalar product is to angle brackets on either side of two vectors separated by a comma.\n\n## Calculus\n\nLimit | Notation\n\nThe syntax for a limit is a the abbreviation \"lim\" followed an expression. Underneath the letters \"lim\" is the value the variable approaches within the expression denoted as the variable with an arrow to the value it is approaching.\n\nDerivative | Notation\n\nThe first derivative of a function is denoted by a apostrophe after the function name. Alternatively the partial symbol can be used to represent the derivative with respect to a variable.\n\nIntegral | Notation\n\nThe notation for an integral in mathematics is a slanted vertical line with a start and end value that describe the range of the integral. This is followed by the function being integrated and the variable with respect to which the integral is being evaluated.\n\nThe nabla symbol is used to represent taking the gradient of a function.\n\n## Computing\n\nArray | Notation\n\nAn array is common data structure used throughout programming and computing.\n\nDivision Computing | Notation\n\nWhen entering an equation into a calculator or the computer the forward slash is used to represent division.\n\nExponent Computing | Notation\n\nWhen computing an expression which contains the exponent operator, the caret symbol is often used in place of what normally would be superscript text.\n\nMultiplication Computing | Notation\n\nWhen entering an expression in a computer multiplication is represented using the asterisk symbol.\n\nSubtraction Computing | Notation\n\nIn computing subtraction is denoted using the hyphen symbol.","date":"2020-01-26 22:24:41","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8688923716545105, \"perplexity\": 821.4468735515675}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579251690379.95\/warc\/CC-MAIN-20200126195918-20200126225918-00457.warc.gz\"}"} | null | null |
Q: Java Event Generation I want to design a system which will generate a specific event at a constant rate and this will continue doing in the background. In the foreground it give output of some other events if I want.
But the background event will not stop. What is the best way to achieve it in java?
A: This is the definition of Threading and it needs to come with some level of understanding.
On a simplest level, make a Thread that sleeps for an amount of time then executes your code. There are lots of other ways to do it, but few are shorter than just overriding the run method of a thread.
If you want something more abstract, look through the concurrent package in the Java docs, there are many methods that do exactly what you want, and java.util.timer is a good one to look at as well.
Be aware of variables and collections that might be accessed by different threads at the same time. Also be aware if you have a GUI that you shuold not update your GUI from this new thread.
Edit to add a Non-thread solution
(I don't think this is really what you want, but in the comments you asked for a non-threaded solution).
If you wish to do this without threads (meaning you really wish to do it in your current thread) you have to occasionally "Interrupt" your current thread to check to see if your other task needs to process. First you need a method like this:
long lastRun=System.currentTimeInMillis();
final long howOftenToRun=60*1000 // every minute
testForBackgroundTask() {
if(lastRun + howOftenToRun < System.currentTimeInMillis()) {
// This will drift, if you don't want drift use lastRun+=howOftenToRun
lastRun=System.currentTimeInMillis()
// this is where your occasional task is.
// The task could be in-line here but of course that would violate the SRP
runBackgroundTask()
}
}
After that, you need to sprinkle testForBackgroundTask throughout your code:
lotsOfStuff....
testForBackgroundTask()
longMethod()
testForBackgroundTask()
morestuff...
testForBackgroundTask()
...
Note that if longMethod() takes a really long time then you will need to put calls to testForBackgroundTask() inside it as well.
I know this is ugly, and the uglyness of this solution is why threads are used. The only advantage is that it will absolutely prevent threading conflicts.
The other single threaded solution--making your code event driven--is even harder and seriously impacts your code (There is a construct called a Finite State Engine made for this purpose).
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District Menu. Provincial Examinations. Grade 10 Provincial Examination Specifications. English 10. 2015/16 Exam Specifications - Effective September 2015 through August 2016. 2016/17 Exam Specifications - Effective September 2016 through August 2017. Top of page. Copyright. Disclaimer. Privacy. Accessibility.
English 10 – Sample. – 1 –. Making Connections Through Reading. Written-Response Rubric. 6. The six response is superior in its depth of discussion and synthesis of ideas. Demonstrates an insightful ... will receive a maximum scale point of 4. This scoring rubric is derived from the BC Performance Standards for Reading. | {
"redpajama_set_name": "RedPajamaC4"
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{"url":"https:\/\/puzzling.stackexchange.com\/questions\/2361\/infinite-troll-tolls","text":"# Infinite Troll Tolls\n\nSo, recently there was a very good riddle posted here: Paying the Troll toll. I would like to propose a similar question which might have been partially answered. But anyways...\n\nSo, same basic idea as the original Paying the Troll toll riddle, however, assume now that there are an infinite number of bridges with an infinite number of trolls. How many pies must I leave with to ensure that I arrive at grandmas' with a whole number of pies? (Edit: assume that you do reach a point at which you arrive at grandmas. This would be when a sequence approaches zero as the number of bridges you go over tends to infinity.)\n\nIn addition, how would this answer change if the ratio of pies given to the trolls were to change. That is instead of half, what about an eighth, etc.?\n\n\u2022 I have answered infinite situation in the original question. for the ratio part it is obvious that no ratio greater than 0.5 of the cakes to trolls will not make the way through infinite bridges. \u2013\u00a0Rafe Sep 29 '14 at 19:21\n\u2022 If an answer helped you, remember you can click the green check next to it to mark it so. This will help future readers who have the same problem as you do, to get good answers quicker. \u2013\u00a0warspyking Oct 27 '15 at 21:39\n\nFirst of all, you'd never reach grandmas.\n\nMoving on from the obvious...\n\n2 pies, get to a bridge, give 1, get it back, now you have 2, keep it going for infinite number of bridges, same solution, but different number of bridges.\n\nThere are n bridges, you have p pies, at each bridge, you have p\/2+1 pies left.\n\nThere are inf bridges, you have 2 cakes, at each bridge, you have 2\/2+1 (2) pies left.\n\nIf you were to give them different ratios it would be something like\n\np\/r+1\n\nIn order to always reach grandmas that equation has to evaluate to p, or p\/r + floor(p\/r\/3) must be less than the total amount of bridges. Since there are infinite bridges, that won't happen, and the only time p\/r+1 = p is when p = 2 and r = 2.\n\np\/r+1 = p\np\/r = p-1\nr(p\/r) = r(p-1)\np = r(p)-r\np+r = p(r)\n\nAnd the only pair of positive integers where when added together are equivalent to when multiplied together are 2, 2.\n\nSimple Answer \"How many pies must I leave with ...?\" There is no unique answer, but most simply, leave home with no pies. Details below.\n\nOriginal conditions\n\nFrom the original question: \"The trolls can't give you half a cake back. It is unhygienic and disgusting.\" Since the trolls can only adjust your number of pies by integer amounts, just start with any number of whole pies to reach grandma's with a whole number of pies. Since the trolls give you back 1 whole pie after taking some number of yours, you will have at least 1 pie to present at grandma's.\n\nFractional pies\n\nIf trolls can give potions of a pie back, and assuming they can subdivide pies into any real number, you can still start with any real number of pies and reach grandma's with a whole number of pies.\n\nSuppose you start with $p$ pies. You leave bridge 1 with $\\frac{p}{2^1} + \\frac{2^1 - 1}{2^0}$ pies, bridge 2 with $\\frac{p}{2^2} + \\frac{2^2 - 1}{2^1}$ pies, and in general, bridge $n$ with $\\frac{p}{2^n} + \\frac{2^n - 1}{2^{n-1}}$ pies.\n\nIn the limit as $n$ tends to infinity, you end up with exactly 2 pies when you reach grandma's, regardless of how many pies you started with, whether positive, negative, fractional, or real. In the case of negative pies (you borrowed them), your pie debt transferred to the trolls, piecemeal ... your friends might not be happy to get those pies back under those conditions.\n\nConveniently, you can still start out with zero pies. If the tolls work the same manner going home, you'll even end up with two pies for yourself when you get home :) .\n\nDifferent tolls\n\nThe final part of the question changes the toll so that crossing the bridge with $p$ pies, you leave the bridge with $\\frac{p}{k}+1$ pies, for some $k$. (Note: previously, $k=2$.)\n\nSolving the power series expansion and leaving home with $p$ pies, you leave bridge $n$ with $\\frac{p}{k^n} + \\frac{k^n - 1}{k^{n-1} (k-1)}$ pies.\n\nIn the limit as $n$ tends to infinity, you end up with $\\frac{k}{k-1}$ pies at grandma's house. If $k \\neq 2$, you don't get to grandma's house with a whole number of pies, regardless of how many pies you left home with. So you might as well leave home with no pies if $k \\neq 2$.","date":"2019-04-24 06:36:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6908512115478516, \"perplexity\": 1221.5246444158981}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-18\/segments\/1555578633464.75\/warc\/CC-MAIN-20190424054536-20190424080536-00352.warc.gz\"}"} | null | null |
Q: Mnemonic for C# generic types I often forget if i have to use in or out when defining covarient and contravarient generic types. In java i have the mnemonic PECS (producer extends consumer super) to help me. Do you know a similar mnemonic for c#?
A: Didn't they do this for us when they called them 'in' and 'out' rather than covariant and contravariant? Just think: am I pushing values 'in', or getting them 'out'? If unsure, try 'out' - it is far more common (and easier to understand).
A: in types are passed in to functions; out types are returned out from functions.
A: When I don't remember, I always refer to IEnumerable<out T> (which means of course I have to remember the signature of that interface...). You can only get instances of T "out" of an IEnumerable<out T>, so it is covariant. If you can only pass instances of T "in" to an interface (or delegate, which is more common), it's contravariant.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,291 |
require 'test_helper'
require 'migration_data/testing'
describe '#require_migration' do
let(:db_path) do
test_dir = File.expand_path(File.dirname(__FILE__))
Pathname(File.join(test_dir, 'db'))
end
it 'loads existed migation' do
Rails.stub(:root, db_path) do
MigrationData::ActiveRecord::Migration.stub(:migration_dir, db_path) do
require_migration 'test_migration'
end
end
end
it 'raises exception on load unexisted migration' do
MigrationData::ActiveRecord::Migration.stub(:migration_dir, db_path) do
assert_raises LoadError, 'cannot load such file -- test_migration2.rb' do
require_migration('test_migration2')
end
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,129 |
Q: Undefined variable _POST if called dynamically I am creating a dynamic form handler and I want to call either $_GETor $_POST based on the forms method of submission.
In this case I am using POST.
However, the following does not work and I can not find out why.
function dumpMethod($method = '_POST') {
var_dump(${$method});
// or
var_dump($$method);
}
Both cases result in:
Notice: Undefined variable: _POST in /path/to/script.php on line ##
But I am sure $_POST is set!
Does PHP not support calling GLOBALS dynamically?
I am running version 7.1
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,415 |
WALT DISNEY ANIMATION Studios unveiled its upcoming slate staying close to the tried and true and keeping with tradition presented stories with heart, captivating audiences by voicing innate objects and maintaining the family theme.
WALT DISNEY ANIMATION Studios has announced RILEY'S FIRST DATE, a working title, is also on the upcoming slate and capitalizes on the overwhelming success of this summer's box office smash INSIDE OUT.
First look scenes from the next chapter in young Riley's life include Anger, Joy, Sadness, Fear and Disgust as our guides who return to chaperon our young charge through the treacherous waters of puberty and beyond.
Pixar Animation Studios heads back inside the mind with an all-new short, Riley's First Date, which made its world premiere this afternoon at D23 EXPO 2015.
Director Josh Cooley, along with INSIDE OUT filmmakers Pete Docter (Up, Monsters, Inc.) and Jonas Rivera (Up), introduced the short, which revisits Riley, now 12, who is hanging out with her parents at home when potential trouble comes knocking.
Mom's and Dad's Emotions find themselves forced to deal with "Riley's First Date?" The short will be included as a bonus feature in the digital HD & Blu-ray releases of Disney·Pixar's INSIDE OUT, which will be available digitally Oct. 13 and on Blu-rayNov. 3, 2015.
With the imagination of WALT DISNEY ANIMATION Studios and the genius of John Lasseter, known for making sequels better with each installment, we can expect Riley and family to tug our emotions, touch our hearts and leave us laughing until long after final credits fade.
An Apatosaurus named Arlo must face his fears and three impressive T-Rexes in Disney/Pixar's THE GOOD DINOSAUR.
T-REXES SPOTTED AT D23 EXPO 2015 – Anticipation for Disney·Pixar's THE GOOD DINOSAUR was amplified when directorPeter Sohn and producer Denise Ream shared breathtaking sequences with fans, including never-before-seen footage of a trio of T-Rexes in action. In theaters onNov. 25, 2015, THE GOOD DINOSAUR features Arlo, a sheltered Apatosaurus who finds himself far from home among a host of intimidating creatures.
Featuring the voices of AJ Buckley, Anna Paquin and Sam Elliott as the T-Rexes, THE GOOD DINOSAUR opens in theaters nationwide Nov. 25, 2015.
GIGANTIC - Things are looking up at Walt Disney Animation Studios – way up. GIGANTIC, Disney's unique take on Jack and the Beanstalk, will feature music from Oscar®-winning songwriters Kristen Anderson-Lopez and Robert Lopez, who greeted D23 EXPO fans in signature style, with song, alongside director Nathan Greno (Tangled) and producer Dorothy McKim (Get A Horse!).
Set in Spain during the Age of Exploration, Disney's Gigantic follows adventure-seeker Jack as he discovers a world of giants hidden within the clouds. He hatches a grand plan with Inma, a 60-foot-tall, 11-year-old girl, and agrees to help her find her way home. But he doesn't account for her super-sized personality—and who knew giants were so down to earth?
Gigantic hits theaters in 2018.
ZOOTOPIA - International superstar Shakira is lending her Grammy®-winning voice to Gazelle, the biggest pop star in Zootopia, D23 EXPO fans learned via a taped message from Shakira today. Ginnifer Goodwin (ABC's Once Upon a Time, Something Borrowed, Walk the Line ), the voice of the film's rookie rabbit officer Judy Hopps, saluted fans alongside directors Byron Howard (Tangled ) and Rich Moore ( Wreck-It Ralph ), and producer Clark Spencer ( Wreck-It Ralph ).
Hilarious new scenes from the film were unveiled, plus a tease of the all-new original song, Try Everything, written by singer-songwriter Sia and songwriting duo Stargate, and performed by Shakira.
MOANA ROCKS D23 EXPO 2015 Dwayne Johnson ( San Andreas, Furious 7, HBO's Ballers ), who lends his voice to the mighty demi-god Maui in Walt Disney Animation Studios' Moana, hit the stage at D23 EXPO 2015 this afternoon.
Renowned directors John Musker and Ron Clements ( The Little Mermaid, The Princess and the Frog, Aladdin ), and producer Osnat Shurer (Oscar®-nominated Pixar shorts One Man Band, Boundin' ) shared dazzling early test footage and revealed plans for the film's inspired music and the extraordinary team behind it.
Tony®-winner Lin-Manuel Miranda (Broadway's hottest new hit Hamilton, Tony-winning In the Heights ), Grammy®-winning composer Mark Mancina ( Speed , Tarzan The Lion King ) and Opetaia Foa'i (founder and lead singer of the world music award-winning band Te Vaka) blend their diverse and dynamic talents to help tell the tale of a spirited teenager who sets out to prove herself a master wayfinder. Foa'i capped the presentation with a show-stopping performance illustrating the magic of the movie's music.
Moana introduces a spirited teenager who sails out on a daring mission to fulfill her ancestors' unfinished quest. She meets the once-mighty demi-god Maui (voice of Dwayne Johnson), and together, they traverse the open ocean on an action-packed voyage. Directed by the renowned filmmaking team of Ron Clements and John Musker ( The Little Mermaid, Aladdin, The Princess & the Frog ).
Moana sails into U.S. theaters on Nov. 23, 2016.
DORY DIVES IN -- Ellen DeGeneres (The Ellen DeGeneres Show), the voice of everyone's favorite forgetful blue tang, took a dip with D23 EXPO attendees this afternoon, celebrating Disney·Pixar's upcoming film Finding Dory.
Joining DeGeneres on stage this afternoon were Ed O'Neill (Modern Family), who lends his voice to Hank, a cantankerous octopus; Ty Burrell (Modern Family), the voice of Bailey, a misguided beluga whale; and Kaitlin Olson (Always Sunny in Philadelphia), who voices Destiny, a kind-hearted whale shark.
Finding Dory swims into theatersJune 17, 2016.
WOODY AND BUZZ MARK TWO DECADES – Pixar Animation Studios' John Lasseter, who's one of Pixar's three founders, was joined on stage by several award-winning artists and filmmakers, including Pete Docter (Inside Out, Up), Andrew Stanton (Finding Nemo, WALL·E), Lee Unkrich (Coco, Toy Story 3), Josh Cooley (Cars 2, Ratatouille) and Galyn Susman (Toy Story 4, ABC's Toy Story OF TERROR!).
Celebrating 20 years since Toy Story transformed the animation industry and the art of storytelling, the Pixar team welcomed iconic two-time Oscar®-winning songwriter and longtime Pixar collaborator Randy Newman, whose Pixar credits include Toy Story, A Bug's Life, Toy Story 2, Monsters, Inc. and Cars. Also joining in the celebration were writers Rashida Jones and Will McCormack ("Celeste & Jesse Forever"), who are part of the Toy Story 4 team, though details of Buzz and Woody's return to the big screen were mysteriously missing.
Toy Story 4, which will be directed by Lasseter and produced by Susman, is slated for release onJune 16, 2017.
Let's be honest, today's trend-focused teens can be hard to please. Sure, well-intended gifters can play it safe and go the gift card route but, in addition to these cards being decidedly boring to receive, they're also easily lost and often forgotten.
Echo Products, from the London based Technology Company and newest British invasion, has finally created a line of very cool, ultra-sleek, distinctive looking collectible and portable power docking stations for Apple and Android Products.
The sixty-fourth edition of the Taormina FilmFest, produced by Videobank and taking place July 14-20 under the new artistic direction of Silvia Bizio and Gianvito Casadonte, has unveiled its full program and attending talent from America, France, Italy and the UK.
President Obama refused to stand alongside the global coalition as world leaders gathered in Paris to lead more than one million marchers in solidarity against terrorism and honor those men and women murdered, choosing instead to remain in Washington.
From Sydney, St. Petersburg, and Dubai to New York, London and Hong Kong, cities throughout the world welcomed 2014 with brilliant, dazzling displays of pyrotechnic majesty, revelry, champagne toasts and midnight kisses.
CANCER 101 and New Yorkers for Charity recently held their annual Black Tie Gala at the historic Loeb Boathouse in New York's Central Park. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,113 |
Horvátország
Križanče falu Bedekovčina községben, Krapina-Zagorje megyében
Križanče falu Cestica községben, Varasd megyében
Križanče falu Podbablje községben, Split-Dalmácia megyében | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,623 |
Q: Sorting multidimensional array for scores using temporary variables Hi guys having some trouble sorting a multidimensional array containing the scores for different users in the leaderboard form of a game I am making. I have tried to sort the scores for each user in descending order using temporary variables and then output this with no success. Any help would be much appreciated thanks. I should add that I have only recently started coding and have to do this project as part of my work for school so I am aware that it may not be very efficient and seem very novice.
Here is my method for sorting
( [i, 2] is the score value stored as a string )
private void sortScore(string[,] sort)
{
bool didSwap;
do
{
didSwap = false;
for (int i = 0; i < userNumber; i++)
{
if (i < (userNumber-1))
{
if (Convert.ToInt32(sort[i, 2]) > Convert.ToInt32(sort[i + 1, 2]))
{
string temp = sort[i + 1, 2];
sort[i + 1, 2] = sort[i, 2];
sort[i, 2] = temp;
}
}
}
} while (didSwap);
for (int j = 0; j < userNumber; j++)
{
rtbScoreboard.AppendText("Name: " + sort[j, 0] + "\t" + "Score: " + sort[j, 2] + Environment.NewLine);
}
}
A: You can use LINQ!
var sortedScores = Enumerable.Range(0, sort.GetLength(0)) //number of items
.Select(x => new
{
Name = sort[x, 0],
Score = Convert.ToInt32(sort[x, 2])
}) //converting to a meaningful structure
.OrderByDescending(x => x.Score) //sort in descending order by score
.ToList();
//building output, I don't know what is userNumber, It is a global variable
// that might store number of players to show in score board.
for (int j = 0; j < userNumber; j++)
{
rtbScoreboard.AppendText("Name: " + sortedScores[j].Name + "\t" + "Score: " + sortedScores[j].Score + Environment.NewLine);
}
You should not store data like that. Using a multidimensional array to store names and score is not a good idea. Try defining a class like Player which has Properties for storing data like Name and Score. Then store instances of Players in a list.
A: You fail to set didSwap to true
bool didSwap;
do
{
didSwap = false;
for (int i = 0; i < userNumber-1; i++)
{
if (Convert.ToInt32(sort[i, 2]) > Convert.ToInt32(sort[i + 1, 2]))
{
string temp = sort[i + 1, 2];
sort[i + 1, 2] = sort[i, 2];
sort[i, 2] = temp;
didSwap = true;
}
}
} while (didSwap);
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,622 |
package net.alcuria.umbracraft.definitions.hero;
import net.alcuria.umbracraft.annotations.Order;
import net.alcuria.umbracraft.annotations.Tooltip;
import net.alcuria.umbracraft.definitions.Definition;
/** Defines a single hero.
* @author Andrew Keturi */
public class HeroDefinition extends Definition {
@Tooltip("The agility function")
public String agiFunc;
@Tooltip("The hero's displayed name.")
@Order(1)
public String name;
@Tooltip("The level this character starts at")
@Order(3)
public int startingLevel;
@Tooltip("The strength function")
public String strFunc;
@Tooltip("A tag for sorting")
@Order(2)
public String tag;
@Tooltip("The vitality function")
public String vitFunc;
@Tooltip("The intelligence function")
public String wisFunc;
@Override
public String getName() {
return name != null ? name : "Hero";
}
@Override
public String getTag() {
return tag != null ? tag : "";
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 713 |
FOUNDATION_EXPORT double LeanCloudVersionNumber;
FOUNDATION_EXPORT const unsigned char LeanCloudVersionString[];
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,056 |
Madral est un nom propre qui peut faire référence à :
Hydronyme
, cours d'eau canal dans la communauté autonome de Catalogne en Espagne.
Pseudonyme
Philippe Madral, de son vrai nom Philippe Léon Gratton (1942-), scénariste, romancier, auteur de théâtre, metteur en scène et décorateur de théâtre français. | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,808 |
Grafisia is een geslacht van zangvogels uit de familie spreeuwen (Sturnidae). Er is één soort:
Grafisia torquata – witkraagspreeuw
Spreeuwachtigen | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,080 |
Theodoros Zagorakis ( ; born 27 October 1971) is a Greek politician and former professional footballer who played as a midfielder. He was the captain of Greece that won UEFA Euro 2004, and was also president of PAOK. He was named the Greek Male Athlete of the Year in 2004. He was elected as a Greek MEP at the May 2014 European Parliament election. He was also the president of the HFF from March to September 2021.
Club career
Kavala
Zagorakis was a central midfielder who could also play on the right side of midfield. He started his career with Kavala in 1988, the club that also produced Zisis Vryzas, with whom he became close friends. He was an important part of helping the team get into the Beta Ethniki.
PAOK
In the 1992–93 season, Zagorakis left Kavala as a winter transfer to join PAOK, part of a string of star transfers made that season. He played for PAOK until December 1997, and also captained the squad in his last two seasons. He rarely missed a league derby and scored important goals, particularly in 1994–95 when he struck four, his best-ever tally.
Leicester City
Zagorakis left for English club Leicester City the following winter and signed for Martin O'Neill's side in 1998. While in England, he took part in two consecutive League Cup finals at Wembley Stadium, losing the first against Tottenham Hotspur in 1999 and defeating Tranmere Rovers in the second, though he was an unused substitute in the latter. Zagorakis will always be fondly known by Leicester fans for his sheer determination in midfield and also his goalkeeping cameo against Crystal Palace, in a 3–3 draw, where he went in goal after Pegguy Arphexad and then Tim Flowers went off injured.
AEK Athens
Zagorakis was disappointed with Leicester manager Martin O'Neill's reluctance to use him regularly and decided to return to Greece in 2000. He moved to AEK Athens, and played alongside fellow countrymen such as Michalis Kapsis, Vasilios Lakis, Demis Nikolaidis and Vasilios Tsiartas. He formed an excellent duo in midfield with both Akis Zikos and Kostas Katsouranis, helping with his game in the defensive part and also in the organization of the offense, where he also competed at times as a captain. A goal with AEK was against Inter Milan at Giuseppe Meazza with a great shot for the round of 16 of the UEFA Cup. With AEK, Zagorakis won the Greek Cup in 2002 against rivals Olympiacos. Zagorakis, during his last season in AEK, has accepted a reduced pay-off which is reported to be £220,000."He showed he genuinely has the club's best interests at heart and is free to take up any of the great offers from teams abroad," said a club statement. In the summer of 2004, having the best moment of his career at the Euro in Portugal and with AEK in a difficult administrative situation, he left the team.
Bologna
On 14 July 2004, he left AEK Athens for Bologna, signing a two-year contract worth €1.5 million per year. Zagorakis–who was voted player of the tournament in Portugal–was released by AEK Athens and joined Bologna on a Bosman free transfer. "Zagorakis is our Greek Baggio", Bologna owner Giuseppe Gazzoni Frascara told the newspaper. In 2004–05, Zagorakis was a regular feature in the squad, but the team found itself relegated to Serie B after a play-out series against Parma. In the following summer, he was released from the team as they couldn't afford his payroll under the new conditions.
Return to PAOK
The Player of the Tournament at UEFA Euro 2004 will arrive in Thessaloniki to sign the deal with PAOK. The midfielder was released by Bologna following the Italian team's relegation from Serie A, and rejoins one of his old clubs. Zagorakis eventually signed a two-year contract with PAOK for €700,000 a year. When Zagorakis landed at Makedonia Airport in Thessaloniki, 7,000 supporters were there to welcome him back. His return however coincided with a turbulent period, with the club many financial and administrative problems. On 28 May 2007, after the fifth Greek Super League All-Star Game, Zagorakis announced his retirement from professional football.
International career
Zagorakis received his first cap for Greece on 7 September 1994, against the Faroe Islands. He scored his first goal against Denmark in 2006 World Cup qualifying, in Athens, while earning his 101st cap. Numbering 120 caps, Zagorakis was Greece's all-time leader in international games played until 12 October 2012, when Giorgos Karagounis made his 121st appearance for the national team. He earned his 100th against Kazakhstan on 17 November 2004, and was the team's longest-serving captain.
Zagorakis played an important role in Greece's win in the 2004 European Championships, and was named the Player of the Tournament by UEFA and consequently he was in the Team of the Tournament. FIFA named Zagorakis as a contender for the 2004 FIFA World Player of the Year award (he finished in 17th place alongside Spain's and Real Madrid's legend, Raúl), as well as UEFA named him for the 2004 Ballon d'Or award (he finished in fifth place), behind the likes of Ronaldinho (third place) and Thierry Henry (fourth place). He holds the record of most consecutive matches (57) (except one in 2006 because of injury) of the national team (due to either injury or not selection) since his first cap in a period of 12 years.
After 12 full playing years as the captain for Greece, Zagorakis announced his retirement from international football on 5 October 2006. More specifically, Greece captain was considering retiring after his team's European championship qualifier against Norway in Athens on 2 October 2006, has also been called up for the match against Bosnia four days later, officials said.
On 22 August 2007, however, he played for the last time with the national side, in a special friendly match against Spain in Toumba Stadium, Salonica. He played for about 15 minutes and was then replaced by Giannis Goumas, receiving an applause by fans singing his name.
PAOK presidency
Following his retirement as a footballer, after many widespread rumours, it was announced that he was to become PAOK's new president, on 18 June 2007.
The club had been in dire financial straits for several years. Zagorakis undertook to sort out finances by attracting investors, increasing revenue and mobilising the club's fan base. In his unifying first statement he said: "The strength of PAOK lies in its supporters, its fans...I would have not made this decision" (to retire from playing and assume administration) "if i did not love my club. The situation for PAOK is very difficult and I will not try to hide its problems; instead I will do all I can to solve the pressing financial issues."
Soon he set about restructuring the club's debt and team, and recruited former teammate Zisis Vryzas as technical director. As of late 2008, PAOK has been able to pay off most of its accumulated debt, seen a steady increase in ticket sales and advertising revenue, and team performance on the field has improved considerably, under the coaching of Portuguese manager Fernando Santos.
On 15 December 2008, Zagorakis announced a major issue of new stock by the club, valued at €22.3 million. He appealed to small investors and stated that, although it is unlikely that the entire sum could be covered in a worldwide crisis environment, he felt confident that the expected influx of capital would enable PAOK finally to put its past problems behind and focus on future growth.
On 8 October 2009, Zagorakis surprised fans and press alike by announcing his decision to quit the presidency. In a short announcement on the club's official website, he cited that personal problems have led him to this decision. He was rapidly replaced by his close friend Zisis Vryzas.
Nevertheless, he kept close contact with his former associates, and also frequently attended the club's home games alongside Vryzas. Zagorakis finally changed his mind and on 20 January he reprised his position as president, with Vryzas stepping down to assume the post of vice-president.
In January 2012 Zagorakis resigned as president after the sale of Vieirinha for financial reasons provoked a supporters' protest.
Style of play
Zagorakis was described by UEFA.com as a "combative, industrious defensive midfielder with a powerful right-foot shot".
Personal life
Zagorakis was featured on the cover of the Greek edition of FIFA 2001.
Political career
Zagorakis is looking to kick-off a new career in politics by becoming a Member of the European Parliament (MEP). He was a New Democracy candidate for the European Parliament in the elections of 25 May 2014. He was elected as an MEP and became a member of the European People's Party (Christian Democrats). He was unveiled as one of the party's hopefuls by the Greek Prime Minister, Antonis Samaras.
In January 2020, a Committee of Professional Sports (with members placed by Sports Minister Lefteris Avgenakis) proposed the relegation of PAOK FC and Xanthi FC. Zagorakis stated that he could no longer be MEP for New Democracy after this. The party's leader and PM Kyriakos Mitsotakis decided his deletion. Several MP's said that Zagorakis was elected in European Parliament to represent Greece, not PAOK.
Career statistics
Club
International
Scores and results list Greece's goal tally first, score column indicates score after each Zagorakis goal.
Honours
Leicester City
League Cup: 1999–2000
AEK Athens
Greek Cup: 2001–02
Greece
UEFA European Championship: 2004
Individual
UEFA Euro 2004: Player of the tournament
UEFA Euro 2004: Team of the Tournament
UEFA Euro 2004 Final: Man of the Match
Greek Male Athlete of the Year: 2004
PAOK MVP of the Season: 1996–1997
See also
List of men's footballers with 100 or more international caps
References
External links
Official website
FootballDatabase profile and stats
1971 births
Living people
Greek Macedonians
AEK Athens F.C. players
Bologna F.C. 1909 players
Premier League players
Greek footballers
Greek expatriate footballers
Expatriate footballers in Italy
Expatriate footballers in England
Leicester City F.C. players
Kavala F.C. players
Serie A players
PAOK FC players
UEFA Euro 2004 players
UEFA European Championship-winning captains
UEFA European Championship-winning players
2005 FIFA Confederations Cup players
Greece international footballers
FIFA Century Club
Super League Greece players
Football League (Greece) players
Greek expatriate sportspeople in England
Greek expatriate sportspeople in Italy
PAOK F.C. presidents
MEPs for Greece 2014–2019
Association football midfielders
MEPs for Greece 2019–2024
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Greek sportsperson-politicians | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,598 |
Жан Менраль Дюпплен (; 1771–1813) — французский военный деятель, бригадный генерал (1809 год), барон (1809 год), участник революционных и наполеоновских войн.
Биография
Начал военную службу 1 июня 1787 года простым солдатом в 89-м пехотном полку. 1 июня 1791 года вышел в отставку, но уже 8 августа того же года был избран сержантом 3-го батальона волонтёров департамента Мёрт. Участвовал в кампаниях 1792-93 годов в рядах Арденнской армии. 16 марта 1792 года стал аджюданом, 15 июля 1793 года – старшим аджюданом. 17 августа 1793 года произведён в капитаны, и в конце 1793 года переведён в Северную армию. 27 декабря 1793 года возглавил гренадерскую роту своего батальона. Сражался в рядах Рейнской и Гельветической армий. 22 июня 1796 года получил два пулевых ранения в сражении при Герсбахе. 20 апреля 1799 года произведён в командиры батальона 106-й полубригады линейной пехоты в Итальянской армии. Отличился при осаде Генуи. 6 апреля 1800 года был ранен при Монтефаччио и 25 декабря 1800 года при Поццоло.
22 декабря 1803 года получил звание майора, и стал заместителем командира 67-го полка линейной пехоты. 1 мая 1806 года переведён в Императорскую гвардию, и возглавил батальон 1-го полка пеших гренадер.
20 октября 1806 года был произведён Императором в полковники с назначением командиром 85-го полка линейной пехоты. Служил в дивизии Гюдена 3-го армейского корпуса Великой Армии. 26 декабря 1806 года ранен в сражении при Пултуске. Был при Эйлау.
28 марта 1809 года произведён в бригадные генералы и в ходе Австрийской кампании 1809 года состоял в 3-м корпусе Армии Германии, отличился в сражении при Регенсбурге, где первым вошёл в город.
29 января 1810 года получил отпуск на три месяца. 10 сентября 1810 года вернулся в строй в качестве командира 2-й бригады 2-й пехотной дивизии Фриана. Принимал участие в Русской кампании 1812 года, и с 26 июля 1811 года командовал 1-й бригадой 5-й пехотной дивизии генерала Компана 1-го армейского корпуса, сражался при Салтановке, Смоленске, Шевардино и Бородино, где получил штыковое ранение во время атаки на Багратионовы флеши. При отступлении армии отличился в боях при Малоярославце и Вязьме. Умер от истощения 25 января 1813 года в Торне в возрасте 41 года.
Воинские звания
Сержант (8 августа 1791 года);
Капитан (17 августа 1793 года);
Командир батальона (20 апреля 1799 года);
Майор (22 декабря 1803 года);
Командир батальона гвардии (1 мая 1806 года);
Полковник (20 октября 1806 года);
Бригадный генерал (28 марта 1809 года).
Титулы
Барон Дюпплен и Империи (; декрет от 15 августа 1809 года, патент подтверждён 19 декабря 1809 года).
Награды
Легионер ордена Почётного легиона (1 апреля 1804 года)
Офицер ордена Почётного легиона (7 июля 1807 года)
Коммандан ордена Почётного легиона (21 сентября 1809 года)
Примечания
Источники
« Jean Duppelin », dans Charles Mullié, Biographie des célébrités militaires des armées de terre et de mer de 1789 à 1850, 1852.
A. Lievyns, Jean Maurice Verdot, Pierre Bégat, Fastes de la Légion-d'honneur, biographie de tous les décorés accompagnée de l'histoire législative et réglementaire de l'ordre tome 4, Bureau de l'administration, 1844, 640 p.
Французские бригадные генералы Наполеоновских и Революционных войн
Бароны Империи (Франция)
Участники Отечественной войны 1812 года
Участники Бородинского сражения | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,851 |
Q: React importing image does not work for CSS I am having problems importing my image for CSS. I have imported the image as a test and it works when using it in the img tag using src={furnitureBG}.
I followed the docs and imported the image and then using the path as the value for url(''). However, no image renders and there is no error message.
I know the path is correct because when I use the code
<img src={furnitureBG} />
the image I desire renders. I am just wondering why this is not working on the following code for CSS background.
import furnitureBG from '../images/furniture-bg-7.png';
const Container = styled.div`
height: 100vh;
width: 100vw;
background: linear-gradient(rgba(0,0,0,.35), rgba(0,0,0,.35)), url('../images/furniture-bg-7.png');
`
A: You can use url(${furnitureBG?.src}) in order to render image using styled component.
import furnitureBG from '../images/furniture-bg-7.png';
const Container = styled.div`
height: 100vh;
width: 100vw;
background: linear-gradient(rgba(0,0,0,.35), rgba(0,0,0,.35)), url(${furnitureBG?.src});
`
A: You can move the /images folder into the public folder:
App.js
const Container = styled.img`
background-image: url("/images/asdf.png");
width: 100vw;
height: 100vh;
`;
function App() {
return (
<div className="App">
<Container />
</div>
);
}
Directory structure:
- project
- public
- images
- asdf.png
- src
- App.js
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,190 |
\section{\label{sec:introduction}Introduction}
Complete understanding of the information processing underlying cognition remains a significant scientific challenge. Progress in neuroscience, computer science, psychology, and neural engineering make this a fruitful time for elucidation of intelligence. Biological experiments and software simulations would be greatly augmented by artificial hardware with complexity comparable to systems we know to be conscious. Intelligent systems implemented with hardware optimized for neural computing may inform us regarding the limits of cognition imposed by the speed of light while providing technological opportunities sufficient to spawn a new domain of the computing industry.
As we will argue, neural computing appears uniquely capable of the distributed, yet integrated, information processing that characterizes intelligent systems. Many approaches to neural computing are being developed, and the maturity of the semicondutor industry makes CMOS a wise initial platform. Yet the central role of communication in neural computing indicates that hardware incorporating different physics may be advantageous for this application. In previous work \cite{shbu2017}, we considered the potential for superconducting optoelectronic hardware to perform neural operations. The principal assumption guiding the design of the hardware platform is that photons are the entities best suited for communication in large-scale neural systems. The hardware platform leverages optical communication over short and long distances to enable dense local fanout as well as distant communication with the shortest possible delay. In a series of papers \cite{sh2018b,sh2018c,sh2018d,sh2018e}, we present details of the design of superconducting optoelectronic neurons and networks that appear capable of achieving the functions required for cognitive computing. In this paper, we summarize the physical reasoning behind the hardware to meet the requirements of cognitive circuits and provide an overview of the operation of the neurons.
The audience we hope to address is broad and includes neuromorphic engineers, perhaps studied in silicon but open to new exploration; the integrated-photonics community, who may see this as a promising application of photonic devices and systems; the superconducting electronics community, who may find benefits to long-standing challenges such as memory, clock distribution, cryogenic I/O, and achieving the voltage necessary to interface with CMOS; neuroscientists, who may utilize this hardware platform to test hypotheses at device and system levels; and the advanced computing community, who may leverage the capabilites of these systems to solve outstanding problems.
\section{\label{sec:cognitiveSystems}Cognitive systems}
The foundational assumption of this work is that light is the physical entity best-suited to achieve communication in cognitive neural systems. To motivate why light is essential for large-scale neural systems, we must describe the systems we intend to pursue.
Broadly speaking, we wish to pursue devices and networks capable of acquiring and assimilating information across a wide range of spatial, temporal, and categorical scales. In a neural cognitive system, spatial location within the network may correspond to information specific to content area or sensory modality, and therefore spatial integration across the network corresponds to integration across informational subjects and types. Information processing must occur across many levels of hierarchy with effective communication across local, regional, and global spatial scales as well as temporal scales. These systems must continually place new information in context. It is required that a cognitive system maintain a slowly varying background representation of the world while transitioning between dynamical states under the influence of stimulus. The objective of this series of papers is to design general cognitive circuits with structural and dynamical attributes informed by neuroscience, network theory, and dynamical systems. Stated generally, systems combining functional specialization with functional integration are likely to perform well for many cognitive tasks \cite{spto2000,spto2002}.
The theme of localized, differentiated processing combined with information integration \cite{toed1998,tosp2003,to2004,seiz2006,bato2008,bato2009,base2011} across space \cite{busp2009,sp2010} and time \cite{sase2001,vala2001,enfr2001,budr2004,bu2006} is central to the device and network designs we present here. In the spatial domain, the demand for integration of information from many local areas requires dense local connectivity (as measured by a clustering coefficient \cite{eskn2015,saki2007,fa2007}), but also connections between these local areas which serve to combine the local information and place it in a larger context at higher cognitive levels \cite{brto2006} (as measured by a short average path length \cite{alba2002}). High clustering combined with short average path length defines a small-world network \cite{wast1998}. For the highest performance, we expect this trend of integration of locally differentiated information to repeat across many scales in a nearly continuous manner \cite{busp2009,sp2010} such that any node in the system is likely to be processing information with local neighbors, but also receiving information from simpler, less-connected units, and transferring information to complex, highly connected units. Networks with this organization across scales are governed by power law spatial scaling \cite{baal1999}.
The patterns are related in the temporal domain where transient synchronized oscillations integrate information from various brain regions \cite{vala2001,sase2001,enfr2001}. Information exchange can occur on very fast time scales, and results of these computations must be combined over longer times. The spatial structure of the network and its operation in the time domain are not independent \cite{spto2000,spto2002,waxu2006}. Fast, local dynamics integrate information of closely related nodes through transient neuronal functional clusters \cite{buwa2012}, while activity on slower scales can incorporate input from larger regions \cite{stsa2000}. Networks with this organization in time are governed by a power law frequency distribution \cite{bata1987,budr2004,bu2006}, characteristic of self-organized criticality \cite{be2007}. Power law spatial and temporal distributions underlie systems with fractal properties \cite{bata1987,bu2006}, and self-similarity across space and time is advantageous for cognition \cite{bu2006,be2007,kism2009,shya2009,ch2010,rusp2011}.
These conceptual arguments regarding information integration across spatial and temporal scales lead us to anticipate networks with hierarchical configuration, with processing on various scales being integrated at high levels to form a coherent cognitive state \cite{brto2006}. The constitutive devices most capable of achieving these network functions are relaxation oscillators \cite{bu2006,st2015}, dynamical entities characterized by pulsing behavior \cite{mist1990} with resonant properties at many frequencies \cite{soko1993,huya2000}. Neurons are a subset of relaxation oscillators with complex operations adapted for spike-based computation \cite{geki2002}.
To illustrate how differentiated processing and information integration are implemented by neurons for cognition, consider vision \cite{laus2011}. In early stages of visual processing, neurons located near each other in space will show similar tuning curves \cite{daab2001} in response to presented stimuli, thus forming locally coherent assemblies selecting for certain features of a visual scene \cite{enfr2001}. These locally differentiated processing units are constructed from architectural motifs \cite{spko2004,onsa2005} and are manifest in biological hardware as mini-columns and columns \cite{mo1997}, which are dedicated to modeling a subset of sensory space \cite{haah2017}. To form a more complete representation of an object within a visual scene, or to make sense of a complex visual scene with many objects, the visual system must combine the information from many differentiated processors. This integration is accomplished with lateral connections between columns \cite{spto2000} as well as with feed-forward connections from earlier areas of visual cortex to later areas of visual cortex \cite{laus2011}. Such an architecture requires some of the neurons in any local region to have long-range projections, motivating the need for local connectivity for differentiated processing combined with distant connectivity for information integration across space.
Temporal considerations are as important as spatial, yet more subtle. To understand information integration in the time domain, consider synchronized oscillations at various frequencies in the context of the binding problem \cite{ro1999,tr1999}. Stated as a question, the binding problem asks how the myriad stimuli presented to the brain can be quickly and continuously organized into a coherent cognitive moment. In the limited context of vision, we ask how a complex, dynamic visual scene can be structured into a discernible collection of objects that can be differentiated from each other and from an irrelevant background \cite{rede1999}. Many studies provide evidence that fast, local oscillations are modulated by slower oscillations encompassing more neurons across a larger portion of the network \cite{vala2001,sase2001,enfr2001,lued1997,stsa2000,budr2004,bu2006,fr2015}. In the case of columns in visual cortex, local clusters tuned to specific stimuli will form assemblies with transient synchronization at high frequencies ($\gamma$ band, 20-80\,Hz \cite{budr2004}). The information from many of these differentiated processors is integrated at higher levels of processing by synchronizing larger regions of neurons at lower frequencies ($\alpha$ band, 1-5\,Hz, and $\theta$ band, 4-10\,Hz \cite{stsa2000,budr2004}). The transient synchronization of neuronal assemblies is closely related to neuronal avalanches \cite{be2007,shya2009}, cascades of activity across all these frequencies. Neuronal avalanches are observed in networks balanced at the critical point between order and chaos \cite{be2007,kism2009,shya2009,ch2010,rusp2011}. Self-similarity in the temporal domain implies operation at this critical point \cite{be2007,kism2009,rusp2011}, and operating at this phase transition is necessary to maximize the dynamic range of the network \cite{shya2009}. Inhibition and activity-based plasticity are crucial for achieving this balance \cite{budr2004,bu2006,siqu2007}.
Networks of excitatory principal neurons interspersed with inhibitory interneurons \cite{robu2015} with small-world characteristics naturally synchronize at frequencies determined by the circuit and network properties \cite{bu2006}. Slower frequency collective oscillations of networks of inhibitory interneurons provide short windows when certain clusters of excitatory neurons are uninhibited and therefore susceptible to spiking \cite{buge2004}. This feedback through the inhibitory interneuron network provides a top-down means by which the dynamical state of the system can provide broad information to the local processing clusters \cite{enfr2001,fr2015}. Regions of cortex with higher information integration focus attention \cite{vala2001} on certain aspects of stimulus by opening receptive frequency windows at the resonant frequencies of relevant sub-processors, providing a mechanism by which binding occurs and background is ignored \cite{lued1997,enfr2001,budr2004,fr2015}. The result of this inhibitory structuring of time is a network with dynamic effective connectivity \cite{brto2006,fr2015}. By constructing a network with small-world, power-law architecture from highly tunable relaxation oscillators, and employing feedback through inhibitory oscillations, we produce a system that can change its effective structural and resonant properties very rapidly based on information gleaned from prior experiences of a large region of the network \cite{budr2004,fr2015}.
This model of binding requires a means by which the resonant frequencies of neuronal assemblies can be associated with certain stimuli, and a means by which the inhibitory interneuron network can learn to associate different assemblies with different frequencies. Plastic synaptic weights make such adaptation possible. Synapses provide a means by which the connectivity of the network can shape dynamics and functionality, and synapses adapt their states based on internal and external activity. As cortex evolves through dynamical states on various temporal and spatial scales, information stored in synapses is integrated. This dynamical state integrates synaptic information across the network, and uses this information as feedback to distributed sub-processors \cite{enfr2001,fr2015}.
For a cognitive system embedded in a dynamical environment to provide adaptive feedback as well as robust memory, the system must comprise a large number of synapses changing on different time scales due to different internal and external factors \cite{fudr2005}. Synapses with many stable values of efficacy can significantly increase memory retention times \cite{fuab2007}, and synapses that adapt not only their state of efficacy but also their probability of state transition are crucial for maximizing memory retention times \cite{fudr2005,khso2017}. Adaptation of probability of state transition is a mechanism of metaplasticity \cite{ab2008}, and many forms appear in biological systems, which employ many techniques for extending memory retention \cite{ab2008}. We expect a cognitive system to utilize differentiated regions of neurons, some with synapses changing readily between only two synaptic states, and other regions with synapses changing slowly between many distinguishable states. We further expect the network to update not only synaptic weights but also the probability of changing synaptic weights. The dynamical state of the system can then sample synaptic memory acquired at many times, in many contexts, while quickly adapting the dynamical trajectory as new stimulus is presented.
To summarize, cognition appears to require differentiated local processing combined with information integration across space, time, and experience. The structure of the network determines the dynamical state space, and the structure of the network adapts in response to stimulus and internal activity. We now ask the question: what physical systems are best equipped to perform these operations?
\section{\label{sec:physicsAndHardware}Physics and hardware for cognition}
The aforementioned insights from neuroscience lead us to emphasize several features of neural systems in hardware for cognition. First, we must use a physical signaling mechanism capable of achieving communication across networks with dense local clustering, mid-range connectivity, and large-scale integration. Second, the relaxation oscillators that constitute the computational primitives of the system must perform many dynamical functions with a wide variety of time constants to enable and maximally utilize information processing through transient synchronized assemblies. Third, a variety of synapses must be achievable, ranging from binary to multistable. The strength of these synapses must adjust due to network activity, as must the update frequency. These neuron and network considerations guide the designs presented in this series of papers.
\subsection{Optical communication}
A principal challenge of differentiated computation with integrated information is communication. The core concept of the superconducting optoelectronic hardware platform is that light is excellent for this purpose. Light excels at communication for three reasons. First, light experiences no capacitance or inductance, so dense local clustering as well as long-range connections can be achieved without charge-based wiring parasitics. Second, it is possible to signal with single quanta of the electromagnetic field, thereby enabling the energy efficiency necessary for scaling. Third, light is the fastest entity in the universe. Short communication delays are ideal for maximizing the number of synchronized oscillations a neuron can participate in as well as the size of the neuronal pool participating in a synchronized oscillation. Light-speed communication therefore facilitates large networks with rich dynamics.
We have argued elsewhere \cite{shbu2017} that the capacitance and inductance of electronic interconnects is not ideal for neural computing. These limitations are ultimately due to the charge of the electron and its mass. Signals in the brain are transmitted via ionic conduction. The operating voltage of biological neurons is near 70 mV, so the energy penalty of $C V^2/2$ is significantly reduced relative to semiconducting technologies operating at 1\,V. Yet the low mobility of ions results in very low signal velocities, severely limiting the total size of biological neural systems \cite{bu2006}. Uncharged, massless particles are better suited to communication in cognitive neural systems. Light is the natural candidate for this operation. It is possible for a single optical source to fan its signals out to a very large number of recipients. This fanout can be implemented in free space, over fiber optic networks, or in dielectric waveguides at the chip and wafer scales. For large neural systems, it will be advantageous to employ all these media for signal routing. The presence of excellent waveguiding materials and a variety of light sources inclines us to utilize optical signals with 1\,\textmu m $\le \lambda \le$ 2\,\textmu m. Additionally, because the energy of a photon and its wavelength are inversely proportional, optoelectronic circuits face a power/area trade-off. Similar circuits to those presented here could be implemented with microwave circuits, but the system size would likely be cumbersome. Operation near telecommunication wavelengths appears to strike a suitable compromise.
\subsection{Superconducting electronics}
The foundational conjecture of the proposed hardware platform is that light is optimal for communication in cognitive systems. The subsequent conjecture is that power consumption will be minimized if single photons of light can be sent and received as signals between neurons in the system. Superconducting single-photon detectors are the best candidate for receiving the photonic signals. In addition to selecting micro-scale light sources and dielectric waveguides, we choose to utilize superconduting-nanowire single-photon detectors \cite{gook2001,nata2012,liyo2013,mave2013} to receive photonic signals because of the speed \cite{yake2007}, efficiency \cite{mave2013}, and scalable fabrication \cite{buch2017} of these devices.
Utilizing superconducting detectors contributes to energy efficiency in two ways. First, because a single photon is a quantum of the electromagnetic field, it is not possible to signal with less energy at a given wavelength. Second, because the device is superconducting, it dissipates near zero power when it is not responding to a detection event.
This choice of employing superconductors has several important ramifications. It requires that we operate at temperatures that support a superconducting ground state ($\approx$\,4\,K), so cryogenic cooling must be implemented. While cooling is an inconvenience, employment of superconducting detectors brings the opportunity to utilize the entire suite of superconducting electronic devices \cite{ti1996,vatu1998,ka1999}, including Josephson junctions and thin-film components such as current \cite{mcbe2014,mcab2016} and voltage \cite{zhto2018} amplifiers. Semiconductor light sources also benefit from low temperature \cite{doro2017}.
We have emphasized that the charge and mass of electrons is a hindrance for communication. Yet the interactions between electrons due to their charge makes them well-suited to perform the computation and memory functions of neurons. In particular, the properties of superconducting devices and circuits make them exceptionally capable of achieving the complex dynamical systems necessary for cognition. To elucidate the specific type of dynamical devices we intend to employ, we now elaborate upon the strengths of relaxation oscillators for cognitive systems.
\subsection{Relaxation Oscillators}
As we have mentioned, a defining aspect of cognitive systems is the ability to differentiate locally to create many sub-processors, but also to integrate the information from many small regions into a cohesive system, and to repeat this architecture across many spatial and temporal scales. A network of many dynamical nodes, each with the capability of operating at many frequencies, gives rise to a vast state space. As computational primitives that can enable such a dynamical system, oscillators are ideal candidates. In particular, relaxation oscillators \cite{st2015,mist1990,soko1993,lued1997,huya2000,bu2006,gile2011,vepe1968,cacl1981} with temporal dynamics on multiple time scales \cite{soko1993} have many attractive properties for neural computing, which is likely why the brain is constructed of such devices \cite{ll1988}. We define a relaxation oscillator as an element, circuit, or system that produces rapid surges of a physical quantity or signal as the result of a cycle of accumulation and discharge. Relaxation oscillators are energy efficient in that they generally experience a long quiescent period followed by a short burst of activity. Timing between these short pulses can be precisely defined and detected \cite{bu2006}. Relaxation oscillators can operate at many frequencies \cite{huya2000} and engage with myriad dynamical interactions \cite{lued1997}. The oscillator's response is tunable \cite{huya2000}, they are resilient to noise because their signals are effectively digital \cite{stgo2005}, and they can encode information in their mean oscillation frequency as well as in higher-order timing correlations \cite{pasc1999,thde2001,sase2001,stse2007,brcl2010,haah2015}.
The relaxation oscillators we intend to employ as the computational primitives of superconducting optoelectronic networks can be as simple as integrate-and-fire neurons \cite{daab2001,geki2002} or more complex with the addition of features such as dendritic processing \cite{thde2001,sase2001,stse2007,brcl2010,haah2015} to inhibit specific sets of connections \cite{budr2004,bu2006,robu2015} or detect timing correlations and sequences of activity \cite{sase2001,haah2015}. While our choice to use superconductors was motivated by the need to detect single photons, we find superconducting circuits combining single-photon detectors and Josephson junctions are well-suited for the construction of relaxation oscillators with the properties required for neural circuits.
\subsection{Neuron overview}
\begin{figure
\centerline{\includegraphics[width=8.6cm]{_general_schematic_small.pdf}}
\caption{\label{fig:general_schematic}Schematic of a loop neuron. Excitatory ($\mathsf{S_e}$) and inhibitory ($\mathsf{S_i}$) synapses are shown, as are the synaptic weight update circuits ($\mathsf{W}$). The wavy, colored arrows are photons, and the straight, black arrows are electrical signals. The synapses receive signals as faint as a single photon and add supercurrent to an integration loop. Upon reaching threshold, a signal is sent to the transmitter circuit ($\mathsf{T}$), which produces a photon pulse. Some photons from the pulse are sent to downstream synaptic connections, while some are used locally to update synaptic weights.}
\end{figure}
We refer to relaxation oscillators sending few-photon signals that are received with superconducting detectors as superconducting optoelectronic neurons. In the specific neurons studied in this work, integration, synaptic plasticity, and dendritic processing are implemented with inductively coupled loops of supercurrent. We therefore refer to devices of this type as loop neurons. The loop neuron presented in the remaining papers in this series is shown schematically in Fig. \ref{fig:general_schematic}. Its operation is as follows.
Photons from afferent neurons are received by superconducting single-photon detectors at a neuron's synapses. Using Josephson circuits, these detection events are converted into an integrated supercurrent which is stored in a loop. The amount of current that gets added to the integration loop during a photon detection event is determined by the synaptic weight. The synaptic weight is dynamically adjusted by another circuit combining single-photon detectors and Josephson junctions. When the integrated current of a given neuron reaches a (dynamically variable) threshold, an amplification cascade begins, resulting in the production of light from a waveguide-integrated semiconductor light emitter. The photons thus produced fan out through a network of dielectric waveguides and arrive at the synaptic terminals of other neurons where the process repeats.
In these loop neurons, a synapse consists of a single-photon detector in parallel with a Josephson junction (which together transduce photons to supercurrent), and a superconducting loop, which stores a current proportional to the number of detected photon arrival events. This loop is referred to as the synaptic integration loop. Within each neuron, the loops of many synapses are inductively coupled to a larger superconducting loop, thereby inducing an integrated current proportional to the current in all its synapses. When the current in this neuronal integration loop reaches a threshold, the neuron produces a current pulse in the form of a flux quantum. This current is amplified and converted to voltage to produce photons from a semiconductor $p-i-n$ junction.
The currents in the synaptic and neuronal integration loops are analogous to the membrane potential of biological neurons \cite{daab2001}, and the states of flux in these loops are the principal dynamical variables of the synapses and neurons in the system. The dendritic processing functions discussed above can be implemented straightforwardly by adding intermediate mutually inductively coupled loops between the synaptic and neuronal loops. Inhibitory synapses can be achieved through mutual inductors with the opposite sign of coupling. Synapses can be grouped on dendritic loops capable of local, nonlinear processing and inhibition, analogous to dendrites \cite{sase2001,bu2006,robu2015}. Dendrites capable of detecting specific sequences of synaptic firing events \cite{thde2001,haah2015} can also be achieved. Neurons with multiple levels of dendritic hierarchy can be implemented as multiple stages of integrating loops. Clustering synapses on multiple levels of hierarchy in this way enables information access at gradually larger length scales across the network through transient synchronization at gradually lower frequencies \cite{stsa2000}. The temporal scales of the loops can be set with $L/r$ time constants, so different components can operate on different temporal scales, enabling relaxation oscillators with rich temporal dynamics. These relaxation oscillators can be combined in networks with dynamic functional connectivity, reconfigurable through inhibition \cite{robu2015,fr2015}. These receiver circuits and integration loops are presented in Ref.\,\onlinecite{sh2018b}.
Synaptic memory is also implemented based on the stored flux in a loop, referred to as the synaptic storage loop. The state of flux in the synaptic storage loop determines the current bias to the synaptic receiver circuit discussed above. This current bias is the synaptic weight. If the synaptic storage loop is created with a superconducting wire of high inductance, the loop can hold many discrete states of flux, and therefore can implement many synaptic weights. In Ref.\,\onlinecite{sh2018c} we investigate synapses with a pseudo-continuum of hundreds of stable synaptic levels between minimal and maximal saturation values, and we show that transitions between these levels can be induced based on the relative arrival times of photons from the pre-synaptic and post-synaptic neurons, thereby establishing a means for spike-timing-dependent plasticity with one photon required for each step of the memory update process.
While synapses with many stable levels are advantageous to extending memory retention times \cite{fuab2007}, it is also important to implement synapses that change not only their efficacy based on pre- and post-synaptic spike timing, but also change their probability of changing their efficacy \cite{fudr2005}. Just as the synaptic weight is adjusted through a current bias on the receiver circuit, the probability of changing the synaptic weight can be adjusted through a current bias on the synaptic update circuit. As in the dendrites, we see a hierarchy can be achieved. In the case of synaptic memory, the synaptic weight and its rates of change are implemented in a loop hierarchy, and the state of flux in the loops can be dynamically modified based on photon detection events. Similar mechanisms can be utilized to adjust the synaptic weight based on short-term activity from the pre-synaptic neuron \cite{abre2004} or on a slowly varying temporal average of post-synaptic activity \cite{bico1982,cobe2012}. The synaptic memory circuits we develop in Ref.\,\onlinecite{sh2018c} are logical extensions of binary memory cells utilized in superconducting digital electronics \cite{vatu1998,ka1999}.
The aspect of superconducting optoelectronic neuron operation that is most difficult to achieve is the production of light. The superconducting electronic circuits that perform the aforementioned synaptic and neuronal operations operate at millivolt levels, whereas production of the telecom photons desirable for communication requires a volt across a semiconductor diode. When a neuron reaches threshold, an amplification sequence begins. Current amplification is first performed, and the resulting large supercurrent is used to induce a superconducting-to-normal phase transition in a length of wire. When the current-biased wire becomes resistive, a voltage is produced via Ohm's law. This device leverages the extreme nonlinearity of the quantum phase transition to quickly produce a large voltage and an optical pulse. The photons of this pulse are distributed over a large axonal network of passive dielectric waveguides. These waveguides terminate at each of the downstream synaptic connections. A downstream synaptic firing event will occur with near-unity probability at any connection receiving one or more photons. Photons of multiple colors can be generated simultaneously or independently, and different colors can share routing waveguides, while being used for different functions on the receiving end, such as synaptic firing and synaptic update. The number of photons produced during a neuronal firing event is the gain of the neuron, and the gain can be manipulated with the current bias to the light emitter. These transmitter circuits are discussed in Ref.\,\onlinecite{sh2018d}, and the network of waveguides that routes the communication events is discussed in Ref.\,\onlinecite{sh2018e}.
To make the analogy to biological neural hardware explicit, synapses are manifest as circuits comprising superconducting single-photon detectors with Josephson junctions. These synapses transduce photonic communication signals to supercurrent for information processing. The dendritic arbor is a spatial distribution of synapses interconnected with inductively coupled loops for intermediate integration and nonlinear processing. The integration function of the soma is also achieved with a superconducting loop, and the threshold is detected when a Josephson junction in this loop is driven above its critical current. The firing function of the soma (or axon hillock) is carried out by a chain of superconducting current and voltage amplifiers that drive a semiconductor diode to produce light. The axonal arbor is manifest as dielectric waveguides that route photonic signals to downstream synaptic connections.
Loop neurons combine several core devices: superconducting single-photon detectors \cite{gook2001,nata2012,liyo2013,mave2013}, Josephson junctions \cite{ti1996,vatu1998,ka1999}, superconducting mutual inductors \cite{miha2005}, superconducting current \cite{mcbe2014,mcab2016} and voltage amplifiers \cite{zhto2018}, semiconductor light sources \cite{shbu2017,buch2017}, and passive dielectric waveguide routing networks \cite{chbu2017,sami2017}. While all the components of these neurons have been demonstrated independently, their combined operation in this neural circuit has not been shown. Yet the physical principles of their operation and the designs presented in this series of papers indicate the potential for loop neurons to achieve complex, large-scale neural systems. The straightforward implementation of inhibition; the realization of a variety of temporal scales through $L/r$ time constants; single-photon-induced synaptic plasticity; and dynamically variable learning rate, threshold, and gain indicate these relaxation oscillators are promising as computational primitives. In conjunction with dense local and fast distant communication over passive waveguides, the system appears capable of the spatial and temporal information integration necessary for cognition and binding.
\subsection{The neuronal pool}
We have argued that light can achieve the connectivity necessary for information integration. There is another quantity that leads us to consider light an ideal messenger in neural systems. This quantity is the total number of neurons that can communicate with one another, referred to as the neuronal pool \cite{bu2006}. The size of the neuronal pool is treated in more detail in Ref.\,\onlinecite{sh2018e}. Here we summarize the salient result.
If we consider networks with predominantly two-dimensional long-range connectivity (as we find in the mammalian cortex and we expect from lithographic fabrication), the number of neurons in the pool scales as the square of the signal velocity divided by the device size, $(v/w)^2$. While devices in the brain are extremely small, signal propagation is not particularly fast (2\,m/s in cortex). Optical signals are seven orders of magnitude faster than this, so even if neural systems employing optics have significantly larger devices, the size of the neuronal pool can significantly exceed what is achievable in biological systems. We estimate the neuronal pool of a superconducting optoelectronic network could comprise as many as a trillion times the number of neurons in the neuronal pool of a biological system.
For cognition, bigger is likely better, as long as new devices represent new information, and the new information can be integrated across the system. Communication and energy efficiency are therefore principal concerns. Optical communication enables massive connectivity, and single-photon detection ensures power density never limits scaling. These considerations illustrate the potential for large-scale cognitive systems utilizing light for communication and superconductors for computation. We take an infinitesimal step toward designing networks of these neurons in Ref.\,\onlinecite{sh2018e}.
\section{\label{sec:discussion}Discussion}
Cognitive systems require differentiated processing and integration of information. Networks with power law spatial and temporal distributions meet these information-processing requirements. Communication is paramount both locally and globally. We conjecture that the requirement of reflecting this significance in hardware suggests we use light for communication. Micro-scale semiconducting devices are ideal light sources for dense neural integration. The requirement of power efficiency steers us to use few quanta of the electromagnetic field as our signals, a possibility enabled by superconducting detectors. This study of superconducting optoelectronic neurons combining semiconducting light sources, single-photon detectors, Josephson junctions, and dielectric waveguides indicates exceptional potential to achieve the neural functions underlying cognition. The large-scale implementation of such systems is particularly intriguing due to light-speed signals and superconductor efficiencies.
We do not propose superconducting optoelectronic networks (SOENs) as an alternative to established neural hardware, but rather as a symbiotic technology. The success of neural CMOS (including optical communication above a certain spatial scale) will contribute to the success of SOENs, as it will be advantageous for SOENs to interface with CMOS via photonic signaling on fiber optic links between cryogenic and ambient environments. SOEN hardware is particularly well suited to interfacing with other cryogenic technologies such as imaging systems with superconducting sensors \cite{alve2015,chsc2017}, as are commonly employed for medical diagnostics \cite{hada2016}, exoplanet search \cite{raca2016,boga1992,kila2016}, cosmology \cite{diad2017}, and particle detectors \cite{le2017}. An intriguing application is in conjunction with other advanced computing technologies such as flux-based logic \cite{li2012,taoz2013,hehe2011} and quantum computers \cite{we2017}. One can envision a hybrid computational platform \cite{deli2017,posc2017} wherein a quantum computer searches the space of network weights, the neural computer learns the behavior of the quantum system, and classical fluxon logic controls the operation of both. A superconducting optoelectronic hardware platform is likely to satisfy the computation and communication requirements of this hybrid technology.
The arguments in this paper are general, and in the subsequent four papers we present the details of the devices, circuits, and networks intended to achieve neural operation. Reference \onlinecite{sh2018b} presents the design of receiver circuits that detect photonic signals and convert them to an integrated supercurrent. We discuss the implementation of inhibition as well as dendritic processing, which are useful for dynamically tuning oscillation frequencies. The short refractory period combined with tunable response frequencies enables dynamic activity across many orders of magnitude in frequency.
In Ref. \onlinecite{sh2018c} we introduce synaptic memory and show that it can be modified on time scales as short as 50\,ps or as long as desired. Memory update can be implemented externally for machine learning or by the internal activity of pre- and post-synaptic neurons, with each step of the memory update process requiring a single photon. We design simple, binary synapses as well as synapses with many internal plastic and metaplastic states, which achieve a balance between quick memory response and long-term recall.
A challenge when integrating superconducting and semiconducting circuits is inducing the $\approx$\,1\,V required to drive semiconductors with low-voltage superconducting circuits. This operation is necessary if signals weighted and integrated in the superconducting domain are to produce optical signals for communication during a neuronal firing event. An amplifier circuit that produces the necessary voltage to drive the light sources is presented in Ref.\,\onlinecite{sh2018d}. A device utilizing the superconductor/metal phase transition achieves the required nonlinearity.
In Ref.\,\onlinecite{sh2018e} we design networks of dielectric waveguides connecting semiconductor optical sources to superconducting synapses. We show that networks of a million neurons firing up to 20\,MHz, hundreds of millions of plastic synapses, and power law degree distribution can be integrated in a single complex network on a 300\,mm wafer. The power dissipated by the network would be 1\,W, a value easily managed by a standard $^4$He cryostat. We close that paper with speculation regarding the limits of neural computing in systems with light-speed communication.
\vspace{0.5em}
This is a contribution of NIST, an agency of the US government, not subject to copyright.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,871 |
HER HUNGER
John Shirley
Start Publishing LLC
New York, New York
_HER HUNGER_ Copyright © 2001, 2014 by John Shirley.
This edition of _HER HUNGER_ published 2014 by Start Publishing LLC.
All Rights Reserved. No part of this book may be reproduced in any manner without the express written consent of the publisher, except in the case of brief excerpts in critical reviews or articles. All inquiries should be addressed to Start Publishing LLC, 609 Greenwich Street, 6th Floor, New York, NY 10014.
Published by Start Publishing LLC
New York, New York
Please visit us on the web at
www.start-media.com
ISBN: 978-1-60977-120-1
_HER HUNGER_ was first published in _Night Visions 10_ ,
edited by Richard Chizmar (Subterranean Press)
Her Hunger by John Shirley
Desire can take any form, can become any fantasy _..._ but it must be fed with life. When a strange woman appears in a small coastal California town, death follows. Only Corey—in love with the beautiful, intelligent, quiet Angela who seems unable to love the boy who so adores her—begins to see what is happening. But no one, not even Corey, can possibly believe what he suspects... and the bizarre murders continue.
* * *
Previously published only in hardcover.
Nominated for the International Horror Guild Award
_Publishers Weekly_ on "Her Hunger": "...supernatural horror with a philosophical twist, Shirley's [novella] is an entertaining tale, scary and full of surprises, of an succubus in California; it features believably off-kilter characters and a neat touch of metaphysics."
Other Fiction by
JOHN SHIRLEY
Novels:
_Transmaniacon_
_Dracula in Love_
_City Come A-Walkin'_
_Three-Ring Psychus_
_The Brigade_
_Cellars_
_(A) Song Called Youth Series Book One: Eclipse_
_In Darkness Waiting_
_Kamus of Kadizar: The Black Hole of Carcosa_
_(A) Song Called Youth Series Book Two: Eclipse Penumbra_
_A Splendid Chaos_
_(A) Song Called Youth Series Book Three: Eclipse Corona_
_Wetbones_
_Demons_
_The View From Hell_
_Her Hunger_
_...And the Angel with Television Eyes_
_Spider Moon_
_Demons (expanded)_
_Crawlers_
_The Other End_
_Black Glass_
_Bleak History_
_Everything Is Broken_
_A Song Called Youth (complete trilogy omnibus)_
_Doyle After Death_
_High_
Fiction Collections:
_Heatseeker_
_New Noir_
_The Exploded Heart_
_Black Butterflies_
_Really Really Really Really Weird Stories_
_Darkness Divided_
_Living Shadows: Stories: New and Pre-Owned_
_In Extremis: The Most Extreme Stories of John Shirley_
Nonfiction
_Gurdjieff: An Introduction to His Life and Ideas_
_New Taboos_
**HER HUNGER**
**PROLOGUE**
_Silbido, California_
She slept deeply but restlessly. Not quite eighteen years old, she twisted and moaned in a comfortable, prettily trimmed bed in a small dark, room with lace curtains on the single window. A single sharp-edged ray of yellow light shot through the room from the window, onto an upper corner of her bed. She breathed deeply, but irregularly; the darkness in her bedroom was breathing along with her. Groaning with her. The air close, warm, scented with her: clinging to her.
The dream shook her; made her undulate, and groan. "No... I don't want that. I don't want it..."
Her fingers, with nails painted the color of onyx, stretched out into the square of light projected onto her bed from the window—stretched out, then clutched the comforter, digging at it. "No uh- _uh-_ no!" Her back arching; her voice was distorted by the increasing thickness of the atmosphere in the dark room, as the other presence, arriving second by second, brought the density of its own hungry being to the very air...
Still asleep—she sat up. She scrambled back, flailing, into a corner of the bed. "Don't!"
Something stirred in the shadows where nothing had been a moment before. Something took substance from the shadows themselves, as if drawing them on for a cloak of sheer black silk. Took form, and proportion...
The silhouette of a woman, tall and willowy. Approaching the bed.
Speaking. _"It's all right. I'll take care of it. We'll do it together..."_
A gentle hand, the color of moonlight, brushed the sleeping girl's hair; the girl's thrashing ebbed, her trembling eased, and she relaxed onto her side, fetal, clutching the pillow to her groin.
**PART THE FIRST: ONE NIGHT IN SILBIDO**
ONE
Summer night, Saturday night, California night: Silbido almost came alive. It was a small town, mall-blighted, smoldering with suppressed discontent like dry leaves blown over dust-hidden embers. Silbido's photocopied tract-homes, split level or ranch style, were arrayed around a small old-town core of Victorian and Spanish-style structures; and there was the fish cannery, along the docks, where the Cruise started.
The cars in the Cruise, that early evening, followed a long paper-clip shape of the Silbido Strip, like a slow-motion race car track, those hot summer nights; cherried old ragtops full of cholos, vans and secondhand Jeeps and SUVs and third-hand pick-ups full of teenagers and college-age kids, parading almost bumper to bumper through the jerky, languid ritual of the Saturday Night Cruise, around and around and around again the same three mile strip, showing off their cars or their parents' cars and their designer clothes or their anti-designer clothes; flipping light sticks and the Bird; honking and hooting; low riders making their raked sixties Chevys and Buicks jiggle and dance in place: a ton of metal imitating Jell-O. From every car, music boomed—usually the same two stations, competing. When the kids drove erratically enough, a few town cops checked for liquor or guns in the cars. Mostly they just watched.
The cruise kids took the La Paz exit off the freeway from the burbs, which dumped them out along Cannery Boulevard; they'd do their perfect circles round the circular dead end where the road became shaped like the bottom of a thermometer, and, cruising slow, they'd drive up Pacific—the main drag—through Old Town, past the strip malls, the fast food stretch, the gleaming little sedans at the GMC dealership; the pickups at the Ford dealership, the endless rows of SUVs at the Dodge dealership, and out to the Grand Silbido Mall, "third biggest Mall in Southern California"; they circled the mall, drove back down past the dealerships, the strip malls, the Burger King, Pollo Loco, McDonalds and the other grease franchises; through Old Town, back down the main drag to the Cannery, around the dead-end loop... and once more to Pacific. The cops tried to enforce a midnight curfew for the cruising. Sometimes it went on till two.
Most of the cruisers pit-stopped at least twice at the Burger King or the Pollo Loco and that's where Brad and Stanny were, in Brad's van, feeling bloated with shakes and fries. Brad was just pulling out of the Burger King parking lot, onto Pacific. It was a campy pseudo-surfer van, with its airbrushed flames and teardrop bubble window near the back. Riding shotgun with his Gibson SG plugged into a little portable battery-operated Pignose amplifier, Stanny was tinkering with riffs at a low volume, trying to play along with the Limp Bizkit on the CD boomer. In the back; representing improbably high hopes, was a mattress covered with imitation-satin black sheets. There was a little pocket cut into the mattress, on the side against the metal wall, where the brass pot pipe and the stash were tucked.
They jerked to a stop at the red light. A small car—a Jetta, Brad thought, craning over Stanny to see—pulling up beside them, packed with girls. At a glance, Brad didn't know any of them.
"There's some quim," Brad said softly. "Quim and trim, dim."
"I'm all over that shit, wicked," Stanny said.
They hesitated, unsure of their strategy, knowing that both of them were less than ideal, despite the van and guitar: Brad with his unfashionable big hair that he couldn't bring himself to get rid of; a pretty face behind all the pimples—teenage girls will tolerate a fair number of pimples, all things considered, but not that many. His triple-pierced ears and nose stud were the shit, but the carefully worn-out leather jacket didn't make it. But he couldn't bring himself to admit that, either. _You need a new set of role models_ , Angela had told him. _I'm the last person to push for trendy but Skid Row is way past twenty minutes ago—more like eight years_.
Stanny, now, was caparisoned in Eminem drag; the same Slim Shady short bleached haircut, the backward visor hat, the same dirty white tank top, baggy pants, more or less the same tattoos; but his nose, which seemed to have taken what should have gone into his weak chin, and his big-puppy over-eagerness, threw his girl-dogging chances into a tail spin.
"You gotta date anyway, Brad," Stanny was saying, "let me talk to them." He was rolling down the window, lurching impulsively into action as usual.
"A date with Angela? She is so _not_ putting out," Brad said. "So what's the point?"
But Stanny was already yelling out the window. "Yo, girls! You bettys are all hella squished in there and like, we got ful-on room in here!"
Brad winced.
The babe driving, the one with shiny-blue eye shadow and sparkles at her temples, came back instantly with, "There's... no way?" Making it sound like a question when it wasn't.
"Hey," Stanny said, relentlessly, conscious that the light was going to change any second, flailing to keep her talking, "you know that sparkle makeup stuff you got is made from fish scales?" He'd offered it in the spirit of a cool believe-it-or-not.
She responded, "You are deeply grotty, you know that?"
The other four girls tittered. She turned to the other girls, "He's all, 'you got fish scales on your face?'"
"Stanny," Brad said, groaning, "don't even try."
But Stanny was a believer in at least trying. "Hey my man here and I, we're in a band, we're gonna jam. He's the, like, lead singer? And I'm the guitar player?"
"Ooh, I'm like _sew_ wicked impressed." Titters became hysterical laughter. She shook her head. "You are just random. Hopelessly random."
"No we're really in a band! Scope it out, I got a Pignose!" He held up the little battery operated portable Pignose amp.
"Don't feel bad about your nose!" yelled the blue-streaked platinum blond out a back window. "You can't help it!"
The girl-car accelerated through the treacherously green light.
"Stanny," Brad said, shifting into first, as an SUV honked behind him, "you are about as smooth as a bleeding pimple."
"Fuck you!" Stanny yelled.
Brad was close to kicking him out of the van for that—he regretted bringing him along in the first place—but then he realized the fuck-you was meant for the jocks honking impatiently from their Jeep behind the van, and Stanny was leaning out the window giving them the finger with both hands. Okay, the fingers.
It wasn't really quite dark out yet, the air sensuously warm; blossom scented. The backyard was giving the heat of the day to the cooling night sky, as Angela shifted on the mossy bench under the flowering plum tree, and turned the page in the old book she was reading in the light from the back porch. They were only a block away from Pacific, and she could hear the Cruisers honking down there. She tried to ignore the sound.
Carrying a book himself, Corey walked up softly, down the mossy lane between the house and the high wooden fence, and paused at the open gate to gaze at her: a tall, long-necked girl stooped over her book. Lush blond hair piled up on top of her head into a disorderly bun, held by an unvarnished wooden pin. Her long white dress was Victorian retro, with only just a suggestion of cleavage showing. The little bump in the otherwise perfect sweep of her long nose set off her profile like a detail in a cameo. A breeze swelled, stirring the scent of plum blossoms, loosing a wedding-train scattering of bluepink petals down over her.
How did she arrange for the petals to fall, just then? he mused, chuckling in his mind. Almost too picture perfect.
Corey was maybe two inches shorter than Angela, and five months younger. Enough younger so it was right on the verge of mattering. He felt, though, like a man, some man, born a hundred andfifty years before, looking at her now.
He padded up behind her, studying the ground to avoid twigs, and got just close enough to look down over her shoulder. To make her jump—to react to him physically. He read from the book of poetry she was reading aloud. " _Lady, three white leopards sat under a juniper tree..."_
Not at all startled, she lifted her head, reciting the next line, _"...in the cool of the day, having fed to satiety..._ "
Corey shook his head. (He'd just cut his hair, the day before—from shoulder length to what Angela called "standard nondescript"—and missed feeling it swish on his collar). "Damn, I was sure I'd snuck up on you that time..."
"I knew you were coming over."
"How come you always know?" He sat down beside her, not as close as he'd have liked to; not even as close as he might've dared.
She turned her head to look at him, pursing away a smile. "I know. How come you know what you know—and do what _you_ do?"
"You mean read over your shoulder? I like the view." He knew she'd take it as a joke on jejune jokes. They knew each other pretty well.
"No, Mister Maturity." She patted the book he'd given her, on her lap. "How come you knew I'd like this poem? How come you always know what books I'd like?"
"I lived across the street from you since you were letting your popsicles melt onto my mom's new carpet. And anyway...I'm a vampire. We can read minds."
He was flipping through the old book in his hands.
"That looks old."
"It's a first edition of _A Voyage to Arcturus_ by David Lindsay. I got it at this antiques-and-old-books shop. I don't know why he let me have it so cheap, it's got to be a collector's item. I read it in a paperback, couple of years ago. The paperback fell apart first time I read it—every page I turned came loose and fell out of the book. Anyway, I wanted to loan it to you. It's a kind of fantasy allegory sort of a deal..."
"That sort of a deal? Is that like a bank loan layaway plan sort of a deal?" she asked distractedly, reading the back cover copy of the Lindsay book.
"I should take you to that shop sometime. You'd like it. It smells like old books in there."
"I love that smell. Pulpy old books. I really do. I like the yellow the pages go around the edges."
"That's why I don't want those e-book things. They don't decay right."
"You," she said, "are wack. Plain old wack. You can have the poetry book back. Gimme this one..."
She took the fantasy novel and admired the cover painting of a fabulous alien world, the image frayed with the edges of the book jacket. He reached over and took the poetry book from her lap, feeling warmth rising from her thighs, his hand trembling slightly with the effort of picking up the book without touching her—
The effort of not touching her.
He pretended to leaf through the book, as if looking for something, and murmured, "So...I called Friday but, um..."
She sighed apologetically. "I know. I wasn't there. Said I'd be. Wasn't. Should've called you. I was visiting my dad. It was excruciating. I can't believe he's a therapist. You'd think a therapist would be sensitive."
"Maybe only when they're paid to be. I went to a therapist for a while after my mom died. He was grotesque. He pretended to be sympathetic. Kept looking at his watch."
He could use words like _grotesque_ and _allegory_ with her; she could say _excruciating_ ; they enjoyed that, like listening to music together that other people—at least other teenagers—didn't like.
As if thinking about that just when he was, she said, "You know what, Corey? You're the only one I can talk to about...a lot of stuff."
He took a deep breath—and the plunge. "Ange...An-gel-a....? Every time we see each other, we talk about stuff...like that. It's easy with you. Like there's something there, like...like a phone line in the air, that makes it easy..."
She hesitated. One, two, three seconds. "I know. It sort of scares me. I just—I like to, um, relate to people like in a dance. Not a slow dance. I mean—you dance with them...but not too close."
"It's not like you don't have boyfriends." Meaning Brad.
"I don't feel anything for Brad. So... I can go out with him without... I don't know how to explain. I guess, with him I don't feel guilty for not getting much closer."
"Oh." He felt stupefied, somehow, by the impenetrability of her attitude toward boys. Stymied.
"You think I'm, like, playing games with people."
"No... I get it, I think. But... aren't relationships supposed to—"
"Angela!" Gail, Angela's mother, calling, from the house. The back porch door opened.
"Here comes the white leopard now," Angela muttered.
Gail Stroheim's flipped hair was platinum; her lips painted blood red. Her wore a silvery blouse, off-white slacks, cream-colored pumps. She favored shades of white; she'd picked out Angela's dress.
"Angela, you're going to be late."
Angela grimaced. "Mom...I think I'm going to cancel..."
Gail looked at Corey, smiling thinly. "I wonder if I could talk to our young Prima Dona alone, Corey?"
Corey stood up, stretching for the excuse of a moment's lingering. "See you later Prima Angela..."
"'Kay."
Corey went back out the way he'd come, waiting till he was in the shadow of the house, to pause for a look back at Angela. Then he crossed the lawn and the street, and trotted up the porch steps of his own house.
Alone with her mom, Angela pretended to read the fantasy novel with a frown of concentration. "Be there in a minute, Mom..."
Gail sat down beside her, reached over, and closed the book.
"Angela, if you make a date, keep it. Be responsible."
"Mom—I don't like the way Brad gets. He's all over me."
"Naturally, because he's got normal healthy urges. You can manage him. I mean, Angela—I don't even like Brad—that's how desperate I am to see you going out with someone. To see you being a normal teenager. You like Corey, he's obviously crazy about you—if you don't like Brad, why don't you—?"
"Because I don't want to spoil my friendship with Corey by getting serious."
"Well go out with Brad. If he drinks and drives or tries to rape you, kick him in the nuts, get to a phone, and call a cab. But go out. Go to the mall, drive around and...whatever. Be a normal girl. Well—" She stood up, letting humor into her voice. "Don't be a completely normal girl: don't get pregnant. But go do _something_. Your dad says you have to just get into normal dating patterns, with whoever, it doesn't have to be serious—"
"If Dad and his opinions are so wonderful, why'd you divorce him?"
"Excuse me? Don't get snotty, Angela. He's right about some things—that's his job. It's time for you to get back into the real world."
She turned and stalked purposefully back into the house. Angela stared after her—and into an expanding nowhere.
"Turn that Slim Shady shit off," Brad said, turning around in the parking lot of the mall.
"It's not Slim Shady—"
"I heard his whiny little voice."
"He's just a guest rapper on it. This is Dr. Dre."
"I can't deal with any of that rap bullshit, except the Beastie Boys. And Bloodhound Gang."
"Bloodhound Gang is tight. But they're not rap, 'cept, you know, now and then they—"
"Stanny? Turn that shit off."
"What, you're gonna make me listen to the Misfits?"
Brad suddenly veered the van to the curb, reached past Stanny and opened the passenger door. "Bail! You pissed me off. Out."
"Hey this rots—the buses stopped running, you could drop me—"
"I'm late. I'm pickin' up Angela, tonight's the night, dude! She's puttin' out tonight, hook or crook! No time to drop you anywhere. Go!"
"Nobody gets anywhere with Angela, Brad, Everybody's tried. She's the fuckin' ice queen."
"Ice queen melts tonight. And you're getting out. Now!"
He shoved Stanny out with a kick of his expensive, high-top, three-color, light-enhanced sneakers, put the car in gear, and roared away.
Stanny landed precariously on his feet, clutching his amp and his guitar; standing in the driveway of the parking lot, shouting after Brad. "This rots, Brad! Fuck you!" He started to add, _Find another guitar player_!
But if he said that, Brad just might.
The van pushed out the envelope, going through the traffic light exactly as it went red, and dwindled down Pacific.
Corey was shooting baskets in the driveway of their sixties-era split-level semi-ranch-style house; the backboard and hoop were on the front of the garage over the garage door: the only place on the house without shingles. The house had been covered, right before Dad bought it, in unpainted wooden shingles. Dad had always meant to coat them with stain, or something, but never found time. Dad was the local sheriff; it was a good excuse for not dealing with shingles. His white Silbido County Sheriff cruiser was parked at the curb.
Corey had the garage door open, so he could shoot hoops in the light from the garage. His movements were self contained, almost ballet graceful; it was the only time he was graceful. He rarely missed a shot.
Corey shot a lay-up, the ball ricocheting back and forth inside the hoop before sinking; he caught it coming down, dribbled back to the imaginary free-throw line and then clasped the ball against his chest as he turned to look toward Angela's house. Toward her bedroom window. The light was on, up there. Maybe her mom had made her put on some makeup. She didn't like wearing makeup.
He turned himself firmly away, went back to shooting baskets with redoubled energy.
Corey's dad, Barry, stepped out onto the front porch, hands in the back pockets of his trousers. He wore civvies, but there was a badge clipped to one side of his belt, a holstered .45 to the other. He was a stocky, thick-bodied man, with a comb-over that wasn't fooling anybody. He'd been shorter than mom, too. His nose was still red from the years he'd spent drinking, after Mom died. He'd been one of those manageable alcoholics, who does everything real carefully, keeps up a front, and chews a lot of breath mints. Corey wondered if his dad had ever driven drunk, back then. Hard to believe, Dad was so set against drinking now. He went to an AA meeting in the next county, once a month.
Barry strolled over, watched his son shoot hoops. "Been two or three blue moons since I saw you miss a hoop."
"I miss sometimes." He shot from the free-throw line. He didn't miss.
Barry rubbed one of his patent leather cop shoes—his one concession to uniform—against a trouser leg, thoughtfully removing a smudge. "You wanted to, Corey, you could score points for the team. And then with the girls."
How come they think they always have to _guide_ you, Corey thought. Like with Angela and her mom. Why can't they give it a rest? Makes you defensive, all the time.
But aloud all he said was, "I don't play for teams. I'd just get tense and screw up. I just shoot baskets because—" He shot. Swisher. "—it helps me think."
Barry nodded. He followed Corey's glance, toward the lit window in the big old house kitty-corner across the street. Corey looked back at the hoop; kept his eyes on the hoop, only the hoop. Barry asked, "You ask her?"
Corey hesitated, then decided not to pretend he didn't know what Dad meant. "She's... she can't tonight. Next week."
Barry nodded again, and walked down to his car. "Dinner's in the warmer."
"'Kay."
Barry unlocked the blocky Chrysler copmobile, got in, started the car, turned on the headlights, waved, and drove slowly off.
Corey set himself, shot, swished the ball through the basket; caught it after one bounce; dribbled, jumped, shot, made it. Then he heard a car, turned to see Brad's van pull up, in front of Angela's house. Turned away but, from the corner of his eye, watched Angela come out of the house, get in the van. The van drive off.
Corey set himself, shot at the basket—
Missed.
TWO
That same summer night...
Two teenagers, a boy and a girl, creeping down a dark hallway...
Angela always refused pot, but she'd been known to drink wine. His father had some good wine. Even good champagne.
Brad and Angela were skulking down the hall of Brad's father's house. Brad lived in the guesthouse, on the other side of the tennis court. He didn't think of this house as his. Brad was skulking along because he didn't want his dad to catch him; Angela was just imitating Brad. Skulking can be infectious.
It was a big house, self consciously modern back in the sixties. Rows of oddly shaped windows; lots of skylights.
Brad glanced up through the bubble skylight. Smog dirt had settled on the outside; the moon, up there, looked warped and grimy through the dirty curved glass.
"All these gold records..." She whispered, looking at the dusty old gold record plaques on the walls of the hallway.
He made a hand sign for "keep your voice down" though she'd only whispered. He whispered in her ear, "Some of them are for being a producer, only a few're for his own stuff. He only had three real hits."
"Three more than most people..."
There was a framed poster of Brad's dad, Mace. In the dark hallway it was hard to see, like Mace's career now.
"Your dad was this big rock star and he buys a house in Silbido? I thought they all went to Malibu or something."
Brad paused at the half open door to the den. "Keep your voice down! He used to have a house in the Bahamas. Wish he hadn't fucking sold it...but when the band burned out..."
She looked over Brad's shoulder, into the den. Pieces of something glittered, yellow and knifelike, on the rug. She gasped softly. "There's a smashed gold record on the floor in there—"
"Will you please, please keep your voice down, baby—never mind the smashed gold records. He gets smashed, he smashes records. He has shit-fits now and then, forget it....Now be quiet and I can get away with this—I'm pretty sure he's sacked out from drinking and snoring somewhere..."
"I don't know—I can feel someone close, Brad."
He gestured for silence, and reached through the partly open door. Opened the liquor cabinet, reached into it—
Yelled, "Ow, fuck!" at the lance of pain as someone slammed the liquor-cabinet door on his wrist. He jerked his hand back, off balance now, falling through the room's door, knocking it inward, wide open.
A desk light switched on, and he saw his father, Mace, looming over him, grinning, swaying.
Shit.
Mace's long gray-streaked black hair was loosely tied back; he had a three-day growth of salt-and-pepper beard; the sweatpants of his red jogging suit tucked were into snakeskin cowboy boots. In the weak, indirect lamplight his wolfish face showed every line he'd ever snorted.
"Braddy boy! What'd she say, she felt someone close by? Braddy boy's the one who feels me! How's that wrist, Brad! You can't do the time, don't do the crime, kid!"
Trying to sniff dismissively, shrug it all away, Brad got to his feet. Why'd she have to see this?
Mace was gazing blearily at Angela. "Whoa-oh! That's the Angela chickie, isn't it? The princess of purity? How you doing, girl thing?"
She shrugged. "Um—'girl thing' is okay."
Mace laughed. "Good sense of humor. You okay, kid?" Brad stepped back from Mace's inquiring hand. Mace made a "whatever" hand gesture and from somewhere produced a joint—big enough, really, to be a spliff. He fired it up with a Zippo, took a hit, and, voice squeaky from holding the hit in, offered it to Angela. "You partake?"
Brad winced. It was different, your dad offering hits. "Mace—shit—" He hadn't called Mace "Dad" in years.
Angela smiled, "Just say no, right?"
"Very good," Mace said, smiling woodenly. "Good attitude. That's what I told Betty Ford, at the clinic. Just say no, Betty, I said. Who said you could fucking steal my liquor, Brad?"
"Okay, I'm sorry, Mace. Thought you wouldn't mind."
"That's why you were sneaking, because you thought I wouldn't mind. You're so full of shit."
"We'd better go, Brad," Angela said. "Sorry, Mister..."
"Mace, just call me Mace. You know—if the problem is you want a man who knows what he's doing, you give me a call. I'm a very understanding 'older man.' All kinds of experience."
Brad groaned. Angela pretended to be interested in a gold record on the hallway wall.
Mace gazed at her body a long, expert moment, sucking on the spliff, then he blew smoke at the ceiling and gestured dismissively. "Why don't you wait out at the boy's chick-magnet, girl thing?"
She blinked, then realized Mace meant the van. She nodded and hurried out.
Brad was glad he hadn't left the keys in it. Freaked out by his father, maybe she'd take the van and split.
Mace grabbed Brad's upper arm, drew him into the den. Brad jerked loose again. "I should kick your ass, Dad, you know that? You do know I could do it, don't you? You just about came on to her."
"No I fucking didn't. I was kidding her. Here..." Mace took a bottle of Jack Daniels from the cabinet, slapped it against Brad's gut. "Take it. You'll get laid this time, kid. This is the shit."
Clutching the bottle, feeling ill for some reason he didn't understand—he should be glad Mace'd given him the shit, shouldn't he?—Brad turned and almost ran out the door and down the hall.
He heard Mace laughing softly, behind him.
Corey's old Impala was scuffed down to the primer. The engine groaned when it started; he'd had it tuned, when he'd worked that summer job at Wendy's for a couple of months, before getting into the argument with the manager, but it hadn't helped much—the whole transmission really needed replacing. You pressed on the accelerator and you waited and when it was in the mood it accelerated.
He drove his friends Wade and Megan down Pacific, moving jerkily in the thick cruising traffic. He looked at them in the rearview, shook his head—the two black teenagers in the back, making out. Megan with her cornrows, her long dangling earrings, her dashiki with the slit up the side; Wade in his basketball jersey and jeans, his hair clipped short. Liplocked with Megan. "You guys are making me ill. It's called 'find a room.' I mean—feeling like a chauffeur is bad enough but I'm not into, like, Corey's Rolling Boudoir."
Why had he come? Corey wondered. Wade's car was broken. That was the excuse. But was he hoping to see Angela out here? Was he afraid to?
"You see that, Megan?" Wade said, turning to Corey but talking to Megan, "How he says, 'boudoir'? I'm impressed."
"Corey got an A in French," Megan said. "I'm sorry, Corey. He's a monster, I can't keep him under control. Raging hormones in this monster."
"You want to talk about monsters," Wade began. "Let us now speak of monsters. It's right here..."
"Don't say it," Corey and Megan said at once, and Wade laughed.
"I can't believe," Corey said, "you talked me into this shit. I hate cruising the strip, it's like some kind of mating ritual for teenagers out here, and you guys are doing all the mating. If I look in the rearview I'm suddenly a goddamn voyeur."
"Pull over, I'll get in the front," Megan offered, pushing Wade firmly away and leaning to put her folded arms on the back of the shotgun seat.
"Meeeeeee-gannnnnnn!" Wade moaned. "Don't leave me, baby! I'll get in the front! You get in the back! But Corey's got to promise not to squeeze my monkey love. You know he wants to squeeze my monkey love."
"Fuck you, Wade," Corey said pleasantly.
"Ooooooh!" Wade and Megan said together. Megan asking, "Is that a first?"
"Just a rarity. But this time we have a winner. I heard him. Mister Proper Guy said fuck you."
"I cuss sometimes, you fucking liar," Corey said.
"That's twice anyway," Megan said. "No you are _not_ getting in front, Wade, I am!"
"You don't have to," Corey said. "But if you could use your tongues for something else—like talking—that'd be nice."
"Hey Corey?" Wade said, looking at the girls in the SUV beside them. "You know you're gonna see her out here somewhere. Brad's going to want to show off his date."
Corey managed to look blank. "Who?"
Megan and Wade looked at each other. Another synchronized comment: " _Who_?"
Wade shook his head. "He's almost believable, isn't he?"
Corey changed the subject. "I've had enough of breathing fumes and listening to designer car-horns. Let's go to, like, I don't know—somewhere else. A movie. Something."
"Too late for a movie," Wade said. "We'd be past curfew. You'd have to tell the cops, 'I'm the sheriff's son.' And I know you don't want to do that shit."
Corey felt a stony anger then. And showed it by stonily saying nothing.
Megan looked at him. Then scowled at Wade. "You know he's sensitive about that. He lost his job because the manager asked him to get his dad to patrol by more often and he told him to go to hell—"
"Sorry, jeez, just kidding. I like Corey's dad. Best cop in the world."
"Wade? Just shut up about it."
Now it was Wade's turn to cast about for a subject change. "How about we go to...the beach! Upper Beach Park!"
Corey shrugged. "Whatever. Anything to get us away from here."
"Somebody waving at us on the sidewalk over there," Megan said. "At least I think it's us he's waving at."
"I hope not," Wade said. "It's that fucking Stanny."
Corey thought, Anyone to buffer him from the sexual heat in the back seat. "What the hell."
He pulled over—Megan and Wade groaning on cue, a groan that wordlessly meant _Not Stanny in the car please_ —and Stanny didn't wait for someone to open the door, he opened it, and crammed himself and his little amp and his guitar in the front seat, bringing in the aromas of sweat and car exhaust, grinning with way too many teeth.
"Dudes! My guardian spirit rescuer dudes!"
"Oh Jesus save us," Wade muttered.
"Fucking Brad stranded me, man. I'm gonna find another band, he can suck my dick."
"Please, don't generate unthinkable images here, Stanny," Megan said.
Stanny looked at her. "You taking Latin, or something in summer school?"
"Forget it."
"Seriously, thanks Corey, I need a ride. Where we going?"
As he spoke Stanny switched on the little amp, began to riff.
"We're going to Pollo Loco, because I want a Dr Pepper or something and—Stanny, I don't want to shout over that thing." Corey reached over and switched the amp off. Then he heard a noise from the back seat and sighed. He nodded toward the back where Wade had once more gathered Megan into his arms.
Corey said, "Welcome to the goddamn Love Connection Limo, Stanny."
"Sick! Can I watch?"
Brad's van was pulled up at a turn-out, overlooking the beach park. In the back of the van, on the mattress, Angela sat with her knees drawn up between her and Brad; he sat across from her. Both of them with their backs against the thin metal walls of the van. Between them, the bottle. Brad had his MP3 player plugged in, playing Metallica: the downloaded _Garage Incorporated_ album. The only light came from a single overhead interior light.
"I just play Metallica on MP3 because it pisses them off," Brad was saying, leaning forward to pour more Jack into her Burger King Chicken Run cup.
"Brad—god—enough."
"You only drank like an old lady tea-sipping drink."
"I don't know why I drank any. I thought maybe it would put me in the mood to be what people want me to be. But it doesn't."
Grudgingly, she took another sip.
"Hey fair is fuckin fair, everybody here's got to be equally drunk. And is everybody drunk here?" He turned to the shadows at the back of the van. "I say, is everybody drunk back there? No?"
She giggled. Brad swilled more Jack, then scooched over beside her and made his move.
She let him kiss her, for a moment—and then turned her head.
Fucking ice queen.
He grabbed her shoulders, pulled her toward him, bent to kiss the swell of her breast—maybe that'd turn her on.
She sighed and simply put up with it.
Excited by the warm silky pressure of her breasts against his face, he tried to press her down on the mattress, spilling her drink. "Brad you're getting me all wet!"
"That's what I want!"
"Your getting this stinky booze on my dress you dipshit!" She shoved hard, and writhed out from under him—she'd had a fair amount of practice, with various guys. She drew her legs up and pushed him back, scrambling away. "I'm not letting you drive if you drink any more, Brad!"
"Your frigid act is bullshit, Angela. No more blue balls! You've been holding out for me—and now I'm here. Admit it! You're not hosin' me on this one! Admit you're waiting for me to make the real move!"
"You are totally full of shit!"
_"Admit it!"_
Then he launched himself at her, and fell onto her, making her shout with pain. She thrashed under him, slapping at his face; he grabbed one her wrists, pulled her blouse down with his other hand, hearing fabric rip, exposing her nipples. The sight of them seemed to send power surging through him. He caught both her wrists, deliberately squeezing them hard enough he could feel the bones straining, and she made a sound that was somewhere between a squeak and a scream, and tried to knee him in the crotch. He pushed his face down against her breasts, so close, he could hear her heart beating like a terrified rabbit's. He dragged his lips up her breasts, her collarbone, heard her whimpering something at him but didn't register exactly what the words were in his turgid excitement—something about how it wasn't real, it wasn't happening, _don't don't don't_ —but he chose not to believe it and he tried force a kiss from her—
She stopped thrashing and let him press his lips to hers—
And she bit his lips, hard.
"Ow fuck! You cunt!" He recoiled from her and she punched him hard in the solar plexus.
He doubled over and she wriggled out from under him and backed against the van's rear door. "Take me home. Now! Or I'll prosecute you, Brad! It's called sexual assault!"
He was kneeling, staring at her, poised on the edge of an abyss. He could jump her. He could...
But there were tears running down her cheeks.
He turned away, crawled up to the front seat and sat there a moment. Then he turned back to her. "Get out of my van."
"What? We're miles from town!"
"Get the fuck _out of my van!"_ He blasted the horn in his anger. "Out!" Another blast. "OUT OF MY VAN!" A long blast on the horn, on and on... Shouting the whole time, his voice blending with the van's horn.
"All _right!_ When you should be apologizing you're kicking me out in the middle of nowhere!"
"I'm gonna have to get stitches in my lip. Get out!"
They glared at one another. Then she turned and fumbled at the back door, found the handle, turned it, and with an effort shouldered the heavy door outward. He waited until she had stepped onto the pavement then he started the van, put it in drive and stomped the accelerator, peeling away without even closing the back door, so that it banged against the latch with the sudden motion.
He looked at her just once in the rearview, a small figure barely visible in moonlight and a faint wash of red taillight. Then that was gone too and he accelerated onto the highway.
THREE
Right then, exactly then, precisely then, Corey and Megan and Wade and Stanny sat in the car parked outside the Pollo Loco, listening to Beck on the radio. Corey was watching a family of Asians inside. Were they Filipino, or Chinese or what? Chinese eating Mexican fast foods. The little boy skimming a tortilla at his sister's head.
"You want anything else, Meg?" Wade asked. "Burrito or something?"
"Yeah would somebody, like, spot me a burrito?" Stanny asked.
"No way," Wade said. "I've sat next to you in class after you'd loaded up on beans."
"Oh please, no," Megan said. "Not for me and definitely not for him."
"Let's go," Corey said wearily. He backed the Impala out and drove through the parking lot to the edge of the street; waited for traffic to open up.
Stanny pointed at two skinheads getting out of their rebuilt US Army surplus jeep in a parking slot to their right. "What're _those_ dudes?" The window was open.
"Stanny—keep your voice down!"
But the skinheads heard him. They were both shaved bald, and the taller one had a swastika tattooed on the side of his head. A head shaped, to Corey's mind, like a peanut; the short one—couldn't be more than five foot two at the most—had two rings through his eyebrows, a cheek piercing, a tattoo on the side of his neck that said _Aryan Nation Auxilary_. Both wore sleeveless levi jackets covered in sloganeering buttons, like the cancel sign over the word _Jew_ ; both had steel-toe boots and chains on their hips.
"Spelled _auxiliary_ wrong," Megan muttered.
Wade hushed her. "Corey?"
"Nobody'll let me into the traffic flow, man."
"Well back up," Megan whispered.
But Corey couldn't. That would be so craven.
The skinheads approached Stanny's side. He gaped through the car window at them. The taller one, whose name was Ron, judging from a tattoo on his forearm, said, "You said _what're'those_ '? You talking like we're..."
"Things," said the other one.
"Shut up, Bumper," said Ron. Ron and Bumper. "Like we're things."
"He's got two _things_ in the back of that car," said Bumper.
Corey could feel Wade go very still.
"Wade?" Megan said softly. "Don't."
"Hey _fuck_ you, you pencil-dick Nazis!" Wade said, rolling his window down.
"Get out the fuckin' car," Ron said. "If you're done makin' _crack babies_ in there..."
Bristling, Wade started to get out. "Mother _fucker_!"
Corey reached back and slapped the lock down on the door, so that Wade couldn't open it for a moment—then put the car in reverse, started the car backwards.
THUNK. "Whoa!" Stanny yelled. "He kicked a big fuckin' dent in your car!" He leaned out the window. "Hey 'Baldy'! You just dented the sheriff's kid's car! Your ass is—"
Corey jerked the car to a stop, and grabbed Stanny's upper arm, yanked him away from the window. "Don't ever fucking say that again, Stanny! I mean it!" Shaking with fury. "Don't say that shit!"
The skinheads were laughing, flipping the bird, shouting _"Nigger lovers!"_
Stanny blinked at him. "What? I was just... I mean, you are the son of the—"
"That's _why_ , dumbshit. Because I am. Just... just forget it. Just shut up."
"What- _ever_. Fuck, people are telling me to shut up all night."
Dutifully, though by now he had thought better of it, Wade said, "I think we should fucking go back there and kick their Nazi asses."
Megan looked a request at Corey.
Corey nodded, and swung the car screeching around the little restaurant, getting angry honks as he forced it into traffic, and headed away from town.
"Where we goin?" Wade asked.
"I don't know. Mexico. Guatemala. Tierra Del Fuego."
"Makeout beach!" Stanny shouted.
"You fuckhead," Brad said to himself. "You fuck."
He stopped the van in the middle of the highway. There was no traffic. The road was so dark. The car trembled in the darkness.
He realized he had been driving without the headlights on. He switched them on, and turned the van around, headed back to get Angela.
He had to apologize and take her home. But fuck. Why couldn't she unbend a little? Why couldn't she reach down inside herself and find _something_ for him. He needed to be inside a girl. He needed to be close to someone. He was so fucking horny. Why couldn't she just...
There was someone in the road up ahead. A ghostly form in the road. A woman. It wasn't Angela—it was a grown woman and it looked like she was almost nude. There was a sort of diaphanous veil around her; her body outlined in voluptuous silhouette—almost cartoon-voluptuous, like the red-haired babe in that old Roger Rabbit movie. _I'm not bad, I'm just drawn that way_.
The woman stepped to the side of the road, about seventy-five feet in front. Seemed to wait for him there.
Then another car was coming toward him, with its brights on, its glare making him cover his eyes...
He had driven past where the woman had waited by the side of the road. He stopped, and turned to look through the back window. He couldn't see her. He put the van in park, got out and looked. She wasn't back there.
"No fucking way. She was just right back there..."
Maybe the other car had picked her up. He got back in the van, turned it around, going slowly back the way he'd come. Didn't see her. Faster and faster, thinking of trying to find the car that had picked her up. Then he saw a flicker of something white trailing after a woman's shape on the road just out of reach of his headlights—he jerked the wheel, yelled, "Shit fucking piss!" as the van fishtailed, almost rolled, then bore down on a telephone pole. He saw the pole rushing toward him; he jerked the wheel over hard, hitting the brakes. The van hissed and ground over gravel that pinged against its underside—and stopped, one wheel in a ditch.
Heart pounding, he thought: I missed the pole. By a fucking inch. The pot and the booze.
He got out, and looked for the woman. "Anyone there? Lady?" No one there, no answer.
_I'm seeing shit. Got to get straight... sober the hell up..._
There was a truck stop, he remembered, about a mile back. Once he got the van back on the road.
The full moon poured its pale reflection over the sea's white-flecked indigo; some of it spilled bluewhite onto the wet yellow sand.
A hundred yards from the surf, Highway 1 ran along the beach, winding with the contours of stony headlands and little bays. A small woods of oak and manzanita and wind-twisted juniper stretched a thatchy finger along a creek that bisected the beach and ran shallowly, with many little islands of sand, into the sea.
Corey and Megan and Wade walked in a small group along the creek, down the little dirt trail toward the beach, Stanny loping about back there, playing with their only flashlight. Spinning with it, sweeping the beam into trees, under bushes. Spotlighting Wade's back: "Jailbreak jailbreak! Ow-oogah!"
"That's hysterical, Stanny."
"I wish I had a laser pointer! I could like, signal people hiding out here..."
"Those things are obnoxious," Megan said.
"What amazed me," Corey said, looking up at the moon, "is that apparently the Pizza Hut company was going to use a giant laser to project 'Pizza Hut' on the moon. Or to cut their logo into the moon."
"Bullshit," Wade said, pausing to skip a stone on the creek.
"No, it's true."
"It's twoo, it's twoo!" Megan said.
"Her mom made her watch _Blazing Saddles_ ," Wade explained.
"I don't know if they were going to cut 'Pizza Hut' into the moon permanently or project it," Corey went on, "but either way, it's insane. I mean, I'm not Walt Whitman or even Annie Dillard—"
"You're not a bearded gay guy from two hundred years ago or a lady journalist?" Megan interrupted. "Coulda fooled me."
"— _but_ ," Corey went on, determined not to laugh at that, "I still find it, like, all—"
"Say it, Corey," Wade said. "Use the word! You know you want to!"
"Okay, fine: it's _fucked up_ that anyone would _want_ to do that. What's next, a giant Amazondotcom billboard on the top of Mount Everest? I mean look at the moon—they'd put a billboard across that? In the middle of the stars? People have no sense of... of..."
"Natural beauty," Megan said, nodding, graver now. "You're right. People'll ruin anything for a few bucks."
"Actually I think I read there was a plan to put some kind of advertising banner in orbit too," Wade said. "Serious. So, like, even the stars—"
"Is that?" Corey squinted. Pointed. "Who is that?"
The other three turned to look. A woman—or a girl—stood silhouetted against the sea. Standing on the beach about a hundred yards away, her back to them. She turned—
"Angela!" Corey burst out.
She walked toward them, slowly and dreamily. Corey noticed that her hair was disheveled; the sleeve of her dress torn. She wasn't wearing shoes or carrying them.
Corey sloshed across the creek toward her, hands in his pockets. She stopped a few yards away, so he did too. "What're you doing out here alone, Angela?"
She looked at the sea. "I came out here alone. I didn't know it. I thought I was here with Brad..."
She sat down on a driftwood log half buried in the sand. Wade and Megan, walking up with Stanny, looked at her, then looked at Corey. Angela just looked at her feet.
"Meg," Wade said, "let's take a romantic walk by the ocean."
"Yeah," Stanny said. "Let's go down by the water and stuff!"
"Not you, dipstick."
"Stanny can go with us," Megan said. "No way I'm going out there alone with Mister Hands—not tonight. Mister Hands is way out of control tonight."
Wade made his hand open and close from the side like a sock puppet. " _It's me, Mister Hands! I Just wanta take a little walk on you, baby!_ "
"Just come on, retard."
The three of them trudged off. Corey sat down beside Angela. "What'd Brad do? Did he—was it... rape?"
She was silent for a long moment. The ocean breathed out its brine; the surf hissed. Finally she said, almost laughing and almost crying, "No. He was right on the edge of it. He was just such a... he was just totally revolting. We're talking... like... intense revulsion, all right? He kept pouring that stuff down me..."
"What stuff?"
"Jack... It was Jack..." Her voice trailed off. She stared at him. She gulped, then she turned suddenly away, bent over double, and vomited behind the log.
Brad felt ill. But he had to get some coffee down him.
He and the counter dude were almost alone in the truck stop. The humming fluorescent lights were too bright, and they made the skin on Brad's hands look sickly white. He stared at them, as he sat at the counter, till the Iranian guy in the grease-stained apron said, "What I get for you?"
"Coffee, bro. Just coffee."
"Sure, yeah. Soon I go out of business with just coffee."
He got the cup for Brad, sloshed coffee in it. There was a little portable TV next to the pipe shelf, behind the guy, with a foreign language channel on it. A lady with a transparent veil was shimmying her head from side to side and singing in some Middle Eastern language. The beat was disco. It was, like, disco Iranian stuff, Brad thought. She was pretty fine, though, what he could see of her in that drapery thing she was wearing.
Then he heard a woman sobbing, somewhere behind him. He turned and saw a woman in a black dress, sitting in a booth, her back to him. Her shoulders shaking with sobs. She wasn't dressed the same. But it could almost be...
He picked up his coffee cup and—wondering at his own nerve—walked over to her. "Um—hi. Are you, all—I mean... you okay?"
She turned and looked at him, almost surprised. Her face was tear streaked but her eyes weren't red or puffy—they were beautiful big black eyes that went with her glossy raven hair, her heart-shaped face, full lips, pointed little chin. She might've been thirty, or thirty-two. Her black dress was low cut; brimming with abundant breasts. Beside her on the seat were black pumps, caked in mud. She sat in the booth barefoot, a cup of coffee in front of her.
He forgot about feeling queasy and drunk, looking at her. Warmth surged up from his crotch and he felt strong again.
"I'm all right," she said at last. Her voice husky. She had a slight accent. Italian? "It's just one of those times when just one thing too many happened..."
"I've been sort of going through some shit like that. You need somebody to, you know, talk to?"
"No, no I'm all right."
Brad nodded. Looked at her cleavage. Made himself turn away. "Okay, well..."
"But—maybe it would be a good idea. Someone to talk things out with."
He was in the seat across from her fast enough to spill his coffee on the table. He muttered, "Sorry," and blotted it with a napkin.
"My name is Ilira," she said.
"Brad." He sipped coffee, for something to do. So he could look into the cup and not at her spillway of cleavage.
"I must look strange, Brad, dressed like this... in a place like this. We were going out—my husband and I—and then..." She shook her head. "I just couldn't stay with him."
Brad grimaced. He thought about Angela. "Well uh—what'd he do?"
"It's..." She glanced at the counter guy, and lowered her voice. "It's a very private thing, really. But maybe it's something I can talk to you about because I _don't_ know you. I can't tell my friends, you see. It would be—"
"Hey, Ilira, I like totally understand." Brad noticed the Iranian counterman wiping down an already clean table nearby. Eavesdropping. "You know what? We'd have more privacy in my van..."
To his surprise, she accepted the offer.
Brad's parked the van at a turnout overlooking a cliff. A place for tourists to take photos. And now...
He couldn't believe his luck. He had a totally luscious babe in the back of his van, drinking his liquor, smoking his dope. Hell, Angela would be all right.
They were sitting up against the metal walls, side by side on the black-sheeted mattress, legs stretched out straight. She'd taken a couple of puffs on his brass pipe, and was drinking Jack from the bottle. He was concentrating on the pipe and now gazing brazenly at her chest, sensing that she didn't mind.
But he remembered you're supposed to talk to girls. Seem interested. "So you walked across all those fields barefoot?"
"I had to get away from my husband. I couldn't stand another second. He would pretend he wanted me and then—he'd reject me. Push me away. Once he literally pushed me out of bed. Just to humiliate me. It was some kind of little game in his head."
"What a fuckin' homo."
"Yes. If he were a real man—he'd have taken me. But not these games..."
"For real."
"Brad—you are very understanding." She gazed into his eyes, her lips gently trembling. Thank you for listening to me."
"Hey, that's all right. I mean—I like listening to you. Then... I can look at you. And you're, you know, beautiful."
"You think so? My husband didn't think so..."
Brad snorted. "Dude's crazy."
She smiled sadly and reached a small white hand to touch his cheek—and then drew her hand back, perhaps afraid, he thought, that he too would reject her.
He reached out and stroked the edge of her lips with his thumb, tracing all the way around. She kissed his thumb. His hand lingered and she kissed the other fingers, one by one. He went for it, kissing her on the mouth, the neck—then down that long, brimming cleavage.
She groaned and wriggled out of her dress—almost shedding it like a snakeskin, but fast, weirdly fast. "So funny to find a real man—in a boy," she said huskily, as she wriggled. "Maybe... I can teach you some things..."
"Yeah," he said. "Why don'tcha do that... that'd be cool..."
She was already nude and they were both yanking his clothes off, panting as they did; he'd never gotten undressed so fast.
Then she drew him into her arms, and rolled till he was on top of her, almost instantly drawing his sex into her—it was like her vagina had reached out and sucked him into her. Like it was a prehensile _thing_ that could grasp and pull and...
She pressed his face into her breasts, and he gasped into the strange smell of her—perfumed and yet there was a faint scent of something else, like a burning leather—and he felt the wet depth of her sending smokestack lightning up his dick and he gasped out, "Jeez, Ilira, shit, Jeez, I don't believe it, this is so..." The pleasure was almost too strong, like an electric shock going all through him. Like it was going to carry his whole life on its current, right out of him. "Oh god I'm going to..."
"Yes, do it. I'd like that. Please. Let it out..."
He arched his back as the orgasm shivered out of him, laughing as he came—
And then he froze, with a click, in place.
His back still arched, his arms holding him up. Locked up, stuck. Tried to talk. Couldn't move his mouth. Couldn't move, not an inch, as she began to _change_ under him; first he just felt her changing, like a nest of writhing animals in a silken sack under him, and then he saw it, saw her rippling, becoming herself, a truer self, and an inhuman mouth locked over his and began to suck the air out of his lungs.
She didn't stop there.
He was screaming, but only inside his skull; only the way a mind can scream. That was all that was left to him.
Angela was sitting up front beside Corey; Stanny was crammed in with Megan and Wade in the back.
"Stanny," Wade said, "You keep your gas to yourself. Just clench it up good."
"Yo I'll try but I'm like only human."
"If you were human I wouldn't worry about it."
Corey, driving down the highway, looked at Angela, then back at the road. "You feel any better?"
"A little," Angela said, hoarsely. "Getting me that water helped. That was nice of you."
"Spend time with Brad and you think it's real nice if somebody gets you water."
She laughed mirthlessly. "I guess."
"So where we going?" Stanny said. "Let's go to the arcade. I want to try my theory."
"What theory?" Megan said.
"You're going to regret asking that, Megan," Wade predicted.
"See, I have this theory," Stanny went on eagerly, "That chicks, if they're, like, playing videogames a lot? That they get all... zoned out. And, like, pliable? And I can—"
"Stanny?" Corey said sharply. "That's not what Angela needs to hear right now."
Angela glanced at Corey with a flicker of irritation. "I can deal with stuff. I'm fine."
Up ahead, Corey saw a van, parked in a turnout. He recognized it, and hoped Angela didn't notice it.
But as they came abreast of it, she said, suddenly, "Pull over! Stop! That's Brad's van!"
"So? You really wanna finish that date?"
"No—something's happened to him! Just stop! Please!"
Corey ground his teeth and the brakes, and skidded the car stutteringly to a stop. He backed up till the road was wide enough for a careful U-turn. The yawning space over the cliff's edge seemed to swing hungrily toward them as he brought the long Impala around. Then they were down the road, and pulling up beside the van.
"He's probably just sleeping it off," Wade said.
"No," Angela muttered. "There's something..."
She stared at the van, and shook her head.
Corey sighed. Brad was such a flaming shithead. He'd abandoned her on the road. And yet she was out here looking for him.
He got out of the car, and walked over to the van. He hesitated, looking out to sea. An unseasonably chill wind was coming off the sea, now, pricking up white caps on the dark waves.
He shivered, and banged on the back door of the van. "Yo..." Shithead, he wanted to say. But he called, "Yo, Brad! You in there?"
Only silence answered.
Corey found the handle, and pulled. It was unlocked. He swung it open and the overhead light came on automatically inside the van.
Between an empty, overturned Jack Daniels bottle and a brass pot pipe, Brad lay on his side, naked, with his spread knees drawn up, his arms sticking out as if fixed in the act of pushing something away, his hips thrust forward, his outthrust penis...quartered: cut as with a kitchen implement into four quarters, like a pickle sliced lengthwise. His head was raked back on his neck, his face like a man in high gravitation, spun in a centrifuge, the flesh itself recoiled toward the back of the head, the teeth and their roots exposed. His eyes were missing. And Corey could see the inside of Brad's skull through them: hollowed out.
Trying to contact some reality in what could not be real, Corey reached out and touched Brad's leg—just the merest touch, but the papery, hollowed out leg crumpled under his fingertips, crackling, and Brad's whole body rocked all at once, wobbling in place like a thing of stiff papier-mâché.
Corey looked closely at the hands, the legs. Each little hair was there. It wasn't papier-mâché.
Corey turned and walked back to the others, finding, somehow, that he was having to work hard at walking straight, like the one time he'd been drunk. Almost afraid to go back to the others; afraid he'd find them turned into dead, stick-dry things the same as Brad.
But they were solid and breathing and murmuring to one another as he returned to them. Stanny saying loudly, as Corey leaned against the car, "Whoa—looks like he's gonna hurl."
The red light flashed on and off, on and off. Hushed male voices. Tires crunching on gravel. Sirens coming and going.
Corey and Angela sat in the back of the patrol car. He was holding her hand—or she was holding his.
"Maybe it was some kind of..." she began.
"No. It wasn't a...dummy or something. You'd have to see it to know. But I don't want you to."
He looked at the ground outside the window: pebbles throwing long shadows in the red flashing light of the patrol car. Red light and streaky shadows, alternating with unbroken blackness.
"Are you really sure?" she asked again. "I mean—Brad might be capable of some kind of...practical joke..."
He looked at her. He took a deep breath. "Be right back." He patted her hand and got out of the car.
It seemed to take a long time to walk to the ambulance. The paramedics were just then loading Brad's body in the back. Corey could see Brad's contorted body—if that's what it really was—frozen in the same position, sideways on the gurney, his arms sticking out from under the sheets, hands hidden in paper sacks that were rubber banded around the wrist. That was something the forensics people did, he knew, to preserve DNA evidence that might be stuck under the fingernails. But it was frighteningly absurd, now. Paper sacks on his dead hands.
The paramedics looked scared.
Corey had been to body retrieval scenes twice before, writing an essay about his dad's work. One car accident, one murder. Both times the cops had made grim little jokes, and talked about football as they went about their work.
This time they hardly spoke. They just stared at the gurney. And looked at the van. And then back to the gurney. Corey had already given a statement, and there was no reason for him to be here, at the ambulance, but no one told him to go away.
Silbido's only plainclothes detective, a tall African-American man with salt and pepper hair, Ganrich, stood with his arms crossed across his gray suit jacket, hands tucked in his armpits. He stared at the state forensics technician, a young Asian woman in a white coat, who was leaning into the van, picking up bits and pieces of...what?...with tweezers, putting them in little jars. The back of her coat went diluted-red, and moonlit-white, and red again in the flashing cruiser light.
Why couldn't they have blue as well as red lights on the bars, like some other cop cars? Corey wondered irritably. He wished they did, now.
Corey felt his father looking at him, probably wondering how to help him through this.
Ganrich cleared his voice, as if to warn people he was going to speak, and said, "Barry?"
"Yeah, Al?" Corey's dad answered.
"How much you think that boy's body weighs now?"
"Maybe an ounce."
"If that. It's all got to be faked."
"That kind of detail?"
Ganrich shrugged. "Then you figure—what?"
"I don't know. A cult? Maybe some kind of chemical drying process. Somebody works for a chemist somewhere, discovered something? Kills you, leaves you like that...? Maybe?"
Corey felt a little better. Maybe this thing could be explained.
Barry went on, "His blood is...gone. But...they weren't after the blood."
"How you know that?"
Barry walked to the back of the van; Ganrich followed. Corey trailed behind. Halfway there, Barry stopped and turned—making Ganrich bump into him.
"Sorry, Al. Corey? Do me a favor, son, wait over with Angela. She's probably scared."
Corey nodded. But when the two men went on to the van, he came close enough to see and hear. The forensics tech nodded that she was done and went to her own vehicle, carrying her jars. Barry grabbed the mattress on the right side, bent it way back, exposing the floor of the truck.
It was all blood. A mass of coagulated blood.
Lots of it had soaked through the mattress, some of it dripping down now from the bent back bottom-side, into the puddle; more blood dripped off the back of the van, forming a new puddle on the ground.
**PART THE SECOND: PERFECT WOMEN**
FOUR
"Girlfriend," Megan said, "We are gonna work it out. This is gonna feel good, for real now. Trust me."
They were in the basement family room, at Angela's house. A Saturday afternoon. Angela sprawled on the sofa, Megan standing by the boombox she'd set up on the pool table. Both of them in leotards and ballet slippers.
"I can't," Angela said. "I feel so weird about Brad. Like I shouldn't have left his dumb ass alone."
"That is such bullshit, and you know it, girl."
"And... if he'd ran his car off a cliff or something at least I could get used to the idea. But no one knows what happened to him. They're saying it was some kind of incineration but it wasn't. I saw them put the body in the ambulance. It wasn't—"
"Angela? You have to get your mind off it. They'll deal with it. We're gonna do our act for the fundraiser. This's going to get you in another space, I swear. You have to put one foot in front of the next and just move on. You'll see why I call you Angel-foot." She smiled. "Now come on..." Megan hit play, a hip hop cut thumped the room like a giant was using the ceiling for a drum; it was a female singer alternating with rap. Megan literally put one foot in front of the next, made it into a dance, and then danced over to Angela. She grabbed her wrists, pulled her to her feet, swung her around—leaning far back herself, and pretending she was about to let go so that Angela squealed and laughed. She pulled her close, spun her around like a fifties dancer, and they separated, both dancing—Angela beginning to smile.
Then the music cut off.
Gail, wearing a gold and white dress, had cut off the sound. "Sorry, girls. Angela, your dad will be picking you up in twenty minutes."
Angela looked blankly at her, thinking that her mom seemed to take some kind of mindless joy in breaking in on her when she was doing things she liked to do. It didn't matter what they were—harmless things—all that mattered was that Mom had the power to interrupt them.
"Mom? I have a guest? We're rehearsing for the benefit?"
"You've had an hour already."
"We just got started."
"She wasn't feeling good," Megan put in. "You know—"
"I mean," Gail said, pointedly ignoring Megan, "why is it I always have to push you into things?"
"That's totally what I'd like to know."
"Angela—we need the child support. We need those checks from your dad. You're going to lunch with your father. It's something you can do for this family."
The room seemed to darken around Angela, when she heard that.
"You _bitch,"_ she said, surprising herself.
Gail stared at her. "I..." She looked at Megan, who was pulling on a shift, and pursing her lips to keep from laughing. "I'll pretend I didn't hear that this one time...since you have a guest. Just—get ready to see your father. Please."
Gail went to the stairs, and hurried up, her face strained.
"Since you have a guest?" Angela muttered. "Since she wants me to go to lunch with Dad, she means. She doesn't want a big confrontation—I'd be less likely to go then. That's all she cares about."
Megan gave Angela a hug—Angela tried to hug her back. "Later, Angel-foot." Angela couldn't quite remember when Megan had started calling her Angel-foot. "It'll be all right. I'll call you."
Corey was sitting on the front steps, spinning the basketball on the tip of his finger, not able to keep it going more than a few seconds. He let it fall between his legs, it bounced on the step and he tried to catch it on the tip of his finger.
Wade came down the sidewalk, up the drive, and scooped up the basketball. "Gimme that thing. You are lame, dog. Watch a master." He spun it on the tip of his finger, then shot at the basket—and missed.
Corey jumped up, caught the rebound, hooked the ball past Wade's block, and made the basket.
"Corey's showing off," Wade said, catching the rebound. "Trying to make me feel bad just because I can spin it better than you."
Corey turned to see Stanny walking up the sidewalk.
Corey exchanged glances with Wade.
"Yeah," Wade said, before Stanny got there. "I admit it. I invited him."
Corey shrugged. They'd barely hung with Stanny before That Night. But they felt a connection with him, now—and they felt a kind of responsibility. Brad had been just about Stanny's only friend.
Corey sighed. "Hi Stanny."
"We going to the arcade!" Stanny said. "You down with that?"
"Am I _down with that_?" Corey asked, laughing.
"He thinks if he hangs out with me, he's black now," Wade said.
"It don't belong to black people no more," Stanny said. "White people took all that shit. You all got to make up some new slang and shit."
"We'll take that under advisement," Wade said.
"The arcade?" Corey said. "That the best we can do?"
"Meeting Megan there," Wade said. "She likes that dance-floor game where you stomp on the lights."
"Megan?" Corey looked at Angela's house. "I thought she was over at Angela's?"
"Angela's Mom tossed her out. She went home to change."
"Must've been when you were inside, like using the bathroom or something," Stanny said.
Corey gave him a sour look. He didn't like Stanny knowing he'd been sitting here watching Angela's house.
"You coming or what?" Wade asked.
Corey swiped the basketball from Wade, dribbled out to his car. Tossed the ball over a shoulder, looking only with his peripheral vision. He could feel it swish through the basket.
"Whoa—" Wade said. "You..."
"I know," Corey said, opening the car door without looking at the basket.
"Now you _are_ showing off."
"I know," he said again. "Come on. The arcade. But you're paying for some gas."
"Man I got to have some money left to play Carn Evil!"
Wearing a powder blue blouse and creased black slacks, Angela met her dad, Marty, in front of the restaurant: a natural foods place called The Curious Quiche.
He was a short, stocky man, with small eyes, a wide mouth; he wore Calvin Klein jeans and a sports jacket, cowboy boots. Her dad's coiffed, medium-long brown hair was thinning even more, she thought, as he walked up; and somehow the long sideburns only made it more noticeable. She wanted to tell him that his haircut was pure 1980, but she never quite did.
"Therrrrre she is!" Marty crowed, arms outstretched. "Sleek, blond, and very very cool. How ya doin' miss Angie?"
She let him kiss her on the cheek. "Hi Dad."
"Your mom just dropped you off, huh? Well she's had her check this month. No need to chat with me, right? Let's go in."
All sensual female perfection, but with weight-lifter muscles, the warrior woman leapt through the castle door into the courtyard of demons, did a cartwheel, coming down on the demon-possessed Flameknight with her dagger-boots—he rebounded, snarling, and struck with his spiked knuckles. She flew back with the impact but flipped in the air, landed neatly on her feet, and immediately fired a soul-blast at him. But the Barbarian whipped three daggers at her in quick succession from his blade-bandolier, and she went down, her breasts heaving and seeping blood.
"Shit," Megan said, "I'm dead."
"You are flat dead, girl," Wade said.
The video game counted down _15, 14, 13, 12_... _Insert Coins to Continue Play_...
"Fuck you, machine," Megan said, "you give me like thirty seconds of play for one token. So, Wade, you get the Coreymeister to come out? Or is he still, like, having a watch-Angela vigil?"
"He's here. But he's still trippin' on her."
"I bet he's playing that western game."
"You bet right. Come on."
The arcade was a roar of competing sound effects, explosion-images, whipping shapes—beeps, blasts, screams, mocking voices, so variegated and numerous they were almost white noise together. You had to listen hard to pick out individual games. Groups of twelve-year-old boys were clustered around Savage Kicker and Sniper 2000; there were two girls flying Star Wars Pod Racers side by side. "I like that Pod Racers," Wade said.
"What?"
"I said I like Pod Racers."
"What?"
"Forget it."
They found Corey firing a plastic light-pulse gun at a wheezing old two-dee pixel-rezzed image of a cowboy who was shooting back with a cartoon six-shooter. "You're dead, podner," the gunfighter was saying. "Better reload and try again."
"How do you reload if you're dead?" Corey asked him.
"Ten seconds, Podner, to keep your score!"
"Yo Corey," Wade said. "You wanna do some air hockey?"
"You just wanna do air hockey because you always beat me."
"And your point is?"
"Yeah okay. Air hockey."
"What?"
"I said yeah okay! Where's Stanny?"
"Stanny? Playing Skull Crazy."
Skull Crazy. Corey shuddered, thinking about Brad's hollowed-out head. An empty skull with all the skin and hair on it.
He pushed his mind away from the thought.
They threaded through the maze of games and kids to the other side of the arcade where air hockey and Bowlerama were. As they went, Corey looked around, half expecting to see Angela. And thinking he saw her, for a moment, two rows of videogames away—but looking closer he saw it wasn't her. It was a teenage girl with Cleopatraesque black hair; tight jeans and a tie dyed T-shirt cut off to show her stud-pierced navel.
How could I think a girl with black hair is Angela? Corey wondered. She hardly even looks like her. Sort of the same figure is all...
But, trailing after Megan and Wade, Corey found himself drawn to watching the girl, as she walked the other way down the farther aisle. She went behind a videogame machine, and when she came out on the other side, she had different clothes on. She had on a skirt now. A short black-leather skirt.
Okay, so there were two girls, twins, one following the other. He dropped back and went down a cross-aisle to the one the girl was in, annoyed at himself for it but unable to keep from it.
He saw her halfway down the aisle, in front of him now—there was only one girl. No twins. The black skirt, the tie dyed top—which, as he watched, seemed to shrivel, to melt, to darken and cling, becoming a red tube top...
Her wrists grew studded leather bracelets. Inverted crucifix ear rings sprouted from her ears. As she turned a corner, turning her profile to him she saw her makeup crawling across her face, going from soft to strident, sexy; her hair redesigning itself, shifting like a living thing, growing shorter...
"No way," Corey said.
The light in here was always shifting with the images on the videoscreens. He'd seen some kind of illusion. Had to be. But he followed her anyway.
Stanny was shooting skulls from a cannon at the dancing topless monster girl in the Martian cemetery. You had to shoot the skulls between her flashing legs, into a grave. He almost had the grave filled up—when he felt someone looking over his shoulder. He turned to see his dream date.
Spike heels, net stockings, short tight black leather skirt, red tube top; her hair teased up rock-n-roll wild. Her makeup a flaring challenge to all comers. Bee-stung lips slightly parted as she gave him an unambiguous look that started in her groin... and ended in his.
"Aren't you Stanny?"
"Yeah uh, how'd you, I mean..."
"I saw you play at a dance. Guitar. You were hot. A _hot_ guitar player. My name's Sira."
"Sira. Whoa. That name's all... it's hella cool. Beautiful as shit. I mean..." He winced. Vaguely aware that Corey was standing to one side, watching and listening with some kind of disapproval. But it didn't matter. It was like he was in a bubble with Sira. He couldn't hear anything, in the arcade, hardly at all, except her. He heard every nuance of her voice in this loud place. And he never took his eyes off her.
"I knew what you meant, Stanny. I get the same way—sometimes I feel dumb, trying to say what I mean. Does that ever happen to you?"
"Does that—? Are you kidding? It abso-fucking-lutely happens to me! _Hell_ yeah!"
"Stanny..." Corey's voice seemed faint; he seemed a long way away.
"It's funny how we can talk...you and me, Stanny," Sira was saying. "What's that song say? 'Lotsa people talk but few of them know.' That's how it is. But we just met a minute ago. I used to play this game a lot so..."
"You too! Whoa! Hey—you wanna play it now? It can be two player!"
"You know what I'd really like? To talk. And... I don't know what else. But it'd be...somewhere we could be... you know... alone."
He gazed at her, swaying, his breath caught in his throat—and that's when that bastard Corey intruded.
Corey stepped in and pulled Stanny two staggering steps back. "Stanny? We got to talk."
Stanny turned angrily to Corey. "What? Fuck off, Corey—I mean no offense, bro, but I'm cruisin' out of here with—" He turned back to Sira. Who was gone. "FUCK! Where is she? She was here! You scared her off, man you fuck!"
"Look I'm sorry but—"
He spun in place, peering between videogames, trying to see her. "Did you even see where she went?"
"No—I sort of saw her melt away out of the corner of my eye—"
"This rots! I'm gonna find her, man!" Stanny jogged off, zigzagging from one side of the aisle to the other, looking for her.
Corey felt a tap on his shoulder. He turned to Sira.
She smiled. Looked at him unblinkingly with big dark eyes. Her eyelashes were waving like the legs of overturned centipedes. "You and me. What do you say?"
"What?"
"We will, you know. When I'm stronger. One thing at a time, though. Right?"
Then she stepped back from him and moving blur fast she slipped off between two games, where a broken one had been removed. He trotted to the gap, and saw a crowd of teenagers, talking and laughing on the other side. And couldn't see her. But was that her—that girl walking away, on the other side of the crowd, with the short blond hair?
He shook his head, and went to find Stanny.
Outside, in the little city park next to the one-story concrete building containing the arcade, standing beside a litter of old cigarette packs and empty beer cans, Ron and Bumper were arguing about whose fault it was they got burned on the blunts. "There's hardly any shit in this fucking dope smoke at all, man," Ron was saying. "You're the one who said you knew a guy—"
"You're the one said you didn't have time to test it, Ron," Bumper protested.
Ron didn't answer. He was staring at a girl. A pale, blue-eyed girl with short, almost crew-cut blond hair, and an iron cross tattooed on her cheek. She wore a Skinhead Nation T-shirt, ripped down to her navel so that her breasts—with a swastika _scar_ between her breasts, just above the sternum—all but spilled free; cammie pants, and a barbarian's furry boots. Faintly, Ron thought, she was glowing.
Why shouldn't an Aryan goddess glow, after all?
She walked boldly up to the two staring skinheads, and gave the power-fist Seig Heil.
Numbly, Ron returned it. Bumper just stared, mouth drooping, almost drooling.
"That swastika scar...Your boyfriend do that?" Ron asked.
"I did it myself. A pocket knife."
"Whoa. Nice," Bumper said.
"You Aryan Nations Auxie or what?"
"White Brotherhood. Just got into town. I'm a traveler, I came in on the train. Had to leave Sacramento. I killed a nigger." She shrugged. "He was puttin' his hands on me."
"No shit?" Ron asked.
"You'll read about it tomorrow, if you can read. I'm Sister Sira." She looked them over critically. "You two just might work out. I need a cell to work out of. There are things to do out there. Out there... and in private. We have to build a new world. But I figure, we ought to get drunk first."
"You haven't touched your wheatgrass shake, babe," Marty said. "It doesn't sound like it'd taste good but it does."
"No it doesn't," Angela said, distantly, looking around the health food restaurant, with its wooden tables and potted plants, so she didn't have to look at her dad.
"It restores the tissues."
"It looks like it should be _wiped_ with tissues."
He laughed. "Okay, whatever. Hey that's my girl, all the way. Come on, eat your quiche, anyway."
"I don't like quiche."
"You know what—I am being so insensitive. I forgot about your friend. Tell me about that—about the...your date. A boy died after you broke up with him?"
"We weren't in any kind of relationship you break up out of. I just went out with him, like, twice. He dumped me somewhere and then...somebody killed him. Or...he got poisoned or...nobody knows."
"So how do you feel about that?"
"I feel...like not talking about it."
He sighed. "You could, you know, make an effort, miss Gloom Rocker."
"I'm not a Gloom Rocker either."
He nodded sagely. "You're probably deep into your self-definition stage."
"Dad—please. I'm not one of your whiners paying you a hundred bucks an hour to tell them what they want to hear."
His eyes narrowed at that and he leaned back in his chair, tapping his fingers on the table. "It might not hurt you to have someone to talk to. I met someone—she talks to me. Helps me."
"Sounds hella cozy."
"Okay. Fine." An infantile pleading crept into his voice as he went on, "I _miss_ you, you know. Maybe you miss me too. We were special-close—not every family—"
She turned to look squarely at him. "Could I go home now? I don't feel good."
"You used that one last time we went out. But yeah, sure, have it your own way."
He stood up, threw his napkin on the table, snatched up the check and went to the cashier.
The refrigerator clicked on, began whirring to itself in the empty house in obedience to its timer.
There was no one in the front hall of Angela's house. It was lit only by a light at the top of the stairs. The shadows of the bannister posts fell across the wall. The shadows quivered, bent, and shifted. Dust swirled up—and settled.
Then the door burst open and Angela rushed in. Her father called from the front steps. "Angela! Wait!"
He came up the stairs; she slammed the door in his face, and locked it.
She stood with her back to the door, hearing his steps recede, outside. She said to the uneasy shadows: "I want him gone. I want him erased from the whole universe."
FIVE
Fifty-seven minutes after midnight, in a basement apartment. Two mattresses on the floor, in opposite corners, where Ron and Bumper slept in sleeping bags. Pile of dirty clothes between. A metal sink and a stove that didn't work, both heaped high with old fast food boxes, pizza boxes from the place where Ron had worked part time before the yellow-niggers he'd delivered to got him fired. The walls were spray painted with slogans: WHITE POWER and BREKE THE ZIONST MACHINE and PURITY MAKES PURITY. There was a naked bulb on a shadeless lamp sitting on the card table where the dope paraphernalia was.
And there were two wooden chairs with a half-naked girl standing on them, one boot on each chair seat.
"You want me, peon?" Sister Sira asked, chuckling. "Then come here."
She had pulled off her T-shirt and was standing on the chairs side by side so her legs were spread far apart, just like in Ron's fantasy, and—also like the fantasy—Ron was kneeling in front of her.
How did she know he wanted this stuff? How did she know he wanted to worship her? How did she know he wanted to stand up and bend over and bury his face in her crotch?
She was wearing jeans, but she reached up and she _parted her jeans at the crotch with her fingers_ , just like opening a curtain. They must've had some kind of hidden clasp in the fabric. And there was her quim, opening up like a red flower, redolent and damp. "Come and get it," she said, like dialogue in porn, and she grabbed the back of his head and pressed his face into her crotch—just as Bumper came down the stairs carrying the gallon bottle of red wine.
Bumper stared and muttered and opened the bottle. He drank, without taking his eyes off them, spilling the red stuff liberally down his chin and chest. He watched as Ron lapped and chewed at her, as Ron pulled his thing free and jerked it.
Suddenly she gripped Ron's head with both hands, and, defying gravity, brought her legs up without taking his face from her crotch, wrapping her legs around his neck so that he was pulled over forward. They crashed—Sister Sira laughing all the time—into the two chairs, smashing them into sticks. And still she kept his face locked in her crotch.
Bumper stumbled down the stairs, unzipping his pants with one hand, the other sloshing a trail of wine. He came to kneel beside her and she grabbed his ass and pulled his crotch to her face and took him in her mouth as she kept Ron half smothered below...
And as Bumper's back arched in ecstasy he saw Ron begin to shake, to squeal, and saw blood squirt out from around the side of Ron's face, upward. Was he biting her? But it wasn't her blood, judging from the way Ron was struggling, flailing to get away, trying to pull free. She only pressed him closer, and something reached out from inside her mouth and grabbed Bumper's dick and got a real firm grip on it, then she started to change and he started to scream. The scream only lasted about ten seconds because after that he wasn't there anymore to scream, there was only the soft wet shell, wobbling in place, emptying out, gushing out from the little vents that'd opened around her mouth, and Bumper was falling with Ron, falling into death, and through it, and on into a place he hadn't believed in until exactly that moment.
It was a shirt-sleeves evening, just after sunset, when Corey drove Wade and Megan up Rancho Road. Megan was sitting up front beside Corey, Wade in back, leaning forward across the back of the front seat. "That's the place," Megan said, pointing. It was a ramshackle, two-story wood-frame house; once a farmhouse before sprawl had engulfed it. The house stood out from the others with its two rusting Chevys on cinderblocks in the front yard; the rusting remains of a Harley, like chromium bones picked over by a tattoo vulture; an entire car engine dangled on a chain from a portable hoist. The yard was stained with oil, cluttered with cans.
"That's him up there," Megan said.
A slack-faced boy of indeterminate age straddled the frame of an open second story window. A flap of cardboard that'd replaced the window was pushed back, elbowed out of his way so he could sit there, shaking something in his hand, over and over. Or just making it look like he had something in his hand. He wore cut-offs, and nothing else. His arms were whipcord lean, and overlong; his lower lip drooped; his eyes were curious the way a raccoon's could be.
Corey stared. " _That's_ the guy you want me to see? You're playing with me!"
"Hush, he'll hear you," Megan whispered. "And his name's Montrose."
"How did I get talked into this?" Corey muttered, looking up at the sickly boy.
"You asked!" Megan said, snorting. "You said you couldn't sleep, you had to know what really happened to Brad, you get all mysterious about something else you saw that you won't tell me about—"
"Yeah what the hell is up with that, Corey?" Wade asked. "Something about the arcade?" "You know what—I was probably just stressed out. I met this crazy girl there... People on drugs babble all kinds of things and she was probably stoned... and I probable didn't see what I thought I saw..."
"Which was what?"
"Which was never mind. Let's just get this over with. We came this far. Come on."
They got out of the car and hesitated, in the middle of the ragged yard, looking up at Montrose. He rattled something in his cupped hand again; Corey thought he glimpsed dice between the bony fingers.
"You remember me, Montrose? I came to ask about my mom's business, when I lived on this block, last year?"
"Whichun?" the boy asked.
Megan tilted her head to one side, at Corey. "Him."
Montrose looked at Corey, and rattled the dice; his bare, grimy foot swinging below the window. Then he stopped his rattling—and turned sharply to look at Corey. He rocked backwards in his seat till he fell into the house, with a thump audible even in the yard and began to shriek, in regularly spaced peals. Between shrieks he'd smack the wall up there, with something; hard enough that the cardboard quivered in the window frame.
The first peal of shrieking would go: "SHEEEEEEEEEEEEEE!" Piercing—the sound carried for blocks. Then in short barks: "SHE-SHE-SHEEEE! SHE-SHE-SHEEEEE!" Then sobbing and thumping. Then again: "SHEEEEEEEEEEEE! SHE-SHE-SHEEEE! SHE-SHE-SHEEEEEE!" Wet sobbing. "SHEEEEEEEEEEEE!"
Across the street, a heavy set old man with an aluminum cane shouted out of his front door: "Who got that boy started! Dammit you pay the woman and shut him up! I've got an invalid wife in here and she can't handle that screechin'!"
Other neighbors shouted. The boy went on, "SHEEEEEEEEEEE! SHE-SHE-SHEEEEE!"
"Jesus Christ," Wade said. "Let's get the hell out of here."
"Not till ya'll paid!" said Montrose's mother, if that's who she was, coming out of the house. Corey thought: Burnt-out biker chick. Then he thought: Don't make snap judgments about people on first sight. He looked at her more closely. And then he thought: Burnt-out biker chick.
"Hi Kizzy," Megan said.
Kizzy was skullish skinny. Her thinning brown hair was tied back in a single braid. She wore dirty white polyester slacks, plastic sandals, a Harley Davidson T-shirt. Homemade, fading tattoos ringed her neck: _Georgie, Bud, Shark, One Wheel, Slim_ , like a choker, as if she'd been making a collar for herself, somehow acknowledging that each one was another link in her chains. Her cheeks and forearms were speckled with tweak marks from picking at imaginary bugs. Her eyes wandered to each face, as she came, and to the sky, and to the neighbors, and to the car, and back to their faces. When she spoke, he saw that most of her teeth were gone.
Speed or crack, Corey supposed. Looking at the tweak marks, he decided it was crack.
_SHEEEEEEEEEEE. SHE-SHE-SHEEEEEE... SHEEEEEEEEEEE..._
Her eyes still bouncing, her hands wringing and then going to her pockets and then coming out and wringing, she said, in a faint southern accent: "Ya'll come to consult my boy, but then you already did, you already spoke, or he wouldn't be wailin' like that, not that he does that every time, he don't, but if you scare him somehow, why he'll start in, that's gonna cost you five dollars right there. Ten dollars, I mean."
_SHEEEEEEEEEEE. SHE-SHE-SHEEEEEE... SHEEEEEEEEEEE_
Ten dollars to talk to a psychic retarded kid, Corey thought. I am now officially a loser.
But he took the money out—he'd come prepared with four five-dollar bills—and passed it to her. There were little twisted gnarls were her fingernails should be.
"Can we go now that we wasted ten dollars?" Wade asked, looking around at the yard.
_SHEEEEEEEEEEE. SHE-SHE-SHEEEEEE... SHEEEEEEEEEEE_
"We didn't learn anything, Kizzy," Megan said.
"Will somebody shut that kid up!" the old man yelled.
"I'm sorry Mistuh Tuscano!" Kizzy called, gesturing spastically at the old man. She walked slowly, dreamily back into the house—then as soon as she got inside the doorway, she darted up the stairs shouting, "Hold on, now, ya'll!" over her shoulder.
_SHEEEEEEEEEEE. SHE-SHE-SHEEEEEE... SHEEEEEE—_
Montrose cut off in mid-shriek, and they thought they heard a slap. A gurgle, a boy laughing, and then, definitely, a slap.
Corey felt ill, and depressed. He had caused this retarded kid to be abused one more time.
Then Kizzy reappeared at the door. "He's ready, ya'll come in."
"Megan—" Wade began.
"You're coming too, Wade."
"This is such a waste of time."
"Yo!" Kizzy barked, making them all jump. "You coming or not? I haven't got all day! I got things to do!"
Yeah, Corey wanted to say, like what?
Megan lowered her voice and said urgently. "He helped my mom. He's got a gift. Now come on you guys."
Corey shrugged at Wade and followed her in.
The stairs were stacked with old magazines of every sort; so many it was hard to get by. Looked as if they'd been looted from other people's trash. They climbed to a short hallway packed to the ceiling with boxes splitting at the seams; oddments of clothing stuck out. There was a two-foot-wide space to sidle by in. The ripe smell of a permanently blocked toilet came from somewhere.
As they passed a padlocked door there came a scrambling of clawed feet, a thump and a febrile barking. Sounded like a Rottweiler, Corey thought. Kizzy slammed the side of her fist into the door, hard, as she passed, and shouted, "Shut up, Obie Wan!" The barking became whining.
At the end of the hall: the boy Montrose's room. It contained a kiddy race-car bed cadged from some junk heap—way too small for Montrose, with a single torn quilt as its only bedclothes—and a scattering of mismatched toys, and pieces of toys. The walls were scribbled with crayons—and every so often an oddly artful crayon image of faces, worked into the scribbles. Random, staring, lonely looking faces.
Montrose was squatting by the window, hugging his knees. At the sight of Corey and Wade and Megan, Montrose began to shriek again. "SHEEEEEEEEEEEEEE—"
Kizzy started toward him, raising her hand—then seemed to think about the witnesses and maybe Child Protection Services, and stopped herself, lowering her hand, bending to speak into his ear. He cringed, and shrieked and pounded the floor with the flat of his hand.
"Montrose-hugabunny? You're gonna get the fuckin' _cops_ down on me and you know what they gone do?"
He screwed up his face to keep the shriek from coming out. "Momsa." He said. "Momsa." Then he opened his hand upward to show an old, yellowed pair of dice, rattling with the trembling of his palm.
"He'll talk now. Ten dollars more and ask him what you want," Kizzy said, straightening.
Megan looked at Corey. "In for a penny...?"
Corey thought about giving Kizzy the money and asking her to make sure she used it to feed the boy. But nothing she'd say to that would be reliable. So he pulled out the last two fives, and passed them over.
"What was it you wanna know, boy?" Kizzy asked, folding the money into tiny little triangles. "I mean is there somethin' certain or you want to try to get a lucky number? He don't do lotto numbers. Don't know any numbers."
Corey took a deep breath, slowly let it out, and finally said: "I met someone... saw something... some things... a guy I knew died... I want to know why... Megan said... I don't know. This is stupid..."
Kizzy looked at Montrose. "You already done wigged his little ass out one time."
Montrose was rocking on his haunches, his eyes squeezed shut, his lips moving soundlessly, as Kizzy bent over and said, "This boy wants to know what happened to his friend that died..."
"He wasn't exactly a friend," Corey mumbled. "A guy I knew. I found his body."
Montrose opened his mouth wide to shriek and Kizzy slapped a hand over it. "Don't you start with that! Now give us some words! You've got some words in you, you little fuck!"
She withdrew her hand slowly, and he rattled the dice in his hand. He never looked at them. Instead he squeezed his eyes shut and after a moment, said, "Corey look at three-thirty-three and no love for breath..."
"He said numbers!" Kizzy said, genuinely surprised. "He never says numbers. He doesn't know numbers. He cain't even count the dots on them dice. That must really mean something. You owe me another ten dollars for that."
"Kizzy," Megan said, "no way."
"It's on you, it's on you, girl! But he said numbers, that means something. And he said 'breath'—when he says 'breath,' he means life. Being alive. You got the rest of that money?"
"It's the deep red, Corey, under the blue. Brad he ain't enough. Your mama told me. Red under the blue glass."
Corey said, "Montrose—" And took a step closer.
Somehow he wanted to soothe the boy, but Montrose recoiled, scrambled back from Corey, and sprinted from the room, bent over double as he went.
"Now you wigged him again," Kizzy said. "He won't say no more, once he's hid hisself."
They were desperately glad to get out of the house, to drive down the street with the windows open, the breeze blowing across them.
"Where now?" Corey asked, abstractedly.
"Burger King," Wade said.
"You can eat after that?" Corey asked.
"He could eat after World War Three," Megan said.
"You told that kid or Kizzy my name, ahead of time, Megan?" Corey asked.
"He knew your name! I never thought of that! Hell no, I didn't tell them. She hasn't got a working phone. She's got a room with about thirty phones in it, most of 'em taken apart, but none of 'em work. No I didn't tell her."
"Come on, she musta heard us say Corey's name," Wade said, as they turned onto Pacific, "and she told it to the boy and gave him some vague crap to recite. They all pull shit like that."
"I don't think anyone said my name," Corey said. "And he knew Brad's name too. I never mentioned who it was, I wanted to consult about."
"Brad's dying was in the papers," Wade said. " _Weird Cult Death_ and all. They figured that it musta been him."
"But how would they connect it with me? My dad..." Corey started to say, _My dad kept my name out of the papers_. But what his dad did was exactly the kind of thing he didn't want people to think his dad would do for him.
"They don't have to connect it with you," Wade said. "It was in the papers that Brad died. Who else died under circumstances people don't understand around here?"
"Listen," Megan said, "I don't call psychic hotlines. I know they're all crooks. I don't believe those mediums they have on TV, I don't buy any of that. But this kid...he knows things. Everybody in that part of town knows he does. The local Holy Rollers were over there trying to exorcise him one time."
"Poor little dude," Corey said.
He thought about what Montrose had said. And he remembered one thing in particular.
_Brad, he ain't enough_.
"Those noises, and then they wouldn't open up, for two days they wouldn't open up," the landlady was saying, as they walked around the outside of the apartment building. She was a stout woman in a flower-print mu-mu; blue mascara and a puff of blazing red hair. "I knew something was wrong, sheriff. But I've been trying to get these crazy shits, excuse my language, out of here for a while. They spray painted on muh walls! And then the smell..."
She was sorting through the keys on an overcrowded ring; going past some of the same keys two and three times. Barry was waiting for her to open the outside door to the basement apartment, or what passed for a basement apartment in this leaning old Victorian broken up into flats. He found his hand straying to the butt of his gun. Why? There was no sound from inside. She hadn't said anything about them being violent.
It was just a feeling.
"They've been arrested more'n once, you know. And they locked me out and I was scared to just use the key and bust in on em but I know they're in there..."
Barry tried the door knob—and it turned. "It's unlocked now."
"Well I'll be gosh darned! I tried it just a little while ago..."
He opened the door—and hit a wall of stink. Gagging, he took a step back, and inhaled a deep breath of the cleaner air outside.
The landlady covered her mouth with a handkerchief and followed him, slowly, inside and down, as he fumbled for the light switch she said was at the bottom of the stairs. He stepped into something that sucked at his shoe as he reached the bottom, and found the circular pillbox shape of the old-fashioned light switch, and threw it. It was connected to the wall-plug circuit, and a lamp—on a card table across the room—came on.
Barry saw he was standing in a small pond of blood; he'd stepped through a coagulated skin of it, and the blood underneath, still wet, was flowing over the top of his shoes.
The landlady was sobbing and running up the stairs. Barry was staring at the dead Nazi punks, one kneeling and the other lying face down, stiff as sculptures; and he could see, even from here, that they were mummified the way Brad had been, like human beings transfigured whole into Styrofoam but with each little hair and wrinkle and all their natural color; and he knew somehow that they were profoundly empty, each one light as a feather.
Looking out the window, hoping to see Corey come home, Angela saw her father arrive instead. She looked up at the sky; she could make out a few stars, beyond the glare of the streetlights. She looked down at her father. She imagined him drawn upward by invisible hands to be flung between the stars, to float away from Earth out there heading for interstellar space, at ever increasing speeds. But all that happened was, he got a big, really big, stuffed-rabbit toy, like something you win at a carnival, out of the trunk of the car. She watched as he pulled transparent wrap off it, and as a woman got out of the car and came to stand beside him. The woman had short blond hair, and wore a sari, though Angela knew she wasn't Indian. That'd be Eden. Angela's dad had told her about him at dinner. Might be in love, he thought, though he'd only met her once. And though he was living with someone else.
Angela could see a crystal glinting at the woman's throat—yeah, definitely Eden. A New Age type, even more than her Dad.
Dad walked up to the front door and rang the bell. Mom wasn't home and Angela wasn't about to answer. He wasn't allowed to just come in. She let the bell ring, over and over, two and a half bonging notes, the song of an idiot; the song of someone who'd bring her a big stuffed rabbit toy. She hoped he looked up and saw her here, too, not answering. But he just carried the stupid stuffed rabbit back to the trunk. Then he thought better of it, and left it on the front steps, as if to wait for her.
She went and sat on her bed, and read the novel Corey had brought her.
Downstairs, Marty was staring gloomily at the front door of the house. Eden put a hand on his arm.
Marty slowly shook his head. "I don't think she's gonna want this rabbit, anyway. I don't know what I was thinking. I mean, you know, you think a teenage girl likes something a little girl would like, now and then. Remember when she was little with Daddy. Just to say, you know, 'You're still my little girl.' But I don't think it'll help any. She just doesn't want to talk to me."
"I know," Eden said gently. Her voice chirpy but infinitely compassionate. "They don't like to talk—except for hours on the phone with their friends." She beamed to show it wasn't really a criticism of his little girl.
"One of her friends died, you know. A guy she dated. Some cult murder."
"I read about that. There are some very dark, sick souls lost in the world—with a lot of bad karma that just keeps getting worse."
"I've been trying to be there for her. But she wouldn't talk about it..."
"Look—you know what?—maybe we should have a cup of coffee or something and share about it?" She touched the crystal hanging from her neck. "I know this place that has those great cappuccinos with the Italian berry syrup in 'em..."
The afternoon sunlight blotted the window of the old bookstore; only when Corey and Angela stood inside, the brass bell on the door still resonating in their ears, did Angela see the name of the store, reading it in the glass backwards. "Does that say _Leatherbound Books_?"
"Hermie's sense of humor," Corey said.
The first thing you saw, when Hermie walked in, Corey reflected, was his beard: it was as if it preceded him into the room. The waist-length gray-streaked beard was piratically braided, threaded with ribbons; his long gray hair, starting halfway back on his head now, was beginning to go dreadlock thatchy by inattention rather than style. He was a short man with a barrel chest, long arms, biker boots, bellbottoms frayed feathery, a Blue Öyster Cult T-shirt from 1979, barely legible now; his teeth were the color of tobacco and as he drifted past them—negotiating moldering, precarious stacks of books with the ease of a seal through coral—the scent of tobacco slipstreamed behind him.
"Don't be talking about my sense of humor," Hermie said, in clipped, rapid-fire syllables, using a singly overgrown fingernail to slit the tape on a box of new books. "You being, after all, Corey, an example of God's sense of humor. Like the platypus."
Angela's eyebrows went up at this; Corey laughed and winked at her. It was just the way Hermie talked. "Least I'm not a refugee from People's Park."
"Haight Ashbury, I'll have you know," Hermie said, putting the new books—all nonfiction from companies like Shambhala, Continuum, and Arkana—into a window display. "I wouldn't have gone near that Marxist People's Park scene over in Berkeley. If I hadn't inherited the bookstore I'd still be in the Haight. Trying to sell books on the street corner there, probably. These new books are it for the rest of the year. All I can afford."
Most of the books in the little store were used; there was a preponderance of fantasy, science fiction, poetry, outdated encyclopedias, old art history books, Tantric sex manuals, beat writers, and "metaphysical books"—the occult.
Hermie stopped what he was doing as if kicked in the pants, and sniffed the air. "I smell perfume. I sense great beauty at hand. Someone out of Spenser's Faerieland."
He turned slowly as if searching for the source of the perfume, and looked at Angela with eyes that would have been monkeylike, so small and brown were they, so set into wrinkled sockets, if not for their glittering intelligence. "And there she is. Corey—believe it or not, there is a celestial guardian, in the shape of a young blond woman, at your elbow. She'll be invisible to you, of course, but you might see her from the corner of your eye."
"She's a long ways from invisible to me, man," said Corey, rolling his eyes. Was Hermie going to embarrass them by coming onto Angela?
But Hermie only looked...and then a troubled flicker passed through his gaze—and he went back to his books.
"Who's this angel he's talking about?" Angela said, pretending to look behind Corey. "Should I be jealous?"
"Every shade of angel is here," said Hermie. "Corey, this is the young lady you gave those books to?"
"Yep. Angela, this is Hermie."
"Hi."
Without looking up from his display, Hermie said. "Angela—don't drift too far from the front desk, here, there are spiders in the back of the store as big as terriers."
"Don't listen to him," Corey said. "Come on, I want to show you some books..."
They browsed for an hour, through dust thick enough to make them sneeze, seeing no spiders but many silverfish. "Lots of science fiction," Angela said, once, "but I don't see any horror."
Corey lowered his voice, glancing at the front of the store. "Hermie says that he's afraid of horror books because he's afraid that, if you read them, your mind is somehow _invoking_ the monsters in them by visualizing them...and then some invisible psychic version might start stalking you. He says it's even worse for the writers—claims something like that is what really happened to Lovecraft. He wasted away because writing that stuff invoked one of these creatures, that—"
"That is _such_ a repellant idea," she muttered. "I mean—riiiiiight."
"Hermie had maybe a little too much acid, back when."
"I don't think I could read horror now anyway..."
He could tell she was thinking about Brad. He could see it in her face when she blocked the thought out of her mind. "Do you like art books?"
Angela was looking at a book of Max Ernst paintings when Corey found himself drifting to the other side of the store—and looking at a locked, antique glass-front bookcase he'd never taken much note of before. The blue glass over the case was tinted light blue. There were numerous old books in the case; but only one of them had a red cover:
_A SPIRITUAL BESTIARY:_
_ASTRAL PREDATORS AND PARASITES_
_by Trevor Housman_.
What had Montrose said? _It's the deep red, Corey, under the blue. Brad he ain't enough. Your mama told me. Red under the blue glass_.
Hermie appeared at his elbow. "Hey—try this on her. _Wuthering Heights_. You got to get your woman dogging act together, man. See, you give her something romantic like this..."
"No, dude, it's got a tragic ending. She couldn't deal with that right now. Some shit has happened to her."
"Okay—this one—E. E. Cummings. It works for Woody Allen. This edition has all his romantic ones..."
"Okay, I'll try it. Yo, man—are these, like, expensive collector's items in this case?"
"Some of them are, yeah. Some are just...unusual."
"How about that red one?"
"You can tell it's red under that blue glass?" He pulled an extendable key ring from a belt loop and unlocked the cabinet. "Have a look. It was a private printing, that book, and there aren't very many copies but it's not a collector's item. It's a hundred years old, you'd think it'd be something they'd want but no one much knows about it."
Corey took the old leather backed book out, and hefted it in his hands. The cover was red-stained leather. "Red under blue glass..."
"What's that?"
"How much for this book?"
"It's not really for sale. But...you can borrow it, if you promise not to lose it...I mean..." Hermie gave him a long, flat look. "If it's important to you."
It was dark out and Momsa Kizzy had been gone for two days and two nights. Montrose was hungry, but he didn't mind being alone. Montrose was never lonely. There were things he longed for, but he was never lonely. He could see the spirits, often enough. Sometimes he saw his granddad—or anyway he took the old man to be his granddad—sighing over Momsa Kizzy, and trying to talk to her, though she couldn't see him at all. Sometimes he saw the imps, whirling through the air, chattering; the imps had no special shape, unless you thought about them, and they'd take the shape you thought about, mock-like. Then there was the see-through worms that whipped through the sky, and settled on people—Momsa had a big one suckin' at her soul—and there was the Fucking Monsters and the other invisible critchers.
Crouching and rocking in his room, he thought about going downstairs to where there might be some of that bright-colored cereal Momsa gave him for most of his meals, but that was in the kitchen and he couldn't go there knowing about the thing that lived under the kitchen sink.
Sometimes Montrose would go to Obie Wan's door, and talk to the dog in its own language, saying someday he'd be free, Montrose has seen it.
Rattling the dice to help figure it, Montrose saw his own death, of course—but not his own dying.
He never saw her coming, though, that night. He was crouching on the car bed, rocking himself and tumbling the dice in his hand to feel the many and few of them and getting set to pee his pants—and then of a sudden he had company. She was just there in his room with him, quick as Momsa flicking a Bic lighter.
He couldn't see into her heart and he couldn't see her future and he couldn't see her past. She was like a hole in space shaped like a woman, in that way.
He thought maybe she was the one he'd seen afar, coming down from the moon, but now he couldn't be sure, for she'd taken pains to hide whatever she was.
But her body, he could see that. She wore a lady's pajamas, and she had very big lips for kissing and very big eyes for looking into. She hadn't bothered with a nose—and in fact he'd never favored noses. But most important, when she opened the front of her lady pajamas, out spilled a single big enormous breast in the middle of her chest, bigger than a big man's head, big as two heads and another, and milk was forming on the nipple, and running down it warm and thick.
All the longing surged out of him, and he wailed to give it a sound as it came, and he opened his arms and she came to him. But instead of coming into his arms, she did something better, she picked him up in hers, and sat him on her lap, and put her breast to his mouth, and let him draw deeply, sweetly on it, and he nestled up to her—dropping the dice on the mattress—and drew on her, drew deep into the warm and soft and giving of that enormous bosom. And only after a long slow ecstasy did he realize that he wasn't really drinking from her, her breast was drinking from him, and his blood was streaming onto the floor as the lightning-fire of his life was drawn away into her, but he didn't care, because he saw some other things now...
Just about that time, while Montrose was dying from this world, Obie Wan felt it happening, and a barking frenzy took him, and a panic beyond bearing, and the dog leapt right through the glass of the window, and fell two stories into the oleander bushes, then ran off down the street, baying and bleeding from broken glass and scratches, with only one eye left, the bushes having taken one out, but not caring, because he—Obie Wan the dog—saw some other things now...
Silbido didn't have a lab of its, but Detective Ganrich had a Bachelor of Science in Forensics, so he had long-established privileges at the county's Forensic Services lab. Still, he preferred to stay out of the way, so often used the facility—with permission—after hours. Ganrich was so comfortable in the place he even watched the little portable TV set up on the counter next to a rack of beakers, when he wasn't looking at slides in the microscope: torn bits of something like fabric and something like skin; the same something. Gathered from the pool of blood that held the desiccated bodies of those two skinheads—one of them had clawed some jean fabric away...
The Comedy Channel was showing old _Evening at the Improv_ re-runs. The standup comic, in a sweater with the sleeves rolled up and khakis, laid out a standard patter. "Another thing that bugs me about airlines..."
"How about not another thing that bugs you about airlines. Christ, airlines, airlines, airlines." Ganrich straightened up, stretching his sore back so that it crackled audibly, then squinted at the clipboard with the notes from the forensics team. Unidentified organic matter. What the hell did that mean? He looked in the microscope again. Seeing for himself wasn't helping.
"...they tell us 'in the event of a water landing'—what the hell is a 'water landing'? They mean in the event of—"
"—a burial at sea," Ganrich said, at the same time as the comedian.
He opened the cabinet containing the glass jars with fresh evidence from the skinhead killings. Peering into the jars: bits of hair, a broken off fingernail, some of the "fabric."
Ganrich took two of the jars to the microscope table, placed them down, yawned—started to put on some rubber gloves—
And stared at the jars. They were empty.
"Whoa. I am not old enough to be having a 'senior moment'." He went back to the cabinet—but the others were empty, too. "What the fuck?"
He went to the microscope. He looked. The slides were empty.
**PART THE THIRD: MAMA TOLD ME NOT TO COME**
SIX
Three dreamers, at three points in Silbido, almost equidistant; many blocks away from one another.
Though dead asleep, Stanny was creeping across his futon, like an alligator moving on its belly over a mud bank. He stopped moving with his head lolling over the edge, and lapsed into a limp, resonant snoring.
His eyes were shut, but he was dreaming the room he was in; he was seeing himself asleep there, in the dream.
If it was a dream.
Sira was there, in the room with him. In his dream.
She was stepping out of the shadows, almost nude; stepping out as if the very shadows were her doorway. She paused, and the waning hooked moon shone through the window and right through her translucent form: he could see the horned moon shining through where her heart should be. The girl he'd seen in the arcade, now appearing to him wrapped only in shadow and thin moonlight and a black leather g-string. From somewhere played a distorted disco-rave tune. She danced slowly, forward, a ripple of lure. "Now I see it," came her sibilant voice, though her lips never moved. "I see more clearly what you want—something like this..."
She opened wide her arms. "Then find me. The beach park on Saturday night."
And she backed into the shadows and as she went her going seemed to pull on him like a magnet, and drew him to all fours, and then to his feet, to stagger toward those shadows, reaching for her—but she was gone.
He lowered his outstretched hands...and they gripped his erect dick...
And he could still hear that music...
...the same music that played in Wade's bedroom, as he lay thrashing on his back in his king-size bed—and the flames at the foot of the bed were moving to the music. In the flames, like Joan of Arc without her stake, was a woman Wade had never seen before.
"Ilira... my name's Ilira..."
And she was naked, this beautiful ebony woman, tall and slender and naked and dark eyed, her head shaven—naked except for the flames that sprouted from her vagina itself, and from the underside of her breasts, to lick up and cover them, red flames within the blue flames of the pyre.
_Red under blue_ , the boy had said...
"Wade? Why don't you do something! Put this fire out! Please help me put it out... I can't bear it... Come to the beach park on Saturday night..."
The flame was sucked backward down into the floor as Wade suddenly sat up, his eyes snapping open. He was alone in the room—with his hands on his crotch.
The music was the same, but more complicated, in Corey's dream. Corey was lying on his side—and no one came into his room. _You come to me..._
Lying in bed, mouth open in sleep, he made walking motions, just the suggestion of them, as the dream deepened...
He was walking through the night, down the beach, alone. Moonlight-edged clouds clashed in the sky as if the wind were blowing from four opposing directions at once; throngs of stars shone through the breaks, and quickly hidden away.
He saw her, then, her back to him: her blond hair and long white dress whipped first this way then that in the skirmishing winds. She was standing with her bare feet in the lacework edge of the surf.
"Angela?" he called.
Angela turned her head to look. Corey couldn't make out her expression in the darkness. Then she looked out to sea again.
That's when the clouds parted exactly like mechanical scenery in an old theatre, and showed a shattered moon—a full moon that had been broken in half, so that its broken edged halves floated next to each other like pieces of an egg shell in oil. And the uneasy light from the broken moon stretched her shadow out across the sand behind her with a ripping sound that tore through the music and the sound of the surf. He stared at the shadow, going cold inside, as it began to change shape, and within the shadow a column of sand geysered up and spun, like a lathe, to shape itself into a woman, _another_ woman...
"Back off, squirming pink worm!" the woman hissed.
Corey forced himself to start toward Angela—but the stars began to fall out of the sky, becoming searing meteors that crashed sizzling into the sea, their impacts making the winds coalesce many into one. The roaring wind from the sea hove up the sand of the beach and the whole beach came _at_ him in a seething wall, and millions of silicon projectiles passed through him and separated every cell of his body from every other...
Until he found himself staring into his own wide-open eyes.
His closet door was open and the mirror on the inside was facing his bed. He was awake, wasn't he? But something squirmed in the shadows of the closet...
"Hey Sleep Hound—front and center!" Barry said, sweeping into the room.
Corey sat up, the dream's palpability fading—but some of it stayed with him. Enough.
Barry opened the curtains, then turned to frown at him. "You sick, boy? You're sweating!"
"Just...a bad dream."
Barry came over, put a palm on Corey's forehead. "Don't feel a fever. Except adolescence."
"I'm fine."
"It's almost eleven. You don't usually sleep this late, pal. You okay?"
"Um—yeah. Sorta had trouble sleeping. Dad—what's up with the investigation?"
Barry moved around the room, picking up dirty clothes strewn across the floor. "Found a couple of more bodies, son. Worse. Nobody you know. It's getting hard to keep the lid on it. Evidence disappearing. The mayor doesn't want the tabloids here...You can hit a basketball hoop from a mile away but you can't hit the clothes hamper?"
"So—what's your theory about...you know. Brad."
"We're in the market for a theory. Tell you the truth the whole thing has us spooked. Anyway—how's Angela?"
"She's all right! Why _shouldn't_ she be?"
"I don't know..." He seemed a little surprised by Corey's reaction to the question about Angela.
Corey suddenly felt lonely; like opening up, just a little. "The only thing, Dad, I don't like about Angela is—she drives me crazy, she's suicidal, she won't commit to me, she won't go out seriously with me, she won't talk to me about what's bothering her, she's so beautiful it hurts me to see her and she lives right across the street from me. But besides that... she's great."
Barry laughed. "Okay, okay."
"Dad—you ever talk to Child Protection Services?"
"Why—turn myself in?"
"Seriously—about people you meet on the job. Taking kids away from abusive parents."
"Someone you're worried about?"
"There's a kid—a retarded kid. People go to him for psychic readings sometimes...His Mother—"
"She's a crack addict? You mean Montrose. We were working on that. But son—that boy, he's one of our bodies."
Corey stared at him. "He's dead?"
"He is, yes. Like the others. In fact, it was Child Protection Services that found him. Why...? Did you know Montrose?"
"I... met him one time. I saw him recently. We—someone was asking him a question..."
"Really. Who?"
Corey blew out his cheeks and admitted sheepishly, "Um... me. I was asking about Brad. I didn't get any clear answer."
"Well since that stuff is bullshit, of course not. Who went with you?"
"Megan and Wade."
"You see anyone else hanging around Montrose's house?"
"No. Just his mother."
"We're looking for her, for questioning. She's dropped out of sight. If there's a cult, maybe she knows about it. You heard anything about a local cult? Your friends talk about anything like that?"
Corey shook his head. Montrose. That kid was dead. Like Brad. Was it because Montrose had tried to warn them? Warn them about what?
_Red under blue... three-thirty-three_...
Barry was watching him, about to ask a question—but seemed to make a conscious choice not to speak his mind. "Hey –I got some French toast on, my man. It's gonna burn unless I get back down there..."
Corey waited till he was out of the room, then he went to his dresser, opened the bottom drawer, and found the book with the red leather cover.
"I am not ready to go back to school," Wade said. "I just am not. I truly am not. No I'm not..." He was talking absently as he sat at Corey's desk, tapping at the laptop, launching mandelbrot sets. "I'm using that program Rudy Rucker made—It's fucking amazing."
"Watch your language," Corey said, vaguely. He was deep in a beanbag chair, looking at the red leather book; his back and shoulders propped against the wall, under a Phish poster. On the far wall was a poster of Stephen Hawking, holding a galaxy in his hand. The portable MP3 player was hooked into the stereo, playing The Toadies. "My dad's home for lunch, he gets annoyed when my friends cuss."
"Hey, I don't say those words sometimes they'll take away my hip-hop license."
"Soon you're gonna start talking about ho's and the bee-atches... 'The Akishra'?"
"The what?"
"Something in this book... Akishra: psychic parasites who suck at addicts... but that doesn't seem to be it... Oh shit. What's the matter with me."
"I've been wondering that for years, dude."
"Shut up—I mean, why didn't I think of it before—he said three-thirty-three..."
"Who...?" Then Wade stopped tapping at the keyboard, and swiveled to look gravely at Corey. "Oh."
"You know he's dead, don't you?"
"Montrose? I heard. I've already been freaked out by that, thanks. I don't need any more of it."
"He said red under blue glass—that's what this book was. Under the blue glass of the case at the bookstore. And he said three-thirty-three. I'm an idiot, of course it's a page number—and look at this!"
Wade hesitated. He looked at the door. Thinking about leaving. But he sighed and came to squat beside Corey. "Okay. Let's see it."
Corey tapped the illustration on page 333. The reproduction of an ancient woodcut. A demonic woman, in two panels. Before and after. A woman—and then the demon. Underneath, the caption said: _Figure 17: SUCCUBUS_.
"That's it, man. She changes appearance, it says, according to what you need to see. 'A shape-shifting demon that takes human female form so she can have sex with men in order to destroy them.' Wade—that's what I saw in the arcade..."
"That was it? Or you just think it is now... Suggestion, my man."
"Listen: '...those who are born with the dark gift, have the gift of summoning darkness. What grandmother has, has granddaughter also. A living darkness summoned with the power of pain; the power of the feminine elemental, from her residence in the barren seas of the moon. And only a conscious love, a living love, a love born without fear, a love awake, can uncall what is called...'"
"What they mean, a conscious love?"
"I've read about it in a book called _Hidden Wisdom_ that Hermie gave me."
"That burned-out old hippie..."
"Hey he knows some shit."
"Watch your language, your dad's home."
"Shut up. This book talks about traditions that hold that, uh—well that most people don't know how to love consciously. They don't know how to _really_ love. A lot of philosophers thought that. If you can really love—for real without any self-interest—that's a kind of power. I mean, you know: paradoxically."
"You getting into your dad's stash of confiscated herb and you're not giving me any? That's what hurts, man."
Corey thought about telling him the rest of it. But he couldn't. If he was wrong...Who knew what people might think? What might they do to her?
"So Corey—you talk to your dad about this?"
"No."
"Right. Because you know what he'd say. And dude, he'd be right. Look—" Wade reached over and closed the book. "—I got to go and do some work for my folks. You going to come to the bonfire Saturday night?"
"What bonfire?"
"I thought Meg told you. We're having a bonfire party. At the little beach park."
"It's like an incredible synchronicity, Eden, you living just down the street from me," Marty was saying. Marty and Eden were walking up to Marty's front door, about seven that night.
Her eyes sparkled when she smiled. "All the beautiful parts of life are coincidences..."
"I'd still like you to meet Angela. Maybe you could help. Jeez, I'm always talking about her—it must be a bore for you."
"Not at all."
"Look—I'd invite you... but..."
"I know," Eden said softly, nodding toward the door. "The girlfriend. That's okay."
"She's not home now—but she'll be back soon. She wouldn't understand."
She stroked his cheek. An electricity between them drew them close; his lips brushed hers. He gasped and took her in his arms.
She pulled back from him, biting her lower lip and looking at him, head tilted down, in that impish way she had. "If you want—we could... talk... a little more... over at my place?"
He felt the temptation like a hot physical caress. But then he saw the headlights of Gwen's car, two blocks down—those distinctive Miata headlights. He stepped back from Eden. "No—Gwen's coming..."
Eden stepped back, and fell into the pose of someone casually parting after a class they'd both taken. "There'll be time, Marty. I'm not ready for you yet, either, perhaps. Can you get away Saturday night?"
"Saturday night. I'll try to manage it..." He looked nervously at the approaching car.
"I'll be in touch." Then Eden breezed calmly away, down the street.
Finally, that day at Hermie's shop, Corey had given Angela two books. The Cummings book, and _Wuthering Heights_.
Now she sat in her room, in an old rocking chair, with the book open on her lap; bending over it to see in the light of the candle. Reading aloud to herself, from time to time. " _'I wish I could hold you,' she continued bitterly. 'Till we both were dead! I shouldn't care what you suffered... why shouldn't you suffer? I do! Will you forget me? Will you be happy when I'm in the earth? Will you say, twenty years hence, 'That's the grave of Catherine Earnshaw. I loved her long ago and was wretched to lose her; but it is past...'"_
A sob caught in Angela's throat. She stopped reading, and picked up the candle. She put her hand on the book's page, and began to drip hot candle wax onto her hand, and onto the page, without knowing why. Outlining her hand in wax. Burning herself just a little. "Why shouldn't you suffer...I do...I do..."
Then her hand contracted over the candlewick, snuffing the flame with her palm; she twitched at the jet of pain—but what she felt, very much more strongly, now, was anger, as the candle crushed in her hand...and she began to shake...
Mace had candles set up around his rehearsal room; waxed dripped down speaker tops, made icicle shapes over the side of a music stand. He wasn't sure why he'd done that—it was just an impulse that came over him. But it was cool, he liked it. He'd smoked some major-league hashish, and the candle-shine looked like something alive, you could almost see living things swirling around it; ghostly worms in the air, and the miscible faces of imps that chattered and departed.
"Freak my own ass out, here," he muttered, and he slapped a power chord on the Telecaster in his hands, almost falling off the padded stool. "Shit. Too much hash, too much tequila. But no no nuh-no. Not enough: still alive."
He looked at the picture of Brad he'd leaned up against the Cuervo bottle on the top of his Marshall amp, next to the glass hash pipe. Brad with his own guitar, a small Gibson he'd given the boy when he'd turned twelve. The boy grinning through the stringy hair hanging in his face.
Mace's stoned gaze wandered; he found himself looking, then, at the old speaker cabinet lying on its back, most of its guts pulled out. It looked damnably like a coffin, right about now.
Fucking kid. What'd he get into? A cult?
Mace'd told that fucking spade cop to check out the little blond bitch, she must've gotten him into whatever it was that killed him.
Fucking kid. Just gone. Abandoning his old man.
Mace took another hit on the hash pipe, then turned to the soundboard to record some kind of demo. If he didn't get another record deal, at least with that German label, he'd have to sell that board and maybe the house.
He put on the earphones, set it up to play the clicktrack he'd recorded earlier, hit _record_ , and as soon as the reel started turning he bashed out a loud, fuzztoned boogie to go over the clicktrack. It sounded all right for a few bars—and then somehow, just disintegrated into noise.
An ichorous fury rose up in him, bringing bile up from that herniated throat of his, and he spat and gagged—and threw his guitar at the amp. Or maybe, really, at Brad's picture. Struck the amp—sparks flew.
"It all all all _all_ sounds like fucking _shit!_ "
"I thought it sounded hot," said the girl in the doorway.
Mace gaped at her. Dressed exactly like Miss Pamela about 1974. Hot pants, the white leather vest showing most of her boobs, the whole nine yards. A girl with long black hair, big dark eyes, pale heart-shaped face. "How'd the fuck you get in here?"
"You called me, in your way. I'm a fan. I've got all your CDs and records. I've got all the vinyl."
"The CDs suck, they don't capture the records. Now fuck off before I call the cops. Where's my fucking tequila? Where's my picture of my boy...shit there's booze all over the fucking picture...goddammit..."
He knelt by the wreckage, the picture of Brad, the fallen guitar; the candlelight making little bright triangles of broken bottle glass and arcs of broken strings curling up.
She stepped over to him, and stood behind him, legs well apart, and drew the back of his head against her soft warm thighs...he could feel the heat of her crotch up there...
"Records, CDs—doesn't matter. I'm into the essence of you, Mace. It's all about essence. That's what gets me off..."
Mace turned to face her, running his hand up her legs. "What the hell."
"What the Hell, Mace." She knelt beside him, and she lay down, without hesitation, in the broken glass and tequila, and her underwear visible beneath the hiked up skirt seemed to just melt away.
SEVEN
Saturday night cruise in Silbido. The cars got off the freeway at the cannery exit, found their way to Pacific, paraded through old town, then out past the fast food places and the car dealerships to the mall, and turned around, retracing back to the old cannery, and then back to Pacific. But a number of cars broke off from the parade, about nine o'clock, Corey's among them, and headed for the beach...
The bonfire was just starting to burn when Corey and Megan and Wade trooped down the sandy trail along the creek to the big half-buried log. In the lee of the log, sheltered from the gusting wind off the sea, Stanny and some teenagers that Corey barely knew had built a pretty good pyramid of driftwood. Someone's boombox was playing the band _They Might Be Giants_.
"You asked Angela?" Megan said, as they found places by the fire.
"Yeah," Corey said. Not wanting to talk about it. "She said she wanted to read."
"Heyyyyy, it's the Meginator!" Stanny said, throwing a stick on the fire so that sparks jumped out at everyone. "It's the Corifier! It's the Waderator!"
"He's doing the Rob Schneiderator," Wade observed.
"He is," Megan said. "Just ignore him."
There were half a dozen other teenagers, drinking beer, laughing. A teenage couple on the other side of the fire roasted a marshmallow together, each with a hand on the stick. Their faces melted together into a kiss; they wriggled against one another; they kissed more deeply; the marshmallow turned black, and incinerated.
Corey might've laughed, another time, but he felt ashen himself, inside. He thought about Angela; he thought about Montrose saying _Momsa!_ He thought about Brad's hands hidden in paper bags. He thought about page 333.
A girl sat down near him: Heather, last year's prom queen, already graduated. A pretty auburn-haired girl with almost catlike green eyes, nearly anorexic-slender; she wore designer jeans and a tie-off blouse that showed her tanned, flat belly. She was carrying a handful of color photos, and as he and Megan and Wade watched, she stuck each one on the end of the sharpened twigs of a branch she'd brought along, and inserted the impaled photos in the growing bonfire, just above the actual flames. She watched with grim satisfaction and a tiny bent smile, turning the photos slowly in the fire, watching the people in the photos turn brown, and crinkle up...and warp out of shape...
"Is that, like, voodoo?" Stanny asked.
"Maybe," Heather said. "My mom kicked me out of the house. But she ever so kindly gave me some family photos to take along."
Now Corey remembered having heard something about Heather's parents finding out she'd gotten an abortion. They were Baptists or something.
Megan shivered visibly, looking at the photos shrivel. "We shouldn't even be out here... really. I mean—after Brad. They never caught whoever did that."
"Would you guys like, lighten up about Brad?" Stanny said, sliding down heavily beside Megan so that sand splashed on her. "I mean I was, like, his guitar player, you see me weeping and shit?"
"I don't know if that says we're too sentimental or you are just amazingly indifferent about your friend dying, dude," Corey said.
Stanny opened his mouth to retort—then cocked his head, and shut his mouth. He stared into the fire. "Wasn't like he was any kind of real friend, man," he said, his voice almost inaudible.
Heather looked at him with a flicker of sympathy.
"Brad's not the first guy I knew to die," Wade said. "My cousin got banged in Compton. Just a no-point drive-by. They weren't even trying to get him."
Megan moved a little closer to Wade, and he put his arm around her as she said, "I just think about dying, sometimes. Just trying to _get it_."
"I know what you mean by that," Corey said, nodding his head. "Death isn't that easy to _..._ to understand..."
Angela's voice came out of the shadows. "Death isn't hard to understand. It's the easiest subject."
Corey turned and saw her standing about thirty feet away, her back to him, staring out to sea, like in the dream. She wore white shorts, a long white shirt, its untucked ends fluttering in the breeze like a skirt.
"Whoa!" Megan said. "You made me jump, girl! I didn't know you were here..."
Corey was about to get up and go to her, but then Angela turned and walked over, sat down nearby, Indian style. "You came after all," Corey said, trying to sound like he hadn't been thinking about it.
"Looks like. I don't know why. Felt like I had to." She shrugged.
A Latino guy with long curly black hair, a friend of Wade's, came up to the fire carrying a red and white plastic cooler. He set it down, grunting, in the sand, and immediately began passing out cans of Coors Light.
"Coors Light?" Wade asked.
"There was a case of it left over in my dad's garage from his diet."
Wade offered a can of beer to Corey.
Corey considered it. Maybe he should unbend and drink—maybe then Angela would drink and unbend. But it didn't feel right. He shook his head at Wade.
Watching him, Angela said, "You know what I like about you?"
"You found something!"
"How you go your own way. You don't let people influence you."
Corey shrugged. Was he like that? "My mom was like that. She was a socialist. Dad hated that. I guess he loved her but he hated it when she talked politics."
"You wanna take a walk?"
"We start now, we could walk to Las Vegas in, say, two or three weeks."
She laughed as he pulled her to her feet, and surprised him by letting him keep her hand in his, as they strolled off out of the circle of firelight.
After a while, when the only sounds they heard came from the bustling sea, Corey took another chance and said, "I was talking to your mom this morning. She said not to bother you. Said you hadn't hardly come out of your room for a couple of days. She figured you needed that, or something."
"I did."
"But—it, like, made me kind of worried. I mean—I know Brad wasn't close to you but... first he assaults you, leaves you out there—then he gets killed by someone...that'd just screw with your head—I mean—maybe not—but—"
Shut up, he told himself, you're babbling.
"You already thought there was something wrong with me before that, I'll bet. Everyone does."
"No. There's something bothering you maybe—that doesn't mean there's something _wrong_ with you."
"Corey—" She stopped, and they watched the surf part around a prow-like rock, in a little jetty of sand. The sky held a broken cloud cover; sometimes the stars shined through. "I just wish I could trust someone. You more than anyone. I'm just all sort of locked up. But I just don't want to talk about why." She turned away.
_You more than anyone_. Hope surged up in him. "You _can_ trust me. Come on...just look at me..." He took her shoulders gently and tried to turn her to face him. Tried to open his heart so much it'd show in his face. "Angela... I love you."
She froze in his hands; he could feel the rigidity under his fingers like a rope suddenly pulled taut.
"Angela—look, I mean it—"
She twisted away, and stepped back; one step and then two. She looked stricken—and almost as if she were in a trance. Then suddenly she hunkered down and hugged her knees, like a little girl trying to make herself small. "No! That isn't happening!" She moaned. "Don't!"
He knelt beside her. "Angela? Baby? What's up? Tell me!"
He touched her shoulder—and she slapped him. _"Leave me alone!"_
His head rang with the force of it; the surprise, the emotional sting was worse.
"Leave you alone," he said, standing. His eyes filling. "You got it."
He turned away and walked toward the distant light of the bonfire.
After fifty yards or so, he turned toward the sea, and sat on a low boulder that stuck up like the back of a turtle in the sand. He knew he was being adolescent-sulky, and he knew that he was hoping she'd come and tap him on the shoulder and apologize for rejecting him when he'd opened up to her and he knew that was a childish, dramatic thing to hope for. But he couldn't help it.
He sat and waited for some minutes. Thinking: This is stupid. Go back to the fire, or go find Angela. She shouldn't be out here alone. Whoever killed Brad could be out here. It might not be any kind of supernatural thing at all—it might just be a psychokiller who got an A in chemistry. And you might be leaving Angela to his tender mercies.
He heard the crunching of feet behind him. He waited. A tap on his shoulder.
He turned—and his smile faded. A woman stood there wearing a long gray coat and sandals. Her hair was long and blond, streaming in the wind, but her face...
It was the girl he'd seen in the arcade with Stanny.
"Who are you?" Corey asked, so softly he probably shouldn't have been audible.
But she said, "Who do you want me to be?"
Her face flickered...and it was Angela, then. And she unbuttoned her coat, opened it, and she was naked underneath.
"Come under with me... Warm me up..."
He found himself reaching for her.
Stanny's marshmallow was melting away, dripping to sputter in the fire. He offered the charred-slug remainder of it to Heather, practically poking her in the eye with the stick. "You want some?"
"Get that drippy stick out of my face!"
Megan and Wade laughed at that. "How _wude_ ," Megan said.
Stanny said, "You sure? Tastes best when it's all burned and blackened and shit."
Then Angela went drifting past them, just out of the firelight, not looking their way, zig-zagging. A drunkard's walk down the beach.
"I want some a what she had to drink," Heather said.
"You get me a ride," Stanny said, thinking he saw his chance with Heather, "I can get you the booze. I'm, like, mature looking."
"Oh bullshit... Really?"
Corey stood just inside the spread-open wings of the coat, not yet embracing her, but letting his hands run down her face, over her breasts; letting his thumb trace a stiffening nipple; stiffening himself, trying to believe in her...
And feeling stoned. Like the time he'd smoked a bong with that girl who'd moved away, Consuela, and he felt like his feet were melting into his shoes, and the top of his head was going soft and flowing upward into the sky.
He hadn't liked it, though. Losing control of himself like that...
She was closing her arms around him...
He looked into her eyes—and saw a screaming void, in there, and faces, lost faces, maybe Brad's face—
"Dammit!" he shouted, in frustration and disgust, backpedaling. Her fingers started to close over his shoulders, to pull him back to her, but he wrenched free, stumbled over the little boulder, and maybe that fall saved him—he thought he saw something close where he'd been like the snapping-shut of an iron wolf-trap, but big enough to engulf a grown man.
He scrambled back and got to his feet. There was just the woman, with her coat hanging open, staring sadly at him. Almost Angela. But not Angela. "You're rejecting me... that hurts..."
He looked away and shouted, into the night, "AN-GEL-AAAAAAA!"
Shouted it with a depth of feeling that flowed from the center of his being...
The woman hissed like a monitor lizard, and recoiled. "So, then." Her voice now sibilant, and very ancient. "You are protected for now. But do you know what happens to unsatisfied desire, boy? It gets stronger and stronger... _and then it comes looking for you."_
She backed away into the darkness, and a sudden wind gusted up, swirling sand into a whirlwind to cloak her. He covered his eyes, remembering the murderous wall of sand from his dreams—but then the wind died away, and she had gone somewhere else.
"Better gimme a pint... uh, make that, what the hell, a _fifth_ , of, uh, tequila," Stanny said.
The cashier was a balding old guy with hooded eyes. Kind of guy who kept a big gun under the counter of his liquor store. Stanny was wearing a battered hard hat he'd found on the seat of an earthmover down the street, and sunglasses.
"Uh huh," the cashier said.
"Yeah," Stanny said, stretching, "got that night time construction work. Make that overtime. Just getting off the old job. Miller time. Tequila time I mean."
"Next time," the cashier said, "try a fake beard. Who knows, it might work."
"Great, fine," Stanny said, "I'll go to the Liquor Barn."
"Uh huh."
Stanny stormed out, trying to slam the door. The air-pressure closer on the door wouldn't let him slam it.
Heather was sitting in her Ford Escort, waiting for Stanny. She rolled down the driver's side window as he strolled up, taking off his hard hat. "You score?"
"Uhhh... no. Come on, we can go to my house, my old man's got some stuff—"
"Like a couple of grade school kids? Forget it. Just get in the car and I'll drop you back at the beach."
"Now that's gross," Sira said, from behind Stanny. "She was just using you to get booze. I mean, God, what an alky bitch."
Stanny turned—gaping. "It's you! From the arcade!"
She was dressed exactly the same way. "I'll buy us some liquor. I've got some fake ID. Then we can walk back to the party. You don't need Miss Debster."
Stanny turned to Heather. "Hey, I don't need a ride, okay?"
Heather blinked. Her lips pursed. Stanny was surprised at her hesitation. Then she said, " _Ex_ -cellent. What- _ever_." She backed the car up and squealed the wheels out of there.
Stanny turned to Sira, beaming. "You're like a total angel from..."
She smiled. "From where?" She went into the liquor store.
"You've got the _most_ wack look on your face, dude," Wade said, as Corey sat down by the crackling bonfire.
Megan was stretched out, sleeping or dozing with her head in Wade's lap. The make-out couple had gone. The Mexican guy was smoking a joint and staring into the flames.
"If I got a wack look," Corey said, feeling unreal; hearing his own words coming from far away, "it's because something seriously wack happened to me. A lot of... really weird stuff. I don't know how to talk about it."
"Dude—I'm getting genuinely worried about you. You using substances you're not telling us about?"
"You seen Angela?"
"She walked by. Down the beach that way. Come on, man. Tell me what happened."
Stanny and Sira were walking along beside the creek, Sira with her hand through his arm, down the dark path through the gulley. Into the smell of brine and wood smoke.
"These paths," Stanny said, "are all mixed up, man, like one of those things that Greek guy goes into with the sword to cut off that minotaur guy's head."
"A labyrinth?" Sira said, passing him the pint of tequila. "Kind of."
"But there's the fire..."
"Actually... is there any major reason we gotta get back to the party, Stanny?"
"Um...no. Fuck it." He sat down abruptly, half falling on his ass with the tequila. "Let's kick it here."
"Yeah lets..." Sira said. "More tequila?"
"Brad gets killed, you meet some drug-crazed girl on the beach, naked under a coat, it's natural, bro, you start to imagine shit. It's in your head, man." Wade tossed a stick onto the dying fire.
"I knew you wouldn't believe me, Wade" Corey said, peering through the darkness, trying to see Angela. He wanted Wade to go with him, to look for her, but Wade didn't want to disturb Megan, who was now snoring. Corey was afraid to be out there with that woman, alone. "Everything's hardware and computer programming with you. Mister Computo-Head. There's more to the world, man. And some of it's invisible."
"Some of the world's invisible? Now I know you're trippin'. Shit I get smoke in my eyes every time the damn wind changes."
"Show me gravity itself, 'homeboy.' Show me magnetism. It's invisible. There's an invisible world and that stuff is just the tip of the iceberg..."
"You been talking to Hermie too much."
Corey shook his head, exasperated. He needed to go look for Angela.
There were gunshots from the beach behind Corey. He turned and saw Heather, setting up tin cans on an up-arching driftwood log about thirty feet away; firing toward a sandbank with a .22 pistol. She fired twice more, the jetting muzzle strobing her pretty face into demonic, glaring red against the darkness. She turned to Corey and asked, her face in shadow now, "Any objection, cop's kid?"
"You wanna be stupid, go ahead." He turned and stared into the fire.
"She went off with Stanny," Wade said, "comes back alone with a gun. Uh oh!"
He was just kidding—but Corey had to turn and ask, "Where's Stanny?"
Heather fired once more, the gun jerking in her hand; the can leaping from the log. "He went off with some rude punk-rock bitch at the liquor store. I never saw her before...streak in her hair like that's wicked anymore..."
"Streak in her hair?" Corey turned to Wade. "Hey Wade—?"
Stanny's ears were filled with the soft rushing of the creek; the sound of his own breathing. Stanny's eyes were shut, while he was making out with Sira, so he didn't see it when her clothes simply evaporated away. But he felt her nakedness, when his hands wandered. He broke the lip lock and stared at her. "Whuh-oh- _man!_ That's all like the Olympic gold medal of getting undressed! Or... Shit I'm drunker than I thought... I mean..."
"Now you," Sira said.
"Me? Oh yeah!" He got up and hopped around, trying to pull off his boot.
"Forget it," Sira said. "Leave 'em on. It's kind of cute..."
She knelt before him, unzipping his pants. "Oh Jeezus," Stanny breathed.
"Yo—Stanny!" Corey's voice, coming from down the trail.
Stanny groaned. "No-o-o! Not again, dude! Please Corey dude, _please_ fuck off, please please pleeeeeeease fuck off! We need some privacy here!" He stalked toward Corey and Wade.
Behind him. Sira was already standing, turning, naked, grinning tigerishly at Corey and Wade as they trotted up into the pool of starlight. Foam formed on her skin, and quickly became clothing.
"You had to have seen that, Wade," Corey said breathlessly.
"Corey—I mean it, man! Go!" Stanny shouted, stamping a half-removed boot. He turned to Sira. "Ignore these fuckheads, baby—Sira?"
She'd slipped into the shadows. Melted into them, it'd seemed to Corey.
Stanny looked in four directions in under a second. "Oh SHIT Corey you did it to me AGAIN!" He started off into the night. Corey grabbed one of Stanny's arms, Wade the other. "Fuckin' lemme go!"
"Stanny, listen—" Wade said. "Corey's probably crazy, and I'm not sure what I saw just now. But, after what happened to Brad, you better hang with us."
"Hey fuck you guys!"
"Come on," Corey said. "Heather was all disappointed you ditched her."
"Bullshit... really?"
Angela found herself alone on the beach, watching a dark figure come toward her... a blue-black silhouette...
How'd she gotten here? She couldn't remember, exactly. A fight with Corey. Then one of the fits...
The figure coming toward her resolved into the shape of—a man. A familiar outline. He got closer and closer.
Her breath caught in her throat as she recognized him. "Dad?"
Marty switched on the flashlight he held in his hands. "Eden? That you?" He shone the flashlight in her face. "Angela? What're you doing out here alone?"
"I'm with some friends. I just took a... a walk. You followed me out here or something?"
"What? No! I had no idea you'd be here! I came here with Eden! Then she said there was an emergency she had to take care of, she'd be right back, but then she wasn't right back, then I went down the beach in the direction she went in—and here you are instead."
"So fine. I'm going back to find my friends. I'll see you later. I know what that girl looks like. If I see her, I'll tell her you're looking for her." Angela turned away.
Marty stepped in and put a hand on her arm. "Baby? This seems like such a... such a synchronicity. One of those coincidences that come from... from the universe. Couldn't we sit down and look at the stars and kind of talk?"
Angela shook loose. "No. We couldn't." She turned and ran into the darkness.
He followed, trying to run up a steep slope of soft sand, after her; the sand slid under his feet, and he stumbled onto one knee, dropping the flashlight. "Shit!" he got clumsily up, clambered up the slope onto the higher beach, and stood.
"Daddy?"
Angela was coming toward him, again. It seemed to him that she was dressed differently, though, and that wasn't possible, was it? Oh but all she'd done was put on that long gray coat. It must've been lying on the beach somewhere.
He shone the flashlight on her, and for a moment the beam—where it fell on her breast—seemed to go right through her, and he saw it illuminate a stone on the beach, behind her. But that was some kind of optical illusion.
She covered her eyes. "Daddy—put out the light. It's hurting my eyes..."
He dropped the flashlight, and she began to unbutton the long coat she was wearing. The flashlight, lying in the sand, shined its beam on the sea behind them, as if looking away.
Ganrich's unmarked car, an old Dodge Dart, pulled up behind Barry's cruiser, at the turnout for the beach park, and he got out.
"Hey, Barry," Ganrich said, waving a flashlight. "What you got?"
"Somebody was up here making out, they heard gunshots. Saw the muzzle flash down on the beach. Called it in on a cellphone." Barry hadn't mentioned noticing his own son's car parked up here.
"Gunshots on the beach. Probably not what we're looking for."
"But what else we got?"
"What else indeed. Let's go..."
They slid down the incline, to the creek through the gulley. Ganrich switched on his flashlight and they followed the weavework of paths toward the weak blotch of firelight. Two silhouettes formed up ahead, coming toward them. Ganrich didn't put the light on them till they were pretty close, so as not to startle them into running.
When he did, Barry saw his son's friends Megan and Wade.
"Hiya sheriff, whatsup?" Wade said cheerfully.
"Heard there was some shooting down here."
"Just that wack Heather," Wade said, "target shooting."
"Wade," Megan said. "You didn't have to say who it was. She's got enough problems."
"Oh, she won't be going to jail anytime soon for that, I wouldn't think," Barry said. "But I'll confiscate that gun."
"Well she already split. She's got her own car. She took Stanny in it somewhere. I don't know where. Home I guess."
"Didn't my son come out here with you?"
"Yeah—he gave me his keys. Megan's not feeling good. I'm taking her home then I'm supposed to come back with a flashlight and help him look for Angela."
Barry looked off into the night. "He's out there looking for Angela?"
"Yup."
"You got a driver's license?"
"I do. You wanna see it?"
"No. You been drinking?"
"Uh... Not to speak of."
Ganrich snorted.
"We'll take your word for it," Barry said. "This time. Take Megan home if you're sure you're sober. I'll find Corey. We'll pick up the car tomorrow."
"Okay, thanks sheriff," Megan said. They waved and, visibly relieved, went past the two cops toward the highway turnout.
"I... don't understand, Angela," Marty said, as Angela dropped the coat, and stood there naked—a silvery outline in the darkness. He couldn't quite see her face. But he could hear his heart beating.
"You don't? But Daddy... when you touched me before... all those times before... you seemed to know exactly what you wanted..."
"I... are you trying to shame me? We both know I made a mistake. I was... I was having a drug problem in those days..."
"It wasn't that. You wanted me. You've always wanted little girls. It's too late to go backwards, Daddy—what's done is done. The black mark is in my soul and can't be erased. It's a hole through my soul now, that you put there—and I want you to fuck it."
She came to him and took his wrists and put his hands on her breasts, pressed his fingers hard into the yielding softness.
He groaned—and she unzipped his pants, and took his stiff member in one hand, and with the other pressed his face to hers for a kiss...
But there was no face there. He'd seen only a glimmer of dark eyes in the shadows...and her blond hair blowing past...
Now there was only the rim, the rind of her face. As if the front of her skull had been sheered off, and everything from the neck up hollowed out like a jack o' lantern. _Her face was gone—and in its place there was only a sucking wound_.
She pressed his face into the hole where hers had been, and—with a powerful, unbreakable viselike grip on his little stiff prick— _imbibed_ him, struggling feebly, into her...face first. The whole essence of him. And spat out what remained.
There was only one, short, muffled scream.
"Angela!" Corey called. And waited. He thought he heard her call from somewhere. Faintly, her voice sucked away by the wind. "Angela! This way!"
Then he saw the pencil of light from a hundred yards away. A little spotlight from the sand, onto the water down there. He could just make out waves in its elliptical glow.
Corey began to jog down the beach, shouting for her. He didn't slow till he got within fifty feet and saw her turning toward him.
Was that her? Was that a long coat—it might be the other one. But no—he was mistaken, it wasn't a long coat. It was Angela's clothing, the shorts and oversized loose white shirt...
"Corey?"
She was walking toward him, her arms open. There was something on the sand, near the flashlight, about ten feet behind her now. Either a man sleeping in the sand, or a piece of flotsam that looked like it. It was hard to tell in the dark.
"You okay?" he asked, reaching out to her.
"Stay away from him!" Angela's voice—but from his left now. He turned, and saw her hurrying toward them, her hair streaming out behind her in the wind off the sea.
The other Angela had stopped—and now her clothes melted away, and she stood there, naked. Then she shifted—and a faint blue phosphorescence picked out her features, her face kaleidoscoping into Sira and Ilira and...
"Eden!" Angela, the real Angela said, coming to stand beside Corey—but out of his reach. And then her face went dark—was just a shadow with a mouth in it.
"Angela," hissed the succubus. "You called me, and I came." They were hearing the words in their minds. Corey could feel it plucking the words from his vocabulary, as if unseen greasy fingers were penetrating his skull, and he cringed at the feeling. "I have taken that revenge you cried out for. Each gave me more strength, in this world; with each I became more substantial. Now I have consumed the prize—the object of your hatred. Your father, little love. He's more than dead! I've erased him from the universe! That's what you asked for, Angela..."
Corey said, "Angela—your dad?"
"He fucked her, Corey! He forced himself on her! And Mama let it happen, didn't she, Angela!"
Angela's shoulders shook with her silent weeping.
"And you know, in Mama's heart, Angela, she knew you had the power—some primeval instinct knew you had the Other Thing, the thing that Montrose had too—not so strong in you, but enough. She knew, somehow, your mama, that you would kill your daddy! And that's what she wanted! So she let it happen—so it would destroy him. And we're all one merry little band of murderers together, aren't we, dear!"
Corey had a strange metallic taste in his mouth; his heart's drumming seemed to've merged into one long roll. "Angela..." He reached for her, the real Angela—and she took a step back from him, uncertainly.
"I don't want her to hurt you. But I don't want you to touch me."
"I just wanted to take you away from here..."
"You can't do that," came the sibilant voice. "She is mine now, Corey. I have, now, enough strength to make her one with me, as so many other women have been, over ten thousand years, since the day I was first summoned. She called me, and sent me out...and sometimes I found those she wanted killed, and sometimes I found others, and each time I grew stronger, and now I have the strength to make her mine. And then, to make you my sustenance, my boy..."
Corey realized his was panting, through his mouth, with fear; he was dizzy, near hyperventilating. He made himself breathe easier. He reached inside himself...
"You can't hurt me," Corey said. "Or you would have by now. There's something stopping you."
Corey started toward the succubus, just to see if it could be intimidated.
It backed away from him, hissing. "I have the strength now! In a moment, when I have her, you will have no more protection!" The succubus turned to Angela. "My dear—join us, and become immortal! Revenge yourself on all men—for every woman who was ever ground under a man's heel! This boy here—he's just like the one who raped you! That's all he wants with you! He's just like your father! Come to me...and _know the secret world_!"
Angela hesitated—and then took a step toward the succubus. "Corey—get out of here. Run!" She opened her arms to the demon.
"Stay away from it, Angela!" Corey shouted. Distantly aware that someone else was running down the beach toward them; two men, one with a bobbing beam of light in his hand.
"I... don't have anything to lose..."
Corey grabbed the flashlight in the sand—pausing only for a moment to stare at the body lying near it. Angela's father—clutching the air, frozen, rolling in the wind from the sea. A thing like Styrofoam, utterly empty, his blood already soaked into the sand.
Corey's stomach lurched. But he turned away—and shone the flashlight on the succubus... which whirled on him, hissing.
Her blond hair fell away, too, just seemed to dissolve, and in its place was a scaly glint, and wet, spiky-tufted ears—or something that looked almost like gills. Her eyes were just holes in her head; her face, except for the missing eyes, was beautiful, the way a statue can be. Heart-shaped and pale. Her breasts... had _beaks_ on them that opened and snapped and laughed. Her crotch was blank. Her feet were those of a giant bird, with three long talons, her hands matching.
"Holy shit," Ganrich breathed, running up, drawing his gun.
The succubus squealed at them like an enraged swine; he and Barry fired at the same time.
The creature's breasts only laughed at them.
Then the succubus raised her arms and the gesture raised the wind, a vortex that leaped up about her and whirled sand like a tornado, so that it flew against them—making Barry and Ganrich and Corey stagger back.
Ganrich dropped his gun—and Corey immediately scooped it up. He staggered through the wind toward Angela and the succubus. Angela was nearly in the thing's arms now. He raised the gun—and pointed it at Angela.
Anything to save her from becoming one of those things...
Then she turned toward him—and he shouted, over the shriek of the sandstorm, "Angela! How much revenge is enough? Is this enough?" He put the gun against his own temple. He heard his father yelling at him to stop, "I'm going to kill _myself_!"
He squeezed the trigger—
But his father had grabbed his wrist from behind, jerked his arm up—the shot went high. He felt a lance of white-hot pain across the top of his head, and his knees stopped working. He fell, and his father caught him...
Angela was staring at him. Astounded. "You really... you do. And I love _you_ , Corey."
" _Goddamnit son of a fucking BITCH!"_ the succubus screamed, as Angela knelt and put her arms around Corey, the succubus began to spin in her own vortex, to get thinner and thinner, longer and murkier and mistier...
The clouds parted, showing the thin scrap of moon that remained...
The substance of the succubus was drawn up, back into the barren seas of the moon whence she'd come... or so it seemed. The winds quieted. Somewhere, a gull shrieked, once...
Sick and weak from the graze, but not badly hurt, Corey stumbled between Angela and Barry, the two of them holding his arms. They followed the muttering Ganrich back to the creek, and the trail, and the world of cars and highways and emergency rooms.
"You want me to do _what?"_ Stanny said.
"Look," Heather said, very reasonable, as she sat behind the wheel of her car in the parking lot of the convenience store where her ex-boyfriend worked, "I've given you a gun, and I'm going to give you my body and my love and my loyalty. All you have to do is fucking rob a little store. I've got a ski mask from when I went skiing and everything. I mean—what else have you got going? Are you going to sit there and lie and say you don't want me?"
"No, but—I mean—it's almost like... like you're... seducing me to get me to rob your ex-boyfriend cause he got you knocked up and then blew you off... or some shit... and..."
"Well—so? You're gonna sleep with me tonight, if we don't get caught, aren't you? Right?"
"I am? For sure?"
She bent over and kissed him, long and hard and deep and wet.
Gasping, he broke free, and looked at her, and then at the gun in his lap; the stiff barrel of the gun was lying right beside the stiff barrel of his dick, its outline showing through his jeans.
Stanny grinned, and picked up the gun, and said, "What the fuck! Let's go!"
-THE END-
About John Shirley
John Shirley is a prolific writer who has published over 30 books and 10 collections. His novels include _Everything is Broken_ , _The Other End_ , _Bleak History_ , and _Demons_ , and seminal cyberpunk works _City Come A-Walkin_ , and the _A Song Called Youth_ trilogy of _Eclipse_ , _Eclipse Penumbra_ , and _Eclipse Corona_. His collections include the Bram Stoker and International Horror Guild award-winning _Black Butterflies_ , _Living Shadows: Stories: New & Pre-Owned_ and _In Extremis: The Most Extreme Stories of John Shirley_. He also writes for screen (The Crow) and television. As a musician, Shirley has fronted his own bands and written lyrics for Blue Öyster Cult and others. In 2013 Black October Records released a two-CD compilation of Shirley's own recordings, _Broken Mirror Glass: The John Shirley Anthology - 1978-2012_. See www.john-shirley.com for more information.
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Q: How to force controller/action to use JsonValueProvider Good day!
I'm using ASP.NET MVC 3 where JsonValueProvider is built-in.
The problem is that it works only if Content-Type: application/json specified.
I'm building an API where JSON is the only supported format and I don't want to force clients setting this header.
Is there any way to force controller/action to use particular ValueProvider?
A: Although this works and is very simple, it seems more of a hack. What you can do is create an attribute deriving from AuthorizeAttribute and in OnAuthorization you set the Content-Type to application/json.
public class JsonActionAttribute : AuthorizeAttribute
{
public override void OnAuthorization(AuthorizationContext filterContext)
{
filterContext.HttpContext.Request.ContentType = "application/json";
}
}
[JsonAction]
public ActionResult JsonOnlyAction(string var1, int var2, ...)
{
...
}
This attribute can also be applied at the controller level.
Originally I tried setting the Content-Type in an action filter but the problem is OnActionExecuting occurs after value providers are selected so setting the content-type there is too late.
OnAuthorization occurs before value providers are selected, and since the JsonValueProviderFactory checks for Request.ContentType.StartsWith("application/json") this will ensure it's selected.
A: I'm sorry if I'm misunderstanding the question, but isn't this what you need to do:
Response.ContentType = "application/json"
You can create an action filter/attribute to avoid repeating this line in multiple actions.
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"redpajama_set_name": "RedPajamaStackExchange"
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# MEET ME IN BUENOS AIRES
_A Memoir_
**Marlene Hobsbawm**
For the joyful grandchildren:
Roman, Anoushka, Wolfgang, Milo, Eve, Rachael and Maxim.
# Contents
1. _Title Page_
2. _Dedication_
3. _Prologue _
4. _Introduction: Claire Tomalin _
5. 1 Vienna Beginnings
6. 2 Wartime Children: The Émigré Child and my Education
7. 3 Secondary School, Teens and Young Adults at Home
8. 4 A Student in Paris
9. 5 Working in London and Secretarial College
10. 6 My Dolce Vita
11. 7 The Congo
12. 8 Back Home and CBC
13. 9 Meeting Eric
14. 10 Being Married
15. 11 A New Family
16. 12 Clapham, Part I
17. 13 Clapham, Part II
18. 14 Our New House
19. 15 Home, Sweet Home
20. 16 Academia
21. 17 Sabbatical in Latin America _en famille_
22. 18 Bourgeois Life in the Seventies
23. 19 Martin's 'At Homes'
24. 20 My Music Career
25. 21 Manhattan
26. 22 Our Italy
27. 23 Mature Lives
28. 24 Age of Glory
29. 25 Hospital Years
30. 26 Death
31. 27 What Remains
32. 28 Eric's Legacy to Me
33. _Acknowledgements_
34. _Photo Credits_
35. _Plates_
36. _Copyright _
# Prologue
Becoming a singleton after fifty years of marriage was an adventure of its own. And starting the endeavour of writing a memoir at eighty-five was quite optimistic, as my vocabulary and memory are diminishing.
I didn't set out to write a memoir: it was a project that unfolded slowly. Once the Cambridge historian Sir Richard Evans expressed his interest in writing my husband Eric's biography, I naturally became very involved. This seven-year undertaking stirred up memories of my own life – from times before, with and after Eric. Once I began, I found myself compelled to continue, but above all, I wanted my grandchildren to know about my life.
Not being a historian, I was more interested in remembering life from another angle – piecing together a record of family, friendship, travel and an unwavering love between two unlikely individuals.
# Introduction
The Hobsbawms – Eric and Marlene – always seemed to me an example of a perfect couple, mutually devoted and at the same time welcoming friends to their house with extraordinary generosity. Invitations to dinners came regularly, to Sunday lunches, to visit them in their cottage in Wales, to celebrate at memorable parties. With Marlene and Eric you always enjoyed yourselves – good talk, good jokes, good arguments, good friends, good food, good wine. And Marlene was cook and organiser, while also teaching music, raising the children and dealing with every sort of practical problem thrown up by family life and marriage to a hard-working and celebrated husband.
I knew Eric by reputation when I was an undergraduate at Cambridge, and my friend Neal Ascherson was taught by him and spoke of him with admiration. While researching my first book in the early 1970s, I therefore consulted Eric on some historical points. My subject was Mary Wollstonecraft, who proclaimed the rights of women in the 1790s, and my impression was that he was not exactly a feminist, while being more kind and helpful in his response than I could have hoped, as indeed he was whenever I asked for advice or help. He also wrote the occasional review for me when I worked on the _New Statesman_ in the mid-seventies, and this was when I went to their house in Hampstead for the first time and met Marlene. I thought she was amazing – and we formed a friendship that has strengthened ever since: forty-five years, I make it.
I realise now how little I knew of her early life, although we were almost the same age and both had continental origins – I French, she Viennese. I understood she had been brought to England as a small child when her father, a successful businessman, saw how dangerous Hitler was. I once asked her about learning a second language and she gave me a charming account of how she had refused to speak a word of English for a long time – I suppose it was a form of protest at being uprooted – but that one day her mother, listening outside the room where she was playing alone, heard her addressing her dolls in perfectly good English. The stubborn, clever child predicted the versatile and charming woman.
Reading her memoir I realise that she and I shared the experience of being sent off to boarding schools we did not always like – she ran away, I fell ill – and that as young women we were both exploring Paris at about the same time, studying French history, reading Gide and Colette, entranced by _Les Enfants du Paradis_ , Prévert, Charles Trenet, and the streets, bridges, parks and paintings in the galleries.
We both also went through the then almost obligatory ritual for young women of learning secretarial skills. After that she, with immense dash, took herself to Italy and found jobs in Rome and Capri, lived a life of utmost sophistication, and moved on to work in the Congo for a year. Adventurous, brave and loyal, she made friends wherever she went, and kept them.
She was twenty-nine when she met and married Eric. He was already known as a scholar, teacher and historian whose books became instant classics, and was greatly in demand as a speaker all over the world. Marlene often travelled with him while also bringing up their son and daughter and establishing a home life that gave him time and space to work. 'In my head I'm a continental woman,' she writes: I think this means she knows how to combine domestic and intellectual life and make it look easy.
But it is not easy. Marlene gave Eric the support he needed to achieve greatness while always remaining a strong and decisive person in her own right. In effect, she taught him by her strength to respect feminism. Her memoir, starting from the sadness of exile and war in childhood, reveals her on every page as enterprising, courageous and warm-hearted – and is a delight to read.
Claire Tomalin
July 2019
Chapter 1
# _Vienna Beginnings_
I was born in Vienna in 1932, the third and youngest child of Louise (Lilly) and Theodore (Theo) Schwarz. My older brothers were Victor Hugo (Vicky), who was five years old and Walter, two years old. On the whole, Mother brought us up mostly as _die Kinder_ – the children – and I liked being lumped together like this even when not keeping up. I felt safe and happy. We had a nanny and a maid, and lived in the leafy suburb of Döbling.
My father, a middle child of ten children, came from Innsbruck. He was lively and not the type to stay put in the Tyrol. As a young man he enterprisingly got himself a job in the luxury hotel business in Paris, where he completely fell for the city, the French, and the cosmopolitan life around him. Later on, he became a businessman in the textile industry. He must have had a flair for it, as we lived well. I believe he had a good reputation and he enjoyed his work, especially travelling and making contacts all over Europe. He spoke several languages and was very interested in politics. Mother had to save all the English newspapers for him when he went away on his business trips. His mother, Grandma Rosa, was our only living grandparent. She regularly came to visit us in Vienna and we had holidays in her house in Igls, up in the hills above Innsbruck. She had a reputation for being difficult, but my mother liked and respected her.
My mother Lilly was born in Vienna, the youngest of five, and was such a late arrival that her siblings were already aged eighteen, seventeen, sixteen, and eleven. Her nearest sister Emmy was responsible for most of her upbringing. Mother remembered a school friend saying, 'You don't have to do what she says – she is only your sister.' Lilly did not have a career and was twenty-one when she married my father. Their roles were pretty clear: Father was the teacher and Mother the pupil. Jumping ahead twenty-five years, my own marriage followed a not entirely dissimilar pattern.
My nursery school was located in the basement of our building, which must have felt cosy, and it was run by my mother's niece. I remember going down long steps willingly and coming back up again. I was about three years old.
During this time, which seems like idyllic family life, my parents had many anxieties, as they were planning to leave Austria for good. My father believed the things that Adolf Hitler was saying in his speeches and he realised the dangers so close in Germany, unlike many of our Viennese friends and family, who never believed Hitler would carry out his plans. My brothers were being prepared for this dramatic change, especially my elder brother Vicky. But nothing was said to me. I was considered too young and was to be protected. My mother and brothers were already having English lessons. My brother Walter (who became a journalist) has written his own memoir _The Ideal_ _Occupation_1, which depicts this early part of our lives extremely well and in more detail.
1 Walter Schwarz, The Ideal Occupation (Brighton: Revel Barker Publishing, 2011), pp. 6–30.
Chapter 2
# _Wartime Children: The Émigré Child and my Education_
We arrived in London as émigrés when I was five. Our home was a flat in a large mansion block in Hammersmith. During the day, the dark, empty garages in the basement were where the children of the flats used to play, and we all joined them. One day, during that horrible game of hide-and-seek (while both my protective brothers were at school), I became very scared when it was my turn to find the other children. Even though I had known so well in advance that I couldn't find them on my own, I just couldn't say it and began sobbing instead. That rather traumatic experience stayed with me, and I got into the habit of often preparing and forestalling to stave off disaster. 'Marlene, why are you worrying about that _now_?' people say. The answer would be: I believe in the power of worry. It works.
I was sent to a Froebel nursery school2 and refused to speak for the whole year I was there. The teachers said I seemed settled and spent my time mainly playing in the sandpit. But I had become a self-imposed mute. Maybe it was a fear of not speaking English well enough (I think I was confused about being a German-speaking child at home), or it could have been stubbornness, or maybe I was upset at not being a proper English girl like the others. Being unprepared meant everything came as a shock, and I think it was the beginning of the University of Life for me.
In I938, a year later, we left London to live in Manchester, where I was enrolled in the junior school of Manchester High School for Girls. However, with the outbreak of war in 1939, all children were evacuated; Vicky was sent to Blackpool and Walter and I to Uttoxeter in Staffordshire. I can see the huge room full of children sitting on the floor with gas-mask cases and belongings. We were supposed to go to our families in pairs, and I was expecting to be with Walter, when suddenly a lady announced that there was a family who would only take one child and 'would anybody volunteer'. My heart sank because I knew my brother Walter's hand would shoot up, and indeed it did. I don't remember any goodbye. And so we were all dispersed. Only Mother and Father were together.
My foster family lived in a small, modest house and another child my age also lodged there. My recollection is that everything seemed brown, both inside and out. I began to lie like a trooper, telling them all sorts of imaginary tales about my real family in Manchester. I invented a baby brother and told them a lot about 'him'. Even though I knew my parents were coming to visit very soon, when I would be unmasked to one and all, I still couldn't stop. 'Should I get nappies?' asked my foster mother, 'Oh yes,' I said, 'Definitely.' Later my mother told me she just couldn't make me out, and I could not explain. Life was incomprehensible to me. I found that praying did no good whatsoever, but instead my rosier fantasies consoled me. And they still do, but now I keep them to myself. Psychology must have been unknown to my folks at that time, despite us being a typical Freudian nuclear family from Vienna.
We could attend a nearby school on alternate days to the local children. I can remember an open shelter outside, but of the inside I remember nothing. I was in a trance. Walter went to his foster family at the end of the same village but we never coincided. One day, on my own, I walked to the large mansion he was staying in (with a Rolls-Royce in the garage) and rang the bell. A servant came to the gate with a very big dog and I asked to speak to my brother. I was told to wait right there. Walter soon appeared and said, 'What do you want?' and I was suddenly struck dumb, answering, 'I don't know.' In my head I had prepared to ask, 'Tell me again exactly why we are here.' He just replied, 'If you don't want anything I will continue what I was doing,' or something along those lines.
Luckily we were not evacuated for long. There was a military miscalculation (not unknown in wartime) and when the Blitz started in earnest – when the bombs really fell – we were all back again in our own homes in the big cities, ready targets for the whistling bombs.
But the Blitz was a good time for me because of the nightlife in the bunks in our well-kitted-out cellar. We were all of us together, and that is the only thing I wanted. We ate delicious food that Mother had prepared during the day and it all seemed very jolly to me, not to mention my lovely new red boiler-jumpsuit (only to be worn during air raids). The long raids provided another advantage. Children under a certain age, like me, did not have to go to school the following day and I spent my time with the women in the house: my mother, our Irish cleaner and one of my father's sisters, Tante Hedi, who used to shriek, 'The veesling bombs!' when she heard a noise. I also played with some of the children of the large Irish family across the road. But I knew my mother was worried about me. She wanted the best for me and so did my father. One of the children in our continental circle of friends, an only child of about ten called George, was apparently happily settled in an English Quaker boarding school in Wigton, Cumberland. It had a good reputation and was known as Brookfield.
Alas, my parents, thinking that an English boarding school was synonymous with the best in life and where I would make many English friends, thought it would be a good idea for me. But it was the worst idea ever.
I was nine when I was sent to the Friends' School, Brookfield. It must have been around this time that I began to switch off from even trying to understand what was going on in my world, which had once again been turned upside down. What the heck was WAR, anyway? Had there been television I probably would have fared better. None of it made sense to me. It was beyond me to figure it out. I turned myself from out to in and became a very un-inquisitive child. I continued with my vague notion that grown-up minds are different and I had better think for myself.
The boarding school was situated in the beautiful Cumbrian countryside and I remember lovely long walks on Sunday, to and from church. But I was very unhappy to be there in spite of it being an interesting school. All letters home had to be shown to our teachers first. But during the holidays I told my parents about desperately wanting to leave and come home. My father said he would find an alternative and meanwhile I should give some letters to the day children to post for me. He sent me stamps, which I managed to hide safely, so I could write freely – I can see one of my letters now: _I carn't bare it_ (sic). And father wrote back to be patient, but I wasn't.
I say I was 'unhappy', but of course children carry on with their lives in unhappiness. I have some letters of mine, which seem quite typical of any boarding-school girl writing home about her activities and most likely I stood in the queue for my skipping-rope turn just like everyone else. But then came the drama.
Towards the end of my second year at Brookfield, I ran away. Not only that, but my worst crime was that I went with a much younger girl! We got very far, reaching the train station, when they grabbed us. In any case the strategy I had in my head would have been doomed: it was to get off the train at any station and ask the way to the nearest police station. I knew my address in Manchester and was convinced that we would be helped – 'That is surely what policemen are for!' I had prepared to say to the police that if they helped me get home, my parents would definitely reimburse the fare. What panic there must have been at the school had we succeeded even a bit. There was an uproar and my parents were immediately summoned; finally they grasped my situation. I think a dog or a cat might have made a better fist of getting back home than I did.
Running away from boarding school brought about the end of my dramas. I returned to the junior school of Manchester High School for Girls as before. My mother later told me that once I was back there and wearing the familiar red-checked dress and panama hat, and living at home, they seemed to have no more trouble with me. Maybe all my difficulties had finally been expelled. I sometimes think my generation of émigré children, who actually lived through this experience, felt better in some ways than the next generation, who could only wonder at the relayed stories of another past in another country. Some parents didn't want to talk about the past at all. My brother Walter, on the other hand, always rejoiced and felt blessed and lucky about being in England, and had nothing but praise for the foresight of our dad, who had the wisdom to get us away in comfort and ease when it was still possible to do so.
Vicky was ten when he came to England, and integrating was different for him. He must have had much deeper memories of Austria than Walter or me. He had the sunniest disposition and was used to being adored by his teachers in Vienna. He and Walter were sent to Colet Court Preparatory School for St Paul's in Hammersmith, and I sometimes imagine how he might have felt on his first day there in the playground. _'WHAT did you say your name was? SCHWARZ?!'_ All conjecture on my part, but maybe around that time a tiny seed was sown which grew with his desire to be like an Englishman and above all, a gentleman. When he became an adult he decided to change his name to Victor Black (by deed poll), as when applying for jobs he was advised against a German name.
2 Friedrich Froebel (1782–1852) established the first kindergartens (and coined the term), using a play-based learning system.
Chapter 3
# _Secondary School: Teens and Young Adults at Home_
I am writing this story of our teenage years now in my ninth decade and as the proud grandmother of five grandchildren who are now around the same age as Vicky, Walter and I were then. It's difficult to compare the lives of these sophisticated Londoners who spend their time on smartphones and other screens, and are connected to the adult world 24/7, with us provincial children in wartime, living at half the pace. I can only hope for their future in a world more dangerous even than ours. Back to 1943, we are still at war and I am eleven, with a Lancashire accent. I went through to the senior school and at sixteen I passed my School Certificate (now GCSE). I can't honestly say I remember any particularly inspiring teachers or even any subjects with enormous enthusiasm but I suppose 'literature' as it was then called would be the strongest candidate. But even so I came away knowing no poetry or familiar books, except _Silas Marner_ , which was permanently on the syllabus. I loathed more than anything in the world having to play hockey in the mud on Thursdays. I considered it barbaric. In the freezing cold our legs turned red, blue or purple. But a school medical found that I had a congenital heart murmur (hurray!) and so mercifully I was released from that immediately. My friends were very jealous.
My best friend was Leila Yael, who was the cleverest girl in the school, and although she eventually moved to Argentina, we have remained lifelong friends. Both of us named our first-born babies Andy. I was never short of friends but having two handsome brothers (who many girls already liked the look of) might have a part in this. Much later on, in my twenties, I was closer to Leila's sister Yvonne, who lived in Europe. Now as grannies, we all three correspond and meet, and any differences are trivial.
My brothers, who had been to 'South' prep school, were now already at Manchester Grammar School, which was not very far away from our Manchester High School for Girls. I think the interaction of the two schools and their pupils was similar to today. But maybe more money was spent, as I remember in addition to school concerts and sports, proper dances were also organised. I looked forward to them with both trepidation and anticipation. Again, having two brothers was a great advantage. I think one of Walter's friends had a crush on me, but I liked a quite different boy. In any case, we were all totally preoccupied with the impression we ourselves were making. Some children were already dating, and the girls would ask each other, 'Did you go all the way?'
Both my brothers were very good at school. Vicky was again popular with teachers and with classmates, and his glory was being the excellent goalkeeper of the school football team. He was eventually offered a place at Trinity College Dublin. For Vicky, having a good time was a high priority. He believed in it seriously and pursued it to make it happen–sometimes it felt like he was curating fun. Maybe this comes from the experience of being taken away from the glorious life that he knew in Vienna.
Walter was the family's scholar and one of those rare children who knew what he wanted to be when grown up: a journalist. He managed to finish his School Certificate exams with half a year to spare and later, after obtaining his Higher School Certificate (A levels), he won an Exhibition to Queen's College, Oxford. Mother's Viennese dreams about English education had indeed come true. She brought it up during one of our last conversations when she was already very frail.
School is all very well, but it is at home, within the family, that children spend so much of their lives and do most of their growing up. What sort of family were we? We were assimilated into British society and did not observe Jewish religious customs, but we were also consciously Jewish. We kept up our German-Viennese language and culture and at the same time fully absorbed all that we liked or disliked about England. Assimilation is usually easier for the children than the parents, and so it was with us: the fact that our Vic was the goalie for the Manchester Grammar School football team completely passed my father by, until one day he suddenly found himself a hero amongst his business colleagues over lunch.
We lived in Moorfield Road, West Didsbury and, after our initial short-lived evacuation, remained there throughout the war until 1953, when we all felt that London was the place to be. The family then moved into a three-storey house in Golders Green, which agreeably backed on to Golders Hill Park.
I think the atmosphere of our household was jolly and united. Passionately hating Hitler played a part in that. Vic, being the eldest child and with the sunniest nature, had, I think, the most admiration from our parents. Mother's favourite was Walter, as she couldn't resist his dreaminess and his earnestness. I was Daddy's girl, no mistake about that. There was rivalry between the boys, who were very different. Sometimes Vicky (who could make me do _anything_ ) and I ganged up on Walter. In turn and with ease, both boys cheated me with the household chores, and I was often teased. The tension between my brothers remained through their adult lives; they were just too different, but became most devoted fathers themselves. Mother's love was practical and father's sentimental. Mother usually understood my struggles, whatever they were, and always helped me. There was solidarity between us; we were a little team of our own, though I wish she had not been so successful at instilling her 'work before play' ethic in me. She was happy when I was, but I had to be useful too and pull my weight.
She was absolutely not a feminist. Her own upbringing had been too traditional. Boys' education was more important. If you wore trousers, you counted double. It didn't really harm me, apart from my education, of course, or rather my lack of it. Eric never quite managed to forgive her for that. When I wanted to give up Latin, it seemed OK by my parents. But now, seventy years later, I find myself the only person in my choir who can't sing in Latin. Luckily all the _Agnus Deis, Glorias, Jubilates, Hosannas in_ _Excelsises, Hallelujahs and Miseres_ can carry one along for pages and pages.
Without television, the wireless was the glue in our lives. We of course followed the news intensely. During the war Father had pinned up a large map with little flags marking the progress and defeats of the Allied troops every day. We loved the BBC, and the beautiful spoken language that came from it. I remember a weekly radio programme, _Monday Night at Eight_ , which we regularly listened to as a family. This show consisted of interviews with all types of people, a detective play, _Dr Morelle_ , a 'Puzzle Corner' and a 'Deliberate Mistake.' I think all England listened, and so the whole country talked and joked about the same people.
As well as the wireless, at home we had books and the gramophone. We also went to the cinema. When later I was allowed to go on my own (what bliss) I saw _Lassie Come Home_ four times. Films were quite different then: the normalities shown revolved around the wholesome family. For example, a couple could not be shown in a bed together unless they were married. And there were no gay people.
Father, from the Tyrol, was an organiser of excursions; what could be better than the Peak District on our doorstep. It felt as if we visited everywhere. We also went on holidays to North Wales, as father was keen on climbing. I climbed up Cader Idris, but went up Snowdon by train. We had a family whistle that was often used – and needed – to keep us together. We also had many holidays in the beautiful Lake District, especially Derwentwater near Keswick.
There was always a large circle of émigré and refugee friends whom we saw regularly. My father was considered a bit of a leader, or more respected because he was better informed. Some were frightened of rumours that Jews were going to be repatriated or such nonsense. My parents became bosom friends with Lilly and Ernst Stiglitz, also from Austria, and very cultured. They were an inseparable couple who had no children and loved spending time with us. They lived about twenty minutes' walk away and every single Sunday they either came to us or we went to them. Such was our cosy and predictable life. We had an allotment not far away, by the River Mersey. All of us émigrés were 'Digging for Victory' with gusto, even though we didn't know much about gardening.
Father was a gregarious man and made new friends easily, including many English ones. My parents knew a wide circle of people, including the historian A. J. P. Taylor, who also lived in Manchester then. My mother was a good housekeeper and produced lovely party food despite the shortages and the rationing. Our house was usually full of people, friends and also many relatives; some from London came for a respite from the bombing there. They also stayed for longer periods with us, like semi-lodgers, and had to bring their ration books.
Our status in England was as 'enemy aliens'. Father should at this point have been sent to the Isle of Man to be interned with many others, but a medical report came back positive for diabetes, so he stayed at home. He was not diabetic; it must have been a urine sample that got mixed up. We were safe but also had to be careful. There was an incident when Vic had left a tiny slit in the blackout curtains in his room and because of this we were reported, then inspected. But luckily it all ended with a smile at a schoolboy's slapdash oversight. Father, by the way, who was politically mostly on the right, idolised Churchill, who was his hero. It was common to support him at the time, as Eric notes in his biography, 'Churchill was associated with heroism.'
Mother was a good employer and usually had help with the housework from girls who were devoted to her. It wasn't a special skill, but her genuine interest and respect for every human being disarmed them. They came mainly from Ireland.
I didn't know until much later in life about my father's infidelity. Some of it only amounted to him flirting with nieces quite openly. But he had a French girlfriend who lived in Switzerland. The business trips must have made it easy for him, but onerous for Mother. I think she took it in her stride and, for all I know, tried to make it fit with her un-feminist views that 'men just are like that'.
When the war was finally over, it seemed obvious that the Victory Day party in 1945 would be held at our home. My father wanted us to wear Austrian dress, me in a dirndl (oh dear), in order to emphasise our gratitude to England, and I obliged.
Classical music played a huge part in our lives, which was the same in both cultures. Mother had a good soprano voice and loved opera best of all. When opera companies came on tour, we often went. My first opera was _The Bartered Bride_ by Smetana. We were regularly taken to the Hallé Orchestra concerts, possibly every week, especially when Sir John Barbirolli himself conducted. This was our Manchester orchestra and we were by now loyal Mancunians. There was a baby grand piano at home and we children all had piano lessons. Walter was the most passionate about music, and from time to time he turned our sitting room – as best he could – into a concert stage. His imaginary orchestra always had their same places – strings by the fireplace, woodwind near the door, and so on. He then proceeded to elaborately conduct from the score of a Beethoven symphony, which he also had playing loudly on the gramophone.
Walter also learned to play the recorder rather well, and had a school friend, Peter Noble, with whom he played duets at home. That was my first introduction to this lovely instrument, which many years later became my own.
Growing older, we were slowly gearing up to leave school and the nest, and thinking about our futures. Walter was called up for his military service after his time at Oxford and was sent to Malaya. Wherever he went, his thoughts were mainly about writing it down on paper in preparation for becoming a journalist. He did this when writing home to us as well. But I recall Mother and I crying our eyes out when we had taken him to Liverpool to board the gigantic steamer for Malaya and war. Walter did indeed become a very well-known journalist and spent his life being a foreign correspondent for the _Guardian_ newspaper in Nigeria, Israel, India and France, all written about in his entertaining memoir. He married a woman named Dorothy Morgan and they had five children together: Habie, Tanya, Ben, Zoe and Zac.
Our Vic's story is different. He not only believed in FUN, he engineered it. Goodness knows where his gift for pleasure sprang from in a cautious, earnest émigré family like ours. When I was about fifteen, I remember a weekend when Vic invited the entire chorus of the D'Oyly Carte Opera, who were on tour in Manchester, to a party at our house. My parents were away on a trip and Walter was also away. I was quite used to Vic's girlfriends being very kind to me and I liked their company, but this party, I felt, was boring, because lots of them went to bed early: I was a late developer and pretty naïve about sexual matters, let alone orgies. Different from some of today's savvy girls. When our parents returned, the house was all tidied up, the laundry done, and everything was in immaculate condition, except one of mum's devoted cleaners was in tears, repeating hysterically 'Only for you, Mrs Schwarz, I stayed in this house, only for you.'
Hedonism got seriously out of control at Trinity College in Dublin, where Vic only stayed one term as he had spent all the money that Father had given him for his three years of study. How foolish was Father – and Vic. All spent on women; the money was gone. He enjoyed being a chivalrous gentleman with his friends. His heart, however, was in politics and he did support work for his local Conservative association and was particularly interested in helping young offenders. He was excellent at this. He later joined and then took over Father's textile business, at which he was very good. I think it was also helpful for Father who, tragically, was hatching Parkinson's disease. The responsibility both changed and suited Vic. And years later, when our father died and mother was alone, he looked after her financial affairs and also continued his gallant ways, such as collecting her from the airport, and making her feel she still had a man to lean on. It gave her such enormous reassurance and pleasure.
Vic came into his own as a family man with the pride and pleasure he took in each and every one. With his wife Oonagh and four children (Emily, Isobel, Charlotte and Humphrey), there was plenty to report, and the news always seemed to be something marvellous; I know he helped with homework and essays because once he mentioned triumphantly on the phone, 'We got an A.' He took his family to Austria for holidays when he could, much more than the rest of us. He hankered after mountains, lakes and scenery, probably liked speaking German and could taste and introduce to his family Po _vidla tascherln_ (cheese dumplings), _Kaiserschmarrn_ (a fluffy shredded pancake) and _Palatschinken_ (jam pancakes) – favourite desserts from early childhood.
Chapter 4
# _A Student in Paris_
I am now sixteen. Father had always talked to me about 'abroad' and I must have got my itch to travel from him. I already loved Paris before I ever went and remember he had taught me the song from the opera _Manon_ , which he loved: ' _Nous irons à Paris tous les deux._ ' After leaving school at sixteen (going on seventeen), I went to Paris to learn French. As this was 1948, I think I might have been one of the first ever _au pair_ girls.
My first charge was four-year-old Irène and we had heaps of fun together every day, the highlight being eating pain au chocolat at four o'clock in the Jardins du Trocadéro. I loved her, but not the family, who called me 'Mademoiselle' all the year I was there, never Marlene. They were rich; we lived in a grand penthouse in the Avenue Kléber, but we didn't really click. Looking back, I think they were my first 'nouveaux riches' acquaintances, and I didn't play my cards right at all. When feeling lonely, I would sometimes chat with the cook in the kitchen, which was very frowned upon.
I had enrolled in the two-year course at the Alliance Francaise to study the French language in the first year and French literature in the second. The first year was all in the Boulevard Raspail and in the second year the course tied up with the Sorbonne University, where many lectures and events were held. At seventeen, and with a bounce, some education began at long last. I was captivated. One of my teachers in the first year, who liked me, was expecting a baby and asked me to commit to being her au pair in my second year, which I did. It was an important and very wise decision. Nadia turned out to be as interested in my French education as I was. I became a student completely engrossed in my studies and the culture around them. She treated me as her daughter.
My duties with the baby were mostly only until lunchtime, and much of that was going to Parc Monceau and reading while he slept. Baby Michel was an absolute angel, and I adored him. If I were an artist I would be able to draw his smile this very day, sixty-eight years since I saw it. I can always conjure him up.
The Hugons lived in the Rue de Rome and Nadia's husband Bernard, who worked in Lille, only came home at weekends. They then liked to be together at home, so I was not expected to do much babysitting even at weekends. I was free to study, go to my classes and wander endlessly around the city. I had plenty of time for gallivanting with my friends and getting glimpses of the so-talked-about Paris by night. Every weekday I took the bus to Boulevard Raspail, not tiring of looking at Île de la Cité, The Louvre, the contours of Notre Dame, the river Seine and its bridges: Pont St Michel, Pont Neuf and Pont des Arts.
The old-fashioned Alliance Française was a serious place with a very good ambience. Most of us were foreigners (several Scandinavians) and we were not all grown up. We were like A-level schoolchildren, but not at home.
Our reading list was not all ancient or difficult. Certainly Racine had to be there, _Andromaque_ and _Phèdre_ , and Molière's _Le Misanthrope_ , but, in an enlightened way, the syllabus concentrated on more contemporary plays like Anouilh's _Antigone_ so that we could engage with criticism on a controversial work and further debate among ourselves. It was wonderful to study the plays people were talking about in cafés. Being in touch seemed very important to Parisians and I wanted to be too. We then went on to study Marcel Pagnol's _César, Fanny_ and _Marius,_ all of which we had to read aloud in class in a _Provençal accent_! Really, they wanted the students to learn about the Midi region and get the _feel_ of it. The Alliance Française's fresh approach to education was quite unique, unusual and certainly new.
A bunch of us took a trip down to Cannes and for a lark we flagged cars, shouting, ' _Sales capitalistes_!' at the tops of our voices. We considered ourselves left wing, though what we knew about politics that sunny day could fit on a postage stamp. We also sang together; our favourites were Edith Piaf and Charles Trenet, and the favourite of all was Yves Montand singing 'Les feuilles mortes' by Jacques Prévert. I sent a letter to Walter full of excitement that I was now on a level to discuss literature and other issues with him. It was meant to impress him with my new sophistication in general. I look at the envelope and smile.
My own personal reading continued with two brilliant tomes of André Gide's journals which I read in daily instalments. For sheer delight and relaxation, there was Colette – the _Claudine_ series, and the _Cheri_ books and _Gigi_ , among others. I was also inspired by Simone de Beauvoir and wanted to copy her self-contained studious ways of being alone in public places. With Sartre, I only liked the existential bit that focused on the absurdity of existence. But at the same time, I gravitated more strongly towards the mood and romance of the city. I was 17 and it was springtime in Paris. I remember the enchantment and feeling overwhelmed by it all. How could one be serious about politics anyway, when you have just seen the film _Les Enfants du Paradis_ for the first time. It remains a timeless romantic epic and has since been voted the best French film of all time.
Chapter 5
# _Working in London and Secretarial College_
I came home to England with good French under my belt but no obvious employable talents, and I was nineteen. Mum and I made the next decisions and I enrolled at the Marlborough Gate Secretarial College in the Bayswater Road London for a year. The skills I learned there led to my first ever full-time paid job.
Who would have thought an unbeliever like myself would be thrust headlong into the ecclesiastical world, even though I had always loved singing Christian hymns? I enjoy some Christian verse, and I have found it useful to have a copy of _Songs of Praise_ as a reference book at hand. In my new job I had become in charge of the subscription and distribution of a Church of England periodical called _The Pulpit Monthly_.
Each issue contained four ready-made articles and ideas to inspire clergymen for their Sunday sermons. Sometimes they panicked when it failed to arrive, as they hadn't yet thought of anything to say. It seemed that I corresponded with all the parish priests, vicars, chaplains, ministers, pastors and rectors of England. They depended on me. They were invariably charming, and sent kind words and little presents too, as though I had written the sermons myself. The office was in the city, right next to St Paul's Cathedral, where I sometimes spent some of my lunchtime, and my lodgings were in Putney, in a beautiful flat overlooking the Thames.
I was consulted by my landlady about which applicant should occupy the second bedroom in her flat. The choice was between three: a Swedish girl, which sounded too unfamiliar, a French boy, which would be useless, as there was only one bathroom, and an Italian girl. I chose her, not knowing she would alter the course of my life. Mariella de Sarzana and I got on very well. We became friends, I invited her to come home to spend Christmas with us, and in the summer, during my holidays, I was invited to visit her in Rome. Now in my early twenties, I went all the way by train, a long journey through France and Switzerland. At the Stazione Termini in Rome, Mariella was waiting for me.
The very first sentence she excitedly and breathlessly said was, 'I need to go to Capri tomorrow and see some people, and find Kirk Douglas. Will you come with me, pleeese?' Of course I said yes. Mariella knew everyone. Gracie Fields was there and I remember meeting her. Mariella was besotted with American films, and was making contacts to get herself, by hook or by crook, to Hollywood, which she eventually did. In Capri we found Kirk Douglas and travelled back to Rome with him – he flirting and playing footsy under the table with both of us.
The island of Capri is surely one of the most beautiful in the world. The higher part is Anacapri, with the villa of San Michele, which the doctor and author Axel Munthe had built at the turn of the century. It is in grandiose, palatial style amidst huge and flamboyant flora, and has its own museum.
The lower part of the island, which descends all the way down to the sea, is a holiday resort with taste: grand hotels with pools, and smaller ones too. There were luxury shops and people running around in Emilio Pucci's 'palazzo pants'. At the end, just off the coast, are situated those two majestic landmark rocks surrounded by sky and sea blues, but not blues I had ever seen before. I was overcome by the colours and felt my life had only been in black and white until that day. It was in part because of this visit that I fell in love with Italy hook, line and sinker. I had the melody 'Isle of Capri' on my brain.
Chapter 6
# _My Dolce Vita_
My first trip to Italy had such an effect on me – the way people interacted with each other with ease and attention, the weather, the endless panoramas and views, the way the architecture was built into the landscape and the language, which I took to easily. In 1955 I decided to live in Italy and this coincided with another bee in my bonnet, which was to become a proper grown-up, someone who could manage to be independent in the adult world. At twenty-three it was high time. I had already secured a job in Rome at the FAO – the Food and Agricultural Organization, a branch of the United Nations. I stayed for almost five years, with regular visits to England in between.
The purpose of the FAO is bold and clear: to eradicate hunger and malnutrition worldwide. To this end, its agricultural, fisheries and forestry programmes are all concerned with the same goal: the sustainability of food production. There are some ideological goals of behaviour and attitudes tucked in as well. It is fair to say FAO is about improving the world and it is highly regarded.
It was miraculous for me, and a privilege and luxury. We were diplomats and VIPs who needed no visas. The salaries were high and the holidays generous. There were also a lot of perks and it was very easy to live within one's means. Probably too easy. It was hardly a fair test of coping alone. Lovely apartments were easy to find and affordable to rent, and many English girls were around to share them with. Although in fact, the first flat I rented belonged to a new Italian friend I was introduced to. She was as beautiful as the day is long, and also so stylishly dressed – as I realised most Roman girls were and I would have to learn.
She wore a yellow cotton dress made of such quality that the fabric had a sheen to it, and open brown sandals. No jewellery – just that. The flat belonged to her brother, an artist who was away in New York being discovered (which he quite quickly was). She was called Mimi Gnoli and was an artist herself. She had many English connections and we are still friends and in contact. The flat in the Via Arenula with a large rooftop terrace was divine, she was a talented gardener. I bought my first car, a tiny second-hand Fiat called Topolino (little mouse). Life in Rome was shaping up to be thrilling and interesting – even more so when I met my first boyfriend.
At the FAO I was hired as a PA and worked mainly in the personnel department – what would be called human resources now. My boss was Italian, a Signor Carlo Buonacorsi with whom I got on very well, and his family spoilt me. I certainly became very knowing about all the staff working there. It was also possible to change to other departments when vacancies occurred, often temporarily. I had a stint in the agriculture department, and the work concerned the dire irrigation problems for the crops in Africa. When I consider this now, more than half a century on, I wonder how it is possible that in certain parts of Africa people are still without enough food and money to live on (this is quite apart from war-torn areas). It is about profits from crops not going to the local people. I expect we, in Europe, probably benefit from this.
Sadly I did not find the expat life at the FAO as fulfilling as anticipated, despite all the comfortable conditions and interesting people. There was no tight culture; everyone had completely different experiences and nationalities. I didn't know where half of them came from on the world map. It was like living in the VIP lounge at the airport, neither at home nor abroad, belonging nowhere. I had come for Italy and the Italian people, not the whole wide world.
Young Italian males in the 1950s, with their sex-obsessed behaviour, were decidedly very tiresome, and their pestering meant a girl could never sit down alone anywhere, no matter how hot and weary. However, apart from this self-imposed rigmarole of chatting up, which was treated like a sport, there was always Rome itself to talk about.
We were all beguiled by the beautiful, imposing buildings, which changed colour so dramatically according to the light and time of day. The luminosity of the stone (travertine) even altered feelings too. Around the hour before sunset, I heard someone say, ' _Questa é la mia ora triste_ ' – this is my sad hour. I can't imagine these words from an Englishman. I know it sounds melodramatic, but I do understand what he meant. Especially in my pet areas around Piazza del Parlamento and the Pantheon.
A few months later, I met my first boyfriend, who was called Osvaldo, a 25-year-old designer in high-fashion menswear, working in his prestigious family firm. His brother, sister and mother also worked there. The business had been started by his father, when the workshop used to be a gathering place for his famous clients and intellectuals, which included Aldous Huxley and the film directors Fellini, Visconti and Antonioni, among others. Always ahead of the fashion, Osvaldo was ambitious and apparently he ended up within the top league with his own collection in collaboration with Giorgio Armani, Nino Cerruti and other international names.
We went around mostly with his friends in a group. We also used to meet in the workshop. He didn't actively 'lead', but he was somehow the one they followed. They were a cheerful lot, interested in culture; he especially always loved looking at architecture wherever we went. It was my first, but not last, penchant for erudite and cultured men. He was neurotic, slim and witty and he was my type. This has remained so – I cannot bear a man who has even a whiff of being pleased with himself. I felt like Beatrice in _Much Ado About Nothing_ , 'There's not a wise man among twenty that will praise himself.'
On our excursions to marvellous places in and outside Rome, I found I had never really _looked_ at buildings before, nor knew basic Bible stories or the inside of churches. Other than frescoes, I had to discover fonts, reliefs, altarpieces, cupolas, triptychs and much more. Was the lack of exposure a fault of my own or the English schooling system?
At weekends we all usually went to the Roman seaside and sometimes on longer trips to Naples, Positano, Sorrento, Pompeii and Herculaneum. Sightseeing in a group nearly all the time seemed strange to me, as in England couples tended to pair off. But slowly I got used to the fact that there was a pack mentality here that I had not encountered before. Of course, I also liked it when we did travel alone. Once I accompanied him on a business trip to Milan.
I was curious to see the Ligurian coast, having till then only been acquainted with the Amalfi one, which I adored, including Naples itself. So we drove all the way back from Milan to Rome, stopping at various places, with huge beaches like Santa Margherita, and also the bijou Portofino. I didn't get to the marvellous city of Genoa until my reincarnation later with Eric – all in the future, and the same was true for Venice. Osvaldo was a wonderful driver and I was never nervous with him, no matter what speed – and _speed_ there was! He also came to London with me and met my family, but his lack of English made for sticky conversation. And my mother was in a fluster and my brothers teased me again. Very unusual and unsophisticated behaviour from my family that day. I don't really know why. Maybe it was a shock that Marlene was now with a man, and no longer the little girl who needed their protection.
It was a perfect time to be in Italy. After the war had ended in 1945, it became part of the West. The Marshall Plan helped rebuild the economy and there was huge optimism. Italians didn't just get back on their feet – they flourished profoundly. Gone were the Fascists, gone was Mussolini and in came free elections, and the traditional political parties were somewhat restored: the Christian Democrats and also a popular Communist Party. Italy became a member of NATO, the UN and, later, the EU, and after all the terrible years of repression and dictatorship, Italians were now free to travel themselves and to welcome the world.
America was their saviour and role model, which launched them into free commerce and enterprise. With the contribution of the Agnelli family and the gigantic car production in the north – of Maseratis, Lancias, Lamborghinis, Alfa Romeos, Ferraris and Fiats – it was called 'Il Boom' aka _il miracolo economico_. For the women, it was a revolution, uprooting entire communities. They no longer wanted to spend their lives on the land, tending animals; they yearned for nice clothes and shoes and to work in offices: they wanted the city. The new freedoms also unleashed a fresh creativity in design, fashion and art. Their furniture and domestic products became international and desired. 'Made in Italy' was about to become all the rage. It still is today.
America's influence permeated all sectors of life, but especially the cinema. So Mariella was not the only one with a passion for Hollywood – Rome too was buzzing with it and soon emerged as a major location for filmmaking, known as 'Hollywood on the Tiber'. The love relationship went both ways; Hollywood had a passion for Italy, creating films there like _Roman Holiday_ with Audrey Hepburn and _Three Coins in the Fountain_. Cinecittà (Rome's film studios) brought much attention to the city and to Italian filmmaking, culminating with Fellini's _La Dolce Vita_. One of the popular songs of 1956 was 'Tu vuò fà l'americano' ('You want to pretend you are an American') sung by Renato Carosone.
Everyone was star-struck and tried to catch a glimpse of Ingrid Bergman, Gina Lollobrigida, Anita Ekberg, Monica Vitti, Marcello Mastroianni and Sofia Loren, to name a few. The film stars were hounded by photographers on Vespas, a new phenomenon they later called the 'paparazzi' (after a surname of one of the riders in _La Dolce Vita_ ). After about three or more years in Rome, I seemed to have widened my circle of acquaintances. It was just as well; at some point there was another girl in Osvaldo's life. The strange thing is that Osvaldo remained as sweet and loving as always with me. I didn't know what or whom to believe, nor what to think. I was not furious with jealousy. I had never expected or really wanted a future with him, though my soft spot for him was always there. Instead I accepted the attentions of a Franco Reggiani. He was distant and rather snobbish, highly educated and keen for us to go out together. He was handsome. He was actually far too chilly for me – I think it was his adorable address that I liked so much: Vicolo del Divino Amore (Little Street of Divine Love) near Piazza Borghese, and now I had an excuse to use it in answer to his little messages.
Franco's sister Alessandra was getting married and I was invited to the wedding in Tuscany. The wedding reception seemed like a cross between a grand opera and a Pieter Bruegel painting. It was held in the grounds of the family farm and in the house.
The staff in colourful clothes were like a chorus, and participated, serving the food and enjoying themselves, albeit knowing their place. The relatives and friends mingled in their city elegance and Alessandra in the simplest white wedding dress with only a decorated veil and no jewellery; she was the most stylish bride I have ever seen, before and since. I talked with an uncle of Franco and the bride, a gentleman called Luxardo Servadio, who said to me that English girls (there were quite a few at the wedding) were too superficial and knew far too little about Italy. Never was there a truer sentence and I liked him so much for coming out with this. It was music to my ears – _so it wasn't just me_. And somehow, in Italy where everything is possible, a group of us arranged to visit Pisa with him, including the famous cemetery and of course right to the top of the Leaning Tower. Luxardo was an adorable guide and a wise and friendly man. He had a passion for history, although he was an industrial chemist by profession. An optimist and most delightful company; we were so very lucky.
Somewhere, sometime later, in a swimming pool, I met one of his daughters, Gaia, who lives in London. She is a writer who has a big heart and feels responsible for Italy - when an important Italian artist or personality is due to arrive in London she prepares to throw a party for them. She is a serious lover of opera and a very colourful person herself. She lives with her second English husband. For me she is a good friend and a link between my girlish and adult Italy.
Chapter 7
# _The Congo_
After my relationship with Osvaldo, I thought it was time to leave Italy, but I was not quite ready to go home just yet. As it happened, there was a crisis in the Congo and the FAO needed staff to work with the UN in Léopoldville, the country's capital, now called Kinshasa. Well, I volunteered. I went home first to see my family in London, and then flew straight to Leopoldville from there. It is a complicated and bitterly tragic story, which I will shorten and omit intricacies, like the synopsis of an _opera seria_. All the main 'characters' are male politicians: the President of the Congo, Joseph Kasavubu (I love pronouncing that surname); Prime Minister Elect, Patrice Lumumba, who was a radical and communist; Secretary General of the United Nations, Dag Hammarskjöld; leader of the Katanga province, Moïse Tschombe; and, lastly, a colonel named Joseph-Désiré Mobutu. After the Congo's independence from Belgium was declared in June 1960, the predictable colonial void appeared, and rebellions began to take place in many provinces. The Belgians sent in troops to restore order, but this only made matters worse and led to even more mutinies. Together, Kasavubu and Lumumba asked the United Nations to send in peacekeeping troops, which Hammarskjöld agreed to do. The Katanga province had the most at stake, as it was the richest province, with the copper belt (and possibly diamonds). Soon the unrest of secessions within secessions and factions within factions began, and then multiplied all over the entire country.
Surely that is enough. But alas not. The situation became much more complicated and dangerous when Lumumba made the decision to seek assistance from the Russians. This of course alarmed the USA and led to that familiar proxy war situation with the Soviet Union. Finally, the unstoppable Colonel Joseph-Désiré Mobutu continued putting himself in charge wherever he could and succeeded in becoming chief of the army.
These are the very basics of the Congo crisis we ouselves spent our evenings playing pontoon in luxury hotels. Were we all involved in an Evelyn Waugh novel, I wondered? I don't know how the journalists managed to file through all the news and piece together the scattered information for their daily articles. Without going into all the tribal factions, I have simplified the complexities of the conflict.
My job was rather interesting. I spent my year there with the UN mission, working in welfare for the UN troops, supplying them with enough books, records and sports equipment, and arranging different leisure activities, film shows and entertainments. I drew the line at umpiring a football match, although I was asked to do that.
The UN personnel and the international press were put up in two hotels in Léopoldville. Mine was the Stanley and the other one was called the Memling.
Once again I found myself in a group. We played cards (for real money, as counters were among the many things you could not buy any more), read books and just hung out together. My main friends in the foreign press were Gerry Ratzin from Reuters and, more vaguely, a Frenchman from Agence France-Presse.
I was keen to see more of Africa. Luckily, Gerry had to go to Johannesburg to cover a story and was glad for me to tag along with him. We had a very pleasant trip; I even had friends from Manchester living there, who had interesting opinions and stories to tell about their lives in South Africa. They had watched me grow up in Manchester and were very excited about my visit. I soon grew tired of gambling and wanted to travel. I then met a very interesting Belgian (Flemish) girl called Monique Geschier, who had a job at the Telecom (I think she was dealing with priority calls) and she lived in a large, charming flat that came free with her job. She asked me to move in with her, the arrangement being I paid for our daily Congolese housekeeper, David. Monique had been living in the Congo for several years, escaping a very sad family past, including an acrimonious divorce, which ended without access to her two little daughters, who lived in Belgium. We got on famously. She was pretty, with real Titian-coloured hair, and all the men looked at her before me, except for Kamal. I don't remember how I met him; he was a very charming (rather short for me) man from the Ismaili clan, who were followers of the Aga Khan. He must have been the sweetest person I've ever known, and he adored me. Not in the usual way, but as though I was a princess. He didn't dream of going to bed with me. In his culture, that happened with quite different women. But together we were very happy to hold hands and cuddle, and we spent many good evenings with Monique and her friends. People were very surprised that we saw so much of each other, and so was I. I didn't learn as much as I should have about Ismaili culture and customs. I wish I had done. Kamal knew I wanted to travel, and he recommended I go to East Africa, to Mombasa in Kenya and Tanzania, where he had relatives. One of his cousins actually lived in an apartment in the Sultan's palace in Zanzibar, and I was invited to visit her. Kamal had already written; we had our tea together, followed by the grand tour of the palace – lots of ornate patterns in refined light colours and gold remains in my memory. These patterns were used to decorate everything, from the carpet to the wall, _objets d'art_ and more. I can see why Matisse wanted to travel there. I spent all my time sightseeing the island, people watching and necklace shopping. The world, as far as I was concerned, felt safe, though I knew it was also terrible. No thoughts, of course, about terrorism. It was just a different era, and a time when I felt completely free and extremely content, perhaps, dare I say, even grown-up. This was just as well, as it turned out to be the last big time on my own. I couldn't peep into the future. Back in the Congo I was soon reminded we were in a war zone. Mobutu, again with his machinations, had pulled off yet another coup and as head of the army (and with the new compliance of Kasavubu) put Lumumba under house arrest. Guarded by Ghanian UN troops, Lumumba escaped but was recaptured and sent to Katanga, where he was tortured and executed under the authority of Tshombe. It was barbaric, as there was absolutely no legal process whatsoever for all this. The deed was just carried out. There were certainly strong worldwide protests and demonstrations, but all to no avail. We were in the middle of the Cold War and no one would speak up for a communist. Even at the United Nations there were not enough votes to take it up. Dag Hammarskjöld was killed in a plane crash, along with his bodyguard, who was also my friend. There were rumours of foul play. With his new alliances, Mobutu became a dictator and stayed in power for years, changing the name of the country to Zaire in 1971. Monique was well connected with the Congolese elite, diplomats and interesting Belgians. I remember she took me to a grand garden party where Mobutu slowly walked towards me and asked me personally if I would be willing to give him English lessons, which I had the common sense and savoir faire to carefully decline.
My year was up and I was ready to go home. Monique came with me and she stayed with us until she found a suitable job. She was very keen on archaeology and it was not long before she found an excellent one working with the team under Kathleen Kenyon – at the time very famous for her excavations in Jericho – on a new expedition. It was the jackpot for her. Later on, she married the director of the Archaeological Institute of America in Chicago, where she still lives. She was reunited with her girls and had a son.
Chapter 8
# _Back Home and CBC_
I came home to London in 1961 and worked for the Canadian Broadcasting Corporation (CBC) in the newsroom as PA to the news supervisor of Europe, Garran Patterson. I got this exciting, well-paid job in central London through my good friend Hannah Horrowitz, who did freelance work at CBC. Hannah was a musical agent and impresario, and very well connected. Our office was in the Langham Buildings, once a hotel, but at the time part of the BBC and situated across the road from Broadcasting House. It is now a hotel again. Our noisy, lively newsroom was always full of journalists staying or passing through; I remember Morley Safer and Roméo LeBlanc particularly. Every day their wisecracks and opinions made me feel we were taking part in an excellent play. I loved my job, starting daily with a telephone conversation to Diana Fowler, a Canadian in the Paris office, about the schedules for the day.
Much later, when I left CBC to prepare for my first child, Diana took over my job and we became friends for life. She married the boss and, as Mrs Patterson, brought two little girls into the world. But after two decades, the marriage ended and she returned to live in Canada. There she eventually married Roméo LeBlanc, who became the governor general, making her the viceregal consort; in other words the First Lady of Canada. No kidding! I still have the video of his installation. There she was – _my_ Diana – on the world stage and doing a very good job of it too.
Chapter 9
# _Meeting Eric_
When I met Eric Hobsbawm he was a groovy single man about town, much in demand socially, as he was known to be good company. He was an offbeat academic and very friendly with his students, teaching history in the evenings at Birkbeck College. As well as writing history books, he was a journalist on all manner of subjects, writing reviews and articles, including a jazz column in the _New Statesman_. For this, he had to hang out in clubs in Soho, including strip bars, and he acquired a pseudonym, Francis Newton, to make sure his students only asked him questions about history. He had been a member of the Communist Party since his schoolboy days in Berlin in 1931.
My brother Walter now had a wife, Dorothy Morgan. They were already raising a family, living in Hampstead Garden Suburb, and I spent a fair amount of my spare time with them. Dorothy was a mature student at Birkbeck College and Eric was her supervisor. Walter and Dorothy gave a dinner party, and 'Walter's sister' and 'Dorothy's supervisor' were both invited. That's how Eric and I met. I think the chemistry between us was there right from the beginning. Neither of us could recollect any of the other guests present, though they were definitely there. Eric had a beautiful tenor speaking voice I was attracted to, and his eyes hardly left me, even when he was engaged in conversation with other guests.
Something definitely changed in me after that evening in November 1961. I remember being bothered that Eric had said he was going off on a trip soon. I was living temporarily in my brother Victor's flat in Mansfield Street, London W1. He'd gone away on a journey and had lent me his gorgeous place until he returned. I shared it with two girlfriends and we decided to organise a dinner party (my idea), each of us inviting a male friend.
While I have a vague recollection of that evening, it is a different memory that still stays with me: I was the first of us to telephone the other, and Eric was very enthusiastic about accepting the invitation. Yes indeed, he was free to come to the dinner in a week's time, but wanted to know what I was doing _now_ this very minute. I dodged that word _now_. This was the early sixties, when nice girls would feel it was too fast to doing something like go along with him to help buy groceries for his flat as a first date. Had it been in the 1970s, after seeing those engaging Woody Allen films, I might have easily have accepted.
The fact is, he was supposed to be in Cuba with a group of intellectuals at a conference also attended by Fidel Castro and Che Guevara. He was looking forward to talking with the other travellers, particularly the activist and writer Arnold Kettle and the theatre director Joan Littlewood. But there was a fault on the plane discovered at Prague and they had to return to London. Eric felt very fed up about his wasted time, about his empty diary and at having no provisions at home. At the exact moment when he gloomily stepped into his flat, I telephoned. It was a lucky start. The stars were aligned. Eric's aborted trip to Cuba did take place in the new year.
When Vic returned from his travels, I moved back into my Paddington Street attic flat over a fishmonger's (now a boutique), and Eric and I began to see each other quite often. What did we do? Like millions of others, we talked about ourselves, and being forty-five and twenty-nine, there was plenty to say. Eric had even been married! Goodness me. His classes were three evenings a week and the other evenings we often tried to meet, going out to supper, cinema, concerts and so on, until simply, 'Your place or mine?'
Eric introduced me to an architect, Martin Frishman and his mother Margaret, a painter. We became firm friends and his mother painted a picture of my father and later another one of Eric. Both were traditional and beautiful large oil portraits. Unlike _La Bohème_ , this was my introduction to _haute bohème_. Martin's large studio flat was in Belgravia, and the previous tenant had been Noel Coward. Martin told me that the difference between bohemians and other people was that bohemians wash their dishes before they eat rather than after. Many of Eric's friends were artists like these.
I wanted to introduce Eric to my cousin Peter Nettl, with whom I had always got on very well and who sometimes invited me on family holidays and business trips. I remember good times in Sardinia and Cairo with them. Peter had left academia to become a businessman, and had written a very fine biography of Rosa Luxemburg, the activist murdered by anti-communist paramilitaries in Germany in 1919. He was thrilled that I was going out with Eric. I gave a small dinner party in my flat at which Peter commented on Eric's telling way of carving a roast leg of lamb. He insisted that despite his fluent German, Eric would be a useless spy, as all would know he could not be anything but an Englishman. Eric, Peter and I could not remain friends for life because, to our perpetual horror, Peter was killed in a Northwest Airlines' plane crash some years later in America. His wife Marietta and three others survived the crash. Their daughter Andrea, now a true culture vulture and a Wagnerian, is and has been my friend since she was seven years old.
Eric was often away, either abroad or at universities around the country, mainly giving papers for seminars. Absences seemed to make the heart grow fonder and our merry dating life continued for almost a year. However, there was a glitch. Eric had won a very generous grant from the Rockefeller Foundation to travel around Latin America for three months to continue his research into 'primitive rebels', a concept that he had used as the title for his second book, but was still investigating. Suddenly this sharpened our minds; to avoid the shock of such a very long separation, things ought to move swiftly between us.
Our relationship was known to Walter and Dorothy, but not as yet to my parents. The person I confided in was Gretl Lenz, who was a very special cousin and also my mother's best friend. Gretl had always played a crucial role in all our lives since we were little in the old Vienna days, and she was very often amongst us, always living nearby; we all loved her and never wanted her to go home to her difficult husband and dachshunds, but she did. I suppose Gretl's name must have been chosen because her brother's middle name was Hans – maybe a touch of Austrian humour. She did not have children, but was very family minded and close to her high-flying international lawyer brother and his very English wife Barbara and, above all, their beloved daughter Patsy, who has always been a close cousin. Gretl's wisdom was always spot on and quick. I took Eric to meet her first. They got on famously and she paved the way for us.
One evening I accompanied Eric to the George Shearing jazz quintet at the Royal Festival Hall, which he was covering for the _New Statesman_. He said something very unromantic like, 'I think we should take out our diaries and find time for a wedding before I have to leave.' That was the proposal and there was certainly no bent knee. It was, however, not so simple, because a register office had to be booked at least three weeks in advance for a wedding. That was a shame, and the only way we could arrange it was to go on our two-week honeymoon _first_ , have our wedding _second_ , and Eric would leave for his research trip _third_. For the honeymoon we had decided on Bulgaria, straight to Sofia via Vienna, which included a night at the opera, and then on to the small, charming resort Golden Sands on the Black Sea. Eric was delighted to see some people there reading Pushkin out loud to one another in the square.
More or less as soon as we got back in October 1962, we married at Marylebone register office. Martin Frishman was our best man and a reception followed at my parents' house. The next day we went on a short second honeymoon. Vic had gracefully lent us his car for the weekend and we drove to Castle Combe in Wiltshire. Eric told me how scared he was of marriage. He snobbishly connected it with having boring holidays in a caravan by the seaside.
Five days after our return to London, Eric was off. Before embarking, a man at the airport asked, 'Is your father also going to Buenos Aires?' It made me laugh. I had not yet started on Eric's wardrobe and that unflattering greenish coloured coat (ugh!) made him look as old as Methuselah.
Leaping ahead nearly half a century, I can tell you another man who must have been there at the airport. He was from MI5, where it was discovered that Eric's research trip was sponsored by the American Rockefeller Foundation. They sent this man to spy on Eric, presumably for his US contacts. All of this was revealed in a fascinating article by Frances Stonor Saunders in the _London Review of_ _Books_3 when Eric's MI5 file became viewable in 2015. Eric had so much wanted to see his MI5 file, but permission was always refused. Martin Jacques, editor of _Marxism Today_ and also an intimate family friend, accompanied me to view the files when they were released. It certainly made dispiriting reading. It seems the spooks disliked Eric intensely. They didn't approve of his looks or his clothes, and there were a few anti-Semitic remarks. More than that, I think they hated him because they had not been able to find anything on him. And that, quite simply, was because there was nothing to find. What did they expect? Surely an open member of the Communist Party, who wore this proudly on his sleeve, would hardly be a spy. But they went on digging.
All throughout our two honeymoons and one wedding, we were in the middle of the Cuban Missile Crisis, with the US and Russia on the brink of war. It was just beginning to settle, but Eric's last words to me before he flew off were, 'Should things go wrong and war does break out, then buy a one-way ticket to Argentina. There is enough money in the bank and I'll meet you in Buenos Aires.'
OH! Heart pumping, I had not reckoned with that. Did I _really_ know my man well enough? But life had fallen into my lap, and I was just going to live it. I think that was how I felt.
3 April 2015, vol. 37, no. 7.
Chapter 10
# _Being Married_
I began my married life as a wife alone, my new husband being somewhere in Latin America, but I was not sure where. I had already moved into Eric's flat in Bloomsbury, Gordon Mansions in Huntley Street, and I continued working in the CBC newsroom. We wrote to each other most days. Letters were crucial as the telephone was too expensive, and in any case our phone was tapped and hardly suitable for the talk of separated lovers. The phone tapping started when Guy Burgess (part of the infamous spy circle, the Cambridge Five) phoned Eric from Moscow as a lark to say he couldn't come to the Cambridge Apostles Dinner and would Eric please make his excuses. After that, there was always the telling click on our phone. A couple of months after Eric left, I threw a party for a few colleagues and friends, which turned out to be quite noisy. Suddenly the phone rang and it was Eric's cousin Denis Preston. Hearing all the voices, he said, 'Are you throwing a party? Sounds very noisy. Ought you to be partying with your husband away?' Oh dear, I thought I had put my foot in it with Eric's family, but he was only teasing. Little did I know then that Denis, who had introduced Eric to jazz in the first place, was the unstuffiest and most swinging relative he had. It was however, a very crucial and timely wake-up call for me. I needed to inspect the flat more sensibly and to sort things out, especially in the kitchen, as it was a stark reminder that fairly soon proper meals would have to be cooked in there every day – by me! Blimey!
For three months I had waited patiently for Eric's return like Cio-Cio San (in _Madame Butterfly_ ) and now it was time to distribute the cherry blossoms. His return was such a relief. We were bursting with joy. And I was pregnant with our honeymoon baby.
Eric and I had already met many of each other's relatives. My parents liked Eric, my mother especially. I knew Eric's sister Nancy, her husband Victor and their three children, Robin, Anne and Jeremy, as well as Eric's close cousin Ronnie, his wife Mary and their daughter Angela. None of them lived in London, so we saw much less of them than my relatives, although when Angela and her family moved to London, they also visited, like all the relations. My brother Vic, always concerned about his little sister, had already made enquiries through his old boys' network about Eric's Communist Party membership, but he was assured that the spooks already knew all about him as an open and declared communist.
Of course, I knew that Eric had been married before, to Muriel Seaman. I also knew that when the marriage dissolved, Eric had had an affair with Marion Bennathan, a mature psychology student, and was the father of her child, Joss. Marion insisted she wanted to keep this a secret, because it would not be fair to her husband, Ezra, and Eric obeyed. Nonetheless Eric would visit Joss from time to time, usually at Christmas to take him to a show, like an uncle.
Eric started to introduce me to his cosmopolitan and English friends and I did the same. I met all manner of remarkable people through Eric. It was one of the best things in my life with him. Our friends were hugely important for both of us. They played a crucial role throughout our married life. They were so close to us and such an important part of our lives even when not present.
When we started meeting each other's friends, Eric was keen that we go to Cambridge first because his friends were very curious about me, having convinced themselves Eric would never marry again. Gabriele Annan and her husband Noel, provost of King's College, had invited us for lunch at the Provost's Lodge. I'm fairly certain the art historian Francis Haskell, a bosom friend of Eric's and of the Annans, was also there. I cannot recollect the others. Later Gaby told me she expected Eric to arrive with a sort of a red battleaxe. The Annans were very chic, both dressed in casual but elegant clothes, with Noel in a bright pink shirt. They welcomed me with huge warmth and friendliness, which immediately lifted my diffidence at university gatherings. Usually when asked what field I was in, I would have to resist the temptation to say, 'The one with the bull in it.' I had no field, nor subject. As yet. A decade later I was able to hold my own as a music teacher. A great relief all round for conversation at formal university dinners when my new label, _Music Teacher_ , solved everything, starting with the card at my place setting.
The sophisticated Annans, of course, could converse on all topics, but many dons were at a complete loss when not talking about college affairs, which really surprised me. We became close to Gaby and Noel; they often invited us and we them. Gaby was very keen on company, partly because she feared boredom; so much so I think it was a phobia. She told me it sprang from her suffering as a lonely child having to endure long adult dinners. Much of our talk was usually about Cambridge friendships or the arts. Noel joked about how he and Eric were both members of political groups, one advocating communism, the other gay rights. In his plummy voice, he said, 'My dear Eric, you were in the Comintern; I was in the Homintern.' Gaby and Noel were a close family with two friendly, high-flying daughters Lucy and Juliet.
We discovered marriage was good for us; at last there was a routine and a rhythm to our lives, which neither of us had had for years. The institution of marriage itself suited us even though it was slightly going out of fashion amongst our artistic crowd. Our friend and poet Erich Fried said to his partner (the sculptor Catherine Boswell), 'Don't worry – I promise to marry you before I divorce you.'
I was head over heels in love with Eric; he was everything to me, and on top of all of that I was in awe of him. Quite a load. Once I apparently put my finger to my lips and whispered to an Italian guest at supper, 'Shh, he is speaking.' I only learned about this gaffe years later, and we both roared with laughter. Eric was in love too, but I think I was perhaps too bourgeois for him to actually let himself go crazy for me. He had never had a steady relationship before, let alone a woman who would produce initialled linen from her mother's trousseau. But his big love did come; it grew little by little until one day, a few years down the line, he said he didn't really enjoy going to places any more without me by his side. I often wondered what it was that bound us together for ever. We were now sophisticated, grown-up cosmopolitans, but in our own sealed selves I think at times we both felt like displaced people, although we never really talked seriously about this to each other. Those who know about Eric's childhood from his autobiography _Interesting Times_ would, I think, understand.
Life in Bloomsbury was very easy for us – being able to walk to most places in the West End, including the Jazz club Ronnie Scott's at night. Not to mention the convenience of living on the same street as the University College Hospital maternity unit. It was a charmed life indeed, but it didn't last that long. Eric was incredibly busy with academia and writing. I discovered I was to share him with the world. Already historians came from many countries to talk to him. He liked it this way, to learn from people he trusted about what was happening in their part of the world. He was a good listener and listened hard. These visitors went mostly to Birkbeck, but old friends and those who knew him well came to the flat, and I got roped in. We soon needed to buy some new crockery. After all, we were living in a single man's flat.
Then Eric's book _The Age of Revolution_ (the first in the _Ages_ series), published by Weidenfeld & Nicolson, came out. It was mostly very well received, both in England and internationally. All at once, publishers had propositions for Eric and professors from all over the place wanted him to come and lecture. Eric's reputation had suddenly shifted. He did not even have a literary agent yet. That's when it all started to be a bit chaotic for him and I gave in my notice at CBC.
Chapter 11
# _A New Family_
Of course we had talked about having children. Actually, Eric adored babies and toddlers. He could not resist making funny faces and noises for them whenever we were travelling on trains or planes. Sometimes it was embarrassing or they wailed when he stopped, which he hadn't predicted. But that was a world apart from the responsibility of raising children.
I discovered that Eric didn't (or, rather, couldn't) lie. It was a revelation. Nothing to do with morality: he was simply not cunning enough. All of us have different levels of feelings and emotions; Eric's seemed fathomless and he wanted to be in touch with what he felt – almost like a dialogue to get at the truth. I think this interfered with being able to simulate or tell a lie easily (naturally, this is only my theory). He told me he would have preferred to have me all to himself, easily able to travel the world together. He also knew, which I did not, how much support serious writers needed. However, having said all this to me, he was also very much Eric the pragmatist _par excellence_. He knew, and had already said, that it was not possible to marry a young and healthy woman, and expect her to compromise hopes of motherhood. He said he was in for the 'whole package'. And in any case, once our children arrived; he was smitten.
In June 1963 our son Andy (Andrew John) decided it was time to enter the world. We were having lunch with cousin Denis in nearby Charlotte Street when I went into labour, and Denis immediately drove us to University College Hospital, pulling my leg as usual – 'Marlene, you are not to mess up my new car' – a joke that had the three of us giggling inappropriately when we arrived at the reception for the maternity unit. Andy was born in the evening, and I became the proud mother of the longest baby measured on the ward. Eric said, 'Oh my God, he looks just like Uncle Sidney.' Mothers normally stayed in hospital for ten days at that time, mainly to learn how to bathe their babies properly, and we all had to be supervised doing this. The NHS was responsible for everything, including daily physiotherapy. They sent us home with equipment and feeling pretty confident. And even then, they came to our homes to see if all was going smoothly.
It turned out Bloomsbury was a lovely place with a baby, especially in June. We were so near the shops and could walk with the pram to Charlotte Street and sit outside at restaurants with our friends as if on the Continent. There was the pretty Gordon Square garden, to which we had a key and where all sorts of interesting mums were also minding their children. I remember the enjoyment of chatting with the writer Antonia Byatt, for one. The only big snag was the many outside steps up to our mansion flat. Every day I was haunted by the image of the pram rolling down the steps like in _Battleship Potemkin_. Later, when Andy was old enough to sit upright, Eric liked to walk around Russell Square with him in a new baby contraption strapped on his shoulders, showing him the world: not surprisingly Andy's first word was, 'Look-at-that,' with his finger pointing. Andy was a healthy and strong baby who soon grew into a very handsome one. He was easy about everything other than accepting new solid food.
We took him to France when he was barely a toddler to stay with Anne and John Willett (man of multiple letters) at their place Le Thiel near Dieppe. To me it seemed that Andy refused to eat anything for the entire week. I went into the garden, clueless as to whether those green tops were carrots or turnips, parsnips or whatever else. I cleaned, cooked and puréed them, but he immediately spat everything out. He was on a hunger strike, I think. In spite of enjoying ourselves so much with the Willetts, whose company we always coveted, Andy's fasting made us mighty glad when our week's stay was up. It was a shame. Our second child was planned. It was Eric who insisted it would be unfair for Andy not to have a sibling. Especially on our travels, the children would need each other. And he was right. It turned out there was only fourteen months age difference between them and they were very close, and have stayed so. This has always been the root of my happiness. Today they both live with their own families, raising their children in houses within a mile from one another. That is, of course, quite normal in families all over the world, but in Jewish families in the twenty-first century I consider it a small miracle. We hoped for a girl and I had only chosen girls' names. In the daytime of 15 August 1964, again across the road in Huntley Street, our daughter Julia Nathalie arrived. She was not at all like Uncle Sidney, and a little rosy. She was named Julia because I so loved the street La Via Giulia in Rome, and Nathalie was a charming French girl, daughter of my parents' Parisian friends, the Jouard family, whom we admired during my teenage years. My day now started with the 5.20 a.m. shipping forecast, pondering over Dogger, Cromarty, Fair Isle, Fastnet Lundy, Finisterre, German Bight and Bailey. I was also reading Doris Lessing's _The Grass is Singing_ , a marvellous account of her life in southern Rhodesia (now Zimbabwe) and I would recommend it wholeheartedly, even without simultaneously feeding a baby in a silent, unheated flat. Life with two children under two years needed some ingenuity. My bright idea was to hire an au pair, a pleasant Dutch girl. She liked to walk as far as Trafalgar Square with Andy in a pushchair because he had become so excited looking at the pigeons and eventually running with them. He insisted on going there every day. He was too little to be very upset or jealous by Julia's arrival, although one day I did see a large toy brick placed in his sister's cot exactly where her head would have been! I spent my afternoons pushing Julia in the pram. With Andy I was told babies should lie on their tummies, which I obeyed, but I didn't with Julia. I preferred her to be on her back so I could see her and she was able to see me too. Much more fun. In all these events, grandmothers who are able to often play a role, and my mother (now Grandma Lilly) was no exception. She would leave wonderfully cooked Gefüllte Paprika (stuffed peppers) or goulash dishes outside our flat door so as not to disturb us. She already had her hands full with my father's Parkinson's disease and his old age, but she made time for us too. At weekends I always had somewhere to go – my parents' warm and welcoming home in Golders Green. The children would play in the garden there and sometimes their little cousins came.
Chapter 12
# _Clapham, Part I Cottage in Wales and Massachusetts_
The pram, the pushchair, the high chair, the playpen: the stuff was crushing us and we began to contemplate buying a house. We had a stroke of luck. Our friends, Alan Sillitoe the novelist and his wife, the poet Ruth Fainlight, wanted to buy a large, handsome Victorian house in Clapham and they were looking for a family to share it with. They found the Hobsbawms. It was in the Old Town of Clapham, the least gentrified area, only just beginning to come up, and the house was cheap enough to enable some fine refurbishments by our architect Max Neufeld, who divided the house into two L-shaped maisonettes. It worked out well because of the generous proportions of the house, with its large windows. Our children were barely three and four when we moved in and David Sillitoe was five. It did not turn out to be a lifetime house, neither for the Sillitoes nor for us. I remember a passer-by asked me when the playschool would open – looking through the window and seeing all the books and toys, it didn't cross her mind that the space could be just for one family. The large garden was a dream for the children, and we shared Doreen the cleaner with the Sillitoes. She was a saint, and did babysitting too.
People also wondered what Alan did for money, because he didn't go out to work. They assumed he had won the Pools. It never occurred to them that he could be a writer successful enough to earn a living. He wrote profusely and brought out a book every year, which were all translated into many languages. Later, two of his books, _Saturday Night and Sunday Morning_ and _The Loneliness of the Long Distance Runner_ , were on the O-level and A-level syllabus for years, and made into successful films, which must have been good news for the family.
During our time in Clapham three tragedies followed one after the other. First, Alan's sister, a mother of four, was diagnosed with cancer. Her youngest child, six-year-old Susan, stayed with Ruth and Alan for a short visit while her mother was receiving treatment. I remember her when she knocked on our garden door once or twice hoping to play with Andy and Julia. She was a lovely and lively little girl. At some stage, after her mother died in 1968, she was adopted by Alan and Ruth.
The Sillitoes decided to go abroad for a year to Mallorca, where they often spent time staying near their great friend, the poet Robert Graves. They rented their Clapham home to an interesting American couple, George and Natalie, who had adopted one little girl called Hannah and were awaiting their second adopted baby. Unfortunately she arrived from America at exactly the same time as Natalie's mother, who expected the full tourist VIP treatment of visits to the theatre, sightseeing and out-of-town places. Neither the mother nor the new baby could now be postponed.
I secretly thought the new baby was sometimes out in her pram in the garden far too long, but I didn't want to interfere. I also knew that too many different people were looking after her while Natalie and her mother went out: Doreen, George and myself were all tiptoeing around this situation. About two weeks later, the little baby girl died in the night. Andy and Julia woke us up with excitement that an ambulance and a police car were outside. George asked me to take the children, including Hannah, out for the day, as the formal proceedings were going to take a long time. With sandwiches, drinks, a frisbee and a ball, I drove them to Battersea Park.
It was a long day sitting on a bench and my nerves were in shreds, my thoughts unable to stop dwelling on the fact that the baby had shown signs of being ill, which had not been followed up. The blame I felt at my silence and non-intervention was a great source of anxiety for me. I thought I was developing a bad cold, but the next day it turned out to be shingles or Bell's palsy. I could not shut my left eye at all and the left corner of my mouth was also paralysed. During this time Eric had been preparing two lectures to deliver in America the following week. But as Mr Pragmatist, he simply phoned the university without a nanosecond's thought and told them his wife was ill and he couldn't come. This was done and settled, but of course the baby's death lingered. I felt so sorry for them all. The police did not delve much and on the certificate 'cot death' was written. Alan and Ruth allowed George and Natalie to stay longer because of their tragedy. The Sillitoes never returned there, and eventually bought a house in Kent.
My palsy lasted a few months, but alas not my regrets, which are still with me. I went around looking like Long John Silver with a black eyepatch. Eric was in the mood to whisk me off to the most famous hospitals in the world – names like the Mayo Clinic, Leningrad and even Japan were glamorously bandied about, but it turned out the Mecca for neurological diseases was in London's own Queen's Square, and one had but to take the 59 bus to get there. It was both a let-down and a relief.
The last tragedy was the death of our good friend Charlotte Jenkins, who was much too young to die, leaving behind her husband Peter (a _Guardian_ correspondent) and her ravishing four-year-old daughter Amy. I had not been to a funeral of someone my age before. I was affected by the beauty of the mourners, many of them young parents, as well as the tributes, especially Jane Miller reading George Herbert's poem 'Life'. Amy was Julia's favourite playmate, and on one occasion when I was collecting her, she turned back and said, 'Goodbye, my beautiful glass mummy.' Amy had recognised her frailty. Children have a way of knowing, even if they don't understand.
Sometimes Eric got exhausted from overwork, and the best thing was to spend a few days away. One of these trips was to North Wales to visit his friend from Cambridge, a Kingsman called Robin Gandy. He was a kind, eccentric and jolly mathematician who spent his holidays in a tiny cottage called Pendomen in the Croesor Valley. It was not far from the foothills of Snowdon on the estate of Clough Williams-Ellis and his wife Amabel, who liked to rent out their houses and cottages to intellectuals and people they knew. Robin had been a friend of Clough and Amabel's son Kitto at Cambridge, and after Kitto was killed in the war, Robin had remained close to his friend's parents.
The famous Italianate village of Portmeirion, which Clough had created, was nearby, and tenants on the estate had the advantage of using it freely and going to the seaside, which had private beaches and a pool, while also living in the wild countryside. Maybe because he now had children, Eric thought how good it would be to have a cottage there too. He asked Robin to recommend him to Clough (a necessity in order to rent) and we hoped something would turn up. Robin must have hit the right tone, giving Eric a glowing review. He soon phoned to tell us a rental cottage had been found and we had first refusal. It was extremely exciting – with short notice at the end of July we managed to farm out the children and the au pair for three days to stay with my mother while we drove up to Merioneth (now Gwynedd) to view the cottage Bryn Hyfryd in the Croesor valley. It was the last cottage in a little row of four. Modest, ordinary and not particularly charming, but you could go out of the back door and the children would be able to run, nearly fly, as far as the eye could see on safe green fields and with magnificent views. There was a little stream nearby. Three bedrooms and bathroom upstairs, with kitchen and living room downstairs.
We were so fixated on arranging to rent the cottage that we almost forgot about picking up the children from Grandma Lilly as promised. We drove fast, suddenly noticing there was no traffic at all. We didn't have a radio. Had something awful happened? A war broken out? Was there a nuclear alert? So engrossed were we in our private affairs, we had forgotten the World Cup at Wembley, where for the first time ever we actually beat the Germans. The TV audience was over 32 million viewers, making it the largest ever (in the UK). It was 30 July 1966. It was to be our first house in Wales and we would have ten years of holidays in it. In the summers, we frequented auctions to buy furniture – Welshmen with their gift of the gab were brilliant auctioneers and knew their role as witty entertainers also made for higher bids. The walks from the house were stunning. Our children learned to climb and loved it. Eric had a ploy – he would tell them they could rest whenever they wanted to, having discovered that they could not sit still for more than a few minutes. There were all sorts of day excursions by car as well (apart from those to Portmeirion), with one memorable visit to Harlech Castle in particular. I don't recommend taking a few children there, as they will inevitably run in different directions shouting, 'Look at me! Look at me!' from the edge of the parapets. I thought I was going to have a heart attack. I enjoyed a long whisky later that night to celebrate coming home with the same number of children that I set off with.
Julia has retained her love of walking and climbing from those days. Andy went in for the more modern running or speed walking.
Then came a surprise. Eric was invited to take a six-month visiting professorship at the prestigious Massachusetts Institute of Technology (MIT) in Boston, USA. We were both very keen to go, especially as this was the time to travel, when the children were not yet in school. Visas for communists were not possible, and a waiver was required. This took time. We carried on with our lives.
I bought a second-hand upright piano in order to refresh my childhood Grade 1-, 2-and 3-level pieces, and to play simple nursery songs and games for the children and their friends (our family baby grand piano had gone to Walter and he deserved it). We were slowly preparing for the trip and the ad we put in the _Guardian_ for an au pair brought us over _sixty_ replies! Eric simplified the decision, saying we should choose an Irish girl, as the Boston area is full of Irish families; indeed, Phil (the girl we chose) did have relatives there and we felt much relieved. It was a success. When I later asked Phil what made her decide to come with us, she said, 'I took one look at Andy's soft brown doe eyes and decided I just had to take care of him.'
MIT wanted students and teachers from all the sciences, including astrophysics, to attend a history course. They were encouraged to improve and expand their minds through interdisciplinary studies. The name of Eric's course, as I recall, was 'Comparative History'. Eric was certainly going to deliver that.
The waiver issue eventually loomed very large in our lives; MIT eased our worries by assuring us that if the worst came to the worst, they would still pay his full salary for the six months. For a day we toyed with the idea that this might not be so bad – 'Where in Europe shall we go?' I think a power battle between MIT and the FBI immigration had begun. MIT were certainly ready to show the bureaucrats that communist backgrounds did not flummox them and they knew precisely which scholars they wanted to be invited.
Eric had already decided ages before that although he very much wanted to visit American universities, for their heterodoxy especially, he would never vow he had not been a communist in order to do so. He would just not go. So when the waiver arrived he was very pleased that MIT had stuck their neck out for him. He could keep his resolution and travel. Academic vanity also raised its head, I suppose. He wanted to prove that he could continue his work successfully while remaining a communist, even during the Cold War. I think these were his internal monologues and remained so.
Around February of 1967, off we went to the real America; quite different from Manhattan, with its hub of all nationalities and lots of characters looking like Albert Einstein. There had been some difficulty in finding us a house, and we ended up in the suburb of Arlington, about thirty minutes by bus from downtown Boston. People there were a mixture of middle- and working-class Americans. Thinking back fifty years, perhaps some of them might have become Donald Trump voters.
Andy went to a playschool for four-year-olds in the mornings, although he spent most of his time on the bus, being the first child to be picked up and the last to be dropped off. But he was happy and loved chatting with all his new bus friends. Julia stayed at home with Phil and me, both of us often in stitches at her determination to do what she had planned to do, aged three. I had time to practise the recorder and there was also a piano.
I enjoyed our modest life in Arlington. It was temporary and no big decisions had to be taken. Eric was very content because academia at MIT was new and different and therefore a novelty – nothing pleased him more than that. And there were many very agreeable colleagues at MIT and also new faces, like Noam Chomsky. I never got my head around his theory of linguistics, but his political views were wonderful and brave. He was a lovely person and said he would visit us in London.
*
We needed to go to California – Eric was pining to see his great friend Ralph Gleason, who lived in Berkeley with his wife Jeannie. Ralph was a thin, wiry New York Irishman whom Eric had admired and befriended in his New York jazz days in the late fifties. He was now a journalist for the _San Francisco Chronicle_ , covering all showbiz and pop music – not a typical friendship match for a middle-aged historian, but you will have gathered that Eric was not typical. When we went to visit, Ralph was reviewing a huge concert, which we were also going along to. At the time, it was the famous 'Summer of Love', which one could write reams about, but I won't. In a nutshell, it was a revolution: an emancipation of the young, who wanted to be done with the forties and fifties, and express themselves differently. They came from across America and beyond to San Francisco, and the epicentre of it all was in the suburb of Haight-Ashbury. The 'hippy trail', as it was called, was also through Europe and the Middle East to India. The hippies dressed in flowery fashion, wanted free love and sex, and took hallucinatory drugs, mainly LSD. Dropping out of conventional society was a new experience, as was their enthusiasm for communal living and a vegetarian diet. This was part of the counterculture that they yearned and strived for.
The music, came out of jazz, but with a different kind of sound, with the most famous performers being the Beatles, the Rolling Stones, Bob Dylan, Jimi Hendrix, Pink Floyd, Ravi Shankar and others. We had come for a very big concert in a huge ballroom called the Fillmore Auditorium, near the Golden Gate Park. It took place just before the gigantic and more famous Monterey Pop Festival. Sitting with the other journalists, Ralph and Eric were further behind me, Jeannie and their teenage children. Aside from the pink psychedelic strobe lighting swirling around (which I did like), the enormous crowds and unbearably loud, slushy music made me nervous.
None of us were on drugs, not even the Gleason teenagers. And so I expect we experienced it very differently from the crowds. I felt a million miles away from Eric, not knowing how he was reacting to all this. When we had a conversation later, I found out that Eric had also not enjoyed it except for the Motown Girls. In his usual way, he had been trying to analyse the evolution of this genre, from jazz, then rock to 'flower-power' music. He said it was probably not possible to get into it without drugs and he also disliked the overamplification. Above all, he felt silly: 'This is for the young. I shouldn't be here.' What a relief. I felt the same.
It was easy to restore my spirits privately. All I had to do was to think of Arlington, where in a couple of days I would see my babes again. We had of course spoken to them, and said we would be bringing presents. But longing always seems more intense towards the very end.
On our return, we decided to do more sightseeing with the children. We had already visited the Boston Aquarium, which was a big hit with Andy. Julia liked it and said some of the teeth were big – Eric was the most enthusiastic of all. I was more fond of the stylish provincial towns in Massachusetts, like Concorde, Lincoln and Lexington; the plain churches were especially impressive. The children were good sightseers, running up and down heaps of museum steps – 'Look here!' and 'Look there!' – and then off to the souvenir shops. They already knew all the culture routines at three and four!
I thought American academics were wonderful people, the best of the best, maybe because they didn't have the same class baggage as in England, and they were modest. But figuring out the Americans I met every day was more difficult. Sometimes there seemed to be a hidden inferiority complex, almost a need to justify themselves, and they expected our impressions of their country to be totally 100 per cent positive. Maybe their sensitivity was due to their families having come from very distant places; a large permanent move, to the US, in their eyes could only have been for the best. The very best.
Chapter 13
# _Clapham, Part II Cambridge-in-the-hills and Schoolchildren_
By September we were all settled back home in London, and our lives contained a schoolboy. Andy, now five, went to the local Macaulay Church of England school, which was much recommended. He filled the house singing, 'Raisin, raisin!', which we thought surely must mean 'praise him, praise him'. And the thing that came next was a best friend. Daniel Letts, whose family we befriended for the usual play dates and chauffeuring but who became real friends as well. John and Sarah Letts were a delightful, quiet couple of many talents. They lived in a beautiful, large Georgian house by Clapham Common. They had four children – Robert, Matthew, Daniel and Vanessa.
John started literary enterprises – the Folio Society and the Trollope Society – was involved in the National Railway Museum in York and the British Empire and Commonwealth Museum in Bristol. Sarah was an artist, an enamellist, who made and sold beautiful jewellery. Apart from her talents in the garden, she also co-wrote (and illustrated) a very good cookery book under her maiden name, Sarah O'Rorke. It seemed there was nothing she couldn't do: she was also a competent amateur recorder player. Sometimes we played in the same group. Our friendship (especially mine and Sarah's) lasted to the end. Although there were gaps when we moved, our friendship flourished again as widows. I treasure her cards and letters, as she didn't use the internet. She died unexpectedly in November 2017, in her sleep, and I am full of sadness and regret; I wish I had seen more of her. It was a huge shock for all who loved her, and even harder for her children.
In Clapham we were accumulating new friends. Among them were Connie and Nick Harman, who lived nearby with their two boys, but the marriage didn't last. Connie, however, remained her gregarious self and we met her at many parties and events. We had friends in common, especially John and Sheila Hale, with whom we were really close, and also Bamber and Christina Gascoigne. Jean Franco, who taught Latin American literature at Columbia University in New York, often came back to her flat in Clapham North when in England, and we saw as much of her as we could. She was English and a delightful jolly petite blonde, as vivacious as could be, and she and Eric could talk Latin America, and other subjects, ad infinitum.
No one could be more hospitable than Paul Wengraf, an art dealer from Vienna, and his wife Gertrude, whom everyone called Dada. They lived in Putney and opened their house to friends most Sunday afternoons. The Dial House, as it was called, had a circular garden and a small, charming pool. The atmosphere made everyone of all generations feel welcome and at home. The house was full of art – African art was their main focus and attraction. High tea was ready around four o'clock, or whenever we had finished making the sandwiches and laying things out. The number of people could range from eight to over twenty, including their children Peter, Tom and Monica. The house also had a splendid mynah bird called Clever, but one day, with the cage door left open, he flew away. As Paul became less mobile – by this time he was in his seventies – Eric usually went to talk to him by his chair in the sitting room. What did they talk about? I suppose art, Vienna, politics and so on. For both of them the time was always too short.
I don't like the expression 'earth mother', but Dada exuded these qualities, as she was open-minded, open-hearted, full of generosity and completely natural. I think she would have made a dreadful actress – there just was no pretence in her. She was good at _fun_ though, and interested in everyone except possibly herself. Dada taught German at Morley College near Waterloo and lived to celebrate her delightful hundredth birthday party in style, organised by her children and grandchildren.
I was not a full-time mum. We had au pair girls – mostly they worked out well – although one was ghastly and one disappeared in the night. An Italian publisher of Eric's asked me to translate into English the letters Antonio Gramsci wrote to his children while he was a political prisoner in Italy. Gramsci was one of the founders of the Italian Communist Party. When sentencing him at a political trial in 1928, the prosecutor said, 'We must stop this brain from thinking.' Gramsci spent the next ten years in prison and various clinics. He died soon after his release in 1937 of a brain haemorrhage in Rome. The Gramsci story is a twentieth-century tragedy of the greatest proportions. I worked very hard on this task and it also provided some respite from hearing my children cry or quarrel downstairs. It was all going rather well, but it turned out someone else was also doing the very same thing – the publishers got it muddled – so I stopped. Actually, it was a bit of a relief. I didn't think Gramsci's letters were all that appropriate for his young children. He criticised too much and one was aware he had not been in children's company for a long while. He was very ill at the time and horribly treated in prison.
We usually went up to Wales during holidays, half terms and all of August. We were always surrounded by friends. I believe that without fail, all of these relationships stemmed from Cambridge. In fact, the whole area should have been called Cambridge-in-the-Hills. How anyone could read any books or do any studying in those undergraduate days there beats me. The Cambridge friendships started before my time, but once I got used to their loud speaking, I fitted in easily.
Already in the valley when we arrived were members of the Bennett family, Kay Herzog and her daughter Anne. The house was also used by relatives, including her sister Liz Eccleshare (daughter of Cambridge dons Joan and Stanley Bennett), who had known Eric since his undergraduate days. She came frequently with her husband Colin and their children.
Historians Edward Thompson and his wife Dorothy and family were neighbours in a farmhouse over the hill. They had many visitors (often historians from the USA), and people liked to say one could hear more typewriters clicking than birds chirping in our valley. I have fond memories of New Year's Eve parties at the Thompsons and that sobering, bracing walk in the early hours against howling winds to reach our car, a magical way to start the New Year.
Plas Brondanw, the home of the Williams-Ellises, was also in the Croesor Valley. Clough himself was at Cambridge, as was his son Kitto. His wife Amabel (née Strachey) was from a different background. She was a writer and a very lively intellectual, always keen to question and discuss many topics with Eric. Unfortunately she was an early riser and Eric was most certainly not. But he grew very fond of her. One April Fools Day, I invented her proximity in order to hurry him out of bed. One of the Williams-Ellis daughters, Susan, who married Euan Cooper-Willis, also educated at Cambridge, was a designer and together they formed and ran the well-known beautiful Portmeirion Pottery.
Needless to say we all visited and ate (rather well) at each other's houses. The middle classes know how to look after themselves, even though we went around in our oldest, moth-eaten clothes and throwaways from London. Eric's Cambridge friend Robin Gandy was enormously good company and invented postprandial games, often requiring one person in turn going out into the kitchen for five minutes (hopefully to begin clearing up). Robin, alas, had the longest laugh known to mankind. It went from pianissimo to fortissimo with much staccato too. Although it seemed funny in the countryside, it was much less so in a London cinema, as we discovered when reciprocating his hospitality.
Later on, Walter and Dorothy and their five children joined the crowd and rented a very small picturesque gatehouse called Gatws, into which they all somehow squeezed.
The odd man out in this harmonious group was the artist Fred Uhlman, a Jewish emigré from Stuttgart, who was lucky that a friend had advised him to move to Paris in 1933 when Hitler became chancellor. Fred got by selling his paintings and drawings to private clients. In 1936, when Hitler became a dictator, Fred was able to cross the Channel to England. Still his luck continued. On a trip to Spain he found true love and met his future wife, Diana Croft. Unbeknown to him, she was a wealthy, top-drawer English aristocrat whose family home was Croft Castle (dating from the fourteenth century and now a National Trust property), halfway between Leominster and Ludlow, about an hour and a half's drive away from the Croesor Valley. Their marriage took place in London and although Diana's parents were dead set against it, particularly her father, the ultra-conservative Lord Croft, there was money to buy a large and beautiful house in Downshire Hill in Hampstead.
Together they became a good and caring couple. They started an artists' refugee committee for stranded Jews who had fled Germany and Austria, trying to find accommodation for them in England. The office was in their new home; I find newlyweds opening their doors for a charity very commendable. Diana's contribution was indispensible. They also founded the Free German League of Culture – while Hitler was attacking and closing down German culture, the League was set up to counteract and enhance it – which attracted many established artists and prominent intellectuals, including Oskar Kokoschka, Alfred Kerr, John Heartfield, Kenneth Clark, Stefan Zweig and more. However, Albert Einstein refused his support, as he felt the project had been taken over by the far left.
Fred adored the North Wales countryside, including its strange industrial buildings, and after the war they went on holiday to Portmeirion. They struck up a friendship with Clough and Amabel. One day, walking with Clough in the Croesor Valley on the slope from Parc Farm, they came across an old disused cowshed behind which was a panorama stretching right down to the sea. Fred asked Clough if he would be able to design and enlarge the cowshed into a house to use as a retreat. Clough actually agreed and it became a spectacular modern home that blended with the land and was very much off the beaten track. Clough called the new home Beudy Newydd (New Cowshed), which he rented out to Fred on a ninety-nine-year lease. Ten years later, when we moved to Parc, the Uhlmans became our neighbours. But Diana felt she was neglecting her responsibilities at Croft Castle, and they came less and less, and passed the house to their son Francis. We were invited over by him and got to know him; nice guy.
We now had two schoolchildren. Julia started in the infants school in September 1969, and she was ready. Before she started school, when we went to collect Andy at the end of lessons, she would peer intensely through the school windows, working out where her classroom would be and where to hang her coat. I think on the day she finally started, she knew exactly where her peg was and rushed in. Other new children were clinging to mums, some were crying. Not our girl. I was the one that didn't know whether to laugh or cry.
Eric was away on his travels much of the time – I can't say I felt settled in Clapham, which I should have been by now. I did try. We started to have some dinner parties – Francis Haskell came with his new bride Larissa, whom he had recently met in Leningrad and who was also an art historian. After supper, she asked us when the dancing would start. This question, among others, continued to charm us for years. For instance, she wanted to know how to survive Christmas in England when you had no car, no children and no television. That problem was immediately solved, as they both came to us for Christmas Day lunch for ever after. It became an established tradition.
Chomsky kept his promise to visit us, even though it was just a fleeting visit. I can see us all now, sitting on the stairs, and Eric explaining to the children about words. Andy asked Chomsky, 'Why do we call blue, blue and red, red? And not red blue and blue red?' 'Aha,' Chomsky replied. 'That's a very important question, and am not entirely sure I know the answer.'
I suppose my Clapham blues seriously began when Eric was in Paris. In May 1968 Eric had to attend a conference on Marx in Paris organised by the UN. He found the prepared papers extremely boring as usual, but what upset him profoundly was the fact that the delegates were impervious to the exciting student demonstrations going on outside in the streets. It was a remarkable time of infectious rebellion against capitalism, leading to the largest general strike in France. The anarchists wouldn't miss out of course; there was a fire and street battles in the Latin Quarter. The Marx delegates were holed up inside, completely missing the entire happenings. I daren't think what was in Eric's mind at this dramatic time. He must have been in silent despair. He wanted change so very badly, but instead the rebellion predictably fizzled out and the Marxist delegates would continue in their same old unbendable ways.
It was then that it struck me there was a jinx on us in that house or in Clapham, and we felt slightly gloomy. I had become a bit superstitious (must have picked that up in Italy). Maybe it was a simplistic way to explain away my strong intuition that we were not in the right place. Around that time, I had started to trust and follow my instincts more confidently than actual logistics. It seemed ridiculous that my mother and family all lived in north London while we were in south London. On the Sundays when Eric was writing, which was most of them, I drove the children right across the city to be with my family.
In reality I didn't have enough close neighbourly friendships. I do not believe women with young children can ever really flourish without this support. The people in the two next-door houses were invisible. We never, ever saw them. The people in the street opposite were in their Caribbean world and were out at work all day. Only on Sunday mornings, when they were cleaning their cars, did we exchange greetings. There was a dangerous main road between us too. That was how it was; our paths didn't cross. Eric was quite content in Clapham but I must have worn him down.
I began to realise that our domestic happiness was in my hands. I started vaguely looking, and one day I found a perfect house in a Hampstead street that led straight into Hampstead Heath and Parliament Hill.
Chapter 14
# _Our New House_
The housing market in the seventies was beginning to boom. All the agents were going mad with gazumping and other legal but disreputable practices, and the estate agents Benham and Reeves were mocked as Benham and Thieves. It was jolly good luck that Mr and Mrs Forbes, owners of number 10 Nassington Road and whose house I was coveting, were old fashioned and sentimental. She insisted on selling to a family with children. They would not budge from that. I made the kids dress nicely with white socks, and brought them along to impress, and it worked. Mrs Forbes said she would save the house for us and she kept her word and kept to the original price. That's women doing business.
The house had three floors, a cellar, a small balcony, a decent-sized garden at the back and pretty flower beds in the front. The next year was all taken up with the move. A fair amount of work was needed, like taking out fireplaces, installing central heating, extending the kitchen and last, but by no means least, putting up bookshelves _everywhere_.
But it was certainly very much the house for us. It felt right. Of course, it was not ready on time – which house is? Dorothy Wedderburn stoically came over on our first night there and we managed champagne on ice and something or other to eat. Also known as 'Doffy' (her baby pronunciation of Dorothy), she was one of Eric's close friends from Cambridge and their friendship lasted a lifetime (they died three weeks apart). She had joined the Communist Party while at Girton College in 1943 and stayed in the party until 1956. Although a member of the great and the good, she refused to accept an honour that was offered to her by the Queen. She always looked so grand and took good care of her appearance – in Eric's obituary of her, he recalled her eightieth birthday and how she had 'reserved a table for the staff of her Knightsbridge hairdressers'.
Moving to Nassington Road was a night to remember; Dorothy became a joint best friend that evening. There is so much hope in the air in a new house. Full of dreams and expectations.
Chapter 15
# _Home, Sweet Home_
Our move from a main road in Clapham to Hampstead, NW3 felt like from polar ice to the Equator. Our house was halfway up Nassington Road towards the Heath. There was so much greenery around, I perceived our home as a country villa. We all loved it; I had not uprooted us for nothing. Eric and I remained there until death did us part.
By chance, Eric already knew our neighbour, Manny Tuckman, a retired GP, and his wife Gita, a sculptor. They ran an artist collective at No. 12. John Southgate, an impressive jazz pianist, was a tenant there who we were friends with – except on the nights he forgot the time and would play way past midnight. The first friend I made lived two doors down: Barbara Zeinau at No. 6 and her German husband Sigurd, a theoretical physicist. It didn't take long to get to know many other neighbours too, but Barbara and I bonded so easily. We are the same age and she was a teacher at St Dominic's, a local primary school. Some say we looked alike.
We have both moved away now, but to this day she still faithfully phones me every Sunday morning for our weekly chat. Sigurd died unexpectedly, and after raising her two children, Varina and Matthew, Barbara became the wife of a delightful architect. He was one of those boat people, restless on land, and was soon off sailing again. Barbara used to come over often, expecting Eric to be away too. During a sweltering heatwave, hot enough to stay outdoors, we talked all through the night in my garden, drinking wine and grumbling about our absent husbands, who were either sailing or lecturing. That long night cemented our friendship for good.
With so much travel and Eric often being away, home traditions were very important for us all. I am not one for surprises. I like to know what's coming and what to look forward to. Christmas is perfect for this; in fact, I ran it like a military operation. It was a Herculean task for me to keep the show on the road for three days, and then pack up for Wales on the fourth.
Christmas Eve (the continental Christmas) was reserved for the four of us only. We had a festive candlelit supper treat on our own for around thirty years – a tradition that all began in Nassington Road, and continues to this day. It was an immovable feast – while I did preparations in the afternoon, Eric would take the children to a museum (in the days when they stayed open on Christmas Eve) or the cinema. My own little ritual was listening to the Festival of Nine Lessons and Carols on the radio from King's College, Cambridge while I stuffed the turkey and scoured the ham to decorate with cloves, both large enough to last over Boxing Day. I enjoyed knowing that so many people would all be doing the same as me. There is a sense of belonging in shared tradition. Recently I heard our friend Hella Pick say on _Desert Island Discs_ that although she felt deeply about being British, 'I still wonder if anyone not born in this country is ever fully accepted and integrated.' My experience is different in that I feel totally integrated, accepted and loyal to Britain, but in my head I'm a Continental woman.
On Christmas Day, the house was a little more full. Grandma Lilly, cousin Gretl, plus a dog always came to us, as well as Francis and Larissa Haskell. Later we acquired the Italian historian and professor of Classics, Arnaldo Momigliano, who also joined the tradition. Eric fetched him and he always arrived, presenting me with a CD of music by the seventeenth-century Italian composer Gesualdo. As I enjoy predictability, this always gave me great pleasure. Sometimes the odd foreign scholar or two who had not reckoned with the British Library being closed and needed Christmas cheer came too. I am not sure how much our children enjoyed these rather intellectual Christmas Day lunches, although they became very fond of the Haskells.
For Boxing Day, Eric's communist friends came for a cold buffet, polishing off food from the past two days. Regulars were Margaret Heinemann, the archaeologist, Tamara Deutscher, the writer and Monty Johnstone, a writer and lecturer, among others – usually about fifteen people. It was totally casual - they were a loud, jolly lot and I'm sorry the names have gone from my memory now. The next day, we would pack up our things, and drive six hours to Wales, where we stayed until after New Year had rolled in.
Chapter 16
# _Academia_
Eric's career had a marvellous and also lucky trajectory, although it began badly and very slowly. Like many Eric's progress was hampered by the Second World War, and for him it was further hindered by the Cold War. As a member of the Communist Party of Great Britain, he was automatically monitored by MI5, who took it upon themselves to hinder as many job possibilities as they could, and the press label 'Stalinist', which haunted him throughout his life, did not help.
Despite achieving a double starred First from Cambridge, among other academic triumphs, he was still turned down from teaching positions at Oxford and Cambridge, which was disappointing and surprising for him. In 1945, his application for a full-time position in educational broadcasting at the BBC was vetoed by MI5, even though he was considered by the BBC 'a most suitable candidate'. A couple of years later, he finally secured a lecturing position in history at Birkbeck College, London.
The acting Master of Birkbeck, a medievalist, Professor Darlington had a personal antipathy towards Eric as well as disapproving of his politics and Communist Party membership. He was heard to have said that Hobsbawm's promotion from reader to professor would only take place over his dead body. As we know, life is stranger than fiction: Professor Darlington died very soon after uttering these words and when Ronald Tress became Master, he overruled the edict, much to the relief and delight of all the staff, who had been clamouring for this promotion for a long time (it was getting to be quasi-scandalous).
By the time he became a professor, Eric was already in his mid-fifties. Better late than never. But he never wasted his time bearing grudges; I tried to learn this from him, but I am still only a beginner. I remember buying a high-waisted Empire-line dress, as was the fashion, for the dinner celebrating his promotion. Our big academic journey had begun and we were on our way – the escalators were going upwards.
From time to time, Eric would tell me he wished he had studied anthropology. At school, he didn't know there was such a subject – his teachers had earmarked him for history. But he did it anyway. He was so interested in peasants and all those who cultivated the land, and his heroes were those who helped fight poverty and oppression: rebels like Robin Hood, and the Mexican revolutionaries Pancho Villa and Emiliano Zapata. I remember Eric talked to me about Quico Sabaté, a Spanish anarchist, and how he admired his bravery in the struggle against Franco and the Spanish state, so much that he had tears in his eyes as he told of Sabaté's killing.
Chapter 17
# _Sabbatical in Latin America en famille_
Eric's passion for Latin America had been ongoing for about twenty years, so when he was offered a six-month sabbatical in 1971, he jumped at it. Firstly, it was a chance to improve his Spanish and secondly, it gave him the opportunity to continue his deeper research into peasants, their rebellions and above all, agrarian reform. Since Fidel Castro in Cuba had apparently shown the way for social revolution in the region, Eric was feeling politically upbeat. And of course he (and all of us) were elated over the failed 1961 American military invasion of the Bay of Pigs. Latin America was, as Eric described in his memoir, a continent 'bubbling with the lava of social revolutions'. But I regret to say that now, in the twenty-first century, we can see this didn't happen the way he had hoped. Though improvements there have been plenty.
Quite soon after moving into Nassington Road, we were preparing to travel. I was kept busy liaising with the teachers at Gospel Oak School, where the children had just nicely settled in. The head teacher, Ron Lendon, was very accommodating. Not only did he give permission for the children to go away, and held their places open in the school, he went above and beyond to keep the classmates in touch. Possibly too much – while in the La Sierra in Peru, Andy received thirty letters all saying exactly the same thing: 'We went to Camden Lock and took boats on the canal.'
Departure day arrived. We would travel to four countries: Mexico, Colombia, Peru and Ecuador. At all passport controls we held our breath. Not because of communism or visas this time, but at some point Julia had drawn a blue ship in my passport and it was too late to get a new one. Reactions were unpredictable – they veered from, 'Which child drew this?' in a friendly manner or disappearing with the passport to check with a higher official. A defaced passport is actually invalid, so very much was at stake.
Our trip began in Mexico City, but our luggage unfortunately did not. We had to travel via Lisbon, where it remained. It turned into a rather undignified April Fool's day arrival, extremely hot, and various academics were summoned to help us buy underwear and other essentials. When this situation lasted over a week, Eric lost his patience and announced that if our suitcases didn't arrive within two days, we were going straight back home. Then in the Latin American way, a VIP, presumably the ambassador, was contacted and, abracadabra, all our suitcases arrived the next morning.
We stayed with a charming French couple, Jacques and Lucero, who had a big house with grounds, and the children liked it very much. There were two dogs – a Dalmatian named Joreck and an Afghan hound named Bingada, as well as two kittens. It was not all play for the children; every day they had some lessons with me, and Andy had to keep a daily diary-cum-scrapbook, which he did very well. I am using it now to write this. Lucero also gave each of them a daily painting lesson.
Ralph Miliband, who was also working in Mexico City, lived in the house with us. We were already friends with him and his wife Marion from London, but it was good to get to know him better in our new exotic surroundings. A warm and interesting man, and we were very glad to have his company.
He fitted so well with us – his children, David and Ed, were of similar ages to ours, and he was a sociologist and Marxist author.
We enjoyed exploring around the nearby Chapultepec Park with the children and we also went on many excursions including the Pyramid of the Sun (which according to Andy's diary had 750 steps, in case you wanted to know), the canals of Xochimilco, with their marvellous coloured gondolas, the city of Cuatla, and Lago de Patzcuaro, to name a few. We were within walking distance of the house where Trotsky had lived and was assassinated (now a museum).
We also went further afield to Cuernavaca, a beautiful city where Eric's friend Ivan Illich lived. We spent the day at his house as the children played and swam in his pool. Originally from Vienna, Ivan was a philosopher as well as a Roman Catholic priest, who wrote many books about 'deschooling', believing all education systems ruined our lives. He also wrote books including _The Powerless Church_ and _Medical Nemesis: The Expropriation of Health_ , among many others. All of them were ultra-controversial and people spoke of him as either a genius or a crackpot. Eric stood on the fence and referred to him as a European ideologue.
Downtown Mexico City there was also the marvellous and world-famous National Museum of Anthropology, which we were all crazy about. For the children it was so engaging and educational, with many replicas of the interiors of ancient indigenous (also referred to as 'Indian') homes. They were like big dolls' houses you could walk into. However, it was in the centre of the city, so it was difficult just to pop in with the children. The area we lived in, Chapultepec, was about fifty minutes away by a bus packed to the brim.
Lectures in Mexico were noisier than those I was used to. Young boys selling chewing gum from trays on straps round their necks loudly shouted, 'Chicklets! Chicklets!' throughout the talks. Eric was nervous at the beginning as his Spanish was rusty, but he mostly managed to cobble together something good. I remember seeing his name on a flyer in a lift, advertising a lecture of his in a vast auditorium the following evening, an event he knew absolutely nothing about. But Eric had learned to wear his flexible hat in Latin America. He was quite a little celebrity there. At this time in England, Eric was not well known and certainly not the intellectual public figure he would become, but in Latin America it was quite different – partly it was his interest in them and their lives.
Everything Eric wrote was translated into Spanish and Portuguese by his fantastic publishers, Critica, in Barcelona, Paz e Terra in Rio de Janeiro, and, later, the larger Companhia das Letras in São Paolo, Brazil.
We were fêted and invited for dinner often by the many people Eric knew (friends, colleagues, pupils and ex-pupils). In Mexico, supper is indeed a night affair for the adults, and a lot of the talk revolved around politics. We were especially good friends with Carlos Fuentes, the well-known Mexican writer, and his wife Sylvia, who was a journalist. Leaping ahead in time, Eric reminisced in his last speech to friends on his ninety-fifth birthday in London about 'waiting for something to eat until almost midnight in Carlos Fuentes' house'. Our time here was soon up, and making our farewells to the warm and exuberant Mexicans was not an easy thing to do.
Two thousand miles and four hours on a plane later (according to Andy – our new trip adviser), we arrived in Bogotá, Colombia. Eric's friend Orlando collected us in a jeep. Eric had been commissioned to write a substantial piece for Bob Silvers in the _New York Review of Books_ on the FARC. These were the Revolutionary Armed Forces of Colombia who were supported by the Colombian Communist Party, which had managed to survive the fierce internal conflicts of the 1960s. Eric was able to meet many of its leaders and supporters. With the World Wide Web several decades off, he had to surround himself with all the local newspapers he could seize.
We stayed in a very pleasant hotel in the centre of Bogotá, but I was worried about the children playing in the lifts and being mistaken for Americans – _gringos_. Much kidnapping was going on and I was on edge about it. Also, I wasn't quite sure how best to keep the children occupied. But as usual, Eric had contacts galore – luckily all his academic friends had maids in their households and help was always offered.
Bogotá's climate is usually rain or cloud, but to my amazement I learned that a one-hour car or train ride would take you to a sunny climate with blue skies and a marvellous swimming pool. Sometimes I was lent a maid to accompany us, or alternatively one of the wives who could speak English came, making a real holiday of it. In fact, all of Colombia was a holiday for me; plenty of time to daydream and think, as well as look around. I should have brought my copy of García Márquez's book _One Hundred Years of Solitude_ to read again.
Apart from many prettily painted, colourful churches with friendly façades, the big attraction in Bogotá was the Gold Museum, which was astonishing. Andy was most taken by being inside the golden strongroom with the golden safe – he was already interested in big money. The museum had corridors paved with gold, like Dick Whittington's imaginary London streets. On the walls, glass cabinets displayed intriguing, mainly pre-Columbian objects, ornaments and jewellery – some primitive, some that one would love to wear today.
After my cushy weeks in Colombia, it was time for a more simple and rustic life in Peru. I expect Eric had not finished enough research for the Bob Silvers piece, but we had to fly off to Lima. I hope my readers are ready for a headful of agrarian reform, because this is the reason we went.
Here it was all about land: who owned it, who cultivated it and who was being exploited since the Spanish conquerors stole it way back in the sixteenth century. From the 1920s onwards, peasant revolts were brewing in Peru and getting stronger towards the late sixties. While we were there, a progressive military government (which sounds implausible, but really existed) under General Velasco Alvarado was in power. He predicted the peasant unrest intensifying and so his strategy was to hand back the large estate farms to the original indigenous owners, but under strict state control. Naturally there were conflicting opinions on this. I think in Eric's mind it was probably the best hope for Peru's impoverished peasants; he wrote in 1970, 'If ever a country needed, and needs, a revolution, it was this.'
These issues of expropriation inspired a group of brilliant young historians to form a research team. This included Joan Martinez Alier (a Spanish man's name, by the way, pronounced 'Huan'), who was then a research fellow at St Antony's College, Oxford, José Matos Mar, who was director of the Institute of Peruvian Studies, and the Peruvian historian Heraclio Bonilla, who came up with the idea that the papers from all the expropriated haciendas (farms) ought to be collected to form an agrarian archive at the University of San Marcos in Lima.
Pablo Macera, a distinguished historian at San Marcos and long-time friend of Eric's, was very much in favour of this scheme and he must have roped Eric in. With Eric's participation, the team got the blessing from the crucial Agrarian Tribunal.
The research team deliberated over the best place to go, and considered the sugar-cane plantations on the coast. But in the end they plumped for the more interesting pastoral and agricultural haciendas in La Sierra, the mountainous area that includes the Andes. So it was to be sheep. Well, we knew about those. But unlike Wales, it was not at all a serene pastoral scene. The Peruvian sheep could not safely graze. They were called _chuscas_ in Spanish and _wakcha_ (meaning orphan or poor) in the indigenous language of Quechua. The _hacendados_ (estate owners) wanted to get rid of the poor-quality sheep belonging to the indigenous shepherds, leading to continuous animosity and sometimes violence between them.
The plan was to start at Huancayo, the capital of the central highlands, where Joan discovered boxes of confidential letters dated from 1920 to the 1950s, before the telephone reached there – a prize for the archives, I should think. We travelled there by train, together with Joan, where the first-class passengers received a supply of oxygen if they needed it at the highest point – the Ticlio Pass, around nearly 5,000 metres (16,000 feet). Eric and Andy did suffer some _soroche_ (altitude sickness), but it didn't affect Julia or me. However, I had my share of mosquito bites, which were huge, including one on my nipple, but they never touched Eric, always right next to me. (I ask you, is that fair?)
Huancayo is a lovely little commercial town surrounded by mountains. But we only stayed one night, as it was more convenient to be in La Sierra with everyone else. We stayed in a cottage. We met the British anthropologist Norman Long from Manchester University, who was living in another part of La Sierra called Matahuasi with his wife Ann, also an anthropologist, and their children Alison and Andrew. Visiting them entailed a fifty-mile journey on a bus with chickens, but my time with them was so special, I didn't mind. Ann and I would work on the children's lessons together, with even a bit of recorder tootling. At 'Sierra playtime' the children all fell on each other with joy; Andy wrote in his diary 'it was super playing with English children'. I felt he was sighing with huge relief. At some point the Longs very kindly looked after Andy and Julia for a whole day, allowing Joan, Eric and me to travel by taxi up to about 4,000 metres (13,000 feet) to look around a new, totally modernised hacienda called Laive.
The taxi driver drove like a demon and we couldn't stop him doing so – he thought it was funny. I was scared out of my wits for the sake of the children if we died. I couldn't breathe with the high altitude and anxiety. I gave Eric a look that embodied: 'Why are we here? Why did I marry you? This is too much.' I was having a wobbly. Thank goodness Joan always stayed calm and kind, because Eric did not.
We did in fact arrive safely. In the main building, there were important papers of value for the archives. We looked around the other buildings, where we saw into the white, pristine future – it was like being at a luxurious high-tech dairy farm. At Laive, the expropriation had been extreme. Much like the eighteenth-century Highland Clearances of the croft people from the Scottish Highlands and Islands, the cruelty was atrocious. The indigenous shepherds and their stock, such as lamas, were pushed out of their grounds. The new _hacendados_ really needed more free land for their contemporary farming practices and above all wanted no contamination between the species. It was terribly cruel but necessary. Joan and his wife Verena went back to Laive at a later date to collect those papers, which they managed to pack and ship to Lima's Agrarian Archive. Joan later went to Cerro Antapongo on his own, among many other places in Peru, collecting papers for the archive.
Back in Lima, Eric's friend Pablo Macera seemed to take charge of our recreation at weekends. He was a man who didn't fit into any pigeonhole – very original, exuberant and domesticated. He was a great cook, but that meant enslaving everybody in the household to this purpose. You can imagine the meals in huge casserole dishes, to which everybody could come – this was his way. Pablo became our leader, always taking charge of making our stay memorable. He was so delighted we had come and I think he temporarily 'expropriated' the university bus for our convenience at weekends. He then invited his family and friends, some servants and even some pets to travel around on the bus with us. We all sang songs together, including a Peruvian vowel song. I found this an ideal way to travel. This is how we saw the Peruvian countryside, stopping where and when we wanted along the way, sometimes chatting with strangers in restaurants or cafés. We didn't go especially to hear music, but there was so much of it around – at parties, in the open air, in the streets. And of course wooden flute music was prominent.
As a family we had already made our expedition to the old Inca capital Cusco and to Machu Picchu; Andy probably gained more out of these historical trips than six-year-old Julia. Like me as a child, I think she was very happy just spending so much time with her family. Here in Peru's Sierra, Eric was in his element. He was doing very much what he wanted. Surprised at feeling so at home in this faraway place, Eric was charmed that on the slopes of Machu Picchu he could find little wild strawberries (berry-picking is an obsession for anyone with a Mitteleuropean background). This is precisely the magic of Latin America, mixing touching European familiarity, with exciting foreign exotica.
Pablo arranged to take us all to a bullfight. I was interested and excited, but didn't enjoy it – I have never liked black humour and this seemed like black sport – however, I wasn't as upset as Pablo's four-year-old daughter, who for the entire time softly cried, ' _Papa, por qué están matando a los toros_?' [Papa, why are they killing the bulls?] Papa took absolutely no notice of this repeated little plaint. It all lasted for what seemed ages. Nevertheless, Andy and Julia, as well as Pablo's son, were totally thrilled and mesmerised. Later Andy wrote about all the rules of this dangerous sporting ceremony in detail.
After a farewell dinner for all of us, it was our last time to say goodbye, and now we had to make our way home. We again had to fly via Lisbon, as the best connection was from Quito – the exquisite little capital of Ecuador. I think we only stayed three days. This was mainly to reconnect with people Eric knew – old students and friends – but also many new people who were apparently keen to meet us. Eric liked nothing better than getting information straight from the horse's mouth. We never did sightseeing alone like tourists, but were always accompanied by local people who showed us the best and knew what to miss out.
Ten miles to the north of Quito is the small town of Otavalo, inhabited by indigenous locals. We went there on a day visit and along the way we crossed the Equator, where the children had photos taken – Julia in the northern hemisphere, Andy in the southern. The 'Indians' all wore short, loose trousers, a black hat and a poncho. They each had one long black plait hanging down their back. They are famous for their handcrafted textiles – weaving, ponchos, tablecloths – and sell these goods around the world, where you can see them in Manhattan, Paris and so on. We were told it was cheaper to buy in the market in Quito, which you had to get up at 5 o'clock in the morning to visit – which we actually did (I bought hand-embroidered tablecloths for Mum and Gretl). One was expected to bargain, which Eric loved, but I loathed. This is always my problem in developing countries. I bought a poncho and later saw the same one on Hampstead High Street.
Chapter 18
# _Bourgeois Life in the Seventies_
Surely everyone is glad to be home, wherever you've been, after a long absence. Home is knowing where the tea bags are. I had a purpose again, organising the whole place and picking up where we left off. But I was easily overwhelmed at such times. My mother used to say she usually felt I had the water right up to my neck. I am afraid she was right.
Eric and the children nestled back into their familiar routines easily while I was dealing with new situations. Our overgrown garden sparked an interest in horticulture for the first time, although I was ignorant of plant names in English or Latin. So I began my own little agrarian reform, making the usual mistakes of planting big things in front of little ones. But I developed a complete new passion for growing old roses, especially French ones like 'Zephirine Drouin', 'Albéric Barbier' and, after my mother died, I planted a 'Louise Odier' near the house. The colours, names and smells intoxicated me. I automatically acquired a new subject of conversation, as it turned out that loads of people loved French roses too.
We also found ourselves with an adorable distraction, having somehow acquired a cat who insisted on belonging to us. She just waltzed into our lives one day and none of us wanted to part with her. We put a notice on a lamp post outside, but nobody claimed her. We decided on the name of Ticlio, after that highest pass in Peru, though when I had taken her to the vet it turned out she was a female and her permanent name became Ticlia. She was a most handsome tabby cat with four snow-white patches on her nose, chin, neck and paws. She had a good character and was not difficult – although she hated laughter, which was a hoot, and was easily annoyed by recorder playing. She knew Eric was the important one because he made the least fuss over her. Ticlia was loved by us all – she never got lost and was with us for fifteen years.
My life seemed to be at a watershed. My family continued their normal lives while I began to do things I'd never done before. I enrolled in an ILEA (Inner London Education Authority) recorder evening class where I met a set of completely new people. Our first teacher, Peter Wadland, was a delight, and extremely talented. His day job was running the Early Music repertoire for Decca Records. It was his ultra enthusiasm that got us going. He would actually jump up a little, saying, 'Yes, that's good. Please let me hear it again. One more time.' It was a long evening for me. Early shepherd's pie for the family, then a dash down to Marylebone Grammar School for the recorder class, over two hours of playing, and then off to the local pub. I did not want to miss out on the instalments of everyone's weekly updates. We made a very satisfactory group.
*
It was wonderful to see our dear friends again after half a year away. Several have already cropped up in previous chapters. As always, many stemmed from Cambridge. They were all sophisticated and could get along well with anyone. In our core circle were Gabriele and Noel Annan, Neal and Isabel Ascherson, Paul and Sally Barker, Leslie Bethell, Nick and Rosaleen Butler, Linda Colley and David Cannadine, Cynthia and Roderick Floud, Roy and Aisling Foster, Michael Frayn, Catherine and Erich Fried, Jack Goody, Richard and Vivien Gott, Felicity Guinness, John and Sheila Hale, Francis and Larissa Haskell, Bruce Hunter and Belinda Hollyer, Ian Hutchinson, Nicholas Jacobs, Martin Jacques, Helena Kennedy, Marina Lewycka, Brenda and John Maddox, Karl and Jane Miller, Juliet Mitchell, Gaia and Hugh Myddleton, Kathy Panama, Ruth Padel, Stuart and Anya Proffitt, Emma Rothschild, Garry and Ruth Runciman, Joseph and Anne Rykwert, Donald Sassoon, Graeme Segal, Stephen and Tia Sedley, Amartya Sen, Leina and André Schiffrin (when in London), Jean Seaton, Yolanda Sonnabend, Vanessa and Hugh Thomas, Claire Tomalin, Marina Warner, Dorothy Wedderburn and John Williams, Lindy and Robert Erskine.
Just looking at this list makes me smile. This is the core group we drew from, but it was interspersed with others all the time, often depending on who was in London.
In the seventies, dinner parties were the norm, usually with eight to ten at the table. Cookery books, especially Elizabeth David, were all the rage. I spent a lot of the time on the telephone, organising who would fit well together. I would think nothing of phoning Jack Goody in India, where he could easily be, if not answering his phone in Cambridge. Eric would have liked the dinners to take place once a month, but then he only saw to the wine. Gradually, towards the naughty nineties, it became more popular to throw larger parties for more people, which were sometimes catered for professionally. I suppose as we grew older, we all made more and more friends.
At our dinner table, we wanted to talk without noise or muzak: conversation is all. If the ambience is right – informal, casual – then people are relaxed enough for freer talk and might come out with things they've never said before or thoughts they are still formulating. In any case, there was much chatter among the chattering classes on whatever was current – I liked it best when one conversation was happening around the table. The food was as _casalingo_ as possible. No posh things _en gelée_ or shaped food en cocotte, just simple, familiar and plentiful dishes, mainly dependent on the weather. Three courses was the usual custom. In the seventies guests still expected cheese or savoury after puddings, but was eventually phased out.
Sometimes I made food that could be laid out for a buffet and people helped themselves. It's nice for guests to get up, and in England everyone feels comfortable queuing. I remember a dinner party when, completely out of character, I said, 'Oh, now we're older we seem to talk more about food; we used to talk about sex.' Gaby's response was: 'Well, Marlene, think of the varieties. Take spaghetti alone... alle vongole, al pomodoro, all'amatriciana, al'arrabiata, bolognese...'
I am no longer sure when the children stopped wanting to play host, mingle and hand out olives to the guests; I suspect Julia (who was more gregarious) enjoyed it for longer than Andy. As they entered their teenage years, their interests were beginning to shift. Julia was spending her time with friends but still singing in the school choir. Andy began learning the guitar and often had an expression on his face that seemed to say, 'Hey, why am I not skateboarding in California?' No longer little children and not yet adults, they were coming into their own.
At some point, Clough Williams-Ellis had decided to restore and refurbish the 16th c farmhouse of Parc Farm. It was part of a larger farm complex consisting of a grand manor house, two big barns and a beautiful small pond beside the farmhouse. It was just about a mile down the Croesor Valley from our cottage Bryn Hyfryd. Waiting for the restoration to finish, it would be two years before we could move into this larger and more romantic house called Parc. Clough had left a tree growing out of an attic room. By now it was so tall, Eric had an extra clause inserted into our lease in case it toppled over in a storm (all of this to the dismay of our lawyer). It was not a cosy home, but we fought the damp and slowly adapted. It had so much character and style. We were still in Snowdonia National Park, and as before, still going on our usual, wonderful walks as well as discovering new ones.
After the commotion of moving house on a fine sunny day, I caught a glimpse of Andy and Samantha Campbell-Jones going for a walk hand in hand. This was a big surprise for me, but their innocence was so sweet. When the children were younger, Eric and I agreed that he would deal with religion and I would deal with sex – but when the time came with Andy, I felt it was Eric's territory and I wanted him to have a hand-on-shoulder chat (even though on this occasion it wasn't necessary). Samantha was one of two lovely daughters of a charming family who were great friends of Dorothy and Walter and often spent their holidays in the Croesor Valley.
We were now a few metres away from Dorothy and Walter, who had moved in to the gatehouse of Parc Farm. Their youngest child, Zac, used to sit outside, where Eric was kept busy chopping much-needed wood for our damp indoors. Many conversations would take place between Eric and the toddler. Once, when Zac repeated something to his dad, Walter asked, 'Who told you that?' to which Zac replied, 'The woodcutter!' pointing towards Eric.
The farmer, Dai Williams, looked after vast numbers of sheep, but no longer any other animals. Whenever coming into conversation with him, Eric would end up unconsciously speaking in a Welsh accent for a while. He had a habit of doing this; the same would happen speaking to the local garage owner, or asking 'Where ya from?' of a New York taxi driver. I didn't mind, but it was entertaining to see how embarrassed the children were. Quite often we three had a good laugh at Eric's droll and eccentric character.
I remember a fine summer weekend when our friends Amartya Sen, the economist, and his wife Eva Colorni came to stay. Eva was for many years my closest friend. She was an Italian economist and teacher, as well as a very domestic and intimate person. She was totally involved with her family and children, Indrani and Kabir. Her death from cancer, when the children were young, was the most shocking thing in my life. Even though Eva was my best friend, I did give Cupid a very small hand in helping Amartya get together with his new wife, Emma Rothschild. I felt Emma was the best thing for him, and the children. In the end, their happiness is what Eva would have wanted most. And so it proved to be.
Before we knew it, both our children were desperate to start life and the 'awkward teenage' years were now upon us. Andy was ambiguous about staying on at school, and continuing with more exams. We were not tiger parents and we never wanted to push them; as Andy recalled, 'I remember that Dad was so keen for us not to feel pressure of following in his footsteps,' but for fear of doing this 'he put no pressure on us at all to go to university'. Clearly, we had taken our eyes off the ball. Eric and I felt higher education could be delayed until later in life if necessary, but we insisted they must at all costs get their A levels.
Eric took the initiative and decided to send a lost and muddled Andy to a sixth-form college called Bransons, split between a first year in Canada near Montreal and a second year back in the UK in Ipswich, which did wonders. He really wanted to go to Canada and their outdoors programme seemed very enticing for him. He came home a confident young man with three A levels under his belt and with many friends of both genders. I remember my mother being very impressed that Eric had put all his principles against private education aside and did what he felt was right for his son. Upon reflection, it could have been in part his original school, William Ellis, who overdid their disappointment that the 'son of Eric Hobsbawm' was falling short of their academic expectations. Andy recently admitted that 'if someone had said to me that university is a unique opportunity for learning and forming memorable life experiences and making lifelong friends, I might have considered it differently' Julia was different: she did want to go to university, but somehow did very badly in her A levels and did not get anything like the required grades to read English literature at Sussex University. This was an enormous shock to her. She was very disappointed indeed, but on reflection she felt that during school she had been intellectually underrated by her teachers, who deterred her from applying to prestigious institutions. Julia admitted this dented her confidence and caused her to 'stop trying a bit' with her A levels.
Instead she attended the Polytechnic of Central London to read Italian and French, but never enjoyed it. Among other reasons, she found herself surrounded by mature students while she herself was only seventeen. She decided not to continue and began working at the student union, where she fell in love with the union president, Alaric Bamping. Andy remembers feeling both concerned and impressed at receiving letters from Julia while he was travelling. She wrote about student protests organised by Alaric (was there perhaps a resemblance to her daddy?), where everyone barricaded themselves into the student union building for days on end. So, the older brother was looking out for his little sister, just like my brother Victor did for me.
Chapter 19
# _Martin's 'At Homes'_
While we were in Latin America, our friend the architect Martin Frishman and his mother had been planning to move to NW3 to a large house in Chalk Farm, which had a huge studio for Martin's collection. It had belonged to Vanessa Redgrave, who had never lived there – it was left to her in a will. By coincidence, they moved in 1971 when we returned, but Martin's dear mother Margaret died a year later.
It was after her death when Martin began his very special 'at homes' for his large circle of friends. These were wonderful evenings, each with their own unique cachet. Mostly his guests were composers, visual artists and writers, including Alice and Georg Eisler, who lived in Vienna but visited London often, Thea Musgrave, Peter de Francia, Elias Canetti, Erich Fried, Catherine Boswell, Max Neufeld, Yolanda Sonnabend and friends, Kathy Panama, Amalia Algueria, Jacob Lind, Frederick Sampson, Lalo and Viviana Fain-Binda, Susie Barry, Estela Weldon and Alexander Goehr. These are the ones I can name now. All were pretty left wing, and in this highbrow artistic and intellectual milieu Eric fitted like a glove. More than that, I like to think he was one of the attractions. They could talk about politics, opera, music, literature and visual arts into the night; to me it seemed they had all read everything. I was hungry to absorb it.
As you entered the house and made your way towards Martin's studio, huge posters of ships, locomotives and trains assailed you. He didn't distinguish between highbrow or lowbrow – you might find an exquisite painting or etching in between two posters of 1930s steamers. Martin's studio was filled with objects from around the world and a big blue rowing boat hung from the ceiling. I think his heart was really all for travelling, observing and bringing the world into his home. He was an unusual collector.
His main passion at that time, however, was Islamic architecture – he had become an expert. Sometimes he showed us the slides he brought home and needed for his teaching at the Royal College of Art, the Architectural Association, and at UCL, which was blissful for Eric, as he was himself crazy about Asian art in general, and Japanese, Chinese and Indian specifically.
The guest list varied, but Martin was a great host and cook, if somewhat ambitious. So these evenings were definitely not to be missed. Martin was a bachelor, with no shortage of girlfriends, but he had to be patient to find his number one. After ten years, his true love appeared, an Argentinian architect of Italian origin, Federica Varoli Piazza. As Martin had been our best man, Eric returned the honour and was best man at their wedding, and then a few years later I was the driver to the Royal Free maternity department when their beautiful baby daughter was on her way into the world. She was named Greta after her paternal grandmother.
Chapter 20
# _My Music Career_
My path to becoming a music teacher was by no means ordinary, and I would describe it as a mix of determination, sheer luck and the right people at the right time.
I knew my first step was to become a member of the Society of Recorder Players (SRP), which has branches in most UK cities. Everyone who can read music and play the recorder is welcome; conducted by professional volunteers, we would sight-read all afternoon.
I enrolled in the Marylebone SRP, a very lucky branch run by Theo Wyatt as chairman and musical director. No one more witty and knowledgeable would ever be found. Theo also ran many other courses off his own bat, each one with a distinctive agenda. His most ambitious course took place over a week every summer near the sea, outside Drogheda in Ireland. The sessions went on all day and were inclusive of all levels, ranging from brilliant (who might even be professionals already) to competent, to those who sometimes lost their place. Grouped by ability into one-to-a-part quartets and quintets, it must have been an enormous task to curate the music, especially before technology as we know it.
The venue was a large, comfortable house and in the seventies we could safely leave our instruments and belongings in unlocked rooms. The sea was less than a ten-minute walk away, close enough for a quick dip in between music sessions. Our communal evening ensembles were hugely enjoyable, and sometimes the more professional players delighted us with a performance of their own. There were also lessons in ceilidh dancing – I remember my partner once being a nun! There were often nuns there, who enhanced the music by playing their fiddles. One didn't sleep very much but it was worth it for the fun we had. I treasure the memories of such a variety of new music, beginning to understand the Irish, making new friends and the many gems Theo would come out with while conducting, such as, 'Speed in music kills just as much as it does on the motorway.'
Back in London, I was enjoying my evening classes at the ILEA (now led by Nancy Winkelman), but I discovered that the one-week residential courses were far and away the biggest help and stimulus for me. There was a new one coming up that I was really interested in, and some of my group (Bob Horsley, Alistair Read, Annie Pegler and Terry Over) had already booked in. It was months away, but I planned to go with them to Herefordshire. Now that Julia and Andy were older, I felt Eric would cope. Also, these residential courses always took place during the relaxed holiday times – relieving Eric of juggling the practicalities of term times.
This course was the jackpot for me. While it offered a variety of areas for study, it was not at the top of my agenda to concentrate on my own performance and virtuosity at this time. My mind was elsewhere – I wanted to work with children. I liked introducing music to them. There was indeed a daily session on teaching recorder in schools, and it was here I found my true calling. I was delighted it was run by the musical director of the mid-Hertfordshire SRP, Herbert Hersom, and I already knew his pull was towards recorder learning at primary level.
Herbert had much fine music up his sleeve and introduced us to beautiful compositions for beginners that included only three to five notes. Here we had our very own English composer, Colin Hand, starting with his _Come and Play Books_ _1_ and _2_. To me they seem like _Lieder_ without words for young children. The titles of the pieces are evocative enough and tell you all you need – 'Waltz', 'Swinging', 'Sailing', 'Fanfare', 'Echo Song' – with delightful piano accompaniments.
On this course I picked up even more tips and insider knowledge over meals and during our free time, as we all talked shop non-stop. No one listened to the news, and the outside world didn't exist. I found amateur musicians were on the whole extremely amiable, modest and jovial people – quite a different species from academics. Coming home was always fantastic. The children were beside themselves with excitement and eager to talk without pausing, even though we were only apart six days. Alone together over supper that evening, Eric sensed this shift in me and suggested that before committing to deeper education in music college (which I was about to investigate), I should consider getting a teaching job, 'Why not see if you like it first?' He was convinced that I knew how to play the recorder well enough to teach in primary school, and he knew my flair with children. I took his comments to heart and I got in contact with the head teacher, Terry Seaton, of the nearby Carlton Junior School. He was ambitious for his school and eager to introduce instrumental music there.
I felt good vibes. Terry made it obvious that he wanted me to join his school. He was straightforward with me and I recall him saying most awkwardly, 'I am not concerned about you not being a good fit for this school, but that you may be shocked or upset by some of the things that come out of the mouths of the children here.' I reassured him that I was a woman of the world and could handle what was coming my way (maybe I should have said I was not a lah-di-dah lady and my husband was a communist.)
As this was my first teaching job, my position would be part-time in the school, with part-time training. I was to have an audition at the music centre in Pimlico and later an inspection whilst teaching the children at Carlton. So Eric was right to steer me this way. It worked out even better than I imagined because it offered plenty of training.
I had already heard along the grapevine that the new head of music for the ILEA, John Stevens (later Sir John Stevens) was a very enlightened man. When hiring music teachers, he was less interested in people with long strings of letters after their names but keen on those with an interest in child development (excellent news for me). It seemed clear he had a vision to lift music in schools out of the Stone Age. He wanted to expand the music curriculum for all children in all his schools and I was eager to learn this new _class music_ – a method that encouraged teaching in a group outside the classroom, in a space where children could move and form a large circle. It focused on learning through musical games, movement, imagination and variations of sound through listening, and also handling instruments. Stevens had certainly picked incredible teachers for our training, they were brilliant, all of them – Leonora Davies, Diana Thompson, Wendy Bird and Stephen Maw spring to mind. It was not just their extensive knowledge of material, but the way they handled the instruments with care and poise was contagious; a bongo would not be banged on but cradled like a newborn and then gently tapped with fingers. Oliver James was the recorder specialist for the ILEA. While I can't recall the exact pieces I played at my audition in Pimlico, I remember it going well. Oliver's inspection at Carlton School took nearly half a term to materialise and so I had time for preparation. Looking around the music shops, I found the recorder section invariably on the lowest possible shelf next to the floor. Tutor books for all other instruments seemed dignified, with a one-to-one tutorial appeal. Although the recorder is often taught in large groups, I could not see why books for beginners had to reflect a classroom cacophony with jokey faces and silly characters. So I made up my own sheets for teaching at Carlton. Later on, I wrote two tutor books for beginners, _Me and My Recorder_ , which Faber Music published in 1989. They were well received (and also translated into Greek) and I am happy to say they are all in print today and am still getting respectable royalties. My daydream now is for a Chinese edition.
In the class inspection, Oliver James had really come mainly to see my interaction with the children, and the ambience I created with and around them. The eight- to nine-year-olds were especially drawn to songs with stories in them, like spirituals once sung by slaves in America. The ten- to eleven-year-olds played a samba, which Oliver noticed I had renamed 'Please Miss, Can We Play the Samba?'. I was certainly learning the importance of repertoire.
I loved my time at Carlton, teaching three mornings a week. Slowly over the years, recorder playing blossomed. A natural progression of trebles and tenors were introduced and Terry the head teacher even bought a bass recorder. Performances, alas, were frequent, but still made me anxious. It took me a while to learn to stay cool. There were many concerts and we appeared in the local newspaper as the Carlton Academy. Eric often made an appearance and got used to sitting on the infant chairs. Some days I wondered if I didn't learn more from those children than they learned from me. Their spontaneity, honesty and inquisitiveness were a constant reminder of life's essentials.
After my ten years there, the junior school was forced to merge with the infant school, creating chaos and tension for children and teachers alike. Sensing this, Terry asked, 'What can I do to entice you to stay, Marlene?' To which, with astonishing confidence, I replied, 'I need an accompanist. I'd like you to hire my friend Don Randall.' I'd met Don in a recorder group. He had retired from a tough boys' secondary school where he had been doing all the music for forty years. A musician through and through, he was the perfect person for the job. He was hired – thank you, Terry. Don was so helpful and useful: it was bliss for me. The children liked him very much, and they respected this small, wrinkly man because they knew quality without realising it. Don was also my 'top and tail' teacher (helping a pupil who was either struggling to keep up or was way above average). So I carried on, and my job at Carlton School lasted another four years – fourteen years in all.
During this time, I was becoming more in demand, as was the status of the recorder. For instance, Angela Mendis, head of music at Fleet School, was helping to run the Saturday-morning 'Young Music Makers' and she asked me to take a group of children and adults. It was good fun because the children outshone the adults so easily and loved being taught alongside them. I had also been teaching one day a week for ten years at a private boys' school, Hereward House. This was a completely different experience. The principal, Leonie Sampson, was so supportive and generous; it definitely raised the level of my work.
I became more ambitious. The recorder players were learning an arrangement of Papageno's song from the opera _The Magic Flute_ and I discovered a boy with an overwhelmingly marvellous voice. I started teaching him Papageno's words and although he was a shy child, lacking in confidence, he surprised himself and agreed to sing it at the concert. I rearranged the recorder part as an accompaniment with piano. His parents were delighted and bought him a costume covered in feathers of such splendour that not even the Royal Opera House could beat it. On the night, Papageno acted and sang his heart out. It was a complete and total triumph – I bless him, Mrs Samson and Mozart.
When I finally decided to leave Carlton, I got my last job at Beckford School in West Hampstead. I was in my late fifties now and decided it was time for a change. I taught there once a week for six years. Musically it was a lively school, with many different instrumental teachers coming and going (strings especially, Suzuki violin and cello). I picked up tips from the Suzuki teacher Jane O'Connor, whose pupils presented themselves so finely. Jane Hills, who became the head, was a recorder player herself. She even played in assembly with the children. This was a huge boost for the recorder's standing and reputation with the children and the parents.
I remember a very talented pupil there, nine-year-old Felicity Squire, who wanted to be a music teacher (her mother was a teacher and also a pianist). I recommended to her parents that Felicity attend a Saturday-morning recorder class run by Angela Rodriguez in Muswell Hill. Angela recognised her talent too and from there she moved on to CYM (Centre for Young Musicians) in Pimlico, where she had the excellent teacher Sue Klein. Finally she made it to Trinity College of Music with Philip Thorby, professor and director of Early Music, where she took her degree. She became a recorder player and music teacher herself. She was also a flautist. Bravo, Felicity.
All this music and all this time and I have not yet mentioned my friend Diane Jamieson, who helped me in my music career more than I can say. She was a natural and also an expert, full of imagination as well as experience. She was a genius on the subject I found the hardest: class music – aka controlling thirty children at one time, with drums and tambourines and more on the loose. Although I had by now substantial class-music material with successful activities, I still sometimes had wobbles on a Sunday night. I would go around to hers, and she would give me a song that would wow the Monday class into obedience and awe.
After my six years were over at Beckford School, I had ideas flowing in and out of my head about making music in another way. I remember it was now or never to embark on a completely new venture of my own: the After-School Recorder Players. I ran it at our home in Nassington Road for both children and parents, and our downstairs furniture was rearranged every Monday. Eric was remarkably flexible – in fact, he now knew how to put up a music stand and fold it back down. There were two groups, beginners, and secondly an ensemble including bass, trebles and tenors. I ran this programme for ten years until our travels abroad became too frequent and our grandchildren would soon be on the way. I roped Don in as pianist and also 'top and tail' teacher, enticing him with dinner and a lift home. I hired Felicity, who came straight to us from her lessons at Trinity College, imparting the latest wisdom and techniques from Philip Thorby, the maestro. By this time, Felicity was already a first-class musician – how luxurious is that for the north London parents to see their treasures making such strides under this direction? Felicity's rapport with the children was a delight to observe. Don was so happy to be involved in this project; he said with frustration, 'My only problem is that none of my friends believe me that every Monday evening I have dinner with Eric Hobsbawm.' Those two got on like a house on fire.
For our concerts, I normally hired a church hall or another suitable venue, but a few times the stage was our hall, with the audience sitting on the staircase. The project as a whole was an original and successful enterprise that ran entirely on word of mouth. I can't deny the pleasure it brought me.
Chapter 21
# _Manhattan_
After Eric's retirement from Birkbeck, New York suddenly became a big part of our lives. The New School for Social Research, and the Graduate Faculty of Political and Social Science there, invited Eric to come and teach for a semester a year, starting in 1984, which carried on for twelve years. Unlike an ordinary university, the New School was on a mission to recruit well-known scholars. In fact, the Graduate Faculty started up in 1933 as the University in Exile in order to rescue academic refugees from Hitler's Germany. How very prescient and smart, as well as kind. Moving ahead to our time, they knew British professors had to retire when they reached sixty-five, and that is when Eric was approached. It was on the initiative of the dean, Ira Katznelson, which quickly won assent from his colleagues. Others were invited from France, Germany and the UK, including Perry Anderson. Some became full-time faculty members; others, like Eric, were invited for one semester each year.
Eric was very excited about the combination of heterodoxy, progressiveness and internationalism at the Graduate Faculty, along with the able students it attracted from all over, including the USA, Latin America, Europe, Russia and China. Eric once counted twenty nationalities in his class. He immediately involved himself in the project of the New School and the role he played in it. I knew he would go above and beyond in his contributions to its agenda, and during the times I was not there, I felt sure his academic and social life coalesced even more strongly.
I really wanted to go with Eric for his first term and I organised wonderful substitutes for my recorder classes. The young and very talented teacher Angela Rodriguez took over at Carlton and my friend Emma Murphy, who was the best recorder performer in Britain at that time, took over at Hereward House School.
Now in their twenties, Andy and Julia were capable of taking on responsibilities and looking after the house while we were away. At the very beginning (for a month or so), Walter's eldest daughter Habie also lived in Nassington Road, until she moved to live with her grandma. Both children had jobs now. Julia was working in television, having zigzagged her way through the ranks in publishing, despite not having a degree. She then worked at Thames Television, which led to a research job at the BBC on the _Wogan_ programme. She then moved on to a small, independent production company making programmes for the fledgling TV company BSkyB.
After spending sixteen months travelling in Australia and California, Andy returned to London. He was working in a combination of advertising sales and music journalism for London's _Alternative Magazine_ – writing a provocative weekly column under the pseudonym Hamish Head – alongside forming a rock band called Tin Gods. Alas, Andy's brief encounter with the recorder was inevitably replaced by the guitar. Both our children visited New York while we were there; in fact, on one of Andy's trips he was on board one of the first ever Virgin Atlantic flights in 1984, where he remembers the unusual experience of 'buskers with guitars playing music up and down the aisle'.
Eric and I arrived in Manhattan for the Fall term in September, together with other newcomers from US universities, Chicago in particular. These colleagues were Charles and Louise Tilly, Ary and Vera Zolberg, and even our dean, Ira Katznelson and his wife Debbie, who had come a little earlier. They all quickly became our friends and we explored Manhattan together, each contributing their new bit of knowledge of a good restaurant, or new brunch place, or where to buy bread from a different country every day.
Eric and I decided we would immediately have a most wonderful time, exploring Lower Manhattan especially, and every visit Eric insisted on his ritual of taking the Staten Island Ferry. We went a few times to the opera, especially for our anniversary in October. I remember seeing Strauss's _Salome_ for the first time (you can't say you liked watching this revolting story, but you can't say you didn't, either), as well as a traditional _Don Giovanni_ , which we never stopped enjoying, however many times we saw it. During our second visit, Bob Silvers (editor of the _New York Review of Books_ ) kindly gave us his stalls seats for the performance of _Rosenkavalier,_ as he and his partner were both indisposed. In turn we gave away our balcony tickets to two students. It was conducted by Carlos Kleiber, with singers Felicity Lott and Anne Sofie von Otter, and it was an electrifying performance. We came out high as kites – neither of us could sleep until dawn. The orchestral players were drawn from all over the world and legend has it that Carlos Kleiber was the strictest of conductors as regards to the number of obligatory rehearsals.) But it was not all classical music, we went to many jazz clubs, and Eric's love of tap dancing brought us to see Jimmy Slyde – the best tap dancer known in New York at that time.
Our first apartment was around the corner from the New School on 9th Street, on the twelfth floor, with a view of the intact Twin Towers. There were an unusual amount of pink mirrors around the place, windows you could see out of but not into – it was like a movie. This extreme backdrop was perfect for wine and cheese parties. We knew so many people because of Eric's global contacts (many of whom found themselves in New York at the same time – you never know who you're going to run into on 5th Avenue). Luckily, my autumn visits always coincided with the charming Thanksgiving parties that dear Eric Foner, the historian, and his wife Lynn gave in their apartment.
I was never short of friends myself. One of these was Leina Schiffrin, my long-standing very close friend from England who moved to New York. Leina is a one-off – gregarious and always speaks her mind if she feels it is the truth. Perhaps her education at the progressive coeducational boarding school Dartington Hall played a role in shaping her free spirit. Her husband André was one of Eric's regular publishers (and an exceptionally good one). They were known for their Sunday brunches out on their penthouse terrace. Leina was up to professional standards in the kitchen and also gave the most delicious and convivial dinner parties, to which, invariably, interesting people came. Not just publishers and writers, but also artists, actors and UN people – André seemed to know everyone in town.
Another dear friend was Betsy Dworkin, whose husband Ronnie spent much of his spare time writing, not unlike my husband. We'd do special things together – she was originally a New Yorker and knew all the best spots – like going to exotic tearooms and walking along the river from Chambers Street to Battery Park, punctuated by small parks of different character. We also liked going shopping (clothes, of course), which in New York is surprisingly relaxing. It's much more lively, with the ease of refunds and exchanges. They know how to do business.
As the spouse of a professor at the New School I had the privilege of being able to enrol in any class that took my fancy, almost for free. Taking advantage of this, I started off with a yoga class, learning new methods with a stylish young teacher, and also dipped my toe into t'ai chi. Of the latter, I still remember being alarmed by the teacher, with his American accent, saying, 'When you do this posture, your ovaries are saying thank you.' It was OK – I was fifty-two years old.
I also took an art-history class, which wasn't free but well worth it. It was called 'Exploring Museums', and we toured the most famous ones, starting with the Metropolitan, Cooper Hewitt, the New-York Historical Society and ending with the Guggenheim. I don't often enjoy reading about the visual arts as written by connoisseurs (who mostly seem to explain how knowledgeable they are in order to impress other art historians), but our teacher Leslie Yudell was crystal clear and very engaging – she helped teach us how to focus and _look_.
The unit where my classes were held was in the original New School for Social Research, which was founded in 1919 for adults to study without any formal entrance requirements. Again I thought, how enlightened that sounds, and as so often in New York, years ahead of their time. I was thrilled to learn that Martha Graham taught dance there in the 1920s and Aaron Copland taught music, among other luminaries.
All I can say is that the time flew by, and Halloween came upon us fast. One cannot get away from it and nor did we want to. There was the big parade, with half of New York migrating from all over the city to Lower Manhattan. The handmade costumes that developed from the numerous art and design schools were fantastically impressive – I remember a skeleton dancing, made bone by bone with intricate detail. Eric and I only bothered with masks. We were very fond of that night, and although there was so much crime and violence in Manhattan, ironically it was always suspended at Halloween.
Slowly I was beginning to see the ups and downs of the city. Both tiring and exciting, the pace of the city never stopped and it felt like always being at a party. Nobody was shy, and strangers would come up and talk to you: 'Hey! I like your sweater, where'd you get it?' Quiet time for myself in public was always interrupted. Even though the children were adults, I still needed time to myself to think and worry over them.
I remember being appalled at the homeless situation, where many could be found living in the subway. There was also a huge area they had chosen to occupy in the deeper east area of Downtown. I have never seen such a thing before, even on television. It all seemed so strange; they didn't appear to wear their sorrow on their sleeves, and the chaos of living on the streets looked structured and supervised, which in fact it was – it was part of the city's programme designed for the homeless. Homeless people often wanted to talk; I felt their loneliness. It was as if they had missed a train, the way you have to wait with despair for another, except this limbo was for ever.
Through Leina's network of friends, I met Inea Engler, who introduced me to the musical household of Lucille Wolff in the West Village – a huge pleasure for me. Luckily, musicians always need rehearsals and audiences, and I was made very welcome at 11 West Street, with its bohemian atmosphere and beautiful young players. I was in my element. In fact, I shall never be able to hear the Andante from Bach's Double Violin Concerto without emotion and the memory of Lucille's daughters playing it. People would often stop on the pavement below the windows to hear this _Hausmusik_. I was so happy to be able to walk there and back in such serenity for my own _soul_ , away from politics and institutions.
Eric did get a lot of satisfaction from the students he taught, and learned from them in return, Latin American students particularly. They enjoyed each other's company as well. He was also pleased to be invited to the Institute for the Humanities at NYU every Friday to listen to special speakers, after which they all went for lunch. It was nearby at the Deutsches Haus in Washington Square. Altogether, it was a tremendous semester.
I would continue to come with Eric each year, but only for two-week periods during school holidays. Over the years, we began to feel at home with our lives, schedules and routines. We'd become New Yorkers. While Julia enjoyed many visits to the city and had her own friendship circle and business connections, Andy actually moved to New York for business in January 1995, working in a pioneering internet start-up he co-founded called Internet Publishing (which turned into Online Magic and later Agency.com) and began dating Kate Ellis on Valentine's Day that same year.
Chapter 22
# _Our Italy_
Many of the highlights in my life happened in Italy: in Rome, in Capri, in Tuscany, in Venice, around the Neapolitan coast and in Milan. You will already know I lived there as a young woman, but Eric also had many past visits and experiences separate from mine. Two different Italian histories were already in place, but now it was going to be our Italy.
In November 1979, Eric and his historian friend Rosario Villari were both invited to the same conference in Venice. Eric was keen I should come, so I phoned Rosario's wife Anna Rosa, whom I had met recently to ask if she was also coming; she agreed to travel provided that I did also. We were all put in the same hotel (a nice one). November weather, often foggy, turned itself into a week of warm sunny Riviera temperatures. Unfortunately the husbands were disappointed in their conference; aside from having to take a fifteen-minute boat ride every day to the island of San Giorgio where it was held, they had grumbles about the organisation and the papers not being up to scratch.
We ladies, on the other hand, had one of our best holidays. We seemed to have so much in common, and I don't only mean that both our historians were paid-up members of the awkward squad. We were compatible –like sisters, and treated each other so. We picnicked on steps, bridges, in quiet, secluded areas away from the tourists. We once found ourselves near the famous orphanage where Vivaldi composed, gave violin lessons and also taught the girls to sing. The most talented ones would become members of the renowned Ospedale della Pietà orchestra and choir, which often toured around Italy and abroad. Not only Vivaldi, but Sammartini was also from Venice, and we probably had a picnic around his plaque too. Famous for being an oboist, he also played the flute and recorder (which was the norm at that time). He was a wonderful composer for the recorder and I have played and taught many of his lovely sonatas and trio sonatas.
Rosario, who was taking many years to write his main book, _Un Sogna di Liberta,_ which covers the Neapolitan Revolution in 1647, really needed to get on with it and research in the British Library. We invited him to come to us whenever he wanted, and he did. That is when our deep friendship began. He was a marvellous guest and when joined by Anna Rosa it was even better. They were practical people, unlike the Hobsbawms. They knew how to fix things with tools – though Rosario always complained about Eric's being blunt and rusty. We all laughed that it sometimes felt as though we had hired them as a working couple. They even entertained us, as Rosario played French and Neapolitan songs (by heart) on the piano and sometimes Anna Rosa sang them. In the kitchen she was a wizard. It seems that two husbands and two wives in a household together works like a dream (why is there not more of this, I wonder?).
When Eric and I travelled abroad it was very practical for them to be in the house alone, not least for feeding and caressing Ticlia, and again Rosario went off every morning to the library. It took a few more years, but he did eventually finish his book in 2001 and it was very well received and reviewed. But the Villaris were addicted to London and this arrangement continued. Their home was in Rome, but each year they spent their holidays in their Tuscan farmhouse Santa Fortunata, beautifully situated and furnished (even with a pool, so they could rent the place out to well-off Americans). It was near the small town of Cetona. Later, our Andy and Kate stayed there as part of their Italian honeymoon, and Julia and Alaric spent time there for peace and quiet, during Julia's pregnancy with Roman.
We had many holidays there, just the four of us. We got on so well – the historians always had much to talk about, but it wasn't all chat about the dreadful politics (we were there when Berlusconi was re-elected). The kitchen, with the patio door to the garden and pool, was the centre of the house. So much time was spent there, and we enjoyed watching a one-act _opera buffa_ unfold. The two protagonists were Pia and Rolando. Pia (soprano) was the maid (who secretly was still in love with her ex-husband Nilo, who ran the main restaurant and gossip hub in the town piazza). She came to work every morning and also cooked our lunch. The gardener Rolando (tenor) was in charge of the grounds that circled the entire property. He looked after the olive trees surrounding the swimming pool and the expansive olive-tree orchard behind; the garden also had pear, plum and two big fig trees, and he was in charge of growing all the lovely produce – courgettes, aubergines, tomatoes and peppers – and also watering the plants, starting with blue agapanthus at the front. The Italian sun makes these flowers much taller and more imposing than they are in England. Hostas are giants there.
Rolando was so attentive with the Villaris, but it was especially sweet and humorous to watch him be attentive to the provocatively dressed Pia, who he clearly had eyes for. Pia must have known, and she teased him by running outside when it suited her, complaining loudly, ' _Dove sono i pomodori_?' (Where are the tomatoes?). He would scurry off to bring them to the kitchen where he would then dawdle, always inventing something to fix. That was the comedy performed at Santa Fortunata every summer morning, except Sundays. Cetona was adorable; I think only the Italians can make piazzas like this. There were no cars, children played safely and there was a communal effort between mothers to watch over the others as well as their own if they stepped into a shop. Of course, we knew everything that was happening behind closed doors in Cetona (thanks to our Pia).
Anna Rosa and I continue to travel back and forth to visit each other, and our beautiful friendship continues today, but without Eric or Rosario, except in most conversation. Because of the the Villaris, we had masses of friends in Italy: theirs and ours – journalist Antonio Polito, historians Corrado Vivanti and his wife Anna, the Perrinis, the Procaccis, Beppe Vacca, the director of the Gramsci Institute, Toni and Giovanella Armellini, and too many more to name. We were also friends with Enzo Crea (a publisher of art books and poetry) as well as a professional photographer. He took the picture of me and Eric at La Foce on the back cover of this book. Benedetta Origo, was also a close friend who often invited us to her two beautiful homes, La Foce or Castel Guliano, in the midst of a glorious landscape of rolling hills and long distance panoramas.
From the beginning of the 1970s, Eric was very popular in Italy, but his peak was in the 1990s onwards. The Italians couldn't get enough of his books, which were all translated into Italian. Eric was first published by Laterza ( _The Age of Capital_ ) and later Giulio Einaudi, a son of Luigi Einaudi who used to be President of the Republic of Italy. Because of this, Giulio was a very grand person and ran a very grand publishing house. Eric was much involved in Giulio's huge project _La Storia del Marxismo,_ working as a principal editor (alongside Vivanti, Haupt, Ragionieri, Marek and Strada, among others) to find British scholars to be approached to write for it. His book on jazz (as well as _Captain Swing_ ) was published by Editori Riuniti, the Communist Party's publishing house.
Eric's next main publisher was Rizzoli (run by Paolo Zanino) and the publicist in charge Anna Drugman was the most competent and attentive lady, who made sure we were treated as VIPs every step of the way. In May 1995, they brought out _The Age of Extremes_ , which was a huge success and presented at the big book fair, Il Salone del Libro in Turin, where he was also given an honorary degree at the University of Turin. I didn't go with him to this but managed to go to the next one, a literary festival in Mantova. I remember the charm of Mantova, and the adorable Giorgio Armani shoes I bought there.
Eric also got prizes, including the Premio Bari. We were invited to the premiere of a new production of the opera _Fidelio_ at La Scala in Milan. The audience was full of stars, luminaries and, unlike in Britain, politicians were there as well – in the royal box was the former President of Italy, who was booed, and his party immediately left the building. In front of us sat Jeremy Irons and his wife Sinéad Cusack. Riccardo Muti was the conductor, and he was marvellous, as was the entire evening. During the interval, I overheard gossip in the ladies' room about Riccardo Muti's current lovers, which made meeting him at the big dinner afterwards – let's say... interesting. Eric was often approached at book fairs and other events by Italian politicians and friends, who would ask if he would write for this or that journal. He was enamoured with the way communism was actually developing in Italy, and how he was able to contribute to the political climate intellectually. Eurocommunism, as it was called, was appealing for Eric as it was softer and wanted more independence from the Soviet Union. This shift started in Prague and also became popular in Spain. Without delving into dialectical Marxism, Eurocommunism arrived rather lukewarm to the Communist Party in Britain, although it's true they did actually publicly condemn the Soviet intervention in Czechoslovakia. Of course, the Communist Party was not important to most people in Britain, and I was always surprised at the lack of awareness at just how active the Communist Party actually was in Italy and other parts of the West. Even in India it was not uncommon to find people wearing small hammer-and-sickle brooches.
When we were back in London, Martin Jacques asked Eric to write for his magazine _Marxism Today_. The successful collaboration of many years led to a long-lasting close family friendship as well. Martin's son Ravi and our grandson Roman continue their good friendship.
Chapter 23
# _Mature Lives_
In the 1990s, Eric and I were not yet old, but enjoying the advantages of maturity, or so _we_ thought. We were at the peak of our abilities and consequently very busy. Our Welsh holidays continued, as we saw our friends and enjoyed the usual outdoor life. Walking was still very good, but getting over stiles was becoming a little challenging for Eric now. A more pressing issue was the rise of Welsh nationalism. It was starting to grow and we didn't like it. Some cottages were even burned down. The five-hour drive from London seemed less appealing. Betsy Dworkin recommended that we make contact with her friends Brenda and John Maddox, who had their second home in the Wye Valley near Hay-on-Wye, to get a feel for the area. We already knew John, who used to be in one of Clough's cottages, but since his second marriage to the Massachusetts-born writer, they had moved to a picturesque large farmhouse near Crickadarn. John was the editor of _Nature_ magazine and Brenda was a biographer, journalist, critic, reviewer, novelist: a real lady of letters. I believe she went to school with Betsy Dworkin, where the connection began. It would continue with the Maddoxes and the Hobsbawms and their children – and eventually grandchildren too. The Hay Festival made a big difference to the area; they were used to foreigners from all over the world and welcomed them with open arms. Eric's agent, Bruce Hunter, had just negotiated a very good deal on the advance for his proposed new book _The Age of Extremes_. We were now in a position to buy a place. It was all very new and exciting for us. About a couple of miles from the Maddox farmhouse in a small hamlet called Gwenddwr (pronounced 'Gwenda'), albeit with its own Anglican church and also an independent Welsh chapel, we found a totally rebuilt and refurbished cottage that looked as though it had been patiently waiting for us. It was named Hollybush and was also built from a ruin, but otherwise it was the diametric opposite of Parc. Listen to this: it had central heating, was newly rewired, the windows would close and it had views nearly as far as England. The house was as cosy as an apartment in Manhattan, though surrounded by sheep. It also had a big garden wrapping around the house and I was able to plant climbing roses. Gaby Annan introduced me to a French rose called 'Papa Meilland'; it was a velvety, deep red, regal rose that didn't fade or wither until it turned black and was ready to die. It was too grand for the cottage but I loved it and it was much commented on in the village. Dorothy Wedderburn, who often came with us, planted a plum tree and there was already a holly bush with the largest red berries, in case the house was used at Christmas. Another difference now was the addition of Louise Nicholson, who came as a cleaner but was essential in so many other ways. For second-homers like us, who come and go, and have people staying when we're not there, you have to have a friend like her. She would prepare for our arrival and take care of things after we left, which made it a real holiday – for all of us. She helped at our parties, she knew everybody and she mingled with charm and ease. Our move to Gwenddwr meant a different style of country living – no more going about in motheaten throwaways from London. The town of Hay became full of stylish garments. The general atmosphere is friendly and content in Wales, especially in Brecon, where we did our food shopping. It seemed as though all the women who worked in the supermarket were happy to be there. The Welsh like talking a lot, and here they could chatter all day long, because they knew the customers and had grown up with many of them. Also, the children behave well in Wales. They obey their parents, who are not on their mobiles all the time. The only thing that gets under my skin in Wales is their bureaucracy, which is dire. They love rules and regulations and go overboard in making them. There's nothing like recycling for inspiration – 'the super-handy scrunch test': if you scrunch it and it doesn't spring back, then it can be recycled. I recall a detailed instruction about comic books being OK, but then pondered if the _Radio Times_ would also be. Walking in the Wye Valley is much easier here than in North Wales. The landscape is softer, so one can walk down the roads and lanes as well as crossing muddy fields. One would happily stroll around in a summer dress and pop in casually to visit friends. Chickens and hens run freely in the village. The annual agricultural show in each parish at the end of July is the big event of the year, and Gwenddwr was no exception. Nearly everyone participates in judging the best cakes, ponies, vegetable arrangements and even babies. The year Eric was president, our granddaughter Eve won the 'best baby under six months' prize. We hoped people would regard it as the coincidence it was, and not that the London Mafia had moved in. As the English second-homers were mostly intellectuals, their house guests usually were too. It was a lovely mix of people who came to the show and you would be as likely to see Melvyn Bragg as the local farmer. Of course the _main_ attraction in these parts is the Hay Festival itself in May. The local second-homer grandees were useful to the director, Peter Florence. John Maddox was made president of the Festival, then Lord Bingham, then Eric, who was succeeded by Stephen Fry (in my time). Peter Florence and Eric admired each other and spent time talking politics and the world. I'm sure Latin America featured. Maybe out of that came Peter's idea to expand the festival, starting with Latin America and eventually many other parts of the world. Eric was hugely popular and always filled the largest tent to capacity. After his death, Peter started a memorial lecture series after him, featuring big names like Joan Bakewell, Amartya Sen and the US politician Bernie Sanders. All this VIP speaking meant I had VIP entry (all the widows of ex-presidents are treated very well) and now I would need it. There is a lot of queuing up at Hay. And not everyone can rest up in the green room like us. I enjoyed being surrounded by friendly people living in Gwenddwr, especially, when I used to come alone, dealing with house affairs. Opposite us was the builder and electrician Idris and his brother Trevor, a shepherd. Next door, there was Marjorie and her dogs, who I would join for walks. She recently started a little open-air shop selling home-made cakes, flans, scones and eggs, which is a great addition to the village. Then there was Verona Atkins, a widow who was born in Gwenddwr, a charming lady with many interesting tales to tell about Gwenddwr's past. Next came Frank and Pam Banks, who were both teachers locally. Frank later became a professor of education at the Open University until his retirement. Now he is a churchwarden and chief bell ringer, and they live there permanently. Frank is very enterprising in starting up theatrical ventures and organising productions. He makes sure there is plenty happening in this quiet corner of the world.
Second-homers George and Phil Littlejohn, who live in Islington, became our close friends. A very joyful and social couple with lots of vivacity who love walking, theatre and anything interesting going on. They are such wonderful hosts, and we were usually there for a New Year's Eve party from which we could stumble home easily. Further down the road in Crickadarn is Jane Birt, separated now from John Birt of the BBC. Jane and I are so close, because she lives in London too but also visits Crickadarn a lot (with her large family). She lives next to Tony and Sarah Thomas, who are also close and hospitable friends. We've had so many memorable dinners there. Sarah is a viola player and Tony is a retired writer and journalist. Eric was a great birdwatcher and he and Dorothy would sometimes go off with their binoculars on excursions to do serious sitting by a lake for long stretches of time. I would drive them there and back. I didn't have the time to sit, and there were already so many birds to see and listen to on our ordinary walks. Every day when I could hear buzzards mewing I would look up and see them soaring high with their flat wings. The area was very well known for kites, and there were many organised opportunities to spot them, but we just kept a really good lookout and it felt like a goal to see one. But it wasn't all staring up at the sky. Eric also wrote a lot in Gwenddwr. He loved working in his study, which was so light, with two windows, each with a different aspect. I was also scribbling. I had been interviewing my housebound mother for a few months and was writing up the memories to make a book for the family. It came out in 1998, with wonderful photos, and I called it _Conversations with Lilly._ It was not a difficult book to write, but it was so convenient having Eric nearby to answer any questions – as Roy Foster said once, 'Eric was a world wide web in and of himself.' This quality made Eric very popular with young female students and many tried to flirt with him. But mostly I called it 'Ph.D. love'. You could almost see their thoughts forming: 'If I was with this guy, imagine how my Ph.D. would turn out.' At one time we found it so agreeable there that we toyed with using Nassington Road as the secondary home. But there was too much happening in London for us to miss. Our two children were now in their thirties. It was around this time they found out they had a half-sibling; without warning, Marion had told Joss's brother Joel, and so we had no option but to tell them about Joss, as we did not want them finding out accidentally. Although it was a big shock for them, they did bond as siblings very soon after. I also got on well with Joss, as we could talk shop – he was a drama teacher. As both our children had partners now, it felt like the right time to tell them. Julia had stuck with her student flame Alaric, and Andy with his New York Valentine. Maybe they didn't find their way to university, but certainly found (and kept) something more important: true love. Andy and Kate's wedding took place at the National Liberal Club by the Thames. It all went off as beautifully and as stylishly as a wedding is dreamt to be. Strawberries and champagne were served on the terrace afterwards, followed by a hired double-decker red bus (destination: 'All Stops To Bliss') to take the guests to the apartment of one of Andy's two best men, Eamon, for the reception and dinner. The building was grandiose and ornate – once the headquarters of the state-owned British Gas. The evening finished with fireworks. The following day, I gave a lunch party in the garden at Nassington Road, luckily on a beautiful day. Some of Kate's relatives had come from Massachusetts and other faraway places and I wanted to welcome them. Four years later, Julia and Alaric decided to tie the knot at Marylebone Register Office. Two of their three children were also there; six-year-old Roman, wearing his first suit, and three-year-old Anoushka, who insisted on climbing onto Julia's lap during the signing. Wolfie was still a twinkle in his father's eye. Alaric's two children, Rachael, aged fourteen, and Max, aged twelve, from his previous partnership with Lesley Campbell were also there, and other members of both families. All helped to fill the large Cinnamon Club, which was chosen for its ability to accommodate their large guest list, which also included many babies and small children. It was such an amazing atmosphere, with the most witty and genuine speeches; much was expected of Eric, as the father of the bride, and he delivered. The Villaris were staying with us at the time, so they were there as well. The wedding programme read: '21 years after they first got together, 7 years after they got back together again, they are getting married.' Their romance was odd at the beginning, not least because of the ten-year age difference between them, but romance is romance, and after a decade of separation they were reunited. Both our newlyweds lived within a mile of each other and we were all close in north London. The family had started growing. I had already discovered that the first-born grandchild feels like being seventeen and falling in love again. If I had worn one of those medical twenty-four-hour heart monitors when Roman was being brought to me, the doctor might well have asked, 'Whatever happened to you at two o'clock? Your heart rate jumped off the charts.' Roman, being our first grandchild, rejuvenated us both. Two years later we found ourselves driving through ice and snow from Hollybush on New Year's Eve as Anoushka had decided to come into the world nine days early. We waited with anticipation and passed the time in a Greek restaurant with our friend and neighbour Liz Eccleshare. The baby was so tiny I thought the name Anoushka was too big for her, but now that she is eighteen it suits her perfectly.
One year after Kate and Andy's wedding, Kate's parents Jane and Bill Ellis flew over to London from New York for another confinement. This time it was very slow. We were all getting anxious about the very long labour, but when we saw him the next day, Milo was as cool as a cucumber and he has stayed that way throughout his childhood. Two years later, there was another arrival expected for Andy and Kate. To my delight I was asked my opinion on names. I was very thrilled and even more so when my choice was the winner. Eve, sometimes called Evie, looked more like an Ellis than a Hobsbawm – hurray! For Julia's third birth, I looked after Roman and Anoushka for the day. Because Roman wanted the new sibling to be a boy, Anoushka did too. They were adamant that it could not be a girl. Well, when I got to the hospital, there was a big, beautiful baby boy. Wolfie (Wolfgang) was the last grandchild to arrive. It was clearly time to roll up our sleeves again. We spent many Sundays together and for Eric the children were unbelievable. He never dwelt on his many years alone, but I know he neither forgot them. Sometimes he still found it difficult to believe that all these beautiful children running around the garden were his family. As a grandfather, Eric was quite a natural with little humans. And jumping ahead, even more impressive with teenagers.
Chapter 24
# _Age of Glory Travels_
When we spent time abroad, we didn't feet like tourists because we were invited guests and Eric was so well connected with the universities, the academic world, journalists and publishers. Our travels began in modest hotels or student lodgings, which we thought would always be the way. Eric loved travelling and academic conferences were a good way of seeing the world. It seemed he knew remarkable people in every city and loved getting his information about the world from his foreign friends. Wherever we went there would invariably appear an old friend, often a historian, who would show us the real local life and tell us truthfully what was going on in their country. Travelling was such a big part of our lives and it's hard to put into words how much fun we had together because Eric was such an enthusiast. Whether it was packed cities or secluded mountains, little rivers or dense forests, the whole of geography was his passion. He never lost interest. With far too many places to name, I've chosen a selection for variety, but also because of the people and our connections that made those places so necessary.
Our trips around Europe were frequent and we had become accustomed to the Continent. Paris had both the familiarity of home and also the buzz of being new and thrilling. Berlin always resonated with me as we spent so much time with German emigré friends in England and of course for Eric, it was part of him. Orphaned at fourteen, he eventually moved to England to live with relatives. A bookish teenager, he joined the student Communist Party while Hitler was coming to power, distributing Marxist leaflets around his area. Being caught doing this would be instant torture and death, but for him there was no choice: you were either red or brown. Prague felt familiar, perhaps because of my background. Could it be the trams, the architecture, the way the streets were cobbled? An added bonus for me was the amount of marvellous recorder music available there. It's a very popular instrument in schools – and beyond – in Germany, Hungary, the Czech Republic, Austria and other countries in Eastern Europe. It was a super place for me to browse in music shops selling sheet music at risible prices. Not to mention how beautiful the cities are with their endless churches, which we loved strolling through together. Because of the short distance, it wasn't uncommon for a friend like Georg Eisler to come and join us from Vienna for a couple of nights.
Gradually our travels became different. People wanted a lecture from Eric, rather than his participation alongside others in a conference. If people wanted him, they knew I would have to be invited too. I measured this upward trajectory by the distance of our hotel to the city's opera house. It was getting nearer and nearer. This was particularly true in Vienna where I could look out of our window in Hotel Sacher Wien and actually see some opera dancers practising next door. Eric had been invited to receive the keys to the city of Vienna, a gesture we thought might (or might not) be connected to some public show for the appreciation of Jews. This nudged the University of Vienna to give him an honorary degree. I was born in Vienna and had two close cousins, Betsy and Patricia Higgins, who had gone back to live there from America and married Austrians after the war. For these ceremonies, I was always asked if I wanted to invite any guests, and indeed I did – I invited Betsy and Patricia for everything and they loved all of it. My cousins are such darlings and made time to take me to all the places of my childhood that I wanted to revisit. I also had plenty of time to myself to daydream in the city, thinking of Orson Welles and Alida Valli in _The Third Man,_ and enjoyed it so much.
While I am talking about Austria, we were invited to the Salzburg Festival several times for Eric to give a talk on culture. As the years passed and Austria was turning right again, it sometimes made us feel really worried and guilty to be there, but we couldn't resist those operas, orchestras, singers, acoustics – there was no such thing as a bad seat. Then during the interval, you see people walking around in dirndls and national costumes and you begin to wonder who is a Nazi and who is not. It was tricky; we did not like Austria as such in these times, but I must say we had very good friends living there.
Another golden oldie was long-time friend Ernst Wangerman, a historian with a brilliant mind, with whom one could have such good conversations about the twentieth century, and his Spanish wife Maria Josefa. We had many wonderful outings with them, spending hours at the Austrian lakes. We would then go back to their house for supper, hoping Maria's marvellous gazpachos were on the menu.
Japan, where we travelled in 1990, was a category of its own. It was totally different in culture from anywhere I had encountered before. I would stay away from Japanese university deans if I were you – their stiffness was so marked. But when you were friends, and they had taken you to a nice familiar place, they behaved quite differently. They had to treat me as though I was a man because it was not customary for women to go and socialise with men in this way, and they didn't know how to treat me otherwise. With a little sake, they relaxed even more and they would tell you ridiculous stories and laugh freely. I often puzzled at whether they were embarrassed or just very jolly.
Eric had a great friend, a Japanese historian, Hiroshi Mizuta, who often came to London, sometimes with his wife Tamae. I'm sure he was responsible for our trip to Japan. Unfortunately his English was not getting better, but rather the contrary. Like all Japanese, he laughed a lot. He has just celebrated his ninety-ninth birthday this year and still corresponds with me from the university care home for elderly academics, where he lives. He has just sent me the Japanese translation of _The Age of Extremes_ , which he oversaw.
I had two completely different trips to India. After Eric retired from Birkbeck, he treated us to a tour of India. Eric had been many times, but for me it was a first. Eric was enthralled by Indian art, which he knew quite a bit about, and wanted me to experience the country with him – no conferences, no lectures, no speeches, no responsibilities, just us with a nice group of strangers going sight-seeing. The tour was my introduction to the country and was beautifully arranged. We were treated so well everywhere. The Indians must be the friendliest people in the world. The only complaint would be the pace – the tour chose too many places to visit in the very short space of two weeks.
My second trip was different because Eric was invited to give talks and we were VIPs, not tourists. India is a tremendously hospitable country and they liked nothing better than to celebrate a prestigious scholar, and the nice thing was that they were scholars themselves. Eric had friends and former students there, and had had an attachment to the country since his Cambridge days. Manmohan Singh himself, prime minister from 2004 to 2014, was a student that Eric had supervised. And finally we could visit our super-scholar dear Romila Thapar in her own home and surroundings. That was a treat; we usually only met at conferences. We also saw our friend Premja, who had spent a month deciding which restaurants would be good enough to bring us to when we came. Luckily he comes often to London. I admired the middle-class academic Indian women, so efficient at organising their lives: they were scholars themselves, perfect hosts, and beautiful-looking with their colourful saris and elaborate jewellery. They did it all. I know they had servants, but all the same, they were remarkable with their vibrant energy.
I'm trying to remember my first trip to Brazil in the 1980s. I was in my mid-fifties and we travelled to Rio de Janeiro. Eric had been invited to speak at a conference after _The Age of Empire_ was recently published. Eric's first publisher in Brazil, Fernando Gasparian, who ran Paz e Terra, insisted we stay in his house surrounded by the most beautiful botanicals. I remember it as a light, palatial home with a dining area and a long table for over twenty-five people. Everything was so spacious, and the family's hospitality boundless. They invited many people to see us – the lyrics of 'The Coffee Song' should have been 'There's an awful lot of kissing in Brazil'. We met Fernando's wife Dalva, who was full of warmth and joy, and their children Helena (studying to be a diplomat), Marcus (who now runs Paz e Terra) Laura and Eduardo. Helena was put in charge of being our chauffeur and guide, and what lovely company she was. It was perfectly clear how highly Fernando thought of Eric; he was always planning new projects and places to see, informing us that we would go by private plane! He was so excited to have us there and seemed to want the whole of Brazil to meet us. I knew Helena and Laura would be coming to London – how would I repay this kind of hospitality? But they were friends, and it was easy.
Another journey to Brazil was ten years later, for the launch of _The Age of Extremes_ published by Luiz Schwarcz of Companhia das Letras in São Paulo. Luiz organised the entire trip. He turned out to be an unusually generous, sensitive person, which is quite rare on this planet. He and his wife Lili looked after us warmly in the bosom of their family in São Paulo. When we first arrived, we managed to get time alone, just the two of us, before all the activity began. I think it was near Paraty, where we stayed at a charming hotel and with a beautiful little pool just for a few days. Then we were ready for the launch, the press, the lunches, the speeches.
Luiz had to take us to Brasília, where it was arranged we would meet President Fernando Henrique Cardoso and a very splendid Brazilian presidential lunch took place. Eric knew him well, but as the women and the men were casually separated, I didn't get to know him. We had a brief amount of time to see the ultra-modern architecture of the city, designed by Oscar Niemeyer and Lúcio Costa, before the next part of the schedule brought us to Rio de Janeiro, where the book would be presented. So here we were again, walking along the great beach of Ipanema, home of the 'The Girl From Ipanema'. Eric did love Brazilian music – the fusion of jazz and bossa nova was very much his thing.
It was on this trip that I saw the favelas. I wondered how people survived in these shantytowns, with the crime especially. By this time, I was ready to go home and for the launch to wrap up successfully without any incidents. My recollection of the book launch is that it filled the 1,000-capacity hall but many still were unable to get tickets. It was organised by Maria Eduarda, of the cultural arm of the newspaper _O Globo_. Sometime later she became the partner of our great friend Leslie Bethell.
Eric had become something of a celebrity in Brazil. Leslie later recalled flamboyant fans shouting, 'Eriky! Eriky!' down the streets at the 2003 International Literary Festival in Paraty, where he and Eric shared a platform. Some women even asked for kisses – ' _Dê-me um beijo_!' – and they wanted to be photographed with him.
This really was the age of glory for Eric. He was awarded so many honorary degrees as well as prizes. One of the most rewarding was the Order of Companions of Honour, which was given to him by the Queen in 1998. Eric was delighted and proud. It was very special for him. As before in his life, it did not involve a rejection of his politics. However, his old friend Dorothy Thompson refused to speak to him indefinitely for accepting this honour. Strong feelings on all sides.
Other awards included the Ernst Bloch Prize, Bochum Historians' Award and the Balzan Prize. The Balzan was one of the most prestigious and his wasn't necessarily given unanimously. It was contentious to give it to a communist. Unfortunately Eric had caught a virus and was unable to collect it from the House of Parliament in Bern in Switzerland. This caused consternation and the Swiss insisted a helicopter, no less, be sent to Hampstead Heath for him. In the end Julia collected the prize and gave his speech undaunted, as she is.
I used to worry about this new success and eminence. Once I remember having to deal with paparazzi. Might it go to his head? Would he still be the offbeat professor from Birkbeck, known as the 'historian from below', famous for championing the poor and dispossessed? But through all the honours and awards, he didn't seem to become any more vain than other successful academics. Although Eric didn't quite hold on to all his resolute beliefs, he still thought that the point of life was to try and make the world a better place.
Now frailer and in his nineties, Eric's last trip would be to a part of the world we'd never visited before. This invitation to Doha, Qatar came nicely, as it also fell on our forty-eighth wedding anniversary. The Egyptian author Ahdaf Soueif, in collaboration with Bloomsbury Publishing, wanted to publish a book on the brand new Museum of Islamic Art, designed by I. M. Pei and opened in 2008. It was built on the end of an artificial peninsula, surrounded by water with intense sun gleaming on it. A fusion of the modern and ancient, it was really a magical place. They invited a wide group of intellectuals and asked them to write on one aspect of the museum for the book. It could be on anything: a painting, an object, the architecture or just the atmosphere itself – the writers were given free rein. Bloomsbury published the book beautifully, but in his very old age Eric's behaviour had at times become too spontaneous and he gave our copy to a lady guest who was admiring it in our drawing room. I was furious with him for a long time. Alas, I don't expect Bloomsbury have any copies left. Needless to say, everything in our visit was first class, or higher, if there is such a thing – the flight, the hotel, the food. Morning, noon and evening there was a constant buffet from all countries with staff attending at all hours. We had such exciting company, nothing but intellectuals in a happy mood, stuffed with good food and drink, which made for stimulating conversation. Our hosts managed to squeeze in a visit to the souks, where we saw falcons being displayed and flying above us. For our anniversary we were recommended a restaurant at the very top floor of a skyscraper, where we saw nothing but twinkly lights. We could have been anywhere. But it turned out to be an intimate and emotional evening where we talked about what we wanted with the time we had left together. Eric was sorry he had spent so much time on his book _Fractured Times_ and he was now going to stop, so we would spend more time together. A lovely thought, all good intentions, but totally naïve. He was a writer, and would no more be able to stop writing than a bird could stop nesting. And why should he? He was the one with little time left for living.
Chapter 25
# _Hospital Years_
The title of this chapter doesn't necessarily refer to the amount of time we spent in hospitals, but is, rather, a metaphor for the constant struggle we faced in going out and about – it felt like preparing to go on the Kon-Tiki expedition each time. We would have everything ready: our tickets for the opera or concert, flowers for the hostess if going to a dinner, and so on. En route to our event he would often turn to me and say, 'I really don't feel too good,' and we would have to turn back. Some years back he had developed leukaemia, which was on the whole not troublesome, but it meant that his immune system could not cope with even the slightest temperature or bug. He was advised to go straight to the hospital if he began to feel unwell. Often this would result in unplanned trips to a (rather nice and quiet) cancer ward. We had our routine of going back and forth to the Royal Free. I used to say, like Laurel and Hardy, 'Another fine mess you have got us into.' We laughed and drank from our bottles of Volvic. Unfortunately after a while, this little cancer ward closed down and we had to endure going to A&E. More often than not, I would phone Julia, who would soon turn up with his favourite fruit sweets and an interesting story from the media world. Of course this cheered him up.
Eric was still mobile in these days, with a stick. He was not in serious pain and had all his marbles. He had problems with swallowing and this was the most depressing part of his condition. What he found so dispiriting was that he needed to eat, but it would take such a long time. He would often have prepared liquid formula, which contained the right substances, but he found them disgusting and like baby food. Keeping his weight up was the main struggle.
At home, I used to ponder the best way to pass the time and keep an old intellectual stimulated and happy. I learned it was best not to dwell on the specifics of medication, pill reminders and all the pragmatic health matters. My ploy was to ask questions like, 'If you could only take one book to an island, would it be Bernard Shaw?' or, 'Do you prefer Bellini or Schubert?' That would bring us back to the memory of a glorious Tuscan day at Benedetta Origo's house, when her cousin made the marvellous remark, 'Bellini was _our_ Schubert.' This had always remained with us. The mind has to be taken away from the illness. The questions had to be light choices that could be easily made, rather than ones that demanded too much thinking. This time period also forced me to slow down and be fully present with him.
We were so close in our last chapter together. It was 24 September when I had to take him to the Royal Free for one of his routine blood transfusions. He was very comfortable with this procedure, which allowed him to read, and he would always be perkier and restored afterwards. Normally, I would come at around six to take him home. But this time, I had a hunch and decided to pop in to see him at lunchtime. I found him worse than when he went in. A staff nurse then advised me, 'I really don't think you should take him home today. He's not well enough and he should spend the night here.' I stayed with him in an annexe to the ward.
The following morning his regular doctors did not appear as usual, and it was clear he was being moved to palliative care. The doctor said, 'We will keep him as comfortable as we can and we will see how we go day by day.' I phoned Andy and Julia and told them these were the last days. Then I phoned his favourite carer, Benny Fernandes, an Indian lady who loved both communism and Jesus. She shaved and cared for him in her way (the nurses had no time for that). He lived another six days and looked pretty good. He had some discomfort but no real pain – all that was taken care of. The grandchildren came, and I phoned some friends and organised the schedule as best I could. Luckily, one of his dearest friends, Leslie Bethell, was in London at that time. These visits did him good. He was lucid and pretty normal. On Andy's last visit, Eric asked, 'Do you think this is the end? Or will there be a final chapter?' to which Andy replied, 'Definitely one more chapter to go, Dad.'
The last evening, I felt that he had begun the process of dying. He was nearly deaf, so I had to get into his bed and talk right into his ear. I knew he didn't want to die, so I told him, 'Another fine mess you've got us into.' And he smiled. When I got out of his bed and stood at the end of it, he pointed at me, looked at me and repeated, 'You, you, you, you.' I knew he was trying to thank me for everything, and then he slept and that would be our final goodbye.
He died in the very early hours of the next morning, in his sleep, on 1 October 2012. The hospital phoned Julia around 4 a.m. and the three of us then went over. I have learned that one does not really expect the expected, and so we were, all three of us, in total shock.
Chapter 26
# _Death_
The news of Eric's death instantly went viral all over the world. He was often the headlines on the main page, not tucked away in the obituary columns. Even _The Times_ leader was about his death. One would think he had been the president of a major country. We bought all the papers, and the private condolence letters arrived at the same time. Home became a 24/7 international mail and newsroom. The phones never stopped ringing at all hours, time zones forgotten by those waking up to the news, and it was difficult to concentrate. It was overwhelming for me because of what was going on inside my head, which was gobbledygook. I seemed quite unable to grasp the fact that Eric was now not existing in the world at all any more, not anywhere, not for anyone. I felt I could have borne the cruelty better if he had gone far away, even left me for someone else, because this way of not existing in the world meant the world itself was now not a thing for me. It was like a form of delirium; I wondered if other bereaved people have had this disorienting experience. Quite paradoxically, I also felt relief. Only a week earlier I was making enquiries about a nearby nursing home, because I sensed the looming disability facing Eric. Soon he would be unable to stand or turn around on his own, however slowly. Our life as we knew it would be over. He would have to be lifted and need two carers during the day and also one at night. Now I didn't have to worry about those things any more, nor most importantly, worry about how Eric was feeling – that was all over. Eric's friend Nicholas Jacobs, from way back in Communist Party days and also our family friend, had recently surfaced again in our lives. He was alone now, and came around often, we called him our _Hausfreund._ With him came interesting talk, and cheer to our house and to our sad predicament. He liked having meals with us and we enjoyed that too. Actually, until the final weakening, before Eric was in hospital, we were still happily going out to the cinema, often the three of us, followed by a restaurant (usually Chinese) for a post-film discussion. Nick kindly came to all our funeral meetings, with his immeasurable knowledge of classical music, to help Andy, Julia and I. Eric had written his wishes for the funeral – which was to be held at Golders Green Crematorium – a long time beforehand and we kept about 80 per cent of them. Andy, a wizard on his mobile phone, found what we wanted in seconds, and Nick knew by heart the best recordings and with the best singers. I wanted the first orchestral bars of Bellini's 'Casta Diva' from the opera _Norma_ to accompany the pallbearers. We were lucky that on the actual day, the number of bars matched exactly with their steps to the resting place of the coffin. Then came music by Beethoven, Mozart and, surprisingly to some, Offenbach and not surprisingly, instrumental jazz – 'Slow Grind' played by the Kenny Barron Trio. Of the many tributes and readings, the highlights were a tour de force eulogy by the historian Roy Foster and also moving memories spoken by Andy. In his written wishes, Eric had specified a non-religious funeral, and yet it was he himself who had previously (out of the blue) asked his friend Ira Katznelson if he would read the Kaddish at his funeral. Perhaps Eric was returning to a memory of when he was fourteen, remembering the advice and wish of his dying mother: that he must never do anything to suggest he was ashamed of being a Jew. When the time came, Ira made a flash visit from New York with his wife Debbie to say the Jewish prayer and then caught the 'red eye' back that same night. Eric had agreed with me that he wanted the melody of 'The Internationale' for the final piece and also exit music, and Andy's mobile searches had come up with a particularly fine French orchestral rendition, which was triumphant with tambourines instead of the usual solemnity. Many in the congregation sang the words out loud and heartily. The funeral ended on an uplifting note, which prevailed at the lunch served with much friendliness by staff at the nearby pub, the Bull and Bush. Andy and Julia came home with me, and some relations too. By six o'clock I was on my own and I knew the best thing for me was my reliable Beethoven therapy: a hot drink, a rug, a good armchair and more or less any Beethoven CD. This time I chose the Ninth Symphony because it begins chaotically and seemingly without a plan (as I found myself that day) but soon it would settle down, and if I had nodded off, the 'Ode to Joy' was sure to wake and revive me. It was in this way that I ended my saddest day.
Chapter 27
# _What Remains_
Beginning with a death certificate, there is a huge amount of paperwork that comes as uninvited as death itself. Lawyers, the bank, the funeral parlour and the cemetery gravediggers – all have to be handled in turn. It is a busy period, unfortunately coinciding with the time needed for quiet reflection. Friends suggested I see a bereavement counsellor, and after deliberating, I decided I would try it. My counsellor was an intelligent and pleasant lady, and although the sessions didn't necessarily lift me out of my gloominess, having a space set aside each week meant there was dedicated time for thinking exclusively about Eric and 'us'. Where should I start? I decided to dwell on both our bad and good traits as individuals, which comforted me. I started with Eric. Although there was never a whiff of any dalliances on my part, he could be very possessive. I can understand this, knowing how much he suffered after his first wife left him for another man. I am convinced that at that time he would have blamed himself, although they were never a good couple and had great difficulties. Surely adding his culpabilities did not help him. His experiences of childhood loss were such that I could never understand how it was that Eric was not more damaged or warped as a person. Nicholas Jacobs has a theory that if you experience good mothering in your first two years, you will cope with whatever life throws at you. But if you are deprived of this, then you are in trouble. I agree. Eric's mother thought the world of him from day one and their relationship was always close. In spite of his mother's death when he was only fourteen and having to work as an au pair, Eric knew about loving – relatives, friends, books, birds and American cars. Fortunately, he did overcome his breakdown after Muriel left him, and without any professional help. With pure strength of character, and his ambition to work and write like hell, he succeeded in conquering his unhappiness. It was really very impressive and admirable. Unfortunately there was no one to impress or to admire him; this was a very long and lonely stretch of time in his life. Many people, friends even, did not realise what an emotional man Eric was by nature. His relationship with Ticlia our cat was a familiar one – the less notice he took of her, the more attention she wanted. This went on for fourteen years. In the end, when she became very ill and could only drag her hind legs, we booked an appointment with the vet to have her put down. Eric said he had better come with me because I probably wouldn't manage the upset. The vet told Eric to hold Ticlia tight while he prepared the fatal injection. There was instant collapse. She did not suffer at all. Eric, however, had huge tears streaming down his face, 'I thought of Auschwitz,' he said. When the bill came he paid it twice by mistake.
A bad trait of mine was that I was mentally lazy. As Eric only took a few seconds to look things up in print and I was slow, he got into the habit of doing it all and that was unfair. I think he must have looked up the time of every train we took in fifty years. He did the heavy lifting of the household paperwork, the car, and all the utility contracts etc., though he did have secretaries to assist through his later years. I am ashamed now how much we all relied on Eric's knowledge. I would sometimes be lazy about the details of a news story and simply ask, 'Whose side are we on'? Officialdom is an alien language to me now. Serves me right. The one particular thing I regret above all else is preventing him from travelling to visit Isfahan in Iran. He was passionate about Oriental and Islamic art, and this was one of the most glorious places to see it. He had been thinking about it for years, but when the time came he was frail and I was struggling to keep his health up. I didn't want to witness it all unravelling. We had contacts there with Iranian scholars and would have been generously looked after, but I was nervous all the same. I had miscalculated badly – all the glories of Istfahan could be visited online, but still he minded terribly not going. He resented it for ever. I felt I had denied him his last supper and still feel rotten about it.
At times I noticed Eric and I had different degrees of loving. When Eric was proud of me, I think his love increased; when I published my recorder books, he was very excited that I had managed to do this. I suppose my love also increased at times, but for opposite reasons, like when he left his briefcase on the plane. I would think to myself, 'He's just a really ordinary Joe, but he's _my_ ordinary Joe.' It seemed as though when I was clever he loved me most, and when he was stupid and silly I loved him most. Our love was an attraction of opposites. During my period of grieving, I also found solace in the writings of Rabindranath Tagore, recommended to me by our friend Amartya Sen. Tagore's words on death were profound – his concept of 'making friends with death' made such an impression on me. Another solace was my choir sessions, where we often sang from John Rutter's _Madrigals_ _and_ _Partsongs_. How happy I was singing John Wilbye's words, 'I hope when I am dead in Elysian plain to meet, and there with joy we'll love again.'
I did actually see Eric one last time. I know that sounds strange, but let me explain. Four months after his death, I was recovering from a hip replacement at the Royal Free Hospital, and was heavily sedated with morphine. Someone had turned on the TV, and there was Eric on the screen as clear as day, wearing one of his very smart suits, the kind he wore when meeting dignitaries; he looked marvellous. I was obviously having some kind of a hallucination, but I remember it vividly and can conjure it up now. He was looking directly at me and had a soft little smile.
Chapter 28
# _Eric's Legacy to Me_
Of all the precious influences Eric left me, inherited friendships were the most important. They were aside from my beloved family and relatives, as well as musician friends. I call these my Team Eric: they were his friends, then became ours, and now they are mine, and I love to have them around. The Aschersons – Neal and Isobel, Leslie Bethell, the Fosters – Roy and Aisling, the Frayns – Claire and Michael, the Gotts – Richard and Vivien, Martin Jacques, Nicholas Jacobs, Jane Miller, Donald Sassoon, Marina Lewycka, Marina Warner, Graeme Segal, Joseph Rykwert, Gaia and Hugh Myddleton. Some are Eric's ex-students, many are mostly now in their sixties and seventies (creeping into their eighties), and many live in north London. The exceptions are Gaia, whom I knew before Eric did; Leslie Bethell, who lives in Rio de Janeiro but also teaches in London; Joseph, who is in his nineties (with all his marbles); Nicholas, whom Eric knew from Communist Party days; and Martin, whom we met together.
They all stayed loyal to me. They knew I had lived with a human encyclopaedia for fifty years, and now I could consult with them instead to sort out my dilemmas. I hope I didn't exploit their kindness too much. It was at my request that many of these Team Eric friends obligingly prepared tributes and delivered wonderful words at the London memorial service for Eric. This was held at Senate House in April 2013, and Birkbeck pulled out all the stops to make it a splendid occasion. Roy Foster kindly hosted and Professors Roderick Floud, Frank Trentmann and Jean Seaton gave tremendous speeches. I also spoke a few words, recalling what a varied and privileged life I had being an academic's wife. Those were the days; we were very lucky and the two of us made a good fist of it. Today I get the impression academic life is much harsher, probably for financial reasons.
I had decided already that I didn't want to stay in our house in Nassington Road any more. I wanted to sell it. It would make the final exodus complete. The children had flown, Ticlia had died and now Eric was gone too. 'Knock knock.' 'Who's there? 'Only Marlene,' and her life within it had disappeared. The house was too big for me on my own. Mainly, the thought of living there without Eric would be like rubbing salt in the wound every day. Leaving my lovely neighbours would be very sad. Rena, Marion and Rob and co., Chris and Dominique Moore, Nick and Ghislaine, the Southgates next door, Berwyn, Charlotte, Jim, and the girls. One evening a letter slid under my door from a couple who lived down Parliament Hill, asking me would I please not sell the house until they'd seen it as they had their heart set on it. They were extremely nice people who still invite me to visit 'my' garden from time to time. Over in Belsize Park, I came across a handsome block of flats and went straight to the estate agent to make enquiries. I was told there was a flat for sale on the first floor, but the owners were in Australia. I would have to wait to see inside until they returned. Very tantalising. I waited two months, then went to inspect with Anna Rosa and Federica Frishman. We all liked the flat immensely, especially the abundance of natural light, even though there were one or two quite radical changes that would have to be made. Despite this, I knew it was right for me; just like with a fella, one knows immediately. No matter what issue, I still thought about what Eric's opinion would be, and when it came to the new flat, I think he would have approved. Being an architect, Federica generously offered her help, and her experienced builder was available, which was indispensable, considering I had last renovated and refurbished a house forty long years ago, and that was before the internet. Now, alas all of the ordering of fittings and furnishings had to be dealt with online. Luckily it suited Federica, who was a marvellous organiser, never without her tape measure to hand. If there was a problem, she would have it sorted before you could say 'Jack Robinson'. With the changes I intended to make, I would have a large music room able to accommodate a dozen recorder players
I had to borrow the money urgently and pay the deposit in order to secure the flat; I was already beginning to plunge into huge decisions like this on my own. It was a tense race, but I made it in time because my unflappable son-in-law Alaric helped me with the transaction of the money. In this new home I altered my way of hospitality, no longer holding dinner parties, since I felt the absence of Eric's partnership in hosting. Always observant and inclusive, Eric had been the main conversationalist, though I grew into discussing world affairs without him. Certainly I had become more daring, developing a new self-confidence to express my opinions. Indeed, the confidence that developed over fifty years in a loving marriage served me well just when I needed it. For entertaining, I've found Sunday lunchtime to be the most relaxed part of the week for guests. Those tending to their gardens or allotments – like the Fosters – can come and go in their wellies. Usually I invite some of the 'Team', as well as integrating new faces. I still prefer, when possible, one conversation for us all around the table, and refer to the meal as brunch, to indicate they should not expect a three-course meal with a duck in the middle. These are jolly occasions. Nicholas is perhaps the most loyal guest and usually arrives earlier in order to help with the drinks. I call him the barista. Nick is a retired publisher who translates, writes book reviews and teaches German literature to adults at U3A. I did exploit him regarding help with the academic mail, which has now finally dwindled. I knew he could keep secrets, and he helped me with my dilemmas in assisting Richard Evans, who was writing a biography of Eric. It took seven years and brought our friendship closer. Luckily for me, Nick is a total culture vulture. He first introduced me to concert performances of early opera and subscribed to those conducted by Ian Page, the artistic director of Classical Opera and the Mozartists. They perform in all the London venues: Cadogan Hall, Royal Festival Hall, Wigmore Hall and St John's Smith Square. I find opera performed in this intimate way intensely moving. Above all, Nick plied me with books. He was amazed I had not read _Madame Bovary_ – so I did and it provided discussion for weeks. Also my first Goethe, which I adored: _The Sorrows of Young Werther_. As soon as I finished it I began at the beginning again. I thought his writing both prosaic and glorious. Then came the most precious of books, _Clean Young Englishman_ , a short autobiography by John Gale, a sensitive and superb prose writer. Books became a lifeline for me. It is very common for new widows to have accidents, and I was no exception. I broke my right ankle in a car crash, then fractured my right wrist in a fall and sprained my left hand at the same time. This often left me stationary at home with a carer.
Unfortunately in life, one never knows what is around the corner; while out shopping one Saturday morning, my phone began to ring – it was Walter's eldest daughter Habie to tell me my brother Walter had died. I dropped everything and put the supermarket in disarray. Again, I was in a state of disbelief. We'd been close ever since our Vienna days and into our adulthood. Because of our shared childhood, he helped, in fact checking the earlier parts of this memoir. Walter was very popular in his family, including with all the children and grandchildren. He made time for everyone, no matter the issue. If funerals are supposed to console, then Walter's provided the most consolation there could be. Aside from his many friends, all the nieces, nephews, children and grandchildren went to such lengths and effort with their songs and music, clever and funny speeches – his son Ben had everyone in tears. Because Walter had begun to develop Alzheimer's, there was also a sense of relief for anyone who really loved him.
Apart from my friends, it is of course my children who really sustain me. They are part of me, and so are their families. 'Next of kin' are three powerful words. Both my children are now absorbed in their various work projects. Julia, a published author, writes and lectures about technology and human connections, and also runs an events and consulting business. Andy is an internet entrepreneur and digital adviser working with a number of start-up ventures and other businesses; he also writes and performs music with his singer/songwriter wife Kate. Despite their hard work and success in their careers, Andy and Julia and their spouses are very good parents. They manage to raise their children so well in these difficult times, with huge dedication and thought. We all stay close. Julia and Andy usually pop in on the weekend. We have a family arrangement to meet on the first Sunday of the month, when both families come around to my place for bagels, cream cheese, smoked salmon, salami and much more. The schedule sometimes has to move when Julia or Andy is travelling for work, but the grandchildren (now all teenagers) like it very much and we all make an effort to keep up the tradition. Luckily, Highgate Cemetery is nearby for us all to visit Eric's grave, which is made of fine Kirkby grey slate from Cumbria, with incisive lettering carved by Annet Stirling. Depending on how you approach the grave (and perhaps your political beliefs) it is situated either a little to the right or a little to the left of Karl Marx.
So there is life after Eric. I still feel his presence, and talk to him – he always was a good listener. In whimsical moments, I daydream about meeting him in Buenos Aires after all.
# Acknowledgements
My thanks especially go to Claire Tomalin for agreeing to do an introduction and for writing such a thoughtful and flattering piece; Jane Miller for being the first friend to read a draft of this book in its raw initial stage; to both my children, Andy and Julia, who helped jog my memory and encouraged me all along the way; Julia for putting me in contact with Sarah and Kate Beal at Muswell Press, who accepted my book and were so warm and kind; helpful friends Leslie Bethell, Martin Jacques, Donald Sassoon, Nick Jacobs and Anna Rosa Villari; Bruce Hunter and Margaret Bluman for their advice; Joan Martínez Alier for helping me through the Agrarian Reform in Peru; Divya Osbon for stepping in during a sticky situation; my assistant Holly Parmley, a writer and artist, who initially came to assist me on the computer and typing, but as our friendship grew, her role changed to help me with the writing of this book, which took us two years to finish.
# Photo Credits
All photos from the Hobsbawm family archive with the exception of :
Marlene in Capri at the Arco Naturale. Photo by Mariella de Sarzana
Eric in Marlene's Paddington Street flat during their early dating. Photo by Marlene
Eric bonding with our first born Andy. Photo by Marlene
Roman, Anoushka, Max, Rachael and Wolfie. Photo by Rachael Campbell
Marlene and Eric. Photo by Enzo Crea
Die Kinder (Walter, Marlene and Victor) in Vienna
Marlene on the wall of her nursery school
My family in the garden in Manchester
Happy emigrés proud to be digging for victory
Marlene in Capri at the Arco Naturale
Eric in Marlene's Paddington Street flat during their early dating
Marlene posing with the United Nations welfare team in the Congo
Grandma Lilly, Julia, Habie and Marlene in Parc Farm
Benedetta, Anna Rosa, Marlene, Eric and Rosario at the Villari's home in Cetona
Eric bonding with our first born Andy
Babes in the woods (Milo and Eve)
Eric sharing a joke with the Queen
Roman, Anoushka, Max, Rachael and Wolfie
Eric with Milo, Eve, Wolfie, Anoushka and Roman at Julia and Alaric's home
Grandma Lilly's 80th birthday party; Isobel, Tanya holding Zac, Julia, Emily, Benjamin, Andy, Charlotte, Zoe, Habie, Grandma Lilly and Humphrey
Andy, Julia, Marlene and Eric
# Copyright
First published by Muswell Press in 2019
Typeset by M Rules
Copyright © Marlene Hobsbawm
Marlene Hobsbawm asserts the moral right
to be identified as the author of this work.
All rights reserved;
no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise without the prior written permission of the Publisher. This book may not be lent, resold, hired out or otherwise disposed of by way of trade in any form of binding or cover other than that in which it is published, without the prior written consent of the Publisher.
No responsibility for loss occasioned to any person or corporate body acting or refraining to act as a result of reading material in this book can be accepted by the Publisher, by the Author, or by the employer of the Author.
eISBN: 9781916129214
Muswell Press
London
N6 5HQ
www.muswell-press.co.uk
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Print at high volumes for an extremely low cost-per-page, and get easy mobile printing. With a reliable, spill-free ink system, you'll be able to print up to 8,000 pages color or 6,000 pages black, and produce exceptional quality.
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"redpajama_set_name": "RedPajamaC4"
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William Ellery Channing, född den 7 april 1780, död den 2 oktober 1842, var en nordamerikansk präst, unitarismens fader. Han var bror till redaktören av North American Review, professor Edward Tyrrel Channing och farbror till poeten William Ellery Channing.
Channing tillhörde först den kongregationalistiska kyrkan i Boston men blev snart misstänkt för att vara irrlärig framför allt i fråga om Kristi gudom och försoningsdöd. Efter ett föredrag i Baltimore där han bland annat påtalade treenighetslärans okristlighet, blev det öppen brytning, varefter en unitarisk kyrka bildades. Channing var även socialt verksam och arbetade för slaveriets upphävande, inom nykterhetsrörelsen, för fångvårdsreformer med mera. År 1881 invigdes i Boston Channing Memorial Church.
Svenska översättningar
Om sjelfbildning (översättning Adolf Regnér, Stockholm, 1848)
Napoleon Bonaparte, skärskådad från christligt religiös synpunkt (Analysis of the character of Napoleon Bonaparte) (översättning And. Hertzman, Linköping, 1848)
Om religionsundervisningen i söndagsskolor och annorstädes: ur ett föredrag om söndagsskolor (översättning Richert von Koch, Stockholm, 1872)
Ett fullkomligt lif menniskans mål: tolf predikningar (The perfect life) (översättning Maria Söderberg, Almqvist & Wiksell, 1881)
Källor
Svensk uppslagsbok. Lund 1930.
Vidare läsning
Amerikanska präster under 1800-talet
Personligheter inom antitrinitarism
Födda 1780
Avlidna 1842
Män
Svensk uppslagsbok | {
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{"url":"https:\/\/www.prioritydetails.com\/2017\/01\/rstudio-best-practice-general-options\/","text":"# RStudio Best Practice: General Options\n\nOver the years RStudio has become the most powerful and versatile IDE of R. I started using RStudio back in 2011, when R was in its version 2.12. Back in the day, RStudio was a better choice for me than the default R-GUI simply because of the auto-completion it provided. Being a C++ and Xcode user, memorizing all those R commands in an early learning stage was impossible for me.\n\nWith R and RStudio becoming more and more popular over the last five years, RStudio keeps providing all kinds of integration to R. It started with other popular software like TeX and Git, then through integration of popular R packages like RCpp. Finally, RStudio hired some of the most talented R developers and provided us with a whole new universe of packages.\n\nHowever, all of those new features sometimes confuse new users, and it seems like we have a lot of them. Besides, when so many features are available, some may not work well with others. This motivated me to write down some of the \u201cbest practice\u201d from my own experience. Just like the style guide on R programming, how to use RStudio or any other tools is merely a personal preference. I don\u2019t think there is a \u201cbest\u201d practice out there, but there are definitely bad ones, and I think many people could benefit from steer clear of those. In each post, I\u2019ll focus on some features of RStudio, and recommend some settings and introduce some tips that I found useful.\n\n## General Options\n\nMost of RStudio\u2019s options lie in the preference panel. These are the global options that apply to the current session, and they are also the default for any projects. For some options, every project could have their own settings in the project options (available under tools\/project-options or the drop down menu at top-right corner). The first sub-panel in preference is the general options and the topic of this post.\n\n### Restore Data\n\nThere is this option called Restore data, and right next to it, Save workspace on exit. These two options, if both selected, save the current environment to a hidden .RData file in your working directory, and restore it when you reopen from that directory.\n\nThis may seem like a convenient feature, but I don\u2019t recommend using them because of the following situations:\n\n1. You don\u2019t always clean your working environment and over the time some large data stayed there in memory. Every time you restart, RStudio have to write that data to file, and then read it back in again, makes it extremely slow at start up.\n2. You relied on the restored data to work, but you let RStudio to ask you on exit whether to save the environment or not. Eventually, you will mistakenly click on No instead of Yes.\n3. The code to generate a tmp data set is accidentally deleted but the generated tmp object is still in memory every time you run the script. Instead of giving an error the first time you restart and run it after you deleted the code, it gives you an error when you happen to overwrite tmp to something else.\n\nAll of the above happened to me. Basically, you can see that these options violate the rule of reproducibility. In order to have a reproducible analysis, you should ensure everything with code, not the tools you use. For example, if you want to save something, add an explicitly call to save it to a file, or set seed before a simulation. Don\u2019t rely on RStudio to keep the record for you.\n\nNo matter you choose to save it or not, I think it\u2019s also good practice to regularly clean (read: clear) the global environment, especially for those who have a very limited imagination for variable names, they will almost always run into case 3 above.\n\n### Save History\n\nAlso because of reproducibility, I recommend checking always save history. When something goes wrong, finding clues in history is way better than finding an object without context in the restored environment.\n\nI also have a little tip on using the saved history. It\u2019s not about an option in the general panel, but since this is the only thing that\u2019s history-related, I\u2019ll put it here.\n\nRStudio provided some really cool shortcuts for command history in console. I think the shortcuts are too good that makes the History pane not useful any more. In console, control\/command + up will bring up the history search. Hit the shortcut without anything, it will bring up the most recent history. Type a few words and hit the shortcut, it will search with partial match. This has become the single most used shortcut when I have to work in the console.\n\nPS: you can see all the shortcuts in Help menu or Alt + Shift + K. There are a lot so I will only mention the ones I find useful.","date":"2020-08-08 02:47:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.17180171608924866, \"perplexity\": 1148.2550773076305}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-34\/segments\/1596439737238.53\/warc\/CC-MAIN-20200808021257-20200808051257-00084.warc.gz\"}"} | null | null |
{"url":"https:\/\/hero.handmade.network\/forums\/code-discussion\/t\/24-alternate_way_of_loading_functions_from_dlls#35","text":"Benjamin Kloster\n48 posts\nHi everyone,\nI found Casey's way of loading functions from DLLs for XInput to be a bit verbose, with lots of repeated names in slightly different capitalizations. For reference, here is the abridged version that just imports XInputGetState.\n\nUsing some C++11, it can be done a little more compact. I hope the comments make most of the idea clear, but for those unfamiliar with C++11, the features used here are:\n\n1. The decltype keyword gives us the type of an expression, in this case, a function signature. Here, it's used to avoid using a typedef for the function signature. We can just use the signature of the original XInputGetState, as provided by the XInput.h header file.\n\n2. Lambda functions. That's this weird thing:\n 1 [](DWORD packetNumber, XINPUT_STATE* state) -> DWORD {...} \n\nIt's basically an alternate way of defining functions. Here, we use it to initialize our XInputGetState_ with a stub function, without needing to declare that stub separately.\n\nNow, the drawbacks of using this approach. First, it requires a bit of C++11 knowledge to understand. Especially decltype is not widely used outside of template libraries. And for the uninitiated, C++'s peculiar lambda syntax might cause high WTFs per minute.\n\nSecond, you probably won't be able to use this with older compilers. It *might* work with Visual Studio 2010, but I haven't tested it beyond the 2013 Community Edition, which handles it fine. I'm pretty sure it works with any reasonably new version of mingw. In any case, I think compatibility is a minor concern here, because it's part of platform-specific code anyway. It's not like this has to compile on Android or Raspberry PI.\n\nThe payoff is more compact code and, more importantly, less name pollution. The original version uses four names per imported function:\n\n1. X_INPUT_GET_STATE for the preprocessor define\n2. x_input_get_state for the function signature typedef\n3. XInputGetStateStub for the stub function\n4. XInputGetState_ for the drop-in replacement\n\nThe C++11 version manages with just one name, XInputGetState_.\n\nNote that I don't think that either approach is objectively \"better\". They both have pros and cons. I mostly felt like sharing.\n\nHappy coding!\nJeff Hutchins\n1 posts","date":"2022-08-16 05:01:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.26053905487060547, \"perplexity\": 2789.767619160777}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882572220.19\/warc\/CC-MAIN-20220816030218-20220816060218-00136.warc.gz\"}"} | null | null |
Thelypteris biolleyi är en kärrbräkenväxtart som först beskrevs av Hermann Christ och fick sitt nu gällande namn av George Richardson Proctor. Thelypteris biolleyi ingår i släktet Thelypteris och familjen Thelypteridaceae. Inga underarter finns listade.
Källor
Kärlväxter
biolleyi | {
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Q: An irreducible polynomial in a subfield of $\mathbb{C}$ has no multiple roots in $\mathbb{C}$
Let $K\subset \mathbb{C}$ be a subfield and $f\in K[t]$ an irreducible polynomial. Show that $f$ has no multiple roots in $\mathbb{C}$.
If I understand this question correctly, I must show that there is no $a \in \mathbb{C}$ such that $(t-a)^n|f$ in $F[t]$ with $n>1$. So suppose $(t-a)^2|f$ and $f=(t-a)^2h$. Then we have $f'=2(t-a)h+(t-a)^2h' \Rightarrow (t-a)|f'$ so $\gcd(f,f'$) is not constant. Therefore $f$ is divisible by some square of non-constant polynomial in $F[t]$, which is a contradiction. Is my argument correct? Thank you.
A: Given a root $\alpha$ of $f(x)$, $f(x)$ (divided by his director coefficient but who cares, we have coefficients in a field) is the minimal polynomial of $\alpha$. Thus there cannot be another polynomial with lower degree such that $p(\alpha)=0$.
If $\alpha$ is a double root for $f(x)$, then $f'(x)$ is again a polynomial in $K[x]$ and of course $f'(\alpha)=0$. But his degree is lower then $f(x)$. We conclude that $f'(x) \equiv 0$. Since we are working in a subfield of $\mathbb C$, thus a field of characteristic $0$, we know that this implies $f(x)\equiv 0$ , absurd.
A: We can assume $f$ has degree $\ge 2$. Suppose that $f$ and $f'$ are relatively prime as polynomial over $K$. Then (Bezout) there exist polynomials $u(t)$ and $v(t)$, with coefficients in $K$ such that $uf+vf'=1$.
But then $f$ and $f'$ cannot have a common root in any field extension of $K$, for such a root would have to be a root of the polynomial $1$. In particular, they cannot have a common root in $\mathbb{C}$. This contradicts the assumption that $f$ has a double root in $\mathbb{C}$.
So $f$ and $f'$ are not relatively prime as polynomials over $K$. Let $g=\gcd(f,f')$. Then $g$ is a polynomial over $K$ of degree $\ge 1$ and less than the degree of $f$, and $g$ divides $f$. This contradicts the irreducibility of $f$ over $K$.
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#import "BLETrigger.h"
#import "BLEKitPrivate.h"
#import "UNCodingUtil.h"
@implementation BLETrigger
- (instancetype) initWithBeacon:(BLEBeacon *)blebeacon
{
if (self = [self init]) {
self.beacon = blebeacon;
}
return self;
}
#pragma mark - BLEUpdatableFromDictionary
- (void)updatePropertiesFromDictionary:(NSDictionary *)dictionary
{
self->_uniqueIdentifier = [dictionary[@"id"] description];
self->_comment = dictionary[@"comment"];
self->_name = dictionary[@"name"];
if (dictionary[@"action"]) {
id <BLEAction> action = [[BLEAction alloc] initWithUniqueIdentifier:[dictionary[@"action"][@"id"] description] andTrigger:self];
if ([action conformsToProtocol:@protocol(BLEAction)]) {
/**
* Called on <BLEAction>, not on custom actions because <BLEUpdatableFromDictionary> is private here
*/
[action performSelector:@selector(updatePropertiesFromDictionary:) withObject:dictionary[@"action"]];
}
self->_action = action;
}
if (dictionary[@"conditions"]) {
NSArray *array = dictionary[@"conditions"];
NSMutableSet *set = [NSMutableSet setWithCapacity:array.count];
for (NSDictionary *conditionDictionary in array) {
BLECondition *condition = [[BLECondition alloc] initWithTrigger:self];
[condition updatePropertiesFromDictionary:conditionDictionary];
[set addObject:condition];
}
self->_conditions = [set copy];
}
}
- (BOOL) validateConditionsWithOccurrence:(BLEEventType)eventType
{
for (BLECondition *condition in self.conditions) {
if (![condition validateForEventType:eventType] || ![condition validateForParameters:YES])
return NO;
}
return self.conditions.count > 0 ? YES : NO;
}
- (BOOL) validateConditionsWithoutOccurrency:(BLEEventType)eventType
{
for (BLECondition *condition in self.conditions) {
if (![condition validateForEventType:eventType] || ![condition validateForParameters:NO])
return NO;
}
return self.conditions.count > 0 ? YES : NO;
}
- (BOOL) validateEventType:(BLEEventType)eventType
{
for (BLECondition *condition in self.conditions) {
if (![condition validateForEventType:eventType])
return NO;
}
return self.conditions.count > 0 ? YES : NO;
}
#pragma mark - NSSecureCoding
- (instancetype)initWithCoder:(NSCoder *)aDecoder {
self = [super init];
if (!self) {
return nil;
}
[UNCodingUtil decodeObject:self withCoder:aDecoder];
return self;
}
- (void)encodeWithCoder:(NSCoder *)aCoder {
[UNCodingUtil encodeObject:self withCoder:aCoder];
}
+ (BOOL)supportsSecureCoding
{
return YES;
}
@end
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\section{Introduction}
The evolution of a chaotic system is unpredictable and difficult to
describe, but its statistical properties are sometime predictable and
(reasonably) simple to be described.
Many of these properties are related to its associated transfer operator, essentially via its spectrum.
This is a linear operator representing the action of the dynamics on
measures space. Let us consider the set $SM(X)$ of finite signed Borel measures
on the metric space $X$. A Borel map $T\colon X\to X$ naturally induces a linear
operator $L_{T}:SM(X)\rightarrow SM(X)$ called the \textbf{transfer operator},
defined as follows. If $\mu \in SM(X)$ then $L_{T}[\mu ]\in SM(X)$
is the measure such that
\par
\begin{equation*}
L_{T}[\mu ](A)=\mu (T^{-1}(A)).
\end{equation*}
Sometimes, when no confusion arises, we simply denote $L_{T}$ by $L$.
In this paper we address two important statistical features of the dynamics: the rate of \emph{convergence to equilibrium} and the \emph{escape rate} in open dynamical systems.
Convergence to equilibrium (see Section \ref{1}) is a quantitative
estimation of the speed in which a starting, absolutely continuous measure
approaches the physical invariant measure.
This is related to the spectrum of the transfer operator, since the speed is exponential in the presence of a spectral gap,
and to decay of correlations. Indeed, an upper bound on the decay of correlations can be obtained
from convergence to equilibrium estimation in a large class of cases, see
\cite{AGP}). An estimation for these rates is a key step to deduce many other consequences:
central limit theorem, hitting times, recurrence rate... (see e.g. \cite{Bo,BS,G07,L2}).
The escape rate refers to open systems, where the phase space has a ``hole''
(see Section \ref{escape}) and one wants to understand
quantitatively, the speed of loss of mass of the system trough the hole.
This is related to the spectral radius of a truncated transfer operator and to the presence of metastable states (see e.g. the book \cite{BBF} for an introduction).
We will present a method which allows to obtain an efficient, effective and quite elementary {\em finite time} and {\em asymptotic} upper estimations for these decay rates. The strategy is applicable under two main assumptions on the transfer operator.
\begin{itemize}
\item the transfer operator is regularizing on a suitable space; it
satisfies a Lasota Yorke inequality (see Equation \ref{1});
\item the transfer operator can be approximated in a satisfactory way by a
finite dimensional one. (see Equation \ref{2})
\end{itemize}
The first item in some sense describes the small scale behavior of
the system. The regularizing action implies that at a small scale we see a kind of uniform behavior.
The macroscopic behavior is then described by a ``finite resolution'' approximation of the transfer operator. This will be represented by suitable matrices, as the transfer operator is linear, and its main features will be computable from the coefficients of the matrix.
In the following we will enter in the details of how this strategy can be
implemented in general, and in some particular system for which we will
present some experiment rigorously implemented by interval arithmetics.
\subsubsection*{Acknowledgements} BS thanks INdAM and the University of Pisa for support and hospitality. SG thanks the Laboratoire de Mathématiques de Bretagne Atlantique for support and hospitality.
IN would like to thank the University of Pisa for support and hospitality, the IM-UFRJ and CNPq.
This work was partially supported by the ANR project Perturbation (ANR-10-BLAN 0106) and by EU Marie-Curie IRSES Brazilian-European partnership in Dynamical Systems (FP7-PEOPLE-2012-IRSES 318999 BREUDS).
\section{Recursive convergence to equilibrium estimation for maps satisfying
a Lasota Yorke inequality\label{sec1}}
Consider two vector subspaces of the space of signed measures on $X$
$$\mathcal{ S\subseteq }\mathcal{ W\subseteq }SM(X),$$
endowed with two norms, the strong norm $||~||_s$ on $\mathcal{S}$ and the weak norm $||~||_{w}$ on $\mathcal{W}$,
such that $||~||_s\geq ||~||_{w}$ on $\mathcal{W}$.
We say that the probability measure preserving transformation $(X,T,\mu)$ has convergence to equilibrium with speed $\Phi$ with respect to these norms if for any Borel probability measure $\nu $ on $X$
\begin{equation}
||L^{n}\nu -\mu ||_{w}\leq ||\nu ||_{s}\Phi (n). \label{wwe}
\end{equation}
This speed can be also estimated by the rate of convergence to $0$ of $||L^{n}\xi ||_{w}$
for signed measures $\xi \in \mathcal{S}$ such that $\xi (X)=0$ or by estimating the decay rate of correlation integrals like $$|\int f\circ T^{n}~g~d\mu-\int f~d\mu \int g~d\mu|$$ for observables $f,g$ in suitable function spaces.
It is important both to have a certified quantitative estimation for this convergence
at a given time (numerical purposes, rigorous computation of the invariant
measure as in \cite{GN2},\cite{H},\cite{I}, see also Remark \ref{cpmp}), or an estimation for its
asymptotic speed of convergence (computer assisted proofs of the speed of
decay of correlations and its statistical consequences).
In the literature, the problem of computing rigorous bounds on the decay of
correlation and convergence to equilibrium rate of a given system was approached
by spectral stability results (see e.g. \cite{L}).
The use of these methods is limited by the complexity of the assumptions and of the a priori estimations which are needed.
In the following we show how the Lasota Yorke inequality, which can be
established in many systems, coupled with a suitable approximation of the
system by a finite dimensional one allows directly to deduce finite time and asymptotic upper bounds
on the convergence to equilibrium of the system in a simple and elementary way.
{\bf Assumptions.} Let us suppose that our system satisfies:
\begin{itemize}
\item The system satisfies a Lasota Yorke inequality.
There exists constants $A,B,\lambda _{1}\in {\mathbb R}$ and $\lambda _{1}<1$ such that
$\forall f\in \mathcal{S},\forall n\geq 1$%
\begin{equation}
||L^{n}f||_s\leq A\lambda _{1}^{n}||f||_s+B||f||_{w}. \label{1}
\end{equation}%
\item There exists a family of "simpler" transfer
operators $L_{\delta }$ approximating $L$ satisfying a certain
approximation inequality: there are constants $C,D$ such that $\forall g\in
\mathcal{S},\forall n\geq 0$:%
\begin{equation}
||(L_{\delta }^{n}-L^{n})g||_{w}\leq \delta (C||g||_s +nD||g||_{w}). \label{2}
\end{equation}
\item There exists $\delta>0$, $\lambda _{2}<1$ and $n_1$ such that, setting
$$V=\{\mu\in \mathcal{S}|\mu (X)=0\}$$
we have $L_\delta (V)\subseteq V $ and
\begin{equation}
\forall v\in V,~||L_{\delta }^{n_{1}}(v)||_{w}\leq \lambda _{2}||v||_{w}.
\label{3}
\end{equation}
\end{itemize}
\begin{remark}
In the following we will consider examples of
systems satisfying such inequality, where $\mathcal{S}$ is the space of
measures having a bounded variation density, $||~||_s$ is the bounded
variation norm and $||~||_{w}$ is the $L^{1}$ one.
We also remark that condition (\ref{2}) is natural for approximating operators
$L_{\delta }$ defined (as it is commonly used), by $\pi _{\delta }L\pi
_{\delta }$, where $\pi _{\delta }$ is a projection on a finite dimensional
space with suitable properties, see Section \ref{appp}, where it is shown how to obtain (\ref{2}) under these assumptions.
We remark that in Equation (\ref{3}) , the condition is supposed on $L_{\delta }^{n}$ which is
supposed to be simpler than $L^{n}$ (e.g. a discretization with
a grid of size $\delta $ represented by a matrix, see Section \ref{appineq}) and its
properties might be checked by some feasible computation.
\end{remark}
Under the above conditions we can effectively estimate from above the convergence to equilibrium in the system
in terms of the matrix
$$M=\left(
\begin{array}{cc}
A\lambda _{1}^{n_{1}} & B \\
\delta C & \delta n_{1}D+\lambda _{2}%
\end{array}%
\right) .$$
Since $M$ is positive, its largest eigenvalue is
\begin{equation}
\rho =\frac{A\lambda _{1}^{n_{1}}+\delta n_{1}D+\lambda _{2}+\sqrt{(A\lambda
_{1}^{n_{1}}-\delta n_{1}D-\lambda _{2})^{2}+4\delta BC}}{2}. \label{eq:rho}
\end{equation}
Let $(a,b)$ be the left eigenvector of the matrix $M$ associated to the eigenvalue $\rho$, normalized in a way that $a+b=1$.
The condition $\rho<1$ below implies that the powers of $M$ go to zero exponentially fast.
Note that the quantities $\delta BC,$ $\delta n_{1}D$ have a chance to be small when
$\delta $ is small, but this is not guaranteed since the choice of $n_{1}$ depends on $\delta $.
If we consider the case of piecewise expanding maps, with $L_{\delta }$ being the
Ulam approximation of $L$, this is the case (see \cite{GN}, Theorem 12 ).
\begin{theorem}\label{prop1}
Under the previous assumptions \ref{1}, \ref{2}, \ref{3}, if $\rho<1$ then
for any $g\in V$,
(i) the iterates of $L^ {in_1}(g) $ are bounded by
\begin{equation*}
\left(
\begin{array}{c}
|| L^{i n_1} (g) ||_s \\
|| L^{i n_1} (g) ||_w
\end{array}
\right)
\preceq M^i
\left(
\begin{array}{c}
|| g ||_s \\
|| g ||_w
\end{array}
\right)
\end{equation*}
Here $\preceq $ indicates the componentwise $\leq $ relation (both
coordinates are less or equal).
(ii) In particular we have
\[
||L^{in_{1}}g||_s \leq (1/a)\rho ^{i}||g||_s,
\]
and
\[
||L^{in_{1}}g||_{w}\leq (1/b)\rho ^{i}||g||_s.
\]
(iii) Finally $\forall k\in \mathbb{N}$
\begin{eqnarray}
\Vert L^{k}g\Vert_s &\leq &(A/a+B/b)\rho
^{\left\lfloor \frac{k}{n_{1}}\right\rfloor }||g||_s. \label{decayrate} \\
\Vert L^{k}g\Vert _{w} &\leq & (B/b) \rho ^{\left\lfloor \frac{k%
}{n_{1}}\right\rfloor }||g||_s. \label{decayrate_weak}
\end{eqnarray}%
\end{theorem}
\begin{proof}
(i)
Let us consider $g_{0}\in V $ and denote $g_{i+1}=L^{n_{1}}g_{i}.$
By assumption \ref{1} we have
\begin{equation*}
||L^{n_{1}}g_{i}||_s \leq A\lambda _{1}^{n_{1}}||g_{i}||_s +B||g_{i}||_{w}
\end{equation*}
Putting together the above assumptions \ref{2} and \ref{3} we get
\begin{equation}
\begin{split}
||L^{n_{1}}g_{i}||_{w}
& \leq ||L_{\delta }^{n_{1}}g_{i}||_{w}+\delta (C||g_{i}||_s+n_{1}D||g_{i}||_{w}) \\
& \leq \lambda _{2}||g_{i}||_{w}+\delta (C||g_{i}||_s +n_{1}D||g_{i}||_{w}).
\end{split}
\label{kkk}
\end{equation}
Compacting these two inequalities into a vector notation, setting $v_{i}=\left(
\begin{array}{c}
||g_{i}||_s \\
||g_{i}||_{w}%
\end{array}%
\right) $
we get
\begin{equation}
v_{i+1}\preceq \left(
\begin{array}{cc}
A\lambda _{1}^{n_{1}} & B \\
\delta C & \delta n_{1}D+\lambda _{2}%
\end{array}%
\right) v_{i} \label{4}
\end{equation}%
The relation $\preceq $ can be used because
the matrix is positive. This proves (i) by an immediate induction.
(ii) Let us introduce the $(a,b)$ balanced-norm as $||g||_{(a,b)}=a||g||_s +b||g||_{w}$. The first assertion gives
\[
||L^{in_{1}}g||_{(a,b)}\leq (a,b)\cdot M^i\cdot \left(
\begin{array}{c}
||g||_s \\
||g||_{w}%
\end{array}%
\right) ,
\]%
hence
\[
||L^{in_{1}}g||_{(a,b)}\leq \rho ^{i}||g||_{(a,b)}.
\]%
{}From this, the statement follows directly.
(iii) Writing any integer $k=in_{1}+j$ with $0\leq j<n_{1}$ , by assumption \ref{1} we have
\[
\Vert L^{k}(g_{0})\Vert_s \leq A\lambda _{1}^{j}\Vert L^{in_{1}}g_{0}\Vert _s
+B\Vert L^{in_{1}}g_{0}\Vert _{w}.
\]%
and the conclusion follows by the second assertion (ii).
\end{proof}
\begin{remark}
We remark that our approach being based on a vector inequality has some
similarity with the technique proposed in \cite{H}. The first inequality used
is the same in both approaches, the second is different. Our inequality
relies on the approximation procedure and is more general.
\end{remark}
\begin{remark} \label{cpmp}
We also remark that our method allows to bound the strong norm of the iterates
of a zero average starting measure. Considering $L^{n}(m-\mu )$ where $m$ is a suitable starting measure (Lebesgue measure in many cases) and $\mu $ is the invariant one, we can understand how
many iterations of $m$ are necessary to arrive at a given small (strong) distance from the
invariant one. This, added to a way to simulate iterations of $L$ with small
errors in the strong norm, allows in principle the rigorous computation of the invariant measure up
to small errors in the strong norm.
\end{remark}
\section{Escape rates\label{escape}}
A system with a hole is a system where there is a subset $H$ such that when
a point falls in it, its dynamics stops there. We consider $H$ to be not
part of the set where the dynamics acts. Iterating a measure by the dynamics
will lead to loose some of the measure in the hole at each iteration,
letting the remaining measure decaying at a certain rate.
In many of such systems a Lasota Yorke and an approximation inequality
(Equations \ref{1}, \ref{2}) can be proved, hence the procedure of the previous section allows to
estimate this rate, which is related to the spectral radius of the transfer operator.
Let us hence consider a starting system without hole, with transfer operator $L$,
consider an hole in the set $H$ and let $1_{H^{c}}$ be the indicator
function of the complement of $H$. The transfer operator of the system with
hole is given by
\begin{equation*}
L_{H}f=1_{H^{c}}Lf.
\end{equation*}
Similarly to the convergence to equilibrium we can say that the escape rate of the system with respect to norms $||\ ||_s ,||\ ||_w $ is faster than $\Psi $ if for each $f$:
\[ ||L^n f||_w \leq \Psi(n) ||f||_s . \]
We will see in the next proposition, that an iterative procedure as the one of the previous section can be implemented to estimate the escape rate.
We will need assumptions similar to the ones listed before.
These are natural assumptions for open systems constructed from a system satisfying a Lasota Yorke inequality.
We will verify it in some example of piecewise expanding maps with a hole (see Section \ref{holely}).
\begin{theorem}\label{prop2}
With the notations of Theorem \ref{prop1} let us suppose
\begin{itemize}
\item the system with hole satisfies for some $\lambda_1<1$
\begin{equation}
\forall n\ge 1,\quad ||L_{H}^{n}f||_s \leq A\lambda _{1}^{n}||f||_s +B||f||_{w}.
\end{equation}
\item $L_\delta ({\mathcal S})\subseteq {\mathcal S} $ and there is an approximation inequality: $\forall g\in \mathcal{S}$%
\begin{equation}
\forall n\ge 1,\quad ||(L_{\delta }^{n}-L_{H}^{n})g||_{w}\leq \delta (C||g||_s +nD||g||_{w}).
\end{equation}
\item Moreover, let us suppose there exists $ n_{1}$ for which $A\lambda_1 ^{n_1} <1$ and $\lambda_2<1$ such that
\begin{equation}
\forall v\in \mathcal{S}, ||L_{\delta
}^{n_{1}}(v)||_{w}\leq \lambda _{2}||v||_{w}.
\end{equation}
\end{itemize}
Then for any $i\ge1$
\begin{equation*}
\left(
\begin{array}{c}
|| L^{i n_1} (f) ||_s \\
|| L^{i n_1} (f) ||_w
\end{array}
\right)
\preceq \left(
\begin{array}{cc}
A\lambda _{1}^{n_{1}} & B \\
\delta C & \delta n_{1}D+\lambda _{2}
\end{array}
\right)^i
\left(
\begin{array}{c}
|| f ||_s \\
|| f ||_w
\end{array}
\right).
\end{equation*}
In particular, as before, we have
\[
||L^{in_{1}}g||_s \leq (1/a)\rho ^{i}||g||_s,
\]
and
\[
||L^{in_{1}}g||_{w}\leq (1/b)\rho ^{i}||g||_s.
\]
\end{theorem}
The proof of the theorem is essentially the same as the one of theorem \ref{prop1},
we remark that the main difference between the two propositions is that now we are looking to the behavior of
iterates of $L_{H}$ and $L_{\delta }$ on the whole space $\mathcal{S}$ and not only on the
space of zero average measures $V$.
\section{The Ulam method}
We now give an example of a finite dimensional approximation for
the transfer operator which is useful in several cases: the Ulam method.
Let us briefly recall the basic notions. Let us suppose now that $X$ is a
manifold with boundary. The space $%
X $ is discretized by a partition $I_{\delta }$ (with $k$ elements) and the
transfer operator $L$ is approximated by a finite rank operator $L_{\delta }
$ defined in the following way: let $F_{\delta }$ be the $\sigma -$algebra
associated to the partition $I_{\delta }$, define the projection: $\pi
_{\delta }$ as $\pi _{\delta }(f)=\mathbf{E}(f|F_{\delta })$, then%
\begin{equation}
L_{\delta }(f):=\mathbf{\pi }_{\delta }L\pi _{\delta }f. \label{000}
\end{equation}
In the literature it is shown from different points of views, that taking
finer and finer partitions, in suitable systems, the behavior of this finite dimensional approximation
converges in some sense to the behavior of the original system, including the convergence of the spectral picture, see e.g. \cite{FCMP, BH, F08, L, B, MFG, B}.
We remark that this approximation procedure satisfies the approximation assumption (\ref{2})
if for example Bounded Variation and $L^{1}$ norms are considered, see Lemma \ref{lemp} .
This allows the effective use of this discretization in Theorems \ref{prop1} and \ref{prop2} for the sudy of piecewise expanding maps.
\section{Piecewise expanding maps and Lasota Yorke inequalities\label{appineq}}
Let us consider a class of maps which are locally expanding but they can be
discontinuous at some point. We recall some results, showing that these systems satisfy the assumptions needed in our main theorems, in particular the Lasota Yorke inequality.
\begin{definition}
A nonsingular function $T:([0,1],m)\rightarrow ([0,1],m)$ is said to be piecewise
expanding if
\begin{itemize}
\item There is a finite set of points $d_{1}=0,d_{2},...,d_{n}=1$ such that $%
T|_{(d_{i},d_{i+1})}$ is $C^{2}$.
\item $\inf_{x\in \lbrack 0,1]}|D_{x}T|=\lambda _{1}^{-1}>2$ on the set
where it is defined.
\end{itemize}
\end{definition}
It is now well known (see e.g. \cite{LY}) that this kind of maps have an
a.c.i.m. with bounded variation density.
\subsection{Lasota Yorke inequality}
Let us consider an absolutely continuous measure $\mu$ and the following norm
\begin{equation*}
||\mu ||_{BV}=\underset{\phi \in C^{1},|\phi |_{\infty }=1}{\sup |\mu (\phi
^{\prime })|}
\end{equation*}
this is related to the classical definition of bounded variation: it is
straightforward to see that if $\mu $ has density $f$ then
\footnote{ If on a interval $I$, $f\geq q$ then consider a function $\phi \in C^{1}$ which
is =-1 on the left of the interval and =1 on the right, and increasing inside, then $\phi^{\prime }\geq 0$ , $\int \phi ^{\prime }dx=2$ and $\int f\phi ^{\prime
}dx\geq 2q$ we can do similarly if $f\leq -q,$\ hence $||\mu ||\geq 2||f||_{\infty }$. The existence of such interval $I$ of course cannot be ensured in general. In this case the use of Lebesgue's density theorem allows to find an interval where the same argument applies up to a small error. }
$2||f||_{\infty }\leq ||\mu ||_{BV}$.
The following inequality can be established (see e.g. \cite{GN}, \cite{L2}) for the transfer operator of piecewise expanding maps.
\begin{proposition}\label{theo:LY}
\label{th8} If $T$ is piecewise expanding as above and $\mu $ is a measure
on $[0,1]$
\begin{equation*}
||L\mu ||_{BV}\leq \frac{2}{_{\inf T^{\prime }}}||\mu ||_{BV}+\frac{2}{\min
(d_{i}-d_{i+1})}\mu (1)+2\mu (|\frac{T^{\prime \prime }}{(T^{\prime })^{2}}%
|).
\end{equation*}
\end{proposition}
We remark that, if an inequality of the following form
\begin{equation*}
||Lg||\leq 2\lambda ||g||+B^{\prime }|g|_{w}
\end{equation*}%
is established (with $2\lambda <1$) then, iterating we have $%
||L^{2}g||\leq 2\lambda ||Lg||+B^{\prime }|g|_{w}=2\lambda (2\lambda
||Lg||+B^{\prime }|g|_{w})+B^{\prime }|g|_{w}...$ and thus%
\begin{equation*}
||L^{n}g||\leq 2^{n}\lambda ^{n}||Lg||+\frac{B^{\prime }}{1-2\lambda }%
|g|_{w}.
\end{equation*}
\subsection{Piecewise expanding maps with holes}\label{holely}
Let us consider a piecewise expanding map with a hole, which is an interval $I$.
Let us see that if the system without hole satisfies a Lasota Yorke inequality and is sufficiently expanding, we can
deduce a Lasota Yorke inequality for the system with an hole.
Let indeed consider a starting measure with bounded variation density, $f$.
If the piecewise expanding map without hole satisfies
\begin{equation*}
||Lg||_{BV}\leq 2\lambda ||g||_{BV}+B^{\prime }||g||_{1},
\end{equation*}
then the action of the hole is to multiply $Lg$ by $1_{I^{c}}$ hence
introducing two new jumps which we can bound by $||Lf||_{\infty }\leq \frac{1%
}{2}||Lf||_{BV}$, hence
\begin{equation*}
||L_{H}g||_{BV}\leq 2||Lg||_{BV}
\end{equation*}%
and the system will satisfy%
\begin{equation*}
||L_{H}g||_{BV}\leq 4\lambda ||g||_{BV}+2B^{\prime }||g||_{1}.
\end{equation*}
This is enough to obtain the Lasota Yorke inequality in interesting examples
of piecewise expanding maps with holes (see Section \ref{escexp} ).
We recall that considering the Ulam discretization for these systems, the needed approximation inequality follows again, by Lemma \ref{lemp}, and the assumptions of Section \ref{escape} applies.
\begin{remark}
We remark that for our method to be applied no conditions on the size of the hole are needed.
\end{remark}
\section{Numerical experiments}
\subsection{Convergence to equilibrium: Lanford map}\label{subsec:Lanford}
In this subsection we estimate the speed of convergence to equilibrium for the map which was
investigated in \cite{Lan}. The map $T:[0,1]\rightarrow \lbrack 0,1]$ is
given by
\begin{equation*}
T:x\mapsto 2x+\frac{1}{2}x(1-x)\quad (\text{mod }1).
\end{equation*}%
To apply Proposition \ref{theo:LY} we have to take into account its second iterate $F:=T^{2}$.
The data below refers to the map $F$ and to the discretization of its transfer operator,
and are some input and outputs of our algorithm.
\begin{center}
\begin{equation*}
\begin{array}{cc}
A=1 & \lambda_1\leq 0.32 \\
B\leq 30.6 & \delta=1/1048576 \\
\lambda_2<0.5 & n_1=18 \\
C \in [1.94,1.95] & D \in [70.992,70.993]
\end{array}
\end{equation*}
\end{center}
\begin{remark}
Using the estimates developed in \cite{GN}, an approximation $f_{\delta}$ for the invariant density $f$ can be computed with an explicit bound on the error. The graph of the map and the computed density are drawn in figure \ref{fig:lanford} and \ref{fig:lanfordinvariant} respectively. In the plotted case $||f-f_{\delta}||_1\leq 0.016$.
\end{remark}
\begin{figure}[!h]
\begin{subfigure}[b]{0.45\textwidth}
\input{lanfdynamic.tex}
\caption{Graph of the lanford map}
\label{fig:lanford}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\input{lanfdensity.tex}
\caption{The invariant density}
\label{fig:lanfordinvariant}
\end{subfigure}
\caption{The Lanford map}
\end{figure}
The matrix $M$ that corresponds to our data is such that
\[M \preceq \left[\begin{smallmatrix} 1.24\cdot 10^{-9} & 30.6\\
1.86\cdot 10^{-6} & 0.5013 \end{smallmatrix}\right].\]
Using equation \eqref{eq:rho} we can compute:
\[\rho \in [0.5013,0.5014].\]
\begin{remark}\label{rem:coeffab}
With a simple computation it is possible to see that the coefficients $a,b$ (see Theorem \ref{prop1}) associated to the leading eigenvalue $\rho$ are:
\begin{align*}
a&=\frac{A\lambda _{1}^{n_{1}}-\lambda_2-\delta n_1 D+\sqrt{(A\lambda
_{1}^{n_{1}}-\delta n_{1}D-\lambda _{2})^{2}+4\delta BC}}{A\lambda _{1}^{n_{1}}-\lambda_2-\delta n_1 D+2B+\sqrt{(A\lambda
_{1}^{n_{1}}-\delta n_{1}D-\lambda _{2})^{2}+4\delta BC}}\\& b=\frac{2B}{A\lambda _{1}^{n_{1}}-\lambda_2-\delta n_1 D+2B+\sqrt{(A\lambda
_{1}^{n_{1}}-\delta n_{1}D-\lambda _{2})^{2}+4\delta BC}}.
\end{align*}
\end{remark}
In this example, hence:
\[a\in [3.6,3.7]\cdot 10^{-6},\quad b\in [0.9999963,0.9999964]\]
Therefore,if $L$ is the transfer operator associated to $F=T^2$ and using \eqref{decayrate} and \eqref{decayrate_weak}, we have the following
estimates:
\begin{align*}
\Vert L^{k}g\Vert_{BV} &\leq (270839)\cdot (0.5014)
^{\left\lfloor \frac{k}{18}\right\rfloor }||g||_{BV}. \label{decayrate} \\
\Vert L^{k}g\Vert _{L^1} &\leq (30.7) \cdot (0.5014)
^{\left\lfloor \frac{k}{18}\right\rfloor }||g||_{BV}.
\end{align*}
We can also use the coefficients of the powers of the matrix (computed using interval arithmetics) to obtain upper bounds
as in the following table:
\begin{align*}
\begin{array}{ccc}
\textrm{iterations} & \textrm{bound for }||L^h g||_{BV} & \textrm{bound for }|| L^h g||_1 \\
h=36 & 5.665\cdot 10^{-5}||g||_{BV}+15.34||g||_1 & 9.279\cdot 10^{-7}||g||_{BV}+2.513\cdot 10^{-1}||g||_1 \\
h=72 & 1.424\cdot 10^{-5}||g||_{BV}+3.855||g||_1 & 2.333\cdot 10^{-7}||g||_{BV}+6.316\cdot 10^{-2}||g||_1 \\
h=108 & 3.578\cdot 10^{-6}||g||_{BV}+9.689\cdot 10^{-1}||g||_1 & 5.862\cdot 10^{-8}||g||_{BV}+1.588\cdot 10^{-2}||g||_1 \\
h=144 & 8.992\cdot 10^{-7}||g||_{BV}+2.436\cdot 10^{-1}||g||_1 & 1.474\cdot 10^{-8}||g||_{BV}+3.990\cdot 10^{-3}||g||_1 \\
\end{array}
\end{align*}
\subsection{Convergence to equilibrium: a Lorenz-type map}
In this subsection we estimate the speed of convergence to equilibrium for a Lorenz type $1$-dimensional map
using the recursive estimate established in section \ref{sec1}.
To do so, we use the estimates and the software developed in \cite{GN,GN2}.
The subject of our investigation is the $1$ dimensional Lorenz map acting on $I=[0,1]$ given by:
\[
T(x)=\left\{
\begin{array}{cc}
\theta \cdot |x-1/2|^{\alpha} & 0\leq x< 1/2 \\
1-\theta \cdot |x-1/2|^{\alpha} & 1/2 < x \leq 1
\end{array}
\right.
\]
with $\alpha=57/64$ and $\theta=109/64$.
Please note that since the Lorenz $1$-dimensional map does not have bounded derivative, the application of Proposition \ref{theo:LY},
is not immediate, but the following is true.
\begin{theorem}[\cite{GN2} Theorem 21]
Let $T$ be a $1$-dimensional piecewise expanding map, possibly with infinite derivative.
Denote by $\{d_i\}$ the set of the discontinuity points of $T$, increasingly ordered.
Fixed a parameter $l>0$ let \[I_l=\bigg\{x\in I \mid \bigg|\frac{T''(x)}{(T'(x))^2}\bigg|>l\bigg\}.\]
We have that $T$ satisfies a Lasota Yorke inequality
\[
||L\mu ||_{BV}\leq \lambda_1 ||f||_{BV}+B||f||_1.
\]
with
\[
\lambda_1\leq \frac{1}{2}\int_{I_l}\bigg|\frac{T''(x)}{(T'(x))^2}\bigg|dx+\frac{2}{\inf |T'|} \quad B\leq \frac{1}{1-\lambda_1}\bigg(\frac{2}{\min(d_{i+1}-d_i)}+l\bigg)
\]
(and in particular it is possible to choose $l$ in a way that $\lambda_1 < 1$ ).
\end{theorem}
We apply our strategy to the map $F:=T^4$, please remark that, once we have the Lasota-Yorke coefficients and the discretization, the approach
is the same as in Subsection \ref{subsec:Lanford}.
The data below refers to the map $F$ and to the discretization of its transfer operator,
and are some input and outputs of our algorithm; to compute the Lasota-Yorke constants we fixed $l=300$.
\begin{center}
\begin{equation}\label{tab:lorenz}
\begin{array}{cc}
A=1 & \lambda_1\leq 0.884\\
B\leq 4049 & \delta=1/2097152 \\
\lambda_2<0.002 & n_1=10 \\
C\in [16.24,16.25] & D\in [11677.3,11677.4]
\end{array}
\end{equation}
\end{center}
\begin{remark}
The graph of the map and its invariant density are drawn in figure \ref{fig:lorenz} and \ref{fig:lorenzinvariant} respectively.
Using the estimates developed in \cite{GN}, if $f$ is the density of the invariant measure and $f_{\delta}$ is the computed density,
we have in this case that $||f-f_{\delta}||_1\leq 0.047$.
\end{remark}
\begin{figure}[!h]
\begin{subfigure}[b]{0.45\textwidth}
\input{dynamic_lorenz.tex}
\caption{Graph of the dynamics}
\label{fig:lorenz}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\input{lorenz_invariant.tex}
\caption{The invariant density}
\label{fig:lorenzinvariant}
\end{subfigure}
\caption{Lorenz $1$-dimensional map}
\end{figure}
The matrix that corresponds to our data is such that
\[M \preceq \left[\begin{smallmatrix} 0.2915 & 4049\\
7.75\cdot 10^{-8} & 0.058 \end{smallmatrix}\right].\]
\begin{remark}
Please note that the Lorenz map we studied has some features that imply a large coefficient $B$ in the Lasota-Yorke inequality therefore
leading to a slower convergence to the equilibrium in the $||\ldotp||_{BV}$ norm.
\end{remark}
Using equation \eqref{eq:rho} we can compute:
\[\rho \in [0.386,0.387].\]
Using remark \ref{rem:coeffab} we have that the coefficients $a,b$ are such that:
\[a\in ([8.12,8.13]\cdot 10^{-5})\quad b\in [0.9999187,0.9999188]\]
Therefore,if $L$ is the transfer operator associated to $F=T^4$ and using \eqref{decayrate} and \eqref{decayrate_weak}, we have the following
estimates:
\begin{align*}
\Vert L^{k}g\Vert_{BV} &\leq (16356)\cdot (0.387)
^{\left\lfloor \frac{k}{10}\right\rfloor }||g||_{BV}. \label{decayrate} \\
\Vert L^{k}g\Vert _{L^1} &\leq (4050) \cdot (0.387)
^{\left\lfloor \frac{k}{10}\right\rfloor }||g||_{BV}.
\end{align*}
We can also use the coefficients of the powers of the matrix (computed using interval arithmetics) to obtain upper bounds
as in the following table:
\begin{align*}
\begin{array}{ccc}
\textrm{iterations} & \textrm{bound for }||L^h g||_{BV} & \textrm{bound for }|| L^h g||_1 \\
h=20 & 1.163\cdot 10^{-1}||g||_{BV}+1.414\cdot 10^{3}||g||_1 & 2.704\cdot 10^{-6}||g||_{BV}+3.469\cdot 10^{-2}||g||_1 \\
h=40 & 1.735\cdot 10^{-2}||g||_{BV}+2.134\cdot 10^{2}||g||_1 & 4.082\cdot 10^{-7}||g||_{BV}+5.025\cdot 10^{-3}||g||_1 \\
h=60 & 2.594\cdot 10^{-3}||g||_{BV}+31.92||g||_1 & 6.105\cdot 10^{-8}||g||_{BV}+7.513\cdot 10^{-4}||g||_1 \\
h=80 & 3.880\cdot 10^{-4}||g||_{BV}+4.774||g||_1 & 9.131\cdot 10^{-9}||g||_{BV}+1.124\cdot 10^{-4}||g||_1 \\
\end{array}
\end{align*}
\subsection{Escape rates}\label{escexp}
In this subsection we estimate the escape rates for a non markov map using the estimates developed in Section\ref{sec1}.
We will study:
\[T(x)=\frac{23}{5}x \quad \textrm{mod } 1,\]
with an hole of size $1/8$ centered in $1/2$.
For the system with holes, the Lasota-Yorke inequality has coefficients
\[A=1 \quad \lambda_1< 20/23 \quad B'\leq 7.08.\]
Below, some of the data (input and outputs) of our algorithm.
\begin{center}
\[
\begin{array}{cc}
A=1 & \lambda_1\leq 20/23\leq 0.87\\
B\leq 7.08 & \delta=1/65536\\
\lambda_2<0.5 & n_1=17 \\
C \in [14.384,14.385]& D \in [20.31,20.32]
\end{array}
\]
\end{center}
The graph of the map and a non rigorous estimation of the conditionally invariant density are drawn in figure \ref{fig:lin_nonmark} and \ref{fig:lin_nonmark_invariant} respectively.
\begin{figure}[!h]
\begin{subfigure}[b]{0.45\textwidth}
\input{dynamic_hole.tex}
\caption{Graph of the dynamics}
\label{fig:lin_nonmark}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\input{hole_invariant.tex}
\caption{The conditionally invariant density}
\label{fig:lin_nonmark_invariant}
\end{subfigure}
\caption{Piecewise linear non-markov map}
\end{figure}
The matrix that corresponds to our data is such that
\[M \preceq \left[\begin{smallmatrix} 0.094 & 7.08\\
2.1\cdot 10^{-4} & 0.506 \end{smallmatrix}\right].\]
Using equation \eqref{eq:rho} we can compute:
\[\rho \in [0.509,0.5091].\]
Using remark \ref{rem:coeffab} we have that the coefficients $a,b$ are such that:
\[a\in ([5.282,5.283]\cdot 10^{-4})\quad b\in [0.9994717,0.9994718]\]
Therefore,if $L$ is the transfer operator associated to $T$ and using \eqref{decayrate} and \eqref{decayrate_weak}, we have the following
estimates:
\begin{align*}
\Vert L^{k}g\Vert_{BV} &\leq (1900.2)\cdot (0.5091)
^{\left\lfloor \frac{k}{17}\right\rfloor }||g||_{BV}. \label{decayrate} \\
\Vert L^{k}g\Vert _{L^1} &\leq (7.09) \cdot (0.5091)
^{\left\lfloor \frac{k}{17}\right\rfloor }||g||_{BV}.
\end{align*}
We can also use the coefficients of the powers of the matrix (computed using interval arithmetics) to obtain upper bounds
as in the following table:
\begin{align*}
\begin{array}{ccc}
\textrm{iterations} & \textrm{bound for }||L^h g||_{BV} & \textrm{bound for }|| L^h g||_1 \\
h=34 & 1.034\cdot 10^{-2}||g||_{BV}+4.241||g||_1 & 1.315\cdot 10^{-4}||g||_{BV}+2.569\cdot 10^{-1}||g||_1\\
h=68 & 6.645\cdot 10^{-4}||g||_{BV}+1.134||g||_1 & 3.513\cdot 10^{-5}||g||_{BV}+6.654\cdot 10^{-2}||g||_1\\
h=102 & 1.559\cdot 10^{-4}||g||_{BV}+2.939\cdot 10^{-1}||g||_1 & 9.111\cdot 10^{-6}||g||_{BV}+1.724\cdot 10^{-2}||g||_1\\
h=136 & 4.025\cdot 10^{-5}||g||_{BV}+7.615\cdot 10^{-2}||g||_1 & 2.361\cdot 10^{-6}||g||_{BV}+4.467\cdot 10^{-3}||g||_1
\end{array}
\end{align*}
\section{Appendix: The approximation inequality from the Lasota Yorke one.\label{appp}}
In this section we see that the approximation inequality (\ref{2} ) directly follows
from the Lasota Yorke inequality and from natural assumptions on the
approximating operator (including the Ulam discretization, which is used in the experiments).
Hence the approximation inequality (\ref{2} ) that is assumed in the paper is a natural assumption in a general class of system, and approximations.
\begin{lemma}
\label{lemp} Suppose $L$ satisfy a Lasota Yorke inequality (\ref{1}) and $L_{\delta } $ is defined by composing with a ''projection'' $\pi_\delta $ satisfying a certain approximation inequality:
\begin{itemize}
\item $L_{\delta }=\pi _{\delta }L\pi _{\delta }$ with $||\pi _{\delta
}v-v||_{w}\leq \delta ||v||_s$ for all $v\in \mathcal{S}$
\item $\pi _{\delta }$ and $L$ are weak contractions for the norm $||~||_{w}$
\end{itemize}
then $\forall f\in\mathcal{S}$%
\begin{equation*}
||L^{n}f-L_{\delta }^{n}f||_{w}\leq \delta \frac{(A\lambda _{1}+1)A}{%
1-\lambda _{1}}||f||_s+\delta Bn(A\lambda _{1}+2)||f||_{w}
\end{equation*}
\end{lemma}
\begin{proof}
It holds
\begin{equation*}
||(L-L_{\delta })f||_{w}\leq ||\pi _{\delta }L\pi _{\delta }f-\mathbf{\pi }%
_{\delta }Lf||_{w}+||\mathbf{\pi }_{\delta }Lf-Lf||_{w},
\end{equation*}%
but
\begin{equation*}
\mathbf{\pi }_{\delta }L\pi _{\delta }f-\mathbf{\pi }_{\delta }Lf=\mathbf{%
\pi }_{\delta }L(\pi _{\delta }f-f).
\end{equation*}%
Since both $\pi _{\delta }$ and $L$ are weak contractions and since $||\pi
_{\delta }v-v||_{w}\leq \delta ||v||_s$
\begin{equation*}
||\mathbf{\pi }_{\delta }L(\pi _{\delta }f-f)||_{w}\leq ||\pi _{\delta
}f-f||_{w}\leq \delta ||f||_s.
\end{equation*}
On the other hand%
\begin{equation*}
||\mathbf{\pi }_{\delta }Lf-Lf||_{w}\leq \delta ||Lf||_s\leq \delta (A\lambda
_{1}||f||_s+B||f||_{w})
\end{equation*}
which gives
\begin{equation}
||(L-L_{\delta })f||_{w}\leq \delta (A\lambda _{1}+1)||f||_s+\delta B||f||_{w}
\label{1iter}
\end{equation}
Now let us consider $(L_{\delta }^{n}-L^{n})f$. It holds%
\begin{eqnarray*}
||(L_{\delta }^{n}-L^{n})f||_{w} &\leq &\sum_{k=1}^{n}||L_{\delta
}^{n-k}(L_{\delta }-L)L^{k-1}f||_{w}\leq \sum_{k=1}^{n}||(L_{\delta
}-L)L^{k-1}f||_{w} \\
&\leq &\sum_{k=1}^{n}\delta (A\lambda _{1}+1)||L^{k-1}f||_s+\delta
B||L^{k-1}f||_{w} \\
&\leq &\delta \sum_{k=1}^{n}(A\lambda _{1}+1)(A\lambda
_{1}^{k-1}||f||_s+B||f||_{w})+B||f||_{w} \\
&\leq &\delta \frac{(A\lambda _{1}+1)A}{1-\lambda _{1}}||f||_s+\delta
Bn(A\lambda _{1}+2)||f||_{w}.
\end{eqnarray*}
\end{proof}
\begin{remark}
From the above Lemma we have in this case that $C$ and $D$ in \eqref{kkk} are given by
\[ C=\frac{(A\lambda _{1}+1)A}{1-\lambda _{1}},\quad D=B\cdot(A\lambda_{1}+2).\]
\end{remark}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,053 |
Q: need to compute change in time between start time and end time.(Python) I need to write a program that accepts a start time and end time and computes the change between them in minutes. For example, the start time is 4:30 PM and end time is 9:15 PM then the change in time is 285 min. How do I accomplish this in Python? I only need to compute for a 24 hour period
A: Here's what you can do:
from datetime import datetime
def compute_time(start_time, end_time):
start_datetime = datetime.strptime(start_time, '%I:%M %p')
end_datetime = datetime.strptime(end_time, '%I:%M %p')
return (start_datetime - end_datetime).seconds / 60
print compute_time('9:15 PM', '4:30 PM')
prints 285.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,452 |
#include "webrtc/video/encoder_state_feedback.h"
#include "webrtc/base/checks.h"
#include "webrtc/video/vie_encoder.h"
static const int kMinKeyFrameRequestIntervalMs = 300;
namespace webrtc {
EncoderStateFeedback::EncoderStateFeedback(Clock* clock,
const std::vector<uint32_t>& ssrcs,
ViEEncoder* encoder)
: clock_(clock),
ssrcs_(ssrcs),
vie_encoder_(encoder),
time_last_intra_request_ms_(ssrcs.size(), -1) {
RTC_DCHECK(!ssrcs.empty());
}
bool EncoderStateFeedback::HasSsrc(uint32_t ssrc) {
for (uint32_t registered_ssrc : ssrcs_) {
if (registered_ssrc == ssrc)
return true;
}
return false;
}
size_t EncoderStateFeedback::GetStreamIndex(uint32_t ssrc) {
for (size_t i = 0; i < ssrcs_.size(); ++i) {
if (ssrcs_[i] == ssrc)
return i;
}
RTC_NOTREACHED() << "Unknown ssrc " << ssrc;
return 0;
}
void EncoderStateFeedback::OnReceivedIntraFrameRequest(uint32_t ssrc) {
if (!HasSsrc(ssrc))
return;
size_t index = GetStreamIndex(ssrc);
{
int64_t now_ms = clock_->TimeInMilliseconds();
rtc::CritScope lock(&crit_);
if (time_last_intra_request_ms_[index] + kMinKeyFrameRequestIntervalMs >
now_ms) {
return;
}
time_last_intra_request_ms_[index] = now_ms;
}
vie_encoder_->OnReceivedIntraFrameRequest(index);
}
void EncoderStateFeedback::OnReceivedSLI(uint32_t ssrc, uint8_t picture_id) {
if (!HasSsrc(ssrc))
return;
vie_encoder_->OnReceivedSLI(picture_id);
}
void EncoderStateFeedback::OnReceivedRPSI(uint32_t ssrc, uint64_t picture_id) {
if (!HasSsrc(ssrc))
return;
vie_encoder_->OnReceivedRPSI(picture_id);
}
// Sending SSRCs for this encoder should never change since they are configured
// once and not reconfigured, however, OnLocalSsrcChanged is called when the
// RtpModules are created with a different SSRC than what will be used in the
// end.
// TODO(perkj): Can we make sure the RTP module is created with the right SSRC
// from the beginning so this method is not triggered during creation ?
void EncoderStateFeedback::OnLocalSsrcChanged(uint32_t old_ssrc,
uint32_t new_ssrc) {
if (!RTC_DCHECK_IS_ON)
return;
if (old_ssrc == 0) // old_ssrc == 0 during creation.
return;
// SSRC shouldn't change to something we haven't already registered with the
// encoder.
RTC_DCHECK(HasSsrc(new_ssrc));
}
} // namespace webrtc
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,334 |
{"url":"https:\/\/gmatclub.com\/forum\/how-many-integers-are-there-between-c-and-d-1-neither-c-115561.html","text":"GMAT Question of the Day - Daily to your Mailbox; hard ones only\n\n It is currently 18 Jun 2018, 18:05\n\n### GMAT Club Daily Prep\n\n#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\n\nevery week, we\u2019ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\n# Events & Promotions\n\n###### Events & Promotions in June\nOpen Detailed Calendar\n\n# How many integers are there between C and D? (1) Neither c\n\nAuthor Message\nTAGS:\n\n### Hide Tags\n\nManager\nJoined: 18 Jun 2011\nPosts: 54\nHow many integers are there between C and D? (1) Neither c\u00a0[#permalink]\n\n### Show Tags\n\n18 Jun 2011, 04:23\n2\n11\n00:00\n\nDifficulty:\n\n45% (medium)\n\nQuestion Stats:\n\n55% (00:52) correct 46% (00:43) wrong based on 400 sessions\n\n### HideShow timer Statistics\n\nHow many integers are there between C and D?\n\n(1) Neither c nor d is an integer.\n\n(2) c \u2013 d = 3\nMath Forum Moderator\nJoined: 20 Dec 2010\nPosts: 1901\nRe: D.S - How many Integers?\u00a0[#permalink]\n\n### Show Tags\n\n18 Jun 2011, 04:47\n1\nguygmat wrote:\nHow many integers are there between C and D?\n\n(1) Neither c nor d is an integer.\n\n(2) c \u2013 d = 3\n\nI got the hint from statement 1 and chose C. If it was not written, perhaps I would have missed.\n\n(2)\nc=5.1, d=2.1; c-d=3; (3,4,5)-- 3 integers between them\nc=5, d=2; c-d=3; (3,4)-- 2 integers between them\nInsufficient.\n\n(1)\nClearly insufficient.\nc=1.1 d=1000000.1\nc=1.1 d=1.1\n\nCombining both;\nc=5.1, d=2.1; c-d=3; (3,4,5)-- 3 integers between them\nOR\nc=55.00001, d=52.00001; c-d=3; (53,54,55)-- 3 integers between them\nAlways 3 integers.\n\nSufficient.\n\nAns: \"C\"\n_________________\nIntern\nJoined: 29 Jun 2011\nPosts: 12\nLocation: Ireland\nRe: D.S - How many Integers?\u00a0[#permalink]\n\n### Show Tags\n\n29 Jun 2011, 09:24\n1\n2\nThe best way to answer these questions is to go through both statements individually and mark them as sufficient or insufficient. Then, only if both are marked as insufficient, you check if the two statements combined are sufficient.\n\nStep 1: Statement 1 tells us neither C nor D is an integer.\nThis is clearly insufficient as it gives us no indication of how many integers are in between C and D.\nInsufficient.\n\nStep 2: Statement 2: C - D = 3.\nIf C and D were both integers (e.g. C = 4 and D = 7), there would be only 2 integers in between them while there would be 3 integers between C = 4.5 and D = 7.5.\nInsufficient.\n\nStep 3: Since both have been marked as insufficient, we now check them combined.\nC - D = 3. C and D are not integers.\nSufficient.\n_________________\n\nhttp:\/\/www.testprepdublin.com\n\nFor the best GMAT, GRE, and SAT preparation.\n\nManager\nJoined: 21 Sep 2008\nPosts: 184\nConcentration: Strategy, Economics\nGMAT Date: 07-17-2015\nGPA: 3.57\nRe: D.S - How many Integers?\u00a0[#permalink]\n\n### Show Tags\n\n29 Jun 2011, 09:51\nQuestion - Since C-D = 3, doesn't that signify that there are indeed 3 integers or whole numbers between C and D? My reasoning is that an integer is a whole number so if the difference is 3 and not, say, 3.34534, there are clearly 3 whole numbers or integers between C and D.\n\nCan someone please tell me where my thinking is off because it clearly is-- am I supposed to be imagining the number line and thinking of an integer as a whole number on the line or..\n_________________\n\nLife with the GMAT:\n\nJerome: Ben, c'est 20 secondes de plus qu'hier sur le meme parcours! C'etait bien le meme parcours la, non?!\nGigi: Mais t'enerve pas, Jerome, je crois que t'as accroche une porte.\nJerome: *$&#(*%&(*#%& Intern Joined: 29 Jun 2011 Posts: 12 Location: Ireland Re: D.S - How many Integers? [#permalink] ### Show Tags 29 Jun 2011, 10:06 1 FatRiverPuff wrote: Question - Since C-D = 3, doesn't that signify that there are indeed 3 integers or whole numbers between C and D? My reasoning is that an integer is a whole number so if the difference is 3 and not, say, 3.34534, there are clearly 3 whole numbers or integers between C and D. Can someone please tell me where my thinking is off because it clearly is-- am I supposed to be imagining the number line and thinking of an integer as a whole number on the line or.. Hi, imagine c=4.2 and d = 7.2. c and d are not integers but their difference is exactly 3. Now count the integers between them. 5, 6 and 7 so 3 integers for non-integer c and d. Now imagine c= 1 and d=4. c and d are integers and their difference is exactly 3. But counting the integers between them (2 and 3) we only get 2 for integer c and d. Is that clearer? _________________ http:\/\/www.testprepdublin.com For the best GMAT, GRE, and SAT preparation. Director Joined: 01 Feb 2011 Posts: 686 Re: D.S - How many Integers? [#permalink] ### Show Tags 29 Jun 2011, 17:47 1. Not sufficient C D 1\/2 5\/2 - 2 integers in between 5\/2 11\/2 - 3 integers in between 2. Not sufficient C D 0 3 - 2 integers in between 1\/2 7\/2 - 3 integers in between Together, Sufficient. C D 1\/2 7\/2 - 3 integers. Answer is C. Manager Joined: 21 Sep 2008 Posts: 184 Concentration: Strategy, Economics GMAT Date: 07-17-2015 GPA: 3.57 Re: D.S - How many Integers? [#permalink] ### Show Tags 29 Jun 2011, 20:49 testprepDublin wrote: FatRiverPuff wrote: Question - Since C-D = 3, doesn't that signify that there are indeed 3 integers or whole numbers between C and D? My reasoning is that an integer is a whole number so if the difference is 3 and not, say, 3.34534, there are clearly 3 whole numbers or integers between C and D. Can someone please tell me where my thinking is off because it clearly is-- am I supposed to be imagining the number line and thinking of an integer as a whole number on the line or.. Hi, imagine c=4.2 and d = 7.2. c and d are not integers but their difference is exactly 3. Now count the integers between them. 5, 6 and 7 so 3 integers for non-integer c and d. Now imagine c= 1 and d=4. c and d are integers and their difference is exactly 3. But counting the integers between them (2 and 3) we only get 2 for integer c and d. Is that clearer? Got it, thanks so much. I was reading way too much into the question _________________ Life with the GMAT: Jerome: Ben, c'est 20 secondes de plus qu'hier sur le meme parcours! C'etait bien le meme parcours la, non?! Gigi: Mais t'enerve pas, Jerome, je crois que t'as accroche une porte. Jerome: *$&#(*%&(*#%&\n\nIntern\nJoined: 16 Jan 2013\nPosts: 30\nConcentration: Finance, Entrepreneurship\nGMAT Date: 08-25-2013\nRe: How many integers are there between C and D? (1) Neither c\u00a0[#permalink]\n\n### Show Tags\n\n28 May 2013, 12:18\nWhat will be the situation if we take one of the values to be -ve ?\nVP\nStatus: Far, far away!\nJoined: 02 Sep 2012\nPosts: 1099\nLocation: Italy\nConcentration: Finance, Entrepreneurship\nGPA: 3.8\nRe: How many integers are there between C and D? (1) Neither c\u00a0[#permalink]\n\n### Show Tags\n\n28 May 2013, 12:22\ndhirsinha wrote:\nWhat will be the situation if we take one of the values to be -ve ?\n\nConsider the case $$c=1.5$$ and $$d=-1.5$$\n\n$$c-d=3$$, c and d no integers\n\nand the number of integers between c and d would be $$3$$ again ($$1.5,1,0,-1,-1.5$$)\n_________________\n\nIt is beyond a doubt that all our knowledge that begins with experience.\n\nKant , Critique of Pure Reason\n\nTips and tricks: Inequalities , Mixture | Review: MGMAT workshop\nStrategy: SmartGMAT v1.0 | Questions: Verbal challenge SC I-II- CR New SC set out !! , My Quant\n\nRules for Posting in the Verbal Forum - Rules for Posting in the Quant Forum[\/size][\/color][\/b]\n\nIntern\nJoined: 16 Jan 2013\nPosts: 30\nConcentration: Finance, Entrepreneurship\nGMAT Date: 08-25-2013\nRe: How many integers are there between C and D? (1) Neither c\u00a0[#permalink]\n\n### Show Tags\n\n28 May 2013, 12:30\n[quote=\"Zarrolou\"] Thanks for the feedback\nMath Expert\nJoined: 02 Sep 2009\nPosts: 46129\nRe: How many integers are there between C and D? (1) Neither c\u00a0[#permalink]\n\n### Show Tags\n\n28 May 2013, 12:46\nguygmat wrote:\nHow many integers are there between C and D?\n\n(1) Neither c nor d is an integer.\n\n(2) c \u2013 d = 3\n\nSimilar questions to practice:\nhow-many-integers-n-are-there-such-that-1-5n-139474.html (OG13)\nhow-many-integers-n-are-there-such-that-r-n-s-131146.html\nhow-many-integers-are-there-such-that-v-n-w-129065.html\nhow-many-integers-n-are-there-such-that-r-n-s-101917.html\n\nHope it helps.\n_________________\nNon-Human User\nJoined: 09 Sep 2013\nPosts: 6995\nRe: How many integers are there between C and D? (1) Neither c\u00a0[#permalink]\n\n### Show Tags\n\n07 May 2018, 06:15\nHello from the GMAT Club BumpBot!\n\nThanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).\n\nWant to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.\n_________________\nRe: How many integers are there between C and D? 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