text:
\begin{document}
\maketitle
\begin{abstract}
	We classify $^*$-homomorphisms from nuclear $C^*$-algebras into uniform tracial sequence algebras of nuclear $\mathcal Z$-stable $C^*$-algebras via tracial data.
\end{abstract}

\renewcommand*{\thetheorem}{\Alph{theorem}}
\section*{Introduction}

Over the last 10 years, the application of von Neumann techniques has been a major theme in the structure theory of simple nuclear $C^*$-algebras through the pioneering work of Matui and Sato \cite{MS12,MS14}.  A starting point for a number of these applications is the following well-known consequence of Connes' revolutionary work on the characterisation of hyperfinite von Neumann algebras \cite{Co76}: maps from a separable nuclear $C^*$-algebra $A$  into $\mathcal R^\omega$ (the ultrapower of the hyperfinite II$_1$ factor) are classified up to unitary equivalence by the trace they induce on $A$ (see for example \cite[Proposition 2.1]{CGNN13}).  Most recently, this result played a key role in Schafhauser's breakthrough new approach to the classification of monotracial separable nuclear $C^*$-algebras which absorb the universal uniformly hyperfinite algebra tensorially (\cite{Sch:Ann}).

For a nuclear $C^*$-algebra $B$ with a unique trace, Connes' theorem allows us to view $\mathcal R^\omega$ as a tracial ultrapower of $B$.  When $B$ has multiple traces a somewhat different reduced product construction is needed in order to be able to handle them all uniformly. This led to the uniform tracial ultrapower $B^\omega$,
formalised in \cite{CETWW}. This, and its precursor in terms of ultraproducts of $W^*$-bundles (\cite{BBSTWW,Oz13}), has been a crucial tool in recent developments.  In particular, in our recent joint work with Winter (\cite{CETWW}), we introduced a new tool --- complemented partitions of unity (CPoU) --- for studying these ultraproducts, and used this to show that Jiang--Su stability and finite nuclear dimension are equivalent in the Toms--Winter conjecture (\cite[Theorem A]{CETWW}).

The principal purpose of this paper is to use the new techniques in \cite{CETWW} to classify maps from separable nuclear $C^*$-algebras into uniform tracial sequence algebras of $\mathcal Z$-stable nuclear $C^*$-algebras, analogous to the consequence of Connes' result for maps into $\mathcal R^\omega$. In the theorem which follows, the uniform tracial sequence algebra $B^\infty$ associated to $B$ is the $C^*$-algebra of bounded sequences in $B$, modulo those converging to zero uniformly over all trace norms. So classifying maps from $A$ to $B^\infty$ up to unitary equivalence is a way of encoding a classification of uniform trace norm approximately multiplicative maps from $A$ into $B$ up to approximate unitary equivalence in uniform trace norm.

\begin{theorem}\label{thm:MainThm}
Let $A$ be a separable nuclear $C^*$-algebra and let $B$ be a separable nuclear $\mathcal Z$-stable $C^*$-algebra with $T(B)$ compact and non-empty.
Then for any continuous affine function $\alpha:T(B^\infty) \to T(A)$, there exists a $^*$-ho\-mo\-mor\-phism $\phi:A \to B^\infty$ which induces $\alpha$.
Moreover, $\phi$ is unique up to unitary equivalence.
\end{theorem}

 Just as the classification of embeddings from separable nuclear $C^*$-algebras into $\mathcal R^\omega$ is vital in \cite{Sch:Ann}, Theorem \ref{thm:MainThm} will form the starting point of the forthcoming joint work of Carri\'on, Gabe, Schafhauser and the last two named authors which will give an abstract approach to the classification of simple separable unital nuclear Jiang--Su stable $C^*$-algebras satisfying the UCT (\cite{CGSTW}).  Our reason for setting up Theorem \ref{thm:MainThm} with the uniform trace norm sequence algebra $B^\infty$ as opposed to the uniform tracial ultrapower $B^\omega$ is so it can be applied exactly as written in \cite{CGSTW}.\footnote{One can obtain an ultrapower version of Theorem \ref{thm:MainThm} by working with an ultrapower $B^\omega$ in place of the sequence algebra $B^\infty$ throughout the paper.}

The role of nuclearity and $\mathcal Z$-stability of $B$ in Theorem \ref{thm:MainThm} is to obtain CPoU from \cite[Theorem I]{CETWW} --- the main technical result of that work.  Although the defining property of CPoU is in the form of the existence of certain partitions of unity for trace spaces, its principal consequence is a local to global tracial type-satisfaction process for $B^\infty$. In less fancy language, this means that if (suitable) properties hold approximately in trace in each tracial GNS-representation of $B^\infty$, then they hold exactly in $B^\infty$.  This gives $B^\infty$ a von Neumann algebra-like flavour (though it is certainly not a von Neumann algebra).  Theorem \ref{thm:MainThm} is obtained in this fashion: we glue together the classification of maps from separable nuclear $C^*$-algebras into finite von Neumann algebras from Connes' theorem over all traces using CPoU.

We also record in Proposition \ref{unitaryexp}, another, somewhat easier, application of CPoU for use in \cite{CGSTW}, which showcases another von Neumann algebra-like property: every unitary in $B^\infty$ is an exponential.

\subsection*{Acknowledgements} The authors would like to thank the referee for their careful reading of earlier versions of this paper and helpful comments.

\numberwithin{theorem}{section}
\section{Preliminaries}

Let $B$ be a $C^*$-algebra.
We let $T(B)$ denote the set of tracial states (which we abbreviate as ``traces'') on $B$.
For $\tau \in T(B)$, we define the associated 2-seminorm on $B$ by
\begin{equation} \|b\|_{2,\tau} := \sqrt{\tau(|b|^2)}. \end{equation}
We let $\pi_\tau:B \to \mathcal B(\mathcal H_\tau)$ be the GNS representation associated to $\tau$, and continue to use $\|\cdot\|_{2,\tau}$ to denote the induced 2-norm on $\pi_\tau(B)''$.

Define the uniform tracial sequence algebra
\begin{equation} B^\infty:=\ell^\infty(B)/\{(b_n)_{n=1}^\infty: \lim_{n\to\infty} \sup_{\tau \in T(B)} \|b_n\|_{2,\tau}=0\}. \end{equation}
We will typically use representative sequences in $\ell^\infty(B)$ to denote elements of $B^\infty$.

The ultraproduct versions of these sequence algebras are obtained using a free ultrafilter $\omega\in \beta\mathbb N\setminus\mathbb N$ in place of $\infty$, and many of their basic properties are the same.  For example, when $B$ is separable and $T(B)$ is non-empty and compact, $B^\infty$ is unital, with the unit represented by an approximate unit $(e_n)_{n=1}^\infty$ for $B$ in just the same way as the ultrapower version of this result (\cite[Proposition 1.11]{CETWW}).

Given a sequence $(\tau_n)_{n=1}^\infty$ in $T(B)$ and a free ultrafilter $\omega$, define the associated \emph{limit trace} $\tau:B^\infty \to \mathbb C$ by
\begin{equation} \tau((b_n)_{n=1}^\infty) := \lim_{n\to\omega} \tau_n(b_n). \end{equation}
We let $T_\infty(B)$ denote the set of all limit traces on $B^\infty$.

On $B^\infty$, one has a uniform $2$-norm\footnote{To see it is a norm, use \eqref{eq:2uNormFormula}.} $\|\cdot\|_{2,T_\infty(B)}$ given by
\begin{equation} \|b\|_{2,T_\infty(B)}:=\sup_{\tau \in T_\infty(B)} \|b\|_{2,\tau},\quad b\in B^\infty. \end{equation}
More explicitly, for $b=(b_n)_{n=1}^\infty \in B^\infty$, one has
\begin{equation} \label{eq:2uNormFormula}
\|b\|_{2,T_\infty(B)}=\limsup_n \sup_{\tau \in T(B)} \|b_n\|_{2,\tau}. \end{equation}

In this paper we will frequently use Kirchberg's $\e$-test, which appears as \cite[Lemma A.1]{Kir06}.
However, we need a slightly different version, as we work with ``sequence algebras'' rather than ultrapowers; we state here the version we need, and remark that the proof is nearly identical to that of \cite[Lemma A.1]{Kir06}.

\begin{lemma}[{Kirchberg's $\e$-test, (\cite{Kir06})}]
\label{lem:EpsTest}
	Let $(X_n)_{n=1}^\infty$ be a sequence of non-empty sets and for $k,n\in\mathbb N$, let $f^{(k)}_n:X_n\rightarrow [0,\infty]$ be a function.  Define functions $f^{(k)}:X_1\times X_2 \times \cdots \to [0,\infty]$ by
	\begin{equation}
	\label{eq:EpsTest1}
	f^{(k)}(x_1,x_2,\dots) := \limsup_{n} f^{(k)}_n(x_n). \end{equation}
	If for every $\epsilon>0$ and $k_0\in\mathbb N$ there exists $x\in X_1\times X_2 \times \cdots$ such that $f^{(k)}(x)<\epsilon$ for $k=1,\dots,k_0$, then there exists $y\in X_1 \times X_2 \times \cdots$ such that $f^{(k)}(y)=0$ for all $k\in\mathbb N$.
\end{lemma}

We next record a useful fact, which is a direct consequence of Cuntz and Pedersen's investigation of traces on $C^*$-algebras (\cite[Proposition 2.7]{CP79}) and the fact that the weak$^*$-continuous functionals on the dual of a Banach space correspond precisely to elements of the Banach space.

\begin{proposition}[{Cuntz--Pedersen}]
	\label{prop:CP}
	Let $A$ be a $C^*$-algebra and let $f:T(A) \to \mathbb R$ be a continuous affine function.
	Then there is a self-adjoint element $a \in A$ such that
	\begin{equation} \tau(a)=f(\tau),\quad \tau \in T(A). \end{equation}
\end{proposition}

In this paper, we will be using CPoU as a property of the uniform tracial sequence algebra $B^\infty$. In \cite[Definition 3.1]{CETWW}, CPoU was originally defined as a property of the uniform tracial ultrapower $B^\omega$, but standard methods allow it to be rephrased as a local property of $B$ instead: see \cite[Proposition 3.2]{CETWW}.
The same methods allow it to be rephrased as a property of $B^\infty$, analogous to the original definition for $B^\omega$, as recorded in the lemma below. For the purposes of this paper, one can take this as the definition of CPoU.

\begin{lemma}
\label{lem:CPoU-Infinity}
Let $B$ be a separable $C^*$-algebra with $T(B)$ non-empty and compact.
If $B$ has CPoU, then for any $\|\cdot\|_{2,T_\infty(B)}$-separable subset $S$ of $B^\infty$, any $\delta>0$, and any $a_1,\dots,a_k \in (B^\infty)_+$ satisfying
\begin{equation} \min\{\tau(a_1),\dots,\tau(a_k)\} < \delta, \quad \tau \in T_\infty(B), \end{equation}
there exist orthogonal projections $e_1,\dots,e_k \in B^\infty \cap S'$ which sum to $1_{B^\infty}$, such that
\begin{equation} \tau(a_ie_i) \leq \delta\tau(e_i), \quad i=1,\dots,k,\ \tau \in T_\infty(B). \end{equation}
\end{lemma}

We will access CPoU through one of the main technical results of \cite{CETWW}, which we recall below. Note that for unital $C^*$-algebras the tracial state space $T(B)$ is automatically compact.

\begin{theorem}
\label{thm:CPoU}
\cite[Theorem I]{CETWW}
Let $B$ be a separable, nuclear, $\mathcal Z$-stable $C^*$-algebra with $T(B)$ compact and non-empty.
Then $B$ has CPoU.
\end{theorem}

\section{Results}

We start by recording an application of CPoU regarding unitaries in uniform tracial sequence algebras of $\mathcal Z$-stable nuclear $C^*$-algebras for use in \cite{CGSTW}.

\begin{proposition}\label{unitaryexp}
Let $B$ be a separable  $C^*$-algebra with CPoU and $T(B)$ compact and non-empty. Let $S$ be a $\|\cdot\|_{2,T_\infty(B)}$-separable subset of $B^\infty$ closed under taking adjoints.
Then every unitary $u \in B^\infty \cap S'$ can be written as an exponential $u = e^{\pi i h}$ for some self-adjoint $h \in B^\infty \cap S'$ of norm at most $1$.
In particular this holds whenever $B$ is separable, unital, nuclear and $\mathcal Z$-stable with $T(B)\neq\emptyset$.
\end{proposition}
\begin{proof}
The final sentence of the proposition follows from the rest by Theorem \ref{thm:CPoU}.   Fix $\e>0$.
By Kirchberg's $\e$-test (Lemma \ref{lem:EpsTest}),\footnote{Since $S$ is $\|\cdot\|_{2,T_\infty(B)}$-separable, testing that a sequence $(b_n) \subset \ell^\infty(B)$ represents an element of $B^\infty \cap S'$ requires only countably many constraints. Indeed, $B^\infty \cap S' = B^\infty \cap S_0'$ for any countable $\|\cdot\|_{2,T_\infty(B)}$-dense subset $S_0 \subset S$, as multiplication is jointly $\|\cdot\|_{2,T_\infty(B)}$-continuous on $\|\cdot\|$-bounded sets.} it suffices to prove that there exists a self-adjoint $h \in B^\infty \cap S'$ of norm at most $1$ such that
\begin{equation} \|u-e^{\pi i h}\|_{2,T_\infty(B)} \leq \e. \end{equation}
For each $\tau \in T(B^\infty)$, using Borel functional calculus, there exists a self-adjoint $x \in \pi_\tau(B^\infty \cap S')''$ of norm at most $1$ such that $\pi_\tau(u)=e^{\pi i x}$.
By the Kaplansky density theorem, we may approximate $x$ by a self-adjoint contraction in $\pi_\tau(B^\infty \cap S')$, which can then be lifted to a self-adjoint contraction $h_\tau \in B^\infty \cap S'$ such that
\begin{equation} \|u-e^{\pi i h_\tau}\|_{2,\tau} < \e. \end{equation}
Set $a_\tau:=|u-e^{\pi i h_\tau}|^2 \in (B^\infty)_+$, so that $\tau(a_\tau)<\e^2$.
By continuity and compactness, there exist $\tau_1,\dots,\tau_k \in T(B^\infty)$ such that for every $\tau \in T(B^\infty)$,
\begin{equation} \min\{\tau(a_{\tau_1}),\dots,\tau(a_{\tau_k})\} < \e^2. \end{equation}
Using CPoU as in Lemma \ref{lem:CPoU-Infinity}, there exists a partition of unity consisting of projections $p_1,\dots,p_k \in B^\infty \cap \{u,h_{\tau_1},\dots,h_{\tau_k}\}' \cap S'$ such that
\begin{equation} \tau(p_ja_{\tau_j}) \leq \e^2\tau(p_j), \quad \tau \in T_\infty(B),\ j=1,\dots,k. \end{equation}
Set $h:=\sum_{j=1}^k p_jh_{\tau_j} \in B^\infty \cap S'$.
Since the $p_i$ are orthogonal and commute with the self-adjoint contractions $h_{\tau_j}$, this is a self-adjoint contraction.
We note that for $j=1,\dots,k$,
\begin{equation} \label{eq:unitaries}
p_je^{\pi i h} = p_je^{\pi i h_{\tau_j}}.
\end{equation}
Using this, for $\tau \in T_\infty(B)$, we compute:
\begin{align}
\tau(|u-e^{\pi i h}|^2)
&= \sum_{j=1}^k \tau(p_j|u-e^{\pi i h}|^2) \\
\notag
& = \sum_{j=1}^k \tau(p_j|u-e^{\pi i h_{\tau_j}}|^2) \\
\notag
&= \sum_{j=1}^k \tau(p_ja_j) \\
\notag
&\leq  \sum_{j=1}^k \e^2\tau(p_j) = \e^2.
\qedhere
\end{align}
\end{proof}

We next turn to the uniqueness aspect of Theorem \ref{thm:MainThm}. Recall the by-now well-known consequence of Connes' characterisation of hyperfiniteness from \cite{Co76}, that if $A$ is separable and nuclear and $\mathcal M$ is a finite von Neumann algebra, then $^*$-homomorphisms $\phi,\psi:A\to \mathcal M$ are strong$^*$-approximately unitary equivalent if and only if $\tau\circ\phi=\tau\circ\psi$ for all $\tau\in T(\mathcal M)$ (see \cite[Proposition 2.1]{CGNN13}, for example\footnote{In the statement of \cite[Proposition 2.1]{CGNN13}, the codomain $\mathcal M$ is required to be countably decomposable; however, this hypothesis is not needed in the proof. It might also be noted that in our application, in the proof of Theorem \ref{thm:Uniqueness}, the codomain $\pi_\tau(B^\infty)''$ is countably decomposable as it has a faithful trace.}).

\begin{theorem}
\label{thm:Uniqueness}
Let $A$ be a separable nuclear $C^*$-algebra and let $B$ be a separable $C^*$-algebra with CPoU and with $T(B)$ compact and non-empty. If $\phi,\psi:A \to B^\infty$ are $^*$-homomorphisms such that $\tau\circ \phi = \tau\circ \psi$ for all $\tau \in T(B^\infty)$ then $\phi$ and $\psi$ are unitarily equivalent.
\end{theorem}

\begin{proof}
Fix $\e>0$ and a finite set $\mathcal F\subset A$.
Since $A$ is separable, by using Kirchberg's $\e$-test (Lemma \ref{lem:EpsTest}), it suffices to prove that there is a unitary $u \in B^\infty$ such that
\begin{equation}
\label{eq:Uniqueness0}
\|\phi(x)-u^*\psi(x)u\|_{2,T_\infty(B)} \leq \e, \quad x \in \mathcal F. \end{equation}
Set
\begin{equation} \eta := \frac{\e}{\sqrt{|\mathcal F|}}. \end{equation}
Fix, for the moment, a trace $\tau \in T(B^\infty)$, and recall that $\pi_\tau:B^\infty \to \pi_\tau(B^\infty)''$ is the corresponding GNS representation.
Then since $A$ is nuclear and $\pi_\tau\circ \phi, \pi_\tau\circ \psi:A \to \pi_\tau(B^\infty)''$ agree on the traces of $\pi_\tau(B^\infty)''$, it follows that these maps are strong$^*$-approximately unitarily equivalent. By Kaplansky's density theorem, the unitaries implementing this can be taken from $\pi_\tau(B^\infty)$ (as done in the proof of Proposition \ref{unitaryexp}). Since the strong$^*$-topology is given by $\|\cdot\|_{2,\tau}$ on bounded sets, it follows that there exists a unitary $u_\tau \in B^\infty$ such that
\begin{equation}
\|\phi(x)-u_\tau^*\psi(x)u_\tau\|_{2,\tau} < \eta, \quad x \in \mathcal F.
\end{equation}
Set
\begin{equation}
\label{eq:Uniqueness1}
a_\tau:= \sum_{x \in \mathcal F} |\phi(x)-u_\tau^*\psi(x)u_\tau|^2 \in (B^\infty)_+, \end{equation}
so that $\tau(a_\tau) < |\mathcal F|\eta^2 = \e^2$.

By continuity and compactness, there exist $\tau_1,\dots,\tau_k \in T(B^\infty)$ such that for every $\tau \in T(B^\infty)$,
\begin{equation} \min\{\tau(a_{\tau_1}),\dots,\tau(a_{\tau_k})\} < \e^2. \end{equation}
Using CPoU as in Lemma \ref{lem:CPoU-Infinity}, there exist orthogonal projections $e_1,\dots,e_k \in B^\infty \cap (\psi(\mathcal F)\cup\phi(\mathcal F) \cup \{u_{\tau_1},\dots,u_{\tau_k}\})'$ which sum to $1_{B^\infty}$ such that
\begin{equation}
\label{eq:Uniqueness2}
 \tau(a_{\tau_i}e_i) \leq \e^2\tau(e_i),\quad \tau \in T_\infty(B). \end{equation}
Set
\begin{equation} u:=\sum_{i=1}^k e_iu_{\tau_i}. \end{equation}
Since the $e_i$ are orthogonal projections summing to $1_{B^\infty}$ and using the fact that they commute with the unitaries $u_{\tau_j}$, it follows that $u$ is itself a unitary.
Moreover, for $x \in \mathcal F$ and $\tau \in T_\infty(B)$, using the fact that the $e_i$ are orthogonal projections which commute with the $u_{\tau_i}$, and both $\phi(x)$ and $\psi(x)$, we have
\begin{equation}
\label{eq:Uniqueness3} |\phi(x)-u^*\psi(x)u|= \sum_{i=1}^k e_i|\phi(x)-u_{\tau_i}^*\psi(x)u_{\tau_i}|. \end{equation}
Hence
\begin{eqnarray}
\notag
\|\phi(x)-u^*\psi(x)u\|_{2,\tau}^2
&\stackrel{\eqref{eq:Uniqueness3}}=& \sum_{i=1}^k \tau\big(e_i|\phi(x) - u_{\tau_i}^*\psi(x)u_{\tau_i}|^2\big) \\
\notag
&\stackrel{\eqref{eq:Uniqueness1}}\leq& \sum_{i=1}^k \tau(e_ia_{\tau_i}) \\
&\stackrel{\eqref{eq:Uniqueness2}}\leq& \sum_{i=1}^k \e^2\tau(e_i) = \e^2.
\end{eqnarray}
Taking the supremum over all $\tau \in T_\infty(B)$, \eqref{eq:Uniqueness0} follows.
\end{proof}

In order to get our existence result into $B^\infty$, we begin with two existence results into von Neumann algebras; these rely on the quasidiagonality of amenable traces on cones established in \cite{BCW16}, which in turn builds on the earlier results of \cite{SWW15,Ga16}.  Recall that a trace $\tau$ on $A$ is \emph{amenable} if given a finite subset $\mathcal F\subset A$ and $\epsilon>0$ there is a c.p.c.\ map $\phi:A\to M_n$ for some $n$ ($\phi$ can be taken to be unital when $A$ is unital) such that \begin{equation}\label{neweq}
\|\phi(ab)-\phi(a)\phi(b)\|_{2,\tr_{M_n}}<\epsilon,\quad a,b\in \mathcal F,
\end{equation}
and
\begin{equation}
|\tr_{M_n}(\phi(a))-\tau(a)|<\epsilon,\quad a\in \mathcal F.
\end{equation}
We write $T_{\mathrm{am}}(A)$ for the set of amenable traces on $A$.
The trace $\tau$ is said to be \emph{quasidiagonal} if (\ref{neweq}) can be strengthened to the operator norm estimate $\|\phi(ab)-\phi(a)\phi(b)\|<\epsilon$ for $a\in\mathcal F$.  We write $T_{\mathrm{qd}}(A)$ for the set of quasidiagonal traces on $A$.  See \cite{Bro06} for details on these approximation properties.

\begin{lemma}\label{l:existence1}
	Let $A$ be a $C^*$-algebra, let $\mathcal M$ be a type II$_1$ von Neumann algebra, and let $\lambda \in T_{\mathrm{am}}(A)$.
	Then given a finite set $\mathcal{F} \subset A$ and $\epsilon>0$, there exist a finite dimensional $C^*$-algebra $F$, a c.p.c.\ map $\theta:A \to F$, and a unital $^*$-homomorphism $\eta:F \to \mathcal M$ such that:
	\begin{align}
	\|\theta(a)\theta(b)\| &< \epsilon \quad && \text{for }a,b \in \mathcal F \text{ satisfying }ab=0\text{, and} \label{l:existence1:eq1} \\
	|\tau (\eta \circ \theta(a))-\lambda(a)|&<\epsilon &&\text{for }a \in \mathcal F\text{ and }\tau \in T(\mathcal M). \label{l:existence1:eq2}
	\end{align}
\end{lemma}

\begin{proof}
	Set $\mathcal G : = \{ \mathrm{id}_{(0,1]} \otimes a \in C_0((0,1]) \otimes A : a \in \mathcal{F} \}$.
	By \cite[Proposition 3.2]{BCW16}, the trace $\delta_1 \otimes \lambda$ is quasidiagonal on $C_0((0,1]) \otimes A$, where $\delta_1$ is the functional of evaluation at $1$ on $C_0((0,1]$.  Thus, there exist a matrix algebra $F$ and a c.p.c.\ map $\phi: C_0((0,1]) \otimes A \to F$ such that
	\begin{align}
\notag
	\| \phi(x) \phi(y) - \phi(xy) \| & < \varepsilon,\quad x,y \in \mathcal G,  \\
	|\tr_F \circ \phi ( x ) - (\delta_1 \otimes \lambda)(x)| &< \varepsilon, \quad x \in \mathcal G.
	\end{align}
	Define $\theta: A \to F$ by $\theta(a):=\phi(\mathrm{id}_{(0,1]} \otimes a)$, so that it immediately follows that \eqref{l:existence1:eq1} is satisfied. As $\mathcal M$ is type II$_1$, $F$ can be embedded unitally in $\mathcal M$. Let $\eta:F \hookrightarrow\mathcal  M$ be any such embedding. By the uniqueness of the trace on $F$, \eqref{l:existence1:eq2} is also satisfied.
\end{proof}

\begin{lemma}\label{l:existence2}
	Let $A$ be a $C^*$-algebra, let $\mathcal M$ be a type II$_1$ von Neumann algebra
and let $\alpha:T(\mathcal M) \to T_{\mathrm{am}}(A)$ be affine and continuous.
	Let $\mathcal{F} \subset A_{sa}$ be a finite set.
	Then given $\epsilon>0$, there exist a finite dimensional $C^*$-algebra $F$, a c.p.c.\ map $\theta:A \to F$, and a unital $^*$-homomorphism $\eta:F \to \mathcal M$ such that
	\begin{equation}
	\label{eq:existence2a}
	\|\theta(a)\theta(b)\| < \epsilon \quad  \text{for }a,b \in \mathcal{F} \text{ satisfying }ab=0
	\end{equation}
	and
	\begin{equation}
	\label{eq:existence2b}
	|\tau(\eta\circ\theta(a))- \alpha(\tau)(a)|<\epsilon, \quad a \in \mathcal F,\ \tau\in T(\mathcal M).
	\end{equation}
	Moreover, if for each $a \in \mathcal{F}$, we are given an element $c_a \in \mathcal M_{sa}$ satisfying
	\begin{equation}\label{eq:existence:neweq} \tau(c_a) = \alpha(\tau)(a), \quad \tau \in T(\mathcal M), \end{equation}
	then for each $a \in \mathcal F$ there exist $x^{(a)}_1,\dots,x_{10}^{(a)},y_1^{(a)},\dots,y_{10}^{(a)}\in \mathcal M$  such that
	\begin{equation}
	\label{eq:existence2c}
	\|\eta \circ \theta(a)-c_a- \sum_{i=1}^{10}[x_i^{(a)},y_i^{(a)}]\| < \epsilon.
	\end{equation}
	If $\alpha(T(\mathcal M))\subset T_{\mathrm{qd}}(A)$, then $\theta$ can be taken to satisfy
	\begin{equation}
	\label{eq:existence2d}
	\|\theta(a)\theta(b) - \theta(ab)\| < \epsilon \quad  \text{for }a,b \in \mathcal{F}.
	\end{equation}
\end{lemma}

\begin{proof}
	The idea is to glue maps from the previous lemma over the centre $Z(\mathcal M)$ of $\mathcal M$, in a manner similar to the proof of \cite[Lemma 2.5]{BCW16}.
	As for $a\in\mathcal F$, the elements $c_a$ satisfying \eqref{eq:existence:neweq} automatically exist by Proposition \ref{prop:CP}, we shall use them throughout the proof.
	Suppose $\mathcal{F}=\{a_1, \ldots, a_n \}$.
	By the structure of commutative von Neumann algebras (\cite[Theorem III.1.5.18]{Bla06}), let $(X,\mu)$ be a locally finite measure space such that $Z(\mathcal M) \cong L^\infty(X, \mu)$.

Let $E: \mathcal M \to L^\infty(X, \mu)$ be the centre-valued trace on $\mathcal M$ (see \cite[Theorem III.2.5.7]{Bla06}).
	Choose natural numbers $C\geq 4$ and $k$ such that $C > \sup_{a \in \mathcal{F}} \|c_a\|$ and $C/k < \epsilon$.
	Set $I :=  \{-Ck+1, \ldots, Ck\}^{n}$ and for $r=(r_1, \ldots, r_n) \in I$, let $p_r \in L^\infty(X,\mu)$ be the characteristic function of the set
	\begin{equation}
	\{ x \in X :  \frac{r_j -1}{k} \leq E(c_{a_j})(x) < \frac{r_j}{k}, \, j=1, \ldots, n\}.
	\end{equation}
	By construction, $(p_r)_{r\in I}$ forms a partition of unity consisting of projections, and, as every trace on $\mathcal M$ factors though $E$ (\cite[Theorem III.2.5.7(iv)]{Bla06}),
	\begin{equation}
	\tau(c_{a_j}) \approx_{1/k} \sum_{r \in I} \frac{r_j}{k} \tau(p_r), \qquad \tau \in T(\mathcal M).\footnote{To improve the readability of this proof, we write $z_1 \approx_\eta z_2$ as shorthand for $|z_1 - z_2| \leq \eta$.}
	\end{equation}
	In particular, for any $r \in I$ and $j = 1, \ldots, n$, we have
	\begin{equation}
	\alpha(\tau)(a_j) \stackrel{\eqref{eq:existence:neweq}}= \tau(c_{a_j}) \approx_{1/k} \frac{r_j}{k}, \qquad \tau \in T(p_r \mathcal M). \label{l:existence2:eq1}
	\end{equation}
	(Note that we implicitly extend $\tau$ to $\mathcal M$, by setting it to be zero on $(1-p_r) \mathcal M$, before we apply $\alpha$ in the previous equation.)

	Let $I_0: =\{r\in I:p_r\neq 0\}$. For each $r\in I_0$, fix $\sigma_r \in T(p_r \mathcal M)$ and set $\lambda_r := \alpha(\sigma_r)$.
	By Lemma \ref{l:existence1}, applied to $p_r \mathcal M$ and $\lambda_r$, there exist a finite dimensional algebra $F_r$, a c.p.c.\ map $\theta_r: A \to F_r$, and a unital $^*$-homomorphism $\eta_r:F_r \to p_r \mathcal M$ such that
	\begin{equation}
	\|\theta_r(a)\theta_r(b)\| < \epsilon
	\end{equation}
	for $a,b \in \mathcal{F}$ satisfying $ab=0$, and
	\begin{equation}
	|\tau(\eta_r \circ \theta_r(a))-\lambda_r(a)|<\frac{\epsilon}{2}  \label{l:existence2:eq2}
	\end{equation}
	for $a \in \mathcal{F}$ and $\tau \in T(p_r \mathcal M)$.
	Set $F:=\bigoplus_{r \in I_0} F_r$. Define $\theta: A \to F$ by $\theta(a) := \oplus_{r \in I_0} \theta_r (a)$ and $\eta: F \to\mathcal  M$ by $\eta((x_r)_{r \in I_0}) := \sum_{r \in I_0} \eta_r (x_r)$. By construction, $\eta$ is a unital $^*$-homomorphism and $\theta$ satisfies \eqref{eq:existence2a}.

	Note that if each $\lambda_r$ is quasidiagonal, this can be used directly in place of Lemma \ref{l:existence1} in the previous paragraph enabling $\theta_r$ to be chosen $(\mathcal{F},\e)$-approximately multiplicative. Therefore, if $\alpha(T(\mathcal M))\subset T_{\mathrm{qd}}(A)$, then $\theta$ can be taken to satisfy \eqref{eq:existence2d}.

	Fix $\tau \in T(\mathcal M)$ for the moment.
	For each $r\in I_0$, set $\tau_r := \frac{\tau(p_r \, \cdot )}{\tau(p_r)} \in T(p_r \mathcal M)$,\footnote{We can choose $\tau_r$ arbitrarily in case $\tau(p_r)=0$.} so $\tau$ can expressed as the convex combination
	\begin{equation}
	\tau = \sum_{r \in I_0} \tau(p_r) \tau_r. \label{l:existence2:eq3}
	\end{equation}
	Thus
	\begin{eqnarray}
	\notag
	\tau (\eta \circ \theta (a)) \hspace*{-5mm} &=& \sum_{r \in I_0} \tau(p_r) \tau_r (\eta_r \circ \theta_r (a)) \\
	\notag
	& \stackrel{\eqref{l:existence2:eq2}}{\capprox{\epsilon/2}}& \sum_{r \in I_0} \tau(p_r) \lambda_r (a) \\
	\notag
	& \stackrel{\eqref{l:existence2:eq1}}{\capprox{2/k}}&  \sum_{r \in I_0} \tau(p_r) \alpha(\tau_r) (a) \\
	\notag
	&=&\alpha(\sum_{r \in I_0} \tau(p_r) \tau_r) (a) \\
	& \stackrel{\eqref{l:existence2:eq3}}{=}& \alpha (\tau)(a) \label{l:existence2:eq4}
	\end{eqnarray}
	for all $a \in \mathcal{F}$.
	Since $\frac2k\leq\tfrac C{2k}<\tfrac\epsilon2$ and $\tau \in T(\mathcal M)$ was arbitrary, this establishes \eqref{eq:existence2b}.

	Now fix $a \in \mathcal{F}$ for the moment and let us explain why $x^{(a)}_i,y^{(a)}_i$ can be found to satisfy \eqref{eq:existence2c}.
	Set $h:= E(\eta \circ \theta (a) - c_a ) \in L^\infty(X,\mu)$, which by \eqref{eq:existence2b} satisfies $\|h\|\leq \epsilon$. Observe that $E(\eta \circ \theta (a) - c_a - h)= 0$, so that by \cite[Theorem 3.2]{FH80} there exist $x^{(a)}_1,\dots,x_{10}^{(a)},y_1^{(a)},\dots,y_{10}^{(a)}\in \mathcal M$ such that\footnote{We can also arrange that $\max_{1\leq i \leq 10}\|x^{(a)}_i\|\|y^{(a)}_i\| \leq 12 \cdot 12 \cdot \|\eta \circ \theta (a) - c_a - h\|$, but we do not need to control the norms of these elements on this occasion.}
	\begin{equation}
	\eta \circ \theta (a) - c_a - h = \sum_{i=1}^{10} [x^{(a)}_i,y^{(a)}_i].
	\end{equation}
Hence,
	\begin{equation}
	\| \eta \circ \theta (a) - c_a - \sum_{i=1}^{10} [x^{(a)}_i,y^{(a)}_i] \| = \| h \| <  \epsilon.\qedhere
	\end{equation}
\end{proof}

We now turn to the existence component of Theorem \ref{thm:MainThm}. For this we will need a so-called `no silly traces' result to show that the limit traces on $B^\infty$ generate all traces on $B^\infty$.  For the purposes of Theorem \ref{thm:MainThm}, we could use (a sequence algebra version) of the original result of this type: \cite[Theorem 8]{Oz13} for $\mathcal Z$-stable exact $C^*$-algebras $B$. This gives a no silly traces result for the $C^*$-algebra ultraproduct, from which it follows that there are no silly traces on the uniform tracial ultraproduct (and this is easily modified to sequence algebras).

However, in Theorem \ref{thm:Existence} below, we prefer not to impose the hypothesis that $B$ is $\mathcal Z$-stable, and instead simply ask that it has CPoU.
Correspondingly, we first show how to obtain a no silly traces result for the uniform tracial sequence algebra just assuming CPoU.  While CPoU is involved to handle possibly non-$\mathcal Z$-stable C*-algebras, to some extent the present result is easier than Ozawa's \cite[Theorem 8]{Oz13} in that we can use the uniform bounds on the number of commutators of a self-adjoint operator in a finite von Neumann algebra which vanishes in all traces from \cite{FH80}, rather than the more delicate growth rate estimates used in \cite{Oz13} which are required to eliminate silly traces from the $C^*$-norm sequence algebra or ultrapower.

We note also that no silly traces for the tracial product $B^\infty$ does not imply no silly traces for the norm product, as demonstrated by the unique trace example by Robert (based on earlier examples by Villadsen) in \cite[Theorem 1.4]{Ro15}.\footnote{Let $A$ be the $C^*$-algebra from \cite[Theorem 1.4]{Ro15} so that condition (iii)(a) from \cite[Theorem 3.24]{ART} fails for $A$.  Then no norm ultraproduct of $A$ can have unique trace (by the equivalence of condition (iii)(a) and the uniqueness of trace on an ultraproduct, which makes up the first part of the proof of (iii)$\Leftrightarrow$(ii) in \cite[Theorem 3.24]{ART}; see paragraph 3 of the proof, which notes this explicitly). Note that $A$ has the Diximer property needed to apply this result by \cite{HZ} as it is simple, unital and has unique trace. Therefore the norm product $A_\infty$ of infinitely many copies of $A$ has traces which are not in the closed convex hull of the limit traces.}

\begin{proposition}\label{prop:nosillytraces}
Let $B$ be a separable $C^*$-algebra with $T(B)$ compact and non-empty and which has CPoU.  Then the weak$^*$-closed convex hull of $T_\infty(B)$ is $T(B^\infty)$.
\end{proposition}
\begin{proof}
Fix a self-adjoint contraction $z\in B^\infty$. Let $\delta:= \sup_{\tau\in T_\infty(B)}|\tau(z)|$. By \cite[Lemma 4.4]{CETWW} (which is extracted from \cite[Theorem 8]{Oz13}), it suffices to prove that $\sup_{\tau\in T(B^\infty)}|\tau(z)|=\delta$.  This will be achieved by producing a self-adjoint $c\in B^\infty$ with $\|c\|\leq\delta$ and contractions $x^{(1)},\dots,x^{(10)},y^{(1)},\dots,y^{(10)}\in B^\infty$ such that $z-c=K\sum_{i=1}^{10}[x^{(i)},y^{(i)}]$, where $K:=12\cdot 12(1+\delta)$.

Choose a representative sequence $(z_n)_{n=1}^\infty$ of self-adjoint contractions for
 $z\in B^\infty$. Then $\limsup_{n\to\infty}\sup_{\tau\in T(B)}|\tau(z_n)|\leq\delta$, and so by rescaling, we may assume $\sup_{\tau\in T(B)}|\tau(z_n)|\leq\delta$ for each $n$.

Fix $n\in\mathbb N$ for the moment. For each $\tau\in T(B)$, let $\pi_\tau$ be its GNS-representation and $\mathcal M_\tau:=\pi_\tau(B)''$.
Then $\sup_{\rho\in T(\mathcal M_\tau)}|\rho(\pi_\tau (z_n))|\leq\delta$. Letting $\tilde{c}_{n,\tau}\in\mathcal M_\tau$ be the result of applying the centre-valued trace in $\mathcal M_\tau$ to $\pi_\tau(z_n)$, we have $\|\tilde{c}_{n,\tau}\|\leq\delta$.  By \cite[Theorem 3.2]{FH80}, there exist contractions $\tilde{x}_{n,\tau}^{(1)},\dots,\tilde{x}^{(10)}_{n,\tau},\tilde{y}_{n,\tau}^{(1)},\dots,\tilde{y}^{(10)}_{n,\tau}\in\mathcal M_\tau$ with $\pi_\tau(z_n)-\tilde{c}_n=K\sum_{i=1}^{10}[\tilde{x}^{(i)}_{n,\tau},\tilde{y}^{(i)}_{n,\tau}]$.  By Kaplansky's density theorem, there exists a self-adjoint $c_{n,\tau} \in B$ with $\|c_{n,\tau}\|\leq\delta$ and contractions $x^{(1)}_{n,\tau},\dots,x^{(10)}_{n,\tau},y^{(1)}_{n,\tau},\dots,y^{(10)}_{n,\tau}\in B$ with
\begin{equation}
\|z_n-c_{n,\tau}-K\sum_{i=1}^{10}[x^{(i)}_{n,\tau},y^{(i)}_{n,\tau}]\|_{2,\tau} < \gamma_n,
\end{equation}
where $\gamma_n < \tfrac1n$.

Let $a_{n,\tau}:=|z_n-c_{n,\tau}-K\sum_{i=1}^{10}[x^{(i)}_{n,\tau},y^{(i)}_{n,\tau}]|^2$. By compactness, there exist $\tau_{n,1},\dots,\tau_{n,k_n}$ such that $\min_{\rho\in T(B)}\{\rho(a_{n,\tau_{n,1}}),\dots,\rho(a_{n,\tau_{n,k_n}})\}<\gamma_n^2$.  As every trace in $T_\infty(B)$ restricts to a trace on $B$, the same minimum holds over $\rho\in T_\infty(B)$. Let $S_n \subset B^\infty$ be the separable subalgebra generated by $z_n$ together with $c_{n,\tau_{n,1}},\dots,c_{n,\tau_{n,k_n}}$ and the contractions $x^{(1)}_{n,\tau_{n,1}},\dots,x^{(10)}_{n,\tau_{n,k_n}}$ and $y^{(1)}_{n,\tau_{n,1}},\dots,y^{(10)}_{n,\tau_{n,k_n}}$.

By CPoU in the form of Lemma \ref{lem:CPoU-Infinity}, there exist pairwise orthogonal projections $e_{n,1},\dots,e_{n, k_n}$ in $B^\infty \cap S_n'$ which sum to $1_{B^\infty}$ and have $\rho(a_{n, \tau_{n,j}} e_{n,j})\leq \gamma_n^2 \rho(e_{n,j})$ for $j=1,\dots,k_n$ and all $\rho\in T_\infty(B)$.

Define $\tilde{c}_n:=\sum_{j=1}^{k_n}c_{n,\tau_{n,j}}e_{n,j} \in B^\infty$, $\tilde{x}^{(i)}_n:=\sum_{j=1}^{k_n}x^{(i)}_{n,\tau_{n,j}}e_{n,j}  \in B^\infty$, and $\tilde{y}^{(i)}_n:=\sum_{j=1}^{k_n}y^{(i)}_{n,\tau_{n,j}}e_{n,j}  \in B^\infty$. Then $\|\tilde{c}_n\| \leq \delta$ and all the $\tilde{x}^{(i)}_n$ and $\tilde{y}^{(i)}_n$ are contractions.
Let $\rho\in T_\infty(B)$. Using the properties of the $e_{n,j}$, we have
\begin{align}
\|z_n-\tilde{c}_{n}-K\sum_{i=1}^{10}[\tilde{x}^{(i)}_{n},\tilde{y}^{(i)}_{n}]\|_{2,\rho}^2=\rho(\sum_{j=1}^{k_n}a_{n,\tau_{n,j}}e_{n,j}) \leq \gamma_n^2.
\end{align}

Taking norm preserving lifts from $B^\infty$ to $\ell^\infty(B)$ and then choosing elements $c_n$, $x^{(i)}_n$, $y^{(i)}_n \in B$ sufficiently far down the representative sequences for $\tilde{c}_n, \tilde{x}^{(i)}_n,\tilde{y}^{(i)}_n \in B^\infty$, we have
\begin{equation}
\sup_{\tau\in T(B)}\|z_n-c_n-K\sum_{i=1}^{10}[x^{(i)}_n,y^{(i)}_n]\|_{2, \tau} \leq \gamma_n
\end{equation}
and $\|c_n\| \leq \delta$.  Assembling these into $c:=(c_n)_{n=1}^\infty$, $x^{(i)}:=(x^{(i)}_n)_{n=1}^\infty$ and $y^{(i)}:=(y_n^{(i)})_{n=1}^\infty$ in $B^\infty$ provides the elements demanded in the first paragraph of the proof.
\end{proof}

We can now give our more general version of the existence aspect of Theorem \ref{thm:MainThm}. The condition that $A$ be nuclear is weakened to the condition that the range of $\alpha$ is contained in the set $T_{\mathrm{am}}(A)$ of amenable traces on $A$.  The second part of the following theorem, regarding the form of a representative sequence for $\phi$, is not needed for Theorem \ref{thm:MainThm}, but we anticipate it playing a role in future nuclear dimension computations. Recall that a c.p.\ map $\phi:A \rightarrow B$ is said to be \emph{order zero} if $\phi(a)\phi(b) = 0$ for all $a,b \in A_+$ with $ab = 0$;
see \cite{WZ09} for the structure theory of these maps.

\begin{theorem}
\label{thm:Existence}
Let $A$ be a separable $C^*$-algebra and let $B$ be a separable $C^*$-algebra with $T(B)$ compact and non-empty which has CPoU and no finite dimensional representations. Given a continuous affine function $\alpha:T(B^\infty) \to T_{\mathrm{am}}(A)$, there exists a $^*$-homomorphism $\phi:A \to B^\infty$ such that
\begin{equation}
\label{eq:Existence1}
 \tau \circ \phi = \alpha(\tau), \quad \tau \in T(B^\infty). \end{equation}
Moreover, $\phi$ can be represented by a sequence $(\phi_n)_{n=1}^\infty$ of c.p.c.\ maps $A\rightarrow B$ each of which factorises as $\phi_n=\psi_n\circ\theta_n$ for a c.p.c.\ map $\theta_n:A\rightarrow F_n$ with $F_n$ finite dimensional, and a c.p.c.\ order zero map $\psi_n:F_n\rightarrow B$.  The maps $\theta_n$ can be taken to be approximately order zero, and if the range of $\alpha$ lies in the quasidiagonal traces on $A$, the $\theta_n$ can be taken to be approximately multiplicative.
\end{theorem}

\begin{proof}
We first note that any c.p.c.\ order zero map $\phi:A \to B^\infty$ satisfying \eqref{eq:Existence1} is automatically a $^*$-homomorphism.
Indeed, let $(e_n)_{n=1}^\infty$ be an increasing approximate unit for $A$. Then
\begin{equation}\label{eq:OZ}
	\phi(a_1)\phi(a_2) = \lim_{n\rightarrow\infty} \phi(e_n)\phi(a_1a_2), \quad a_1,a_2 \in A,
\end{equation}
as a consequence of \cite[Corollary 4.1]{WZ09}. It therefore suffices to prove that $\lim_{n\rightarrow\infty} \phi(e_n) = 1_{B^\infty}$ in $\|\cdot\|_{2,T_\infty(B)}$. We compute that
\begin{align}
\notag
	\|1_{B^\infty}-\phi(e_n)\|_{2,T_\infty(B)}^2 &\leq \sup_{\tau \in T(B^\infty)} \tau(|1_{B^\infty}-\phi(e_n)|^2)\\
\notag
	&\leq \sup_{\tau \in T(B^\infty)} \tau(1_{B^\infty}-\phi(e_n))\\
\notag
	&\leq \sup_{\tau \in T(B^\infty)} 1 - \tau(\phi(e_n))\\
	&= \sup_{\tau \in T(B^\infty)} (1 - \alpha(\tau)(e_n))\rightarrow 0,
\end{align}
as by Dini's Theorem, $\alpha(\tau)(e_n)$ converges to 1 uniformly on $T(B^\infty)$.

Notice that due to Proposition \ref{prop:nosillytraces}, it is enough to establish equation \eqref{eq:Existence1} for limit traces.
Fix $\e>0$ and finite sets $\mathcal F\subset A$ and $\mathcal G \subset A_{sa}$. We will prove that there is a finite dimensional $C^*$-algebra $F$, a c.p.c.\ map $\theta:A\rightarrow F$ and a c.p.c.\ order zero map $\psi:F\rightarrow B^\infty$ such that for $\phi=\psi\circ\theta$ we have
\begin{align}
	\|\theta(a)\theta(b)\| &\leq \epsilon \quad && \text{for }a,b \in \mathcal F \text{ satisfying }ab=0\text{, and} \label{eq:Existence2a} \\
	|\tau (\phi(a))-\alpha(\tau)(a)|&\leq \epsilon &&\text{for }a \in \mathcal G \text{ and }\tau \in T_\infty(B). \label{eq:Existence2b}
\end{align}
In the special case that $\alpha(T(B^\infty))\subseteq T_{\mathrm{qd}}(A)$, we will show that we can additionally replace (\ref{eq:Existence2a}) by the stronger condition
\begin{equation}\label{eq:Existence2c}
\|\theta(a)\theta(b)-\theta(ab)\|\leq\epsilon,\quad a,b\in\mathcal F.
\end{equation}
Once this is achieved, an application of Kirchberg's $\e$-test (in the form of Lemma \ref{lem:EpsTest}) can be used to obtain the required $\phi$ (and $\psi_n,\theta_n$) in a very similar fashion to \cite[Lemma 7.4]{BBSTWW}. We set this out for the passage from (\ref{eq:Existence2a}) and (\ref{eq:Existence2b}) to the required $\phi$ such that $(\theta_n)$ are approximately order zero in  the next paragraph. The passage from (\ref{eq:Existence2c}) and (\ref{eq:Existence2b}) to obtaining a $\phi$ such that the maps $(\theta_n)_{n=1}^\infty$ are approximately multiplicative is similar (and slightly easier).

For each $n$, let $X_n$ denote the set of triples $(F_n,\theta_n,\psi_n)$, where $F_n$ is a finite dimensional $C^*$-algebra $F_n$, $\theta_n:A\to F_n$ is c.p.c.\ and $\psi_n:F_n\to B$ is c.p.c.\ order zero.\footnote{Although all the $X_n$ represent the same set, we use the subscript $n$ for direct comparison with the notation of the $\e$-test.}  Fix a countable dense subset $(x_k)_{k=1}^\infty$ of $A_{\mathrm{sa}}$. Noting that the collection of pairs $(a,b)$ in $A_+$ with $ab=0$ is a subspace of the separable metric space $A \times A$, we may also fix a countable dense subset $(a_k,b_k)_{k=1}^\infty$ of these orthogonal pairs.  Set $X:=\prod_{n=1}^\infty X_n$ and define functions $f^{(k)}_n:X_n\to[0,\infty]$ by
\begin{align}
\notag
f^{(k)}_n(F_n,\theta_n,\psi_n):=\max_{j\leq k}&\big(\|\theta_n(a_jb_j)\| \\
&+\sup_{\tau\in T(B)}|\tau(\psi_n(\theta_n(x_j)))-\alpha(\tau(x_j))|\big),
\end{align}
for $(F_n,\theta_n,\psi_n)\in X_n$.  Given $\e>0$ and $k_0\in\mathbb N$, let $\mathcal{F}:=\{a_k,b_k:k\leq k_0\}$ and $\mathcal G:=\{x_k:k\leq k_0\}$ and take $F,\theta,\psi$ satisfying (\ref{eq:Existence2a}) and \eqref{eq:Existence2b}.  Then $\psi$ lifts to a sequence $(\tilde\psi_n)_{n=1}^\infty$ of c.p.c.\ order zero maps $F\to B$ by projectivity of c.p.c.\ order zero maps with finite dimensional domains (\cite[Proposition 1.2.4]{Wi09}, which rephrases Loring's work \cite[Theorem 4.9]{Lo93} on projectivity of cones over finite dimensional $C^*$-algebras to this setting).  The sequence $\big(F,\theta,\tilde\psi_n\big)_{n=1}^\infty$ in $\prod_{n=1}^\infty X_n$ satisfies
\begin{equation}
\limsup_n f^{(k)}_n\big(F,\theta,\tilde\psi_n\big)\leq 2\epsilon,\quad k=1,\dots,k_0.
\end{equation}
Applying the $\e$-test gives a sequence $(F_n,\theta_n,\psi_n)$ in $\prod_{n=1}^\infty X_n$ with
\begin{equation}
\limsup_nf^{(k)}_n(F_n,\theta_n,\psi_n)=0,\quad k\in\mathbb N.
\end{equation}
Defining $\phi_n:=\psi_n\circ\theta_n$ and $\phi:A\to B^\infty$ to be the map induced by $(\phi_n)_{n=1}^\infty$ gives the required $\phi$.

With the $\epsilon$-test in place, we now commence the construction of maps satisfying (\ref{eq:Existence2a}) and \eqref{eq:Existence2b}. By Proposition \ref{prop:CP}, for each $a \in \mathcal G$, we may choose a self-adjoint element $c_a \in B^\infty$ such that
\begin{equation} \label{eq:existence.new}
\tau(c_a) = \alpha(\tau)(a), \quad \tau \in T(B^\infty).
\end{equation}
Also, set \begin{equation} \delta := \frac\e{\sqrt{|\mathcal G|}}. \end{equation}

We work with the weak$^*$-closure $\overline{T_\infty(B)}$ of $T_\infty(B)$ which is weak$^*$-compact. Fix $\tau \in \overline{T_\infty(B)}$ for the moment. Consider the finite von Neumann algebra $\mathcal M_\tau:=\pi_\tau(B^\infty)''$.

We claim that $\mathcal M_\tau$ is type II$_1$. As $T(B)$ is compact, any approximate unit $(e_n)_{n=1}^\infty$ for $B$ satisfies $\inf_{\rho\in T(B)}\rho(e_n)\rightarrow 1$ by Dini's Theorem.  Then
\begin{equation}
\inf_{\rho\in T_\infty(B)}\rho(e_n)=\inf_{\rho\in \overline{T_\infty(B)}}\rho(e_n)\rightarrow 1,
\end{equation}
and hence $\tau(e_n)\rightarrow 1$.  Thus $\|1_{\mathcal M_\tau}-\pi_\tau(e_n)\|_{2,\tau}\rightarrow 0$ and $\pi_\tau (e_n)$ converges $^*$-strongly to $1_{\mathcal{M}_\tau}$. Suppose that $\mathcal M_\tau$ has a non-zero type I$_k$ summand for some $k\in\mathbb N$, with corresponding central projection $p$, so that $p\mathcal M_\tau$ has a separating family of finite dimensional representations.  Since $B$ has no non-zero finite dimensional representations, we must have $p\pi_\tau(B)=0$.  On the other hand $p\pi_\tau(e_n)\neq 0$ for sufficiently large $n$. This contradiction proves the claim.

Let $T(\pi_\tau): T(\mathcal{M}_\tau) \to T(B^\infty)$ be the map induced by $\pi_\tau$.
By Lemma \ref{l:existence2} (applied to the map $\alpha\circ T(\pi_\tau):T(\mathcal M_\tau)\rightarrow T_{\mathrm{am}}(A)$), there exist a finite dimensional $C^*$-algebra $F_\tau$, a c.p.c.\ map $\theta_\tau:A \to F_\tau$ and a unital $^*$-homomorphism $\eta_\tau:F_\tau \to \mathcal M_\tau$ as well as $x^{(a)}_1,\dots,x^{(a)}_{10},y^{(a)}_1,\dots,y^{(a)}_{10} \in \mathcal M_\tau$ for $a \in \mathcal G$, such that
\begin{gather}
	\label{eq:Existence3a}
	\|\theta_\tau(a)\theta_\tau(b)\| < \e \quad  \text{for }a,b \in \mathcal{F} \text{ satisfying }ab=0,\text{ and} \\
	\label{eq:Existence3b}
	\|\eta_\tau \circ \theta_\tau(a)-\pi_\tau(c_a)- \sum_{i=1}^{10}[x_i^{(a)},y_i^{(a)}]\| < \delta\quad \text{for }a \in \mathcal{G}.
\end{gather}

By the Kaplansky density theorem, at the cost of replacing the norm estimate in \eqref{eq:Existence3b} by a $\|\cdot\|_{2,\tau}$-estimate, we may assume that the elements $x^{(a)}_i,y^{(a)}_i$ belong to $\pi_\tau(B^\infty)$, and thus lift to elements $x^{(\tau,a)}_i,y^{(\tau,a)}_i \in B^\infty$.
Using the order zero Kaplansky density theorem (\cite[Lemma 1.1]{HKW12}) we may approximate $\eta_\tau$ in $\|\cdot\|_{2,\tau}$ by an order zero map $F_\tau \to \pi_\tau(B^\infty)$, and then this can be lifted to an order zero map $\psi_\tau:F_\tau \to B^\infty$ by \cite[Proposition 1.2.4]{Wi09}.
Starting from \eqref{eq:Existence3b}, we can perform these approximations and lifts so that
\begin{equation}
\label{eq:Existence4b}
	\|\psi_\tau(\theta_\tau(a))-c_a- \sum_{i=1}^{10}[x_i^{(a,\tau)},y_i^{(a,\tau)}]\|_{2,\tau} < \delta,\quad a \in \mathcal{G}. \end{equation}
Let us now set
\begin{equation} s_\tau := \sum_{a \in \mathcal G} \Big|\psi_\tau(\theta_\tau(a))-c_a-\sum_{i=1}^{10} [x_i^{(a,\tau)},y_i^{(a,\tau)}]\Big|^2 \in (B^\infty)_+, \end{equation}
so that by \eqref{eq:Existence4b} we get $\tau(s_\tau) < |\mathcal G|\delta^2 = \e^2$.

By continuity and compactness of $\overline{T_\infty(B)}$, there exist $\tau_1,\dots,\tau_k \in \overline{T_\infty(B)}$ such that for every $\tau \in \overline{T_\infty(B)}$,
\begin{equation} \min\{\tau(s_{\tau_1}),\dots,\tau(s_{\tau_k})\} < \e^2. \end{equation}
Set \begin{align} \notag S&:= \psi_{\tau_1}(F_{\tau_1}) \cup \cdots \cup \psi_{\tau_k}(F_{\tau_k}) \cup \{c_a:a \in \mathcal G\} \\
&\qquad\cup \{x^{(a,\tau_i)}_j,y^{(a,\tau_i)}_j: a \in \mathcal G,\ i=1,\dots,k,\ j=1,\dots,10\}, \end{align}
a separable subset of $B^\infty$. Using CPoU as in Lemma \ref{lem:CPoU-Infinity}, there exist orthogonal projections $e_1,\dots,e_k \in B^\infty \cap S'$ which sum to $1_{B^\infty}$ such that
\begin{equation}
\label{eq:Existence5}
 \tau(s_{\tau_i}e_i) \leq \e^2\tau(e_i),\quad \tau \in T_\infty(B). \end{equation}
Set $F:= \bigoplus_{i=1}^kF_{\tau_i}$, and define $\theta:A\to F$ and $\psi:F\to B^\infty$ by
\begin{equation}
\theta(a):=(\theta_{\tau_1}(a),\dots,\theta_{\tau_k}(a)),\quad a\in A,\label{eq:defTheta}
\end{equation}
and
\begin{equation} \psi(x_1,\dots,x_k):=\sum_{i=1}^k e_i\psi_{\tau_i}(x_i),\quad (x_1,\dots,x_k)\in F. \end{equation}
Then \eqref{eq:Existence2a} is an immediate consequence of \eqref{eq:Existence3a}. Since the $e_i$ are orthogonal positive elements commuting with the images of the c.p.\ order zero maps $\psi_{\tau_i}$, it follows that $\psi$ is c.p.\ and order zero.
Moreover $\psi$ is contractive since $\psi(1_A) \leq \sum_{i=1}^k e_i = 1_{B^\infty}$.

Finally, for $a \in \mathcal G$ and $\tau \in T_\infty(B)$, writing $\phi=\psi\circ\theta$, we compute
\begin{eqnarray}
\notag
&&\hspace*{-6em} |\tau(\phi(a))-\alpha(\tau)(a)|^2 \\
&\stackrel{\eqref{eq:existence.new}}=&|\tau(\psi(\theta(a))-c_a)|^2\notag
\\&=&\Big|\tau\Big(\sum_{i=1}^ke_i\Big(\psi_{\tau_i}(\theta_{\tau_i}(a))-c_a-\sum_{j=1}^{10} [x^{(a,\tau_i)}_j,y^{(a,\tau_i)}_j]\Big)\Big)\Big|^2\notag \\
\notag
&\leq& \tau\Big(\sum_{i=1}^k e_i\Big|\psi_{\tau_i}(\theta_{\tau_i}(a))-c_a-\sum_{j=1}^{10} [x^{(a,\tau_i)}_j,y^{(a,\tau_i)}_j]\Big|^2\Big) \\
\notag
&\leq& \tau\Big(\sum_{i=1}^k e_is_{\tau_i}\Big) \\
&\leq& \sum_{i=1}^k \e^2\tau(e_i)  = \e^2,
\end{eqnarray}
where on the third line, we use the fact that the $e_i$ make up a pairwise othogonal partition of unity of projections commuting with $S$. This proves \eqref{eq:Existence2b}, and completes the proof with $\theta_n$ approximately order zero.

When $\alpha(T_\infty(B))$ consists of quasidiagonal traces, \eqref{eq:Existence3a} can be replaced by an $(\mathcal{F},\e)$-approximate multiplicativity condition using the last clause of Lemma \ref{l:existence2}. The map $\theta$, as defined in \eqref{eq:defTheta}, will then satisfy \eqref{eq:Existence2c}, and this completes the proof with $\theta_n$ approximately multiplicative.
\end{proof}

We end by recording how Theorem \ref{thm:MainThm} follows as special cases of the existence and uniqueness results of this section.
\begin{proof}[Proof of Theorem \ref{thm:MainThm}]
This is a consequence of Theorems \ref{thm:Uniqueness} and \ref{thm:Existence}. The CPoU hypothesis on $B$ needed in both these theorems is automatic for separable nuclear $\mathcal Z$-stable $C^*$-algebras by Theorem \ref{thm:CPoU}; moreover $\mathcal Z$-stability is an obstruction to having finite dimensional representations. The hypothesis in Theorem \ref{thm:Existence} that $\alpha$ takes values in the amenable traces on $A$ is automatic as all traces on a nuclear $C^*$-algebra are amenable, essentially by Connes' theorem.\footnote{In fact, all traces are uniformly amenable by \cite[Theorem 3.2.2(5)$\Rightarrow$(1)]{Bro06} and Connes' theorem, and uniformly amenable traces are amenable: see \cite[Section 3.5]{Bro06}.  (Note that \cite{Bro06} works with unital $C^*$-algebras throughout, so to apply these results one should unitise both $A$ and all its traces).}
\end{proof}

\newcommand{\etalchar}[1]{$^{#1}$}

config: arxiv


TITLE: If length in a frame moving with respect to a stationary frame contracts, how do we define the speed of the moving frame?
QUESTION [0 upvotes]: Under Galilean transformations, the speed of a moving system $k$ with respect to a stationary system $K$ is straightforward. Assuming the origins of $k$ and $K$ coincide at t=0, the spatial coordinates of $k$ are related to those of $K$ with time as follows:
$$ x' = x-vt $$
To demonstrate the speed of $k$ with respect to $K$ within a Galilean system, we can find how the distance between any pair of points from $K$ and $k$ coinciding at $t=0$ varies per unit time. Let $x_0$ be a stationary point in $K$ and $x_k$ be a point in $K$ which moves along the path of a stationary point $x'_0$ in k. Let us set the stationary points to have an equal and constant value: $x_0= x'_0 = C$. Let the moving point $x_k$ in $K$ be described as follows:
$$ x'_0 = x_k - v t \implies x_k = x'_0 + vt  = C + vt$$
The speed of $k$ with respect to $K$ is then:
$$\frac{x_k - x_0}{t} = \frac{C + vt - C}{t}  = v $$
This shows that no matter which choice of coinciding initial points we make in our analysis, the speed of $k$ with respect to $K$ is $v$. In the special theory of relativity, it seem this is only true for the origin:
$$ x'_0 = \gamma(x_k - v t) \implies x_k = \frac{x'_0}{\gamma} + vt  = \frac{C}{\gamma} + vt$$
$$\frac{x_k - x_0}{t} = \frac{\frac{C}{\gamma} + vt - C}{t}  = \frac{ C (\frac{1}{\gamma} - 1)}{t} + v = \frac{ C (1 - \gamma)}{\gamma t} + v$$
Therefore, only if we choose to analyze the origin does the speed of $k$ with respect to $K$ remain $v$. But, the origin can be chosen to be anywhere arbitrarily so this shouldn't be so. Do the laws of physics really depend on my choice of origin?

REPLY [0 votes]: EDIT: I misunderstood the original question. The actual question is just a math one, why do we get the same speed regardless of which point we choose in the moving coordinate system. The answer is that both time and space need to be transformed, i.e. we need to apply the full Lorentz transforms.
For any arbitrary fixed $x_0$ in the "moving" frame, the path it traces out in its rest frame is the set $(t, x_0)$. which corresponds to the path $\gamma(t+vx_0, x_0+tv)$ in the "stationary" frame. But note that we assume $v$ and $x_0$ are constant, so let us re-parameterize the path with $T = \gamma(t + vx_0)$; then:
$t = \frac{T}{\gamma} - vx_0$
and so
$\gamma(x_0 + tv) = \gamma x_0 + \gamma(\frac{T}{\gamma} - vx_0) v
 = Tv + C$ for some constant $C$
(remember that $\gamma$, $v$, and $x_0$ are all constant; we could work out $C$, but it's tedious and irrelevant).
The path $(t, x) = (t, x_0)$ in moving coordinates is thus expressed as $(T, X) = (T, vT + C)$ in stationary coordinates; in other words it obeys the equation $X = vT + C$ where $T$ is the (Earth) time. The slope is $v$, and so we see that the initial choice of $x_0$ affected only the offset $C$ and not the speed that we calculate.
Original answer:
You can define the speed in any coordinate system, and it's relative to that coordinate system. The speed is simply $\frac{\Delta x}{\Delta t}$ where $x$ and $t$ are as measured in the specific coordinate system in question. Let's call this the "stationary" coordinate system. It's irrelevant that there is length contraction in  the "moving" coordinate system, because that's not the system we're using to define the speed: we're defining it relative to the "stationary" coordinates. In fact the speed relative to the "moving" coordinates (moving along with the object) is always 0.


TITLE: How does the solution of ODE $y'=F(t,y)$ extend to an open interval?
QUESTION [0 upvotes]: I'm trying to solve the above problem from Taylor's PDE I, and I'm supposed to use compactness of $K$. But how does it work?

REPLY [1 votes]: Let me first motivate why do you need compactness for the statement. Essentially, what the Fundamental theorem tells us is that if F is locally Lipschitz on an open subset $U$, then for any points $y\in U$, you can find a neighbourhood $V$ of $y$ and a time $\tau$ such that there exists a unique solution for every initial condition $y_0$ in $V$. So, all these neighbourhoods will form a cover of $U$ (in particular $K$), and here is where compactness comes in: the finiteness of the subcover will then allow you to extend the interval. Note that you also need to assume your solution stays in the compact set $K$, otherwise your solution will cease to exist.
Proof: By the Fundamental theorem, for all $y\in K$, there exists an open neighbourhood $V_y$ of y and a time $\tau_y$ such that the solution is defined for all initial condition $y_0\in V_y$ and $|t|\le\tau_y$. The set $\{V_y\colon y\in K\}$ form an open cover of K, so by compactness of K, there exists a finite subcover $\{V_{y_j}:j=1,\ldots,n\}$. Define $\tau$ to be $\min_j\tau_{y_j}$, then $\forall y_0\in K$, the solution $y(t)$ exists for $|t|\le\tau$. But since we assume $y(t)\in K$ for $t\in[T_0,T_1]$, then $y(t)$ is defined for at least $t\in[T_0-\tau,T_1+\tau]$.

As a Corollary, if there exists a compact subset $K$ such that once inside $K$, the solutions can't leave it, then those solutions are defined for all $t>0$.


TITLE: Why does $\sin(\cos x)=\cos(\sin y)$ result in a lattice of circles?
QUESTION [8 upvotes]: Using Desmos graphing, I made an equation $\sin(\cos x)=\cos(\sin y)$ (here) which resulted in a strange lattice of symbols. I know that the trigonometric functions relate a triangle's sides to its angles (SOHCAHTOA), so how does this equation create circles?

REPLY [5 votes]: I do like Desmos.  I want to like it more, but it does have limitations.  In this particular case, you can see how the loops arise quite clearly by plotting the 3D graphs of $z=\sin(\cos(x))$ and $z=\cos(\sin(y)$ on the same set of axes.  The result looks like so:


TITLE: Continuity of the joint distribution function given continuity of marginals
QUESTION [0 upvotes]: Suppose $X$ and $Y$ are continuous random variables such that $F_X$ and $F_Y$ are the respective distribution functions. Suppose $F_X$ is continuous at $x_0$ and $F_Y$ is continuous at $y_0$. Then show that $F_{X,Y}$ is continuous at $(x_0,y_0)$ where $F_{X,Y}$ is the joint distribution function.

I am highly doubtful about the rigor of my method but here is my attempt:

Consider any sequence $\{x_n,y_n\}$ converging to $(x_0,y_0)$. Then $x_n\to x_0$ and $y_n\to y_0$. Hence $F_X(x_n)\to F_X(x_0)$ and $F_Y(y_n)\to F_Y(y_0)$.
Now fix $m$. Then consider the corresponding subsequence $(x_n,y_m)_n$ of $(x_n,y_n)$. Then $F_{X,Y}(x_n,y_m)=P(X\leq x_n,Y\leq y_m)\to P(X\leq x_0,Y\leq y_m)=F_{X,Y}(x_0,y_m)$ by continuity property of $F_X$, and noting that $m$ is fixed. Now letting $m\to\infty$, we obtain $\lim_{m\to\infty}F_{X,Y}(x_0,y_m)=F_{X,Y}(x_0,y_0)$ and hence $\lim_m\lim_nF_{X,Y}(x_n,y_n)=F_{X,Y}(x_0,y_0)$.
By taking limits the other way, we similarly obtain $\lim_n\lim_mF_{X,Y}(x_n,y_m)=F_{X,Y}(x_0,y_0)$. So both the iterated limits exist and are equal, concluding that $\lim_{m,n}F_{X,Y}(x_n,y_m)=F_{X,Y}(x_0,y_0)$

This is true for every sequence $(x_n,y_n)\to(x_0,y_0)$ so the problem is solved.
EDIT: While conversing with Dilip Sarwate, I realized that the problem does not make sense if $X,Y$ are not jointly continuous. So I throw in this supposition: $X,Y$ are jointly continuous as well.

REPLY [0 votes]: So I thought it over and decided to post a solution to this problem. Please check if this is correct. This approach differs from what I wrote in the question only in terms of rigor.

Define the events $A_n=\{X\leq x_n\}$ and $B_m=\{Y\leq y_m\}$, where $n,m\in\mathbb N$. Also define $A=\{X\leq x_0\}$ and $B=\{Y\leq y_0\}$.
Now, it is given that $P(A_n)\to P(A)$ as $n\to\infty$ and $P(B_m)\to P(B)$ as $m\to \infty$.
Fix $m$. Then $P(A_n,B_m)$ is a sequence in $n$. Then,  $\lim_{n\to\infty}P(A_n,B_m)=a_m$ exists. I want to show that $a_m=P(A,B_m)$.

The proof to this, however, is non-rigorous, but I think it works.

After fixing $m$, we have restricted the movement of $Y$, so basically we are looking at movements of $X$ now. $Y$ does not get out of $B_m$, while $X$ may go on dancing, provided that $x_n\to x_0$. So essentially, $P(A_n,B_m)\to P(A,B_m)$ because $\{A_n,B_m\}\to\{A,B\}$ and $P(A_n)\to P(A)$.

This is a terrible way of writing it but sorry, this is the best I can do!!
Anyway, after this:

Similarly $\lim_{m\to\infty}P(A,B_m)=P(A,B)$. Also for each $m$, $\lim_{n\to\infty}P(A_n,B_m)$ exists (due to the horrible argument above).
Thus I show the existence of the final limit $\lim_{m\to\infty,n\to\infty}P(A_n,B_m)=P(A,B)$ using the theorem in Double Sequences which states that if $s_{n,m}$ be a double sequence such that $\lim_m\lim_ns_{n,m}=a$ and $\lim_na_{n,m}$ exists uniformly for $m$ then $\lim_{m,n}s_{n,m}=a$.

I am just not sure if I can apply these in the above ways I have shown.


text:
\section{Proposed Algorithm}
\label{methods}

Let $x^*\in\R^n$ be the signal that we are trying to reconstruct, $A\in\R^{m\times n}$ be the measurement matrix, and $\eta\in\R^m$ be independent noise. Given the measurement matrix $A$ and the observations $y=Ax^* + \eta$, we wish to reconstruct an $\h{x}$ that is close to $x^*$.

A generative model is a deterministic function $G(z;w){:} \; \R^k\rightarrow\R^n$ which takes as input a seed $z\in \R^k$ and is parameterized by ``weights'' $w\in\R^d$, producing an output $G(z;w)\in\R^n$. These models have shown excellent performance generating real-life signals such as images~\cite{goodfellow2014generative,kingma2013auto} and audio~\cite{wavenet}. We investigate \emph{deep convolutional} generative models, a special case in which the model architecture has multiple cascaded layers of convolutional filters~\cite{krizhevsky2012imagenet}. In this paper we apply a DCGAN~\cite{radford2015unsupervised} model and restrict the signals to be images.

\subsection{Compressed Sensing with Deep Image Prior (CS-DIP)}
Our approach is to find a set of weights for the convolutional network such that the measurement matrix applied to the network output, i.e. $AG(z;w)$, matches the measurements $y$ we are given. Hence we initialize an \emph{untrained} network $G(z;w)$ with some fixed $z$ and solve the following optimization problem:
\begin{equation}\label{eqn:w_star}
    w^*=\argmin_w  \| y - AG(z;w) \|^{2}.
\end{equation}

This is, of course, a non-convex problem because $G(z;w)$ is a complex feed-forward neural network. Still we can use  gradient-based optimizers for any generative model and measurement process that is differentiable.
Generator networks such as DCGAN are biased toward smooth, natural images due to their convolutional structure; thus the network structure alone provides a good prior for reconstructing images in problems such as inpainting and denoising~\cite{ulyanov2017deep}.
Our finding is that this applies to general linear measurement processes. We restrict our solution to lie in the span of a convolutional neural network. If a sufficient number of measurements $m$ is given, we obtain an output such that $x^*\approx G(z;w^*)$.

Note that this method uses an untrained generative model and optimizes over the network weights $w$. In contrast previous methods, such as that of Bora et al.~\cite{bora2017compressed}, use a trained model and optimize over the latent $z$-space, solving $z^*=\argmin_z  \| y - AG(z;w) \|^{2}$. We instead initialize a random $z$ with Gaussian i.i.d. entries and keep this fixed throughout the optimization process.

In our algorithm we leverage the well-established total variation regularization~\cite{rudin1992nonlinear,wang2008new, liu2018image}, denoted as $TV(G(z;w))$. We also propose an additional learned regularization technique, $LR(w)$; note that without this technique, i.e. when $\lambda_L = 0$, our method is completely unlearned. Lastly we use early stopping, a phenomena that will be analyzed theoretically in Section~\ref{theory}.

Thus the final optimization problem becomes
\begin{align}
\label{eqn:y-AG}
    w^*=   \argmin_w  &  \| y - A\, G(z;w) \|^{2} +  R(w; \lambda_{T}, \lambda_{L}).
\end{align}
The regularization term contains hyperparameters $\lambda_{T}$ and $\lambda_{L}$
for total variation and learned regularization:
$R(w; \lambda_{T}, \lambda_{L})= \lambda_{T} TV(G(z;w)) + \lambda_{L} LR(w)$. We now discuss this $LR(w)$ term.

\subsection{Learned Regularization}\label{sec: learned regularization}

Without learned regularization CS-DIP relies only on linear measurements taken from one unknown image. We now introduce a novel method which leverages a small amount of training data to optimize regularization.
In this case training data refers to measurements from additional ground truth of a similar type, e.g.~other x-ray images.

To leverage this additional information, we pose Eqn.~\ref{eqn:y-AG} as a Maximum a Posteriori (MAP) estimation problem and propose a novel prior on the weights of the generative model. This prior then acts as a regularization term, penalizing the model toward an optimal set of weights $w^*$.

For a set of weights $w\in\R^d$, we model the \emph{likelihood} of the measurements $y=Ax, y\in\R^m,$ and the prior on the weights $w$ as Gaussian distributions given by

\[
p(y|w) = \frac{\exp\left(-\dfrac{\|y-AG(z;w)\|^2}{2\lambda_L}\right)}{\sqrt{(2\pi \lambda_L)^m}}; \quad \quad \quad
p(w) = \frac{\exp\left(-\frac{1}{2}\left(w-\mu\right)^T\Sigma^{-1}\left(w-\mu\right)\right)}{\sqrt{(2\pi)^d|\Sigma|}},
\]
where $\mu\in\R^d$ and $\Sigma\in\R^{d\times d}$.

In this setting we want to find a set of weights $w^*$ that maximizes the posterior on $w$ given $y$, i.e.,
\begin{eqnarray}\label{eqn: y-AG + R}
    w^*& = & \argmax_w p(w|y) \equiv  \argmin_w \|y-AG(z;w)\|^2 + \, \lambda_L \left(w-\mu\right)^T\Sigma^{-1}\left(w-\mu\right).

\end{eqnarray}

This gives us the learned regularization term
\begin{equation}\label{eqn: lr}
LR(w) = \left(w-\mu\right)^T\Sigma^{-1}\left(w-\mu\right),
\end{equation}

where the coefficient $\lambda_L$ in Eqn.~\ref{eqn: y-AG + R} controls the strength of the prior.

Notice that when $\mu=0$ and $\Sigma=I_{d\times d},$ this regularization term is equivalent to $\ell_2$-regularization. Thus this method can be thought of as a more strategic version of standard weight decay.

\subsubsection{Learning the Prior Parameters}\label{sec:learning_mu_sig}
In the previous section, we introduced the learned regularization term $LR(w)$ defined in Eqn.~\ref{eqn: lr}.
However we have not yet learned values for parameters $(\mu,\Sigma)$ that incorporate prior knowledge of the network weights. We now propose a way to estimate these parameters.

Assume we have a set of measurements $S_Y = \{y_1,y_2,\cdots,y_Q\}$ from $Q$ different images $S_X = \{x_1,x_2,\cdots,x_Q\}$, each obtained with a different measurement matrix $A$. For each measurement $y_q,q\in \{1,2,...,Q\},$ we run CS-DIP to solve the optimization problem in Eqn.~\ref{eqn:y-AG} and obtain an optimal set of weights $W^*=\{w^*_1,w^*_2,\cdots,w^*_Q\}$. Note that when optimizing for the weights $W^*,$ we only have access to the measurements $S_Y$, not the ground truth $S_X$.

The number of weights $d$ in deep networks tends to be very large. As such, learning a distribution over each weight, i.e. estimating $\mu \in \R^d $ and $\Sigma \in \R^{d\times d}$, becomes intractable. We instead use a layer-wise approach: with $L$ network layers, we have $\mu \in \R^L $ and $\Sigma \in \R^{L\times L}$. Thus each weight within layer $l\in \{1,2,...,L\}$ is modeled according to the same $\mathcal{N}(\mu_l, \, \Sigma_{ll}) $ distribution. For simplicity we assume $\Sigma_{ij} = 0 \; \forall \, i \neq j$, i.e. that network weights are independent across layers. The process of estimating statistics $(\mu,\Sigma)$ from $W^*$ is described in Algorithm~\ref{algo:DIP-musig} of the appendix.

We use this learned $(\mu,\Sigma)$ in the regularization term $LR(w)$ from Eqn.~\ref{eqn: lr} for reconstructing measurements of images. We refer to this technique as \emph{learned regularization}. While this may seem analogous to batch normalization~\cite{ioffe2015batch}, note that we only use $(\mu,\Sigma)$ to penalize the $\ell_2$-norm of the weights and do not normalize the layer outputs themselves.

\subsubsection{Discussion of Learned Regularization}

The proposed CS-DIP does not require training if no learned regularization is used, i.e. if $\lambda_L = 0$ in Eqn.~\ref{eqn:y-AG}.
This means that CS-DIP can be applied only with measurements from a single image and no prior information of similar images in a dataset.

Our next idea, learned regularization, utilizes a small amount of prior information,  requiring access to measurements from a small number of similar images (roughly $5-10$). In contrast, other pre-trained models such as that of Bora et al.~\cite{bora2017compressed} require access to ground truth from a massive number of
similar images (tens of thousands for CelebA). If such a large dataset is available, and if a good generative model can be trained on that dataset, we expect that pre-trained models \cite{bora2017compressed,grover2018amortized,kabkab2018task,mardani2017recurrent} would outperform our method. Our approach is instead more suitable for reconstructing problems where large amounts of data or good generative models are not readily available.


text:
\begin{document}

\maketitle

\emph{The Authors have addressed almost completely all the issues I pointed out in the original version of the
paper. Moreover, they have answered to all the questions I wrote in my first report. The paper is
complete and well written. Although the contribution of this work is not completely innovative because
it relies on “techniques” already know in our community, I think it should be published after two
additional minor revision.} \\

\subsubsection*{Section 1:}
\begin{enumerate}
 \item \emph{Page 3: The Authors wrote “In this work, we identify suited embedded DIRK schemes [25, 26,
27] ...........and efficient method.”. First of all, it is not correct to say “suited methods” because
what has been done is to search in the literature and use the schemes that seem to be good.
“Suited” is not the appropriate word. I am asking the Authors they stress more how they have
selected these implicit RK schemes and state that clearly in the manuscript in the introduction
and when the schemes are again presented (Section 3.2). Secondly, I would not say to loudly
“efficient methods”. This is not a manuscript where preconditioning, smoothers or any other
kind of algorithm that makes a temporal solver efficient have been investigated.
}\\
We changed ``suited'' and we also deleted ``efficient'', because we think you are right. Furthermore, we have added a sentence in the introduction that states that this work is ``an intermediate step toward an efficient solver for for compressible fluid flows.'' Furthermore, in the beginning of Section 3.2, we added a paragraph about why we chose these particular DIRK schemes:
``As (22) constitutes a differential algebraic equation (DAE), special care has to be taken in choosing suitable methods. Desirable properties are A-stability to allow for large time steps; and the DIRK schemes should have no explicit stage, which is a necessary condition for convergence in the case of DAEs, see also Remark 9.
A-stable methods that fulfill this condition can be rarely found in the literature, we use those presented in [25, 26, 27].''
Furthermore, in Remark 9, we again cite necessary literature on the discretization of differential algebraic equations and explain that our methods have to be A-stable.

\end{enumerate}

\emph{I think the Authors should really highlight more how they have chosen the RK methods (not just because
they are reported in literature) and they should say more explicitly the goal of the paper, i.e. combine
known techniques in a new context and analyze the results.} \\
In the introduction, it now states explicitly the goal of this paper: "The goal of this paper is to demonstrate that the combination of two well-known ingredients (embedded DIRK and HDG) is possible in the context of compressible fluid flows. In this sense, this work constitutes an intermediate step toward demonstrating that an efficient solver for unsteady aerodynamic problems can also be based on HDG.``

\end{document}


TITLE: What does this $[\mu]$ mean here?
QUESTION [2 upvotes]: Here's what those notation means

This $"[\mu]"$ makes the setence weird and I don't understand what it means
Please help
Here's the link of the whole paper, and this is at page 151 https://projecteuclid.org/download/pdf_1/euclid.acta/1485892007

REPLY [1 votes]: This "$[\mu]$" is used as a part of the following equivalent constructions:
-"almost all $[\mu]$"
-"almost everywhere $[\mu]$"
-"almost surely $[\mu]$"
Their meaning is the following:
If we say, that almost all $[\mu]$ elements of a measurable space $(X, \Omega, \mu)$ have some property, that means that the measure $\mu$ of the set of all elements of it that do not have this property is equal to zero.
We write $[\mu]$ here, to explicitly say what measure were we talking about, as having several distinct non-equivalent measures can otherwise lead to confusion:
For example, almost all real numbers are non-zero under Lebesgue measure, but almost all of them are zero under the corresponding Dirac measure.


text:
\begin{document}

\title{Polyharmonic Daubechies type wavelets in Image Processing and Astronomy, II}
\author{Ognyan Kounchev, Damyan Kalaglarsky, Milcho Tsvetkov}
\maketitle

\textbf{Abstract: }\emph{We consider the application of the polyharmonic
subdivision wavelets to Image Processing, in particular to Astronomical
Images. The results show an essential advantage over some standard
multivariate wavelets and a potential for better compression.}

\textbf{Key words:} \emph{Wavelet Analysis, Daubechies wavelet, Image
Processing, Astronomical Images.}

\section{Introduction}

In \cite{KounchevKalaglarsky2010} we have constructed the basic elements of
the polyharmonic Wavelet Analysis. There we have provided all details of the
construction of the filters for the whole infinite family of father and mother
wavelets. Here we will finalize the multivariate construction and provide
experimental evidence of the power of the new wavelets.

\section{Polyharmonic Wavelet Analysis \label{SpolyharmonicSubdivision}}

The \emph{Polyharmonic Wavelet Analysis} arises in the context of the
polyharmonic subdivision. In general, it is obtained when the
(one-dimensional) polynomials of degree $2N-1$ are replaced by polyharmonic
functions of degree $N.$ The main element of the construction is the
application of the Fourier transform in $n$ variables in $\mathbb{R}^{n+1}.$
For simplicity we assume that the Image which we consider is a function
$u\left(  t,y\right)  ,$ for $t\in\mathbb{R}$ and $2\pi-$periodic in
$y\in\mathbb{R}^{n}.$ We have
\begin{align}
u\left(  t.y\right)   &  =\sum_{\eta\in\mathbb{Z}^{n}}\widehat{u}_{\eta
}\left(  t\right)  e^{i\left\langle \eta,y\right\rangle }\qquad\text{where the
Fourier coefficients are}\label{FourierTransform}\\
\widehat{u}_{\eta}\left(  t\right)   &  =\frac{1}{\left(  2\pi\right)  ^{n}
}\int_{0}^{2\pi}\cdot\cdot\cdot\int_{0}^{2\pi}u\left(  t.y\right)
e^{-i\left\langle \eta,y\right\rangle }dy.\nonumber
\end{align}
For every fixed $\eta\in\mathbb{Z}^{n}$ and $\xi=\left|  \eta\right|  $ we
make Wavelet Analysis which is based on the non-stationary father and mother
wavelets $\left\{  \varphi_{m}^{\xi}\left(  t\right)  \right\}  _{m\geq0}$ and
$\left\{  \psi_{m}^{\xi}\left(  t\right)  \right\}  _{m\in\mathbb{Z}}$ of
Theorem 4. Recall that they have refinement coefficients $M_{j}^{\left[
m\right]  ,\xi}$ and $\left(  -1\right)  ^{j}M_{1-j}^{\left[  m\right]  ,\xi}$
respectively, where the polynomial $M^{\left[  m\right]  ,\xi}\left(
z\right)  $ was defined at the end of \cite{KounchevKalaglarsky2010}. This
gives the wavelet expansion
\begin{equation}
u\left(  t.y\right)  =\sum_{m,j\in\mathbb{Z}}\sum_{\eta\in\mathbb{Z}^{n}
}\gamma_{\eta,m,j}\psi_{m}^{\left|  \eta\right|  }\left(  t-j\right)
e^{i\left\langle \eta,y\right\rangle }. \label{waveletExpansion}
\end{equation}

We summarize the results in the following theorem, for the proofs we refer to
\cite{DynKounchevLevinRender}.

\begin{theorem}
For every $\xi\geq0$ the functions $\psi_{m}^{\xi}\left(  t\right)  $ generate
a non-stationary MRA of $L_{2}\left(  \mathbb{R}\right)  $ by means of the
definitions
\begin{align*}
V_{m}^{\xi}  &  :=clos_{L_{2}\left(  \mathbb{R}\right)  }\left\{  \varphi
_{m}^{\xi}\left(  t-j\right)  :j\in\mathbb{Z}\right\} \\
L_{2}\left(  \mathbb{R}\right)   &  =clos_{L_{2}\left(  \mathbb{R}\right)
}\left(  \bigcup_{m\in\mathbb{Z}}V_{m}^{\left|  \xi\right|  }\right)  .
\end{align*}
Here $clos_{L_{2}\left(  \mathbb{R}\right)  }$ denotes the usual closure in a
linear and topological sense with respect to the space $L_{2}\left(
\mathbb{R}\right)  .$ Respectively, the Wavelet Analysis spaces $W_{m}
^{\left|  \xi\right|  }$ defined by means of
\[
W_{m}^{\xi}\bigoplus V_{m}^{\xi}=V_{m+1}^{\xi}
\]
are generated as
\[
W_{m}^{\xi}=clos_{L_{2}\left(  \mathbb{R}\right)  }\left\{  \psi_{m}^{\xi
}\left(  t-j\right)  :j\in\mathbb{Z}\right\}  .
\]
\end{theorem}

As in the polyspline Wavelet Analysis studied in \cite{okbook}, the
one-dimensional MRA $V_{j}^{\left|  \eta\right|  }$ which is generated for
every $\eta\in\mathbb{Z}^{n}$ is related to a ''polyharmonic MRA'' $PV_{j}$ as
$PV_{j}=\bigoplus_{\eta\in\mathbb{Z}^{n}}V_{j}^{\left|  \eta\right|  }$ by
means of the Fourier transform formula (\ref{FourierTransform}). Respectively,
we have also the polyharmonic Wavelet Analysis defined by $PV_{j}\bigoplus
PW_{j}=PV_{j+1}$ and reduced to infinitely many one-dimensional Wavelet
Analyses for every $\eta\in\mathbb{Z}^{n}$ in the form $PW_{j}=\bigoplus
_{\eta\in\mathbb{Z}^{n}}W_{j}^{\left|  \eta\right|  }$ by means of the Fourier
transform formula (\ref{FourierTransform}).

Respectively, the space
\[
L_{2,per}\left(  \mathbb{R}^{n+1}\right)  =\left\{  f\left(  t,y\right)
:t\in\mathbb{R},\ y\in\mathbb{R}^{n};f\text{ is }2\pi-\text{periodic in
}y\right\}
\]
is generated by an infinite amount of \textbf{orthogonal (non-stationary)
mother wavelets}
\begin{equation}
\left\{  \psi_{m}^{\left|  \eta\right|  }\left(  t\right)  e^{i\left\langle
\eta,y\right\rangle }:\eta\in\mathbb{Z}^{n},\ m\in\mathbb{Z}\right\}  .
\label{wavelets}
\end{equation}
Although these wavelets do not have a compact support in direction $y,$ it is
important to note that they have ''elongated support'' in the $y$ direction,
which plays an essential role for the sparseness of the ''edge
representation'' in the images, a point of view nicely described in
\cite{Donoho}, see also \cite{mallat}.

\section{The algorithm}

From the above it follows that the algorithm for Decomposition and
Reconstruction of the \textbf{Polyharmonic Subdivision Wavelet Analysis
(PhSdWav)} consists of a Fourier transform in direction $y$ and consecutive
application, for every $\eta\in\mathbb{Z}^{n},$ of one-dimensional Wavelet
Analysis in direction $t.$ It consists of the following steps:

\begin{enumerate}
\item  As usually we take as \textbf{first approximation} to the image the
image itself where the expansion is in terms of the scaling functions
$\left\{  \varphi_{m}^{\left|  \eta\right|  }\left(  t\right)  \right\}
_{m\geq0}.$

\item  Transforming to Wavelet Domain - this is the main difference with the
usual approach. Roughly speaking, we obtain $\widehat{u}_{\eta}\left(
t\right)  $ in formula (\ref{FourierTransform}) and make its Wavelet Analysis
for every $\eta\in\mathbb{Z}^{n}$ which gives us the coefficients
$\gamma_{\eta,m,j}$ in the wavelet expansion (\ref{waveletExpansion}). Let us
note that these coefficients are in general complex numbers.

\item  Filtering the small (insignificant) coefficients by using a threshold parameter.

\item  Quantizing the remaining coefficients.

\item  Encoding by arithmetic coder.
\end{enumerate}

The algorithm was written in Matlab.

\section{Experiments with images}

We apply our Wavelet Analysis to different classes of images -- synthetic and
real. We compare the results of our method  PhSdWav with parameter $N=9$ (see
\cite{KounchevKalaglarsky2010}) against the $2d$ wavelet transform with
Daubechies wavelets $db9$ (which are tensor product wavelets and correspond to
the one-dimensional Daubechies wavelets with parameter $N=9$). We use the
standard function $dwt2$ in Matlab which contains a large class of $2d$
wavelets, and $db9$ is one of the most used representatives. \textbf{In all
Figures below} we first display on top-left the original image, on top-right
is the result of PhSdWav compressed representation obtained by wavelet
thresholding, and on the bottom is the result of the $db9$ compressed
representation obtained by  wavelet thresholding. In order to make comparison
between our method PhSdWav and $db9$ we perform  thresholding which results in
equal PSNR.

\subsection{Elongated support tests on $64\times64$ pixel edges\label{Sedges}}

For the proper compression and representation of images it is of fundamental
importance to understand how the method applies to ''edges''. For that reason
we will first show the performance of our method to some synthetic edge images.

\subsubsection{Vertical edge}

In this test the image is a \textbf{vertical edge} which is a function
$f_{vert}\left(  t,y\right)  $ defined in the rectangle $\left[  0,1\right]
\times\left[  0,1\right]  $ by
\[
f_{vert}\left(  t,y\right)  :=\left\{
\begin{array}
[c]{c}
0\qquad\text{for }0\leq t\leq1/2\\
1\qquad\text{for }1/2<t\leq1
\end{array}
\right.  .
\]
We take a sampling of $f\left(  t,y\right)  $ by a $64\times64$ pixel image.
The performance of the polyharmonic Wavelets and the $db9$ wavelets is
provided in \textbf{Table 1} and \textbf{Figure 1}  below.

\begin{align*}
&  \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{Table
1.}\\
&
\begin{tabular}
[c]{||l||l||l||l||}\hline\hline
& Num. coeffs & PSNR & Entropy\\\hline\hline
Pol. Wav. & \textbf{48} & \textbf{120.7379} & \textbf{0}\\\hline\hline
db9 & \textbf{960} & \textbf{121.7412} & \textbf{1.01e+003}\\\hline\hline
\end{tabular}
\end{align*}

\begin{figure}[th]
\centering
\includegraphics[width=0.80\textwidth
]{Result_vertical.jpg}
\caption{Vertical Edge}
\label{fig:Result_vertical}
\end{figure}

As we said above the wavelets of our basis (\ref{wavelets}) show elongation in
direction $y$ and this makes the above image $f_{vert}$ perfect for
representation and compression with the polyharmonic subdivision wavelets. We
see that the quality of representation by means of PhSdWav (measured by the
PSNR) is the same as the quality of representation by means of the standard
$2-$dimensional Daubechies wavelet $db9$ but the number of coefficients is
$20$ times less!

The image $f_{vert}$ would correspond to the one-dimensional image provided by
the Heaviside function, $f\left(  t\right)  =0$ for $0\leq t\leq1/2$ ,
$f\left(  t\right)  =1$ for $1/2<t\leq1.$ The wavelets have an excellent
performance for such images.

Just the opposite is the next example.

\subsubsection{Horizontal edge}

In this test the image is a \textbf{horizontal edge}, which is a function
$f_{hor}\left(  t,y\right)  $ defined in the rectangle $\left[  0,1\right]
\times\left[  0,1\right]  $ by
\[
f_{hor}\left(  t,y\right)  :=\left\{
\begin{array}
[c]{c}
0\qquad\text{for }0\leq y\leq1/2\\
1\qquad\text{for }1/2<y\leq1
\end{array}
\right.  .
\]
We take a sampling of $f_{hor}\left(  t,y\right)  $ by a $64\times64$ pixel
image. The performance of the polyharmonic Wavelets and the $db9$ wavelets is
provided in \textbf{Table 2} and \textbf{Figure 2} below.

\begin{align*}
&  \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{Table
2.}\\
&
\begin{tabular}
[c]{||l||l||l||l||}\hline\hline
& Num. coeffs & PSNR & Entropy\\\hline\hline
Pol. Wav. & \textbf{2560} & \textbf{121.4740} & \textbf{2.52e+003}
\\\hline\hline
db9 & \textbf{960} & \textbf{121.7412} & \textbf{1.01e+003}\\\hline\hline
\end{tabular}
\end{align*}

\begin{figure}[th]
\centering
\includegraphics[width=0.80\textwidth
]{Result_horizontal.jpg}
\caption{Horizontal Edge}
\label{fig:Result_horizontal}
\end{figure}

We see now how inefficient the PhSdWav are for the representation of $f_{hor}$
compared to $db9$ wavelets, since for the same PSNR we need almost $3$ times
more coefficients. This result is expected since PhSdWav has a very strong
vertical orientation.

\subsubsection{Skewed edge}

Somewhere in the middle is the case of a skewed edge. In this test the image
of \textbf{skewed edge}, which is a function $f_{skew}\left(  t,y\right)  $
defined in the rectangle $\left[  0,1\right]  \times\left[  0,1\right]  $ by
(this is an approximate definition)
\[
f_{skew}\left(  t,y\right)  :=\left\{
\begin{array}
[c]{c}
0\qquad\text{for }0\leq t\leq\frac{1}{6}\left(  y+2.5\right)  \\
1\qquad\text{for }\frac{1}{6}\left(  y+2.5\right)  <t\leq1
\end{array}
\right.  .
\]
We take a sampling of $f_{skew}\left(  t,y\right)  $ by a $64\times64$ pixel
image. The performance of the polyharmonic Wavelets and the $db9$ wavelets is
provided in \textbf{Table 3} and \textbf{Figure 3 }below.

\begin{align*}
&  \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad\qquad\qquad\text{Table
3.}\\
&
\begin{tabular}
[c]{||l||l||l||l||}\hline\hline
& Num. coeffs & PSNR & Entropy\\\hline\hline
Pol. Wav. & \textbf{512} & \textbf{126.8204} & \textbf{ 386.7158}
\\\hline\hline
db9 & \textbf{940} & \textbf{121.9320} & \textbf{977.3171}\\\hline\hline
\end{tabular}
\end{align*}

\begin{figure}[th]
\centering
\includegraphics[width=0.80\textwidth
]{Result_skewed.jpg}\caption{Skewed
Edge}
\label{fig:Result_skewed}
\end{figure}

We see that although the edge is prettily skew (the slope is approximately
$\tan\alpha=6$ ) the PhSdWav have a very nice performance compared to $db9.$

\subsection{Lena image}

Let us make a more detailed analysis of the seminal image of Lena in
$128\times128$ pixels. We provide the Compression Ratio and the Bits Encoded
for both methods in \textbf{Table 4} and \textbf{Figure 4 }below.
\begin{align*}
&  \qquad\qquad\qquad\qquad\qquad\qquad\quad\qquad\qquad\qquad\qquad
\qquad\qquad\qquad\qquad\text{Table 4.}\\
&
\begin{tabular}
[c]{||l||l||l||l||l||l||}\hline\hline
& N. coeffs & PSNR & Entropy & Cmpr. rat. & Bits Enc.\\\hline\hline
Pol. Wav. & \textbf{4,150} & \textbf{77.7338} & \textbf{5.15e+004} &
\textbf{5.9815} & \textbf{42,802}\\\hline\hline
db9 & \textbf{5,187} & \textbf{77.6742} & \textbf{5.65e+004} & \textbf{5.9562}
& \textbf{44,012}\\\hline\hline
\end{tabular}
\end{align*}

\begin{figure}[th]
\centering
\includegraphics[width=1.00\textwidth
]{Result_Lena.jpg}\caption{Lena Image}
\label{fig:Result_Lena}
\end{figure}

We see that for the same PSNR our method PhSdWav is better -- it needs $20\%$
less coefficients. This is an astonishingly good result since as we have seen
in the experiments with the synthetic edges in Section \ref{Sedges}, our
method of PhSdWav is adapted to the vertical direction $y$ and the Lena image
does not have a dominating orientation of the edges.

\subsection{Astronomical Images}

Hereafter we analyze different types of astronomical images.

\subsubsection{The Pleiades star cluster, a part with faint objects only,
sized $128\times128$}

First we consider an image which is a part of the Pleiades image containing
only faint stars. The results are provided in \textbf{Table 5} and
\textbf{Figure 5 }below.
\begin{align*}
&  \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\qquad\qquad\qquad\text{Table 5.}\\
&
\begin{tabular}
[c]{||l||l||l||l||l||l||}\hline\hline
& N. coeffs & PSNR & Entropy & Cmpr. rat. & Bits enc.\\\hline\hline
Pol. Wav. & \textbf{2,663} & \textbf{77.5805} & \textbf{2.93e+004} &
\textbf{7.6028} & \textbf{33,456}\\\hline\hline
db9 & \textbf{5,187} & \textbf{76.3970} & \textbf{4.52e+004} & \textbf{6.9317}
& \textbf{37,818}\\\hline\hline
\end{tabular}
\end{align*}

\begin{figure}[h]
\centering
\includegraphics[width=1.00\textwidth
]{Result_astro_128.jpg}
\caption{Pleiades star cluster, faint objects, 128x128 pixels}
\label{fig:Result_astro_128}
\end{figure}

We see that for (almost) the same PSNR we have much less coefficients, the
compression ratio is higher, and our entropy is much less, which shows a
better potential for compression.

\subsubsection{The Pleiades star cluster, a part with bright objects, sized
$256\times256$}

Next we consider a part of the Pleiades containing several bright objects. The
results are provided in \textbf{Table 6 }and \textbf{Figure 6} below
\begin{align*}
&  \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\qquad\qquad\qquad\text{Table 6.}\\
&
\begin{tabular}
[c]{||l||l||l||l||l||l||}\hline\hline
& N. coeffs & PSNR & Entropy & Cmpr. rat. & Bits enc.\\\hline\hline
Pol. Wav. & \textbf{15,159} & \textbf{50.0717} & \textbf{2.89e+005} &
\textbf{13.8338} & \textbf{73,750}\\\hline\hline
db9 & \textbf{18,502} & \textbf{49.4214} & \textbf{3.57e+005} &
\textbf{8.5379} & \textbf{122,814}\\\hline\hline
\end{tabular}
\end{align*}

\begin{figure}[h]
\centering
\includegraphics[width=1.00\textwidth
]{Result_astro_256.jpg}
\caption{Pleiades star cluster, bright objects, 256x256 pixels}
\label{fig:Result_astro_256}
\end{figure}

Again we see that for the same PSNR we have not only less coefficients but
also a much higher compression ratio.

\subsubsection{The Pleiades star cluster, chain, sized $128\times128$}

Here we consider a chain image of a part of the Pleiades image. The results
are provided in \textbf{Table 7} and \textbf{Figure 7 }below
\begin{align*}
&  \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad
\qquad\qquad\qquad\qquad\text{Table 7.}\\
&
\begin{tabular}
[c]{||l||l||l||l||l||l||}\hline\hline
& N. coeffs & PSNR & Entropy & Cmpr. rat. & Bits enc.\\\hline\hline
Pol. Wav. & \textbf{2,764} & \textbf{88.6818} & \textbf{2.4849e+004} &
\textbf{8.8706} & \textbf{28,528}\\\hline\hline
db9 & \textbf{5,340} & \textbf{87.5730} & \textbf{5.7468e+004} &
\textbf{8.4636} & \textbf{30,973}\\\hline\hline
\end{tabular}
\end{align*}

\begin{figure}[h]
\centering
\includegraphics[width=1.00\textwidth
]{Result.jpg}
\caption{Pleiades star cluster, chain image, 128x128 pixels}
\label{fig:Result}
\end{figure}

For the same PSNR, the number of coefficients of our method PhSdWav is twice
less than the number of  $db9$ coefficients. This result is expected since the
image is vertically oriented. However the compression ratio is not much bigger
apparently due to more noise in the background of the image.

\textbf{Acknowledgement.} The first named author was sponsored partially by
the Alexander von Humboldt Foundation, and all authors were sponsored by
Project DO--2-275/2008 ''Astroinformatics'' with Bulgarian NSF.

\newpage


TITLE: Hodge theory on semi-Riemannian manifolds [reference request]
QUESTION [2 upvotes]: I need to learn a bit of Hodge theory on manifolds and I am looking for a reference which covers the case where the metric has arbitrary signature $(p,q)$. Most books I have found seem to focus on the Riemannian case alone.
Could anyone point me in the right direction? I know some basic sheaf theory and sheaf cohomology, if that matters.

REPLY [1 votes]: I am extremely skeptical that one could do Hodge theory on a semi-Riemannian manifold without extreme changes, because the Laplacian in signature $(p,q)$ is $\partial/\partial x_1^2 + \dots + \partial/\partial x_p^2 - \dots - \partial/\partial x_{p+q}^2$, which is not an elliptic operator, and ellipticity is completely crucial to the Hodge theory. This is probably not the only issue, but it's certainly first that comes to mind.
A great source for Hodge theory on Riemannian manifolds in the appropriate generality (elliptic complexes of differential operators on vector bundles) is Wells, Differential analysis on complex manifolds. A different approach (less general and different in flavor) can be found in both Griffiths and Harris and Warner; both are more or less the same proof.


TITLE: What makes a good mathematical theory?
QUESTION [2 upvotes]: I recently read about Galois connections, and that they show up in a lot of different places in mathematics. Given there apparent ubiquity, I thought they might have a rich theory. However, when asking other people in my department, I got varying answers as to why they were not particularly interesting and/or were not worth studying. It is hard to tell if this is due to a bias or whether there could be an objective argument made as to why mathematicians invest a lot of effort in characterizing and classifying one class of objects such as groups or topological spaces and not another.
I am interested in answers that either address the particular theory of Galois connections (and possibly arguing that they are in fact worth studying) or attempt to give a general criteria for what makes a good mathematical theory.

REPLY [1 votes]: In my opinion, a good mathematical theory should move a particular area one step forward (overcome established traditions, provide further conceptual clarity). This is a given. It should extend what has already been proven, perhaps, provide more light on a subject which was previously in the dark.
Yet, I also think a great or good mathematical theory should spark one's imagination. This is crucial for me. It is important that a mathematical theory, once it is generally accepted - triggers a general collective consciousness into action, in order for the area under inspection to progress. This can only be done if the theory presented contains a certain amount of rigor and divine creativity, (what in Germany, they would call a Geist i.e. invoking some sort of creative spirit, so to speak).
I would like to reference a passage from both Frege and Hilbert's correspondence:
In a letter to Frege, Hilbert wrote:

...every theory is only a scaffolding or schema of concepts together
  with their necessary relations to one another, and the basic elements
  can be thought of in any way one likes.

Frege, G. 1980. Philosophical and mathematical correspondence, ed. B. McGuinness.


TITLE: How to draw heart shape (3D) using Maple
QUESTION [3 upvotes]: I'm drawing a heart shape (3D) using Maple, but I think it is not correct. Please hint to me how to draw it correctly.

with(plots);
implicitplot3d((2*x^2+y^2+z^2-1)^3-(1/10)*x^2*z^2-y^2*z^3, x = -1.2 .. 1.2, y = -1.2 .. 1.2, z = -1.2 .. 1.3, numpoints = 80000);

REPLY [1 votes]: with(plots);

implicitplot3d((2*x^2+y^2+z^2-1)^3-(1/10)*x^2*z^2-y^2*z^3, x = -1.2 .. 1.2,
                 y = -1.2 .. 1.2, z = -1.2 .. 1.3, numpoints = 80000);

it is true. no problem in plot.


TITLE: Upper bound of the sum $\sum_{n=1}^\infty \tan^{-1}\left(\frac{x}{1-(x-n)^2}\right)$ as $x\to \infty$
QUESTION [3 upvotes]: I am stuck at finding a Big-O upper bound of the following sum as $x\to \infty$ $$\sum_{n=1}^\infty \tan^{-1}\left(\frac{x}{1-(x-n)^2}\right)$$
I tried: Since $$ \sum_{n=1}^\infty\tan^{-1}\left(\frac{x}{1-(x-n)^2}\right)=\sum_{n=1}^\infty O\left(\frac{x}{n^2}\right) $$
Now we have for $x_0>0$ and $M>0$ $$ \sum_{n=1}^\infty\tan^{-1}\left(\frac{x}{1-(x-n)^2}\right)=\left(M \sum_{n=1}^\infty \frac{1}{n^2}\right)x $$
Now since $\sum_{n=1}^\infty\frac{1}{n^2}$ converges so can we write the following ? $x\to \infty$ $$ \sum_{n=1}^\infty\tan^{-1}\left(\frac{x}{1-(x-n)^2}\right)=O(x) $$
Thnak you for your valuable time and comments.

REPLY [2 votes]: Strating from @Svyatoslav's answer, we can compute
$$S(N)\sim\int_0^\infty\arctan\Bigg[\frac{N}{1-(n-N)^2}\Bigg]dn$$ (have a look here for the antiderivative). At the upper bound, the result is $0$. Expanded as a series for large values of $N$, this gives , using $t=\pi  \sqrt{2N}$,
$$S(N)\sim t-1+\frac{\pi ^2}{t}+\frac{\pi ^4}{2 t^3}-\frac{16 \pi ^4}{15 t^4}-\frac{\pi ^6}{2
   t^5}+O\left(\frac{1}{t^6}\right)$$
Comparing for a few values on $N$
$$\left(
\begin{array}{ccc}
N & \text{approximation} & \text{solution} \\
 100 & 43.4288 & 38.2745 \\
 200 & 61.8319 & 61.9841 \\
 300 & 75.9530 & 76.0780 \\
 400 & 87.8577 & 87.9663 \\
 500 & 98.3459 & 98.4454 \\
 600 & 107.828 & 107.915 \\
 700 & 116.548 & 116.643 \\
 800 & 124.664 & 124.728 \\
 900 & 132.286 & 132.504 \\
 1000 & 139.496 & 138.660
\end{array}
\right)$$


TITLE: hyperbolic 3-manifold of finite volume
QUESTION [3 upvotes]: Is there a complete description of hyperbolic 3-manifold of finite volume ?
Or similarly a classification of finitely generated torsion free subgroups of $PSL(2,\mathbf{C})$ with finite covolume?
Is it true that any such hyperbolic 3-manifold of finite volume can be obtained as $S^{3}-K$ where $K$ is some knot?

REPLY [8 votes]: One can regard the geometrization theorem as a classification of hyperbolic 3-manifolds of finite volume. These are the interiors of ($\mathbb{P}^2$-)irreducible compact 3-manifolds $M$ with incompressible boundary, Euler characteristic $=0$, and atoroidal (every map $\mathbb{Z}^2 \to \pi_1(M)$ factors through a map to $\pi_1$ of a boundary component).
In principle, there are algorithms for listing hyperbolic 3-manifolds with bounded volume. Some version of this algorithm is implemented for small enough volume e.g. here: https://arxiv.org/abs/0910.5043. What such a list entails is a list of finitely many cusped manifolds, and a description of Dehn surgery coefficients that give all manifolds bounded by a fixed volume (although there's some thorny issues here about exactly how one computes or compares volumes). If you want to list hyperbolic manifolds, say by rank (the number of matrices one needs to multiply and take inverses to generate the full group), then things become much trickier (see e.g. https://arxiv.org/abs/1708.01774). Listing by the minimal number of tetrahedra in a triangulation should be more feasible.
The third part of your question is addressed here: https://mathoverflow.net/a/277968/1345. But more fundamentally, finite volume hyperbolic 3-manifolds may have multiple ($>1$) or zero cusps, and hence could not be a knot complement.